Affine Algebraic Geometry : Geometry of Polynomial Rings 9789811280085, 9789811280092, 9789811280108

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Affine Algebraic Geometry

Geometry of Polynomial Rings

SERIES ON UNIVERSITY MATHEMATICS ISSN: 1793-1193 Editors: Wu-Yi Hsiang Tzuong-Tsieng Moh Ming-Chang Kang S S Ding M Miyanishi

University of California, Berkeley, USA/ Hong Kong University of Science and Technology, Hong Kong Purdue University, USA National Taiwan University, Taiwan (ROC) Peking University, China University of Osaka, Japan

Published Vol. 11 Affine Algebraic Geometry: Geometry of Polynomial Rings by M Miyanishi Vol. 10 Linear Algebra and Its Applications by T-T Moh Vol. 9

Lectures on Lie Groups (Second Edition) by W-Y Hsiang

Vol. 8

Analytical Geometry by Izu Vaisman

Vol. 7

Number Theory with Applications by W C Winnie Li

Vol. 6

A Concise Introduction to Calculus by W-Y Hsiang

Vol. 5

Algebra by T-T Moh

Vol. 2

Lectures on Lie Groups by W-Y Hsiang

Vol. 1

Lectures on Differential Geometry by S S Chern, W H Chen and K S Lam

Series on University Mathematics – Vol. 11

Affine Algebraic Geometry

Geometry of Polynomial Rings

Masayoshi Miyanishi Osaka University, Japan & Kwansei Gakuin University, Japan

World Scientific NEW JERSEY

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LONDON

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Miyanishi, Masayoshi, 1940– author. Title: Affine algebraic geometry : geometry of polynomial rings / Masayoshi Miyanishi, Osaka University, Japan & Kwansei Gakuin University, Japan. Description: New Jersey : World Scientific, [2024] | Series: Series on university mathematics, 1793-1193 ; vol. 11 | Includes bibliographical references and index. Identifiers: LCCN 2023031983 | ISBN 9789811280085 (hardcover) | ISBN 9789811280092 (ebook for institutions) | ISBN 9789811280108 (ebook for individuals) Subjects: LCSH: Polynomial rings--Textbooks. | Geometry, Affine--Textbooks. | Geometry, Algebraic--Textbooks. Classification: LCC QA251.3 .M593 2024 | DDC 516/.4--dc23/eng/20231016 LC record available at https://lccn.loc.gov/2023031983

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Printed in Singapore

To the late Professor Masayoshi Nagata

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Preface

One of the innovations brought into algebraic geometry by A. Grothendieck ´ ements de G´eom´etrie Alg´ebrique is through his publications including El´ the establishment of a bijective correspondence between affine schemes and commutative rings, by which one can introduce algebro-geometric methods to commutative algebra and create vice versa a new field in algebraic geometry where one studies geometry of affine domains over a field. An affine domain A over a field k is the quotient ring of a polynomial ring k[x1 , . . . , xn ] by an ideal I. A geometric approach to study an affine variety X = Spec A with coordinate ring A is considered to study geometrically a big ring like A which is not a finite union of local rings.1 Study of polynomial rings has been an important subject in Mathematics, and there are various approaches depending on which area of Mathematics polynomials or polynomial rings are considered in. Frankly speaking, what is decisively understood about polynomials in algebraic geometry is limited to the case of one variable. Many partial results are obtained in the case of many variables, but comprehensive understanding is not decisive. During the same period, but away from influences of A. Grothendieck, mathematicians including M. Nagata, Sh. Abhyankar, T.T. Moh, M. Suzuki and others have started their own study of polynomials in two variables over the complex number field C through precise analysis of how the affine plane curve defined by a polynomial behaves at infinity, i.e., out of the affine plane. Their contribution culminated in Abhyankar-Moh-Suzuki theorem, which states that a polynomial f (x, y) whose zero locus {f = 0} is isomorphic to an affine line can be taken as one of coordinates, i.e., C[x, y] = C[f, g] for some polynomial g(x, y). This result revived interest in the automorphism 1 We

would like to call them global rings if there are no fear of misunderstandings.

vii

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Affine Algebraic Geometry

theorem of C[x, y] by Jung-Van der Kulk and geometric problems on affine surfaces including the Cancellation Problem.2 This happened from the late 1960s to the early 1970s. Around the same period there was also a worldwide revival of interest toward the Enriques-Kodaira classification of algebraic surfaces. In the mid-1970s, S. Iitaka introduced the logarithmic Kodaira dimension of noncomplete algebraic varieties and proposed a project of classifying noncomplete surfaces with expectations that logarithmic Kodaira dimension should work as Kodaira dimension did in the classification of smooth projective surfaces. Iitaka’s students including S. Tsunoda and Y. Kawamata, and the people including T. Fujita and the author showed that Iitaka’s expectation did work to a certain degree and bring some results beyond the expectation.3 Their approach is now developed into logarithmic geometry, which is a study of pairs (V, D) of a complete variety V and an effective divisor D. In most cases, by Hironaka’s resolution of singularities, V is made to be smooth and D a divisor with simple normal crossings. The study so far shows that geometry changes according to what kind of singularities is admitted on V and D. For an affine variety X, we find easily such a pair (V, D) by embedding X into a projective space via the embedding X ,→ An ,→ Pn and taking the closure X as V and D = V \ X. By this approach, one is able to observe the geometric behavior of X at infinity, i.e., on D or near D.4 There is the famous Jacobian Conjecture which asserts that if polynomials f1 , . . . , fn ∈ C[x1 , . . . , xn ] have a nonzero constant as the Jacobian determinant ∂(f1 , . . . , fn ) ∂(x1 , . . . , xn ) then C[x1 , . . . , xn ] = C[f1 , . . . , fn ]. Under this assumption, the mapping φ : An → An , (x1 , . . . , xn ) 7→ (f1 , . . . , fn ) induces a local analytic isomorphism between every point of the origin An and its image of the target An . So, the conjecture asks if these local analytic isomorphisms are induced by a polynomial isomorphism. Unfortunately, 2 It

is also called the Zariski’s Problem. the author’s book on Open algebraic surfaces [59]. 4 This approach was and hopefully still is successful as exemplified by a solution due to M. Koras and K. Palka of the long-standing Coolidge-Nagata conjecture which asserts that a complex irreducible curve on P2 homeomorphic to a line P1 is mapped to a line by a birational automorphism of P2 (see [47]). 3 See

Preface

ix

the conjecture is not verified even in the case n = 2. A formal (or analytic) inverse mapping of φ is rather easy to construct, but it is very difficult to show that the inverse mapping is a polynomial mapping. This is partly due to the fact that total (or whatever) degree of polynomials are not reliable as a measure to control the behavior of polynomials because the degree changes easily as the set of coordinates {x1 , . . . , xn } is replaced by another one having changes of nonlinear terms. It is not clear if there exists a geometric approach which enables to replace this method of formal inverse mapping, though some success is obtained by such approaches. One honest impression the author had through various geometric challenges is that the affine space (even the affine plane) is so immaculate that one cannot find clues to get a geometric study started with. Affine algebraic geometry emerged from these backgrounds and problems. The subjects we treat therein are probably more biased on the affine spaces and polynomial rings, but nothing more than noncomplete varieties treated in standard algebraic geometry. As long as we want to use geometric approach we cannot avoid a minimum background of algebraic geometry. Hence this volume begins with an introduction to algebraic geometry. The present author tried to start explanations from the beginning without omission of proofs so that the readers with knowledge of algebra and geometry taught at the third year level of the undergraduate study can understand, though some important results are not given proofs which are mostly involved and the author expects are provided by more advanced textbooks with established reputation. Precise references are given with few exceptions in such cases. The first chapter of the present volume is based on the lectures on algebraic geometry which the author gave at Kwansei Gakuin University for graduate courses over several years. Some parts are taken from the author’s books on higher algebra and algebraic geometry which are written in Japanese and have never been translated into foreign languages [60, 78]. One can consider the first chapter as a quick introduction to algebraic geometry and commutative algebra, and skip it if one has some background on the subject. Enriques-Kodaira classification of projective algebraic surfaces as well as the theory of logarithmic Kodaira dimension is explained in [59], although the referred book is an advanced one for specialists and graduate students (perhaps Ph.D students). We are reminded to make this volume as accessible as possible for the beginning students in algebraic geometry. So, we tried not to make heavy use of advanced results. Topics specialized in affine algebraic geometry begin from Chapter 2

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Affine Algebraic Geometry

onward. The first topic is a proof of the AMS theorem which uses the linear pencil of curves on the projective plane P2 and the elimination of base points. In fact, these results symbolize the dawn of affine algebraic geometry. We then explain generalizations of the Jacobian conjecture in dimension two. The readers will see how effectively affine algebraic geometry is used to this conjecture. There is also a wish of the author to reveal contributions of hidden ramification at the infinity to (not necessarily) finite ´etale coverings of noncomplete varieties. The presentation of section 1.4, subsections 1.8.4 and 1.8.5 is based on the contents of [78]. The author would like to express his belated thanks to late Professors Masayoshi Nagata and Masaki Maruyama for the joint authorship of the book. The present book is dedicated to Professor Nagata, who is one of the founders of the research area of studying geometry of rational surfaces and polynomials. The contents of Chapters 2 and 3 are partly based on the author’s lecture notes [58] and [62]. Last but not the least, the author would like to express his indebtedness to the editor Ms. Kwong Lai Fun of World Scientific Publ. Co. for the opportunity to write a book on affine algebraic geometry and constant encouragement during the writing of this book.

June, 2023 M. Miyanishi

Contents

Preface 1.

vii

Introduction to Algebraic Geometry 1.1

1.2

1.3

Review on basic results in commutative algebra . . . . . . 1.1.1 Ring of quotients and local ring . . . . . . . . . . 1.1.2 Spectrum of a ring and Zariski topology . . . . . . 1.1.3 Irreducible decomposition of a topological space . 1.1.4 Prime ideal decomposition of radical ideals . . . . 1.1.5 Generic point, closed point and Krull dimension . 1.1.6 Hilbert basis theorem . . . . . . . . . . . . . . . . 1.1.7 Integral extension and Noether normalization lemma 1.1.8 Lying-over theorem and Going-up theorem . . . . 1.1.9 Krull dimension of affine domains . . . . . . . . . Review on finitely generated field extensions . . . . . . . . 1.2.1 Transcendence basis and transcendence degree . . 1.2.2 Regular extension and separable extension . . . . Schemes and varieties . . . . . . . . . . . . . . . . . . . . 1.3.1 Affine schemes of finite type and affine varieties . 1.3.1.1 Irreducible decomposition of an affine scheme of finite type . . . . . . . . . . . 1.3.1.2 Density of the set of closed points . . . . 1.3.1.3 Affine varieties and function fields . . . . 1.3.1.4 Structure sheaves . . . . . . . . . . . . . 1.3.2 Morphisms of affine schemes . . . . . . . . . . . . 1.3.2.1 Intersection of affine open sets . . . . . . 1.3.2.2 Open immersion and closed immersion . xi

1 1 1 3 5 7 9 11 12 16 19 20 20 24 27 27 27 28 30 31 33 34 35

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Affine Algebraic Geometry

1.3.2.3

1.4

1.5

1.6

1.7

1.8

Behavior of structure sheaves under a morphism . . . . . . . . . . . . . . . . . 1.3.3 Schemes and varieties . . . . . . . . . . . . . . . . 1.3.3.1 Definition and examples of schemes . . . 1.3.3.2 Morphism of schemes . . . . . . . . . . . 1.3.3.3 Fiber products of schemes . . . . . . . . 1.3.3.4 Separated schemes . . . . . . . . . . . . 1.3.3.5 Rational maps of algebraic varieties . . . Graded rings and projective schemes . . . . . . . . . . . . 1.4.1 Graded rings and projective spectrums . . . . . . 1.4.2 Projective schemes and projective varieties . . . . 1.4.2.1 General properties of projective schemes 1.4.2.2 Projective varieties . . . . . . . . . . . . 1.4.2.3 Projective closure of an affine variety . . Normal varieties . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Discrete valuation rings and normal rings . . . . . 1.5.2 Normalization of affine domains . . . . . . . . . . 1.5.3 Normal varieties and normalization of algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Unique factorization domains . . . . . . . . . . . . 1.5.5 Weil divisors and divisor class group . . . . . . . . 1.5.6 Zariski’s main theorem . . . . . . . . . . . . . . . Smooth varieties . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 System of parameters and regular local ring . . . . 1.6.2 Regular sequence and depth of a local ring . . . . 1.6.3 Jacobian criterion . . . . . . . . . . . . . . . . . . 1.6.4 Sheaf of differential 1-forms and canonical sheaf . Divisors and linear systems . . . . . . . . . . . . . . . . . 1.7.1 Invertible sheaves . . . . . . . . . . . . . . . . . . 1.7.2 Cartier divisors . . . . . . . . . . . . . . . . . . . . 1.7.3 Linear systems . . . . . . . . . . . . . . . . . . . . 1.7.4 D-dimension, Kodaira dimension and logarithmic Kodaira dimension . . . . . . . . . . . . . . . . . . Algebraic curves and surfaces . . . . . . . . . . . . . . . . 1.8.1 Serre duality and Euler-Poincar´e characteristic . . 1.8.2 Riemann-Roch theorem for a curve . . . . . . . . 1.8.3 Algebraic curves . . . . . . . . . . . . . . . . . . . 1.8.4 Intersection theory on algebraic surfaces . . . . . . 1.8.5 Riemann-Roch theorem for surfaces . . . . . . . .

35 37 37 39 40 41 43 45 45 50 50 53 56 59 59 64 66 69 74 76 84 84 94 97 100 110 110 112 114 117 120 120 121 123 126 135

Contents

Fibrations and relatively minimal models of surfaces . . . . . . . . . . . . . . . . . . . . . . 1.9 Appendix to Chapter 1 . . . . . . . . . . . . . . . . . . 1.9.1 Primary decomposition of ideals . . . . . . . . 1.9.2 Tensor products of algebras . . . . . . . . . . . 1.9.2.1 Construction . . . . . . . . . . . . . . 1.9.2.2 Flat modules . . . . . . . . . . . . . 1.9.3 Inductive limits and projective limits . . . . . 1.9.3.1 Inductive limits . . . . . . . . . . . . 1.9.3.2 Projective limits . . . . . . . . . . . . 1.9.3.3 Ideal-adic completion . . . . . . . . . 1.9.4 Fiber products of schemes . . . . . . . . . . . 1.9.5 Reviews on sheaf theory . . . . . . . . . . . . . ˇ 1.9.6 Cech cohomology of sheaves of abelian groups ˇ 1.9.6.1 Cech cohomology . . . . . . . . . . . 1.9.6.2 Coherent sheaf cohomologies over projective varieties . . . . . . . . . . 1.10 Problems to Chapter 1 . . . . . . . . . . . . . . . . . .

xiii

1.8.6

2.

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

136 144 144 148 148 150 155 155 156 156 158 162 169 169

. . 172 . . 177

Geometry on Affine Surfaces

191

2.1 2.2

191

2.3

2.4

Characterization of the affine plane . . . . . . . . . . . . . Admissible data for an affine curve with one place at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Euclidean transformation associated with admissible data . . . . . . . . . . . . . . . . . . . 2.2.2 (e, i)-transformation associated with admissible data . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Irreducible affine curves with one-place at infinity 2.2.4 Abhyankar-Moh-Suzuki theorem . . . . . . . . . . 2.2.5 Theorem of Gutwirth and pathological A1 -fibrations . . . . . . . . . . . . . . . . . . . . . 2.2.6 Abhyankar-Moh problem on embedded lines in positive characteristic . . . . . . . . . . . . . . . . Automorphism theorem of the affine plane . . . . . . . . . 2.3.1 Linear pencils of rational curves and field generators . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Proof of automorphism theorem by Jung and van der Kulk . . . . . . . . . . . . . . . . . . . . . . . Algebraic group actions on the affine plane . . . . . . . . .

198 199 206 210 212 213 216 220 220 226 232

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2.5 2.6

2.7

2.8 3.

2.4.1 Algebraic groups, actions and quotient spaces 2.4.2 Finite subgroups of Aut k[x, y] . . . . . . . . . 2.4.3 Finite group actions and invariants . . . . . . 2.4.4 Quotient singularities on surfaces . . . . . . . Birational automorphisms of rational surfaces . . . . . 2.5.1 Noether factorization theorem . . . . . . . . . Boundary divisors of affine surfaces . . . . . . . . . . . 2.6.1 Quantitative criterion of SNC divisors . . . . . 2.6.2 Shift transformation on the boundary divisor . 2.6.3 Theorem of Ramanujam-Morrow . . . . . . . . Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . 2.7.1 Unramified morphism . . . . . . . . . . . . . . ´ 2.7.2 Etale coverings . . . . . . . . . . . . . . . . . . 2.7.3 Riemann-Hurwitz formula for curves . . . . . . 2.7.4 Inverse and direct images of divisors and the projection formula . . . . . . . . . . . . . . . . 2.7.5 Amalgamated product of two groups . . . . . . 2.7.6 Quotient varieties by finite group actions and ramification of the quotient morphism . . . . . Problems to Chapter 2 . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . 282 . . 286 . . 290 . . 300

Geometry and Topology of Polynomial Rings — Motivated by the Jacobian Problem 3.1

3.2

3.3

Plane-like affine surfaces . . . . . . . . . . . . . . . . . 3.1.1 Simply connected algebraic varieties . . . . . . 3.1.2 Unit group, unit rank and independence of boundary divisors . . . . . . . . . . . . . . . . 3.1.3 Gizatullin surfaces and affine pseudo-planes . . 3.1.4 Affine pseudo-planes — more properties . . . . 3.1.5 tom Dieck construction of affine pseudo-planes 3.1.6 Platonic A1∗ -fiber spaces . . . . . . . . . . . . . 3.1.7 Homology planes . . . . . . . . . . . . . . . . . Jacobian conjecture and related results . . . . . . . . . 3.2.1 Jacobian conjecture and its variants . . . . . . 3.2.2 Partial affirmative answers . . . . . . . . . . . Generalized Jacobian conjecture — affirmative cases . 3.3.1 Results in arbitrary dimension . . . . . . . . . 3.3.2 Results for surfaces . . . . . . . . . . . . . . . 3.3.2.1 Surfaces having A1 -fibrations . . . .

232 236 238 248 256 256 263 263 266 269 276 276 280 281

319 . . 319 . . 320 . . . . . . . . . . . . .

. . . . . . . . . . . . .

322 324 329 332 336 342 345 345 349 354 354 356 356

Contents

3.4

3.5

3.6 4.

3.3.2.2 Surfaces having A1∗ -fibrations . . . . 3.3.2.3 Case of κ = 1 . . . . . . . . . . . . . Generalized Jacobian conjecture for various cases . . . 3.4.1 Case of Q-homology planes of κ = −∞ . . . . 3.4.2 Counterexamples . . . . . . . . . . . . . . . . . Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . 3.5.1 Makar-Limanov invariant . . . . . . . . . . . . 3.5.2 The fundamental group at infinity . . . . . . . 3.5.3 Algebraic surfaces and log Kodaira dimension 3.5.4 Logarithmic ramification formula . . . . . . . . Problems to Chapter 3 . . . . . . . . . . . . . . . . . .

Postscript 4.1 4.2 4.3

xv

. . . . . . . . . . .

. . . . . . . . . . .

357 363 369 369 372 378 378 381 385 387 390 407

AMS theorem and thereafter . . . . . . . . . . . . . . . . 407 Suzuki-Zaidenberg formula . . . . . . . . . . . . . . . . . . 408 Cancellation problems . . . . . . . . . . . . . . . . . . . . 409

Bibliography

411

Index

417

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Chapter 1

Introduction to Algebraic Geometry

1.1

Review on basic results in commutative algebra

All rings treated in this book are commutative and unitary unless otherwise specified. 1.1.1

Ring of quotients and local ring

A ring R is noetherian if every ascending chain of ideals I0 ⊆ I1 ⊆ · · · ⊆ In ⊆ In+1 ⊆ · · · ceases to increase. Namely, there exists an integer N such that In = In+1 for every n ≥ N . This condition is called the ascending chain condition for ideals (ACC, for short). The ACC is equivalent to the condition that every ideal is finitely generated. An R-module M is said to be finitely generated over R, or simply a finite R-module if M = Rm1 + · · · + Rmn for a finite set of generators {m1 , . . . , mn }. The quotient module M/N of a finite R-module M by an R-submodule N is a finite R-module, and a submodule N is also finite if R is noetherian (see Lemma 1.1.11). Conversely, if an R-submodule N of M and the quotient module M/N are finite, then so is the R-module M . A subset S of a ring R is a multiplicative set (or a multiplicatively closed set) if (i) 0 ̸∈ S, 1 ∈ S, and (ii) s, t ∈ S implies st ∈ S. If p is a prime ideal of R, then the complement S := R \p is a multiplicative set. In fact, the definition of prime ideal implies, by contrapositive, ab ̸∈ p if a ̸∈ p and b ̸∈ p. If S is a multiplicative set of R, the ring of quotients 1

2

Affine Algebraic Geometry

(or ring of fractions) S −1 R is defined as the set na o S −1 R = | a ∈ R, s ∈ S , s where a/s denotes the equivalence class in the product R × S under the equivalence relation (a, s) ∼ (b, t) if and only if u(at − bs) = 0 for some u ∈ S. Hence a/s is considered as a usual fraction whose numerator is an arbitrary element a ∈ R and whose denominator is an element s ∈ S. But a/s = b/t if (a, s) ∼ (b, t). If R is an integral domain, the above equivalence relation holds if and only if at = bs. The set S −1 R has a ring structure for which addition and multiplication are defined respectively by a2 a1 s2 + a2 s1 a1 a2 a1 a2 a1 + = , · = . s1 s2 s1 s2 s1 s2 s1 s2 There is a natural ring homomorphism i : R → S −1 R defined by i(a) = a/1. The kernel of i is an ideal I0 = {a ∈ R | as = 0 for some s ∈ S}. With this ring homomorphism i, there is a bijective correspondence between the ideals I of R such that I ⊇ I0 and I ∩ S = ∅ and the set of ideals of S −1 R. Here we consider only proper ideals, where I is a proper ideal of R if I ⫋ R. The bijective correspondence is given by o na | a ∈ I, s ∈ S , J 7→ i−1 (J). I 7→ I(S −1 R) = s This correspondence restricts to a bijection between {p | a prime ideal of R such that p ∩ S = ∅} and {P | a prime ideal of S −1 R}.

We denote the ideal i−1 (J) by J ∩ R by abuse of notation. This observation implies that S −1 R is noetherian if so is R. For example, let S = R \ p for a prime ideal p of R. We denote (S −1 R) by Rp and p(S −1 R) by pRp . Then pRp is the biggest prime ideal with respect to inclusion. Hence pRp is a unique maximal ideal1 of Rp . A ring R is called a local ring if it contains a unique maximal ideal m in the sense that any proper ideal I of R is contained in m. By the notation (R, m) we 1 An

ideal m of R is a maximal ideal if I is a proper ideal of R such that I ⊇ m then I = m.

Introduction to Algebraic Geometry

3

mean that R is a local ring with maximal ideal m. A maximal ideal m is a prime ideal. In fact, suppose that ab ∈ m. Let I = m + aR, which is an ideal such that I ⊇ m. Hence either I = R or I = m. If I = m then a ∈ m. Suppose that I = R. Then 1 = ax + z with z ∈ m. Then b = abx + bz ∈ m. So, a ∈ m or b ∈ m. Let R be an integral domain. Then S := R \ {0} is a multiplicative set. The ring of quotients S −1 R is now a field, which we call the field of quotients or the field of fractions and denote by Q(R). Let I be an ideal of R. An ideal of the quotient ring (or the residue ring) R = R/I is written in a form J/I, where J is an ideal of R such that J ⊇ I. If R is noetherian, so is the ring R. 1.1.2

Spectrum of a ring and Zariski topology

Let R be a ring. The set of prime ideals of R is called the spectrum (spec, for short) of R and denoted by Spec R = {p | a prime ideal of R}. Given a ring R, the spectrum Spec R is also called an affine scheme with the coordinate ring R. We can define a topology, called the Zariski topology, in Spec R by assigning closed sets satisfying axioms of topology. A closed set is V (I) for an ideal I of R, where V (I) = {p ∈ Spec R | p ⊇ I}. Axioms of topology for closed sets require (i) Spec R and ∅ are closed sets. T (ii) λ∈Λ V (Iλ ) is a closed set, where {Iλ | λ ∈ Λ} is possibly an infinite set. S (iii) A finite union j∈J V (Ij ) is a closed set. For (i), we have Spec R = V ((0)) and ∅ = V (R), where (0) is the zero T P ideal. For (ii), we have λ∈Λ V (Iλ ) = V ( λ∈Λ Iλ ). For (iii), it suffices to show that V (I1 ) ∪ V (I2 ) = V (I1 I2 ) = V (I1 ∩ I2 ), where I1 I2 is the ideal of R generated by {a1 a2 | a1 ∈ I1 , a2 ∈ I2 }. Since I1 ∩ I2 ⊇ I1 I2 and since p ⊇ I1 I2 implies either p ⊇ I1 or p ⊇ I2 , we have V (I1 ) ∪ V (I2 ) ⊇ V (I1 I2 ) ⊇ V (I1 ∩ I2 ) ⊇ V (I1 ) ∪ V (I2 ), which proves the assertion. The Zariski topology is T0 , but not necessarily T1 . Namely, if two distinct prime ideals p, q are given, there is a closed set

4

Affine Algebraic Geometry

V (I) which contains either one of p or q but not the other. One cannot choose p or q. In fact, p ⊂ q if and only if p ∈ V (I) always imply q ∈ V (I) for a closed set V (I). √ For an ideal I, define the radical of I, denoted by I, by √ I = {a ∈ R | an ∈ I for some n > 0}. √ An ideal I is a radical ideal if I = I. Then we have the following theorem. Theorem 1.1.1. For an ideal I of R we have \ √ I= p. p∈V (I)

√ p ⊇ I. We show the √ T opposite inclusion. If there exists an element s ∈ ( p∈V (I) p) \ I, then S = {sn | n ≥ 0} is a multiplicative set of R such that S ∩ I = ∅. Hence I(S −1 R) is a proper ideal of S −1 R. By Zorn’s lemma (see Corollary 1.1.4 below) there exists a maximal ideal M of S −1 R such that I(S −1 R) ⊂ M. Let m = R ∩ M(= i−1 (M)). Then m is a prime ideal of R such that m ⊃ I T and m ∩ S = ∅. Namely m ∈ V (I) and m ⊃ p∈V (I) p, whence s ∈ m. This contradicts m ∩ S = ∅. √ Corollary 1.1.2. For ideals I, J of R, V (I) = V (J) if and only if I = √ J. √ √ Proof. By Theorem 1.1.1, √ I = J follows if V √(I) = V (J). The converse is clear because V (I) = V ( I) and V (J) = V ( J). Proof. If p ∈ V (I) then p ⊃

√

I, whence

T

p∈V (I)

Let S be a partially ordered set. It is called an inductive set if every totally ordered subset has an upper bound. The following result is called Zorn’s lemma. Lemma 1.1.3. Let S be an inductive set. Then S has a maximal element. This result yields an important result. Corollary 1.1.4. Let I be a proper ideal of a ring R. Then there exists a maximal ideal m such that m ⊇ I. Further, a maximal ideal is a prime ideal. Proof. Let S be the set of proper ideals of R containing I and define a partial order in S by setting J1 ≥ J2 if and only if J1 ⊇ J2 .

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Let J1 ≤ J2 ≤ · · · ≤ Jn ≤ Jn+1 ≤ · · · be a totally ordered subset of S. Set S J = n≥1 Jn . Then J is a proper ideal of R containing I, and J is clearly an upper bound of the totally ordered subset. Hence S is an inductive set, and S has a maximal element m, which is a maximal ideal of R such that m ⊇ I. Let m be a maximal ideal. Suppose that ab ∈ m with a, b ∈ R. Then aR + m is an ideal containing m. Hence aR + m = m or aR + m = R. If aR + m = m then a ∈ m. If aR + m = R then ax + m = 1 for x ∈ R and m ∈ m. Then b = b(ax + m) = abx + bm ∈ m. So, either a ∈ m or b ∈ m. This implies that m is a prime ideal. 1.1.3

Irreducible decomposition of a topological space

A topological space X is noetherian if a descending chain of closed sets F1 ⊇ F2 ⊇ · · · ⊇ Fn ⊇ Fn+1 ⊇ · · · stops always to decrease, i.e., there exists N > 0 such that Fn = Fn+1 for every n ≥ N . If R is a noetherian ring then X = Spec R is a noetherian √ T space. In fact, write Fi = V (Ii ) with Ii = Ii = p∈Fi p. Then the descending chain of the Fi corresponds to an ascending chain of radical ideals of R I1 ⊆ I2 ⊆ · · · ⊆ In ⊆ In+1 ⊆ · · · and the termination of the ideal chain implies the termination of the chain of closed sets. A topological space X is quasi-compact if any open covering U = {Uλ }λ∈Λ of X has a finite open sub-covering X = U1 ∪ U2 ∪ · · · ∪ Un , where Ui = Uλi with λi ∈ Λ. Lemma 1.1.5. A noetherian topological space is quasi-compact. Proof. Let U = {Uλ }λ∈Λ be an open covering of X. We may assume S that for any µ ∈ Λ, λ∈Λ\{µ} Uλ ̸= X. Consider a well-ordering on Λ and suppose that λ1 < λ2 < · · · < λn < λn+1 < · · · and identify λi with i ∈ Z. Let Fn = X \ (U1 ∪ U2 ∪ · · · ∪ Un ), which is a closed set of X satisfying F1 ⊃ F2 ⊃ · · · ⊃ Fn ⊃ Fn+1 ⊃ · · · .

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Affine Algebraic Geometry

Since X is noetherian, the descending chain of closed sets ceases, i.e., there exists N > 0 such that Fn = Fn+1 for all n ≥ N . Then FN = ∅. Hence X = U1 ∪ U2 ∪ · · · ∪ Un . A topological space X is reducible if there is a decomposition X = F1 ∪F2 for two closed sets F1 , F2 with Fi ⫋ X. Otherwise, X is called irreducible. If F is a closed subset of X, we can say that F is reducible (or irreducible) with respect to the induced topology on F . Lemma 1.1.6. Let F be a closed subset of a noetherian topological space X. Then there exists a finite set of irreducible closed subsets F1 , . . . , Fn such that F = F1 ∪ F2 ∪ · · · ∪ Fn , Fi ̸⊂ Fj for all i ̸= j. These closed subsets Fi are determined by the subset F uniquely up to permutations. The set Fi is called the irreducible component of F and the decomposition F = F1 ∪ F2 ∪ · · · ∪ Fn is called the irreducible decomposition. Proof. We prove first the existence of a decomposition. Let S be the set of closed subsets F of X such that F is not a finite union of irreducible closed subsets. Then S is an inductive set with respect to an order defined by reverse inclusion of subsets. Namely, F ≤ F ′ for F, F ′ ∈ S if F ′ ⊆ F . Given an (ascending) totally ordered subset of S, there exists an upper bound by the noetherian condition of X which ceases descending chains of closed subsets of X. Hence S has a maximal element, say F0 . Then F0 is reducible. Write F0 = F1 ∪ F2 for proper closed subsets F1 , F2 of F0 . Then F1 > F0 and F2 > F0 . Since F0 is a maximal element of S, F1 and F2 are written as finite unions of irreducible closed subsets. Write the decompositions as F1 = F11 ∪ F12 ∪ · · · ∪ F1r

F2 = F21 ∪ F22 ∪ · · · ∪ F2s ,

where Fij is irreducible for i = 1, 2. Then we have F0 = F1 ∪ F2 = (F11 ∪ · · · ∪ F1r ) ∪ (F21 ∪ · · · ∪ F2s ), which is a finite union of irreducible closed subsets. This contradicts the assumption that F0 ∈ S. We prove next that a decomposition is unique up to permutations. Let F = G1 ∪ G2 ∪ · · · ∪ Gm

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be another irreducible decomposition of the same kind as in the statement. Then we have G1 = G1 ∩ F = (G1 ∩ F1 ) ∪ (G1 ∩ F2 ) ∪ · · · ∪ (G1 ∩ Fn ). Since G1 is irreducible, G1 = G1 ∩ Fi for some 1 ≤ i ≤ n. After a permutation of indices, we may assume that i = 1. Then G1 ⊆ F1 . Similarly, we have F1 = (F1 ∩ G1 ) ∪ · · · ∪ (F1 ∩ Gm ). Hence F1 = F1 ∩ Gj and hence F1 ⊆ Gj . This implies that G1 ⊆ Gj , whence j = 1. Namely F1 = G1 . We can argue as above by replacing F1 by Fj , and show that n = m and Fi = Gi after a suitable permutation of indices. 1.1.4

Prime ideal decomposition of radical ideals

Lemma 1.1.7. Let R be a noetherian ring and let X = Spec R. Let I be a radical ideal of R and let F = V (I). If F is irreducible then I is a prime ideal. Conversely, if I is a prime ideal then V (I) is irreducible. Proof. Suppose that ab ∈ I. Then, for any p ∈ F , ab ∈ I ⊆ p. Hence a ∈ p or b ∈ p. This implies that F ⊂ V (a) ∪ V (b), where V (a) = V (aR) and V (b) = V (bR), and F = (F ∩ V (a)) ∪ (F ∩ V (b)), where F ∩ V (a) = V (I + aR) and F ∩ V (b) = V (I + bR). Since F is irreducible, F = F ∩ V (a) or F = F ∩ V (b), i.e., F ⊆ V (a) or F ⊆ V (b). This implies that \ √ aR ⊆ p = I = I, or bR ⊆ I. p∈F

Hence a ∈ I or b ∈ I. So, I is a prime ideal. We prove the converse. Suppose that V (I) is reducible, and write V (I) = V (I1 ) ∪ V (I2 ) with V (I1 ) ⫋ V (I) and V (I2 ) ⫋ V (I). Since V (I1 ) ∪ V (I2 ) = V (I1 I2 ), it follows that I1 I2 ⊆ I. Since I is a prime ideal, either I1 ⊆ I or I2 ⊆ I. Then either V (I) ⊆ V (I1 ) of V (I) ⊆ V (I2 ). This is a contradiction. Corollary 1.1.8. Let R be a noetherian ring and let F = V (I) for a radical ideal I. Then there exists a uniquely determined set of prime ideals {p1 , . . . , pn } such that

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Affine Algebraic Geometry

(i) I = p1 ∩ p2 ∩ · · · ∩ pn , and (ii) pi ⊂ ̸ pj for any pair (i, j) with i ̸= j. The closed subsets V (p1 ), . . . , V (pn ) correspond bijectively with irreducible components F1 , . . . , Fn of F = V (I). Proof. There is an irreducible decomposition of F which is uniquely determined up to permutations of components F = F1 ∪ · · · ∪ Fn , where Fi = V (pi ) for a prime ideal pi . Then we have V (I) = V (p1 ) ∪ · · · ∪ V (pn ) = V (p1 ∩ · · · ∩ pn ), where p1 ∩ · · · ∩ pn is a radical ideal. Then it follows by Corollary 1.1.2 that I = p1 ∩ · · · ∩ pn . The decomposition I = p1 ∩ p2 ∩ · · · ∩ pn in Corollary 1.1.8 is called the prime decomposition2 of the radical ideal I. For a fixed 1 ≤ i ≤ n, write \ ∨ pj = p1 ∩ · · · ∩ pi−1 ∩ pi ∩pi+1 ∩ · · · ∩ pn j̸=i

and Y j̸=i

∨

pj = p1 · p2 · · · pi−1 · pi ·pi+1 · · · pn ,

Q T ∨ where pi shows that the ideal pi is omitted. Then j̸=i pj ⊆ j̸=i pj , and T hence j̸=i pj ̸⊂ pi because pi ̸⊂ pj for any pair (i, j). Let ai be an element T of ( j̸=i pj ) \ pi . For an element a ∈ R, the subset (I : a) = {x ∈ R | ax ∈ I} is called an ideal quotient of the ideal I. It is an ideal of R containing I. For the ideal quotient (I : ai ) it holds that (I : ai ) = pi . In fact, if x ∈ pi T then ai x ∈ ( j̸=i pj ) by the choice of ai and ai x ∈ pi because x ∈ pi . Hence ai x ∈ I and pi ⊆ (I : ai ). Conversely, if x ∈ (I : ai ) then ai x ∈ I and 2 Later

we need a finer decomposition of ideals, called the primary decomposition of ideals. We develop the theory in the appendix.

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ai ̸∈ pi . Hence x ∈ pi . This shows that pi = (I : ai ). Furthermore, if an ideal quotient (I : a) is a prime ideal p, then p⊇I⊇

n Y

pi .

i=1

Hence p contains some pi . We say that the ideal quotient (I : a) is a prime divisor of I if (I : a) is a prime ideal. The set of all prime divisors of I is denoted by Ass (R/I). Then each pi is a minimal among prime divisors of I with respect to the inclusion order. A non-minimal prime divisor of I is called embedded. √ The radical n = 0 of the zero ideal (0) of R is called the nilradical. Let √ n = 0 = p1 ∩ p2 ∩ · · · ∩ pn be the prime decomposition. Since any prime ideal p of R contains n, the above argument shows that p ⊇ pi for some 1 ≤ i ≤ n. This implies that X = Spec R = V (p1 ) ∪ V (p2 ) ∪ · · · ∪ V (pn ) and each V (pi ) is an irreducible component of X. 1.1.5

Generic point, closed point and Krull dimension

Let R be an noetherian ring and let X = Spec R. We have a prime ideal px identified with each point x ∈ X. For a subset S of X, we denote by S the closure of S with respect to the Zariski topology. A point x of X is a closed point if {x} = {x}. A point x is a generic point if X = {x}. Lemma 1.1.9. The following assertions hold. T (1) S = V (I(S)), where I(S) = x∈S px . (2) If S consists of a single point x then the closure {x} is irreducible. We have {x} = {y} if and only if x = y. (3) x is the generic point of an irreducible √ component of V (I) if and only if px is a minimal prime divisor of I. √ (4) X is irreducible if and only if the nilradical 0 of R is a prime ideal. (5) x is a closed point if and only if px is a maximal ideal. Proof. (1) If a closed set V (I) contains S then px ⊇ I for every p x ∈ S. This implies that I ⊆ I(S), where I(S) is a radical ideal, i.e., I(S) = I(S). This implies that I(S) defines the smallest closed subset of X which contains S, i.e., the closure of S.

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Affine Algebraic Geometry

(2) By (1), {x} = V (px ) which is irreducible by Lemma 1.1.7. We have {x} = {y} if and only if V (px ) = V (py ), which is equivalent to px = py , i.e., x = y. (3) and (4) The assertion follows from Corollary 1.1.8. (5) If {x} = {x} then px is a maximal ideal of R. For, otherwise, there is a maximal ideal m ⫌ px and m ∈ {x}. The converse is clear by definition. A series of irreducible closed subsets of X = Spec R X0 ⫌ X1 ⫌ . . . ⫌ Xn corresponds to a sequence of prime ideals p0 ⫋ p1 ⫋ . . . ⫋ pn . The index n is called the length of the sequence. If a sequence of maximal length (which could be ∞) exists, the length is called the Krull dimension of X (or R) and denoted by K-dim X (or K-dim R). Later we will see that if R is an affine domain over a field k then K-dim X is equal to the transcendence degree tr.degk K, where K = Q(R) is the field of quotients of R. Let p be a prime ideal of R. We consider, in particular, a descending chain of prime ideals with p as the beginning term p = p0 ⫌ p1 ⫌ · · · ⫌ pr .

(1.1)

If there is a finite chain such as (1.1) with maximal length r, the integer r is called the height of p and denoted by ht p. If there are chains like (1.1) of length arbitrarily big, we define ht p = ∞. Let I be an ideal of R and let √ I = p1 ∩ p2 ∩ · · · ∩ pm √ be a prime decomposition of I. Then we define the height of I by ht I = min ht pi . 1≤i≤m

Let Ω(R) denote the set of all maximal ideals of R. We define the dimension of R (or X = Spec R) by dim R = dim X = max ht (M). M∈Ω(R)

Let I be an ideal defining a closed subset V (I) of X, we define the codimension of V (I) by codim V (I) = ht I.

Introduction to Algebraic Geometry

1.1.6

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Hilbert basis theorem

The following result is called Hilbert basis theorem. Theorem 1.1.10. Let R be a noetherian ring and let A = R[x1 , . . . , xn ] be a polynomial ring in n variables over R. Then A is also a noetherian ring. Proof. It suffices to prove the assertion in the case n = 1. Let I be an ideal of R[x] and let a = {a ∈ R | there exists f (x) = axn + (terms of lower degree) in I}. Pr Then a is an ideal of R. Write a = i=1 ai R. Choose fi ∈ I so that fi = ai xni + (terms of lower degree)

and let m = max{ni | 1 ≤ i ≤ r}. We may assume that n1 = · · · = nr = m. Let M = R · 1 + R · x + · · · + R · xm−1 , which is an R-submodule of A, and N = I ∩ M . Then N is an R-submodule of M . Since M is a finite Ps R-module, so is N by Lemma 1.1.11 below. Write N = j=1 Rgj . Then we claim that r s X X I= R[x]fi + R[x]gj . i=1

j=1

Since fi , gj ∈ I, one inclusion ⊇ is clear. We prove the reverse one. Let f ∈ I. Write f = axn + (terms of lower degree) ∈ I. Then a ∈ a. Pr Hence a = i=1 ai bi . If n ≥ m, let ! r X ′ f =f− bi fi xn−m ∈ I. i=1

′

Then deg f < n. By induction on deg f we can write f=

r X

fi hi + g,

i=1

g ∈ M.

Ps Since g ∈ I, we have g ∈ N . Hence g = j=1 cj gj with cj ∈ R. So, we have r s r s X X X X f= fi hi + cj gj ∈ fi R[x] + gj R[x]. i=1

j=1

i=1

j=1

Lemma 1.1.11. Let R be a noetherian ring and let M be a finite R-module. Then an R-submodule N of M is finite.

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Affine Algebraic Geometry

Proof. Write M = Rm1 + · · · + Rmr with generators m1 , . . . , mr . We prove the assertion by induction on the number r of generators. Suppose r = 1. Then M = Rm1 . Let a1 = {a ∈ R | am1 ∈ N }. Then a1 is Pr1 Ra1i . Let an ideal of R. Since R is noetherian, we can write a1 = i=1 ni = a1i m1 ∈ N . Take z ∈ N and write it as z = am1 . Then a ∈ a1 and Pr1 Pr1 Pr1 Rni . b1i ni . Hence N = i=1 b1i a1i . Then z = i=1 hence a = i=1 Suppose that r > 1. Let M2 = Rm2 + · · · + Rmr and N2 = M2 ∩ N . Let a = {a ∈ R | there exists an element u = am1 + a2 m2 + · · · + ar mr ∈ N }. Ps Then a is an ideal of R. Hence a = i=1 Rci . Let ui be an element of N such that r X ui = ci m1 + cij mj . j=2

Let z ∈ M and write z = am1 + a2 m2 + · · · + ar mr ,

a=

s X

bi c i .

i=1

Then z−

s X i=1

bi u i ∈ M 2 ∩ N = N2 .

By induction hypothesis, we can write N2 = Ps Pt i=1 Rui + j=1 Rvj .

Pt

j=1

Rvj .

Then N =

Let k be a field. A finitely generated k-algebra is the quotient ring A = k[x1 , . . . , xn ]/I of a polynomial ring k[x1 , . . . , xn ]. If A is further an integral domain, i.e., I is a prime ideal, we call A an affine domain over k or simply an affine k-domain. Corollary 1.1.12. Let A be a finitely generated k-algebra over a field k. Then A is noetherian. 1.1.7

Integral extension and Noether normalization lemma

Let R be a ring and S a subring of R. An element a ∈ R is integral over S if a is a root of an equation f (x) = 0 for a monic polynomial f (x) = xn + b1 xn−1 + · · · + bn ∈ S[x].

(1.2)

Lemma 1.1.13. The following conditions for an element a of R are equivalent.

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(1) a is integral over S. (2) The subring S[a] of R generated by a over S is a finite S-module. (3) There is a subring T of R such that S ⊂ T ⊂ R, a ∈ T and T is a finite S-module. Proof. (1) ⇒ (2). If r ≥ n with n = deg f (x) in the equation f (a) = 0 (see (1.2)), ar is a linear combination of 1, a, . . . , an−1 with coefficients in S. So, S[a] is generated by 1, a, . . . , an−1 over S. Hence the S-algebra S[a] of R is a finite S-module. (2) ⇒ (3). Take T = S[a]. (3) ⇒ (1). Write T = Sm1 + Sm2 + · · · + Smn . Since a ∈ T and T is an S-algebra, we have ami = bi1 m1 + bi2 m2 + · · · + bin mn ,

bij ∈ S.

Let B be an (n × n)-matrix (bij )1≤i,j≤n . Then we have   m1  m2    (aEn − B)  .  = 0,  .. 

(1.3)

mn where En = (δij )1≤i,j≤n is the (n × n) identity matrix with the Kronecker delta δij . Let D = aEn − B and let D∗ be the cofactor matrix of D. As an important result in linear algebra, we have equalities D∗ D = DD∗ = dEn , where d = det D. Then, by multiplying D∗ to both hand sides of the equality (1.3) from the left, we have Since we have we have Hence we have

dm1 = dm2 = · · · = dmn = 0. c1 m1 + c2 m2 + · · · + cn mn = 1,

ci ∈ S,

d = c1 dm1 + c2 dm2 + · · · + cn dmn = 0. a − b11 . d = −bji . . −bij a − bnn

= an − tr(B)an−1 + · · · + (−1)n det B = 0.

Namely, a is integral over S.

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Affine Algebraic Geometry

Corollary 1.1.14. Let S ⊂ R be the same as in Lemma 1.1.13. If a1 , a2 ∈ R are integral over S then so are a1 + a2 and a1 a2 . Proof. The subalgebra S[a1 , a2 ] of R is a finite S-module because a1 , a2 are integral over S. In fact, S[a1 , a2 ] are generated by monomials ar11 ar22 over S, and monomials ar11 ar22 with higher degree r1 , r2 are linear combinations of monomials of a1 , a2 with lower degree over S. Hence a1 + a2 and a1 a2 as elements of S[a1 , a2 ] are integral over S by Lemma 1.1.13. Let Se be the set of all elements of R which are integral over S. By Corollary 1.1.14, Se is a subalgebra of R containing S. We call Se the integral closure of S in R. If Se = R, i.e., every element of R is integral over S, we say that R is an integral extension of R. If Se = S, we say that S is integrally closed in R. If S is an integral domain and integrally closed in the field of quotients Q(S), we simply say that S is an integrally closed domain. Further, if S is noetherian and integrally closed in Q(S), S is called a normal domain. Lemma 1.1.15. Let R be an integrally closed domain and let S be a multiplicative set. Then S −1 R is also an integrally closed domain. Proof. Let ξ be an element of Q(R) = Q(S −1 R) such that ξ n + (a1 /s1 )ξ n−1 + · · · + (ai /si )ξ n−i + · · · + (an /sn ) = 0, ai /si ∈ S −1 R. We may assume that s1 = · · · = sn = s ∈ S. By multiplying sn to the above monic equation of ξ, we obtain (sξ)n + a1 (sξ)n−1 + · · · + ai si−1 (sξ)n−i + · · · + sn−1 an = 0. Hence sξ is integral over R, and sξ ∈ R because R is integrally closed. Hence ξ ∈ S −1 R. Corollary 1.1.16. Let T1 ⊃ T2 be subalgebras of R containing S. If T1 is integral over T2 and T2 is integral over S then T1 is integral over S. Proof. Let a ∈ T1 . Then a satisfies a monic equation am + c1 am−1 + · · · + cm = 0,

c1 , . . . , cm ∈ T2 .

Then S[c1 , . . . , cm ] is a finite S-module because ci is integral over S. Further, S[c1 , . . . , cm ][a] is a finite S-module because it is a finite S[c1 , . . . , cm ]module. This implies that a is integral over S.

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Let k be a field and let A be an affine domain over k. Elements a1 , . . . , an of A are algebraically dependent over k if there exists a nonzero polynomial f (x1 , . . . , xn ) ∈ k[x1 , . . . , xn ] such that f (a1 , . . . , an ) = 0. If there are no such polynomials, we say that a1 , . . . , an are algebraically independent. In particular, any nonempty subset of generators {x1 , . . . , xn } of the polynomial ring k[x1 , . . . , xn ] consists of algebraically independent elements. The following result is called Noether normalization lemma, but we state it as a theorem. Theorem 1.1.17. Let k be a field and let A be an affine domain over k. Then there exist elements x1 , . . . , xn of A satisfying the following conditions. (i) x1 , . . . , xn are algebraically independent over k. Hence A contains a polynomial ring k[x1 , . . . , xn ] as a k-subalgebra. (ii) A is integral over k[x1 , . . . , xn ]. (iii) The field of quotients L of A is an algebraic extension of k(x1 , . . . , xn ) which is the field of quotients of k[x1 , . . . , xn ]. Proof. Since A is an affine k-domain, we can write A = k[Y1 , . . . , Ym ]/P, where P is a prime ideal of a polynomial ring k[Y1 , . . . , Ym ] over k. Let yi = Yi + P be the residue class of Yi . Then A = k[y1 , . . . , ym ]. If P ̸= 0, then there exists a nonzero polynomial f (Y1 , . . . , Ym ) ∈ P such that f (y1 , . . . , ym ) = 0. Let zi = yi − y1ri (2 ≤ i ≤ m).

If we take the ri so that then we have

0 ≪ r1 ≪ r2 ≪ · · · ≪ rm ,

f (y1 , . . . , ym ) = f (y1 , z2 + y1r2 , . . . , zm + y1rm )   terms in y1 of degree < N with = by1N + coefficients in k[z2 , . . . , zm ] = 0, where b ∈ k\{0}. Hence y1 is integral over k[z2 , . . . , zm ]. Since yi = zi +y1ri , A = k[y1 , . . . , ym ] is integral over A2 := k[z2 , . . . , zm ]. Write again A2 = k[Z2 , . . . , Zm ]/P2 , where P2 is a prime ideal. If P2 ̸= 0 we can apply the same process as above and reduce the number of generators by one. Repeating this process, we eventually reach to a subalgebra k[x1 , . . . , xn ] such that

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Affine Algebraic Geometry

(1) A = k[y1 , . . . , ym ] is integral over k[x1 , . . . , xn ], and (2) there are no nontrivial algebraic relations among x1 , . . . , xn . Then k[x1 , . . . , xn ] is a polynomial ring over k in n variables. The third condition is now obvious. 1.1.8

Lying-over theorem and Going-up theorem

Our aim is to show that if A is an affine k-domain as in Theorem 1.1.17 then K-dim A is equal to the number n obtained in the theorem. For this purpose we need two theorems on integral extensions of integral domains. We begin with a lemma. Lemma 1.1.18. Let R′ ⊃ R be an integral extension of integral domains. Then the following assertions hold. (1) If R′ is a field, so is R. (2) If R is a field, so is R′ . Proof. (1) Let a ∈ R\{0}. Then there exists the inverse a−1 in R′ because R′ is a field. So, a−1 is integral over R, and hence satisfies a monic relation (a−1 )n + a1 (a−1 )n−1 + · · · + an = 0,

ai ∈ R.

With an multiplied to the relation, we have 1 + a1 a + · · · + an an = 0, which is written as
a −(a1 + · · · + an an−1 ) = 1. So, a−1 = −(a1 + · · · + an an−1 ) ∈ R. This implies that R is a field. (2) Let a′ ∈ R′ \ {0}. Then a′ satisfies a monic relation (a′ )n + a1 (a′ )n−1 + · · · + an = 0,

ai ∈ R, an ̸= 0.

Since R is a field and an ̸= 0, there exists the inverse a−1 n in R. So, the monic relation is written as  ′ n−1 1 = (a′ ) −a−1 − · · · − a−1 n (a ) n an−1 . Hence  ′ (a′ )−1 = − an−1 (a′ )n−1 + · · · + a−1 n an−1 ∈ R .

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17

Theorem 1.1.19 (Lying-over theorem). Let R′ ⊃ R be an integral extension of integral domains. Then the following assertions hold. (1) For any prime ideal p of R, there exists a prime ideal p′ of R′ such that p′ ∩ R = p. We then say that p′ lies over p. (2) If two prime ideals p′1 , p′2 of R′ lie over p, there is no inclusion relation between p′1 and p′2 , i.e., p′1 ̸⊆ p′2 , p′2 ̸⊆ p′1 . Proof. (1) Let p ∈ Spec R. Set S = R \ p. Then S is a multiplicative subset in R and R′ , and S −1 R ⊂ S −1 R′ . If e p′ ∈ Spec S −1 R′ satisfies ′ −1 −1 ′ ′ ′ e p ∩ (S R) = p(S R), then p := e p ∩ R satisfies p′ ∩ R = (e p′ ∩ S −1 R) ∩ R = p(S −1 R) ∩ R = p.

Hence, by replacing R and R′ by S −1 R and S −1 R′ , we may assume that R is a local ring with maximal ideal p. Let M′ be any maximal ideal of R′ and let q = M′ ∩ R. In fact, M′ exists as a maximal ideal of R′ containing pR′ by Corollary 1.1.4. Then R′ /M′ ⊃ R/q is an integral extension. By Lemma 1.1.18, R/q is a field. Hence q is a maximal ideal of R, i.e., q = p. Now we can take p′ = M′ . (2) Suppose that p′1 ⊇ p′2 , p′1 ∩ R = p′2 ∩ R = p for p′1 , p′2 ∈ Spec R′ . It suffices to show that p′1 (S −1 R′ ) = p′2 (S −1 R′ ). In fact, this implies that p′1 = p′2 because p′1 (S −1 R′ ) ∩ R′ = p′1 ,

p′2 (S −1 R′ ) ∩ R′ = p′2 .

Hence we may assume that (R, p) is a local ring. Then R′ /p′2 ⊃ R/p is integral, whence R′ /p′2 is a field by Lemma 1.1.18. So, p′1 /p′2 = 0, i.e., p′1 = p′2 . Theorem 1.1.20 (Going-up theorem). Let R′ ⊃ R be an integral extension of integral domains. Given any sequence of prime ideals of R pn ⫋ pn−1 ⫋ · · · ⫋ p1 , there exists a sequence of prime ideals of R′ p′n ⫋ p′n−1 ⫋ · · · ⫋ p′1 such that p′i ∩ R = pi for 1 ≤ i ≤ n. Proof. By induction on the length n of a given sequence, it suffices to prove the result for n = 2. So, suppose that we are given a sequence p2 ⫋ p1 of prime ideals of R. By Theorem 1.1.19 there exists a prime ideal p′2 of R′ such that p′2 ∩ R = p2 . Let S1 = R \ p1 . If we find a prime ideal e p′1 of

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S1−1 R′ such that p′2 (S1−1 R′ ) ⊆ e p′1 and e p′1 ∩ (S1−1 R) = p1 (S1−1 R), we can ′ ′ ′ take p1 := e p1 ∩ R which satisfies p′2 ⊆ p′1 ,

p′1 ∩ R = e p′1 ∩ (S1−1 R) ∩ R = p1 (S1−1 R) ∩ R = p1 .

Since p2 ⫋ p1 , p′2 ⫋ p′1 follows. So, replacing R and R′ by S1−1 R and S1−1 R′ , we may assume that (R, p1 ) is a local ring. Under this assumption, choose any maximal ideal p′1 of R′ containing p′2 . Then R′ /p′1 ⊃ R/(p′1 ∩ R) is an integral extension. Since R′ /p′1 is then a field, R/(p′1 ∩ R) is a field by Lemma 1.1.18. Hence p′1 ∩ R is the maximal ideal, which implies that p′1 ∩ R = p1 . We have the following consequence of these theorems. Corollary 1.1.21. Let R′ ⊃ R be an integral extension of noetherian integral domains. Then we have: (1) K-dim R < ∞ if and only if K-dim R′ < ∞. (2) Suppose K-dim R < ∞. Then dim R = dim R′ . Proof. (1) Suppose n = K-dim R < ∞. Then there exists a chain of prime ideals of R p0 ⫌ p1 ⫌ · · · ⫌ pn . By Lying-over theorem, there exists p′n ∈ Spec R′ such that p′n ∩ R = pn . By Going-up theorem there exists a chain of prime ideals of R′ p′0 ⫌ p′1 ⫌ · · · ⫌ p′n such that p′i ∩ R = pi for 0 ≤ i ≤ n. Hence K-dim R′ ≥ n. If R′ has a chain of prime ideals of length n′ p′0 ⫌ p′1 ⫌ · · · ⫌ p′n′ , set pi = p′i ∩ R. Then, by Lying-over theorem, p0 ⫌ p1 ⫌ · · · ⫌ pn′ . Since K-dim R = n < ∞, this implies n′ ≤ n. Hence K-dim R′ = n < ∞. By a similar argument, K-dim R′ < ∞ implies K-dim R < ∞. (2) If K-dim R < ∞ then dim R = K-dim R. By (1), we have dim R = dim R′ .

Introduction to Algebraic Geometry

1.1.9

19

Krull dimension of affine domains

Let A be an affine domain over a field k. By Noether normalization lemma, A is an integral extension of a polynomial ring k[x1 , . . . , xn ]. We understand that n = 0 if and only if A is an algebraic field extension of k. Theorem 1.1.22. With the notations and assumptions as above, we have K-dim A = dim A = n. Proof. We proceed by induction on n. If n = 0 then A is a field and (0) is the unique ideal of A which is not equal to A. Hence K-dim A = 0 = dim A. We suppose that the assertion holds for the case of the number of polynomial generators being less than n. Set B = k[x1 , . . . , xn ]. Let (0) = p0 ⫋ p1 ⫋ p2 ⫋ · · · ⫋ pr be a chain of prime ideals of B. Let B = B/p1 = k[x1 , . . . , xn ], where xi = xi + p1 . Then a nonzero polynomial in p1 gives rise to a nontrivial algebraic relation among x1 , . . . , xn . Hence, by the proof of Theorem 1.1.17, B is integral over a polynomial ring over k[y1 , . . . , ym ] with m ≤ n − 1. On the other hand, we have a chain of prime ideals in B, (0) = p1 /p1 ⫋ p2 /p1 ⫋ · · · ⫋ pr /p1 . Hence, by the induction hypothesis, we have r − 1 ≤ K-dim B ≤ n − 1. So, r ≤ n. On the other hand, we have a chain of prime ideals in B (0) ⫋ (x1 ) ⫋ (x1 , x2 ) ⫋ · · · ⫋ (x1 , . . . , xn ). Hence K-dim B ≥ n. This implies that K-dim B = n < ∞. By Corollary 1.1.21, we have K-dim A = K-dim B = n. This gives the equality dim A = dim B = n.

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Affine Algebraic Geometry

1.2

Review on finitely generated field extensions

1.2.1

Transcendence basis and transcendence degree

Let k be a field and let L/k be a field extension. We say that the extension L/k is finitely generated if L is the field of quotients Q(A) of an affine kdomain A. Write A = k[a1 , . . . , ar ] and L = k(a1 , . . . , ar ). If a1 , . . . , ar are variables over k, i.e., k[a1 , . . . , ar ] is a polynomial ring, we say that L = k(a1 , . . . , ar ) is a purely transcendental extension (or a rational function field). By Noether normalization lemma (see Theorem 1.1.17), there exists a set of variables {x1 , . . . , xn } such that L is an algebraic extension of Q(k[x1 , . . . , xn ]) = k(x1 , . . . , xn ). Let L/k be a field extension and let ξ ∈ L. If ξ is not algebraic over k we say that ξ is transcendental (or algebraically independent) over k. Let ξ1 , . . . , ξr be elements of L. Then ξ1 , . . . , ξr are transcendental over k if, for every 1 ≤ i ≤ r, ξi is transcendental over k(ξ1 , . . . , ξi−1 ). Lemma 1.2.1. Let ξ1 , . . . , ξr be elements of L. Then the following conditions are equivalent. (1) ξ1 , . . . , ξr are transcendental over k. (2) For any permutation σ of {1, . . . , r}, ξσ(1) , . . . , ξσ(r) are transcendental over k. Proof. It suffices to show that (1) ⇒ (2). Since every permutation is a product of transpositions (i − 1, i),3 it suffices to show that if ξj is transcendental over k(ξ1 , . . . , ξj−1 ) for every 1 ≤ j ≤ i, then ξi−1 is transcendental over k(ξ1 , . . . , ξi−2 , ξi ). Suppose that ξi−1 is algebraic over k(ξ1 , . . . , ξi−2 , ξi ). Then there exists an algebraic relation a0 (ξi−1 )N + a1 (ξi−1 )N −1 + · · · + aN = 0,

(1.4)

where aj ∈ k[ξ1 , . . . , ξi−2 , ξi ], a0 aN ̸= 0 and gcd(a0 , . . . , aN ) = 1. Note that k[ξ1 , . . . , ξi−2 , ξi ] is a polynomial ring over k by the hypothesis. Since it is a UFD (see subsection 1.5.4 below), we can consider gcd(a0 , . . . , aN ). The hypothesis implies that there exists aj such that aj ̸∈ k[ξ1 , . . . , ξi−2 ]. For every 1 ≤ j ≤ N , we can write aj as a polynomial in ξi with coefficients in k[ξ1 , . . . , ξi−2 ], Nj

aj = aj0 ξi 3 It

+ (terms of lower degree in ξi ), aj0 ∈ k[ξ1 , . . . , ξi−2 ],

is well-known that any permutation is a product of transpositions (i, j), and if i < j then (i, j) = (i, i + 1)(i + 1, i + 2) · · · (j − 2, j − 1)(j − 1, j)(j − 2, j − 1) · · · (i, i + 1).

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where aj0 ̸= 0 if aj ̸∈ k[ξ1 , . . . , ξi−2 ] and Nj = 0 if aj ∈ k[ξ1 , . . . , ξi−2 ]. Let M = max Nj . 0≤j≤N

Then the equation (1.4) becomes   X  aj0 (ξi−1 )N −j  ξiM + (terms of lower degree in ξi ) = 0. (1.5) Nj =M

Since ξi−1 is transcendental over k(ξ1 , . . . , ξi−2 ), X aj0 (ξi−1 )N −j ̸= 0. Nj =M

Hence the equation (1.5) shows that ξi is algebraic over k(ξ1 , . . . , ξi−1 ). This is a contradiction. Let L/k be a finitely generated field extension. A set of elements {ξ1 , . . . , ξn } of L such that ξ1 , . . . , ξn are algebraically independent over k and L is algebraic over k(ξ1 , . . . , ξn ) is called a transcendence basis. The following result shows that L has a transcendence basis and every such basis has the same number of elements, which we call the transcendence degree of L/k and denote it by tr.degk L. Lemma 1.2.2. Let L = k(y1 , . . . , yr ) be a finitely generated extension of k. Then the following assertions hold. (1) If ξ1 , . . . , ξn are elements of L which are transcendental over k then n ≤ r. Hence a transcendence basis of L exists. (2) Two transcendence bases of L/k have the same number of elements. (3) We can take a transcendence basis {x1 , . . . , xn } as a subset of the system of generators {y1 , . . . , yr } of L/k. Proof. We choose a maximal set of transcendental elements {x1 , . . . , xn } as a subset of the set of generators {y1 , . . . , yr }. If y1 is transcendental over k, we keep it. If y1 is algebraic over k, we discard it. Inductively, if yi is transcendental over k(y1 , . . . , yi−1 ), we keep it. If yi is algebraic over k(y1 , . . . , yi−1 ), we discard it. After finishing this process up to the element yr , let {x1 , . . . , xn } be the set of elements of {y1 , . . . , yr } which we kept through these processes. It is then clear that n ≤ r and {x1 , . . . , xn } is a maximal set of transcendental elements of L over k, i.e., a transcendence basis of L/k.

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Affine Algebraic Geometry

Suppose that ξ1 , . . . , ξs are transcendental elements of L over k. Since ξ1 is algebraic over k(x1 , . . . , xn ), there is an algebraic relation a0 ξ1N + a1 ξ1N −1 + · · · + aN = 0,

(1.6)

where ai ∈ k[x1 , . . . , xn ] and a0 aN ̸= 0. Since ξ1 is transcendental over k, some aj ̸∈ k. It follows that some aj contains a nonzero monomial term in x1 , . . . , xn . After a permutation of indices {1, 2, . . . , n}, we may assume by Lemma 1.2.1 that this monomial term contains x1 . Write N

aj = aj0 x1 j + (terms of lower degree in x1 ), where the coefficients of aj including aj0 are elements of k[x2 , . . . , xn ] such that aj0 ̸= 0 if aj ̸= 0 and Nj = 0 if aj = 0. Let M = max Nj . 1≤j≤N

Then the equation (1.6) is written as   X N −j  aj0 ξ1  xN 1 + (terms of lower degree in x1 ) = 0, Nj =M

where 

 X

 Nj =M

aj0 ξ1N −j  ̸= 0.

Hence x1 is algebraic over k(ξ1 , x2 , . . . , xn ), and {ξ1 , x2 , . . . , xn } is a transcendence basis of L over k. Since ξ2 is algebraic over k(ξ1 , x2 , . . . , xn ) and ξ2 is transcendental over k(ξ1 ), an algebraic relation for ξ2 with coefficients in k[ξ1 , x2 , . . . , xn ] has a nonzero coefficient which contains some variables among x2 , . . . , xn , say x2 . By the same argument as above, ξ2 is replaced by x2 . Thus x2 is algebraic over k(ξ1 , ξ2 , x3 , . . . , xn ) and {ξ1 , ξ2 , x3 , . . . , xn } is a transcendence basis of L over k. By induction on s, suppose that {ξ1 , . . . , ξi , xi+1 , . . . , xn } is a transcendence basis of L over k, where i < s. Then ξi+1 is algebraic over k(ξ1 , . . . , ξi , xi+1 , . . . , xn ). Since ξi+1 is transcendental over k(ξ1 , . . . , ξi ), we may replace by the above argument ξi+1 by one of xi+1 , . . . , xn , say xi+1 . Then {ξ1 , . . . , ξi+1 , xi+2 , . . . , xn } is a transcendence basis of L over k. Thus we can replace x1 , . . . , xs by ξ1 , . . . , ξs . Hence s ≤ n. If {ξ1 , . . . , ξs } is a transcendence basis of L over k, we can convert the roles of {x1 , . . . , xn } and {ξ1 , . . . , ξs } and conclude that n ≤ s.

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By Noether normalization lemma and Theorem 1.1.22, we obtain the following result. Theorem 1.2.3. Let A be an affine domain over a field k. Then dim A = tr.degL, where L is the field of quotients of A. We prove the following result. Theorem 1.2.4. Let L/k be a finitely generated field extension and let M be an intermediate field extension, i.e., k ⊂ M ⊂ L. Then M/k is a finitely generated field extension. Further, tr.degk M ≤ tr.degk L. Proof. (1) Case: M/k is an algebraic extension. Let {x1 , . . . , xn } be a transcendence basis of L/k. Since M (x1 , . . . , xi−1 )/k(x1 , . . . , xi−1 ) is algebraic, xi is transcendental over M (x1 , . . . , xi−1 ). Hence {x1 , . . . , xn } is transcendental over M . Let {m1 , . . . , mt } be elements of M which are linearly independent over k. We then claim that {m1 , . . . , mt } are linearly independent over k(x1 , . . . , xn ). In fact, suppose we have a linear relation f1 m1 + · · · + ft mt = 0, fj ∈ k(x1 , . . . , xn ). Replacing f1 , . . . , ft by gf1 , . . . , gft with g ∈ k[x1 , . . . , xn ], we may assume that f1 , . . . , ft ∈ k[x1 , . . . , xn ]. Write X α fj = a(j) α x , α α

1 xα 1

(j)

n where α = (α1 , . . . , αn ), x = · · · xα n and aα ∈ k. Then we have   t X X   xα = 0. a(j) α mj

α

j=1

Since x1 , . . . , xn are transcendental over M , we have t X

a(j) α mj = 0.

j=1

Hence

(j) aα

= 0 for all j and α. Namely fj = 0 for all j. Hence t ≤ [L : k(x1 , . . . , xn )] < ∞.

So, M/k is a finite algebraic extension. (2) Case: M is not necessarily algebraic over k. Choose a transcendence basis {x1 , . . . , xn } of L/k such that {x1 , . . . , xm } is a maximal system of transcendental elements of M/k. Then m ≤ n. In fact,

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Affine Algebraic Geometry

{x1 , . . . , xm , . . . , xn } is a maximal system of transcendental elements of L/k which extends the system {x1 , . . . , xm }. Then L/k(x1 , . . . , xm ) is a finitely generated field extension and M/k(x1 , . . . , xm ) is algebraic. By the case (1), [M : k(x1 , . . . , xm )] < ∞. Hence M is finitely generated over k. It is now clear that {x1 , . . . , xm } is a transcendence basis of M/k. This shows that tr.degk M ≤ tr.degk L. 1.2.2

Regular extension and separable extension

We begin with the following result. Lemma 1.2.5. Let L/k be a finitely generated field extension. Then the following conditions are equivalent. (i) For every finite algebraic extension k ′ /k, the tensor product L ⊗k k ′ is an integral domain. (ii) For every algebraic extension k ′ /k, the tensor product L ⊗k k ′ is an integral domain. (iii) For every algebraic extension k ′ /k, the tensor product L⊗k k ′ is a field. (iv) For an algebraic closure k, the tensor product L ⊗k k is a field. Proof. (i) ⇒ (ii). If L ⊗k k ′ is not an integral domain for an algebraic extension k ′ /k, then there exist two elements ξ ′ , η ′ of L ⊗k k ′ such that ξ ′ η ′ = 0 and ξ ′ = u1 ⊗ α1 + · · · + um ⊗ αm ,

η ′ = v1 ⊗ β1 + · · · + vn ⊗ βn ,

where ui , vj ∈ L and αi , βj ∈ k ′ . Let k ′′ = k(α1 , . . . , αm , β1 , . . . , βn ). Then k ′′ /k is a finite algebraic extension and ξ ′ , η ′ ∈ L ⊗k k ′′ . So, L ⊗k k ′′ is not an integral domain. (ii) ⇒ (iii). L is a subalgebra of L ⊗k k ′ , and L ⊗k k ′ is integral over L. By Lemma 1.1.18, L ⊗k k ′ is a field if L ⊗k k ′ is an integral domain. (iii) ⇒ (iv). This is clear because k/k is an algebraic extension. (iv) ⇒ (i). Let k ′ /k be a finite algebraic extension. Then k ′ is isomorphic to a subfield of k. Then L ⊗k k ′ is a subalgebra of L ⊗k k. Since L ⊗k k is a field, the subalgebra L ⊗k k ′ is an integral domain. A finitely generated field extension L/k is a regular extension if it satisfies the equivalent four conditions of Lemma 1.2.5. If L/k is a regular extension, then k is algebraically closed in L. In fact, if α ∈ L is algebraic over k which is not in k then k ′ := k(α) is a finite algebraic extension. Then

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the inclusion k ′ ,→ L induces the inclusion k ′ ⊗k k ,→ L ⊗k k, while k ′ ⊗k k is not an integral domain. So, L ⊗k k is not an integral domain. Lemma 1.2.6. Let A = k[a1 , . . . , an ] be an affine domain over a field k and let L be the field of quotients of A. Then the following conditions are equivalent. (1) A ⊗k k is an integral domain, where k is an algebraic closure of k. (2) L/k is a regular extension. Proof. (1) ⇒ (2). Let S = A \ {0} and let S = S ⊗ 1 = {a ⊗ 1 | a ∈ S} ⊂ A ⊗k k. Then S is a multiplicative set of A, S is identified with S and S is −1 hence a multiplicative set of A ⊗k k. Since k is integral over k, S (A ⊗k k) −1 is integral over S −1 A = L. Since S (A ⊗k k) is an integral domain, it is a field by Lemma 1.1.18 and hence identified with L ⊗k k. Thus L/k is a regular extension. (2) ⇒ (1). Since k/k is a flat extension (or since k is a direct summand of a k-module k), the natural inclusion A ,→ L induces an injection A ⊗k k ,→ L ⊗k k. Since L ⊗k k is an integral domain, so is A ⊗k k. The following is an example of an affine domain A/k such that A ⊗k k is not an integral domain. Example 1.2.7. Let A = R[x, y]/(x2 + y 2 ) over the real number field R. We denote the residue classes of x, y modulo (x2 + y 2 ) by the same letters. Hence they satisfy x2 + y 2 = 0. Let S = {xn | n ≥ 0}, which is a multiplicative set of A, and S −1 A = R[x, x−1 , y/x], where (y/x)2 = −1. Since R[y/x] ∼ = C the complex number field, we have S −1 A ∼ = C[x, x−1 ]. −1 On the other hand, the natural homomorphism A → S A is injective, A is an integral domain. Hence A is an affine domain over R. Meanwhile, √ √ A ⊗R C ∼ = C[x, y]/(x2 + y 2 ) has a relation (x + −1y)(x − −1y) = 0. Hence A ⊗R C is not an integral domain. Example 1.2.8. Let p be a prime number and let Fp = Z/pZ which is the prime field of positive characteristic p. Let k = Fp (t) with an indeterminate t and let k ′ = k(τ ), where τ p = t. Let A = k[x, y]/(y p − txp ) and let S = {xn | n ≥ 0} as in the above example. Then the natural homomorphism A → S −1 A ∼ = k[x, x−1 , (y/x)] ∼ = k ′ [x.x−1 ] is injective, where y/x p is identified with τ because (y/x) = t = τ p . This implies that A is an integral domain. Meanwhile, A ⊗k k ′ ∼ = k ′ [x, y]/(y p − txp ) has a nonzero nilpotent element y − τ x because (y − τ x)p = y p − txp = 0.

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Let L be a finitely generated field extension with tr.degk L = n. Then there exists a transcendence basis {x1 , . . . , xn } such that L/k(x1 , . . . , xn ) is algebraic. Hence L/k(x1 , . . . , xn ) is a finite algebraic extension. If k has characteristic zero, then L/k(x1 , . . . , xn ) is a separable algebraic extension. But it is not necessarily the case if k has positive characteristic. We say that the field extension L/k is a separable extension if there exists a transcendence basis {x1 , . . . , xn } such that L/k(x1 , . . . , xn ) is a separable algebraic extension. It is known that L/k is a regular extension if and only if L/k is a separable extension and k is algebraically closed in L (see [74, Theorem 3.5.2]). For Example 1.2.8, the field of quotients L of A is not a separable extension of k.

Introduction to Algebraic Geometry

1.3

27

Schemes and varieties

1.3.1

Affine schemes of finite type and affine varieties

Let k be a field, which we consider as a base field or a ground field. This means that algebras, schemes and varieties to be considered in the present section are considered over k. 1.3.1.1

Irreducible decomposition of an affine scheme of finite type

If A is a finitely generated k-algebra, the spectrum Spec A is called an affine scheme of finite type over k . If √ n := 0 = p1 ∩ p2 ∩ · · · ∩ pn is the prime decomposition of the nilradical n, Spec A/pi is an irreducible component of the topological space Spec A with respect to the Zariski topology. Recall that the nilradical n is the set of nilpotent elements, i.e., the set of elements a ∈ A such that am = 0 for some m > 0. The residue ring A = A/n has no nonzero nilpotent elements. Then √ Spec √ √ A = Spec A as the topological spaces because V (I) = V ( I) and I ⊃ 0 for every ideal I of A. When we write X = Spec A we denote Spec A by Xred and call it the reduced form of X. If A = A, i.e., n = 0, we say the X is a reduced affine scheme. Write X = Spec A. For every point x ∈ X, there exists the corresponding prime ideal px of A. We consider the point x attached with the local ring Apx and call X an affine local-ringed space or a local ringed space which is affine. We often denote the local ring (Apx , px Apx ) by (OX,x , mX,x ). The residue ring OX,x /mX,x = Apx /px Apx does not contain a proper ideal because mX,x is a maximal ideal. Hence we call OX,x /mX,x the residue field and denote it by k(x) (or κ(x)). Lemma 1.3.1. We have the following assertions. (1) k(x) = Q(A/px ), which is the field of quotients of the residue ring A/px . (2) k(x)/k is a finitely generated field extension with tr.degk k(x) = dim A/pi − ht (px /pi ), where pi is a minimal prime divisor of A contained in px . (3) k(x)/k is an algebraic extension if and only if x is a closed point, i.e., px is a maximal ideal of A.

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Proof. (1) Consider the natural ring homomorphism ι : A → Apx /px Apx given by a 7→ a1 + px Apx . Then Ker ι consists of elements a ∈ A such that sa ∈ px for some s ∈ A \ px , whence a ∈ px . Namely, Ker ι = px and ι induces inclusions A/px ,→ Apx /px Apx and Q(A/px ) ,→ Apx /px Apx . Since Apx /px Apx consists of elements of the form as + px Apx , it is clear that Q(A/px ) = Apx /px Apx = k(x). (2) We may assume by replacing A by A/pi that A is an integral domain. The residue algebra A/px is an integral domain which is finitely generated over k, i.e., an affine domain over k. Hence k(x)/k is a finitely generated extension with tr.degk k(x) = dim A/px . It is a standard fact in commutative algebra (see [54, 72]) that if A is an affine domain over k then the following equality holds for a prime ideal p of A, ht (p) + dim A/p = dim A. (3) Suppose that k(x)/k is a finite algebraic extension. We have inclusions k ⊂ A/px ⊂ k(x). Since k(x) is a finite k-module, so is the ksubmodule A/px . Hence A/px is integral over k. By Lemma 1.1.18, A/px is a field. So, px is a maximal ideal, and x is a closed point. Conversely, suppose that x is a closed point, i.e., px is a maximal ideal. Then A/px is a finitely generated k-algebra and a field as well. If d := tr.degk k(x) > 0, then A/px contains a polynomial ring k[y1 , . . . , yd ] as a k-subalgebra, and A/px is integral over k[y1 , . . . , yd ]. Lemma 1.1.18 implies that A/px is not a field. This is a contradiction. 1.3.1.2

Density of the set of closed points

Let X = Spec A be the same as above. We denote by Ω(X) (or MSpec (A)) the set of all closed points of X (or maximal ideals of A). As a subset Ω(X) ⊂ X, we consider the induced topology on Ω(X). We can consider Ω(X) as a ringed space by associating the local ring Apx to every point x ∈ Ω(X). For x ∈ Ω(X) we denote the associated prime ideal px by mx to emphasize that px is a maximal ideal. For a closed set V (I) of X for an ideal I of A, we denote by VΩ (I) the intersection V (I) ∩ Ω(X). Theorem 1.3.2 (Hilbert’s Nullstellensatz). Let A be a finitely√generated algebra over a field k. Then, for any ideal I of A, the radical I is written as \ √ I= mx . x∈VΩ (I)

Hence the topology on X is recovered by the induced topology on Ω(X).

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Proof. For every x ∈ VΩ (I), it holds that I ⊂ mx . Hence we have the inclusion \ √ I⊆ mx . x∈VΩ (I)

√ T We show the opposite inclusion. Suppose that I ⫋ x∈VΩ (I) mx and let √ T s ∈ ( x∈VΩ (I) mx ) \ I. Then S := {sn | n ≥ 0} is a multiplicative set of A such that S ∩ I = ∅. Let M be a maximal ideal of S −1 A such that M ⊃ IS −1 A and let m = M ∩ A, i.e., m is the inverse image of M by the natural homomorphism A → S −1 A, a 7→ a/1. Since S −1 A = A[s−1 ] is a finitely generated k-algebra, S −1 A/M is a finite algebraic extension of k. Hence A/m, as a k-subalgebra of S −1 A/M, is a finite algebraic extension of k. This implies that m is a maximal ideal of A by Lemma 1.3.1. Further, m ∩ S = ∅ by the argument in subsection 1.1.1 and I ⊂ m. Namely, m ∈ VΩ (I). Then s ∈ m, which is a contradiction because m ∩ S = ∅. Theorem 1.3.3. Let A be an integral domain, and let K be the field of quotients of A. Then \ \ A= Apx = Amx , x∈X

x∈Ω(X)

where Apx and Amx are viewed as local rings contained in K and the intersections are considered inside K. Proof. It suffices to show the equality \ A= Amx . x∈Ω(X)

T One inclusion is clear. So, we prove the other inclusion A ⊇ x∈Ω(X) Amx . T Let ξ ∈ x∈Ω(X) Amx and let J = {a ∈ A | aξ ∈ A}. Clearly J is a nonzero ideal of A, and J = A if and only if ξ ∈ A. Suppose that J ̸= A. Then there exists a maximal ideal m ⊇ J by Corollary 1.1.2. Since ξ ∈ Am by the choice of ξ, there exists an element t ∈ A \ m such that tξ ∈ A. Hence t ∈ J ⊆ m. This is a contradiction. Corollary 1.3.4. With the same assumptions as in Theorem 1.3.3, the following equalities hold for an element s ̸= 0 of A and the open set D(s) = {x ∈ X | s ̸∈ px }, \ \ A[s−1 ] = Apx = Am x . x∈D(s)

x∈D(s)∩Ω(X)

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Proof. Let S = {sn | n ≥ 0} be the multiplicative set generated by the element s. Then S −1 A = A[s−1 ] is an integral domain and D(s) = Spec (S −1 A). Since Apx = (S −1 A)px (S −1 A) for x ∈ D(s), the equalities follow from Theorem 1.3.3. 1.3.1.3

Affine varieties and function fields

Let A be an affine domain over a field k and let K be the field of quotients of A. If K is a regular extension of k, Ω(X) (or X := Spec A) is called an affine variety defined over k. Write A = k[x1 , . . . , xn ]/P, where k[x1 , . . . , xn ] is a polynomial ring over k. Since the ideal P is finitely generated, write P = (F1 , . . . , Fm ), where F1 , . . . , Fm are polynomials in x1 , . . . , xn with coefficients in k. Assume that k is algebraically closed. Then every closed point x ∈ Ω(X) has the residue field k(x) = k because k(x)/k is an algebraic extension, and the corresponding maximal ideal mx is the maximal ideal Mx of k[x1 , . . . , xn ] modulo P, k[x1 , . . . , xn ] → A = k[x1 , . . . , xn ]/P → A/mx = k. Since k[x1 , . . . , xn ]/Mx = k, the maximal ideal Mx is written as Mx = (x1 −α1 , . . . , xn −αn ), where α1 , . . . , αn ∈ k. Write An = Spec k[x1 , . . . , xn ]. Hence the affine variety Ω(An ) is identified with the set k n which consists of n-tuples (α1 , . . . , αn ) of elements of k. The correspondence is to identify the maximal ideal (x1 − α1 , . . . , xn − αn ) with a point (α1 , . . . , αn ) of k n . The affine scheme X is a closed subset of An defined by P, and hence Ω(X) is identified with the closed subset of Ω(An ) = k n . We have Ω(X) = {(α1 , . . . , αn ) ∈ k n | Fi (α1 , . . . , αn ) = 0, 1 ≤ i ≤ m}. Let I be an ideal of A. Then I is a prime ideal of A if and only if V (I) = Spec (A/I) is a √ reduced irreducible closed subset, i.e., the prime ideal decomposition I = I = p1 ∩ · · · ∩ pr has only one prime ideal. Hence a prime ideal p of A corresponds to a reduced irreducible closed subset V (p), which we call a closed subvariety of X and p is the generic point of V (p). If k is algebraically closed, Ω(V (p)) is a closed subvariety of Ω(X). We often confuse Spec A with an affine variety Ω(X) with all irreducible closed subvarieties of Ω(X) considered together. Let A be an affine domain defined over an algebraically closed field k and let K = Q(A) be the field of quotients of A. An element ξ ∈ K is written as ξ = a/b with a, b ∈ A and b ̸= 0. If b ̸∈ mx for a point x ∈ Ω(X) then we can evaluate the element ξ at x by setting ξ(x) = a(x)/b(x) ∈ k. Since the open set DΩ (b) := D(b) ∩ Ω(X) contains the point x, which we

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then call an open neighborhood of x, we can evaluate ξ alike at all elements x′ ∈ DΩ (b). So, ξ is a rational function defined on an open set DΩ (b). We note here that a(x′ ), b(x′ ) ∈ A/mx′ = k(x′ ) = k and hence ξ is a k-valued function on DΩ (b). We say that an element ξ ∈ K is defined or regular at x ∈ Ω(X) if I(ξ) := {b ∈ A | bξ ∈ A} ̸⊂ mx , i.e., ξ = a/b with a, b ∈ A and b ̸∈ mx . Then the local ring OX,x at x ∈ Ω(X) is the set of ξ ∈ K which is regular at x, and the maximal ideal mX,x is the set of ξ ∈ K which is regular at x and ξ(x) = 0. We extend this definition to a non-closed point x ∈ X. The closure {x} ∩ Ω(X) is a closed subvariety and x is the generic point of {x} ∩ Ω(X). We say that ξ ∈ K is defined at x if ξ is defined at a closed point, say y, of {x}∩Ω(X). In fact, if ξ = a/b with a, b ∈ A and b ̸∈ my then b ̸∈ px because px ⊂ my . Conversely, if ξ = a/b as above with b ̸∈ px , take a maximal ideal M of A[b−1 ] containing px A[b−1 ] and its restriction m = M ∩ A. Then m is a maximal ideal of A containing px . So, m corresponds to a closed point in {x} ∩ Ω(X) and b ̸∈ m. Hence the local ring OX,x = Apx is the set of rational functions ξ ∈ K which is defined on {x} ∩ Ω(X). Since X is assumed to be an affine variety, the field K is the set of rational functions which is defined at the generic point of Ω(X) (corresponding to the zero ideal (0) of A). We say that K is the rational function field of X. 1.3.1.4

Structure sheaves

Let U be an open set of an affine variety X. We define the subring Γ(U, OX ) by Γ(U, OX ) = {ξ ∈ K | ξ is defined at every point x ∈ U }. Let U = {Ui }i∈I be an open covering of U , i.e., each Ui is an open set conS tained in U and U = i∈I Ui . Then we have the following exact sequence ρ Y ρ′ Y 0 → Γ(U, OX ) −→ Γ(Ui , OX ) −→ Γ(Ui ∩ Uj , OX ), (1.7) i∈I

i̸=j

where ρ assigns to ξ ∈ Γ(U, OX ) the restriction of ξ onto each Ui , and ρ′ Q assigns to an element {ξi }i∈I of i∈I Γ(Ui , OX ) the difference ξi |Ui ∩Uj − ξj |Ui ∩Uj for every pair (i, j) with i ̸= j. The sequence (1.7) being exact means that the following properties are satisfied: (i) ρ is injective. Namely, for ξ ∈ Γ(U, OX ), if ξ|Ui = 0 for every i ∈ I then ξ = 0. Equivalently, two functions ξ, η ∈ Γ(U, OX ) are equal if and only if ξ|Ui = η|Ui for every i.

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Q (ii) If ρ′ maps {ξi }i∈I to the zero of i̸=j Γ(Ui ∩ Uj , OX ), i.e., if ξi |Ui ∩Uj = ξj |Ui ∩Uj for every pair (i, j) with i ̸= j, then there exists ξ ∈ Γ(U, OX ) (uniquely by (i)) such that ξi = ξ|Ui for every i ∈ I. Note that every nonempty open set contains the generic point of X. The above two properties are easily verified. The sequence (1.7) being an exact sequence says that the assignment U 7→ Γ(U, OX ), i.e., the functor Γ(·, OX ) (or simply OX ) from the collection of all open sets U to rings Γ(U, OX ), is a sheaf. We call OX the structure sheaf of the scheme X. In the construction of OX , we used the assumption that X is an affine variety and the property T that Γ(U, OX ) = x∈U OX,x in the function field K of X for every open set U . But we can extend the construction as follows for an arbitrary affine scheme X = Spec A. S Let U be an open set of X. Then U = i∈I D(ai ), where D(ai ) = Spec A[a−1 i ] with ai ∈ A. In fact, write U = X \ V (I) for an ideal I of A. For any point x ∈ U , px ̸⊃ I. Choose a ∈ A such that a ∈ I and a ̸∈ px . Then x ∈ D(a) ⊂ U . Note that D(ai ) ∩ D(aj ) = D(ai aj ). We define Γ(U, OX ) as the kernel of Y Y ρ′ : A[a−1 A[(ai aj )−1 ], i ]→ i∈I

i̸=j

where ρ′ is a mapping (ξi )i∈I 7→ (ξi |D(ai aj ) − ξj |D(ai aj ) )(i,j)∈I×I,i̸=j . −1 Here ξi |D(ai aj ) is the image of an element ξi ∈ A[a−1 ] by the i ] in A[(ai aj ) −1 −1 natural ring homomorphism A[ai ] → A[(ai aj ) ]. If U = D(a) then it can be shown that Γ(U, OX ) = A[a−1 ] (see Problem 1 of Chapter 1). Given a point x ∈ X = Spec A and any open neighborhood U of x, i.e., U is an open set of X containing x, there exists an open neighborhood D(a) with a ∈ A such that x ∈ D(a) ⊂ U , where a ∈ A \ px . Note that the set S(x) of all open neighborhoods of x induces an inductive system {Γ(U, OX ) | U ∈ S(x)}, where S(x) satisfies the conditions:

(i) If U, U ′ ∈ S(x) then there exists U ′′ ∈ S(x) such that U ′′ ⊂ U ∩ U ′ . Hence we have the natural restriction homomorphisms ρU U ′′ : Γ(U, OX ) → Γ(U ′′ , OX ) and ρU ′ U ′′ : Γ(U ′ , OX ) → Γ(U ′′ , OX ). (ii) Let T(x) be the set of open neighborhoods of the form D(s) with s ∈ A \ px . Then T(x) ⊂ S(x), and S(x) and T(x) are cofinal in the sense that for any U ∈ S(x) there exists D(s) ∈ T(x) such that D(s) ⊂ U .

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Hence we have inductive limits and the equalities between them, lim Γ(U, OX ) = −→

U ∈S(x)

lim −→

D(s)∈T(x)

Γ(D(s), OX ) = lim A[s−1 ] = Apx = OX,x . −→ s∈A\px

We call OX,x the stalk of the sheaf OX at the point x. For the last description on inductive limits, see the appendix. 1.3.2

Morphisms of affine schemes

Let φ : B → A be a ring homomorphism. For a prime ideal p ∈ Spec A, the inverse image φ−1 (p) is a prime ideal. In fact φ induces an injective ring homomorphism φ : B/φ−1 (p) → A/p. Since A/p is an integral domain, so is B/φ−1 (p). Hence φ−1 (p) ∈ Spec B. Thus we have a mapping a φ : Spec A → Spec B, which assigns φ−1 (p) to p. We call a φ the morphism associated to φ. Let R be a ring and let φ : B → A be an R-algebra homomorphism. Namely, φ is a ring homomorphism such that φ(αb) = αφ(b) for α ∈ R. Then the morphism a φ : Spec A → Spec B satisfies πA = πb ◦ a φ, where πA : Spec A → Spec R (resp. πB : Spec B → Spec R) is the morphism associated to the natural ring homomorphism R → A (resp. R → B) which makes A (resp. B) an R-algebra. We then say that a φ is an R-morphism. If R is the base field k and A, B are k-algebras, we say that a φ is a kmorphism. The morphism a φ : Spec A → Spec B is a continuous mapping with respect to the Zariski topology on Spec A and Spec B. In fact, if J is an ideal of B, the inverse image a φ−1 (VB (J)) is equal to {p ∈ Spec A | φ−1 (p) ⊇ J}, where the condition φ−1 (p) ⊇ J is equivalent to the condition that p ⊇ φ(J)A. Hence a φ−1 (VB (J)) is equal to VA (φ(J)A), where the subscript A or B indicates that the closure is considered in Spec A or Spec B. Set f = a φ, X = Spec A and Y = Spec B. Consider how f behaves with local ringed space structures on X and Y . Let x ∈ X and y = f (x). Then x and y correspond respectively to prime ideals p and q of A and B such that q = φ−1 (p). Hence φ extends to a local homomorphism Bq → Ap such that b/t is mapped to φ(b)/φ(t), where t ̸∈ q implies φ(t) ̸∈ p. The term local homomorphism means, by definition, that it sends the maximal ideal to the maximal ideal. Hence a φ induces a local homomorphism f ∗ : OY,y → OX,x by f ∗ (b/t) = φ(b)/φ(t) for every x ∈ X and y = f (x) ∈ Y . Then we say that f : X → Y is a morphism of local ringed spaces f : (X, OX ) → (Y, OY ).

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Further, if ψ : C → B is a ring homomorphism, set Z = Spec C and g = a ψ : Y → Z. Then a (φ ◦ ψ) = a (ψ) ◦ a (φ) = g ◦ f . 1.3.2.1

Intersection of affine open sets

Let X = Spec A be an affine scheme. An open set U of X is called an affine open set if U itself is an affine scheme. We have the following result. Lemma 1.3.5. Let U1 , U2 be nonempty affine open sets of X = Spec A. If U1 ∩ U2 ̸= ∅ then U1 ∩ U2 is an affine open set. Proof. We treat only the case A is an integral domain. A general case is proved by making use of the fact that the diagonal morphism ∆X : X → X × X is a closed immersion (see subsection 1.3.3). We refer the readers also to [57]. Under the assumption, any nonempty open set contains the generic point of X, hence U1 ∩ U2 ̸= ∅. Write U1 = Spec B1 and U2 = Spec B2 . Let B3 be the ring generated by B1 and B2 in K = Q(A). Then we have inclusions A ⊂ Bi ⊂ B3 for i = 1, 2. If x ∈ U1 ∩ U2 then Apx = (B1 )px B1 = (B2 )px B2 = (B3 )px B3 holds. Verification is left to the readers as an exercise. Conversely, let p3 be a prime ideal of B3 . Let pi = p3 ∩ Bi for i = 1, 2 and let p = p3 ∩ A. Then p corresponds to a point of X. Since Ui is an open set of X, the local ring (Bi )pi is equal to OX,x = Ap for i = 1, 2. Since (B3 )p3 contains (Bi )pi for i = 1, 2 and is generated by these two local rings, we conclude that (B3 )p3 = Ap = OX,x . This implies that U1 ∩ U2 = Spec B3 . An open set of an affine scheme is not necessarily an affine scheme. Proposition 1.3.6. Let X := A3 = Spec k[x1 , x2 , x3 ] be the affine space of dimension three. Let V = V ((x1 , x2 )) be the closed set defined by the ideal (x1 , x2 ). Hence V is isomorphic to the affine line A1 = Spec k[x3 ]. Let U = A3 \ V . Then the open set U is not an affine open set. Furthermore, Γ(U, OX ) is isomorphic to A := k[x1 , x2 , x3 ] and U = D(x1 ) ∪ D(x2 ). Proof. We have Γ(U, OX ) =

\ x∈U

OX,x ⊂ k(x1 , x2 , x3 ).

In fact, a rational function ξ ∈ Γ(U, OX ) is written as ξ=

f (x1 , x2 , x3 ) , f (x1 , x2 , x3 ), g(x1 , x2 , x3 ) ∈ k[x1 , x2 , x3 ], g(x1 , x2 , x3 )

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where the polynomials f, g are uniquely determined up to constants in k if we assume that f and g have no common divisors in the unique factorization domain k[x1 , x2 , x3 ]. If g(x1 , x2 , x3 ) ̸∈ k then the function ξ is not defined on U ∩ V (g), where V (g) consists of points x ∈ X such that g(x) = 0. It can be shown that U ∩ V (g) ̸= ∅. We leave the proof to the readers as an exercise. So ξ ̸∈ Γ(U, OX ). This is a contradiction. It is clear that U = D(x1 ) ∪ D(x2 ). In fact, D(x1 ) = Spec k[x1 , x−1 1 , x2 , x3 ] and D(x2 ) = Spec k[x1 , x2 , x−1 , x ]. 3 2 In subsection 1.3.3 we define a scheme structure on U in Proposition 1.3.6. 1.3.2.2

Open immersion and closed immersion

Suppose that X = Spec A is irreducible and reduced. For an affine open set U = Spec B of X, by Theorem 1.3.3, the ring B is obtained as \ B= OX,x . x∈U

So, we can write B = Γ(U, OX ). The inclusion map iU : U → X is induced by the ring homomorphism ρU : A = Γ(X, OX ) → Γ(U, OX ), which is intuitively the restriction of regular functions on X to regular functions on the open set U . Hence the induced local ring homomorphism i∗U : OX,x → OU,x is an isomorphism for every x ∈ U . Note that iU = a ρU is a morphism of affine schemes. We say that iU is an open immersion. Let V = V (I) be a closed set of X = Spec A defined by an ideal I of A. The quotient homomorphism q : A → A/I, a 7→ a + I induces a morphism of schemes jV : V = Spec A/I → X = Spec A defined by p 7→ q −1 (p) for p ∈ Spec A/I. The morphism jV is called a closed immersion or a closed embedding. 1.3.2.3

Behavior of structure sheaves under a morphism

Let φ : B → A be a ring homomorphism and let f : X → Y be the associated morphism of affine schemes, where X = Spec A, Y = Spec B and f = a φ. We define a sheaf f∗ OX on Y by Γ(W, f∗ OX ) = Γ(f −1 (W ), OX ) S for an open set W of Y . In fact, if W = j∈J D(bj ) is a covering by affine

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open sets of W then f −1 (W ) is an open set of X such that f −1 (W ) = S j∈J D(φ(bj )), hence the exactness of ρ Y ρ′ Y Γ(W, OY ) −→ Γ(D(bj ), OW ) −→ Γ(D(bi bj ), OW ) j∈J

i̸=j

follows from the exactness of the sequence ρ Y ρ′ Y Γ(f −1 (W ), OX ) −→ Γ(D(φ(bj ), OX ) −→ Γ(D(φ(bi bj )), OX ). j∈J

i̸=j

The sheaf f∗ OX is called the direct image of OX by f . Then there is a homomorphism φ e : OY → f∗ OX defined as follows. For any open set W of Y , the natural ring homomorphism φ(W e ) : Γ(W, OY ) → Γ(W, f∗ OX ) = Γ(f −1 (W ), OX ) is induced canonically on the kernels from the following commutative diagram of exact sequences Q Q ρ ρ′ Γ(W, OY ) −→ j∈J Γ(D(bj ), OY ) −→ i̸=j Γ(D(bi bj ), OY ) ↓ ↓ ρ Q ρ′ Q −1 Γ(f (W ), OX ) −→ j∈J Γ(D(φ(bj )), OX ) −→ i̸=j Γ(D(φ(bi bj )), OX ), where the vertical arrows are the products of the ring homomorphisms induced by φ, −1 B[b−1 ], j ] → A[φ(bj )

B[(bi bj )−1 ] → A[φ(bi bj )−1 ],

b φ(b) 7→ bsj φ(bj )s b bsi btj

7→

φ(b) . φ(bi )s φ(bj )t

Hence, attached to a ring homomorphism φ : B → A, there is a pair (f, φ) e of the morphism of topological spaces f : X → Y and the homomorphism of sheaves of rings φ e : OY → f∗ OX . We denote it by (f, φ) e : (X, OX ) → (Y, OY ). We often abbreviate it as f : X → Y and call it the morphism associated to φ. Note that φ e : OY → f∗ OX recuperates the ring homomorphism φ as φ = φ(Y e ). Hence there is a bijection which is order-reversing under the composition ∼

Homalg (B, A) −→ Hommor (X, Y ), φ 7→ (f, φ). e

(1.8)

Suppose that A is a k-algebra, where k is an algebraically closed field. Let X = Spec A, which has the structure morphism πA : X → Spec k. We denote by Homk−mor (Spec k, X) the set of k-morphisms f : Spec k → X

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such that πA ◦ f = a (idk ), where idk is the identity homomorphism of k. The morphism f is determined by a k-algebra homomorphism α : A → k such that a α = f and α|k = idk , where α is determined by Ker α. Then Ker α is a maximal ideal mα of A and it corresponds to a closed point xα . Since Spec k has only one point, say o, corresponding to the ideal (0), the closed point xα is nothing but the image f (o). By these arguments, we have a bijection ∼

Homk−mor (Spec k, X) −→ Ω(X), f 7→ f (o).

The bijection (1.8) shows that a morphism f : X → Y is an isomorphism, i.e., there exists a morphism g : Y → X such that g ◦ f = idX and f ◦ g = idY , if and only if the corresponding algebra homomorphism φ : B → A is an isomorphism. 1.3.3 1.3.3.1

Schemes and varieties Definition and examples of schemes

A local ringed space is, by definition, a pair (X, OX ) of a topological space X and a sheaf of local rings OX . We say that (X, OX ) is a scheme4 if there is an open covering U = {Ui }i∈I such that (Ui , OX |Ui ) is an affine scheme, where OX |Ui is the restriction of OX to Ui , i.e., Γ(W, OX |Ui ) = Γ(W, OX ) for every open set W contained in Ui . Hence (X, OX ) is a collection {(Ui , OUi ) | i ∈ I} of affine schemes such that (Ui , OUi ) and (Uj , OUj ) are patched together along the open set Ui ∩ Uj for every pair (i, j) with i ̸= j. Each open set Ui is called a local chart or an affine local chart. Suppose that Ui ∩ Uj is an affine scheme for every (i, j) with i ̸= j. This last condition is guaranteed by the separatedness condition on (X, OX ), which is to be considered later. Write Ui = Spec Ai and Uij := Ui ∩Uj = Spec Aij (resp. Spec Aji ) when considered as an open set of Ui (resp. Uj ). Hence Spec Aij = (Uij , OUi |Uij ) and Spec Aji = (Uij , OUj |Uij ). Then there exists an isomorphism θji : Spec Aij → Spec Aji corresponding to an algebra isomorphism σij : Aji → Aij , i.e., θji = a σij , such that (i) θij ◦ θji = idSpec Aij and θji ◦ θij = idSpec Aji . (ii) θℓi = θℓj ◦ θji for distinct three indices i, j, ℓ, where Uijℓ = Ui ∩ Uj ∩ Uℓ , and θji (resp. θℓj or θℓi ) is for the restriction θji |Uijℓ (resp. θℓj |Uijℓ or θℓi |Uijℓ ). 4 It

used to be called a prescheme and a separated prescheme a scheme. But it is now standard to use the term scheme for prescheme.

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Example 1.3.7. Let Ui = Spec k[xi , yi ] be the affine plane for i = 0, 1, 2. Patch these three pieces by the following data: ∼

−1 θ01 : D(x1 ) −→ D(x0 ), σ10 (x0 ) = x−1 1 , σ10 (y0 ) = x1 y1 ∼

θ12 : D(y2 ) −→ D(y1 ), σ21 (x1 ) = x2 y2−1 , σ21 (y1 ) = y2−1 ∼

−1 θ02 : D(x2 ) −→ D(y0 ), σ20 (x0 ) = x−1 2 y2 , σ20 (y0 ) = x2 .

Then X := U0 ∪ U1 ∪ U2 is the scheme isomorphic to the projective plane P2 over k to be observed in the next section. In fact, if (X0 , X1 , X2 ) is a system of homogeneous coordinates on P2 , the patching data is given as follows: Ui = {(a0 , a1 , a2 ) ∈ P2 | ai ̸= 0}(i = 0, 1, 2) X2 X0 X2 X0 X1 X1 , y0 = , x1 = , y1 = , x2 = , y2 = , x0 = X0 X0 X1 X1 X2 X2 D(x1 ) = U10 , D(x0 ) = U01 , D(y2 ) = U21 , D(y1 ) = U12 , D(x2 ) = U20 , D(y0 ) = U02 . Further, U0 ∩ U1 ∩ U2 ∼ = D(xi yi ) in Ui for i = 0, 1, 2. Example 1.3.8. Let Ui = Spec k[xi ] be the affine line over k for i = 0, 1. We consider to patch U0 and U1 by identifying D(x0 ) = U0 \ {0} and ∼ D(x1 ) = U1 \ {0} by an isomorphism θ : D(x0 ) −→ D(x1 ). Since D(x0 ) = −1 −1 Spec k[x0 , x0 ] and D(x1 ) = Spec k[x1 , x1 ], θ is the morphism associated ∼ −1 ∗ with a k-algebra isomorphism σ : k[x1 , x−1 1 ] −→ k[x0 , x0 ]. Denote by A −1 ∗ ∼ ∗ 5 the multiplicative group of units in a ring A. Then (k[x0 , x0 ]) = k × Z, where k ∗ = k \ {0} and Z signifies an abelian group {xn0 | n ∈ Z}. Similarly, ∗ ∼ ∗ (k[x1 , x−1 1 ]) = k × Z, and the isomorphism σ induces an isomorphism ∼ σ ∗ : k ∗ × Z −→ k ∗ × Z such that σ ∗ is the identity homomorphism on the ∗ factor k ∗ and σ ∗ (x1 ) = ax±1 0 , where a ∈ k . −1 ∗ ′ (1) If σ (x1 ) = ax0 , replace x0 by x0 = a−1 x0 . Thus we may assume that σ ∗ (x1 ) = x−1 0 . Then the scheme X obtained by the above patching is isomorphic to the projective line P1 to be considered in the next section and hence separated. (2) If σ ∗ (x1 ) = ax0 , we may assume by replacing x0 by x′0 = ax0 that σ ∗ (x1 ) = x0 . The scheme X = U0 ∪ U1 obtained by the patching is not separated.6 In fact, if k is the complex number field C and the usual 5 If u, v are units, the product uv and the inverse u−1 are also units of A. Hence A∗ is a group with respect to the multiplication and the inverse of A. A ring isomorphism ∼ ∼ φ : A −→ B induces a group isomorphism φ∗ : A∗ −→ B ∗ . 6 The reasoning is to be read after reading the later part which treats the separatedness condition.

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metric topology is considered on X, there is a convergent series of points {Pi }i∈N≥0 lying on U0 ∩ U1 , which converges to the origin (0) on U0 and U1 . On the scheme X it can converge to two points, i.e., either (0) on U0 or (0) on U1 . This never happens if X is separated because the diagonal subset ∆X (X) ⊂ X × X is then a closed set. For this example, the intersection of any two affine open sets is again an affine open set. This shows that even if the condition in Lemma 1.3.5 is satisfied, the scheme is not necessarily separated. If X is an affine scheme, its property is described by the coordinate ring Γ(X, OX ). If X is a scheme in general, its local property can be described by affine open charts Ui . The structure sheaf OX is given also by patching the ∼ ∗ structure sheaves OUi by the isomorphism θji : OUj |Ui ∩Uj −→ OUi |Ui ∩Uj . But the intersection Ui ∩Uj is not necessarily an affine open set in the affine ∼ scheme Ui . So, we need to understand what the isomorphism θji : Uij −→ Uji is. Namely we have to define a morphism of schemes. An open set U of a scheme X has the induced scheme structure (U, OX |U ). In fact, if U = {Ui }i∈I is an open covering of affine local charts, then {U ∩ Ui }i∈I is an open covering of U and each U ∩ Ui is covered by affine open sets in the affine chart Ui . With this induced structure, U is called an open subscheme. A closed subset V of X is a closed subscheme if there is a local ringed structure (V, OV ) satisfying the following conditions: (c-i) V is a closed subset of X. ^ (c-ii) With the above affine local charts U = {Ui }i∈I , OV |V ∩U = (A i /ai ) i

for each open set Ui = Spec Ai and an ideal ai of Ai .

So, V is reduced if and only if 1.3.3.2

√

ai = ai for every i ∈ I.

Morphism of schemes

Let X and Y be schemes defined by local affine charts U = {Ui | i ∈ I} and V = {Vj | j ∈ J} of X and Y , respectively. A continuous mapping of underlying topological spaces f : X → Y is called a morphism of schemes if U and V satisfy the conditions: (m-i) For each i ∈ I, there exists j ∈ J such that f (Ui ) ⊂ Vj and the restriction fji := f |Ui : Ui → Vj is a morphism of affine schemes. (m-ii) If f (Ui ∩ Ui′ ) ⊂ Vj ∩ Vj ′ then fji and fj ′ i′ commute with patching isomorphisms. Namely, θjY′ j ◦ fji = fj ′ i′ ◦ θiX′ i holds on each affine open set of Ui ∩ Ui′ such that f (Ui ∩ Ui′ ) is contained in an affine

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open set of Vj ∩ Vj ′ , where the superscript X and Y signify that the patching isomorphisms are considered on X and Y . The second condition (m-ii) implies that f induces a homomorphism of sheaves OY → f∗ OX , which is locally defined by φ eij : OY |Vj → (fji )∗ (OX |Ui ),

where φij : Γ(Vj , OY ) → Γ(Ui , OX ) is the ring homomorphism giving rise to fji . We denote by Hom(X, Y ) the set of all morphisms from X to Y . A scheme X is called an S-scheme if a morphism πX : X → S is fixed for a certain scheme S. The fixed morphism πX is called the structure morphism of the S-scheme. If both X and Y are S-schemes then a morphism f : X → Y is an S-morphism if πX = πY ◦ f . The set of S-morphisms from X to Y is denoted by HomS (X, Y ). If S = Spec R for a ring R, we write HomR (X, Y ) instead of HomSpec (R) (X, Y ). In the subsequent sections, we consider the case where R is a field k. We then say that X and Y are defined over k. For example, if A is a k-algebra, the structure morphism πX : X → Spec k is induced by the canonical injection k ,→ A. 1.3.3.3

Fiber products of schemes

We introduce the notion of fiber product of schemes. For this purpose we need a categorical point of view. Let S be a scheme and let (Sch /S) be the category whose objects are S-schemes and morphisms are S-morphisms. For two S-schemes X, Y we consider the set-theoretic product HomS (T, X) × HomS (T, Y ) for all T ∈ (Sch /S). If there exists an S-scheme Z and S-morphisms pX : Z → X and pY : Z → Y such that the induced mapping αT : HomS (T, Z) → HomS (T, X) × Hom(T, Y ),

(1.9)

h 7→ (pX ◦ h, pY ◦ h)

is a bijection for all T ∈ (Sch /S). Further, the following diagram is commutative for all S-morphism g : T → T ′ , α

′

HomS (T ′ , Z) −−−T−→ HomS (T ′ , X) × HomS (T ′ , Y )   y(◦g,◦g) ◦g y HomS (T, Z) −−−−→ HomS (T, X) × HomS (T, Y ), αT

where the left vertical arrow ◦g is the composition with g, i.e., h′ 7→ h′ ◦ g and the right vertical arrow (◦g, ◦g) is also the composition with g, i.e.,

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(f, f ′ ) 7→ (f ◦ g, f ′ ◦ g). When such an S-scheme Z exists, we write it as X ×S Y and call it the fiber product of X and Y over S. The morphisms pX : X ×S Y → X and pY : X ×S Y → Y are called the projections. For the construction, we refer to the appendix. Important is the fact that if X, Y, S are affine schemes Spec A, Spec B, Spec R, respectively with Ralgebras A, B, then X×S Y is isomorphic to the affine scheme Spec (A⊗R B). Let f : X → Y be a morphism of schemes and let U be an open set of Y . Let i : U → Y be the canonical open immersion. Then the base change iX : X ×Y U → X is an open immersion and hence identified with the inverse image (f −1 (U ), OX |f −1 (U ) ). Similarly, let V be a closed set of Y and let j : V → Y be the canonical closed immersion. Then the base change jX : X ×Y V → X is a closed immersion. See Problem 6 for the proof. 1.3.3.4

Separated schemes

In the above bijection αT (see the mapping (1.9)), set X = Y = T . Then there exists a morphism ∆X : X → X ×S X such that αX (∆X ) = (idX , idX ), hence ∆X satisfies p1 ◦ ∆X = p2 ◦ ∆X = idX , where p1 : X ×S X → X and p2 : X ×S X → X are the projections to the first and the second factors, i.e., p1 = pX and p2 = pY with Y = X in the above notations. An S-scheme X (more precisely with a morphism πX : X → S) is separated over S if ∆X is a closed immersion. If X = Spec A and S = Spec R with an R-algebra A, then the morphism ∆X is determined by the multiplication homomorphism µA : A ⊗R A → A, a ⊗ a′ 7→ aa′ ,

which is surjective. Hence ∆X is induced by an isomorphism A ∼ = (A ⊗R A)/Ker µA . Namely an affine scheme X is separated over an affine scheme S. If X is separated over an affine scheme S = Spec R, the intersection U ∩ U ′ of two affine open sets U = Spec A, U ′ = Spec A′ is also an affine ′ scheme. In fact, U ∩ U ′ = ∆−1 X/S (U ×S U ), which is a closed subset of the ′ ′ affine scheme U ×S U = Spec (A ⊗R A ). Remark 1.3.9. The topology of X ×X Y is not a product topology of those on X and Y . In fact, let S = Spec k with a field k and let X ∼ =Y ∼ = A1 . 2 1 ∼ ∼ Then X ×S Y = Spec k[x, y] = A . The proper closed sets of A are finite sets of closed points. Hence the product topology on X ×S Y has closed sets which are either unions of lines like {P } × A1 and A1 × {Q} with P, Q are

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closed points on A1 or finite sets of closed points like (P, Q). However there are closed sets on A2 defined by irreducible equations f (x, y) = 0. Hence the Zariski topology on X ×S Y is stronger than the product topology. Example 1.3.10. (1) Let f : X → Y be a morphism of S-schemes and let T be an S-scheme. Let pi and qi for i = 1, 2 be the first and second projections from X ×S T and Y ×S T respectively. By the bijection ∼

αX×S T : HomS (X ×S T, Y ×S T ) −→ HomS (X ×S T, Y )×HomS (X ×S T, T ) there exists an S-morphism fT : X ×S T → Y ×S T such that q1 ◦fT = f ◦p1 and q2 ◦ fT = p2 . The second relation implies that fT is a T -morphism, and the first relation shows that the following diagram is commutative: fT

X ×S T −−−−→ Y ×S T   yq1 yp1 f

X −−−−→ Y. We say that fT is the base change of f . If f : X → Y is a k-morphism and if k ′ /k is a field extension, take S = Spec k and T = Spec k ′ . Then fT is often denoted by f ⊗k k ′ . The notation is justified as the scheme-theoretic fiber product is the tensor product if schemes are affine schemes. (2) Let f : X → Y be a morphism of schemes. For y ∈ Y there exists a natural morphism ιy : Spec k(y) → Y . In fact, let V = Spec B be an affine open neighborhood of y. Then y corresponds to a prime ideal qy of B and k(y) = Q(B/qy ). So the homomorphism B → B/qy ,→ k(y) gives a morphism Spec k(y) → V , which, composed with an open immersion ιU : U → Y ,7 gives a morphism ιy : Spec k(y) → Y . The fiber product X ×Y Spec k(y) is denoted by Xy and called the fiber of f over the point y. If the set f −1 (y), which is the topological inverse image of f , is the empty set, we write Xy = ∅. If Xy ̸= ∅, it is a scheme defined over the field k(y). A scheme X is irreducible if the topological space X is irreducible, and reduced if the stalk OX,x is reduced for every x ∈ X. Let X be an irreducible and reduced scheme. If two open sets Ui (i = 1, 2) are disjoint from each other, then V1 ∪ V2 = X, where Vi = X \ Ui . This contradicts the 7 Definition

of open immersion was given for an affine open set of an affine scheme. It is easily generalized to the case of an open set of a scheme. Let U be an open set of a scheme X. We choose an affine open covering U = {Ui }i∈I of X in such a way that U1 = {Ui }i∈I1 is an affine open covering of U . Then define a morphism ιU locally as ιU |Ui = idUi for i ∈ I1 . This ιU is an open immersion and the definition is independent of the choice of such an affine open covering.

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irreducibility of X. Hence U1 ∩U2 ̸= ∅. If Ui = Spec Ai for i = 1, 2, then A1 and A2 have the common field of quotients K = Q(A1 ) = Q(A2 ) because, for x ∈ U1 ∩ U2 , we have OX,x = (A1 )p1 = (A2 )p2 , where pi is a prime ideal of Ai giving rise to the point x. This implies that K = Q(OX,x ) for every point x ∈ X. We call K the function field of X. If X is a k-scheme, we denote X by k(X). The above argument shows that the generic points of U1 and U2 coincide, hence X has a unique generic point. Let k be a field and let X be a k-scheme. We say that X is a kscheme of finite type if X has a finite affine open covering U = {Ui }i∈I , where I is a finite set, and each Ui has the coordinate ring Ai which is a finitely generated k-algebra. Since Ai is then a noetherian ring each Ui is a noetherian space. Hence X is quasi-compact as a topological space. A kscheme X of finite type is geometrically irreducible (resp. and geometrically reduced) if X ⊗k k ′ is irreducible (resp. reduced) for every algebraic extension k ′ /k, or equivalently for an algebraic closure k of k. Definition 1.3.11. A k-scheme X of finite type over k is an algebraic variety defined over k if the following two conditions are satisfied. (i) X is geometrically irreducible and reduced. Equivalently, the function field k(X) is a regular extension of k (see Lemma 1.2.5). (ii) X is separated over k. Namely, the diagonal morphism X → X ×k X is a closed immersion. An affine variety is an algebraic variety which is an affine scheme. 1.3.3.5

Rational maps of algebraic varieties

Let X and Y be algebraic varieties defined over k. We consider a pair (U, φ) of an open set U ̸= ∅ of X and a morphism φ : U → Y . Two of such pairs (U, φ) and (U ′ , φ′ ) are equivalent, (U, φ) ∼ (U ′ , φ′ ) by notation, if there exists a pair (U ′′ , φ′′ ) such that U ′′ ⊆ U ∩ U ′ and φ′′ = φ|U ′′ = φ′ |U ′′ . An equivalence class of (U, φ) is called a rational map and denoted by φ : X 99K Y . In the above definitions, if (U, φ) ∼ (U ′ , φ′ ) then φ|U ∩U ′ = φ′ |U ∩U ′ . In fact, take a closed point P ∈ U ∩ U ′ . Then there exist an affine open neighborhood UP = Spec AP of P and an affine open set V = Spec B of finite type in Y such that UP ⊆ U ∩ U ′ and φ|UP (resp. φ′ |UP ) is defined by a k-algebra homomorphism σ : B → A (resp. σ ′ : B → A). Since UP ∩ U ′′ ̸= ∅, there is an open set D(s) ⊂ UP ∩ U ′′ so that φ|D(s) = φ′ |D(s) . Namely i ◦ σ = i ◦ σ ′ , where i : A → A[s−1 ] is an inclusion. Since B is finitely generated over k and A is an integral domain, this implies that

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σ = σ ′ . So, φ = φ′ in an open neighborhood of P ∈ U ∩ U ′ . Since P is arbitrary, we have φ = φ′ on U ∩ U ′ . This implies that if a rational map φ : X 99K Y is given, there exists the greatest open set Umax where φ becomes a morphism. The open set Umax is called the domain of definition of φ. Suppose that φ : Umax → Y is dominant, i.e., φ(Umax ) contains a nonempty open set of Y . Then φ induces an inclusion of the function fields φ∗ : k(Y ) ,→ k(X). In fact, if an inclusion of function fields σ : k(Y ) ,→ k(X) is given, σ is induced by a pair (U, φ) with a dominant morphism such that σ = φ∗ . So, σ is induced by a rational map φ : X 99K Y . In particular, ∼ if there is an isomorphism σ : k(Y ) −→ k(X), there exist rational maps φ : X 99K Y and ψ : Y 99K X such that φ and ψ induce an isomorphism and its inverse between nonempty open sets U ⊂ X and V ⊂ Y . In such a case, we call a rational map φ birational.

