Algebraic Surfaces (C.I.M.E. Summer Schools, 76)

G. Tomassini ( E d.) Algebraic Surfaces Lectures given at a Summer School of the Centro Internazionale Matematico Esti...

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G. Tomassini ( E d.)

Algebraic Surfaces Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Cortona (Arezzo), Italy, June 22-30, 1977

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11086-3 e-ISBN: 978-3-642-11087-0 DOI:10.1007/978-3-642-11087-0 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 st Reprint of the 1 ed. C.I.M.E., Ed. Liguori, Napoli 1981 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(c.I.M.E.)

SURFACES ALGEBRIQUES COMPLEXES ARNAUC BEAUVILLE

SURFACESA L G ~ B IRQUES COMPLEXES

ARNAUD BEAUVILLE

INTRODUCTION

Cet expos6 camprend deux parties. La premisre est un sumo1 assez rapide de la classification d3Enriquesdes surfaces algdbriques. On s'est inspird, bien entendu, Be la litt6rature classique sur le sujet, et en particulier du s6minaire Chafarevitch [Ch.2].

On a essays d'ttre aussi 616mentaire que possi-

ble, en supposant toutefois connue la cohomologie des faisceaux cohgrents. On renvoie 8 [Be] pour une exposition plus dBtaill6e ainsi que pour .des exemples.

La seconde partie comprend des indications

la d6monstration par

Chafarevitch et ~iatechki-~hapiio du th6orPme be Torelli pour les surfaces

K 3 3CtC.PPl.

$1. Notations e t rappels.

Nous dirons sirdplement surface au l i e u de surface projective e t l i s s e sur

C.

Soit

-D

3

S une surface,

s i D e t . D'

D'

D, D'

deux diviseurs sur

- B S ( ~ ) l e faisceau inversible associ6 b

.

. On note

sont lin6airemelrt Lquivalents !i.e.

diviseur d'une fonction rationnelle sur

- H~(s, bS(D))

S

ou simplement $(Dl

S) D

:

D-D'

est l e

.

.

s ' i l n'y a pas de confusion possible.

l e s espaces de cohomologie dufaisceau

BS(?>

.

- h l ( ~ =) a h e H'(D) - ; ( ( ~ , ( D I ) = h O ( ~-j $(Dj + h2b) - D = espace p r o j e c t i f des diviseurs e f f e c t i f s lindairement lquivalents B D. ts

( s i h0(5)

espace p r o j e c t i f associ6

=

2

, on

dit que

ID \

H*(D)

.

e s t un @.';iceau).

- KS ou K = diviseur canonique = un diviseur t e l - Pic(S) = groupe des diviseurs modulo dquivdence = groupe des classes d'isomorphisme r)

qua

eS(K) = nS2

IinEaire

de faisceaux inversibles

En vertu de t h 6 o r h e s g6n€rawr,,on a : pic@) = ~ ' ( 6 ,

03

19g) =

HI^,^@:)

d6signe l e faisceau des fonctions holoraorphes s u r

come

S

, considbr6e

v a r i 6 t 6 anslytipue. Cette dernisre i n t e r p r e t a t i o n .permet de consid'e-

r e r l a s u i t e exsctc! :

d'oa l ' o n d6duit l a s u i t e exacte importante :

0 -3 ~ ' ( 6 . 2 ) j H1(s, BS) ---9 P ~ C ( S+H2(5J) ) Posons : p i c O ( s ) = H'

H2(S, o S )

-9

(a)

(s, e s ) / s l @,a)

2 2 EIS(S) = ~ e r ( H( 6 , ~ )-4H (s, 6 S ) )

Le groupe

Pic(S) s p p a r a i t come une extension :

de d e w groupes de nature diffgrente:

- le pue

groupe p i c O ( s ) e s t un groupe d i v i s i b l e ; l a th6orie de Hodge rnontre 1 iI ( S . 1 ) e s i un r6seau aana H'(s, Bs) , autrement d i t p i c o ( s ) a

m e s t r u c t u r e n a t u r e l l e de t o r e complexe

- e t rnsme,

'

en f a i t , de vari'et6

ab6lienne. -.

l e groupe PJS(S)c H'(s,'B) Le cup-produit sur

e s t un groupe de type f i n i .

2 H (s,z)

symEtrique 5 valeurs dans

ZI

sont dexx aibiseurs, on note

indait sur

,l e

NS(S) une f o m e b i l i n 6 a i r e

produit d ' i n t e r s e c t i o n ; s i

D

et

( D - D ' ) l e prorluit de 'leurs c l a s s e s dans

D' NS(S).

On obtient a i n s i une f o m e b i l i n 6 a i r e sym6trique s u r l e groupe d e s d i v i s e u r s

qui joue

LIE

r 5 l e fondmental, dans l a th'eorie des surfaces. S i

C, C '

sont d e w courbes irr6ductibles d i s t i n c t e s . on a : (c.c' ) =' nanbre de points d'intersection de C leur multiplicitb.

et

C'

, compt6s

avec

Rappelons quelques th8orhes fondsmentaux :.

,

Dudit6 de Serre : hi(IC-I)) =

OSiL2

.

On utilisera t r b souvent Riemm-Roch sous la forme suivante, qui utilire 18 d ~ i t i d,e Serre : ~O{D)+ ~P(K-D) ( eS) + 1/2 ($ D.K)

-

7

.

Formule du jzenre : Soit C m e courbe irr6ductible sur S

Ce ncmbre est le genre de C

, not6

. Si

g(C)

C

est singulisre, son

genre est strictement plus grand que celui de sa n o d i s 6 e ; en particulier, on a &(c) = 0. si et seulement si C l f ? 1

.

Invariants num&riaues On pose :

pg(~)= dim H ' ( s ,

BS) = dim H'(s,

a:)

= dim H'(K)

(par

dualit6 ds Serre)

On notera simplement q, pg, Pn s'il n'y a pas de confusion possible. Tous ces invariants sont des invariants birationhels. On a :

T ( esl On pose bi

3

b4 = bo = 1 que b, = 2% On pose

dimQ:H'(X,G)

.

et b3 = b,

Jtop(S)

= 1

-q

+

Pg

; par dualit6 de Poincar6, on a

. De plus, il rgsulte de la thgorie ae

' Z (-1)

i

bi

2

- 29

+

b2

Hoage

Ces invariants sont r d i 6 s par $18 :

Formule de N. Noether : 12 X ( BS) =

8 '+ )!

top

(s)

Mous u t i l i s e r o n s l a varigtg d'Albanese d'kqe surface S ; rappelons i c i l e s propriSt6s. qui nous. i n t l r e s s e n t :

I1 existe une v a r i g t i ebglienne Alb(S) ;de morphisme d : S 4 ~ l b ( s ) t e l s que :

-Si

a_ & 1

,

d (s)

ahension p

,-e

n ' e s t pas rBfiuit 2 un ooint;

-S i d (s)

e s t m e courbe B , cette,'courbe e s t l i s s e de' genre l a f i b r a t i o n p : S --j,B a ses f i b r e s ,connexes. On d i r a que p est l a f i b r a t i o n d'Albsnese de S

qet

.

Nous u t i l i s e r o n s 6galement l e t h g o r h e classique s d v &

:

"ThdorZme d e b e r t i n i " : S e t S une surface, C une courbe, g : S --+ C un morphisme s u r j e c t i f . I1 e x i s t e une courbe l i s s e B e t un dia~ramme cornmutatif:

t e l que l e mo,+hisme

p : S-+

B a i t s e s f i b r e s connexes;

Notons o_ue l a f i b r e g$n'eriqxe du morpbhme p e s t a l o r s l i s s e e t irr&luctible, puisque l a f i b r e gdndrique d'u-n morphisme ae v a r i 6 t 6 s lisses sur E

est toiSjours l i s s e ( t h g o r h e ae Smd)

.

Enfin l a rmarque s u i v a t e e s t t r i v i a l e mais extrhement u t i l e i

Remarque u t i l e : Soient

une courbe irrdductible sur S

C

aiviseur e f f e c t i f . A l l s D6monstration

;

e t ' nbO ; d o r s

(D.c)+O

.

, oil

oli .&-it D = D r + nC

,telle D'

.

2

(D.,c) = (D'.c) + n ( ~ )+O

2

-

0

,

D

ne contient pas

C

,

que C

52. Applications birationnelles.

On peut classifier: l e s surfaces 8 isomorphisme prss, ou, plus grossibrement, 8 isomorphisme birationnel p r b . ~ e ' p r o b l & e ne se pose pas pow l e s courbes, puisque toute application birati6nneU.e drune courbe l i s s e dans une autre e s t partout d6flnie. Pour l e s surfaces, on va voir,que toute application birationnelle s'obtient 8 p a r t i r de trans,fomations " b l b e n t a i r e s " , l e s &latement s Rappel

.

:

Soit

S une surface,

phisme birationnel

-

-

£ restreint 8 p

C

E :S

--$

S

£-'(~-p) e s t

.

pt S

. il e x i s t e une surface

tels w e :

un isomorphisme sur S-p ;

e s t une courbe isonorphe 2

E

lp'

, aui

s r i d e n t i i i e neturellenent

3 llensembl= des directions tangentes 1 S & p On d i t que E' e s t 1',8clatement de t i o m e l l ? ae'lr6cla&ment. On

a:

ns(s")

A

S e t un m o r

ns(s)

S .en p

,et

. E l a &mite excep-

c~ a [EJ

(b1

A

La forme &'intersection sur S &axit d o n d e par l e s fonrmles : (f*

D

.t * D 1 )

(L*D.E) =

3 = -1

o

f

(D.D')

D,D1

aiviseurs mr

S

S o i t C une courbe irrdductible. sur S

.

, passant

par p

avec m u l t i p l i -

A

c i t d m On dEf6nit l e transform6 s t r i c t C de C cornme 1'adhbrence On v h i f i e immbdiatement gue : dans 2 de E - ~(d-p)

.

-

EtC=C^+mE

c2 m2 ;.2 = c.x + m

&*oill l a n 1i8duit : $

=

et

(cl

XOus adnettons sans d h o n s t r a t i o n l e s t h d o r h e s s d v a n t s (cf. par exemple [ch. 1 1 ) : Thborhe d'glimination des ind6terminations. Soient -

S une surface,

V une varibt8 alah-i'aue,

une application rationnelle. I1 e x i s t e un momhisme d'une s u i t e f i n i e d'bclatements, e t un mowhisme le diaiqramme :

'P : S-+V a "

:S 4S

, comuosb

N

s

s o i t commutatif. ThEorhe de *structure des morphismes birationnels. TouCv morphisme birationnel (d.'une surface dans une autre) e s t com~osd d vune s u i t e f i n i e d'bclatements. Corollaire Sofi

7

: S ---3S T

m e application birationnelle. Il existe une

surface S e t un diagrarmne cornmutatif :

a' Bchtements .." XS(S~ f @ 8 21 en

Soit f : S 4 k' un mrphisme birationnel..

nmbre n

dont il e s t compos6 e s t d6tenuin6 par is formule ES (s) particulier, tout morphisme birationnel de S dans e v e - m h e est un ismorphisme.

Pour toute surface S, notons B(s): ltensemble des classes dtisomorpIrisme de surfaces b i r a t i o n n e l l e m ~ tjsomorphks 3 S S i S, , S26 B(S) , on d i t que Sl domine S2 s l i l existe un morphisme birationnel (i.e. un composd d tEclatem&ts) 'de S, dam S2 :D1ap&s ce qui prbdde, on intro&uit d n s i

.

une relation d'ordre s u r B(S)

. On d i t qu'une

surfece S est minimale s i

e l l e e s t minimde dans B(S) , c'est-&dire s i - t o u t morphisme birationnel de. S dam m e surface S1 e s t un isomorphisme. Propdsition Toute surface domine une surface minimale. Dhonstration : soit S , une surface. S i . S n t e s t pas miniinale, 21 existe UI morphisme birationnel S 4 S1 qui n t e s t pas un isomorphisme. S i S1 n t e s t pas 'miaimale ,il existe de m k e S 4 S2 , e t ainsi de sGite; c 3mme :

on arrive- n6cessairement B une surface minimale. aomin6e par

S

.

Disons qu'une courbe Ec S e s t exceptionnelle s t i l existe un &latement E : S 3 S' (St surface l i s s e ) t e l que E s o i t la droite ~ t c e p t i o n nelle de S. ; il rgsulte du th&r&e de structure des morphismes birationnels qu'une surface e s t minimale si e t seulement s i eZle ne contient D ~ Sae

courbe exceptionnelle

.

,Les courbes exceptionnelles sont c a r a c t 6 r i s 6 e ~ ' ~ a rl e ,th&or&nesuivant que nous admettrons : CritSre de contsaction de Castelmovo Une courbe E e s t e x c e ~ t i o n n e l l es i e t seulement s$ E,JP'

e t E~ = -1

L'ensemble B(S) sera en principe connu. dss que l'on connaitra ses 616ments minimaw tous l e s autres &ant obtenus 8 p a r t i r de cew-18 par des kclatements. Deux cas peuvent s e presenter : il y a.un seul mod& minimal, ou il y en a plusieurs.

-

Dkfinition 1 : Une surface S e s t rkp;lEe s i e l l e e s t birationnellement iso~ o r p h e1 C'XP' ,051 C e s t une courbe l i s s e . S i de D ~ U = C =lP1 ,t -0

sue

S

e s t rationnelle.

ThGorSme des modsles ninimaux Soient

S, S f deux surfaces minimales,

birationnelle.

Q.: S 9 S t

une a m l i c a t i o n

S n ' e s t pas rg~lke., '-f e s t un isomomhisme.

En p a r t i c u l i e r ,

B(S)

a un s e u l 616ment minimal, e t t o u t automorphisme

birationnel de S e s t un automorphisme. La d6monstration u t i l i s e l e lemme fondamental de l a thdorie des surfaces-:

Leame-&

(~nri~ues-~astelnuovo)

Soient S une surface minimale non r6ql6e, AZss :

K.C 9 0

C

une courbe irrQductible.

.

Ce lemme s e r a dkmontr6 plus t a r d ( 58)

.

Dgmonstration du t h 6 o r h e : Par l e t h b o r h e a'dlimination 8es indetermina-

tions, il existe un diagramme cammutatif :

cn

ail y =

.. -

!

e s t un compos8 de n 'edatements e t f

un norphhne

birationnel. Pami tous l e s d i a g r m e s possibles, choisissons-en un t e l que n

soit minima; il s'sgit de montrer que n = 0 Soit E l a droite exceptionnelle de ltBclatement El , Si f ( ~ )6 t a i t r'eduit 2 un point, l e morphisme f se factoriserait en f ' o ;L, et l'on contredirait l a minimditd de n

.

.

Donc f ( E ) est . m e courbe C Come r Q s d t e de l a formule (c) p.7 que :.

.

f

,*est un campos'e df8clatements, il

d'03 m e contradiction avec l e lemme-cl6. classifi,ication des surfaces se divise donc en deux branches : d'un c6t8 l e s surfaces r6gl8esS qu'oh peut consid6rer come connues du point de vue birationnel, mais dont or cherchera l e s modhles minimam; de l ' a u t r e l e s surfaces non rSglQes, pour lesquelles l a classii3cation "bir6guli&reV' reirient essentiellement au mgme qhe l a classification birati'onnelle : il s u f f i r a de classer Zes surfaces minimales.

53. Surfaces r6~18eSe t

Soit *b

C

rationnelles

C une courbe l i s s e . Une surface n&rnitriauement r6@e de est. une surface S , munied'un moiphisme l i s s e . p ; S -+ C a t

l e s f i b r e s s o a t isomorphes. 2 P '

.

I1 n'est pes Svident a pri'ori. qu'une surface g60m6triquenent r6gihe (D~P. 1 ) ; c e l a r g s u l t e du :

e s t rdgl6e

Soient

S une surface,

un mor~hisnede S

p

On suppose q u ' i l e x i s t e x t C

t e l clue l a f i b r e

Alors il e x i s t e un ouvert U _de :C

(xi $ x) e t un isomorphisme de E r U .jections pss,:pg(s) = 0

.

, de

s u r une courbe l i s s q C. -1 1 p ( x ) . s o i t isomorp~e8 8

l a forme U = C

p1 (v) u

U x P'

- {x, .. .

, comutant

.

x

nt avec l e s pro-

.

F = pl(x) On a F~ = 0 k t F.K = -2 (fornule, du genre), done si DE \K( on doit avois D.F = -2 mais a u s s i D.F & 0 par l a i-emarpue u t i l e (p. 6 ) , e t par s u i t e IKJ = @ Notons

.

pas 2 : il e x i s t e un diviseur -

R

sur S t e 1 que

(H.F).= I .

o , 1s flkche

pic(^! 4 H ~ ( s , z ) e s t s u r j e c t i v e ((.a)p.3). 2 I1 s u f f i t donc de montrer q u ' i l e x i s t e une c l a s s e k c H (s,z) t e l l e que 2 h.f = 1 , en notant f l a classe de F dans H (s,z) Pour a v a r i a 5 l e Come p (s) = g

dam

H ~ ( s , z,)l ' e n s e ~ b l edes

.

entiers

(8.f)

e s t un i d e a l de ZI

, ae l a

forme

B.Z ( d ? ~ ) . L'application a l/d (a.f) e s t une forme l i n E a i r e 2 s u r H*(s,z) ; par d u a l i t 6 de ~ o i n c a r g ,il e x i s t e un e l h e a t . f ' G H. C S , ~ )

t e l que : ( a . f q ) = I / & (a.f) e t donc

f = d.ft

modulo t o r s i o n clans

2 H (SJ)

.

pourtout

2 ~&H(s,z)

Considikons l a s u i t e exacte :

On en d6duit l a s u i t e exacte longud de cohomologie :

La s u i t e des espaces quotients pour

r

suite

a r e S ( ~ + r F ) ), de dimension 2

H'(s,

1 H (S, 6 ( ~ i + r F ). d o i t t t r e s t a t i o n n a i r e

assez 'grand; .donc il e x i s t e un

s

r % e l que b

soit bijectif r s u r j e c t i f . Choisissons un sous-espace v e c t o r i e l V de

, tel

que

e t par

a ? ( ~ )= H'(F, 6F(l)) ; notons P

-

l e pinceau correspondant. I1 peut avoir des composantes f i x e s , mais e l l e s doivent s t r e contenues dam c e r t a i n e s fi'bres t i n c t e s de

F

,

F

X1 (puisque P n'a pas de points f i x e s s u r

F

Xk

F)

de p

dis-

. De msme l e s

points f i x e s de l a p a r t i e mobile de P sont contenus d'ans des f i b r e s F, , F d i s t i n c t e s de F notons F , Fx l e s k+ 1 X1 X1+l ' m f i b r e s de p . qui ne sont pas irr6ductibles. Posons 'J = C - {xl, , , x,] e t notons P f ' l a r e s t r i c t i o n de P 2 p - l ( ~ ) Le pinceau P' e s t sans

,. ..

,...

.

..

.

points f i x e s ; t o u t e courbe Ct

de P'

e s t rgunion d'une section de p e t ne contient pas de f i -

6ventuenernent.de c e ~ t a i n e s ~ f i b r e smais ; en f a l t bres, sans quoi on a u r a i t e s t f o r d de sections

Ct

pour t .# t ' de Za f i b r a t i o n . p

Ct n C t I f @

(ct)

,

te!P1

. Donc l e pinceau . Come il e s t ,

P' sans

points f i x e s , il d 6 f i n i t un morphisme g : *-'(u) +'?t de f i b r e g l ( t ) = C t ; -1 on en dgduit un morphisme h = ( P , ~ ) de p (u) s u r U S P' Comme h

-1( ( x , t ) )

= FXn Ct

,

h

.

e s t un isomorphisnle, d'o3 l e t h g o r b e .

Remarque 4 Lorsque p e s t l i s s e , S e s t donc un fZbr6 en &oites projectives audessus de C , localement t r i v i a l (pour l a topologie de ~ a r i s k i ) .L'ensemble des c l a s s e s d'isomorphisme de t e l s f i b & s i i d e n t i f i e 2 liensemble H'(C,PGL(P,B C ) )

. Oreon d6duit de l a s u i t e exacte :

l a s u i t e exacte de cohomologie :

Or

2 ( C , BE) = 0

, par

exemple parce que

en d r o i t e s projectives s u r fibr6 vectoriel

C

E de rang

e s t une courbe; done t o u t f i b r 6

C

e s t l e fibrd projectif 2

sur

C

. Les f i b r 6 s

P (E) associ6 B un

C L P ~ ( Ee) t l P C ( ~ ' ) sont

isomorphes si e t seulement s i .il e x i s t e un f i b & i n v e r s i b l e que E 1 2 E@L

.

L

sur

La c l a s s i f i c a t i o n des surfaces gBom6triquement r6gl6es s u r v e c t o r i e l s de rang 2

ramenge 2 c e l l e des fib:&

sur

C

C

tel

e s t aonc

C :Celle-ci e s t l o i n

d'^etre t r i v i a l e , mais peut 8 t r e consid6r6e come bien comprise

- cf.

.

[R]

Leme 5 Soient

S

une surface,

C

une courbe l i s s e ,

dont l a f i b r e ~ 6 n 6 r i s u ee s t isoinorphe 2 P'

. -e

p : SJC

unmorphisme

f i b r e de

p

n ' e s t pas

i r r g d u c t i b l e , e l l e contient une d r o i t e exceptionnelle. ~6monnstration: S o i t pour t o u t du genre

i

(car

.

LC.>-1

F

n. C? = Ci (3' 1

Soit -

C x P'

1

-

si

, 1'6galit6.n'ayant

t i o n n e l l e . Par s u i t e s i trouverait

une f i b r e r6ductible,

C

n. C. 1

1

. On a

2 Ci

<0

n j c j ) < 0). Bone par l a formule.

l i e u que s i

C.

e s t une courbe excep-

F ne contenait pas tie courbes exceptionnelles, on

K.F & O ,. ce qui c o n t r e d i r a i t

C

F =

K.F = -2'

.

une courbe l i s s e non rationnelle. Les modsles minimam de

sont l e s surfaces g60m6triquement r6gl6es ae base

C

.

D6monstration : I1 e s t c l a i r qulune surface g6om6triquement rdglLe ne contient pas de d r o i t e s exceptionnelles, c a r celles-ci devraient s'envoyer surjective-

ment s u r

C

, ce

qui e s t impossible.-Soient /S

application b i r a t i o n n e l l e de sur

S

sur Cxl? 1

,

. Considdrons l ' a p p l i c a t i o n r a t i o n n e l l e

C

une surface minimale, p

la projection de

f

une

C * P' .

p o f : S - I ~ C ; par l e th6o-

r h e d'blimination des ind6teminations il e x i s t e un diagramme cornmutatif :

05 l e s f. sont des dclatenents, e t i '

.

e x c e p ~ i o n n e l l eae l'bclatement

%(E,)

,%

un morphisme. Notons

g n a C o ~ eC

e s t r6duit 2 un p o i n t , de s o r t e que

%

1

l a cboite

n ' e s t pas r a t i o n n e e e , . s e . f a c t o r i s e en

En continuant l e proc6d6, on v o i t f i n d e m e n t que pf. e s t f i b r e gdndricpe isomorphe 2 IP

E

.

u q

%-I

"n'

morphisme, de

D'aprss l e lemme 5, l e s f i b r e s de pf sbnt i r r 6 d u c t i b l e s , donc rationn e l l e s l i s s e s (puisqu'elles v 6 r i f i e n t 'F = 0 , P;K = -2 , d'o6 g ( ~ )= 0 ) : par s u i t e

S

e s t une surface gkom6triquement r6gl6e de base

C

.

Nous nous contenterons d'bnoncer sans d h o n s t r a t i o n l e - r 6 s u l t a t analogue pour l e s surfaces rationnelles. On s a i t quk t o u t f i b r 6 v e c t o r i e l s u r P'

est

s o m e de f i b r d s - i n v e r s i b l e s ; il en r 6 s u l t e que l e s surfaces g60m6triquemeat r6glkes de base

IF'

sont l e s surfaces : pour

ng0

.

Thkorhe Les surfaces r a t i o n n e l l e s minimales sont p2 n # 1 . -

e t l e s surfaces

Fn pour

Remarque

7,

I1 e s t f a c i l e de c a l c d e r l e s invariants numdriques des surfaces rdglkes : e s t b i r a t i o h e l l e m e n t Qquivalente 2 C x P' , on trouve :

s i S

S e s t minimale

S i de plus

KCs = 8 (I-q)

lQ2

, on

a :

b , ( ~ )= 2

$ =9

tandisque

#

et

,P2

b2(s2)=1.

Ious allons v o i r qu'inversement l'annulation des

Pn- c a r a c t d r i s e l e s surfaces.

r6gldes.

$4. Caractdrisation des surfaces r a t i o n n e l l e s e t .r&lldes, Rappelons que nous ddmontrerons plus l o i n Lemme-cl6 :

tif sur S

Soit

.e

une surface minimde non r e a c e ,

S

s

($8) l e :

D.K 8 0

D un diviseur effec-

.

Lemme 8

Soit

D

S une surface m i n i m a l e e c m : Soit

X

. Uors

une section hyperplane de

applique l e lemme-cl6; de m h e s i e f f e c t i f pour

2 K < 0

(H.K) r 0

, en

S

K ) =~X2 +

-8

>

0

et

. Si

prenant

n assez grand. On peut donc supposer

( H + r 0 K ) . ~= 0

e s t r6glCe.

S

(H.K)< O ,on

D = nH+K qui. e s t

.

( K . K )0~

, de

s o d e que s i

r

e s t un r a t i o n n e l 7 r

0

Dm = m(H+rK), pour t o u t m

Posons

ro

e t suffisamqnt voisin'de

(K-Dm) ,H

devient ndgatif pour

.

\<mi

grand l e syst3ne

grand

m

e s t ton vide; c o m e

a :

. Par Riemann-Roch,

mr t 2

t e l que

o n a : h O ( ~ m ) + , h O ( ~ - ~ m ) + quand o, m--+Comme

, on

, on

m

v o i t que pour

, on

Dm,K < 0

conclut encore

par l e lemme-cl6. Theorhe 9 (~hstelnuovo) & S

S

une surface avec

q = P2 S ' O

.

Alors

e s t rationnelle.

S

2 ~ k m o n s t r a t i o n: On peut supposer S minimale. Si CK < 0

.S

l e e 8

0

,l e

donne : hO(-K) 2 1 + K2*1

(K.H) 4 0

, d p o 3l e

, on

applique l e

t h d o r h e de,Riemann-Roch (compte tenu de P2 = 0 ) donc s i H

e s t une section hyperplane, on a

r d s u l t a t par l e lemme-clb.*

Notre but e s t maintenant de c a r a c t h i s e r l e s surfaces rgg16es par l p a n nulation des plurigenres

,celle

Si qal

de p

8

Pn

. Si

q = 0

, l'annulation

de P2

suffit;

n l e s t pas l o i n de s f l i r e :

Leme 10

&S 2 K 4 0

S . m e surface minimde avec p ( e t donc

S ' e s t rdglde) sauf s i

g

=0

g = 1

,

q

a

;$'2,

1

.

On a a l o r s = O

I?

.

) s'ecrit i c i : D h o n s t r a t i o n : l a formule de Noether (p. 5 2 io 8q = K + b2

-

1 1 . s u f f i t donc de v 6 r i f i e r qu'on ne peut avo* a s r e pour c e l a l a f i b r a t i o n dpAlbanese p : S +

q-= 1

B

,

, 03

b2 = ?

consi-

3 (= ~ l b ( ~ ) )e s t une

courbe e l l i p t i q u e ; il e s t c l a i r qupune s e c t i o n byperplane Se b r e g6n6rique de p sont lincairement ind6pendantes.dans

. On

S

NS(S)

et

, de

%me ti-

sohe

que b2 >/ 2. P r o ~ o s i t i o n11 S&t A l s

= 0 , q = 1. g sont des courbes l i s s e s de nenre > / I ,

S une surface minimale non r & l & e avec

,

S = ( Cx F)/ G

C & F

un Rroupe f i n i d'a~tomorphismes de

, opErant

C

(de manisre compatible avec l a projection s u r

3

C ou

F

p

sans p o i n t s f i x e s s u r C)

,

C/.G e s t e l l i p t i w e ,

e s t elliptique. I

D6monstration (rapide):On c o n s i d b e l a f i b r a t i o n dVAlbanese ' p : S 02

B = A l b ( ~ ) e s t une courbe e l l i p t i q u e . On dgsigne par

-1(b)

p g

pour

= g(F7

Pas 1 -

1

:

G

C xF

be B

. g>2

,

,,e t p

par

une f i b r e g6n6rique de

F

3

est lisse;

l i s s e s , s o i t de l a forme

nE

g = 1

,&

E

, les

F b

p

-+ B,,

l a fibre

. On pose

f i b r e s ae p

sont s o i t

e s t une courbe e l l i p t i q u e l i s s e .

En premier l i e u l e f a i t que b2 = 2

e n t r a i n e que l e s f i b r e s sont i r r 6 -

auctibles. On u t i l i s e ensuite' l a formule topologique suivante : top

(s) =

QP(B)

-

;Itop(~w,I . +

Lorsqu'une f i b r e gkn6rique (irrgductible)

Fb

, un

GB

Fr

( ;(top(~b)

-

>toi(~.l

1)

.

(a1

s e s p 6 c i d i s e en une f i b r e s i n g u l i s r e

c e r t a i n nombre de

1-cycles s u r F

Cvanescents") disparaissent dam l'homologie de

( l e s "cycles

Fb ; autrement dit, on a

.

, e t donc Ttop(Fb) > Ytop(Fr ) Or par h ~ p o t h h e (13) = 0 ; l e formule (d) montre donc q u ' i l ne peut Ji. a v o i r Ttop(s) = 1 top., de f i b r e s singulleres. Enfin s i F = nC , on h o u v e : b 1 1 1 = 2 1 ( e ' C ) = (C.K) = ; ( F ~ . B )= -(F1 .I() i; ; Jtop(~,., ) 1 t o p (Fb) = ;(top(c) n

bl(~bz ) bl(F1 )

Pas 2: Si -

p

e s t l i s s e , il e x i s t e un revgtement & a l e

f i b r a t i o n image r 6 c i ~ r o ~ u e

= S < BC

-4

C

C

soit t r h i a l e

-+ B

t e l sue l a U

( i . e . S g C u,F7

1.

La f i b r a t i o n p Quitte

d k f i n i t une femille de /courbes de genre

passer 5 un revstement s t a l e C

de

B

cohomologie (mod n) de ces courbes, e'est-&dire localement constant des S i . n ,3

, on

P : Un,g

'Tn,$

A ' ( F ~ ,l / n l )

, a cohomologie

-,

S4C

s e l de

C

.

B.

rendre constant l e systkme

e t en c h o i r i r une-base symplectique. g

(mod n) r i g i d i f i k e , qui e s t universeile; f :C J T n

,g

t e l l e que l a fibra-

s e dkduise de P par image rdcipro@ue. O r l e revctement univer-

est

e t c e l u i de

d

un domaine born6

, donc

T e s t l T e s p a c ede Teichmiiller T nyg e qui e s t l e morphisme f e s t t r i v i e l (i.e. f ( c ) e s t un p o i n t ) ,

ce.qui implique que l a f i b r a t i o n l e rev^etement C avec

g sur

peut " r i g i d i f i e r " l a

s a i t - ([GI) q u ' i l e x i s t e une f a i l l e de courbes de genre

, c t e s t - & d i r e q u ' i l e x i s t e un morphisme tion

, on

4

S

-+ C

e s t t r i v i a l e . On peut supposer

B g a l o i s i e n de groupe

G

, de

s o r t e que

S = ( C KF)/G,

elliptique.

C

(AUl i e u d ' u t i l i s e r l e thkorbme d i f f i c i l e de s t r u c t u r e de l'espace de Teichmiiller, on peut considsrer l'espace

$ des.modules des vari6tbs ab6n,g liennes principalement polaris&es, 2 cohomologie (mod n) Pigidifi6e; son revstement univei-sel e s t l ' e s p a c e de Siege1 X $

, qui

e s t un domaine born6

pratiquement par construction; I1 f a u t ' a l o r s u t i l i s e r l e the^or8me de Torelli.) Pas 3: Si ramifis tion -

p

C *B

4

Soit

C

a des f i b r e s multiples (donc g = I ) , il e x i s t e un revctement J t e l sue s i S d&gne l a normalisbe de S a g C , 1 2 f i b r a dkduite de p

soit triviale.

B'' une .oourbe l i s s e sur S

l a norm6liske de- S

choix convenable d e 5 B ' , l a projection p 1 : S' -+B3

dkduite de p

C

de B'

sur

C

C 3B

St+ S

Sf

on montre que pour un e s t &ale. La f i b r a t i o n

p'

e s t l i s s e . Q u i t t e h passer h un revzte-

a a l o r s come pr6ckaement un morphisme

e s t une courbe a f f i n e (c ' e s t un revstenent T n, 1 C ;v i a l ' i n v a r i a n t j) , donc f e s t - t r i i r i a l e t l a f i b r e t i o n

f : C 3 Tn,, (n a3) ramifis de

, on

. Or

que p ( 3 ' ) = B ; notons

posssae une s e c t i o n : e l l e n ' a donc pas de f i -

bres multiples, autrement d i t ment Btale

, telle

. Alors S! e s t l i s s e ;

xg B'.

d6duite de

p'

e s t t r i v i a l e . On pe-at supposer l e revctement

g a l o i s i e n de 'groupe

G

, dd

s o r t e que

S = ( CK F )/G Y

, avec

La. pi-oposition $st donc dgmontrge.

C o r o l l a i r e 12

a t

5

une s u r f a c e minimale non r6plke avec

( i ) I1 e x i s t e . u n e&r

s

(ii) (iii)

a

,&

= (E*F)/G

S

t e l que Pn

na 0

I

E

e_t

F

# 0 ;

pg = 0 ,

q 31

a

n ~ oe ;

'

sont elliptiques, L

n'e'st pas du type pr6cddent,

.

,

Pm ne s o n t mas boxnss.

&S

D6monstration : I1 r g s u l t e de l a p r o p o s i t i o n q u ' i l e x i s t e un rev^etement Btdle de degr6 ,n -rt : C x F 3 5 , avec g ( ~ ), g ( ~ )a 1. On a p g ( c x F) = g ( ~ ) . g ( ~ )7 , , donc i1 e x i s t e un d i v i s e u r e f f e c t i f DF COUUXL~' 7T e s t d t a l e , on s a i t que K zTX-% , de s o r t e que : CxF . r4 D 6 17. f K S I = / n . K S / dl05 pn(s)>l

' I K ~ ~; ~

.

rn (s)> p r ( c x F ) ', c e qui montre que l e s

Le msme argument montre que P sont non born& dss que t i q u e s , on a

g ( ~ ) ou

g ( ~ & ) 2. Enfin s i

0 ; donc si D E

KC&= n u l , c e qui s i g n i f i e que D = 0

,

i.e.

n%

1 ,l n.KS

C

et

e diviseur

0

.

F

sont 'ellip-

nf D

est

Du t h e o r h e de Castelnuovo, du lemme 10 e t du c o r o l l a i r e 12 r g s u l t e le : Th6orsme 13 [Enriques) Une s u r f a c e Remarcue -

S

e s t r6gl6e s i e t seulement s i P

condition dgEnriques : S Les surfaces du t y p e

,

n>O

.

14

( C X F ) / G , donne l a

U n e Btude p l u s approfondie des s u r f a c e s du t y p e

pg = 0

= 0 pour t o u t

n

e s t r6glge s i ' e t seulement s i S , = (E ~ F ) / G, 06 E

et

F

PI2 = 0

.

sont e l l i p t i q u e s e t

sont appelses s u r f a c e s b i e l l i p t i q u e s (ou p a r f o i s hy-perelliptiques,

mais c e t t e terminologie p r g t e 2 confusion). On peut en donner une c l a s s i i i c a -

$5.. Dimension de Kodaira Les r 6 s u l t a t s qui prbcsdent montrent l'importance des

Pn

dans l a

c l a s s i f i c a t i o n des surfaces. 11s Londuisent 2 poser l a : Dkf i n i t i o n 15

Soit l e de

S une surface; E r

systzme

n )/ 0

ynX l ' a p p l i c a t i o n

, notons-

rationnel-

dans un espace' p r o j e c t i f (kventuellement vide) dgfinie a a r l e

S

. L_a "dimension de Kodaire" &

1 n~

S

k ) e s t l a plus grande dimension des images des

(On convient de poser

, n&e K (s) (ou simplercent ynK pour n 2 0 .

dim(@)=-1).

Explicitons l a d e f i n i t i o n :

-

K

(s) = -1

.(jPn

= 0 pour t o u t

nW S

rbgl6e (par l e th&or&ne

dlEnriques).

-

k

-

(s) = 0 9 Pn

= 0 ou 1

11 e x i s t e N

(S) = 1

de

-

1L (s)

=2

,

(Pa

, et

il e x i s t e N t e l que PN = 1.

t e l que P m t2 ; e t pour t o u t

n

, l'image

e s t au plus une courbe.

I1 e x i s t e N

YNX

t e l que l'image de

s o i t une surface.

Une d k f i n i t i o n analogue peut s t r e donnee pour l e s courbes; on v o i t aussi1 tstque K ( P ) = - 1 , K(C) = 0 s i e t seulement s i C e s t e l l i p t i q u e , et

k (c)

= 1

sf e t seulment s i

=(c)

2.

Exemole 16 1/

S = Cr C"

.

- s i - . C ou -

s i .C e t

-

si

-si

C

. On v k r i f i e immgdiatement que C' = p l

C'

,

K(S) = -1 ;

sont e l l i p t i q u e s , K.(s) = 0 .;

e s t elliptique, e t

g ( ~ )e t

:

g ( C 1 ) >2

,

g(c1)>,2, K(s) = 2 .

K(S) = 1 ;

2/

ST= Vd

l,..w,P = i n t e r s e c t i o n complhe

faces de degr6s

dl

.

hy-persqr-

(x di

- r - 3 ) ,~ 03

H

e s t une

I1 en r d s u l t e que :

p u r s = V2

K(5)=-1

de r

lpr'2

.

,... , dr

Un calcul f a c i l e montre que KS z section m e r p l a n e de S

dans,

, V3 ' v2,2

K(S) = O

(et e n f a i t

K(s) = 2

pour l e s autres.

3/ ,Toute surface t e l l e que

KSzO)

pour

S = V 4 , %,3

n. K s 0 pour un e n t i e r

n

V2,2,2

, en

particulier

t o u t e surf ace b i e l l i p t i q u e (remarque 14) ou abdlienne, a dimension de Koaaira z6ro.

56. Surfaces avec

ic

=

o

.

Ce sont l e s surfaces avec

Pn = 0 ou

1 pour t o u t

Pn = 1 pour au moins un

n

, et

.

Lemme 17

Dhonstration :

a/ On a

2>,0

par l e lemme 8 ; supposons

2 K > 0.. On a

par Riemann-Roch : h O ( n ~+) h O ( ( l - n ) ~ )-+ s Pour

n& 2

, le

syst&ne

\ ( 1 - n ) ~ )\

quand

n 3

co

ne peut contenir

. un diviseur

E

,

sans quoi .on a u r a i t

(E.K) >/ 0 - (~emmg-616) e t done

donc que

quand

Pn 7 0

b/ Comme K'

., 1s formulb

,

. Posons

4

K~ S 0 ; on trouve

qui c o n t r e d i t

k = 0

.

de Noether s 1 6 c r i t :

; posons 'm = mld,

Ec

,

D = n' A

dans

E = A -dK

E = 0 puisque

Soit

, ce

+8

mn K I , &'oh \ -2

m'D = n'E E

effectif d'os

.

D t 1 nK\

c/ Soient on a

=0

n

(ml,n') = 1 ; donc

S une surface minimale avec

n = nl&. Comme

A

6 1d1(

1

1,

a pour un diviseur

E = m' A

P~C(S; ) on a '

Prim -=

m'E et

= n'L. = 0 Pa = 1

.

,

.

K = 0. Une des 4 p o s & b i l i t & s

suivantes e s t r 6 d i s 6 e : 1/

pg = 0

,

2/

pg = 0

,

q = 1

3/

Q

= 1

,

q =0

4/

pg = 1

,

q

q = 0

t

2

. ,-., h l o r s 2K r 0 dqEnriques" :

. .

s

.

. On d i t que

edt une "surface

S

e s t une surface b i e l l i p t i q u e (remarque 1 4 ) K S

s0

. On d i t que .-

S e s t une "surface

esC une surfece abblienne.

D6monstration :, 1/ Si pg = 0 , q = 0 , on a P2& l p a r l e t h 6 o r h e de Castelnuovo, d'oh p a r Riemann-Roch : hO(-2K) Comme

p = 0 B

, on

+. hO( 3 ~9 )1

doit avoir

.

Pg = 0 p a r - l e lemme 17. cf

hO(-2~p ) 1 ; par s u i t e

2K 3 0

2/ Les surfaces minimales avec

.

p. = 0 t3

,

, donc

q d 1 ont bt6'

a".

classifi6es

(S4); il r 6 s d t e du c o r o n a i r e 12 que c e l l e s

qui v g r i f i e n t

sont l e s s u r f d e s b i e l l i p t i q u e s .

= 0

- Supposons maintenant a

1

q = O ,

3/ S i

ou

, Riemann-Roch

q = 0

d;oii 3'/ S i

2 .

~ O ( - K )= 1

, il

q = 1 2E

Soit

DCIK-£1,

0

et

lemme

, on

17 b/

donne hO(-K) + h0(2K)t 2

o

K

.

e x i s t e un diviseur

. Appliquons-hi

mais

5

. Par l e

p = 1 g

E t e l que

E

,

P

0

Riernann-Roch :

~ O ( E ) + ~ O ( K - F . )dl03 ~ I hO(rc-s)ti. Koa I K 1 ; p u i s q u e P 2 = 1 , o n a

, ce

e t donc D = KO

2D = 2Ko

qui c o n t r e d i t

I1 e x i s t e donc pas de surface minimale avec

% = I y

C $0

&= 0

.

,

q = l =

d h o n t r e r . qu'une surface v g r i f i a n t k = 0

4/ I1 r e s t e

,.

pg = 1 , q = 2 e s t une surface ab6lienne. C'est l ' o b j e t de $a proposition 20.

-

Soient -

S une surface,

s u r j e c t i f 3 f i b r e s connexes,

fibre A &

F~ Q g 2

,

D G 0

D =

,&

D b o n s t r a t i o n : Posons

Eliminons l e s

c2i

3 une courbe l i s s e , '

C,

ni Ci D2 = 0 Fb =

,...', Cr

p : S 4

s i e t seulement si D = .r.F

,

un morphisme

.

(ni6 Z )

mi Ci.

B

l e s composantes i r r 6 d u c t i b l e s d i m e

m.

en u t i l i s a n t . l e f a i t que

7

0

. On a

:

( F ~ . c ~=) 0 :

b

( r EQ).

U ci

Come

n. n. 2 = -2 pour touk m. m

e s t connexe, on n t a QgalitE que d i

dl03 le lanme.

1

i .,jl

j

P r o p o s i t i o n 20 Sdt

S une surface minimale avec

,

X= 0

pg = 1,

q = 2

. Aloys

S e s t une surface ab6lienne. Ddmonstration : Notons A = A l b ( ~ ) e t d : .S On distinguera quatre cas, suivant pue et

+A

&(s)

Le morphisme d'Albanese.

e s t une courbe ,ou une surface,

e s t t r i v i a l ou non.

KS

'

K $ 0 L

K

Notons

come

K~ =

o

2

le diviseur e f f e c t i f de e t K.C. 3 0 pour t o u t i

0 = L C i = n. Ci

donc ou bien bien

..

C:

= O

+

=

j#i 2 = -2 e t Ci Ci.Cj

, on

a K.Ci

K =

ni Ci ;

= 0 ; par s u i t e :

i.(C..C.) J 1 J

C,

,

. Eci-ivons

f $(

s.0

e s t r a t i o n n e l l e l i s s e (car K.Ci pour t o u t

j

# i

= 0)

, ou

,

5

, 03 l e s D~ sont d e s aiviseurs K = D, 2 e f f e c t i f s 1 supports connexes e t d i s j o i n t s , on a' Dq = 0 pcmr t o u t 4

On conclut que s i l ' o n Q c r i t et :

-

ou bien

e s t une courbe i r r e a u c t i b l e de genre 1 ( i . e . une courbe

D,

e l l i p t i q u e l i s s e ou une courbe. r a t i o n n e l l e avec un point double)

- ou bien 1 a/

------

K $0

e s t r6union 'de courbes r a t i o n n e l l e s l i s s e s .

Dd et

d

(S)

e s t une courbe.

On considsre l a f i b r a t i o n d9Albanese p :/S 4 B de genre

,les

2

p a r l e lemme 19 rd L $+ e t

dbs que n

diviseurs

, on

biCB

. Pour

n

est

B

so&. contenus dans des f i b r e s de

Dd

doit avoir

. Come

Dd

= r,

p ;

, avec

pour t o u t d

Fbo(

convenable on a u r a donc :

e s t assez grand o n . a u r a , P

2

, ce

qui contredit

K=

0

Come

A

de

contenant des dourbes r a t i o n n e l l e s e s t contract6e s u r un p o i n t ; o r

K

ne c o n t i e n t pas de courbes r a t i o n n e l l e s , t o u t e composante

.

p a r un th6or&ne de M;cmford,tout d i v i s e u r e f f e c t i f un p o i n t v e ' r i f i e

, 05

n E

D2< 0

. Les

s e u l s 'Dd

t e l que

D

(D) s o i t

p o s s i b l e s sont donc de l a forme

e s t une courbe e l l i p t i q u e l i s s e , t e l l e que

E

d

D,

courbe. On s a i t qu12 t r a n s l a t i o n p r s s , t o u t morphisme de

d ( E ) ' s o i t une

E dans A e s t

un morphisne de v a r i d t d s abdliennes; de s o r t e qu'en c h o i s i s s a n t convenablement l ' o r i g i n e de A

on peut supposer que

abslienne de A. Posons

A' = A/E'

, et

. P a r l e th6orZme de B e r t i n i

f : S -+A'

d ( ~ =) E'

e s t une sous-varidt6

consid6rons l a f i b r a t i o n (p. 5

.)

, il

e x i s t e un diagramme

conmutatif :

oii

B

p(2)

e s t une courbe l i s s e , e t oil l e s f i b r e s de p e s t r d d u i t ?A un p o i n t , e t

E = !I /qlF n

qE

b y assez grand.

2 a/

Posons Soit

0

K c(

et

(s) = B

S' =

s

zr. on. conclut

o(

, or? d d d c i t

du lemme 19

que

a l o r s comme prSc6demment que Pn&2

pour

(s) e s t une courbe.

, et

xg B'

E~ = 0

s o n t connexes. Come

choisissons un revgtenent S t a l e de d e g r 6 a 2

; S'

B'

e s t connexe, e t nunie a l u n revttement 6 t a l e

-9

B

;C

Ona

T: s ' + s . .

p g ( s t ) = 1; on en t i r e

(eS,)

et

=a

K ~ ,z ~ * $ s . o , 6'03

Q(s') = 2 . Mais on a 1 XO(~~,)

g ( ~ f ) b 3e t

-+ HO(S', Q:,

( p a r exemple parce que l a f l s c h e

~(s')&~(B')

)

eat injective),

d'oc contradiction.

Pour ddmontrer l a proposition, il s u f f i t de montrer que . e e s t Btdle, puisque . . t o u t revstement B t d e d'une v a r i d t 6 ab6lienne e s t lui-m&ue une vari6td ab6lienne; autrement dit, s i

w e s t une

s u f f i t de montrer que

w

d4

riquement d t d e , l a forme 0 ;elle

KS

$7.

e s t donc p a r t o u t

Surfaces avec

K = 1 et

2-forme,partout

. e s t partout

# 0

K

.

=2

# 0 sur S

# 0 sur A

, il

. Come

e s t gQn6-

H

n ' e s t pas identiquement n u l l e ; puisque

.

Proposition 21 Soit S -

une surface minimale.

59

conditions suivantes sont dpuivalentes.

a/ K = 2 b/

,2

K 2

0

S . non r a t i o n n e l l e

c/ I1 e x i s t e un e n t i e r

pM

: S--3

dl

no

t e l que t o u t

D6monstration : I1 e s t c l a i r que une section hyperplane d i S

c/ + a /

. Come

donne : hO(nK-H) + h O ( ( l - G ) K + X) Puisque

((1-n)K + H) .K

ntno

, l'application

rationnelle

est birationnille.

IC2> 0

-+-

.

Montrons que b/ r S c / : s o i t

,l e thdorhe qusnd

devient n6gatif pour

de Riernann-Roch

n 3 e . .

n grand, il r g s u l t e au

leme-cl6 que l e s y s t h e \ ( l - n ) ~+ HI e s t vide, donc pour n&no . I1 est' 1nK - H 1 contient un diviseur ef f e c t i f . En : on a nK = H + En

-

.

H

Soit n&l

S m e surface m i n l n d e avec

, le

systkme

fnK!

8 = 0 . S u ~ w s o n sq

ne s o i t pas rbdtiit

a pour zmn

uii sevi d i ~ i s e u rde ~ s o s t e qxe

1

,0 5 -Z

e s t l a ~ a r t i ef i x e de InK et; \MI n'a pas Be c o q santes fixes. On e a,lors K.2 = K.M = z2 = Z.M = I ? = 0; l e s y s t h e M

nK = Z

t.

M

n 1a pas Be points f i x e s , e t d 6 f i n l t un momhisme ( 6 g d ir

qnK)9

S

stir une cousbe. Dbonstration : S

e s t non sgglge, donc K.25

(lemme-cl6); cornme :

ni? = X.Z

on a K . 2 = K.M = 0 I -Puisque le diviseur . M et

M.Z

0

K.M

sont

+0

e s t mobile, on .a b?&0

; or :

$=M(.~K.-z)=-M.Z e t par s u i t e

et

+ K.M

z2 = Z.(~K-& =

Come h? = 0

,le

0

systhe

tmc

.

~?=M.z=o

IM ! e s t sans points f i x e s , e t '&init (s) e t a i t une surface, on a u r a i t

donc un clorphisme $' : S --.3 plVI ; si M~ 2 0 , donc rp (3) e s t une courbg!

.

