Algebraic Surfaces by Arnaud Beauville (auth.), G. Tomassini (eds.)

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

Lectures: A. Beauville: Surfaces algébriques complexes.- F.A. Bogomolov: the idea 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 ok² =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 in regards to the "Nullstellensatz".

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B=) ( ~ * a . e * ~=f (a 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' ) . 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 .

I1 r g s u l t e a u s s i t d t de l a proposition 8 que lorsque f e s t l a f i b r e d'une fi: 4 bration de Kummer, s X ( f ) e s t isomorphe come module quadratique b (DL) . Proposition 10. if i b r e s con- =, f l a c l a s s e dans P ~ C ( Xd'une ) f i b r e de q. ; a l o r s , X est: une surface de Kummer sp'eciale,, & q e s t une f i b r a t i o n de Kummer. ~6monstration: I1 r 6 s u l t e a u s s i t d t du l e m e 19, l s r e p a r t i e , que l a forme quadratique sur SX(f) e5t n6gative non d6gGn8r6e.

The'dual of tde vector,space H1(c,L) 0 to the space H space. (c 'I ,QD~ L-~ ), The point is that is isomorphic (via differentials on C) so that we ask about the vanishing of this last Now recall that a nor,-zero meromorphic function on a curve C (proIf f E H0(C,% @ L-l) jective:) has as many zeros a:. poles. s a contradiction if - deg (wC L-I) = deg L - (2g - 2) . 3) if f # 0, 0 deg div(f) of yC @ L-I is 0. 3) is a form of Kodaira's vanishing theorem for the curve C. 4) that we proved ,dth the ~reviousargument.

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