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 quite a lot of algebraic geometry. specific cognizance is paid to raised dimensional kinds, the minimum version software, and surfaces of the final variety. a listing of Francia's courses is integrated. participants comprise mathematicians from Europe, the USA, Japan, and Brazil

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Japan 20 (1968), 52–82. -P. Jouanolou, Théorèmes de Bertini et Applications, Birkhäuser, Boston–Basel– Stuttgart 1983. [KaSa] J. Kajiwara and E. Sakai, Generalization of a theorem of Oka-Levi on meromorphic functions, Nagoya Math. J. 29 (1967), 75–84. Formal functions, connectivity and homogeneous spaces 23 [K] S. L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. [Lü] M. Lübke, Beweis einer Vermutung von Hartshorne für den Fall homogener Mannigfaltigkeiten, J.

More generally, a presheaf in MHS∞ , with ﬁnite ﬁltrations presheaves, can be sheaﬁﬁed to an A-mixed sheaf, in a canonical way, by applying the usual sheaﬁﬁcation process to the ﬁltrations together with the presheaf. 3. Say that an A-mixed sheaf H is ﬂasque if HA is a ﬂasque sheaf. For a given Q-mixed sheaf H we then dispose of a canonical ﬂasque Q-mixed sheaf x∗ (Hx ) x∈X where the product is taken over a set of points of X. 3. Hodge structures and Zariski cohomology We show that, if H is a Q-mixed sheaf then there is a unique Q-mixed Hodge structure on the sections such that (−, gr (†)) = gr (−, †).

We ignore the geometrical meaning of these exotic (1, 1)-classes. It will be interesting to produce concrete examples. The conjectural picture is as follows. 52 L. Barbieri-Viale Let π : X •,q · → X be an hypercovering. 2 for r = 1. Now E1 = H q (Xi , HX1 i ) = 0 for q ≥ 2 (where Xi are the smooth components of the hypercovering X of X) and all non-zero terms are pure Hodge structures: therefore the spectral sequence degenerates at E2 . 2, we get an extension · · 0 → H 1+i ((H 0 (H 1 ))• ) → H1+i (X , HX1 ) → H i ((H 1 (H 1 ))• ) → 0 · (16) in the category of mixed Q-Hodge structures.