Classification Theories of Polarized Varieties 0521392020, 9780521392020

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London Mathematical Society Lecture Note Series. 155

Classification Theories of Polarized Varieties Takao Fujita Professor, Department of Mathematics, Tokyo Institute of Technology

The right of1he

Vnioceiry (Cambridge Sc pool and eeg eli manner of boo St

net granted by Henry VIII in 1534. The Univeceiry has poled endpabuiehed condnaouely since 1584.

CAMBRIDGE UNIVERSITY PRESS Cambridge

New York Port Chester Melbourne Sydney

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 I RP 40 West 20th Street, New York, NY 10011, USA 10, Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1990

First published 1990 Library of Congress cataloguing in publication data available British Library cataloguing in publication data available

ISBN 0 521 39202 0 Transferred to digital printing 2003

Contents

Introduction Chapter 0. Summary of Preliminaries §1. Relative viewpoint

1

§2. Singularities

6

§3. Intersection theories

P. Semipositive line bundles and vanishing theorems §5. Birational classification of algebraic varieties Chapter I.

§1. §2. §3. §4. §5.

9

14

19

d-genus and Apollonius Method

Characterizations of projective spaces Basic notions in the Apollonius method Iteration of the Apollonius method Existence of a ladder Classification of polarized varieties of d-genus zero

22 25 28 31

38

§6. Polarized varieties of d-genus one: First step

43

V. Results of Lefschetz type §8. Classification of Del Pezzo manifolds §9. Polarized varieties of d-genus one: remaining cases §10. Polarized manifolds of d-genus two

55

62 77

83

Sectional Genus and Adjoint Bundles §11. Semipositivity of adjoint bundles §12. Polarized manifolds of sectional genus s 1

107

§13. Classification of polarized manifolds of a fixed sectional genus: higher dimensional cases

108

§14. Classification of polarized surfaces of a fixed sectional genus

114

§15. Polarized manifolds of sectional genus two

122

Chapter II.

Chapter

I.

93

Classification Theories of Projective Varieties

§16. Castelnuovo bounds

139

§17. Varieties of small degrees §18. Adjunction theories

147

Related Topics and Generalizations §19. Singular and quasi- polarized varieties §20. Ample vector bundles

151

Chapter N.

164 171

§21. Computer-aided enumeration of ruled polarized surfaces of a fixed sectional genus

176

References

184

Subject Index

202

Introduction

By a polarized variety we mean a pair ing of a projective variety on it.

(V, L)

consist-

and an ample line bundle

V

L

We will classify such pairs and describe their

structure as precisely as possible.

Needless to say, algebraic varieties are the main In this book, however, we

object in algebraic geometry.

mainly consider the pair itself.

rather than the variety

(V, L)

There are several reasons of taking this viewpoint.

First of all, polarization (or the linear system defined by it) is very important for describing the structure Fn is of a variety. For example, the projective space

described by a homogeneous coordinate system, namely a linear parametrization of

But it is by no

H0(IPn, 0(1)).

means easy to recognize a projective space without being For beginners it takes some thought

given a polarization.

to see that a twisted cubic in

is isomorphic to

F3

and this is because the polarization priori.

V

IF1,

is not given a

0(1)

Another example is the space parametrizing linear

F2's contained in a smooth hyperquadric in actually isomorphic to

IF5.

This is

Is that obvious to you ?

i3.

There are polarizations which are not very ample but Fn For example, let f: V -+ are useful for this purpose. be a finite double covering. but not very ample.

To recognize

to considering the pair map defined by

f*10(1)1.

L = f 0(1)

Then

V

(V, L), since

via

f

f

is ample,

is equivalent

is the rational

In this case the graded algebra

viii

Introduction

G(V, L) = ®t20H0(V, tL) (V, L)

is a hypersurface in a weighted projective space.

Although mL G(V, mL)

has a very simple structure and

is very ample for

is not as simple as

m >> 0, the graded algebra

Another classical

G(V, L).

example is Weierstrafi's normal form

y2

=

4x3 - g2x - g3,

which exhibits an elliptic curve as a weighted hypersurface of degree six in the weighted projective space In general, by the ampleness of G(V, L)

by taking

Proj.

L, V

IP(3,

2, 1).

is recovered from

This provides us with an

algebraic approach to algebraic varieties.

Thus, the study

of polarized varieties is indeed an algebraic geometry (but of course geometric approaches are also useful).

Second, we note that most interesting examples of algebraic varieties carry natural polarizations.

In classi-

cal projective geometry the hyperplane section gives such polarizations. 0-divisors.

In case of Jacobian varieties we have the

On the other hand, abelian varieties can be

recognized by their periods.

The interplay of these two

approaches yielded the theory of 6-functions.

We should

also observe the importance of polarizations in Torelli type theorems for many types of varieties, e.g.

K3-surfaces.

In moduli theories you find many examples of moduli spaces which carry polarizations constructed in a natural way. In my opinion, God did not make abstract varieties, but polarized varieties.

Third, we notice the relationship with the theory of singularities.

The vertex of a cone over a polarized varie-

ty gives a typical example of a singularity.

Thus, the

category of polarized varieties can be viewed as a subcate-

Introduction

ix

gory of singularities.

Moreover there is an evident simi-

larity of notions and theorems between them.

It is not easy

to formulate this analogy in a logically strict way, but nevertheless it is useful for heuristic purposes to have Recently classification theory

this parallelism in mind.

and the theory of minimal models of algebraic varieties has made remarkable progress (cf. [KMM], [Mor5]), where the study of certain singularities (terminal, canonical, log-terminal etc.) plays an important role.

I believe that

these objects, namely, polarized varieties, singularities and algebraic varieties, should be and will be studied together more and more in the future.

OK, let us now agree that we will study polarized varieties here.

But how ?

We want to recognize each polarized variety what it is.

(V, L)

as

However, there are too many types of polarized

varieties and there is no almighty method for all of them. We must employ various tools according to the types of (V, L).

Therefore we need first to classify polarized

varieties, so that we can distinguish those objects for which a certain approach works well.

Thus, philosophically,

each approach should have its own classification theory. This is why I wrote

"Theories"

in the title of this book.

Here I will present two such theories.

The first one may be called Apollonius method. given a polarized variety ILI

LD

pair

such that

(D, LD)

is the restriction of (D, LD)

(V, L), we take a member

Namely, D

of

is also a polarized variety, where L

to

D.

We first study the

which is of lower dimension, and then proceed

Introduction

x

to

(V, L)

type.

by making use of various results of Lefschetz

Thus, we use induction on

dim V

for proofs.

This

approach has a long history in classical geometry (this idea can be traced back to Apollonius, so I would like to call this method 'Apollonius method').

However, since we do not

assume that

L

is very ample, there is not always such a

good member

D

of

in general. The theory of

ILI

which is defined by

4(V, L) = n + Ln - h0(V, L)

d-genus,

where

n = dim V, provides several sufficient conditions for the existence of such a member

D.

Moreover, d-genus turns out

to be a powerful invariant for characterizing d

(V, L)

if

Chapter I is devoted to this theory.

is small enough.

In Chapter U we present the theory of adjoint bundles K + mL, where

K

is the canonical bundle of

a positive integer.

defined over

Here

V

K + mL in

Z

is

V.

has a special

(V, L)

is not nef, i.e.

(K + mL)Z < 0

for

The description of such a structure

is very precise when n - m is always nef.

m

is assumed to be smooth and

theory on minimal models, we see that

some curve

and

By a polarized version of Mori-Kawamata's

C.

structure when

V

is small, while

K + (n + 1)L

Using this theory we can classify polarized

manifolds by their sectional genus, which can be defined by the formula

2g(V, L) - 2 = (K + (n - 1)L)Ln-l.

We remark

that adjoint bundles have long been studied (especially when n = 2, m = 1) by the Apollonius method. many beautiful results in this way.

assumption is weaker (L

Sommese obtained

In our case the

is just ample) and we need

different techniques, but the conclusions are very similar to classical ones.

Introduction

xi

In Chapter I we survey Ionescu's classification theory of projective varieties (= varieties embedded in projective spaces) and Sommese's theory on adjunction process (when

L

is ample and spanned), but technical details of the proofs are omitted.

In Chapter N we discuss several further developments and generalizations, and at the end we present a computeraided enumeration of ruled polarized surfaces of a fixed sectional genus.

In addition, numerous related topics are mentioned in various parts, which will be useful especially for advanced readers who want to know about the present state of investigations.

There are many interesting and important theories which

we do not treat in this book - e.g. moduli theories amongst others.

However, I hope that our studies will provide a

good starting point for moduli theories in many cases.

The reader is assumed to have some knowledge of algebraic geometry, as can be found in Hartshorne's book [Ha4] for example.

For the sake of convenience, in Chapter 0,

I

give a brief summary of matters which are often used freely in this book.

For their proofs the reader should consult

appropriate references.

This book is not quite self-

contained in the sense that we usually present a result with an outline of proof of it, and refer to other papers for technical details.

I hope that this rather helps the reader

to have a good idea of what is most important.

I would be

happy if young people can enjoy the experience of applying modern general theories in concrete problems.

xii

Introduction

Acknowledgements

I would like to express my hearty thanks to all persons who have contributed to the preparation of this book.

I am

especially grateful

to Professors K. Kodaira, S. Iitaka and K. Ueno who introduced me to algebraic geometry, an area with numerous interesting problems;

to members of the Department of Mathematics at Komaba,

University of Tokyo, where I was able to make most of the studies in this book, enjoying excellent facilities with spiritual support;

to Professors L. Badescu, M. Beltrametti, R. Hartshorne, E. Horikawa, P. Ionescu, Y. Kawamata, A. Lanteri, R. Lazarsfeld, M. Miyanishi, Y. Miyaoka, S. Mori, H. Popp,

J. Roberts, F. Sakai, M. Schneider, T. Shioda, A. J. Sommese and many other mathematicians for comments, advice,

encouragement, support and/or correspondence with many interesting preprints and offprints;

to Professor Oshima, whose excellent word processor has made it possible for me to prepare the manuscript;

to the editor and the staff of the Cambridge University Press, who gave me the opportunity of publishing this book and gave numerous valuable comments on the first version of the manuscript; and to all persons who are willing to read this book. TO_

k 0- 0

Fj

ta,

Department of Mathematics Tokyo Institute of Technology Oh-okayama, Meguro, Tokyo 152 Japan

Introduction

x ii i

Notation, Convention and Terminology Basically we employ the customary notation in algebraic geometry as in [Ha4].

In most cases we work in the category of algebraic spaces defined over a fixed algebraically closed field A

They are assumed to be proper and of finite type over

By a variety we mean

unless specifically stated otherwise. an irreducible reduced space. variety.

A.

Manifold means a non-singular

Point means a .R-rational point, while scheme-point

means a scheme theoretical point.

Vector bundles are often identified with the locally free sheaves of their sections, and these words are used interchangeably.

Line bundles are identified with linear

equivalence classes of Cartier divisors, and their tensor products are denoted additively, while we use multiplicative notation for intersection products in Chow rings. The pull-back of a line bundle morphism

less.

L

via a or

LY, but

when confusion is impossible or harm-

This convention applies for pull-backs of other

objects too.

For example, if

Cartier divisor on of

is denoted usually by f L

f: Y -+ X

sometimes just by

on X

L

X, by DY

f

is birational and

D

is a

we mean the total transform

D (we never mean the strict transform). To avoid possible confusion, the canonical bundle of a

manifold M ary notation

is denoted by KM.

or

K(M)

Similarly

©(M)

normal bundle of a submanifold XCcM'

and

M

tangent and cotangent bundles of C

KM, unlike the customQ(M)

denote the

respectively.

But the

in M is denoted by

xiv

Introduction

Now we list notation used very often. Sing(X): the set of singular points of a space

X.

Supp(Y): the support of a subspace (or subscheme) Y

of

X.

OX: the structure sheaf (or the trivial line bundle) of

X.

hl(X, T): the dimension of the F

coherent sheaf

on

Y[L] (or Y(L)): = Y ®0 L

i-th cohomology group of a

X.

for a line bundle

L.

ILI: the complete linear system associated with

L.

[d]: the line bundle associated with a linear system

A.

BsA: the set-theoretic intersection of all the members of PA: the rational map defined by

A.

(Z): the Chow homology class of an algebraic cycle 8V: the dual bundle of a vector bundle Sk8: the

k-th symmetric product of pr-l-bundle

IP(8), 1X(8): the

p

A point

over

Ny

subspace

x

on

space

Ex/Ny.

Ha, H. Fa, 1P

:

,

X.

8.

of codimension one in

p E P(8)

at

over

Z.

8, r = rank 8.

corresponds to a linear

H(8): the tautological line bundle on H(8)p

8

associated with

X

A

EX = Ar

IP(8).

The fiber

is identified with the quotient

O[tH(8)]

is often denoted by

0(t).

the (pull-backs of) 0(l)'s of projective spaces indicated by the same Greek letters.

Chapter 0. Summary of Preliminaries

The matters in this chapter are well known among We review

experts and will be freely used in this book.

them for the convenience of the reader and hope that it serves as a sort of 'dictionary'. The reader is advised to skip this chapter at first, and then to refer to it if necessary. A paragraph (x.y) will be referred to as (O.x.y) in later chapters, but as (x.y) within this chapter. §1. Relative viewpoint

Many notions and theorems in algebraic geometry can be formulated in a relative situation. Formally this is a very simple (apparently trivial) process, but it provides a The 'Bible' of powerful approach for many applications. this philosophy is [EGA].

X

(1.1) Let

be an algebraic space defined over an

algebraically closed field any sheaf

d

For

char(t) = p 2 0.

OX-algebras of finite type, we can define

of

the relative spectrum

morphism

with

St

a: Z -a X

Z = 9'ftecX(d)

such that

with a natural affine

n*OZ = d (cf. [EGA;II,Sl] or

[Ha4;II, Ex. 5.17]).

(1.1.1) Example (Stein Factorization). be a proper morphism of spaces. coherent sheaf of

morphism for

0: Y - Z

f = n

-

0

0

.

'

is connected.

and

(1.1.2) Remark.

are varieties and if

X

is a

is a finite

O*OY = OZ.

Hence

Such a factorization

is determined uniquely by

Stein factorization of

= f*OY

f: Y -b X

Moreover there is a morphism

Z = SafecX(d).

f = n

.

r: Z

OX-algebras, so

such that

every fiber of

Then

Let

f

and is called the

f.

If

Y

f

is surjective, both

is normal, then

Z

X

and

is just the

Y

0: Summary of Preliminaries

2

X

normalization of (1.2) Let

in the rational function field of be a sheaf of graded

J = ®d20 91d

algebras of finite type over a space

OX-

Then we obtain a

X.

with a projective morphism

S = 3to/X(9)

space

Y.

n: S -+ X Here we

in a natural way (see [EGA;II,S3] for details).

give a few important examples. (1.3) Scroll of a vector bundle. Let let

be the symmetric algebra of

9'

a 1Pr-l-bundle over

on

n*XOt

and the pair

quotient bundle natural

of

H(8)

of the total space of the dual

Thus

corresponds to a quotient space

of dimension one.

n E -+ H(8)

is the

E, minus the zero section, modulo the

y E F(6)

En(y)

8.

P(8)

E,

Gm-action via the scalar multiplication.

each point

X

and is denoted by

IP(8)

for some vector bundle

(EV - O(X))/Gm

EV

is

This is

is called the scroll of

(P(8), H(8))

8 = OX[E]

If

and

or

1PX(8)

The corresponding line bundle is denoted by

0(1).

*

X

9hojX(9J)

- St8 for any t e Z.

called the tautological sheaf on

of

Then

on

Moreover there is an invertible sheaf

such that

IP(8)

S.

X, and will be denoted by

from now on.

IP(8)

r

be a locally free sheaf of rank

6

There is a surjection

of vector bundles on

identified with H(8) y.

naturally isomorphic to

Hy

IP(8), and

Hy

is

The kernel of this homomorphism is QIP(8)/X

the relative cotangent bundle of

0 H(8), where n.

S1F(8)/X

is

In particular we have

the following canonical bundle formula: K1P(8)

=

*(KX + det 8) - r H(8).

Remark. The word 'scroll' is used in different meanings

0.1: Relative viewpoint

3

in several papers.

(1.4) Fact (Relative rational map).

E be a vector bundle on X

a morphism,

line bundle on

f E --i L.

f= n

. p

f: Y -* X be

Let

and let

be a

L

Suppose that there is a surjection

Y.

Then there is a morphism p: Y - IP(E) * and p H(E) = L.

such that

For a proof, see [EGA;II,(4.2.3)] or [Ha4;II,7.12].

L

is said to be relatively very ample with respect to

f

(or

f-very-ample for short) if there is a vector bundle

E

as

above such that f-ample if

is a closed embedding.

p

mL

f-very-ample for some

is

is said to be

L

m > 0.

(1.5) Blowing-up (cf. [EGA;II,S8] or [Ha4;p.163]). Let

subscheme of

in

i

be a coherent sheaf of

i C

OX

X' = 9 oi.X(g)

of a variety and set

Let

X.

OX-ideals defining a 3d

Then

30 = OX.

9' = ED d20'gd, where

is called the blowing-up of X

is called the center of this blowing-up.

d-th power

be the

along

X

a: X'

C.

C

is a

birational morphism with the following properties. (1.5.1) The inverse image ideal the image of the natural homomorphism

3' on n*3

X', which is

-' OX

principal ideal defining a Cartier divisor E E

is called the exceptional divisor of

(1.5.2) n(E) = C

and

on

x'.

a.

X' - E = X - C.

(1.5.3) For any morphism

f: Y --' X

*

image of the homomorphism f 3 --' OY is a unique morphism

is a

g: Y -. X'

such that the

is invertible, there

with f = n

.

g.

This is called the universal property of the blow-up. (1.5.4) Let

Z

be a closed subvariety of X not in

C

0: Summary of Preliminaries

4

and let

be the image of

3Z

blowing-up

of

Z'

subvariety of

i

via

OX --+ OZ.

with respect to

Z

is a closed

3Z

X'.

is called the strict (or proper) transform of

Z'

of the blowing-up

ifold of

X, then

is smooth along

C

(1.6) Fact (Blowing-down).

Cartier divisor on

Xb

I

is a subman-

r = codim C.

E be an effective X

is smooth along E

is isomorphic to the scroll of a

(E, [-E] E)

vector bundle

C

The relative canonical

Let

Suppose that

X.

and

is isomorphic to the scroll

(E, OE[-E])

of the conormal bundle of C in X. KX'/X is [(r - 1)E], where bundle

and that

Z

X' -i X.

X

(1.5.5) If

space

Then the

over a manifold

containing

the blowing-up of Xb

C

Then there exists a

C.

X

as a submanifold such that

along

C, where

E

is

is identified

with the exceptional divisor.

For a proof, see [Ar] (or [Moi2], [Nkn] if

A = C).

This criterion is not true in the category of schemes.

(1.7) The relative version of the sheaf cohomology is the higher direct image.

For any continuous map spaces and for all

f: X -i Y

of topological

i s 0, we have natural functors

between the categories of sheaves of abelian groups which have the following properties.

Rif* : Xb (X) - l1b (Y)

(1.7.0) R0f* = f*. (1.7.1) For any exact sequence in

21b(X)

and for all

homomorphisms

0

8 -> S

%

0

i 1 0, there are naturally defined

8i: Rif*` - Ri+1f*d

such that

0.1: Relative viewpoint

0 -+ f*8 -, f*T - f*`

5

Rlf*8

-* ... -,

8

is exact in

-+ R1f*F -p R1f*C4 -ii Ri+lf*8 (1.7.2) If

is a point, then

Y

Ri-1 f* 8i-+ R1f*t8

Rif*.T ti Hi(X, S).

(1.8) Fact (Leray Spectral Sequence). and

g: Y

Rp+q(g.f)*3.

converging to

This means that there are objects

p, q

1 s r g -

and

drq: Er'q -

r

p+r,q-r+l

in

for

Zfb(Z)

such that and

Ker(dpq)/Im(dp-r,q+r-1)

r

Ep'q

together with homomorphisms

1) dp'q . dp+r,q-r+l = 0

Ep,q = r+l

Then

Y e Z(b(X).

E2'q = Rpg*(RqfY)

there is a spectral sequence with

all

f: X -+ Y

Let

be continuous maps and let

Z

Xb(Y).

r

p, q, r e Z,

for any

2) EP ,q -- Rpg*(Rqf*`3),

3) for each that

dp'q = 0,

p, q

there is an integer

dp-r,q+r-1

= 0

and

r(p, q)

Ep'q -- Ep'q

such

for any

r a r(p, q), and 4) there is a descending filtration of that

and

FO,k o Fl,k D

... D Ft,k = 0

Ep'q = EP'q

for all

p

If

f: X --+ Y

algebraic spaces and if Rif*.T

>> 0

T

at a point

Er

terms.

is a proper morphism of

is a coherent

is coherent on Y for every

(R1f*Y)y = 0

£

q, we say that the

and

spectral sequence degenerates at the (1.9) Fact.

for

EP,q

Fp,p+qIFp+l,p+q When

such

y

on

Y

i a 0.

if

For a proof, see [EGA;X,(4.2.2)].

OX-module, then

Moreover

dim f(y) < i.

0: Summary of Preliminaries

6

f: X -> Y

(1.10) Fact (Serre's Vanishing Theorem). Let

Then a line

be a proper morphism of algebraic spaces.

bundle

on x

L

f-ample if and only if, for every

is

coherent sheaf X

on

X, there is an integer for any

Rif*(S ® [tL]) = 0

that

Let

(1.11) Fact.

f: X - Y be a proper morphism as

Y be a coherent sheaf on

above and let for every point

x

on

i > 0.

and

t i N(3)

such

N(3)

X, the stalk

0Y f x -module via the mapping

0Y

gx

Suppose that,

X.

is a flat

f(x)

Then

0X x'

x(Xy, Y ) _ 2i(-1)1hi(Xy, Yy) is a locally constant y function in y e Y, where Xy = f 1(y) and 3 is the

restriction of Y

to

Xy.

Moreover

upper-semicontinuous function in is locally constant at vector bundle

E

hi(Xy, Sy)

for each

y

y E Y, then

I.

Rif*Y = 0[E]

over a neighborhood of

identified naturally with

is an If this

for some Ey

y, and

is

Hi(Xy, 3y ).

For a proof, see [EGA;1Q,S7] or [Ha4;1Q,12.9]. (1.12) We have the following criterion for flatness.

f: X -i Y

Fact (cf. [EGA;1V,(6.l.5)]). Let

morphism of algebraic varieties.

and X

is locally Cohen Macaulay.

Suppose that Then

f

Y

is smooth

is flat if and

dim f 1(y) = dim X - dim Y for every

only if

be a proper

y E Y.

§2. Singularities (2.1) Fact (Serre Duality).

dim M = n Exti(Y, WM) ,n-i(M,

Y)

and let

M be a manifold with

Let

wM = Dn be its canonical sheaf.

Then

is naturally isomorphic to the dual space of for every coherent sheaf

S

on

M.

Here we give a generalization of this fact on singular

0.2: Singularities

7

spaces for the convenience of later use.

For a more

complete treatment, see e.g. [Ha2].

X

(2.2) Let

and P

dim P = N (both X

with

may be non-complete).

Suppose that any irreducible component of dimension 0) S)X

n.

P

be a closed subspac6 of a manifold

We define

X

is of the same

qX = dxtp+q(OX, c)p).

is an invariant of X

Then

and is independent of the

choice of an embedding X (; P. unless

1) qX = 0

0 s -q s n.

and

2) dim Supp(EbXq) s q

3) When X

Supp(cXn) = X.

OXn = wX

is smooth,

and

OXq = 0

for

q # n.

The proofs are easy (see e.g. [F8]). (2.3) A subvariety

X

component of

if

of X

Z

is called an embedded

dim Z < dim X

some coherent subsheaf

,N

of

and if

for

Z = Supp(.N)

Here, as usual, Supp

OX.

denotes the set-theoretical support. Fact (cf. [F8;(1.11)]).

an embedded component of X (2.4) Definition.

on X

x

point

any component tion means

Z

if

of

X

q = dim Z.

Set

x g Supp(gxq)

for

0 s q < n

for any

with x e Z. q < k.

X

and

This condi-

is said to be

k-Macaulay if it is so at every point on

locally

is

k-Macaulay at a

is said to be

Supp(qXq)

Z

Z c Supp(cXq).

if and only if

dim Z s q - k

Then

X.

Then we have the following facts. 1) X

is locally 1-Macaulay if and only if X has no

embedded component. 2) Let

D

such that

be a subscheme of x JD = Ker(OX -+ OD)

of pure dimension n - 1 is invertible.

Then

D

is

0: Summary of Preliminaries

8

x e D

(k - l)-Macaulay at 3) Let D

D

Then

be as above.

k-Macaulay at

is

If X X

k-Macaulay at

is

x.

k-Macaulay at

is

x

if

x.

These are easily proved.

From these facts we deduce

further the following. (2.5) Fact.

X

depth(OX'r) 2 Min(k, height p)

with x E

x

k-Macaulay at

is

if and only if

for any scheme-point

p

C X.

This is called usually 'Serre's condition Sk'. 0X

Thus,

is a Cohen Macaulay local ring if and only if

x

x, namely

n-Macaulay at

(9xq)X = 0

for

q # n.

X

is

9 xn

is

the dualizing sheaf in this case. Let

(2.6) Fact.

assume that some

X

k < n.

k-Macaulay on

be a linear system on X

A

k-Macaulay on an open set

is

Then any general member

of

D

and

U c X d

for

is

D n U n (X - BsA).

This follows from the observation in (2.4). (2.7) Fact.

Let

X be a locally

k-Macaulay variety

and let H be an ample line bundle on locally free sheaf and

£

8

on

X.

Then, for any

X, H1(X, 8[-tH]) = 0

i < k

for

>> 0.

For the proof, use the spectral sequence with Ep,q = Hp(X, 9X ® a")

H-p-q(X, S)v,

converging to

we obtain by using (2.1).

A variety

(2.8) Fact (Serre's criterion).

normal if and only if it is locally singular locus is of codimension

V

is

2-Macaulay and its

> 1.

which

9

0.3: Intersection theories

For a proof, see e.g. [EGA;W,§5.8].

As a corollary we

obtain the following. (2.9) Fact (cf. [Sei]).

V such that

a normal variety that

Let

char(t) = 0

or that

A

A be a linear system on Suppose further

BsA = 0.

Then any

is very ample.

connected component of any general member of

A

is normal.

(2.10) Finally we recall a result in Hironaka's desingularization theory [Hirn]. Definition.

n: X' - X be a blowing-up of X

Let

This is said to be admissible if the center

in (1.5).

is smooth and

X

is normally flat along

as

C

C.

We omit the precise definition of normal flatness.

Roughly it means that the algebraic nature of the singularities of

X

relative to Let

(2.11) Fact.

closed subset of

V.

C

varies continuously along

char(l) = 0.

is a finite sequence V' = Vk - Vk_l of admissible blowing-ups such that inverse image of

-

V'

Then there

-+ V1 - VO - V is smooth and the

V' is a divisor whose support has

in

S

S be a

V be a variety and let

Assume that

C.

only normal crossing singularities.

§3. Intersection theories (3.1) Let A.

19

be the category of algebraic spaces over

We use a theory which has the following features:

a) There is a contravariant functor

A'(*)

category of graded rings with unit.

Thus, for any morphism

from

18

to the

*

f: X --i Y

in

A'(Y) -p A'(X)

19, we have a ring homomorphism

such that

The multiplication in

f

= A'(f):

f*(Ad(Y)) c Ad(X), f*lY = 1X.

A'(*)

is called the cup product.

0: Summary of Preliminaries

10

b) There is a covariant functor

A.(*)

has a natural structure of graded A'(X)

action of

on A.(X)

A.(f): A.(X) --> A.(Y)

is a homomorphism of

where

A.(X)

d) Ad(X) = Ad(X) = 0 Y

unless

in

T, the map

via the map

f*

def

A'(Y)-modules, A'(f).

0 s d s dim X.

is a point, there is an isomorphism

we have

The

is called the cap product.

f: X - Y

acts on

X,

A'(X)-module.

c) For every morphism

A'(Y)

to the

Moreover, for each space

category of abelian groups. A.(X)

T

from

Moreover, if

deg: A0(Y) z Z.

degX = deg > AO(f): AO(X) -p Z, where

So

f: X -+ Y

is the trivial map.

e) For each complete variety natural element

f: V -- W

V

morphism with

in

An(V).

n = dim V, there is a

Moreover, for any morphism

of varieties of the same dimension

f*(V) = d(W) f) If

(V}

V with

for the mapping degree

d

of

is a smooth complete variety with

8: Ad(V) - An_d(V)

(V)

is bijective for any

is said to be Poincare dual of

the inclusion

n = dim V, the

a e Ad(V)

d.

a

if

and

8(a)

A subvariety

t(Z) = b(a)

for

c: Z g V.

(3.2) Example.

H2d(Xan; Z)

f.

defined by the cap product

are said to be Poincarb duals of each other. Z

n, we have

and

When

.

= C, by setting

Ad(X)

=

Ad(X) = H2d(Xan; Z), we obtain a theory

with the above property.

(3.3) There exists a theory as in (3.1) such that Ad(X)

X

is the group of algebraic cycles of dimension

modulo rational equivalence (see [Ful]).

A'(X)

is called the Chow ring of

X.

d

In this case

on

0.3: Intersection theories

11

(3.4) Given a theory as in (3.1), we have a theory of Chern classes of vector bundles as follows. E

For every vector bundle

in A'(X)

c(E)

X, an element

on a space

is defined in a natural way so that the

following 'axioms' are satisfied. a) If

c.(E) = 0

and

for

Chern class while

c3(E) a A1(X), then

with

c(E) = 21 ci (E)

j > rank(E).

is called the total

c(E)

is called the

ci (E)

c0 = IX

j-th Chern class.

b) If 0 --+ E - F - G - 0 is an exact sequence of vector bundles on X, then c(F) = c(E)c(G). for any morphism

c) c(f E) = A'(f)(c(E))

d) For any Cartier divisor 2 µiZi

on a variety

D

inclusion map

(Zi)

A.(ti)(Zi)

stands for Then

ti: Zi c, V.

cI([D]), where

D =

is the Poincarg dual

(D)

cycle in X bundles on

Then

Z = 2 µiZi

Let

of dimension X.

For example

for the dual bundle

(3.5) Definition.

d

EV.

be an algebraic

L1, -1 Ld

and let

degX((cl(L1) A

(Z) = 2 µi(ci)*(Zi)

D.

a) - d) we can deduce well known

formulas for Chern classes, as in [Hirz]. (-1)ici (E)

for the

is the line bundle defined by

[D]

From these properties

where

V, let

be the prime decomposition as a Weil divisor and set

(D) = 2 µi(Zi), where

of

f: Y --p X.

cup and cap products respectively.

the intersection number of

L1,

be denoted by

If

a Z,

A

as above, and

A

and

denote

This integer is called

Ld with X

be line

Z, and will

is a variety and

we often write simply L1 Ln, where

n = dim X.

Z = X,

0: Summary of Preliminaries

12

We have If

(3.4.d).

for any Z

is a curve, then

of the line bundle

on

Lz

(3.6) Definition.

total Segre class s(E)c(EV) = 1X s1(E) = cl(E), (3.7) Fact.

bundle

E

of rank

r

n-j

(V)

for any

E

on

X, the

is defined by the formula

By easy computation we get

S2(E) = C1(E)2 - c2(E),

be the natural map. R

Z.

A'(X).

Let

by

ILnI

is the usual degree

L(Z)

For any vector bundle

s(E)

in

D E

(P, H)

be the scroll of a vector

on a variety

V

Therefore, for any C1(H)J+r-1)(P)

a n si(E)(V) = (x*a n

x: P - V

and let

x*(C1(H)J+r-1(PJ)

Then

j s 0.

and so on.

= si(E){V} E a E An-j(V), a Z.

Using this formula and (1.5.5), we can calculate

various intersection numbers on the blowing-up

X

of

X'

as in (1.5).

(3.8) We have the following theory of Todd classes for any reasonable theory having the properties in (3.1). For each coherent sheaf element

Y

on a complete space

r(S) = 21 r (Y) E A.(X)OQ

is defined in a natural

way so that the following 'axioms' are satisfied. called the total Todd class of

X, an

T(Y)

Y.

a) If 0 -i T -- % --a X --p 0 is an exact sequence, then T ('S) b)

= T (Y) + T(X).

idim

c) If

for any complete variety

V(0V) _ (V)

f: X - Y

is a proper morphism, then in

V.

f*r(S) _

A.(Y)@ = A.(Y) 0 Q.

(3.9) The following 'theorems' follow from the above

is

0.3: Intersection theories

13

'axioms' in (3.8).

1) degX(i0(f)) = X(Y) der

2jz0(-1)Jhj(3).

2) r(3 0 L) =

for any line bundle

More generally, we have vector bundle

E

for any

:(3' 0 E) =

and its Chern character

a A'(X)@,

See [Hirz] for its precise definition.

r = rank(E).

3) If M is.a smooth complete variety, then expressed explicitly by the Chern classes tangent bundle of

Note that

ch(E).

ch(E) = rX + c1(E) + (cl(E) 2 - 2c2(E))/2 +

where

L.

M.

i(OM)

ci

is

of the

For example, if n = dim M, then

sn = (M),

c(M)/2,

to-1

sn-2 = (c12 +

(Noether's formula),

to-3 =

and so on.

See [Hirz] for a general formula for

t1(0M).

4) Combining 1), 2) and 3), we obtain a formula, which is called the Riemann-Roch-Hirzebruch formula, expressing X(OM[E])

of a vector bundle

E

on a manifold M

in terms

of Chern classes.

(3.10) The above theory was established by Hirzebruch

Grothendieck generalized the theory to

in the case (3.2).

an abstract context. (3.11) Let

L

results in (3.9),

polynomial in (d/n!)tn

for

t

be a line bundle on a variety

V. By the

X(V, tL) = 2J>0(-1)ihJ(V, tL)

is a

d = Ln{V).

is a polynomial in the Ln.

n = dim V, and its main term is

of degree

More generally,

ti's

X(V,

for any line bundles

L1,

This fact can be proved by an elementary argument

0: Summary of Preliminaries

14

without using the intersection theory as in this section. Hence we can define the intersection number

by

using the coefficients of the above polynomial and deduce various properties of it (cf. [Kl]).

§4. Semipositive line bundles and vanishing theorems (4.1) Definition.

An element of

a Q-bundle on

X.

is denoted by

[L]Q, or often just by

For a line bundle

Pic(X) 0 Q

is called

L, the Q-bundle L

L ® 1

when confusion is

impossible or harmless.

Div(X) 0 Q

Similarly, an element of

Q-divisor (or IQ-Cartier divisor), where

group of Cartier divisors on Q-divisor

D

is denoted by

The

X.

[D]@,

is called a Div(X)

is the

Q-bundle defined by a or just by

[D]

D.

The notion of algebraic Q-cycles is defined similarly. (4.2) For any

@-bundle

L

and a curve

define the intersection number LC e Q

in

C

X, we

in a natural way.

L

is said to be numerically semipositive (or nef, for short) if

for any curve

LC 2 0

C

in

X.

'nef' is NEVER

Thus,

an abbreviation of 'numerically equivalent to an effective Q-divisor'.

positive

L

m E Q

is said to be ample if

and an ample line bundle

is said to be semiample if some line bundle line bundle BsjmFj = 0

F

L = m[H]@

H

L = m[H] Q

such that

Similarly, L

H.

for some

BsIHI _ 0.

for some

m > 0

and

In particular, a

is semiample as a c-bundle if and only if

for some positive integer

m.

The following results are well known for line bundles,

and they are obviously true for any Q-bundle

L.

(4.3) Theorem (Nakai's criterion, cf. [Nk], [Ha3]).

L

0.4: Semipositive line bundles and vanishing theorems

Ldim W{W} > 0

is ample if and only if

W of X with

15

for every subvariety

dim W > 0.

(4.4) Theorem (Kleiman's criterion, cf. [K1], [Ha3]). L

is nef if and only if

subvariety W of

X.

(4.5) Remark.

A + L

If

A

L

is a limit of ample

is nef, then

L

Hence any nef

Q-bundles:

This is a useful observation.

L = lima-++0(L + mA).

(4.6) Index Theorem.

nef

is ample and

is ample by the results above.

Q-bundle

for every

Ldim W{W) 2 0

v-bundles on a variety yij

711722 s 7122, where

Lit L2, Al, ***1 An-2

Let

with

V

be

Then

dim V = n.

is the intersection number

LILjA1...An-2(V). Outline of proof.

Ak's and

assume that the

may assume that the members of

By the above limit method, we may Li's

are all ample.

Moreover we

Taking general

Ak's are very ample.

we can reduce the problem to the case

IAkI

n = 2, which is the classical index theorem on surfaces. (4.7) The following simple lemma is useful in various situations. Lemma.

Let

Lit Hi (i - 1,

on a variety X Li - Hi = [Ei]

,

n) be nef '-bundles

with n = dim X. Suppose that, for each for some effective

Q-divisor E.

i,

Then

Ll ... Ln a H1 ... Hn, Proof.

Then

Let

Ei = 2 PaWa

L1 ... LiHi+l ... Hn - L1 ... Li-1Hi ... Hn =

L1 ... Li-lHi+1 ... Hn{Ei) = a 0

be the prime decomposition.

by (4.4).

So

L1...Ln 2

pa L1 ... Li-1Hi+1 ... Hn(Wa) 2

...

a H1...Hn.

0: Summary of Preliminaries

16

(4.8) Definition. L = m[F] Q

A Q-bundle

for some

m > 0

is said to be big if

L

F

and some line bundle

that the rational map defined by

JF)

its image (but we do not require

BsIFI = 0).

such

is birational onto Of course,

ample Q-bundles are nef and big, but the converse is not true in general.

We have the following criterion for the

bigness.

A Q-bundle

(4.9) Theorem (Kodaira's Lemma).

projective variety is big if and only if L - E for some effective Q-divisor

L

on a

is ample

E.

.For a proof, see [F8;(2.8)] or [KaMM;0-3-3]. (4.10) Theorem.

of dimension

n.

Let

Then

L

be a nef @-bundle on a variety

is big if and only if

L

Ln > 0.

For a proof, see e.g. [Fl5;(6.5)]. (4.11) We recall vanishing theorems of Kodaira type. Theorem ([Kodi]).

Let

A be an ample line bundle on a

compact complex manifold M whose canonical bundle is then

for

H1(M, K + A) = 0

K.

i > 0.

This famous result has been generalized by several authors.

Here we present one such generalization.

(4.12) Theorem (cf. [Kaw2], [Vi2]).

bundle on a smooth variety n = dim M,

M, defined over

whose canonical bundle is

there is an effective

Q-divisor D

decomposition D = 7 piDi i) 0 s µi < 1

for each

ii) Supp(D) = U Di

K.

A be a line

Let

C

and with

Suppose that

on M with prime

such that

i,

has only normal crossing singularities,

iii) A - D is nef and big.

0.4: Semipositive line bundles and vanishing theorems

Then

Hi(M, K + A) = 0 Remark.

for any

17

i > 0.

There are generalizations of the above result

in which the pair

is allowed to have certain mild

(M, D)

singularities (cf. [KMM] = [KaMM]). (4.13) Corollary.

A be a nef and big line bundle

Let

on a manifold M defined over

Then

C.

H'(M, -A) = 0

for

i < dim M. (4.14) Corollary.

Let

V

normal projective variety

be a nef line bundle on a

L

over

for some ample line bundle H on

C V.

such that Then

Outline of proof. We may assume that H

L2Hn-2 > 0

H1(V, -L) = 0. is very ample.

Using (2.7) and (2.9) we reduce the problem to the case n = 2, where (4.13) applies by the technique in [Muml]. (4.15) The vanishing theorem (4.12) is the most important technical tool in Kawamata's theory on minimal models (cf. [KMM]).

The following result, which originated from an

idea in [Rd2], is very useful for our study. Fibration Theorem.

and suppose that morphism

L = K + A

0: M --+ W

on W such that

M, K, A, D

Let

is nef.

be as in (4.12)

Then there is a

together with an ample line bundle

0 H = L

and

H

0*0M = OW.

This is just a reformulation of the Base Point Free Theorem (see e.g. [KMM;3-1-1]).

In the usual formulation,

it is asserted that

for every sufficiently

large integer

£

BsJLLl = 0

under the assumption of the theorem.

Obviously this follows from the existence of as above.

0, W

and H

The proof of the converse is as follows.

Take p > 0

such that

BsJpLJ = 0.

Let

M -* X c ]PN

0: Summary of Preliminaries

18

be the morphism defined by

Stein factorization of it and let 0X(1)

to

Then

W.

Every fiber

P

q and

defined by

p

are coprime.

Then

JqLj.

q

through W since W

is normal.

*

pH

(pH) = pL =

Thus we have

Hence

*

LF

Hence H

and H

0, W

factors

p

since p and

is numerically equivalent to

P.

is

comes from

qL

L = 0 H for some H E Pic(W)

are coprime.

BsjgLI = 0

is a point since

From this we infer that

So

0 P = pL.

such that

numerically trivial.

Pic(W).

is finite.

p: M -+ Y be the mapping

Let

p(F)

be the

be the restriction of

is connected and

0: M ---+ W

Now, take another positive integer and

X

is ample since W -+ X

P

F of

M -+ W

IpLI, let

is ample since

P

q

since is ample.

P

as desired.

The above argument is found in the proof of the Contraction Theorem (cf. [KMM;3-2-1]).

Combining this

theorem and the Cone Theorem (cf. [Mor3;(1.4)], [KMM;4-2-1]) we obtain the following. (4.16) Theorem.

suppose that

morphism

L = K + A

0: M -- W

1) 0*0M = 0W

and

2) for any curve Z

is not nef.

and a curve

from

z

in

F e Pic(M),

Pic(W), and Remark.

The ray

R

Then there is a

on M

such that

LR < 0, m,

is a point if and only if

O(Z)

is numerically equivalent to

3) for any

be as in (4.12) and

M, K, A, D

Let

mR

m,

if and only if

FR = 0

FR > 0

for some

if and only if

F

F

comes O-ample.

is

Such a fibration is not unique in general.

