Algebraic Geometry 9781400876686

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ALGEBRAIC GEOMETRY

PRINCETON MATHEMATICAL SERIES

Editors: MAHSTON MORSE and A. W. TUCKER 1. The Classical Groups. By HERMANN WEYI. 2. Topological Groups. By L. PONTRJAGIN. Translated by EMMA LEHMER. 3. An Introduction to Differential Geometry. EISENHART.

By LUTHER PFAHLER

4. Dimension Theory. By WITOLD HUREWICZ and HENRY WALLMAN. 5. Analytical Foxindations of Celestial Mechanics. By AUREL WINTNER. 6. The Laplace Transform. By DAVID VERNON WIDDER. 7. Integration. By EDWARD J. MCSHANE. 8. Theory of Lie Groups. By CLAUDE CHEVALLEY.

9. Mathematical Methods of Statistics. By HARALD CRAMER. 10. Several Complex Variables. By S. BOCHNER and W. T. MARTIN. 11. Introduction to Topology. By SOLOMON LEFSCHETZ.

12. The Topology of Surfaces and their Transformations. By JAKOB NIELSEN and WERNER FENCHEL. 13. Algebraic Curves. By ROBERT J. WALKER.

14. The Topology of Fibre Bundles. By NORMAN STEENROD. 15. Foundations of Algebraic Topology. By S. EILENBERO and N. STEENROD.

16. Functionals of Finite Riemann Surfaces. By M. SCHIFFER and D. C. SPENCER.

17. Introduction to Mathematical Logic, Vol. I. By ALONZO CHURCH. 18. Algebraic Geometry. By SOLOMON LEFSCHETZ.

19. Homological Algebra. By HENRI CARTAN and SAMUEL EILENBERG.

ALGEBRAIC GEOMETRY

By SOLOMON LEFSCHETZ

PRINCETON, NEW JERSEY · 1953

PRINCETON UNIVERSITY PRESS

Published : 1953 by Princeton University Press London : Geoffrey Cumberlege, Oxford University Press L.C. CARD 52—13158

PRINTED BY THE PITMAN PRESS, BATH, ENGLAND

Preface The present volume grew out of a set of lithoprinted Lecture Notes issued in two parts in 1935-38 and long since out of print. The material of the Notes has been amplified considerably in places, and Chapters II and IV in parts, Chapters III and IX are new. In the main however the general program of the Notes has been preserved. In Chapters II, III, IV, on algebraic varieties the groundfield is generally merely taken infinite. In Chapters V to IX, which except for Chapter IX, are devoted to the classical study of algebraic curves, the groundfield is prudently taken to be algebraically closed and especially of characteristic zero. It is no secret that the literature on algebraic geometry, now nearly a century old, is as indigestible as it is vast. This field is now undergoing an extensive process of recasting and reorganization in which the most advanced arsenal of modern algebra is playing a fundamental role. At all events one cannot write on algebraic geometry today outside of the general framework of algebra. On the other hand many have come to algebraic geometry and have been attracted to it through analysis, and it would seem most desirable to preserve this attraction and this contact. A common ground for algebra and analysis is found in the method of formal power series which was adopted in the earlier Lecture Notes and is utilized here again to the full. This method has made it possible for example to operate with a general groundfield of characteristic zero, and yet to provide for algebraic curves a treatment surprisingly close to the classical treatment of Emile Picard's Traite d'Analyse, vol. 2, or of Severi's Vorlesungen uber algebraische Geometrie. It is not too much to say that the whole of the classical theory in which the periods play no role may be dealt with by means of formal power series. By way of preparation it is assumed that the reader is in possession of the rudiments of modern algebra (rings, fields, ideals, polynomials and their factorization) such as are amply developed for our purpose in any recent text. On the geometric side he should also possess elementary information on affine and projective spaces. In point of fact, the topics required along these lines in the book have been summarized in the first chapter and the first few pages of the second. We wish especially to give our thanks to Ernst Snapper, who read most carefully the whole manuscript and made many exceedingly valuable suggestions for improvement and corrections. We could scarcely exaggerate our debt to him. S. LEFSCHETZ Princeton, New Jersey

Contents Preface

γ

Chapter I. Algebraic Foundations § 1. Preliminaries § 2. Resultants and elimination § 3. Algebraic dependence. Transcendency § 4. Extension of the groundfield § 5. Diiferentials (characteristic zero)

3 5 6 7 10

Chapter II. Algebraic Varieties: Fundamental Concepts § 1. § 2. § 3. § 4. § 5.

Affine and projective spaces Algebraic varieties and their ideals General points. Dimension. Function field .... Projections Differentials. Singular points. Tangent spaces (ground­ field of characteristic zero) § 6. Some intersection properties

16 21 26 33 35 40

Chapter III. Transformations of Algebraic Varieties § 1. § 2. §3. § 4. § 5. §6. § 7.

Rational transformations Birational transformations Normalsystems Product spaces Algebraic correspondences Complementsonintersections Appendix: Groundfield of characteristic ρ

φO

.

.

.

47 49 53 61 64 72 74

Chapter IV. Formal Power Series §1. Basicconceptsandtheorems § 2. Algebroid varieties § 3. Local properties of algebraic varieties § 4. Algebraic varieties as topological spaces vii

78 81 89 96

viii

CONTENTS

Chapter V. Algebraic Curves, their Places and Transformations § 1. § 2. § 3. § 4. § 5. § 6. § 7. § 8.

Formal power series in one and two variables .... Puiseux's theorem The places of an algebraic curve Valuations Multiple points, intersections and the places . . . . Rational and birational transformations and the places . Space curves Reduction of singularities

98 99 104 108 112 117 121 127

Chapter VI. Linear Series § 1. Divisors and their classes § 2 . L i n e a r series: F i r s t p r o p e r t i e s § 3. Birational models and linear series § 4. Rational, elliptic and hyperelliptic curves § 5. Adjoint curves and series § 6. The theorem of Riemann-Roch

135 137 142 144 145 149

Chapter VII. Abelian Differentials § 1. Preliminary questions § 2. The divisors of the differentials. Differentials of the first kind § 3. Elliptic and hyperelliptic differentials. Canonical model . § 4. Differentials of the second and third kinds . . . . § 5. Jacobian series

155 158 160 165 172

Chapter VIII. Abel's Theorem. Algebraic Series and Corre­ spondences §1. § 2. § 3. § 4. § 5.

Abel's theorem Algebraic series Algebraic correspondences between two curves Algebraic correspondences of a curve with itself Products of correspondences

. .

. .

. .

176 178 181 189 194

Chapter IX. Systems of Curves on a Surface § 1. Generalities on the curves on a surface § 2. Differentials of the surface Φ § 3. Simple differentials § 4. Double differentials

196 201 203 205

CONTENTS

ix

§ 5. Algebraic dependence of curves on a surface according to Severi 216 § 6. Surface product of two curves. Application to corre­ spondences 220 § 7. Birational invariance 222 Appendix

224

Bibliography

226

List of symbols most frequently used in the text

229

Index

230

ALGEBRAIC G E O M E T R Y

J. Algebraic Foundations §1. PRELIMINARIES 1. The reader is expected to be familiar with the elementary concepts of modern algebra: groups, rings, ideals, fields, and likewise with the customary notations of the subject. Multiplication is supposed to be commutative throughout. To avoid certain awkward points appeal is made to the well known device of an all embracing field Ω which includes all the elements of rings, • · · , under consideration. (1.1) Notations. Aggregates such as x 0 , · · · , x n or Ot1, · · · , am will often be written χ or a, the range being generally clear from the context. Accordingly the ring or field extensions K[xn, • • • , xn\ or K(OLv · · · , am) will be written K[x\ or K(a), with evident variants of these designations. Similarly for example for the functional notations: f(x) or φ(a) for f(x0, · · · , xn), or 99(04, · · ·, am). In this connection the "partial" extensions K\x0, • • • , xr], K(a.y, · • • , as), will also be written Kr[x\, Ks(a), with meaning generally clear from the context. The following symbols of point-set theory will also be utilized throughout: C: is contained in; D: contains; Π: intersection, U union; G : is an element of. (1.2) The groundfield. Very soon a certain fundamental field K, the groundfield will dominate the situation and all rings and fields will then be extensions of K. When K is of characteristic ρ the universal field Ω is also supposed to be of the same characteristic. The groundfield is always assumed to be infinite and perfect (irreducible polynomials have no multiple roots in an algebraic extension of K). Often also K is supposed to be algebraically closed (polynomials with coefficients in K have all their roots in K). The unique algebraic closure of a field Φ is denoted by φ. All rings will have a unit element and will always be integral domains (without zero divisors) and with unity element. (1.3) Noetherian rings. This all important class of rings includes all those considered in the book. Consider the following two properties of a ring Si: (a) Every sequence of distinct increasing ideals of H: (I1 C Ci2 * * * , is necessarily finite. (b) Every ideal a of Oft has a finite base.

ALGEBRAIC

4

FOUNDATIONS

That is to say there is a finite set base of the ideal, such that every

[CHAP. I of elements of a, the

a satisfies a relation:

One refers to (a) as the ascending chain property, and to (b) as the Hilbert base property. And now: (1.4) The ascending chain property and the Hilbert base property are equivalent. A Noetherian ring is a ring which possesses one or the other of the two properties, and therefore both. (1.5) The polynomial ring

is Noetherian.

(1.6) Every ideal a of a Noetherian ring and hence of K\x\ admits a canonical decomposition ideals. If:

where the

are primary

then the

are all distinct

is the prime ideal associated with

and unique. For a detailed treatment of the above properties see van der Waerden [1], II, Ch. X I I . (1.7) Homogeneous rings, ideals, and fields. B y a, form ^ is meant a homogeneous polynomial.

Let the quantities

be

such that the only relations between them are of type „ where the

are forms with coefficients in K.

of forms I

Consider now a collection

such that

(b) if moreover - also.

g and g' have

In other words '

if

then:

the same

degree

(a) then

behaves like a ring save that

addition is restricted to forms of the same degree. W e refer to homogeneous ring and denote it by

as a

Homogeneous ideals

a H of JJjy are defined in the usual way save that addition is again restricted to forms of equal degree.

Homogeneous integral domains, Noetherian

rings, prime ideals, primary ideals, are also defined in the usual way and properties (1.4),

(1.5), (1.6) hold with all ideals homogeneous.

quotients of forms of

^

The

of the same degree make tip a subfield of

called a homogeneous Held and written The extension to multiforme in say n sets of variables

homogeneous separately

t

_

is quite automatic. The rings and

fields are written 2. W e shall now recall a certain number of properties of polynomials in indeterminates xv referring mainly to their factorization. (2.1) If

and

an infinite number of sets

there exists

, such that

(2.2) Factorization in the ring

is unique to within a

factor in K. (2.3) Letf,

_

_

__

If f is divisible by g in

i.e., when both are considered as polynomials

§ 2]

RESULTANTS

in y with coefficients in

,

AND ELIMINATION

5

and if g has no factor free from y (i.e. in K[x])

then f is divisible by g in (2.4) Forms.

The following properties may be stated: Let us describe

the sets

i of numbers of K as essentially

and

distinct whenever not all the a,:, nor all the number

such that

(2.5) Property (2.1) for n>

are zero and there is no i. Then:

1 holds for forms when the infinite sets

I under consideration are restricted to sets essentially distinct in pairs. (2.6) The factorization properties polynomials are forms. A polynomial or form in if it contains a term in

(2.2) and (2.3) hold when all the of degree s is said to be regular

(2.7) Given a polynomial or form

it is

always possible to find a non-singular linear transformation

which changes f into a new polynomial or form

) regular in

some or all the variables

§2.

3. W e

RESULTANTS AND ELIMINATION

shall recall some elementary properties of resultants and

elimination theory.

For further elaboration and proofs the reader is

referred to treatises on algebra, and notably to van der Waerden, [1], II, Chapter X I , and E. Netto, [1], II. Consider first two polynomials in one variable x:

where the

are indeterminates.

homogeneous form in the

The resultant

) is a doubly

, whose coefficients are integers and whose

explicit expression is well known but will not be required here.

Let 5R

be the rational field, ! any finite algebraic extension of 51 and let and fiff[a; b] be the associated doubly homogeneous rings of the Then the only pertinent facts as to the resultant are: (3.1) (3.2)

i is of degree n in the at and m in the br is irreducible in every ring

One of its terms is (This is so-called

"absolute irreducibility.") (3.3) There exist unique polynomials A and B of degrees at most n — 1 and (3.3a)

with coefficients in

, such that

ALGEBRAIC

6

FOUNDATIONS

(3.4) Let the coefficients a{, b} and the roots of a field K.

[CHAP. I

off and rjj of g be elements

Then

(3.5) Let f and g have their coefficients in a field K. J then

common factor

If they have a

Conversely if

then f and g have a common factor Let now / , g be forms of degrees m, n in denote the resultant as to

Let J

i.e. as if / and g were polynomials in xr

Then: (3.6) Let f or g have indeterminate coefficients. Then R(f, g; xr) is a form of degree mn in

, and in (3.3a) A and B are forms in all

the xt and of degrees

Moreover a n.a.s.c. in order

that - - - - -

- fajtj! containing

andone

Xr

xr have a common factor containing xr is that

of them regular in When it exists

the common factor is in Consider now r + 1 forms in. and let

. with indeterminate coefficients

be the degree of

.

There exists a multiform

) in the sets of coefficients of the _

whose coefficients are

integers, the resultant of the / , , and with the following properties: (3.7)

_ is of degree mjmt in the coefficients of

(3.8) ~

is absolutely irreducible in the ring of multiforms with integral

coefficients. (3.9) There take place identities where the ., _ are multiforms with integral coefficients in the coefficients of the

and in the (3.10) If one takes for the fi forms of

is a n.a.s.c. in order thai the system admit a solution with the x.t not all zero and in . _ (3.11) If

'•> the highest degree term in ;

then ,

contains a

term More generally given any set of forms b

^ with indeterminate

coefficients there exists a resultant system where each B%H is an irreducible multiform such as

above and now:

(3.12) Same (3.10) with DEPENDENCE. ~ as the n.a.s.c. § 3 . as ALGEBRAIC TRANSCENDENCY

4. Let . be a field over K.

The elements

are said

to be algebraically dependent over K whenever they satisfy a relation If the term oc1 is

EXTENSION

§ 4]

OF THE OROUNDFIELD

7

actually present in P we say that o^ is algebraically dependent on over K.

As the groundfield K is generally clear from the context the

mention "over K" is usually omitted. A transcendence base { that:

> for

over K is a set of elements of

(a) no finite subset of the i

element

is algebraically dependent;

such

(b) every

is algebraically dependent upon some finite subset of

(4.1) If the number p of elements in one transcendence base is finite {only such cases arise in the sequel) then it is the same for all other such bases. The n u m b e r p of elements in a transcendence base is called the transcendency of

over K, written transc K

, or merely transc

when the

particular K is clear from the context. One may manifestly definethe transcendency p over K of any set of elements of

as the maximum number of elements

which are algebraically independent over K. is a field between K and

so that

(4.2) If transc^

1

». It is readily shown that transc

where all three are fields then transc^-

_

- transc K L.

(4.3) Rational and homogeneous bases.

These two concepts will be

found very convenient later. Given a field L over K we will say that a elements of L is a rational base for L over K whenever

set {

A set of elements

_

of some field

over L is known as a homogeneous base for L over K whenever

If _

then this condition is equivalent to .

) whereis

now fixed. (4.4) If L has a finite rational base then r = transc^ L is finite. Another noteworthy property is: (4.5) Let K have zero characteristic. Then if

is a trans-

cendence base for L over K and L is a finiteextension there exists an element L.

of L such that

of

is a rational base for

Hence if M is a field over L and

' are such that

is a transcendence base for L and L is a finite extension of exists an geneous base for §L. 4.

w

i

t

h

L

such t

h

a

t

there i

s

a homo-

EXTENSION OF THE GROUNDFIELD

5. In many questions arising naturally in the study of algebraic varieties it is necessary to replace the groundfield K by a finite pure transcendental extension, that is to say by an extension by a finite number of indeterminates.

ALGEBRAIC

8

FOUNDATIONS

[CHAP. I

W e are particularly interested in what happens then to the polynomial ideals and their mutual relations. ideal a =

Since all questions are trivial for the

1, consisting of all polynomials of the ring

we assume throughout Let us suppose that the ideal a has the base replacing K by any field

Upon

' the fi will span in

referred to as the extension of a.

i new ideal

Let in particular .

be an extension by indeterminates u , . I f disregarding a common denominator

Then /

is equivalent to:

e

v

| and

, we may write

e

r

y

I

f

the extension is

by a single variable u we will write (5.1) The extension operation

has the following properties:

(5.2) It preserves the relations of inclusion, sum, intersection and product. (5.3) If p, q are a prime and a primary ideal then so are (5.4) If p is the prime ideal of q then

r

,

is the prime ideal
' ' ' > fs}· Let the forms Ji span a polynomial ideal a. If / vanishes at all algebraic points of V, then it fulfills the conditions of Hilbert's zero theorem relatively to a. Hence there is a relation f = ^Pifv

'Pi

€

K\x\.

Upon replacing throughout the coordinates X by tx} and equating powers of t, there results a relation such as just written but with the Cpi G Ku[x] and hence fp £ aH. The converse is obvious. We shall now consider some elementary properties of varieties. (5.2) The variety V is independent of the coordinate system. (5.3) The union of a finite number and the intersection of any number of varieties are varieties. The proof for the infinite case follows from Hilbert's base property. Since distinct ideals may give rise to the same variety, it will be convenient to associate with each variety a unique ideal. By the ideal αJ1 of a variety V will be meant the ideal of all forms of K rr[x] which vanish identically on V. This is clearly the largest ideal giving rise to F. In particular, a form vanishes on the variety F of a prime ideal if and only if it is in Pji. A variety whose ideal is prime is said to be irreducible. (5.4) The variety of a primary ideal qn is the irreducible variety of its prime, ideal pH. Let the ideal aH of a variety F admit a canonical decomposition aH = Qiff Π " ' ' Π QsH anc^ let ViH t^ie prime ideal of q lH and Vi its variety. Among the piH discard those which contain the intersection of the rest. Let the remaining set properly numbered be ρ 1H, · · · , ρrH. This set is unique. Going back to the varieties this yields: (5.5) There is a unique decomposition V = F1 U · · · U Vr where the Vi are irreducible and none is contained in the union of the rest. The irreducible varieties F1, • · • , Vr are called the components of F. Clearly: (5.6) If V has a single component it is irreducible and conversely. The following irreducibility criterion is sometimes taken as definition of irreducibility: (5.7) The variety V is irreducible if and only if it is not the union of two varieties F', F" neither of which contains the other. Let F = F' (J V where F' contains a point M' which is not in F" and V" contains a point M" which is not in F'. Let F1, · · • , V'„ and F1, · · - , F" be the components of F' and V". Thus F = j

§ 2]

ALGEBRAIC

VARIETIES

AND THEIR IDEALS

23

Upon suppressing in the union the contained in the union of the rest, there will still remain a containing M', and a r , say 1 containing M", and we will have a decomposition , into irreducible varieties none contained in the union of the rest. Thus V has at least the two components 1 and so it is reducible. Conversely if V is reducible it has at least two components.

Let

V be one and the union of the rest. Then " where neither of contains the other. If ptH is again the ideal of the component of V then any form containingV contains also t and hence is in p jJZ and therefore in The converse is clear and so a H is the ideal of V. Thus: (5.8) The ideal of a variety is the intersection of the ideals of its components. 6. A subvariety V of a variety V is a variety contained in V but distinct from (6.1) If V is a subvariety of V then . , ' is non-vacuous. Hence if V is a variety distinct from (6.2) If

there are points of KPm not contained in V.

' then each component of V is contained in a component

ofV. It is clearly sufficient to assume that V is irreducible and V is not. Let s be the components of V. Then

By (5.7) some (

contains the others and therefore contains V.

But this implies (6.2). (6.3) If aB, a'n are the ideals of then V is a subvariety of V if and only if aH is a proper subideal of _ _ This follows at once from the uniqueness of the ideal of avariety combined with the fact that the inclusions and are equivalent. 7. The following are noteworthy special ideals and varieties. I. Ideal zero. This ideal consists of the single element zero of KH[x] and its variety, which is irreducible, is the space KP™ itself. II. Ideal unity. This is KH[x] and its variety, likewise irreducible, consists (formally) of the null set. III. Linear ideal. W e mean thereby an ideal spanned by linear forms of If the coefficient matrix of the l { is of rank p and p = 0 we are in case I and if we are in case II. For , the variety is a linear subspace

md any

subspace may be obtained in this manner. These subspaces are all irreducible varieties. For the ideal " may be generated by

24

ALGEBRAIC VARIETIES

[CHAP. II

ρ of its forms. In a suitable coordinate system we may assume that is

general it does not contain Vr. Hence Xm is transcendental on . . Hence Vr contains a general point and hence such a point with Hence dim^ .

_

,, relative to K, with _ _.

W is a subvariety of . .

Then

_ transcendental and trans c^ M

On the other hand since . Hence d i m a n d

is not in so dim

W

This completes the proof of (16.1).

§4.

PROJECTIONS

17. The two classical operations on algebraic varieties are projections and intersections.

W e deal here and there with intersections and shall

now consider projections. Take a fixed point 0 strictly in containing 0.

If M is any point of

and let S be a hyperplane not other than 0 the line

cuts

ALGEBRAIC

34

S in a unique point M'.

VARIETIES

The operation Pr on

is called a projection of

such that ]

onto S, and 0 is the center of the projection.

This is projection in the classical sense. defined in

[CHAP. I I

KIJm

The same operation may be

. A s a special case the center is at infinity and the pro-

jecting lines MM'

are parallel to a fixed direction.

These operations

are so well known that it is not necessary to dwell at length upon them. A natural generalization is as follows: spaces _ in

and /

and M

.

Take in .

which do not intersect. span a subspace

1

two fixed sub-

Given any point M not which intersects Sk

tJ

in M' (see below) and the projection is now 1

', The space

is again called the center of the projection. Since Sk and ~

do not intersect one may select a fc-simplex

and

simplex

such that A0 • • • Am is an m-simplex.

in

I t is to be chosen as the simplex

of reference. The points of S k are now defined by (17.1) and those of

by

The assertion made regarding M not in

lent to the following:

If M

is in ^

t is equiva-

take

is a unique point M' of Sk such that M'M For M not in Sk nor in

and

"it

otherwise there

meets

Alternatively:

, there is a unique pair of points M' in

such that the line.

are the coordinates of

contains j

. In fact if

, we must have relations

where X and ji are not both zero.

Since

necessarily and our assertion follows. Notice that M' is merely the point of Sk whose first

coordinates are the same as those of M .

This is a simple rule

of operation for obtaining the projection. The extension to a KAm

is automatic and may be left to the reader.

18. Projection of a variety. i.e., the forms of a H lacking the

The forms of

define a variety Vk of the space of V onto Sk from the center ,

contained in 1

make up an ideal a|f o f . _

. They

" called the projection * "

W

e

verify

at once: (18.1) The points of V not in (18.2) If aH is prime so is
1 has no point of multiplicity > d — 1. For if d > 1 at a d-tuple point the "tangent cone" consists of d lines.

A property of some interest, likewise a direct consequence of the definitions is: (24.7) At a k-tuple point of f the polars of order h < k have multiplicity at least k — h and some have exactly that multiplicity.

Since the polars are projectively invariant we may state: (24.8) Projective transformations do not affect the multiplicities of the points of a hypersurface.

In the applications it is convenient to have direct information regarding the behavior in affine coordinates. We find at once that a n.a.s.c. for the point (1, 0, ··· , 0) to be a £-tuple point of / is that /=

• • • ' X m ) + 4~ 1 ~%+ΐ( Χ 1> ' ' · ' X m) H

·

Hence in affine coordinates with F ( X ) =/(1, X 1 , · · · , X m ) the origin is a fc-tuple point when and only when F = Fk(X) + FM(X) + ••• + Fd(X)

where F h is a form of degree h . One verifies also directly that F k ( X ) (the set of terms of lowest degree in F) is the tangent cone at the origin (tangent hyperplane for k = 1). Thus for the curve in the X, Y plane: Fk(X, Y) + Fk+1(X, Υ) + - " = 0

the equation of the tangents at the origin is F k ( X , Y ) = 0. In particular for the curve aX + bY+ F2(X, Y) + · · · = 0

where a and b are not both zero, the origin is an ordinary point and aX + bY — 0 is the tangent at the origin. At a point (CT1, · · · , a m ) in the affine space KA m the situation is the same as above save that the Xi must be replaced by Xi — at throughout.