Remaroues 23 11 On peut m s l i o r e r notablement l'assertion c/ : s i S e s t une surface minimale, l e s y s t h e 1 nK 1 &t sans pbihts fixes ass que n $4 ; dbs que est un isomorphisme en dehors de certaines courbes n 3 5 , l e morphisme T,K rationnelles,. qiii- qont contrsctkes en des points singuliers d'un type trss simple ("singularit+ rationnelles"). On renvoie 2 [B) complbte de Is situation.

pour m e ktude t r b

21 Malgr6 l e peu qu'on en a dit,les sui-faces avec k = 2 (appelges aussi "surfaces de type g6nkral" sont celles que l'on rencontre l e plus frsquemment; il s u f f i t pour s'en convaincre de regarder quelques exemples :

-

Toutes l e s surfaces intersections compibtes, sauf l e s V2, V3,'.V4, sont de' type g6n6ra.l (exemple 16.2) ;-

V2,2,

V2 ,3, V2,2,2

- Tout pmduit

( ~ l u sgknkrdement, toute surface fide courbes de genre,* br6e sur m e courbe de genre ) 2 , avec fibre:g&&ique de genre a 2 ) . e s t de type gknbkt (exentple 16.1 );

- Toute S'

surface contenue dans une variEt6 ab6lienne est.de type g6n6rd; e s t surjectif, e t s i S e s t de type gkn8rsl, alors e s t de type gbn6rd.

Si

f : S' 3 S

Passons mainten&nt :aux surfaces avec IC = 1 : fh6or&ne 24 '

,

K~ = 0 , u S une suiiace'dnimale avec k = I . A l l e s t une courbe 1-e, existe &I moehisme. surjectif p : S 4 B , & B l a f i b r e &ndrique de ' p t a n t une courbe elliptique l i s s e .

soit -

l

Inversement, soient S m e surface minimale, B une courbe l i s s e , un mornhisme' surjectif dont l a f i b r e @nQique e s t ellip&e. U o r s l'une des 3 possibilit6s suivantes e s t r6alis6e : p : S-B

(i)

S e s t une surface rdglke de base ellip%ique.

(ii)S e s t une surface avec

t iii')

& a

K.= 0

K = 1 ; a l o r s dans

.

P~c(s.)

Q :

DQonstratipn : ~ o i ' t S une surface minimale avec

.

K =

1 ; il r k s u l t e

S o i t n t e l quo Pn%2; du lemme 8 k t de l a proposition 21 que K2 = 0 notons Z l a p a r t i e f i x e au s y s t b e \&I , M s a p a r t i e mobile. Le lemme 22 montre que M

d 6 f i n i t un morphisme

l e th6orbme de B e r t i n i (P. 5

06' B

@ S t une

)

, il

9 : S --3C

. D'aprh

e x i s t e une f a c t o r i s a t i o n :

courbe l i s s e , e t 03 l a f i b r e gCri6rique F

de p

est l i s s e

e t irrkductible. Comme F

e s t contenue dans un diviseur de \ M \ e t que K.F = 0 , d'o6 K.M = 0 , on ddduit du l e m e cl'! que g ( ~= ) 1 guisque

F~ =

o

.

2 P; on a donc D & 0

D =

ri Fb A

Examinons maintenant l a place .de S e s t r6&'!e

de base

.,

, l n 6 g a l i t 6 n16tant rZalisde

C

,la

fibre F

que s i

.avec ri tQ+ (1-e

19)

dans l a c l a s s i f i c a t i o n . , Si, S

s?envoie surjectivenent s w C

, d0nc

e s t e l l i p t i q u e ou rationnelle. S i S k t a i t r a t i o n n e l l e on a u r a i t \ = 8 ou 9 (remarque 7 ) , a q b 3 1 -K I # 0 p a r Riemann-Roch; mais a l o r s 2 on t r o u v e r a i t (-K) $ 0 ataprbs ce qui pr6cSde, ce qui e s t impossible.

C

8

S i S n q e s t pas rdglde, on a n6cessairement

= 0

, donc

k c2

.

La d e r n i h e a s s e r t i o n r s s u l t e de c e qui prSc%de. Corollaire 25 , '

Soit

S une surface minimale avec

t e l sue l e s y s t b e

ri Fbi ,

D = e r . = mi 1

Pour

e

.

n t e l que Pn # 0

soit entier

d . Soit

ri

avec

, et

assez grand, l e systsme

Soit

S une surface minimale,

I

Pn(S) = 0

1 sur B I ~ 1K , avec

(p.

6

t e l que

1

e s t sans points d = en.

une courbe s u r .S - t e l l e que C

2

.

0

nBO ; c a r s i D E lnK\

pour t o u t

par l a remarque u t i l e

>

que

VIJ

:

zi bi

. La formule du genre montre que

a / On a D.C >, 0

C

on e n t i e r

i ;.alors

pour t o u t

; on a

De e

f i x e s . I1 en va donc de m h e poirr l e s y s t h e

(c.K) c 0

d L.1

s o i t sans points f i x e s .

\ dK \

D6monstration : Soient

. I1 e x i s t e un e n t i e r

= 1

)

, d'oa

K.C 3 0

.

, on

aurait

1

b/ S i car si

C'

morphe s u r

qk1

l'image de S

Btait une surface S'

dans s a v a r i b t b dSAlbanese e s t une courbe :

, on p o u r r a i t

A l b ( ~ ) ayant u e r e s t r i c t i o n

2 S'

trouver une 2-forme holonon identiquement.ndle;

par image rgciproque, on en dbduirait une 2-forme holomorphe non nulle sur S. Si

q>l

, i l . existe

courbe l i s s e de.genre' q

donc une f i b r a t i o n

.

5 0

.

On va supposer que

S

Premier cas

: K

2

, oil

B

e s t une

n ' e s t pas rgglBe, e t a r r i v e r 2 une contradiction.

c/ I1 e x i s t e une courbe En e f f e t l e produit

p :' S 4 B

C

t e l l e gue

C.K

0

, 1 C+K !

(c+~K).C devient nggatif pour

donc par l a remarque u t i l e il e x i s t e w

n

t e l que :

= 6

.

n assez grand,

Si D =

'x

ni Ci'

IC

C.

e x i s t e une courbe d/ Kt

$ q22 C

IC

+ (n+l)~ 1. = (d et-'\,E+~[ = $ ; donc il

f c + n ~ \# J d , on a D.R 4 0

+ nK \

Ci.K

vkrifiant

<0

,et

une-courbe d r i f i a n t

C

e s t m e s e c t i o n de l a f i b r a t i o n

C.KCO

6 .

s

g(C) = q;

,

p : S-3 B ;~i q = 1

.

I

+, K \ =

.m

, \ c + K ~= 0

C

e s t un r e v ~ e u e n t s. t a l e de B

(ci

on a

~ ~ ~ l i ~ Riemann-Roch, u a n f on trouve en e f f e t :

Si

q = 0

, on

trouve

g(C)

. Si

0

,

qal

C

ne peut, S t r e contenue

m e f i b r e Fb

dans

de

p

, sans

quai on auraik F = nC (puisque C &O : l e m e 19) , 4'03 g ( ~ I )1 5 0 b p a r l a formule du genre, e t S s e r a i t rdglge. Donc C e s t un r e v s t e u e n t raI

mifie de B , de degrd de .Rienann-Hurwitz :

2

, avec

d

f

p o i n t s de r a m i f i c a t i o n . La fo&ule

2~3(C) - 2 = d(2q-2) + r montre a l o r s que

,et

r = 0

e/ I1 e x i s t e une courbe C ' e s t c l a i r si ,q = 0 IC+K] = 0

, on

, car

C sur

exceptionnelle. Supposons aonc Soit

C

t e l l e que

n

t e l que :

.

\ 2 +~ nK On.

Ci

(par

dl)

\

. Posons

a D.K c -1 vkrifient

(c.K)

Ci.K
.

4

0

,

.

IC+K 1 =

t e l l e que

C

@

D =

. On a a u s s i

n. C. 1

1

,-Ic+K( =

C.K<-1

# -1

C.K

. C.K< 0

sans quoi

C

..

'on v o i t comme en

c o n t i e n t un d i v i s e u r e f f e c t i f

Supposons q u r i l e x i s t e un p a r Riemann-Roch :

, donc

q a1

= 1

vkrifiant

S

pour t o u t e courbe

a g ( ~ =) 0 par, d/

e x i s t e un

sauf si

d = 1

$

.

,

serait c/ q u r i l

D ; ID+K

\=

$

.

; on peut supposer que t o u s l e s

ICi + K

1= 0 ,e t

i t e l que ni>2

par s u i t e

; alors

g(ci) = q

\K+2Ci] =

,,d r o 5

0 = h0(ir+2ci) & 1-q + 1/2 (LC: + 2ci.IC)

= 3(q-1)

-

(ci.K)

c e q u i e s t impossible.

r ,2

Supposns que

I I C + C ~ 1+ C = ~0

;dors

, dl03

:

0 = h o ( ~ + c 1 + c 2 )= h 1(IC+cl+c2) + I - p + I / ~ ( C ~ + +C I~/ Z) (~C ~ + C ~ ) . K

.

= hl(i(+c +C ) + q-1 +(c1.c2) Ceci e n t r a i n e C .n C2 = 1

1 et

0

2

1

h (K+c,+c~) = 0 ; mais l a s u i t e exacte :

+ es + 8

'0 4 es(-c,-c2) donne

e 0 -+ o

1 C2 = h (K+C1+C 2) a 1 , d'os c o n t r a d i c t i o n . D e s t une courbe i r r d d u c t i b l e C , qui v g r i f i e

1

1

h (-c1-C2.)

A i n s i l e di*seur

f/ 0 s l ' o n a r r i v e 2 une c o n t r e d i c t i o n Soit

C

une courbe v E r i f i a n t h0(c)>/1-q + 1/2

Si

q = 0

, on

(c.K) < -1

2 (C -C.K) .=

a donc sur S

-

IC+K 1

et

, i.e.

(c.K)

qkl

, supposons

d'abord que

p o i n t d o i t bouger lin6airemen-t s u r que

S

h O ( c ) &2

.

S e s t rationnelle.

F

e s t &onc un p o i n t , e t c e

F : c e c i e n t r a i n e que F e s t r a t i o n n e l l e , et

d a m c e c a s on c o n s i d s r e l a s u r f a c e p

:

e s t rdglge.

Reste e n f i n l e c a s 03 q = 1 ddduite de

. On a

s o i t une s e c t i o n de l a f i b r a t i o n

C

dfAlbanese. Sa t r a c e s u r une f i b r e gdnGique donc

0

un p i x e a u de courbes r a t i o n n e l l e s : on

conclut p a r l e th&or&ne de Yoether-Enriques b e Dans l e c a s

=

C

e s t un r e v s t a n e n t d t a l e de

S' = S rcSC

e t l a fibration

. E l l e posssde une s e c t i o n n a t u r e l l e

l a projection

T:

S' 9 S

e : C *St

B ;

p ' : S7--X ;

e s t b t a l e , d e s o r t e que :

(e(c).ICS,) = d e g C ( e x $ , )

= degC(ICS)

J

= (c.K) < -1

et

7

y ( 8 s , ) = d e g ( ~ ) . (es) = o Le thdorsme de Riemann-Roch donne a l o r s de s o r t e que S'

c o m e prdcddement

e s t r d g l d e e t donc a u s s i

2 Dewigme c a s : K > 0 g/ Le lemme 10 montre a l o r s que

q = 0

S

.

. Comme

P2 = 0

ho (S ' ,e (C ) ) &2,

,l e

th6orhc

de Riemmn-Roch donne 2

hO(-K) 3 l+K r/ 2 Supposofis que 1-K\ , on peut dcrire que (C.K).d 0 et E un on concluLtcornme en f/

.

contienne un diviseur r6ductible R Come , 03 C est une courbe irr6ductible telle diviseur effectif; mais alors IC+K I = I-E I = 0 et que S est rationnelle. est i'rr6ductible. On peut donc supposer d6somais que tout diviseur de 'I-K\ 2 1 Comme (-K) > ;O , la remarque utile montre que pour tout diviseur D

R.K< 0

R = C+E

1

effectif sur S il existe un

n tel que :

.

Soit E G 1 D + nK \ , et,supposonsd'abord, E # 0 Soit C une courbe irrdductible contenue dans E ; on a 1C+K 1 = 0 , et C.K 4 0 puisque 1-K I contient au moins un pinceau de courbes irr6ductibles. On en d6duit come pr6c6dermnent que S est rationnelle. Reste le cas 05 quel que soit le diviseur D sur S , il existe un n tel que D + nK E 0 ; autrement dit, 6i P~C(S)= VZ.[K]. Comme p = 0 g la suite exacte((a) p. 3 ) mbntre pu'on a kors H~CS,Z) = z. LKJ ; la

.

dualitd de Poincard entraine alors K~ = 1

7 C@J

. Mais on a

:

= 1

et il y a donc incompatibilitd avec la formule d? Noether.

$1.

Structures de Hodge

.

r, on notera rR = r z R et rC =.I' z C . L'espace vectoriell complexe rc est le complexifi6 de rR; il est donc muni Four tout 2-module

une conjugaison canonique, not6e x

@

M

Z

Dgfinition 1 : Une structure de Hodge de

r

@

d'

. D O ~ n ~ S sur

un 2-module de type fini

est une d6comuosition :

r

=

et HP,~ (p,q r 0) p+q=n 09 les HPyq sont des sous-espaces vectoriels complexes.de rC, vgrifiant

e

Soit X une vari6t6 projective et lisse sur C; pour tout n, la thlorie de Hodge d6finit une structure de Hodge de-poids n sur H~(x,z). Le sous-espsce ~p.9 de H~(x,c) est alors canoniquement isomorphe h H~(X,Q'); on identifie ~o(d,C$jr)

2 un sous-espace de Hn(x,C!)

en particulier

(6gal 2 IInyO) via la coho-

mologie. de de R h q : led 'n-formes holomorphes sont f e d e s , donc &finissent des classes de cohomologie dans * ( X , C ) ; Exemple 1 : Structure de Hodge de poids 1. Rappelons que Si V est un espace vectoriel &el,

il est gquivalent de se

donner une structure complexe sur V (2.e. un endomorphisme J de V tel que 2

J = -lV)

ou m e d6composition de V eR C en deux sous-espaces comylexes con-

jugugs : si V est muni d'une structure complexe, on d6compose V aR @ su'ivant

l e s 2 sous-espaces .propres de J i.el&ifs aux valeu,rs propres + i e t 4; inversement; s i V aR C = W

i, on

d g f i n i t J s u r V p a r I d formule J(w+;)

= i(w-w)

pour t o u t w E W. 11 en r k s d t e que l a donnee d'une s t r u c t u r e de Hodge de poids 1 s u r

I'

est

6 q u i v a l e d t e . 1 l a donnee d'une s t r u c t u r e complexe sur TRY ou encore s u r l e I 1 tore r6el S i T = H (x,z), l e t o r e complexe a i n s i obtenu e s t l a v a r i ' l t 6

.

de Picard associEe 1 X.

.

Exemple 2 : S t r u c t u r e de Hodge de poids 2 C ' e s t une d6comeosition H1"

=

rC = H~~~

6H1"

te

H O ' ~

, aTec

. On ne s a i t pas a s s o c i e r B c e t t e d&omposition

i1

3

H'"=

P y 2e t

un o b j e t &omktri-

que a u s s i simple qu'un t o r e complexe. ~ k f i n i t i o n2 : On a p p e l l e r a p o l a r i s a t i o n d'une s t r u c t u r e de Hodge d e o o i d s n ( r , ~ ~ l' a~ donn6e ) d'une forme b i l i n g a i r e non d6g6n6rle sur

rC,

sym6trique

s i n e s t p a i r , a l t e r n 6 e s i n e s t impair, prenant d e s v a l e u r s e n t i b r e s sur

r,

e t v6rifiant :

(cr.8) = 0

s i ci

E

HPyq

,B

E . H ~ ' ~ ' 'e t p+pl

(a.;)

> 0

si a

E

H""

,n

pair ;

'>

si a

E

H ~ "

,n

impair

i(a.6)

0

;

.

Si.X e s t une varigt'6 de dimension n;le s a t i o n s u r l a s t r u c t u r e de Hodge de H"(x,z)

# n

cup-produit d k f i n i t une p o l a r i -

.

Exemple 1 : ( s u i t e ) . Une p o l a r i s a t i o n correspond 2 m e forme a l t e r n g e s u r I', qui 6tend6e

2

rR

L6rifie : (Ja.J$) = (a.B)

(JU.~)> 0

et

/

pour a,$

E

rR

.

C'est ce .qu1onappelle une fonne de Riemann (ou une polarisation) sur le tore

. A partir d'une

complexe T = rR/T

telle polarisation, la thkorie des fonc'

tions thsta' fourmt un plongement projectif de T, qui est donc une varidt6 ab6lienne

.

Exemple 2 : (suite) Soit (r,~"~) m e structure de Hodge' polarisee de poids 2, de sorte que

rC

H' ,' e

= H~~~ e

H O * ~ .Les

(a.G) > 0 pour tout a

E

.sous-espaces H ~ " et

H O * sont ~

isotropes; come

f12'0, la restriction de .la-formebiiinsaire 2

H2'0 e Hoy2 est non ddgen6r6e, de sorte que H' 'l

est lforthogonalde

H~~~ @ Hos2 d a m TC. I1 en.r6sulte que la donn6e sur

r,

muni ,d'un produit sca-

laire, dlune structure de Hodge polaris6e (par le produit scalaire donn6) 6quivaut 2 celle du sous-espace isotrope H~~~ dans rC, vdrifiant (a.a) > 0 pour tout a c

~

~

~otons~ h2*0 de

.

9

'

( rla~,grassmannienne ) des sous-espaces isotropes de dimension

rC; c'est

pour tout a r H2'0

une varidt6 alg6brique projective. La condition (a.;)

> 0

d6finit un ouvert (analytique) GO(I'~) de G ( I ' ~ )dont , les

points correspondent,bijectivementaux structures de Hodge polarisdes sur (muni d'un produit scaiaire fixd) Soit X

-+

.

T un morphisme lisse de vari6t6s al&briques,

sont des surfaces, et soit 0 un point de T. Posons

r

E

dont les fibres

= H2 (x~,z), muni de la

forme d'intersection. Supposons que l'action de rl(T,o) sur il existe alors pour tout t

r

T un isomorphisme canonique

r soit triviale; -+

2 H (xt,Z),

respectant les formes d9intersect30n. En transportant par cet isomorphisme la 2 structure de Hodge ~olarisdede H (Xt,Z), on obtient,pour tout t une structure de.Hodgepolarisee de poids 2 sur

r,

dl08 une application T

+

G O ( r C ) . L8 thdo-

rie.de Hodge permet de prouver que cette application est analyticye; on dit que c'est~19"applicationdes pdriodes" de la faille de surfaces X

-+

T.

est justifide par la raison suivante. La structure:de. 2 Hodge dlune suAface X est ddterminde par 1a.position de 112'0 dans H (x,c); or Cette te&inolbgie

si 1'0;

i4entifie.H2,~5 HO(X,~~)et H2(x,e),au anal de B~(x,F), la flsche

i : XO(X,Q~) -t H (x,c)* correspondant & 1' inclusion est donnde per 2 2 = w pour w E HO(X,S? ), Y E H~(X,Z). Si lton fixe m e base (wJ '

.

de,Ii0(~,$)

Y

4

et une base (yi) de H2 (x,z), la struktuke de Hodge est ddtemiinbe

par l a "matrice des pdriodes"

(jl;%) -

.

,Si par exemple p (x).= i , la vari6t6 G ( I ' ~est ) la quadrique dans CI; 2 [P(H2:X,~)) d6finie par l'znnulation de la forme qzadratique sur B (x,c); si l1on choisit une base (y.) de H2(x,z), le point de..cettequadri~uecorresponw , 03 o est une dant % la structure de Hodge de H2 (x,z)a pour coordonnBes Yi 2-fome #U sur X.

I

Proble'me cle Torelli R. Tcrelli

.

'a ddmontrd ([TI) que la donnse* des pdriodes d'une surface de

Riemann X (i.e. de la structure de Hodge polarisde sur H

7

(x,z),ou encore de

la jacobienne polaris6e JX) caractdrise X 5 isomorphismepr$s. On appelle maintenant "probl~mede ~orelli"la question de savoir si les structures de Hodge polarisdes sur la cohomolo~ied'une varidt6 ddterminent ia vari6td. La solution de ce probldme trss difficile n'est .connue que dans de r&es

oas.

Le cas des surfaces K3

8

'et'e r s s o l n r&emment par Chafarevitch e t Piatechki/

Chapiro ( [ ~ h - P I ) ; on va donner i c i une indication de l e u r d6monstration. On renvoie pour l e s d k t a i l s ( e t des compl6ment.s) & l ' a r t i c l e CCh-PI. Etudions d'abord l e probl?me de T o r e l l i pour l e s surfaces ab'eliennes. La s t r u c t u r e de Hodge de poids un d'une t e l l e surface A dktermine sa variktk de Picard, c ' e s t

a

d i r e 1 8 vari6t'e abglienne duale

l ' o n r6cupSre A come v a r i k t 6 abklienne duale de

2.

x;

il e s t bien connu m e

On a kgalement un th6orS-

me de T o r e l l i pour l e s s t r u c t u r e s de Hodge de poids 2 : ? r o ~ ' o s i t i o n3 : Soient A, A' deux surfaces abdliennes. Si l e s s t r u c t u r e s de Hodge oolar i s k e s s u r H2 (A,z) D6monnstration :

& H2 (A' ,Z) .sent

isomor~hes,A

A' sont isomorohes.

2 1 Rappelons q u ' i l e x i s t e un isomorphisme canonique de h H (A,Z)

2 2 sur H (A,z); avec c e t t e i d e n t i f i c a t i o n , l a dBcomposition de HoQe de H (A,@) e s t donn6e par :

2 1 K*(A,c) z A H (A,c) = n2ti1y0

e ( H ~ , O ~ He~ A~~ H ' )O , '

.

1

11 f a u t donc montrer que l a p k i t i o n de R " ~ dam H (A,c) e s t carsctkriske par c e l l e de

A~H"O

dans A%' ( A , & ) ; o r , c ' e s t 12 un r 6 s u l t a t bien connu: il

exprime que l e "morphisme de PlCcker", qui va d.e l a grassmannienne des droi.t e s de P-3 d a m p5, e s t un plongement.

92. Sfarfaces de K m e r et surfaces K3

.

Rapelons (Thbor8me 18,lbre partie) qu'une surface K3 est une sur'face vbrifiant'K I ,0, q = 0. Les intersections comple'tes V4, V2,3, V2

sont des 0

surfaces K 3 (cf. exemple

16, 18re partie);

0

plus gdndrakment, pour tout g r 3

il existe une lfamiile de surfaces K3 de degrb 2g-2 dans pg. Nous allons indiquer

cons%hction particuli8re de surfaces K3, '8 partir de surfaces abb-

~e

.

liennes

Soit h une surface abslienne. L1involutior9 de A,,d8finie par @(a) = -a, admet 16 poihts fixes isol6s a, Notbns

E :

-1 (ai) (1

E

2+A S

,...,a16, qui sont les points d'grdre

2 de A.

118clatementde ces 16 $oints, Ei le diviseur exceptionnel

i s 16). L' involution 9 se prolonge en une involution a de

On d8signe par X la vari6tk quotien"c/u, par n :

-+

2..

X l'application canoni-

.

que; on pose Li = n(~$

,Proposition4. La .surfaceX = $0 est une surface IC3, av~el6esurface de K m e r associde

h A.

D6monstration : Hontrons d'abrd que X est lisse. C1est clair en dehors des L ~ ;solent p , E~ ~

,q

= ~ ( p ) on . peut trouver des coordonndes locales (x,y)

sur A au voisinage de ai telles que :

eRi=-x et que x et t

-

*

e y r y

= $ foment un

syst?me de coordonn6es locales dans

2 au voisi-

nage de p. Comme cr*t = t, on conclut que t et u = x2 foment un systbe de coordonn8es locales sur X au voisinage de q; en particulier X est lisse en q. Soit w une 2-forme holomorphe non nulle sur 4. La forne &*w est inva-

r i a n t e par

a; e l l e e s t donc de l a forme

~ * 605 , a

e s t une 2-forme m6romorphe

sur X. I1 e s t c l a i r que a est holomorphe e t part&t fO en dehors des Li; au voisinage de, ai, on a : @=kdxidy

k a C

d'oit, au vdisinagje de p :

= k dx

€*:*o

A

d ( t x ) = kx.8.x

ce qui montre que h = 5 - d u

A

d t e s t holomorphe

2-forme holomorphe partout

#

0 sur X, donc

dt =

A

k a* (du

A

at),

# 0 s u r Li. Ainsi o e s t une

%3

0. S ' i l e x i s t a i t une 1-forme

holomorphe non n u l l e sur X, on en dgduirait une 1-fome holomorphe

0 sur A

invar'iante par 8, c e qui e s t impossible : donc q ( ~ =) 0 e t X e s t une surface K3

. IA surface de K-er

r i f i a n t Li.L.

J

X conti'ent 16 d r o i t e s r a t i o n n e l l e s L1,.. .?LI6, 6-

= 0 pour i f j . Inversement :

~ e & e 5. Soient x une surface K3, L~ ,..., L , ~16.courbes l i s s e s r a t i o n n e l l e s t e l l e s que Li n L. = 0 .J' 21 = ZLi Picl~).

.=

i#j. Sup~osonaqu'il e x i s t . ~1 c: p i c ( ~ )t e l que X e s t .me surface de Kmmer.

D6monstration .: Botons L l e f i b r 6 en d r o i t e s de c l a s s e 1; s o i t s une section

.

e2 du f i b r k L dont l e ' , d i v i s e u r des zdros e s t ELi.

Posons :

i=

(x c:

L, xe2 = sI

.

La projection de L sur X a s f i n i t un revstenlent double long des Li. On a T * L ~ = 2Ei

, 03 Ei

plus : 2 IE= ~ ( a * ~ =~ 2) L~ ~

--

?T

:

+

X, ramifis l e

e s t une courbe r a t i o n n e l l e l i s s e ; ae

4

d * 0 3:E

=

-

I.

Le c r i t k e decontraction de Castelnuovo montre q u ' i l e x i s t e un ?ior~hisme birationnel E :

2

+

A qui contracte l e s Ei

. Prouvons que A e s t une surface

abblienne. On a : KA A E E*KA + C E z ~ n

* +~C E~ ~,

X ( q ) , . On v s r i f i e

Calculons ~ ( 0 =~ )

suite ~(8,) = ~(0,) + x(~-')

=b+

4

a h

K~ n

facilement que s, %

o

;

$ e f'; par

l2 par Riemann-Roch.

-

( c L ~ ) ~ 8 ; on a don? ~ ( 8 =~ 0,) d'oa q ( ~ =) 2. La c l a s s i f i c b t i o n des surfaces (Th6ore'me

18, le're p a r t i e ) montre que A e s t

une surface abblienne. Notons a l ' i n v o l u t i o n qui &change l e s deux f e u i l l e t s du revdtement ramif i 6 a; comme u ( E ~ = ) Ei

, eLle

provient d'une, involution 0 de A. S o i t

ai = E ( E ~l'un ) des points f i x e s de 0; 0 o p h e s u r l'espace tdngent

A en

ai par multiplication par -1. On en d&duit a u s s i t 8 t que 0 e s t -1'involution a

t-+

-a de A, pour l a s t r u c t u r e de groupe de A pour l a q u e l l e ai e s t l ' o r i g i n e .

Par s u i t e , . X e s t l a surface de Kummer associge & A. Lemme 6.

Avec l e s notations pr&&ientes', dssignons par L l e sous-groupe X 2 de H (x,z)engendr& par l e s Li. Le sous-groupe ?T*E*H2 (A,z) c '$(x,z) e s t 1'

-

orthogonal de

4

$(x,z).

Dbonstration : Au cours de l a d&monstration, on notera pour abrsger H'(T) = R'(T,z)

pour t o u t e vari6t6 T.

I1 e s t i m g d i a t que n,e S o i t x c (L

X

1';

comme $x

s*x = €*a. Come x =

2

*Hg (A) c ($1';

montrons l ' i n c l u s i o n contraire.

e s t orthogonal a m Ei, il e x i s t e a

E

H2(A) t e l que

-21 n *11.*x, il suff!it de prouver que a = 2a' pour

a ' c H ( A ) ; ou encore, $ar d u a l i t & de Poincar6, 'que (a.b) e s t p a i r pour t o u t b

6

H~(A).

Posons k' =

- mi,

X' =

On ve montrer plus bas qTle

T'"

2 - Fi -;nofons

s'

:

AT

+

Xi-la restriction

2

-

e s t s w j e c t i f ; on en ddduit que E: ( A ) e s t en-

gendrd pe;r lm(ax) e t Ips- Ei. Posons e*b =

+ CniEi, avec y

E

2 I! ( x ) ,

ni€%ona:

*

*

(a.b) = ( ~ * a . ~ * b=) ( ~ * a . e * ~=f (a x . y) ~ = 2(x.y) d'o3 l e r d s u l t a t . 11 r e s t e

2 montrer que n f * e s t s u r j e c t i f . f i i s q u e 7i' e s t un revstement k t a l e

de degr6 2, il e x i s t e une s u i t e s p e c t r e l e :

9 = I i P ( Z l ( 2 ) , H ~ ( A) )'

,gH*~(x' )

.

A' ) s u r ~ ~ ( 4).,' e t Notons q l l a c t i o n de l ' i n v o l u t i o n cr ( r e s t r e i n t e 9 La thgorie de lb.cohomologie des groupes f i n i s donne : ,E = (-1 ) p 2 d P H ~ ( A)' -q9 ,~ e r ( u-E ) . / $oq+cp) pour p # 0 9 P Pour q s 2, l a r e s t r i c t i o n Iiq(A) -c H ~ ( A 'e) s t un isomorphisme; par s u i t e

.

.

-

o

.

E

p r q r 2. On en dgduit en p&ic&ier

EE*'

= E": = 0 ; il en re9 9 2 s v l t e que Ifwedge-homomorphisme" H (x') + . E ; ' ~ de l a s u i t e s p e c t r a l e e s t sur=

j e c t i f . Mais c e t homomorphisme n f e s t a u t r e que nl*; c e c i acheve de ddmontrer

Proposition 7.

Soient X @ X i deux surfaces dc Kumaer,

y:

2 7 B (x,z) + X-(x' , Z ) un isomorphisme Be s t r ~ c t u r e sde i i o d ~ e~ o l a r i s g e s ,f fi cue

~(5 = 5; ) .

X

g

X' sont isomoruhes.

~6monstration: Notons A,,A' l e s surfaces abbliennes associges 2 X,XV. I1 e s t

im&i&t que (sr*c*a.a*c*b) = 2(a.b) pour a,b e s t injectif:-?Xi$sulte

a l o r s du lemme

E

2.

H (A,z); en p a r t i c u l i e r a,€*

6 que 'f i n d u i t un isomorphisme JI de

I~'(A,z) sur I i 2 ( ~ ',z) 4 respectant. l e cup-produit

. I1 e s t c l a i r que $ i n d u i t un

I

isomorphisme de s t r u c t u r e s de Hodge; l a proposition 3 montre a l o r s que A e t A' sont isoqorphes. E l l e s sont par consgquent isomorphes comme v a r i 6 t 6 s abb-

liennes, ce qui entrarne que X e t X' sont isomorphes.

$3. Sui+faces de Kummer sp6c;ales. On d i t qu'une surface de K w e r associde 2 une surface abblienne. A e s t sp6ciaJ.e s i A e s t r6ductible, i . e . ab6liennes :

0

4

E'

-

A

s ' i l e x i s t e une s u i t e exacte de varidt6s

--L E +'o.

02 E e t El sont des c w r b e s e l l i p t i q u e s . Notons R l a courbe ( r a t i o n n e l l e ) quotient de E par l l i n v o l u t i o n x

On d i r a que l a f i b r a t i o n q : 'X

+

W

-x; on

R eSt une " f i b r a t i o n de Kummer" pour X. I1

e s t c l e i r que 9 e s t l i s s e en dehors des

4 points de ramification r,

,..., r 4

Proposition 8. Soient e un p o i n t d ' o r d r e 2

& E,

t e l s que *(ai) = e ; oh posk r = u(=).

a,,

...,ah l e s 4 o o i n t s d ' o r d r e 2 & A

On

:

= 2 F~ + L.~+ L~ + L~ + L~

03 Fr -

e s t une courbe r a t i o n n e l l e l i s s e e t Fr.Li = 1 (1 5

D6monstration : On a :

*r

q

oil

Fe e s t

=

21 T*T *q*r

=

T*E

*p*e =

l e transform6 s t r i c t d a m

-

A

(Fe +

i

5

4).

ir

C E ~ ), i=1 -1 de l a courbe e l l i o t i q u e Fe = p ( e ) . T*

L ' i n v o l u t i o n o i n d u i t sur Fe une i n v o l u t i o n du t y p e x W - x ; p a r s u i t e , .rr*Pe = 2Fr

, 03 Fv =

?,/(a)

e s t isonorphe 2 P

.

1

C o r o l l a i r e 9. Soient X (resp. X' ) une s u r f a c e de K m e r s o d c i a l e , f

-

(m.f ' )

l a fibre

2 q6nCrique d'une f i b r a t i o n de ~ummer. ~ o i t q un isonornhisme de H (x,z) 2

H (x'

,z), i n d u i s a n t un isomor~hismed e s s t r u c t u r e s de Rodge p o l a r i s g e s ,

fo?mant c y c l e s e f f e c t i f s en c y c l e s e f f e c t i f s , e t t e l que'f'(f) X

g X'

= f'.

sont i s o m o r ~ h e s .

trans-

Ddmonstration': Compte t e n u d e l a p r o p o s i t i o n 7, il s u f f i t d e prouver que

q(%)

= LX,

. On a

4

f = 2Fr

+ I= .C Li 1

& a m ~ i c ( ~ ) , ' d ' o i if' = 2 % ~) + z % L ~ )

r

dans p i c ( X q ) . Comme l e s s e u l e s dEcompositions d e f ' en s o m e de d i v i s e u r s e f f e c t i f s s o n t de . l a forme f ' = 2Fr, l e rdsultat.

+ ZL; , on

en d6c?uit que Y ( L ~ )= L!

J

, d'oa-

Pour f E ~ i c ( X ) ,on not'era gX(?) l e SOLS-groupe de P ~ C ( Xengendrg ) par U

l e s Elements x t e l s que x2 = -2, x.f = 0; s i f c s X ( f ) , on pose ~ i ( f =) gx(f ) / ~ . f , e t on l e munit de l a forne quadratique induite. Supposons qye f s o i t l a f i b r e d'une f i b r a t i o n q: X, + P',

5 . f i b r e s connexes; s o i t x un

Elkment de P'ic(x) t e l que x2 = -2 e t x.P = 0. Le thgorsrne de Riemann-Roch montre que l ' u n e des c l a s s e s x ou -x contient un diviseur e f f e c t i f , qui e s t n6cessairement une some de composantes des f i b r e s de q. On en d'eduit que dans ce c a s j l e module z X ( f ) est.engendr6 par l e s composantes des f i b r e s d e q .

I

Notons par a i l l e u r s DL l e module eo.ei= I e:

= -2

ei.ej = 0

2 nun? du produit

pour

1 < i < 3

poui

o

-<

i s 3

pour

1

S

i

S

j

s c a l a i r e d g f i n i par:

,'

S

3

.

I1 r g s u l t e a u s s i t d t de l a proposition 8 que lorsque f e s t l a f i b r e d'une fi:

4

bration de Kummer, s X ( f ) e s t isomorphe come module quadratique b (DL)

.

Proposition 10. 1 Soient X une surface K3, q: X + IP un morohisme s u r j e c t i f ?if i b r e s con-

=,

f l a c l a s s e dans P ~ C ( Xd'une ) f i b r e de q. Supposons que

sX(f)

4

(~~1.;

a l o r s , X est: une surface de Kummer sp'eciale,, & q e s t une f i b r a t i o n de Kummer. ~6monstration: I1 r 6 s u l t e a u s s i t d t du l e m e 19, l s r e p a r t i e , que l a forme quadratique sur SX(f) e5t n6gative non d6gGn8r6e. Le module quadratique

- s X ( f ) se d6compose donc

de manisre unique come s o m e de sous-modules i r r 6 -

ductibles; chacun de ces sous-modules e s t engendr6 par l e s composantes d'une f i b r e rgductible. Sous l'hypothsse de l'Enonc6, on conclut que q admet quatre f i b r e s r6-

d u c t i b l e s , de t y p e D4; c ' e s t

2 d i r e que chaque f i b r e r 6 d u c t i b l e s ' b c r i t

L

Fk

1

2Fi +

iZl

k = O,.

L4k+i

On o b t i e n t en p a r t i c u l i e r

.

.,3

16 c o u r b e s . ~ r a t i o n n e l l e sl i s s e s Li, deux

point commun, e t t e l l e s que

Zi

3

:

deux sans

-2(CFt k ) + bf. On d6duit a l o r s du lemme 5

que X e s t l a s u r f a c e d e Kummer a s s o c i e e B une s u r f a c e a b s l i e n n e A. ~ o n s i d 6 r o n sl k revgtement r a m i f i 6 s :

t

+

X, e t posons EG =

-1

.IT

(FA)

.

1

C ' e s t un r e v e t m e n t double de P r a m i f i s e n . 4 p o i n t s , donc une courbe e l l i p t i que, qui coupe transversalement l e s d r o i t e s e x c e ~ t i o n n e l l e sE l ,

...,Eb.

Par

s u i t e ; .E(EA) = E' e s t une courbe e l l i p t i q u e s u r A; en c h o i s i s s a n t une o r i g i n e convenabie s u r Er e t A, on peut ~ o ' n s i d 6 ~ eErr comme,une s o u s u a r i 6 t d abglienne de A. Notons E le courbe e l l i p t i q u e A/E1; on 'a un diagrarrme cornmutatif :

x-

B

>

X

1

oi3 r e s t un r e v e t m e n t double, r a m i f i g au-dessus d e s 4 p o i n t s de B au-dessus desquels q n ' e s t pas l i s s e . On en c o n c l u t que q e s t une f i b r a t i o n de Kummer pour X. Lemme 11

.

S o i t X une f-f'

pour m

s ~ f a c K3, e

et

soit f

E

P ~ C ( Xune ) c l a s s e ~ r i m i t i v e( i . e .

e n t i e r ' e n t r a r n e m=kl j , t e l l e aue f 2=O

v i s e u r e f f e c t i f x.

( f.xf 2 0 Dour t o u t d i -

f e s t l a f i b r e d'un mornhisze q: X -+ P ' 2 f i b r e s con-

nexes . Dgmonstration : Comme ( f . h ) 2 0 s i h e s t l a c l a s s e &'une s e c t i o n h n e r p l a n e ,

-f ne peut ttre la classe d'un diviseui. effectif. 'Le th6orsme de Riemann-Roch montre alors que ho(f) 2 2. Ecrivons f I Z+M, 03 Z est fixe et M n"a pas de composantes fixes. Lfargument du lemme 22, Isre partie, montre que !Z2=$=0. On en + h i t que Z=O (sans quoi Z serait mobile par ~iemann-~och) et que est un pinceau sans points fixes. I1 dkfinit donc un morphisme q : X qui s'e factorise (thbor&e

thsse

On

&

ftlest la iibre iie p et d le degr6 du morphisme r. &is

I entraine &=I,

+ P',

de ~ertini)en? q : X.A C 5 P' , 03 p est 2

fibres connexes. C o m e q(~) = 0, la courbe .C est isomorphe 2 PI.

i = a.ff ,;oil

/MI

donc

Ithypo-

d,'oil le r6sultat.

Thkorhe 12. Soient X une surface de Kul~uuzrsodciale, X"' use surface K3, ? :

z2(x,&) -+

H2 (x' ,z) un isomorphisme Be structures de H0dp.e ~olariskes,trans-.

formant cycles effectifs en cycles e f f e c t i f s . , a X

e X' sont isomorphes;

D6monstration : Notons f la classe dans P~C(X) ae la fibre dfune fibration de Kunrmer; considdrons lt61dment, f' = '?(f)

c

P~C(X~ 1. I1 vQrifie les hypothsses

du l e m e 11; il 6Gfinit donc une fibration q: X * i?';B fibres connexes, et 4 on a SXl(ff) C: ( D ~ ). 11 rdsulte alors de la proposition TO que XI est une surface de Kummer sp6ciale et q une fibration de Kummer. On conclut avec le carollstre 9. Remarque 13. Soient X, XI devx surfaces K3,

v: F~C(X)

+

pic(xl)

un isomor-

~hisrnerespectant le cu$-$roduit, h.l& ciasse dans pic(^) dfune section hyperplane de X. Les conditions suivantes sont 6quivalentes : a) \p transforme les cycles effectifs en cycles effectifs;

correspond 5 une surface de Kummer sp6ciale. On montre alors que l'ensemble des p(x3, 03 x correspand 2 une surface de Kummer spgc'iale, est dense dans

P. L' injectivft6 de p en rgsulte aussi'tdt

.

CBI

[Be]

E. BOMBIERI A. BEAWILL23

: Canonical models of surfaces of general type.

:

Publ. Math. I.R.E.S.

42 (1973)

Surfaces al&riques Ast6risque.

conolexes. A parartre dam

CCh11 I.R. CHAFAREVITCH : Foundations of aleebraic ~eometry.Springer-Verlag CCh21 CHAFAREVITCH et ale. : Algebraic surfaces. Proc. of Steklov inst. of

Math. n075 '( 1965). CCh-PI CHAFAREVITCH et PIATECHKI-FRO : A Torelli theorem for algebraic surfaces of tyoe K3. Math. USSR Izvestja Vo1.5

.

(19711, n03. [GI

A. GROTHENDIECK : Techniques de construction en &om$trie

analytique.

I-x S6m. Cartan, t.13 (1960/61). [R]

M. RAYEIAUD

:

CTI

R. TORELLI

: "Sulle variets di Jacobi"

Familles de fibr6s vectoriels sur une surface de Riemann. sh. Bourbaki 11'316 ( 1966/67).

. Rendiconti dells R.

dei Lincei, sbri* 5a, Vo1.22 ( 1913). '

Arnaud Beauville Universitd d'Angers Fac. 'des Sciences BE. Levoisier

Acc:

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(c.I.M.E.)

METHODS OF ALGEBiUIC GEDMETRY I N CHAR. p

AND THEIR APPLICATIONS

ENRICO BOMBIERI

METHODS OF ALGEBRAIC GEOMETRY IN CXAR. p AND THEIR APPLICATIONS

TO ENRIQUES' CLASSIFICATION Enrico Bombieri Scuola Normale Superiore, Pisa I.

The aib of these lectures is to illustrate some of the algebraic

techniques needed for algebraic geometry in char. p, with a particular view to the theory of algebraic surfaces and knriques' classification. We shall study the new char. p features of Kodaira's vanishing theorem, the completeness o'f the characteristic system and the theory of the Picard variety, togethsr with the study of elliptic and quasi-elliptic fibrations, the study of Enriques' surfaces in char. 2 and the characterization of abelian surfaces by means of their numerical invariants. Our basic list of notations will be as follows:

x

= an algebraic variety = Albanese variety of X

hlb

pic X = Picard scheme of X 0 Pic X = connected component of 0 F laic X q = dim Pic X = dim Alb X,

the irregularity of X

K = y[ = the canonical class of X Bi = ith Betti number hPt = dim H~(x,~) 3

h6?' = h2f0 if dim X = 2

Pg

%=

O:

if dim l = 2

and X is smooth = in general, the

dualizing sheaf for Gorenstein varieties hq(p)

= dim H~(x,F)

11.

The Kodaira vanishing theorem.

The origin of Kodaira's vanishing theorem goes back to the theory of Riemann surfaces: the problem there is the construction o f meromorphic functions with prescribed zeros and poles.

For example, it is a well-known

fact in the theory of elliptic functions that a meromorphic non-constant doubly periodic function of a complex variable nust have at least two poles or

a double pole, that is a non-constant meromorphic function on a curve of genus 1 has at least two poles. Let us consider in more detail the case of a projective non-singula~ curve C over an algebraically closed field k. let PI,. gers.

..,Pr

Suppose g

be r points on C, and let ml,.

..,mr

=

genus (C) and

be arbitrary inte-

We ask what is the dimension of the vector space of meromorphic

functions on C, regafar outside PlSH.,Pr plicity at most ml,... ml + * * *

+ mz

,&

and having their poles of multi-

The Riemann-Roch theorem shows that. if 2 .

is sufficiently large then dim = rn1

(2.1)

+

+nr+ 1

and in fact sufficiently large means here ml fashioned language, one says that this space and eq. (2.1) arises:

%+1 -

+

0.-

g

+ mr > 2g -

2.

In old-

g is the postulated dimension of

is referred to as a postulation formula.

The problem

when is the dimension of this space actually equal to what it is

postulated to be? This is-betterstated'in terms of invertible sheeves. divisor D = C.qiPi

, and if we set

deg D =

mi and

If D is the

if L is the invertible

sheaf

L P@(D)

= sheaf of germs of functions on C with poles not.

worse than -D N ,

= sheaf of gems of sections of the line bundle [D]

determined by D 0

then clearly our vector space is H (C,L)

and our postulation formula becomes

0

dim H (C,L) = deg D

+ 1 -. g

As we have seen, this is not universally true; what is true is dim H'(c,L) dim H1(C,L) = deg L '1 g (2.2).

-

+

- .

The question of the validity of the postulation formula now becomes that of the'vanishing of the cohomology group H'(c,L).

In order to handle this group,

the Roch ;art of the Riemann-Roch theorem comes to help. the'dual of tde vector,space H1(c,L) 0

to the space H space.

(c

'I ,QD~ L-~ ),

The point is that

is isomorphic (via differentials on C)

so that we ask about the vanishing of this last

Now recall that a nor,-zero meromorphic function on a curve C (proIf f E H0(C,% @ L-l)

jective:) has as many zeros a:. poles. = 0

deg div(f) but alsb the assumption f

E H (cpluC @ L-l) shows that

This i.s a contradiction if

- deg (wC L-I) = deg L - (2g - 2) . deg L > 2g - 2, hence in this case every section 2

@

This shows indeed that'

H'(c,L)

(2,.3)

if f # 0,

0

deg div(f)

of yC @ L-I is 0.

we thus obtain

= 0

if deg L > 2g

- 2;

the result (2.3) is a form of Kodaira's vanishing theorem for the curve C. Another form of this result can be stated as Hd{C,~-l) = 0 if deg L > 0

(2.4)

and it is indeed (2.4) that we proved ,dth the ~reviousargument. In the case of surfaces, we deal with functions having zeros and poles along @yen

curves.

Assuming X a projective non-singular surface, L an

invertible sheaf on X. (2.5)

x(L)

we have the Riemann-Roch theorem 2 = C (-l)i dim $(x,L) i-0 = hO(L)

- hX(L)

+ h2(L)

where (AWB) denotes the intersection number and where c,[X]

a

vzlue on X

of the second,Chernclass of the tangent bundle

= C (-1)'~~ = Euler class

i=O = self-intersection of the diagonal in X X X.

The additional basic results used together with the Riemann-Roch theorem are:

A.

Serre's duality.

If n = dim X

then $(x,L)

is dual to

(x*%@L-I) ; g.

The theorem of Enriques-Severi-Zariski-Serre. If L is ample (i.e.

if there i s an integer

V

such that

y

> 0 and L @ ~ @(D),

D hyperplane

section of X in some projective embedding) then for every coherent sheaf F we have

d(x,FCOL"') for all sufficiently large m, i,e.

-C.

Kodaira"~vanishing theorem.

m

=

o

2 mo(F,~),

and all i > 0.

(char = 0)

(first form) If L is ample then $(x,,,,~L)=O

for i > O .

(second form) If L is ample then

H~(x,L-')

= 0 for i < dim X

.

.More generally, we have

2'. The theorem.of ~odaira-Akizuki-~akano.(char =

0)

If L is ample then L-l) = 0 for p 3. q < dim X

tIq(X,@@ If we compare

2

and

2

.

(first form) we see that Kodaira's vanishing

theorem is. the sharpest form of the theorem of Enriques-Severi-Zariski-Serre in the special case in which F = 9 On the other hand, the condition char = 0 is really required and additional conditions are needed in char for .;hi validity of

#

2.

Now we shall give Ramanujads proof of the result of Kodaira-Akizuki-'

takano.

0

L ' i s very ample, hence t h e r e i s a

Let us consider f i r s t a special case: section

s

E

H'(x,L)

with

D = div ( s ) = a smooth hyperplane section of

We choose l o c a l coordinates

are l o c a l coordinates on D equation f o r

where

3,

df >

D,

I

is I

xl,...,x

and

P

a t a point

on

.n

D.

,X;-~,X,)

70

xl,...,x

n- 1

is a l o c a l

We have t h e exact sequence of sheaves

t h e sheaf of i d e a l s of

i s the multiplication by

such t h a t

..

xn s f(xl,.

E

X

X.

rD

D (thus

df

and

YO(-D)),

res

qe@X/rD, It i s

i s t h e r e s t r i c t i o n map.

a useful exercise t o v e r i f y t h e asserted exactness, and the f a c t t h a t the mappings above a r e w e l l - d e f i n d (i.e. ing well together). a r e supported a t

independent of l o c a l coordinates and patch-

We do t h i s a s follows,

D,

Since t h e sheaves i n question

we nee* v e r i f y exactness .onLy a t points

of a l l , we check t h a t the map l o c a l equations f o r

D,

a 1-cocycle defining

D,

i s well-defined.

df

If

(fi,ui]

or. ui n U so that j 5' we see t h a t the i d e a l sheaf

fi = 6. .f

[,gij,Ui

D.

First

i s a s e t of

0

sj]

is

iD i s an i n v e r t i b l e

ZJ

sheaf ,with t r a n s i t i o n functions

P

-1

[gij,Ui

n uj3:

dfi. = g i j d f j mod f

Now j

hence t h e map r$-l@ ' i s well-defined,

To v e r i f y exactness, we have t h a t

of d i f f e r e n t i a l forms d,.

c(%

= dxi

1

... c

.

