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

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Of the Our p u r p o s e A2(X) in terms is to of the • AI(s) . 10) = ~(f,~l) and Therefore nonsingular divisor system f I~ We say that be seen of L look at the e m b e d d i n g -i TX Tx we can find an i r r e d u c i b l e belonging (i)~ ~ X ' to the c o m p l e t e linear by use of B e r t i n i ' s is a t a u t o l o g i c a l in the sequel X ÷ ~(f,~xl~i) our a r g u m e n t s divisor of X do not d e p e n d Theorem. 4). 1. Y = f-l(c) U = X\Y Let in this i be the d i s c r i m i n a n t be the ruled We d e n o t e U , by C by surface of the s i n g u l a r s e c t i o n we w a n t T to d e s c r i b e the r e s t r i c t i o n the i n c l u s i o n T + U curve of of Tx and we set X and let fibres.

To C x mapping a 0 and hO(sl2(~)) = O, nonsingular on a or the g e n e r a l hO(sl2(~))- there bundle surface. to f a m i l y Suppose to a c o n i c a O, the A l b a n e s e curve Furthermore belongs equivalent bundle on a surface S birationally equivalent pl ; exists S is a D e l a morphism Pezzo the anticanonical surface sheaf X ~ C such with -i ~S that Pic(S) the ~eneric = Z @enerated Moreover 2 1 ~ ~S ~ 6. 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.

0 is a l g e b r a i c a l l y [B], and involution. - T*). 2 is n a m e d and an e l e m e n t to Pic°(C) is not n 40 trivial. If {h }~ is a s y s t e m of local equations of C, then the e q u a t i o n ~ox~+ 4+ 4 ~0 locally defines a conic bundle the d o u b l e in Pic O (C) . in the p r o j e c t i v e X having covering scheme ~((gs(D) C as d i s c r i m i n a n t ~:C ÷ C d o e s n ' t split curve. since • £9S(n) @ C9S) Moreover ~ is not zero 41 §2. The C h o w group Let f:X + S A 2(x) of a conic be a conic bundle.

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