(mR)m>O

generated by

R

in the group of numerical

equivalence classes of 1-cycles with real coefficients is called the extremal ray represented by

R, and

0

is called

19

0.5: Birational classification of algebraic varieties

the contraction morphism of this exteremal ray.

R

is

called an extremal curve belonging to this ray.

§5. Birational classification of algebraic varieties (5.1) Definition (cf. [Iii], [1i2], [U], [F8]).

be a line bundle on a variety if

m > 0, and

for any

ImLI = 0

otherwise, where

PImLI

x = Maxm>0(dim PImLI(v))

is the rational map defined by ImLI.

is sometimes denoted just by

x(L, V)

KV

smooth and if

x(KV, V)

V.

x(L, V) k 0, we have

If

is

and is denoted by

V

This is a birational invariant of (5.2) Remark.

V

If

x(L).

is its canonical bundle, then

is called the Kodaira dimension of x(V).

L

x = x(L, V) = --

We set

V.

Let

x(L, V) =

tr.deg(®tz0H0(V, tL)) - 1 = limsupt,_(log(h0(V, tL))/log t). (5.3) Fact (Iitaka's Fibration Theorem).

line bundle on a normal variety there is a birational morphism 0:

-> W such that

with

V

such a fibration

0

F

Then

and a morphism

dim W = x(L, V)

for any generic fiber

x(LF, F) = 0

be a

L

x(L, V) k 0.

a: 1`---> V

0*0V = OW,

Let

of

0.

and

Moreover

is determined uniquely up to a

birational equivalence. 0

is called the Iitaka fibration of

smooth and

L

L.

When

is

V

is its canonical bundle, it is called the

canonical fibration of

V.

(5.4) litaka's classification Program.

Given a smooth variety n = dim M.

If

fibration by (5.3).

M, we have

0 < x < n, M

x(M)

0,

1,

admits a non-trivial

We hope that the investigation of such

manifolds can be reduced to the study of lower dimensional

20

0: Summary of Preliminaries

cases.

Hence we will study the three cases

x =

0, n.

When n = 1, this corresponds to the classification of Riemann surfaces by their universal coverings. If

x < 0, Mori-Kawamata theory will be useful.

If

x = 0, the Albanese mapping gives more precise information on the structure of

M (cf. [Kawl]).

The case

x = n

is

the most difficult to study.

(5.5) Classification of smooth algebraic surfaces [Beal], [BaPV], [112], [Zr], [Sha], [Mum2], [B0M],

...

(cf. ).

In the birational classification theory of surfaces, we

may consider only relatively minimal surfaces, i.e. surfaces having no (-1)-curve.

For such a surface

K

1) The canonical bundle

S

we have:

is nef if and only if

x(S) k 0.

2) If

x(S) < 0, then

over a curve.

3) If BsjmKI = 0 4) If in

In particular

S

x(S) 2 0, then K for some x(S) = 0,

Pic(S), then

If K $ 0

S = F2

in

S

or

S

is a

P1-bundle

is ruled.

is semiample, which means

m > 0.

K

If K = 0

is numerically trivial.

is an abelian variety or a K3-surface.

Pic(S), then

S

hyperelliptic surface, unless

is an Enriques surface or a char(St) = 2

or

3.

There are several ways of defining the notion of a K3-surface.

Roughly speaking, it is a surface having the

same numerical invariants as smooth quartics in

An Enriques surface (when the form

where

char(.) # 2) is a surface of

is a K3-surface and

point free involution of .

P3.

t

is a fixed

0.5: Birational classification of algebraic varieties

A hyperelliptic surface (when surface of the form and

char(&) # 2, 3) is a is an abelian surface

2S

is a cyclic group acting on

points.

Sf/, where

21

By the Albanese mapping, S

without fixed

has the structure of an

etale fiber bundle over an elliptic curve where every fiber is isomorphic to a fixed elliptic curve.

example in [Beal]) S 5) If

Sometimes (for

is called bi-elliptic surface.

x(S) = 1, S

is an elliptic surface (or possibly

a quasi-elliptic surface when

char(SF) = 2, 3).

The ellip-

tic fibration is unique and is the canonical fibration. 6) There are many types of

S

with

x(S) = 2.

They

are called surfaces of general type.

(5.6) We have many results which can be viewed as 3-dimensional versions of those in (5.5). [Vil], [W], [Mor4], [Miy2],

See [U], [Ii2],

.

When n k 4, it is still dificult to get satisfactory results.

However, there are many interesting problems.

Chapter I. d-genus and the Apollonius Method In this chapter we present the classification theory by The main technique is the hyperplane section method using induction on the dimension. d-genus.

§1. Characterizations of projective spaces First of all, as the most typical example of the Apollonius method, we recall the proof of the following. (1.1) Theorem (cf. [Gor], [KobO]).

Proof.

(V, L) = (1n, 0(1)).

We use the induction on

is easy, so we consider the case of

Then

ILI.

and

V has only Cohen Macaulay

Assume that

Then

singularities.

be a

(V, L)

n = dim V, Ln = 1

polarized variety such that h0(V, L) 2 n + 1.

Let

The case n = 1

n.

Take a member

n k 2.

Ln-1D = Ln = 1.

If

D = D1 + D2

D

for

Ln-1Di

Di, then

non-zero effective Weil divisors Ln-1D 2 2

since

L

is ample.

and reduced as a Weil divisor.

for the homomorphism

By the above observation scheme

D

has a

On the other hand, D

V

such that

8: OV[-L] - OV Supp(D)

and

is irreducible

D

Therefore

natural structure as a subscheme of OD

> 0

Coker(8)

defining

D.

is irreducible and the

is reduced at its generic point.

Cohen Macaulay singularities, hence so does

V D.

has only Therefore

D

has no embedded component and is reduced everywhere, so

D

is a variety.

0V -+ OD

0.

Now we use the exact sequence Taking

®L

and then

0 --i OV[-L]

HO, we get an exact

___+

sequence

0 --+ HO (V, OV ) -* HO (V, L) - HO (D, L D ).

h0(D, LD)

z

h0(V, L) - 1 s n.

hypothesis to

Hence

Applying the induction

(D, LD), we obtain

(D, LD) - (lpn-1, 0(1))

23

§ 1: Characteristics of projective spaces

Therefore the restriction map h0(V, L) = n + 1. 0 H (V, L) - H 0(D, LD) is surjective. For any point x and

D, there is

H0(V, L).

from

Hence

T(x) # 0, and

with

T E H0(D, LD)

its global sections. such that

So

L = p*0(1). If

the restriction of

L

to

p-1(y)

Therefore

So

L.

p: V _ n y e Pn

is trivial, which is a finite

p

deg(p) = Ln = 1, hence

We also have

birational.

is spanned by

for some

dim p-1 (y) > 0

contradicts the ampleness of morphism.

OV[L]

gives a morphism

ILI

comes

T

Since D E ILI, this

x % BsILI.

BsILI = 0, or equivalently,

implies

on

p

is

is an isomorphism by virtue of

p

Zariski's Main Theorem.

The exact sequence

(1.2) Technical remark.

OV[-L] - OV If

V

OD --> 0

0 ->

is important in the above proof.

is a non-reduced scheme, it may have embedded compo-

nents and

OV[-L] -+ OV

is not always injective.

Without

the Cohen Macaulay assumption, we have the above exact sequence at the first step, but then

may have embedded

D

components, and we cannot go further.

However, working

carefully in a more general setting, we can eliminate this assumption (cf. §5).

(1.3) The theorem (1.1) yields the following interesting result. Corollary (cf. [KobO]).

complex manifold such that

K

Let

-(K + nL)

is the canonical bundle and

be a polarized

(M, L)

is ample, where

n = dim M.

Then

(M, L) =

(]pn, 0(1)). Proof.

Hi(M, tL) = 0

By Kodaira's vanishing theorem (0.4.11) we have for

i > 0

and

t 2 -n.

Therefore

X(t) =

I. A-genus and the Apollonius method

24

2i(-1)'hi(M, tL) = 0

is a polynomial in

x(t)

Moreover

vanishing theorem.

and

by the Riemann

Ln = 1

So

.

Since

x(0) = 1.

n, this implies

of degree

t

(t + n)/n!

x(t) = (t + 1) Roch theorem.

-n s t < 0

for

h0(M, L) = x(1) = n + 1

by the

Hence (1.1) applies.

(1.4) Corollary.

manifold such that

Let

be a polarized complex

(M, L)

Ln = 1

KLn-l < -n.

and

(M, L) =

Then

(1Pn, 0(1)). We have

Proof.

H'(M, -tL) = 0

by Kodaira's vanishing theorem. for

ho (M, K + tL) = 0

Hence

X(-t) = 0

t t n

x(t) = (t + 1) ... (t + n)/n! (-1)-nx(-n-1) = 1.

Then

Ln-1D

Let

since

hn(M, -tL) =

(K + tL)Ln-1 < 0.

h0(M, K + (n+l)L)

Hence

.

be a member of

D

i < n

and

This implies

KLn-1 + n + 1 < 1.

=

D = 0, so

implies

Moreover

0 < t s n.

for

t > 0

for

Since

K = -(n + 1)L

in

_

IK + (n+l)LI.

is ample, this

L

Pic(M).

Therefore

(1.3) applies. Remark.

In Chapter II, we will generalize these results

by applying Mori-Kawamata theory. (1.5) Corollary.

If a K&hler manifold

M

is a

deformation of IPn' then M = IPn. Clearly

Proof.

Hodge theory. Let

H2(M; Z) _ Z

Therefore

M

that there is a deformation family x e X

such that

x # o.

Then

LX = 0(1) h0(M, L)

M. = M

L = Lo

for any a

for some

Pic(M). (Mx) o

and

extends to a family x # o.

We have

h0(Mx, LX) = n + 1

by

h0'2(M) = 0

is projective and

be the ample generator of

L

and

Pic(M) = Z.

We may assume

parametrized by MX = Fn (Lx)

Ln = LXn = 1

for any

such that and

by the upper semicontinuity

§2: Basic notions in the Apollonius method

25

Hence (1.1) applies.

theorem (0.1.11).

The rigidity of

(1.6) Remark.

under arbitrary

IP3

deformation was proved by Nakamura [Nkm] by a similar Of course, his argument is much more complicated

method.

than ours, since he cannot assume the ampleness of

L.

§2. Basic notions in the Apollonius method (2.0) Let n

D

and let

be a polarized variety of dimension

(V, L)

be a member of

Suppose that

ILI.

irreducible and reduced as a subscheme of is called a rung of

D

case

a polarized variety of dimension (V, L)

study

is reflected in that of via

(V, L)

n - 1. (D, LD),

(D, LD)

is

The structure of

hence we can

using induction on

(D, LD)

is

In such a

V.

The pair

(V, L).

D

This is

n.

the main idea of the Apollonius method. (2.1) Let

D

be a rung of

have an exact sequence

®[tL], we obtain

is determined by

xJ .

e Z

xJ de=1

= 1, we write

since x,7(V,

X(V, tL).

x(t) E Z

L)

j a 0.

Then

X(V, tL) =

for each

t.

These integers

are important invariants of

t[J] - (t - 1)[J] = jt[J-1],

for every

we have

(V, L).

Since

xj(D. LD) = xj+l(V' L)

By the Riemann Roch Theorem we have

xn(V, L) = Ln, which is called the degree of denoted by

X(D, tLD) _

t[J] = (t+j-l)!/(t-l)! = t(t+l) ... (t+j-1)

Setting t101

Then we

Thus, the Hilbert polynomial of

x(V, tL) - X(V, (t - 1)L).

and

as above.

0 -+ OV[-L] - 0V -+ OD --' 0.

Taking cohomologies after

(D, LD)

(V, L)

d(V, L).

We set

(V, L)

g(V, L) = 1 - xn-1(V, L)

call this the sectional genus of

(V, L).

Thus

and is and

d(D, LD) =

I.0-genus and the Apollonius method

26

If

g(D, LD) = g(V, L)

and

d(V, L)

for a rung

D

of

has only Gorenstein singularities and K

V

(V, L).

is the

canonical bundle, we have the following 1)L)Ln-1.

2g(V, L) - 2 = (K + (n -

sectional genus formula:

This follows from the Riemann Roch Theorem.

When

n = 1, we

g(V, L) = h1(V, 0V).

have

(2.2) The d-genus of

is defined by the formula

(V, L)

4(V, L) = n + d(V, L) - h0(V, L). We have for a rung

d(V, L) - 4(D, LD) = dim Coker(r) t h1(V, 0V) D

(V, L), where

of

H0(V, L) -. H0(D, LD).

is the restriction map

r

In particular

and the equality holds if and only if such a case

tivity of

4(D, LD) t d(V, L) is surjective.

r

is said to be a regular rung.

D

In

The surjec-

is important from

rt: H0(V, tL) -+ HO(D, tLD)

the ring theoretic viewpoint. (2.3) Theorem (cf. [F4;(3.1)] & [F2;Prop. 2.2]).

be a polarized variety and let

(V, L) JaLl

defined by

D

be a member of

for some

6 E H0(V, aL)

Let

a > 0.

Let. 1,

k be homogeneous elements of the graded algebra G(V, L) = ED tk0H0(V, tL)

images in

G(D, LD)

is generated by

Proof.

Let

generated by Then the

8

A

8

and the

j's.

Set

Then

G(V, L)

At = A n H0(V, tL).

G(D, LD)

Hence it suffices to show

exact sequence

Suppose that

j's.

be the subalgebra of

and the

be their

nj's as an algebra.

rt(At) = HO(D, tLD), for r/j's.

nk

,

nl'

via the restriction.

is generated by the

G(D, LD) G(V, L)

and let

is generated by

Ker(rt) C At.

The

H0(V, (t - a)L) -* H0(V, tL) -* H0(D, tL)

§2: Basic notions in the Apollonius method

Ker(rt) =

implies

(2.4) A line bundle

t.

is said to be simply generated

L

In this case the rational map

H0(V, L).

is generated by

G(V, L)

if the graded algebra

just the map

Therefore we infer

(t - a)L).

by induction on

At = H0(V, tL)

27

defined by

pItLI

is

ItLI

followed by the corresponding Veronese

pILI

Hence

L

is very ample if it is ample and

simply generated.

L

is sometimes said to be normally

embedding.

generated in such a case.

Using this notion we get the following useful result. (2.5) Corollary.

surjective and if

LD

If

-

H0(V, L)

H0(D, LD)

is

is simply generated, then

is

L

simply generated.

(2.6) In the situation in (2.3), G(D, LD) phic to the quotient of

modulo

G(V, L)

is isomor-

Moreover we

6.

have the following.

Theorem (cf. [F4;(3.2)] & [F2;Prop.2.4]).

D, a, j and

be as in (2.3).

,j

G(D, LD)

are derived from

gl(')1,

'

.

V, L,

aj = deg j =

Set

Suppose that all the relations among the

deg i)j.

'(k) =

r(j's in _

,7k) = 0, where the gi's are weighted homogeneous

gr('(1'

'

polynomials in

Y11 *'*1 Yk

with

, Xk

such that

with

deg X0 = a

f1(0, Y1'

'

Yk)

and all the relations among are derived from

and

Then, there

deg Yj = aj.

exist weighted homogeneous polynomials X1,

Let

fi,

,

deg Xj = aj

gi(Y1'

6, 11

f1 = ... = fk(d' E1,

'

Yk)

, k in

fr

for

in

X0,

j > 0

for each G(V, L)

, k) = 0.

Thus, roughly speaking, any relations among the

nj's

i

I. A-genus and the Apollonius method

28

extend to relations among

and the

8

The proof of

Ed's.

this theorem uses a graded version of Nakayama's Lemma and is due to Mori, who used this method to prove the following. If

(2.7) Corollary ([Morl]).

is a weighted

(D, LD)

complete intersection in addition, then so is (2.8) Remark.

ampleness of

exact sequence

In theorems (2.3) and (2.6), the

is not necessary.

L

(V, L).

Moreover, if we have the

0 -+ OV[-aL] -+ 0V -- OD -+ 0

h0(D, 0D) = 1, we do not need to assume that

and if D

is

irreducible and reduced.

§3. Iteration of the Apollonius method (3.1) Definition.

of subvarieties of is a rung of

Vi

L

V = Vn

Vn-1 D

is called a ladder of for each

(Vi+1, Lj+l)

the restriction of each rung

V

A sequence

to

V1

j k 1, where

V

if

(V, L) L1

is

It is said to be regular if

VJ.

is regular in the sense (2.2).

Of course we have

dim Vi = j

Moreover, g(V, L) = g(VJ, L

)

and

d(V3, L) = d(V, L).

= g(Vi, L1)

is the arithmetic

i

genus of

Vi, which is why

g(V, L)

However, such a ladder does not always exist, while

genus.

g(V, L)

is always defined.

As for the d-genus, we have 2

is called the sectional

2 d(V1, L1)

d(V, L) 2 .. 2 d(VJ, Li)

and the equalities hold if and only if

the ladder is regular.

Letting

r

H0(Vi, L) -+ H0(Vi_i, L), we have

be the restriction map 4(V J, L

- 4(Vj_1, Lj_1)

)

i

= dim Coker(r), which reflects the gap in the structures of (VJ, Li)

and

(Vi _11 LJ_1)

as we saw in §2.

called 'deficiency' in classical geometry.

This was

Similar

29

§3: Iteration of the Apollonius method

phenomena can be observed for a divisor curve

d(V1, L1) k 0

Obviously

V1.

equality holds if and only if

4(V1, L1)

on the

IL1J

in this case and the

H0(V1, L1) -> H0(D, LD) = sd V1 = 11.

This is in turn equivalent to

is surjective. Thus

D E

is also a kind of deficiency.

d(V, L)

is

the sum of these deficiencies and hence is sometimes called However,

'total deficiency'.

is well-defined even

4(V, L)

if there is no ladder.

(3.2) Using the Apollonius method successively, we can generalize many results on curves to results on First of all, we have

having a ladder. Remark.

We conjecture that

(V, L)

g(V, L) a 0.

g(V, L)

for arbitrary

0

polarized variety, even if there is no ladder.

If

is

V

char(s) = 0, this is proved by using Mori-

smooth and

Kawamata theory on minimal models (see Chapter II).

However,

the problem is unsolved in general (see §19 in Chapter N), especially when

char(s) > 0.

(3.3) Proposition. ladder. 0

and

so

L

(V, L)

We have

Hence

g(V, L) = g(V1, L) = 0

When

4(V1, L) = 0, and

L1

is

By induction using (2.5), we infer that

is simply generated for each Remark.

g(V, L)

4(V, L) a 4(Vj, L) 2 4(V1, L) 1 0.

A(V, L) = 0, the ladder is regular and V1 = 1P1.

has a

is simply generated.

simply generated. Lj

if

If in addition the equality holds, then

Proof.

When

4(V, L) 2 0

j.

d = 0, we can prove the assertion

without assuming that a ladder exists (see (4.12)). (3.4) Proposition.

Let

(V, L)

be a polarized variety

I. A-genus and the Apollonius method

30

with

Suppose that

g(V, L) = 0.

Then

variety.

itself) is a normal

for any

H1(V3, -tL) = 0

j 2 2,

char(s) = 0, this follows from Mumford's

When

t > 0.

(Vi )

4(V, L) = 0.

We claim

Proof.

V

Vi (including

such that each rung

has a ladder

(V, L)

vanishing theorem (0.4.14).

To prove the claim in general,

we use the induction on

Since

Therefore

V1 = IP1.

is true for

j = 2

h1(VJ, -tL)

induction hypothesis.

We have the exact

is normal.

V i

s

H1(VJ-l, -tL)

--i

h1(V3, -(t+l)L)

On the other hand

by the

H1(Vj, -tL) = 0

by Serre's vanishing theorem and Serre duality,

t >> 0

since

we have

is a rational surface and the claim

V2

by [Ek2;Th.1.6].

Hence

j > 2.

for

g(V, L) = 0,

H1(Vi, -(t+l)L) -i H1(Vi, -tL)

sequence for

J.

Hence

h1(Vj, -tL) = 0, as desired.

From this claim we infer that

hl(V, 0)

s

h1(Vn_1, 0)

h1(V1, 0) = 0, so the ladder is regular and

4(V, L) _

4(V1, L1) = 0.

The assertion is not true without the assump-

Remark.

For example, let

tion of normality.

V

be the non-normal

variety obtained by identifying two points on

H

plane

V,

g(V, [H]) = 0

and

(V,

[H])

has a ladder,

d(V, [H]) > 0.

(3.5) The following result is useful when Theorem. ladder.

Let

(V, L)

Assume that

2) BsILI = 0

if

J

is small.

be a polarized variety having a

g = g(V, L) 2 4(V, L) = J.

1) the ladder is regular if

3) L

A hyper-

containing neither of these points is an ample

divisor on but

1P n.

Then

d = d(V, L) k 2d - 1,

d s 2d,

is simply generated,

g = d

and

Hq(V, tL) = 0

for

§4: Existence of a ladder

t, q with

any

31

0 < q < n, if

d > 2d.

Proof (cf. [F2;Th. 4.1] & [Fll;(3.6)&(3.8)]). assertion is more or less known when

The

n = 1, possibly under

For example, the assertion

different formulations.

g = d

in the case 3) is a part of Clifford's theorem (at least on See [F2;Sl] for details.

smooth curves). For

we use induction on

n k 2

Since

n.

g(Vn_1, L)

g(V, L) 2 d(V, L) 3 4(V n_1, L), the theorem applies to (V

n-11

L).

Assume that d k 2d'+l.

Therefore

So

d > 4(V n_1, L) = d'

d = d'

if

ILIV n-1

ILn-lI

g = d

by 1) and by the induction

(2.5) proves the simple generation.

G(V, L) -+ G(Vn_1, L) g hq(V, tL)

for

is surjective, we have

hq-1(Vn_l, tL) = 0

Hence

hq(V, tL) s

Since

hq(V, (t-l)L)

This is true also for

q = 1.

since

1 < q < n,

by the induction hypothesis. hq(V, LL) = 0

Serre's vanishing theorem. 0 < q < n.

BsILI _

and so

This proves 2).

d a 2d.

In case 3), clearly hypothesis.

g i d > 4'.

proving 1).

By 1) we have B5ILn-1I _ 0

by 3). This contradicts

g = 4'

Then

in case 1).

Therefore

for

£

>> 0

hq(V, tL) = 0

by for

This proves 3).

Remark.

Thus a linear basis of

generator system of

G(V, L)

H0(V, L)

in case 3).

When

becomes a d z 24 + 2,

by induction using (2.6), we can show that the relations among them are derived from those of degree two (see [F2] for details).

§4. Existence of a ladder Here we will establish sufficient conditions for the

I.0-genus and the Apollonius method

32

existence of a ladder of (V, L). Technically this is one of the most difficult part of the theory. (4.1) To begin with, let us consider the easiest case.

is smooth and

Suppose that

char(A) = 0, V

base points.

Then a general member D

D

is a rung.

of

ILI

It is connected since

by Bertini's theorem. So

ILK

L

has no is smooth is ample.

Proceeding similarly we get a ladder.

In general, the main obstruction for the existence of a BsILI, and the singularity of

ladder is the base locus When

V.

char(.) > 0, we have additional technical troubles. The result below is very useful when 4(V, L) > dim BsILJ

(4.2) Theorem.

variety

dim 0

(V, L), where

particular

d z 0

is small.

4

for any polarized

is defined to be

always, and

d = 0

implies

-1.

In

BsILI = 0.

For the proof, we use the following facts. (4.3) Theorem.

variety

Y

Let

such that

A

Bs A = 0.

the rational map defined by

member of

A.

be a linear system on a normal

A

Let

and let

W be the image of D

be a generic

Then

1) D

is irreducible if

2) D

is reduced if A

dim W k 2. is complete.

Proof. 1) is a famous result due to Zariski [Zr;p.30]. When char(.) = 0, 2) is clear by Bertini's theorem. char(.) > 0, we need to be a little bit more careful.

When See

e.g. [Fll;(2.7)] for details. (4.4) Lemma.

a variety

D1 U D2 Then

Y with

Let

D

be an effective ample divisor on

dim Y = n

and suppose that

for some proper closed subsets

dim(DI n D2) a n - 2.

Di

of

Supp(D) _ Supp(D).

§4: Existence of a ladder

33

This is clear when

Proof.

divisor is connected. When

2, since every ample

n

n > 2, take a general hyperplane

section and use induction on

See [Fl;Lemma 6.1] for

n.

details.

(4.5) Proof of (4.2), Step 1.

We will prove the

following more general assertion by induction on 4(V, A) > dim Bs A [A] = L

for any linear system

A

such that

is ample.

Here

Assume

n = 1.

v: V' - V

The normalization

n 2 2.

Hence

finite morphism.

n + Ln - (1 + dim A).

is defined to be

4(V, A)

This assertion is proved easily if

is ample and

v L

We claim Indeed, if

dim Bs A

dim Bs(v A) _

is generated by

JAI

B5IAI = A n Bs A, so

*

v A.

dim B5IAI s dimjAJ - dim A.

A

- dim A = 6 = 1, there is a member

dimJA

such that

dim BSIAI

A

and

A.

2 dim Bs A - 1

In general we use the induction on

ample.

is a

So it suffices to prove the inequality for

dim Bs A.

IAI

n = dim V.

of

Therefore since

A

is

6.

By these observations we reduce the problem to the case in which

is normal and

V

(4.6) Step 2.

homomorphism

Let

3

A

is complete (so

be the image of the natural

BsILI.

Let

n: Y -* V

V

with center

the normalization of the blowing-up of Then

be J.

is the principal ideal of an effective Cartier

a i

divisor

This ideal defines

OV(H°(V, L)) ® [-L] -> 0V.

a natural scheme structure of

*

A = ILI).

E

surjection

Y

on

such that

n(E) = BsILI

0y(H0(V, L)) 0 [-n*L] -- n*3

system M on

Y

such that

Bs M = 0

and

The

as a set.

gives a linear *

E + M = n ILI.

I. A-genus and the Apollonius method

34

p: Y -i1PN (N = dimILI)

Let

W

and let

Then

be its image.

M = p*IHI

H = 0W(1),

for

is the rational map defined by

p.n-1

and

be the morphism defined by M

We treat

ILK.

the following five cases separately. This means

(4.7) Case 1, dim BsILI = n.

4(V, L) = n + d > n

so

by the ampleness of

(4.8) Case 2, dim BsILI = n - 1

X

Let

be a general fiber of

dim W = 1.

p: Y -> W

and set

w =

Ln = LYn= L n-lE + L n-1M = L n-lE + wL n-1 X. Y Y E X is ample on n(E) and dim n(E) = n - 1, we have

L

Ln-lE > 0.

Ln-1X x 1, so

Similarly

HO(1PN00(1)) Hence

and

L.

Then

deg W.

Since

h0(V, L) = 0,

--+

h0(V, L)

HO s

The bijection

d > w.

factors through

(Y, 0(M))

H0(W, H).

Thus we have

h0(W, H) t 1 + w t d.

A(V, L) Z n, as desired. (4.9) Case 3, dim BsILI < n - 1

and

dim W = 1.

The proof is similar to that in case 2. have

Ln-lE Z 0

and

d 2 w.

Therefore

(4.10) Case 4, dim BsILI < n - 1 Let

S

be a generic member of M

corresponding member of of

V, since

V

is normal.

Hence D

D

dim BsILI So

S

ILI.

This time we

4(V, L) 2 n - 1. and

dim W > 1.

Then D = n(S) Note that

< n - 1.

D

and let

as subsets = M

(MI

1-Macaulay since

V

is normal.

So

no embedded component, and is reduced everywhere. 4(V, L) 2 4(D, ILID)

and

induction hypothesis to for

ILI.

since

is irreducible and reduced by (4.3).

is irreducible and generically reduced.

is locally

be the

BsILI = BsILID.

Moreover D

has

We have

Applying the

SLID, we get the desired inequality

35

§4: Existence of a ladder

(4.11) Case 5, dim BsILI = n - 1 Let

be a generic member of

S

corresponding member of

D = F + G Then

for

this implies

Hence

be the

D

H0(G, LG))) = 1

We have

ILIG).

by the normality of

Moreover

4(V, L) > 4(G,

F n G c BsILIG,

Since

n - 2 t dim BsILIG < 4(G,

-

and

is the fixed part of ELI.

by ( 4 . 4 ) .

dimILIG = dimILI - 1.

d(G, LG).

F

G = n*S, where

dim(Ker(H0(V, L)

Hence

M

dim W > 1.

As a Weil divisor, we have

ILI.

dim(F n G) 2 n - 2

and

V.

Ln = Ln-1F + Ln-1G > Combining these

ILIG).

4(V, L) k n, as desired.

inequalities we get

Thus we have completed the proof of (4.2). (4.12) Corollary.

very ample if Proof.

is simply generated and hence

L

d(V, L) = 0.

We use induction on

be the normalization.

n = dim V.

Let

v: Y --+ V

Then

h0(Y, L y ) a h0(V, L) and Since LY is ample, we have the

4(Y, L ) s 4(V, L) = 0. Y Any generic member equality by (4.2). irreducible and reduced by (4.3).

D

of

ILY1

Then we have

is

4(D, LD)

4(Y, L

Y ) = 0, so LD is simply generated by the induction hypothesis. Therefore LY is simply generated by (2.4). Thus

hence

) is generated by H0(Y, LY ) = H0(V, L) and Y is surjective. The injectivity G(V, L) --+ G(Y, L ) Y

G(Y, L

is clear and the assertion follows from this. Remark.

Thus

4(V, L) = 0

implies that

Actually we will classify all such pairs

(V, L)

(4.13) Next we consider the case in which finite set.

V

is normal. in §5. BsILI

By (4.2), this is indeed the case when

d = 1.

p: Y -+ W be as in (4.6), and

Let

a: Y -+ V, E, M

set

d = d(V, L), 4 = 4(V, L).

and

is a

Then we have the following.

I. A-genus and the Apollonius method

36

Proposition.

each point on when

is

has only Cohen Macaulay singularities at

V

finite and that

dim W = n, B = BsILJ

Suppose that

Suppose in addition that

B.

Then

char(s) > 0.

d k 2d - 1

has a ladder.

(V, L)

For a complete proof, see [F2;Prop.3.4] and [Fil; (3.3)].

Here we sketch the outline of it when

Let

S

be a generic member of

corresponding member of

M

Moreover

irreducible and generically reduced.

at

B, D

and let

be the

D

Then, as in (4.10), D

ILI.

reduced at any point not on

char(St) = 0.

B.

Since

D

is

is Cohen Macaulay

Therefore it is

has no embedded component.

is a rung of

D

D

reduced everywhere.

Thus

check that

satisfies the hypothesis of the

(D, LD)

is

We can

(V, L).

proposition, so we can proceed to obtain a ladder. Let the notations be as above and

(4.14) Lemma.

suppose that

point of of

ILI

point of

B

is finite, V

dim W < n, B

Then a generic member

d z 2d - 1.

and that

is a rung of

(V, L),

and

complete proof, see [Fll;(3.2)]. Y

the Chow ring of

and let D

f: V' - V Ep

is smooth at each

D

since

For a

char(s) = 0.

LE = 0

We remark that

B = n(E)

f ILI

p

be the blowing-up at a point

if and only if

for some

m > 1.

in

is a finite set.

p

be the exceptional divisor lying over p.

is singular at

part of

D

B.

Here we sketch the proof when

Let

is smooth at each

mEp

Assuming

on

Then

is in the fixed m > 1, we will

derive a contradiction. *

First of all, we show that

L'

= f L - mEp

B

is nef on

§4: Existence of a ladder

V.

Indeed, if

Z ¢ BsIL'I

is a curve on

Z

and

37

L'Z Z 0.

Therefore

E Z < 0 since [Ep]E = 0(-l). p P In either case L'Z a 0, as desired. Y

is a morphism is a part of E)n-1

L(L -

n': Y -+ V'

= Ln > 0

(0.4.7), we have

= mwEpX k mw 2 2w.

Y

Mn

=

(LY)n

=

LYMn-1

EpX a 1.

Hence

On the other hand

B.

The irreducibility of

when n > 2.

E'

with

Ep(L - Ep - E') = -Ep2 = 1,

-

LMn-1

mE*AMn

I

Combining them we get d s 2d - 1.

is smooth at each point on

D

follows from

When n = 2, we have

effective divisor

LMn-1

d =

d k 2w k 2(d - d + 1), which contradicts D

MMn-lEA

0 g 4(W, 0W(1))

n - 1 + w - h0(V, L) = w - d + d - 1.

Thus we conclude that

By

dim W = n - 1.

k

=

LMn-1

Note that

and hence

mEp

Then

= f.n'.

it

Ep = (n') Ep. on

d -

such that

*

Y, so

on

L'Z > 0.

if necessary, we may assume that there

E, where

= d - mwE*X

LZ = 0

by the normalization of the graph of the

f-1.n

rational map

Ep, then

Z c Ep = IPn-l, then

If

and

Replacing

not in

V'

E =

n'*E' = 0. w = 1.

so

dim W = n - 1

E*

p + E'

Hence

for some *

*

wEAX = EpM =

Therefore

n*(E + X) = n*X, which is irreducible, as asserted.

D = The

rest of the assertion is obvious. (4.15) Now we obtain the following. Theorem. d a 2d - 1.

Let

Suppose that

empty) and that (V, L)

V

be a polarized variety such that B = BsILI

is finite (possibly

is smooth at each point on

B.

Then

has a ladder.

Proof. (V, L)

(V, L)

(4.13) applies if

has a rung

same hypothesis as

D

dim W = n.

by (4.14) and

(V, L).

If

(D, LD)

dim W < n,

satisfies the

So we can complete the proof by

I. A-genus and the Apollonius method

38

induction on Remark.

n.

The ampleness of

fact it is enough to assume

is not necessary.

L

In

d(V, L) > 0 (see [Fll;(3.4)]).

On the other hand, the smoothness of

V

at

B

is

indispensable in general (cf. [F25]), but we have the following partial result. (4.16) Theorem.

such that that

Let

4(V, L) = 1

be a polarized variety

(V, L)

and

g(V, L) < d(V, L).

Suppose

char(.) = 0, V has only 2-Macaulay singularities (cf.

(0.2.4)) and that

point on

B.

V

Then

is locally Cohen Macaulay at each (V, L)

has a ladder. There

For the proof, consult [F25;(1.2)].

V

is

assumed to be locally Macaulay everywhere, but it is easy to generalize as above. changed.

If

n = 2, nothing need be

Indeed, if

D

n > 2, any general member

of

is a

ILI

rung by the argument in [F25].

By (0.2.6),

2-Macaulay at any point not on

B, so we get a ladder by

induction.

D

is locally

The details are left to the reader.

§5. Classification of polarized varieties of d-genus zero We will classify all the polarized varieties of d-genus zero by the Apollonius method. The result is well-known in classical geometry.

(5.1) Throughout this section

variety with

4(V, L) = 0.

ample by (4.12). Hence

g(V, L) = 0

Moreover and

L

(V, L)

is a polarized

is simply generated and very

g(V, L) k 0

Hq(V, tL) = 0

and (3.5) applies.

for any

q,

0 < q < n. Clearly

(V, L) = (1P, n 0(1))

if

d = Ln = 1.

t

with

§5: Classification of polarized varieties of A-genus zero

When Pn+l

d = 2, V

is a possibly singular hyperquadric in

L = 0V(1), since

and

39

L

is very ample.

(5.2) Now we will study the case where first we assume that

and let K

is a smooth manifold M

V

At

d s 3.

be its canonical bundle. (5.3) Claim. Proof.

h0(M, K + nL) = d - 1.

This is obvious when

general member

D

of

KD

is

[K + D]

h1(M, K + (t - 1)L) = hn-1(M, (1 - t)L) = 0 Moreover

duality and (3.5.3).

is very We have

D'

by Serre

since

IK + (t - 1)LI _ 0

(K + (n - 1)L)Ln-l = 2g - 2 < 0.

Therefore we obtain

H0(M, K + nL) = H0(D, KD + (n - 1)LD) sequence of cohomologies.

n > 1, any

L

is smooth since

ILK

Its canonical bundle

ample.

When

n = 1.

by using the exact

Thus we complete the proof by

induction.

(5.4) dim BsIK + nLI < n - 1.

D n BsIK + nLI = BsIKD + (n - 1)LDI

Since

above argument, this is proved by induction on (5.5) Claim.

unless

(F2,

(M, L)

Proof.

since

When n = 2, we have

(K + 2L)2 a 0

(K + 2L)2 = 0

This implies

by (5.4).

Moreover pg = 0

We have

since

h1(M, 0) = 0

smooth rational curve with

K2 = 9

by

C

We have

of

ILI

is

is a

K2 + c2(M)

=

by Noether's formula (cf. (0.3.9.3)), and

c2(M) = 2 + b2(M). unless

C2 > 0.

K2 s 8

Actually M

KL < 0.

a rational surface since a general member

12X(M, 0) = 12

n.

0(2)).

KL + L2 = 2g - 2 = -2.

(3.5.3).

by the

and

Since

b2 > 0, we have

b2 = 1.

(K + 2L)2 = 0

In the latter case

M = P2

by

I. A-genus and the Apollonius method

40

Moreover

the classification theory of rational surfaces. and hence

K + 2L = 0(1)

L = 0(2).

(5.6) Claim. (K + nL)2Ln-2 = 0

Let

be a general member of

S

(K + 3L)2L = (KS + 2LS)2(S) = 0

sequence

ELI. Then, by (5.5),

unless

(S, LS)

We will rule out this exceptional case.

(F2, 0(2)).

Set

n a 3.

By induction we reduce the problem to the case

Proof. n = 3.

if

H = K + U.

Then

HS = 0S(1).

By the exact

h1(M, -tL - 2H) s h1(M, -(t + 1)L - 2H)

that since s

for any

h1(S, sH) = 0

(M, L) # (F2, 0(2))

(5.7) BsIK + nLI = 0.

rational mapping f

Ln-2A1A2 > 0

Hence

and (5.6).

Moreover

A

1

This yields

Pic(M).

IK + nLI

by (5.4).

is ample.

n A2 = 0

and

is a curve.

Al, A2 of If

of the

W

A

1

IK + nLI.

n A2 # 0, then

This contradicts (5.5) BsJK + nLI _ 0.

dim W = 1.

(5.8) W = IP1

fiber X

L

since

>> 0

Hence

Moreover, the image

defined by

dim(A1 n A2) t n - 2

i

in the sequel.

Take two general members

Proof.

for

Similarly we obtain

h0(M, 2H - L) > 0. Hence L = 2H in LS2 = 4 = L3 = 8H3, which is absurd. We assume

t,

h1(M, -2H) t

h1(M, -LL - 2H) = h2(M, K + 2H + £L) = 0

h0(M, L - 2H) k h0(S, 0S) > 0.

of

and

(X, LX) - (]Pn-1, 0(1))

for every

f.

Set

w = deg W.

0 s 4(W, 0W(1)) = 1 + w - (d - 1), hence

w i d - 2.

Proof.

Then

for any

Therefore

s.

by Serre duality and the vanishing theorem.

Then

we infer

0M[-2H] -* OM[L - 2H] --' OS -> 0

0

We have

W c IPd-2

by (5.3).

§5: Classification of polarized varieties of 0-genus zero

41

On the other hand wLn-1X = (K + nL)Ln-1 = d - 2. Ln-1X = 1

and

Ln-1X = 1

implies

(X, LX)

1P1, while

since

(1Pn-1, 0(1))

ti

W

This implies

4(W, 0(1)) = 0.

Therefore

L

is

very ample.

is a locally free sheaf of rank n

(5.9) 8 = f*(0M[L])

on W by (5.8) and (0.1.11). 8, namely

scroll of

(M, L)

Moreover ti

is the

(M, L)

(F(8), H(8))

H(8)

where

is

the tautological line bundle. F1

It is well-known that every vector bundle on direct sum of line bundles (cf. (Grol]). sum factor that

Al

LZ = A1.

of

For each direct

S. there is a section

Hence

is a

of

Z

f

such

deg Al > 0.

Summing up, we now obtain the following. Let

(5.10) Theorem.

with n = dim M a 2

and

be a polarized manifold

(M, L)

4(M, L) = 0.

Then

is

(M, L)

isomorphic to

1) (1Pn, 0(1)), 2) a hyperquadric in

pn+1

with

L = 0(1),

F1, which is a direct

3) the scroll of a vector bundle on

sum of line bundles of positive degrees, or 4) (1?2, 0(2)) (this is often called the Veronese surface).

When n = d = 2, (M, L)

Remark.

be of type 3) as well as type 2).

can be considered to

Otherwise the four

classes above are disjoint.

Every scroll of an ample vector bundle actually of 4-genus zero. Roch Theorem.

only if

8

We have

8

on

F1

is

This follows from the Riemann-

d a n

and the equality holds if and

is the direct sum of

0(1)'s.

(5.11) We now study the case in which

V

is singular

I.0-genus and the Apollonius method

42

4(V, L) = 0.

and

by

We embed

V

We introduce the following notation.

ILI.

x, y

(5.12) Given two points denote the line passing them. X * Y

P = FN (N = n + d - 1)

in

X, Y

A point on it is called a

X c P, let

For any

be a linear subspace of

T

P

Ridge(X n T) _ 0

R # 0, X

such that

if

dim T =

R = 0).

See e.g. [Fl;54].

is said to be a generalized cone over

(5.14) Lemma.

X

Let

be a subvariety of

be a point of X not in

Ridge(X).

Then

The equality holds only if X

deg X - 1.

R = Ridge(X)

X = (X n T) * R.

and

The proof is elementary and easy.

x

by

This is a

N - 1 - dim R and T n R = ¢ (hence T = P

If

x e X,

X.

(5.13) Proposition.

Then

P, let

of

such that

x * X = X}.

Ridge(X) = 1.x e X

linear (possibly empty) subset.

and let

x * y

x * y

We define x * x = x and X * ¢ = X

y e Y and x # y. convention.

vertex of

P, let

For subsets

be the union of all lines

We set

on

X n T.

and let

P

deg(x * X) s

is smooth at

x.

This is well-known and easy to prove. Let

(5.15) Theorem.

for

d(V, L) = 0 unless

over a submanifold

x

Moreover h0(V, L).

of

V.

M with

Then

Then V

P

with

Sing(V) = Ridge(V)

is a generalized cone

4(M, LM) = 0.

x % Ridge(V)

dim W = n + 1

deg W < deg V - 1 So

be a subvariety of

Moreover

Suppose that

Proof.

point

L = 0V(1).

is linear.