§ 6]

SOME INTERSECTION

PROPERTIES

43

25. Let us discuss now intersections of m hypersurfaces in The central property is: (25.1) Theorem of Bezout. Let v,, , _ be hypersurfaces of ivhich only intersect in a finite set [• of points, and let , be the degree of t There may then be assigned multiplicities to the independent of the coordinate system, such that counted with these multiplicities the number of intersections is Introduce a hyperplane with indeterminate coefficients u(. Referring to (1, 3.7) the resultant is a form of degree d of , This form admits the factorization in K(u) (see van der Waerden [1], II, p. 17) (25.2) By (I, 3.10) the. intersect if and only if one of the factors I i.e. as one might have expected if and only if l(u; x) contains an M}. The multiplicity of M t is by definition the exponent Oj at the right. Since their sum is d, and they evidently do not depend upon the coordinate system the theorem follows. Under the assumptions of the theorem let the coordinate system be such that none of the points M} is on the simplex of reference. Hence if M (£) is any one of them none of the is zero. Take now and such that ^ , >. Thus we may assume that * Since we have then the non-trivial relation

Thus and similarly all the other coordinate ratios are in K. (25.3) If the hypersurfaces • intersect in a finite number of points, these points are strictly in KPm. (If K is not algebraically closed one must replace it by its closure). (25.4) Application. If the hypersurface f has only a finite number of singularities these are strictly in . i f K is not algebraically closed). For the singularities are the intersections of all the first polars. Hence one may choose m intersecting only in the singular points and (25.4) follows then at once. As a complement to Bezout's theorem we may prove: (25.5) For almost all sets of hypersurfaces in Bezout's theorem, and hence for a general set there are actually d distinct intersections. Assuming again as in (I, 3) that the coefficients of the fi are indeterminates, let L be the closure of the field which they generate. Then represents a hypersurface in which is in fact a collection of hyperplanes. Now a n.a.s.c. in order that some of the

is that the

44

ALGEBRAIC

VARIETIES

[CHAP. I I

discriminants of the binary forms i.e. the resultants of their first partial derivatives vanish. These discriminants are multiforms in the coefficients of the fi with integral coefficients, and if they vanish identically they will do so for all special choices of the However upon taking for (and every i) a binary form in x0, without multiple factors and without factor in alone, one verifies at once that there are d distinct intersections. Hence our discriminants are not all identically zero and in particular they are not so when the are general. Thus for g e n e r a l t h e r e will actually be d distinct intersections. 26. It so happens that for the intersections of a line and a hypersurface / , and for those of two plane curves there are actually alternate methods to obtain the intersections and their multiplicities. We shall therefore examine each of these two special cases and show that the two available methods yield the same result. Consider first the intersection of a line X and a hypersurface / , where is not in / . The two methods of (24) and (25) are independent of the coordinate system. Let us apply the (w)-method by considering X as the intersection of ... - hyperplanes. We choose coordinates such that these hyperplanes are . Thus X becomes the line . of the simplex of reference, i.e. joining . ' V These two points are also chosen not on / . By the first method the intersections correspond to the binary factors of J That is to say by that method if in . (26.1) then the intersections are the points are their multiplicities. We have theorem. According to the (w)-method we take we have the identity

and the as prescribed by Bezout's and

(26.2)

where if d is the degree off then Thus cr3- is the multiplicity of f according to the (w)-method and we must show that Let , , , ; be any point of the line X and let l(u) contain x, M . Thus ' and we may take Then Hence (26.2) for i = 0 yields a relation

§ 6]

SOME

INTERSECTION

PROPERTIES

45

Since the product at the right is of the same degree d as j v „ u; „, , „, and the latter is not divisible by x0, we have . v v , ± / Hence from (26.1) at once This establishes the concordance of the two methods for the intersection multiplicities of a line and a hypersurface. Since the first method is independent of the choice of the hyperplanes which intersect in A, this holds also for the second method. 27. Consider now the intersections of two plane curves f(x) and g(x) of degrees m and n. We choose coordinates such that / and g are regular in and take the resultant „ „ _ It is a binary form 1 of degree mn. ' If and g have a common factor, the number of intersections is infinite and conversely. W e suppose then that ~ so that there is only a finite number of intersections. Choose now a triangle of reference such t h a t / , g are regular in the xit that no M} is on the triangle, and that no two are collinear with the vertex (0, 0, 1). W e may then choose for . coordinates Since is a n.a.s.c. for / , g to intersect we have in R (27.1) Suppose that explicitly

Then according to (I, 3.1)

will hold as stated if they hold for and 2. Some examples. First example. T is the projective transformation

If the rank of the matrix of the coefficients is < n + 1, T is a projective transformation of onto an s-subspace of . and there is a fundamental variety which is a subspace S of of dimension m — s—1. If s = m = n we have a non-singular projective transformation of one space onto the other.

§2]

BIRATIONAL

TRANSFORMATIONS

49

Second example. Let Q be an irreducible quadric of which is not a cone, A a point of Q, P a plane of The projection of Q from A onto P is a rational transformation with A as fundamental variety. 3. We assume now definitely that the rational transformation T: satisfies conditions (1 abc) and discuss some of its properties. (3.1) A n.a.s.c. for the existence of a rational transformation T of onto WB is that the function field be isomorphic with a subfield L of the function field in an isomorphism which preserves the elements of the groundfield K. If T exists as above the relations deduced from (1.1) give rise to an isomorphic imbedding which is a suitable Conversely let such a exist and identify the elements of with their images under Assuming the varieties in general position we will have We may then write where the have properties (1 abc). Since T given by (1.1) transforms Vr into an irreducible variety with the same general point N as Ws, Ws is TV r and (3.1) follows. (3.2) A rational transformation T of Vr does not increase the transcendency of the general points and hence dim (3.3) Rational transformation of onto Wr. The degree. When dim has the same transcendency r as and so is merely a finite algebraic extension of The degree of this extension depends solely upon T and is known also as the degree of T. It is in fact the degree and does not depend upon the representations of the two function fields through general points. §2.

BIRATIONAL TRANSFORMATIONS

4. Let V be an irreducible variety in formation given by

and

a rational trans-

(4.1) Then the relations (4.2) define a rational transformation: written and called the product of T and This product is associative. Suppose in particular that r = s and that is a transformation of onto Then is also a rational transformation Let us assume that transforms one general point of into itself: Then T'T, operating on M assumes the form and hence its general form is i.e. it is the identity transformation on If TM = N, so that N is a general point of then from

50

TRANSFORMATIONS OF VARIETIES

[CHAP. Ill

TT'TM — TM there follows T T'N = N. Hence TT' is the identity transformation on Wr. Under the circumstances T' is known as the inverse of T, written J1-1, and the operation T is described as a birational

transformation. A simple example of a birational transformation is a non-singular projective transformation of a space onto another of equal dimension. Another is the transformation of a KPf referred to coordinates X to KP™ given by 1

Pyi = (Tlxj ) I x t , i = 0, 1, · · · , m

whose inverse reads the same way with χ and y interchanged. The fundamental varieties are the spaces of the faces of dimension less than m — 1 of the simplexes of reference in the two spaces. The simplest case is the quadratic transformation from plane to plane PVo = X1X2,'

PVi = X2X0> PV2 =

From the properties of rational transformations we deduce at once the following properties: (4.3) A n.a.s.c. for the existence of a birational transformation T: Vr —*• Wr is that their function fields be isomorphic in an isomorphism preserving K. If rational transformations T, T' induce isomorphisms θ, Θ' with the appropriate subfields of the function fields then T'T induces θ'θ, and if T is the identity so is Θ. Hence if T' = T_1, θ'θ and 66' are both the identity. Hence a birational transformation T induces a θ which is an

isomorphism. Conversely if θ is an isomorphism the identification of corresponding elements of ΚΗ(ξ) and ΚΗ(η), where M(ξ) and Ν(η) are as before, induces a T which has an inverse. Hence T is a birational transformation and (4.3) follows. (4.4) Ifthere is a birational transformation T: V^-W, the two varieties have the same dimension (see 3.2). (4.5) A n.a.s.c. for a rational transformation T: Vr -> Wr to be bi­ rational is that the degree of T be unity. For this is also the n.a.s.c. that Κ Η ( ξ ) coincide with Κ Η ( η ) . (4.6) Birational equivalence. Two irreducible varieties V, W are said to be birationally equivalent whenever there exists between their function fields K r , K w an isomorphism τ which preserves the groundfield K . Each variety is also called a birational image or birational model of the

other. It is clear that birational equivalence is a true relation of equi­ valence in the customary sense. Furthermore, since dim V = transc K v , and likewise for W, we have: (4.7) Birationally equivalent varieties have the, same dimension. We may rephrase (4.3) as: (4.8) A n.a.s.c. for the birational equivalence of V and W is that there exist a birational transformation of one onto the other.

§2]

BIRATIONAL TRANSFORMATIONS Vr

Wr

51

(4.9) If T is a birational transformation — sthen there is a fundamental subvariety F0 for T in Fr and a fundamental subvariety W0 for T-1 in Wr. The transformation T operates on Vr — F0 and T-1 on Wr — W0. Moreover T and hence T"1 is one-one between the sets of general points of the two varieties. 5. Birational geometry. By the birational geometry of an irreducible variety is meant the study of those properties of the variety which are preserved under birational transformations. In essence they are then the properties which depend solely upon the function field Kv. The simplest are the transcendencies of the points and the dimension but there are others to be discussed in later chapters. It is a familiar fact that projective equivalence of figures has the advantage over aifine equivalence of bringing under one category appar­ ently dissimilar geometric objects, for instance ellipses and hyperbolas. A similar advantage is possessed by birational over projective equivalence. Thus it may be shown that a non-ruled surface of order three in KP3 is birationally equivalent to a projective plane. Therein lies one of the basic reasons for the importance of birational geometry within the whole domain of geometry. Examples of birational equivalence. We have already observed that non-singular projective transformations between two spaces KPm are birational transformations. Let us examine some more sophisticated examples. Example I. Let V r of KP m have the following property: the secants of Vr, or lines of KPm which meet the variety in two or more points, lie in a variety Vs, s < m. It may be shown that this holds in fact when m > 2r -f- 1, and will be proved explicitly later for algebraic curves. Let then O be a point of KPm not in Vs and S a hyperplane not passing through 0. If M is a general point of the variety the line OM will meet S in a single point N and M -*• N defines a birational transformation T of Vr into a variety F'r contained in S. As is well known T is merely a central projection of center 0. Notice that without information about the locus of the secants one could merely affirm that T is rational. Take for instance a cone Vm '1 of vertex 0 and its intersection F m-2 with a hypersurface/ of degree μ > 1 not passing through the vertex. Assuming Fm-2 and the base V'm~2 of the cone in S both irreducible, the projection is now merely a rational transformation of Fm-2 onto F'm_2. On the other hand if / is a quadric through O the projection is again birational. Example II. Rational varieties. If the function field K v of Fr is a pure transcendental extension of transcendency r over K then Fr is birationally equivalent to a projective r dimensional space and is said to be rational. The field Kv is then in fact isomorphic with the field KH(y)

52

TRANSFORMATIONS

of the space formation T of by relations:

OF VARIETIES

[CHAP. ILL

Under the general theory, there is a birational transonto the space and T and its inverse are represented

The variety being assumed in general position there results the associated affine relations

Example III. Planar duality. Dual curves. To give differentials free play the groundfield is assumed of characteristic zero. Consider first a line in the plane

Let the be considered as coordinates of a point (u) of The correspondence between the points of and the lines of is one-one and lies at the root of the classical duality of plane projective geometry. Owing to this correspondence the are also referred to as line-coordinates for We merely recall that the lines l(u) through a given point. of are imaged into the points of a given line of and conversely with the two planes interchanged. One must also bear in mind that the relationship between the two planes is symmetrical. Thus the are line-coordinates for Consider now an irreducible curve / of With any non-multiple point A off associate the point B of which representsthe tangent to / at A. Let T denote this operation on the curve / . I f i s a general point of / then N(rf) = TM is given by

Hence T is the rational transformation represented by (5.1) If m is the degree of / then Euler's relation yields Hence (5.2)

§3]

NORMAL SYSTEMS

53

By differentiation and since Zftdii = ο we find (5.3)

= °>

= °-

Now the /i( cannot all be zero since M would then be a multiple point of / and hence transc M = 0. It follows that N is not the null-set. Suppose that transc N = 0, i.e. the ratios ηι/η} €. R- Then (5.2) yields, with the OC2 G R and not all zero: Soci^i = 0. As a consequence 1(a) = SajXi must be divisible by /, hence m = 1 and / is a line. Conversely when / is a line the point N is fixed. Let us suppose then that / is not a line. Thus Tf is an irreducible curve g of KP\ and we will have ρ(η) = 0. Upon treating now g like / and taking its transform in KP% the image of N will be a point M'(ξ'). Here again M' is not the null-set. Then (5.3) with M', N in place of M, N yields (5.4)

^drii = 0.

Since transc N = 1, the (Irji can only satisfy essentially one linear relation. Comparing (5.4) with (5.3) we see that the ξζ and the ξ'{ are proportional and hence M' = M. Thus the relationship between / and g is symmetrical and T is one-one, hence birational The curves / and g are then birationally equivalent and each is known as the dual of the other. From the definition of g we deduce at once the following results: (5.5) There is an almost one-one correspondence between the points of g and the tangents of f. In this correspondence the tangents to f through a point of KP1x are imaged into the intersections of g with a line of KP\ and conversely. Similarly with f and g interchanged. The order η of g, or number of tangents to / through a general point of KP^ is known as the class of /. The order m of / is the class of g. (5.6) Remark. The extension to space duality and hypersurfaces is quite automatic and may be left to the reader.

§3. NORMAL SYSTEMS 6. Let us drop for a moment the assumption that Vr is irreducible, so that its ideal an is for the present arbitrary. If {φν · · · , (prj] is a base for aH then the points of V consist of all the solutions of the system 3- is not zero when unique solution

and the determinant of the first degree The system has a

(2.4b) where the are non-units of (Bochner-Martin, p. 7.) (2.5) Noteworthy special case: The relations (2.5a) where the gt are of degree at least two, have a unique solution of the form (2.4b). Moreover in this solution the determinant of the coefficients of the first degree terms in the The asserted property of the is readily obtained by substituting their series in (2.5a) and comparing coefficients of like powers of the We may think of (2.4b) as defining a transformation T from the variables to the variables A transformation (2.5a), with is said to be regular. If T, T' are regular so is and property (2.5) implies also the regularity of Hence regular transformations form a group. In point of fact the properties which will mainly interest us are those which are invariant under regular transformations. (2.6) A non-unit is said to be reducible whenever it is associated with a product of two non-units: , where are nonunits. When this does not hold / is said to be irreducible. (2.7) The unique factorization theorem. Every non-unit f(u) admits a finite decomposition where g, • • • , h are irreducible and unique to within unit factors (See Bochner-Martin, p. 193). (2.8) Real or complex fields. Suppose that the groundfield K is one or the other of these two fields. Then if the initial functions in the three propositions which we have just considered are convergent in

2]

ALGEBROID

VARIETIES

81

a certain neighborhood of the origin, all the functions that occur in their statements are likewise convergent in a suitable neighborhood of the origin. This provides of course powerful analytical content for these theorems. In the case of the real field it is also useful to bear in mind that the series which occur are all real.

2.

ALGEBROID VARIETIES

3. As a basis for the local study of algebraic varieties we shall develop a fairly parallel theory of algebroid varieties. The treatment will only be carried far enough to cover our later requirements. While we shall deal with the same basic set of parameters ui as before, the only properties that will interest us are those left invariant under regular transformations. One may introduce in ideals, prime and primary ideals as in all rings. Such ideals will be referred to as algebroid. The first important property is: (3.1) Theorem. Every algebroid ideal a is Noetherian. (See I, 1.3.) We will show that the Hilbert base property holds. Let o and let be its leading form. It is seen at once that the set is a homogeneous ideal of the homogeneous ring By the base theorem for the latter has a finite base w h e r e i s the leading form of an Thus

and is relation of We with where of the degree verify hence degree infinite v,degree if is the with then is degree. of leading the at once and degree then Similarly Hence form more that andofof ifisgenerally sone ofisthen ,degree may the and Thus degree the there form forms is is of a first finite for f a )Since each then are basevso Moreover for anchosen a. and if that a

82

FORMAL

POWER

SERIES

It follows from (3.1) (see I, 1.6) that every ideal a of canonical decomposition

[CHAP. IV has a

(3.2) where the are primary ideals such that if is the prime ideal of then the are distinct and unique. We shall only consider ideals of non-units and no further mention of this assumption will be made later. All our ideals are then subideals of the maximal ideal rrt of all non-units. (3.3) We will now define the algebroid concepts. Take a ring and a prime ideal b of are residue classes of non-units of mod b, then the set is defined as an algebroid point or merely M, and the are the coordinates of M. The totality of all algebroid points is the algebroid space Let and let us define what we mean by f(M) — be an element of the class of mod b, and let Let also admit the decomposition (1.1). Then and its degree with h. Hence Thus the class of mod b is fixed and it is by definition It may be thought of as the value of the series / at the algebroid point M . In particular if then and / is said to contain the algebroid point M. Consider now an ideal a of itself. The totality of all algebroid points contained in every series of a is the algebroid variety of the ideal a. The variety is defined as irreducible if where is a prime ideal, and it is defined as reducible otherwise. The qualification "algebroid" is often omitted where the meaning is otherwise clear. Suppose that the ideal a has the canonical decomposition (3.2) and among the let be a set such that none is contained in the intersection of the rest. Then (3.4) The are the components of In view of (3.4) we shall mainly concentrate upon irreducible algebroid varieties. The following property is immediate: (3.5) The algebroid ideal consisting of all the series containing a given algebroid point M is prime. We will denote this ideal by and its (irreducible) variety by 4. Consider now a prime ideal p and its variety We shall discuss more or less together the notions of general point and dimension for The treatment will rest upon normal systems patterned after those of (III, 6, 7, 8) for algebraic varieties.

2]

ALOEBROID

VARIETIES

83

Since p consists of non-units, its elements have a least degree Let be of degree Then / is irreducible. For if then since p is prime one of the factors say g p and as g is of degree this is ruled out. Let be the leading form of f(u) and apply a linear transformation to the such that becomes regular in Then

where the right hand side is a special polynomial in and irreducible in and is of degree (2.3, II). Let , and Here is a prime ideal of We may repeat the same reasoning as above for etc. At each step there will be required an appropriate linear transformation and they combine to a transformation of the initial coordinates u { with the following property due to Riickert: At the step there is obtained a special polynomial

where the are non-units of is of degree irreducible in The process stops at an

and is such that

The points of the variety will all satisfy the normal system analogous to (III, 6.1) for algebraic varieties: (4.1) As a first application consider any series g(u). We have upon dividing

where

and the degrees of

and

with k. From this follows (4.2) where

and etc., we will arrive at a relation

Proceeding similarly with the

(4.3) Note that if one applies this process to a series (4.3) j runs only up to h and

then in Applying this

FORMAL

84

POWER SERIES

[CHAP. IV

to the coefficients of in we find that we may reduce them mod To sum up we may state the following two properties: (4.4) One may assume in (4.3) that (4.5) Every series g{u) of is equal mod to an element of the ring

(4.6) Let now denote the residue class mod of the coordinate Uj. The are the coordinates of an algebroid point M. As M annuls every series of p and only these, M is a general algebroid point of the variety in the same sense as for algebraic varieties: if the series g(u) is annulled by M then it is annulled by every algebroid point of Since satisfies (4.1). Since is a special polynomial in of we may state: (4.7) The coordinate of M is integral algebraic over Hence all the coordinates are integral algebraic over 5. By making use of the general algebroid point one may paraphrase the argument of (III, 7, 8) and obtain most of the following comprehensive results: (5.1) Theorem. There exists in a prime ideal a set of series (5.1a)

with the following properties:

An equivalent form of II is: II'. To within a factor in is a polynomial in with leading coefficient not in and of least degree in It divides therefore in every other such polynomial of III. If then a n.a.s.c. in order that a series g is that it satisfy a relation (5.1b) as so (5.1c) does aIV. normal every Sincesystem point of of satisfies the equations variety the system for For this reason we refer to this system

2]

ALOEBROID

VARIETIES

85

The only points in the statement which are not covered in (III, 7, 8) are the assertions as to and the necessity in property III. Regarding one arrives at first by obtaining with a leading coefficient Since d(u) is integral algebraic over one may multiply throughout by the product of the conjugates of d(u) and as a result the new leading coefficient hence has the property asserted under I. Passing now to the necessity of property III, let and let denote this necessity for a Since there is no in holds trivially, so we assume and prove We have now a relation (4.2). Upon dividing the polynomials of (4.2) by as to (note that by II', is divisible by as polynomials of there results a relation (5.2) where this follows

and e is the degree of

From

(5.3) Now (4.5) yields Since the degree of (5.3) in by the irreducibility property II, hence Applying now to the and substituting in (5.2), follows. Remark. In the preceding argument we have used in an essential manner property (4.7), i.e. in the last analysis the existence in of the special polynomials of (4). 6. One will surmise that the number depends solely upon the ideal p and its variety 93, and is in some sense the dimension of This is indeed the case and the most convenient way to prove it is by means of E. Noether's property as it appears in (III, 11.1). Let "chain", "length of a chain", etc., have the same meaning for algebroid as for algebraic varieties (III, 11). (6.1). Theorem. The length of the longest chain beginning with the variety of (5.1) is Thus no matter how the general point arises the number is the same and depends solely upon the variety It is therefore natural to refer to r as the dimension of or of its ideal. It is indicated by denoting the variety as As in the algebraic case r is the number of algebroidally independent coordinates among those of any general point of the variety. Let where is irreducible and and let be the ideal of Thus It follows that some series g(u) is in but not in

FORMAL

86

POWER

The notations being those of (5.1), hence it satisfies an irreducible equation

SERIES is algebraic over

[CHAP. IV and

Thus Since

and necessarily and since is in p' but not in p. This means that the prime ideal It follows that upon applying to the normalization of (4.1) we will come to a last function with at most r variables. Hence if is the least r for any choice of variables necessarily

Evidently

whatever By the result just proved if is a chain beginning with 93 then 1 and so hence Let denote property (6.1) for r. Since for the system (5.1c) reduces to we have then and the origin. The only possible chain is then the origin itself and so holds. Consider now The ideal hence Since it contains 0, they form together a chain of two links, the maximum possible. Thus (6.1)! holds. Assume now and that (6.1 ),,_! holds. Let the variables undergo if necessary a linear transformation such that is regular in all of them. Then Take now indeterminate and Then Hence one may solve one at a time the relations

for the They will not all be zero and they will be the coordinates of a certain algebroid point For the variety clearly Since we infer from (5.1b) that if then Hence the prime ideal of and so Since is in but not in and so By there is a chain of length r beginning with . Hence there is one of length beginning with This completes the proof of (6.1). (6.2) When is reducible its dimension r, also the dimension of its ideal, is by definition the largest dimension of the components of A pure variety is one whose components all have the same dimension. It is a mere exercise to prove: (6.3) If are distinct irreducible varieties then dim dim

2]

ALOEBROID

VARIETIES

87

(6.4) The Hilbert zero theorem. If a series f(u) contains the variety 93(a) of the ideal a then some power In view of the decomposition (3.2) it is sufficient to prove the theorem for a primary ideal Let be the prime ideal of q and M a general point of the variety Since hence some which proves (6.4). (6.5) Differentials. When K is of characteristic zero the definition of differentials in the field K((u)) or in the quotient field of the integral domain a prime ideal, follows essentially the same pattern as for the similar algebraic situation and need not detain us here. (6.6) Monoidal systems. If the characteristic of K is zero, it will be seen by reference to (III, 7) that the reduction to a monoidal system utilized for algebraic varieties (see III, 8) may be carried out for algebroid varieties. With a suitable choice of variables the monoidal system will assume the form

(6.7)

where g(u) is an irreducible special polynomial with coefficients in and Of course (5.1) still holds with (6.7) in place of (5.1a) and (5.1b) replaced by (6.8) as the condition that Even when the groundfield K has a characteristic the algebroid variety may possess a monoidal representation. It will then be called monoidal. Thus when the characteristic is zero every algebroid variety is monoidal. Consider now a fixed choice of the parameters and relative to this choice series where as in is a special polynomial in of the ring Let b be the ideal generated by the and let us consider only ideals over b. Suppose that a is such an ideal with the monoidal representation (6.7). Since any series h(u) is, mod b, an element h'(u) of and since is an integral domain, referring to (III, 10.11) we may apply with but few changes, the reasonings of (III, 10) and obtain the following result: (6.9) Let the ideal a contain the ideal b. Let be monoidal with representation (6.7), and let g admit a decomposition where the g{ are irreducible special polynomials of the same nature as g

FORMAL

88

POWER SERIES

[CHAP. IV

itself. The series h which satisfy a relation (6.8) with g replaced by gt, and those which satisfy a relation (6.8) with g replaced by form an associated pair of a prime ideal and primary ideal There take place the relations where the are all distinct. Since they all have now the same dimension r, is a pure r-dimensional algebroid variety. (6.10) Remark on the complex field. If K is the complex field and one only admits convergent series, everything goes through as before. Owing to the base theorem the variety of an ideal a is determined by a finite system of relations (6.11) Hence there will be a region of convergency common to all the For (u) in the region we will thus obtain analytical points of namely the points of the affine space which satisfy (6.11). The variety is called a complex algebroid variety. 7. Local rings. A local ring is a Noetherian ring with unit element and hence with units (elements whose inverses are in whose non-units form an ideal m (necessarily maximal and prime). These rings whose importance is growing constantly, will be barely touched upon here and merely insofar as they are required later. For more ample information see Krull [2] and Chevalley [1], whose writings we follow in substance. Immediate properties are: (7.1) The ideals of admit a canonical decomposition. (7.2) If a is an ideal of then is a local ring with as ideal of non-units (see van der Waerden [1], II, p. 21). Evidently also: (7.3) A power series ring is a local ring. Let m be as above. As a basis for topologizing the ring we prove: (7.4) Let us show first that (7.5)

n • m = n.

We have at once so that we only need to show that There is a canonical decomposition

and it is sufficient to show that be the prime ideal of There are two possibilities: (a) Then for some and hence, owing to the definition of n; (b) Since m is maximal, we do not have Hence there is an element m of rrt not contained in If n is any element of n, then mn and since m is not in necessarily hence

LOCAL PROPERTIES

3] Thus always Let now

89

and hence which proves (7.5). be a base for n. From (7.5) follows

Hence if

then

for every i. It is seen at once that d is a unit. Therefore and hence which is (7.4). Let now and define its norm as follows: for a unit a, then by (7.4) there if a is a non-unit and is a such that a is in but not in and we take The usual norm properties are readily verified and as a consequence metrizes which is thus turned into a metric space. In this topology: (7.6) Every ideal of (he local ring is closed. Thus in particular every ideal of the power series ring is closed. Referring to (7.2) and in its notations by (7.4): This means that if is in every ideal then which implies (7.6). This completes the general material that we shall require in the sequel regarding local rings. 3.