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Graded rings and projective schemes Graded rings and projective spectrums

Let R be a ring. An R-algebra A is a graded ring over R if A is a direct L sum n∈Z An such that An is an R-module and Am An ⊆ Am+n under the multiplication of A for arbitrary m, n ∈ Z. By this condition, A0 is closed under the multiplication, whence an R-algebra and An is an A0 -module. An element of An is called a homogeneous element of degree n. L An ideal I of A is called a homogeneous ideal if I = n∈Z In , where In = I ∩ An . For an ideal I of A, I is homogeneous if and only if a = P n∈Z an ∈ I implies an ∈ I for every element a of I and every n ∈ Z. Except for the case of rings of quotients or unless otherwise specified, we consider only the case where A0 = R and the condition “An = 0 if n < 0” L holds. Under the last condition “An = 0 if n < 0”, A+ := n≥0 An is a homogeneous ideal of A, which is called the irrelevant ideal. Example 1.4.1. (1) Let A = R[x1 , . . . , xr ] be a polynomial ring in r variables over R. Let An be an R-free module consisting of all monomials of degree n αr 1 α2 cxα 1 x2 · · · xr , c ∈ R, αi ≥ 0,

r X

αi = n.

i=1

Then A is a graded ring such that A0 = R and An = 0 if n < 0. (2) Let A be a graded ring over a ring R and let I be a homogeneous ideal of A. Then the quotient ring A = A/I is also a graded ring, which is L a direct sum A = n∈Z An with An = An /In . L An A-module M is a graded A-module if M = n∈Z Mn and Am Mn ⊆ Mm+n for arbitrary m, n ∈ Z. An A-submodule N of M is a graded L submodule of M if N = n∈Z Nn , where Nn = N ∩ Mn . Whenever we consider a ring of quotients A[a−1 ] for a nonzero homogeneous element a of degree d, we assume that a is not a nilpotent element and A[a−1 ] is the ring of quotients S −1 A with respect to the multiplicative set S = {an | n ∈ Z}. Then, by setting deg(b/an ) = m − nd for b ∈ Am , A[a−1 ] is a graded ring. We consider the subring A[a−1 ]0 , which is also denoted by A(a) . Lemma 1.4.2. If a ∈ Ad and b ∈ Ae then A[(ab)−1 ]0 = A[a−1 ]0 [(bd /ae )−1 ].

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Proof. An element of A[(ab)−1 ]0 is written as x/(ab)k with x ∈ Ak(d+e) . If we choose integers h, ℓ so that k + ℓ = dh, then we have    e h xbℓ a x . = · (ab)k ak aeh bd Conversely, an element of A[a−1 ]0 [(bd /ae )−1 ] is a finite sum of elements of the form (y/ak )(bd /ae )−ℓ with y ∈ Adk , which is written as ya(d+e)ℓ bk . (ab)dℓ+k This proves the assertion. Lemma 1.4.3. The following assertions hold for a graded ring A such that An = 0 for every n < 0 and a graded A-module M . (1) A is noetherian if and only if A0 is noetherian and A is a finitely generated A0 -algebra. (2) Suppose that A is a finitely generated A0 -algebra. Then M is a finite A-module if and only if (i) Mn is a finite A0 -module for every n. Further, Mn = 0 for every n smaller than some integer n0 . (ii) There exist integers n1 and d > 0 such that Mn+d = Ad Mn for every n ≥ n1 . Proof. (1) The if part follows from Hilbert basis theorem (see Theorem 1.1.10). We prove the only if part. Since A is noetherian, the irrelevant homogeneous ideal A+ is finitely generated and A0 = A/A+ is noetherian. Let {a1 , . . . , ar } be a set of generators of A+ which we assume to be homogeneous elements. We show by induction on n that An Pr is generated by a1 , . . . , ar over A0 . Let x ∈ A1 . Then x = i=1 bi ai , where deg(bi ) = deg(x) − deg(ai ) ≤ 0, whence bi = 0 if deg(ai ) > 1 and bi ∈ A0 if deg(ai ) = 1. So, x ∈ A0 [a1 , . . . , ar ]. Suppose that Pr A1 , . . . , An ⊆ A0 [a1 , . . . , ar ]. Let x ∈ An+1 . Write as above x = i=1 bi ai with homogeneous elements bi of degree (n + 1) − deg(ai ) ≤ n. Since every bi ∈ A0 [a1 , . . . , ar ], we have x ∈ A0 [a1 , . . . , ar ]. (2) The if part is easy to verify. In fact, the union of generators of A0 -modules Mn for n0 ≤ n ≤ n1 + d − 1 generates M as an A-module. We prove the only if part. By the hypothesis, A = A0 [a1 , . . . , as ], where Pt the ai are homogeneous elements with deg ai = di , and M = j=1 Amj , where the mj are homogeneous elements with deg mj = ej . Then, for

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αs 1 every n ∈ Z, Mn consists of linear combinations of elements aα 1 · · · as mj Ps with coefficients in A0 , where n = ( i=1 αi di ) + ej . Since αi ≥ 0 and di > 0, there are finitely many choices of (α1 , . . . , αs , j). Hence Mn is a finite A0 -module. Let n0 = min1≤j≤t deg mj . Then Mn = 0 for every n < n0 because d/d Ar = 0 if r < 0. Let d be the lcm of d1 , . . . , ds , and let bi = ai i for 1 ≤ i ≤ s. Consider the set S of all elements of M of the form αs 1 aα 1 · · · as mj ,

0 ≤ αi ≤ d/di (1 ≤ i ≤ s), 1 ≤ j ≤ t.

Since there are finitely many such elements, we denote the set by {xµ }1≤µ≤ℓ . Let n1 = maxµ deg xµ . If n ≥ n1 , then every element of Mn+d with d > 0 is written as X fµ (b1 , . . . , bℓ )xµ , fµ (b1 , . . . , bℓ ) ∈ A0 [b1 , . . . , bℓ ] \ A0 . µ

Hence Mn+d ⊆ Ad Mn . The inclusion Ad Mn ⊆ Mn+d is clear. L Let A = n≥0 An be a graded ring. We consider the set Proj A of all homogeneous prime ideals of A which do not include the irrelevant ideal A+ . We call Proj A the projective spectrum. If I is a homogeneous ideal, we set V+ (I) = {p ∈ Proj A | p ⊇ I} and D+ (I) = (Proj A) \ V+ (I). If I is a principal ideal aA with a homogeneous element a, we write V+ (I) = V+ (a) and D+ (I) = D+ (a). Lemma 1.4.4. With the V+ (I) as closed sets of Proj A for all homogeneous ideals I, Proj A is a topological space. Namely we have the following relations. (1) V+ (0) = Proj A, V+ (A) = V+ (A+ ) = ∅. S T (2) V+ ( λ∈Λ Iλ ) = λ∈Λ V+ (Iλ ). (3) V+ (I1 I2 ) = V+ (I1 ) ∩ V (I2 ). The set {D+ (a) | a ∈ An , n > 0} is a basis of open sets. Proof. The assertions (1), (2), (3) are proved in the same way as in subsection 1.1.2. Only the homogeneity must be taken into account. As for the second statement, let p be a homogeneous prime ideal such that p ̸⊇ A+ . Then there exists a homogeneous element a ∈ (A+ ) \ p. Hence p ∈ D+ (a). Note that a is not a nilpotent element. Otherwise, ar = 0 for r > 0, and hence a ∈ p.

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The topology on Proj A is also called the Zariski topology. The topological space Proj A has a scheme structure by the following result. Lemma 1.4.5. The following assertions hold. (1) Let a be a homogeneous element in A+ . For p ∈ D+ (a), the ideal p′ := pA[a−1 ] is a homogeneous prime ideal of A[a−1 ] and p′0 := p′ ∩ A[a−1 ]0 is an element of Spec A[a−1 ]0 . (2) The mapping ψa : D+ (a) → Spec A[a−1 ]0 ,

p 7→ p′0

is a homeomorphism, where the topology on D+ (a) is the one induced by the topology of Proj A. Hence we can consider the affine scheme structure of Spec A[a−1 ]0 transferred on D+ (a) via ψa . (3) Let a, b be homogeneous elements of degree d, e, respectively. Then the affine scheme structures of D+ (a) and D+ (b) are compatible on D+ (a) ∩ D+ (b). Hence Proj A is a scheme. Proof. (1) The result follows from the argument in subsection 1.1.1. (2) ψa is a continuous map. Let d = deg a. For b ∈ And , b/an ∈ ψa (p) if and only if am b ∈ p for some m > 0. Since a ̸∈ p, this is equivalent to b ∈ p. Hence ψa−1 (D(b/an )) = D+ (a) ∩ D+ (b). So, ψa is continuous. ψa is injective. Let p, q ∈ D+ (a) such that p ̸= q. Then there exists a homogeneous element b ∈ A+ such that b ∈ p and b ̸∈ q. Replacing b by bd if necessary, we may assume that b ∈ And . Then ψa (p) ̸∈ D(b/an ) and ψa (q) ∈ D(b/an ). Hence ψa (p) ̸= ψa (q). ψa is surjective. Let p′0 ∈ Spec (A[a−1 ]0 ). Set  {b ∈ An | bd /an ∈ p′0 } if n > 0 pn = {b ∈ A0 | ba ∈ pd } if n = 0 L and set p = n≥0 pn . Then p is a homogeneous ideal of A. We show that p is a homogeneous prime ideal such that ψa (p) = p′0 . Suppose that x ∈ Am , y ∈ An and xy ∈ pm+n . Then a2 xy ∈ pm+n+2d , (a2 xy)d /am+n+2d = ((ax)d )/am+d )·((ay)d /an+d ) ∈ p′0 . Hence (ax)d /am+d ∈ p′0 or (ay)d /an+d ∈ p′0 . Suppose that (ax)d /am+d ∈ p′0 and m > 0. Then xd /am ∈ p′0 and hence x ∈ pm . If m = 0 then ax ∈ pd and hence x ∈ p0 . If (ay)d /an+d ∈ p′0 the same argument applies. As a consequence, xy ∈ pm+n implies x ∈ pm or y ∈ pn . This shows that A/p = ⊕n≥0 An /pn is an integral domain., whence p is a homogeneous prime ideal. ψa−1 is a continuous map. For b ∈ An with n > 0, it holds that ψa (D+ (ba)) = ψa (D+ (a)) ∩ D(bd /an ) (see also Lemma 1.4.4). This shows that ψa−1 is continuous.

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(3) D+ (a) ∩ D+ (b) is an affine scheme Spec (A[(ab)−1 ]0 ) which, by Lemma 1.4.2, is an open set of D+ (a) induced by A[a−1 ]0 → A[(ab)−1 ]0 = A[a−1 ]0 [(bd /ae )−1 ] and an open set of D+ (b) induced by A[b−1 ]0 → A[(ab)−1 ]0 = A[b−1 ]0 [(ae /bd )−1 ]. Hence D+ (a) and D+ (b) are patched together via the identification D+ (a) ∩ D+ (b) = D+ (ab) = Spec (A[(ab)−1 ]0 ). Since Proj A is covered by {D+ (a) | a ∈ An , n > 0}, Proj A has a scheme structure. Proj A is called a projective scheme and A is called the homogeneous coordinate ring. Since A[a−1 ]0 is an A0 -algebra, Proj A is viewed as a scheme over Spec A0 . Example 1.4.6. (1) Let A = k[x0 , x1 , . . . , xn ] be a polynomial ring in n + 1 variables over a field k. We make A a graded k-algebra by setting A0 = k and deg(xi ) = 1 for 0 ≤ i ≤ n. For any homogeneous prime ideal p ∈ Proj A, since p ̸⊇ A+ = (x0 , x1 , . . . , xn ), there is i such that xi ̸∈ p, i.e., p ∈ D+ (xi ). This implies that Proj A = D+ (x0 ) ∪ D+ (x1 ) ∪ · · · ∪ D+ (xn ). For any i ̸= j, D+ (xi ) ∩ D+ (xj ) = Spec A[(xi xj )−1 ]0 and −1 A[(xi xj )−1 ]0 = A[x−1 ] i ]0 [(xj /xi )

= k[x0 /xi , . . . , xi−1 /xi , xi+1 /xi , . . . , xn /xi ][(xj /xi )−1 ] = k[x0 /xj , . . . , xj−1 /xj , xj+1 /xj , . . . , xn /xj ][(xi /xj )−1 ]. Set Ui = D+ (xi ). Then every Ui is isomorphic to the affine n-space An and the patching isomorphism θji : Uij → Uji with Uij = Ui ∩ Uj ⊂ Ui is given by a k-algebra isomorphism φij : k[x0 /xj , . . . , xj−1 /xj , xj+1 /xj , . . . , xn /xj ][(xi /xj )−1 ] ∼

−→ k[x0 /xi , . . . , xi−1 /xi , xi+1 /xi , . . . , xn /xi ][(xj /xi )−1 ]

φij (xℓ /xj ) = (xℓ /xi ) · (xi /xj ) (0 ≤ ℓ ≤ n).

The projective scheme Proj A in Example 1.4.6 is called the projective space of dimension n and denoted by Pnk , where the subscript k is put if one emphasizes the base field k. (2) Assume that k is algebraically closed. With A as in (1) above, we e = Spec A which is the affine space An+1 . The set of closed points set X e is identified with k n+1 the product set of (n + 1)-copies of k, where Ω(X)

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the point of origin (0) := (0, . . . , 0) corresponds to the irrelevant ideal b =X e \ (0) and define an equivalence relation A+ = (x0 , . . . , xn ). We set X n+1 b on X(k) = k \ (0) (a0 , a1 , . . . , an ) ∼ (b0 , b1 , . . . , bn ) ⇔ bi = cai for some c ∈ k ∗ , 0 ≤ i ≤ n.

Then the equivalence class containing (a0 , a1 , . . . , an ), which we denote in the form of a continuous ratio (a0 : a1 : · · · : an ) (or simply by (a0 , a1 , . . . , an ) or [a0 , a1 , . . . , an ]), is represented by (a0 /ai , a1 /ai , . . . , ai−1 /ai , ai+1 /ai , . . . , an /ai ) if ai ̸= 0, which is a closed point of Ui = D+ (xi ). So, Pn (k) is identified with b the set of equivalence classes X(k)/(∼). In view of this construction, the e is called a homogeneous coordinate coordinate system (x0 , x1 , . . . , xn ) of X system of the projective space Pn . A closed point, say P , of Pn is a closed point of some open chart Ui = D+ (xi ). Hence P is given as xj /xi = αj ∈ k for 0 ≤ j ≤ n, where αi = 1. Hence P corresponds to a homogeneous prime ideal P = (x0 − e = An+1 α0 xi , . . . , xn −αn xi ) of A. Namely P corresponds to a line LP on X given by LP = {(α0 t, α1 t, . . . , αn t) | t ∈ A1 (k) = k} e which passes the origin (0) of X. Given a homogeneous polynomial e The F (x0 , . . . , xn ) of degree d, we regard F as a regular function on X. d restriction of F on LP is t F (α0 , . . . , αn ). Define a regular function f on Ui by f (x0 /xi , . . . , xn /xi ) = x−d i F (x0 , . . . , xn ). Hence we have i

∨

i

∨

td f (α0 , . . . , 1, . . . , αn ) = F (α0 t, . . . , t , . . . , αn t) = td F (α0 , . . . , αn ). Hence the function f vanishes at the point P if and only if the function F vanishes on LP . Further, the function F vanishes on LP if and only if F vanishes at a point, say P , on LP which is different from (0). 1.4.2 1.4.2.1

Projective schemes and projective varieties

General properties of projective schemes L Lemma 1.4.7. Let R be a ring and let A = n≥0 An be a graded ring over R. Then we have the following results.

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(1) Proj A is a separated scheme over Spec R. (2) Let n be the nilradical of A and let n+ = A+ ∩ n. Then both n and n+ are homogeneous ideals and Proj (A/n+ ) = (Proj A)red. (3) Proj A is reduced and irreducible if A+ has no zero divisors as a ring without the identity element. Proof. (1) Set X = Proj A and Ua = D+ (a) for a homogeneous element a ∈ A+ . Then U = {Ua }a∈A+ is an affine open covering of X. Let ωab,a : A[a−1 ]0 → A[(ab)−1 ]0 be the canonical ring homomorphism for homogeneous elements a, b ∈ A+ . Then {Ua ×R Ub = Spec (A[a−1 ]0 ⊗R A[b−1 ]0 )}a,b∈A+ is an open covering of X ×R X. Since the ring homomorphism A[a−1 ]0 ⊗R A[b−1 ]0 → A[(ab)−1 ]0 , x ⊗ y 7→ ωab,a (x)ωab,b (y) is surjective, the associated morphism of affine schemes defines a closed immersion ∆a,b : Uab → Ua ×R Ub . Since the restriction of the diagonal morphism ∆X/R : X → X ×R X to Uab splits as ∆a,b

∆X/R |Uab : Uab −→ Ua ×R Ub ⊂ X ×R X, and since {Uab }a,b∈A+ is also an open covering of X, this shows that ∆X/R is a closed immersion. (2) Let a = ai1 +ai2 +· · ·+air be a nilpotent element with aij ∈ Aij \{0}, where i1 < i2 < · · · < ir . If an = 0 then ani1 is the homogeneous part with the smallest degree. Hence ani1 = 0. Then (a − ai1 ) is nilpotent. Hence ai2 is nilpotent by the same argument. Repeat this argument to show that ai1 , ai2 , . . . , air are nilpotent. Hence n is a homogeneous ideal. Then it is clear that n+ is a homogeneous ideal. Let A = A/n+ which is a graded L ring such that A = n≥0 An , where An = An /An ∩ n+ . Let a ∈ Ad with d > 0. Consider the residue homomorphism φa : A[a−1 ]0 → A[a−1 ]0 which maps x/ak with x ∈ Adk to x/ak , where x = x + n+ . If x/ak is a nilpotent element, then ar x ∈ n+ for some r ≥ 0. Hence ar x = ar x = 0 and x/ak = ar x/ar+k = 0. This implies that A[a−1 ]0 is reduced. In fact, A[a−1 ] itself is reduced. Further, if φa (x/ak ) = 0 then as x ∈ n+ for s ≥ 0 and hence (as x)t = 0 for some t > 0 and (x/ak )t = 0 in A[a−1 ]0 . So, Ker φa is contained in the nilradical of A[a−1 ]0 . The converse follows from the fact that the nilradical of A[a−1 ]0 is zero. Namely, φa induces an −1 ∼ isomorphism A[a−1 ]0 /φ−1 ]0 , and the associated isomorphism a (0) = A[a

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θa = a φa : D+ (a) −→ D+ (a)red. If b ∈ A+ is a homogeneous element, then we have a natural commutative diagram A[a−1 ]0  y

φa

−−−−→

A[a−1 ]0  y

A[(ab)−1 ]0 −−−−→ A[(ab)−1 ]0 , φab

which induces a commutative diagram of schemes θ

D+ (ab) −−−ab−→ D+ (ab)red   ιab,a y y(ιab,a )red D+ (a) −−−−→ D+ (a)red , θa

where ιab,a and (ιab,a )red are open immersions. Hence θa |D+ (ab) = θab . This implies that the morphisms in {θa }a∈A+ patch together to give a morphism ∼ θ : Proj (A) −→ Proj (A)red. (3) Suppose that A+ has no zero divisors. For a homogeneous element a ∈ Ad with d > 0, we show that A[a−1 ]0 is an integral domain. Suppose that x/am ̸= 0, y/an ̸= 0 and (x/am ) · (y/an ) = 0 for x ∈ Amd and y ∈ And . Hence ar xy = 0 for some r ≥ 0. Since ar x ̸= 0, we have a contradiction by the hypothesis if m > 0 and n > 0. So, either m = 0 or n = 0. For arbitrary s, t > 0, we have as x ̸= 0 and at y ̸= 0 because x/am ̸= 0 and y/an ̸= 0 in A[a−1 ]0 . Since deg(as x) > 0, deg(at y) > 0 and (as x/as+m ) · (at y)/at+n ) = 0, we have also a contradiction. Hence A[a−1 ]0 is an integral domain. In order to show that Proj A is irreducible, it suffices to show that Proj A is connected. In fact, suppose that Proj A is connected. If Proj A = X1 ∪ X2 with proper closed subsets, X1 ∩ X2 ̸= ∅ because Proj A is connected. Let x ∈ X1 ∩ X2 , which corresponds to a homogeneous prime ideal px of A. Since px ̸⊇ A+ , there exists a homogeneous element a ∈ A+ which is not in px . Then x ∈ D+ (a) and D+ (a) = (X1 ∩ D+ (a)) ∪ (X2 ∩ D+ (a)), where X1 ∩ D+ (a) ̸= ∅ and X2 ∩ D+ (a) ̸= ∅. But this is a contradiction because D+ (a) = Spec A[a−1 ]0 is irreducible. Now suppose that Proj A is ` disconnected. Write Proj A = X1 X2 , where X1 and X2 are nonempty open and closed subsets. Then there exist homogeneous elements a, b such that ∅ ̸= D+ (a) ⊆ X1 and ∅ ̸= D+ (b) ⊆ X2 . Let d = deg a and e = deg b. Then bd /ae ̸= 0 by the assumption that A+ has no zero divisors. Hence D+ (ab) = Spec A[a−1 ]0 [(bd /ae )−1 ] ̸= ∅. Since D+ (ab) = D+ (a) ∩ D+ (b) ⊆ X1 ∩ X2 = ∅, we have a contradiction.

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Remark 1.4.8. In the assertion (3) of Lemma 1.4.7, the converse does not hold as shown by the following example. Let A = k[x0 , x1 , x2 ] be a graded algebra over a field k such that A0 = k, deg(x0 ) = 1, deg(xi ) = 2 for i = 1, 2, x20 = 0 and x1 , x2 are variables. Then Proj A = D+ (x1 )∪D+ (x2 ), where D+ (x1 ) = Spec k[x2 /x1 ] and D+ (x2 ) = Spec k[x1 /x2 ]. In fact, there are no elements x0 /xr1 in A[x−1 1 ]0 with r > 0 because deg(x0 /xr1 ) = 1 − 2r < 0. Similarly, there are no ∼ 1 elements x0 /xs2 in A[x−1 2 ]0 with s > 0. Hence Proj A = P . By Lemma 1.4.7, the projective space Pnk is an algebraic variety of dimension n, and it is a variety different from affine varieties. Theorem 1.4.9. The ring of regular functions on Pnk defined over a field k consists of constants. Namely we have Γ(Pnk , OPnk ) = k. Proof. With notations in Example 1.4.6, Γ(Ui , OPnk ) = k[x0 /xi , . . . , xn /xi ] for 0 ≤ i ≤ n. Further, we have n \ Γ(Pnk , OPnk ) = Γ(Ui , OPnk ) ⊂ k(x0 /xi , . . . , xn /xi ). i=0

Suppose that f (x0 /xi , . . . , xn /xi ) ∈ Γ(Pnk , OPnk ) has total degree d. Then there exists a homogeneous polynomial F (x0 , . . . , xn ) of degree d such that F (x0 , . . . , xn ) = xdi · f (x0 /xi , . . . , xn /xi )

and F (x0 , . . . , xn ) contains a nonzero monomial M which is not divisible by xi . Suppose that d > 0 and M is divisible by some xj with j ̸= i. Then the function f (x0 /xi , . . . , xn /xi ) is not regular on the closed subset V (xi /xj ) of Uj = D+ (xj ). This is a contradiction, and hence d = 0. 1.4.2.2

Projective varieties

Lemma 1.4.10. Let f : X → Y be a separated morphism of schemes and let ι : V → X be a closed immersion. Then the composite f ◦ ι : V → Y is a separated morphism. In particular, if Y = Spec k with an algebraically closed field k, X is an algebraic variety and V is an irreducible and reduced closed subset of X, then V is an algebraic variety. Proof. We have the following commutative diagram V  ∆V /Y y

ι

−−−−→

X  y∆X/Y

V ×Y V −−−−→ X ×Y X, ι×ι

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where the morphism ι and the diagonal morphism ∆X/Y are closed immersion by the hypothesis. The morphism ι×ι is determined by the morphisms pi ι ι ◦ pi : V ×Y V −→ V −→ X for i = 1, 2 (see the construction of fiber product in the appendix). We have to show that ∆V /Y is a closed immersion. By the construction of fiber products, we may assume that Y is an affine scheme Spec B. Let U = {Uλ }λ∈Λ be an affine open covering of X such that Uλ = Spec Aλ . The scheme V as a closed subset of X is covered by the affine open sets Vλ := Uλ ∩ V = Spec Aλ /Iλ , where Iλ is an ideal of Aλ . Since X is separated over Y , Uλ ∩Uµ = ∆−1 X/Y (Uλ ×Y Uµ ) is an affine scheme Spec (Aλµ ) (see comments before Remark 1.3.9), where Aλµ is the image ∆∗X/Y (Aλ ⊗B Aµ ). We have Iλ Aλµ = Iµ Aλµ for any λ, µ ∈ Λ, which we set Iλµ . This implies that the affine schemes Vλ := Uλ ∩ V = Spec (Aλ /Iλ ) and Vµ := Uµ ∩ V = Spec (Aµ /Iµ ) are patched together by the patching ∼ isomorphism θµλ : Uλµ −→ Uµλ to form the closed set V . The above commutative diagram considered on an affine open set Uλ ×Y Uµ of X ×Y X induces a commutative diagram ι

−−−−→

Vλµ  ∆V /Y y

Uλµ  y∆X/Y

Vλ × Vµ −−−−→ Uλ × Uµ , ι×ι

where Vλµ = Vλ ∩ Vµ = Spec (Aλµ /Iλµ ), which is obtained from a commutative diagram ι∗

←−−−− Aλ ⊗R Aµ x ∗ ∆X/Y

Aλµ /Iλµ x  ∆∗ V /Y

(Aλ /Iλ ) ⊗R (Aµ /Iµ ) ←−−−∗− Aλ ⊗R Aµ . (ι×ι)

∗

◦ ∆∗X/Y

∆∗V /Y

∗

∗

Since ι = ◦ (ι × ι) and ι ◦ ∆∗X/Y is surjective, it follows that ∆∗V /Y is surjective. Hence ∆V /Y : Vλµ → Vλ ×Y Vµ is a closed immersion. This implies that ∆V /Y : V → V ×Y V is a closed immersion, and hence V is separated over Y . The second assertion follows immediately. Since k is algebraically closed, V is geometrically irreducible and reduced. By the first assertion V is separable over k. Hence V is also an algebraic variety. In the situation of the second assertion of Lemma 1.4.10, we say that V is a closed subvariety of X since V becomes an algebraic variety embedded into X as a closed set.

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L Let k be a field. Let A = m≥0 Am be a noetherian graded ring which is generated by A1 over A0 = k. By Lemma 1.4.3, A1 is a finite k-module and there is a surjective homomorphism φ : Rn+1 = k[x0 , . . . , xn ] → A such that dimk A1 = n + 1, Rn+1 is a polynomial ring with (n + 1)-variables which is viewed as a graded k-algebra with deg xi = 1 for 0 ≤ i ≤ n, {φ(x0 ), . . . , φ(xn )} is a k-basis of A1 and φ is a graded homomorphism in the sense that φ maps degree r elements of Rn+1 to degree r elements of A.8 P Let I = Ker φ. Then I is a homogeneous ideal. In fact, if f = m≥0 fm ∈ I P then 0 = φ(f ) = m≥0 φ(fm ) with φ(fm ) ∈ Am . Hence φ(fm ) = 0 for every m ≥ 0. Then fm ∈ I. Let X = Proj A. Then X is identified with a closed set V+ (I) of Pnk . Write I = (F0 , . . . , Fr ), where the generators Fi are homogeneous polynomials. Suppose that k is algebraically closed. By the explanations given in Example 1.4.6, the set X(k) of closed points contained in X consists of (α0 , . . . , αn ) ∈ Pn (k) such that Fi (α0 , . . . , αn ) = 0 for 1 ≤ i ≤ r. We call X a projective algebraic set. By Lemma 1.4.7, the set X is irreducible and reduced if A is an integral domain, i.e., the ideal I is a prime ideal. Since X is separated over Spec k by Lemma 1.4.10, X is an algebraic variety. It is called a projective algebraic variety. As with Theorem 1.4.9, we have the following result. Theorem 1.4.11. Let V be a closed integral (i.e., irreducible and reduced) subset of a projective space Pnk over a field k. Then Γ(V, OV ) = k. L Proof. Write V = Proj A with a graded k-algebra A = m≥0 Am which is a surjective image of a polynomial ring Rn+1 by a graded homomorphism φ : Rn+1 → A of degree 0, where n + 1 = dimk A1 . The function field k(V ) of V is the field of quotients Q(A[a−1 i ]0 ) for some (and every, in fact) 0 ≤ i ≤ n. Let ξ ∈ Γ(V, OV ). Since ξ is regular on the open set −1 Ui ∩ V = Spec A[a−1 i ]0 , ξ ∈ A[ai ]0 can be written as ξ=

F (a0 , . . . , an ) , adi

where F (x0 , . . . , xn ) is a homogeneous polynomial of degree d in Rn+1 such that some nonzero monomial is not divisible by xi . Suppose that this 8 A ring homomorphism φ : A → B is said to be a graded homomorphism of degree d if L L A= n≥0 Bn and φ(An ) ⊆ Bn+d for every n ≥ 0. The integer d can n≥0 An , B = be a negative integer.

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nonzero monomial is divisible by, say x0 . Then we can write   −d  F (a0 , . . . , an ) ai . · ξ= a0 ad0 Then ξ is not defined on the closed set V (ai /a0 ) of V ∩ D+ (a0 ) unless d = 0 or ai /a0 is a nonzero constant c ∈ k on V ∩ D+ (a0 ). In the latter case, xi − cx0 is an element of the homogeneous ideal I = Ker φ of Rn+1 . By the choice of φ so that φ induces a k-isomorphism on degree one parts, we have I1 = 0 and accordingly xi = cx0 , which is a contradiction. Hence d = 0 and ξ is an element of k. 1.4.2.3

Projective closure of an affine variety

We consider to embed a given affine variety into a projective space and take the closure of the embedded variety in the projective space. We change the notations to make the presentation as natural as possible. We set Sn = k[x1 , . . . , xn ] be a polynomial ring in n variables over a field k and Rn+1 = k[X0 , . . . , Xn ]. The affine space Ank is embedded into the projective space Pnk = Proj Rn+1 as Ank = D+ (X0 ) ⊂ Pnk . Hence we identify xi with Xi /X0 for 1 ≤ i ≤ n. Let Y be an affine variety embedded into the affine space Ank as a closed subset. Hence Y = Spec B and there is a surjective k-algebra homomorphism ψ : k[x1 , . . . , xn ] → B such that Y = V (J) for J = Ker ψ. Write J = (f1 , . . . , fr ). Define homogeneous polynomials Fi (1 ≤ i ≤ r) by Fi (X0 , . . . , Xn ) = X0di f (X1 /X0 , . . . , Xn /X0 ), di = deg fi and set I = (F1 , . . . , Fr ). Then I is a homogeneous ideal of Rn+1 . Let A = Rn+1 /I, which is a graded k-algebra and there is a surjective graded homomorphism φ : Rn+1 = k[X0 , . . . , Xn ] → A. Let X = Proj A. Then we have the following result. Lemma 1.4.12. With the above notations and assumptions, the following assertions hold.

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(1) X is a projective algebraic set in Pn := Proj Rn+1 . (2) X ∩ U0 = Y , where U0 = D+ (X0 ). (3) Let H0 := V+ (X0 ) which is identified with Pn−1 with the homogeneous coordinate ring k[X1 , . . . , Xn ]. Then X ∩ H0 is a projective algebraic set isomorphic to Proj Rn+1 /(X0 , F1 , . . . , Fr ). Proof. Straightforward. The proof is left to the readers. With the above notations, a closed set V+ (h) of Pn defined by a homogeneous polynomial of degree one h = a0 X0 + a1 X1 + · · · + an Xn , (a0 , . . . , an ) ∈ k n+1 \ {(0)} is called a hyperplane. The defining equation h is determined up to nonzero constants. Namely, V+ (h) = V+ (h′ ) if and only if h = ch′ with c ∈ k ∗ . The if part is clear. For the proof of the only if part, we consider only the case V+ (h) ̸= V+ (X0 ). Then we have V+ (h) ∩ D+ (X0 ) = V (a0 + a1 x1 + · · · + an xn ),

V+ (h′ ) ∩ D+ (X0 ) = V (a′0 + a′1 x1 + · · · + a′n xn ).

Since V+ (h) ∩ D+ (X0 ) and V+ (h′ ) ∩ D+ (X0 ) are irreducible and reduced, we have V+ (h) ∩ D+ (X0 ) = V+ (h′ ) ∩ D+ (X0 ) if and only if the principal ideals (a0 + a1 x1 + · · · + an xn ) and (a′0 + a′1 x1 + · · · + a′n xn ) coincide. The last condition is equivalent to a0 + a1 x1 + · · · + an xn = c(a′0 + a′1 x1 + · · · + a′n xn ), c ∈ k ∗ . So we obtain the assertion. This means that the set of all hyperplanes {V+ (h)} corresponds bijectively with the set of equivalence classes of {(a0 , a1 , . . . , an ) ∈ k n+1 \ {(0)}} modulo the equivalence relation (a0 , a1 , . . . , an ) ∼ (a′0 , a′1 , . . . , a′n ) ⇔ ai = ca′i (0 ≤ i ≤ n), c ∈ k ∗ .

The latter set (k n+1 \ {(0)})/(∼) is the projective space Pn by Example 1.4.6. To distinguish it from the projective space Pn in which we con∨

sider the hyperplanes, we denote the latter projective space by Pn and call it the dual projective space. Example 1.4.13. Let x, y be the coordinates of the affine plane, i.e., A2 = Spec k[x, y], where k is an algebraically closed field. Embed A2 into the projective plane P2 = Proj k[X0 , X1 , X2 ] via the identification A2 = D+ (X0 ) by x = X1 /X0 and y = X2 /X0 . Let f = f (x, y) be an irreducible polynomial in k[x, y], which defines an irreducible and reduced

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curve C = V (f ). In fact, the principal ideal (f ) is a prime ideal of k[x, y] e be the (see subsection 1.5.4 for UFD) and dim k[x, y]/(f ) = 1. Let C 2 e projective closure of C in P . Then C = V+ (F ), where F (X0 , X1 , X2 ) = X0d f (X1 /X0 , X2 /X0 ), d = deg f. We call the homogeneous polynomial the homogenization of f . The points e \ C are called the points at infinity of the affine curve C. in C

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1.5 1.5.1

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Normal varieties Discrete valuation rings and normal rings

We begin with the following important result. Lemma 1.5.1 (Nakayama’s lemma). Let (R, m) be a local ring and let M be a finite R-module. If an R-submodule N of M satisfies M = N +mM then it holds that M = N . Furthermore, one can take a system of generators {z1 , . . . , zn } of M so that the system of residue classes {z 1 , . . . , z n } with z i = zi + mM is an R/m-basis of the vector space M/mM . Proof. The condition implies that (M/N ) = m(M/N ). So, it suffices to show that M = mM for a finite R-module M implies M = 0. Write Pr M = i=1 Rmi for a system of generators {m1 , . . . , mr } of M . Since M = mM , we have relations for 1 ≤ i ≤ r mi =

r X j=1

αij mj , αij ∈ m,

which are written as a matrix equation   m1   (Er − A)  ...  = 0, A = (αij )1≤i,j≤r . mr

Let D = (Er − A) and let D∗ be the cofactor matrix of D. Then D∗ D = dEr , where d = det D. Then, as in the proof of Lemma 1.1.13, we have dmi = 0 for 1 ≤ i ≤ r. Since d − 1 ∈ m, d is invertible in R. Hence mi = 0 for 1 ≤ i ≤ r. This shows that M = 0. To prove the second assertion, take a system of elements {z1 , . . . , zn } so that {z 1 , . . . , z n } is a basis of the vector space M/mM over the field R/m Pn and set N = i=1 Rzi . Then M = N + mM . By the first assertion, we then have M = N . Let R be an integral domain and let K = Q(R) be the field of quotients of R. We say that R is a valuation ring if, for every element ξ ∈ K, either ξ ∈ R or ξ −1 ∈ R. Since ξ = a/b with a, b ∈ R, R is a valuation ring of K if, for any a, b ∈ R, either a ∈ bR or b ∈ aR. Lemma 1.5.2. Let R be a valuation ring and let K = Q(R). Set m = {a ∈ R | a−1 ̸∈ R}. (We consider 0 ∈ m.) Then (R, m) is a local ring. Further, R is integrally closed in K.

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Proof. We show first that m is an ideal of R. In fact, suppose that ax ̸∈ m for a ∈ m and x ∈ R. Then (ax)−1 ∈ R by the definition of m. Then x(ax)−1 = a−1 ∈ R. This contradicts that a ∈ m. Suppose a, b ∈ m. Then a + b ∈ m. In fact, either a ∈ bR or b ∈ aR. If a ∈ bR then a + b ∈ bR ⊂ m. If b ∈ aR then a + b ∈ aR ⊂ m. Hence a + b ∈ m. Thus m is an ideal of R. Let a ∈ R \ m. Then a−1 ∈ R, and a ∈ R∗ . So, (R, m) is a local ring. We show that R is integrally closed. Suppose that ξ satisfies a monic equation ξ n + a1 ξ n−1 + · · · + an = 0, ai ∈ R.

If ξ ̸∈ R them ξ −1 ∈ m, and the above equation becomes 1 + a1 (ξ −1 ) + · · · + an (ξ −1 )n = 0.

Then 1 ∈ m. This is a contradiction. Hence ξ ∈ R. Let R be a valuation ring and let K = Q(R). Set K ∗ = K \ {0} and R the unit group of R. Then K ∗ /R∗ is a totally ordered abelian group by defining an order and the addition as follows: ∗

• ηR∗ ≥ ξR∗ for ξ, η ∈ K ∗ if ηξ −1 ∈ R. • (ξR∗ ) · (ηR∗ ) = (ξηR∗ ). Then ξ ≥ η and η ≥ ξ if and only if ηξ −1 ∈ R∗ . The group K ∗ /R∗ is called the value group of the valuation group. We denote it by Γ and write the multiplication additively, i.e., [ξR∗ ]+[ηR∗ ] = [(ξη)R∗ ]. We further consider the extra element ∞ and assume that ∞ > α for any element of Γ. Define a map

∗

v : K → Γ ∪ (∞)

by v(ξ) = [ξR ]. Then v satisfies the following conditions: (i) v(ξ · η) = v(ξ) + v(η). (ii) v(ξ + η) ≥ min{v(ξ), v(η)}. (iii) v(ξ) = ∞ ⇔ ξ = 0. The mapping v is called the additive valuation attached to the valuation ring R. Furthermore, R = {ξ ∈ K | v(ξ) ≥ 0} and m = {ξ ∈ K | v(ξ) > 0}, where we note that, since the element 0 in Γ corresponds to R∗ , v(ξ) ≥ 0 if and only if ξ ∈ R, and v(ξ) > 0 if and only if ξ ∈ R and ξ −1 ̸∈ R. Lemma 1.5.3. Let (R, m) be a local ring. Then the following three conditions are equivalent.

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(i) (R, m) is a noetherian valuation ring. (ii) (R, m) is a principal ideal domain (PID, for short). (iii) R is a valuation ring such that the value group Γ is isomorphic to Z. Proof. (i) ⇒ (ii). We show that any proper ideal I is a principal ideal. Since R is noetherian, I is finitely generated, i.e., I = (a1 , . . . , ar ). For any pair (i, j) with i ̸= j, either ai ∈ aj R or aj ∈ ai R. After replacing the order of a1 , . . . , ar , we may assume that aj ∈ ai R if j > i. Then I = a1 R. In particular, m = tR. (ii) ⇒ (i). A PID is a noetherian ring because every ideal is principal. Write m = tR. We show that every element a ∈ R is written as a = utn for n ≥ 0 and u ∈ R∗ . If a ̸∈ m then a ∈ R∗ . So, we set n = 0 and u = a. Suppose that a ∈ m. Then a = ta1 . If a1 ̸∈ m then a1 ∈ R∗ . So, set n = 1 and u = a1 . If a1 ∈ m then a = t2 a2 . If a2 ̸∈ m, we argue as above. If a2 ∈ m, set I1 = a1 R and I2 = a2 R. Then I1 ⫋ I2 because a1 = ta2 and t ̸∈ R∗ . We show that a = tn u for some n > 0 and u ∈ R∗ . Otherwise, if we write a = tn an with an ∈ R, then an ∈ m. Set In = an R. Since an−1 = tan , we have In−1 ⫋ In . Hence we have an ascending sequence of ideals I1 ⫋ I2 ⫋ · · · ⫋ In ⫋ · · · .

This contradicts that R is noetherian. Given a, b ∈ R, write a = tn u, b = tm v with u, v ∈ R∗ . Then a ∈ bR (if n ≥ m) or b ∈ aR (if m > n). (ii) ⇒ (iii). The argument in “(i) ⇒ (ii)” shows that a = tn u with n ≥ 0 and u ∈ R∗ . Hence Γ is generated by the class tR∗ , hence isomorphic to Z. (iii) ⇒ (ii). Since every element a ∈ R is written as a = tn u with n ≥ 0 and u ∈ R∗ , where t is an element of R such that v(t) = 1, every ideal I of R is principal. In fact, given an ideal I of R, take an element a ∈ I such that v(a) is the smallest. Such an element exists because v(b) ≥ 0 for every b ∈ R. For any element a′ ∈ I, v(a′ a−1 ) = v(a′ ) − v(a) ≥ 0, whence a′ a−1 ∈ R, i.e., a′ ∈ aR. This shows that I = aR. A valuation ring (R, m) satisfying the three conditions of Lemma 1.5.4 is called a discrete valuation ring (DVR, for short). A DVR is characterized in simpler terms. Lemma 1.5.4. Let (R, m) be a noetherian local ring. Then the following two conditions are equivalent. (i) (R, m) is a DVR. (ii) R is an integral domain and m is a nonzero principal ideal.

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Proof. (i) ⇒ (ii). Clear by the definition of DVR. (ii) ⇒ (i). Write m = tR. We show that any nonzero element a ∈ R is written as a = tn u with n ≥ 0 and u ∈ R∗ . If a ̸∈ m then a ∈ R∗ . So, take n = 0 and u = a. If a ∈ m, write a = ta1 . By induction, suppose that a = tn an with an ∈ R. If an ̸∈ m, we are done. Otherwise, T T a = tn+1 an+1 with an+1 ∈ R. Set J = n>0 mn = n>0 tn R. Then J = 0 by Krull’s intersection theorem (see Example 1.9.12(1)). We can also show it by Lemma 1.5.1 since J = mJ clearly and J is a finite R-module as an ideal of a noetherian ring. This proves that if a ̸= 0 then an ̸∈ m for some n > 0. Let I be a proper ideal. Write I = (a1 , . . . , am ). Then ai = tni ui as above for every 1 ≤ i ≤ m. Let n0 = min1≤i≤m ni . Then I = tn0 R. Then (R, m) is a DVR by Lemma 1.5.3. A generator t of the maximal ideal m = (t) of a discrete valuation ring R is called a uniformisant of R. The following result will be used in the proof of Theorem 1.5.6 below. Lemma 1.5.5. Let R be a noetherian integral domain and let p be a prime ideal of R. Suppose that Rp is not a valuation ring. Then, for any nonzero element a ∈ p and any element b ∈ (aR : p), the quotient b/a is integral over R, and (b/a)p ⊆ p. Proof. We may assume that b ̸= 0. For x ∈ p, bx = ay for some y ∈ R. We claim that (aR : b) ⊆ (xR : y). In fact, for z ∈ (aR : b), bz = as for some s ∈ R. Since bzy = asy = bxs, we have zy = xs because b ̸= 0. So, z ∈ (xR : y). The claim is proved. Suppose y ̸∈ p. Then (xR : y) ⊆ p, and p ⊆ (aR : b) ⊆ (xR : y) ⊆ p. Hence p = (xR : y). This implies that pRp = xRp . In fact, for p ∈ p and (v/t) ∈ Rp , we have p · (v/t) = (pyv)/(yt) ∈ xRp as yt ̸∈ p. The converse is clear because x ∈ p. It then follows by Lemma 1.5.4 that (Rp , pRp ) is a DVR. This contradicts the hypothesis. Hence y ∈ p. This implies that (b/a)p ⊆ p. Since p is a finite R-module, it follows that b/a is integral over R. A noetherian integral domain R which is integrally closed in the field of quotients Q(R) is called a normal ring. Theorem 1.5.6. Let (R, m) be a noetherian local ring. Then the following two conditions are equivalent.

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(i) (R, m) is a DVR. (ii) R is a normal ring of dim R = 1 Proof. (i) ⇒ (ii). By Lemma 1.5.2, R is integrally closed in Q(R). Since R is noetherian, R is a normal ring. By Lemma 1.5.3, every ideal of R is a power mr of the maximal ideal. This implies that (0) ⫋ m is a maximal sequence of prime ideals. Hence dim R = 1. (ii) ⇒ (i). Suppose that R is not a DVR. For a nonzero element a ∈ m, m is a minimal prime divisor since dim R = 1. By explanations after Corollary 1.1.8, m = (aR : b) for some b ∈ R. Then b ∈ (aR : m). By Lemma 1.5.5, b/a is integral over R. Since R is integrally closed, b/a ∈ R. Then b ∈ aR, hence (aR : b) = R. This is a contradiction. Hence R is a valuation ring. Since R is noetherian, R is a DVR. Corollary 1.5.7. Let R be a normal ring and let p ∈ Spec R such that ht p = 1. Then Rp is a DVR. Proof. By Lemma 1.1.15, Rp is integrally closed. Since R is noetherian and ht p = 1, Rp is a normal ring of dimension one. By Theorem 1.5.6, Rp is a DVR. For the definition of prime divisor in the next result, see the explanations after Corollary 1.1.8. Lemma 1.5.8. Let R be a normal ring and let a be a nonzero element of R. Then any prime divisor p of aR has height one. Hence aR has no embedded prime divisors. Proof. Write p = (aR : b) for an element b ∈ R. Then b ∈ (aR : p). If Rp is not a DVR, then b/a is integral over R by Lemma 1.5.5. Since R is integrally closed, b/a ∈ R. Hence b ∈ aR and (aR : b) = R, which is a contradiction. Thus Rp is a DVR. This implies that ht p = 1. The second assertion is clear. In fact, if p is an embedded prime divisor then p contains a minimal prime divisor p0 of aR. Then ht p ≥ 2. This is a contradiction. The following result plays an important role in algebraic geometry. Theorem 1.5.9. Let R be a normal ring and let K be the field of quotients of R. Then we have \ R= Rp , ht p=1

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where p moves over all prime ideals of height one of R and Rp is considered as a subring of the field K. T T Proof. It suffices to show that ht p=1 Rp ⊆ R. Let ξ ∈ ht p=1 Rp and write ξ = b/a with a, b ∈ R. If a ∈ R∗ , it is clear that ξ ∈ R. So, assume that a ̸∈ R∗ . Namely, aR ̸= R. Let aR = q1 ∩ · · · ∩ qn be an irredundant primary decomposition (see the appendix, Theo√ rem 1.9.4) and let pi = qi . Then ht pi = 1 for every 1 ≤ i ≤ n, and hence there are no inclusion relations among the pi . This implies that qj Rpi = Rpi if j ̸= i. Hence we have   n n \ \ aRpi =  qi  Rpi = (qj Rpi ) = qi Rpi . j=1

j=1

Since (qi Rpi ) ∩ R = qi by Lemma 1.9.3, we have b∈

n \

i=1

(aRpi ∩ R) =

n \

i=1

(qi Rpi ∩ R) =

n \

qi = aR.

i=1

It then follows that ξ ∈ R. 1.5.2

Normalization of affine domains

Let R be a normal affine domain over a field k and let K be the field of quotients. Let L be a finite algebraic extension of K and let S be the integral closure of R in L. Under the assumption that L/K is a separable extension, we prove that S is a finite R-module and hence that S is an affine domain. As a corollary, we will obtain a result that if A is an affine e of A domain over the field k of characteristic zero, the integral closure A e is a finite A-module. We call A e the in Q(A) is an affine domain and A normalization of A. Since L/K is a sequence of simple extensions, we may assume that L/K is a simple extension L = K[α] such that α is a root of an irreducible equation f (x) = 0, where f (x) = a0 xn + a1 xn−1 + · · · + an , ai ∈ R, a0 an ̸= 0. Since L = K[a0 α] and a0 α is a root of an−1 f (x), we may assume that f (x) 0 e be the smallest extension of L is a monic polynomial, i.e. a0 = 1. Let L Qn such that f (x) decomposes into linear factors, i.e., f (x) = i=1 (x − αi ),

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e where α = α1 . Then L/L is a Galois extension and αi is integral over R for every 1 ≤ i ≤ n because f (αi ) = 0. Let D be the Wronskian determinant 1 α1 α12 · · · α1n−1 1 α2 α22 · · · α2n−1 Y = (αj − αi ). D = ··· ··· j>i 1 α α2 · · · αn−1 n n n We call d = D2 the discriminant of f (x). We have the following result (see [72, Theorem 10.15]). Lemma 1.5.10. With the above notations and assumptions, let R′ = R[x]/(f (x)) = R[α]. Then dS ⊆ R′ . Hence S is a finite R-module. Proof. By the assumption, d ̸= 0. As the field extension L/K is simple, L = K[α] and {1, α, . . . , αn−1 } is a K-basis of L. There exists a e→L e such that φi (α) = αi , where 1 ≤ i ≤ n. We K-automorphism φi : L set φ1 = id. Since α is integral over R and φi (f (α)) = f (αi ), αi is integral over R. Let b ∈ S. As an element of L, we can write n−1 X b= uj αj , uj ∈ K. j=0

Then, for 1 ≤ i ≤ n, we have φi (b) =

n−1 X

uj αij .

j=0

Write these equations in a matrix form as follows:      φ1 (b) u0 1 α1 · · · α1n−1  φ2 (b)     1 α2 · · · α2n−1      u1  .  . =  ···   ..   · · · · · · n−1 un−1 1 α · · · α n n φn (b) Denote the Wronskian matrix by W and its cofactor matrix by W ∗ . Since αi (1 ≤ i ≤ n) is integral over R, all entries of the cofactor matrix W ∗ are integral over R, by multiplying W ∗ to the above matrix equation from the left, we know that Duj (0 ≤ j ≤ n−1) is integral over R, where D = det W and W ∗ W = DEn . Note that d = D2 is invariant by every φi for 1 ≤ i ≤ n. Namely d ∈ K and integral over R. Since R is normal, d ∈ R. Further, duj = D(Duj ) is an element of K which is integral over R. So, duj ∈ R. Pn−1 Pn−1 It then follows that db = j=0 (duj )αj ∈ R′ = j=0 Rαj . So, dS ⊆ R′ . Pn−1 j We have S ⊆ d1 R′ = j=0 R αd . Namely, S is an R-submodule of a finite R-module d1 R′ . Since R is noetherian, S is a finite R-module by Lemma 1.1.11.

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Corollary 1.5.11. Let A be an affine domain defined over a field k of e be the integral closure of A in the field of quotients characteristic zero. Let A e e is an affine domain L = Q(A). Then A is a finite A-module, hence A over k. Proof. By Noether’s normalization lemma (see Theorem 1.1.17), there exists a polynomial ring R = k[x1 , . . . , xn ] of dimension n = dim A such that R is a k-subalgebra of A and A is integral over R. Let K = Q(R). Then R is a normal ring as R is a unique factorization domain (see the next subsection), and L is a finite separable extension of K as the characteristic of e By k is zero. Let S be the integral closure of R in L, which is equal to A. e Lemma 1.5.10, S is a finite R-module. Since we have inclusions R ⊆ A ⊆ A, e it follows that A is a finite A-module. Remark 1.5.12. Corollary 1.5.11 is valid without the restriction on the characteristic of k. For the reference, see [98, 1, vol. 1, Ch. V, Theorem 9]. A morphism of affine schemes f : Spec A → Spec B associated with a ring homomorphism φ : B → A is called a finite morphism if A is a finite Be → Spec A module via the homomorphism φ. Then the morphism ν : Spec A e defined by the inclusion A ,→ A is a finite morphism. The morphism ν is called the normalization morphism. 1.5.3

Normal varieties and normalization of algebraic varieties

Let X be a connected scheme of finite type over a field k. We say that X is a normal scheme if the local ring OX,x is normal for every point x ∈ X. Then X is irreducible. In fact, if X is reducible, let x be a point in the intersection of two irreducible components. Then the local ring OX,x has at least two minimal prime divisors corresponding to the irreducible p components. Hence (0) ̸= (0) in OX,x . This is a contradiction because OX,x is an integral domain. Further, for any affine open set U = Spec A, the coordinate ring A is integrally closed. This follows immediately from T the equality A = Γ(U, OU ) = x∈U OX,x (see Theorem 1.3.3). In view of the explanations given in subsection 1.3.1 we assume in this subsection that k is an algebraically closed field and X is separated over k. Reformulation of the subsequent arguments in the case where k is not algebraically closed is left to the readers. Under the additional assumption, the function field k(X) = Q(A) is a regular extension of k, hence X is an algebraic variety. We call it a normal variety.

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Lemma 1.5.13. Let X be a normal variety over k and let x be a closed point. A rational function ξ ∈ k(X) is regular at x if and only if ξ is defined on every subvariety D of codimension one passing through x. Proof. Let U = Spec A be an affine open neighborhood of x. Then x corresponds to a maximal ideal m of A. Let R = Am . A closed subvariety D of codimension one with D ∩ U ̸= ∅ corresponds to a prime ideal p of height one, and x ∈ D is equivalent to p ⊆ m. Then ξ is regular at x if and only if ξ ∈ R, and ξ is defined on D if and only if ξ ∈ Ap = RpR . Then the assertion follows from Theorem 1.5.9. Corollary 1.5.14. With the same assumptions as in the above lemma, if a rational function ξ ∈ k(X) is not regular on X then ξ is not regular on a subvariety of codimension one. Let Z be the set of points of X at which ξ is not defined is a finite union of subvarieties of codimension one. Proof. Since X has a finite affine open covering X = U1 ∪ · · · ∪ Ur , it suffices to show the assertion in the case where X is an affine variety. Let X = Spec A. Then Z is a union of the p codimension one components of the closed set V ((A : ξ)). Since the ideal (A : ξ) has finitely many prime divisors of height one, Z is a finite union of subvarieties of codimension one. Our next goal is to prove the following theorem. We need some definitions on morphisms of algebraic varieties. A morphism of algebraic varieties f : X → Y is a finite morphism if the following conditions are satisfied (see also Remark 1.5.12 for a special case): (i) There exists a finite affine open covering U = {Ui }i∈I of Y such that f −1 (Ui ) is an affine open set for every i ∈ I. (ii) Write Ui = Spec Bi and f −1 (Ui ) = Spec Ai . Then Ai is a finite Bi module. A morphism f : X → Y is dominant if the image f (X) contains an open set of Y . As a special case, f is birational if f is dominant and induces an isomorphism of the rational function fields, i.e., f ∗ : k(Y ) → k(X) is an isomorphism. Note that a dominant finite morphism is surjective by Lying-over theorem (see Theorem 1.1.19). Theorem 1.5.15. Let X be an algebraic variety defined over k. Then there e and a birational finite morphism ν : X e → X. exist a normal variety X e Such a pair (X, ν) is determined uniquely up to an isomorphism. Namely,

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if (Z, µ) is another pair with Z a normal algebraic variety and µ : Z → X ∼ e −→ a birational finite morphism, then there exists an isomorphism λ : X Z such that µ ◦ λ = ν. Proof. By Definition 1.3.11, there exists a finite affine open covering U = ei be the {Ui }i∈I such that Ui = Spec Ai and Ai is an affine k-domain. Let A 9 e ei . integral closure of Ai in the function field L = k(X) and let Ui = Spec A ei → Ui is a finite By Corollary 1.5.11, the normalization morphism νi : U morphism. For i, j ∈ I with i ̸= j, Uij = Ui ∩Uj is an affine scheme Spec Aij , eij = U ei ×U Uij , where Aij is an affine domain (see Problem 5). Let U i e e eij , where which is an open set of Ui (see Problem 6). Then Uij = Spec A eij = A e ⊗ A is a finite Aij -module. Since U ei is a normal variety, A T i Ai ij e ei is normal. Aij = xe∈Ueij OUei ,ex , which is integrally closed in L because U eij is the integral closure of Aij in L. Namely, νi | e : U eij → Uij Hence A Uij

is the normalization morphism. With Ui ∩ Uj viewed as an open set Uji eji → Uji is the normalization morphism. of Uj , the restriction νj |Ueji : U ∼

On the other hand, there is an isomorphism θji : Uij −→ Uji with which we patch Ui and Uj along Uij and Uji . This isomorphism is induced by a ∼ k-algebra isomorphism σji : Aji −→ Aij . This isomorphism is extended to ∼ e eji −→ a k-algebra isomorphism σ eji : A Aij and a k-isomorphism of schemes ∼ eij −→ U eji . Then the collection {U ei , θeij | i, j ∈ I} and the collection θeij : U ei → Ui | i ∈ I} define an algebraic variety X e and a birational finite {νi : U e → X. morphism ν : X To prove the second assertion, take the above finite affine open covering U = {Ui }i∈I . Since µ : Z → X is a finite (hence affine) morphism, the open set Zi := µ−1 (Ui ) is an affine open set for every i ∈ I (see Problem 7). Write Zi = Spec Bi . Since Z is normal and Bi is a finite Ai -module, Bi is ei . Namely, the integral closure of Ai in Q(Ai ), hence Bi is isomorphic to A ∼ e there is a k-isomorphism λi : Ui −→ Zi . One can show that λi and λj map eij to Zij = Zi ∩ Zj and U eji to Zji so that (λj | e ) ◦ (θeji ) = λi | e . Hence U Uji Uij ∼ e −→ {λi }i∈I patch together to give an isomorphism λ : X Z. e the normalization of X and the finite morphism ν : X e →X We call X the normalization morphism. Let X be an algebraic variety and let K be a finite algebraic field extension of the function field k(X). With the notations as in the above proof, 9 There

∼

are k-isomorphisms of fields σji : k(Uj ) −→ k(Ui ) induced by the patching isomorphisms given before Example 1.3.7. We fix some index i and take L = k(Ui ).

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ei be the integral closure of Ai in K and let U ei = Spec A ei . Then the let A e ei }i∈I and normalization morphism νi : Ui → Ui is a finite morphism, and {U e → X. We call X e the {νi }i∈I patch together to give a finite morphism ν : X normalization of X in the field K. If L is a finitely generated field extension of k(X) then the algebraic closure K of k(X) in L is a finite algebraic extension (see Theorem 1.2.4). Hence we can consider the normalization of X in K. We call it the normalization of X in L. 1.5.4

Unique factorization domains

Let R be an integral domain. Recall that all units of R make a multiplicative group R∗ . Two elements a, b ∈ R are concomitant (a ∼ b) if b = ua with u ∈ R∗ . Then it is clear that a ∼ b if and only if aR = bR. A nonzero element a ∈ R is reducible if a = bc for some b, c ∈ R \ R∗ . A nonzero element a ∈ R is irreducible if it is not reducible. A prime element is an element p ∈ R such that the principal ideal pR is a prime ideal. A unique factorization domain (UFD, for short) is an integral domain R satisfying the following two conditions. (i) Every nonzero element a ∈ R is written as a product a = ua1 · · · am , where u ∈ R∗ and all ai is an irreducible element. (ii) If a has two decompositions into products of irreducible elements, say a = ua1 · · · am = vb1 · · · bm , u, v ∈ R∗ , then m = n and bi ∼ aσ(i) for every 1 ≤ i ≤ m and a permutation σ of the set {1, 2, . . . , m}. Lemma 1.5.16. The following assertions hold. (1) In a noetherian integral domain, every nonzero element has a decomposition into irreducible elements. (2) A prime element is an irreducible element. (3) If R is a UFD, an irreducible element is a prime element. Conversely, if every irreducible element is a prime element, then R is a UFD provided R is noetherian. (4) A UFD is an integrally closed domain. (5) A PID is a UFD. Proof. (1) Let S be the set of principal ideals aR such that a does not admit a decomposition into irreducible elements. It suffices to show that S = ∅. Suppose S ̸= ∅. We define a partial order by inclusion, i.e.,

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bR ≤ aR if and only if bR ⊆ aR. Since R is a noetherian domain and hence an ascending chain of principal ideals terminates, any totally ordered subset of S has an upper bound. Hence S is an inductive set. By Zorn’s lemma, S has a maximal element, say a0 R. Then a0 is reducible. Write a0 = a1 a2 with a1 , a2 ̸∈ R∗ . Then a0 R ⫋ ai R for i = 1, 2, and hence a1 R, a2 R ̸∈ S. This implies that ai has a decomposition into irreducible elements, ai = ui ai1 · · · aini ,

ui ∈ R∗ ,

(i = 1, 2).

Then a0 = (u1 u2 )a11 · · · a1n1 a21 · · · a2n2 is a decomposition into irreducible elements. This is a contradiction. So, S = ∅. (2) Let p be a prime element of R. Suppose p = ab. Then either a = pa′ or b = pb′ with a′ , b′ ∈ R. Since R has no zero divisors, it follows that either a′ b = 1 or ab′ = 1. Namely, b ∈ R∗ or a ∈ R∗ . Hence a is an irreducible element. (3) The first assertion is clear. In fact, let a be an irreducible element and let bc ∈ aR. Hence ad = bc. Since R is a UFD, b, c and d have decompositions into irreducible elements, b = ub1 · · · bm ,

c = vc1 · · · cn ,

d = wd1 · · · dr ,

u, v, w ∈ R∗ .

Hence wad1 · · · dr = (uv)b1 · · · bm c1 · · · cn . Then a is concomitant with either bi (1 ≤ i ≤ m) or cj (1 ≤ j ≤ n). Namely, either bi ∈ aR or cj ∈ aR. Hence either b ∈ aR or c ∈ aR. This shows that a is a prime element. We prove the second assertion. Since R is assumed to be noetherian, the condition (i) in the definition of UFD is satisfied. We show that the condition (ii) is satisfied as well. Suppose that we have two decompositions into irreducible elements a = ua1 · · · am = vb1 · · · bn , u, v ∈ R∗ . Then a1 , . . . , am and b1 , . . . , bn are prime elements by assumption. We proceed by induction on min(m, n). Since a1 is a prime element, one of b1 , . . . , bn belongs to a1 R, say b1 ∈ a1 R after a permutation of {1, . . . , n}. Since b1 is also a prime element, we have a1 ∼ b1 and u′ a2 · · · am = v ′ b2 · · · bn , u′ , v ′ ∈ R∗ .

By induction, we conclude that m − 1 = n − 1 and ai ∼ bi after a suitable permutation of {2, . . . , n}. Thus the condition (ii) is verified, and R is a UFD.

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(4) An element ξ of the field of quotients Q(R) is written as a fraction ξ = b/a of elements a, b of R such that, when a, b are decomposed into products of irreducible elements, a and b have no common irreducible elements up to concomitance relation. Suppose that ξ satisfies a monic relation ξ n + c1 ξ n−1 + · · · + cn = 0, ci ∈ R. Then we can write it as bn = −a(c1 bn−1 + · · · + cn an−1 ). It shows that if a is not a unit there is at least one irreducible element shared by a and b. Hence a ∈ R∗ . So, ξ ∈ R. Thus R is integrally closed in the field of quotients Q(R). (5) Suppose that R is a PID. Let a ∈ R be a non-unit. Then aR ⫋ R. Hence there exists a maximal ideal m1 containing aR. Since m1 is a prime ideal, m1 = p1 R. Then a = p1 a1 . If a1 ̸∈ R∗ , we find a prime element p2 such that a1 = p2 a2 . Note that aR ⫋ a1 R ⫋ a2 R. If this process does not terminate, we will find an infinite ascending series of proper ideals aR ⫋ a1 R ⫋ a2 R ⫋ · · · ⫋ an R ⫋ · · · . This is a contradiction because a PID is a noetherian ring. Hence a is written as a = up1 p2 · · · pn with prime elements pi and u ∈ R∗ . Namely R is a UFD. In view of the assertion (3), a noetherian UFD is called also a factorial domain. In the case of decomposition into prime elements, the uniqueness of decomposition up to concomitance follows automatically. Given elements a1 , . . . , an of a UFD R, after writing them as products of irreducible elements, we can find an element obtained as a product of as many irreducible elements as possible which are common up to concomitance relation among a1 , . . . , an . It is called the greatest common divisor and denoted by gcd(a1 , . . . , an ). By gcd(a1 , . . . , an ) = 1 we signify that there is no irreducible element common to a1 , . . . , an . We can also consider the least common multiple. The detail of how to define it is left to the readers as an exercise. The following result is important. Theorem 1.5.17. Let R be a factorial domain. Then a polynomial ring R[x] is also factorial. In particular, a polynomial ring k[x1 , . . . , xn ] over a field k is a factorial domain. Hence it is a normal domain.

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Proof. (1) Let K be the field of quotients of R. A polynomial f (x) ∈ R[x] is called primitive if f (x) = a0 xn + a1 xn−1 + · · · + an with a0 ̸= 0 and gcd(a0 , . . . , an ) = 1. Every polynomial F (x) ∈ K[x] is written as F (x) = ξf (x) with ξ ∈ K and a primitive polynomial f (x) ∈ R[x]. Since (R[x])∗ = R∗ (see Problem 8), this decomposition is determined up to multiples of R∗ . Namely, if F (x) = ηg(x) with η ∈ K and g(x) a primitive polynomial in R[x] then g(x) = uf (x) with u ∈ R∗ , i.e., g(x) ∼ f (x). (2) We prove that if f1 (x), f2 (x) ∈ R[x] are primitive polynomials then f1 (x)f2 (x) is a primitive polynomial. In fact, write f1 (x) = a0 xm + a1 xm−1 + · · · + am and f2 (x) = b0 xn + b1 xn−1 + · · · + bn . Let p be a prime element of R. Since f1 (x), f2 (x) are primitive, we find integers r, s ≥ 0 such that p | ai (i < r), p ∤ ar

p | bj (j < s), p ∤ bs .

The coefficient of the monomial x(m+n)−(r+s) in f1 (x)f2 (x) is written as X X X ai bj = ar bs + ai bj + ai bj , i+j=r+s

i+j =r+s i < r, j > s

i+j =r+s i > r, j < s

where the second and the third terms of the right-hand side in the equality are divisible by p, though the first term is not. Hence f1 (x)f2 (x) is primitive. (3) A polynomial f (x) ∈ R[x] is a prime element of R[x] if and only if either f (x) = a ∈ R which is a prime element of R or f (x) is primitive and f (x) is an irreducible polynomial in K[x]. In fact, suppose that f (x) is a constant a. Then R[x]/(a) = (R/(a))[x], which is an integral domain if and only if so is R/(a). This verifies the above assertion in the case where f (x) = a. So, suppose that f (x) ̸∈ R and f (x) is a prime element. It is clear that f (x) is primitive. Suppose that f (x) is reducible in K[x]. Write f (x) = G1 (x)G2 (x) with Gi (x) ∈ K[x] and deg Gi (x) > 0. Write Gi (x) = ξi gi (x), where ξi ∈ K and gi (x) is a primitive polynomial in R[x]. Since f (x) is primitive, it follows that ξ1 ξ2 ∈ R∗ , i.e., f (x) ∼ g1 (x)g2 (x). Comparison of x-degrees shows that f (x) is not a prime element. Hence f (x) is an irreducible polynomial in K[x]. Conversely, suppose that f (x) is a non-constant primitive polynomial and an irreducible polynomial of K[x]. Suppose that g(x)h(x) = f (x)ℓ(x) in R[x]. Since K[x] is a PID (and hence a UFD by Lemma 1.5.16), either g(x) ∈ f (x)K[x] or h(x) ∈ f (x)K[x]. Suppose that g(x) ∈ f (x)K[x], i.e., g(x) = f (x)Q(x) with Q(x) ∈ K[x].

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Write Q(x) = γq(x), where γ ∈ K and q(x) is a primitive polynomial. Then, writing γ = a/b with a, b ∈ R and gcd(a, b) = 1, we have bg(x) = af (x)q(x), where f (x)q(x) is a primitive polynomial. This implies that a | g(x) and g(x)/a ∼ f (x)q(x), whence g(x) ∈ f (x)R[x]. If h(x) ∈ f (x)K[x], we have h(x) ∈ f (x)R[x] in a similar way. So, f (x) is a prime element of R[x]. (4) Let f (x) be an element of R[x], and decompose f (x) in K[x] into a product of irreducible polynomials f (x) = G1 (x) · · · Gr (x). For every 1 ≤ i ≤ r, write Gi (x) = (ai /bi )gi (x) with ai , bi ∈ R, gcd(ai , bi ) = Qr 1 and a primitive polynomial gi (x). Then, with a, b ∈ R such that a| i=1 ai Qr and b| i=1 bi , we have bf (x) = ag1 (x) · · · gr (x), where g1 (x) · · · gr (x) is a primitive polynomial. This implies that a = bc for some c ∈ R, i.e., b ∈ R∗ . Write c = up1 · · · ps with u ∈ R∗ and prime elements p1 , . . . , ps of R. Hence f (x) = up1 · · · ps g1 (x) · · · gr (x). This is a decomposition of f (x) into a product of prime elements of R[x]. Namely, R[x] is factorial. Let R = k[x1 , x2 , . . .] be a polynomial ring in infinitely many variables over a field k. Then R is a UFD, but not noetherian. We prove the following result. Theorem 1.5.18. The following two conditions about a noetherian integral domain R are equivalent. (i) R is a UFD. (ii) Every prime ideal of height 1 of R is a principal ideal. Proof. (i) ⇒ (ii). Let p be a prime ideal of height 1. Take a nonzero element a ∈ p. Let a ∼ p1 · · · pn be a decomposition into prime elements pi . Since a ∈ p, it follows that pi ∈ p for some i. Then pi R ⊆ p. Since pi R is a prime ideal and ht p = 1, we have p = pi R. (ii) ⇒ (i). Let a ∈ R be a nonzero element. We show that a is concomitant to a product of prime elements. Suppose that a ̸∈ R∗ . Let p1 be a minimal prime divisor of aR. By Krull’s principal ideal theorem (see Lemma 1.6.3 below), p1 has height 1, and hence p1 = p1 R by the assumption. Write a = p1 a1 . If a1 ̸∈ R∗ , then aR ⫋ a1 R. By the same process for a1 , we can write a1 = p2 a2 with a1 R ⫋ a2 R unless a2 ∈ R∗ . This process

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must terminate in finitely many steps. For otherwise, we obtain an infinite ascending chain of ideals aR ⫋ a1 R ⫋ · · · ⫋ an R ⫋ · · · . This contradicts the assumption that R is noetherian. Corollary 1.5.19. Let R be a UFD and let S be a multiplicative set of R. Then S −1 R is a UFD. Proof. Let s be an element of S and let s = up1 · · · pr be a decomposition −1 into prime elements. Then it is clear that R[s−1 ] = R[p−1 1 , . . . , pr ]. Hence we may assume that S is generated by prime elements of R. To prove the assertion, it suffices to show that a prime ideal P of S −1 R of height one is principal. Since P = p(S −1 R) for a prime ideal p of height one of R such that p ∩ S = ∅ and since p is a principal ideal pR, P = p(S −1 R). 1.5.5

Weil divisors and divisor class group

Let X be a (connected) normal scheme defined over a field k. We say that a scheme is integral if it is irreducible and reduced. An irreducible divisor D on X is an integral closed subscheme of codimension one. Namely, if U = {Ui }i∈I is a finite affine open covering of X with Ui = Spec Ai , D ∩ Ui (if not the empty set) is defined by a prime ideal pi of Ai whose height is equal to one. We consider a free abelian group Div (X) generated by all irreducible divisors on X. Hence an element of Div (X) is a finite sum P i ni Di , where ni ∈ Z and Di an irreducible divisor. We call such an element of Div (X) a divisor. More precisely, it is called a Weil divisor. A P divisor D = i ni Di is effective if all ni ≥ 0. Let D be an irreducible divisor with the generic point ξ, i.e., {ξ} = D. Then the local ring OX,ξ , which we denote also OX,D , is a discrete valuation ring by Theorem 1.5.6. We denote by vD the associated valuation vD : k(X) → Z ∪ (∞). Let f ∈ k(X) be a nonzero rational function on X. ∗ and n ≥ 0, where mX,D = (t). If f ∈ OX,D then f = utn with u ∈ OX,D −1 Then vD (f ) = n. If f ̸∈ OX,D then f ∈ OX,D and vD (f ) = −vD (f −1 ). P We define the divisor (f ) of f by setting (f ) = D vD (f )D, where D ranges over all irreducible divisors of X. This is a finite sum. In fact, X has a finite affine open covering U = {Ui }i∈I of X such that Ui = Spec Ai with a normal affine domain Ai . Since f ∈ Q(Ai ), we can write f = ai /bi with ai , bi ∈ Ai . Then (f )|Ui = (ai ) − (bi ), where (ai ) and (bi ) are finite sums of divisors because ai R and bi R have only finitely many prime divisors. If

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f = ai /bi = ci /di with ci , di ∈ Ai we have (ai ) − (bi ) = (ci ) − (di ) because ai di = bi ci . Hence, with Ui ranging over finitely many open sets, (f ) is a finite sum of irreducible divisors. We call (f ) a principal divisor defined by f . If we write (f ) = (f )0 − (f )∞ with no common irreducible divisors between effective divisors (f )0 and (f )∞ , we call (f )0 (resp. (f )∞ ) the zero part (resp. polar part). Since vD (f g) = vD (f ) + vD (g), there is an abelian group homomorphism v : k(X)∗ → Div (X), whose kernel is Γ(X, OX )∗ . In fact, if vD (f ) = 0 for all irreducible divisor ∗ D on X, then f, f −1 ∈ OX,D . It then follows by Theorem 1.5.9 that −1 f, f ∈ OX,x for every point x ∈ X. Then f, f −1 ∈ Γ(X, OX )∗ . The cokernel of v is denoted by Cℓ (X) and called the divisor class group of X. Hence we have the following exact sequence 0 → Γ(X, OX )∗ → k(X)∗ → Div (X) → Cℓ (X) → 0. If X = Spec A, we denote Cℓ (X) by Cℓ (A). Divisors D, D′ are linearly equivalent if D, D′ have the same image in Cℓ (X), i.e., D − D′ = (f ) for f ∈ k(X)∗ . Theorem 1.5.20. Let A be a normal affine domain over a field k. Then A is a factorial domain if and only if Cℓ (A) = 0. Proof. Suppose that A is a factorial domain. By Theorem 1.5.18, every irreducible divisor D on X = Spec A is principal. Namely there exists an element p ∈ A such that D = (p). In fact, D = V (p) for a prime ideal p with ht (p) = 1. Since p = pA, vD (p) = 1. Meanwhile, if D′ is an irreducible ∗ divisor on X different from D then p ∈ OX,D ′ . Hence vD ′ (p) = 0. So, the homomorphism v is surjective and Cℓ (A) = 0. Conversely, if Cℓ (A) = 0 then any irreducible divisor on X is written as D = (f ) for f ∈ Q(A). Since vD′ (f ) = 0 for D′ ̸= D, we know that f ∈ A by Theorem 1.5.9 and f A = p as p is a unique prime divisor of f and pAp = f Ap . Hence A is factorial by Theorem 1.5.18. The following result is called Nagata’s lemma and quite useful in concrete problems. For the original result, see Nagata [70, Lemma 1]. Lemma 1.5.21. Let A be a noetherian integral domain and let p be a prime element of A, i.e., A/pA is an integral domain. If A[p−1 ] is a UFD then A is a UFD.

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Proof. By Theorem 1.5.18, it suffices to show that every prime ideal p of A of height one is a principal ideal. If p ∈ p then p = pA because ht p = 1. Suppose that p ̸∈ p. Since A[p−1 ] is a UFD, there exists an element a ∈ p such that pA[p−1 ] = aA[p−1 ]. Since p is a unit in A[p−1 ] we may assume that a ̸∈ pA. Let x be an element of p and let n be the smallest integer ≥ 0 such that pn x = ay with y ∈ A. If n > 0 then y ∈ pA, whence y = py ′ . This implies that pn−1 x = ay ′ , which is a contradiction. So, n = 0 and x ∈ aR. Hence p = aA. If A is a normal ring, the divisor class group Cℓ (A) is generated by Cℓ (A[p−1 ]) and the irreducible divisor of Spec A defined by pA which is a principal ideal. Hence Cℓ (A) = Cℓ (A[p−1 ]) = 0. So, A is a UFD. 1.5.6

Zariski’s main theorem

Let X be a scheme of finite type over a field k. A point x ∈ X is an isolated point if the set {x} is a closed and open set. Hence the set {x} is a connected component of X. On the other hand, let A be a noetherian k-algebra. We say that A is an artinian ring if dim A = 0. Lemma 1.5.22. The following assertions hold. (1) Let A be a noetherian algebra over a field k. If dim A = 0 then A is a finite direct product of local artinian rings. (2) Let X be a scheme of finite type over a field k. If dim X = 0 then X has a finite set as the underlying set, and every local ring OX,x is a local artinian ring. Hence X is an affine scheme. Proof. (1) Let (0) = q1 ∩· · ·∩qn be an irredundant primary decomposition of the zero ideal. Since dim A = 0, every primary ideal qi belongs to a maximal ideal mi . In particular, there are no inclusions among {m1 , . . . , mn }. Hence qi + qj = A if i ̸= j. By the Chinese remainder theorem (see Problem 2), we have a ring isomorphism A∼ = A/q1 × · · · × A/qn ,

(1.10)

where A/qi is a local ring with maximal ideal mi /qi . Under the isomorphism (1.10), the maximal ideal mi corresponds to A/q1 × · · · × A/qi−1 × mi /qi × A/qi+1 × · · · × A/qn . Hence the multiplicative set Si = A \ mi corresponds to A/q1 × · · · × A/qi−1 × (A/qi − mi /qi ) × A/qi+1 × · · · × A/qn .

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Hence the localization Si−1 A = Ami corresponds to A/qi because the localization in the direct product of rings is done component-wise and the i-th component is a local ring for which (A/qi − mi /qi ) is the set of units. We have thus an isomorphism Ami ∼ = A/qi and an isomorphism A∼ = Am 1 × · · · × Am n . (2) By definition, X has a finite affine open covering U = {Ui }i∈I such that Ui = Spec Ai and Ai is a finitely generated k-algebra. By the assertion (1), Ui is a direct sum of Spec (Ai )mij with 1 ≤ j ≤ ni . Since I is a finite set, X is a direct sum of finitely many Spec (Ai )mij , where some of Spec (Ai )mij might belong to different Ui ’s. A morphism of schemes f : X → Y is said to be of finite type if there exist affine open coverings U = {Ui }i∈I of X and V = {Vj }j∈J of Y such that, for every i ∈ I and j = σ(i) ∈ J, f (Ui ) ⊂ Vj and Ai is finitely generated over Bj , where σ : I → J is a mapping of sets, Ui = Spec Ai and Vj = Spec Bj . If X and Y are schemes of finite type over a field k, any morphism f : X → Y is a morphism of finite type. If f : X → Y is a morphism of finite type, the base change fT : X ×Y T → T is a morphism of finite type for a morphism T → Y . In particular, the fiber Xy over a point y ∈ Y is a scheme of finite type over the residue field k(y). A morphism of finite type f : X → Y is a quasi-finite morphism if either Xy = ∅ or dim Xy = 0 for every point y ∈ Y . A finite morphism is a quasi-finite morphism. The following result is an important result. The given proof is due to C. Peskine [85]. Theorem 1.5.23 (Zariski’s main theorem). Let f : X → Y be a birational, quasi-finite morphism of algebraic varieties defined over a field k. Assume that Y is a normal variety. Then f : X → Y is an open immersion. Proof. Step 1. Let x ∈ X and let y = f (x). Then there exist affine open neighborhoods U = Spec A of x and V = Spec B of y such that U ⊂ f −1 (V ), where we may assume that A and B are affine domains over k. The restriction f |U : U → V is induced by a k-algebra homomorphism φ : B → A. Since f is birational, φ extends to a k-isomorphism Q(φ) : ∼ Q(B) −→ Q(A). Hence φ is injective. We may assume that B is a ksubalgebra of A. Since B is normal and Q(A) = Q(B), B is integrally closed in A. Let q (resp. p) be the prime ideal of B (resp. A) corresponding

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to the point y (resp. x). Hence p ∩ B = q. Since x is isolated in the fiber f −1 (y), p is a minimal prime divisor of qA and p(A ⊗B Q(B/q)) is a maximal ideal of A ⊗B Q(B/q). Step 2. Under the above settings we prove the following assertion. Claim 1. Under the above setting there exists an element s ∈ B \ q such that A[s−1 ] = B[s−1 ]. If the Claim 1 is proved, it follows that the open set DY (s) = Spec B[s−1 ] coincides with DX (s) = Spec A[s−1 ], which is mapped to DY (s) by f . Namely, X is an open set of Y near the point y = f (x). Since x is arbitrary in X, this implies that f is an open immersion. The proof of this claim is divided into several claims. Claim 2. Let A be an integral domain and let B be a subdomain. Suppose that B is integrally closed in A and A is a finitely generated B-algebra. For a prime ideal p of A and q = p ∩ B of B the following two conditions are equivalent. (i) There exists an element s ∈ B \ q such that A[s−1 ] = B[s−1 ]. (ii) Ap = Bq . Proof. (i) ⇒ (ii). We have Ap = (A[s−1 ])pA[s−1 ] = (B[s−1 ])qB[s−1 ] = Bq . (ii) ⇒ (i). Write A = B[a1 , . . . , ar ] and let S = B \ q. Then Bq = S −1 B ⊆ S −1 A ⊆ Ap . Since Ap = Bq by assumption, we have Bq = S −1 A. Hence ai = bi /si for bi ∈ B and si ∈ S. Set s = s1 · · · sr . Then sai ∈ B for every 1 ≤ i ≤ r. Then, setting b′i = (bi /si )s, we have A[s−1 ] = B[b′1 /s, . . . , b′r /s, s−1 ] = B[s−1 ].

The Claim 2 asserts that, in order to prove Claim 1, we may replace B by Bq and A by A[S −1 ] = A ⊗B Bq and assume that (B, q) is a local ring, B is integrally closed in A, and p is a minimal prime divisor of qA as well as a maximal ideal of A. Under this additional assumption, we prove the following assertion. Claim 3. If A = B[x] then Ap = B, where (B, q) is a local ring. Proof. Since A/qA = B[x]/qB[x] ∼ = (B/q)[x] with x = x + qA, which is an algebra generated by a single element x over a field B/q, it follows that

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dim((B/q)[x]) ≤ 1. If dim((B/q)[x]) = 1, (B/q)[x] is a polynomial ring in one variable x over B/q. Then, since p is a minimal prime divisor of qA, p = qA, which is not a maximal ideal. Hence dim((B/q)[x]) = 0. So, the element x is algebraic over a field B/q. Namely, there is a relation for some n > 0, xn + b1 xn−1 + · · · + bn ∈ qB[x], b1 , . . . , bn ∈ (B \ q) ∪ {0}. Set y = xn + b1 xn−1 + · · · + bn + 1.

(1.11)

Then A = B[x] is integral over B[y], and q′ := p ∩ B[y] is isolated over q. In fact, it follows by Going-up theorem (Theorem 1.1.20) that q′ is a maximal ideal of B[y]. Suppose that q′ is not a prime divisor of qB[y]. Then there exists a prime ideal q′′ such that q′ ⫌ q′′ and q′′ ∩ B = q (since p ⊃ q′′ and p ∩ B = q). Since B[x] is integral over B[y] there exist prime ideals p′ , p′′ such that p′ ⫌ p′′ , p′ ∩ B[y] = q′ and p′′ ∩ B[y] = q′′ . This contradicts the condition dim((B/q)[x]) = 0. Further, B[y]/qB[y] is integral over B/q (i.e., y := y + qB[y] is algebraic over B/q), and y is a unit in B[y]/qB[y]. In fact, if y is not a unit, there exists a maximal ideal n of B[y] such that y ∈ n. Note here that y ̸∈ qB[y], i.e., y ̸= 0, since y ̸∈ qB[x]. By Lying-over theorem, there exists a maximal ideal m of B[x] such that n = m ∩ B[y]. Then y ∈ m, though y ≡ 1 (mod qB[x]). This is a contradiction. Hence y satisfies the following algebraic relation y m + cm−1 y m−1 + · · · + c0 ∈ qB[y], cm−1 , . . . , c0 ∈ B, c0 ∈ B \ q. Write this relation as y m + cm−1 y m−1 + · · · + c0 = q0 + q1 y + · · · + qt y t , q0 , . . . , qt ∈ q. Then (c0 − q0 ) ̸∈ q, whence y is invertible in B[y] ⊆ A. Let s = max(m, t). Then we have (c0 − q0 )(1/y)s + · · · + (cs − qs ) = 0. Hence 1/y is integral over B. Since B is integrally closed in A, 1/y ∈ B. On the other hand, 1/y ̸∈ q, for otherwise 1 = (1/y) · y ∈ qB[y] which is a contradiction. Hence y ∈ B because 1/y is a unit in B as (B, q) is a local ring. Thus we have B[y] = B. By the equation (1.11), x is then integral over B, and hence x ∈ B as B is integrally closed in A. So, we have A = B.

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Before going to the next claim, we introduce the conductor ideal of a ring extension. Let R be a ring and let S be a subring. Then the conductor ideal of R over S is the set c = {z ∈ R | zR ⊆ S}. Note that c ⊂ S because 1 ∈ R. Since c is closed under addition of elements and multiplication of elements of R and S, c is an ideal of R and S. We denote c also by c(R/S). Though the conductor ideal is zero in general, c ̸= 0 if R and S are integral domains, Q(R) = Q(S) and R is a finite S-module. Claim 4. Suppose further that there exists an element t ∈ A such that A is a finite B[t]-module. Let c be the conductor ideal of A over B[t]. Suppose that there exists a nonzero polynomial F (x) = b0 xn + (terms of lower degree) ∈ B[x] such that F (t) ∈ c. Then br0 ∈ c for some r > 0. Proof. Case 1. F (x) is a monic polynomial. We then show that A = B[t]. If deg F (x) = 0 then F (x) = 1 and F (t) = 1 ∈ c. Hence A = B[t]. Suppose deg F (x) > 0. If F (t) = 0, then t is integral over B and hence t ∈ B. So, A is a finite B-module, and A = B as B is integrally closed in A. Suppose that F (t) ̸= 0. For any element a ∈ A, set G(t) := aF (t) ∈ B[t]. Since F (x) is monic, we can write by polynomial remainder theorem in B[x] G(x) = Q(x)F (x) + R(x), deg R(x) < deg F (x), where G(x) ∈ B[x] is obtained from G(t) by replacing t by x. Set a′ := a − Q(t). Then a′ · F (t) = R(t). If a′ = 0 then a = Q(t) ∈ B[t]. So, assume that a′ ̸= 0. Then a′ is integral over B[t] and t is integral over −1 B[a′ ]. In fact, A is integral over B[t], whence a′ is integral over B[t]. Since a′ · F (t) = R(t) and deg F (x) > deg R(x), F (t) = (1/a′ )R(t) shows −1 −1 that t is integral over B[a′ ]. Therefore a′ is integral over B[a′ ]. So, a′ ′ is integral over B, and a ∈ B by the initial hypothesis that B is integrally closed in A. So, a = a′ + Q(t) ∈ B[t]. Thus A = B[t]. Case 2. F (x) is not monic. (1/b0 )F (x) is a monic polynomial, A[b−1 0 ] −1 −1 is a finite B[b−1 ][t]-module and B[b ] is integrally closed in A[b ]. By 0 0 0 −1 the Case 1, we have A[b−1 ] = B[b ][t]. Since A is a finite B[t]-module, 0 0 br0 A ⊆ B[t] for some r > 0, i.e., br0 ∈ c. Claim 5. With the same assumptions as in Claim 4, suppose that the prime ideal p of A is isolated over q, i.e., p is a maximal ideal of A and a minimal prime divisor of qA. Then it holds that Ap = B.

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Proof. With c as in Claim 4, we consider two cases (1) c ̸⊆ p and (2) c ⊆ p. Case (1). Let a ∈ (B[t] \ (p ∩ B[t])) ∩ c. Such a exists since c ⊂ B[t]. Then A[a−1 ] = B[t][a−1 ]. Hence B[t]p∩B[t] = Ap . This implies that p ∩ B[t] is a minimal prime divisor of qB[t]. Since A is integral over B[t], p ∩ B[t] is a maximal ideal. Hence p ∩ B[t] is isolated over q. By Claim 3, we have Ap = B. Case (2). Let p′ be a minimal prime divisor of c in A such that p ⊇ p′ . Let A = A/p′ , B = B/B∩p′ , where B∩p′ ⊆ q, t = t+p′ and p = p/p′ . Then t is algebraically independent over B. In fact, if t is algebraic over B, there exists a polynomial g(x) in one variable x with coefficients in B\(p′ ∩B) such √ that g(t) ∈ p′ . Since p′ = ( c : a′ ) for some a′ ∈ A \ p′ (see the argument r after Corollary 1.1.8), we have (g(t)a′ )r = g(t)r a ∈ c with a = a′ and r r > 0. So, g(t) aA ⊆ B[t]. Since B[t][aA] is a finite B[t]-module (as A is a finite B[t]-module), Claim 4 implies that bs0 (aA) ⊆ B[t] for s > 0, where b0 is the coefficient of the highest degree term of g(x). Namely, bs0 a ∈ c. Since a ̸∈ p′ , we have b0 ∈ p′ ∩ B. This is a contradiction to the choice of the polynomial g. By virtue of the next Claim 6 which deals with a bit more general situation, p is not isolated over q = q/(p′ ∩ B). Hence p is not isolated over q. So the case c ⊆ p cannot occur. Claim 6. Let A be an affine domain and let B be an affine subdomain. Suppose that an element t ∈ A is algebraically independent over B and A is a finite B[t]-module. Then p is not isolated over q := p ∩ B for a prime ideal p of A. Proof. We may assume that p is maximal over q. Let m = p ∩ B[t]. Then m is also maximal among prime ideals of B[t] lying over q. Further B is integrally closed in B[t]. If m is minimal over q as well and hence isolated over q, then (B[t])m = Bq by Claim 3. This contradicts the hypothesis that t is algebraically independent over B. Thus m is not minimal over q. This implies that there exists a prime ideal n of B[t] such that n ∩ B = q and m ⫌ n. If B is normal, by Going-down theorem (see Theorem 1.5.24) applied to A ⊇ B[t], there exists a prime ideal p′ of A such that p ⫌ p′ and p′ ∩ (B[t]) = n. Hence p is not isolated over q. e be the integral closure of B in Q(B), If B is not a normal ring, let B which is a normal ring and a finite B-module by Corollary 1.5.11. Let e = A ⊗B B e = B[A]. e e is integrally closed in A e10 and t is an element A Then B 10 Let

e be integral over B. e Since Q(B) e = Q(B), there exists an element b ∈ B ξ ∈ A e then ξ is integral over B. Hence bξ such that bξ ∈ A. Note that if ξ is integral over B

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e which is algebraically independent over B. e Let e e of A p be a prime ideal of A e such that e p ∩ A = p, and let e q=e p ∩ B. Since e p is maximal over e q, e p is not isolated over e q by the foregoing case. Hence there exists a prime ideal e p′ e =e such that e p⫌e p′ and e p′ ∩ B q. Let p′ = e p′ ∩ B. Then p ⫌ p′ by Lying-over ′ theorem and p ∩ B = q. So, p is not minimal over q. Claim 7 (Proof of Claim 1). Since A and B are affine domains over k, A is a finitely generated B-algebra. By Claim 2, replacing B by Bq and A by A⊗B Bq , we may assume that (B, q) is a local ring. Write A = B[x1 , . . . , xt ]. For 0 ≤ i ≤ t, let Bi = B[x1 , . . . , xi ], let Ai be the integral closure of Bi in A, let pi = p ∩ Ai and let qi = pi ∩ Bi . We proceed by descending induction on i to show that Ap = (Ai+1 )pi+1 implies Ap = (Ai )pi . In fact, since A = At and p = pt it holds that Ap = (At )pt . For i = 0, it holds that Aq = B because A0 = B0 = B as B is integrally closed in A. By Claim 2, there exists s ∈ (Ai+1 ) \ (pi+1 ) such that (Ai+1 )[s−1 ] = −1 A[s ]. Hence xi+2 , . . . , xt ∈ (Ai+1 )[s−1 ]. Write xj = aj /snj with aj ∈ Ai+1 for i + 2 ≤ j ≤ t. Then A[s−1 ] = Bi+1 [(aj ), s, s−1 ], where (aj ) stands for a set {ai+2 , . . . , at }. Let q′i+1 = p∩Bi+1 [(aj ), s]. Since s ̸∈ q′i+1 , we have Ap = (Bi+1 [(aj ), s])q′i+1 . Since aj (i + 2 ≤ j ≤ t) and s are integral over Bi+1 , Bi+1 [(aj ), s] is a finite Bi+1 -module. Set Ai = Ai ∩ Bi+1 [(aj ), s] and pi = p ∩ Ai . Since Bi+1 = Bi [xi+1 ], we have Bi+1 ⊆ Ai [xi+1 ]. Therefore, we have Bi+1 [(aj ), s] = Ai [xi+1 ][(aj ), s], which is a finite Ai [xi+1 ]-module. Meanwhile, q′i+1 is isolated over pi . In fact, since Ap = (Bi+1 [(aj ), s])q′i+1 , q′i+1 is a minimal prime ideal over pi . Further, q′i+1 is maximal over pi . In fact, since B/q ⊆ Bi+1 [(aj ), s]/q′i+1 ⊆ A/pA, and since p is isolated over q, A/pA is an algebraic extension of B/q. Hence q′i+1 is maximal over q. Namely, pi is a maximal ideal. The assumptions for Claim 5 are satisfied with Bi+1 [(aj ), s] ⊃ Ai [xi+1 ], where we take t = xi+1 . In fact, note that Q(Bi+1 [(aj ), s]) = Q(Ai [xi+1 ]). Since Bi+1 [(aj ), s] is a finite Ai [xi+1 ]-module and every element of Bi+1 [(aj ), s] not belonging to Ai [xi+1 ] is expressed as a fraction of elements of Ai [xi+1 ], there exists a polynomial F (x) ∈ Ai [x] such that F (xi+1 ) is in the conductor c of Bi+1 [(aj ), s] over Ai [xi+1 ]. By Claim 5, we have (Bi+1 [(aj ), s])q′i+1 = (Ai )pi since Ai is integrally closed in Bi+1 [(aj ), s]. Hence Ap = (Ai )pi . Meanwhile, is integral over B. Since B is integrally closed in A we have bξ ∈ B. This shows that e Hence ξ ∈ B. e ξ ∈ Q(B).

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since Ai ⊆ Ai ⊆ A, pi = pi ∩ Ai , we have (Ai )pi = (Ai )pi = Ap . So, the induction step from i + 1 to i is proved. The proof of Zariski’s main theorem is completed. We only state the following result which was used in Claim 6 in the above proof. For the proof we refer to [72, Theorem 10.13] or Matsumura [54, Theorem 9.4]. In both references, proofs use Galois extension (or normal extension) of a normal ring. See also Problems 9 and 10 for the substantial part of the proof. Theorem 1.5.24 (Going-down theorem). Let A be an integral domain and let B be a subdomain of A. Assume that B is a normal ring and A is integral over B. Let q1 ⊇ q2 be prime ideals of B and let p1 be a prime ideal of A such that p1 ∩ B = q1 . Then there exists a prime ideal p2 of A such that p1 ⊇ p2 and p2 ∩ B = q2 .

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Smooth varieties

In the present section we abuse the notations and denote Krull dimension K-dim (R) of a ring R by dim R. We also assume that dim R < ∞. 1.6.1

System of parameters and regular local ring

We extend Lemma 1.5.22, the assertion (1). Lemma 1.6.1. Let (R, m) be a noetherian local ring. Then the following two conditions are equivalent. (1) dim R = 0. (2) The ideal m is nilpotent, i.e., ms = 0 for some s > 0. Proof. (1) ⇒ (2). Let (0) = q1 ∩ · · · ∩ qr be an irredundant primary de√ composition and let pi = qi . Since dim R = 0, each of the pi is a maximal ideal. Since p1 , . . . , pr are pairwise distinct p and R is a local ring, it follows that r = 1 and p1 = m. Hence m = (0). Since m is finitely generated, n write m = (a1 , . . . , at ). For 1 ≤ j ≤ t, aj j = 0. Take s > 0 so that Pt s ≥ j=1 nj . Then ms = 0 for s > 0. p (2) ⇒ (1). Since m is nilpotent p by assumption, we have m = (0). If p is a prime ideal of R then m ⊇ p ⊇ (0). Hence p = m. So, dim R = 0. A ring R satisfies the descending chain condition (d.c.c. for short) if all descending chain of ideals of R I0 ⊇ I1 ⊇ · · · ⊇ Ii ⊇ Ii+1 ⊇ · · · stops to decrease, i.e., there exists an integer N > 0 such that Ii = IN for all i ≥ N . For a noetherian local ring (R, m), if R satisfies the d.c.c., then R is an artinian ring. In fact, we have a descending chain m ⊇ m2 ⊇ · · · ⊇ mi ⊇ mi+1 ⊇ · · · .

Since it stops, we have mn = mn+1 for n ≫ 0. Since mn is a finite p R-module, n m = 0 by Nakayama’s lemma (see Lemma 1.5.1). Then m = (0), whence dim R = 0.11 Given a finite R-module M , we say that M has a composition series of submodules M = M0 ⊃ M1 ⊃ · · · ⊃ Mi ⊃ Mi+1 ⊃ · · · ⊃ Mn−1 ⊃ Mn = 0 11 The

following is a theorem of Akizuki (see [72, Theorem 9.1]): Let R be a ring satisfying the d.c.c. if and only if R is noetherian and dim R = 0.

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such that, for every 0 ≤ i < n there is no R-submodule N in between Mi ⊃ Mi+1 which is different from Mi and Mi+1 . The integer n is the length of M as any composition series of M has the same length [98, III, §11, Theorem 19], which we denote by ℓ(M ). Lemma 1.6.2. Let R be an artinian ring. Then the following assertions hold. (1) Every finite R-module M has a composition series. α

β

(2) Let 0 → M1 −→ M2 −→ M3 → 0 be an exact sequence of finite Rmodules, i.e., M1 is a submodule of M2 and M3 ∼ = M2 /M1 . Then ℓ(M2 ) = ℓ(M1 ) + ℓ(M3 ). In particular, ℓ(M1 ) ≤ ℓ(M2 ), and ℓ(M1 ) < ℓ(M2 ) provided M3 ̸= 0. (3) R satisfies the descending chain condition. Proof. (1) By Lemma 1.5.22, R is a direct product of local rings R∼ = R/q1 × R/q2 × · · · × R/qr , where (0) = q1 ∩ q2 ∩ · · · ∩ qr is an irredundant primary decomposition. Then M is a direct product M ∼ = M1 × M2 × · · · × Mr , where Mi is a i

∨

finite R/qi -module. In fact, let ei = (0, . . . , 0, 1, 0, . . . , 0) for 1 ≤ i ≤ r. Then 1 = e1 + · · · + er is the idempotent decomposition of the unity, which means ei ej = δij ej . Set Mi = M ei . Then M ∼ = M1 × · · · × Mr . Since an (R/qi )-module is naturally an R-module and M has a composition series if and only if Mi has a composition series for every 1 ≤ i ≤ r, it suffices to prove the assertion in the case (R, m) is a local ring. Since ms = 0 for some s > 0, we have a sequence of R-modules M ⊃ mM ⊃ m2 M ⊃ · · · ⊃ ms−1 M ⊃ ms M = 0, where the subquotient module mi M/mi+1 M (0 ≤ i < s) is an R/m-module. Since mi M is a finitely generated R-module, mi M/mi+1 M is an R/m-vector space of finite rank ni , (R/m)v1 + · · · + (R/m)vni . Let Mij be the inverse image of (R/m)vj+1 +· · ·+(R/m)vni by the quotient homomorphism mi M → mi M/mi+1 M for 0 ≤ j ≤ ni . Then we have a chain of R-submodules mi M = Mi0 ⊃ Mi1 ⊃ · · · ⊃ Mini = mi+1 M

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such that Mi(j−1) /Mij ∼ = (R/m)vj . Hence there is no R-submodule between Mi(j−1) and Mij . So, M has a composition series of length ℓ(M ) =

s−1 X

rank (mi M/mi+1 M ).

i=0

In the case R is a product of local rings as above, we have the equality r X ℓ(M ) = ℓ(Mi ). i=1

(2) Suppose M1 and M3 have composition series M1 = M10 ⊃ M11 ⊃ · · · ⊃ M1n1 = 0, M3 = M30 ⊃ M31 ⊃ · · · ⊃ M3n3 = 0.

Let M2i = β −1 (M3i ) for 0 ≤ i ≤ n3 and M2(n3 +j) = α(M1j ) for 0 ≤ j ≤ n1 . Then a sequence M2 = M20 ⊃ · · · ⊃ M2n3 = α(M10 ) ⊃ · · · ⊃ M2(n3 +n1 ) = α(M1n1 ) = 0 is a composition series of M2 . Hence ℓ(M2 ) = n1 + n3 = ℓ(M1 ) + ℓ(M3 ). (3) Let I = I0 ⊇ I1 ⊇ · · · ⊇ Ii ⊇ Ii+1 ⊇ · · · be a descending ideal chain. Since every Ij is a finite R-module, the assertion (2) implies ℓ(I) ≥ ℓ(I1 ) ≥ · · · ≥ ℓ(Ii ) ≥ ℓ(Ii+1 ) ≥ · · · . Since ℓ(M ) ≥ 0, the above sequence of length is bounded from below. Further, ℓ(Ij ) = ℓ(Ij+1 ) if and only if Ij = Ij+1 . This implies that the sequence stabilizes, i.e., ceases to decrease. Let R be a ring and let p be a prime ideal. For an integer n > 0, set p n (n) p = (p Rp ) ∩ R, which is a primary ideal such that p = p (see Problem 11). p(n) is called the symbolic nth power of p. We need a few results which we borrow together with proofs from [72]. The first result is Krull’s principal ideal theorem. (n)

Lemma 1.6.3 (Krull’s principal ideal theorem). Let R be a noetherian integral domain and let a be a nonzero element of R such that aR ̸= R. Let p be a minimal prime divisor of aR. Then ht p = 1. Proof. It suffices to show that a prime ideal q ⫋ p must be the ideal (0). For the purpose, we may assume by replacing R by Rp that (R, p) is a local ring. Set ai = q(i) + aR for i > 0. Then ai ⊇ ai+1 because q(i) ⊇ q(i+1) . Hence we have a descending chain of ideals a1 ⊇ a2 ⊇ · · · ⊇ ai ⊇ · · · .

(1.12)

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In the quotient ring R/aR, the above chain gives a descending chain a1 ⊇ a2 ⊇ · · · ⊇ ai ⊇ · · · ,

(1.13)

where ai = ai /aR. Meanwhile, since p is a minimal prime divisor of aR, dim(R/aR) = 0, i.e., R/aR is an artinian ring. By Lemma 1.6.2, the descending chain (1.13) stabilizes, i.e., ai = an for i ≥ n for some n > 0. Then q(n) + aR = q(n+1) + aR. Note that a ̸∈ q because p is a minimal prime divisor of aR and q ⫋ p. This implies that q(n) = q(n+1) + aq(n) . In fact, any element x ∈ q(n) is written as x = y + az for y ∈ q(n+1) and z ∈ R. Hence az ∈ q(n) and q(n) is a primary ideal belonging to q. Since a ̸∈ q, it follows that z ∈ q(n) . So, q(n) ⊆ q(n+1) + aq(n) . The converse is clear. We then have q(n) /q(n+1) = a(q(n) /q(n+1) ) in a noetherian local ring R/q(n+1) with (a + q(n+1) ) ∈ p/q(n+1) which is the maximal ideal. By Lemma 1.5.1, we have q(n) /q(n+1) = 0, i.e., q(n) = q(n+1) . Note that q(i) Rq = qi Rq for every i > 0. Hence qn Rq = qn+1 Rq . Again, by Lemma 1.5.1, we have qn Rq = 0. Since R is an integral domain, q = (0). Lemma 1.6.3 implies that with the notations therein we have dim R/aR ≤ dim R − 1, where dim R/aR = supp dim R/p with p ranging over all minimal prime divisors of aR.12 In geometric terms, this result implies that on an affine variety X = Spec A all irreducible components of a subvariety V = V (f ) defined by a single element f ∈ A (called later a hypersurface section), i.e., V = {P ∈ X | f (P ) = 0}, have codimension one. Theorem 1.6.4 (Krull’s height theorem). Let R be a noetherian ring Pr and let I = i=1 ai R be an ideal of R generated by r elements a1 , . . . , ar . Let p be a minimal prime divisor of I. Then ht p ≤ r. Proof. Let p = p0 ⊃ p1 ⊃ · · · ⊃ ps be a chain of prime ideals. It suffices to show that s ≤ r. If there are prime ideals in between p0 and p1 , we consider the set S = {q ∈ Spec R | p0 ⫌ q ⫌ p1 }. 12 If R is an affine k-domain, the equality holds since dim R is equal to the length of any maximal chain of prime ideals. In fact, dim R = dim Rm for any maximal ideal m of R, and Rm is a catenary ring (see several lines before Lemma 1.6.10).

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Since R is noetherian, S is an inductive set with respect to the order defined by inclusion. Hence, by Zorn’s lemma, there exists a maximal element with which we replace p1 . Thus we may assume that there is no prime ideal in between p0 and p1 . For the purpose, we may replace R by Rp and assume that (R, p) is a local ring. Since p is a minimal prime divisor of I, it holds that I ̸⊂ p1 . We may assume that a1 ̸∈ p1 . Then there is no prime ideal containing p1 + a1 R except for p. Hence p is a minimal prime divisor of p1 + a1 R. Since p is a maximal ideal by assumption, p1 + a1 R is a primary ideal belonging to p. So, there exists an integer t > 0 such that ati ∈ p1 +a1 R for every 2 ≤ i ≤ r. Write ati = a1 bi + a′i with bi ∈ R and a′i ∈ p1 . Let Pr ′ ′ I′ = divisor of I ′ contained in i=2 ai R√and let p be a minimal prime √ ′ ′ p1 . Since ai ∈ I + a1 R, it follows that I ⊆ I + a1 R and hence p is a minimal prime divisor of I ′ +a1 R. Since I ′ +a1 R ⊆ p′ +a1 R, p is a minimal prime divisor of p′ + a1 R. This implies that p/p′ is a minimal prime divisor of a1 (R/p′ ) in the ring R/p′ . Since p/p′ is a maximal ideal, ht (p/p′ ) = 1 by Lemma 1.6.3. Since p′ ⊆ p1 ⫋ p, we have p′ = p1 . Since I ′ is generated by r − 1 elements, ht (p1 ) ≤ r − 1 by induction. Hence s − 1 ≤ r − 1, i.e., s ≤ r. Corollary 1.6.5. Let (R, m) be a noetherian local ring. Then dim R ≤ dimR/m m/m2 . Proof. Note that m/m2 is a vector space of finite rank over the field R/m. Let n = rank (m/m2 ) and let a1 , . . . , an be elements of m such that {a1 , . . . , an } is a basis of m/m2 , where ai = ai + m2 . Then we have Pn Pn Pn m = i=1 ai R + m2 . Hence we have m/( i=1 ai R) = m(m/( i=1 ai R)). Pn By Nakayama’s lemma (Lemma 1.5.1), it follows that m = i=1 ai R. Now, by Theorem 1.6.4, we have dim R ≤ n. For a noetherian local ring (R, m), rank R/m m/m2 is called the embedding dimension of R. Lemma 1.6.6. Let (R, m) be a noetherian local ring of dimension n and let {a1 , . . . , an } be a set of elements of m. Then the following conditions are equivalent. Pn (i) ht ( i=1 ai R) = n. pPn Pn (ii) m= i=1 ai R is a primary ideal belonging to m, whence i=1 ai R. Pn (iii) m is a unique minimal prime divisor of the ideal i=1 ai R. Pn (iv) ℓ(R/ i=1 ai R) < ∞.

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Pn Proof. (i) ⇒ (ii). Let p be a minimal prime divisor of i=1 ai R. By the condition (i), ht (p) ≥ n. Since ht (p) ≤ n by Krull’s height theorem, we have ht (p) = p n. Since m is a unique prime ideal of height n in R, we have Pn p = m. Then i=1 ai R = m. (ii) ⇔ (iii). Clear. Pn (iii) ⇒ (iv). There exists an integer t > 0 such that mt ⊆ i=1 ai R. Pn Hence ℓ(R/ i=1 ai R) ≤ ℓ(R/mt ) < ∞. Pn (iv) ⇒ (i). Let i=1 ai R = q1 ∩ · · · ∩ qr be an irredundant primary Pn decomposition. Since R/ i=1 ai R is artinian, we have R/

n X i=1

ai R ∼ = R/q1 × · · · × R/qr .

Hence R/qi is artinian for 1 ≤ i ≤ r. Since a minimal prime ideal of Pr Pn √ √ qi , it follows that qi = m. Hence ht ( i=1 ai R) = i=1 ai R is one of the n. A set {a1 , . . . , an } of elements of m satisfying the equivalent conditions in Lemma 1.6.6 is called a system of parameters of the local ring (R, m). The existence of such a system in a noetherian local ring (R, m) is guaranteed by the following result. Lemma 1.6.7. Let R be a noetherian ring and let p be a prime ideal of R with ht p = n. Then there are elements a1 , . . . , an of p and a prime ideal chain p = pn ⫌ pn−1 ⫌ · · · ⫌ p1 ⫌ p0 such that ai ∈ pi \pi−1 for 1 ≤ i ≤ n. If (R, p) is a local ring, then is a primary ideal belonging to p.

(1.14) Pn

i=1

ai R

Proof. Since ht p = n, there is a prime ideal chain p = pn ⫌ pn−1 ⫌ · · · ⫌ p1 ⫌ p0 . Then ht pi = i for every 0 ≤ i ≤ n. We construct a1 , . . . , an inductively in this order so that, for any 1 ≤ s ≤ n, s is the minimum of ht q for Ps minimal prime divisors q of i=1 ai R. Assume that we have constructed a1 , . . . , as−1 so that s − 1 is the minimum of ht q for minimal prime divisors Ps−1 q of i=1 ai R. By Krull’s height theorem, every minimal prime divisor of Ps−1 Ps−1 i=1 ai R has height ≤ s − 1. So, every minimal prime divisor ofP i=1 ai R s−1 has height s − 1. Let q1 , . . . , qm be all minimal prime divisors of i=1 ai R. If s − 1 < n then qj ̸= ps for any 1 ≤ j ≤ m. Hence we can find an element

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as ∈ ps such that as ̸∈ qj for every 1 ≤ j ≤ m. In fact, since q1 , . . . , qm , ps ∨

are pairwise distinct, qj ̸⊃ ps ∩ q1 ∩ · · · ∩ qj ∩ · · · ∩ qm for 1 ≤ j ≤ m. Take Pm ∨ bj ∈ (ps ∩ q1 ∩ · · · ∩ qj ∩ · · · ∩ qm ) \ qj and set as = j=1 bj . Then every Ps minimal prime divisor of i=1 ai R has height s by Theorem 1.6.4. Hence Ps ps is a minimal prime divisor of i=1 ai R. By the choice of a1 , . . . , an , p is Pn Pn a unique prime divisor of i=1 ai R if (R, p) is a local ring. Hence i=1 ai R is a primary ideal belonging to pn = p. By the definition, if {a1 , . . . , an } is a system of parameters of (R, m) then {aσ(1) , . . . , aσ(n) } is also a system of parameters for any permutation σ of {1, 2, . . . , n}. Let us explain the geometric meaning. Let X = Spec A be an affine variety of dimension n over a field k and let P be a closed point of X corresponding to a maximal ideal m. Then there exist rational functions f1 , . . . , fn such that f1 , . . . , fn are regular at P , f1 (P ) = · · · = fn (P ) = 0 and the hypersurface sections V (f1 ) ∩ · · · ∩ V (fn ) of X has dimension 0 at P as the local ring Am /(f1 , . . . , fn ) is an artin local ring. Corollary 1.6.8. Let (R, m) be a noetherian local ring of dim R = n. Define a non-negative integer s(R) by ( !! ) r X s(R) = min r | ℓ R/ ai R < ∞, ai ∈ m . i=1

Then s(R) = dim R. Pr Pr Proof. Suppose that ℓ(R/( i=1 ai R)) < ∞. Let I = i=1 ai R. As in the proof of Lemma 1.6.6, m is a (unique) minimal divisor of I. By Krull’s height theorem (Theorem 1.6.4), n = ht m ≤ r. So, n ≤ s(R). On the other hand, there exists a system of parameters {a1 , . . . , an } of (R, m) by Lemma 1.6.7. Hence s(R) ≤ n. Therefore s(R) = dim R. The next result will suggest how to construct a system of parameters. Lemma 1.6.9. Let (R, m) be a noetherian local ring of dimension n and let a1 , . . . , ar be elements of m, where 0 ≤ r ≤ n. Then we have an inequality dim R/I ≥ dim R − r, I =

r X

ai R,

i=1

where the equality holds if and only if {a1 , . . . , ar } is a part of a system of parameters.

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Proof. In order to prove the inequality, by induction on r, it suffices to show that dim(R/aR) ≥ dim R − 1 for any element a ∈ m. In fact, if dim R < ∞ then dim R/aR < ∞. If a2 , . . . , am is a system of parameters Pm for R/aR with ai = ai + aR, then ℓ(R/(aR + i=2 ai R)) < ∞. By Corollary 1.6.8, we have 1 + (m − 1) ≥ dim R, while m = dim R/aR + 1. Hence we have dim R/aR + 1 ≥ dim R. Suppose that the equality holds in the statement. Note then that dim R = n and dim(R/I) = n − r. Let {ar+1 , . . . , an } be a system of parameters of R/I, where ai = ai + I for r + 1 ≤ i ≤ n. Then !! !! n n X X ai R = ℓ (R/I)/ ℓ R/ ai (R/I) < ∞. i=1

i=r+1

Hence {a1 , . . . , ar , ar+1 , . . . , an } is a system of parameters by Lemma 1.6.7. So, {a1 , . . . , ar } is a part of system of parameters of R. Conversely, if {a1 , . . . , ar , ar+1 , . . . , an } is a system of parameters of R, then we have !! n X ℓ (R/I)/ ai (R/I) < ∞. i=r+1

Hence n−r ≥ dim(R/I) by Corollary 1.6.8. Since it holds that dim(R/I) ≥ n − r, we have dim(R/I) = n − r. A noetherian local ring (R, m) is called a geometric local ring if it is a localization of an affine algebra over a field k. If it is an integral domain, we call it a geometric local domain. A local ring (R, m) is catenary if, for any prime ideals p ⊂ p′ , any maximal prime chain p = p0 ⊂ p1 ⊂ · · · ⊂ pr = p′ has length equal to ht (p′ /p). It is known that a geometric local ring is catenary (see [72, (34.3)]). The following result shows that for any system of parameters for a geometric local domain, we can find a prime ideal chain like (1.14). Lemma 1.6.10. Let (R, m) be a noetherian catenary local domain of dimension n. Let {a1 , . . . , an } be a system of generators of m. Then there exists a prime ideal chain m = pn ⫌ pn−1 ⫌ · · · ⫌ p1 ⫌ p0 = (0) such that ai ∈ pi \ pi−1 for 1 ≤ i ≤ n.

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Proof. Let p1 be a minimal prime divisor of a1 R. Then ht (p1 ) = 1 by Lemma 1.6.3 and ht (m/p1 ) = n − 1 by the catenarian property of (R, m). We construct a prime ideal chain by induction on n. Let R = R/p1 Pn and m = m/p1 . Let ai = ai + p1 for 2 ≤ i ≤ n. Since R/ i=2 ai R = Pn Pn R/( i=2 ai R + p1 ) which is a surjective image of R/ i=1 ai R, we have Pn Pn ℓ(R/( i=2 ai R) ≤ ℓ(R/ i=1 ai R) < ∞. This implies that {a2 , . . . , an } is a system of parameters of (R, m). By induction hypothesis, there exists a prime ideal chain m = pn ⫌ pn−1 ⫌ · · · ⫌ p2 ⫌ (0) such that ai ∈ pi \ pi−1 for 2 ≤ i ≤ n. Let pi be the inverse image of pi in R for 2 ≤ i ≤ n. Then, together with p1 , we obtain a required prime ideal chain. Let (R, m) be a noetherian local ring and let {a1 , . . . , an } be a system Pn of parameters of R. Further, if m = i=1 ai R, the set {a1 , . . . , an } is called a regular system of parameters of R, and R is a regular local ring if (R, m) has a regular system of parameters. A ring R (not necessarily local) is a regular ring if Rp is a regular local ring for every p ∈ Spec R. In fact, in view of Theorem 1.6.12, it suffices that Rm is a regular local ring for every maximal ideal of R. Let R be a noetherian ring and let I be an ideal of R. For n ≥ 0, I n /I n+1 L is an R/I-module, where I 0 = R. The direct sum grI R := n≥0 I n /I n+1 is a graded ring over R/I for which the degree n part is I n /I n+1 and the multiplication I m /I m+1 × I n /I n+1 → I m+n /I m+n+1 is given by the canonical multiplication I m × I n → I m+n . The algebra grI R is called the associated graded ring of R with respect to I. Since R is noetherian, the ideal I is finitely generated as I = (a1 , . . . , ar ). Then we have a natural surjective (R/I)-homomorphism of graded algebras φ : (R/I)[x1 , . . . , xr ] → grI R,

xi 7→ (ai + I 2 ), 1 ≤ i ≤ r,

where (R/I)[x1 , . . . , xr ] is a polynomial ring of dimension r over R/I. If (R, m) is a noetherian local ring, we have with k = R/m 2

φ : k[x1 , . . . , xn ] → grm R,

where n = dimk m/m . We have the following characterizations of a regular local ring (see [98, VIII, §11, Theorem 25]). Theorem 1.6.11. Let (R, m) be a noetherian local ring with dim R = n. Then the following conditions are equivalent.

