New PDF release: Algebraic Geometry--Open Problems

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

ISBN-10: 3540123202

ISBN-13: 9783540123200

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T i* D E T*D that V = HO(c,~(D+9*A)) the e n d o m o r p h i s m where , equivalence). d. 2) ÷ 0 = x PX ii) A2(X) iii) p ( k e r 2 ~ x) iv) the m o r p h i s m Proof. 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).

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 .

Three as U T and ~ ( f , t ) %* e e C2( T), 8 E C 1 (T). % f,f ~ = O and f,s, = id. 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 .

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


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