By Mauro C. Beltrametti, Fabrizio Catanese, Ciro Ciliberto

The papers during this quantity disguise a large spectrum of algebraic geometry, from explanations idea to numerical algebraic geometry and are as a rule interested in better dimensional kinds 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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On algebraic 1-motives related to Hodge cycles 39 It is reasonable to expect that such a statement will follow from the original Grothendieck-Hodge conjecture and abstract Hodge theory. Grothendieck-Hodge conjecture (for coniveau p and degrees 2p, 2p + l) can be reformulated as follows (cf. Grothendieck's remark on motives in [19]). Consider gr~, CHP(X) and A~;i ~ gr ~m CHP+ 1(X) (fork = C this is the algebraic part of J P+ 1(X)) as l-motives with rational coefficients. The Hodge realization of these algebraically defined 1-motives is NP H 2P(X) and NP H 2P+ 1(X) respectively.

Lect. Math. Bombay 1968. [R] Z. Ran, On subvarieties of abelian varieties, Invent. Math. 62 ( 1981 ), 45H79. [Ro] H. Rossi, Continuation of subvarieties of projective varieties, A mer. J. Math. 91 ( 1969), 565-575. [RVdV] R. Remmert and A. Van de Ven, Zur Funktionentheorie homogener komplexer Mannigfaltigkeiten, Topology 2 (1963), 137-157. -P. Serre, Geometrie algebrique et geometrie analytique, Ann. Inst. Fourier 6 (1955), 1-42. [Se] F. Severi, Alcune proprieta fondamentali dell'insieme dei punti singolari di una funzione analitica di piu variabili, Mem.

CHP on each component X; of X. ' I . ' yields a complex . -4 F,~ CHP(X;) ~ F~ CHP(X;+J)--+ · · · af : where~: is the alternating sum of the pullback along the face maps X;+ 1 --+ X; for 0 ::: k ::: i + 1. The complex of Chow groups (CHP)•, induced from the simplicial structure as above, is filtered by sub-complexes: 0 ~ (F,~ CHP)• ~ ... ~ (F,:, CHP)• ~ (F,~ CHP)• = (CHP)•. Define F*(CHP)•~(F* m m CHP)• · The extension (1) given by the filtration F:, CHP(X;) on each component X; of the simplicial scheme X.