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(i) (R, m) is a regular local ring. (ii) The homomorphism φ : k[x1 , . . . , xn ] → grm R is an isomorphism, where k = R/m. (iii) m/m2 is a vector space over k of rank n. Proof. (i) ⇒ (ii). Let a be the kernel of the above homomorphism φ : k[x1 , . . . , xn ] → grm R. If a ̸= 0, then dim(grm R) ≤ dim k[x1 , . . . , xn ] − 1 by the remark after Lemma 1.6.3. Meanwhile, dim R = dim(grm R) = n by [54, Theorem 13.9]. This is a contradiction. Hence a = 0 and φ is an isomorphism. (ii) ⇒ (iii). Since m/m2 is isomorphic to a k-vector space k · x1 + · · · + k · xn , its rank is equal to n. (iii) ⇒ (i). Let {a1 , . . . , an } be a set of elements of m such that Pn {a1 , . . . , an } is a k-basis of m/m2 , where ai = ai +m2 . Then m = i=1 ai R. Since dim R = n, {a1 , . . . , an } is a regular system of parameters of R. Hence (R, m) is a regular local ring. Regularity of a noetherian local ring (R, m) is kept under various ringtheoretic operations, but proof is rather difficult and, in subsequent discussions, we only make use of these results without going into proofs. So, we just state some of these results, and give necessary references for the readers who are interested in proofs. Theorem 1.6.12. Let (R, m) be a regular local ring of dim R = n. Then the following assertions hold. (1) Let p be a prime ideal of R. Then (Rp , pRp ) is a regular local ring of dimension equal to ht p. (2) R is a UFD. (This result is called a Theorem of AuslanderBuchsbaum-Nagata.) Hence R is a normal ring. b of R is (3) Suppose that R contains a field. Then the m-adic completion R isomorphic to a formal power series ring k[[x1 , . . . , xn ]] in n variables, where k = R/m. For assertion (1), see [72, Corollary 28.3] or [54, Theorem 19.3]. For assertion (2), see [72, Theorem 28.7] and see also [72, p. 217] for a history of the result. Assertion (3) follows from a structure theorem of complete local rings [72, Theorem 31.1].

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1.6.2

Affine Algebraic Geometry

Regular sequence and depth of a local ring

Let (R, m) be a noetherian local ring. An element a of R is a regular element of R if a is not a zero divisor of R. A sequence of elements {a1 , . . . , ar } of m is called a regular sequence of length r if a1 is a regular element of R and ai is not a zero divisor of R/(a1 R + · · · + ai−1 R) for 2 ≤ i ≤ r. A regular sequence {a1 , . . . , ar } is maximal if there is no regular sequence of R containing {a1 , . . . , ar } as a proper subset. If there exists a regular sequence {a1 , . . . , ar } of the largest length, r is called the depth of the local ring R and denoted by depth (R). Lemma 1.6.13. Let (R, m) be a noetherian local ring. Then the following assertions hold. (1) Let a1 , . . . , ar be elements of m. Suppose that {a1 , . . . , ar } is a regular sequence. Then {a1 , . . . , ar } is a part of a system of parameters of R. (2) A regular sequence {a1 , . . . , ar } is maximal if and only if Pr HomR (R/m, R/I) ̸= 0, where I = i=1 ai R. (3) depth (R) ≤ dim R. Pr−1 Proof. (1) Proof proceeds by induction on r. Let R = R/( i=1 ai R). By induction, we may assume that {a1 , . . . , ar−1 } is a part of a system of parameters of R. By Lemma 1.6.9, it holds that dim R = dim R − (r − 1). Pr−1 Meanwhile, since ar = ar + i=1 ai R is not a zero divisor of R, ar ̸∈ p for every prime divisor of (0) in R (see Theorem 1.9.4). By Krull’s height theorem (Theorem 1.6.4), we have ! r X dim R/(ar ) = dim R/ ai R = dim R − 1 = dim R − r. i=1

By Lemma 1.6.9, {a1 , . . . , ar } is a part of a system of parameters of R. (2) We prove first the only if part. Since the regular sequence S {a1 , . . . , ar } is maximal, Theorem 1.9.4 shows that m = p p, where p ranges over prime divisors of I. This implies that m is a prime divisor of I, i.e., m = (I : b) for some b ∈ R (see the proof of the assertion (1) of Theorem 1.6.14). Then the R-homomorphism R → R/I defined by 1 7→ b + I induces a nonzero R-module homomorphism R/m → R/I. Conversely, suppose that there exists a nonzero R-module homomorphism Pr h : R/m → R/( i=1 ai R). Let b be the image of 1 + m by h. Then the

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annihilator ann (b) := {x ∈ R | bx = 0} is equal to m. So, every element of m/I is a zero divisor, and the sequence {a1 , . . . , ar } is maximal. (3) This follows from the assertion (1). It is known that every maximal regular sequence of R has the same length (see [54, Theorem 16.8]). The proof depends on the argument using cohomology theories of modules. Theorem 1.6.14. With the above notations and assumptions, the following assertions hold. (1) depth (R) = 0 if and only if m is a prime divisor of (0). (2) If an element a ̸= 0 is not a zero divisor, then depth (R/aR) = depth (R) − 1. (3) We have inequalities depth (R) ≤ inf dim(R/p) ≤ dim R, p

where p ranges over prime divisors of (0). Proof. (1) Suppose that depth (R) = 0. Then every nonzero element a ∈ m has a nonzero element b such that ab = 0. Let {p1 , . . . , pr } be the set of all maximal prime divisors of (0) in R. Namely, pi is a maximal element of Ass (R) := {p | prime divisor of (0)} with respect to the inclusion order. Note that every pi (1 ≤ i ≤ r) is contained in m. Suppose that pi ⫋ m for every 1 ≤ i ≤ r. Then there exists an element a of m such that ∨

a ̸∈ pi for every 1 ≤ i ≤ r. In fact, since pi ̸⊃ p1 · · · pi · · · pr there exists an element ai ̸∈ pi such that ai ∈ pj for every j ̸= i. Set a = Pr Sr i=1 ai . Meanwhile, i=1 pi is the set of all zero divisors of R. In fact, consider the set S of all ideal quotients (0 : b) of the ideal (0), which we consider with an order defined by the inclusion relation. We know that Ass (R) ⊂ S, and a maximal element of S is a prime ideal. This S Sr shows that p∈Ass (R) p = i=1 pi is the set of all zero divisors of R. Since Sr a ∈ m \ ( i=1 pi ), this contradicts the assumption depth (R) = 0. Hence m = pi for some 1 ≤ i ≤ r. The converse is clear because m = (0 : a) for some a ∈ m by the assumption. (2) Let {a1 , . . . , ad } be a regular sequence of the largest length of R/aR, where ai = ai + aR. Then {a, a1 , . . . , ad } is a regular sequence of R. Hence depth (R) ≥ depth (R/aR) + 1. Let R = R/aR. Since {a1 , . . . , ad } is a maximal regular sequence of R, by !!Lemma 1.6.13, we have d !! d X X HomR R/m, R/ aR + ai R = HomR R/m, R/ ai R ̸= 0. i=1

i=1

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So, {a, a1 , . . . , ad } is maximal. Hence depth (R) = depth (R) + 1. (3) Let d = depth (R). Proof proceeds by induction on d. If d = 0 the assertion is clear. Suppose d > 0. Let a be a regular element of m. Then depth (R/aR) = d − 1 by the assertion (2). For a prime divisor p of (0) in R, let p′ be a minimal prime divisor of p + aR, whence p′ /aR is a prime divisor of (0) in R/aR. Then p′ /aR is a minimal prime divisor of (0) in R/aR. In fact, since p′ /aR is a minimal prime divisor of (p + aR)/aR, write p′ /aR = ((p + aR)/aR : x) with x = x + aR. Meanwhile, if we write p = (0 : y) then (p + aR)/aR = (0 : y) with y = y + aR. Then it holds that p′ /aR = (0 : xy). Since a = a + p is a nonzero element in R/p, p′ /p is a prime divisor of a(R/p). Hence ht (p′ /p) = 1 by Krull’s principal ideal theorem (Lemma 1.6.3), and therefore dim(R/p′ ) = dim(R/p) − 1. By the inductive hypothesis applied to R/aR, we have depth (R/aR) ≤ dim((R/aR)/(p′ /aR)) ≤ dim(R/p′ ). Hence depth (R) ≤ dim(R/p). Remark 1.6.15. We note the following facts about a regular local ring (R, m) with a regular system of parameters {a1 , . . . , an }. (1) R is an integral domain, i.e., a local domain. In fact, if ab = 0 for nonzero elements a, b ∈ m, by Krull’s intersection theorem (Theorem 1.9.10), there exist integers r, s > 0 such that a ∈ mr , a ̸∈ mr+1 , b ∈ ms and b ̸∈ ms+1 . Then a ∈ mr /mr+1 and b ∈ ms /ms+1 are nonzero elements in grm R. Hence ab = ab ̸= 0 by Theorem 1.6.11. This is a contradiction. Hence R is an integral domain. (2) {a1 , . . . , an } is a regular sequence. We prove this assertion by induction on n. Suppose that {a1 , . . . , ai } is a regular sequence. Let Pi Pi R = R/( j=1 aj R), m = m/( j=1 aj R) and let ai+1 , . . . , an be the residue classes of ai+1 , . . . , an in R. Then dim R = dim R − i by Lemma 1.6.13, and m = (ai+1 , . . . , an ). Hence (R, m) is a regular local ring of dimension n − i. Hence R is an integral domain and ai+1 ̸= 0. So, {a1 , . . . , ai , ai+1 } is a regular sequence. A noetherian local ring (R, m) is called a Cohen-Macaulay local ring if depth R = dim R. If (R, m) is a regular local ring, it is a Cohen-Macaulay local ring. Lemma 1.6.16. Let (R, m) be a normal local ring with dim R ≥ 2. Then depth R ≥ 2. Hence, if dim R = 2 then R is a Cohen-Macaulay local ring.

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Proof. Since a normal local ring is an integral domain, any nonzero element a1 is not a zero divisor. Any prime divisor of a1 R has height 1 by S Lemma 1.5.8. So, m ⫌ p p, where p moves over all prime divisors of a1 R. Hence there is a nonzero divisor a2 of R/a1 R. The following result is called Serre’s criterion of normality. Theorem 1.6.17 (Serre’s criterion of normality). Let (R, m) be a noetherian local domain. Then R is normal if and only if the following two conditions are satisfied. (i) For each prime ideal p of height 1, Rp is a discrete valuation ring. (ii) depth Rp ≥ min(2, ht p) for every prime ideal p of R. Proof. Suppose that R is normal. Then the condition (i) follows from Corollary 1.5.7 and the condition (ii) from Lemma 1.6.16. Conversely, we prove that the conditions (i) and (ii) imply that R is integrally closed in Q(R). Suppose that b/a is integral over R, i.e., (b/a)n + c1 (b/a)n−1 + · · · + cn = 0, T where a, b ∈ R, a ̸= 0, c1 , . . . , cn ∈ R. Let aR = i qi be an irredundant √ primary decomposition and let pi = qi . Suppose that ht pi ≥ 2 for some i. By the condition (ii), we have depth Rpi ≥ 2. Since pi = (aR : x) for some x ∈ R, pi Rpi /aRpi consists of zero divisors. Meanwhile, by a remark before Theorem 1.6.14, a maximal regular sequence of Rpi starting with the element a/1 has length ≥ 2. Namely, pi Rpi /aRpi contains an element which is not a zero divisor. This is a contradiction. This implies that ht pi = 1 for every i. Now, since Rpi is a normal ring by the condition (i), T b/a ∈ Rpi . So, b ∈ aRpi ∩ R ⊆ qi Rpi ∩ R = qi . Then b ∈ i qi = aR. Namely, b/a ∈ R. 1.6.3

Jacobian criterion

Let X = Spec A be an affine variety defined over a field k. We say that X is regular if A is a regular ring, and that X is geometrically regular if X ⊗ k = Spec (A ⊗k k) is regular, where k is an algebraic closure of k. A geometrically regular affine variety is also called a smooth affine variety. Since an algebraic variety is a finite union of affine open sets which are affine varieties, we can extend all these definitions to an algebraic variety defined over k. Namely, an algebraic variety X is smooth if the local ring

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(OX,x , mX,x ) is geometrically regular for all closed point x ∈ X. An algebraic variety X is Cohen-Macaulay if OX,x is a Cohen-Macaulay ring for every point x ∈ X. We assume that k is algebraically closed. An affine variety X = Spec A over k is a closed set of the affine space An defined by an ideal I = (F1 , . . . , Fr ), where F1 , . . . , Fr are elements of the polynomial ring e = k[X1 , . . . , Xn ]. Namely, we have A = R/I. e R If r = 1, we say that X is e then a hypersurface. If r ≥ 1 and {F1 , . . . , Fr } is a regular sequence of R X is called a complete intersection. Lemma 1.6.18. Assume that an affine variety defined over k is a complete intersection. Then X is Cohen-Macaulay. Proof. Take a point x ∈ X, and let p (resp. P) be the corresponding e Then Ap = R eP /I R eP . By Lemma 1.6.9 and prime ideal of A (resp. R). eP − r. By Theorem 1.6.14, Lemma 1.6.13(1), we have dim Ap = dim R eP − r. Hence dim Ap = depth Ap . So, X is we have depth Ap = depth R Cohen-Macaulay. Let P ∈ X be a closed point. As a closed point of An , P is given by coordinates as P = (α1 , . . . , αn ) with every αi ∈ k. Let Xi′ = Xi − αi . Hence Xi = Xi′ + αi . Given a polynomial F (X1 , . . . , Xn ) ∈ k[X1 , . . . , Xn ]. The Taylor expansion of F is F (X1 , . . . , Xn ) = F (X1′ + α1 . . . . , Xn′ + αn ) n X X ∂F ∂ i1 +···+in F i i (P )Xi′ + (P )X1′ 1 · · · Xn′ n . = F (P ) + i1 X · · · ∂ in X ∂X ∂ 1 n i i=1 i1 +···+in ≥2

Let R = OX,P and m = mX,P . Let xi be the image of Xi′ in the local ring R. Then {x1 , . . . , xn } is a system of generators of m. We will find a k-basis of the vector space m/m2 . The generators {F1 , . . . , Fr } of the ideal I give rise to the following linear equations in R/m2 with respect to xi = xi + m2 for 1 ≤ i ≤ n.  n X ∂F1   (P )xi = 0 F1 (x1 + α1 , . . . , xn + αn ) =   ∂X  i  i=1   n  X  ∂F2  F (x + α , . . . , x + α ) = (P )xi = 0 2 1 1 n n ∂Xi i=1    ··· ··· ··· ···    n  X  ∂Fr   F (x + α , . . . , x + α ) = (P )xi = 0.  r 1 1 n n ∂X i i=1

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∂Fi (P ), where 1 ≤ i ≤ r and 1 ≤ j ≤ n. So, we obtain an Write aij = ∂X j (r × n)-matrix   a11 a12 · · · a1n  a21 a22 · · · a2n    J = . . . .  .. .. · · · .. 

ar1 ar2 · · · arn

We call J the Jacobian matrix of the embedding X ,→ An at the point P . We consider all the minors of J, and define an integer s ≥ 0 by s = max{t | t ≤ min(r, n), some tth minor of J is nonzero}.

Lemma 1.6.19. With the same notations and assumptions, we assume that dim R = d. Then the following assertions hold. (1) s ≤ n − d. (2) If s = n − d then R is a regular local ring. Hence X is smooth at P . (3) If s < n − d, R is not a regular local ring, and X is not smooth at P . Proof. (1) Suppose that there exists a nonzero tth minor of J. By changes of numbering for the Fi and Xj , we may assume that the tth minor a11 a12 · · · a1t a21 a22 · · · a2t . . .. .. · · · ... a a ··· a t1

t2

tt

is nonzero. Then x1 , . . . , xt are expressed as homogeneous linear polynomials in xt+1 , . . . , xn . Namely, rank m/m2 ≤ n − t. Since d ≤ rank m/m2 by Corollary 1.6.5, we have d ≤ n − s. So, s ≤ n − d. (2) If s = n − d, then m/m2 has a k-basis xn−d+1 , . . . , xn . Then m = (xn−d+1 , . . . , xn ). This implies that {xn−d+1 , . . . , xn } is a regular system of parameters of (R, m). (3) We need more than d elements to generate the maximal ideal. Hence (R, m) is not a regular local ring. Lemma 1.6.19 is called a Jacobian criterion of smoothness. This is a very simple and convenient criterion in the case where X is a hypersurface X = V (F ). If   ∂F ∂F (P ), . . . , (P ) ∂X1 ∂Xn is a nonzero row vector, P is a smooth point of X. The converse also holds.

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1.6.4

Sheaf of differential 1-forms and canonical sheaf

Let A be a B-algebra and let M be an A-module. A B-derivation of A to M is a B-module homomorphism δ : A → M such that δ(a1 a2 ) = a1 δ(a2 ) + a2 δ(a1 ) for a1 , a2 ∈ A. In particular, δ(b · 1) = bδ(1) = 0 for b ∈ B. Set Der B (A, M ) the set of B-derivations of A to M . We can define addition and multiplication by elements of A in Der B (A, M ) by (a1 δ1 + a2 δ2 )(a) = a1 δ1 (a) + a2 δ2 (a). Thus Der B (A, M ) is an A-module. We construct an A-module Ω1A/B with a B-derivation d : A → Ω1A/B such that ∼ HomA (Ω1A/B , M ) −→ Der B (A, M ), f 7→ f ◦ d is an A-isomorphism for every A-module M . P Let µ : A ⊗B A → A be the multiplication of A, i.e., µ( i ai ⊗ a′i ) = P ′ i ai ai . Let I be the kernel of µ. Since µ is surjective, we can identify (A ⊗B A)/I with A via the identification a ⊗ 1 + I with a. Then I/I 2 is an A-module by a(x + I 2 ) = ((a ⊗ 1)x + I 2 ), where x ∈ I. We set Ω1A/B = I/I 2 and call it the module of differential 1-forms of A over B. Lemma 1.6.20. With the above notations, the following assertions hold. (1) Define d : A → Ω1A/B by d(a) = (a ⊗ 1 − 1 ⊗ a) + I 2 . Then d is a B-derivation of A to Ω1A/B such that

(2) (3) (4)

(5)

Φ : HomA (Ω1A/B , M ) → Der B (A, M ), Φ(f ) = f ◦ d is an A-module isomorphism for every A-module M . Ω1A/B is generated by d(A) as an A-module. If A is finitely generated over B then Ω1A/B is a finite A-module. If S is a multiplicative set of A then Ω1S −1 A/B ∼ = S −1 Ω1A/B . Let (R, m) be a noetherian local ring over a field k and let m = Pn 1 i=1 ai R. Suppose that R/m = k and ΩR/k is a finite R-module. P n Then Ω1R/k = i=1 Rdai . If R = Ap and m = pAp for an affine domain A over k and a maximal ideal p of A, then Ω1R/k is a finite R-module.

Proof. (1) For a1 , a2 ∈ A, we have a1 a2 ⊗ 1 − 1 ⊗ a1 a2

= (a1 ⊗ 1)(a2 ⊗ 1 − 1 ⊗ a2 ) + a1 ⊗ a2 − 1 ⊗ a1 a2

= (a1 ⊗ 1)(a2 ⊗ 1 − 1 ⊗ a2 ) + (1 ⊗ a2 )(a1 ⊗ 1 − 1 ⊗ a1 )

= (a1 ⊗ 1)(a2 ⊗ 1 − 1 ⊗ a2 ) + (a2 ⊗ 1)(a1 ⊗ 1 − 1 ⊗ a1 ) − (a2 ⊗ 1 − 1 ⊗ a2 )(a1 ⊗ 1 − 1 ⊗ a1 ),

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whence d(a1 a2 ) = a1 d(a2 ) + a2 d(a1 ). It is clear that d(b) = 0 for b ∈ B. Hence d : A → Ω1A/B is a B-derivation.13 Let f ∈ HomA (Ω1A/B , M ) and let δ = f · d. If δ = 0 then f (da) = 0 for every a ∈ A. By (2) below, Ω1A/B is generated by d(A) as an A-module, whence it follows that f = 0. So, Φ is injective. Conversely, let δ ∈ Der B (A, M ). Define δe : A ⊗B A → M e 1 ⊗ a2 ) = a1 δ(a2 ). Since δ is B-trivial, δe is well-defined to be an by δ(a e ⊗ 1 − 1 ⊗ a) = −δ(a) and A-module homomorphism. Since δ(a e 1 ⊗ 1 − 1 ⊗ a1 )(a2 ⊗ 1 − 1 ⊗ a2 )) = −a2 δ(a1 ) − a1 δ(a2 ) + δ(a1 a2 ) = 0, δ((a δe induces an A-module homomorphism fδ : Ω1A/B → M such that δ = −fδ · d. Hence Φ is surjective. P P (2) An element i ai ⊗ a′i belongs to the ideal I if and only if i ai a′i = 0. Then we have ! X X X ′ 2 ′ ′ ′ ai ⊗ ai + I = (ai ⊗ 1)(1 ⊗ ai − ai ⊗ 1) + ai ai ⊗ 1 + I 2 i

i

=−

i

X

ai da′i .

i

Hence Ω1A/B is an A-module generated by d(A). Suppose that A = B[a1 , . . . , an ]. Then an element of A is written as a = f (a1 , . . . , an ), where f (X1 , . . . , Xn ) is a polynomial in X1 , . . . , Xn over B. Then we have n n X X ∂f Adai . (a1 , . . . , an )dai ∈ da = ∂Xi i=1 i=1 So, Ω1A/B is a finite A-module generated by {da1 , . . . , dan }. (3) By (2), Ω1S −1 A/B is an S −1 A-module generated by d(S −1 A), where  a  sda − ads ∈ S −1 Ω1A/B . = d s s2 So, Ω1S −1 A/B ⊆ S −1 Ω1A/B . Conversely, it is clear that S −1 Ω1A/B ⊆ Ω1S −1 A/B . Hence Ω1S −1 A/B = S −1 Ω1A/B . (4) We note first that Der k (R, k) ∼ = Homk (m/m2 , k). In fact, let δ ∈ Der k (R, k). Since k ∼ = R/m, any element a ∈ R is written as a = α + ξ, where α ∈ k and ξ ∈ m. Since δ is k-trivial, δ(a) = δ(ξ). So, δ is uniquely determined by δ|m . If ξ, η ∈ m then δ(ξη) = ξδ(η) + ηδ(ξ) = 0. This shows that ∼ Homk (m/m2 , k). Der k (R, k) = 13 d : A → Ω A/B , denoted also by dA , is called the universal derivation of the B-algebra A in the sense that a B-derivation δ : A → M with an A-module M is obtained as δ = φ · d for a uniquely determined A-module homomorphism φ : ΩA/B → M .

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Since Der k (R, k) ∼ = HomR (Ω1R/k , k), it follows that Ω1R/k ⊗R R/m ∼ = m/m2 . Pn Write m = (a1 , . . . , an ) and let Ω = i=1 Rdai . Then Ω is an R-submodule of Ω1R/k such that (Ω1R/k /Ω) ⊗R (R/m) = 0. By Nakayama’s lemma (Lemma 1.5.1), we have Ω1R/k = Ω. Here we note that Nakayama’s lemma is available since we assume that Ω1R/k is a finite R-module. (5) Let B = k and S = A \ p in the above (3). Then we have Ω1R/k ∼ = −1 1 S ΩA/k , where Ω1A/k is a finite A-module by (2). Hence Ω1R/k is a finite S −1 A-module, where R = S −1 A. Lemma 1.6.21. The following assertions hold. β

α

(1) Let C −→ B −→ A be ring homomorphisms with which B (resp. A) is an C-algebra (resp. B-algebra). Then we have an exact sequence of A-modules u

v

Ω1B/C ⊗B A −→ Ω1A/C −→ Ω1A/B −→ 0. (2) Let B be a C-algebra, let J be an ideal of B and let A = B/J. Then we have an exact sequence of A-modules d

u

J J/J 2 −→ Ω1B/C ⊗B A −→ Ω1A/C → 0.

(3) Let A be an integral domain, and let B be a subring of A such that A is finitely generated over B. If Ω1A/B = 0 then the field extension Q(A)/Q(B) is a separable algebraic extension. Proof. (1) We have the following commutative diagram with two exact rows: µB

0 −−−−→ IB −−−−→ B ⊗C B −−−−→ B −−−−→ 0    yα′ yα⊗α yα µA

0 −−−−→ IC −−−−→ A ⊗C A −−−−→ A −−−−→ 0,

where α′ is a C-module homomorphism induced by α ⊗ α such that α′ ((b ⊗ 1)x) = α(b)α′ (x) for b ∈ B and x ∈ IB .

2 2 ) ⊗B A → IA /IA , So, α′ induces an A-module homomorphism α : (IB /IB 1 1 which is u : ΩB/C ⊗B A → ΩA/C . There is a natural surjective ring homomorphism A ⊗C A → A ⊗B A, which induces an A-module homomorphism v : Ω1A/C → Ω1A/B . Using the isomorphism in Lemma 1.6.20(1), the A-module homomorphisms u and v yield the homomorphisms u∗ and v ∗ in the following sequence for an A-module M , where u∗ (δ) for δ ∈ Der B (A, M ) is the same

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derivation δ viewed as a C-trivial one, and v ∗ (δ ′ ) for δ ′ ∈ Der C (A, M ) is the restriction of δ ′ to B. It is then clear that u∗

v∗

0 −→ Der B (A, M ) −→ Der C (A, M ) −→ Der C (B, M )

(1.15)

is an exact sequence for every A-module M . The exactness of the sequence in the assertion (1) follows from the exactness of the sequence (1.15) (see Problem 12). (2) Let dJ be the restriction of the universal derivation dB : B → Ω1B/C to the ideal J. Then dJ (J 2 ) ⊆ JΩ1B/C . Hence dJ induces an A-module homomorphism dJ : J/J 2 → Ω1B/C ⊗B A = Ω1B/C /JΩ1B/C , which we denote also by dJ . The A-module homomorphism u : Ω1B/C ⊗B A → Ω1A/C is induced by the canonical surjection ρ : B → A = B/J. In order to prove the exactness of the sequence in the statement, it suffices to show that the dual sequence u∗

d∗

J 0 −→ Der C (A, N ) −→ Der C (B, N ) −→ HomA (J/J 2 , N )

is exact for all A-module N (see Problem 12). Let δ ∈ Der C (A, N ). If u∗ (δ) = δ · ρ = 0 then δ = 0 because ρ is surjective. Hence u∗ is injective. Let D ∈ Der C (B, N ). Then D = φ · dB , where φ : Ω1B/C → N is a Bmodule homomorphism with N viewed as a B-module via ρ. Suppose that d∗J (D) = 0. Then D|J = 0 and hence D induces a C-module homomorphism D′ : A = B/J → N such that D = D′ · ρ. Since D is a C-derivation, D′ is a C-derivation of A. It is clear that d∗J · u∗ = 0. (3) By Lemma 1.6.20(3), we have 0 = S −1 ΩA/B ∼ = ΩS −1 A/B ∼ = Ω1Q(A)/B ∼ = Ω1Q(A)/Q(B) , where S = A \ {0}. Let K = Q(A) and K0 = Q(B). By the hypothesis, K is a finitely generated extension of K0 . Let {ξ1 , . . . , ξn } be a transcendence basis of K over K0 . Then, by the assertion (1), we have an exact sequence 1 Ω1K0 (ξ1 ,...,ξn )/K0 ⊗K0 (ξ1 ,...,ξn ) K → ωK/K → Ω1K/K0 (ξ1 ,...,ξn ) → 0. 0

Thus we obtain Ω1K/K0 (ξ1 ,...,ξn ) = 0. Replacing K0 by K0 (ξ1 , . . . , ξn ), we consider the case tr.degK0 K = 0, i.e., K/K0 is a finite algebraic extension. Suppose that K/K0 is not separable. Then the characteristic of K is a prime p > 0, and there exists an intermediate field K0 ⊆ K1 ⊆ K such that K = K1 (u) with up = a ∈ K1 . By the assertion, Ω1K/K0 = 0 implies Ω1K/K1 = 0. Meanwhile, Ω1K/K1 = Kdu ̸= 0. So, K/K0 is a separable algebraic extension. Conversely, if K/K0 is a separable algebraic extension, then Ω1K/K0 = 0.

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In fact, K is a simple extension K0 (θ) of K0 . Let f (x) = 0 be the minimal (monic) equation of θ over K0 . If δ is a K0 -derivation of K into K, then f ′ (θ)δ(θ) = 0. Since f ′ (θ) ̸= 0, we have δ(θ) = 0. This implies that δ = 0. Consider the case tr.degK0 K = n > 0. Let ∂i be the partial derivative ∂/∂ξi of K0 [ξ1 , . . . , ξn ], which is extended to a K0 -derivation δi of K0 (ξ1 , . . . , ξn ). By the foregoing step, we know that K/K0 (ξ1 , . . . , ξn ) is a separable algebraic extension. Then K = K0 (ξ1 , . . . , ξn )(θ). Let f (x) = 0 be a minimal equation of θ over K0 (ξ1 , . . . , ξn ). Let δ = δi for some 1 ≤ i ≤ n. Then we have f ′ (θ)δ(θ) + fδ (θ) = 0, where fδ (x) is a polynomial obtained by applying δ to each coefficient of f (x). By the minimality of f (x), we know that f ′ (θ) ̸= 0. Hence δ = δi is extended to K by setting δ(θ) = −fδ (θ)/f ′ (θ). So, Der K0 (K, K) ̸= 0 and Ω1K/K0 ̸= 0. This contradicts the condition that Ω1K/K0 = 0. So, tr.degK0 K = 0. This argument shows that if K/K0 is a separable extension with separating transcendence basis {ξ1 , . . . , ξn } we have Ω1K/K0 =

n X

Kdξi .

i=1

Theorem 1.6.22. Let X = Spec A be an affine algebraic variety of dimension n over an algebraically closed field k 14 and let (R, m) be the local ring (OX,x , mX,x ) at a closed point x of X. Then the following assertions hold. (1) (2) (3) (4)

Ω1A/k is a finite A-module. Ω1R/k ∼ = Ω1A/k ⊗A R. If (R, m) is a regular local ring, Ω1R/k is a free R-module of rank n. If X is smooth then Ω1A/k is locally free. Namely there exists a finite affine open covering U = {Ui = Spec Ai | i ∈ I} of X such that Ω1Ai /k is a free Ai -module of rank n for every i ∈ I.

Proof. (1) This is a restatement of Lemma 1.6.20(2). (2) This also follows from Lemma 1.6.20(3) with S = A \ px . (3) The assumption in Lemma 1.6.20(4) is satisfied. The condition R/m = A/p = k, in particular, is satisfied because k is algebraically closed. 14 We

can drop the condition that k is algebraically closed, but we assume this condition to avoid the technical complicatedness.

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Pn Hence Ω1R/k = i=1 Rdai , where {a1 , . . . , an } is a regular system of parameters of (R, m). Then there exists a surjective R-module homomorphism ρ : Rn → Ω1R/k such that ρ(ei ) = dai , where {e1 , . . . , en } is a free basis of Pn Rn with which we identify i=1 ci ei with (c1 , . . . , cn ). Let N = Ker ρ. Let K = Q(R). Since K is a flat R-module, we have an exact sequence ρ⊗K 0 −→ N ⊗R K −→ Rn ⊗R K ∼ = Ω1K/k −→ 0, = K n −→ Ω1R/k ⊗R K ∼ Pn where ρ ⊗ K is an isomorphism because Ω1K/k ∼ = i=1 Kdξi (see the end of the proof of Lemma 1.6.21(3)). So, N ⊗R K = 0. This implies that N = 0, for otherwise R has torsion elements. This argument is the same as in [31, Theorem 8.8]. (4) Let x be a closed point of X corresponding to a maximal ideal p of A. Write R = Ap = S −1 A with S = A \ p. Since (R, m) with m = pR is a regular local ring, we have a regular system of parameters {a1 , . . . , an } Pn such that Ω1R/k = i=1 Rdai . Replacing A by a ring of quotients, we may assume that a1 , . . . , an ∈ A. Define an A-module homomorphism ρ : An → Ω1A/k by ρ(ei ) = dai for 1 ≤ i ≤ n. Let M = Coker ρ and N = Ker ρ. Then we have an exact sequence S −1 ρ

0 −→ S −1 N −→ (S −1 A)n −→ S −1 Ω1A/k −→ S −1 M −→ 0, where S −1 ρ is an isomorphism. Hence S −1 M = S −1 N = 0. By the −1 argument in (3) above, we have N = 0, and S ′ M = 0 for S ′ = {sr | r ≥ 0} −1 −1 with s ∈ S because M is a finite A-module. Then (S ′ A)n ∼ = S ′ ΩA/k and x ∈ D(s). Note that this isomorphism occurs for an open set D(s) of an arbitrary closed point x ∈ X. Since X is covered by finitely many open sets D(s), it follows that Ω1A/k is a locally free A-module. In addition to the explanations in section 1.3.2, we give some more definitions on a sheaf of modules on a scheme, which is also written as a Module with “m” in the capital letter. Let X be a separated scheme over a field k and let OX be the structure sheaf. Namely, for any affine open set U = Spec A of X, we assign the ring Γ(U, OX ) which coincides with A. If U is a (not necessarily affine) open set of X, consider an affine open covering V = {Vj | j ∈ J} with Vj = Spec Aj . Note that Vj ∩ Vj ′ is an affine open set Spec Ajj ′ because X is separated over k. We define the ring Γ(U, OX ) as the kernel of the ring homomorphism Y Y ρ′ : Aj −→ Ajj ′ , j∈J

j,j ′ ∈J

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where ρ′ ({ξj }j∈J ) = {(ξj |Vj ∩Vj′ − ξj ′ |Vj ∩Vj′ )}j,j ′ ∈J . Then Γ( · , OX ) is viewed as a functor U 7→ Γ(U, OX ) from the category of open sets of X to the category of k-algebras satisfying the exactness of the sequence Y ρ Y ρ′ 0 −→ Γ(U, OX ) −→ Γ(Ui , OX ) −→ Γ(Ui ∩ Ui′ , OX ), i,i′ ∈I

i∈I

where U = {Ui | i ∈ I} is an open covering of U and ρ(ξ) = (ξ|Ui )i∈I and Γ(U, OX ) is independent of the choice of the covering U. f on Similarly, for an A-module M , we can define a module sheaf M f X = Spec A by defining Γ(U, M ) as the kernel of the homomorphism Y Y ρ′M : Mj −→ Mjj ′ , j∈J

j,j ′ ∈J

where V = {Vj }j∈J is the same as above, Mj = M ⊗A Aj , Mjj ′ = M ⊗A Ajj ′ and ρ′M is defined in the same way as for ρ′ . For a (not necessarily affine) f) is defined by open set U , Γ(U, M ρ′M Y ρM Y f) −→ f) −→ f), 0 −→ Γ(U, M Γ(Ui , M Γ(Ui ∩ Ui′ , M i∈I

i,i′ ∈I

f is called where U = {Ui | i ∈ I} is an affine open covering of U as above. M f is a a quasi-coherent OX -Module. If M is a finite A-module, we say that M coherent OX -Module. We say that F is obtained by patching together local fi | i ∈ I}. If F is a quasi-coherent OX -Module, the stalk of F at pieces {M x ∈ Ui is Fx =

lim −→

D(s)∈T(x)

Mi [s−1 ] = (Mi )p = Mi ⊗Ai (Ai )p ,

where p is a prime ideal of Ai corresponding to x ∈ Ui and T(x) is the set of open neighborhoods of x, and the last equality follows from the fact that tensor product and inductive limit commute. For an algebraic variety defined over k, we can define the structure sheaf OX and a quasi-coherent (or coherent) OX -module sheaf F by applying the above construction for a finite affine covering U = {Ui | i ∈ I}, where fi for an Ai -module Mi Ui = Spec Ai and the restriction of F onto Ui is M (or a finite Ai -module Mi ). Now assume further that X is a smooth algebraic variety of dimension n defined over k. Then there exists a finite affine open covering U = {Ui }i∈I of X such that Ui ∩ Uj is an affine open set and Ai = Γ(Ui , OX ) is an e 1 , which is a coherent affine k-domain (see Problem 5). Let Ω1Ui /k = Ω Ai /k 1 OUi -Module defined by ΩAi /k . For i, j ∈ I, the stalks of Ω1Ui /k and Ω1Uj /k

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at a point x ∈ Uij := Ui ∩ Uj are equal to Ω1R/k by Lemma 1.6.20(3), where R = OX,x . By Lemma 1.6.23 below, we have Ω1Ai /k ⊗Ai Aij = Ω1Aj /k ⊗Aj Aij ,

which implies that OUij -Modules Ω1Ui /k |Uij and Ω1Uj /k |Uij are equal to Ω1Uij /k . Hence the local pieces {Ω1Ui /k }i∈I patch together and form a coherent OX -Module Ω1X/k . We call it the sheaf of differential 1-forms. By construction, Ω1X/k is a locally free OX -Module of rank n since X is smooth and has dim X = n. Lemma 1.6.23. Let M, M ′ be finite A-modules such that Mp = Mp′ for every p ∈ Spec A. Then M = M ′ . Proof. Let {m1 , . . . , mn } and {m′1 , . . . , m′n′ } be systems of generators of M and M ′ , respectively. For each p ∈ Spec A, Mp = Mp′ implies that ′

n n X X mi = (a′ij /s′ )m′j and m′j = (aji /s)mi j=1

i=1

′

for some s, s ̸∈ p. So, we may assume that M [s−1 ] = M ′ [s−1 ] for some s ̸∈ p. Since Spec A is quasi-compact, we find a finite affine open covering ′ −1 {D(si ) | 1 ≤ i ≤ r} of X such that M [s−1 i ] = M [si ] for every 1 ≤ i ≤ r. We show that the sequence is exact ρ

0 −→ M −→

n Y i=1

ρ′

M [s−1 i ] −→

Y

−1 M [s−1 i , sj ],

(1.16)

i,j

where ρ(m) = (m, . . . , m) and r

r

r

ρ′ ((mi /sri i )) = (mi sj j /sri i sj j − mj sri i /sri i sj j ). Sn Suppose that ρ(m) = 0. Then stii m = 0. Since i=1 D(si ) = Spec A, we Pn Pn have a relation i=1 stii ai = 1. Then m = i=1 ai stii m = 0. Hence ρ is injective. It is clear that ρ′ · ρ = 0. Suppose that ρ′ ((mi /sri i )) = 0. For 1 ≤ c r c +r c i, j ≤ n, we have sci i sjj (sj j mi − sri i mj ) = 0, i.e., sci i sjj j mi = sci i +ri sjj mj . Then we have c sjj mj sci i mi = c +r . sci i +ri sjj j Here the choice of integers ci , cj depends on the choice of a pair (i, j). If we choose an integer N ≫ 0 then we have N +rj

(sN i mi )sj

N +ri = (sN . j mj )si

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The last equation is obtained from the former by multiplying siN −ci sj j . If N ≫ 0, the last equality holds for all pair (i, j). If we replace mi , ri by Qn ri −1 sN i mi , N + ri , we may assume that the element (mi /si ) of i=1 M [si ] satisfies the equation r

mi sj j = mj sri i for every pair (i, j). Write Then we have sri i m =

Pn

n X

ri i=1 si ai

= 1 with ai ∈ A. Let m =

Pn

i=1

ai mi .

  n X r aj sri i mj =  aj sj j  mi = mi .

j=1

j=1

Hence we have mi = m/sri i . So, (mi /sri i ) belongs to Im ρ. Similarly, we have an exact sequence ρ

0 −→ M ′ −→

n Y i=1

ρ′

M ′ [s−1 i ] −→

Y

−1 M ′ [s−1 i , sj ].

(1.17)

i,j

In (1.16) and (1.17), the second and the third terms are the same by the choice of the si . Hence it follows that M = M ′ . We summarize the descriptions on the structure of Ω1X/k for a smooth algebraic variety X of dimension n defined over an algebraically closed field k. For a closed point x ∈ X, the stalk Ω1X/k,x is given as Ω1R/k , where U := Spec A is an open neighborhood of x and R = Ap with a maximal ideal p of A corresponding to x. Let {a1 , . . . , an } be a regular system of parameters of (R, m) with m = pR, where we may assume that a1 , . . . , an ∈ A by replacing U by a smaller affine neighborhood. Since Ω1X/k is locally free, we may e 1 , where Ω1 = Pn Adai . If x′ is a assume that Ω1 |U ∼ = Ω1 ∼ =Ω X/k

U/k

A/k

A/k

i=1

closed point of U , ai −ai (x′ ) ∈ mU,x′ with ai (x′ ) ∈ k and dai = d(ai −ai (x′ )) for every 1 ≤ i ≤ n. Namely, {a1 − a1 (x′ ), . . . , an − an (x′ )} is a regular system of parameters of a regular local ring OX,x′ . Thus {a1 , . . . , an } gives a regular system of parameters of every closed point x ∈ U after a necessary change ai 7→ ai − ai (x). So, we call {a1 , . . . , an } a system of coordinates on U , and call U a coordinate neighborhood. If {a1 , . . . , an } is a system of coordinates on U ⊂ X and {a′1 , . . . , a′n } is a system of coordinates on U ′ ⊂ X, at a point x ∈ U ∩ U ′ , both systems {ξ1 , . . . , ξn } with ξi = ai − ai (x) and {η1 , . . . , ηn } with ηi = a′i − a′i (x) are regular systems of parameters of the local ring OX,x . Hence {ξ 1 , . . . , ξ n }

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with ξ i = ξ + m2X,x and {η 1 , . . . , η n } with η i = ηi + m2X,x are k-basis of the vector space mX,x /m2X,x . Hence we have     η1 ξ1  η2   ξ2       .  = J  . , .  .   ..  ηn

ξn

where J is an (n × n)-matrix in GL (n, k)   α11 α12 · · · α1n  α21 α22 · · · α2n   J =  ··· ··· ··· ··· . αn1 αn2 · · · αnn In fact, we can write     dη1 dξ1  dη2   dξ2       .  = J  . ,  ..   ..  dηn

dξn

where J is an (n × n)-matrix in GL (n, OX,k )   a11 a12 · · · a1n  a21 a22 · · · a2n   J =  ··· ··· ··· ···  an1 an2 · · · ann

and J = J(x) = (aij (x)). By Theorem 1.6.12,(3), we have bX,x = k[[ξ1 , . . . , ξn ]] = k[[η1 , . . . , ηn ]] O

bX,x . So, J or J is called the Jacobian matrix of the and aij = ∂ηi /∂ξj in O coordinate changes {ξ1 , . . . , ξn } → {η1 , . . . , ηn }. Pn With the above notations, Ω1U/k ∼ = i=1 OU dξi , which is a free OU Vn 1 Module of rank n. Take its exterior product (ΩU/k ) = OU (dξ1 ∧· · ·∧dξn ), which is a free OU -Module of rank 1. We denote it by ΩnU/k . Similarly, we Vn 1 can consider ΩnU ′ /k = (ΩU ′ /k ) = OU ′ (dη1 ∧· · ·∧dηn ). On the intersection ′ U ∩ U , the change of bases is given by dη1 ∧ · · · ∧ dηn = det(J)dξ1 ∧ · · · ∧ dηn .

By patching together ΩnU/k for coordinate neighborhoods U of X, we have an OX -Module ΩnX/k of rank 1 such that ΩnX/k |U = ΩnU/k and an isomorVn 1 phism of OX -isomorphism ΩX/k ∼ = ΩnX/k . The OX -Module ΩnX/k is called the canonical sheaf of X.

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1.7

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Divisors and linear systems

Let X be a normal algebraic variety of dimension n defined over an algebraically closed field. We gave in subsection 1.5.5 the definitions of Weil divisors, the free abelian group Div (X), the linear equivalence of Weil divisors and the divisor class group Cℓ (X) which is the quotient group of Div (X) modulo linear equivalence. The notion of divisors is very important in studying the structure of a given algebraic variety. In the next chapters we are mainly interested in smooth algebraic varieties where we need a different approach to divisors, that is, Cartier divisors. 1.7.1

Invertible sheaves

An invertible sheaf is a coherent OX -Module L which is a locally free OX Module of rank 1. Namely there exists an open covering U = {Uλ | λ ∈ Λ} of X such that L|Uλ = OUλ eλ ∼ = OUλ for each λ ∈ Λ, where eλ is a free basis. Since X has, by definition, a finite affine open covering, we may assume that Λ is a finite set. For λ, µ ∈ Λ, we have OUλµ eλ = OUλµ eµ on Uλµ = Uλ ∩ Uµ , which implies that eµ = fµλ eλ , fµλ ∈ Γ(Uµλ , OX )∗ .

(1.18)

We can easily deduce the following properties (i) fλλ = 1 for every λ ∈ Λ, −1 (ii) fλµ = fµλ for all λ, µ ∈ Λ, (iii) fνλ = fνµ fµλ on Uλ ∩ Uµ ∩ Uν for λ, µ, ν ∈ Λ. The set {fµλ | λ, µ ∈ Λ} is called the transition functions of L with respect to the open covering U. The conditions (i) and (ii) follow from (iii). In fact, if we take λ = µ = ν in (iii), we obtain fλλ = 1. If we take λ = ν in (iii), we obtain (ii). If L is free, we can take fµλ = 1 for all λ, µ ∈ Λ. Conversely, if an open covering U and the transition functions {fµλ } with respect to U are given, we can patch together free OUλ -Modules OUλ eλ by the relation (1.18) and obtain an invertible sheaf. Let V = {Vj | j ∈ J} be an open covering of X. We say that V is finer than the open covering U = {Uλ | λ ∈ Λ} if there exists a mapping σ : J → Λ such that Vj ⊂ Uσ(j) for every j ∈ J. We say also that V is a refinement of U (U ≤ V for the notation). If the open covering U makes an invertible sheaf L locally free, then a refinement of U also makes L locally free. Given two invertible sheaves L, M on X, which becomes free on open coverings U = {Uλ | λ ∈ Λ}, U ′ = {Uλ′ ′ | λ′ ∈ Λ′ }, respectively, then

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U ∩ U ′ = {Uλ ∩ Uλ′ ′ | λ ∈ Λ, λ′ ∈ Λ′ } is an open covering of X which is finer than U and U ′ . Further, L and M become free on U ∩ U ′ . So, given a finite number of invertible sheaves L1 , . . . , Lr , there is an open covering of (i) X which makes L1 , . . . , Lr locally free. So, if {fµλ | λ, µ ∈ Λ} are transition Qr (i) functions for Li with respect to U, then { i=1 fµλ | λ, µ ∈ Λ} are transition −1 functions for L1 ⊗ · · · ⊗ Lr with respect to U. The set {fµλ | λ, µ ∈ Λ} defines an invertible sheaf L−1 such that L ⊗ L−1 = OX . This is the reason why L is called invertible. ∼ Let φ : L → M be an OX -isomorphism of invertible sheaves. Namely, ∼ for any open set U ⊂ X, φ(U ) : Γ(U, L) → Γ(U, M) is a Γ(U, OX )isomorphism of Modules, and if V ⊂ U is a smaller open set, then U U (ρU V )M · Γ(U, φ) = Γ(V, φ) · (ρV )L , where (ρV )L : Γ(U, L) → Γ(V, L) is U the restriction map and (ρV )M is similarly defined. To simplify the notations, we write φ(U ), L(U ), M(U ) instead of Γ(U, φ), Γ(U, L), Γ(U, M). We choose an open covering U = {Uλ | λ ∈ Λ} such that L|Uλ = OUλ eL,λ and M|Uλ = OUλ eM,λ for every λ ∈ Λ. Then φ(Uλ )(eL,λ ) = hλ eM,λ with hλ ∈ Γ(Uλ , OX )∗ for λ ∈ Λ because φ(Uλ ) is an isomorphism. Let {gµλ } be transition functions of M with respect to U. Then we have hµ fµλ = gµλ hλ . Namely we have a relation on Uµλ = Uµ ∩ Uλ , gµλ = (hµ |Uµλ )fµλ (hλ |Uµλ )−1 ,

λ, µ ∈ Λ.

(1.19)

Conversely, if the transition functions {fµλ } and {gµλ } for L and M are related as in (1.19) for {hλ } with hλ ∈ OX (Uλ )∗ , then the OUλ homomorphisms φUλ : LUλ → MUλ defined by φUλ (eL,λ ) = hλ (eM,λ ) ∼ patch together to define an OX -isomorphism φ : L → M such that φUλ = φ|Uλ . We then say that {fµλ } is cohomologous to {gµλ } and write {fµλ } ∼ {gµλ }.

ˇ With the notations of Cech cohomologies explained in the appendix, the group of transition functions modulo relations (1.19) with an open covering ˇ replaced by finer ones if necessary coincides with the Cech cohomology 1 ∗ ˇ (X, O ). On the other hand, the set of the isomorphism classes group H X of invertible sheaves is an abelian group under the tensor product, which we denote by Pic (X). By the above explanations, we have an isomorphism of abelian groups ∗ ∼ ˇ 1 (X, OX H ) = Pic (X). ∗ ∗ The sheaf OX is defined as Γ(U, OX ) = Γ(U, OX )∗ for all open set U ⊂ X. ∗ Hence OX is a sheaf of abelian groups on X written multiplicatively.

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1.7.2

Affine Algebraic Geometry

Cartier divisors

Let L be an invertible sheaf on X which becomes free on an open covering U as above. Let {fµλ } be the transition functions of L. Fix an index λ0 ∈ Λ and let fλ = fλ0 λ . Then fλ is a rational function on Uλ . Let Dλ = (fλ ) on Uλ (see Section 1.5.5). For µ ∈ Λ, define Dµ on Uµ in the same way with fµ = fλ0 µ . On the intersection Uµλ = Uµ ∩Uλ , we have fµ = fλµ fλ , whence Dµ |Uµλ = Dλ |Uµλ because fµλ ∈ Γ(Uµλ , OX )∗ . This implies that the set of local Weil divisors {Dλ | λ ∈ Λ} defines one and the same Weil divisor D, which we write D(L) and call it a Cartier divisor associated to L. Suppose that L ∼ = M, where {fµλ } and {gµλ } are transition functions of L and M. Then we find rational functions {hλ } which satisfy the relation (1.19). Replacing (µ, λ) in (1.19) by (λ0 , λ), we have gλ = hfλ h−1 λ , where gλ = gλ0 λ and h = hλ0 . Since hλ ∈ Γ(Uλ , OX )∗ , we have D(M)λ = D(L)λ + (h) on Uλ . Hence it holds that D(M) = D(L) + (h), i.e., D(M) ∼ D(L). We say that a Weil divisor D on X is a Cartier divisor if there exists an open covering U = {Uλ | λ ∈ Λ} such that D|Uλ is a principal divisor (fλ ). Then fµλ := fµ−1 fλ ∈ Γ(Uµλ , OX )∗ because (fλ ) = (fµ ) on Uµλ . It is clear that {fµλ } satisfies the three conditions of transition functions. Hence, by setting L|Uλ = OUλ · (1/fλ ), {fµλ } define an invertible sheaf ) for every λ ∈ Λ. Namely, we L on X such that D(L)|Uλ = (fλ ) + (fλ−1 0 have D(L) ∼ D. We denote L by OX (D). Suppose that D, D′ are Cartier divisors which become principal on the same open covering U as above. Write D|Uλ = (fλ ) and D′ |Uλ = (gλ ) for λ ∈ Λ. Suppose that D′ = D +(h). Then hλ := gλ−1 (fλ h) ∈ Γ(Uλ , OX )∗ . Hence we have gλ = h−1 λ fλ h, and −1 −1 ′ ∼ gµλ = gµ gλ = hµ fµλ hλ . This implies that OX (D ) = OX (D). There exists a mapping σ : Pic (X) → Cℓ (X) which assigns a Cartier divisor D(L) to an invertible sheaf L. By the construction it is clear that σ is a homomorphism of abelian groups. If D(L) ∼ 0 then L ∼ = OX . Hence σ is an injection. We have the following result. Theorem 1.7.1. If X is a smooth algebraic variety, then the homomorphism σ is an isomorphism. Pr Proof. Let D = i=1 ni Di be a Weil divisor on X with irreducible components Di . To prove that the linear equivalence class of D is in the image of σ, it suffices to show that D is locally principal, that is to say, for all

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closed point P ∈ X there exists an open neighborhood U of P so that D|U = (f ) for some rational function f . Decompose D = D1 + D2 so that every irreducible component of D1 (resp. D2 ) passes (resp. does not pass) through the point P . Let C be an irreducible component and m the coefficient of C in D1 . In the local ring R = OX,P which is a regular local ring, hence a factorial domain by Theorem of Auslander-BuchsbaumNagata (Theorem 1.6.12(2)), the prime ideal p of height 1 defining C is Ps principal by Theorem 1.5.18. Let D1 = i=1 mi Ci and let fi ∈ OX,P be Qs an element such that Ci = (fi ). Set f = i=1 fimi . Then f is a rational function of X such that none of the irreducible components of the divisor D1 − (f ) passes through P . In fact, there is an open neighborhood U of P such that D|U = (f ). Since P is an arbitrary closed point of X, this shows that D is a Cartier divisor. If X is smooth, there is no distinction between Weil divisors and Cartier Pr divisors. So, we use the term divisor. For a divisor D = i=1 ni Di with Sr every ni ̸= 0, the support is the union i=1 Di . In the previous section, we defined the canonical sheaf ΩnX/k which is a locally free OX -Module of rank 1, hence an invertible sheaf. The divisor (more precisely, the linear equivalence class) associated to ΩnX/k is denoted by KX and called the canonical divisor of X. Here we give a rather simple way of obtaining a divisor D which represents the canonical divisor class Vn 1 KX . Let K = k(X) and let ωK/k = ΩK/k which is a free K-module of rank 1. We choose a nonzero element ω of ωK/k . Let U = {Ui }i∈I be an open covering of X such that each Ui is a coordinate neighborhood with (i) (i) a system of coordinates {x1 , . . . , xn }. Let fji be the determinant of the (i) (i) Jacobian matrix with respect to the coordinate change {x1 , . . . , xn } → (j) (j) {x1 , . . . , xn }. Then {fji } is the transition function of ΩnX/k with respect (i)

(i)

to the covering U. On Ui , write ω|Ui = gi dx1 ∧ · · · ∧ dxn . Then gi = fji gj on Ui ∩ Uj . On the other hand, if KX is a divisor such that OX (KX ) ∼ = −1 n ΩX/k , we have OX (KX )|Ui = hi OUi for hi ∈ K, whence hi = fji hj . This implies that hi /gi = hj /gj on Ui ∩ Uj . Hence there exists an element ξ ∈ K such that hi = ξgi for every i ∈ I. Let D be a divisor on X such that D|Ui = (gi ). Since KX |Ui = (hi ), we have KX = D + (ξ). Namely, KX ∼ D. So, we have only to compute D on each coordinate neighborhood Ui . We write D as (ω).

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1.7.3

Affine Algebraic Geometry

Linear systems

Let X be a normal algebraic variety defined over a field k and let D be a Cartier divisor on X. Let U = {Uλ | λ ∈ Λ} be an open covering of X such that D|Uλ = (fλ ). Let L := OX (D) be the associated invertible sheaf. Then L|Uλ = OUλ · (1/fλ ). We set M (D) = {f ∈ k(X) | D + (f ) ≥ 0} ∪ {0}. Lemma 1.7.2. The following assertions hold. (1) M (D) is a k-vector space. (2) M (D) ∼ = Γ(X, OX (D)). (3) Let X be a normal projective variety defined over a field k. Then dimk M (D) < ∞. Proof. (1) Let f1 , f2 ∈ M (D). Assume that f1 f2 ̸= 0. For c1 , c2 ∈ k ∗ , we have to show that D + (c1 f1 + c2 f2 ) ≥ 0. Let vi be the discrete valuP ation attached to the DVR OX,Di , where D = i ni Di be the irreducible decomposition of D. Then ni + vi (fj ) ≥ 0 for j = 1, 2. Then we have ni + vi (c1 f1 + c2 f2 ) ≥ min(ni + vi (c1 f1 ), ni + vi (c2 f2 )) ≥ 0. Hence c1 f1 + c2 f2 ∈ M (D). The rest of the proof is left to the readers. (2) Let s ∈ Γ(X, L) with L = OX (D). Since L|Uλ = OUλ (1/fλ ), the restriction sλ := s|Uλ ∈ Γ(Uλ , L) is written as sλ = hλ /fλ with hλ ∈ Γ(Uλ , OX ). Then, for λ, µ ∈ Λ, we have hλ /fλ = hµ /fµ on Uλµ . So, {hλ /fλ | λ ∈ Λ} defines a rational function, say h ∈ k(X) such that (h)+D|Uλ = (hλ ) ≥ 0 on Uλ . So, (h)+D ≥ 0, and h ∈ M (D). If f ∈ M (D) then hλ := f fλ ∈ Γ(Uλ , OX ). Hence f = hλ /fλ is independent of λ ∈ Λ. Namely, {hλ · (1/fλ )} gives an element of Γ(X, L). This correspondence is clearly bijective. (3) Note that an invertible sheaf OX (D) is a coherent OX -Module. Then H 0 (X, OX (D)) ∼ = Γ(X, OX (D)) has finite rank by Theorem 1.9.22. Hereafter we assume further that X is projective. Let dimk M (D) = d+ 1. If f, g ∈ M (D), the equality D + (f ) = D + (g) holds if and only if g/f ∈ H 0 (X, OX ) = k (see Theorem 1.4.11) because the principal divisor (g/f ) = 0 implies that the rational function g/f is regular at every closed point of X (see Theorem 1.5.9). Suppose that d ≥ 0. Let {f0 , f1 , . . . , fd } be a k-basis of M (D). An element f ∈ M (D) is written as f = α0 f0 +α1 f1 +· · ·+αd fd with (α0 , α1 , . . . , αd ) ∈ k d+1 . Express g ∈ M (D) as g = β0 f0 + β1 f1 + · · · + βd fd . Then g/f = c ∈ k ∗ if and only if (β0 , β1 , . . . , βd ) = (cα0 , cα1 , . . . , cαd ).

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Since D + (f ) ≥ 0 for f ∈ M (D) implies f ̸= 0, D + (f ) corresponds bijectively with the point (α0 : α1 : · · · : αd ) of Pd = k d+1 /(∼). We denote this projective space by |D| or L(D) and call it a complete linear system. If we take a nonzero linear k-subspace M of M (D), we have a linear subspace L(M ) := P(M ) of |D|, where L(M ) := P(M ) = {D + (f ) | f ∈ M }.

We call L(M ) a linear system parametrized by the k-vector space M , and dim M − 1 the dimension of L(M ). We also write the linear system by L without mentioning to the k-module M , and the dimension of L by dim L. If dim L = 1, we call L a linear pencil. By abuse of notation, by D′ ∈ L, we mean that D′ is an effective divisor corresponding to a point of L. Lemma 1.7.3. With the notations as above, let L ⊂ L(D) be a linear system of dimension m ≥ 0. Then the following assertions hold.

(1) If D′ = D + (f ) then L(D′ ) ∼ = L(D). So, replacing L by the isomorphic ′ image in L(D ), we may assume that D ≥ 0. (2) Suppose that m > 0 and D ≥ 0. Let F be the greatest common divisor of L. Namely D′ ≥ F for every D′ ∈ L and there is no common irreducible component in D′ − F when D′ moves in L. Write L′ = {D′ − F | D′ ∈ L}. Then L′ is a linear system such that L = L′ + F and dim L′ = dim L. S (3) Suppose that m > 0. We then have D′ ∈L Supp D′ = X. Proof. (1) It is easy to see that the correspondence M (D′ ) ∋ g 7→ gf ∈ M (D) is a k-isomorphism of vector spaces. Hence L(D) and L(D′ ) are accordingly isomorphic. So, by replacing D by a member D′ of L(D), we may assume that D ≥ 0. (2) Clear. P (3) Let D = i ni Di be the irreducible decomposition with ni > 0. We remark that if D′ = D + (f ) is a member of L(D), then the polar part (f )∞ satisfies 0 ≤ (f )∞ ≤ D. Let M be a subspace of M (D) which gives rise to L and let {f0 , . . . , fm } be a k-basis of M . By (1), we may assume that D ∈ L. Since D ≥ 0 we may assume that f0 = 1. An element f ∈ M is written as S f = α0 f0 + α1 f1 + · · · + αm fr . In order to show that D′ ∈L Supp D′ = X, it suffices to show that any closed point P ∈ X lies on some D′ ∈ L. Since D ∈ L, we may assume that P ̸∈ Supp D. Hence f0 , . . . , fm are regular at P by the above remark. Then we can find the coefficients α0 , . . . , αm such that α0 f0 (P ) + · · · + αm fm (P ) = 0 and (α0 , . . . , αm ) ̸= (0, . . . , 0). So, P lies on some D′ ∈ L.

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The divisor F in the assertion (2) is called the fixed part of L and L′ the movable part of L. Replacing L by L′ we often assume that L is fixedcomponent free, i.e., F = 0. Even if L has no fixed components, there might exist subvarieties of codimension > 1 which are common to all members of L. We set Bs (L) = {P ∈ X | P ∈ Supp D′ , ∀D′ ∈ L}. Then Bs (L) is a closed set, and the support of the fixed part F is the part of Bs (L) of codimension one. Suppose that L is a linear system of dimension m > 0 such that L has no fixed part. Let {f0 , . . . , fm } be a k-basis of the vector subspace in M (D) which defines L. Consider a rational mapping ΦL : X 99K Pm , P ∈ X 7→ (f0 (P ), f1 (P ), · · · , fm (P )). Clearly, ΦL is defined on an open set U := X \ (Supp (D) ∪ Bs (L)). Let φ = ΦL |U . Let {x0 , x1 , . . . , xm } be a system of homogeneous coordinates of Pm and let Hα be a hyperplane defined by α0 x0 +· · ·+αm xm = 0, where α = ∨

(α0 , . . . , αm ) is a point of the dual projective space Pm . Let U be the domain of definition of ΦL . Suppose that either X is smooth or D is a Cartier divisor. Let P ∈ U and let fP be a defining equation of D near the point P . Then (Φ|L )−1 (Hα ) = D + (α0 f0 + · · · + αm fm ) on U since we can identify the coordinates (f0 (P ), . . . , fm (P )) with ((fP f0 )(P ), . . . , (fP fm )(P )). Hence Dα := D + (α0 f0 + · · · + αm fm ) is obtained by taking the closure of (ΦL |U )−1 (Hα ) in X. Suppose further that Bs (L) = ∅. Then ΦL : X 99K Pm is defined on X \ (Supp D). If P ∈ Supp D. With the above notations, ((fP f0 )(P ), . . . , (fP fm )(P )) ̸= (0, . . . , 0) because Bs (L) ̸= ∅. So, ΦL is defined also at P . This implies that ΦL : X 99K Pm is a morphism if Bs (L) = ∅. A dominant morphism Φ : X → C from X to a smooth projective curve C is called a pencil. If C is rational, i.e., C ∼ = P1 , then Φ is called a rational pencil. If C is not rational, Φ is an irrational pencil. We show that a rational pencil is a linear pencil. Lemma 1.7.4. Let Φ : X → P1 be a surjective morphism from a normal projective variety X. Let {x0 , x1 } be a system of homogeneous coordinates of P1 . Let x = x1 /x0 and y = x0 /x1 . Then x ∈ k(X). Let P0 (resp. P∞ ) be the point (1, 0) (resp. (0, 1)) of P1 and let F0 := Φ∗ (P0 ) (resp. F∞ := Φ∗ (P∞ )) be the fiber (X, Φ) ×P1 Spec k(P0 ) (resp. (X, Φ) ×P1 Spec k(P∞ )). Then the following assertions hold. (1) If C is an irreducible component of Φ∗ (P0 ) then the local ring OF0 ,C ,

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which is the local ring at the generic point of C, is an artin local ring of length vC (x). Hence Φ∗ (P0 ) is identified with the zero part (x)0 of the function x on X. Similarly, Φ∗ (P∞ ) is identified with the polar part (x)∞ . (2) Let Pα be a point (1, α) of P1 with α ∈ k. Then (x − α) = Φ∗ (Pα ) − Φ∗ (P∞ ). Hence all fibers of Φ constitute members of a linear pencil. Proof. (1) Since X is normal, the local ring OX,C is a DVR with a uniformisant t. Let ξ be the generic point of C. Then OF0 ,ξ = OX,ξ /(x), ∗ where x = tr u with r = vC (x) the value of x at ξ and u ∈ OX,ξ . Then there is a composition series of OX,ξ /(x)-modules, OX,ξ /(x) ⫌ tOX,ξ /(x) ⫌ · · · ⫌ tr−1 OX,ξ /(x) ⫌ (0). Hence ℓ(OX,ξ /(x)) = r. So, we can let OX,ξ /(x) correspond to rC. Hence P Φ∗ (P0 ) corresponds to an effective divisor i vCi (x)Ci , where Ci moves P over all irreducible components of Φ∗ (P0 ). It is clear that i vCi (x)Ci = (x)0 . By the same argument, Φ∗ (P∞ ) corresponds to (x)∞ . Namely, we have (x) = (x)0 − (x)∞ . (2) It is easy to show that (x − α)0 = Φ∗ (Pα ) and (x − α)∞ = (x)∞ . Since Φ∗ (Pα ) = (x − α) + Φ∗ (P∞ ), the morphism Φ is given by a linear system Λ = {Φ∗ (P∞ )+(f ) | f ∈ M }, where M is a k-vector space generated by 1 and x. We say that a linear system L ⊆ L(D) of dimension m is composed of a pencil if ΦL : X 99K Pm is factored by a smooth projective curve C ΦL : X 99K C → Pm and ΦL : X 99K C is dominant. 1.7.4

D-dimension, Kodaira dimension and logarithmic Kodaira dimension

We begin with the following result. Lemma 1.7.5. Let f : X 99K Y be a rational map of algebraic varieties defined over k. Suppose that f is represented by a pair (U, φ) of an open set U ⊂ X and a morphism φ : U → Y . Then there exists an irreducible closed subset F ⊂ Y such that φ(U ) ∩ F contains an open set of F and F is the closure of φ(U ) ∩ F .

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Proof. Replacing U and Y by affine open sets, we may assume that φ is induced by a morphism a h : Spec A → Spec B, where h : B → A is a kalgebra homomorphism. Let B ′ = Im h. Then B ′ = B/Ker h, B ′ is finitely generated over k and B ′ ⊂ A. Let F be the closure in Y of the closed set V (Ker h). Then F is irreducible. Replacing Y by F , we may assume that h is injective, hence φ is dominant. Let K be the field of quotients Q(B), which is the function field k(Y ). We consider only the case where AK := A ⊗B K is an integral domain, i.e., the case where general fibers of f are irreducible and reduced. Since AK is an affine domain over K, by Noether normalization lemma (see Theorem 1.1.17), there exists a polynomial ring SK := K[x1 , . . . , xd ] in AK such that AK is integral over SK , where d = dim AK . We can choose {x1 , . . . , xd } as elements of A. Since A is finitely generated over k, there exists an element s ∈ B such that A[s−1 ] is integral over a polynomial ring B[s−1 ][x1 , . . . , xd ]. Let U ′ = φ−1 (D(s)) = Spec A[s−1 ]. Then φ|U ′ splits as σ

p

φ|U ′ : Spec A[s−1 ] −→ Spec B[s−1 ][x1 , . . . , xd ] = D(s) × Ad −→ D(s),

where σ is a finite surjective morphism by Lying-over theorem (see Theorem 1.1.19) and p is the projection to the factor D(s) = Spec B[s−1 ]. Hence φ(U ′ ) = D(s). So φ(U ) contains an open set D(s) of Y . In a general case, let P1 , . . . , Pm be the minimal prime divisors of AK and let pi = A ∩ Pi . Let Ai = A/pi . Then Ai is an affine domain over k, B ,→ Ai → AK /Pi and (Ai )K = AK /Pi . By the above case, we find an Qm element si ∈ B so that φ(Spec Ai [s−1 i ]) = D(si ) ⊂ Y . Let s = i=1 si ∈ B. Then U ′ := φ−1 (D(s)) is an open set of U and φ maps U ′ surjectively onto D(s). With the notations in the statement, the closed set F is the closure of the image of f , and dim F is the dimension of the image of f . We denote dim F by dim f (X). Let X be a normal projective variety of dimension n defined over a field k and let D be a Cartier divisor on X. We consider complete linear systems |mD| for all m ≥ 1. If |mD| = ̸ ∅ for some m > 0, we set κ(X, D) = max{dim Φ|mD| (X) | |mD| = ̸ ∅, m ≥ 0}.

If |mD| = ∅ for all m > 0, we set κ(X, D) = −∞. We call κ(X, D) the Ddimension of X. The notion was introduced by Iitaka [38]. By definition, κ(X, D) takes values −∞, 0, 1, . . . , dim X. Assume that X is a smooth projective variety. Let KX be its canonical divisor. The KX -dimension κ(X, KX ) is denoted by κ(X) and called

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the Kodaira dimension of X, which gives a basic classification of smooth projective varieties according to the values of κ(X). In order to define logarithmic Kodaira dimension, we have to introduce some necessary definitions. Let D be an effective divisor on X. We call D reduced if all irreducible components of D have coefficient one. Write P D = i Di . For the sake of simplicity, we assume that each irreducible component of D is smooth, hence each Di passes through one and the same point at most once. Let P ∈ Supp D and let Dij (1 ≤ j ≤ m) be all irreducible components passing through P . We say that D has normal crossings if there exists a regular system of parameters {x1 , . . . , xn } at P such that xij = 0 is a local defining equation of Dij at P for 1 ≤ j ≤ m. If this condition is effective for all closed point P ∈ X, we then say that D is a divisor with simple normal crossings. Here the term simple is to signify that each irreducible component Di is smooth. We also call D a simple log divisor and call a pair (X, D) a log pair. We define the sheaf Ω1X/k (log D) of differential 1-forms with simple poles along D (or simply sheaf of logarithmic differential 1-forms along D) by setting the stalk at P as X X dxj Ω1X/k (log D)P = OX,P dxj + OX,P , xj j̸∈{i1 ,...,im }

j∈{i1 ,...,im }

where {x1 , . . . , xn } is a regular system of parameters at P . The nth exterior Vn 1 product ΩX/k (log D) is an invertible sheaf ΩnX/k (log D) whose stalk at P is     ^ ^ dx j , ΩnX/k (log D)P = OX,P  dxj  ∧  xj j̸∈{i1 ,...,im }

j∈{i1 ,...,im }

which is therefore isomorphic to ΩnX/k ⊗ OX (D) = OX (D + KX ). We call D +KX the logarithmic canonical divisor of a pair (X, D) (or simply the log canonical divisor). The (D + KX )-dimension of X is denoted by κ(X \ D) and called the logarithmic Kodaira dimension (or simply the log Kodaira dimension) of X \ D. It is known, in fact, that if X \ D ∼ = X ′ \ D′ for ′ ′ ′ ′ another log pair (X , D ) then κ(X, D + KX ) = κ(X , D + KX ′ ) (see [38] or [59]).

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Affine Algebraic Geometry

Algebraic curves and surfaces

The purpose of this section is to recall several basic results on smooth algebraic curves and surfaces. An algebraic variety X defined over k is a geometrically integral scheme of finite type over k, i.e., X ⊗k k ′ is an irreducible and reduced scheme of finite type over k ′ , where k ′ /k is an algebraic extension. In particular, the function field k(X) is a regular extension over k. If dim X = 1 (resp. dim X = 2) we call X an algebraic curve (resp. algebraic surface). A locally free sheaf F of rank r over an algebraic variety X is a coherent OX -Module such that F|Uλ ∼ = ⊕r OUλ (∀λ ∈ Λ) for an open r covering U = {Uλ }λ∈Λ of X, where ⊕ OUλ is a free OUλ -Module of rank r. In the present section the ground field k is assumed to be algebraically closed. Hence all closed points of an algebraic variety are k-rational points, i.e., the residue field is k. 1.8.1

Serre duality and Euler-Poincar´ e characteristic

The following result is called the Serre duality (see [31, Chapter III, §3] or [57, Theorem 6.29]). Theorem 1.8.1. Let X be a smooth projective variety of dimension n defined over k and let ωX := ΩnX/k be the canonical sheaf. Then, for a locally free sheaf F on X and its dual sheaf F ∨ := Hom(F, OX ), there is an isomorphism H i (X, F) ∼ = H n−i (X, F ∨ ⊗ ωX ), 0 ≤ i ≤ n. In particular, if F = OX (D) for a Cartier divisor, we have an isomorphism H i (X, OX (D)) ∼ = H n−i (X, OX (KX − D)), 0 ≤ i ≤ n.

Let X be a projective algebraic variety defined over k and let F be a coherent OX -Module. By Theorem 1.9.22, dimk H i (X, F) < ∞. So, we set hi (X, F) = dimk H i (X, F). By Theorem 1.9.19, H i (X, F) = 0 if Pn i i i > n := dim X. So, χ(X, F) = i=0 (−1) h (X, F) is defined, which we call the Euler-Poincar´e characteristic of F. Quite often, hi (X, F) or χ(X, F) are written as hi (F) or χ(F) with the mention to the variety X omitted. Let 0 −→ F −→ G −→ H −→ 0 be an exact sequence of coherent OX -Modules. Then we have a long exact sequence in Theorem 1.9.18. By splitting the long exact sequence into short

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exact sequences 0 → H 0 (X, F) 0→ L0 0→ L1 0→ L2 ···

→ → → → ···

H 0 (X, G) → H 0 (X, H) → H 1 (X, F) → H 1 (X, G) → ··· ···

L0 L1 L2 L3

→0 →0 →0 →0

we can show that χ(X, G) = χ(X, F) + χ(X, H). 1.8.2

Riemann-Roch theorem for a curve

P Let D be a divisor on a smooth projective curve C. Write D = i ni Pi P for distinct points Pi . Then we define deg D = i ni .15 If D ≥ 0 then ` D is identified with a scheme i Spec (OC,Pi /tni i ) which we denote also by D, where ti is a generator of mC,Pi . Then D is a finite direct sum of the spectrums of artin local rings OC,Pi /(tni i ). Hence dim D = 0. If D ≥ 0 then we have an exact sequence 0 −→ OC (−D) −→ OC −→ OD −→ 0.

(1.20)

Lemma 1.8.2. Let D, D′ be divisors on a smooth projective curve C. If D ∼ D′ then deg D = deg D′ . Proof. By Theorem 1.7.1, D ∼ D′ if and only if OC (D) ∼ = OC (D′ ). If D = P P ni 0 i ni Pi ≥ 0 then deg D = i ℓ(OC,Pi /(ti )) = h (D, OD ) = χ(D, OD ). The scheme D is viewed as a closed subscheme of C with a closed immersion ι : D → C. Then h0 (D, OD ) = h0 (C, ι∗ OD ). Tensoring OC (D) to the exact sequence (1.20), we have an exact sequence 0 −→ OC −→ OC (D) −→ OD −→ 0,

(1.21)

where OD ⊗OC OC (D) ∼ = OD because OC (D) is locally free. Hence deg D = χ(OC (D)) − χ(OC ). If D ≥ 0 and D′ ≥ 0, we have deg D = deg D′ because OC (D) ∼ = OC (D′ ). P In a general case, write D = D+ − D− , where D+ = ni >0 ni Pi and P P D− = ni 0. Then the following assertions hold. (1) If d > 2g − 2 then dim |D| = d − g. (2) If d > 2g − 1 then φ := Φ|D| : C 99K Pdim |D| is a morphism and each member D′ ∈ |D| is of the form φ∗ (H) for some hyperplane H of Pdim |D| . Hence d = deg(φ)·deg φ(C), where deg(φ) = [k(C) : k(φ(C))] and deg φ(C) is the number of intersection points of the curve φ(C) and a general hyperplane H. (3) If d > 2g then Φ|D| : C → Pdim |D| is a closed immersion. Proof. (1) Since deg(KC − D) = (2g − 2) − d < 0, KC − D is not linearly equivalent to an effective divisor by Lemma 1.8.2. Hence h0 (OC (KC − D)) = 0. By the Riemann-Roch formula, we have h0 (OC (D)) = 1 − g + d. So, dim |D| = d − g. (2) Let P be a closed point of C. Let |D| − P be a linear system {D1 ≥ 0 | D1 + P ∈ |D|}. Then dim(|D| − P ) = d − g − 1 because deg(D − P ) > 2g − 2. This implies that there is a member D′ ∈ |D| such that P ̸∈ Supp D′ . Hence Bs |D| = ∅ and φ := Φ|D| : C 99K Pdim |D| is a morphism such that φ∗ (H) ∈ |D| and any member of |D| is of this form. For the remaining formula on degrees, let φ(C) ∩ H = {Q1 , . . . , Qe } with e = deg(φ(C)).16 If H is a general hyperplane, all points Q1 , . . . , Qe are distinct, and over each of Q1 , . . . , Qe , the inverse image φ−1 (Q) consists of f distinct points for f = [k(C) : k(φ(C)] the extension degree. So, d = f · e. (3) Let P, Q be closed points of C. By the Riemann-Roch formula, we have dim(|D|−(P +Q)) = d−g−2 because deg(D−P −Q) = d−2 > 2g−2. Since dim(|D|−P ) = dim(|D|−Q) = d−g−1, there exist divisors DP , DQ of |D| such that P ∈ DP , Q ̸∈ DP and P ̸∈ DQ , Q ∈ DQ . Since DP = φ∗ (HP ) and DQ = φ∗ (HQ ) for hyperplanes HP , HQ of Pdim |D| , φ(P ) ̸= φ(Q) if P ̸= Q. Hence φ : C → φ(C) is bijective as sets. If we take P = Q, a member DP of |D| is defined near P by g ′ := α0 f0 + · · · + αm fm = 0 if P ̸∈ Supp D (or h := fP (α0 f0 + · · · + αm fm ) = 0 if P ∈ Supp D), 16 Let V be a closed subvariety of dimension m embedded into a projective space Pn . The intersection of V with a linear space L of dimension n − m is a finite set of points if L is general. We define the degree of V as the maximum of the number of points in V ∩ L when L moves in such a way that V ∩ L is a finite set. The degree of V , denoted by deg V , depends on the embedding V ,→ Pn .

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where m = dim |D| and fP = 0 is a defining equation of D at P (see the arguments before Lemma 1.7.4). Note that (α0 x0 +· · ·+αm xm )(φ(P )) = 0. Hence g ′ ∈ mC,P if P ̸∈ Supp D (or h ∈ mC,P if P ∈ Supp D). Since dim(|D| − P ) = dim(|D| − 2P ) + 1, f (or g) moves by one dimension in mC,P /m2C,P . This implies that mφ(C),φ(P ) /m2φ(C),φ(P ) → mC,P /m2C,P is an isomorphism. Hence deg φ = 1 and φ : C → φ(C) is an isomorphism. A Cartier divisor D on a smooth projective variety X is very ample if there exists a closed immersion ι : X → Pn such that OX (D) ∼ = ι∗ OPn (1), where OPn (1) = OPn (H) for a hyperplane H. If nD is very ample for some n > 0 then we say that D is ample. Corollary 1.8.5. Let C be a smooth projective curve defined over k. Then the following assertions hold. (1) A divisor D is ample if deg D > 0. (2) Let F be a finite set of closed points of C. Then C \ F is an affine curve. (3) There is a bijection between the set of closed points of C and the set of discrete valuation ring of the function field k(C). The correspondence is given by P ∈ C 7→ OC,P . Proof. (1) By Lemma 1.8.4, (2g + 1)D is very ample since deg(2g + 1)D = (2g + 1) deg D > 2g. Hence D is ample. (2) Let F = {P1 , . . . , Pd } with d ≥ 1. Identify F with a divisor F = P1 +· · ·+Pd . If nd > 2g then OC (nF ) ∼ = ι∗ OPm (1) for ι = Φ|nF | : C ,→ Pm , where m = dim |nF |. Hence n(P1 + · · · + Pd ) = ι∗ (H0 ) for some hyperplane H0 of Pm . Since C \ F is a closed set of Am = Pm \ H0 , C \ F is an affine curve. (3) Since C is embedded into Pm as a closed set, C is isomorphic to Pm Proj A for a graded ring A over k generated by A1 = i=0 kai . Let (O, m) be a DVR of k(C) and let v be the associated valuation of k(C). Since (A[a−1 0 ])0 ⊂ k(C), we can compare the values v(ai /a0 ) (0 ≤ i ≤ m). Suppose that v(a1 /a0 ) = min1≤i≤n v(ai /a0 ). Then v(aj /a1 ) = v(aj /a0 ) − v(a1 /a0 ) ≥ 0 for j = 0, 2, . . . , m, and O ⊃ A[a−1 1 ]0 . Let P be a point of −1 −1 Spec A[a1 ]0 ⊂ C corresponding to m ∩ (A[a1 ]0 ). Then O ≥ OC,P , which means that O ⊃ OC,P and m ∩ OC,P = mC,P .17 Since OC,P is a DVR as a normal local ring of dimension 1, it is easy to show that O = OC,P . In fact, 17 We

write it as O ≥ OC,P and say that O dominates OC,P .

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if b ∈ O \ OC,P then b−1 ∈ mC,P ⊂ m as OC,P is a DVR, whence 1 ∈ m, a contradiction. Suppose that C has genus g = 0. Take a point P ∈ C. Since deg KC = −2 < 0, we have dim |P | = 1. It is then clear that Bs |P | = ∅. Hence φ := Φ|P | : C → P1 is a morphism such that φ is surjective and 1 = deg |P | = deg φ · deg φ(C). Hence deg φ = 1, which implies that φ is birational. Then φ : C → P1 is an isomorphism by Zariski Main Theorem. An elliptic curve is, by definition, a smooth projective curve defined over k such that the genus g is 1. Let P ∈ E. Then dim |3P | = 2, dim |2P | = 1 and dim |P | = 0. Hence φ := Φ|3P | induces a closed immersion C ,→ P2 . By Lemma 1.8.4(2), we have 3 = deg 3P = deg φ · deg φ(C). If deg φ(C) = 1 then φ(C) is a line, which is defined by α0 x0 + α1 x1 + α2 x2 = 0 with (α0 , α1 , α2 ) ̸= (0, 0, 0). Meanwhile, φ is given as Q ∈ C 7→ (f0 (Q), f1 (Q), f2 (Q)) ∈ P2 with respect to a k-basis {f0 , f1 , f2 } of H 0 (C, OC (3P )). Hence α0 f0 + α1 f1 + α2 f2 = 0. This is a contradiction. Thus deg φ(C) = 3 and deg φ = 1. Namely, C is isomorphic to a hypersurface of degree 3 in P2 by Lemma 1.8.4(3). So, C is identified with a cubic hypersurface of P2 . Theorem 1.8.6. Let C be a smooth projective curve of genus g defined over k. Then the following assertions hold. (1) g = 0 ⇔ κ(C) = −∞. (2) g = 1 ⇔ κ(C) = 0 (3) g > 1 ⇔ κ(C) = 1. Proof. If g = 0 then deg KC = −2. Hence |nKC | = ∅. So, κ(C) = −∞. If g = 1 then deg KC = 0 and |KC | ̸= ∅. This implies that KC ∼ 0. This implies that κ(C) = 0. If g > 1 then deg KC > 0. Hence KC is an ample divisor. This implies that κ(C) = 1. The converse then holds. Let X be a smooth affine curve defined over k. Then X is realized as a closed set of the affine space An . Write An = Pn \ H0 , where H0 is the hyperplane at infinity. Let X be the closure of X in Pn . Then X may not be smooth on X \ X. Let C be the normalization of X (see Theorem 1.5.15). Then C is smooth because C is a normal curve, and every DVR of the function field k(X) is given as OC,P for P ∈ C. In fact, X = Proj A for a graded affine domain defined over k such that A Pn is generated by A1 = i=0 kai . Let O be a DVR of k(X). Then, as in the proof of Corollary 1.8.5(3), O dominates a local ring OX,Q . By the

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construction of the normalization C of X, O dominates a local ring OC,P , where OC,P is a DVR. Hence O = OC,P . Actually we can show that C is projective, but the detail of proof is omitted. Hence X is realized as an open set C \ F , where F is a finite set of closed points. Under this preparation, we can state the following result. Theorem 1.8.7. Let X = C \ F be a smooth affine curve defined over k, where F ̸= ∅. Then the following assertions hold. (1) κ(X) = −∞ if and only if X ∼ = A1 . ∼ (2) κ(X) = 0 if and only if X = A1∗ , where A1∗ = A1 \ {one point}. (3) Otherwise κ(X) = 1. Pd Proof. (1) Let F = {P1 , . . . , Pd } with d > 0 and let D = i=1 Pi . Then deg(D + KC ) = d + 2g − 2. If deg(D + KC ) > 0 then κ(X) = 1 by the Riemann-Roch formula. Hence, if κ(X) = −∞ then d + 2g − 2 ≤ 0, which implies that 2g − 2 ≤ −d < 0. Then g = 0 and d = 1 or 2. If g = 0 and d = 2 then D+KC ∼ 0 because any two points of P1 are linearly equivalent. So, κ(X) = 0. If g = 0 and d = 1 then X ∼ = P1 \ {one point} ∼ = A1 . (2) By the above observation, κ(X) = 0 if and only if g = 0 and d = 2. Hence X ∼ = A1∗ . (3) Clear. 1.8.4

Intersection theory on algebraic surfaces

Let X be a smooth projective surface defined over a field k which is assumed to be algebraically closed. Then there is no distinction between Weil and Cartier divisors, and the correspondence D 7→ OX (D) gives a bijection between Cℓ (X) and Pic (X), i.e., the linear equivalence class of a divisor corresponds to the isomorphism class of the invertible sheaf OX (D). Given two irreducible curves C1 , C2 on X, if C1 ̸= C2 , then C1 and C2 meet in finitely many points P1 , . . . , Pr . We evaluate precisely the local intersection multiplicity i(C1 , C2 ; Pj ) for 1 ≤ j ≤ r as dimk OX,P /(f1 , f2 ), where fi is a local defining equation of Ci (i = 1, 2) at P = Pj and the interPr section number (C1 · C2 ) as the sum j=1 i(C1 , C2 ; Pj ), and extend the intersection multiplicity to the case of divisors D1 , D2 and also the case of self-intersection (D · D). The theory will have various applications to the surface theory. We will make the description as concise as possible. So, if it is possible to refer to published books of the author, we omit the proof.

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Let L1 , L2 be invertible sheaves on X. We define the intersection number (L1 · L2 ) by −1 −1 −1 (L1 · L2 ) = χ(OX ) − χ(L−1 1 ) − χ(L2 ) + χ(L1 ⊗ L2 ).

The following properties of the intersection number is proved in [57, Chapter 8, Theorem 8.2]. Lemma 1.8.8. The following assertions hold on the intersection number. (1) The intersection number is a Z-valued bilinear form on Pic X. Namely, it satisfies the properties. (i) (ii) (iii) (iv) (v)

(L1 · L2 ) depends only on the isomorphism classes of L1 and L2 . (L1 · L2 ) = (L2 · L1 ). (L1 ⊗ L′1 · L2 ) = (L1 · L2 ) + (L′1 · L2 ). (L−1 1 · L2 ) = −(L1 · L2 ). (OX · L2 ) = 0.

(2) Suppose that H 0 (X, Li ) ̸= 0 for i = 1, 2. Write Li = OX (Di ) for an effective divisor Di for i = 1, 2. Suppose further that D1 and D2 have no common irreducible components. Then we have X (L1 · L2 ) = dimk OX,x /(fx , gx ), x∈X

where x runs over the closed points of X and fx , gx are local equations of D1 , D2 at x. (3) If D1 ∼ D1′ and D2 ∼ D2′ then (D1 · D2 ) = (D1′ · D2′ ). Proof. (1) Proof is omitted. (2) Under the given conditions we have an exact sequence 0 → OX (−D1 − D2 ) → OX (−D1 ) ⊕ OX (−D2 ) → OX → OD1 ⊗ OD2 → 0. In fact, the exactness is verified from the exactness of the sequence of stalks α

β

γ

0 → f gR −→ f R ⊕ gR −→ R −→ R/(f, g) → 0, where R = OX,x , f = fx , g = gx and α(f gu) = (f gu) ⊕ (−f gu), β(f v ⊕ gw) = f v + gw, γ(z) = z + (f, g)R (see Problem 15). We note that R is a UFD by Theorem of Auslander-Buchsbaum-Nagata (Theorem 1.6.11). Taking the Euler-Poincar´e characteristics, we have χ(OD1 ⊗ OD2 )

= χ(OX ) − χ(OX (−D1 )) − χ(OX (−D2 )) + χ(OX (−D1 − D2 )) = (OX (D1 ) · OX (D2 )).

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Meanwhile, since Supp (D1 ) ∩ Supp (D2 ) is a finite set of closed points, we have X χ(OD1 ⊗ OD2 ) = h0 (OD1 ⊗ OD2 ) = ℓ(OX,x /(fx , gx )). x∈X

(3) Since OX (Di ) ∼ = OX (Di′ ) for i = 1, 2, the assertion is clear by the definition. We abbreviate (D · D) as (D2 ) and call it the self-intersection number of D. Let X be a smooth projective surface and let P be a closed point. We construct a smooth projective surface X ′ with a birational morphism ∼ σ : X ′ → X so that σ induces an isomorphism σ −1 (X \ {P }) −→ X \ {P } and σ −1 (P ) ∼ = P1 . We take a coordinate neighborhood U of P and a system of coordinates {u, v} such that u(P ) = v(P ) = 0. Let ΓU be a closed subset of U × P1 such that ΓU = {(Q, (α0 , α1 )) | Q ∈ U, (α0 , α1 ) ∈ P1 , α0 v(Q) = α1 u(Q)}.

∼ P1 and σ induces an Let σ : ΓU → U be the first projection. Then σ −1 (P ) = isomorphism between ΓU \σ −1 (P ) and U \{P } because (u(Q), v(Q)) ̸= (0, 0) if Q ̸= P , whence (α0 , α1 ) = (u(Q), v(Q)). We can patch X \ {P } and ΓU along the open sets U \ {P } and ΓU \ σ −1 (P ) by this isomorphism. The obtained surface is X ′ and the birational morphism σ is extended canonically to a birational morphism σ : X ′ → X. We say that X ′ is obtained by the blowing-up of a point P . We denote X ′ by BlP X. The point P is called the center of the blowing-up σ : X ′ → X. The inverse image E := σ −1 (P ), which is isomorphic to P1 , is called the exceptional curve. A point P1 of the exceptional curve E is called an infinitely near point of order 1 of the point P0 := P . If we blow up the point P1 of E1 ⊂ X1 := X ′ , we obtain the exceptional curve E2 ⊂ X2 = BlPi−1 Xi−1 . A point P2 ∈ E2 is an infinitely near point of P of order 2 (and an infinitely near point of P1 of order 1). Iterating this construction, we can define an infinitely near point Pn of order n of P0 , which is also considered as a sequence (Pn , Pn−1 , . . . .P1 , P0 ), where Pi ∈ Ei ⊂ Xi := BlPi−1 Xi−1 for 1 ≤ i ≤ n and Ei is the exceptional curve. We call the point P0 an ordinary point of X. Suppose that U = Spec A, where A is an affine k-domain. Let I = (u, v) L be the ideal which defines the closed point P . Let S = i≥0 I i be a graded ring over A, where I 0 = A. We show that ΓU = Proj S. In fact, Proj S = D+ (u)∪D+ (v), where D+ (u) = Spec A[v/u] and D+ (v) = Spec A[u/v]. Let t0 , t1 be variables and let φ : A[t0 , t1 ] → S be a surjective homomorphism

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of graded rings over A defined by φ(t0 ) = u and φ(t1 ) = v. Then Proj S is identified with a closed subscheme of Proj A[t0 , t1 ] = U × P1 defined by the ideal Ker φ = (vt0 − ut1 ), which is therefore isomorphic to ΓU . It can be shown that BlP X is a smooth projective surface over k. Let C be an irreducible curve on X through the point P . By shrinking U if necessary, we may assume that C is defined by f = 0 in the open set U , b be the m-adic completion of OX,P (see the appendix), where f ∈ A. Let O b = k[[u, v]], and f is written as where m = mX,P . Then O f = fµ + fµ+1 + · · · , fi ∈ k[[u, v]]i , i ≥ µ, fµ ̸= 0, b of degree i. Since f (P ) = 0, where k[[u, v]]i is the homogeneous part of O we have µ > 0. We call µ the multiplicity of C at P and denote it by µ = µ(C, P ). It is clear that C is smooth at P if and only if µ(C, P ) = 1. Let σ : X ′ → X be the blowing-up at P . We define the curve σ ∗ (C) by patching the curve C \ {P } and the curve defined by σ ∗ (f ) = 0 along the curve cut on the open set σ −1 (U \ {P }). Let x = v/u and y = u/v. Then σ ∗ (f ) is given as σ ∗ (f ) = uµ fµ (1, x) + uµ+1 fµ+1 (1, x) + · · ·

= v µ fµ (y, 1) + v µ+1 fµ+1 (y, 1) + · · · .

The exceptional curve E = σ −1 (P ) is defined by u = 0 on D+ (u) = Spec A[x] because σ ∗ (u, v) = (u, ux) = (u) in A[x] and by v = 0 on D+ (v) because σ ∗ (u, v) = (vy, v) = (v) in A[y]. The equation σ ∗ (f )/uµ = fµ (1, x) + ufµ+1 (1, x) + · · · defines the closure of the curve σ −1 (C \ {P }) in X ′ ∩ D+ (u). Similarly, the closure of the curve σ −1 (C \ {P }) in X ′ ∩ D+ (v) is defined by σ ∗ (f )/v µ = fµ (y, 1) + vfµ+1 (y, 1) + · · · . These two curves are, in fact, the same if restricted on D+ (uv), and hence defines an irreducible curve, which we denote by σ ′ (C) and call it the proper transform (or strict transform) of C on X ′ . We call σ ∗ (C) the total transform of C on X ′ . As explained above, we can identify σ ∗ (C) with a divisor σ ′ (C) + µE. The intersection of σ ′ (C) and E is given by a homogeneous Qr equation fµ (u, v) = 0. Note that fµ (u, v) = uµ i=1 (x − αi )mi , where α1 , . . . , αr are distinct roots of fµ (1, x) = 0 and the point (u, v) = (0, 1) apPr pears with multiplicity m∞ = µ − ( i=1 mi ). Here the point (u, v) = (0, 1) corresponds to a root y = 0 of fµ (y, 1) = 0. Let Pi′ be the point of ′ X ′ defined by u = x − αi = 0 and P∞ the point v = y = 0. Then

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′ E ∩ σ ′ (C) = {P1′ , . . . , Pr′ , P∞ } and mi = i(E, σ ′ (C); Pi′ ) for 1 ≤ i ≤ r and i = ∞. Hence (E · σ ′ (C)) = µ. If an irreducible curve C does not pass through the point P then P σ −1 (C) ∼ = σ ′ (C) with C. If D = i ni Ci is = C. So, we identify σ ∗ (C) P P a divisor on X, we define σ ∗ (D) = i ni σ ∗ (Ci ) and σ ′ (D) = i ni σ ′ (Ci ). We call σ ∗ (D) and σ ′ (D) the total transform and the proper transform of D, respectively.

Lemma 1.8.9. Let σ : X ′ → X be the blowing-up of a point P and let E be the exceptional curve. Then the following assertions hold. (1) Let D be a divisor on X. Then there exists a divisor D′ such that D ∼ D′ and Supp D′ ̸∋ P . (2) Let D1 , D2 be divisors on X. Then (σ ∗ D1 · σ ∗ D2 ) = (D1 · D2 ). (3) (σ ∗ D · E) = 0 for any divisor D on X. (4) (E 2 ) = −1. (5) Let Ci (i = 1, 2) be irreducible curves on X and let µi = µ(Ci , P ). Then (C1′ · C2′ ) = (C1 · C2 ) − µ1 µ2 , where Ci′ = σ ′ (Ci ). (6) Let KX ′ be the canonical divisor of X ′ . Then KX ′ ∼ σ ∗ KX + E, hence (KX ′ · E) = −1. Proof. (1) Since X is a projective surface, X is a closed subvariety of Pn . Let H be a hyperplane of Pn such that P ̸∈ H. Then X \ (X ∩ H) is a closed set of An = Pn \ H. So, X \ (X ∩ H) is an affine variety Spec R. We consider the case where D is an irreducible curve C passing through P . Then C is defined by f = 0 near the point P , where f is an irreducible element of OX,P . Since OX,P = Rm for a maximal ideal m of R, one can write f = a/s with a, s ∈ R and s ̸∈ m. Then C is the reduced irreducible component of the divisor (a) passing through P . Consider the divisor D′ = (C)+(f −1 ) = (s)−((a)−C). If D′ is considered as a divisor on X, D′ may contain the components in X ∩ H as zeroes or poles, but those P components lie off P . Hence D′ ∼ C and P ̸∈ Supp D′ . If D = i ni Ci is an arbitrary divisor, apply the above construction to each irreducible component Ci passing through P . Then we find a required divisor D′ . (2) By (1), write Di ∼ Di′ for i = 1, 2, where P ̸∈ Supp (Di′ ). Then ∗ σ (Di ) ∼ σ ∗ (Di′ ). Since Supp (σ ∗ Di′ ) ∩ E = ∅, it is clear that (σ ∗ D1 · σ ∗ D2 ) = (σ ∗ D1′ · σ ∗ D2′ ) = (D1′ · D2′ ) = (D1 · D2 ). (3) With D′ in (1), we have (σ ∗ D · E) = (σ ∗ D′ · E) = 0 because Supp (σ ∗ D′ ) ∩ E = ∅.

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(4) Let C be an irreducible curve on X with µ(C, P ) = µ > 0. We have σ C = σ ′ C + µE and (σ ′ C · E) = µ. Since (σ ∗ C · E) = 0, we deduce that (E 2 ) = −1. (5) The intersection number is a bilinear form on Pic X. Then the result follows from the foregoing results. (6) With the above notation, let ω = du∧dv, which is a nonzero element of ωK/k , where K = k(X). Let Pe0 be the point on E = σ −1 (P ) which is given as Pe0 = (1, 0) with respect to the homogeneous coordinates (t0 , t1 ) on E. Let x = v/u. Then v = ux and ω = udu ∧ dx. Namely we have u−1 ω = du ∧ dx. Since σ ∗ KX is represented by (ω)X ′ = σ ∗ ((ω)X ) and OX ′ (E) = u−1 OX on an open neighborhood of Pe0 , it follows that (ω)X ′ + E = (du ∧ dx)X ′ ∼ KX ′ . This gives KX ′ = σ ∗ (KX ) + E. Since (σ ∗ (KX ) · E) = 0, we have (KX ′ · E) = −1. ∗

Let Λ be a linear pencil on X (see subsection 1.7.3). It is a linear subsystem of dimension 1 in |D|, which is parametrized by a k-module M = k · 1 + k · f , where we assume that D0 := D and D∞ := D + (f ) are members of Λ. We say that Λ is generated by D0 and D∞ . For an element α0 + α1 f ∈ M , the member D + (α0 + α1 f ) is sometimes denoted by D(α0 ,α1 ) , which corresponds to a point (α0 , α1 ) ∈ P1 . We assume that Λ has no fixed components, but Bs Λ might not be the empty set, which is given as the intersection points D0 ∩ D∞ . Since X (D2 ) = (D0 · D∞ ) = i(D0 , D∞ ; P ), P ∈Bs Λ

2

Bs Λ has at most (D ) points. Let P ∈ Bs Λ and let f0 , f∞ be local defining equations of D0 , D∞ , respectively. Then f∞ is concomitant with f0 f in OX,P . Hence we have i(D0 , D∞ ; P ) = ℓ(OX,P /(f0 , f∞ )) = ℓ(OX,x /(f0 , f0 (α0 + α1 f ))) = i(D0 , D(α0 ,α1 ) ; P ). Similarly, we have i(D0 , D∞ ; P ) = i(D1 , D2 ; P ) for any two distinct members D1 , D2 of Λ. Let µ0 = µ(D0 , P ) and µ∞ = µ(D∞ , P ) be the multiplicities at P , which P P is defined by µ(D, P ) = i ni µ(Ci , P ) if D = i ni Ci is the irreducible decomposition. Let σ : X ′ → X be the blowing-up of P . Then (σ ′ D0 · σ ′ D∞ ) = (D0 · D∞ ) − µ0 µ∞ . Since µ0 > 0 and µ∞ > 0 if P ∈ Bs Λ, we find µP = min(α0 ,α1 )∈P1 µ(D(α0 ,α1 ) , P ). Set σ ′ Λ = {σ ∗ D(α0 ,α1 ) − µP E | (α0 , α1 ) ∈ P1 }.

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We call σ ′ Λ the proper transform or strict transform of Λ by σ. Note that σ ′ Λ has no fixed components, and the self-intersection number of a general member of σ ′ Λ is smaller than the one for Λ. Hence we have the following result. Proposition 1.8.10. Let Λ be a linear pencil on X without fixed components. Then there exists a birational morphism τ : V → X such that (i) τ is a succession of blowing-ups, (ii) the proper transform τ ′ Λ has no base points. Hence Φτ ′ Λ : V → P1 is a surjective morphism. Let C be an irreducible curve on X which is not necessarily smooth. We define the arithmetic genus pa (C) of C by the following formula (see Problem 16) 1 pa (C) = (C · C + KX ) + 1. 2 For an effective divisor D, we can define pa (D) by this formula. We use later this generalization. A point P ∈ C is a singular point if the multiplicity µ(C, P ) > 1. If C ′ is the proper transform of C by the blowing-up at P , C ′ may have singular points on the exceptional curve E. Such points are called infinitely near singular points of C. There might appear singular points of C as the intersection points of the proper transforms of C and the exceptional curves of higher order. Those singular points are also called infinitely near singular points of C. Lemma 1.8.11. Let σ : X ′ → X be the blowing-up of a closed point P and let E be the exceptional curve. For an irreducible curve C on X, we have the following assertions. (1) (2) (3) (4) (5)

Let C ′ = σ ′ C. Then pa (C ′ ) = pa (C) − 12 µ(µ − 1), where µ = µ(C, P ). χ(OC ′ ) = χ(OC ) + 12 µ(µ − 1). pa (C) = h1 (C, OC ). χ(OX (C)) = 21 (C · C − KX ) + χ(OX ). e be the normalization of C and let g be the genus of C. e Then we Let C have X1 µP (µP − 1), g = pa (C) − 2 P

where P runs over all singular points of C including infinitely near singular points and µP is the multiplicity of C at P .

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Proof. (1) Since KX ′ = σ ∗ (KX ) + E and C ′ = σ ∗ (C) − µE, the arithmetic genus formula for pa (C ′ ) yields the result. (2) The exact sequence 0 −→ OX (−C) −→ OX −→ OC −→ 0 gives an equality χ(OC ) = χ(OX ) − χ(OX (−C)). Similarly, we have χ(OC ′ ) = χ(OX ′ ) − χ(OX ′ (−C ′ )). On the other hand, we have an exact sequence 0 → OX ′ (−E) → OX ′ → OE → 0.

Taking tensor products of the terms of the exact sequence with OX ′ (−C ′ − (i − 1)E), we have an exact sequence for 1 ≤ i ≤ µ 0 → OX ′ (−C ′ − iE) → OX ′ (−C ′ − (i − 1)E) → OE (−µ + i − 1) → 0,

which yields χ(OX ′ (−C ′ − iE)) = χ(OX ′ (−C ′ − (i − 1)E)) − χ(OE (−µ + i − 1)) = χ(OX ′ (−C ′ − (i − 1)E)) + (µ − i),

where χ(OE (−µ + i − 1)) = −µ + i by the Riemann-Roch formula for E∼ = P1 . Hence we have χ(OX ′ (−C ′ − µE)) = χ(OX ′ (−C ′ − (µ − 1)E)) χ(OX ′ (−C ′ − (µ − 1)E)) = χ(OX ′ (−C ′ − (µ − 2)E)) ······ ········· χ(OX ′ (−C ′ − 2E)) = χ(OX ′ (−C ′ − E)) χ(OX ′ (−C ′ − E)) = χ(OX ′ (−C ′ ))

Adding up both sides we obtain an equality

+ +

0 1 ··· + µ−2 + µ − 1.

1 χ(OX ′ (−σ ∗ (C))) = χ(OX ′ (−C ′ )) + µ(µ − 1). 2 We can show that χ(OX ′ (−σ ∗ (C)) = χ(OX (−C)) and χ(OX ′ ) = χ(OX ) (see [57, pp. 193–194]). Then we have χ(OC ′ ) = χ(OX ′ ) − χ(OX ′ (−C ′ ))

1 = χ(OX ) − χ(OX ′ (−σ ∗ (C))) + µ(µ − 1) 2 1 = χ(OX ) − χ(OX (−C)) + µ(µ − 1) 2 1 = χ(OC ) + µ(µ − 1). 2

e → C be the normalization morphism which is birational. (3) Let ν : C Let Pe be a point of ν −1 (P ). Then O e e is a DVR of k(C). Let w be the C,P

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associated valuation. We argue with the notations before Lemma 1.8.9, With coordinates u, v of P viewed as functions on C, the values w(u), w(v) are evaluated. Suppose that w(v) ≥ w(u). Then w(v/u − α) > 0 for some α ∈ k. Hence OC, eP e dominates the local ring OC ′ ,Pα of a point Pα = (1, α) on E ∩ C ′ . If w(u) > w(v) then OC, eP e dominates OC ′ ,P∞ . Hence the morphism splits as ν1 ν2 e −→ C. C ′ −→ ν:C

This argument shows that the normalization of C is obtained by blowing up the surfaces at singular points of C and its proper transforms. This is called the embedded resolution of singularity of C. By the assertions (1) and (2), we have e χ(OC ) + pa (C) = χ(OC ′ ) + pa (C ′ ) = · · · = χ(O e ) + pa (C), C

e = h0 (O e ) − h1 (O e ) + pa (C) e = 1 because pa (C) e = where χ(OCe ) + pa (C) C C 1 0 1 h (OCe ) = g (see Problem 16). Since h (OC ) = 1 we have h (OC ) = pa (C). (4) The proof of Theorem 1.8.3 shows that if C is an irreducible curve and ∆ is a Cartier divisor on C then we have χ(OC (∆)) = χ(OC ) + deg ∆.

Apply this formula to the case ∆ = OX (C) ⊗OX OC = OC (C). Then we have χ(OC (C)) = χ(OC ) + deg OC (C),

where deg OC (C) = (C 2 ). On the other hand, we have an exact sequence which gives

0 → OX → OX (C) → OC (C) → 0,

χ(OX (C)) = χ(OX ) + χ(OC (C)) = χ(OX ) + χ(OC ) + (C 2 ) = χ(OX ) + 1 − pa (C) + (C 2 ) 1 = χ(OX ) + 1 − (C · C + KX ) − 1 + (C 2 ) 2 1 = χ(OX ) + (C · C − KX ). 2 (5) Let P be a singular point of C and let C ′ = σ ′ C. Then we have e → C is factorized by C ′ , and shown that the normalization morphism ν : C 1 ′ pa (C ) = pa (C) − 2 µP (µP − 1). We apply the same argument to a singular point P ′ of C ′ if such P ′ exists. Then ν is factorized as e → C ′′ → C ′ → C, ν:C

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where C ′′ is the proper transform by the blowing-up of X ′ at P ′ . Since 2 2 2 (C ′ ) = (C 2 ) − µ2P and (C ′′ ) = (C ′ ) − µ2P ′ , etc., the morphism ν is factorized by only finitely many proper transforms of C e → C (n) → C (n−1) → · · · → C ′′ → C ′ → C ν:C e → C (n) is obtained as the proper transform of and the final morphism C (n) C . This observation gives the stated formula. 1.8.5

Riemann-Roch theorem for surfaces

Theorem 1.8.12. Let X be a smooth projective surface defined over an algebraically closed field k and let D be a divisor on X. Then we have the following formula χ(OX (D)) =

1 (D · D − KX ) + χ(OV ). 2

Proof. The assertion (4) of Lemma 1.8.11 gives the case D is irreducible. P P Write D = i ni Ci . The case D ≥ 0 can be proved by induction on i ni . P In the case D is general, write D = A−B with A ≥ 0 and B = j bj Cj ≥ 0. P A proof is given by induction on j bj . The readers are referred to [57, Theorem 10.6] for details. Since X is projective, it is a closed subvariety of Pn for some n. Let H be a hyperplane of Pn and OPn (H) ∼ = O(1). Let OX (A) = O(1) ⊗ OX . Then a Cartier divisor A is very ample (see the definition before Corollary 1.8.5). For general two hyperplanes H1 , H2 , let O(Hi ) ⊗ OX = OX (Ai ) for i = 1, 2. Since X, H1 , H2 meet in finitely many points transversally, i.e., for an intersection point P , the defining elements ai of Ai in OX,P give a regular system of parameters (a1 , a2 ), the number of intersection points is equal to (A1 · A2 ) = (A2 ) > 0, which is the degree of X in Pn . For any irreducible curve C on X, we have (C · A) > 0. If B is an ample divisor the inequalities (B 2 ) > 0 and (B · C) > 0 hold because mB is a very ample divisor for some m > 0, i.e., Φ|mB| gives a closed immersion X → Pn for some n. The following result shows that the above inequalities are necessary and sufficient conditions for a divisor to be ample. Theorem 1.8.13 (Nakai criterion of ampleness). Let D be a divisor on a smooth projective surface X. Then D is ample if and only if

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(i) (D · C) > 0 for all irreducible curve C on X. (ii) (D2 ) > 0. A divisor D on X is numerically equivalent to 0 if (D · C) = 0 for any irreducible curve C. We denote it by D ≡ 0. For divisors D, D′ , D ≡ D′ if D − D′ ≡ 0. It is clear that if D ∼ D′ then D ≡ D′ . Let Div 0 (X) = {D ∈ Div (X) | D ≡ 0}. Then Div 0 (X) is a subgroup of Div (X). We set NS (X) = Div (X)/Div 0 (X) and call it the N´eron-Severi group of X. It is known that NS (X) ⊗ Q is a vector space over Q of finite rank. Its rank is denoted by ρ(X) and called the Picard number of X. The following result is called the Hodge index theorem (see [57, Theorem 10.9]). Theorem 1.8.14. Let A be a divisor on X such that (A2 ) > 0. If (A·D) = 0 for a divisor D then (D2 ) < 0 or D ≡ 0. 1.8.6

Fibrations and relatively minimal models of surfaces

There are still too many results to be stated in the surface case which we need to develop research on affine algebraic surfaces. But the part on introduction to algebraic geometry is already too heavy, and necessary results are written in [57, Chapter 9] and [59, Chapter 1]. So we leave the readers to these references for the remaining results, and state only basic and frequently used results. As in the previous subsections, X is a smooth projective surface defined over an algebraically closed field k. In subsection 1.7.3, a pencil is a dominant morphism Φ : X → C from X to a smooth projective curve. We extend the definition so that a pencil of curves is a dominant rational mapping Φ : X 99K B, where B is a smooth projective curve. We used the letter C for the base curve, but we need the letter C for many other purposes. So, we use B to mean a base curve. Definitions like rational (or irrational) pencil are extended to the case of rational mappings. Let ρ : X 99K B be a pencil of curves. Then ρ is given by a morphism ρU : U → B, where U is the domain of definition of ρ. Let Z be the closure of the set {(u, ρU (u)) | u ∈ U } in X × B. For a point b ∈ B, we set Xb := (X × {b}) ∩ Z which is identified with a subset of X and called the fiber of ρ over the point b ∈ B. A pencil ρ : X 99K B is irreducible if the field extension k(X)/k(B) is a regular extension. Hence if ρ is a morphism the generic fiber Xη := X ×B Spec k(B) is geometrically integral. If ρ is a morphism and irreducible, we say that ρ is a fibration.

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Lemma 1.8.15. (I) Let ρ : X 99K B be a pencil of curves. Then the following assertions hold (see Lemma 1.7.4 and [57, Lemma 9.5]). (1) If ρ is irrational then ρ is a morphism. (2) If ρ is rational then Λ := {Xb | b ∈ B} is a linear pencil on X without fixed components. (II) Suppose that the characteristic of k is zero and ρ is not irreducible. Let e be the algebraic closure of k(B) in k(X). Then K e is the function K e and ρ is factored as field of a smooth projective curve B ρ e ν e −→ ρ : X 99K B B,

e is an irreducible pencil and ν : B e → B is a where ρe : X 99K B finite morphism. The factorization is called the Stein factorization (see [57, p. 178]). (III) Suppose that ρ : X 99K B is a morphism. If ρ is irreducible then ρ∗ OX = OB and every fiber is connected. (See [57, Lemma 9.3].) In the following result, the first assertion (resp. the second assertion) is called the first theorem of Bertini (resp. the second theorem of Bertini). See [57, Theorems 9.6 and 9.11]. Theorem 1.8.16. Let X be a smooth projective surface defined over an algebraically closed field of characteristic zero and let L be a linear system on X such that dim L > 0. Then the following assertions hold. (1) Suppose that L has no fixed components and is not composed of a pencil of curves. Then a general member of L is irreducible and reduced.18 (2) Suppose that Bs L is a finite set. Then a general member of L is smooth outside the set Bs L. In the case of a fibration we have the following result (see [57, Lemma 9.10]). Lemma 1.8.17. Suppose that k has characteristic zero. Let ρ : X → B be a fibration of curves. Then the following assertions hold. (1) The generic fiber Xη of ρ is geometrically normal. Namely, Xη ⊗k(B) k(B) is a smooth projective curve, where k(B) is an algebraic closure of k(B). 18 The members of L are parametrized by a projective space Pm . If some property holds for all members of L corresponding to points of an open set U ̸= ∅ of Pm , we say that a general member of L satisfies the property.

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(2) There exists an open set U ̸= ∅ such that Xb is a smooth projective curve over k for all b ∈ U . A smooth rational curve C is called a (−n)-curve if (C 2 ) = −n < 0. Such a curve does not move in the sense that dim |C| = 0. In fact, if C ∼ D with C ̸= D then C ̸⊂ Supp D, and hence (C · D) ≥ 0 while (C · D) = (C 2 ) < 0, a contradiction. The exceptional curve of a blowing-up is a (−1)-curve. The converse also holds. Theorem 1.8.18 (Contractability criterion of Castelnuovo). Let X ′ be a smooth projective surface and let C be a (−1)-curve on X ′ . Then there exists a birational morphism τ : X ′ → X such that X is a smooth projective surface, τ (C) = Q is a point and τ is the blowing-up of the point Q with τ −1 (Q) = C. The above morphism τ is called the contraction or blowing-down of C. Let D1 , D2 be divisors on X. We say that D1 is algebraically equivalent to D2 (D1 ≈ D2 by notation) if there exist a smooth irreducible curve B, points b1 , b2 ∈ B and an invertible sheaf L on X × B such that OX (D1 ) ∼ = L|X×{b1 } and OX (D2 ) ∼ L| . Then (D ·D) = (D ·D) for any divisor = X×{b2 } 1 2 D on X (see [57, Lemma 9.13]). If ρ : X → B is a fibration, then Γρ := {(x, ρ(x)) | x ∈ X} is an irreducible subvariety of codimension one, hence OX×B (Γρ ) is an invertible sheaf such that OX (ρ−1 (b)) ∼ = OX×B (Γρ )|X×{b} . Hence two fibers ρ−1 (b1 ) and ρ−1 (b2 ) are algebraically equivalent on X. Lemma 1.8.19. Let ρ : X → B be a fibration. Then the following assertions hold. (1) Every fiber F of ρ has a constant arithmetic genus. P (2) If F is reducible, F is written as a connected divisor F = i ni C i , where Ci is an irreducible curve and ni = ℓ(OF,Ci ). Here F is identified with a scheme Xb := X ×B Spec k(b), where b = ρ(F ). P (3) (F 2 ) = 0. If F = i ni Ci is reducible, then (F · Ci ) = 0 and (Ci2 ) < 0 for every i. Proof. (1) Let F be a fiber of ρ. Then, by the arithmetic genus formula (see the definition after Proposition 1.8.10), we have pa (F ) = 12 (F ·F + KX )+1, which is invariant under algebraic equivalence. (2) By Lemma 1.8.15, F is connected. Hence if F is reducible, every irreducible component of F is a curve. The local ring OXb ,C of F at the generic point of Ci is an artin local ring, and its length is equal to the multiplicity ni .

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(3) Since (F 2 ) = (F · F ′ ) for a different fiber F ′ , we have (F 2 ) = 0. P Similarly, (F ·Ci ) = (F ′ ·Ci ) = 0. Since (F ·Ci ) = ni (Ci2 )+ j̸=i nj (Cj ·Ci ) P and j̸=i nj (Cj · Ci ) > 0 because F is connected, it follows that (Ci2 ) < 0 for all i. Let σ : X ′ → X be the blowing-up of a point P and let E = σ −1 (P ). Since σ ∗ (D) = (f )X ′ for f ∈ k(X ′ ) = k(X) implies D = (f )X , it follows that σ ∗ : Pic X → Pic X ′ is injective, and Pic X ′ = σ ∗ (Pic X)⊕Z[E], where [E] is the linear equivalence class of E. Then E is numerically independent of σ ∗ (Pic X). Hence ρ(X ′ ) = ρ(X) + 1. This implies that if C0 is a (−1)-curve and τ : X → X1 is the contraction of C0 then ρ(X1 ) = ρ(X) − 1. If there exists a (−1)-curve on X1 then its contraction τ1 : X1 → X2 makes ρ(X2 ) = ρ(X1 ) − 1. Since the Picard number ρ(Z) is positive for a smooth projective surface Z, the above contractions cannot continue endlessly. Hence, by contracting (−1)-curves, we find a birational morphism φ : X → Y such that Y is a smooth projective surface and Y has no (−1)-curves. Such a surface Y is called a relatively minimal surface. It should be noted that the contractions starting from the same X may lead to non-isomorphic relatively minimal models. Lemma 1.8.20. Let ρ : X → B be a fibration such that a general fiber Xb has genus 0. Then the following assertions hold. (1) The generic fiber Xη is isomorphic to P1k(η) , where k(η) = k(B). Hence general fibers are isomorphic to P1k . We call the fibration ρ : X → B a P1 -fibration.19 Furthermore, there is a curve S on X such that ρ|S : S → B is an isomorphism. We call S a section (or cross-section) of ρ. A section S satisfies (S · F ) = 1 for a fiber F of ρ. P (2) Let F = i ni Ci be a reducible fiber. Then the following assertions hold. (i) Every Ci is isomorphic to P1 , (Ci2 ) < 0 and (Ci · Cj ) = 0 or 1 if i ̸= j. Further Ci ∩ Cj ∩ Cℓ = ∅ if i, j, ℓ are distinct. (ii) One of the components, say Ci , is a (−1)-curve. Let τ : X → X1 be ρ1 τ the contraction of Ci . Then ρ splits as ρ : X −→ X1 −→ B, where ρ1 is a P1 -fibration. (iii) If ni = 1 for a (−1)-curve Ci then there exists another component Cj which is a (−1)-curve. 19 A

fibration ρ : X → B is called an F -fibration if general fibers are isomorphic to an algebraic variety F .

