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Algebraic Geometry: A Volume in Memory of Paolo Francia ( De by Mauro C. Beltrametti, Fabrizio Catanese, Ciro Ciliberto

By Mauro C. Beltrametti, Fabrizio Catanese, Ciro Ciliberto

The papers during this quantity hide a large spectrum of algebraic geometry, from explanations idea to numerical algebraic geometry and are regularly taken with greater dimensional forms and minimum version software and surfaces of basic kind. part of the articles grew out of a convention in reminiscence of Paolo Francia held in Genova in September 2001 with nearly 70 contributors.

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Simplicial presheaves) of Q-mixed Hodge structures on X (resp. on X. ) such that the filtrations are finite as filtrations of subpresheaves on X. These presheaves can be sheafified to sheaves having finite filtrations and preserving the above conditions on the stalks. Make the following working definition of sheaves (or simplicial sheaves) of mixed Hodge structures. Let A, Q and C denote as well the constant sheaves on X (or X. ) associated to the ring A, the rationals and the complex numbers. 1.

50 L. Barbieri-Viale Case i = p and j = 2p. Let X be a smooth C-scheme. If j = 2p we then have N; H 2 P(X)(t) = 0 fori > p and NP H 2 P(X)(t) = cg;/. Moreover, from (13) there is an induced edge map sl~ : HP(X, Jt;(p)) ~ H 2 P(X)(p) which is a map of oo-mixed Hodge structures and whose image is NP H 2 P(X)(p). This equal the image of the classical cycle class map clP : CHP(X) ~ H 2 P(X)(p). Z) xeXP coincide with NS P(X), the group of algebraic cycles of codimension pin X modulo algebraic equivalence.

Considering the Leray filtration LP H 2P(X, Q) a natural question formulated in [7] is if FP n LP H 2 P(X, Q) will be given by higher Chern classes. However, this would not be true without some extra hypothesis on X and does not tell enough about the algebraic part of all H 2 P(X, Q). 2. An outline of the conjectural picture Let X be a proper integral C-scheme. Let H ~ H 2 P+i (X) be our mixed Hodge structure on H 2 P+i (X, Z) 1(torsion) for a fixed pair of integers p ::::: 0 and; E Z. First remark that we always have an extension 0--+ grt_ 1 H--+ W2pH/W2p-2H--+ grtH--+ 0.

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