V

V

for some singular

for W = x * V.

by (5.14) and

h0(W, °W(1))

d(W, 0(1)) < d(V, L) = 0, which is absurd.

§6: Polarized varieties of 0-genus one: First step Sing(V) = Ridge(V).

Thus we get

Next take

as in (5.13) and set

T

M

4(M, LM) = 0, for

Then

M = V n T.

is obtained by taking hyperplane

sections successively.

Ridge(M) = 0

So

implies that

M

Now the assertion is clear.

is smooth.

(5.16) Remark. (V, L)

43

(M, LM)

is determined uniquely by Theorems (5.15) and (5.10)

modulo isomorphism.

together give a complete classification of polarized varieties of d-genus zero and describe precisely their structures. They are known as 'varieties of minimal degrees' in classical projective geometry (cf. [GH], [EH]).

The

results (5.10) and (5.15) have been known from ancient times, though classical case. old ones.

is assumed to be very ample in the

L

Our argument is a little different from the

I hope it is more easily understood by those with

modern mathematical background - thanks to the power of the sheaf cohomology theory. (5.17) Corollary.

V

is normal and has only rational

Cohen Macaulay singularities if Remark. general.

The singularity of

L

V

is not Gorenstein in

Ridge(V) # 0, then

Indeed, if

generated by

4(V, L) = 0.

Pic(V)

and hence the canonical sheaf

invertible unless

is

wV

is not

d = 2.

§6. Polarized varieties of 4-genus one: First step Throughout this section let variety with

4(V, L) = 1

(6.1.1) B = BsILI

(6.1.2) if

V

and

(V, L)

n = dim V.

be a polarized Then we have:

is at most a finite set by (4.2).

is smooth at

B, then

(V, L)

has a

I. A-genus and the Apollonius method

44

ladder by (4.15).

(6.1.3) Suppose that B

point of Then

and that

(V, L)

is locally Macaulay at each

V

is locally 2-Macaulay everywhere.

V

has a ladder if

Ln = 1.

This is proved by the argument in (1.1).

(6.1.4) Let the singularity of

V be as in (6.1.3).

has a ladder if

Then, by (4.16), (V, L)

and

char(St) = 0

g(V, L) < Ln (6.2) In the rest of this section we assume that has a ladder.

(V, L)

This is not always true, but we only

have partial results without this assumption. g(V, L) = g 2 1.

assume

We also

This is in fact true if

only 2-Macaulay singularities.

V

has

The proof is based on (3.4)

and can be found in [F11;(5.2)]. By these assumptions we can apply (3.5).

In particular

we have: 1) (V, L)

has a regular ladder.

2) BsILj _ 0

if

3) g(V, L) = 1

d = Ln x 2.

and

L

(6.3) Definition.

is simply generated if

A polarized variety

d k 3.

(X, L)

is

called a Del Pezzo variety if the following conditions are satisfied: (a) 4(X, L) = 1.

(b) g(X, L) = 1. (c) X

has only Gorenstein singularities and the canonical

sheaf

wx

is

0x[(1 - n)L], where

(d) Hq(X, tL) = 0

for any

t, q

n = dim X.

with

0 < q < n.

In classical geometry, 'Del Pezzo surface' means an

I.0-genus and the Apollonius method

45

algebraic surface whose anti-canonical bundle is ample (or very ample, in ancient times).

Our Del Pezzo variety is a

natural higher dimensional polarized version of this notion. (6.4) We have the following relationships between the conditions (a) " (d) in (6.3).

1) (b) follows from (c).

2) When X

is smooth and

char(.) = 0, (d) follows

from (c) by Kodaira's vanishing theorem (0.4.11). 3) (a) follows from (C) and (d). Hence

(X, L)

may be called a Del Pezzo manifold if

is smooth, char(St) = 0

KX = (1 - n)L.

and

To prove 3), we use the method in (1.3). X(X, tL)

for

is a polynomial in

dtn/n!, where

term

0 > t > 1 - n

of degree

t

Moreover

d = Ln.

and

n

X(t) aer

with the main

X(0) = 1, X(t) = 0

X(1 - n) = (-1)n.

Therefore

X(t) = (t + 1)...(t + n - 2)(dt2 + (n - 1)dt + (n - 1)n)/n! and

h0(X, L) = X(1) = n + d - 1.

4) (c) implies (d) if

L

This implies (a).

is very ample.

The proof is technically complicated and can be found in [Fll;(5.12)].

5) (a) and (c) follow from (b) and (d). For a proof, see [F11;(5.9)].

6) (c) and (d) follow from (a) and (b) if (X, L)

d z 3

and

has a ladder.

Indeed, we get (d) by (3.5.3) and then use

5) above.

7) (c) and (d) follow from (a) and (b) if x locally 2-Macaulay and if

X

(X, L)

has a ladder.

is

I. A-genus and the Apollonius method

46

The proof is complicated, but is found in [Fll;(5.11)].

A polarized variety

(6.5) Corollary.

4(X, L) = g(X, L) = 1

with

(X, L)

is a Del Pezzo variety if one of the

following conditions is satisfied. 1) L

is very ample.

2) X

is smooth.

3) char(.) = 0

(X, L)

is locally Cohen Macaulay.

In case 1) we have

Proof.

In case 2),

X

and

(X, L)

d z 3, so (6.4.6) applies.

has a ladder by (4.15).

has a ladder by (6.1.3) or (6.1.4).

In case 3),

So (6.4.7)

applies in these cases.

(6.6) Here we provide a 'cheaper' proof of (6.5.2) in the case where induction on

A

ILI

C'

By (6.2) any general member

n = 2.

is smooth if

Indeed, if

This is true even if

d > 1.

is another general member, then

a finite set by (4.2).

Since

C

C'

by Bertini's theorem.

C n C'

is

Hence

C

Thus

C

is a smooth elliptic curve

d.

We claim while the map

bijective.

Indeed, otherwise, b1(X) 2 2

H1(X, 0) = 0.

h: H1(C; Z) - H1(X; Z)

Lefschetz theorem.

Since

Therefore

is surjective by the

H1(C; Z) = Z ® Z, h

Alb(C) = Alb(X).

fiber of the Albanese map

a: X --* Alb(X).

1P1

for any general

F

must be

F

be any

Then

F n C

Let

a simple point by the above reasoning and thus

F ti

d = 1.

B = C n C' = BsILI, so it is smooth everywhere

is smooth at

So

C

= d = 1, this is a

simple point and the intersection is transverse.

for any

We use

C.

n.

Suppose first of

is the complex number field

is

FL = FC = 1.

with F n BSILI = 0.

More-

§6: Polarized varieties of A-genus one: First step

47

over every fiber is irreducible and reduced since ample.

From this we infer that

(X, L)

is the scroll of the vector bundle However, since

Alb(X) = C.

C1(8) = L2

and

our assumption and we conclude

(K + L)L = 2g - 2 = 0

,

H0(X, K + L)

K + L = 0

Pic(X).

in

and

on

h0(X, L)

This contradicts

H1(X, 0) = 0.

and by Serre duality we infer

H1(X, K)

d = a*OX[L]

is ample, we have

8

is

is a 1P1-bundle, and

a

4(X, L) = 2 (cf. (At)).

By the exactness of

L

H0(C, K(C)) -+ IK + LI

Since

# 0.

is ample, this implies

L

Thus we prove (6.3.c), and by

Kodaira's vanishing theorem we get (6.3.d). n > 2.

Now we consider the case member of

D

be a general

By (6.2.2) and by (4.14), we infer that

ILI.

We can easily check that

is smooth.

Let

(D, LD)

D

satisfies the

conditions (a) and (b) in (6.3), so it is a Del Pezzo manifold by the induction hypothesis. the sequence --+

H0(X, K + (n - 1)L) --* H0(D, K(D) + (n - 2)L)

H1(X, K + (n - 2)L)

we get

IK + (n - 1)LI n = 2.

case

By the exactness of

and by Kodaira's vanishing theorem, # 0.

This implies (6.3.c) as in the

We obtain (6.3.d) by the vanishing theorem.

(6.7) Let

(V, L)

be as in (6.2) and we will study

dividing the cases according to the value of

(V, L)

Suppose first

d = 3.

is very ample by (6.2.3).

Then Hence

h0(V, L) = n + 2 V c Pn+l

and

(6.8) Suppose that (6.2.3). one.

So

d = 4.

A one-dimensional rung (Vi, L)

Again V1

L

and

L

is a hypercubic.

Any polarized manifold (M, L) n+1. 4(M, L) = 1 is a hypercubic in Corollary.

Ln.

with

d = 3

is very ample by

is of arithmetic genus

is a complete intersection of type (2, 2)

I. A-genus and the Apollonius method

48 in

By induction using (2.6), we infer that

1P3.

(V, L)

is a complete intersection of two hyperquadrics in Any polarized manifold

Corollary.

in

with

(M, L)

d = 4

is a complete intersection of type

4(M, L) = 1

and

Fn+2

(2, 2)

Ipn+l. is very ample also in the case

(6.9) L

Moreover

d > 4.

is a Del Pezzo variety by (6.4.6).

(V, L)

is never a complete intersection.

But it

The classification of

such varieties will be treated in §8 and §9. (6.10) Now we study the case

p: V -,Fn with

by (6.2.2), we have a morphism Clearly

Since

d = 2.

p 0(1) = L.

is a finite morphism of degree two.

p

there is no upper bound for

g(V, L)

BsILI = 0

Note that

in this case.

(6.11) We study the structure of such a double covering

f: X -+ Y

OY-algebra and

a sheaf of

f

Suppose that

Note that

char(.) # 2.

assuming

d = f*OX

is

X = Jfte-c(d).

Then

is flat.

d

is locally free of

rank two.

So we have a trace homomorphism

t: d -> 0Y.

Then

is twice the identity, where

is the natural

t.t

homomorphism d = 0y ® A cation of

OY

A = Ker(t).

gives a homomorphism

algebra structure of this homomorphism = 0Y[-F], P

char(&) # 2, this implies

0Y-modules where

as

4

Since

d.

f.

The

B

of

12FI.

F

A

such that

and

A

This is the

X is determined by the triple (Y, F, B)

up to an isomorphism over

will be denoted by

No2 , 0Y.

For the line bundle

defines a member

branch locus of

A:

The multipli-

is determined by the sheaf

4

0.

c

Y.

RB,F(Y).

Hence such a double covering Since

B E 12FI, F

is often

49

§6: Polarized varieties of A-genus one: First step

determined by

is in fact flat if

f

Remark.

In such a case we write

B.

RB(Y)

also.

is smooth and if

Y

X

has only Cohen Macaulay singularities (cf. (0.1.12)). (6.12) We describe the structure of precisely by using local coordinates. open set C

of

Y

of

U

such that

An

OY-base

gives a fiber coordinate of the total space of

'NU

the line bundle

On the other hand

FU.

.NU c dU

So we have a natural map

C E H0(f-1(U), OX). over

Take a small enough

is trivial.

.NU

more

RB F(Y)

and hence

f-1(U)

- FU

This is an embedding onto the image, which is

Y.

defined by

C2 = p, where

is the function

q,

on U.

j4(CeC)

The above local structures are patched globally as Let

follows.

on

C2 = fAlt CY

defining ('2

Let

as above.

U2

E H0(U2, OY)

the divisors

2

be a fiber coordinate of

B

Then

FU . 2

(f2p)

U2 n Uµ, where Let

F.

T2 = f2'U 2P µ

be a covering by small open sets

Y = U2U2

be defined by

is a 1-cocycle T E H0(Y, 2F)

and let

be the corresponding function such that U2 n Uµ

on

Then X

X2 = (C22 = P2)

in

is obtained by patching FU . 2

From these observations we infer:

1) The relative canonical bundle 2) If

K(X/Y)

is

f F.

is smooth, X has only hypersurface singularities

Y

and they lie exactly over the singular points of (6.13) Corollary.

with

d(V, L) = 1

ladder,

Then 2g + 2

V

(V, L)

d = 2.

be a polarized variety

Suppose that

is locally Cohen Macaulay and

V = RB(IPn)

and

and

Let

L

for some hypersurface is the pull-back of

B.

B

0(1).

has a

(V, L)

char(A) # 2.

of degree (V, L)

is a

I. A-genus and the Apollonius method

50

weighted hypersurface of degree projective space

1P(2, 1,

V

1).

,

over the singular locus of

2g + 2

is singular exactly

B.

(6.14) Now we consider the case

D1, , Dn

members

of

a finite set by (4.2).

We get a ladder by setting n: V' -, V

n L - E = H, where

that P.

E

Since

1P n-1 and

isomorphism and E on

V'.

is the line bundle

is ample on

X

cally reduced.

point

x

of

ladder of

p.

V'

X

If

such

0(1)

on

is an

Thus, for every

x

1

X = p- (X).

for

i

EX

>> 0.

Since

is ample on

X.

is irreducible and generi-

is everywhere reduced for any general

since it is the one-dimensional rung of a Clearly

(V, L).

Remark.

flat and

P

X

be

1P n-1

[L - E]E = 0(l), PE: E --i P

Moreover

dim X = 1.

E

and

P

p: V'

LX - EX = [H]X = 0, this implies that Hence

B, let

In L - El

is a simple point for

iL - E

Recall that

n Dn-J.

Then D a

is a section of

P, E n X = EX

B.

and let D be the proper

B

H

is

is smooth at

V

be the blowing-up at

Hence we have a morphism

_ 0.

(1 D'.

on

DJ

So

n Dn

is a simple point

VJ = D1 n

the exceptional divisor over transform of

B = D1 n

Ln = 1, B

Since

Take general

d = 1.

Then

ILI.

and the intersection is transverse.

Let

in the weighted

V

g(X) = g(V, L).

is locally Cohen Macaulay, then

is locally Macaulay for every fiber

X.

f

is

So

X

has no embedded component and hence is reduced everywhere. (6.15) Now we assume that Macaulay.

V

is locally Cohen

In subsections before (6.18), we also assume

g(V, L) = 1. Set

9 = OV,[2n*L]

and

Y = p*9.

Then

h0(Xx, 9 x)

2

§6: Polarized varieties of 0-genus one: First step

for every fiber

and X

Xx

over

x E P

51

deg 0x = 2LX = 2

since

is of arithmetic genus one. Moreover

by global sections.

Hence, by (0.1.11),

surjective.

morphism

f: V'

T, we have a

f H _ 9 (cf. (0.1.4)).

such that

---' W

is

p 3" -+ 0

is the scroll of

(W, H

So, if

is locally free

3'

of rank two and the natural homomorphism

is spanned

9x

fx

This is a finite double covering since the restiction

is so over every

Xx -> Wx = F1

The natural homomorphism

on

x

P.

p*OV,[2L - E] - p*OE[2L - E]

is a zero map since

E n X

for every fiber

Therefore, using the exact sequence

X.

is the base point of 1[2L - E]XI

0 -+ OV, [2L - 2E] --p OV, [2L - E) -+ OE[2L - E] -> 0,

we get

P*OV,[2L - E] = p*OV,[2L - 2E] ti p*OV, 0 [2H] ti OP[2H ].

Next we use 0 -+ OV , [ 2L - E] -+ 0 - OE -* 0.

We have

R1p*OV,[2L - E] = 0

for every

fiber so

0 - OP[2H ] -+ 3 -+ Op -

Hence

Xx.

S = 0P(2) a Op

since

(6.16) Now we assume (6.11).

Then

We have

S c B

V'

*

and is

IHC - 2p HEI

p: W -+ P.

a section of

char(.) # 2

RB,F(W)

V'

S

so that we can apply B E

for some

since f: x Xx -. W x

is smooth along

B = S + B1

is exact,

0

Note that

H1(P, 0(2)) = 0.

is the unique member of

S = f(E)

B

HI(Xx, 2L - E) = 0

since

since

V'

12FI, F e Pic(W).

is smooth along

for some other component

B1

with

generated by

H.

[B1] = bH

and p H .

H

+

S n B1 = 0.

Then

c = 0

since

general

[B] = (H. - 2HE) + 3HC

x.

So

since

is

[B1]S = 0.

is ramified at four points for

Moreover b = 3

fx

B1.

Pic(W)

since

Hence

E.

is singular exactly over the singular locus of We may set

E n Xx

is ramified at

and

F = 2HC -

H.

I. A-genus and the Apollonius method

52

Thus the structure of

Conversely, for any member

explicitly.

® 0)

W = 1P(2H

(V, L)

is described very

(V, L)

B1

RS+B1,2HCH (W)

S.

(6.17) Remark.

In the above case (6.16),

weighted hypersurface of degree six in

..., 1).

using (2.7).

e(V) + n - 1 = e(V') = 2e(W) - e(S) - e(B1).

is a divisor in

W, so the total Chern class is computed

as in the case of hypersurfaces in obtain

is a

is smooth, the topological Euler number can be

V

computed by B1

(V, L)

1P(3, 2, 1,

n

This can be proved by induction on If

on

as above, we obtain a polarized variety

by blowing down the divisor on

lying over

13Hj

of

when n = 3.

e(V) = -38

1PN.

In particular, we

This is used in

[F5-1] to

prove the following fact:

M be a smooth threefold such that

Let

ample and the cohomology ring H(M; Z)

H (1P3; Z) = Z(h]/h4.

is isomorphic to

Then M = F3.

This result is generalized in the case (cf. [F6],

[L.

is

-K(M)

n = dim M g 6

W]).

(6.18) Now we study the case

g = g(V, L) s 2.

Unfortunately the general case is difficult and we do not have a complete answer, even if sketch our result when every fiber

Xx

p

V

is smooth.

Here we

is hyperelliptic, which means,

is a double covering of

F1.

See [F5-11I]

for details.

(6.19) Let

restriction 8 = p*w

wx

w to

be the canonical sheaf of

Xx

V'.

is the canonical sheaf of

is locally free of rank

g

and

p 6 -, w

Then the

X

x

is

§6: Polarized varieties of A-genus one: First step surjective.

We have a morphism

f H(s) = w.

Its restriction

canonical map of

fx: Xx -*IPg-1

that

is hyperelliptic.

p

is a 1P1-bundle over P = 1Pn-l

is a finite double covering.

f: V' --i W

OP[eH ]

is the

This is a double covering onto a

Xx.

Hence the image W = f(V') a section of

such that

f: V1 --,IP(6)

Veronese curve by the assumption that

and

53

W = F(Y)

0, where

and

is

and we have an exact sequence

p: W --> P

S --> Op -

S = f(E)

is a vector bundle such

Y

is the integer with

e

0 -+

[S] S = 0(-e).

Thus we are in a similar situation to the case where g = 1, but this time there are several possibilities since f

is not always ramified over (6.20) Now we assume B e

for some

S.

Then

char(A) # 2.

V' = RB F(W)

We have three possibilities:

12FI.

'

a) S c B.

b)SnB=0. c) S ¢ B and S n B# 0. (6.21) Case a).

B = S + B1

for some member

V

S n B1 = 0. of

B1.

Similarly as in (6.16), e = 2 of

B1

such that

1(2g + 1)H,l

is singular exactly over the singular locus

(V, L) is a weighted hypersurface of degree

in IP(2g+1, 2,

e = 1

and

*

effective divisor addition to

4g + 2

1, , 1).

(6.22) Case b).

E

such that

E, we can blow down

points by (0.1.6).

double covering

f

z ¢ B

*

E*

f S = E +

for some

*

E n E

_ 0, E

*

In

= S.

*

and

E

S

to smooth

By these blow-downs we get a finite V1'

- W'

phically onto a hypersurface such that

and

where

z

ti

Bb

IPC.

B

is mapped isomor-

of degree

2g + 2

is the image point of

on S.

Fn So

I. A-genus and the Apollonius method

54

V

is the blowing-up of

z

and the exceptional divisor is identified with

have

L = [H.]V - E .

at a point

RB,(ff'n)

Thus

zl

lying over We

E .

is a birational

(V, L)

modification of a polarized variety of the type in (6.13).

if

n a g + 2, there is a line

in

£

that the intersection multiplicity of point on

is even.

i n Bb

where the smoothness of

zl a Z1.

the proper transform the ampleness of

that

Bb

at each

LZi = (H. - E )Zi t 0 V

on

such that HCZ = 1.

of

Z1.

for

This contradicts

The arguments in [F5-E;§18] apply at

is smooth at any point on

B

1(2g + 2)Ha + 2bH,l

H(kH ® 0)

bundle

Z1, Z2

B n S.

n = 2, P = Ek = IP(kH$ ® 0), S e

particular B E

and

such

z

L.

(6.23) Case c).

first, since

£

passing

Now, by the evenness,

Then

Zi

Fn

Indeed,

is assumed, but the proof is

B

has two components

We may assume

g + 1.

This is Lemma (17.5) in [F5-III],

valid without this assumption. (fb)-1(j)

n

(V, L) exists only if

Such a pair

on

where

Ek

Ha

JHa + aH,I

and

is the tautological line

b, a

and

In

are integers such

-e = k + 2a, gk + (g - 1)a + b = 1.

Moreover

a 2 0.

However, unlike the case in [F5-III;(l8.5)], we do not have

2b 2 -k

since some big multiple of the unique member may possibly be contained in

JHa - kH,)

In this exceptional case C n S = 0

and

B

B.

is singular along

B

So

is reduced (or equivalently if

k > 0.

V

is normal),

we can rule out the above possibility and we obtain b = 1, a = 0

C.

a = 0, but we cannot go further in general.

Such an example does really exist even if If

of

C

and

e = 0

as in [F5-IQ;§18].

k = Of

Therefore

§7: Results of Lefschetz type X F ,

P = 1P

P -+1P1 a

V

Thus

is

55

B E 1(2g + 2)Ha + 2H

j

and

is the fiber of

S

over a branch point of the double cover is rational since any general fiber of

B -*1P1.

V' - 1P

1P

(6.24) Problem.

Study the case in which

A = d = 1

is not a hyperelliptic fibration.

and

p

§7.

Results of Lefschetz type We review several useful results which relate the

structure of

V

and

when

D

is a rung of

D

To

(V, L).

begin with, recall the following classical theorems. (7.1) Theorem.

Let

A

be an effective ample divisor

on a compact complex manifold M with A -- M be the inclusion. 1) wi(t): :ti (A) -i ai(M) a n-1(t)

is bijective for

is bijective for

i < n - 1

is bijective for

i < n - 1,

is injective and its cokernel is torsion free.

4) Hp'q(t): Hq(M, QM) - Hq(A, QA) if

A

is bijective for

is smooth or if p = 0.

tive for p + q = n - 1 A

and

is surjective.

3) H1(t): H1(M; Z) -* H1(A; Z)

< n - 1

i < n - 1

is surjective.

Hn-1(t)

Hn-1(t)

t:

Then

2) Hi(t): Hi(A; Z) -i Hi(M; Z) and

Let

dim M = n.

HP q(,)

p + q

is injec-

under the same condition.

is assumed to be smooth in the following assertions.

5) Pic(t): Pic(M) --' Pic(A)

is bijective if

n > 3.

This

is injective and its cokernel is torsion free if n = 3.

6) Pic(t): Pic0(M)

Pic0(A)

is an isomorphism if

n > 2

and is injective if n = 2. 7) Alb(t): Alb(A) -+ Alb(M) is an isomorphism if n > 2.

It

I.0-genus and the Apollonius method

56

is surjective and its general fiber is connected if

n = 2.

Outline of proof. Using Morse theory [Mil] we prove 1). 2) and 3) follow from 1).

A

Hodge theory yields 4) if

is

When p = 0, we use Kodaira's vanishing theorem.

smooth.

5) and 6) follow from 3) and 4). Remark.

7) follows from 2) and 4).

There are several generalizations of the above

results (see [Gro2], [Gro3], [Ha3], [Badl]).

(7.2) The structure of a variety is often described by a morphism defined on it having a special property.

Such a

morphism can be defined by the pull-back of a very ample linear system on its image.

relation between

V

Thus, in order to study the

D E ILI, we need various extension

and

theorems for morphisms, linear systems and line bundles.

For this, several vanishing theorems are very important for the technical background. found in [Sol].

Such a systematic approach is

We review here some slightly improved

versions of results in [Sol]. (7.3) Lemma (cf. [Sol;Lemma IB] & [F4;(2.1)]). and let

be an ample effective divisor on a scheme S a coherent sheaf on where

3"(t)

such that any

t > 0.

denotes

S

such that F ® 0S[tA].

Let

be

be an integer

i 1 0

is surjective for

Hi+1(S, 3) = 0.

For the proof, use the exact sequence Hi(A

T

A

Ker(Y - FA) = 3'(-l),

H1(S, F(t)) - H1(A, 3"(t)A) Then

Let

Y(t)A) -* Hi+l(S, 3(t - 1))

H1(S, 3'(t)) -*

Hi+l(S, S(t))

and

Serre's vanishing theorem as in (3.5.3). (7.4) Lemma (cf. [F4;(2.2)]).

ample divisor on a locally

Let

A

be an effective

k-Macaulay variety

V

and let

§7: Results of Lefschetz type

6

57

be a locally free sheaf on i < k

such that Then

and

Let

V.

be an integer

i

H1(A, S(t)A) = 0

for any

t g 0.

H1(V, 6) = 0.

H1(V, d(-i)) = 0

Proof.

for

by duality and by

>> 0

i

Using the exact sequence

Serre's vanishing theorem.

we prove

H1(V, 8(-t-1)) --b H1(V, 6(-t)) --, H1(A, 6(-t)A),

the assertion by similar arguments.

A

Let

(7.5) Lemma.

be an ample effective divisor on

a locally

2-Macaulay variety

bundle on

V

H1(A, (F - tA]A) = 0

such that

H0(V, F) -> HO(A, FA)

Then

t 2 1.

that the conclusion is equivalent to

Proof.

ILAI.

ILIA =

t 2 1.

Remark.

FA = 0

Suppose that

h0(V, F)

and similarly

if

Note

Then

Pic(V) -+

is injective.

Pic(A)

Then

by (7.4).

be as in (7.5) and suppose

for any

HI(A, [-tA]A) = 0

that

V, A

Let

for any

is surjective.

H1(V, F - A) = 0

This is clear since

(7.6) Corollary.

F be a line

and let

V

2

h0(A, FA) = 1

I-FI

by (7.5).

F E Pic(V).

Hence

# 0,

IFI

F = 0.

So

# 0.

for some

The assumption of this corollary is satisfied and if

char(.) = 0, dim V > 2

A

is normal.

However,

unlike the case in (7.1.5), the cokernel is not always torsion free, even if (7.7) Lemma.

dim V > 3.

Let

and

S

A

be as in (7.3) and let

be a linear system of Cartier divisors on

S

such that

dim pA(A) < dim A

A n BsA = 0.

Suppose that

morphism

pA

defined by

AA.

morphism

PA

defined by

A

Then

BsA = 0

for the and the

is an extension of

PA

such

A

I. A-genus and the Apollonius method

58 that

PA(V) =

PA

(A).

This is proved by a similar argument to that in (5.7). See [F4;(2.7)] for details. Let

(7.8) Corollary.

A

be an ample effective divisor

on a locally 2-Macaulay variety Let

F be a line bundle on

V

V

with

n = dim V a 3. BsIFAI = 0,

such that

Fn-lA = 0

and

H1(A, [F - tAJA) = 0

BsIFI = 0

and

PIFI(V) = PIFI(A).

for any

Then

t > 0.

For the proof, use (7.5) and (7.7). Let

(7.9) Corollary.

assume in addition that

A

A

and

V

be as in (7.8) and

is smooth and

char(A) = 0.

Let

f: A --' W be a surjective morphism onto a projective scheme

W with

dim W g n - 3.

FA = f H for some

Suppose that

H on

and some ample line bundle

F E Pic(V)

W.

Then f

V - W.

extends to a morphism Outline of proof.

By an argument using Kodaira's

vanishing theorem and Leray spectral sequence, we infer that H1(A, F - tA) = 0

for any

t > 0, so (7.8) applies.

See

[F4;(2.9)] for details. (7.10) Corollary.

and suppose that morphism

V

Proof.

V

Let

is smooth.

Then

f

extends to a

W.

We may assume

For a very ample line bundle such that

and W be as in (7.9)

A, V, f

dim W > 0 H

on

W,

and hence

we have

n > 3.

F E Pic(V)

FA = f H by (7.1.5), so (7.9) applies.

(7.11) Remark.

We can sometimes extend a morphism by a

similar method using vector bundles instead of line bundles,

since a vector bundle generated by its global sections gives

§7: Results of Lefschetz type

59

a morphism to Grassmann varieties.

See [F3,7] & [Mukl,2].

(7.12) Now we give another application of (7.5). Lemma (cf. [F4;(3.5)]). A E JaLl

variety and let is locally

projective space

,

in

dr)

V

is a weighted complete d

,

r

in the weighted

)

and that

, wn+r)

]P(a, w1,

H1(A, tLA) = 0

Proof. so

F(wl,

Suppose that

a > 0.

n = dim V 1 3.

is a weighted complete intersection of type

(V, L)

(dl,

(dl,

be a polarized

(V, L)

for some

2-Macaulay, (A, LA)

intersection of type

Then

Let

H0(V, tL) -f H0(A, tLA)

wn+r).

,

for any integer

by [Morl],

t

is surjective by (7.5).

Hence

(2.6) applies.

(7.13) Corollary. Let

A be an ample effective divisor

on a locally 2-Macaulay variety (.n-1,

that

0(1))

n = 2

some integer =

When

and

F e Pic(V).

F = [A].

Suppose that

Proof.

A n-1{A)

for some

Then

a.

Hence

a nFn.

n = 2

and

Then

H1(V, 0V) = 0.

a = Fn = 1

F = [A], V V

Hence

This implies

n > 2, or

Suppose that

Set

[A]A = 0(a)

by (7.6), so

[A] = aF

larities since it is smooth along so

(A, FA)

(V, F) = (Ipn, 0(1)).

n > 2.

Moreover

Noether's criterion.

such that

V

for

an-1

=

and (7.12) applies.

is a rational surface by has only isolated singuA.

Hence

V

is normal,

4(V, F) = 4(A, FA) = 0.

(V, F) = (]P2, 0(1)).

(7.14) Now we consider the following problem: Let

A be an ample effective divisor on a locally

2-Macaulay variety vector bundle

8

V

and suppose that

on a smooth variety

an exact sequence 0 -> q

9'

A = ]P(8) W.

for some

Does there exist

---> 8 -* 0 of vector bundles

I. A-genus and the Apollonius method

60

on

such that (V, A) = (IP(Y), F(6)) ?

W

(7.14.1) Remark.

under one of the following hypotheses:

Yes

is

extends to a morphism

is locally Cohen Macaulay, the answer

V

and if

f: V -i W

fA: A -+ W

If

a) [A]A

Ax

for any fiber

= 0(1)

of

fA.

x

b) r = rank(s) 2 3

and

Indeed, for any

x E W, Ax = fAl(x)

divisor on the locally Macaulay variety (V

X,

[Ax]) = (Fr00(1))

(8 ® 1V

is an ample Vx = f-1(x).

Hence

by (7.13).

Then

3 E Pic(W).

for some

is surjective.

Pic(V) --+ Pic(Ax)

So

(fA)*0A[A] =

Y = q 0 f*0V[A]

has

the desired property.

(7.14.2) We have the affirmitive answer if smooth, char(t) = 0 Indeed, fA tion map

and if

extends to

Pic(V) -* Pic(A)

V

is

r = rank(s) k 3.

f

by (7.10) and the restricHence

is surjective by (7.1.5).

(7.14.1.b) applies.

(7.14.3) When

r = 2, the problem is more subtle, but

we have several partial results [Bad2-5], [FaSS), [Fas2l, [SatS], [Satl].

(7.15) Now we consider the extension of blow-down. Lemma (cf. [F4;(5.5)]).

Let

A

be an ample effective

divisor on a compact complex manifold

Suppose that

M.

is the blowing-up of a projective manifold C

which is a submanifold of B

M

center

and

bundle over

r k 3.

as a divisor

is isomorphic to the blowing-up of N with

such that

Proof.

with center

of codimension

Then there is a manifold N containing B

C

B

A

A

is the strict transform of

The exceptional divisor C.

E

on

B

A

on is a

N. IPr-1

Using the Leray spectral sequence we obtain

§7: Results of Lefschetz type

H(E, [E - tA]E) = 0 H1(A, [-tA])

-

61

for any

H1(A, [E - tA]) - H1(E, [E - tA]E)

We have

t > 0.

Moreover

(7.1.5).

E = DA

E -+ C

very ample line bundle H L E Pic(M)

over we get

on

LE

by (7.1.5).

a

x

HA = LA

for some

for any

Hence

t > 0

E --> C

Ex

and

C

on

by using

extends to

is a hyperplane in

Since

C.

More-

HC.

Applying (7.14.1.b), we infer that

Fr- bundle over

over any point

D, we take a

is the pull-back of

H1(E, LE - tA) = 0

by (7.8).

Then

by

by (7.5).

IFI

extends to

B.

the Leray spectral sequence.

D -* C

D E

for some

Now,, in order to show that

F E Pic(M)

for some

[E] A = FA

and by

H1(A, [E - tA]) = 0

Kodaira's vanishing theorem, we get for

By the exact sequence

t k 0.

D

is

Dx = Fr

= 0(-1), we have

[E]E

x

= 0(-l).

[D]D

Hence, by (0.1.6),

D

can be blown down

x

smoothly along the direction manifold

N

and we obtain a

as desired.

Remark. r = 2.

D --> C

There are several partial results in the case

See [Fa],

[So5], [FaSl].

(7.16) The divisor

on N

B

is not always ample in

case (7.15), but we have the following criterion. Lemma.

Let

M

along a submanifold HC

such that

M

for some

over

C.

be the blowing-up of a manifold N Let

C.

Suppose that

is ample.

y > 0, where

Then H

H be a line bundle on N

E

is ample on

B

is ample on

is the exceptional divisor N.

For a proof, see [F4;(5.7)].

situation (7.15), then

ftHM - E

If

C

is a point in the

is ample by this criterion.

(7.17) We have seen that various conditions are imposed

I.0-genus and the Apollonius method

62

on the structure of such that variety.

(V, L)

if

ILI

contains a member

D

is isomorphic to a given polarized

(D, LD)

Sometimes these conditions turn out to be so

restrictive that no such pair

(V, L)

exists where

V

is

smooth, or even no normal pair at all, except the cone over (D, LD).

See

[Sol], [Bad2], [F9, 14], [Lviti3].

It often happens that, because of such restrictions,

the induced polarization for a given variety

LD

must be of very special type

D, provided

V

In his

is smooth.

series of papers [Badl "S. 4], Badescu observed such phenomena

for a number of varieties Fn-1,

1Ps x ]pt, IP1-bundles

D, for example, in the cases over curves, etc.

D =

Here we just

present the following sample result. (7.18) Theorem.

fold M D

Let

D = 1Pn-1

as an ample divisor.

be contained in a maniM .

If n > 2, then

n

and

is a hyperplane on it. Indeed, if

A = C, Pic(M) --* Pic(D) = Z

by (7.1.5), so (7.13) applies.

is surjective

By Grothendieck's Lefschetz

theorem (see e.g. [Gro3], [Ha3]), the same argument works unless

n = 3 = char(.).

This final exceptional case was

proved by Tango [Ta] using a direct approach, and later by Badescu [Bad2] using techniques of lifting to char(.) = 0. See (8.12) ti (8.15) for the details of this method.

§8. Classification of Del Pezzo manifolds Throughout this section let (M, L) be a smooth Del Pezzo variety as in (6.3) with n = dim M a 2 and d = Ln. (8.1) First we review the classical result concerning the case

n = 2.

if and only if

A polarized surface

K = -L

(S, L)

is Del Pezzo

for the canonical bundle

K.

63

§8: Classification of Del Pezzo manifolds If S'

S

by contracting

L' = L + E

E

to a point.

comes from

Pic(S')

for the canonical bundle ample by (7.16).

we have If

E, we get another surface

contains a (-1)-curve

Thus

K'

and

of

LE = -KE = 1,

Since

K' = -L'

Moreover

S'.

(L') 2 = L2 + 1.

S

is relatively minimal, then

or a Hirzebruch surface

Z'1

S = L'0.

or

S = IP2

S

S = Z0 z IP1

by the ampleness of

From these observations we infer that IP2

is

L'

is a Del Pezzo surface and

(S', L')

IP1-bundle over a curve. In the latter case

from

Pic(S')

in

is a X

F1

-K.

is obtained

S

by blowing up several points successively, unless Note that these points cannot be infinitely near

since any Del Pezzo surface contains no (-2)-curve. F2 and no six of Similarly no three points are colinear on them lie on a conic.

Note that every (-1)-curve

E

namely a smooth rational curve with

on

S

is a 'line',

LE = 1.

Conversely,

any such 'line' is a (-1)-curve.

(8.2) In the rest of this section, we study the case n k 3.

We assume

d x 5

in §6.

Therefore

L

lifting. [Fnlti3],

was studied

d s 4

is simply generated and very ample.

We first consider the case (8.12) to the case

since the case

char(St) = 0, and proceed in

char(3t) > 0

by using the technique of

The results are known to Fano and Iskovskih (cf.

[Isl, 2]), at least when n = 3

and

char(.) = 0.

Their method is based on the observation that a Del Pezzo 3-fold contains many lines and is covered by them. approach here is slightly different from theirs. (8.3) Claim.

d ;g

8.

Our

I. A-genus and the Apollonius method

64

Assuming

d k 9, we will derive a contradiction.

Taking general members of problem to the case

Then a general member

n = 3.

is a Del Pezzo surface.

ILI

d a 9.

since

Then any general member

n = 3.

S

The restriction mapping

is injective by the Lefschetz Theorem

r: Pic(M) ---, Pic(S)

In order to study

(7.1.5).

of

By (8.1), (S, L) = (IP233H)

is a Del Pezzo surface.

ILI

S

This contradicts (7.18).

(8.4) Suppose that of

if necessary, we reduce the

ILI

Im(r), we follow the argument

in [F5].

Recall that P

M c IPd+1

since

L

is very ample.

be the space parametrizing hyperplanes in

Let and set

IPd+l

P*

((x, h) E M X

F

corresponding to

x e Hh), where

I

h e P*.

Then

F

Hh

is the hyperplane

is a IPd-bundle over

M

P*

f: F --

and the fibers of

be the open subset of

B e Pic(S)

If

such that

is

be the image

comes from

and hence is

Pic(F)

G

S = f- 1 (0).

U

Let

be the monodromy

Im(ju).

Pic(M), then

comes via

B

On the other hand, if

G-invariant.

G-invariant, then

Let

ILI.

parametrizing smooth members of

n1(U, o) - Aut(H2(S; Z)) = Aut(Pic(S))

representation and let

B

are members of

o E U

and take a point

ILI /.c:

P

*

C1(B)@ E H2(S; Q)

comes from

H2 (F; IQ)

by Deligne's invariant cycle theorem [D;(4.1.1)].

H2(F; Q)

is generated by

0 *(l)

and

H2(M; Q)

since

F

P

is a

IPd-bundle over

H2(M; Q)

M.

since the restriction of

comes from

c1(B)Q

Therefore 0

(1)

to

S

is

P

trivial.

B E Im(r)

Hence

B

is mapped to a torsion in

Coker(r), so

by (7.1.5).

Thus, B E Im(r)

if and only if

B

is

G-invariant.

§8: Classification of Del Pezzo manifolds

65

In particular the canonical bundle of Moreover

G.

is stabilized by

S

preserves the intersection products of

G

Pic(S).

(8.5) Suppose now that case 1

A general member

n = 3.

We first consider the

d = 8.

P1

S

of

E

be the (-1)-curve on

is

ILI

1P1

x

by (8.1).

1

Assume that

Let

S = E1.

is the unique element in

[E]

Therefore

[E]

in (8.4).

Hence

Pic(S)

is invariant under the monodromy action as for some

= FS

[E]

F E Pic(M).

It is

H1(S, E - tLS) = 0

for any

E = DS

D E

Therefore we have

for some

(D, LD) = (1P200(1))

JFI

by (7.5).

and

[D] D = 0(-l)

(L + D)3 = (L + D)2L = (L + D)L2 = 9. where

[L + D] S = 3HS

the birational map Pic(M), so

L + D = 3H

HS = Ha + Hp

(M, H)

ti

of

S = F1 x F1.

and K = -2L = -4H

ruled out, and

Therefore

in

by (7.1.5).

More-

Therefore

Pic(M).

by (1.3). n = 4.

This contradicts (7.18).

n > 4

A general member Hence

(T, LT)

This case is thus

cannot occur either.

(8.6) Suppose that n = 3.

via

LS = 2Ha + 2H

is a Del Pezzo 3-fold as above.

(F3, 0(2)).

0(1)

comes from

HS

Then

H E Pic(M)

for some

(1P3, 0(1))

ELI

On the other hand

H E Pic(M).

Secondly we consider the case T

Hence

which is absurd.

Thus we conclude

L = 2H

by (7.13).

By (7.1.5)

for some

Hence

t > 0.

is the pull-back of

HS

S --'1P2.

9 = (L + D)3 =

over

S.

E2 = -1 = KE.

with

easy to see that

so

or

A general member

d = 7. S

of

First we consider the case ILI

is a Del Pezzo surface

of degree seven, and hence is the blowing-up of

F2

at two

I. A-genus and the Apollonius method

66

points. The proper transform

E

these points is a (-1)-curve on E

contracting

]P2

We get

X IP1

S.

IP1

S, namely

are exactly three (-1)-curves on

E

and those Since the

which are blown up.

IP2

by

It is easy to see that there

to a point.

over the two points on

passing

of the line on

monodromy action in (8.4) preserves the configuration of (-1)-curves, we infer that Hence, as in (8.5),

on M so

such that

M

E = DS

a Pic(S)

is

L + D

G-invariant.

for some effective divisor

(D, LD) = (1P200(1))

and

is the blowing-up of another manifold

comes from

by (7.16).

a Pic(M1)

Lt

We have

at a point

Mb

Moreover

(IP3, 0(2))

Kb

by (8.5).

of

Thus

and

Therefore

Mb.

M

is ample on

Lb

= d + 1 = 8

(L1')3

the canonical bundle

and

D

[D]D = 0(-1),

is the exceptional divisor (cf. (0.1.6)).