1s

pq X, p

consists of germs

Is i,, I n y ' I OD)p i s i d e n t i f i e d with elements

dxI, I = , ( i 1 9 - . p i

, (3%

ndxi

9 L' . and the k e ~ e of l t h e r e s t r i c t i o n map i s made up of

mod (xn))dxIv

elements

qj% +?mD

qD@

.(cp,

mod ( x n ) ) d x l ~dxn which v i a

df

is identified

n & r card I=q-1 with

g n#.?I card I=q-1

r e s u l t follows.

(< mod (xn) )dxl,

i. e. with

( n i - b 960

and t h e

A s we a r b dealing with a l g e b r a i c a l l y closed f i e l d s of d e f i n i t i o n , of

char. 0 we s h a l l suppose t h a t of

X,

Now, s i n c e D i s a hyperplane section

k = C.

t h e Lefschetz theorem shows t h a t .the n a t u r a l map

i s an isomorphism i f

-

i < n

1 and i n j e c t i v e i f ' i = n

H~(x,c)> -

- 1.

$(D,@)

By the Hodge

decomposition

p

i s an isomorphism i f I f we s e t

+ q
- -1 and is i n j e c t i v e i f

rD

Lil = O D @

- 1.

+q=n

LD i s a very ample i n v e r t i b l e sheaf on

then

being associated with t h e hyperplane s e c t i o n of

D,

p

By an induction

D,

hypothesis, we thus have

-

H P ( X , ~ - ' 0 L;)'

(2.8) if 0

p

+ q < n.

The exact cokomology sequence of (2.6)

- 1 @ L-'

D

ef $63

(2.9)

+ q2n

p

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

4-

(which we w r i t e as

G1, 5@ --+ 0 ) now y i e l d s

0') 3 H ~ ( X , C $ )

Hp(x,$8

. i s an isomorphism i f

(2.9)

=O

-

1 and i s 'injective i f

3 -+$@

f a c t o r s through

$i

- 1.

p-+ q = n

4

cBD -+

Since from

and (2.7) we conclude t h a t ' t h e n a t u r a l map

(2.10)

HP(x,Q

i s an isomorphism i f

?,+ q < n

-4H~(X,C$B

- 1.

(n,)

F i n a l l y we look a t t h e cohomology

sequence of

b

.O--+~@~-~,rg~~oq--+ and deduce from (2.10) t h a t

i f ptq < n-1.

-

L-'1 = o

HP(x,~$o

-

The l a s t piece of t h e cohomology sequence f o r pt-q = n-1 y i e l d s

o

HP(x,B ~ 1")

HP(x@

and t h e l a s t arrow i s i n j e c t i v e by (2.7) I & x , C ~ @L-')

= 0 if

p

+9=n

-

1,

--+H P ( x , ~ @oBD )

and (2.9).

It follows t h a t

and t h e pmof of

C'

i n case

L

is

very ample is completed by induction on I n t h e general case i n which

n = dim.X.

i s ample but not very ample we use the

L

method of branched coverings, already introduced by Mumford i n t h i s type of questfons. Say Lm i s very ample and l e t of d section of let

L

D

lLml

be a smooth divisor of zeros

L ~ . With respect t o a s u i t a b l e a f f i n e covering

n Uj)

6; determined by the 1-cocycle (gij,Ui

system of local equations f o r X' C A

s i d e r now t h e v a r i e t y

F i r s t of a l l ,

X'

hence

D,

1

fi = gyjfj

and l e t on

Ui

[Ui)

of

(fi.uii]

n Uj.

X

be a .

We con-

x X defined by

i s a smooth projective variety, since a t a si'ngular point

we have

hence

z

i

= 0, f i = 0, dfi = 0

and

fi = 0

is not a smooth divisor, contra-

d i e t i n g our assumptions. Let

ii : X

* n f

we ccnsider

a smooth d i v i s o r

..

,X

be t h e covering map.

then l o c a l l y

i s a section of

z = [zi)

xl,.

' 4 X

n

on X

D'

on X',

such t h a t

ic;: local coordinates on

( i ) .D' (is) (iii) (iv)

D'

X'.

*fi

m = fi = zi and

satisfying

L'

zi = g

If

,

z hence ij j . The section z defines

a s one can s e e by taking l o c a l coordinates

fi = x l , ,and-now z, x2,. Finally

D'

..

,x

n

a r e corresp'ond-

i s very ample, because

i s smooth

22

i s very ample on X;

however, we w i l l not prove t h i s f a c t here.

* = n L,

* zm = rr f.

i s ample.

belongs t o an ample system

dim X' L~

* rr L

*

*

Then L' = l7 L

If

L' ' i s the i n v e r t i b l e sheaf

we see t h a t by the special case considered before we have

HP(X*,~,@ L**l) = 0 if

p

+ p < n.

Everything n o r follows from t h e f a c t t h a t

a d i r e c t f a c t o r of t h e previous group.

H ~ ( x , ~@ $ f l )

is

To show t h i s , we have t o produce a

s p l i t t i n g of

which i n t u r n w i l l follow i f and only i f

4 -=-Tr&' In order t o e x h i b i t the required s p l i t t i i ~ g ,we use a t r a c e map.

splits.

U

i s open i n X and. l e t

E N~'(u),$,

.r Let

E

where

: X'

--3-

X'

@ L*-')

X'

be t h e automorphisn of

Say

.

determined by

zi

+ ezi

ern =; 1 and l e t

*.

ff

w = p m ;

t h e r e i s a unique element Tr(m) ( P ( U , ~ B L-~I such t h a t #

&*(~r u)) = w

,

.,,

and f i n a l l y

$.~r

i s t h e required s p l i t t i n g .

To s e e t h i s ye have t o check t h a t

1

*

q = 9 T ~ ( F$,

which i s e a s i l y done with a computation i n l o c a l coordinates, W e conclude with t h e remark t h a t t h e condition

twice:

t h e f i r s t time, t o prove (2.7)

by t h e i n t e g e r

(2.7)

m

already when

--

It is, however, the f a i l e r e of

2 which is responsible for the fzilzlre of

Kodaira's vanishing theorem i n char

3 0.

Now we analyze i n inore d e t a i l t h e s i t u a t i o n i n case t h e c r u c i a l cohomology group t o study i s

D E

l ~ f s,

i s a s e c t i o n of

has 6een used

and t h e second t i n e , allowing d i v i s i o n

t o define the splitting. dim X

char = 0

dim X = 2.

Here

I f there i s a divisor H'(x,L-I).

i with d i v ( s ) = D,

we have an exact

sequence (2.11) It i s c l e a r t h a t constant;)

0 H (X, r D ) = 0

( t h e only regular functions on X

hence the cohomology sequence of (2.11)

+

1 dim8k e r ( ~(X,

LOX)

We a r e l e d J t o Problem 1.

Compute hO((nD).

Problem 2.

Compute

1 u(D) = dim h r [ ~ (x, Lemma (~anianujam). -

operation P

If either

i s i n j e c t i v e on H1&,

We show t h a t i f

Dl

char(k) = 0 OX),

< D2

-

H'(D, UD)]

.

.

OX)4H1(D, OD)]

o r t h e Frobenius cohomology

we have

a(D) = a(Dred), Proof. -

yields

+ hO(mD)

dim H ' ( X ~ L ' ~ = ) -1

(2.12)

are the

5 2D1

. then

a(Dl) = a(D2);

the 1-a

w i l l f o l l o w by induction on the m u l t i p l i c i t i e s of t h e various scn?pmezts of 3. If

char(k) = 0 ,

t h e i d e a l sheaf E

-+t

1

+E

we use a truncated exponential sequence as follows.

i s of square 0 (because DL < D2

Xl1"D2

i s a group homomorphism

PI

(3

exact sequence

* 40

D2

5

Since

2D1) the map

which f i t s i n an

D2

'

and we get a 'cohomology sequence H~{D~,@*)-%'H~(D~, )&P~C(D~)-P~C(D~). We bhall :how fact, say except f o r

Q'

t h a t t h e kernel of ker(pic(D2->

pic(D2) +pic(D1)

pic(lll)]

and

p($) = a and now

hence

~

,

i

In

(we use additive notation

nd. = 0

1). T h e r e i s 8 E ~

has no torsion.

~

~ such , r

~

~

@

~

~

)

f@

E ~HO(D~,B* )

0 * H (Dl,0 ) is such that

If y

.

aye@,

a(yl'm)

then

'

=

B

* H~(D~, )

provided we can take an m-th root of y In this case however B € aH0(Dl,@ 1 = ker I)

*

1

hence a =

p(B)

with

= 0 and there is no torsion in ker{Pic(~~)>

Now y, being a global

* section of @ ,

P~C(D~)].,

is written uniquely .as

Y=Yo+E where

Yo

is locally constant and non-zero on Dl, while E

order N jmax multiplicity of components of Dl.

is nilpotent of

In order to take m-th roots,

we may suppose yo = 1 and, as we are in char =: 0, we may use the binomial series

is nilpotent)

(E

1

+l

N- 1 m = g (IF) 3-0

&3

Now look at

and let:

K(D~) = ker(picO(X) Since ker[pic(D2)-->

pic(D1)]

has no torsion either.

pic(Di))

.

has no torsion, we also see that K(D1)/K(D2)

If Ai is the connected component at 0 0

is a subgroupscheine of the abelian variety Pic (X)

then Ai

of K(D~)

and, as we are

0, Ai is reduced and is'an abelian variety by Cartier's theorem.

in char Now A1/h2

has only torsion of finite order, since K(D~)/A~

group and K(D~)/K(D dim A1 = dim A2. ker[-il(x,

-

Ox)-+

is a finite

is a finite group and 2) has no torsion, hence A1,'A2 Finally, as we are in char = 0, we see that 1 ) ] is the tangent space at 0 of A; (recall H (Di, i'=

that K(Di)

is reduced, -beinga groupscheme in char

a(Di) = dim Ai, and a(Dl)

9

0) hence

= b(~~).

If char(k) = p we proceed in a different and much simpler way.

We have

the exact sequence

o--'gDl@D2-+ 0

- 4 C o

D2

hence a cohomology sequence -

-

If we apply the;Frobenius cohomology operation F to this sequence we obtain the exact sequehce

Y

Y

@ ) = 0. In fact,' F acts on H 1 b 0 ) by r&ng D2 ' D2 cocycl& to the p-th power, and now p 2 2 while %,uD. is of .Square 0. l L Thus we have proved that F kills the kernel of H (DZY%)4 because PH'(X,

Now look a t

If u is an element of H1(x,@~) with a(")

40

1

and ~ ( u )= 0, then

W )--+ til(D1 # )) and m(u) = 0 by our previous remark, D2 ' hence a ( m ) = 0 while a(u) $ 0. Since F ker(h) C ker(a), we see that F 1 This completes the proof of ~amanujam's cannot be injective on H (x,@?,).

a(u)

E ker{~'(D~

'

Iemma. The next result is very important and useful in dealing with curves on a surface. Let C . > 0 be an effective divisor on a smooth surface 0 X and let L be an invertible sheaf on C with H (c,L) $ 0. Proposition 1.

Then there is a decomposition C = C1

such that

+ 5,C1 2 0, C2 > 0

.

(C1*c215 deg

(L@@ C2 c2 Moreover, this will hold for every C1 such that there is a section s of L vanishing on C2 but not on any C' Proof. -

Let s € H0(c,L)

with C1 C C* C C.

be a non-zero section of L.

of s to sn irreducible component of

c

If the restriction

is never identically 0, then,the '

degree of L on this component is non-negative and a fortiori degC L

2 0.

Now suppose that 0 < CL < C is a maximal divisor stich that s vanishes on C1.

Then we have two exact sequences of sheaves 0-fl 0-

>0

-%L-+Z.L/S~

-+=-0 . C1

F

c L / s u C r L@O

where F is supported at finitely many points.

Taking Chern classes on X

we,find

whence the equation

=

(1 + C2 + c22)(?

in the Chow ring of X.

- .length F)(:

+ Cl + C12 -

dekl L)

The equation in degree 2 is simply (CleC2)

+ length F = deg

L

C2

and Proposition 1 is proven. Definition. An effective divisor D on a non-singular surface, X is numerically contiected if for every decomposition D = Dl + D2, Di > 0 we have

(

~

D~~ ) 0 > 0.

Now x+?e can solve Problem 1 as follows. Proposition 2. Proof.' -

0

If D is numerically connected, then' h fQD) = 1.

We apply Proposition I with C = D, L = @D.

Now

deg (0 ) = 0 for all C2 and if D = C1 + C2 with Ci > 0 we cannot C2 C2 have (Cl0CZ) 5 0. It follows that, by the last clause of Proposition 1, we

must have Cl = 0 hence no section of OD can vanish on components of D. This sh-

that H'(D,@~)

consists only -of constants atid the required con-

clusion follows. Now we . prove

Ramanujam's vanishing the'orem. Assume that char(k) = 0 or the Frobenius 1 cohomolo~y~operation is injective on H (x,Ox). Let D be a numerically con,

nected divis~ron X and let L = . ) D ( $ ? f Suppose also that for large' n the linear system of a pencil,and has no fixed components. H1(X,Ci) Proof. N

IL I

lLnl

is not composite

Then we have

= 0

.

The assumptions about L imply that there is N > 0 such that

has no base points and determines a birational map 9 : X

+Y

where

Y is a normal ,surface. We shall show that, taking a larger N if necessary, N we have H'(x,L-~) = 0,- hence (2.12) yields a(C) = 0 for every C 6 1L Since ND E ILN we,have ~ ( K D )= 0 and nc;w 'namartujam' s leshows that

I.

I,

1

+ h0(QD) + "(D) 0 = -1 + h (UD)= 0,

a(D) = 0. , Using (2.12) again we get h (L-~)= -j

by ~ro~osition 2, and our theorem is proven. It ,reriainsto check the vanishing of H~(X,L*'~)for sufficiently large N.

If L is ample, this is a consequence of the Enriques-Severi-Zariski-

Serre theoqem. be as before.

In the general case we proceed as follows. Let 3 : X--+ Y 2 We want to show that @ LNn) = 0 for all lirge n and

hl(x,(k

the required result will be a consequence of Serre's duality.

@ L~"))

for large n

era^

i

shows that H1(x,4 @ LNn) = 0 if 1 2 Nn 1 = 0 and if R P*((DX @ L ) = 0 The vanishing of the H .

spectral Fequence for the map

H'(Y,+,(~;

Now the

is a consequence of the Enriques-Severi-Zariski-Serre theorem,

because LNn))- = H1(y, 1 2 In order to check the vanishing of R 9*(%@ H1(~,+,(G@

$*(g) B &b)) . 1;

Nn, ) we note that, since this

sheaf is concentrated at the finitely many points P

(&*(-<

has dimension 1, it is sufficient to prwe th&

such that $"(P)

Y

@ L"))~

= 0.

By

Grothendieck's "~hkorSmefondamentale des morphismes propres" it is sufficient to prove

