Algebraic Geometry--Open Problems by C. Ciliberto, F. Ghione, F. Orecchia

Algebraic Geometry--Open Problems by C. Ciliberto, F. Ghione, F. Orecchia

By C. Ciliberto, F. Ghione, F. Orecchia

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H O ( ~ v~m), where ~V sheaf. We d e n o t e by ~(V) the K o d a i r a d of V and by X(O v) = ~ (-l)ihi(c9 v) the Euleri=O characteristic of V. let sr(~PV) be the r - s y m m e t r i c p-forms closed we m e a n a c o m p l e t e F on a d - d i m e n s i o n a l the m - p l u r i ~ e n u s dimension Poincare By v a r i e t y sheaf results fix an a l g e b r a i c a l l y zero. the d i m e n s i o n is the c a n o n i c a l and p r e l i m i n a r y Finally, tensor for all p, O < p < d, of the sheaf of r e g u l a r ~P.

At Geometry", 35 In the first results where s e c t i o n we r e c a l l on conic bundles: the case S = ~ 2 some r e l a t i o n s the m a i n is studied. reference Moreover between conic bundles negative Kodaira In S e c t i o n for this is [B] we p o i n t out and t h r e e f o l d s with dimension. 2 we find a d e c o m p o s i t i o n up to i s o g e n y , as d i r e c t the P r y m v a r i e t y of the g r o u p A2(X), sum A 2 (S) @ A 1 (S) S PX' w h e r e associated s u r f a c e S w i t h q(S) w i t h X.

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. the C h o w group of the c y c l e - c l a s s e s rational classes equivalence, by algebraically find a d e c o m p o s i t i o n Prym variety Aq(x) First of all, We denote cq(x) q modulo the s u b g r o u p to zero.

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