D

and

[E]

Kb

=

Mb

-2L6

for

(Mb, Lb) =

is the blowing-up of

IP3

a point.

Secondly we consider the case n = 4. member

T

(7.15), M

of

is a Del Pezzo 3-fold as above.

ILI

is the blowing-up of another manifold

point, and N contains

as a divisor.

IP3

Such a pair

ample by (7.16). (7.18).

Thus the case

n 2 4

(8.7) Suppose that

(N, IP3)

d = 6.

Again

n = 3

member

S

of

at three non-colinear points

p1' p2' p3.

lying over

at first.

of

is the blowing-up

ELI

Let pi

Then a general

Ei

be the (-1)-curve

and let

Zi

be the

proper transform of the line on

IP2

N

By at a

Moreover it is

does not exist by

cannot occur.

assume

IP2

Any general

at

§8: Classification of Del Pezzo manifolds

passing the two points other than on

67

Then the (-1)-curves

pi.

are exactly these six curves, which form a cycle.

S

Aut(Pic(S))

analyzing the structure of

By

and using (8.4), we

infer that there are the following three possibilities (see [F5;S5] for details):

1) The pull-back H

of

2) H - E.

comes from

3) Pic(M)

is generated by

for some

Pic(M)

Take

Case (8.7.1).

comes from

0 2(1)

Pic(M).

i.

L.

F E Pic(M)

FS = H.

such that

LS - FS = 2H - E1 - E2 - E3 = f 0(1), where

Then

F2

birational map onto

for any

H1(S, FS - tLS) = 0 Therefore

(7.5).

finite, so

which contracts the t > 0, so

S n BsIF1 _ 0

and

BsIFI

IL - F1

yield a morphism

p Ha = F

p H

and

L = p (Ha + H

is ample.

of

on

JaHa + bHPI

Hence

p: M -->F2 x ]P2

= L - F.

p

(L - F)3 =

tions of the mappings

such that

and

is a member

W = p(M)

for some

M ---, Fa

a, b.

The restric-

M - F2 to

b > 0. We have

and

= 3(a + b)deg(p), so

IP2,

some

W

r.

W

and

is defined by the equation

Here

H3{W}

a = b = deg p = 1.

012

and

r > 0

Hence

6 = L3{M} =

homogeneous coordinates (a:a:a)

since

W

are

S

both birational morphisms, so they are surjective. a > 0

and

IFI

is a finite morphism since

Therefore

Fa X F2

by

is at most

BsIF1 = 0 = BsIL - Fl, so

This implies

Hence

Zi's.

0 s (L - F)3, but

(L - F)3 + F3 = L3 - 3LF{S} + 3F2{S} = 0. F3 = 0.

is the

(FIS = IFS1

Similarly we have

F3 2 0.

f

For suitable (#0: 1:p2) :Er=0ajij

is irreducible.

= 0

of

Pa 2

for

Therefore

is normal and hence M . W by Zariski's Main Theorem.

Hence

r = 2

since

W

is smooth, so

(M, L)

is the scroll

I.0-genus and the Apollonius method

68

of the tangent bundle

and

F2.

Then

F E Pic(M).

is the contraction of

As in the case (8.7.1) we have

Z1.

Moreover, by (7.8),

by (7.5).

LS - FS = 2H - E2 - E3 =

f: S -- P1 X P

f (H77 + He), where E1

of

By symmetry we may assume that H - E1

Case (8.7.2).

extends to some

0

IFIS = IFSI

extends to

pIH-E I: S *--b F 1

a morphism

have and

IL - FIS = ILS - FSI

Therefore

Then

IL - FI.

S n BsIL - FI = 0

Therefore

(L - F)3 = 0

dim W t 2

F1 X P

77

and

p

such that

77

yield a morphism

p

L = 0(H + H77 + He).

L

0: M --+ P

We have

is birational.

0

Therefore

is ample.

P1 X Ft

X

6 = L3 =

Hence

H77 + HC)3 =

It is finite since

p

(L - F)3 = 0.

since

On the other hand W D p(S) = P1 X Fl, so W Now,

We also

Let W be the image of the map

BsIL - FI = 0.

defined by

by (7.5), so

is at most finite.

BsIL - FI

implies

F = p H,.

such that

(L - F)3 = L3 - 3LF{S) + 3F2(S) = 0.

and

F3 = 0

: M -i P

p

0

is an

isomorphism by Zariski's Main Theorem. Case (8.7.3). We will show that this case cannot occur. M c P7

Recall that

and

For any general hyperplane Su = M n Hu

infer that

open condition and

S

is a line in

E1

Hu

in

P

containing

E1.

lying on M lines.

Then

Each

is smooth.

and meeting dim A 2 2.

a > 0.

E1, we

Su

is a Del Pezzo E1, two of

Hence there are infinitely many lines in E1.

Since

Let

A

Pic(M)

every non-zero effective divisor on M for some

P7 d=

is smooth, since smoothness is an

surface, and contains five (-1)-curves besides which meet

P

So

A

is ample and

P

be the union of such is generated by is a member of A n Z1 0 0.

L,

IaLl

Hence

§8: Classification of Del Pezzo manifolds

there is a line

Therefore

on M which meets both

L

and

E1

Z1.

for the hyperplane H with M n H = S.

£ c H

Clearly

69

£ c S

and

is a (-1)-curve on

l

is no such curve meeting

and

E1

But there

S.

Thus we get a

Z1.

contradiction, as desired.

Next we consider the case

(8.8)

n = 4, any general member

on

t > 0.

T

Hence

So

If

T

and

f

is a member of

LT = [Ha + Hf]T.

extends to M

L.

Moreover

Similarly

f: M 2

P2 a

x lP2

(Ha + H4(Pa X P2) = 6,

Since

f

comes

for any

by (7.8).

is finite since

L

is

is an isomorphism by Zariski's Main Theorem.

is of the type (8.7.2), then

LT = H

[Ha]T

H1(T, Ha - tLT) = 0

Thus we get a morphism

f*[Ha +

is birational.

ample.

and

T -->lP2

extends.

F2

such that

f

Pa X F2

by (7.1.5) and

Pic(M)

from

is a Del Pezzo 3-fold

ILI

is of the type (8.7.1), T

T

IHa + HMI

of

When

It is of the type (8.7.1) or (8.7.2).

as in (8.7). If

T

d = 6, n s 4.

+ H77 + HC.

Each map

by (7.9).

Thus we get a morphism

such that

L = f (H

+ H77 + He).

T = P

T - P1

X P17

x lP

extends to M X Fl X F1

f: M --+ P

f has positive

But

dimensional fiber, contradicting the ampleness of

L.

Thus

this case is ruled out.

When n = 5, any general member D isomorphic to a morphism

Pa X F2

of

is

ILI

by the above observation.

f: M -+ Pa X F2

such that

(7.9), which yields a contradiction.

ruled out, and the case n > 5

L = f*(Ha + H

by

Thus this case is

cannot occur either.

(8.9) It now remains to consider the case case is studied fully in [F5-II].

We obtain

d = 5.

This

But the proof is terribly

I. A-genus and the Apollonius method

70

Here we just sketch the results.

complicated. Let

linear

parametrizing

be the Grassmann variety Gr(5,2)

C

C5 (or equivalently lines in

C2's in

embedded in

by the PlUcker embedding, and

1P9

is a Del Pezzo 6-fold of degree any Del Pezzo manifold with this variety

5.

This is

F4).

(G, 0(1))

Our main result is that is a linear section of

d = 5

In particular n g 6.

G = Gr(5,2).

The proof in [F5-II] is based on a certain birational

correspondence between M

and

described as follows. F2 C P5 Let C = F1 X F5

D =

along

JPa

theoretically in

Let

be the blowing

P#

be the exceptional divisor over

be the proper transform of

D#

and let

E

C,

When n = 6, this is

be the Segre embedding and let

be a linear embedding.

C 1Pa

up of

Fn.

D

on

C

Scheme-

P#.

is the intersection of three hyperquadrics

C

The linear system generated by these hyperquadrics

D.

yields a morphism D# --+F2.

Actually this is a

1P3-bundle

can be blown down smoothly along this direction.

and

D

Let

P# -> M6

be the blow-down.

Then

2Ha - E = L

for

P

some

L E Pic(M6), and it turns out that

Pezzo 6-fold of degree

(M6, L)

is a Del

5 (cf. [F5-II;(7.12)]).

A similar construction can be carried out in the cases n = 4 C

and

n = 4, we start from a Veronese curve

When

5.

of degree three in

rational scroll

C

case we blow up

Fn

transform of

D

D = 1P3.

When n = 5, we start from a

of degree three in along

suitably.

D = ]P4.

In either

C, and then blow down the proper See [F5;(7.8) & (7.10)] for

details.

The situation is a little different when

n = 3.

There

71

§8: Classification of Del Pezzo manifolds

is a certain birational transformation between M

and

Q3

but another transformation from a smooth hyperquadric

of

By blowing up

Q3.

be a smooth hyperplane section

D

Let

is more important.

]P3,

Q3 along a certain curve

D

in

C

D, we get a

and then blowing down the proper transform of

birational transformation onto a Del Pezzo 3-fold of degree See [F5;(7.4)].

five.

In the proof of the main result, we reconstruct birational transformations as above from the side of

This is done inductively on n

M.

and requires a lot of

computation.

From this we infer that every Del Pezzo n-fold

of degree

is isomorphic to the others for each fixed

5

C c D c Fn (or

Indeed, the triples

as above are all

Q3)

Therefore, from this

projective equivalent to each other.

very observation, it follows that

n.

M

is a linear section of

Gr(5,2). below.

See [F7] for technical details.

We may assume ILK

on

is ruled out by the argument outlined

n > 6

The case

must be

Then any general member

n = 7.

on

8

t > 0

and

(7.5).

We have

M.

for any

t k 0.

H0(M, 6) = H0(G, 6G)

is surjective for any

H0(M, 8[tL]) 0 H0(M, L)

for any as in

H0(G, 6[tL]G) 0 H0(G, LG) -+

We also see that

H0(G, 8[(t + 1)L]G)

to a vector

G

Hi(G, 8[-tL]G) = 0

Hence

i = 0, 1.

of

By using various vanishing theorems

Gr(5,2).

G, we extend the universal bundle on

bundle

G

--'

HO(M, 8[(t + 1)L])

This implies that

6

global sections, so we get a morphism extending the identity of

t a 0.

G.

Hence

is surjective

is generated by p: M --> Gr(5,2) = G

As in (8.8), this yields a

I. A-genus and the Apollonius method

72

contradiction by an argument in [Sol;Lemma I-A].

Likewise the Grassmann variety

Remark.

Gr(n, r)

cannot be an ample divisor on another manifold unless r = 1, n - r = 1

(n, r) = (4, 2).

or

(8.10) Remark.

See [F7;(5.2)].

d = 5, there is a completely

In case

different approach due to Mukai.

His method uses the

restriction of the universal bundle on

M y Gr(5,2)

embedding

Gr(5,2).

The

can be recovered by reconstructing

This approach works for many other

this vector bundle.

types of Fano manifolds (cf. [Mukl,2]). (8.11) Summing up, we now establish the following. Theorem.

Let

(M, L)

n = dim M k 3

and

d = Ln.

be a Del Pezzo manifold with When

char(A) = 0,

(M, L)

is

of one of the following types. 1) d = 1 6

and

(M, L)

is a weighted hypersurface of degree

in the weighted projective space

IP(3, 2, 1,

,

It

1).

has a structure as in (6.14) - (6.16). 2) d = 2

and

(M, L)

is a double covering of

along a smooth hypersurface of degree

pull-back of

branched

is the

L

4.

IPn

0(1).

3) d = 3

and

M

is a hypercubic in

4) d = 4

and

M

is a complete intersection of two hyper-

quadrics in 5) d = 5

Ipn+2.

and

M

is a linear section of

6) d = 6 and M is either for the tangent bundle

6

]pl x IP1 of

and

M

8) d = 8

and

(M, L) = (]P3, 0(2)).

L

Gr(5,2) c 1P9.

x Ipl,

1P2

x

F2

or

IP(0)

F2

7) d = 7

Remark.

L = 0M(1).

IPn+l.

is the blowing-up of

is determined by

M

IF3

since

at a point.

K = (1 - n)L.

73

§8: Classification of Del Pezzo manifolds

char(A) = p > 0.

Now we study the case where

(8.12)

The conclusion will be the same as in (8.11) if

d

3.

Following Badescu's idea [Bad2], we use the technique of lifting to characteristic zero.

Here we review the argument

in [F11].

(8.13) Key lemma.

d = L3 a 5

with

Then the pair

Let

D

and let

be a Del Pezzo 3-fold

(M, L)

be a general member of

is liftable.

(M, D)

This means that there is a smooth morphism over the ring

Spec(W(.))

W(it)

9

A( -p S

a =

of Witt vectors together

on

A(

such that the fiber of

over the closed point of

S

is isomorphic to

with an effective divisor (A(, 9)

JLJ.

Note that

(M, D).

is a discrete valuation ring of

W(R)

characteristic zero with residue field

A.

The proof consists of several steps. 1) By (8.1), D at

points with

r

F1 X P1

is

0(D)

2) A general member We obtain

Hence

r = 9 - d .g 4.

for the tangent bundle

curve.

or the blowing-up of

C

of of

H1(D, 0(D)) = 0

D.

is a smooth elliptic

ILDI

H1(C, 0(D)[tL]) = 0

for any

using the exact sequence 0 -* OC -- 0(D)C Hence

h1(D, 0(D)[tL])

this implies

t

3) We have an exact sequence -s 0

of vector bundles on

bundle of step 2).

So

M.

Hence

4) Let

3

for any

0(M)

H1(D, O(M)[tL]D) = 0 H2(M, 0(M)) = 0

be the

by

By step 1),

t a 0.

O(M)D -+ LD

0 --> 0(D)

D, where

t > 0

0C[L] ---> 0.

h1(D, 0(D)[(t - 1)L]).

H1(D, 0(D)[tL]) = 0

1P2

is the tangent

for any

t a 0

by

by (7.3).

OM-ideal defining

D.

0(M)

can be

I. A-genus and the Apollonius method

74

identified with the sheaf of derivations of a natural homomorphism 0(M, D)

0(M) -- Yt*mM(3/32, 0M/3).

Let

M

This is locally free on

be its kernel.

and we

0 - 0(M, D) - 0(M) -+ OD[L] --> 0.

have an exact sequence Hence

OM, so we have

H2(M, 0(M, D)) = 0

by step 3).

5) By the obstruction theory for the liftability due to Grothendieck [Gro2;expose IQ],

formal completion

is liftable over the at the closed

S = Spec(W(.))

of

S^''

(M, D)

point.

This formal lift is algebraizable since

ample.

Thus the proof of the Key lemma is complete. Let

(8.14)

The generic fiber field

of

A'

over

Let

*

(M , D

3-fold with

L

polynomial implies H1(M, L) = 0

)

[D*], (ML*) is

(M', L')

a Del Pezzo

implies

S.

The invariance of the Hilbert g(M', L') = 1.

and

Moreover

h0(M, L) = rank f*Y = h0(M', L'),

= g(M', L') = 1

(8.15) Lemma.

is a smooth family of

(4, Y)

(L')3 = d

d(M', L') = d(M, L) = 1. g(M , L

=

q = [9],

polarized manifolds over

*

be the algebraic closure of

Si

d = (L*) 3.

Indeed, for

*

as above.

be the scalar extension of

)

Then, for

Si*.

S

is

is defined over the quotient

(M', D')

W(St). *

and let

St'

be the family over

(Al, 9)

D

Hence and

(L*)3

=

*

*

4(M , L

so

(L')3 = d, )

= d(M', L') = 1.

There is a natural commutative diagram

Pic(M)

--> Pic(D)

1*

1*

Pic(M ) -' Pic(D ), where the vertical maps are bijective. The proof consists of several steps. 1) Let

m

be the maximal ideal of

W(A).

We have an

75

§8: Classification of Del Pezzo manifolds

exact sequence

0 --> OM --> 0 j

x

--> O j_ i- 0,

where

Ojx

is the sheaf of multiplicative groups consisting of invertible sections of

for any

j.

Similarly 2) 2

So H1jx

0jM = 0A/mj0A.

Hence

Pic(AZ)

Pic(O)

Pic(D)

for

ti Pic(M) for

AZ = A x S

rb = A xS S.

So, for any

is ample on

F e Pic(AZ), there

is an exact sequence A --* B --> F -f 0 on and 8

B

Replacing 2

bundles

d

extends to

and

0

by

on

y: 4 - R.

A

St, we extend

A.

3' = Coker(y)

Pic(A) -4 Pic(AZ)

is obvious.

Similarly

3) The map is smooth.

to vector

B

is an extension of So

M.

Pic(g) -+ Pic(g)

It is injective since

F

3 e Pic(A).

is bijective.

is surjective since

A - M' = M

divisor defined by the principal ideal

4) We have

206

is surjective, while the injectivity

Pic(A() -+ Pic(M')

Pic(g) -* Pic(D')

and

A

Then the homomorphism A - B

and is invertible in a neighborhood of

So

such that

AZ

are direct sums of line bundles of the form

e Z.

Thus

H1(Oj_i

m0 A.

A(

is a Cartier Similarly

is bijective.

Pic(D') = Pic(D)

by the preceding steps.

rank(Pic(D')) = 10 - d = rank(Pic(D )).

Moreover, the

above bijection preserves the intersection pairings by its definition 1) '. 3).

Hence the determinant of the matrix

formed by the intersection numbers with respect to a Z-base of

Pic(D')

is

Therefore

(-1)9-d.

Pic(D') -> Pic(D

)

is

bijective.

*

5) Pic(M ) --i Pic(D * step 4), G = Gal($ /St')

acts trivially on

* )

is injective by (7.1.5). *

By

acts trivially on Pic(D ). So * * Hence Pic(M') _- Pic(M ). Pic(M ).

G

I.0-genus and the Apollonius method

76

Combining these observations we prove (8.15). Let

(8.16) Lemma.

with n > 3

and let

be a Del Pezzo n-fold

(M, L)

D be a smooth member of

Then

ILI.

Pic(M) = Pic(D).

for any

Hi(D, tLD) = 0

Pic(D), where Moreover

11

t E Z, i = 1, 2.

theorem (cf. [Gro3]).

Suppose

Note also

at three points.

IP2

H - Ei comes from Pic(M)

H, Ei E Pic(D)

this we infer that

D.

Then, by

or F(e).

Fl X F1 X IP1

IPl X IP1 X IP1,

by (8.15), where

along

d = 6.

be as in (8.14).

is the blowing-up of

When M*

M

This completes the proof.

Let M

n = 3.

is either

D

that

Pic(111) =

by Grothendieck's Lefschetz

(8.17) Let us now consider the case

(8.7), M

So

is the formal completion of

Pic(M) = Pic(1)

first that

Hence

is also a Del Pezzo manifold.

(D, LD)

Proof.

are as in (8.7).

M = F1 X F1 X F1

From

by the same argument

as in (8.7.2).

When M M * P(©)

*

ti F(6), H

by (8.15), so

Pic(M)

by the same reasoning as in (8.7.1).

Next suppose Then

ILK.

comes from

n = 4.

(D, LD)

Let

D

be a general member of

is a Del Pezzo 3-fold as above.

(8.16) in place of (7.1.5), we obtain M *

F2

X

F2

Using by the

argument in (8.8). The case

true when

n > 4

is easily ruled out.

Thus (8.11) is

d = 6.

(8.18) Similarly we establish (8.11) also in the case d k 7.

The proof is simpler than in the case

details, see [Fli].

d = 6.

For

§9: Polarized varieties of A-genus one: Remaining cases

d = 5, the problem is more subtle.

(8.19) When

It is easy to show that

Suppose that

n = 3.

generated by

L

char(.) = 0,

the normal bundle

is

0 ® 0

77

Pic(M)

and that there are many lines on

is

If

M.

M

of a general line on

.N

by the generic smoothness theorem.

This is a key

observation in the classical theory of Fano-Iskovskih.

By

blowing-up along such a line we recover a birational correspondence with a smooth hyperquadric as in (8.9).

If 3

I= 0(-I) ® 0(l), we still get a birational map onto

,

but its structure is more complicated than in the case X

0 ® 0. When

char(A) > 0, however, it may be possible that

N ti 0(-l) ® 0(1)

for every line on

M.

As a matter of

fact, we can rule out this possibility by analyzing the

structure of lines on M

very precisely, but the proof is

very lengthy and is omitted here.

We encounter new problems in higher dimensions too. However, with some effort, we can adapt the argument in [F5-II] and get the same result as in the case

I hope that

char(.) = 0.

Mukai's method (8.10) works also in the

positive characteristic case and gives a simpler proof. Thus, (8.11) is true even if few cases in which

char(.) > 0, except for a

char(.) = 2 a d.

§9. Polarized varieties of A-genus one: remaining cases Throughout this section let (V, L) be a polarized variety with n = dim V a 2 and A(V, L) = 1. Set d = Ln and g = g(V, L). The results are summarized in (9.18). (9.1) We assume that

indeed true either if

V

(V, L)

has a ladder.

This is

is smooth at each point of

BsILI

I.0-genus and the Apollonius method

78

g < d, char(s) = 0

(cf. (4.15)), or if

Cohen Macaulay at each point of

is locally

V

and

and 2-Macaulay

BsILI

everywhere (cf. (4.16)).

There are examples which have no ladder though

A = 1.

But such polarized varieties are not treated in this book. (9.2) In §6 we assumed that

g s 1.

This is true if

is locally 2-Macaulay (cf. [Fll;(5.2)] and (3.4)). consider the case

assumption Let

g s 0, so

j > b.

that

and

(5.17).

n Dn-j

and the

Vb, so

L)

Dn-b a

for each Vb

v-1(Vb) = Vb.

is locally Macaulay by

Therefore

V'

Vb.

By (0.2.8), V

Now, perturbing

is not normal if

for the normalization

such

In particular

j 2 b.

little if necessary, we get a ladder of

Thus, V

ILI

Moreover,

dim(Sing(V) n Vb) s dim Sing Vb

since

Di's are ample.

each rung is normal.

Then

is surjective for

is locally Macaulay along

V

dim Sing(V) < n - 1 < b - 1

s 4(V, L) = 1.

D1,

On the other hand

V = Vn D by the

A(Vb, L) = 0.

H0(V

--+

Therefore we have

Hence

V1 = 1P1

0 = 4(V1, L1) s

H0(V, L)

Vi = D1 n

BsILJ c Vb.

along

Then

(V, L).

be the maximal number with

b

d(Vj, L) = 1 any

be the normalization and let

be a ladder of

D V1

Here we

g s 0.

v: V' -+ V

Let

of

(9.3) We studied the case

(V', L

4(V', L) = 0

g s 0, and

is normal

v-1(Di)

a

) such that V by (3.4).

4(V', L) = 0

V.

g k 1

and

d s 4

in §6.

In the remainder of this section we study the case where is very ample.

V

By (6.2.3), L

is very ample if

L

and

g k 1

d 2 3 (provided a ladder exists, as always). (9.4) We will review the theory in [F17].

Since

L

is

§9: Polarized varieties of A-genus one: Remaining cases

very ample, we have V

that

V c

Fn+d-2

L = 0V(1).

We assume

is singular, since the smooth case was studied in

We further assume that

§8.

and

79

is neither a hypercubic nor

V

a complete intersection of two hyperquadrics.

Using (5.13), we reduce the problem to the case where Ridge(V) = 0.

This is equivalent to saying that

is not

V

This condition is assumed

a cone over any other variety. in the rest of this section.

(9.5) Take a singular point

on

v

which is the union of all the lines passing point on

V (cf. (5.12)).

Ridge(V) _ 0.

Hence

for

section M Moreover

of

Ha

let

R = Ridge(W)

ifi

by (5.14).

By (5.15), we have

and r

v e R

and

=

det

dim R = 0.

4(M, 0(1)) = 0.

be the blowing-up of

be the pull-back of

scroll of

since

and for some smooth linear

Note that

W.

and another

dim W = n + 1

and w = d - 2.

dim M = n - r

(9.6) Let

v

Moreover w = deg(W) s d - 2

A(W, 0(1)) = 0

W = R * M

Then

W = v * V,

Set

V.

Then

°W(1).

6 = 0M(1) ® 0M (r+l) on

W

M.

along

R

is the

(S, Ha)

The unique member

*

of

f

where a

is the exceptional divisor of

IHa - f 0M(1)1

and

is the map

f

a:

T - W

give an isomorphism D= R x M.

aDI

R

LJ -L M al

W

f*BI

B E Pic(M).

For a proof, see [F17;(8)].

Fn+d-2

C

(9.7) The proper transform for some

D

Note that the restrictions of

TiJ --b M.

D

IHa +

and

a

?

of

V

Moreover

is a member of BsIBI = 0.

I.0-genus and the Apollonius method

80

(9.8) Key Lemma.

contains the fiber over

V

If

contains no fiber of

11

n : D -* R. D

x E R, we can show that

x e Ridge(V), contradicting the assumption in (9.4).

See

[F17;(9)] for details of the proof of the Key Lemma.

(9.9) The type of the polarized manifold is classified in (5.10).

There are three cases:

is a Veronese curve of degree

(c) M

(M, 0M(1))

d - 2.

(v) (M, 0M(1)) = (IP2, 2Hso d = 6

r = n - 2.

and

is the scroll of a vector bundle

(s) (M, OM(1))

on

6

IP77.

We will proceed further by case-by-case arguments.

M

type of

depends on the choice of

v E Sing(V).

(9.10) In the case (9.9.c) where we have

(d - 2)H P, IPn-l-bundle

over

infer that

V E

M

IHa + 2f H

The

T, and

on

OM(1) _

and

M = IP1

is a

T/

Using (9.8), we

and hence smooth.

is the normalization of

Moreover

V.

L) = 0.

The divisor

nD

defined by an equation where

(#O:8l)

on

is

D = R x M = pa-1 x IP1

£0(a)P02 + £1(a)fl0fl1 + £2(a),612 = 0,

is the homogeneous coordinate of

M

and the

Gi's are linear forms in the homogeneous coordinate (a0: :ar)

of

implies

Then

R.

This

n - 1 = r t 2.

The singular locus of

singularities along

R.

V

is

R

and

V

turns out to be normal and

Pezzo variety.

V

has non-normal

See [Fl7;p.153] for details.

(9.11) In the case (9.9.v), we see Moreover

by (9.8).

(GO = tl = 12 = 0) _ 0

V E

IHa + f HPI.

(V, L)

See [F17;p.154] for details.

is a Del

§9: Polarized varieties of A-genus one: Remaining cases

The divisor

n D

D = 1Pa x F2

on

is defined by Li's are linear

10(a)fl0 + L1(a)fll + t2(a)P2 = 0, where the

forms in

n - 2 = r t 2

Hence

ar.

a0,

81

by (9.8).

For

further details see [F17;p.155].

(9.12) In the case (9.9.s), set is generated by

Pic(M)

of the map

M --1P77.

HC

HC = 0M(1).

and the class

H77

Then

of a fiber

It turns out that there are the

following three cases:

(so) B = 2Hwhere B

is the line bundle as in (9.7).

(si) B = HC - (d - 4)H27.

(su) B = 2H

- (2d - 6)X17.

For the proof, see [Fl7;p.156]

(9.13) In the case (9.12.so), V

Thus the situation is that of (9.2).

g = 0.

unlike the case (9.9.c),

1

Analyzing the divisor r t 2

is not normal and

in this case.

However,

is not the normalization of [/ n D

on

D

V.

as before, we get

See [F17;pp. 157-160] for further

details.

(9.14) In the case (9.12.si), (V, L) Pezzo variety. follows.

The vector bundle

In each case

r

8

is a normal Del

is classified as

is bounded by (9.8) as before.

1) 8 = 0(4, 2) (def [4H,] ® [2H72]), n - 2 = r = 0 and d = 8. 2) 6 = 0(3, 2), n - 2 = r = 0 and d = 7. 3) 6 = 0(3, 1), n - 2 = r s 1 and d = 6. 4) S 0(2, 2), n - 2 = r s 1 and d 6. 5) 8 0(2, 1, 1), n - 3 = r = 0 and d = 6. 6) 8 = 0(2, 1), n - 2 = r s 2 and d = 5. 7) 8 = 0(1, 1, 1), n - 3 = r g 2 and d = 5.

I.0-genus and the Apollonius method

82

The type of singularity of precisely.

V

is described very

See [F17;pp. 161-170] for details.

(9.15) In the case (9.12.su), it turns out that 8

0(1, 1)

and M = 1P1 x 1P1

with

HC = H77 + H.

changing the role of the two rulings of

d = 4, Hence,

M, we have

(9.12.so).

(9.16) Summing up, we obtain the following. Theorem.

d(V, L) = 1

V

Let

for

be a subvariety of

L = 0V(1).

1PN

Suppose that

V

such that is singular,

is not a cone over another variety, and is neither a hypercubic nor a complete intersection of two hyperquadrics. Then

(V, L)

zation

V'

is a normal Del Pezzo variety, or the normaliof

V

is a smooth variety with

Indeed, (V, L)

4(V', LV,) = 0.

is Del Pezzo in the cases (v) and (si),

while the latter is true in the cases (c), (so), (su). the case (c), (V, L)

In

is a non-normal Del Pezzo variety.

(9.17) Theorem (cf. [F17;(2.9)]).

normal Del Pezzo variety as in (9.16).

Let Then

(V, L)

be a

(n,d) = (2,8),

(2,7), (2,6), (2,5), (3,6), (3,5), (4,6), (4,5) or (5,5).

This follows from the results in (9.11) and (9.14). Now we summarize the results in the chart of the next page.

§ 10: Polarized manifolds of A-genus two (9.18) Chart.

83

Classification of singular polarized d-genus one.

varieties of

(V, L)

YES

has a ladder ?

I

1N0

g(V, L) > 0 ?

???

g s 1

g = 0

I

(6.14) ti

d-4

(9.2)

k5

(6.10) (6.7)

ti

(6.23)

d-3

d- 2

d- 1

L

(6.8)

is very ample

(6.13) (9.9)

(c)

(9.11) g=l

non-normal Del Pezzo

normal Del Pezzo

non-normal

is not very ample

(s)

(v)

(9.10) g=1

(so) (9.13) g=0

L

generalized cone

1

(9.12)

(su)

(si) (9.14) g=1

(9.15) d=4 same as (so)

normal Del Pezzo

§10. Polarized manifolds of 4-genus two Here we study the case d = 2. Since the technical difficulties increase as d becomes larger, we consider only smooth polarized varieties for the sake of simplicity. Moreover we assume char(s) = 0. Our method does not work The results are summarized fully without these assumptions. in the chart (10.12). (10.1) Key Lemma.

with

Let

(M, L)

dim M = n 1 3, d = Ln 2 2

and

be a polarized manifold d(M, L) = 2.

Then any

I.0-genus and the Apollonius method

84

general member

D

of

regular rung, namely

4(D, LD) = 2 D

The smoothness of

Moreover D

is smooth.

ILI

and

ILDI = ILID

can be proved by the method in

§4, but the argument is far more complicated. [F12;(4.1)] for details.

4(D, L) s d(M, L)

See

As for the last assertion, we have

in general.

d(D, L) S 1, then

If

Hence

H1(D, OD) = 0

by the preceding results.

h1(M, -L) = 0

by Kodaira's vanishing theorem.

ILID =

ILDI

is a

hI(M, OM This implies

4(D, L) = 4(M, L).

and

By virtue of this lemma we reduce many problems to two-dimensional cases by the Apollonius method. (10.2) Theorem.

Let

be a polarized manifold

(M, L)

with n = dim m 2 2, d(M, L) = 2

and

g(M, L) s 1.

is the scroll of a vector bundle

(M, L)

elliptic curve

Then

over an

6

C.

A proof can be found in [F16;(0.2)].

Another proof can

be obtained by using the theorems (12.1) and (12.3) in the next chapter. Remark.

We obtain

h0(M, L)

by using Atiyah's theory [At].

=

h0(C,

48)

Therefore

= c1(6) = Ln

2 = 4(M, L) = n

in this case.

(10.3) In the rest of this section let polarized manifold with g = g(M, L) k 2. case

d = 1

n = dim M k 2, d(M, L) = 2

We assume further

be a and

d = Ln > 1, since the

is difficult to study.

(10.4) When results.

(M, L)

dim BsILI

> 0, we have the following

See [F16;(1.14) & (1.17)].

1) Y = BsILI

is a smooth rational curve with

LY = 1.

§ 10: Polarized manifolds of 0-genus two

n: M' --p M be the blowing-up along

2) Let

be the exceptional divisor over 3) Let

85

M' -

W be the image of the morphism *

defined by

in

Then

L - El.

E

Bsin*L - El _ 0.

Then

Y.

and let

Y

Fn+d-3

dim W = n - I, deg W = d - 1

4(W, 0(1)) = 0.

and

4) E

is a section of

5) p

is flat and every fiber of

6) If

s 1.

y

is a polarized manifold of the above

type for any general member

D

of

ILI.

7) There exists a morphism

0: M --> Y = ]P1

y E Y, the fiber

over

any

(My, Ly)

y

L

Y 8) d a n.

such that, for is a polarized

d(My, Ly) = 4(My, L ) = 1 y is the restriction of L.

and

Moreover, if

is a trivial

variety with where

is determined

y

g = (d - 1)y. (D, LD)

n a 3,

is an irreducible

p

reduced curve of arithmetic genus by the formula

E = W.

and

p: M' --> W

fibration and

d = n, then

is the Segre product of

(M, L)

and a fixed polarized manifold

FXN and

0

g(My, Ly) = y,

(N, A).

(1P27 ,

This means

H7)

M

L = [HM + AM.

(10.5) By (10.4.7), we can proceed further as in the d = A = 1 (cf. §6).

case

Theorem (cf. [F16;(2.4)].

and suppose in addition that covering

f: M' -> P

B

of

l3H.I

f

on

IHc - 2HI, where

member of

is

with

y = 1.

of the scroll

8 = 0E ® [-2EJE

bundle

Let things be as in (10.4)

S + B1, where

E = W.

Then there is a double

(P, H) of the vector S = f(E)

H, = [-E]p. B1

is the unique

The branch locus

is a smooth member of

S n B1 = 0.

(see the diagram in the next page)

1. 0-genus and the Apollonius method

86

Y-E

S

n

n

n n

M

f--

x

W

f

V

M' -* P - W

This corresponds to the results in (6.15) and (6.16). If

y a 2

and if every fiber of

is hyperelliptic, we

p

have similar structure theorem to those in (6.21), (6.22) See [F16] for details.

and (6.23).

There are also cases in which general fibers of

are

p

These cases seem more difficult to study

non-hyperelliptic.

and are left to the reader.

(10.6) In the rest of this section we assume that

We will study the cases

dim BsILI 9 0.

d > 4, d = 4, 3, 2

separately.

(10.7) Suppose (4.15) and L

L

d > 4.

Then

has a ladder by

(M, L)

is simply generated by (3.5.3). In particular Moreover

is very ample.

g(M, L) = 2

and

H1(M, tL) = 0

for any t E Z, 0 < i < n. Next we claim

bundle K Indeed, if

of

D

ample.

We have

This is proved by induction on

M.

of

ELI

such that

p E D, since

KD

n.

of

L

is very

h1(M, K + (n - 2)L) = hn-1(M, (2 - n)L) = 0

IK + (n - 1)LID = IKD + (n - 2)LDI

bundle

for the canonical

p E BsIK + (n - 1)LI, we can find a smooth

member

and

BsIK + (n - 1)LI = 0

D.

by the choice of

p

for the canonical

is a base point of this linear system

D, which contradicts the induction hypo-

thesis.

By induction we have also we have a morphism Let

X

f: M - 1P1

be any smooth fiber of

h0(M, K + (n - 1)L) = 2. with f.

So

K + (n - 1)L = f H Then we have

Ln-1X

.

=

§ 10: Polarized manifolds of A-genus two

87

(K + (n - 1)L)Ln-1 = 2g(M, L) - 2 = 2.

is very

The latter does not occur since the

]Pn-1's.

X

canonical bundle of h0(X, LX) = n + 1

of rank

L

is a hyperquadric or a disjoint union of two

ample, (X, Lx) linear

Since

Kx = (1 - n)LX.

is

f*0M[L] _ 8

and

Therefore

is a locally free sheaf

n + 1.

There is no multiple fiber F = 2Y, [Y]Y

F

of

Indeed, if

f.

would be numerically trivial and we would have

1 = x(OX) = x(OF) = x(OY) + x(OY[-Y]) = 2x(OY), which is absurd.

Thus, every fiber

hyperquadric since

is an irreducible reduced

Mx

Ln-1Mx = 2

is very ample.

L

and

From this we infer that the natural homomorphism *

f 8 - OM[L]

is surjective, so we have a mapping

with pH = L

(P, H

onto the scroll

For every point x on IP1, the

map

the embedding of the hyperquadric of

12HC + bH,J

we have

on

for some

P

d = HnM = 2Hn.+1 + bHn_H

hand, we have

p

of

x : Mx

'

-- Px

Thus

Mx.

b e Z.

M

b + e = 3, which implies

ti ]Pn

= 2e + b.

e = d - 3

We can further classify

and

IP

is

is a member e = cl(S)

On the other since the

-(n + 1)HC + (e -

is

P

over

Setting

K = [(1 - n)HC + (b + e - 2)H ]M

canonical bundle of

p: M --i P

Hence

b = 6 - d.

as follows.

(M, L)

(10.7.2) When n = 2, one of the following conditions is satisfied.

0) M

JP

x ]P

- E, where

E

is the sum of the (-1)-curves

L = 2H27 + 3H

over these points. 1) M

(12 - d) points and

is a blowing-up of

is

at

d g 12.

I1 (def IP(H

® 0)), or is a blowing-up of

a point lying on the (-1)-curve on

1'1.

I1

at

L = 2Ha + 2H - E,

I. A-genus and the Apollonius method

88

where

Ha

E

0 (if

d

is

is the line bundle

on

0(1)

M = 1) or the last (-1)-curve.

ti 0(4, 3, 2) (deP [4H ] ® [3H ] ® [2H J)

and

® 0)

1P(H

Note that

or

0(4, 3, 1)

in this case.

and

® 0)

2) M ti 12 = 1P(2H

(10.7.3) When

L = 2Ha + H.

8 = 0(5, 3, 1).

n = 3, there are five cases.

1) d = 5, 6 = 0(1, 1, 0, 0). 2) d = 6, d ti 0(1, 1, 1, 0). of

1Pa

x 1P1

M

is a finite double covering

and its branch locus

is a smooth member of

B

12Ha +

3) d = 7, 6 = 0(1, 1, 1, 1).

along a smooth curve

C

M

is the blowing-up of

1P3

which is a complete intersection of

two hyperquadrics.

4) d = 8, d = 0(2, 1, 1, 1).

smooth hyperquadric

Q3

in

5) d = 9, 8 = 0(2, 2, 1, 1).

M IF4

is the blowing-up of a along a smooth conic curve.

M = 1P1 x 11.

(10.7.4) When n > 3, (M, L) of

(IP1, 0(1))

and

(Q3, 0(1)).

must be the Segre product Thus

n = 4, d = 8

and

6 = 0(1, 1, 1, 1, 1).

For a proof of these facts, see [Iol].

We can obtain

another proof by using the facts in (15.20) in Chapter II,

which were proved in [F21].

Thus, we have a complete classification when (10.8) Suppose and (3.5.2).

Let W = p(M) w = 4

d = 4.

So we have and set and

We have

BsILI _ 0

p: M -+IFa+l Then

w = deg W.

deg p = 1, or

such that

d > 4.

by (4.15) p H6 = L.

4 =

w = deg p = 2.

(10.8.1) In the former case, p

is birational and W

89

§10: Polarized manifolds of A-genus two

is a hypersurface of degree

g(M, L)

So

4.

W

The equality holds if and only if

g(W, Ha) = 3.

is normal, and

in this case by Zariski's Main Theorem.

If

g(M, L) = 2,

has a similar structure to those in (10.7).

(M, L)

M = W

In

fact, using-(10.2) further, we obtain the following. Theorem.

Let

be a polarized manifold with

(M, L)

n = dim M z 2, d(M, L) = 4 that

and

d(M, L) = 2

defines a birational morphism

ELI

and suppose

p: M -.,n+1

Then one of the following conditions is satisfied. a) M = p(M)

is a smooth hyperquartic.

b) g(M, L) = 2, n = 2, M 8

points and

is a blowing-up of

L = 2X77 + 3H

c) g(M, L) = 2, n = 3

and

to those in (10.7.3) with d) g(M, L) = 1, n = 2 bundle

and

X

(M, L)

has a similar structure

8 = 0(1, 0, 0, 0). (M, L)

is a scroll of a vector There is an exact

C.

sequence (#): 0 -- T1 -+ 8 -4 T2 -> 0 for some with

deg(.T

di) Fl dii) Fl

at

IP

1P17

- E (cf. (10.7.2.0)).

on an elliptic curve

8

g(M, L) = 3.

T

e Pic(C)

= 2, and we have either 3"2

and the sequence

(#)

splits, or

and the sequence

(#)

does not split.

g2

The proof uses the results in X15 and will be given in (15.22).

When n = 2, the corresponding result can be

deduced also from the classification theory of quartic surfaces (cf. [Umz], [Ur]). (10.8.2) When

w = deg p = 2, p

of a possibly singular hyperquadric

is a double covering W.

The structure of

such double coverings are studied in [F13].

If

n s 3, W

turns out to be smooth and the observations in (6.11) and

90

I. A-genus and the Apollonius method

In particular the branch locus is a smooth

(6.12) apply.

hypersurface section and is connected.

n = 2, there are

If

several possibilities, as follows. If

Let

v

W

is singular, W

is a cone over a conic curve.

Then

be its vertex.

is an isolated fixed

p-1(v)

point of the sheet changing involution of branch locus consists of surface section of

W

locus of

p

over

The

W.

and a smooth connected hyper-

v

(M, L)

In particular

(*11) in [F13].

If

W.

M

is said to be of the type h1(M, 0M) = 0.

is smooth, we have

W = 1P1 X 1P1.

The branch

is either a smooth ample divisor (a:

type

(1(1,1)+) in [F13]), or the sum of several fibers of one ruling (def type (1'(1,1)0)

In the latter case

for some hyperelliptic curve

M = 1P1 x C

C.

deg p = 1, there is no upper bound for

Unlike the case g(M, L)

in [F13]).

in the above cases where

(10.9) Suppose

d = 3.

deg p = 2.