LOCAL PROPERTIES OF ALGEBRAIC VARIETIES

8. We shall now apply the algebroid theory to the study of local properties of algebraic varieties. This will be done by associating a definite algebroid space with the individual points of the variety. Given a point P in let the coordinates xt be so chosen that P is the point (1, 0, • • • , 0), i.e. the origin for the affine coordinates Everything will revolve around the properties of the ring and of its ideals. We are only interested in the properties of which are invariant under regular transformations of the coordinates Now a projective transformation preserving the coordinates of the point P has the general form The associated transformation of the affine coordinates

is

and so it is regular in the Thus it falls under the allowed category. As a consequence our results will not depend upon the choice of coordinates for provided that those of the point P remain (1,0, • • • ,0). The algebroid varieties of the algebroid space will now be referred

90

FORMAL POWER SERIES

[CHAP. IV

to as branches, written S, or S if dim 33 = r. A general algebroid point of 23 will be called a parametric point of the branch. The term "branch" is borrowed from the classical case where K is the real field and one has branches of a curve, of a surface, etc. A S1 is generally merely a curvilinear arc (K real) through P. The "places" of a curve discussed at length in the next chapter, fall under this category. There is nothing to prevent us from considering the algebroid points as points of the affine space KAr^, As a consequence the algebroid space Xm is embedded in KAr^, or for that matter in the projective space KP'£. This gives meaning to statements such as: the parametric point M is in the algebraic variety V" or "33 C V." Let now Vr be an algebraic variety of KPm through the point P and let α be its ideal in K\X\, i.e. as a variety of KA™. Since the poly­ nomials of α all vanish at the origin they are non-units of iT[[X]]. Hence they span in i£[[X]] an algebroid ideal a* and the latter is merely the closure of α as a subset of _fiT[[X]]. We apply to a* the canonical de­ composition (3.2) into primary algebroid ideals, with p;, qt as loc. cit. Among the p; let Ip1, · · · , ρσ} be the set of those such that none is contained in the intersection of the rest. To the pf, i < a, there correspond the branches Si = S(pj; P) whose set 33( Fr. P) is the neighborhood of the point P in Fr. The branches Si are the local components of Vr at the point P. When V r is a complex variety the ordinary points of S(Fr, P) (see 6.10) make up a true neighborhood of P in V r . Simple examples where 33( Fr, P) has more than one branch are readily given. Thus at a multiple point of a plane curve Γ where there are Jc distinct tangents there are k branches (see below). The dimension of the algebroid variety S(Fr, P) is by definition the local dimension of Vr at the point P. It may well vary from point to point. Thus let S and I be a plane and a line in KP3, where I is not in S and let Q be their intersection. If V = I U S, then dim F = 2. At every point of S the local dimension is likewise two. However at the points of I other than Q the local dimension is unity. It may be noted that at the point Q there are two local components of F, but only one at the other points. 9. Now let V r be irreducible and as before let α be its ideal in [X] and a* the closure of a in _K"[[X]]. While a is prime this need not be the case regarding a*. The central property regarding the local components is the following: (9.1) Theorem. Every branch of the irreducible variety V r over an algebraically closed groundfield is r-dimensional. Hence V r has the local dimension r at every point. We shall actually prove the more complete property: (9.2) The ideal a* of V r in ϋΓ[[.3Γ|], K algebraically closed, has a canonical r

3]

LOCAL PROPERTIES

91

decomposition where the prime ideals associated with the are all r-dimensional. If the groundfield is of characteristic zero then and so the canonical decomposition is Property (9.2) also implies: (9.3) The number of local components of . at the point P is s. Hence the ideal of in K[[X]] is certainly not prime when As an example let the groundfield K be algebraically closed and of characteristic zero. We will show in the next chapter that an irreducible curve r with a /c-tuple point P with distinct tangents has then k local components at P. Examples with any value of k are readily produced, and they correspond to a non-prime algebroid ideal although the algebraic ideal a is prime. We shall require two normal forms for V, both in affine coordinates, that of (III, 6): (9.4) where is irreducible and regular in of (III, 8.1):

and the monoidal form

(9.5)

Moreover F is irreducible and regular in We recall also that by (III, 8.5a) the ideal a consists of those and only those polynomials which satisfy a relation (9.6) An important observation must be madehere when the groundfield K is of characteristic Namely if are the initial coordinates then (see III, 22) the of (9.5) are obtained from them by relations (9.7) where is "almost" any such matrix of elements of K. On the other hand the of (9.4) are derived from a succession of m — r transformations of types (9.8) where again the matrix of the coefficients at the right in (9.8), is almost any such matrix of elements of K. Since is almost any we conclude that in (9.4) and (9.5) one may assume that the coordinates are the same.

FORMAL

92

POWER

SERIES

[CHAP. IV

10. We will first require a certain new ring. For convenience let us set The new ring is needed because while is a local ring is not. In fact the units of 91 are merely the elements of K and its non-units are all the polynomials of positive degree. Thus and are non-units. Hence 1 is an element of the ideal spanned by the non-units of 9?, and so 91 is not a local ring. The basic observation is that the elements of the field K(X) of the form where A and B are polynomials and do form a ring This is precisely the ring that we need. Notice that its elements may also be written more conveniently (10.1) Now if a is an ideal of 9?, its elements span in 9J' an ideal Conversely if is an ideal of 91' and R above is in a' then R also. Thus a' contains polynomials and their totality constitutes an ideal a of such that As a consequence a base for a is likewise one for a'. Thus is Noetherian. It also has an element unity. Furthermore if m is the ideal of 9? then the ideal is the set of all the non-units of , and incidentally is the ideal of the non-units of Hence is a local ring. Returning again to the representation (10.1), using the relation in

we may imbed isomorphically in by identifying the element R of 9?' with the element This imbedding is seen at once to assign to 9T its natural local ring topology as defined in (7). Henceforth is identified with its image in We will have and the imbeddings are topological. Let us observe finally that is in a sense the smallest local ring over 11. We are now well equipped for the proofs of our theorems. Referring to (6.9), to prove Theorem (9.1), and also the part of (9.2) which corresponds to an arbitrary groundfield, it is sufficient to prove the analogue of (9.6) or rather of (III, 8.5) for series: (11.1) A n.a.s.c. for a series 0(X)

to be in the ideal

that it satisfy a relation (11.1a) where p is the fixed exponent of (9.5) (see III, 7.7). The proof rests upon the following lemma, in which the rings are as in (10):

is

3]

LOCAL PROPERTIES

93

(11.2) Lemma. Let and let b be an ideal of The elements OL of such that COL form an ideal a. The n.a.s.c. in order that be in the ideal is that To obtain (11.1) all that is necessary is to choose and

We proceed to the proof of the lemma. Sufficiency is clear by a limiting process, so that we only need to consider necessity. Preliminary remark: Since is a local ring an ideal in is closed in From this follows that i.e. the elements of extended to which are in make up Suppose now that Since may be indefinitely approximated by polynomials, one may choose, for any a polynomial Then such that

Thus the polynomial is in the extension of the ideal of into It follows that there is a polynomial of 5R of the form such that Hence there is a such that Therefore Since e is a unit of we have Thus is arbitrarily near and since .. is closed it contains This completes the proof of the lemma, hence also of (11.1). As we have seen this proves Theorem (9.1) and the part of (9.2) referring to a general groundfield K. Let generally F(X) admit in the following factorization into distinct irreducible factors: (11.3) The pair corresponds to By reference to (III, 10) it will be seen that when and only when Now if K is of characteristic zero the irreducible polynomial F(X) of has only simple roots in in any field over . Hence every and the rest of (9.2) follows. 12. The results just proved may be completed in certain points. (12.1) The parametric points of the branches of an irreducible are general points of the variety. The branches of center P are in one-one correspondence with the factors of (11.3). We may suppose that corresponds to and it will then have the normal representation (12.2) Since F is regular in

the

are special polynomials in

of the

94

FORMAL

POWER SERIES

[CHAP. IV

ring The parametric point will have for coordinates a solution of the above system in which the are indeterminates. Hence the algebraic transcendency of is at least r. On the other hand if the prime ideal of in then from (9.6) and since divides F and there follows Hence Therefore transc and is a general point of the variety. (12.3) If a hypersurface contains a local component of the irreducible variety then it contains V itself. For if W contains it contains and hence by (12.1). 13. The simplest type of branch corresponds to a representation (12.2) in which and is of degree one in Such a branch will have a representation (13.1) The branch is then said to be linear. (13.2) The intersection of a linear branch with a hypersurface G which is not in the prime ideal is pure (r — 1)-dimensional. In view of (13.1) one may replace G mod by a series and assume 0\ irreducible. At the cost of a linear transformation on the one may also assume that is regular in and hence that it is a special polynomial in Thus the intersection will have the monoidal representation

Hence it is a and (13.2) follows. The point P is said to be non-singular whenever its neighborhood consists of a single linear branch and it is singular otherwise. Let P be non-singular and let as before denote the ideal of in The fact that there is a single linear branch and that it is represented by the system (13.1) means that (13.3) Conversely let (13.3) hold. Then .. is prime since its quotient ring is an integral domain. Hence there is only one branch 93r centered at P. In view of (13.3) the coordinates of a parametric point of 33r satisfy a system (13.1) so that 23r is linear. Hence P is non-singular. Thus: (13.4) A n.a.s.c. in order that the point P be non-singular for is that with a proper numbering of the coordinates there take place the relation (13.3). Let again P be non-singular. Upon applying a regular transformation from (13.1) is replaced by (13.5)

3]

LOCAL PROPERTIES

95

where and the Jacobian matrix (coefficient matrix of the terms of the first degree)

is of rank In view of this one may solve (13.5) by the implicit function theorem as (13.6) and (13.5) and (13.6) together imply the relation (13.7) One refers to (13.5) as a parametric representation at the point P and to the as the parameters of this representation. It is not difficult to see that the behave like local coordinates in the following sense. Let be an ideal of defining an algebroid variety 2B. If then mod The generate an ideal of X[[i;]] and its algebroid variety is the intersection Thus the "ideal theory" of is merely theideal theory of the intersections of with the algebroid varieties of From (13.2) we deduce the following noteworthy result: (13.8) Theorem. The intersection of an irreducible without singularities with a hypersurface G which is not in the ideal of is a pure For the intersection is dimensional in each point and hence it has no components of dimension Remark. The extension of the "local coordinate" scheme to products of varieties is quite automatic. We will prove however the following interesting result: (13.9) Let r and be irreducible and let A, B be nonsingular points of with parameters Then (A, B) is a nonsingular point of and it has for parameters the and together. Let be affine coordinates with origins at A, B in the spaces of the two varieties, and let be their prime ideals in Fronl (13.3) and

follows readily (13.10) where is the ideal spanned by in the ring Hence (A, B) is an ordinary point of From (13.10) follows that are local parameters for the point and so the same holds for the and the (13.11) Remark. One will compare the present characterization of non-singular points wholly independent of differentials with the characterization of (II, 21) in terms of differentials for characteristic zero. For

96

FORMAL POWER SERIES

[CHAP. IV

the latter the prime ideal α of V in K [ X ] contains polynomials Fr+i, i = 1, 2, · · · , m — r such that D(Fr+1,

· , Fm)

D{Xr+1, • • • , XJ ^ By the implicit function theorem the system ^+,(X) = 0,

i = 1, 2, · · · , m — r

has then a unique solution for the Xr+i such as (13.1). Hence there is a unique branch Sr of Vr centered at P and it is linear. Thus the earlier definition implies the present one. The converse also holds but is not so easily proved and we shall not stop to do so here.

§ 4. ALGEBRAIC VARIETIES AS TOPOLOGICAL SPACES 14. When the groundfield K is the complex field topology allied with analysis plays a fundamental role. In this direction there are very important modern investigations notably by W. D. Hodge, for which the reader is referred to his book [1]. For earlier contributions see also S. Lefschetz [1, 2], We shall merely discuss here a few of the fundamental concepts closely related to power series representations. We first define a parametric η-cell as an η-cell U n (topological image of the open spherical region in Euclidean w-space) together with a topo­ logical mapping (parametrization) t of Un into a spherical region of an Euclidean number space referred to coordinates U1,··· , un or parameters of the cell. A temporary convenient notation is (Un, t, Ui). An analytical n-manifold is a connected space with a covering by parametric w-eel Is u a i)} with the following property: If X is in both {(^«> and U", 0 then the uai of the points of a certain neighborhood of x0 are analytical functions of the corresponding with non-vanishing Jacobian at x 0 . From well known properties of the Jacobian follows that this condition is symmetrical in the two overlapping cells. An analytical M1 is known as an analytical arc. It is an analytical arc of M n whenever the arc may be covered by parametric cells (U), t}, υλ]) each contained in a L™ and so that Vi j is one of the parameters Moti. If the parameters uai can be chosen throughout so that the Jacobians of the definition of Mn have constant sign, then Mn is orientable·, otherwise it is non-orientable. Let η be even, say η = 2m and let a complex parametric 2m-cell Vm be now defined as the topological image of a spherical region in the space Sm of m complex variables V1, · • • , vm. Except for this let mani­ folds be defined as before. The resulting structures are called complex m-manifolds written M™. Let {(V"\ ta, v^)} be the coverings which serve to define M f and let

§ 4] ALGEBRAIC VARIETIES AS TOPOLOGICAL SPACES

97

V0.^ = Vrii + iv'ly Then tx is a topological image of a spherical region of the Euclidean space of the variables v'aj, v'^. The real Jacobians of these mappings are up to a constant factor

-dKl * ' ' » »J!2 >0. 1>(νβυ -",vt

Hence: (14.1) A complex m-manifold is a real analytical orientahle 2m-manifold. Let us notice finally that: (14.2) If the manifolds are compact one may assume that the coverings are finite. For further information on all the concepts just developed the reader may consult the author's books [3] and [4]. 15. Returning now to varieties let us merely observe that when the groundfield is complex, all the properties of power series discussed earlier in this chapter remain true even if one imposes the additional restriction that they have a positive radius of convergency. A complex projective m-space Sm is the continuous image of a closed spherical region in an Euclidean 2m-dimensional space and hence Sm is compact. Moreover Sm is arcwise connected: any two points may be joined by an arc (in this case a rectilinear segment). Finally it may certainly be covered by complex parametric 2m-cells turning Sm into a complex m-manifold Mf. Its topology turns all the varieties of Sm into topological spaces under the topology induced by that of Mf. Let us now prove: (15.1)r An irreducible Vr of Sm is connected. Since K is algebraically closed an irreducible F0 is a point so that (15.1)0 holds. Now, referring to Picard [1], p. 429, (15.1)] holds: irreduc­ ible complex algebraic curves are connected. (His proof only given for plane curves is readily extended to curves in any space). Assuming now (15.1)r_1; r > 1, we shall prove (15.1),.. By (III, 20.1) one may find a hyperplane I which intersects Vr in an irreducible Vr"1 which is therefore connected by (15.1)^. If P is any point of Vr a hyperplane V through P intersects Vr in a pure r — 1 dimensional variety one of whose com­ ponents say Vr-1 contains P. Now F'r_1 is intersected by I in some point Q. Since F'r_1 is connected, P is in a connected set which meets the fixed connected set Fr-1. Hence Vr is connected. Suppose now that Vr has no singularities. Referring to (13) every point of Vr is contained in a complex parametric 2r-cell. The totality of these cells gives rise to a covering under which Vr becomes an Mrc. It is also readily shown that Vr is a closed set. Since Sm is compact so is Fr. Hence: (15.2) A complex irreducible Vr without singularities is a compact complex r-manifold.

V, Algebraic Curves, Their Places and Transformations With the present chapter we take up the thorough study of algebraic curves. An intensive use of formal power series in one variable will enable us to carry over to the algebraic domain a large part of the technique and classical results of algebraic functions of a single complex variable. There are also noteworthy contacts with the theory of valuations whose importance, owing to the work of Zariski, is increasing by leaps and bounds in algebraic geometry. For a more complete treatment of the subject matter of this and the next chapter, the reader is referred to the excellent book of R. J. Walker [2], Hereafter the groundfield K is assumed to be algebraically closed and of characteristic zero. 1.

FORMAL POWER SERIES IN ONE AND TWO VARIABLES

1. If f(x,y) is an element of the power series ring

then

(1.1)

that is to say the series is regular in y. The Weierstrass preparation theorem assumes therefore the form: (1.2) Whatever

we have

(1.2a) where the

are non-units of

and n is the degree of

in

y in (1.1).

The special polynomial

is an element of the integral domain In this ring factorization to within units is unique. We have at once: (1.3) The factors of a special polynomial g{x, y) are equivalent to special polynomials. By combining with (1.1) we find: (1.4) A finite system of relations

98

2]

PUISEUX'S

THEOREM

99

is equivalent to a single relation where g is a special polynomial in y. 2. Formal power series in one variable present a shift of emphasis from rings to fields. For while no simplification is possible for a quotient of series in two or more variables, this is not at all the cases for quotients of series in one variable. Indeed a quotient of two elements of may be written

where r may well be negative. Thus rather than mere series in nonnegative powers, we are led here naturally to consider series of the form (2.1)

with the obvious laws of combination. The natural collection which they form is the field K((x)) and it comes at once to the fore. Consider now the algebraic extension by the solution y of Instead of y we write as usual and thus also for K((y)). Thus is the field of all the formal power series in a;1'" with coefficients in K. Let denote the union of all the fields K((xlln)) for Let y, say Then both are in and there follow definitions for zero, in the obvious way and readily shown to be unique. Hence is again a field over K and since it contains K it is also of characteristic zero. (2.2) Differentials. Let dx be an indeterminate. Then if where

we define its differential in the natural way and show that these differentials behave throughout in the usual manner. 2.

PUISEUX'S THEOREM

3. This is the central proposition regarding formal power series in one variable. Its statement is: (3.1) Theorem. The field is algebraically closed. Let that is to say

where the are elements of The theorem asserts that

i.e. fractional power series in

100

ALGEBRAIC

CURVES

[CHAP. V

where the are likewise fractional power series in x. By a classical argument all that is necessary is to prove that has a root in y which is in In carrying out the proof there is no restriction in assuming that: (a) the have no negative exponents; (b) (c) the are integral power series in x. The assertion as to (a) is obvious, and as regards (b) and (c), they may always be achieved by a suitable transformation We may then take (3.2) It is clear that Puiseux's theorem will follow if we can prove: (3.3) With f as in (3.2) if /(0, a) then f(x, y) has a root such that Upon making the change of variables the situation will be as before save that Thus (3.3) will be reduced to (3.4)„ With f as in (3.2) if /(0, then has a root such that This is the property which we shall now prove. Since (3.4)2 is obviously true we assume for every and prove Moreover if holds and so we assume We are seeking then a solution of the form (3.5) where and are and the exponents positive fractions with the same denominator. Let

are

(3.6) where the least is zero and the greatest is n. If (3.5) is to be a solution upon substituting y from it in lowest degree terms in x must cancel in

the

(3.7) Let us mark in a cartesian plane the points The lowest degree terms in (3.7) correspond to the points M on some line I: such that no point M is below I. Moreover to have some cancellation of terms I must contain at least two of the points. This leads to the classical Newton polygon (Fig. 1). This is a polygonal line resting upon the two axes, concave towards the origin, whose vertices are points M and such that no such points are below it. The condition that the line I must satisfy is merely to be one of the sides of the polygon. Let us choose for I the side QR of the figure. If are the vertices on QR we will have

2] where

PUISEUX'S is the least

Moreover if

and

THEOREM

101

It is clear that

are any two vertices on QR then

hence

FIG. 1

where the last fraction is merely reduced to its simplest expression. It follows that mod q. Hence in particular

We must now choose for t a root of g(t) = 0 and by I it is the substitution and replacing x by we find

Making

To find the exponent fj,x one must deal with as previously with / . There are now two possibilities: Case I. Then and the least degree nx of a term in alone in is certainly By the preparation theorem where is a special polynomial of degree Hence by has a solution such that and this implies (3.4)„. Case II. Then q = 1 and so is an integer. Upon treating like / , etc., either: (a) we only have case II throughout, so that the are all integers and a root is obtained in yielding or else (b) at some stage case I occurs and holds once more. This completes the proof of (3.4) and hence of the theorem.

102

ALGEBRAIC

CURVES

[CHAP. V

4. We shall now consider a certain number of special cases required later. At the outset let

where and has no negative powers. One may then write the Taylor expansion where the Bk behave like the One must also bear in mind that/(a, y) has the A;-tuple root when and only when and (4.1) If f(a, y) has b for k-tuple root then f(x, y) = 0 has exactly k solutions in y which are in and whose series begin with b. Hence if all the n solutions of which are in are of non-negative order. Setting and referring to (3.3) there is one root such that Then behaves like / with k — 1 in place of k. The proof is then completed by an obvious induction. (4.2) Let the coefficients B} all be in and let f(a, b) = 0, Then by (4.1) has a single solution such that y±(a) = b. This solution is actually in and is merely the usual Taylor expansion of y(x) such that f(x, y) = 0, y(a) = b. By direct substitution of a series in f(x, y) = 0, it is found that the coefficients can be calculated uniquely one at a time and these coefficients turn out to be those of the Taylor series. (4.3) The equation has a unique solution in (Implicit function theorem for one variable.) (4.4) The equation has n distinct solutions in and they are all units. If one of them, is then all the others are represented by where is a primitive nth root of unity in K. (4.5) The equation has n distinct solutions in such that y(a) = b. Setting hence and applying (4.4) yields (4.5). (4.6) The equation has in the n solutions where is a primitive n-th root of unity in K. Setting hence and applying (4.5) the required result follows.

PUISEUX'S

§2]

THEOREM

103

(4.7) If is irreducible the equation in y: has no multiple roots in Suppose that there is a multiple root Then is also a root of Hence / and fy have a common factor By applying Euclid's algorithm if / and fv have a common factor it is in Now we may assume that the degree m of / in and is of degree m — 1 in y. Hence the common factor would have to be of smaller degree than m in y and consequently / would be reducible in hence also in contrary to assumption. Hence (4.6) holds. 5. Application to the complex field. Let the groundfield K be the

complex field and let

be a polynomial in y with coefficients analytic and holomorphic at It is then known from function theory that there are m roots analytic in x in the neighborhood of a and tending to the m roots Moreover as x describes a small circle around the value a, the are distributed into circular systems of the following nature: if the notations are properly chosen then will be one of the circular systems and its elements undergo a cyclic permutation as x turns once around a. Furthermore as the tend to the same say It follows that the of the cyclic system are holomorphic functions of in the neighborhood of a, and hence

where the series is convergent for small x — a. Thus the m roots are represented by m convergent fractional power series in a; — a for x near a. Since this is precisely the number of formal power series solutions known to exist from Puiseux's theorem, we conclude that the Puiseux's series represent actual analytic solutions. If and a is say a root of multiplicity k then we would replace y by and proceed as before. We may thus state: (5.1) When the groundfield is the complex field the Puiseux series solutions represent analytic solutions. Application. Let

where all the coefficients are real. As we have seen the unique solution with x is given by the McLaurin expansion y(x) of f(x, y) — 0 which calculated in the customary way and whose coefficients are now all real. Thus the solutions o f / = 0 near the origin are all given by a real power series convergent in a certain vicinity of the origin. That is to say on the real

ALGEBRAIC

104

CURVES

[CHAP. V

curve f(x, y) = 0 the origin has an analytic arc for neighborhood. This will hold of course if A(a, b) is any point of the curve such that The expansion will then express y — & in powers of Notice that the situation is the same in the complex domain save that the neighborhoods are now 2-cells (simply connected regions). 3.

THE PLACES OF AN ALGEBRAIC CURVE

6. Let f(x) be an irreducible algebraic curve of degree m in and A (a) any point of the curve. Choose the coordinates such that / i s regular in and that Upon passing to affine coordinates we have a curve F(X, Y) where F is of degree m and contains a term in The point A is now in and has coordinates written (b, c). In particular F(b, Y) = 0 has finite roots, one of which is c. By reference to (3.3) F(X, Y) = 0 has a certain number of solutions (6.1) where have no common factor. Setting say that the pair of expressions

we may

(6.2)

are the coordinates of a point t. Since transc transc The projective coordinates of

off depending upon the indeterminate is a general point of the curve. are given by

(6.3) 7. The result just obtained suggests that we consider general points Such a of / with coordinates in the integral power series field point will have coordinates xi given by a system (7.1) It is manifestly no restriction to assume that the are not all zero. Thus is a point of As a matter of fact

Since all the terms of the series at the right must vanish so that A is a point of the curve. Since is a general point of / we have: (7.2) A n.a.s.c. for a curve g to contain f is that From analytical considerations we are justified in considering as equivalent two points such that the representation analagous to(7.1)for is derived from (7.1) by a regular transformation Since these transformations form a group we have here a true relation of equivalence. We will also say that the representations corresponding to are equivalent. A class of equivalent representations is

§ 3]

PLACES OF AN ALGEBRAIC CURVE

105

known as a place of the curve/. The point A is the same for all equivalent representations of a given place, and is referred to as the center of the place. We shall now reverse the procedure and go back from (7.1) to (6.3). We will suppose the coordinates so chosen that / is not a line through one of the vertices of the triangle of reference and that furthermore α0 φ 0. Since x0(t) is then a unit we may divide the xt(t) by it and thus replace (7.1) by a representation (7.3)

px0 = 1, pxx = b + tmE(t),

px2 = c -f tm'E1(V),

where m, m' > 0. By (4.4) we can write E (t) = E ™(t). Then by (4.3) if τ is a new indeterminate the relation τ = tE 2 (t) has a solution t = TE 3 (T). Substituting in (7.3) and writing t for τ we obtain a representation equivalent to (7.3) (7.4)

px0 = 1,

PX1 = b + tnd,

px2 = c + IcjVh'1.

Here n, n v · · · are all positive and have no common factor. If we had permuted the role of X and X or X we would have arrived at a similar representation with some of the coordinates permuted. Any one of these is known as a normal representation. Whenever M1 at ττ. The line AB is known as the tangent to the place π. Its equation is (14.2)

I xi> am> aa I = 0,

(i = 0,l,2).

For the normal representation (13.5) we see at once that A = 1, and A and B have the respective coordinates (1, a, b) and (0, 1, m j. The two points are thus manifestly distinct. The equation of the tangent to π reduced to affine coordinates is found to be Y - b = τητ{Χ — a) and as it has the same slope as the tangent to / to which 77 is attached, the two tangents coincide. Upon combining with the results of (13) we have: (14.3) Theorem. To a q-tuple tangent I at the p-tuple point A there corresponds a set of places centered at A, tangent to I, and whose orders have for sum the multiplicity q of the tangent. The sum of the orders of all the places centered at A is the multiplicity ρ of the point itself. Various corollaries and subordinate results may be mentioned. They are all immediate consequences of what precedes. (14.4) To a simple tangent I at A there corresponds a single linear place 7τ centered at A and tangent to I. If m is the slope of the tangent then the corresponding solution Y(X) of F = 0 is Y(X) = m X + bX2 + * ' ' • This is also a representation for 77. (14.5) A p-tuple point A with distinct tangents is the center of ρ distinct places, all linear, and each tangent to one of the tangents to the curve at A. By (13.3) the place TT1 corresponding to the tangent slope mt has a representa­ tion with parameter X : Y = m{X + atXz + ·· · . (14.6) A n.a.s.c. for a point A to be the center of a place of the curve f is that A be a point of f. (14.7) The order of a place π of center A is the order at π, of every line through A other than the tangent to 7r itself. (14.8) Remark on the ordinary points. If A is an ordinary point F = *X + β Υ - \

,

β#0.

