By Arnaud Beauville (auth.), G. Tomassini (eds.)

Lectures: A. Beauville: Surfaces algébriques complexes.- F.A. Bogomolov: the speculation of invariants and its purposes to a couple difficulties within the algebraic geometry.- E. Bombieri: tools of algebraic geometry in Char. P and their applications.- Seminars: F. Catanese: Pluricanonical mappings of surfaces with okay² =1,2, q=pg=0.- F. Catanese: On a category of surfaces of normal type.- I. Dolgacev: Algebraic surfaces with p=pg =0.- A. Tognoli: a few feedback concerning the "Nullstellensatz".

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Serre's duality. If n = dim X then $(x,L) is dual to (x*%@L-I) ; g. The theorem of Enriques-Severi-Zariski-Serre. 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 .

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(=). 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 ) . ,/(a) e s t isonorphe 2 P . 1 C o r o l l a i r e 9. Soient X (resp. f ' ) l a fibre 2 q6nCrique d'une f i b r a t i o n de ~ummer. 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'.

D' * be t h e covering map. i s a section of z = [zi) xl,. ' 4 X If L' ' i s the i n v e r t i b l e sheaf we see t h a t by the special case considered before we have HP(X*,~,@ L**l) = 0 if p + p < n. Everything n o r follows from t h e f a c t t h a t a d i r e c t f a c t o r of t h e previous group. H ~ ( x , ~@ $ f l ) is To show t h i s , we have t o produce a s p l i t t i n g of which i n t u r n w i l l follow i f and only i f 4 -=-Tr&' In order t o e x h i b i t the required s p l i t t i i ~ g ,we use a t r a c e map.