Algebraic Geometry: Proceedings of the International Conference held in L’Aquila, Italy, May 30–June 4, 1988 [1 ed.] 9780387522173, 0-387-52217-4 [PDF]

Zitiervorschau

Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and E Takens

1417 III

A.J. Sommese A. Biancofiore E.L. Livorni (Eds.)

Algebraic Geometry Proceedings of the International Conference held in L'Aquila, Italy, May 30-June 4, 1988 I IIIlll I I

Springer-Verlag Berlin Heidelberg NewYork London ParisTokyo Hong Kong

Editors

Andrew John Sommese Department of Mathematics, University of Notre Dame Notre Dame, Indiana 46556, USA Aldo Biancofiore Elvira Laura Livorni Dipartimento di Matematica, Universit~ degli Studi di L'Aquila 67100 L'Aquila, Italia

Mathematics Subject Classification (1980): 14J99, 14N05, 14M99, 14C99 ISBN 3-540-52217-4 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-52217-4 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-VerlagBerlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

Introduction The question of how the geometry of a projective variety is determined by its hyperplane sections has been an attractive area of algebraic geometry for at least a century. A century ago Picard's study of hyperplane sections led him to his famous theorem on the 'regularity of the adjoint '. This result, which is the Kodaira vanishing theorem in the special case of very ample line bundles on smooth surfaces, has led to many developments to this day. Castelnuovo and Enrique~ related the first Betti number of a variety and its hyperplane section. This and Picard's work led to the Lefschetz hyperplane section theorem and the modern work on ampleness and connectivity. A large part of the study of hyperplane sections has always been connected with the classification of projective varieties by projective invariants. Recent new methods, such as the adjunction mappings developed to study hyperplane sections, have led to beautiful general results in this classification. The papers in this proceedings of the L'Aquila Conference capture this lively diversity. They will give the reader a good picture of the currently active parts of the field.The papers can only hint at the friendly 'give and take' that punctuated many talks and at the mathematics actively discussed during the conference. The success of this conference was in large part due to the Scientific and Organizing Committe: Professor Mauro Beltrametti (Genova), Professor Aldo Biancofiore ~'Aquila), Professor Antonio Lanteri (Milano), and Professor Elvira Laura Livorni (L'Aquila). The publication of this proceedings would not have been possible except for the efforts of Professor E.L.Livorni. Andrew J. Sommese

T a b l e of c o n t e n t s B a d e s c u L.

I n f i n i t e s i m a l d e f o r m a t i o n s of n e g a t i v e weights a n d h y p e r p l a n e s e c t i o n s ............................. 1

Ballico E.

O n k - s p a n n e d projective surfaces .......................... 23

B e l t r a m e t t i M . / S o m m e s e A.J.

O n k - s p a n n e d n e s s for projective s u r f a c e s .......... 2 4

Biancofiore A.

O n t h e h y p e r p l a n e s e c t i o n s of r u l e d s u r f a c e s ..... 52

C a t a n e s e F.

F o o t n o t e s to a t h e o r e m of I.Reider ........................ 6 7

C l e m e n s H.

A n o b s t r u c t i o n to m o v i n g m u l t i p l e s of sul~arieties ......................................................................

Decker W . / P e t e m e l l T./ Le Potier J . / S c h n e i d e r M.

75

H a l f - c a n o n i c a l s u r f a c e s i n P4 .................................... 91

Ellia P . / P e s k i n e C.

G r o u p e s de p o i n t s de p2; c a r a c t ~ r e e t position unfforme......................................................... 111

F u j i t a T.

O n s i n g u l a r Del Pezzo varieties ............................... 117

H u l e k K.

A b e l i a n s u r f a c e s i n p r o d u c t s of projective spaces ...............................................................................

129

l o n e s c u P.

E m b e d d e d projective v a r i e t i e s of s m a l l invariants. Ill.................................................................. 138

Livomi E.L.

O n the existence of some s u r f a c e s ........................ 155

Oliva C.

O n t h e p l u r l a d j o i n t m a p s of polarized n o r m a l Gorenstein surfaces ..................................................... 180

PaUeschi M.

O n t h e a d j o i n t line b u n d l e to a n a m p l e a n d s p a n n e d one ...................................................................

184

Reid M.

Q u a d r i c s t h r o u g h a c a n o n i c a l surface ................... 191

Reid M.

I n f i n i t e s i m a l view of e x t e n d i n g a h y p e r p l a n e section- deformation theory ..................................... 2 1 4

Reider I.

T o w a r d A b e l - J a c o b i t h e o r y for h i g h e r d i m e n s i o n a l v a r i e t i e s a n d T o r e l l i t h e o r e m ........... 2 8 7

Sakai F.

R e i d e r - S e r r a n o m e t h o d o n n o r m a l s u r f a c e s ........ 301

List of s e m i n a r s held d u r i n g the conference..............................................................................................................................

320

I N F I N I T E S I M A L D E F O R M A T I O N S OF N E G A T I V E WEIGHTS AND H Y P E R P L A N E SECTIONS

Lucian B~descu

Introduction

C o n s i d e r the following: Problem.

Let

(Y,L) be a normal p o l a r i z e d v a r i e t y over an algebra-

ically closed field k, i.e. a normal p r o j e c t i v e v a r i e t y Y over k together w i t h an ample line bundle L on Y. T h e n one may ask under w h i c h conditions

the f o l l o w i n g statement holds:

(#) Every normal p r o j e c t i v e v a r i e t y X c o n t a i n i n g Y as an ample Cartier d i v i s o r such that the normal bundle of Y in X is L, is isomorphic to the p r o j e c t i v e cone over

(Y,L), and Y is e m b e d d e d in X as the

infinite section. Recall that the p r o j e c t i v e cone over p r o j e c t i v e v a r i e t y C(Y,L)

(Y,L)

is by d e f i n i t i o n the

= Proj(S[T]), w h e r e S is the graded k - a l g e b r a

S(Y,L) S[TJ

= ~ H°(Y,L i) associated to (Y,L), and the polynomial S - a l g e b r a i=o (with T an indeterminate) is graded by deg(sT i) = deg(s)+i when-

ever s E S

is homogeneous.

tion the s u b v a r i e t y V+(T),

The infinite section of C(Y,L)

This problem has classical roots hints).

In

[IJ,

[23,

is by defini-

and it is isomorphic to Y. (see [3J for some historical

C33 and C4J, among other things, we produced seve-

ral examples of p o l a r i z e d v a r i e t i e s

(Y,L) s a t i s f y i n g

(#). If Y is smo~

oth of d i m e n s i o n ~ 2, and if Ty is the tangent bundle of Y, Fujita subs e q u e n t l y proved in [6~ the following general criterion: fies

(Y,L)

satis-

(#) if H I ( y , T y ~ L i) = o for every i < o . In this paper we prove two m a i n results.

in the spirit of

E43)

is a c r i t e r i o n for

The first one

(which is

considers the case w h e r e Y has singularities,

(Y,L) to satisfy

(#). This criterion

and

(see theorem I

in §1) improves a r e s u l t of C4] and involves the space of first order i n f i n i t e s i m a l d e f o r m a t i o n s of the k - a l g e b r a S(Y,L).

In §2 we apply it

to check that the singular Kummer v a r i e t i e s of d i m e n s i o n symmetric products of certain v a r i e t i e s satisfy ample line bundle.

~3

and the

(#) w i t h respect to any

In §3 we m a k e a few remarks w h e n Y is smooth and sta-

te an open question. the Schlessinger's

It should be noted that in the first two sections

deformation

theory

(see [18],

[19J) plays an essen-

tial role. The second main result a pn-bundle

(n ~I)

(see theorem 6 in §4) shows that if Y is

over a smooth projective

curve B of positive genus,

and if X is a normal sin@ular projective variety containing Y as an ample Cartier divisor,

then X is isomorphic

B = p1 was discussed

in

arbitrary

[1] and [2]. Putting these results together, we

genus),

in

to the cone C(Y,L).

~3J, while the case when X is smooth

get a complete description

The case (and B of

of all normal projective varieties

ning a pn-bundle over a curve as an ample Cartier divisor

contai-

(see theorem

7 in §4). Unless otherwise

specified,

the terminology

and the notations

used are standard. §I. The first main result In the set-up and notations algebra S = S(Y,L) [8], chap.

III).

of the above problem,

is finitely generated because L is ample

= deg(ai)

TI,...,Tn,

that deg(T i) =

Then S is isomorphic

(as a graded

to kit I .... ,Tn]/I in such a way that a i corresponds

for every i = 1,...,n

gene-

the polynomial k-algebra in

graded by the conditions

= qi for every i = 1,...,n.

k-algebra) Timod I

(see e.g.

Let al,...,a n be a minimal system of homogeneous

rators of S/k, and denote by k[T1,...,Tn] n indeterminates

the graded k-

to

(where I is the kernel of the homomor-

phism m a p p i n g T i to ai). Let fl,...,fr be a minimal system of homogeneous generators

Theorem

of I, and set: (I)

d = max(dl,...,dr),

I.

In the above notations

i) HI (Y,L i) = o for every i 6 2 where S+ is the irrelevant maximal ii) T = TI(s/k,S)

where d i = deg(fi)assume the followin@:

, Or equivalently,

(-i) = o for every I ~ i ~ d, where d is ~iven by is the space of first order infinitesimal

the k-algebra S, and T sI =

Then the property

(#)....holds for

(I), T S =

deformations

~ T sI(i) is the d e c o m p o s i t i o n ie~ the Gm-aCtion of the ~raded k-algebra S (see ~ 8 ] , ~ 7 3 ) .

Proof.

d e p t h ( S s + ) ~ 3,

ideal of S.

(Y,L).

Let X be a normal projective variety containing

Y as an

ample Cartier d i v i s o r such that O x ( Y ) ~ O Y ~ L. Let t ~ H ° ( X , O x ( Y ) ) global equation of Y in X, i.e. divx(t)

of

arisin 9 from

be a

= Y. Denote by S' the graded

k-algebra

S (X,Ox(Y))

= ~ H ° (X,Ox(iY)) . T h e n u s i n g the s t a n d a r d i=o

exact

sequence 0''

~ Ox((i_1)y) ~

the h y p o t h e s i s HI(X,Ox(iY))

t

"'qn r e s p e c t i v e l y ,

and deg(T)

the s u r j e c t i v e

S' h o m o g e n e o u s

elements

= ai,

D e n o t e by P the p o l y n o m i a l

indeterminates

s a y i n g that

sees that S'/tS' ~

S

w h e r e deg;~t) = 1).

s u c h t h a t b.modl tS'

= k[b I ..... b n , t ].

= 1,...,n,

~O,

i), and a t h e o r e m of S e v e r i - Z a r i s k i - S e r r e

Then choose bl,...,bnE

.~,n,

~ Li

= o for e v e r y i < < o, o n e i m m e d i a t e l y

( i s o m o r p h i s m of g r a d e d k - a l g e b r a s ,

in n+1

~ Ox(iY)

T I ,... ,Tn,T,

of d e g r e e s

i & 1,...,n. k-algebra

ql''"

T h e n S' =

kit I .... ,Tn,T]

g r a d e d by d e g ( T i) = qi'

i = I,..

= 1. For e v e r y m > I set S m = S ' / t m s ', and c o n s i d e r

homomorphism

and 8m(T)

Bm:P

= t', w h e r e

~ Sm s u c h that

8m(T i) = b'i, i =

we have denoted

by b'

the e l e m e n t b m o d tms '. Let F I , . . . , F s be a s y s t e m of h o m o g e n e o u s

gene-

r a t o r s of the i d e a l J = K e r ( S m ) , Now,

a c c o r d i n g to

The s m - m o d u l e

-

~8],

§I

Ex(Sm/k,S)

for e v e r y b ~ S '

and p u t e i = d e g ( F i ) , (or also

t i o n of S m / k by S is a k - a l g e b r a p h i s m of k - a l g e b r a s of E, i s o m o r p h i c

E

- The s m - m o d u l e

T1(Sm/k,S)

c l a s s e s of e x t e n -

S = sm/t'S m . Recall

that an e x t e n -

E together with a surjective

~ Sm w h o s e

as an s m - m o d u l e

~ 4 ] ) , we c a n c o n s i d e r :

of all i s o m o r p h i s m

tions of S m o v e r k by the s m - m o d u l e

i = 1,...,s.

kernel

homomor-

is a s q u a r e - z e r o

ideal

to S. defined

by the f o l l o w i n g

exact

sequen-

ce (2)

Derk(P,S)

where Derk(P,S) is d e f i n e d

u

is t h e s m - m o d u l e

in t h e f o l l o w i n g way:

ment of Hom m(j/j2,S) vanishes

~ Homsm(j/j2,S)

defined

of all k - d e r i v a t i o n s if D 6 D e r k ( P , S )

o f P in S, and u

t h e n u(D)

by t h e r e s t r i c t i o n

D/J

~o,

is the e l e -

(which n e c e s s a r i l y

on S J 2 ). It turns out that T 1 (Sm / k ,S) is i"n d e p e n d e n t of t h e

c h o i c e of the p r e s e n t a t i o n Now, the p o i n t dules

.....~.... T..I ( s m / k , S )

(see

~8],

P/J of S m.

is t h a t t h e r e

theorem

(3)

I, p a g e

is a c a n o n i c a l 12, or also

~:Ex(Sm/k,S)

S i n c e S m is a g r a d e d k - a l g e b r a , t i o n TI (Sm/k,S)

=

(see

19).

~7],

page

~

~

g i v e n by the e x a c t s e q u e n c e

[14], p a g e

consider

of s m - m o -

41o):

m T 1 (Sm/k,S).

Tl(sm/k,S)

T I ( s m / k , S ) (i) a r i s i n g

C o m i n g b a c k to o u r s i t u a t i o n ,

isomorphism

has a n a t u r a l

from the G m - a c t i o n

gradaof S m

the e l e m e n t of E x ( S m / k , S )

(am )

~ S N= tms'/ tm+Is,

o

We need to c o m p u t e

~(am) E TI(sm/k,S)

t i o n of the i s o m o r p h i s m

~ (see

tive d i a g r a m

rows

with exact

o

ei-m.

v[

~tms'/tm+Is

' ~ S

By the d e f i n i -

from

the c o m m u t a -

P o

~ Sm

=o

Bm. Thus v ( F i m o d

j2)

= tmGi(bl,..-

m o d tm+ls '

with Gi(b I , b n , t ) ~ S' h o m o g e n e Q u s of d e g r e e • ,... ei- m H O m s m ( J / j 2 ,S) c o r r e s p o n d s to the v e c t o r (G{, .... Gs),

Then wove

ling the e x a c t

sequence

According TI(sm/k,S) degree

in

j correspond

~73,

.,s.

Since deg(Gl) (4)

page

to those

(hl,...,h s) w i t h

ii).

But the t r i v i a l

o

o

there

Assume

now that we k n o w that

t' = t m o d tls ' . T h e n

recall

~(a I) = o by hypois >S[T]/(T)~S

~S~T]/(T)

~o,

=~S

I

. . . . . .

in the m i d d l e

,o

S

o

maps T m o d ( T 2) into t.~

for some m, 2~ o, i = 1,...,n,

tem of weights

~-graded

= o for eve(as a ~raded

to the p o l y n o m i a l S - a l g e b r a SIT3 in such a way that t is

m a p p e d into T.

§2. Applications of t h e o r e m I

The tools for v e r i f y i n g hypotheses of type ii) of theorem ] have been d e v e l o p e d by S c h l e s s i n g e r in s e n t i a l l y due to Schlessinger) polarized varieties

D J~. The lemma 1 below

(which is es-

provides examples of s i n g u l a r normal

(Y,L) s a t i s f y i n g the c o n d i t i o n ii) of t h e o r e m I.

Start w i t h a smooth p r o j e c t i v e v a r i e t y V and a finite group G acting on V. Denote by Y the q u o t i e n t v a r i e t y V/G and by f:V canonical morphism.

~ Y the

Let L be an ample line bundle on Y and set M =

= ~ (L). Since f is a finite morphism, M is also ample.

Let S = S(Y,L)

and A = S(V,M)

be the graded k-algebras

associated to

(Y,L) and (V,M)

respectively. L e m m a 1. In the above notations assume the following: ' i) D i m ( V ) ~ 3 and char(k)

is either zero, or p r i m e to the order

/~/ of G. ii) G acts freel[ on V o u s i d e some closed G - i n v a r i a n t subset of V of c o d i m e n s i o n

~ 3.

iii) H I ( v , M -i)

= o for e v e r y i ~ I

(in c h a r a c t e r i s t i c zero this

is always fulfilled by Kodaira's v a n i s h i n 9 t h e o r e m ) . iv) H I ( V , T v ~ M -i) = o for every i T I, w h e r e T v is the tangent bundle of V. Then T~(-i) Proof.

= o for every i ~ I.

Since lemma I is not given in [193 in this form, we inclu-

de its proof for the c o n v e n i e n c e of the reader.

From ii) we infer that

the singular locus of Y, Sing(Y)~, is of c o d i m e n s i o n ~tale outside Sing(Y).

~3,

and that f is

Using this, the n o r m a l i t y of Y and

follows that f,(Mi) G = L i for every i ~ o .

~63,

§7, it

This shows that G acts on A

by a u t o m o r p h i s m s of graded k-algebras and that the k ~ a l g e b ~ a of invariants A G coincides w i t h S. Consider the c a r t e s i a n d i a g r a m Spec(A)-(A+)

= W

~

V

, U

= Spec(S)-(S+)

= W/G

~ Y = V/G

w i t h q and p the canonical p r o j e c t i o n s of the Gm-bundles W and U resp e c t i v e l y (see [83, chap. II, §8). If F is the r a m i f i c a t i o n locus of f, -I then q (F) is the r a m i f i c a t i o n locus of g, and hence g acts freely on W ouside a closed G - i n v a r i a n t subset of W. In particular, locus Z of U is of c o d i m e n s i o n

>13 in U. Then by

the s i n g u l a r

[19J and [200 we get

that T U = g,(Tw)G , w h e r e T U is the tangent sheaf of U. T a k i n g into account of hypothesis

i) we infer that T u is a direct summand of g, (Tw) ,

and in p a r t i c u l a r (6)

H I (U,T U) is a direct summand of H 1 (U,g,(T W)) ~- H I (W,T W)-

On the o t h e r hand, it is w e l l k n o w n that there is a canonical exact s e q u e n c e

(see e.g.

o

[14] or

~ 0w

E21~) m TW

2 q*(Tv)

-~ o

w h i c h yields the exact s e q u e n c e (7)

H I (W,O W)

~ H I (W,T W)

, H I (W,q*(T v))

One has the n a t u r a l H I (W,q* (Tv)) ~ H I (W,O W) middle

~

in

H I (W,O W) ~-- ~ H I (V,M i) and i£~ give n a t u r a l g r a d i n g s on

H I ( V , T v ~ L i) , w h i c h

and o n

space

isomorphisms

H I (W,q*(Tv)) (7)

respectively.

has also a n a t u r a l

O n the o t h e r hand,

gradation

H I (W,T W)

the

=

= ~ H I (W,T W) (i) a r i s i n g f r o m the G - a c t i o n on W, and all t h e s e t h r e e iEZ m g r a d a t i o n s are c o m p a t i b l e w i t h the m a p s in (7). T h e r e f o r e , u s i n g h y p o theses

iii)

and iv) we get t h a t H I (W,T W) (i) = o for e v e r y i < o. T h e r e

is also a n a t u r a l the G m - a C t i o n

gradation

on U , a n d this

g r a d a t i o n of H I (W,T W). (8)

H I (U,Tu) gradation

Consequently

~

H I (U,T U) (i) a r i s i n g

is c o m p a t i b l e

for e v e r y

logy shows

singularities

in c o d i m e n s i o n

d e p t h z (Tu)>I 3. T h e n the e x a c t

that the r e s t r i c t i o n

(6) w i t h the

i< o

[203 we i n f e r that all the s i n g u l a r i t i e s

in p a r t i c u l a r ,

via

from

we get:

H I (U,T U) (i) = o

S i n c e U has o n l y q u o t i e n t [I 93 and

=

~ 3, by

of U are rigid,

sequence

m a p H I (U,T U)

and

of local c o h o m o -

• H I (U-Z,T U) is an

isomorphism. Finally, singularities

s i n c e U has o n l y q u o t i e n t and e o d i m u ( Z ) ~ 3, by

H I (U-Z,Tu).

Recalling

w e get the c o n c l u s i o n

(8) and the i s o m o r p h i s m

of l e m m a

N o w we i l l u s t r a t e s o m e examples.

how t h e o r e m

~ 3. R e c a l l

- V defined

of o r d e r

isolated

I can be a p p l i e d

-via

I to the s i n g u l a r

that a singular

is the s u b g r o u p by u(x)

se of x in the g r o u p - l a w points

H I (U-Z,T U) ~ H 1 (U,T U)

Q.E.D.

2 on V

2 generated

= -x for e v e r y x £ V

of V).

(see

singularities

of o r d e r

If char(k)

lemma

I - on

Kummer varie-

Kummer variety

form V/G, w h e r e V is an a b e l i a n v a r i e t y

and G C A u t ( V ) u:V

I.

First we apply theorem

ties of d i m e n s i o n r i e t y of the

(and h e n c e C o h e n - M a c a u l a y ) 1 [197 and C2oJ w e get T S -

Y is a v a -

of d i m e n s i o n

(where -x is the inver-

~i>2, t h e r e

are e x a c t l y

~6~)~, and h e n c e Y = V / G has e x a c t l y

(which are all q u o t i e n t

d~2

by the i n v o l u t i o n

singularities).

22d 22d

Now we

have: Theorem

2. Let Y be a s i n g u l a r

and let L be an a r b i t r a r y the p r o p e r t y

(#) holds

Proof. very i~I,

We

satisfied,

line b u n d l e

lemma

Indeed,

on Y. If char(k)

while

iii)

I implies

vanishing

that T~(-i)

the h y p o t h e s e s and iv)

b u n d l e of an a b e l i a n v a r i e t y

fact t h a t the K o d a i r a ' s

of d i m e n s i o n

d~3

~ 2 then

(Y,L).

first s h o w that

w i t h S = S(Y,L).

I are c l e a r l y the t a n g e n t

ample

for

Kummer variety

theorem

f o l l o w u s i n g the

is t r i v i a l , holds

= o for e-

i) and ii) of l e m m a fact that

t o g e t h e r w i t h the

for an a b e l i a n v a r i e t y

in arbitrary c h a r a c t e r i s t i c

(see

[16~, §16).

It remains to check that HI(y,L i) = o for every i c ~ the first hypothesis of t h e o r e m I). If f:V morphism, char(k)

then by

(which is

• Y is the canonical

[19], L i is a direct summand of f,f*(L i) because

@ 2 = /G/, and hence HI(y,L i) is a direct summand of

H1(y,f,f*(Li))

~ HI(v,f*(Li)).

for every i @ o because f*(L) according to S c h l e s s i n g e r

By

~6],

§16 the latter space is zero

is ample. On the other hand, if i = o,

[192, page 24, we infer

= H I ( V , O v ) G , and G acts on HI(v,o~7)~ by t HI(y,Oy)

~t

that HI(y,Oy) ~

=

It follows that

= o. A p p l y i n g theorem I we get the conclusion.

Q.E.D.

Further examples of s i n g u l a r normal v a r i e t i e s s a t i s f y i n g

(#) w i t h

respect to any ample line bundle are the symmetric products of c e r t a i n smooth p r o j e c t i v e varieties.

Let Z be a smooth p r o j e c t i v e variety of

d i m e n s i o n d ~ 3, and let Y be the symmetric product Z (n) = V/G, where: n~2

is a fixed integer, V = Z n

times),

(the direct product of Z w i t h itself n

and G is the symmetric group of degree n acting on V by

g- (z I ..... z n) = (Zg(1), .... Zg(n )) for every g ~ G and

(z I .... ,z n) ~ V .

T h e n the r a m i f i c a t i o n locus of the c a n o n i c a l m o r p h i s m f:V codimension d = dim(V)~3

> Y has

in V.

T h e o r e m 3. Let Z b ~ a smooth p r o ~ e c t i v e v a r i e t y of d i m e n s i o n d ~ 3 such that H I ( z , M ) =

o for every line b u n d l e M on Z, and let n ~ 2

integer such that either char(k)

= o, or n < c h a r ( k )

for every ample lin e bundle L on Y = z(n)

be an

if char(k) > o. Then

the p r o p e r t y

(#) holds for

(Y,L). Note. The simplest examples of v a r i e t i e s Z s a t i s f y i n g the hypotheses of theorem 3 are all s m o o t h h y p e r s u r f a c e s

in pd+l w i t h d ~ 3 .

Proof of theorem 3. The hypotheses imply in p a r t i c u l a r that HI (Z,Oz) = o, and then the see-saw p r i n c i p l e implies that f*(L) ~ p ~ ( L 1 ) ~ . . . ~ p ~ ( L n ) , P1:V

(see

~6~,§5)

immediately

w i t h L I , . . . , L n E Pic(Z)

and

~ Z the p r o j e c t i o n of V onto the i-th factor. Since L is am-

ple on Y and f is finite,

f*(L)

is ample on V, and hence L i is ample on

Z for every i = 1,...,n. As in the proof of theorem 2, it will be sufficient to check the following: HI(v,f*(Li))

= o

HI(V,Tv~f*(Li)) in order to deduce satisfied. formulae, theorem,

for every i ~ = o

, and

for every i < o,

(via lemma I) that the hypotheses of theorem 1 are

But these v a n i s h i n g s

are easily checked using the K~Inneth'S

the fact that T v = p ~ ( T z ) ~ . . . ~ p ~ ( T z ) ,

the hypotheses of the

and the fact that L i is ample for i = 1,...,n

(which implies

10

that H°(Z,L~)

= o for every j ~ o and i = 1,...,n).

of the theorem follows from theorem I.

Then the c o n c l u s i o n

Q.E.D.

§3. A few remarks w h e n Y is smooth

In this section we shall assume that Y is smooth and char(k) Then it is known that the space T~(i) way

= o.

can be computed in the f o l l o w i n g

(see ~23J, page 337 and theorem 3.7). First, there is an exact se-

quence of v e c t o r bundles o

~ Oy

~ M

-~ Ty

~ o,

w h i c h is the dual of the exact s e q u e n c e 0

~f~l

>F

>Oy

','

0

c o r r e s p o n d i n g to the image of L in H I (Y,fl 1) via the canonical m a p H I (Y,O~) ~ Pic(Y) given by f

~ / ~ yI

• df/f. Then it is proved in loc. cit. that T SI (i) = Ker(H I ( y , M ~ L i) ......

(9)

w h e r e S = S(Y,L) ning of §I. Using theorem,

- H I (Y,fll) induced by the map O~

• H I (Y,~Li+qj)) for every i E ~ , j=l and ql .... 'qn have the same m e a n i n g as at the begin-

(9), the first exact sequence and the Kodaira's v a n i s h i n g

it follows that the c o n d i t i o n "T~(-i)

a c o n s e q u e n c e of the c o n d i t i o n " H I ( y , T y ~ L -i) Y is smooth and char(k)

= o for every i ~ ~" is = o for every i ~ I

. If

= o, one can get rid of the u n p l e a s a n t hypo-

thesis i) of t h e o r e m I because of the following: Theorem 4 (See ~63). Let dimension

(Y,L) be a smooth p o l a r i z e d v a r i e t y of

~ 2 such that H I ( y , T y ~ L -i)

= o. Then the p r o p e r t y

(#) holds for

= o for ever Y i ~ I and char(k)

=

(Y,L).

T h e o r e m 4 is proved in C6J; via a q u i c k argument,

it is also a

c o n s e q u e n c e of t h e o r e m 2 in E22~. Using theorem 4 and the m a i n result of E223 we prove the following: T h e o r e m 5. Let char(k)

linear system holds

(y,L) be a smooth Polarized v a r i e t y such that:

= o, d i m ( Y ) ~ 2, H I ( y , T y ~ L -i)

for

= o for i = I and i = 2, and the

~LI contains a smooth divisor.

Then the p r o p e r t y

(#)

(Y,L).

Proof.

By t h e o r e m 4 it will be sufficient to show that

H I ( y , T y ~ L -i) = o for every i ~ I. Let H 6 ~LI be a smooth d i v i s o r of IL~. Since d i m ( Y ) ~ 2, H is also connected.

If we d e n o t e by L H the res-

t r i c t i o n L ~ O H and by T H the tangent bundle of H, we have the c a n o n i c a l

11 exact sequence o which yields

~TH~

LH i

~ (Ty~ L-i)~ O H

~ o,

the exact sequence

~Io i)

H°(H,TH~LHi)

~ H°(H,(Ty~L-i)~OH

For every i ~ 2 the last space is zero. main result of

~223

(which extends

first space could be ~ o only if in which case it follows property

~ L~ -i

(#) holds

for

(H,L H) ~

(pi,o(I))

(Y,L) ~

in this case.

by the the

(and then i = 2),

(p2,o(I)),

Thus we

H ° ( H , T H ~ L H I) = o for every i ~ 2. Then by in the middle

On the other hand,

a theorem of Mori-Sumihiro),

easily that (Y,L)

I-i ~ H°(H,L H ).

)

and hence the

may assume that

(Io i) we get that the space

is zero for every i ~ 2. Finally,

using this and the exact

sequence (11 i)

O

~ T y ~ L -i-1

(Ty~) L - i ) ~ OH

-~ T y ~ L-i

H I (Y,Ty~)L -i ) is injec-

we infer that the map H I ( Y , T y ~ L - i - I ) tive for every i>~2. Corollar[.

Therefore

Let

~o,

= o for every i>I I. Q.E.D.

H I ( Y , T y ~ L -i)

(Y,L) be a smooth polarized

d>~ 2 such that there is a smooth divisor

v~riet[

of dimensio n

H ~ IL~ for which the exact

sequence (12)

O

is not split char(k)

~ TH

~Ty~O

(in particular,

H ~

~

H 1 (H,TH~LHI)

LH

~ O

~ o). Assume moreover

= o and H 1 ( Y , T y ~ L -1) = o. Then the property

(#) holds

that

for

(Y,L). Proof.

According

to the proof of theorem

5, the exact sequence

(1115 shows that it is sufficient to prove that H ° ( H , ( T y ~ L - 1 ) ~ O H = o. The exact sequence (lo I ) yields the exact sequence (13)

5 =

H° ( H , T H ~ LHI )----~ H° (H, ( T y ~ L-I )~ OH) ---~H° (H ,OH ) ~ ~H1 ( H ' T H ~ L H 15" By ~22~, the first space could be ~ o only in one of the follo-

wing cases:

either

first case

(Y,L) ~

this case; and hence then

(H,L H) ~

(pd-1,0(1)),

(pd,o(1)),

~(I)

(12) splits.

Therefore

•

~

~

is the obstructlon

we get the result. Remark. better

(H,L H) ~

(p1,o(2))-

In the

(Y,L) has the property (#) in 1 -I the Second case is ruled out because then H ( H , T H ~ L H ) = o, we may asstume H°(H,T

(13) shows that H ° ( H , ( T v ~ L - 1 ) ~ O u )

Since

or

and hence

,

I

H

~ L -I) H

= o if and only if "

-1

in H ( H , T H ~ L H ) such that

= o, and ~(I)

~ o.

(125 be split,

Q.E.D.

In a more special

result than theorem

situation,

L'vovskii

5 and its corollary.

that Y C P n is a smooth non-degenerate

projective

proved

in

More precisely, subvariety

L15] a assume

of pn of

12

I ~ 2 and d e g r e e ~ 3, such that H (Y,Ty(-I)) n+1 n+1 = o. Let X C P be an i r r e d u c i b l e s u b v a r i e t y of P

dimension

= Y, and X is smooth along Y and t r a n s v e r s a l ded in pn+l

as a hyperplane.

skii has an even w e a k e r His proof

Let

conditions

ensuring

corresponding

exact

tuations

(12)

when

A necessary swer

by its global

that there sequence is split

sections.

is not split.

Or, e n u m e r a t e

condition

such that this q u e s t i o n

H I ( H , T H ~ L H I) = o for H E ~LI

if and only known

(H,LH)

classical

(p2,0(I)), Cp5

and R i e m a n n - R o c h if

~

result,

the p r o j e c t i v e

general,

(Y,L) is i s o m o r p h i c

(the latter

over

on B, w i t h n ~ 1 .

Theorem positive

section

Y = P(E)

to the p r o j e c t i v e where

Denote

section of p l x p 2 ~

are all i s o m o r p h i c

curve

and

as h y p e r p l a n e

let E be a vector

by Y = P(E)

the p r o j e c t i v e

~ B the canonical

to

projection.

sections

bundle bundle The m a i n

is the following:

cone

C(Y,L) bundle

assume

that the genus of B is

and Y is e m b e d d e d

of t h e o r e m

from

~3,

over

variety

T h e n X is i s o m o r p h i c in X as the infinite,

of Y in X.

[23,

6 lies in the fact that, and

of all normal p r o j e c t i v e

are pn-bundles

to one of the following:

as an ample C a r t i e r divisor.

The m o t i v a t i o n w i t h some results

happens

= o. Let X be a s i n ~ u l a r normal p r o j e c t i v e

L is the normal

description

curve,

6. In the above notations,

and char(k)

containin~

gets that this

Indeed,

2, or 3. And by a well-

surfaces

an i r r a t i o n a l

to E, and by p : Y

result of this

Is it also suffi-

for w h i c h

or any smooth h y p e r p l a n e

Let B be a s m o o t h p r o j e c t i v e

associated

H one

w i t h i=I,

an-

blown up at a point).

§4. p n - b u n d l e s

of rank n+1

(Y,L)

has a p o s i t i v e

can be easily enumerated.

on the curve

embedding

plane

the pairs

(p1,0(i))

(plxp1,0(1,1)),

via the Segre

the si-

for H ~ ~LI general.

In the case of surfaces,

by d u a l i t y

Find s u f f i c i e n t

H E ILl such that the

is that H I ( H , T H ~ L H I) ~ o for H 6 IL~ general.

cient?

ons

v a r i e t y of d i m e n s i o n

is a s m o o t h d i v i s o r

(12)

cit.).

(loc.

w e m a y ask the following:

(Y,L) be a s m o o t h p o l a r i z e d

d ~ 2 such that L is generated

= o

L'vov-

techniques.

C o m i n g back to the above corollary, Question.

pn is embed-

Y. In fact,

than HI(y,Ty(-1))

different

such that x ~ p n =

to pn, w h e r e

T h e n X is a cone over

assumption

uses c o m p l e t e l y

= o and char(k)=

a curve:

combining

[3~, we get the f o l l o w i n g varieties

whose

hyperplane

it

complete secti-

13

Theorem trar y

genus,

thermore

7.

Assume

that

B is a s m o o t h p r o j e c t i v e

and let Y = P(E)

that char(k)

ning Y as an ample

be a p n - b u n d l e

over

B

curve of arbi-

(n~1).

= o. Let X be a n o r m a l p r o j e c t i v e

Cartier

divisor.

Assume

variety

T h e n one has one of the

fur-

contai-

following

possibilities ! a) X ~ p3,

y ~ plxp1 ' and Y is e m b e d d e d

b) X is i s o m o r p h i c Y is e m b e d d e d

in X as the i n t e r s e c t i o n

c) T h e r e

is an exact o

sequence

~ 0B

~ F

-

~

for some L ' ~ P i c ( B ) ,

and Y ~ P(E')

normal

bundles

~ E'

bundle

X ~ P(F),

in p4, y ~ p1xp1,

of X w i t h a h y p e r p l a n e

of v e c t o r

such that F is an ample v e c t o r

d) X is i s o m o r p h i c

in X as a quadric.

to a s m o o t h h y p e r q u a d r i c

of p4.

of B of the f0rm

~ o

in the sense 0 f

to the p r o ~ e c t i v e

and

~o3,

is e m b e d d e d

cone C(Y,L},

b u n d l e of Y in X, such that Y is e m b e d d e d

E' = E ~ L ' '

in X v i a ~. where

L is the

in X as the i n f i n i t e

section. Remarks.

I) In certain

in ~3], t h e o r e m

orems

(see

of the result given

(cf.

Another

also

theorem

6 was proved

lemma 2 in

proof of t h e o r e m

6 above).

L. B~descu,

in

~23,

[33

of a long c a s e - b y - c a s e

[22, theorems

Atti

I, 2 and

bundle.

Accad.

the-

case is w h e n

and X is s m o o t h is c o m p l e t e l y

Ligure

and its proof,

p l a n e b l o w n up at

Sci.

Lettere,

subsequently

of the g e n e r a l

rays

38

(1981),

in the case X is singular).

7 in case X is smooth was

as an a p p l i c a t i o n

~,

Note that the p r o o f

The B r o j e c t i v e

t h e o r y of e x t r e m a l

3, and

The m o s t d i f f i c u l t

in case Y = P ( O p 1 ~ O p 1 ( - 1 ) )

using Mori's

traction

as the result

5,

E is a rank two v e c t o r

as an ample divisor,

by P. I o n e s c u ping,

i.e.

in our short note,

a point 3-7

7 is o b t a i n e d [I], t h e o r e m

3, 4 and 5, and t h e o r e m

Y is a surface,

cases

6.

2) T h e o r e m discussion

(but not all)

given

adjunction

and K a w a m a t a - S h o k u r o v

map-

con-

theorem.

Proof of t h e o r e m rem and the A l b a n e s e

6. A c c o r d i n g

mapping

yield

Y

to

~2~ and

[3~, the Lefschetz

the c o m m u t a t i v e

theo-

diagram

mU~X

B

where

U is an o p e n n e i g h b o u r h o o d

= Xreg).

Then X has

finitely many

of Y in X

(in fact, we can take U =

singularities,

and by H i r o n a k a

[11],

14 there is a d e s i n g u l a r i z a t i o n f:X" ties: X"

~ X w i t h the following proper-

f induces an i s o m o r p h i s m f-1(U) ~ U, the rational map q" = qof: ) B is in fact a morphism,

divisors of normal crossings

and the exceptional

fibres of f are

(i.e. w i t h smooth components of c o d i m e n -

sion one i n t e r s e c t i n g transversely).

T h e n the normal b u n d l e

X" is L, and since L is ample, L is in p a r t i c u l a r p-ample.

of Y in One of the

m a i n point in the proof of theorem 6 is the following lemma, w h i c h is e s s e n t i a l l y the r e l a t i v i z a t i o n of t h e o r e m 4.2, chap. Lemma 2. Let q":X"

III of CJJ.

~ B be a s u r j e c t i v e m o r p h i s m b e t w e e n

the normal p r o ~ e c t i v e v a r i e t i e s x" and B, and let Y be an e f f e c t i v e C a r t i e r d iyisor on X" such that the r e s t r i c t i o n p:y ....... surjective.

> B of q" is

Assume that the normal bundle L of Y in X" is p-ample.

Then

there is a canonical c o m m u t a t i v e d i a g r a m

X"

yr

-~X'

IP B

w i t h X' a n o r m a l p r o j e c t i v e variety, q':X'

; B a morphism, v a

b i r a t i o n a ! m o r p h i s m such that v is an . i s o m o r p h i s m i n a n e i j h b o u r h 0 o d of Y, and v(Y)

is a q ' - a m p ! e e f f e c t i v e C a r t i e r d i v i s o r on X'°

Proof of lemma 2. First we are going to show that for i > ~ o the following three conditions

are satisfied:

i) L i is p-very ample. ii) The canonical map q "*~" ~,(Ox. (iY))

~ OX,, (iY) is surjective.

iii) The canonical m a p q~ (Ox,, (iY)) Indeed, since L is p-ample, the exact s e q u e n c e q~ (Ox. (iY))

i) holds.

~ p, (L i) is surjective. Now we prove iii). C o n s i d e r

(iT I ) ~ p, (Li)

R1q~ (Ox. ((i-I)Y) ) ~ R l q ~

(0X,,(iY))

, R I p, (L i) induced by

o

~Ox.((i-1)y )

20x.(iY)

> Li

>o.

The last sheaf is zero for i>> o because L is p-ample chap. III,

(2.2.1)). Hence the m a p ~i is s u r j e c t i v e for every i ~ j

some j > o. Since q" is a p r o j e c t i v e morphism,

for

RIq~(Ox,,(jY)) is coherent

on B, and t h e r e f o r e ~i becomes an i s o m o r p h i s m for i > > o , holds.

(see [8],

i.e. iii)

15

TO p r o v e ii), o b s e r v e that by

[82, chap. II,

(3.4.7), ii) is

e q u i v a l e n t to the fact that for every affine o p e n subset D = Spec(A) of B, the sheaf Ox.(iY)/q"-I(D) i >>o.

is generated by its global sections

for

But by iii), the natural map H°(D,q~(Ox,,(iY))) ~ H°(q"-I(D),

Ox.(iY))

~ H°(D,p~(Li)) ~ H ° ( p - I ( D ) , L i) is s u r j e c t i v e for

i >~ o b e c a u s e D is affine.

U s i n g the fact that L i is p-very ample, it

follows that Li/p-1 (D) is generated by its global sections, (by the above surjectivity),

Ox.(iY)/q"-I(D)

and hence

is generated by its glo-

bal sections. Now w e fix an i > >

o such that i), ii) and iii)

are fulfilled.

From ii) i~ follows that there is a unique B - m o r p h i s m v 1 : X = P(q~(Ox,,(iY))) such that v~(Op(1)) ~ Ox.(iY). = v~(Y).

~ P :=

Set X I = v I (X")and YI =

Since L i is p - v e r y ample, we know that V l / Y : Y

~ Y1 is an

i s o m o r p h i s m and that iY 1 is a B-very ample Cartier d i v i s o r on X I . Furt h e r m o r e Y = v~1(Y1 ) because a global e q u a t i Q n of the e f f e c t i v e C a r t i e r d i v i s o r iY on X" separates points x and x' such that x 6 Y and x ' ~ X"-Y. Then c o n s i d e r the Stein f a c t o r i z a t i o n of v I

V

Xl I

X'

:= Spec(vl,(Ox.))

X1

Since v,(Ox.) ~ OX, v/Y:Y

and X" is normal, X' is also normal. Notice that -I (Y'), so by Zaris-

~ Y' = v(Y) is an i s o m o r p h i s m and Y = v

ki's m a i n t h e o r e m

(see ~83, chap.

III,

(4.4.1)), v is an i s o m o r p h i s m in

n e o g h b o u r h o o d of Y in X". Since w is a finite m o r p h i s m and YI is B-ample, Y' = w * ( Y I) is q ' - a m p l e on X' , w h e r e q' is the c o m p o s i t i o n X'

w

~ Xl .

~ p

~ B. L e m m a 2 is proved.

Note. The above proof of lemma 2 is an a d a p t a t i o n of the proof of t h e o r e m 4.2, chap.

III in

E93

to the r e l a t i v e case.

Proof of t h e o r e m 6, continued. We apply lemma 2 to the d e s i n g u l a r i z a t i o n X" of X such that q" = qof is a morphism,

and get the normal

p r o j e c t i v e v a r i e t y X' w i t h all p r o p e r t i e s stated in lemma 2

(in parti-

cular, Y is an effective q ' - a m p l e Cartier d i v i s o r on X'). Notice that v blows d o w n to points o n l y s u b v a r i e t i e s of X" that are contained in the e x c e p t i o n a l locus of f, and since X' is normal, by [8J, chap. (8.11.1) we infer that there is a unique m o r p h i s m u:X'

II,

~ X such

that qou = q' and f = uov. N o t i c e also that the c o n s t r u c t i o n of u and X' is c a n o n i c a l and depends only on X, Y and the r a t i o n a l map q, and not on the choice of the d e s i n g u l a r i z a t i o n

f:X"

~X.

16

With this construction = P(E)

in hand, we can proceed

and L is a p-ample line bundle,

sitive integer s>11 (14)

L ~ Oy(s)~p*(M-1),

Replacing E by E ~ N ,

where Oy(S)

with N E P i c ( B )

(14) M has a s u f f i c i e n t l y

According to the Lefschetz that F ~ O y

=~ Oy(1)

and since X" is smooth,

we may assu-

high degree. there is an F ~ P i c ( U )

such

2). Set U" = f-1 (U)

', we may consider the sheaf F on U",

F extends

(non-uniquely)

to a line bundle on

X", still denoted by F. Since the map Pic(U) (14) can be translated

high degree,

In other words,

theorem,

-

= Op(E) (s)-

of sufficiently

(cf. e'g'uC2J'~u proof of theorem

and U' = u -I (U). Since U" ~

tive,

Since Y= and a po-

such that

we get that L = O p ( E ~ N ) ( s ) ~ p * ( N - S ~ M - 1 ) . me that in

further•

there is an M E Pic(B)

into FS/u" ~

~ Pic(Y)

(Ox.(Y)~q"*(M))/U"~

fore there is a divisor D supported by the exceptional such that F s ~ O x . ( Y ) ~ q " * ( M ) ~ O x . ( D ) .

is injecThere-

fibres of f,

If D = D+-D_, with D+ and D_>/o,

after replacing F by F~Ox,,(D_)

(which still has the restriction Oy(1)

to Y), we may assume that D>/o•

Furthermore,

since M is of sufficiently

high degree,

for a general divisor b 1 + . . . + b m ~ IMI (with b. ~ b. for -I l 3 i ~ j) , the fibres X"i = q'' (b i) are all smooth and transverse to all

components

of D as well as to all their possible

intersections.

