Algebraic Geometry — Open Problems: Proceedings of the by E. Ballico, Ph. Ellia (auth.), Ciro Ciliberto, Franco

By E. Ballico, Ph. Ellia (auth.), Ciro Ciliberto, Franco Ghione, Ferruccio Orecchia (eds.)

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Additional resources for Algebraic Geometry — Open Problems: Proceedings of the Conference Held in Ravello, May 31 – June 5, 1982

Example text

P l o ~ et ; on p o s e g : V~U est -1 Pi = g (pi) une fibration est ia cohomologie eonsid~r~e en droites r~union exceptionnel ce qui suit Le m o r p h i s m e tandis se coupant est toujours 2. de X a u - V= g-l(U). projectives, de d e u x p l a n s dans d'ordre que en un point. La ~ coefficients entiers. a) Le g r o u p e isomorphe H4(G) contient ~ H4(G)~H2(R), et un ~l~ment cl(N) est d'ordre 2 : il un 61~ment est d'ordre en effet 2 dans e2(~). ) ~ cause )H4(G) = O, H 4 ( P . ) = O. 1 d) Le g r o u p e H 4 ( U ) c o n t i e n t un ~l~ment d'ordre 2.

4 of previous the The birational 2 ~S = 5 does case Proposition. to C x p 2 . construction due In This is to E n r i q u e s not occur in (c) 2 if ~S = 5 t h e n fact, a consequence (see family [E], of §8 a n d X is a classical also [Co], p. 4 7 3 - 4 7 4 ) . We about Ch. summarize conic I in here bundles the case without on a surface S = ~2. 5 Let (i) The morphism (2) Let ~X be f:X Since of + S be some well S. They all questions contained case f is in sheaf is a l o c a l l y free There a curve C i__nn S w i t h - for - if every f -I s is (s) to involve [B], local extended in surface.

Lieu Nous a l l o n s une cette un description ~ cette que une Pour donc pour loin que cela dis- de s a v o i r pour classe prou- a montrer les vari6- de v a r i 6 t 6 s pour existe. t 6. FI BRES EN CONIQUES. D~finition : existe une fibres sont I1 tion On d i t surface des est coni~ues facile rationnelle sur (ou, let de c o u r b e s un qui fibr~ Sous qu'il - - si en les - sip est pest On d i t cependant le Proposition surface est Soit rationnelle la tout r~sultat la une s'il dont une les une une applica- courbe rationnelle birationnellement radu ~quivalente classe de vari~t~s rationnelles.

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