Hence F = 0 has a single Puiseux solution Y ( X ) such that Y(O) = 0 and it is merely the McLaurin expansion of Y(X) obtained in the customary way: Y = - ^ X+yX2+···. P The place 77 of center A referred to as an ordinary place, has the repre­ sentation X = t,

Y=—

~ P

and it is clearly linear.

r • • •

§5]

MULTIPLE POINTS, INTERSECTIONS

115

(14.9) General places. Up to the present we have only admitted places whose centers are fixed points, i.e. of transcendency zero. There is no reason however to be so strict about it. For let M (ξ) be a general point of the curve. Upon replacing the groundfield K by L = ΚΗ(ξ), M becomes a fixed point. Moreover / continues to be irreducible in L and since L is algebraically closed the general theory applies. Since M is an ordinary point there is a single place v of center M and it is linear. We refer to 77 as a general place of the curve. Notice that the process of (14.8) to obtain a representation for π does not require the introduction of elements other than those of ΚΗ(ξ). The algebraic closure was only needed to justify the procedure. 15. Let now g be a form prime to / so that it intersects / in a finite number of points. Let A be one of these. (15.1) The multiplicity of A as an intersection of f and g is equal to the sum of the orders of g at the places of f centered at A. Take again affine coordinates with the origin at A and in particular such that F and G are both regular in Y. It is no restriction to assume furthermore that their leading coefficients in Y are unity. If d is the degree of F there will be d Puiseux solutions Yt(X), i = 1, 2, • • · , d, all in K{X), and none with negative exponents. Taking the resultant of F and G relative to Y there is obtained R { X ) = X " E ( X ) = ± Π£(Χ, Y i ) . The axes having been properly chosen, G will not pass through the intersections, other than the origin, of F with the Y axis. We substitute again for each Yi its series Yt(X) and inquire for the lowest degree term in G(X, Yi(X)). If Yi begins with a constant term, i.e. corresponds to places of F whose centers are on the Y axis but not at the origin, then G(X, Yi(X)) will also begin with a constant term. On the other hand if Yi(X) is represented by (13.3) with 77 of (13.5) as its associated place, the result will be σ(Χ) = G ( X , m , t X + I M i X n ' " 1 ) . Let S(X) be the lowest degree term in σ(Χ). Since there are η factors G(X, Yt(X)) for the place π, they will contribute to the lowest degree terms in R(X) the same power as (δ(Χ))η, which is the same as d(tn), i.e., in the last analysis the order ρ of G(X, Y ) a t π . By combining (15.1) with Bezout's theorem there follows: (15.2) If g is prime to f and d, d' are the orders of f, g then the sum of the orders of g at the places off is dd'. (15.3) Remark. Owing to our concentrating upon the curve /, the present treatment of intersection multiplicities at a point is not sym­ metrical with respect to the two curves / and g. A symmetrical treatment may be given on the following pattern. First of all let us drop the condition

116

ALGEBRAIC

CURVES

[CHAP. V

that one or the other curve is irreducible and let a place of a curve be defined as a place of any irreducible component of the curve. Then curves without common components have no common places. Suppose now that / and g are relatively prime. Let the affine origin A be an intersection of / and g and let and be places of / and g of center A. There will correspond to each a circular system of solutions of F(X, Y) and G(X, Y) say for and F:

and for co and G:

where and are primitivep-th and g'-th roots of unity in K. Since and a) are distinct places whatever Consider now the product

Since is symmetrical in both the and the it contains only powers of X itself and they are all positive. Hence The number is positive and readily shown to be independent of the choice of axes. We define it naturally as the intersection multiplicity of the two places and at A. Now let and be the distinct places of / and g with A for center. Under our hypothesis The place occurs in an irreducible factor of / which occurs in a factorization of / into prime powers, say to the power Similarly there is a related to If we set (15.4) then, on the strength of (I, 3.4) and (II, 27) we readily see that the number v is precisely the intersection multiplicity of the two curves at A. In other words the intersection multiplicity is the sum of the intersection multiplicities of the pairs of places of the two curves centered at A, where each place is to be taken with the same multiplicity as the component of / or g to which it belongs. In point of fact the treatment just given is quite general. It would hold for instance about as well for F and G special polynomials of the power series ring K[[X, F]]:

Only unimportant and obvious modifications in the concept of place would be required to cover this case. In a different direction we are now in a position to complement the

§ 6 ] RATIONAL

AND BIRATIONAL

TRANSFORMATIONS

117

theorem of Bezout in a noteworthy way. Suppose that a n d a r e as before and that the axes X, Y are chosen not tangent to or Then have the same orders on and on Therefore the expansions assume the forms

The tangents to We find now:

and

at the origin are then

Hence if i.e. if and of orders p, q have distinct tangents then and conversely. Thus (15.5) If two places of orders, p, q and with the same center are not tangent, their intersection multiplicity has its least value pq and conversely. Upon combining with (15.4) we obtain: (15.6) If the point A is r-tuple for the curve f and s-tuple for the curve g, then its intersection multiplicity for f and g is at least rs. It is exactly rs when and only when the two curves have no common tangent at the point A. §6.

RATIONAL AND BIRATIONAL TRANSFORMATIONS AND THE PLACES

16. Let / and g be irreducible curves in and and suppose that there exists a rational transformation T off onto g. Under our general theory T will be given by a system (16.1)

where (a) the P { are forms of of the same degree, without common factor and hence not all divisible b y / ; (b) if is a general point off and then is a general point of g. We are in the particular case where T maps a variety onto one of the same dimension. It induces then an isomorphic imbedding making Kf an algebraic extension of of a certain degree fi. As one may expect the role of fi will be dominant here. We infer at once from our general theory (III, 19.5): (16.2) If T is a rational transformation of f onto g and fi is its degree, then for almost all points N of g, and for all its general points, the set T_1N consists of fi distinct points of f, and they are all general when N is general. Referring to (III, 16) since the groundfield is of characteristic zero, the transformation may be represented by a monoidal system:

(16.3)

118

ALGEBRAIC

CURVES

[CHAP. V

Moreover if is a general point of g then is irreducible as a form of and its degree is p. 17. We shall now extend the transformation T to the places. Let the place of / have the irreducible representation (17.1) The point (16.1) for T we find (17.2)

is the center of

Upon substituting in the relations

where the bi are not all zero so that is a point of Since the parametric point is a general point of / , the point is a general point of g. Hence (17.2) is a representation of a place of g whose center is B. Hence B is a point of g. It is seen at once that if one replaces (17.1) by any other representation for there results merely another representation for Hence is uniquely determined by T and It is natural therefore to extend T to the set of all the places of / by defining , If A is not fundamental then the are not all zero. Hence Thus we may state: (17.3) One may extend the rational transformation to a singlevalued transformation of the places. If is a place of f whose center A is not a fundamental point of T then the center of : is TA. One may extend T to all the points but it may cease then to be singlevalued in a finite set of points o f / . This is done as follows. For every of center A take and if its center is B then define TA as the set of points B. Since the number of places of center A is finite, the set TA is always finite. If A is not a multiple point then TA is unique, and if A is not fundamental then TA is the same as before. Hence the extension of T is consistent with the original definition of T. However, T as extended may well be multiple valued at the singular points, i.e., at a finite set of points. Returning to our main problem let us take the coordinates so that and choose as in (7) the parameter t so as to have a normal representation for equivalent to (17.2): (17.4) where are positive and relatively prime. As we have seen (8.2) the number d is independent of the normal representation and represents the lowest order of the elements of at 7T, also the H.C.F. of the orders of the elements of K° at the place tx. We call d the local degree of T at

§ 6 ] RATIONAL for

AND BIRATIONAL

TRANSFORMATIONS

Upon making in (17.4) the change of parameter ie normal irreducible representation

119

there results

(17.5) Suppose that T is birational. Then hence We may then take This time Since v and are uniquely determined by and extended to the places becomes one-one. As already noted in the proof of (8.7), the identification of associated with T leaves the orders of the elements unchanged. with Hence the valuations , and are the same, so that the one-one correspondence between the places is the one which is associated with (8.7) or (10.2). We may also complete (8.7) in the following manner: (17.6) The one-one correspondence between the places determined by a birational transformation is such that for almost all places the center oj in f is imaged into that of TTT in g and conversely for g and 18. Returning to our earlier situation suppose again that where is given by (17.1) and by (17.5). Thus is one of the places making up We assign to 77 the multiplicity d where as before d is the local degree at We shall now prove: (18.1) Theorem. If is a place of g then consists of fi places off where fi is the degree of T and each place 7r of is to be counted d times, where d is the local degree of T at the place We may also add the following complement: (18.2) If B is the center of an ordinary place of g and (B is the sole image of A), where A is not a fundamental point of T, then A is the center of a place of Hence in view of (16.2), for almost all places the fi places of are distinct. (18.3) The places such that consists of fewer than fi distinct places are known as the branch-places of T on g. An example of a branchplace is the following: / is of degree and without singularities; T is the projection off from a point 0 not in / onto a line I which does not pass through 0. The branch-places are the places of I centered at the intersections with I of the tangents from 0 to / . (18.4) Let us first dispose of (18.2). Let be given in the representation (17.5) and let be its parametric point. The set will consist of fj, points whose coordinates are obtained by solving the system (16.3) for the with the replaced by the from (17.5). Setting the first relation (16.3) assumes the form

where is a polynomial of degree fi in with coefficients in fi distinct roots in which by Puiseux's theorem are in

and , say

120

ALGEBRAIC

CURVES

For each there is a single solution for the second relation of (16.3) is linear in

where coordinates (18.5)

[CHAP. V since

Accordingly we will have for the /jl points

It is not ruled out that a, b m a y b e negative. However, after clearing negative powers and setting we obtain representations 1, 2 of the usual types for fj, places imaged into by T. It is also a consequence of the Puiseux theory that if . then there is a series beginning with the coordinate of A and so (18.2) will hold. 19. Passing now to the proof of (18.1), suppose then that is such that . Since and the places have birational character we may replace our curves by birationally equivalent curves. This will now be done in a definite way in relation to and As we have seen (8.1) there is an element of . of order one at Then there exists another such that We may assume that the order of at is positive. For if it is negative we replace by and if is of order zero then and we replace It is also desirable to have of degree [i over If it is not then some combination will be so and continue to be of order one at It may then take the place of Let be the irreducible equation satisfied by Since is birationally equivalent to / and is a general point of F. This means that we can so choose / that it has a general point with such that is of order one at and of positive order there. Hence we may choose the element of as parameter for and will then have a representation (19.1) Furthermore since is still of degree over the system (16.3) will have preserved all its former properties. A similar treatment will now be applied to g and We first require an element of whose order at is unity and whose order at is positive. At all events there is a of order one at Its expansion on will be and we may choose r so that then

will have the required behavior

§ 7]

SPACE CURVES

121

Operating as above we may select g and a representation for (19.2) Since Hence on Let now

and is divisible by d, it is a number ed, be of order s at . Thus Hence at Thus all the orders are divisible by ed. Hence Hence by (4.6) there are d distinct solutions

(19.3) where is a primitive eZth root of unity in K. Hence from (19.1) the relations for the representative point of

These are then the relations (18.5) corresponding to and we see that there are exactly d such sets. Each corresponds to one of the points mh. Hence the number of places such as each counted d times is This proves the theorem. (19.4) Involutions. Let us consider as a single object the set of places of where each place is taken as often as its multiplicity. Take any place of / . The place is uniquely determined and so is the set which contains tt. Thus any one of the places of y enables one to determine all the other places of the set y to which itbelongs. The set is thus in one-one correspondence with the set of the places of g. We refer to the collection as an involution on the curve/, and to fj, as its degree. The usual designation for such an involution will be §7.

SPACE CURVES

20. Practically everything that precedes is directly applicable to an irreducible curve F in The few deviations of interest later are discussed below. A place of T may be taken in a general representation (20.1)

the center being the point This representation may be reduced to a normal irreducible representation in a suitable coordinate system (20.2)

where , have no common factor.. The order questions are dealt with as before.

122

ALGEBRAIC

CURVES

[CHAP. V

As in (14) one may assume the representation such that the the coordinates of a point Then a hyperplane through A :

are

is of least order n or at and of greater order if it contains B or equivalently the line AB. For this reason the line AB is defined as the tangent to r at the place 21. Projections. We shall only consider those from a point. We take the point as the vertex of the simplex of reference and the projection is into a hyperplane chosen as If F has the representation (21.1)

then the projection F' has the representation

(21.2) Since T is a curve the are not all zero and hence is a point or a curve. If is a point T is a line through If F is not a line through as we assume until further notice, the projection operation Pr is a rational transformation of a certain degree / 2, and contains some of that transcendency. Hence dim V = 2 and so F is a F2. Upon augmenting F2 by the tangents Tj we still have a F2 containing all the tangents and this is (28.1). 3

1

§ 8]

REDUCTION

OF

SINGULARITIES

133

29. Once in possession of (27.1)one may assume in proving (27.2) that r is a non-singular curve in and that it is not in a plane, for then the ultimate reduction would already be achieved. The proof will rest here also upon the choice of a projection center avoiding certain loci of secants. We first prove: (29.1) The multisecants of F (secants meeting T in more than two points) are on a variety of dimension at most two. Let us observe at the outset that if N, are points of T and the secant meets V in further points N", • • •, N{s) then transc (N, • • •, Niiy) = transc for For by (25.3), the number of points is finite and hence they are algebraic over K(N, N1). Let be copies of F in spaces If is the prime ideal of T l e t b e the corresponding ideals of associated with Writing let

Consider now the matrix and let • • • , denote in some order its determinants of orders two and three. The forms of together with the span an ideal a of which determines a variety W of the product space The forms of a together with the span an ideal of the same ring, which determines a variety contained in W. If is a point of W and are the images of in r , then N, will be collinear. However, will only be a true multisecant of F if the three points are distinct, that is to say if the point In view of (27.7) it is clear that we merely need to prove: (29.2) dim or equivalently transc In proving this we may further assume that is not in W', i.e. that is a true multisecant of I\ We proceed with (29.2). Since transc transc we must show that the latter At all events it is Suppose then transc This implies transc transc Then also transc For the alternative transc implies that N is fixed. This is ruled out as follows. Projecting F from N on a plane H not containing N there results a plane curve g which is a rational transform of r . If M is the common projection of and 9Jt the field obtained by adjoining the coordinate ratios of M to K, then those of are in the algebraic closure of Since both 501 and its closure have transcendency unity over K, we must have transc

[CHAP. V

ALGEBRAIC CURVES

134

whereas by assumption it is 2. This contradiction proves that the three points Ν, N', N" are general for Γ. Let L denote the algebraic closure of the field K(N), and similarly for L' and N'. Since N is general it is not in the hyperplane xn, and so in the projection just considered from N into H we may choose H = x . The equations of the projection will then be g(x x , s) = 0, X = 0, where g G L Ix , x , X ]. The multiple points of the projection g are in LP2 and hence they have transcendencies zero over L. Prom transcg- L= 1 and transc i c (N', N") = 2, there follows transc^ (N', N") = 1. Hence if M is as above, transc^ M = 1 and M is general for g. It follows that M is not a multiple point. As a consequence the tangent to g at M is the common projection of those to Γ at N' and N". Hence these two tangents are coplanar and so they intersect. Hence N" is one of the points in finite number whose tangents meet the tangent to Γ at the point N'. This means that the coordinates of N" are in L' or transc£, N" = 0 and therefore transcg- (N', N") = 1. Thus our original assumption is untenable and we must have transcA, (N', N") < 1. This proves (29.2) and hence also (29.1). To complete the argument we still require: (29.3) The secants NN' such, that the tangents at N and N' intersect are contained in a variety of dimension < 2. As above let N be general and let the projection be as before from N on x0. If P is the intersection of the tangent to Γ at N with X0, then the tangents to Γ which meet NP are projected into tangents to g from P, and the points such as N' are projected into the points of contact, in finite number, of the tangents to g from P. It follows once more that transcy- N' = 0, transc/c (N, N') = transc^- N = 1 and as before this yields (29.3). Proof of (27.2). Let V 1 be the variety containing all the multisecants, V2 the variety of (29.3), V3 the variety containing all the tangents to Γ. Thus the union V of the Vi is of dimension < 2 in KPz. Hence one may take a center of projection O not on V and project Γ from O on a plane not containing 0. Let g be the projection. Since Γ has only linear places and O is not on F 3 , g has likewise only linear places. Since 0 is not on V 1 , no multisecants pass through 0, and so g has only double points for singularities. Finally, since O is not on V2, the tangents at the double points are distinct. Moreover, g is a birational transform of Γ and hence of /. For, if g were not, almost all points of g would be the projections of at least two points of Γ. Thus g would contain an ordinary point M projection of two points N, N' of Γ and, as we have seen, the tangents to Γ at N, N' would then meet, which requires O G F2 contrary to assumption. Thus g satisfies all the requirements of (27.2) whose proof is now complete. 0

lt

11

1

2

3

2

x

0

VI. Linear Series The notion of linear series arises in substance when one considers the collection of the intersections with the basic plane curve / , of the curves of a linear family, for instance all the lines or conics of the plane. The major birational properties of algebraic geometry center around this fundamental concept. Needless to say throughout the chapter the groundfield K continues to be algebraically closed and of characteristic zero. § 1.

DIVISORS AND THEIR CLASSES

1. As we shall be concerned mainly with rational functions and their orders, properties which have birational character, we may as well take as birational model a plane irreducible curve / . We denote its order by m. Let and let • be a place of the curve. We call a zero [a pole] of order v of R whenever R is of order v [of order Take a representation of iJ where P, Q are forms of equal degree /j, of Let be the orders of P and Q at where each is zero at almost all places. Then (sum of the orders of the zeros) — (sum of the orders of the poles) since each sum is m/i. Hence: (1.1) For any non-zero element R of the sum of the orders of the zeros is equal to the sum of the orders of the poles. A complementary result is: (1.2) When R has no zeros, and hence no poles, then R is in K. Under the hypothesis and in the above notations P and Q have the same zeros and with equal orders 00(77-3) such that

Let

be a place which is not a so that Then is a form of order with as a zero of order at least a>(77,), and in addition with the zero of order at least one. Hence the sum of the orders of the zeros of , i.e. of the multiplicities of its intersections with F exceeds m[i. Therefore mod F. Hence

From (1.2) follows: (1.3) Two elements of with the same zeros and poles and the same orders in each differ only by a factor in K. 135

136

[CHAP. VI

LINEAR SERIES

2. Consider now all the places ττ of / as generators of an additive group (5. Any element D of © is an expression, called a divisor; D = P1TT1 + · · · +

P n TT n =

Σρ(π)TT,

where ρ(ττ) is integral-valued and zero at almost all places. The meaning oi D ± D', D = 0 is obvious. The number η = Σρί is the degree of D and the divisor D is effective whenever every > 0. Noteworthy divisors are the following: I. Corresponding to R G K1 and not in K: (a) D' the divisor of its zeros; the Tri are the zeros of R and the p{ their orders; (b) D" the divisor of the zeros of R~l or divisor of the poles of R; (c) the divisor D = D' — D" or divisor of R itself. Notice that D' and D" are effective but D is not. II. Let P(x) G Ku[x\ be a form not divisible by /. Let Tri be its intersection places with / with intersection multiplicities pt. Then D = EpjTri is the divisor of the zeros of P and it is effective. If μ is the order of P then the degree of D is τημ. III. The divisors of an involution / on /. Referring to (V, 19.4) I results from a rational transformation T: / —g, wherein Ka C Kf and Kf is of degree μ over Kg. The set of places πν · · · , ιτα of / making up T7-1TT (tt a place of g) with multiplicities p{ have for sum an effective divisor D = Σρ8ττ{ of degree μ. The salient property of these divisors is that each is uniquely determined by any one of its places. One may also remember that almost all the divisors of I consist of μ distinct places (the multiplicities are then unity). An arbitrary collection of places and multiplicities is not very promising from the standpoint of geometry. Hence we can have but little interest in the full additive group of the divisors. A much more fruitful line of attack appears when divisors are compared in relation to the elements of the function field. We shall say that two divisors D, D' are equivalent written D •—- D', whenever D-D' is the divisor of an element R of Kf. It is at once seen that this is a true equivalence relation. It gives rise therefore to equivalence classes (I and then to an automatic definition of the additive group of the classes. In point of fact we need only to retain that the operations CjzK' have a meaning. Note also this consequence of (1.1): (2.1) All the divisors of the same class have the same degree n. The number η is also referred to as the degree of the class. Since everything introduced in the section rests upon the function field we may state: (2.2) The divisors and their classes as well as the operations +, — on the divisors and, classes have birational character.

§2]

LINEAR §2.

3. Let and let (3.1)

SERIES:

FIRST

PROPERTIES

137

LINEAR SERIES: FIRST PROPERTIES

linearly independent elements of indeterminate).

Let be the least order of any atthe place This number is zero for almost all places. The order of itself at is where and again at almost all places. We refer to the effective divisor

as the variable divisor of Take any fixed effective divisor and set The divisors belong to a fixed class. The collection is known as a linear series. The degree n of is the degree of the series and r is its dimension. Following Brill and Noether the series is denoted by The correspondence is one-one. Sinceone may consider as a point of this correspondence turns into a projective space and it is in fact this space which we have in view when we consider Upon replacing the set linearly independent linear combinations of the is unchanged. The effect on is a change of coordinates. One may evidently write where Then is the variable intersection of (3.2) with / . Upon multiplying the by the same Q intersecting in a divisor containing and still calling the new polynomials wewill have the following situation: has for intersection divisor where A is fixed and variable; One may then say that is generated by It may also be observed that if one considers the as forms in the indeterminates then they are linearly independent mod / . A series represented by a subspace of is said to be contained in A series which is contained in no other series but itself is said to be complete. Since series and their inclusions are defined in terms of the function field we may state: (3.3) Under a birational transformation a series [a complete series] grn goes into a series [a complete series] grnfor the transformed curve. 4. Consider now a grn generated say by (3.2).

138

LINEAR

SERIES

Corresponding to any consider h points / and in relation to these the system

[CHAP. V I

of the curve

If we take for any point which is not fixed for _ there results a system whose coefficient matrix is of rank one and it defines in the space KP\ a linear subspace Suppose that there have been obtained points such that the corresponding system has a coefficient matrix of rank and so defines an We may then select not an and not contained in all the satisfying The resulting system will have a coefficient matrix of rank h. In particular for we may select r points such that the coefficient matrix of is of rank r and so determines a single point S of It is clear that in the above construction one may choose the so as to avoid any preassigned finite set of points of / . In particular, one may choose for the ordinary points of / . Thus will be the center of a single place and the r places thus obtained will be distinct. We have thus proved the following useful property: (4.1) Corresponding to and its generating system (3.2) one may select r distinct points , where each . is the center of a single place and with the following two equivalent properties: (a) the matrix is of rank r;(b) there is a unique divisor of containing all the places Let now be a place of / with parameter t and let p be the least order of any Thus

where the are not all zero. If is to be of o r d e r a t must satisfy the linear system of a relations

, the

Notice that among these relations the first certainly has coefficients not all zero. Let as before __ denote the variable part of the intersection divisor of Pl. (4.2) If the variable divisor is to contain a preassigned divisor B of degree v the must satisfy a system of v linear relations whose coefficient matrix is not zero. Hence in the space . these special _ will correspond to the points of a subspace S of dimension s where In fact for at least one divisor of B there will result a system of relations such as above with the not all zero, and the asserted property follows.

§2]

LINEAR

SERIES:

FIRST

PROPERTIES

139

An immediate consequence of (4.2) is: (4.3) If the places are not in the fixed divisor of then there exists a divisor of ~ which does not contain the For if is generated by the containing make up a hyperplane in and all that is necessary is to t a k e e x t e r i o r to all the (4.4) For any necessarily This is trivial for and when by (4.1) some divisor of contains r distinct places. (4.5) Application. An irreducible curve f of degree m possesses at most

distinct double points with distinct tangents.

The proof will rest upon the series cut out by the curves degree passing through the double points. Since (4.5) holds manifestly for , 2 we will assume so that there are curves of degree A form of degree has a number of coefficients equal to

of ,

Since passage through a point imposes a single linear relation upon the coefficients there will exist a curve passing through any preassigned set of points. In particular if there are more than N double points there will exist curves through of them say . Among these curves there will be , linearly independent say and they will also be linearly independent mod / since / is irreducible and the degree of the According to is the center of two linear places Let , so that is of degree Thus has an intersection divisor where is of degree Since is already ruled out and so (4.5) holds for cubic curves. Assuming then generates a with which contradicts (4.4) and this proves (4.5). (4.6) Let us make here a remark useful later. If there are exactly N double points with distinct tangents say then defining as above we find this time by the same calculation , hence That is to say an irreducible / with N double points contains 5. (5.1) If havea common divisor D then both are contained in where evidently If neither of the given series contains the other then Let the two series be generated by

140

LINEAR

[CHAP. V I

SERIES

We may suppose the so chosen that D is cut out by That is to say D is the corresponding to and similarly for the second series. Since we may replace the and the we may assume at the outset that and hence that the have the same degree. Among the a certain number may be linear combinations mod These linear combinations will form with a set of forms linearly independent mod / . We may choose the notations so that these forms are and that Then the forms will be linearly independent of the Pi mod / . If we define then generates a containing the two given series. For corresponds to the such that and to the such that If neither of the two series contains the other and hence This completes the proof of (5.1). (5.2) Every determines a unique complete linear series which contains it. In particular every effective divisor D (since it is determines a unique complete linear series of which it is an element, and which is written

(5.3) A complete consists of all the effective divisors of an equivalence class. All the divisors of are in a fixed class Suppose that contains an effective divisor D x which is not in Let Since is the divisor of an element of the function field and the divisor of zeros of Hence the divisor of zeros of . where is effective. Hence with both and as elements. Clearly is also excluded since is complete and is ruled out since ioes not contain Dv Hence there is containing a contradiction which proves (5.3). (5.4) Sum and difference of linear series. Let be two linear series and their classes. The class is known and its complete for the difference series is called the sumSimilarly provided that the class _ contains effective divisors. This will certainly be the case whenever there exist effective divisors and A such that 6. (6.1) Theorem of Bertini for algebraic curves.