Thus,

replacing D by D" = D+D', with D' = X~' + •"" + X"m' we get (15)

F s ~= Or, , ( Y ) ~ O x . (D") ,

where D" is a normal crossing divisor on X" such that D" = D+D', with D>~o and Supp(D) sum of distinct i ~

contained

in the exceptional

fibres of q"

put F (i) = Fi~Ox,,([-iD"/s]),

divisor on X"

(with

every i # j),

E/~J denotes

every real number a, if i = js +

fibres of f, and D' a

(and hence D' reduced).

/ki irreducible

where if

~

=~aj~j is a ~ 3 and reduced, and /ki @ ~j for

the integral divisor ~ a ~

[a] denotes

r is an arbitrary

Then for every

/~,

where

for

the largest integer ~ a. JNotice that

integer such that o ~ r ~ < s - 1 ,

then by

(15) we get: (16)

F (i) ~ OX,,(jY)~F(r)

Now, the second main point in the proof of theorem 6 is the following: Lemma 3.

Rbq"(F (-i))

Proof of lemma 3. consisting

= o for every i>/I

The proof follows

in using global v a n i s h i n g

(i~e. local)

ones

(see e.g.

the well known philosophy

theorems

~5], appendix

and b = o,1.

I).

in order to get relative

17 Let N be a sufficiently N~DRbq~(F(-i))is

generated

Ha ( B , N ~ Rbq~(F (-i)) the Leray spectral

ample

line bundle on B such that

by its global

= o for a ~ l ,

sections

and such that

b = o,1 and i ~ I

(i fixed).

Consider

sequence

Ea,b Ha ( B , N ~ R b q ~ (F (-i)) 2 =

~ H a + b (X,,,q,,, (N) ~ F (-i))

a,b By the choice of N, we have E 2 = o for a > o , which implies that H ° ( B , N ~ R b q ~ ( F (-i)) =~ Hb(x " , q " * ( N ) ~ F (-i)). Since N ~ R b q ~ ( F (-i)) is generated

by its global

sections,

it is sufficient

side is zero,

hand-side

is zero. TO this end, using the fact that N is sufficiently

ample,

or by the above isomorphism,

to show that the

left-hand

by Bertini we can choose

a divisor C l + . . . + C e ~

if i ~ j) such that X"i = q.-1 (c i) is smooth, and transverse

~N~ (with c i ~ cj

not included

with Z = X~+...+X".e

, Hb(x ,,,q,,.(N)~F(-i)) Notice

on X, OX. (iY) is generated self-intersection

number

that since

f*(Ox(Y))

by its global (Ox.(Y)'dim(X"))

~ Hb(z,F(-i)) ,

='~Ox,,(Y) and Y is ample

sections

for i>> o, and the

is strictly

positive

the divisor Y is nef and big in the terminology

refore,

recalling

(15) and the definition

zero by the Kawamata-Viehweg

vanishing

ce of D"),

disconnected)

variety

and hence the middle

Corollar~

of

(in par-

~1]).

The-

of F (i) , the first space is

theorem

space is also zero by the same vanishing (but possibly

in Supp(D"),

to D. Then we have the exact sequence

Hb(x .,F(-i))

ticular,

that the right-

([133 , [ 2 ~ ) .

theorem

applied

The third

on the smooth

Z (taking into account of the choi-

space

is zero.

(to lemma 3). For ever~ i ~

Q.E.D. ........set . G i = v,(F(i)).

Then Rbq~(G_i ) = o for every i ~ I and b = o,1. Proof.

Consider

by lemma

sequence

= Rbq~(Rav, (F(-i)) E a,b 2

(17) From

the Leray spectral

(17) we get Rbq~(G

~.Ra+bq~(F(-i)).

i) = Rbq~(v,(F (-i)) ~ R b q ~ ( F ( - i ) ) ,

3 the last sheaf is zero for every i ~ I a~d b = o,I.

Remarks.

1) In the

(final part of the)

shall use only the above corollary.

However,

proof of theorem

varieties

2) From the definition

of the F(i)'s

H°(U,F (j))

and hence one has a natural

6 we

vanishing

theorem

X" and Z. one immediately

infers

for every open subset U C X" and i,j ~ o one has natural maps H°(U,F(i)) ~

Q.E.D.

we needed to prove first

lemma 3 because we had to apply the Kawamata-Viehweg on the smooth projective

and

structure

~ H0(U,F(i+J)), of a graded k-algebra

on

that

18

~H°(V,Gi), where V is an arbitrary open subset of X'. i=o 3) At this point we want to thank the referee who kindly pointed out to us that in the earlier version of this paper we had incorrectly defined the sheaves F (i) by F (i) = F i ~ O x , , ( - ~ D " / s ] ) (instead of F (i) = Fi~Ox,,([-iD"/s3)).-- W i t h this (incorrect) definition statements similar to lemma 3 and its corollary still hold, but remark 2 (which shall be needed below) ted definition

fails to be true.

Fortunately,

with the correc-

of the F(i)'s we had to make only minor changes

in the

proof of lemma 3 and its corollary. Proof of theorem 6, continued.

Having the corollary of lemma 3,

we can finally conclude the proof as follows.

Recalling

(14), we dis-

tinguish two cases: Case s = I. Replacing E by E ~ M -1 , we may assume that L "--'Oy = (I). Then by the corollary of lemma 3, Rbq~(Ox, (-Y))

= o for b = o,1. Now,

the exact sequence o

~ Ox, ((i_l)y)

(where t ~ H ° ( X ' , O x , exact sequence

t

~ OX, (iY)

~Oy(i)

>o,

(Y)) is a global equation of Y in X') yields the

(i~ o)

R1q~ (Ox, ((i-I)Y) ) Since by ~83, chap.

~ R1q~ (Ox, (iY))

III

(2.1.15),

R1p,(Oy(i))

~ RIp, (Oy (i)) . = o for every i ~ o ,

and since we know that R I q,(Ox, ' (-Y)) = o, by induction on i we get that R Iq,(Ox, ' (iY)) = o for every i ~ o . sequence yields (18 i)

o

for every i ~ o

~ q ~ ( O x , ((i-1)Y))

By E83, chap.

III

In particular,

the above exact

the exact sequence

(2.1.15)

t

~ q ~ ( O x , (iy))

~p,(Oy(i))

~o.

again,

~p,(Oy(i)) ~ S(E),where S(E) i=o is the symmetric OB-algebra of E. Denoting by S = ~ q ~ ( O x , (iY)), from i=o (18 i) we get S/tS ~ S(E). Since S(E) is generated by its homogeneous part of degree one, and since deg(t) OB-algebra

= I, it follows that the graded

S is generated by S I = q~(Ox, (Y)).

ral homomorphism

S(F)

On the other hand, since q~(Ox,(-Y)) by induction on i in

In particular,

~ S is surjective,

the natu-

where F = q~(Ox, (Y))-

= o and

~p,(Oy(i)) ~ S(E), i=o (18 i) we infer that S i is a locally free O B- mo-

dule of rank ( x n + ii +l ~ for every i ~ o . It follows that the surjective maps Si(F) > S i are all isomorphisms because Si(F) and S i are vector bundles of the same rank.

Thus S ~ S(F),

is a q'-ample Cartier divisor on X'

and recalling that Y

(lemma 2), we get that X' is iso-

19

morphic

to t h e p r o j e c t i v e

bundle

associated

to F. T h e

exact

sequence

(18 I) b e c o m e s (1 9)

O

a) S u p p o s e a result

first

of G i e s e k e r

ther with bundle

the

fact

locus m u s t

of t h e o r e m

surjection

= Spec(S(E)) c

§8).

open

~X'

exact

whose

vector

and this

toge-

OX, (Y)=

is a d e s i n Y, this

= X'

theorem,

~ X has f is an iso-

contradicts

the

hy-

is i m p o s s i b l e . is split. the

zero

second

Then

F ~" E~)OB,

section

projective

is the

(see C8~,

on B, by G r a u e r t ' s (3.5)),

B 'l ;V(E)

morphism

is Y = P(E)

~Io2,

to get the

= X"

intersect

f:X"

main

the the

bundle

(see

X'

not

words,

(19)

complement

b~ndles

down

where

or e q u i v a l e n t l y ,

Since

does

~ O B yields

E is an a m p l e

can be b l o w n

ample,

a)

sequence

~ P(F),

for v e c t o r

(4.16))

the v e c t o r

x is n o n s i n g u l a r ,

E@)O B ,X'

immersion

Since

ampleness

the

Then

[5],

locus

case

split.

that

Zariski's

6. T h e r e f o r e

is not

or also

In o t h e r

by the

(19)

(which m e a n s

q'-ample).

exceptional

and hence,

2.2,

F is also

be z e r o - d i m e n s i o n a l .

b) T h e r e f o r e

tural

theorem

that

• O.

sequence

is a m p l e

(and not o n l y

In p a r t i c u l a r ,

potheses

and the

show

~ E

the exact

Oy(1)

of X w h o s e

fibres,

morphism.

~ F

(see C7J,

is a m p l e

gularization

that

that

E is a m p l e ) ,

= Op(F)(I)

finite

• OB

the

zero

variety

chap.

II,

criterion section

=

na-

of

i(B)C

Proj (~) H ° (B, • i=o

Si(E))~T])

(with T an i n d e t e r m i n a t e

p, (Oy(i)) C(Y,L). the

Now,

curve

a morphism tly

for e v e r y

the m o r p h i s m

Case with

C(Y,L) a) w e

theorem

formula

Since

latter

I).

variety

Since

= X' ~ P(F)

on X),

-~ X. S i n c e

Y is a m p l e

on b o t h

that

iE

~

this

morphism

in c a s e

s = 1.

be an a r b i t r a r y

but

hhe

cone

m X has to c o n t r a c t

Y is a m p l e

infer

SI(E)

is n o t h i n g

(since

6 is p r o v e d

s >12. Let

o.~ r ~ s-1.

tion's

f:X"

i (B) to a p o i n t

as in case

and h e n c e

i>/ o, the

of d e g r e e

and

is in

fact

integer,

v*(Ox, (jY)) ~ OX,, (jY) , by

hence

C(Y,L)

(16)

one

gets

and X,

exac-

an i s o m o r p h i s m ,

and set

i = js + r,

and the p r o j e c ~

we get

(2o)

G i ~ Ox, ( j Y ) ~ G r

Furthermore,

by

(14),

(15)

, with

G o = Ox,

and the d e f i n i t i o n

of the

sheaves

Gi

w e have (21) where (21),

Gi~O

M o , . . . , M s _ I are for e v e r y o

Y

line

Oy(i) ~ p * bundles

i>z o w e have • Gi_s

Let D = Spec(A)

be any

the

(M-3~Mr)

on B

exact

t , Gi affine

,

(MO ~ O B).

Then

by

(2o)

and

sequence ~Oy(i)~p,(M-J~Mr open

subset

of

)

B such that

~o. E/D "&

20

=

On+1 -D

,

•

,

=

q,-1

M / D q 0 D and M r / D ~ 0 D for e v e r y o ~ r ~ s-1 Set X D (D) -I ! and YD = p (D). T h e n the a b o v e e x a c t s e q u e n c e r e s t r i c t e d to X D b e c o m e s O

t

~ Gi_ s

S i n c e by the c o r o l l a r y = R1q~(Gi_s)/D

= o for e v e r y

~ Gi

(i) ~ o. DI of l e m m a 3 w e h a v e H (X~,Gi_ s) = i/o.

Cartier divisor

that X D' is i s o m o r p h i c

sequence

= o, an e a s y i n d u c t i o n o n i im-

weighted

(I ,...,I ,s).

to t h e c a n o n i c a l

In p a r t i c u l a r ,

pro-

Furthermore,

projection

for e v e r y b 6 B ,

of

X~ =

q,-1 (b) is i s o m o r p h i c to the w e i g h t e d p r o j e c t i v e s p a c e P ( 1 , . . . , 1 , s ) -I o v e r k and Yb = p (b) is c o n t a i n e d in X ~ as t h e i n f i n i t e s e c t i o n (i.e. the subvariety

V+(T)

of P(1,...,1,s)).

S u m m i n g up, w e s h o w e d t h a t t h e r e is a c l o s e d that q' d e f i n e s Proj (ALTO),

of B' on B, B ' ~ X 6

and B ' n X~ is p r e c i s e l y

~ P(1,... ,1,s) and let L(y)

an i s o m o r p h i s m

(s>1 2) for e v e r y b E B .

be the g e n e r a t i n g

subset

B' of X' s u c h

= V+(To,...,%)~

the v e r t e x x b of the c o n e X~ Let Y C Y

be an a r b i t r a r y p o i n t ,

line of the cone X' P(Y)

j o i n i n g t h e po-

21

ints y and Xp(y). Then X'-B' is the disjoint union of all Ly-Xp(y) (y E Y), and hence we get a well-defined function B:X'-B' putting B(x) = y if X E L y - X p ( y ) .

~ Y by

The above discussion shows that B is

in fact an algebraic morphism defined in a neighbourhood V (or in X) which is a retraction of Y C V . also [6J,

Then using lemma 3 in [33 (cf.

(3.1)), we infer that X ~ C(Y,L)

Theorem 6 is completely proved.

of Y in X'

also if s ~ 2.

Q.E.D.

R E F E R E N C E S

I. L. B~descu, On ample divisors,

Nagoya Math. J., 8_~6 (1982), 155-171.

2. L. B~descu, On ample divisors:II, Algebraic Geometry Proceedings, Bucharest 198o, Teubner-Texte Math.

40, Leipzig 1981, 12-32.

3. L. B~descu, Hyperplane sections and deformations, Proceedings,

Algebraic Geometry

Bucharest 1982, Springer Lect. Notes Math.

1o56, 1-33.

4. L. B~descu, On a criterion for hyperplane sections, Math. Camb. Phil. Soa., 103

Proc.

(1988), 59-67.

5. T. Fujita, On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan, 32 (198o), 153-169. 6. T. Fujita, Rational retractions onto ample divisors, Coll. Art. Sci. Univ. Tokyo, 7. D. Gieseker,

Scient.

Papers

33 (1983), 33-39.

P-ample bundles and their Chern classes, Nagoya Math. J., 43 (1971), 91-116.

8. A. Grothendieck & J. Dieudonn~,

El~ments de G~om~trie Alg~brique,

chap. II, III, Publ. Math. 9. R~ Hartshorne,

IHES, 8, 11 (1961).

Ample subvarieties of algebraic varieties, Lect. Notes Math.

1o. R. Hartshorne, Ample vector bundles,

156

Publ. Math.

Springer

(197o).

IHES, 29

(1966), 63-94.

11. H. Hironaka,

Resolution of singularities of an algebraic variety

over a field of char. zero, Annals Math., 79

~1964), lo9326.

12. P. Ionescu, Generalized adjunction and applications, Math. Camb. Phil.

Soc., 99

Proc.

(1986), 457-472.

13. Y. Kawamata, A generalization of Kodaira-Ramanujam's theorem~ Math. Ann., 261

vanishing

(1982),

43-46~

14. S. Kleiman & J. Landolfi, Geometry and deformations of special Schubert varieties,

Compos. Math., 2_~3 (1971), 407-434.

22 15

S.M. L'vovskii,

Prolongation of projective manifolds and deforma-

tions, VINI~I Preprint, Moskow University, 16

1987

(in Russian).

D. Mumford, Abelian varieties, TATA Lecture Notes Math., Bombay, 1968.

17

H. Pinkham, Deformations of algebraic varieties with Gm-action,

18

M. Schlessinger,

Ast~risque,

20

(1974), Soci~t~ Math. France.

Infinitesimal deformations of singularities, Thesis, Harvard Univ., 1964.

i9

M. Schlessinger,

Rigidity of quotient singularities,

Invent. Math.,

14 (1971), 17-26. 20

M. Schlessinger, On rigid singularities,

Rice Univ. Studies, 5 1

21

E. Viehweg, Vanishing theorems, Journ.

22

J. Wahl, A cohomological characterization of pn, Invent. Math.,

23

J. Wahl, Equisingular deformations of normal surface singularities:

(1973), 147-162. reine angew. Math.,

355

(1982), I-8. 72 I, Annals Math.,

(1983), 315-322. Io4

(1976), 325-356.

INCREST Bucharest, Dept. of Mathematics B-dul P~cii 22o, 79622 Bucharest, RUMANIA

On k-spanned projective surfaces Edoardo Ballico Dip. di Matematica, Universit~ di Trento, 38050 Povo (TN),Italy

This note can be considered as an app

na

to [BS], since here we give an improvement of [BS], th.2.4.

First we recall a few notations. We work over the complex number field. Let T c P N be a scheme of dimension 0; T is called curvilinear ff it is contained in a smooth curve, or equivalently if it has embedding dimension at most 1. Let X be a complete variety embedded in a projective space by a linear subspace W of H0(S,L), L~ Pic(X). (X,W) (or X if there is no danger of misunderstanding) is called k-spanned if for all curvilinear subschemes T of X with length(T) = k+l, the restriction map from W to H0(T,LT) is surjective. L is called k-spanned if (X,H0(X,L)) is k-spanned. Here we prove the following resulL Theorem Let (S,W) be a k-spanned smooth surface with k~_3. Then dim(W)~k+5. Proof. Set P:= P(W), hence ScP. Assume w:= dim(W):gk+4. Take a general hyperplane H of P and set C:= Hr'~S. Then C is a smooth, k-spanned curve in H. It is easy to check that the projection from a point of a smooth m-spanned curve, rn~.2, is a smooth (m-1)-spanned curve in the appropriate projective space. After (k-2) general projections, we find a smooth 2-spanned curve Z in a projective space U, dim(U)_q4. If dim(U) R,

L to a fi6re

f

p. We say that

of

p

(S,L)

is a ~e~oP_l (resp. a co~c bua~d~e) over a nonsingular curve R i f there is a surjective morphism with connected fibres p : S m> R, with the property that L is relatively ample with respect to

p and there exists some very ample line bundle

26

M on R such that is a g e o m ~ Z ~ restriction of

KS ~ 2L ~ p M (resp. KS ~ L ~ p M). We also say that

~ u ~ conic b~nd~e i f

L to a fibre

We denote the rational

f

p1

of

if

F0 = p1 x ~1 with

0f(2).

p1

F n, the Hirze-

bundle which is not a scroll in the above

L = 0~0(1,1). We say that

S is a Pe~ Pezzo su.~face

-KS is ample.

(0.4) Caste]nuovo's bound. Let S

p is

(S,L)

bundle p : S - - > R and the

bundle P(0p1 ~ 0p1(n)), n ~ 0, by

br~ch ~u)~face. Note that the only sense is

S is a p1

and l e t

L

be a very ample line bundle on a smooth surface

C be a general element in

projective space pN and l e t

d =

ILl. Assume that

ILl

embeds S in a

L.L. Then g(L) = g(C) and Castelnuovo's Lemma

says that (see e.g. [ 1])

(o.4.1) where [ x l means the

greatest

integer

~ x.

From (0.4.1), writing

d-2/N-2

=

= (d-2-¢)/(N-2), 0 ~ c ~ N-3, we find that

N Z2), )2

d ~ N/2 + /2(N'2)g(L)+(( ~4 this leading to

d ¢ ~

N/2 + /2(N-2)g(L)+1/4

if

N-4 is odd;

if

N-4 is even.

(0.4.2)

L N/2 + /2(N-2)g(L) (0.5)

k-spannedness. Let

C). We say that

L

L

be a line bundle on

S (resp. on a nonsingular curve

is k-spcu~ncdfor

k ¢ 0 i f for any distinct points z I . . . . . zt t on S (resp. on C) and any positive integers k l , . . . , k t with Z ki = k+l,' the i=1 map r(L) --> F(L e 0Z) is onto, where (Z,OZ) is a 0-dimensional subscheme defined by the ideal sheaf

I Z where IzOs, z

IzOs,zi is generated by (xi,Y~i) on S,

at

is 0S, z (resp. zi , with

0C,z) for

(xi,Y i )

i = 1, . . . . t (resp. IzOc,zi is generated by Yiki 'Yi

on C). We call a 0-cycle

local

z~ {z I . . . . . zt} and coordinates

at

Zi

local coordinate at

zi

Z as above adm~ible.

Note that 0-spanned is equivalent to

L

being spanned by

F(L)

and 1-spanned

is equivalent to very ample. (0.5.1)

If

L

is k-spanned on

S, then L.C ~ k

for every effective curve C on

27

S,

with

equality only

if

C : pl.

Further

either

C : p1

or

Pa(C) = 1

if

deg LC = k+l. The fact that

L.C ~ k

is clear from the d e f i n i t i o n , as well as

Now, looking at the embedding of N ~ k, so that if

C in FN given by r(L c)

C = F 1 whenever deg LC = k or

one has

deg Lc = N = k+l

hO(Lc) a k+l. deg LC =deg C and Pa(C) = 1

deg LC = k+1.

(0.5.2)

Let

say that

L

be a k-spanned line bundle on either

get(L)

V ~

k-~paJ~ L (or

S or a smooth curve

a k-~pan~Ln9 s~t of

C. We

L) i f for a l l admis-

sible O-cycles (Z,OZ) with length(Oz) = k+l, the map g - - > F(L e 0z) is onto. For a given admissible O-cycle (Z,Oz) on S we say that a smooth curve C

comp#Jt~ble with (Z,OZ)

is

if:

-CDZ -

red ; for any point

meters at at

z

z E Zred , where IzOs,z = (x,yn) , x,y local para-

z, then

and mz

f - x E mn where f is the local equation of z is the maximal ideal of Os,zo

C

Thus we get the following characterization of k-spannedness, we need to prove the key-Lemma below. Y c F(L)

k-spans L

compatible curves

on

S

C on

i f and only i f for a l l smooth connected

S, Im(V--> r(Lc)) k-spans the restriction

Lc • (0.5.3) LEMMA. L ~

C

and l ~

Li be k i - s P ~ n e d J ~ e b u n d ~ eYJth~ an S or on a smooth c~ve

vi c F(Li )

V1 e . . . 0 Vm ~

ki-spoJ~

Li

for

i

= 1. . . . . m.

Then the

~mage g

of

r(L 1 e . . . e Lm) (k I + . . . + km) - ~paJ~6 L1 e . . . e Lm.

Proof. In view of the characterization of k-spannedness given in (0.5.2) we easily see that one can reduce to the curves case. Further the result is c l e a r l y reduced t to the case m = 2. Thus we have to show that, given a O-cycle Z = ~ niP i on C t i=1 where ni > O, Z ni = k+l, the map V - - > r(L 1 e L2 e 0Z) is onto. i=1 To see this f i x an index = aj+bj, j ~ i , where aj > O, t t Z1 =

~ arPr, r=l

e]ements order for

Z2 =

sI . . . . . Sia i j ~ i

i . Write

ni = ai+b i where ai > O, bi > 0 and nj = t t bj > O. Then ~ a = k1+I, ~ br = k 2 + l and l e t r=l r r=l

Z brPr. By the fact that r=l

gI

kl-spans

L1

we can choose

of V1 whose images in F(L 1 e OZl) vanish at pj to the aj.-th

and which have prescribed

choose elements uI . . . . . U.lbi of

ai-1

V2 whose images in

j e t at

Pi" Similarly we can

r(L 2 o OZ2) vanish at

pj

to

28 the bj-th order for tensor

powers of

vanish at

pj

j }~ i and which have prescribed bi-1 j e t at Pi" Note that the these sections give a space of sections

to the

nj-th

j e t at Pi" Now W1. . . . . Wt

order for

j ~ i

Wi

of

L1 0 L2

which

and which have prescribed

hi-1

clearly generate r(L 1 o L2 ® OZ), so we are done. Q.E.D.

Let us recall the following numerical characterization of k-spannedness. (0.6) THEOREM (/31, (2.1)). S a~leJt

Let

L be a nef

L-L ~ 4k+5. Then e~e)~

d~v~o~ D ~uch t h a t

and big

Zine

bu~d£e on a ~ f a c e e ~ t ~ an e f f e ~ u e

Ks+L /6 k-~pe~ned o ~ e ~ e

L-2D ~ ~-effec;~ive,

D contains ~om~ admissible O-e~CE~

o~ degree t+l ~ k+l wh~e the k-~pa~oted~e~ fo~Lt~ and L.D-t-1 s D.D < L.D/2 < t + l .

We need the following consequence of the result above (compare with (5.3.4)). (0.7) PROPOSITION. Leot S be a Ve~ Pezzo ~uptface wk~ch ~ a blo~x~g up of F 1. Let

L

be the ~panned ~ e

pcEEback of

0pi(1 )

to

bundle on S, o b ~ e d

F 1 ~nd~ the

bundle projection

FuputheJt a~ume Ks.KS =1. Then Kit e Lq ~ Proof. Let

M = Kit-1 g Lq.

(k+1)2+4(k+1)q since

N o t e that

k-apaam~if

M is

F0

by p u ~ n g back to Fi __> p l ,

q ~ i

ample and

t ¢ k, KiI.L = 2. Thus M-M ¢ 4k+5. I f

o~

S the i = 0,1.

and t ~ k.

M-M = (t+1)2+4(t+1)q KS g M = Kit e Lq

is not k-spanned, by the Theorem above one has M.D-k-1 ~ D-D < M.D/2 < k+l for some effective divisor

D on S. Now, since t ~ k, M-D/2 < k+l gives KiI-D

1

and hence D-D ~ M.D-k-1 ~ qL-C ~ O. Since Ks.D = 1, D.D ~ 1

by the genus formula. Then by the Hodge index theorem we

get

Ks'KS = D.D = 1 and also K~1 - D. This leads to the contradiction 2q ~ D-D = 1.

qL.D = Q.E.D.

Note that (0.7.1)

g(Kst ® Lq) = t ( t - 1 ) / 2 + (2t-1) q + 1.

29 To the reader's convenience we recall here the following result from [ 3 ] , we use several times. (0.8) PROPOSITION ( [ 3 l , (2.6)). Le~t S be a Pe~ Pezzo surface. Them Kst ~

k-~p~-

n ~ for k~ 0 i f a n d o ~ i f : if

(0.8.1)

t z k/3

(0.8.2)

t z k/2 X~ S = p1 x p1;

(0.8.3)

t a k+2 i f

Ks.KS = 1;

(0.8.4)

t a k

Ks.KS ~ 3 or

i(

i~

Ks.Ks = 2,

(O.9).k-reductlon. Let k-red~e.tion of

(S,L)

S = ~2;

KSt L

~

Ks.KS = 2 and k ~ 1. 2.

ueyuj ampZc i f f t

be a llne bundle on

S. A pair ( S ' , L ' ) is said to be a

i f there is a morphism ~ : S --> S'

with a f i n i t e set F blown up and L - ~*L'-k~-I(F). Note that Apart from some cases where k ~ 2

expressing

S as

S'

K~ e L ~ ~*(K~, e L').

is e x p l i c i t l y needed, we carry out for com-

pleteness most results for k-spanned line bundles with

k ~ 1, even though in the

"classical" case k = 1 they don't give something new. In § 4 we use extensively the results of [13] . We refer d i r e c t l y the reader to [ 13| instead of reporting here the results we need. Through the paper we also use well known results describing polarized pairs

(S,L)

with

L

of sectional genus

g(L) = 0,1; for this we refer e.g. to [5] a n d [ l ] .

§ 1. k-spannedness on curves.

Throughout this section we denote by C a nonsingular irreducible curve of genus g(C) and by

Kc

the canonical divisor of

C. Our aim is to express the k-spanned-

ness on C in terms of some useful numerical conditions. (1.1) LEMMA. Le~t L be a ~ e (1.1.1)

L X~ k-~panned i f

(I.1.2)

if

bundle on C. Then: deg L ~ 2g(C) + k;

deg L = 2g(C)+k-l, t

~ k-mpomned i f and om2~ i f hO(L-Kc) = 0 .

Proof. (1.1.1) follows from the definition. Indeed, l e t points on C and l e t

kI . . . . . kr

r

z I . . . . . z r be

non negative integers such that

r

distinct

r i~ 1 ki = k+l.

30 r Then hO(Kc-L+ ~ kizi) = O, so that we have a surjective map r(L) --> r(L (~ OZ), i=l r where Z is the O-cycle defined as Z = Z kizi; this means that L is k-spanned. i=1 To prove (1.1.2), note that, since deg L = 2g(C)+k-1, we can write L = KC 0 L for some line bundle L of degree k+l. Then hl(L) = 0 and hence L is k-spanned i f and only i f *)

hi(L-D) = hO(D-L) = O,

for every effective divisor *) is equivalent to

D on C with

deg D = k+l. We claim that condition

hO(L) = hO(L-Kc) = O. In fact, for any divisor

D as above,

deg(L-D) = 2g(C)-2; hence hi(L-D) )~ 0 implies that L-D = KC that is so hO(L) ¢ O. Viceversa, hl(L-L) = hl(Kc ) =0

L - D,

if hO(L) ~ O, a contradiction. Q.E.D.

The following plays a relevant role in the sequel. (1.2) THEOREM. Let L be a k-spa~ned Z./~¢ bund.Ce on C and l e t (1.2.1)

KC /~ k-sp~ed;

(1.2.2)

hO(L) ~ i

(1.2.3)

g(C) ~ 2k+1.

hl(L) ~ O. Then:

for any Zi~te bund.Ee L with deg L ~ k+l;

Proof. First, we can assume hi(L) = 1. Indeed, i f

hi(L) = hO(Kc-L) ~ 2, we can

write Kc-L ~ A+M where hO(A) = 1 and the moving part M is base points free. Then L' = L+M is k-spanned by (0.5.3) and hl(L') '= hO(A). To prove (1.2.1), note that KC is k-spanned i f and only i f hl(Kc-Z) = h1(KC) = I, for every length exact sequence

k+l

O-cycle Z

on

C. This easily follows by looking at the

0 - - > KC 00c(-Z) --> KC - > Now, i f

hl(Kc-Z) ~ 2, clearly

KC 0 0Z - >

O.

hO(Kc-L+Z) ~ 2 since Kc-L is effective and hence

by duality hl(L-Z) ~ 2-. Again, the k-spannedness of t can be expressed as hi(L-Z) = hl(L), this leading to a contradiction. Thus KC is k-spanned and hl(Kc-Z) = hO(z)=1 for every length

t ~ k+l

O-cycle Z on

C. This gives (1.2.1) and (1.2.2). From

the Existence Theorem (see [1 ], p. 206) we know that for any integer t ~ (g(C)+2)/2 there exists a line bundle L on C of degree t

and with

hO(L) ~ 2. Therefore

i t has to be k+1 < (g(C)+2)/2, which gives (1.2.3). Q.E.D. (1.3) KEY-LEMMA. Let

L

be a k-speu~ned ZY.~.e bccnd.Ze on C.

Then hO(L) > k+2if

31

g(C) > 0. Proof.

Since

L

is k-spanned one sees that the

Jk(C,L) is spanned by the image of Since J0(C,L) = L and Tc ® L

r(L)

k-th

holomorphic j e t

bundle

under the natural map Jk : L n > Jk (C'L)"

are ample vector bundles we see from the exact

sequence

*(k)

0 - - > TC that

e L - - > Jk(C,L) --> ak.1(C,L) --> 0

is an ample vector bundle of rank k+l. Then h0(L) a r k Jk(C,L) +

Jk(C,L)

dim C = k+2. (1.4) COROLLARY. Let

L be a k-~pwnncd ZJ.ne bundle on C ~x~th g(C) > 0 and l e t

d = deg L. The: (1.4.1)

d ~ k+2;

(1.4.2)

d ~ 2k+2 i f

d < 2g(C) with

eq~

onZ# i f

¢Lthe~ d = 2g(C) oh

L ~ Kc , k = 1, g(C) = 3. Proof.

If

hl(L) = 0, then d-g(C)+1 = h0(L) > k+2

hl(L) ~ 0 C l i f f o r d ' s theorem and (1.3) yield

gives d .> k+g(C)+l ~ k+2.

If

d/2 +I ~ h0(L) ~ k+2, whence d ~ 2k+2

and (1.4.1) is proved. Note that C l i f f o r d ' s inequality holds true also i f hl(L) = 0 whenever d $ 2g(C). Therefore d > 2k+2 by (1.4.1). Now, d = 2k+2 gives the equality in the C l i f f o r d ' s theorem, so we find that

d = 2g(C) i f

hl(L) = 0, and either

hyperelliptic curve with

L a multiple

of

If

the

unique g~

L ~ Kc

on

C if

or C is

a

hl(L) # 0.

L ~ KC , d = 2k+2 = 2g(C)-2 and g(C) > 2k+1 by (1.2.3), this leading to k = I ,

g(C) = 3. In

the

positive integers

remaining case d .< 2g(C)-2,

L ~ ng~ and

KC ~ mg~

for

some

m,n, m ~ n. Then Kc ~ L+(m-n)g~ would be very ample, a contra-

diction to h y p e r e l l i p t i c i t y . This proves (1.4.2). Q.E.D. (1.5) REMARK (compare with § 6). Note that i f with

L

is a k-spanned line bundle on

S

pg(S) > 0, then for a general element C E ILt the r e s t r i c t i o n L = LC v e r i f i e s hl(L)(= h0(KsIc )) ~ 0, so deg L = L.L ~ 2g(C)-2. Hence g(L) > 2k+I

the condition

by (1.2.3) and L-L > 2k+3 i f

k ~ 2 by (1.4.2).

§ 2. A lower bound for hO(L).

Let

L

be a k-spanned line bundle on

S. In this section we show that the

32 k-spannedness condition forces

S

to be embedded by

ILl

in a projective space

of dimension at least 5. First of a l l , note that from Lemma (1.3), we have (2.1)

h0(L) ~ k+3,

so the claim is clear i f k ~ 3. (2.2) Let

L

be a k-spanned line bundle on S with k ~ 2. Take a point

l e t V2 c r(L) denote the space of the sections of at

x. We c/~im that after

L

chosing a t r i v i a l i z a t i o n of

L at

can be written in the form s I = q1~O(3). . . . . st = qt+O(3) quadratic functions in the local parameters at

x

not ( i d e n t i c a l l y ) zero and have no common factors, Indeed, set

I = { i , q i ~ 0}, J = { j , q j = 0)

X

x, a basis of

where the

q's

and at least 2 of the

V2 are

qa's are

~ = i .... ,t. and assume that the q i ' s have

a linear common factor, say u. The maximal ideal mx to be of the form m = (u,v)

x E S and

that vanish to the 2-nd order

of

0S, x

can

be

assumed

for some linear factor v.

We can also assume that on the open set

U0 = {x E S, s0(x) ~ O, s0E r(L)}

a basis for r(S,L) on U0 consists of the h0(L)-I elements {u,v . . . . . si . . . . . sj . . . . }. Now L is 2-spanned by the assumption so that the map p : r(L) --> r(L e Oxl(U,V3)) is onto. Since clearly

u

and the s i ' s , s j ' s

h0(L)-2, which contradicts

belong to Ker ~ we find dim Ker p

dim F(L 8 0x/(U,V3)) = 3.

Note that the claim we proved here shows that the k~nel of the evaluation map J2,x : (S x r ( L ) ) x - - > J2(S,L)x at

x /nd~ce.d by J2: L - - > J2(S,L)

has dimension

at mo~t h0(L)-5. We can now prove the following general result. (2.3) THEOREM. I f the

2-th

jets bundl~ of a

k >. 2 spanned l i n e

on S don't ~pan J2(S,L)

at at least one point, then:

(2.3.1) Cl(S) 2 = 2c2(S)

and the tangent bundle of either

double cov~ of

as a direct ~m of line bund/~;

S ~p~

(2.3.2) Cokernel (J2 : S x r(L) --> J2(S,L)) -- KS e L. Proof. Consider the commutative diagram J2 S x r(L) J

> J2(S,L) l

~

r

JI(S,L)

S

b~ndle

o r an u n i f i e d

L

33

where ~ denotes the surjective restriction map, whose kernel is L

is very ample, Jl is onto, the restriction

has image contained in

j

of

J2

T~(2) ® L.

to the kernel

Since

K of

Jl

T~(2) e L, so one has a morphism j : K ~ > T~(2) e L.

Fix a point

x E S, local coordinates (z,w) at

By the earlier argument (2.2), the image of

x and a t r i v i a l i z a t i o n of L at x.

j

in

T~ (2)~ 8 L at x

is of the form

(Imj) x = {~¢(dz,dw)+u~(dz,dw); x,uE ¢, ¢,~ homogeneous quadratic functions without common factors}. I t is easy to see that any such a special pencil has precisely 2 distinct elements which are squares, e.g. the map FI __> p l

given by (¢,~) has degree 2 and has

precisely two branch points by Hurwitz's theorem. Thus the pencil is given at x by

{x~ + u~; ~,u E ¢, ~i,~2~ TS,x}. Hence two directions on TS are determined at

x.

I t is easy to check that they

vary holomorphically and give a submanifold A c P(Ts) = [ TS-$I/¢ , Sthe 0-section of TS --> S, which is a two to one unramified cover of S under ~A' the restriction to A of P(TA)

~ : ~(Ts) --> S. Now, either A is a union of 2-sections of

~ or qAI(A) c

is a union of 2 sections of F(TA) --> A, where qA: P(T~) --> P(Ts)

induced map. In the former case TS = L1 e L2

is the

for 2 line bundles L1, L2 on S; in

the l a t t e r case TA = L~ e L~ for 2 line bundles L~,L~ on A. Note that by a well known result of

Bott [ 4 ] ,

former case and

ci(A)2 = 2c2(A)

2-sheeted cover

in

the

Cl(Li)2 = Cl(L~)2 = 0.

Thus Cl(S)2 = 2c2(S) in the

in the l a t t e r case. Since ~A is an unramified

latter

case,

it

follows

that

c1(A)2 = 2c1(S)2,

c2(A) = 2c2(S). Thus in any case c~(S) = 2c2(S), which proves (2.3.1). Note that the image j(K) in Ts and hence the cokernel is

e L in the former case is (L e L) e (L e L)

LI e L2 e L = KS e L.

In the l a t t e r case L1, L2

are

]ocally s t i l l well defined as in the former case. Nonetheless chosing any open set U such that

LI,L 2 are well defined,

identified with

TU = LI e L2 and LI e L2

KS. Thus the cokernel of

j

is

canonically

is always KS e L. So we are done by

noting that coker j = coker J2" Q.E.D.

34 As a consequence, we get the result claimed at the beginning of this section. (2.4) THEOREM. Le~t L be a k-~poJ~ncd~e 6undZe on S, k ~ 2. Then hO(L) ~ 6. Proof. From (2.1) we know that hO(L) z 5 and and assume hO(L) = 5.

hO(L) ~ 6

if

k ~ 3. So let

k = 2

Then previous argument (2.2) shows that the evaluating

map J2 : S x r(L) --> J2(S,L)

induced by J2 : L--> J2(S,L)

is injective. Hence

we have an exact sequence of vector bundles 0 ~ > S x r(L) ~ > J2(S,L) ~ > KS e L ~ > 0 by the

above Theorem. Thus det(J2(S,L)) ~ Ks 8 L. Now a direct computation, by

looking at the exact sequences, t = 1,2,

(2.4.1)

0 - - > T~(t) e L - - > Jt(S,L) --> Jt_l(S,L) --> 0

shows that

det(J2(S,L)) = K~ e L6. Therefore K~ e L5 ~ 05

a line bundle M on so p(t) = x(Mt)

S such that

and hence there exists

M-5 ~ KS, M3 ~ L. Since

M3 ~ L,

M is ample

is a non degenerate degree 2 polynomial. But M-5 ~ KS implies, by

Kodaira's vanishing theorem, p(t) = 0 for

t = -i,-2,-3,-4,

a contradiction. This

proves that hO(L) ~ 6. Q.E.D. Next we show that i f hO(L) = 6, then J2(S,L) is generically spanned. (2.5) PROPOSITION. WX~h the notation ~ i n (2.4), t h e e ~ x ~ S ~uch that

J2,x : (S x r(L)) x ~ > J2(S,L)x

Proof. Note that i f J2(S,L)

(S,L) = (p2, 0p2(2) )

at l ~ t

one p o i ~

~,s onto.

i t is well known that

J2: S x r(L) m>

is onto (see e.g. [ 8 1 o r [11]). Thus we can assume (S,L) ~ (p2,0p2(2))

and let us suppose J2,x

to be not onto for any x ~ S. Then by (2.3.1) there is

an exact sequence of vector bundles 0 - - > Ker j 2 - - > S x F(L) --> J2(S,L) --> KS e L - - > 0 and hence Ker J2

has rank 1, since rk(S x r(L)) = rk J2(S,L) = 6. The total Chern

classes verify the relation, where K = Ker J2' (2.5.1)

(I+K).c(J2(S,L)) = I+Ks+L.

We know that cI(J2(S,L)) = det(J2(S,L)) = 4Ks+6L while a putation, by using sequences (2.4.1), gives us

l o n g but standard com-

35 (2.5.2)

c2(J2(S,L) ) = 5c2(S)+5Ks.Ks+20Ks.L+ISL.L.

Furthermore from (2.5.1) we obtain K.Cl(J2(S,L)) + c2(J2(S,L)) : 0 and hence (2.5.3)

c2(J2(S,L)) : (3Ks+5L).(4Ks+6L) : 12Ks.Ks+38Ks.L+30L.L.

By combining (2.5.2) with (2.5.3), and noting that we find (2.5.4)

Cl(S) 2 = 2c2(S) by (2.3.1),

L.L : 3c2(S)+12(g(L)-1).

Note also that Ks+L is nef. Otherwise (S,L) would be either (p2,0(2)), (p2,0(1)), (pIxpI,0(1,1)) or a scroll, contradicting (S,L) ~ (p2,0(2)) or the fact that L is at least 2-spanned. Therefore

Ks. Ks+4(g(L)-I) ~ L-L; hence (2.5.2) and Ks.KS =

2c2(S) lead to (2.5.5)

c2(S)+8(g(L)-I) ~ O.

Clearly g(L) ~ 0 since k ~ 2 and (S,L) # (p2,0(2)). Similarly g(L) ~ 1 : otherwise (S,L) would be either a scroll, contradicting again k ~ 2, or a Del Pezzo surface, contradicting c2(S) ~ O. Thus g(L) ~ 2, so 2c2(S) = Ks-KS < 0 and therefore x(OS) < O. This implies that S is birationally ruled, so Ks-KS ~ 8(1-q(S)), and the Riemann-Roch theorem yields (2.5.6)

c2(S) ~ 4-4q(S).

Hence from (2.5.5), (2.5.6) we infer that g(L) ~ (q(S)+1)/2. Now, since (S,L) is neither (p2,0(1)), (p2,0(2)), (p1 x pi,0(1,1)) nor a scroll, i t has to be g(L) > q(S) (see e.g. [12|). So we get q(S) = O, contradicting x(OS) < O. This proves the Proposition. Q.E.D. Now, certain arguments that we have not been able to make rigorous, together with the fact that (S,L) : (p2,0p2(2)) whenever J2 is an isomorphism by a result due to Sommese111], suggest the following (2.6) Conjecture.

Let L be a k-spanned line bundle en S,

i f and only i f (S,L) : (p2,0p2(2)).

k ~ 2. Then hO(L) = 6

36

§ 3. k-spannedness on

geometrically

Throughout this section, S a nonsingular curve self-intersection

ruled surfaces.

is assumed to be a geometrically ruled surface over

R of genus g(R). As usual,

E2 = -e

E,f

denote a section of minimal

and a f i b r e of the ruling. Here we find some s u f f i c i e n t

numerical conditions for a line bundle

L

on

S

to be k-spanned. In some case,

such conditions come out to be also necessary. First we consider the case g(R) = O. be a HXvtzeb~u~eJ~~uptfac~ of i n v ~ t r handle on S. Then L /6 k-apoJu~ed i f and o ~

r ~ 1 and l e t

(3.1) PROPOSITION. Le~t S = F L ~ aE+bf be a ~ e

if

a ~ k

and

b ~ ar+k. Proof.