In this case

BsILI

may and

may not be empty. (10.9.1) When M ->IPn.

If

BsILI _ 0,

ILI

gives a triple covering

n 3 4, such a triple covering is of triple

section type by [Lazi] (or [F20]).

This means that

embedded in the total space of a line bundle over triple section.

Fn

as a

which are not of triple section type.

Their structures have yet to be studied.

See also [Mirl],

for triple coverings.

(10.9.2) When

dim BsILI = 0, we have the following

results (cf. [F23]). 1) BsILI

IPn

is

However, when n s 3, there are many smooth

triple covers of

[F20], [To1-3]

M

consists of one simple point

p.

§ 10: Polarized manifolds of 0-genus two 2) Let

x: M' -+ M

of

curve such that 4) S = p(E)

p

with

Then

p.

yields a morphism

Every fiber X

p

be the blowing-up at

the exceptional divisor over 3) jx*L - El

91

and let

Bsln L - El _ 0.

p: M' - Fn

dim X > 0

E be

of degree two.

is an irreducible

EX = 1.

is a hyperplane in

Pn

Furthermore, it turns out that there are the following three possibilities: *

a) p S = 2E + D

for some effective divisor D

with

dimp(D) < n - 1. p*S

E*

+ D, dimp(D) < n - 1, and * * divisor such that p(E ) = S, E n E _ 0. * * * b)

= E +

E*

is a prime

C) P S = E + E + D, dimp(D) < n - 1, p(E ) = S and E n E * is a hyperplane in

E 2:

Fn-1.

The above cases are studied thoroughly in [F23] and it is found that the situation is similar to that of the corresponding cases in (6.20). In particular we have g(M, L) + 3

in case b) and

n

3

2n s

in case c), while case

a) can occur in any dimension. (10.10) When

d = 2, we have only partial results.

consists of two points, which may be infinitely near.

BsILI

When

g(M, L) = 2, we can say a little more (see [Lan)), and

there are several interesting examples.

However the general

case is still difficult to study.

(10.11) Thus we have classified polarized manifolds (M, L)

with

4(M, L) = 2 (assuming char(.) = 0) except the

following cases, which are yet to be studied.

1) d = L n = 1.

92

I.0-genus and the Apollonius method

2) d = 2

and

is a finite set.

BsILI

Fn

3) Triple coverings of 4) dim BsILI = 1

not of triple section type.

and a general fiber of the mapping

p

in

(10.4) is non-hyperelliptic.

(10.12) Chart. Classification of polarized manifolds of d-genus two.

A (M, L) = 2

g

dim M = 2

&

2

(10.2)

elliptic scroll

d 1 2

dim BsILI=1

dim BsILI;90

d=2

d=3

d=4

(10.7) [i01]

triple cover Fn of

(10.4)

da5

Is

p

1

type (10.9.2) [F23]

(10.5) [F16]

(10.8.1)

birational to a hyperquartic

(10.8.2)

double cover of a hyperquadric [F13]

hyperelliptic ?

Chapter II. Sectional Genus and Adjoint Bundles Throughout this chapter we assume that K is the complex number field C, but most of the results are true if char(A) = 0. How they can be generalized in positive characteristic cases is an interesting problem. The contents of this chapter are relatively independent of those in Chapter I. It is not necessary to have read Chapter I, but it could be helpful. §11. Semipositivity of adjoint bundles (11.1) Throughout this section let dim M = n s 2.

ized manifold with bundle of

nef for

Let

K

be the canonical

We will consider whether or not

M.

be a polar-

(M, L)

K + tL

is

t > 0.

The theory of adjoint bundles was first developed by BsILI = 0 (cf. §18 in Chapter B) using

Sommese in the case

However, our approach is technically

Apollonius method.

independent of his, since we do not assume

BsILI = 0.

Our main tool is Mori-Kawamata theory, and the result Thus, if

(0.4.16) is especially important.

nef, there is an extremal curve the contraction morphism (11.2) Theorem. (IPn, 0(1)).

of

0: M --* W

K + nL

and

(cf. (0.4.16)).

R

is nef unless

In particular K + tL

is not

(K + tL)R < 0

with

R

K + tL

(M, L) =

is always nef if

t > n.

The proof will be given in (11.6). Remark.

The nefness of

from Mori's theory [Mor3].

extremal curve [Mor3;(1.4)].

Z

with

K + tL

follows

Indeed, if not, there is an

(K + tL)Z < 0

This is impossible since

(11.3) Lemma.

t > n

for

Suppose that

K + tL

and

-KZ g n + 1

by

LZ > 0.

is not nef and let

II. Sectional Genus and Adjoint Bundles

94

0, W, R

be as in (0.4.16).

such that any

X be a subscheme of M with

Let

birational.

Suppose in addition that

x = O(X)

Then

is a point.

B E Pic(M)

and any

q s dimO-1(x)

Hq(X, B

with

q = dim 0-1 (x) = dim X.

ample line bundle H

on W

subsystem of

X

Dl, , Dn_q

general members

V0 = M.

is a subscheme of HP(VJ, B + sH) = 0

Vn_q

Then if

for

(B - K)R 2 0.

It is enough

Take a very

Take

x.

Vi = GD1 n

and set

codim VJ = J.

Moreover X

is large enough.

t

for any

0 A

of

= 0

be the linear

A

We claim

p > 0, j 2 0, s >> 0.

We prove this claim by induction on

is nef and big on M

B - K + sH

)

consisting of members passing

JHI

n iDa, and

and let

is

dim X > 0

Proof (cf. [F18;(2.3)], [Mor3;(3.25.1)]). to consider the case

0

for

j.

By (0.4.16.3),

s >> 0.

So the

assertion follows from the vanishing theorem (0.4.12) when j = 0.

0 --' OV

When

i-1

j > 0, we have an exact sequence

[-iH] -b OV -p OV j-1 j

-i 0

since the defining

Di's form a regular sequence at each point.

equations of

Hence the assertion follows from the induction hypothesis. Having proved the claim, we next use the exact sequence -' OX -p 0,

0 --> 3 - OV

where

i

is the ideal defining

n-q

X

in

Vn_q.

Since

dim(Supp(i)) s q, we have

hq(X, B + sH) g hq(Vn_q, B + sH) = 0 above claim. point.

Thus

On the other hand HX = 0 hq(X, B

(11.4) Lemma.

X

)

with

s >> 0

since

by the O(X)

is a

= 0, as desired.

Assume the hypotheses of (11.3) and

suppose in addition that 0-1(x)

for

X

is an irreducible component of

dim X = dim 0-1(x) = q > 0.

be a desingularization of

X.

Then

Let

a: Y - X

Hq(Y, BY) = 0

for any

§ 11: Semipositivity of adjoint bundles

line bundle

on M

B

such that

Proof (cf. [F18;(2.4)]).

95

(B - K)R z 0.

We have an exact sequence

0 --i OX -+ 9*0Y - W -f 0 of sheaves on X where supported on

dim(Supp(T)) < q.

Hence

Sing(X).

hq(g*0y[B]) s hq(X, B) = 0

by (11.3).

{x e X

I

Next consider the

H1+j(Y, By) (cf. (0.1.8)).

dim g

(x) > 0)

Therefore

E2'J = H1(X, Rig*OY[B])

Leray spectral sequence with converging to

Let

S =

Then

E = g 1(S).

and set

dim(Supp(RJ9*0Y)) g dim E - j

< q - j

by (0.1.9).

j = q - i > 0.

We have

E2'0 = 0

E2 'J

for

= 0

preceding observation, so i + j

Let

(11.5) Lemma.

by the

with

i, j

0 be as in (11.3) (in particular

birational) and suppose that x

for any

= 0

Ex1,'3

Hence

which yields the desired assertion.

= q,

point

is

1W

on

Then

W.

for some

q = dim 0-1 (x) > 0

(K + qB)R z 0

B E Pic(M)

for any

with BR > 0. Proof (cf. [F18;(2.5)]).

Assuming

(K + qB)R < 0, we

will derive a contradiction. Take a component X as in (11.4) and let by (0.4.16.3), so Similarly

hand

-[K + qB]y

is nef and big. for

i < q for

hq(Y, K + tB) = 0

x(Y, K + tB) = 0

for any

and

t 2 0

Hence

for at most

x = 0

is nef and big.

Therefore t s q.

On the other Hence

By the Riemann-Roch of degree

t

q

0-ample

is

by (11.4).

0 a t s q.

Theorem, this is a polynomial in BY > 0.

BY

is ample and

BX

h1(Y, [K + tB]Y) = 0

be its smooth model. B

Y

0-1(x)

of

values of

q

since t,

contradicting the above observation. (11.6) Proof of Theorem (11.2). is not nef.

We take

R

and

0

with

Suppose that K + nL (K + nL)R < 0

as

II. Sectional Genus and Adjoint Bundles

96

is birational, we have

0

If

above.

n > dim 0-1(x) > 0

x e W, but this contradicts (11.5).

for some Thus

F

Let

dim W < n.

be a general fiber of

Its canonical bundle is the restriction of -(K + nL)

F

is ample on

Moreover

K.

By (1.3), this

by (0.4.16.3).

(F, LF) . (,n, O(1)), F = M

implies

0.

W

and

is a point.

Thus we have completed the proof.

This result is still valid if

Remark.

M

is allowed

to have certain mild singularities, e.g. rational Gorenstein singularities (cf. [F18]), or even arbitrary log-terminal See [KMM] (or [Rd4], [Mor4])

singularities (cf. [F24;§3]).

for the notion of such singularities.

Similarly the

subsequent results in this section can be generalized to the singular case, but we usually need stronger assumptions on K + tL

singularities for

(11.7) Theorem.

K + (n - 1)L

with smaller

Suppose that

K + nL

Then

is nef.

is nef except the following cases: Fn+l

is a hyperquadric in

a) M

t.

L = 0(1).

and

b) (M, L) = (F2. 0(2)).

is a scroll over a smooth curve.

c) (M, L)

Suppose that

Proof. R

and

0

(11.6), 0

(K + (n - 1)L)R < 0

with

general fiber Hence

for every fiber reduced.

is not nef.

Take

As in

as before.

cannot be birational by (11.5). Then

dim W > 0.

Suppose that

(11.6).

K + (n - 1)L

F

0, so

of

W X

We have

0.

h0(X, LX)

opr

(F, LF) =

is a curve. of

dim F g n -

Moreover

Therefore a

X

,

0(1))

Ln-1X

=

as in

Ln-1F = 1

is irreducible and

h0(F, LF) = n

semicontinuity theorem (0.1.11), so

for any

1

by the upper

d(X, LX) 4 0

and

§ 11: Semipositivity of adjoint bundles

Hence

(X, LX) = (1Pn-1, 0(1)).

free sheaf on W

and

97

is the scroll of

(M, L)

M

dim W = 0, every curve in

proportional to

If

R.

L ti aA

Pic(M), then K + (n - 1)aA

(M, A) = (F200(1)).

is numerically

for some

and A E

a 2 2

is not nef and

This is possible only when

by (11.2).

Thus

S.

c) is satisfied.

the condition When

is a locally

= 0*0M(L]

48

(n - 1)a s n

n = a = 2

and

b) is satisfied.

Thus

If neither of the above cases occur, every line bundle on

M

is numerically an integral multiple of In fact

infer K ti -nL.

K = -nL

Hence we

L.

by (0.4.16.3), so

(M, L)

a) by a result in [Kob01, the proof of which is

is of type

similar to that in (1.3). (11.8) Theorem. that

Suppose that

Then K + (n - 2)L

n > 2.

K + (n - 1)L

is nef and

is nef except in the

following cases.

a) There exists an effective divisor (Fn-1,

(E, LE)

and

0(l))

E on M such that

[E] E = 0(-l).

bO) (M, L)

is a Del Pezzo manifold with b2(M) = 1, or

(F3, 0(j))

with

in

F4

with

j = 2 or 3, (1P4, 0(2)), or a hyperquadric

L = 0(2).

bl) There is a fibration

0: M --* W over a smooth curve

w

with one of the following properties: for every fiber

bl-V) (F, LF) = (1P2, 0(2))

bl-Q) Every fiber in

Pn

F

of

0

member of

is locally free, M 12H(8) + BPI

restriction of

H(8).

of

0.

is an irreducible hyperquadric

having only isolated singularities.

0*0M[L]

F

is embedded in

for some

Moreover, P = F(8)

B E Pic(W) and

L

8 =

as a is the

II. Sectional Genus and Adjoint Bundles

98

b2) (M, L)

is the scroll over a smooth surface.

For the proof, we use the following fact. (11.8.0) Lemma.

Let

onto a projective variety

X

algebraic variety

f: X --* Y be a morphism from an

be a Cartier divisor on X there is a curve f(z)

in

Z

such that

codim f(E) > 1.

EZ < 0

such that

Supp(E)

E Then

and

is a point.

We use induction on n = dim X

tive.

X

We may assume that

Outline of proof.

is projec-

m = dim f(E).

and

m > 0, we take a general hyperplane section

If

Let

Y.

X

and replace

and

Y

f-1(A)

by

and

A.

n > 2, we take a general hyperplane section replace

X

by H and

Y

and

problem to the case where

f(H).

If

A

on

m = 0

on X

H

Y and and

Thus we reduce the

and m = 0, and use the

n = 2

classical index theorem.

This lemma is true even if

Remark.

X

is a non-KAhler Moishezon variety.

lemma only in the case where

f(E)

char(.) > 0, or if

Here we use this

is a point, but we give

a general version for the convenience of later use. (11.8.1) Beginning of the proof of (11.8). (n - 2)L 0: M --' W

and suppose that K + A

for some

a prime divisor such that is a curve

C

by (0.4.16.2). O(Z)

in

is birational.

dim W = n, 0

= n - 1

45-

E

Hence

is a point, so

x E W by (11.5).

such that

Z c E.

R

and

EC < 0.

0

Let

Therefore

for any curve Thus

Moreover E

be

By (11.8.0), there

E c '-1(x).

EZ < 0

Take

is not nef.

A =

(K + A)R < 0.

as in (0.4.16), so

(11.8.2) When

Set

Z

contracts

ER < 0

such that E

to a

§ 11: Semipositivity of adjoint bundles

99

point, but nothing else.

By (11.5) we have

(K + (n - 1)L)R k 0.

the equality must hold. B e Pic(M)

some

since

Indeed, otherwise, 0 < BR < LR (K + (n - 2)L)R < 0.

for

LR = aHR

So

This contradicts (11.5) since

a 2 2, H E Pic(M).

for some

We claim that

0 > (K + (n - 2)L)R = (K + a(n - 2)H)R. K + (n - 1)L

By this claim [K + (n - 1)L] E = 0.

hi (M, K + tL + £H)

have

(11.3), where on

For any

W.

Hence

hi(E, tLE) = 0 X(E, tLE) = 0

H

=

comes from

i > 0,

t k 0

hl(E, K + tL) = 0 for for

for

i > 0,

t = 1 - n,

as in

since

(1Pn-1,

(E, LE) =

Thus we have

0(1))

and by (1.1).

[E]E = 0(-l)

by the

a) of the theorem.

(11.8.3) When w = dim W < n, let Then

x(t) _

is a

x(t)

Ln-lE = 1

Therefore

n - 1.

so

t z 0,

x(0) = 1. Hence

and

-1

[K + (n - 1)L] E = 0, we get

adjunction formula.

(K + (n - w + 1)L)x

X

be a general is nef by (11.2).

n - w + 1 >n-2 and ws 2.

Hence

(11.8.4) When w = 0,

H

>> 0, we

This implies

i > 0, t z 1 - n.

h0(E, LE) = x(l) = n, so

0.

£

is the pull-back of an ample line bundle

polynomial of degree

fiber of

and

Hence

hi (M, K + tL + £H - E) = 0

x(t) = (t + 1)...(t + n - 1)/(n - 1)!

Since

Pic(W).

Pic(M) = Z

K = kH, L = £H.

be the ample generator and set

k + (n - 1)i a 0, k + (n - 2)1 < 0 k k -(n + 1) unless

by (11.2).

Therefore

by assumption. L = 1

and

Then

Moreover

k = 1 - n

(n, -k, £) = (3, 4, 2), (3, 4, 3), (3, 3, 2)

(4, 5, 2).

Thus we have

Let

by (0.4.16.3).

or

b0).

(11.8.5) When w = 1, take

A E Pic(M)

attains the smallest positive value.

Then

such that

LR = LAR

AR and

100

II. Sectional Genus and Adjoint Bundies

KR = kAR

for some

k + (n - 2)t < 0 Hence

k,

by applying (11.2) to

k = 1 - n

(X, AX) = (F200(1))

A

0, since

of

and

L

x

As in the case

= 0(2).

for every fiber

F

Thus we have

O-ample by (0.4.16.3).

is

by (0.4.16.3),

[K + 3A]x = 0

(F, AF) = (F2, 0(1))

(11.7;c), we have

X.

unless (n, -k, t) = (3, 3, 2).

(5-i) In the latter case so

k + (n - 1)t k 0,

We have

E Z.

k k -n

and

and

t = 1

t

bl-V).

in

Fn

comes from

as in (11.7;a).

L = 0X(1)

and

F

of

most two prime components. then

F1C > 0

for some curve

F = 2Y, then

YR = 0

4(F, LF) t d(X, LX) = 0, so Y = OM(L)

Thus

*

homomorphism

n + 1

on

p: M -+ P 6.

such that

is irreducible

p

B E Pic(W).

the tangent space of

M

p

in

y

PX = Fn.

on

P.

is

I2H + BPI

Therefore

PX

x

Hence

I = Sing(FX),

and the tangent space of

coincide in the tangent space of

(P, H)

over each point

is a member of

At every point

M

The natural

H = L, where

FX = 0-1(x)

itself is an embedding and

is a

6

By (0.1.4), we have

The restriction of

is the embedding of

for some

is

comes from

F

W.

is surjective.

0 6 -, Y

the scroll of

p

C

By (0.1.11),

*

on W

F1 # F2,

is a hyperquadric.

(F, LF)

6 = 0*1.

and

locally free sheaf of rank

a morphism

has at

By the upper semicontinuity theorem we have

and reduced.

Set

But

F2.

E Pic(M)

[Y]

This cannot occur either.

Pic(W).

for

for

and hence this cannot occur.

R

and

Ln-1X = 2, F

=

in

C

K + (n - 1)L

Since

F = F1 + F2

If

numerically proportional to If

Ln-1F

Since

0.

is a hyperquadric

[K + (n - 1)L] F = 0

by (0.4.16.3),

Pic(W)

every fiber

k = 1 - n, X

and

t = 1

(5-li) When

§ 11: Semipositivity of adjoint bundles

[e(P)/e(M)]y = [e(P)/e(PX)]Y

101

for any curve

This contradicts

(2H + B)Y = [F]Y = 0.

Y

X, hence

in

HY > 0.

Thus

I

cannot contain a curve, namely every fiber has only isolated Now we have

singularities.

(11.8.6) When

w = 2, we have

X

for any smooth fiber F

for every fiber

bl-Q).

of

of

0.

D

in

C

We claim

dim F = n - 2

Indeed, otherwise, k(D)

0.

point for some prime divisor is a curve

(X, LX) = (Fn-2, 0(1))

D

such that

on

is a

By (11.8.0), there

M.

Hence

DC < 0.

DR < 0

by

(0.4.16.2). However, there are curves in other fibers which do not meet Thus of

F.

D, so this case cannot occur. dim Fi = n - 2

for any irreducible component

Take a large integer

ample and take general members

Then N = D1 n

n Dn-2

t

such that

scheme of dimension zero consisting of di = Ln-2Fi.

of

JGLl.

Moreover, by applying

is smooth.

Fi, we infer that

Bertini's theorem on

is very

£L

D1, -1 Dn-2

Fi

is a smooth

N n F.

to-2di points, where

Now, let us consider the restriction map

0N: N - W, which is proper and finite over

x = O(F).

If

we take a small enough neighborhood (with respect to the metric topology) U of

x, then each connected component

contains only one point of

ON-1 (U)

The restriction Set

of

(]Pn-2, 0(1))

Then

So

F

is proper finite for every A. deg(ON) _ 2Z mA 3 2. in-2di since

.fin-2di.

On the other hand (X, LX)

for any nearby fiber

Combining these we obtain A.

= N n F.

0Z: UZ --* U

mz = deg(OA).

##((,1)) _ #(N n F)

ON-1 (x)

UZ

X, so

2i di = 1, and

is irreducible, Ln-2F = 1

and

deg(ON) _ to-2

ml = 1 0l

for every

is an isomor-

phism by (an analytic version of) Zariski's Main Theorem.

II. Sectional Genus and Adjoint Bundles

102

Hence

is smooth.

U

Therefore

d(X, LX) = 0

,J (F, LF)

is locally free on

0*OM[L]

scroll of

S.

Hence

by (0.1.11).

for every fiber

(Fn-2, 0(1))

is flat by (0.1.12), and

0

F.

Thus we have

is smooth and

W

Clearly

W.

(F, LF) = 6 _

is a

(M, L)

b2).

Thus we have completed the proof of (11.8).

1) The result (11.8) was obtained by

(11.9) Remark.

Ionescu [Io3] and by the author [F18] independently by different methods.

2) The argument in (11.8.6) works to prove that (M, L)

is a scroll over

W, if

equidimensional, W

is normal and if

(M, L)

smooth and

over an open dense subset of

0

W

is

is

is a scroll

See [F18;(2.12)].

W.

(11.10) Theorem (11.8) can be viewed as a polarized version of the following classical fact (cf. (0.5.5)). Theorem.

The canonical bundle

algebraic surface a) S

S

K

of a smooth

is nef except the following cases:

contains a (-1)-curve.

bO) S = F2 bl) S

IP1-bundle over a smooth curve.

is a

(11.11) As a polarized version of the theory of rela-

tively minimal models of surfaces, we have the theory of minimal reduction as follows.

This notion was introduced by

Sommese in a slightly different context.

A polarized manifold

(M, L)

E

minimal if there is no divisor a divisor

manifold

E M'

exists, then at a point

M

will be said to be as in (11.8;a).

If such

is the blowing-up of another

x, and

E

is the exceptional

§ 11: Semipositivity of adjoint bundles

L'

E Pic(M'), and

L + E = L'M

Moreover

divisor (cf. (0.1.6)).

is ample on

L'

103

for some

M' (cf. (7.16)).

Note

and K + (n - 1)L = [K' + (n - 1)L'] M for the canonical bundle K' of M'. In such a case we say that

(L' )n = Ln + 1

that

(M, L)

is the simple blow-up of

such that

(E', L') = (Fn-1, 0(1))

for the proper transform

(L' - E)Y f 0

So

x ¢ E'

and

E'

lifts isomorphically on

for some

since

positive

LY =

E Y > 0 i

and

i

since

Ei Y < 0

[Ei]E

Ei

since

of the

Indeed, if

= 0(S)

Y C E..

L.

M.

# Ej, there is a curve

Clearly

n a 3.

6, so

= 0(-l)

[Ei]E

E.

We

of a line in

Y

type (11.8;a) are disjoint from each other.

E. n E.

E'

[E']E, = 0(-l).

If in addition n 3 3, exceptional divisors

E. n E. # 0

x.

x, contradicting the ampleness of

passing

E' = 1P n-1

and

x E E', we would have

Indeed, if

x % E'.

claim

at

is not minimal, there is a divisor

(M', L')

If

(M', L')

Y

in

for some

On the other hand

Y C Ei.

Thus we get a

contradiction.

When n = 2, it is possible that contract

Ei

to a point, the other

Ei n Ej # 0.

Ej

If we

does not remain a

(-1)-curve.

When

n a 3, let

n: M - M0

contraction of all the divisors Then M0.

L + 2i Ei = n LO

Ei

be the birational of the type (11.8;a).

for some ample line bundle

By the above observation we infer that

minimal.

This process

reduction of

(M, L).

M -> M0

L0

(M0, L0)

on is

will be called the minimal

Of course the map

n

is the blowing-

up at several points.

When

n = 2, we get similarly a map

n: M - M0

such

H. Sectional Genus and Adjoint Bundles

104

that

is a sequence of simple blowing-ups and

a

In fact

is minimal.

is the blowing-up at several points

n

(not infinitely near) on

M0

However, unlike the

again.

n k 3, such a process

case

(M0, L0)

is not unique in general.

a

So we should say 'a minimal reduction of

(M, L)' in this

Unfortunately, this theory is not powerful enough to

case.

An improved version will be

study polarized surfaces. discussed in §14.

(11.12) As a polarized version of Mori's theory on

threefolds [Mor3], we have a result describing the case in which

K + (n - 3)L

is not nef and

In general, if have

R

moreover

and

K + tL

is not nef for some

as in (11.1) with

0

dim W = n

n > 3 (see [F18;Th.4]). t > 0, we

(K + tL)R < 0, and

dim W g n - t (see [F18;(3.3)]).

or

(11.13) Survey. There is a beautiful application of the theory of adjoint bundles to the structure of projective manifolds containing many rational curves (cf. [Lane])' Let

(M, L)

of dimension

n.

be a polarized manifold defined over For a positive integer

Hilbert scheme of smooth rational curves that 6

LZ = 6.

X E M, let

For

for a general point curves in

X6

on

pa a 0

I

surjective at a general point on SZcF

IV

M

such

is covered by such

ZCM

> 0.

6

x e Z).

is surjective, the differential map

of normal bundles

M

p6 a 0.

for some

((x, Z) E M X Xa

in

pa = dim C6(x)

Set

x.

Thus,

M.

if and only if

Suppose that flag scheme

x

Z

be the

X a

be the subscheme of

C6(x)

consisting of curves passing

6, let

C

F.

Let

Since

F

be the

n: F

M

0(F) - a 0(M)

is

Hence the homomorphism

is surjective at any

105

§ 11: Semipositivity of adjoint bundles

general point on a general fiber

Z

of F ---b QCs, where

is identified with the corresponding curve Clearly

A

is trivial, so

NZcF

a=

NZCM

global sections at any general point on

a(Z)

in

Z

M.

is generated by Hence it

Z = F1.

is a direct sum of line bundles of non-negative degrees.

So

(n - 1) + c1(,N) = h0(.N) = dim Xa = dim F - 1 = n + pa - 1 and hence of

pa = cl(.N) = -KZ - 2

K

for the canonical bundle

M.

From this, by (11.2), we obtain (M, L)

ti

(IPn, 0(1)).

the property

Thus

pa s na - 2

is characterized by

(Fn, 0(1))

pa = (n + 1)6 - 2.

unless

Using (11.7) and (11.8),

we get similar characterizations of other varieties too. See [LanP] for details.

[Sat2] studies similar

problems in positive characteristic cases.

(11.14) We have a similar situation in the study of Severs varieties (cf. [Zk2], [LazV]). 1N Let M be a submanifold in with that

where

M

dim M = n

is not contained in any hyperplane and

union of all secant and tangent lines of

This last

M.

M

condition is equivalent to saying that

is mapped isomor-

phically onto its image under the projection from a general point on such cases.

M

IPN.

Sec(M) # IPN

M, namely, the

is the secant variety of

Sec(M)

such

Zak proved

-

IPN

IPN-1

3n s 2N - 4

in

is called a Severs variety if the equality

holds, and Zak classified all such varieties.

It turns out

that there are only four such varieties up to a projective equivalence, and their dimensions are is a Veronese surface

2, 4,

8

and

16.

in

fact

(M, 0(1))

when

n = 2, which is a classical result due to Severs.

(IPa, 2Ha)

F5

In

II. Sectional Genus and Adjoint Bundles

106

Zak's ingenious proof is based on various geometric Among them, the following fact is of funda-

observations.

mental importance.

For a general point {x E M

I

The line passing

tangent line of

Mj.

and

x

Then

Sec(M), let

on

u

Qu

QU

be the set

is a secant line or a

u

(n/2)-dimensional

is an

smooth hyperquadric.

A proof of this fact can be found in [FUR) too. Actually, Zak proved the above assertion for all points on

Sec(M) - M, and this stronger version plays a crucial

role in his theory.

However, the weaker version given above

is enough to study the case

n s 4.

Let us consider the case are many quadrics on

M

notation in (11.13).

Hence

(F200(2)) If

u

for example.

in this case, so KQ = -3.

is a scroll, Q

(K + 2L)Q = 1.

In view of

is the scroll of h0(M, L) = 5

and

Thus we conclude

is a section of

over

So

under the

C = 1P1.

is

(K + L)Q < 0. since

M --' C

Q2 = 1, we infer that

0(2) ® 0(1)

There

By (11.7), (M, L)

(M, L)

But then

M c 1P4, contradicting the assumption. (M, L)

(1Pa, 2Ha).

n = 4, M contains many

(1P1

but more delicate arguments, we infer case.

p2 = 1

or a scroll over a curve since

(M, L)

When

n = 2

(M, L)

(1P2

a

x

1P2, Ha + H,)

X 1P1)'s. K + 3L = 0

By similar in this

by the classifica-

tion theory of Del Pezzo manifolds (cf. (8.11)).

See [FUR]

for details. Remark.

The four Severi varieties are closely related

to the four standard

I-algebras: A, S[i]/(i2 + 1), the

quarternion algebra and the Cayley algebra. This was

§ 12: Polarized manifolds of sectional genus S 1

107

discovered by Roberts [Rob;pp.13-18]. §12. Polarized manifolds of sectional genus (12.1) Theorem. g(M, L) z 0

for any polarized manifold

The equality holds if and only if

(M, L).

s 1

A(M, L) = 0.

Suppose that

g(M, L) s 0.

Then

(K + (n - 1)L)Ln-1, hence

K + (n - 1)L

is not nef.

Proof.

So

In the cases a) and b), clearly we have

(11.7) applies.

A(M, L) = g(M, L) = 0.

In the case c),

scroll of a vector bundle

of rank

d

is the

(M, L)

n

over a curve

and K = -nL + a*(det 6 + KC), where

Ln = c1(6)

Hence

0 > 2g - 2 =

is the canonical bundle of

C

and

n

is the map

C.

KC

M -> C.

2g(M, L) - 2 = (- L + n*(det 8 + KC))Ln-1 = deg KC.

Hence

g(M, L) = g(C)

Thus

and

C = F1,

so

Thus we have completed the proof.

d(M, L) = 0.

(12.2) Remark.

is a scroll over a curve

(M, L)

If

g(M, L) = g(C).

C, we have always

(12.3) Theorem.

with

implies

g g 0

g(M, L) = 1.

Let

Then

(M, L) (M, L)

be a polarized manifold is either a Del Pezzo

variety or a scroll over an elliptic curve. Proof.

If

K + (n - 1)L

is not nef, (11.7) applies.

g(M, L) = 1, we must have

Since (M, L)

(11.7;c).

By (12.2),

is a scroll over an elliptic curve.

If K + (n - 1)L is nef, we have K + (n - 1)L = 0 *H for some

0: M -> W

(0.4.15).

assumption.

W

and an ample line bundle

On the other hand, Since

is a point.

L

Hence

H

HMLn-1 = 2g - 2 = 0

on W by by

is ample, this is possible only when (M, L)

is a Del Pezzo manifold.

(12.4) Thus we have satisfactory results in the case

II. Sectional Genus and Adjoint Bundles

108

where

I suspect that they are valid under

g(M, L) s 1.

much weaker assumptions.

See §19 for details.

§13. Classification of polarized manifolds of a fixed sectional genus: higher dimensional cases Using the theory of adjoint bundles, we will establish an algorithm for classifying polarized manifolds (M, L) with n = dim M and g = g(M, L) fixed. By this method we can prove the following result for example. (13.1) Theorem.

n

For any fixed

and

only finitely many deformation types of

g, there are except

(M, L)

scrolls over curves.

Here, 'deformation type' is defined as follows.

By a

deformation family of polarized manifolds we mean a proper surjective smooth morphism

f: .M -* T

possibly non-compact complex manifold f-ample line bundle the fiber Mt.

Then

f-1(t)

Y

on

and let

(Mt, Lt)

onto a connected but T

together with an

X.

For each

Lt

be the restriction of

t e T, set

Mt

be

L

to

is a polarized manifold for each

t,

which is said to be a member of this family. Polarized manifolds

(M, L)

and

are said to

(M', L')

be deformation equivalent to each other if there is a chain (M0, L0) _ (M, L), (M1, L1),

polarized manifolds such that

of

(Mk, Lk) _ (M', L') (Mj_l, Lj_1)

and

L.)

(M j,

belong to the same deformation family for each

j.

A defor-

mation equivalence class will be called a deformation type. (13.2) The Hilbert polynomial deformation invariant of

hand, for any fixed deformation types of

(M, L)

x(t) = x(M, tL) by (0.1.11).

is a

On the other

x(t), there are only finitely many (M, L)

with

X(M, tL) = x(t).

§13: Classification of polarized manifolds of a fixed sectional genus:

109

higher dimensional cases

This is a consequence of Matsusaka's Big Theorem [Mat]. Indeed, there is an integer such that = X(t).

mL

m > 0

depending only on

is very ample for any

Such pairs

(M, L)

with

x(t)

X(M, tL)

are parametrized by a certain

(M, L)

subset of the corresponding Hilbert scheme, which has only finitely many connected components.

Thus, (13.1) is equivalent to the following assertion.

For any fixed n polynomials

g, there exist finitely many

and

xl(t),

,

such that the Hilbert poly-

xk(t)

nomial of any polarized manifold g = g(M, L)

and

is one of the

(M, L)

xj's

with n = dim M

unless

is a

(M, L)

scroll over a curve.

We actually need to exclude scrolls

(13.3) Remark.

over curves.

n

rank of

Indeed, for any ample vector bundle

over a curve satisfies

6

C

n = dim M

and

there is no upper bound for is ample for any

A E Pic(C)

are infinitely many possible

We may assume

n 3 2

and

Then, by (11.7), K + (n - 1)L scroll over a curve. assume that

S

However,

Hence there

deg A > 0.

with

6 ® A

xj's.

(M, L)

g k 2

with fixed

n

and

by the results in §12.

is nef unless

(M, L)

is a

In the rest of this section we will

K + (n - 1)L

(13.5) Suppose that Ln

g(M, L) = g.

(M, L)

d = Ln = c1(.), since

(13.4) Now we will classify g.

g, the scroll

of genus

of

'

is nef.

K + (n - 2)L

is nef.

Then

0 < d

(K + (n - 1)L)Ln-1 = 2g - 2, so there are only

finitely many possible values for the following general result.

d.

Hence we can apply

II. Sectional Genus and Adjoint Bundles

110

(13.6) Theorem (cf. [Ko1M]). For any fixed

n, d and g,

there are only finitely many deformation types of

(M, L).

Proof (a slightly modified version of that in [KolM]).

We may assume that A = K + (n + 1)L ak = AkLn-k

We claim that

, n

if

n, d

and

is ample by (11.2).

is bounded for each

This is clear for

are fixed.

g

k = 0,

2 for each k by the ak+lak-l S ak Index Theorem (0.4.6). Hence the claim is proved by

k = 0, 1.

Moreover

induction on

k.

This claim implies that so

(K + mL)n

KkLn-k is bounded for each

is bounded for each

Therefore

m > n.

h0(K + mL) = n + (K + mL)n - A(M, K + mL)

k,

is bounded too.

By Serre duality and Kodaira's vanishing theorem we X(M, -mL) = (-l)nh0(M, K + mL)

infer m > n.

Since

X(M, tL)

is bounded for each

is a polynomial of degree

coefficients are determined by its values at -(2n + 1).

polynomials for

n, its

t = -(n + 1),

Hence there are only finitely many possible X(M, tL).

This is enough by the

observation in (13.2). (13.7) Remark.

(13.6) is enough to prove (13.1) in

the case (13.5), and assures us, at least philosophically, of the existence of certain classification theories of polarized manifolds of this type.

Unfortunately, however,

this method is not powerful enough from the practical viewpoint. It yields a list of possible Hilbert polynomials, but this list is usually too big and far from being best possible.

Moreover it gives very little information on the

deformation type itself. q = h1(M, 0M)

For example, the irregularity

should be bounded since it is a topological

§ 13: Classification of polarized manifolds of a fixed sectional genus: higher dimensional cases

111

invariant, but we need another method to accomplish this (as a matter of fact, we conjecture that

arbitrary polarized manifold

for

q a g(M, L)

(M, L)).

Thus, we should not be satisfied, but should not be too disappointed.

No method has almighty power, and the case in

which K + (n - 2)L

is nef is a kind of 'general type'.

The above method at least provides us with a starting point for a classification theory (cf. [BeBS]). (13.8) In the rest of this section, we consider the

case in which K + (n - 2)L

So (11.8) applies when n > 2.

is nef.

have the classical result in (11.10). case (11.8;a), we do not have in the case (11.10;a).

K + (n - 1)L

is not nef while

LE = 1

When n = 2, we However, unlike the for the (-1)-curve

E

Therefore the theory of minimal

reduction as in (11.11) is not enough to study polarized We study the case

surfaces.

Here we assume

n = 2

n > 2.

(13.9) In the case (11.8;a), let minimal reduction as in (11.11).

K0 + (n - 1)L0

and

K + (n - 1)L.

Moreover

number fixed

(M, L) v.

(M0, L0)

Clearly

g(M0, L0) = g

L0n M0

= d + v, where

The deformation

is blown up.

is determined by that of

(M0, L0)

g, and then decide how many points on (M0, L0).

Clearly

problem is reduced to the case where

is

is the

v

Hence we should first classify

blown up for each

be the

is nef since its pull-back on M

number of points at which type of

in the next section.

and the

(M0, L0)

M0

for

can be

v < L0n, so the (M, L)

is minimal.

By (13.5) and (13.6), it suffices to consider the cases b0), bl) and b2) in (11.8).

In the case b0) we use the

II. Sectional Genus and Adjoint Bundles

112

classification theory of Del Pezzo manifolds in §6 and §8. The other cases will be studied separately.

(13.10) The case bl) is further divided into the two cases

and

bl-V)

In case

f: M -+ W

bl-V), we have for every fiber

(IP2, 0(2))

locally free sheaf of rank of

8

and

*

L = 2H + f B

=

0 g H3 = e

H

since

6g - 6 = d + 4e.

b = (g - 1)/2 - e

on

Of course

B E Pic(W).

Let

q

be

e = c1(6), b = deg B.

Moreover

is nef.

2g - 2 = HL2

and

d

So

e

are bounded.

Hence

Moreover H = K + 2L =

by the canonical bundle formula for

q

Thus

is also bounded.

W.

Hence

In fact we

4q - 4 = 2e - 2(g - 1) < g - 1.

have

Enumerating all the possible triples obtain a classification of

(M, L)

(q, e, b), we

of the type

deformation type is clearly determined by (13.11) In case on a scroll where of

is a

is the scroll

(M, H)

is the canonical bundle of

0 = 2q - 2 + e + 2b.

n = 3.

8 = f*(0M[H])

W.

is bounded, too.

KW

(F, LF)

d = L3 = 8H3 + 12H2B = 8e + 12b

H + f (KW + det 8 + 2B) scrolls, where

3

such that

f.

and

and set

4H3 + 4H2B = 4e + 4b,

and

of

for some

W

the genus of the curve Then

F

Then HF = 0(1)

H = K + 2L.

Set

bl-Q).

W

(P, H)

B E Pic(W).

and set

of a vector bundle Moreover

L = HM.

6

Let

e = c1(6), b = deg B. Since

the canonical bundle

KP

of

P

is

The

(q, e, b).

is a member of

bl-Q), M

bl-V).

12H + BPI

over a curve q

W,

be the genus rank 8 = n + 1,

[KW + D] P - (n + 1)H

for D = det 8, so K = AM - (n - 1 )L for A = KW + D + B. Hence 2g - 2 = Ln-lA = 2 deg A = 4q - 4 + 2e + 2b, so

e + b = g + 1 - 2q.

113

§ 13: Classification of polarized manifolds of a fixed sectional genus: higher dimensional cases

We claim

s

Indeed, taking

2e + (n + 1)b s 0.

def

determinants of the symmetric matrices giving the quadric Mx

form defining

Px = Fn

in

over

x e W, we obtain a

2D + (n + 1)B

holomorphic section of

which vanishes

exactly at the points over which Mx

Hence

is singular.

s a 0. We have

d = Ln{M} = Hn(2H + B){P} = 2e + b.

Hence

(n - 1)d + s = 2n(e + b), so (n - 1)d + s + 4nq = 2n(g + 1). Therefore

d, s

and

are bounded, so

q

e = ((n + 1)d - s)/2n the possible values of

are also bounded. q, e

and

Enumerating all

b, we obtain a classifi-

and

The deformation type of

cation.

b = (s - d)/n

(M, L)

is determined by

(q, e, b). Eventually we can further determine the vector

Remark.

bundle

This is indeed the case if

8.

(13.12) In case bundle

8

of rank

b2), (M, L)

n - 1

is the scroll of a vector

on a smooth surface

ample by definition, and hence

A

det 6

def

(KS + A)A{S}

S.

for the canonical bundle

is

8

By

is ample.

2g - 2 = (KS +

the canonical bundle formula we have =

g = 2, see §15.

KS

of

S.

A)MLn-1 Hence

g(S, A) = g. For any rational curve

Z

on

decomposable on the normalization of rank(8) = n - 1 2 2.

In particular

is ample and

8

S,

So

Z.

(S, A)

AZ = c1(8Z) Z cannot be a

scroll over a curve.

We should first classify the polarized surface This is done using the method in the next section.

after we classify ample vector bundles for each pair

(S, A).

We have

8

with

(S, A).

There-

det 8 = A

d = Ln = s2(8) = A2 - c2(8)

U. Sectional Genus and Adjoint Bundles

114

by (0.3.7), so and

d

since

A2 1 2

by [B1G]. Moreover

c2(.) > 0

are bounded for any fixed

c2(d)

proof of (13.1) is reduced to the case

Thus the

(S, A).

In order to

n = 2.

get a good classification, however, we must work harder ad hoc

using

techniques.

§14. Classification of polarized surfaces of a fixed sectional genus (14.1) In this section we will classify polarized surfaces

M

E

L-weight of to a point.

bundle

a (L

of weight

)

be the contraction of

LZ k m

Note that

m.

for any (-1)-curve

is said to be a

(M, L)

for the canonical bundle

+ mK ) If

L

is called

for some ample line

In such a case

(M ,

M, m = LE

on

L + mE = n L

Then

blowing-up of

E

a: M -+ M

Let

E.

on M .