H'(D,

(r; o

~~~)/y~c@ = o

for all effective divisors D with support @ ' ( P ) . Now the dualizing sheaf -1 2 2 bn D is hence by duality we have to show that

7,4%

H~(D,L-~%Y,'/@~) . = 0

By Proposition 1, if this last group were non-zero, we could write D = Dl

+ D2, DL 2 0,

D2 > 0 and get

whence Nn(L0D2) Now ( L . D ~ ) = 0, because

+

5

2

(D2)

.

contracts D to a point, and

the intersection quadratic form of

i;

(u:)

< 0 becsuse

contracted divisor is negative definite;

this contradiction proves what we wanted. Finally we mention the following very precise result.

-

Theorem, Let L be an invertible sheaf on X,

char(k) = 0 ,

and

suppose

(ii)

(L-C) 2 0 for all C > 0.

Then we have Hi(x,L")

= 0

for i < 2.

Conversely, assume (L~)> 0 n sufficiently large.

and tliat H~(x,L-~)= 0 for i < 2 and

Then we have (2-C)2 0

for all C > 0.

This theorem, due to Ramanujarn, gives a necessary end sufficient condition for the vanishing of H1(x,L'~)

for invertible sheaves with (L2) > 0 ,

in terms of numerical intersection properties of L.

111.

The

completeness of the characteristic system. The idea of char-

acteristic system is intimately related to the concept of deformations of a curve on a' surface.

Intuitively speaking, if [ c ~ ) is a continuous family of

curves on X, it defines a section of the normal bundle N of Co in X as followso The normal bundle of Co in X

is nearly isomorphic with a neigh-

borhood'of Co in X, and for ,small a

the curve C is a section of this a neighborhdod; as a -+ 0, the curves Ca will be more and more identified with a section of the normal bundle N

of Cg.

In terms of divisors of C0'

nowili

be a certain divisor ya on. Co and yo = lim y is the a-0 a divisor of the section of the normal bundle of Go. If S is a family of CQ

deformations of Co, what we have is a characteristic map

The fundamental question is now whether p 0 whether dim S = dim 9 (CO,N).

is bijective and, even better,

In order to put this in rigorous form, let L = Spec k[r]/r2 be a curve on X.

and let D

h e defines the normal sheaf of D in X by ND

=~p)/% ;

if D .is non-singular, it is also the sheaf of germs of sections of the normal bundle of D in X. Proposition. The set of curves

aC X x I

which extend D C X is

0

naturally isomorphic with H (x,N~). Proof. -

Let {fi,ui]

are given by

on Ui

n vj.

define D. ,If Q extends D,

+ rgi,Ui]

(fi

where r2 = 0 and (fi

Now let (fi.+'egi) = (aif.+ rbij)(fj

on

Ui

n Uj

+ rgi)

; this gives

f .= aijfj i pi = a'. .g. f bijfj ZJ

J

+

€gj)

local equations for

-

(unir)(fi 3

+ rg.)J

hence gi = aijgj mod (fj) which shows that [gi mod fi, U ~ ' J transforms by means of the 1-cocycle [aij,"

i

"011,

i. e. is a section of the sheaf @(D)/YD

2 ND.

It is also

clear that every section of ND gives rise in this way to a unique 8 c X x I and our result follows. Given a family of curves & C X x S and a closed point s

Corollary.

S

there is a canonical homomorphism

I

1

Zariski tangent space Ts to S at s

Moreover, if

CX x S

L HO(X,N,)

.

is a universal family, p is an isomorphism

The proof of the corollary is obtained by noting that given t there is a canonical morphism f : I+ to I, we obtain

OS)f

-S

with image s.

TS, By base extension

C X x I which extends Ds and apply the previous pro-

Linear
position.

The above corollary shows that if S is the universal family of defonnations of D, then Zarisu tangent and the ~robiemof deformation becomes that of the smoothness of S at D. The fundamental theorem ii the'following.

-

If H'(x,@(D))

= 0, 'the following are

is reduced if char(k)

= 0 (Cartier) or if p = 0 g

Theorem (Grothendieck-Cartier).

equivalent: (i)

S is non-singular at D

(ii)' S is reduced at D (iii) Pic0 (X) is reduced (iv) Pic0 (X)

is non-singular.

Moreover, picO(x) (Mumford).

We shall not prove this result here and refer to Mumford's book for proof.

IV.

Elliptic

or quasi-elliptic fibrations.

We are interested in surfaces X containing a one-parameter family of elliptic curves or more generally curves of genus 1.

By blowing up the base

points of the family, taking axstein factorization and blowing down exceptional curves of the first kind which appear as components of a fibre, we are led to study gelatJivelyminimal fibrations.

A

fibration f : X

+B

of a smooth

surface over a non-singular curve B, with f*Ux =OB, with a11 fibres of arithmetic genus 1 and such that no exceptfonal curve of the first kind is a component of a fibre, is called a relatively mininal elliptic fibration or quasi-elliptic fibration. Since the function field k(X)

is separable over k(B),

almost all

fibres are generically smooth, and since we deal with the case in which &.f

=@;

we see that almost all fibres are irreducible.

It follows that

they 'can'beonly of the following kinds: (a)

non-singular elliptic

(b)

rational with a node.

(c)

rational with a cusp.

Now case (b) cannot occur, and case (c) may occur only in characteristics 2 and 3.

We shall distinguish between (a) and (c) by talking about elliptic

fibrations and quasi-elliptic fibrations respectively.

A curve D = G n.E. on X is said to be of canonical type if 1 I

C K ~ E ~ )=

(D.E.) = 0 for all components Ei.

If D is not a multiple of

enother curve of canonical type, D is said to be indecomposable. fibre of f : X

4 B is of canonical type.

Now every

In fact, if C is a general

fibre we have

(D-E~)= (c-E~)-=o because C and the fibre D = C n.E are disjoint; to prove that (K*E~)=' 0 1 i 2

too, we note that if D is reducible then (Ei) 2 . 0 'because

as

Ei

and

are distinct for

E

j

two o r more components, because for a t l e a s t one index j

$

Dred

and i n f a c t

.- 2

an excepti'onal curve of the f i r s t kind,

and

with

( D ~ =) 0

has

Dred

( E i * ~ . )> 0 J (K-Ei) < 0, for otherwise

would be

Ei

against the hypothesis

t h a t our f i b r a t i o n i s r e l a t i v e l y minimal. Ei

if

+ (Ei)2 5 -2

= (K*Ei)

2 (K*Ei) = -1, (Ei) '-1

f o r a l l components

(Ef), < 0

i s connected and hence

Now we cannot have

i.

-2 5 2p(Ei) and wemust have

i, j,

It follows t h a t

and f i n a l l y t h e assumption

( K * E ~2) 0

p(D) = 1 together

yields

= C ni(K'~i) 2 0 and a l l

( K - E ~ )must be

0.

~t f i n i t e l y many points

with

\22

and

PX

b,.. b

B

the fibre

f-l(bl)

i s multiple,

indecomposable of canonical type.

We meed:

Letma. irreducible.

Let

D =

Assume t h a t

Then every d i v i s o r

Z

only i f . D =~ 0 and Proof. -

C ni Ci

Write

.r

be an e f f e c t i v e d i v i s o r on X

(C~OD 5)0

C miCi

for a l l i s a t i s f i e s Z2 5 0

and t h a t

with each

Ci

D i s connected.

and e q u a l i t y holds i f and

Z = I D , X E Q. xi = mi/ni.

We have

I f e q u a l i t y holds everywhere we have e i t h e r

xi = x

j or

(Ci*c.) = 0

J

for a l l

i, j and since D is connected, xi is constant, i.e.

mi = Xni,.X E Q .

Corollary* If D is indec~mposableof canonical type, then D is numerically connected, hence dim H '(D,%) Proof. -

Let D = Dl

+ D2,

= 1.

2 Di > 0. NOW 2(D1-~2) = (D )

- (Dl)2 - (D:)

by txe previous lemma and the fact that D is indecomposable, hence Dl, D2 cannot be proportional to D. The last clause.of the corollary now follows from Proposition 2 of Ch. 11. We return to the question of the nature of the fibres of f.

We have

where L is an invertible sheaE on B and T is supported at the points

dim~O(f''(b),@)22; 1 r +b) in fact, by EGA 111 7.8, R f,UX is locally free at b if and only if OX (3.2)

is cohomoJ.ogically flat at b in dimension 0. cdrollary, the fibre jb('f multiple fibre.

In vicx cf the precedicg

must be decmposable and, in particular, is a

Since not every multiple fibre verifies (3.2),

Definition.

this suggests:

The multiple fibres over supp T are called wild fibres.

A fundamental result in the theory of elliptic or quasi-elliptic surfaces

determines the canonical divisor on X in terms of the fibres of f.

We have

B be a relatively minimal'ellipticor quasi1 elliptic fibration and let R f @X = L @ T. Then Theorem 1.

Let

f : X+

*

* -1 cyl ='f (L @

13.3)

uQ@O(C a P 1 ai

%here (i)

m P

(ii)

where ?(B)

are the multiple fibres

X X

05 a <

x

"k

is the genus of B.

Proof. -

hence i f

For any non-multiple f i b r e

yl,...,yr 0

4

a r e d i s t i n c t points of

r

B

.

C f-L(Yi)) -+

Wx 4

i=l

yields

we have

the cohomology sequence of r 63 -1 +o is1 . f (Y,)

r

di+p

r

provided we take

@(

l a r g e enough.

( ~ . f - ' ( ~ ) ) = 0 therefore

have

f-l(y)

r

i=l

If

1 2o

fl(yi))

i s i n the l i n e a r system above, we

D

cannot have components transversal t o

D

f i b r e s and we can w r i t e KX I (sum of f i b r e s )

where

A

is contained i n a sum of f i b r e s and does not contain whole

0

f i b r e s of

f.

we have A

v5 2

Av

If 0

i s 'a. connected component of

with e q u a l i t y i f and only i f

of t h e f i b r e containing

h.

,

2 (A,,) = 0,

= avPv

have. (<)

2

(h)2 0. the fibre

(A') =

L

containing' ,A,

f o r some i n t e g e r

by the previous Lemma

i s a r a t i o n a l submultiple

A,,

0

2 av

(4)5

0

2 0 = (KX*AV)= (A*%) = (AY))

Moreover, we have Cv

A,

It follows t h a t

(
+A

i s multiple,

CV = mvPV,

We have proved t h a t

X = ~*%(A)Q @(raAPx)

E

div(B)

and we deduce

.

f+% = UB(A) Now the d p a l i t y theorem f o r a map s8y.s t h a t

f*% = H

1

O ~ Rf&,

ug)

L - ~ @WB because t h e dual of a t o r s i o n sheaf i s

0,

and i t follows t h a t

* ux = f (L-I @ uB)@ o tc a,PX)

The s p e c t r a l sequence of ' t h e map

f

and

< C. We have a l s o shown t h a t we must

= 0.

for a suitable A

hence

yields

.

by the Riemann-Roch theorem on B.

Since deg

UJB

= 2p(B)

-2

we obtain state-

ment (iv) of our theorem. It remains to prove (iii), and this follows from Proposition 3.

5 , P,I

Let

aX be as in Theorem 1 and let

~i)

vx = order is the sheaf of ideals of P

where of Pi.

a'

be the order of the normal sheaf

Then we have:

( i ) VX

divides

5

an6 a X

In particular, if a& < % the multiple fibre m P

in writing.

-1

then vX <

end this is equivalent to

being wild.

X X

Proof.

+1

In what follows we drop the suffix i, everywhere for simplicity If r

2 s 2-

1

the restriction map @rp

hence h1(Or*)is non-decreasing with r.

Since

dosPis surjectsve, x(urp)= 0,

this proves

that h0(@ ) is non-decreasing too. rP We have an isomorphism

-

-

aad via this isomorphism we get an exact sequence res ' ~ V P O QP @(Vt.l)P 0 where res is the restriction. Since constants Tn B (P,@ (VC.l)P)

-'

into constanis in HO(P,@~~),

0

dim H (p,qvtlIp)

tpj-mP

2 2.

are mapped

the cohomology sequence shows that

Finally

V

divides both m and a

(trivial) and GpQ J -'-'

1%

In order to check the isomorphisn? top %' Qp9

D

OP

+ 1,

becacse

.

we no&

& a t since

v,(Vp)= 0 and

h o p = 1, we have h0(mp) = hl(q) = 1, hence mP has a non-trivial secNow m has degree 0 on every component of P and P is numericalP ly connected. By Proposition 1 of Ch. 11, we see that s cannot vanish tion s.

identically on any component of P.

Also, s cannot have isolated zeros,

because wp has degree 0 on each component. We deduce that s never vanishes and hence multiplication by s-1 yields the required isomorphism wp% Qp, This proves Proposition 3. Corollary. V

X

5

If in addition h1

1 we have either a

X

f

1 = mX

or

+ a 4-1,

X

Proof.

= mk* = 0 "d

Since X(@(yt.llP)

0 h

((i(vtl)P)= 0,

using duality we

find '~(~1)~)2 2

dimP('H Now the cohomology'sequenceof 0

+ mx -JyV-l

yields dim H*(x,

Q % j ( _ m(vtl)p

r-v-'@ 3)

1 1 since dim H (X w ) = dim H (X,Ox)

'X

dimension is possible onLy if fore (a+l)/v = m)\,

or m/v

V

5

5

--+ 0

dim H'(X,W~)

1 by hypothesis, This increase in

+ a + 1 2 m,

or 1

+ (a+l)/V

2 m/v.

- 1.

In some cases St may happen that there is a maltiple fibre mP

v = 0, i.e.

There-

P has 8 trivial normal sheaf.

such that

By the results of Ch. I11 we now

see that the universal deformation space of P consists of a single point but with Zariski tangent space of dimension 1; although there are non-trivial first-order deformations of P, we find obstructPons to extend these deformations to higher order. The occurrence of wild fibres is a'phenmenon peculiar to char. p. fact Kodaira has shown that in char. 0 we must have and more generally Raynaud has proved that mX/yk istic p.

V

= mk, aX = mX

In

-

1

is a power of the character-

Now we s h a l l examine i n more d e t a i l t h e case i n which B = P

1

,

the pro-

j e c t i v e line.

B = pL, a l l f i b r e s a r e , r & i o n a l l y equivalent and t h e canonical

If

bundle .formula takes a simpler form. Theorem 2.

If

B = pl,

We have

then

= r f -1(y) + C aXP, X " + length T. Moreover KX

where

r = L2

unless

+ X(@x)

Z+C-zo X mx i s r d t i o n a l o r ruled and t h e plurigenera of

X

X

a r e ' t h e n given by

the formula

[I

where

denotes the

Proof. -

The f i r s t formula i s obvious, because a l l f i b r e s a r e r a t i o n a l l y

equivalent, i.e. The inequality

X

unless

i n v e r t i b l e sheaves on P' r

.='.-

+z A

a~ --?

0

a r e c l a s s i f i e d by t h e i r degree.

a l s o comes from

(5.~ 2) 0

h

f o r a l l ample

i s r a t i o n a l o r ruled (Castelnuovo, Enriques, Mumford).

H

Since

we have

where

0 j b (n) < rnX

1

, hence dim H'(x,&~

= dim I40(B,@~(nr+ n

provided

nr

+E x

p] a"'

2 -1,

I n any case, we have t h a t

x "x

r +1

and our r e s u l t follows. Pn

cannot grow more than l i n e a r l y with

I f one defines t h e Kodaira dimension t o be

we have

&)I ))

n.

x = -1

a l l plurigenera

x = 0

Pn

x = 1

Pnwyn7

5

1 for a l l

% =

X

X = 0,

and

Pn

0

X

i s e l l i p t i c o r q u a s i - e l l i p t i c (Enriques, Mumford). X

need not be e l l i p t i c .

i s e l l i p t i c t h e canonical bundle formula and t h e Riemann-Roch

X

B

f o r some n

o ..

some Y >

t h e s i t u a t i o n i s more complicated, s i n c e

However, i f theorem on

n,

i s r a t i o n a l o r r u l e d (Castelnuovo, Enriques, Mumford),

K = 1 then

while i f If

then

-1,

vanish

some y > 0

x = 2 -pnrJm, If

Pn

shows t h a t

where. y

Pn N yn

2

0

-

We deduce t h a t Theorem 3. o r 1.

Let - X be a n e l l i p t i c o r q u a s i - e l l i p t i c s u r f a c e with K = 0

Then i f

f : X-+

B

i s a r e l a t i v e l y minimal f i b r a t i o n we h a v e ' t h a t a . - 2 + length T + C 2

2p(B)

docs not depend on t h e f i b r a t i o n

x ="A

f.

Theorem 3 i s meaningful, s i n c e some s u r f a c e s may have more than one fibration. Going back t o s u r f a c e s w i t h x = G o r 1 we s e e t h a t ninimal models of them a l l have dimln5l

(<)

In fact, i f

+ d i m [ - ( n - l j s l --+

quadratically with %

= 0.

= -1)' or

n.

= 0 o r 1 then

as

n

-+ =,

by t h e Riemann-Roch theorem,

This implies that e i t h e r

Pn& yn2, Y > O

(5)5 0. 2

o,

> 0 then

(<)

(hence If

=.2),

(<) < 0

and

by Mumford i n a l l c h a r a c t e r i s t i c s , we have t h a t

P, = 0

for a l l

(hence

s o t h a t we deduce t h a t i f X X

i s minimal then, a s proved i s r u l e d and U' = -1.

T h i s being done, t h e Riemann-Roch theorem f o r t h e sheaf OX y i e l d s

and we o b t a i n t h e

n

Basic formula. --

If (<) = 0 we have 1 10 + 12pg = 8 dim H (x,Ux) + 2(2 dim H1(x,OX)' 2q) +.B2

-

.

It ,is k n o w that dim H1(X,OX) = q if char(k) = 0 and more precisely dim H

1

(x,@~) 2 q

in any characteristic, with equality if and only if ~ic(X)

is reduced; we

have also (~ombieri-~umford) 1 dim H (X,@*)

5 q + p8

.

the basic formula shows that we have only 1 finitely many possibilities for h1((4y), B2 and A = 2h 29. In

If we give an upper bound for p g'

(3) -

particular we have Table of ~nvariantsif --

(K')

= 0, pg

2 1.

The first four invariants are deformation invariants, since they are determined by the Betti numbers, while the last three

$

are not always deformation invariants if char(k)

0. This suggests

that Enriques' classification by means of plurigenera should be replaced in char(k) = p by means of a classification,using the Betti and Chern numbers, together with the Kodaira dimension. We mention here a new phenomenon of char. p. classical theorem of Castelnuovo shows that if

If char. = 0, then a

x((3X)

< 0 then X is ruled.

This is no longer true in char. p, and Raynaud has constructed non-ruled

p >0

surfaces with f : X--+ ties.

B

g

and

X(@x) < 0.

These surfaces admit f i b r a t i o n s

with f i b r e s of p o s i t i v e arithmetic genus having cusp singulari-

The following r e s u l t of W. Lang shows t h a t t h i s i s intimately related

with the f a i l u r e of Kodaira's vanishing theorem i n Theorem 4.

Let

f :X--4

char. p.

be a f i b r a t i o n such t h a t a l l f i b r e s a r e

B

i r r e d u c i b l e and have p o s i t i v e a r i t h m e t i c genus, admitting a section C. NC = @*(c)/%

be t h e normal sheaf on 2

Then i f

C >0

(and

C = B

+o .

H+X,H-'J

X = 0.

We cannot have

pg 2 2, f o r i n-i s1s2

sl, s2 a r e l i n e a r l y independent sections of

i = 0, 1,.

..

for a l l

n,

,n

a r e l i n e a r l y independent sections hence

x

2 1.

Thus surfaces with

ing t o the previous table. rJ

f :X > -

X

Let

Now ~((9%)= nx(OX) and

5

5

2

a t once t h a t t h e case i n which B1 = 2

implies

@(5) then of (5)and

H = 0

,

Pn

2n +

1

have 2nvariants accord-

be such a surface and suppose t h a t We have

1H ~ ( x , o ( ~ ) ) hence ,

x(@x)

n = 1 i f . x ( @ ~= ) 2, n

because

X

i s an 6teI.e covering of degree n.

H ~ ( ? , o ( ~ )Z ' ) HO(x,~,o(M(x))

that

i s a section).

C

£ * N ~ i s ample

Now we s h a l l consider surfaces with if

since

we have t h a t

( i ) H = oX(c). (ii)

C

Let

2 if

X =

*

= f WX

0

for

and a l s o N

X

too.

and from the t a b l e of i n v a r i a n t s we see x(%)

= 1 or

B2 = 14, B 1 = 2

dim ~ l b ( X )= 1,

hence

~ ( 0 =~ 0.)

cannot occur i f

X

This shows X

= 0

has connected & t a l e

coverings of a r b i t r a r i l y t h i g h order, obtained by pull-back of Q t a l e coverings of

Alb(X).

This leads t o t h e following preliminary c l a s s i f i c a t i o n of

minimal models.

(b)

x = 0, B 2 = 22, B1 = 0. % = 0, B2 = 10, B1 = 0.

(c)

%

(a)

= 0, B2 = 6, B1 = 4.

The surface i s c a l l e d a K 3 surface. We c a l l these Enriques' surfaces. We s h a l l prove t h a t t h i s characterizes

abelian surfaces. (d)

K

= 0, B2 = 2, B1 = 2.

These surfaces a r e called h y p e r e l l i p t i c or

quasi-hyperelliptic. We shall not deal here with surfaces in (d), ing facts.

If X

is as in (d),

then f : X

but mention only the follow-

+ Alb(X)

tion and there exists a second structure tr : X + P '

is an elliptic fibraof X as an elliptic

For a detailed analysis of these fibrations and

or quasi-elliptic surface.

the corresponding finer classification we refer to Bombieri-Mumford.

V.

Enriques' surfaces in char. 2.

In this chapter we study Enriques'

surfaces. Classically, they are defined by means of plurigenera: Pg = 0, P2 = 1, q = 0, x = 0. In our table, we find however two distinct types, one with p = 0, q = 0, h 1(O ) = 0 and another with p = 1, q = 0 but g X g h1(uX) = 1 (hence Pic(X) is not reduced aad cl;ar(k) 4 0). It.is reasonable to keep the name Enriques' surfaces for both types, because it can be shown that deformations of classical Enriques' surfaces in char. 2 may lead to surfaces of the second type. Theorem 4. Non-classical Enriques' surfaces (i.e. with p

g

--

1)

can

occur only in char. 2. Proof. -

Let X be a non-classical Enriques' suiface.

theorem of Murnford guarantees that theorem we have dim1251

Y(

I0

+ dim[-51 >- 0

Since x = 0, a

(in fact, by, the Riemann-Roch

and

diml25[ = 0 because

-

X

= 0

1-si

1 = 0, hence dirnl-K,,I 2 0; since both and Pg I are non-empty, we must have E 0 ). We have dim H (X ' @X ) = 1; hence let 1 [a .) E Z (OX) be a non-trivial 1-cocycle and consider the Ga-bundle i~ 1 1 ,and n :W c X defined locally as A X Ui, coordinate ziS on A and dim151 =

5

,

glued by

zi = zj + aij

.

There is a nowhere vanishing 2-form m on X and q = d z i ~ m is a regular 3-form on W,

because d a . . ~u = 0 identically.

Clearly q never vanishes

1J

and we conclude that

$e

0. Now consider the Frobenius cohomology operation

1

F on H ( ~ , 6 ~ ) A. t t h e cocycle l e v e l , we s t a r t with (a. .}, and o b t a i n t h e

[.b].

new 1-cocycle

Since

13 H ' ( X , ~ ~ )i s one-

i s n o n - t r i v i a l and

}

(a. 13

dimensional, t h e r e i s a constant

Ek

)i

such t h a t

(ap.], =3

[aij}

a r e coho-

mologous, hence we can w r i t e

- bj '

+ bi

= lail Now

f = d e f i n e s a g l o b a l f u n c t i o n on

+ bi -

a h = laij

b Let j' 1 = 1 and . iz

may assume n :Y

*X

Hence bi

zi = z. 3- a J ij

be t h e s u r f a c e

Y

zi

+ a,

a

f = 0

P P

3-manifold,

and

W.

in

If

1 $ 0 we

shows t h a t t h e p r o j e c t i o n

i s an & t a l e p-covering, of p r t i n - S c h r e i e r type with group

for a l l

E 0:

in

because

W,

i s a non-singular (connected) surface.

Y

identically

Y

- xzi - bi

Z:

0.

i,

Thus

otherwise

Y

(aij)

= C

If

P

P'

we cannot have

would be t r i v i a l , hence

df

i s not

i s reduced and a complete i n t e r s e c t i o n i n a smooth

hence a Gorenstsin surface.

Now

$,r 0

and t h e normal sheaf of

i s a l s o t r i v i a l and we s e e t h a t i n any c a s e

W

9" OY i s t h e t r i v i a l sheaf.

We deduce t h a t ~(0,)

2 h0 (0,)+ h2(@,) 0 = h (Uy)

On t h e o t h e r hand,

n

: Y*

X

+ h0 (wy)

= 2

.

i s f i n i t e and f l a t and

n+%

i s f i l t e r e d by

subsheaves

6, C [Ox tB

C [6x@ zi%

ziaX]

zi

- zj,E$,

and we have

z:@~]

C n,ay

***

because

e

p

q u o t i e n t s a l l isomorphic t o

@x-

Finally x(Y,eY) = x(X,n,Coy)

=

C x(X, q u o t i e n t sheaves)

= p x(XPx)X) = p and

p

52

follows from r h e previous i n e q u a l i t y

~ ( 6 6~ 2.)

Hence

3 = 2

a s asserted. We a l s o deduce t h a t i s non-singular then Thebrem 5. Proof. i n char;

(1

X(@y) = 2, wy is a

Y

Oy and

h 1(my) = 0,

i.e.

K3-surface while i n any case it i s

if

Y

K3-like.

Every Enriques surface i s e l l i p t i c o r quasi-elliptic.

We s h a l l not deal with t h e c l a s s i c a l case, since the known proofs

extend e a s i l y t o chat. p.

i s non-classical.

X

So assume t h a t

Everything follows (as i n char. 0) from t h e following result. Pro osititm-

Let

-E-T

be an i r r e d u c i b l e curve on X with

C

Ic~

Then t h e l i n e a t system

contains a reducible d i v i s o r

2 (C ) > 0.

D which i s not

t h e sum of 'two' non-singular r a t i o n a l curves. 'We postpone the pl'oof of t h i s r e s u l t ' a n d go f i r s t t o t h e proof of our theorem. Let

'

=-f(C2) + 1

p

and l e t

(c2) > 0,

'Step .l. I f hence

be t h e smallest genus of i r r e d u c i b l e curves with

D' be a s i n t h e Proposition. i s an i r r e d u c i b l e component of

E

D

then

p(E) < p(C)

p(E). = 0 o r 1. I n f a c t , let

D = E

+ Dl.

We have

(D*Df) = (C-Dl)

0

because

C

is

I

i r r e d u ~ i b l eand and

(E

(C ) > 0, ,hence (DOE)J ' ( ~ 2 ) . Now

--

2

= (DOE,) ( E - D ) )< (c2),

Step 2.

= -2

23

1

2

(Ei) .= -2

and

+ E1

for a l l

Ei

1 2 1.

irreducible.

2 (E )

< 0 and i n both

p(E) < p(C).

Ei

i,

1.

i s of canonical type, hence we may

non-singular rational.

Finally,

has s e l f - i n t e r s e c t i o n

dimlei I- E j

iie.

or

+ (E*D1)

a r e non-singular r a t i o n a l curves with

p(Ei).= 1 then

previous proposition. Ei

Ei

( E ~ O)E5 2

?nd

(E-D*) > 0

(D.E) = (E')

contains a curve of canonical type o r D = C miEi

6nd t h e

I n fact, i f suppose

X

Either

where C mil E

,

i s connected, hence e i t h e r

Dred

cases

2

g

mi-2 3 (Ei*E-) 2 3

Also C

(Ei-E ) 5 2. For i f then j J 2 and the Riemann-Roch theorem y i e l d s

Note a l s o t h a t t h e general element of

On the other hand

by t h e

I E +~ E.1 I

is

'

and D

-

*EjI Ei

- Ej.> 0,

+ (D - Ei

- Ej) C IcI

because C m. > 3; this clearly contradicts the result 1

=

i n S t e p 1.

-step. Final

X contains a curve of canonical type.

Consider D = C miEi

+

in Step 2.

If (Ei*E.) = 2 then E. E is of canonical type. Hence we J 1 1 nlsy assume (E .E ) 5 1 for all i, j. Consider the connected graph with i j vertices Ei and edges connecting Ei, Ej if (Li-E.) = 1. If this graph J N N N N N r4 contains a complete Dynkin diagram An, Dl1,E6, E;., E8 (e.g. An is a loop) AS

then by attaching suitable mgltiplicities to the vertices of this subgraph we get a curve of canonical type.

Otherwise the graph itself is a Dynkin diagram

and the associated self-intersection quadratic form is negative definite, which contradicts ( D ~ )> 0. Finally X

is elliptic or quasi-elliptic since

it contains a curve of canonical type. It remains to prove the proposition. Let L be an invertible sheaf on X, L s

E r(L)

q$

and

I L ~ 4 0;

If

and C = div(s)

we have the exact sequence s O-+@x > L+@~QL.+ 0

F(L)~as the vector subspace of r(L)

and we define sections s E

I'(L)

such that

A H'(x,L)

H'(X,OX) is the zero map;

consisting of those

1 ~ will 1 ~ be the associated (possibly non-complete) ,

linear system. Lema 1. -

Assume that there exists C

E

ILI

0 with h (0,)= 1.

Then we

have: either a) hl(L)

= 0 and

dirnl~l= dimlLl0

= T1

2 (L )

or

b)

1 h (L) = 1 and dimlLl0 =

Moreover, in case b) we have D E

l?lO

(L2). if and only if hO(oD) = 2, hence

lLIO i s r e d u c i b l e and i s not t h e sum of two r a t i o n a l curves.

every element of Proof. -

The Riemann-Roch theorem y i e l d s d i m l ~ l=

(L2)

+ hl(L)

.

The cohomology sequence o f

O---S#~ where

D = div(s)

gives

1 H (x,oX)& and now chooring

1 H (x,L)

D = C,

-

hence w i t h

L

+ y,->

1 H ( D , w ~> -)

h

1

0

H ~ ( x , % )4

(y,)= h ' ( ~ ~ )=

H ~ ( x , o ~ )i s an isomorphism, t h e r e f o r e

H1(o,uD)-->

1 h (I.)

s u r j e c t i v e and

= 0 o r 1.

o

1 we s e e t h a t 1 H'(X ' CoX ) > H (x,L)

The same exact sequence, f o r any

D,

is

also

yields h l ( ~ )= -1

1

+ h0 (eD) + dim

1

1 m f H ( x , @ ~ )-% H ( x , U )

and a ) , b) follow a t once. Lemma 2. curve

If

h 1(L@ 2) = 0

then

C

Proof. Let 2 s

a l o n e and

C i s i r r e d u c i b l e and 0 h (@2C) = 1.

and

C = div(s) E

and

ILI

1

= 2(I. 2 ).

@2

diml~

IL~

1 H (X,L) = 0, C

2

> 0,

thus

contains an i r r e d u c i b l e

be i r r e d u c i b h .

i s t h e 0-nap,

H'(x,L@')

H'(X,@,) t i o n by

2 I (L ) > 0, h (L) = 0

s i n c e i t f a c t o r s through m u l t i p l i c a -

s2 E 1-(Le2).

hence 2C

Clearly

On t h e o t h e r hand,

i s numezically connected and

Now Lemma 2 follows from L e m 1.

I n t h e proof of our proposition, by Leuana 1 we may assume 1 82 h + ~ ) = , h (L ) = 0, 2 @2 (L j = 2n > 0, d i m ( ~ 1= n, d i m l ~

Let

1 [ a . j E Z (DX) be a n o n - t r i v i a l 1-cocycle. 13

1 = 4n . We have a l r e a d y seen

t h a t t h e Frobenius cohomology o p e r a t i o n y i e l d s a r e l a t i o n 2 ai) with

1E

k;

(OX)*

1

- Laij

= bi

- bj

we s h a l l f i x once f o r a l l such a datum

{bi}.

3 EZ be a 1-cocycle r e p r e s e n t i n g t h e c l a s s o f L il {si} i s a s e c t i o n of L, t h e image of t h e c l a s s of {aij] {f

NOW l e t

in by

Pic(X).

If

H 1(x,CX) -%D

H~(x,L) is represented by the 1-cocycle s i a i j and since this is a 1-coboundary:

hl(~) = 0

The corresponding datum

(Oil

is uniquely determined modulo sections of L, and extending by linearity.

and we fix it by choosing it on a basis of r(L) We use if of -

and o

b

to construct sections of ~'L

are associated

data, then

0 , ~

L@*.

ST

if

as follows:

+ Ot

o

2

s,t E r(L),

+ 10s + bs2 are sections

The verification of this fact is strai~htforvard but clearly uses

in an essential way the fact 'that we work in char. 2. Now consider sections of the form

+ Xos + bs 2 + sr + at

(A'

o2

(A")

m+Ot

in T(L@~). There are some obvious cases in which (A')

and (A") vanish

identically, namely: (A'):

if

s = o = . O or

(A''):

if

s = pt, o =

or

s=t=O,

stO,

O =

t

with p E k,

or t = 7 = O . We call these the trivial relations.

Now one shows that there exists a non-

trivial relation, by a simple dimension argument and the fact that the set of sections we have defined forms a closed subvariety of the vector space

r(~@~). Let

(A') = 0

(A") = 0 be a non-trivial relation; then div(s)

or

satisfies the conclusion of our proposition.

s = o = 0 and the relation would be trivial.

First of all,

s

$

0

otherwise

Suppose for example that the

relation is 0

and suppose

div(s)

2

+ 10s + bs 2 + ST + at

irreducible. S(T

and since

div(s)

= 0

We have

+ XO + bs)

+

= ~ ( t 0)

is supposed to be irreducible we easily see that we may

deduce that Ui = sigi for suitable regular functions gi.

-

Now

-

fijuj yields siaij = sigi - fijsjgj = s ~ ( gj)~ and ~ Siaij = Oi aij - gi gj, i. e. [aij] would be a I-coboundary, contradiction. A

-

similar argument applies for a relation of type (A").

+ E",

div(s) = E' that 'if s H~(x,("x)if E'

E

s'

'

with E'

$

E"

and E', E"

'

Finally, if

irreducible curves we see

is a corresponding decomposition we may assume that

S'S"

H'(x,o(E'))

is the zero map.

-

This however can never happen

is non-4ingular rational, as the cohomology sequence S - ) HO ~(X , NE' x )( ' H

H'(X,MEI

shows, because the normal bundle of E'

1)

is negative, hence H0(X,NE,) = 0.

This completes the proof, We shall nct dwell further on the study of these interesting snrfaces and refer to Bombieri-Mumford for the construction of explicit examples and for additional results. VI:

Characterization of abelian surfaces.

We have encountered in Ch, B1 = 4 .

mi minimal

surfaces with H = 0, B2 = 6,

We now characterize these surfaces by

Theorem 6. --

A minimal surface with H = 0, B2 = 6, BI = 4

is abellan.

In order to prove this result we shall make use of still different char. p techniques.

,

0 The surface X lus invariants h1(flX) = q = 2, pg = 1, so that Pic (X) is reduced and ALb(X) dur?

151 =

0,

is then 2-dimensional. Moreover,

Pg = shows hence either y( e 0 or there is on X a curve of canonical

type, namely the curve in

151.

In this case however X would be elliptic

or quasi-elliptic and the canonical bundle formula would show again that

y[

=0

or K = 1, Hence IS( s 0; we deduce that the same holds for every

&tale covering of X. 'Let f : X

e Alb X

be the ~ibanesemapping.

First of all, if f were

not s u r j e c t i v e then

would f a c t o r i z e a s 9 : X +C with

f

Now C has & t a l e coverings p

and

'x'

X'--4

while s i n c e X'--+ q = dim Alb(X1) certainly increases;

g'

dth

2 g'

Now

xi

. i s again a surface of t h e same type a s

-

This shows t h a t f

a s t h e degree of t h e covering

f : X*

Alb X

i s a f i n i t e map.

Then we must have

This contradicts Alb X-

a

Alb X

i s surjective.

Suppose

E

C S - 3 C-

A s i m i l a r argu-

i s an i r r e d u c i b l e curve

( E ~< ) 0 and considering & a l e coverings Alb X

same c l a s s , with a r b i t r a r i l y many curves

Let

X

dim Alb
induced by & t a l e coverings of

disjoint.

together with a

i s s u r j e c t i v e we c e r t a i n l y have

increase t o

f(E) = pt.

X ' e X

C'.

C'

ment now shows t h a t

of a r b i t r a r i l y high degree prime t o

' d C

determines an & t a l e covering X 1 - 4 X

= X XC C'

s u r j e c t i v e map

n :C

of genus 2.

C

Ei

we obtain X',

s t i l l i n the

2 w i t h . (E.) < 0 and the

Ei

B ~ ( X ' )= 6. be m u l t i p i i 6 a t i o n by

j,'

xAlb

XL = X

let

Alb

and consider Alb X

Alb X

f , and hence

f

a r e f i n i t e morphi&

a

which we s h a l l henceforth assume,

.

4 maps of degree 1 q

If

and deg f

XA-a

X

L

= deg f.

and Alb X

% 5 0.

'

(&p) = 1,

e Alb X

are Qtale

f , i s a separable map we can argue a s follows.

be a t r a n s l a t i o n i n v a r i a n t regular 2-form on Alb X.

separable,

a

If

f*q i s a ' r e g u l a r 2-formron X and

It follows a t once from t h i s t h a t

* f 9

f : X-

Since f

Let

is

never vanishes becauie Alb X

i s everywhere an

Q t a l e morphism; s i n c e e v e w & t a l e covering of an abelian v a r i e t y i s an abelian v a r i e t y , we conclude t h a t

X

i s abelian.

I n char. p t h e r e i s t h e unpleasant p o s s i b i l i t y t o consider t h a t

f

may

5

be inseparable; now down.

f*q

i s i d e n t i c a l l y 0 and t h e previous argument breaks

I n t h i s case we use t h e following powerful char. p technique.

93 X

The s u r f a c e

l i e s i n a smooth and proper a l g e b r a i c family of s u r f a c e s

defined over a f i n i t e f i e l d and, i f we can prove t h a t t h e s u r f a c e s over t h e i t s e l f w i l l be a b e l i a n ; t h i s i s e s s e n t i a l l y a

X

closed p o i n t s a r e abelian, s p e c i a l i z a t i o n argument.

Now suppose t h a t

* fkL'

Le,=

is a f i n i t e field.

k

Fix

L ample on

Alb X.

Then

i s ample on XA w i t h H i l b e r t polynomial X(L?)

= (deg fi) X(~:n)

..=. (deg f ) X ( ~ @ n ) which i s independent of there is

depending only on t h e H i l b e r t polynomial, hence independent of

N,

such t h a t

4,

k-varieties since

By a fundamental r e s u l t of Metsusaka-Muntford

1.

i s very ample f o r a l l

a X

a.

Therefore t h e i n f i n i t e s e t of

cail a l l be embedded i n a f i x e d

a

M , w i t h f i x & degree.

P

i s a f i n i t e f i e l d , t h e r e a r e o n l y f i n i t e l y many v a r i e t i e s i n pM

k

with f i x e d degree, and i t follows t h a t a l l t h e p a i r s over

a r e isomorphic

(X&,L?)

t o f i n i t e l y many of them, . Now we have:

k

(1) given 'X and an amp1.e L, X

Now

%

with

t h e group of automorphisms

f

of

i s an a l g e b r a i c group, hence w i t h only f i n i t e l y &ny com-

f LN L

ponents (Matsusalca) ,;

.

(ii) X ,

t h e group

and each

g 5 A

a

a c t s on

Let

Xn.

such t h a t

has

g " ~ Ln. n ~ Then

A.

Gn

Gn

i s p o s i t i v e dimensional.

i s abelian.

A& a r e i n

under

g LAIV L

a*

0

Gn,

0

Gn.

Also

AI

Xn

@N

a' La

).

Then g

i s a n a l g e b r a i c group and A C Gn

a

of

A&

Xn for

Xn

Now

0

Gn

0 cannot c o n t a i n a n o n - t r i v i a l l i n e a r

Gn

would make

Xn

ruled, and

n (2/A2)lt and subgroups of f i x e d bounded index of

0 2 2, Xn c o n s i s t s of only one o r b i t 0 i s a c o s e t space Gn/H, hence Xn i t s e l f i s an

We deduce t h a t

hence

(X

A f i r s t consequeace i s t h a t t h e connected component

subgroup f o r t h e a c t i o n on

0

a c t s on

4

C ~ u t ( X ~ be ) t h e group of automorphisms

Gn

9-

i n f i n i t e l y many Gn

a

@N (x,,L~ ) be isomorphic t o i n f i n i t e l y =any

Let

of

of t r a n s l a t i o n s by p o i n t s o f o r d e r

A

dim Gn

abelian variety.

Finally we have &tale

xn. and Xn, Alb X

a r e both abelian v a r i e t i e s .

t h a t we may assume t h a t t h e composition morphism.

If

K

Xn-

By choosing an origin, we see Alb X

i s a s u r j e c t i v e homo-

i s t h e kernel, it i s now easy t o see t h a t

a s u i t a b l e subgroup VII.

3AlbX

>X

K' C K,

openproblems.

hence

X

Xn/K1 f o r

X

i t s e l f i s an abelian variety.

We mention here r a t h e r b r i e f l y some i n t e r e s t i n g

open problems of t h e geometry of surfaces i n

char.. p.

F i r s t of a l l , t h e Kodaira vanishing theorem i n

char. p

i s not yet well

understood; f u r t h e r work i n t h i s d i r e c t i o n ,is c e r t a i n l y desirable. I n t h e theory of e l l i p t i c and q u a s i - e l l i p t i c surfaces, i n

char. 2 and 3,

one should study how an e l l i p t i c f i b r a t i o n may become quasi,-elliptic; a l s o one should t r y t o find functic~nali n v a r i a n t s , l i k e the j-invariant i n the e l l i p t i c case, associated t o q u a s i - e l l i p t i c fibra'iions. One needs a l s o a deeper study of families of curves i n

&as.

p

with

generic s i n g u l a r i t i e s , and a b e t t e r understanding of unirational surfaces i n char. p would be a n i c e achievement. The.,relations (and computation) of t h e de Rham cohomology and Hodge cohomology i n

char. p

should a l s o be worked out, i n o r d e r % o see how the

various d e g r e e s o f faillure of Hodge theory a r e r e f l e c t e d i n t h e geometry. From t h e point: of view of Enriques' c l a s s i f i c a t i o n , s t i l l much more work has t o be done i f

X'=

1 and 2.

The appearance i n these dimensions of the

Raynaud surfaces shows t h a t many i n t e r e s t i n g new aspects may have s t i l l t o come. REFERENCES D. Mumford, Lectures on Curves on an Algebraic Surface, Annals of Math.

Studies No. 59, Princeton University Press, 1966.

E. Bombieri and D. Mumford, Enriques' Classification of Surfaces in Char. p , 11, in Complex Analysis and Algebraic Geometry, Iwanami Shoten, 1977,

pp. 23-42.

,

idem, 111. Inventiones Math. 35' (1976, pp. 197-

232. D. Mumford, idem, I. in: Global Analysis, Princeton University- Press, 1969. E. BombieriandD. Husemoller, Classification and embeddings of surfaces, Ptoc. Symposia Pure Math. XXIX, (19741, pp. 329-420.

CEN TRO INTERNAZIONALE MATIMATIC0 E S T I V O

Td the memory of

(c.I.M.B)

Lucien Godcaux /1887-1975/

ALGEBRAIC SURFACES WITH

'

q

= 'pg = O.

.

Igor Dolgachev

CONTENTS

Introduction Chapter I. Classical examples 51.

The Enriques surface

52.

The Godeaux surface

53. The Campedelli surface Chapter 11.

Elliptic surfaces

51. Generalities 12.

Torsion

53.

The fundamental group

Chapter 111.

Surfaces of general type

51.

Some useful leriunas

52.

Numerical Godeaux surfaces

53.

Numerical Campedelli surfaces

54.

Burniat's examples

85.

Surfaces with p'l)

06.

Concluding remarks

Bibliography Epilogue

9

Introduction

1. Notations. Let F be a complex algebraic surface.

We will use

.thefollowing Standard notations:

OF

:

the structure sheaf of F

OF(^) %=

uF

:

the invertible sheaf associated with a divisor D on F

- cl(F)

= 0I

hi (D)

: minus

the'first Chern class of F or a

. F .

canonical divisor on F

"

:

.

KP1

:

the canonical- sheaf of

the dimension of the space Hi (F.OF(D)

)

.

0 2 p (F) = h (KF) = h (OF) ; the geometric genus of F 9 1 1 q(F) = h = h (OF) : the irregularity of F

5

.

(5)

:

the self-intersection index of

2 p ci) .(F) = $,

+

.

1 , where F'

rational surface F

5?

is a minimal model of a non;

the linear genus of F

.

c2(F) : the topological Euler-Poincare characteristic of F P,(F)

2 + c2(F) ) = -q(~)+ p (F) = . 1/12 (% 9

genus. P,(F)

=

h

0

(nK$

:

the n-genus of F

.

.

- 1 : the arithmetical

.

NS(F)

:

the Neron-Severi group of F , the quotient of the Picard group Pic@')

by the subgroup of divisors

algebraically equivalent to zero

(=

Pic iF)

if q

=

0).

If not stated otherwise F nil1 be always assumed to be non-singular and projective.

2. Historical. It is easily proved that for a rational suxface F

(that is birationally equivalent to the projective plane invariants q(F)

lFL)

the

and p ?F) are zero. The interest to non-rational g

surfaces with vanishing ,q and p was born in 1896 when Castelnuovo g had established the necessary and sufficient col~ditionsfor a surface to he rational.

Clebsh had proved earlier that a aurve of genus 0 is ratj.onal

The question whether a surface with q = nat;lral problem.

= O is rational was a pg In 1101 Castelnuovo had shown that the answer is nega-

tive in general proving that one must add also the condition P2 = 0 and constructing an example of a non-rational surface with q = p = g

0

.

In the same paper he also exhibited other examples of such surfaces due to Enriques.

The latter were of particular destiny, as it turned cut

later they play a special role in the general classification of algebraic surfaces representing one of the four classes of surfaces with vanishing Kodaira dimension (see Ill, [ 6 ] )

.

Both examples of Enriques and

Castelnuovo belong to the class of elliptic surfaces, that is they contain

a pencil of elliptic curves.

In particular, we have for these

surfaces p (1) = 1

.

Later Enriques gave another construction of his

surfaces and also presented other non-rational surfaces with q = p = 0 g [17'1 They were also elliptic s&aces.

.

The first examples of surfaces of general type.with q = pq = 0 appeared only in 1931-32 when Godeaux had constructed a surface with q =. p '= 0 ,and p")

9 surface wLth p ( ' ) =

with ) ' ( p

4

=

= 2 [I81 and Campedelli had constructed (Dl) a

3

.

Later Godeaux constructed some other examples

1201.

3, Modern develop!=.

The new interest to the surfaces under the

title is related to the general problem of the existence of surfaces wieh given topological invariants which became of the main concern after the period of the reconstruction of Enriques* classification results had happii? 'ended. The particular interest to the surfaces with p(l)=

2

and 3 (numerical Godeaux and Campedelli surfaces) Is due to Bombieri's paper [4] where for ail other surfaces it was settled the question of the birationality of the 3-canonical map

(t3K

.

Now due to works of

Bomhieri-Catanese (5,111, Miyaoka [32] and Victor Kulikov (non-published) we know that

+3K

is birational for these surfaces, but I do not include

the corresponding proofs in this survey refering

to the paper of Catanese

in these proceedings. In Chapter 11, I expose in more detailsthe results of my paper 1141 which deals with elliptic surfaces with q

=

p = 0 g

.

The theory of

Y

~ o d a i r a - ~ ~ ~ - ~ a f a rallows e v i r to classify all such surfaces. In Chapter 111, we study more interesting case of surfaces of the l surfaces are divided into nine classes corresponding general type. ~ l such

t o t h e p o s s i b l e values o f s u r f a c e s with t h e same

p (1)- 2, 3 , . .

p'l)

., 10

.

To d i s t i n g u i s h t h e

one may consider t h e group

g e n e r a l l y t h e whole fundamental group

s (F) 1

.

Tors(F)

o r more

It can be shown (see

Chapter 111, 56) t h a t t h e r e a r e only a. f i n i t e number of p o s s i b l e

?r

lVS

f o r s u r f a c e s of t h e same c l a s s , and hence one may ask about some e x p l i c i t e s t i m a t e o f t h e o r d e r o f ' Tors(F) f o r t h e cases

p'l) = 2

.

Unfortunately, t h i s i s known only

(Bombieri) and 3 (Beauville,Reid) and only i n

t h e f i r s t case t h i s estimate i s the b e s t possible. know whether t h e c l a s s e s with o f s u r f a c e s with

4 5 p(l)( 7

p ( l ) = 8 and 10 a r e empty* a r e due t o Burniat [7,81

here a new versior? o f h i s c o n s t r u c t i o n (171, us t o calculate

f o r such s u r f a c e s .

. .

=he examples We p r e s e n t

[ 3 7 ] ) which enables

The examples o f s u r f a c e s

p(l)= 9 a r e due t o Xuga [293 and Beauville 131

with

4.

Tors(F)

Moreover, we do n o t

.

Acknow1edgemer.t~. - This work owes very much t o many people with

whoin I had a coilversation on t h e s u b j e c t a t d i f f e r e n t p e r i o d s o f my life.

It would be impossible t o mention them a l l .

I am e s p e c i a l l y

indepted t o Miles Reid and F a b r i z i o Catanese whose c r i t i c a l remarks were very valuable.

I t i s a l s o a g r e a t p l e a s u r e t o thank C.I.M.E.

f o r t h e i r support during t h e p r e p a r a t i o n o f t h i s paper.

*

see. Epilogue.

and M.I.T.

CHAPTER I.

51.

CLASSICAL EXAMPLES.

The a r i q u e s s u r f a c e .

Let

IP3

x i , , i z.0, and

b e ' t h e p r o j e c t i v e 3-space with homogeneous coordinates

...,3 .

let 4

Consider t h e coordinat,e t e t r a h e d r o n

be a s u r f a c e i n

..,, 6 ) of

Ei (i = 1,.

a l s o assume ' t b + t common m i n t s with

X

p3

T : x x x x = 0 0 1 2 3

which passes t w i c e l y through, t h e edges

T ' , t h a t is, h a s

E i a s its ordinary double l i n e s .

has, no ' o t h e r s i n g u l a r p o i n t s o u t s i d e T

.

Since t h e s e c t i o n o f

double r e d u c i b l e cubic curve, we s e e ' t h a t F e x p l i b i t l y we may consider

F

We

T and o t h e r

by a coordinate plane i s t h e

must be o f o r d e r 6

.

More

F ' a s , g i v e n ' b y t h e equation:

Let ' F ' be.t h e , n o a i a l i z a t i o n of

X

.

Then

F

is a-non-singular surface:

To s e e it one h a s t o look l o c a l l y a t t h e normalization of t h e a f f i n e coordi?

n a t e c r o s s : xyz = 0 i n A

3

.

Here t h e normalization w i l l be j u s t t h e d i s -

j o i n t union:of t h r e e planes, t h e i n v e r s e image of t h e s i n g u l a r l o c i w i l l be t h e union

of

s i x l i d e s l y i n g by p a i r s i n t h e s e planes.

Two l i n e s in each

'

of t h e planes correspond t o t h e two a x i s l y i n g - i n t h e same coordinate plane. The i n v e r s e image of ' t h e o r i g i n w i l l be t h e t h r e e points,. each of them i s t h e i n t e r s e c t i o n p o i n t o f t h e two l i n e s i n one o f t h e planes. t h e p i c t u r e is a s . f o l l o w s :

So, l o c a l l y

Let p

:'

F + X be the projection.

Then the local analysis above shows

that for any edge E. of the tetrahedron T we have

1

I

where Ci and Ci

are non-singular rational curves meeting each other

transversally at two points arising from the two pinch-points of X

lying

on each of the edges. Ci and C Ci and

C

3

do not meet if

and E are not incident, otherwise j meet transversally at one point, j

Ei

C. n C. n C = $3 for distinct i, j, k 1 3 k

.

NOW we use the classical formula for the canonical sheaf of the

normalization of a surface of degree n

where H

in P~ :

is the inverse image of a plane section of X and A

conductor divisor (= the annulator of the sheaf p

X

appendix to Chapter I11 of [ 4 3 ] )

where C = C 1

+

... + C6 .

.

is the

(0F/O X ) (see Mumford's

In our case we easily find that

7

The global sections of

wF correspond to quadrics in

passing through the edges of the tetrahedron T

.

B-

Since by trivial

reasons such 'quadricsdo not exist we have

Next, taking for 2H

the inverse image of the union of two faces of

the tetrahedrqn, we obtain that

where Ci

is the common edge of these faces, and

C j

is the

opposite edge. Taking for 4H the inverse image of the union of all faces (=

the tetrahedron T f we get

Thus we have

and hence F ,Since K

is non-rational. is numerically equivalent to zero, we have

By the adjunction formula we get

Thus, Ci is a reducible curve of arithmetical genus 1 2Ci

%

2C

.

Since

and C does not meet C we infer that the linear: i j [2C. contains a pencil of curves of arithmetical genus 1

j

system

I

Since there are no base points of 2Ci we obtain by BertJni's theorem that almost all curves form this penci2,arenon-Singular elliptic curves.

Note also'that this pencil ctmtains t m degenerate

durves, 2Ci and

2C

j

.

NOW ,we may use the formula expressing ,'cZ(F) in termsof the

Euler-Poincare characteristic of degenerate cumes of the elliptic pencil (see 111, Ch. IV) :

where ,BI are all singular curves of'thepencil.

Since

we deduce that

Since $i

2

= 0

we Get by the Noether formula 12(I

This obviously implies that

q(~j

= 0

.

- q(F))

= c2(Fl > 0

.

32..

The Godeaux surface.

Consider the projective involution o. of

I!?%

of order 5 given

in coordinates by the formula:

C being a prjmitive 5-th root of unity. This invdiution acts freely outside the vertices of the coordinate tetrahedron, Let F'

be a

non-singular quintic which is invariant under a m d does not pass through these vertices.

For example, we may take Tmr

F 1 a quintic

with .the equation:

(For a general surface F'

with the properties almm one has ta add 2 2

to the left side 8 invariant monomials xOx2x3,.-J cyclic group of order 5 generated 5y Consider the quotient F = F1/G

(3

.

, acting *mly

, the projection

Let G be the on F'

p : F' + F

. is a

finite non-ramified map of non-singular surfaces, Lemma. Ther.

Let p : F' -> F be a finite non-ramifiedmp of degree n

.

Proof. -

The first relation easily follows from the equality pX (wF) = wF"

since p

is smooth and finite. The second one follows from the

Noether formula and the relation c (F') = n c2(F) 2

,

which can be

proved either by topological arguments or using the equality

9

1

being the sheaf of 1-differentials, and standard properties ofs-

Chern classes. Since we have for F',

2

KF, = 5 , pa(Fr) = 4 we get from the

lemma

Since, obviously, q(F) 5 q(F')

,

we obtain

Next, note that F is minimal, that is there .curvesof the first kind lying on it.

we

no exce~tional

Indeed, the inverse image of

such curve under,. p would be the disjoint union of five exceptional carves of the,firstkind on F"

.

the minimality of P and the fact of,generaltype.

However, F'

52

is minimal.

From

1 it fallows that F is

Another way to show ?his is to use the property of H

ample sheaves: p (aF) is ample implies

%

is ample.

Since F' .is simply-connected we obtain that the map p universal covering. In particular, Tors(F) = T~(F) = 2/52

is the

.

93.

The Campedelli surface.

his

is a double ramified covering of the projective plane

IP

2

branched along some curve of -the 10-th degree ( more ~reciselyit is a minimal non-singular model of such covering). Let

where

Ci

W

be the following reducible curve of the ~ o - t hdegree

are non-singular conics

and

quartic with the following properties:

D

is a nan-singulhr

To see that such configuration of curves exists one'may take for

C2

and C3 ewo concentric circles lying in the complement to the line at infinity, the points P5 and P6 will be the two cyclic points. The existence of a quartic D touching the conics Ci easily' follows from the consideration of the net XC C + P C 1 2

C

1 3

+ vC2C3

= 0

.

Lemma 1. Let X be a non-singular surface and W a reduced curve on it.. Suppose that there exists a divisor D on X such .that.

