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

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

ISBN-10: 3540123202

ISBN-13: 9783540123200

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O. 1 d) Le g r o u p e H 4 ( U ) c o n t i e n t un ~l~ment d'ordre 2. En e f f e t ~ la suite e e x a c t e de G y s i n p o u r la fibration en spheres g : V~U s'~crit HI(u) C --~H4(U) C o~ e d ~ s i g n e le celle-ci annul~e est l'~l~ment d'ordre 2 de H 4 ( U ) . c e) Puisque de H4(X) cup-produit par H 3 ( Q i ) = O, de m~me de H3(X) avec 2. 2 de H4(V) c ~ celui-ci le contient par g ~ H4(V) C dualit~ Si la ) H2(V) C classe Im(e)] provient , d'Euler du f i b r ~ O~ l ' a s s e r t i o n (d'apr~s groupe H4(U) est c done un ~l~ment de P o i n c a r ~ et est c)) d'un isomorphe d'ordre par claire ~ sinon ~l~ment d'ordre a un sous-groupe 2, la en spheres et formule il enest des coefficients universels.

1. cokernel Denote of the m a p by ÷ G(X) q the c o m p o s i t i o n PROPOSITION cylinder Proof. 2 map exists classes a cycle A2(X) ~:J(C) closure belongs to A2(U) #u:A2(V) for > + G(X) all {j~:A2(X) by L e m m a 8 belonging simplicity, in the that 8 E ker ~ and J(C) Zariski morphism to p r o v e a cycle x £ J(C) y £ A 2 (X) projection G(X) . induced by the is s u r j e c t i v e . Therefore of For the The m o r p h i s m It s u f f i c e s ¢Im#. 1 ÷ A2(U)} all via set u = JU" We p*, of u*y in X and as y ~ A 2 (X) .

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. Then Then sheaf a quadratic s e S\C two an ordinary is in we have: the of direct rank 3 and form. at m o s t ordinary that: s is a n o n s i n @ u l a r i_~s i s o m o r p h i c - if such point results proved c a n be o n X.

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

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