If

L

is k-spanned, then L.f = a a k and L.E = -ar+b ~ k. To show the con-

verse, write L ~ k(E+(r+1)f)+(a-k)(E+rf)+(b-(ar+k))f and note that

E+(r+1)f

is very ample and

E+rf, f are spanned (see e.g. [ 6 I, p.

379, 382). Then we are done by (0.5.3). Q.E.D. (3.2) REMARK. On a quadric F0 = p1 x p1 (a,b)

is k-spanned i f and only i f

is clear that a l l n e bundle L

a a k,

b a k.

of type

Indeed, Op1xpl(a,b)

is

k

=

min(a,b)-spanned. Thus we can assume g(R) > O. Recall that is the invariant of

S.

(3.3) PROPOSITION. Let and q(S) > O. Let

KS ~ -2E+(2g(R)-2-e)f where e = -E-E

S

be a 9eome/~cz~3Jj ~ e ~

L ~ aE+bf be a ~ e

a~face with X~u~GIn~ e ~ 0

bumdle on S. Then L ~

k-~pa~nnedif

a a k; b a ae+2q(S)-2+max(k+2,e). Proof.

First note that

E+ef is nef; indeed we see that

i r r e d u c i b i l e curve B on with

a > O, B ~ ae

(E+ef)-B ~ 0

S, recalling that for such a curve B,

B ~ E,f,

for every B ~ aE+Bf

({6 ~ p. 382). Now l e t M = L-KS ~ (a+2)E+(b+e-2(q(S)-l))f.

Then

M-M = (2b-4(q(S)-l)-ae)(a+2)a2(k+2)2

Further M is nef; indeed

and hence

M-M ~ 4k+5

for

k a I.

37 M ~ (a+2)(E+ef)+(b-ae-e-2(q(S)-l))f and b o t h E+ef, f are nef. Thus i f

L

is not

say that there exists an effective divisor

k-spanned, Theorem (0.6) applies to

D

such that

M.D-k-1 ~ D-D < M-D/2 < k+l. We can write D ~ xE+yf where x=D.f ~ O, y=D.(E+ef) ~ O. Now M'D=x(b-ae-e-2(q(S)-1))+ y(a+2), t h e n f r o m M.D/2 < k+l

and

the

assumptions made on

a

and

b we get

y(k+2) < 2(k+1) which leads to y = 0,1. I f y = O, D.D = -ex2+2xy ~ M.D-k-1 yields

-ex+x(k+2)-k-1 ~ -ex2

and

x ~ 1

y = O. Hence ex(x-1)+2x ~ O, a contradiction.

since

If

y = 1,

D.D ~ M.D-k-I gives 2 -ex+l ~ -ex that is

xe(x-1)+l ~ O, again a contradiction. Q.E.D.

(3.4) PROPOSITION. Let q(S) > O. Let

S be a g e o m ~ ~

r u l e d s u r f a c e of i n v a ~

L ~ aE+bf be a li.ne bundle on

S. Then

e- 2r+2 by (3.1) and the

r genus formula 4 = (Ks+L).L = 2(b-2-r) gives b = 4+r. Then r < 2 and we are done.

Q.E.D. (5.2) PROPOSITION. Let L be a k-spanned l i n e

bundle on S w i t h k > 2 and s e c t i o n a l

genu~ g(L) = 4. Then e i t h ~ : (5.2.1)

k = 3, S = p I x p I

(5.2.2)

k = 2,

(5.2.3)

k = 2, e£ther

o v e r an e l l i p t i c

and

L ~ 0S(3,3);

S i ~ a cubic ~ r f a c e i n p3

S= •

r

r < 3, L - 2E+(5+r)f

with

c u r v e of i n v a r i a n t

and L ~ 0S(2); or,

e = -i

Proof. One has hO(L) >. 6 by (2.4) then

and

or

S is a pl

bund~e

L --- 2 E + 2 f .

d = L-L > 8

by Castelnuovo~s bound (0.4.2).

Therefore the genus formula and the Riemann-Roch theorem give us hO(Ks+L) = ×(Ks+L) = 4-q(S). Then i f

Ks+L is very ample, i t

has to be

q(S) = 0 and IKs+L] embeds S as a

43

surface of degree d' = (Ks+L)2 in p3. Hence KS ~ Os(d'-4) and L ~ Os(5-d'). Now since pg(S) = 0 and L is at least 2-spanned the only possible cases are I f d'=2 we get class (5.2.1). I f d' = 3, S is a cubic in p3 that L is 2-spanned since

L-/ = 2 for a line

I

and

d' = 2,3.

L ~ 0S(2).

Note

on S, so we find class (5.2.2).

I f Ks+L is not very ample (S,L) is a geometrically ruled conic bundle by (4.1) and q(S) = 0,1

by (3.6). Let L - 2E+bf.

I f q(S) = O, S = F r and b > 2r+2 by (3.1). The genus formula 6 = (Ks+L)-L = 2(b-2-r) gives b = 5+r. Then r .< 3. I f q(S) = 1, KS =-2E-ef, e = -E2, b - 2e > 4 by (1.4.1) 6 = (Ks+L).L = 2(b-e)

and the equality

yields b = 3+e, hence e = -1. An easy check by using (0.6)

shows that L --- 2E+2f is 2-spanned (see also (3.5.1)). Q.E.D. In the remaining case g(L) = 5, Theorem (2.4) plays a relevant role. (5.3) PROPOSITION. Let L be a k-spoJtne~L ~ne b u ~ e on S wX~th k > 2 and ~ e e ~ o ~ ge~

g(L) = 5. The~ ei~dteJt:

(5.3.1)

k = 2 and

IL(

embed~ S ~

p5

o~ a K3 ~cutface of degree 8, a e~mpl~te

i~tersect~on of three quadric~ ; (5.3.2)

k = 2, (S,L) = (FI,3E+5f);

(5.3.3)

k = 2, S /~ a l)e~ Pezzo scutface, L ~-2K S,

(5.3.4)

k = 2,

(5.3.6)

k

Ks-KS = 4;

blo~/Jcg up ~ : S - - > F r o( F r, 7 d~tine.t poXJ~vi~ Pi ' L ~ G*(4E+(2r+5)f)-2 Z Pi' Pi = ~ ' l ( p i ) ; i=1 (5.3.5) k = 2, (S,L) -- (~r,2E+(6+r)f), r < 4; or,

Proof.

S X~ the

2, S i~ a p1

S i n c e hO(L) ~ 6

r = 0,1,

along 7

bundle oue~t oat e~pyCic e_u~u¢, L -- 2E+(e+4)f, e = O, -1. by (2.4),

Castelnuovo's

inequality

(0.4.2)

gives now

d=L-L>8. F i r s t , l e t us assume Ks+L very ample. We distinguish two cases, according to the value of If

pg(S).

pg(S) > 0

i t has to be d = 8

by the genus formula and hence Ks-L = 0 so

that KS ~ O. From |12|, § 3 we know that 5 = g(L) ~ hO(L)+q(S)-1 and hence hO(L)=6, q(S) = O. Thus ILl

embeds S as a degree 8

view of (1.2.3). Note that

K3 surface in p5.

Further

S is a complete intersection of three quadrics. Indeed,

i f not, i t is known that a general element C E ILl

contains a g~ ( s e e e.g.

p. 142). Now KC ~ LC is 2-spanned and hO(D) ~ 1 for any divisor deg D ~ 3 by (1.2), this contradicting If

q(S) = O,

{2),

D on C with

C to be trigonal.

pg(S) = O, the Riemann-Roch theorem yields

which gives

k = 2 in

hO(Ks+L) = X(Ks+L) = 5-q(S),

hO(Ks+L) = 5. Then !Ks+LI embeds S in

F4

as a surface

44

of degree d' = (Ks+L)2 and one has (see [ 6 ] , p. 434) (5.3.7)

d'2-5d'-lO(g(L) - I) + 12×(0S) = 2Ks.KS.

Now the usual Hodge index theorem yields

dd' ~ [L.(Ks+L)]2 = 64, so that

d'. 4, j = l . . . . ,6. I t thus follows that

o(A)

contains the cubics

Cj's

and t h i s clearly contradicts

L-A= L.L = 9. Thus we can assume Ks+L not very ample. Then (S,L) is a geometrically ruled conic bundle by (4.1) with i r r e g u l a r i t y

q(S) = 0,1

or

2

in view of (3.6).

Note that the case q(S) = 2 does not occur. Indeed the equalities (Ks+L)-L = 8, If

Ks.KS = 8(i-q(S)) give d = 8

if

q(S) : 2, a contradiction.

q(S) = O, the genus formula 8 : (Ks+L)'L = 2(b-2-r)

L -- 2E+bf, r = -E 2. Then, since b ~ 2r+2

(Ks+L)2 = O,

yields b = 6+r

by (3.1), we find

r < 4

and

where

we are

in class (5.3.5). If

q(S) = 1, by using again the genus formula one has

• = -E 2

and

deg LE = b-2e ~ 4

b = 4+e, where L - 2E+bf,

by (1.4.1). Thus we find either

e = -1, b = 3. Note that in both cases

L

is

e = O, b = 4

or

2-spanned in view of (3,3), (3,4)

and we are in class (5.3.6). Q.E.D. (5.5) REMARK. attempted

I f the conjecture (2.6) is true,

without

success to show that the

then (5.3.1) does not occur. We

restriction

equal to the complete intersection of three quadrics in p5,

L of

should be noted that there exist such S which contain a l i n e , Z, these, since L.Z = i < 2, i t follows that

L

is not

0p5(1) to

S, S

is only 1-spanned. I t of

p5 and for

2-spanned. In general though

there are no lines on such an intersection of quadrics. (5.6) REMARK(compare with § 6). Let L be a k-spanned l i n e bundle on S with k ~ 2

46

and assume pg(S) ¢ 2. Then g(L) = 2k + 1 by (1.2.3). In the extremal case g(L) = = 2k + 1 the inequality pg(S) s k - 3 holds true, hence in particular

x(Os) s k - 2. To see t h i s , recall that L.L ¢ 2k+3

by (1.5), so

the

genus formula

reads Ks.L ~ 2k - 3. Thus we are

showing that

Ks-L ~ pg(S) + k. Indeed, hO(Ks-L) = 0

hO(KsIc) ~ pg(S). Now i f the Pi'S

are

since

done after

(Ks-L)-L < 0 so that

pg(S) - 2 different points, on

S, we have

hO(KslC-~ pi ) ~ 2. Therefore deg KSIC-pg(S)+2 = K s . L - p g ( s ) + 2 ¢ k + 2 by (1.2.2).

§ G. Geography of surfaces and k-spannedness.

In this section we study the relation between k-spannedness of a line bundle L on

S

and the birational geometry of

S. We aim for a broad picture. The

arguments we use clearly give much sharper bounds in particular cases. Since the case of very ample line bundles is Well studied we make the blanket assumption that k ~ 2. Through this section we shall use repeatedly almost a l l the results we stated in § 1 as well as the genus formula (0.2) and property (0.5.1). We also use a number of well known results on the birational classification of surfaces for which we refer to [21. We shall write (6.1)

Let

~ : S--> S'

d

instead of

be a morphism of

L.L. S to a minimal model S'.

= (~,L)** = [~(C) ] where C is a smooth element of

Let L' = r ILJ. Note Ks ~ ~*Ks,+ i~ 1"= nip i

where the Pi'S are the irreduciblecomponents of the positivedimensional fibres of ~, ni > I , r = e(S)-e(S'). Further

ni = I for a l l

is a simple blowing up of a f i n i t e set of

r

i

i f and only i f

x : Sin> S'

points. From this we easily obtain

the following simple lemme. (6.1.1) LEMMA. One has L.KS > k(e(S) - e(S')) + L.x*KS, , wX~ e q u ~

if

(S',L')

~

a

k - r e d ~ o n of

(6.1.2) COROLLARY. If

i f and on~

(S,L).

~(S) => O, then L.KS > k(e(S) - e(S')). I f fu~th~

~(S) > 1

and hO(K~) > 0 for ~om¢ t > 0 thcn L-KS > k(e(S) - e(S')) + (k + 1 ) / t . Proof. I t follows from (6.1.1) by noting that A of

Ix*tKS,I

KS,

is nef and the general element

has positive arithmetic genus, so that

L.A >. k + 1. Q.E.D.

47

Then d ~ 2k + 3. Fu,~the~ g(L) ~ 2k + 1 u ~ e ~

(6.2) THEOREM. A~ume ~(S) ~ O.

po~ibZy i f

S

X~ m ~ a ~ ,

pg(S) = 0

and

or

q(S) = 0

~(S) = 2 then q(S) = O, 1 ~ Ks.KS s 9, d ~ (5k + 10)/2 Proof. Let

C be a general element o f

i t follows that d that

hl(Lc ) = 0

2k + 3. I f

ILl • Since ~(S) ~ O,

hl(Lc ) ~ 0

and therefore

1. I f

then

g(L) ~ 2k

and

and g(L) ~ (3k + 8)/2. d ~ 2g(L)

-

2

so

g(L) ~ 2k + 1. Thus we can assume

pg(S) = O. Since

x(Os) ~ 0 we conclude that

q(S)

= 0 or 1. Further, by the Riemann-Roch theorem (6.2.1)

d = hO(Lc) + g(L) - 1 = 2hO(Lc) + Ks.L

whence

(6.2.1)' If

S

g(L) - i = hO(Lc) + Ks.L. were non minimal,

Ks.L ~ k

by (6.1.1). Hence g(L) ~ k+2+k

Therefore we can assume further that

by (6.2.1)'

S is minimal. Now, l e t g(L) ~ 2k. I f

~(S)=2,

then Ks.KS ~ 1 and ×(Os) > O, while pg(S) = 0 implies q(S) = 0 and hence x(Os)=I. Thus

Ks-KS ~ 9

hO(K~) ~ 2.

by the

I t thus

Miyaoka-Yau i n e q u a l i t y .

follows that

The Riemann-Roch theorem gives

Ks.L ~ (k + 1)/2

by (6.1.1).

Actually

(k + 2)/2 since otherwise we would have a pencil of rational or e l l i p t i c

Ks-L curves

on S. Then by (6.2.1), (6.2.1)' we find d ~ (5k + 10)/2 and g(L) ~ (3k + 8)/2. Q.E.D.

(6.3) THEOREM. q(S). Then

Let

S be a

p1

bundle

p : S--> R

(6.3.1)

d > 2k2 and

g(L) ~ (k - 1) 2 i f

(6.3.2)

d > k(k + 2)

and g(L) > k(k + 1)/2

(6.3.3)

d ~ 2k + 4 and g(L) ~ 2k + 1 i f

Proof. Let p, so

E

be a section o f

p

o u ~ a eu,~ve R

of" g e ~

q(S) = O;

if

q(S) = 1;

q(S) > 2.

of minimal s e l f - i n t e r s e c t i o n and

f

a fibre of

L --- aE + bf.

If

q(S) = O, E2= - r

> k(b + k) >. 2k2. Let

q(S) = 1,

and b > ar + k, a > k

Similarly

by (3.1). Hence d = L . L = a ( 2 b - a r ) _ >

g(L) > 2(k - 1) 2 .

E2 = -e. Here

a > k

and e i t h e r

b .> ae + k + 2 i f

e > 0 or

2b - ae .> k + 2 i f e = -1 (see (3.5)). So d = L.L = a(2b-ae) ~ 2(k + 2)k i f and d ~ k(k + 2) i f e = -1. if and

e -> 0

and

2g(L) - 2 => (k - 1)(k + 2)

g(L) .> k(k + 1)/2.

e > 0

Further 2g(L) - 2 = (a - 1)(2b - ae) > 2(k - 1)(k + 2) if

e = -I.

In e i t h e r case d .> k(k + 2)

48

Let q(S) ~ 2. We know from (4.3.3) that kKS + 2L is nef. Hence k2Ks. KS+4kKS-L + 4L-L ~ O. Now Ks.KS = 8 - 8q(S), then 4k(2g(L) - 2) ~ (4k - 4)d + (8q(S) - 8)k 2 and also g(L) - i z (k - 1)d/2k + (q(S) -1)k. If

hI(Lc ) ¢ 0 we are done. Hence we can assume hl(Lc ) = O, so t h a t

d = hO(Lc) +

g(L) - 1 z g(L) + k + 1. Thus + (q(S) - l ) k

g(L) - 1 ~ (k - 1)(g(L) + k + 1)/2k which gives (k + 1)(g(L) - 1)/2k z (k - 1)(k + 2)/2k

+ (q(S) - l ) k

or g(L) a ( k - 1 ) ( k + 2 ) / ( k + l ) Finally

d ~ g(L) + k + 1 y i e l d s

+ 2k2/(k+1)

+ 1 = 3k -1 ¢ 2k + 1.

d ~ 4k > 2k + 3. Q.E.D.

(6.4) THEOREM. I f

Ks'K S ~ - x < 0

and

S /~ not a

pl

d ~ 2k + 3; g(L) ~ k(1 + x/4) + Proof.

Now kKS + L

bundle then

3/4k.

is nef by ( 4 . 4 ) , so we f i n d -k2x + 2k(2g(L) - 2) - (2k -1)d ~ 0

and also

(6.4.1) If

g(L) - 1 ~ kx/4 + (2k - l ) d / 4 k .

d > 2g(L) -2

we get

g(L) - I ~ kx/4 + (2k - l ) ( 2 g ( L ) - 2)/4k + (2k - l ) / 4 k or (g(L) - 1)/2k ~ kx/4 + (2k - 1)/4k and also g(L) > k2x/2 which gives by ( 6 . 4 . 1 ) ,

d > k2x + 4k - 1 > 2k + 3. I f

+ k + I/2 d < 2g(L) - 2, then d .> 2k + 3

g(L) >. k(1 + x / 4 ) + 3 / 4 k . Note that k2x/2

and,

+ k +1/2 > k ( l + x / 4 ) + 3/4k. Q.E.D.

49

(6.5) THEOREM. If

S is not a

x(Os) < 0 and

d ~ 2k + 3;

•1 bundle then

g(L) > k(2q(S) - 1) + k/4.

Proof. Since x(Os) 1 and Ks.KS < 8(1-q(S)) < O. Use Theorem (6.4) with x = 8q(S)-7. Q.E.D. It mainly remains to consider rational and e l l i p t i c surfaces. (6.6) LEMMA. I f Proof.

S is e i t h e r

hO(K~lr) -> 9. of

S i s r a t i o n a l and Ks.K S > 0 then

the

p2 or a blowing up of

An easy c a l c u l a t i o n

shows t h a t

Each time a point is blown up on a surface, the number of sections

anticanonical

line

bundle

hO(K~lr) - # where # denotes KS• Ks < 8

~:r"

hO(Ks1) > O.

decreases

by at most

the number of blowing ups.

i,

so t h a t

hO(Ksl ) >

Thus since #= K~:r. KFr -

the Lemma is proven. Q.E.D.

(6.7) PROPOSITION. ASSume S is not a (6.7.1)

d ~ k 2 and

(6.7.2)

d ~ 2k + 3 and

(6.7.3)

d ~ 4k

Proof.

Since

and

p1

g(L) ~ k(k - I ) / 2 + i g(L) ~ 2 k - i

kKS + L

if

if

if

g(L) a 2k + I

bundle. Then:

Ks-K S ~ 0 and

Ks.L ~ -4.

is nef by (4.4) one has

(6.7.4)

d ~ -kKs-L.

I f S is rational and Ks'Ks a O, Lemma (6.6) gives so d ~ k2. Further (6.7.5) whence

S is rational;

Ks-K s ~ -4;

hO(Ks1) > O,

hence -Ks-L ~ k

2g(L) - 2 ~ - ( k - I)Ks.L ~ k(k - I ) g(L) ~ k(k - 1)/2 + i . This proves ( 6 . 7 . 1 ) .

Now (6.7.2) follows from (6.4) while (6.7.4) and (6.7.5) y i e l d ( 6 . 7 . 3 ) .

Q.E.D. (6.8) REMARK. Note that i f

kKs + L is

nef,

by writing

(kKS + L) 2 = k2Ks-KS +

(2k - 1)Ks.L + 2g(L) - 2 ~ 0 we find 2g(L) - 2 ~ -k2Ks. Ks - (2k - I)Ks.L. Therefore i f

Ks.L ~ 0 and Ks.KS < 0 one has g(L) ~ k2/2 + I

(6.9) THEOREM. If

and d( ~ -k2Ks. KS - 2kKs.L) a k2.

S is rat/ona/, d ~ k2. Fuptther g(L) ~ k(k-1)/2

+ min(1,k-2)

50

if

Ks.L ~ 0 and g(L) > 5k/4

Proof. I f

S is a

pl

if

Ks.L > O.

bundle use (6.3). I f

i f Ks.K s ~ O; (6.8) i f and Ks.L > O.

Ks.K s < 0

and

S is not a

pl

Ks-L ~ O; (6.4) with

bundle, use (6.7.1) x = 1 if

Ks.KS < 0

Q.E.D. (6.10) THEOREM. Le~C S be an e l l i p t i c ruled surface but not a d > k2,

g(L) > (k2+2)/2

un~;ess Ks-L > O. If

pl

bundle. Then

then d > 2 k

Ks.L > 0

+ 3

and

g(L) > 5k/4. Proof.

We know that

kKS + L is nef by (4.4). Then 2g(L) - 2 ~ -k2Ks.Ks - ( 2 k - 1)Ks.L

as in (6.8) with d >. 2g(L) - 2 ~ k2,

Ks-KS < O. So i f If

Ks.L < 0

we find

and also

g(L) > (k2+2)/2

Ks.L > 0 we use (6.4). Q.E.D.

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[I]

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[2 I

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[3I

M. Beltrametti, P. Francia, A.J. Sommese, On Reider's method and higher order

/4]

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T. Fujita, On polarized manifold~ whose adjoint bundles ace not semiposX~iue,

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R. Hartshorne, Algebraic G e o m , , G.T.M. 52, Springer-Verlag (1977). P. lonescu, Embedded projective varieties of smaJ.~Z inva/c/a~_~, Proceedings of the Week of Algebraic Geometry, Bucharest 1982, Lectures Notes in Math., Springer-Verlag, 1056 (1984).

[8 ]

A. Kumpera, D. Spencer, Lie equations, Vol. I: General theory,

Princeton

Univ. Press, New Jersey, 1972. [9 I

E.L. Livorni, Cl~sification

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51

[ 101

E.L. Livorni, C,b~z~,L~,E~on 04 a£geb~d.c Sw'c~ae~ wi..th aee.~ona2, ge.~u~ £e~,s than o~ equa£ ,to ~J~c I l l " R~ex( ~u~t~ace~ wX~thdim ¢KxeL(X)=2, Math. Scand., 59 (1986), 9-29.

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Sommese, Compact Complepc Man.i.fo.ZdA Po~e~J.ng a Ldae BuncLZe ~.Cth a

T~J:u~ Jet BundLe, Abh. Math. Sem. Univ. Hamburg, 55(1985), 151-170. [ 121

A.J. Sommese, Amp~.ecLLuY.J~o.~ on Go~cn~tex:n uaJu2ct~Le~, Proceedings of Complex

[ 131

A.J. Sommese, A. Van de Ven, On Zhe adjuatc.tJ~on mapping, Math. Ann., 278

Geometry Conference, Nancy 1985, Revue de l ' I n s t i t u t E. Cartan, 10 (1986). (1987), 593-603. [ 141

E. Ballico, A eJ~ct~nJ.za;Ci.on o4 the Ve~on~e ~u~ace, to appear in Proc. A.M.S.

[ 151

E. Ballico, On k-,6panned projecti.v¢ zu,'tfac~, in this volume.

Note. Very recently some improvements have been obtained by E. Ball[co. s l i g h t modification of Conjecture (2.6) is proved. Furthermore, l e t ned line bundle on a smooth surface with

L

In [ 1 4 | a

be a k-span-

k > 3. Then in [15] the better lower bound

hO(L) a k+5 is given (compare with (2.1)).

In both the papers [141 and [15] our

results of Section 5 are also used. In our new paper "Zero cycles and k-th order embeddings of smooth projective surfaces" we define a k-very ample line bundle on a smooth projective surface, S, as a line bundle, L, such that given any length k+l zero dimensional subscheme ( Z , OZ) on S the restriction map F ( L ) --. F (L ® O z ) is onto. This definition is stronger than that of k-spannedness, but in the paper mentioned above we show that the key criterion for k-spannedness, Theorem (0.6) of this paper, holds for k-very ampleness. This means that all the results in this paper hold for k-very ampleness. We are currently preparing a sequel to this paper where we give other new and stronger consequences of k-very ampleness.

ON THE HYPERPLANE SECI~ONS OF RULED SURFACES

Aldo Biancofiore Dipartimento di Matematica, Universit~ degli Studi de L'Aquila Via Roma Pal.Del Tosto, 67100 L'Aquila, Italia

Introduction Let L be a line bundle on a connected, smooth, algebraic, projective surface X. In this paper we have studied the following questions: 1) Under which conditions is L spanned by global sections? I.e. if OL:X--oPN denotes the map associated to the space F(L) of the sections of L, when is ~L a morphism? 2) Under which conditions is L very ample? I.e. when does g~Lgive an embedding? This problem arise naturally in the study, and in particular in the classification, of algebraic surfaces (see [3],[5],[6],[8],[9],[10]). In this paper we have restricted our attention to the case in which X is gotten by blowing up s distinct points Yl ..... ysEY, where Y is a geometrically ruled surface. If we denote by P1,...,Ps the corresponding exceptional curves then a line bundle L on X is of the form L----'~*(L^) - ~

t;P~ where n:X---~Y is the blowing up morphism with center Yl .....Ys, and L ^ is j=l....,s J J

a line bundle on Y. Partial answers to the questions (1) and (2) in the case in which X is a Hirzebruch surface are in [1] when t 1.... =ts=l. In [4] it was studied the very ampleness of L ^. In §0 we explain our notation and collect background material. In §1 we give sufficient conditions under which L is spanned or very ample. In §2 we find some special properties of rational ruled surfaces. In §3 we refine the results found in §1 for rational ruled surfaces under the hypothesis of general position of the points y 1,'",Ys" We would like to thank A.J.Sommese for very useful discussions.

§0 Background Material. (0.0) Let L be a line bundle on a smooth connected projective surface X. Let M=L-K X, where KX is the canonical line bundle on X, (0.1) In order to semplify our notations we give the following definitions: Let X and L be as in (0.0). 1. We say that L is "0-very ample" if L is spanned by global sections.

58

2. We say that L is "l-very ample" ifL is very ample.

(0.2) Definition: For every meN, denote by D M the set of all divisors Ec-X, such that E~0 and mE is effective. Moreover we set D =LImcNDm and Din= { EED l I M-2EED }. (0.3) Theorem (Reider): Let X,L and M be as in (0.0). Assume that: I) MED; 2) M2>5+4i; 3) (M-E)-E>__2+ifor any EED 1 and i=0,1. Then L is i-very ample. Proof: See [21. (0.4) Throughout this section we will always assume that X,L and M are as in (0.0). The following results have been proved in [2,§1]. Let EED 1- Then E=EI+...+Ek where Ej, j=t ..... k are all the irreducible and reduced components of E. Denote by E , i=0,1, the set of all EED 1 such that either 1

k=l or if k~.2 then the following inequalities must be satisfied (0.4.1) and (0.4.2)

~ j = 1,...,kEj.(E-Ej)_>(k-1)(2+i)+ 1

E"E">2

if E=E'+E"

and

E',E"ED 1"

(0.4.3) If any EE E.t'-~DM, i=0,1, verify the inequality 1

(0.4.4)

(M-E).F_2_2+i

then (0.4.4) holds also for any EED M"

(0.4.5) Lemma: Let EEE., i=0,1. Then g(E)_>0, where g(E)=I+(E+Kx).E/2. 1

(0.4.6) Remark: Let EED 1" Then 1) (M-E)'E=L.E-2g(E)+2; 2) If g(E)=0 then EEE if and only if E is smooth. Moreover if L is i-very 1

ample then L-F~_i. (0.4.7) Lemma: Let EED M"g(E)=l and L be very ample. Then L'E>3. (0.4.8) Let EEDM.Since (0.4.9) M2--4E-(M-E)+(M-2E) 2, then E.(M-E)_>2+i if and only if M2>5+4i+ (M-2E)2. Moreover from (0.4.9) assuming ~ M2>5+4i (0.4.10) it follows that (0.4.11)

L (M-E)'E 1.

54 (0.4.12) Lemma: Let EED M, i=O,1. Assume that E2_>0,(M-2E)-E.20 and that (0.4.10) holds. Then one of the following is satisfied: 1) i=0, E2=0, M-E=I; 2) i=l, E2=0, M-E=I,2; 3) i=l, E2=l, M=3E. (0.4.13) Lemma: Let M2>5+4i and E2_>- t for any Ee E.c'u9 M such that g~)--0. If there is 1

Ee E.c"a9 M such that g(E)=l, E2=0 and I_...>ts>l. Since

j=l,,..,S J J

Pj it follows that M=L-Kx-(a+2)e0+(b-2q+2+e)f-~..

J= l,...,S

J=l,...,S

(ti+l)Pi a

a

where q=g(C)=hl,0(C)=hl,0(X) denotes the irregularity of X. Any divisor E on X is such that E=-xC0+Yf-~ a.P.. For all the notations about ruled surfaces see [7]. j=l,....s J J" Throughout this section X,L and M are supposed to be as in (1.0). (1.0.1) Lemma: Let M2>0 and a__>0.Then MED.

Proof: From h2(0tM)=h0(Kx-aM)=0 and from the Riemann-Roch theorem it follows that h0(aM)>i~(O X)+(1/2)(a2M2-aM.Kx)>0, for a>>0.

/.

A={(x,e)eZxZ[x_>0and e_>-q},Al={(x,e)eA Ix_>2and -q_0. Assume that E'(M-2E)=x(w-ze/2)+ z(y-xe/2)-~., cq~,~ze/2. If E-~=xC0+Yf-~. ,

cqPjEE.

J= l,...,S -~

then: 1) Ifx=0then 0_ " J j=l,...,k ~ J L

k(2+i)+t. Therefore (0.4.1) is satisfied. Let E' and E" be effective divisors on X such that E=E'+E". In order to prove the claim it is enough to prove (1.2.2)

0~'+Pt)-E">_2.

If E"'Pt_>0 then (1.2.2) is verified since E"E_>2. So we can assume E"'Pt__

1

if ((tj+l)/2)>_ ~ and tj is even

I

0 if ((tj+l)/2)> ctj and tj is odd. Now it is easy to check that

1

57 if (z,e)EF 0 PJ< { z+w z+w-ze/2

if (z,e)EF l.

/.

(1.2.3) Definition: We say that L-=,aC,+bf-~] t.P. satisfies the property (P i),i--0,1, if for any ~' j=l,...,s J J E---f-]~

j=l,...,s

0t.P..~D1 with 0_3 for any E---f-]~ -

'

j=l,...,s

~.E.ED J J

1.

Denote by T i the set of all E~xC0+Yf-]~

j=l,...,s

o~.P:E~ c'u9 ,, such that J J i iv,

1) t_

f

0

if (x,e)e F 0

xe/2 if (x,e)E F 1

Max{0,(tj+l-z-w)/2} if (z,e)e F 0 3) Min {x,(tj+l)/2} >~j_>

L Max {0,(tj+ 1-z-w+(ze/2) }/2 if (~,e)~ F1

(1.2.5) Theorem: Let i=0,1. Assume that M2>5+4i, L satisfies property (P i) and that (M-E)'E2_2+i for any E~ T i such that E2(a+2-2x)y+(x-1)xe+(2+i)x+ ./

'~

J

~'~.. (xtj-tx;(tj+ 1-ct;)). If x=0 then from (1.2) it follows that 02+i when x_>l,

/.

(1.3.2) Remark: The bound (1.3.1) is sharp. It can be improved if not all yj, j= 1,..,s lie on D - C 0. (1.3.3) Corollary: Assume e>0 and s=0. If (1.3.4) a_>i and b_>ae+2q+i then L is i-very ample. Moreover if q3

8i=~i(L) =

where sk is the number ofj~{1 ..... s} such that tj=k. We note that: 1) if a=0 then i=0 and s=0; 2) if a=l and i=l then s=0; 3) if a=2 and i=l then s2=0. (1.4) Theorem: Let e_5+4i ; 2) (M-E)-F2_2+i for any EEE i ~

M-

1) We have M2=2(a+2)(b+2-2q-ae/2)-~..

tj-

J= 1 , . . . , s

(tj+l)2->2(a+2)(2+i+gi+~ . J= 1 ,...,S

(1/2) ~j=l,..,,s ((tJ+1)2/(a+2)))" If a=i then s=0 and M2>_2(2+i)(2+i+8i)>5+4i. If a=l+i then tj =1 for j=l ..... s and M2_>2(3+i)(2+i+Si+s(l+i)/(3+i))>5+4i since 8i-> { -I-t/6-s/3

if i=0

-2+1/8-s/2

if i=l.

If a=2+i then l5+4i. Since

f

-11/8-s/2-3s2/8

if i=0

5i--- ~.-3/2-S+Sl/2

if i=l.

Assume now a2>3+i. Since (1/(a+2)) _>_2+i+~Si-ae/2>2+isince 8i2ae/2. Assume now x_>2.Then a2>2x-1_>3,y_>xe/2, ~jx(2+i+Si+~[~ . •

J=l,...,S

(ti-(ti+l-o~i)o~.,/x)).Thus (M-E)-E>2+i. "J

J

J

/.

(1.4.2) Corollary: Let s=0 and eiand (1.4.3) b>ae/2+2q+i+Max {ae/2,- 1-i/2 } then L is i-very ample. On the other hand ifL is i-very ample and either q=l, a_~>land e=-I or q=2, a=l and e3 and L'C0=-ae+b>3 i.e. b_>-a+3 and b>-a/2+3/2. Thus (1.4.3) holds. If q=2 and a=l, then L-C0=-e+b_>5which implies (1.4.3).

/.

60 §2 Rational Ruled Surfaces. (2.0) Throughout this section we will always assume that X,Y,L and M are as in (1.0) and we will always let q=0 and M2->5+4i, i=0,1. (2.0.1) Lemma: Let Ew'x=C°+Yf'~j=I ,...,s~XJPJeEicxO 'M . Assume x>l. Then either x=l and y=0 or (2.0.2)

Y> I xe

if either x=l and e:20 or x>2 and e2 and e2_2.

Proof: Let E^-=xC0+Yf be an effective divisor on Y. Let p.j(E^) denote the multiplicity of E n at yj. Then I.tj(E^)>~xjand E---~*(EA)-E .

CtjPj.Assume now that (x,y)~(1,0). By [7, Prop.V.2.20

J--l...,S

p.382] we have that E ^ is not irreducible when y~.xe-1. Let x=l. If y_2+i for any Ee Si such that E2(x-1)e- 1 then h I (D')=0. Hence (2.1.3) implies (2.1.4) also in this case. Thus the claim is proved. /. From (2.1.4) it follows that h0(D)=h0(yf)+)-~-k=0,...,x-ih0(0)-kC0) I C0)= hO(O pl(Y))+Ek=l,..., xh0(O pl(y-ke)). which together with h0(O p1(5))= I 5+1

L0

ifS>0 if 5_0. Now it is easy to see that E-(E-Kx)_> j=l,...,s .I J EA-(E^-Kx).

(3.1.3) Lemma: Let Yl,'",Ys be in general position w.r.t.L. Let E~--xC0+yf-Y~ cxjPj ~ ~i ~ j=l,...,s be such that (x,y)~(1,0). Then (3.1.4)

E-Kx5+4i,i=0,1, and let Yl ..... Ys be in general position w.r,t. L. Suppose that ifi=l then for any E~ElC-a9 M such that g(E)= ~ 2

if x__3then (3.4.4) implies (3.4.3). Proof of theorem (3.4): Assume that there is E~--xC0+Yf-]~.. J=l,...,S

~qPiEE.fqD M i--0,1, such that J a

1

L-El and that (2.0.2) holds. We have (3.4.10)

g(E)l, g(E)--0 and xl; 2) Ife>l and either g(E)=0 and 0__2. From (3.4.8) and (3.4.9) it follows that (3.4.12)

b k given by local duality .

has a natural decreasing filtration , given by the powers of the

maximal ideal of p , and the last non zero term of this filtration is called the socle of R , and shall be denoted by S = S p The condition that R be a Gorenstein ring implies that S is a 1-dimensional k-vector space. (1.3) W e recall m o r e o v e r that the pairing (1.1) is compatible with the algebra structure on R , i.e. , for f,g e R , < f,g>=< 1 , fg>,and therefore the socle S is just the annihilator of the maximal ideal A p of R = R p . In the sequel , given a k-vector space V , we shall denote by V v its dual.

Theorem 1.4 Let X be a Gorenstein surface and Z a 0-cycle on X ; let L be a Cartier divisor on X and [L] the invertible sheaf associated to the Cartier divisor . If JZ denotes the ideal sheaf of Z , we may consider the exact sequence (*) H 0 ( [ K + L ] )

r .... > H 0 ( [ K + L ] I z

)

" ' H 1 ( JZ [ K + L ] ) ,

and consider an isomorphism of the middle some local trivialization of [ K + L] ) . Then there is an isomorphism between

term with R(Z)

i) the group of extensions 0

~ JZ [ L ]

~(9x

modulo the subgroup of extensions 0

>E ) Ox

) E'

( given by

~- O, ) [L ]

( giving E as the subsheaf of E' defined as the preimage of JZ [ L ] ) .

; O

69

ii) the group of linear forms H 0 ( [K + L] ).

et e R ( Z ) v

vanishing

on

the image of

Moreover , in the above isomorphism , E is locally free if and only if Z is a 1.c.i. and ,writing ap for the restriction of cz to Rp , a p does not vanish on the socle

Sp of R p .

P r o o f . Dualizing the exact sequence (*) , we obtain that the group of linear forms o~ as in ii) is isomorphic to the space H I ( J Z [ K + L ] ) v modulo the subspace H I ( [ K + L ] )v , and we conclude for the first assertion since these two vector spaces are naturally isomorphic to Extl ( JZ [ L ], 0 X ), resp. to Ext 1 ( [ L ], (9X ). We denote by

a*

an extension

in E x t l ( j z

[L],(9 X)inducing

a.

We have to see when does the extension ct* give a locally free sheaf E. First of all , since E has rank 2 , if E is locally free , then Z is locally defined by two equations , so Z must be a 1.c.i.. Moreover , the local to global spectral sequence for Ext provides a natural map : Extl(jz[L],0X (9X )

)

) H0(E×tl(jz[L],0X

_= Ext 2 ( ~ g z [ K + L ] , [ K ] )

) -_- H 0 ( E x t 2 ( 0 z [ L ] ,

-_- H 0 (

[K+L]Iz)V

=_ R ( Z ) v

( the last two isomorphisms being respectively given by Serre duality on X and by the chosen trivialization of [ K + L ] around Z ). The given extension local extension

o~*

0

) CgX,p

) Ep

) Jz,p

Rp v

with R p ,

Using local duality we can identify function g around p

whose class in

Rp

Moreover , since Z is a l.c.i. , the ideal functions

h 1, h 2 , and

morphism

of

then

free sheaves

maps to

thus naturally

Ep

et , with

represents JZ

) 0 hence

CXp giving a as follows. we can pick a

,Xp.

is locally generated

by two

is given as the cokernel of the homo-

associated to the transpose of the row

( g , h 1, h 2 ) so that we have an exact sequence 0

> 0X, p

)CgX,p3

) Ep

in E_ and the embedding of (9 X,p ~ 0X,p with the first factor of (gX,p 0 X,p

, if

) 0 is induced by the isomorphism of ( hence the quotient of Ep by

h is the column with coefficients h 1 , h 2

, is isomorphic

to

70

c9X,p 2 / h(9 X,p

, and thus to Jz,p

It is now clear that

Ep

as desired) .

is locally free

if and only if

p , i.e. its class does not annihilate the socle

g does not vanish at

Sp of Rp . Q.E.D,

Remark

1.5.

If H I ( [ K + L

unique extension

t~* inducing

]) = 0,then

for each

cxe R ( z ) V

there is a

~.

Example t.6 If Z is a cycle of length 2 supported at a smooth point p of X , then there do exist local coordinates ( x, y ) such that JZ is generated by ( x 2 , y ) . The socle S coincides with the maximal ideal of R , and such a locally free extension exists if and only if S is not contained in the image of the restriction map r from H 0 ( [ K + L ] ) . I.e., either p is a base point and Im ( r ) = 0 , or p is not a base point and r is not onto. Example 1.7.

If Z consists of

m

distinct smooth points , Pl ' "" Pm , then

E is locally free iff ct p is non zero for each p = Pl , " P m • In this case we have a non trivial extension ( by which we mean, not obtained from an extension 0 ; ~9X ) E' ) [L ] ) 0 ) if and only if the points Pi are projectively dependent via the rational map associated to the linear system IK + LI , or ,more precisely, if the linear functionals e i , for i= 1, ..m, given by evaluation at Pi ( and in fact only defined up to a scalar multiple ) are linearly dependent ; this is in fact the condition that r be not surjective. We obtain a locally free sheaf if no Pi is a base point o f l K + L l a n d i f , q i being the image point of Pi ,

there

does exist

among the qi's a relation of

linear d e p e n d e n c e with all the coefficients different from zero. T o understand what this geometrical condition means, we m a y assume that q l , ' " qh is a maximal set of linearly independent elements among the qi's

: then , since the given field k is infinite , such a relation o f linear

dependence exists if and only

if

h < m

and

the remaining qj's do not all

lie in one o f the c o o r d i n a t e h y p e r p l a n e s of dimension ( h - l ) spanned by the points q l , "'qh"

the

projective

space

of

R e m a r k 1.8, The following observation came out in a conversation I had with Mauro Beltrametti . Assume that X is smooth and that Z is a 0-cycle for which the restriction map r is not onto , whereas for each subscheme Z' of Z the restriction map r' is onto . Then the image of r is a hyperplane in R ( Z ) , hence there is a unique nonzero linear form a vanishing on Im ( r ),

71 and a corresponding extension E is locally free ( implying that Z must be a l.c.i. ). In fact , otherwise E is contained in its double dual E' which is locally free , and gives an extension 0 - - - - ~ @X ~ E' ~ JZ'[ L ] - - ~ 0 where now Z' is a proper subscheme of Z . By assumption this sequence is split locally at Z , hence also the extension giving E is locally split, a contradiction. The following lemma is essential in order to be able adjoint linear systems I K + L I give embeddings of X . Lemma 1.9. If p is a smooth point of X and H 0 ( [ L ] )

to prove that the

surjectsonto~9 Z for

each 1.c.i. 0-cycle Z of length 2 supported at p , then I L t gives an embedding at p . Proof. Let Mp be the maximal ideal of the local ring ~gX,p : i f H 0 ( [ L ] ) does not surject onto ~gX, p / A p2 2-dimensional

and

, by our assumption , the image is

intersects A p / A p 2

Thus we obtain a contradiction by )tp 2 and by W .

in a 1 -dimensional subspace W.

by considering the length 2 cycle Z defined

Q.E,D. Remark 1.10 fact,if

The lemma does not hold already for a

H0 ( A p [

L ] ) does not surject onto

contained in a 2- plane

W

in

)t p / A p 2

A 1 singularity . In

Ap/Ap2

, then the image is .Unfortunately W and A p 2

generate a length 2 , but not a l.c.i, cycle , because if the line W v in the Zariski tangent space is tangent to X , then JZ is not locally generated by two elements .

2 PLURICANONICAL EMBEDDING$ OF SURFACES OF GENERAL TYPE

In this section k is an algebraically closed field of characteristic 0 and X is the canonical model of a surface of general type : thus X is a normal Gorenstein projective surface with o~X ample , and if S is a minimal resolution of singularities of X , S is a minimal surface of general type. To a singular point p of X there corresponds a divisor E on S , called a fundamental cycle , and consisting, with suitable multiplicities , of all the curves mapping down to p ( hence these are all curves which have 0 intersection number with K ) The main property we want to mention

72 here ( cf. [ Ar ] for more details ) is that there is a natural isomorphism ( given by pull -back ) between 0X, p /3tp 2 and H 0 ((9 2E ) = H 0 ( (9 2E ( mK ))

, and therefore

a pluricanonical system

I ¢0xm I gives an

embedding at p if and only if the sequence

(2.1.)