In

without fixed places almost all the divisors consist of n distinct places. Let be generated by (3.2) where the Pi are linearly independent mod / . The relations (6.2)

§2]

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SERIES:

FIRST

PROPERTIES

141

define an irreducible correspondence For / is irreducible in and if is a general point of / then is irreducible over i since it is of degree one in the , Given there correspond to it, without exception a finite number of points o f / , namely its intersections with which is always prime t o / . Hence by (III, 18.6) to almost every there correspond distinct points of / each counted with multiplicity unity, and hence distinct places each counted with the same multiplicity one (V, 15.1) and this is precisely (6.1). (6.3) Application to the class. Let as before / be irreducible and let its order The first polar of the point. is

To begin with the are linearly independent. For suppose for some point Then this will hold in every coordinate system. Let the system be so chosen that is the point (1, 0, 0 ) . T h e n , Thus / does not contain Since ifc is of the form and irreducible it must be of degree one, contrary to assumption. Since the are of degree they are also linearly independent mod / . Let be the variable part of the intersection divisor of and n its degree. Thus generates a The points on / where the all vanish are the double points In order t h a t h a v e in one of the places of center order above the minimumthe will have to satisfy a certain linear relation. Hence for almost all the divisor will not contain a place of center and, furthermore by Bertini's theorem its n places willbe distinct. Hence (III, 5, Example III) for almost all points of. their centers will be ordinary distinct points of / and they are the contacts of the tangents from to / . Hence n is the class of the curve / . To calculate it take one of the double points B and let be the places of center B. Choose coordinates such that B is (1, 0, 0) and the tangents at B the lines Then

Hence for almost all. the curve is not tangent to / at B. It follows that is of order one at Hence each enter with coefficient unity in the fixed part of the intersection divisor of Hence

This is a special case of one of a classical set of formulas given by Pliicker and relating various characters of an algebraic curve: order, class, number of inflexions, of bitangents, of double points and of cusps.

LINEAR

142 § 3.

SERIES

[CHAP. V I

BIRATIONAL MODELS AND LINEAR SERIES

7. Given a set of forms of the same degree and not all divisible by / , we may on the one hand consider the linear series generated by (7.1) and on the other the associated rational transformation (7.2) To replace the by another set of linearly independent linear combinations of themselves is the same as to apply a projective transformation in and will not affect our treatment. Taking advantage of this choose the so that the first r + 1 are linearly independent mod / and the rest divisible by / . This amounts to selectingcoordinates in such that is in the subspace At the same time it assigns to the generating system In other words one may as well take Since the case offers no interest we assume also Finally since the fixed part of the divisors of plays no role in the sequel we shall suppose that there is none, or in our earlier notations that . Since T is a rational transformation of / onto T it has a certain degree A n.a.s.c. for T to be a birational model is that We wish to translate this property into a property of Let the place of V have in its inverse the places o f / , where has multiplicity Thus is a divisor of the involution on / associated with the rational transformation T. Referring to (V, 17), and recalling that for the local degree is , one may select parameters t for and for such that Since has no fixed places there exists a _ whose divisor contains no places of . and in particular none of the Thus defines an element which is also in s i n c e i s of the form

It is clear that is the divisor of zeros of . as an element of . On the other hand since it has a certain order a a t the place of . As a consequence on Since is a zero of wehave Hence enters in with the positive coefficient and contains Let now be the distinct divisors of the involution which contain the places of We have just shown that if . contains

§ 3] the place of then no common places

BIRATIONAL contains

MODELS

143 Since d i s t i n c t h a v e

(7.3) That is to say every divisor of grn is an exact sum of divisors o f T h e series is then said to be compounded with the involution When is merely the collection of places of / and / and r are birationally equivalent. The series is thus not compounded with any Such a series is said to be simple. Thus if r is not a birational model o f / , is compounded with an Conversely let this be the case and suppose that nevertheless T is birational. Let be an ordinary place o f / . Then the divisors of containing IT make up with TT as its sole fixed part. Hence T cannot be compounded with an To sum up we may state: (7.4) Theorem. A n.a.s.c. in order that the curve T associated with be a birational model of f is that be simple. is compounded with an involution , Returning to the case where and referring to (7.3) the divisor has for degree Since the Bt have the common degree fx we have As we have a in r , by (4.4): Hence the useful inequality (7.5) which limits the degree of an involution with which might be compounded. (7.6) Normal curves. Every algebraic curve T of order n which is a m o d e l o f Kf in KI>r may be generated by a system (7.1) such that cuts out a simple We say that T is a normal curve of KPT whenever: (a) T is not contained in a I there is no curve of order n in some KPr contained in no whose projection into KPr is T. If such a curve exists coordinates may be so chosen for that is the space Thus if is the curve (7.7) then r is the curve represented by the first r + 1 relations (7.7), and by Now on the image of is the series cut out by the hyperplane and it is contained in the cut out by all the hyperplanes of Thus when T is not normal is not complete. The converse is immediate. Hence: (7.8) A necessary and sufficient condition in order thata curve T of be normal is that the hyperplanes of the space cut out on complete

144

LINEAR SERIES

[CHAP. VI

§4. RATIONAL, ELLIPTIC AND HYPERELLIPTIC CURVES 8. Rational curves. (8.1) Theorem. A n.a.s.c.for the rationality of a curve f is that it possess a linear series of type g". Then all its complete series are of that type. A series gnn can only be complete since its dimension is the largest possible for its degree (4.4). Supposing / rational we may assume that it is the line x0. Then the forms ψη{χχ, x2) order η cut out a series of degree n. Since the η + 1 monomials are manifestly linearly independent mod x0, the dimension of the series is η and so it is a g*. Since there is a ψη cutting out any preassigned divisor of degree n, the family {g™} thus obtained is the totality of all the complete series on a rational /. Thus all but the sufficiency condition is proved. Suppose now that / possesses a series (/", η > 0. Since the dimension is n, the series is complete. Let π be an ordinary place not. fixed for the series. The divisors of 1 the existence of a gr™ is shown to imply that of a g1}. Assume then η = 1, and let the series be generated by Px = /I0-P0 Jr Then (8.2)

ρλ0 = P

1

(X),

ρλχ = — P0(x)

is a rational transformation T of the curve / onto the line KP1. If Mx is the point (A0, A1) of the line, there corresponds to Mx a divisor Dx of g\ consisting of a single place π and the center Nx of π is the inverse T-1Mx. Hence T is of degree one, and so it is birational. Therefore the curve / is rational. (8.3) Application. The following irreducible plane curves of degree m are rational: (a) curves with a point of multiplicity (m — 1); (b) curves

with the maximum number N = (m — 1 )(m — 2)/2 of double points with distinct tangents. Since a line is rational one may assume to > 1. Regarding (a), the lines through the (m — l)-tuple point are linearly independent absolutely and also mod / since their degree < to. Their intersection divisors each contain one variable point. Hence the lines generate a g\ and so (a) follows. In particular then irreducible conics are rational. Regarding (b), we may then assume m > 2. By (4.6) / contains a g^, hence it is rational. (8.4) Rational involutions. If the curve/ possesses a rational involution I there is a related rational transformation T of / onto a line I. If I is the line KPf, T is given by relations (8.2) and the divisors of Iμ are the variable divisors of g1 generated by Px — A0P0 -j~- A1P1. The converse is obvious. Hence: (8.5). The rational involutions on a curve f are merely the collections

of divisors of its one dimensional linear series.

§ 5]

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145

(8.6) Another fruitful approach to rational involutions is as follows. If X is a n affine coordinate for the line I then K f is algebraic over K(X), and hence Kf is a simple extension K(X, Y). If F(X, Y) = 0 is the irre­ ducible equation satisfied by Y, F is an affine birational model of / and in this model the divisors of I in the affine plane are the intersections with F of the lines X = const. Thus F is of degree μ in Y. The converse is obvious. As an application of (8.5) one will prove readily: (8.7) Conies and cubics carry an infinity of rational involutions of degree two. 9. Hyperelliptic curves. We have just shown (8.6) that / possessss a rational I2 when and only when it is reducible to the type (9.1)

F = A ( X ) Y * + 2 B ( X ) Y + C( X ) = O ;

A,B,C£K[X].

A further reduction is obtained by means of the birational transformation X-+X, A(X)Y +B(X)-> Y to the form (9.2)

Y2 = F Q ( X ) ,

F G K [ X \ and of degree q.

= G2H

Suppose that F where the factors are in K \X\ and H has no s q u a r e f a c t o r s . T h e n t h e b i r a t i o n a l t r a n s f o r m a t i o n Y — > G ( X )Y , X - > X operates the reduction to the same form (9.2) but so that F has no square factors. Suppose q = 2r, and let F have the root a. The birational transformation 1 Y X— + a, Y X^ ' Xr will operate the reduction to the form (9.2) with q odd. Upon combining with (8.5) we have then: (9.3) A necessary and sufficient condition in order that the curve f possess a rational involution I 2 is that it be birationally equivalent to a curve of type (9.4), Y 2 = F 2p+1 (X) ivhere F has no square factors. The involution is then the g\ cut out by the lines X = constant. The conics (p = 0) and cubics (p = 1) have already been discussed. The conics are rational. The curves reducible to the type (9.4)x, are known as elliptic, and those reducible to the type (9.4),, ρ > 1, are known as hyperelliptic. We shall see later that ρ is a birational character: the genus, so that curves corresponding to distinct values of ρ are birationally distinct.

§ 5 . ADJOINT CURVES AND SERIES 10. Let us adopt henceforth as birational model a curve / which has no other singularities than double points with distinct tangents. An

146

LINEAR

SERIES

[CHAP. V I

adjoint to the curve is a curve that passes through the double points. An adjoint of order q will be denoted by We recall that if there are double points then

is non-negative (4.5). The number is known as the genus of the curve/. We shall show that p is a birational invariant of the curve and its importance will manifest itself presently. Let us suppose that adjoints do exist. Referring to the proof of (4.5) this is certainly true for and hence for every , when In point of fact adjoints exist provided that

Thus for there exist which clearly holds if T ' be the double points and let adjoints of order Let " — be the two places of center Dt. Set also If the adjoints exist each i intersects / in a divisor effective. We say that G is cut out by r The totality of the divisors G constitutes a linear series , the series cut out by the adjoints effective, we say that is cut out by the adjoints through and is the linear series cut out by the adjoints through The divisor is also said to be a residue of 1 relative to the series cut out by the The following is the basic property of the adjoints: (10.1) Max Noether's residue theorem.

Let G, G' be equivalent

effective

divisors and let there exist an adjoint cutting out . effective. Then there exists likewise an adjoint cutting out For the proof we require three preliminary lemmas. As before the curve / is assumed of order m. (10.2) Lemma. If a line I cuts f in m distinct points and the curve g contains each then mod / , where of course I, g, Select a triangle of reference none of whose vertices are on / and such that I is the line Thus the B i are on and distinct from and . Since / is regular in we may divide g by / as to and obtain ivhere and is of degree Thus where is of degree vanish at the points Since at the B i it follows that vanishes at the m points. Thus the curve g" of degree in m points. Hence m o d / , i.e. mod/. (10.3) Lemma. If I intersects f in the double point D and in m — 2

§ 5]

ADJOINT

CURVES AND SERIES

147

further points and if g contains the and has a double point in D then again mod / , where as before Select the triangle of reference so that is D, the are on and the vertices . are not on / . The same relations will take place as before so that:

This time since g and / have a double point in A1 they are of degree in Hence this must also hold for , Thus Now of degree must intersect in the distinct points Bt. Hence and the conclusion is the same as before. (10.4) Lemma. Let be the two places whose center is the double point D. If g has a zero of order at least two in both and then it has at least a double point in D. Take a triangle of reference with and not tangent to / at D. Passing to affine coordinates we will have

where is a form of degree i. The two places TT, TT' of center A have then representations (V, 14.5)

Since C is of order at least two in each we must have Since is of degree one with two distinct roots, 1 and G has multiplicity at least two at D. hence Proof of the residue theorem. Since there is an element 1 of the function field whose divisor is Let be the zeros of Q and choose coordinates so that the triangle of reference bears no special relation to More explicitly the are not to be on fPQ, not on a line joining two double points, nor on any tangent at a point or issued from a point , and each side of the triangle is to intersect / in m distinct points. Since Q and / are regular in upon taking their resultant as to x2 we obtain a relation

Since the lines _ they are not tangent L are the lines to / and do not intersect on the curve. Multiplying then both P and Q by B the function R will be determined by a quotient of forms of type: (10.5)

148

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SERIES

[CHAP. V I

Let now be an adjoint cutting out a divisor effective. Upon setting, , we find that instead of (10.5) R will be likewise determined by a quotient (10.6) Let now M be one of the intersections of with / and let be a place of center M. Given any form we denote by co(g) its order at If A is any divisor then will designate the coefficient of in A. Since is not tangent to / at and like all places of / is linear we have We also have:

Now on the one hand the order of as a pole of R is at most since the do not intersect on / . the other hand it is Hence

; on Hence

We have now two possibilities: is not a double point. Then , hence . Therefore if contains the point M . (b) M is a double point. Then hence if S is of order at least two at Thus S is of order at least two at each of the two places of center M and therefore (10.4) i f is a double point for S. We conclude that if then by reference to (10.2) and (10.3) S fulfills all the conditions for divisibility by m o d / . Beginning then with one may replace by another form still called S at the cost of replacing by _ throughout. The orders of S at the intersections of with / will merely have been lowered by one will remain the same. unit, and those at the intersections with Hence we will have (10.2) and (10.3) save that will have been replaced by The same process may be continued until all the are reduced to zero. At the end we shall have a determination of R by a quotient where is the same adjoint as before. Since the divisor of R is and the divisor of zeros of is the same for is H e n c e i s an adjoint of the same order q as and the theorem is proved. 11. (11.1) Given any effective divisor G let H be a residue of G as to the adjoints . Then the complete series determined by G is cut out by the adjoints through H. (11.2) The adjointsthrough a given divisor H cut out a complete linear series. In particular for (11.3) The adjoints cut out a complete linear series.

§ 6]

THE RIEMANN-ROCH THEOREM

149

If the curve / has no singularities all the curves of the plane are adjoints. Hence: (11.4) I f f h a s n o s i n g u l a r i t i e s a l l t h e c u r v e s o f a g i v e n d e g r e e c u t o u t o n f a complete linear series. As an application we will prove a special case of a classical result on cubic curves: (11.5) Let f, Z1 be two plane cubics where one, say f, is non-singular. Suppose that the two curves intersect in nine distinct points M1, · · · , M9. Then any cubic passing through eight of the points also passes through the ninth. Let TTi be the place of center Mi and let H = TT1 + · · · + τr8. The cubics φ through H have a variable intersection divisor D such that the collection {D} is a complete series. Since the full intersection divisor of φ is of degree 9 and has a fixed part of degree 8 the series is a g\. Thus r < 1. If r — 1 then / is rational and all its complete series must be of type g% (8.1). Now the forms x0, X1, x2 are linearly independent mod /. Hence the series which they generate is a g\ and it is complete since it is the series cut out by all the lines. Hence / is not rational. It follows that r = 0 and so ψ generates a glt i.e. all its places are fixed. Since Z1 is a special φ, ττ9 is in the intersection divisor of φ with /. Hence if φ contains H, i.e. it passes through M1, · • • , M8 it also contains π9, i.e. it passes through M9. As a matter of fact (11.5) is true even when / has a double point or is singular or even reducible, but this is not readily proved by means of linear series. On the other hand our argument holds whatever the fixed part H, of degree 8, of the intersection divisor of fv and it asserts that if φ contains H then its intersection divisor is the same as that Oif1-

The following interesting property is an immediate consequence of (11.5). Let A1A2A3 and B1B2Bz be two sets of collinear points of f, where the six points are distinct and the lines AiBi are not tangent to /. Then AiBi intersects / in a third point Ci and C1C2C3 are collinear. The restrictions as to distinct points and non-tangency are of course readily removed. Let us also recall finally that (11.5) with the irreducibility restriction removed implies Pascal's theorem. For these and other related properties of cubics the reader will profitably consult Robert Walker [1], p. 191 and van der Waerden [2], p. 87.

§6. THE THEOREM OF RIEMANN-BOCH 12. This fundamental theorem deals with the difference η — r for a complete grn. In view of (11.1) it is natural to calculate first the difference nq — rg for the complete series g^ generated by the adjoints φα. If δ is the

150

LINEAR

[CHAP. VI

SERIES

number of double points then the divisor centers are the double points, is of degree

, sum of the places whose Hence

Among the adjoints a central role is played by those of order The curves and the adjoint series which they generate are known as canonical curves and canonical series. It turns out to be convenient to set where That is to say only adjoints are to be considered. We have then: (12.1) In particular the degree of the canonical series is (12.2) Hence its dimension A divisor and the complete series are said to be special whenever there exists a canonical curve through G. Let G be special and let H be a residue of relative to the canonical series. Set also

Then | G | may be generated by all the through H and by all the through G. By Noether's theorem G may be replaced by any and H by any Hence in particular is a character of The number is known as the speciality-index of It is the maximum number of linearly independent adjoints through any divisor G of the series. W h e n i s not special its speciality-index is We may now state: (12.3) Theorem of Riemann-Roch. Let be the speciality-index of a complete Then (12.3a) In particular if the series is non-special (12.3b) Since for is certainly non-special, (12.3b) holds for n sufficiently large. Thus as a consequence of the theorem: (12.4) p is the maximum of n — r for any complete linear series and this maximum is actually reached. Since complete linear series have birational character, a consequence of the theorem of Riemann-Roch is (12.5) The genus p is a birational invariant. Owing to this property one refers sometimes to p as the genus of the function field Kf.

§6]

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THEOREM

151

13. For the proof of the Riemann-Roch theorem we shall require: (13.1) Lemma. Let be a curve of degree n and points of the plane. Then for n the conditions expressing that yi passes through the s points are linearly independent. Let t be any integer It is sufficient to show that there is a curve of degree n passing through any t of the points but not through the rest. To that end take t lines where lt contains but no other „ and a curve containing no point The product curve answers the question. We shall now proceed with the proof of the Riemann-Roch theorem essentially along lines laid down by Max Noether. Let again be the number of double points. The number of terms in a form of order q is

If the form is to vanish at the double points, then the coefficients satisfy . linear relations which are linearly independent when q exceeds Hence the adjoints make up a linear family among which are linearly independent where for q sufficiently high. The family contains a linear subfamily composed of adjoints divisible by / , i.e. of form Among these there are linearly independent and no more where Hence for q sufficiently great there are curves linearly independent mod / . Under the circumstances the dimension of the complete series cut out by the is, with

where Therefore the adjoints of order s sufficiently high cut out a complete series Thus for the complete series cut out by the curves above a certain value, we have i.e. the theorem of Riemann-Roch holds. Bearing in mind that for q arbitrary one may have to replace aa by we have (13.2)

In particular . Hence: (13.3) The canonical series exists for md it is a Both (13.2) and (13.3) are temporary since the theorem of RiemannRoch asserts that the numbers s are all zero.

152

LINEAR

SERIES

[CHAP. V I

(13.4) Consider now any complete whatsoever and let G be a divisor of the series. By (13.2) one may choose q so high that exceeds the degree of G. Then by (4.2) some contains 6?. Let H be the residue of G as to Thus H is of degree ^ Since is cut out by the containing H its dimension Hence and therefore (13.5) Notice that at this stage it is already known that for any complete series and that the maximum p is reached. That is to say (12.4) is already proved and the genus p shown to be birationally invariant. In particular if / c o n t a i n s i t is rational (8.1) and so as for a line. 14. Further progress in the proof of the Riemann-Roch theorem will rest upon: (14.1) Max Noether's reduction lemiua. If G is a special divisor, contains as a fixed place. for almost all places tt the complete series Let be a maximal linearly independent set of canonical curves through G. They are also linearly independent mod / since Hence the fixed intersections of with / make up a divisor effective. Take a place not in In particular then the center of is not a double point. The excluded choices for are manifestly finite in number. Since is not in there is a containing R but not Since is not a double point we may pass a line I through meeting / in m distinct points Thus At is the center of one and only one place Now containing with residue Hence the complete series is cut out by all the curves through . Since such a curve meets I in distinct points is divisible by I and so the divisor which it cutsout as a residue to contains Thus is a fixed place of and the lemma is proved. 15. The proof of the theorem of Riemann-Roch will now follow rapidly. (15.1) A complete hence any -such that is necessarily special. Suppose we have which is ruled out. If the series is a hence / is rational and so p = 0 (13.5), a contradiction. Thus Let then and suppose Then certainly and so the condition is satisfied. Let G be a divisor of the series. Since the canonical series is of dimension there is containing G. If H is the residual divisor of G as to is generated by the canonical curves through H. Thus (15.1) holds whenever We may therefore suppose and use induction on n. Supposing then (15.1)

§6]

THE RIEMANN-ROCH

THEOREM

153

true for sarily

we shall prove it for n. Since necesTake a place not in nor fixed for There is a divisor containing If G is special the same reasoning as above will prove special.Suppose then G non-special. Since is not fixed for is a and since , this is special. Since the through do not contain and is not in the reduction theorem may be applied as between • and (14.2). Hence has as fixed place. This contradiction proves (15.1). Since for every grn we may state: (15.2) If is complete non-special That is to say all complete non-special series are of type This holds in particular for theseries cut out by adjoints Thus if the series is of type The result just stated is the so-called Riemann part of the RiemannRoch theorem. Roch's complement is the statement regarding the dimension of complete special series to which we now turn our attention. Let have the speciality-index Thus if G is any divisor of there exist a linearly independent canonical curves and no more through If 77 is not a fixed place of the linear series which they generate, nor in by the reduction theorem is a and its speciality-index is Proceeding thus we shall obtain finally a complete of speciality-index zero, i.e. non-special. By (15.1) This is precisely the Riemann-Roch theorem whose proof is now complete. 16. Some noteworthy complements follow: (16.1) Law of reciprocity of Brill-Noether.

Let the complete

series

9rn> 9n> have for sum the canonical series. Then

By Noether's residue theorem is cut out by Let G be a divisor of the through G. Hence is the speciality-index of grn and that of Consequently from which to (16.1) is but a step. (16.2) The canonical series is it is without fixed places and it is the only completeof f. Hence the canonical series is a birational invariant. We have seen that the canonical series is Since its specialityindex , we have , and hence Thus the series is a Suppose that there existed another complete series of dimension and degree Since is special. Since its n is the same as for the canonical series the two must coincide.

154

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SERIES

[CHAP. V I

Suppose now that the canonical series has a fixed place Then is a complete Its and so we must have which is false. Hence the canonical series has no fixed places. (1G.3) The adjoints cut out series without fixed places. This is already true for i Consider any The only fixed divisor of the curves of this type is Hence this is also true for all the

VII. Abelian Differentials Historically speaking abelian integrals came first and arose as the natural generalization for the complex field of trigonometric and elliptic integrals. However for a general field one may only consider differentials, since integration in such fields is at best not a simple matter. On the other hand the differentials provide us with additional and very powerful means for a direct attack on birational geometry. Our present program is to develop the general theory of the differentials and to discuss some applications. As in the preceding two chapters the groundfield is assumed algebraically closed and of characteristic zero.

§1. PRELIMINARY QUESTIONS 1. The abelian differentials of an irreducible curve / are merely the elements of the vector space S of the differentials of the field Ki. Since dim /=1, Φ is spanned by any differential dR of a transcendental element R of Kt. Thus an abelian differential of/ is reducible to the form RdS, R and ,S' £ K i , S n o t i n K . (1.1) Since the abelian differentials are defined in terms of the function field Kf alone they have birational character. Hence we may choose for our curve any convenient birational model. Unless otherwise stated the model selected will be the same as before: a curve / of degree m in KP^. which is not a line through a vertex of the triangle of reference. In par­ ticular since / φ x0 we will be able to pass freely to the affine model F(X, Y) =/(1, X, Y)· Moreover for a general point the ratios X1IrX0 and x2lx0, i.e. X and Y, will be transcendental. This property is decidedly convenient when dealing with differentials. (1.2) Let the Xi and hence also X, Y be the coordinates of a general point of /. Since X is transcendental at a general point, one may write (1.3)

du = R(X, Y)dX.

In homogeneous coordinates the related form is (1.4)

du — R(x0, x v x2)

x,,dx, — x ,dx„ s xO

which is homogeneous and of degree zero in the Xi, dxt combined. It should be noted that in one or the other form d u is merely a con­ venient standard designation for the abelian differential and it does not

A BE LI AN

156

DIFFERENTIALS

[CHAP. VII

imply that u is an element of such that In the background there is the fact that when K is the complex field and R, S are considered as analytic functions say of X then there exists always an analytic function such that However, as the example of . shows, it may not always be possible to choose At a general point Since a general point is not multiple, t h e / ^ are not all zero. Hence the preceding relations yield (1.5) Under our hypotheses the ratios are all transcendental. Hence one may replace in (1.4) by any pair Since the are not all zero one may choose a pair such that At the cost of renumbering the coordinates we suppose Hence we may write (1.6)

where A, B are forms of and the ratioA/B is of degree m — 3. Let be any point strictly in i.e. let the ci be elements of K which are not all three zero. Then

For otherwise the general point (x) would be one of the finite intersections of / with the first polar of C and the ratios would be in K. In view of (1.5)

This yields for du the symmetric form (1.7) It is an elementary matter to verify that under a projective transformation the second fraction, and hence also preserves its form. That is to say du may be written in the form (1.7) in any coordinate system whatever. Herein lies the advantage of the form (1.7).