L

is

P2 or a 1P1-bundle over a curve.

(14.2) For a (-1)-curve the

K

in this case, unless

E

contains a (-1)-curve

is either

M

is nef, we can

K

If

Hence we consider the case where

argue as in (13.5). not nef.

g(M, L) = g.

with

(M, L)

K

of

M

on

M,

n

L + mK =

Z

is said

to be admissible. (14.3) If

n

is admissible, L + mK

otherwise, (L + mK)Z < 0 theory [Mor3].

Moreover

0: M -+ W

A = L + MK

is not nef.

L + mK

L + (m - 1)K

L + (m - 1)(L + mK).

*

Indeed,

by Mori's

Z

This contradicts the admissibility.

maximal number such that

fibration

for some (-1)-curve

L + (m + 1)K

other hand,

is nef.

m

Thus

On the

is the

is nef.

m(L + (m - 1)K)

is ample since

Hence, by (0.4.15), there is a

onto a normal variety W

for an ample line bundle

A

on

such that W.

Note that

115

§ 14: Classification of polarized surfaces of a fixed sectional genus 0

is determined uniquely by

cases: 0) dim W = 0. In case

1) dim W = 1.

and M

L = -mK

0),

F2

r

at

unless

M = F2, we obtain a

taking

(r - 1)

In case so

KF < 0

and

M

Thus

in the classical sense.

or a blowing-up of

There are now three

(M, L).

2) dim W = 2.

is a Del Pezzo surface is either

points (0 s r g 8).

F1

X

Hence,

F1-bundle over a curve by

times admissible contractions of weight

F

for every fiber

(L + mK)F = 0

1),

and any general fiber is

0 < LZ = -mKZ

We have

1P1.

for any component

of

m. 0,

Z2 < 0

of a singular

Z

Hence any singular fiber is of the form

fiber.

F1

Z'0 =

where each component is a (-1)-curve of

Z + Z

L-weight

m.

Contracting one of them for every singular fiber, we get a 1P1-bundle over the curve

This process is a sequence of

W.

admissible contractions of weight

is birational and

2), 0

In case

any (-1)-curve

with

E

point for some curve Hence

= -mKZ.

m.

Conversely, if

LE = m.

M, we have

on

Z

Hence

Ek

El,

W

L + mK Lb + mKb

of 0

L-weight

on

Lb

0

contractions of weight

on

m

Z Z

i j

> 0

M, which are

W.

L + 2 mEj =0L *[L

Note that

for some

+ mK] =

of

W

and hence

is a sequence of

k

admissible

for the canonical bundle is ample.

Moreover,

is the contraction of all of

is smooth and

ample line bundle

0 < LZ

Thus, there are finitely many

mutually disjoint, and them.

and

is a

(Zi + Zj)2 k 0, contradicting

Zi, Z,, then

the index theorem.

O(Z)

m.

Indeed, if

such curves are mutually disjoint.

(-1)-curves

Z2 < 0

is a (-1)-curve of weight

Z

for some curves

is a point for

O(E)

m.

Kb

II. Sectional Genus and Adjoint Bundles

116

0) - 2), the map

In any of the above cases

factors

0

M -> M .

through any admissible contraction

(14.4) Taking admissible contractions successively we such that

(M', L')

get a polarized surface

]P1-bundle over a curve, or its canonical bundle unless

is nef,

K'

Such a sequence of admissible contractions

M = IP2.

will be called an admissible

Mr = M'

M = M0 -+ Ml -

minimalization of

(M, L).

M' = El, although

We stop when

there is still a (-1)-curve in this case.

m = (mr,

be the

mj

is called the weight sequence of

ml)

,

Let

aj:(Mj_1, Lj_1) -> (M., L.).

weight of the contraction Then

is a

M'

this minimalization.

mr a . a ml.

Note that

canonical bundle of curve of

J J

J

M.

since its

J

+ mJKJ]MJ-1 = Lj_l + mjKj_1

[L

be the

K.

be the exceptional

E.

is nef on

L. + m.K.

J

pull-back

and let

M.

Then

a..

To see this, let

is nef on

Mj_1.

0 f (L. + mjKj)Ej+l = mj+1 - mj.

Hence

(14.5) Definition. The weight sequence of an admissible minimalization of spectrum of

(M, L)

will be called the adjoint

(M, L).

This is because it does not depend on the choice of the minimalization.

Indeed, in the case (14.3;0) and 1), the

weight sequence

m

is

m)

(m,

for every minimaliza-

In the case (14.3;2), every minimalization factors

tion.

through spectrum

W, so mb

m = (mb, m, of

(W, Lt').

proved by induction on Remark.

,

m)

for the adjoint

Thus the independence of

m

b2(M).

This well-definedness of

m

reason why we stop when we reach the case

is the main M'

= 1 1

in

is

§ 14: Classification of polarized surfaces of a fixed sectional genus

117

(14.4).

The adjoint spectrum is a deformation

(14.6) Theorem.

invariant of

(M, L).

We use induction on

Proof.

It is enough to

b2(M).

prove the invariance under small deformations.

f: A - T

Let

with an

be a deformation family of surfaces

f-ample line bundle

for some member over

such that (M, L) = (Mo' L o)

Z

o e T.

Let

admissible contraction of a (-1)-curve (14.2).

p

0

on M

E

as in

By the stability of (-1)-curves (cf. [Kod2]), we is a member of a family

M

may assume that and

be an

po: M --> M

For each

tion of a

(-1)-curve

:

A: A -+ A , by shrinking

is a member of

necessary.

f

pt: Mt -> Mt

t E T,

Et

Mt, and

on

A - T T

if

is the contrac-

Et = 8M

for some t

effective divisor in (14.3), L

0

on

8

+ (m - 1)K0

This implies that

is not a scroll. any is

t e T.

pt

Hence the adjoint spectrum

(m (t), m),

(Mt, Lt).

Ht = Lt + (m - 1)Kt

Lt + mKt is nef since

Then

t E T.

where

Therefore

m (t) m(t)

induction hypothesis on (14.7) Lemma.

Let

m(t)

m = (mr,

,

(M, L)

Then

1) K2 = K'2 - r, + mi,

of

(Mt, Lt)

t

by the

The proof is now complete.

(MC, LC) -i ... --> (Mr, Lr) = (M', L')

2) KL = K'L' + mr +

(Mt, Ht)

is admissible for

is independent of

b2(M).

(M, L).

is

is the adjoint spectrum of

spectrum of a polarized surface

minimalization of

As we saw

is ample by the admissibility of

po, so we may further assume that ample for any

LtEt = m.

Clearly

A.

and

m1)

be the adjoint

and let

(M, L) _

be an admissible

II. Sectional Genus and Adjoint Bundles

118

mi2,

3) L2 = L'2 - mr2 where

is the canonocal bundle of

K'

M'.

4) g(M', L') - g(M, L) = 2 mj(mj - 1)/2 k 0, and the ml =

equality holds if and only if

= mr = 1.

The proof is easy.

(14.8) The deformation type of

by the deformation type of

and the adjoint

(M', L')

We will classify the possible types of

spectrum.

for any fixed

M

is not ruled.

is nef for an admissible minimalization as in

K'

(14.7), so 2g - 2 = L2 + KL 2 d + mr + and

d

(M', L').

of

+ m1

by

Hence there are only finitely many possibilities

(14.7.2). for

(M, L)

g = g(M, L).

First we consider the case in which Then

is determined

(M, L)

m.

After enumerating them, we study the type By (14.7), d'

are determined by

d, g

and

=

L'2, K'L'

and

m = (mr,

g(M', L')

m1).

By (13.6),

this is already enough to prove (13.1) in this case. We have

(K')2.d'

s

by the index theorem.

(K'L')2

Since

K'2 Z 0, we can enumerate all the possible values of

K'2.

M

x(M) = 0

is of general type if and only if if and only if

K'L' > 0, M

K'2 = K'L' = 0.

is an elliptic surface with

K'2 > 0.

When K'2 = 0 and x(M) = 1.

In

order to proceed further we need various ad hoc techniques. In lucky cases we can determine the possible deformation types, but usually we get just a list of possible values of numerical invariants like q = h1(M, 0M), Pg = h2(M, 0M) etc. (14.9) Now we consider the case in which We assume C

of genus

M # F2.

Thus, M'

q, where

is a

(M, L) --->

M

is ruled.

1P1-bundle over a curve

(M', L')

is an admissible

119

§ 14: Classification of polarized surfaces of a fixed sectional genus

minimalization.

for some vector bundle

M = M' = )P(d)

that

where

general fiber of M -i C.

H2

since we assume that

(M, L)

F

is a

K2 = 8 - 8q,

and

and K ti -2H + (E + 2q - 2)F.

E

L ti xH + yF,

z = Ex + 2y, we

Then, setting

KL = 2x(q - 1) - z

L2 = xz, =

and

denotes the numerical equivalence and

ti

get

of rank two such

d

H = H(d)

cl(S) = 0 or 1. Set

e d=

Hence

r = 0.

Now, until (14.12), we study the case

Note that

since

x > 1

Therefore

is not a scroll.

2g - 2 = (x - 1)z + 2x(q - 1) > 2x(q - 1) Z 4q - 4, since z > 0

by

Moreover both

Thus

d = xz.

2q < g + 1

is bounded.

(x - 1)(z + 2q - 2) = 2(g - q) > 0

x

and

and hence

z = ex + 2y, we can

Using

are bounded.

z

enumerate all the possible values of fixed

q

and

E, x, y

q,

for any

The deformation type is determined by these

g.

numerical invariants, and thus we get a classification. is nef if and only if

2H - EF

(14.10) Remark.

semi-stable (cf. [Miy2;§3]).

ample in this case since

Hence

xH + yF

there are many semi-stable vector bundles on values

(q,

If

take

8 = 0 ® 0.

have

y > 0

q =

solutions Set

imply

E

But if

(E, x, y)

b = -KL.

and all the

= 0, the same is true since we can q = 0

and

= 1, we also need to

E

since there is a section

(14.11) Remark.

C

q > 0,

enumerated by the above method do

E, x, y)

really occur.

is

is actually

When

z = ex + 2y > 0.

6

d

with

HA g 0.

Let us consider how many possible g, q

we have for a fixed Then

z = 2x(q - 1) + b

2(q - 1)x2 + bx - d = 0.

one positive solution for

x, so

If

x

and

and

d.

d = xz

q > 0, there is at most and

z

are determined

U. Sectional Genus and Adjoint Bundles

120

uniquely.

z = ex + 2y,

Since

unless both

x

and

z

0,

provided

1

one positive solution positive solution

s

z > x.

However, sometimes we have two

x.

and

a

are even since

In such a case both

P.

a + P = b/2

and

addition, occasionally we have

Hence we

a§ = d/2.

(1, a,

In

- a/2) and/or

The existence of the two solutions

(1, P, a - 9/2).

and

and

b

(e, x, y) = (0, a, fi) or (0, §, a).

have two solutions

two

are also unique

q = 0, the above is true when there is only

Even if

(0, a, §)

y

In the latter case

are even.

can take the both values

d

and

a

reflects the fact that

(0, 5, a)

has

BO

1P1-rulings.

(14.12) Now we study the case where

(13.9), we reduce the problem to the case

L + 2K

is nef and

r > 0.

Then

m = ml > 1.

0 s (L + 2K)L = 4g - 4 - d.

this is already enough to prove

As in

By (13.6),

(13.1).

In order to obtain a more precise classification, we proceed as follows.

0 s (L + 2K)2 =

First we have

4K2 + 8g - 8 - 3d = 8g - 32q - 4r - 3d + 24. q, r

and

are bounded.

d

(14.13) We may fix claim

KL < 0

when

Indeed, if t >> 0

K

x(M, L + tK) > 0

KL > 0, then

# 0 If

for any

since

We

We have

for any

h2(L + tK)

((1 - t)K - L)L < 0.

t >> 0, which is absurd since

KL = 0, then

K2 < 0

is not numerically trivial.

either.

K2 = 8 - 8q - r.

since

K2 i 0.

h0((1 - t)K - L) = 0

ruled.

K2

by the Riemann-Roch Theorem.

IL + tKI

Therefore

_

Therefore

M

by the index theorem,

is

for

Hence this case cannot occur

Thus we have proved the claim.

§ 14: Classification of polarized surfaces of a fixed sectional genus

(14.14) m

is bounded for any fixed

Indeed, L + tK (14.3).

Hence

is nef for any

0 S (L + mK)2

L2, KL

0 s t t m

121

and

K2.

as in

and the assertion is clear

when K2 < 0. When K2 a 0, we have

KL < 0

(L + tK)2 = 0

for some positive real number

determined by

L2, KL

wise

and

K2.

would be ample.

in.

For any fixed

(14.15) Claim.

t, which is

Then m t t, since other-

m(L + tK) = (m - t)L + t(L + mK)

Thus we get a bound for

Therefore

by (14.13).

L2, KL, K2

and

there exist at most finitely many adjoint spectra

r

This is proved by induction on (14.16) For any fixed values of

(14.7).

and

Then we compute

Setting

and

e, x

m.

using (14.14).

g, we enumerate all the possible

q, r, d, KL, K2

above method.

r,

m = (mr,

m1)

d' = L'2 and for

y

by the

b = K'L'

(M'. L')

by

as in (14.9),

we enumerate all their possible values by the method of (14.9).

These invariants

determine the type of possible types of (11.11).

reduction

General pair

(14.17) Remark.

tion invariants, but

e, x, y, r

(M, L).

(M, L)

(M0, L0)

q,

and

m

together

Thus we can classify the

which is minimal in the sense

(M, L)

is obtained from a minimal

by simple blow-ups. q, r, d, KL, K2 e, x

and

y

and

m

are deforma-

depend eventually on the

choice of admissible minimalization. If

(Lr + mrKr)2 > 0, then the minimalization is a

composition of processes of the type of (14.3.2) only. Hence (Mr, Lr) = (M', L'), and

e, x

and

y

are determined

U. Sectional Genus and Adjoint Bundles

122

uniquely.

(Lr + mrKr)2 = 0, the final process of

But if

the minimalization is of the type (14.3.0) or (14.3.1), and e

depends on the choice of (-1)-curves.

may assume

e

In this case we

for the purpose of classification.

= 0

(14.18) Based on the above observations, we have

produced a computer program to enumerate all the possible deformation types of ruled polarized surfaces for any fixed sectional genus

g (see §21).

§15. Polarized manifolds of sectional genus two

We apply the methods of §13 and §14 to classify polarized manifolds of sectional genus two. Details and proofs can be found in [F21] and [F22]. (15.1) Throughout this section let ized manifold with

K + (n - 1)L

be a polar-

g(M, L) = 2, n = dim M a 2.

As in (12.3), (M, L) two if

(M, L)

is a scroll over a curve of genus

is not nef.

Hence we consider the case where K + (n - 1)L

is nef.

(15.2) When n = 2, we have the following result. Theorem (cf. [BeLP], [F21]). (S, L)

with

g(S, L) = 2

Any polarized surface

satisfies one of the following

conditions.

0) There is another polarized surface (S, L)

is a simple blowing-up of

1) The canonical bundle K to

L

1') S

2) K

and

of

S

(S', L')

(S', L')

such that

at a point.

is numerically equivalent

L2 = 1.

is a minimal elliptic surface and is numerically trivial and

3) There is an ample vector bundle

KL = L2 = 1.

L2 = 2.

T

on an elliptic curve

123

§ 15: Polarized manifolds of sectional genus two C

c1(S) = 1, S = F(3'), L = 3H(Y) - AS

such that

A E Pic(C)

with

deg A = 1.

So

4) There is a vector bundle such that Pic(C)

and

S = 1P(Y)

with

Y

on an elliptic curve

Y

is the blowing-up of

P

is the (-1)-curve over

Ep

bundle on

C

60 ) S = 1P1

X 1P1

deg A = 1.

with

E

is a line

.

= 12.

L2

at a point.

P2

is the pull-back of

L2 = 12

and

and

0 2(1)

is the exceptional curve.

62) S = T2 = F(3) 1P1

A

and

L2 = 1.

61) S = E1, the blowing-up of

H

S

L = 5H(f)S - 2AS - 2Ep, p

and L = 2Ha + 3H

L = 4H - 2E, where

C

c1(3) = 1,

such that

and

p

at

L2 = 4.

on an elliptic curve

P = F(f)

on

p

C

B E

for some

(c1(S), deg B) _ (0, 1) or (1, 0).

together a point

where

L2 = 3.

L = 2H(T) + BS

5) There is a vector bundle

for some

L = 2H(S) + H .

and

7) -K

for the vector bundle

is ample, K2 = 1

up of

S"

9) (S, L)

pl, p2

on

L2 = 12.

and

L = -2K.

8) There are a Del Pezzo surface and two points

3" _ [2H ] ® 0

on

S"

at these points and

L2 = 4.

(S", L")

such that

with

(L")2 = 1

is the blowing

S

L = 3L" - 2E1 - 2E2. L2 = 1.

is a scroll over a curve of genus two.

The proof is outlined in (15.3) ti (15.6) below.

(15.3) Suppose that Hence

K

is nef.

Then

0 £ KL = 2 - L2.

d = L2 = 1 or 2. If

implies

d = 2, then K ti 0 (".

index theorem, so If

KL = 0.

K

is nef, this

denotes numerical equivalence) by the 2) is satisfied.

d = 1, we have

1) is valid, or

Since

(K - L)L = 0.

(K - L)2 < 0

Hence

K " L

by the index theorem.

and Since

II. Sectional Genus and Adjoint Bundles

124

(K - L) 2 = K2 - 1, the latter implies

K2 = 0, so we have

1').

K

(15.4) Now we suppose that Clearly

is nef.

b) there is a (-1)-curve

E

on

B E Pic(C).

Then, as in (14.9), K2 = 8 - 8q

Moreover

for

e

= c1(Y) = 0 or 1.

KL = 2x(q - 1) - z

When

= 1, y = -1

d = 4,

q = 0, F

and

z = ex + 2y.

Moreover is a fiber

F

so we have 4).

has a direct sum factor of degree

In either case

In

In the second case we have

is decomposable.

4 = (x - 1)(z - 2)

q t 1.

d = 3.

L + F, where

3H '

Hence we have 3). and

and

(x, z) _ (3, 1) or (2, 2).

is ample since

S -- C.

e + y = 1

e

C, or

for some

y = deg B, q = h1(S, OS) and

q = 1, we have

H def H(3)

and

L = xH(Y) + BS

We may assume

L2 = xz,

the first case we get

T

on a curve

2g - 2q = (x - 1)(z + 2q - 2) > 0, so

When

of

Y

S.

(15.5) In case a) we have and

K + L

Therefore either

S # IP2.

for some vector bundle

a) S = F(Y)

x E Z

is not nef, but

implies

d = xz = 12

Hence g 0.

y > 0

Hence

since

z k x

(x, z) = (2, 6) or (3, 4).

and we can easily obtain

60),

61) or 62).

(15.6) In case (15.4.b), we may assume (-1)-curve

E, since otherwise we have

is nef as in (14.3), so we have If

we have If

d = 4, we get

0).

LE a 2 Then

for any

L + 2K

0 S (L + 2K)L = 4 - d.

2K + L ti 0

by the index theorem, so

7).

d = 3, we have

0 2 (3K + L)2 = 9K2 - 3 K2 S 0, which implies

KL = -1

and

(3K + L)L = 0.

by the index theorem.

Hence

Therefore

(2K + L)2 s -1, contradicting the

125

§ 15: Polarized manifolds of sectional genus two

nefness of

Thus this case is ruled out.

2K + L.

d = 2, we have

If

Hence

theorem.

KL = 0

(2K + L)2 t -2, which is absurd.

The remaining case is

(mr,

bundle of

Hence

S

K2 = -1.

Then

ml = 2.

x(K1) < 0

L12 = 5, K1L1 = -1

q = 1, S1

L1 = xH + BS

and

z = 1, x = 5,

When Hence

2 2

0 = (L

2

+ 3K2)2.

mj = 3

Thus

is a

y = -2.

for

m2 = 2

= -3

Hence

=

K12 = 0.

Since

1P1-bundle over an elliptic curve

e, y

-1 = K1L1 = -z

q = 0, we have

K22 = 1, K L

0 g (L + 2K)2

(q, r) = (1, 1) or (0, 9).

and

= 1

e

and

as in (15.5). Let

Then

be as before.

Then

is ruled.

S

(L + 3K)2 = -2 < 0, we infer

Since

K12 = 9 - 8q - r, we have

When

and

m1 a 2,

Since

is the canonical

K.

On the other hand

by (14.13).

5 + 4K2, so

(S, L) --> (S1, L1)

be the adjoint spectrum.

K1 L1 = KL - m1 < 0, where

we have

K2 < 0

m1)

,

Let

d = 1.

be an admissible minimalization and let

-> (Sr, Lr)

-+

by the index

K2 g -1

and

and

Thus we have since

z = ex + 2y

5 = L12 = xz, so 5).

(L1 + 3K1)2 = -1.

L22 = 9, so

L2 ti -3K2

j a 3

and

and

(L2 + 3K2)L2 =

by the index theorem.

and we have

8).

Thus we have completed the proof of (15.2). (15.7) In the cases

For the case (15.2.2), we have the following.

more to say. Theorem. K '

0

and

3) ti 9) in (15.2), we have nothing

Let

L2 = 2.

(S, L)

be a polarized surface such that

Then one of the following conditions is

satisfied. 1) S

is the Jacobian of a curve of genus two and

class of a translation of the 1') S = C1 X C2

L

©-divisor.

for some elliptic curves

Cl, C2

and

is the

II. Sectional Genus and Adjoint Bundles

126

L = [F1 + F2], where

is a fiber of

F.

L = [Z + F], where

is a hyperelliptic surface and

2) S

S -+ C

F

is a fiber of the Albanese fibration

z

is a section of

a: S -b Alb(S)

a.

3) There is a finite double covering

branched

f: S -fIP2

along a smooth curve of degree six and

L = f 0(1).

is an Enriques surface and its K3-cover

4) S

Fa X FT

finite double covering of

member of

and

I4Ha + 4Ht1.

is a

branched along a smooth

is the pull-back of

L

H6 + Hr.

Proof.

By the classification theory of surfaces (see

(0.5.5)), S

is either an abelian surface, a hyperelliptic

surface, a K3-surface or an Enriques surface. abelian surface, we have If

S

is K3, we get

is an

1) or 1') by the classical theory.

is hyperelliptic, we obtain

S

S

If

h0(S, L) = 3

and

2) by [F22;(2.15)]. 4(S, L) = 1

If

by using

the Riemann-Roch theorem and the vanishing theorem.

Hence

3) is satisfied by (6.13). If

is an Enriques surface, let

S

Then

K3-cover.

with

n*0

h0(S, L)

h0(S, 2L + N) = 5

and

dim BsILI = 1, S

BsILI

2;2

0

Since

h0(S, 2L)

Hence

= 4

dim BsII a 0 and

2L = f*0(1).

H0(S, L)

Let and

4(^S`,

2;)

=

d(S, L) = 2.

would be ruled by (10.4.8).

by (4.15) and (3.5.2).

be linear basis of ly.

h0(S, 4L) = 17.

= 4, h0(S, 2L)

with

f:

N E Pic(S)

h0(S, L + N) = 2,

=

is at most finite, so

Moreover

for some

be the

By the Riemann-Roch theorem and the vanishing

2N = 0.

theorem we have

If

= 0s ® 0[N]

S

:r:

for = 2.

Therefore *

= n L.

Therefore

Thus we have a morphism (C0, C1)

and

H0(S, L + N)

(to, t1)

respective-

h0(S, 2L) = 5, we have a linear relation among

§ 15: Polarized manifolds of sectional genus two

127

E12 E H0(S, 2L). Therefore 0c1' C12' E02 is a hyperquadric in 1P3, and hence deg f = 2. W = f('`) 02'

The structure of such double coverings was studied in In particular

[F13].

[Fl3;§4]), so E

W

must be smooth since

W = {Z0Z1

for

Z0 1}

if the bases

2L)

Therefore

with

`1 = alai, 0 = a0a1

Z0 = a0a0,

covering transformation and

c

Thus

points, over which

acts on

t

f

and

c

W

1

by

The

a0.

Z.

t

ai = (-1)lai

= Z.

and

W with four fixed

f

is described as follows.

H0(S, 2L + N), but

Hence we have

Z1 =

is unramified.

The branch locus of Recall that

are

)

1

acts on

' _ -Zi, hence we may assume (-1)Ja

CJ its

n

ni

x IP1 = (01 0:01),(t0:T1

W = lP1

of

t

= 0 (cf.

Zi = n*Ci' Zi

and

chosen suitably.

1

a H0(S, 2L + N)

form a basis of

h0(S, 2L + N) = 5.

such that

H0(S, 2L + N).

i

and the

Counting the

dimensions we see that there is a linear relation among

the nCJ i's, and monomials in degree -

n n' Z0' Z1' Z0' Z1

is generated by

and we have a quadratic equation of

whose discriminant defines the ramification locus of Of course described.

c

77

_ -V .

Thus the structure of

In particular we have

(S, L)

f.

is well

4).

Thus we have completed the proof.

It is easy to find

examples of such polarized surfaces. (15.8) In the case (15.2.1), we have the following. Theorem.

type (15.2.1).

Let

Then

2,

C0' 1' 0 and 1 of

The graded algebra

4.

77

(S, L)

be a polarized surface of the

q = h1(S, OS) = 0.

the following conditions is satisfied.

Moreover, one of

128

II. Sectional Genus and Adjoint Bundles

1) Pg d=

h2(S, OS) = 0.

L # K

in

2) pg = 1

10

in

and

(S, L)

if

and

is a weighted complete intersection

IP(3, 3, 2, 2, 1).

(S, L)

is a weighted hypersurface of degree

IP(5, 2, 1, 1)

For a proof, see [BeLP;p.193]. S

h0(S, L) = 1

Pic(S).

of type (6, 6) in 3) pg = 2

Furthermore

In the case 1) above,

is called a numerical Godeaux surface.

We have several

examples and partial results on numerically Godeaux surfaces (cf. [Miyl], [Barl, 2]), but a complete classification of them is not yet known.

The surfaces of the above types 2) and 3) were studied by Catanese [Cat] and by Horikawa [Hor2-H] respectively. have

4(S, L) = 1

We

in the case 3),and (6.21) applies, since

the other cases in (6.20) are ruled out by (6.22) & (6.23). (15.9) In the case (15.2.1'), let elliptic fibration.

f: S -> C

be the

Then there are the following

possibilities (cf. [BeLP;p.192] and [Sern; Th. 2.1]). q = 1

and

pg = 0.

b) x(OS) = 0, g(C) = 1, q = 1

and

pg = 0.

a) x(OS) = 0,

c) x(OS) > 0,

C = IP1,

C = IPq = Pg = 0.

In the case

equivalent to

f

a) above, Serrano shows that

(D x E)/G --> DIG, where

G

is a finite

abelian group acting faithfully on smooth curves He further classified all such pairs

is

(S, L)

D

and

and found

several examples.

In fact, Serrano's method works also in the case and we can classify such pairs and construct examples.

b),

E.

§ 15: Polarized manifolds of sectional genus two However,

129

is not abelian in this case (cf. [F28]).

G

c), we have some results in [BeLP], but a

In the case

complete classification is not yet known. example of such

(S, L)

either.

(15.10) In the case (15.2.0), let Then

minimal reduction. curve

Z

(Mb, L1')

We have no

(L')2

passing a point on

Mb

and

1

>

be the

L

(Mb

LbZ > 1

for every

which is blown-up.

Hence

must be of one of the types 2), 3), 4), 6i) or 7).

(15.11) Now we study the case

n = dim M k 3.

First we

have the following.

be a polarized manifold with

Theorem.

Let

(M, L)

n = dim M k 3

and

g(M, L) = 2.

Then one of the following

conditions is satisfied. and

1 ) K -. (3 - n)L

d = Ln = 1.

f: M - Fn

2) There is a finite double covering along a smooth hypersurface of degree 2') (M, L) (M', L')

3) (M, L)

6, L = f 0(1), d = 2.

is a simple blowing-up of a polarized manifold of the above type

2) at a point. Moreover

n = 3.

is a scroll over a smooth surface.

4) There is a hyperquadric fibration smooth curve 5) (M, L)

branched

W

f: M -> W

onto a

as in (11.8;bl-Q).

is a scroll over a smooth curve of genus two.

The proof is outlined in (15.12) ti (15.13).

(15.12) We may assume that otherwise

is nef, since

5) is satisfied.

Suppose that = 2.

K + (n - 1)L

K + (n - 2)L

is nef.

Then

d

2g - 2

Moreover, by the fibration theorem (0.4.15), we have *

K + (n - 2)L = f A

for a morphism

f: M -

W

and an ample

H. Sectional Genus and Adjoint Bundles

130

line bundle

A

When

on

W.

and K = (2 - n)L.

Hence we have

Using

Using vanishing theorem as in (1.3) and

d = 1, we have

Thus

f.

A ". L

dim W > 0

deg A = 1

Hence

for any fiber

dim W > 1,

A2Ln-2 k 1

so

1).

The case

b2) we have

HF = 0(1)

K + (n - 2)L

and

is not nef.

b0) is easily rule out.

bl-V) as follows.

L = 2H + f B

Then In the

bl-Q) we have

3), and in the case

We rule out the case

4).

Set

H = K + 2L.

for some

B E Pic(W).

2 = (K + 2L)L2 = 4H3 + 4H2B, which is absurd. Finally we consider the case (11.8.a).

be the minimal reduction of

(M, L).

1), 2), 3), 4) or

the conditions

1) is ruled out since

arguments.

3), 4) and 5) there is a curve for every point (15.10).

infer

X

by a version of the index theorem (cf. [F21;(1.5)]).

(11.8) applies.

Hence

if

= 1.

(A - L)Ln-1 = 0, this implies

Since

(A - L)2Ln-2 2 0.

(15.13) Suppose that

Then

n-1

(6.14) ti (6.16), we can derive a contradiction by the

Thus we have

case

Ln-1X = 1

and

AML

since

is a Del Pezzo variety of degree one.

(X, LX)

argument in [F21;(1.4)]. and

dim W = 0

2) by (6.13).

dim W = 1, then of

Hence

= 0.

d(M, L) = 1 (see [F13;(1.11)] for details).

(6.4.3), we get

When

AMLn-1

d = 2, we have

p

Therefore

n = 3

on

Z

Let

(M', L')

This satisfies one of

5)

by the preceding

(L')n > 1.

with

In the cases

L'Z = 1

and

Z a p

This is impossible as in

M'.

(M', L')

is of the type

by the argument in [F21].

2), and we

Hence we have

2').

Thus we have completed the proof of (15.11). (15.14) In the cases

2), 2') and

5) in (15.11), we

§ 15: Polarized manifolds of sectional genus two

have nothing more to say.

In the cases

131

1), 3) and

have more precise results (cf. [F21], [F22]).

4) we

Here we

review them briefly.

(15.15) In case (15.11.1), we have

K = (3 - n)L

in

Pic(M)

H1(M, 0M) = 0

by [F21;(2.1)].

and

This is obvious

by Kodaira's vanishing theorem when n > 3.

When n = 3,

however, we need some deep results for the proof, such as Yau's solution of Calabi conjecture [Y], Miyaoka's inequality for Chern numbers [Miy2], or Kawamata's theory on Albanese mappings [Kawl].

1) If in addition

hypersurface of degree IP(5, 2, 1,

10

as in (15.8.3).

By using (2.6) we infer that

jective spice

IP(3, 3, 2, 2, 1,

In the above cases

This is true when

a=

4) If 111

Moreover

L

M

1

is a weighted

in the weighted pro-

---o 1) (cf. [F21;(2.4)]). 2), M

is simply connected. n = 3

Pic(M)) s 5.

a1(M) = Z/5Z

and the universal

is a hypersurface of degree five in

IP4.

= 0 (1).

t = 4, then

5) If

covering

#(torsion part of

of

as in

We have instead the following results.

s = 5, then

covering

1) and

(6, 6)

(M, L)

(M. L)

n > 3, but not always true when

d(M, L) 2 3. 3) r

is a weighted

in the weighted projective space

complete intersection of type

and

(M, L)

4(M, L) = 2, we have a ladder of

2) If (1.1).

1)

,

4(M, L) = 1,

a1(M) = 2;/4Z

and the universal

is a weighted complete intersection of type

(4,4) in the weighted projective space

IP(2,

2, 1, 1, 1, 1).

Thus, the results are higher dimensional versions of

132

II. Sectional Genus and Adjoint Bundles

The proof is similar to that of the case

those in (15.8). (15.7.4).

See [F21;(2.5),(2.6) & (2.8)] for details.

(15.16) The following problems are still unsolved.

1) Find examples and classify polarized pairs such that

n 3 4, K = (3 - n)L, Ln = 1

(M, L)

J(M, L) > 2.

and

2) Find examples and classify polarized threefolds L3 = 1, K = OM, 4(M, L) = 3

such that

(M, L)

and

g 3.

s

An example will be a polarized version of those in [Barl, 2].

is the scroll of

(15.17) In the case (15.11.3), (M, L) an ample vector bundle Set

A = det 8.

the types

on a surface

Then, by (13.12), we have

A2 = Ln + c2(6) a 2.

rational curve

n - 1

of rank

6

in

Z

Therefore

S.

2), 3), 4), 60), 61) or

S

is the

Hence

of genus two.

C

the cases 1') " 4) in (15.7) cannot occur.

S

for any

7) in (15.2).

Jacobian variety of a smooth curve

on

and

is of one of

(S, A)

1) In case (15.2.2), it turns out that

8n_1(C, o) 0 N

g(S, A) = 2

AZ 3 n - 1 k 2

Moreover

S.

Moreover

6 =

N

for some numerically trivial line bundle

and for some point

o

on

Here

C.

6n_1(C, o)

is

the Jacobian bundle which is described as follows. Let

Pk = C x

product of

C

...

and let

X C / Gk

be the

ak: Pk -> S = J(C)

induced by the Albanese mapping of known that

blowing-up at a point.

i-th projection

(C, o).

1k-2-bundle for

ak is a

C x

Let x C

k > 2

(P

k'

D)

be the morphism It is well and

n2

is the

o

of the

Di be the fiber over -* C

and set

This is an ample effective divisor on Moreover

k-th symmetric

D

Pk such that

Di)/6k.

Dk = 1.

is the scroll of a vector bundle on

S,

§ 15: Polarized manifolds of sectional genus two

which is defined to be

6k_1(C, o).

The proof of this fact

8 = 8n_i(C, o) 0 N

2) If

is of the type (15.2.4), we have

(S, A)

for any fiber

0(1) e 0(1)

F *

Therefore

n = 3

and

bundle

on

of rank two.

C

F(3) = S

and

8 = a

over

(0, 1) or (1, 0), and stable.

F

C

of

M

is the fiber product of

Moreover Both

L3 = 3.

AF = 2.

for some vector

(c1(3), c1(6)) _ are semi-

%

and

Y

c1(%) = -1.

such that

AF = 3

n: S --> C, so F = 0(1) e 0(1) e 0(1)

for some vector bundle

0 H(3)

2c1(%)H(3) + 3H(3)2 = 1, stable.

0 H(3)

n = 4

In the former case we have

0(2) e 0(1).

8 = a

a: S - C, since

of

is of the type (15.2.3), we have

(S, A)

for any fiber

*

C.

8F =

See [F22;(2.4)] for a proof.

3) If

or

is very

For details, see [F22;(2.1l)'(2.23)].

lengthy.

1P('S)

133

and

of rank three on

%

Moreover we have

c2(8) =

L4 = A2 - c2(6) = 2

and

19

is

See [F22;(2.6)].

When F = 0(2) a 0(1), we have an exact sequence 0 -+ 0[n*G + 2H(3)] --> 8 - 0[n*T + H(3)] - 0 G, T E Pic(C).

for some

Moreover we have (cf. [F22;(2.7)])

i) deg T = 0, deg C = -1, c2(6) = 1

and

L3 = 2, or

ii) deg T = 1, deg G = -2, c2(8) = 2

and

L3 = 1.

4) If S = Pa X

1P 1

n: S --b1P1 'J

def

and A = 2Ha + 3HV

n8[-Ha]

2HP a H

c2(8) = 3

For any fiber

AF = 2, so F = Ha a H

we have

By restricting

have

is of the type (15.2.60), we have

(S, A)

is locally free on 8

to a fiber of

Hence and

8

[H

L 3 = 9.

IP1

and

S - IP a

+ 2H

F

of

Therefore 8 = n*W 0 Ha.

we infer that '

a [Ha + H

and we

II. Sectional Genus and Adjoint Bundles

134

for any fiber 6

c9 0 H

ting

6

F

to the section

E

on

Hence

By restric

IP

n, we infer

of

and AE = 2.

and

c2(8) = 3

`

AF = 2

H1 = H - E.

n: S --> P1, where

of

for some vector bundle

HE = 0

since

is of the type (15.2.61), we have

(S, A)

If

= HP ® HA

8 = [2H - E] ® [2H - E],

Hence

L3 = 9.

The pairs

in these cases

(M, L)

Indeed, M

fact isomorphic to each other.

the fiber product of

and

® H1)

iP(2H

60) and

61) are in

is isomorphic to

W H

IP(H

over

IPA

and there are two scroll structures. 5) If

(S, A)

AE = 2

since

n = 3

is of the type (15.2.7), we have

E

for any (-1)-curve

on

It turns out

S.

that there are two possibilities: i)

(8

ti

[-K] ® [-K], c2(8) = 1

ii) c2(6) = 3

and

and

L3 = 3;

or

L3 = 1.

The latter case does really occur and we can describe the structure of

8

more precisely.

See [F22;(2.8)].

(15.18) In case (15.11.4), we employ the notation in (13.11).

Then

6n = 4nq + (n - 1)d + s k 4nq + (n - 1)d, so

q s 1. (15.19) When

i) d = 1,

q = 1, there are three possibilities:

ii) d = 2,

id) d = 3 = n.

or

If d = 1, we have s = n + 1, so b = 1 Moreover rank one.

deg Q k 0

for any quotient bundle

This case does occur for any

and Q

n k 3.

of

e = 0. 8

of

See

[F21;(3.8) & (3.9)] for details. If

d = 2, we have

case occurs for any

s = 2, so

n k 3.

b = 0

and

e = 1.

See [F21;(3.10) & (3.12)].

This

§ 15: Polarized manifolds of sectional genus two

135

If d = 3 = n, we have s = 0, so b = -1 Moreover every fiber of ample vector bundle.

is smooth, and

f: M --> W

Let

7L

is a member of

be a polar-

(M, L)

Assume that

ized manifold of the type (15.11.4).

b e

is an

See [F21;(3.13) & (3.14)].

Theorem (cf. [F21;(3.30)]).

M

8

q = 0, we have the following result.

(15.20) When

Then

e = 2.

and

I2H(6) + bH,I

on

1P(')

W = lP1.

for some

and one of the following conditions are satisfied.

1) d = 1, b = 5

and

[-HJ ® [-H]

6 = 0(-1, -1, 0, 0) (a=

®0®0). 2)

d = 2, b = 4

and

8

3)

d = 3, b = 3

and

8 = 0(0, 0, 0, 0).

3') d = 3, b = 3, 6

4)

d = 4, b = 2,

0(-1, 0, 0, 0).

0(-1, 0, 0, 1) and 6

BsILI _ 0. BsILI

is a point.

0(0, 0, 0, 1).

6 = 0(0, 0, 1, 1). 6) 6 = 0(0, 1, 1, 1). 7) d = 7, b = -1, 6 = 0(1, 1, 1, 1). 8) d = 8, b = -2, 6 = 0(1, 1, 1, 2). 8 ) d = 8, b = -2, 6 = 0(1, 1, 1, 1, 1). 9) d = 9, b = -3, 6 = 0(1, 1, 2, 2). 5)

d = 5, b = 1, d = 6, b = 0,

The proof is elementary but lengthy. Remark.

In all the above cases we have

so the results overlap with those in (10.7). the pair

(M, L)

in the above case

d(M, L) = 2,

Note also that

9) is isomorphic to the

pair of the type (15.17.4).

(15.21) Thus, by the results of this section, polarized

manifolds of sectional genus two are classified almost completely (see the chart in the next page).

II. Sectional Genus and Adjoint Bundles

136

Maeda [Mae] classified polarized surfaces of sectional genus three by the method in §14.

When

L

is very ample, we have more precise results.

This topic will be discussed in §17 in the next chapter. Classification of polarized manifolds of sectional genus two (n = dim M a 3, d = Ln, q = hl(M, 0M)) Chart.

1

2') d=1

2)

)

d=2 q=0

d= 1

q=0 n=3

double cover of

(15. 15)

3

4

)

5)

)

scr oll ove r S

scrol l over C q=2 (15. 18)

IPn (

15. 17)

q= 0

1)

2 )

3)

4)

5)

q=2 d=l

q= l d= 3 n= 3

q=l

q=0 d=9

q=0 n=3

n=3I

n=4

n=3

d=2

1)

2)

3)

4=1

4=2

J k3

3-i )

5-ii)

d=3

d=1

q=1 (15.19) 1

I

I

I

nk4

n=3

-I

I

3-ii) d=l

d=2

5-i)

q=0 (15.20)

-1

i)

I

I

I

I

ii)

n=3

n=4

d=l

d=2

iii) d=3

dIO

d=8

n=3 i=5

(15 16 1) .

i=4

t;g3

.

(15.15.4) (15.15.5) (15.16.2)

(15.22) Now we prove the theorem in (10.8.1). g(M, L) S 1,

If

bundle

8

(M, L)

is the scroll of a vector

over an elliptic curve

C

by (10.2).

Then

§ 15: Polarized manifolds of sectional genus two since

= cl(8)

h0(C, (8)

It is easy to show that such that

is ample (cf. [At]), so we have

6

rank(8) = n = d + h0(L) - d = 2.

Hence

h0(M, L) = Ln = 4.

has a sub-bundle

8

Let

deg(Y) k 2.

137

be the quotient bundle

2

This is identified with the restriction of section corresponding to Therefore

is spanned.

= 0

H1(Y (9 2v)

splits.

Hence

2.

L = H(8)

deg(2) a 2

d-i) in (10.8.1).

