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

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

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L²-invariants: theory and applications to geometry and K-theory

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N o w we p o i n t o u t some p r o p e r t i e s ~A ' ~ CLAIM Proof. The n o t a t i o n s i. 1. = 1 , 3. ~ Since last = f a + f b-i,T x equality is Proposition f b . i , T x) ,. we the same ker~u Hence r~ e,j,p*2~ has . 3 u*' I,T x = i,T so that the . 1b)). = O since Thus = f 2a. d. Let now T*:J(C) ÷ J(C) be the canonical involution rU induced by the double etale covering :C ÷ C Then we 55 have CLAIM 2. The equality p,j*e*~A(e) holds, for Proof. 3 X the Let discriminant of Let P:Px map ~A set 9:C Prym ÷ A2(X) curve C be be .

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.

L'EXEMPLE D'ARTIN-MUMFORD. La m ~ t h o d e ne s ' a p p l i q u e de J A r - M ] pas sort aux vari~t~s u n p e u du c a d r e de F a n o ~ elle a cependant puisqu'elle l~avantage 29 de f o u r n i r bas~e un crit~re sur le r~sultat Proposition : de t o r s i o n d'irrationalit~ suivant Pour est La d ~ m o n s t r a t i o n mais plus I1 pour une s'agit si une G la droite congruence de R d a n s G . f-l(q) cation f se des droites de R e ~ e dans X. proposition du § 5 , X la de t o r s i o n . X Voici Soit projective suivantes S, : pas droites contenues une (cf.