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(3) X is obtained by a successive blowing-ups from a relatively minimal surface Y with a P1 -fibration ρY : Y → B. Proof. (1) The generic fiber Xη is a normal projective curve of arithmetic genus 0. In fact, we have an exact sequence ρ∗ (Ω1B/k ) → Ω1X/k → Ω1X/B → 0, which is derived from the exact sequence in Lemma 1.6.21(1). Since Ω1B/k is an invertible sheaf on B, the homomorphism ρ∗ (Ω1B/k ) → Ω1X/k is injective. Taking the second exterior product of Ω1X/k , we have Ω2X/k ∼ = ρ∗ (Ω1B/k ) ⊗ 1 ∗ 1 ΩX/B , i.e., KX ∼ ρ (KB ) + KX/B , where OX (KX/B ) = ΩX/B . Restricting both sides to the generic fiber Xη , we have KX |Xη ∼ KX/B |Xη = KXη . More precisely, if U is an open set such that KB |U ∼ 0, we have KX |ρ−1 (U ) ∼ Kρ−1 (U )/U . It follows that deg((KXη ⊗k(B) k(B)) = −2, where k(B) is an algebraic closure of k(B). Set Xη = Xη ⊗k(B) k(B). Then Xη ∼ = 1 Pk(B) . It follows that H 0 (Xη , O(KXη )) ∼ = H 0 (Xη , O(KXη )) ⊗k(B) k(B). Hence Φ|−KXη | = Φ|−KXη | ⊗k(B) k(B). Since the image of Xη by Φ|−KXη | 2 , it follows that the image of Xη by is a smooth curve of degree 2 in Pk(B)

Φ|−KXη | is also a curve C of degree 2 in P2k(B) which is defined over the field k(B). Now we use Theorem of Tsen which says that the function field k(B) of an algebraic curve defined over an algebraically closed field k is a C1 -field.20 Then the curve C has a k(B)-rational point Q whose homogeneous coordinates in P2 are elements in k(B). Then C is isomorphic to P1 over k(B). In fact, the set L of lines in P2k(B) passing through Q is a linear pencil, and each line ℓ ∈ L meets the curve C in the point Q and a point Q(ℓ) since deg C = 2. A k(B)-rational point Q(ℓ) is viewed also as a k(B)-rational point of Xη . Hence the mapping ℓ 7→ Q(ℓ) gives an isomorphism P1k(η) ∼ = Xη . Let S be the closure of {Q} in X. Then S is an irreducible curve in X, and ρ|S : S → B is a birational morphism. Hence it is an isomorphism because B is a smooth curve. So S is a section. It is clear that (S · F ) = 1. (2) Since F is connected and (F · Ci ) = 0, (Ci2 ) < 0 for every i (see Lemma 1.8.19). On the other hand, by the arithmetic genus formula, 20 A field K is a C -field if an equation F = 0 defined by a homogeneous polynomial of 1 degree d in N variables F (x1 , . . . , xN ) with coefficients in K has a root (α1 , . . . , αN ) ̸= (0, . . . , 0) if d < N .

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(KX · F ) = −2. Hence we have −2 = (KX · F ) =

X

=

X

i

i

ni (KX · Ci ) ni {2pa (Ci ) − 2 − (Ci2 )},

(Ci2 )

where 2pa (Ci ) − 2 − ≥ −1 with the equality occurring if and only if (Ci2 ) = −1 and pa (Ci ) = 0, i.e., Ci is a (−1)-curve. Hence there is some Ci which is a (−1)-curve. If ni = 1 there is another component Cj which is a (−1)-curve. Suppose that Ci is a (−1)-curve. Let τ : X → X1 be the contraction of Ci . Since ρ(Ci ) is a point b on B, ρ1 := ρ · τ −1 is defined at P1 and ρ1 : X1 → B is a P1 -fibration. If ρ−1 1 (b) is reducible, we find a (−1)-curve among the components of ρ−1 (b) and contract it. Continuing 1 1 this process with (−1)-curves appearing in the P -fibration induced by ρ, we find a birational morphism φ : X → Y and a P1 -fibration ρY : Y → B such that φ is a composite of blowing-downs of (−1)-curves, ρ = ρY ◦ φ and every fiber of ρY is irreducible. Suppose that F = mC is an irreducible fiber. Since (F · S) = m(C · S) = 1 for a section S of ρY , we have m = 1. Hence every fiber of ρY is irreducible and reduced. The assertion (3) follows from the above observation because ρ is obtained from ρY by a succession of blowing-ups. Note that each (−1)-curve appearing in an intermediate blowing-up is contained in a fiber of the corresponding P1 -fibration. Given a P1 -fibration ρ : X → B, a reducible fiber F is called also a singular fiber or a degenerate fiber. An irreducible fiber is called a smooth fiber. A P1 -fibration without reducible fibers is called a P1 -bundle. The forthcoming arguments are quite sketchy, and we only state results which we need in the next chapters. For the details see [57, Chapter 11]. Let ρ : X → B be a P1 -bundle. Let S be a section of ρ (see Lemma 1.8.20(1)). Then we have an exact sequence of OX -Modules 0 → OX → OX (S) → OS ⊗OX OX (S) = OS (S) → 0. Applying ρ∗ , which is a left-exact functor, we have an exact sequence 0 → ρ∗ OX → ρ∗ OX (S) → OS (S) → 0, ∼ OB (see Lemma 1.8.15(III)) and ρ∗ (OS ⊗O OX (S)) ∼ where ρ∗ OX = = X OS (S) because ρ|S is an isomorphism. The left homomorphism is surjective

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because R1 ρ∗ OX = 0. Let F = ρ∗ OX (S) and let L = OS (S). We have an exact sequence 0 → OB → F → L → 0. Since L is an invertible sheaf, F is a locally free OB -Module of rank 2. In fact, since Xη ∼ = P1k(η) and OX (S) ⊗OX OXη ∼ = OP1 (1), a local free basis {e1 , e2 } of F is considered as homogeneous coordinates of degree 1 if restricted to Xη . Then there exists a birational mapping X 99K Proj (S • (F)) over B, where S • (F) is a symmetric OB -Algebra generated by F and Proj (S • (F)) is a projective scheme over B constructed locally by making use of a local free basis as homogeneous coordinates. We have the following result (see [57, Lemma 12.7]). Lemma 1.8.21. Let ρ : X → B be a P1 -bundle. Then there exists a locally free OB -Module F of rank 2 such that X is B-isomorphic to P(F) := Proj (S • (F)). Theorem 1.8.22. Let ρ : X → B be a P1 -bundle over B = P1 . Then X = P(F), where F ∼ = OB ⊕ OB (n) with n ≥ 0. Proof. For locally free OX -Modules F and G of rank 2 over a smooth projective curve B, P(F) ∼ = F ⊗ L for L ∈ Pic (B) = P(G) if and only if G ∼ (see [57, Corollary 12.9]). So, we have only to show that if B ∼ = P1 then F is decomposable to a direct sum of two invertible sheaves F ∼ = L1 ⊕ L 2 ∼ (see [57, Lemma 12.2]). Since Li = OB (ni ) with ni ∈ Z for i = 1, 2, it suffices to replace F by F ⊗ OB (−n1 ) ∼ = OB ⊗ OB (n), where we assume n1 ≤ n2 and set n = n2 − n1 . The P1 -bundle P(OP1 ⊕ OP1 (n)) with n ≥ 0 is named a Hirzebruch surface of degree n and denoted by Fn or Σn . Let ρ : X → B be a P1 bundle and let S be a section. Let P be a point on S and let σ : X ′ → X be the blowing-up of P . Let E be the exceptional curve and let ℓ be the fiber of ρ through P . Then σ ∗ ℓ = ℓ′ + E with ℓ′ = σ ′ ℓ and σ ∗ S = S ′ + E with S ′ = σ ′ S. Contract ℓ′ which is a (−1)-curve. Then we obtain a −1 P1 -bundle ρ′ : Y → B such that ρ′ (ρ(ℓ)) = E and S ′ is a section with 2 (S ′ ) = (S 2 ) − 1. The operation of changing ρ to ρ′ is called an elementary transformation with center the point P . If we take the point P off the section S but on the same fiber ℓ, the self-intersection number of S ′ increases by 1. Note that elementary transformation is defined for a curve B of any genus.

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We collect the properties of Fn which we make frequent use of in the subsequent parts of the present book. For the proof see [57, Theorem 12.10]. Lemma 1.8.23. Let X = Fn with n ≥ 0 and a P1 -bundle structure ρ : X → B = P1 . Then the following assertions hold. 2

(1) There are two sections M, M ′ on X such that (M 2 ) = −n, (M ′ ) = n and (M ·M ′ ) = 0. M is called a minimal section in the sense that (M 2 ) is minimal among all sections of ρ. There is a unique minimal section if n > 0, but exist infinitely many if n = 0. In fact, F0 ∼ = P1 × P1 . ∼ (2) Pic (X) = Z[M ] ⊕ Z[ℓ], where ℓ is a fiber of ρ. (3) KX ∼ −2M + (−n − 2)ℓ. Hence (KX 2 ) = 8. (4) By an elementary transformation with center a point on (resp. off ) the minimal section, Fn changes to Fn+1 (resp. Fn−1 ).

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1.9 1.9.1

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Appendix to Chapter 1 Primary decomposition of ideals

In the present subsection, we assume that a ring R is noetherian. A proper ideal I of R is reducible if we can write I = I1 ∩ I2 with proper ideals Ii such that I ⫋ Ii for i = 1, 2. An ideal I is irreducible if it is not reducible. Whenever we say of reducible or irreducible ideals, the ideals are proper ideals. Lemma 1.9.1. An arbitrary ideal I of R has a decomposition I = I1 ∩· · ·∩ In with irreducible ideals Ii (1 ≤ i ≤ n). Proof. The proof is similar to the proof of Lemma 1.1.6. Let S be the set of proper ideals of R which are not written as the intersections of irreducible ideals. Suppose that S ̸= ∅. Define an order I ≤ J by the inclusion I ⊆ J. Then S is an inductive set because R is noetherian. By Zorn’s lemma, S has a maximal element, say J. Since J is not irreducible, J = J1 ∩ J2 for proper ideals Ji ⫌ J for i = 1, 2. Since J is a maximal element of S, Ji ̸∈ S for i = 1, 2. Then we have expressions Ji = Ji1 ∩ · · · ∩ Jini , i = 1, 2

with irreducible ideals Jij . Then J is written as the intersection of irreducible ideals J = J1 ∩ J2 = (J11 ∩ · · · ∩ J1n1 ) ∩ (J21 ∩ · · · ∩ J2n2 ) .

This is a contradiction.

√ An ideal q is called primary if ab ∈ q for a, b ∈ R and b ̸∈ q implies √ a ∈ q. In other terms, q is primary if ab ∈ q and a ̸∈ q implies b ∈ q. Let √ p = q. Then p is a prime ideal if q is a primary ideal. In fact, suppose ab ∈ p and b ̸∈ p. Then (ab)n ∈ q for some n and bn ̸∈ p. Hence an ∈ q by √ definition. So, a ∈ p. If p = q we say that q is a primary ideal belonging to the prime ideal p. Lemma 1.9.2. An irreducible ideal I is a primary ideal. √ Proof. Suppose that ab ∈ I and b ̸∈ I. For every n > 0 define the ideal quotient Jn := (I : bn ) = {x ∈ R | bn x ∈ I}.

Then these ideal quotients make an ascending chain

J1 ⊆ J2 ⊆ · · · ⊆ Jn ⊆ Jn+1 ⊆ · · · .

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Since Jn = R if and only if bn ∈ I, by the hypothesis that bn ̸∈ I for every n > 0, Jn is a proper ideal. Since R is noetherian, the ascending chain terminates, i.e., Jm = Jm+1 = · · · for some m > 0. Set I1 = I + aR and I2 = I + bm R. Then I ⊆ I1 ∩ I2 . We show that the inclusion I1 ∩ I2 ⊆ I also holds. An element x ∈ I1 ∩ I2 is written as x = α + ay = β + bm z, α, β ∈ I, y, z ∈ R.

Since ab ∈ I, we have

bm+1 z = aby + b(α − β) ∈ I.

Hence z ∈ Jm+1 = Jm . So, bm z ∈ I and x ∈ I. This shows that I = I1 ∩ I2 . Meanwhile, I2 ̸= I because bm ̸∈ I. Since I is an irreducible ideal, I = I1 and a ∈ I. So, I is a primary ideal. Lemma 1.9.3. The following assertions hold true. (1) Let m be a maximal ideal of R. If an ideal I contains mn for some n > 0 then I is a primary ideal. (2) If q1 , . . . , qr are primary ideals belonging to the same prime ideal p then q1 ∩ · · · ∩ pr is a primary ideal belonging to p. (3) Let S be a multiplicative set of R and let q be a primary ideal of R such that q ∩ S = ∅. Then qRS is a primary ideal of RS (= S −1 R) such that √ √ qRS = qRS . Further, it holds that qRS ∩ R = q. Conversely, if Q is a primary ideal of RS then Q = qRS with q = Q ∩ R. √ Proof. (1) Note that I = m. Suppose that ab ∈ I and b ̸∈ m. Then I + bR = R. In fact, if I + bR ̸= R then there exists a maximal ideal m′ containing I + bR. So, mn ⊆ I ⊆ m′ . This implies that m = m′ and b ∈ m, which is a contradiction. Hence 1 = x + by with x ∈ I and y ∈ R. Then a = a(x + by) = (ab)y + ax ∈ I. This shows that I is a primary ideal belonging to m. √ (2) Let x ∈ p. Since qi = p we have xni ∈ qi for some ni > 0 (1 ≤ i ≤ T Qr r r). Then xn ∈ i=1 qi for n = i=1 ni . This implies that √ √ p ⊆ q1 ∩ · · · ∩ qr ⊆ qi = p, √ whence p1 ∩ · · · ∩ qr = p. If ab ∈ q1 ∩ · · · ∩ qr and b ̸∈ p then ab ∈ qi and √ b ̸∈ qi for every 1 ≤ i ≤ r. It follows that a ∈ qi for every 1 ≤ i ≤ r. So, q1 ∩ · · · ∩ qr is a primary ideal belonging to p. √ √ √ √ (3) We show that qRS = qRS . The inclusion qRS ⊆ qRS is clear. We show the opposite inclusion. Suppose (a/s)n ∈ qRS . Then (a/s)n = q/t with s, t ∈ S. So, s′ (an t − sn q) = 0 for some s′ ∈ S, i.e.,

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an s′ t = qsn s′ . Hence (as′ t)n = q(ss′ )n tn−1 ∈ q. This implies that as′ t ∈ √ √ q and (a/s) = (as′ t)/(ss′ t) ∈ qRS . We next show that qRS is a primary √ ideal. Suppose that (a/s)·(b/t) ∈ qRS with s, t ∈ S and b/t ̸∈ qRS . Since (ab)/(st) = q/u with q ∈ q and u ∈ S, we have s′ (abu − qst) = 0 for some √ s′ ∈ S. This implies that a(bs′ u) = qss′ t ∈ q and bs′ u ̸∈ q. Hence √ a ∈ q and (a/s) ∈ qRS . Thus qRS is a primary ideal belonging to qRS . Finally we show that qRS ∩ R = q. In fact, q ⊆ qRS ∩ R is clear. Suppose (a/1) ∈ qRS . Then sa ∈ q for some s ∈ S. Note that if q ∩ S = ∅ then √ √ q ∩ S = ∅. So, s ̸∈ q. Then a ∈ q. This implies that qRS ∩ R ⊆ q. By Lemmas 1.9.1 and 1.9.2, any proper ideal I of R is written as an intersection of primary ideals I = q1 ∩ q2 ∩ · · · qn .

(1.22)

If some of q1 , . . . , qr happen to belong to the same prime ideal, we may replace them by their intersection by Lemma 1.9.3 and assume that in the decomposition (1.21) the following conditions hold: √ √ (i) qi ̸= qj if i ̸= j. ∨

(ii) For every 1 ≤ i ≤ n, I ⫋ q1 ∩ · · · ∩ qi ∩ · · · ∩ qn .

If these conditions are satisfied, we call the decomposition (1.21) the irredundant primary decomposition of I. To obtain such a decomposition, we √ √ start with the decomposition (1.21). If qi1 = · · · = qir then consider qi1 ∩ · · · ∩ qir as a single primary ideal and decrease the number of primary ∨

ideals by r − 1. If I = q1 ∩ · · · ∩ qi ∩ · · · ∩ qn then drop qi from the components and decrease the number by one. Thus we obtain a part of the following results. Theorem 1.9.4. Let I be a proper ideal of a noetherian ring R. Then the following assertions hold.

(1) I has an irredundant primary decomposition √

I = q1 ∩ · · · ∩ qn .

(2) Let pi = qi for 1 ≤ i ≤ n. Let {pi1 , . . . , pir } (i1 < i2 < · · · < ir ) exhaust all minimal members of {p1 , . . . , pn } with respect to inclusion. Then we have √ I = pi1 ∩ · · · ∩ pir , √ which is a prime decomposition of the radical ideal I. Furthermore, pi (1 ≤ i ≤ n) is a prime divisor of I, and every prime divisor of I appears among {p1 , . . . , pn }.

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(3) If I has two irredundant primary decompositions I = q1 ∩ · · · ∩ qn = q′1 ∩ · · · ∩ q′m , we have (i) n = m. q √ (ii) If qi and q′j are the same minimal prime divisor of I then qi = q′j . (4) Ass (R/I) = {p1 , . . . , pn }. Proof. (1) The proof been given. √ has √ √ √ (2) It holds that I = q1 ∩ · · · ∩ qn . Let pi = qi (1 ≤ i ≤ n). With the minimal members {pi1 , . . . , pir } among {p1 , . . . , pn }, we have √ I = pi1 ∩ · · · ∩ pir , √ which is a prime ideal decomposition of I (see Corollary 1.1.8). We show that pi (1 ≤ i ≤ n) is a prime divisor of I, i.e., pi = (I : ai ) for some ai ∈ R. It suffices to consider the case i = 1. Since q2 ∩ · · · ∩ qn ̸⊆ q1 , take an element b ∈ (q2 ∩ · · · ∩ qn ) \ q1 . Then (I : b) = (q1 : b). In fact, (I : b) ⊆ (q1 : b) is clear. Suppose bx ∈ q1 . Then bx ∈ I because b ∈ q2 ∩ · · · qn . Hence (q1 : b) ⊆ (I : b). We admit for a moment a claim √ that (q1 : b) is a primary ideal belonging to p1 = q1 . If p1 = (q1 : b) we have only to take a = b to show that p1 = (I : a). If p1 ̸= (q1 : b) and hence p1 ⫌ (q1 : b), there exists an integer m > 0 such that pm−1 ̸⊆ (q1 : b) 1 m and p1 ⊆ (q1 : b) because p1 is a finite R-module. Take an element c ∈ pm−1 \ (q1 : b). Since ((q1 : b) : c) is a primary ideal belonging to 1 p1 by the above claim and p1 ⊆ ((q1 : b) : c), we have p1 = ((q1 : b) : c) = ((I : b) : c) = (I : bc) = (I : a), where a = bc. We have to prove the above claim. Suppose that xy ∈ (q1 : b). Then bxy ∈ q1 . If x ̸∈ (q1 : b), then bx ̸∈ q1 . Since q1 is a primary ideal, y r ∈ q1 r for some r > 0 and hence by r ∈ q1 . So, shows that p y ∈ (q1 : b). This (q1 : b) is a primary ideal. Suppose x ∈ (q1 : b). Then bxs ∈ q1 for some √ s > 0. Since b ̸∈ q1 we have (xs )t ∈ q1 for some t > 0. Hence x ∈ q1 = p1 . We have therefore p √ q1 ⊆ (q1 : b) ⊆ (q1 : b) ⊆ q1 . p √ Hence we have (q1 : b) = q1 = p1 .

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We prove that any prime divisor p = (I : b) appears in {p1 , . . . , pn }. With the above notations, we have p = (I : b) = (q1 ∩ · · · ∩ qi : b) =

n \

(qi : b) =

i=1

s \

(qij : b),

j=1

where {qi1 , . . . , qis } exhaust all qi such that (qi : b) ̸= R, i.e., b ̸∈ qi . Note √ that (qi : b) is then a primary ideal belonging to pi = qi , i.e., pni i ⊂ (qi : b). So, we have  n s s Y \  pij  ⊆ (qij : b) = p, j=1

j=1

where n = max{nij | 1 ≤ j ≤ s}. Then pij ⊆ p for some ij . Namely there exists pi (1 ≤ i ≤ n) such that pi ⊆ p and b ̸∈ qi . On the other hand, (I : b) ⊆ (qi : b) ⊂ pi . Hence p = pip . p √ √ (3) Since { q1 , . . . , qn } and { q′1 , . . . , q′m } coincide with the set of prime divisors of the ideal I, it follows q that m = n. Let p be a minimal √ prime divisor of I. Then p = qi = q′j . Since qk ̸⊆ p for k ̸= i as p is minimal, we have IRp = qi Rp . Similarly, IRp = q′j Rp . Then we have by Lemma 1.9.3 qi = qi Rp ∩ R = IRp ∩ R = q′j Rp ∩ R = q′j . (4) For a finite R-module M and a nonzero element m ∈ M , we set (0 : m) = {a ∈ R | am = 0}. If (0 : m) is a prime ideal p, we call p an associated prime ideal of M . The set of all associated prime ideals of M is denoted by Ass (M ). It is known that Ass (M ) is a finite set [54, Theorem 6.5]. If M = R/I then Ass (R/I) is the set of all prime divisors of I. 1.9.2 1.9.2.1

Tensor products of algebras Construction

Let R be a ring and let A, B be R-algebras. We define a tensor product A ⊗R B and state some of formal properties of tensor products. Let A × B be the product set of the underlying sets of A and B, and let F be a free R-module with basis A × B. Namely F is a finite sum n X i=1

αi (ai , bi ),

ai ∈ A, bi ∈ B, αi ∈ R,

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P which is also written as (a,b)∈A×B α(a,b) (a, b), where the coefficient α(a,b) is zero except for a finite number of pairs (a, b). The addition of two such finite sums and the scalar multiplication are defined by X

α(a,b) (a, b) +

(a,b)∈A×B

β(a,b) (a, b) =

(a,b)∈A×B

 α

X

X

(α(a,b) + β(a,b) )(a, b)

(a,b)∈A×B

 X

α(a,b) (a, b) =

(a,b)∈A×B

X

αα(a,b) (a, b).

(a,b)∈A×B

Let G be the R-submodule generated by   (a1 + a2 , b) − (a1 , b) − (a2 , b), a1 , a2 ∈ A, b ∈ B (a, b1 + b2 ) − (a, b1 ) − (a, b2 ), a ∈ A, b1 , b2 ∈ B  (αa, b) − (a, αb), α ∈ R, a ∈ A, b ∈ B. We denote the quotient R-module F/G by A ⊗R B, and the residue class (a, b) + G by a ⊗ b. Then there are the following relations (a1 + a2 ) ⊗ b = a1 ⊗ b + a2 ⊗ b, a1 , a2 ∈ A, b ∈ B a ⊗ (b1 + b2 ) = a ⊗ b1 + a ⊗ b2 , a ∈ A, b1 , b2 ∈ B αa ⊗ b = a ⊗ αb, α ∈ R, a ∈ A, b ∈ B.

We define the multiplication on A ⊗R B by

(a′ ⊗ b′ ) · (a ⊗ b) = (a′ a) ⊗ (b′ b), a, a′ ∈ A, b, b′ ∈ B.

This is well-defined because the multiplication (a′ , b′ ) · (a, b) = (a′ a, b′ b) on F preserves the submodule G as shown by (a′ , b′ ) · {(a1 + a2 , b) − (a1 , b) − (a2 , b)}

= (a′ a1 + a′ a2 , b′ b) − (a′ a1 , b′ b) − (a′ a2 , b′ b),

(a′ , b′ ) · {(a, b1 + b2 ) − (a, b1 ) − (a, b2 )}

= (a′ a, b′ b1 + b′ b2 ) − (a′ a, b′ b1 ) − (a′ a, b′ b2 ),

(a′ , b′ ) · {(αa, b) − (a, αb)} = (αa′ a, b′ b) − (a′ a, αb′ b). Thus A ⊗R B is an R-algebra. Its identity element is 1 ⊗ 1 and the zero element is a ⊗ 0 = 0 ⊗ b, a ∈ A, b ∈ B. There are the canonical R-algebra homomorphisms ιA : A → A ⊗R B, a 7→ a ⊗ 1

ιB : B → A ⊗R B, b 7→ 1 ⊗ b.

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The following result is used in the construction of fiber product. Let A, B, C be R-algebras. We define HomR (A, C) as the set of R-algebra homomorphisms for R-algebras A and C, and we consider the following mapping of sets Ψ : Hom(A ⊗R B, C) → HomR (A, C) × HomR (B, C), φ 7→ (φ ◦ ιA , φ ◦ ιB ). Proposition 1.9.5. With the above notations, Ψ is a bijection. Proof. Suppose that Ψ(φ) = Ψ(φ′ ) for φ, φ′ ∈ HomR (A ⊗R B, C). Then we have φ(a ⊗ 1) = φ′ (a ⊗ 1) and φ(1 ⊗ b) = φ′ (1 ⊗ b) for a ∈ A and b ∈ B. Then φ = φ′ because φ(a ⊗ b) = φ(a ⊗ 1)φ(1 ⊗ b) = φ′ (a ⊗ 1)φ′ (1 ⊗ b) = φ′ (a ⊗ b). Hence Ψ is injective. We show that Ψ is surjective. In fact, given Ralgebra homomorphisms φA : A → C and φB : B → C, define an R-algebra homomorphism φ : A ⊗R B → C by setting φ(a ⊗ b) = φA (a)φB (b), a ∈ A, b ∈ B. Then φ ◦ ιA = φA and φ ◦ ιB = φB . 1.9.2.2

Flat modules

Let R be a ring and let A be an R-algebra. For an R-module M , by the same construction as above, we can define a tensor product M ⊗R A, which we can make an A-module by the multiplication from the right. If f : M → N is a homomorphism of R-modules, then f ⊗R A : M ⊗R A → N ⊗R A is a homomorphism of A-modules. It is a standard fact (see [9, Algebra, §3, 6, Proposition 5]) that if α

β

0 −→ M1 −→ M2 −→ M3 −→ 0

(1.23)

is an exact sequence of R-modules and R-module homomorphisms then we have an exact sequence α⊗A

β⊗A

M1 ⊗R A −→ M2 ⊗R A −→ M3 ⊗R A −→ 0. We simply say that the tensor product ⊗R A is a right exact functor. If the homomorphism α ⊗R A is injective for all exact sequence (1.23), then A is said to be flat over R. Further, if M ⊗R A = 0 implies M = 0 for an R-module M , then we say that A is faithfully flat over R.

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Instead of an R-algebra A we take an R-module N . By the same construction, we can define a tensor product M ⊗R N which is an R-module, and the tensor product ⊗R N is a right exact functor. Namely, we have an exact sequence of R-modules α⊗N

β⊗N

M1 ⊗R N −→ M2 ⊗R N −→ M3 ⊗R N −→ 0. An R-module N is flat if α ⊗ N is injective for all exact sequence (1.23). Further, a flat R-module N is faithfully flat if M ⊗R N = 0 for an R-module M implies M = 0. A free R-module of finite rank is faithfully flat. A finite projective R-module, which is a direct summand of a free R-module of finite rank, is also faithfully flat. A ring of quotients T −1 R for a multiplicative set T of R is a flat R-module. This is an easy exercise. In particular, Rq is a flat R-module for every q ∈ Spec R. But T −1 R is not necessarily faithfully flat. We consider flatness in a geometric situation. Let f : X → Y be a morphism of schemes. We say that f is a flat morphism if OX,x is a flat OY,y -module for every point x ∈ X and y = f (x). If f is a surjective (i.e., f (X) = Y ) flat morphism we say that f is a faithfully flat morphism. We consider only the case where X and Y are affine schemes, and refer the readers to [31] for a general case. We set X = Spec A, Y = Spec R and f = a φ, where φ : R → A makes A an R-algebra. Lemma 1.9.6. The following two conditions are equivalent. (i) A is a flat R-module. (ii) For every p ∈ Spec A and q = φ−1 (p), Ap is Rq -flat. Namely, f : X → Y is a flat morphism. Proof. (i) ⇒ (ii). Let S = A \ p and T = R \ q. Then φ(T ) ⊂ S. So, φ induces a local homomorphism φp : T −1 R = Rq → S −1 A = Ap . Let M be an Rq -module. Let M[R] be the module M viewed as an Rmodule. Then M[R] ⊗R Rq ∼ = M as an Rq -module. In fact, if t ∈ T then tM = M because the multiplication of t is an automorphism of M . So, m ⊗ (z/t) = t(t−1 m) ⊗ (z/t) = (t−1 m) ⊗ z = (zt−1 m) ⊗ 1. Hence M ⊗ R q Ap ∼ = (M[R] ⊗R Rq ) ⊗Rq Ap ∼ = M[R] ⊗R Ap ∼ = (M[R] ⊗R A) ⊗A Ap . Note that if 0 → M1 → M2 → M3 → 0 is an exact sequence of Rq modules then 0 → (M1 )[R] → (M2 )[R] → (M3 )[R] → 0 is an exact sequence of R-modules. Since the tensor products ⊗R A and ⊗A Ap are both exact functors, the tensor product ⊗Rq Ap preserves exact sequences. Hence 0 → M1 ⊗Rq Ap → M2 ⊗Rq Ap → M3 ⊗Rq Ap → 0

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is an exact sequence. So, Ap is Rq -flat. (ii) ⇒ (i). Let α

β

0 −→ L1 −→ L2 −→ L3 −→ 0

be an exact sequence of R-modules. Then it suffices to show that α ⊗R A : L1 ⊗R A → L2 ⊗R A is injective. Since α ⊗R Ap = (α ⊗R Rq ) ⊗Rq Ap for q = φ−1 (p) and p ∈ Spec A, α ⊗R Ap is injective for all p ∈ Spec A. Suppose that a nonzero element z ∈ L1 ⊗R A is mapped to 0 by α ⊗R A. Let a = {a ∈ A | az = 0}. Then there exists a maximal ideal m of A such that a ⊆ m. For p = m, α ⊗R Ap is injective. Hence sz = 0 for s ̸∈ m. This is a contradiction because s ∈ a. So, A is R-flat. Lemma 1.9.7. By abuse of notation, let φ : (R, q) → (A, p) be a local homomorphism of local rings. Suppose that A is R-flat. Then the following assertions hold. (1) φ is injective. (2) A is faithfully flat over R. Proof. (1) Let a = Ker φ. Consider an exact sequence 0 → a → R → R/a → 0. By taking tensor products with ⊗R A, we have an exact sequence of A-modules 0 −→ a ⊗R A −→ A −→ A/aA −→ 0.

Hence we have a ⊗R A = φ(a)A = 0. We show that this implies a = 0. Suppose a ̸= 0. Take a nonzero element z ∈ a and let a′ = zR. Since A is R-flat, it follows that a′ ⊗R A = 0. Hence we have 0 = (a′ ⊗R A) ⊗A (A/p) = (a′ ⊗R (R/q)) ⊗(R/q) (A/p).

Since A/p is an extension field of R/q, it follows that a′ ⊗R (R/q) = 0. Namely we have a′ = qa′ . Since q is the maximal ideal and a′ is a finite R-module, it follows by Nakayama’s lemma (Lemma 1.5.1) that a′ = 0. This is a contradiction. So, a = 0 and φ is injective. (2) Let M be an R-module such that M ⊗R A = 0. In order to prove that A is faithfully flat over R, we have to show that M = 0. Suppose that M ̸= 0. Let M ′ be a finite R-submodule of M . Since A is R-flat, this implies that M ′ ⊗R A = 0. Hence M ′ = qM ′ by the same argument as above. By Nakayama’s lemma, we have M ′ = 0. So, it follows that M = 0. Theorem 1.9.8. Let f : X → Y be a flat morphism defined by φ : R → A. Then the following conditions are equivalent.

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(i) A is faithfully flat over R. (ii) f : X → Y is surjective. Proof. (i) ⇒ (ii). Let q ∈ Spec R. We show that qAq ⊊ Aq . Suppose that qAq = Aq . Since L ⊗R/q A/qA ∼ = L ⊗R A for an R/q-module L, it follows that A/qA is faithfully flat over R/q. We have (A/qA) ⊗R/q Q(R/q) ∼ = A ⊗R (Rq /qRq ) ∼ = Aq /qAq = 0. Hence Q(R/q) = 0, which is a contradiction. So, qAq ⊊ Aq . Let m be a maximal ideal of Aq containing qAq . Then m ∩ Rq ⊇ qRq . Since qRq is the maximal ideal of Rq , we have m ∩ Rq = qRq . Let p = m ∩ A. Then p ∩ R = (m ∩ A) ∩ R = (m ∩ Rq ) ∩ R = qRq ∩ R = q. This shows that f is surjective. (ii) ⇒ (i). It suffices to show that M ⊗R A = 0 for an R-module M implies M = 0. Let p ∈ Spec A and let q = p ∩ R. We have (M ⊗R A) ⊗A Ap = (M ⊗R Rq ) ⊗Rq Ap = 0. Since Ap is faithfully flat over Rq by Lemma 1.9.7, it follows that M ⊗R Rq = 0. By the assumption that f is surjective, q moves over all prime ideals of R. We show that M = 0. Otherwise, let m be a nonzero element of M , and let a = {c ∈ R | cm = 0}. Then a is a proper ideal of R. Let m be a maximal ideal such that m ⊇ a. Then M ⊗R Rm = 0. So, sm = 0 for an element s of R \ m. Then s ∈ a ⊆ m. This is a contradiction. So, M = 0. Remark 1.9.9. Suppose that a ring homomorphism φ : R → A makes A a flat R-module. Then the following assertions hold. (1) If A is faithfully flat over R then φ is injective. In fact, let a = Ker φ. By tensoring an exact sequence 0 → a → R → R/a → 0

with ⊗R A, we have a ⊗R A = 0 because (R/a) ⊗R A ∼ = A/φ(a)A ∼ = A. Since A is faithfully flat over R, it follows that a = 0. So, φ is injective. (2) Suppose further that R is noetherian and A is finitely generated over R. With the same notations as above, f = a φ : X → Y is then an open morphism. Namely, for every open set U ⊆ X the image f (U ) is an open set of Y . In order to prove this assertion, we take an arbitrary point x ∈ U and let y = f (x). Then we can find affine open neighborhoods x ∈ U ′ ⊆ U and y ∈ V ′ ⊆ Y such that U ′ ⊆ f −1 (V ′ ), Γ(V ′ , OY ) is noetherian and Γ(U ′ , OX ) is flat and finitely generated

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over Γ(V ′ , OY ). Namely, we have the same situation for f |U ′ : U ′ → V ′ as for f : X → Y . So, it suffices to show that f (X) is an open set of Y . We will be done if we find a morphism of affine schemes g : Y ′ = Spec R′ → Y = Spec R such that g(Y ′ ) = Y \ f (X). Let y ∈ f (X). We only show that if the morphism g exists then y is an interior point of f (X). Let q be the prime ideal of R corresponding to y. Since y ∈ f (X) there exists a prime ideal p of A such that φ−1 (p) = q. By Lemma 1.9.7 and Theorem 1.9.8, the morphism Spec Ap → Spec Rq is surjective. Namely Spec Rq ⊂ f (X). Since f (X) ∩ g(Y ′ ) = ∅, Y ′ ×Y Spec Rq = ∅. So, R′ ⊗R Rq = 0. Since Rq = limt∈R\q Rt with Rt = R[t−1 ], we have −→ R′ ⊗R Rq = limt∈R\q Rt′ = 0 with Rt′ = R′ [t−1 ], we have Rt′ = 0 for some −→ t ∈ R\q.21 In fact, the identity 1 becomes 0 in the course of limt∈R\q Rt′ . −→ Then D(t) ∩ g(Y ′ ) = ∅. Namely, y ∈ D(t) ⊂ f (X) = Y \ g(Y ′ ). See also [31, Exercise 9.1, p. 266]. (3) Flatness is related to geometric properties of morphisms of schemes. Openness as stated above is one of these properties. Let f : X → Y be a morphism of finite type of algebraic varieties defined over a field k. Suppose that f is flat. Then, for every point y ∈ f (X), each irreducible component of the fiber Xy := X ×Y Spec (k(y)) has dimension equal to dim X − dim Y . This property is called the equidimensionality of fibers. See [EGA] for details. (4) There are several results which are useful in research. Let φ : (R, q) → (A, p) be a local homomorphism of noetherian local rings. Then we have the following results. (i) Suppose that A is regular and A is R-flat. Then R is regular (see [EGA, Vol. 24, Prop. 6.5.1]). (ii) Suppose that A is normal and A is R-flat. Then R is normal (see [EGA, Vol. 24, Cor. 6.5.4]). (iii) Suppose that R is regular, A is Cohen-Macaulay and dim A = dim R + dim(A/qA). Then A is R-flat (see [EGA, Vol. 24, Prop. 6.1.3]). (iv) Suppose that R is a DVR and A is an integral domain. Then A is R-flat. b be the n-adic comple(5) Let (R, n) be a noetherian local ring. Let R tion (see below the subsection for ideal-adic completion). Then there b by which R b is a flat is a natural local ring homomorphism R → R, 21 See

the next subsection for inductive limits.

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b is a faithR-module (see [72, Corollary 17.11]). By Lemma 1.9.7, R fully flat R-module. 1.9.3 1.9.3.1

Inductive limits and projective limits Inductive limits

A partially ordered set (I, ≤) is an inductive set if, for any i, j ∈ I, there exists k ∈ I such that i ≤ k and j ≤ k. A collection A consisting of rings {Ai | i ∈ I} and ring homomorphisms {φji : Ai → Aj | i, j ∈ I, i ≤ j} is an inductive system indexed by an inductive set I if the following conditions are satisfied: (i) If i ≤ j ≤ k then φki = φkj ◦ φji . (ii) For any i ∈ I, φii = idAi . We then construct the inductive limit, denoted by limi∈I Ai , as the −→ L quotient ring of the direct sum i∈I Ai by the equivalence relation (∼) generated by the following relation: ai ∼ aj for ai ∈ Ai and aj ∈ Aj if there exists k ∈ I such that φki (ai ) = φkj (aj ). Then limi∈I Ai is a ring −→ with a ring homomorphism fi : Ai → limi∈I Ai which satisfies the following −→ conditions. (1) fi = fj ◦ φji whenever i ≤ j. (2) Let B be a ring and let {gi : Ai → B}i∈I be a collection of ring homomorphisms such that gi = gj ◦ φji if i ≤ j. Then there exists a unique ring homomorphism g : limi∈I Ai → B such that gi = g ◦ fi for −→ every i ∈ I. In fact, we define fi : Ai → limi∈I Ai by setting fi (ai ) to be the equivalence −→ class containing ai . Then the above two conditions are clear. In the case of OX in subsection 1.3.1, the set S(x) is partially ordered by U ≤ U ′ if and only if U ′ ⊆ U , and an inductive set because U ∩ U ′ ⊆ U and U ∩ U ′ ⊆ U ′ . If D(s) ∈ T(x), Γ(D(s), OX ) = A[s−1 ]. Supposepthat D(t) ⊆ D(s). Then V ((s)) ⊆ V ((t)), which is equivalent to (t) ⊆ (s), i.e., tn is divisible by s for some n > 0. Write tn = su with u ∈ A \ px . Then 1/s = u/tn . Hence there exists a ring homomorphism φD(t),D(s) : A[s−1 ] → A[t−1 ] by mapping 1/s to u/tn . Thus limD(s)∈T(x) A[s−1 ] is the −→ ring A with 1/s added to for all s ∈ A \ px , which is the ring of quotients Apx . Inductive limits are defined in the same way for inductive systems of

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sets, groups, R-modules, etc. We will tacitly use the following result. Let {Ai }i∈I be an inductive system of rings. Suppose that for each i ∈ I we are given an exact sequence of Ai -modules βi

α

i 0 −→ Li −→ Mi −→ Ni −→ 0

and for i < j we have a commutative diagram βi

α

i 0 −→ Li −→ Mi −→ Ni −→ 0 ↓ fji ↓ gji ↓ hji

0 −→ Lj

αj

−→ Mj

βj

−→ Nj

−→ 0.

Then the inductive limits give an exact sequence of A-modules β

α

0 −→ lim Li −→ lim Mi −→ lim Ni −→ 0, −→ −→ −→ i∈I

i∈I

i∈I

where A = limi∈I Ai , α = limi∈I αi and β = limi∈I βi . Namely, limi∈I is −→ −→ −→ −→ an exact functor on the category of modules. For the proof, it suffices to represent an element of the limit (e.g., limi∈I Ni ) by an element of the ith −→ term (e.g., Ni ) for some i ∈ I. 1.9.3.2

Projective limits

Let I be anew a partially ordered system. A projective system of rings is a collection of rings {Ai | i ∈ I} and ring homomorphisms {ψij : Aj → Ai | i, j ∈ I, i ≤ j} such that (i’) ψik = ψij ◦ ψjk for i ≤ j ≤ k. (ii’) ψii = idAi for every i ∈ I. We construct the projective limit, denoted by limi∈I Ai , as the subset of ←− Q the product i∈I Ai consisting of elements (ai )i∈I such that ai = ψij (aj ) whenever i ≤ j, i.e., ( ) Y Ai | ai = ψij (aj ), i ≤ j . lim Ai = (ai )i∈I ∈ ←− i∈I

i∈I

The projective limit is also called the inverse limit, and it is again a ring. 1.9.3.3

Ideal-adic completion

As an example of projective limit, we consider a linear topology on a ring and its completion. Let A be a ring and let a be an ideal of A. Let I = N≥0 be the set of non-negative integers. Then, with the natural order of integers, I is a totally ordered set. Let An = A/an and let ψnm :

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Am → An be the natural quotient homomorphism if n ≤ m. A collection {An (n ∈ I), ψnm (n ≤ m)} is a projective system. The projective limit b In order to describe limn∈I An is the a-adic completion and denoted by A. ←− b we consider the a-adic topology on A by making the set {a + an | n ∈ I} A,

a generating family of open neighborhoods of an element a for every a ∈ A. In fact, we have (a + an+m ) ⊂ (a + am ) ∩ (a + an ). The a-adic topology is also called a linear topology on A. It is Hausdorff, i.e., two distinct points are separated by disjoint open neighborhoods of respective points, if and T T only if n>0 an = (0). In fact, if n>0 an ̸= (0) take a nonzero element T c ∈ n>0 an . Then, for any a ∈ A, the elements a and b := a + c are T not separated. Suppose n>0 an = (0). For distinct elements a, b ∈ A, b − a ̸∈ an for some n > 0. Then (a + an ) ∩ (b + an ) = ∅. If either A is a noetherian ring with a contained in the Jacobson radical J which is the intersection of all maximal ideals of A or A is a noetherian domain, T the condition n>0 an = (0) follows from the following result (see [72, Theorem 3.11]). In fact, if a is contained in J and 1 + a is a zero divisor with a ∈ a then there exists a maximal ideal m such that 1 + a ∈ m. Since a ∈ a ⊂ m, we have 1 ∈ m, a contradiction. If A is a domain, 1 + a with a ∈ a is a nonzero element, hence it is not a zero divisor. Theorem 1.9.10 (Krull’s intersection theorem). Let M be a finite module over a noetherian ring A and let a be an ideal of A. Then T n n>0 a M = (0) if and only if no element a of A such that a − 1 ∈ a is a zero divisor with respect to M . Consider a sequence {an } = {a0 , a1 , . . .} of elements of A indexed by N≥0 . It is a Cauchy sequence if, for any integer r > 0,  there exists an integer N > 0 such that am − an ∈ ar (∗) for all m, n ≥ N . Two Cauchy sequences {an }, {bn } are equivalent, {an } ∼ {bn } by notation, if, for any s > 0, there exists an integer M > 0 such that an − bn ∈ as for all n ≥ M . We say that a Cauchy sequence {an } is refined if, for any n > 0, an+1 − an ∈ an . Given a Cauchy sequence {an }, define a Cauchy sequence {a′n } by setting a′r = aN (r) for r ≥ 0, where N (r) is the smallest positive integer N satisfying the above condition (∗). Then {an } ∼ {a′n } and {a′n } is refined. Two refined Cauchy sequences {a′n } and {b′n } are equivalent if and only if a′n − b′n ∈ an for every n > 0. Let C(A) (resp. C ′ (A)) be the set of Cauchy sequences (resp. refined Cauchy sequences) with elements in A. Then, by

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termwise addition and multiplication, C(A) (resp. C ′ (A)) becomes a ring with zero {0} = {0, 0, . . .}, and N := {{an } ∈ C(A) | {an } ∼ {0}} (resp. N′ = N ∩ C ′ (A)) is an ideal of C(A) (resp. C ′ (A)). Further C ′ (A)/N′ ∼ = C(A)/N, which we denote by c A. Then the following results hold. Theorem 1.9.11. We have an isomorphism c A ∼ (A/an ). There = lim ←−n≥0 is a canonical ring homomorphism ι : A → limn≥0 (A/an ) whose kernel is ←− T n n>0 a . Proof. We construct a ring homomorphism θ : C ′ (A) → limn≥0 An by ←− Q setting θ({an }) = (an ) ∈ n≥0 (A/an ), where an = an + an is the residue class of an modulo an if n > 0 and a0 = a0 . Since an + an = an+1 + an for every n > 0, this is a well-defined mapping of sets, and it is, in fact, a surjective ring homomorphism. The kernel of θ consists of refined Cauchy sequences {an } such that an ∈ an for every n > 0. Hence {an } ∼ {0}. The converse is clear. Hence c A ∼ (A/an ). = lim ←−n≥0 There is a ring homomorphism i : A → C ′ (A) such that i(a) = {a}, where the nth term of the Cauchy sequence is a for every n ≥ 0. Hence ι = θ ◦ i gives a ring homomorphism from A to limn≥0 (A/an ), and Ker ι is ←− T clearly n>0 an . Example 1.9.12. (1) Let (A, m) be a noetherian local ring with maximal n b where T ideal m. Then we denote the m-adic completion A, n>0 m = (0) by Krull’s intersection theorem. Hence there is an injective ring homomorb phism ι : A ,→ A. (2) Let A = k[x1 , . . . , xn ] be a polynomial ring in n variables x1 , . . . , xn over a field k and let a = (x1 , . . . , xn ) be the ideal generated by b is the ring of formal power series x1 , . . . , xn . Then the a-adic completion A k[[x1 , . . . , xn ]] which consists of infinite sums of monomials X f (x1 , . . . , xn ) = cα xα , cα ∈ k, α

αn 1 where α = (α1 , . . . , αn ) with αi ≥ 0 and xα = xα 1 · · · xn .

1.9.4

Fiber products of schemes

We prove the following result (see [78]). Theorem 1.9.13. Let S be a scheme, and let f : X → S and g : Y → S be S-schemes. Then there exist an S-scheme h : Z → S and S-morphisms p : Z → X and q : Z → Y such that Z is a fiber product of X and Y over S.

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Our proof consists of several steps. Step 1. Let Z be a fiber product of X and Y over S. Let U (resp. V ) be an open set of X (resp. Y ) and let W = p−1 (U ) ∩ q −1 (V ). Then (W, p|W , q|W ) is a fiber product of (U, f |U ) and (V, g|V ) over S. Proof. If α : T → U and β : T → V be S-morphisms. Then there exists an S-morphism γ : T → Z such that p ◦ γ (resp. q ◦ γ) is the composite of α (resp. β) with the canonical open immersion U → X (resp. V → Y ). ιW Z, where ιW is the canonical open Then γ factors as γ : T → W −→ immersion. This implies that (W, p|W , q|W ) is a fiber product of (U, f |U ) and (V, g|V ). Step 2. Let Z be an S-scheme, and let p : Z → X and q : Z → Y be S-morphisms. Let U = {Ui }i∈I and V = {Vj }j∈J be open coverings of X and Y , respectively. Let Wij = p−1 (Ui )∩q −1 (Vj ) for (i, j) ∈ I ×J. Suppose that (Wij , p|Wij , q|Wij ) is a fiber product of (Ui , f |Ui ) and (Vj , g|Vj ) for every (i, j) ∈ I × J. Then (Z, p, q) is a fiber product of (X, f ) and (Y, g) over S. Proof. We prove that the mapping HomS (T, Z) → HomS (T, X) × HomS (T, Y ),

γ 7→ (p ◦ γ, q ◦ γ)

is a bijection for any S-scheme T . We prove that the mapping is injective. Suppose that two morphisms γ1 , γ2 ∈ HomS (T, Z) satisfy p ◦ γ1 = p ◦ γ2 and q ◦ γ1 = q ◦ γ2 . For each Wij we have γ1−1 (Wij ) = γ1−1 (p−1 (Ui ) ∩ q −1 (Vj )) = γ1−1 p−1 (Ui ) ∩ γ −1 q −1 (Vj ) = γ2−1 p−1 (Ui ) ∩ γ2−1 q −1 (Vj ) = γ2−1 (Wij ).

S Hence γ1 |Wij = γ2 |Wij by the hypothesis. Since Z = (i,j)∈I×J Wij , it follows that γ1 = γ2 . We next prove that the mapping is surjective. Suppose that we are given S-morphisms α : T → X and β : T → Y . Set Tij = α−1 (Ui ) ∩ β −1 (Vj ). By the hypothesis, there exists an S-morphism γij : Tij → Wij such that p ◦ γij = α|Tij and q ◦ γij = β|Tij . If Tij ∩ Ti′ j ′ ̸= ∅ for (i, j), (i′ , j ′ ) ∈ I × J, we have γij (Tij ∩ Ti′ j ′ ) ∩ γi′ j ′ (Tij ∩ Ti′ j ′ ) ⊆ Wij ∩ Wi′ j ′ , and

Wij ∩ Wi′ j ′ = p−1 (Ui ) ∩ q −1 (Vj ) ∩ p−1 (Ui′ ) ∩ q −1 (Vj ′ ) = p−1 (Ui ∩ Ui′ ) ∩ q −1 (Vj ∩ Vj ′ ).

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By Step 1, Wij ∩ Wi′ j ′ is a fiber product of Ui ∩ Ui′ and Vj ∩ Vj ′ over S, and it holds that p ◦ (γij |Tij ∩Ti′ j′ ) = p ◦ (γi′ j ′ |Tij ∩Ti′ j′ ), and

q ◦ (γij |Tij ∩Ti′ j′ ) = q ◦ (γi′ j ′ |Tij ∩Ti′ j′ ). Hence it follows that γij |Tij ∩Ti′ j′ = γi′ j ′ |Tij ∩Ti′ j′ . S Since T = (i,j)∈I×J Tij , the set of morphisms {γij }(i,j)∈I×J defines an S-morphism γ : T → Z such that γ|Tij = γij for every (i, j) ∈ I × J. It is now clear that p ◦ γ = α and q ◦ γ = β. These arguments show that Z is a fiber product of X and Y over S.

Step 3. Let U = {Ui }i∈I and V = {Vj }j∈J be respectively open coverings of X and Y . Suppose that a fiber product of (Ui , f |Ui ) and (Vj , g|Vj ) over S exists for every (i, j) ∈ I × J. Then a fiber product of (X, f ) and (Y, g) over S exists. Proof. Set Λ = I × J. For λ = (i, j), let (Zλ , pλ , qλ ) be a fiber product of (Ui , f |Ui ) and (Vj , g|Vj ), which exists by the assumption. For λ′ = (i′ , j ′ ), −1 ′ ′ by Step 1, Wλλ′ , which we define to be p−1 λ (Ui ∩ Ui ) ∩ qλ (Vj ∩ Vj ), is a fiber product of Ui ∩ Ui′ and Vj ∩ Vj ′ over S. Further, Wλ′ λ := p−1 ′ λ (Ui ∩ Ui′ ) ∩ qλ−1 ′ (Vj ∩ Vj ′ ) is also a fiber product of (Ui ∩ Ui′ ) and (Vj ∩ Vj ′ ) over S. Meanwhile, by the definition of fiber product, these two fiber products are isomorphic over S by a uniquely determined S-isomorphism. Hence ∼ there exists an S-isomorphism θλ′ λ : Wλλ′ −→ Wλ′ λ . These isomorphisms {θλ′ λ }λ,λ′ ∈Λ satisfy the conditions (i) θλλ′ ◦ θλ′ λ = (idZλ )|Wλλ′ , (ii) θλ′′ λ′ ◦ θλ′ λ = θλ′′ λ on Wλλ′ ∩ Wλλ′′ . Hence, by patching these local data, there exist an S-scheme Z, its open ∼ covering W = {Wλ }λ∈Λ and S-isomorphisms {τλ : Wλ −→ Zλ }λ∈Λ such that θλ′ λ = τλ′ ◦ τλ−1 on Wλλ′ . On the other hand, since pλ′ ◦ θλ′ λ = pλ and qλ′ ◦ θλ′ λ = qλ on Wλλ′ , we have pλ ◦ τλ = pλ′ ◦ τλ′ and qλ ◦ τλ = qλ′ ◦ τλ′ on Wλ ∩ Wλ′ . This implies that there exist S-morphisms p : Z → X and q : Z → Y such that p|Wλ = pλ ◦ τλ and q|Wλ = qλ ◦ τλ . fλ = p−1 (Ui ) ∩ q −1 (Vj ). We show For every λ = (i, j) ∈ I × J, set W ∼ f that Wλ = Wλ . Since Wλ = Zλ , by Step 2, it then follows that (Z, p, q) is a fiber product of (X, p) and (Y, q). For any λ′ = (i′ , j ′ ), we have −1 fλ ∩ Wλ′ = (τλ′ )−1 (p−1 W λ′ (Ui ) ∩ qλ′ (Vj )) −1 ′ ′ = (τλ′ )−1 (p−1 λ′ (Ui ∩ Ui ) ∩ qλ′ (Vj ∩ Vj )).

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fλ ∩ Wλ′ is a fiber product of Ui ∩ Ui′ and Vj ∩ Vj ′ . Hence, by Step 1, W fλ ∩ Wλ′ = (τ ′ )−1 (Wλ′ λ ), and hence that W fλ ∩ Wλ′ = This implies that W λ fλ = Wλ . Wλ ∩ Wλ′ . Since λ′ ∈ I × J is arbitrary, it follows that W Step 4. Let S = {Sa }a∈A be an open covering of S, let Xa = f −1 (Sa ) and let Ya = g −1 (Sa ). If there exists a fiber product of (Xa , f |Xa ) and (Ya , g|Ya ) over Sa for every a ∈ A then a fiber product of (X, f ) and (Y, g) over S exists. Proof. If T, T ′ are Sa -schemes, they are considered as S-schemes with Sa structure morphisms composed with the natural open immersion Sa → S. This implies that Xa ×Sa Ya ∼ = Xa ×S Ya . Since {Xa }a∈A and {Ya }a∈A are open coverings of X and Y , respectively, it suffices by Step 3 that Xa ×S Yb exists for arbitrary a, b ∈ A. Set Xab = Xa ∩ Xb = f −1 (Sa ∩ Sb ) and Yab = Ya ∩ Yb = g −1 (Sa ∩ Sb ). By Step 1, there exists a fiber product Zab := Xab ×S Yab . We show that Zab ∼ = Xa ×S Yb . For S-morphisms α : T → Xa and β : T → Yb , we have f (α(T )) = g(β(T )) ⊆ Sa ∩ Sb , whence α(T ) ⊆ Xab and β(T ) ⊆ Yab . So, there exists an S-morphism γ : T → Zab such that α = pab ◦ γ and β = qab ◦ γ, where pab : Zab → Xab and qab : Zab → Yab are projections. Hence Zab ∼ = Xa ×S Yb . Step 5. Suppose that X, Y, S are affine schemes with coordinate rings A, B, R, respectively. Let Z be the affine scheme Spec (A ⊗R B) and let p : Z → X (resp. q : Z → Y ) be the S-morphism induced by the natural R-algebra homomorphism A → A ⊗R B (resp. B → A ⊗R B) defined by a 7→ a ⊗ 1 (resp. b 7→ 1 ⊗ b). Then Z ∼ = X ×S Y . Proof. Let T be an S-morphism. We show that the mapping Φ(T ) : HomS (T, Z) → HomS (T, X) × HomS (T, Y ),

γ 7→ (p ◦ γ, q ◦ γ)

is a bijection. Let T = {Ti }i∈I be an affine open covering of T . Then Φ(Ti ) is a bijection for every i ∈ I by Proposition 1.9.5. Let γ, γ ′ ∈ HomS (T, Z) such that p ◦ γ = p ◦ γ ′ and q ◦ γ = q ◦ γ ′ . Then γ|Ti = γ ′ |Ti since Φ(Ti ) is a bijection, which implies γ = γ ′ . So, Φ(T ) is an injection. Suppose that (α, β) ∈ HomS (T, X) × HomS (T, Y ). Then, for every i ∈ I, there exists γi ∈ Hom(Ti , Z) such that p ◦ γi = α|Ti and q ◦ γi = β|Ti . For arbitrary i, j ∈ I, γi |Ti ∩Tj = γj |Ti ∩Tj because Φ(Ti ∩ Tj ) is injective. Hence there exists an S-morphism γ ∈ HomS (T, Z) such that γi = γ|Yi for every i ∈ I. Now it is clear that Φ(T )(γ) = (α, β).

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1.9.5

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Reviews on sheaf theory

Sheaves on affine schemes are introduced in subsection 1.3.1 for an explanation of structure sheaves OX , and quasi-coherent and coherent OX -Modules are also introduced in subsection 1.6.4. But the presentation is simple and limited. To prove the finite dimensionality of cohomology groups of coherent OX -Modules over a projective (or more generally, complete) algebraic variety X, we need some basic definitions and results of the sheaf theory. In this subsection, we give them in a way as efficient as possible. The readers who would not be satisfied with this presentation or like to learn the theory more deeply are referred to [31] or [EGA]. Let X be a topological space and let TX be the category of open sets of X. Namely, an object of TX is an open set of X and a morphism is the ′ ′ inclusion iU U ′ : U → U which exists if U ⊆ U . A presheaf P on X is a contravariant functor P : TX → (Sets). Namely, P assigns a set P(U ) to ′ ′ each open set U of X and the restriction map ρU U : P(U ) → P(U ) to a U morphism iU U ′ . The assignment satisfies the conditions that (i) ρU = idP(U ) U ′′ U′ U ′′ ′ ′′ and (ii) ρU = ρU ◦ ρU ′ if U ⊆ U ⊆ U . If the category (Sets) is replaced by the category (Ab) of abelian groups or the category (Rings) ′ of commutative rings, the morphism ρU U should be understood to be not only a mapping of sets but also a homomorphism of abelian groups or commutative rings. A presheaf P is a sheaf if for any U ∈ TX , i.e., an open set U of X, and an open covering U = {Uλ | λ ∈ Λ} of U , the following sequence is exact: (1)

P(U ) −→

Y λ

P(Uλ )

(2) −→ Y Y (3)

−→

λ

µ

P(Uλµ ),

where the arrows (1), (2) and (3) are respectively Y (1) P(U ) ∋ x 7→ (ρλ (x)) ∈ P(Uλ ) λ

(2)

Y λ

(3)

Y λ

∈

YY

P(Uλ ) ∋ (xλ ) 7→ ρµλµ (xµ ) ∈

YY

P(Uλ ) ∋ (xλ ) 7→

ρλλµ (xλ )

λ

λ

µ

µ

P(Uλµ ) P(Uλµ ).

Uλ λ We employ here the abbreviations ρλ = ρU Uλ and ρλµ = ρUλµ . The exactness of the sequence means that the arrow (1) is injective, that is to say, x = y if Q ρλ (x) = ρλ (y) for all λ and that for (xλ ) ∈ λ P(Uλ ), there exists x ∈ P(U ) such that xλ = ρλ (x) if and only if xλ |Uλµ = xµ |Uλµ for all λ, µ ∈ Λ.

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If P, Q are presheaves (or sheaves) on X, a morphism (or a homomorphism) of presheaves (or sheaves) is a morphism of functors u : P → Q, i.e., for every U ∈ TX , a mapping u(U ) : P(U ) → Q(U ) is given so that ′ ′ U′ ′ ρU U,Q ◦ u(U ) = u(U ) ◦ ρU,P for U ⊆ U . For a point x ∈ X, let TX,x = {U ∈ TX | x ∈ U }. Let P be a presheaf ′ ′ ′ on X. Then {P(U ), ρU U | U, U ∈ TX,x , U ⊆ U } is an inductive system. The inductive limit Px := limU ∈T P(U ) is the stalk of P at x ∈ X. We −→ X,x define a presheaf a P on X as follows. Let U be an open set of X and let U = {Uλ }λ∈Λ be an open covering of U . For the open covering U, define P(U) by ( ) Y P(Uλ ) | (aλ )x = (aµ )x , ∀x ∈ Uλµ , λ, µ ∈ Λ , P(U) = (aλ ) ∈ λ∈Λ

where (aλ )x is the image of aλ in Px . If an open covering U ′ = {Uλ′ ′ }λ′ ∈Λ′ of X is a refinement of U such that Uλ′ ′ ⊆ Uf (λ′ ) with a mapping f : ′ Λ′ → Λ, define a mapping pU U ′ : P(U) → P(U ) by (aλ ) 7→ (af (λ′ ) |Uλ′ ′ ). U Then {P(U), pU ′ } is an inductive system indexed by the Cov(U ) of all open coverings of U ordered by refinement. The inductive limit lim P(U) is −→ a

U ∈Cov(U )

P(U ). Then one can show that a P is a sheaf, called the sheafification of P. There is a morphism of presheaves α : P → a P which maps a ∈ P(U ) to the image of ({U }, a) in a P(U ), where {U } is an open covering of U . The morphism α induces the isomorphism Px ∼ = (a P)x for every x ∈ X. One can show that if β : P → F is a morphism of presheaves with a sheaf F on X then β splits as β = γ ◦ α, where γ : a P → F is a uniquely determined morphism of sheaves. In particular, a F = F if F is a sheaf. e=` There is a more intuitive construction of a P. Set P x∈X Px . For an open set U of X and a ∈ P(U ), define a subset V (U, a) = {ax | x ∈ U } e where ax is the image of a in Px . Then the collection {V (U, a) | of P, e In U ∈ TX , a ∈ P(U )} forms a neighborhood basis of a topology on P. fact, if V (U1 , a1 ) ∩ V (U2 , a2 ) ̸= ∅, there exists x ∈ U1 ∩ U2 such that (a1 )x = (a2 )x . Hence there exists an open set U3 ⊂ U1 ∩U2 such that x ∈ U3 U2 1 and ρU U3 (a1 ) = ρU3 (a2 ) := a3 . Then V (U3 , a3 ) ⊂ V (U1 , a1 ) ∩ V (U2 , a2 ). e → X be the projection, i.e., Px ∋ ax 7→ x ∈ X. Then π is Let π : P locally homeomorphic. Define a presheaf P ′ by assigning to U ∈ TX the e | s is continuous and π|U ◦ s = idU }. Then P ′ is set P ′ (U ) = {s : U → P a sheaf and coincides with a P. The elements of P ′ (U ) are called sections e over U . (or local sections) of P

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Let (X, A) be a ringed space with a structure sheaf A of rings. A presheaf P on X is a presheaf of A-modules if P(U ) is an A(U )-module and ′ the restriction ρU U ′ : P(U ) → P(U ) is a module homomorphism compatible U with the restriction ρU ′ : A(U ) → A(U ′ ) for U ′ ⊆ U . The sheafification a P is then a sheaf of A-modules. We often abbreviate a sheaf of A-modules as an A-Module with the capital letter M. f g We say that a sequence of presheaves of A-modules F1 −→ F2 −→ F3 is exact if, for every open set U , f (U )

g(U )

F1 (U ) −→ F2 (U ) −→ F3 (U )

is exact, i.e., Ker g(U ) = Im f (U ). Then, for all x ∈ X, the induced sequence of Ax -modules fx

gx

(F1 )x −→ (F2 )x −→ (F3 )x

is an exact functor, i.e., it maps is exact. This fact holds because limU ∈T −→ X,x exact sequences to exact sequences. Let f : F → G be a homomorphism of A-Modules. Define a presheaf K by K(U ) = Ker f (U ). Then K is an A-Module, which we denote by Ker f . Define a presheaf H by H(U ) = Coker f (U ). The presheaf H is not necessarily a sheaf. Its sheafification a H is the cokernel Coker f . It follows that a sequence of A-Modules f

g

0 −→ F −→ G −→ H −→ 0

is an exact sequence of A-Modules if and only if the sequence of stalks fx

gx

0 −→ Fx −→ Gx −→ Hx −→ 0

is an exact sequence of Ax -modules for all x ∈ X (see Problem 14). This further entails that a long sequence of A-Modules fi−1

fi

fi+1

· · · −→ Fi−1 −→ Fi −→ Fi+1 −→ Fi+2 −→ · · ·

is an exact sequence of A-Modules if and only if the sequence of stalks (fi−1 )x

(fi )x

(fi+1 )x

· · · −→ (Fi−1 )x −→ (Fi )x −→ (Fi+1 )x −→ (Fi+2 )x −→ · · ·

is an exact sequence of Ax -modules for all x ∈ X.

If F is an A-Module, a homomorphism u : A → F of A-Modules is given by an element s ∈ F(X) in such a way that u(U ) : A(U ) → F(U ) maps a ∈ A(U ) to a(s|U ) ∈ F(U ). Conversely, s ∈ F(X) gives a homomorphism u : A → F. We denote by A(I) a free A-Module such that A(I) (U ) = ⊕I A(U ) for an open set U . The following definitions for an A-Module M appear frequently in the text.

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(1) M is quasi-coherent if, for every x ∈ X, there exists an exact sequence (A|U )(J) −→ (A|U )(I) −→ M|U −→ 0,

for an open neighborhood U of x and (not necessarily finite) sets I, J. (2) M is finitely generated if, for every point x ∈ X, there exists an exact sequence (A|U )n −→ M|U −→ 0, for an open neighborhood U of x and n > 0. (3) M is coherent if (i) M is finitely generated and (ii) for any open set U , any integer n > 0 and any (A|U )-homomorphism φ : (A|U )n → M|U , the (A|U )-Module Ker φ is finitely generated. Let F and G be A-Modules over a ringed space (X, A). If U ∈ TX then we can consider a presheaf U 7→ Γ(U, F) ⊗Γ(U,A) Γ(U, G), but it is not necessarily a sheaf on X. By the sheafification we obtain the tensor product F ⊗A G which is an A-Module. We also consider the functor U 7→ HomAU (F|U , G|U ), which is an A-Module. We denote this sheaf by HomA (F, G). For details, the readers are referred to [31, Chapter II, Section 5] or [EGA]. We state the following result. Lemma 1.9.14. Let (X, A) be a ringed space. Then the following assertions hold for A-Modules. (1) Let f

g

0 −→ F −→ G −→ H −→ 0

(2) (3) (4) (5)

be an exact sequence of A-Modules. If two of F, G and H are coherent then the rest is coherent. A-Modules F, G are coherent if and only if F ⊕ G is coherent. For an A-homomorphism of coherent A-Modules f : F → G, Ker f, Coker f and Im f are coherent. For coherent A-Modules F, G, F ⊗A G and HomA (F, G) are coherent A-Modules. For A-Modules F and G, if F is coherent, then HomA (F, G)x ∼ = HomAx (Fx , Gx ).

Proof. We only prove the assertion (1). Case G, H are coherent. Since G is finitely generated, for all x ∈ X, u there exist an open neighborhood U of x and an exact sequence (A|U )p −→

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G|U −→ 0, where p > 0. Since H is coherent, the kernel of g|U ◦ u : (A|U )p → H|U is finitely generated. Hence, after replacing U by a smaller open neighborhood of x, we have the following commutative diagram g|U ◦u

v

(A|U )q −→ (A|U )p −→ H|U −→ 0 ↓w ↓u ∥

0 −→ F |U

f |U

g|U

−→ G|U

−→ H|U −→ 0,

where w is a homomorphism induced by u and w is surjective because it is so on each stalk. Hence F is finitely generated. Meanwhile, the condition (ii) for the coherence of F is clear because F is a subsheaf of a coherent sheaf G. So, F is a coherent sheaf. Case F, G are coherent. Since G is finitely generated, so is H. In order to show the condition (ii) of coherence, consider a homomorphism u : (A|U )n → H|U . We may replace U by a smaller open set to show that Ker u is finitely generated. Since u maps the free basis of (A|U )n to a finite set of elements of H(U ) which are in the image of g : G(U ) → H(U ) if U is replaced by a smaller one, we may assume that u splits as g|U

∃v

u : (A|U )n −→ G|U −→ H|U −→ 0. Since F is finitely generated, we may assume that there exists a surjective e homomorphism (A|U )m −→ F|U → 0. Consider the following commutative diagram

0 −→

0 ↑

F |U ↑e

0 −→ (A|U )m

f |U

−→ h

G|U ↑t

g|U

−→ ↖v k

H|U ↑u

−→ 0

−→ −→ (A|U )n+m s (A|U )n −→ 0 r ←− ←− ↑w ↑kw (A|U )p

id

−→ (A|U )p

,

where h, k, r, s satisfy r◦h = 1(= id), k◦s = 1, k◦h = r◦s = 0, h◦r+s◦k = 1, and t = f |U ◦ e ◦ r + v ◦ k. Hence f |U ◦ e = t ◦ h, g ◦ t = u ◦ k, v = t ◦ s. Since G is coherent, Ker t is finitely generated, the homomorphism w exists after shrinking U if necessary. Therefore, the top two lines and the left two columns are exact. This implies that the right column is exact. Hence H is coherent.

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Case F, H are coherent. We show first that G is finitely generated. For x ∈ X there exist an open neighborhood U of x and surjective hoe u momorphisms (A|U )m −→ F|U and (A|U )n −→ H|U . As in the previous commutative diagram, define t : (A|U )m+n → G|U . By making use of the five lemma, it is easy to see that t is surjective. We prove next the condition (ii) of coherence. Let U be an open set and let u : (A|U )n → G|U be a homomorphism of (A|U )-Modules. We show that Ker u is finitely generated. If we take the composite (g|U ) ◦ u : (A|U )n → H|U , Ker (g|U ◦ u) is finitely generated because H is coherent. Hence after shrinking U if necessary, we may assume that there exists a surjective homomorphism of (A|U )-Modules v : (A|U )m → Ker (g|U ◦ u). Then there exists a homomorphism of (A|U )Modules w : (A|U )m → F|U . Since F is coherent, after shrinking U again if necessary, we may assume that there exists a surjection p : (A|U )ℓ → Ker w. Thus we have the following commutative diagram 0 −→

F |U ↑w

f |U

−→ v

g|U

G|U ↑u

−→ H|U −→ 0 ∥

g|U ◦u

(A|U )m −→ (A|U )n −→ H|U ↑p ↑v ◦ u id

(A|U )ℓ −→ (A|U )ℓ

.

By the diagram chasing, we know that the middle column is an exact sequence. Let f : X → Y be a continuous map of topological spaces. Let TX (resp. TY ) be the category of open sets of X (resp. Y ). For a presheaf of abelian groups P on X, let f∗ (P) be a presheaf on Y defined by V ∈ TY 7→ f∗ (P)(V ) = P(f −1 (V )).

If F is a sheaf of abelian groups on X, f∗ (F) is a sheaf of abelian groups on Y . In fact, if V = {Vλ | λ ∈ Λ} is an open covering of an open set V ∈ TY then f −1 (V) = {f −1 (Vλ | λ ∈ Λ} is an open covering of f −1 (V ) ∈ TX . Since F is a sheaf on X, the sequence (1)

F(f −1 (V )) −→

Y λ

F(f −1 (Vλ ))

(2) −→ Y (3)

−→

λ,µ

F(f −1 (Vλ ) ∩ f −1 (Vµ ))

is exact, hence the sequence (1)

f∗ (F)(V ) −→

Y λ

f∗ (F)(Vλ )

(2) −→ Y (3)

−→

λ,µ

f∗ (F)(Vλ ∩ Vµ )

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is exact. So, f∗ F is a sheaf on Y , which we call the direct image of F by f . Let U be an open set of X. Let IU = {V ∈ TY | f (U ) ⊆ V }. With the reversed order of inclusion, i.e., V ≤ V ′ ⇔ V ′ ⊆ V , IU is an inductive set, and IU ⊆ IU ′ if U ′ ⊆ U . Let Q be a presheaf of abelian groups on Y . Define a presheaf f • Q on X by U ∈ TX 7→ f • Q(U ) = lim Q(V ). −→ V ∈IU

′

If U ⊆ U , we have a natural restriction homomorphism

f • Q(U ) = lim Q(V ) −→ lim Q(V ) = f • Q(U ′ ). −→ −→ V ∈IU

V ∈IU ′

If G is a sheaf of abelian groups on Y , f • G is not necessarily a sheaf. So, we denote by f ∗ G the sheafification of f • G. We call f ∗ G the inverse image of G by f . Let Xe,ab be the category of sheaves of abelian groups on X. Then it can be shown that HomXe,ab (f ∗ G, F) ∼ = HomY e,ab (G, f∗ F). In fact, if φ : G → f∗ F is a homomorphism of sheaves, we have φ(V ) : G(V ) → F(f −1 (V )), V ∈ TY .

Fix U ∈ TX and let V run in IU to obtain lim φ(V )

→−→ φ′ (U ) : f • G(U ) = lim G(V ) − −→ V ∈IU

lim F(f −1 (V )) −→ F(U ), −→

V ∈IU

which commutes with the restriction morphism. Thus we obtain a homomorphism of presheaves φ′ : f • G → F, which factors as φ′ : f • G → a

φ′

f ∗ G −→ F since f ∗ G = a (f • G). We denote a φ′ by φ♯ . Conversely, if given ψ : f ∗ G → F, then we have for V ∈ TY ψ ′ (V ) : G(V ) →

lim −→

V ′ ∈If −1 (V )

G(V ′ ) = f • G(f −1 (V ))

ψ(f −1 (V )

−→

F(f −1 (V ),

which gives ψ ′ : G → f∗ F. We denote ψ ′ by ψ♭ . Let (X, A) and (Y, B) be ringed spaces. A morphism of ringed spaces Φ = (f, φ) : (X, A) → (Y, B) is a pair consisting of a continuous map f : X → Y and a sheaf homomorphism of rings φ : B → f∗ A. If F is an A-Module then f∗ F is an f∗ A-Module on Y . Hence f∗ F is considered as a B-Module via φ, which we denote by Φ∗ F and call the direct image of F by Φ. If G is a B-Module, then f ∗ G is an f ∗ B-Module. Since there is a morphism of sheaves of rings φ♯ : f ∗ B → A, we set Φ∗ G := f ∗ G ⊗f ∗ B A

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which is an A-Module. We call Φ∗ G the inverse image of G by Φ. We have a bijective correspondence HomA (Φ∗ G, F) ∼ = HomB (G, Φ∗ F). In subsections 1.3.2 and 1.6.4, we observed the structure sheaf OX on an affine scheme X = Spec A, and quasi-coherent and coherent OX -Modules attached to A-modules. We summarize the results without proofs which we need in the present book. For the proof, see [31] or [EGA]. Theorem 1.9.15. Let X = Spec A be an affine scheme. Then the following assertions hold. f is quasi-coherent. If F is a quasi(1) Let M be an A-module. Then M ^ coherent OX -Module, then Γ(X, F) is an A-module and F ∼ F). = Γ(X, ∼ Γ(X, OX ). The correspondence M 7→ M f gives an In particular, A =

equivalence of the category of A-modules and the category of quasiβ α coherent OX -Modules. In particular, if 0 −→ L −→ M −→ N −→ 0 is an exact sequence of A-modules, then we have an exact sequence of quasi-coherent OX -Modules α e f β e −→ e −→ 0. 0 −→ L M −→ N e

(2) Assume that A is noetherian. Then the above correspondence induces an equivalence of the category of finite A-modules and the category of coherent OX -Modules. 1.9.6 1.9.6.1

ˇ Cech cohomology of sheaves of abelian groups ˇ Cech cohomology

In subsection 1.3.1, we defined a structure of local ringed space on an affine scheme X = Spec A. Generalizing this notion, the notion of ringed space is introduced as a pair (X, A) of a topological space X and a fixed sheaf of rings A on X. When A is a sheaf of local rings, we denote it by OX instead of A. We consider mostly sheaves of A-modules (abbreviated as AModules). If A is a constant sheaf Z, the Z-Modules are sheaves of abelian groups. Let U = {Ui | i ∈ I} be an open covering of X and let F be an AModule. For an (n + 1)-tuple αn = (i0 , i1 , . . . , in ) ∈ I n+1 , set Uαn = Ui0 ∩ Ui1 ∩ · · · ∩ Uin . Define C n (U, F) and dn : C n (U, F) → C n+1 (U, F) as

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follows: n

C (U, F) = {f = (fαn ) ∈

(dn f )αn+1 =

n+1 X j=0



Y αn ∈I n+1

Γ(Uαn , F) | fβn = sgn

αn βn

 fαn

if βn is a permutation of αn }

(−1)j (f(αn+1 ,∨j) |Uαn+1 ), ∨

where (αn+1 , ∨j) = (i0 , . . . , ij , . . . , in+1 ) if αn+1 = (i0 , . . . , in+1 ). It is easy to show that dn+1 ◦ dn = 0. Hence we have a complex d0

d1

dn−1

dn

0 −→ C 0 (U, F) −→ C 1 (U, F) −→ · · · −→ C n (U, F) −→ · · · . Its nth cohomology group Ker dn /Im dn−1 is denoted by H n (U, F). In d0

particular, H 0 (U, F) = Γ(X, F) because C 0 (U, F) −→ C 1 (U, F) coincides with Y −ρ′ Y Γ(Ui , F) −→ Γ(Uij , F), − ρ′ ((fi )) = (fj |Uij − fi |Uij ), i,j

i∈I

′

with Ker ρ = Γ(X, F) and H 0 (U, F) = Ker (−ρ′ ). Let V = {Vj | j ∈ J} be an open covering of X such that U ≤ V, i.e., V is finer than U. Then there exists a mapping τ : J → I such that Vj ⊆ Uτ (j) for every j ∈ J. Define τ n : C n (U, F) → C n (V, F) by τ n (f )j0 ,...,jn = fτ (j0 ),...,τ (jn ) |Vβ , where β = (j0 , . . . , jn ). Then we have a commutative diagram of complexes with a homomorphism of complexes (τ n )n≥0 : C • (U, F) → C • (V, F), d0

d1

dn−1

dn

d0

d1

dn−1

dn

0 −→ C 0 (U, F) −→ C 1 (U, F) −→ · · · −→ C n (U, F) −→ · · · ↓ τ0 ↓ τ1 ↓ τn

0 −→ C 0 (V, F) −→ C 1 (V, F) −→ · · · −→ C n (V, F) −→ · · · .

Then (τ n ) induces a homomorphism of cohomologies τ n : H n (U, F) → H n (V, F) for n ≥ 0. This homomorphism seemingly depends on the choice of τ . If τ ′ : J → I is another mapping such that Vj ⊆ Uτ ′ (j) for every j ∈ J, define σ n : C n+1 (U, F) → C n (V, F) by σ n (f )j0 ,...,jn =

n X

(fτ (j0 ),...,τ (jk ),τ ′ (jk ),...,τ ′ (jn ) )|Vj0 ,...,jn .

k=0

Then we have 0

n

τ ′ − τ 0 = σ 0 · d0 , τ ′ − τ n = σ n · dn + dn−1 · σ n−1 (n ≥ 1).

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n

Namely, (σ n )n≥0 is a chain homotopy from (τ ′ )n≥0 to (τ n )n≥0 . Then the n homomorphisms τ n , τ ′ : H n (U, F) → H n (V, F) coincides with each other for every n ≥ 0. This shows that the homomorphism τ n does not depend on the choice of τ . So, denote this homomorphism by τV,U . Let Cov(X) be the partially ordered set of open coverings of X with order defined by refinement U ≤ V. Then {H n (U, F) | U ∈ Cov(X)} is an inductive system, ˇ n (X, F) and hence limU ∈Cov(X) H n (U, F) exists. We denote the limit by H −→ ˇ call it the nth Cech cohomology of F over X.

Let X be an algebraic variety defined over a field k, which is a local ringed space with the structure sheaf OX . For a quasi-coherent OX -Module F, we can define a cohomology theory (H n (X, F))n≥0 defined by an injective resolution of F in the category of OX -Modules (see [31, Chapter III]). We state some related results without further explanations. Theorem 1.9.16. Let X be an algebraic variety defined over k and let F be a quasi-coherent OX -Module. Let U be an affine open covering of X. Then H n (X, F) and H n (U, F) are Γ(X, OX )-isomorphic for every n ≥ 0. ˇ n (X, F) for all n ≥ 0. In particular, H n (X, F) ∼ =H

Theorem 1.9.17 (Theorem of Serre). Let X be an affine scheme ˇ n (X, F) = and let F be a quasi-coherent OX -Module. Then H n (X, F) = H 0 for all n ≥ 1. Conversely, if X is a quasi-compact, separated scheme such that the underlying topological space of X is a noetherian space and if H n (X, F) = 0 for all n > 0 and all quasi-coherent OX -Module F then X is an affine scheme. Proof. For the first statement, see [31, Chapter III, Theorem 3.5]. For the second statement, see [ibid., Chapter III, Theorem 3.7]. In the subsequent arguments we need the following two theorems, which we state for an algebraic variety, though the results are valid for more general topological spaces. Theorem 1.9.18. Let X be an algebraic variety over a field k and let f

g

0 −→ F −→ G −→ H −→ 0 be an exact sequence of OX -Modules.

Then there exists a long exact

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sequence of cohomologies 0

∂

→ H 0 (X, F) → H 0 (X, G) → H 0 (X, H) → H 1 (X, F) → ∂

∂

∂

∂

H 1 (X, G) → H 1 (X, H) → H 2 (X, F) → H 2 (X, G) → H 2 (X, H) →

H 3 (X, F) → H n+1 (X, F) →

··· ···

→ H n (X, F) → H n (X, G) → H n (X, H) → ··· .

Further, H 0 (X, F) = F(X), etc. Theorem 1.9.19 (Vanishing theorem of Grothendieck). Let X be an algebraic variety of dimension n over a field k and let F be an OX Module. Then H i (X, F) = 0 for all i > n. Proof. See [22] and [31, Chapter III, Theorem 2.7]. 1.9.6.2

Coherent sheaf cohomologies over projective varieties L Let k be a field and let A = k-algebra such that n≥0 An be a graded L A0 = k and A+ is generated by A1 . Let M = n∈Z Mn be a graded Amodule. Namely M is a direct sum of the set Mn (n ∈ Z) of elements of degree n and becomes an A-module with the condition that Aℓ Mn ⊆ Mℓ+n for every ℓ, n ∈ Z. Here we assume that Aℓ = 0 if ℓ < 0. Let f : M → N be a homomorphism of A-modules. Then f is a graded homomorphism of degree d if f (Mn ) ⊆ Nn+d for every n ∈ Z. Denote by HomA (M, N )d the set of graded A-homomorphisms of degree d. For a ∈ Aℓ and f ∈ HomA (M, N )d , we have (af )(Mn ) ⊆ aNn+d ⊆ Nn+(ℓ+d) , whence Aℓ HomA (M, N )d ⊆ HomA (M, N )ℓ+d . This shows that HomA (M, N ) = L can define a graded Ad∈Z HomA (M, N )d is a graded A-module. We L module structure on M ⊗A N as a direct sum n∈Z (M ⊗A N )n , where L (M ⊗A N )n = p+q=n (Mp ⊗A Nq ). In fact, if a ∈ Aℓ , x ∈ Mp , y ∈ Nq then a(x ⊗ y) = ax ⊗ y = x ⊗ ay ∈ (M ⊗A N )ℓ+n . The graded k-algebra A as above is noetherian if and only if A1 is a finite k-vector space (see Lemma 1.4.3). We assume that A is noetherian, and all graded A-modules are finite A-modules. Then M is generated by finitely many homogeneous elements {m1 , . . . , mr }, where di = deg mi . For an integer ℓ, let A(ℓ) be a graded A-module such that A(ℓ)n = Aℓ+n . An A-module homomorphism f : A → M which maps 1 to a homogeneous element m of degree d belongs to HomA (A, M )d . We can view this homomorphism f as an A-module homomorphism A(−d) → M of degree 0. In fact, the element 1 in A(−d) has degree d and hence f is viewed to have

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degree 0. By this observation, there is a graded A-module homomorphism of degree 0 φ:

r M i=1

A(−di ) → M → 0,

which is surjective. Ker φ is a graded finite A-module. By the same argument, we have an exact sequence s M j=1

φ′

A(−ej ) −→

r M i=1

φ

A(−di ) −→ M −→ 0,

where φ and φ′ have degree 0. PN Assume that A is noetherian and write A1 = i=0 kai with a k-basis {a0 , . . . , aN } of A1 . Then A = k[a0 , . . . , aN ] and hence there is a surjective graded homomorphism θ of degree 0 from a polynomial ring k[x0 , . . . xN ] to A such that θ(xi ) = ai . Let I = Ker θ. Let X = Proj A. Then θ induces an isomorphism ι : X → V+ (I) ⊂ PN = Proj (k[x0 , . . . , xN ]) such that ι−1 (D+ (xi )) = D+ (ai ). Hence ι is a closed immersion. Let M be a graded A-module. For a homogeneous element a ∈ A+ ^ let (Ma )0 = M [a−1 ]0 , which is an A[a−1 ]0 module. So, (M a )0 is a quasicoherent sheaf on D+ (a) = Spec A[a−1 ]0 . In particular, we have quasifi | 0 ≤ i ≤ N } with Mi = M [a−1 ]0 , each of which is coherent sheaves {M i SN defined over D+ (ai ), where X = i=0 D+ (ai ) is an affine open covering of X. For 0 ≤ i, j ≤ N , D+ (ai aj ) = D+ (ai ) ∩ D+ (aj ) and there exists an isomorphism of A[(ai aj )−1 ]0 -modules ∼

∼

θji : Mi ⊗A[a−1 ]0 A[(ai aj )−1 ]0 → M [(ai aj )−1 ]0 → Mj ⊗A[a−1 ]0 A[(ai aj )−1 ]0 i

j

such that, for m ∈ Md , m m m 7 ⊗ θji : d ⊗ 1 7→ d → ai ai adj



aj ai

d .

∼ fi |D (a a ) → fj |D (a a ) This implies that the isomorphisms θeji : M M + i j + i j fi } and form a quasi-coherent OX patch together quasi-coherent sheaves {M f. If M is a finite A-module then M f is a coherent OX -Module. In module M −1 fact, coherence is a local condition and M [a−1 i ]0 is a finite A[ai ]0 -module if M is a finitely generated, graded A-module (see Theorem 1.9.15).

Lemma 1.9.20. The following assertions hold. g Then OX (ℓ) is an invertible sheaf on X. (1) Let OX (ℓ) = A(ℓ).

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(2) Let ℓ, m be integers. Then OX (ℓ) ⊗OX OX (m) ∼ = OX (ℓ + m) and 22 ∼ HomOX (OX (ℓ), OX ) = OX (−ℓ). (3) Let F be a quasi-coherent OX -Module. For all ℓ ∈ Z, set F(ℓ) = L F ⊗OX OX (ℓ). We set Γ∗ (F) = ℓ∈Z Γ(X, F(ℓ)), which is a graded Γ∗ (OX )-module, hence a graded A-module. We then have F ∼ = Γ^ ∗ (F). If F is a coherent OX -module, Γ∗ (F) is a finite graded A-module.

−1 −1 ℓ ∼ Proof. (1) Since A(ℓ)[a−1 i ]0 = A[ai ]0 ·ai = A[ai ], OX (ℓ) is an invertible sheaf with transition functions fji = (aj /ai )ℓ with respect to the open SN covering X = i=0 D+ (ai ). (2) Since we have

Γ(D+ (ai ), OX (ℓ)) ⊗Γ(D+ (ai ),OX ) Γ(D+ (ai ), OX (m)) = A[a−1 ]0 · aℓ ⊗ −1 A[a−1 ]0 · am ∼ = A[a−1 ]0 · aℓ+m , i

i

A[ai ]0

i

i

i

i

it follows that OX (ℓ) ⊗OX OX (m) ∼ = OX (ℓ + m). Since OX |D+ (ai ) = −1 −1 −1 ^ ^ ^ ℓ m A[ai ]0 , OX (ℓ)|D+ (ai ) = A[ai ]0 · ai and OX (m)|D+ (ai ) = A[a i ]0 · a i , we have HomOX |D+ (ai ) (OX (ℓ)|D+ (ai ) , OX (m)|D+ (ai ) )

−1 ℓ m = HomA[a−1 ]0 (A[a−1 i ]0 · ai , A[ai ]0 · ai ) i

m−ℓ ∼ . = A[a−1 i ]0 · ai

∼ OX (m − ℓ). Hence it follows that HomOX (OX (ℓ), OX (m)) = (3) For the first assertion, see [31, Chapter II, Proposition]. For the L second assertion, we use a criterion that a graded A-module M = n∈Z Mn is a finite A-module if and only if (i) dimk Mn < ∞ for all n ∈ Z and there exists an integer n0 such that Mn = 0 for all n < n0 and (ii) there exist integers n1 and d > 0 such that Mn+d = Ad Mn for all n ≥ n1 . Since X = ∪ni=0 D+ (ai ), we can show that if F is a coherent OX -Module, Γ∗ (F) satisfies these two conditions (i) and (ii). See also [31, Chapter II, Theorems 5.17 and 5.19]. If there is a surjective k-homomorphism θ : k[x0 , x1 , . . . , xn ] → A such that θ(xi ) = ai , there is a closed immersion ι : X → Pn such that f ⊗O N e ∼ our situation, we have M ⊗A N ), and if M is a finite A-module, = (M^ X ^ ∼ f e HomO (M , N ) = HomA (M, N ). 22 In

X

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D+ (ai ) = ι−1 (D+ (xi )). So, the open covering UX := {D+ (ai ) | 0 ≤ i ≤ n} is the inverse image of the open covering U := {D+ (xi ) | 0 ≤ i ≤ n}. For a ˇ coherent OX -Module F, the Cech cohomologies H i (UX , F) and H i (U, ι∗ F) ˇ are the same for all i ≥ 0. This is because the Cech complexes C • (UX , F)

and C • (U, ι∗ F) are the same. Furthermore, ι∗ (F) is a coherent OPn Module. By Theorem 1.9.16, if dimk H i (Pn , ι∗ (F)) < ∞, it follows that dimk H i (X, F) < ∞, i.e., the k-vector space H i (X, F) has finite dimension, In order to show that dimk H i (Pn , ι∗ (F)) < ∞, we need the following result (see [31, Chapter III, Theorem 5.1]). Lemma 1.9.21. Let X = Pnk . Then the following assertions hold. L 0 (1) ℓ∈Z H (X, OX (ℓ)) is isomorphic to k[x0 , x1 , . . . , xn ] as graded kalgebras. In particular, H 0 (X, OX (ℓ)) = 0 if ℓ < 0. (2) H i (X, OX (ℓ)) = 0 for 0 < i < n and all ℓ ∈ Z. (3) H n (X, OX (ℓ)) ∼ = H 0 (X, OX (−n − 1 − ℓ)) for all ℓ ∈ Z. i (4) dimk H (X, OX (ℓ)) < ∞ for all 0 ≤ i ≤ n and ℓ ∈ Z. We prove the following result after all the above results. Theorem 1.9.22. Let X be a projective algebraic variety defined over a field k and let F be a coherent OX -Module. Then dimk H i (X, F) < ∞ for 0 ≤ i ≤ dim X and H i (X, F) = 0 if i > dim X. Proof. By the remark before Lemma 1.9.21, we may assume that X = Pn and A = k[x0 , . . . , xn ]. Let M = Γ∗ (F). Then M is a finite graded Amodule. By the argument before Lemma 1.9.20, there exists a graded homomorphism of degree 0 φ:

r M i=1

A(−di ) −→ M −→ 0.

Taking the associated OX -Modules, we have an exact sequence 0 −→ F1 −→

r M i=1

f

OX (−di ) −→ F −→ 0,

(1.24)

where F1 = Ker f . By Lemma 1.9.14, F1 is a coherent OX -Module, and by Lemma 1.9.21, the kernel and the cokernel of the homomorphism in the long exact sequence of cohomology groups for (1.23) ∂j : H j (X, F) −→ H j+1 (X, F1 ), 0 ≤ j ≤ n

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have finite ranks over k. We then say that ∂j is an isomorphism modulo k-vector spaces of finite rank. Since F1 is coherent, we can apply the same argument as above to construct a coherent OX -Module F2 and an isomorphism modulo k-vector spaces of finite rank ∂j+1 : H j+1 (X, F1 ) −→ H j+2 (X, F2 ), 0 ≤ j ≤ n. Repeat this argument until H n+1 (X, Fn+1−j ) = 0 by Theorem 1.9.19. This shows that H j (X, F) is a k-vector space of finite rank.

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1.10

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Problems to Chapter 1

1. Let A be an integral domain and let U = D(a) be an open set of X = Spec A. Show that Γ(U, OX ) = A[a−1 ]. S Answer. Γ(U, OU ) is defined as follows: Let U = i∈I D(ai ) be an open covering of U . Then Γ(U, OU ) = Ker ρ′ , where Y Y ρ′ : A[a−1 A[(ai aj )−1 ], i ] −→ i∈I

i̸=j

(ξi )i∈I 7→ (ξi |D(ai aj ) − ξj |D(ai aj ) )(i,j)∈I×I,i̸=j . −1 Replacing A by A[a−1 ] by A[(aai )−1 ], we may assume that i ] and A[a Si P U = X = Spec A. Then X = i∈I D(ai ) if and only if i∈I ai A = A. P Then there exists a finite subset J ⊂ I such that j∈J aj A = A. So, Q we may assume that I is a finite set. Let (bi /ani i )i∈I ∈ i∈I A[a−1 i ]. S We may assume that ni = n for all i ∈ I. Then i∈I D(ani ) = X. Replacing ai by ani , we may assume that n = 1. Then we have a P relation i∈I ai zi = 1. Further, (bi /ai )i∈I ∈ Ker ρ′ is equivalent to ai bj = aj bi for all i, j with i ̸= j. Here we use the assumption that A P is an integral domain. Let c = i∈I bi zi . Then we have   X X X ai c = ai bj zj = bi aj zj = bi  aj zj  = bi . j∈I

j∈I

j∈J

Hence c = bi /ai for all i ∈ I. 2. Prove the following assertions. (1) Chinese remainder theorem. Let R be a ring and let I1 , . . . , In be pairwise prime ideals of R, i.e., Ii + Ij = R for i ̸= j. Then there exists a ring isomorphism R/(I1 ∩ · · · ∩ In ) ∼ = (R/I1 ) × · · · × (R/In ), where I1 ∩ · · · ∩ In = I1 · · · · · In . √ (2) Suppose that Ii = mi with a maximal ideal for 1 ≤ i ≤ n, where m1 , . . . , mn are pairwise distinct maximal ideals. Then Ii + Ij = R for i ̸= j. ∨

Answer. (1) We first show that I1 · · · Ii · · · In + Ii = R for 1 ≤ i ≤ n. (j) In fact, fix i. Since Ij + Ii = R for j ̸= i, we find bj ∈ Ij and ai ∈ Ii (j) such that bj + ai = 1. Then we have (1)

(i−1)

1 = (b1 + ai ) · · · (bi−1 + ai

(i+1)

)(bi+1 + ai

= b1 · · · bi−1 bi+1 · · · bn + ci , ci ∈ Ii .

(n)

) · · · (bn + ai )

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This implies that I1 · · · Ii · · · In + Ii = R. Define a ring homomorphism φ : R → (R/I1 ) × · · · × (R/In ) by a 7→ (a + I1 , . . . , a + In ). Then Ker φ = I1 ∩ · · · ∩ In . We show that ∨

φ is surjective. For every 1 ≤ i ≤ n, since Ii + I1 · · · Ii · · · In = R, ∨

there exists an element bi such that bi ∈ I1 · · · Ii · · · In and 1 − bi ∈ Ii . For an element (a1 + I1 , . . . , an + In ) ∈ (R/I1 ) × · · · × (R/In ), set a = b1 a1 + · · · + bn an . Since b1 , . . . , bi−1 , bi − 1, bi+1 , . . . , bn ∈ Ii , we have a − ai = b1 a1 + · · · + bi−1 ai−1 + (bi − 1)ai + bi+1 ai+1 + · · · + bn an ∈ Ii . Hence φ(a) = (a1 + I1 , . . . , an + In ). For the last assertion, suppose that we have ideals I, J of R such that I + J = R. Then a + b = 1 for a ∈ I and b ∈ J. Then, for x ∈ I ∩ J, we have x = ax + bx ∈ I · J. This implies that I ∩ J ⊆ I · J. The opposite inclusion I · J ⊆ I ∩ J is clear. So, I ∩ J = I · J. For n ≥ 2, we proceed by induction on n. Assume that I2 ∩ · · · ∩ In = I2 · · · In . Since I1 + I2 · · · In = R, we have I1 ∩ I2 ∩ · · · ∩ In = I1 ∩ (I2 · · · In ) = I1 · (I2 · · · In ) = I1 · I2 · · · In .

So we are done. (2) Suppose that Ii + Ij ̸= R. Then there p exists a maximal ideal m √ such that Ii + Ij ⊆ m. Since Ii = mi and Ij = mj , it follows that mi , mj ⊆ m. Then mi = m = mj . This is a contradiction. Hence Ii + Ij = R. 3. Let R = k[x, y] be a polynomial ring in two variables over a field k and let I = (x2 , xy). Verify the following assertions (see the appendix for primary decomposition). (1) I is not a primary ideal. (2) p I = (x) ∩ (x2 , y) is pthe irredundant primary decomposition. (3) (x) = (x) and (x2 , y) = (x, y). (x) is a minimal prime divisor and (x, y) is an embedded prime divisor. Answer. (1) xy ∈ I, x ̸∈ I but y n ̸∈ I for all n > 0. So, I is not a primary ideal. (2) The inclusion I ⊆ (x) ∩ (x2 , y) is clear. Let f ∈ (x) ∩ (x2 , y). Then f = xg = x2 h1 + yh2 with g, h1 , h2 ∈ k[x, y]. Since x | yh2 , we can write h2 = xh3 . Hence f = x2 h1 + xyh3 ∈ I. Here (x) is a prime ideal,

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p and (x2 , y) is a primary ideal because (x2 , y) = (x, y) is a maximal ideal and (x, y)2 ⊆ (x2 , y). Further it is clear that the decomposition is the irredundant one. (3) It is easy to show that (x) = (I : y), (x, y) = (I : x) and (x) ⊂ (x, y). 4. Prove Krull’s intersection theorem (see Theorem 1.9.10) according to the following steps. T (1) Let N = n≥0 an M . Then aN = N . (2) By the argument in the proofs of Lemmas 1.1.13 and 1.5.1, show tat N = 0 if the condition on the elements a − 1 for a ∈ a holds. In the proof, one can use the following result (see [72, Theorem 3.7]). Lemma of Artin-Rees Let A be a noetherian ring, let M be a finite A-module and let N, N ′ be A-submodules of M . Let a be an ideal of A. Then there exists an integer r > 0 such that an N ∩ N ′ = an−r (ar N ∩ N ′ ) for every n ≥ r. Answer. (1) Use Lemma of Artin-Rees to N = M and N ′ = N . Since an M ∩ N = N and an−r M ∩ N = N , we have N = an−r N . Now take n = r + 1. Then aN = N . (2) Let {z1 , . . . , zm } be a system of generators of N . Then we have zi =

m X j=1

aij zj , aij ∈ a,

which we can write by using a matrix representation   z1   (Em − A)  ...  = 0, A = (aij )1≤i,j≤m . zm

Let D = (Em −A), let D∗ be the cofactor matrix of D and let d = det D. Then D∗ D = dEm , d = 1 − a with a ∈ a and dzi = 0 for 1 ≤ i ≤ m. Since d is not a zero divisor of M by the assumption, we have zi = 0 for every 1 ≤ i ≤ m. Namely, N = 0. 5. Let X be a scheme defined over a field k and separated over k. Let Ui = Spec Ai (i = 1, 2) be an open set such that Ai is a finitely generated k-algebra. Show that U3 = U1 ∩ U2 is an affine open set such that the coordinate ring A3 is a finitely generated k-algebra.

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Answer. Let ∆ : X → X ×k X be the diagonal morphism which is a closed immersion by the definition of X being separated over k. Then U3 = U1 ∩ U2 = ∆−1 (U1 × U2 ), hence ∆|U3 : U3 → U1 ×k U2 is a closed immersion. Since U1 ×k U2 = Spec (A1 ⊗k A2 ), A3 is the surjective image of A1 ⊗k A2 . Since A1 ⊗k A2 is a finitely generated k-algebra, so is A3 . 6. Let f : X → Y and g : Z → Y be morphisms of schemes. Let W = X ×Y Z and let p : W → X and q : W → Z be the projections. We denote p by gX and call it the base change of g. Prove the following two assertions. (1) If g is an open immersion then gX : W → X is also an open immersion. (2) If g is a closed immersion then gX : W → X is also a closed immersion. Answer. (1) Let w = (x, z) ∈ W with x = p(w) and z = q(w). Let y = f (x). Choose affine open neighborhoods U = Spec A, S = Spec B and T = Spec R of x, z, y, respectively such that U ⊂ f −1 (T ) and S ⊂ g −1 (T ). By the construction of the fiber product W , the local ring OW,w is a localization of A ⊗R B with respect to a prime ideal, hence a localization of OX,x ⊗OY,y OZ,z . Since g is an open immersion, Z is identified with an open set g(Z) of Y . In particular, OZ,z ∼ = OY,y . Hence OW,w ∼ = OX,x because OX,x ⊗OY,y OZ,z is a local ring isomorphic to OX,x . This implies that W is identified with an open set p(W ) = f −1 (g(Z)) of X. Namely, gX is an open immersion. (2) With the notations for (1) as above, S is taken as the closed set V (J) = Spec R/J of T since g is now a closed immersion. Then OW,w is a localization of A ⊗R (R/J) ∼ = A/JA, hence isomorphic to OX,x /JOX,x . This implies that W is obtained by patching closed sets of the form V (JA) of U . Hence gX is a closed immersion. 7. A morphism of schemes f : X → Y is an affine morphism if there exists an affine open covering U = {Ui }i∈I such that f −1 (Ui ) is an affine open set of X for every i. Show that if f : X → Y is an affine morphism then f −1 (U ) is an affine open set for every affine open set U of Y . Answer. For any i ∈ I, let Ui = Spec Bi and f −1 (Ui ) = Spec Ai . Let U ′ ⊆ U ∩ Ui be an affine open set of the form D(s) with s ∈ Bi . Then f −1 (U ′ ) = Spec Ai [s−1 ]. Since U ∩ Ui is covered by affine open sets like U ′ and {U ∩ Ui }i∈I is an open covering of U , by replacing Y by U , it

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suffices to show that X is affine if there exists an affine open covering U = {Ui }i∈I of Y such that f −1 (Ui ) is an affine open set of X for every i ∈ X. We may further assume that Ui = D(si ) for si ∈ B. fi , For every i ∈ I, we have f∗ (OX )|Ui = (f |f −1 (Ui ) )∗ (Of −1 (Ui ) ) ∼ = A which is a quasi-coherent OUi -Module. Since quasi-coherency of an OY -Module is a local property, we know that f∗ (OX ) is a quasicoherent OY -Module, which is an OY -algebra. Let A = Γ(Y, f∗ (OX )) = Γ(X, OX ), which is a B-algebra. Then the morphism f splits as β

α

f : X −→ Spec A −→ Y. In fact, since there is a natural ring homomorphism ρi : A = Γ(X, OX ) → Γ(f −1 (Ui ), OX ) = Ai , there is a morphism αi = a ρi : f −1 (Ui ) → Spec A. It is then easy to show that αi |f −1 (Ui ∩Uj ) = αj |f −1 (Ui ∩Uj ) for i, j ∈ I. Hence {αi }i∈I patch together and give a morphism α : X → Spec A. The morphism β : Spec A → Y is associated to the canonical homomorphism B → A. We restrict the morphism f over Ui = D(si ), and we have α

βi

i −1 f |f −1 (Ui ) : f −1 (Ui ) = Spec Ai −→ Spec A[s−1 i ] −→ Spec B[si ].

−1 ^ f Since f∗ (OX ) is quasi-coherent, f∗ (OX )|Ui = A[s i ] = Ai , whence α|f −1 (Ui ) = αi is an isomorphism for every i ∈ I. So, α : X → Spec A is an isomorphism, and X is an affine scheme. 8. Let R be an integral domain and let R[x] be a polynomial ring in one variable over R. Prove the following assertions.

(1) R[x] is an integral domain. (2) (R[x])∗ = R∗ . Answer. (1) Let f = a0 xm + · · · + am and g = b0 xn + · · · + bn be nonzero elements of R[x], where a0 b0 ̸= 0. Then we have f · g = a0 b0 xm+n + (a0 b1 + a1 b0 )xm+n−1 + · · · + am bn . Then f · g ̸= 0 since a0 b0 ̸= 0. Hence R[x] is an integral domain. (2) With the same f, g as above, assume that f · g = 1. Then a0 b0 = 0 if m + n > 0. This contradicts the assumption that a0 b0 ̸= 0. Hence m = n = 0 and a0 b0 = 1. This proves the assertion. 9. Let R be a normal domain, let K = Q(R) be the field of quotients of R, let L/K be a finite Galois extension with Galois group G, and let S be the integral closure of R in L. Prove the following assertions.

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(1) The group G maps S into S. Thus G is viewed as an automorphism group of S which leaves elements of R invariant. (2) Let S G denote the set of G-invariant elements of S. Then we have S G = R. (3) Let p be a prime ideal of R and let S = {P1 , . . . , Pr } be the set of all prime ideals of S such that Pi ∩ R = p. Then G acts transitively on the set S. Answer. (1) Let ξ ∈ S and let f (X) be a monic polynomial of R[X] such that f (ξ) = 0. Let σ ∈ G. Then σ f (ξ) = f (σ ξ) = 0. Hence σ ξ ∈ S. This proves the assertion. (2) Let ξ be a G-invariant element of S. Since LG = K, ξ ∈ K and ξ is integral over R. Since R is a normal ring, ξ ∈ R. The inclusion R ⊆ S G is clear. (3) Let P, P′ be prime ideals of S such that p = P ∩ R = P′ ∩ R. Suppose that P′ ̸∈ {σ P | σ ∈ G}. By Lying-over theorem (Theorem 1.1.19), there are no inclusions among elements of {σ P | σ ∈ G} ∪ {P′ }. Hence there exists an element α ∈ P′ such that α ̸∈ σ P for every σ ∈ G. In fact, since   Y σ τ  P ̸⊃ P′ ·  P , τ ̸=σ

Q

 τ σ there exists an element α(σ) ∈ P′ · τ ̸=σ P such that α(σ) ̸∈ P. P ′ σ Let α = σ∈G α(σ). Then α ∈ P and α ̸∈ P for every σ ∈ G. Let Q a = σ∈G σ α. Then a ∈ R, a ∈ P′ ∩ R and a ̸∈ P ∩ R. This is a contradiction. Hence P′ = σ P for some σ ∈ G. 10. Under the additional assumption that A is a finite B-module and the field of quotients K = Q(B) has characteristic zero, prove Going-down theorem (Theorem 1.5.24) according to the following steps. (1) Let M = Q(A) be the field of quotients, which is a finite separable extension of K by the additional assumption. Let L/K be the smallest Galois extension of group G containing M . Since M/K is a simple extension, write M = K(θ). Then L is obtained by adjoining K with all conjugates of θ, i.e., if f (X) is the minimal polynomial of θ over K, then L is the extension field of K adjoined with all roots of the equation f (X) = 0. (2) Let C be the integral closure of B in L. Then A ⊆ C, and G is a group of automorphisms of C such that C G = B.

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(3) Let q1 ⊇ q2 be prime ideals of B, and let p1 be a prime ideal of A such that p1 ∩ B = q1 . Let P1 be a prime ideal of C such that P1 ∩ A = p1 . By Going-up theorem (Theorem 1.1.20), find prime ideals P′i (i = 1, 2) of C such that P′i ∩ B = qi and P′1 ⊇ P′2 . By Problem 9, P1 = σ P′1 for σ ∈ G because P1 ∩ B = p1 ∩ B = q1 and P′1 ∩ B = q1 . Let P2 = σ P′2 and let p2 = P2 ∩ A. Show that p1 ⊇ p2 and p2 ∩ B = q2 . Answer. (1) and (2) are clear. (3) Since P′1 ⊇ P′2 , we have P1 ⊇ P2 as P1 and P2 are translates of P′1 and P′2 by an automorphism σ of C. Hence p1 = P1 ∩ A ⊇ p2 = P2 ∩ A and p2 ∩ B = P2 ∩ B = P′2 ∩ B = q2 . 11. Let R be a ring and let p be a prime ideal. For an integer n > 0, set p(n) = (pn Rp ) ∩ R. Verify the following assertions. p (1) p(n) is a primary ideal such that p(n) = p. (2) p(n) Rp = pn Rp . T (3) Suppose that R is noetherian. Then n>0 p(n) = Ker (ι), where ι : R → Rp is the canonical homomorphism a 7→ a/1. Answer. (1) Let amp∈ p(n) for m > 0. Then am t ∈ pn for t ∈ R \ p. Then a ∈ p. Hence p(n) ⊆ p. p On the other hand, if a ∈ p then an ∈ppn Rp ∩ R = p(n) . Hence p ⊆ p(n) . Suppose that ab ∈ p(n) and a ̸∈ p(n) = p. Then a ∈ R \ p and ab ∈ pn Rp . Hence b ∈ pn Rp ∩ R = p(n) . So, p(n) is a primary ideal belonging to p. (2) Clearly, pn ⊆ p(n) . Hence pn Rp ⊆ p(n) Rp . Conversely, p(n) Rp = (pn Rp ∩ R)Rp ⊆ pn Rp . T T (3) If x ∈ n>0 p(n) then x/1 ∈ n>0 pn Rp by the assertion (2). Krull’s intersection theorem (Theorem 1.9.10) then implies that x/1 = 0. Conversely, if x/1 = 0 then x/1 ∈ pn Rp for every n > 0. Hence x ∈ p(n) . T So, x ∈ n>0 p(n) . 12. Let A be a ring. Show that a sequence of A-modules f

g

M1 −→ M2 −→ M3 −→ 0

is an exact sequence if and only if the dual sequence g∗

f∗

0 −→ HomA (M3 , N ) −→ HomA (M2 , N ) −→ HomA (M1 , N )

is an exact sequence for every A-module N .

Answer. We first show the only if part. Suppose that g ∗ (γ) = γ ◦g = 0 for γ ∈ HomA (M3 , N ). Since g is surjective, it is clear that γ = 0.

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For every γ ∈ HomA (M3 , N ), (f ∗ ◦ g ∗ )(γ) = γ ◦ (g ◦ f ) = 0 because g ◦ f = 0. Suppose that f ∗ (β) = β ◦ f = 0 for β ∈ HomA (M2 , N ). Since Ker g = Im f , β |Ker g = 0. Hence there exists γ ∈ HomA (M3 , N ) such that β = γ ◦ g = g ∗ (γ). This implies that Ker f ∗ = Im g ∗ . We prove the if part by following the argument in [9, Chap. II, §2, Theorem 1]. Take N = Coker g = M3 /Im g and let ρ : M3 → N be the quotient homomorphism. Then g ∗ (ρ) = ρ ◦ g = 0. Since g ∗ is injective, ρ is the zero map. Hence N = 0 and g is surjective. Next take N = M3 and an element idM3 ∈ HomA (M3 , M3 ). Then (f ∗ ◦ g ∗ )(idM3 ) = 0. Hence g · f = 0. This implies that Im f ⊆ Ker g. Finally take N = Coker f and the quotient morphism φ : M2 → N . Then f ∗ (φ) = φ ◦ f = 0. By the exact sequence (the second sequence), there exists a homomorphism ψ : M3 → N such that φ = ψ ◦ g. This implies that Ker g ⊆ Ker φ = Im f . Hence Ker g = Im f . 13. Let (R, m) be a local ring (OX,x , mX,x ) for an algebraic variety X of dimension n and a closed point x ∈ X such that R/m = k. Show that the following isomorphism exists Ω1R/k ⊗R k ∼ = m/m2 . Answer. In Lemma 1.6.20(2), take B = R, C = k, A = R/m to obtain an exact sequence m/m2 −→ Ω1R/k ⊗R R/m −→ 0. By Lemma 1.6.19(1), there is an R-isomorphism Φ : HomR (Ω1R/k , m/m2 ) ∼ = Der k (R, m/m2 ). Consider a mapping D : R → m/m2 defined by D(a) = a − α = (a − α) + m2 , where α ∈ k and a − α ∈ m. Then D is clearly a k-derivation. Hence, by Φ, we have a commutative diagram m/m2 −−−−→ Ω1R/k ⊗R R/m −−−−→ 0   yid yD m/m2

m/m2

Hence D is surjective. Now, comparing dimensions of m/m2 and Ω1R/k ⊗R R/m, we obtain the stated isomorphism. 14. Show that on a ringed space (X, A), a sequence of A-Modules f

g

0 −→ F −→ G −→ H −→ 0

(1.25)

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is an exact sequence of A-Modules if and only if the sequence of stalks fx

gx

0 −→ Fx −→ Gx −→ Hx −→ 0, ∀x ∈ X

(1.26)

is an exact sequence of Ax -modules for all x ∈ X.

Answer. The sequence (1.25) is an exact sequence of A-Modules if K = Ker f = 0, Im f = Ker g and L := Coker g = 0, where Im f = Coker (K → F). Suppose that these equalities hold. Since K(U ) = Ker (f (U )) = 0 for any open set U , we have an exact sequence f (U )

0 −→ F(U ) −→ G(U ).

Since limx∈U is a left exact functor, we have an exact sequence −→ fx

0 −→ Fx −→ Gx .

Since then F ∼ = Im f ∼ = Ker g, we have an exact sequence for any U ∈ C, where C is the category of all open sets of X, f (U )

g(U )

0 −→ F(U ) −→ G(U ) −→ H(U ).

This implies the exactness of the sequence fx

gx

0 −→ Fx −→ Gx −→ Hx .

On the other hand, L is the sheafification of a presheaf P defined by g(U )

G(U ) −→ H(U ) −→ P(U ) −→ 0, U ∈ C.

Since limx∈U is an exact functor, we have an exact sequence −→ Gx −→ Hx −→ Px −→ 0,

where Px = Lx = 0 because L = 0. Hence we have an exact sequence (1.26) fx

gx

0 −→ Fx −→ Gx −→ Hx −→ 0, ∀x ∈ X.

Conversely, we show that (1.26) implies (1.25). We use the above notations K, L, etc. Since then Kx = 0 for all x ∈ X, K(U ) = 0 e = ` because K(U ) is the set of local sections of K x∈X Kx . Hence f : F → G is injective. The image Im f is then isomorphic to F. Since (g ◦ f )x = gx ◦ fx = 0, we have g ◦ f = 0. Hence f splits as ∼

p

f : F −→ Im f −→ M := Ker g −→ G,

where we identify F and Im f . Then px : Fx → Mx is an isomorphism for all x ∈ X. Then the kernel and the cokernel of p are zero by the above argument. Hence p is an isomorphism. The surjection of g : G → H is proved in a similar fashion. Show by the above argument that L := Coker g = 0. So, we obtain the exactness of the sequence (1.25).

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15. Let R be a UFD and let f, g ∈ R such that gcd(f, g) = 1. Show that the following sequence is exact: α

β

γ

0 → f gR −→ f R ⊕ gR −→ R −→ R/(f, g) → 0,

where the homomorphisms α, β, γ are those given in the proof of Lemma 1.8.8. Answer. It is clear that the homomorphism γ is surjective and that Ker γ = Im β. Suppose that f u + gv = 0. Since f, g are coprime in a UFD R, it follows that u = gw and v = −f w. Hence (f u) ⊕ (gv) = (f gw) ⊕ (−f gw) = α(f gw). The homomorphism α is clearly injective. So, the sequence is exact. 16. Let X be a smooth projective surface defined over an algebraically closed field k and let C be a smooth irreducible curve on X. Prove the following assertions. (1) Let L be an invertible sheaf on X. Then (L · OX (C)) = deg(L|C ), where L|C = L ⊗OX OC and deg(L|C ) is the degree of a divisor ∆ on C such that L|C ∼ = OC (∆). (2) Let g be the genus of C. Then g is given by the formula23 (C · C + KX ) + 1. g= 2 Answer.

(1) By the formula in subsection 1.8.4, we have

(L · OX (C)) =

χ(OX ) − χ(L−1 ) − χ(OX (−C)) + χ(L−1 ⊗ OX (−C)).

Meanwhile we have an exact sequence

0 −→ OX (−C) −→ OX −→ OC −→ 0.

Tensoring L−1 to the above sequence we have an exact sequence 0 −→ L−1 ⊗ OX (−C) −→ L−1 −→ L−1 ⊗ OC −→ 0,

which yields an equality

χ(L−1 ⊗ OX (−C)) = χ(L−1 ) − χ(L−1 ⊗ OC ).

Plugging this equality into the first formula we have

(L · OX (C)) = χ(OX ) − χ(OX (−C)) − χ(L−1 ⊗ OC ) = χ(OC ) − χ(L−1 ⊗ OC ).

23 It

is the convention that the intersection number (C · C + KX ) should be understood as (C · (C + KX )).

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The Riemann-Roch formula for the curve C gives χ(OC ) − χ(L−1 ⊗ OC ) = 1 − g − (1 − g + deg(L−1 ⊗ OC )) = deg(L|C ).

Hence we have (L · OX (C)) = deg(L|C ). (2) Let i : C ,→ X be the closed immersion. By Lemma 1.6.21(2), we have an exact sequence 0 −→ J /J 2 −→ i∗ Ω1X/k −→ Ω1C/k −→ 0, V2 where J = OX (−C). Taking of this sequence, we have an isomorphism 2 ^ OX (KX ) ⊗OX OC ∼ = OC (KC ) ⊗ J /J 2 . = (i∗ Ω1X/k ) ∼ = Ω1C/k ⊗ J /J 2 ∼ From the second exact sequence in (1) we have an exact sequence

0 −→ OX (−2C) −→ OX (−C) −→ OX (−C) ⊗OX OC −→ 0, which gives J /J 2 ∼ = OX (−C) ⊗O OC . Hence we have X

OX (KX )|C ∼ = OC (KC ) ⊗ (OX (−C)|C ). Taking degrees of divisors on C and noting that deg(KC ) = 2g − 2, we have (KX · C) = 2g − 2 − (C 2 ),

whence we have the equality (C · C + KX ) + 1. g= 2 n ∼ 17. Show that Pic (P ) = Z, which is generated by a hyperplane H∞ . Answer. Let (X0 , X1 , . . . , Xn ) be the homogeneous coordinate system. Then H∞ is defined by X0 = 0, and Pn \ H∞ is the affine space An = Spec k[x1 , . . . , xn ], where xi = Xi /X0 for 1 ≤ i ≤ n. Let F be an irreducible hypersurface of degree d defined by a homogeneous equation F (X0 , . . . , Xn ) = 0. Then an irreducible divisor F satisfies F = dH∞ + (f ), where f = F (1, x1 , . . . , xn ). Thus F ∼ dH∞ , and Pic (Pn ) ∼ = Z. 18. Prove Bezout’s theorem which asserts that for curves C, D on P2 having no irreducible common components we have X (deg C)(deg D) = (C · D) = i(C, D; P ). P ∈C∩D

Answer. Let ℓ∞ be the line at infinity. Then C ∼ (deg C)ℓ∞ and D ∼ (deg D)ℓ∞ . Since (ℓ2∞ ) = 1, we have the above equality. We also use Lemma 1.8.8(2).

