By Paolo Francia, Fabrizio Catanese, C. Ciliberto, A. Lanteri, C. Pedrini, Mauro Beltrametti

Eighteen papers, many drawing from shows on the September 2001 convention in Genova, disguise a variety of algebraic geometry. specific realization is paid to better dimensional kinds, the minimum version application, and surfaces of the final variety. an inventory of Francia's guides is integrated. members contain mathematicians from Europe, the U.S., Japan, and Brazil

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**Sample text**

The extension (1) given by the ﬁltration Fm∗ CHp (Xi ) on each component Xi of the simplicial scheme X , for a ﬁxed p ≥ 0, yields the following short exact sequence of complexes def · 0 → gr 1Fm (CHp )• → (CHp )• /Fm2 → gr 0Fm (CHp )• → 0 (3) p Note that gr 1Fm (CHp )i contains the group of k-points of the abelian variety AXi /k and, moreover gr 0Fm (CHp )iQ is the ﬁnite dimensional vector space of codimension p cycles on Xi modulo homological (or numerical) equivalence. From (3) we then get a long exact sequence of homology groups and, in particular, we obtain boundary maps λim : H i (gr 0Fm (CHp )• ) → H i+1 (gr 1Fm (CHp )• ).

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