§1]

PRELIMINARY QUESTIONS

157

(1.8) Let 77 be a place of/ and let xi

=

i = 0,l,2

be a normal irreducible representation of π. The parametric point Jft(9?(i)) is a general point of/ and hence the function field Kf is isomorphic with KH((du, (3.1) at almost all places. Since X is transcendental at the general point ( X , Y) one may write Now at almost all places one may take as parameter Thus and hence at all places where R and and where one may choose t as above. Since the excluded places are in finite number (3.1) follows. By virtue of (3.1): is a divisor o f / . It is known as the divisor of the differential du. Consider two differentials At If

we will have

, are the orders of R, Rx and those of du, are Hence the coefficient of in is the order at of the element of Hence is the divisor of or That is to say the divisors of the differentials are all equivalent. Their class is the differential class. Its effective elements make up a complete linear series called the differential series. (3.2) Theorem. The differential series is the canonical series. The proof will come after a further study of the differentials. 4. Let du and 77 be as before and let We will say that is a zero of order o> of du when is a pole of order—00 of du when A differential which has no poles is said to be of the first kind (abridged as dfk). If du, are dfk so is and Hence the dfk are the elements of a vector space with scalar domain K. It follows at once from the definitions just given that: (4.1) A birational transformation sends a dfk into a dfk and transforms

§2]

DIVISORS

OF THE

DIFFERENTIALS

159

2B into an isomorphic space. Hence the maximum number of linearly independent dfk is a birational invariant. We will prove in fact : (4.2) Theorem. A necessary and sufficient condition for du to be an abelian differential of the first kind is that it be reducible to the projective form: (4.2a) where

is a canonical curve. The cartesian form is

(4.2b) From this follows at once: (4.3) The dimension of the vector space 2B of the differentials of the first kind is equal to the dimension p of the system That is to say there can be found p, but no more, linearly independent differentials of the first kind. For the complex field this is a classical result due to Abel. (4.4) Rational curves have no differentials of the first kind. This last result could be proved directly. For when the differentials are of the form R(X)dX, . If K is not constant there are always poles, while dX has as a pole. Proof of (4.2). Referring to the proof of the residue theorem (VI, 10) one may write

where the lt are lines not tangent to/and do not join two double points o f / , and A is a form of degree since du is of degree zero in the x.t and Let intersect the curve / in M. Suppose first that M is not a double point. If 77 is the place of center M we must have and hence A(x) passes through M. On the other hand if M is a double point and the center say of and then and lt are each of order one on and so A is of order two there. Consequently it has M as double point at least (VI, 10.4). To sum up then, A passes through all the intersections of with / and if passes through a double point of / , A has at least a double point there. Since is not tangent to / nor joins two of its double points, A is divisible by mod / (VI, 10.2, 10.3). The same reasoning enables one to suppress the factors one at a time. Thus du will be reducible to the projective type (4.2a) except that so far it is only known that is a form of degree w — 3, but not necessarily an adjoint.

160

ABELIAN DIFFERENTIALS

[CHAP. VII

Returning, however, to the double point M, we have this time the affine from (4.2b). As above FY is of order one at Trl, hence the same holds for ym_3. Hence (pm_3 must pass through M and so it is an adjoint. This proves the necessity of the condition of the theorem. Suppose now du in the projective form (4.2a). Let π be any place of F and let M be its center. Referring to (V, 14.4) when M is not a double point one may put du in the affine form (4.2b) with X as the parameter and FY φ 0. Since φ,η_3 is a polynomial du will not have π as a pole. On the other hand when M is a double point the cartesian form for du may be taken such that on η (one of πν π2 above) X is still the parameter and Fy is of order one (V, 14.5). Since 1, coincident tangents at the multiple point Λ7(0, 0, 1). Hence the dfk and the genus cannot be obtained from our general theory and require a special treatment. We will suppose that ρ > 1 and also that Ji1(O) φ 0. This last con­ dition is not a true restriction, for if it is not fulfilled one may apply an affine transformation X —> X φ a, Y-^Y where F(a) Φ 0. As a result the curve will retain the form (5.1) but we will have F(O) φ 0. (5.2) Theorem. The curve (5.1) has ρ and no more linearly independent dfk. From (5.2) together with (4.3) we conclude: (5.3) The genus of the curve (5.1) is p. Remembering the reduction to the form (5.1) we may deduce from (5.3): (5.4) The genus of the curve Y 2 = F 2p+2 (X), where F has no square factor, is again p. Proof of (5.2). It will be by direct reduction of the differentials. Now any abelian differential can be written in the form (5.5)

du •

A(X) + YB(X) dX C(X) Y

§ 3] ELLIPTIC

AND HYPERELLIPTIC

DIFFERENTIALS

161

where K[X\- We shall examine the behavior of du at the various places of the curve. The homogeneous representation of the curve is (5.6) where

. Since

we see that because / has no square factor, the point N(0, 0, 1) is the only multiple point (actually an ordinary point for p = 1) and at the same time the only point on (at infinity). Thus the ordinary points are all in the affine plane. However if M(a, b) is such a point the representation for the place of center M will not be the same for Thus as regards the representations of the places we have to examine three distinct cases. We shall examine each separately and consider the corresponding behavior of du. First case. The center of the place is a point M(a, b) with hence Since M is a simple point and is not the tangent at M, we may take for the parameter Then and can only be apole if >. Suppose then that this holds and let One may then have poles at and at the place whose center is (a, —b), and a is a suitable parameter for also. It follows that if du is to be of the first kind must be of order . Since the order of Y is zero this requires that both A(X) and B(X) be divisible by Thus the factor may be suppressed in A, B, G. Second case. The center of tt is a point. Thus now This time thereis only one place it to be considered. Since a is a simple root of and so (5.1) considered as an equation in X has a unique solution in if[[y]] (V, 4.2). Hence we may take Thus it has the representation Notice that at

and since

is of order zero at

Thus we only need to

consider the order of its coefficient in du. Suppose this time G factorable as before and

We are only interested of course in the case Let first ^ , Then du is of order at and so we must ! may be suppressed as before. have hence Thus

162

A BE LI AN

DIFFERENTIALS

[CHAP. VII

If the order of du is and again with the same conclusion. To sum up, if du is a dfk then it is reducible to the type (5.7) where A, B are polynomials of degrees q, r. Third case. The center of is the multiple point N(0, 0, 1). As we shall see this case reduces essentially to the preceding. Consider the curve (5.8) where G has no square factor. At the cost of a possible change of the coordinate X we may assume that , Then the birational transformation,

whose inverse has the same form reduces (5.8) to a curve (5.1) and therefore also (5.1) to the type (5.8) with 0(0) = 0. Notice in passing that under T the only points of (5.1) which go to infinity are those on the line X — 0. Since by assumption there are exactly two such points and as they are simple for (5.1) we conclude that the hyperelliptic curve (5.8) has two places with centers at infinity. On the other hand under T the only point of (5.8) imaged into the point at infinity N is the origin. As the latter is a simple point of (5.8) it is seen that the point at infinity N of the curve (5.1) is the center of a single place. With evident meaning of the symbols T reduces du to

The order must be non-negative at the unique place of center in the preceding case a suitable representation is (here a = 0):

, As

and we merely need to verify that the bracket is of non-negative order. We must have hence since otherwise there is a term of odd negative order. Then the first term is of order and as this must not be negative we must h a v e H e n c e when du is of the first kind it is reducible to the form (5.9) Conversely when du is in this form, whatever , du has no poles and so it is of the first kind. Notice also that the representation is unique.

§ 3] ELLIPTIC

AND HYPERELLIPTIC

DIFFERENTIALS

163

For if it were not some with but mod which is manifestly ruled out. To sum up, we have: (5.10) Theorem. Every dfk of the curve (5.1) may be uniquely represented in the form (5.9) and conversely every differential (5.9) is a dfk for the curve. The maximum number of linearly independent dfk is then manifestly the number of terms in polynomials i.e. it is p. This proves theorem (5.2). (5.11) Noteworthy special case. In the elliptic case and the only dfk is, up to a constant factor

and this is the well known Weierstrass differential. When K is the complex field X as a function of u is the Weierstrass elliptic function p(u). (5.12) Canonical series. Let us suppose The divisor H(du) will be merely the divisor of the zeros of du. To find the zeros take du in the form (5.9). Suppose also that is relatively prime to F(X) and has only simple roots. Let a be such a root. As we are under case I, we may take . Since

Hence the place tt is a zero of order one and likewise for The sum of the orders of the zeros of du is which is the maximum possible. Hence if we write upon varying the the variable intersections generate the full canonical series of our hyperelliptic curve. It may be observed that the lines or in affine coordinates X = const., cut out on our curve a rational involution , with which the canonical series is compounded. 6. A certain amount of classification using the genus and the canonical series is already possible. Unless otherwise stated the curve f is again as in the earlier part of the chapter. (6.1) All curves of genus zero are rational, that is to say they are birationally equivalent to the straight line, and hence to one another. This is notably true regarding non-singular conics. It is sufficient to point out that when p = 0 the adjoints s sufficiently high, cut out a (VI, 13) making the curve rational (VI, 8.1). (6.2) All curves of genus one are elliptic. In this case there is a single canonical curve andit intersects / in the divisor of the double points Special case: and is a constant. The adjoints cut out a Those containing

164

A BE LI AN

DIFFERENTIALS

[CHAP. V I I

suitably chosen places will cut out a residual series Hence the curve may be reduced to where F3 has no multiple roots. (6.3) All curves of genus two are hyperelliptic. For the canonical series is a (6.4) A necessary and sufficient condition for a curve f to be hyperelliptic is that its canonical series be composite. The series is then compounded with a rational involution which is unique. Necessity has already been proved (5.12). As for sufficiency suppose that the canonical series (hence and is compounded with an involution

According to

and since

we have Therefore / is hyperelliptic Suppose now that there exist two distinct involutions . Then the canonical series must be compounded with each. Hence a divisor containing a place must contain at least two more places. Select now the places in succession so that is not one of the fixed places of the canonical divisors containing the Under our assumption the special divisor containing the contains at least more (each counted with its multiplicity). Hence the degree of the series which is ruled out unless And in fact (6.5) Rational and elliptic curves contain an infinite number of rational involutions I2. Take the curve in the reduced form for rational curves, I for elliptic curves. In the first case the lines through any point M of the plane, in the second the lines through any point N of the curve generate for each M or N the series g\ of an infinite family which gives rise to an infinity of distinct involutions of the asserted type. 7. The canonical model. When / is not hyperelliptic the system (7.1) where is a maximal linearly independent set of canonical curves, generates a birational model F o f / known as the canonical model of the field Kf which is a curve of order On T the canonical series is cut out by the hyperplanes of Since the set is unique to within a linear transformation the canonical model F is unique to within a projective transformation. Moreover since the canonical series is complete T is normal (VI, 7.8). Let us show that T has no singularities. Assume that I i s a singular point. The hyperplanes through M cut out a series without fixed points. If are the places of center M, the hyperplanes cut out in addition a fixed divisor where Hence one may add to the series a fixed divisor. so as to produce

§ 4]

THE SECOND AND THIRD KINDS

165

a special series = | #'+ SfiT^_«|· This series is of dimension < ρ — 1 and > ρ — 2, and so it is a special grfρ~_?4. The residual divisors relative to the canonical series make up a g% and by Brill-Noether's reciprocity law: 2ρ — 6 = 2(ρ — 2 — s), hence s = 1 and the series is a g\. This is ruled out however since Γ is not hyperelliptic. To sum up we may state: (7.2) Whenever the canonical model exists (i.e. ρ > 2 and nonhyperelliptic case) it is normal, non-singular and unique to within a pro­ jective transformation of KPv~l. Remark. The simplest canonical models are the plane quartics without singularities. They correspond to ρ — 3 and their canonical series is cut out by the straight lines of the plane.

§4. DIFFERENTIALS OF THE SECOND AND THIRD KINDS 8. Let du = RdS be an abelian differential with a pole at the place π. If t is a parameter for π we will have (du)n = R(t)dS(t) =

+ · · · + -j + b0 + bjt + · · · ^ dt.

As already observed (3) under the change of parameter t —>• tE(t), the order η remains fixed. It is seen at once that the coefficient a1 is likewise unchanged. Thus both η and a1 depend solely upon du and π. We refer to U as the residue of du at the place rr, and whenever Ct φ 0, du is said to have a logarithmic singularity at the place tt and π to be a logarithmic place of du. The justification of these terms is evident. We now define du as a differential of the second kind (abridged as dsk) whenever it has no logarithmic places, and as a differential of the third kind (abridged as dtk) otherwise. In particular differentials of the first kind are also of the second kind. Furthermore: (8.1) Under a birational transformation an abelian differential of a given kind goes into one of the same kind. 9. One may also deal with the differentials from a different point of view. For later purposes we shall take the curve in the affine plane, not a line X = const., (9.1) F(X, Y)= 0 but otherwise wholly unrestricted. Then if F is of degree TO in F and a £ K there are m roots Yv · · · , Ym of (9.1) in the field of fractional power series K{(X — a)} obtained by means of Puiseux's theorem. The roots break up into a certain number of circular systems of conjugate roots. Let them be so numbered that Y v · · · Yq is one of the systems. If Y1 has the expansion (9.2) Y1 = b + (X - α)*/«(α0 + αχ(Ζ - α)1'· + • · · ) 1

1

then the expansion of Y k , k < q is obtained by replacing (X — a)llv by

166

roots is on the line

A BE LI AN

DIFFERENTIALS

[CHAP. VII

where is a primitive q-th root of unity in K. To the q there corresponds a unique place tt of F whose center and which has the representation

(9.3) If one replaces say by then (9.3) is merely replaced by the equivalent representation obtained by the change of parameter Let be the distinct places thus obtained. The value may also be included in the above considerations in the following manner. The corresponding places are on the line in projective coordinates. The projective change of coordinates which yields for the affine coordinates (9.4) will bring, relative to ' and the same situation as before, so that we may now view like any other value. Since the X axis is an irreducible curve without singularities the point is the center of a single place of the line whatever a. For , N is merely the point (0, 1, 0). The places are said to be over It may be noted that all the places of F fall under the category "over ' just considered. Since X is transcendental every differential may be put in the form . We have at once (9.5) With S(X) we associate the differential of the X axis, i.e. of the field K(X) (9-6) . . (9.7) When du, dv are related as above then: (a) If du has no pole over then is not a pole of dv: (b) the residue of dv at is the sum, of those of du at the poles over ; (c) the sum of all residues is the same for du and dv. Let first a be finite and let us return to the solutions Yt of (9.1) where the first q are grouped in the circular system represented by (9.2). This circular system gives rise to the place tt represented by (9.3) and we have

§ 4]

THE SECOND AND

Hence du has a pole at 77 if and only if residue is On the other hand to the roots the sum

THIRD ^

KINDS

167

and the corresponding

there corresponds in dv

Now (9.8) yields for

Hence if

is not a pole of cr, while if

it is.

The corresponding residue of a is merely the coefficient equal to that of du for This proves (a) and (b) in the present instance. For the transformation (9.4) is applied and leads to the same conclusion relative to (9.9) and places on the line Since the behavior of (9.8) at the place of center is that of dv at the place of center (0, 0, 1), properties (9.7a) and (9.7b) hold without exception. Since all the places of F are over some (9.7c) is a consequence of (9.7b) and (9.7) is proved. 10. Differentials of the third kind. It is more expedient to begin with these. The basic result is: (10.1) Theorem. Let be given places and A n.a.s.c. for the existence of a dtk with the as the residues at the places 77^ and with no other logarithmic places is that (10.1a) Select coordinates such that the curve is not a line so that X is a transcendental element of Then one may apply (9.7). Accordingly to prove necessity one may replace du by a differential , Breaking S into partial fractions we find

where the sum is finite and the exponents may be positive or negative. Henceit is only necessary to deal with a differential Now if dv can only have a residue at infinity. The change of variables (9.4) shows at once that this residue is zero. As for it has the residue The change of variables (9.4) replaces it by with residue Thus dv has the two residues Hence the necessity of (10.1a) holds for it and therefore in all cases.

168

A BE LI AN

DIFFERENTIALS

[CHAP. VII

To prove the sufficiency of (10.1a) it will be necessary to proceed in a roundabout way. We first treat the following special case: (10.2) Given any two places, there is an abelian differential du with as poles of order one and no other poles. Furthermore it has residues at and Suppose first and take F = Y, i.e. the curve is the X axis. One may even choose this axis so that the centers of are L. It is then clear that

behaves as asserted. Henceforth we assume Referring to (2.1) one may choose F such that intersects the curve in no special way and also such that the centers are ordinary points, in the affine plane and such that i , and that the lines intersect F in points and which are all distinct. Consider now the differentials of the form

where as before (pk denotes an adjoint of order k. It is readily shown that du has no pole at infinity. Let be the (unique) places of centers By the same reasoning as in (4) one will show that the only possible poles of du are the places Since are suitable parameters for one will find that they are poles of order , Hence the divisor of the poles of du is

Let

and choose an adjoint cutting out H. Observe now that: (a) there are adjoints linearly independent mod F, and hence also absolutely; (b) to impose upon to cut out H is to impose upon its coefficients linear relations with coefficients in K. Hence there are at least linearly independent curves cutting out H. Among them there are the p linearly independent curves Hence there is at least one, and is to denote it henceforth, which is not of this type. Let us show that does not contain nor Suppose that it contains Then intersects in the m distinct points Mt. Hence And now intersects in the distinct points Hence and consequently which is not the case. Hence oes not contain Mx and likewise not Mx.

§ 4]

THE SECOND AND THIRD KINDS

It follows that du has ing residues are

169

for sole logarithmic places. If the correspondi behaves in accordance with (10.2).

The sufficiency proof for the condition in Theorem (10.1) is now immediate. Assume (10.1a) and for each let dul have the logarithmic places with residues Then has residues

at the places

and

at

This completes the proof of the theorem. 11. Differentials of the second kind. We have first the following important property : (11.1) If R is any transcendental element of the function field , then dR is a dsk. Moreover if is the divisor of the poles of R then the divisor of the poles of dR is This shows in particular that dR is not a dfk. This follows at once from the fact that if t is a place and its parameter then

One may now ask whether there are any differentials of the second kind which are not of type dR. The answer is found in the following theorem: (11.2) Theorem. There exist exactly 2p dsk such that: (a) no linear combination is a dR, (b) every other dsk dv satisfies a relation

Moreover the dvi may be chosen in the following manner: (a) Given any set of p distinct places whose sum is not special, then has just one pole of order two at and no other; (b) the dvII+i are p linearly independent dfk. (11.3) Suppose first _ Let again . so that the curve is the X axis. Thus

170

ABELIAN

[CHAP. VII

DIFFERENTIALS

Breaking R(X) into partial fractions yields where the

since dv is of the second kind. Hence

which is a differential of a rational function as it should be when (11.4) Henceforth we assume By (2.1) given any place of center one may choose F such that is an ordinary point of F. The affine axes may be taken with as the origin and with the X and Y axes intersecting the curve in m distinct points and none at infinity. Taking this time

we show as before that may be so chosen that is the only pole of du and is of order two. Its residue is necessarily zero and so du is of the second kind. Thus: (11.5) Given a place it of F there exists a dsk having at a pole of order tivo and no other pole. (11.6) Suppose now that du has a pole of order k > 2. We may dispose considered above. of the situation so that the place in question is Take k — 1 distinct lines in KP2 through Mx, each intersecting F in m distinct points. Let be a form of degree k — 1 not passing through any intersection of the I, with / . Then

defines a certain element R of It is clear that R has as pole of order k — 1 and the places whose centers are the other intersections of the /, with / as poles of order one. Hence dR has a pole of order k at and one of order two at If t is a parameter for we have then

where the coefficients are all in K and has a pole of order

As a consequence In addition it has acquired

new poles of order two but the order of no pole already present has been increased. By repetition of this operation, i.e. by adding to du, assumed a dsk, a suitable dR, one may replace it by a dsk, still written du, which has only poles of order two.

§ 4]

THE SECOND AND

THIRD

KINDS

171

12. Select now p distinct places whose sum is not a special divisor. This may be done as follows: In accordance with (VT, 4.1) the first may be chosen such that they belong to just one divisor of the canonical series. Taking then not in that divisor will achieve the desired result. Let now be a dsJc with as pole of order two and no other pole. Let be p linearly independent dfk. (12.1) No non-trivial linear combination K, can be a dR, R For if J? existed its divisor G(R) would be of the form Hence the complete series would have positive dimension. However since A is not special and of degree p, dim and this contradiction proves (12.1). To complete the proof of (11.2) it is sufficient to show that if du is a dsk, then As shown in (11.6) modulo a suitable dR one may reduce du to a differential still called du whose poles are all of order two. We shall now reduce in this same manner du to a differential having no other poles than some of the and those of order two. Suppose that is not a Then is a divisor of degree p + 1 and hence dim . Therefore there exists an having for divisor . It has necessarily as pole since otherwise R1 would have a divisor H — A and this possibility has already been excluded. If t is a parameter for irv we have then

where the coefficients are in K and

Hence

ceases to have as a pole and has acquired no other poles, if any, than some of the with order two. After a finite number of steps then du will have been reduced to the desired type. Suppose then that du has no other poles than some of t h e a n d those of order two. If is a parameter for we have

where all coefficients are in K and

Hence

since the left hand side has no poles and hence is a dfk. This completes the proof of theorem (11.2).

172

A BE LI AN §5.

DIFFERENTIALS

[CHAP. VII

JACOBIAN SERIES

13. Let A be a linear series without fixed places. By Bertini's theorem only a finite number of divisors of the series have multiple places, so that the multiple places form a finite set If is an element of the divisor of A, let be its coefficient in y. The divisor is known as the Jacobian divisor of A. As an example let / have ordinary singularities and let it not pass through the point (0, 0, 1). Let the series A be cut out by the lines and suppose that the tangents among them have order two (ordinary contact) at the places of tangency , • • • , . Then the Jacobian divisor and its degree n is the class of the curve. We have then (VI, 6.10): Hence here (13.1) and this is one of the results which we propose to generalize. Returning to the general case since the Jacobian divisor A T manifestly has birational character relative to A we may choose any convenient birational model for / . Let / be so chosen that the centers of the are all ordinary points and not on Take now two simple divisors of A and let be an adjoint of some order k cutting out but not passing through any point The residual divisor is disjoint from and is cut out by an adjoint of order k through Thus A is generated by Set also through

Since Hence

Let finally the order of is the order of dR at

follows that the Jacobian divisor of dR. The divisor of poles series we have

be the curve is the same as that and conversely. It

is merely the divisor of zeros Hence if C is the canonical

and consequently, passing to the complete series (13.2)

„ ,

From this follows that if is a complete series without fixed places and A is a one dimensional subseries likewise without fixed places then Aj is a member of a complete series, the Jacobian series given by (13.3)

§ 5]

JACOBIAN

SERIES

173

By comparing degrees there follows for the degree v of the Jacobian series the relation (13.4) which is the desired extension of (13.1). (13.5) Remark. To the relation (13.2) there is attached an interesting definition of the canonical series due to Enriques. Let A, B denote linear series of positive dimensions without fixed places. One may prove directly the relations

From this follows that if exists, it is a fixed series and it is this series which is to be defined as the canonical series. The parallel with the differential series is patent: the fixed series under the definition of Enriques is merely the fixed series of the divisors | which are all equivalent members of the class of the divisors of the abelian differentials. (13.6) Let us return to the place We may choose coordinates so that its center is neither on x0 nor on and then will have a representation

Since

is of order zero at

the order of dR will be that of

Now by differentiating / we find

Since

dR has the same order at

as

(13.7) Consider now the Jacobian From Euler's relation follows that the order of J at

is the same as that of

i.e. again that of (13.7). In other words the Jacobian divisor is the divisor cut out outside the double points by the Jacobian curve J. It is from this property series." that there stem the terms "Jacobian divisor" and "Jacobian

174

A BE LI AN

DIFFERENTIALS

[CHAP. VII

14. We shall now apply the results obtained so far to the proof of an interesting formula due to Zeuthen. Suppose that T is a rational transformation of degree of an irreducible curve / into another and the other notations be applied as before to / and let , be the analogues for Referring to (V, 18.1) if are the branch places of T then: (a) when is not a then consists of /x distinct places; consists of say places, Each is assigned a certain multiplicity as a member of and We assign to the branch place multiplicity In point of fact the number which will matter here is (14.1) It is known as the number of branch-places, each being counted with its multiplicity As a consequence of T one may view the function field as a subfield of Hence an element R of is also in Let R be chosen in the following manner: Let us assume as we may that the curve has only ordinary singularities.By Bertini's theorem almost every first polar of will cut out on a divisor consisting of ordinary distinct places, none centered at a double point and none a One may in particular assume the coordinates such that does not go through (0, 0, 1) and that the first polar of this point behaves as indicated above. This first polar however cuts out the Jacobian group A T of the cut out by the pencil We may furthermore assume that the divisor cut out by is also free from the and we will set Under the circumstances A j is merely the divisor of zeros G*(dR), and the divisor of poles (dR) is free from the Let us consider now o n / . Referring to (V, 17) it will be seen that if and has multiplicity d among the fi images of then the order of R at is d times the order of R at From this we conclude that: (a) if is not a branch place , i.e. if then the order of R at and is the same; (b) if is a and has the multiplicity d then has multiplicity d — 1 at Hence the Jacobian divisor on / consists of the following: the fj, places of the inverse of each place of A T each counted once; the places of , where is to be taken with the multiplcity

Since the two sets of places are disjoint On the other hand if

denote canonical divisors of

we have

§ 5]

JACOBIAN

SERIES

Hence From this follows (14.2)

Hence passing to the degrees we find finally (14.3)

which is Zeuthen's formula.

175

VIII. Abel's Theorem Algebraic Series and Correspondences Abel's theorem is one of the great classics of our subject. The study of algebraic series will provide a converse of the theorem and bring out its role in the comparison with linear series. Algebraic series are followed by the general study of correspondences between two curves, of a curve with itself and of the related fixed point questions. The general notations and assumptions are the same as in the preceding three chapters. §1.