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19. Let X be an algebraic variety defined over a field k and let L be an invertible sheaf on X. Let F be a locally free OX -Module of rank 2 defined by an exact sequence 0 −→ OX −→ F −→ L −→ 0. Suppose that H 1 (X, L−1 ) = 0. Then the above exact sequence decomposes as F ∼ = OX ⊕ L. Answer. Let U = {Ui }i∈I be an open covering of X such that L|Ui = OUi ei for all i ∈ I. Then F|Ui = OUi e ⊕ OUi ei , where we identify ei with a lift of ei in F|Ui and e is a base of OX independent of i ∈ I. Then, for i, j ∈ I, we have ej = αji ei + βji e, where {αji } is a transition function of L and βji ∈ Γ(Uji , OX ). For distinct indices i, j, k ∈ I, we have αkj αji = αki and βki = αkj βji + βkj on Uijk = Ui ∩ Uj ∩ Uk . Let {e∗i }i∈I be the dual basis of L−1 , i.e., L−1 |Ui = OUi e∗i and e∗i (ei ) = 1. ˇ Then it is easy to see that e∗i = αji e∗j on Uij . In the Cech cohomology 1 −1 H (U, L ), we have βji e∗j − βki e∗k + βkj e∗k = 0.

In fact, if we replace e∗j by the relation e∗j = αkj e∗k , we have (βji αkj − βki + βkj )e∗k = 0

ˇ by the above relation. Namely {βkj e∗k } is a Cech 1-cycle. By the 1 −1 1 −1 assumption that H (X, L ) = H (U, L ) = 0 (see Theorem 1.9.16), there is {γi e∗i }i∈I ∈ C 0 (U, L−1 ) such that βkj e∗k = γj e∗j − γk e∗k on Ujk . This implies that βkj = γj αkj − γk on Ujk . Then we have the relation ek +γk e = αki ei +βki e+γk e = αki ei +(γi αki −γk )e+γk e = αki (ei +γi e). This shows that {ei +γi e}i∈I defines a lift of L in F so that F = OX ⊕L. 20. Let ρ : X → B be a P1 -bundle over a smooth projective curve B. Let S be a section of ρ. Show that Pic X ∼ = ρ∗ (Pic B) ⊕ Z[S]. Answer. We introduce some definitions in the case where ρ : X → B P is a P1 -fibration. Let D = i ni Ci be a divisor on X. An irreducible component Ci is transversal if ρ|Ci : Ci → B is dominant, while Ci is vertical if Ci is contained in a fiber of ρ. Let ρ now be a P1 -bundle. Write D = Dt + Dv , where every irreducible component of Dt (resp. Dv ) is transversal (resp. vertical). Since Xη ∼ P1 , let d = (Dt · P= k(B) Xη ). This is an abuse of definition. If Dt = j mj Tj with irreducible P P components Tj , define Dt,η := j mj Tj,η and d = j mj ℓ(Tj,η ), where

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Tj,η = Tj ×B Spec k(B) which is an algebraic scheme of dimension 0 over k(B) and ℓ(Tj,η ) is its length. Then Dt,η − dSη = (f ) on Xη for f ∈ k(X). Hence D−dS −(f ) = (Dt −dS −(f ))+Dv has no transversal component. Thus D = dS + (f ) + ρ∗ (∆) for ∆ ∈ Div B. 21. Let ρ : X → B be a P1 -bundle, where B ∼ = P1 . Show that there is at most one negative section on X, where S is a negative section if S is a section of ρ with (S 2 ) < 0. Answer. Let S, T be distinct sections of ρ. By Problem 20, T ∼ S + nℓ, where ℓ is a fiber of ρ. Set (S 2 ) = −m < 0. We may assume that n ≥ 0. If n < 0 then exchange the roles of S and T . We have (T 2 ) = −m + 2n < 0. Hence m > 2n. Suppose n > 0. Then 0 ≤ (S · T ) = (S 2 ) + n = −m + n < 0, a contradiction. Hence n = 0. Then 0 ≤ (S · T ) = (S 2 ) < 0, a contradiction.

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Chapter 2

Geometry on Affine Surfaces

2.1

Characterization of the affine plane

Let k be an algebraically closed field, which we fix throughout Chapter 2 as the ground field unless otherwise mentioned. Based on a general theory on algebraic varieties, we develop a theory which is more inclined to various structures and properties of affine varieties. To make obtained results more decisive, we mostly consider the case where dimension is two. We begin with the following result which we use frequently and tacitly in the subsequent arguments (see Hartshorne [32, Theorem 4.2 and its proof, p. 69]). Lemma 2.1.1. Let V be a projective surface and X an affine open set of V . Then there exists an effective ample divisor D on V such that Supp D = V \ X. In particular, Γ := V \ X satisfies the following conditions. (i) Every irreducible component of Γ has pure codimension 1. (ii) Γ is connected. (iii) X contains no complete curves.1 An A1 -fibration on an affine variety is considered in many respects as an affine analogue of a P1 -fibration on a projective variety. The following result will relate an A1 -fibration with a P1 -fibration in the surface case. 1 An algebraic curve C is complete if, for any DVR (O, m) of k(C) there exists a point P ∈ C such that O dominates OC,P , i.e., O ⊇ OC,P and m ∩ OC,P = mC,P . If C is projective, it is complete. Since all DVR of k(C) are parametrized by a smooth projective e which is the normalization of C, we have curve C,

e O e ) = k. k ⊆ Γ(C, OC ) ⊆ Γ(C, C Hence Γ(C, OC ) = k. 191

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Lemma 2.1.2. Let X = Spec A be a smooth affine surface and f : X → B be an A1 -fibration over a smooth affine curve B, i.e., there exists an open set U ⊂ B such that for all closed point b ∈ U the fiber f −1 (b) is isomorphic to A1 . Suppose that the characteristic of k is zero. (1) Let F be a fiber of f . Then Fred is a disjoint union of the affine lines. (2) X contains an affine open set W of the form W = f −1 (U ) ∼ = U × A1 , where U is an open set of B. Proof. (1) By Lemma 1.4.12, there is a projective algebraic surface V such that X = V \D, where D is an effective reduced divisor on X. By resolution of singularities due to Zariski [99] or Hironaka [34], we may assume that V is smooth. Then the A1 -fibration f is extended to a dominant rational mapping ρ : V → B, where B is a smooth projective curve containing B as an open set. If B is irrational then ρ is a morphism by Lemma 1.8.15, and if B is rational then f is given by a linear pencil Λ, which has no fixed components. The base point locus Bs Λ if it exists consists of a unique point P∞ lying outside X. In fact, since a general fiber Xb is isomorphic to A1 , its closure Vb in V is a rational curve whose normalization is P1 . Thus Vb \ Xb is dominated by a single point P1 \ A1 . By Proposition 1.8.10, the base points of Λ are eliminated by the blowing-ups of P∞ and possibly its infinitely near points. Then, changing V by a new surface obtained by the blowing-ups, we may assume from the beginning that Bs Λ = ∅. By Theorem 1.8.16, general fibers of ρ are smooth and hence isomorphic to P1 . So, ρ : V → B is a P1 -fibration and there is a unique cross-section S ⊂ V \X such that all irreducible components of V \ (X ∪ S) are contained in fibers of ρ. If a fiber F = ρ−1 (b) is a singular fiber, its configuration, i.e., how all the irreducible components meet each other, is depicted by Lemma 1.8.20. In particular, the following properties should be noted. (i) Every fiber of ρ is connected and meets the section S in one point transversally. (ii) There is no circle contained in F , i.e, any connected sequence of successively meeting irreducible components in F , say Ci1 , Ci2 , . . . , Cir , never meet the starting component Ci1 . We then say that the configuration of F is a tree. (iii) Only terminal component can meet the open set X, where a component is terminal if it meets only one other component of F . For otherwise, a complete component will be left in X.

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(iv) There is a unique component, say C0 , of F which meets the section S. All other components are connected to the component C0 via a connected path of components. The assertion (1) follows from the above observations. (2) By the construction of a P1 -bundle P(F) (see Lemma 1.8.21), there exists an open set U such that F|U ∼ = OU ⊕ OU . Then ρ−1 (U ) ∼ = P1 × U . −1 −1 −1 −1 Since f (U ) = ρ (U )\(S ∩ρ (U )), it follows that f (U ) ∼ = A1 ×U . Let X be an affine open set of a smooth projective surface V . Let D be P an effective reduced divisor such that Supp D = V \X. Write D = i Di be the irreducible decomposition and G(D) be the subgroup of Pic V generated by the class [Di ]. Then Pic X = Pic V /G(D). We have the following result. Lemma 2.1.3. Let f : X → B be an A1 -fibration over an affine smooth rational curve. Suppose that the characteristic of k is zero. Then the following assertions hold. (1) Let {F1 , . . . , Fr } exhaust all singular fibers of f , and let Fi = P (i) (i) (i) ∼ 1 j mj Cj , where Cj = A . Then we have * + r M M X (i) (i) (i) ∼ Z[C ]/ m [C ] | 1 ≤ i ≤ r . Pic X = j

i=1

j

j

j

j

(2) Suppose that Pic X = 0, i.e., the coordinate ring of X is a UFD. Then f : X → B has no singular fibers. Namely, f is an A1 -bundle and X∼ = B × A1 . (3) Write X = Spec A. Then A∗ = k ∗ if and only if B ∼ = A1 . Proof. (1) With the notations in the proof of Lemma 2.1.2, we may assume that the fibers ρ−1 (b) with b ∈ B \ B are smooth fibers and any fiber ρ−1 (b) with b ∈ B is a smooth fiber if f −1 (b) is integral. Hence singular fibers of ρ are exhausted by F i := ρ−1 (bi ) with bi = f (Fi ) for 1 ≤ i ≤ r. By Lemma 1.8.20, the abelian group Pic V is then generated by the classes of a smooth fiber ℓ of ρ which we take outside of X, the section S and all the irreducible components of singular fibers F 1 , . . . , F r with relations F i ∼ ℓ. In the quotient group Pic X = Pic V /G(D), the classes of ℓ, S and the irreducible components of F i ∩ (V \ X) (1 ≤ i ≤ r) become zero. Hence we have the presentation of Pic X in the assertion (1). (2) Suppose that Pic X = 0. It follows from (1) that there are no singular fibers of f : X → B. By the assumption made in the proof of the assertion (1), this implies that ρ is a P1 -bundle and S is a section.

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Since a P1 -bundle is locally free, there exists an open covering U = {Ui }i∈I such that ρ−1 (Ui ) ∼ = Ui × P1 . In fact, if we write V = P(F), then F|Ui = OUi e0 ⊕ OUi e1 and hence P(F|Ui ) = Ui × Proj (k[e0 , e1 ]) ∼ = Ui × P1 . The section S is given by the homogeneous ideal (s0 e0 − s1 e1 ) of Ri [e0 , e1 ], where Ui = Spec Ri and s0 , s1 ∈ Ri with s0 Ri + s1 Ri = Ri . Hence there are t0 , t1 ∈ Ri such that s0 t0 − s1 t1 = 1. Let e′0 = s0 e0 − s1 e1 and e′1 = −t1 e0 +t0 e1 . Then the section S|Ui is given by e′0 = 0. Let xi = e′1 /e′0 . Then f −1 (Ui ) ∼ = Ui × Spec k[xi ] ∼ = Ui × A1 . On Uij = Ui ∩ Uj , we have xj = αji xi + βji with αji ∈ OB (Uij )∗ and βji ∈ OB (Uij ). On Uijℓ = Ui ∩ Uj ∩ Uℓ for distinct i, j, ℓ ∈ I, we have αℓi = αℓj αji and βℓj = αℓj βji + βℓj . Hence {αji } is a set of transition functions for an invertible sheaf L on B. Since Pic B = 0, we have {αi ∈ O(Ui )∗ }i∈I such that αji = αj αi−1 . Replacing xi by αi−1 xi , we may assume that αji = 1 for all i, j ∈ I with i ̸= j. Then βℓi = βji + βℓj . By Theorem 1.9.17, we have H 1 (U, OB ) = 0. This implies that βji = βj − βi for βi ∈ OB (Ui ). Set x′i = xi − βi . Then x′j = x′i for every pair (i, j) ∈ I 2 . Set this element x. Then X = B × Spec k[x] ∼ = B × A1 . (3) Since B = Spec R is a smooth affine rational curve, B ⊆ A1 = 1 P \ {one point}. If B = A1 \ {P1 , . . . , Pn }, the points Pi is given by Qn x = ci ∈ k, where A1 = Spec k[x]. Then R∗ = k ∗ × i=1 Z[x − ci ] because Qn a unit of R is written as c i=1 (x − ci )ni with c ∈ k ∗ and ni ∈ Z. Hence R∗ = k ∗ if and only if B = A1 . Note that, by (2), A∗ = R∗ . Lemma 2.1.4. Let X be a smooth affine surface defined over k of characteristic zero. Suppose that Pic X = 0. Then the following conditions are equivalent. (i) There is an A1 -fibration f : X → B to a smooth affine curve B. (ii) There is an open set W of X such that W ∼ = U × A1 , where U is an affine curve. Proof. (i) ⇒ (ii). Clear from the above observations. (ii) ⇒ (i). We can find a smooth projective surface V such that X is an open set of V , each fiber of the projection W → U , which is isomorphic to A1 , extends to a smooth projective curve Fb for b ∈ U and Fb ∩ Fb′ = ∅ if b ̸= b′ . Suppose that U is rational. Then Fb ∼ Fb′ . Let Λ be a linear pencil on V generated by Fb and Fb′ . Since Fb ∩ Fb′ = ∅, Bs Λ = ∅. Hence ρ := ΦΛ : V → P1 restricts to an A1 -fibration f : X → P1 . If f (X) = P1 then the class of a fiber ℓ remains as a nonzero element in Pic X, which

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contradicts the assumption Pic X = 0. So, B := f (X) is an affine open set of P1 . Suppose that U is irrational. Let B be a smooth projective curve such that k(B) = k(U ). Such a curve B is uniquely determined by U as a curve parametrizing all DVRs of k(U ). Let Γ ⊂ V × B which is the closure of the graph Γ = {(x, b) | x ∈ W, b ∈ U, b = p(x)}, where p : W → U is the projection. Let ρ : Γ → B be the projection. It is clear that if b ∈ U then ρ−1 (b) = Fb . Let q : Γ → V be the projection. Then q is quasi-finite. In fact, if dim q −1 (v) = 1 for v ∈ V then ρ(q −1 (v)) = B, which contradicts the above construction. Then q : Γ → V is an isomorphism by Zariski’s main e → V is a e → Γ. Then q ◦ ν : Γ theorem. In fact, take the normalization ν : Γ quasi-finite birational morphism between normal algebraic surfaces. Hence e∼ q ◦ ν is an open immersion, indeed, an isomorphism. So, Γ =Γ∼ = V . Then 1 ρ : V → B restricts an A -fibration onto B. Since Pic X = 0, this case does not occur. Summarizing the above lemmas we obtain the following characterization of the affine plane. Theorem 2.1.5. Let X be a smooth affine surface defined over k with characteristic zero. Then X is isomorphic to the affine plane if and only if the following three conditions are satisfied by X. (i) X contains an open set W isomorphic to U × A1 , where U is an affine curve. (ii) A∗ = k ∗ if X = Spec A. (iii) Pic X = 0, i.e., A is a UFD. An open set like W ∼ = U × A1 is called a cylinderlike open set or an A -cylinder. In the case of char k = 0, a k-derivation δ on a k-algebra A is locally nilpotent if, for every a ∈ A, δ n (a) = 0 with n ≫ 0. For δ, define a k-algebra homomorphism exp(tδ) : A → A[[t]] by exp(tδ)(a) = P 1 n n 0 n≥0 n! δ (a)t , where t is a variable over A, δ = idA and 0! = 1. Then δ is locally nilpotent if and only if exp(tδ) splits via A[t] ,→ A[[t]]. 1

Lemma 2.1.6. Let X = Spec A be an affine k-scheme. Suppose that char k = 0 and dim A = 2. Then the following conditions are equivalent. (i) X contains a cylinderlike open set W = U ×A1 such that A∩k(U ) ̸= k. (ii) A has a nonzero locally nilpotent k-derivation δ.

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Proof. (i) ⇒ (ii). Let W = U × A1 be a cylinderlike open set. By Lemma 2.1.4 and its proof, there exists an A1 -fibration f : X → B, where B is a smooth algebraic curve. If B is projective, then k(U )∩A = k. In fact, let h ∈ k(U ). If h ̸∈ OB,b for b ∈ B, then h = τ −m h1 with a uniformisant τ of a DVR OB,b , m > 0 and h1 ∈ OB,b . Hence h is not regular along f −1 (b). T This implies that if h ∈ k(U ) ∩ A then h ∈ b∈B OB,b = k because B is projective. By the assumption, U is affine. Write U = Spec R. We may assume that U = D(s) = Spec A0 [s−1 ], where B = Spec A0 and s ∈ A0 . Hence W = Spec R[t] with a variable t. Let δ = sn (∂/∂t), which is an R-trivial locally nilpotent derivation of R[t]. Since A is an affine k-domain, write A = k[a1 , . . . , as ]. Since A[s−1 ] = R[t], δ(ai ) = sn−ni a′i with a′i ∈ A. Hence if n ≥ max1≤i≤s ni then δi is a nonzero locally nilpotent k-derivation of A. (ii) ⇒ (i). Let A0 = Ker δ := {a ∈ A | δ(a) = 0}. Then A0 is a ksubalgebra of A such that A0 = Q(A0 ) ∩ A. For any element a ∈ A \ A0 , there exists n > 0 such that δ n (a) ∈ A0 \ {0} and δ n+1 (a) = 0. Let s = δ n (a) and u = δ n−1 (a)/s ∈ A[s−1 ]. Then A[s−1 ] = A0 [s−1 ][u], where u is algebraically independent over A0 [s−1 ]. (See [58, Lemma 2.2.1], [27, Lemma 2.2.1] or [19].) Hence tr.degk Q(A0 ) = 1. By Zariski’s finiteness theorem (see [27, Theorem 2.2.4]), A0 is an affine domain over k with dim A0 = 1. Let B = Spec A0 and let f : X → B be the morphism induced by the canonical inclusion A0 ,→ A. Then f is an A1 -fibration, and f −1 (D(s)) is a cylinderlike open set. Remark 2.1.7. Let V = Fn be a Hirzebruch surface of degree n ≥ 0. With the notations in Lemma 1.8.23, let S be a divisor such that S ∼ M + mℓ with m > n. By Nakai criterion (Theorem 1.8.13), S is an ample divisor. One can show that dim |S| = 2m − n + 1 and dim |S − ℓ| = 2m − n − 1.2 This shows that the linear system |S| has no fixed components. By the first theorem of Bertini (Theorem 1.8.16), a general member of |S| is irreducible. We denote a general member by S again. Since rS is very ample if r ≫ 0, 2 We

have an exact sequence 0 → OV (mℓ) → OV (M + mℓ) → OP1 (m − n) → 0

because OM ⊗OV OV (M + mℓ) = OP1 (m − n). Since OV (mℓ) ∼ = ρ∗ (OP1 (m)), we have an exact sequence 0 → H 0 (P1 , OP1 (m)) → H 0 (V, OV (M + mℓ)) → H 0 (P1 , OP1 (m − n)) → 0. So, dim H 0 (V, OV (M + mℓ)) = 2m − n + 2. Here we used the cohomology long exact sequence and the result that H i (V, ρ∗ (OP1 (m)) ∼ = H i (P1 , OP1 (m)) for i = 0, 1.

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X := V \ S is a smooth affine surface, and f := ρ|X : X → B := P1 is an A1 -fibration parametrized by a projective curve B. The surface X contains a cylinderlike open set, but the A1 -fibration is not obtained by a locally nilpotent derivation. We say that an A1 -fibration f : X → B is of complete type if B is a projective curve and of affine type if B is an affine curve. Finally we consider a singular fiber of an A1 -fibration. Lemma 2.1.8. Let f : X → B be an A1 -fibration from a smooth affine surface X to a smooth algebraic curve B and let F0 be a singular fiber of f . Then (F0 )red is a disjoint union of components isomorphic to A1 . Proof. There exist a smooth projective surface V , a smooth projective curve B and a P1 -fibration f : V → B such that X (resp. B) is an open set of V (resp. B) and f |X = f . Let Γ = V \X. Then Γ is a connected closed set of pure dimension one (see Lemma 2.1.1), and X contains no complete curves. Further, since f is an A1 -fibration there exists an irreducible component, say S, such that S is a cross-section of f . In fact, such a cross-section is unique in Γ. Let F 0 be the fiber containing F0 . By Lemma 1.8.20, F 0 is a connected divisor consisting of smooth rational curves whose dual graph is a tree, and F 0 is connected to the section S. These observations imply that F 0 \ F0 is a connected sum of smooth rational curves and hence F0 consists of components X ∩ Ci (1 ≤ i ≤ n), where the Ci are terminal components `n of F 0 . Hence X ∩ Ci ∼ = A1 and (F0 )red = i=1 (X ∩ Ci ) (a disjoint sum of the affine lines).

198

2.2

Affine Algebraic Geometry

Admissible data for an affine curve with one place at infinity

e→C Let C be an irreducible algebraic curve defined over k and let ν : C be the normalization morphism. Let P be a point of C. We say that P is a one-place point if ν −1 (P ) consists of single point Pe. Since OC, eP e is a DVR of the field k(C), the local ring OC,P is dominated by one and only one DVR of k(C).3 Let X be a smooth affine rational surface defined over k and let C0 be a smooth irreducible curve on X. We say that C0 has only one place at infinity if there is one and only one DV R of the function field k(C0 ) which does not dominate the local ring of any closed point of C0 . More e be a smooth projective curve containing C0 as an open intuitively, let C e \ C0 consists of a single point, say Pe0 , for set. Then the complement C which OC, eP e0 is the DVR of k(C0 ) which does not dominate any closed point of C0 . Fix a pair (X, C0 ). We say that (X, C0 ) has admissible data D = {V, U, C, ℓ0 , Γ, d0 , d1 , e} if the following conditions are satisfied: (1) V is a smooth projective surface, U is an open set isomorphic to X and Sn V \ U = ℓ0 ∪ i=1 Γi . Note that Γi has dimension one by Lemma 2.1.1. Sn (2) Γ is an effective divisor such that Supp Γ = i=1 Γi satisfies the following conditions. (i) Γi ∼ = P1 for every i. (ii) (Γi · Γj ) ≤ 1 if i ̸= j. (iii) Γi ∩ Γj ∩ Γℓ = ∅ if i, j, ℓ are distinct. Sn (iv) i=1 Γi contains no cyclic chains, i.e., there is no chain {Γi1 , . . . , Γir } such that Γij ∩ Γij+1 ̸= ∅ (1 ≤ j ≤ r − 1) and Γir ∩ Γi1 ̸= ∅. (3) C is a closed irreducible curve on V such that C ∩ U ∼ = C0 and C \ C0 =

{P0 }, which is a one-place point at infinity of C0 . (4) C meets only one irreducible component ℓ0 of V \ U at the point P0 . We set d0 = i(C, ℓ0 ; P0 ) = (C · ℓ) and d1 the multiplicity of C at P0 . Hence d0 ≥ d1 . (5) As a divisor on V , C ∼ d0 (eℓ0 + Γ), where e ≥ 1 and Γ is an effective divisor on V such that Supp Γ = V \ (U ∪ ℓ0 ). By identifying U with X, we use X instead of U . Example 2.2.1. Let C0 be a closed irreducible curve in X = A2 := 3A

DVR of k(C) defines a valuation, which is sometimes called a place.

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Spec k[x, y] defined by an equation f (x, y) = 0. Embed A2 into P2 as the complement of the line ℓ0 at infinity. Let C be the closure of C0 in P2 . We assume that C \ C0 consists of only one one-place point. Set V = P2 and P0 := C ∩ℓ0 . Let d0 = (C ·ℓ0 ) and d1 = µ(C, P0 ) the multiplicity at P0 . Let D = {P2 , A2 , C, ℓ0 , 0, d0 , d1 , 1}. Then the pair (A2 , C0 ) has admissible data D. The following result is obvious, but helpful to find the multiplicity of a given curve at a one-place point. Lemma 2.2.2. Let C be an irreducible closed curve on a smooth projective surface V and let P be a one-place point of C with multiplicity d1 = µ(C, P ). Let B be an irreducible curve passing through the point P simply, i.e., P is a smooth point of B. Then i(C, B; P ) = d1 unless σ −1 (P ) ∩ σ ′ (B) = σ −1 (P ) ∩ σ ′ (C), where σ : V ′ → V is the blowing-up of the point P . Proof. Let E be the exceptional curve of σ. The proper transform σ ′ (B) meets E in the point which corresponds to the tangent direction of B at P . Meanwhile, σ ′ (C) meets E in a single point because P is a one-place point of C. Suppose that σ ′ (C) ∩ E ̸= σ ′ (B) ∩ E. Then σ ′ (C) ∩ σ ′ (B) ∩ E = ∅. Hence we have X (C · B) = i(C, B; P ) + i(C, B; Q) Q∈C∩B,Q̸=P

= i(C, B; P ) + (σ ′ (C) · σ ′ (B)).

Write σ ∗ (C) = σ ′ (C) + d1 E and σ ∗ (B) = σ ′ (B) + E. Then we have (C · B) = (σ ∗ (C) · σ ∗ (B)) = (σ ′ (C) · σ ′ (B)) + d1 . Hence d1 = i(C, B; P ). 2.2.1

Euclidean transformation associated with admissible data

Let (X, C0 ) be a pair with admissible data D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} such that d0 > d1 ≥ 1. Find integers d2 , . . . , dα and q1 , . . . , qα by the Euclidean algorithm with respect to d0 and d1 . d0 = q1 d1 + d2 d1 = q2 d2 + d3 ········· dα−2 = qα−1 dα−1 + dα dα−1 = qα dα

0 < d2 < d1 0 < d3 < d2 ··· 0 < dα < dα−1 1 < qα .

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Let N = q1 + · · · + qα . The Euclidean transformation ρ : VN → V0 = V associated with the admissible data D is a composition of blowing-ups ρ = σ1 ◦ · · · ◦ σN , where σ1 : V1 → V0 is the blowing-up of P0 , C (1) = σ1′ (C), (1) (1) Γ(1) = σ1′ (Γ) = σ1∗ (Γ), ℓ0 = σ1′ (ℓ0 ), ℓ1 := ℓ1 = σ1−1 (P0 ) and P1 = (1) (1) ℓ1 ∩ C (1) . Then P1 = ℓ0 ∩ ℓ1 , d1 = (C (1) · ℓ1 ) and d0 − d1 = (C (1) · ℓ0 ). Then µ(C (1) , P1 ) = min(d1 , d0 − d1 ) is the multiplicity of C (1) at P1 . (2) Let σ2 : V2 → V1 be the blowing-up of P1 , C (2) = σ2′ (C (1) ), ℓ0 = (1) (2) (2) −1 σ2′ (ℓ0 ), ℓ1 = σ2′ (ℓ1 ), ℓ2 := ℓ2 = σ2 (P1 ) and P2 = ℓ2 ∩ C (2) . For 1 ≤ i ≤ N , define σi : Vi → Vi−1 inductively as follows. σi is the (i) blowing-up of Pi−1 := ℓi−1 ∩ C (i−1) on Vi−1 . Let ℓi := ℓi = σi−1 (Pi−1 ), (i) (i−1) ℓj = σi′ (ℓj ) for 0 ≤ j < i, C (i) = σi′ (C (i−1) ), Γ(i) = σi′ (Γ(i−1) ) = ∗ (i−1) σi (Γ ) and Pi = C (i) ∩ ℓi . For 0 ≤ i < N , set ri = ds if q0 + q1 + · · · + qs−1 ≤ i < q0 + q1 + · · · + qs , where we set q0 = 0. Lemma 2.2.3. For 0 ≤ i < N , the point Pi+1 is an infinitely near point of Pi , and ri = µ(C (i) , Pi ) which is the multiplicity of C (i) at Pi . Proof. The first assertion is clear and just for a reminder. We prove the second assertion. We have (i)

i(C (i) , ℓq0 +···+qs−1 ; Pi ) = ds−1 − tds (i)

i(C (i) , ℓi ; Pi ) = ds ,

where t = i−(q0 +· · ·+qs−1 ). Since 0 ≤ t < qs , it follows that ds < ds−1 −tds if i ̸= N − 1 and ds−1 − tds = ds if i = N − 1. Since Pi is a one-place point of C (i) , we have µ(C (i) , Pi ) = min(ds , ds−1 − tds ). (N )

The curves ℓi (0 ≤ i ≤ N ) are the proper transforms of exceptional curves which appear in consecutive steps of blowing-ups in ρ. To elucidate (N ) the steps of blowing-ups, we rename ℓi in the following way. (N )

E0 := ℓ0 , (N )

E(s, t) := ℓi

if i = q0 + q1 + · · · + qs−1 + t, 1 ≤ s ≤ α, 1 ≤ t ≤ qs .

Intersection relations of these curves are well exhibited by drawing a configuration picture of these curves. More simply, it is shown by drawing the dual graph which consists of vertices and edges, where one assigns a vertex (denoted by a circle ◦) to each irreducible curve and an edge (denoted by a line segment) to each intersection point of two curves which is a transversal one.

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Lemma 2.2.4. Let (X, C0 ) be a pair with admissible data D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} such that d0 > d1 ≥ 1. Let ρ : VN → V be the Euclidean transformation associated with D. Then the following assertions hold. (1) ρ−1 (ℓ0 ) is an effective divisor supported by E0 and E(s, t) for 1 ≤ s ≤ α and 1 ≤ t ≤ qs whose self-intersection numbers are given as follows.  2 (ℓ0 ) − q1 − 1 if α > 1 (E02 ) = (ℓ20 ) − q1 if α = 1 2 (E(s, qs ) ) = −2 − qs+1 if 1 ≤ s < α − 1 (E(α − 1, qα−1 )2 ) = −1 − qα (E(α, qα )2 ) = −1 (E(s, t)2 ) = −2 for 1 ≤ s ≤ α and 1 ≤ t ≤ qα − 1. (2) The dual graph of ρ−1 (ℓ0 ) is a linear chain as shown in a figure in the next page. Proof. Tedious but straightforward. With the above notations d0 , d1 , . . . , dα and q1 , . . . , qα , define an integer a(s, t) for 1 ≤ s ≤ α and 1 ≤ t ≤ qs inductively by a 0 = d0 a(1, t) = t(a0 − d1 ) for 1 ≤ t ≤ q1 a(2, t) = a0 + t(a(1, q1 ) − d2 ) for 1 ≤ t ≤ q2 ········· a(s, t) = a(s − 2, qs−2 ) + t(a(s − 1, qs−1 ) − ds ) for 1 ≤ t ≤ qs , 2 ≤ s ≤ α. Lemma 2.2.5. The above-defined integers a(s, t) satisfy the following conditions. (i) If α = 1, i.e., d2 = 0, then a(1, q1 ) ≥ d0 . Otherwise, a(1, q1 ) ≥ d2 . More precisely, if α > 1 and q1 ≥ 2 then a(1, q1 ) > d0 . (ii) If α ≥ 2 then a(s, qs ) > ds−1 > ds for 2 ≤ s ≤ α. Hence a(α, qα ) > dα . (iii) For 2 ≤ s ≤ α, a(s, 1) > a(s − 1, qs−1 ). (iv) For 1 ≤ s ≤ α and 1 ≤ t ≤ qs − 1, a(s, t + 1) ≥ a(s, t) > 0. (v) dα | a(s, t). (vi) a(α, qα )dα = d0 (d0 − d1 ).

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Affine Algebraic Geometry

(1) α : even (E02 )

E0

(2) α : odd (E02 )

E0

−2

E(2, 1)

−2

E(2, 1)

−2

E(2, q2 − 1)

−2

E(2, q2 − 1)

−(q3 + 2) −2

E(2, q2 ) E(4, 1)

−2 −(qα−1 + 2)

−(q3 + 2) −2

E(2, q2 ) E(4, 1)

−2 E(α − 2, qα−2 )

−(qα−2 + 2)

E(α − 3, qα−3 )

−2

E(α, 1)

−2

E(α − 1, 1)

−2

E(α, qα − 1)

−2

E(α − 1, qα−1 − 1)

−1

E(α, qα )

−(qα + 1)

E(α − 1, qα−1 )

−2

−2 −(qα−2 + 2)

E(α − 1, 1) E(α − 3, qα−3 )

−(q2 + 2)

E(α − 1, qα−1 )

−1

E(α, qα )

−2

E(α, qα − 1)

−2

E(α, 1)

−(qα−1 + 2)

E(α − 2, qα−2 )

−2

−2

−2

−(qα + 1)

E(3, 1) E(1, q1 )

−2 −(q2 + 2)

E(3, 1) E(1, q1 )

−2

E(1, q1 − 1)

−2

E(1, q1 − 1)

−2

E(1, 1)

−2

E(1, 1)

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Proof. (i) By definition, a(1, q1 ) = q1 (d0 − d1 ) = q1 d0 − d0 + d2 = (q1 − 1)d0 + d2 . Since d0 > d1 , either q1 ≥ 2 or q1 = 1 and d2 > 0. If q1 ≥ 2 then a(1, q1 ) ≥ d0 + d2 ≥ d0 . If q1 = 1 and d2 > 0 then a(1, q1 ) = d2 . If α = 1 then q1 ≥ 2 and hence a(1, q1 ) ≥ d0 . If α ̸= 1 then d2 ̸= 0 and a(1, q1 ) ≥ d2 . If α > 1 and q1 ≥ 2 then a(1, q1 ) > d0 . (ii) If α ≥ 2 then a(1, q1 ) ≥ d2 by (i). Since a(2, q2 ) = d0 + q2 (a(1, q1 ) − d2 ), we have a(2, q2 ) ≥ d0 . Hence a(2, q2 ) > d1 > d2 . If α ≥ 3 we prove a(s, ds ) > ds−1 > ds by induction on s. For s = 3, a(3, q3 ) = a(1, q1 ) + q3 (a(2, q2 ) − d3 ) > d2 + q3 (d2 − d3 ) > d2 . By induction on s(≥ 4), suppose that a(s − 2, qs−2 ) > ds−2 and a(s − 1, qs−1 ) > ds−1 . Then a(s, qs ) = a(s − 2, qs−2 ) + qs (a(s − 1, qs−1 ) − ds ) > ds−2 + qs (ds−1 − ds ) > ds−2 > ds−1 . Hence, if α ≥ 2, a(s, qs ) > ds−1 for 2 ≤ s ≤ α. In particular, a(α, qα ) > dα−1 > dα . (iii) For s = 2, a(2, 1) − a(1, q1 ) = d0 − d2 > 0. For s ≥ 3, a(s, 1) − a(s − 1, qs−1 ) = a(s − 2, qs−2 ) − ds > 0 by (ii). (iv) For s = 1, a(1, t + 1) − a(1, t) = d0 − d1 > 0. Hence a(1, t + 1) > a(1, t) > 0. For s ≥ 2, a(s, t + 1) − a(s, t) = a(s − 1, qs−1 ) − ds ≥ 0, where the equality holds if s = 2 and > 0 holds if s ≥ 3 by (ii). Thus a(s, t + 1) ≥ a(s, t) ≥ · · · ≥ a(s, 1) > a(s − 1, qs−1 ) ≥ · · · > a(1, q1 ) > 0 by (iii). (v) Note that dα divides d1 , . . . , dα . Since a(1, t) = t(d0 − d1 ), dα | a(1, t). Since a(2, t) = d0 + t(a(1, q1 ) − d2 ), dα | a(2, t). Suppose that dα | a(s′ , t) for s′ < s and 1 ≤ t ≤ qs′ . Then a(s, t) = a(s − 2, qs−2 ) + t(a(s − 1, qs−1 ) − ds ), and dα | a(s, t). (vi) We have a(α, qα )dα = a(α − 2, qα−2 )dα + a(α − 1, qα−1 )qα dα − qα d2α = a(α − 2, qα−2 )dα + {a(α − 1, qα−1 ) − dα }dα−1

= a(α − 2, qα−2 )dα

+ {a(α − 3, qα−3 ) + a(α − 2, qα−2 )qα−1 − qα−1 dα−1 − dα }dα−1

= a(α − 2, qα−2 )(dα + qα−1 dα−1 ) + a(α − 3, qα−3 )dα−1 − dα−2 dα−1

= a(α − 3, qα−3 )dα−1 + {a(α − 2, qα−2 ) − dα−1 }dα−2 . Assume by induction that a(α, qα )dα = a(j − 2, qj−2 )dj + {a(j − 1, qj−1 ) − dj }dj−1 . Then, since a(j − 1, qj−1 ) − dj = a(j − 3, qj−3 ) + qj−1 a(j − 2, qj−2 ) − qj−1 dj−1 − dj = a(j − 3, qj−3 ) + qj−1 a(j − 2, qj−2 ) − dj−2 , we have a(α, qα )dα = a(j − 2, qj−2 )(dj + qj−1 dj−1 ) + a(j − 3, qj−3 )dj−1 − dj−2 dj−1 = a(j − 3, qj−3 )dj−1 + {a(j − 2, qj−2 ) − dj−1 }dj−2 .

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Thus we have a(α, qα )dα = a0 d2 + {a(1, q1 ) − d2 }d1 = d0 d2 + (q1 − 1)d0 d1 = d0 (d0 − d1 ).

Define positive integers c(s, t) for 1 ≤ s ≤ α and 1 ≤ t ≤ qs inductively as follows. c(1, t) = t c(2, t) = 1 + tc(1, q1 ) ······ c(s, t) = c(s − 2, qs−2 ) + tc(s − 1, qs−1 )

for for ··· for

1 ≤ t ≤ q1 1 ≤ t ≤ q2 ··· 1 ≤ t ≤ qs , 2 ≤ s ≤ α.

Then we have the following result. Lemma 2.2.6. c(α, qα )dα = d0 . Proof. We compute as follows: c(α, qα )dα = c(α − 2, qα−2 )dα + c(α − 1, qα−1 )qα dα

= c(α − 2, qα−2 )dα + c(α − 1, qα−1 )dα−1

= c(α − 2, qα−2 )dα + {c(α − 3, qα−3 ) + c(α − 2, qα−2 )qα−1 }dα−1 = c(α − 3, qα−3 )dα−1 + c(α − 2, qα−2 )(dα + qα−1 dα−1 )

= c(α − 3, qα−3 )dα−1 + c(α − 2, qα−2 )dα−2 . As in the proof of Lemma 2.2.5, we can show c(α, qα )dα = c(j − 2, qj−2 )dj + c(j − 1, qj−1 )dj−1

for

3 ≤ j ≤ α.

Hence we have c(α, qα )dα = c(1, q1 )d3 + c(2, q2 )d2 = q1 d3 + (q1 q2 + 1)d2 = d0 .

Lemma 2.2.7. Let (X, C0 ) be a pair as above with admissible data D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} such that d0 > d1 ≥ 1. Let ρ : Vb := VN → V be the b := C (N ) = ρ′ (C), Euclidean transformation of V associated with D. Let C (N ) b b ℓ0 := ℓN , d0 := dα , and let   a(α, qα ) d0 eb := + (e − 1)c(α, qα ) dα dα

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205

and b := e(d0 /dα )E0 Γ qs α X X + {a(s, t)/dα + (e − 1)c(s, t)d0 /dα }E(s, t) + (d0 /dα )ρ∗ (Γ) − ebℓb0 , s=1 t=1

where a(s, t) and c(s, t) are integers defined previously. Let db1 be the mulb at Pb0 := C b ∩ ℓb0 . Then the following assertions hold. tiplicity of C (1) The pair (X, C0 ) is a pair with admissible data b = {Vb , X, C, b ℓb0 , Γ, b db0 , db1 , eb} D such that db1 ≤ db0 ≤ d1 < d0 and eb ≥ 4e − 2. b contains no (−1)-curves provided Γ contains no (−1)(2) (ℓb20 ) = −1, and Γ 2 curves and (ℓ0 ) ̸= q1 if α > 1 and ℓ20 ) ̸= q1 − 1 if α = 1. b be the linear pencil on Vb spanned by C b and db0 (b b Then Λ b (3) Let Λ eℓb0 + Γ). is the proper transform by ρ of the linear pencil Λ on V spanned by C and d0 (eℓ0 + Γ). Proof. By a straightforward computation, we know that C (N ) ∼ d0 E0 +

qs α X X

a(s, t)E(s, t) + d0 ∆(N ) ,

s=1 t=1

where ( ∆(N ) = ρ∗ ((e − 1)ℓ0 + Γ) = (e − 1) E0 + (N )

qs α X X

) c(s, t)E(s, t)

+ ρ∗ (Γ).

s=1 t=1

(N )

Further, (C (N ) · ℓN ) = dα , (C (N ) · ℓj ) = 0 for 0 ≤ j < N and b ℓb0 , db0 , eb and Γ b defined as above, we have (C (N ) · ρ∗ (Γ)) = 0. Then, with C, −1 b b b b b = C ∼ d0 (b eℓ0 + Γ). Note that ρ (X) is isomorphic to X and Supp (Γ)

b Hence (X, C0 ) has admissible Vb \ (X ∪ ℓb0 ). Further Vb \ X = ℓb0 ∪ Supp (Γ). b b b b b b b data D = {V , X, C, ℓ0 , Γ, d0 , d1 , eb}. The inequalities db1 ≤ db0 ≤ d1 < d0 holds clearly. Let d0 = b0 dα and d1 = b1 dα . Then gcd(b0 , b1 ) = 1 and b0 > b1 ≥ 1, whence b0 ≥ 2. Since eb = b0 (b0 − b1 ) + (e − 1)b20 by Lemmas 2.2.5 and 2.2.6, it follows that eb ≥ 4(e − 1) + 2 = 4e − 2. The assertion (2) follows from Lemma 2.2.4 and the assertion (3) is easy to prove.

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2.2.2

(e, i)-transformation associated with admissible data

Let D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} be admissible data for a pair (X, C0 ) such that d0 = d1 ≥ 1. Let P0 = C ∩ ℓ0 and let σ1 : V1 → V be the blowing-up at P0 . Let C1 = σ1′ (C), let ℓ1 = σ −1 (P0 ) and let P1 := C (1) ∩ ℓ1 . Let (1) (0) (0) (1) d1 be the multiplicity of C1 at P1 . Set d1 = d1 . If d0 = d1 = d1 , let σ2 : V2 → V1 be the blowing-up at P1 . Define σj : Vj → Vj−1 , C (j) , (j) (j) (j) ℓj := ℓj , ℓt (0 ≤ t < j) and d1 inductively as follows if 1 ≤ j ≤ e (0)

(j−1)

and d0 = d1 = · · · = d1 . σj is the blowing-up at Pj−1 := C (j−1) ∩ (j) (j−1) (j) (j) ′ (j−1) ℓj−1 , C = σj (C ), ℓj = σj−1 (Pj−1 ), ℓt = σj′ (ℓt ) and d1 is the (1)

multiplicity of C (j) at Pj := C (j) ∩ ℓj . For 1 ≤ i ≤ e, if d0 = d1 = (i−1) · · · = d1 , define the (e, i)-transformation ρ of V associated with D as the composite ρ = σ1 ◦ · · · ◦ σi : Vi → V . It should be reminded that the Euclidean transformation of V associated with admissible data is defined if do > d1 and the (e, i)-transformation is (1) (i−1) defined if d0 = d1 = d1 = · · · = d1 . Lemma 2.2.8. Let D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} be admissible data for (X, C0 ) such that d0 = d1 ≥ 1. Suppose that the (e, i)-transformation ρ : Vi → V is defined for 1 ≤ i ≤ e. Then the following assertions hold. (1) The proper transforms of C, ℓ0 and the exceptional curves on Vi have the following dual graph: −1 C (i)

(i)

ℓi

−2

(ℓ20 ) − 1

−2

(i)

(i)

ℓi−1

(i)

ℓ1

ℓ0

(i)

(i)

(2) If 1 ≤ i < e and d1 < d1 = d0 , then Di = {Vi , X, C (i) , ℓi , Γi , d0 , d1 , (e − i)} is admissible data for (X, C0 ), where (i)

(i)

Γi := ρ∗ (Γ) + (e − i + 1)ℓi−1 + · · · + eℓ0 . If (ℓ20 ) ̸= 0 and Γ contains no (−1)-components4 then Γi contains no (−1)-components. The linear pencil Λ(i) on Vi spanned by C (i) and d0 ((e − i)ℓi + Γi ) is the proper transform by ρ of the linear pencil Λ on V spanned by C and d0 (eℓ0 + Γ). 4 Namely,

a (−1)-curve which is a component of Γ.

Geometry on Affine Surfaces

(3) If i = e we have C (e) ∼ d0 Γe , where

(e)

207

(e)

Γe := ρ∗ (Γ) + ℓe−1 + · · · + eℓ0 .

The linear pencil Λ(e) spanned by C (e) and d0 Γe is the proper transform by ρ of the linear pencil Λ on V spanned by C and d0 (eℓ0 + Γ). The pencil Λ(e) is irreducible and free from base points. Proof. All the assertions are clear from the construction. Here we need the following result. Lemma 2.2.9. Let the characteristic of k is p, which is either zero or positive. Let V be a smooth projective surface and let f : V → B be a surjective morphism from V to a smooth projective curve B such that general fibers are irreducible curves. Assume that a fiber f ∗ (b) satisfies the following conditions. (i) f ∗ (b) = d∆, where d is the multiplicity and ∆ is the reduced form, Pn i.e., when we write f ∗ (b) = i=1 di ∆i with irreducible components ∆i Pn then d = gcd(d1 , . . . , dn ) and ∆ = i=1 (di /d)∆i . Sn (ii) Supp (∆) = i=1 ∆i satisfies the conditions: (ii-1) (ii-2) (ii-3) (ii-4)

Each ∆i is isomorphic to P1 . ∆i intersects ∆j (if at all) transversally in at most one point. ∆i ∩ ∆j ∩ ∆ℓ = ∅ for distinct i, j, ℓ. Supp (∆) contains no circular chains.

Then the multiplicity d of f ∗ (b) is a power of the characteristic p. Proof. Proof consists of three steps. Step 1. Let Z = ∆red . We show that Z is simply connected (see Appendix to Chapter 2). Let φ : W → Z be an ´etale covering. For each component ∆i of ∆, the restriction φi := φ|∆i : Wi := W ×Z ∆i → ∆i is an ´etale covering. Since ∆i ∼ = P1 , by Example 2.7.4, ∆i is simply connected and Wi is a disjoint union of m copies of ∆i , where m = deg φ. We prove the assertion for Z by induction on the number n of irreducible components. For n = 1 the assertion follows from the above observation. If n > 1, choose Pn an irreducible component ∆1 such that ∆1 meets Z ′ := i=2 ∆i in only one point. Since the dual graph of ∆ is a tree, i.e., no cyclic chains contained, `m (j) by the assumption, such ∆1 exists. Write W1 = j=1 ∆1 and W ′ := ` (j) (j) ∼ m W ×Z Z ′ = Z ′ , where Z ′ = Z ′ . This decomposition is possible j=1

by the induction hypothesis because Z ′ satisfies the same conditions as Z.

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Let P = ∆1 ∩ Z ′ and let φ−1 (P ) = {P (1) , . . . , P (m) }. We can name the (j) (j) (j) (j) above components ∆1 , Z ′ so that P (j) = ∆1 ∩ Z ′ for 1 ≤ j ≤ m. Set ` (j) (j) d Z (j) = ∆1 ∪ Z ′ . Then Z (j) ∼ = Z and W = j=1 Z (j) . So, Z is simply connected. Step 2. Suppose that d is not a power of p and write d = pα d′ with gcd(d′ , p) = 1. Let t be a uniformisant of B at b, i.e., mB,b = (t). Let B ′ be ′ a smooth projective curve such that k(B ′ ) = k(B)(t1/d ) and let ψ : B ′ → B be the morphism corresponding to the injection k(B) ,→ k(B ′ ). Then ψ ramifies totally over b. Namely there exists a unique point b′ ∈ B ′ such d′ ′ ′ ′ that ψ(b ) = b and mB ′ ,b′ = (t ) with t = t . We then say that B ′ ramifies totally over b. Let V ′ be the normalization of V ×B B ′ and let ψe : V ′ → V and f ′ : V ′ → B ′ be the canonical projections to V and B ′ . We have a commutative diagram e ψ

V ′ −−−−→ V   yf f ′y ψ

∗

B ′ −−−−→ B.

Let W ′ = f ′ (b′ ). Let v be a closed point of V lying on ∆ but not the intersection point of two irreducible components and let x be an element of mV,v such that x = 0 is a local equation of ∆ as a divisor near the point v. We show that (i) V ′ is smooth near the fiber W ′ , (ii) ψe maps W ′ onto pα ∆ (considered as a closed subscheme of V and α defined by xp = 0 near the point v), and e W ′ : W ′ → pα ∆ is an ´etale covering. (iii) φ′ := ψ| ′

′

α

∗ . Hence t1/d = (u1/d )xp in In fact, we have t = uxd with u ∈ OV,v ′ ′ the field k(V ) = k(V ) ⊗k(B) k(B ), which is a field because the extension k(V )/k(B) is a regular extension as f : V → B being an irreducible pencil. α ′ d′ Hence t′ = uxd is an irreducible equation with t′ = t1/d . Let τ ′ = t′ /xp . ′ d Then τ ′ = u. Hence ψe−1 (v) has d′ points. We have

V ′ ×V Spec (OV,v ) = the normalization of (V ×B B ′ ) ×V Spec (OV,v ) = the normalization of B ′ ×B Spec (OV,v ) = Spec (OV,v [τ ′ ]),

d′

∗ . Hence the assertions (i), (ii) and (iii) hold for all where τ ′ = u ∈ OV,v ′ points of V lying over v. Since this observation holds for every closed point

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209

v ∈ ∆ except for the intersection points of ∆, the above three assertions follow by making use of purity of branch loci (see Lemma 3.2.9). Moreover, ∗ ψe∗ (pα ∆) = W ′ = f ′ (b′ ) and W ′ is connected. e W ′ : W ′ → pα ∆. Let Z = (pα ∆)red , let W = Step 3. Let φ′ = ψ| ′ W ×pα ∆ Z and let φ = (φ′ )Z be the base change of φ′ . Then φ : W → Z is a connected ´etale covering of degree d′ > 1. In fact, deg ψ = d′ and ψ −1 (v) has d′ points for v ∈ Z. This is a contradiction to Step 1, and this completes the proof of the lemma. Corollary 2.2.10. Let D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} be admissible data for (X, C0 ) with d0 = d1 > 1. Assume that d0 is not divisible by p. Then there exist admissible data D′ = {V ′ , X, C ′ , ℓ′0 , Γ′ , .d′0 , d′1 , e′ } for (X, C0 ) such that: (i) D′ are obtained from D by an (e, i)-transformation of V associated with D, where 1 ≤ i < e. (ii) d′0 = d0 , d′1 < d′0 and e′ < e. 2 (iii) (ℓ′ 0 ) = −1, and Γ′ contains no (−1)-components if (ℓ20 ) ̸= 0 and Γ has no (−1)-components. Proof. If the (e, e)-transformation is possible, then d0 is a power of p by Lemma 2.2.9. This contradicts the assumption that d0 is not divisible by p. Hence, after the (e, i)-transformation with 1 ≤ i < e, the multiplicity of C (i) at the point Pi = C (i) ∩ ℓi is smaller than d1 = d0 . By Lemma 2.2.8, D′ are admissible data for (X, C0 ). As in the above corollary, if we obtain admissible data D′ for (X, C0 ) with d′1 < d′0 , the Euclidean transformation associated with D′ is possible. Note that the dual graph of the (irreducible) exceptional curves obtained by the Euclidean transformation is a linear chain as exhibited in Lemma 2.2.5, meanwhile the dual graph of the (e, i)-transformation is a slanted linear chain sprouting from the last exceptional curve E(α, qα ) of the previous Euclidean transformation and the last exceptional curve is the terminal component of the sprouting linear chain which is located on the other end of the component meeting E(α, qα ). Proposition 2.2.11. Let D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} be admissible data for (X, C0 ) such that d0 = d1 ≥ 1. Assume that the (e, e)-transformation (1) (e−1) ρ : Ve → V is defined, hence d0 = d1 = d1 = · · · = d1 hold. Let Λ be the linear pencil on V spanned by C and d0 (eℓ0 + Γ). Then the generic

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member of Λ has only one place outside of X which is a purely inseparable place. Hence a general member of Λ has only one place outside of X. Proof. Let Λ(e) be the proper transform of Λ by ρ, which is then spanned by C (e) and d0 ∆, where ∆ = Γ(e) , and is an irreducible linear pencil free from base points. Set S = ℓe . Then (C (e) · S) = d0 (∆ · S) = d0 , and d0 is a power of the characteristic p by Lemma 2.2.9. Let f : Ve → P1 be the fibration defined by Λ(e) . Let S = S \ (S ∩ ∆) and T = P1 \ f (∆). Then the restriction of f onto S induces a surjective morphism φ : S → T of degree d0 . Since S and T are isomorphic to A1 , choose inhomogeneous coordinates s and t on S and T respectively so that the point C (e) ∩ S is defined by s = 0 and the point f (C (e) ) is defined by t = 0. Then φ is given by a polynomial t = g(s) ∈ k[s] of degree d0 such that g(0) = 0. Since the point C (e) ∩ S is a one-place point of C (e) and Λ(e) has no base points, g(s) is written as g(s) = asd0 with a ∈ k ∗ . We may assume that a = 1 by replacing a1/d0 s by s. This implies that f := f |S : S → P1 is the αth iteration of the Frobenius endomorphism of P1 , where d0 = pα . (e) The generic fiber Vη := V (e) ×P1 Spec k(P1 ) is a projective normal (e) 1 curve defined over k(P ), and the curve S gives a point Pη of Vη which is (e) purely inseparable over k(P1 ). Hence Pη is a one-place point of Vη . Thus, the generic fiber of f has one place outside of X, and so do general fibers of f . 2.2.3

Irreducible affine curves with one-place at infinity

Let (X, C0 ) be a pair as above. We prove the following two lemmas. Lemma 2.2.12. Let D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} be admissible data for (X, C0 ) such that either d0 or d1 is not divisible by the characteristic p. e = {Ve , X, C, e ℓe0 , Γ, e 1, 1, ee} satisfying the Then there exist admissible data D following conditions. (i) There exists a birational morphism ρ : Ve → V , which is an alternating composite of the Euclidean transformations and the (e, i)transformations associated with admissible data. e = ρ′ (C), and ρ−1 (X) = X, i.e., the centers of blowing-ups in each (ii) C step of ρ lie outside of X. e spanned by C e and eeℓe0 + Γ e is the proper transform (iii) The linear pencil Λ by ρ of the pencil Λ spanned by C and d0 (eℓ0 + Γ). e = D. Suppose Proof. If d0 = 1 then d1 = 1, and we will be done with D

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211

d0 > d1 ≥ 1. Then, by the Euclidean transformation associated with D, b = {Vb , X, C, b ℓb0 , Γ, b db0 , db1 , eb} we obtain, by Lemma 2.2.7, admissible data D b b b such that d1 ≤ d0 ≤ d1 < d0 . Since d0 = gcd(d0 , d1 ), db0 is not divisible by p. If db0 > db1 , we apply the Euclidean transformation b If db0 = db1 , we apply the (e, i)-transformation. By associated with D. Lemmas 2.2.8 and 2.2.9, we have 1 ≤ i < e and in the obtained admissible data D′ = {V ′ , X, C ′ , ℓ′0 , Γ′0 , d′0 , d′1 , e′ }, we have d′1 < d′0 , where d′0 is not divisible by p. Then the Euclidean transformation associated with D′ is applicable. Thus, by decreasing the value of d0 by the Euclidean transformation and the (e, i)-transformation applied in an alternating way, we b such that db0 = db1 = 1. obtain admissible data D Lemma 2.2.13. Let D = {V, X, C, ℓ0 , Γ, 1, 1, e} be admissible data for the pair (X, C0 ). Let ρ : Ve → V be the (e, e)-transformation of V associated with D, let Λ(e) be the linear pencil on Ve spanned by C (e) and Γ(e) = ρ∗ (Γ)+ (e) (e) ℓe−1 + · · · + eℓ0 (see Lemma 2.2.8 for the notations) and let f : Ve → P1 be the fibration defined by Λ(e) . Then the following assertions hold. (1) C (e) is an irreducible curve which is smooth at the point C (e) ∩ ℓe . (2) The last exceptional curve ℓe is a section of the fibration f . (3) Let D be a fiber of f other than Γ(e) . Then D is an irreducible curve, D is smooth at the point D ∩ ℓe , D0 := D ∩ X has only one-place outside of D0 . Furthermore, pa (D) = pa (C (e) ). If C0 is smooth then pa (D) is equal to the genus of C0 . (4) Suppose that C0 ∼ = A1 . Then f0 := f |X : X → A1 = P1 \ {f (Γ(e) )} is 1 an A -bundle over A1 . So, X ∼ = A2 . Proof. We prove the assertions (3) and (4). Other assertions are clear from the foregoing arguments. (4) Suppose that D is reducible. Write D = D1 + D′ , where D1 is the component meeting the section ℓe . So, D1 has the coefficient 1 in D as viewed as a divisor, and D′ is the sum of other components. Then Supp D′ ⊂ X0 . Since X0 is affine by assumption, this is a contradiction as no complete curves are contained in an affine variety. So, D is irreducible and reduced. Since two irreducible fibers of f have the same arithmetic genera, the rest of the assertion is clear. (5) By (4), if C0 ∼ = A1 then pa (D) = 0, whence D ∼ = P1 . An A1 -bundle 1 1 1 ∼ 2 ∼ f0 : X → A is trivial, i.e., X = A × A = A by Lemma 2.1.3 provided the characteristic of k is zero. In the case of positive characteristic, we refer the readers to [59, Chapter 3].

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Lemma 2.2.14. Let D = {V, X, C, ℓ0 , Γ, d0 , d1 , e} be admissible data for (X, C0 ) and let Λ be the linear pencil spanned by C and d0 (eℓ0 +Γ). Suppose that the coordinate ring A := Γ(X, OX ) is a UFD with A∗ = k ∗ and that the curve C0 is defined by f = 0 for f ∈ A. Let Cα be a curve on X0 defined by f = α for α ∈ k. Then, for any member D ∈ Λ other than d0 (eℓ0 + Γ), the affine curve D0 := D ∩ X coincides with Cα for some α ∈ k. Conversely, any Cα is of the form D0 for some D ∈ Λ. Proof. The principal divisor (f ) is written as (f ) = C − ∆, where Supp (∆) ⊆ ℓ0 ∪Supp (Γ). Since C ∼ d0 (eℓ0 +Γ), write (g) = C −d0 (eℓ0 +Γ) for g ∈ k(V ). Then (f /g) = d0 (eℓ0 + Γ) − ∆, and f /g ∈ A∗ = k ∗ . Hence f = cg with c ∈ k ∗ . So, (f ) = C − d0 (eℓ0 + Γ). Namely Λ is a linear subsystem of |d0 (eℓ0 + Γ)| defined by a k-module k · 1 + k · f . So, D0 for D ∈ Λ and Cα with α ∈ k correspond bijectively. 2.2.4

Abhyankar-Moh-Suzuki theorem

We prove several theorems which are derived from the above arguments. Abhyankar-Moh-Suzuki theorem mentioned in Preface is one of these theorems. We work in the settings of Example 2.2.1. The first one is due to Moh [66] and called the irreducibility theorem of Moh. Theorem 2.2.15. Suppose that either d0 or d1 is not divisible by p. Then the curve Cα defined by f = α is an irreducible curve with only one place outside of X for every α ∈ k. Proof. Note that X = Spec A ∼ = A2 implies that A is a UFD and satisfies ∗ ∗ A = k . The assertion follows from Lemmas 2.2.12 and 2.2.13(2), and Lemma 2.2.14. The next theorem due to Ganong [21] deals with the case for which there is no condition on non-p-divisibility of d0 or d1 . It is called the generic irreducibility theorem of Ganong. Theorem 2.2.16. The generic member of Λ is an irreducible curve with one purely inseparable place outside of X. Hence the curve Cα has only one place outside of X for a general value α of k. Proof. Starting with admissible data D for (X, C0 ) in Example 2.2.1, we apply the Euclidean transformation and the (e, i)-transformation as long as the resulting proper transform Λ′ of Λ has base points. The last step of these processes is the case treated in Proposition 2.2.11, where we have

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213

a surjective morphism f : V → P1 induced by the proper transform of Λ and an irreducible curve S on V such that the restriction f |S : S → P1 is a purely inseparable morphism. Then the assertion follows from this observation. Note that not all fibers of f are curves with one place outside of X. The third theorem is due to Abhyankar-Moh [2] and independently to Suzuki [93] and is called the embedding line theorem or Abyyankar-MohSuzuki theorem. Theorem 2.2.17. Assume that C0 ∼ = A1 and that either d0 or d1 is not divisible by p. Then the curve Cα is isomorphic to A1 for every α ∈ k. Furthermore, A = Γ(C, OX ) = k[f, g] for some element g ∈ A. Proof. Starting with admissible data D for (X, C0 ) in Example 2.2.1, we can reach to the surjective morphism f : Ve → P1 in Lemma 2.2.13. By the assumption, we have pa (C) = 0. Hence pa (D) = 0 for every fiber D of f other than Γ(e) . This implies that D0 = D ∩ X ∼ = A1 , and hence 1 1 1 f0 : X → A is an A -bundle over A . By Lemma 2.2.13, X ∼ = A2 and f is one of coordinates of A, i.e., A = k[f, g] with g ∈ A. 2.2.5

Theorem of Gutwirth and pathological A1 -fibrations

Nagata [76] gave the following result originally due to Gutwirth. We follow the notations in [76] which is a bit different from our previous notations. Let ℓ0 be a line in V = P2 and let X = P2 \ℓ0 ∼ = A2 . Let C be an irreducible curve of degree d on V such that C ∩ X ∼ = A1 . Let P = C ∩ ℓ0 which is a one-place point. Define all singular points P1 , . . . , Pn of C lying over the point P such that P1 = P and Pi is infinitely near to the point Pi−1 for 2 ≤ i ≤ n and the multiplicities mi of C at these points. More precisely, m1 is the multiplicity of C = C (1) at P1 , let σ1 : V2 → V1 := V be the blowing-up of P1 , let C (2) = σ1′ (C (1) ), let E2 = σ1−1 (P1 ), let P2 = C (2) ∩ E2 and let m2 be the multiplicity of C (2) at P2 . Inductively, let σi : Vi+1 → Vi be the blowing-up of the point Pi , let Ei+1 = σi−1 (Pi ), let C (i+1) = σi′ (C (i) ) and let Pi+1 = C (i+1) ∩ Ei+1 . The multiplicity of C (i+1) at Pi+1 is mi+1 . Here mi ≥ 2 for 1 ≤ i ≤ n and mi = 1 if i > n. A theorem of Gutwirth states the following. Pn Theorem 2.2.18. Let L be the linear system |dℓ0 | − i=1 mi Pi . Namely, L consists of members of |dℓ0 | which pass through all the points P1 , . . . , Pn with multiplicity ≥ mi . If dim L ≥ 1 then d − m1 divides d.

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Nagata conjectured the following. Conjecture 2.2.19. Suppose that d is not divisible by p. Then dℓ0 ∈ L. Our results (Lemmas 2.2.12 and 2.2.13 in particular) prove that the conjecture holds. The assumption that p ∤ d is important. If this assumption is violated, something pathological in the case of positive characteristic occurs. Namely there exists a fixed-point free (linear) pencil whose fibers all have singular points outside X. Hence the assertion in the conjecture does not hold if p | d. Nagata [76] gave the following Example 2.2.20. Suppose that the characteristic p is positive. Let a be an integer such that p > a > 1 and gcd(a, p) = 1. Consider an affine plane curve C0 on A2 = Spec k[x, y] with parametrizations in t x = tp

2

and y = tap + t + b,

b ∈ k.

Then the defining equation of C0 is 2

2

f (x, y) = y p − (xap + x + bp ). The curve Cα for α ∈ k defined by f = α is also isomorphic to A1 since Cα is smooth and has parametrizations x = tp

2

2

and y = tap + t + b + α−1/p .

Let D = {P2 , A2 , C, ℓ0 , 0, p2 , p2 − ap, 1} be the admissible data for (A2 , C0 ). By the process of the Euclidean transformations and the (e, i)transformations, we reach to a surjective morphism f : V → P1 such that f is induced by the proper transform of Λ = ⟨C, d0 ℓ0 ⟩ and the proper transform C has a one-place singular point outside of A2 . Hence the assumption of Theorem 2.2.18 is not satisfied. In fact, general irreducible members have one-place singular points outside of A2 . So, the morphism f has moving singular points. We exhibit this phenomenon in a special case p = 3 and a = 2. Proposition 2.2.21. If p = 3 and a = 2 then the phenomenon described in the above example occurs. More precisely, the fibration f has only one e in Problem 7 of singular fiber 3Γ, where Γ has the same dual graph as Γ Chapter 2, and all other irreducible fibers have a cusp singularity of type ξ 3 = η 2 outside of A2 .

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Proof. Let C α be the closure in P2 of the curve Cα defined by y 9 − x6 − x − (b9 + α) = 0. So, C α is defined by X29 − X03 X16 − X08 X1 − (b9 + α)X09 = 0. The point P = C α ∩ ℓ0 is given by (X0 , X1 , X2 ) = (0, 1, 0). Let u = X0 /X1 and v = X2 /X1 . Then C α is defined near P by v 9 − u3 − u8 − (b9 + α)u9 = 0.

(2.1)

So, (A2 , C0 ) has admissible data D = {P2 , A2 , C, ℓ0 , 9, 3, 1}, where C = C 0 . With the previous notations, d0 = 9, d1 = 3, q1 = 3. Perform the Euclidean transformation associated with D by introducing parameters u = vu1 , u1 = vu2 and u2 = vu3 . We have the following configuration of exceptional curves: ′

ℓ′0

Cα −1

u1

u2 E1

−2

−2

−2

u3

E3

v

E2 ′

After the Euclidean transformation, the proper transform C α is defined by the following equation obtained from the equation (2.1) by setting u = v 3 u3 and dividing it by v 9 1 − u33 − v 15 u83 − (b9 + α)v 18 u93 = 0. Set w = u3 − 1. The above equation is rewritten as w3 = −v 15 (w + 1)8 − (b9 + α)v 18 (w + 1)9 .

(2.2)

We perform the blowing-ups wi−1 = vwi for 1 ≤ i ≤ 5, where w0 = w. We then have the equation w53 = −1 − (b9 + α)v 3 + v 5 w5 − v 10 w52 + · · · − (b9 + α)v 48 w59 . Set z = w5 + 1. Then we have z 3 = −(b9 + α)v 3 + v 5 (1 + terms in v and z of total degree ≥ 1). The blowing-up z = vz ′ gives an equation 3

z ′ = −(b9 + α) + v 2 (1 + terms in v and z ′ of total degree ≥ 1).

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The last equation implies the stated assertion. The reducible fiber is written as 3Γ = 9ℓ + 6E1 + 12E2 + 18E3 + 15E4 + 12E5 + 9E6 + 6E7 + 3E8 e = 3Γ, where the exceptional curves Ei for 4 ≤ i ≤ 8 come from the curves with parameters wi for 1 ≤ i ≤ 5. Suggested by Proposition 2.2.21, we pose the following definition in the case of positive characteristic. Definition 2.2.22. Let ρ : X → B be a surjective morphism from a smooth affine surface X to a smooth affine curve B. We call ρ a pathological A1 fibration if general fibers are isomorphic to the affine line but X ∼ ̸ B × A1 . = Our interest lies in the extension of ρ to a morphism q : V → B from a smooth projective surface V containing X as an open set and a smooth projective curve B containing B as an open set such that ρ = q|X . Such a morphism q : V → B exists. Then there exists a unique curve S on V such that S ⊂ V \ X and q|S : S → B is purely inseparable. Furthermore, for any irreducible fiber F of q, F ∩S is a one-place point of F , and F ∩X ∼ = A1 if F ∩ X is rational and smooth. 2.2.6

Abhyankar-Moh problem on embedded lines in positive characteristic

In the present subsection we assume that k has positive characteristic p. The following problem is called a problem of Abhyankar-Moh, which is yet unsolved. Problem 2.2.23. Let f (x, y) ∈ k[x, y] be a polynomial such that the curve f = 0 is isomorphic to A1 . Is the curve f (x, y) = α isomorphic to A1 for every α ∈ k? There is the following result (see [61, Lemma 6.3.4.1]). Lemma 2.2.24. Let f (x, y) ∈ k[x, y] be as in Problem 2.2.23, let C0 be the curve defined by f = 0 in A2 = Spec k[x, y] and let C be the closure of C0 in P2 when A2 is embedded into P2 as A2 = P2 \ ℓ0 in the standard way. For α ∈ k, let Cα be the curve f = α. Let D = {P2 , A2 , C, ℓ0 , 0, d0 , d1 , 1} be admissible data for (A2 , C0 ) (see Example 2.2.1). Let Λ be the linear pencil spanned by C and d0 ℓ0 . Let σ : V → P2 be the shortest succession

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of blowing-ups at the base points of Λ including infinitely near points of P0 = C ∩ ℓ0 and let q : V → P1 be the morphism defined by the proper transform σ ′ Λ. Then the following assertions hold. (1) σ is a composite of the Euclidean transformations and the (e, i)transformations. The last step is the (e, e)-transformation whose last exceptional curve S induces the bijection q|S : S → P1 . Hence q|S is a purely inseparable morphism of degree, say pn . The generic fiber Vη := V ×P1 Spec k(P1 ) as well as the general fibers of q are irreducible curves with one place at the intersection point with S. (2) Let T → S be a purely inseparable covering such that T ∼ = P1 . Let W be the normalization of (V ×P1 S) ×S T and let qeT : W → T be the composite of the normalization morphism W → V ×P1 T and the base change qT : V ×P1 T → T . We choose T to satisfy the conditions: (a) The generic fiber Wζ := W ×T Spec k(T ) is a complete smooth curve. Namely, a possible singular point of Vη is resolved by the normalization after the field extension k(T )/k(P1 ). (b) The closure T ′ in W of the point Wζ which lies over the point Vη ∩S is a section of qeT . Let θ : W → V be the normalization morphism W → V ×P1 T followed by the first projection V ×P1 T → V . Then θ is a purely inseparable morphism of degree pm := [k(T ) : k(P1 )] with m ≥ n, where k(P1 ) is identified with k(f ). (3) For every α ∈ k, the fiber C α := q −1 (α) is an irreducible and reduced curve such that C α ∩ A2 = Cα . (4) The following three conditions are equivalent. (i) The surface W \ (e qT−1 (∞)) is smooth. (ii) Let z be an inhomogeneous coordinate of T such that the point z = ∞ lies over the point of P1 which corresponds to the member d0 ℓ0 . Then m the hypersurface z p = f (x, y) in the affine 3-space Spec k[x, y, z] is isomorphic to A2 = Spec k[z, w] for some element w. (iii) The curve Cα defined by f = α is isomorphic to A1 for every α ∈ k. Proof. (1) By the iteration of the Euclidean transformations and the (e, i)transformations, whose composite is the birational morphism σ : V → P2 , we reach to the situation treated in Lemma 2.2.9, where the proper transform σ ′ Λ is free from base points and hence defines a surjective morphism q : V → P1 . The construction of σ shows that the last exceptional curve S is a (−1)-curve on V such that q|S : S → P1 is a purely inseparable morphism of degree, say pn .

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(2) The generic fiber Vη := V ×P1 Spec k(P1 ) has a purely inseparable place outside of Vη ∩ A2 . Its singularity can be resolved by taking the normalization of Vη ⊗k(P1 ) k(T ) for a suitable purely inseparable extension k(T )/k(P1 ) of degree pm with m ≥ n. Hence the condition (a) is satisfied. Let Wζ be the generic fiber of qeT : W → T . Then the place of Wζ over the point Vη ∩ S is taken to be k(T )-rational. Let T ′ be the closure of the point over Vη ∩ S in W . Then there is a birational mapping γ : T → T ′ such that qeT ′ ◦ γ = idT . Then γ is an isomorphism and hence the curve T ′ is a section of qeT . Since k(W ) = k(V ) ⊗k(P1 ) k(T ), it follows that [k(W ) : k(V )] = [k(T ) : k(P1 )] = pm . So, θ is a purely inseparable morphism of degree pm . (3) For a point P ∈ P1 , let νP be the number of irreducible components of q −1 (P ). Since V \ (S ∪ q −1 (P∞ )) ∼ = A2 and rank Cℓ (P2 ) = 1, the construction of q implies that Cℓ (V ) ⊗ Q is generated by S and irreducible components of q −1 (P∞ ). Hence we have X X rank Cℓ (V ) = 1 + νP∞ = 2 + (νP − 1) = 1 + (νP − 1) + νP∞ . P ̸=P∞

P ∈P1

1

This implies that νP = 1 for every P ∈ P such that P ̸= P∞ . So, C α is irreducible for all α ∈ k. Suppose that Cα = {f = α} is not reduced. Let (Cα )red be defined by g = 0. Then f − α = g µ with µ ≥ 2. Since C α := q −1 (α) = µG with the closure G of (Cα )red in V , we have (q −1 (α) · S) = pn = µ(G · S). ′

n′

−n′

n′

So, µ is a power of p, say pn with n′ > 0. Then f = g p +α = (g+αp )p . This is a contradiction since f is an irreducible polynomial. So, Cα is reduced.5 (4) (i) ⇒ (ii). Write the fiber q −1 (0) as C 0 which is the closure of C0 = {f = 0} in V . Let F 0 be the fiber qe−1 (0) which lies over C 0 . Since C0 ∼ = A1 . Since = A1 , it follows that θ−1 (C0 ) = F 0 \ (F 0 ∩ T ′ ) ∼ −1 ′ W \ (e qT (∞)) is smooth and T is a section of qeT , it follows that F 0 ∼ = P1 . By the arithmetic genus formula and by the fact that all closed fibers are linearly equivalent to F 0 , it follows also that all irreducible fibers of qeT are isomorphic to P1 and the generic fiber Wζ has arithmetic genus 0. Since the section T ′ gives a k(T )-rational point of Wζ , we have Wζ ∼ = P1k(T ) (see the proof of Lemma 1.8.20(1)). This implies that qeT : W → T is a P1 -fibration 5 In [61, p. 325], there is an argument using the projection formula for θ : W → V which is a finite, purely inseparable morphism and W is normal. A short explanation on the projection formula is available in the Appendix.

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e := W \ (e with a unique singular fiber qeT−1 (∞). Hence U q −1 (∞) ∪ T ′ ) ∼ = A2 . 1 m Since T → P is a purely inseparable morphism of degree p and since T ′ \ (e q −1 (∞) ∩ T ′ ) = Spec k[z] and P1 \ {∞} = Spec k[f ], it follows that m e is the hypersurface z p = f (x, y). By the construction of W , we have U pm z = f (x, y). e → T \ {∞} is an A1 -bundle U e = {z pm = (ii) ⇒ (iii). Since (e qT )|Ue : U f (x, y)} → A1 = Spec k[z], the curve Cα = {f (x, y) = α} is isomorphic to −m −m (e qT )−1 (αp ) \ ((e qT )−1 (αp ) ∩ T ′ ) which is isomorphic to A1 . (iii) ⇒ (i). Since all fibers but one singular fiber of qeT is isomorphic to P1 under the condition (iii) as T ′ is a section of qeT , qeT : W → T is a P1 -fibration. Suppose that W \ (e qT−1 (∞)) has a singular point Q on a fiber F . Since W is normal, there are finitely many singular points, and the resolution of singularity at Q is obtained by replacing Q by a tree of smooth rational curves R whose irreducible component has self-intersection number ≤ −2. In fact, the exceptional curves are part of a singular fiber of a P1 fibration on a smooth projective surface. By Lemma 1.8.20, they form a tree of smooth rational curves. By the same lemma each irreducible component E has negative intersection number. If (E 2 ) = −1, E is contractible by Theorem 1.8.18, and we may replace E by the image E by the contraction. If F has singular points Q1 , . . . , Qr , let E1 , . . . , Er be the exceptional loci Sr having no (−1)-curves. Then (F \ {Q1 , . . . , Qr }) ∪ ( i=1 Ei ) supports a singular fiber of a P1 -fibration on a smooth surface. Since F is reduced and a unique (−1)-curve in this fiber, this contradicts Lemma 1.8.20(2)(iii). This shows that W \ (e qT−1 (∞)) is smooth. 2

2

Example 2.2.25. Let f (x, y) = y p − (xap + x + bp ) be as in Exam2 ple 2.2.20. Then a hypersurface z p = f (x, y) in A3 = Spec k[x, y, z] is isomorphic to A2 . In fact, set u = y p − xa − bp − z p and v = y − ua − b − z. Then x = up , u = v p and y = v + v ap + b + z. Hence k[x, y, z] = k[v, z]. It seems that the hypersurface z p = f (x, y) is not isomorphic to A2 . Indeed, with the notations in Lemma 2.2.24, the morphism q|S : S → P1 has degree p in the present case (see the case p = 3 and a = 2 in Proposition 2.2.21). So, we need a morphism T → P1 of degree p2 to obtain the section T ′ of qeT : W → T . Namely, taking q|S : S → P1 is not sufficient to produce a section to qeS : W → S (the case T = S).

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2.3

Affine Algebraic Geometry

Automorphism theorem of the affine plane

In this section we look into the automorphism group of the affine plane and clarify the structure of the automorphism group. The base field k is an algebraically closed field of characteristic p. An influential reference in this topic is Nagata’s lecture notes [77]. 2.3.1

Linear pencils of rational curves and field generators

Let V be a smooth projective surface defined over k and let Λ be an irreducible linear pencil on V satisfying the following conditions: (i) General members of Λ are rational curves. (ii) Let B be a smooth projective curve which parametrizes the pencil Λ, whence Λ defines a dominant rational mapping φ : V → B. The generic member Vη of Λ, i.e., a normal projective curve defined over k(B) whose function field k(B)(Vη ) is k(V ). Then Vη is geometrically smooth. Namely, Vη ⊗k(B) k(B) is a smooth projective curve, where k(B) is an algebraic closure of k(B). The condition (ii) is equivalent to e := σ ′ (Λ) has no (ii’) Let σ : Ve → V be a birational morphism such that Λ base points. Then general fibers of the morphism σ ◦ φ : Ve → B are smooth. If k has characteristic zero, the condition (ii) follows from the condition (i) by Theorem of Bertini (see Theorem 1.8.16). If k has positive characteristic, this is, however, not the case because of the moving singular points. Further the curve Vη has genus 0 and is defined over the C1 -field by Theorem of Tsen. Hence Vη ∼ = P1 over k(B) by Lemma 1.8.20. Since Λ is a lin1 ∼ ear pencil, i.e., B = P over k, it follows that k(V ) is a purely transcendental extension k(x, y). The field k(B) is identified with a subfield k(f ) as the curve B is rational. Since Vη ∼ = P1k(B) , we have k(V ) = k(B)(Vη ) = k(f, g) with g ∈ k(V ). It is clear that if k(V ) = k(f, g) then Vη ∼ = P1 over k(B) = k(f ). The following result is a corollary of Lemma 1.8.20. Lemma 2.3.1. Let V be a smooth projective surface and let Λ be an irreducible linear pencil on V satisfying the above conditions (i) and (ii). Let

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Pr Bs (Λ) be the set of base points of Λ.6 Let F = i=1 ni Ci be a degenerate member of Λ. Then the following assertions hold. ∼ P1 and (C 2 ) < 0. If Ci ∩ Bs (Λ) = ∅ then Ci = i If Ci ∩ Cj ̸= ∅ for i ̸= j and Ci ∩ Cj ∩ Bs (Λ) = ∅ then (Ci · Cj ) = 1. If Ci ∩ Cj ∩ Cℓ ∩ Bs (Λ) = ∅ for distinct i, j, ℓ then Ci ∩ Cj ∩ Cℓ = ∅. Suppose that (Ci2 ) < 0 whenever Ci ∩ Bs (Λ) ̸= ∅. Then the set S consisting of irreducible components Ci of F with Ci ∩ Bs (Λ) = ∅ is not an empty set, and there is a (−1)-curve in S. (5) If a (−1)-component, say C1 , of S has n1 = 1 then there exists a (−1)-component in S which is not C1 . (1) (2) (3) (4)

Proof. Let σ : Ve → V be the shortest succession of blowing-ups at the base points of Λ and infinitely near base points such that the proper transform e has no base points. Hence Λ e defines a P1 -fibration fe : Ve → B, and Λ e corresponding to F is a degenerate fiber of fe. The the member Fe of Λ components of F not meeting Bs (Λ) are hence untouched by the birational morphism σ. The first three assertion follow from Lemma 1.8.20. e 2 ) ≤ −2 for the proper transform C ei of Ci (4) If Ci ∩ Bs (Λ) ̸= ∅ then (C i because (Ci2 ) < 0 and the blowing-up at a base point P0 lying on Ci drops (Ci2 ) by at least −1. Suppose that there are more base points which are infinitesimal to P0 . If there is a sequence of points P0 , P1 , . . . , Ps such that P1 is infinitesimal to P0 , P2 is infinitesimal to P1 , and Ps is infinitesimal to Ps−1 , the last exceptional curve Es arising from the blowing-up of Ps is a (−1) curve, but it is not contained in the degenerate fiber Fe. The proper transforms on Ve of the intermediate exceptional curves belong to Fe, but have self-intersection number ≤ −2. So, the assertion follows from Lemma 1.8.20. (5) The same argument as for (4) is applied to prove (5). An element f ∈ k[x, y] is called a field generator if k(x, y) = k(f, g) for some element g ∈ k(x, y). This implies that the linear pencil Λ on P2 defined by the inclusion k(f ) ⊂ k(x, y) gives a P1 -fibration after eliminating the base points of the pencil Λ. The constraint that f is a polynomial implies the following result due to Russell [87]. Embed A2 into P2 by an open immersion ι : A2 → P2 (see Problem 8). Let C 0 be the closure in P2 of the affine curve C0 = {f = 0}, and let C α be the closure of Cα = {f = α} 6 Bs (Λ)

is a finite set of ordinary base points of Λ and does not contain infinitely near base points.

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for α ∈ k. Let d0 = deg C 0 . Then the set {C α }α∈P1 with C ∞ := d0 ℓ0 is a linear pencil Λ(f ). Write F (X0 , X1 , X2 ) = X0d0 f (x, y),

x = X1 /X0 , y = X2 /X0 .

Then C 0 ∩ ℓ0 consists of points {(0, a1 , a2 ) | fd0 (a1 , a2 ) = 0} and it is the set of ordinary base points of the pencil Λ(f ). Theorem 2.3.2 (Russell). Let f ∈ k[x, y] be a field generator. Then there are at most two points (including infinitely near points) of C 0 lying on ℓ0 .7 In particular, the highest degree form fd0 of f has at most two distinct linear factors. Proof. Suppose that C 0 meets ℓ0 at three points P0 , Q0 , R0 of P2 . Let σ : X1 → X0 := P2 be the blowing-ups of these three points and let (j) E1 (i = 1, 2, 3) be the inverse images of P0 , Q0 , R0 respectively. Then the proper transform ℓ′0 has self-intersection number −2. When we eliminate the base points of Λ by blowing up base points (ordinary or infinitely near), the resulting exceptional curve E remains untouched in the procedure of further elimination of base points if and only if the exceptional curve E does e which is free from not belong to any member of the proper transform Λ base points and E lies transversally to general fibers of the P1 -fibration fe : Ve → P1 . This implies that the fiber Fe∞ which contains the proper transform ℓe0 of ℓ0 contains no (−1)-curves. This is a contradiction by Lemma 1.8.20. Hence C 0 meets ℓ0 in at most two ordinary points. Even if C 0 meets ℓ0 in at most two points, we are led to a contradiction if there are at least three points on ℓ0 including infinitely near base points of Λ. The same argument as above applies. The following result will give an example of field generators. Proposition 2.3.3. Let d, e be positive integers such that gcd(d, e) = 1. Let f = xd y e − 1. Then f is a field generator. Proof. Since gcd(d, e) = 1 we find positive integers a, b such that ad−be = ±1. We consider the case ad − be = 1. The other case can be treated in a similar fashion. Define a polynomial g = xb y a − 1. Then we have x=

xad y ae (f + 1)a = . xbe y ae (g + 1)e

7 This means that if P is an infinitely near point of order i, P lying on ℓ is equivalent 0 i i to saying that Pi lies on the proper transform of ℓ0 on the surface where Pi is an ordinary point.

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Hence x ∈ k(f, g). Similarly, we have y=

xbd y ad (g + 1)d = . bd be x y (f + 1)b

So, y ∈ k(f, g). Hence k(x, y) = k(f, g), which shows that f is a field generator. See Problem 10. If a polynomial f (x, y) has three ordinary points on the line at infinity ℓ0 , f is not a field generator. It is shown by the next easy result. Proposition 2.3.4. Suppose that the characteristic p is not equal to 3. Let f = xy(x − y) + 1. Then the projective curve Cη defined over k(f ) such that k(f )(Cη ) = k(x, y) is a geometrically regular curve with arithmetic genus 1. Hence the fibration fe : Ve → P1 defined by the elimination of base points of the pencil Λ(f ) is an elliptic fibration. So, f is not a field generator. Proof. Let t be a variable over k. The curve defined by xy(x − y) + t = 0 is birational to the curve Cη defined over k(t) = k(f ). Homogenization of the equation xy(x − y) + t = 0 is X12 X2 − X1 X22 + tX03 = 0.

By the Jacobian criterion of smoothness (see a remark after Lemma 1.6.19), we know that the projective curve in P2k(t) defined by the above homogeneous equation is a geometrically regular curve with arithmetic genus 1 (see Problem 16 of Chapter 1) if p ̸= 3. It meets the line at infinity ℓ0 in three points {(0, 0, 1), (0, 1, 0), (0, 1, 1)} transversally. So the induced fibration fe : Ve → P1k is an elliptic fibration, and f is not a field generator. Qn Remark 2.3.5. Let f = i=1 (x − ai y) + 1, where (a1 , . . . , an ) is a general point of k n . Suppose that p ∤ n. Then the projective curve Cη defined over k(f ) such that k(f )(Cη ) = k(x, y) is a geometrically regular curve of arithmetic genus 12 (n−1)(n−2) (see Problem 11). The fibration fe : Ve → P1k defined by the elimination of base points of the pencil Λ(f ) is a fibration whose general fibers have genus 12 (n − 1)(n − 2). The affine curve f = 0 has exactly n one-place points at infinity. Hence, the bigger the number of points at infinity n is, the higher genus the fibration fe has. Now we turn to the case where the curve f = 0 has one ordinary point at infinity. One can find a similar statement in [77, (1.8), p. 21]. Lemma 2.3.6. Let f (x, y) ∈ k[x, y] define a closed irreducible curve C0 on A2 = Spec k[x, y] such that C0 is isomorphic to A1 . Fix an open immersion

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ι : A2 → P2 and let ℓ0 = P2 \ ι(A2 ). Let C be the closure of ι(C0 ) in P2 , let P0 = C ∩ ℓ0 , let d0 = (C · ℓ0 ) and let d1 be the multiplicity of C at P0 . Then d0 > d1 . Assume that f is a field generator. Then d0 and d1 are divisible by d0 − d1 . If either d0 or d1 is not divisible by the characteristic p then f is a field generator, and d0 and d1 are divisible by d0 − d1 . Proof. We show first that d0 > d1 . Suppose that d0 = d1 . Let σ1 : V1 → V0 := P2 be the blowing-up of the point P0 and let E1 = σ1−1 (P0 ). The proper transform ℓ′0 = σ1′ (ℓ0 ) meets E1 in a point P1 . Since E1 parametrizes the lines of P2 passing through P0 , there exists a line ℓ1 on P2 such that ℓ′1 = σ1′ (ℓ1 ) passes through P1 . Then (C · ℓ1 ) > d1 = d0 . Since (C · ℓ1 ) = deg(C) = d0 this gives a contradiction. Hence d0 > d1 . Let σ e : Ve → V0 = P2 be the shortest succession of blowing-ups at the base points of the linear pencil Λ = Λ(f ) including infinitesimal ones. Since d0 > d1 , the morphism σ e comprises first the Euclidean transformation σ : V → V0 associated with admissible data D = {P2 , A2 , C, ℓ0 , 0, d0 , d1 , 1}. The inverse image F = σ −1 (ℓ0 ) has the configuration given in Lemma 2.2.5. Note that the proper transform σ ′ Λ has base points on the component E(α, qα ) and not on any other component of σ −1 (ℓ0 ). Lemma 2.3.1 then implies that the component E0 = σ ′ (ℓ0 ) is a (−1)-curve. We consider three cases separately. Case 1. α = 1, i.e., d0 = q1 d1 with q1 ≥ 2. Then the dual graph of F is depicted as 1 − q1

−1

−2

E0

E(1, q1 )

−2

−2 E(1, 1)

Hence −1 = (E02 ) = 1 − q1 , i.e., q1 = 2. Then d0 − d1 = d1 , and d0 − d1 divides d0 and d1 . Case 2. α = 2, i.e., d0 = q1 d1 + d2 and d1 = q1 d2 with q2 ≥ 2. The dual graph is given as −q1

−2

E0

E(2, 1)

−2

−1

−(q2 + 1) −2

E(2, q2 − 1) E(2, q2 ) E(1, q1 )

−2 E(1, 1)

Hence q1 = 1 and d0 − d1 = d2 . So, d2 divides d0 and d1 . Case 3. α ≥ 3. As above, (E02 ) = −q1 = −1. In the dual graphs

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given in Lemma 2.2.5, we contract components E0 , E(2, 1), . . . , E(2, q2 − 1) of the member F in this order; these components are contractible because they have no base points of σ ′ Λ. Let V ′ be a smooth surface obtained by these contractions and let E be the image of E(2, q2 ) on V ′ by these contractions. Then (E 2 ) = −q3 if α = 3 and (E 2 ) = −(q3 + 1) if α ≥ 4. Note that q3 ≥ 2 if α = 3 and q3 ≥ 1 if α ≥ 4. Hence (E 2 ) ≤ −2. Let Λ′ be the proper transform of Λ on the surface V ′ . The proper transform F ′ of F on V ′ is a member of Λ′ , and F ′ contains no (−1)-components. This is a contradiction by Lemma 2.3.1. The following result due to Abhyankar-Moh [2] follows from Lemma 2.3.6. Corollary 2.3.7. Let k be a field of characteristic p. Let φ(t) and ψ(t) be elements of a polynomial ring k[t] such that m = deg φ(t) > 0, n = deg ψ(t) > 0 and the k-algebra homomorphism θ : k[x, y] → k[t] defined by θ(x) = φ(t) and θ(y) = ψ(t) is surjective, i.e., k[t] = k[φ(t), ψ(t)]. Let f (x, y) be an irreducible polynomial such that Ker θ = (f ). Further, assume that f (x, y) is a field generator. Then either m divides n or n divides m. If gcd(m, n) = 1 then f is a field generator, and either m divides n or n divides m. Proof. We may assume that k is algebraically closed. Fix an open immersion ι : A2 → P2 , and let (X0 , X1 , X2 ) be homogeneous coordinates on P2 such that ℓ0 := P2 \ ι(A2 ) is defined by X0 = 0. Hence we set x = X1 /X0 and y = X2 /X0 . Let ρ : A1 → A2 be a closed immersion defined by ρ = a θ, i.e., ρ maps t to (x, y) = (φ(t), ψ(t)). Then ρ(A1 ) is the curve C0 = {f = 0}. Let C be the closure of C0 in P2 . Then C ∩ ℓ0 = {P0 }. Write φ(t) = am tm + · · · + a0 ,

ψ(t) = bn tn + · · · + b0 , am bn ̸= 0.

By exchanging the roles of x and y if necessary, we may assume that m > n. Then the curve C near the point P0 is expressed by  m  X0 = τ X = am + am−1 τ + · · · + a0 τ m  1 X2 = τ m−n (bn + bn−1 τ + · · · + b0 τ n ), where τ = 1/t. Making τ = 0, it follows that P0 = (0, am , 0), (C · ℓ0 ) = m and the multiplicity of C at P0 is m − n. In fact, let u = X0 /X1 and bP2 ,P = k[[u, v]] and C is defined by u = τ m and v = X2 /X1 . Then O 0

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v = τ m−n up to unit elements. This shows that C is expressed analytically by an equation um−n = v m . Since ℓ0 is defined by u = 0, it follows that (C · ℓ0 ) = m and µ(C; P0 ) = m − n. Now the assertion follows from Lemma 2.3.6. 2.3.2

Proof of automorphism theorem by Jung and van der Kulk

We state the theorem and give a proof. Let k be a field of characteristic p which need not to be algebraically closed. Let Aut k (k[x1 , . . . , xn ]) be the group of k-automorphisms of a polynomial ring k[x1 , . . . , xn ]. If σ : ∼ ∼ k[x1 , . . . , xn ] −→ k[x1 , . . . , xn ] is a k-automorphism then a σ : An −→ An is a k-automorphism of the affine n-space An . The mapping σ 7→ a σ induces ∼ an anti-isomorphism Aut k (k[x1 , . . . , xn ] −→ Aut K An , i.e., a (τ ◦ σ) = a σ ◦ a τ . We define a linear (resp. affine or de Jonqui`ere) transformation as an element ξ (resp. σ or τ ) of Aut k (k[x1 , . . . , xn ) having the following property. (1) There exists a matrix A(ξ) ∈ GL (n, k) such that σ(x1 , . . . , xn ) = (ξ(x1 ), . . . , ξ(xn )) = (x1 , . . . , xn )A(ξ). (2) There exist A(σ) ∈ GL (n, k) and a row vector ⃗a = (a1 , . . . , an ) ∈ k n such that σ(x1 , . . . , xn ) = (x1 , . . . , xn )A(σ) + ⃗a. (3) There exist (a1 , . . . , an ) ∈ (k ∗ )n and polynomials fi ∈ k[xi+1 , . . . , xn ] for 1 ≤ i ≤ n with fn ∈ k such that τ (x1 , . . . , xn ) = (a1 x1 , . . . , an xn ) + (f1 , . . . , fn ). All linear (resp. affine or de Jonqui`ere) transformations forms, under composition, the subgroup of Aut k (k[x1 , . . . , xn ]), which we denote GL (n, k) (resp. An or Jn )). An automorphism σ is tame if σ is an element of the subgroup generated by An and Jn . Otherwise we call σ wild. To simplify the notations, we consider the case n = 2. An automorphism φ ∈ Aut k[x, y] is given by φ : (x, y) 7→ (f (x, y), g(x, y)),

(2.3)

where f (x, y), g(x, y) ∈ k[x, y] such that k[f, g] = k[x, y]. This implies that a k-automorphism φ of k[x, y] maps generators x, y to f (x, y), g(x, y) respectively. The corresponding automorphism a φ of A2 maps a

φ : (α, β) 7→ (f (α, β), g(α, β)),

(2.4)

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where (α, β) is a (closed) point of A2 . If we consider (x, y) as a generic point of A2 , the expression (2.3) is considered as the expression of (2.4). But the composition of two automorphisms φ, ψ matters. If ψ ∈ Aut k[x, y] is given by (h(x, y), ℓ(x, y)) with k[h, ℓ] = k[x, y]. Then the composition ψ ◦ φ as k-algebra automorphisms is given by ψ ◦ φ : (x, y) 7→ (f (h(x, y), ℓ(x, y)), g(h(x, y), ℓ(x, y))), but the composition a φ◦ a ψ has the same expression, while the composition ψ◦ a φ has a different expression. Although we have a risk of this confusion, we dare to abuse the expression (2.3) for the expression of a φ. In the above definition, for the composition of two de Jonqui`ere transformations to be de Jonqui`ere, the order x1 , . . . , xn must be fixed. For example, the transformations τ, τ ′ ∈ Aut k[x, y] given by   ′ τ (x) = ax + f (y) τ (x) = bx + c, and τ (y) = by + c τ ′ (y) = ay + f (x). a

Then τ ′ = ι ◦ τ ◦ ι, where ι(x) = y and ι(y) = x. So, if x, y are considered in this order, τ ′ is not a de Jonqui`ere transformation, but it is a de Jonqui`ere transformation if the order y, x is used. For the transformations of type (1) and (2), any change of the order x1 , . . . , xn brings the transformation to a transformation of the same kind. We need some preparatory results. If a de Jonqui`ere transformation τ is given by τ (x) = αx + f (y) and τ (y) = βy + γ, where f (y) = bn y n + · · · + b0 with bn ̸= 0, we say that τ is a de Jonqui`ere transformation of degree n. Assume that n ≥ 2. τ gives a birational automorphism T : P2 99K P2 such that T |A2 = a φ. In terms of homogeneous coordinates (X0 , X1 , X2 ) such that x = X1 /X0 and y = X2 /X0 , T is given by T : (X0 , X1 , X2 ) 7→ (X0n , αX0n−1 X1 + F (X0 , X2 ), βX0n−1 X2 + γX0n ), where F (X0 , X2 ) = X0n f (X2 /X0 ) = bn X2n + bn−1 X0 X2n−1 + · · · + b1 X0n−1 X2 + b0 X0n . Since T |A2 is an automorphism, it is clear that (i) The point P0 := (0, 1, 0) is a unique point of the source P2 where T is not regular. We call P0 the fundamental point of T .

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(ii) The line ℓ0 \ {P0 } is sent to the point P0′ = (0, 1, 0) of the target P2 . We call ℓ0 := {X0 = 0} the fundamental curve of T . (iii) Since the inverse τ −1 of τ is given by τ −1 (x) = α−1 x−α−1 f ((y−γ)/β) and α−1 (y) = β −1 (y − γ), the inverse birational mapping T −1 has also the fundamental point P0′ = (0, 1, 0) and the fundamental curve ℓ′0 , where the prime indicates that the point and the curve are considered on the target P2 . We find a smooth projective surface V and the birational morphisms q : V → P2 and q ′ : V → P2 such that T = q ′ ◦ q −1 . The morphism q (resp. q ′ ) is the shortest succession of blowing-ups of the fundamental points of T (resp. T −1 ) including infinitely near fundamental points which appear subsequently after blowing-ups. To begin with, let q1 : V1 → V0 := P2 be the blowing-up of the fundamental point P0 . Then the birational mapping T ◦ q1 : V1 → P2 has the fundamental point P1 := E1 ∩ q1′ (ℓ0 ), where E1 = q1−1 (P0 ). Let q2 : V2 → V1 be the blowing-up of P1 . Then T ◦ (q1 ◦ q2 ) has the fundamental point P2 := E2 ∩ q2′ (E1 ), where E2 = q2−1 (P2 ). For 2 ≤ i < n, we set Pi := Ei ∩ (q2 ◦ · · · ◦ qi )′ (E1 ), where qi : Vi → Vi−1 is the blowing-up at Pi−1 and Ei = qi−1 (Pi−1 ). The points Pi is the fundamental point of T ◦ (q1 ◦ · · · ◦ qi ). On the curve En−1 , there is a unique fundamental point Qn of T ◦ (q1 ◦ · · · ◦ qn−1 ) which is not the intersection point Pn of En−1 and the proper transform of E1 . Including the point Qn , we further blow up the (n − 1) points to eliminate the points where T is not regular. In terms of local coordinates, the above blowing-ups are explained as follows. Let u = X0 /X1 and v = X2 /X1 . Then (u, v) is a system of coordinates such that P0 is given by (u, v) = (0, 0) and ℓ0 is defined by u = 0. The transformation T is given in terms of (u, v) by T : (u, v) 7→ (un , αun−1 + un f (v/u), βun−1 v + γun ). Introduce u1 by u = vu1 . Then u1 is a coordinate of E1 such that P1 is given by (u1 , v) = (0, 0) and the transformation T is given as follows by replacing u by vu1 and dividing three coordinate terms by v n−1 , T : (u1 , v) 7→ (vun1 , αun−1 + v(bn + bn−1 u1 + · · · + b0 un1 ), un−1 v(β + γu1 ). 1 1 The blowing-up of P1 is given by introducing v1 = v/u1 and the transformation T is changed to T : (u1 , v1 ) 7→ (v1 un1 , αu1n−2 + v1 (bn + · · · + b0 un1 ), un−1 v1 (β + γu1 ). 1 The point P2 is given by (u1 , v1 ) = (0, 0). The blowing-ups at points P2 , . . . , Pn−1 are done by introducing v2 = v1 /u1 , v3 = v2 /u1 , . . . , vn−1 =

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229

vn−2 /u1 . For 2 ≤ i < n, the transformation T is given as T : (u1 , vi ) 7→ (vi un1 , αu1n−i−1 + vi (bn + · · · + b0 un1 ), un−1 vi (β + γu1 )). 1 For i = n − 1, T is given by T : (u1 , vn−1 ) 7→ (vn−1 un1 , α + vn−1 (bn + · · · + b0 un1 ), u1n−1 vn−1 (β + γu1 )). Now the fundamental point is Qn = (u1 = 0, vn−1 = −α/bn ). Resuming the blowing-up of Qn , we further have to blow up (n − 1) fundamental points Qn , Qn+1 , . . . , Q2n−2 . These are the processes to decrease the u1 degree of the third coordinate un−1 (β + γu1 ). Determination of the points 1 Qn , . . . , Q2n−2 depends on the coefficients of the polynomial f (y). The morphism q : V → P2 is the composition of all these blowing-ups. The proper transform q ′ (ℓ0 ) is a (−1)-curve, and the last exceptional curve is denoted by L, which is a (−1)-curve. The component with weight (i.e., self-intersection number) −n is the proper transform of q1−1 (P0 ). The configuration of irreducible exceptional curves with self-intersection number is exhibited by the following dual graph. −1

−2

q ′ (ℓ0 )

|

−2 {z

n−2

−2

}

−2 |

{z

n−2

−2

−1

}

L

−n In the picture, we can contract the component q ′ (ℓ0 ) first and all components except for L. The composition of these contractions gives the morphism q ′ : V → P2 such that T = q ′ ◦ q −1 . By the above construction, it is clear that V, q, q ′ are uniquely determined up to isomorphisms. The birational transformation T has 2n − 1 fundamental points including infinitely near ones. The inverse T −1 has the ordinary fundamental point (0, 1, 0) on ℓ′0 and totally 2n − 1 fundamental points. An affine transformation σ given by σ(x) = ax + by + α1 and σ(y) = cx + dy + α2 is extended to an automorphism S of P2 by   1 α1 α2 S : (X0 , X1 , X2 ) 7→ (X0 , X1 , X2 )  0 a c  . 0 b d Hence A2 ∩ J2 consists of those elements with c = 0 with respect to the above expressions. If σ ̸∈ J2 then a (τ ◦ σ) as a birational transformation of

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Affine Algebraic Geometry

P2 has the ordinary fundamental point (0, 1, 0) and its inverse a (τ ◦ σ)−1 has the ordinary fundamental point (0, d/(ad − bc), −c/(ad − bc)) which is not (0, 1, 0) because c ̸= 0. The automorphism theorem due to Jung [40] and van der Kulk [50] is stated as follows. Theorem 2.3.8. Let k be a field of characteristic p. Then Aut k[x, y] is generated by the subgroups A2 and J2 . Furthermore, Aut k[x, y] is an amalgamated product of A2 and J2 . Proof. Step 1. Let φ ∈ Aut k[x, y]. (φ(x), φ(y)), where

Then φ is given by (x, y) 7→

φ(x) : = f (x, y) = f0 (x, y) + f1 (x, y) + · · · + fm (x, y) φ(y) : = g(x, y) = g0 (x, y) + g1 (x, y) + · · · + gn (x, y),

where fi (x, y) and gj (x, y) are homogeneous parts of polynomials f (x, y) and g(x, y) of degree i and j, respectively and fm (x, y)gn (x, y) ̸= 0. We set m = deg f  and n= deg g. Let ξ be a linear automorphism given by ab ξ(x, y) = (x, y) . Replacing φ by ξ ◦ φ if necessary, we may assume cd that fm (x, 0) ̸= 0 and gn (x, 0) ̸= 0. Then we can define a closed immersion ρ : A1 → A2 by ρ∗ (x) = f (x, 0) and ρ∗ (y) = g(x, 0).

Since C0 := ρ(A1 ) is the image of the line {y = 0} by an automorphism a φ : A2 → A2 , it follows that C0 ∼ = A1 , k[ρ∗ (x), ρ∗ (y)] = k[x] and the ′ defining equation f (x, y) of C0 is a field generator. By Corollary 2.3.7, it follows that either m | n or n | m. On the other hand, since f (x, y) = 0 defines a curve isomorphic to A1 , it has one place outside A2 . Namely, ∗ fm (x, y) = αλm 1 with α ∈ k and λ1 ∈ k[x, y] is a linear form with coefficient in k, which is an algebraic closure of k. Similarly, gn (x, y) = βλn2 with λ2 ∈ k[x, y]. We may assume that mn > 1. If mn = 1 then φ ∈ A2 . By Bezout’s theorem (see Problem 1 of Chapter 1), the closures of the curves f (x, y) = 0 and g(x, y) = 0 intersect in one point of A2 with multiplicity 1 and in points of ℓ0 with the summed-up intersection multiplicities equal to mn − 1 > 0. This implies that the linear forms λ1 and λ2 give the same point on ℓ0 . So, we may assume that λ1 = λ2 , which we write by λ. Suppose that n | m. Write m = nd. Define an automorphism τ ∈ J2 by τ (x) = x − γy d with α = β d γ and τ (y) = y. Then φ1 := φ ◦ τ satisfies φ1 (x) := f1 (x, y) = f (x, y) − γg(x, y)d and φ1 (y) := g1 (x, y) = g(x, y).

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Then m1 := deg f1 (x, y) < m and n1 := deg g1 (x, y) = n. We can now proceed by induction on (m, n), where we consider the lexicographic order (m, n) > (m′ , n′ ), i.e., either m > m′ or m = m′ and n > n′ . By decreasing the degrees of f and g, we reach to the case deg f = deg g = 1 which is the case where the automorphism φ = (f, g) ∈ A2 . Step 2. We show that Aut k[x, y] is an amalgamated product (see Appendix to Chapter 2) of A2 and J2 . For this purpose, it suffices to show that (1) If σi ∈ A2 \ J2 (1 ≤ i < r) and τj ∈ J2 \ A2 (1 ≤ j ≤ r) then τ1 ◦ σ1 ◦ τ2 ◦ σ2 · · · τr−1 ◦ σr−1 ◦ τr ̸∈ A2 . (2) If τi ∈ J2 \ A2 (1 ≤ i < r) and σj ∈ A2 \ J2 (1 ≤ j ≤ r) then σ1 ◦ τ1 ◦ σ2 ◦ τ2 · · · σr−1 ◦ τr−1 ◦ σr ̸∈ J2 . (1) Let T1 = a (τ1 ◦ σ1 ◦ τ2 ◦ σ2 · · · τr−1 ◦ σr−1 ◦ τr ) and T2 = a (τ2 ◦ σ2 · · · τr−1 ◦ σr−1 ◦ τr ). Hence we have T1 = T2 ◦ a (τ1 ◦ σ1 ). As remarked before Theorem 2.3.8, the ordinary fundamental point of a (τ1 ◦σ1 )−1 is different of the ordinary fundamental group of a τ2 which is (0, 1, 0). Hence in the process of eliminating fundamental points of T1 , the ordinary fundamental group (0, 1, 0) is viewed as an infinitely near fundamental point. By induction, the birational transformation T2 has totally Pr − 1) fundamental points. Hence T1 has fundamental points as i=2 (2niP r many as i=1 (2ni − 1) ≥ r > 0. If T1 is an affine transformation, it has no fundamental point. So, the assertion follows. The shortest succession of blowing-ups for eliminating fundamental points of T1 produces a smooth projective surface W and birational morphisms q, q ′ : W → P2 . The configuration of exceptional curves on W is exhibited by the following dual graph with weights −1 q ′ (ℓ0 )

∆1

−3

∆2

−3

−3

∆r

−1 L

where ∆i for 1 ≤ i ≤ r is the dual graph before Theorem 2.3.8 with q ′ (ℓ0 ) and L removed and with n replaced by ni . (2) Let T1 = a (σ1 ◦ τ1 ◦ σ2 ◦ τ2 · · · σr−1 ◦ τr−1 ◦ σr ) and T1′ = a (τ1 ◦ σ2 ◦ τ2 · · · σr−1 ◦ τr−1 ◦ σr ). Then T1 = T1′ ◦ a (σ1 ). Since σ1 ̸∈ J2 , there is a point P0 on ℓ0 such that a σ1 (P0 ) = (0, 1, 0) and P0 ̸= (0, 1, 0). Then the point P0 is the first fundamental point of T1 . Hence σ◦ τ1 · · · σr−1 ◦τr−1 ◦σr ̸∈ J2 .

232

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Affine Algebraic Geometry

Algebraic group actions on the affine plane

In this section, the base field k is an algebraically closed field of characteristic p. We consider an algebraic group action on the affine plane A2 . The problem is to be considered in a broader context, i.e., not only A2 but also the affine n-space An with n ≥ 2, but we are ignorant of the structure of Aut k An if n ≥ 3, which is a main obstacle to the problem. 2.4.1

Algebraic groups, actions and quotient spaces

We first quickly review necessary definitions and basic results. An algebraic group G is an algebraic variety defined over k with structure of a group such that its product mG : G × G → G, its inverse iG : G → G are given by morphisms of varieties. The identity element e of G is a k-rational point. Instead of an algebraic group, we can consider an algebraic group scheme which is a scheme G with group structure. If k has characteristic zero, then a group scheme G is reduced (Theorem of Oort), but if k has positive characteristic, there are non-reduced group schemes. In particular, an infinitesimal group scheme is a nontrivial (not equal to Spec k), onepointed (i.e., Gred = Spec k) group schemes, and it plays an important role in algebraic geometry in positive characteristic. Let G be an algebraic group. Then G is smooth because any k-rational point g has a translate hg belonging to the smooth locus of G, where the translation by h : G → G, x 7→ hx induces an automorphism of G and the smooth locus of G is non-empty. Hence G is a disjoint union of connected components. A connected component G0 containing the identity e is closed under group operations. It is called the identity component. In fact, G0 is a normal subgroup and G/G0 8 is a finite group. Let G be an algebraic group. As we have identified an algebraic variety X with the set of closed points X(k), we can identify G with the set G(k) of closed points, which becomes a group. Let G, H be algebraic groups defined over k. A k-morphism f : G → H is called a homomorphism of algebraic groups if f commutes with multiplications and inverses, i.e., mH ◦ (f × f ) = f ◦ mG and f ◦ iG = iH ◦ f . The morphism f is determined by its restriction f (k) = f |G(k) : G(k) → H(k). We say that an algebraic group G acts on an algebraic variety X if there is given a morphism σ : G × X → X which satisfies axioms of group 8 Let

G be an algebraic group and H an algebraic subgroup. Then the quotient G/H is given a structure of algebraic variety.

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233

action, i.e., two morphisms σ ◦ (1G × σ) and σ ◦ (mG × 1X ) from G × G × X to X coincide and the morphism σ ◦ (e × 1X ) from Spec k × X to X coincides with 1X , where 1G and 1X are the identity morphisms of G and X. Since σ(k) : G(k) × X(k) → X(k) induces σ : G × X → X, we can consider the set-theoretic action σ(k). In this section, we mostly consider an algebraic subgroup of Aut k A2 , hence there is a natural action of G on A2 as automorphisms. We consider the case G is a finite group, which is considered to be an algebraic group by setting G0 = Spec k. Let G be a connected algebraic group. According to a theorem of Chevalley, there is a normal subgroup N such that N is an affine algebraic group and the quotient G/N is a projective algebraic group, which is necessarily commutative and called an abelian variety. We will be interested only in an affine algebraic group which is also called a linear algebraic group. Let G be a connected linear algebraic group. The identity component of the maximal normal solvable subgroup R of G exists and is called the radical. Then G/R is a semi-simple algebraic group. A connected solvable algebraic group R has a normal algebraic subgroup U such that U is a unipotent algebraic group and R/U is a torus. The subgroup U is called the unipotent radical and a normal subgroup of G. If U = {e} then we call G a reductive group. The quotient group G/U has the trivial unipotent radical and hence is a reductive group. For the details, we refer the readers to any standard textbook on algebraic groups, e.g., Humphreys [36]. We discuss below more about the additive group Ga , the multiplicative group Gm and unipotent groups. Let G = Spec A be an affine group scheme. We can view G as a group functor from the category of k-algebras to the category of groups. Namely, for a k-algebra R, we denote by G(R) the set Homk (Spec R, G). The group operations mG , iG and the identity element e : Spec k → G give the group structure on G(R). Namely, by setting T = Spec R, we have mG (R) : G(R) × G(R) = Homk (T, G × G) −→ Homk (T, G) = G(R), (α, β) 7→ mG ◦ (α, β)

iG (R) : G(R) = Homk (T, G) −→ Homk (T, G) = G(R), π

α 7→ ιG ◦ α

e

eR : T −→ Spec k −→ G, where π is the structure morphism of k-scheme T . The group operations mG , iG , eG are given respectively by k-algebra homomorphisms µG : A → A ⊗ A, ιG : A → A, εG : A → k, which are called the comultiplication, coinverse, augmentation.

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Affine Algebraic Geometry

Example 2.4.1. (1) Let A = k[t] be a polynomial ring in one variable. Define µ, ι and ε by µ(t) = t ⊗ 1 + 1 ⊗ t, ι(t) = −t, ε(t) = 0.

Then Spec k[t] has a group scheme structure with m = a µ, i = a ι and e = a ε. We call the group scheme the additive group scheme and denote it by Ga . For T = Spec R, Ga (T ) = Homk (k[t], R) = R and the induced group operation is the addition of two elements in R. (2) Let A = k[t, t−1 ] be a ring of Laurent polynomials. Define µ, ι and ε by µ(t) = t ⊗ t, ι(t) = t−1 , ε(t) = 1.

Let Gm = Spec k[t, t−1 ] with the induced group operations. We call Gm the multiplicative group scheme. For T = Spec R, Gm (T ) = Homk (k[t, t−1 ], R) = R∗ and the induced group operation is the multiplication of two units in R. (3) A group scheme G is called a unipotent group scheme if G has a normal series of subgroup schemes {e} = G0 ◁ G1 ◁ · · · ◁ Gn−1 ◁ Gn = G

such that Gi /Gi−1 is a subgroup scheme of Ga for 1 ≤ i ≤ n. If k has characteristic zero, Ga has no nontrivial subgroup schemes, Gi /Gi−1 ∼ = Ga for every 1 ≤ i ≤ n. This entails that the underlying scheme of G with group structure forgotten is isomorphic to the affine space An . (4) A group scheme G is a torus if G is isomorphic to a direct product of n copies of Gm , i.e., G = Gnm . Lemma 2.4.2. Let σ : Ga × X → X be an action of Ga to an affine algebraic scheme X = Spec R and let φ : R → k[t] ⊗k R be a k-algebra homomorphism such that a φ = σ. Define k-linear endomorphisms δi of R for i ≥ 0 by X φ(z) = δi (z)ti . i≥0

Then δ0 = idR and δ1 is a k-derivation of R with values in R. Suppose that k has characteristic zero. Then δi = i!1 δ1i and δ1 is a locally nilpotent derivation (see section 2.1). In terms of δ := δ1 , we have RGa = Ker δ and X1 φ(z) = exp(δ)(z) = δ i (z)ti . i! i≥0

By Lemma 2.1.6, X contains a cylinderlike open set U × A1 if δ ̸= 0.

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235

Suppose that an algebraic group G acts on an affine variety X = Spec R. For a ∈ R, g ∈ G(k) and a point x ∈ X, define g a by g a(x) = a(g −1 x), where gx is a translation of x by g. Thus the coordinate k-algebra R of X has a G(k)-action. We say that a ∈ R is G-invariant if g a = a for all g ∈ G(k). The set RG of G-invariant elements is a k-subalgebra of R. Let k(X) be the function field of X. Since k(X) = Q(R) we define a G(k)action on k(X) by g ξ = g a/g b for ξ = a/b ∈ k(X) with a, b ∈ R. It is clear that RG = k(X)G ∩ R. This implies that if R is integrally closed then so is RG . Lemma 2.4.3. Let σ : Gm × X → X be an action of Gm onto an affine algebraic scheme X = Spec R and let ψ : R → k[t, t−1 ] ⊗ R be a k-algebra homomorphism such that a ψ = σ. Define k-linear endomorphisms ψi for all i ∈ Z by X ψ(z) = ψi (z)ti , i∈Z

P

where ψj ◦ ψi = δji and i∈Z ψi = idR . Hence R = ψi (R) is a Z-graded k-algebra. Further RGm = R0 .

