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

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

ISBN-10: 3540123202

ISBN-13: 9783540123200

ISBN-10: 3540400478

ISBN-13: 9783540400479

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I) = Im# + p(Px ) ; ~ Im~ We have n p(Px ) ; ~+p:A2(S)@AI(s)@Px to s h o w that there ÷ A2(X) exists is an i s o g e n y . 3. p*~* = f r. ]" m e a n s the (see hence [f r . j*e*(f find e') in G(X) . p*2e) AS , relations ker some diagram , A 2 (X) ]ij (~) - T * commutes. the = 2(T*~-e) give, for 58 p,p*~*r*~' Since r*:A2(S) 2T*e = 2e ii), iii). iv). , = 2(T*a-~) ÷ Al(c) that is They is the immediately ¢+p by i). is s u r j e c t i v e (a,b,c) c ker(¢+p) , so t h a t Moreover ker2~x is a f i n i t e i) , t h e n we o b t a i n 2~ E k e r { ~ j ( C~) -T*) follow The m o r p h i s m zero map, iii) by ii).

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 . the Then of one restriction + A 2 (X) order double PX = I m ( ~ j ( ~ ) - T * ) the = e,j,p*:J(C) of points ÷ C variety n of etale , to and PX covering of has ker (~j ~ -T*)=v*J (C) (C) Proof. since The T*r*a To prove relation = ~*a the ~*J(C) for all converse, c ker(]ij ~ -T*) (c) is c l e a r a ~ J(C) take a =Cg~(D) ~ J(C) such that 56 a = T*a , that We can choose has positive induced of by is a divisor Put B of D H 9*(B-A) C that .

GRIFFITHS : The intermediate Jacobian the cubic t h r e e f o l d , Ann. of Math. 95 (1972), 281-356. [E] 21 309-391. Prym varieties Math. CASTELNUOVO Math. : are Vari~t~s 10 ( 1 9 7 7 ) , A. F. G. On p e r i o d 75-95. : A. BEAUVILLE Inventiones [B3] which 25 ( 1 9 7 2 ) , A. BEAUVILLE Ann. S. [B2] : curves, 189-238. EAr-M] M. ARTIN e t tional A. MAYER algebraic F. ENRIQUES : Sopra una involuzione non r a z i o n a l e d e l l o s p a z i o , Rend. Acc. Lincei, s . 5 a, 51 (1912), 81-83. of 33 IF1] G.

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Algebraic Geometry — Open Problems: Proceedings of the Conference Held in Ravello, May 31 – June 5, 1982 by E. Ballico, Ph. Ellia (auth.), Ciro Ciliberto, Franco Ghione, Ferruccio Orecchia (eds.)


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