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

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

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Notations, definitions, and p r e l i m i n a r y §2. The C h o w group A2(X) §3. The a l g e b r a i c representative §4. The c l a s s i c a l intermediate §5. Some o p e n q u e s t i o n s . results. of a conic bundle. of A2(X) . Jacobian of a c o n i c b u n d l e . References. INTRODUCTION In this paper we s t a t e some bundles study X on n o n s i n g u l a r describe, with negative the C o n f e r e n c e A reason for this of We m a i n l y of cycles equivalent of to zero J(X) . v e r s i o n of a talk on June conic in the c l a s s i f i c a t i o n dimension.

We o b t a i n %f,y = %f , ( A~* 8 - s , ( T ) ) h e n c e ~*~ £ A 2(UT). = 8"[,s,(T) Therefore = ~. s*~*~ It f o l l o w s 8 e A 1 (T), = ~ E A 2(T) . Then one has A 2 ( U T ) -~ ~*A 2(T)(D~*A I(T) "s,AO(T) = ~*A 2(T)~)s,A I(T) . d. 2. Proof. We have written in the (a,b) to s h o w form all elements f*a + f * b - i , T , of A2(U) can be for some since £ A 2 (T) • A 1 (T) by L e m m a (~,8) f,f*~ = O and Moreover f,w = n'f, 8 = n*8' with B' = f,Y ~*A2(U) Recalling 2) that ~U is s u r j e c t i v e .

To prove the converse, the classes of divisors let Picn(y ') be the group of numerically equivalent to zero. The 50 quotient group Num(Y') by C o and F where Choose shows = Pic(Y')/Picn(y F is a f i b r e of an ample divisor that D o n S. the d i v i s o r s n o w y E K o. as e l e m e n t Then (e0j),y of A 2 ( X ) . ~ E Pic(X). The so t h e c l a i m implies and algebraic a direct to t h e over ~. ~ + C and C O a section. (e0j)*~ = O, = O for all ~ E N u m ( Y ' ) . equivalent Suppose in Y' Since on the ruled That is numerical surface Y', y E A 1 (Y').

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