W s'2D , then there is a double covering

.

branched exactly along W

Moreover,

Y

is normal and non-singular

.

over the complement to the sihgular focus of W

-

Proof. Assume firstly that W

is non-singular. Let F be the line

bundle corresponding to the divisor D and of X such that F I U

j

equation {c

; .

IU.)

3

a coordinate covering

is trivial and W is given by the local

0) on Uj

.

Let gij be a system of transition

j

functions for F

, then

ci = g . . ~ on 11 j

ui n

O

5

and we may consider 2

the subvariety Y of F given by the equations x = j It is obviously xj is a fibre coordinate of F U j' the projection '. Y .+X satisfies the properties stated

I

c , where j checked that in the lemma.

If W is sikgular we apply the arguments above to X replaced by X' = X

-S

locus of W X

.

and W

by W' = W

- S , where

Then it suffices to take for

in the double covering

Y' -+ X'

Y

S

is the singular

the normalization of

constructed as above.

Remark.

The sheaf

L

= OX(D)

can be c h a r a c t e r i z e d a s t h e subsheaf

of a n t i i n v a r i a n t s e c t i o n s of t h e d i r e c t image then t h i s sheaf i s determined uniquely by by a n element o f order 2 i n

Pic(X))

.

f,(Oy)

.

If

q(X) = 0

( s i n c e they d i f f e r

W

This shows.that i n t h i s

c a s e any double covering with p r o p e r t i e s from lemma 1 can be obtained by t h e construction of t h e lemma. ~ p i l ~ i nt hgi s lemma t o t h e plane curve

W

along

W

Pi

.

we may c o n s t r w t a double coveripg

.

non-singular model o f

p : X + lP2

t h e curve

W

L

with

Si =

2

.

-

2,

Y

F

IP2

branched

.

b e t h e minimal r e s o l v ~ t i o nof s i n g u l a r i t i e s of

IP2

12 ' Si = -1

W

i s given by

,

.

Now we apply t h e lemma t o t h e s u r f a c e curve

of

w i l l be o b t i i n e d a s t h e minimal

The proper transform of

i s a l i n e on

where

Y

This s u r f a c e has s i x s i n g u l a r p o i n t s l y i n g over t h e p o i n t s

The Campedelli s u r f a c e

Let

and t h e 10-th degree

IP2

X

and t h e non-singular

and consider the corresponding double,covering r

5,

compute the canonical class Lemma 2.

Let g

:

Proof.

.

.

F' + X

To

we use the following:

V' + V be the double covering of non-singular

surfaces branched along the curve W on V'

:

,

gf(W)

-

25 for some. divisor D

Then

First, note that our double covering can be obtained by

the constructi0:1 from lemma 1

.

In fact, consider the splitting

into invariant .md anti-invariarit pieces.

Then clearly

is contained in the invariant piece that is in 0 v

(7v -Algebra gx ( 0v ,)

L

Syonr(L) = Qv B

L82

- Thus

is the quotient algebra of the symmetric algebra @2 J, where B L~~ B, by the Ideal generated by L

j is an ideal :;heaf i n

...

-

.

Ov

Taking the spectrums we get that

V' = Spec(g (0 , ) ) .,isisomorphic to the closed subscheme of the x v -Y line bundle J? V(L) = Spec(Synm(L)) Looking locally we easily

-

.

identify J with the,sheaf OV(-W) and obtain that V'

is

constructed with the help of a divisor D corresponding to F in the same way as in lemma '1

.

Now, the formula for K

,,, can be proved very simply. In

notations of lemma 1 we consider a 2-form w on V in local coordinates c

j

and some other function -t

j '

Then we use the

relation dc.Adt. = 2x dx hdt to obtain that (.gX(w)) = gx( (w)) I j j j + (xj = 0) (the brackets ( ) dei-lotesthe divisor of a 2-'form).

+

This pkovds the lemma. ~hus,we have

Asstlme that D t IKF,

I , then we see fro3 above that

D .= rx(~')

for some

The latter linear system is equal to the inverse transform of the system of conics on 3P2

passing through a l l points Pi

that the latter does not exist we argue as follows. P1

and P2 the two cyclic infinite points

equations for C1

and C2 i n the form:

and the equation for Cj

on the form:

(1,

.

To show

Taking for

Pi, 0) we get the

The points P3, P4, P5, P6 will have the coorciinates (0,

fb, 1) respectively.

Now let C be a conic with an equation

which passes through the points Pl,. PI and

(+a, 0, 11,

..,P6 .

P2 we may assume that a3 = 0

Since it passes through

and a = a2 = 1

.

Since

it passes through P3 and Pg we get the equations

which give a4 = 0 and and

a = -b 6 ~hus

.

a = -a 6

.

Similarly we get a5 = 0

This contradiction shows that C does not exist.

$,=PI

and P'(F') = 0 . g

Since r is branched along Si, i = 1

i' = 1, 2, 3

, we

,..., 6

-1 and p (ci),

see that

-

for some curves Si and

Ei

on F'

.

- 2 1 x 2 1 2 1 S. =$r (S.) ) =-(2s) =-(-4) 4 i , 4

Also, we notice that

=-I

,

zi are exceptional curves of the 1st kind. F 1 + F be the blowing down of all % . We will show is a minimal model of F' . We have

This shows that Let

a

that

F

:

since

This shows that

If E is an exceptional curve of the 1st kind on F (E I$)

Si

.

= -1

, then

-

and hence E must coincide with one of the curves C. or

However, neither of them is an exceptional curve, because

To compute

<

we use that

It remains to noirice that F is a surface of general type, since it is minimal and has positive

<.

(see Chap. 3, 51, .lemma 3)

.

In particular, we have q(F) 5 p (F) = 0 4 ~ l s onote that 2% is determined by

the net of quartjcs .XC1C2 f pC1C3

+ vc2C3

and is of dimension 2

.We also have the following obvious torsion divisoxs of order

2on F :

It is immediately checked that

.

and

Th'is shows that

1t.will be shown ip Chapter 111,

93 that, i n f a c t , we have the

equality

Tors (F) = (Z/2Z)'

.

118 CHAPTER 2.

1.

ELLIPTIC SURFACES.

Generalities.

projective non-singulur surface X

A

f

exists a morphism general fibre X

is called elliptic if there

onto a non-singular curve B whose

: ,X -t B

is a smooth curve of genus 1

an elliptic fibration on X

.

.

Such f is called

From general properties of morphisms

of schemes we infer that almost all fibres are non-singular elliptic curves over the ground fieid k

(as everywhere in this paper we

or algebraically closed of characteristic 0

assume that k = C

A n elliptic surface X

)

.

is called minimal if there exist an elliptic

fibration withoat exceptional curves of the 1st kind in its fibres

(

(such fibration will be called minimal). Let f : X + B be an elliptic fibration on an elliptic surface

L and

Xb

a fibre over a point b

divisor on X

, then

6

B

.

Consider )6 as a positive

according to Kodaira [261 it is one of the following

types : mLO

:

1 m 1

:

1 m 2

Xb = mE 0 , m 2 1

, where

Eo

is a non-singular elliptic curve;

5 = mEo, m k 1 , where Eo is a rational curve with a node; : 5 = mE + mE1, r n l 1 , where Eo and El are non-singular 0 rational curves meeting transversally at two points;

1 m b

:

%=

mEo

$...+

m 2 1, where

curves with E. fl E . n Ek = pl 1

3

(EiEi+l) = 1, i = O,.. 11 : Xb = Eo

,a

., b

- 1,

Ei

are rational non-singular

for distinct i, j, and k and assuming Eb = Eo (b,

rational curve with a cusp;

3)

.

111 : X

b

= E

0

+

El, where Eo and E are non-singular rational .I

curves with simple contact at one point;

IV

:

X = E' b 0

+

E 1

+

E2, where

Ei are non-singular rational curves

fransversally meeting each other at one point p = E0nE1 f l E2' X

Ig3: X

b

= E

0

+

E 1

+

E 2

+

~g + 2E4+br where all

Ei are non-singular

rational curves transversally interesecting as shown on the picture

+ 2E1 + 3E2 + 4E3 + 5E4 C 2E5 + 4336 + 3E7 + 6E8, where 0 are non-singular rational curves interesecting as shown on

1 1 ~ : Xb = E

E~

the &ture

11?

= E 0

:

+ 2E1 +

3E 2

+

E3

+

2E4

+

3E5

+

2E6

+

4E7, where

Ei

are non-singular rational curves and the picture is

1vX :

%=

E + 2E + E 0 1, 2

+'

2E 3

+ E4' +

2E + 3E6, where Ei 5 .

are

rational non-singular curves and the picture is

A singular fibre of type mlb

multiple of multiplicity m Let f fibre X

rl

:

X

+

,

b2 0

,

m 2 2 , is called

.

B be an elliptic fibration, then its general

is a sooth curve of genus 1 over the field K

functions of B

of rational

and it is an abelian varlety over K -ifand only if

it has a K-rational point.

In geometric terns the latter is equivalent

to the existence of a global section s : B -+ X

of the morphism

, this

Consider the jacobian variety J of X.

n

is again a

smooth culxe over K of genus l'with a rational point over K For any 'extension K1/K such that X

.

f

.

has a k'-rational pint

there exists a na'tural.isomorphism of K'-curves X Q K' r J 5 K' rl K K Accordihg'to general properties 'of birational transformations of

.

two-dimensional schemes there exists the unique minimal elliptic, ,fibration j'.: A

such that

B

-+

A

rl

=

J

.

This surface is called

the jacbbian shzface of ithe elliptic surface X section,'all singular fibres of For any b e B

Proposition 1.

X

fibre the fibrations f

:

over the henselization

Zb

Proof. Let zb : %(b) -ijb , and jb : i\(b)

-+

+

+

B

. ,Since j

has a

j are non-multiple. such that Xb

is a non-multiple

and,j : A

are isomorphic

+

B

of the local ring 0 Brb

.

spec-Gb be the restriction of f over

Spec

Zb

the same ior j

.

Since

Zb

is

smooth at some point of a component of multiplicity 1, there exists

%. .

a section'of

This implies that the general fibre :(b),,

an abelian curve over the fraction field

f$,

of

ijb .

is

From this we

= z(b),, and hence in virtue of the uniqueness rl of the minimal models we get %(b) " %(b) infer easily that W(b)

.

proposition 2. type mlb Proof. b'

E

B'

.

Let b e B

Then the fibre

Let B'

-+

over h

such that Xb

%

of

j :.A

is a multiple fibre of -+

B

is of type

1%

B be a overing of B ramified at some point

with the ramification index equal to m

.

Let f;,

X1(b')

:

+

change map X x B'

-t

B

-

Spec OB,,b,..be the restriction of the base

. Denote by : X'(bl) +- Spec 0

B' over the local ring OB,,b,

-

X'(bl) the normalization of X' (b') and let f;,

B',bl

be the composite map. Let x

c

at x

the map m t = (xy)

by

.

X be a double point of the fibre Xb f

.

:

X + B

is isomorphic to the map

This shows that X1(b')

A

Then 'formally 2

+

A1

given

formally at the point x'

m 3 lying over x is isomorphic to the hypersurface tm = (xy) in. A

.

Taking the normalization we observe that there are exactly m points xi,

..., r'm

c

X'(b')

is given by the equation t = xy

-

the fibre X(bV),

5 (b')

lying over x1 and formally at'each x k

.

Looking qlobally we infer that

is of type l1mb

.

Performing the same base change for j

:A +

and'resolving

B

the singulariti5s of the obtaiped surface A S ( b ' ) , we will get the with the closed fibre,o£ type l1mb (Proposition 1). scheme over 0 B1,b' case we'find that it can be only if the fibre of j 1 over b is of type 1 b Checking case

.

Let j

,:

section, W(j) B

for which

from W(j) (p.h.s)

A +

B

be a minimal elliptic fibration with a global

be the set of all minimal elliptic fibrations over j ser,vesas the jacobian fibration.

the general fibre XK

For any

f

As it is well known the set of all p.h.s.

of minimal models for A

K

.

X +B

is a principal homogeneous space

for AK over the field K of rational functions on B

1 cohomology group H (K, Ax)

:

for AK

.

forms the Galois

In virtue oftheexistence and uniqueness

.1 the map W j ) + H (K,

AK)

is bijective.

To compute Let i

A ) K

= Spec K-B

:

we argue as follows ( € 3 4 , 38, 411).

be the inclusion of the general point.

Identify,AK with the etale sheaf which it represents arid let A = -

The sheaf 5 is representable by'the commutative group

i.X .X

K

'

scheme over B which is obtained by throwing out all points of the surface 'A where

.

f is non-smooth (the Neron model of" +?. The

Leray spectral sequence for i gives the exact cohomology sequence:

For any closed point b a B we have

where

ijb

is the fraction field of'the henselization

local ring

oBSb ,

$ = yKi$, .

suffices to comgute for ail n

periodical).

we get

Bib

TO compute H'(%,%~

1 the subgroup H.

elements killed by muLtiplication by

5

n (since ' H

Using the Kummer exact sequence

(%, A%)

of the

it of the

is always

Now, since

0

coincides with its Picard variety

pic

we have

where

is the strict localization of the morphism Spec 5 B~b Since Spec 6 is cohomoloqical.ly trivial B,b

jb : %(b)

.

j over b

where

ib

where

%=

+

spec8 B,b It remains to add that

%

:

specQ+

2 in case

%

is the inclusion of the general point.

is of type

1 is of,type 1 1, 1 2 1 , db = 0

Let f eleinent x e

: X

,

db = 1

in case

in the remaining cases.

-+ B be the elliptic fibration representing an

i H (K, A k ) ' .

Then we interprete the composite map

as follows. The general fibre of the strict localization +

spec

8B,b

represents a p.h.s.

over the field

: z(b)

%

and,

+

1 hence, an element of H ("

Also, it can be checked that

qb(X) equals.the class of the normal sheaf of the reduced fibre

$

0 in the Picard group Pic(X ) b

1 witb .It (ri, A

1

= lim H (%,1.1,)

..Kb.

, whose

torsion part is identified

1-0

= lim H (srpn)

-5-

f

-

From this observation we immediately obtain the followihg 1

(x) # 0 if and only if the b fibre Xb of,the corresponding elliptic fibration is multiple.

Proposition 3 .

For any x

The multiplicity of

K

6

H (K, & , A

IJ

equaIs.'the order of +,,(x)

in H ' ( % ,

A... )

%

The last assertion follows from the proof of Proposition 2. 1 Now we shall compute the kernel H (B,&) the Tate-Shafarevich group of AK)

. of the map (so called

.. First, we have the following

exact sequence:

which comes froa the identification of A with its Picard K 0 1 variety Ker ( (R jxbm)K, + l 1 (see the details in 1241 ) K

%=

.

Since j has a global section, the exact cohomology sequence gives the isomorphism

Next, considering the Leray s--~ectral sequence for j and G m, A 1 = 0 , i > 1 we get using that Rij G = H x m m

(BIG

and

.

I n v i r t u e of b i r a t i o n a l invariance o f P r o p o s i t i o n 4.

Assume t h a t

Br (1241) we o b t a i n

i s a r a t i o n a l s u r f a c e , then

A

I n p a r t i c u l a r , any ninimal e l l i p t i c s u r f a c e without m u l t i p l e f i b e r s whose jacobian s u r f a c e

A

i s r a t i o n a l i s isomorphic t o

The l a s t t h i n g t o do i s t o i n v e s t i g a t e t h e group Let

$

A

be t h e subsheaf of

A

.

2

H (B,$.

which i s r e p r e s e n t a b l e by t h e connected

component o f t h e u n i t of t h e group scheme

c , ( e q u a l t o t h e surface

A

minus a l l i r r e d u c i b l e components o f ' t h e f i b e r s which do n o t meet some fixed section of fibers)

.

j

and a l s o minus s i n g u l a r p o i n t s o f i r r e d u c i b l e

We have t h e "Kummer e x a c t sequence"

which g i v e s t h e e x a c t sequence

1 0 H (B, $ 1 - - H

The q u o t i e n t sheaf

2

(B,,&

0

) j

n

2 H (B,

0

& ) -+O.

,A/&0 h a s f i n i t e support, hence

Applying the, g l o b a l d u a l i t y theorem 1121, we g e t

2 0 0 ^O H (B,,A, ) = Hom(H (B,& )

, Z/nZ) .

The dual sheaf

8 coincides with

n-

duality of the jacobian variety

e. Suppose that

q(A) = 0

.

AK

A n-

in virtue of the auto-

.

Now we use the following.

Then

0 0 Proof. Any element of the group H,(B,,E)

represents a sectiog.

.of j of ordeb dividing n which meets the same irreducible component of a fiber as the fixed zero section.

Moreover, any two such sections

do not meek each other, since for any point b homomorphism

A,(%)

-t

0 equality H (S~ecr)B,b:

<)

# 0

,

A&) = H0 (k, n-A

which is a particular case

b

cohomology

(

I121)

.

Suppose that

Then

is a divisor supported in some fiber Fi

Since S and

.

and let S be a section from this group different

from the zero sxtion SO

where F;

the reduction

The latter follows, for example, from the

of some general property of &ale H'(B,

E

is an isomorphism on the subgroup of

.%(kf

points of finite order.

8

of j

.

So meet the same component of fibers we get immed-

e A

iately that ,F! e = 0 for each component of Fi lemma below we get that F; = Fi and hence

.

Applying the main

where F is any fiber fuse that since

'

q(A)

= 0 we have

and hence all fibers'are linearly equivalent).

B = P1

Now from the computation

.

(see again below) we get for any section ( S K 1 = -1 A 1 2 since S = P we get S = -2 + 1 p (A) = -1 - p g ( A ) < 0 of KA

-

+ p3' (A) <

..

9

However

This contradiction proves the lemma. From this lemma we get the following.

.

Proposition 5.

Suppose that q(A) = 0

is surjective.

In particular, for any finite set of closed points

bll...,br h > 0) i

B

E

scch'that the fiber

%i

Then the map

1 is of type 1 hi

and any collection of positive numbers

y,.. .,m

(is1

r;

there

exists a ninimal elliptic fibration f : X - + B whose jacobian fibration equals j and whose fibers- % are of typ. mi1hi i i=ll'.i.,r

.

.

Now we shall compute the canonical class surface X

.

5

of an elliptic

We restrict ourselves for the simplicity to the case

of regular surfaces X k.e. we assume that q(X) = 0) general case we refer to [6] or [ 2 7 ]

.

. ' z

For the

In particular, we may assume

that the base B of any elliptic fibration f : X + B projective line

.

is the

0

.

Main lemma.

([6]). Let

C =

surface X with each Ei

In C be an effective divisor on a i i

irreducible. Assume that

, all

(Ci,D) j 0

i

and.that D is connected. Then every divisor Z = xmici satisfies Z2 g 2

hclds if and W l y if D = Proof.

Write xi = mi/ni

z2

and

0

Z = rD

,r

0

and equality

Q..

and consider the equality

n n (CiWC.) i j i j 3

= lx x

Jf equality holds everywhere, then we have either xi = x (C.C ) = 0 2.

j

for all i;j ; since D Hence xi

does not occur.

or j is connected the last possiblity

is constant, that means that mi = m i ,

r€Q Theorem. Let surface

X

5 :X + P '

with q(X) = 0

be an elliptic fibration of an elliptic

.

Then

e r e I is any fibre of f , Fi = m FO all multiple fibres of i i multiplicity m. L

.

Proof. For any non-singular fibre -

]6 we have

Taking a sufficiently large number of distinct "generaln points and considering the exact sequence

bl,...,b

we get

If

D is a divisor in the linear system above, we have

( D - F )= 0

,

for any fibre F

.

This implies that we can write

5 wnere

r

g 0

Q,

(sum of fibfes)

r ,

is contained in a union of fibres and does not contain

. Let Xb . If

fibres of f

r0

the fibre

Xb = In E then ii

and

+

be a connected component of

r

contained in

2

(Eil < 0 if Xb

This shows that

-

(Kx:XEi)= 2g(Ei)

2

- ( ~ f =)

curve of the 1st kind. Thus, if Xb

0

is reducible, that implies that

, since

Hence, we have

Ei cannot be an exceptional (KXEi)

= 0

.

is reducible, then

[Po- Ei)

= 0

, all

componenks Ei of

Applying the main lemma we get that

.

3

r 0 = xXb , r f i Q

.

So, we have proved that

q(

%

nF

+ l a i 0~ i,

u (a. < m 1

and it remains to show that n = pg(Z)

- 1, and

i

ai=m

For this we note, firstly, that the divisor 1a.p' i

part of the linear system 0

KX

.

i

i

-1, is the fixed

Indeed, any rational function belonging

to the space If :X,O (K 1)

must be constant on the general fibre of f X X and hence, it is induced by a rational function on IP' But then it:

,

.

0 is either regular on the divisor Fi 0

to mi at F.

1 -

Thus we have

that proves 'the assertion about n Next, by Riernann-Fach

.

, or

has the pole of order multiple

and t h i s shows t h a t

(using again t h e arguments above1

a

i

+ l a m

.

This, obvioulsy, i m p l i e s ' t h a t

i '

Corollary 1. For any minimal e l 1 , i p t i c surface

Furthermore, i f

Proof.

,

then t h e plurigenils

The f i r s t a s s e r t i o n follows e a s i l y f r o m t h e proof of t h e

theorem. KX

q(X) = 0

X

Indeed, we have provea without assungtion

q(X) = 0

that

is numerically equivalent t o a r a t i o n a l l i n e a r combination of

fibres. TO prove t h e second assereion, we use t h a t

where

0

ai a mi

.

theorem, we g e t t h a t

This, of c0urse;proves

Again, using t h e arguments of t h e proof of t h e

iaiF:i

equals

the a s s e r t i o n .

t h e fixed part of

lnICxl

.

An elliptic surface with q = p = 0 is rational g if and only if its minimal elliptic fibration contains at most one

Corollary 2.

(1131).

multiple 'fibre. In fact, P (X) = 0

implies that the number of multiple fibres

2

r(1

.

In another direction the assertion follows immediately.

Corollary, - 3.

(~odeaux). Suppose that q(X) = p (X) = 0 9

where Fbi , i = I,...,

r

, are

.

Then

all multiple fibres.

Next, we want to compare the numerical invariants of an el3iptic surface and its jacobian surface, Proposition 6. Let by EP(Z)

k # C

f

:

X + B be an elliptic fibration.

Denote

the topological ~uler~~oincare characteristic (in case

, the

field of complex numbers, we consider 1-adic etale

cohomology). Then

For the proof we refer to 111, Ch. 4 (k =

C)

or [12] (arbitrary k)

Note that we use here the assumption char(k) = 0

.

.

In the general

case there is some additional term depending on the wild ramification. Corollary. Let X be a minimal elliptic surface, -. surface.

Then

A

its jacobian

The first equality follows fr~~propositions 1 and 2 , the second one follows from the first and the Noether formula. Proposition 7. Let

f

:

that for some fiber Xb

X + B

be an elliptic fibration. Suppose

the reduced curve

is singular.

Then

Proof. The hypothesis implies that under the Albanese map alb: X the fiber curves).

Xb

+

Alb(X)

goes to a point (since all of its components are rational

This shows that in the canonical commutative diagram alb X

$

Alb(X)

is a finite surjective map and -hence dim Aib(X) = dim J ( B ) = genus(B)

Corollary.

Suppose that the jacobian fibratio?

singular fiber.

Then for any elliptic surface

surface equal to A

j X

: 4 -+

B has a

with the jacobian

we have

.

be an elliptic surface with q = p = 0 Then g its jacobian surface is rational. Conversely, any elliptic surface.with :orollary.

Let X

rational jacobian surface has q = p

9

= 0

.

.

Thus .all elliptic surfaces with

q = p = 0 are obtained from 9

rational jacobian elliptic surfaces by choice of some fibres of type 1. 1 h and Some element of finite order of the Picard group of each of these,fibres. All rational elliptic surfaces can be described with the help of so called Halphen pencils on the projective plane ([131)

.

These

are the p'encils of curves of degree 3m with 9 multiple points of multiplicity m lying on a cubic. The case m = 1 corresponds to jacobian surfaces. To find a place in the above classificatjon of elliptic surfaces with q = p = o for the Enriques surfaces cc,nstructed in Chapter 1 g we note that for such surfaces P2 = 1 In virtue of the fir&

.

corollary to the theorem in 51 we get the following relation for the multiplicities m of multiple fibres i

This, of course, can occur only in the case

Applying the,formula for the canonical class of elliptic surfaces we see that, on the contrary, for any minimal elliptic surface X with. q = p = 0 and two multiple fibres of multiplicity 2 we have 9 Notice also that the following result holds: 2% = 0

.

Theorem ( ~ n r i q u e s ) . Any a l g e b r a i c s u r f a c e 2K

X

= 0

X

with

q = p = 0 and g

i s an e l l i p t i c surface.

The proof is too long t o reproduce h e r e ( s e e I l l , Ch. 9, and a l s o 161). There is a l s o a theorem (again due t o Enriques) which s t a t e s

,

t h a t any s u r f a c e with

q = p

its b i r a t i o n a l model.

Again t h e proof is t o o long t o be reproduced

g

= 0

h e r e ( s e e 111, Ch. 9 and a l s o 121).

2K = 0

has a s e x t i c s u r f a c e a s

The p a r t i c u l a r form of t h i s

s e x t i c passing through t h e edges o f a t e t h r a e d r o n corresponds t o a p a r t i c u l a r Enriques s u r f a c e .

2.

Torsion.

In .this section we shall prove that any finite abelian group can be realized as the torsion group of an elliptic surface w.ith q = p = O .

g

Lemma.

Suppose that D

surface X w j ' h

q = 0

is a torsion divisor on an elliptic minimal

.

Then D

is linearly equivalent to a

rational lineak combination of fibres of some elliptic fibration on X 2

Proof.

Since h (K X

.

Let D' f IKX +DI

.

+

0

D) = h ('-Dl = 0, by Rien,ann-Rochwe get

does not intersect a general fibre

Since D'

of any elliptic fibration (because

K~

does riot), it equals some

linear combination of components of 'fibres. boreover, D' does not intersect any component (because K X lemma from 51 we get thpt D' of fibres.

Thus, D'

- Ij(

%

does not).

Applying the main

is a rational linear combination

D is also a rational linear combination

of fibres. Theorem.

Let f : X

q(X) = 0

.

fibres.

Then

+

B be a minim&

elliptic fibration with

0 Let F - = m F i b i I i = l,!.., bi

r be all'its multiple

r Tors (Pic(X)) = Ker ( where

m = ml... mr, $(al1...,a

$

)) =

r

Z/m,

i i

'4 Z/m)

J

mod m

,

, mi

=m/mi

.

.

Proof. Using the lemma we may write any torsion divisor D in the form

where 0

5

ai < mi,

F any non-multiple fibre.

Intersecting the both sides with some transversal curve C we obtain

and hence

%&is shows that 1 is uniquely determined by ai and, moorewer,

Now we know (see the proof of the theorem in 51) that the divisor F

is in the fixed part of any linear system contnining it.

Hence the coefficients ai are determined uniquely by the divisor class of D

.

is injective.

This shows that the map

...,a

Now for any (al,

)

satisfying condition ( x ) the divisor

has zero intersection with any transversal curve and any component

of fibres. This shows that D is numerically equivalent to zero, and, hence, D is a torsion divisor.

of a

This proves the surjectivity

.

Corollary 1. kn notations above

Corollary 2.

For any finite abelian group G

elliptic surfac~? with q = p = g

Prcof.

.-

0

there exists an

such that

Applyincj Proposition 4 we may find such an elliptic 'surface

with multiple fibres of any prescribed multiplicities. Let -

be the primary decomposition of G

.

Consider a surface X with

the following collection of multiplicities:

Then applying the theorem we easily see that

Corollary 3 -

(1141)

.

There exists an elliptic surface with

q = p

3

= 0

which is not a rational surface and has no torsion divisors. Just take a surface with multiple.£ibres of coprime multiplicities And apply Corollary 1 and Corollary 2 to the theorem of 31

.

§3.

~undknentalgroup.

Here following to Kodaira

[281'

and Iithaka I251 we shall compute

the,fundamentalgroup of an elliptic surface, over the field.of complek numbers. Let f

:

X + B

be an elliptic fibration.

I

c B be

Lema.1. Let U

of f over U

fU : XU + U

point p 0

an open set such that the restriction

6

X

u

has no multiple fibres.

lying in a non-singular fibre.

Choose a

Then the following

exact sequence holds

proof.

Consider the inclusion map

map

:

fU

X

U

a U

an6 the projection

XU

and the correspondent homororphisms of fundanlefital

groups. Then the image of Tr (X ,p 1 E(pO) 0

is clearly contained in the

in the kernel of the second homomorphism, and we have to show that it coincides with the kernel and the second homomrphism is surjective. Restricted over sufficiently small U

the map f

is a differentiable

2-torus fibre bundle, and the corresponding sequence is the exact homotopy sequence. This obviously proves.the surjectivity of the second h~~1;~mxphism. Let y be a loop with the origin at po fibre of

fU

, there

. Let

XU

be a singular

0

exists a local section D +

small disc centered at f(po)

(since X u

0

s, D

beinga -0 is not a multiple fibre): 0

Assuming t h a t a loop on

X

goes t o zero under

y

keeping t h e p o i n t

f (pol

po

y

it allows t o deform

?x &

y

fixed.

to

This proves

t h e lemma. Next, bl,

...,b

let

Dlr...,D

be some open d i s c s around t h e p c i n t s

r

f o r which t h e f i b r e

i s m u l t i p l e o f m u l t i p l i c i t y . mi

5: I

Assume t h a t over the punctured d i s c s o f smooth.

Let

...- Dr

U=B-Dl-

.

V" = f-'(~:)

i

D; X

'

u

t h e morphism

=f-l(IJ),

is

f

V . = ~ " ( D ~ ) ,

We s h a l l apply van Kampen's theorem t o compute

a 1 (x)

.

Let

6,

0

be some loops on generates

originated a t

11 (*f(p0)

tll..-,t2 be t h e loops on

...,a g ;

bl,

B

starting a t bl,

...,b!I another loop m i g i n a t e d al(U;

ti,

with-the origin a t

a f , b; po

canonical g e n e r a t o r s of

Lemma 2.

The group

a , . a , b , . . , b g relations:

9

.

f(po)

...,br

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

Denote by

lhen asstuning t h a t

nl(XU;

£(pol)

pol

( g = genus o f

, we

XU

a

;

at

£(pol generate

l y i n g over bi

t., a., bi 1 1

a r e chosen a s t h e

g e t t h e following.

i s generated by B)

ti

and

;

some loops on

n1( U ;

which

Po)

going arount? tlie p o i n t s al,

po

6,

0,

ti ,...,tr'

with t h e following b a s i c

1

(ii) the group ( 6, a) nl(XU

;

p0)

generated by 6, a

is normal in

i

(iii) aq.ya*-lbi-l* *a*b,a ,-lb.-lt, 1 11 g g g g l.--t: (iv) some relation between 6 and a

e 16, a)

:

(may be trivial)

.

This follows immediately from Lemma 1 and the known structure of

Choose 'somepoints p., i = 1,..., r lying over :D X

and

X

some loops iji, ui in the fibre X generating n (X 1 f(pi) 'Pi) f (pi) Let Ti be a 14,op going around bi and passing through f (pi) ,

-

-

ti some loop on 'V Lemma 3. The gnmp

''

lying over t. which passes through p

i

nl(v:,

pi)

-

is generated by

X

IS^,

i'

x -

ai, tf with

the following basic relations: X X -

(i) diai (ii) :6 (iii) (iv)

-

x x

a 6

X X and ai generate a normal subgroup in n1 (Vi ; pi)

i-ri6 i'= 6yFi

yiu:

i

;

h. X 1 x= tji ai ti

'5

, if

i

Proof. --

;

is of type m1h ii'

?+plying Lerma 1 we will prove the first assertion and find

the first two relations. To obtain another pair of relations w e will use the following description of :V

which is due to Kodaira [27]

There exists an unramified covering F group is a cyclic group of order mi

.

+.vX i

.

whose covering transformation

The space F is represented

in the form

where in the first case

r

is the discontinuous group of analytic

automorphisms

m.

m

m (j(z

i,

is a holomorphic function of z

r

In the second case x

with

Im j ( z

n

I I ~ ( T3) 0

exponential map T -+ exp(2~ i.c)

covering transformation group then t

7

.

with the upper

and the covering map with the

, we

the universal, covering space of :V

0

0 )

g'enerated by the automorphism

Identifying the universal covering space of :D

I£' hi =

>

is the infinite cyclic group of analytic

automorphisms of D.xC

half plane H = {T

')

H

get that in t h e both cases is equal to HxC and

the

may be described as follows:

consists of analytic automoprhisms

If hi > 0

, then 7

c o n s i s t s of a n a l y t i c automoIphisms

Identifying i n t h e usual way the loops originated at pi

with

covering transformations, we may assume t h a t ti corresponds t o t h e element of

Si

7

corresponds t o the element of

with

(1, n2, nJ

= (1, 0, 0 )

,

with

(nl, e2, n ) = (0' 1' 01

,

3

ai corresponds t o the element of 7 with

(I$, n2,

n3) = ( 0 , 0, 1)

The r e l a t i o n s fiii)and (iv) a r e irerified aow immediately. To use van Kampen's theorem we consider homom~rphisms

w h i c h correspon9 t o t h e n a t u r a l inclusions

of so= paths cdnnecting t h e points

pi

viW w x and t o a choice

asd po

.

and a l s o the

natural surjections

Applying t h e same arguments a s i n t h e proof of Proposition 2 from 91 we may assume t h a t t h e c y c l i c covering F of t o an e l l i p t i c f i b r a t i o n over degree

rn

,w i t h

implies that

f i b r e of type

Di

, tha

<

can he prolonged

cycle a w e r i n g o f

over the arigin.

Di

of

This e a s i l y

.

Moreover, 'if

hi > 0 we get t h a t

Collecting everything together we obtain: Theorem.

The fundamental group

6, a, al,-..,

ag, bl,

r

1

(X)

is generaced by l e t t e r s

..., bg, tl, -.,t

.

r

.

The basic relations are

ii) ( 6 , o}

some relation between

V)

Corollary.

is a normal subgroup

Let

f

: X

6 and

,

0

(may be t r i v i a l )

+ 3P1 be an e l l i p t i c fibration.

.

Then r l ( X )

is abelian i f and only if it has at h s t 2 multiple fibres.. In f a c t ,

rl(X)

given by generators

has a s its quotient the group G t 3 ,

tl,

...,t

f

...,mr 1

and relations

These groups are well known i n the theory of automorphic functions. Namely, there e d t natural representations of these groups as a

discrete subgroup of the automrphism group of one of the three standard planes:

the Riemannian

and the Lobachevsky H = { z c

@

1 lp ( 5 )

1 Im(z)

, the

> 0)

.

Euclidean

5

,

W e have (see [301) t h a t

each of these cases corresponds t o the sign of t h e number

We have the case

6:

..,mr)

iff

e > 0 and

iff

G(q,.

.iff

e = 0

and

iff

G(ml,

...,mr)

i s non-commutative

and

iff

G(ml,

...,m

is infinite

pL (IT)

is finite;

nilpotent; iff

H

e < 0

non-nilpotent

Thus,

vl(X)

Gb18

m2' mj)

)

.

csn be abelian only i n the case

e > 0

.

In t h i s case,

~ ~ ( 2 , @ ) / { +, 1 )t h a t is the

.is a f i n i t e subgroup of

rotation group of some regular polyhedron (r = 3) o r a cyclic group

(r = 2)

.

This, of course, proves the corollary.

.

be an e l l i p t i c surface with' q = p = 0 , 9 which admits an e l l i p t i c fibratioil f : X + PL w i t h exactly two multiple Corollary.

(1141)

Let

X

f i b r e s of coprime multiplicity.

Then. X

i s a simply connected non-

rational surface. In f a c t , its fundamental #oup being abelian has t o coincide w i t h t h e homology group H1(X.,Z)

.

Since q[X) = 0

It remains t o apply Corollary 1 of 52

.

,H

~ ( X = ~ )Tors(i11(X,7&)

.

Remark. correct.

In I143 the argument that

lrl(X)

is abelian was not

So, i n f a c t , it was proven there only that there e x i s t

q =

= 0 with no torsion divisors. Pg This was the original question of F. Severi. non-rational surfaces with

SURFACES OF GENERAL TYPE

CHAPTER 111.

51. Some useful lemmas.

Lemma 1. Let X be a scheme and T is a finite subgroup of the Picard group Pic(X) t :X + X T

.

Then there exists a finite etale Galois covering

uniquely determined by the properties T = Ker(Pic(X) f=Pic(XT))

and the Galois group of f is isomp~hicto the'character group ChartT) Proof.

For any E

t

T

let OX(E)

L

be the corresponding invertible sheaf.

+3 0 (€1 has a natural structure of an &tT X 0 -Algebra corresp~ndingto the isomorphisms 0 (E) 5 Ox(€') -+ OX(€+€') XX

The locally free sileaf

Put XT = Spec(L) flat.

.

=

Then the projection f

It is also etale, since det(L1 =

G = Char(T)

~(€1 ,

x

:

X T

0

(€1

E6T X

+

= f (0

.

= OX

The group

acts naturally on XT multiplying each summand OX(€) by

c G

.

.

Clearly, the invariant subalgebra

Then

?(L)

.

LG

OX

=

L

E

, hence

Ker(Pic(X)

f" -

OX and fxfXc~) =,. f (0 ) = L P3 OX(€) = T X~ This implies that L 5 OX(&) = 0 (El) for some

=

= E$TOX(f:) X~ E' c T and hence 1 = OX(&

- E')

X

€

T

.

The inclusion T C Ker is

obvious. To prove the uniqueness note that for any finite Galois covering f : X'

-t

.

X is finiteand

rp/G = X and f ?.s a Galois covering. Assume that Pic (XT))

,

X with the Galois group G we have

This immediately follows from the'Hochshild-Serre spectral sequence or from direct considerations.

Now f (0 ,) x X

must split into eigen subsheaves corresponding to

characters of G

Let

LX

be the invertible sheaf corresponding to a character in virtue

of the above identification of Char(G) with the subgroup of Pic(X) Then

L

f (0 ,) x X

X

being lifted onto X f is trivial, thus it is embedded into and is ismorphic to one of its suwnands (namely, f K ( OX ,)

This shows that X' = Spec(f (0 , ) ) x

X

is isomorphic to XT

Corollary.

In the above notations

i

=H T : E

B (X .O

i

(XI Ox(€))

X~

More generally, for any locally free sheaf

We

have

L

x

)

-

constructed

above.

proof. -

.

on X we have

It remains t o applytheLeray s p e c t r a l sequence which dzgenerates

because Lemma 2. with

is finite.

f

(Bombieri 141).

Let

q(F) = 0 , m = Tors(F)

F be a swrfac'e of general type the order of the torsion group.

Then PJFI

5

1

2

and

i f there e x i s t a f i n i t e abelian unramified covering of

F of

irreg-

u l a r i t y a t l e a s t one. Proof. group

Let

-

f : F

Tors(F)

-t

F

be the covering corresponding t o the torsion

i n v i r t u e of Lemma 1. By the lemma of

52, Chapter 1

-we know t h a t

Now apply the following c l a s s i c Noether theorem (see 141. Th. 9) :

a@

consider separately the two i)ossible cases: a)

any order

n

q(i?);)> 0

.

Let

: Then

P(n) +

pic (F) contains a finite subgroup of

be the corresponding e t a l e covering.

W e have

+ q(F(n))-

P ~ ( F ( ~=) )n (1 + pa@))

dividing by

n

Now dividing by

b)

and letting n +

m

w

Proof.

Since

2

+

2

,

we g e t

we obtain

q(F) =

0 : Then

and it suffices t o divide both sides by m Lemma 3.

1

15 z n K-F

.

Let F be a surface of general t y p .

'Ihen

By Noether's formula

52 > 0

and cZ(P) > 0

Th. 13) we get the inequality.

(otherwise,

F

worrld be ruled, 141,

Lemma 4.

Let F be a surface of general type and

numerically equivalent to mKF

,m 2

1

.

D be a divisor

Then

1 . A (F, OF(D+KF)) = O .

Proof.

This immediately follows from the following Ramanujam's

form of Kodaira's Vanishing theoren (C. Ramanujam,. 3. Indian Math. Soc., 38 (1974), 121-124) surface, (C

L

:

Let X be a complete non-singular

and invertible sheaf on X

1 (L) C) 2 0 for any curve on X

.

such that

(cl(L)2 ) > 0

i

Then E (X, L-l) = 0

for

i = 0 , 1 .

Corollary. The m-th plurigenus Pm of a surface of general type F

is given by

in particular

Use Reimann-Roch and Lemma 4 Lemma 5.

Let

f

:X +

applied to D = (m

- 1)s.

Y be a double covering of non-singular

surfaces branched along a reduced curve W C Y

.

Then

and

Proof. The subsheaf Oy is naturally identified with the subsheaf of

f,(O

invariant under sheet-interchange. Since the ckaracter-

X

istic is assumed to be zero (or at least prime to 2 a direct summand of &(OX)

, the complement being a

anti-invariant sections. The sheaf of the invariant subsheaf, that is

sheaf

JC

Oy

.

This shows that X

of the vector bundle (ia2

- 1) .

53 show that

v(L)

),

L" oy

.

this sheaf is

sheaf

'L

of

is obviously a subsheaf thus iB2z 3

f&

some Ideal

is isomorphic to the subscheme l?)n

= S ~ ~ C ( ~ @) ~ L defined

by the ideal

Now, the local arguments of the proof of emm ma 2, Ch. 1,

:;=

0 (-W)

Y

and a = f*(ay k 3 L-l) X

Corollary. Let F be an invertible sheaf.on Y..

. Then

52.

Numerical Godeaux surfaces.

By this we mean any surface of general type F

with

In virtue of Lemma 3 and corollary to Lemma 4 of gl we get moreover that

qfF) = 0

and P,(F)

-1

= 2 m(m

- 1) + 1 .

We will distinguish these surfaces by the value of its torsion group TorsCF)

.

First of all, by Lemma 2 of 1-wehave the

following. Proposition 1. If m

=

Tors(F)

then

For any abelian unramified covering F' + F we have

Bmposition 2.

(Bombieri).

with Tors(F) = 6

Proof.

There are no numerical Godeaux surfaces

.

Assume that Tors(F) = Z/2Z

cB 2/32

.

Then there exists an

unramified covering 'F' + F of order 2 with Tors(F')

Z/3Z

.

~y the lemma of Chapter 1, 92 we have

p

1 = 3

and

By proposition 1 q(F') p

(1)-

- 3, pg = 1,q = 0

-q(F1l

= 1

.

0 and hence we obtain a surface with

.

and tke torsion group %G/B Bowever

this contradicts Theorem 15 of 141 Remark.

+ p9(F')

.

Since the previous proof is a simple application of

Theorem 15 of 141, which in its turn is provad using other nontrivial. results of [43, it is better to give an independent proof. As suggested by Miles Reid we can argue as follows. Let Y ba the covering of X corresponding to the group of torsion of ordx 6. Then p (Y) 9 Now we will us+ Lema (E. Horikawa).

(5)= 2pg(Y) - 4 . 2

- < 5,

= 6 = 2p (1)

g

-4.

Let Y be a surface of general type with There

151

is an irreducible linear system whose

general member is a hyperelliptic curve. Proof. Suppose that

where F

is a fixed part.

pencil, say C

a[Co]

Assuem that

, where

Ic~

a > 1 and

is composed of a

[%I

is an irreducible

Then p (Y) a. + 1 and the equality holds if fCOl is linear 9 0 We have K O F 2 0 ,therefore (i-e. dimR (Y, O ( C o ) f = 2 ) 'I 2 2 5 P and since :C / O we get c0 2 2 ,because 42 2 2

pencil.

.

5

5

.

Hence

and we have a contradiction.

Thus we may assume that

IcI

is not

composed of a pencil. Now the analysis of the proof of Noether's inequality 1 K 2 + 2 (see [ 4 ] , p. 209) shows that in the case of the p (Y)5 2

g

equality y =

Y

151

is an irreducible non-singular curve C of genus

2 (5) + l .

Now the exact sequence

0

shows that dim H (C, restriction of

= 2p (Y) 9

OC(

K C)) = pg(Y) Y

1 ~ on ~ 1C

.

Then 2D

- 2 = %2 + 2 = deg D + 2 .

.

-

1

%

KC and

Let D denotes the 0

2 dim H (C, OC(D)) =

Now by a classical Clifford's

theorem on special divisors it follnws that C is hyperelliptic (see, for example, H. Martens. J. &ine

Angen. Math. 233, (1968),

89-400).

After we have proven the lemma the arcjument is very simple.

If

is an automorphism of the covering

Y +

X then a acts

freely on Y and hence on a general member C of

1 ~ .~ But 1 this

is obviously impossible (any automorphism of a hyperelliptic curve

has a fixed p i n t ) .

.

Lemma.

lReid 1391).

Let F be a minimal numerical Godeaux surface.

Then (i) For any non-zero

g c Tors(F)

positive divisor D 9 (ill

t

if g # g w then D

1%

there exists a unique

+ g1

and D

;

have no common

9'

9

components; (iii) if g, 9' and gn are distinct non-zero elements of Tors(F) then D and D do not meet. 9. Dg' 9" Proof. -

xi)

By Riemann-Roch

By Serrets duality. h e e

a

n

h2($

hl($

t g) = f h0(-g) = 0

+ g)

-

h1 (-4 = 0

corollary to Lemma 1, 51 and Proposition 1 (ii) If one of obvious.

D

9

Qr

D 9'

, since

g # 0

.. By

in virtue of tE.2

.

is irreducible the result is

Suppose that

is the decomposition into irreducible components with C and chosen so that

If C = C'

(D

C) =

(q C')

then D = D'

= 1

,because

(recall that

(D = $1

C'

=

there are no relations

between fundamental curves (that is, curves with no intersection

with

5)other than equality

( 141,

Prop. 1)

.

Let. E be the comon part of D and D'

,.

then E~ < 0

and even, since it is a positive combination of fundamental curves. Thus ( D - E I 2 = D 2 (D

- 2(D-E) +

E ~ =1 + E

2

5-1. But

- E ) =~ (D - E) (D' - E)

must be non-negative, since D

-E

-E

and D'

have no common

components. (iii) Since

52 =

1 each two D

and D

4

9'

,g #

g' meet

transversally at a non-singular point for both curves. The fact that three distinct D

9'

and

D g'

alent to the fact that 0 (D

- Dg

D g"

meet at a point is equiv-

being restricted on D is g" Write the exact isomorphic to the structure sheaf of D F

9

,)

9"

'

sequence

and the corresponding cohomology sequence

Since D zero.

-

D is a non-zero torsion divisor, the first term is 4 ¶' I By duality, the third term is equal to h (51 for some

torsion divisor.

That is also zero (see the proof of (i))

.

This

contradicts the non-triviality of the middle term. Propasition 3.

(Bombieri-Catanese, Reid).

Godeaux surfaces with Tors(F) = 2/22 8 2/22 Proof. Let -

There are no numerical

.

F be such a surface. Then we have the three distinct

non-zero t o r s i o n d i v i s o r s of order 2..

Let

t h r e e d i v i s o r s constructed i n Reid's lemma. 2D, 2DS and

D, Dg and

men

2D" belbng t o t h e l i n e a r system

.r;he

1-1

P CF) 2

= dim I 2 a l+ l = 2

.

divisors

and by t h e

property ( i i i ) they cannot be members of a pencil.?.

However, we know t h a t

be t h e

D"

Thus, dim 12$[

2 2

.?&is cont-adiction

proves t h e asse-don. Remark.

The proof of Bambieri-Catanese 157 uses other more el&orate

arguments.

The p m f frem I 3 2 J . i ~not conplete.

Thus, we have the

following possible cases:

,We know examples af surfaces with 9;/52 (the Godeaax surfaces of I

52, e p t e r I)

.

of such surfaces.

Let us show t h a t these a r e e s s e n t i a l l y a l l e-amples The proof.&low

i s Clce t o Miles %id

[391.

be the unramified covering of order 5 corresponding t o

Let

t h e torsion group Tors(F)

.

Then by the ccarollarj t o I R r l of

51 we have

We know form Reid's lemma f i ) t h a t

0

h (K + g) = 1. g # 0

.

Let

x1,x 2 ,x3,x 4 be non-zero elements corresponding to t h e four non-zsro elements of

TcrsfF)

.

generating this space. zero an

F

, therefore

We may consider them as elements of

-

0

31 (F, %Il$)j

Since by Reid's lemma t h e x 's have no conmn

1

on

they define a morphisra f

3

: F + H)

.

Since

dF =

and the degree of f must divide 5 we get that f

5

is birational onto a surface F'

of degree 5

.

This quintic F'

must be a normal surface, since the arithmetic genus of its hyperplane sections coincides with the genus of its inverse images (=canonical

- .

divisors) on F

f

Thus F'

coincides with the canonical model of

and as such has only double rational points as singularities. The group G = Char(Tors(Ff) = 2/52

-

acting on F acts by

0 functoriality on the canonical model F' = Pr~j(~_$&i (F, O$mE$) i multiplying x . by some 5 (6 a 5 t h root of unity). Thus F

is "almostn the quotient of a quintic by 2/52

.

More exactly, the

canonical model of F is isomorphic to such quotient. We refer to 1111 and [321 for the study of pluricanonical maps

of numerical Godearn surfaces. Also in 1321 it can be found the facts concernbg the moduli space of surfaces with Tors = 2/52 Surfaces with Tors(F) =

Z/4Z

(Reid-Miyaoke)

.

.

To construct such surfaces we will pull ourselves by.shoe-strings. Assume that such surface F

exists.

As for the Godeaux surfaces we

0 :2 consider the elements x e H (F,OF(KF + gi)) , where i g1 * g2 g1 3 2 g3 = g1 are non-zero elements of Tors(F) Then x1x3 and x2

.

0

fork a basis for H (F,0F(2KF follows from Reid's lemma) 0

H (F,OF (2KF + gl))

(x2x3,y1) and Proposition.

0

.

+ g2) )

(their linear independence

Let ,yl and yj be sections of

and H (F,0F(2%

+ g3) )

respectively such that

(x1x2*y3) form bases. (Reid).

pluricanonical ring

The above elements xi, yi generate the 0A(F) = m=o $ H (F,c+(~K-) P F

w

0

= m=oH (F,O~(~% + 9)) gt Tors

#

of the surface

which is the unramified covering of F

ing to the torsion.grpup Tors(F)

.

correspond-

There are two basic relations

of degree 8 between these generators. Proof. -

The monomials

However, by the corollary to %emma 4, 91 we find that

Thus there is a linear dependence between these 8 monomidls, which we will write

In the same way the 8 monomials,

0

and h (4%

+ g2)

= 7

.

Hence we have the second relation

Both these relations of degree 4 considering xi,yi the graded canoni3al ring

A@

* 0= S O H (F,O$%-)

as elements of

-

kext, let

be the quotient polynomial ring. Grade B by the condition deg(Xi) = 1, deg(Y.1

= 2

, then we

have the morphism of graded algebras

The proposition is equivalent to the assertion that $ is an isomorphism. NOW, the ~oincare'function (compare 1151 )

PB(t) =

1 dim Bi ti =

In xirtue of the formula for P. 1

4 2

3

)

)

*

(l+t2) (1-t)

this coincides with

- (t) = 1pi(F>

'A(F)

2 2 * -

(1-t) (1-t

i t

.

~hus; it suffices to check that $ is injective.

ff $

is not injective then the image of the rational map

w i l l be a proper closed subscheme of

Let

j

be the e e d d i n g

~

1 -P B~ ' ~ )~= iE3B2i 1

.

canonical model M

v-z7

,a : P * M

Q

V

.

corresponding t o the surjection

the canonical m a p of

I

onto its

The composition

i s easily t o be seen coincides with the 2-canonical map

In virtue of Reid's le-

¶J

%-

is regular (see the analogous

argment i n the previous case of the Godeaux surfaces), thus

is a l s o regular. Let

=Q

(F) ,

2%-

subscheme of Since

V Q

This shows t h a t

V

.

By om assumption,

2T

is i n f a c t a morphism.

is a proper closed

V spans 'a its degree is a t l e a s t 6 . and 1 5 1 has no fired p a r t it implies t h a t deg V = 8 o r 16 . Horecver,. i n the f i r s t case,

Since

( 2 % ~= ~16

i s a surface and

2T

$

defines a 2-sheet covering

and in the second case g i s a birational morphism.

Since

deg j ( V ) = 16

(this follpws from the equality of the Poincare functions for

-

and B) we get that in.the second case V = V

.

~ ( 3

SO, we may assume

-.

that B2= is a 2-sheeted covering onto its image V Let C r F be a non-singular curve, the map gl-C equals the canonical map of C and since it is %sheeted

C must be a hyperelliptic curve and

g

Ic

its hyperelliptic involution. Now, notice that the canonical map

.

also factors through and hence through g Then % cuts 1 1 out on C a ,g4 which is composed with hyperelliptic 92 ' This implies that K F ~is~not a complete linear system. But the

Q

5

letter contradicl:~the vanishing of

i-?(~,

.

Corollary. Let F be a numcrical Godeaux surface with Tors(F) = Z/&

-F

,

its unramified covering corresponding to the torsion group. ''Then

the canonical model M of

F

intersection V4,4(1,1,1,212).

is isomorphic to a weighted complete The action of the group Char(Z/rlZ) = p4

on M is induced by the action of this group on the weighted projective space IP (1,1,1,2,2)

which multiplies the first three coordinates by

5 , 5*, 53 accordingly and the fourth and the fifth coordinate by

GI '5 model

accordingly (5 a primZtive 4-th root of 1) M

of F is obtained by dividing

.

The canonical

M by this action.

This corollary prompts to us the way to construct F

.

For this

one may take a non-singular F = V (1,1,1,2,2) invariant under the 414 above action on P(1,1,1,2,2) and not containing the fixed point of this action.

Using the general properties of weighted complete

intersection (which are quite analogous to the ones of usual nonsingular complete intersections) we find (see, for example, 1151):

Dividing F by the free action of