L

i∈Z

Ri with Ri =

Related to Theorem of Abhyankar-Moh-Suzuki (Theorem 2.2.17), we have the following Theorem of Rentschler (case of characteristic zero). For the case of char (k) = p > 0, see [55]. Theorem 2.4.4. Let σ : Ga × A2 → A2 be an action of Ga on the affine plane A2 = Spec R. Then there exists a system of coordinates {x, y} of A2 such that (1) If char (k) = 0 then we can take x, y in such a way that y ∈ RGa and φ(x) = x + f (y)t, where f (y) ∈ k[y]. (2) If char (k) = p > 0 then there exists a system of coordinates {x, y} of A2 such that y ∈ RGa and n

φ(x) = x + f0 (y)t + f1 (y)tp + · · · + fn (y)tp ,

where f0 (y), . . . , fn (y) ∈ k[y]. Proof. We prove only the case of char (k) = 0. Since R is normal, the ring R0 of Ga -invariants is also normal. Further R0 is finitely generated over k (see [27, Theorem 2.2.4]) and (R0 )∗ = k ∗ . In fact, R0 is a UFD. Then R0 = k[f ] (see Lemma 2.1.3). Since A2 contains a cylinderlike open set U × A1 with an open set U of Spec k[f ], Theorem of Abhyankar-MohSuzuki implies that R = k[f, g] for an element g ∈ R. We can choose

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a system of coordinates {x, y} of A2 so that f = y and g = x. Since char (k) = 0 the Ga -action σ is given by a locally nilpotent derivation δ on R such that δ(y) = 0. Then δ(x) ∈ R0 for otherwise δ is no more locally nilpotent. Let δ(x) = f (y). Then φ(x) = x + f (y)t. Remark 2.4.5. In the case char (k) = 0 the additive group scheme Ga has no nontrivial finite groups, but in the case of char (k) = p > 0, Ga contains n a subgroup of order pn as the set of roots of the equation xp − x = 0, where Ga = Spec k[x]. The group {α ∈ k | αp − α = 0} is equal to the set {0, 1, . . . , p − 1} ∼ = Z/pZ. 2.4.2

Finite subgroups of Aut k[x, y]

We prove the following theorem, which follows from Theorem 2.3.8 with the help of Theorem 2.7.10. Theorem 2.4.6. The following assertions hold true. (1) Let F be a finite subgroup of order n of Aut k k[x, y]. Then a conjugate of F is contained in either A2 or J2 . (2) Suppose that either char (k) = 0 or char (k) = p > 0 and p ∤ n, then there exists an element ρ of Aut k k[x, y] such that ρ−1 F ρ is in GL (2, k). (3) If char (k) = p > 0 and p | n, the assertion (2) does not necessarily hold. Proof. (1) The assertion follows from Theorems 2.3.8 and 2.7.10. (2) We may assume that F is a subgroup of A2 or J2 . Case F ⊂ A2 . It suffice to show that there exists a point P = (α, β) such that g P = P for every g ∈ F , i.e., P is a fixed point. In fact, define an affine transformation ρ of A2 by   1 00 ρ(x, y, 1) = (x, y, 1)  0 1 0  . αβ1 Since ρ(0, 0, 1) = (α, β, 1), ρ−1 F ρ ⊂ GL (2, k) if and only if P = (α, β) is a fixed point of F . Each element σ ∈ A2 is expressed by a matrix representation   a(σ) c(σ) 0 σ(x, y, 1) = (x, y, 1)  b(σ) d(σ) 0  . α(σ) β(σ) 1

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237

Let r(σ) = (α(σ), β(σ)) be a point of A2 viewed as a row vector and let A(σ) be an invertible matrix   a(σ) c(σ) A(σ) = . b(σ) d(σ) For σ1 , σ2 ∈ A2 , we have A(σ1 ◦ σ2 ) = A(σ1 )A(σ2 ), r(σ1 ◦ σ2 ) = r(σ1 )A(σ2 ) + r(σ2 ). Then r0 = (α0 , β0 ) is a fixed point of F if and only if r(σ) = r0 − r0 A(σ), P where r(σ) = (α(σ), β(σ)). Set r0 = n1 σ∈F r(σ). This division by n is possible by the assumption. Since ! 1 X 1 X r(σ1 ◦ σ2 ) = r(σ1 ) A(σ2 ) + r(σ2 ), n n σ1 ∈F

σ1 ∈F

we have r(σ2 ) = r0 − r0 A(σ2 ). Hence r0 gives a fixed point of F . Case F ⊂ J2 . Each element τ ∈ J2 acts on A2 by τ (x) = a(τ )x + fτ (y), τ (y) = b(τ )y + c(τ ), a(τ ), b(τ ), c(τ ) ∈ k, a(τ )b(τ ) ̸= 0, fτ (y) ∈ k[y]. The multiplication τ1 ◦ τ2 gives a(τ1 ◦ τ2 ) = a(τ1 )a(τ2 ), fτ1 ◦τ2 (y) = a(τ2 )fτ1 (y) + fτ2 (b(τ1 )y + c(τ1 )), b(τ1 ◦ τ2 ) = b(τ1 )b(τ2 ), c(τ1 ◦ τ2 ) = c(τ2 ) + b(τ2 )c(τ1 ).

Let J20 = {τ ∈ J2 | c(τ ) = 0}, which is clearly a subgroup of J2 . Let P d = n1 τ ∈F c(τ ). Summing up the equation c(τ1 ◦ τ2 ) = c(τ2 ) + b(τ2 )c(τ1 ) over τ1 ∈ F and dividing the sum by n, we have c(τ ) = d − b(τ )d. Then τ (y − d) = b(τ )(y − d). Hence, by replacing y by y − d, we may assume that F ⊂ J20 . We look for a polynomial g(y) ∈ k[y] such that τ (x + g(y)) = a(τ )(x + g(y)) for every τ ∈ F . If such a polynomial exists, define an automorphism ρ ∈ Aut k[x, y] by ρ(x) = x + g(y) and ρ(y) = y. Then (ρ−1 ◦ τ ◦ ρ)(x) = a(τ )x and (ρ−1 ◦ τ ◦ ρ)(y) = b(τ )y. Hence ρ−1 ◦ F ◦ ρ ⊂ GL (2, k). A polynomial g(y) satisfies τ (x + g(y)) = a(τ )(x + g(y)) for every τ ∈ F if and only if fτ (y) = a(τ )g(y) − g(b(τ )y) for every τ ∈ F . Since c(τ ) = 0 for τ ∈ F , by the relation fτ1 ◦τ2 (y) = a(τ2 )fτ1 (y)+fτ2 (b(τ1 )y +c(τ1 )), fτ (y) writes in the form fτ (y) =

a(τ ) 1 fτ ◦τ ′ (y) − fτ ′ (b(τ )y). a(τ ◦ τ ′ ) a(τ ′ )

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Summing up the relation over τ ′ , we have X fτ ◦τ ′ (y) X fτ ′ (b(τ )y) nfτ (y) = a(τ ) − . ′ a(τ ◦ τ ) a(τ ′ ) ′ ′ τ ∈F

τ ∈F

τ (y) Set g(y) = n1 τ ∈F fa(τ ) . Then fτ (y) = a(τ )g(y) − g(b(τ )y) for every τ ∈ F . So, we are done. (3) Let F = Z/pZ. Define an F -action on A2 by σ(1)(x) = x + 1 and σ(1)(y) = y, where σ : F → A2 is a group homomorphism and 1 is a generator of F (see Remark 2.4.5). Then σ(F ) ⊂ J2 , but σ(F ) has no fixed point. So, σ(F ) ̸⊂ GL (2, k).

P

2.4.3

Finite group actions and invariants

We consider a finite group action on an affine variety. Let σ : G×X → X be a G-action on X. We here denote a finite group by the letter G instead of F . A G-action is given by a finite family of k-automorphisms {σg | g ∈ G} such that σgh = σg ◦ σh and σe = idX . The automorphisms σg induce k-algebra ∗ = σh∗ ◦ σg∗ , σ ∗ automorphisms σg∗ of the coordinate ring R of X. Since σgh ∗ does not keep the order of product, we set ρg = σg−1 . Then ρ : G → Aut k R is a group homomorphism. We say that the G-action σ is faithful if σg = id q

σ

implies g = e. Since σ is factored as σ : G −→ G = G/Ker σ −→ Aut k X and σ is injective, we tacitly assume that σ is injective. For a ∈ R, we denote ρg (a) by g a or g(a). The ring of G-invariants of R is the subring RG = {a ∈ R | g a = a, ∀g ∈ G}. Lemma 2.4.7. The following assertions hold under the above settings. (1) If R is finitely generated over k then so is RG . (2) Q(R)G = Q(RG ). (3) If R is normal then so is RG . Proof. (1) Write R = k[a1 , . . . , am ], i.e., {a1 , . . . , am } is a system of generators of R over k. Write G = {g1 , . . . , gn }. For each 1 ≤ i ≤ m, write Y Fi (t) = (t − g ai ) = tn + αi,1 tn−1 + · · · + αi,n , g∈G

αi,1 = (−1)

X

g

ai , . . . , αi,j = (−1)j

g∈G

. . . , αi,n = (−1)

X ℓ1 0 and 0 ≤ nr − ds < n. If G ⊂ SL(2, k) the G-action is ζ · (x, y) = (ζx, ζ −1 y). Then RG = k[xn , xy, y n ] ∼ = k[X, Y, Z]/(XZ = Y n ). Example 2.4.20. Consider the case where G is a binary dihedral group of order 4n. As a subgroup of SL(2, k), G is generated by matrices     ζ2n 0 0i a= and b = , −1 0 ζ2n i0 where ζ2n is a primitive 2nth root of the unity and i2 = −1. Let H = ⟨a⟩ ∼ = Z/2nZ. Since bab−1 = a−1 , H is a normal subgroup of G such that G/H ∼ = Z/2Z. By Example 2.4.19, RH = k[X, Y, Z]/(XZ = Y 2n ), where X = x2n , Y = xy and Z = y 2n . Let b be the residue class of b in G/H. Since b(x) = iy and b(y) = ix, we have b(X) = (iy)2n = (−1)n Z, b(Y ) = −Y, b(Z) = (ix)2n = (−1)n X. (i) Case n ≡ 0 (mod 2). Then b(X) = Z, b(Z) = X and b(Y ) = −Y . Hence RG = (RH )G/H = k[X + Z, Y 2 , (X − Z)Y ].

Set u = X + Z, v = Y 2 and w = (X − Z)Y . Since w2 = (X − Z)2 Y 2 = {(X + Z)2 − 4XZ}Y 2 = (u2 − 4v n )v, we have RG = k[u, v, w]/(w2 − u2 v + 4v n+1 = 0).

Geometry on Affine Surfaces

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(ii) Case n ≡ 1 (mod 2). Then b(X) = −Z, b(Y ) = −Y and b(Z) = −X. Hence RG = k[(X − Z), Y 2 , (X + Z)Y ]. Set u = X − Z, v = Y 2 and w = (X + Z)Y . By a computation similar to the case (i), we have w2 = (u2 − 4v n )v. So, RG = k[u, v, w]/(w2 − u2 v + 4v n+1 = 0). In other cases of finite subgroups of SL(2, k), the invariant subring RG has the following expressions (see Durfee [15]). Proposition 2.4.21. Let G be a binary tetrahedral, binary octahedral or binary icosahedral group in SL(2, k). Then the invariant subring RG is given as the following hypersurface.  2 3 4  k[u, v, w]/(w + u + v = 0) if G is binary tetrahedral G 2 3 R = k[u, v, w]/(w + u + uv 3 = 0) if G is binary octahedral  k[u, v, w]/(w2 + u3 + v 5 = 0) if G is binary icosahedral. e → X be a resolution of singularity at the point O of X. Let Let ν : X −1 E = ν (O) and let E = E1 + · · · + Er be the irreducible decomposition of E whose irreducible components are all smooth rational curves. We say that the resolution ν is minimal if the number of irreducible components r is the smallest. We have the following result. Theorem 2.4.22. Let G be a small finite subgroup of GL (n, k) such that either G is cyclic or G is a subgroup of SL(2, k) and let X = X/G. Let e → X be a minimal resolution of singularity at O and let E = ν −1 (O). ν:X Then the weighted dual graph Γ of E is given as follows. (1) G is a cyclic group of order n and acts on A2 by ζ · (x, y) = (ζx, ζ d y) with 0 < d < n and gcd(n, d) = 1. Let n 1 = [α1 , α2 , . . . , αr ] = α1 − d 1 α2 − .. 1 .− 1 αr−1 − αr be an expression of n/d as a continued fraction, where αi ≥ 2 for 1 ≤ i ≤ r. Then Γ is −α1

−α2

−αr−1

−αr

E1

E2

Er−1

Er .

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Affine Algebraic Geometry

If d = n − 1 then n/(n − 1) = [2, . . . , 2]. | {z } n−1

(2) If G is a subgroup of SL(2, k), then E consists of (−2) curves. So we omit to write the weights, i.e., self-intersection numbers in the dual graph Γ. If G is BD2n , then Γ is

Type Dn+2 . |

{z n

}

(3) G is BT, BO or BI. Then Γ is (i) G = BT Type E6

(ii)

(iii)

G = BO Type E7

G = BI Type E8 .

The quotient singularity with G ⊂ SL(2, k) is called a Du Val singularity or rational double singularity of type A, D, E. The subscript in the notation An , Dn , Er (r = 6, 7, 8) is for the number of irreducible components. Remark 2.4.23. If G ⊂ GL (2, k) the dual graph Γ of the exceptional curve E is similar to the above graphs, but the weights are not necessarily −2. For the detail, see Brieskorn [10]. Let R = k[x, y] be the same as above. A k-subalgebra A of R is said to be cofinite if the extension R over A is integral. So, R is a finite A-module. The following results are obtained by a different idea based on logarithmic Kodaira dimension of X ◦ , where X = Spec A and X ◦ = X \ Sing X. For the first result see [59, Theorem 2.4.8] and for the second one see [59, Theorem 2.5.1].

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255

Theorem 2.4.24. Let A be a cofinite k-subalgebra of R = k[x, y]. If either A is regular or equivalently R is A-flat then A is a polynomial ring over k in two variables. Theorem 2.4.25. Let A be a normal cofinite subalgebra of R = k[x, y] and let X = Spec A. Suppose that X is not smooth. Then X is isomorphic to A2 /G for a small finite subgroup of GL (2, k). Further, the following assertions hold. (1) A is a UFD if and only if X is a hypersurface w2 + u3 + v 5 = 0 in A3 = Spec k[u, v, w]. (2) X ◦ = X \ Sing X contains a cylinderlike open set if and only if G is cyclic.

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2.5 2.5.1

Affine Algebraic Geometry

Birational automorphisms of rational surfaces Noether factorization theorem

This is the subject studied by many people and there are many references, e.g. Nagata [71] and Blanc-Hed´en [8]. We discuss the subject from a viewpoint of affine de Jonqui`ere transformations which played a main role in the previous sections. The base field is in this section an algebraically closed field of characteristic p. We begin with complementing standard results in algebraic geometry which we use in forthcoming arguments. Let X be an irreducible projective variety defined over k. Then we consider a birational morphism ρ : V → X such that V is a smooth projective e be the normalization of X in the function variety. For the existence, let X e If char k = 0, this holds field k(X). Then resolve the singularity of X. by Theorem of Hironaka [34]. If char k = p > 0 and dim X = 2, V exists by a theorem of Zariski. If char k ̸= 2, 3, 5 and dim X = 3, V exists by a Theorem of Abhyankar. Let φ : V → W be a birational morphism of smooth projective surfaces V, W . Then φ is a product of blowing-ups of points (see [31, Corollary 5.4]). Let Vi (i = 1, 2) be a smooth projective surface defined over k. Let φ : V1 99K V2 be a birational mapping. Then φ induces the k-isomorphism φ∗ : ∼ k(V2 ) −→ k(V1 ). If φ∗ = id, we call φ a natural birational automorphism. Let U (φ) be the complement of the fundamental points and the exceptional curves of φ, which is an open set of V1 . Let Γ(φ) ⊂ V1 × V2 be the closure of the graph of φ|U (φ) : U (φ) → V2 , i.e., Γ(φ) = the closure of {(P, φ(P )) | P ∈ U (φ)}. Then there exists a surjective morphism qi : Γ(φ) → Vi (i = 1, 2) which is the restriction to Γ(φ) of the projection from V1 × V2 → Vi . Let σ : Veφ → Γ(φ) be a resolution of singularity of Γ(φ) and let pi = qi · σ. Then pi : Veφ → Vi is a birational morphism of smooth projective surfaces, whence pi is a product of blowing-ups of closed points. Veφ p1

V1

φ ······

p2

V2

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If (P1 , P2 ) ∈ V1 × V2 is a pair of points in Γ(φ) such that P1 ∈ U (φ) and P2 ∈ U (φ−1 ) then φ(P1 ) = P2 and φ−1 (P2 ) = P1 . We consider mostly the case where V1 = V2 , which is a rational surface V , and φ is natural, i.e., φ is the identity on U (φ). If V is either P2 or a Hirzebruch surface Fn (n ≥ 0), the group of all birational automorphisms of V is denoted by Bir(V ). If V = P2 , a birational automorphism of P2 is called a Cremona transformation, and Bir(P2 ) is denoted by Cr2 . If φ : V1 99K V2 is a birational automorphism of the same variety V = V1 = V2 , we call V1 (resp. V2 ) the source (resp. target) of φ. Lemma 2.5.1. Let φ : P2 99K P2 be a Cremona transformation. Let Λ2 = |ℓ| be a complete linear system of lines on the target P2 . With the above e = {p∗ (ℓ) | ℓ ∈ Λ2 } and let Λ1 = {p1∗ p∗ (ℓ) | ℓ ∈ Λ2 }. notations, let Λ 2 2 Then Λ1 is a linear system possibly with base points, whose k-module is M (Λ1 ) = k · φ∗ (x) + k · φ∗ (y) with a system of inhomogeneous coordinates {x, y} of P2 \ ℓ0 ∼ = A2 for ℓ0 ∈ Λ2 , and φ is the rational mapping associated to Λ1 . Proof. Clear. A biregular automorphism of the projective space P2 is a linear, which is given by an element of PGL (3, k) (see Problem 18). The following result is well-known. For the reference, see [71] and [8]. But the proof follows from Corollary 2.5.7 and Lemma 2.5.8. Lemma 2.5.2. The group Cr2 = Bir(P2 ) is generated by linear transformations and natural Cremona transformations. We consider examples. Example 2.5.3. Let (X0 , X1 , X2 ) be a system of homogeneous coordinates on P2 and let P0 = (1, 0, 0), P1 = (0, 1, 0) and P2 = (0, 0, 1). Let p1 : Ve → P2 be the blowing-ups of points P0 , P1 , P2 and let Ei = p−1 1 (Pi ) for i = 0, 1, 2. Let ℓ0 (resp. ℓ1 and ℓ2 ) be the lines connecting points P1 , P2 (resp. P0 , P2 and P0 , P1 ).9 Then the proper transforms ℓ′i of ℓi on Ve is a (−1) curve. So, blow down ℓ′0 , ℓ′1 , ℓ′2 by the morphism p2 : Ve → P2 . Let φ = p2 · p−1 1 . It is a Cremona transformation. 9 We call the triangle △P P P the coordinate triangle of the system of homogeneous 0 1 2 coordinates (X0 , X1 , X2 ). When we say simply a triangle we consider a triangle similar to the coordinate triangle.

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Affine Algebraic Geometry

E2

P2 ℓ1 P0 ℓ2

ℓ′1

p1

ℓ0 P1

ℓ′0

Q1 p2

E1

E0

E2

E0

Q0 E1

Q2

ℓ′2 In terms of homogeneous coordinates, φ is given by  (X0 , X1 , X2 ) 7→

1 1 1 , , X0 X1 X2

 = (X1 X2 , X2 X0 , X0 X1 ).

In terms of linear systems, Λ1 consists of conics passing through the points P0 , P1 , P2 . If we choose three points P0′ , P1′ , P2′ arbitrarily so that they are not colinear, i.e., not lying on the same line, we can construct a birational transformation ψ by means of blowing-ups of P0′ , P1′ , P2′ and contractions of the proper transforms of three lines connecting two of P0′ , P1′ , P2′ . Then ψ = β · φ · α−1 , where α is a linear transformation mapping (P0 , P1 , P2 ) to (P0′ , P1′ , P2′ ) and β is the one mapping (Q0 , Q1 , Q2 ) to (Q′0 , Q′1 , Q′2 ), where Q′0 , Q′1 , Q′2 are the points obtained from the blowing-downs for ψ. We call ψ a quadratic transformation and φ the standard quadratic transformation. Let V be a Hirzebruch surface Fn (n ≥ 0) with a P1 -fibration ρ : V → P . We denote by ℓ and Mn a fiber of ρ and a section with (Mn2 ) = −n. Let P ∈ ℓ and let σ : Ve → V be the blowing-up of P and let E = σ −1 (P ). Then the proper transform ℓ′ of ℓ is a (−1) curve. Let τ : Ve → V ′ be the contraction of ℓ′ , and let ρ′ : V ′ → P1 be the P1 -fibration induced by ρ. Then the fibers of ρ′ are the same as those of ρ except for the fiber (ρ′ )−1 (ρ(ℓ)), which is the image τ (E). The birational mapping τ ◦ σ −1 : V → V ′ is called an elementary transformation of the ruled surface V with center at P . We denote it by elmP : V 99K V ′ . Suppose that V ∼ = Fn . It ′ ∼ ′ ∼ is then clear that V = Fn+1 if P ∈ Mn and V = Fn−1 if P ̸∈ Mn . If P1 , . . . , Pm are distinct points, we denote by elmP1 ,...,Pm the composition elmPm ◦ · · · ◦ elmP1 . The mapping elmP1 ,...,Pm is the identity morphism Sm outside the union of fibers i=1 ρ−1 (ρ(Pi )). Hence it is natural. 1

A quadratic transformation on P2 can be defined for a triplet of points {P0 , P1 , P2 } which are not necessarily ordinary points.

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Example 2.5.4. Consider a triangle consisting of three lines ℓ0 , ℓ1 , ℓ2 and three intersection points P0 , P1 , P2 which are not colinear. Consider a triplet of points {P0 , P2 , P2′ }, where P2′ is the infinitely near point of P2 of the first kind which corresponds to the direction of the line ℓ0 . Let p1 : Ve → P2 be the blowing-ups of P0 , P2 , P2′ and let p2 : Ve → P2 be the contraction of ℓ′0 , ℓ′1 , E2 in the following configuration. E2

P2 ℓ1 P0 ℓ2

p1

ℓ0 P1

ℓ′1

E0

−1 −1

−2

0 ℓ′2

−1 −1

E2′ ℓ′0

Q2 p2

E0 Q0 ℓ′2

′

E2 Q1

The linear system Λ1 on the source P2 corresponding to Λ2 = |ℓ| on the target P2 consists of conics C such that C passes through the points P0 , P2 and C and C ′ are tangent at P2 with i(C, C ′ ; P2 ) = 2, where C and C ′ are general members of Λ1 . We call this transformation p2 ◦ p−1 1 a quadratic transformation of the second kind along P0 , P2 and ℓ0 . The inverse of a quadratic transformation of the second kind along P0 , P2 and ℓ0 is a quadratic transformation of the second kind along Q0 , Q2 and E 0 . Lemma 2.5.5. Let C be an irreducible curve of degree d on P2 which has a one-place singular point P2 of multiplicity d − 1, where d ≥ 2. Then the following assertions hold. (1) There exist s line ℓ0 such that P2 ∈ ℓ0 and (C · ℓ0 ) = i(C, ℓ0 ; P2 ) = d. (2) C is a rational curve with only one place at P2 , and C0 := C \ {P0 } ∼ = A1 . (3) There exists a system of coordinates {x, y} of P2 \ ℓ0 ∼ = A2 such that d ∗ C0 is defined by y + ax = 0 for a ∈ k . (4) There exists a quadratic transformation T of the second kind along P2 and ℓ0 such that the image T (C) is an irreducible curve of degree d − 1 with a point Q2 such that the multiplicity of T (C) at Q2 is d − 2. If d = 2 the image T (C) is a line on P2 . Proof. (1) Let σ : V → P2 be the blowing-up of P2 and let E = σ −1 (P2 ). Then σ ∗ (C) = σ ′ (C) + (d − 1)E and σ ∗ (KP2 ) = KV − E (see Lemma 1.8.9).

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It then follows that 1 ((KV · σ ′ (C)) + (σ ′ (C)2 )) + 1 = 0. 2 Hence σ ′ (C) is a smooth rational curve and σ ′ (C) meets E in a single point Q2 with i(σ ′ (C), E; Q2 ) = d − 1. Let ℓ0 be the line such that P2 ∈ ℓ0 and Q2 ∈ σ ′ (ℓ0 ). Then i(C, ℓ0 ; P2 ) = d. (2) Since C \ {P2 } ∼ = σ ′ (C) \ {Q2 } and σ ′ (C) ∼ = P1 , it follows that 1 ∼ C \ {P2 } = A . (3) We can choose a system of homogeneous coordinates (X0 , X1 , X2 ) on P2 such that P2 and ℓ0 form a part of the coordinate triangle. Let ξ = X0 /X2 and η = X1 /X2 . Then the defining equation of C \ (C ∩ ℓ0 ) is written as ξ d−1 + g(ξ, η) = 0, where g(ξ, η) is a P2 homogeneous polynomial in ξ, η of deξ η gree d. As a homogeneous equation in ℓ1 ℓ0 X0 , X1 , X2 , it is written as X0d−1 X2 + F (X0 , X1 ) = 0, pa (σ ′ (C)) =

where F (X0 , X1 ) = X2d g(X0 /X2 , X1 /X2 ). Let x = X1 /X0 and y = X2 /X0 . The defining equation of C \ P2 in the affine plane Spec k[x, y] is then y + f (x) = 0 with f (x) = X0−d F (X0 , X1 ) ∈ k[x] and deg f (x) ≤ d. Write f (x) = axd + h(x) with a ∈ k ∗ and deg h(x) < d. Set anew y + h(x) as y. (4) Let T : P2 99K P2 be a quadratic transformation of the second kind along P0 , P2 and ℓ0 , where P0 is the point given as C ∩ ℓ1 = {P0 , P2 } and i(C, ℓ1 ; P0 ) = 1. As in Example 2.5.4, blow up the point P2 and its infinitely near point Q2 to produce exceptional curves E2 , E2′ , where (E22 ) = −2 and 2 (E ′ 2 ) = −1. The proper transform C ′ of C is a smooth rational curve touching E2 at E2 ∩ E2′ with multiplicity d − 2. Then we have 2

(C ′ ) = d2 − (d − 1)2 − 1 − 1 = 2d − 3,

where d2 − (d − 1)2 − 1 is the reduction of the self-intersection number by the blowing-ups of P2 , Q2 and the last −1 is due to the blowing-up of the point P0 . Hence the image C of C ′ by the blowing-downs of ℓ′1 and E2 has 2

(C ) = 2d − 3 + (d − 2)2 = (d − 1)2 .

Hence C is an irreducible curve of degree d − 1 on P2 such that it has a one-place singular point of multiplicity d − 2 provided d ≥ 4. Lemma 2.5.6. Let C be an irreducible plane curve of degree d with oneplace singular point P2 of multiplicity d − 1. Assume that either char k = 0 or d is not divisible by p = char k. Then the following assertions hold.

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(1) Let ℓ0 be a line such that P2 ∈ ℓ0 and i(C, ℓ0 ; P2 ) = (C · ℓ0 ) = d. Then C and dℓ0 span a linear pencil Λ whose base points are P2 and its infinitely near points. (2) Let σ e : Ve → P2 be the shortest succession of blowing-ups such that the e of Λ is free from base points and hence induces a proper transform Λ 1 e e be the last exceptional curve appearing P -fibration π e : V → P1 . Let E e containing the proper in the process σ e. Then the member Fe∞ of Λ e is a section of π transform of ℓ0 gives a unique reducible fiber of π e, E e, ′ and the proper transform ℓe0 := σ e (ℓ0 ) is a unique (−1) component of the fiber Fe∞ . (3) Starting with the contraction of ℓe0 we can contract all irreducible come Since ponents of Fe∞ except for the component meeting the section E. 2 2 e ) = −1, we finally contract E e to obtain P . Let τe : Ve → P2 be the (E composition of all these contractions. Then τe ◦ σ e−1 : P2 99K P2 is a 2 natural birational automorphism of P such that (i) the point P2 is a fundamental point and the line ℓ0 is a fundamental curve, (ii) the image of C is a line on the target P2 . Proof. By Lemma 2.5.5, the assertions (2) and (3), one can apply the results in section 2.2 in the proof of Theorem of Abhyankar-Moh-Suzuki. We therein use the assumption that either char k = 0 or d is not divisible by p. The rest of the assertions follow from these arguments. The birational transformation τe ◦ (e σ )−1 of P2 is called a Cremona transformation of de Jonqui`ere type of degree d. The fiber Fe∞ in the case d = 3 is exhibited by the following dual weighted graph e E

E5

E4

−1

−2

−2

E3 −2 −3

E2

ℓe0

−2

−1

E1

where Fe∞ = 3ℓe0 + E1 + 3E2 + 3E3 + 2E4 + E5 with the proper transform Ei on Ve of the exceptional curve of the ith blowing-up. By the assertion (2) of Lemma 2.5.6, we have the following result. Corollary 2.5.7. Let ψ : P2 99K P2 be a Cremona transformation of de

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Jonqui`ere type of degree d. Then ψ is a product of (d − 1) quadratic transformations of the second kind and an element of PGL (3, k). Proof. By Lemma 2.5.6 there exists a quadratic transformation of the second kind T1 such that (T1 ◦ ψ −1 )(ℓ) is an irreducible curve of degree d − 1 with a one-place singular point of multiplicity d − 2. Hence there exist quadratic transformations of the second kind T2 , . . . , Td−1 such that (Td−1 ◦ · · · ◦ T1 ◦ ψ −1 )(ℓ) is a line. Hence Td−1 ◦ · · · ◦ T1 ◦ ψ −1 is a biregular automorphism α of P2 which is an action of an element of PGL (3, k). Hence ψ = α−1 ◦ Td−1 ◦ · · · ◦ T1 . The following result due to Nagata [71, Theorem 6] is essential. Lemma 2.5.8. Let φ : P2 99K P2 be a Cremona transformation. Then φ is a product of Cremona transformations of de Jonqui`ere type. By summarizing the previous results, we obtain Noether factorization theorem.10 Theorem 2.5.9. Let φ : P2 99K P2 be a Cremona transformation. Then φ is a product of quadratic transformations and linear transformations.

10 Noether

factorization theorem is due to Max Noether, who is father of Emmy Noether. Noether normalization lemma is due to Emmy Noether.

Geometry on Affine Surfaces

2.6

263

Boundary divisors of affine surfaces

As in the previous sections, the ground field k is an algebraically closed field of characteristic p. Let V be a smooth projective surface and let D be Pn a reduced effective divisor. Namely D is a sum D = i=1 Ci of irreducible curves Ci with coefficients 1, hence D is identified with a reduced curve Sn i=1 Ci . As defined in subsection 1.7.4, such a divisor D is a divisor of simple normal crossings (an SNC divisor, for short) if every Ci is smooth and the components Ci meet each other transversally at the intersection points. We say that the divisor D is the boundary divisor of X := V \ D if V is the log smooth completion of X by D. 2.6.1

Quantitative criterion of SNC divisors

We begin with the results borrowed from [56]. Pn Lemma 2.6.1. Let V be a smooth projective surface and let D = i=1 Ci be a reduced effective divisor. Let m be the number of connected components of D and define e(D) by X e(D) := m − n + (Ci · Cj ). i 0. the nth symmetric products, that mny /mn+1 = mnx /mn+1 x y Since we have an exact commutative diagram 0 → mny /mn+1 → Oy /mn+1 → Oy /mny → 0 y y ↓ ↓ ↓ n+1 n → O /m → O /m 0 → mnx /mn+1 x x x x x → 0, we deduce, by induction on n, that φn : Oy /mny → Ox /mnx is an isomorphism. So, by taking projective limits, we have an isomorphism ∼ b by −→ φ b:O Ox .

Hence the condition (i) implies the condition (ii). by → O bx is an isomorphism. Then we Conversely, suppose that φ b: O have a commutative diagram φ

Oy −−−−→ Ox   yιx ιy y by −−−−→ O bx , O φ b

where φ b is an isomorphism, and ιy and ιx are faithfully flat. Let α

β

0 −→ L1 −→ L2 −→ L3 −→ 0 be an exact sequence of Oy -modules. Then we have an exact sequence of Ox -modules α⊗O

β⊗Ox

L1 ⊗Oy Ox −→x L2 ⊗Oy Ox −→ L1 ⊗Oy Ox −→ 0. If α⊗Ox is not injective, let K = Ker (α⊗Ox ). Since φ◦ι b y is faithfully flat, bx = 0. Since ιx is faithfully flat, we have K = 0. it follows that K ⊗Ox O This is a contradiction. So, Ox is a flat Oy -module. A morphism f : X → Y of algebraic k-schemes is an ´etale morphism at a closed point x ∈ X if f is unramified at x and OX,x ia a flat OY,y -module for y = f (x). f is an ´etale morphism if f is unramified and flat.

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2.7.2

Affine Algebraic Geometry

´ Etale coverings

A morphism f : X → Y as above is an ´etale covering if f is an ´etale and finite morphism. Lemma 2.7.3. Let f : X → Y be an ´etale covering of algebraic varieties of dimension n. Then the following assertions hold. (1) X is smooth if and only if Y is smooth. (2) Suppose that Y is smooth. Then we have: (2-1) For each closed point y ∈ Y , f −1 (y) consists of d points, where d = [k(X) : k(Y )]. Hence the extension k(X)/k(Y ) is an algebraic separable extension. (2-2) Ω1X/k ∼ = f ∗ Ω1Y /k . In particular, if X and Y are projective, KX/k ∼ ∗ f KY /k . Proof. (1) Let x ∈ X and y ∈ Y be closed points such that y = f (x). Then Ox is a flat Oy -module. If Ox is regular, then Oy is regular by Remark 1.9.9(4)(i). So, the only if part holds. If Oy is regular, my is generated by a regular system of parameters {η, . . . , ηn }. Since mx /m2x ∼ = Pn my /m2y , we have mx = O η . Hence {η , . . . , η } is also a regular 1 n i=1 x i system of parameters of Ox . So, Ox is regular, and the if part holds. (2-1) Let V = Spec B be an affine open neighborhood of y and let U = f −1 (V ) = Spec A. Then A is a finite B-module which is flat since f is a flat morphism (see Lemma 1.9.6). Then A ⊗B Oy is a finite flat Oy -module, hence it is a free Oy -module of rank d = [k(X) : k(Y )] (see [54, Theorem 7.10]). By Lemma 2.7.2, we have A ⊗B Oy /my ∼ = A/m1 × · · · × A/md .

Since A ⊗B Oy is a free Ou -module, the number of points in f −1 (y) is equal to [k(X) : k(Y )]. Note that the extension k(X)/k(Y ) is a finite algebraic extension. If it involves a purely inseparable extension, the number of points of f −1 (y) for a general point y ∈ Y is strictly less than d. (2-2) We have an exact sequence of coherent OX -Modules f ∗ Ω1Y /k → Ω1X/k → Ω1X/Y → 0,

where Ω1X/Y = 0 since f is an unramified morphism. Meanwhile, since X and Y are both smooth varieties of the same dimension n, f ∗ (Ω1Y /k ) has no torsion elements. So, f ∗ Ω1Y /k → Ω1X/k is injective and hence an isomorphism. By the nth exterior products, we have KX/k ∼ f ∗ KY /k if X and Y are projective and the canonical divisors are defined.

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We say that an algebraic variety X is simply connected if any ´etale `d covering f : X → Y is decomposed into d copies of Y , i.e., X = i=1 Xi , where Xi ∼ = Y for all 1 ≤ i ≤ d and d = deg f = [k(X) : k(Y )]. An easy example is given by Example 2.7.4. The projective line P1 is simply connected. In fact, let f : X → Y be an ´etale covering with Y ∼ = P1 . Since the restriction of f to a connected component of X is an ´etale covering of Y , we may assume that X is a connected smooth projective curve. By Lemma 2.7.3, we have KX/k = f ∗ (KY /k ). Hence 2gX − 2 = d(2gY − 2), where d = deg f and gX (resp. gY ) is the genus of X (resp. Y ). So, gX = d(0 − 1) + 1 = 1 − d ≥ 0. This implies that d = 1 and f is an isomorphism. 2.7.3

Riemann-Hurwitz formula for curves

In order to consider an ´etale covering of a non-complete algebraic variety we need to consider a dominant finite morphism f : X → Y from a normal projective variety to a smooth projective variety. Since f is then surjective, f is called sometimes a finite covering of Y . Given a smooth projective variety Y , a finite covering f : X → Y gives rise to a finite algebraic extension k(X)/k(Y ). Conversely, given a finite algebraic extension K/k(Y ), take the normalization X of Y in K and the normalization morphism f : X → Y . Then f is a finite covering (see Theorem 1.5.15). The variety X is necessarily projective. Hence there is a one-to-one correspondence between the set of finite algebraic extensions of k(Y ) and finite coverings of Y . We consider the case dim Y = 1 and the extension k(X)/k(Y ) is separable, i.e., p ∤ d. Then X is also smooth, and we have an exact sequence of coherent OX -Modules u

0 −→ f ∗ Ω1Y /k −→ Ω1X/k −→ Ω1X/Y −→ 0,

where u : f ∗ Ω1Y /k → Ω1X/k is injective. In fact, otherwise an element z in the kernel is a torsion element of f ∗ Ω1Y /k , i.e., there exists locally an element a ∈ OX (U ) such that az = 0. This is because uξ : (f ∗ Ω1Y /k )ξ → Ω1X/k,ξ for the generic point ξ of X is an isomorphism of k(X)-modules Ω1k(Y )/k ⊗k(Y ) k(X) → Ω1k(X)/k . But there are no torsion elements in a locally free OX -Module f ∗ Ω1Y /k . Let y ∈ Y be a closed point. The local ring (Oy , my ) is a DVR with my = (t) for a uniformisant t. Let f −1 (y) = {x1 , . . . , xr }. By the argument in the proof of Lemma 2.7.3, we have A ⊗B Oy /my ∼ = Ox /me1 × · · · × Ox /mer , 1

x1

r

xr

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Affine Algebraic Geometry

where d = e1 + · · · + er . Here V = Spec B is an affine open neighborhood of y and U := f −1 (V ) = Spec A. The integer ei is called the ramification index of f at xi . It is clear that f is unramified at xi if and only if ei = 1. Let τi be a uniformisant of mxi . Then t = vτ ei with v ∈ Ox∗i . Then Ω1X/Y,xi ∼ = k[τi ]/(τiei −1 ) if p ∤ ei because dt = ei (τi )ei −1 dτi . In fact, Ω1X/Y has stalks 0 except for points with ramification index > 1 provided we assume that all ramification indices are not divisible by the characteristic p. By taking the Euler-Poincar´e characteristics of the exact sequence, we have χ(f ∗ Ω1Y /k ) + ℓ(Ω1X/Y ) = χ(Ω1X/k ). By the Riemann-Roch theorem (Theorem 1.8.3), we have χ(f ∗ Ω1Y /k ) = 1 − g + deg(f ∗ KY ) χ(Ω1X/k ) = 1 − g + deg(KX )

P and ℓ(Ω1X/Y ) = x∈X (e(x) − 1), where g = gX and e(x) is the ramification index of f at x. Hence we have X d deg KY + (e(x) − 1) = deg KX . x∈X

Thus we obtain the following theorem, called the Riemann-Hurwitz formula. Theorem 2.7.5. Let f : X → Y be a finite covering of smooth projective curves such that p ∤ d := [k(X), k(Y )] and p ∤ e(x) for all x ∈ X if e(x) > 1. Then we have the following formula X 2gX − 2 = d(2gY − 2) + (e(x) − 1). x∈X

2.7.4

Inverse and direct images of divisors and the projection formula

Let X be a normal algebraic surface defined over an algebraically closed field k. Then the group Cℓ (X) which is by definition the quotient group of Div (X) by the subgroup of divisors in Div (X) which are linearly equivalent to zero. Note that Cℓ (X) is an abelian group, which may have torsion elements. The Q-vector space spanned by Cℓ (X) is Cℓ (X)Q := Cℓ (X) ⊗Z Q, where the torsion elements in Cℓ (X) disappear because [D] = (1/n)(n[D]) = 0 if n[D] = 0 in Cℓ (X). Since the set Sing X of singular points is a finite closed set, let X ◦ = X \ Sing X. Then Pic (X ◦ ) ∼ = Cℓ (X ◦ ) ∼ = Cℓ (X). The last isomorphism is given by a mapping

Geometry on Affine Surfaces

283

P

P ni Di 7→ i ni Di , where Di is the closure in X of an irreducible divisor Di in X ◦ . This mapping clearly preserves the linear equivalence. Let f : X → Y be a finite morphism of normal algebraic surfaces, which we assume is dominant. We may replace Y (resp. X) by Y ′ = Y \ (Sing Y ∪ f (Sing X)) (resp. X ′ := f −1 (Y ′ )) and assume that X and Y are smooth. Then f∗ (OX ) is a flat OY -Module (see Remark 1.9.9(4)(iii)). For an invertible sheaf OX (D), f∗ (OX (D)) is a locally free OY -module. Let Pn (j) V = {Vj } be an open covering of Y such that f∗ (OX (D))|Vj = i=1 OVj ei is a free OVj -Module. Then, on Vj ∩ Vj ′ , we have i

   (j) (j ′ ) e1 e1    .   .  = Aj ′ j  ..  ,  .   .  (j) (j ′ ) en en 

where Aj ′ j ∈ GL (n, Γ(Vj ∩ Vj ′ , OY )). We call {Aj ′ j | j, j ′ ∈ J} the transition matrices of f∗ (OX (D)). Then {det(Aj ′ j ) | j, j ′ ∈ J} defines an invertible sheaf det(f∗ (OX (D))) on Y , which we denote by f∗ (D) and call the direct image of D. If B is a divisor on Y , the inverse image f ∗ OY (B) is an invertible sheaf on X, which we call the inverse image of B. The direct and inverse images are defined in more geometric terms as follows. For an irreducible divisor D of X, we define the direct image f∗ D by f∗ D = [k(D) : k(D)]D, where D is the image f (D) and [k(D) : k(D)] is the degree of the field extension k(D)/k(D). Note that f (D) is a closed irreducible divisor of Y and the restriction f |D : D → D is also a finite P morphism. If D = i ni Di is a divisor on X, we define f∗ D by f∗ D = P i ni f∗ Di . For an irreducible divisor B on Y , the local ring OY,B is a e be the normalization of OY,B in DVR of the function field k(Y ). Let O e k(X). Then O is the intersection of finitely many DVRs O1 , . . . , Or of k(X) such that Oi = OX,Di for an irreducible divisor Di for 1 ≤ i ≤ r. We define the ramification index ei of Di over B as the integer vi (τ ), where τ is a uniformisant of OY,B and vi is the valuation of k(X) associated to the Pr valuation ring Oi . We define the inverse image f ∗ B as f ∗ (B) = i=1 ei Di . P P For a divisor B = j mj Bj , we define f ∗ B = j f ∗ (Bj ). Suppose that B − B ′ = (b) with b ∈ k(Y ). If an irreducible divisor Di of X lies over an irreducible divisor Bj on Y . Let ti (resp. τj ) be a uniformisant of OX,Di (resp. OY,Bj ) and let vi (resp. wj ) be the associated valuation of k(X) (resp. k(Y )). Then vi (b) = wj (b)vi (τj ). This implies that if B − B ′ = (b) on Y then f ∗ B − f ∗ B ′ = (b) on X. Hence f ∗ : Cℓ (Y ) → Cℓ (X) is defined.

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Both definitions of f∗ and f ∗ coincide with each other by [EGA, IV, Prop. (21.10.17)]. The direct and inverse images define Q-linear homomorphisms f∗ : Cℓ (X)Q → Cℓ (Y )Q and f ∗ : Cℓ (Y ) → Cℓ (X) satisfying the following formula, called the projection formula. Theorem 2.7.6. With the above notations, we have an equality f∗ (f ∗ B) = (deg f )B for a divisor B ∈ Cℓ (Y ). In order to prove the result, we need to prove that f ∗ and f∗ are defined as Q-linear homomorphisms and then the formula holds. We prove it only in the case k(X)/k(Y ) is a separable extension. If the field extension k(X)/k(Y ) is a Galois extension with Galois group G, then G acts on X. In fact, V = Spec S is an affine open set then f −1 (V ) = Spec A is an affine open set because a finite morphism f is an affine morphism. Hence A is an integral closure of B in k(X). So, G acts on A (see Problem 9 of Chapter 1). Further, B is the set of G-invariant elements. In fact, k(Y ) ∩ A = B. We then say that X is a Galois covering of Y and Y = X/G. For any point y ∈ Y , G acts on the set f −1 (y) transitively. Lemma 2.7.7. Suppose that k(X)/k(Y ) is a separable algebraic extension. Then f∗ : Cℓ (X)Q → Cℓ (Y )Q is defined, and the projection formula holds. Proof. Suppose that k(X)/k(Y ) is a Galois extension. Then the Galois group G acts on X and Y = X/G. Let D be an irreducible divisor and let P g g∈G D is the sum of translates of D by G. Let H be the stabilizer of D, i.e., H = {g ∈ G | g D = D}. It is clear that the stabilizer of g D is gHg −1 and [G : H] = [G : gHg −1 ]. Let D = f (D). Since [k(g D) : k(D)] = [k(D) : k(D)], we have   X X g  f∗  D = f∗ (g D) = |G|f∗ D. g∈G

g∈G

P P P If D = i ni Di is a divisor, we can define g∈G g D by i ni ( g∈G g Di ). Suppose that D − D′ = (h) with h ∈ k(X). Then we have g D − g D′ = (g h) for g ∈ G and hence X X Y g g ′ g D− D = (N (h)), N (h) = h. P

g∈G

g∈D

g∈G

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285

By applying f∗ on both sides of the above equality, we have

|G|(f∗ D) − |G|(f∗ D′ ) = f∗ (N (h)) = |G|(N (h)).

Hence we have f∗ D − f∗ D′ = (N (h)) in Div (Y ) ⊗ Q. So, f∗ : Cℓ (X)Q → f∗ (Y )Q is defined. If B is an irreducible divisor on Y , it is clear by the above definitions that ! X X X ∗ f∗ (f B) = f∗ ei Di = ei f∗ Di = ei [k(Di ) : k(B)]B = |G|B. i

i

i

We prove the last equality by the same arguments as in used in the proof of Lemma 2.7.2. Hence, by the linearity, the projection formula holds for a general divisor B on Y . Suppose next that k(X)/k(Y ) is a separable algebraic extension. Let L be a minimal Galois extension of k(Y ) such that k(X) is a subfield of L. Let Z be the integral closure of Y in L, and let fZ be the normalization morphism. Then fZ : Z → Y splits as q

f

fZ : Z −→ X −→ Y, where, if G is the Galois group of L/k(Y ), there exists a subgroup H of G such that q : Z → X is the quotient morphism by H. For a divisor B ∈ Cℓ (Y )Q , it is clear that f ∗ (B) ∈ Cℓ (X)Q . By the case of Galois extension applied to q : Z → X, we have q∗ q ∗ (f ∗ (B)) = |H|f ∗ (B). Hence we have 1 1 f∗ f ∗ (B) = f∗ (q∗ q ∗ (f ∗ (B))) = (f ◦ q)∗ (f ◦ q)∗ (B) |H| |H| |G| 1 (fZ )∗ (fZ )∗ (B) = B = [G : H]B. = |H| |H| So, the projection formula holds. Since 1 1 1 f∗ D = f∗ (|H|D) = f∗ (q∗ q ∗ D) = (fZ )∗ (q ∗ D) |H| |H| |H|

for D ∈ Cℓ (X)Q , f∗ : Cℓ (X)Q → Cℓ (Y )Q is a linear homomorphism.

For the sake of later use, we state the following result, called the projection formula. We refer the readers to Hartshorne [31, Appendix A, A.4]. Theorem 2.7.8. Let f : X → Y be a finite morphism of smooth projective surfaces. Let C (resp. D) be a divisor on X (resp. Y ). Then the following formula holds. (C · f ∗ D) = (f∗ C · D).

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Proof. To understand the above formula intuitively, suppose that both C and C are smooth irreducible curves and that D is an effective divisor such that Supp D has no intersection with the branch locus B(φ) of φ := f |C : C → C. Here B(φ) is the image by φ of the ramification locus R(φ) of φ, where the ramification locus R(φ) is the set of points of C at which the morphism φ ramifies. The ramification locus and the branch locus are closed set of C and C, respectively. For each point Q ∈ D ∩ C, there are m points P1 , . . . , Pm of C such that φ(Pi ) = Q, where m = deg φ. Since P1 , . . . , Pm lie outside of R(φ), we have isobC,P ∼ b morphisms O O i = C,Q for each i. Write D = (a) locally near the point Q with a ∈ mC,Q . Then a is the defining equation of (f ∗ D)|C at every point Pi . Hence i(f ∗ D, C; Pi ) = i(D, C; Q). This implies that Pm ∗ i=1 i(f D, C; Pi ) = m · i(D, C; Q) = i(D, f∗ (C); Q). This observation implies the projection formula under the assumption. Proof in a general case is reduced to the special case. 2.7.5

Amalgamated product of two groups

The definition of free product and amalgamated product of finitely many groups can be found in any standard textbook on group theory. But, in order to make the presentation of the book self-contained as much as possible, we give a definition according to Hall [30] in the case of two groups G, H such that G, H have subgroups A ⊂ G and B ⊂ H with both A and B isomorphic to a group C by fixed isomorphisms. By these isomorphisms, we identify A, B with C and call C the subgroups of G, H. ` Let X = G H be the set-theoretic disjoint sum of G and H. An ordered finite set (x1 , x2 , . . . , xn ) of elements of X is called a word, and let W be the set of words. We include the empty set ∅ as a word. Let w1 = (x1 , . . . .xn ), w2 = (y1 , . . . , ym ) ∈ W be two words. Then a set w1 w2 = (x1 , . . . , xn , y1 , . . . , ym ) is a word. By assigning to a pair of words (w1 , w2 ) the union w1 w2 , we can define a multiplication in the set W which is associative and for which the empty word ∅ plays a role of identity. We define an equivalence relation by the following conditions. (i) If w = (x1 , . . . , xn ) has xi = 1 (the identity element of G or H), then w ∼ (x1 , . . . , xi−1 , xi+1 , . . . , xn ).

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(ii) If w has two adjacent elements xi , xi+1 belonging to the same group G or H, then w ∼ (x1 , . . . , xi−1 , xi xi+1 , xi+2 , . . . , xn ). (iii) If xi = c ∈ C then w ∼ (x1 , . . . , xi−1 c, xi+1 , . . . , xn ) ∼ (x1 , . . . , xi−1 , cxi+1 , . . . , xn ). (iv) If w1 ∼ w2 and w2 ∼ w3 then w1 ∼ w3 . By the condition (i), a word (1, 1, . . . , 1) ∼ ∅. The equivalence class of words containing w = (x1 , . . . , xn ) is denoted by x1 x2 · · · xn . The set W/(∼) of the equivalence classes of words is denoted by G ∨C H and called the amalgamated product of G, H over C. It is a group with multiplication induced by the one in W, which is compatible with equivalence relation, and the identity element induced by ∅. The inverse of the class of w = −1 (x1 , . . . , xn ) is w−1 = (x−1 n , . . . , x1 ). If the group C is trivial then G ∨C H is a free product of G, H and denoted by G ∗ H. An element g of G ∨C H has the shortest expression (as a word) if g is represented by a word (x1 , . . . , xn ) of length n and not represented by a word of shorter length. Every element G ∨C H has such an expression. By the above construction of equivalence relation, a word with the shortest length to express a given element of G ∨C H is uniquely determined modulo the equivalence given by the condition (iii). Namely, if (x1 , . . . , xn ) is one expression of the shortest length then, for every c ∈ C, (x1 , . . . , xi c, c−1 xi+1 , . . . , xn ) represents the same element of G ∨C H and has the same length n. Lemma 2.7.9. Let Γ be a group generated by its subgroups G and H. Then Γ is isomorphic to G ∨C H if and only if the following three conditions are satisfied. (a) Let r ≥ 1, let gi ∈ G \ H (1 ≤ i ≤ r) and let hj ∈ H \ G (1 ≤ j ≤ r − 1). Then g1 h1 g2 h2 · · · gr−1 hr−1 gr ̸∈ H. (b) Let r ≥ 1, let gi ∈ G \ H (1 ≤ i ≤ r − 1) and let hj ∈ H \ G (1 ≤ j ≤ r). Then h1 g1 h2 g2 · · · hr−1 gr−1 hr ̸∈ G. (c) G ∩ H = C. Proof. Suppose that Γ ∼ = G∨C H. Then every element x of Γ is represented by a word of the shortest length, say, x = g1 h1 · · · gm hm gm+1 , where either gi ∈ G \ H and hi ∈ H \ G for (1 ≤ i ≤ m) or gi ∈ G \ H (2 ≤ i ≤ m + 1) and hi ∈ H \ G (1 ≤ i ≤ m). Then the conditions (a) and (b) are satisfied for otherwise the word is replaced by a word of shorter length. By the

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condition (iii) for equivalence relation, an element x of G or H such that (g, x, h) ∼ (gx, h) ∼ (g, xh) is an element of C. Hence G ∩ H = C. If the conditions (a), (b) and (c) are satisfied, we can show that Γ ∼ = G ∨C H. The proof is left to the readers. We prove the following well-known result. Theorem 2.7.10. Let Γ = G ∨C H be an amalgamated product of G and H over C = G ∩ H. Let x be an element of finite order of Γ. Then x is conjugate to an element of G or H. Proof. The following argument is due to Kraft-Schwarz [49]. Let x be an element of order n. We may assume that n > 1. Further, we may assume that, by replacing x by its conjugate, x has an expression of the smallest order among the conjugates of x. Write x in either form of the shortest length, (i) x = g1 h1 · · · gr−1 hr−1 gr , gi ∈ G \ H, hi ∈ H \ G (1 ≤ i < r), (ii) x = h1 g1 · · · hr−1 gr−1 hr , hi ∈ H \ G, gi ∈ G \ H (1 ≤ i < r). We consider the case (i). The other case can be handled in the same fashion. Since x has finite order and x2 = g1 h1 · · · gr−1 hr−1 (gr g1 )h1 · · · gr−1 hr−1 gr , we have gr g1 ∈ H; for, otherwise, the above expression is the one for x2 of the shortest length and x could not have finite order. Then we have g1−1 xg1 = h1 g2 · · · gr−1 (hr−1 gr g1 ),

where hr−1 gr g1 ∈ H, and g −1 xg1 has the expression of length smaller than the length of x. This is a contradiction to the choice of x. More generally, there is a theorem due to Serre [90]. Independently, Igarashi and the author [37] proved Theorem 2.4.6, and its proof contains a proof of Lemma 2.7.12. The proof given below is based on our proof. Theorem 2.7.11. With the same setting as in Theorem 2.7.10, let F be a finite subgroup of A := G ∨C H. Then F is conjugate to a subgroup of G or H. Proof. We use the following Lemma 2.7.12. If there exists an element x ∈ F \(G∪H), we have a−1 xa ∈ G∪H and |a−1 F a∩(G∪H)| > |F ∩(G∪H)|. Hence, by replacing F by a suitable conjugate of F , we may assume that F ⊂ G ∪ H. If F ̸⊂ G and F ̸⊂ H, there exist elements g, h of F such

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that g ∈ G \ H and h ∈ H \ G. By Lemma 2.7.9, the element gh of F does not have finite order. This is a contradiction. Hence either F ⊂ G or F ⊂ H. Lemma 2.7.12. With the same notations as in Theorem 2.7.11, let x be an element of F . Then there exists an element a ∈ A such that a−1 xa ∈ G ∪ H and a−1 (F ∩ (G ∪ H))a ⊂ a−1 F a ∩ (G ∪ H), where G ∪ H is the set-theoretic union of G and H in A. Proof. Since A is an amalgamated product of G and H over C = G ∩ H, the element x is written in one of the following ways: (i)r x = g1 h1 · · · gr−1 hr−1 gr hr with gi ∈ G \ C (1 ≤ i ≤ r), hj ∈ H \ C (1 ≤ j < r) and hr ∈ H, ′ (ii)r x = h′1 g1′ · · · h′r−1 gr−1 h′r gr′ with h′i ∈ H \ C (1 ≤ i ≤ r), gj′ ∈ G \ C ′ (1 ≤ j < r) and gr ∈ G. We prove the following assertions for every r ≥ 1. (1)r If x is written in the way (i)r then there exists an element a ∈ A such that a−1 xa is written in the way (ii)r−1 and a−1 (F ∩ (G ∪ H))a ⊂ a−1 F a ∩ (G ∪ H); (2)r If x is written in the way (ii)r then there exists an element a ∈ A such that a−1 xa is written in the way (i)r−1 and a−1 (F ∩ (G ∪ H))a ⊂ a−1 F a ∩ (G ∪ H). The assertion (1)1 and (2)1 are understood to be x ∈ G and x ∈ H, respectively. In fact, if x = g1 h1 with g1 ∈ G \ C and h1 ∈ H \ C then x does not have finite order. Hence h1 ∈ C. So, x ∈ G. By a similar argument, x ∈ H in the case (2)1 . Proof of the assertion (1)r . Since xn = 1 for some n > 0 we have xn = (g1 h1 · · · gr hr ) · · · (g1 h1 · · · gr hr ) = 1. {z } | n-times

Since (h1 · · · gr hr )(g1 h1 · · · gr hr ) · · · (g1 h1 · · · gr hr ) = g1−1 ∈ G, Lemma 2.7.9 implies that hr ∈ C = G ∩ H. If r = 1 then x = g1 h1 ∈ G, i.e., (1)1 holds. Suppose that r > 1. By Lemma 2.7.9, we have gr hr g1 ∈ C. In fact, we have (h1 · · · hr−1 )(gr hr g1 )(h1 · · · hr−1 ) · · · (gr hr g1 )(h1 · · · hr−1 ) = (gr hr g1 )−1 ∈ G.

Let g = gr hr g1 . Then gr hr = gg1−1 ∈ G and

g1−1 xg1 = h1 g2 · · · gr−1 hr−1 g.

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Hence g1−1 xg1 has an expression as in (ii)r−1 . We prove that g1−1 (F ∩ (G ∪ H))g1 ⊂ g1−1 F g1 ∩ (G ∪ H). Let x0 ∈ F ∩ (G ∪ H). Since xx0 ∈ F and hence (xx0 )m = 1 for some m > 0, we have (xx0 )m = (g1 h1 · · · gr hr x0 ) · · · (g1 h1 · · · gr hr x0 ) = 1. {z } | m-times

Since (h1 g2 · · · gr hr x0 )(g1 h1 · · · gr hr x0 ) · · · (g1 h1 · · · gr hr x0 ) = g1−1 ∈ G and since gr hr ∈ G \ C, Lemma 2.7.9 implies that x0 ̸∈ H \ C, i.e., x0 ∈ G. Since this is the case for every element x0 of F ∩ (G ∪ H), it follows that F ∩ (G ∪ H) = F ∩ G. Then g1−1 (F ∩ G)g1 ⊂ g1−1 F g1 ∩ (G ∪ H) because g1 ∈ G. So, take g1 as a. Proof of the assertion (2)r . Since xn = 1 for some n > 0, gr′ ∈ C by Lemma 2.7.9. If r−1 then x = h′1 g1′ ∈ H, hence (2)1 holds. Suppose that r > 1. As in the case (1)r , h′r gr′ h′1 ∈ C by Lemma 2.7.9. Set h′ = h′r gr′ h′1 . −1 −1 −1 ′ h′ . So, h′1 xh′1 has Then h′r gr′ = h′ h′1 and h′1 xh′1 = g1′ h′2 · · · h′r−1 gr−1 an expression as in (i)r−1 . Next we show that h′1

−1

(F ∩ (G ∪ H))h′1 ⊂ h′1

−1

F h′1 ∩ (G ∪ H).

Let x0 be an element of F ∩ (G ∪ H). Since xx0 ∈ F and (xx0 )m = 1 for some m > 0, a similar argument as for the case (1)r shows that x0 ∈ H. −1 −1 Hence F ∩ (G ∪ H) = F ∩ H. Then h′1 (F ∩ H)h′1 ⊂ h′1 F h′1 ∩ (G ∪ H) because h′1 ∈ H. So, take h′1 as a. 2.7.6

Quotient varieties by finite group actions and ramification of the quotient morphism

Let X be a smooth algebraic variety defined over an algebraically closed field k. We assume that a finite group G acts on X and char k is prime to the order of G. If X is an affine variety Spec A, we defined the quotient variety Y as a normal algebraic variety Spec AG and quotient morphism q : X → Y as the morphism defined by the inclusion AG ,→ A. The morphism q has the following properties. (1) For each closed point y ∈ Y , the finite set q −1 (y) is G-transitive. (2) For an affine open set V = Spec R of Y , the inverse image U = q −1 (V ) is an affine open set Spec S and R = S G .

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In fact, (1) follows from Problem 9 in Chapter 1. Since q is a finite morphism, it is an affine morphism. Hence U is an affine open set (see Problem 7, Chapter 1). Note that R consists of an element h ∈ k(Y ) which is regular at every closed point of V . Similarly, f ∈ k(X) is regular on U if and only if f ∈ S. For h ∈ R, q ∗ (h)(P ) = h(q(P )) is defined at q(P ) if P ∈ U . So, q ∗ (h) is regular on U . Hence R ⊆ S ∩ q ∗ (k(Y )) = S G . Conversely, let f ∈ S G = S ∩ q ∗ (k(Y )). Write f = q ∗ (ξ) with ξ ∈ k(Y ). For a closed point Q ∈ V , there exists a point P ∈ U such that q(P ) = Q. Then f (P ) = ξ(q(P )) = ξ(Q) is defined. So, ξ ∈ R. Hence S G ⊆ q ∗ R. Thus S G = q ∗ (R). We simply denote this by R = S G . If X is a projective variety, we have the following result. Lemma 2.7.13. Let X be a smooth projective variety with an action of a finite group G. Then there exists a normal algebraic variety Y and a morphism q : X → Y such that, for any affine open set V of Y , the inverse U = f −1 (V ) is a G-stable affine open set and V = U/G. Proof. For a closed point P ∈ X, let O(P ) = {g(P ) | g ∈ G}, which is a finite set. Then there exists a hypersurface section F of X such that O(P ) ∩ F = ∅.13 Let U = X \ F , which is an affine open neighborhood of T O(P ). Then g∈G g(U ) is a G-stable affine open neighborhood of O(P ). S So, X is covered by G-stable affine open sets, say X = i∈I Ui with Ui = Spec Ai . Let Bi = AG i and Vi = Spec Bi . For i, j ∈ I, Ui ∩ Uj is a G-stable open set Spec Aij . Since Ai is an affine domain by the above construction, the open set Ui ∩ Uj is an affine variety. The observation before the statement of the lemma shows that Vi ∩ Vj = (Ui ∩ Uj )/G in Vi . This equality holds also as an open set of Vj . Then {Vi | i ∈ I} patch S together to define Y = i∈I Vi . The quotient morphisms qi : Ui → Vi also patch together and give the quotient morphism q : X → Y . Since qi is a finite morphism, so is q. If V is an affine open set, V ∩ Vi is affine as well for i ∈ I. The inverse U = f −1 (V ) is covered by Ui′ = qi−1 (V ∩ Vi ) (i ∈ I). Since V ∩ Vi = Ui′ /G for every i ∈ I, it follows by a similar argument that V = U/G. Remark 2.7.14. It is shown in [24, Expos´e V, Prop. 1.8] that if a finite 13 Use the argument in the answer to Problem 9, Chapter 1. Write X = Proj R. Let M and Pg be the irrelevant ideal S and the homogeneous prime ideal of the point g(P ). among It suffices to show that M ⫌ g∈G Pg . Since there is no inclusion relation Q Q {Pg | g P ∈ G}, Pg ̸⊃ h̸=g Ph . Take S a homogeneous element ag ∈ Pg \ ( h̸=g Ph ) and let a = g∈G ag . Then a ∈ M \ ( g∈G Pg ).

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group G acts on a normal algebraic variety, the quotient morphism q : X → Y = X/G exists if and only if X is covered by G-stable affine open sets. Suppose that a finite group G acts faithfully on a smooth algebraic variety X of dimension n and the quotient Y = X/G exists. For a closed point P ∈ X, let GP = {g ∈ G | g(P ) = P } the stabilizer group or the isotropy group of P and let Γ be the set {P ∈ X | GP ̸= (e)}, where e is the identity element of G. We call Γ the stabilizer locus in X and denote it also Γ(G, X). A point P is a fixed point if GP = G. The set of fixed points in X is called the fixed point locus, which is a closed set of X by the next Lemma 2.7.15(1). If we are given a morphism f : X → X ′ of varieties endowed with actions of a group G such that f (gx) = gf (x) for g ∈ G and x ∈ X, we say that f is G-equivariant or simply G-morphism. We have the following result. Lemma 2.7.15. With the setting as above, the following assertions hold. (1) Γ is a closed set of codimension > 0. The function on ∆ : P ∈ ∆ 7→ |GP | is upper semicontinuous. (2) Let P be a closed point of X, GP the stabilizer group and m = mX,P b of the maximal ideal of the local ring OX,P . Then the completion O the local ring OX,P has a natural GP -action as with the k-vector space b m/m2 endowed with the natural linear representation of GP , and O b is isomorphic GP -equivariantly to the completion Om/m2 ,O , where O is the point of origin. Namely, if {ξ1 , . . . , ξn } is a regular system of parameters of O and if we denote by ui the residue class ξi + m2 , there Pn b is a linear representation of G on the k-vector space i=1 k · ui and O is GP -isomorphic to a ring of formal power series k[[u1 , . . . , un ]] with the GP -action extended from the linear representation. (3) Let P be a closed point of X and let H = GP . Let O = OX,P and let OH be the H-invariant subring of O, which is a local ring. Then d H. b H∼ (O) =O Proof. (1) Let Φ : G × X → X × X be the graph morphism (g, x) 7→ (g(x), x). Let ∆ : X → X × X be the diagonal morphism x 7→ (x, x) which is a closed immersion. Let G∆ be the fiber product (G×X, Φ)×X×X (X, ∆). Then G∆ is a disjoint union of closed subsets, each of which is a closed subset {x ∈ X | g(x) = x} ⊆ (g) × X for a fixed g ∈ G. Then Γ is the image of G∆ \ ((e) × X) by the projection p2 : G × X → X. Hence Γ is a closed set of X. Since the G-action is faithful, Γ is a proper closed subset of X. The second assertion is clear by the construction.

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(2) By assumption, O = OX,P is a regular local ring. For g ∈ GP , ∼ the action of g induces a local ring isomorphism. g ∗ : O −→ O. Hence ∼ it induces a linear isomorphism g ∗ : m/m2 −→ m/m2 . Namely, there is a natural linear representation of G on the k-vector space m/m2 . Here we use a big result due to Luna [51, III, Lemme]: Slice Theorem of Luna. Suppose that GP is a reductive algebraic group. Then there is a GP -equivariant morphism φ : X → (m/m2 )∗ , which is ´etale at P lying over O = φ(P ), where (m/m2 )∗ is the dual vector space of m/m2 . The assertion (2) follows from this result by Lemma 2.7.2(2). (3) Since H is a finite group, O is a finite OH -module. In fact, there exist an affine k-domain A with an H-action and an H-stable maximal ideal M of A such that O = AM . Then B := AH is an affine k-domain and A is a finite B-module. Let m = M ∩ B. Then Am = A ⊗B Bm is a finite Bm -module. Since M is a unique prime ideal lying over m, it follows that Am is a local ring, i.e., AM = Am . Since OH = Bm , it follows that O is a finite OH -module. For g ∈ H, consider a homomorphism of OH -modules g − id : O → O,

a ∈ O 7→ g(a) − a ∈ O. T d H is a flat Let Og = Ker (g − id). Then OH = g∈H Og . Meanwhile, O d d H , we have an exact sequence of O H -modules b∼ OH -module and O = O⊗ H O O

d H −→ O b g−id b 0 −→ Og ⊗OH O −→ O, d H . Since ∼ Og ⊗OH O bg = whence we have O \ \ \ d d d H) ∼ H ∼ H, bH = bg ∼ O (Og ⊗OH O O g ) ⊗O H O O =( = OH ⊗OH O = g∈H

g∈H

g∈H

d H. bH = O we have O Lemma 2.7.16. With the same setting as in the previous lemma, we additionally assume that n = 2. Then the following assertions hold. (1) Suppose that the stabilizer locus Γ(G, X) contains a one-dimensional component Γ1 . Then Γ1 is a smooth curve. (2) There exists an open set U1 of Γ1 such that GP is constant for P ∈ U1 . We call GP for P ∈ U1 the stabilizer group of Γ1 and denote it by GΓ1 . Let G1 = GΓ1 . Then G1 is a cyclic group.

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(3) Let Γ1 be an irreducible component of Γ(G, X) and let Γ1 be the imPr age of Γ1 by the quotient morphism. Then q ∗ (Γ1 ) = |G1 | i=1 gi (Γ1 ), where {gi | 1 ≤ i ≤ r} is a complete set of representatives of G/G1 , `r i.e., G = i=1 gi G1 . (4) Suppose that Γ(G, X) is an irreducible curve with GP constant for all P ∈ Γ(G, X). Then the quotient surface Y = X/G is smooth. (5) Suppose that a closed point P of Γ(G, X) is an isolated point, i.e., there are no curves of Γ(G, X) passing through P . Then the point q(P ) is a quotient singular point of Y .14 Proof. (1), (2) By construction of Γ(X, D), each irreducible component is a finite union of closed sets with groups (Γ1 , G1 ) ⫌ (Γ2 , G2 ) ⫌ · · · · · · ⫌ (Γr , Gr ), where Γ1 ⫌ · · · ⫌ Γr and G1 ⫋ · · · ⫋ Gr . It is clear that the stabilizer group Gi is constant for each point of an open set Ui = Γi \ Γi+1 . In the case n = 2, the sequence has length at most 2. Let P be a point of U1 = Γ1 \ Γ2 . b with O = OX,P is isomorphic By Lemma 2.7.16(2), the GP -action on O to k[[u1 , u2 ]] with a linear representation of GP on m/m2 ∼ = k · u1 + k · u2 for m = mX,P . The Γ1 corresponds to a line ℓ through the origin and GP consists of pseudoreflections around ℓ. Hence GP is a cyclic group. bX,P and k[[u1 , u2 ]], we may (3) Under the GP -equivariant isomorphism O assume that Γ1 corresponds to the u2 -axis and the action of G1 corresponds b G1 ∼ to the mapping u1 7→ ζu1 , where ζ s = 1 with s = |G1 |. Then O X,P = s k[[u1 , u2 ]]. Further, the translate gi Γ1 has the stabilizer group gi G1 gi−1 and mapped to the same image Γ1 on the quotient surface. (4) Let P ∈ X be a closed point. If GP = (e), then the quotient bY,Q , bX,P ∼ morphism q : X → Y is unramified over Q = q(P ). Hence O =O s ∼ b and Q is a smooth point of Y . If P ∈ Γ(G, X) then OY,Q = k[[u1 , u2 ]], where s = |GP |. Hence Q is a smooth point of Y . (5) Write H = GP . Passing to the faithful H-action on k[[u1 , u2 ]] via the linear representation of H on m/m2 (with the previous notation), H contains no pseudoreflections for otherwise there is an irreducible component of Γ(G, X) passing through P . Hence H acting on m/m2 is a small bX,P )H ∼ b 2 subgroup of GL (2, k). Hence (O = (k[[u1 , u2 ]])H ∼ =O A /H,O . Hence q(P ) is a quotient singular point. 14 A singular point Q of a normal algebraic surface Y is called a quotient singularity if bY,Q is isomorphic to the completion O b 2 the completion O A /H,O , where O is the quotient singular point.

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We construct a finite morphism f : X → Y from a normal projective surface to a smooth projective surface such that a cyclic group G of order n acts on X and f is the quotient morphism with Y = X/G. Our construction is a copy of the one due to Horikawa [35] for a double covering. We assume that n is not divisible by char k. Let B be an effective reduced divisor on Y such that B ∼ nF for a divisor F , i.e., OY (B) ∼ = OY (F )⊗n in Pic (Y ). Let V = {Vi }i∈I be an affine open covering of Y such that OY (F )|Vi = OVi vi and vj = hji vi on Vi ∩ Vj with hji ∈ Γ(Vi ∩ Vj , OY∗ ). We may also assume that OY (B)|Vi = OVi ui and uj = bji ui on Vi ∩ Vj with bji ∈ Γ(Vi ∩ Vj , OY∗ ). By replacing V by a finer affine open covering of Y , we may assume the following conditions: (1) B|Vi is defined by bi = 0 with bi ∈ Γ(Vi , OY ) for every i ∈ I. (2) The assumption that B ∼ nF implies that there exists a set {ci }i∈I with ci ∈ Γ(Vi , OY∗ ) such that bji = cj (hnji )c−1 i . Since we can take 1/bi as ui , we have bi = bji bj . Let ξi be the dual base of vi and let Zi = Spec k[ξi ] × Vi ∼ = A1 × Vi . Then patch together {Zi }i∈I by ξi = hji ξj . Thus we obtain an A1 -bundle over Y . Define a closed set Xi of Zi by the equation ξin = bi ci . Then {Xi }i∈I patch together because n n n n ξin = bi ci = bji bj ci = cj hnji c−1 i bj ci = (cj bj )hji = ξj hji = (ξj hji ) .

We denote by X the surface thus obtained. The projection of the A1 bundle Z over Y restricted to X gives a finite morphism f : X → Y . By Serre’s criterion of normality (see Theorem 1.6.17), X is normal. In fact, the conditions (i) and (2) hold since X is locally an irreducible and reduced hypersurface in a smooth variety of dimension 3. The cyclic group G, identified with {ζ i | 0 ≤ i < n} for a primitive root ζ of unity, acts on X by ξi 7→ ζξi for every i. Then the G-action is faithful and X/G = Y . Theorem 2.7.17. In addition to the above notations and assumptions, we assume that the divisor B is either a disjoint union of smooth irreducible curves or a divisor with simple normal crossings. Then the following assertions hold. (1) X has at worst a rational double point of type An−1 . (2) The canonical divisor KX is a Cartier divisor such that KX ∼ f ∗ (KY ) + (n − 1)R ∼ f ∗ (KY + (n − 1)F ), where R is a reduced effective divisor on X which is defined by ξi = 0 on the open set f −1 (Vi ) for i ∈ I. We have f ∗ (B) = nR.

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The divisor R is called the ramification divisor and the divisor B is called the branch divisor of f . We call X together with f an n-ple covering with branch divisor B. Proof. (1) Let P be a point of X and let Q = f (P ). If Q ̸∈ B then f −1 (Q) has n points because ξi (P )n = bi (Q)ci (Q) ∈ k ∗ . These distinct n points are given according to distinct n values of ξi (P ). Suppose that Q is a smooth point of B. Then there exists a regular system of parameters {x, y} of OY,Q such that x = bi . Then {ξi , y} is a regular system of parameters of OX,P . Hence X is smooth at P . Suppose that Q is an intersection point of two irreducible components of B, say B (1) and B (2) , whence B (1) ∩ B (2) = {Q}. Then bi = bi1 · bi2 , where B (1) and B (2) are defined by bi1 = 0 and bi2 = 0, respectively. Then {bi1 , bi2 } is a regular system of parameters of OY,Q . Hence X is defined by ξ n = bi1 bi2 ci near the point P . Hence P has the same singularity as the quotient singularity of A2 /G with G acting on A2 = Spec k[x, y] by (x, y) 7→ (ζx, ζ −1 y). This is a rational double point of type An−1 (see Theorem 2.4.22). (2) Let R be a Cartier divisor on X such that OX (R)|Ui = OUi (1/ξi ) for i ∈ I, where Ui = f −1 (Vi ). Suppose that B is a disjoint union of smooth irreducible curves. It then holds that f ∗ (B) = nR and f induces an isomorphism fR : R → B. We say that f ramifies totally over B. Since f ◦ := f |X\R : X \ R → Y \ B is an ´etale finite morphism, we have KX\R ∼ (f ◦ )∗ KY \B . For a point P ∈ R and Q = f (P ), we can take regular systems of parameters {bi , y} for OY,Q and {ξi , y} for OX,P . By differentiating the equation ξin = bi ci we have nξin−1 dξi = ci dbi + bi dci . Taking the wedge product with dy, we have nξin−1 dξi ∧ dy = ci dbi ∧ dy + bi dci ∧ dy. Since dci = αdbi + βdy with α, β ∈ OY,Q , we have

nξin−1 dξi ∧ dy = ci (1 + c−1 i bi α)dbi ∧ dy.

∗ ∗ Since ci (1 + c−1 i bi α) ∈ OY,Q , we have KX ∼ f (KY ) + (n − 1)R near the point P , where we note that OX (R)P = OX,P (1/ξi ). So, these observations imply that KX ∼ f ∗ (KY ) + (n − 1)R. Since OX (R) and OY (F ) have the same transition functions with respect to V, we have R ∼ f ∗ (F ). This gives the last piece of the stated formula. Suppose that B is a divisor with simple normal crossings. Let S be the set of intersection points of B. Set Y ◦ := Y \ S, X ◦ := f −1 (Y ◦ ) and

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f ◦ := f |X ◦ . Since X has only rational double points lying over points of S, it is known by Watanabe [95] that X is Gorenstein, i.e., Ω2X/k is an invertible sheaf. Then KX is the closure of KX ◦ . Applying the argument in the case B is smooth to f ◦ : X ◦ → Y ◦ . We easily obtain the stated formula. Example 2.7.18. Consider a coordinate triangle in Example 2.5.3 consisting of three lines ℓ0 , ℓ1 , ℓ2 on Y = P2 . Set B = ℓ0 + ℓ1 + ℓ2 , which is an effective reduced divisor with simple normal crossings such that B ∼ 3ℓ. We denote the intersection points of ℓ0 , ℓ1 , ℓ2 by Q0 , Q1 , Q2 instead of P0 , P1 , P2 . Let f : X → Y be a triple covering with branch divisor B. Then there exists e →X three rational double points P0 , P1 , P2 of type A2 on X. Let σ : X −1 be a minimal resolution of singularity. Then σ (Pi ) = E1i + E2i for i = 0, 1, 2, where E1i and E2i are (−2) curves intersecting each other in one point transversally and meet one of the proper transforms Ci of the P2 C i = f −1 (ℓi ). The weighted dual graph of i=0 (Ci + E1i + E2i ) is given as follows. E21 E22

C2

C1

E12

E01

E11 E02

C0

The self-intersection number of Ci is computed as follows. We exhibit the computation by taking C1 as an example. Write C = C1 , E1 = E22 , E2 = E21 , F1 = E01 and F2 = E02 . By Sakai [88], we have the following equation in Cℓ (e(X)) ⊗ Q 2 1 σ ∗ (C) = C + (E1 + F1 ) + (E2 + F2 ). 3 3

Namely, (σ ∗ (C) · Ei ) = (σ ∗ (C) · Fi ) = 0 for i = 1, 2. Since f ∗ (ℓ) = 3C, we 2 have (C ) = 1/3. Then we have 4 1 2 = (C ) = (σ ∗ (C)2 ) = (C 2 ) + . 3 3

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e → Z be the contraction of C1 , E22 , C0 , E02 , Hence (C 2 ) = −1. Let τ : X C2 , E12 and let L0 = τ (E01 ), L1 = τ (E11 ) and L2 = τ (E21 ). Then (L2i ) = 1 for i = 0, 1, 2. The Z ∼ = P2 and {L0 , L1 , L2 } is a coordinate triangle. It is clear by the above construction that the mapping f ◦ σ ◦ τ −1 is a rational mapping of degree 3, but it is not a morphism. If this mapping is restricted to U = P2 \ (L0 ∪ L1 ∪ L2 ), it gives an ´etale finite endomorphism of U of degree 3. We treat the same example by a different method. Example 2.7.19. Let Y = P2 and B = ℓ0 + ℓ1 + ℓ2 as in Example 2.7.18. Let µ1 : Y1 → Y be the blowing-ups of points P0 , P1 , P2 . Let ℓ′i = µ′1 (ℓi ) and Ei = µ−1 1 (Pi ) for i = 0, 1, 2. These six curves form six edges of a hexagon as in Example 2.5.3. Let Q01 = ℓ′0 ∩ E1 , Q02 = ℓ′0 ∩ E2 , Q10 = ℓ′0 ∩ E0 , Q12 = ℓ′1 ∩ E2 , Q20 = ℓ′2 ∩ E0 , Q21 = ℓ′2 ∩ E1 . Let µ2 : Y2 → Y1 be the blowing-ups of these six points. Let Eij = µ−1 2 (Qij ), where (ij) ∈ I := {(01), (02), (10), (12), (20), (21)}. Let µ = µ2 ◦ µ1 : Yb → Y , where Yb := Y2 . Set ℓ′′i = µ′ (ℓi ), Ei′ = µ′2 (Ei ) for i = 0, 1, 2 and Eij = µ−1 2 (Qij ) for (ij) ∈ I. Then we have µ∗ (B) =

2 X i=0

(ℓ′′i + 2Ei′ ) + 3

X (ij)∈I

Eij ∼ 3µ∗ (F ), F = ℓ,

we can write it as   2 2   X X X b := B (ℓ′′i + Ei′ ) ∼ 3 µ∗ F − Ei′ − Eij .   i=0

i=0

(ij)∈I

b → Yb be the triple covering with branch divisor B. b It is then easy Let fb : X to ascertain the following assertions. bi , where ℓbi and E bi are (−1) curves with (1) fb∗ (ℓ′′i ) = 3ℓbi and fb∗ (Ei′ ) = 3E 2 2 b b b (ℓi ) = (Ei ) = −1 for i = 0, 1, 2. Meanwhile, Eij := fb∗ (E)ij is a (−3) curve for (ij) ∈ I. (Use the projection formula (see Theorem 2.7.8).) b and S (2) The irreducible components of the divisor B (ij)∈I Eij has the following weighted dual graph:

Geometry on Affine Surfaces

b02 E

299

ℓb0

b01 E

b1 E

b21 E

−3

−1

−3

−1

−3

−1

−1

−3

−1

−3

−1

−3

b12 E

ℓb1

b10 E

b0 E

b2 E

ℓb2

b20 . E

b0 , E b1 , E b2 , we have the same weighted (3) By blowing down (−1) curves E graph as in Example 2.7.18.

300

2.8

Affine Algebraic Geometry

Problems to Chapter 2

1. Let V be a smooth projective surface defined over an algebraically closed field k of characteristic zero and let C be a smooth curve isomorphic to P1 . Assume that X := V \ C is an affine factorial surface. Show that V ∼ = P2 and C is a line, hence X is isomorphic to A2 . Answer. Since X is factorial, any irreducible divisor D(̸= C) on V restricted on X is a principal divisor (f ) with f ∈ Γ(X, OX ). Hence D − (f ) has support on C. Namely D ∼ mC with m > 0. Hence Pic V ∼ = Z[C]. In particular, KV ∼ aC with a ∈ Z. Note that some positive multiple of C is very ample on V . Hence we have (C 2 ) > 0. By the arithmetic genus formula we have (C · C + KV ) = (a + 1)(C 2 ) = −2. Hence either (C 2 ) = 1 and a = −3 or (C 2 ) = 2 and a = −2. We show that χ(OV ) = 1. In fact, H 2 (V, OV ) ∼ = H 0 (V, KV ) = 0 because 1 KV ∼ aC with a < 0. Since χ(OV ) = 1−h (OV ) it suffices to show that h1 (OV ) = dim H 1 (V, OV ) = 0. In fact, it is known that if Pic V is a discrete group then H 1 (V, OV ) = 0 (see [59, p. 6]). A fact is that Pic 0 V which consist of elements of Pic V algebraically equivalent to zero is a connected algebraic group defined over k and the Lie algebra of Pic 0 V at the identity element is isomorphic to H 1 (V, OV ). So, χ(OV ) = 1. In the case (C 2 ) = 1 and a = −3, we have dim |C| = 2. Then Φ|C| : V → P2 is a quasi-finite birational morphism. Hence it is an isomorphism, and Φ∗ (ℓ) ∼ = C for some line ℓ on P2 . In the case (C 2 ) = 2 and a = −2, the Riemann-Roch formula shows that χ(OV (C)) =

1 (C · C − KV ) + χ(OV ) = 3 + 1 = 4. 2

Since h2 (OV (C)) = h0 (OV (−3C)) = 0, we have dim |C| = 3 + h1 (OV (C)) ≥ 3. Let P be a point on C and let P ′ be an infinitely near point to P lying on C. This implies that if E is the exceptional curve by the blowing-up σ : V ′ → V at P , P ′ is the intersection point of E and the proper transform C ′ of C. Then |C| − (P + P ′ ) is a linear system whose member consists of a curve B such that B passes the point P and its proper transform σ ′ (B) passes the point P ′ . Then Λ := |C|−(P +P ′ ) is a linear pencil. Let ρ : W → V be the composite of blowing-ups at the points P and P ′ , let E1 be the proper transform of the exceptional curve of the blowing-up at P and let E2 be the exceptional curve of the

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blowing-up at P ′ . Then (E12 ) = −2, (E1 · E2 ) = 1, (E22 ) = −1, and the proper transform ρ′ Λ consists of fibers of a P1 -fibration π : W → P1 such that E2 is a cross-section and all members of ρ′ Λ are irreducible curves with self-intersection number zero. Meanwhile, E1 is contained in a fiber of π. This is a contradiction (see Lemma 1.8.20). 2. Let X = Spec A be an affine algebraic variety defined over k. Define Γ∗ (X) to be the quotient group A∗ /k ∗ , where A∗ is the multiplicative group of units of A. Prove the following assertions. e be the normalization of A in k(X) and let X e = Spec A. e Then (1) Let A ∗ ∗ e Γ (X) is a subgroup of Γ (X). e is a free abelian group of finite rank, and so is Γ∗ (X). (2) Γ∗ (X) P (3) Let V be a smooth projective surface and let D = i Di be an effective reduced divisor on V such that X = V \ D is affine. If the irreducible components of D are linearly independent, then Γ∗ (X) = 0. In particular, if D is irreducible then Γ∗ (X) = 0. Pn (4) Let V be a smooth projective variety and let D = i=1 Di be an effective divisor on V such that X := V \ D is affine. Then there exists an exact sequence of abelian groups 0 → Γ∗ (X) →

n M i=1

Z[Di ] → Pic V → Pic X → 0.

e induces an injective homomorAnswer. (1) The inclusion A ,→ A ∗ ∗ e e phism A ↣ A . Hence we have an injection Γ∗ (X) ↣ Γ∗ (X). (2) By the assertion (1) we may assume that X is normal. By projectivization, we may assume that X is an affine open set of a normal projective variety V . Let D := V \ X. Then D has pure codimension P one by Lemma 2.1.1. Let D = i Di be the irreducible decomposition. An element f ∈ A is a unit if and only if the principal divisor (f ) has support on D. Namely the zeroes and poles of (f ) are components of D. Let Div (D) be a free abelian group generated by irreducible components Di . Then we have an injective homomorphism Γ∗ (X) ↣ Div (D). Hence Γ∗ (X) is a free abelian group of finite rank. (3) and (4) Clear from the arguments for the assertion (2) and the fact that Pic (X) is the quotient group of Pic V by the subgroup generated by irreducible components of D. 3. Let (R, m) be a noetherian local ring containing a field k such that k → R → R/m is an isomorphism. Show that Ω1R/k ⊗R k ∼ = m/m2 .

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Answer. Apply Lemma 1.6.21(2) to the case B = R, J = m and A = C = k and obtain a surjection δ

m/m2 −→ Ω1R/k ⊗R k. δ∗

We show that the dual homomorphism, Der k (R, k) −→ Hom(m/m2 , k) is an isomorphism of k-vector spaces. In fact, for d ∈ Der k (R, k), d is determined by d|m since any element a ∈ R is written as a = α + a′ with α ∈ k and a′ ∈ m and d(α) = 0. Meanwhile, d(m2 ) = 0 since d(R) ∈ k = R/m. So, δ ∗ is an isomorphism. Hence the surjection δ is an isomorphism. 4. Let k be an algebraically closed field of characteristic zero. Show that the affine line A1k is then simply connected. Answer. Set Y := P1 containing A1 in the standard way. Let f : X → Y be a finite covering such that f |f −1 (A1 ) is an ´etale covering. Let f −1 (y0 ) = {x1 , . . . , xr }, where P1 = A1 ∪ {y0 }. Let ei (1 ≤ i ≤ r) Pr be the ramification index of f at xi and let d = deg f . Then d = i=1 ei Pr and hence i=1 (ei − 1) = d − r. By the Riemann-Hurwitz formula, we have 2gX − 2 = −2d + (d − r). So, 2gX = 2 − (d + r) ≥ 0. Since d ≥ 1 and r ≥ 1, we have d = r = 1 and gX = 0. This implies that X ∼ = P1 −1 1 ∼ 1 1 and f (A ) = A . So, A is simply connected. 5. Let f : X → Y be an ´etale covering. For a morphism g : Z → Y , let XZ = X ×Y Z and let fZ : XZ → Z be the projection. Show that fZ is an ´etale covering. Answer. Since fZ is the base change of f by g : Z → Y , fZ is a flat and finite morphism. Let x′ ∈ XZ be a closed point, and let z = fZ (x′ ) and y = g(z). Choose affine open neighborhoods V = Spec B of y and V ′ = Spec B ′ of z so that V ′ ⊆ g −1 (V ). Write f −1 (V ) = Spec A and (fZ )−1 (V ′ ) = Spec A′ . Then A′ ∼ = A ⊗B B ′ , and Ω1A′ /B ′ ∼ = Ω1A/B ⊗A A′ . In fact, if N ′ is an A′ -module, then we have ∼ Der B ′ (A′ , N ′ ) HomA′ (Ω1A′ /B ′ , N ′ ) = ∼ = Der B (A, N ′ ) ∼ = HomB ′ (Ω1 [B]

A/B

⊗B B ′ , N ′ ),

′ where N[B] is the B ′ -module N ′ viewed as a B-module via B → B ′ . Hence Ω1A′ /B ′ ∼ = Ω1A/B ⊗B B ′ . If ΩA/B = 0 then Ω1A′ /B ′ = 0. So, ′ fZ : X → Z is unramified. Thus fZ is an ´etale covering if so is f . 6. Let k be an algebraically closed field of characteristic zero. Show that the affine plane A2 is simply connected.

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Answer. Let f : X → Y := A2 be a connected ´etale covering and let ρ:Y →B∼ = A1 be an A1 -bundle. If we choose a system of coordinates {u, v} on Y then ρ is the projection (u, v) 7→ u. For each closed point b ∈ B, let Lb = ρ−1 (b) which is isomorphic to A1 . Then the restriction of f onto f −1 (Lb ) is a finite ´etale covering fb : f −1 (Lb ) → Lb by Problem 5 above. `d (i) e be the By Problem 4, f −1 (Lb ) = i=1 Lb , where d = deg f . Let B normalization of B in the function field k(X). By Stein factorization e such that (see Lemma 1.8.15(II)), there exists a morphism ρe : X → B the following diagram is commutative

f

X −−−−→ Y   yρ ρ ey ν e −−− B −→ B,

e → B is the normalization morphism. The morphism ρe has where ν : B (i) e is an A1 -bundle. fibers Lb as a connected fiber and hence ρe : X → B e → B is an ´etale covering. Since B e is connected, This implies that ν : B 2 it follows by Problem 4 that d = 1. So, A is simply connected. 7. Let C be a smooth plane curve defined by F = (X0 −X2 )3 +X13 +X23 = 0 in P2 = Proj k[X0 , X1 , X2 ] defined over an algebraically closed field of characteristic p ̸= 3. Prove the following assertions. (1) C is a smooth elliptic curve, i.e., g = 1, and meet the line at infinity ℓ0 = {X0 = 0} at the point P0 = (0, 0, 1) with (C · ℓ0 ) = 3. (2) Let C0 be an affine plane curve defined by f = x3 + (1 − y)3 + y 3 in A2 = Spec k[x, y]. If we set x = X1 /X0 and y = X2 /X0 . Then C0 = C \ {P0 }. (3) D = {P2 , A2 , C, ℓ0 , 0, 3, 1, 1} are admissible data for the pair (X, C0 ). (4) Let Λ be a linear pencil spanned by C and 3ℓ0 and let fe : Ve → P1 be the linear pencil free from the base points which is obtained from V by a succession of the Euclidean transformations and the e be the fiber of fe containing the proper (e, i)-transformations. Let Γ e transform of ℓ0 . Then Γ has the following dual graph:

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Affine Algebraic Geometry

2 4 ℓ′0 3

6

5

4

3

2

1

where each vertex represents a (−2)-curve and the number attached to each vertex signifies the coefficient of the corresponding compoe nent in Γ. (5) There is exactly one more fiber D of fe such that D ∩A2 is defined by f = 1/4, which is an irreducible rational curve with a cusp (x, y) = (0, 1/2). The component with coefficient 1 meets the section of fe. Answer. (1) Note that KP2 ∼ −3ℓ0 and C ∼ 3ℓ0 . By the arithmetic genus formula (after Proposition 1.8.10), we have pa (C) = 1. Since C is smooth by the Jacobian criterion of smoothness (see Lemma 1.6.19), C has genus 1 and hence is an elliptic curve. The intersection point P0 := C ∩ ℓ0 is (0, 0, 1). Hence P0 is a flex, where the tangent line ℓ0 meets C with local intersection multiplicity 3. The rest of the assertions is clear and easy to show. 8. Let ι : A2 → P2 be an open immersion. Show that ι is the standard embedding A2 ,→ P2 as the complement of a line ℓ0 . Answer. By Goodman’s theorem (see [32, Theorem 4.2]), the closed set P2 \ ι(A2 ) is a connected curve supporting an ample divisor. Write Sr P2 \ι(A2 ) = i=1 Ci be the irreducible decomposition. Since Pic (P2 ) = Z[ℓ0 ], Ci ∼ di ℓ0 for every i, where di > 0. Suppose that r = 1. Then Pic (ι(A2 )) ∼ = Z/d1 Z. Hence d1 = 1, and the immersion ι : A2 → P2 is the standard one. Suppose that r ≥ 2. Then d1 C2 − d2 C1 ∼ 0. Hence there exists a function f ∈ k(P2 ) such that (f ) = d1 C2 − d2 C1 . Then f ∈ Γ(A2 , OA2 )∗ = k ∗ . This is a contradiction. Hence ι is the standard open immersion. 9. Let F0 be a Hirzebruch surface of degree 0 with a fiber ℓ and a minimal section M . Hence Pic (F0 ) = Z[ℓ] ⊕ Z[M ] with (ℓ2 ) = (M 2 ) = 0 and (ℓ · M ) = 1. Let ι : A2 → F0 be an open immersion. Let F0 \ ι(A2 ) = C1 ∪ · · · ∪ Cr with irreducible curves Ci . Prove the following assertions.

Geometry on Affine Surfaces

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(1) r = 2. (2) C1 and C2 are smooth rational curves such that (C1 · C2 ) = 1. (3) Write Ci ∼ ai M + bi ℓ. Then ai ≥ 0, bi ≥ 0, a1 b2 + a2 b1 = 1 and a1 b2 − a2 b1 = ±1. (4) Exchanging the roles of C1 , C2 , we may assume that a1 b2 −a2 b1 = 1. Then a1 b2 = 1 and a2 b1 = 0. Hence a1 = b2 = 1 and either a2 = 0 or b1 = 0. So, either C1 ∼ M + b1 ℓ and C2 ∼ ℓ, or C1 ∼ M and C2 ∼ a2 M + ℓ. The pencil |ℓ| in the first case and the pencil |M | in the second case gives an A1 -fibration on ι(A2 ). (5) Let ∆ be the diagonal of F0 , which is, by definition, the image of the closed immersion (id, id) : P1 → F0 = P1 × P1 , whence ∆ ∼ M + ℓ. Let P0 ∈ ∆ and let Λ = |M +ℓ|−2P0 . Since dim |∆| = 3, dim Λ = 1. In fact, every member except for M + ℓ, where we assume that M ∩ ℓ = {P0 }, is smooth. Hence irreducible member of Λ is a smooth rational curve meeting ∆ twice at the point P0 . This pencil Λ cuts on ι(A2 ) = F0 \∆ an A1 -fibration. Answer. (1) On the real (a, b)-plane, consider the vectors ⃗vi connecting the origin and the point (ai , bi ). Since Pic (F0 ) is a free abelian group of rank 2 and ι(A2 ) is a factorial affine open set of F0 , we have r ≥ 2. Suppose that the vectors ⃗v1 and ⃗v2 are independent. If r ≥ 3 then we have (a3 , b3 ) = q1 (a1 , b1 ) + q2 (a2 , b2 ), qi ∈ Q, qi ≥ 0. This implies that n3 (a3 , b3 ) = n1 (a1 , b1 ) + n2 (a2 , b2 ), ni ∈ Z, ni ≥ 0, n3 > 0. Hence we have n1 C1 + n2 C2 − n3 C3 = (f ) with f ∈ k(F0 ) and f ∈ Γ(A2 , OA2 )∗ , which is a contradiction. (2) We use Lemma 1.8.20 which describes the structure of a degenerate fiber of a P1 -fibration on a smooth projective surface. Since ι(A2 ) has an A1 -fibration which is extended to a linear pencil Λ on F0 having base points in general. If Λ has no base points or base points lie on a point of C1 or C2 but not of C1 ∩ C2 , then it is clear that (C1 · C2 ) = 1. In fact, there exists a birational morphism σ : V → F0 such that the e of Λ is free from base points. We take σ as the proper transform Λ shortest succession of blowing-ups at base points including infinitely near ones. The proper transforms of C1 , C2 are then contained in a e The unique degenerate fiber F∞ of the P1 -fibration ρe induced by Λ.

306

Affine Algebraic Geometry

assumption implies that the point(s) C1 ∩ C2 are not blown up by σ. Hence (C1 · C2 ) = 1. Even if one point of C1 ∩ C2 is blown up, the other points are not, and the exceptional curves, by which we mean the proper transforms of irreducible ones on V , are components of F∞ . Since F∞ contains no circular chains, it follows that C1 , C2 meet in ei be the proper transform of Ci in V for one point P0 of F0 . Let C i = 1, 2. Since each irreducible exceptional component arising by σ has self-intersection number ≤ −2 except for the last one which is a e1 or C e2 is a section of ρe and not contained in F∞ . Hence either C e1 is a (−1)-curve. Then contract C e1 and (−1)-curve. Suppose that C e1 . the component which becomes (−1)-curve after the contraction of C We continue this process until the image of F∞ becomes irreducible. If either one (or both) of C1 , C2 has singularity at P0 , or C1 , C2 are smooth at P0 but (C1 · C2 ) > 1, we have in the midway of the above contractions three irreducible components meeting a (−1)-curve, which is impossible. This shows that (C1 · C2 ) = 1. (3) The equality a1 b2 +a2 b1 = 1 follows from (C1 ·C2 ) = 1. On the other hand, the linear equivalence classes of C1 , C2 form a basis of Pic (F0 ). We have the relation     C1 M =A , C2 ℓ   a1 b1 where A = . Then the matrix A is unimodular, i.e., det(A) = a2 b2 ±1. This gives the second equality. (4) and (5). Proof is straightforward. 10. Let d, e be positive integers such that gcd(d, e) = 1 and let f = xd y e +1 be a polynomial in k[x, y]. Verify the following assertions. (1) f is an irreducible polynomial. (2) Let d = 2 and e = 3. Let Λ be a linear pencil on P2 generated by e be 5ℓ0 and the closure C of the affine curve C0 = {f = 0} and let Λ the proper transform of Λ by the shortest succession of blowing-ups e has no at the base points of Λ, say σ e : Ve → V0 := P2 , such that Λ e defines a P1 -fibration fe : Ve → P1 which has base points. Then Λ two degenerate fibers. Answer. (1) We make use of the arguments in the proof of Theorem 1.5.17. The polynomial f is concomitant to x−d f = y e + (1/xd ) which is a irreducible polynomial in k(x)[y] because gcd(d, e) = 1. Since

Geometry on Affine Surfaces

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gcd(xd , 1) = 1, f is a primitive polynomial. Hence f is a prime element of k[x, y]. Hence f is an irreducible polynomial. (2) The pencil Λ is equal to Λ(f0 ), where f0 = x2 y 3 . The closure C 0 of the curve {f0 = 0} is equal to 2ℓ1 + 3ℓ2 as divisors, where ℓ1 (resp. ℓ2 ) is the line defined by X1 = 0 (resp. X2 = 0). Hence λ = ⟨2ℓ1 +3ℓ2 , 5ℓ0 ⟩, and Bs Λ consists of points P1 := (0, 0, 1) and P2 := (0, 1, 0) and their suitable infinitely near points. Let σ1 : V1 → V0 := P2 be the blowingup of P1 and let E1 = σ −1 (P1 ). Then σ1′ Λ = ⟨2ℓ′1 + 3ℓ′2 , 3E1 + 5ℓ′0 ⟩, and the point ℓ′1 ∩ E1 is a base point of σ1′ Λ, where ℓ′i = σ1′ (ℓi ) for i = 0, 1, 2. We perform the blowing-ups at the base points of the proper e transforms of Λ until we reach to the base-point-free linear pencil Λ. The following dual graph will give two degenerate fibers, where S1 , S2 are two last exceptional curves which are sections of the P1 -fibration fe : Ve → P1 . −3

−2

−1

−3

−2

ℓe0

−1 S1 −2

−1

ℓe1

ℓe2

−2

−1

S2 −3

11. Let t be a variable over an algebraically closed field of characteristic p. Let n ≥ 3 be an integer such that p ∤ n. Let a1 , . . . , an be disQn tinct nonzero elements of k, let f (x, y) = i=1 (x − ai y) + t and let F (X0 , X1 , X2 ) be a homogeneous polynomial given by F (X0 , X1 , X2 ) = X0n f (X1 /X0 , X2 /X0 ) n Y = (X1 − ai X2 ) + tX0n . i=1

Let C be a curve on P2 = Proj k(t)[X0 , X1 , X2 ] defined by F (X0 , X1 , X2 ) = 0. Show that C is a smooth curve defined over k(t) with arithmetic genus 12 (n − 1)(n − 2). Answer. We apply the Jacobian criterion of smoothness (see a remark after Lemma 1.6.19). Partial derivatives of F are given as

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follows: ∂F = ntX0n−1 ∂X0 n Y

 n X (X1 − ai X2 ) 

∂F = ∂X1 i=1



1  X − aj X2 1 j=1   n n Y X ∂F −aj . = (X1 − ai X2 )  ∂X2 X − aj X2 1 i=1 j=1 Let Q = (α0 , α1 , α2 ) be a singular point of C if it exists, where k(t)(α0 , α1 , α2 ) is an algebraic extension of k(t). Since n ̸= 0 by the assumption, we have α0 = 0. Since F (Q) = 0, we have α1 = ai α2 for some i. We may assume that i = 1. Then we have n Y ∂F (a1 − ai ) = 0 (Q) = α2n−1 ∂X1 i=2 n Y ∂F n−1 (Q) = −α2 a1 (a1 − ai ) = 0. ∂X2 i=2

This implies that α2 = 0. Hence α1 = 0. So, the point Q does not exist. The (arithmetic) genus of C follows from the arithmetic genus formula. 12. Let C be a closed curve on A2 which is isomorphic to A1 . Let G be a subgroup of Aut k A2 consisting of elements g which preserves the curve C. By choosing a system of coordinates {x, y} of A2 such that C is defined by y = 0, show that G is conjugate to a subgroup   ρ(x) = ax + f (y) G0 := ρ ∈ Aut k k[x, y] . ρ(y) = by Answer. Let τ be an automorphism of A2 such that τ (C) is the line {y = 0}. By this identification, we have G0 := τ Gτ −1 is the group of automorphism which maps y to cy with c ∈ k ∗ . Take any element −1 ρ ∈ G0 . Then ρ(y) = by with b ∈ k ∗ . Let σ be a linear trans1 0 formation such that (σ −1 (x), σ −1 (y)) = (x, y) −1 . Note that 0b σ −1 ∈ G0 . Then ρ′ := σ −1 ◦ ρ satisfies ρ′ (y) = y. Hence ρ′ induces a k[y]-automorphism of k[x, y]. So, ρ′ (x) = ax + g(y) with a ∈ k[y]∗ and g(y) ∈ k[y]. Since ρ = σ ◦ ρ′ , we have ρ(x) = ax + g(by) and ρ(y) = by. Now put f (y) = g(by).

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13. Show that there are no nontrivial group scheme homomorphisms between Ga and Gm . Answer. (1) Let h : Gm → Ga be a homomorphism of group schemes over k. Write Ga = Spec k[u] and Gm = Spec k[t, t−1 ]. Then h is given by a k-algebra homomorphism α : k[u] → k[t, t−1 ] such that P (α ⊗ α) ◦ µGa = µGm ◦ α and εGa = εGm ◦ α. Write α(u) = i∈Z ai ti with ai ∈ k. The condition (α ⊗ α) ◦ µGa = µGm ◦ α gives a relation X X X a i ti ⊗ 1 + 1 ⊗ ai ti = a(t ⊗ t)i . i∈Z

i∈Z

i∈Z

This relation implies that ai = 0 if i ̸= 0. Hence α(u) = a0 ∈ k. The condition εGa = εGm ◦ α gives 0 = a0 . Hence α = 0. This implies that h is trivial. (2) Now let h′ : Ga → Gm be a homomorphism of k-group schemes. Then h′ = a β for a k-algebra homomorphism β : k[t, t−1 ] → k[u] such that (β ⊗ β) ◦ µGm = µGa ◦ β and εGm = εGa ◦ β. Then β(t) is a unit in k[u]. So, β(t) = b ∈ k. Since εGm = εGa ◦ β, we have 1 = b. Namely, h′ is trivial. 14. Let G be a connected algebraic group acting on an affine algebraic variety X = Spec R such that R is a UFD. Suppose further that G has no nontrivial homomorphisms to the multiplicative group Gm and that R∗ = k ∗ . Show that the ring R0 of G-invariants of R is a UFD. Answer. Note that R0 is not necessarily finitely generated over k. We prove that every nonzero element a ∈ R0 has a decomposition to a product of irreducible elements of R0 and the decomposition is unique up to permutations and concomitance equivalence relation. Let a = a1 a2 · · · an be a decomposition into irreducible elements in R. Up to concomitance relation, the action of G induces permutations of these elements. Hence there is a group homomorphism G(k) → Sn , where Sn is the symmetric group of n letters. Since G is connected, this homomorphism is trivial. Namely, for g ∈ G(k) and 1 ≤ i ≤ n, g ai = ui (g)ai , where ui (g) ∈ k ∗ by the assumption. It is clear that the correspondence g 7→ ui (g) induces a group homomorphism ui : G → Gm . By the assumption, ui = 1, i.e., g ai = ai for 1 ≤ i ≤ n. Hence a has a finite decomposition into a product of irreducible elements in R0 . Suppose that a = a1 · · · an = b1 · · · bm has two decompositions into irreducible elements. By the uniqueness of decomposition in R, we have n = m and ai ∼ bσ(i) for some σ ∈ Sn . Since ai , bσ(i) ∈ R0 , we

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Affine Algebraic Geometry

have ai /bσ(i) ∈ R0 ∩ R∗ . Since R∗ = k ∗ , we have ai /bσ(i) ∈ k ∗ . So, the uniqueness of decomposition in R0 follows. 15. Suppose that char (k) = 0. Let δ be a locally nilpotent derivation of a polynomial ring R := k[x, y]. Show that there exists an automorphism ρ ∈ Aut k[x, y] such that δ = ρ−1 ◦ (f (y)(∂/∂x)) ◦ ρ for f (y) ∈ k[y]. Answer. Let σ : Ga × X → X be an action of Ga onto X = Spec k[x, y] such that the coaction is given by exp(δ). Then RGa = k[η] and R = k[ξ, η]. Further, by Theorem 2.4.4, δ(ξ) = f (η). Let ρ ∈ Aut k[x, y] be an element such that ρ maps (ξ, η) to (x, y). Then δ := ρ ◦ δ ◦ ρ−1 is a locally nilpotent derivation of R such that ρ◦δ◦ρ−1 (x) = (ρ◦δ)(ξ) = ρ(f (η)) = f (y), ρ◦δ◦ρ−1 (y) = (ρ◦δ)(η) = 0.

Hence ρ ◦ δ ◦ ρ−1 = f (y)(∂/∂x). So, we have the assertion. 16. Let Ri (i = 1, 2) be an A-algebra and let R = R1 × R2 be a direct product of A-algebras. Namely, define addition and multiplication in the direct product R of the sets R1 , R2 by (x1 , x2 ) + (y1 , y2 ) = (x1 + y1 , x2 + y2 ) and (x1 , y1 ) · (x2 , y2 ) = (x1 x2 , y1 y2 ). Then R is an Aalgebra by a · (x1 , x2 ) = (ax1 , ax2 ) with the zero element (0, 0) and ` the identity element (1, 1). Show that Spec R = Spec R1 Spec R2 , which is the direct sum of the sets Spec Ri (i = 1, 2). By making use of this fact, show that given a finite group F and a field k, F is viewed as a finite group scheme over k whose coordinate ring is L P R = g∈F k·eg , where 1R = g∈F eg is the decomposition of the unity, i.e., eg ·eh = 0 (g ̸= h), eg ·eh = eg (g = h), and whose comultiplication µ : R → R ⊗k R, coinverse ι : R → R and augmentation ε : R → k are k-linear homomorphisms given by X µ(eg ) = eh ⊗ eℓ , h,ℓ∈F,g=h·ℓ

ι(eg ) = eg−1 , ε(eg ) = 1 (g = 1F ), 0 (g ̸= 1F ), where 1F is the identity element of F . Answer. Set e1 = (1, 0) and e2 = (0, 1). Then 1R = (1, 1) = e1 + e2 is the decomposition of the identity element. In particular, e1 · e2 = 0. Let p be a prime ideal of R. Since e1 · e2 ∈ p, either e1 ∈ p or e2 ∈ p. If e2 ∈ p then p corresponds bijectively to a prime ideal of R/e2 R = R1 . Hence {p ∈ Spec R | e2 ∈ p} is identified with Spec R1 . Similarly, {p ∈ Spec R | e1 ∈ p} is identified with Spec R2 . Note that

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no prime ideal p ∈ Spec R contains both e1 and e2 . Hence Spec R = ` Spec R1 Spec R2 . For the second assertion, note that the set of krational points (Spec R)(k) consists of maximal ideals which are the kernels of k-algebra homomorphisms ρg : R → k given by ρg (eh ) = 0 if h ̸= g and ρg (eg ) = 1. Then the product of two rational points corresponding to ρh , ρℓ is given by µ ρh ⊗ρℓ R −→ R ⊗k R −→ k ⊗k k ∼ = k,

which is ρg by the definition of µ. Hence h · ℓ = g. The rest is argued in a similar fashion. 17. Let k be an algebraically closed field and let an affine algebraic group G = Spec B act on an affine algebraic scheme X = Spec A. For an arbitrary element a ∈ A, show that the k-linear subspace W (a) := P g g∈G(k) k( a) is a k-vector space of finite dimension. Answer. Let σ : G × X → X be the k-morphism giving the action of G on X and let φ : A → B ⊗k A be the k-algebra homomorphism such Pn that a φ = σ. Write φ(a) = i=1 bi ⊗ ai , where ai ∈ A and bi ∈ B. For g ∈ G(k), we have g

a = (ρg ⊗ idA ) · φ(a) =

n X

ρg (bi )ai ,

i=1

where ρg : B → k is a k-algebra homomorphism giving rise to the Pn element g ∈ G(k). Set V := i=1 kai . Then g a ∈ V . Hence W (a) ⊆ V . Since dim V ≤ n, it follows that dim W (a) ≤ n. 18. Let φ : P2 → P2 be a biregular automorphism. Then φ is given by an action of an element of PGL (3, k). In fact, a biregular automorphism of Pn is given by an action of an element of PGL (n + 1, k). The proof is essentially the same as the one given below if we use the intersection theory on a higher-dimensional variety. Answer. Let ℓ be a line on P2 and let D = φ∗ (ℓ). Then (φ∗ D)2 = (ℓ2 ) = 1. Hence φ∗ (ℓ) ∼ L, where L is a line on the source P2 . This implies φ induces a k-isomorphism of the associated k-modules ∼ φ∗ : M (ℓ) −→ M (L). Let {X0 , X1 , X2 } be a system of homogeneous coordinates on P2 . Then φ∗ induces a k-linear automorphism of the k-vector space k · X0 + k · X1 + k · X2 , which is given by an matrix A ∈ GL (3, k) as (φ∗ (X0 ), φ∗ (X1 ), φ∗ (X2 )) = (X0 , X1 , X2 )A, where the following identification is made: φ∗ (ℓi ) = {φ∗ (Xi ) = 0}, where ℓi = {Xi = 0} for i = 0, 1, 2 and hence the pull-back by φ∗ of a line

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Affine Algebraic Geometry

{a0 X0 + a1 X1 + a2 X2 = 0} is {a0 φ∗ (X0 ) + a1 φ∗ (X1 ) + a2 φ∗ (X2 ) = 0}. It is clear that A is determined by φ up to the scalar multiplication of elements of k ∗ . 19. Let P0 be a point of the Hirzebruch surface F0 = P1 × P1 and let L0 (resp. M0 ) be the fiber (resp. the section) through the point P0 of the first projection p1 : F0 → P1 . Let σ : V → F0 be the blowing-up of P0 and let L′0 (resp. M0′ ) be the proper transform of L0 (resp. M0 ). Then L′0 and M0′ are (−1) curves. Let ρ : V → P2 be the contractions of L′0 and M0′ . Then the image by ρ of the exceptional curve E = σ −1 (P0 ) is a line on P2 . Let θ = ρ ◦ σ −1 which is a birational mapping F0 99K P2 . Answer the following questions. (1) Let ι : F0 → F0 be the interchanging transformation which maps (P, Q) 7→ (Q, P ), where (P, Q) ∈ F0 = P1 × P1 . Then φ := θ ◦ ι ◦ θ−1 : P2 99K P2 is a Cremona transformation. Write φ in terms of homogeneous coordinates (X0 , X1 , X2 ). (2) Let ⟨ι⟩ be the subgroup generated by a biregular automorphism ι. Show that the quotient surface F0 /⟨ι⟩ is isomorphic to P2 . Answer.

(1) The rational mapping θ : P1 × P1 99K P2 is given by (U0 , U1 ) × (V0 , V1 ) 7→ (U1 V0 , U0 V1 , U1 V1 ),

where (U0 , U1 ) and (V0 , V1 ) are the systems of homogeneous coordinates of P1 . In fact, if P0 has coordinates (1, 0) × (1, 0) then L0 (resp. M0 ) is defined by (1, 0) × (V0 , V1 ) (resp. (U0 , U1 ) × (1, 0)) and the above mapping sends L0 (resp. M0 ) to (0, 1, 0) (resp. (1, 0, 0)). The exceptional curve E corresponds to the line X2 = 0. The inverse θ−1 is given by (X0 , X1 , X2 ) 7→ (X1 /X2 , 1) × (X0 /X2 , 1). Hence ι is given in terms of (X0 , X1 , X2 ) by (X1 /X2 , 1) × (X0 /X2 , 1) 7→ (X0 /X2 , 1) × (X1 /X2 , 1). Hence, by applying θ, we have (X0 , X1 , X2 ) 7→ (X1 /X2 , X0 /X2 , 1) = (X1 , X0 , X2 ), because for (U0 , U1 ) × (V0 , V1 ) = (X0 /X2 , 1) × (X1 /X2 , 1), we have U1 V0 = X1 /X2 , U0 V1 = X0 /X2 and U1 V1 = 1. So, φ is realized by (X0 , X1 , X2 ) 7→ (X1 , X0 , X2 ). (2) Apply Lemma 2.7.16 to the present setting. The stabilizer locus Γ(⟨ι⟩, F0 ) is the diagonal ∆ = {(P, P ) ∈ P1 × P1 | P ∈ P1 }, which

Geometry on Affine Surfaces

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is linearly equivalent to L0 + M0 . Let q : F0 → W := F0 /⟨ι⟩ be the quotient morphism. By (4) of Lemma 2.7.16, W is a smooth projective surface. Let C = q(∆) and ℓ0 = q(L0 ) = q(M0 ). Then q ∗ (C) = 2∆ and q ∗ (ℓ0 ) ∼ L0 + M0 . Then 2(C 2 ) = (q ∗ (C))2 = 4(∆2 ) = 8, whence (C 2 ) = 4, and 2(ℓ0 )2 = (q ∗ (ℓ0 )2 ) = (L0 + M0 )2 = 2, whence (ℓ20 ) = 1. We have also 2(C · ℓ0 ) = (q ∗ (C) · q ∗ (ℓ0 )) = (2∆ · L0 + M0 ) = 4, whence (C · ℓ0 ) = 2. In fact, C meets ℓ0 in one point with multiplicity 2. By the assertion (1), ι is given by φ : (X0 , X1 , X2 ) → (X1 , X0 , X2 ) on P2 . Then W is birational to P2 = Proj k[Y0 , Y1 , Y2 ], where Y0 = (X0 + X1 )X2 , Y1 = X0 X1 and Y2 = X22 . In fact, C is given by Y02 = 4Y1 Y2 and ℓ0 by Y2 = 0, which is tangent to C at the point (0, 1, 0). 20. Assume that char k ̸= 2. Let C be a smooth conic15 As C ∼ 2ℓ, one can consider a double covering f : V → P2 with branch locus C. Then prove the following assertions. (1) V is isomorphic to the Hirzebruch surface F0 = P1 × P1 so that the ramification curve R is the diagonal ∆ ∼ M +L, where L and M are fibers of the two projections p1 and p2 from F0 to P1 , respectively. (2) Let Q be a point of C and let ℓQ be the tangent line of C at Q. Then f −1 (ℓQ ) = LP + MP , where f (P ) = Q, LP = p−1 1 (p1 (P )) and MP = p−1 (p (P )). 2 2 (3) Conversely, let ι : F0 → F0 be the interchanging automorphism, i.e., (P1 , P2 ) 7→ (P2 , P1 ), and let ∆ be the diagonal. Then the quotient F0 /⟨ι⟩ ∼ = P2 and the image of ∆ by the quotient morphism is a smooth conic. Answer. (1) and (2). Since f ∗ (C) = 2R and (f ∗ (C)2 ) = 2(C 2 ) = 8, we have (R2 ) = 2. Let ℓQ be the tangent line of C at Q. Since f |V \R : V \ R → P2 \ C is a finite ´etale covering and ℓQ \ {Q} ∼ = A1 −1 is simply-connected, f (ℓQ ) = C1 + C2 for irreducible curves C1 , C2 on V such that C1 ∩ C2 = {P }, (C1 \ {P }) ∼ = (C2 \ {P }) ∼ = A1 and 2 2 (C1 ) = (C2 ) because f is a Galois covering with Galois group Z/2Z and C1 , C2 are translates of each other by the Galois group action on V . Since 2 = ((f ∗ ℓQ )2 ) = (C1 + C2 )2 = (C12 ) + (C22 ) + 2(C1 · C2 ) = 2((C12 ) + (C1 · C2 )) 15 A

conic is a curve of degree 2 on P2 . If it is not smooth, C is either a sum of two lines ℓ1 + ℓ2 or a line counted twice 2ℓ.

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and (C1 · C2 ) > 0, it follows that (C12 ) = (C22 ) = 0 and (C1 · C2 ) = 1. Hence |C1 | and |C2 | define two P1 -fibrations such that they intersect fiberwise transversally. It follows then that V ∼ = F0 . (3) It is clear that the quotient morphism q : F0 → F0 /⟨ι⟩ is a double covering with ramification divisor ∆ and branch divisor ∆/⟨ι⟩. Furthermore, for each point P ∈ ∆, the fibers LP and MP of p1 and p2 mapped to the same curve on F0 /⟨ι⟩. These facts imply that F0 /⟨ι⟩ ∼ = P2 , q(∆) is a conic and q(LP ) = q(MP ) is the tangent line of F0 /⟨ι⟩ at the point Q = q(P ). 21. Let V be a smooth rational projective surface and let D be an SNC divisor with the weighted dual graph as below: −2

0

−3

C1

C2

C3 .

