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
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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
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
/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
-
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 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
(&*(-
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 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
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 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.-- 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
.
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+ $3')
n s s o c i n t e d t o t h e l i n e a r system
Ym : P(T) -+ '?!I
K*
- e > 0 , then
K
fi-
( x , ~ ) has
the trivial bundle
F= 0
s a t i s f i e s our c r i t e r i o n and so ne have the following Lempa.. -
If
K2
-e
-
x -17 ~ ~ .
i s an algebraic family since i t i s the v-nion of aige-
Then
b r a i c f a m i l i e s f o r a f i n i t e number of -he bundles So Re have proved t h e following theorem: Theorem. The family of curves of genus
g
Fi
.
c o n s t i s t i n g of dl
i s m algebraic f m i l y . 1 C o r o l l a ~ r .There a,re only a f i n i t e number of homoloqica~ ~
curves ses -
X
with
f(X)$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 ) = - ( 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.
_ 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