Notice also that we have n

1

(F)

v4

we get the surface F with

= 0 and thus

An explicit exanple of V4,q(1,1,1,2,2)

For a more general example see 1321

with the properties above:

.

Surfaces with Tors(F) = Z/32. Here the same method of Miles Reid shows that the covering $of such surface F

P (1!1,2,2,2,3,3),

is embedable into the weighted projective space unfortunately, not as a complete intersection.

There are not any explicit constructions of

ithe example in I391

does not work) and, thus, the question of the existence of such surfaces

F is still open*

*

see Epilogue.

.

Surfaces with Tors(F) = 2/22

(Campedelli-Kulikovdort).

The main idea here belongs to Campedelli, who proposed to construct a surface with p(ll = 2 as a double plane branched along a 10th order curve with 5 triple points of type x point. below).

3

+ y6

= 0

and an ordinary 4-ple

Unfortunately, his construction of such a curve is false (see Victor Kulikov (non-published) proposed to modify the Campedelli

curve, taking the union of two conics and two cubics such that one of the cubics has a double point, both conics pass through this point and touch both the cubics at other points. pf this configuration

( [353):

It is easily checked that

Oort gave an explicit construction

Let W = C1

u Z2

L) Dl C) D2

, where

where

the point

P6

is an ordinary double point of

D2

, and

the combination

of the points above is considered as a divisor on any non-singular curve taking part in the intersection.

Let F be the minimal non-singular model of the double plane branched along the curve W 'Assertion 1:

.

Proof.

This is s i m i l a r t o t h e proof used a t t h e c o n s t r u c t i o n o f t h e

c l a s s i c a l Campedelli s u r f a c e from Chapter 1, 93 Let

where

with

L

divisor

W

i s a l i n e on

2 S. = -2

Let

be- t h e minimal r e s o l u t i o n o f s i n g u l a r p o i n t s

p : X + IP2

of t h e branch curve

.

, 1L i

.

Then the s t r i c t i n v e r s e transform of

3P2

W

,

2

5 ; 5 . = -1

,i

= 6, 7; 5'

r : I?' + X b e t h e double covering o f 5 -1 p (W) si , then

2

i

X

= -2,

9 ';

= -1

.

branched along t h e

+ 1

.i=1

1,

D r IK ; then w e s e e from above and c o r o l l a r y F 5 to Lemma 5. 51 t h a t D = r * ( ~ ' ), where D1 t I~~'(L) S; S; Assume t h a t

- 1 - - s61 i=l

and hence e q u a l s t h e proper i n v e r s e image y d e r

p

o f a c o n i c passing

througll the points Pl,

...,P6 .

not situated on a conic.

However, obviously these points are

This shows that

Since r is branched along

IKp ,I = j3

Si, i=l,.. .,5

and

, rx(p-l(~i>) =

2 3

and thus

-1

p

(ci)

, i=1,2,

we see that E*(s.) 1. = 2Zi)

for some curves Si and ??' on i

Fs

.

Also, we have

- 2 1 x 2 1 2 1 Si = $r . (Si) ) = z(2S. j = -$-4)

= -1

- 2 1 t - 1 2 1 - 1 2 1 C; = ~ ( (P r (Cif) ) = q(2(~ (Ci)) 1 = ~ t - 8 ) = -2

This shows that 6 : F' + F

i

are exceptional cunVes of the 1st kind.

be the blowing dor-m of a11

the minimal model of

a d hence

-S

F'

.

.

We h a w

si .

X e will show that

Let

F

is

is an exceptional curve of the 1st kind on

Assuming that E we get that

-

curves C. and also

( E *2%)

, S;

or

zi2= r

X

= -2

- S"

E

+

1 = 2

si2 + 1

,

coincides with one of the

However, we sew above that

5 '

(s;)

and hence

F

= -4

+1

= -3

F~ = F;~= i 2 ,S;

-2

= r

2

= -2.

Now

and the assertion is proven. Assertion 2.

Proof.

In the proof of Assertion 1 we have found already a torsion

divisor of order 2, this is

In virtue of the analysis of the torsion of numerical Godeaux surfaces we know that Tors(F) = 2/22

or 2/42

.

Let us exclude the secand

possiblilty. Assume that involution

.

g

is a torsion divisor of order 4

6 of F corresponding to its rational projection onto

6r (g) % g , then 2g divisors on P2 Thus, 6n(g)

le2

If

.

,. since there are no torsion r\, -g , because 6. defines an auto%

.O

morphism of the torsion group 2/42 from

Consider the

I $ + gl

.

Then

.

Let

D

g

be the unique curve

1251

The bicanonical system

is a pencil, generated by the two

curves

and

We see that

12$1

has the fixed component, namely

to be contained in both the curves D

9

and

D-

S

-S; , which

has

.

D and D However, by Reid's lemma 9 -9 has no common compnents. This contradiction

proves the assertion. Remark.

Campedelli proposed to construct the branch curve W

union of.3 conics Cl, C2, Cj. and a quartic C2

are bitangent to Cg

, touch each other

D

as the

such that C1 and

at a point,

D has a

node at one of the two ordinary intersection points of Cl

an8 C2

,

passes through the five contact points of the conics with the same tangent direction (see 191)

.

The arguments similar to the one used above show that the bicanonical system of the corresponding double plane is equal to the inverse image of the pencil of quartics on

with C1 U

C2

U Cj

IP2 touching

D at the p i n t s of cpntact

and having a node at the node of 2

the two curves from this pencil

C1+ C2

and

D we

krll

,

Considering

find two

torsion divisors of order 2.

This contradicts Proposition 3.

Thus

the Campedelli construction does not exist. Surfaces with Tors(F) =

0

.

There are no examples of such surfaces. consider

Maybe it is worth to

a version of the example above with the branch curve W

equal to the union of two conics and two cubics forming the following configuration (Kulikov):

where

C1

and

C2

are co~ics,and D D cubics. 1' 2-

Arguing as above we would show that the bicanonical system is equal to the inverse image of the pencil of quartics passing through

P1,...,P5 at

P6

with the same tangent direction as W

.

and having a node

It is seen that there are no' members of this pencil composed

. elements of order 2 . of components of W

This easily proves that .there are no torsion

Of course, the existence of thts configuration is not easy to justify.

3.

Numerical Campedelli surfaces.

These are surfaces with distinguished by the order m Beauville I31 and Reid t h a t

.

p

= 0 and pC1'= 3 g of its torsion gorup.

m 5 10

numerical Campedelli surfaces with

.

They are It was proved by

Here we exhibit examples of

m = 2, 4, 7 and

.

8

no examples of such surfaces with other possible value of

There arem*,

moreover there a r e no examples of numerical Campedelli surfaces with

Tors (F) = 2/42 a)

.

Sldsgi~al- camnedel&i-sggage=.

know (Chapter 1, 53) t h a t

Tors(F) 3 (z/2z)3

For -them we already

.

We w i l l prove now

t h a t we have the equality. Proposition (Miyaoke 1321, Reid 1391)

.

Let

-+ F

r :F

be the unram-

i f i e d covering of the c l a s s i c a l Campedelli surface corresponding t o the subgroup

T = (Z/22)

canonical system F$

of the torsion group Tors (F)

defines t h e b i r a t i o n a l morphism of

intersection of 4 quadrics i n

a6

.

Then the onto the

.

Proof. We know (Chpater 111, 81) t h a t

Let us show t h a t

since h 0 ( 2 9 = 3

, we

0 get that h (s+g) 52

equality, then 1 2 ~ ~ is1 composed of the pencil

*

see Epilogue.

.

If xs have the

-1

.

Considering the restriction of /KF

+ PI

-

onto

zl, we see that

this pencil has a base point on S1

.

also this point as its base point.

However, the curves

2z

+

2-d

+

2S

+

2 S and

3 4 -S 3at two2 distinct points.

1

This shows that 12$1

has

+ 2 c3 + 2z5 + 2z6 from 125-1 intersect 1 This contradiction proves the needed

2-d

assertion. Denote the lements of T by 000, 100, 110, 010, 001, 011..101, and 111

.

Let

be non-zero sections.

..

0

-

.

Clearly, rx(xi) = yi, i=O,. ,6 , generate H (F,%(%)) All 2 0 0 squares x belong to H (F,O ( 2 1) and, since h (2KF) = 3 , there i F KF must be 4 relations among them. This shws that there are 4 relations 2

between yi in H~(~%(K$

.

1

Now we can find explicitly these

relations. We know that the bicanonical system 1 2 5 1

is represented,

by the net of quartics

(in notation of Ch. I, 53).

Up to a permutation we easily find that.

x,,2 corresponds to C.1C2 X

X

2

a

2 2

3

n

3"'2' 3"'1'

2 x4

2 corresponds to C 1 2

2 5

I

2 6

n

X

X

i where

*

c2x;

"

C&

rn

D

(resp.R2, resp. t3) is the line through the points P5

and P6

(resp. Pg and P4

a

resp.

p1

an6 P2

)

.

This gives the following relations among y i

for some non-zero constants a, b

,..., g, h .

Thus we obtain that the canonical image the complete intersection V

-aK(F)

is contained in

of t+he four quadrics given above.

It

is easily checked that V has only isolated singular points (in fact '

24 double ordinary points) and hence being a complete intersection is

@E(F)

an irreducible surface. .

dim

X

= 2

-

This implies that @j$)

Assume that

@-(p) K

is a curve.

is isomorphic to the projective line

= V

if only

Then its normalization

1 P (since g(F) = 0

in view

of the corollary to Lemma 1, Ch. 111, 51 and the remark above concluding 0

that h

($

+ E)

= 1

acts faithfully on

for any 7

E E

P = P($(F,o-( F

T)

.

%) )

Clearly the group T = (2/2)

and hence on the image Ci(F1

this shows that T is isomorphic to a subgroup of Aut(lP

is impossible.

3

1

)

, but

this

.

Thua we obtain that

is a complete intersection of four quadrics. Remark. A@)

=

Computing the Poincare function of the canonical ring

% HO(i)-'0-F( % m=O

-))

we see that it coincides with the Poincare w

fkction of its subring , g ; H

O-

(F,o$K$)

m

.

This shows that these

.

rings are isomorphic and V is the canonical node1 of

In

particular V has exactly 24 double ordinary points corresponding to the inverse images of the three (-2)-curves on F

:

- -

-

C1, C2 and C3

.

Also we get that the canonical model of F is the quotient of V by the group

(Z/2)

.

In this way it is easily to get the moduli space

of the classical Campedelli surfaces.

It is a unirational variety of

dimension 6 (look at the coefficients of the four equations of V above).

See the details in [32J.

Corollary.. Let F be a classical Campedelli surface. Then

In fact, the surface F F

obtained as the unradfied covering of 3

corresponding to the subgroup (Z/2Z) C TorstF)

is simply-connected

(because it is isomoprhic to a minimal resolution of double rational points of a complete intersection). b)

-GodGodeau~'-s~rfa~eg.These surfaces were constructed by

Godeaux as the quotients of suitable intersections of four quadrics

in lP6

by cyclic group of order 8 acting freely

( 1201)

.

Consider four quadrics given by the equations:

where a generator

of

6 = Z/&

quadrics by the formulas:

(xo'x1'~2tx3,x4.x5x6) 'where

6

= exp(2 i/8)

.

-

a c t s on t h e intersection

X

of.these

2 3 4 5 6 (xOIsx1,1; x2,c X 3 4 X4,5 X5'5 x6)

The same argument a s i n the case of c l a s s i c a l Godeaux surfaces ,shows t h a t the quotient

X/G

i s a numerical Campedelli surface with

Tors (Pic( X / G ) = nl (X/G) = Z/EZ

C)

.

-GodGode&u~-$&dds~r~a~e=. These are a l s o quotients of t h e

intersection of four quadrics by other groups of order 8 ( [ 3 9 1 ) .

First, consider t h e group 6

3e

- by

t h e formulas:

G = (Z/2Z)

.

Define the action of

G

on

It is clear that for any fixed point (i.e. a Mint with non-trivial isotropy subgroup) at least three of its coordinates must be zero.

This

shows that G acts freely on the surface given by the equations 2 2 2 pixi = P.x. = Ic.k2 - d.x. = 1 1 l i - 1 1 1

o ,

where all minors of maximal order of the matrix

are non-zero. Second, consider the group G = 2/22 @ X / 4 Z

.

Let gl = (1,O)

be its generators. Define the action of

g2 = (0.1)

the formulas

(

ni/2 = e )

G

on lP6

by

:

Now .notice that any fixed point is £ixed either under gl or 2

under g2

.

Thus, the set of the fixed point in P6

to the action of G is the set

This shows that the surface

X

given by the equations

with respect

,

2 2 2 aOxO+alx2+ax + a x x + a x x = O 2 5 3 1 3 4 4 6 2 2 2 bx. + b x +b2x5+bxx + b x x = O 0 0 1 2 3 1 3 4 4 6 2

COXl

+

C1X3

2

+

s0x21 + a1x32 +

2

C2X6

+ C3X0X5 +

2 d2x6

+ d3x0x5 + a4X:

9 x2 4 4

G i n v a r i a n t we may consider t h e quotient

X/G

=

.

i s e a s i l y can be chosen not passing through F

= 0

o

Since it i s obviously

, which

is a numerical

Campedelli s u r f a c e with n1 IX;/G) = Z / Z

@ 2/42

.

The l a s t example i s more i n t e r e s t i n g 1401 be t h e quaternion group.

sin&

g2 =

Consider t i s action on

-I f o r a l l g f. 1

, any

X

Let

Q

= (21, +i,+j, +k)

P6 by the formulas:

fixed p o i n t is fixed by

This shows t h a t t h e s e t of fixed w i n t s

Now, t h e surface

.

given by t h e equations:

-1

.

=o

c x x + c x x + c x x 0 0 2 1 3 6 2 4 5

d x2 + d x2 + d x2 + d ( x2 + x2 + x2+ x2) = O 0 0 1 1 2 2 3 3 4 5 6 is G-invariant and obviously can be chosen to be non-singular and not passing thro,ugh F

.

Taking the quotient V = X/G

we obtain

a numerical Campedelli surface with

sl(v) =

Q8

,

Tors(V) = 2/22

d) g u ~ f ~ c ~ s - w ~ zo~s-=-ZL7Z t&

.

$

.

2/22

It: is proven by Godeaux 1211

and Reid 1391 that if such surface F exists then the canonical model

F

of its covering corresponding to the torsion group is given by seven

.

cubical equaticms in the surface X c 'P

More precisely, it is shown by Reid that

given by the equations

-.

is a very good candidate to be such surface P

It is certainly

5 invariant with respect to the involution 6 of IP

where 5 = exp(2+/7)

.

Also, this involution acts freely on X

.

It has the same H i l b e r t polynomial a s

-F .

The only thing t h a t has

t o be proven i s t h a t - - - X i s non-singular and canonically embeded. e) -CamCamp=dgl~i~ob_r~-&u~i~o~ ~rf~c~sA The h i s t o r y here is

t h e same a s i n t h e case of s i m i l a r surfaces with

p(')=

2

. - Kulikov

proposed t o modify t h e c l a s s i c a l Campedelli surface replacing the branch curve new W

W

by another curve a l s o of t h e 10th order.

i s constructed a s the union

P are non-singular cubics,

C

and

More precisely, t h e

W = E U F U C LJ D

D

, where

.E and

a r e conics, which i n t e r s e c t each

o t h e r accordinc: t o t h e following picture:

E/

Oort gave t h e e x p l i c i t equations ( i n a f f i n e coordinates):

The same arguments a s i n the case of a l l o t h e r double planes considered above show t h a t the bicanonical system of the surface

equals the inverse image of the linear system of quartics passing through Pi with the same tangent direction as W

.

Also, in the

same manner it can be shown that the minimal nor?-singular model of the corresponding double plane is a numerical Campedelli suyface. The curves C

IJD

,

C

u 2~ ,

line given by the equation x

D U 2L'

+

1= 0

, where

L

(resp. x

- 3 = 0)

(resp. L') is the

the bicanonical Sivisors effectively divisible by 2

.

determine

Thus, they

define three torsion divisors of order 2, whose sum is, in fact, linearly equivalent to zero. This shows that

.

Tors (F) 3 ( 2 / 2 ~ ) ~ It is easy to see that there are no more torsion divisors ~f order 2

.

Applying Beauville's estimate of #Tors we get that

Unfortunately, I cannot see h w to exclude the second possibility. But it is conjectured that it can be done. Remark. We have two different constructions of surfaces with Tors = (2/2Z)3

, these are the

classicla Campedelli surfaces and the Godeaux-

Reid surfaces. It is easy to see (using the proposition from this

section) that the Godeaux-Reid surface is a deformation of the classical Campedelli surface (see the.details in 1361)

.

4. Budat's surfaces

These. surfaces were constructed in [7,8] as certain (2,2)-covers of the projective plane.

The linear genus p(l)

takes value '3, 4,

5, 6, and 7 for them. Later this construction was reproduced in a modern way by C. Peters I371

.

Here I give some other version of

*

this construction which allows to compute the torsion group.

First, we consider a minimal rational elliptic surface V + 3P1 with two exceptional fibres F = 2E + E + E + E3 + E4 and 0 0 1 2 n F' = 2E' + E' 9- E' + E' + Ei of type lo (see Ch. 11, 91). We 0 0 1 2 3 also suppose that there exist 4 sections S1, S2, S3, S4 nonintersecting each other with the properties:

To construct such a surface V one may consider the ruled surface P2, that is a IF1 -bundle over P ' for which curves 2s0

2 (so) = -2

+ R1 +

nonintersecting

, an

fi2

elliptic pencil on it generated by the

+ R3 + g4

so and

with a section so

and

2s

,

s being any section

Ri any four distinct fibres of F2

The minimal resolution of the base points of this pencil s provides the needed elliptic surface V Next, let F1

*

nRi

.

and F2 be any two distinct non-singular fibres

, consider the pencil P generatedby the divisors F1+2S3.+E3+E'3 F ~ + ~ s ~ + E ~ +.E :It is easily seen that P has 2 base points

of V and

.

See Epilogue

of multiplicity 2, namely, Q1 = Flfl F1

Q2 = F2

S4

I2 S3

.

Moreover,

(resp. F2)-touchesnon-singular curves of the pencil at Q1

.

(resp. Q ~ )

Let Dl and D2 be two curves of p

without common components.

Consider the following five possible cases (it will be shown later that all of them can be realized): A)

Di are both non-singular;

Dl = E + D; , where Di is non-singular, D2 as in A) ; 1 C) D~ as in B), D2 = E i + D; , where Di is non-singular; B)

+

D)

D = E 1 1

E)

Dl as in D),

E' 2

+

D;

D = E 2

The following ?roperties 2

(Di) = 4 D;

,

, where 2

Di is non-singular, D2 as in C) ;

t E'

1

+ Di , where

is non-singular.

are easily checked:

(Dis) = -(DiF) = -2

touches Di at Q1 and Q~

Di does not meet any of E

j

(Dl *El = 2

D;

, where

(F any fibre)

,

(D;

or E'

j

-

DI)

2

= 4

, ,

'

E denotes any other irreducible

component of D i' The Burniat surfaces will be constructed as minimal non-singular models of the double covering of V branched along the curve where in each of the cases A)-E) the curve W A).

W =

Bl

W =

is as follows:

W

,

The following pictures represent

in cases

(the thick curves denote the components of

A)

W )

and D)

:

.

To get a minimal non-singular model of this dozlble covering we

proceed as in the case of the classical Campedelli surfaces. Let

u Ri

: V' + V

be the birational morphia which blows up the curves

and R i at the points Qi (i-1.2)

, where we assrne that

Then the divisor

is 2-divisible and non-singular. Thus we may form a double covering

r

: X'

+ B'

model of X

branched along this divisor which will be a non-singular

.

where B = rX(px(E1 =

+

E;)

rx(px())

= 0 3E

= -r (pXf~i) )

,

in case

,

in case B)

,

,

in case

C)

,

, in ease

D)

,

A)

= -rx (px (E2 + El)) , in case E)

t

.

x Now notice that p (Ri) are exceptional curves of the 1st kind taken X x with multiplicity 2 The same is true also for r (p (Ei)) or

.

rX(px(E;))

K if r is branched along p (Ei) or px(q)

g : X' + X

be the blowing down these exceptional curves.

-D .= O~(~~($(D)) ) Then, we get

for any divisor D on V

,a

d also

.

Let

Put

i = 0 x frx(~i))

.

where

-E2

- E;

,

i n cases A ) , Bf , C)

,

i n case

D)

,

i n case

E)

,

,

Since

-F 'L 2E

0

%

2E;

2E0 +

'L

2E;)

2E + E1 1. 2E'0 + Ei 2Z0

+.EL

2z

+ z1 + E2 'L

0

and

'L

2E; + Ei + Ei 2 3 0

+ zi + E;

,in

case A)

,

,in

case B)

,

, in

case C)

,

, i n case D) , , in

case E)

,

This implies a)

1 52 = a((2Kx)

= 6

,

in case

A)

,

= 5

,

in case B)

,

= 4

, ,

in case C)

,

in case D)

,

= 2 , in case E)

.

= 3

is non-rational (since 2 5 is positive)

.

b)

X

C)

X is a minimal model (since for any exceptional curve of

the 1st kind C

< 0 and this implies that C is

(2K.f)

one of the curves

Po ,

"I

Fo

,

-Ei

- , but it is easily

or Ef

checked that neither of them is an exceptional curve 0-fthe 1st kind). It remains to

For simplicity

we

show that

will prove it only in the case

A)

.

In other cases

the proof is similar. Suppose that 1KX[ # $ we

get

This implies that

.

Then taking its inverse transform on X'

This means t h a t t h e r e e x i s t s a p o s i t i i r e d i v i s o r

which p a s s e s through t h e p o i n t s

Q1

and

.

Q2

Now n o t i c e t h a t

moreover,

D

2

= 0

, and

ID' 1

moving p a r t

( D I % ) = -2

ID!

of

.

If

we must have

dim

ID[

> 1 then f o r t h e

.

( D ~ >~ 0)

~hus

ID[

h a s some f i x e d p a r t which c l e a r l y c o n s i s t s o f com-ponents o f 4

Eo

+

El0

+

4

-1Ei + 1=2 .I E; 1=2

.

p e n c i l o f r a t i o n a l curves) n

o f t h e s e compaents t o i n t e r s e c t i o n index.

IE 1 + E i

(since

El

+ Ei +

[E

1

ID1

.

does n o t c o n t a i n t h e p o i n t s

Q1

and

+

Ei

is an urreducible

does n c ~ ti n c r e a s e t h e s e l f -

2S1

This shows t h a t

[E 1

2Sll

However, it can be seen t h a t adding any

e q u a l t o t h e moving p a r t o f

a r e no curves i n

+

+

2s11

+

IS*

1

+

2s.

/

1

is, i n f a c t ,

Thus, s i n c e the f i x e d p a r t o f Q2

,we

D

have t o show t h a t t h e r e

p a s s i n g ttxough

and

Q1

Q2

.

B u t t h i s is easy, because t h e only curve l i n e a r l y e q u i v a l e n t t o

El

+ E'.1+

2S1

passing thrugh

whibh does n o t p a s s through

Qi

Q1

i s t h e curve E3

+

E' 3

+

2S3

.

The o n l y t h i n g hanging on u s is t h e proof o f t h e e x i s t e n c e o f t h e cases

A)-E)

.

member o f t h e p e n c i l

Of course, f o r

P

use a representation of

A) it

is non-singular. V

is easy, s i n c e t h e g e n e r a l To c o n s t r u c t o t h e r c a s e s we

a s a double p l a n e which comes from t h e

inversion involution of t h e g e n e r a l e l l i p t i c f i b r e of V

by this i n v o l u t i o n we g e t t h e s u r f a c e

V

.

Dividing

Z o b t a i n e d from t h e q u a r d i c

p1 x JP1

by blowing up 8 p o i n t s , t h e f o u r o f them

a r e s i t u a t e d on a f i b r e Pi, Pa, Pi, P i

F

on a f i b r e

N1,

,

V

Pi

and

-t

N

, Pi

Pi Z

P2

-+

The

e q u a l s t h e union o f t h e proper

Z

o f t h e curves

These

P' i '

t h e r a t i o n a l mag -t

V

F

,

, and

F'

four f i b r e s passing

Ni

correspond t o t h e s e c t i o n s

i

correspond t o t h e curves

LO, LA

from t h e p o i n t s

Z

o f t h e same p r o j e c t i o n .

F' # F

of t h e second p r o j e c t i o n , each o f them

N2, Nj, N4

through

Z

Eo, E;)

, and

correspond t o t h e curves

Si

on

t h e l i n e s blown up

.

, Ei

Ei

Consider

which i s t h e composition o f t h e b.ldwing down

and t h e l i n e a r p r o j e c t i o n o f t h e q u a d r i c onto

p1 X lP1

,

o f t h e f i r s t p r o j e c t i o n , and o t h e r 4

branch l o c u s o f t h e p r o j e c t i o n i n v e r s e transforms' onto

PI, PI, PI, P4

c e n t e r a t some p o i n t l y i n g o u t s i d e t h e branch l o c u s o f

V

+

JP2

Z

with

+ ,pl %

I ? '

Then t h e image of t h e branch locus w i l l be e q u a l t o t h e union o f s i x l i n e s , two o f them passing through some p o i n t f o u r o f them passing .through o t h e r p o i n t The p e n c i l of e l l i p t i c curves on l i n e s through

ml

and

pencil

on

All A2, B1 coni;

V

and

curves

.

througil

m2

P

, the

Al

2

, say

bo, R;)

# A1

, say

nl,

F1 and F2 correspond t o some l i n e s B~ = m

Let

.

1

n

%I, B

fln1

and

Dl

"

2

= m2

n R4

To g e t t h e c a s e B) we j u s t t a k e f o r

as in

B)

, and

p e n c i l p a s s i n g through t h e p o i n t Dl

n2, n3, n

i s obtained from t h e p e n c i l of

from this p e n c i l p a s s i n g through t h e p o i n t

take for

, and

.

The

corresponds t o t h e p e n c i l o f c o n i c s p a s s i n g through B2

c a s e C) we t a k e

E0

V

A

Al

2;)

for

D2

I nl

.

go

n

nL

Dl ;

a

i n the

t a k e a c o n i c from t h i s To g e t t h e c a s e D) we

a conic from t h e p e n c i l passing through t h e p o i n t s

; ,n2

o f t h e l i n e s ) , and

D2

( t h a t can be done .only f o r some s p e c i a l choice as in C).

F i n a l l y , t o g e t t h e c a s e E) we t a k e

4'

for

Dl

the same conic a s in D ) , and f o r

through the points

0

ll.n2

$6 n nl

and

the conic passing

D2

(also take some special

.

choice of the l i n e s )

Now we w i l l compute the torsion of Burniat's surfaces.

Obviously,

we have the following torsion divisors of order 2:

-

Case A):

gl=Eo-

-E;

, g2

- - -S3 - -R2

g4 = S2 n

CaseC):

-

-

6

-

= Eo - F 1 -

4

-

glzF1+q-F2-R2

-

,

-

g 3 = E o - F2

-

- R2

- F1 - S3 -- ,g2=S2-S3-R2, A

h

g5 = D2

#

5

- -

g 3 G2 - S 4 - R p g 4 = $ + l - E ; -

-

-

-s3

- R 2 , g 2 = S 3 + R2 - S4 - RL , - g 3 = D i + r ( l + I ( 2 - S2 - Eo - . - - - - . gl = F1 + 3 - F2 - I(2 , g2 = S3 + R2 - Sq - R1 A

Case D):

-

g l = F1 + R 1 -

A

,

F2

A

v

A

Case E):

A

We w i l l show t h a t , i n f a c t , these d i v i s o r s generate the whole torsion group.

-.

Let

Tors(X)

.

2Tors(X)

denote the subgroup of elements of order 2 i n

Then Tors (X) = 2 T o r s ( ~ )

.

,

Proof. Let 6: X +X

be the involution of the second order induced by the

rational double projection of X onto V

. Then 6 induces an automorphism of

Tors(X) of order 2 6": Tors(X) +Tors (X)

.

For any g (Tors(X) the divisor g ~ X ( g ) is invariant with rkspect to 6 and hence being taken twicely comes from a torsion divisor on V. Since V is rational, we get that the latter is linearly equivalently to zero. Thus

@ Tors (X) # Replacing g by 2g we,\that

qTors(x)

implies the existence

s of a non-trivial torsion divrsor g such that g +d (g)%O.

Let D be an effective divisor from the linear system I s + g 1 9

where g as above

,

. Then

Using the computation of 2 5 on the page 90 we get that there exists a

curve

C

(

c I Fo + FA

+ 4S1

+ 2E1 + 2Ei + F I

E i s a linear combination of other Ei ,Ei

D~ +t?(og)

=pcrX(pxccrrr

.

)

sach that

Since pX(~) splits under the covering r -1 touch the branch curve W' = p (WI

+

RV1 + R;

:

.

X' -+ V'

, it must

Counting the inter-

section indices we easily find that px(~) touches the curves p-1 ( F ~ ) and p-l(~~)at one point P1 and Pa respectively. and touches the -1 -1 curves p (Dl) (or p (D;) and p-l(~1 (or p-l(~*)) at two 2 2 points Pj, Pi and P4, Pi respectively. Also, it does not touch the components Ei or E;

of

low notice that both W'

to the automorphism h of

V'

Wv

.

and FJ"(C) are invariant with respect induced by the inversion automorphism

of the elliptic pencil. This shows that the points P1 and P2 are fixed under h (and hence are situated on one of the sections 1 p (Sill , and the points P3 and Pi (resp. P4, Pi ) are conjugate

-

with respect to h

.

Using this we observe that any curve C'

which passes through P1 and P2 and touchcs pX(c)

%

pK(~f

at Pq and P4

will necessarily touch pX(~) at all 6 points Plr P2, Fj, Pi, P4, Pi X Since dim [p (C) = dim (251 = 6 we always can choose such C'

I

fi

Considering r (C').

.

we get the contradiction in view of the fol.lowing:

Let F be a non-singular projective surface with q(F) = 0

Suhlenuna.

Dl and D2 effective divisors such that D

I D1+D21

torsion divisor. Then for any D with bl

+ D2

Proof.

Assume the contrary, let D

there exists a point P C F

satisfy the assertion of the lemma. by the divisors D and Dl

+ D2

.

- D2

,

is a non-trivial

with no common component such that

ID^ + D21

(D. Dl)p f (D. D2fp

which does not

Consider the linear pencil generated Resolving its base points we get a

morphism f : F v + P1 of a surface F' onto p1

.

birationally equivalent to F

with a fibre containing two numerically equivalent components.

'Themain-lemma of Chapter 2, 81 shows that it is possible only in the case when the general fibre of f is disconnected. Moreover, in this case f has to factor through f'

: F ' h B

, where

B

is a non-rational

curve. This of course, contradicts the assumption q(F) = 9 Theorem.

Let X be a Burniat -face Tors(X) =

Proof. -

of linear genus p'l)

(z/z)'

.

. Then

(1)

We already know that Tors[X) = 2Tors(X)

and, even more,

that any torsim divisor class is invariant with respect to the involution induced by the projection r morphism

f :

)!

-+

t

X'

+

1'

.

Consider the

which is defined by the inverse imago of the.

Ipl

elliptic pencil on V'

.

We have the following multiple fibres of

this morphism: Case

A) :

Case B): Case Let Torsf(X)

C)

- -

2E0, ZE;), 2B1

-

2E1, 2g1

, D) , E) :

+

+

2 5 , 2P2

2 5 , 282

2e1

+ 2R1,

+ 2R2

2G2

be the subgroup of Tors(F)

of fibres of f

.

+

+

2R2 ;

;

2R2

.

generated by components

Using the main lemma from Chapter 2, .§1we see that

and can be generated by the first three (resp. two, resp. one) divisars gi indicated on page 94

.

Let

X

11

be t h e g e n e r a l f i b r e of

Pic(X)

-+

Pic (X )

where

Pic(XI1)

f

.

The r e s t r i c t i o n homomorphism

induces the imbedding

1

r denotes the subgroup of d i v i s o r s on

X

which a r e

i n v a r i a n t with respect t o t h e automorphism induced by t h e projection

r

:X

-+

11 11 covering Dl

n

n

r

and

D2

(or

The

.

Di)'

This shows t h a t each

four generates

.

Pic(V,,))

can be represented by a

D 6 P i c (X,,)

l i n e a r combination of the curves

. . * - - - -

Dl# D2. S1t S2. S3, Sq

Using the r e l a t i o n s on

2S1

-I,

2Sj rnodulo Ei, E;

Di

-I,

2s

we f i n d t h a t each d i v i s o r p c ( x 11

.

i s ramified along the two p o i n t s defined by the-curves

11

(or D i )

being t h e general e l l i p t i c f i b r e on V

V

V

5

2.1

(the l a t t e r

V

modulo Ei, E;

- -Sj' -Si - -j

defines an element of

S

,

Now we notice t h a t t h e covering

r

: X'

-+

V'

is defined by t h e

l i n e bundle corresponding t o t h e d i v i s o r I

p (3F+2S1) (see p. 8 8 ) .

- R i - Ri - 3R1 - 3R2

mod.

Eit E' j

This implies t h a t

+

i4 +

2g2 modulo components of f i b r e s of

There i s a l s o a r e l a t i o n between

zi

f

.

-

S 1

+ F2% F3 + %

modulo components of fibres of f

because Si defines the 4 points of order 2 on V Summariz+ng we get that 2Pic(X 9 divisors

-S3 - -S2 , -S4 - -S2

r

.

is generated by the three

C

and Dl

- s3

which as it is easily checked are independent. The arguments above show that any element of Tors(X)(Torsf(X) can be represer.ted by a sum of the above divisors plus a combination of components of fibres of f cases

A)-E)

.

It is easy to find in each of the

the corresponding torsion divisors.

In fact, we obtain

that these divisors are combinations of divisors gi (i=4, 5 , 6 in case A)

, i=3, 4, 5 in case B),

i=2, 3, 4 in case C), i=2, 3 in

Case D), i=2 in case E)) indicated on p. 73. This proves the theorem. Remark.

As we observed above the morphism f

fibres of multiplicity 2 in case

A).

Let B

covering of P1 by an elliptic curve B

+

:X

+

lP1

P1 has 4 multiple be the 2-sheeted

branched at the four points

corresponding to the multiple fibres. The normalization. X'

of the

surface X x B is a double covering of X non-ramified outside the a1 two points Q1 = Rl and Q = g2 n R;! Also, X being mapped 2

sln

onto B

.

-

has the infinite fundamental group, the points

lying over Q1 and Q2 are ordinary double points. the complement X

- (Q1, Q21

fundamental group, hence X

5I

'i;i2

This shows that

has a non-ramified covering with infinite itself has infinite fundamental group.

Another way to prove that the fundamental group of the Eurniat surface with p(')=of Chapter 111, 81

7. is infinite is based on the corollary to Lemma 1

.

Consider the surface XT

torsion group T of X

.

corresponding to the

Then we have

.

Consider the inverse image of the pencil P onto X The divisor 0 A - 2E1 belongs to this pencil and h ( ~ D ~ + R ~ +=R-2~ ) Now

.

2 ( q +fi2 + 2(F+ 2z1

h!:D 1 - K X )

A

I)

2(2D1

- 5 - 2z1 + E0 + E 0( + 2%

- - 2F- - 2z1 + E +E; - %R-1- 2R2 0

This shows that 2El

+

3 + x2

+

g

+

*) See Epilogue

+

2fi2)

b '

2z2)

2(E;

-To)

and hence

This, of course, implies that XT and thus X group. *I

2%

+

has infinite fundamental

0

.

55.

Surfaces with

.

p(l)= 9

Such surfaces were constructed by M. Kuga 1291 and A. Beauville 131. Kuga's - - -construction: -----Let group

H = { z E.

(C

:

Im(z) > 0}

IP G L(2,R) = SlL (2,m ) / k l

be t h e upper h a l f plane.

The Lie

i s i d e n t i f i e d i n a n a t u r a l way with

i t s group of a n a l y t i c automorphisms.

r

Let f r e e l y on

be a d i s c r e t e subgroup of I P G ~ ( 2 . 1 ~x ) P G L(2.B) a c t i n g with H x H \=act quotient V = 11 x H / r By hlatsushima-Shimura

.

1311 we have hl'O(v)

= h0.1 (V) = q(V) = 0 ;

hl.l (V) = 2p (V) 9

+

2

.

'ilierefore,

Next, n o t i c e t h a t

V

has no exceptional curves of the f i r s t kind

(and more generally, no r a t i o n a l curves), because t h e p r o j e c t i o n

H x H

-t

H x Hn =V

s p l i t s over such curve, b u t

H x H

does not

contain any complete curves. Thus, t o f i n d t h e needed surfaces with

it s u f f i c e s t o choose such

r

that

p (V) = 0

9

and

p(l) =. 9

By the Gauss-Bonnet formula c, (V) =

1: vol (V)

,.

47r2

where the volume vol(V)

is computed by integration of the invariant

volume element

( (zl,z2)

= (xl+iyl, x 2+iy2) being the coordinates on H x H)

.

Now, let k = Q(&)

be a real quadratic field, d

A = A(k,e)

the discriminant;

be the division quatercion algebra with the center

k

and with the discriminant 6 = p1p2...p2r

to he totally indefinite (that is,

N

:A + k

-0

be the reduced norm of

be the maximal order of

A

k

~ ( 2 )be the group of all units of

-

a discrete subgroup of V

H

X

R/r

I

.

M2(R) )

(D

A ;

equals 11 ;

2

;

t

and the A + A e, JR = M2 (IN@' M(P) ~

Q

j . : 6L2(P)

Q

: Nfg) = 1')

Consider the natural injection i projection

A B W = K2(R)

(unique up to conjugation if

the class number of

Y = Is r E(CJ

assumed

Xc

GL2 ( P )

+

PGL(2,lR)

X

pGL(2,lR)

.

Let

r

(T)

= j (i

.

be

PGL(2,lR) x pG~(2,lRj ..with compact quotient

we note that

'l

is isomorphic to the image of

According to. Simizu ( 142)) the volume vol(H x H/r) through the zeta function Ck(s) of k by the formula:

-r

into ~

can be expressed

~

.

/

k

( lpl

denotes t h e norm o f prime i d e a l

where

5(s)

p

of

is t h e Riemann z e t a f u n c t i o n and

k)

.

L(s,x)

D i r i c h l e t L-function a s s o c i a t e d w i t h t h e c h a r a c t e r . X n

;if

if

X(n) =

(n2-1) /8

,i f

The v a l u e of the Riemann z e t a a t 2 e q u a l s

71'16

d = 8m1, m

.

The v a l u e

a-1

~ h u s ,we have

, the

d

d . = 4m, m E 3 mod

where C x ( n ) ehrin/d n=l

mod

d l 1 mod 4

a t 2 equals

T(X) =

i s the

Gauss' sum

4

1 mod 4

L(s,X)

has absolute value 11 (X) I = d'I2

Since the Gauss* sum -c (XI C2 =

-

v01

is positive

, we

and

get

4,

Next, we have to be assured that the group

acts freely on HxH, and hence

is smooth. Since the stabilizator group of any point is a finite

HxH/r

subgroup of

r , that can be

if and only if

r

has no elements of finite

order. Let g E r

.

be an element of order N,

in 7 We have g aigebra

A

=,

1

, and

thus

96T g

2N

some of

=I

. Then the quaternion

has to contain a subfield.isomorphic to the field

Conversely! if the class number h (k) = 1 , then that

its preinages

A>

(l(e2 dlN)

implies

has an element of order N.

Since the maximal subfield of

A

has degree 2 over k

, we- have

Thus the only possible orders for N are

Obviously, an element of order 2 in

NOW, if

7 defines the unit element of

r

cb (N) = 2 (N=3,4,6) then the maximal subfield K of A coincides

with k fe2*i/N), if &(~)=4 quadratic subfield of Q(e

then K = 9(eZTim)

2Ti/N

1

and k is the real

-

Let K be a quadratic extension field of k; then the local arguments show that K is enbedable into A = A(,k,8) if and only if pig

does

not decompose in K. Now weare ready to give an explicit example. Example. k = Q (J2), d = 8, 8 = p2p5

, where

p2 and p5

lie over 2 and 5

accordingly. We compute

we observe ,thatthe only cyclotomic

To check the smoothneSs of HxHf

field containig k is Q(e ""1,

and in this case p2 and p5 do not

decompose. Thus it suffices to consider the cases

3 =

3,4,and 6

the second case K = g( d ,i), and in the first and the third, K

. In

=QV 2 J

.

-3)

In -theboth cases we easily verify that p2 and p5 do not decompose. Notice that other examples can be also ohtained by taking instead of some other

'

discrete subgroups in

2 , for example,

= {gc? E (2) : N(g) is a totally positive unit of k}

.

We refer to [29] for the examples of the corresponding surfaces H To compute the torsion group Tors(Hxk/r)

For any maximal two-sided ideal pO_ in in g/p (= M (IF ) or IF 2 q q

2

e

.

we note that

we may consider the image $ ( f )

, q=No~/~(p),depending on whether pte

or pie).

Moreover, by the Eichler approximation theorem we have

where U = { a E P

(a) = 1) is a cyclic group of order qil

: NF

9

.

q2 q

This innnediately shows that it is always Tors(V) =

-r

/(tl)[J;,E $ 1 .

The more detailed analysis gives the following result:, Thcorem([291).

There exists a subgroup I of

containing fr,rl such that

where qi = NkiQ(Pi)

I

,0

if (2) = P~P;, p2 # P; and pZ)8

2

a = 1

if p2(2

0

is true for

(p2 = 2

, P2)8

but other divisor of 2 dividesee

if (3) = P~P;r p3 # P; and ~ ~ ,$P$+O 8

1 if (3)

M

I

,

,P;~O

otherwise

0.

2

Moreover,

= P1. .Pr ;

2

p3

, p310

or p3 13 and other divisor of 3 divides 8

otherwise 5

[xq

if the congruence subgroup conjecture of Bass-Serre

. Also, - I t M

if and only if one of q i g I mod 4

.

In the above example we have

T/M= Z / 3

@ 2/6

Beauville' - - - - -s -examples ---

, where

V = CxD/G

.

( 131 ). .These

surfaces are constructed as the quotients

C and D are complete non-singular algebraic curves of

genus g at least 2, G is a finite group acting freely on the product. To construct the quotient with the needed properties Beauville proposes

to take for G a finite group of order (g(C)-1) (g(D)-1)

acting on the both

C and D with the rational quotients. In order to get a free action on CxD he puts

where

is tin automorphism of G

u

non-freely on

U(g)

C

such that for all

acts freely on D

In virtue of the lemma of chap.1~52

gtG

acting

.

we have

Moreover, V does not contain any rational curves, since the projection CxD

*V

m.

has to split over such curve and there are no rational curves on

his

implies that V is a minimal model.

It remains to prove that the irregularity q(V) = 0

but, since C/G

and D/U(G)

Example 1. C = D

. We have

are rational curves, the both summands are zeros.

is the plane curve with the equation:

x5+y5+z5=0, G = (z/5)

acts on C by the formulas:

-

cr

(p,q) (x,Y,~) = (SPx,5%,z) , 5 e2ri/5 is the automorphism of G given by (1,Ol-c (1,1),(0,1)+

The set of elements of G which act freely G =

ill'"

A\IU(A)

F x a m p l ~2. C =

I)

is the curve of genus 4 given by the equation in lp3:

acts m C

+

t3 = 0

,

xy

+ zt =

automorphism given by (1,O) +(1,1),

The set of elements of G

and G = {l)"

0

.

by the formulas:

(p,d (x,y tztt) = (S~X,S-~Y ,Sqz,#-gt)

a is the

-1 (p,q) ,pfq) and

.

x3 K Y 3+ z3 G = (2/3l2

is A

(1,2).

1

5=

e

,

2ri/3

(0,ll-t. (lt2)

.

acting freeiy on C is the set A =C(p,q) ,p:q#0)

A V U(A)

Applying the well known Hochshild-Serre exact sequence:

ue see that Tors(CxD/G) 3 G/[G,G]

.

In particular, in the above examples the torsion group is non-trivial.

.

6. Concluding renarks

It would be very optimistic to expect the complete classification of all surfaces of general type with p =O. g

ow ever,

there are still many problems

to answer in the visible future. One of the most interesting from my point of view is the following: Problem 1. Is there a simply connected surface of general type with p =O? 9

Or more weak Problem 1'. Is there a surface of general type with p =O and trivial 4 torsion group? !Consider the class of all surfaces of general type with p =O and fixed 9 P2- p(l). Then there exists s number N such that the k~canonicksystem defines a birational morphism for all such surfaces([4]).

Thus the set

,ofits N-canonicla models can be parametrized by an open subset of the tIilbert scheme corresponding to some Hilbert polynomial. since the latter is of finite type, this open subset consists of finite number~of connected components. The surfaces parametrized by a connected Wriety are diffeomorphic, and,in particular,have the same fundamental group. This argument shows that there are only finite number of possibilities for the fundamental group of a surface. In particular, the order of the

.

torsion group is bounded by a constant depending only on p (1)

Problem 2. Find a bound for the ordeir of the torsion group of surfaces with the fixed p")

(as always of general type and with p =O) 9

.

We remind that it is done in the cases of numerical Godeaux and Carnpedelli surfaces. Consider the class of all surfaces with the fixed value

the torsion group T. Denote it by Problem 3. Can

M(~,T)

F4 (a,T).

be parametrized by a comected variety? In

particular, are the elements of !d(a,~)

diffeomorphic to each other?

For the start it would be very interesting to knrinr the answer at least

in the cases M(2,Z/2), M(2,Z/3) and M(3,2/262/2)

,Recall that in the

last case we know two (and possibly even three) dizferent constructions of surfaces from this class. In some cases the znswer is positive (e-g. fA(2,&/4), M(~,z/:/!,), fd(3,abelian of order 8 ) ) . We still do not know if all possible values of p('' Problem 4. Are there surfaces with p ( ' ) =

are realized

)

.

8 and 10 ?

There is much hope to solve the following Problem

5.

Find all possible

torsion groups of numerical Godeaux and

Campedelli,surfaces.* 1 The validity of the following assertion is observed in all known examples : Problem 6. Prove that the fundamental group is Snfinite in the case 27

and finite otherwise.

BIBLIOGRAPHY

111 Algebraic surfaces, ed.I.R.Shafarevich. Proc.Steklov Math.Inst.,vol.75( 1965(Engl.transl., AMS,1967)

.

I21 Artin M., On Enriques' surface. Harvard thesis,l960. I31 Beauville A., A letter to the author of September 25,1977. 141 Bombieri E., Canonical models of surfaces of general type,Publ,Math.I.H,E.S., 42(1973),171-219. 151 161

, Catanese F., , Mumford

A non-published manuscript.

D., Enriques' classification of surfaces in char.p.11, "Complex ~nalysis'and Alg.Geometry",Cambridge Univ.Press,l977,p. 23-42.

[71 Burniat P., Sur les surfaces de genre P12W

, Ann.Math.Pura

et Appl.(4),

71(1966),1-24. 101

, Surfaces algdbriques r&uli&res

dc genre g&cnktrique

p =0,1,2,3 et de genre lineairc? 3,4,.. .,8p c7. Colloque de g

9

dorn.~l~&br.du C.B.R.M. ,Bruxelles,1959,129-146, [91 Campedelli L.,Sopra alcuni piani doppi notevoli con curva di diramazione del decimo ordine. Atti Acad.Naz.LinceiflS(,I932),358-362. [10]Castelnuovo G., Sulle superficie di genere zero. Memorie Soc.Ital.Scienze, (3) ,I0 (1896),103-l.23. 2 [IlICatanese F., Pluricanonical mappings of surfaces with K =1,2 and q=p =O, g

these proceedings, pp. [12lCohomoloqie etale

(SGA 4%

2

, ed.P.Deligne,

Lett-Notes Math., :vo1,569,1977

.

(131 Dolgachev I., On rational surfaces with a pencil of elliptic curves. 1zv.Akad.Nauk SSSR(ser.math.),30(1966),1073-1100(in

, On

[14!

russian).

Severi's conjecture on simply connected algebraic

surfaces. Dokl.Akad,Nauk SSSR,170(1966),249-252(Sov.Math.

.

Doklady,7(1966) ,1169-1172) I151

, Weighted projective varieties (?reprint).

[16] Enriques F., Le superficie algebriche. Bologna.1949. [I73

,

Un' osservazione relativa alle superficie di bigenere uno. Rendiconti Acad.Sci.Bologna,

[la] Godeaux L.,

Sur une surface al&brique

1908,40-45.

de genere zero et de bigenre

deux. Atti Acad.Naz.Lincei,l4(1931),479-481. [I91

,

Lea surfaces alg&riques

non rationnelles de genres

arithmetique et g&om&rique

nuls (Actualites scient. ,NO 1230,

Paris,1933. 1201

,

Sur la construction de surfaces non rationnelles de genres zero. Bull.Acad.Royale de Belgique,45 (19491,668-693.

[21]

Les surfaces al&briques

de genres nuls a courbes bicanoniques

irreductibles. Rendiconti Circolo Mat.Palermo,7(1958),309-322. 1221

Recherches sur les surfaces non rationnelles de genres arithmetiques et g&om&riques nuls.Journ.Math.Pures

et Appl.,

4 $ (1965),25-4l. [231

, Surfaces non

rationnelles de genres zero, Bull.Inst.Mat.Acad.

Bulgare,12 (1970),45-58. [24] Grothendieck A.,

Le groupe de Brauer, I1 ,111, "Dix exposes sur la

cohomologie des sch&nasn,Amsterdam-Paris,North-Holland, 1968,pp.66-188.

I251 Iithaka S

., Deformations of compact complex surfaces, 111.

;[ourn .~ath.

Soc.Japan,23(1971).692-705.

I261 Kodaira K., On compact analytic surfaces,II. Ann.Math.,77(1963),563-626.

, On

1271

the structure of compact complex analytic surfaces,I.

Amer.Journ.Math.,86(1964),751-798.

, On

1281

homotopy Krsurfaces, nEssays on Topology and related

topicsn,Springer,1970,pp.58-69.

1291 Kuga M., preprint. 1301 Magnus W., Noneuclidean tesselations.Acad.Press.1974. 1311 Matsushirna Y.,Shimura G., On the cohomoloyy groups attached to certain vector valued differential forms on the product of the upper half planes.Ann.Math., 78(1963') ,417-449. I321 Miyaoke Y., l'ricanonical maps of numerical Godeaux surfaces. Inv.Math., 34 (1976),99-111.

1321 '

, Gn

numerical Campedelli surfaces."Complex analysis and

Algebraic geometrym,Cambridge Univ.Press,l977,pp.112-118. 1331 Mumford DL, Pathologies ,111. Amer.Journ.Math.,89(1967),94-104. 1341

-.,

Cohomology of abelian varieties over function fields. Ann.Math.,76(1962),185-212.

t351 Oort F., A letter to c.Peters of February 1976. t361 Peters C., On two types of surfaces of general type with vanishing geometric genus. Inv.?bth.,32(1976),33-47. t371

, On

some Burniat's surfaces with p =O., g

preprint

t381 Raynaud M.,

Caracteristique de ~uler-Poinqardd-un faisceaux constructible et cohomologie des vari&es

abgliennes.

"Dix exposes s3r la cohomologie des sch&mas",~orth-~olland, 1968.pp.12-31. 1391 Reid M.,

Some new surfaces with p =0, preprint.

f401

A letter to D.Mumford of December 22,1975.

Zi

f41J Shafarevich I., Principal homogeneous spaces fiefined over a function field. Trudy Steklov Math.:Enst.,64(1961),316-346 &ransl.AM~,vol. 37,85-115)

.

[42] Shimizu H., On discontir.uous groups operdting on the product of the

upper half planeS.Ann.Math.,77(1963),33-71. 1431 Zaxisky O., Zrlgebraic surfaces. Springer, 1974.

EPILOGUE

After this work has been almost done the author was informed in many new results. 1. Numerical Godeaux surfaces with Tors = E/3

have been constructed by

Miles Reid [45]. The construction is very delicate. 2. The final version of Peters' preprint 1371 has been published [44].It

can be found there the result about the torsion of Burniat's surfaces(the proof is not complete). Also it is proven'there that the fundamental group is ipfinite in case p(1)=7. Phis result is also refered to M.Reid. 3+ F.Oort and C.Peters also have proven that the torsion of Campedelli-

-0ort-Xulikov surfaces with p("-2

is equal to 2 / 2 (1511).

and also calculated the 4.~.'Inoue has constructed surfaces with p(1) 4 fundamental group for Burniat's surfaces ( [ 4 6 ] ) . 5. ~ . ~ e ihas d cmiputed the canonical ring of numerical Godeaux surfaces

with Tors=Z/2

( I461)

.

6. %.Reid has proven that # Torss9 for numerical Campedelli surfaces. He conjecixres that 9 can be replaced by 8 and the surfaces with thetqrsion group of-order 8 are the Godeaix-Reid surfaces .Another conjecture:

# Tors < 30 for surfaces with p(11=4 (1471). 7.-Using the nonarchimedean unifomization theory D.Mumford has construced a surface with p(l)=10

( I481 )

.

8.-Wng people have discovered independently a surface with Tors = Z / 5 and p(')=3

([46]).

As it was explained to me by Fabrizio Catanese it can be

constructed in the foloowing way. Let F be a quintic surface in P3 which is invariant under an involution of order 5 and posseses 20 ordinary double points.Also assume that there exisad a quartic surface 3 tangent to F along a curve C which passes through these double points and smooth at them. The

existence of such surfaces F and B is proven in Souble points to the surface

F , then

Blow up F at these 20

the sum of the twenty exceptional

is linearly equivalent to the strict inverse transform

-2-curves on

of C taken twicely. Let curves,

! be

the double covering of

branched at those

V the blowing down of the strict trnasfoms of the branch

locus. Then it can be easily shown that

52 =

10, pg(W) = 4

. The Z/5-action

on F extends to a free action on V and the quotient defines the needed surface X.

By Reid's result (see

the surface

? can be realized as

6.) we get

Tors(X) = Z/5

. Moreover,

a noh-singular compactification of

a quotient o f the upper half planes by a discrete gronp of Hilbert's type

([SO]), this implies that

.

group of X is 2 / 5

9. C.Peters

t is simply connected,

and hence the fundamental

conjectures that-forany double plane of gene~altype kith

pq=o the torsion group consists of elements of order 2 (1443 '1

.

Supplementary bibliography.

1441 Peters C.

, On

certain examples of surfaces with p =O due to Burniat. 4

Nagoya Math.Journ.,66(1977).209-1191441 '

-

t451 Reid M.,

,A

letter to the author of March 9, 19?8. 2

Surfaces with p =O,K =1.Journ.Toky~~th.Facultyfto appear). g

1461 Reid M.; A letter to the author of January 18, 1978.

, I481 Mumford D.,

A letter to thz author of April 24,1978.

An algebraic surface with

I(

2

ample,(K )=9,pg=q=0(preprint).

6491 Gallarati D., Ricerche sul contatto di superficie algebricie

cilrve.Nemoires des Acsd.RoyaPe de Belgique.,t.32,P.3, 150! Vanderge G . ,Zaqier D., Hilbert modular group for field Q(/J-3)

.

Inv.Math. 42(1977),93-131. [%I] Dart F.,Peters C.,

- A ~ ~ m p e d e l surface.with li torsiongroup Z i 2 .

.

(preprint)

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (c.I.M.E.)

THE THEORY OF INVARIANTS AND I T S APPLICATIONS TO SOME PROBLEMS I N THE ALGEBRAIC GEOMETRY

THE THEORY OF IIWARIAETS AND ITS APPLICATIONS

TO SOTIE PROBLEXS IB THE ALGEBRAIC ,IEOiC~TRY F. A . Bogomolov

Steklov I n s t i t u t e