0

-" HO ( [ m K - 2 E ] )

H0 ( [ m K ] )

;H 0 (02E(mK))

.........)0

is exact. Assume that m > 1 : then H 1 ( [ m K ] ) =0(cf. [ Bom] ),and of (2.1.) amounts to the vanishing H 1 ( [ mK -2E ] ) = 0 .

the exactness

Lemm__a.2.2.__: If E is a fundamental cycle on a minimal surface of general type S , then H 1 ( [ mK -2E ] ) = 0 , provided m > 3 , or m = 3 , K 2 > 2. Proof, denoted -2E ] )

At page 188 of [ Bom ] ( proof of theorem 3 , where E is though Z ) , it is shown that the desired vanishing holds if H 0 ( [ ( m - l ) K is not zero , and one has moreover m 2 K 2 > 9 , m+K 2>4.

We can therefore assume that H 0 ( [( m -1) K -2E ] ) = 0 . Since also H2( [( m -1) K -2E ] ) = 0 ( in fact the dual space is H 0 ( [ (2-m)K + 2E ] ) , which is zero for m > 2, otherwise we would have an effective divisor with negative intersection number with K ) , the conclusion is that, by the Riemann-Roch formula , 1/2 (m-1)(m-2) K 2 -4 + Z is non-positive. Since K 2 > 2 , m > 2 , ;¢>0,theonlypossibilityisthatm=K 2=3, If H I ( [ mK -2E ] ) is non zero, recalling that m = 3 , we have a non-trivial extension (@) 0

> 0S

)E

~ 0S (2K-2E)

We obtain immediately that 4 and c 2 ( E )

(#) 0

) 0.

H 0 ( E ) has dimension 1 , whereas

= 0 , hence E is numerically unstable ( c f .

we have a Bogomolov ~ 0s(M

)

destabilizing ~E

Cl 2 ( E ) =

[Bog] , [Rei] ) and

extension

) Jz ( D )

Z=I.

~ 0,

where Z is a 0-cycle, and the divisor M - D is in the positive cone. Recall also that M + D is linearly equivalent to 2K - 2E . Therefore K ( M - D ) > 0 , a n d KM+ KD=2K 2 =6,henceKM.>3,

73 while K E < 3 . As a consequence we get H0 ( [ - M ] ) = 0 : tensoring both exact sequences ( @ ) and (#) by (9S ( - M ), we obtain that H0 ( E (- M)) is at least 1-dimensional and is a subspace of H0 ( [ D ] ) , so that we may assume D is an effective divisor. Recall though that by our assumption H 0 ( [ M + D ] ) = 0 ,hence H 0 ( [ M ] ) = 0 too. We noticed that 3 = K 2 < K M , h e n c e H 0 ( [ K - M ] ) = 0 , a n d dually H 2 ( [ M ] ) = 0 , so that the Riemann-Roch formula gives us that 1/2(M 2-MK) + 1 isanonpositivenumber,i.e.M 2< KM- 1. We have ( M + D ) 2 = 4 ( K - E ) 2 = 4 ( K 2 + E 2 ) = 4 = D 2 + 2 M D + M2< D2+2MD +KM-1;sinceKM+KD=6,weobtain (2.3)

KD tgS ( 2 K - 2 E )

~E

~ 0 S

~0.

Since the first term of the above sequence has no global sections, by our assumption, the above sequence is easily seen to give a splitting of ( @ ), a contradiction. Q.E.D. for Lemma2.2

Corollary 2.5. then the m th

If X is the canonical model of a surface of general type , pluricanonical system I ~ X m I gives an embedding of X

whenever m > 4 , o r m = 4 , K

2 >l,m=3

and K 2 > 2 .

74 proof. The proof follows theorem 1 of [ Rei ], lemma 1.9., and lemma 2.2.. Q.E.D.

REFERENCES.

[ Ar ] - Artin,M. : On isolated rational singularities of surfaces, Amer. J o u r M a t h , 88, ( 1966 ), 129 - 136. [ Bog ] - Bogomolov, F.A. : Holomorphic tensors and vector bundles on projective varieties, (translated in) M a t h U , S . S R l z v e s t i j a , 1 3 , (1979),499-555. [Bom] - Bombieri,E.: Canonical models of surfaces of general type ,Pub/. M a t h . / H E.S. ,42,( 1973 ), 171-219. [ Cat ] - Catanese,F. : Canonical rings and "special" surfaces of general type, P r o c . o f Symp. in P u r e Math.,46, ( 1987 ) , 175-194. [ Ek ] - Ekedahl,T. : Letter to the author, october 1986. [ G-H ] - Griffiths,P.-Harris,J. : Residues and zero-cycles on algebraic varieties, A n n o f Math.(2) ,108 ,( 1978 ) , 461-505. [ Rei ] - Reider,I. : Vector bundles of rank 2 and linear systems on algebraic surfaces, A n n of Math. , 127, ( 1988 ), 309-316. [ Tju ] - Tjurin,A.: Cycles,curves and vector bundles on an algebraic surface, Duke Math. J o u r . , 54 ,( 1987 ), 1-26.

AN O B S T R U C T I O N

TO M O V I N G M U L T I P L E S OF S U B V A R I E T I E S

Herbert Mathematics

Clemens

Department.,

Salt L a k e City,

Univsity

Utah

of U t a h

84112,

USA

~0. Introduction In t h i s paper, a

reduced,

we c o n s i d e r the

irreducible

following

complex analytic

situation.

variety

Let X be

a n d let

Y cx be a c o m p a c t dimension

subvariety which

q

and c o n n e c t e d ,

generic point

of Y.

is reduced,

equidimensional

of

a n d s u p p o s e that X is s m o o t h at e a c h

W h e n we say t h a t

we m e a n t h a t we h a v e a c o m m u t a t i v e Z

a multiple

of Y moves

in X,

diagram

~ vW

~A v

c .....

where

~ is

W is

proper

reduced,

dominates

and

~

flat

and

the

unit

normal,

disc

and

A with

every

in t e r m s

component

obstructions

of the d o m i n a n t of Y in X.

map

to the e x i s t e n c e i

a n d the h i g h e r

We b e g i n differential

define

of

Z

neighborhoods

operators

to Y" at

into the

the h i g h e r - o r d e r

have a deformation

considering

then

"normal

"good points."

f o r m a l i s m of local

obstructions

(0.i),

do not d e p e n d on

of Z in W.

in ~I by g e o m e t r i c a l l y

all t h i s

In §2,

formally.

it gives

operators

integer

r.

By the

"symbol"

of this o p e r a t o r must be the r - t h p o w e r

formula

for r = I.

if we

"normal

Z for e a c h p o s i t i v e

for h i g h e r d e r i v a t i v e s

The o b s t r u c t i o n s

and

Intuitively,

r-th order

to Y" d e f i n e d a l o n g

we

cohomology

differential

the o p e r a t o r

of a d i a g r a m order

The idea is to get s o m e o b s t r u c t i o n s

can be f i l t e r e d so that the g r a d e d p i e c e s

higher-order

translate

Z = g*({O}),

Y.

neighborhoods which

onto

irreducible

We w i s h to c o n t r u c t (0.i)

X

measure

(see(l.2)),

of the

symbol

whether

this

the of

76

relation

on s y m b o l s

obstructions

is p o s s i b l e .

in the case

v a r i e t y P and a d e f o r m a t i o n We see t h a t condition

these

that the

appropriate

order

In ~3,

(0.I)

obstructions

first two on some

w i t h X r e p l a c e d by P is given.

reduce

image of Z u n d e r as Z m o v e s

The a u t h o r w o u l d

we c o m p u t e the

in w h i c h X is a C a r t i e r d i v i s o r

in t h i s f

case e x a c t l y to the

remain

i n s i d e X to

in W.

like to t h a n k

J. K o l l ~ r

for m a n y h e l p f u l

discussions.

§1.

"Fibering"

To f r a m e the p r o b l e m that,

in a d d i t i o n

is a p r o j e c t i v e neighborhood this, let

in g e o m e t r i c terms,

to the a s s u m p t i o n s

variety.

made

s u b - s y s t e m to X.

linear

For e a c h y E Y,

let By be the b a s e

IDI c o n s i s t i n g

in d i v i s o r s

Y transversely

at

F~Y.

y

F varies

singular points

in a a l g e b r a i c of Y and the

If U is a s u f f i c i e n t l y

Y.

To do

space

for

y

locus

of the y.

l y i n g in a

IDI in the o r i g i n a l system whose only

singular points

small o p e n set

and

linear

which pass through

except

By c h a n g i n g the q - p l a n e

system,

lie on Y.

regular

of a g e n e r i c q - d i m e n s i o n a l

in

ample

we a s s u m e t h a t X

s y s t e m on p r o j e c t i v e

T h e n By m e e t s

are the

in ~0,

of Y in X, and we w i s h to "fiber" X' o v e r

tDf be t h e r e s t r i c t i o n

divisor

we b e g i n by s u p p o s i n g

We let X' d e n o t e a s m a l l

we t a k e a v e r y a m p l e

hyperplane

a neighborhood

very

fixpoints

of X w h i c h

in X'

containing

Y' = Y - F, then there

is an a n a l y t i c

fibration

p: U whose

f i b r e at

containing

y

is the u n i q u e

)Y'

component Dy of the u n i q u e By ~ U

y: ,,,,,

yv

I

77

Suppose X.

Then

that the q - c y c l e m ' Y m o v e s

for f i x e d y E Y' a n d small

of d e g r e e m',

and,

by the

a n a l y t i c m a p x(t),

implicit

s = t m',

be a n o n - t r i v i a l

~c

scheme

Ys N D y

function theorem,

such that

f(t)

fibre

s, the

f a m i l y Ys in

x(t)E Ys~Dy

is f i n i t e

there

is an

for all

t•

Let

(i. I) L e m m a :

polydisc

in an a n a l y t i c

0

to

mapping

f r o m a d i s c A to the

(0,...,0).

Let ~ I

of t h e s h e a f of d i f f e r e n t i a l

be the g e o m e t r i c

differential

(xl(t) .....xc(t))

complex-analytic

A c which takes

at t = 0

=

fibre

operators

at

be the g e o m e t r i c

operators

on A,

a n d let

(Xl,...,x c) =(0,...,0) of the s h e a f of

on ~c,

and let

f.: ~l-----)~c be the m a p i n d u c e d by first-order

f.

Then there

is a n o n - t r i v i a l

al~/~x I + ... +

ac~/~x c

in t h e

i m a g e of f. all of w h o s e p o w e r s

Proof:

Let D = ~/~t.

(i • 2)

where

Recall that

~ r ~ ci I=(i 1..... i x) the u s u a l

summation

runs o v e r all p a r t i t i o n s notational which

simplicity,

c = 2.

m

into

.(DirXkr) . .

is u s e d r

Dr ~ X k l .. -~x kr

in the

positive

k's

and

integers.

I For

in the case

in

= a't m + h i g h e r p o w e r s

y(t)

Then f,(~m/~tm)

are p r o d u c t s

convention

=

We w r i t e

by l i n e a r

f.(~rm/~trm),

f. (~m/~t m)

(Dil Xkl) .

of

lie in G r ( i m a g e f.) .

we do the rest of the p r o o f

x(t)

where,

homogeneous

operator

change =

(~mx/~tm) "~/~x.

the of

= b't n + h i g h e r p o w e r s of c o o r d i n a t e s ,

coefficients s

terms

Also,

we can a s s u m e

in the e x p r e s s i o n

of o p e r a t o r s

of the

that m < n.

~s /2(x,y) S

form ~mix/~tmi

for

for s > r

or ~ m i y / ~ t m i w i t h

78

~imi be

= s and

zero.

each

Similarly,

must

be

this

coefficient

zero

Note system,

unless

that then,

corresponding the

m i > 0.

rule

the the

only

So some m i < m, coefficients

coefficient

occurs

if we m o v e at

least

operators

for the

has

q-plane

IDI

smooth

points

al~/Dx I + ... +

normal

that

by

bundle

X is a r e d u c e d

is a c o n n e c t e d X.

We

assume

component is,

restating

(reduced that

of Y.

we h a v e

and

the

our

operators form

in o u r

linear

X, t h e

acD/Dx c p a s t e

according

on Y ~ X .

analytic

at the

that

to

"a m u l t i p l e

We a s s u m e

only

variety,

and that

compact

subvariety

generic

point

Y of

of e a c h

of Y m o v e s

in X,"

that

diagram:

v ~W

(2.1)

~A v

i

c

Y

But

D r /Dx r.

original

of Y a n d

hypotheses

commutative

Z

D r /~(x,y) r

(Dmx/~tm) r.

operator

and equidimensional)

assume

must

of the s u b m a n i f o l d

irreducible

following

coefficient

of Y in X.

X is s m o o t h

We

the

of t h e

$2. M o v i n g a m u l t i p l e

We b e g i n

of a l l

in f r o n t

the

at

a n d the

--

~x

where i) W is n o r m a l , and

flat

with

2) t h e

reduced

reduced

fibres

f

is a g e n e r i c a l l y

4)

i:

Z

section

)Y

W ^ denote

so the

group

finite

formal

of ~ W / ~ Z s i n d u c e s

a disc

variety

is g e n e r i c a l l y

the

of d i m e n s i o n

over

analytic

3)

Let

and

irreducible,

A=

q+l,

{t 6 C:

Z = K*(0)

a n d K is p r o p e r Itl < i}.

ideal-theoretically,

morphism,

finite

completion

an ~ W - m O d u l e

on e a c h

component

of W a l o n g endomorphism

Z. of

of

Any

Z.

global

@~/~Z s,

79

Ext I (~,W/~Z s, (~,W) has the

structure

of an H0(f~,W^)-module.

Let

~: Ext I (~,W/@Z s, (~,W) .......;...Ext .. I ((~.W/~]Z, ~,W ) be such that

the n a t u r a l

morphism

Sxtl { ~ / ~ Z ' composes

~)

w i t h ¢ to give the

~sxt~ ( ~ / ~ z s, ~'w)

identity

on

Ext I ((~.W/~.~Z, (~.W) = H 0 (Nz/w*) = C . We

"differentiate"

f E H0(@,W^)

~ E ExtI(~,W/~Z s, (~,W) on

f

by d e f i n i n g

to be given

the a c t i o n

by the

of

formula

~(f-~). In the

case

in w h i c h

differentiation action

Z is a smooth

can be i d e n t i f i e d

point

on a curve

(non-canonically)

W, this with the

of E r < s ar 8 r /St r

on

(formal)

functions

Let m = max{m': natural

map

discussed

f*~y-~Z

f*~Y-~Z"

in ~I.

m' } where

Then,

Z(k)

scheme

denote

is induced

by the

for r < s, we have m o r p h i s m s

f.(~yr/~yS) Let

the a r r o w

a the k-th order

with

functions

given

is a d o u b l e

complexes

of sheaves

) ~zrm/~zSm. neighborhood

by ~ / ~ z k + l . whose

of Z, that

Recall two

is, the

that

that

filtrations

have

there

E2-terms

• and ~wq(~e~Bp(;,~ respectively, occurring

and that,

Z (k)), @~)

~ k " , the o n l y n o n - z e r o if ~ = ~ Z k ' /--Z

at E 2 in e i t h e r

complex

occur

when

q = I.

So,

terms

80

~ W

1 (~W~BP (~Z k ' / ~ Z k''' ~'Z (k)) ' ~ ) ~wP(~zk'/aZ

Let A be an open Z' = B ~ Z . in A,

and s i m i l a r l y

for Z'

(~%g)) •

let B = f-l(A)

the k-th order

neighborhood

"~/)cZ-Q~"

and of Y'

isomorphism

maps

is an i s o m o r p h i s m

diagram

and

U s i n g the above

ExtBi(f*~, which

k'', ~ w q ( ( ~ , Z ( k ) ,

set in X and Y' = A ~ Y ,

We let Y' (k) denote

and the n a t u r a l

=

~)

;ExtAi(~,

for i = 0 ,

f*~),

we c o n s t r u c t

the

following

for k > sm-rm:

ExtlA ( (~.~ /--~ {So71,

O-A )

=

Extlw (

O.W/..~ Zm+l,

O,W )

1 I sm+l ExtB( ~'B / ~ Z' ' ~'B )

!

(2.2)

/

1

/

*

sm+l

.... r m ~ sm+l nOmB{ ~ Z,I~z, ,

, k+rm-sm~__ k+l ~x Z' l ~ Z' )

i HOmA(~

r/.~s+l ~k+rm-sm/.~k+l f* r/.~s+l Y"-- Y' ' f* -- Z' --- Z' ) = H O m B ( ( ~ Y ' / ~ ' ~ Y ' )

1

' "-~

k+rm-sm/ .k+l) Z' "~-~ Z'

1

HOmA('~ Y'/~'~ r.~s+l Y' ' f*(~" Z' (k) )

r/.Q s+l H°mB( f*(~ Y"-- Y' ) ' (~"z' (k))

=

If r = S and k = 0, the map Extwl(~w/~zrm+l, induced map,

by the d i a g o n a l

and we will

complete

denote

intersection

~W )

arrow

)Homy(~yr/~y

above,

it by ~.

will

Notice

r+l,

be c a l l e d that,

i * ~ z) the

if Y is a local

in X,

Homy(~yr/~y

r+l,

i , ~ Z) = H0(Z;

symbol

i*SrNy/x ) .

81

Since ~ z k / ~ z

k+l = ~ Z

from the d i a g o n a l

We next point

where

lies e n t i r e l y

Y' is a dense the

Y'

which

there

that

smaller the

f

disc t r a n s v e r s e

to some

of Z for which

to a single open

fibre

subset

of Y w h i c h

Dy over

are also

is a f i b r a t i o n

of a that

of Z.

m = max{m':

smooth

Y' is

in some n e i g h b o r h o o d

at p o i n t s

E z of f-l(Dy)

a

of Y which

We also assume

we have

component

can be c o m p u t e d

k ~ 0 and r = s.

is finite

if necessary,

component

any

points

as in §I.

small

symbol

Zariski

of Y' and W is smooth

z E f-l(y),

a component

smooth

constructed

sufficiently

by m a k i n g

using

this m a c h i n e r y

inside

of each point

any

above

of X, and a r o u n d

neighborhood chosen

arrow

restrict

y E Y',

points

for all k ~ 0, the

of f-l(y,).

that

Again,

for y e Y' and for

containing

Choose

z

such a

is a z

f*~y-~z,,m'}.

in Z",

We have

maps 1 H{z}(Ez;~E which

are

= z)

isomorphisms

1 > H{z} (f-l(Dy) ;~f-l(Dy)) since

the

fibration

= ) H~0}(U; ~ U ) ~ has

reduced

fibres

by

assumption.

We n o w examine

GrrmH{~}(A; ~ A )

the c o m m u t a t i v e

(

GrrmH0(~(W;

diagram

~ W )) --0-->

H0(i*SrNy/x )

T 1 GrrmH{zi (E z; ~ E z) = ~ E z ~

and we interpret differential We are then

the top

operators in the

at

Grrm~

(W; ~ W )

left and lower 0

situation

right

in A and at of Lemma

y

(I.I).

) ~Ez~

i*(SrNy/x)

groups

as symbols

in Dy respectively. So the o p e r a t o r s

~ r m / ~ t rm on A give

sections

of (~'E Z (~ i*(SrNy/x)

which

are

just

the

powers

of the

section

given

by the

image of

of

82 ~m /~t m " Now i . ~ z is t o r s i o n - f r e e elements

by our a s s u m p t i o n

of H o m y ( ~ y r / ~ y r+l,

g e n e r i c points

of generic

finiteness,

i . ~ Z) which are r-th powers

must be r-th powers everywhere.

at

So we can conclude

the following:

(2.3)Theorem: suppose

Suppose that a m u l t i p l e

we have a d i a g r a m

a non-trivial positive

(2.1)

of Y moves

satisfying

in X, that

i) - 4).

is,

Then there

is

section p of H o m y ( ~ y / ~ y 2, i . ~ Z) such that every

power of p is the "symbol"

of an element

of

H 1 (A; ~ A ) = H 0 (~-~ (W; ~ w ) ) . {0}

(2.4)Corollary: obstruction

to splitting 0

Then,

Let ~ e Extx I ((~.X/--~yr, ._~yr/~yr+l) the sequence

) ._~yr/._~yr +i

under the h y p o t h e s e s

non-trivial

> (~,X/.-~y r+l of the theorem,

> ~.X/~-~yr

~(~r)

= 0 in Extxl((~,X/~yr , f*(~'Z(rm) ) "

Note:

By filtering

r+l'

f*(~'Z(rm) )'

f*(~'Z(rm) we obtain g r a d e d q u o t i e n t s so a sequence

values

in Extx I ((~.X/~y r, i.~,Z).

Proof:

Apply the functor RHomw(

to the sequence

for

i.~.Z) --Homy (..~yr/._~yr+l, i.~zrm/._~zrm+l ) ~H°mx(~yr/~Y

i,~zn/~zn+l--i.(~,Z,

> 0.

there must be a

~ E Homy(~y/.J~y 2, i.(~,Z) such that,

~r e Horny (~.~yr/~yr+l,

be the

of o b s t r u c t i o n s

• (~,Z(k))

of the form

w h i c h take their

83

0

)~zrm/~zrm+l

) ~_W/~ Z rm+l

) ~.W/~Z rm

)0

to get a map Homw(~zrm/~zrm+l, Let 6r be the element

)Extw I(o'W/~Z rm' O'Z(k) )"

~Z(k))

of Extw I (@qN/~Z rm+l, O,W )

which maps to the operator ~ 2 m / ~ t 2 m corresponding

element

on U, and let C 2 be the

in

H°mw(~zrm/~zrm+l' Since this element

O'Z(k) ) "

comes from an element of H ° m w ( ~ W / ~ Z rm+l' ~'Z (k))

as long as k _> rm, it must go to

0

in

Extw I (@'W/~Z rm' @'Z (k)) "

Let ~r be the image of £r in Homx(~yr/~yr+l

H°mx(~yr/~yr+l'

, f . ~ z r m / ~ z rm+l))

f*~'Z(rm) )

Then ~r must go to zero in Extxl (~,X/~y r, f*~'Z (rm)) ) " since the dia g r a m rml_D rm+l H ° m B ( ~ Z''~ z' , ~" z'(k) )

)

1 rm ExtB (f~"B/'/~ Z', ~'z' (k))

1

1

r ~+i ) H ° m A ( ~ y . / ~ y . ' f*~'Z'(k) is commutative.

z ' (k)

So ~r must go to zero in Extx I (~'X/~Y r' f*@'Z (rm)) "

But the above t h e o r e m says that ~r in Homy(~yr/~yr+l,

i.~zrm/~zrm+l ) ) = Homy(~yr/~yr+l,

must be the r-th power of a non-trivial

section of

H o m y ( ~ y / ~ y 2, i,~- Z) ) •

i.~,Z))

84

When X and Y are both smooth, H o m y ( ~ y r / ~ y r+l, H0(i*(SrNy/x)), H0(i*Ny/x)

i,~z))

we have the i d e n t i f i c a t i o n

= Homz(i*(~yr/~yr+l),

~Z))

=

and ~r must be the r-th power of the section of

w h i c h is the symbol of ~m/~t m.

~3. Computing the obstruction Suppose now that we have a

(reduced)

connected

equidimensional

subvariety Y ~P a n d are given a motion

(3.1)

of a m u l t i p l e §2.

z i~

~w

Y

~P

~A ~f

of Y satisfying

I) - 4) listed at the b e g i n n i n g

of

As in §2, define m = max{m':

f*~y~zm'}.

Let Z# be some c o n n e c t e d union of components f*~y

of Z for which

= ~Z m

at the generic points of Z# and let Z ^ denote the union of the other c o m p o n e n t s

Next

of Z.

suppose that we have a

(reduced)

divisor

X~P d e f i n e d by the section F of a line bundle ~ .

Also assume that X is smooth at the generic

point of each component (3.2)

Now suppose that

F-f E F ( J z # 2 m ~ d z ^ 2 m + l

an a s s u m p t i o n order"

of Y.

w h i c h implies that Y and the "first n o n - t r i v i a l

deformation

the v a n i s h i n g

~)

of a multiple

of the o b s t r u c t i o n

of Y lie in X. in C o r o l l a r y

We will show that

(2.4)

is equivalent

to the condition that F°f vanish to order 2m+l along Z.

85 The non-trivial element ~ E H o m p ( ~ y / ~ x 2 , i,~z ) given by this motion as in §2 has the property that ~ 2 E H o m p ( ~ x 2 / ~ X 3, f*~Z(2m)) is the image of an element yE H o m p ( ~ p / ~ y 3, f,~Z(2m)) under the natural restriction map.

Notice that y must be

surjective at each generic point of z# in order to have restriction ~2.

Furthermore,

y comes from

y'E Homw(~w/Jz2m+I, ~Z(2m)).

The assumption

(3.2) on F.f tells us that

~' (F,f) E Homw(f*~.-i/jz2m+if*~.-l,

(~'Z(2m))

is actually an element of

Homw(f*~.-i/..~z2m+if*~ -I, ~z2m/']Z 2m+l ) c Homw(f*~.-i/~z2m+if*~. -I, ~'Z(2m) ) "

To relate this situation to the computation in §2 for the obstruction to moving a multiple of Y in X, we let 0

) f*~'Z (2m) ---9 ~"

be an injective resolution of f*(~'Z(2m) as an (~.p-module and apply the functor

to the diagram below, where ~, = @,p and "]y = .]y/p:

88

o

o

0

nt + 1 et montrons que le groupe de points E' d6crit dans la proposition contredit le principe de position uniforme. C o m m e E' est contenu dans une courbe de degr6 t, il suffit de v6rifier deg(E') _> h ° ( O p 2(t)). Mais le caract~re de E ° est (n o ..... nt_l) avec nt_ 1 -> n t + 2 z s + 2 _> t + 3. Donc: t-1

deg(E') _> ~

o

(t+3-i) = t(t+3) - t(t-1)/2 = t(t+7)/2 _> ( t + 2 ) ( t + l ) / 2 .

115

Corollaire

2: Soit E un g r o u p e de points plan de degr6 d. Soit -r = m a x {n,

H ( E , n ) < d}. Soit s u n conditions

suivantes

entier tel que s ~- d/s et que -r _~ s - 3 + d/s. L ' u n e des

est v6rifi6e:

(i) E est i n t e r s e c t i o n c o m p l e t e d ' u n e c o u r b e de d e g r 6 s e t degr6 d/s et -r = s - 3 +d/s.

d ' u n e c o u r b e de

(ii) il existe t, avec 0 < t < s, et un sous g r o u p e de points E' de E c o n t e n u dans une c o u r b e de degr6 t tel que : t[q: + (5-t)/2] .~ d e g ( E ' ) -~ t[-r - t + 3]. D~m:

Soient

(n o .... ) le caractbre de E et t le plus grand entier tel que (n o . . . . . .

n t _ l ) est c o n n e x e . R e m a r q u o n s d ' a b o r d que t ~_ s. S-1

E n effet n o = q -

+2z

s- 1 +d/set

~

(s - 1 + d/s - 2i) = d. C o n s i d 6 r o n s alors

o

le g r o u p e de points E' de caract~re groupe

d6crit

dans

la

proposition.

(n o . . . . . nt_ 1) qui est soit E soit le sous Les

t-1

d6montrent:

~

in6galit6s

-r + 2 _~ n i _~ t: + 2 - i

t-1

(q: + 2 - i ) _ ~ d e g ( E ' )

_~ ~

o

('r + 2-

2i)

soit

t['r

+ (5-t)/2] _~

o

deg(E') _~ t[q" - t + 3]. s-I

I1 reste h traiter le c a s t

= s. Dans ce cas les in6galit6s d -~ d e g ( E ' ) _~ o~ (ni-i) _~

s-1

-~ ~

o

(s - 1 + d/s - 2i) = d entrainent E = E' et n i = s - 1 + d/s - i p o u r i = 0,

.... s-1. P o s o n s s' = d/s. Soit A le c 6 n e projetant de E. C o n s i d 6 r o n s la r6solution m i n i m a l e , c o m m e R - m o d u l e , d'un A - m o d u l e dualisant f2 A : 0 ---~So-lR[i-2] --~@So-lR[s+s'-l-i] Le R-module

~A

a d o n c un s y s t 6 m e m i n i m a l

-~ Q A -o 0

de g6n6rateurs

(cx o . . . . . c~ s - l )

avec d e g ( c x i ) = 1 + i - s - s'. Consid6rons la matrice, ~ coefficients dans R, dont les c o l o n n e s sont les c o o r d o n n 6 e s de une m a t r i c e

triangulaire

xJ2CXo par rapport ~ (cx o . . . . . CXs_l). C'est

~ termes diagonaux

de d e g r 6 s 0.

Si Fun de c e s t e r m e s d i a g o n a u x est nul, d i s o n s c e l u i c o r r e s p o n d a n t ~ la c o l o n n e d ' o r d r e t, il est clair que, s i t est m i n i m a l , le sous R - m o d u l e de Q A e n g e n d r 6 p a r (x o . . . . . c~ t-1 est i d e n t i q u e

au sous R - m o d u l e e n g e n d r 6 p a r cx o,

x 2 ~ o ' "'" x ,-1 2 (x o et q u e ce sous R - m o d u l e est un A - m o d u l e . U t i l i s a n t le l e m m e , il existe un sous g r o u p e de points E' de E de caract6re (n o . . . . . n t _ l ) , t-1

d o n c de d e g r 6

~ o

t-1

(ni-i) = ~

(~r + 2 - 2i) = t('r

- t + 3), c o n t e n u dans une

o

c o u r b e de d e g r 6 t. Si t o u s l e s termes d i a g o n a u x sont inversibles

la m a t r i c e est inversible et ~ A

= A.c~ o = A [ s + s ' - l ] . Dans ce cas A est un anneau de G o r e n s t e i n de d i m e n s i o n

116

un quotient de K[x o, x 1, x2],

donc

une intersection

complete

d'apr~s

un

th6or~me de Serre. Mais la r6solution minimale: 0 ~ @~-lR[-s'-i] --> i,

general

irreducible

and

hence

reduced

L)

has

a ladder

in

this

the

case

dim

= 1

and

W

any

since

member

V

is

member of

locally

H

ILl. Cohen

case. w

= deg(W)

> i.

We

will

a contradiction. Let

wX.

X

So

2 + d w

effective

of

points. Let

It

some

points.

p 0~(i).

are

M --~ V

be d

- ~(V,

- d.

a general

= L(E L)

Hence

+ wX)

- 1 = d w

= d,

fiber

= wLX and

~ w, 0

LX = 1

of

f:

whale

~ A(W,

and

~(W,

Since

1 = L X = (E + H ) X

= EX

tion

of

is

= ~ ~iEi

= 0

for

= 0,

we

e~d~2.

E i have

of

# 0. 0

the We

form set

E

e = -E02.

a (L - E 0 - H) 2 = d

M --~ W.

for

dim

0(1))

H

=

with

by

A

= 0. [A], ~0

numerically,

= hO(v,

~ 1 + w

0(1))

Since - e

Then,

L)

- 1 =

-

(I

+ dim

So

W

= pl.

the = i,

prime E0X

L(L

- E 0 - H)

the

index

[A]

A)

=

decomposi-

= 1

= L(E

theorem.

and

EiX

- E O) Hence

120

Let i.

Note

if

KE i

K

be the canonical

that

Ei

< 0.

2

< 0

Then

since

i # 0, E i X

contradicts

the minimality

implies

~ KE 0

=

that

KX

~

=

+

L)L

KE

Assume 2g(M, this

L)

-

2

(K

contradicts

e

-

of

that

f,

Since

LY

=

= 0.

So

Ei

= 0

and

~.

KL

= K(E

g(V,

< 0.

Then

two

Indeed,

= i, w e h a v e

phisms

free

~ --~ 0 v [ L ] ,

one defines

the

Coker(~,~[-L] length

= 0 = A(M,

~ ®

by Nakayama's

dim(Supp(~))

singular

points

criterion.

This

a curve

in

We claim Then

0v(H0(V,

E +

- ~ 9)

Thus

g(M,

~

L)

= -2.

KX

We

is a s i n g u l a r

further fiber

Y

of multiplicZ.

But

~. E

is a s e c t i o n since

of

0 = LE

f. =

E2

So

M

+ wEX.

the unique

L)).

= 1 0v/m

base

point

injective of

Then we have natural

and IHI,

of the natural

~[-L]M

--~ 0 M.

its cokernel of

Since

is

for the maximal = ~/m9

last

Therefore

which ideal

and hence

homomorthe

0 E.

~,0E,

ILl.

is o f m

of

~ = m~.

So

Lemma.

< I, t h e n

are

L)

f o r o n e o f them,

is a s u b s h e a f

{(~/mg)

v.

= 0

d = e

v = ~(E),

system

implies

and

be the cokernel

I-L] --~ 0 V

the point

v ~ Supp(~)

= F.

sheaf

g(V,

LM).

~

Set

linear

This

defining

F

let

= ~ ® ~[-L]

one.

If

2 e. M o r e o v e r

~ V --~ 9.

he the

E = E0

and hence

or a (-l)-curve

LZ

and

~

which

< d.

(-l)-curves

see that

homomorphism

the claim,

if t h e r e

Now we easily

9 = ~,0 M and

This

Since

X ~ F1

of

Set

a (-l)-curve

z e - 2

+ wX)

L)

by the minimality

g(M,

be

for any

~ 0

1

is a point.

Thus we prove

is i m p o s s i b l e

that

KE.

must

f ( E i)

~ d + e - 2 ~ 2 d - 2.

LX

LM)

x(Y)

LE i

Then

either

Note

the

We claim

is a ~ l - b u n d l e .

surface

0v

M.

2.

KX

is a H i r z e b r u c h

Let

of

the hypothesis

it c o n t a i n s

i t y two. this

f

of

O.

Thus we conclude claim

bundle

finite.

contradicts

V

is n o r m a l

Therefore

0 = d(M,

L M)

V

off

finite

is n o r m a l

< d(V,

L)

a curve

Y

points.

So

by Serre's

= i.

Thus we have

Supp(~). LF

~ d.

T o s e e this,

Y n E = ~

since

take

v ~ F.

So

YX

> 0

in and

M

such that Y

meets

a

121

general

member

of

f.

On the other

by

~

F

ILl,

meets

ILIM = E + hand,

and hence

a general

IHI

points

on different

are mapped

member

at points

of

on

fibers

to different

]LI

at

d

d

different off

points

points.

E

on

f i b e r s of

are separated V.

Therefore

Thus we prove

the

claim. Now we have is a n i n t e g e r [(~[tL]),

~(~[tL])

with

and we

The case

n > 2

f: M --~ W c F

set,

LE = 0

This

H.

Then

L)

A(D,

LD)

If then

LD)

has

Before

a ladder,

a polarized

and

A(W,

Note that

H

H)

is v e r y

p: P = F W ( ~ )

the unique a general B1 for

member member

is s m o o t h F = bH¢

branch

locus

S of

be the

of

B

and

H)

(W, H) = 0.

0p(1)

of

A

is l o c a l l y L D)

applies.

If

In either

Q.E.D.

an example

showing

d i m W = n - I, H n-I be a scroll

Set

double

of

positive Since

covering

f,O M = 0 p • 0 p ( - F ) .

to

~.

H u = p H.

and set

S n B 1 = @.

over

F I.

~ = 0W • 0w[2H]

associated

is a s e c t i o n

for s o m e

is a f i n i t e

such that

D

[F4]).

L).

0~(i) d > 0.

S

A(D,

and

or

of

.....

member

L)

such that

Fl-bundle

IH¢ - 2Hal

l(2b - I)H¢I

there

= L2H n-2

we exhibit

let

line bundle

and connected,

- Hu,

(V,

= E + A

is a f i n i t e

hypothesis

so d o e s

g(W,

--~ W

be the tautological

= g(V,

[FI]

(W, H)

and

H~

the

is i n d i s p e n s a b l e .

ample

let

x*ILl

as b e f o r e ,

(cf.

For example

and

L D)

ample

hence

manifold

= 0.

g(D,

r

tL) +

contradicts

the pull-back

any general

Similarly

applications,

g < d

Take

Hence

t, w h e r e

= ~(V,

= BslL [

LH n-I

= i, t h e i n d u c t i o n

is v e r y

giving

=(E)

and

This

x: M --~ V,

We denote

Pic(M)

in

n = 2.

Let

Since

D = x,S.

LD)

LD

that the hypothesis

= d

in

+ r ~ d.

in case

M.

terms

tL M) ~ ~ ( Y [ t L ] )

L M)

easy.

ring of

We have

~(D,

~ 0,

(1.3)

the proof

and so is

= I.

(D,

= g(M,

d i m W ~ n - 1 > I.

and reduced.

~(V,

L)

constant

~(M,

be as before.

L = E + H

is i r r e d u c i b l e Macaulay

modulo

Since

is rather

in the Chow

implies

case

g(V,

finish

and

by

r ~ d.

we obtain

hypothesis

~ rt

p.

Then

Let

integer

Let

B1 b.

Then

B = B1 + S e f: M --~ p

We have

be

12FI

with

f S = 2E

for

122

some

prime

-2H,

we

divisor

have E

to

rational Pic(V)

LE

see

sections.

of

> 0

on

(E + H a ) Z

implies

S

is

n X

EZ

next

L E = 0,

we

calculate

Ln

g,

infer

=

(K +

(n - I ) L ) L n - I

K

= f * ( K P + F) canonical

H~L n-I

since

we

= d

and

is

the

2H¢

E + Ha)

h0(M, ~(V, b

claim

of

W,

L)

> 1.

~ 1. Hence

If

#:

component

M -~

W

Thus,

Ha)

~ < 1,

~

= 1.

Alternately, fixed

L)

when of

and

F

(V,

L)

isoit

is

from

the

must

When b

b

> 1,

we

a curve

satisfies

of the

+

and

transform and the

p.

Since

of

- 2)H¢

= -2

= d.

To

2g

- 2

In view + Ha) M

- n

KWLn-I[M} n-2

since

SO

M.

(2b

This

Indeed,

K

of

for

- l)d

- 2

= 21-nKW~ -

=

ampleness.

= Hn-I{w]

=

LC

But

the

normal.

(b

to

= f-l(x).

is

KL n-I

(n

-I

- 2)d.

computation. b

= i.

H a ) = hO(w, have

g

we

can

of

proving

V

(K W

= d

= 0.

Z

global

suffices

~ 0

= (b - l)d.

= 22-nKW(2H)

= 1,

EZ

X

= EH n-I a

infer

easy

~ hO(p,

L)

=

proper

and

> 0,

by

the

So

bundle

unless

[LMI , s i n c e is

with

it

a fiber

since

= Ln

by

= 1

we

in

F) M

we

H uL n - I

- 1)d

d(V,

~ h0(M,

M --~ V

Hence

LV,

EZ = HaZ

LH n-I a

+

g

we

H).

generated

be

E.

if

canonical

Thus

(b

in

LC

LM)

+ 2H u

Z

irreducible

.....

a

Let

= g(V,

g

= 22-nKWsn_2(~){W}

Now

x:

formally

example

of

conclude

g = g(M,

Kw

=

an

ampleness

only

f-l(x)

= 2 L n = 2d,

obtain

[S] S =

L = E + f H a comes

is

contained

= 22-nKW~-I{P~ we

(W,

is

contained

Thus

-

bundle

is

over

= f He

V.

holds is

for

(K W

=

in

not

= Ln-IH

have

2L M the

C

is

point,

Ln

L)

since

morphism

singularity

that

(V,

prove

f(Z)

show

Moreover,

cone

infer

that

equality

= 0.

we

to

Z

that

This

pair

curve

Then

a simple

contradicts We

order

The

We

= W.

a birational

projective

first

any

is

point.

the

= S

> i.

note

for

latter

b

E

there

This

if

~ O.

So

of

= 0.

in

M.

and

Macaulay.

this, So,

LC C

the

vertex

property

To

M

a normal

Cohen

since

show

to

the

and

desired

on

[E] E = -H.

contracting morphic

E

H)

= 0,

have

Indeed,

which

in

directly

EF = 1

for

arithmetic assumptions

= n

is

L)

check

that

E

general

the

b

=

Thus

impossible

A(V,

genus

L)

- 1 + d.

fact

any

of

hO(v,

when

= 0. is

fiber

the F

- 1. theorem

(1.2)

of

123

except

g < d.

Indeed,

when

But it does not have a l a d d e r n = 2, any m e m b e r of

some d i v i s o r

D

of d e g r e e

is not prime.

Thus there

member

Y

ILl,

(V, L)

constructed

of

find a s e q u e n c e till

one.

(1.4)

V

assume

g < d

in (1.2).

d = 2

and

W

(1.5) Then

1) any l a d d e r HO(vj,

3) g = 1 4) L

L) > 0 of

embedding

L

Then

Hq(v,

see

see

~ , ( E + p D)

for any general

of the same type as

W.

Therefore

dimension

two,

we can

but never

ILl, we need not

[F2] or

is the u n i q u e

[F4;

(3.4)].

base p o i n t

It is an o r d i n a r y

(V, L)

be a p o l a r i z e d

is regular,

w h i c h means,

is s u r j e c t l v e

V

generated

presented IF2;

means

HO(v,

L).

if

Theorem

of

double

[LI

and

p o i n t when

variety

as in

(1.2).

the r e s t r i c t i o n

map

j,

for each

ILl, L

tL) = 0

f o r any

q,

Corollary.

or

[F4;

IF4;

(5.2)

algebra L

& (5.3)].

~ta0H0(V,

is v e r y

ample

of d e g r e e

t

(3.8)].

Here, is

tL)

in this case.

two u n d e r the

is said to be q u a d r a t i c a l l y (V, L)

by

4.1]

by e q u a t i o n s

Let

Del Pezzo v a r i e t y

d • 3, and

d ~ 4.

In p a r t i c u l a r

is d e f i n e d

see

if

that the g r a d e d

Corollary.

For a proof,

(1.7)

V.

is simply

g i v e n by

(1.6)

until

of

for

d = L n ~ 2,

"simply generated"

If f u r t h e r m o r e

V

E + p D

and

(V, L)

For a proof,

by

x(E)

Let

is q u a d r a t i c a l l y

generated

variety

b > i.

is a hyperquadric.

if

and

n > 2,

at each base p o i n t of

p o i n t of

L) --~ H O ( v j _ I , L)

2) BslL I = @

When

section

For a proof,

(1.3),

Corollary.

g = g(V,

of

and

So the m e m b e r

is a p o l a r i z e d

is s m o o t h

is a s i n g u l a r

W = pl.

from a h y p e r p l a n e

d > 1

is of the form

is no ladder.

(Y, Ly)

In the e x a m p l e this

on

of s u b v a r i e t i e s

dimension If

d

ILMI

if

presented.

be as a b o v e and a s s u m e with

0 ~ q ~ n.

Thus,

in this case,

d ~ 3.

(V, L)

is a

(0.3). If in a d d i t i o n

d = 3, V

is a hypercubic.

If

124

d = 4, V

~2.

is a c o m p l e t e

Classification (2.0)

and

(2.1) d = 1,

L)

= 1. For

Pezzo

When

d = i,

is a Del

L) = A(V, Indeed,

of

ILl

L)

Pezzo

variety

with

= 1.

(1.2)

applies

if

d > 1.

is i r r e d u c i b l e

and reduced

each

is a l s o

If since

a ladder. of

(V, L),

assertion

n > 1, w e t a k e

(Vj,

When

double

degree

4

The

covering

L)

a Del

When

(1.7)

applies.

particular,

if

(1.5;

If is n o t

(2.5)

(V,

over

(V,

To see this, Then (2.6)

L = HV This

...,

n.

is a w e i g h t e d P(2,

1,

of d e g r e e

n = 1, the

genus

[M]

one.

(or see

branched

1).

$2]).

of d e g r e e

Moreover

along

In case

IF2;

hypersurface

...,

6

I).

In c a s e

argument

L) L)

similarly

there

a hypersurface

4

is a of

is v e r y

ample

a cone over But

this

is a l s o

and

trouble

by

that g(C,

(2.2).

another

The

always

second

= g(V,

c a n be a v o i d e d

d = 3 or 4, in

IF5].

its

structure

t h e case.

then

In is

Indeed:

the projective

cone

variety.

is i d e n t i f i e d H)

If

variety,

variety,

a De1Pezzo V

(1.5).

the t h e o r y

is n o t

is a D e 1 P e z z o

note

as

2).

d z 5, we c a n use

described.

(C, H)

I,

is of a r i t h m e t i c

is p r o v e d

precisely If

V

space

d ~ 3, L

V

on

hypersurface

L = f Op(1).

from

(2.4)

induction

f: V --~ p n

assertion

follows

2,

(V, L)

projective

first

assertion

the

F(3,

and use Mori's

d = 2,

such that

is a w e i g h t e d

space

since

a ladder

in the w e i g h t e d finite

we use

is w e l l - k n o w n

(2.3)

(V, L)

projective

F o r a proof,

IHI.