If

3" = 2, the

1P1

X

M = C x F1 Thus we

F1.

d-ii).

have

Now we consider the case

n = 2, and let

that

(M, L).

Then

be a minimal reduction of Hence

4), 6i), 7)

(M0, L0)

of a vector bundle h0(8)

=

h0(M0, L0)

deg(2) = 2

is of the type (15.2.9), it is a scroll over a curve

8

2

map

H0 (2)

as in the case

-

H1 (fl

C

of genus two.

Since

h0(M, L) = 4, there is an exact

such that

deg(g) _

g(C) = 2, both

Since

g = 1.

are the canonical sheaf

and

In case

b) in (10.8.1).

(#): 0 -> 3 --> 6 - 2 -* 0

sequence

satisfies one

(M0, L0)

9) in (15.2).

or

6i), we have clearly the condition If

Suppose first

g(M, L) = 2.

(M0, L0)

d0 = L02 2 4.

of the conditions

3"

Moreover, the natural

w.

vanishes, which is the Serre dual of

the extension class of p

L

Y -* 8 --> 2 - 0

0

would be a double covering of

p

to the

3 # 2, then

above sequence does not split, because otherwise and

8/T.

since

deg(f) = deg(2) = 2. If

and the sequence

Thus we have

of rank one

F

(#).

is a double covering of

Hence

(#)

F' X F1.

splits, but then

Thus this case is

ruled out. If

(M0, L0)

(M0, L0)

since

is of the type (15.2.4), we have d0 = 4.

Hence

M = F(Y)

(M, L)

for some vector

U. Sectional Genus and Adjoint Bundles

138

bundle

over an elliptic curve

3'

for some (1, 0).

such that

B E Pic(C)

C, and

(c1(3), deg(B)) = (0, 1) or

For any quotient bundle Therefore

deg B > 0.

S2(Y) ® B

Hence

is ample.

L = 2H(.T) + BM

2 3'

3

of

we have

is semistable and

h0(M, L)

9

aer

h0(C, 9) = C1(9) = 3

=

by (At], which contradicts the assumption. If

(M0, L0)

(M0, L0).

is of the type (15.2.7), we have

Moreover, any member

of

Z

ble, reduced and of arithmetic genus one.

the restriction of p

p

to

cannot be birational.

Z

I-KI

is irreduci-

Since

LZ = 2,

is a double covering.

In fact,

p

(M, L)

Hence

is a double covering

of a singular hyperquadric in this case.

Thus this case is

ruled out by assumption.

Combining the preceding arguments we complete the proof

when

n = 2.

When n k 3, we apply (15.11), and trace the

chart in (15.21). case

n = 2.

the condition

Since

Note that

q = 0

by the result in the

d = 4, we must reach (15.20), and have

c) in (10.8.1).

Thus we have completed the proof of (10.8.1).

Chapter 1. Classification Theories of Projective Varieties In this chapter we are mainly concerned with the cases in which L is very ample and char(.) = 0. We will survey the theories due to Castelnuovo, Ionescu, Sommese, et al. §16. Castelnuovo bounds P =

FN

which is not contained in any hyperplane, Castelnuovo has found an upper bound for the genus of C in terms of the degree. Here we provide a higher dimensional version of Castelnuovo bound using the hyperplane section method. For any curve

C

in

(16.1) Given positive integers 95c,d(µ) = (d - c - 1)µ - cµ(µ - 1)/2

where

q

and

c

and

we set

d,

y(c, d) _ 0c,d(q)

is the largest integer such that

cq

d - 1.

Then we have the following. Fact (Castelnuovo bound).

reduced curve of degree N + 1.

Then

d

in

Let

FN

h1(C, 0C) g y(N-1, d)

C

be an irreducible

such that unless

For a proof, see [GH;p.252] and [Rat]. C

h0(C, 0(1)) C

_

is strange.

Here, a curve

is said to be strange if all the tangent lines at smooth

points of

C

pass through a common point.

no curve is strange except lines.

Even if

If

char(g) = 0,

char(t) > 0,

smooth curves are not strange except conic curves in case char(3) = 2 (cf. [Ha4;p.312]). (16.2) Remark.

1) y(c, µc + 1) = cµ(µ - 1)/2 in

d

in the interval

µc + 1

y(c, d)

d s (µ + 1)c + 1.

is linear Hence

v = [(d - 2)/c], the largest integer

y(c, d) _ 0c'd(v)

for

with

We also have

cv g d - 2.

and

y(c, d) = Maxµ2O(Oc d(µ))'

IH: Classification Theories of Projective Varieties

140

For any fixed function in c

and

d

Hence

c.

for any fixed

µ,

is a decreasing

Oc,d(y)

is a decreasing function in

y(c, d)

On the other hand, y(c, d)

d.

increasing function in

d

if

c

is fixed.

2) In (16.1), it suffices to assume that is injective (or

H0(C, 0 C(1))

This is because

dim V = n

V

with

p

defined by

Let

h0(C, 0 C(1)) a N + 1).

BsILJ = 0

char(.) > 0, suppose in addition that

and

and the morphism

is birational onto its image.

for

L

d = Ln

d = 4(V, L).

When

Proof.

V

that

member

n = 1,

D

of

(16.1) applies since we may assume

When

is not strange.

n > 1, we take a general

and use induction on

ILI

n, using (16.2),

and further (0.2.9) if

d(D, LD) s d(V, L)

char(.) > 0.

This fact is very useful for classifying very ample line bundle (16.4) Corollary.

its canonical bundle.

M

Let

be a manifold and let

Suppose that

possibly not a morphism).

n = dim M

where

Let

Proof.

rational map

and M'

plKl.

Then

d

d

on

M'

char(.) = 0,

K

K

be

is nef

is birational (but

IKI

Kn > (n + 1)pg - n(n + 2),

pg = h0(M, K).

be a smooth model of the graph of the Then we have a birational morphism

r: M' --> M, an effective divisor

system

with

(V, L)

Here is another application.

L.

and the rational map defined by

Thus

When

is normal and

V

g(V, L) f y(d - d - 1, d)

Then

c.

be a line bundle on a variety

L

such that

ILK

is very ample.

H0(P, 0(1))

is a decreasing function in

y(c, d)

(16.3) Theorem.

is an

such that

gives a morphism

E

n IKI

on

M'

= E + A

p: M' - P =

1PN

and a linear and

BsA = 0.

with N = dimIKI

141

§ 16: Castelnuovo bounds

M' - p(M')

such that

W

def

is birational.

Set

L = [A],

d = Ln, g = g(M', L), 4 = 4(M', L), c = N - n = d - 4 - 1. v = [(d - 2)/c] s n.

Assume that

Then we claim

Indeed, this is true if

nd/2 > y(c, d).

d = (/c + 1 )c + 1 .

d = uc + 1

In the interval µc + 1 s d s (y + 1 )c + 1

both sides of the claimed inequality are linear in Hence the claim is always true, so

(16.2.1).

(R + E + nL)Ln-l 2 nd,

where

R

2g < nd. 1)L)Ln-1

=

is the ramification

v > n.

Now we have

d - 2 k (n + 1) c = (n + 1)(p9 - 1 - n), so

d k (n + 1)pg - n(n + 2) + 1.

Moreover

Kn = Ln

by (0.4.7)

since

K

Hence

Kn I (n + 1)pg - n(n + 2) + 1, as desired.

and

by

This contradicts the above estimate, so we

a.

conclude that

d

2g - 2 = (K' + (n -

On the other hand we have

divisor of

or

L

are nef and

E = x K - L

(16.5) Remark. 1) We may allow M

is effective.

to have canonical

singularities.

2) When

n = 2, (16.4) yields the famous estimate

c12 2 3pg - 7.

3) The birationality of (16.4).

For example, if

p

is essential in (16.3) and

p: V -> W

there is no upper bound for

is a double covering,

g(V, L).

(16.6) Now we consider the case where maximum

y(c, d).

g

attains the

The results will not be used in the later

sections of this book, but they are worth mentioning.

The

two-dimensional case was investigated by several authors in the study of surfaces of general type with (cf. [ASK], [Hor2], [Mir2]). Theorem.

c12

= 3pg - 7

Here we present the following.

Let the hypothesis be as in (16.3) and

III: Classification Theories of Projective Varieties

142

suppose in addition that

g(V, L) = y(c, d).

has a regular ladder and

L

Proof.

The case

n.

classical (see [GH; p.253]), so we assume

ladder and

LD

of

Then

ILI.

If

since

y(c, d)

is

Take a

has a regular

(D, LD)

is a decreasing function in

(V, L).

ample by (16.6).

L

is simply generated.

is ample, then

1),

is very

L

(V, L)

will be

There are three types of such

2) d = 2d.

1) d > 2d.

is a regular

D

Such a polarized variety

called a Castelnuovo variety.

In the case

L

by (16.3)

c = d - d - 1.

Therefore

Hence, by (2.5),

(16.7) If in addition

varieties:

n > 1.

g(D, LD) < y(c, d)

This contradicts the assumption. rung of

n = 1

is simply generated by the induction hypoth-

4(D, LD) < d, then

esis.

(V, L)

is simply generated.

We use induction on

general member D

Then

3) d < 24.

v = [(d - 2)/c] = 1

and

y(c, d) = A.

This case is treated in (3.5.3). For the case

2), we have the following.

(16.8) Theorem.

such that

(V, L)

BsILJ = 0, d = 2d

is simply generated, V

i) L

ities and ii)

Let

ILI

wV = (2 - n)L

in

and

be a polarized variety g > A.

has only Gorenstein singularPic(V), or

gives a finite double covering

polarized variety

(W, H)

Then either

such that

p: V

W onto a

4(W, H) = 0

and

p H = L. For a proof, see [F13;(1.4)].

Polarized varieties of the type (16.8.1) are our Castelnuovo varieties of the second kind.

They are

polarized higher-dimensional version of K3-surfaces and Fano

143

§ 16: Castelnuovo bounds

3-folds (cf. [Sail,2], [Isl,2], [Mukl,2]).

Polarized varieties of the type (16.8.1) are called hyperelliptic polarized varieties and were classified Important examples of such double

completely in [F13].

In fact, a minimal

coverings were studied by Horikawa.

surface of general type with

K2 = 2pg - 4

double covering over a variety with this case

K

admits such a However, in

d = 0.

is not always ample, and the branch locus may

have singularities, so the arguments are more difficult than in [F13].

Such surfaces often provide interesting examples

in deformation theory.

(16.9) For a Castelnuovo variety kind (the case

d < 2d), let

the hyperquadrics in dim X = n + 1

P = FN

4(X, H

and

X

)

X be the intersection of all containing = 0, where

This is proved by induction on

n.

V.

H

is a Castelnuovo variety.

and set

Then we have

H = 0(1).

The case

When

due to Castelnuovo (see [GH;pp.527-533]). a general hyperplane

of the third

(V, L)

D = V n H.

(D, LD)

Moreover, by (2.6), X n H

dim(X n H) = n

and

4(X n H, 0(1)) = 0

hypothesis, so

dim X = n + 1

and

is

n > 1, take

Then

intersection of hyperquadrics in H containing

C.

n = 1

D.

is the

Hence

by the induction

4(X, H) = 0.

Note that deg X = N + 1 - dim X = N - n = d - d - 1 V lies on X as a Well divisor. (16.10) Now we assume in addition that

We divide the cases according to the type of and (5.15), there are the following cases. 1) (X, H) = 2) (X, H)

(,Pn+l,

0(1))

is a hyperquadric in

p . n+2.

V X.

is smooth.

By (5.10)

III: Classification Theories of Projective Varieties

144

3) (X, H)

F1, or a generalized cone over

is a scroll over

such a scroll. 4) (X, H)

Is a Veronese surface

(IP2, 0(2)), or a

generalized cone over such a surface. (16.11) In case (16.10.1), V degree

Conversely, for any hypersurface of degree

d.

we have

is a hypersurface of

= d - 2, c = 1, a = [(d - 1)/c] = d - 1

1

d,

and

Thus the bound is in fact

g = (d - 1)(d - 2)/2 = y(c, d). attained.

(16.12) Suppose that (16.10.2) holds. P1 X IP1

two-dimensional hyperquadric over it, then

quadric

I = Sing(X)

for

divisor and let

I3`

Then

n E.

P

E

Q.

s = dim I

is smooth.

E

y e I n V, let

be the exceptional

be the proper transforms of V n E

and

This is impossible since

cone over

Hence we

where

Indeed, if

y, let

at

and IPn+l

X

if

V n 1 = 0.

be the blow-up of

V.

or a generalized cone

n - 1 - s > 2,

s = -1

and

We first claim

and

is a

is a generalized cone over a smooth hyper-

of dimension

0

X

can be viewed also as type 3).

X

X

assume that

If

I

is a linear n E

]Pn-1

X in

is a generalized

Thus the claim is proved.

By this claim

V

is a Cartier divisor on

a complete intersection in

P.

Moreover

X

X

and hence

has at most

isolated singularities.

Conversely, for any complete intersection of type

(2, a), we have d = 2a, d = 2a - 3, c = 2, y = [(d - 1)/c] a - 1

and

g = (a - 1)2,

so the bound

y(c, d)

is

attained.

(16.13) In case (16.10.3), we first study the case in

§ 16: Castelnuovo bounds

which

X

Then

is smooth.

8 = 0(a1,

145

43n+1)

def

V E

we have

a > 1

and

a, b

h0(V, L)

since

V

Hence

h0(X, Ha)

=

d = (aHa + bHP)Han = alai + b

161,

= 26j.

161

g(V, L) > 0.

since

= deg X = d - d - 1 = c,

161

n + 1 +

Then

X.

H. = H(8)

Set

where the

1PI,

for some integers

JaHa + bHfI

is a divisor on

over

[ajH,]

E)

6j's are positive integers. We have

is the scroll of

(X, 0(1))

and

g = a(a-1)61/2 + (a-1)(b-1) = (a-1)(d-c-1) - (a-1)(a-2)c/2. Thus the bound

is attained if and only if

y(c, d)

_ [(d-l)/c] or [(d-2)/c],

or equivalently,

K

a = n + 2

of

V

if and only if

L

This case provides examples in

b = 2 - 161.

and

is

1 - c t b s 1.

Note also that

There are many examples of this type. the canonical bundle

a - 1

which the bound (16.4) is attained.

Next we consider the case in which Sing(X) = Ridge(X)

have

smooth subvariety

det

with

S

R.

X

X = R * S

and

4(S, 0(1)) = 0

taking linear section (cf. (5.13), (5.15)). and suppose that

is singular.

(S, 0(1)) is a scroll of

for some

obtained by r = dim R

Set 0(61,

6n-r)

'

def

8

over

1P1.

such that

X

exceptional set of

n*OX(1) = Ha, a

and

be the proper transform of subvariety of

k

Suppose that R

is the vertex.

for some

a > 0.

V a R, then

(I, Ha)

and let

be

Then we have a birational morphism

the scroll of

n:

= 6 ® 0®(r+l)

Set

We

such that r = 0. If

D

D a

V

a=

n-1 (R)

F(0®(r+1))

a IF

on

is the Let

x R.

3`, namely, the unique

n(!`) = V.

Then

X

V I R, then

is a cone over a V

and

V E

S

10X(a)l

There are many examples of this type.

E n X' = S

contains linear

]P n-1

and

as in

If

146

III: Classification Theories of Projective Varieties

(16.12), where

denotes the proper transform on the

'

blowing-up at

and

R

n = 1 or E n V' = n-1(R)

case

is the exceptional divisor.

E

is a fiber of

1Pn-l

In either

S --+ F 1 .

Therefore

is a point.

n 11

is finite,

V

so it is an isomorphism by Zariski's Main Theorem. case we have

for some

on

JaHa + HftI

E

Hence

In this There

a > 0.

are many examples of this type also. Now suppose that

on X

By taking hyperplane sections

r > 0.

and using induction, we infer that either is isomorphic over any general point of

! --> V

V 0 R In

R.

either case the codimension of the exceptional set of is greater than one, so Moreover

E

1V

b = 0, then

section of

Hence

V.

11 E

have a non-constant morphism r = 1.

11D = F ]P1

i1D

L = [Ha ]V,

b = 1

when

'

Sn-r) ® 0®(r+l))

where

Ha

is

03t (1).

-* V

is a

D

However we

.

since

with

If

1.

D -* R, which

a > 0.

and this is a member of

V =

13` = F(0(S11

and

or

There are many examples of this type (with

Summary.

on

b = 0

JaHa + HP1, and

In particular

D -> R.

11

is smooth.

V

contains a fiber of

= V n D

contradicts

for

on

JaHa + bH D

since

11 = V

or

Hence

r = 1).

(aHa + bH,I

r = -1, 0, 1

Moreover

r = 1,

0 s b s 1 when r= 0, and 1- 161 sb;g 1 when r=-1. (16.14) Case (16.10.4) can be treated similarly. have line.

R n V = 0 Hence

dim R g 0.

bound

We

since the Veronese surface contains no

V E I0X(a)I

for some positive integer

a

and

Conversely any variety of this type attains the

y(c, d).

§ 17: Varieties of small degrees

147

There is no smooth polarized surface of this

Remark.

type (16.14) with K = L.

However, if we allow

an ordinary double point at

resolution, we can obtain a minimal surface

cl

2

= LV by choosing

K

type such that = 3pg - 7

for

of general

iV

suitably.

a

to have

be the minimal

`

and let

R

V

We have

V. These surfaces and their deformations

were studied by [Mir2]. §17.

Varieties of small degrees

We will classify projective varieties V c IPN by the value of d = deg V. This theory is due to Ionescu, who completed the classification for small degrees up to 8. (17.1) We will classify pairs

L

is

This is equivalent to classifying projective

very ample.

subvarieties

V

in

IPN

such that the restriction mapping

0: HO(IPN, 0(1)) _ HO(V, °V(1)) 0

such that

(V, L)

is injective if and only if

hyperplane in

IPN.

Note that

is bijective.

V

is not contained in any

is said to be linearly normal if

V

is surjective.

(17.2) We have 2 0.

The classification is obvious if In case

d = 3, we have

the former case latter case

(V, L)

(V, L)

is

d

2.

(c, 4) = (1, 1) or (2, 0). IP'

(IP1, 0(3)), X(1, 2),

denotes the scroll of the vector bundle F1.

.

is a hypercubic in

a generalized cone over one of them, where

over

c= N - n

since

0 s 4(V, L) s d - 1

E(1,

In

In the 1, 1)

or

1(611 ***1 6r)

0(S1) ® ... ® 0(b r)

These facts were pointed out in the paper [*],

which is, according to a rumor, due to Well. (17.3) In case

d = 4, we have

(c, 4) = (1, 2),

(2, 1)

111: Classification Theories of Projective Varieties

148

or

(3, 0).

is a hyperquartic in case

(V, L)

is a complete intersection of type

When

(cf. (6.8)).

A = 0,

(2, 2)

(1, 2).

It

A = 1

in case

is isomorphic to

(V, L)

(F100(4)), 1(1, 3), £(2, 2), Z(2, 1, 1), E(1, 1, 1, 1), (1P200(2))

This

or a generalized cone over one of them.

classification is due to Swinnerton-Dyer [Sw]. (17.4) As for the case

d = 5,

6

and

7, we refer the

reader to the well written paper [Iol] of Ionescu. was treated in [ioi-2] and [ioi-3].

d = 8

the strategy of his investigation.

The case

Here we outline

Many results hold for

cases of any characteristic, but we assume

char(.) = 0

for

the sake of simplicity.

When

c = N - n = 1, (V, L)

is obviously a hypersur-

We have some results and conjectures in case

face.

c = 2,

but this case is difficult to study in general. On the other hand, if is small.

c

is large, then

We have a complete classification in case

and a satisfactory theory in case

When

c

and

A

A = 0,

d = 1 (see Chapter I).

are in the intermediate range, the

problem is usually difficult. is bounded by (16.3).

(V, L)

A = d - 1 - c

The sectional genus For each fixed

g

of

(d, A, g), we

can classify smooth pairs having these invariants by the method in Chapter U, using adjoint bundles.

In fact, a

variant of this method described in the next section X18 is very useful. since

L

We can use the Apollonius method in addition

is very ample.

However we should be careful since

the very ampleness is not always preserved under various reduction steps in Chapter U, which often yields very subtle problems (see §18).

§ 17: Varieties of small degrees

149

It is sometimes useful to divide the cases according to the values of

q = h1(V, 0V).

We have

q s g

by Kodaira's

vanishing theorem.

Thus, when

char(.) = 0, Ionescu succeeded in obtaining

a classification theory of smooth projective varieties which is roughly summarized as follows. (17.5) The case

d = 5.

c = 1: a hypersurface of degree

5.

c = 2, 4 = 2: g s y(c, d) = 2. g = 0: impossible since this would imply g = 1:

(V, L)

4 = 0.

is an elliptic scroll.

g = 2 = n: simple blow-ups (seven times) of a polarized surface of the type (15.2.61). g = 2 < n: (V, L) c = 3, 4 = 1:

(V, L)

is of the type (15.20.5),(10.7.3.1). is a Del Pezzo manifold and hence a

linear section of

Gr(5, 2) c 1P9 (cf. (8.9)).

c = 4, 4 = 0: similar to the case (17.6) The case

d g 4.

d = 6.

c = 1: a hypersurface of degree c = 2, 4 = 3: g s y(c, d) = 4.

6.

A difficult case to study.

See [Iol] for details.

c = 3, 4 = 2: g s 2. g = 1: an elliptic scroll.

n = 2

since

4 = 2.

g = 2 = n: simple blow-ups (six times) of a polarized surface of the type (15.2.61). g = 2 < n: (V, L) c = 4, 4 = 1: (V, L)

is of the type (15.20.6),(10.7.3.2). is a Del Pezzo manifold.

See (8.1)

III: Classification Theories of Projective Varieties

150

and (8.11.6) for further results. c = 5, 4 = 0: similar to the case (17.7) When

d = 7, the situation is similar to, but

more complicated than, the case

d = 6.

The cases

(c, 4) =

are the most difficult to study.

(2, 4) and (3, 3) (17.8) Remark.

In the course of his study Ionescu

succeeded in classifying (L

d s 5.

is very ample, V

(V, L)

in the following cases

is smooth and

char(.) = 0

as above):

1) g(V, L) = 2 ([Iol;§3]). 2) g(V, L) = 3 ([Iol;S4]). 3) 4(V, L) = 2 ([Iol;Th.3.12]). 4) 4(V, L) = 3 ([Iol;Th.4.18]). 5) g(V, L) = 4 ([Iol-2;Th.1]). 6) g(V, L) = 5 ([Iol-2;Th.2]). 7) 4(V, L) = 4 ([Iol-2;Th.3]). 8) d(V, L) = 8 ([Iol-2]).

In [Io2], he gives also a rough classification in the case where

d 2 24 + 1.

There is no theoretical limit to such a classification theory of

(V, L)

with small invariants, but the technical

troubles increase rapidly beyond these ranges.

How can we generalize these results when singular and/or

V

is

char(t) > 0 ?

This would be an interesting problem. (17.9) The case

n = 2, i.e. the classification of

embedded surfaces, has a long history of investigation.

The

notion of sectional genus is also almost as old as the problem itself.

In fact, Ionescu's theory mentioned above

151

§18: Adjunction theories

is based on classical theories on surfaces as a starting n = dim V.

point for induction on

Recently this topic has begun to regain the attention of many mathematicians (cf. [BeLP], [B1L1".4], [Livl 4]).

Okonek [Okl-3] has introduced vector bundle techniques in such studies.

§18. Adjunction theories (18.1) As we have seen in §17, the sectional genus is an important notion in the study of projective varieties.

The main technical tool for such study is the adjunction theory.

Here we make a brief survey of it and its

applications.

We have a classification theory of L

and

for any fixed V

with ample

(V, L)

g = g(V, L), at least when

char(A) = 0

Why then do we want another theory to

is smooth.

classify pairs with very ample

L ?

There are in fact several reasons. 1) The methods in §13 and §14 yield a list of possible types of

(V, L)

with ample

but they do not say when

L

L

for any fixed

g = g(V, L),

can be very ample.

This is

often a non-trivial and very interesting problem. 2) Under the additional assumption that ample, the list of possible types of drastically cut off.

L

is very

is usually

(V, L)

Therefore, in order to classify such

pairs, it would be nicer to have a short-cut method which does not involve unnecessary and sometimes troublesome classification of pairs with general ample

L.

3) It is difficult to extend the method of Chapter II to

III: Classification Theories of Projective Varieties

152

the case in which

has bad singularities.

V

In positive

characteristic cases the situation is even more hopeless.

However, there is a good chance that the method described in this section works, at least partly, in such more general cases too.

(18.2) For simplicity we assume

char(s) = 0 n = 2

As in Chapter II, the cases

section.

in this n a 3

and

need separate treatments. We first consider the case Let

Theorem.

such that

L

n = 2.

be a smooth polarized surface

(S, L)

is very ample.

Then the following conditions

are equivalent to each other: 1) BsIK + LI

for the canonical bundle

# 0

K

of

S.

2) q = hl(S, 0S) = g(S, L). 3) (S, L)

a

is a scroll over a curve or

for some

(1P2, 0(a))

2. For the proof, see [V] and [So3;(1.5)&(1.5.2)].

except for the well-understood case above, a morphism

Thus,

defines

JK + LI

0, which will be called the adjunction map.

(18.3) We now divide the cases according to the nature of

¢.

Y

Let

When

be the image of

dim Y = 0, we have

classical Del Pezzo surface.

¢.

K = -L When

and

is a

(S, L)

dim Y = 1, 0

is a conic

bundle over a curve. In these cases the structure of

(S, L)

is easy to understand.

When

dim Y = 2, let

torization of

0.

Then

S --' S' --f Y S --, S'

be the Stein fac-

is the same map as

(14.3), which contracts all the (-1)-curves that

LE = 1.

E

on

in

0

S

Moreover there is an ample line bundle

such L'

§ 18: Adjunction theories

on

where

is ample and

H = K' + L'

such that

S'

153

is the canonical bundle of

K'

K + L = HS,

S'.

A natural method here is to study the pair and proceed inductively. ample.

However, H

(S', H)

is not always very

In order to make this method work, we need to know

the structure of

(S, L)

when H

is not very ample.

For

this purpose we have the result below. Let the hypothesis be as in (18.3),

(18.4) Theorem.

Then

dim Y = 2.

H

is very ample except in the following

cases. 1)

L = -2K

2) L' = -2K' and 3)

K2 = 2,

so

d = 8, g = 3.

(K')2 = 2, so

d = 7, g = 3.

and

L = -3K

K2 = 1,

and

4) (S, L) = (iP(8), 3H(8))

so

d = 9, g = 4.

for an ample vector bundle c1(8) = 1, so

an elliptic curve such that

6

on

d = 9, g = 4.

This was proved independently by Serrano, Ionescu [Io4]

and also by Sommese-Van de Ven [SV] after several partial results [V], [So3;3.1], [Einl].

Here we outline a proof

from [Io4] which is based on the following remarkable result due to Reider-Beauville (cf. [Rdr], [Bea2], [Sak3]). (18.5) Theorem.

surface

Let

D

be a nef divisor on a smooth

X with canonical bundle

1) BsJK + DI _ 0 divisor 2) K + D

Z

if

such that

Z

Then

D2 > 4, unless there is an effective DZ = 0, Z2 = -1

is very ample if

effective divisor

K.

DZ = 1, Z2 = 0.

D2 k 9, unless there is an

satisfying either

i ) DZ = 0

and

Z2 = -1 or -2,

ii) DZ = 1

and

Z2 = 0 or -1,

iii) DZ = 2

and

Z2

= 0, or

or

III: Classification Theories of Projective Varieties

154

and D

iv) Z2 = 1

is numerically equivalent to

(18.6) Outline of proof of (18.4).

When

not very ample.

have

R),

LZS = 1

so

ZS2 = -1

ZS2

and

with respect to

Z

L.

and

Hence we have a divisor

or iv) in (18.5.2).

iii)

is

Hence

= 0 or -1. 2

Thus the case

S'

ZS

is a line

(S, L) is not a scroll,

since

# 0

S

on

Z

In the case R), we

This contradicts

is a (-1)-curve.

ZS

0 < HZ = (K + L)ZS.

H

Suppose that

d = L2 3 9, we can apply (18.5) by

setting X = S', D = L'.

satisfying

3Z.

By a

ii) is ruled out.

similar (and slightly longer) argument we rule out also the case

In the case

iii).

Moreover

S = S'.

LZ = 3

occur.

implies

Hence

L2

=

L'2 = 9

and hence

is ample, irreducible and reduced.

Z

g(Z) s 1

implies

g(Z) = 0

iv), we have

since

KZ = -3

and

g(Z) = 1, so

L

is very ample.

However

g(S, L) = 1, which cannot

q = h1(S, OS) t g(Z) = 1.

It

is now easy to see that we have (18.4.3) (resp. (18.4.4)) if q = 0 (resp. q = 1).

When

d g 8, we use case-by-case arguments as in X17.

See [Io4;pp.355-356].

(18.7) Combining these tools we classify pairs with very ample line bundle may assume

BsJK + Ll = 0

Suppose that a: S -> S0

of

S

S

L

and

as follows.

L.

is not ruled.

Then the minimal model

is determined uniquely, and the problem

LO

is not

Such a classification can be done in

various cases, e.g. when number.

(S0, L0), where

However, we must be careful since

always very ample.

By (18.2), we

g(S, L) 3 2.

can be reduced to the classification of LO =

(S, L)

g = g(S, L)

is a fixed small

In general, however, the problem is not easy, since

§ 18: Adjunction theories

155

the classification theory of minimal surfaces is not so precise as we want.

When

S

is ruled, we study the problem by iterating

the process in (18.3).

dim Y < 2

If

ample, the structure of

(S, L)

wise we proceed to study

or if

H

is not very

is well understood.

Other-

Since the adjunction

(S', H).

process must terminate in finite steps for ruled surfaces,

we can establish a classification theory by a suitable induction, as in §14.

This method works for any fixed

g = g(S, L), provided

the classification theory of minimal surfaces is good enough for our purpose, and also for any fixed for any fixed

d = d(S, L)

and

However, there are many

d = L2, as in §17.

technical troubles.

(18.8) In the rest of this section we consider the case where

n k 3.

The simplest philosophy here is to reduce the

problem to lower dimensional cases by taking hyperplane sections.

This is indeed a powerful approach and has been

utilized by many mathematicians.

Here we review another

important theory, i.e. Sommese's theory on adjunction processes, which he developed in a series of papers, and which is analogous to our theory in §11. (18.9) In Sommese's theory

Moreover

larities.

is allowed to have certain mild singu-

V

Usually

is assumed to be

BsILI = 0, but is not necessarily very

spanned, which means ample.

L

V

is normal, Q-Gorenstein at every

point, and has only rational Gorenstein singularities except at finitely many points. Remark.

In many cases

'rational Gorenstein'

in the

III: Classification Theories of Projective Varieties

156

final assumption can be replaced by

'log-terminal'.

See

[F24].

(18.10) To begin with, we generalize (18.2) in the higher dimensional cases (compare (11.7)). Suppose that

Theorem. ample.

K

is smooth and

BsIK + (n - 1)LI = 0

Then

unless

V

L

is very

for the canonical bundle

is either a scroll over a curve or

(V, L)

g(V, L) = d(V, L) = 0.

A proof can be obtained by combining [So4;(3.1)] and Note that the exceptional cases are charac-

[So10;(3.6)).

g(V, L) = q def h1(V, 0 ). Here we provide a direct proof using induction on n. By (18.2), terized by the condition

the assertion is true when n = 2. x E BsIK + (n - 1)LI.

which contains We see

x.

D

Assume that

Take a general member is smooth since

L

D

n > 2 of

and

ILI

is very ample.

x E BsIK + (n - 1)LID = BsIKD + (n - 2)LDI,

so

is of the exceptional type by the induction

(D, LD)

hypothesis.

Hence

(18.11) Remark.

is also of the exceptional type.

(V, L)

It is an interesting but subtle

problem whether or not we can remove the assumption of very ampleness, assuming

BsILI = 0

instead.

a few counter-examples even in the case there are very few such examples.

very difficult to show that

Ln s 4

In fact there are

n = 2, but probably

In particular, it is not for any such counter

example.

Example 1. A2 = 1

and set

K + L = A

Take a Del Pezzo surface L = 2A.

is not spanned.

Then

L

(S, A)

such that

is ample and spanned, but

§18: Adjunction theories

157

Take an ample vector bundle

Example 2.

on an elliptic curve with

cl(8) = 1.

Let

of rank two

8

(P, H)

be the

scroll of

and set L = 2J_ Then L ii 2MPIS assd

spanned, but

K + L

is not spanned.

(18.12) In Chapter II, we classify

the nefness of

K + tL

for

K + tL

K + (n - 2)L

t > 0.

especially important in §13.

according to

(V, L)

is

However, the spannedness of

is. not easy to handle even if

t = n - 1.

The key

idea here, due to Sommese [Solo], is to study the semiThe main result in the case where

ampleness (cf. (0.4.2)). t = n - 1

is the following. Let

(18.13) Theorem.

K be a canonical

(V, L)

be as in (18.9) and let

Q-Cartier Well divisor of

V.

Then the

following conditions are equivalent. 1) h1(V, 0V) = g(V, L). 2) (V, L)

is a scroll over a smooth curve or

3) h0(m(K + (n - 1)L)) = 0

for every

J(V, L) = 0.

m > 0.

4) h0(OV(K + (n - 1)L)) = 0. 5) K'+ (n - 1)L

is not nef as a Q-bundle on

6) K + (n - 1)L

is not semiample as a Q-bundle.

V.

For the proof, see [SolO;(4.1),(4.2),(4.4)], where the following result plays an important role. (18.14) Theorem.

Let

normal variety and let divisor of

V.

ample divisor on

Let V.

A

V

be a locally

K be a canonical

Q-Gorenstein Q-Cartier Well

be an irreducible reduced effective Suppose that

V

has only log-terminal

singularities except at finitely many points and that is Cartier in a neighborhood of

A

for some positive

mK

III: Classification Theories of Projective Varieties

158

integer

m.

t E Q with

for some V

Suppose further that

unless

dim A = 1

(K + tL)A

t 2 2 - m-1.

Then

K + tL

A

is nef on

is a scroll over

(V, [A])

and

is nef on

A.

For a proof, see [F24;(5.7)], which slightly improves upon the result in [SolO;(2.1)] for the case t k 2.

m = 1

and

Because of this improvement we can weaken the

assumption on the singularity of

a little in (18.13)

V

(cf. Remark to (18.9)).

(18.15) To prove (18.13), we also use the following. Theorem ([SolO;(2.2)]).

K

(18.14) and suppose that

V, L

Let

Then

is Cartier.

semiample if it is nef and

A

and

be as in K + tL

is

is semiample.

[K + tL] A

Using these results, we prove (18.13) by induction on n = dim V.

(18.16) Thus, in general, we want results describing K + tL

the cases in which

is not nef (or not semiample).

If it is semiample, we study the fibration Im(K + tL)l

m > 0.

for

If

0

0

obtained by

is not birational, it gives

non-trivial information on the structure of

(V, L).

If

0

is birational (this is the general case), we study K + sL for smaller

s, after suitable birational contractions if

necessary. (18.17) Definition.

spectral value

a(V, L)

smallest real number

a

positive integers

and

u

Let of

(V, L) (V, L)

such that

be as in (18.9).

The

is defined to be the JuK + vLJ = 0

v, for which

uK

for all

is Cartier and

v < (n + 1 - o )u. This notion, introduced by Sommese, is very convenient

§ 18: Adjunction theories

159

He first proved

for formulating results systematically.

(V, L) (cf. [SolO;(1.2)]) and described the

for all

a k 0

structure of

with

(V, L)

a(V, L) g 3 (ibid. (4.4),(5.3)).

His results correspond roughly to ours in §11. a(V, L) E Q

whether or not

for all

He asks

This is an

(V, L).

interesting question, to which an affirmitive answer was given by Sakai [Sak2] when

is a normal surface.

V

But

A similar

this seems to be a difficult problem in general.

invariant can be defined for singularities (cf. (18.22)). Now we outline Sommese's results. (18.18) Fact.

Then 1)

Let

be as in (18.9) and

(V, L)

is of one of the following types.

(V, L)

a = 0.

(1Pn, 0(1)).

2) A hyperquadric in

n+1.

a = 1.

3) A scroll over a smooth curve.

a = 1.

4) (A generalized cone over) (F1, 0(e)), e k 3. and

a s 2.

a = 2 - 2/e

V has a quotient singularity.

5) (A generalized cone over) (F200(2)). 6) There is a surjective morphism variety

W with

= K + (n - I)L Warning.

dim W s 2

0: V -> W

such that

for some ample

a = 3/2.

onto a normal

Q-bundle

A

on

0 A

and

0*0V = 0W W.

a = 2.

The meaning of 'scroll' in [Solo] is

different from ours.

(18.19) In order to study the case

2 < a s 3, Sommese

introduced the following notion. Definition.

A normal polarized variety

said to be a reduction of of

V'

at a finite set

V', and if

L = L'V - E

(V, L)

F

if

V

(V', L')

is

is the blowing-up

contained in the smooth part of

for the exceptional divisor

E

III: Classification Theories of Projective Varieties

160

over

As in the smooth case, the minimal reduction of

F.

is well-defined.

(V, L)

(18.20) Fact.

Let

be as in (18.9) and suppose

(V, L)

Suppose in addition that

2 < a s 3.

codim(Sing(V)) 3 3

and further that

singularities when n s 3. reduction of

Let

V

is Gorenstein,

V

has only rational be the minimal

(V', L')

Then one of the following conditions

(V, L).

is satisfied.

0: V' - C

1) There is a surjective morphism curve

C

such that

fiber

F

of

onto a smooth

for any general

(F, LF) = (IP200(2))

a = 5/2.

0.

2) (V', L') = (Q3, 0(2)), where

T4.

is a hyperquadric in

3

a = 5/2.

3) (V', L') = (1P4, 0(2)).

a = 5/2.

4) (V', L')

a = 8/3.

(IF3, 0(3)).

5) There is a surjective morphism variety

W

with

dim W g 3

0: V' - W

such that

onto a normal

O*OV, = OW

*

0 A = K' + (n - 2)L'

where

K'

for some ample line bundle

is the canonical bundle of

For details and proofs see [Solo]. if and only if

K' + (n - 2)L'

V'.

and

A

on

W,

a = 3.

Note that

a

> 3

is nef and big.

(18.21) Using the above theories we can develop classification theories of polarized varieties of the type (18.9) according to the sectional genus. similar to those in Chapter U.

following result when Theorem.

such that

Let

In particular we have the

g = 1.

(V, L)

g(V, L) = 1

The results are of course

and

be a normal polarized variety BsJLJ = 0.

Then one of the

§18: Adjunction theories

161

following conditions is satisfied.

has only rational Gorenstein singularities and

1) V

is a Del Pezzo variety.

(V, L)

is a (generalized cone over a) scroll over

2) (V, L)

an elliptic curve.

d

vector bundle

k k 0

This means that there exist an ample on a smooth elliptic curve

and a birational morphism

(M, f *L)

f: M

such that

V

(0Cek).

d ®

is isomorphic to the scroll of

When n = 2, we can

For a proof, see [F24;(5.4)].

BsILI _ 0. Instead

obtain a similar result without assuming we assume that

C, an integer

is locally Gorenstein and that the

V

canonical bundle is negative (cf. [Bre], [HidW]).

Such a

surface is sometimes called a 'Del Pezzo surface', but it is not always a Del Pezzo variety in our sense in (6.3), since the condition (6.3;d) is satisfied only when

has no

V

irrational singularities. In the case

g a 2, we have the results [BeSiti4],

[FaS3], [LiS] etc. (18.22) Remark.

We can define a similar invariant for

singularities. Let

p

speaking, X Let

be a point on a normal variety

should be considered as a germ of a variety. S = {x E X

be a desingularization and set

n: M --> X

dim n-1(x) > 0),

We say that

E = a-1(S).

desingularization if the set

E

in

M

always exists by Hironaka's theory. M.

to be admissible if

is a divisor having

Let

An effective Q-divisor Supp(R) c E

and if

is a nice

a

Such a map

only simple normal crossing singularities.

cal bundle of

Strictly

X.

K

n

be the canoniR

on

K - R

M

is said

is n-nef,

III: Classification Theories of Projective Varieties

162

E

and

= -1 - InfR(v(R)), where

sible

such that

runs

i

n(Ei) a p, and then set

runs through all the admis-

R

Using (11.8.0) we can show that

Q-divisors.

n(C)

Ei's are components

v(R) = Mini(r), where

We set

ri a Q.

through all the suffix such that e

M

in

C

R = 2 riEi, where the

is a point. Thus of

for any curve

(K - R)C ? 0

i.e.

is

e

independent of the choice of the nice desingularization so it is denoted by of the singularity If

(X, p)

n,

and will be called the energy

e(X, p) (X, p).

Q-Gorenstein singularity, this cor-

is a

responds to the invariant introduced by Shokurov [Sho2].

Note also that log-terminal singularities can be defined as Q-Gorenstein singularities with negative energy. e(X, p) k - dim X

We conjecture that

equality holds if and only if

p

is a smooth point.

an interesting question whether or not (if

and that the

when

e Q

e

e

< 0

We also hope

> 0, there are many counter examples).

e

It is

that there is a classification theory of singularities such that

e

+ dim X, which corresponds to Sommese's spectral

value, is small.

The energy of a polarized variety by

e(V, L) = e(X, p) - 1, where

at the vertex

p

and

K

When

V

is defined

is the singularity

(X, p)

of the projective cone

(V, L) (cf. (19.12)).

(V, L)

X

associated with

has only mild singularities

is the canonical Q-bundle, we have another invariant

E'(V, L) = Sup{t E

and we have

e

a

e'

and

K - tL

IR

is nef),

a i e' + 1 + n, where

Sommese's spectral value and

If

is

The equalities

n = dim V.

hold in many cases, but not always.

a

V

has only

§ 18: Adjunction theories

163

log-terminal singularities and if Mori-Kawamata theory.

e'

< 0, then

by

E Q

E'

It would be an interesting (but

extremely difficult) problem to determine the set

Sn c IR

consisting of the values of energies of all the polarized varieties of a fixed dimension

Presumably

n.