1. I n t r o d u c t i o n t o i n v a r i a n t theory

F i r s t I viant t o r e c d l t h e c;assical

c o n s t r u c t i o n of

ci

locally

t r i v i a l f i b r e bundle. In t h e simplest case, t h e f i b r e bundle

"5 F and base X may be constructed a s an a s s o c i a t e d bundle t o any p r i n c i p a l f i b r a t i o n X X with a group G as a f i b r e : with f i b r e t h e group a c t s on

G

a c t s on t h e f i b r e

F on t h e r i g h t

,

a ~ dF x = X

F

and on

X

x T G

F

, where

on t'ne l e f t . A simple, but

i m p o r t a t , remark: t h i s const;ruction i s twice f m c t o r i z l f u n c t o r i a l on t:e

h :F

f o r any f i b r e bundles h y :F3

4

goh

- it

is

base aqd f u n c t o r i a l on t h e f i b r e , i n t h e sense

t h a t t o a n y morphism f o g =

G

-

of G spaces (such t h a t g r G ) vie have the morphism of corresponding -4

Q

. The p a r m e t e r ' T h e r e i s a cocycle HI (X, rG) where rG i s some subbundle o f t h e sheaf of func t i o n s on X with v a l u e s i n . These f a c t s a r e proved d i r e c t l y Qa

$P

G

from t h e diagram:

i s well defined because

h

( 4 x h)

0

g = g o (1 x h )

and

h

i s the

o r b i t space. Renark. If F there is

2

i s a norphism of

Q

G - i n m i x a t subspaces, then ,

.

~ b b u i l d l e,FT+ QT

Row suppose t h a t ea,ch o b j e c t i n t h e previpus construction h a s come a d d i t i o n a l s t r u c t u r e and t h a t thews strv.ctures are cornpr-tiblc with each o t h e r . Thel; the constructed morphism t i b l e 7 d t h t h e s t r u c t u r e on t h e spaces F x and. For example i f

Xg

, X,.F,

Q, G

an a l g e b r a i c group a c t i n g on X x h : F 4,Q

and

Qx

.

are algebraic v a r i e t i e s ,

, and

a r e a l g e b r a i c , then

r e of a l g e b r a i c v a r i e t i e s and

w i l l be compa-

k

the projections

F T and

h 8 : FT

*Qr

Q

p :X

G is

pX

have t h e stmctu_

i s a n a l g e b r a i c mor-

phism. Eow, we can u s e t h i s ~ o n s t ~ r r c t i oinn t h e foll.owing siixation: suppose t h a t

G

i s a l i n e a r a l g e b r a i c group, P, Q a r e vecto:r

spaces with a l i n e a r a c t i o n of group G and

X

get

h , : F1

, both

of f i n i t e dimension,

i s a compact a l g e b r a i c v a r i e t y ( o r perharps a complex m a n & f o l d ) . Then T E H 1 (x, 6 (5)) and f o r every G-morphi'sm F -+ Q we

p

4

Q I. and we can transposk t h e s e c t i o n from

, u s i n g t h e morphism

0

h*. : H (x,F) I

-9

HO

(x, Q)

to

P

. This morphism

i s non-linear i f .h i s non-linear and u s u a l l y we cannot d e f i n e my analogous morphism f o r t h e higher cohomology group (x, F )

.

E~ a s s o c i a t e d t o a cocycle is E E ' (X, 6 iGL(k) ) ) Then any t e n s o r proiiuct (Ek ) @ m of Example. Consider t h e v e c t o r bundle

'd

.

cor?stm.cted by mean of t h e same cocycle

T

@

m: EL(^)

--+ ~ ~ ( k m.)

and repre:,eotaticn

-

O f course t h e same i s t r u e f o r any o t h e r a l g e b r a i c representa-

t i o n of

GL(k), Ve have a h a t u r a l morphism, c a l l e d t h e ~ e r o n e s e k , defined by y "m morphism, k.: Ex -) ) :E( It commutes with t h e a c t i o n o f G L ( ~ ) on both spaces and we g e t t h e Veronese morphism

.

.

yk - 4 E k e p r n .

Eon our problem i s a follovrs. Suppose t h a t we c m c o n s t r u c t a

s e c t i o n of

E

om w h e ~ e in i s Zarge enoqgh. Fhat

c m be d e b c e d

t h e p r o p e r t i e s of E ? For this we have t o f i n d a l o t of morphisms i'mm : E

into

!'a simpler complex G-spacef1. These simple G-spaces v,ill be c a l l e d modeis and we s h a l l look a t i n v a r i a n t theory as a theory of morphisms i n t o models. The cl-assical i n v a r i a n t theory, from such point of view, i s a theory of rnorphisms i n t o one model, the trivi'al. Let u s take the group

G reductive f o r b e g i n n i ~ g . Then we d e f i , ne the t h r e e kinds of models: I ) t h e t r i v i a l G-modul

C

,

1 1 ) t h e homogeneou$ spaces G/H where of t h e group .G ,

H

i s a reductive subg~oup

111) t h e HV-mmif 016s. Let u s define an

W-manifold.

presentation of a group maximal tor&" if

T

X = w 2' f o r

ma any

G

I r e c a l l t h a t an i r r e d u c i b l e re-

i s defined by a. character

k c h two r e p r e s e n t a t i o n s C x wcV(G)

where

Then t h e r e i s a highest v e c t o r The closure of the o r b i t

Gx

W(G)

X (T)

of a

, C 7,coincide

. i s t h e Yl'ejrl group of G.

x y e T - such t h a t

T x

r=

X(T)X

,

i s c a l l e d an HV-manifold, and w i l l

be denoted A X .

We w i l l r e f e r t o the s e t of k k: as the s e t of the models 'of t h i r d tyye. Tbe A Y could be described -in another way. Take m y parabolic subgroup P c+ v e c t o r bundle racter

E

G

axid the homogeneous one diniensional

on t h e space

G/P

. constructed

by mean of a ch,

2

(P). Now suppose t h a t we can blovi-down t h e zero-section of E t o a point, i , e . E i s negative. The manifold t h a t we g e t a f t e r this blow-down i s an a f f i n e HV-manifold and a l l HV-manifold can be obtained in t h i s way. Recall t h e -in' r e s u l t s of t h e i n v a r i a n t theory (i.e. t h e theory of morphisms i n t o the f i r s t model, i n our d e f i n i t i o n ) . A 1 1 p o i n t s in affine spaces divide i n t h r e e types: a) S t a b l e points: they have t h e m a x i m a l dimensional closed o r b i t

natural projection f o l d of

*

that i f

phism

H

X

Ei -unstable p o i n t s i n

Nx

X

t h e submanifold

'G/H

p : G @ B -+ G/H

Looking a t t h e o r b i t s i n

f(x) = f(y) t h e n p : P! + G/H

.

G/H

. But

i s the subnmi-

YJx

f(Y5)

c o i n c i d e s with

i s e f i n i t e roorphism f r o n

m-d, f

WGiI3

, tvhere

\Itx

a.nd

p(x) = pjy)

to

\

we can conclude

\7G,,H,

and so

p

incl-~~coa a mog

Remark. T h i s i s a hol omorphic morphism a.n& i t 'can . b e p r o d & t h a t i t i s r e g u l a r , b u t we s h a l l n o t need such d e t a i l s f o r o u r purposes.

2. U n s t a b i l i t y of bundt e, s u f f i c i e n t c c n d i t i o n s .

We need t h e f o l l o w i n g thsorem f o r o u r prograr! of c o n s t r u c t i n g G-morphisms i n t o t h e model spaces.

.

Theohm 3 (~ogomolov) F o r anx lyinz

G-invarirdlt e f f i m v a r i e t y

t h e manifold of u n s t a b l e p o u

t r i v i a l S-morphism

x -f o r

f :X 4 A

NSG

some model

X

t h e r e e x i s t s a non 12

X'

A X a r e thr? only a f f i n e connected G-va-

Remark, The v z r i e t i e s

r i e t i e s which c o n s t i s t of two o r b i t s one o f which i s a p o i n t . The proof of t h i s p r o p o s i t i o n r e q u i r e s some n e w ~ p r o p o s i t i o n s . t h e v a r i e $ y of u n s t a b l e ESG In view of t h e Tieoren 1 BSG = GKST f o r

R e c a l l t h a t we w i l l denote by p o i n t s i n G-module

61N

.

-the f i x e d maximal t o r u s conjugate. The space

TC G

@*

,

because t h e maxi~nal tori,

a s a T-space c m be decomposed i n t o a

d i r e o t sum of so c a l l e d % e i g h t subspacesrt where

T

a c t s on O:

X

@? a r e non zero,

X v c cN

by means o f

7c Definitiar,. For any v e c t o r

f i n i t e s e t of

char T

are all

$

=

1 char T %

we denote by

Supp v

such t h a t t h e p r o j e c t i o n s o f

v

the into

224

By (supp v> we denojte the convex envelope i n t h e l i n e a r - r e a l

X(T)z @ IR' of

space

t * ~ es e t

Supp v. It i s a polytope.

Now we d e s c r i b e t h e y a r i e t y Lemma 1. Any v e c t o r

s

vector

with s e

ITSG

.

v e f3SG i s eouivalent uncler

(~upps

t o some

G

>'.

Proof. This follows a t once from Theorem 1 2nd from t h e f a c t that

.

GNS4 = NSG

A

So f o r each

1c

X (T)

we can take t h e half-space

X'T) Q B and d e f i n e t h e subspace z

i3

space which c o n t a i n s a1.l v e c t o r s ce (1, x)

Then

? 0:

is finite

ITSG =

$4view

v

Supp v

such t h a t

u G I l . The number I

(1,x) > 0

as t h e v e c t o r ,sub-

VIC F'

half-spa-

of d i f f e r c n t spzce VL

N

of t h e f i n i t e n e s s of .Supp C

.

Lemma 2. Suppose t h e t t h e r e i s a Linear P-ariant

zubspce

P i s a p a r a b o l i c s u b m o u n of G. Then t h e spec

where

V C gN

i s closed i n

Q!N

GV

.

Proof. I n f a c t

i s t h e image of t h e bvndle

G @ V with P f i b r e V on t h e compact manif o l d G/P The morphim h : G ~3V 4 GV I? 13 proper and h-' ( x ) i s compact f o r m y x ; t h u s GV i s closed. GV

.

< ,> on

For t h a t l e t u s t a k e a s c a l a r product

X(T)

p o s i t i v e l y d e f i n i t e and i n v a r i e n t under t h e a c t i o n X ( T ) (23 ll? 7~

. A s u s u a l i f t h e Lie a l g e b r a of

.

GV1

Now we d e f i n e some submanifold which s l i g h t l y g e n e r a l i z e

2 IR

which i s

Yi(G)

on

i s simple then i t

G

i s uniquely defined up t o m u l t i p l i c a t i o n by a s c a l a r . For each

E

2 (4) & B

(x-e(;

Q

o()?

Define

0

Q

.

4,

d e f i n e t h e subset

as t h e s e t

d e r only those v a l u e s

(x- 4 , 4 ) = 0

oC f o r which

as t h e space of v e c t o r s

ne and

V

'd 01

C Supp

Ad

spect t o the t o m s

T

r

by

x E Sup*

x

iff

6

cN

. Collsi-

9 A d i-s non empty. Now d e f i

v

such t h a t

a s t h e space of t h e v e c t o r s

Lemma 3. Consider t h e r o a

,

8

v

Supp v C

such t h a t

of t h e Lie a l g e b r a

. The s p a c e V A,

Ad

,

mpp v c 9 A 4 .

9vrLth r e -

i s i n v z r i a n t under-

p z . r ~ ~ b o 1 isubgroup. c

Pd vhoae t h e L i e a l g e b r a i s generated by t h e e i ~ e n v e c t o r sof r o o t s r such t h a t ( r , d )+O We denote them by

.

the sam l e t t e r . The proof i s a d i r e c t consequence of t h e f o r n u l a f o r a c t i o n of N t h e g e n e r a t o r s of a L i e group on C

.

So by Lemma 2,

GVd

j

.

s a closed space f o r a l l oC

Remark. There a r e only a f i n i t e number of d i f f e r e n t submanifold in

GV,

cC

.

N

The group

P

c o n t a i n s a non t r i v i a l u n i p o t e n t r a d i c a l

t h i s group i s generated by

and H

( 4 ,ri)

>. 0

. In t h e

exp ( t r i ) , where

which i s generated b y r o o t s

t0rrt.s

P

sane way

.

ri

r

i

U(P);

i s a r o o t of

7

contains a reductive m b g ~ c u p

with

,ri) = 0

(

snd

the

' I

The group

P

can be represented as an e x t e n s i o n O+U(P)+P+M-+O

u cs. e.--<,.- -'

F o r any m

such t h a t

G

m

o(

€ 2( T ) ,

. m

i s n c h a r a c t e r of

d

. Define Ho = k e r d an& consicler t h e r i i l g s S of t h e Eo-invariant r e g u l a r f u n c t i o n s on V a nd . V,A, i s an H-invariant subspace of ' C . Thus S h a s a d i r e c t decomposition i n t o homogeneous subspaces S-' . The group H' a c t s on sm by mean t h e r e d u c t i v e group

H

of

md(H). Let ua remark t h a t i n t h e above decomposition we have m m 7 0 because f o r any v e S (V, A, ), supp v l i e s i n t h e s e t we have n (Supp ,V bd ) and f o r any p o i n t i n x e Supp V

.

(x,d ) = ( g ,d).O

9 Aa

Now we can c o n s t r u c t a G-norphism from t h e f i b r e bundle G @ Pr

fact

V

a+

fm

and, as m

p h i m as

2

&to

A

ms~

by meen of t h e f u n c t i o n s

, f

€

smL.In

g i v e s u s a P-morphi s m

?

l i e s i n a V e y l c h a b e r n of

G-extension o f

fmA

Pd

. We c a l l it

, we

%.

get t h e mor-

Lemma 4. There i s a commutative diagramm of G-morphisms

Proof. Let u s take a Bruhat decomposition of any element

.

, where

gaG, TO prove the 'Lem-

, u G U(Pd .) , w c w ( G ) , p e P m a we need t o check t h a t i f v 4 Vdd and gv~c, f o r some g = pwu

4Pd

, then 3 (gv) = g 7)/, (v) o r , more precisely, t h a t = 0 , b e c a s e the l i n e s CmA and g cm, i n Am& means t h a t only a t the ,origin. We remmk t h a t (p w u ) v c g

"YA( v )

w(uv)GV

A;

, but

*&(uv)

. So,

=T(v)

we have t o prove the foll06ing: i f

i f we put

cross

uv =;,

then

f o r any w # w ( P ) , t h e mbgroup of the W e ~ 1group leaving P i n v a r i a n t , then Y m : sN-+, N s m and the = 0. Take a Veronese morphism

6 E VAd

(v)

induced niorphim

Y, :

-+

. The function

sm(vmA, )

f,

can

y m and t h e l i n e a r projecfion of weight (m 4 ) i n on one dimensioral i n v a r i a n t subspace Gm, .S ' ( V ~ ~ , ) We cen consider the Weyl transformation as a l i n e a r

be obtpineii as the composition of

(T) and of %he space X (T) 63 R E z For every v e C , w(Supp v) = Supp (w). Thus, i f v , w e V b , , then Supp wv c AX For any y we have Smy€ m Supp y There((supp v).. A s w i s an isof o r e , i f fmd (v) # 0 , we have transformation of the l a t t i c e

.

metry on Supp w E

x(Tj @

(I!

7L

, we

have ( d

,4) =

inf

x s Supp v

(x,

; but

IT

.

inf (x, d ) 3 (d,k), So we have the q u a SUPP V ( d , d ) = (wd.4) and t h i s implies t h a t w d = d and i n view of the f a c t t h a t the set of tho r o o t s of P r W(P) and

x

lity

.

that i s w-invaria3l.t.

h

Thus we g e t a contradiction 4

proves t h a t t h e r e i s a morphism

vd such

( f d (v)

thzt

# O! ). This R ?Ya r\

% '< =

.

Remark. T[e can prove t h a t f o r any f , , f o r m l a r g e , the morm phism induced by , f = fmd i s r e g u l a r m o r p h i a GVo( -+ Amo,

.

.

Wow consider the subset of

where all morphisms Td

GVd

are

trivial. Lemmz 6.

r\

?-'

f€Sm

m*

2

(0)

Woof. The s e t where all of

Ho-unstable points i n Y"

i s a f i n i t e union of s e t s

GVd

fmd

are zero i s exactly the manifold but i t can be described a s

.

A

s ( ( T n ~ (see ~ )Lemma I ) So the f u l l ' m m i f old where are zero i n Vg, i s Ho VsV V and the corresponding submanifold i n ''c may be described as G(Vlcl ) n G V d

HoVs

all

f,

,

(a,d)70

This proves the existence of the .morphim f : X

A

i s an irreducible G-manifold.

The case when by

X =

u X.

iz,O 1

X vihen X

m y be reduced t o the previous one

using the l i n e a r G-space

of the regular functions on X

which are t r i v i a l on u X. . The, image /3: (X) and i t s closure i>O are irrkducible G-manif >ids. So, i n view of the 7revious J(x) 1

construction we have a morphism

which defines the morphFsm

This proves our main theorem. Example. Consider the case near- SL(2,E)-space

sig2

G

=: SL(2,E)

of dimension i + l .

uld an irreducible li-

SL(2,C)

a c t s on t h i s

space as on the space of the homogeneous polynomials of two varig bles

x,y

of degree (i-kl).

A s i t i s well known (see D. Eumford: Geometric invariant .theory)

the unstable points correspond to the polynomials i n some other coordinates

p(x, y )

which

x * , y l can be w r i t t e n . a s p ( x * , y * ) =

1

i s defined = xik p(x,y) where k > [i/2]. The l i n e { x' = 0 uniquely by t h i s property f o r any p(x,y) # 0. So our manifold ITS i n ci+l can be described as an h a g e of a linezr BoSL(2,C) mogeneousbundleon lpl = s L ( ? , @ ) / L I ( A = ( a ?-, ) ) where

tpe f i b r e i s the l i n e a r space degree l e s s t h m Ci/23.

E ' ~ ) of polynomials

The functions

p ( x t ,xy) of

can be choosen as the

f,

corresponding t o t ? ~c? o e f f i c i e n t of the

l i n e a r f u n c t i o n s on

monomial of highest degree i n y

.

3. Subbu.ndles i n cotangent * w d l ' e of a surface. Now we apply the previous r e s u l t s t o studjr t h e vector bundles k on compact manifolds. Let u s take a r e c t o r bundle E We may gg

.

sumz t h a t i t i s associated t o a p r i n c i p a l f i b r a t i o n

.

X y vdth f i Then f o r ang r e p r e s e n t ~ t i o n

GL(k). , x E K' (X, 6 (GL(k)) ) : GL(B) --+ GL(B), we can construct a bundle

bre

ted t o

. Suppose t h a t

X

9

Zg

on X

i s an i r r e d u c i b l e represenfation

-.

has some non trivial s e c t i o n s Row, rl c. GL(k)-invarismt function f on $ z E x we

and t h a t t h e burrdle % ? Laking any r e g u l a r o

,

.

s Because X i s compact, s must be constant and so, u s i n g oix? r e s u l t s of the theory of

g e t a r e g u l a r function on X f

associa

f o

t h e morphisms i n t o models, we g e t t h e fallowing .Proposition 2.

s H*(x,%'

)

%ss t a b l e a t a p o i n t

, i.e. a"., then x

sx i s s t a b l e under t h e a c t i o n of GL(k) s i s s t a b l e a t a'lsro t h e r p o i n t y : itloreover t h e r e a r e t w o bases

t h e point ( e l,

e 1

lynomial . P

such t h a t

Corollary 1. 'ole f o r any

...

,

If

ycX

Corollary 2.

.

& Ex sx = P ( e l , .

sx = 0

If sx

d

..,ek],

f o r a point

i s s t a b l e aod

dim GL(~) then t h e bundle

E

Y

r e s p e c t i v e l y end a po-

& s

XGX

Y

then

dim GL(k)s,

..,h).

= P(I,, , s

Y

i s unsta-

i s equal. t o

E .taken on a f i n i t e unramified cove-

e

ring X of X becomes

a product

k

(@

6)8 F

1

dimension&.

where F -

i s one

These r e s u l t s can be obtained using the morphim into the f i r s t kind mostels, vie get the following Proposition 3.' then -

Lf s

i s semi-stable a t

G ax

XE X

2

s z G/E

s i s everywhere semi-stable md the structure group of-

bundle

E

reduces t o

H

.

Row suppose t h a t any triangular

GL(k)-invariant function

w ~ a i s h i n ga t the origin also vanishes a t b3e a t aay point

x

X

. 110 we

8

X

.

Thus

f

sx i s unsta-

can use the morphisms i n the t h i r d

kind moficls. We suppose only t h a t

X

i?noma3. and connected al-

sx l i e s i n a non l i n e a r subbundle subfibration of unstable CL(k) points in B

gebraic,variety. Then A

eN ,

EjGL(')

,

0 7

and any non t r i v i a l morphi~mof t h i s f i b r a t i o n i n t o

A X gives u s a section of the bundle A*, on X This bundle i s the bundle. of highest weight vectors the l i n e a r bundle E Our previous X 95 assumptions on the i r r e d u c i b i l i t y of Y gives u s t h a t f ( z ) ,

.

in

Zz C* being the central subgroup of i s t r i v i a l on

the character

C*

the following way. Take a torus

T

.

G L ( k ) , i s t r i v i a l and> that

. So i t , c m be expressed i n

, the

maximal torus of

GL(k)

W e order them i n increxsing way so we have a natural f l a g of subspaces i n

Ex.j Ex = E o 3 El

Supp Ei-1 \ Supp Ei f o r the character. Supp Ei

.

...

2 E k ; Supp Ei

c Supp Ei-l

and

i s the s e t of the points with the same I n i = (nj ,%) and ni i s 'minimal on

,...

2

This gives u s a representation f o r a point i n A (det E)

no

D (det El)

@

...

@ (det Ek)

?E

or, i n a more usual way, a s the product of volume elements

3

-as

n

k.

of f l a g spaces. We remark that the parabolic subEi moup l e a v i n g t h i s flag i n v a r i z n t i s j u s t the group Pd of the c o n s t m c t i o n of yo(

.

The tensor

(s)

a t those p o i n t s

d e f i n e s a flag of subspaces

where

x

.

b o l i c subgroup of So on X\Y

where

yd( s )

... 2 %

E

. This

h a s a . a~d d i t i o n a l p r o ~ e r t ~Our . tensor k r. can be expressed as yd ( s ) H'(x, d e t E$ @ d e t E-'i k 1 r. ) 0 . and li = ri rank ~ . / r a n i i 2 I n f a c t the point 1

.

1

f i x e s a l i n e bv.n&e

A%,

'b

a n d a point on i t . This rne~me

t h a t we h ~ v ea t r i v i a l l i n e b m d l e

6

h: @-+A

l i e s i n the one cEm~nsiona1

%t%

. The image

f l a g subbxndle i n As

where

2%

because it d e f i n e s iia para-

0

t h e r e i s defined a f l a g of su'ubum3.les i n

Eo> E1

T(s)

#

..,

GL(k) That parabolic snbgroup i s defined unifrom a subvariei;y of codimension g r e a t e r .than two.

quely away flag

vd( sX )

Eo> El

tini = 0

ri

-- ni

A2

h(& )

03

X

and a homomorphism

defined by the f l a g E 2 6,

...>Ek .

we g e t a r e p r e s e n t a t i o n

- ni,l

0.

This product equals t o the product of formal bundles (rank E dot E~ @ ( r d E~ d e t E)-')

.

In f a c t , when we change our bundle one dimensional bundle

E

k @ i=l

by a niultiplication of

t h i s product does nct change. Let u s tA raak E take i n s t e a d of F t h e f o r m 1 birndle F s ~ c h t h a t (F ) k r Then ( 1 ) i s equal t o t h e product TT d e t (E=€3 F) = det & , i =I d e t ( E ~ Q??)ri = ri ranlc Ei (det FiO P) and F r rmi E det~-'. F

' -

Remark. This formal c a l c u l a t i o n wi-th foz%nzCL bundles, which does n o t e x i s t s i n nature, h e l p u s t o understand what happens i n the general case.

@=

@ F We may a l s o consider a bundle [E as t h e t e n s o r product o f (E @ F/rank x)@

1

'=

where

rank F = 1

SO, even i f t h e cog

responding bundle does not e x i s t , i t s tensor product exists. Fiow we give the definition of unstable bundle. Definition. Let me say t h a t

E

i s unstable i f there e x i s t s a bundle

E

ted t o the same principal f i b r a t i o n than

*

and there i s a section

H'(X,E ) f 0 a HV-subf ibration of

.

be a vec3sor bundle 0 n . a smooth variety X

3

*

md

E

, such

E

'

associg

that: d,et ' E

-0,

s c HO(x,EP)

sx = 0

which l i e s i n a t every point.

In view of the previous consideration it can be proved the foll o wing Theorem 4. If there e x i s t s n bundle Er

i s not t r i v i a l and 0

sE H

(x, E

-y

. )

I.IO(X,E' ) # 0

E '

, but

which i s t g i v i p l i n some point

trivial. br~ndlesassociates to

such t h a t

det E P = 0,

there i s a section

xrX

, then

'

a1.l not

are unstable.

E'

Remark. I n the case of curves t h i s definition coincides wi+h t h a t one given by Seshadri, i n view of the f a c t t h a t actually and therefore we may s e l e c t a special f l a g

Pic x / I ? i e 0 ~=

containing one element only. When

rank E = 2

mensional bundle

the f l a g consist of one element, the one di-

F wi.th a homomorphism h :P -+ E which i s a

mononorphisn away from a submanifold of codimension two. So t h a t 1 ) h i s a I-lolomorphic morp'riism F

2) f o r some positive integer n

4

E

, the

divisor

n -(2F

- det E)

i s effective.

Remark - t h a t i f such subbundle

F

e x i s t s then I t i s uniquely

determined. W e s h a l l use t h i s property i n all our considerations. The proof of these f a c t s becomes evident if we represent our bundle

E

, away

from a f i n i t e number of points, a s an extension

s2mE B (-m det E) can be expressed as a sequence of 2m extensions and L correspolzds to a subbundle F 8 (-m(F + E)) Then

.

It follows t h a t

U

I f t h e r e i s another sach a bundle F+F d d e t E dim H'(x, if

s

X

and

, then

there. i s a morphism

. - F + P z E . I t follows t h a t

d i r n ~ O ( ~ , 2 d e t ~ - 2 ( ~ + 5 ) ) >A0s

2(F+g)-2detE)>O

=0

F

we have

a t every point then

F

i s uniquely determined.

4. Symmetric t e n s o r s on a surface.

I n t h i s l e c t u r e I should l i k e t o d i s c u s s some

viaSrS

t o f i n d ten-

s o r s of s p e c i a l kind on a. v a r i e t y . We r e s t r i c t s ourselves t o the case

X

i s a smooth algebraic 'surfece V

.

E and i t s syrmetric powers s%. We can calcul a t e t h e Euler-Poincard c h a r a c t e r i s t i c of t h i s bundle: (v,s%) = Take a bundle

X

.

2

By the Riemann-Roch formula t h i s i s a (-I )i dim B' (v,s%) i =O polynonial of m of degree k+l , where k = rank E and the coef

=

f i c i e q t s of t h i s polynomial depends only on the Chem c l a s s e s of

t h e surface and the bundle

E

. Noreover t h i s formula implies:

1 ) the c o e f f i c i e n t s of the highest power o f

m. does n o t depend on

i n v a r i a n t of the surf zce , 2

multiplied by some constant s(k) cl c2 which depends only on t h e rvztr of X

2) t h i s coefficient i s

-

.

?Ye give a brief survey of these calculations. Recall t h a t

X m = X(S%)

Row we g e t t h e expression f o r

We remark . t h a t t h i s expression n a t u r a l l y d i v i d e s i n t o t h r e e p a r t s . The growth of the f i r s t when- m 01.

equal t h a n c . 2 " .

tends t o i n f i - n i t y i s l e s s

For t h e second {21 we "ve

inequality:

. Only t h e t h i r d p a r t can grow as c m does n o t depend on t h e i n v a r i a n t s of V , c , ( s m ~ ) and c2(srnZ) nay be expressed i n terms of c,(E) and c2(E) . The d i r e c t c a l c u l a t i o n 12 3

5 c mk

k+f

(and i t could be simply reduced t o the case. of bunriles of r-anlr two k+? 2 b y f o r n ? l a r g u n e n t s ) give Xm#s(?i)rn (el-cp)

-

Corollary. If I.,+ 1 ascm -

.

c;(E) = O

and i f

c~(E) 0

then

We wish t o ded.uce the existence of s e c t i o n s of

. Tm g r o w

smZ from the

growth of Lemma. If -

m

,

tends t o i n P i n i t y

then e i t h e r 1)

dim H'(v, s ~ E ~grows ) a s a mk+l

2)

m,k* d i n HO(?f,sis )

or

Proof.

f

r;l

k+ 1 prows 'as b E

5 dim H'(v, smE)

+

. *

dim H'(V,S~E @ K )

b.iEilft;r we g e ~$(v, SmZ) X ~ ( V , S ~@E K*)

t h e groups ELI&.

H'(V,S~E' @ K )

a d

. By S e r r e

. But t h e climensions of

H0(v,srnE*

)

don't d i f f e r verx

Pur t h e r e i s an e f f e c t i v e d i v i s o r D with D-K

c t i v e and exact sequence.

Consider t h e corresponding cohomology sequence

a l s o effe-

The growth of t h e r i g h t hand .term i s l e s s tFan c mk

as i t i s

l e s s then symmetric power of a bu.~dle on a. curve. So dim

HO

(v, S ~ E @ * K-D)

If

Corollary.

face

2 c,

i s unstable.

V

t

dim

-2k c2 > k-1

.

(v, s ~ E * )+ O ( k )

HO

0

then the b u d l e

E~

on a w -

Proof. In f a c t t h e i n e q u a l i t y i s equivalent t o i n e q u a l i t y k-I 2 det E C @ (12 )>Lo 2 2B c1 4 0 which i s equivalent t o c E @ (- a'' This h n d l e 'Eo ) does not e x i s t but t h e r e exf,st

.

--

i t s symrncti-ic powers

CISm Ek @. ( - d e t

sd'(g0)

e a s i l y conclude from our i n e q u a l i t y t h a t

o

E B r n ) I and we

ariif

C&

dim H'(v, s ~ E +~ )

. Now we can define a restriction of (v, s ~ E ~ ) a t any fibre Ex . Then, a s dim s ~ -- Edim ~SX ~5 c mk-I , we can conaluae tha$ t h e r e e x i s t many s e c t i o n s of H ~ ( V , S * E ~o) r vanishing a t the poi.2t x V . A 1 1 these secl ions s (v, s".$) pxe unstable s e c t i o n s and they gives u s u n s t a b i l i t y f o r bundle E~ on v . idim

H'(v, s%:)

HO

HO

C

Corollary. subbundle

F

f o r some

If,

rank E = 2

&

2 c 1 > 4 c2

dim

of rank 1 such t h a t

.

n >0

HO

then there exists a

-

(v,(2F d e t E ) @ n ) > 0

Now we s h a l l have a deal with a S e v e r i l s Problem: Problem. Does t h e r e e x i s t s a I-connected s u r f a c e with non zero group of holomorphic symmetric tensor

H' (v, s

i d ). for

some, i ?

The main reason f o r i n v e s t i g a t i o n of t h i s question was t h a t the 1

0

group R (V, fi.) i s a homotopy i n v a r i a n t of surface because t h i s group i s isomorphic t o a f i r s t cohomology group of V

.

Lemma,. -

If

K

>7 f o r

2

3

a surface

E?d) z c m + A ,

dim aO(v,

c

ro

of general type the: , A some c o n s t m t . V

Proof. I n f a c t

but f o r

s : srn.fll* 63 K

s e (m-1) K we have i n c l u s i o n

(simply t e a s o r i n g by s ).

3

srnd

d i m A O ( V , S ~>~dim ) H ~ ( v , s ~ ~ ~ @ K

So

T h i s proves t h e l e m a .

Remark. L a t e r we shall: prove t h a t

o 2x1 1 d(-mk)) H (v,S

=0

f o r a surfa.ce of generill type. ITOWwe s h a l l constrv.ct mmy e x a a p l e s o f s u r f a c e s H ' ( v , s ~ ~ ' )f 0

f o r sone

i

Toke a ?-connected v a r i e t y k

11yperm.rface s

BT~'~ .

1

.

> ~ ' - 2 $7~

of dimension

h

Li

with

V

1r-1-2 and

which have t r a n s v e r s a l i n t e r s e c t i o n w i t h .k+2 i,!~ Then vrc have z s u r f a c e V C ,

.

I t i s a connected s u r f a c e i n

.

$t2 If t h e d e g r e e s o f Li w e lri2 l a r g e enough 8and dim Ei ;., 4 t h e n a simply c a l c u l a t i o n g i v e s u s 2k t h a t 1
<

.

metric tensors.

-Remark.

Ye can g e t a more e x a c t r e s u l t . With t'nis procedure we 4 1 can o b t a i n s u r f a c e s with HO (V,S R ) # 0 i f dim l;Ikf2 9 26

.

1

Another d i r e c t way of g e t t i n g s u r f a c e s ~ L t h H ~ ( V , S ~)Rf 0 f o p a ~ yi 7 I abelian variety

x

W

-

on A 3 1) V

2) V

i s t h e following. L e t u s t a k e a t h r e e dimensional

. Consider an i n v o l u t i o n

M

on

A

3

w i t h f i n i t e number of f i x e d g o i n t s , p o i n t s of o r d e r 2

, vi

. Let u s t a k e a burface

V*

2

i s a smooth hyperplane s e c t i o n o f A

2

cqntains a l l points

vi

in

3

so t h a t

,

and a t every p o i n t

cal e q u a t i o n i n a l o c a l p l a n e c o o r d i n a t e s

vi has a l o -

It can e a s i l y be seen t h a t on the desingultirisation 'H V2/&

of

we have the following:

1 ) there e x i s t s a tensor 2 2 ) &'Ii s I-connected,

da2 = 0

2 E H O ( K 2 , S2

1

,

)

0

3) a l l tensors .which e x i s t on X2 can be expressed i n a form k ~ a i j k ~ ~ ~ @ w : 2 @ wwhere 2 3 kl>k2+k x k i = 2 1 , ai j k

3 '

a r e constants.

1

Now we can even construct a family of surfaces

2

Vt

mch t h a t

1) the group

H ~ ( v ~1 , si s~ not ~ ~t r )i v i a l Cur. s;ly i - 2 2 - 1

2) the group

H ~ ( V ~ , S =~0~ f~o'r )any

i

We use the preeious construction f o r

Vo

ging only the l o c a l representation a t p o i n t s ( d z o + Pz, then

+ Xz2)

H'(v,s~fil)

=

t pi

= 0

.

z:

t

0(m3)

t#O

.

and f o r

Vt

and

. ~f

v

i

d

,

by c

h

to

,/J,

7 a r e general

Remark. The analog of t h i s procedure give u s a tensor

3

1)

on a 1-comec-trfi surface.

S ~ H O ( ~ T , S-CL

fi

5. F i n i t e n e s s Theorems f o r surfaces of general type. I n t h i s l e c t u r e we s h a l l prove t h e following Theorem 1.

-

Let

be a surface and

V

Then f o r m y Ene-bundle

h: E

+

we have

E

i t s cotangent bundle.

with a not t r i v i a l morphism

dim HO(V, n E ) < c n

+/)

.

Proof. The proof c o n s i s t oz several steps. Step 1. Let section

S E H ~ ( V , S ) be a section. We w i l l associate a

G E H0(r,

p*E)

t o t h i s section

some f i n i t e ramified covering of s(V)c nE

and l e t

s where

p :7 -+ V

is

V. Consider t h e subvariety

-+ nE be t h e n a t u r a l norphism of l i n e bundle (which i s given l o c a l l y by z zn, z being the coordina

9

:E

~

r/

t e on a f i b r e ) . Let stricted to

V, = ( p i 7 (s(V)) c I3

. m e morphism

reramified

-

d e f i n e s a f i n i t e covering

V,

qn

9' : V V 1 Let r : V., V1 be t h e along t h e zero l o c u s of tine s e c t i o n s i n v e r s e image of E or, V corresponding t o t h e p r o j e c t i o n 1 N r Qn p:vI V Ta have a non t r i v i a l norphism h* : 5 --t V1 c 0 and a s e c t i o n h$E) c Elo(?, ) where s c H (v,, B) is. t h e n s

- .

-

N

.

turaK. s e c t i o n of sor

p"s

E

,nl-v

3 I

&

E corre7ponilil:g t o

can be expressed 'as

r e the projection

'

U

5: V 1

--+V

Step 2. The l i n e bundle S

Of

on V

E

-

E

-n s

s

. Ve rzmark t h a t t h e ten4

ln'ca.11~ at- a p o i n t

.

V,

x

whe-

i s no3 ramified,

d e f i n e s a one dimensional. f o l i e t i o n

w i t h a . f i n i t e ntmber

of s i n g u l a r p o i n i x , Any section

g i v e s u s a holorno~yhicf o m on s u r f a c e

two holomorphic s e c t i o n & ~ dw = s,/s2

, as ve

V

. If

s,, s2 a r e

d w = 0

have

t h e f o n c t i o n w . i s cons%ant zlong the f i b r e s of

Es

. Eow t a k e

i :X

-+ V

the

i* 3 The r e s t r i c t i o n m o r p h h HO(V,, nE) --t H'(x,

nE)

is

ang curve

X

t:hich

is transversal to

n a b r a l anbedding. *-S t e o

a'?

ES and l e t

a monomorphism. I n f a c t i f s t j = 0 then f o r any s e c t i o n "2 which i s n o t equal t o zero on X we have sl/s2 i n constant a l o n g f i b r e s of E s i b u t s1/s2 = 0 on X , t h e r e f o r e sl/s2 = O on V , So we g e t t h a t i * i s a monomorphisnl. But f o r any l i n e '

bxnble

on a curve

F

X

we have

/3

dim ~ ~ ( ~ , . n P ) < c n + and

t h i s proves the thecrem. N O ~ Ywe s h a l l give a g e n e r a l i z a t i o n o.f t h e theorem f o r v a r i e t i e s

of m y dimension. Theorem 2. ,et

$

be & p r o j e c t i v e v a r i e t y and l e t

t h e i - e x t e r i o r Dower of i t s cotangent bunale xhere f o r m y l i n e bundle we have

E

d i n $ (&I, nE)

kz,i

with n o n - t r i v i a l homomorphisme

< eni +

.

fi

-

. Then

h : F,

Proof. The proof i n c l u d e s t h e same s t e p s as t h e previous one and s t e p

q.

d

-

Ste~ 0. We reduce our s i t u a t i o n t o the case when

dim Id = i i l

dil

.

For t h i s we have t o consider a pencil of v a r i e t i e s through t h e one general enough Q, of dim Q = i. 'Phis v a r i e t y hzs t o be chosen t r a n m e r s a l l y t o the f o l i a t i o n induced on general v a r i e t y ui+l by a bundle Fs

.

Step 1. We remark t h a t i f

fii($1) 6

.

F

C,

ai(t'i)

dim El = i + l

, then

= T(Y) Thus F d e f i n e s a f 01iat1.m of dimension one on ?IFor' any s s H O ( & , n ~ C) Ho (hk , S n i ) we construct i n (-K)

.

n

t h e same manner a ramified covering v a r i e t y a r o o t of degree n , p*s = -n s , Z G HO ,

(c,

Step 2. Again

for

s

N

Iil,

vih3re p*s

has

p ~ .j

sic ~O(l$,nz),

i s a r a t i o n a l fun-

c t i o n which .is constant along the f i b e r s of one dimensional f o l i a tion since

dgi = 0

.

S t e ~3. A s we supponed i n the s t e p 0, t h e submanifold t r a n s v e r s a l t o t h e f o l i a t i o n Fs HO

.

Row we have a mono~n~rphism

(M,~F)-% K O ( Q , ~ F )and by s i m i l a r arguments,

dim HO(Q,?F)Cc ni

dim H0(?4,nF)4

.

+P

is

Q

Remark. Y. Miyaoha used t h e procedure of "extracting a r o o t n It can be described as produf o r a geoeral symmetric tensor s

.

1

cing a family { D ~ , of

CV

n l i n e bundles on ramified covering V,

,.

on V1 (p : -+V) can be described as p*s ( HO ( Di) c s"nl k i t h t h e help of symmetric product of i=I homomorphisms P : Pi -4.d Using t h i s procedure and some c& c u l a t i o n s f o r symmetric t e n s o r s with coefficients. i n one dimensio. 2 L c n a l bundles and t h e Bott-Eaum i n e q u a l i t y f o r subbundle DiK D.12 he proves t h e Bombieri's i n e q u d i t y f o r surfaces of general type: C2 4 3 Ep such that t h e $ensor

p*s

.

-

.

1

Now we consider t h e case of v a r i e t i e s of dinension more then two. Theorem 3. L e t

M

bq a_ v a r i e t y

dimension n

whose canonical class is non negative. Then i f

h

with

~

~

i- 5 polariza-

~

h

-

t i o n o_f

EI

, we

have

inequality

of Lascoux-Bergey ---

In f a c t we can construct a surface

with the following pro-

VCEn

perties: ( i ) Pic V =

and i s generated by the r e s t r i c t i o n of the gene-

r a t o r of ( i i ) for

Pic B

.

~vlybundle

H'(v,

(-nh) 0

dl],

n 70

have the equality = IiO(i%, (-nh)@ Q ~ )

dJv)

( i i i ) A O ( V , L ~ =~ HO(N, ~~)

, we

.

ni) .

From the i n s t a b i l i t y c r i t e r i o n we obtain the i n s t a b i l i t y of the

dly

bundle face

V

.

i f the inequality

c21

-n-1 2n c 2

o

holds for a sq

I r e c a l l t h a t there e x i s t s a f l a g of subb~ndles

.,.

Fk

-+ E vfnere hi : Pi

- 'i+l

is

R

,ho

k;1

I?, -> Po monomorphiain out of

...

a fini-fa number o f points and for some integer vector (I; ,%) 0' rank Pi the divisor K ) i s effective. A s the @ ni (det Fi n

-

i

Picard group of iL i s rank Pi ni (det Fi ) n

we see t h a t f o r some

-

i

,

i s effective with

n.1 . 0. But the bud2e i Further det Pi is det Ti has a non t r i v i a l moqhism i n t o fi ample snd from the previous theorem there i s an inequality dim $(ii, nl?)+ c n i + The condition ( i i ) titlls us t h a t a homoraorphim det Pi -+ i(y along V extends to M So we obtain + a contradiction and get required inequality f o r EI

.

.

ail

. .

R e m a r k . Our r e s u l t i s s l i g h t l y weaker than the Lascoux-Berger

conjecture f o r the case the inequality

(c,

.

Pic Frl = Indeed vie have obtained only 2n c2)nhn-'50 butnot 2 0 . n-1

--

The sane argument can be used in the case i s an~pleif we assurne t h a t

Pic Is

-K i s a generator of

Pic M

and

. In

-1C

fact

taking the surface

which s z t i s f i e s the p-operties ( i ) , ( i i ) , rank F i ( i i i ) we get t h a t the divisor ni (det Fi K) i s e f n fective. This implies d e t Fib 0 i n Pic E3. From ( i i ) and ( i i i ) HO(X,*)

we get t h a t have

s u lt

.

H0(1d,

V C IJ

-

f 0

, but

ni)z H ~ ( x(3) , =0

f o r coniugation isomorphism we

for

i70

and t h i s proves the re-

6. Foliations on surfaces. I n t h i s l e c t u r e we w i l l study the following problem. Let

a surface of general type and consider the s e t of a l l curves of geometric genus

which can be birationally embedded into

g

be

V

V

.

These curves l i e i n a number of connected algebraic families. We

wmt t o describe these families under sone homological r e s t r i c t i o n s . We s h a l l deal with a surf ace V , K 27 0 and rcwk Pic V 3 2 HO(V, sill!B 3')

Consider the group bundle on V = (

.

where

F

i s some l i n e a r

. A s usual we consider the formal equzlity

i

)

. From a

S~D'QF =

d i f f e r e n t point of view the above grouup

i s isomorphic t o the group H'(P(T), i ( +~~ / i ) ) where the projeotivization of the tangent bundle

and

T(v)

P(T)

-D

is

is a

d i v i s o r called the Grothendieck generator of the group of divisors Pic P(T) over

Pic V

bundle on P(T).

For any smooth curve

mor-phism f

into

phism

-+ P(T)

tf : X

(f ( x ) , (df),)

. It corresponds to the tautological l i n e

V

X

which has a non t r i v i a l

we have the corresponding t a n g t n t i a l morwhich maps a point

i n P(T)

. Because of

.

x

X

dim X = 1

t o the p a i r

,

tf is a holo-

morphic morphism i n t o P(T) Consider now a l i n e bundle P 2 F < 0 and F E L 0 , where we take a product i n the group

wit11

Pic V R and suppose t h a t f o r some r a t i o n a l q the expression 2 c1 c2 f o r a formal. bunCie fil@ qF i n positive. Using the

-

formula f o r Chern classes, we get where

--

K~

- e + 3X

i s the EuLer c h a r a c t e r i s t i c of

V

.

2 2

F q; 3F q z 0 Then we have

ul

asymptotic i n e q u a l i t y

~ " (@ d qF))

dim

HO(V,

Suppose a l s o t h a t

L e ~ 2 a . The group

H'(v,s"(T

+ 2qF) 2 > 0

(1:

.

H ~ ( V , S " ( T6 (-$8)))

+

0

homonorp11ism i n t o

= 0

i s t m s t a b l e on

So t h e r e e x i s t s a subburdle

A'

.

i s trivis.1.

H ' ( V , S ~ ( T 69 (-@)) )

S G H ~ ( V , S ~@ (T (-@)))

> cm3

and c o n s i d e r t h e group

B (V, n(-K--qF))

Proof. I n f a c t

@ (-@)))

. Thus any t e n s o r V

.

of rank one w i t h non t r i v i a l

L

dim H ' ( v , ~ L O (-IC-2qF)n2@ - n , W 0 1 2 (nl . n 2 , ~ 3 ) . Then H'(v,~L) > c n2 ?a13we o b t a i n

f o r sane p o s i t i v e

mcll t h a t

a contradiction. So

a i m #(V,~"(fl

1

rem vie g e t , f o r some

@ @))

m

grows a s cm3. Bg t h e I i t a l r a theo-

1a.rge enough t h a t r a t i o n a l ~cor:~liism

i s a ' o i r a t i o n a l embedding. Suppose t h a t s o r o f ).'

X

Definizion. A curve Le-t X

i s said regular i f

tf(X)cB

In f z c t

.

be a r e g u l w curve. Then t h e degree o f Vm t f ( X )

l e s s Chan t h e degree of m-symmetric t e n s o r s on

{ D nF

m e t r i c genus

3 < ng

g

and

a r e mapped v-nder a mo~phism I?'

X

if

X

FSO

ym

. T l ~ n st h e x l i e i n an a l g e b r a i c

f o r which we can f i n d such bundle Pic V

X F 5 0 and K2

F

on

. We have t o c o n s i d e r o n l y

- e + 3K

F/m

+ 3~~

Pic V 0 R 76

m

70

SO(m)

are i n v a r i m t under vrhich l e a v e s t h e

i n v a r i a n t a d v e c t o r bundle

xed. The corresponaing q u a d r a t i c form on the 2-plane t h e form '~-'x

. If

f X

i s tvo d i m e m i o n a l because b o t h i n e q u a l i t i e s

t h e compact group of t r a n s f o r m a t i o n form (x,x>

.

t o a s e t of

family. We now c o n s i d e r t h e v a l u e s of t h e h o o o l o g i c a l c l a s s t h e c z s e when

is

. So a l l r e g u l a r c u r v e s o f geg

F X 0

c u l v e s of bouncied degree i n

)

i s ' k e base d i v i -

B = BF

.

m

H0 (v,~(D+ $3')

n s s o c i n t e d t o t h e l i n e a r system

Ym : P(T) -+ '?!I

K*

- e > 0 , then

K

fi-

( x , ~ ) has

the trivial bundle

F= 0

s a t i s f i e s our c r i t e r i o n and so ne have the following Lempa.. -

If

K2

-e<0

of geoxetric genus

g

P7ow suppose t h a t

Pic V

K*

then the f m i l g of r e 9 l a . r curves f o r

j?

=O

i s ?Aqebraic.

- o <0

. We attempt

to f i n d t h e s e t i n

f o r which t h e r e e x i s t s such a, bundle, perharps d i f f e r e n t

f o r d i f f e r e n t curves. Geometrically we have an equation of a hyper bola i n plane:

Thc hyperbola h z s as as asymptotic l i n e s t w o l i n e s p a r a l l e l t o the cone

1x2

'

0

3,

L,, L2, and the cone 1:

1

of bandle

B

with

t h e i n c p d i t y (1) i s tcmgent cone a t the origin. So i f we take in K we can f i n d a r a t i o n a l nlmber q w.ch t h n t I I g 2 + 3 K * P q+3pZq2- 2, 0 and f o r X which h z s F * X f 0 T11us 3'

.

we have t h e following LewLa. The f m i l y --

of a l l r e g ~ l a rcurves

don 't l i e i n a h a l f ~ l m eP X > 0 Consider the f a r i l y

o~arer

which

g

is algebraic.

of a l l r e g u l a r curves of genus

g

who-

se horiological c l a s s I";

f (X)E H2(M, Z ) i s n o t contained i n a cone E 2 2 which i s d u a l t o a smzller cone I<, = ( 3 / ~ C ) (K +22aF') >

-

x -17 ~ ~ .

i s an algebraic family since i t i s the v-nion of aige-

Then

b r a i c f a m i l i e s f o r a f i n i t e number of -he bundles So Re have proved t h e following theorem: Theorem. The family of curves of genus

g

Fi

.

c o n s t i s t i n g of dl

i s m algebraic f m i l y . 1 C o r o l l a ~ r .There a,re only a f i n i t e number of homoloqica~ ~

curves ses -

X

with

f(X)$KD

[f(~)] where geometric genus of X 9 g L a r f o r any F i n the s e t P i e V B R \ KL ,

{

n

X

1 2 ~ -

i s rem-

1

This e v e s u s the l o r d e l l problem f o r the surfaces of g e n e r d

. Suppose t h a t tf(x) c Bp a base l o c u s d i v i s o r on P(T) f o r + . Now % which d e f i n e s a f i n i t e number of d i r e c t i o n a t point . Thus theye does n o t l i e i n a l i n e a r component of d i v i s o r type. ITow me d e s c r i b e t h e i r r e g u l a r curves on

V

i(D

any

@)

xCV

BF

which nap s u r j e c t i v e l y Bi C BF corresponds a l i n e a r subbundle

e x i s t s a f i n i t e number of s u r f a c e s via

p :P(T)

V

li

of the t:mgent bundle of t h e [email protected]

such a way t h a t i f C

on

Bi

V

. Any

to

tf(X)cRi,

of t h e .bundle li

P ro imaition. -

All

Bi

j

then

tf(X)

surface

.,

Bi

in

is an i n t e g r a l curve

. So we o b t a i n t h e f o l l o w i n g p r o p o s i t i o n

r r e 9 l n . r F-curvca ?=-:the

i n a r y s of c u m e s

which corm f r o m i n t , ~ : ~ - p curve I of finit_? ix:.mber of f o l i a t i c n s on

So we h ~ v et o describe t h e i n t e g m b l e cv.rvcs of folin.tioi1 i f we vtmt

t o d e s c r i b e a l l c u r v e s of genus

7 . The "11

o

g

.

s u r f a c e s with b2=3.

I n t h i s l e c t u r e w e s h a l l d e s c r i b e F - i r r e g u l a r curves. I t f o l l o w s from t h e p r e v i o u s l e c t u r e t h a t a l l such curves l i e i n a f i n i t e nug b e r of f o l i a t i o n s o n r a n i f i e d 'covering

V

of: our s u r f a c e 76

.

So we a r e i n t e r e s t e d i n i n t e g r a l curves of a f o l i a t i o n . Yle r e c a l l t h a t a one-dimensional f o l i a t i o n on a s u r f a c e i s t h e sa:ne a s a o r of t h e t a n g e n t bundlc of

ra,nlc one subbundle of

fll

pose t h a t two curves

X I , X2

i s de2ined by

h: P ->

fi'

p o i n t s of t h e s e two curves

Xi

. Sup-

a r e i n t e g r a l f o r a f o l i a t i o n which

, where

at a f i n i t e number of p o i n t s

V

pi

ran!:

. Then

h

i s l e s s t h m 1 only

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

can be t h e p o i n t s

pi

. The main

theorem which we s h a l l prove i n t h e f 0 l l o w i ~ g Theorem. Let u s take a f o l i a t i o n

i t must s a t i s f y e i t h e r

FS on a s u r f a c e V

. Then

-

1) there e x i s t s a r a t i o n a l nor~hisrn p :V X p n t o z cv.rve mch t h a t Fs i s t l n m n t t o t h e o s n ? r i c f i b r e of p

X

.

2 ) t h e r e e x i s t s only a f i n i - t e nurajer of a l ~ e b m i cc u r v e s vzhich

integm.1 f o r

F

grc

S

The proof i s based on t h e l o c a l d e s c r i p t i o n o f t h e f o l i a t i o n . Loczll_y we may suppose t h a t t h e f o l i a t i o n can be g i v e n by a hol-omorg'nic form

&I

= P dx + Q dy

with o n l y one sin-wlar p o i n t namely

t h e o r i g i n . L e t u s take l o c a l a l g e b r a i c curve i n a small neighbourhood

rn/x

U =0

of t h e o r i g i n which a r e i n t e g r a l f o r

.

Renark. --

GJ

. Thus we hawz

T h i s d e f i n i t i o n i s somenhat d i f f c r e n t frorc t h c norna.

defini4;ion of i n t e g r z l curve of a f o l i a t i o n bclt t h e y d i f f e r only on a d i v i s o r . We shlzll c a l l such a l o c a l l y i r r e d u c i b l e curve a l o c a l alcebz&r i n t ec r a l and denote -

it

Cl.a.i.1.

Eow we u s e t h a procedvwe of the

monoidzl .lransformation. T d c e f o r ned by a number of N

U

U

which 4.s o b t ~ &

d - p r o c e s s e s above t h e o r i g i n . Then i f

i s t h e corresponding morph-ism, we can d e f i n e t h e indu-

p . U --PU

ced f o l i a t i o n on

N

U

by means of t h e form

p W L . , . Again i t has

[ 1.a.i.

only f i n i t e number of s i n g d a r p o i n t s . The ced form

4

a surface

$*

Q

a r e t h e same as t h a t o f w

1of

t h e indu-

except f o r t h e excep-

t i o n a l components. We have t h e f o l l o w i n g Theoren. There e x i s t s a s u r f a c e [

i

.

p

X

on~

N

U

N

U

over

which d o n ' t l i e i n

U

such t h z t z?y two

p-i(0)

(p :

a -+

U)

are disjoint. The proof o f t h i s i s r a t h e r t e c h n i c a l and vie d o n ' t go through. We a p p l y the- theorem t o t h e compact s u r f a c e

F. P,laking a f i n i t e number o f

V

and f o l i a t i o n

-processes we g e t a s u r f a c e

on it such t h a t t h e non-exceptional

and a f o l i a t i o n

Z

V

integral

FS a l g e b r a i c curve n o t have no i n t e r s e c t i o n . Thus for any two such c u r v e s we have

Xi-X

j

=0

.

Let us take i n t e g r a l curves with

i s l e s s than rank 2 X. =O 1

(x,,s)

then

N

Pic V

2

X 1. = 0

. The nunber of

ti~em

and i t i s f i n i t e . ALI the curves rni-tf?

m e l ~ o m o l o g i c ~ l llinearly y equ.ivalent bbcc~use on the plane = 0 the f o r n i.s n e g a t i v e l y d c f i n i t a and i f y 2 = O

g = tX

. So if

x2

PicOV= 0

containing a l l our curves. If

then we g e t a p e n c i l n,X,- n 2X 2 ~ i c ~ \ l 0. f thcn we g e t a p e n c i l w i t h

t h e base curve of ,yenus more thzn 0 , So e i t h e r our f o l i a t i o n i s

tan

gent t o t h e p e n c i l of curves o r our i n t e g r a l clwves l i e i n a dirisor of t h i s p e n c i l and t h e r e f o r e coincide with thsm. This proves t h e theorem.

C EN TRO INTERNAZIONALE MATEMATICO ESTIVO

(c.I.M.E.)

PLUMCANONICAL MAPPINGS OF SURFACES WITH

K ~ =1,2, q=p =O g

F. C A T M E S E

FLURICAWOMICAL MAPPIRGS OF SUIGnACES WITH

8'

1,2

, q=p -o(*) g-

F. Catanese Pisa (**I

I. Introduction.

This lecture i s a continuation of Dolgacev's ones on surfaces with q=p =0, ana considers those minimal models of such suy g faces f o r which g2=2 (numerical Carnpedelli surfaces) and those for which K ~ = I (numerical Godeaux surfaces): they are of general type by c l a s s i f i c a t i o n of surfaces. The Plain Theorem of [I] ( t o which we will r e f e r a s ~ c N ] ) asserted among other things t h a t f o r a minimal surface of general type, 3 mX denoting the r a t i o n a l map associated t o the complete l i n e a r system se exceptions

*

I

mKS

1 , 9 mK i s b i r a t i o n a l , f o r m

,

3,

with th_e

This seminar i s an e-%position of j o i n t work of E . BombieriF. Catanese,

(**) This author i s a member of G.N.S.A.G.A.

of C.N.R..

2

a)

K 1 ,

b)

K 2=2,

pg=2 pg=3

c)

A 1 .

Pg=O

,

1~4.3

, ,

-3

and, possibly,

w . 3

, K 2=2 , pg=O , -3

.

~t was l a t e r shown t h a t the exceptions of c ) don't r e a l l y oc_

.

pg=O ms? was proven by ~ o m b i e r i(unpublicur: the case K 2=l shed) and subsequently by us along a simpler l i n e of proof ([4]), =0, m=3 by Miyaoka f61 and subsequently by Kulit h e case K ~ = I Pg kov and u s (along a d i f f e r e n t l i n e of proof, unpublished), the K 2=2,pg=0, m=3, by P e t r s ( [lq ) i n the p a r t i c u l a r case of case a Campedelli double plane, i n the general one by us ([3]) and 12 t e r , independently by X Benveniste (.mpublished). The main goal of t h i s lecture i s by one side t o prove these r e s u l t s i n the simplest fashion and by the other one t o exhibit the application of some new lemmas (of [3] ) which allow one t o handle reducible curves i n nearly the same way than non singular ones. We w i l l give- our proof f o r the f i r s t case, f o r the second we w i l l give the main s t e p s ( i n which our d i f f e r s from Miyaoka's proof): f o r numerical Campedelli surfaces, f i n a l l y , we remmk t h a t the proof appearing here i s a combination of our with an ar_ gument of Benveniste's proof.

.

11. Some a u x i l i a r y r e s u l t s .

K2( 2, q=O t h e r e i s only a f i n i t e nwnber of irreducible curves C with Lemma 1 . On a surface

S

of general type with

K*C f 1.

2 Observe t h a t i f c < 0 C i s i s o l a t e d i n i t s c l a s s of numerical equivalence, hence i n t h i s case i t s u f f i c e s t o show t h a t the number of such classes i s f i n i t e . Here ve use the index Proof. -

theorem ( [9] page 128) t o the e f f e c t t h a t on the subspace of numerical c l a s s e s orthogonal t o K the i n t e r s e c t i o n form i s n e g a t i ve definite: i P K.C=O c2= -2 ( a s c2 < 0 and KC + c2= = 2 p(C)-2 5 -2) a d the number of such c l a s s e s is f i n i t e (moreover such curves are numerically independent, see [53 pag. 177, [c.M.~pag. 174-5). If K*C=I , then (K-(K 2 )c) is orthogonal t o 2 2 2 2 2 and so C 2( I/$; however K, hence 0, (K-(K )C) = K ( ( K )C -1)

--