{Vj}

L)

variety.

in the w e i g h t e d

very

g(V,

D

So o n e o b t a i n s

(2.2)

then

(V,

a ladder.

member

any ladder

section

So

has

any general

Ln-ID

this

d = L n.

(V,

o f two h y p e r q u a d r l c s .

of Del P e z z o v a r i e t i e s

Throughout

dim V = n

intersection

L). by the

with So

a general

(1.6)

and

following

member

(0.3)

of

apply.

125

S u p p o s e that

Lemma.

d ~ 5

i n t e r s e c t i o n singularities. Proof.

Let

D

Then

assuming

that

vertex

is l o c a l

complete

complete Del

V

intersection.

Pezzo

variety

(2.7)

unless

If

table

only

F I ( N 1, H ( C 1 ) ) n = 3.

of

is a c o n e

ILl.

over

We will

D.

intersection,

(D,

be t h e c a s e

derive

Indeed, L D)

since

must

since

a

(D,

the

be a g l o b a l LD)

is a

d ~ 5.

Let

(V, L)

be as above.

is n o t normal, shows

or

a non-normal

Then

V

is normal

R,

degree we g e t

hO(v,

cone

d - 2.

terminology

Let

L)

[F5]). with

R

line

bundle

is the p u l l - b a c k

bundle

H u.

The unique of

of

V

is l o c a l

(C 1)

n = 2, o r

(V,

M

Thus member

~: ~ --~ W.

V

V

over

three

complete

with

n = 1,

p2(N1,

Q(CI))

precisely L)

with

the

with

structure

local

is a p l a n e V

times

is e m b e d d e d

with

of

complete

of d e g r e e

set o f v e r t i c e s P

that

along

R

any point

sections

is of t y p e

being

of

For

is,

[FS;

let

~

- 2)H~ • 0 • © • 0)

M = F1

of d e g r e e

one.

0p(1)

f: ~ -~ M D

of

Moreover

via is a

tautological

~ --~ W c P,

in the

be t h e

~ = PM((d

of

W

(6)]).

Then

The

of

of

(c)

which

will

where line be

p3-bundle.

IH a - f

(d - 2)H~I

~(D)

= R, D = M

v

the g e n e r a l i z e d

(see and

by

is a 4 - f o l d

hyperplane

W = M * R,

the

v

d - 2 (V

p = pd+l

P.

vertex

general

in

in

Moreover

W. on

and

of

be t h e b l o w i n g - u p

is the

divisor

R

curve

transform

by

or

describe

threefold

= d + 2

locus

proper

denoted

we will

Pezzo

Taking

of M

~

[F5],

W = v * V

a Veronese

over

with

of

The classifi-

n ~ 3.

Del

singular

the

(N 1)

applies.

singularities.

We h a v e The

(2.7)]

the singularity

F I ( N 1, Q ( C I ) )

Following

intersection

that

[FS;

if it is of t y p e

In p a r t i c u l a r

(2.8)

cone

V

there

intersection

on

has o n l y local c o m p l e t e

is not a cone.

member

cannot

of d e g r e e

V

n ~ 3.

cation

ILl.

V

This

Corollary.

Proof.

such

V

be a g e n e r a l

contradiction of

and that

is t h e × R

exceptional

a n d the

second

H~

126

projection

is t h e r e s t r i c t i o n

The proper

transform

member

of

of

M

and

on

D

is d e f i n e d

fQ:

Q --~ M

covering M

x plo

pl

IH a + f 2H~I. (a0:al:a2)

is a

with

~.

of

V

of

R

locus

~ * Ha. ]Q = H E + Ha"

and

see

(7)

(8)

•

•

~.

It is a

coordinates

then the divisor

(E0:fll)

Q = ~ n D

a 0 ~ 0 2 + a l E O E 1 + a 2 E I 2 = 0.

while

defined

on

homogeneous

suitably•

by the equation

branch

is a d i v i s o r

If w e c h o o s e

pl-bundle,

For proofs,

•

~

of

~Q: by

Q --~ R aI

2

is a f i n i t e

= 4 a 0 a 2.

where

H~

(c.l)•

(c.3)

This

[FS]

double

implies

is t h e p u l l - b a c k in

Hence

of

Q

0(1)

of

•

O

The map over

~ --~ V

is a

p2-bundle

exact

sequence

that

~(~)

cokernel

(2.9)

case

V

8,

Since

0, O) --~ ~ - ~ about

--~ O ( d

V.

More-

IH u + 2 H E I, w e h a v e a n

~ •

observation 0(-2)

0

on

Q, w e i n f e r

- 2, O,

From this we obtain

the normal (2.8)•

is n o r m a l

n ~ 6

When

~3.

above

i).

In particular

case

(resp.

0(1,

Unlike

When

The

to

M.

of

O,

M that

0) --~ 0(0,

~ = 0(d

such

- 2,

the

0, 0)

i,

I) =

H E ~ HE.

above

a cone.

By the

a n d is t h e n o r m a l i z a t i o n

--~ O ( d - 2, O,

of the composition

2)Hfl •

in the

over

0 --~ 0 ( - 2 )

= ~.

is i s o m o r p h i c (d -

is f i n i t e

V 6,

where

V

d ~ 8 (resp.

does not occur

5) if

and

there

is a s s u m e d

and singular,

is s m o o t h 5,

case below,

if

[FS;

6,

6,

not to be

(2.9)] 5) if

bound

of

d

a cone•

applies

unless

n = 2 (resp.

V

is

3, 4,

5).

d a 5.

d a 5, w e h a v e

n = 2 (resp.

is n o u p p e r

3, 4,

n ~ 6.

5,

6).

Moreover

See

[F3].

in

[B2],

d ~ 9

Applications Here we recall

necessary

is n o t nef. as i n

birational

improve

a few results

making

some

supplements.

(3.1)

Here,

and will

Let

X

Let [B2],

be

a smooth

¢: X --~ Y we will

and contracts

4-fold

whose

canonical

be a contraction

be interested an effective

bundle

of an extremal

in the case where

divisor

E

¢

to a point•

K = Kx ray. is

127

(3.2)

In t h e a b o v e

such that b. E

We have

and

a = b = 1

is i s o m o r p h i c

For

there

unless

to e i t h e r

(3.3)

sequel

In the

Pezzo

apply

the

which

is n o t

results

(3.4)

So,

by

E

[BI;

further

LE)

the

Lemma

(see

= 0.

2.3]

(1.2)

positive

integers

a,

singular)

hyperquadric.

2.4

(0.3).

instead

X

when

a = b = I. and

on

Moreover,

of Prop.

case

L

in

~ = 0,

[B2].

Then

(E,

Therefore

of Prop.

2.3

L E)

we can

in

[B2],

(1.3)).

is normal,

f r o m n o w on,

for s o m e

of the p r o o f

We u s e

true

line bundle

or a (possibly

we s t u d y

in S2.

always

When

[B2]. We

variety

A(E,

p3

see t h e b e g i n n i n g

is an a m p l e

= -bL E

[E]E

a proof,

is a Del

in

= -aL E

KIE

case

we c a n c l a s s i f y

we assume

assume

d ~ 5, s i n c e

applies

and

that

E

(E,

LE)

is n o t

otherwise

by the method

normal.

(2.2),

(2.3)

or

(1.7)

applies. Thus

(2.7)

particular

the

Let

~: ~ --~ X

exceptional the

proper

(2.8)

Thus

divisor

of

E

too.

[Z]Q

Let

~v

and

[-Z] Z

H~.

We h a v e

other

hand

Pic(~).

~

of

along

the

~Z

Suppose

E

type

E

on

is i s o m o r p h i c

of

~

X

along

R

~ = ~. bundle

R

of

Q

p2. let

Z

be the

of the blowing-up,

E.

to the

This

abstract

applies

to

Z n ~ = Q ~ P ~1 × F1a"

Furthermore of

In

to

and

is i s o m o r p h i c

subscheme

(2.8).

in

~,

R

in

which

is

[D]Q

=

(2.8).

H~Ha[Q~

this

is of the

By the universality

bundle

is t h e t a u t o l o g i c a l

So

- a).

of

R.

be t h e c o n o r m a l

b = 1 in c a s e

0R(3

over

is the n o r m a l in

R

locus

Therefore

H a + (3 - d ) H ~

LE)

be the b l o w - u p

transform

blowing-up in

singular

(E,

of

line b u n d l e ,

= - ( H ° + (3 - d ) H ~ ) ( H

is e q u a l = -bHa

to

+ 2H¢

which

Then will

Z = FR(~V) be d e n o t e d

+ H E) = d - 4.

H~Ha[~]z[Z) , while when

X.

On the

~ = ~ E - 2Z

[E]R = 0 R ( - b ) .

by

Actually

in we have

(3.3). that

So

KXl E = - a L E .

H~2H

{Z~ = 3 - a

H ¢ H a ( 2 H ~ - b H a) = 6 - 2a - b.

Then and

KX[ R = ~ R ( - a )

and

d - 4 = H~Ha[~][Z}

Cl(~V)

=

=

128

In our p a r t i c u l a r case we have Moreover

~ ~ ~ M ( 5 H ~ • H~ • H~)

a = b = 1

d = 7.

by (2.8).

It is u n c e r t a i n w h e t h e r this case

Remark.

and h e n c e

d = 7

does r e a l l y

o c c u r or not. (3.5) Any way,

thus we have shown that

This i m p r o v e s u p o n Prop.

d ~ 8

in case (3.3).

2.4 in [B2].

Bibllography [BI] M. Beltrametti,

On d-folds whose canonical b u n d l e is not

n u m e r i c a l l y effective,

a c c o r d i n g to Mori and Kawamata,

to appear in Annali Mat. [B2] M. Beltrametti,

Pura e Appl.

C o n t r a c t i o n s of n o n - n u m e r i c a l l y e f f e c t i v e extremal

rays in d i m e n s i o n 4, in Proc. Conf. on Alg. Geom. pp. 24-37,

T e u b n e r Text zur Math.

[FI] T. FuJita, zero,

On the structure of p o l a r i z e d v a r i e t i e s w i t h d-genera

J. Fac. Sci. Univ.

[F2] T. FuJita,

of Tokyo,

22 (1975),

103-115.

D e f i n i n g e q u a t i o n s for c e r t a i n types of p o l a r i z e d

varieties, Iwanami,

B e r l i n 1985,

92, 1987.

in C o m p l e x A n a l y s i s and A l g e b r a i c Geometry,

Tokyo,

pp.165-173,

1977.

[F3] T. Fujita, On the s t r u c t u r e of p o l a r i z e d m a n i f o l d s of total d e f i c i e n c y one,

I, II and III, J. Math.

709-725,

33 (1981), 415-434 & 36 (1984),

ibid.,

Soc. J a p a n 32 (1980), 75-89.

[F4] T. Fujita, On p o l a r i z e d v a r i e t i e s of small d-genera, J.

[F5] T. FuJita,

[M]

T 6 h o k u Math.

34 (1982), 319-341. P r o j e c t i v e v a r i e t i e s of A-genus one,

in A l g e b r a i c and

Topological Theories --

to the m e m o r y of Dr. T a k e h i k o MIYATA,

pp.

1985.

149-175,

Kinokuniya,

S. Mori, On a g e n e r a l i z a t i o n of c o m p l e t e intersections, K y o t o Univ.

Note.

15 (1975),

J. Math.

619-646.

Here I w o u l d llke to correct an error in [F5,

(2.11)].

There I c l a i m e d "the p o s s i b l e type of s i n g u l a r i t i e s are s u b g r a p h s of Dynkin diagram

... ", but this is not true.

I s h o u l d have w r i t t e n

"the p o s s i b l e type of s i n g u l a r i t i e s are graphs w h o s e c o r r e s p o n d i n g root systems are s u b s y s t e m s of the root system of the D y n k i n d i a g r a m

... ".

I w o u l d like to thank Dr. T. Urabe who p o i n t e d out this mistake.

ABELIAN

SURFACES

IN P R O D U C T S

OF P R O J E C T I V E

SPACES.

Klaus Hulek Mathematisches Institut, Universit~t Bayreuth P o s t f a c h 10 12 51, D - 8 5 8 0 B a y r e u t h F e d e r a l R e p u b l i c of G e r m a n y

0. I n t r o d u c t i o n It is w e l l degree

ten.

problem the

known

[HM],

[R],

existence

problem there exist. not

of

falls is

only

This

faces

in For

more

subtle

future does

such

In this

brief

surfaces

in ~2x

Here

class

mostly

surface

of

parts.

possible

from

a consequence

really

than

not

seem

exists,

the

pursue

point.

not

abelian

note and show

that

for

which

an

abelian

that

Although

our

results

are

to

quite

cases

surface

can

formula, abelian of

but surcubic

The

proof

which

are

a lot

of

this

note.

to

it

back

easy

this

both

products

methods

come

Again

in

only

candidate.

harder

investigate

~3"

considerations

hope

in the

the

namely

needs

but

to be a r e f e r e n c e

~ix

self-intersection

elementary

here,

we s h a l l

P2

follows

necessarily

a much

shall

however,

rather this

has is

we

the

it

in ~¢

surfaces

P x ? are the o b v i o u s ones, 2 2 PI x ~3 we e x h i b i t a p o s s i b l e

a surface

shall

two

follows

such

We

[HL].

abelian

one

As

every

the e x i s t e n c e

ILl,

into

entirely.

curves.

that

To p r o v e

to

at

prove,

that

some there

literature.

i. P r e l i m i n a r i e s If

Z = XxY

is a p r o d u c t

we d e n o t e

the c a n o n i c a l

projections

by p

and q r e s p e c t i v e l y : Z = XxY

oj-, X

If

Z

and

•

In p a r t i c u l a r ,

are

Y

line

denote

resp.

h2

the

on X resp.

if X = Yk and Y = ~ m we O(a,b)

We

bundles

classes

:= O~k(a)

of

~

Y we set

set

O~m(b).

OZ(I,0) , resp.

OZ(0,1)

in

H 2 (Z,Z)

by

hI

130

Lemma not

I.I:

Let

contain

curve

an

and

X = C

Proof.First the

Case

i:

g(C)

X

~

C

g(C)

= 0.

x~

a,b

Since

X

is

By

on

×

~2

an

does

elliptic

curve.

assertion and

C is

is

this

obvious

would

since

imply

X.

and

1

= O(a,b)

the

adjunction

formula

= Ox(a-2'b-3)"

abelian

(ah1+bh2)

the

C

2

> 0. ~X

product

cubic

surjective

1-form

C = ~ (X)

a smooth

Then be

the

unless

is

2.

must

Then

i some

z

Then

X D

a non-constant

O~ for

a curve.

where

assume

of

be

surface

x D

projection

existence

C

abelian

~X

= OX

and

this

((a-2)h1+(b-3)h2)hi

implies

= 0

(i = 1,2)

i .e. b(b-3) a(b-3)+b(a-2) It

follows

that

b = 0

an

exact

~

(i = i) (i = 2).

3

and

a = 2.

h I (0 x)

2:

and

the

K~nneth

g(C)

= 0,

we

= I.

Here

can

= h I (0)

from

= 0.

our

argument = Pic

C

is x Pic

(X)

= Z

s,

O(b)

2 some

get

formula

write OCx ~

for

we

a contradiction.

P i c ( C x ? 2) i.e.

other

~ HI(OX ) ~ H2(0(-2,-3)).

h 2 (0(-2,-3))

Case

the

sequence

Serre-duality

Hence

On

-~ O ~ O x ~ 0

0(-2,3)

HI(O) By

= 0 = 0

b

> 0 and

Z ~ Pic

~Cx~2

= 0C

C. ~

Since 0(-3)

very ~2

similar.

By

[Ha,p.292]

the

131

the

adjunction

formula ~X

Let

a = deg

Z.

ab i.e.

b = 3,

gives

= £

then b(b-3)

= 0

+ a(b-3)

= 0

a = 0.

follows

that

Finally

we

For

any

before

we

find

[]

0(3))

recall

and

the

= H°(Z) the

® H°(O(3))

assertion

is

self-intersection

embedding

i

:

X

~

Z

of

then

obvious.

formula

from

codimension

d

[F,p.103] with

.

normal

NX/Z:

bundle

i for

as

Since

Z = OC

regular

O(b-3)[X.

arguing

H°(Z it

[]

all

i,[~]

= Cd(Nx/Z)

n

[a]

~ G A,(X).

In p a r t i c u l a r ,

if X is a s m o o t h s u r f a c e 2 = c2(Nx/Z) n Ix] .

in

a 4-manifold

Z then

IX]

2.

Abelian

surfaces

In

section

this

Proposition X = C

x D

Proof. of

We

X

is

2.1:

we

C

shall

integers

D

apply

fact

X

i_n_n 72 x ?2

smooth

the

self-intersection

+ Sh~

a 0.

From

a,H,7

x? 2

the

surface are

= ah~

0 ~ T x ~ T?

and

2 prove

is

formula.

that

Tx

is

+ ~hlh 2 the

normal

IX ~ N X / ? 2

trivial

bundle

×? 2

one

of

the

form

cubics.

form

[X] with

x ?

abelian

and

first

the

2

shall

Every

where

of

in ?

finds

~ 0 2

sequence

The

class

132

C(Nx/t2~2)

= c(T~2~21x)

Hence c 2 ( N x / ~ 2 x ~ 2)

=

3 a + 3B + 97.

Since [X] 2

=

the s e l f - i n t e r s e c t i o n

(I)

Claim

i:

(ii)

B = 0

projection

+

formula

3 a + 3B + 97

(i)

a = 0 or

Clearly

72

or

2a~ implies

=

7

2

+ 2a8.

a z 6

B z 6.

it

is

onto

enough

the

to

second

prove

factor

(i).

gives

Assume

that

a surjective

a

>

0.

Then

map

qlx : X ~ P2 of d e g r e e negative

a.

Since

an abelian

self-intersection

Nakai-Moishezon

criterion

it

surface

does not

follows

implies

that

contain

q~X is

curves

finite.

with

Hence

that

ex(0,1) = (qlx)* e~ (I) 2 is a m p l e .

By K o d a i r a

and Riemann-Roch

vanishing

h°(OX(0,1))

Since

h1(OX(0,1))

= h2(OX(0,1))

= 0

gives

h°(OX(0,1))

=

[OX(0,1)]2

a h°(Op

(i))

=

= 3

[ X ] h 2 = ~.

we

find

a z 6.

2 Claim

2:

Assume

a = 0 a,S

or

B = 0.

> 0.

From

72-97 By Claim

1 we have

the

other

hand

= a(3-B)

a , B z 6,

72-9T On

(1) w e g e t + #(3-a). hence

• - 36

72-97

z

8, 4

for

all

7

~

R,

a

the

133

contradiction. In case

order

to

prove

B = 0

being

factor

gives

a map

where

D

the

analogous.

X ~D

is

a

normalization

c

map.

Let

set

of

from

the

singularities

X be

X which

its

~

In

we

this

singular)

D

can

case

now

assume

projection

a

onto

=

0

the

the

second

~

D

be

the

let

X

be

the

2

(possibly

open

Let

proposition

be

the

over

D

O

lies of

D we

Zariski-closure.

curve.

smooth .

o

Since

can

Let

part idxv

consider

Then

we

have

v

of

X

D

is

: D and

an

O

isomorphism

away

a subset

?

to b e

o a commutative

of

2

x D.

diagram

2 g

~ idxu

X By

construction

follows

from

lemma

I.i

where

C is

and

3.

g [S,

it

X = C

~ is

5,

that

a smooth

Abelian

x D. 2 finite and

theorem

follows

× D as

?

D is

cubic.

surfaces

we

?i

class x 73

in

3.1: is

is

Proof.

of

But

?

a,B ~ > 0

the

z

0

of

the

that

finite. > 0.

as

Since the

already

abelian

surface

is

smooth

it

isomorphism.

and

D must

X.

map

in ?

that

have

X=

C

been

By

x

smooth

8hih2+~h~._

X

to

Every

in

?i

x

abelian

73

then

surface

in

a product.

be

q

induces

of

a

and

B is

as

follows:

a map

~ 73

Then

the 3

an

interpretation

e = deg Note

curve

X

an

= ~ h i h 2 + Sh~.

projection

a surface

=

isogenous

class

and

the

is [X]

:-- ql x x ~ onto

then

is

3

X

form

[X]

If

elliptic

Since g

x ?

I_ff the

necessarily

Let

Then

an

that

prove

Proposition its

115]

claimed.

I

Here

birational.

p.

does

q'deg

proof not

of

X. proposition

contain

abelian

2.1

the

map

surfaces

it

X

~

X

follows

must that

be B

134

is s u r j e c t i v e As

and

before

we

the

fibres

want

to

are

make

space

use

curves

of

the

of d e g r e e

B.

self-intersection

formula

From C ( N x / ? Ix~3)

=

c(Tp1 × T~31X)

=

(l+2hl)(l+4h2+6h~+4h:

) I[X]

we get c2(Nx/~

x~ I

) =

(8hlh2+6h~) "[X]

3

=

6a

+

8S.

Since [ X] 2 =

we

2a/3

find 2a~

(2) Since

Claim

B > 0

i:

Since

By l e m m a Since

q

this

6a + 8B.

=

also

shows

that

a > 0.

a z 8. a > 0

I°I

the p r o j e c t i o n

X

cannot

is f i n i t e

q

induces

be a plane,

the line

OX(0,1)

=

hence

a finite

X spans

map

~3"

bundle q e~

(1) 3

is

ample.

By K o d a i r a

vanishing

h°(OX(0'l))

=

and

Riemann-Roch

![OX(0'I)]22

this

= ~. ih22 [X]

shows

= e2.

Since h°(OX(0,1))

z

h °

(e?

(i))

= 4

3

we

find

Claim

2:

a z 8.

4 s ~ ~ 6.

We can r e w r i t e (3)

Since

a z 8 we get

(2)

as a(6-~)

=

B • 6. A l s o

B(a-8). from

(3)

resp.

(2) we

find

that

B ~ 2

135

implies

a < 0 and B = 3 leads

This

leaves

us w i t h

the

(a,B) We

have

to

consider

or

5.

Since fibre

following

possibilities

(16,4),(10,5)

the

first

Hence

or

two

4 s ~ s 6.

for ~ a n d

(8,6).

possibilities.

To

do

the f i b r e

every Xt

X t is a s p a c e

morphism

is

curve

of d e g r e e

we

B = 4

from ?

reduced

to X is c o n s t a n t it f o l l o w s t h a t i irreducible. Moreover for a g e n e r a l

and

~

= O X t b y the a d j u n c t i o n formula, i.e. p is Xt f i b r a t i o n . W e h a v e a l r e a d y s e e n t h a t this f i b r a t i o n has fibres.

By

it can,

therefore,

canonical

this

fibration

t ~ ~I

every fibre

exclude

the

For each

=

to a c o n t r a d i c t i o n .

Kodaira's

bundle

classification

not

have

formula

any

[BPV,

_, ~X = p O~

of

singular

singular Corollary

(-2)

fibres

fibres

at

(12.3),

p.162]

an

elliptic

no

multiple

(BPV,

all.

But

p.151]

then

the

implies

= Ox

I a contradiction. Hence to

show

look

As

the o n l y that

again

plane

remaining is

at the

before

bundle.

X

the

It

is

cubics.

Poincare's

possibility

isogenous

to

is

(e,B)

a product.

=

(8,6).

In o r d e r

to

It r e m a i n s see

this

we

fibration

general either In

any

theorem

fibre an

case

on

must

be

elliptic X

smooth

sextic contains

complete

with

curve an

reducibility

trivial

or

the

elliptic X

is

canonical

union

of

curve

and

isogenous

two by

to

a

product. 3.2

Remark:

general

fibre

As

we

of

the p r o j e c t i o n

have

seen

in

the

proof

of

proposition

of

two p l a n e

3.1

the

: X ~ P is e i t h e r first I n the

case

an e l l i p t i c can

second

fibration

again case

we

i sextic

be

curve

excluded

can use

or a u n i o n using

Stein

the

canonical

factorisation

cubics.

bundle

to g e t

The

formula.

an e l l i p t i c

136

over

an elliptic

par£icular

curve

X is a f i b r e

3.3

Problem:

IX]

= 8 h l h 2 + 6h~

We want a good ?3"

curve. with

Do

Let

2Qo=

for

Po b e

this

of

natural

2 resp. X

The

E be

a point

The

~2:

degree

in 71

with

smooth

x ?3

plane

cubics.

can

which be

seems

embedded

to b e in ? I x

an elliptic

order

2 and

let

Qo

be

a point

of

order

projections := E / < P o >

E ~ E2

:= E / < Q o >

We

:= E 1 x

which

set

E2.

map

is a n e m b e d d i n g

(-i,=2)

: E ~ X

kern

I n ker

since

= deg

E'EI

E.E 2 = deg

~2 =

{0}.

Moreover

~2 = 4 "I

= 2.

set H 2 := E + E 1 .

Note

that H22 = 8 ,

The

following

(I) T h e

linear

H 2 . E 2 = 3.

statements system

are

easy

IH2~ is b a s e

to c h e c k : point

free

and defines

a map

: X ~ ?3" (2)

Under

mapped

the

map

the

a map

translates

to s m o o t h

of

plane

the

of d e g r e e

El 2.

~ 71 This

induces

elliptic

cubics.

let P':

be

q

isomorphically

Finally

In

with

an e x a m p l e

surface

E ~ EI

4.

=

We

X

section

abelian

let

I:

have

all

C.

surfaces

an

purpose

Po"

over

are

exist?

to c o n c l u d e

this

fibres

bundle

abelian

candidate

For

C whose

a map

curves

E 2 are

4

137

p

3.4 Problem:

: X-~

I"

Can one choose p such that the map (p,q)

: X ~ P1x P3

is an embedding? It

is

easy

to

check

that

the

construction

is

such

that

numerical conditions are fulfilled.

Acknowledgement:

The

author

would

like

to

thank

the

DFG

for

support under grant HU 337/2-1.

References

[ BPV]

Barth,W., surfaces.

Peters,C., Van de Ven,A.: Springer Verlag 1984.

[ F]

Fulton, W.:

[ H]

Hartshorne,

[ HM]

Horrocks, G., Mumford, D.: A rank 2 bundle on ~ 15,000 symmetries. T o p o l o g y 12, 63-81 (1973).

[ HL]

Hulek,K.,

I n t e r s e c t i o n theory. R.:

J. Reine Angew. Math.

complex

Springer V e r l a g 1984.

A l g e b r a i c Geometry,

Lange,H.:

Compact

Springer Verlag 1977.

Examples of abelian surfaces in ~ . 363,

201-216

(1985).

[L]

Lange, H.: Embeddings of J a c o b i a n surfaces in Y¢. J. Reine Angew. Math. 372, 71-86 (1986).

[ R]

Ramanan, S.: Ample divisors on abelian surfaces. London Math. Soc. 51, 231-245 (1985)

IS]

Shafarevich, Verlag 1977.

I.R.:

with

Basic

algebraic

geometry.

Proc. Springer

all

ERBEDBED PROJECTIVE VARIETIES OF SMALL I N V A R I A N T S . I I I P a l t l n Ionescu U n i v e r s i t y o f Bucharest~ Department o f H a t h e m a t i c s , s t r . Academiel 14, 70109 Bucharest , ROMANIA Introduction S e v e r a l y e a r s ago we have s t a r t e d cation

of

values of

their

numerlcal

[8],

[10],

Ill],

[9],

classification

the

Invariants cessary.

tool

of our

found o u t

the c l a s s i f i c a t i o n

first

task

is

investigated for

d~6 t h e l i s t

the e x i s t e n c e o f

following

a maximal

g~7, tL~5 ( c f . [ 1 4 ] ) . llst;

series,

Thus,

the

from the

settles

list

types was l e f t

given

(see the t a b l e

in [ 1 1 ] in [ 1 2 ]

for for

varieties

d=7 t u r n s

vlng

the e x i s t e n c e of

Finally, settled

let

of

us p o i n t

by o t h e r

existence [21])

systematically

two t y p e s o f

surface with

J,

two p a r t s .

really

occurs.

for

Thus, [11])

d=8 the u n d e c i d e d is

the

last

in

these s i t u a t i o n s .

to be e f f e c t i v e ,

while

have to be e x c l u d e d

c o m p l e t e d the c l a s s i f i c a t i o n

this

The

In c o n t r a s t

to I l l ]

of whe-

time we t o o k the o p p o r t u n i t y for

pro-

embedded m a n i f o l d s . out

t h o s e cases w h i c h ,

in

the meantime, were

n a m e l y : A.Buium ( c f . [ 2 ] )

surface of

surfaces

recently

out types

up to d e g r e e 8.

proved the e x i s t e n c e

from [ 1 5 ] ,

to

t h e few g e n e r a l methods a v a i l a b l e

authors,

of a certain

[25],

For d=7 (see [ 1 0 ] ,

open, w h i l e

d=8 t h r e e

re we used m a i n l y ad hoc methods, to present

[9],

[24], inherent

each case has to be

it

p r o b l e m in a l l

b e l o w ) , We have thus

smooth p r o j e c t l v e

of other

On t h e o t h e r

into

This paper, which

the e x i s t e n c e

given

llst

splits

secondly,

cf.[8]p

cases were more numerous ( c f . [ 3 2 ] ) . this

(cf.

llm'itations,

to d e c i d e w h e t h e r or n o t

four

in

was the a d J u n c t ] o n m a p p i n g ,

understood completely

[13])

was e f f e c t i v e ,

enough (see

genus g and the A-genus A) became ne-

problem naturally

to o b t a i n in o r d e r

the

interested

the d e g r e e d, c o n s i d e r a t i o n

the

t h e method e m p l o y e d : d~8 ( c f . hand,

a classifi-

following

assumed t o be s m a l l

investigation

were r e c e n t l y

We. g r a d u a l l y

( o v e r ¢)

A l t h o u g h we were p r i m a r i l y

a c c o r d i n g to

The b a s i c

varieties

invariants,

[12]).

(namely t h e s e c t i o n a l

whose p r o p e r t i e s [14]).

a program a i m i n g at

embedded smooth p r o j e c t i v e

d e g r e 8 in p 5 ;

of a certain

3-fold

o f d e g r e e 8 in p 4

first

C. Okonek ( [ 2 0 ] ,

of

degree 7 in p 5 and

finally,

as we can j u d g e

A l e x a n d e r showed the e x i s t e n c e o f

d=8, g=5 in p 4

a seemingly subtle

proved the

case.

a rational

139 The

reader may

consult

~4~,

~11i ,

~4j , ~ I ~

for further

references

on

the subject. AE~nowled~Ement. sation.

Spec|al

on v e c t o r

thanks

as

that

in

the

thus,

the

are due

Basically

first

the

a (smooth)

-

the

- ~

-

is

sectional

- TX(O ~) - ~X o r Pg

genus of

X.

is

the

dual

of

the

OX(K)

= h°(~X

tangent

is

the

valence) -

If

YCX

-

I

denotes

and

~

tn

~

. Let

linearly

section of

over

normal

codim

X=s, of

us

and n o t a -

recall

was r e p l a c e d

notations:

from

~i~

by " s c r o l l " =

a curve".

For

and n o n - d e g e n e r a t e

degree of

clo-

X=d.

X.

X.

is

of

the of

bundle

(cotangent)

E.

bundle

bundle

denotes

of

of

X.

X.

the

sheaf

of

o z . L is

equivalence

Oiy denotes ideals

variety table

submanifolds

blowing-up

linear

(resp.numerical

equi-

dlvlsors.

a subvarlety,

The f o l l o w i n g degenerate)

a vector

canonical

- A smooth projective

locus

discussions

)

- O1=O 2 ( r e s p . Dl~O 2)

X is

helpful

conver-

)

- Ev d e n o t e s

-

useful

same d e f i n i t i o n s

the

dim X=r,

hyperplane

the

became " s c r o l l

connected,

the~-genus

q=hl(Ox

for some

for many

~

some o f

sed s u b v a r i e t y ; - H ts

~il~ ~j u s e d

from

we r e c a l l a smooth,

we e m p l o y

fibratlon

term "scroll"

n ~£ i s

g is

to C . B ~ n i c ~

to I. C o a n d ~

two p a r t s

term "linear

convenience -X

indebted

bundles.

Conventions. tions

I am

of

is

Y with in F

of

2

•

called the

degree

center

of

a divisor

(class).

Y.

also

presents XCP;

a line

of

restriction

Z;

list 8.

a manlfold. of

(llnearlv

Notation

E denotes

the

normal,

Oz:X-Y

means

exceptional

nonthat

140

r I 2-8

Abstract structure of X pJ

H or Ox(H) 0(8)

s c r o l 1 over ~pt

_pl

x

P~=]

0(2,2) o* (3L) -E O(z)

-oi~ : X--~P 2

~3 g=2

-Op

, P4:X-'Ip2

O*(4L)-2Eo-EI-

Om,,o

-

,.,-E4

e~O I~ Co+4F e=2 H~-Co+5F

scroll over an e l l i p t i c

cu rye XCP]x Q3 as a hyperplane ,,,

section, Q3(P4 the hyperquadric

~1 x

Q3.

. . . . . . . .

Segre embedding

g=3 -

OPl . . . . .

P8:X-Fe

e~3

o*(He)-EI-,..-E

8

- Op],...,P8=X--p2

He=2Co+(4+e)F o*(4Lt-EI-,, .oE 8

- f : x - - P 2 double c o v e r i n g

H=-2K

- scroll

over an e l l i p t i c

curve

. p1xp3 ~ Q6, Q6c P 7

a

hyperquadr ic - f : X ~ Z C p1 x P3C e 7 d o u b l e c o v e r i n g , Z a hyp e r p l a n e s e c t i o n of plxp3 - P ( E ) , E r a n k - 2 v e c t o r bundle on p 2 , g i v e n by

tautological

0-0 2-E'ICpl,.q4!, F° 8 f:X-~lplxp 3 double covering, discrimlnant dTvisor DelO(2,2) l scrol I, e = -2, q=2 C x ~l C C p 2 curve of degree 4 geometrically rured e l l i p t i c surface, e=-I

!

....

g=4

f*O(J,3)

Segre embedding H~-2C +F o

I

141

Abstract structure of X

8 or Ox(H)

_

°P],...,PlO :X_q2c p3

o* (38q)-E 1-...-E 10

:X-SOp 3 "°P1,..,,P h S cubic surface

o* (2Hs)-E l - . . . -E/4

-°P l . . . . . P]2:X-Fe, e:~4

o*(He)-Ez-...-EI2 He=2_2.Co+(5+e)F

P~(E), E rank-2 vector bundle on the quadrlc Q, given by O.--OQ.-~E-,..I (3,3) {P],,,.,P]0

I 2

>3

| 2

1 2

}

-0

g=5 KJ surface complete intersection

(2,2,2)

g=5 : X.,4p2 OPo,...,P]o

o*(7L)-Eo-2EI-o..-2E 1

g=6 -Op:X-S, SK) surface "OPI . . . . 'PIG

:X-.ip2

o*(6L)-EI-...-EI2-

g=7 "----~], • X minimal,elliptic, IKI q=0 flH+KI:X-'tP! with fibres complete

l

_>1 _>1

tautological

Intersections (2,2) complete intersections

(214)

I

,

-2E13

...-

16

142 I.

The R u m f o r d - F u j l t a

The f o l l o w l n g lizes

the f a m i l i a r

g r e e ~2g+1

result fact

criterion due to Mumford [ ] 8 ]

and F u j t t a

Theorem A ( R u m f o r d - F u J I , a ) . variety

A2r,

we show t h a t

[I])

d-n+t.

gives

Conversely~ for

any i n t e g e r s

b u n d l e E on C h a v i n g

c](F)=a.

Assume

follows

Assume we have a l r e a d y

sequence o f

w i l t h gt. 7 d~2r.

Prop.3.1]. A=r

so n e c e s s a r i l y

) b u n d l e F on C w i t h

some n o n - s p l i t

o f such a ~ s c r o l l

and B a r t h ~ s

an ample v e c t o r

i_nn F 2 d - I

r and de~ree d o v e r an

to t h e n u m e r l c a l

is a contradiction, last

dimension

the e x i s t e n c e

A=r

In Pn t w i t h

there

Cl(E)=a.

A=r;

of

Is e q u i v a l e n t

The e q u a l i t y

This

a,b>O,

d,r

we w o u l d g e t

q(X)=O.

ple

the Segre embedding o f P l x j p d - I

Proposition.

v6n

If

of

rank b and f o u n d an am-

By Rlemann-Roch t h e r e

Is

the form

O--Oc-E-F~-O. By a r e s u l t ly,

due to G l e s e k e r

c I (E)=a,

ple vector

rk(E)=b.

(see [ 5 ] ,

Th.2.2),

The above argument

b u n d l e E l on C w i t h

rk(E1)=r,

E i s ample and,

shows t h a t

we may f i n d

cl(E])=d-2r

obviousan am-

and an e x a c t

se-

quence 0--0 c-E l - - F - 0 . Take some L e P l c ( C )

with

ve c l ( E ) = d

be enough to prove

it

will

For any two p o i n t s ~) O c ( - p - Q ) ) = H I ( E I

c](L)=2.

Let E = E ] ( ~ L ,

P,Q¢C we l e t ~

that

(5.3)

the

V

L1)=H°(E~)

L]}=O s i n c e

=0 f o r

simple

Lemma ( c f . [ 2 ]

m_oorphlsm o n t o vertible

following

is v e r y ample on X,

LI=L(~ 0c(-P-Q)~Pic°(C),

g r e e O. Thus we have H|(Ox(1)(~m: O c ( - P - Q ) ) = O sequence o f

X,dP(E) ~ C. S i n c e we ha-

0X(1)

We g e t H I ( E ( ~

E 1 is ample and L 1 has deand t h e

result

is a con-

lemma.

Lemma 3 . 4 ) ,

Let X be a m a n i f o l d

and K:X--C

some smooth c u r v e .

Le__t X p = ~ ' t ( P ) ~ P¢C. I f R I~ an I n RIXp _is v e r y ample and H I ( R ( ~ ) O x ( - X p - X ~ ) ) =

s h e a f on X such t h a t

any P,QcC, t h e n R i s v e r y ample. 6.

The e f f e c t i v e

(6,1)

Theorem,

is effective, For a p r o o f , tence of

the four

llst

of m a n l f o t d s

The l l s t

apply types

of m~nifolds

(5,2), left

of

(l,3),

undecided

degree

7 an__dd 8

of de~re~ 7 qiven

(1,8) in Ill],

and ( 2 . 2 )

In [11]

to show the e x i s -

152 (6.2.) fibratlon with

Consider

over P]

having

dim X-2 a r e g o t

ded S e g r e bie.

Into ~5.

We s t a r t

~:X-C the

of

F for

a useful

dimension

following

r,

of

d-8,

that

of

a fibre

~ and O y ( L ) - : O y ( 1 ) .

sections.

Indeed,

Is

for

(4p2)

Examples

on p ] x ~ 2,

dim X~3 a r e n o t

hyperquadric

C, L e t

of ~,

us f i r s t

the exact

embedpossl-

fibratlons introduce

E-~.(Ox(H)),

Then we c l a i m

consider

a hyperquadrlc

(cf,[12]).

type

the c a s e s

remark valid

Q for

H)

g-~.,4

divisors

o v e r any b a s e c u r v e

notations:

a fibre

(7)

Invariants

by t a k i n g

N e x t we p r o v e

with

ned by g l o b a l

Y,.IP(E) ~ C,

that

Oy(L)

Is

span-

sequence

0--O x (H-Q)--0 X (H)-OQ ( H ) - O .

Since is

now t h e c a s e when i X ,

Q is

a hyperquadrlc

surjective,

i n iP r ,

h°(Ox(H-O.))-h°(Ox(H))-r-l.

It

h°(Oy(L-F))..h°(Oy(L))-r -1 0-'Oy(L-F)-Oy(L)~OF(L)"0 for

any f i b r e

Now r e t u r n

the

restriction

map H ° ( O x ( H ) ) - - H ° ( O Q ( H ) )

so we g e t follows

that sequence

and t h e e x a c t

shows

that

0y(L)

is

spanned,

since

OF(L). i s

so

F.

to our

c a s e when dim X " 3 ,

d=Zg..8,

l

C=4P

We F i n d H I ( O x ( H ) ) - O a n d , u s i n g ( 7 ) , H I ( O x ( H - Q ) ) = O . Thus we must h a v e 4 4 E= ( ~ ) O ( a l ) , w i t h a . 2 0 . M o r e o v e r , we g e t X ¢ I 2 L + 2 F I and e l ( E ) ]: a l - 3 , i=,1 i i-l S i n c e a t l e a s t o n e a i i s z e r o , t h e map T t L t : Y - - P 6 maps Y o n t o a c o n e o f degree

3.Thus

sections nal

and u s i n g

curve

tradicts

of the

(6.3) surface

of

curve.

i)

Bertini's

t h e o r e m we F i n d

X lies

such a c o n e .

on a t w o - d i m e n s i o n a l

following

Passing that

to hyperplane

some smooth s e c t i o -

cone o f

d e g r e e 3.

This

Lemma. L e t C be a smooth c u r v e o f . d e g r e e d c o n t a i n e d degree

con-

lemma.

b which

Is a c o n e w l t h

l_f6 P~C,

b divides

d,

b divides

d-l.

PcC,

blow-up

geometrically (6.4)

P and compute

ruled

Consider

vertex

P over

(6.2)

we f i n d

intersection

in a

some smooth

in p4 o f

fibrations

(cf[12]).

they

numbers on t h e

resul-

surface.

now s u r f a c e s

are hyperquadric We show t h a t

in

in

tf

For a proof,

which

contained

Then: il)

ting

X is

cannot

exist.

(L3)-4,X¢I2L+~*BI

for

Using some

degree the

8 with

notations

degree-zero

g-5

and q-1

introduced divisor

on t h e

153

elllptic

c u r v e C, As remarked In

Since

(L3)-4,

d e g r e e 4, bility

either

(6.2)

we have a morphism T I L I

T I L I maps Y b l r a t l o n a l l y

say Z, o r X Is c o n t a i n e d

:y-dp 4 .

onto e hypersurface of

in a h y p e r q u a d r l c , T h i s

last

Dossi-

is a b s u r d s i n c e o t h e r w i s e X w o u l d be a c o m p l e t e i n t e r s e c t i o n .

Next we show t h a t to p l a n e s ,

it

Z Is a cone,

is enough to f i n d

a r e no such c u r v e s , we f l n d

Indeed,

Now, i f

the f i b r e s

a curve contracted

L is ample and i t

hl(0x(H))=].

since

follows

F a r e mapped

by T I L t .

If

there

H](E)=H](0x(H))=0.

T i s a c u r v e such t h a t

(L.T)=0.

But

we have

T ~ X=~ (because LIx=H is v e r y ample we c a n n o t have T C X ) . As a c o n s e q u e n c e , no d i v i s o r say D~GL+~F is c o n t r a c t e d ,

on Y is c o n t r a c t e d It

by F I L I ,

Indeed,

if,

follows:

0=(DoL2)=4~ + p and,

since D~X=~,

ved t h a t FIL l of

If

(D.X,~=2~=0,

StILl

is a f i n i t e ,

birattonal

d e g r e e 4 in p 3 .

using

of

e~-I

[6]

infinitely

implies

secting of

many p a i r s

in a l i n e .

planes,

its

if

ly

ruled

clearly

elliptic absurd.

was f i r s t

of

that

necessarily

fibres

Taking

with its

pullback

C o n s i d e r now the r a t i o n a l

existence

of

Now, and u s i n g (6.4),

(5.2),

(6.6) given

type of surfaces.

in

and

(1.3), (3.2)

that

(1.2),

followinq

such a p a i r

auadric,

besi-

a geometricalwhich

this

surfaces

recently This

(4.2),

we see t h a t

inter-

class of

dlfferent

J.

llst

(1.7),

for

(].12).

result

of manifolds

..th e i n t r o d u c t i o n .

subtle

can be a p p l i e d .

p r o p o s e d in [ 1 2 ]

(1.14),

g=5

A l e x a n d e r proved the

seems t o be a r a t h e r

the f o l l o w i n g

is

argument.

s u r f a c e s o f F 4 h a v i n g d=8,

the maximal l i s t

Theorem. The e f f e c t i v e

the t a b l e

of planes

there

to a quadric,

the methods d e s c r i b e d so f a r

looking over

(6.5)

so b-e=2 and

we deduce t h a t

by a c o m p l e t e l y

AS we u n d e r s t o o d from [ 1 5 ] , this

b=2,

by T I L I , we f i n d

s h o u l d be p o i n t e d o u t

s i n c e none o f

surface,

4=2b-e and b-e=

Z containes a certain

(C~[12]), case,

From t h i s

s u r f a c e mapped b l r a t i o n a l l y It

e=0,

surface

elliptic

F mapped to p a i r s

e x c l u d e d by Okonek in [ 2 1 ]

(6°5)

Thus we p r o -

we t a k e a h y p e r p T a n e o f F 4 c o n t a i n i n g

intersection

des t h e two p l a n e s .

ruled

Oh.V, we have L t s ~ C o + b f ,

Co i s mapped two t o one o n t o some l i n e . are

Is a b s u r d .

morphlsm between S and a c e r t a i n

S i n c e S is a g e o m e t r i c a l l y

the n o t a t i o n s

=(Co+bf.Co)~2. Mbreover,

so ~=~=0 w h i c h

is a g e n e r i c member, t h e map i n d u c e d by r e s t r i c t i n g

degree 8 (6,2),

was p r o v e d .

o f d e g r e e 8 i s as

154 References 1. 2.

3. 4. 5. 6. 7. 8. 9. I0. If. 12. 13. 14.