ABEL'S THEOREM

1. (1.1) Let / be a curve of genus and let be a point whose coordinate ratios are in a field of transcendency If is an abelian differential then we will understand by (du)M the differential of the field (1.2) Let now be a linear series cut out on / by (1.3) Let us set The general divisor of the series, written conveniently , consists of n places whose centers have their coordinate ratios in a finite algebraic extension L of Since L is a differential field and its space of differentials is spanned by the if du is a has a meaning. We then have (1.4) A sum such as in (1.4) will be referred to as an abelian sum (understood for a dfk). We may now state: (1.5) Abel's theorem. The abelian sums taken at the places of a general divisor of a linear series are all zero. Let be the series. Since is trivial we assume Referring to (1.4) we must show that all the are zero. Suppose that any say We may then choose and still have To disprove this merely requires to prove the theorem for the cut out by Changing slightly our notations and setting, contrary to our general usage the theorem for the

cut out by

const. Since the 176

we must prove are linearly

§1]

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THEOREM

177

independent mod / , X is transcendental. Hence there is a such that . If is the irreducible equation relating X and Y, F is a birational model of / and (We depart here for a moment from our standard meaning for F.) Referring now to (VII, 9) we have (1.6) By (VII, 9.7a) since du has no poles, dv has none on the X line. Hence However the change of variables has no poles and so shows that if > then dv has a pole at infinity. Hence and the theorem is proved. (1.7) Remark. If du = RdS, R and S and with as in (1.4), Abel's theorem is equivalent to

The expression in the sum is conveniently written for the theorem the formulation:

, giving

(1.8)

2. As an application let K be the complex field and consider the special cubic in the classical Weierstrass form of elliptic functions:

so that the roots of are distinct. The genus case, and up to a constant factor there is just one dfk

in the elliptic

Let be three collinear points of the curve, where k is a constant so that the third point is fixed. We have then

(2.1)

and by Abel's theorem (2.2)

It follows that (2.1) defines a solution with an arbitrary parameter k of the differential equation (2.2). It is therefore the general solution of this equation.

178

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AND §2.

CORRESPONDENCES

[CHAP. VIII

ALGEBRAIC SERIES

3. Let be an irreducible correspondence between an algebraic variety in and the curve / and let be the graph of and n the degree of on / . T o a general point N on there correspond n distinct general points of / (III, 17.3) and hence also the correspondence is uniquely determined by any one of the general points Since is a general point it is the center of a single place of / . Let We refer to as an irreducible algebraic series. The numbers n, r are the degree and the dimension of the series and the latter is conveniently denoted by Since together with N any one of the points determines the full correspondence one may say that the pair likewise suffices to determine Consider now a finite set of irreducible algebraic correspondences between and / and let • • • , correspond to in the obvious way. Let be positive integers and set (3.1) The may also include exceptionally correspondences whose divisor consists of a single fixed place. The symbol

is called an algebraic series. The degree of the series is its dimension is r, and the series is once more written It is clear that a is an irreducible whose associated variety is a It will be convenient to designate by K(N) the field obtained by adjoining to K the coordinate ratios of N. We shall be concerned presently with the possibility that the general divisor D of be contained in a divisor of a linear series Suppose that the latter is generated by (3.2) Upon writing that

contains

there is obtained a system of relations

where . Since these relations are symmetrical in the they may be replaced by an equivalent system

Hence the total system asserting that D is contained in a divisor of takes (3.3) the form

§ 2]

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179

There is evidently no restriction in assuming that the rank of the system is Then since it is known that there is a non-trivial solution in the Upon solving for a of the in terms of the rest and substituting in (3.2) there results a system (3.4)

where

is a linear combination of the y)j with coefficients in and Thus generates the complete linear series where D is fixed. Hereafter we drop the index h and write • • • , instead of • • • . It is to be understood that for i n s t a n c e i s found times among the 4. Since the coordinate ratios of the are in a finite algebraic extension of K(N) they are in a differential field. Hence if du is any dfk the abelian sum has a definite meaning. We propose to prove the following proposition whose second part (sufficiency) is a converse of Abel's theorem. (4.1) Theorem. A n.a.s.c. for an algebraic series to be contained in a fixed linear series is that all the abelian sums > of the general divisor D of vanish. The theorem holds for p = 0 since on the one hand there are then no dfk, on the other is contained in the unique made up of all the divisors of degree n. We will therefore assume at the outset that To prove necessity let of (3.2) generate The through D make up the and as the system is non-empty exists. The curve is a certain where the Hence defines a rational transformation r of V onto a variety be the point of whose coordinates are the We may suppose that and introduce the affine coordinates Similarly we may introduce affine coordinates for N. The coordinates . Now by (1.8):

and this proves necessity. Turning now to sufficiency let of (3.2) represent a system of adjoints of order k so high that those through D cut out a complete residual series without fixed places and of degree Thus is non-special and so it is a Its generating system may be taken to be (3.4). The general divisor, i.e. with the indeterminates, consists of t ordinary distinct places. Let us specialize now the point N of to a point N'

180

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AND

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[CHAP. VIII

such that the linear system (3.3) preserves its rank. The divisor D will become D' and the system .(3.4) will become

the being now linearly independent over K. The form generates a series which is the residual of D' as to the series cut out by the adjoints of order k. By Bertini's theorem almost all the divisors cut out by consist of t distinct places. Furthermore almost none have their centers on Hence we may select the with ratios in K such that the centers of the places of the divisor which they cut out are ordinary distinct points of / , in the affine plane and not in Let be a set of p linearly independent canonical curves. Since is non-special, the matrix is of rank p. One may suppose the

so numbered that the determinant

Consider now the fixed divisor and let divisor (of degree p) cut out by the adjoints of order k through If are the centers of the places of H then

be the

For if we specialize the general point N of becomes and hence is ruled out. Furthermore since the are not in i.e. they are in the affine plane and not in , the same holds for the Hence their coordinates exist and Moreover since they are in K(N) they have differentials. Since is a divisor of the fixed series cut out by the adjoints of order k through by the necessity condition of the theorem Hence which reads here

Since necessarily Hence the are all fixed and D is a divisor of the fixed linear series cut out by the adjoints of degree k through This completes the proof of theorem (4.1). Notice that down to the moment when we utilize the abelian sums what we have said applies to any Denoting then by the series whose general divisor is H, we have proved the following result useful below:

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181

(4.2) Corresponding to the algebraic series γ' η there exists a series γτρ such that if D, D* are the general divisors of the two series then D* is nonspecial, D -)- D* is in a fixed linear series, and hence (necessity of the condition in (4.1)) for every differential of the first kind: S(du, D)

-)-

S(du, D*) = 0.

Extension of algebraic series. If we allow in the basic relations (3.1) negative coefficients vh, the divisors D may cease to be effective. Whatever they are the new correspondence will be referred to as a generalized algebraic series and the same designations as before will be applied to it. Upon replacing "linear series" by "divisor class" Theorem (4.1) still holds. That is to say: (4.3) A n.a.s.c. for the divisors of yrn to be contained in a fixed divisor class is that every abelian sum S(du, D) = 0. Suppose that in (3.1) Vi = — Vi < 0. By (4.2) there exists a yrv with divisor Df such that S(du, Df) = — S(du, D1). Hence by (4.1) D t -)- Df is in a fixed linear series. Therefore as regards (4.3) one may replace D i by — Df and hence Vi by v] > 0. By applying the same treatment to all the negative coefficients v}, one will reduce the present case to (4.1).

Thus (4.3) is a consequence of (4.1).

§3. ALGEBRAIC CORRESPONDENCES BETWEEN TWO CURVES 5. This topic belongs to one of the most interesting and suggestive parts of algebraic geometry. Especially noteworthy contributions have been made by A. Hurwitz [1] and by F. Severi, (see [5], Ch. VIi where numerous references and historical indications are given). There are many interesting geometric applications. An ample supply will be found in H. G. Zeuthen [1]. We consider then algebraic correspondences between two irreducible curves / and / *. The first curve is the same as we have dealt with so far. The curve/* is in KP^., and obeys the same restrictions as /. Elements related to / * will be denoted by the same letters as the corresponding elements of / but with complementary asterisks. In particular places and parameters are denoted by π, t for / and by π*, t* for /*, and the related representative points by M v Mt*. Special designations for correspondences will be as follows: The correspondence itself is denoted by £, its indices by μ and μ*, its graph (in the product surface / X /*) by Φ. A place of Φ or of a component of Φ when it is reducible will be denoted by π and its parameter by Θ. The transformations induced by G are T: f —>f* and Τ~λ·. /*—>/. Consider now a correspondence (£ which is irreducible, i.e. its graph Φ is an irreducible curve of the surface / X /*, and non-degenerate,

182

SERIES

AND

CORRESPONDENCES

[CHAP. VIII

i.e. is not a curve where A and are points of / and This means that neither Tf nor is a singlepoint. If is a general point of the graph then M and are general points of / a n d / * associated under the correspondence and conversely. There are also two rational transformations defined by The degrees of S and S* are and By the definition of the indices TM consists of distinct general points of and of /i distinct general points o f / . The operations S and cause isomorphic imbeddings of the function fields and as subfields of the function field of the graph The degrees relative to are for and [j, for As in the case of rational transformations the correspondence is extended to the places. Let be a place of the graph and its parameter. The representative point of is of the form and and are parametric points of two places and which are associated under We have then Let and d be the local degrees of S and S* on and If are parameters for the two places we have then (5.1) If we make the regular change of parameter (5.1) assumes the simpler form

or

(5.2) The correspondence is now extended to the places by pairing off and under The place of the graph is then considered as one of the images of the pair For it is not ruled out that the same pair n, may have several images in the graph. Let of (5.2) satisfy (5.3) The numbers have no common factor. For if c were one could be replaced by hence As a result would not be the local degrees of 6. We must now assign multiplicities to and as elements of

Let M t and be the representative points of and . Given and therefore there correspond to it as many distinct points associated with the same place in relation to as there are values of obtainable from (5.2). These values are (6.1)

ALGEBRAIC

§ 3]

CORRESPONDENCES

where is a primitive dth. root of unity in K. the point whose parameter t is given by

183

To (6.1) there corresponds

(6.2)

Since of (5.2), (5.3) have no common factor, the points thus obtained are all distinct. We assign then naturally to the multiplicity d in relation to and we denote it henceforth by Similarly the place is assigned the multiplicity as place of in relation to 7. Let the birational models of / and still called , be so chosen that A and the centers of and , areordinary points of / and Suppose that there are several places of the graph with the same center In particular say The total multiplicities and assigned to and as elements of and of are Since

and

are the degrees of

and S we will have

8. Branch places and Zeuthen's formula for correspondences.

The

branch places of the irreducible correspondence on / are the places such that in there are places with multiplicity Let us suppose that is such a place and has been assigned the multiplicity in relation to Then is assigned in relation to the branch place multiThe total branch place multiplicity of is plicity

extended this time to all the places with center such that A is the center of Thus for a given there may now be several points and several places , each bringing its contribution to Notice that if is not a branch place then Thus is defined for all places off. The total number of branch places on/, each counted with its multiplicity is by definition

That this number is finite will appear at once. Consider in fact the rational transformation multiplicity of as element of is the

_ of (5). The of the orders of

the elements of on and it is the same as the .. multiplicity of as element of From this follows that contributes the same amount to the branch place multiplicity of relative to S as does relative to and the correspondence itself. Therefore e is also the number of branch places of S as defined in (VII, 14). Since the number of distinct branch places of S is finite and each has a finite multiplicity the number is finite.

184

SERIES

AND

CORRESPONDENCES

[CHAP. VIII

Needless to say there is a number of branch places for and it has the same properties as e, with _ _ in place o f / , S. The analogues of B of (VII, 14) for S and are the divisors of given by where the sums are extended to all the places of , If C, : and canonical divisors of (VII, 14.2) yields on

and

are

(8.1)

By comparing degrees this yields Zeuthen's formula for irreducible correspondences (8.2)

For a rational transformation of degree u, i.e. a irreducible correspondence and the formula reduces to (VII, 14.3). The e of that formula is the same as the present 9. Correspondences between rational curves.

As one would expect

when the two curves are rational one may proceed much more directly. Let the models of the curves be the projective lines . and

referred to the coordinates and Take again f o r _ an irreducible non-degenerate correspondence. The field is now algebraic over and hence there is a relation (9.1) where is irreducible. The graph is merely the curve If is a general point of the line / then the points of TM are those whose coordinates satisfy (9.1). Hence is of degree in the , and for evident reasons of symmetry it is of degree in If are associated under and have the centers and the reader will readily verify that the multiplicity of as element of Ttt is the multiplicity of as factor of the form

(9.2) General remark. One may say that an irreducible non-degenerate correspondence between two lines is completely determined by a single relation, here (9.1). In this case then the ideal of the graph is merely the principal ideal of generated by the element . Such correspondences may well exist in the general case, namely when the graph is defined by a single irreducible relation The ideal is then again the principal ideal One may also characterize this special case by the property that in the product space , the graph is then the complete intersection of the s u r f a c e w i t h a hypersurface 10. Once multiplicities have been assigned to the elements of one may introduce the divisor which

§ 3]

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CORRESPONDENCES

185

we call the image divisor of under T, written The degreeof is Similarly there is an image divisor under and its degree is These divisors will play a central role in the comparison with linear series. Just as in the case of algebraic series it is of interest to combine irreducible correspondences into composite correspondences. This is done formally as follows: Given the irreducible non-degenerate correspondences with associated elements we define a reducible correspondence as an expression

where the are arbitrary integers. Intuitively corresponds to the "reducible" graph consisting of the taken with the multiplicities When the are all positive they are essentially the multiplicities of the components of When the a t are negative however, their geometric interpretation would require the extension of the analogue of the divisor concept to curves on a surface, here the p r o d u c t ( S e e the concept of cycle in the next chapter: IX, 5.) Pursuing the treatment of a composite the associated transformations are by definition the operations

If

we define

If are correspondences one defines and way. A function is said to be additive wherever

in the obvious

We also say that a property is additive whenever if it holds for and then it holds also for Thus the transformations , the divisors the indices the multiplicities of the places are all additive. The transformation T is said to be linear whenever the divisor is in a fixed divisor class. If is effective this means also that it is in a fixed linear series of degree Similarly for and D. It is clear that linearity is additive. If both T and are linear itself is said to be linear. (10.1) All correspondences between rational curves are linear. Owing to the additive property we only need to consider an irreducible correspondence _ Assuming then / and both rational we are in the situation of (9). With as there defined we may write

SERIES

186

Hence the divisors forms

AND

CORRESPONDENCES

[CHAP. VIII

are elements of t h e c u t out on / by the

Thus is linear. Similarly of course for T, and so (10.1) holds. The central theorem regarding linearity is: (10.2) Theorem. If one of the transformations is linear so is the other. Hence in order that the correspondence itself be linear it is sufficient that one of be linear. We will assume linear and prove that T has the same property. Let us set where both and are non-negative. Then are both effective, and Under our assumption is in a fixed equivalence class of / where and . are effective. Thus on / . There exists then an such that if is a general point of has on / the divisor We call C and D the fixed and the variable divisors of R on / . Suppose that has the same variable divisor D as R. Then has only a fixed divisor H on / . Hence there is an which has H as its divisor on / . Hence has the divisor zero on / . Since is a rational function of , where it is equal to an element of L o n / . Since this element can only be in the intersection of L with , it reduces in fact to an element on / . Thus Hence R is unique to within a factor where and

In what follows it will be convenient to put R in the explicit form

where Upon substituting for the their expressions in the represenfixed tation its of mination the order (10.3) variable (10.4) divisor coefficient for i and one of An The the divisor to at does interesting variable fixed prove . of not Then isdivisor athat given precisely becomes contain divisor observation TC. place isofThe linear an D. . element Thus problem oIn may fwe / ithe nthe merely is O bereduces .same aorder made Let function need manner to regarding be is to the an such such prove: determination element weas that the can S whose deterof there ,form and

§ 3]

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CORRESPONDENCES

187

exists an element _ whose product by R suppresses as part of the fixed divisor. If has the same property then is not part of the divisor of hence and have the same order (10.5) Let us suppose that the places of correspond under Upon substituting the

in

we obtain

where we have written The numerator and denominator are elements of the ring of formal power series Upon applying (V, 1.2a) to them one obtains a relation (10.6) (10.7) (10.8)

where p, q are special polynomials in t and relatively prime. 11. Corresponding to the place and the associated relations (6.2) introduce the expression (11.1) where for the present are considered as independent indeterminates. Since the product is symmetric with respect to the and each term except is of positive degree in , we have

where the right hand side is in . Furthermore Since

is of degree

in

and is a special polynomial in t. the preparation theorem yields

(11.2)

where Now the relation

and is a special polynomial in

has a solution in t given by

Upon solving for in terms of t we see that the special polynomial in has the conjugate roots in

188

SERIES

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[CHAP. VIII

It has already been observed that these roots are distinct (end of 6 with t and interchanged). Since their number is the degree of in (degree understood as for polynomials), and since the coefficient of in is unity we have (11.3) Thus is the analogue of with the roles of and reversed. of in the expression of 12. Suppose now that the coefficient is positive. If designate representative points of , then through there is found in exactly one general point M t for each of the d expressions given by (6.2), i.e. for each of the factors of (11.1). Since each of these points is repeated times, is a factor of Similarly if is a factor of Since all the possible points Mt arise in the way just described, and are the products of the powers thus obtained. We have therefore the following situation: Let denote all the places of all the graphs with the same center where A and are the centers of and For define as the coefficient of the particular such that is a place of Let denote the corresponding to Then (12.1)

As a consequence we also have in view of (11.2) for each (12.2) The degrees of in t and of in are the numbers Therefore the coefficient of in is and it is the coefficient of as a variable place in the divisor D of R for fixed. On the other hand the degree of the product in (12.2) in ' is and this is the coefficient of as a variable place of the divisor of R for fixed. However this degree is precisely the coefficient of in Hence • is linear. This proves (10.3) and hence Theorem (10.2). 13. In connection withcorrespondences Abel's theorem leads to noteworthy relations. Let be the genus of and let be sets of p and linearly independent dfk for / and Suppose first that the correspondence is irreducible and let M be a general point of / and the points of TM. Since the are general points of the abelian sums (13.1) all have meaning. Since the coordinate ratios of the are in a definite algebraic extension of K{M) (the field of the ratios of the coordinates of M), and since the sum in (13.1) is symmetric in the is in fact a differential of the field K(M), i.e. in the last analysis of the curve / . Now

4]

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189

by little more than a paraphrase of the proof of (VII, 9.7) one may show that since du* has no poles the same holds for dvk. Thus dvk is a dfJc of the curve/. If G = Σσβ·{ where the Ki are irreducible then we define for (£ dv k = Σσβ(άη*, D*) and the conclusion is the same. Since dv k is a dfk of / we have a linear relation (13.2) dv k = lX k i d U i , λ Μ € Κ. We may consider the system (13.2) as a linear transformation τ of the linear systems of the dfk into one another which we will write (13.3)

τdu* = IX k Su l .

By reversing the roles of the two curves we find likewise a transformation (13.4)

τ-Hu k = Σλ*4η*,

λ* Η G Κ.

Here as in the case of T, T-1 one must not read into the designations τ, r_1 the usual "inverse transformation" relationship. They are merely convenient notations to underscore their operating in opposite directions. Let us agree to write τ = 0 if every Xki = 0 and likewise for τ-1. According to theorem (4.1) 27-1 is linear if and only if τ — 0 and likewise for T and τ-1. Hence theorem (10.2) is equivalent to the following: (13.5) If one of τ, τ - 1 is zero so is the other. Notice that in the usual sense of linear transformations if T , T f correspond to τ, τ-1 for Ci, then i

(13.6)

τ = SffiTi,

1

τ-1 = Iai τΓ1.

§ 4 . ALGEBRAIC CORRESPONDENCES OF A CURVE WITH ITSELF 14. Everything that has been said so far continues to be valid when /* = /, i.e. for correspondences of the curve /with itself. There arises how­ ever the interesting problem of determining the number of fixed places of the correspondences, i.e. of the places π which are self-corresponding or such that 77 is a place of TN, or equivalently, of T~1 Π. We shall only be in position to deal with this problem for certain special correspondences to be defined presently. Let 3 denote the identity correspondence, i.e. the correspondence consisting merely of all the couples (π, π). The special correspondences (E alluded to above are characterized by the property that there exists an integer γ such that C + is linear. That is to say the divisors TTT -)- γττ belong to a fixed equivalence class. The integer γ is known as the valence of (£ and (I is referred to as a valence correspondence. If γ = 0, £ is linear. Henceforth instead of "linear correspondence" we shall say "correspondence with valence zero." This is in keeping with the accepted terminology.

190

SERIES AND CORRESPONDENCES

[CHAP. VIII

When £ is the identity correspondence both T and reduce to the identity transformation. Hence according to (8.1) if £ has the valence γ then T^1 π γπ is likewise in a fixed divisor class. Hence the concept of valence is perfectly symmetrical with respect to T and T-1. 15. It is convenient to apply here also the technique of products and graphs. To that end, assuming as before/C KP% we introduce a copy /* of / in KP\», i.e. /* is the curve of ΚΡ%* represented by /(«*) = 0. The points, places, · · · of / have then obvious images in /* and a correspondence £:/f gives rise to a correspondence ff* which we continue to denote for convenience by (I. The graph of an irreducible correspondence £ is then an irreducible curve Φ C / X /*· The transformations τ, τ-1 of (13) on the differentials may likewise be introduced here. They are both of course linear transformations of the system {du{} of the dfk into itself. Since to £ = 3 there corresponds the identity transformation I on the dfk we may state as a consequence of (13.5): (15.1) If one of τ -(- γΐ, τ-1 -(- γΐ is zero so is the other. By means of the differentials we also prove readily: (15.2) If the genus ρ of the curve f is positive and if a correspondence £ has a valence then this valance is unique. For if there are two distinct valences γ, γ' then (γ — γ')Ι = 0. Hence (γ — γ'){άη)π = 0 = {du)n, i.e. du = 0. Thus/has no dfk and so T-1

p = Q.

One may state more precisely: (15.3) A correspondence £ of a curve of genus zero with itself may be assigned any valence. For by (10.1) £ has valence zero. Since £ + y3 has likewise valence zero one may also assign to £ the valence γ. The property just proved indicates that on rational curves valences are immaterial. This will be confirmed when we come to the coincidence and fixed point formulas (see 18 and 20). For in these formulas γ always enters in the combination ργ, and hence it does not appear when the genus ρ is zero. 16. Let us turn our attention now to the fixed places. We must first define the fixed place multiplicity of an irreducible correspondence ¢. Since on the surface / X /* the fixed places are among the intersection places of the graph Φ of (£ with the graph Γ of the identity the multiplicity can only have meaning if Φ is not Γ, i.e. if £ φ 3· Suppose then that the correspondence £ is not the identity. Let π, π*, tt be related as usual and let it be a fixed place. Let also / be so chosen that the center A of IT is an ordinary point of/. Then the center A* of TT* will also be an ordinary point off* and consequently (A, A*) will be an ordinary point of / X /* (IV, 13.9). Furthermore if t is a

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191

parameter for A, i.e. a parameter for π, then t* (the same as t) will be a parameter for ττ* and t, t* will be parameters for (A, A*) and / χ /*. Let us observe now that the graph Γ of the identity is a birational model of / and hence irreducible. For relative to the identity the pro­ jection S: Γ ->/ is a rational transformation of degree unity. If the irreducible representation of ττ is PXi = Ia11V

that of n* will be

ax* = These represent coordinates of parametric points M t , M*, of π, ττ*. Hence (Mt, M*) will be a parametric point for the place ω of the graph Γ of the identity centered at (A, A*). Thus ω is represented in the coordinates t, t* by the sole relation

t = t*.

(16.1)

This shows among other things that Γ has a single place ω centered at {A, A*), its representation in the coordinates t, t* being (16.1). The place π of Φ is represented in the same coordinates by

t — n M*t* d *l d E(rft* Vd ) = 0, or in K\\t, 3. This does not exclude the plane as a surface but if it occurs it is supposed to be a subspace of K P . Since we are assuming Φ without multiple points all its branches are linear. Let A(CL ) be any point of Φ and let the coordinates be so chosen that A is in the affine space. Then two of the coordinate differences r

i

i

0

3

3

2

0

i

z

1

r

0

3

2

r

i

0

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GENERALITIES ON THE CURVES

197

X i — Cb i , say u = X 1 — O 1 , ν = X 2 — a 2 may be chosen as parameters for A and the linear branch S2 of center A will be represented by relations =

(1

1¾ = U,

+ 9i(u, V),

s is said to be of the second kind for Φ whenever it is of the second kind at every curve C of Φ. If one defines an ω0 as merely an element R of Κφ, W1 is of the second kind at C if it is regular at C to within a dR, R £ Κφ. This is quite in keeping with the definition for curves. A differential of the third kind is one which is not of the second kind. 1

s

2

SIMPLE

§3]

DIFFERENTIALS

203

We note at once that: (6.6) If is of the first kind it is also of the second kind. (6.7) is of the second kind. §8.