By the shift transformation, make the left terminal component C1 have weight 0. Similarly, make the right terminal component C3 have weight 0. In both cases, show the results by writing the obtained weighted dual graphs. Furthermore, verify the following assertions for X = V − D which we assume to be an affine surface. (1) X has two distinct A1 -fibrations ρ′ and ρ′′ over the base curve isomorphic to A1 . (2) Both fibrations ρ′ and ρ′′ have unique singular fibers which consist of five disjoint components isomorphic to A1 . We have Pic (X) ∼ = Z⊕4 . 2 (3) X contains A as an open set. (4) The component C2 in D gives a P1 -fibration p0 such that p0 |X is an A1∗ -fibration ρ : X → A1 , where A1∗ = P1 \ {0, ∞}. With the notations in the proof below of the assertion (2), the (−1) curves Ei′ on V ′ and Ei′′ on V ′′ give the five pairs of two (−1) curves, say Ei′ + Ei′′ for 1 ≤ i ≤ 5, such that Ei′ , Ei′′ meet in one point of X transversally and Ei′ + Ei′′ is a fiber of p0 . Answer. The weighted dual graph with the left terminal component C1′ of weight 0 is

Geometry on Affine Surfaces

0

0

−5

C1′

C2′

C3′

315

and similarly, the weighted dual graph with right terminal component C3′′ of weight 0 is −5

0

0

C1′′

C2′′

C3′′ .

(1) The obtained divisor D′ = C1′ + C2′ + C3′ lies on a smooth rational projective surface V ′ . By the Riemann-Roch theorem for surfaces, we have 1 1 χ(V ′ , OV ′ (C1′ )) = (C1′ · C1′ − KV ′ ) + χ(OV ′ ) = − (C1′ · KV ′ ) + 1 = 2, 2 2 where χ(OV ′ ) = 1 because pg = q = 0 and (C1′ · KV ′ ) = −2 because pa (C1′ ) = 0 and (C1′ )2 = 0. Hence dim |C1′ | = 1 and the linear pencil |C1′ | defines a P1 -fibration p′ : V ′ → P1 for which C1′ is an irreducible fiber and C2′ is a cross-section. Hence p′ induces an A1 -fibration ρ′ : X → A1 . The argument is completely similar in the case of C3′′ . (2) Since X is affine by the assumption, X has no complete curves. In the case of ρ′ , the curve C3′ is contained in the singular fiber F3′ = P5 C3′ + j=1 Ej′ , where Ej′ is a (−1) curve with (C3′ · Ej′ ) = 1 and hence Ej′ ∩ X ∼ = A1 . This fiber is a unique singular fiber in the fibration P5 P5 p′ . Since C1′ ∼ C3′ + j=1 Ej′ , j=1 Ej′ ∩ X ∼ 0. This implies that can argue in the same way in the case of C3′′ . Pic (X) ∼ = Z⊕4 . We P5 (3) If we remove j=2 Ej′ ∩ X from X, the open set U is isomorphic to A2 because the open set has an A1 -bundle structure over A1 . (4) Clearly, Ei′ and Ei′′ are disjoint from C2 , hence they are fiber components of p0 . They form pairwise five fibers Fi of p0 . For three of the fibers Fi , say i = 1, 2, 3, contract the components Ei′′ and for the remaining two fibers, contract the components Ei′ for i = 4, 5. Then the resulting surface is the Hirzebruch surface F0 and the image D0 of D consists of one fiber of the projection p1 : F0 → P1 and two fibers of the projection p2 : F0 → P1 . 22. Let X = F0 \ ∆, where F0 is the Hirzebruch surface of degree 0 and ∆ is the diagonal, i.e., ∆ = {(P, P ) | P ∈ P1 }. Prove the following assertions.

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(1) X is isomorphic to an affine hypersurface xy = z 2 − 1 in A3 = Spec k[x, y, z].16 (2) For each (P, P ) ∈ ∆, let LP (resp. MP ) be the fiber of the first (resp. the second) projection from F0 = P1 × P1 to P1 . Then ∆ and LP + MP generate a linear pencil ΛP which defines on X an A1 fibration fP : X → A1 with a unique singular fiber (LP + MP ) ∩ X. For two distinct points P ̸= Q of P1 , general members of fP and fQ meets in two points transversally. Further there is an automorphism σ e of F0 such that σ e(∆) = ∆ and σ e∗ (ΛQ ) = ΛP . Answer. (1) Let (X0 , X1 ) be a system of homogeneous coordinates of P1 and let (Y0 , Y1 ) be its copy. Set U0 = X0 Y0 , U1 = X0 Y1 , U2 = X1 Y0 , U3 = X1 Y1 . Then a mapping (X0 , X1 ) × (Y0 , Y1 ) 7→ (U0 , U1 , U2 , U3 ) defines a closed immersion of F0 into P3 (called the Segre embedding), and the image is a quadric hypersurface U0 U3 = U1 U2 . The diagonal ∆ is the intersection of this image by a hyperplane U1 = U2 . Set V = U1 − U2 . Then the image is written as U0 U3 = U1 U2 = U2 (V + U2 ). Then the complement X = F0 \ ∆ is defined by   U0 U3 U2 U2 · = 1+ . V V V V Set x = 2 · UV0 , y = 2 · UV3 and z = 2 · UV2 + 1. Then the above equation for X becomes xy = z 2 − 1. (2) A general member of ΛP is an irreducible curve CP meeting ∆ at (P, P ) with local intersection multiplicity 2. Let CQ be a general member of ΛQ . Since CP ∼ LP + MP ∼ LQ + MQ ∼ CQ , we have (CP · CQ ) = 2. Let σ be an automorphism of P1 such that σ(P ) = Q and let σ e be the product (σ, σ) : P1 × P1 → P1 × P1 . It is then clear that σ e(P, P ) = (Q, Q) and σ e∗ (ΛQ ) = ΛP . 23. Show that the following assertions hold. (1) Let P1 = (a0 , a1 , a2 ), P2 = (b0 , b1 , b2 ), P3 = (c0 , c1 , c2 ) be three points of the projective plane P2 . Then P1 , P2 , P3 are colinear if and 16 This is the simplest example of what is called a Danielewski surface. It gives a counterexample to the cancellation problem, which asks if X × A1 ∼ = Y × A1 implies X ∼ =Y for algebraic varieties X, Y .

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only if the determinant of the matrix A is zero, where   a0 a1 a2 A =  b0 b1 b2  . c0 c1 c2 (2) Suppose that P1 , P2 , P3 are not colinear. If Q1 = (α0 , α1 , α2 ), Q2 = (β0 , β1 , β2 ), Q3 = (γ0 , γ1 , γ2 ) are not colinear, there exists an element g ∈ PGL (3, k) such that g(Pi ) = Qi for i = 1, 2, 3. (3) This problem can be generalized to the case n ≥ 3 as follows. Points P1 , P2 , . . . , Pn+1 of Pn are coplanar if there exists a hyperplane H such that Pi ∈ H for 1 ≤ i ≤ n + 1. Show that P1 , . . . , Pn+1 are coplanar if and only if the determinant of the (n+1)×(n+1)-matrix whose rows are homogeneous coordinates of points P1 , . . . , Pn+1 is zero. Suppose that P1 , . . . , Pn+1 are not coplanar. If Q1 , . . . , Qn+1 are non-coplanar points of Pn there exists an element g ∈ PGL (n + 1, k) such that g(Pi ) = Qi for 1 ≤ i ≤ n + 1. Answer. (1) A line ℓ on P2 is defined by an equation d0 X0 + d1 X1 + d2 X2 = 0, where (d0 , d1 , d2 ) ̸= (0, 0, 0). Then three points P1 , P2 , P3 lie on ℓ if and only if     d0 0    A d1 = 0  . d2 0 Such equality holds if and only if |A| = 0. (2) It suffices to find an element g ∈ GL (3, k) such that   α0 α1 α2 g · A =  β0 β1 β 2  . γ0 γ1 γ2 Then we have only to take 

 α0 α1 α2 g =  β0 β1 β2  · A−1 . γ0 γ1 γ2 (3) Generalization to the case n ≥ 3 is left to the readers.

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Chapter 3

Geometry and Topology of Polynomial Rings — Motivated by the Jacobian Problem 3.1

Plane-like affine surfaces

The ground field k is an algebraically closed field of characteristic zero. A main theme in this chapter is to consider an ´etale covering of a given normal algebraic variety X, which is, by definition, a finite morphism f : Y → X which is flat and unramified at every point of Y . In particular, f is surjective and Y is normal. An ´etale covering f is irreducible if Y is irreducible. If f is obtained as the quotient morphism of a faithful action of a finite group G on an irreducible algebraic variety Y , we say that f is a Galois covering (see subsection 2.7.4 for the case of algebraic surfaces). Each closed fiber of a Galois covering f is then a G-orbit. A normal algebraic variety X is called simply connected if there are no irreducible ´etale coverings f : Y → X. e together with a finite ´etale covering If an irreducible algebraic variety X e → X is called a universal covering if the following two conditions are fe : X satisfied: e is simply connected. (i) f is a Galois covering with a group G and X (ii) If g : Y → X is an irreducible finite ´etale covering with a normal algebraic variety Y then there exists a subgroup H of G such that e e → Y is the quotient morphism. and fe = g ◦ q, where q : X Y ∼ = X/H e ′ → X satisfies these two conditions with a finite group G′ , then If fe′ : X ′ e →X e ′ and h′ : X e′ → X e such G∼ = G and there exist isomorphisms h : X ′ ′ that h ◦ h = idXe and h ◦ h = idXe ′ . An irreducible ´etale covering is often mentioned as an ´etale covering in the forthcoming explanations. Modulo e → X is called the universal covering and the these isomorphisms, fe : X group G the fundamental group. We denote G by π1 (X). If k = C the field of complex numbers, then X(C) has a structure of a complex variety, and 319

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π1 (X) coincides with the topological fundamental group π1 (X(C)). In the e as well as π1 (X) does not necessarily exist. present algebraic case, X 3.1.1

Simply connected algebraic varieties

Lemma 3.1.1. Assume that char k = 0. Then the affine space An is simply connected. Proof. Step 1. Consider the case n = 1. Let f : C → A1 be an ´etale e → P1 , covering of degree d. Then f extends to a finite morphism fe : C e where C is a smooth projective curve containing C as an open set and e be the genus of C. e By the Riemann-Hurwitz formula fe|C = f . Let g(C) (see Theorem 2.7.5), we have X e − 2 = −2d + 2g(C) (ePe − 1), e∈fe−1 (∞) P

where ePe is the ramification index. Let m be the number of points of P fe−1 (∞). Then we have e − 1) = d − m. Hence we have e∈fe−1 (∞) (eP P e e = 0 and hence d = m = 1. 2g(C) − 2 = −(d + m) < 0. It implies that g(C)

Hence f : C → A1 is an isomorphism. Step 2. Let X = A2 and let f : Y → X be an irreducible ´etale covering of degree d. The finite morphism f is extended to a morphism e = P2 , where Ye is a smooth projective surface and X is embedded fe : Ye → X e into X = P2 in the standard way as X = P2 \ ℓ with a line ℓ. Namely, e in k(Y ) and then Ye as a resolution of we take the normalization Yb of X singularities of Yb . With a system of coordinates {x, y} on A2 , consider the inclusion of the fields k(x) ,→ k(X) ,→ k(Y ), which defines a linear pencil ρ : Ye → P1 . Let {ℓα | α ∈ k} be the A1 -fibration on X induced by the projection (x, y) 7→ x. Then f −1 (ℓα ) is a disjoint union of d copies of A1 because A1 is simply connected by the Step 1. By the Stein factorization (see Lemma 1.8.15), ρ splits as ρ e

ν

e −→ P1 , ρ : Ye 99K B e is a smooth projective curve and ν is a finite morphism. Since where B 1 ∼ ℓα = A and f |f −1 (ℓα ) : f −1 (ℓα ) → ℓα is an ´etale covering, f −1 (ℓα ) consists of d irreducible components, each of which is isomorphic to A1 , it follows e \ ν −1 (∞). that ν −1 (β) consists of d points for β ∈ P1 \{∞}. Let B = B 1 1 Then ν|B : B → A is an irreducible ´etale covering of A . This implies that d = 1. Hence f : Y → X is an isomorphism.

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Step 3. We proceed by induction on n. Suppose that An−1 is simply connected. Let f : Y → X = An be an irreducible ´etale covering. As in e = Pn , where Ye is a the Step 2, we extend f to a morphism fe : Ye → X smooth projective variety containing Y as an open set. Let {x1 , x2 , . . . , xn } f∗

be a system of coordinates on An and let k(x1 ) ,→ k(X) ,→ k(Y ) be the inclusion of fields, which defines a linear pencil ρ : Ye 99K P1 . By the Stein factorization,1 ρ splits as ρ e ν e −→ ρ : Ye 99K B P1 ,

e is a smooth projective curve. Let {Hα | α ∈ k} be the set of where B n 1 linear hyperplanes Hα of An defined by p−1 1 (α), where p1 : A → A is the −1 e e projection (x1 , . . . , xn ) 7→ x1 . Since f |fe−1 (Hα ) : f (Hα ) → Hα is an ´etale covering, fe−1 (Hα ) is a union of d irreducible components, each of which is isomorphic to An−1 . Here we use the induction hypothesis that An−1 is simply connected. We can finish the proof by arguing in the same way as in the Step 2. Corollary 3.1.2. Assume that char k = 0. Let X be a smooth algebraic variety which contains an open set U isomorphic to An . Then X is simply connected. Proof. Let f : Y → X be an irreducible ´etale covering of degree d. Then f |f −1 (U ) : f −1 (U ) → U ∼ = An is an irreducible ´etale covering of degree n d. Since A is simply connected and f −1 (U ) is irreducible, it follows from Lemma 3.1.1 that d = 1. So, f is an isomorphism. Corollary 3.1.3. Assume that char k = 0. Let X be a smooth algebraic variety which has an An−1 -bundle structure ρ : X → B over a smooth algebraic curve B. Let f : Y → X be an irreducible ´etale covering. Then there exists an ´etale covering ν : C → B such that Y ∼ = C ×B X. Proof. Consider the composite ρ ◦ f : Y → B. By the Stein factorization, there exist morphisms ν : C → B and g : Y → C such that ρ ◦ f = ν ◦ g. By the argument in the proof of Lemma 3.1.1, ν is a finite ´etale morphism and g defines an An−1 -bundle structure on Y . Then the natural morphism Y → C ×B X over C is an isomorphism because it is an isomorphism restricted on the fiber over all point of C. 1 Though

the Stein factorization is explained only for surfaces in Lemma 1.8.15, it holds for a smooth projective variety of arbitrary dimension.

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Remark 3.1.4. If char k = p > 0, there is a finite ´etale covering f : A1 → A1 of degree p. In fact, let x be a coordinate of A1 and let f be a morphism defined by x 7→ xp − x. This is an ´etale Galois covering with group Z/pZ. The action is given by x 7→ x + 1. The following result shows a noteworthy property of a simply connected algebraic variety. Theorem 3.1.5. Let X be a smooth algebraic variety. Assume that X is simply connected. Then the Picard group Pic X has no torsion elements of order prime to p = char k. Proof. Let F be a divisor on X whose class in Pic X has order n > 0 with gcd(n, p) = 1. Namely we have nF ∼ 0. We construct an ´etale Galois covering f : Y → X of degree n with group G ∼ = Z/nZ such that X = Y /G. Construction is the same as the one given before Theorem 2.7.17. Let U = {Ui }i∈I be an affine open covering of X such that OX (F )|Ui = OUi ui ∗ ). Since nF ∼ 0, for every i ∈ I. Then uj = fji ui with fji ∈ Γ(Ui ∩ Uj , OX ∗ n there exists a family {ci ∈ Γ(Ui , OX ) | i ∈ I} such that fji = c−1 j ci for all 1 i, j ∈ I if the covering U is fine enough. Let Z be the A -bundle over X which is associated to F . Namely, Z|Ui = Spec k[ξi ] × Ui and ξi = fji ξj for i, j ∈ I. Define an algebraic variety Y with a morphism f : Y → X locally n n n by f −1 (Ui ) = Spec k[ξi ]/(ξin − ci ) for i ∈ I. Since ξin = fji ξj = c−1 j ci ξj , Y together with the morphism f is well-defined. Since gcd(n, p) = 1, f |Ui is an ´etale covering of degree n. In fact, the group G of nth roots of unity ∗ ) and acts on Y as ξi 7→ ζξi . If f is an isomorphism, then ξi ∈ Γ(Ui , OX −1 fji = ξj ξi . Hence F ∼ 0. So, X is not simply connected if Pic X has a non-trivial torsion element whose order is prime to p. 3.1.2

Unit group, unit rank and independence of boundary divisors

Let V be a smooth algebraic variety and D an effective reduced divisor on Pn V . Let D = i=1 Di be the irreducible decomposition and let F (D) = Ln i=1 ZDi be the free group generated by the Di . Let X = V − D. Then we have an exact sequence ∗ 0 → Γ(V, OV∗ ) → Γ(X, OX ) → Ker φ → 0, φ

0 → Ker φ → F (D) −→ Pic (V ) → Pic (X) → 0, Pn Qn ∗ where φ( i=1 ai Di ) = i=1 OV (Di )⊗ai . We denote Γ(X, OX ) by U (X) and call it the unit group of X. We denote Ker φ by U (X) and call it the

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reduced unit group. If X is affine and X = Spec R then U (X) is the multiplicative group R∗ of units. Further, if V is projective then Γ(V, OV∗ ) = k ∗ and U (X) = R∗ /k ∗ . Since Ker φ is a subgroup of a free abelian group of finite rank, U (X) is also a free abelian group of finite rank. We call its rank the unit rank of X and denote it by γ(X). If we assume that V is projective and X is affine, the group U (X) does not depend on the choice of an open embedding X ∼ = V \ D ,→ V . We say that the irreducible components Di of D are independent modulo linear equivalence, or linearly independent for short, on V if φ is injective. The following result is clear by the definition. Lemma 3.1.6. With the above notations, the Di are linearly independent on V if and only if the unit rank of X is zero. The following result follows from Lemma 2.1.1 and Theorem 2.6.11. Lemma 3.1.7. Let X be a smooth affine surface satisfying one of the following conditions. (i) X contains an open set U isomorphic to the affine plane A2 . (ii) There is a surjective A1 -fibration ρ : X → B with B isomorphic to A1 or P1 . Let V be a smooth projective surface which contains X as an open set. Then the complement D = V \ X is a divisor whose irreducible components are rational curves which are linearly independent. Hence U (X) = k ∗ . If σ : V ′ → V is a birational morphism such that D′ := σ ∗ (D)red is an SNC divisor. Then D′ is a tree. Proof. Suppose that X contains an open set U which is isomorphic to A2 . By Lemma 2.1.1, V \ U is a union of irreducible divisors and V \ X is a partial union of these divisors. Hence each irreducible component of X \ U has codimension 1. By blowing-ups with centers on the divisor V \ U , we can make it an SNC divisor, and by blowing-down the SNC divisor, we can obtain a minimal log smooth completion of U ∼ = A2 . Since the boundary 2 divisor of a minimal log smooth completion of A is a linear chain of rational curves by Theorem 2.6.11, D is a divisor of rational curves, and D′ is a tree. Suppose that X has a surjective A1 -fibration ρ : X → B such that B is isomorphic to A1 or P1 . Then there exists an irreducible linear pencil Λ on V such that every member F of Λ gives a fiber F ∩ X of ρ if F ∩ X ̸= ∅. By blowing-ups of points on V \ X, we may assume that Λ defines a P1 fibration f : V → C, where C ∼ = P1 . In particular, D is an SNC divisor.

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Then V \ X consists of one cross-section S and fiber components of f . Sn Pn Write V \ X = S ∪ ( i−1 Ci ). Suppose that i=1 ai Ci + bS ∼ 0. Then the intersection with a general fiber ℓ of f gives b = 0. Then, after blowing down singular fiber contained in V \ X to a smooth fiber if such singular Pn fiber exists, the divisor i=1 ai Ci is rewritten as aℓ +

ri m X X i=1 j=1

aij Cij ∼ 0,

S ri C is contained in a singular where ℓ is a smooth fiber in V \ D and j=1 Sri ij fiber Fi (1 ≤ i ≤ m). Since X is affine, j=1 Cij is connected. Its intersection matrix is negative definite because Fi ∩ X ̸= ∅. Hence aij = 0 for 1 ≤ i ≤ m and 1 ≤ j ≤ ri . Then aℓ ∼ 0, which implies a = 0. Thus the irreducible components of D are linearly independent. The assertion U (X) = k ∗ follows from Lemma 3.1.6. 3.1.3

Gizatullin surfaces and affine pseudo-planes

The ground field k is assumed to have characteristic zero. Elimination of this hypothesis will give new research problems. Geometric properties of polynomial rings can be studied via A1 fibrations on the corresponding affine spaces. We here consider A1 fibrations on affine surfaces. If an A1 -fibration is of affine type, i.e., if the base curve is an affine curve, the Picard group is described in Lemma 2.1.3. We generalize the formula in the case containing the case of A1 -fibrations of complete type. Lemma 3.1.8. Let ρ : X → B be an A1 -fibration from a smooth affine surface to a smooth algebraic curve B. The following assertions hold. (1) If F is a singular fiber of ρ, then Fred is a disjoint union of affine lines, ` i.e., F = j mj Cj , where mj is the multiplicity of Cj ∼ = A1 in F . (2) Let F1 , . . . , Fr exhaust all singular fibers of ρ, and write Fi = P (i) (i) j mj Cj . With these notations, there is an exact sequence   r M M X (i) (i)  0 → Pic B → Pic X → Z[Cj ]/⟨ Z[mj Cj ]⟩ → 0. i=1

j

j

(3) Pic X is a finite group if and only if B is a rational curve and every singular fiber Fi is irreducible.

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Proof. (1) is proved in Lemma 2.1.2. (2) The natural homomorphism ρ∗ : Pic B → Pic X is injective. In fact, we extend the morphism ρ to a P1 -fibration f : V → B, where V (resp. B) is a smooth projective surface (resp. curve) containing X (resp. B) as an open set and f |X = ρ. Let D = V \ X. Then D contains a unique cross-section S of f and other irreducible components are fiber components of f . Let ∆ be a divisor such that ρ∗ (∆) ∼ 0. By linear equivalence on the curve B, we may assume that Supp ∆ contains no points b ∈ B such that f −1 (b) is a singular fiber. Then f ∗ (∆) ∼ 0 on V , whence, restricted on S, f ∗ (∆)|S = ∆ ∼ 0. Let Fi = ρ∗ (bi ) with bi ∈ B and let F i = f ∗ (bi ). Then f ∗ (bi ) ∼ 0 P (i) (i) (mod Pic B). Hence ρ∗ (bi ) = j mj Cj ∼ 0 (mod Pic B). We may and shall assume that every fiber of f lying outside of X is a smooth fiber. Since Pic V is generated by f ∗ (Pic B), S and all irreducible components of (i) F i (1 ≤ i ≤ r), Pic X is generated by Pic B and the components Cj for 1 ≤ i ≤ r and all j. It is now easy to check that the sequence is exact. (3) If B is non-rational, i.e., g(B) > 0, then Pic B is an abelian variety of dimension g(B). Then Pic B is an infinite group. So, if Pic X is a finite group then B is a rational curve. If some singular fiber, say Fi , has two or more irreducible components, the fiber Fi contributes ni − 1 to rank Pic X, where ni is the number of irreducible components of Fi . So, if Pic X is finite, every singular fiber Fi is irreducible. The converse is easy to verify. By Lemma 2.1.4 and Theorem 2.1.5, the affine plane A2 is characterized as a smooth affine surface X such that U (X) = k ∗ , Pic X = 0 and X has an A1 -fibration ρ : X → B of affine type. In considering problems concerning the affine plane like the Jacobian conjecture to be treated later, we need to consider affine surfaces which are close to A2 ; in other words, affine surfaces which share some properties with A2 , but some properties of characterizing A2 are relaxed. A theorem of Ramanujam (see Proposition 2.6.10) asserts that a minimal log smooth completion of A2 has a rational linear chain as the boundary divisor. Let X be a smooth affine surface which is an open set of a smooth projective surface V as the complement of a minimal SNC divisor D which is a linear chain of rational curves. We call such a surface X a Gizatullin surface. Remark 3.1.9. Details on Gizatullin surfaces are given in [27, Chapter 2, §2.6]. As noteworthy properties, we can list the following by writing D = D1 + · · · + Dn :

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(1) If n = 1, i.e., D is irreducible, then (D2 ) > 0 and D ∼ = P1 . Hence (D · KV ) < 0. By Enriques criterion of ruledness (see [59, Chapter 1, Theorem 4.5.1]), V is a ruled surface because P12 = dim |12KV | + 1 = 0. If the irregularity q > 0 every curve isomorphic to P1 is a fiber component of the unique P1 -fibration, which is a contradiction. Hence q = 0 and V is a rational surface. Since D is linearly independent, γ(X) = 0. If Pic X is a finite group, V has Picard number 1. Hence V ∼ = P2 and D is either a line or a conic. (2) We assume that n ≥ 2. Since D supports a very ample divisor, there is an irreducible component Di with non-negative weight. By shift transformations or a succession of blowing-ups, we can assume that the left terminal or right terminal component has weight 0. Then it follows that V is a rational surface. Two terminal components give two different linear pencils without base points. Hence X has two different A1 -fibrations whose general fibers intersect. By Theorem 3.5.3, X is either an ML0 -surface (if γ(X) = 0) or X ∼ = A1 × A1∗ (if n = 3 and the left and right terminal components have weight 0). See Appendix for ML0 surfaces. (3) Let X be a Gizatullin surface with γ(X) = 0, i.e., an ML0 -surface. Then there exists an A1 -fibration ρ : X → A1 satisfying the conditions. (i) There exist a log smooth completion X of X and a P1 -fibration f : X → P1 such that ρ = f |X . (ii) The boundary divisor D = X \ X is written as D = ℓ + S + A, where ℓ is a smooth fiber of f , S is a cross-section of f and A + A′ supports a unique singular fiber of f with a divisor A′ . Further, every irreducible component of A has weight ≤ −2 and every irreducible component of A′ restricted on X is a curve isomorphic to A1 and one of the irreducible components of A′ is a (−1) curve. Every irreducible component of A′ is called a feather of X. Pr (iii) Write A′ = the multiplicity of Ai in the i=1 Ai and let mi be Lr Pr singular fiber of f . Then Pic X ∼ = i=1 Z[Ai ]/( i=1 mi [Ai ] = 0). So, Pic X has rank r−1 and the torsion subgroup (Pic X)tor ∼ = Z/dZ, where d = gcd(m1 , . . . , mr ). This is [27, Chapter 2, Lemma 2.6.8].

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(4) An analogue of a Theorem of Abhyankar-Moh-Suzuki (see Theorem 2.2.17) holds for a Gizatullin surface X with γ(X) = 0. To wit, let C be a curve on X isomorphic to A1 . Then there exists an A1 -fibration ρ : X → B such that B ∼ = A1 and C is a fiber of ρ. This is [27, Chapter 2, Theorem 2.6.12]. This result shows that a Gizatullin surface with γ(X) = 0 is a surface close to the affine plane. For an affine surface with an A1 -fibration, the Picard group Pic X is determined by the following result. Lemma 3.1.10. Let X be a smooth affine surface with an A1 -fibration ρ : X → B, where B ∼ = A1 or B ∼ = P1 . Then the following assertions hold. P (1) If B ∼ = A1 , then rank Pic (X) = P ∈B (nP − 1), where nP is the number of irreducible components of the fiber ρ−1 (P ). If B ∼ = P1 , then P rank Pic (X) = 1 + P ∈B (nP − 1). (2) Assume that B ∼ = A1 and nP = 1 for every P ∈ B. Then Pic (X) ∼ = Q −1 Z/m Z, where m is the multiplicity of the fiber ρ (P ). In P P P ∈B particular, if there is only one multiple fiber mF , then Pic (X) is a cyclic group of order m. Proof. (1) Embed X into a smooth projective surface V with a P1 -fibration f : V → B such that D := V \ X is an SNC divisor. Since D contains a unique cross-section, say S, we know that D consists of S and some connected fiber components of f which meet S. Since X contains no complete curves, it follows from what was already explained that any fiber of f which contains irreducible components of D intersects with X at some of the terminal components of the tree D. This means that any fiber of ρ is a disjoint union of the affine lines with multiplicities. Note that rank Pic (Fn ) = 2 for a Hirzebruch surface Fn of order n and each blowing-up increases the rank of the Picard group by one. Then P rank Pic (V ) = 2 + P ∈B (n′P − 1), where n′P is the number of irreducible P components of the fiber f −1 (P ). Since D = S + P ∈B (D ∩ f −1 (P )) and D∩f −1 (P ) consists of (n′P −nP ) components, we obtain the asserted result. (2) If B ∼ = A1 , one point of B, say P∞ , is not included in B. Let ℓ∞ be the fiber of f over P∞ which we may assume to be smooth. Then every fiber f −1 (P ) is linearly equivalent to ℓ∞ . Hence if mP CP is the component of f −1 (P ) left in X with multiplicity mP , then we have a relation mP FP ∼ 0. In fact, there are no other relations on the FP . Hence we have the asserted result.

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Definition 3.1.11. A smooth affine surface X is an affine pseudo-plane if X satisfies the following properties. (i) X has an A1 -fibration ρ : X → B such that B ∼ = A1 and ρ has a unique multiple fiver dF , which is irreducible. (ii) Pic X is a finite cyclic group of order d ≥ 2. We say that X is an affine pseudo-plane of type (d, n, r) if the following condition is satisfied. (iii) X has a log smooth completion V with D = V \ X such that the dual graph of D is as given below, where n ≥ 1 and r ≥ 1. Furthermore, F is the closure of F in V and S ′ is the unique cross-section contained in D. −d E1 0 ℓ′∞ −1 F −2

−2 Ed+r−1

Ed+1

−2

−2 Ed

Ed−1

−2

−2

E2

ℓ′0

S

′

−n

Remark 3.1.12. (1) Let X be an affine pseudo-plane of type (d, n, r). It is obtained from the Hirzebruch surface Fn by the following construction. Let M be a minimal section and ℓ0 , ℓ∞ two fibers. Choose a point P0 ∈ ℓ0 which is not the intersection point ℓ0 ∩ M . Blow up P0 to obtain the exceptional curve E1 and the proper transform ℓ′0 of ℓ0 . Further blow up the point P1 := E1 ∩ ℓ′0 and its infinitely near points P2 , . . . , Pd−1 which lie on the proper transforms of E1 . The exceptional curve by the blowing-up of Pd−1 is Ed , which has multiplicity d in the total transform of ℓ0 . We construct a side chain Ed+1 , . . . , Ed+r−1 , F by blowing up a point Pd which is not E1 ∩Ed nor Ed ∩Ed−1 and its infinitely near points Pd+1 , . . . , Pd+r−1 . Hence an affine pseudo-plane of type (d, n, r) exists. (2) By performing elementary transformations (see section 1.8) on the fiber ℓ∞ , we can assume that n = 1 without changing the affine part X. We then say that X has type (d, r), where d represents the length of the principal chain and r represents the length of the side chain. If r ≥ 2 the side chain exists, while X is a Gizatullin surface if r = 1.

Geometry and Topology of Polynomial Rings

3.1.4

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Affine pseudo-planes — more properties

We consider an affine pseudo-plane X of type (d, r) and denote it by X(d, r) to remind of its type. Lemma 3.1.13. Let X be an affine pseudo-plane of type (d, r). Let n = |r − d|. We have the following assertions. (1) Assume that r < d. Then X is isomorphic to the complement of M0 ∪Cd in the Hirzebruch surface Fn , where M0 is the minimal section and Cd is an irreducible member of the linear system |M0 + dℓ0 | with a fiber ℓ0 such that (M0 · Cd ) = i(M0 , Cd ; P0 ) = r and F = ℓ0 ∩ X, where P0 = M0 ∩ ℓ0 . (2) Assume that r ≥ d. Then X is the complement of M1 ∪ Cd in a Hirzebruch surface Fm for some m ≥ 0, where M1 is a section with (M12 ) = n and Cd is a section with (Cd2 ) = 2d+n which is an irreducible member of |M1 + dℓ0 | such that (M1 · Cd ) = i(M1 , Cd ; P1 ) = r and F = ℓ0 ∩ X, where P1 = M1 ∩ ℓ0 . If we assume that the minimal section M0 of Fm does not pass through the point P1 then m = n. Proof. (1) In the weighted dual graph in Definition 3.1.11 where n = 1, contract the curves S ′ , ℓ′0 , . . . , Ed , Ed+1 , . . . , Ed+r−1 in this order to obtain the birational morphism σ : V → Fn , where n = d − r, M0 = σ(E1 ), Cd = σ(ℓ′∞ ) and ℓ0 = σ(F ). Since (σ(F ) · σ(ℓ′∞ )) = 1 and (σ(ℓ′∞ )2 ) = d+r = 2d−n, we have Cd ∼ M0 +dℓ0 . In fact, M0 +dℓ0 is a unique reducible member of |M0 + dℓ0 | and other members are all smooth irreducible curves meeting each other only at the point P0 . (2) The contraction of S ′ , ℓ′0 , . . . , Ed , Ed+1 , . . . , Ed+r−1 in this order gives a morphism σ : V → W . Since rank (Pic V ) = 2 + d + r, we have rank (Pic W ) = (d + r + 2) − (d + r) = 2. Hence W is a Hirzebruch surface Fm of order m. Let M1 = σ(E1 ), Cd = σ(ℓ′∞ ) and ℓ0 = σ(F ). Then we have (M12 ) = r − d = n, (Cd2 ) = d + r = 2d + n, (M1 · Cd ) = r, (M1 · ℓ0 ) = (Cd · ℓ0 ) = 1.

Hence we have to show that m = n under the assumption that P1 ̸∈ M0 . In fact, let τ : V → Fd be the contraction of S ′ , ℓ′0 , . . . , Ed−1 and F , Ed+r−1 , . . . , Ed+1 . Then τ (E1 ), τ (ℓ′∞ ) and τ (Ed ) are respectively the minimal section, a section disjoint from the minimal section and a fiber. We apply an elementary transformation with center Q1 of Fd which is the image of Ed+1 and not the intersection points τ (E1 ) ∩ τ (Ed ) and τ (ℓ′∞ ) ∩ τ (Ed ).

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Then we obtain a Hirzebruch surface Fd−1 . By this transformation, we regain Ed+1 and contract Ed . We continue this kind of transformations. Regain Ed+2 and contract Ed+1 , regain Ed+3 and contract Ed+2 , and finally regain F and contract Fd+r−1 . Thus we obtain the Hirzebruch surface Fm from Fd . The inverse elementary transformations are centered at P1 and its infinitely near points P2 , . . . , Pr−1 which are the intersection points of the proper transforms of M1 and Cd . Hence the image M0′ of M0 on Fd is disjoint from the minimal section τ (E1 ), and (M0′ )2 = −m + r. Hence r − m = d, i.e., m = r − d = n. The case P1 ̸∈ M0 is possible. In fact, let A be a section of Fd which is disjoint from the minimal section τ (E1 ) and different from τ (ℓ′∞ ). Take the points P1 , . . . , Pr−1 so that they lie on the proper transforms of A. Then the image A′ of A on Fm is the minimal section and P1 ̸∈ A′ . Lemma 3.1.14. Let X be an affine pseudo-plane of type (d, r) with r ≥ 2. Then the following assertions hold. (1) X has one and only one A1 -fibration ρ : X → B such that B ∼ = A1 and ρ has a unique irreducible multiple fiber ρ∗ (b0 ) = dF , where b0 ∈ B. (2) Let α be an automorphism of X. Then α preserves the A1 -fibration ρ. Namely there exists an automorphism β of B such that ρ ◦ α = β ◦ ρ, where β fixes the point b0 ∈ B. (3) The correspondence α 7→ β induces a group homomorphism φ : Aut X → Aut (B, b0 ) ∼ = Gm , where Aut (B, b0 ) consists of all automorphisms of B fixing the point b0 . Proof. (1) By the definition X has an A1 -fibration ρ : X → B with B ∼ = A1 and the weighted dual graph of the boundary divisor D in Definition 3.1.11. By elementary transformations on ℓ′∞ , we may assume that n := −(S ′ )2 ≥ 2. Then the smooth projective surface V with the boundary divisor D is a minimal log smooth completion of X. Since D is not a linear chain, X is not an ML0 -surface. So, ρ is a unique A1 -fibration of X over A1 . (2) The morphism ρ ◦ α defines an A1 -fibration with a unique multiple fiber on X. Since such an A1 -fibration is unique by the assertion (1), ρ ◦ α = β ◦ ρ with β : B → B. It is then clear that β is an automorphism of B fixing the point b0 . (3) For automorphisms αi (i = 1, 2) of X, we have automorphisms βi of B such that ρ ◦ αi = βi ◦ ρ. Then we have ρ ◦ (α1 ◦ α2 ) = (ρ ◦ α1 ) ◦ α2 = (β1 ◦ ρ) ◦ α2

= β1 ◦ (ρ ◦ α2 ) = β1 ◦ (β2 ◦ ρ) = (β1 ◦ β2 ) ◦ ρ.

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Hence we have a group homomorphism φ : Aut X → Aut (B, b0 ), where B = Spec k[x] with b0 defined by x = 0. Then Aut (B, b0 ) consists of automorphisms x 7→ λx with λ ∈ k ∗ . Hence Aut (B, b0 ) ∼ = Gm . Lemma 3.1.15. Let X be an affine pseudo-plane of type (d, r) with r ≥ 2 and let ρ : X → B be a unique A1 -fibration. Then there does not exist a Gm -action σ : Gm × X → X such that σ induces a trivial Gm -action on B (see Problem 3). Proof. Suppose that X has a non-trivial Gm -action. There exists a Gm equivariant log smooth completion W of X. Namely, by a theorem of Sumihiro [92], there exists a smooth projective surface W satisfying the following conditions. (1) W contains X as an open set and the divisor H = W \ X is an SNC divisor. Furthermore, the A1 -fibration ρ extends to a P1 -fibration ρ : W → B, where B ∼ = P1 . (2) There exists a Gm -action τ : Gm × W → W which induces the Gm action σ : Gm × X → X by the restriction, i.e., τ |Gm ×X = σ. Hence Gm acts fiberwise on W , and the divisor H is Gm -stable and hence each irreducible component of H is Gm -stable. (3) There exists an irreducible component, say H0 , of H such that H0 is a crosse-section of ρ. Other components of H are fiber components of ρ. The (−1) fiber components can be contracted without losing the Gm -action. Hence we may assume that the fiber ρ ∗−1 (b∞ ) is a smooth fiber and ρ−1 (b0 ) has a side chain, where {b∞ } = B \ B and ρ−1 (b0 ) is the multiple fiber dF . Let P be a fixed point on H. By Slice Theorem of Luna (see the proof of Lemma 2.7.15), there exists a Gm -morphism φ : U → TW,P := (m/m2 )∗ from a Gm -stable affine open neighborhood U of P to the tangent space TW,P of W at P such that φ(P ) is the origin O, φ is ´etale at P and the tangential linear representation of Gm on TW,P is diagonalizable. Let Hi be the irreducible component of ρ−1 (b0 ) ∩ H with P ∈ Hi . Let x be a coordinate of B such that b0 is defined by x = 0. Near the point P , the component Hi is defined by ξ = 0 such that x = uξ d , where ξ ∈ m := mW,P ∗ and u ∈ OW,P . Let η be a local parameter of Hi such that η ∈ m. Then we may assume that m = (ξ, η)OW,P , g ξ = α(g)ξ (mod m2 ) and g η = β(g)η (mod m2 ) for g ∈ Gm (k) = k ∗ , where α and β are multiplicative characters of Gm . Since x is Gm -invariant and u − u(P ) ∈ m, the condition x = uξ d

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implies that α(g) = 1 for g ∈ Gm . If β(g) = 1 for all g ∈ Gm then Slice bW,P is Theorem implies that the induced Gm -action on the completion O trivial, whence the Gm -action on OW,P and k(W ) = Q(OW,P ) is trivial. This is a contradiction. Hence every component of ρ−1 (b0 ) has non-trivial Gm -action. Meanwhile, the fiber ρ−1 (b0 ) is contracted to a smooth fiber together with the Gm -action, and we get it back by blowing up Gm -fixed points. It is clear that a curve isomorphic to P1 with a non-trivial Gm -action has two fixed points which are the intersection points of fiber components or the cross-section H0 . Then we cannot produce a side chain. Corollary 3.1.16. Let X be an affine pseudo-plane of type (d, r) with r ≥ 2. Assume that there exists a non-trivial Gm -action on X, which we view as a group homomorphism σ : Gm → Aut X. Then the group homomorphism φ : Aut X → Aut (B, b0 ) in Lemma 3.1.14 is surjective. Proof. By Lemma 3.1.15, σ is not contained in Ker φ. Hence φ◦σ : Gm → Gm is a nontrivial homomorphism. Then it is surjective, whence so is φ. 3.1.5

tom Dieck construction of affine pseudo-planes

Affine pseudo-planes of type (d, r) are observed in tom Dieck [14] as affine surfaces without the cancellation property. His construction is based on a result corresponding to Lemma 3.1.13. In order to state tom Dieck’s result which enables us to write the universal covering of an affine pseudo-plane of type (d, r) as an affine hypersurface in A3 , we write the Hirzebruch surface Fn of order n ≥ 0 as the quotient space of A2∗ × P1 under a Gm -action, where A2∗ = A2 \ {(0, 0)}. Let (x0 , x1 ) be a system of coordinates of A2 and let (w0 : w1 ) be a system of homogeneous coordinates of P1 . Then the Gm -action on A2∗ × P1 is given by λ((x0 , x1 ) × (w0 : w1 )) = (λx0 , λx1 ) × (λn w0 : w1 ),

λ ∈ Gm (k).

Then Fn ∼ = F := (A2∗ × P1 )/Gm . Since A2∗ × P1 is not an affine variety, the quotient space must be constructed by patching the quotient spaces of affine Gm -stable open sets of A2∗ × P1 modulo the Gm -action. For the details, see Problem 5 at the end of the chapter. The standard P1 -fibration ρ : Fn → P1 is obtained by taking the Gm -quotients of the first projection p1 : A2∗ × P1 → A2∗ ,

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which is Gm -equivariant, i.e., the projection commutes with the Gm -actions and p1 induces a morphism p1 /Gm : Fn → P1 . With the above notations w1 = 0 (resp. w0 = 0) determines the minimal section M0 (resp. a section M1 such that M0 ∩ M1 = ∅). Lemma 3.1.17. Let X be an affine pseudo-plane of type (d, r). Let n = |d − r|. Then the following assertions hold. (1) Assume that r < d. Then X = Fn \(M0 ∪Cd ), where Cd is an irreducible member of |M0 + dℓ0 | defined by w1 (a0 xd1 + a1 x0 xd−1 + · · · + ad xd0 ) + ad+1 xr0 w0 = 0, 1

(3.1)

where (a0 , a1 , . . . , ad , ad+1 ) ∈ Pd+1 with a0 ̸= 0. (2) Assume that r ≥ d and m = n (see Lemma 3.1.13). Then X = Fn \ (M1 ∪ Cd ), where Cd is an irreducible member of |M1 + dℓ0 | defined by w0 (a0 xd1 + a1 x0 xd−1 + · · · + ad xd0 ) + ad+1 xr0 w1 = 0, 1

(3.2)

where (a0 , a1 , . . . , ad , ad+1 ) ∈ Pd+1 with a0 ̸= 0. Proof. With the notations of Problem 5, the coordinates of the open sets F10 ∪ F11 of Fn are exhibited in the following picture: ℓ0 P1

x0 x1

M1

w0 xn 1 w1

xn 1 w1 w0

P0

Cd M0 .

x0 x1

(1) Assume that r < d. Set n = d − r. By Lemma 3.1.13, Cd is an irreducible member of |M0 + dℓ0 |. In the open neighborhood F10 , Cd is defined by an equation (  d )  r xn1 w1 x0 x0 x0 · a0 + a1 + · · · + ad + ad+1 = 0, w0 x1 x1 x1

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where (a0 , . . . , ad ) ∈ Pd with a0 ad+1 ̸= 0. In fact, the equation obtained from the above one by multiplying xr1 is the equation (3.1) w1 (a0 xd1 + a1 x0 xd−1 + · · · + ad xd0 ) + ad+1 xr0 w0 = 0. 1 It defines the inverse image q −1 (Cd ) by the quotient morphism q : A2∗ × P1 → Fn , and hence it is Gm -homogeneous with respect to the weights (w(x0 ), w(x1 ), w(w0 ), w(w1 )) = (1, 1, n, 0). Since (a0 + (x0 /x1 ) + · · · + ad (x0 /x1 )d is a unit near the point P0 = M0 ∩ ℓ0 , we have a0 ̸= 0. We have the last term ad+1 (x0 /x1 )r with ad+1 ̸= 0 because i(M0 , Cd ; P0 ) = r and M0 + dℓ0 ∼ M1 + rℓ0 . (2) The case r ≥ d is treated in a similar way. Theorem 3.1.18. The following assertions hold true. e r) is isomorphic to an affine hypersurface (1) The universal covering X(d, in A3 = Spec k[x, y, z] defined by an equation xr z + (y d + a1 xy d−1 + · · · + ad−1 xd−1 y + ad xd ) = 1. (3.3) e r) → A1 (2) The projection (x, y, z) 7→ x induces an A1 -fibration f : X(d, such that every fiber except for f −1 (0) is smooth and the fiber f −1 (0) consists of d copies of A1 with multiplicity one. e r) defined by (3) There is a Ga -action on X(d, t · (x, y, z) = (x, y + txr , z − x−r {((y + txr )d + a1 x(y + txr )d−1 + · · · + ad xd )

−(y d + a1 xy d−1 + · · · + ad−1 xd−1 y + ad xd )}), where t ∈ Ga = k. (4) Let ω be a d-th root of unity. Then there exist uniquely determined polynomials pω (x), qω (x) ∈ k[x] satisfying the following conditions. (i) (ii) (iii) (iv)

deg pω (x) ≤ r − 1. pω (0) = ω. xr qω (x) + pω (x)d + a1 xpω (x)d−1 + · · · + ad−1 xd−1 pω (x) + ad xd = 1. pλω (λx) = λpω (x), qλω (λx) = λ−r qλ (x) for any d-th root λ of unity.

By making use of these polynomials, we define the morphism e r), (x, t) 7→ t · (x, pω (x), qω (x)) φω : A2 ∼ = A1 × Ga → X(d, which is an open immersion onto an open set Uω which is the comple` ment of γ̸=ω Ga · (0, γ, 0). The inverse morphism is defined by  y − pω (x)   ) if x ̸= 0   (x, xr (x, y, z) 7→     (0, −z + qω (0) ) if x = 0. dω d−1

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e r) is obtained by glueing together the d-copies of the affine plane (5) X(d, A2 by the transition functions   pω (x) − pλ (x) 1 1 1 1 . gλω := φ−1 ◦φ : A ×A → A ×A , (x, t) → 7 x, t + ω ∗ ∗ λ xr (6) The Galois group is a cyclic group H(d) := Z/dZ of order d and acts as λ · φω (x, t) = φλω (λx, λ1−r t). Proof. (1) The surface X(d, r) is the complement in Fn of the curves M0 (resp. M1 ) and Cd defined by the equation (3.1) (resp. (3.2)) if r < d (resp. r ≥ d). We consider the case r < d. The other case r ≥ d can be treated in a similar fashion. Since M0 is defined by w1 = 0, the surface X = X(d, r) satisfies the condition
a0 xd1 + a1 x0 xd−1 + · · · + ad−1 x0d−1 x1 + ad xd0 + ad+1 xr0 (w0 /w1 ) ̸= 0. 1 Namely there exists a morphism g : (A2∗ × P1 ) \ q −1 (M0 ∪ Cd ) → A1∗ which assigns to each point (x0 , x1 ) × (w0 : w1 ) the value of the left-hand side of the above relation. Since the left-hand side has weight d with respect to the Gm -action on A2∗ × P1 , the morphism g must be surjective. The fiber g −1 (1) is defined by (a0 xd1 + a1 x0 xd−1 + · · · + ad−1 x0d−1 x1 + ad xd0 ) + ad+1 xr0 (w0 /w1 ) = 1, 1 where a0 ad+1 ̸= 0. Further g −1 (1) has an induced action of H(d) := Z/dZ and g −1 (1)/H(d) ∼ = X(d, r). Since w1 ̸= 0, by setting x = x0 , y = x1 , z = w0 /w1 , we can normalize the above equation to the equation (3.3). The assertion (2) is easily checked. If we remove all components of f −1 (0) but one component, we obtain an open set isomorphic to A2 . Hence the affine hypersurface defined by the equation (3.3) is simply connected, e r) of X(d, r). H(d) is the Galois hence it is the universal covering X(d, e group of the covering X(d, r) → X(d, r) acting as λ · (x, y, z) 7→ (λx, λy, λn z) for λ ∈ H(d). In terms of φω ’s, it is written as in the assertion (6). If r ≥ d, e r) is given we take x = x0 , y = x1 , z = w1 /w0 . So the H(d)-action on X(d, by λ · (x, y, z) 7→ (λx, λy, λ−n z) for λ ∈ H(d).

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e r) such that (3) Let δ be a derivation on the coordinate ring of X(d, δ(x) = 0, δ(y) = xr , δ(z) = −(dy d−1 + (d − 1)a1 xy d−2 + · · · + ad−1 xd−1 ).

Then δ is a locally nilpotent derivation. Hence δ defines a Ga -action on e r) by the proof of Theorem 2.1.5, which is as given in the assertion. X(d, (4) Write pω (x) = ω + c1 (ω)x + · · · + cr−1 (ω)xr−1 , where the coefficients are to be determined by the relation xr qω (x)+ d

pω (x) + a1 xpω (x)

(3.4) d−1

+ · · · + ad−1 x

d−1

d

pω (x) + ad x = 1,

which is obtained from the equation (3.3) above by substituting pω (x), qω (x) for y, z. By the condition (i), it is easy to see that pω (x) is uniquely determined. Namely the coefficients c1 (ω), . . . , cr−1 (ω) are uniquely determined by putting the coefficients of the terms xi (1 ≤ i ≤ r − 1) to be zero in the left-hand side of the equation (3.4). Then qω (x) is uniquely determined as well. By multiplying λd = 1 to the relation (3.4), we obtain (λx)r λ−r qω (λ−1 (λx)) +(λpω (λ−1 (λx)))d + a1 (λx)(λpω (λ−1 (λx)))d−1 + · · · + ad (λx)d = 1.

Replace λx by x in the above relation. Then the uniqueness of the polynomials pλω (x), qλω (x) imply that pλω (x) = λpω (λ−1 x) and qλω (x) = λ−r qω (λ−1 x). Now replace λx by x. Then we obtain the relation (iv). Note that φω : A2 → Uω is injective and Uω ∼ = A2 . Hence φω is an isomorphism by [6]. The other assertions are verified in a straightforward manner. 3.1.6

Platonic A1∗ -fiber spaces

As explained in section 2.4, the affine normal surface Z := A2 /G for a small finite subgroup G of GL (2, k) has a non-trivial Gm action induced by the standard Gm -action on A2 . The Gm -orbits on X define an A1∗ -fibration on the minimal resolution Ze parametrized by a curve isomorphic to P1 , where A1∗ = Spec k[x, x−1 ]. Let f : X → B be an A1∗ -fibration. Namely, all but finitely many fibers of f are isomorphic to A1∗ . We summarize results on A1∗ -fibrations. For details, the readers are referred to [27]. Lemma 3.1.19. Let f : X → B be an A1∗ -fibration such that B is a smooth projective curve and f is surjective. Let φ : V → B be P1 -fibration from a log smooth completion V of X such that f = φ|X . Then the following assertions hold.

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(1) Let D be the boundary divisor of X in V . Then D contains either two cross-sections S1 , S2 or an irreducible component S such that (S ·ℓ) = 2 for a general fiber ℓ of φ. Let Xη be the generic fiber of φ, i.e., Xη = X ×B Spec k(B). In the first case, S1 ∩ Xη and S2 ∩ Xη give two k(B)rational points of Xη . In this case we call f an untwisted A1∗ -fibration. In the second case k(S) is a quadratic algebraic extension of k(B), i.e., [k(S) : k(B)] = 2. We then call f a twisted A1∗ -fibration. In the untwisted case, there is an open set U of B such that f −1 (U ) ∼ = U ×A1∗ . Furthermore, we may assume that S1 ∩ S2 = ∅. In the twisted case, there is an open set U of B such that (f |S )−1 (U ) is a smooth affine curve and X ×B (f |S )−1 (U ) ∼ ̸ U ′ ×A1∗ = (f |S )−1 (U )×A1∗ , but X ×B U ′ ∼ = ′ for any open set U of B. (2) Let Fj (1 ≤ j ≤ r) be all singular fibers of φ. Assume that every Fj is a linear chain with a unique (−1) component Ej which satisfies Ei ∩ X ̸= ∅. Then we have D + KV ∼ (r − 2)ℓ −

r X

Ej ,

j=1

where ℓ is a general fiber of φ. Let mj be the multiplicity of the component Ej ∩ X in the fiber Fj ∩ X. Write ℓ ∼ Fj = mj (Ej + ∆j ), where ∆j is an effective divisor with Q-coefficients. Then    r r   X X 1  ∆j , ℓ+ D + KV ∼Q (r − 2) −   mj j=1

which implies with m =

j=1

Qr

j=1 mj    r r   X X 1  m(D + KV ) ∼ m (r − 2) − ∆j ℓ+   mj j=1

j=1

as divisors with Z-coefficients. (3) With the above notations assume that m1 ≤ m2 ≤ · · · ≤ mr . Then the logarithmic Kodaira dimension κ(X) has value −∞, 0 or 1 if and only if Pr (r − 2) − j=1 1/mj is < 0, = 0 or > 0, respectively. Hence, if r = 1, 2 then κ(X) = −∞. Suppose that r ≥ 3. Then κ(X) = −∞ if and only if r = 3 and {m1 , n2 , m3 } = {2, 2, n} (n ≥ 2), {2, 3, 3}, {2, 3, 4} or {2, 3, 5}, κ(X) = 0 if and only if either r = 3 and {m1 , m2 , m3 } =

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{2, 3, 6}, {2, 4, 4}, {3, 3, 3} or r = 4 and {m1 , m2 , m3 , m4 } is one of the following: {2, 3, 7, 42}, {2, 3, 8, 24}, {2, 3, 9, 18}, {2, 3, 10, 15}, {2, 3, 12, 12} {2, 4, 5, 20}, {2, 4, 6, 12}, {2, 4, 8, 8}, {2, 5, 5, 10}, {2, 6, 6, 6}.

Otherwise, κ(X) = 1.

Proof. (1) For a general point b ∈ B, the fiber f −1 (b) is isomorphic to A1∗ and the fiber φ−1 (b) isomorphic to P1 . So, two points of φ−1 (b) lie outside of X. This implies that φ−1 (b) meets two irreducible components S1 , S2 of D or a single irreducible component S. Hence we have the untwisted or twisted case. The rest of the assertion can be verified straightforwardly (see [59, §1.6, Chapter 3]). (2) By contracting irreducible components of singular fibers Fj which are (−1) components or become subsequently (−1) components, we have a birational morphism σ : V → V0 := Fn the Hirzebruch surface of order n. Let S 1 = σ(S1 ) and S 2 = σ(S2 ). We can assume that (S 1 )2 = (S1 )2 . Namely, S1 is not affected by the above contractions. Then S 1 ∩ S 2 = ∅, whence one of the following two cases occurs: (1) One of S 1 and S 2 is a minimal section of Fn , i.e., (S 1 )2 = −n or (S 2 )2 = −n. (2) n = 0 and (S 1 )2 = (S 2 )2 = 0. This follows from Hodge index theorem (see Theorem 1.8.14). Reversing the contraction morphism σ, we obtain a sequence of blowing-ups which begin with the blowing-ups of r points P1 , . . . , Pr of S 2 , perform the blowing-ups with centers at infinitely near points of the points Pj and end up with the surface V . Let ℓj be a fiber of Fn through the point Pj . We show by induction on the number of blowing-ups with centers on Fj (and its infinitely near points) that the increment to D + KV by the special fiber Fj is σ ∗ (ℓj ) − Ej . We assume that j = 1. Let σs

σi+1

σ

1 1 Vs1 −→ Vs1 −1 −→ · · · −→ Vi+1 −→ Vi −→ · · · −→ V1 −→ V0

be the sequence of blowing-ups to regain the singular fiber F1 from the smooth fiber ℓ1 . Note that the fiber F1 is a linear chain by the assumption. Pr Let D0 = S 1 + S 2 + j=1 ℓi . Note that KV0 = −S 1 − S 2 − 2ℓ. Hence D0 +KV0 ∼ (r−2)ℓ. Let Ai be the exceptional curve of σi : Vi → Vi−1 . Note that KV1 = σ1∗ (KV0 ) + A1 . Since σ1 is the blowing-up with center S1 ∩ ℓ1 or S2 ∩ ℓ1 , we have D1 = σ1∗ (D0 ) − A1 if A1 ⊂ D1 and D1 = σ1∗ (D1 ) − 2A1

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if A1 ̸⊂ D1 . Hence D1 + KV1 = σ1∗ (D0 + KV0 ) ∼ (r − 2)ℓ if A1 ⊂ D1 and D1 + KV1 ∼ (r − 2)ℓ − A1 if A1 ̸⊂ D1 , where σ1∗ (ℓ) is identified with ℓ. Suppose that Di + KVi ∼ (r − 2)ℓ if Ai ⊂ Di and Di + KVi ∼ (r − 2)ℓ − Ai if Ai ̸⊂ Di . If Ai ̸⊂ Di the blowing-up process ends at this step. Suppose that Ai ⊂ Di . Since the blowing-up σi+1 has center at an intersection point ∗ of Ai and Di − Ai and Di+1 = σi+1 (Di ) − Ai+1 if Ai+1 ⊂ Di+1 and Di+1 = ∗ ∗ σi+1 (Di ) − 2Ai+1 if Ai+1 ̸⊂ Di+1 and since KVi+1 = σi+1 (KVi ) + Ai+1 , ∗ we have Di+1 + KVi+1 = σi+1 (Di + KVi ) ∼ (r − 2)ℓ if Ai+1 ⊂ Di+1 and Di+1 + KVi+1 ∼ (r − 2)ℓ − Ai+1 if Ai+1 ̸⊂ Di+1 . This process continues to the step i = s1 . Hence Ai ⊂ Di for 1 ≤ i < s1 , and at the step i = s1 we have As1 ̸⊂ Ds1 and hence Ds1 + KVs1 ∼ (r − 2)ℓ − E1 , where As1 = E1 . The above process is applied in the same way to other fibers ℓ2 , . . . , ℓr . So, we obtain the stated formula for D + KV . Pr (3) The divisor m( j=1 ∆j ) is an effective divisor whose support has negative-definite intersection form, the fixed part of the linear system    r r X X 1  N (r − 2) −  ∆ ℓ + j mj j=1 j=1 Pr with m | N contains m( j=1 ∆j ). Hence we have   r X 1  ℓ . dim |N (D + KV )| = dim N (r − 2) − mj j=1 The assertion (3) follows from this observation. For a precise computation, see Problem 6. Lemma 3.1.20. With the same notations as in Lemma 3.1.19, we assume that B ∼ = P1 , r = 3, (S12 ) ≥ −1 and κ(X) = −∞. Further, write Fj = Γj + mj Ej + ∆j (j = 1, 2, 3), where Γj , ∆j are connected and (S1 · Γj ) = 1 = (S2 · ∆j ). Then the following assertions hold. (1) Let dj be the absolute value of the determinant of the intersection matrix of ∆j . Then dj = mj and it is also the absolute value of the determinant of the intersection matrix of Γj . Furthermore, mj is equal to the number of irreducible components of Fj . P3 (2) The divisor S2 + j=1 ∆j,red is contracted to a quotient singular point Q in the sense of [10]. Denote by Vb the normal projective surface P3 obtained by contracting S2 + j=1 ∆j,red.

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P3 (3) Let X be the open set Vb \(S1 + j=1 Γj,red) of Vb . Then X is isomorphic to A2 /G with a small finite subgroup G of GL (2, k). Proof. (1) Let the weighted dual graph of Γj be a linear chain L with weights −a1 , . . . , −as from the left to the right. Then it corresponds to a continued fraction [a1 , . . . , as ] which is written as d(L) = [a1 , . . . , as ], d(L1 ) where d(L) (resp. d(L1 )) is the absolute value of the determinant of the intersection matrix of the linear chain L (resp. L1 ) and L1 is a subchain of L with the vertex of weight −a1 removed (see [59, Chapter 1, Lemma 3.3.1]). Assume that the linear chain L = Γj determines a fractional number nj /dj with gcd(nj , dj ) = 1 when the left most component of L meets the cross-section S1 . Then the linear chain ∆j determines a fractional number nj /(nj − dj ) when the right most component of ∆j meets the cross-section S2 . Here is a conceptual (not strict, but correct) proof. Consider W = A1 × P1 with A1 = Spec k[x] and P1 = Proj k[u, v] and set y = v/u. Let G = Z/nZ act on W by ζ(x, y) = (ζx, ζ d y). Then the projection p1 : W → A1 is given by (x, y) 7→ x. Then the G-action preserves the fibration p1 : W → A1 . Hence p1 induces a P1 -fibration p1 : W := W/G → A1 = Spec k[xn ]. Let q : W → W be the quotient morphism and let Q1 (resp. Q2 ) be the image by q of the point (x = 0) × (1 : 0) (resp. (x = 0)×(0 : 1)). Since (x, 1/y) is a system of local parameters at the point (x = 0) × (0 : 1), G acts as ζ(x, 1/y) = (ζx, ζ n−d (1/y)). Hence the point Q2 has a quotient singularity of type (n, n − d). Let F = p−1 1 (x = 0) and n F = p−1 (x = 0). Then F passes two quotient singular points Q1 and Q2 . 1 f be the minimal resolution of singularity of these points. Then W f Let W 1 n 1 f inherits the P -fibration p1 , which we denote by pe1 : W → A = Spec k[x ]. Then the fiber Fe looks like the fiber F1 . Namely Fe consists of Γ, mL, ∆. Let A1 , . . . , As be the irreducible components of Γ such that (A2i ) = −ai and let Bt , . . . , B1 be the irreducible components of ∆ such that (Bj2 ) = −bj , where n/(n − d) = [b1 , . . . , bt ]. Write Fe =

s X i=1

αi Ai + mL +

t X

βj B j ,

j=1

where (L2 ) = −1 because ai ≥ 2 and bj ≥ 2 for all i, j. Let S0 (resp. S∞ ) be the section of pe1 . Since (S0 · Fe) = (S∞ · Fe) = 1, and since (S0 · A1 ) =

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1 = (S∞ · B1 ), αi and βj are determined uniquely by the conditions that α1 = 1, αi−1 − ai αi + αi+1 = 0 (1 ≤ i ≤ s), α0 = 0, αs+1 = m, β1 = 1, βj−1 − bj βj + βj+1 = 0 (1 ≤ j ≤ t), β0 = 0, βt+1 = m, αs − m + βt = 0.

Then the theory of peeling (see [59]) shows that m = n. (See Problem 6 for a special case.) P3 (2) By Brieskorn [10, Satz 2.11], the intersection matrix of S1 + j=1 Γj P3 is not negative definite by the assumption (S12 ) ≥ −1. Since S1 + j=1 Γj P3 is disjoint from S2 + j=1 ∆j , Hodge index theorem implies that the interP3 section matrix of S2 + j=1 ∆j is negative definite. Hence (S22 ) ≤ −2 and P3 S2 + j=1 ∆j is contracted to a quotient singular point. (3) Let X ◦ be the smooth part of X, i.e., X ◦ = X \ {Q}. Then it is shown that the fundamental group π1 (X ◦ ) is a small finite subgroup G of GL (2, k) and the universal covering is A2 \ {(0)} → X ◦ (see [59, Chapter 3, Theorem 2.5.3]). This implies that X ∼ = A2 /G. The algebraic surface X ◦ with an A1∗ -fibration ρ := φ|X ◦ : X ◦ → P1 is called a Platonic A1∗ -fiber space after the Platonic triplets {2, 2, n}, {2, 3, 3}, {2, 3, 4}, {2, 3, 5}. The fibration ρ has three irreducible multiple fibers whose multiplicities form one of the Platonic triplets up to permutations. The following result is one of the fundamental theorems in affine algebraic geometry (see [59, Chapter 3, Theorem 2.5.4]). Theorem 3.1.21. Let V be a smooth projective surface and let D = Pr i=1 Ci be an SNC divisor such that the intersection matrix ((Ci · Cj ))1≤i,j≤r is not negative definite. Let X = V \ D. Assume that κ(X) = −∞ and X has no cylinderlike open sets. Then there exists an open set U of X and a proper birational morphism φ : U → T ′ onto a smooth algebraic surface T ′ such that (i) Either U = X or X \ U has pure codimension one, (ii) T ′ is an open set of a Platonic A1∗ -fiber space T with dim T \ T ′ ≤ 0. There are many applications of Theorem 3.1.21. Among them, we pick up one result which will be used in the next section. Theorem 3.1.22. Let the additive group scheme Ga act non-trivially on the affine 3-space A3 . Then the Ga -invariant subring Γ(A3 , OA3 )Ga is an affine domain, and isomorphic to a polynomial ring in dimension

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two. Namely, the quotient space A3 /Ga exists and the quotient morphism q : A3 → A3 /Ga is surjective. 3.1.7

Homology planes

In the present subsection the ground field k is the complex number field. Given an algebraic variety X defined over C, the set X(C) is viewed as an analytic variety in the natural way. We denote X(C) by the same letter X. With the complex metric topology on X, we can view X a topological space and use the homology or cohomology theory. See Spanier [91] for the knowledge of topological terms and results. Let X be a smooth affine surface defined over C. X is a homology plane (resp. Q-homology plane) if Hi (X; Z) = 0 (resp. Hi (X; Q) = 0) for every i > 0. Since Hi (X; Q) = Hi (X; Z) ⊗Z Q, a homology plane is a Q-homology plane, but not vice versa. If X is a normal affine surface admitting at worst quotient singularities and satisfying the same homology conditions as above, we call X a log homology plane (resp. log Q-homology plane. These are important classes of plane-like algebraic surfaces, and there is an account for this topic in [59, §4, Chap. 3]. We only extract some results which are related to the forthcoming topics. The following is a Theorem of Gurjar-Pradeep-Shastri [28]. Theorem 3.1.23. A log Q-homology plane is a rational surface. Lemma 3.1.24. Let X be a Q-homology plane and let V be a log smooth completion of X with the boundary divisor D. Then the following assertions hold. (1) The irregularity q(V ) = 0 and the geometric genus pg (V ) = 0. This follows from Theorem 3.1.23. (2) D is simply connected, i.e., every irreducible component is isomorphic to P1 and the dual graph of D is a tree. ∗ ) = C∗ and Pic (X) ∼ (3) Γ(X, OX = H1 (X; Z) is a finite group. (4) Homology groups are obtained as follows. H0 (X; Z) = Z, H1 (X; Z) ∼ = H 2 (X; Z) ∼ = Coker (H 2 (V ; Z) → H 2 (D; Z)), Hi (X; Z) = 0 (i ≥ 2).

The higher homology group Hi (X; Z) (i ≥ 3) vanishes by Theorem of Kaup-Narasimahan-Hamm [27, Theorem 1.1.4].

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Consider a smooth projective plane curve C of degree d on P2 . Let X = P2 \ C. For a pair (P2 , C), we have the following exact sequence of integral cohomologies, i.e., cohomologies with Z-coefficients 0 → H 0 (P2 , C) → H 1 (P2 , C) → H 2 (P2 , C) → H 3 (P2 , C) → H 4 (P2 , C)

→ H 0 (P2 ) → → H 1 (P2 ) → → H 2 (P2 ) → → H 3 (P2 ) → → H 4 (P2 ) →

H 0 (C) H 1 (C) H 2 (C) 0 0.

∼ Z if i = 0, 2, 4 and H i (P2 ) = (0) if i = 1, 3 and that Noting that H i (P2 ) = H 0 (C) ∼ = H 2 (C) ∼ = Z and H 1 (C) ∼ = Z2g , we deduce that H 0 (P2 , C) ∼ = H 1 (P2 , C) = (0), H 2 (P2 , C) ∼ = Z2g , H 3 (P2 , C) ∼ = Z/dZ, H 4 (P2 , C) ∼ =Z, where g = 21 (d − 1)(d − 2) is the genus of the curve C. Then, by the Lefschetz duality Hi (X) ∼ = H 4−i (P2 , C), 0 ≤ i ≤ 4, we obtain H0 (X) ∼ = Z, H1 (X) ∼ = Z/dZ, H2 (X) ∼ = Z2g , H3 (X) ∼ = H4 (X) ∼ = (0). Hence X is a homology (resp. Q-homology) plane if and only if d = 1 (resp. d = 2). Furthermore, it is known by Zariski’s theorem (see [80, Corollary 2.8]) that the complement X = P2 \ C of a curve C on P2 has an abelian fundamental group if the irreducible components of C meet each other transversally; we allow the irreducible components to have the nodal singularities.2 Hence π1 (X) ∼ = H1 (X) ∼ = Z/dZ. If we admit the curve C to have singularities or to split into several components, the homology groups of X = P2 \ C change accordingly. We exhibit the change in the case d = 3, i.e., the case of degeneration of an elliptic curve. Since H0 (X) ∼ = Z and H3 (X) ∼ = H4 (X) ∼ = (0) for each case of degeneration, we only trace the change of H1 (X) and H2 (X) in the following table: 2 An

irreducible curve C has a nodal singular point P if C has two local branches at P meeting each other transversally. Meanwhile, a singular point on C has a cuspidal e → C has only one point lying over P . In the nodal singularity if the normalization ν : C case, ν −1 (P ) has two points.

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degeneration smooth nodal cuspidal 3 lines 3 confluent lines 2ℓ1 + ℓ2 3ℓ H1 (X) H2 (X)

Z/3Z Z/3Z Z2

Z

Z/3Z

Z2

Z2

Z

0

0

Z

0

0

0

Here ℓ or ℓi signifies a line. Note that X is a smooth affine complex surface, hence Hi (X) ∼ = (0) for i > dim X by Theorem of Kaup-NarasimahanHamm. The logarithmic Kodaira dimension of a Q-homology plane X can be any value of −∞, 0, 1, 2. Except for the case κ(X) = 2, we know a rough structure theorem. We refer the interested readers to the references [59], [81]. Theorem 3.1.25. Let X be a Q-homology plane with κ(X) = −∞. Then X has an A1 -fibration ρ : X → A1 such that for every point P ∈ A1 the fiber ρ−1 (P )red is isomorphic to A1 . If ρ has no multiple fibers then X ∼ = A2 . Otherwise, let F1 = µi Ci (1 ≤ i ≤ r) exhaust all multiple fibers, where Qr µi ≥ 2 and Ci ∼ = A1 . Then H1 (X; Z) ∼ = i=1 Z/µi Z. Note that an affine pseudo-plane is a Q-homology plane, and the quotient space A2 /G for a small finite subgroup G ⊂ GL (2, C) is a log Qhomology plane.

Geometry and Topology of Polynomial Rings

3.2

345

Jacobian conjecture and related results

3.2.1

Jacobian conjecture and its variants

In this section, the ground field is an algebraically closed field k of characteristic zero. In subsection 3.2.1, we gather together well-known results on the Jacobian conjecture. In subsection 3.2.2, we generalize the Jacobian conjecture to a problem for all algebraic varieties, which is to be called the Generalized Jacobian conjecture. In subsection 3.2.3, we explain some of affirmative results. Whenever we consider an endomorphism φ : Xu → Xℓ , we tacitly assume that φ is a dominant morphism. The following is a renowned Jacobian Conjecture. Conjecture 3.2.1 (JCn ). Let F = {f1 , . . . , fn } be a set of elements of a polynomial ring k[X] := k[x1 , . . . , xn ] in n variables X = {x1 , . . . , xn }. Suppose that the Jacobian determinant ∂f 1 · · · ∂f1 ∂x1 ∂xn ∂f2 ∂f2 · · · ∂x ∂x n ∈ k∗ . 1 J(F ; X) = ··· ··· ··· ∂fn ∂f ··· n ∂x1

∂xn

Then k[x1 , . . . , xn ] = k[f1 , . . . , fn ]. We also denote the Jacobian determinant by   f1 , . . . , f n J . x1 , . . . , x n In geometric terms, the conjecture can be expressed as follows. Define a polynomial endomorphism F : An → An by (x1 , . . . , xn ) 7→ (f1 (x1 , . . . , xn ), . . . , fn (x1 , . . . , xn )). We also denote this morphism by F = (f1 , . . . , fn ). If F is an automorphism of the affine space An , there exists another polynomial endomorphism G = (g1 , . . . , gn ) : An → An such that G ◦ F = id and F ◦ G = id. By a theorem of Ax (see Theorem 3.2.3), a polynomial endomorphism F = (f1 , . . . , fn ) : An → An is an automorphism if there exists a polynomial endomorphism G = (g1 , . . . , gn ) : An → An such that G ◦ F = id. In fact, the last relation implies that F is injective, hence an automorphism. Denote by J(G ◦ F ; F ) the Jacobian determinant J(G; X) with the set of polynomials F = (f1 , . . . , fn ) substituted for the variables X = (x1 , . . . , xn ) in J(G; X).

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Affine Algebraic Geometry

Partial differentiation of a composite function then yields the following relation J(G ◦ F ; X) = J(G ◦ F ; F ) · J(F ; X) = 1. Hence the Jacobian determinant J(F ; X) is an invertible element of C[x1 , . . . , xn ]. Since k[x1 , . . . , xn ]∗ = k ∗ , it follows that if a polynomial endomorphism F : An → An is an automorphism, then the Jacobian determinant J(F ; X) is a nonzero element of k. We often say that an element of k ∗ is a nonzero constant when an element f ∈ k[x1 , . . . , xn ] is viewed as a polynomial function on An . The Jacobian Conjecture asserts that the converse of this result holds. Conjecture 3.2.2 (the second form). A polynomial endomorphism F = (f1 , . . . , fn ) : An → An is an automorphism if the Jacobian determinant J(F ; X) is a nonzero constant. The following is a Theorem of Ax [6]. Theorem 3.2.3. An injective endomorphism φ : X → X of an algebraic variety X is an automorphism. The conjecture (JC1 ) holds because the condition ∂f /∂x ∈ k ∗ for a polynomial in one variable x f = a0 xm + a1 xm−1 + · · · + am ,

a0 ̸= 0

implies that deg f = 1. In the case n ≥ 2, the conjecture remains unsolved. Set yi = fi (x1 , . . . , xn ) (i = 1, . . . , n), where we assume that fi (0, . . . , 0) = 0. With the condition J(F ; X) ∈ k ∗ , one can express x1 , . . . , xn as formal power series in y1 , . . . , yn . In fact, we can determine the coefficients of the terms of a formal power series φi by the method of undetermined coefficients xi = φi (y1 , . . . , yn ) =

∞ X

cα y α

|α|=0

after substituting fi (x1 , . . . , xn ) for yi , where y α = y1α1 · · · ynαn for α = Pn (α1 , . . . , αn ) and |α| = i=1 αi . If k = C, by the inverse mapping theorem, the formal power series φi (y1 , . . . , yn ) is viewed as a holomorphic function in variables y1 , . . . , yn in a small open neighborhood of the origin in the Y = (y1 , . . . , yn )-space. Namely, the polynomial endomorphism F induces a local isomorphism between the small open neighborhoods of the origin in bX,0 ∼ bY,0 . This holds even if the X and Y spaces. Hence it follows that O =O

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k ̸= C. Let P = (a1 , . . . , an ) ∈ An and let Q = F (P ) = (b1 , . . . , bn ). After the change of variables xi = x′i + ai , yi = yi′ + bi , ′ ′ ′ we have yi = Fi (x1 , . . . , x′n ) and J(F ′ ; X ′ ) = J(F ; X) ∈ k ∗ . Hence the polynomial endomorphism F : An → An induces a local isomorphism between the completions of the local rings OAn ,P and OAn ,Q . This also follows from the observation that if J(F ; X) ∈ k ∗ , the polynomial endomorphism F : An → An induces an isomorphism TF : TAn ,P → TAn ,Q of the tangent spaces or an isomorphism dF : Ω1An ,Q → Ω1An ,P of the cotangent spaces. This implies that the morphism F is quasi-finite. Namely, for any point Q ∈ An , the inverse image F −1 (Q) = ∅ or dim F −1 (Q) = 0. In fact, if there is a subvariety Z ⊂ An with dim Z > 0 such that F (Z) = Q then take P ∈ Z as a smooth point. Then TZ,P ⊂ TAn ,P is mapped to 0 by TF , which is a contradiction. Hence the condition J(F ; X) ∈ k ∗ is equivalent to the condition that F : An → An is unramified (or even more ´etale3 ). Although F is locally an isomorphism, one cannot conclude that the endomorphism F gives an isomorphism between An and its image F (An ). There might be two or more points P1 , . . . , Pd of An mapped to the same image Q. If one takes a point Q to be a general point in F (An ) the number d is equal to the degree of the field extension k(x1 , . . . , xn )/k(f1 , . . . , fn ). A polynomial endomorphism F : An → An is called a local isomorphism. We also say that F is unramified or ´etale at every point of An . With this terminology, the Jacobian conjecture asserts equivalently the following: Conjecture 3.2.4 (the third form). A polynomial endomorphism F of An is an isomorphism if it is unramified everywhere on An . If F : An → An happens to be a finite morphism then F is an ´etale covering. Since An is simply connected, F becomes an isomorphism. So, the Jacobian conjecture is true if the following conjecture is true for X = An . Conjecture 3.2.5 (Generalized Jacobian Conjecture (GJC)). Let φ : X → X be an unramified endomorphism of an algebraic variety X. Then φ is a finite morphism. The Jacobian conjecture has no confirmed affirmative answer though there are many approaches and connections with difficult problems in unthinkably remote research areas. In this volume, we take the integral cloe = Spec A and sure A in k(x1 , . . . , xn ) of the ring k[f1 , . . . , fn ]. Let X 3 Since

F is a quasi-finite morphism between two smooth varieties, F is flat by Remark 1.9.9(4)(iii).

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e → An = Spec k[f1 , . . . , fn ] be the normalization morphism. Since A Fe : X is a subring of k[x1 , . . . , xn ], the finite morphism Fe splits the morphism F as e ι F e −→ F : Xu −→ X Xℓ ,

where X = An and the morphism ι is given by the inclusion A ,→ k[x1 , . . . , xn ] is an open immersion by Zariski’s main theorem (see Theorem 1.5.23). In order to distinguish the source from the target in an endomorphism F : X → X, we denote the source X by Xu and the target X by Xℓ . The subscript “u” (resp. “ℓ”) is for the upper (resp. the lower) e is a normal affine variety and X e \ X is pure one-codimensional. X. Then X e If X \ X ̸= ∅ then the conjecture (JCn ) fails to hold. Furthermore, some e \ X might ramify over Xℓ . We call this pheirreducible components of X nomenon the hidden ramification at infinity of the morphism F : Xu → Xℓ . The (GJC) aims to look at this phenomenon for an affine algebraic variety X in more general categories. We will see by examples that there are, in e \ X. With these setting we fact, the cases where X has the nonempty X can pose the following conjecture concerning the hidden ramification in the integral closure. Conjecture 3.2.6 (Extended Jacobian Conjecture (ExJC)). Let φ : X → X be an unramified endomorphism. Then the normalization e → X is an unramified morphism. morphism φ e:X The following implications are clear: (GJC) =⇒ (ExJC) =⇒ (JC). There are many partial results on (JCn ), especially on (JC2 ). We list some of well known results below, and the readers are referred to the book of Essen [17] for more comprehensive treatments. The following conjecture gives a link between (JC) and (GJC). Conjecture 3.2.7 (Equivariant Jacobian Conjecture (EqJC)). Let e be a simply connected smooth affine surface with Γ(X, e O e )∗ = k ∗ . Let X X e and let φ e → X e be an G be a finite group of automorphisms of X e : X ´etale endomorphism such that φ e ◦ σg = σg ◦ φ e for every g ∈ G, where e is the canonical injective group homomorphism. Then φ σ : G → Aut (X) e is an automorphism.

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e Let X = X/G. Then φ e induces an endomorphism φ : X → X, and φ e is an automorphism if and only if φ is an automorphism (see Lemma 3.3.4). e ∼ There is a partial affirmative result in the case X = A2 and G is a small finite subgroup of even order (see [63]). 3.2.2

Partial affirmative answers

We begin with a general result for (JCn ). Theorem 3.2.8. The following assertions hold for a polynomial endomorphism F : An → An with J(F ; X) ∈ k ∗ . (1) The image of the polynomial mapping F : An → An is a Zariski open set and contains all codimension one points. (2) If F is birational, i.e., k(x1 , . . . , xn ) = k(f1 , . . . , fn ) holds additionally, then (JCn ) holds. (3) More generally, if k(x1 , . . . , xn ) is a Galois extension of k(f1 , . . . , fn ), then (JCn ) holds. Proof. (1) If φ : Xu → Xℓ is an unramified endomorphism of a smooth algebraic variety, then φ is also a flat morphism (see Remark 1.9.9). Hence φ is an ´etale morphism and an open morphism (see Remark 1.9.9(2)). Furthermore, if Γ(X, OX ) is a UFD and Γ(X, OX )∗ = k ∗ , the image of φ contains all points of codimension one in Xℓ (see Problem 7). The assertion (1) follows from these observations. e be the normalization of Xℓ in the function field (2) Set X = An . Let X ι φ e e −→ k(Xu ). We have a splitting of the morphism Xu ,→ X Xℓ , where φ e is the normalization morphism. If φ(Xu ) contains all points of codimene \ ι(Xu ) does not contain any irreducible component of sion one of Xℓ , X codimension one because φ is birational. In fact, if D1 is an irreducible e \ ι(Xu ), then φ(D divisor of X e 1 ) has codimension one in Xℓ , hence there exists an irreducible divisor D2 in Xu such that the closure of φ(D2 ) is φ(D e 1 ). Namely there exist two points of codimension one of Xu lying over e = Xu because the generic point of φ(D e 1 ). This is a contradiction. Then X e \ Xu has pure codimension one. This means that φ : Xu → Xℓ is a finite X morphism, hence φ is an ´etale covering. Then φ is an isomorphism because X is simply connected. See also the proof of Lemma 3.3.1, the assertion (1). e → (3) Set X = An and consider the normalization morphism φ e : X Xℓ . Let G be the Galois group of the field extension k(Xu )/k(Xℓ ). Then e and φ the Galois group G acts on X, e is the quotient morphism, where

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e Xℓ = X/G. By the assertion (1), the image φ(Xu ) is a Zariski open set of Xℓ containing all codimension one points of Xℓ . Let P be a point e and let Q = φ(P ). Then G acts on φ of codimension one of X e−1 (Q) transitively (see Problem 9 of Chapter 1) and one point, say P ′ , of φ e−1 (Q) ∼ belongs to Xu . Then OX,P = OXu ,P ′ . Since OXu ,P ′ is ´etale over OXℓ ,Q , so e over OXℓ ,Q . Hence φ e is ´etale over Xℓ at all points of codimension is OX,P e e → Xℓ is one. By the purity of branch loci (see Lemma 3.2.9 below), φ e:X

an ´etale covering. Since X = An is simply connected, it follows that φ e is an isomorphism, hence so is φ.

Lemma 3.2.9 (Purity of branch loci). Let f : X → Y be a finite separable morphism4 from a normal algebraic variety X to a smooth algebraic variety Y . Suppose that f is unramified over Y at every point of codimension one of X. Then f is unramified over Y at every point of X. For the proof the readers are referred to [72, Theorem 41.1]. The (JC2 ) has affirmative results in more special cases (see van den Essen [17]). Theorem 3.2.10. Consider (JC2 ). Let deg f1 = m and deg f2 = n, where deg stands for the total degree with respect to variables x1 , x2 . Then the following assertions hold. (1) In one of the following cases, k[f1 , f2 ] = k[x1 , x2 ] holds (Nakai-Baba [79]): (i) Either m or n is a prime number. (ii) Either m = 4 or n = 4. (iii) m = 2p > n and p is an odd prime. (2) If either m or n is a product of at most two prime numbers, then k[x1 , x2 ] = k[f1 , f2 ] holds (Applegate-Onishi [4]). (3) If max(m, n) ≤ 100, then k[f1 , f2 ] = k[x1 , x2 ] holds (Moh [66]). (4) If C[x1 , x2 ] ⫌ C[f1 , f2 ], gcd(deg f1 , deg f2 ) ≥ 16 (Heitmann [33]).5 (5) With the affine plane A2 embedded into the projective plane P2 in the standard way, we denote the line at infinity by ℓ∞ . Denote by V (f1 ) the closure in P2 of the affine plane curve V (f1 ) defined by f1 = 0. Similarly, we define V (f2 ). Suppose that V (f1 ) ∩ ℓ∞ as well as V (f2 ) ∩ 4f

: X → Y is a separable morphism if k(X)/k(Y ) is a separable extension. reproved Moh’s criterion as above by a different method. See an interesting review by van den Essen in Mathematical Reviews MR1055020. 5 Heitmann

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ℓ∞ consists of a single point for every pair (f1 , f2 ) satisfying J(F ; X) ∈ k ∗ . Then (JC2 ) holds (Abhyankar [1]). Remark 3.2.11. (1) Nakai-Baba’s criterion (i), (ii) and (iii) is included in Applegate-Onishi’s criterion. In their proof, total degree of f1 and f2 with weights of x1 and x2 which are different from the standard weights w(x1 ) = w(x2 ) = 1 is used. Such weights are determined by looking at the slopes of the Newton polygon of f1 and f2 .6 With such weights, the terms of f1 and f2 with smaller total degree than the top ones in the standard weights appear as the top degree terms. Hence one can observe how the Jacobian condition affects the middle terms. Criteria (1) and (2) excludes many pairs of (deg f1 , deg f2 ) from the list of candidates of counterexamples. If deg f1 ≤ deg f2 ≤ 30, 45 cases among the total 465 cases remain undetermined. (2) If deg f1 ≤ deg f2 ≤ 100, the four cases (deg f1 , deg f2 ) = (48, 64), (50, 75), (56, 84), (66.99) are undetermined by Moh and Heitmann criteria. Moh showed by hand calculation that the (JC2 ) holds in these four cases. For a set of polynomials F = (f1 , . . . , fn ), define a derivation δi of the polynomial ring k[x1 , . . . , xn ] by 

i

∨



 f1 , . . . , h, . . . , fn  h 7→ δi (h) = J   x1 , . . . , x n by substituting h for fi . Proposition 3.2.12. Let F = (f1 , . . . , fn ) be a set of elements of a polynomial ring k[x1 , . . . , xn ] satisfying J(F ; X) = 1. Then the following conditions (1) and (2) are equivalent. (1) k[x1 , . . . , xn ] = k[f1 , . . . , fn ]. 6 n = 2, we often use x, y in place of x1 , x2 . Given a polynomial f = PIn the icase j i,j cij x y , the set Supp (f ) = {(i, j) ; cij ̸= 0} is called the support of f . In the first quadrant of the plane with coordinates (i, j), consider the smallest convex polygon containing the origin (0, 0) and the points of Supp (f ). We call it the Newton polygon of f and denote by N (f ). The following result is due to Abhyankar [1]. Lemma The Jacobian conjecture is equivalent to the following condition: For any pair of polynomials F = (f, g) satisfying the condition J(F ; X) ∈ k∗ , the Newton polygon N (f ) (or N (g)) is a triangle with three points (0, q), (0, 0), (p, 0) as summits, where p, q are non-negative integers.

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(2) There exists some i (1 ≤ i ≤ n) such that (i) δi is locally nilpotent. (ii) Ker δi is a polynomial ring k[y1 , . . . , yn−1 ] in (n − 1) variables i

containing a subring

∨ k[f1 , . . . , fi , . . . , fn ],

i

∨ k[f1 , . . . , fi , . . . , fn ]

and k[y1 , . . . , yn−1 ] ⊇

is an unramified extension by the condition

J(F ; X) = 1. i

(iii) (JCn−1 ) holds for k[y1 , . . . , yn−1 ] ⊇

∨ k[f1 , . . . , fi , . . . , fn ].

Proof. (1) ⇒ (2). We can take F as a set of coordinates of k[x1 , . . . , xn ]. Hence we may assume that xi = fi for 1 ≤ i ≤ n. Then δi = ∂/∂xi and i

∨

Ker δi = k[x1 , . . . , xi , . . . , xn ]. Then the three conditions (i), (ii), (iii) follow obviously. (2) ⇒ (1). We may assume that i = n. We set δ = δn . We can put Ker δ = k[y1 , . . . , yn−1 ] because Ker δ has dimension n − 1. Since δ(fn ) = 1, by the proof of Lemma 2.1.6 or Problem 8, we have Ker δ ⊇ k[f1 , . . . , fn−1 ],

k[x1 , . . . , xn ] = k[y1 , . . . , yn−1 , fn ]. Since the ring extension k[x1 , . . . , xn ] = k[y1 , . . . , yn−1 , fn ] ⊇ k[f1 , . . . , fn ] is unramified, it follows that the extension k[y1 , . . . , yn−1 ] ⊇ k[f1 , . . . , fn−1 ] is unramified.7 By the condition (iii), k[y1 , . . . , yn−1 ] = k[f1 , . . . , fn−1 ]. Then k[x1 , . . . , xn ] = k[y1 , . . . , yn−1 , fn ] = k[f1 , . . . , fn ]. Corollary 3.2.13. Let F = (f1 , f2 , f3 ) be a set of polynomial ring k[x1 , x2 , x3 ] such that J(F ; X) = 1. Define δi as in the above proposition. Then the following conditions are equivalent. (1) k[x1 , x2 , x3 ] = k[f1 , f2 , f3 ]. (2) δ3 is locally nilpotent and (JC2 ) for Ker δi ⊂ k[f1 , f2 ] holds. Proof. It suffices to show that the condition (2)-(ii) holds. This follows from Theorem 3.1.22. A is a B-algebra and x is a variable over A, then Ω1A[x]/B[x] ∼ = Ω1A/B ⊗A A[x]. 1 1 Hence ΩA/B = 0 if and only if ΩA[x]/B[x] = 0. So, A ⊃ B is unramified if and only if A[x] ⊃ B[x] is unramified. See subsection 2.7.1. 7 If

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The following result appears in Bass-Connell-Wright [7]. Corollary 3.2.14. Let F = (f1 , f2 ) be a pair of elements in k[x1 , x2 ] such that J(F ; X) = 1. We then call the pair a Jacobian pair. Then k[x1 , x2 ] = k[f1 , f2 ] if and only if δ1 or δ2 is a locally nilpotent derivation. Proof. The assertion follows from Proposition 3.2.12. In fact, Ker δ is a polynomial ring in one dimension over k if δ is a locally nilpotent derivation of k[x1 , x2 ]. In fact, Ker δ is an affine domain of dimension 1 which is a UFD with only constant units. Such a ring is necessarily a polynomial ring in one variable over k. So the condition (2)-(ii) is satisfied. The condition (2)-(iii) is also satisfied since (JC1 ) holds. Remark 3.2.15. (1) The Jacobian conjecture can be observed over a nonclosed field, e.g.,the real number field R. In the real case, by Pinchuk [82], there is an example of a pair of polynomials (f1 , f2 ) in R[x1 , x2 ] such that the morphism F = (f1 , f2 ) : R2 → R2 is a local isomorphism but not a homeomorphism, while the Jacobian determinant is not invertible but has no solutions in R2 . (2) There are many approaches to (JCn ) by observing the conditions on a set of polynomials F = (f1 , . . . , fn ) brought by the restriction J(F ; X) ∈ k ∗ . Since we are interested in the behavior of the morphism F at the boundary at infinity, we do not touch upon such results. The references [7, 17, 41, 62, 65] will give more informations on the conjecture.

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Generalized Jacobian conjecture — affirmative cases

The ground field k is an algebraically closed field of characteristic zero. We consider the generalized Jacobian conjecture (GJC) (see Conjecture 3.2.5) for a smooth algebraic variety X. 3.3.1

Results in arbitrary dimension

We begin with the following result, of which the first assertion is fully or partially given in Theorems 3.2.3 and 3.2.8. Lemma 3.3.1. Let φ : X → X be an ´etale endomorphism of a smooth algebraic variety. Then φ is an automorphism if one of the following conditions are satisfied: (1) φ is injective or birational. (2) The logarithmic Kodaira dimension κ(X) = dim X. (3) k = C, X is projective and has nonzero Euler number e(X), where P e(X) = i≥0 (−1)i dimQ Hi (X(C); Q) (see [59, Chapter 1, §1.9]). Proof. (1) If φ is injective, the assertion follows from Ax’s theorem (see Theorem 3.2.3). If φ is birational, Zariski’s main theorem (see Theorem 1.5.23) implies that φ is injective because X is smooth and φ is quasifinite. Hence the assertion follows from Ax’s theorem. (2) The assertion follows from [38]. (3) Let d := deg φ. Since φ is a finite morphism if X is projective (see Problem 8), by the topological argument using a triangulation of the complex space X(C), we have e(X) = de(X), where d is the degree of the morphism φ, i.e., d = [k(Xu ) : k(Xℓ )]. If e(X) ̸= 0 then d = 1. Namely φ is birational. Hence φ is an automorphism by (1). The following result due to Iitaka [39, Theorem 2] indicates that the (GJC) is mostly related to the case of κ(X) = −∞. Theorem 3.3.2. Let X be a smooth algebraic variety with logarithmic Kodaira dimension κ(X) ≥ 0. If φ : X → X is a dominant morphism, then φ is an ´etale endomorphism. Lemma 3.3.1, the assertion (2) and Theorem 3.3.2 are called Theorem of Iitaka. Example 3.3.3. If G is a commutative group variety with the product written additively, then the multiplication by m endomorphism for a

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positive integer m > 1 is a finite ´etale morphism, and it is an automorphism if and only if G is a vector group. Furthermore, κ(G) = 0 if and only if G has no unipotent subgroups. It is well-known that G has no unipotent groups if and only if G is an extension of an abelian variety by an algebraic torus. G is then called a semi-abelian variety. If k = C, appearance of a finite ´etale endomorphism as the multiplication by m endomorphism for an abelian variety is related to the group structure on G = Cn /L which comes from a lattice structure L on the universal covering space Cn . The following lemma allows us to assume that X is simply connected in (GJC) in many cases. Lemma 3.3.4. Let X be a normal algebraic variety and let φ : Xu → Xℓ be e→ an ´etale endomorphism. Assume that X has a universal covering f : X X with a finite group G as the fundamental group. Then φ lifts to an ´etale e →X e such that f ◦ ψ = φ ◦ f and ψ(gx) = χ(g)ψ(x) endomorphism ψ : X e for g ∈ G and x ∈ X, where χ : G → G is a group homomorphism. e →X e is given and satisfies Conversely, if an ´etale endomorphism ψ : X e where χ : the condition that ψ(gx) = χ(g)ψ(x) for g ∈ G and x ∈ X, G → G is a group homomorphism, then there exists an ´etale endomorphism φ : X → X such that f ◦ ψ = φ ◦ f . Furthermore, deg ψ = deg φ, and φ is a finite morphism if and only if ψ is a finite morphism. e and let Z be a connected component of Proof. Let W = (X, φ) ×X X W . Since the first projection p1 : W → X is a finite ´etale morphism, so is the restriction p1 |Z : Z → X. By Problem 11, there exists a finite ´etale e → Z such that f = (p1 |Z ) ◦ q. Let ψ = (p2 |Z ) ◦ q, where morphism q : X e is the second projection. Since p2 is an ´etale morphism, so p2 : W → X e → X. e By the construction, it is clear are p2 |Z and the composite ψ : X e Then (f ◦ ψ)(gx) = that φ ◦ f = f ◦ ψ. About the G-action, let x ∈ X. (φ ◦ f )(gx) = (φ ◦ f )(x) = (f ◦ ψ)(x), whence ψ(gx) = χ(g, x)ψ(x) for some χ(g, x) ∈ G. Since G is a finite group and x 7→ χ(g, x) induces a morphism χ(g, ·) : X → G, χ(g, x) is independent of x. We can write ψ(gx) = χ(g)ψ(x). Since χ(g1 g2 )ψ(x) = ψ((g1 g2 )x) = χ(g1 )ψ(g2 x) = χ(g1 )χ(g2 )ψ(x), it follows that χ(g1 g2 ) = χ(g1 )χ(g2 ). Namely χ : G → G is a group homomorphism. e →X e satisfies the condition Conversely, if an ´etale endomorphism ψ : X that ψ(gx) = χ(g)ψ(x) for some χ(g) ∈ G, then define φ : X → X by

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φ(x) = f (ψ(x)), where x is a point of X such that f (x) = x. If x′ ∈ X satisfies f (x′ ) = x, then x′ = gx for g ∈ G. Then f (ψ(x′ )) = f (ψ(gx)) = f (χ(g)ψ(x)) = f (ψ(x)). So, φ is well-defined. The endomorphism φ is ´etale because we have the following local isomorphisms ∼ bX,x ∼ be ∼ be bX,φ(x) . O =O =O X,x = OX,ψ(x) Since φ ◦ f = f ◦ ψ and deg f = |G|, we have |G| · (deg ψ) = deg(f ◦ ψ) = deg(φ ◦ f ) = |G| · (deg φ), whence we have deg ψ = deg φ. Suppose that φ is a finite morphism. Since φ ◦ f is then a finite morphism, it follows that ψ is a finite morphism. Conversely, suppose that ψ is a finite morphism. Then so is φ ◦ f = f ◦ ψ. Let U = Spec A be an affine open set of X such that A is a noetherian ring and ψ −1 f −1 (U ) e Since f is an ´etale Galois covering, the imis an affine open set Spec A. −1 −1 age f (ψ f (U )) is identified with ψ −1 f −1 (U )/G which is an affine open e Since A e is a finite Aset Spec B. We then have inclusions A ⊂ B ⊂ A. module and A is noetherian, B is a finite A-module. Hence φ is a finite morphism. 3.3.2 3.3.2.1

Results for surfaces Surfaces having A1 -fibrations

Now we consider several affirmative results for smooth surfaces X according to κ(X) though they tend to be more partial in the case κ(X) becomes lower. If κ(X) = −∞ we have the following result. Theorem 3.3.5. Let X be a smooth affine surface with κ(X) = −∞. Then any ´etale endomorphism φ : X → X is an automorphism in one of the following cases: (1) X is irrational but not elliptic ruled. (2) Γ(X, OX )∗ ̸= k ∗ , and rank (Γ(X, OX )∗ /k ∗ ) ≥ 2 if X is rational. Proof. (1) Since κ(X) = −∞, by Theorem 3.5.9 in the Appendix below, there exists an A1 -fibration f : X → B, where B, by the assumption, is a smooth curve of positive genus g > 1 and we may assume that B = f (X). Let φ : X → X be an ´etale endomorphism. Then there exists an endomorphism β : B → B such that f ◦ φ = β ◦ f . In fact, for a fiber component C of f , f (φ(C)) is a point of B because C is rational (see

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Lemma 2.1.8) and B is not rational.8 Since the genus g of B is greater than 1, κ(B) = 1. Hence β is an automorphism by Theorem of Iitaka (see Lemma 3.3.1). The automorphism group Aut (B) is a finite group (see [31, Chap. IV, Ex. 2.5]). Let N = |Aut (B)|. Note that φN is an automorphism of X if and only if so is φ. We may assume, after replacing φ by φN , that β = idB . Hence f = f ◦ φ. Now consider the generic fiber Xη of f with the generic point η of B. The endomorphism φ induces an ´etale endomorphism φη : Xη → Xη . Since Xη is isomorphic to A1K with the function field K = k(B), φη is an automorphism. This implies that φ is a birational morphism. Hence φ is an automorphism by Lemma 3.3.1. (2) Let A = Γ(X, OX ). Since κ(X) = −∞, X contains a cylinderlike open set U0 × A1 = Spec R[x], where U0 = Spec R is an affine normal curve. We have A ⊂ R[x] and A∗ ⊂ R∗ . Let R0 be the k-subalgebra of A generated by all elements of A∗ . Since rank A∗ /k ∗ < ∞ (see subsection 3.1.2), the algebra R0 is finitely generated over k. Let S be the normalization of R0 in A. We have S ⊆ R. Let B = Spec S and let f : X → B ⊆ B be the morphism induced by the injection S ⊂ A, where B = f (X). Since S ∗ ⊇ A∗ ⊋ k ∗ , it follows that κ(B) ≥ 0 (see Problem 17). Let F be a general fiber of f . By Kawamata’s addition theorem (see Theorem 3.5.13), we know that κ(F ) = −∞. Namely, F ∼ = A1 . Hence f is an A1 -fibration. Let φ : X → X be an ´etale endomorphism. Since B is irrational or B is a rational curve with at least three places at infinity, φ induces an endomorphism β such that f ◦ φ = β ◦ f . In fact, β is an automorphism by the Riemann-Hurwitz theorem. Namely, if d = deg β, we have −2 ≥ −2d + 3(d − 1), which gives d = 1. Note that if B is rational then B ∼ = P1 and B \ B has three or more points. An automorphism β extends to an automorphism β of B which permutes the finite set B \ B. Further, if β fixes pointwise the set B \ B then β = id. Hence β has finite order. The rest of the proof is the same as in the case (1). 3.3.2.2

Surfaces having A1∗ -fibrations

Consider next the case where f : X → B is an A1∗ -fibration. We need a result on what a singular fiber of such a fibration looks like. 8 Theorem

of L¨ uroth (see [57, Theorem 1.36]) implies that if there is a dominant morphism from a rational curve C to a curve B then B is a rational curve.

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Lemma 3.3.6. Let X be a smooth affine surface with an A1∗ -fibration f : X → B. Let F0 be a singular fiber of f . Then F0 has a decomposition F0 = Γ + ∆, where (1) Γ ∩ ∆ = ∅. (2) Γ = 0, Γ = mΓ1 with m ≥ 1 and Γ1 ∼ = A1∗ , or Γ = m1 Γ1 + m2 Γ2 , where mi ≥ 1, Γi ∼ = A1 for i = 1, 2, and Γ1 and Γ2 meet each other in one point transversally. (3) ∆ ≥ 0 and Supp ∆ is a disjoint union of curves isomorphic to A1 . Proof. Consider the untwisted case. Namely, in a log smooth completion V of X which has necessarily a P1 -fibration f : V → B such that f |X = f , the boundary D := V \ X contains two cross-sections S1 , S2 of f . We may −1 assume that the singular fiber F 0 = f (b0 ) with b0 = f (F0 ) meets S1 , S2 in two distinct points. We may assume that D contains no (−1)-curves whose contraction does not break the SNC condition of D. If F 0 links S1 , S2 by components contained in D then Γ = 0 and ∆ is a disjoint union of affine lines. Otherwise, (F 0 )red ∩ D is disconnected and connected by components having non-empty intersection with X. Write (F 0 )red ∩ D = D0u ∪ D0ℓ , where D0u (resp. D0ℓ ) is the connected component linked to S1 (resp. S2 ). We call D0u (resp. D0ℓ ) the upper part (resp. the lower part) of the boundary part of F 0 . Since X contains no complete curves, there are two cases; (1) there is a single irreducible component C of F 0 such that C meets both D0u and D0ℓ , (2) there are two irreducible components C1 , C2 of F 0 such that (C1 ·C2 ) = 1, C1 ∩ D0u ̸= ∅ and C2 ∩ D0ℓ ̸= ∅. In the case (1), Γ = mΓ1 , where Γ1 := C ∩ X and in the case (2), Γ = m1 Γ1 + m2 Γ2 , where Γi = Ci ∩ X for i = 1, 2. Now all other components of F 0 either meets or linked to D0u or D0ℓ . So, the components with non-empty intersection with X are isomorphic to A1 and form a disjoint union ∆. In the twisted case, there is a 2-section T instead of two cross-sections which is a smooth irreducible curve such that a fiber F of f meets T either two distinct points or in a single point with multiplicity 2, i.e., f |T : T → B is a double covering of smooth curves. If a singular fiber F 0 meets T in two points, we can argue as in the untwisted cases. If F 0 meets T in a single point P , blow up P and a point infinitely near to P so that the total transform of T + F 0 is an SNC divisor. We can show with ease that F0

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has a decomposition F0 = Γ + ∆, where Γ = 0. See also [59, Chapter 3, Theorem 1.7.3]. Lemma 3.3.7. Let f : X → B be an A1∗ -fibration from a smooth affine surface X to a smooth curve B and let φ : X → X be an ´etale endomorphism such that f ◦ φ = f and the image φ(Xu ) contains all codimension e → Xℓ be the norone point of Xℓ , i.e., codim X (X − φ(X)) ≥ 2. Let ν : X malization of Xℓ in the function field of Xu . Then the following assertions hold. e such that φ = ν ◦ ι. (1) There exists an open immersion ι : Xu ,→ X e (2) ν : X → Xℓ is an ´etale Galois covering of Xℓ with a cyclic group G of e is smooth. order n as the Galois group, where n = deg φ. Hence X (3) Let F0 = Γ + ∆ be a singular fiber of f . Then φ is finite over Γ, i.e., φ∗ (Γ) is G-stable, and φ is totally decomposable over ∆, i.e., the stabilizer group of each connected component of ∆ is trivial. Proof. First we assume that f : X → B is untwisted. (1) Let K be the function field of B and let Xη be the generic fiber of f . Then Xη = Spec K[x, x−1 ] because f is untwisted (see Lemma 3.1.19), and φ induces an ´etale K-endomorphism φK : Xu,η → Xℓ,η . Clearly, φK is given by a K-algebra endomorphism θK : x 7→ ax±n of K[x, x−1 ], where a ∈ K and n = deg φ and “±” means that either sign is possible. Let G be the group of the nth roots of the unity in k, which is a cyclic group of order n. The group G acts on Xu,η by (ζ, x) 7→ ζx, where ζ ∈ G, and Xℓ,η is clearly the quotient variety Xu,η /G. The G-action on Xu,η is extended e and Xu , by Zariski main theorem, to an action on the normalization X, e e is a G-surface, i.e., an is embedded into X as an open set. Note that X algebraic surface with a G-action. (2) Let b be a closed point of B such that the fiber F := f −1 (b) is a smooth fiber. Let O := OB,b and let XO := X ×B Spec O. Then we can choose the element x above so that XO = Spec O[x, x−1 ] and the induced endomorphism φO : XO → XO is given by an O-endomorphism x 7→ ax±n , where a ∈ O∗ . In fact, since f is untwisted, XO is viewed as the complement of two disjoint sections of P1O . So, the G-action extends eO = XO , where over XO and Xu,O /G = Xℓ,O . In this case, we have X e e XO = X ×B Spec O. Since X is smooth and codim X (X − φ(X)) ≥ 2, e → Xℓ is ´etale and purity of branch loci (see Lemma 3.2.9) shows that ν : X e is smooth. hence X (3) Now let F := f −1 (b) be a singular fiber and write F = Γ + ∆ as

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in Lemma 3.3.6. Then φ(F ) ⊆ F . Since φ preserves two points at infinity of general fibers of f , hence of the singular fiber F and since codim X (X − φ(X)) ≥ 2 by the hypothesis, it follows that φ∗ (Γ) = Γ and φ∗ (∆) = ∆ with multiplicities. In particular, φ is surjective, i.e., φ∗ (F ) = F . We claim that if Γ ̸= 0 then Γ is G-invariant. In fact, take a log smooth completion b →X e be a G-equivariant resolution p : V → B of f as follows. Let µ : X e b and that X b \ Xu of singularities of X such that Xu is an open set of X 1 b b is an SNC divisor. Then X has an A∗ -fibration f ◦ µ : X → B. We can find a G-equivariant log smooth completion p : V → B of this A1∗ -fibration b is an SNC divisor, B is a smooth such that V is G-equivariant, V \ X projective curve containing B as an open set and p|Xb = f ◦ µ. The Gequivariant completion is possible by Sumihiro’s theorem. Let Σ = p−1 (b), where b = f (F ). Then Σ ∩ Xu = F and Σ is G-invariant. If Γ were not G-invariant then the translation g ∗ Γ of Γ by some element g of G would be a divisor disjoint from Γ and Σ would therefore contain a loop. This is impossible. So, Γ is G-invariant. Now suppose ∆ ̸= 0, and let ∆1 be an irreducible component of ∆. Since ∆1 and φ(∆1 ) are isomorphic to A1 and since φ∆1 : ∆1 → φ(∆1 ) is an ´etale morphism, it is an isomorphism. Note that G acts transitively on the components of ν −1 (φ(∆1 )), where e → X is the normalization morphism, and that the isotropy group of ν:X ∆1 is trivial by the previous remark. Hence g(∆1 ) ̸= ∆1 for any non-unit e → X is ´etale above the singular fiber F . Thus, element g of G and ν : X e → X is a finite without using purity of branch loci, we know that ν : X ´etale morphism. Now we assume that f : X → B is twisted. Then there exists a double covering τ : B ′ → B such that fX ′ : X ′ → B ′ is an untwisted A1∗ -fibration, where X ′ is the normalization of the fiber product X ×B B ′ and fX ′ is a composite of the normalization morphism X ′ → X ×B B ′ and the projection −1 p2 : X ×B B ′ → B ′ . In fact, we take the restriction to f (B) ∩ T of the double covering f T : T → B in the proof of Lemma 3.3.6 as the double covering τ : B ′ → B. If τ is unramified over a point b ∈ B, it is clear that X ×B B ′ is smooth along fibers over the fiber f −1 (b). If τ ramifies over b, the inverse image of T on the normalization Ve ′ of V ×B T splits to two smooth irreducible components which cross normally at the point Q lying over b and the fiber of the induced P1 -fibration on Ve ′ passing through Q is the same as its image on V which is the fiber of f . Hence Ve ′ as well as X ′ is a smooth surface. The ´etale endomorphism φ : Xu → Xℓ induces an ´etale endomorphism φ′ : Xu′ → Xℓ′ such that fX ′ = fX ′ ◦ φ′ . f′ be the Furthermore, deg φ′ = deg φ and codim X ′ (X ′ −φ′ (X ′ )) ≥ 2. Let X

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f′ normalization of Xℓ′ in the function field of Xu′ . It is readily verified that X ′ ′ e ×B B . More precisely, X f has is the normalization of the fiber product of X f′ → X f′ induced by the involution of the double covering an involution i : X e is the quotient surface X f′ /⟨i⟩. As shown in the untwisted B ′ → B,9 and X ′ f f′ → X ′ is an case, X is smooth and the normalization morphism ν ′ : X ℓ ´etale Galois covering with a cyclic group G of order n = deg φ′ as the Galois group. Since the involution i commutes with the Galois group action, the assertions for φ : Xu → Xℓ follow from the corresponding assertions for φ′ : Xu′ → Xℓ′ . Let f : X → B be an A1∗ -fibration on the smooth affine surface X. Let F0 = Γ + ∆ be a singular fiber of f . We call F0 an A1∗ -singular fiber of the first kind, (of the second kind, or of the third kind) if Γred ∼ = A1∗ , 1 1 ∼ (Γred = A ∪ A or Γred = ∅), respectively. We also say that a singular fiber F0 is of simple type (resp. of non-simple type) if Γ = mC with C ∼ = A1∗ and m > 1 and ∆ = 0 (resp. otherwise). Lemma 3.3.8. Let X be a smooth affine surface with an A1∗ -fibration f : X → B and let φ : Xu → Xℓ be an ´etale endomorphism such that fXu = fXℓ ◦ φ. Then φ is an automorphism if ρ has an A1∗ -singular fiber of the second kind. Proof. We use the notations of Lemma 3.3.6 and its proof. Suppose that f has a singular fiber of the second kind F0 = Γ + ∆. Write Γ = a1 C1 + a2 C2 with C1 ∼ = A1 . Then C1 and C2 meet each other transversally in a = C2 ∼ −1 single point P . Since φ∗ (Γ) is G-stable, the fiber S := fX (fXℓ (F0 )) is an u ′ ′ 1 A∗ -singular fiber of the second kind S = Γ + ∆ , and Γ′ = b1 D1 + b2 D2 with D1 ∼ = D2 ∼ = A1 and D1 , D2 meeting transversally in a single point Q. Furthermore, the group G acts on D1 ∪ D2 freely because the morphism φ induces an ´etale finite morphism from D1 ∪ D2 onto C1 ∪ C2 . Nevertheless, it is clear that the point Q is fixed under the G-action. This implies that the order n of G equals to one. Since n = deg φ, φ is a birational morphism. e is isomorphic to Xℓ . Hence φ : Xu → Xℓ is an open immersion. Then X Then φ is an isomorphism by a theorem of Ax (see Theorem 3.2.3).

field extension k(B ′ ) ⊃ k(B) is an algebraic extension of degree 2, hence it is a Galois extension with Galois group ⟨i⟩ ∼ = Z/2Z. The birational automorphism i extends to an automorphism of B ′ such that B ′ /⟨i⟩ = B. An automorphism of order 2 is often called an involution. 9 The

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The following lemma is preparatory to Theorem 3.3.10. Lemma 3.3.9. Let π : X → Y be an ´etale finite Galois covering of a smooth algebraic variety Y with a cyclic group G of order n as the Galois group. Then there exists an invertible OY -module L such that L⊗n ∼ = OY Ln−1 ⊗i ∗ and X ∼ Spec ( L ) as Y -schemes. Furthermore, we have π (L) ∼ = = i=0 OX . Proof. Since the assertion is of local nature we may assume that Y is affine. So, let Y = Spec R and let X = Spec A, where we view R as a subalgebra of A. Since G is the kernel of the nth power morphism x 7→ xn of the multiplicative group scheme Gm,k , the group G is written as a group scheme in the following form: G = Spec k[t] with tn = 1, µ(t) = t ⊗ t, ε(t) = 1 and η(t) = t−1 , where µ, ε and η are respectively the comultiplication, the augmentation and the coinverse (see Example 2.4.1(2)). The action of G on X is given in terms of the following coaction: ∆ : A → A[t],

a 7→ ∆(a) =

n−1 X

∆i (a)ti .

i=0

The property that ∆ is a coaction is equivalent to the following properties: (1) The mapping ∆i defined by a 7→ ∆i (a) is a k-linear endomorphism of A. Pn−1 (2) ∆i ∆j = δij ∆j , where δij = 1 if i = j and 0 if i ̸= j, and i=0 ∆i = idA . (3) ∆i (a1 )∆j (a2 ) ∈ ∆i+j (A) for a1 , a2 ∈ A, where we replace i + j by an integer ℓ with 0 ≤ ℓ < n and ℓ ≡ i + j (mod n) if i + j ≥ n. Set Ai = ∆i (A) for 0 ≤ i < n. Then Ai is an R-module and A0 = R, which is the G-invariant subalgebra of A. In view of the above properties, we have Pn−1 A = i=0 Ai and Ai ·Aj ⊆ Ai+j . Now the property that π is ´etale and finite implies that A1 is a projective R-module of rank 1, Ai ∼ = A⊗i 1 (1 ≤ i < n) ⊗n ∼ and A1 = R. Conversely, if A1 is a projective R-module of rank 1 such Pn−1 ∼ that A⊗n etale R-algebra structure = R then A := i=0 A⊗i 1 1 is given an ´ ∼ if an isomorphism θ : A⊗n → R is assigned. The group G acts on A as 1 Pn−1 P n−1 and ζ is a primitive nth follows: ζ · ( i=0 ai ) = i=0 (ζ i ) · ai if ai ∈ A⊗i 1 ∗ ∼ ∼ root of the unity. Clearly, π L = OX because A1 A = A.

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Theorem 3.3.10. Let X be a smooth affine surface with an A1∗ -fibration f : X → B. Assume that Pic X = (0) and Γ(X, OX )∗ = k ∗ . Then any ´etale endomorphism φ : X → X is an automorphism provided f ◦ φ = f . Proof. Since Pic (X) = (0) and Γ(X, OX )∗ = k ∗ , it follows that the A1∗ fibration f is untwisted. In fact, suppose that f is twisted. Let p : V → B be a log smooth completion of f : X → B. Let D = V \ X. Note that D contains only one irreducible component H which is a 2-section of the P1 -fibration p, i.e., (H · F ) = 2 for a general fiber F of p, and all other components are fiber components of p. Since Pic (X) is the quotient group of Pic (V ) modulo the subgroup generated by irreducible components of D, the class [S] of the 2-section S of p remains in Pic (X) as a nonzero 2-torsion element, and hence Pic (X) is not zero. Since Pic (X) = (0) by the hypothesis, f must be untwisted. Furthermore, codim X (X − φ(X)) ≥ 2. This follows from Problem 7 of this chapter. e be the normalization of Xℓ in the function field of Xu . By Let X e → X is an ´etale finite Lemma 3.3.7, the normalization morphism ν : X Galois covering with a cyclic group of order n as the Galois group. By Lemma 3.3.9 there exists an invertible sheaf L such that L⊗n ∼ = OXℓ and Ln−1 e ∼ X = Spec ( i=0 L⊗i ). Since Pic X = (0) the invertible sheaf L is trivial. e is a disjoint union of n-copies of X. Since X is an dense open set Then X e we know that n = 1. Hence φ is an automorphism. of X, 3.3.2.3

Case of κ = 1

We begin with the following result. Lemma 3.3.11. Let X be a smooth affine surface and let φ : Xu → Xℓ be an ´etale endomorphism of X. Suppose that κ(X) = 1. Then there exist an A1∗ -fibration ρ : X → B and an automorphism β : B → B such that β ◦ ρ = ρ ◦ φ. Proof. There exist log smooth completions (Vu , Du ) and (Vℓ , Dℓ ) of Xu and Xℓ , respectively and a morphism Φ : Vu → Vℓ such that φ is induced by restricting Φ to the open set Xu of Vu . For every integer m > 0, the log ramification formula (see Appendix) implies |m(Du + KVu )| = |mΦ∗ (Dℓ + KVℓ )| + mRΦ ,

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where RΦ is the log ramification divisor of Φ which is an effective divisor. Since κ(X) = 1, the associated morphisms Φ|m(Du +KVu )| and Φ|m(Dℓ +KVℓ )| define respectively A1∗ -fibrations ρu : Xu → Bu and ρℓ : Xℓ → Bℓ by a structure theorem of Kawamata (see Theorem 3.5.11) if m ≫ 0. In fact, ρu and ρℓ coincide with the same A1∗ -fibration ρ : X → B. Let M (m(Dℓ + KVℓ )) be the k-vector space associated to the linear system |m(Dℓ + KVℓ )| and let {f0 , f1 , . . . , fN } be its k-basis. Then Φ|m(Dℓ +KVℓ )| is given by a rational mapping P ∈ Vℓ 7→ (f0 (P ), . . . , fN (P )) ∈ PN and its image is the curve B ℓ , which is the closure of Bℓ . Similarly, the associated vector space M (mΦ∗ (Dℓ + KVℓ )) has a k-basis {φ∗ (f0 ), . . . , φ∗ (fN )}, which might differ from the basis {f0 , . . . , fN } of M (m(Du + KVu )) by an automorphism β of PN . Thus we obtain the relation β ◦ ρ = ρ ◦ φ. Given an ´etale endomorphism φ : Xu → Xℓ with Xu ∼ = Xℓ ∼ = X, where e → Xℓ of Xℓ in the X being as above, consider the normalization ν : X e with function field of Xu . Since X is smooth, Xu is a Zariski open set of X e an open immersion ι : Xu ,→ X and φ = ν ◦ ι. We are interested in the e \ Xu . complement X Lemma 3.3.12. Let X be a smooth affine surface with κ(X) = 1 and e → Xℓ be the let φ : Xu → Xℓ be an ´etale endomorphism. Let ν : X normalization morphism as above. Then the following assertions hold: e is a smooth affine surface with κ(X) e = 1 and Xu is an open set of (1) X e Furthermore, ν : X e → Xℓ is an ´etale Galois covering with Galois X. group G ∼ = Z/nZ, where n = deg φ. e → Bu (2) The A1∗ -fibration fu : Xu → Bu extends to an A1∗ -fibration fe : X such that β ◦ fe = fℓ ◦ ν, where β : Bu → Bℓ is an automorphism of B. e \ Xu is a disjoint union of affine lines. The number N of irreducible (3) X e \ Xu is zero or given by the formula: components of X r r X X N= (ndi − d′i ) = (n − 1) di , i=1

i=1

where r is the number of singular fibers of f and di (resp. d′i ) is the number of irreducible components in f −1 (bi ) (resp. (β ◦ fu )−1 (bi ))) isomorphic to A1 with {b1 , . . . , br } exhausting all points of B such that f −1 (bi ) is a singular fiber. (4) Suppose that f : X → B has no singular fibers of non-simple type. Then φ is a finite ´etale endomorphism. Hence (GJC) holds for such a surface X. In particular, if X is an A1∗ -bundle over a smooth curve B then (GJC) holds.