15. 16. 17.

18. 19. 20. 2]. 22. 23. 24. 25.

Barth,

W. T r a n s p l a n t i n g cohomology c l a s s e s in c o m p l e x - p r o j e c t i v e s p a c e . Amer. J. M a t h . 9 2 ( 1 9 7 0 ) , 9 5 1 - 9 6 7 . Bulum, A. On s u r f a c e s o f degree at most 2n+] In p n in P r o c e e d i n g s o f the Week o f A l g e b r a i c Geometry, B'uucharest 1982, S p r i n ger Lect. Notes Math., I056(1984). Fujita, T. Definlng equations for certain types of polarized varieties, Complex Analysis and ATgebraic Geometry, Tokyo, lwanami, (1977), 165-173. Fujita, T. Classificatlon of polarized manifolds of sectional genus two, Preprint. Gieseker, D. P't'~'~ple bundles and their Chern classes, Nagoya Math~J.43, (19717, 91-116. Hartshorne, R. A~,~ebraic Geometry, Springer (1977). litaka, S. Algebraic Geometry: an introduction to birational geometry of a l g e b r a i c v a ~ e t l e s , " S p r l n g e r (1982). lonescu, P. An enumeration of a'll smooth r p r o j e c t i v e v a r i e t i e s of degree 5 and 6, INCREST P r e p r l n t Series Math., 74, (1981). lonescu, P. VariEt~s pro~ectives l i s s e s de .de~r~s 5 et 6, C.R.Acad. S c i , P a r i ~ , 293, ( 1 9 B l ] , ~ 5 - 6 8 7 . Ionescu, P. Embedded p r o j e c t i v e v a r i e t i e s of small i n v a r i a n t s , INCREST P r e p r i n t Series Math., 72 (1982). Ionescu, P. Embedded pro~ective v a r i e t i e s of small i n v a r i a n t ~ in Proceedings of the Week of Algebraic Geometry, Buchar e s t 1982, Springer Lect. Notes Math., 1056 (1984). Ionescu, P. Embedded pro~ective v a r i e t i e s of small i n v a r i a n t s I I , 'Rev. Roumaine Math. Pures A--ppl.,31 (1986), 539-544. Ionescu, P. V a r i e t i e s of small de~ree, An. St. Univ. A . I , C u z a , lassy, 31 s . l . a ( 1 9 8 5 ) ~ 17-19. Ionescu, P. Ample and very ample d i v i s o r s on surfaces, Rev. Roumaine Math. Pures Appl., 33(T98~), 349~35~. Katz, S. Hodge numbers of linked surfaces in P , Duke Math. J., 55(1987), 89-95. Kleiman, S. Geometry on ~ras!mannians and applications to splitting bundles and smoothin~ cycles, Publ.Math. IHES, 36 (1969/, 281-297. Le Potier, J. Stab[l itE et amplitude s'ur P2(E), in Vector bundles and

differential equations, Proceedings,N4ce, 1979, Progress in Math.7, Birkh~user. Mumford, D. Varieties defined by quadratic equations, in Questions on algebraic varieties (CIM£ Varenna 1969) Ed. Cremonese, Roma 1970. Newstead P.E. Reducible vector bundles on a quadric surand Schwarzenberger R.L.E. ~ace, Proc. Camb. P h i l . Soc.,60, (1964), 42]-424. Okonek, C. 0ber 2-codimenslonale U n t e r m a n n i g f a l t i g k e i t e n vom Grad 7 in ~4 und pS, Math. Z., I87(1984), 209-219. Okonek,C. F'F~I'~chen vom G,rad 8 im ~4, Math. Z., 191 (1986), 207-223. Peskine, C. and Liaison ~es var-r~t~s aIq~briques I, Inv.Math. 26(1974) S~)iro, L. 271-302. Reider, I . Vector bundles of rank 2 and l i n e a r systems on a l g e b r a i c surfaces, Ann. Math.127 (1988), 309-316. Serrano, F. "The ad~unction mapping and h y p e r e l I i p t i c d i v i s o r s on a s~rface, J. relne angew,' Math. 3Bl (19~7), 90-109. Sommese, A, and ~n the ad~unction mapping, Math. Ann. 278 (1987), Van de Ven, A. 594-603.

ON THE EXISTENCE OF SOME SURFACES Elvira Laura Livorni Dipartimento di Matematica,Universith degli Studi de L'Aquila Via Roma Pal.Del Tosto, 67100 L'Aquila, Italia To my children Luca and Fabio Introduction

The problem of classifying algebraic, projective, surfaces with small projective invariants i.e. degree or sectional genus is an old problem. It was started by Picard and Castelnuovo, see references. Roth in [33], [34], [35], [36] and in [37], gave a birational classification of connected, smooth, algebraic, projective surfaces with sectional genus less than or equal to six. Classically, the adjunction process was introduced by Castelnuovo and Enriques [12] to study curves on ruled surfaces. Recently, after Sommese and Van de Ven study of the adjunction mapping, see [39], [40], [41], the problem of giving a biregular classification of smooth, connected, algebraic, projective surfaces with either small degree or small sectional genus has been studied again by various authors, see references. We started the study of such surfaces while we were Sommese's student in Notre Dame in 1981. The main tool for the identification of the numerical projective invariants and for the description of a minimal pair of (X,L) were the iterated adjunction mappings, for the definition see [5]. Actually it turned out, see [7], that we really need to iterate the adjunction mapping only for g=g(L) _>8. The reason for writing this paper is that after Reider's results, [31], it has been possible to answer to the often subtle question if the pairs (X,L) determined in the previous papers, see references, do really exist i.e. if the line bundle L on X is very ample. We like to call those surfaces "the candidate surfaces'. Using Biancofiore's results, see [3], [4], and Buium's results, [9], we have been able to answer for most of the cases when the Kodaira dimension •(X)=-oo. When K(X)_>0, unfortunately Reider's method doesn't help us. In this case we haved used again Buium's results in [9] but there are still open some very interesting cases for example the existence of elliptic surfaces either with n(X)---0or n(X)=l. See cases 9, 14, 17, 20, 21 and 23 in §4. It was our intention to quote all the works of mathematician who gave contribution to this nice classification started by the italian classical school. We apologize if we have forgotten someone. The organization of the paper is as follows: In §0, we collect background material and explain the conditions we have to impose on the points that we have to blow up on a minimal model in order to guarantee the very ampleness of L. See (0.3),(0.t 1) and (0.19). In §1, we determine the existence of surfaces with g < 7 and whose minimal model is p2. In §2, we determine the existence of surfaces with g < 7 and whose minimal model is a rational ruled surface. In §3, we determine the existence of surfaces with g < 7 and whose minimal model is an irregular ruled surface. In §4, we determine the existence of surfaces with g < 7 and whose minimal model has non-negative Kodaira dimension.

156

I would like to express my gratitude to Andrew J. Sommese for his constant encouragement to continue in my work although the hard job of being a mother. I woul like to thank A.Lanteri for useful conversation regarding surfaces with ~(X)=I.

§0 BackgroundMaterial. (0.1) Let X be an analytic space. We let O X denote its structure sheaf and hi,0(X)=dimHi(X,O X)" If X is a complex manifold, we let K X denote its canonical bundle.

(0.2) Let X be a smooth, connected, projective surface. Let D be an effective Cartier divisor on X. Denote by [D] the holomorphic line bundle associated to D. If L is a holomorphic line bundle on X, we write [ L ] for the linear system of Cartier divisors associated to L. Of course if ] L I is non empty, then [D]=L for De [ L 1. Let E be a second holomorphic line bundle on X. Then L'E denotes the evaluation of the cup product, Cl(L)^c2(E) on X, where Cl(L) and c2(E) are the Chem classes of L and E respectively. If De ] L [ and Ce ] E 1, it is convenient to write D'C=D'E=L'C=L-E. We let g=g(L)= (L'L+Kx'L+2)/2, which is called the adjunction formula. If there is a smooth De [ L ], then g=g(L)= hl'0(D). We let q--dimH 1(X,O X) and pg=dimH2(X,O X)"

(0.3) Definition: Let Pl ..... Ps be a finite set of points of p2. We say that Pl,--',Ps are in general position if no three are collinear and no ten lie on a conic, and furthermore after any finite sequence of admissible transformations, the new set of s points also has no three collinear. It is easy to check that if the points satisfy this definition then: 1) No six of them lie on a conic 2) No eight of them, one double, lie on a rational cubic 3) No nine of them, three double, lie on a rational quartic 4) No nine of them, six double, lie on a rational quintic. This definition is an easy consequence of the definition in [15,pg.409 ex.4.15]

(0.4) Let m e N and X be a smooth surface. We write D m for the set of all divisors EC_X such that E * 0 and mE is effective. (0.5) Definition: Let L be a line bundle on a smooth, connected, projective surface X. Let M=L®Kx-1. A divisor E on X is said to be a Reider divisor if E~ D 1 and (M-E)-E 2p + 2 and is necessarily nonspecial, and C is an intersection of quadrics imposing exactly 2n + p + 1 conditions on quadrics. Therefore, if E -- C n Q is the intersection of C with a quadric then Y. consists of 2n + 2p points, and imposes 2n + p conditions on quadrics. (iii)

The bound p < n - 2 is also best possible, since the intersection E -- C c~ Q of a

quadric with a canonical curve C c IPn of genus n + 1 has degree 4n and imposes 3n - 1 conditions on quadrics. This example suggests that d > 4n and f < 3n - 1 imply either E is not generically an intersection of quadrics, or equalities throughout and Y. is ideal-theoretically an intersection of quadrics. (iv)

The conjecture is an easy exercise if E is contained in a linearly normal curve C c p n

of degree < 2n - 1; in §2 I will prove that it also holds if Z is contained in a surface F c IPn of degree n - 1, a result presumably known to Fano and Castelnuovo. (vi)

It might be interesting to formulate an analogous conjecture in the style of Mark Green

for the higher syzygies: E of small degree and an intersection of quadrics

=¢,

higher syzygies aren't generated in lowest degree?

(1.6) Conditional results. Conjecture 1.5 implies the following. (i)

Let C be a curve and D -- gr d a divisor with 2D special such that the rational map

tpD: C ~ F c IPr is a birational embedding; suppose that the image tPD(C) = F is generically an

197

intersection of quadrics in the sense of (1.1); then d > h0(2D)-I (ii)

and

h0(2D) > 4 r - 5 .

Let X be a surface of general type for which the l-canonical image ~PK(X) c IPPg- 1

is generically an intersection of quadrics; then X satisfies K2 > 4pg+q-12.

Proof. (i) This is exactly the same as (1.4, ii). Without loss of generality I can assume IDI is free, and set d=

degD

and

p = I~--~]-n,

where n = r - 1 as in (1.2). Then since deg tPD(C) >_2n + 2p + 1, Conjecture 1.5 would imply that h0(0c(2D)) > 3n + 1 + min (p, n - 2). However, Clifford's theorem for the special divisor 2D gives d > h0(0c(2D))-1, and it follows from these inequalities that p > n - 2, and thus h0(2D) > 4n - 1 = 4r - 5. For (ii), note that (1.2) and (i) give 1-q+Pg

+ K 2 = P2 > p g + 4 ( p g - 2 ) - 5 .

Q.E.D.

Remark. The cases p = 1 and p = 2 of (1.5) are contained in (1.3), proved in §2, so that it follows for example that for a surface X with K 2 = 3pg - 6 and birational cpK, the image q~K(X) c ppg-1 is contained in a 3-fold W of degree < pg - 2. When pg > 12 the only possibilities for W are a rational normal 3-fold scroll or a double cone over an elliptic curve, or linear projections of these.

(1.7) Relation with the free pencil trick. Let C c p n + l

be a curve. The free pencil trick is a

classical method of giving a lower bound on the rank of the restriction map PC: H 0 ( 0 P (2)) --~ H0(0C(2)): fix n + 2 general points P0,'" Pn+l ~ C, and choose the coordinates x0,.. Xn+1 so that xi(Pj) -- 8ij. Then x0, x 1 span the pencil of hyperplanes through FI = n then h0((gc(A)) = 1, and this implies that the 3n + 3 monomials {x02, X0Xl, Xl 2, x0xi, XlXi, xi2}i = 2,.. n+l are linearly independent, so h0((gc(2H)) >_3(n + 1). This is the same result as (1.4, ii), and the proof is just a mutation of the proof in (1.2) and (1.3, i): it boils down to saying that after taking a general hyperplane section Z: (x0 = 0) C, the only quadrics of rank 2 containing Z with vertex in the linear subspace (x I = 0) are of the form Xl.k where k is a linear form vanishing at Z - A.

(1.8) It's important to understand the weakness of this argument: the bilinear problem of estimating the rank of PC is reduced to a linear one, but at the expense of considering only the (2n+3)-dimensional subspace of forms involving x0 and x 1.

§2. Rational normal scrolls and the proof of (1.3) The remaining assertions (ii) and (iii) of (1.3) will follow from the two following theorems. (2.1) Theorem. Let Z c IPn be a set of d > 2n + 2p + 1 points, linearly general and uniform with respect to quadrics. Suppose that p < n - 2 and that Z imposes f < 2n + p conditions on quadrics. Then Z is contained in a p-dimensional rational normal scroll F (possibly a cone; if p = 1, F is a rational normal curve). (2.2) Theorem. Suppose that Z c F c p n is contained in a rational normal surface scroll of degree n - 1. Then Conjecture 1.5 holds for Z.

199

(2.3) Plan of proof of (2.1). Write II = < P 1 , . . P n _ I > --- IPn - 2 for the codimension 2 space spanned by n - 1 elements of Z. A dimension-count shows that H u Z is contained in at least n - p linearly independent quadrics, and the intersection of these will consist of H together with the required scroll F. This is a classic construction for rational normal scrolls, and the only possible way it can degenerate would contradict the fact that X is linearly general. The tricky part is to show that the initial points P 1"' Pn-1 are also contained in the residual intersection F, and this is where the uniform assumption on Z is used. (2.4) Dimension count. Since quadfics of H = p n - 2

vanishing at the n - 1 given points

{P1,'" Pn-1 } form a vector space of dimension (n 2 1), it follows that H imposes at most this number of conditions on quadrics of p n

through E. Therefore the vector space of quadrics

through H and Z has dimension > ( n +22 ~] - ( 2 n + p ) - (x n z: 1) = n - p , as required.

(2.5) Classic construction for scrolls. Consider the blow-up o: F 0 ~ [pn of [pn in H; with its projection x: F 0 ---) [P 1, this is the n-dimensional scroll F 0 = [P(EO), where E 0 is the rank n vector bundle over [P 1

E0 = 0(1) ~) (9~(n-t). F 0 contains anegative divisor B 0 = (~-lyI ~, ~1 x ~ n - 2 , and I write A 0 for the fibre of ~. The morphism o: F 0 - - ) [pn is given by the complete linear system ]A0 + B01. Let Q1," Qn-p be the linearly independent quadrics of [pn through FI provided by (2.4). Then since Qi D H, it follows that each (~*Qi = B 0 + Q i ' , where Qi ' ~ 2 A 0 + B 0. Now for k = 1,.. n - p, set

Fk =

k I1 Qj'cF j=

0.

By induction on k, suppose F k is irreducible and is a ~ n - k - l _ b u n d l e over [pl, having degree k + 1 under the morphism (~ to ~ n defined by the divisor A k + B k (where I write A k and B k for the restriction of A 0 and B 0 to Fk). Then Fk+ 1 c F k is in the divisor class 2 A k + B k, and so has degree k + 2 under (~, and has a unique component that is a [pn-k-1 -bundle over ~ 1 Thus if reducible, Fk+ 1 could be written as a sum of A k with a divisor in IAk + Bkl; then

200

O(Fk+l) c (~(Fk) c IPn would be contained in the union of two hyperplane sections, of which the one containing O(Ak) can be chosen to pass through II. Since Z is certainly contained in C(Fk+ 1) to II, this contradicts the assumption that Y. is linearly general. Therefore F = O(Fn_ p) c p n is a p-dimensional scroll as required. Remark. If irreducible, B k c F k is itself a scroll mapping birationaUy to c(B k) c FI if k >- 1. It can certainly happen that B k is reducible, but in any case ~(B k) c 11 has codimension k - 1 and degree k. (2.6) Key technical point: £ c F. The points of £ other than {P1,'" Pn-1 } belong to each quadric Qi and not to H, and hence are in the residual component F of ~ Qi = FI vo F. I now prove that the points Pi for i = 1,.. n- 1 also belong to F, following an argument kindly supplied by Eisenbud and Harris. For this, let Pn, Pn+l ~ y" be two more points, and choose the coordinates Xn, Xn+ 1 so that x n vanishes on < H , Pn+l > and Xn+1 on < H , Pn>; then of course H: (x n = Xn+ 1 = 0).

Lemma,

After possibly reordering {P1,." Pn-1 }, the equations of F can be written in the

determinantal form rk [ xn

Xn+t

X1 k2 .- kn-p]

]

_< 1,

(*)

~1 ~2 -. ~n-p

where the ki are linear forms such that ki(Pj) = 8ij for i, j = 1,.. n-p. Here the n - p quadrics through Z u II of (2.4) are given by

I Xn Qi = det

Xn+l

~t

for i = 1,.. n - p .

(**)

First of all, the lemma implies £ c F, and thus proves Theorem 2.1. In fact, the remaining quadrics of the determinantal are

Qij = det

for i , j = 1 , . . n - p

with i ~ j .

gi Now Qij vanishes on F, hence on all the points of Z except for the Pi, and the ever-vigilant reader will be able to see from the form of Qij that it also vanishes at Pk for k = 1,.. n-p, k ~ i, j. Thus in total it vanishes at

201

> 2n+2p+l-(n-1)+(n-p-2)

= 2n+p

points of Y. Hence by the uniform assumption on Z, Qij vanishes at all points of 5"..

Proof of the lemma. Every quadfic through H is of the form Xnkt - Xn+lX, so the equations of the n - p quadrics Qi can certainly be put in the form (**) for some linear forms Xi and Ixi. It follows that F is given by equations (,). Now I claim that the ~'i restrict to n - p linearly independent forms on H. For otherwise some nonzero linear combination of the Q i would be of the form

with )~ a linear combination of x n and Xn+ 1 ; but Q(Pn) = 0 implies that ~. would have to be a multiple of x n, since xn(P n) = 0 and Xn+ I (Pn) ~ 0. This is absurd, since I~ is not contained in a quadric of rank 2. Suitable linear combinations of the ~ i achieve the statement of the lemma. Q.E.D.

(2.7) Plan of proof of (2.2). The proof of (2.2) considers the linear system L with assigned base points cut out on F by quadrics of p n

through Z (the case of F

a cone causes no

problem, just resolve the vertex). Firstly, since h0(F, 0(2)) = 3n and Z imposes n - p ,

that is,

dimL > n-p-l;

write L = M + D with M mobile and D the fixed part. Clearly since F is a scroll, 0(2) has degree

2

relative to the projection

~: F --~ IP 1,

and there are

3

possibilities" for the

decomposition L = M + D: Case 1. M has degree 2 over ~ 1, and D is a union of [3- 0 fibres of n. Case 2. M and D both have degree 1 over [P 1. Case 3. M is a union of > n - p - 1 fibres of ~ and the base locus D has degree 2 over P 1 In Case 3, there is nothing to prove: the base locus D D E and is the intersection of all quadrics through E; clearly deg D _ 2 and E has b base points, so M 2 = kE 2 > 4E 2 >_ 4b > 4 ( 2 n + 2 p + l - 2 1 3 ) , giving 13 > n + 2p + 2, which is absurd. Step 2. Write F for the normalisation of the general element of M; consider the free linear system M F cut out on F by M (after subtracting off the base locus). Then degM F < M2-b

< 4(n- 1)-413-(2n+2p+l n - p - 2 , which contradicts Clifford's theorem ff M F is special. Step 3. Therefore M F is nonspecial, so I get a bound on the genus of g from RR, which will lead to a contradiction. I intend to use the classical language, writing ]~ miP i for the actual base locus of M, including infinitely near points; the reader who is unduly distressed by this can perform the easy exercise of translating the following argument in terms of successive blow-ups of the base locus. First, degM F = M 2 - ~ m i 2

= 4n-4-4[~

- ]~mi 2

and

g(r)

= n-2-[3

- ~ ( 2 i),

Here n - 2 - ~ is the genus of the general element of 12H - 13AI on F (exercise using the adjunction formula). Also, since M F is a nonspecial linear system, RR gives n-p-1

< h0(F,M F) = 1 - g ( F ) + d e g M F = 3n-3-313-

Y ~ ( m i ; 1).

However, M has at least b > 2n+2p+l-2[~ assigned base points, and hence 2n+2p+l-213

h0(F, I E . O ( 2 ) ) - I

-> n - p - 1 .

Therefore n-p-1

_ 2n+p+213.

However, since M ~ H - 13A moves in an irreducible linear system on

F

it follows that

h0(F, OF(M)) = Z(0F(M)) = n + 1 - 213, and it is obvious that D ~ 2H - M imposes h0(F, 0(2)) - h0(F, OF(M)) = 3n - n - 1 + 213 = 2n - 1 + 213 conditions on quadrics. Therefore the a points of X n D

impose dependent conditions on

quadrics, and by the uniform assumption, this implies that X c D. The remaining assertion, that deg D < n + p follows from the fact that 213 < p + 1.

Q.E.D.

§3. A vector bundle a p p r o a c h

(3.1) This section starts off with some numerology. The following material on quadrics of small rank or containing large linear subspaces is well-known and will be used throughout.

Proposition. The (projective) space of quadrics of rank _ = IP n c IPn + l .

(2)

In view of 0C(1) -- E + E', obviously dim IEI >_- n - dim 2'¢+2. This is a version of (1.3) in which the condition of linear generality is replaced by the weaker uniform condition: if E has a subset {P 1,'" Pk ) of k points spanning a ~-plane IP~, then every point P ¢ E is contained in such a subset. This and other uniform properties of E come from the fact that C is a fairly general hyperplane section of X, and E general in a pencil on C; so it is appropriate to work directly on X.

(3.7) Construction and properties of E, I use the notation of the start of (3.5). First of all, how does one get pleasure or profit out of a quadric of rank

5? I view the nonsingular quadric

Q c Ip4 as a hyperplane section of the Klein quadric Gr = Gr(2, 4) = Q6 c [p5 Let g 0 be one

208

of the two tautological quotient bundles on Gr, and dimV = 4

(8)

H0(OGr(1)) = A 2 V

(9)

V = H0(Gr, g0), so that Gr = Gr(2, V); then fgGr(1) = A 2 g0, and a nonsingular hyperplane section of

Gr

corresponds to the isotropic spaces of a non-

degenerate skew bilinear form ~t: V x V ---+k. Now under the assumptions of (3.5), X c Q5 and X c~ Sing Q5 = ¢" This means that X has a projection morphism to the nonsingular Q c ~4, so that writing g for the restriction to X gives the following.

Proposition. X c Q 5

and X c ~ S i n g Q 5 = la gives rise to avector bundle ~ of rank 2 on X

such that A 2 ~ -= K X,

that is,

K X®~*

~' ~,

(10)

and V a 4-dimensional space of sections spanning ~: 0-+ ~F --+ V ® O ~

~ -+0

(11)

V has a nondegenerate skew bilinear form ~7: V × V --+ k such that at every point, the fibre of ~F is isotropic for ~, inducing an isomorphism

V: ~ - ~ g* = I~(-Kx).

(12)

(3.8) Lemma. (i) Any general section s ~ H0(~) defines a short exact sequence 0-+ 0 ---+ ~ ---+ IE.Ox(K X) -+ 0,

(13)

where E c X is a reduced set of points, so that E=c2(~)

and C l ( ~ ) - - K X.

(14)

2E = K 2

as 0-cycles of X

05)

(ii)

(modulo rational equivalence). (iii) If s ~ V c H0(~) construction of (3.5), (1).

is a general element then E is just a reinterpretation of the

209

Proof. The construction on X is the pull back of a tautological construction on Q. The choice of a tautological bundle T0 on Gr(2, 4) determines one family of generators of Gr = Gr(2, 4) = Q6 c p5. A general element s ~ V = H0(Gr, T0) defines an exact sequence over Gr 0--) 6 G r --~ E0 ---) I~z.6Gr(1) -)0,

(16)

where 7z = c2(T0) is a generator of Gr in the given family. If 7t 1 and ~2 are generators of Gr in the two families then ~z1 + rc2 is a codimension 2 linear section of Gr; on the other hand, each of them restricts down to a generator of Q, so that on Q, the class of a codimension 2 linear section is twice a generator. (ii) follows from this. The generators of Q move around in a free system, so that the nonsingularity in (i) follows from the separability of X ----)Q. (iii) is an exercise for the reader. Problem. Assume that the projection X---* Q c ~ 4 is a birational embedding; (presumably this is the main case?). If K is the field of definition of the generic section s ~ H0(T), then is it true that the Galois group Gal(K(E)/K) is the full symmetric group on E? This would imply that E is uniform, and would be an analogue for vector bundles of the Lefschetz-Harris principle for very ample divisors (compare (1.2)).

(3.9) Lower bound for H0(T). If follows immediately from (13), orfrom RR that Z(T) = 2~((9 X) - deg E.

(17)

Assuming K 2 < 4 p g - 1 2 and q = 0, this gives ~(T) = 2 p g + 2 - ½K 2 > 8.

(18)

Now Serre duality together with (10) gives h2(T) = h0(T*(Kx)) = h0(T), so h0(T) = h2(T) = p g + 1 - ¼ K 2 + 2 h I ( T )

-> 5.

(19)

On the other hand, the number of hyperplanes of [ppg-1 containing E is h0(IE.6X(K)), and from (13), this is h0(IE.6x(K)) = h0(~) - 2

(20)

1 K 2 - 1 < n-2. d i m < E > -- p g - h 0 ( E ) < "Z

(21)

so that (19) implies

This shows that E is not linearly general.

(3.10) E is not quadratically general. (11) and (12) give an exact sequence

210

0 ~ V --~ H0(E) ~

HI(E *) -+0.

(22)

Hence hl(~ *) = hl(E(K)) = h 0 ( E ) - 4 > 0.

(23)

Now tensoring (13) with K X gives 0-+ O --+ ~ --+ IE-(gx(K X) -+0,

(24)

leading to the cohomology exact sequence H I ( K x ) = 0-+ H I ( ~ ( K x )) --+ HI(IE.(gx(2K)) --+ (25) - + H2((gx(Kx )) = k -+ 0

Therefore hl(IE.OX(2K)) = h l ( ~ ( K x )) + 1 = h0(~) - 3.

(26)

This proves that E imposes (27)

fE = deg E - h0(~) + 3 conditions on quadrics. (3.11) C u r i o u s c o n c l u s i o n . E satisfies

< E > = I P '¢, d e g E > 2 ~ ) + 2 x + 1 ,

fE = 2 V + x ,

where 1 < ~ < '~. That is, the numerical assumptions of Conjecture 1.5 hold for E (except possibly for the cases ~ = '~-1, ~)).

Proof. Set = IP'~; then by (21), (15) and (27),

Now set

,~ = n + 3 - h 0 ( E ) ;

(28)

1 2., deg E = ~-K

(29)

fE = deg E - h0(E) + 3;

(30)

211

= degE-n-V;

(31)

= fE - 2~;

(32)

adding 2n - 21-K2 to both sides of (31) gives n-v

= h0(E)-3

(33)

1 2.

(34)

= ~+2n-

~K

The~ d e g E = ½K2 = n + v + r ~

= 2"~+~+h0(E)-3

= 2 ~ + 2 7 ~ + ( 2 n - ~1K 2 ) > 2~1+2;t+ 1

(35)

fE = 2v + ft.

(36)

and

The final inequality ~ _< v comes easily from r~ = deg E - n - V and deg E < 2n - 1, since (1, 2) imply that two copies of = IP~1 span [pn, so that 2"~ + 1 > n. Remark. Theexceptionalcases ~ = "v-l, ~1 of (3.11) only occur if deg E is close to 2 n - 1 and ~ close to (n-1)/2; by (19-26), this correspondsto h0(E), h l ( E ) , hl(IE.~X(K)) and hl(IE.OX(2K)) close to their maximum. It's quite likely that there are cleverer bounds on these groups than the trivial one using (1,2) I have used.

(3.12) F i n a l r e m a r k s . My feeling is that there is a vague analogy between Conjectures 0.2 and 1.5 and the subject of special linear systems on curves on K3 surfaces, another area of research that has lain dormant for around 10 years, and has recently been opened up again by Lazarsfeld's ideas using vector bundles [Reid1, Lazarsfeld, Green and Lazarsfeld]. In fact, an important part of the original motivation for [Reidl] was the idea (which goes back to Petri [11]) of trying to capture a special linear system on a curve C in terms of a variety of small degree C c V c N g - 1 through the canonical curve; then if C is a hyperplane section of a K3 surface X c Ng, the problem is to extend V to a variety W c Q3g containing X, and intersections of quadrics are very relevant to this.

212 References

[Andreotti] A. Andreotti, On a theorem of Torelli, Amer. J. Math 80 (1958), 801-828. [4 authors] E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of algebraic curves, vol. I, Springer, 1985. [Babbage] D.W. Babbage, A note on the quadrics through a canonical curve, J. London Math Soc. 14 (1939), 310-315. [Castelnuovo] G. Castelnuovo, Ricerche di geometria sulle curve algebriche, Atti R. Accad. Sci. Torino 24 (1889), 196-223. [Ciliberto] C. Ciliberto, Hilbert functions of finite sets of points and the genus of a curve in a projective space, in Space curves (Rocca di Papa, 1985), F. Ghione, C. Peskine and E. Sernesi (eds.), LNM 1266 (1987), pp. 24-73. [Eisenbud] D. Eisenbud, letter and private notes, c. 1979. [Fano] G. Fano, Sopra le curve di dato ordine e dei massimi generi in uno spazio qualunque, Mem. Accad. Sci. Torino 44 (1894), 335-382. [Harris] J. Harris, Curves in projective space, S6minaire de Math Sup. 85, Presses Univ. Montr6al, 1982. [Horikawal] E. Horikawa, Algebraic surfaces of general type with small c 12, II, Invent. Math 37 (1976), t21-155. [Horikawa2] E. Horikawa, Algebraic surfaces of general type with small c12, V, J. Fac. Sci. Univ. Tokyo, Sect IA Math 28 (1981), 745-755. [Green and Lazarsfeld] M. Green and R. Lazarsfeld, Special divisors on curves on a K3 surface, Invent. math 89 (1987), 357-370. [Lazarsfeld] R. Lazarsfeld, BriU-Noether-Petri without degenerations, J. diff. geom. 23 (1986), 299 -307. [Mumford] D. Mumford, Varieties defined by quadratic equations, in Questions on algebraic varieties (CIME conference proceedings, Varenna, 1969), Cremonese, Roma, 1970, pp.29-100. [Petri] K. Petri, ¢0ber Spezialkurven I, Math. Ann. 93 (1925), 182-209. [Reidl] M. Reid, Special linear systems on curves lying on a K3 surface, J. London math soc. 13 (1976), 454-458. [Reid2] M. Reid, Surfaces with pg = 0, K 2 = 2, unpublished manuscript and letters, 1977. [Reid3] M. Reid, ~1 for surfaces with small K 2, in LNM 732 (1979), 534-544. [Reid4] M. Reid, Surface of small degree, Math Ann. 275 (1986), 71-80. [Sommese] A.J. Sommese, On the birational theory of hyperplane sections of projective three-

213

folds, Notre Dame preprint, c. 1981. [Tyurin] A.N. Tyurin, The geometry of the Poincar6 theta-divisor of a Prym variety, Izv. Akad. Nauk SSSR, 39 (1975), 1003-1043 and 42 (1978), 468 = Math USSR Izvestiya 9 (1975), 951986 and 12 (1978), No. 2. [Segre] B. Segre, Su certe variet~t algebriche intersezioni di quadriche od a sezioni curvilinee normali, Ann. Mat. Pura App. (4) 84 (1970), 125-155. [Xiao] Xiao Gang, Hyperelliptic surfaces of general type with K 2 < 4X, Manuscripta Math 57 (1987), 125-148.

Miles Reid, Math Inst., Univ. of Warwick, Coventry CV4 7AL, England Electronic mail: miles @ UK.AC.Warwick.Maths

Infinitesimal view of extending a hyperplane section - deformation theory and computer algebra Miles Reid 1), University of Warwick

§0. Alla marcia (0.1) The extension problem. Given a variety C c IPn - l , I want to study extensions of C as a hyperplane section of a variety in Ipn: C c ~n-1 f~

f~

Xc

pn

with C = IPn-1 ~ X;

that is, C: (x 0 = 0) c X, where x 0 is the new coordinate in IPn. I will always take the intersection in the sense of homogeneous coordinate rings, which is a somewhat stronger condition than saying that C is the ideal-theoretical intersection C = IPn - 1 ~ X.

(0.2)

Some cases of varieties not admitting any extension were known to the ancients: for

example, the Segre embedding of lP 1 × ip2 in IP5 has no extensions other than cones because all varieties of degree 3 are classified ([Scorzal-2, XXX], compare systematic obstructions of a topological nature to the existence of around

[Swinnerton-Dyer]); and X

were discovered from

1976 by Sommese and others (see [Sommesel], [Fujital], [B5descu], [L'vovskiil-2]).

More recent work of Sommese points to the conclusion that very few projective varieties C can be hyperplane sections; for example, Sommese [Sommese2-3] gives a detailed classification of the cases for which K C is not ample when C = p n - 1 c~ X is a smooth hypersection of a smooth 3-fold X; this amounts to numerical obstructions to the existence of a smooth extension of C in terms of the Mori cone of C.

(0.3)

The infinitesimal view. Here I'm interested in harder cases, for example the famous

problem of which smooth curves C of genus g lie on a K3 surface C c X; the infinitesimal 1) Codice Fiscale: RDE MSN 48A30 Z114K

215

view of this problem is to study the schemes

C c 2C c 3C c..

which would be the Cartier

divisors kC : (x0k = 0) c X if X existed. Here each step is a linear problem in the solution to the previous one. For example, assuming that C is smooth, the first step is the vector space N(1) = {2C c ~n extending C} = H 0 ( N ~ n - I l c ( - 1 ) ) or dividing out by coordinate changes, ~'1(_1) = {2C extending C} = coker {H0(Tlpn-I(-1))---*H0(NIpn-Ilc(-1))}. Singularity theorists know this as the graded piece of degree (-1) of the deformation space ~-1 of the cone over C. However, the extension from (k-1)C up to kC is only an affine linear problem (there being no trivial or cone extension of 2C); in particular 1st order deformations may be obstructed.

(0.4) This paper aims to sketch some general theory surrounding the infinitesimal view, and to make the link with deformation theory as practised by singularity theorists. My main interest is to study concrete examples, where the extension-deformation theory can be reduced to explicit polynomial calculation, giving results on moduli spaces of surfaces; for this reason, I have not taken too much trouble to work in intrinsic terms. It could be said that the authors of the intrinsic theory have not exactly gone out of their way to make their methods and results accessible. The indirect influence on the material of § 1 of Grothendieck and Illusie's theory of the cotangent complex [Grothendieck, Illusie2] will be clear to the experts (despite my sarcasm concerning their presentation); §1 can be seen as an attempt to spell out a worthwhile special case of their theory in concrete terms (compare also [Artin]), and I have groped around for years for the translation given in (1.15, 1.18, 1.21) of the enigma [Illusiel, (1.5-7)]. Thus even a hazy understanding of the Grothendieck ideology can be an incisive weapon, which I fear may not pass on to the next generation.

(0.5) Already considerations of 1st and 2nd order deformations lead even in reasonably simple cases to calculations that are too heavy to be moved by hand. An eventual aim of this work is to set up an algorithmic procedure to determine the irreducibility or otherwise of the moduli space of Godeaux surfaces with torsion Z/2 or {0}, suitable for programming into computer algebra (although this paper falls short of accomplishing this); see §2 for this motivation and §6 for a 'pseudocode' description of a computer algebra algorithm that in principle calculates moduli spaces of deformations.

(0.6) Acknowledgements. The ideas and calculations appearing here have been the subject of

216

many discussions over several years with Duncan Dicks, and I must apologise to him for the overlap between some sections of this paper and his thesis [Dicks]. I have derived similar (if less obviously related) benefit from the work of Margarida Mendes-Lopes [Mendes-Lopes]. I am very grateful to David Epstein for encouragement. I would like to thank Fabrizio Catanese for persuading me to go the extremely enjoyable conference at L'Aquila, and Prof. Laura Livorni and the conference organisers for their hospitality. This conference and the British SERC Math Committee have provided me, in entirely different ways, with a strong challenge to express myself at length on this subject.

Contents Chapter 1. General theory §1. The Hilbert scheme of extensions. Precise definitions; graded rings and hyperplane sections; the Hilbert scheme [H of extensions of C. Deformation obstructions and the iterated linear structure [H(k) ---, ~(k-1) __, .. [H(0) = pt. Relation with ~1

and versal

deformations. §2. Examples, comments, propaganda. Pinkham's example; applications by Griffin and Dicks. Motivation, possible future applications.

Chapter II. Halfcanonical curves and the canonical ring of a regular surface {}3. The canonical ring of a regular surface. If there exists an irreducible canonical curve C ~ IKl, then R(X, KX) is generated in degrees 1, 2, 3 (except for 3 or 4 initial families). Coda to §3. Canonical map of an irreducible Gorenstein curve and 'general' divisors. {}4. Graded rings on hyperelliptic curves. Easy result that should be better known: if C

is hypereUiptic and

D

a divisor made up of Weierstrass points then the ring

R(C, Oc(D)) has a very concrete description. Chapter III. Applications {}5. Numerical quintics and other stories.

The classification and moduli theory of

numerical quintics from the infinitesimal point of view. Flexible form of equations, and determinantal formalism. §6. Six minuets for a mechanical clock. A 'pseudocode' description of a computer algebra algorithm to compute moduli spaces of deformations.

References

217

Chapter I. General theory

§1. The Hilbert scheme of extensions

This overture in the French style is mainly formalism, and the reader should skip through it rapidly, perhaps taking in the main theme Definition 1.7 and its development in Theorem 1.15; Pinkham's example in §2 gives a quick and reasonably representative impression of what's going on.

(1.1) Let C, OC(1) be a polarised projective k-scheme (usually a variety), and S = R(C, 0 f ( 1 ) ) = t~) H0(C,~gc(i)) i_>0 the corresponding graded ring. Suppose given a ring ~ c S of finite colength, that is, such that S / R is a finite-dimensional vector space. Often R = S, but I do not assume this: for example, if C c p n - 1 is a smooth curve that is not projectively normal, its homogeneous coordinate ring P, = k[Xl,.. Xn]/IC is of finite colength in R(C, tgC(1)) (the normalisation of R). Throughout, a graded ring R is a graded k-algebra R = ~)R i i_>0 graded in positive degrees, with R 0 = k.

Main Problem. Given a graded ring R and a 0 • Z, a0 > 0. Describe the set of pairs x 0 • R, where R is a graded ring and x0 • Ra0 a non-zerodivisor, homogeneous of degree a 0, such that = R/(x0).

Notice that since x 0 is a non-zerodivisor, the ideal (x 0) -- x0R = R. If R is given, then I write R (k) = R/(x0k+l), and call R (k) the kth order infinitesimal neighbourhood of 1~ = R (0) in R.

218

This notation and terminology will be generalised in (1.8), (1.2) The hyperplane section principle. Let R be a graded ring and x 0 e R a homogeneous non-zerodivisor of degree deg x 0 = a0 > 0; set R = R/(x0). The hyperplane section principle says that quite generally, the generators, relations and syzygies of R reduce mod x0 to those of R. = R/(x0), and in particular, occur in the same degrees. In more detail:

Proposition (i) Generators.

Quite generally, let R = • R i be a graded ring, and

R = R/(x0), where

x 0 e Ra0. Suppose that R is generated by homogeneous elements xl,.. x n of degree deg x i = ai; then R is generated by x 0, x 1,.. x n. That is, = k[Xl,.. Xn]/I =~ R = k[x0,.. Xn]/I, where 7 c k[Xl,., x n] and I c k[x0,.. Xn] are the ideals of relations holding in R and R. (See (1.3, (3)) for the several abuses of notation involved in the xi.) (ii) Relations. Keep the notation and level of generality of (i). Suppose that f(x 1,.. x n) e i is a homogeneous relation of degree

d

holding in

~

then there is a homogeneous relation

F(x0,.. x n) e I of degree d holding in R such that F(0, x 1,'" Xn) ---"f(xl,'" Xn)" Let fl,'" fm ~ i be a set of homogeneous relations holding in R that generates i, and for each i, let Fi(x0,.. Xn) e I be a homogeneous relation in R

such that

Fi(0, x 1,.. x n)

fi(xl,.. Xn). Now assume that x 0 is a non-zerodivisor. Then F1,.. F m generate I; thatis, i = (fl,-- fn) ~

I = (F 1,.. Fn) with F i ~ fi"

(iii) Syzygies. Quite generally, let F 1,'" Fm e k[x0,'" Xn] be homogeneous elements, and consider the ideal I = (F1,..Fm)

and the quotient graded ring R---k[x0,..Xn]/t.

fi = Fi(0, Xl,'" Xn) e ldXl,'- Xn], and set ~ = (fl,'" fm) and = R/(x 0) = k[xl,., x n l / i Then the following 3 conditions are equivalent: (a) x 0 e R is a non-zerodivisor in R; (b) (x 0) ~ I = x0I c ldx0,.. Xn];

For each i, write

219

(c) for every syzygy

or: Ei ~ifi E 0 e ktXl,., x n] between the fi there is a syzygy E: ~ i L i F i = 0 ~ k[x0,., x n] between the F i with Li(0,x 1,.. Xn) -= gi(Xl,.. Xn). (1.3) Remarks. (1) This is standard Cohen-Macaulay formalism, see for example [Mumfordl] or [Saint-Donat, (6.6) and (7.9)]; everything works just as well if the non-zerodivisor x 0 is replaced by a regular sequence (~ 1,'" ~k)" (2) Recall the general philosophy of commutative algebra that 'graded is a particular case of local'. The assumption that R is graded and a0 > 0 is used in every step of the argument to reduce the degree and make possible proofs by induction. In the more general deformation situation x0 ~ R or x 0 ~ H 0 ( 0 X ), one must either assume that R or O X is (x0)-adically complete (for example (R, m) is a complete local ring and x 0 ~ m); or honestly face the convergence problem of analytic approximation of formal structures. This is the real substance of Kodaira and Spencer's achievement in the global analytical context, and, in the algebraic setup, is one of the main themes of [Artin]. By (ii), R is determined by finitely many polynomials of given degree, so it depends a

priori on a finite-dimensional parameter space. Morally speaking, rather than graded and degree < 0, the right hypothesis for the material of this section (and for the algorithmic routines of §6) should be that "IT1 and 71"2 are finite-dimensional. (3) Abuse of notation. There are two separate abuses of notation in writing xi: (a) the same x i is used for the variables in the polynomial ring k[x 1,.. x n] and for the ring element x i = im x i e R = k[x 1,.. Xn]/I; there is no real ambiguity here, since I usually write = for equality in R and --- for identity of polynomials. (b) I identify the variables in the two polynomial rings k[x0,., x n] and k[Xl,.. Xn]; this means that there is a chosen lifting kIx 1,.. x n] q k[x0,., x n] of the quotient map k[x0,., x n] ~ k[x 1"" Xn] = k[x0,'" Xn]/(x0)" Notice that from a highbrow point of view, I always work in a given Irivial extension of a (smooth) ambient space (with a given retraction or 'face operator'), thus sidestepping the unspeakable if more intrinsic theory of the cotangent complex [Grothendieck, Illusiel, Illusie2, Lichtenbaum and Schlessinger]. (4) Higher syzygies for R extend to R in a similar way; in fact (1.2, ii-iii) can be lumped together as a more general statement on modules. (5) The notation of (1.2) will be used throughout §1. I'll write d i = deg f i and sj =

220

deg O'j.