SIMPLE DIFFERENTIALS

7. We are especially interested in closed differentials (7.1) (7.2) The relation (7.2) expresses the fact that is closed. Let C be a non-polar curve of Then determines on C an abelian differential written If C is we shall write however more simply for Let us revert now to the situation of §1 with f a s a certain projection of the surface into an affine space with coordinates x,y,z. Let C be the same irreducible curve as in (4) and let R be the rational function of (4). Suppose explicitly that Then on

(7.3a) then has an R of the same form as (7.3) but with n replaced by . Hence modulo a suitable dW not affecting the kind of we may assume that on (7.4) where

is the same as before. Now on

204

CURVES ON A SURFACE

[CHAP. I X

where in [ • • • ] only the term of order — 1 has been written. In view of (7.2) the differential of

is of the second kind on and so it has no logarithmic place on Hence its logarithmic residue on must be zero, or

at the general point

of 0, and hence on C. Therefore

Thus the logarithmic residues of at the places have the common value K. We call the logarithmic residue of relative to the polar curve C. This will be fully justified upon our showing, as we shall do presently, that does not depend upon itsderivation by means of At all events upon subtracting a suitable our initial will assume a form such that on (7.5) We observe now that dR, where R is as in (7.3) is of the form but with an R in which there is no term in Hence if _ with the R of (7.5) is of the second kind at the curve C we must have and the converse is obvious. Thus a n.a.s.c. for a^ to be of the third kind at C is that the logarithmic residue Suppose and let U be any element of of order unity on C. The closed differential of the surface has the same logarithmic residue as the reduced with R in the form (7.5), and so is regular at C. To sum up we may state: (7.6) Corresponding to any closed a)1 and polar curve C of a>1 and for any U of order unity cm C there exists an element such that the closed differential

is regular at C. The element K is the logarithmic residue of relative to C and depends solely upon and C. A n.a.s.c. for to be of the second kind at C is that The only point to be proved is the uniqueness of for given and C. This is immediate. For if behaves like then has no logarithmic residue as to C and so

§4]

DOUBLE DIFFERENTIALS

205

If C is said to be a logarithmic curve of Let be the set of logarithmic curves of a>1 with Let F be so chosen that the are all in general position as to If is of degree then has on the polar places centered at the intersections of ,vith Since is general all these intersections are distinct. Since the residue o f a s to is and the sum of all residues is zero we have

It follows in particular that a closed differential of the third kind has at least two logarithmic curves. The complete result is embodied in the following all important: (7.7) Theorem of Picaid. Corresponding to the algebraic surface there exists a number the Picard number of such that: (a) there are sets of p irreducible curves which cannotbe the logarithmic curves of any closed differential of the third kind of (b) any irreducible curves of are the logarithmic curves of some closed differential of the third kind of (See Picard-Simart [1], II, p. 241). The proof of the theorem will be based upon certain properties of the double differentials of the second kind and so we must turn our attention to these. §4.

DOUBLE DIFFERENTIALS

8. Consider then a double differential (8.1)

If we have then

and this is the general form of a dcov (8.2) The differentials of the first kind, the derived dcov and those of the second kind constitute vector spaces with scalar domain K. (8.3) and are subspaces of The dimension of is known as the geometric genus of the surface and generally denoted by It is the maximum number of linearly independent differentials of the first kind. The dimension of mod is denoted by It is the maximum number of differentials of the second kind of which no linear combination is a derived differential. There is a very extensive theory of the differentials of the first kind but we shall not go into it here. Regarding the second kind there is again this basic result due to Picard:

206

CURVES ON A SURFACE

[CHAP. I X

(8.4) Theorem. The number is finite (see Picard-Simart, [1], II, p. 186). This property (actually only its derivation) will be utilized in proving the theorem on the number p and so we proceed to establish it. 9. Let us start with any say (8.1) and let C of (3) be one of its irreducible polar curves. With F as in (4) we will have the representation (4.1) on 7T(B). The associated abelian differential of the curve C (9.1) is called the residue of on C. (9.2) A n.a.s.c. in order that a double differential be of the second kind relative to the irreducible curve C is that its residue be a derived differential for C, that is to say a differential dS, If is of the second kind relative to C some is regular at C. Hence ct)2 and dm1 have equal residues on C. A s s u m i n g d x let us calculate its residue. We have

Hence

where the only term written at the right is the term in It follows that the residue of and hence also of is i.e. it is a dS,, Thus the condition is necessary. To prove sufficiency suppose that the residue of With as in (7.3a) we find at once that ) is such that the corresponding R is of order at least i and this is achieved by merely augmenting Proceeding in the same way with a • , and setting s now an whose integrand R is of we will find that order Therefore we may assume that this holds already for the initial By hypothesis (9.3) Hence (9.4) is a differential of the proof of (9.2).

' such that

is regular at C. This completes

§4]

DOUBLE DIFFERENTIALS

207

10. Consider again the differential of (8.1), and let it be of the second kind. Let be its polar curves. We may suppose the projection F so chosen that they are all in general position relative to and that none is a component of the intersection of with F. Let be identified with the earlier curve C and consider the relation (7.3) relative to We propose to calculate To that end consider the expression

Notice that since the are all finite for y finite their integral symmetric functions are polynomials in y. Hence

where zero on

is the projection of We also have

in the xy plane. Evidently Q is of order

Hence

Upon forming now of (7.3a) we see that: (a) all the remain finite for a value y which does not correspond to an intersection of with a (b) all the products in all the denominators of the and hence itself, remain finite for any y such that the corresponding remain distinct. Hence is of the form

where G has no other zeros than those of type (a) or (b). Thus has no other poles than these and possibly also Now behaves like o>2 relative to but with n replaced by the same treatment yields a still of type (10.1) such that behaves like relative to but with instead of n, etc., down to in which the corresponding n is unity. If still denotes the appropriate coefficient of then since co2 is of the second kind a relation (9.3) holds, where again S only has poles of types (a), (b) and oo. If we set

CURVES ON A SURFACE

208

[CHAP. I X

11. Let us set (11.1)

The functions U, V are infinite on the intersection of with F, and on certain curves already specified. Thus in addition to and these special they are also infinite on the residual intersection ~ of

with F. We propose to replace by a similar differential but with eliminated as polar curve. Applying then almost any affine transformation to coordinates we will have (11.2)

We may furthermore suppose that O is regular in If resultant of and . as to we have

is the

Hence multiplying in (11.2) A, B, C by a we will replace G by Thus we may assume that , If still denotes the projection of in the plane then where and are relatively prime. Since is almost any coordinate system is regular in Let be the roots in of They all remain finite when is finite. We have now the reduction into partial fractions represented by

where all the new quantities are polynomials with coefficients in K. Let represent the curve Upon multiplying both sides of (11.3) by taking the appropriate derivatives and making we find that has for only poles those of type (a), (b), co, relative this time to and to the intersections of with Now if we suppress in 1 /G the term beyond the sum we do not affect the behavior of relative to This means that we may replace G in (11.2) by where has zeros of types (a), (b). Writing again x, y, z for we may sum up the final result as

DOUBLE DIFFERENTIALS

§4]

209

follows: There is a certain curve E h associated with C h and a differential ω

"

B h { x ,y,z)dy — A h (x,y,z)dx =

; 4ί

""εΙΙ*'«'"1

where ψΛ has no other zeros than those of types (a), (b), relative to the intersections of C h with E h , and ω 2 + da> l h is regular on C h . 12. The same treatment may be applied to all the curves C h and yields the following result: Let βν • • · , β3 be the values of y such that the planes y = consist of the plane at infinity and of the planes y — const., through the intersections of two curves Ch, Eh, of those tangent to Ch, and of those through place centers with the same pro­ jection on the xy plane. There exists an OJ1 such that ω2 + άω1 is regular at all curves C h but may have acquired new polar curves Hp k . Let now b v · · • , b r E K be the values of y such that the planes y = b{ consist of the planes y = const, which are tangent to F or to Δ or pass through its multiple points or through the points where the planes tangent to Δ coincide, or finally intersect F in a reducible curve. Referring to (III, 20.1) the number of these last planes is certainly finite if {Hy} is almost any pencil of plane sections. Let us show that for almost no pencil {H y } does a value B i include a /S3-. First of all one may always choose on Φ the pencil {Hy} so that the points projected into the exceptional points on Δ, or the contacts with F of planes y = const., are neither on a Ch nor on an Eh. For with general {Hy} all these points have transcendency two, whereas the points on the Ch, Eh have at most transcendency unity. Moreover the planes through the former points do not contain generally an intersection of a Ch with an Eh, and are not tangent to a Ch nor pass through a multiple point of a Ch nor finally through a point of a Ch which has on the plane xy the same projection as another point of the same Ch. Let us show finally that in general the plane y = const, through an intersection A of C h and E h does not intersect F in an irreducible curve. Indeed since A is not a multiple point of F it is a point where the tangent plane through F contains the original ζ axis. Thus A is almost any point of Ch and the plane y = const, (new coordinate y) is merely almost any plane. Hence it intersects F in an irreducible curve. We conclude then that for a general {H y } and hence for almost every {Η υ } the values b t will not include a value β ύ . 13. Let us set then as we may for the reduced ω2 (13.1)

ω2 =

A ( x , y, z) 1 dx dy —- · — ; B ( x , y, z) G(y) Fz

, ^ A, B, G £ K[x, y, z],

where G has no other roots than the β ί and A/B has no other polar curves than perhaps H x.

210

CURVES ON A SURFACE

[CHAP. IX

In addition to everything else we may evidently assume F ( x , y , z ) regular in its three variables. Applying then reductions such as those of (VI, 10) we will have on Hy (as curve of K(y)P2) A D(x, y,z) —= , B E(y)

,, „ „ T D , E € K\x, y, zl

and hence one may assume that in (13.1) B = E ( y ) . Now if y 0 is a root of E(y) and its multiplicity is v, A(x, y, z)j(y — y0)v is of order zero on Hya- Hence the quotient is a polynomial mod F. It follows that ω2 can be put in the form (13.2)

Oj2

=

P(x, y, z)dx dy \ Hy)* ζ

Ρ , E G K[x, y, z],

Since P j F z is regular at the intersection places of Δ and general H v , it is shown as for abelian differentials that P is adjoint to F. We may now apply certain s t r i c t l y algebraic reductions given by Picard (see Picard-Simart [1] II, pp. 163-181) to the effect that one may subtract from ω2 a certain άωΎ, where ωΛ = 1

L(x, y, z)dy — M(x, y, z)dx N(y)F,

;

L, Μ, Λ, G Κ\χΛΐ,ζ\,

where L and M are adjoint to F, and where ω2 — da>1 is like (13.2) but with G= 1 and P of bounded degree. In other words to within a do>1 the initial differential of the second kind is reducible to the type ,,00, (13.3)

ω2 =

P(x,y,z)dxdy

where P is an adjoint polynomial whose degree does not exceed a certain μ. Since among the differentials of type (13.3) the number of those linearly independent is finite, theorem (8.4) on the finiteness of the number p0 is proved. 14. Strictly speaking Picard's argument is algebraic except where he utilizes property (9.2). Since our proof of (9.2) is algebraic, the whole reduction is algebraic and valid for our general choice of groundfield. One may observe also that Picard devotes considerable effort to the case where for some root β of G(y) the curve Ηβ contains a singularity of the curve Δ. Since we have shown that this eventuality may be avoided, this part of Picard's argument may be omitted. The treatment of Picard leans heavily upon a certain theorem of Castelnuovo. In Castelnuovo's proof (Picard-Simart [1], II, pp. 72-74) continuity considerations play an important part. We shall therefore state the special case of the theorem which is required and give the full algebraic adaptation of Castelnuovo's proof.

§ 4]

DOUBLE

DIFFERENTIALS

(14.1). Theorem of Castelnuovo.

Let T,

211

be two curves of degrees c, d

in KP3. Let G be a surface going simply through T and II a plane intersecting r and in c and d distinct points and Bp where the are ordinary points of G and G is not tangent to II in these points. Let be the intersection lines with II of the tangent planes to G at the Av Then the complete linear system of the surfaces of degree q sufficiently high satisfying conditions A: they pass through T and and are tangent to G along P, cuts out on II the complete linear system of the curves of II of degree q satisfying conditions they pass through the A0 B} and are tangent to the lines at the points Av 15. Notice first that one may extend lemma (VI, 13.1) to curves y) satisfying as follows: The number of linearly independent conditions on the coefficients of for of sufficiently high degree is merely All that is necessary is to replace in the proof the lines through some of the by conics which are tangent to the in some of the points and not in the others. Setting now we will rest the proof upon the following property: The system is spanned by a finite number of systems where I is a fixed line (systems consisting of any satisfying plus I taken q — m times). The curves for all I, span a linear s y s t e m a n d we shall first show that Let be the dimensions of . If I, I' are two distinct lines then the curves satisfying through the intersection of I, V form a linear system of dimension This system contains the subsystems made up of all the and of all the Since the two have in common we have From the relations

we infer that

Since

the last inequality holds also in reverse. Hence as asserted. Since can be spanned by a finite number ot systems holds. 16. Proceeding step by step we find that is spanned by all the systems distinct fixed lines. To prove (15.1)a it is sufficient therefore to prove:

212

CURVES

ON A SURFACE

[CHAP. I X

(16.1) The set of all systems consisting of a line of II taken r times spans the complete system of the curves of degree r. Let II be referred to coordinates and let E denote the system of all curves It is sufficient to show that every monomial is in E. We have whatever

Taking for the

. distinct values of a2 and writing these relations then solving we find that they are in E. Thus

Taking now distinct values of and solving for we find that it is in E. This proves (16.1) and hence (15.1). 17. We will now take projective coordinates for so chosen that I! is Consider the change of coordinates where and are indeterminates. The plane where v is an indeterminate, intersects T and in c and d distinct points and and the tangent planes to the surface O at the M i in lines The points and the lines depend algebraically upon and v. However let us make a curve vanish at the and have the tangents in the Sincethese conditions are symmetrical in the the and the tangents we will have

where the are indeterminates and is a form of degree m in the with coefficients in Let r be the degree of in v. Then the surface

intersects II in plus r times the line That is to say the surface satisfying A intersects II in the general plus r times I. The number r, the degree of in v may be lowered only when satisfy a certain number of relations. In other words for almost every i.e. for almost every pencil is fixed. 18. Reverting to the initial coordinates we may suppose that is almost any pencil through II and we will have (18.1)

The

are not all divisible by

for then one could lower r to

§ 4]

DOUBLE DIFFERENTIALS

213

Let be such that no linear combination is divisible by and let there exist a linear combination divisible by but not by One may evidently replace by

and is a linearly independent set. Proceeding thus we shall obtain a set such that no linearcombination is divisible by More precisely will be a linearly independent set. Theassociated linear system (18.1) will thus cut out on II the system where I is the line Taking now v arbitrary pencils (almost any pencils) through II we obtain for each surfaces which satisfy A and cut out on II the complete system where is a fixed line. By (15.1) the full set and j variable, spans a linear system which intersects II in This proves Suppose now that has been proved for some Thus cuts out on II. If H is any plane and cuts out on II. By (15.1) then with q in place of m, the set of all for all H, cuts out a system which spans . Hence a fortiori cuts out Since holds it holds for all 19. For our purpose it is necessary to complement one of the results of Picard. He showed (Picard-Simart [1], II, p. 182) that if in a differential co2 of type (13.3) the degree q of the polynomial exceeds a certainthen one may lower it to fj, by subtracting a da>1 where (19.1) In this reduction one may suppose that the curve at infinity is irreducible and has only ordinary singularities (double points with distinct tangents). How if H is a non-singular irreducible hyperplane section of the surface one may choose the projection of to a surface F of KP3 such that H only acquires ordinary singularities. On the strength of this remark we may show that the Picard result is equivalent to the following: (19.2) Let H be a non-singular irreducible hyperplane section of and let OJ2 have H as its sole polar curve and to order q. Then there exists a number [i such that if g > fi, there can be found an with the sole polar curve H such that has H as sole polar curve and to order /x. The Picard property evidently implies (19.2) with H as the curve at

214

CURVES ON A SURFACE

[CHAP. IX

infinity. To prove the converse choose coordinates with H as the curve at infinity and F regular in z. Thus F will be of degree m in z. Then (19.3)

ω 2 - άω χ =

Q 1 (χ, y, z) dx dy -=Λ F1

Q 0 (z, y, ζ)

Q1 € K[x, y, ζ],

where Q 1 IQ 0 is of degree μ and is regular except at infinity, and where Q 1 is adjoint to F. As before one may assume that Q0 = G(y) £ K[y\. On the other hand since F is regular in ζ one may divide Q1 by F as a poly­ nomial in z, and replace Q1 by its remainder which is like Q1 but of degree < m in z. Let b be a root of C(y). Since Q 1 Ky — b) is regular on H b , Q 1 ^x, b, ζ) — 0 is a consequence of F(x, b, ζ) = 0. AVe shall prove in a moment that this last curve has no multiple component. Admitting this property for χ an indeterminate, the m roots in ζ of F(x, b, z) are distinct and they must all be roots of Q^x, b, z). Since Q1 is of degree < m, Q1^x, b, z) — 0 identically and hence Q^x, y, z) is divisible by y — b. It follows that it is divisible by G(y). Hence the relation (19.3) holds with Q0 = 1. That is to say, modulo a άων ω2 is equal to a differential of the same form with q replaced by q — 1. Hence (19.2) is equivalent to the Picard property. There remains to show that F{x, b, z) has no square factors. Assuming almost any system of coordinates no tangent to H x , of the form y = b, will be multiple, i.e. will have two or more points of contact and this rules out the eventuality under consideration. It may be noted that the asserted property of the tangents is equivalent to the following for the dual plane curve Γ of Hx: a general line of the plane of Γ does not pass through a multiple point of the curve. Returning now to (19.2), and operating in the space of Φ, consider a general hyperplane H = Συ,χ, of that space. Upon replacing K by K u (u) the irreducibility properties of Φ, Hy, etc. are preserved. Applying now (19.2) we obtain a certain μ{Η) for the hyperplane H. Since the property under consideration is expressible by an algebraic system in the Ujiij, μ(Η) preserves its mean­ ing when the general hyperplane H is replaced by almost any other. That is to say (19.4) There is a fixed number μ which is related as in (19.2) to almost every hyperplane section of Φ. We will denote by N

(T)

the number of coefficients of a form of degree μ in four variables. The importance of this number will appear in a moment.

§ 4]

DOUBLE

DIFFERENTIALS

215

20. We come now to the crux of the argument on the number p. We return to the curve C, of degree d and in general position of (4) and let Let us view for a moment as a curve of As such it is still irreducible (III, 13.1). Referring to (VII, 10.1) there is a differential of the third kind (as curve of LP2) whose only poles are the and and they are all of order one with logarithmic residues at the and The coefficient L(x, z), i.e. it is representable as a rational function in x, z with coefficients in an algebraic extension Thus Now if are the conjugate values of 8 then

has the same poles as K(x, y, z), or more accurately

with equal residues there and It follows that

(20.1)

is an abelian differential of which has for sole poles at finite distance the and with multiplicity two in each. That is to say (20.1) behaves like a differential of the second kind at finite distance on The reductions of (VII, 4) (proof of VII, 4.2) are applicable and will yield here a function of some field where is algebraic over K(y), such that

where jugates finally a relation

and is adjoint to Replacing by its consumming and dividing by s there results

We may note here that where has for logarithmic curves G with residue and possibly some hyperplane sections. Applying now the Picard reduction to relative to suitable new coordinates with still almost any pencil, there is found an whose polar curves are merely hyperplane sections and such that where has no other logarithmic curves than C and some irreducible hyperplane sections, and where has a single polar curve H of Moreover His almost any hyperplane section of and the order of as to

CURVES ON A SURFACE

216

[CHAP. I X

21. Consider now any N + 1 curves We may form for each C\ relative to a common II and in a suitable coordinate system

where is adjoint of degree to F. Since there are nomials they are linearly dependent. Thus there exist not all zero such that

polyand

Hence and therefore is a closed differential with the polar curves and logarithmic residues not all zero. In addition it may have some polar curves which are irreducible hyperplane sections. Let be the residue as to H ( . If are the equations of in affine coordinates of the space of then

has thesole logarithmic curves H with residues for and for H. Consider now any N + 1 irreducible curves whatsoever. We form and as above with residues for the first and for the second. Since the are not all zero we may always dispose of the situation so that If one of is zero we already have a closed differential with some of the C{ alone for logarithmic curves. If both are then

has the same property. This completes the proof of Picard's theorem (7.7) and shows that §5.

ALGEBRAIC DEPENDENCE OF CURVES ON A SURFACE ACCORDING TO SEVERI

22. We shall now sketch the relation between the Picard p theorem and Severi's all important notion of algebraic dependence of curves on a surface. It has been proved by Chow and van der Waerden (see van der Waerden [2] p. 157) that there is a one-one correspondence between the curves of a given degree on a surface, here and the points of a finite

§5]

ALGEBRAIC DEPENDENCE

217

set of algebraic varieties Vv · · • , VQ. The collections of curves corres­ ponding to the points of the same variety Vi are said to form an algebraic system of curves of Φ. If C, D are two curves in such a system we write C = D. This is Severi's algebraic dependence between curves. From this it is but a step to relations of the form ΣλΑ = 0, where the sum is finite and the Xi are integers. The maximum number p* of algebraically independent curves on the surface is the Severi base n u m b e r . I t s m e a n i n g i s t h a t t h e r e e x i s t p * i n d e p e n d e n t c u r v e s C v · · · ,C which form a base in the sense that given any curve C whatever there takes place a relation (22.1)

XC = IXiCi,

X φ 0.

In the complex domain it has been proved by Severi that p* = p, or explicitly (Severi [2]): (22.2) Theorem. The maximum, number of algebraically independent curves o n Φ i s equal t o the P i c a r d n u m b e r p . More precisely Severi has proved (see Severi [2]): (22.3) The relation (22.1) is equivalent to the existence of a closed differential ω1 with C and the Ci as polar curves and with the logarithmic residues — X, Xi with respect to them. It is understood of course that while Co1 may have other polar curves, there mil be no corresponding logarithmic residues. The central geometric result here is then the fact that there is a finite base. Actually Severi showed that: (22.4) There exists a system of ρ + a — 1 curves Cv · · · , Cp, Dv · · · , Dc-V such that the C1 are independent and that every curve C satisfies a relation (22.5)

C = I X l C i + τ μJ J j .

This may also be formulated as follows: (22.6) Theorem of Severi. The additive group of the cycles of the surface Φ may be finitely generated. (Severi [3].) We may also recall in this connection, that upon turning Φ into a four dimensional manifold (in the sense of topology) then as shown by the author ([2], p. 145): (22.7) The Picard number ρ = B? — p0 where R2 is the second Betti number of Φ. Hence in particular ρ < R2. Furthermore if D0 denotes the identity then D0, ·•· , Da^1 generate an isomorph of the torsion group of the surface. The basis for (22.7) is the author's proposition ([2], p. 81): (22.8) Theorem. One may associate with each algebraic curve C on Φ a

218

CURVES ON A SURFACE

[CHAP. IX

definite two dimensional cycle of Φ in the sense of topology. Then there is complete equivalence between a relation of dependence ΣΑ A = O and the. homology (in the sense of topology) ^XlCt --ο 0. 23. Severi's proof of his central theorem (22.6) rests upon Picard's theorem on the number p, the theorem of Chow-van der Waerden and the following property: (23.1) Corresponding to any cycle C on F one may calculate a certain function "/(C) of C (its virtual dimension) ivith the property that if χ(ΰ) > 0, there is some curve (effective cycle) C = C. Now (23.1) has been proved by Severi by purely algebraic arguments. We may therefore consider Severi's theorem (22.6) as established for an algebraically closed groundfield of characteristic zero. 24. Let C be a fixed irreducible curve of degree d of the surface Φ and D a complete intersection of Φ with a hypersurface G of degree μ. The sum of the intersection multiplicities of C with D is the same as for C with G, i.e. it is equal to μά. Let it be denoted by [CD], The same result will follow if D1 is merely the residual intersection of Φ with a hypersurface G of sufficiently high degree through a fixed curve E of Φ and we will thus have a related number [C-D1]. It is at once seen that [CD] whenever defined is additive relative to both C and D. Prom this it is but a step to defining [CD] for all cycles C, D. Now an essential part of Severi's theory may be expressed as: (24.1) Seven's equivalence criterion. A n.a.s.c. in order that a cycle XC = Ο, X Φ 0, is that \CD] = O whatever the cycle D. (24.2) Corollary. A n.a.s.c. in order that the cycles C1, · · • , Cp form a base is that the intersection determinant Δ = I [CtCJ IΦ 0. From this we may readily deduce with Severi [3]: (24.3) A base [Ci] for which the integer ] Δ ] is minimum has the property that every cycle C satisfies a relation XC = XIXiCl, X Φ 0. In other words C — Sxl Ci is an element of finite order of the additive group of the cycles. Such a base is said to be minimal. Noteworthy special case. If Δ = ^ 1 then (Ci) is a minimal base. Let C, C' be any two curves, and let (Ci) be a base. Thus XC = ZXtCi,

X'C = ZXfiCi

and therefore XX'[CC'] = IXtX1ICtCi]

§ 5]

ALGEBRAIC

219

DEPENDENCE

where the bilinear form at the right is non-degenerate of rank p. From this follows readily: (24.4) Let be independent curves so that One may then find curves such that is a base. For any two curves ' let If whatever then is a base. (24.5) Applications. I. Plane. If £ is a line then every curve C is in the multiple where d is the order of C. Hence is a minimal base, there are no cycles of finite order and II. Ruled surface. Let H be a hyperplane section and G a generator of and let as before m be the degree of Then

That is to say: almost any two distinct hyperplane sections intersect in m points; a hyperplane section and a generator intersect in one point; distinct generators do not intersect. It follows that there is no relation with For intersection withG yields a = 0, and intersection with Suppose and let be a base. Let

We wish to

find

such that

This requires that which gives a unique solution for the integers It is clear that is likewise a base. Let now be any two curves on We will have

If are the orders of the two curves and intersections with a generator then

the number of their

Now Corrado Segre has given a classical formula according to which is a bilinear form in d, and Hence by (24.4) and is a base. Since

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CURVES ON A SURFACE

[CHAP. IX

is a minimal base. Hence (24.6) This time

Hence and this is precisely Segre's formula. Consider in particular a quadric. There are two systems of generators and intersecting each in a point. We also have Let be the numbers of intersections of , with a Then Hence the well known formula Since is a minimal base so is It may be proved in fact that for a ruled surface the additive group of the cycles has no elements of finite order. Hence in (24.6) §6.

SURFACE PRODUCT OF T W O CURVES. APPLICATION TO CORRESPONDENCES

25. Let be two irreducible curves and let the situation and 1 are assumed nonnotations be those of (VIII, §3). In particular singular in suitable spaces. The general theory which has been developed is not directly applicable to the surface but it may be carried over without major modifications. We shall merely outline the various steps omitting most details. Let be points of and parameters for the places , Since these places are linear one may assume that are actually affine coordinates for and the point Let be affine coordinates with as origins for the space of Together they form affine coordinates for the product space with as the origin. If are the elements of ' determined by then Since are ascending power series in there is an isomorphic imbedding of as a subring of the power series ring , From (IV, 13.9) we infer that there is only one

§6]

SURFACE PRODUCT OF TWO CURVES

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branch 23 of