Sn

will

look like, at least partly, as the energy spectrum of the n-body problem in quantum mechanics. r E

the set of

such that

IR

singular polarized surface

Alekseev [Al] studied for some possibly

K(S) ti rL

Even this special case

(S, L).

is very hard to study.

Given a pair tive divisor

D

(M, D)

on M

larities, we define E

If

= Sup{t E

M - D

IR

e(M, D) I

and an effec-

by

x(K(M) - tD) = dim M}.

is affine, this has a similar nature as the

M = 1n

and

D

Thus we conjecture that

is a hyperplane on it.

a classification theory of such pairs small.

M

having only normal crossing singu-

energies of other objects. unless

of a manifold

(M, D)

e

k -n

We will have with

E(M, D)

Moreover, the classification theory of these three

objects (polarized varieties, singularities and affine pairs) will be unified in future.

This unified theory, I

dream, might be related to (some non-linear version of) the quantum mechanics.

Chapter IV. Related Topics and Generalizations 519. Singular and quasi- polarized varieties (19.1) In the preceding theories on polarized varieties we usually needed some conditions on singularities.

This

may look like just a technical matter, but it often involves very delicate problems.

For example, let us consider the

following.

g(V, L) k 0

(19.2) Conjecture.

variety

for every polarized

(V, L).

This assertion is verified in the following cases. 1) (V, L)

has a ladder.

The proof is very easy.

Thus there is no problem if

L

is spanned by global sections.

is smooth and

2) V

char(s) = 0 (cf. (12.1)).

may have certain mild singularities (e.g.

In fact, V

log-terminal, see [F18], [F24;§31). 3) V

is a normal surface.

For a proof in this case, see [Sak2].

Despite these partial answers, the conjecture is still unproved in general.

(19.3) Here we present an approach for the study of singular polarized varieties.

As an application, we prove

Conjecture (19.2) in the case

n = dim V g 3.

A pair

Definition.

line bundle variety if in

V

and

L

L

on

V

(V, L)

of a variety

and a

will be called a quasi-polarized

is nef and big (e== LC a 0

Ln > 0).

V

Two such pairs

for any curve

C

(V1, L1) and (V2, L2)

165

§ 19: Singular and quasi-polarized varieties

are said to be birationally equivalent to each other if there is another variety morphisms

f

i

:

together with birational

V'

fl L1 = f2 L2.

V' --i Vi (i = 1, 2) such that

Under these notions, we can use various techniques in birational geometry. (19.4) Conjecture. g(V, L) a 0 variety

for any quasi-polarized

(V, L).

This assertion is logically stronger than (19.2), but sometimes easier to prove.

In the rest of this section, we assume

char(.) = 0.

We have

g(M, LM) g g(V, L)

M - V.

Hence, in order to prove (19.4), we may assume that

V

In the sequel we outline a proof of (19.4) in

is smooth.

the case

n = dim V g 3.

(19.5) Theorem.

Let

variety with n = dim V polarized variety to

for any desingularization

See [F24] for details. (V, L) 3.

be a quasi-polarized

Then there exists a quasi-

(V', L') which is birationally equivalent

(V, L), has only

Q-factorial terminal singularities,

and satisfies one of the following conditions: 1) K' + (n - 1)L'

divisor

K'

of

is nef for the canonical

Q-Cartier

V'.

2) d(V', L') = 0. 3) (V', L') Remark.

is a scroll over a smooth curve.

We omit the precise definition of

Q-factorial

terminal singularities, which can be found in [KMM] for example.

They are very mild singularities.

(19.6) Outline of proof of (19.5). V

is smooth and K + (n - 1)L

We may assume that

is not nef. By Mori-Kawamata

N: Related Topics and Generalizations

166

0: V -> W

theory (cf. (0.4.16)) there is a contraction an extremal ray

with

R

(K + (n - 1)L)R < 0.

unlike the ample case, it is possible that

When Hence set

LR = 0, we have

L = 0 A

is birational since

0

(X E W

dim T-1(x) > 0)

I

LR > 0,

If

2) or 3) is satisfied as in (11.7).

the condition

codim E = 1, it turns out that

L

terminal singularities, and we set codim E > 1, W

0+: V+ --* W

S

be the

E = 0-1(S).

has only

If

Q-factorial

(V11 L1) = (W, A).

has bad singularities.

we can take the flip

A E Pic(W).

Let

is big.

W

However,

LR = 0.

for some

and let

of

of

If

However, by [Mor5], and we set

0

(V+.

(V1. L1) =

(-P+)*A).

Next we examine the model

It has only mild

(V1, L1).

singularities so that Mori-Kawamata theory applies. we can replace (V1, L1)

(V1, L1)

by another model

(V2, L2)

if

1), 2), 3).

satisfies none of the conditions

In the above process the Picard number if

Hence,

p

codim E > 1, while it decreases by one if

is preserved codim E = 1.

By Shokurov's result (cf. [Shot] or [KMM;5-1-15]), flip processes cannot continue infinitely, so finite steps.

Since

terminate at some setting

(V

p

decreases after

is finite, the whole process must

p

Lk), and the proof is completed by

k'

(V', L') = (Vk' Lk).

Remark.

The above argument works in any dimension if

the Flip Conjecture I,

(19.7) Proof of g(V', L') 3 0.

II

in [KMM] is true.

(19.5) - (19.4).

It suffices to show

The argument used in (12.1) works here,

since we have the following result. (19.8) Theorem (cf. [F24;(1.1)]).

Let

(V, L)

be a

167

§ 19: Singular and quasi-polarized varieties

quasi-polarized variety with

i(V, L) g 0.

4(V, L) = g(V, L) = 0.

and

Let

Proof.

be the normalization of the graph of

G

the rational mapping defined by tional morphism

n*ILI

n: G --' V, a morphism

= E + p*IHI

E

H = OP(1).

for

F

Let

S = deg W.

and

(0.4.9), A = L - D

4(V, L) g 0

implies

Ln = Ln-wHw = S = Hn. for all

Ln-jHj = 8

dim p(E) < n - 1. curve

Hence

Remark.

Then

by (0.4.7). Ln a b

Hence k

Ln-wHw =

h0(V, L) - w,

d(V, L) = 0 = A(W, H), w = n Ln 2 Ln-1H a

...

and

. Hn, we get

In particular Hn-lE = 0.

Hence

E # 0, we apply (11.8.0) to find a

EC < 0

and

p(C)

But

is a point.

case.

Moreover

g(V, L) _

The above argument works well even if

(19.9) Remark. g(V, L)

BsILI = 0.

Thus we have completed the proof.

char(.) > 0, or if

V

is a non-KAhler Moishezon variety.

Quasi-polarized varieties with small

seem to be classified in a similar way to the ample First of all we conjecture

normal and

= 2

If

E = 0, so

g(W, H) = 0.

n

p.

LC = (E + H)C = EC < 0, contradicting the nefness of

then

L

W = p(G), w = dim W

is effective on any general fiber

Since j.

such that

C

such that

Now, by Kodaira's Lemma

Therefore, by (0.4.7),

SLn-wF a S. so

DF

Ln-wF 2 An-wF > 0

F, we have

Set

G

is an ample c-bundle for some effective

Since

D.

on

p: G - p = FdimILI

be a general fiber of

0 g d(W, H) ¢ w + 8 - h0(V, L).

Q-divisor

Then we have a bira-

ILI.

and an effective Cartier divisor

L.

BsILI = 0

Then

g(V, L) = 0.

4(V, L) = 0

if

V

is

This and (19.4) are true when

g 2 (cf. [F24;52]), or if

n g 3

by (19.5) (the case

was previously studied by Sakai).

n

IV: Related Topics and Generalizations

168

(19.10) Until now we have assumed that

What can be said if

variety.

V

V

is a

is allowed to be

non-reduced or reducible ? This is a natural, important and interesting question, but we do not have simple answer.

1) There is no lower bound for

4(V, L).

To produce examples, we add a large number of embedded points into

h0(V, L)

Then

V.

Hence

remains the same.

becomes large, while

4(V, L)

Ln

becomes arbitrarily

There are also many other ways of producing such

negative. examples.

2) There is no lower bound for

g(V, L).

We can construct examples similarly to the above. 3) When

V

is reducible, it is not obvious how we

should define invariants like

d(V, L), g(V, L).

Thus, there are numerous difficulties, but there are also encouraging positive results. 4) Let

M

manifold

be a surjective morphism from a

f: M --* C

onto a smooth curve

line bundle on

The triple

M.

Del Pezzo fibration if any general fiber

F

of

Let

(f, M, L)

(F, LF) f.

C.

L

be an

f-ample

will be called a

is a Del Pezzo manifold for

A singular fiber in such a

fibration can be viewed as a sort of

'Del Pezzo scheme'.

In [F26] all the possible types of such singular fibers are classified.

because

M

The result is surprizingly simple, mainly is assumed to be smooth.

If we allow

M

to

have mild singularities, we get many more examples of such Del Pezzo schemes.

§19: Singular and quasi-polarized varieties

169

(19.11) It should be mentioned that singular polarized varieties (or even schemes) appear in a natural way in the theory of Mori-Kawamata.

0: M - W

Let

tremal ray (x E W

Suppose that

R.

dim $-1(x) > 0)

I

codim E = 1

we assume and

be the contraction morphism of an ex-

E = '-1(S).

and S

and

Set

S =

For simplicity ER < 0

Then

is a point.

Thus, the classifi-

is a polarized variety.

[-E])

(E,

is birational.

0

cation of such contractions is very closely related to that See [Mor3], [Bell], [Be12],

of polarized varieties. [F25;§3].

(19.12) Here we discuss another important relationship between the theory of polarized varieties and the theory of singularities.

Given a normal polarized variety be the scroll of of

6 = L ® OV

over

defines the unique member

iS

v: P -> V.

a section of

HD = 0

(V, L), let

The subbundle

V.

D0

(P, H)

L

JH - LPI, which is

of

and the normal bundle

0

[DO]D = -L

is negative.

So, by Grauert's criterion, there

0

is a birational contraction point

on

v

0: P - C When

C = C(V, L).

very ample and the image of

V

is just the projective cone over this case

C - O(D_)

is another section of OV

of

8.

L

of

is simply generated ((-4

is projectively normal), C V

with vertex

is the affine cone over x

to a normal

D0

v.

In

V, where

D_

corresponding to the subbundle

With these observations in mind, by conical

singularity of type isomorphic to

C

at

(V, L) v.

we mean a singularity which is

Many properties of such a

singularity can be translated to those of

(V, L).

For

IV: Related Topics and Generalizations

170

example we have:

is Cohen Macaulay if and only if

1) v

Macaulay and

Hq(V, tL) = 0

0 < q < n = dim V.

2) v

q, t E Z

such that

It is Gorenstein if and only if a E Z

for some

WV = 0V[aL]

for any

is locally

V

in addition.

is a rational singularity if and only if

Hq(V, tL) = 0

for any

q > 0

and

t k 0, provided

V

has

only rational singularities. 3) If

is simply generated, then the multiplicity of

L

the singularity

v

dimension of Remark.

d(V, L), and the embedding

is equal to

v

h0(V, L).

is

Taking conical singularities after choosing a

(V, L), we can produce various examples of singu-

suitable

larities with the desired properties. (19.13) In view of (19.12.3), we define d + e - h - 1

multiplicity proved

for a

d-dimensional local ring

of embedding dimension

e

4(R, m) ? 0

A(R, m) =

[Sal l'6] further studied the cases in which

Ooishi [001,2] defined the notions of

of

Abhyankar [Ab]

is Cohen Macaulay.

(R, m)

if

h.

(R, m)

d

Sally

is small.

4-genus and sectional

genus for noetherian rings and studied their properties. (19.14) The smoothability of conical singularities is

closely related to the existence of a polarized manifold (M, A)

such that

If such tions of A to

and

(D, AD) = (V, L)

exists, let

(M, A)

µ: IP(A (D OM) --* M

0M

respectively.

M0 + It D.

Let

A

for some M0

and M

D E

JAI.

be the sec-

corresponding to the subbundles

Then

M_

is linearly equivalent

be the pencil containing them.

Let

§20: Ample vector bundles

A'

be its image on the cone

of

A'

is isomorphic to

171

C(M, A).

Any general member M0 + µ D

M, while the image of

C(V, L), at least set-theoretically.

is isomorphic to

This

is also true in the scheme-theoretical sense under some additional assumptions.

Thus, C(V, L)

is smoothable in a

certain sense (see [F9;(3.11)]).

On the other hand, if the pair smoothable to a pair ation, the rung

D

(M, D)

is a deformation of

polarized manifold.

(V, L)

is

under a flat polarized deform-

very roughly speaking, C(V, L) small deformation of

(C(V, L), q5(D_))

D_ = V.

Hence,

is not smoothable if any cannot be a rung of another

This was the basic idea of [So2].

See

also [F9], [F14], [Sol], [Bad2'5], [Lvl,2]. §20.

Ample vector bundles

A pair (V, 8) consisting of a variety V and an ample vector bundle 8 on V can be viewed as a generalization of the notion of a polarized variety. We review a few theories on such pairs. Throughout this section we assume n = dim V a 2, r = and V is smooth.

rank 8 a 2, char(s) = 0

First, following the idea of [YZ), we give vector bundle versions of the results in §11 on adjoint bundles. (20.1) Theorem (compare [YZ;Theorem 1]). Set and assume r . n. ®n+10(1)),

Then

K + A

is ample unless

A = det 8 (V, 8) =

where K is the canonical bundle of

(lPn

V.

For the proof we need the following lemmas.

For any extremal ray

(20.2) Lemma (cf. [Mor3;(l.4)]).

V, there is a rational curve

R

on

R

such that

-KZ s n + 1.

Z

in

V

representing

IV: Related Topics and Generalizations

172

(20.3) Lemma.

Let

R

be an extremal ray of E

Suppose that there is a subset

there is a rational curve -KZ g n + 1 -

such that any

V

is contained in

R

extremal curve representing

of

representing

Z

V.

Then

E.

such that

R

E.

For a proof, see [Io3;(0.4)]. (20.4) Proof of (20.1). nef.

If not, there is an extremal rational curve

that

(K + A)Z < 0

and

-KZ g n + 1

since the pull-back of

r > n

K + A

First we prove that

such

Z

by (20.2).

is

But

AZ k

to the normalization of

8

is the direct sum of line bundles of positive degrees.

Z

Thus

we get a contradiction, as desired.

Now, by (0.4.15), we have a fibration

on W such that

H

an ample line bundle

this is not ample, there is a curve of

0.

We may assume that

(20.3).

Therefore

contradicts

and

0 H = K + A.

if

contained in a fiber

Z

is an extremal curve as in

Z

-KZ s n - 1

0 = HZ = (K + A)Z

is not birational and

0: V -* W

0

if

dim W < n.

Hence

AZ k r > n.

and

dim W > 0,

If

This

is birational.

can be

Z

viewed as an extremal rational curve of a general fiber of

0, so

KZ = KFZ k - dim F - 1 k -n

and

-KZ t n + 1 and Let

(P, H)

canonical bundle

Moreover

K + A = 0. AZ

of

P

is

6

r = n + 1

over

-(n + 1)H

bundle formula in (0.1.3) for scrolls. negative.

since

0: P -> X

V.

Then the

by the canonical

In particular it is

Hence there is an extremal ray

contraction morphism

This

by (20.2).

r.

be the scroll of KP

F

Thus we conclude that

again yields a contradiction. dim W = 0

0

R

of

P

whose

is different from the map

§20: Ample vector bundles

By (0.4.16.2), no curve is contracted by both

n: P --+ V.

maps

and

a.

Assume that

Hence

0-1(x).

n(Z)

x

Take a point

(KP + qH)R k 0

Then

q a n + 1, but then

in

Z

is birational.

dim 0-1(x) = q > 0.

that

173

such

by (11.5).

is a point for some curve

This case is ruled out by the above

observation.

We have

dim X < dim P.

contracted to a point by n

F

:

0, so

0, hence

F

Since the canonical bundle of have of

(F, HF) = (1Pn, 0(1))

again, we obtain

nF

V = 1Pn

V.

Hence

F

dim X = dim F = n.

KP F = -(n + 1)HF, we

is

Using the finiteness

by (1.3).

Now it is easy to see that line in

Similarly

dim X a n.

n g dim F = 2n - dim X, so

is

a

is a finite morphism for any general fiber

F --' V

of

No curve in a fiber of

by [Laz2]. 6(-1)

is trivial on any

is trivial by [OSS;3.2.1], so

t(-l)

8 . ®n+10(1), as desired.

Using similar arguments we can also prove the following results.

See [YZ] for details.

(20.5) Theorem (cf. [YZ;Theorem 2]). is nef unless

(V, 8)

(1Pn

If

®n0(1)).

(20.6) Theorem (cf. [YZ;Corollary 1]).

non-zero effective divisor on crossing singularities.

V

1Pn

and

D

be a

V having only simple normal

If the locally free sheaf

is a hyperplane on

(20.7) Theorem (cf. [YZ;Theorem 3]).

K + A

D

Let

of vector fields with logarithmic poles along then

r = n, K + A

D

0(V, D) is ample,

V.

If

r = n - 1,

is nef except in the following cases.

1) There is a line bundle

L

on

V

such that

(V, L)

is a

IV: Related Topics and Generalizations

174

scroll over a smooth curve vector bundle 2) (V, 8)

on

for some

C.

®n-10(1))

(1Pn,

3) (V, 8)

4) V

3

8 = 3® ® L

and

C

n-20(1)

(1Pn,

is a hyperquadric

(D 0(2)). 1Pn+1

in

Qn

and

®n 10(1).

8

Furthermore, by using the results and techniques in [Mor2], [Laz2;S4] and [CS], we can prove the following result, as conjectured in [Muk3]. (20.8) Theorem (cf. [F27]).

If

r = n, K + A

is ample

except in the following cases. 1) (V, 8)

ti

(D n0(1)).

(]P',

2) There is a line bundle scroll over a smooth curve vector bundle 3) (V, 8) =

7

on

C

V

such that

and

8 ti 3® ® L

is a

for some

® 0(2)).

is the tangent bundle of 5) (V, 8) = (Qn, en,(,)). 4) V = Fn and

(V, L)

C.

®n-10(1)

(]Pf,

on

L

8

V.

(20.9) Now we review vector bundle versions of the classification theory by sectional genus. Clearly

(V, A)

is a polarized variety for

Its sectional genus is called the The pairs with

(V, 8).

A = det S.

c1-sectional genus of

g(V, A) g 2

are classified in

[F22] as follows.

(20.9.0) g(V, A) = 0

if and only if

V = 1P2

and

8 =

0(1) ® 0(1).

(20.9.1) g(V, A) = 1

exactly in the following cases:

1) (V, 8) = (]P200(2) e 0(1)). 2) V =

T2

and

8

is the tangent bundle of

V.

§20: Ample vector bundles

3) (V, 8) =

175

0(1) ® 0(1) ® 0(1)).

(1P2

2

Of course

4) (V, 8) = (Q2, 0(1) ® 0(1)).

X 1F1.

= 1P1

5) (V, 8) = (1P3, 0(1) (B 0(1)).

(20.9.2) If

g(V, A) = 2, then

Thus

n = 2.

8

is of

one of the types classified in (15.17).

The above results are based on the classification of the polarized manifold (20.10) Let

H

(V, A).

(P, H)

is ample by definition.

called the cation of

be the scroll of

The sectional genus

0(1)-sectional genus of (V, 8)

over

8

with small

(V, 8).

V. Then

g(P, H)

is

The classifi-

0(l)-sectional genus is in

fact a part of the classification of polarized manifolds.

Indeed, for each type in the classification list of polarized manifolds, we simply check whether or not it admits a structure of a scroll of a vector bundle.

The result is as

follows (cf. [F22;S3]). (20.10.0) g(P, H) = 0

if and only if

V = pn

and

8 =

0(1) ® 0(1).

(20.10.1) g(P, H) = 1

exactly in the following cases:

1) (V, 8) = (1P2, 0(2) ® 0(1)).

2) (V, 8) = (1P2, 0(1) ® 0(1) ® 0(1)). 3) V = F2 4) (V, 8)

and (Q2

6

is the tangent bundle of

V.

0(1) ® 0(1)).

(20.10.2) g(P, H) = 2

exactly in the following cases:

1) (V, 8) = (Q3, 0(1) ® 0(1)). 2) (V, 8)

is the type of (20.9.2).

(20.11) We have assumed

n 3 2

since, when

V

is a

N: Related Topics and Generalizations

176

curve, both

g(V, A)

and

g(P, H)

are the genus of

V.

At present, we do not have a good vector bundle version of the theory of

d-genus.

§21. Computer-aided enumeration of ruled polarized surfaces of a fixed sectional genus (21.1) Below is the core subroutine of our computer program "Classification of Rational Polarized Surfaces" written in N88-Japanese-BASIC.

This subroutine is designed

to enumerate, for any fixed values of

L2, KL

and

the possible deformation types of polarized surfaces such that

L + 2K

S # 1P2, S

is rational and either

K2 = 8

K2, all (S, L)

or

is nef.

Using this subroutine and by the observation in §14, we can produce programs for various purposes. 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410

'LL, KL, KK is fixed IF KK0 THEN 1240 IF C=0 THEN GOSUB 1510 IF I>1 THEN I=I-1:G0TO 1290 ELSE RETURN M(I)=M:B(I)=B:C(I)=C IF KK+I>0 THEN 1300 IF KK+I=O THEN 1280 I=I+1:B(I)=B:C(I)=C:M(I)=M:GOTO 1260 IF B>0 THEN D=B*B:GOTO 1320 M=M(I)+l:B=B(I)-KK-I+1:C=C(I)-B(I)-B:GOTO 1210 IF B 0, then we blow down further.

i = 1.

mi-l

0

defined by

mi.

Hence we are done

i > 1, we should consider the case in which

is one bigger.

1240-70: If

1280: When

Ki2

=

K2 + i < 0, we blow down further.

Ki2 = 0, bi

should be positive by (14.13).

1290: Here we study the case in which than as it was.

Let

bi' = bi - Ki-12 and since

is of the type

This case is treated in 1510-30.

is not nef for larger

If

A i

bi

'

mi is one bigger

denote the new values. Then we have

ci' = ci - 2bi + Ki-12 = ci - bi - bi'

-Ki-1(Li-1 + miKi-1) and

ci = (Li-1 + miKi-1)' 2

IV: Related Topics and Generalizations

178

1300: Here

Ki2 > 0, so

bi g 0, we have with

(Li + tKi)2 g 0

Hence

t s m.

should change

bi

for some real number

is not nef for larger

A i

When

should be positive.

mi

t

and we

mi-1'

1310: Since

K1. 2 > 0, we should have

Otherwise we change

the index theorem. 1320-30: If

Ki2 = 8,

S

is a

bi2

ciKi2

by

mi.

Otherwise

]P1-bundle.

we continue to blow down. 1340: Set

x2e + 2xy = c

Then

r = 0).

Ar = Lr + mrKr = xH + yF (set

A0 = L

ex + 2y + 2x = b.

and

this subroutine we study this Diophantine equation. that

m > 0

r > 0

if

Note also that and

c > 0.

Hence

c = xz.

while

x(b - 2x) = c.

equation and then consider 1350: Since Hence

m = 0

In

Note

r = 0 (cf. 1180).

if

z = b - 2x

Then

z = ex + 2y.

Set

if

We first solve this

e, y.

c > 0, both

x

and

are positive.

z

b = z + 2x k 3.

1360-80: We first look for the smallest solution 1390: No solution is found such that remains to consider the case y a 0

and

x > 1, we have

b = 2x + 2 = c + 2, so Therefore we may assume x = 3

and

y = 0.

z s 3 z > 1.

x = 1

If

so it

4x s c,

Since

x > 1.

and

x.

z = 2, then

should have been a solution. Then

z = ex + 2y = 3.

This is possible only when

e = 1,

b = c = 9.

Thus this step 1390 is justified. 1400-20: Here equation. odd.

e = 0

Hence

If

x

is the smallest solution of the

z = ex + 2y e = 1

is possible.

and

is odd, then

z - x = 2y k 0.

e

If

and z

x

must be

is even,

§21: Computer-aided enumeration of ruled polarized surfaces

means

1430: m = 0

m > 0, Ar

If

ample.

1440: If

K2 = 8,

is just nef and

is odd, x

z

y > 0

so

1450-60:,If both

since

L

is

is possibly

y

0.

is the unique solution.

and

z

179

x

are even, then

possible provided the corresponding

e = 1

is

is positive (or

y

non-negative when r > 0). 1470-1500: We now consider whether or not there is another solution for b

only if

and

c

x.

By (14.11), such a solution exists If so, there are possibly

are even.

four cases: (e, x, y) = (0, a, P), (0,

with

fi,

or

a)

x = a

solution

(1,

fi,

(1, a, P - a/2),

with

a - P/2)

a < P.

The solutions

is treated in the preceding steps.

(0, P, a)

is essentially the same as (0, a,

Hence we should consider the solution This is actually the case if

P

(1,

in a situation as in (14.3;0) or 1).

Moreover, if

is a Del Pezzo surface.

fi,

is even and

1510-30: Here we consider the case

b = -KiLi a 0.

The fi).

a - X4/2).

8 f 2a.

c = 0.

Thus we are

In this case we have

b = 0, then

A. = 0

and

This is possible only if

2

Ki

S.

> 0.

If these conditions are satisfied, we have a solution (e, x, y) = (0, 0, b/2).

Recall that we may assume

e = 0

(cf. (14.17)).

1540: In the subroutine beginning here, the data of the

new type of polarized surfaces thus found is handled appropriately.

(21.3) We have produced similar programs for the classification of irrational ruled surfaces too.

The

algorithm is rather simpler than that in the rational case. (21.4) Here we exhibit several samples of results of

IV: Related Topics and Generalizations

180

First we

our computer experiments by the preceding program. present a list of

(S, L)

such that

is not a scroll, g(S, L) = 4 S

g

meaning of

q, mj, e, x, y

1) 0: 2) 0:

1 1

3) 0: 1 4) 0: 5) 0: 6) 0:

1 1

1

too.

-1: -1: -1: -1: -1: -2:

0 0 0 0 0 0

1

-2: 0 -2: 0 -2: 0 -2: 0 -2: 0 -2: 0 -2: 0 -2: 0 -3: 0 -3: 0 -3: 1 -3: 0 -3: 0 -4: 0

21) 0: 2 22) 0: 2 23) 0: 2

-1: 0 -1: 0 -1: 0

7) 0: 8) 0: 9) 0: 10) 0:

11) 0: 12) 0: 13) 0: 14) 0:

1 1 1

1 1

1 1 1

15) 0: 1 16) 0: 1 17) 0: 1 18) 0: 1 19) 0: 1

20) 0:

24) 0: 2 25) 0: 2 26) 0: 2 27) 0: 2 28) 0: 2 29) 0: 2 30) 0: 2 31) 0: 2 32) 0: 2 33) 0: 2 34) 0: 2 35) 0: 2 36) 0: 2 37) 0: 3 38) 0: 3 39) 0: 3 40) 0: 3 41) 0: 3 42) 0: 3 43) 0: 3

-1: 0 -1: 0 -2: 0 -2: 0 -2: 1 -2: 0 -2: 0 -2: 0 -2: 0 -3: 0 -3: 1 -3: 0 -4: 0 -1: 0 -1: 0 -1: 0 -1: 0 -2: 0 -2: 0 -2: 0

x 38 26 22 22 21 22 16

g < 4.

.. ,

m1)

19 13 11 11 10 11

19 13 11 11 10 11

19 13 11 11 10 11

19 13 11 11 10 11

19 13 11 11 10 11

19 13 11 10 10 11

19 12 11 10 10 11

18 12 10

8

8

8

8

7

7

7

8 7

7 7

7

7

8 7

14

16: 15:

20

20: 10 10 10 10 10 10 10

14 12 14 12

14: 13: 14: 12: 11: 14: 10: 4: 9: 8: 7:

11

14 10 9

8 8 6

7

7

7

7

7

7

6

6

6

6

6

6

7

7

7

6 5

6

7 6

7

6 5

7 5

5 7 5

5

7 5 4 4 4 3

5 7 5

7 5 5 7 5

4 4 4 3

4

4 4 3 3

4 4 3 3

4

4 4 3

7 5

4 4 3

6

9

10 10 6 9

6 6

6

7

4 5 5

5 5

5

7

6

4 4 4 3 3

4 4 3 3 2

28 20 20

28: 14 14 14 14 14 14 14 13 20: 10 10 10 10 10 10 9 9 20: 10 10 10 10 10 10 10 7

18 17 16 12 11 12 10 10

18: 17: 16: 12: 5: 12: 11: 10: 9: 8: 3: 8: 5:

9 8

7 6 5

20 16 14 14 12

10 8

The

are as in §14.

y: (mr

38: 26: 23: 22: 21: 22:

is nef or

Similar lists are obtained

See [F21], [Mae] for

for larger

K2, e,

L + 2K

and either

is a F1-bundle over a curve.

q:L2

is ruled, (S, L)

S

6

7 8 9 9

4 5 6 3

2

4 5 4 4 5 2 3 4 3 3 2

3 3 4 4 4 2 2 2 3 3 2

6

5

5 5 6 5

4 4 4 3 3 2

4 4 3 3 3 2

7 5 5 4 5 4 4 3 3 3 2

20: 10 10 10 10 10 10 10

9

7 8 3 4 5 3 3 4 4 2 3 3 2 4

16: 15: 14: 12: 10: 9:

9

9

9

8 8 6

8 8

8 8

8 8

8

6 5 6 5 5

6

6

5

5

6

6

5 5

5

8 6 5 6 5

5

5

4 4 3 3 2

4

4 4 3

5

6 5 5

4 4 3 3 2 8

4

4 3 3 2

4 3 3 2

8 7

8

7

5

8 7 7 6 5

5

7 6 5

4

4

4

4

7 7 6

6

7

3 2 8 7 7 6 5

4

5 6

8 8

2 2 2 2 3 2

2

7

8 8 8

9

3

5 6

8 8 8

9

2 2

8 7

8

6

5

7

7

5

6 6

6

6

5

2

5

6 6 5

4

4

3 4

3 3

2 2 2 3 3 3 4 2 2 3 2

2 2

2

2 2 2

2

2

§21: Computer-aided enumeration of ruled polarized surfaces

44) 0: 3 45) 0: 3 46) 0: 4 47) 0: 4 48) 0: 4 49) 0: 4 50) 0: 4 51) 0: 4 52) 0: 5 53) 0: 5 54) 0: 5 55) 0: 6 56) 0: 7 57) 0: 8 58) 0: 9

-2: 0 -3: 0 -1: 0 -1: 0 -1: 0 -2: 0 -2: 0 -3: 0 -1: 0 -1: 0 -2: 1 -1: 0 0: 0 0: 0

59) 60) 61) 62) 63) 64) 65) 66) 67)

0:12 0:18 0:20 0:20 1: 1 1: 1 1: 1 1: 1 1: 1

68) 69) 70) 71) 72)

1: 1: 1: 1: 1:

1

73) 74) 75) 76) 77) 78) 79) 80) 81) 82) 83) 84) 85) 86) 87) 88) 89) 90) 91) 92) 93) 94) 95) 96) 97)

1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1:

2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4

3: 8: 8: 8: -1: -1: -2: -2: -2: -2: -2: -3: -3: -3: -1: -1: -1: -2: -2: -2: -2: -2: -3: -3: -1: -1: -1: -1: -2: -2: -2: -2: -3: -1: -1: -1: -2: -2: -3: -1: -1: -1: 0:

98) 99) 100) 101)

1 1 1 1

1: 5 1: 5 1: 6 1: 7

1: 0

0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 1 1 1 1 0 1 0 1 0 1 0 0 1

8 6

14 12 10 8 6

4 10 8 5

6 8

4 6 4 3 2 2

8: 6: 14: 12: 11: 8: 7: 6: 10: 8: 2: 6: 8: 5: 6:

4 3 7 6

4: 3:

2

5

4 3 2 5 4 2 3 4 2 3

6

25

1:

7

21 -10:

4

15 19 13 11 13

5 3

1:

-9: 1:

-4: -6:

4 4 2

1: 2:

3 3

27 -13:

5 6

7

19 17 15 11 10 10

7 6 5

4 3 2 5 4 2 3 4 2 3 2

4 3 7 6

5

5

4 3 2 5 4 2 3 4 2 3

4 3 2 5 4 2 3 4 2 3

2

2

1:

-7: -7:

2 2

3 3 4 2 2 3

2 2

3

7 3 4 3 3 4

2 2 3 3

1: 2:

2

2

2

3

3

2

19 14 14 13 11

-9:

4

1:

5 5 6

8 8 7

1:

9

7 6

5

13 10 10 6 6

1: 1:

-4: -3:

-6: -5: -5: -3: -2: -1: -6: 1:

-4: 1:

-2:

4

2:

9

-4:

7

1: 1:

5 7

-3:

3

2

2

3 3 3 2

2

2 3 2

2

2 2 2

2

3

4 4 2 2 2 2 3 2

4 3 7 6

5: 4:

37 -18:

9

4 3

4 3 7

3 2 7

6 5

6 5

4 3

4 3 2

2 5

4 2

3 4 2 3 2

5

4 2 3 4 2 3

3 2 7 6 5 4 3 2 5 3 2 2 4 2 3

3 2 6

4 5

2 3 2 4 3 2 2 3 2

181

3 2 3 4 4 2 2 2 2 3 2 2

3

2

2

2 2 2

2

2

IV: Related Topics and Generalizations

182

102) 103) 104) 105) 106) 107) 108)

1: 8 1: 8 1: 9 1:12 1:12 2: 4 2: 4

0: 0 0: 1 0:

1

0: 0: -8: -8:

0 1 0 1

4 4 3 2 2 2 2

1:

-1: 0: 3: 2: 1: 0:

(21.5) Now we can calculate the number of deformation types of ruled polarized surfaces with

g(S, L) = 4

Suppose that the minimal reduction

follows.

of one of the types in the above list.

(S0, L0)

(S0, L0) ?

The number of (-1)-curves of

S --' S0

d0 = (L0)2, so there are

deformation types of such

polarized surfaces

d0

(S, L), including

However, occasionally, (S0, L0) by

(S, L).

x = 2

for

is smaller than

(S0, L0)

itself.

is not determined uniquely

This happens to be the case if and only if (S0, L0).

in the case of types

In the above list (21.4), this occurs 61), 62), 105), 106), 107), 108).

Thus, for example, types 61) and 62) together yield types of

is

How many types of

do we have whose minimal reduction is

(S, L)

as

(S, L), instead of

we just add up

d0

2 x 20 = 40

types.

21

Otherwise

to conclude that there are 346 types.

The existence of these types is uncertain.

If the

following conjecture is true, all the types of ruled polarized surfaces enumerated by this method do exist (for larger

g

too).

Conjecture.

surface

S

effective S

Any ruled surface can be deformed to a

such that the cone in

H2(S; IR)

of pseudo-

1R-divisors is generated by all the (-1)-curves on

and curves of positive self-intersection numbers.

(21.6) The table below shows the number of possible types

(S0, L0)

with given

(g, q, d0)

as in (21.4).

Here

§21: Computer-aided enumeration of ruled polarized surfaces

183

5 s g s 7. g

5

q

0

1

32 18

9

72 41 35 18 8 7 4 3 2

10

1

1

4 3

11 12 13 14 15 16 17 18

0

0

2

2

1

1

3

0 0 0

0 0 0 2

1 1

3 0 0

0

1

1

d0 1

2 3

4 5 6 7 8

6

0

1

2

3:

0 :221 0 :146 1 94

58 62 42 36 19

0 0 0 0

0 0 0 2

2:

19 15

:

0

72

:

0: 35 0: 19 0: 17 2: 10

4 4 4 3 1

7

0

1

2

3

:525 :384 :257 :146

103 116 111

1

51

1

0 0 0 0

0

:100

9

1

10

0 0 0 0 0

: : :

65 41

33 34 23

34

21

0 0 0 0

:

16

9

1

7 8

4

5

5

2 2 1

1 1 1

0 0 0 0 0 0

2

2

2

1

0

1 1

1

9 2 1

2

0 0

20

0 0 0

0 0 0

1

2

21

1

0

24 27 28

2

0 2 0 2

0

1

5

0 0

0 2

0 0 2

1 1

32

0 0

2

A similar table is obtained for larger

g

too.

(21.7) By the method (21.5), we can calculate the number

t(g)

of types of ruled polarized surfaces of

sectional genus

g

2

3

4

5

0

18

1

9

69 40

199 142

567 329 12

q 2 3 4 5

t(g)

g g 10

The result for

g.

5

7

6

is as follows.

8

9

10

1836 865

4699 1923

12273 3631

30965 7760

70975 14876

19

31

5

9

60 16

64 23

107 30

5

9

13

15985

38821

86006

5

27

Problem.

109

346

908

2725

6962

Study the asymptotic behaviour of

t(g)

when

184

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202

Subject Index adjoint bundle: 39,40,86,93,96,97,104,152,153,156,157,158, 165,171,173,174 adjoint spectrum: 116,121,125,177,180 adjunction theory: §18 admissible minimalization: 116 ample vector bundle: 41, 113,132 adjoint bundle of 171,173,174 c -sectional genus of 174 175 0i1)-sectional genus of :

:

:

ampleness criterion for blow down: 61 Nakai's 14 6 Serre's Bertini theorem for irreducibility, reducedness: 32 for k-Macaulayness: 8 for normality: 9 for smoothness: 36,62 : :

bi-elliptic surface:= hyperelliptic surface big: 16,164 blowing-down: 4 extension of 60 :

blowing-up: 3 canonical bundle formula for Chow ring of 12 simple 103 canonical bundle formula for blowing-up: 4 for double covering: 49 for scrolls: 2

:

4

:

:

Castelnuovo bound: 139,140 Castelnuovo variety: 142,143 characterization of In (and

Qn): 22,23,24,52,59,62,93,104, 159,171,173,174,175

Chern class: 11 Chow ring: 10 of scroll: 12 Clifford's theorem: 31

Cohen Macaulay singularity: 22,36,38,43,44,49,50,56,170 deformaton type: 108,110,118,182 (generalized) cone: 42,79,159,161,169,170 degree: 4 Del Pezzo fibration: 168

203

Subject Index

Del Pezzo surface: 62,115,161 of degree one: 123,134,153,156 of degree two: 153 Del Pezzo variety (or manifold): 44,83 45,46,82,107,161 characterization of classification of 72 of degree 1: 50 2: 49 3: 47 4: 47 5: 69,77 6: 66,76 7: 65 8: 65 in positive characteristic case: 73 d-genus: 26 :

:

desingularization: 9 double covering: 48,49,51,53,85,89,91,126,129,142,143 elliptic ruled variety: 123,133,134,153,157 elliptic scroll: 47,84,89,107,137,517,161 embedded component: 7,23,168 energy: 161 Enriques surface:. 20,126 exceptional divisor: 3 extension theorem (of Lefschetz type) of blowing-dowm: 60 of line bundle: 55,76 of linear system: 57 of morphism: 57,58 of scroll structure: 60 of vector bundle: 71 extremal ray (extremal curve): 18,93,166,171,172 Fibration theorem: 17,115,172 flatness criterion: 6 flip: 166 higher direct image: 4 Hilbert polynomial: 13,24,25,45,95,99,108,110 hyperelliptic fibration: 52,86 hyperelliptic polarized variety: 143,90 hyperelliptic surface (:= bielliptic surface): 21,126 hyperquadric fibration: 87,100,112,129,134,135,159,160 index theorem: 15,98,110,167 intersection number: 11,13 Jacobian vector bundle: 132

204

Subject Index

K3-surface: 20,126 Kleiman's criterion of nefness: 15 Kodaira's Lemma for bigness: 16 ladder: 28 regular existence of

:

28,30,141 30,32,36,37,38,43,44 :

Lefschetz theorem: 55 Leray spectral sequence: 5 lift to characteristic zero: 73 log-terminal singularity: 96,156,157,162,164 k-Macaulay singularity: 7,8,44,57,58,59 Cohen Macaulay singularity: 22,36,38,43,44,49,50,56,170 minimal reduction: 103,11,121,129,130,159,160 Nakai's criterion of ampleness: 14 nef: 14,15,16,17,18 and big: 16,164 normal singularity: 8,9,17,30,43,78,80,81,82,164,169 polarized manifold of d-genus 0: 41,107 1: 47-55,72 2: 92,150 150 fixe d of sectional genus 0: 30,41,107 1: 72,84,107 2: 86,122,129,150 3: 136,150 :

fixed: 108'

polarized varieties of 4-genus

0: §5,42,79,142,143,167 1: §6,§9,82,83

proper transform: 4 Q-bundle: 14

Q-divisor: 14 quasipolarized variety: 164 rational scroll: 41,80,107,145,§17 relatively (very) ample: 3 relative rational map: 3 ridge: 42,79,145 Riemann-Roch-Hirzebruch formula: 13 rung: 25 regular

:

26,27

,114ti ,150

205

Subject Index

scroll: 2,102,160,172 over curves: 96,107,108,109,122,129,152,156,157, 158,159,165,174 surfaces: 98,101,113,129,132,159 12 Chowring of :

e llip ti c ext ens i on o f

47 , 84 89 , 107 , 137 ,§ 17 161 s truc t ure: 60 41 , 80 , 107 145 ,§ 17

:

rati ona l

,

:

,

,

sectional genus: 25,26

c-

:

Oil)-

174 :

175

Segre class: 12 semiample: 14,157,158 Serre's condition Serre duality: 6

Sk: 8

Serre's normality criterion: 8 vanishing theorem: 6 Severi variety (in the sense of Zak): 105 simple blow-up: 103 simply generated: 27,30,35,38,44,86,142,170 spectral sequence: 5 spectral value (in the sense of Sommese): 158,159,160,162 Stein factorization: 1 strict transform: 4 Todd class: 12

triple cover: 90 upper-semicontinuity theorem: 6 vanishing theorem Kawamata-Viehweg's 16 Kodaira's 16 Mumford's 17 Serre's 6 Sommese's 56 :

:

:

:

:

varieties of minimal degrees: 43 varieties of small degrees: §17 varieties with many rational curves: 104 Veronese surface: 39,41,80,96,97,106,112,146,159 (weighted) hypersurface (or complete intersection): 28,47, 48,50,52,53,59,89,128,131,144