(1.4)

P r o o f of (1.2, i).

Easy:

mod x 0,

g ~ R

every homogeneous

can be written as a

polynomial in x 1," Xn, so that g = g0(xl,., x n) + x 0 g ' , where g' ~ R is of degree deg g - a 0 < deg g, and induction.

(1.5) P r o o f of (1.2, ii). It's traditional at this point to draw the commutative diagram

(x0) n I ~

4,

4, ~

]

$

(x 0)

~

$

k[x0,..x n] ---* k[Xl,..x n]

$ 0-~

0

I

$ 0 ~

0

$

x0R

--~

-* 0

$

R

--~

R

4,

4,

4,

0

0

0

~ 0

with exact rows and columns. Now I ~

] is surjective by the Snake Lemma. Take any f ~

homogeneous of degree

with

d

and

F~ I

F~---~f. Then

f= F-x0g

(this uses the lift

f e k[Xl,., x n] c k[x0,.. Xn]). If I take only the homogeneous piece of F and g of degree d then f = F - x0g still holds, so F ~

f.

Now suppose {F1,.. F n} are chosen to map to a generating set {fl,'" fn } of I, and let G ~ I be any homogeneous element. Then since G ~

g ~ I = (fl,-- fn), I can write

g = ~gifi with homogeneous ~i ~ k[Xl,.. Xn], so H = G-~giFi

~ (x 0) h I .

Claim. If x 0 is a non-zerodivisor of R then (x0) n I = x0I. Because H=x0H'~I

=~

x0H'=0

in R

=~

H'=0

Thus G = ~ £iFi + x 0 G ' with G' ~ I, so I'm home by induction.

in R

~

H'~I.

221

(1.6) Proof of (1.2, iii). (a) ~ (b) has just been proved, and ~

is just as elementary. I prove

(b) ~ (c). Write F i ~ fi + x0gi, and suppose the syzygy of ]~ is c: ~ tif i ~- 0. Then I ~ ~£iFi ~ x o ~ t i g i ~ (x0), so that (b) implies that ~ £igi e I, and so ~ rig i --- ~ miF i. Then ~ L i F i -= 0,

where

L i = (t i - x 0 m i ) .

Conversely, assume (c) and let g ~ ldx0,., x n] be such that x0g -= ~ tiF i. Then ~ £ifi - 0 so that by (c) there exist L i ~ t i with ~ LiF i - 0, and x0g ~ x 0 ~ (t i - Li)Fi, so cancelling x 0 gives g ~ L

Q.E.D.

(1.7) The Hilbert scheme of extensions of P,. This solves Problem 1.1: the set of rings R, x 0 ~ Ra0 such that R -- R/(x 0) can be given as the set of polynomials F i extending the relations fi of P, such that the syzygies cj extend to Ej. To discuss this in more detail, fix once and for all the ring R, its generators x 1,.. Xn, relations fi and syzygies ~j. I also fix the polynomial ring k[x0,., x n] overlying R and discuss the set of extension rings R together with the data {Fi, Ej) of relations and syzygies as in (1.2). Then

{k/R, {Fi, Ej}IR/(x0)~,R}

< bij> B H :

( Fi = fi+x0gi [~LijFi_=01. Zj: Lij -- tij+x0mij

The set on the right-hand side has a natural structure of an affine scheme BH = BH(R, a0), the

big Hilbert scheme of extensions of R. For the gi ~ kIx0,.. Xn] and mij ~ k[x0,., x n] are finitely many polynomials of given degrees, ;o their coefficients are finite in number, and can be taken as coordinates in an affine space; the conditions ~ LijF i ~ 0 are then a finite set of polynomial relations on these coefficients. Remark. The (small) Hilbert scheme

II-Iff~~) : {VR, ~lR/(x O) :,< ~.}

222

is part of primeval creation, so can't be redefined: it parametrises ideals I c ldx0,., x n] such that (x0) n I -- x0I

and I / x 0 I = I, and is a locally closed subscheme of the Grassmannian of

I_ 2, and D a Cartier divisor on C such that 2D ~ KC; assume that C and D are not in the 4 exceptional cases (i-iv) below. Then the graded ring R(C, D) is generated in degrees < 3 and related in degrees < 6. Exceptional cases: (i)

C is hyperelliptic of genus g ~ 2 and h0(O C (D)) = 0; in this case R(C, D) is

generated in degrees < 4 and related in degrees _ g + 1 on an irreducible curve C. (3.5) Set-up for the proof of (3.4). This section waltzes through the major case of a nonhyper-

247

elliptic curve C; the tr/o section §4 covers the relative minor case when C is hyperelliptic in much more detail; (see (3.11) if you don't know what it means for an irreducible Gorenstein curve C to be hyperelliptic). When g = 3 and C is nonhyperelliptic then either (iv) holds, or h0(C, 0 c ( D ) ) = 1;

I offer the reader the lovely exercise of seeing that in this case, which

corresponds to a plane quartic with a bitangent line, R(C, Oc(D)) is a complete intersection ring R(C, D) = k[x, Yl, Y2, z]/(f, g)

with

deg(f, g) = (4, 6).

Thus I suppose throughout this section that C is nonhyperelliptic and g > 4. Introduce vector space bases as follows: x 1,.. x a E H0(D); Yl,'" Yg E H0(2D) = H 0 ( K c ); z 1,.. Z2g_ 2 ¢ H0(3D). Write I(m, n) for the kernel of the natural map qgm,n: H 0 ( m D ) ® H0(nD) ~

H0((m+n)D),

and V~;m,n: H0(~D) ® I(m, n) ~

I(f+m, n)

for the natural map.

(3.6) Main L e m m a . (I) 9m,2 is surjective for every m > 2;

(II) I(m+2, 2) = im ~2;m,2 + i m Vm;2,2. This result is similar to [Fujita2, Lemma 1.8]; the proof occupies (3.8-10) together with a technical appendix.

(3.7) L e m m a 3.6 ~

Theorem 3.4. (I) implies by induction that if m = 2~ > 2 is even, then

H0(mD) is spanned as a vector space by the set St(y) of monomials of degree £ in the Yi; and if m = 2£+I > 3 is odd then H0(mD) is spanned as a vector space by the set z ® S f-1 (y) of monomials of the form zj times a monomial of degree ~-1 in the Yi. This obviously implies that R(C, D) is generated in degrees < 3. The relations in low degrees can be written

248

deg 2

xixj = Lij(Y )

(linear forms)

deg 3

xiYj = Mij(z )

(linear forms)

deg 4

xizj = Nij(Y )

(quadratic forms)

deg 6

zizj = Pij(Y)

(cubic forms).

These relations clearly allow any monomial of degree m in the x i, yj, z k to be expressed as a linear combination of S£(y) if m = 2_£ o r o f z ® S£-l(y) ff m = 2£+1. For the relations, suppose that

Fm: fm(X, y, z) -- 0 • R m is a polynomial relation of

degree m between the generators x, y, z of R(C, D). I must show that F m is a linear combination of products (monomial) x (relation in degree < 6). Any term occuring in F m can be expressed as a linear combination of monomials S £(y) or z ® s~-l(y)

by using products of the relations just tabulated. Therefore I need only deal with

linear dependence relations between these monomiats in R m (for m >_7). By just separating off one Yi in each monomial in an arbilrary way, a linear combination of these monomials in R m can be written as the image of an element ~ ~ Rm_ 2 ® R2; to say that it vanishes in R m means that ~ • I(m-2, 2). But then I_emma 3.6, (II) says that e im ~g2;m-4,2 + im ~gm-4;2,2. This means that the relation in degree m corresponding to ~ is a sum of relations in degrees m - 2 and 4 multiplied up into degree m. By induction, the result follows.

Q.E.D.

(3.8) Proof of (3.6, I), and notation. Let A -- P3+.. Pg be a divisor on C made up of g-2 general points. Since C is nonhyperelliptic, K C is hirational, so that 12D - AI = IKC - At is a free pencil (by general position [4 authors, p.109]); hence the free pencil trick gives the exact sequence 0~

H0((m-2)D+A)

---.

H0(2D-A)®H0(mD)

~

H0((m+2)D-A)

m =2

1

2xg

2g-1

m _>3

(m-2)(g- 1)-1

2 x (m- 1)(g-l)

m(g-1) + 1;

the indicated dimension count shows that the right-hand arrow

249

H0(2D - A) ® H0(mD) ~

H0((m+2)D - A) ~ 0

is surjective. Let t m ~ H0(mD) be an element not vanishing at any of P3,'" Pg, and, as in the Pelri analysis, choose the basis Yl,'- Yg of H 0 ( K c ) such that Yl, Y2 bases H0(2D - A), and yi(Pj) = ~ij for i, j = 3,.. g (Kronecker delta). Then by the free pencil trick, H0((m+2)D-A) = H0(mD)y l ~ H 0 ( m D ) y 2 , and, obviously, troy i for i = 3,.. g forms a complementary basis of H0((m+2)D). This proves (3.6,I). Similarly, H0((m+4)D-A) = H0((m+2)D)Yl + H0((m+2)D)Y2 and tm+2Y i for i = 3,.. g is a complementary basis of H0((m+4)D). (3.9) As u runs through H0(mD+A), the relations p(u) = uy 1 ® Y2 - uY2 ® Yl ~ I(m+2, 2) express the fact that H0((m+2)D)y 1 n H0((m+2)D)y 2 = H0(mD+A)yly 2 c H0((m+4)D - A), which is part of the free pencil trick. The key to (3.6, II) is to prove that for m > 3, p(u) ~ im ~2;m,2

for all u e H0(mD+A);

since ~ ® p(v) = p(~v) for ~ e H0(2D) and v e H0((m-2)D +A), this follows trivially from Claim.

H0(2D) ® H0((m-2)D +A) ~

H0(mD +A) ~ 0

is surjective. (3.10) Proof of (3.6, II). Claim 3.9 is proved in (3.15), and I first polish off (3.6, II) assuming it. Suppose that m > 3. Step 1. The subspace {p(u)} = p(H0(mD+A)) is the kernel of H0((m+2)D) ® H0(2D-A) ~

H0((m+4)D-A) c H0((m+4)D),

250

and the tmYi ® Yi map to a complementary basis. Therefore, a subset S c I(m+2, 2) = ker {H0((m+2)D) ® H0(2D) --+ H0((m+4)D)} will span I(m+2, 2) as a k-vector space provided that (i) and (ii)

S contains the p(u); S spans a subspace complementary to H0((m+2)D) ® H0(2D-A) ~ ~ k'tmYi ® Yi,

in other words, any r I ~ H0((m+2)D) ® H0(2D) can be written r 1 = rlS+112D_A+rl t

(*)

where rl2D_ A ~ H0((m+2)D) ® H0(2D-A) and 11S, rlt are linear combinations of S and of the tmYi ® Yi respectively. Step 2. Now set S = im ~g2;m,2 + im ~gm;2,2- By (3.9), im ~q2;m,2 contains p(u) for u H0(mD+A). Therefore, it is enough to verify (,) for any T1 E H0((m+2)D) ® H0(2D). Break up H0((m+2)D) ® H0(2D) as a direct sum of the following 4 pieces: V 1 = H0((m+2)D) ®H0(2D-A); V 2 = H0((m+2)D-A) ® ~ k'Yi; V 3 = ~ k-tm+2Y i ® yj

summed over i, j = 3,.. g with i ~ j;

V4 = ~ k'tm+2Yi ® Yi for i = 3,. g. For V 1 and V 4 there's not much to prove. Also since H0((m+2)D-A) = H0(mD)y 1 + H0(mD)Y2 and R(2, 2) contains Yl ® Yi - Yi ® Yl and Y2 ® Yi - Yi ® Y2 for i = 3,.. g, it follows that V 2 c V 1 + im ~gm;2,2" Finally, for the summand V3, note that for i, j = 3,.. g and i ~ j, YiYj ~ H0(4D-A) = H0(2D)Yl + H0(2D)Y2 , so that I(2, 2) contains the Petri relation Yi®Yj - a i j ® Y l - b i j ®Y2

with aij,bij e H 0 ( K c ).

Therefore also tm+2Yi ® yj ~ V 1 + im ~gm;2,2.

251

This completes the proof of (3.6, II), modulo Claim 3.9.

Coda to §3. 'General' divisors and the proof of (3.9) (3.11) L e m m a (the hyperelliptic dichotomy). Let C be an irreducible Gorenstein curve of genus g =PaC>2. (i) (ii)

The canonical linear system IKcI is free; K C is very ample unless 9 K is a 2-to-1 flat morphism to a normal rational curve.

Proof (See [Catanese, §3] for a discussion of a more general problem; however, my proof of (ii) seems to be new even in the nonsingular case!). (i) Suppose P • C is a base point of IKcI; then h0(mp.~c(Kc)) = g and by RR h l ( m p . 0 c ( K c ) ) = 2, so by Serre duality the inclusion Hom(0c, 0C) = k c Hom(mp, 0C) is stlict. A nonconstant element of Hom(mp, 0 C) is a rational function h • k(C) such that h . m p c 0 C. Since d e g h - m p - - d e g m p = - l ,

it is easy to see that

h.mp = m Q

for some

P 6 Q • C, and it follows that P and Q are Cartier divisors on C, hence nonsingular points, and as usual h defines a birational morphism C ~ [P 1, necessarily an isomorphism. (ii) If 9K: C ~ p g - 1

is not birational then it is clearly 2-to-1

to a normal rational

curve. Suppose it is birational to a curve of degree 2g-2. If A = P3+.. Pg is a divisor on C made up of g - 2 general points then IKC - AI is a free pencil by general position, and arguing as in (3.8), sd(H0(Kc)) ~ H 0 ( d K c )

is surjective; thus the ring R(C, KC)

is generated by

H0(Kc). Therefore the ample divisor K C is very ample. Q.E.D.

(3.12) Claim 3.9 will also follow from the free pencil trick, once I prove that the divisor A = P1 +'' P g - 2 made up of g - 2 general points is 'general enough' for

ID+AI to be free and

birational.

Proposition. Let C be an irreducible Gorenstein curve of genus g, and D a divisor class such that 2D ~ K C. Let A = P3 +'" Pg be a divisor on C made up of g-2 general points. Then (i)

Suppose that g > 3 , andthat C is nonhyperelliptic if g = 3 ;

then h 0 ( O c ( D ) ) < g - 2 ,

so that H0(C, 0 c ( D - A ) ) -- 0 and h0(C, •c(D+A)) = g - 2. (ii)

Suppose that g > 4, and that C is nonhyperelliptic if g = 4; then ID + AI is free; it's

a free pencil if g = 4.

252

(iii)

Suppose that g > 5, and that C is nonhyperelliptic if g = 5; then (PD+A is

birational. (3.13) Proof of (3.12, ii). It's enough to prove Hom(mp, Oc(D-A)) = 0 for every P e C, since then by duality and RR, h0(mp.0c(D+A)) = g - 3 < h0(Oc(D+A)) = g - 2, and ID + AI is free. Case H00~) = 0. Then h0(mp-Oc(D) ) -- 0 for every P • C, so by RR, hl(mp.0c(D)) = t. By Serre duality, dimHom(mp, 0c(D)) = 1 for every P • C; hence there is just a 1-dimensional family of effective divisors A (of any degree) with Hom(mp, 0 c ( D - A ) ) ~ 0 for any P. Since A varies in a family of dimension g-2 > 2, it can be chosen to avoid this set. Case H 0 ( D ) ~ 0. By RR and duality, the inclusion H0(Oc(D)) c Hom(mp, Oc(D)) is strict only for P in the base locus of IDI; therefore I ca0 assume that the general divisor A imposes linearly independent conditions on each of the vector spaces Hom(mp, 0c(D)) (there are in effect only finitely many of them). So if Hom(mp, Oc(D-A)) ~ 0 for P • C then dim Hom(mp, Oc(D)) > g - 1. Using RR and duality as usual, this is the same as h0(mp.0c(D)) _> g - 2. This contradicts (a singular analogue of) Clifford's theorem: by the linear-bilinear trick, the map S2H0(mp.Oc(D)) .--¢ H 0 ( m p 2 . 0 c ( K c ) ) has rank > 2h 0 - 1 (with equality if and only if the image of C under the rational map defined by H0(mp-Oc(D)) is a normal rational curve), so g > h 0 ( m p 2 . 0 c ( K c )) + 1 > 2h0(mp.0c(D)) > 2 ( g - 2), that is, g < 4 and C is hyperelliptic in case of equality. This contradiction proves (ii). The reader can do (i) as an exercise in the same vein.

253

(3.14) Proof of (3.12, iii). This is very similar: I prove that there exists a nonsingular point Q such that Hom(mp, Oc(D+Q-A)) = 0 for every P e C; as before, RR and duality imply that h0(mp.0c(D+A-Q)) = h0(0c(D+A)) - 2, so that q~D+A is an isomorphism near Q. Case h0(D) < 1. Then h0(D+Q) = 1 for a general point Q, and fixing such a point, RR and duality imply that dim Hom(mp, 0c(D+Q)) = 2 for every P • C; hence the family of effective divisors A with Hom(mp, 0c(D+Q-A)) ~ 0 for any P has dimension 2, and as A varies in a family of dimension g - 2 >_ 3, I can choose it to avoid this. Case h0(D) > 2. I pick a general Q, so that h0(D+Q) = h0(D); then, as before, the inclusion H0(0c(D+Q) ) c Hom(mp, ~gc(D+Q)) is strict only for P a base point of ID+QI; so that there are only finitely many distinct vector spaces Hom(mp, (gc(D+Q)), and I can assume that the general divisor A imposes linearly independent conditions on each of them. Thus Hom(mp, 0c(D+Q-A)) ~ 0 implies dim Hom(mp, Oc(D+Q) ) > g - 1,

that is,

h0(mp.Oc(D+Q)) > g - 2.

As before, the linear-bilinear trick gives rk {S2H0(mp.0c(D+Q)) ~ H0(mp2.0c(Kc+2Q)) } > 2h0(mp.0c(D+Q) ) - 1, Now [K C + 2QI is free and H0(0c(Kc+2Q)) = g + 1, so g + 1 > h0(mpZ-0c(Kc+2Q))+ 1 > 2h0(mp-0c(D+Q)) > 2 ( g - 2); that is, g < 5 and C is hyperelliptic in case of equality. Q.E.D. (3.15) Proof of Claim 3.9. h0(D+A) = g - 2. If g _>5 then qOD+A is birational, so that I can choose a divisor B = Q 1 +'" Qg-4 made up of general points, and sections si e H 0(D+A) such that si(Qj) = ~ij" Then using the free pencil trick in the usual way shows that H0(2D) ® H0(D + A - B) --~ H0(3D + A - B) is surjecfive; if t e H0(2D)

doesn't vanish at Q1,.. Q4

then sit

for

i = 1,..g-4

is a

complementary basis of H0(3D+A). The statement for m _>4 is an easy exercise using the same

254

method. Q.E.D.

§4. Graded rings on hyperelliptic curves

(4.1) Notation, introduction. A nonsingular hyperelliptic curve of genus g is a 2-to-1 cover 7t: C - 4 [pl

branched in

2g+2

distinct points

{Q1,..Q2g+2}c[Pl,

lifting to points

{P1,-" P2g+2 } c C (see the picture below); the Pi • C are the Weierstrass points, the points of C for which 2P i • g 12. If D = ~ dip i is a divisor on C

made up of Weierstrass points, or

equivalently, invariant under the hyperelliptic involution t: C - 4 C, I am going to describe an automatic and painless way of writing down a vector space basis of H0(C, 0 C (D)),

and a

presentation of the ring R(C, 0c(D)) by generators and relations. In a nutshell, the method is the following. Fix a basis (tl, t2) e H 0 ( p 1, 0(t)) = H0(C, g12) of homogeneous coordinates on IP 1 For each i = 1,.. 2g+2, let ui: 0 C c, 0c(Pi) be the constant section. Since 2P i • g 12, it follows that ui 2 • H0(C, g 12), SO that I can write ui 2 = £i(tl,t2),

(*)

where £i is the linear form in t 1 and t 2 defining the branch point Qi • p 1. Now it is moreor-less obvious that any vector space of the form H0(C, 0 c ( D ) )

has a basis consisting of

monomials in the ui, and that the only relations between these are either of a trivial monomial kind or are derived from (.). (4.2) Easy preliminaries. (i) The decomposition of n . 0 C into the (+l)-eigensheaves of t is ~ , 0 C = 01131 ~9 Olpl(-g-1), and the algebra structure on

~t,0 C

is given by a multiplication map

f: S2(01pl(-g-1)) =

0~1(-2g-2) --~ 01p 1, which is a polynomial f2g+2(tl, t2) vanishing at the 2g + 2 branch points Qi; (ii)

the Weierstrass points add up to a divisor in [(g+l)g121, that is PI+"P2g+2

(iii)

~ ( g + 1)g12;

locally near a branch point, 7t.tVc(P i) = Oip 1 • Olpl(Qi)-u i.

255

Remark. For any partition {P1,'" Pa} u {Pa+l,'" P2g+2 } of the Weierstrass points into two sets, P 1 +'" Pa + (g+l-a)g 12 ~ Pa+l +'" P2g+2, as follows from (ii) and 2Pi ~ gl 2. This will be important in what follows (see (4.5)); it corresponds to passing between the (+l)-eigensheaves of n.Oc(P 1 +.. Pa + kgl2)"

P1

P2

Pa

__

P2g+2

C

Q1 0

Q2 O

Qa 0

pl

0

%g+2 0

Proof. (i) is standard; one affine piece of C is C: (y2 = f2g+2(t))" It's easy to see that y/tg +1 is a rational function on C with div (y/tg +1) = P 1+.. P2g+2 - (g+l).g 12; this proves (ii). For (iii), if t is a local parameter on P 1 at a branch point Q • P 1 and u2 = t-(unit), then u is a local parameter at P • C, so 1/t has a simple pole at

Q and the

(-1)-eigensheafof n,OC(P) is O~l-u/t--Olpl(Q)-u. Q.E.D. (4.3) Simplest examples of graded rings. (a) Let D = g12; then H0(0c(D)) = (t 1, t2), and

H0(0c(kD)) = H0([p 1, 0[pl(k)) (3 H0(p 1, •[pl(k-g-1)); thus for k < g all the sections of 0c(kD) are in the (+l)-eigenspace, so no new generators are needed, and I get the final generator w e H0(0c((g+I)D)) in degree g+l satisfying w2= f2g+2(tl, t2). So R(C, g12)=k[tl, t2, w]/F, with deg(tl,t2, w, F) = l, l, g+l, 2g+2 ,

256

and C = C2g+2 c P(1, 1, g+l). (b) Let D = P with P E C a Weierstrass point; write P = P2g+2

and P 1,'" P2g+l

for the

remaining Weierstrass points, and u:OC~Oc(P)

and

v:OC~0c(PI+..P2g+I

)

for the two constant sections. Since u 2 : 0 C ~ Oc(2P) -- 0 c ( g l 2) is the constant section, I can choose the coordinates (t 1, t 2) so that u 2 = t 1, and t 2 e H0(C, 0c(2P)) is a complementary basis element. Now n.OC((2k)P ) = ~.Oc(kgl2) = •lpl(k) ~ O l p l ( k - g - 1 ) , and by (4.2, iii), ~,0C((2k+I)P) = X.0c(P) ® 01pl(k) -- 0[pl(k) (3 0[pl(k-g), so that monomials u£,u£-2t2,., base H0(0c(tP)) for £ _< 2g; but in degree 2g + 1 there is a new section z in the (-1)-eigenspace. Under the linear equivalence (2g+l)P ~ P2g+2 + g'gl2 "" P1 +'' P2g+l, z is the constant section v: 0 C c, ~gc(PI+.. P2g+l); in more detail, if y is chosen as in (4.2, ii) then diV(tlg+l/y) = (g+l)(2P) - (P1 +'" P2g+2) = (2g+l)P -- (P1 +'" P2g+l) so that z = v t l g + l / y . If f = f2g+l(tl,t2) is the form defining the 2g+l branch points in p l , then z 2 = f(u 2, t2), so R(C, P) = k[u, t2, z]/F,

with

deg(u, t2, z, F) = 1, 2, 2g+1, 4g+2,

and C = C4g+2 c P(1, 2, 2g+1). Remark. The ring R(C, g12) of (i) can be obtained by eliminating the elements of R(C, P) of odd degree; that is, R(C, 2P) = R(C, p)(2). This means replacing u by t l = u 2,

z by w = u z ,

and F: z 2 = f4g+2 by F': w 2 = u2f4g+2(u, t 2) = f'2g+2(tl, t2).

(4.4) Lemma, Let D be a divisor on C. Equivalent conditions: (i) (ii)

the divisor class of D is invariant under t, that is D ~ t ' D ; D -,, D' with D' = t*D';

257

(iii)

D is made up of Weierstrass points, that is (after a possible renumbering), D ~ P l + . . P a + b g l 2.

Proof. The implications (ii) ~

(iii) ~

(i) are trivial, so assume (i). By adding on a large

multiple of gl 2 if necessary, I assume that D is effective. If t*D ~ D but t*D ~ D then IDI is a nontrivial linear system. I pick one Weierstrass point, say P1; then the divisor class D - P1 is invariant under t, and ID - P11 contains an effective divisor, so that induction on deg D proves (ii). Q.E.D. Remark. Since D + t*D ~ (deg D).gl 2 for any divisor D on a hypereUiptic curve, a 4th equivalent condition on D is (iv)

2D ~ (deg D)-gl 2.

This set of divisors includes of course all divisor classes with 2D ~ 0 or 2D ~ KC, etc. Useful fact: each 2-torsion divisor on a hyperelliptic curve is (up to renumbering) of the form P1 +'" P2a - a'gl2 ~ P2a+l +.. P2g+2 - (g+l-a)'gl2 • Go on, check for yourself that there are 22g of these!

(4.5) Theorem. (I) For an invariant divisor D = P1 +'" Pa + bgt2 , set D' = Pa+l +'' P2g+2 + (a+b-g-1)gl2 , so that D ~ D' by Remark 4.3. Write u: 0 C ~ Oc(P 1 +.. Pa) and v: 0 C ~ 0C(Pa+ 1 +.. P2g+2) for the constant sections. Then ~,0c(D)

=

0[pl(b).u • 0[pl(a+b-g-1).v

and H0(0c(D)) = H0(0(b)).u ~ H0(0(a+b-g -1))-v. In other words, i f I write sk(tl , t2) for the set of (k+l) monomials tl k, tlk-lt2,., t2k (or ~ if k < 0) then H0(0c(D)) has basis sb(t 1,t2).u,

s a + b - g - l ( t l ,t2).v.

(II) Write fa(tl, t2) and g2g+2 -a(tl , t2)

for the forms defining

Q 1 +'' Qa and

Qa+l +'' Q2g+2 in [P 1. Then the graded ring R(C, 0 c(D)) is generated by monomials in R(C, 0c(kD)) for suitable initial values of k, and related by monomial relations together with relations deduced from

258

u 2 = fa(tl, t2),

Proof of (I). ~ . 0 c ( D )

v 2 = g2g+2_a(tl, t2).

has a uniquely determined 7 / 2

action compatible with the inclusion

0 C c., 0 c ( P 1 +.. Pa), and the (+l)-eigensheaf is clearly 0 p l(b)-u. Multiplication by the rational function Y/tl g+l ~ k(C) described in the proof of (4.2, ii) induces an isomorphism Oc(D ) ~, Oc(D'), and since

Y/t1 g+l

is in the

(-1)-eigenspace,

the isomorphism interchanges the

(+l)-eigensheaves. This proves (I). (4.6) I will regard (II) as a principle, and not go into the long-winded general proof, which involves introducing notation k0+-, kl -+ for the smallest even and odd values of k for which each eigensheaf of ~x,0c(kD) has sections, and a division into cases according to which of these is smaller. I now give a much more precise statement and proof of (II) in the main case of interest. Suppose that, in the notation of Theorem 4.5, b > 0 Note that

2D = (a+2b).gl 2,

and

so that

a+2b < g+l

_< 2 a + 3 b .

~ .Oc(2D) = 0(a+2b) •

0(a+2b-g-1).uv,

where

uv: 0 C c, 0 c ( P 1 +.. P2g+2) is the constant section. Write V + to denote the (+l)-eigenspaces of a vector space on which t acts; the point of these inequalities is just to ensure that H0(D) + = H0(0(b)).u ~ 0, H0(2D) - = H0(0(a+2b-g-l)).uv = 0, (so also H0(D) - = 0) H0(3D) - = H0(0(2a+3b-g-1))-v ~ 0. Notice that this case covers all effective halfcanonical divisors on a hyperelliptic curve of genus g > 4, for which a + 2b = g - 1. Theorem. The graded ring R(C, D) is generated by the following bases: (x0, xl,..Xb)

=

sb(tl,t2).u

=

tlbu, tlb-lt2u,..t2bu

~

H0(D)+;

(Y0, Yl,'" Yd)

=

sd(tl, t2)

=

tld, t l d - l t 2 ''" t2 d

~

H0(2D)+;

(z 0,z 1,..z c)

=

sc(tl ,t2).v

--

tlcv, tlc-lt2v,..t2Cv

~

H0(3D) -.

where I set d = a+2b = deg D and c = 2a+3b-g-1

for brevity. The relations are given as

259

follows: P

rk/X0

Xl

..

Xb-1

Y0

Yl

[

x2

..

Xb

Yl

Y2

..

Yd-1

z0

Zl

•-

Yd

Zl

z2

o.

Zc-1]

-k.

The divisor A p = P(1) + k.gl 2 is a Cartier divisor of degree of degree 2k + 1 on C, and plays the role of 2k + 1 coincident Weierstrass points. Node-like points.

In local analytic coordinates, y2 = x2k. At such a point P e C there is a

unique nonzero Cartier divisor P ( 0 ) - - d i v p (y/x k) of degree 0 such that t ' P ( 0 ) = P(0). This satisfies 2P(0) = 0, and the pull-back of P(0) to the normalisation is 0 (since y / x k = +1 is invertible at the two points), but P(0) + igt2 is effective

~

i>k.

261 The divisor A p = P(0) + k.gl 2 is a Carder divisor of degree of degree 2k on C, and plays the role of 2k coincident Weierstrass points. The divisors ~ A p summed over distinct branch points P and of degree a < g - 1 are characterised as the Carder divisors on C invariant under t and with h 0 = 1, in complete analogy with sums of distinct Weierstrass points. Now by analogy with Lemma 4.4, it can be seen that any Carder divisor (or divisor class) on C invariant under t is a sum of divisors of the form P(1) for cusp-like P, of divisors of the form P(0) for node-like P, and of a multiple of gl 2. Any effective Carder divisor D invariant under t is of the form D = ~ A p + b g l 2 , with b > 0 , summed over a subset of the branch points P, and as in Theorem 4.5, if I set a = deg ~ Ap and write ~ ' Ap for the complementary sum, then D ~ D' where D' -- ~ ' A p + (a+b-g-1)gl2 , The statement and proof of Theorems 4.5-6 now go through with only minor changes.

Chapter III. Applications

§5. Numerical quintics and other stories

(5.0) Preview. In this toccata section I work out in detail the deformation theory in degree _ 1. Observe, by definition of Tz, we have

(3.11)

M°z = H ° ( K x + L) ® Op z

0

Definition 3.12. Let [Z] e F~ and Pz as in (3.1). The filtration M z :-- { M ~ D M ~ D ".. D M ~ } will be called the period filtration for Z with respect to Ox(L). The index of the last piece of AAz will be called the weight of the filtration. R e m a r k 3.13. By definition, M ~ ' s are torsion free sheaves on Pz (in fact, they are locally free in codim > 3, since they are 2-nd syzygy sheaves (see [4])). Put mi = rk(Ad~) and consider the flag variety: k

F(k,~; n°(Kx + L)) = {[M] e I I a~(m,,n°( A'X + L)) i=0

M={H°(Kx+L)=M

° DM 1 D'"DM}},

[M i] C G r ( m i , U ° ( K x + L))}

where (3.14)

where k is the weight of M z and rfi = ( m 0 , m l , . . . ,ink). If there is no ambiguity we omit the indices in the notation F(k, rfi; H ° ( K x + L)). So we obtain the desired map

az : ez ....

~ F(k, rfi; g ° ( K x + L))

(3.15)

294

by sending ~ E Pz to the point of the flag variety defined by Adz,~. R e m a r k 3.16. 1) It is straightforward to define the period filtration and the corresponding map into the flag variety for families of 0-cycles. O

2) If [Z] E P~, then the filtration Adz = {Ad}} is trivial and hence is az.

2. A polarization of (a, Y, F) We turn now to the canonical map ~ whose existence was asserted in (1.7). Recall the morphism e : $ ' T z ) H ° ( O z ) ® COpz as in (3.8). This induces the morphism

~z : Z x e z

, e(:f~)

t:(~S///q'(Z)

(3.17)

PZ where t2(Z) and q(Z) are the natural projections. The map ~ can be defined by setting /¢

for every [Z]

(5

t~l([z])

=

/~Z

F (refer to (1.6) and (1.7) for notations).

Finally, to define the divisor D z in P(q-z) we dualize the image of az. precisely, consider T = z x Pz xp~ P(Tz)

~

~ / Z x Pz

More

(3.18) P(q'z)

Consider the morphism of sheves on T: r1 *

¢;)(-1

))

' , OT

g;)(-1)

where Op(q-j)(1) (resp. Op(¢z)(1)) is the tautological sheaf on P(q-~) (resp. on P(7"z)) such that (q}),(Op(¢~)(1)) = q-z (resp. (qz),(Op(q-z)(1)) = q-z. Put DT = (s = O) and define D z = (v2),(DT). Again we can define Dy/r as in Definition 1.5 by setting D v / r l q - l ( p z ) = D z , where q : 17" = P(q'z) , Y is the natural projection. 3. T h e infinitesimal p o l a r i z a t i o n of (a, Y, F) We begin by explaining the morphism p in (1.8), 2). The differentiM of a z in (3.15) induces the morphism +1 ®

, % ®

i-

295

where U is the Zariski open subset of Y where .M~'s are locally free. This morphism factors M~÷ 1 ® o r ........., a b ® M~+ M~ 1 (3.19) (see §4, for details). Combining (3.19) with inclusions Jk4~+1 ~ 3,4~ we deduce

MW * o~ ~ fib ® M~ which is equivalent to

(M~

Ov

' Homou ~ M ~ + , ,

M'~-~) A/it

(3.20)

The restriction of the morphism in (3.20) to O u/r is the morphism pt as in (1.8), 2). To describe the morphism p0 in (1.8) 1), we recall the morphism Resz in (3.6) which gives rise to the following diagram 0

l H ° ( K x + L)* ® flez(1)

l H°(Oz) ® Op z

"~ H ° ( K x + L)* ®

H ° ( O z ( K x + L)) ® Oez H ° ( K x + L)

(3.21)

l H ° ( K x + L)* ® Op z (1)

l 0

This induces the morphism

Cz

, H ° ( K x + L)* ® ~tpz(1 )

(3.22)

Observe: the subsheaf Op z ~ H°(Oz)®Of, z (see (3.5)) is in ker(r) of (3.21). This implies that the morphism in (3.22) factors through Tz = _Tz_ to give the morphism Or z Tz

) H ° ( K x + L)* ®apz(1)

296 which is equivalent to , H ° ( K x + L)* ® T~ ® Opz(1 )

p°:Oe z

We can d e f n e p as in (1.8) 1), by setting

p°l~_,([z] ) = p°l. z

=po

for every [Z] E F.

§4. Geometry of 0-cycles and the Infinitesimal polarization A reason that the InfinitesimM polarization is related to the geometry of the underlying~ 0-cycles comes from being able to identify Ty with O Y/r' on the Zariski open subset Y of Y corresponding to the locally free sheaves. LEMMA 4.1.

There is a natural morphism hr : 7"y

, 0 v/r

o

which is an isomorphism on Y, the subset of Y corresponding to the locally free sheaves. Proofi

We show how to define h z = hr[pz. From (3.5) we have

Tz "~ ker (R:p,. (E') L ,,,here :rz = ~

OP z

(4.2)

H l ( O x ) ® Opz )

(see (3.6) and (3.7)). Tensoring (3.2) with p~Ox(Kx) and t~ing its

direct image under p] we deduce

H i ( K x ) ® Opz(1)

; Rlpl*(e ®p~Kx)

>H ; ( f f z ( K x + L)) ® Op z

, H 2 ( K x ) ® OF, z(1)

,0

(4.3)

Dualizing (4.3) yields 0

, Opz(-1 ) --÷ Hi(yz(I(x

+ L))* ® Op z

, H'(Ox) ®

> (R]pl-(g "® p ~ K x ) ) *

Opz(-1 )

,

(4.4)

The last two terms on the left in (4.4) are part of the Euler sequence twisted by Op z ( - 1 ) (recaIi: H I (Jz(I(x +L))* = E x t I (Jz(L), Ox) and (3.1) for definition of P z ) . So (4.4) implies O p z = k e r ( R l p p ( £ ® p f l i. x. .). ® Opz(1) 5 >H l ( O x ) ® O p z ) (4.5)

297

Consider the pairing

Rip1° (E') @ Rip1. (E ® p;Kx) .....,R2p,. (det C ® p ; O x ( - L + Kx))

Ildef

II

R'pl. (E ® p ; O x ( - L ))

(4.6)

H2( Kx ) ® Op z (1)

This pairing is non-degenerate at y C Pz such that Eu = El{y}xx is locally free: E'v = Ey ® O x ( - L ) = E; and the pairing (4.6) at y becomes Ha(E;) ® HI(Eu ® Ox(Kx)) , H2(Kx) which is the duality pairing. Rewrite (4.6) as follows: (4.7)

Rlpl*(~ ') "~ (R pl*(~" @p2I{x)) ® Opz(1) and consider the diagram R l p l . ( ~ ')

¢

,,

Rlpl.($®p~(Kx)) * ® O c t ( l )

H I ( O x ) Q Op z - -

H l ( O x ) Q Opz

where 7 and 6 as in (4.2) and (4.5) respectively. This yields the morphism

(4,2) hz : Tz = ker 7

~ ker 6 (4.J) O PZ

O

which is an isomorphism on Yz, the Zariski open subset of P z corresponding to the O

locally free sheaves, because ¢ is an isomorphism on Yz. Q.E.D. 0

R e m a r k 4.8. Let Y z be the Zariski open subset of Pz where hz as in the proof of Lemma 4.1 is an isomorphism. This induces an isomorphism

hz:9-z®Ooyz --~ [

- H ~ - Z } + L))]*®O{'z(1)

which follows from the diagram 0

,

OPz

,

0

~

Op z

~

q-z

,

Tz

Opz(1 )

+

Opz

....... ,

0

....... >

0

1 H ®

(4.9)

298

[ H°(Oz(Kx+L)) ] *

o

w h e r e g = E x t I ( f l Z ( K x + L ) ) = H I ( f l Z ( K x + L ) ) * = [ HO(gx+n).....J . F o r y 6 Y z the isomorphism [ H ° ( O z ( K x +L)) ]*

hz(y): Tz,~

t H~-~

r:

'

where Tz,y ~f "]'z ® k(y), is given by the residue map as in [2] (also [1]). Using (4.9) we can reinterpret the morphisms pi's in (1.8). Consider the pairing

7"z ® H°(Kx + L)

, H ° ( O z ( K + L)) @ Oez

defined by the multiplication. By (4.9) and definitions of M } ' s in (3.9) 2), we have ,

H ° (flz(Kx + L)) ® O o

Yz

This combined with inclusions M ~+1 ~-+ M ~ yields

®z® j~+l

j~

which factors through -'

Tz@-and is equivalent to

.=

,Uom(M

We claim (but omit a proof here) that this coincides with the differential of a z in (3.15) via the isomorphism (4.9). The above gives a simple way to compute p i I~.z as in (1.8)

2). Turning to the morphism P0 [Yz we recall

),2: S2q-z

>H2(-L) * ® Oe,(1) = (M})* ® Oez(1)

which factors through

S2Tz

, (M~,)* ® Cgpz (1)

This yields "Tz

, (.M°z) * (9 "T~ ® Opz (1)

Using (4.9) we arrive to (1.8), 1). The meaning of the fact that the Infinitesimal polarization is geometric is illustrated by the fact that it detects the decompositions of cycles in F~(L) into special subcycles. More precisely, we have the following

299 PROPOSITION 4.10 ( = THEOREM 2).

Let [Z] E P(i), then Z has special decomposition with respect to O x ( L ) (see Det~nition 1.12; for P(i), see the statement of Theorem 2).

Proofi

Let y E P z be such that hz(y) is an isomorphism, where hz a £ i n the proof

of L e m m a 4.1 and hz(y) its value at y. Set Tz,y = Tz ® k(y), the fibre of Tz at y, and Tz 0 the subspace of Tz,y isomorphic to ker(p~) (p~ as in the statement of Theorem 2) via hz(y). P u t Tz, y

~'z ® k(y), the fibre of Tz at y, and ,b(0 "- Z,y , the inverse image of

T Z,y (/) under the projection Tz,y

) Tz,y.

Observe: "b(/) ~z,y D Ho((gx) and it is a subring of H°(Oz).

To see this is enough

to show that ~(0 "~z,y to show that Z,y is closed under the multiplication: let f , g E ~(i). f • g E ~b(i) "z,y is equivalent to ( f . g ) . m E SO(Jz(Kx+i)) Mk~ for every m E M Z,y, i where M Z,y i is the fibre of M ~• at y and we use the interpretation o f p vi discussed in Remark 4.8. i

Mz'~vfor every m E M),y. Since g E ,-~(i) ~z,u we have: g . m E Ho(flz(K.t_L)~--~ +(i) mg = g • m and using the fact that f E * z,y we obtain:

( f . g ) . m = f ( g . m) = f . mg E

Putting

M Z,y i H ° ( J z ( K + L))"

The inclusion of rings 4~(0*z,vC H°(Oz) induces surjective morphism of schemes f :Z

~ Z' = S p e c /\' ~ (z,y]" 0

Take Z~, a proper subscheme of Z', and Z~ = Z' \ Z~. Seting Zi = f*(Z:) we obtain the decomposition Z = Z1 + Z2. This decomposition is special (in a sense of Definition 1.12) since

Pz, : H ° ( K x + L) ----* H ° ( O z , ( K x + L)), the restriction map, is not surjective for i = 1,2 (this can be seen as follows: there exists gi E ,~(i) * z,v such that gi ~ flz, and gi E H°(Oz\z~ ® flz\z,); this implies o = (Resz

¢I =

¢)

(4.11)

for every ¢ E H ° ( K x +L), where Resz as in (3.6). Since (Resz,)* = Pz, it follows from

(4.11) gi E ker(Resz, ) = [coker (Pz,)]*

Q.E.D.

300

R e m a r k 4.12. If Z in Proposition 4.10 is reduced, then for every p' E Z' the subcycle Zp, = f - l ( p , ) is special with respect to O x ( L + K x ) (i.e. h i (,.7"zp,(Kx + L))" 7~ O) for every p' E Z ~ and the special decomposition for Z is as follows:

Z= E

Zp,.

p' E z '

In particular, if O x ( L + K x ) is base point free, then deg(Zp,) > 1, for every p' E Z' (since F°(L) = 0; see (2.3) for notations); if O x ( L + K x ) i s very ample, then deg(Zp,) > 2, for every p' e Z' (since F°(L) = 0).

REFERENCES [1] F. Catanese, Footnotes on a theorem of I. Reider, preprint. [2] P. Grii~ths, et al, Topics in transcendental algebraic geometry, Annals of Math. Studies, Princeton University Press (1984). [3] P. Griifiths, J. Harris, Principles of algebraic geometry, John Wiley, New York (1978). [4] C. Okoneck, M. Schneider and H. Spindler, Vector bundIes on complex projective spaces, Progress in Math. 3 (1980), Brikhauser.

301

R E I D E R - S E R R A N O ' S METHOD ON NORMAL SURFACES

Fumio Sakai Department of Mathematics,

Dedicated

to Pro/.

Saltama University,

Dr. F. Hirzebrueh

on his 60th birthday

Let Y be a normal projective surface over C, visor on Y.

Urawa 338, Japan

and let D be a Well di-

G e n e r a l i z i n g the methods of Relder and Serrano, we prove a

criterion for the very ampleness of the adJolnt linear system

IKy + DI.