C% 3 , so ~2 < o un_ 2 p ( ~ ) - 2= I + C? -2 implies c2 OM, 2 2 l e s s only i f i t i s K =I r C =I , K homologous t o C Note t h a t (K-( K ~ ) c ) belongs t o a numerical c l a s s orthogon a l t o K with s e l f i n t e r s e c t i o n bounded from below by -14, hence can belong only t o a f i n i t e number of classes, and the same then

.

occurs f o r C ; f i n a l l y i f C i s homologous t o K ~ O ( U ~ ( C ) ) = I (compare [67 o r Dolgacevvs l e c t u r e ) , and,the surface being regul a r , there i s a f i n i t e number of such curves. Q.E.De

W e r e f e r t o [3]

f o r the proof of the following lemmas

A,B,B'

Lemma S,

2

Ae

Let

C

be a positive divisor on a smooth surface

an i n v e r t i b l e sheaf on C i)

ii)

with

ho(C, 2 )5 1 : then e i t h e r

there e x i s t s a section S not vanishing i d e n t i c a l l y on any component of C , and degc 2 0, equality holding i f f 2 = OC or

oe

there e x i s t s a section o, C1 ,C2 > 0 such t h a t c & ~ +c2$ a - 0 but o J C t $ O i f C 1 c C 1 f C , a n d

lcl=

Lemma B. -

r

i s an irreducible Gorenstein curve and l w , ~ # $ , then / w , I has no base points. r4ore generally a reduced point p of a curve C on a smoth xzrface is not a base point of IwC 1 if e i t h e r

i)

p

If

i s simple on C

and belongs t o a component

with

P(P) 2 1 or ii)

p

i s singular and f o r evexy decomposition

(ci > 0)

C&

+

1 one has Cl *C2 > (cl = c * ) ~ =intersection

t i p l i c i t y of

C,

, C2

at

p

C

.

2 mul-

Remark. If C i s given by two e l l i p t i c curves meeting tral?, s v e r s a l l y a t a point p , p is a base point of I wC I , and i n f a c t condition i i ) i s violated; however i f C i s given by more than three l i n e s i n the projective plane a l l meeting i n the. same point p , condition i i ) i s violated but p i s not a base point. On a numerical Godeaux surface i f E i s a torsion c l a s s . # c , (D,

denoting the unique curve o f % I I+E I ) , C=DE + DWE has a s a base point of I w , and i n f a c t DE OD

b r n ~ D-,~

C

-

-6

=l

=

= ( D ~ * D -(compare ~)~ Dolgacev's l e c t u r e and the following of

this). Lemma B' . -

is a reduced singular point of a curve C the maximal i d e a l of lying on a smooth surface, denote by p i n C , and l e t x:: C be a normalization of C a t p Then ~ o r n ( n, ) can be embedded i n the r i n g A of regular P *P functions of a t x-' ( p ) If

uC

p

mP

-

.

.

(x. ~ e n v e n i s t e l . Let

S be a numerical Campedelli a positive integer: then the family 5 , of i r r e ducible curves E such t h a t I - E 5 m and 1 E 1 = {EJ i s a f i n i t e one.

Lemma 2

surface and m

ProoP.

If

E

E%

, h2(E)=h0(~-~).h L ( ~ ) = 0 . hence

R.R. gi2 E Associate orthogonal t o I:

ves I + ( ~ ~ - r s= )2 ( b ( ~ 5 ) )1 , so -(#.) t o E E 3, the following numerical c l a s s

4,

-

= a (x~E)x. Here, a s i n lemma 1 , we use the inaex theorem t o i n f e r 2 2 2 and t h i s , together ( 0. But = ZE* = 4E2- Z(K*E)

aE

CE

YE

-

.

-

with the already used inequality E? XE 2 , gives the result that 1_ z(K*E)'- 8 4K-E > -(8+4m+2m2) ; i n t u r n t h i s

gg -

implies t h a t $B

may belong only t o a f i n i t e s e t of numerical

classes. Suppose now

$

-6

: i f we

prove t h a t then e i t h e r

2

wK+E we are done (the surface being regular each 1 2 c l a s s can be given by a t most 2m such curves where m is the

El-

E

or

-

-

2

E

order of the torsion subgroup T

can be read a s

of

P~c(s)).

2 ( ~ ,- E ~ ) - ( K E -KE2)x ~

and we can

2 asswne F =KE -KE > 0 ; more over F i s an even number, because' 1 2 ~ ( K * E ~ ) = ~ ( E ~ -(Er ~ odd ) * Ewould imply K*E,.K*E2 t o be even, i which i s obviously absurd). This equality, i n turn, when i = l , can be read a s 2E12 = = 2E E

+ r K*E1

1 2

and the f a c t t h a t

El * E g 0

(#) El2 5 XE1 soon imply

111. B i r a t i o n a l i t y of

Let

S

, Pi

$ 4K

f o r a numerical Godeaux surface.

be a numerical Godeaux surface ( a minimal model, a s

we always assume). We r e c a l l t h a t f o r any and m ,

2

~ l l yZ ( E -E*)-~K ~

pm=h0(6(mr)) =

1

5

S

of general type,

IU(R-I)K*+ ~ ( 0(C5J ) pag. 184,Ec.q

pag. 1 8 5 ) ~ so t h a t i n t h i s case p2=2, p 3 4 , p4=7. Take U t o be the Zariski open set whose complement i s given by t h e union of t h e curves D such t h a t K *D 5 1 , with t h e locus of base points of \ 2K I and of singular points of curves C G 12~~1. Remark indeed t h a t r e s t r i c t e d t o U i s a regular .map, then we claim Theorem 1

.

is an i n j e c t i v e morphism.

Proof. Suppose t h a t

g4K(~) P

4K

a r e two points of U such t h a t (y) ; by our choice of 0. arc may choose a curve

D e \ 2 ~ sot. \ y #D,

x,y

and the unique curve ~ ~ 1 s.t. 2 ~ x1 E C:

is a curve of 1 4 K j passing through and x,y are simple p o i n t s of C Given a sheaf 3 , denote by 3 (-x)=

now C+D

t h e i d e a l sheaf of the point

.

x,

3m.

x) , and by Mx

hence (where

y E C

nxis

the sheaf suppor-

,

ted a t

.

x with s t a l k M

t h e exact sequence

0 4

We obtain t h a t

-

0

H (c, O ~ ( K ) ) =by~

0

K

K

to-

1

H ( bS(-K) ) ( t h i s

i s by the vanishing theorem of EIwnford, C8 , a s s e r t i n g t h a t i f is an i n v e i s spanned by global s= t i b l e s h e d such t h a t f o r l a r g e n 2 gether with the vanishing of

1

c t i o n s and has three algebraically independat sections, then HI ( 2 " )=o : it can be applied t o bS(mK)

, m 2 1,

and f o r l a t e

use ue observe t h a t by duality a l s o gives hi( 6 S ( ~ ) ) =for? ~

, m22).

i 5 1

?7e w i l I derive a contradiction by shoving t h a t 6 C ( ~ )is isomorphic t o 6 (x+Y) Consider f o r t h i s the following exact sequences:

.

Then from t h e i r cohomology sequences one g e t s 1 I =hl ( bS (4~-x-Y))=h ( bC(4~-X-y)) , and by Serre d u a l i t y on the '

curve

C

h0(c,z)=l

,

where b = oC @ O ~ ( ~ K - X - ~ )O- ~~(=X + ~ - K ) .

, a f t e r observing t h a t occurs d?E 6,. what we wanted

IrTe a r e ready t o apply lemma

degc< = 2-K *C-0 ; then i f

i)

A

t o show. Case i i ) cannot occur: i n f a c t one would have

cl -C2 2

deg

b

@

2 = -KC2+

(number of points of

C=C,+C2

C2 n p . y ) )

C2

~ u by t our choice of KC

2

2 2, so i n

any case

u , C1 *C2

if

x, o r

y

~

c

then ~

.

,

,

should be non positive, and t h i s

contradicts the following r e s u l t of Bonbieri about connectedness of divisors homologous t o pluricanonical divisors ( fc.1.1.I pag. 181) : i f D --mK , m L 1 and D S 1 - t D 2 , Di > 0 , then Dl *D2 >_ 2 , except when ~ ~ k 4 2 but , then Dl*2-K. Q3-D.

IV. B i r a t i o n a l i t y of @

3K

f o r a n w e r i c a l Campedelli surface.

-

By the j u s t quoted formula here

p2=3

, p3=7.

From now on we suppose t h a t x,y are two points such t h a t $ 3 K ( ~ )= @3r(Y): as p2=3 there e x i s t s a curve C E \ 2K 1 c o n t a i ning them both.

Proof, -

The cohomology sequences of

0 -7\6,(31-C)

)

Os(3K-~-~)-2

o c ( 3 ~ - ~ - y)-) 0

\\I

K give

1

h ( b C ( 3 ~ - x - y ) ) = h'(C?S(3~-x-y))

+ h2 (6(1))= 1+1

.

as

1

P r o ~ o s i t i o n4. For general x, y , and C E 25-x-y 1 , x and y a r e simple points of the curve C which i s hyperelliptic having %= bC(x+y) as its hyperelliptic i n v e r t i b l e s h e d . Proof. The second p a r t of the statement is an easy consequence of the f i r s t p a r t , ledma 3 plus Serre d u a l i t y on C , the f i r s t w i l l be proven i n two steps.

Step I:

x

belongs

t o only one irreducible component of

t h e same holds f o r y

.

.

C

r r2

,

In f a c t i f , say, x belongs t o two components of C , by lemma I 8 . Pi >_ 2, hence is equal t o 2. and by lermna 2

h O ( b S (p i ) ) )-- 2.

that

r

E

mite

C=

fl +

r2. I n f a c t i f r;

r2+ F

i s an irreducible curve

there e x i s t by the previous remark

rl+Y i =r; +r;, t

HI 3 H,

,

HI *K

3

which cannot hold range i n a f i n i t e take, by what has t&bugh x , and x

r;, r;

rl # r

E

E

) r21such

r;=r; + H;, r;=r, +H, 1

0 ; but t h i s implies

I f it were

Step 11:

and so

(K*F=o) and consider

*

HI =HI

and so

H I

would be a base point of

1 rl /

that

, would

1f l ]

rl

,

f o r genwal x (the numerical c l a s s of can s e t ) ; while i f it were one could = just been said, a curve E not passing then consider C 1 = + +P

r r'

r,

and y

a r e simple points of

r; 1 r31 C

.

.

For t h i s we can use lemma 3 and Grothendieck d u a l i t y t o infer t h a t dim h. ~om(W~7'R~. bC)=2. Taking ? a normalization of

i s connected exactly 11 , lemma 3) h0(6_)=1 C we can apply lemma B' : x,y both singular would imply h0(0_)2 C 2 2, a contradiction, while i f however x i s simple, y s i n e at as C

C

x,y we observe that by s t e p I , hence by Ramanujamls r e s u l t ([I

.

l a r , one g e t s h0( ~ _ ( x ) ) = 2 so x belongs t o a r a t i o n a l curve; C S

being of general type, t h i s cannot occur f o r general

x

.

Q.E.D.

mom now on we suppose C E rements of proposition 4.

1 2K-x-y 1

t o s a t i s f y the requi-

Lemma 5 , w Proof.

If

and

H~(c,%)

-= hC636 S,

, one

div(Si) = ai+bi

vanishing a t

A'(%)

..

, .S 6 are s i x d i s t i n c t general sections of a,,

...a

has that a section oE

vanishes a t

6

b,

,

*,b6 too. Q.EeDe

W e can pass now t o the proof

OF

Theorem 2. For a wwnerical Campedelli surface i s a b i r a t i o n a l map. Proof. Consider -

@ :S

i n an./ hyperplane so t h a t V

- IP6 : V

d = deg V

=

t5

@ ( s ) i s nct containeZI

.

i s not a curve, otherwise t h e general element oP

be decomposable i n more than

curves D

with

Ken 5

?,

d

1 3K f

would

elements, while we kllow that t h e

are a F i n i t e number.

By Theorem 5.1 of 123 ( a l s o [7J Th. A >

has no base points, hence i f

,

m=deg @

we know t h a t 13I

1

2 dm = ( 3 ~ = ) 1 8.

W e must then prove t h a t i t i s impossible tc have e i t h e r d e g $ = 2 or d e g @ = 3 . deg @ = 2

Case I,

7

There i s defined on S that

y=cr(x) i f

@ (x) = @

a b i r a t i o n a l involution (y) ; S

of

-9

C,

and

(a) =

b

o(C) = C:

i n fact if

i s the point of'

(b) (by lemma 5 ) .

C

such

being a m i i i i m a l modei

i s an automorphism, hence o*(O (K)) = Rernak t h a t

o

6 (K! a

a

,

i s a general p o i r ~ t

s - t b 6 ( a 4 - b ) ~hC

I

H ' ( ~ ( K)) --) The exact sequence 0 -4H'(G(-X) I-> H' ( b , ( x ) ) o implies hO(OG(K)) = 0 which is impass&

-+

.

--+

ble by v i r t u e oP the following. Lemma --

C p )" = $32

6.

Proof of the lemma. By lemma 5 bC(3K)*. h : t o prove t h a t , for instance, b C ( 8 ~ ) 3 : But

.

base points, and we may pick up zeros on =(a, ) ,

c ,

...o(aI6)

a?,

...

a

16'

so i t suPPices

1 4K (

has no

DS(4K)) with 16 simple has 16 simple zeros too,

S 6 ('H

Tnen

G*s

, therefore oX(S)*S 1C i s a section of O C ( 8 r )

whose divisor i s l i n e a r l y equivalent t o

h@ 16

c

Q.E.D.

Case 11:

deg @ =3 .

Let

a

be a general point of

(a

C

, and

-a

be such t h a t

a+; s l h C I : $ ( a ) =
Q

-'{c)=c

U N'

and

N'

.

is a r a t i o n a l curve ( t h e r e is a bira-

t i o n a l map from NP=locus{a']

and

I hCl"~').

So

S

contains

a continuous family of r a t i o n a l curves, which is though absurd. Q.E.D.

V.

g i r a t i o n a l i t y of Denote by

@= $

Lemma 7. -

V

f o r a numerical Godeaux surPace.

3Ks

and by

is not a cwve.

V =

9 (s) ,

d=deg V.

Proof. -

Otherwise d would be L 3 and the moving p a r t of 13K 1 would be decomposable in more than d elements, which, by lemma 1 , have intersection with K at least 2: then one would have 3 K . K 2 26.2 6. Lemma 8. 1 3K I has no fixed part. See [6 3 , pag 4 03 for the proof.

.

However 1 3K 1 can have base points, and they are characterized by the following proposition, in which we denote, as in the following, by T = Tors (?icfs)) and by D , if E # 0 , E E T , the unique curve in 1 Kt E 1, Proposition 9. IE b is a base point of 13K I there exists E GT, E f E , such t h a t c=D~+D - ~is the unique e m ve in 1 2K / passing through b : moreover D E and D, E: have b as the unique point of transversal intersection. Conversely, if E f & , D E 0 D - E gives a base point of 1 3 ~ 1 .

-

-

Proof. Ve recall first Niles Reid's lemma (see Dolgacev's lecture) which asserts that if g $ z 'are non zero torsion classes, D and D T have no coirmon component, hence intersect transversally in only one point. Take C E 1 21 ~such that b E C : then bC( 3)g ~ uC and b is a base point of ( uC 2p(c)-2

1 (because h1 ( 0 (K) )=o, so

= (c+K)*C=6=) p(c)=4

lC=1wC / ) .

13~1

and by lemma B C must be reducL

ble; moreover if P is a component of C containing b, K * P 2 1 ( K * f =O 3 P is rational non siagulzr,kwe mat every sg ction of 6(31), as it vanishes at b , vanishes on the whole

of f" : this would contradict lemma 8 ) , so tha.t

b belongs to

at most two cozponents. Thea pick up an irreducible component

Po

of

such t h a t , i f

C

one component of

coo o >-

By S e r r e l s duality

belongs t o one and only

Consider the exact sequence: e ( K + Po) --j6(31)

i=0,1

hi(6(&

?r6(- Po)

.

3 0 ( 3 ~ )->a c,

P , ) ) = ~ ~ ) - ~ ( OPo))=O (£or i s a consequence of t h e exact sequence

(i=l

and t h e i r r e d u c i b i l i t y of f ,) is a base point f o r (6 (3B;)

-$ b ---? and one obtains t h a t b

0

C= Po+Co, b

-4 0

1.

C,

Then one has the exact sequence

and, dualizing,

B'

First,

b

implies

(

must be a simple point of 6 (- )

C,,

otherwise lemma

and using Lemma A plus the al-

I

Ce

ready quoted comectedness theorem FOP pluricanonical divisor,

-

6C0 of

,

has degree 5 -1

P

Co = C1+C

2

o >- ( po+CI ) *C2,

- r o * C 2 3 @,C2:

such t h a t and

c=(

hence t h e r e e x i s t s a decomposition however then

,+c, )+c2 i s not numericaily m e -

f'

cted. The same reasoning gives the vanishing of

and

b

being simple,

Again lemma A such that

I+)

amounts t o

gives e i t h e r

- Poc2+l 2 C, C2

6

Go

( F ,)r

iF b

hG(ec (-Po)), 0

h O ( O (- pO+b))=l

6

c0

c0

(b)

or

.

Co=C, 1-C

2

E C2

This l a s t i s impossible, the other two p o s s i b i l i t i e s imply C=G'+Cll where Cr--K-Cw, again by the connectedness theorem

,

and 6 ,(c")= 6 ,(b) C C

by lemma A

again.

Finally the exact sequence 0

-+ H'((~(c"-c'))

-4 K'(~{C"))

-)

H'(o,,(c")\

-+ 0

.

-

implies t h a t C' # C" , so C f = D S f o r a s u i t a b l e E # E Conversely, i f b=DEA D & we claim t h a t b i s a base point 1 1 f o r 1 3 ~ 1 . In f a c t h (6(3K-DE ) ) = h (-D )=o ( t h e D;S

-

-E

1E'=

a r e -4 hence numerically connected), s o t h a t ( 3~

I

= 16 (3QI: then. as OD ( 3 ~ ) "wDE@ODE(b) h l ( O D ( 3 ~ ) )= D& E € = hO( G (-b)) = 0 , s o b is a base point of ( 3 K ( , and =€'

1%

1 3 X 1.

therefore of

Q.E.D.

138

D -E

I

Lemma 10. -

If t h e r e e x i s t s

€

eT

sot. E f

-

then

C ,

i s spanned by the two l i n e a r subsystems D -I-12~E 1, E +12K+&\. Proof. -

Note t h a t by

R.R.

nolz zero torsion c l a s s

T

h O ( b( 2 ~ + % ) ) = 2 ,while

p =4, hence it s u f f i c e s t o show t h a t 3 these two subsystems have no common element. This i s c l e a r , how-

-

ever, since i f one should have D + M = D, c: + M E M E D E -E would be a p o s i t i v e divisor z K ( i n f a c t DE and DW6 have no $

common component), contradicting

p =O. B C)*F,.De

Two general curves of f 3KI are sinple a.t oP 13K , and have there a transversal i n t e r -

P r o ~ o s i t i o n1 1 .

a base point section. Proof. b a E0 D-

= b+\% j E

b

I

If a general curve of

,

, so

f

3K

1

would be singular a t

would then be a double base point or" 13~11 =

b

D.

b

would be a base point of the canonical system

of

D E ; so i f

p(f')=~

f i s the component oP

D

and, by Lemma A, Ds = Cl+C2,

where ..

.

,

- t o which

b

belongs

p,

C,I' 2

C;C2

.

5

c-C (K+D - ) = 2oc ~ =O -(as C is-made of curves E s .t - 2 E 2 2 K*E=O): t h i s hovever,contradicts the numerical connectedness of

.

This r e a s o n i n g ' t e l l s a l s o t h a t a general curve is not tan'D gent a t b neither t o D , nor t o D m S If b is not a ba-h& a base point of I 2K+ E ) , we a r e s e point o'f 12~: i'(',through: but in' the :contrary 'case; by lehma 1'0, b would be a , . . -. w e have j u s t 'shown t o be singular base p&nt. ,.of. . , 1. 3 ~1. . , vrhich

.

or'is

I .

j.

,

,

impossible .Li. Q.E.D.

'

Denote by.? n , - the order of the t o r s i o n group thebrem 14 of

.HJ (pag

. 21 4-5)

+Z ,p:54 S,

~ o r s ( ~ i c ( ~; )I L)j Z ZZ ble .-amif ( [C

.n

.] lemma 1 4 , pag . 21 2),

either Hence

ied cover

0

n

or

Z/

223

of

T

n 1 6 ; moreover

'

S : by

n=6

'=) , 3 hence t h e r e would e x i s t a douwith

%2,

)( ( 6+2. <

. .

4(g)=0

but then ?=Tors ( p i c (3)) should be

([c.M~ t h . 1 5 , pag. 215).

5 5 , and z / ~ + z / ;~ Z cannot be t h e t o r s i o n

group, by Miles Reid's lemma '(compare ~ o l g a c e vs; lecture). Combining these with the previous r e s u l t s , we obtain. Corollary 12.

There a r e no i n f i n i t e l y near base points i o r

1 3 ~ 1 ,and the number

b of

them

is

0

if

TZ0,Z/2Z

I

if

T Z L / ~ =

2

if

TZL/5TL..

.

Theorem 3 . i j is birational. W e r e f e r the reader t o [6], pag. 107-1 08 f o r the proof of t h i s last: p a r t . we only remark t h a t one has t o prove t h a t cannot be more than I , and so one must show t h a t (as m=deg 9 = md + b , and d 2 2) the following cases cannot hold

3

For case i ) i t saCfi c e s t o consider t h a t

V, a quadric, con

t a i n s a pencil sf reducible hyperplane s e c t i o m , a d by taking inverse images one c o n t r a d i c t s Lemma 1 Case i i ) is managed showing t h a t

.

V

canrrot have a douole

l i n e (by a similar argument t o the preceding one ), hence i t i s a normal cubic, and then t h a t t h e r e e x i s l a pencil of quadrics c u t t i n g on V t h e images of curves i n 1291: however t h i s gives rise t o a numerical contradiction. F i n a l l y case i i i ) makes d i r e c t use of t h e existence of t h e divisors D homologous t o K {guaranteed by corollary 1 2)

.

R E F E R E N C E S [C .MJ

[I]

Bombier'i, E.

Canonical models of Surfaces of Gene-

r a l Type, Publ. Math, I .H.E 3 . 4 2

L.23

, pp

. 171-21 9.

Bombieri, E. The ~ l u r i c a n o n i c a lMap of a Complex Surface, Several complex Variables I, Maryland 1970 s p r i n g e r Lecture Notes 155, pp, 35-87. Bombieri,E .-Catanese,F.

f131

surface with

2

K =2, pg=O

The t r i c a n o n i c a l map of a

,

Preprinr ~ i s a( t o appear

i n a volume dedicated t o C .P.

k1

amn nu jam)

.

Bombieri,E-Catanese,F. B i r a t i o n a l i t y of t h e quad+ canonical map f o r a numerical Godeaux surface (to appear i n

.

B .u.?~,I)

Kodaira,K,

Pluricanonical systems on algebraic

smfaces OF general type, J, Math. Soc. Japan, 20 (1968) pp. 170-192, Miyaoka,Y.

Tricanonical maps of numerical Godeaux

Surf aces Inventiones Math. 34 (1 976), pp C73

Miyaoka,Y.

.

99-1 1I

.

On numerical Campedelli Surfaces, Corn-

plex ~ n a l y s i sand Algebraic Geometry, Cambridge Univ. Press 1977, pp. 113-718.

182

L9 1 Eo

I

Mumford,D. The canonical r i n g of an algebraic s w f a c e , Annals oS Math., 76 (1 962), pp. 61 2-61 5. Mumford,D. Lectures on Curves on an a l g e b r a i c surf a c e , Annals of Math. Studies, 59 (1 966). Peters,C.A.M.

On two types oE surfaces oE general

type with vanishirig geometric genus, Inventiones Math. 3 2 , (1976). pp. 33-47,

Ramanujam,C .P , Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. 36 (1 9 7 2 ) , pp .4l-51

.

CENTRO INTERNAZIONALE M A T W T I C O ESTIVO

(c.I.M.E.)

ON A CLASS OF SURFACES OF GENERAL TYPE

F . CATANESE

ON A CLASS OF SURFACES OF GENERAL TYPE

I. Introduction.

This lecture contains an exposition, without many d e t a i l s 2nd proofs, (they w i l l appear i n a fixture paper), of a joint res e s c h of E .Bombieri-F .Catanese , dealing whith surf aces having 2 the following numerical invariants: K =2, p =q=? (of course they g

are of general type). The i n t e r e s t about the existence 02 these surfaces was mot& vated by thefollowing remark: i f

i s a minimal surface of ge2 neral type, and one considers the numerical characters K ,

x =X (*)

S

(0,) = 1 -Wg , then the following i n e q u a l i t i e s hold

Research made when t h i s author was a member of G .N .S .A .G .A, of C.N.R..

( ~ o e t h e rs inequality) ( ~ ~ g o m o l o v - ~ i ~ asoinequality) ka~

K

2

5 9X

( a consequence of castelnuovols c r i t e r i o n ) A

Moreover, i f

is irregular,

S

S

X

. 5

1

.

admits m a m i f i e d covers A

of any order m ; a s these two i'nvariants f o r S are those of S multiplied by m , and Noether's inequality must hold f o r S

then, i t t u r n s out t h a t

KS2 >_ 2 K

(US).I n

the

f~ ,K2)

plane

then, t h e minimal i r r e g u l a r surfaces of general type l i e i n a convex ragion whose lower vertex corresponds t o surfaces with 2 K =2

,;y

=I.

, by the r e s u l t s q=pg=' (pag. 21 2) , and one is conducted t o check i P these ni-

Finally i n t h i s case one must have of

.MJ

nimal values of

K2,

x

a r e r e a l l y a t t a i n e d , trying t o construct

such surfaces.

W e have proved f i r s t l y an existence and u n i c i t y t h ~ o r e m about these surfaces, namely t h a t there e x i s t double ramified covers of t h e double symmetric product of an e l l i p t i c curve vhich have these numerical invariants, and moreover t h a t a l l such surfaces a r i s e i n t h i s way. It has been then possible t o show t h a t t h e i r canonical mod e l s a l l belong t o a family with non singular base space, hence by t h e r e s u l t s of [ll] they a r e a l l deformation of each other (with non singular base), and are a l l diffeomorphic; t h e i r funds mental group i s proven t o be abelian exploiting t h i s remark and a s u i t a b l e degeneration of the branch locus: Finally i t i s poss2ble t o prove t h a t the constructed family coincides with t h e Kur a n i s h i family (when t h e canonical models are nonsingular)

.

11. Geometry of t$e_double symme_tric product of an e l l i p t i c

curve. -Let

E

be an e l l i p t i c c m ;

ble symmetric product of

volution taking

(x,y)

The quotient nap

sj

i s t h e q u o t i e n t oP

E

to

'E

the d o ~ -

g2= E x B : E(2), E~

by t h e in-

(y,x).

n: E2 -4E")

has the diagonal OF a s r a m i f i c a t i o n locus, and E (2) i s non s i n g u l a r (campare ) . Fixing a base point i n

law on

E

mapping

E

as t h e zero element of a group

s.t, p l ( ( x , y ) ) = x-ty ind~cesa p:E(2)--4 13 (note t h a t a d i f f e r e n t choice of t h e base

, the map

point a l t e r s

p

p1:~2->E

only up t o a t r a n s l a t i o n on E)

.

The following hold:

.

Proposit ion I over E

.

Proposition

1

p :E'2)---+

bis.

S

makes

E( 2 )

I

i n t o a P -bundle

possesses a I-dimensional E m i l y oP

p

c r o s s s e c t i o n s , two of vhich i n t e r s e c t t r a n s v e r s a l l y i n only one point. Proof.

For

oa(z) = (a,z-a).

'at

a c I: consider t h e s e c t i o n %:E Denoting by

meet t r a n s v e r s a l l y i n

4 E(2)

f a t h e irnageof o a ,

f a

set. and

(a,a8). g.s.0.

Denoting by

t h e homology c l a s s oE a s e c t i o n and by

f t h a t of a f i b r e , we knovi & a t t h e second homology group oP E (21 i s f r e e i y generated by 0'

,

f

and

8

*f = 1

,(j2=

I.

£L0

(the

f i r s t a s s e r t i o n i s by t h e Leray-Hirsch theorem).

If D

i s a d i v i s o r , denote by

deg D

t h e i n t e r s e c t i o n nw;

+

ber of D with a fibre; one has Proposition 3. The following sequence is exact 0 7\ P~C(E)

P*

Pic(E('))

2

4 O

.

The proof of the preceding assertion is by diagram chasing, while, by the fact that every morphism of pn into an abelian variety is constant, one gets. proposition 4. If D is a positive divisor on E(2), D\O. we come now to the canonical bundle oP E(2): considering the we obtain the equality map n and the triviality of K ~2 * /r fi -2K = A , where = n+ a. For z E E denote by F ~ = ( z ) ; using prop. 3 and the structure of Pic (E) we can show that for a suitable choice of the base point in E K= -2 + F,. AS for positive divisor D, deg (D) 2 0 , one can sharpen prop.

r

4 to 2

Proposition 5 . If B is an irreducible curve with B =0, either i) B is a fibre or ii) B is homologous to -nK , n >_ 1

.

There are on E ( ~ )two rational pencils of curves, which will be of par.ticularuse in the seque1,that ure are just going to describe. Consider on E* the following Family of curves, {C a a e E'

I

{

given by Ca = (x, x+a) x c E] : these curves are isomorphic to E and are "translatedN of = C,. Observe that C n Cb = $ if a+b , and the symmetry a involution takes C to C-a (which is a different curve if a

2a

#

0). Denote by

that

2a=0: the symmetry operation operates t r i v i a l l y on

af0

not on c, ~f A

so Ci

i=1 ,2,3, one of the three points such

ai,

Zi ci

.

= "(c,.

b,

A

is a double unramified cover, ai ai A i s an e l l i p t i c curve not isomorphic t o E , ~ , ( c A . ) = z c ~

)

x:CA 4 Ci

"1

while f o r the other values of a n :c j p(c ) a

a

is an isomorphism.

We can summarize t h i s i n the following Proposition 6. There e x i s t s on E a r a t i o n a l pencil of curves l i n e a r l y equivalent t o -2K, whose elements-are a l l isomorphic t o E , but t h r e e particular members which a r e Brice a non singular e l l i p t i c curve not isomorphic t o E Piroof.

The family is given by

fx+(~~)f,,

. .

1

= X,(C ) the Family can be parametrized by P : then -a * r e c a l l t h a t -2K Z h Z n,(c,). As

r,(ca)

Q.-E -0.

BY an e n t i r e l y equal argument w e have

Proposition 7.

1 2 PI contains the r a t i o n a l pencil given

~ r o p o s i t i o n8. On E (2) the three curves curves homologous t o Sketch of proof.

-K If

.

C

is such a curve,

fi

C

i

a r e the only

8 *C=O.

SO

fi

b n

C = $ : by an easy numerical argument one g e t s t h a t N

i s a connected curve

N

However,

C.

C

(C)

TC"

being homologous t o

and d i s j o i n t from i t , it m 7 u t be the graph of a t r a n s l a t i o n on E ( c f . [lo] ) so one of the curves Ca: f i n a l l y the condition

+

TC

N

.

N

N

x , ( c ) = ~ c implies C=Cn

f o r some

i~{1,2,3].

ai Corollary 9. Froof. -

I - 3 ~ 1 # $ , h O ( - n ~ )= h1 ( - n ~ ) ,

Consider that

those t h r e e curves i s

Z

Ei s 2 V

,

-FA

ai

-3K.

The other a s s e r t i o n comes from R.R.

Poi? n

,

0.

so t h a t the sum of

plus duality.

Proposition 10. h0 (6( - r ~ )) =

Proof.. -

m

m+l

if

r=2m

m

if

r=2m+l

The proof of the f i r s t a s s e r t i o n uses induction on

i n the cohomology sequence of

t o obtain

\ -2mK I 3

h 0 ( 6 (-2mK)) m+l , and e x p l o i t s the f a c t that t o deduce t h a t h 0 ( 6 ( - 2 m ~ ) ) >_ m+l ml-2x1

.

For the second one we apply Ramanujam's theorem A ( [ 9 ] [C .Mg ) : I-2mK ] being composed of t h e r a t i o n a l pencil \ -2K 1 h1 ( 6 (2mK))=m-I

.

,

Then Serre d u a l i t y and corollary 9 a r e enough t o f u l f i l l the proof. Q .E .D

If

7

is an automorphism of

defines an automorphis

r(r) of

E

,

E(2' ;

(x,y)

.

( ~ ( x, T)( Y ) )

any autoin~rphism g

of

moreover has t h e e f f e c t of permuting t h e f i b r e s of

E

so induces an autornorphism ~ ( g ) of shows

H ( K ( T )(x) ) = T(x)+z(o)

that

.

E

and

A simple c a l c u l a t i o n

that

SO

p

Ker

i s given

HoK

by t h e four t r a n s l a t i o n s of period two. Then we observe t h a t , a s

pa 0 f

( a , b) =

pic(E(')),

g

and

\ pal =iraf ,

i s t h e i d e n t i t y on

must be t h e i d e n t i t y ; so i f

* * g ( F ~ ) = F g~ ,(K)=K, and * Hence g ( P ) = ? -

g*(2

* F )=g (-K+F,)=z

KeF H,

g

P

.

.

and Ker H has order 4 ai i s i n j e c t i v e , Ker HoK=Ker H , and

Fa+ F p

Then, noticing t h a t HoK

if g*

K

i s Ontotone g e t s .

Proposition 1 1 ,

is an isomorphism of

H

A U ~ ( E )onto

AUt(E'2)).

1x1, Existence of surfaces v i t h

Given a l i n e bundle with a cover

1

LJi{

=

@

S

(x,zi))

~ = d i v ( ~i)s smooth, then if

D

suppose we a r e given

( r i ) C H' (

q ,o*)

defined by a s e c t i o n

* : then one may take i n

(with coordinates

, J

D

king t h e square r o o t s of

2, p --=I. g

on E('2)

L

U ,a cocycle

and a p o s i t i v e d i v i s o r L

gL=

L

the surface

, that is

S'

S'

{siJ

of

defined by ta-

i s defined i n

by the equation S'

S=

defining L,

2

si(x)=zi.

i s smooth t o o , and

St

Uix@

I£ i s normal

has no multiple components,

Supposing f h a t D e x i s t s , smooth, one obtains, using t h e sign2 t u r e formula of 4 ;a

iff

p+

and other computations, t h a t ,

( (b-1 )F,+ Fx),

a=3,

b=l.

L

being E

S' has the d e s i r e d numerical i n v a r i a n t s

So we are left with the task of showing that the generic element of I D I is non singular if D

r-

2FxG -3KcF

-2x : in view of th6 result of prop. 11 one can restrict himself to consider only the divisor D Z -3K+F0. E

6

Proposition 12. h1(6(D))=o , / D I has dimension 6 and its generic element is irreducible non singular. Sketch of proof.

as

ID/

21-2~l+

A

ci c F-2

(i=1,2,3), i 101 3 ~1+?2$3+~o it is easily seen that ID / has neither fixed part nor base points, and Bertini's theorem applies. The first two assertions follow from R .R., Serre duality and Ramnujam's 1

(hl(0(D)) = h (~K+F,)). By the results of r5] , when D has no multiple components, St has at most rational double.points as its singulaxities iff D has no singular points of multiplicity greater than three, and any triple point has no infinitely near singularities 02 mu2 tiplicity greater than two; moreover if this conditions are satisfied , a minimal desingularization g:S -4 S ' of S' has the desired numerical invariants. \Je observe that the condition is fulfilled in our linear is irreducible (by the genus formula), and it is system if D possible to describe explicitly which are the reducible curves in theorem

1 -3x+F0 I , and which the Itbad"ones:

we omit this point for the

sake of brevity, as well as the verification of Proposition 1 3: i)

S

is a minimal surface

ii) S ' is the canonical model of S (see [C .M.] finition and properties)

Por its dg

4 E(*)

if f : s t

iii)

h=pof og : S

i s the projection induced Prom L, i s t h e Albanese mapping of S

.

-+E

W e have moreover Proposition 14.

Let S, So

the above described way: i f

be

two surfaces obtained i n

W: S -+So

i s an isomorphism,

EZ~ l b ( s ) Zb(so)ZE0 ~l and under t h i s i d e n t i f i c a t i o n of E and E, Ji( i s induced by an automorphism y oP E taking I t o Lo, D

to

Do.

Sketch of proof. and suppose that of

Y

By proposition 13 you may i d e n t i f y

E

, Eo

commutes with the respective Albanese maps

S, So.

induces an isomorphism A

reover the f i b r e s two, and bundle of

of the canonical models

mc

i f f @$(x+y) i s the unique hyperelliptic

$: t h i s remark enables us t o define

2

IV. Surfaces with

S ',

of the Albanese map are curves of genus

F

f (x)=f(y)

y '

K =2,

;y = I ,

q=l

ly on

E (2)

.

a r e double covers of the

symmetric product o f t h e i r Albanese variety. Let for

9 :S

--+E

= Alb(~) be the Albanese map of -1

(@I.)

, xu=

-

S

, and ,

u E X , s e t G~ = p K + F~ F,. Ire r e f e r t o [G.M.] £or the proof of the following Pacts i)

yu

c s

hO( s

que curve i n ii) iii)

CU

,6 ( I ~)=I ) 1 KU 1

)

.

(denote then by

i s generically irreducible

hO( 6 ( K + K ~ ) ) = 3

'+v

e B

.

p (cU)=3

cU the uni-

Fixing

c

v e E , !f u

we have the divisor

G E

.

CU+CV-u

E

J K + K ~ ~

The mapping u

Proposition 1 5.

lornofphic mapping \YV of ~ r o p o i i t i o n16'.

into

E

>- Cu+CV-u 1 K + X2~P/ 2

.

A

The image

defines an ho-

of

Yv

so

y V i s invariant

is an irreducible

r a t i o n a l curve. -C V-U +C v - ( ~ - ~ ') CU+CV-u-

PrOoEo

the involution which takes 1

isomorphic t o IP mapping)

.

u --4v-u,

and whose quotient is

yY: JP1 --+

(denote now by

by

(K+Kv

1

the induced Q.E.D.

Froposition 17.

For general

0

v

i s non singular.

Sketch of proof. An uniformizing parameter on the univers a l cover of E induces a derivation D on each vector bundle, and, given a section a , we use the c l a s s i c a l notation o ' f o r ~ ( 00 )

{ Ui)

on which 6 (K ) i s t r i v i a l i z e d f o r U u : then, Ou being the section of 6 (Ku) defining where a depends holomorphically on u iu

Take a cover u near

The conditton t h a t read a s

of

=

independent of

i

"

.

not of maximal -rank a t

(&)

'i (v-^u)

Now, for general

yi

.

yv is

-a iG

that

S

A

v

+ A

A,

, if u

for

+ v-u

i s a regular function on

A

Ui

A

,

.

A

u=u can be

a s u i t a b l e constant

t h i s e q u a 1 i . t ~implies

is a cocycle deiining 6 (K~), one f; ( 4 = f)ihence gets a f t e r a simple computation t h a t

~ u then, t if f

. .(u)

rj,

1J

1J

i t i s a coboundaxy i n

HI( S , 6 ) . A similar conclusion can be i s not of maximal rank a t any point.

yv

drawn i n general i f Denoting by A . .(u)

(vi\

covering

f

j($1

=

P*

a cocycle f o r bE(u-O) ( r e l a t i v e t o a

1J

compatible with

(

1;

.(G)

A,&)

1

pi]

t e for a l l

u

E

have that

: @*H'(E,

0 ) --+H'(S,@) A; .(9)

an isomorphism (compare f3-j ), so the homogeneity of

, we

)

A, j (3

is however

i s a coboundary. By

under translation, it i s possible t o w r i -

x: .(u) -&LJ--A . .(u) = Yi(u)

- Yj(u) ,

yi ( ~ 1 r6

with

0).

13

??OW,

Ax.J.(u) Aij'O)

-

integrating on any path from exp( , ~ ; ~ ( t ) d t )

u e ~ p ( yj ~ s( t ) d t

and so

0

0

to

and u

u

, one

gets

should be two

'

l i n e a r l y equivalent points, which i s c l e a r l y absurd. Q.E.D.

The f a c t t h a t

fcU

Cu*CV=2 and the i r r a t i o n a l i t y of the pencil

1 implies t h a t t h i s pencil has no base points hence

Froposition 18. Y v (KcKv 1 has no base points and bv i s --

an irreducible conic. Froof.

It suf£ices t o show that

6ucible), and yet only t h a t non sincgulm)

. Take then

deg

Cu,CU,

h

deg dv=2(5

being i r r e -

2 2 ( bir being generally

general, meeting i n two points

x,y : Cu+CV-u, CUI+CV_u, represent two points i n the plane

IK+K~I

by

which l i e i n the intersection of

IK+Kv-X /

.

with the l i n e Q.E .D.

Proposition 19. point

, there e x i s t s a y c cU fi cut.

For general y e S

(u,u*)a ~ ( 2 )(with

+*I,

s.t.

wique

Proposition 20. The cor~espondenceoP prop. 19 extend t o a holomorphic mapping f ' :S -4E( 2, Moreover f ' f a c t o r s as

.

g s ->sf

where a) S t i s the canonical node1 of S,g the canonical happing b) f i s a F i n i t e map of degree two c) t h e r e e x i s t s a line bundle L on E (2) and a section a. c H*(L ) such t h a t S is isomoqhic t o the surfa-%\e(2),

.

@

ce of the square r o o t s of

o

in L

V. Some r e s u l t s on the structure of these surfaces anclon-their

deformations. Sharpening the r e s u l t oE prop. 20 one can show t h a t i-t i s possible t o choose

-

~ a 3 r F, ( e s s e n t i a l l y by using

AU~(E(~))).

We w i l l sketch very rapidly the construction of a family containing a l l the canonical models

+R

St

of our surfaces: take

the l o c a l universal family of e l l i p t i c curves over the

Siege1 upper halfplane, and form i t s syrmnetr-ic f i b r e p ~ o d u c t (2)-

)H

.

A l l the i n v e r t i b l e sheaves

6,(2)

(3 f

- F,)

fit to-

I 2

gether t o form an i n v e r t i b l e sheaf (on

( 2,

and o m can chog

se sections

...

so,

St =

to...t6,

5 (2)+

H

4Hx p6 i s

S

2tiSi

~

(*)

2,

such t h a t f o r a l l defines a r e l a t i v e divisor 6 on

l'(e B

@ ,

(we use the terminology of 18J )

.

Then on

(2)x

p6-

defined azz i n v e r t i b l e sheaf t h a t we s t i l l denote

6

and a r e l a t i v e divisor l i n e a r l y equivalent t o 2 d?. by by the square Define I--) 4 t o he the variety defined i n

5

6

r o o t s of a section defining : then we have a f ami3y 3 Hx p6 and an open dense subset V of iIx,?r6 such t h a t the f i b r e s over points of V m e a l l the canonical models of the surfaces we a r e considering. B y using the r e s u l t of Tyurina about l o c a l resolution oE s i n g u l a r i t i e s of such Earnilks ( ) we deduce t h a t our surfaces a r e a l l diffeomorphic.

fi 11

=osition

21

.

Sketch of proof.

n, (s)

h*

XI

(E) *

By the above remark we may consider the A

f

i

A

p a r t i c u l a r surface £or which D=Fo+C1+C 2+C 3 ' -

Then we observe t h a t by Van Kampenfs theorem

n, (s)%, (S ) ; c a l l

n

fi

F ~ = £ - ~ ( F ~and ) , observe t h a t St-F, - E -{o] i s a d i f f e r e n t i a ble fibye bundle with f i b r e a smooth curve of genus two. Take U t o be a small disk around 0

in

E : (pof)" ( u ) =

s

IU

is cont r a c t i b l e t o F, , which i s simply connected, a d we can apply agair, Van Xampents theorem t o the open s e t s S , S 1 -F, i n S ', A

lu

E , together with the exact honotopy sequence A of a bundle (where u e U- {O [ , FU i s t h e f i b r e of S over

u

and

E-{o~ i n

u) , t o obtain the Poll.owing diagram, comnmutative, exact i n t h e columns sad the rows , 'and which gives e a s i l y omt r e s u l t .

Froposition 22. then

If

0

denotes the tangent bundle of

S,

h1(GS)=7. Sketch of the proof ( i n the simpler case when

D

is

smooth). Using the Hirzebruch Riemann-Roch theorem one sees that O 1 2 i t i s equivalent t o show t h a t h (QS cg QS)=l. One e x p l o i t s f i r s t one f a c t : you have an involution on S determined by 1 f 1 : s - - + ~ = ~ ( 2 ) , so B O (OS @ Q2 S )= H 4- x H , where H* i s the

-

subspace of invariant, H- of a n t i invariant sections oE 1 2 $2, e, Qs, and by one side the sections of H+ correspond t o 1 2 1 sections of RX x QX x L , Z Q ~ 6 (),f by the other, sections X

of

H-

vanish on the r a n i f i c a t i o n locus R I * RS1 Q ( K ~ - R ) Z 52, x f (K,)

from sections of

6

,

hence they come

, Secondly one

1 e x p l o i t s the uniqne sectioa of Q (wllich defines a t r i v i a l subX 1 I bundle T of QX) and the f a c t t h a t QX o ~ ( P )52; ^.b(-f +F,)

has no non zero sections, t o i n f e r t h a t every section oP 1 (P) 2 RX @ O(P) comes Prom a section of' the subbundle T

@6

rG(r),

which, though, possesses only one section. The sane

method can be applied t o shov t h e vanishing of

HO

(Q:

@ £*(K

))

Finally, when D is non singular, one cal take the constructed family a d show the injectivity of the Kodaira-Spencer map: then l o c a l l y our family is t h e Kuranishi family (compare

[6J

9

[17J 1

.

References

CENTRO INTERNAZIONALE MATEMATICO E S T I W

( C.I.M.

E.

SOME REMARKS ABOUT THE "NULLSTELLENSATZ"

A.

TOCNOLI

SOME

REMARKS

ABOUT

THE

"NULLSTELLENSATZ"

.

Tognoli

A.

(Pisa)

Introduction.

and Let

K

be a field

Vr

the ring of the regular func-

K" .-

tions (in the sense of F.A.C.) on

Kn

.

Two problems are now natural: 1)

to carachterize the ideals of definition of

2)

to carachterize the ideals of definition of

r Kn

We have the following results: Theorem I.

-

An ideal

I

of

r

is of definition K if and only if is intersection of maximal ideals.

Theorem 11.

-

An ideal

of

I

. . . ,xn]

K[X~,

is of de-

finition if and only if for any homogenous polynomial

P

va-

nishing only in the origin we have :

In the following we shall prove theorem I1 (see 141 ) , theorem I is an easy consequence of proposition 1

.

Preliminary remarks Let

I

be an ideal of

K [xi,. .. ,Xn]

, in what follows

.

ideal shall mean "proper idealtt

We shall call set of zeroes of

I

the set:

An ideal I is called ideal of definition if

Let

K

pK

be a field, we shall denote by

of the homogeneous elements

P(Y1,Y ) = 0 2

P

of

K [yl, Y

2

1

if and only if

the set

such that

Y = Y = 0 1

2

.

Definition 1.

- An

to satisfy condition

ideal

KF1,

of

I

if for any

e(

p(gl ,g2) Q I

I/

...,xn]

PKwe have:

P G

.

g1 G I,gg E I

It is a natural problem to see if

is said

$

is empty. The

following proposition gives the solution:

Proposition sed field and

N

q

- Let K

be a non algehraicaliy

such that : 0

1)

P(Y1, ...,Y ) = 0 if and only if 4

2)

the coefficient of

Proof.

,

Ks 3 K

Let

K C

d Y1.

is purely inseparable on G

P12

...

2

1

,

= Y = 0 q

cl = deg P

)

Ks

that

K

P = y

K

. Let

K

s

-

K

.

and K

exists and

S

.

K

and sup-

the Galois group of a Galois extension of

containing W If

=

1

be the algebraic closure of

It is well known (see

pose

P

Y

the maximal separable extension of

S

G-

. Then there exists a homogeneous poly-

. . .Y91

P E K[Y~,

nomial

1.

K

. 1

+

GJ

Y2

is invariant under

and G

Moreover we have: if The polynomial

P12

P12=

, and x

bK

2

g(Yl

+

W Y 2 ) then

g e G

hence then P

(x)= 0 12

Y =Y = 0

1

is homogeneous of degree, say

2

dl

.

dl Y1

and the coefficient of

is

1.

In a similar way we define:

and after

q

-1

steps we obtain ,the desired result.

New suppose

,

K = K

then there exists b\ B

-

I?

KS

is purely inseparable on K and hence has m a minimal polynomial of the form xP = 0 . In this case By hypothesis

G)

P

we define

P

12

=

(Y1 + cJY2)pm

and by the above arguments re

end the proof.

Corollary.

Lf

K

is not algebraically closed

not empty. The proof of the theorem.

We wish to state the following

Theorem an ideal

1.

Let

I

of

K

be a non algebraically closed field then

K[x~,

. . .,xn]

is an ideal of definition if

and only if it satisfies the condition d

Proof. T he part:

I

of definition

. 3

1

is clear. Now we shall prove the other implication. Step. I.

(reduction to prime ideals).

satisfies oL

F i r s t we s h a l l prove t h a t if Let pose

Y

d 1

We have: then

P(Y ,O) = yd 1

d g E I

=>

I

gE I, if

a?d hence

n>?O,

=n I

j

j

satisfies

I

(we remark t h a t

I =

pne

b u t we c a n u s e

be t h e decomposition of

PK

1.

a s in-

I

j

deals.

W e wish t o prove t h a t

.j

fi .

l e t u s sup-

and t h i s p r o v e s t h a t i f

1

t e r s e c t i o n , o f priiiie

I

then I =

a

h a s 1 a s c o e f f i c i e n t , d = deg P ( s e e p r o p o s i t i o n 1 ) .

Let now

any

pK and

P E K [yl ,y2] be an element o f

nd =9 g G I

g q H

satisfies

I

a i f and o n l y if

satisfies

I

satisfies.

It is clear that i f a l l the

I

satisfy

j

O(

then

I

h a s t h e same p r o p e r t y .

On t h e c o n t r a r y suppose

By t h e f a c t t h a t hence

gbI1,

is impossible.

or

I

g1411

I

satisfies

satisfies ( Il

Ed

& and

we have

I1

gglEI1

does

and

i s a prime i d e a l ) and t h i s

So we have proved t h a t i t i s enough t o v e r i f y t h e theorem i n t h e c a s e

Step 2 . Let

I = K[x~,

(The c a s e

d

. We

Z(1)

Z(1j

# @

= g

d

).

.

K [ x ~ , .. ,x,]

and suppose

wish t o prove ' t h a t , i f

.. .,xn].

(proper) ideal have

i s a prime i d e a l t h a t s a t i s f i e s

be an i d e a l o f

I

satisfies

I

Z(1)

= @

,

I

then

I t is s u f f i c i e n t t o prove t h a t f o r any

,

I

maximal among t h o s e t h a t s a t i s f y

b(

we

.

B y t h e above r e s u l t s w e can suppose

I

prime.

B y t h e proof o f t h e n o r m a l i z a t i o n theorem ( s e e [I]

know t h a t t h e r e e x i s t

t h e n t h e c l a s s {Y

\

I

in

d2,

...,dn E N

. .'n3

K [ x ~ ,.

R e l a t i o n s ( 1 ) a r e i n v e r t e d by

such t h a t i I " :

/I

i s i n t e g r a l on

) we

.

5 : K[X1, . .. , Xn3

Hence we have an isomorphism

>-

.PI,. ..

,yn].

Let t r a s c

.

[XI,. . . ,X ]

K

{W13

fn-q

%-q

, . . . ,{Wn-,\

then a f t e r

~4,...,x,]

n

-

q

steps

,...,wn'1

+K[w,

. .., I W n ] J

K [{Wn-,[,

i s a b a s e o f t r a s c e n d e n c e of

(K[X~,...,XJ) '

"n-q(~)

...

Let u s now i d e n t i f y and

:

,

a r e i n t e g r a l on

, . . . ,{wn]

and [Wn-,+,\

= q

'I

we have an isomorphism where

4

I

with

4

n-q

(I)

Let u s d e n o t e

2 p,\,... , Bn-q =

We remark t h a t

p(a,b)

= 0

~ . t

if,

n

: K[X

Z

K I x ~ , ,xn]

with

K[w~,

. . . ,Wn I

. =

{ w ~ - .+, ~Z q . =

{wn

1

and

{wn-,\.

= I

satisfies

and o n l y i f ,

a

d

=

xnJ -+KC%;

be t h e n a t u r a l p r o j e c t i o n and

is t h e i d e a l g e n e r a t e d by

Z1,

=

i f and o n l y i f , f o r

b = 0

.

,..., Bn-q , zI,....,z 9 j n-'(ml),

...,Z 'I .

where

m'

I

We wish to prove that the maximality of

I

satisfies d and hence, by

, we deduce that q

Before we prove that

and therefore

.

K

gebraic on

= 0

satisfies condition

dL.

in

use All the monomials of are in

m1, hence

1 2 P(ao, ao) = 0

if and only if

g

1, 2

that have positive degree

)

if, P(gl ,g2) 6 % ' I , ' and only if,

where

Kte', ..., en -q 1.

P(g

and

a '

0

a2

1 2 hence we have: P(ao,a )

satisfies oC

I

are element of

0

=O

0

a '

=

a2 = 0 0

This proves that 'We

and in these hypothesis

satisfies the condition

ol

in

K[x~~...~x~J, Let now suppose

I h l y h2 6

P(hl,h2)

~

b

xn3~, ,

?n, then p((hl1

jhil is the class of

hi

modulo

I

Em

h21) Q

where

.

But, by the above prodf, we have: h.

~ BK-and let ~ us

P E

hi&

%' and hence

(because ~ D I ) .

We can now suppose In particular

X1

mod I

K.[x~,

. .. .Xn-)

/I is algebraic on

is algebraic on

K , hence there

K

.

9

exists

p(Xl)

ai X;

=

i =o We may suppose

,

€1

P(X1)

a t e K

.

h a s no z e r o i n

K

,

b e c a u s e , on

t h e c o n t r a r y , we s h o u l d be a b l e t o f i n d an element of ( s e e [I] pag.

Kn

in

Z(I)

1%').

Let u s c o n s i d e r t h e polynomial:

From t h e f a c t t h a t A

that

P(Y1,Y2)

We have

,

E

P

E

p r o v e s t h a t we can n o t suppose

Let g E.

I

Suppose

g

#

Let US d e n o t e by and Let

a n d , hence

1 tZI

and t h i s

.

Z(1) = $

be a prime i d e a l t h a t s a t i s f i e s

K [ x ~ ,.. . , X n 3

I

I

( ~ e n e r a lc a s e ) .

such t h a t

We must prove t h a t

by

it follows

BK .

P(X1) = P ( X l , l J

S t e p 3.

K

h a s no z e r o i n

\Q

I tu

I

%(I)'

g CI

aC

and

O

.

and l e t u s d e f i n e t h e i d e a l of

K[xl.

(P =

~ - X ~ + ~.S. .x,+~ E K3 F ~ ,

.. . , X n + 2

generated

.

A

1 = { h c ~ [ ~ ~ , . . . , X ~ +h ~€] 1l g f~o r some

It i s not d i f f i c u l t t o v e r i f y t h a t we wish t o prove t h a t

'. I

satisfies

a!

h

I

.

t

EN).

i s an i d e a l and

Let

e PK

P

and l e t u s suppose

g1 . g Z

Kkl,.

. . ,Xn + l1

such t h a t :

(1) g s ~ ( g l , g 2 ) =

q(X1, i

where

h

i

hence

X

n+l- g

a r e some g e n e r a t o r s of

- - '

where t h e

=

n

. Taking

a r e o b t a i n e d from t h e

aCq

Xn+l

I

....X

+

= 0

,

and

from ( 1 ) we deduce:

1

tution

...,Xn i l )hi(X1.

g

Wi

by t h e s u b s t i -

and by m u l t i p l i c a t i o n o f a power o f

g

t o e l i m i n a t e t h e denominators. From ( 2 ) and t h e f a c t t h a t /r

~ , E I,

d

satisfies

oC

we have

*,

g 2 e 1 .

The i d e a l sfies

I

,

A

I

hence

h

h a s no z e r o ( I 1 G I

h a s no z e r o ) and s a t i -

.

We have t h e r e f o r e :

Now we r e s t r i c t o u r s e l v e s t o

lf =

0

and we o b t a i n

and the theorem is proved because

Remark 1.

If we denote by

is prime.

I

6)

the set of homogeneous

K

polynomials, in one or two variables, zero only in the origin, theorem 1 is true for any field

. In the following

K

we shall assume this hypothesis.

Remark 2. the cone in

Associating to any affine variety

ICnfl

Theorem.

-

OC

C

Kn

we obtain the following:

A homogeneous ideal

I C K

an ideal of definition if and only if dition

V

I

El,.. . ,Xn+ll satisfies con-

.

Remark 3.

Let

V

C K~

be an affine variety and

the ideal of all polynomials zero on

Iv

.

V

W e have:

Theorem.

- An

ideal

ideal of definition in condition d

Proof.

I PV

K , .

KIX1,

X I

v

if, and only if,

= Py

is an

I

satisfies

. Let V : K[x~,

.. .,xJ---+

projection, it is easy to see that in

. .,

PV

be the natural

~ ~ ' ( 1 ) satisfies

. . . ,Xn1 and the thesis follows.

d

Remark 4.

I be

Let

that satisfies d

n prime ideal of

I

then

K[x~,

. .. ,XnI

is absolutely prime (id est

K E .~..,,xnI, K

N

the extension of C1

K

,

I

in any

subfield of

is prime). A#

I@K

Proof.

defines the completion of the variety

K

V

I

defined by

. If

is irreducible the completion is

V

also irreducible (see C 2 3 ) and the result is proved.

Remark 5.

Let

IC K

satisfies condition d

fX1,. . . ,Xn7

then

is absolutely unranified

I

N

(id est for any field

K 3K

be an ideal that

we have

I@

=

m). K

I @ i? defines the completion o f the variety

Proof.

K

V

I

defined by

(see

l2j

)

and hence coincides with its

radical.

Remark 6. generated

K

-

Let

K

,

K #

,

be a field, then a finitely

algebra 4 , is isomorphic to the algebra of

the polynomial functions restricted to an affine variety if, and only if,

Remark 7. ideal of

Let

K

be a real closed field and

...,xn3 . Then

K[x~,

I

I

an

is an ideal of definition

if, and only if:

Proof.

Clearly if

I

is of definition

satisfied. We wish to prove that if

p)

d

is true. It is easy to verify that if

P

then

I =

P)

is

is true then I

satisfies

fi ; using the arguments of theorem

1 we

deduce that it is enough to prove the remark in the case is a prime ideal. Let

P E K[Y~,Y~~be an homogeneous polynomial of

can suppose

P

is irreducible of degree two (see

@K

133

)

If 2 P ( Y , Y ) = ~ Y ~ + ~+Y Yc 1 2 1 1 2 2

, ~P c B ~

have : b2

Let us suppose

-

4

ac 4 0

p(glrg2) € I

from (1) it follows that

and hence :

and

e)

gl C I, g2 G I

true and the remark is proved.

is true, then and hence

& is

Y

.

References

fi]

L .BERETTA and A.TOGNOL1. "Some basic facts i n algrebraic geometry on a non alge'oraically closed f i e l d " , ~ . Scuola Norm. --

[23

D,W,DVsOIS.

J .J.RISLER.

3

31 9761,

34-359.

"A nullstellensatz for ordered f i e l d s n , m k i v

for Mat.,

131

m, ~ e r IV,

~uq,

8 (1969). iil-114.

* m e caracterisation des ideat= dees vmietes

algebriques r e e l e s " , ~ .

Acad, Sci. P e i s 271 (1 970);

1171-1173. [4'3 W .A.ADKINS, P.GIANI1, A,TOGNOLI. "A Nullstellensatz for an algebraically non-closed fieldN. To appear on B,U&f.I..

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