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Proof. By Lemma 3.3.11, there exist an A1∗ -fibration f : X → B and an automorphism β of B such that f ◦ φ = β ◦ f . We set Y = Xu with an A1∗ -fibration g = β ◦ fu : Y → B. Then φ : Y → X satisfies g = f ◦ φ. We can apply the same proof as for Lemma 3.3.7, the assertion (2) to prove the assertions (1) and (2). (3) Let F0 = Γ + ∆ be a singular fiber of f lying over a point b ∈ B. Then fe−1 (b) = ν −1 (Γ) + ν −1 (∆) is a singular fiber of fe such that ν −1 (Γ) = φ−1 (Γ) as it is G-stable and ν −1 (∆) is totally decomposable. Let q := β −1 (p) and write f −1 (q) = Γ′ + ∆′ . Then Γ′ = ν −1 (Γ) and ν −1 (∆) is the G-translate of ∆′ . Hence we conclude by a fiberwise argument that e \Xu consists of connected components each of which is isomorphic to A1 . X Pr Now, by the above argument, we have N = i=1 (ndi − d′i ). Furthermore, since φ is ´etale and ν is finite and ´etale, φ induces a bijection between the connected components of ∆′ and ∆ by assigning to a component C ′ of ∆′ the subset {g(C ′ ) | g ∈ G} of ν −1 (∆) and then the image of these components by ν which is a single component of ∆. So, d′i = di . Since β is an automorphism, β induces a permutation of the set {b1 , . . . , br }. (4) By the equality in (3), we have N = 0 because all di is zero. Hence e = X. Namely, φ is a finite morphism. X The existence of an automorphism in Lemma 3.3.11 is sometimes annoying. The following result shows that we can assume β = idB by replacing φ by its power. We can allow this replacement since our goal is to show that φ is an automorphism and φ is an automorphism if and only if so is φn for some n > 0. Lemma 3.3.13. Let X be a smooth affine surface with κ(X) = 1, hence with an A1∗ -fibration f : X → B, and let φ : Xu → Xℓ be an ´etale endomorphism such that fℓ ◦ φ = β ◦ fu for an automorphism β of B. Then the following assertions hold: (1) φ sends a singular fiber of non-simple type to a singular fiber of nonsimple type. (2) After replacing φ by its iteration, we may view the automorphism β of B as the identity morphism in one of the following cases: (i) κ(B) = 1. (ii) κ(B) = 0 and there exists a singular fiber in the fibration ρ. Proof. There is no non-constant morphism from A1 to A1∗ (compare the unit groups). Hence it follows that φ sends any singular fiber of non-simple

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type to a fiber of non-simple type. Suppose that there exists a singular fiber of non-simple type. Let Tns be the set of points b of B such that f −1 (b) is a fiber of non-simple type. Then β induces a permutation of the set Tns . By replacing φ by φn (hence β by β n ), we may assume that β induces the identity automorphism on the subset Tns . If there are only singular fibers of simple type, the same argument works well for the set Ts because φ sends a multiple fiber to a multiple fiber, where Ts is the set of points b ∈ B such that f −1 (b) is a singular fiber of simple type. Consider the assertion (2). If κ(B) = 1 then B is a curve of log general type. Hence β is an automorphism of finite order. So, replacing φ and β by φn and β n , we may assume that β = idB . Suppose that κ(B) = 0 and there exists a singular fiber of non-simple type. Namely there exists a point b ∈ B such that f −1 (b) is a singular fiber of non-simple type. We may assume that β(b) = b. Then β induces an automorphism of B ′ := B \ {b}. Since κ(B ′ ) = 1 we are done by the previous case. If there are only singular fibers of simple type, we are done by the above remark. Lemma 3.3.14. With the same assumptions as in Lemma 3.3.13, we further assume that B is a rational curve and that rank Pic (X) = 0 and Γ(X, OX )∗ = k ∗ . Then the following assertions on the A1∗ -fibration f : X → B hold: (1) B is isomorphic to P1 or A1 . (2) If B ∼ = P1 then f is untwisted and every fiber of f is irreducible. Hence every fiber is isomorphic to A1∗ or A1 if taken with the reduced structure. (3) If B ∼ = A1 and f is untwisted, then every fiber of f except for only one fiber F0 is irreducible and isomorphic to either A1∗ or A1 if taken with the reduced structure. The unique reducible fiber F0 consists of two irreducible components C0 , C1 such that (i) Γ = C0 + C1 with C0 ∼ = C1 ∼ = A1 , (ii) Γ = C0 ∼ = A1∗ and ∆ = C1 , or (iii) Γ = 0 and ∼ ∼ ∆ = C1 + C2 with C1 = C2 = A1 . If B ∼ = A1 and f is twisted, every 1 1 fiber is isomorphic to A or A∗ if taken with the reduced structure. Proof. (1) Since B is rational by the hypothesis and since Γ(X, OX )∗ = k ∗ , we have B ∼ = P1 or B ∼ = A1 . (2) and (3) Let f : V → B be a log smooth completion of f : X → B, where V is a log smooth completion of X, B is a smooth projective curve which is a completion of B and f is a P1 -fibration such that f |X = f . If −1 B ∼ = A1 , we may assume that the fiber f (∞) is a smooth fiber, where {∞} = B \ B if B ̸= B. For a point b ∈ B, let Nb (resp. nb ) be the number

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−1

(b) (resp. f −1 (b)). Then we have X (Nb − 1). rank Pic (V ) = 2 +

of irreducible components of the fiber f

b∈B

Let D = V \ X and d be the number of irreducible components of D. If f is untwisted, X d=2+ε+ (Nb − nb ), b∈B

where ε is equal to 0 (resp. 1) if B = B (resp. B ̸= B). If f is twisted, X d=1+ε+ (Nb − nb ). b∈B

Since rank Pic (X) = rank Pic (V ) − d = 0 by the hypothesis Γ(X, OX )∗ = k ∗ , we conclude that if f is untwisted X (nb − 1) = ε, b∈B

and if f is twisted 1+

X b∈B

(nb − 1) = ε.

If B = B then ε = 0, whence f is untwisted and every fiber is irreducible. This is the assertion (2). If B ̸= B then ε = 1, whence either f is untwisted and the assertion (3) holds, or f is twisted and all fibers are irreducible. We refer to [53, Theorem 3.2] for the proof of the following result. Lemma 3.3.15. With the same assumptions as in Lemma 3.3.14, suppose further that B ∼ = P1 . Let m1 Γ1 , . . . , ms Γs , ms+1 ∆s+1 , . . . , ms+t ∆s+t exhaust all singular fibers of f , where Γi ∼ = = A1∗ (1 ≤ i ≤ s) and ∆j ∼ 1 A (s + 1 ≤ j ≤ s + t). We set r = s + t. Let φ be an ´etale endomorphism of X such that f · φ = β · f for an endomorphism β of B. Suppose r ≥ 3. Then β is an automorphism except when the multiplicity sequence is one of the following: {m1 , . . . , mr } = {2, 2, 2, 2}, {2, 3, 6}, {2, 4, 4}, {3, 3, 3}. Let pi = ρ(Γi ) and qj = ρ(∆j ) with the above notations. If β is an automorphism, then β induces a permutation on the finite set {p1 , . . . , ps , qs+1 , . . . , qr }. Hence, by replacing φ by its suitable power, we may assume that β is the identity morphism. Namely, we have f ◦ φ = f . Then Lemma 3.3.7 yields the following result.

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Theorem 3.3.16. Let X be a smooth affine surface with an A1∗ -fibration f : X → P1 . Suppose that κ(X) = 1, Pic (X) = 0 and Γ(X, OX )∗ = k ∗ . Suppose further that there are at least three singular fibers and that the multiplicity sequence of the singular fibers of f is none of the following: {m1 , . . . , mr } = {2, 2, 2, 2}, {2, 3, 6}, {2, 4, 4}, {3, 3, 3}. Then any ´etale endomorphism φ of X is an automorphism. Proof. Since κ(X) = 1, an ´etale endomorphism φ of X preserves the A1∗ fibration f . Namely there exists an automorphism β of the base curve P1 such that f ◦ φ = β ◦ f . After replacing φ by its power, we may assume that β is the identity morphism. The rest of the proof is the same as in Theorem 3.3.10.

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Generalized Jacobian conjecture for various cases

3.4.1

Case of Q-homology planes of κ = −∞

We assume that the ground field k is C. Recall that a Q-homology plane is a smooth affine surface X defined over k such that Hi (X; C) = 0 for i > 0. If κ(X) = −∞ then X has an A1 -fibration ρ : X → B (see Theorem 3.5.9). By Theorem 3.1.25, B ∼ = A1 and every fiber of ρ is irreducible. A fiber mA is called a multiple fiber if A ∼ = A1 and m ≥ 2. Let m1 A1 , . . . , mn An exhaust Qn all multiple fibers with mi ≥ 2 then Pic (X) ∼ = H1 (X; Z) ∼ = i=1 Z/mi Z. To develop the observations on (GJC) for Q-homology planes X of κ(X) = −∞, we need the following result of Bundgaard-Nielsen [11] and Fox [18]. Lemma 3.4.1. Let C be a smooth projective curve of genus g. Let n ≥ 1 and let b1 , b2 , . . . , bn be distinct points in C and m1 , m2 , . . . , mn arbitrary e → C such integers ≥ 2. Then there exists a finite Galois covering τ : C e that the ramification index at any point of C over bi is mi for i = 1, 2, . . . , n except when (1) g = 0 and r = 1, (2) g = 0, r = 2 and m1 ̸= m2 . If C ∼ = P1 , n = 2 and gcd(m1 , m2 ) = d > 1 then we can construct a finite e → P1 with the ramification index d at b1 , b2 . cyclic covering τ : C Using this lemma, we can show the next result. Lemma 3.4.2. Let ρ : X → B be an A1 -fibration on a Q-homology plane X. Suppose that ρ has at least two singular fibers, i.e., n ≥ 2. Let g : A1 → X be a non-constant morphism. Then the image of g is a fiber of ρ. Proof. Let mi Fi (1 ≤ i ≤ n) be all the singular fibers of ρ and let bi = ρ(Fi ). Then (Fi )red ∼ = A1 for each i and mi > 1. By assumption, n > 1. Using Lemma 3.4.1 to a smooth completion C := B, we can construct a e → B such that the ramification index at any point finite Galois covering B e→C over bi is mi . In fact, let C be a smooth completion of B and let τ : C −1 e e be the Galois covering in Lemma 3.4.1. Let B = C \ (τ (C \ B) and e → B. Since n > 1, it is easy to see that consider the restriction τ |Be : B e is a non-rational curve or an affine rational curve with at least 2 either B e be the normalization of the fiber-product X ×B B e places at infinity. Let X

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e → B e is also is an ´etale covering of X.10 The induced morphism ρe : X 1 an A -fibration. By the general theory of covering spaces the morphism e Since B e is either non-rational or has g lifts to a morphism ge : A1 → X. non-constant regular invertible functions, the image of ge is a fiber of ρe. It follows that the image of g is a fiber of ρ. Lemma 3.4.3. For i = 1, 2, let ρi : Xi → Bi be an A1 -fibration on Qhomology planes Xi . Let φ : X1 → X2 and β : B1 → B2 be dominant morphisms such that ρ2 · φ = β · ρ1 . Let mΓ2 be an irreducible fiber of ρ2 lying over a point b2 ∈ B2 with m ≥ 1 and Γ ∼ = A1 , and let b1 ∈ B1 be a ∗ point such that β(b1 ) = b2 . Suppose that ρ1 (b1 ) = ℓΓ1 , where Γ1 ∼ = A1 and ℓ ≥ 1. Suppose furthermore that φ is an ´etale morphism. If the ramification index of β at b1 is e, then ℓe = m. In particular, if m = 1 then ℓ = e = 1. Proof. Let ti be a uniformisant of OBi ,bi . Then β ∗ (t2 ) ∼ te1 . Choose points Pi ∈ Γi such that φ(P1 ) = P2 . Then there exists ξi ∈ OXi ,Pi such that Γi is defined by ξi = 0 at Pi . We have φ∗ ◦ ρ∗2 (t2 ) = ρ∗1 ◦ β ∗ (t2 ) ∼ ρ∗1 (te1 ) ∼ (ξ1ℓ )e ∼ φ∗ (ξ2m ) ∼ ξ1m ,

because φ is ´etale at P1 and maps Γ1 to Γ2 . Hence ℓ · e = m. Applying these lemmas, we have the following result. Lemma 3.4.4. Let X be a Q-homology plane with an A1 -fibration ρ : X → B. Let m1 A1 , . . . , mn An exhaust all multiple fibers of ρ, where Ai ∼ = A1 . Let φ : Xu → Xℓ be an ´etale endomorphism. Then the following assertions hold: (1) If n ≥ 2, then there exists an endomorphism β of B such that ρ ◦ φ = β ◦ ρ. (2) The above endomorphism β is an automorphism if either n ≥ 3 or n = 2 and {m1 , m2 } = ̸ {2, 2}. Proof. (1) The assertion is an immediate consequence of Lemma 3.4.2. (2) Note that β : B → B is a finite morphism because B is the affine line (see Problem 18). Let bi = ρ(Ai ). By Lemma 3.4.3, β maps the 10 Let t be a uniformisant of O e e t be a B,bi . Let b ∈ B be a point lying over b and e m , where m = m . Let P be a point of ρ−1 (b) e uniformisant of OB, . Then t = ( t ) i ee b and Γ := ρ−1 (b)red is defined by ξ = 0 at P . Then we have (e t)m ∼ ξ m . Then in the normalization, the inverse image of Γ splits into m copies of Γ because (ξ/e t)n = 1.

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set {b1 , . . . , bn } to the same set. Suppose furthermore that q1 , . . . , qs exhaust all the points, none of which belongs to {b1 , . . . , bn }, are mapped to {b1 , . . . , bn }. Then, by Lemma 3.4.3, the ramification index of β at qj , say ej , is larger than 1. In fact, if β(qj ) = bi then ej = mi . Then β induces an ´etale finite morphism β : B − {b1 , . . . , bn , q1 , . . . , qs } −→ B − {b1 , . . . , bn }. Hence the comparison of the Euler numbers yields an equality 1 − (n + s) = d(1 − n),

(3.5)

where d = deg β. On the other hand, by summing up the ramification indices, we have an inequality 2s + n ≤ dn.

(3.6)

So, by combining (3.5) and (3.6) together, we have an inequality 2(d − 1)(n − 1) = 2s ≤ (d − 1)n.

(3.7)

Suppose d > 1. Then n ≤ 2. Hence, if n ≥ 3 then d = 1 and β is an automorphism. Suppose that d > 1 and n = 2. Then the equality occurs in (3.7), and hence the equality occurs in (3.6). Namely, the ramification index ej at qj is 2 for all j, and s = d − 1. Since d > 1 implies s > 0, we may assume that q1 is mapped to b1 . Then m1 = e1 = 2. Suppose d ≥ 3. Then 2s = 2(d − 1) > d. Hence one of the qj is mapped to b2 , . . . , bn , say b2 . Hence m2 = 2. In this case, after a suitable change of indices, one of the following two cases is possible: (1) s = s1 + s2 = d − 1, and q1 , . . . , qs1 , b1 (or b2 ) (resp. qs1 +1 , . . . , qs , b2 (or b1 ) are mapped to b1 (resp. b2 ). (2) s = s1 +s2 , d = 2s1 = 2s2 +2, and q1 , . . . , qs1 (resp. qs1 +1 , . . . , qs , b1 , b2 ) are mapped to b1 (resp. b2 ). Finally, suppose that d = n = 2 and s = 1. Then we may assume that β(q1 ) = b1 and β(b1 ) = β(b2 ) = b2 . Then m2 = 2 as well by Lemma 3.4.3. So, if {m1 , m2 } = ̸ {2, 2}, then d = 1 and β is an automorphism. As a consequence of Lemma 3.4.4, we prove the following result. Theorem 3.4.5. Let X be a Q-homology plane with an A1 -fibration ρ : X → B. Let m1 A1 , . . . , mn An exhaust all multiple fibers of ρ. Suppose that either n ≥ 3 or n = 2 and {m1 , m2 } = ̸ {2, 2}. Then any ´etale endomorphism φ : X → X is an automorphism.

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Proof. By Lemma 3.4.4, there exists an automorphism β of B such that ρ◦ φ = β ◦ ρ. Since β is an automorphism, Lemma 3.4.3 implies that β induces a permutation of the finite set {b1 , . . . , bn }. By replacing β by its suitable power β r , we may assume that β fixes pointwise the set {b1 , . . . , bn }. Since n ≥ 2 and β (or rather the induced automorphism of a smooth completion B∼ = P1 of B) fixes the point at infinity b∞ , β is the identity automorphism. Let K = k(B) be the function field of B and let Xη be the generic fiber of ρ. Then Xη is isomorphic to the affine line over K, and φ induces an ´etale endomorphism φK of Xη . Since φK is then finite, φK is an automorphism. Hence φ is birational. Then Zariski’s main theorem implies that φ is an open immersion. Note that Pic (X)Q = 0 and Γ(X, OX )∗ = k ∗ . Suppose that X ̸= φ(X). Then X \ φ(X) has pure codimension one. Since Pic (X)Q = 0, there exists a regular function h on X such that the zero locus (h)0 of h is supported by X \ φ(X). Then φ∗ (h) is a non-constant invertible function on X, which contradicts the property Γ(X, OX )∗ = k ∗ . So, φ is an automorphism. 3.4.2

Counterexamples

A counterexample to the (GJC) is a pair of a smooth algebraic variety X and a non-finite ´etale endomorphism φ of X. It is clear that if (X, φ) is a counterexample to (GJC) then so is a pair (X × T, φ × idT ) for any smooth variety T . The following result gives a counterexample to the (GJC). Theorem 3.4.6. Let C be a smooth cubic curve in P2 and let X = P2 \ C. Then the following assertions hold: (1) κ(X) = 0 and Pic (X) ∼ = Z/3Z. (2) There exists a surjective, non-finite, ´etale endomorphism φ : Xu → Xℓ of degree 3. e be the normalization of Xℓ in the function field of Xu . Then (3) Let X e X is a smooth affine surface containing Xu as a Zariski open set, and e \ Xu is a disjoint union of six affine lines. X e → Xℓ be the normalization morphism. Then ν is ´etale and (4) Let ν : X ν|Xu = φ. So, the (ExJC) holds. Proof. We prove the assertion (2). The other assertions are easy to prove. Let π : W → P2 be a triple covering which ramifies totally over the curve C (see Theorem 2.7.17). Then W is a smooth projective surface with KW ∼ π ∗ (KP2 + 2H),

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where H is a hyperplane in P2 , i.e., a line. Then KW ∼ −π ∗ (H), hence 2 −KW is ample and (KW ) = 3. So, W is a del Pezzo surface of degree 11 3. The 27 lines on W are obtained as follows. The curve C has 9 flexes P1 , . . . , P9 .12 Let ℓi (1 ≤ i ≤ 9) be the tangent line of C at the point e be the inverse image of C on W and let Pei be a point on C e Pi . Let C (j) −1 lying over Pi . Then π (ℓi ) consists of three lines Li (j = 1, 2, 3) such (1) (2) (3) e meet each other transversally only in the point that Li , Li , Li and C (j) e Pi . Then {Li | 1 ≤ i ≤ 9, 1 ≤ j ≤ 3} are the 27 lines on W . Choose six disjoint lines L1 , . . . , L6 among them and contract them to obtain a b = ρ(C). e Then P2 \ C b is isomorphic birational morphism ρ : W → P2 . Let C to X 13 and ρ induces an isomorphism ∼ = e + L1 + · · · + L6 ) −→ b ρ′ : W \ (C P2 \ C. Let φ be the composite of ρ′

−1

and π|W \(C+L . Then φ : X → X e 1 +···+L6 ) e e e → X is is a required ´etale endomorphism. Let X = W \ C. Then ρ|Xe : X e \X the normalization morphism of X in the function field of W . Hence X consists of six disjoint affine lines.

We have another counterexample. Theorem 3.4.7. Let C be a smooth projective curve of genus g and let T = Spec C[ξ, ξ −1 ] be a one-dimensional algebraic torus. Let Q1 and Q2 be the points of T defined by ξ = 1 and ξ = −1, respectively. Let P1 and P2 be two distinct points of C. Let Y = C × T , let Ci = C × {Qi } and 11 A smooth projective surface W is a del Pezzo surface if −K W is an ample divisor. The 2 ). It is known by Manin [52] that 1 ≤ d ≤ 9 degree is the self-intersection number (KW and W is isomorphic to F0 (d = 8) or the blowing-up of P2 at 9 − d points in a general position and the number of (−1) curves is finite. If d = 3, W is isomorphic to a cubic hypersurface in P3 , a (−1)-curve in W is a line in P3 contained in W when W is identified with a cubic hypersurface, the number of (−1)-curves on W is 27, and the contraction of mutually disjoint six (−1)-curves on W gives back P2 . 12 A flex on a cubic curve C in P2 is a point P ∈ C such that the tangent line ℓ meets P C only at P , whence (ℓP · C) = i(ℓP , C; P ) = 3. 13 A closed immersion of an elliptic curve C into P2 is given by Φ 2 |3P | : C ,→ P with ′ ′ a closed point P . If f : C → C is an isomorphism of elliptic curves with P := f (P ), then there exists an automorphism g : P2 → P2 such that g · Φ|3P | = Φ|3P ′ | · f . Suppose that C (resp. C ′ ) is written as Y 2 Z = 4X 3 − g2 XZ 2 − g3 Z 3 (resp. Y ′ 2 Z ′ = 4X ′ 3 − g2′ X ′ Z ′ 2 − g3′ Z ′ 3 with respect to a system of homogeneous coordinates (X, Y, Z) (resp. (X ′ , Y ′ , Z ′ )) such that the point P (resp. P ′ ) is given as (0, 1, 0). Then the j-invariant of C (resp. C ′ ) is given as j = g23 (g23 − 27g32 )−1 (resp. j ′ = g ′ 32 (g ′ 32 − 27g ′ 23 )−1 ). Since j = j ′ , we have (g/ g2′ )3 = (g3 /g3′ )2 . Namely, g2 = cg2′ and g3 = dg3′ with c, d ∈ k∗ . p By a change of coordinates (X, Y, Z) 7→ (cX, c3 /dY, dZ), we may assume g2 = g2′ and g3 = g3′ . Then the complements P2 \ C and P2 \ C ′ are isomorphic to each other.

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let Ti = {Pi } × T for i = 1, 2. Let σ : Z → Y be the blowing-up of the points (P1 , Q1 ) and (P2 , Q2 ) and let Ei = σ −1 ((Pi , Qi )) (i = 1, 2). Let X = Z \ (σ ′ T1 + σ ′ T2 ), where σ ′ Ti is the proper transform of Ti by σ. Let q : X → T be a morphism induced by the projection C × T → T . Let g : T → T be the endomorphism defined by g ∗ (ξ) = ξ n for odd n > 2, let e = X ×T (T, g) and let qe : X e → T be the canonical projection. Then qe X ∗ has 2n singular fibers L1j = qe (Q1j ) and L2j = qe∗ (Q2j ) (1 ≤ j ≤ n), where Q1j and Q2j are defined respectively by ξ = ω j−1 and ξ = −ω j−1 with ω being a primitive nth root of unity. The fibers L1j and L2j have the same forms as the fibers L1 := q ∗ (Q1 ) and L2 := q ∗ (Q2 ), respectively. Write ∼ A1 and M1j L1j = M1j + ∆1j and L2j = M2j + ∆2j , where ∆1j ∼ = ∆2j P2= Pn e and M2j are considered as open sets of C. Let X1 := X \ ( i=1 j=2 ∆ij ). Then the following assertions hold: (1) X1 is isomorphic to X and the composite φ of the open immersion e and the canonical projection X e → X is a surjective nonX1 ,→ X finite ´etale endomorphism of degree n. (2) We have κ(X) = 1 if g > 0 and κ(X) = 0 if g = 0. If g = 0 then X∼ = F0 \ (D1 + D2 ), where F0 ∼ = P1 × P1 and D1 , D2 are the curves of type M + ℓ, where ℓ and M are fibers of two projections from F0 to P1 . (3) In the case g = 0, X is isomorphic to Spec k[x, y, z, z −1 ]/(xy = z 2 − 1). A surjective ´etale endomorphism φ : X : = Spec k[x′ , y ′ , z ′ , z ′

−1

2

]/(x′ y ′ = z ′ − 1)

→ X := Spec k[x, y, z, z −1 ]/(xy = z 2 − 1) is given by x = x′ , y = y ′ (z ′

2(n−1)

+ z′

2(n−2)

2

n

+ · · · + z ′ + 1), z = z ′ ,

where n is a positive integer. (4) X has an untwisted A1∗ -fibration ϕ : X → C induced by the projection Y → C. Proof. (1) The restriction q1 |X1 : X1 → T is a C-fibration with two reducible fibers L11 and L21 lying over the points ξ = 1 and ξ = −1, respectively. It is then clear that X1 ∼ = X. (2) Apply Theorem 3.5.12 to the fibration ϕ : X → C which is induced by the first projection p1 : Y → C. Then ϕ is an untwisted A1∗ -fibration. Since ϕ−1 (P ) ∼ = A1∗ for a general point P ∈ C, we have κ(X) ≤ κ(ϕ−1 (P ))+

Geometry and Topology of Polynomial Rings

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dim C = 1. Then, by Theorem 3.5.13, we have κ(X) ≥ κ(ϕ−1 (P )) + κ(C). Hence κ(X) = 1 if g > 0 and κ(X) ≥ 0 if g = 0. If g = 0 then X contains an open set U ∼ = A1∗ × A1∗ , whence κ(X) ≤ κ(U ) = 0. So, κ(X) = 0. Two curves D1 , D2 meet in two points P, Q. Blow up these two points and then remove the proper transforms of D1 , D2 and two exceptional curves. The resulting surface is isomorphic to X. (3) With the notations in Problem 22 of Chapter 2, the curve z = 0 is defined by 2U2 + V = 0, i.e., X1 Y0 + X0 Y1 = 0. This defines a curve D2 linearly equivalent to M + ℓ. Meanwhile, the surface Spec k[x, y, z]/(xy = z 2 − 1) is isomorphic to F0 \ ∆, where the diagonal ∆ is defined by X1 Y0 − X0 Y1 = 0. Call this curve D1 . Then the surface Spec k[x, y, z, z −1 ]/(xy = z 2 − 1) is isomorphic to F0 \ (D1 + D2 ). The rest is straightforward. (4) Clear. Remark 3.4.8. (1) The non-finite ´etale endomorphism given in the assertion (3) in Theorem 3.4.7 can be generalized to the universal coverings of some classes of affine pseudo-planes. Let X(d, r) be tom Dieck’s e r) affine pseudo-plane of type (d, r) (see Theorem 3.1.18) and let X(d, r d 3 be its universal covering which is defined by x z + y = 1 in A . Let X = Spec k[x, y, z, y −1 ]/(xr z + y d = 1). Define a surjective ´etale en−1 r d domorphism φ : X = Spec k[x′ , y ′ , z ′ , y ′ ]/(x′ z ′ + y ′ = 1) → X = Spec k[x, y, z, y −1 ]/(xr z + y d = 1) by   n d(n−1) d(n−2) d x = x′ , y = y ′ , z = z ′ y ′ + y′ + · · · + y′ + 1 . Then φ has degree n, and there are n2 − n affine lines missing in the above X for φ to be finite. (2) Dubouloz and Palka [16] showed that there are counterexamples to (GJC) of affine pseudo-planes of non-Gizatullin type, i.e., those with side-chains. We consider a Q-homology plane X with an A1 -fibration ρ : X → B such that ρ has two multiple fibers 2A0 , 2A1 . This is the case which we excluded in Theorem 3.4.5. Let f : V → B ∼ = P1 be an extension of ρ to a 1 P -fibration on a log smooth completion V of X. Let F0 , F1 be the fibers f −1 (b0 ), f −1 (b1 ), where bi = ρ(Ai ) for i = 0, 1. If the closures Ai are unique (−1)-components in F0 and F1 such that Ai ∩ X = Ai , then Fi,red \ Ai is a linear chain consisting of three rational (−2)-components, and Ai meets the middle component at points Ri which are not the intersection points with other components. Even though the fibers F1 and F2 are similar, the

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positions of the points R1 and R2 on the middle components might not be similar in the fibers F1 and F2 . If ρ′ : X ′ → B ′ is a Q-homology plane with two multiple fibers 2A′0 , 2A′1 , the surface ρ′ : X ′ → B ′ might not be isomorphic to ρ : X → B provided the positions of R1 and R2 are different between two surfaces. We say that multiple fibers 2A0 and 2A1 are equilateral if there exists a birational automorphism α of V such that α preserves the P1 -fibration f , α is regular on an open set f −1 (U ) of V containing F0 and F1 with an open set U ⊂ B, the restriction α = α|(X∩f −1 (U )) is an automorphism of X ∩ f −1 (U ) and α(F1 ) = F2 . Example 3.4.9. Let △P0 P1 P2 be the coordinate triangle of P2 with respect to a system of homogeneous coordinates (X0 , X1 , X2 ) (see Example 2.5.3). let C be a smooth conic defined by X0 X1 = X22 . Then C touches the lines ℓ0 , ℓ1 at the points P1 , P0 , and C intersects the line ℓ2 at P0 , P1 . Let P0′ (resp. P0′′ ) be infinitely near points of P0 (resp. P0′ ) lying on the curve C. Define P1 , P1′ , P1′′ in a similar way. Let σ : V → P2 be the blowing-ups of seven points P0 , P0′ , P0′′ , P1 , P1′ , P1′′ , P2 and let E0 , E0′ , E0′′ , E1 , E1′ , E1′′ , E2 be the (proper transforms) of the exceptional curves arising from the blowingups of P0 , P0′ , P0′′ , P1 , P1′ , P1′′ , P2 , respectively. Let F0 = ℓ′1 + E0 + 2E0′ + 2E0′′ and F1 = ℓ′0 + E1 + 2E1′ + 2E1′′ , where ℓ′i is the proper transform of ℓi for i = 0, 1. Then there exists a P1 -fibration f : V → ℓ′2 = σ ′ (ℓ2 ) induced by the linear pencil of lines on P2 through the point P2 such that F0 , F1 are singular fibers and the curve E2 is a cross-section. The proper transform C ′ := σ ′ (C) meets E0′′ and E1′′ in single points. Let D = (ℓ′1 + E0 + E0′ ) + (ℓ′0 + E1 + E1′ ) + E2 + ℓ′∞ , where ℓ′∞ is a smooth fiber and X = V \ D. Then ρ := f |X : X → B ∼ = A1 is an A1 -fibration with two multiple fibers 2A0 and 2A1 , where Ai = Ei′′ ∩ X for i = 0, 1. These multiple fibers 2A0 , 2A1 are equilateral by an automorphism α of X induced by the involution (X0 , X1 , X2 ) 7→ (X1 , X0 , X2 ) on P2 . Let ρ : X → B be an A1 -fibration with two equilateral irreducible ′ multiple fibers 2A0 and 2A1 in the above example. Let g : B → σ ′ (ℓ2 ) be a double covering branched over the points b0 := F0 ∩ σ ′ (ℓ2 ) and b∞ := ′ ℓ′∞ ∩ σ ′ (ℓ2 ), and let V ′ be the normalization of the fiber product V ×ℓ′2 B . Let h : V ′ → V be the composite of the normalization morphism ν : V ′ → ′ ′ V ×ℓ′2 B and the projection p1 : V ×ℓ′2 B → V . Let Y := h−1 (X). There ′

exists a P1 -fibration f ′ : V ′ → B induced by f . ′ ∗ Let b′0 be a point of B over b0 . Then the fiber f ′ (b′0 ) = h∗ (F0 ) is shown in the following left configuration, where h∗ (E0′′ ) splits to the sum of

Geometry and Topology of Polynomial Rings

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e1 + A e2 . The inverse image h∗ (F1 ) is a sum of two copies two (−1)-curves A of F1 , which are equilateral by the covering involution. ℓ′0 −1 ℓe′0 −2 −1 −2 −2 −2 −2 −1 −1 −2 −2 −1 ℓ′∞ ℓe′∞ −4 −1 h −2 −1 −2 −1 e′ −1 E0 −2 −2 ℓ′2 −2 −2 e0 −2 E

(f ′ )−1 (b′0 )

−1

ℓe′2

Fe1

Fe1′

F0

F1

e1 , A e2 in the fiber (f ′ )−1 (b′ ), Throwing out one of two (−1)-components A 0 ′ ′ we obtain a Q-homology plane X like X. If X is isomorphic to X then we have a counterexample to (GJC). But this is an open problem.

378

3.5

Affine Algebraic Geometry

Appendix to Chapter 3

We assume throughout the present section that the ground field k has characteristic zero. 3.5.1

Makar-Limanov invariant

Let X = Spec A be an affine algebraic variety of dimension n defined over k. Let δ be a nonzero k-derivation of A into A. Take a smooth point x ∈ X. Set R = OX,x and m = mX,k . Then (R, m) is a regular local ring of dimension n. Let {a1 , . . . , an } be a regular system of generators. By subsection 1.6.4, we have Ω1A/k ⊗A R ∼ = Ω1R/k =

n M

Rdai .

i=1

Further, about the stalk of HomOX (Ω1X/k , OX ) at x, we have HomOX (Ω1X/k , OX )x

∼ = HomR (Ω1R/k , R) ∼ =

n M

R∂i ,

i=1

where {∂1 , . . . , ∂n } is the dual basis of {da1 , . . . , dan }. By Lemma 1.6.20 and its proof we have HomR (Ω1R/k , R) ⊗R (R/m) ∼ = Hom(m/m2 , k). The k-derivation δ, extended naturally to a k-derivation of R, gives a kderivation δx from R to k, i.e., δx (a) = δ(a) (mod m). If x is a general point of X then δx ̸= 0. We call Hom(m/m2 , k) the tangent space of X at x and denote it by TX,x . Let ai ≡ ai (mod m2 ). Then m/m2 = ⊕ni=1 k · ai . Hence {a1 , . . . , an } is viewed as a system of coordinates of TX,x . A vector δ(a1 , . . . , an )(x) := {δ(a1 )(x), . . . , δ(an )(x)} is a tangential direction of δ at the point x, where δ(ai )(x) = δ(ai ) (mod m). If δ is locally nilpotent then exp(tδ) : A → A[t] defines a Ga -action σ : Ga × X → X (see Lemma 2.4.2). Then δ(a1 , . . . , an )(x) is a tangential direction of the Ga -orbit Ga (x) := σ(Ga , x) at x. Lemma 3.5.1. With the above notations, we have the following assertions. (1) Let δ be a locally nilpotent k-derivation of A and let B = Ker δ. Then δ is naturally extended to a k-derivation δK of K = Q(A), and K0 = {ξ ∈ K | δK (ξ) = 0} is equal to Q(B).

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(2) Let δ1 , δ2 be locally nilpotent k-derivations of A. Then Ker δ1 = Ker δ2 if and only if aδ2 = bδ1 for a, b ∈ A \ {0}. In fact, we can take a, b ∈ Ker δ1 = Ker δ2 . We then say that δ1 and δ2 are similar and write δ1 ∼ δ2 . (3) Assume that n = 2 and Ker δ1 ̸= Ker δ2 . Then Ker δ1 ∩ Ker δ2 = k. Proof. (1) Let ξ ∈ K0 and write ξ = a2 /a1 with a1 , a2 ∈ A. Since   a1 δ(a2 ) − a2 δ(a1 ) a2 = = 0, δK a1 a21 we have ξ ∈ Q(B) if δ(a1 ) = 0 and ξ=

a2 δ(a2 ) = a1 δ(a1 )

otherwise. By the same argument, ξ=

δ m (a2 ) , δ m (a1 )

where δ m (a1 ) ̸= 0 and δ m+1 (a1 ) = 0, whence δ m (a1 ) ∈ B. Then δ m (a2 ) ̸= 0 and δ m+1 (a2 ) = 0 as δK (ξ) = 0. So, ξ = δ m (a2 )/δ m (a1 ) ∈ Q(B). The other inclusion Q(B) ⊆ K0 is clear. (2) Suppose that B = Ker δ1 = Ker δ2 . By Lemma 2.1.6, there exist s1 ∈ −1 B and a1 ∈ A such that δ1 (a1 ) = s1 and A[s−1 1 ] = B[s1 ][a1 ]. This implies that A ⊗B K0 = K0 [a1 ], where K0 = Q(B). Similarly, A ⊗B K0 = K0 [a2 ] with a2 ∈ A such that δ2 (a2 ) = s2 ∈ B. So, we put A ⊗B K0 = K0 [u], where u is transcendental over K0 . As the locally nilpotent derivations δ1 and δ2 of K0 [u], δ1 (u) and δ2 (u) are elements of K0 , and δ1 and δ2 are determined by the images δ1 (u) and δ2 (u) in K0 . So, δ1 (u)/δ2 (u) = a0 /b0 for a0 , b0 ∈ B \ {0}. Then a0 δ2 = b0 δ1 on A ⊗B K0 . Since a0 δ2 − b0 δ1 is a k-derivation of A and A ,→ A⊗B K0 is injective, it follows that a0 δ2 = b0 δ1 . The converse is clear. (3) Let Bi = Ker δi for i = 1, 2. Let Ki = Q(Bi ) and let K0 = K1 ∩ K2 . We show that K1 ̸= K2 , whence K0 ⫋ Ki for i = 1, 2. In fact, suppose that K1 = K2 = K0 . Then δ1 and δ2 are locally nilpotent derivations of K0 [u], whence, by the argument of the assertion (2), a0 δ2 = b0 δ1 for a0 , b0 ∈ A \ {0}. This implies that Ker δ1 = Ker δ2 , which is a contradiction to the assumption. If tr.degk K0 = 0 then K0 = k, and B1 ∩B2 = k because k ⊆ B1 ∩ B2 ⊆ K1 ∩ K2 = K0 = k. Suppose that tr.degk K0 = 1.14 Since 14 If

dim A = 2 then B is an affine domain of dimension 1 by Zariski’s finiteness theorem (see [19]), which is stated as follows:

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tr.degk Ki = 1 for i = 1, 2, Ki is a separable algebraic extension of K0 . Write K1 = K0 (θ), where θ is a root of an irreducible equation f1 (t) = 0 with f1 (t) ∈ K0 [t]. Applying δ2 to f1 (θ) = 0, we have f1′ (θ)δ2 (θ) = 0, where f1′ (θ) ̸= 0. Hence δ2 (θ) = 0. This implies that K1 ⊆ K2 . Similarly, we have K2 ⊆ K1 . So, K0 = K1 = K2 . This contradicts the assumption. For an affine algebraic variety X = Spec A defined over k, we define T the Makar-Limanov invariant ML(X) as the k-subalgebra δ Ker δ of A, where δ ranges over the set of locally nilpotent k-derivations of A. Corollary 3.5.2. Let X = Spec A be an affine algebraic surface. Then the following assertions hold about ML(X). (1) ML(X) is an affine domain over k. (2) ML(X) = A if and only if there are no nontrivial locally nilpotent kderivations of A. (3) dim ML(X) = 1 if and only if there is a unique nontrivial locally nilpotent k-derivation up to similarity. (4) ML(X) = k if and only if there are two non-similar nontrivial locally nilpotent k-derivations. A smooth affine surface X is said to be an ML0 -surface if ML(X) = k. Theorem 3.5.3. Let X be a smooth affine surface, let V be a minimal log smooth completion of X and let D = V \ X. Then X is an ML0 -surface if and only if D is a rational linear chain and γ(X) = 0, whence X is a Gizatullin surface with γ(X) = 0. Proof. We prove only the if part by referring the proof of the only if part to [27, Chapter 2, Theorem 2.6.2]. By the explanation after Lemma 3.1.8, V is a rational surface. By shift transformations or a succession of blowingups applied to the linear chain L which consists of the components of D, the resulting linear chain, say Lℓ on a projective surface Vℓ , has the left terminal component, say Cℓ , of weight 0. The linear system Λℓ := |Cℓ | is a linear pencil and defines an A1 -fibration of affine type ρℓ : X → B, where B is an affine curve isomorphic to A1 because γ(X) = 0. By Lemma 2.1.4, ρℓ is the quotient morphism by the Ga -action given by a locally nilpotent Theorem. Let k be a field and let A be a normal affine domain over k. Let L = Q(A) and let K be a subfield of L such that k ⊆ K and tr.degk K ≤ 2. Then B := A ∩ K is an affine domain over k. By this theorem, Ker δ1 and Ker δ2 are affine domains of dimension 1 over k.

Geometry and Topology of Polynomial Rings

381

derivation associated to the A1 -fibration ρℓ . By a similar process, we can change the linear chain L to a linear chain Lr on a projective surface Vr whose right terminal component, say Cr , has weight 0. Then Λr := |Cr | gives a Ga -action on X. By the birational mapping φ : Vℓ 99K Vr which changes Lℓ to Lr , the image of Λℓ is a linear pencil with base points on Dr , while Λr has no base points. So, the A1 -fibrations on X cut out by these two linear pencils are different. Hence the associated Ga -actions, or locally nilpotent derivations, are different. Hence X is an ML0 -surface by Lemma 3.5.1. Remark 3.5.4. With the notations in the above proof, the unique exceptional case for which φ∗ Λℓ has no base points is the case where L consists of three vertices such that two terminal components have weight 0. Then Λℓ = Λr and the middle component is a cross-section of the P1 -fibration defined by the pencil Λℓ = Λr . In this case γ(X) = 1. 3.5.2

The fundamental group at infinity

Let X be a smooth affine surface defined over C and let V be a log smooth completion of X with the boundary divisor D. Then V and X are viewed as complex manifolds, and D is a union of complex submanifolds. With the natural complex metric on V , we take a small real number ε > 0 and a set Uε = {P ∈ X | ∥ P, D ∥< ε}, where ∥ P, D ∥ is the minimum of the distance ∥ P, Q ∥ of two points P, Q when Q moves on D. U is an open neighborhood of the boundary divisor. Let Sε = U ε \ Uε . The fundamental group π1 (Sε ) does not depend on the choice ε if it is small enough. It is called the fundamental group at infinity of X and denoted by π1∞ (X). We refer the readers to Ramanujam’s paper [83] for detailed explanations. It is an analogue of the local fundamental group of a singular point on a normal algebraic surface V defined by Mumford [69]. The description of these fundamental groups are given in terms of generators and relations by making use of the weighted dual graph Γ(D). In the case of the local fundamental group, D is the exceptional curves for a desingularization of P ∈ V such that they form an SNC divisor. Suppose that the divisor D is a tree of rational curves. Extracting the definition from [69] and [83], we define the group π1 (Γ) associated to a weighted graph Γ, which is generated by as many generators as the vertices {v1 , . . . , vn } of Γ which are subject to the following relations.

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Affine Algebraic Geometry

First, order the vertices in some fixed manner as v1 , v2 , . . . , vn . For each vertex v with weight −d, let vi1 , vi2 , . . . , vir be all the vertices connected to v such that i1 < i2 , . . . < ir . As the convention, we assume that v occurs among these vertices, say v = via . For each vertex v, we consider a relation vi1 · · · via−1 · vi−d · · · vir = e. a Next, if v, w are any two vertices which are connected by an edge, then we impose the relation v · w = w · v, i.e., the generators corresponding to v, w commute. The quotient of the free group generated by the vertices of Γ modulo these relations is the group π1 (Γ). If Γ = Γ(D) for a log smooth completion (V, D) of X, π1 (Γ(D)) is the fundamental group at infinity π1∞ (X). It is known that π1∞ (X) is independent of the choice of a log smooth completion in the case char k = 0. We call this construction of π1∞ (Γ) the Mumford-Ramanujam presentation. This group π1∞ (X) modulo the commutator subgroup is the first homology group at infinity H1∞ (X). Let Γ be a weighted graph. A drip of Γ is a pair of vertices {u, v} such that u has weight 0 which is linked only to the vertex v and v is a vertex linked to u and at most one other vertex of Γ. An operation of removing vertices u, v and the edge linking the vertices {u, v} and an edge linking v to a third vertex (if it exists) is called the removing of the string {u, v} from Γ. Lemma 3.5.5. Let Γ be a connected weighted tree. Then the following assertions hold. (1) Let {u, v} be a drip of Γ such that the weight w(v) of v is −d and let Γ′ be obtained from Γ by removing the string {u, v}. Then π1 (Γ) = π1 (Γ′ ). (2) Suppose that Γ is a linear chain. Then π1 (Γ) is a cyclic group Z/|∆|Z, where |∆| is the absolute value of the determinant of the intersection form on the free abelian group generated by the vertices of Γ. Hence ∆ is non-zero if and only if π1 (Γ) is finite cyclic. Proof. (1) Let {u, v} be a string of Γ with w(u) = 0 and w(v) = −d. If Γ consists only of vertices u, v and an edge linking them, it is easy to see that π1 (Γ) = {e}. Otherwise, let w be a third vertex linking to v. Let the ordering on the vertices of Γ be u, v, w, . . .. Since u0 v = e and uv −d w = e, we have v = e and u = w−1 . Hence it follows that π1 (Γ) = π1 (Γ′ ).

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(2) Let the ordering on Γ be v1 < v2 < · · · < vn , which accords with the left-to-right order of vertices on a line: −d1

−d2

−d3

−dn−1

−dn

v1

v2

v3

vn−1

vn .

By the Mumford-Ramanujam presentation we have the following relations (1 ≤ i < n)

vi+1 vi = vi vi+1 v2 =

v1d1 , . . . , vi−1 vi−di vi+1

= e, . . . , vn−1 = vndn .

It is then easy to show that the generator corresponding to v1 generates π1 (Γ). Let ∆1 = d1 and d1 −1 0 · · · · · · 0 −1 d −1 0 · · · 0 2 0 −1 d3 −1 · · · 0 ∆i = . . . . . . .. . . . . . . . . .. 0 · · · 0 −1 di−1 −1 0 · · · · · · 0 −1 di for 2 ≤ i ≤ n. Then it holds that vi+1 = v1∆i for 1 ≤ i < n. In particular, the relation vn−1 = vndn yields ∆

d ∆n−1

v1 n−2 = v1 n

.

Since ∆n = dn ∆n−1 − ∆n−2 , we have a relation v1∆n = e. This proves the assertion because ∆n is the absolute value of the determinant of the intersection form. Lemma 3.5.6. Let X be an affine pseudo-plane of type (d, r). Then π1∞ (X) is a group generated by x, y with relations xr = y d = (xy)d . If r = 1 then π1∞ (X) is a finite cyclic group of order d2 . Proof. With the notations of the weighted dual graph of D in Defini′ tion 3.1.11, let vi (1 ≤ i ≤ d+r−1), v0′ , s′ , v∞ be the generators correspond′ ′ ′ ing to Ei (1 ≤ i ≤ d+r −1), ℓ0 , S , ℓ∞ , respectively. Let x = vd+r−1 , y = ℓ′0 and z = v1 . Then we have vi = xd+r−i for d ≤ i ≤ d + r − 1, s′ = 1, ℓ′∞ = y −1 , vi = y i for 2 ≤ i ≤ d, and vd = z d . Furthermore, xr = y d = z d

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and zy d−1 xr−1 = z 2d . Thence we obtain z = xy. So, π1∞ (X) has generators and relations as described in the statement. If r = 1, i.e., D is a linear chain, then π1∞ (X) is an abelian group generated by y. So, π1∞ (X) = H1∞ (X). A direct computation shows that H1∞ (X) is a finite group for r general, and isomorphic to Z/d2 Z if r = 1. Lemma 3.5.7. Let V be a smooth projective surface and let D be a linear chain of smooth rational curves. Write D = D1 + · · · + Dn . Let σ : V ′ → V be the blowing-up of a point P of D and let D′ = σ ∗ (D)red . Then π1 (Γ(D)) = π1 (Γ(D′ )). In particular, π1 (Γ(D)) is unchanged by the shift transformations. Proof. The weighted dual graph Γ(D) is given by the graph given in the proof of Lemma 3.5.5, where the vertex vi corresponds to the irreducible component Di and (Di2 ) = −di . If P is a point on Di which is not the intersection points Di−1 ∩ Di nor Di ∩ Di+1 . Then Γ(D′ ) has a new vertex w with weight −1 corresponding to σ −1 (P ) linked to the vertex vi and the weight −di is changed to −di − 1. Then we have vi = w,

vi−1 vi−di −1 vi+1 w = e,

vi vi+1 = vi+1 vi ,

where, by the abuse of notations, the vertices in Γ(D′ ) corresponding to the proper transforms are denoted by the same symbols as in Γ(D). From these relations, it follows that π1 (Γ(D)) = π1 (Γ(D′ )). If P = Di−1 ∩ Di , Γ(D′ ) has a new vertex w with weight −1 which is linked to the vertices vi−1 and vi and the weights of vi−1 and vi are changed to −di−1 − 1 and −di − 1. In the group π1 (Γ(D′ )), we have vi−1 w = wvi−1 , vi w = wvi −d

vi−1 w−1 vi = e, vi−2 vi−1i−1

−1

w = e, wvi−di −1 vi+1 = e.

From these relations, we have w = vi−1 vi . Substituting w by this relation in the other relations in the above second line, we get back the relations in π1 (Γ(D)). So, π1 (Γ(D′ )) = π1 (Γ(D)). Since a shift transformation is a composite of blowing-ups on the linear chain and the blowing-downs of components of the linear chain, it does not change the group π1 (Γ(D)). The following result is a famous theorem due to Ramanujam. Theorem 3.5.8. Let X be a smooth affine surface defined over C. Then X is isomorphic to the affine plane A2 if and only if π1 (X) = π1∞ (X) = {e}.

Geometry and Topology of Polynomial Rings

3.5.3

385

Algebraic surfaces and log Kodaira dimension

In this subsection the ground field k is an algebraically closed field of arbitrary characteristic. We explained several results, e.g., Theorem 3.1.21, related to log Kodaira dimension. Since structures of smooth affine surfaces were clarified according to values of log Kodaira dimension, affine algebraic geometry has made a remarkable progress. Here we summarize significant structure theorems of affine algebraic surfaces in terms of log Kodaira dimension. Precise treatment on this subject is given in [56] and [59, Chapter 3]. So our treatment here becomes more intuitive and explanatory. Let X be a smooth affine surface (or a non-complete algebraic surface) and let (V, D) be a log smooth completion. Namely V is a smooth projective surface and D is an SNC divisor on V such that X = V \ D. If X is affine D supports an ample divisor on V , and vice versa. As explained in subsection 1.7.4, the log Kodaira dimension κ(X) is independent of choices of log pairs (V, D). If X contains an open set U ⊂ X such that X \ U is an SNC divisor on X, we may assume, after replacing (V, D), that Γ := V \ U is an SNC divisor on V . Then Γ + KV ≥ D + KV , whence κ(U ) ≥ κ(X). In fact, this inequality holds without the above affine restriction on X. If X is a product S × T of two smooth curves S, T then κ(X) = κ(S) + κ(T ). It is our convention that n + (−∞) = −∞ for a nonnegative integer n. If T ∼ = A1 , it is easy to compute κ(T ) = −∞. Hence, if X contains a cylinderlike open set U = S × A1 , the above explanation shows that κ(X) = −∞. Since an affine variety cannot contain a complete curve, the projection ρS : U → S extends to an A1 -fibration ρ : X → S, where S is a projective or affine curve containing S as an open set. We say that X is affine ruled if X contains a cylinderlike open set U . A structure theorem in the case of κ(X) = −∞ is stated as follows. A smooth algebraic surface X is said to be connected at infinity if D is connected for a log smooth completion (V, D) of X. Theorem 3.5.9. Let X be a smooth affine surface (or more generally, a smooth algebraic surface which is connected at infinity). If κ(X) is −∞ then X is affine ruled. This is a natural generalization of Enriques criterion of ruledness that a smooth projective surface V is ruled if and only if κ(V ) = −∞. In fact, it suffices that P12 := h0 (V, OV (12KV )) = 0. A log smooth completion (V, D) of X is N C-minimal if there is no

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(−1)-curve E such that either E ⊂ D and (E · D) = 0, 1 or E ̸⊂ D and (E · D) = 1. Given a log smooth completion of X, if there is a (−1)-curve E which makes the given pair not N C-minimal, one can contract E. A succession of contractions of (−1)-curves of this kind changes the given pair to an N C-minimal pair. The case κ(X) = 0 was studied in details in [20, §8]. In particular, the following results limits, as in the projective case, to the four classes under the assumption bi (X) := hi (X; Q) = 0 for i = 1, 2, i.e., X is a Q-homology plane (see [59, Lemma 4.4.2, Chapter 3]). Theorem 3.5.10. Let X be a Q-homology plane with κ(X) = 0. If X is N C-minimal, X is isomorphic to one of the four surfaces H[k, −k] (k ≥ 1),

Y {3, 3, 3},

Y {2, 4, 4},

Y {2, 3, 6}.

For more details on the above surfaces, see [ibid.]. In the case κ(X) = 1, the following is a structure theorem of Kawamata (see [56]). Theorem 3.5.11. Let X be a smooth affine surface with κ(X) = 1. Then there is an A1∗ -fibration f : X → B. The rest of affine surfaces belong to the case κ(X) = 2. Notwithstanding, the conjecture (GJC) holds for this class of surfaces. In other words, one can say that (GJC) holds for almost all smooth affine surfaces. Thus the counterexamples if exist are limited to the classes for which the structures are known more precisely. Finally we quote results from [59] which we use in the text. Let X, Y be algebraic varieties. A morphism f : X → Y is called a fibered variety if general fibers of f are irreducible and reduced. Theorem 3.5.12 (Easy addition formula). Let f : X → Y be a fibered variety with smooth X. Then, for a general point y ∈ Y , we have κ(X) ≤ κ(f −1 (y)) + dim Y. Theorem 3.5.13 (Addition theorem of Kawamata). Let f : X → Y be a fibered variety of relative dimension ≤ 1 between smooth algebraic varieties X, Y . Then, for a general point y ∈ Y , we have κ(X) ≥ κ(f −1 (y)) + κ(Y ).

Geometry and Topology of Polynomial Rings

3.5.4

387

Logarithmic ramification formula

Let φ : X → Y be a dominant morphism of smooth algebraic varieties. Then there exist log smooth completions (V, D) and (W, ∆) of X and Y respectively and a dominant morphism Φ : V → W such that Φ|X = φ. Hence Φ−1 (∆) ⊆ D. Let KV and KW be the canonical divisors of V and W . Then we have D + KV ∼ Φ∗ (∆ + KW ) + R for some effective divisor R on V which is called the log ramification divisor. We show this in the case of dimension 2. Lemma 3.5.14. With the above notations, we assume that dim X = dim Y = 2. Then the divisor R is an effective divisor supported by the union of curves C on V such that C is contracted by Φ, Φ|C : C → Φ(C) is ramifying, or C is a component of D with ϕ(C) ̸⊂ ∆. The curve C has coefficient zero in R if C is a component of D which is mapped onto a component of ∆ by Φ or possibly if C is a component of D which is contracted to an intersection point of two irreducible components of ∆. Proof. Suppose that C is not contracted by Φ. Let C = Φ(C). Choose a general point P on C and let Q = Φ(P ). (1) Suppose that C is not a component of D. We choose a system of parameters {t, u} of W at Q so that C is locally defined by t = 0 and u is a parameter along C. Similarly, we choose a system of parameters {ξ, η} of V at P so that C is defined by ξ = 0 and η is a parameter along C. Then Φ∗ (t) = f (ξ, η) and Φ∗ (u) = g(ξ, η), where f, g ∈ OV,P . We fix a nonzero, rational, differential 2-form ω of W and express it as ω = bdt ∧ du, where b ∈ k(W ). Similarly, write Φ∗ (ω) = adξ ∧ dη with b ∈ k(V ). The divisor D +KV (resp. ∆+KW ) is determined near the point P (resp. Q) by evaluating a (resp. b) in terms of the normalized discrete valuation vC (resp. vC ) associated to C (resp. C). By the assumption, C is not a component of ∆ as well. By a simple calculation, we obtain   f, g a = Jb, J =J . ξ, η Hence, near the point P , D + KV differs from Φ∗ (∆ + KW ) by vC (J)C. Note that Φ|C : C → C is unramified at P if and only if J(P ) ̸= 0. In fact, if vC (f ) = r, then vC (J) = r − 1.

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(2) Suppose that C is a component of D and C is not a component of ∆. Then ω = bdt ∧ du and Φ∗ (ω) = a(dξ/ξ) ∧ dη. Hence a = bξJ with J as above. Write f = ξ m f1 with m > 0 and ξ ∤ f1 . Then we have   ∂g f1 , g m ξJ = ξ (mgη + J1 ), gη = , J1 = J . ∂η ξ, η So, vC (ξJ) ≥ m > 0. C appears in R if m > 0, i.e., C → C. (3) Suppose that C is a component of D and C is a component of D. Express ω and Φ∗ (ω) as dξ dt Φ∗ (ω) = a ∧ dη. ω = b ∧ du t ξ Then we have   bξ f, g a= J . f ξ, η Hence, near the point P , D + KV differs from Φ∗ (∆ + KW ) by vC (ξJ/f )C which is zero. In fact, if we write f = ξ m f1 with m > 0 and ξ ∤ f1 in OW,P then we have ξJ mξ m f1 gη + ξ m+1 J1 ξJ1 = = mgη + , m f ξ f1 f1 where J1 is the Jacobian of f1 , g with respect to ξ, η and vC (gη ) = 0 for a general point P of C. (4) Now suppose that C is contracted by Φ. Then, choosing a system of parameters {t, u} of V at the point Q = Φ(C), one can write Φ∗ (t) = ξ r f1 (ξ, η) and Φ(u) = ξ s g1 (ξ, η) with ξ ∤ f1 g1 , r > 0 and s > 0. Hence we have       f1 , g1 ∂g1 ∂f1 f, g = ξ r+s−1 rf1 − sg1 + ξ r+s J J ξ, η ∂η ∂η ξ, η   f1 , g1 ∂g1 ∂f1 = ξ r+s−1 I with I = rf1 − sg1 + ξJ . ∂η ∂η ξ, η This implies that ω = bdt ∧ du = aξ r+s−1 Idξ ∧ dη.

If C is not a component of ∆, then ∆ + KW differs from Φ∗ (D + KV ) by vC (ξ r+s−1 I)C which has a positive coefficient. If C is a component of ∆, three cases are possible. Namely, the point Q = Φ(C) lies in V \ D, lies on a single irreducible component D1 or is the intersection point D1 ∩ D2 . In the first case, dξ ω = adt ∧ du = aξ r+s I ∧ dη ξ

Geometry and Topology of Polynomial Rings

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and the coefficient of C in (D + KV ) − Φ∗ (D + KV ) is vC (ξ r+s I) which is positive. In the second case, we choose a system of local parameters {t, u} so that D1 is defined by t = 0 and u is a parameter along D1 . In the third case, we choose {t, u} so that t and u are respectively parameters along the curves D2 and D1 . In the second case, we have ω=a

aξ s I dξ dt ∧ du = ∧ dη, t f1 ξ

where f1 ̸= 0 on a general point of C. In the third case, we have ω=a

dt dη aI dξ ∧ = ∧ dη. t η f1 g1 ξ

In the second case, the component C appears with a positive coefficient in the difference (D + KV ) − Φ∗ (∆ + KW ). In the third case, the coefficient is non-negative with C possibly not appearing in the difference.

390

Affine Algebraic Geometry

3.6

Problems to Chapter 3

We assume that the ground field k has characteristic zero unless otherwise specified. 1. A locally nilpotent k-derivation δ on an affine k-domain is called reduced if the ideal δ(A)A of A generated by the set {δ(a) | a ∈ A} has no prime divisors of height 1. Let A be a polynomial ring in two variables over k. Prove the following assertions. (1) Given any reduced locally nilpotent derivation δ on A, there exists a system of coordinates {x, y} such that δ = ∂/∂x. (2) For an element σ ∈ Aut k (A) and a reduced locally nilpotent derivation δ of A, let σ ∗ (δ) = σ ◦ δ ◦ σ −1 . Then σ ∗ (δ) is a reduced locally nilpotent derivation of A. (3) Given two reduced locally nilpotent derivations δ, δ ′ of A, there exists σ ∈ Aut k (A) such that δ ′ = σ ∗ (δ). Answer. (1) It follows from Theorem of Rentschler (see Theorem 2.4.4) that there exists a system of coordinates {x, y} such that Ker δ = k[y] and δ = f (y)(∂/∂x) for f (y) ∈ k[y]. Since δ is reduced, we can take f (y) = 1. (2) Note that (σ · δ · σ −1 )n = σ · δ n · σ −1 for n > 0, whence σ ∗ (δ) is a locally nilpotent k-derivation. It is also reduced. (3) Write δ = ∂/∂x (resp. δ ′ = ∂/∂u) with respect to a system of coordinates {x, y} (resp. {u, v}). Define an element σ ∈ Aut k (A) by u = σ(x) and v = σ(y). Write an element a ∈ A as a = f (u, v) = f (σ(x), σ(y)). Then we have ∂ (f (u, v)) = fu (σ(x), σ(y)) ∂u   ∂ −1 −1 ·σ (f (σ(x), σ(y)) (σ · δ · σ )(a) = σ · ∂x ∂ =σ· (f (x, y)) = σ(fx (x, y)) = fu (σ(x), σ(y)). ∂x Hence δ ′ = σ ∗ (δ). 2. Let X be an affine pseudo-plane with the Picard group of order d. Show the following. δ ′ (a) =

(1) The unit rank of X is zero, i.e., U (X) = k ∗ .

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e → X of order d such that (2) There exists a cyclic Galois covering π : X e X is simply connected. Answer. Let ρ : X → B be an A1 -fibration with a unique multiple fiber F of multiplicity d, where B ∼ = A1 . Choose a coordinate x on B such that the fiber F lies over the point x = 0. Write the fiber F = dC. Let Q ∈ X be a point on C. Let ξ = 0 be a local defining equation ∗ e → B be the of C at Q. Then x = uξ d , where u ∈ OX,Q . Let σ : B 1 ∗ d e e morphism, where B = A = Spec k[t] and σ (x) = t . Let Y = X ×B B e be the normalization of Y . Over the point Q, the scheme Y is and let X e is given locally given as the spectrum of OX,x ⊗k[x](x) k[ξ](ξ) . Hence X d e lying by OX,Q [t/ξ], where (t/ξ) = u. Hence there are d points of X e → B is ´etale over B \ {x = 0}, it follows over the point Q. Since σ : B e → X, which is the composite of the that the canonical morphism π : X

first projection Y → X composed with the normalization morphism e → Y is a finite Galois ´etale covering of degree d. The inverse image X of the multiple fiber F = dC by π is a disjoint union of d copies of the affine line, each of which has multiplicity 1. Further, the A1 -fibration e \ π −1 (F ) → B e \ {t = 0}, which extends to ρ lifts to an A1 -fibration X ∗ e e ρe : X → B such that π (F ) is the fiber over the point t = 0. Remove (d − 1) copies of A1 leaving one copy, we obtain an open set U of e such that U ∼ e is simply connected. Further, since X = A2 . Hence X π|U : U → X is dominant, it follows that U (X) = k ∗ . 3. Show that X is an affine pseudo-plane of type (d, d−1) is isomorphic to P2 \C, where C is a curve defined by X0 X1d−1 = X2d , where d ≥ 2. Note that X is a Gizatullin surface if d = 2 because the boundary divisor consists only of the main chain. Answer. By Lemma 3.1.13 and its proof, X is isomorphic to the complement of M0 ∪ Cd in the Hirzebruch surface F1 , where (M0 )2 = −1, (M0 · Cd ) = d − 1 and (Cd2 ) = 2d − 1. Further, the multiple fiber dF is given by a fiber ℓ0 of F1 such that M0 + dℓ0 ∼ Cd . Now contract the curve M0 to obtain P2 . The image C of Cd on P2 has a cusp of multiplicity d − 1 and (C 2 ) = 2d − 1 + (d − 1)2 = d2 . Hence C is a curve of degree d. The image L of the curve ℓ0 has (L2 ) = 1 and (C · L) = 1 + (d − 1) = d, whence L is a line. This implies that X is isomorphic to P2 \ C, where C is defined by X0 X1d−1 = X2d and L is defined by X1 = 0. 4. Let C be a rational curve on P2 defined by X0 X1d−1 = X2d with d > 2

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and let X = P2 \C. Let x = X0 /X1 , y = X2 /X1 and f = x−y d . Verify the following assertions. (1) Let Λ be a linear pencil on P2 generated by the curves C and dℓ1 , where ℓ1 is defined by X1 = 0. Then Λ defines an A1 -fibration ρ : X → B, where B ∼ = A1 . This is a unique A1 -fibration on X. (2) Define a Gm -action τ : Gm × X → X on X by (X0 , X1 , X2 ) 7→ (λd X0 , X1 , λX2 ), where λ ∈ k. Then τ defines a torus subgroup T ∼ = Gm of Aut (X) such that φT : T → Aut (B, b0 ) ∼ = Gm is the d d-th power mapping λ 7→ λ . (3) There exists a locally nilpotent derivation δ on Γ(X, OX ) such that δ(x) = dy d−1 f −1 and δ(y) = f −1 . Then δ defines a Ga -action σ : Ga × X → X associated with δ. Furthermore, τλ−1 δτλ = λd+1 δ for λ ∈ k ∗ , where σλ = σ|{λ}×X . (4) Let G be the kernel of φ : Aut X → Aut (B, b0 ). Then G consists of automorphisms α such that α(y) = cy + f −m (a0 f r + a1 f r−1 + · · · + ar )

α(x) = x − y d + α(y)d ,

where cd = 1, ai ∈ k (0 ≤ i ≤ r) with a0 ar ̸= 0 and m > r ≥ 0. Hence Aut (X) is not an algebraic group. Answer. (1) Let Y = P2 \ (C ∪ ℓ1 ) = X \ ℓ1 . Then Y is an affine surface with Γ(Y, OY ) = C[x, y, f −1 ], where f = x − y d . Furthermore, Y has an A1 -fibration {f = λ | λ ∈ k ∗ } which extends to a unique A1 -fibration ρ : X → B. In fact, the A1 -fibration on Y is obtained by removing a unique multiple fiber dℓ1 from ρ. The uniqueness of ρ follows from Lemma 3.1.14. (2) It is clear that λ (f ) = λd f and B \ {b0 } = Spec k[f, f −1 ]. The assertion (2) follows from this remark. (3) If there is a locally nilpotent derivation δ on Γ(X, OX ), X has an associated Ga -action. Since φ : Aut X → Aut (B \ {b0 }) maps Ga to the identity, we have δ(f ) = 0. Define a locally nilpotent derivation δ0 of Γ(Y, OY ) = k[x, y, f −1 ] by δ0 (x) = dy d−1 and δ0 (y) = 1. Note that B = Spec k[f ] and B1 := B \{b0 } = Spec k[f, f −1 ]. Further, ρ−1 (B1 ) ∼ = B1 × A1 . Hence, by Lemma 2.1.6, Γ(X, OX ) has a locally nilpotent derivation δ = f −s δ0 . We show that if s = 1 then δ(Γ(X, OX )) ⊆

Geometry and Topology of Polynomial Rings

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Γ(X, OX ). In fact, since C − dℓ1 = (f ), we have dF = (f −1 ) on X, where F = ℓ1 ∩ X. Then we have (dy d−1 · f −1 ) = (d − 1)(ℓ2 ∩ X) − (d − 1)(ℓ1 ∩ X) + d(ℓ1 ∩ X) (1 · f

−1

= (d − 1)(ℓ2 ∩ X) + (ℓ1 ∩ X) > 0

) = d(ℓ1 ∩ X) > 0,

where ℓi is the line on P2 defined by Xi = 0. The automorphism σλ for λ ∈ k is given by σλ (y) = exp(λδ)(y) = y + f −1 λ and σλ (x) = exp(λδ)(x) = x−y d +σλ (y)d because σλ (f ) = f . Noting that τλ (f −s ) = λ−ds f −s , we have τλ−1 δτλ (x) = τλ−1 δ(λd x) = τλ−1 (λd dy d−1 f −s ) = λd dλ−d+1 λds y d−1 f −s = λds+1 (dy d−1 f −s ) = λds+1 δ(x) τλ−1 δτλ (y) = τλ−1 δ(λy) = τλ−1 (λf −s ) = λsd+1 f −s = λds+1 δ(y). Hence we obtain the stated formula for s = 1. (4) Let α be an automorphism in G. Let t = X1 /X2 and u = X0 /X2 . Since α induces the identity on B, it acts along the fibers of ρ. Since Y = X \ ℓ1 = Spec k[y, f, f −1 ] as x = y d + f and α(f ) = f , α is written as α(y) = cf n y + f s (a0 f r + a1 f r−1 + · · · + ar )

α(x) = x − y d + α(y)d ,

where c ∈ k ∗ , ai ∈ k (0 ≤ i ≤ r) with a0 ar ̸= 0 and n, s, r ∈ Z with r ≥ 0. Conversely, given such an automorphism α of Y as written above, we consider when α extends to an automorphism of X. Note that C ∩ ℓ1 = (1, 0, 0) and ℓ1 ∩ X is contained in the open set P2 \ ℓ2 . Let Z = P2 \ (C ∪ ℓ2 ) be an open set of X containing ℓ1 ∩ X. Since t = 1/y, u = x/y and f = (utd−1 −1)/td , it follows that td f is a nowhere vanishing function on Z \ ℓ1 . We write
α(y) = c∗ t−dn−1 + t−(s+r)d a∗0 + a∗1 td + · · · + a∗r trd , where c∗ , a∗i (0 ≤ i ≤ r) are nowhere vanishing functions on Z. Since α(t) is divisible by t and not divisible by t2 along the curve ℓ1 ∩ Z and since d > 2, it follows that n = 0 and s + r ≤ 0. Hence c∗ = c. Then we can write x − y d + α(y)d α(u) = α(y) 2

=

u + (cd − 1)t−(d−1) + dcd−1 g ∗ t−d(s+r)−(d−2) + · · · + g d t−d c + t1−d(s+r) (a∗0 + a∗1 td + · · · + a∗r trd )

(s+r)+1

.

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Affine Algebraic Geometry

Since 1 − d(s + r) ≥ 1 in the denominator, the numerator of α(u) must be regular along the curve ℓ1 ∩Z. Namely, we have cd = 1 and s+r < 0. Set s = −m. Then we obtain a formula for α as stated in the assertion (4). 5. The Hirzebruch surface Fn = Proj (OP1 ⊕ OP1 (n)) with n ≥ 0 is obtained as the quotient space Z/Gm of Z := A2∗ × P1 with respect to the following Gm -action λ((x0 , x1 ) × (w0 : w1 )) = (λx0 , λx1 ) × (λn w0 : w1 ),

λ ∈ Gm (k),

where A2∗ = A2 \ {(0, 0)}, and (x0 , x1 ) is a system of coordinates on A2 and (w0 : w1 ) is a system of homogeneous coordinates on P1 . The standard P1 -fibration ρ : Fn → P1 is obtained from the projection p1 : A2∗ × P1 → A2∗

by taking the Gm -quotients

ρ = p1 /Gm : Z → A2∗ /Gm = P1 .

Show this fact by proving the following assertions.

(1) Let U0 = {x0 ̸= 0} and U1 = {x1 ̸= 0} be the open sets of A2∗ , and let W0 = {w0 ̸= 0} and W1 = {w1 ̸= 0} be the open sets of P1 = Proj k[w0 , w1 ]. Then the open sets   −1 w1 , U0 × W0 = Spec k x0 , x1 , x0 , w0   w1 , U1 × W0 = Spec k x0 , x1 , x−1 1 , w0   w0 U0 × W1 = Spec k x0 , x1 , x−1 , , 0 w1   w0 U1 × W1 = Spec k x0 , x1 , x−1 , 1 w1 are Gm -stable open sets and cover the variety Z. (2) The quotient spaces of the above four affine surfaces are given as follows:   x1 xn0 w1 , , F00 := (U0 × W0 )/Gm = Spec k x0 w 0   x0 xn1 w1 , , F10 := (U1 × W0 )/Gm = Spec k x1 w 0   x1 w 0 F01 := (U0 × W1 )/Gm = Spec k , , n x0 x0 w1   x0 w 0 F11 := (U1 × W1 )/Gm = Spec k , n . x1 x1 w1

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(3) These four quotient spaces are isomorphic to the affine plane and patched together to form an algebraic surface F by the following identifications, where we identify Fij (i, j = 0, 1) with the open sets of F : (xn1 w1 )/w0 = (x1 /x0 )n · ((xn0 w1 )/w0 ) on F00 ∩ F10 w0 /(xn0 w1 ) = ((xn0 w1 )/w0 )−1 on F00 ∩ F01 n n n −1 w0 /(x1 w1 ) = (x0 /x1 ) · ((x0 w1 )/w0 ) , (x0 /x1 ) = (x1 /x0 )−1 , on F00 ∩ F11 . (4) The restriction of ρ on Fij (i, = 0, 1) is given as follows:     x1 xn0 w1 x1 ρ|F00 : Spec k , −→ V0 := Spec k , x0 w 0 x0     x0 x0 xn1 w1 −→ V1 := Spec k , , ρ|F10 : Spec k x1 w 0 x1     x1 w 0 x1 ρ|F01 : Spec k , , −→ V0 := Spec k x0 xn0 w1 x0     x0 x0 w 0 ρ|F11 : Spec k , −→ V1 := Spec k . x1 xn1 w1 x1 This shows that ρ−1 (V0 ) = V0 × P1 and ρ−1 (V1 ) = V1 × P1 . (5) We can write OP1 ⊕ OP1 (n) as follows: (OP1 ⊕ OP1 (n))|V0 = OV0 w0 ⊕ OV0 (xn0 w1 ),

(OP1 ⊕ OP1 (n))|V1 = OV1 w0 ⊕ OV1 (xn1 w1 ). This identification shows that F ∼ = Fn as P1 -bundles over P1 . (6) The curve defined by w1 = 0 (resp. w0 = 0) defines a cross-section M0 (resp. M1 ) of ρ such that (M02 ) = −n (resp. M12 ) = n). Answer. To verify each assertion, we need some computation, which is not difficult. So verification is left to the readers. 6. With the notations in the proof of the assertion (1) of Lemma 3.1.20, obtain the coefficient m of the component L when n = 10 and d = 3. Answer. Since 10/3 = [4, 2, 2] and 10/7 = [2, 2, 4] the fiber Fe has the weighted dual graph −4

−2

−2

−1

−4

−2

−2

A1

A2

A3

L

B3

B2

B1 .

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Write Fe as Fe =

3 X (αi Ai + βi Bi ) + mL, i=1

where α1 = 1, − 4α1 + α2 = 0, α1 − 2α2 + α3 = 0, α2 − 2α3 + m = 0,

β1 = 1, − 2β1 + β2 = 0, β1 − 2β2 + β3 = 0, β2 − 4β3 + m = 0. Since α1 = β1 , the above equations have a solution α1 = 1, α2 = 4, α3 = 7, β1 = 1, β2 = 2, β3 = 3, m = 10.

Hence we obtain m = n = 10. 7. Let R be an affine k-domain such that R is a UFD and R∗ = k ∗ . Let φ is an injective k-homomorphism. Let X = Spec R and let Φ = a φ : X → X. Suppose that Φ is a quasi-finite morphism, i.e., for any point z ∈ X, the inverse image Φ−1 (z) is either a finite set or the empty set. Show that the image of Φ contains all points of codimension one. Answer. We prove the assertion by contradiction. We write Φ as Φ : Xu → Xℓ to distinguish the source X from the target X, where Xu = Xℓ = X. Suppose that a point p ∈ Xℓ of codimension one is not in the image of Φ, where p is a prime ideal of height one. Since R is a UFD, p = f R for f ∈ R. Suppose that α(f ) ̸∈ k. Then α(f ) is not a unit of R. Let P be a prime divisor of α(f ), whence ht (P) = 1. Note that α(f ) ∈ P if and only if f ∈ α−1 (P). Since p is not in the image of Φ, it follows that ht (α−1 (P)) ≥ 2, whence Φ is not quasifinite over α−1 (P). This contradicts the assumption. So, α(f ) ∈ k ∗ . Namely, α(f − c) = 0 for some c ∈ k. Since f ̸∈ k ∗ , this contradicts the assumption that α is injective. 8. Let A be a B-algebra domain. Let δ be a B-trivial derivation of A. Assume that δ is locally nilpotent and there exists an element u of A such that δ(u) = 1. Show that A = R[u], where R = Ker δ. Answer. Since δ is locally nilpotent, there exists a ring homomorphism φ = exp(tδ) : A → A[t], which is defined by X1 δ i (a)ti , a ∈ A, φ(a) = i! i≥0

where t is a variable over A and φ(a)|t=0 = a. For an element a ∈ A, define degδ (a) as degt φ(t). Let R = Ker δ. We prove the inclusion

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a ∈ R[u] for every a ∈ A by induction on degδ (a). If degδ (a) = 0 then a ∈ R ⊂ R[u]. Let n = degδ (a). Then δ n (a) ∈ R because δ n+1 (a) = 0. Let a′ = a − (1/n!)δ n (a)un . Then degδ (a′ ) < n. By the induction hypothesis, a′ ∈ R[u]. Then a ∈ R[u] because (1/n!)δ n (a)un ∈ R[u]. 9. Let f : X → Y be a quasi-finite morphism of algebraic varieties. Assume that X is a normal projective variety. Show that f is a finite morphism. Answer. Let ν : Ye → Y be the normalization morphism of Y in the function field k(X). Then we have a splitting f ν f : X −→ Ye −→ Y, e

where fe is the morphism induced by f . If we take affine open coverings U = {Ui }i∈I of X and V = {Vj }j∈J in such a way that f |Ui : Ui → Vα(i) (i ∈ I) for a mapping of sets α : I → J. Let Ui = Spec Ai and Vj = Spec Bj . Since f is quasi-finite, k(X) is a finite extension of k(Y ) via f ∗ . So, Bα(j) is viewed as a k-subalgebra of Ai . Since Ai is eα(i) of Bα(i) integrally closed in Q(Ai ) = k(X), the integral closure B in k(X) is a k-subalgebra of Ai . Hence we have a splitting fg |U

ν

] f |Ui : Ui −→i Veα(i) := Spec B α(i) −→ Vα(i) . Patching the local morphisms fg |Ui (i ∈ I), we obtain a morphism fe : X → Ye . Since X is projective, by [31, II, Cor. 4.8], fe is a proper morphism.15 Since fe is a quasi-finite, birational morphism and Ye is a normal variety, by Zariski’s main theorem, fe is an open immersion. Hence fe is an isomorphism. Since the normalization morphism ν : Ye → Y is a finite morphism, so is the morphism f . 10. Let X = Spec A be a smooth factorial affine surface with A∗ = k ∗ . Let C = V (f ) be an irreducible curve on X such that κ(Y ) = −∞, where Y = X \ C. Show that X ∼ = A2 = Spec k[x, y] and one of the coordinates x, y can be chosen as f . Answer. Since Y = Spec A[f −1 ], Y is a smooth affine surface. By the assumption, κ(Y ) = −∞. By Theorem 3.5.9, Y is affine-ruled. Hence there is an A1 -fibration ρ′ : Y → B ′ , which is clearly extended to an A1 -fibration ρ : X → B such that C is contained in a fiber of ρ. By an 15 A morphism f : X → Y of k-schemes is a proper morphism if f is separated, of finite type and universally closed, i.e., for any k-scheme T , fT : X ×k T → Y ×k T is a closed mapping.

398

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algebraic characterization of the affine plane (see Theorem 2.1.5), X is isomorphic to the affine plane, B ∼ = A1 , C ∼ = A1 and f is a coordinate of B, i.e., B = Spec k[f ]. Hence f can be taken as a one of the coordinates of X. 11. Let X be a normal algebraic variety. Assume that there exists a unie → X with the fundamental group G which is versal covering f : X a finite group by the convention of this book. Let g : Y → X be an e irreducible finite ´etale covering. Show that Y ∼ for a subgroup = X/H g h e H of G and f splits as X −→ Y −→ X. e f ). Since Answer. Consider a fiber product W := (Y, g) ×X (X, g : Y → X is a finite ´etale morphism, so is the base change p2 = e which is the second projection. Let Z be a connected gXe : W → X, e is also a finite ´etale covering. component of W . Then p2 |Z : Z → X e is simply connected, p2 |Z is an isomorphism. Hence we obtain Since X p1 | Z e∼ a morphism h : X = Z −→ Y such that f = g ◦ h, where p1 is the first e induced projection. Consider the field extensions k(X) ⊂ k(Y ) ⊂ k(X) e by the morphisms g and h. Since k(X)/k(X) is a Galois extension with e H. group G, there exists a subgroup H of G such that k(Y ) = k(X) e → X/H e Then the quotient morphism q : X splits the morphism h as q

α

e −→ X/H e h:X −→ Y,

where α is a birational morphism which is, further, finite because so e is h. Since Y and X/H are normal varieties, Zariski’s main theorem e implies that α : X/H → Y is an isomorphism. 12. Let Y = Spec k[x, y, x−1 ] = A2 \{x = 0}. Prove the following assertions (see [3, Theorem 1.2]): (1) Let φ : Y → Y be an ´etale endomorphism. Then the associated algebra homomorphism α := φ∗ : k[x, y, x−1 ] → k[x, y, x−1 ] is given by xr y + g(x) , α(x) = cxn , α(y) = xm c ∈ k ∗ , r ∈ Z, r ≥ 0, g(x) ∈ k[x]. (2) φ is a finite morphism. Hence (GJC) holds for Y . Answer.

(1) The algebra homomorphism α is given by f (x, y) α(x) = cxn , α(y) = , xm c ∈ k ∗ , n, m ∈ Z, n ̸= 0, m ≥ 0, f (x, y) ∈ k[x, y].

Geometry and Topology of Polynomial Rings

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Then we have the following matrix relation for differentials      dα(x) ncxn−1 0 dx = . dα(y) (fx xm − mf xm−1 )x−2m fy x−m dy Since φ is ´etale by the assumption, the Jacobian matrix is invertible in k[x, y, x−1 ], whence we have (ncxn−1 )(fy x−m ) = dxℓ , Namely we have fy =

d ℓ+m−n+1 x , nc

d ∈ k ∗ , ℓ ∈ Z.

ℓ + m − n + 1 ≥ 0.

This implies that f ∼ xr y + g(x),

r = ℓ + m − n + 1 ≥ 0, g(x) ∈ k[x],

where ∼ means that f is a multiple of an element of k ∗ of the term in the right side. (2) It suffices to show that k[x, y, x−1 ] is integral over k[α(x), α(y), α(x)−1 ]. By (1), we have k[α(x), α(y), α(x)−1 ] = k[xn , (xr y + g(x))x−m , x−n ]. Note that k[x, (xr y + g(x))x−m , x−1 ] = k[x, y, x−1 ]. Hence k[x, y, x−1 ] is integral over k[α(x), α(y), α(x)−1 ]. Note that α is an automorphism if and only if n = 1. 13. Let f : X → Y be a finite dominant morphism between smooth algebraic varieties X, Y and let d = deg f . Suppose that the inverse image f −1 (y) has d points for every closed point y ∈ Y . Show that f is an ´etale covering. Answer. We may assume that X = Spec A and Y = Spec R, where A and R are affine k-domains, A is an R-algebra and A is a finite R-module. Note that f is R-flat because X and Y are smooth (see Remark 1.9.9(4)(ii)). It suffices to show that for every closed point x ∈ X and y = f (x), we have mX,x = mY,y OX,x . Replacing A and R by A ⊗R OY,y and OY,y , we may assume that R is a local ring (O, m) and A is a finite, flat O-module. Then A is a free R-module of rank d. In fact, since Q(A) = A ⊗R Q(R) is a Q(R)-module of rank d, the rank b be the m-adic completion of O. of the free O-module A is d. Let O Ld b ∼ b b Then A = A ⊗O O = i=1 OX,xi , where f −1 (y) = {x1 , . . . , xd }, which ∼ b bX,x ∼ b b implies that O i = O for every i. Hence OX,xi = O for every i. This bX,x = mO bX,x . Since mX,x ⊇ mOX,x and O bX,x implies that mX,x O i

i

i

i

i

is a faithfully flat OX,xi -module, it follows that mX,xi = mOX,xi .

i

400

Affine Algebraic Geometry

14. Let C := {X03 +X13 +X23 = 0} be a cubic curve on P2 and let S := P2 \C. Let T := {X03 + X13 + X23 + X33 = 0} be a cubic hypersurface in P3 and let π : P3 → P2 be the projection π([X0 , X1 , X2 , X3 ]) = [X0 , X1 , X2 ].16 Denote by T0 the curve T ∩ {X3 = 0}. Verify the following assertions.

(1) π : T \ T0 → S is a Galois covering of group Z/3Z. (2) T is a del Pezzo surface with KT = OT (−T0 ). There exist six points in general position on P2 , say P1 , . . . , P6 , such that T is obtained from P2 by blowing up these points. Let Ei be the exceptional curve lying over Pi . Then each Ei meets T0 transversally in exactly one point and hence each Ei is a straight line in P3 . Let τ : T → P2 be the blowing-up morphism and let C ′ = τ (T0 ). We have an isomorphism T \ (T0 ∪ E1 ∪ · · · ∪ E6 ) → P2 \ C ′ .

∼ C ′ , there is an isomorphism C ∼ Since T0 = = C ′ . Hence there exists an automorphism of P2 which maps C onto C ′ . Clearly, the morphism π : T \ (T0 ∪ E1 ∪ · · · ∪ E6 ) → S is an ´etale but nonfinite morphism. Thus we obtain a non-finite ´etale endomorphism fe := π ◦ τ −1 : S → S. (3) Consider the following action of Z/3Z = ⟨σ⟩ on T induced by a Z/3Z-action on P3 , σ([X0 , X1 , X2 , X3 ]) = [θX0 , θ2 X1 , θX3 , X4 ]. Here θ is a primitive cubic root of unity. This action has no fixed points on T \ T0 and it commutes with the projection π : P3 → P2 and the covering transformation h : T − T0 → T − T0 , where σ([X0 , X1 , X2 ]) = [θX0 , θ2 X1 , θX3 ],

h([X0 , X1 , X2 , X3 ]) = [X0 , X1 , X2 , θX3 ]. (4) There exist six disjoint lines F1 , . . . , F6 on T such that σ keeps this set of lines stable. We can use these six skew lines on T 17 to get a morphism g : T → P2 such that these six lines are the exceptional curves for the blowing up morphism g. Then σ acts fixed-point freely on T \ (T0 ∪ E1 ∪ . . . ∪ E6 ) and commutes with the projection π : T \ T0 → S. Thus we get an induced non-finite ´etale endomorphism from the quotient space S/(σ) to itself. 16 The projection π is defined on P3 \ {[0, 0, 0, 1]}. Since T ̸∋ [0, 0, 0, 1], π is defined on T as a morphism. 17 Two lines L and L′ on P3 are in a skew position if L ∩ L′ = ∅.

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Proof. (1) The morphism π : T → P2 is a finite morphism of degree 3 such that π −1 (P ) consists of 3 points if P ̸∈ C and only one point if P ∈ C. For example, π|U0 : U0 → π(U0 ) is defined by (x, y, z) 7→ (x, y), where U0 = {X0 ̸= 0}, x = X1 /X0 , y = X2 /X0 , z = X3 /X0 and x3 + y 3 + z 3 + 1 = 0. Hence z has three roots if x3 + y 3 + 1 ̸= 0 and only one root z = 0 if x3 + y 3 + 1 = 0. By Problem 13, this shows that π|T \T0 : T \T0 → P2 \C is an ´etale covering. The group G = ⟨θ⟩ of cubic roots of unity acts on T \ T0 by [X0 , X1 , X2 , X3 ] 7→ [X0 , X1 , X2 , θX3 ]. Hence π|T \T0 is a Galois covering. (2) Let F be a hypersurface of degree d in Pn . It is known that KF ∼ (KPn + F )|F ∼ OF ((−n − 1 + d)H), where H is a hyperplane. In the present case, n = d = 3. Hence KT ∼ −HT , where OT (HT ) = OP3 (H) ⊗OP3 OT . Since HT is an ample divisor by Nakai criterion of ampleness (see Theorem 1.8.13), KT is a del Pezzo surface. By Manin [52], T is obtained by blowing up six points P1 , . . . , P6 in general position.18 Since (KT · Ei ) = (−HT · Ei ) = (−H · Ei ) = −1, it follows that Ei meets a hyperplane in one point in P3 . This implies that Ei is a line in P3 which is contained in T . The converse holds, i.e., a line E in P3 contained in T is a (−1)-curve in T . The remaining assertions are proved in Theorem 3.4.6. (3) Straightforward. (4) Let F1 := {X0 + X1 = 0 = X2 + X3 }. Then F2 := σ(F1 ) = {X0 + θX1 = 0 = X2 + θ2 X3 } and F3 := σ 2 (F1 ) = {θX0 + X1 = 0 = X2 + θX3 }. It is easy to see that these three lines are mutually disjoint. Let F4 := {X0 +θX3 = 0 = X1 +X2 }. Then F5 := σ(F4 ) = {X0 +X3 = 0 = θX1 + X2 } and F6 := σ 2 (F4 ) = {θX0 + X3 = 0 = X2 + θX3 }. Then F4 , F5 , F6 are mutually disjoint and F1 is disjoint from each of them. It follows that the six lines Fi are mutually disjoint and σ preserves the set of these lines. 18 The six points P , . . . , P are said to be in general position if (1) no three points lie 1 6 on the same line and (2) six points do not lie on the same conic. If the points P1 , . . . , P6 satisfy these conditions, (−1)-curves on T are (i) six exceptional curves of the blowingup, (ii) fifteen proper transforms of the lines passing through two of six points P1 , . . . , P6 and (iii) six proper transforms of conics passing through five points of P1 , . . . , P6 . Hence there are 27 of them. If three of P1 , . . . , P6 lie on the same line, say P1 , P2 , P3 lie on a line ℓ, then the proper transform ℓ′ of ℓ is a (−2)-curve. If six points lie on the same conic C the proper transform C ′ is a (−2)-curve. Hence these points being in general position requires that the surface obtained by blowing-up of these points does not have (−n)-curves with n ≥ 2. It is also known that one can choose six (−1)-curves E1 , . . . , E6 among 27 of such curves so that E1 , . . . , E6 are disjoint from each other. The contraction of these mutually disjoint six (−1)-curves gives back P2 .

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15. Let C be a cuspidal plane cubic curve in P2 defined by f ((X0 , X1 , X2 ) = X02 X1 − X23 = 0.

Hence the cusp is P∞ = (0, 1, 0). Let ℓ0 be the line X0 = 0. Then C ∼ 3ℓ0 and (C · ℓ0 ) = 3. Set X = P2 \ C. Verify the following assertions. (1) X has an A1 -fibration ρ induced by the linear pencil Λ on P2 spanned by C and 3ℓ0 . The fibration ρ has a unique multiple fiber 3(ℓ0 ∩ X). Hence X is an affine pseudo-plane. e →X (2) Pic (X) = Z/3Z. There exists a triple cyclic ´etale covering X e which is a Galois covering. The surface X is a surface defined by an equation x2 y = z 3 + 1. e is a Danielewski surface of another kind,19 and simplyHence X connected. (3) X has a non-finite ´etale endomorphism of degree d > 1 if d ≡ 1 (mod 3).20 Answer. (1) A similar case was already treated in the text. So, the detail is left to the readers. (2) A triple cyclic covering q : W → P2 ramifying totally over the curve C is constructed by taking a hypersurface W in P3 defined by an equation X33 = X02 X1 − X23 .

e = W \ q −1 (C). Since q −1 (C) is the intersection of W with the Let X e is defined by an equation hyperplane X3 = 0, X x2 y = z 3 + 1, e is a where x = X0 /X3 , y = X1 /X3 and z = X2 /X3 . Hence X e Danielewski surface, which is simply-connected because X contains A2 e → X is the universal covering. as an open set.21 Hence q|Xe : X 19 A smooth affine surface X is a Danielewski surface by definition if X has an A1 fibration f : X → A1 = Spec k[x] such that all fibers of f except for f −1 (0) are isomorphic to A1 and f −1 (0) is a disjoint sum of irreducible components A1 with multiplicity 1. 20 This question is fairly hard to answer. So, the readers might well treat it as a reminder of a result. 21 Consider the projection f : X e → A1 defined by (x, y, z) 7→ x. If x ̸= 0 then the fiber is isomorphic to A1 . The fiber f −1 (0) is a disjoint union of three affine lines. If we throw two components from the fiber f −1 (0), the complement is isomorphic to A2 .

Geometry and Topology of Polynomial Rings

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(3) Let Γ = ⟨θ⟩ ∼ = Z/3Z with θ3 = 1. Then Γ acts on W by

(X0 , X1 , X2 , X3 ) 7→ (X0 , X1 , X2 , θ2 X3 ). e by Hence Γ acts on X X1 X2 X0 ,= y = , z= θ · (x, y, z) = (θx, θy, θz), x = X3 X3 X3 e e is, in and the quotient surface X/G is isomorphic to X. The surface X e fact, the surface S(3, 2) with an θ-action of weight a = 2 in the termie is the surface S(3, 2, 2). The nology in [16]. Hence the surfaceX = X/Γ assertion follows from the construction of counterexamples in [ibid.].22 16. Let ℓ0 , ℓ1 , ℓ2 be non-concurrent three lines in P2 and let X = P2 \ (ℓ1 ∪ ℓ2 ∪ ℓ3 ). Prove the following assertions. (1) X is isomorphic to A2 \ (x and y axes), which is isomorphic to A1∗ × A1∗ . (2) An ´etale endomorphism φ of X is a finite morphism. (3) Let A be the coordinate ring of the n-dimensional algebraic torus Gnm . Then every quasi-finite endomorphism φ∗ of A is a finite homomorphism, which is factorized by some N th power endomorphism.

Answer. (1) There exists an automorphism α ∈ PGL (3, k) which maps the lines ℓ0 , ℓ1 , ℓ2 to the coordinate lines {X0 = 0}, {X1 = 0}, {X2 = 0}, respectively. Hence, by setting x = X1 /X0 and y = X2 /X0 , we may assume that ℓ0 is the line at infinity of A2 , ℓ1 is the y-axis and ℓ2 is the x-axis. Hence P2 \ (ℓ0 ∪ ℓ1 ∪ ℓ2 ) is A2 \ ({x = 0} ∪ {y = 0}), which is isomorphic to A1∗ × A1∗ . (2) The assertion follows from a more general assertion (3). (3) Write A as an n-dimensional Laurent polynomial ring ±1 k[x±1 1 , . . . , xn ]. Then one can write φ∗ (xi ) = αi xa1 i1 xa2 i2 · · · xanin

(1 ≤ i ≤ n), αi ∈ k ∗ .

The Jacobian determinant can be computed as follows.   ∗ φ∗ (x1 ) · · · φ∗ (xn ) φ (x1 ), . . . , φ∗ (xn ) = · |R|, J x1 , . . . , x n x1 · · · xn

22 A similar construction is possible. Let C = {f (X , X , X ) = 0} be a rational curve 0 1 2 of degree n ≥ 3 with a cusp of type (n, n − 1), i.e., a cusp of type xn−1 = z n . Let X = P2 \C. Then X is an affine pseudo-plane with a unique multiple fiber of multiplicity n. The fundamental group π1 (X) is Γ = Z/nZ = ⟨ω⟩ with ω n = 1 and its universal e = {xn−1 y = z n + 1}, where Γ acts as covering is obtained as a Danielewski surface X ω (x, y, z) = (ωx, ωy, ωz). Hence X = X/Γ e = S(n, n − 1, n − 1), and it has a non-finite ´ etale endomorphism of degree d if d ≡ 1 (mod n(n − 2)).

404

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where a11  a21 R=  ··· an1 

a12 a22 ··· an2

··· ··· ··· ···

 a1n a2n  . ··· 

ann

Let R∗ = (bij ) be the adjoint matrix of R. Then R∗ R = N En , where N = det R. We show that if φ∗ is quasi-finite then N ̸= 0. Suppose that N = 0. Then there exists a non-zero integral row vector (c1 , . . . , cn ) such that (c1 , . . . , cn )R = (0, . . . , 0). Then we have  c  ∗ c  c1  ∗ φ (xn ) n x1 · · · xcnn φ (x1 ) 1 ∗ = 1. ··· =φ α1 αn α1c1 · · · αncn Meanwhile, if φ∗ is quasi-finite, then φ∗ is injective. This is a contradiction. Hence N ̸= 0. It is then easily verified that  ∗ b  ∗ b φ (x1 ) i1 φ (xn ) in ··· = xN (1 ≤ i ≤ n). i α1 αn This implies that we have inclusions ±N ∗ ±1 ±1 C[x±N , . . . , φ∗ (xn )±1 ] ⊆ C[x±1 1 , . . . , xn ] ⊆ C[φ (x1 ) 1 , . . . , xn ].

Hence both inclusions are finite homomorphisms. 17. Let B be a smooth projective curve and let U = Spec A be an affine open set. Let S = B \ U . Suppose that A∗ ⊋ k ∗ . Prove the following assertions. (1) If B is irrational, then S contains more than two points. (2) If B is rational and rank A∗ /k ∗ ≥ 2 then S contains more than two points. (3) In both cases (1) and (2), we have κ(U ) ≥ 0. Answer. (1) Since U is affine, the set S is not the empty set. Let a ∈ A∗ \ k ∗ . Then the divisor (a) has support on S. Hence S contains two or more points. Suppose that S = {b1 , b2 }. Since deg(a) = 0, we may assume that (a) = n(b1 − b2 ). If (a) = n(b2 − b1 ). Consider a−1 instead of a. This implies that the rational mapping q ′ : B 99K P1 defined by b 7→ a(b) ∈ P1 defines an nth cyclic covering q : B → P1 , i.e., a cyclic group G = Z/nZ acts on B and the quotient morphism q : B → P1 is given by the rational mapping q ′ which ramifies totally over the points q(b1 ) and q(b2 ). Then, by the Riemann-Hurwitz formula, the genus g of B is given by 2g − 2 = −2n + 2(n − 1), whence g = 0. This is a contradiction since g > 0. Hence S contains more than two points.

Geometry and Topology of Polynomial Rings

405

(2) Let S = {b0 , . . . , br }. We can choose an inhomogeneous coordinate x of B\{b0 } so that b0 and bi (1 ≤ i ≤ r) are given by x = ∞ and x = ci . Qr Qr Then A = k[x, i=1 (x − ci )−1 ], whence A∗ ∼ = k ∗ × i=1 Z[(x − ci )−1 ]. So, rank A∗ /k ∗ = r. If r ≥ 2, then S contains points {b0 , b1 , b2 }. (3) Let D be a reduced effective divisor of B such that Supp D = S. If B is irrational then |KB | = ̸ ∅. Hence |D + KB | = ̸ ∅ and κ(U ) ≥ 0. Suppose that B ∼ = P1 . Then KB ∼ −2b with a point b ∈ B. By assumption, D ∼ (r + 1)b ≥ 3b and D + KB ∼ (r − 1)b ≥ b, whence κ(U ) = 1 > 0. 18. Let B be a smooth algebraic curve and let φ : B → B be a dominant endomorphism. Prove the following assertions. (1) If B is projective then φ is a finite morphism. (2) If B is affine and has only one place at infinity then φ is a finite morphism. (3) If B ∼ = A1 then φ is a finite morphism. Answer. (1) Since dim B = 1, φ is a quasi-finite morphism. A proper and quasi-finite morphism is a finite morphism. (2) Let B be a smooth completion of B. Then B is a unique projective curve such that k(B) = k(B). Then B is isomorphic to the normalization of B ℓ in the function field k(Bu ). Hence φ extends to a dominant endomorphism φ : B → B. Let b ∈ B \ B. Then φ−1 (b) ⊂ B \ B. Since B \ B = {b}, it follows that φ−1 (b) = B \ B. Hence φ is a finite morphism as φ = φ ×B B. (3) Write Bu = Spec k[x] and Bℓ = Spec k[y]. Then φ is defined by a k-algebra homomorphism α : k[y] → k[x]. Hence α(y) = a0 xn + a1 xn−1 + · · · + an−1 x + an , where a0 ̸= 0. This implies that x is integral over k[α(y)]. So, φ is a finite morphism.

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Chapter 4

Postscript

The subject treated in this volume is a part of affine algebraic geometry, where there are many results which prompted its developments. We refer to an article [64] for quick reviews on the developments. Here, for the readers who are willing to study more in affine algebraic geometry, we review important results and developments of research in this area. The other references for the details are [27, 59]. 4.1

AMS theorem and thereafter

The theorem was originally proved by Abhyankar-Moh [2] and Suzuki [93] in early 1970s. Since then there are 13 different proofs published including the proof in this volume. Every proof, geometric or topological, uses information at infinity given by a curve C isomorphic to A1 . The following theorem, called Theorem of Lin-Zaidenberg, is one of definitive results. Theorem 4.1.1. Let C be an irreducible algebraic curve in A2 defined over the field C of complex numbers. Assume that C is topologically contractible. Then there exists a system of coordinates {x, y} of A2 such that C is defined by xm = y n , where gcd(m, n) = 1. Next goal is perhaps to determine the standard forms of an irreducible algebraic curve C in A2 which is isomorphic to A1∗ . There are several trials (see [12]). There were big leaps in a research of projective plane curves with only cuspidal singularities by [47] and [48]. The first one is the Coolidge-Nagata conjecture. Theorem 4.1.2. Let C be a projective rational curve in P2 defined over the complex number field C with only cuspidal singularities, which is equivalent 407

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Affine Algebraic Geometry

to saying that there exists a birational bijective morphism θ : P1 → C. Then there exists a Cremona transformation α ∈ Bir(P2 ) such that α(C) is a line. A key ingredient is a refined application of the log minimal model program applied to a pair (V, 21 D), where φ : (V, D) → (P2 , C) is a birational morphism such that D = φ−1 (C) is an SNC divisor. The argument used in the proof is a refinement of the one used to prove Theorem 3.1.21. The second one also settles a long standing problem on the number of singular points of a cuspidal rational curve [48]. Theorem 4.1.3. Let C be a complex projective plane rational curve with only cuspidal singularities. Then the number of cuspidal singular points is at most four. In the proofs of both theorems, the use of the logarithmic version of Miyaoka-Yau inequality due to Kobayashi [45], which we call the MiyaokaYau-Kobayashi inequality, is essential to control the boundary divisor on the (almost) minimal model. See [59, Chapter 2, Theorem 6.6.2] for the inequality. 4.2

Suzuki-Zaidenberg formula

Topological methods and tools work well for affine algebraic varieties defined over C. If (V, D) is a log smooth completion of a smooth affine variety X, homology theory or cohomology theory for a pair (V, D), together with long exact sequences and Poincar´e and Lefschetz dualities, give necessary information on topological properties of X. The alternating sum X eX) := (−1)i bi (X) i≥0

of the Betti numbers bi (X) := rank Hi (X; Q) for 0 ≤ i ≤ dim X is the topological Euler number of X. The following formula due to Suzuki [94] and Zaidenberg [97] and it is very useful to detect and determine the singular fibers of a fibration on algebraic surfaces. Theorem 4.2.1. Let f : X → C be an affine surjective morphism from a smooth algebraic surface X onto a smooth algebraic curve C such that a general fiber F is irreducible. Let F1 , . . . , Fr exhaust singular fibers of f (this is the smallest set of fibers of f outside which f is a C ∞ locally trivial

Postscript

409

fiber bundle). Then we have e(X) = e(C) · e(F ) +

r X i=1

(e(Fi ) − e(F )).

Further, e(Fi ) ≥ e(F ) for all i and if the equality holds for some i then F is either A1 or A1∗ and Fi is isomorphic to F if taken with the reduced structure. For the other usage of topological methods, the readers are referred to [27, Chapter 1]. 4.3

Cancellation problems

Let X and Y be algebraic varieties defined over k, and let T be an affine variety. The Zariski cancellation problem asks if X × T ∼ = Y × T implies n 1 ∼ X = Y . If Y = A and T = A , the problem asks if X ∼ = An . If n ≤ 2 this problem is affirmatively answered. For more details, the readers are referred to [59]. But, if X and Y are Danielewski surfaces of the type xn y = z m − 1, the answer is negative by Danielewski. His result is not published. For the case Y = A3 and T = A1 over an algebraically closed field of positive characteristic, Neena Gupta [26] gave a negative answer by giving a counterexample based on Asanuma’s threefold [5]. There are many important results about this problem, and it is almost certain that this problem is one of the leading problems in affine algebraic geometry in future.

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Index

ˇ Cech cohomology, 171

(e, i)-transformation, 206 2-section, 358 C1 -field, 140 D-dimension, 118 G-action faithful, 238 G-invariant, 235 G-morphism, 292 A1 -cylinder, 195 A1 -fibration multiple fiber, 369 equilateral, 376 of affine type, 197 of complete type, 197 pathological, 216 A1∗ -fibration, 336 twisted, 337 untwisted, 337 A1∗ -singular fiber, 361 of non-simple type, 361 of simple type, 361 of the first kind, 361 of the second kind, 361 of the third kind, 361 P1 -bundle, 141 P1 -fibration degenerate fiber, 141 singular fiber, 141 smooth fiber, 141 a-adic topology, 157 n-ple covering, 296

addition theorem of Kawamata, 386 additive valuation, 60 admissible data, 198 affine domain, 12 affine pseudo-plane, 328 principal chain, 328 side chain, 328 type (d, n, r), 328 type (d, r), 328 affine scheme, 3 of finite type, 27 reduced, 27 algebraic curve, 120 algebraic group, 232 abelian variety, 233 identity component, 232 linear, 233 radical, 233 reductive, 233 semi-simple, 233 unipotent, 233 unipotent radical, 233 algebraic surface, 120 algebraic variety, 43 projective, 55 algebraically independent, 15, 20 amalgamated product, 287 arithmetic genus, 132 formula, 132 417

418

Affine Algebraic Geometry

Artin-Rees’s lemma, 179 automorphism tame, 226 wild, 226 base change, 42, 180 birational automorphism natural, 256 blowing-down, 138 blowing-up, 128 center, 128 oscillating, 273 branch divisor, 296 branch locus, 286 canonical sheaf, 109 Cauchy sequence, 157 refined, 157 chain homotopy, 171 closed subvariety, 30 codimension, 10 cofinite, 254 cohomologous, 111 colinear, 317 complete intersection, 98 completion a-adic, 157 component terminal, 192 transversal, 188 vertical, 188 composition series, 84 length, 85 concomitant, 69 configuration tree, 192 contraction, 138 Coolidge-Nagata conjecture, 407 coordinate ring, 3 homogeneous, 49 coordinate triangle, 257 coplanar, 317 cross-section, 139 cubic hypersurface, 125 curve complete, 191

cylinderlike open set, 195 Danielewski surface, 316, 402 decomposition primary, 144 prime, 8 degree, 123 del Pezzo surface, 373 degree, 373 depth, 94 derivation, 100 locally nilpotent, 195 reduced, 390 similar, 379 universal, 101 descending chain condition, 84 dimension, 10 Krull, 10 direct image, 283 discriminant, 65, 242 divisor, 74, 113 ample, 124 boundary, 263 canonical, 113 Cartier, 110, 112 direct image, 283 inverse image, 283 irreducible, 74 linearly equivalent, 75 linearly independent, 323 logarithmic canonical, 119 normal crossings, 119 numerically equivalent, 136 principal, 75 reduced, 119 simple log divisor, 119 SNC, 263 minimal, 269 support, 113 very ample, 124 Weil, 74 with simple normal crossings, 119 principal polar part, 75 zero part, 75 divisor class group, 75

Index

domain factorial, 71 integrally closed, 14 normal, 14 drip, 382 removing, 382 easy addition formula, 386 element irreducible, 69 prime, 69 reducible, 69 regular, 94 elementary transformation, 142 elliptic curve, 125 embedded resolution of singularity, 134 embedding dimension, 88 endomorphism ´etale, 347 multiplication by m, 354 unramified, 347 Enriques criterion of ruledness, 385 equidimensionality, 154 ´etale covering, 280, 319 irreducible, 319 Euler number, 354 Euler-Poincar´e characteristic, 120 example 1.4.1, 45 exceptional curve, 128 feather, 326 fiber product, 41 fibration, 136 F -, 139 P1 -, 139 degenerate fiber, 141 singular fiber, 141 field rational function, 20, 31 field extension finitely generated, 20 purely transcendental, 20 field generator, 221 field of fractions, 3 field of quotients, 3

419

finite covering, 281 finite subgroup of GL (2, k) small, 249 first homology group at infinity, 382 fixed point, 236 fixed point locus, 292 flat, 150 faithfully, 150 flex, 304, 373 free product, 287 function defined, 31 rational, 31 regular, 31 function field, 43 fundamental curve, 228 fundamental group, 319 at infinity, 381, 382 fundamental point, 227 Galois covering, 284, 319 generic point, 30 genus, 121 geometric genus, 263 Gizatullin surface, 325 Going-up theorem, 17 graded homomorphism, 55 graded ring, 45 associated, 92 graph admissible, 267 branch number, 267 circle, 264 edge, 264 linear chain, 266 maximal, 267 loop, 264 rational, 267 rod, 267 star-shaped, 267 terminal, 264 tree, 264 vertex, 264 branching, 267 central, 267 weighted dual, 264

420

Affine Algebraic Geometry

greatest common divisor, 71 group scheme additive, 234 augmentation, 233 coinverse, 233 comultiplication, 233 multiplicative, 234 unipotent, 234 height, 10 Hilbert basis theorem, 11 Hilbert’s Nullstellensatz, 28 Hirzebruch surface, 142 Hodge index theorem, 136 homogeneous coordinate system, 50 homogeneous element, 45 homogenization, 58 homology plane, 342 Q-, 342 log, 342 log Q-, 342 homomorphism graded, 172 hyperplane, 57 hypersurface, 98 hypersurface section, 87 ideal associated prime, 148 conductor, 80 homogeneous, 45 irreducible, 144 irrelevant, 45 primary, 144 quotient, 8 radical, 4, 9 reducible, 144 inductive set, 155 inductive system, 155 integral, 12 integral closure, 14 integral domain geometric local, 91 integral extension, 14 integrally closed, 14 intersection multiplicity

local, 126 intersection number, 126 inverse image, 283 invertible sheaf, 110 transition functions, 110 involution, 361 irreducible component, 6 irreducible decomposition, 6 irregularity, 263 isotropy group, 292 Jacobian Conjecture, 345 Equivariant, 348 Extended, 348 Generalized, 347 Jacobian criterion of smoothness, 99 Jacobian matrix, 99, 109 Jacobian pair, 353 Kodaira dimension, 119 log, 119 logarithmic, 119 Krull’s height theorem, 87 intersection theorem, 157 principal ideal theorem, 86 length, 10 limit inductive, 155 inverse, 156 projective, 156 linear pencil proper transform, 132 strict transform, 132 linear system, 115 complete, 115 composed of a pencil, 117 dimension, 115 fixed part, 116 fixed-component free, 116 movable part, 116 pencil, 115 linear topology, 157 local chart, 37 affine, 37

Index

local homomorphism, 33 local isomorphism, 347 local ring catenary, 91 dominate, 124 geometric, 91 local-ringed space affine, 27 log canonical divisor, 119 log minimal model program, 408 log pair, 119 log ramification divisor, 387 log ramification formula, 387 log smooth completion, 269 N C-minimal, 385 minimal, 269 Lying-over theorem, 17 Makar-Limanov invariant, 380 method of undetermined coefficients, 346 Miyaoka-Yau-Kobayashi inequality, 408 Module, 164 direct image, 168 inverse image, 169 module faithfully flat, 151 flat, 151 graded, 45 module of differential 1-forms, 100 morphism, 33, 39, 279 G-equivariant, 292 R-, 33 S-, 40 affine, 180 birational, 67 closed embedding, 35 closed immersion, 35, 41 dominant, 67 faithfully flat, 151 fiber, 42 finite, 66, 67, 278 flat, 151 isomorphism, 37 local ringed spaces, 33

421

locally finite, 278 normalization, 66, 68 of finite type, 77 open, 153 open immersion, 35 proper, 397 quasi-finite, 77 quotient, 239 separable, 350 separated, 41 structure, 40 unramified, 276, 278 multiplicatively closed set, 1 multiplicity, 129, 207 Mumford-Ramanujam presentation, 382 N´eron-Severi group, 136 Nagata’s lemma, 75 Nakai criterion of ampleness, 135 Nakayama’s lemma, 59 neighborhood coordinate, 108 open, 31 Newton polygon, 351 nilradical, 9 Noether factorization theorem, 256, 262 Noether normalization lemma, 15 norm, 239, 243 normal ring, 62 normalization, 64, 68, 69 one-place, 198 at infinity, 198 open covering finer, 110 refinement, 110 open set affine, 34 pencil, 116 curves, 136 fiber, 136 irrational, 116 irreducible, 136

422

Affine Algebraic Geometry

rational, 116 Picard number, 136 place, 198 Platonic A1∗ -fiber space, 341 Platonic solids, 250 point closed, 9 fixed, 292 generic, 9 infinitely near, 132 infinitely near of order 1, 128 isolated, 76 ordinary, 128 points at infinity, 58 polynomial alternating, 242 monic, 12 primitive, 72 prescheme, 37 presheaf, 162 primary decomposition, 8 irredundant, 146 prime divisor, 9 embedded, 9 minimal, 9 projection, 41 projection formula, 284, 285 projective algebraic set, 55 degree, 135 projective general linear group, 249 projective space, 49 dual, 57 projective system, 156 pseudoreflection, 244 group, 244 purity of branch loci, 350 quotient singularity, 252, 294 Du Val, 254 rational double, 254 quotient space, 239 ramification total, 208 ramification divisor, 296 ramification index, 282, 283

ramification locus, 286 rational map, 43 birational, 44 dimension of the image, 118 domain of definition, 44 reduced form, 27, 207 reflection, 243 group, 244 regular extension, 24 regular sequence, 94 maximal, 94 residue field, 27 resolution of singularity minimal, 253 Riemann-Hurwitz formula, 282 Riemann-Roch theorem for a curve, 122 for a surface, 135 ring artinian, 76 local, 2 regular, 92 regular local, 92 ring of fractions, 2 ring of quotients, 1 ringed space, 169 local, 27 scheme, 37 S-, 40 geometrically irreducible, 43 geometrically reduced, 43 group scheme, 232 integral, 74 irreducible, 42 normal, 66 of finite type, 43 projective, 49 reduced, 42 section, 139 minimal, 143 negative, 189 Segre embedding, 316 self-intersection number, 128 semi-abelian variety, 355 separable extension, 26

Index

separatedness condition, 37 Serre duality, 120 Serre’s criterion of Normality, 97 set inductive, 4 sheaf, 32, 162 coherent, 106, 165 direct image, 36, 168 finitely generated, 165 homomorphism, 163 inverse image, 168 local section, 163 morphism, 163 quasi-coherent, 106, 165 section, 163 sheafification, 163 stalk, 33 structure, 32 sheaf of differential 1-forms, 107 sheaf of logarithmic differential 1-forms, 119 shift transformation, 269 singular point, 132 cuspidal, 343, 407 moving, 214 nodal, 343 source, 257 space irreducible, 6 noetherian, 5 quasi-compact, 5 reducible, 6 spectrum, 3 projective, 47 stabilizer group, 292 stabilizer locus, 292 stalk, 163 Stein factorization, 137 structure theorem of Kawamata, 386 subscheme closed, 39 open, 39 subvariety closed, 54 support, 351 surface

423

affine ruled, 385 connected at infinity, 385 relatively minimal, 139 symbolic nth power, 86 symmetric polynomial, 239 elementary, 239 fundamental theorem of, 240 system of coordinates, 108 system of parameters, 89 regular, 92 tangent space, 378 tangential direction, 378 target, 257 Taylor expansion, 98 Theorem of Abhyankar-Moh-Suzuki, 213 Akizuki, 84 Auslander-Buchsbaum-Nagata, 93 Ax, 346 Bertini the first, 137 the second, 137 Bezout, 187 Chevalley, 233 Chevalley-Shephard-Todd, 244 Ganong, 212 Grothendieck, 172 Gurjar-Pradeep-Shastri, 342 Gutwirth, 213 Iitaka, 354 Lin-Zaidenberg, 407 Luna, 293 Moh, 212 Oort, 232 Rentschler, 235 Russell, 222 Serre, 171, 288 Sumihiro, 331 Tsen, 140 torus, 233 trace, 243 transcendence basis, 21 transcendence degree, 21 transcendental, 20

424

Affine Algebraic Geometry

transform proper, 129, 130 strict, 129 total, 129, 130 transformation affine, 226 Cremona, 257 de Jonqui`ere type, 261 de Jonqui`ere, 226 de Jonqui`ere of degree n, 227 elementary, 258 Euclidean, 200 interchanging, 312 linear, 226, 257 quadratic, 258 quadric standard, 258 the second kind, 259 transition matrices, 283 unique factorization domain, 69 unit group reduced, 323

unit rank, 323 universal covering, 319 valuation ring, 59 discrete, 61 uniformisant, 62 value group, 60 variety affine, 30, 43 Cohen-Macaulay, 98 fibered, 386 geometrically regular, 97 normal, 66 regular, 97 simply connected, 281, 319 smooth, 97 word, 286 Zariski topology, 48 Zariski’s finiteness theorem, 379 Zariski’s main theorem, 77 Zorn’s lemma, 4