Introduction.

Recently,

Relder([Rd])

and Serrano

[Se] have discovered new methods

to study linear systems on a smooth projective surface. argument goes back to Mumford's proof theorem.

On the other hand,

surface

divisor

[Mi]

We will observe that by either

one can prove the following:

Proposition tive

[Mu] of the R a m a n u J a m v a n i s h i n g

Serrano has applied Miyaoka's version

of the R a m a n u J a m v a n i s h i n g theorem. method,

Relder's

1.

Let

D be a big

d{v£sor

If

HI(x,O(-D))

~ 0,

X.

E suoh

that

(1)

v{th then

(D - E ) E £ O,

D2 > 0 on a s m o o t h there

(li)

is

a nonzero

D - 2E i s

a big

projec-

elfeotive divisor.

With the aid of this result, we study adjoint linear systems on a normal surface Y. HO(y,~)

Let ~ be a line bundle on Y.

defines a complete linear system

For a point y E Y, let m that

Y

surjective.

We say that

H0(y,~) ~ ~ ® ( 0 / m y S ~/my,)

I~I and a rational map ¢~ of Y.

denote the maximal

I£] has no base points

IEI

The vector space

ideal sheaf of y.

if the maps H0(y,z) ~ £®(@/my)

separates

We say

Yy E Y are

rue distinot points if the maps

Yy ~ Vy'E Y are surjective.

We also say that

302

I~]

separates

Yy e Y a r e ls

tangent

surjective.

a morphlsm,

to-one

vectors

We k n o w t h a t

(ll)

morphlsm,

I£1

(ill)

separates [~[

i s a l o c a l embedding.

i f the maps H0(y,£)

(i)

]~l

has

two dlstlnct

separates

no base

points

tangent

~ ~®(0/m~)

points

¢=> @£

¢=> @~ l s

vectors

a one-

everywhere

¢=> @£

I~I) i s very a~p~e I f ¢~ g i v e s an

F i n a l l y , ~ (or

embedding o f Y, i . e . , ding.

everywhere

¢~ i s a o n e - t o - o n e morphlsm and i s a l o c a l embed-

Our maln r e s u l t i s t h e f o l l o w i n g :

Theorem

1.

d£v£sor

on Y .

d~v~sor. there

Let

tf

em£sts

Y be a normal Assume

that

the

D2 > 8 + ~ ( Y ) a nonzero

project£ve adjo~nt

+ y(Y),

e:fective

0 ~ DE < 4 + (p(Y)

surface, d~v£sor

then

and

Ky + D { s a C a r t i e r

}Ky + DI

d£u£sor

Let D be a nef

£~ v e r y

E on Y such

unless

ample

that

+ y(Y))/2

(~(Y) + y ( Y ) ) / 4 ~ E2 ~ (DE)2/D2, and

DE - 2 -

E2 < 0 £f DE = O.

Compared

with

the

smooth

involves

the

nonnegative

measures

the

sum o f

larities,

while

singularity.

Choose

Take

that

construct

the

an

a smooth

cohomology

Ideal

contributions the

dlvlsor

Theorem terms from

contribution

here

sheaf

d wlth

~ ~®(0/d) dlvlsor surface

J on X such

X, that

2 in

n(Y), the

from

h o w we p r o v e

effective

projective

(cf.

compensation

map H 0 ( y , ~ )

a nonzero

an effective

is

We s k e t c h

~(Ky + D). Suppose

y(Y)

the

case

the

Sect.3), Y(Y).

not

The

the

worst

above

non-rational

~ = ~.~(-J)

Set

support

surJectlve.

a blrattonal

as

~ =

o n Yo In

order

~:X ~ Y and

Conslder

the

sequence:

H0(X,~*~®0(-J))

HO(y,~®~)

~ H0(X,~*Z)

~ H0(X,~*~®(~/0(-J))

~

~

HO(y,~)

g®(O/~)

to

follows.

morphlsm c d.

n(Y)

singu-

theorem.

E o n Y, we a r g u e

result

term

non-Gorensteln

0-dimenslonal is

thls

~ HI(x,~*~®0(-J))

303

Since

@/J ~ e / M i s

jectlve, If

It

If

D is

big,

1 to

that

from

above

~*£®O(-J)

Proposition such

the

follows

we w r i t e O.

surjectlve, the

= e(KX + 9),

e.g., D and

if

one finds

c a n make E = ~ . E n o n z e r o . following

result.

Let

index

r.

If

ample,

(n(Y)

+ y(Y))/2.

The content

then

of this paper

faces.

In Sect.2 we define

maximal

ideal

provide

two proofs

Theorem

i.

I would

like

results

several

on normal

of P r o p o s i t i o n

i.

me by his p r e p r l n t

for

Sect.l

singularity.

Sect.4

E on X

J carefully, 1,

contains

we

we s h o w t h e surface

of

n Z 4 + r-l+

is devoted on normal

connected

to

surwith the

In Sect.8 we the proof of

1 to p l u r l - c a n o n l c a l

and plurl-

In Sect.6 we discuss

the

i. for encouragement,

on this topics,

[AS],

ample

~ O.

can apply

projective

invarlants

surfaces.

to thank A . S o m m e s e

of his papers

of Theorem

very

local

be sur-

HI(x,@(-D))

dlvlsor

about divisors

surface

of P r o p o s i t i o n

slons and c o r r e s p o n d e n c e

is

one

we c h o o s e

is the following.

sheaf of a normal

version

If

that

then

Q-Gorensteln

InrKyl

and technical

systems

we s e e

effective

big.

In Sect. S we apply T h e o r e m

antlcanonlcal

prints

2E i s

Y be a normal

notation

relative

by duality,

a nonzero

cannot

that HI(x,~*g®@(-J))

sequence

As a n a p p l i c a t i o n

prepare

~ g®(O/~)

~2 > 0 ( c f . Lemma 3 ) ,

(D - E ) E ~ 0 a n d D -

Ky i s

map H O ( y , g )

[SV].

helpful

discus-

and for sending me the pre-

I also

thank T . F u j l t a

for I n s p l r l n g

[F].

i. Preliminaries.

We r e f e r

to

be a normal There the

is

for

Molshezon

a Q-valued

numerical

reducible

[Sa3]

the

intersection

C o n Y,

results

theory

A dlvlsor and

on normal

By a d £ v i s o r

surface.

equivalence.

curves

basic

is

we m e a n a W e l l

on divisors. D is

surfaces.

neI if

pseudoeffeot~ve

if

Let

Y

divisor.

We d e n o t e DC 2 0 f o r DP a 0 f o r

all all

by irnef

304

divisors

P o n Y.

HO(y,O(mD))

= 0 for

The

Lemma 1.

(1)

D£s

(ll)

D belongs

divisor

Define

K(D,Y)

all

m > O.

foLLowing

biu

are

= tr,deg. We s a y

• HO(y,O(mD)) mkO

that

big

D is

-

If

1,

and

K(D,Y)

-~

if

= 2.

equivalent:

a n d D2 > O. to

the

positive

cone

(i.e.,

D2 > O, PD > 0 f o r

a nef

P).

Lemma 2 ( e f . [ R a ] ,

p.44).

D = D1 + D 2 v h e r e

both

Let DI a r e

D be a nef

divisor

pseudoefYective

~ith

If

D2 > 0 o n Y.

a n d D t ~ O,

then

we h a v e

D1D 2 > O.

Let

Lemma 3 .

surfaces.

Let

~ Y be a b{rationaL

f:Y

9 be a divisor

(i)

If

D ~s big,

(il)

If

9 2 > 0 and

Proof.

(i)

Since

f*(D)

K(D,Y) big.

If

flo~ahezon

so

£ s D.

if

D is

big,

2.

~ ~(9,Y).

blg,

Let

s urfacea.

big.

then

= D + G+, we i n f e r

-D w e r e

and set

Set

then

f:Y

Let

Since be big,

~ Y be a birationaL 9 be a d~visor

effective D = f.D,

divisor

D2),

then 0 ~ DE < c(/2 DE - ct/4 ~ E2 £ (DE)2/D 2, E2 < 0 £ f DE = O.

with

floishezon

Then

G+ ~ O, G_ ~ O. (6.2)

9 2 > O, which

morphism

and

that

in

either

is

on Y v£th

E such

~ = (D 2 -

normal

D £s big.

from Theorem

(il)

G would

of

D = f.D.

f * D = D + G a n d G = G+ - G

~s a nonzero

- 2E i s > O,

+ G

then

= ~(D + G+,Y)

Proposition

there

Write

on Y,

~orphis~

D or

absurd.

of

E = f.E.

that -D i s

Q.E.D.

two norma$

9 2 > O. (a)

[Sa3]

Suppose

(D - E ) E £ O,

If

D is

hey

(b)

and E

305

Proof. = ~2 f*E

It

follows

+ ~ > 0. = E

+ r.

V 2 + GV.

from

By

Lemma

Then

have

The

then

2,

the

index

E 2 < 0.

Local

~

~ 0,

=

that

D

- 2E

(D - 2 E ) E

(f*(D

- E)

is big.

> 0.

- G

Note

Write

+ r)(f*E

f*D

- r)

=

that

= D

D2

+ G,

(D - E ) E

-

(V - ½ G) 2 _ ~1 G 2 $ - ~1 G 2 = ~ / 4 .

inequality:

D 2 > ~,

Let

2,

DE

(i)

Thus

Hodge

Since

3,

(D - E ) E

(D - E ) E

We

Lemma

DE

theorem

in

the

- ~/4

on Y

above

$ E 2 < DE/2,

gives

range

the

of

DE,

so

that

inequality:

0 $ DE

E2 $

< ~/2.

( D E ) 2 / D 2.

(DE)2/D 2 £ DE/2.

If D E

= 0,

Q.E.D.

invariants.

(V,y) be a germ o f a normal s u r f a c e s l n g u l a r l t y .

maximal i d e a l s h e a f o f y. w i l l be u s e d i n S e c t . 4 .

Let m be t h e

We now p r e p a r e some l o c a l i n v a r l a n t s , which L e t ~:U ~ V be t h e m i n i m a l r e s o l u t i o n o f y.

There e x i s t s an e f f e c t i v e Q - d l v l s o r A s u p p o r t e d on - l ( y ) ~*KV = KU + A.

Note t h a t Supp(A)

double p o i n t .

Cf.[Sa3].

= -l(y)

such t h a t

unless y is a ratlonal

Define

8 = _A2

The num~r£caL sA i s In

integral.

this

then larity,

£nd~x of y is

case,

that

that

y Is

to

be the

Gorenste{n

s = 1,

Let

Z be the

m m ~.e(-Z)

and

0/m m m . ( ~ / ~ ( - Z ) ) .

Artin

To c o v e r a nonzero

Recall

defined

[A] the

non-rational

effective

(*)

A -

(**)

~.¢(-F)

showed that

Y is

divisor m-nef, c m2 .

if

fundamental

mn ~ n . 0 ( - n Z ) case,

least

consider

integer

0 ( K V) cycle

If

y is

ls

- F)C ~ 0

trivial

of y,

that

near

y

One k n o w s

a rational

singu-

Y n ~ 1. the

following

V on -l(y). 1.e.,(A

s such

Y C on -l(y),

conditions

of

306

If F satisfies There

(~),

then RIK.o(-F)

is a unique minimal

the

adjo~nt

dim

R l n . @ U , Za Z Z,

stein.

fundamental

Set

ma

divisor

cycle

sa

= O.

of

Z a satisfying y.

Z Za Z ~.

= n.e(-za).

So O/K.@(-F)

It

In

then

(~).

turns

particular,

One has

a

m K.(@/@(-F)). We call this Z a

out Za

canonical

that

= a

hl(Oz a)

if

y

surJection

is

=

Goren-

@/m a ~ @ / m .

Define u = -(Z a Among effective such

that

4) 2.

divisors

-(Z t

surjection

-

-

4) 2

e/n.O(-z

F satisfying

takes

t)

the

(.)

and

minimum.

~ e/m 2.

In

(~),

view

let of

Zt

(++),

be

the

there

one is

a

Define

= -(Z t _ 4) 2 - p. In g e n e r a l , i t n.0(-ns4).

i s not easy t o f i n d Zt .

To e s t i m a t e p, v, s e t ~n =

I f y i s not a r a t i o n a l double p o i n t , then ~n c m.

the l e a s t p o s i t i v e i n t e g e r e such t h a t ~e ¢ m2"

Let e be

T h e n esA s a t i s f i e s (+)

and (*+), hence £

(s

-

1)25,

~ + ~ ~

(es

-

i)25,

To indicate+ y, we w r i t e as my, may, 4y, Zy, Zay, Zty, 5y, py, Vy.

Lemma 4.

If

y

not

double

e ~ 3

e ~ 2

(ii)

~f y £s G o r e n s t e £ n ,

then

the

map

4 m -K U on

this.

Suppose

@(Kv)®Yn Y2~2

po£nt,

£f

s

= 2 and

then

e

~

then

e ~ 5 £f

s = 1,

4.

By Theorem 3.2 in [L2], i f both -ps4 - 2KU and -qs4 - 3KU are

~-nef, Since

s ~ 3,

a rat£onaL

(i)

Proof.

£f

is

.

Example

i.

® ]q

n-l(y)

and

y is Gorenstein.

~ In" Hence

~p

The ~4

result

c m 2.

Suppose

of

~ fp+q -4

is

surJectlve,

is ~-nef, We

Laufer

the

and

assertion

have

~.0(nK

[L3]

then

U)

5p+q

= 5p~q

(i)

follows

= ~.@(~*(nK

asserts

that

c m

V) ]4

by a single

smooth

.

from - n4)

= ]lJ3

Q.E.D.

that y is resolved

2

curve C of

= +

307

genus

g Z

2.

is n o r m a l l y In

Set

Z = 0c(-C),

generated,

this

case,

Z = C,

(d + 2 g

- 2)C

(if

Remark In

i.

case

y

(Laufer

3~

If

Za

[LI],

Reid

two

- 2),

is G o r e n s t e l n ,

method

give

if

Z = -C 2.

d Z 2g

= Z t = 2C,

d > 2g

is m i n i m a l l y

Reider's

We

y

e.g.,

d = deg

+ i,

A =

((d

If

then + 2g

= 2C

(if

then

p = 0,

v ~

if

6 ~ 3,

then

elliptic,

d = 2g

d Z 2g

2C

- 2 and

satisfies

- 2)/d)C, - 2).

(*),

and

Cf.

if

sA

(**).

=

[MR].

(e - i ) 2 6 ,

so

u ~ 96.

e = 2 and

so

p =

[Re]).

and

proofs

Serrano's

of

method.

Proposition

Proof by Mumford-Relder's method.

1 in

the

Introduction.

The non-vanishing of HI(x,e(-D))

gives r i s e to a v e c t o r bundle ~ with a n o n t r l v l a l e x t e n s i o n : 0 * ~ * ~ * O(D) * O. C l e a r l y , c i ( ~ ) 2 = D2, c2(~ ) = O, so t h a t c1(~)2 > 4c2(~).

One deduces

from the Bogomolov theory t h a t t h e r e i s an extension: 0 -~ 0 ( Q ) ® ~ where

£

scheme

and

an

Z and

0 ~ ~ ~ @(E).

is

IZZ It

hence

Q

~ ~; ~

lz~

~

Invertlble

sheaf,

is a b i g

divisor

is n o t

follows (D - E ) E

zero, that =

D

one

O, I Z is with

finds

= Cl(~)

(E + Q ) E

Since D i s big, i t

aiaO, ~i E Q, and l e t

Q2

an

ideal > 0.

effective

= Q + 2E,

< 0.

Proof by Mlyaoka-Serrano's method. D = P + N.

an

sheaf Since

of the

divisor

0 = c2(Y)

a 0-dlmensional composition

map

E such

~ -~

-- (E + Q ) E

that

+ deg

.Z

Q.E.D.

Consider the Z a r l s k l decomposition:

i s known t h a t p2 > O.

Write N = ~ I E I ,

[N] denote the i n t e g r a l p a r t of N.

I t follows

from the Mlyaoka-RamanuJam vanishing theorem [MI] t h a t HI(x,~(-(D-[N])) =

O.

So i f HI(x,O(-D)) ~ O, we must have [N] > O.

Consider a

308

sequence: exact

DO = D -

[N] .....

Dk_lEjk

> 0 for

HI(x,O(-D))

= O.

k < n

such

[N].

Set

(D P(D

that

-

k,

One

can

DkE j

E = D -

2E)

= p2

2.

(D -

Dk .

(111)

the

There

is

an

2

projective (i)

nonzero

(Relder

D2 > 4 ,

effective DE = O,

(ii)

nonzero

because

IID

construct

obtain

g 0 for

D -

the

the

a

sequence

E)E

2E) 2 + 4(D 2E i s

following

all

components

matrlx

EtE j

of

vanishing: DO . . . . .

components

(D -

D 2 = (D -

O.

obtains

irreducible

By L e m m a 1 ,

projective

surface,

tf

all

~

big.

Ej

~ O. -

One

E)E.

of

all

D -

Dk

that

finds

One has

also

Q.E.D.

properties Ej

of

Dk w i t h

of

E:

E,

irreducible

components

of

E

definite.

smooth

Theorem

(-Dk_ 1)

Jk

inductively

therefore

intersection

negative

~ eE

By c o n s t r u c t i o n ,

> O.

E)Ej

one

~ 0 for

We f u r t h e r

(i)*

For

~ e(-Dk_ 1)

all

2 E ) 2 m D 2 > O,

Remark

is

Dn = D.

sequence: 0 ~ e(-D k)

If

Dk = D k _ 1 + E J k . . . . .

2 > 8,

effective

surfaces,

[Rd],

and then

cf.

[BL],

Let

D be a nef

[K x

divisor

+ D[

IK x

d~visor

or

+ D[

[SV],

has

E such

E 2 = -1,

then

we h a v e

E 2 = -1,

DE = 1 ,

E 2 = 0 or

DE = 2 ,

E 2 = O,

D ~ 3E,

E 2 = 1.

or -2, -1,

following

Let

[Se]).

divisor

criterion.

X be a s~ooth

on X.

points

untess

there

ex£sts

that

very

E satisfy~ny

DE = O,

[F],

no base

DE = 1 ,

is

the

E 2 = O.

a~pge one

of

ungess

there

exists

the

:og~o~iny:

a

a

309

Proof.

This

is n o w a c o r o l l a r y

we i l l u s t r a t e

how Proposition

that D 2 > 4 a n d t h a t blowing

HI(x,e(K~

that D is b i g

(Lemma

zero effective E = ~.E. gives

We

divisor

4. A d j o i n t

linear

consider

the

divisor.

that E > 0.

and

systems

Let Y be a n o r m a l

Let Rat(Y)

non-ratlonal)

singularities

~(Y)

=

~

o n Y.

6y

+

and

denote

(D - E ) E K 0. 2 applies,

So D E = 0 or i.

let D be a d i v i s o r divisor

o n Y.

the l o c u s of r a t l o n a l

(resp.

y E Irr(Y)

{ ( ~ y - 8) + }

If

Irr(Y)

=

if

Irr(Y)

~

y e Irr(Y)

see

~

= max { ~ , 0 }

Sect.2.

taken

over

Theorem 3. d~v~aor

Note all

Let

for

that

~ ¢ ft. 8 + v(Y)

y e Irr(Y).

If

Y be a nor~a~

on Y a u c h

that

For

the

definition

= max { 8 , Y is

Mo{shezon

of

py} w h e r e

Gorenstein,

Ky + D £s a C a r t i e r

d£v~sor.

6y,

the

then

surface.

We

K y + D is a C a r t i e r

+

Here

If

Q.E.D.

= max

Set

and

~y

0 ~(Y)

t h a t ~2 > 0, so

Define

~

y E Rat(Y)

( D E ) 2 / D 2.

the a d J o l n t

Irr(Y))

By duality,

surface.

surface

in w h i c h

We

1 to D, w e f i n d a n o n -

then E 2 = 0.

on a n o r m a l

(resp.

- 2L.

Then Proposition

if D E = i,

Assume

c u r v e o f ~.

D 2 > 4 implies

DE - 1 ~ E 2 N

Moishezon

situation

Set D = ~ * D

Proposition

(1)

L e t ~ : X ~ X be the

that D - 2E is b i g a n d

E such

K E 2 < DE/2,

t h e n E 2 = -i,

~ 0.

Applying

x.

the e x c e p t i o n a l

The hypothesis:

3).

see e a s i l y

: DE-1

DE = 0,

~ 0.

- 2L))

By proving

this k i n d o f r e s u l t s .

a n d let L d e n o t e

+ ~*D

w e get H I ( x , 0 ( - D ) )

1 yields

3 in S e c t . 4 .

[KX + DI h a s a b a s e p o i n t

up of X at x,

must have

of Theorem

Let

~y a n d

Uy,

maximum i s

~(Y)

= O.

D be a nef

310

(i)

If

exists

D2 > 4 + ~ ( Y ) ,

a nonzero

then

effective

]Ky + DI

divisor

has no base

E such

points

unless

there

that

1 0 I DE < 2 + 5 n ( Y ) DE - 1 - ~1 ~ ( Y ) E2 < 0 if (li)

If

exists

tangent

a nonzero

(DE)2/D 2,

and

DE = O.

D2 > 8 + ~ ( Y ) ,

Y and separates

E2 I

i

then

I Ky ÷ DI s e p a r a t e s

vectors

effective

everyuhere

divisor

two distinct

oYI

E such

Irr(Y)

points

unless

on

there

that

0 & DE < 4 + ½ ~ ( Y ) DE - 2 - ¼ n ( Y ) E2 < 0 if (ill)

If

unless

there

+ V(Y), exists

then

DE - 2 - ¼ ( ~ ( Y ) E2 < 0 if

of T h e o r e m

(il)

and

Proof

of

Let

(iii)

~:X ~ Y be

= ~ ay,

+ £(Y))

~ E2 g

1 in the I n t r o d u c t i o n .

3.

the

Then we have

(DE)2/D 2,

We o m i t minimal

the

proof

resolution

of of

(i). the

Write

~

follows

Ay

the f o l l o w i n g :

-

~

(Z ay - Ay).

y E Irr(Y)

~ = e(Ky

singularities

= - ( a - A) 2, b e c a u s e

y E Rat(Y) prove

that

from

3.

za Y y E Irr(Y)

A - A =

E such

and

The assertion

~

~(Y)

divisor

vectors

+ v(Y))

and h =

We f i r s t

effective

tangent

DE = O.

of T h e o r e m

Theorem

]Ky ÷ DI s e p a r a t e s

a nonzero

0 ~ DE < 4 + ½ ( n ( Y )

Proof

and

DE = 0 .

D2 > 8 + n ( Y )

everyuhere

~ E 2 ~ (DE)2/D 2,

of

+ D). Y.

Set

A

311

Let

Lemma 5.

E be a nonzero

( 4 - h - E ) E ~ O.

Proof.

Assume

effective

Then E = n.E

d~u~sor

on X s u c h

that

> O.

to the contrary

that F, is exceptional

for ~.

Decompose

as E = ER + EI where ER (resp. EI ) are supported on ~-l(Rat(Y)) (Irr(Y))). We

also

We

have

(A

~-l(Irr(Y)). tion.

prove

y,

y'

be

H0(y,~)

~

(A

- E)EI

infer

both

y but

(f)

y,

y'

The

other

the

two

first

and

y'

y'

these

be

the

(a),

are

(d) We

can

be

let

~:X

we

H0(~,@*£)

~

because - A

A

- E)F.

- A

> 0,

(if

is

ER

> 0).

~-nef

on

a contradic-

done

from

~ X be

and

D -

big.

2.

that

We

smooth

with

(e)

the

the

canonical

divide

but

E Rat(Y),

the

into

y'

y

cases

blowing

ups

respectively.

is

surjective, Set

Since

It

E > 0.

D = ~*D

D is

effective is

easy

The

it

D = D2 -

Lemma 3 that

a nonzero

we f i n d

deal

is

the

six

cases:

E Rat(Y),

E Rat(Y), (a),

map

(c) y'

(d),

y

is

E Irr(Y)

and

(f).

similarly.

~ 0. ~ 0.

is

Suppose

D 2 > 8 + n(y).

at Set

y @

and

y',

and

= ~ o ~.

let

L,

Letting

~®(0/~)

we o b t a i n 2E

y'

y

that

U

~

HI(x,0(-D)) We s e e

Y.

surJective. (b)

here

Assume

H0(~,O*~®(0/0(-A-L-L')))

and

tion

+ ER)E R ~ 2

have

@/myeO/my,

~

not

curves,

HI(x,~*~®O(-A-L-L'))

5,

(A

(il). on

y,

?b

H0(y,~) 0/~

of

smooth,

exceptional

= @.0(-A-L-L'),

> 0.

= -(K x

F'I > 0), that

points is

E Irr(Y),

cases

case

have

- E)E R

(if

part

distinct

6 Irr(Y).

In

Since

- A

> 0

from

~®(e/myo0/my,)

smooth

L'

that

Q.E.D.

we

(a)

- A

We

Now Let

see

(resp.

follows + ~ -

~(Y)

big.

see

properties

A 8,

2L under

We a p p l y

divisor to

-

therefore

~.E

of

E then

2L'.

Then

our

hypothesis

Proposition

E on X such

that

that

> 0,

that

we

1 to (D -

and

hence

follows

from

by

E)E

~2 D, ~ 0

Lemma

Proposi-

312

In c a s e tively. K*D

(d),

As

let

Z,

in c a s e

Z'

(a),

+ A - A - Z - Z'.

~(Y)

8, we

In c a s e

can

(f),

surjective.

rest

is the

In order the

to

D2 > ~ ( Y )

+ ~

map H 0 ( y , z )

Y

is

Let

~ 2 : X * X1 b e

Let

up o f

we h a v e

the

and

blowing

that

~2

By using

Lemma

2 that

0 ~ DE

sition We

pass

Z t = Z yt° ~.O(-A-P). From

the

to t h e Set

cohomology

(1)

5, w e

can

The

consider

Assume that

Suppose

that

a smooth point. (u,v)

M = (u,v2). the

the

a point

We r e g a r d

at

Let

exceptional

y,

so that

~l:Xl

* X be

curve

Yl on L1,

curve

of

Set

¢ = ~ o ~1 o ~2" 2)

c ~,

By Proposition

prove

(ll)

that

E > O.

DE

In

we m u s t

flnd

D - 2E We

Let

~ g®O/~

have

implies

is blg. infer

Z = Zy,

be

Set

from

~ E2 ~

cannot

this

an effective

= Z t - Z a if y E I r r ( Y )

HO(y,~)

~2"

By d u a l -

assumption

- 2 - n(Y)/4

y is s i n g u l a r .

the map

the

i, w e

~ O,

and

P

8,

y.

corre-

L1.

¢.0(-A-L1-2L

over

which

exceptional

= D 2 _ ~(y)

if y E R a t ( Y )

sequence:

y E Y.

to

D = ¢*D + A - A - 2L 1 - 4L 2 .

(D - E ) E

in w h l c h

~ c m y' 2

be

that

the

of

< 4 + ~(Y)/2,

case

P = 2Z

Since

is blg.

that

cannot

L2 b e

Since

~2

~E.

let

~ M.

Since

E > 0 such

= D2

= D 2 _ ~(y).

coordinates

L1 be

transform

HI(x,O(-D))

divisor

Set D =

~2

we p r o c e e d

a point

where

and

ity,

that D

(iii),

y is

local

let

Set

so

~2

+ 8 otherwise.

in which

~ O.

~ O.

and

up o f X 1 a t

8/8v

¢.0(-L1-2L2)

Since

deduce

+ a - A,

Take

surJectlve,

strict

~ O.

o/maeo/m a,y Y

~

respec-

surJectlve.

HI(x,¢*~®O(-A-LI-2L2))

> O,

(ll)

> ~(Y)

case

y,

HO(y,~)

D = ~*D

We c a n c h o o s e not

~ O.

• O/m~, w e yEIrr(Y) J

of

not

the

Y at

the

the m a p

vectors.

ls

direction

L1 denote

case,

wlth

o n X.

the

rest

y E Irr(Y),

blowing

to

the

y',

(a).

if

the

sponds

in c a s e

tangent

* ~®0/~

(a).

~

of y,

HI(x,K*Z®O(-A-Z-Z'))

in c a s e

set

of

deal

a point

H0(y,z)

If w e

* Z®(0/m~)

We f i r s t y as

as

that

cycles

HI(x,@(-D))

~.(O/O(-A))

prove

separation

as

fundamental

we have

may c my,

~ O.

same

the see

Then

since

HI(x,~*~®O(-A))

we

proceed

Since

be

E =

Propo-

( D E ) 2 / D 2.

Z a = Zy,a •

Set ~ = surJectlve.

313

HO(x,n*~)

~

HO(x,n*g®(e/e(-A-F))) o

HO(y,£)

we d e d u c e duality,

~

that

e/Y

HI(x,~*Z®e(-A-C)))

we have

HI(y,~(-D))

~ O.

~ O.

Letting

D = ~*D

If y E Rat(Y).

then we have

D 2 _ ~2 = _(~ _ A - F) 2 = -(~ - A) 2 - 4 ( K X + Z)Z If y 6 Irr(Y), D 2 _ ~2 In e i t h e r sition (D

-

= _(a

case,

Remark

_ A)2

we

1 to this

in the f o r m e r

3.

In the a b o v e

ordeT

is the G o r e n s t e l n

Cartier (i)

Let

divisor If

÷ 8,

case,

See

We

effective

we get E = K.E the c o n d i t i o n s

(ill)

for

the

its

trick

Propo-

divisor

E with

> 0.

By P r o p o -

in

(li)

of u s i n g

systematic

+ Uy.

then a p p l y

if y is n o n - r a t l o n a l .

I learned

[BFS]

= n(Y)

study

if y is Q.E.D.

the and

ideal for

the

analysis.

If D is a C a r t i e r

Theorem 4,

in

proof,

(u,v 2) f r o m A . S o m m e s e . higher

a nonzero

that E s a t i s f i e s

the c o n d i t i o n s

_ (Z a _ ay)2}

so that D is big.

D, and we find

As

and

_ {(Z t _ &y)2

get ~2 > O,

2, we a s s e r t

rational,

= n(Y)

t h e n we have

E)E N O.

sition

+ ~ - A - C. by

divisor,

locus

one

can s t u d y

IKy + D] on Gor(Y),

which

Y.

Y be a normat

Moishezon

surlaee,

and

tet

D be anel

on Y .

D2 > 4 ,

then

IKy + DI

h a s no base

points

on G o r ( Y )

unless

there exists a nonzero effec~tve divisor E suoh that DE = 0 , (1I)

untess

If

D2 > 8 ,

-1

then

~ E2 < 0,

or

DE = 1 ,

IKy + DI s e p a r a t e s

0 ~ E2 ~ 1/D 2,

tuo

dist~not

points

on Gor(Y)

there exists a nonzero effective divisor E sat~sly~ng one ol the

fo~towin~ conditions:

314

DE = 0 ,

-2 K E2 K 0,

DE = 1,

-1

DE = 2 ,

0 g E2 ~ 4 / D 2,

D m 3E,

Corollary. Cartier

Let divisor

Gor(Y)

for

g E 2 ~ 1/D 2 ,

E2 = 1.

Y be a normal on Y .

Then

projective IKy + nH]

surface. separates

We s t u d y

plurlcanonical there

is

divisor.

least

such

The

i)

5.

r,

Let

such

InrKyI

1 + 3/(2r) (ii)

for

a positive r

is

points

on

Ky i s

no b a s e

+ ~(Y)/4

the

r

ample.

such

index

Q-Corenstein

systems. that

When Y i s

rKy is

a Cartier

o f Y,

projective

surface

uith

Then

points

for

n ~ 2 + r-l+

n(Y)/2,

and

for

n >

2 Ky ~ 2 / r .

if

separates

SYstems.

integer

called

Y be a normal

that

has

lnrKyl

Irr(Y)

two d i s t i n c t

and pluri-anticanonical

Q-Gorenstein,

index

H be an a m p l e

n ~ 4.

6. Plurlcanonlcal and plurl-anticanonlcal

Theorem

Let

tuo

n ~ 4 + r -1

distinct

+ n(Y)/2,

points and for

on Y a n d

is

very

n > 2 + 3/(2r)

ample

off

+ ~(Y)/4

if

2 Ky ~ 2 / r . (lli)

InrKyI

is

n > 2 + 3/(2r)

Proof. have If

(1)

Set

KyE ~ i / r

n ~ r-l+

satisfying

very

ample

+ (n(Y)

D = (nr for

all

2 + n(Y)/2, (1)

of

for

n ~ 4 + r-l+

+ r(Y))/4

-

1)Ky.

dlvlsors then

Theorem

3.

if

Since

rKy is

E > 0 o n Y,

D2 > 4 + n ( Y ) If

(~(Y)

÷ r(Y))/2,

and

for

2 Ky ~ 2 / r .

Cartier

and ample,

a n d we g e t

DE ~ n -

and

n > 1 + 3/(2r)

there

would

+ ~(Y)/4

we r

-1

be no E

and ~

~ 2/r,

315

then

D2 > 4 + ~ ( Y ) .

1/r.

In

would

have

Theorem

case

(a),

DE

- 1

3

cannot

Corollary

We d i v i d e we w o u l d

- ~(Y)/4

(cf.[B],

surface,

such

(i)

InKyl

separates

(ii)

InKyI

is

We can here

T h e o r e m 6.

4 -

r,

very

Let

such

r-l+

2.

the

-Ky is

+ y(Y))/2,

KyE ~ 2 / r , In

either

proofs

of

case

case, (ii),

(b)

(b),

we

E with (iii).

Corenstein

KyE =

(I)

of

Q.E.D.

projective

Then points

for n Z 5.

1 n ~ 5 + ~ y(Y).

result

ampleness

In

Y be a normal

two distinct

Y be a normal

that

(~(Y)

omit

for

(a)

DE Z 2 + n ( Y ) / 2 .

is ample.

a~pLe

the v e r y

two c a s e s :

(DE)2/D

Let

s h o w an a n a l o g o u s

state

index

We

[Sal]).

that K y

have >

exist.

into

for

the a n t i c a n o n i c a l

divisor.

We

part.

O-Gorenstein

projective

a~pLe.

Then

and

n > 2 - 1/(2r)

for

I-nrKyI

is

surface

very

a~p~e

+ (~(Y)

with /or

n

+ y(Y))/4

if

2 Ky a 2 / r .

Corollary.

-Ky i s

Let Y b e a normal

ampte.

Then

[-nKy[

is

Oorenstein

projective

surface,

very

for

and

ample

n ~ 3,

that

such

for

n ~ 1 if

2 Ky Z 3.

Proof. double very

In this points,

ample

if

case, or

2 below.

Example

2.

degree the

We over

d > 0.

contraction

that

(b) Y is an e l l i p t i c

2 2 (n + i) K y > i0.

Example

P(0e0(-b))

it is k n o w n

Cf.

either

(a) Y has o n l y

cone.

In case

[SV].

For

(a),

the case

rational

l-nKyI

(b),

is

see

Q.E.D.

consider a smooth There

the G o r e n s t e i n curve

cones.

B of g e n u s

is a u n i q u e

of b to a v e r t e x

section

Let X be a p l - b u n d l e

g, w h e r e

b is a d i v i s o r

b w i t h b 2 < 0.

z of the cone Y.

of

Let ~ : X ~ Y be

We f i n d

that

6

z

=

316

(2g

-

2

the

following

(b)

g

=

In Set

+

D

Since

=

Ky

(2g

cases:(a)

i,

d

~_ i,

case

(a),

=

-

(n

there

o n X, w e

2

d)2/d,

(c)

Ky

is

l)Ky. are

2

g

>

2,

=

0,

ample

If

no

infer

g

-

n

~

curves

d

=

2.

if

q

>

with

(Remark

I),

In c a s e

(b),

very ample

off

conditions

(ii)

we h a v e

KyC This

Thus

as

see

In case

7.

Let

(c),

canonical

there

be of

ea{sts

is

= max

0

very

qb,

if

q

ample

is

Gorensteln

where

q

=

Suppose

I.

off

z

=

2 that

{8,pz}.

(2g

2)/d,

q

>

InKyl

D 2 Z i0.

is an e f f e c t i v e Suppose

b

tangent

+ q)2

is v e r y

l-nKy[

[-nKy[ Since

divisor

is v e r y

E satisfying Since

C is a r u l i n g

a

germ An

Let

a

every and

~

~

is

sheaf

~

tangent

off

z.

B y the

vectors

Pz ~ 9d

singularity, on

U

is

effective

on U, d~v~sor

(= d if d Z 3), d > 3.

K:U

~-gen~rated

~ if

£ I R I ~ . ~ ( K U + D) ~ O, E supported

same

at z if D 2

V

a the

surjeetive.

D be a d£u£sor

ir-

so

for n ~ 2.

surface

the

DE Z n + 1

line,

for n > 3, for n > 1 a n d

normal

invertible

a nonzero

(D - E)E ~ O.

of

is

s i n c e E 2 ~ 4 / D 2 = i/d.

is v e r y a m p l e

separates

ample

l-nKyl

to see that C 2 Z i/d for

D 2 = (n + l)2d,

ample

Then

s u c h E exists.

if C 2 = l/d,

Thus

(a),

[-nKy[

Suppose

a n d we get E 2 = l/d,

is v e r y

K*~.Z

3.

i.

than

separates

u z ~ 9d(l

that

-

5).

other

InKyI

Since

in

(Theorem

self-intersection

- i) 2, we c o n c l u d e

3.

and

{8,pz}.

y.

map

Proposition

~

~

Y

case.

(V,y)

resolution

Ky

It is e a s y

E 2 Z l/d,

l-nKyl

Relative

that

= -i.

C o n Y,

= max

that

i,

negative

there

in T h e o r e m

in the c a s e

> 8 + y(Y) we

z unless

is a c o n t r a d i c t i o n .

reason

KB

set D = -(n + l)Ky.

curve

= -2.

2,

that

f o r n ~ 5 a n d g > 4d + I.

n = i, K y E

reducible

- l)2(q

see

-

3 and R e m a r k

at z if D 2 > 8 + y(Y)

for n > Ii,

We

InKyl

vectors

ample

d)2/d. dl2g

5,

from Theorem

D 2 = d(n

-

on - l ( y )

then

suoh

317

Proof,

According

([Sa2]), on ~ Ej

-1

of

(y), E.

[Sa2], that

one

to

can write

(ii)

P is

One h a s

Appendix. the

relative

D : P + N where ~-nef

the

version

and PEt

vanishing

One c a n p r o v e

theorem the

of

Proposition

Theorem 7.

Let

y,

~,

D be

~-generated

un~es~

that

Proof.

U,

there

ezists

(i)

the

the

N is all

1.

Zariskl

irreducible

in

+ D -

Q-divisor components

[N])

a similar

= 0.

Cf.

manner

to

Q.E.D.

~ame a s a b o v e .

a nonzero

decomposition

an effective

R1K.0(Ku

assertion

proof

V,

of

= 0 for

second

such

in

the

efYec£tve

T h e n 0 ( K U + D) d~v£sor

E on - l ( y )

(D - E ) E ~ 1 .

Assume V i s S t e i n , and c o n s i d e r a p o i n t x on U.

Let ~:U - U be

the blowing up o f U a t x, and l e t L be the e x c e p t i o n a l curve. ~o~.

is

Set ¢ =

I f x I s a base p o i n t o f IKU + DI, then we i n f e r e a s l l y t h a t

HI(u,~(K~ + ~*D - 2L)) ~ 0.

Applylng P r o p o s i t i o n 3 t o t h e d i v i s o r D =

~*D - 2L, we f i n d an e f f e c t i v e d i v i s o r E supported on ¢-1(y) such t h a t (D - E)E ~ O.

Corollary exists

Then E = ~.E > 0,

([Sh]).

an e f f e c t i v e

(D - E)E ~ 1 ( P r o p o s i t i o n 2).

In part6euLar, d~u~sor

0 ( K U)

E u~th

is

~-yenerated

unless

Q.E.D.

there

E2 = - 1 .

References:

[AS]

A n d r e a t t a , M . , Sommese,A.:On the a d j u n c t l o n mapping f o r s i n g u l a r proJectlve v a r l e t l e s .

[A]

Preprlnt

Artln,M.:On i s o l a t e d r a t l o n a l s l n g u l a r l t l e s o f s u r f a c e s . Amer. J . Math. 88, 129-136 (1966)

[BFS] B e l t r a m e t t l , M . , F r a n c l a , P . , Sommese,A.:0n R e i d e r ' s method and h i g h e r o r d e r embeddings. [BL]

Preprint

B e l t r a m e t t l , M . , L a n t e r l , A . : O n t h e 2 and 3 - c o n n e c t e d n e s s o f ample d i v i s o r s on a s u r f a c e . Manuscrlpta Math. 58, 109-128 (1987)

318

[B]

Bombieri,E.:Canonical Math.

IF]

IHES

42,

Fujlta,T.:

models

172-219

Remarks

on

of

surfaces

of

general

type.

Publ.

(1973) adjoint

bundles

of

polarized

surfaces.

Preprint [I]

Ionescu,P.:Ample

ILl]

Laufer,H.:On

ample

divisors.

elliptic

Preprint

singularities.

Amer.J.Math.

99,

(1977)

Laufer,H.:Weak stein

very

minimally

1257-1295 [L2]

and

surface

simultaneous

resolution

singularities.

Proc.

for

deformations

Symp.

Pure

of

Math.

40,

Goren1-29

(1983) [L3]

Laufer,H.:Generation larities.

[Mi]

In

[Mu]

and

Morrow,J., 547-565

the

forms

571-590

G~om.

for

surface

singu-

(1987) vanishing

Alg.

d'Angers,

theorem pp.

on

a sur-

239-247.

1980

Rossi,H.:Canonleal

embeddings,

Trans.

A.M.S.

261,

RamanuJam.

In:

(1980)

Mumford,D.:Some

footnotes

Math.

to

- a tribute,

Ramanujam,C.P.:Remarks Indian

[Re]

de

Noordhoff,

C.P.RamanuJam [Ra]

109,

Mumford-Ramanujam

:Journ~es

Sijthoff [MR]

4-plurlcanonical

Amer.J.Math.

Miyaoka,Y.:On face.

of

Soc.

Reid,M.:Elliptie

pp,

on 36,

the

the

41-51

work

of

247-262, Kodaira

C.P.

Springer-Verlag, vanishing

1978

theorem.

J.

(1972)

Gorenstein

singularities

of

surfaces.

linear

systems

Preprint

(1975) [Rd]

Relder,I.:Vector braic

[Sal]

surfaces.

Ann.

Sakai,F.:Enrlques Amer.J.Math.

[Sa2]

bundles

of

Math.

rank 127,

2 and 309-316

classification 104,

1233-1341

Sakai,F.:Anticanonical

models

of

on

alge-

(1988)

normal

Gorensteln

surfaces.

(1982) of

rational

surfaces.

Math.

Ann.

269, 389-410 (1984) [Sa3] S a k a l , F . : W e i l d i v i s o r s on normal s u r f a c e s . Duke Math. J . 877-887 (1984)

5|,

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[Se]

Serrano,F.:Extension Ann.

[Sh]

[SV]

277,

of

395-413

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Thesis,

Sommese,A.,

Van

593-603

on

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Math.

(1987)

Shepherd-Barron,N.l.:Some

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Math.Ann.

3

List of seminars held during the conference

F.Catanese: Components of the moduli spaces of surfaces. M.Chang: Buchsbaum subvarieties of codimension 2 in pn. C.Ciliberto: Hyperplane sections of K3 surfaces. H.Ciemens: The use o f D - modules in the study of deformations of submanifolds: I - II. L.Ein: Some special Cremona transformations.

T.Fujita: Classification of polarized varieties by sectional genus and A-genus: I - II - III. K.Hulek: Abelian surfaces in p4 and their moduli. J.Murre: Height pairing of algebraic cycles. C.Peskine: Remarks on the normal bundle of smooth threefolds in p5 _ Remarks on Noether theorem for smooth surfaces in p3.

Z.Ran: Monodromy of plane curves. M.Reid: Infinitesimal view of extending a hyperplane section - The quadrics through a canonical surface. I.Reider: Toward Abel-Jacobi theory for higher dimensional varieties and Torelli Theorem.

F.Sakai: Reider-Serrano's method on normal surfaces. M.Schneider: Compactifications of C 3. A.J.Sommese: The Classical Adjunction Mapping - The Adjunction Theoretic Structure of Projective Varieties - Some Recent Results On Hyperplane Sections - Projective Classification of Varieties.