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 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 comments in regards to the "Nullstellensatz".

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Que; on pose Li = n(~$ ,Proposition4. surfaceX = K m e r associde h A. $0 est une surface IC3, av~el6esurface de D6monstration : Hontrons d'abrd que X est lisse. 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.

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 . 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. 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. J ' 21 = ZLi Picl~). = i#j. Sup~osonaqu'il e x i s t . me surface de Kmmer. : 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 .

I-x S6m. 13 (1960/61). [R] M. RAYEIAUD : CTI R. TORELLI : "Sulle variets di Jacobi" Familles de fibr6s vectoriels sur une surface de Riemann. sh. Bourbaki 11'316 ( 1966/67). Rendiconti dells R. 22 ( 1913). ' Arnaud Beauville Universitd d'Angers Fac. 'des Sciences BE. ) METHODS OF ALGEBiUIC GEDMETRY I N CHAR. p AND THEIR APPLICATIONS ENRICO BOMBIERI METHODS OF ALGEBRAIC GEOMETRY IN CXAR. p AND THEIR APPLICATIONS TO ENRIQUES' CLASSIFICATION Enrico Bombieri Scuola Normale Superiore, Pisa I. The aib of these lectures is to illustrate some of the algebraic techniques needed for algebraic geometry in char.

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