# 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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Discrete differential geometry is an energetic mathematical terrain the place differential geometry and discrete geometry meet and engage. It presents discrete equivalents of the geometric notions and strategies of differential geometry, akin to notions of curvature and integrability for polyhedral surfaces.

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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.