By Mauro C. Beltrametti, Fabrizio Catanese, Ciro Ciliberto

The papers during this quantity hide a large spectrum of algebraic geometry, from reasons conception to numerical algebraic geometry and are regularly curious about better dimensional kinds and minimum version application and surfaces of common variety. part of the articles grew out of a convention in reminiscence of Paolo Francia held in Genova in September 2001 with nearly 70 members.

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EOAIII] A. Orothendieck and J. Dieudonne, Elements de Oeometrie Algebrique III, Inst. Hautes Etudes Sci. PubI. Math. II ( 1961 ). [EOAIV] A. Orothendieck and J. Dieudonne, Elements de Oeometrie Algebrique IV (2), Inst. Hautes Etudes Sci. PubI. Math. 24 ( 1965). [SOA I] A. Orothendieck, Revetements etales et groupe fondamental, Lecture Notes in Math. 224 , Springer-Verlag, Berlin 1971. [SOA2] A. Orothendieck, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux, North Holland, Amsterdam 1968.

On algebraic 1-motives related to Hodge cycles 41 The category MHS 00 is abelian and MHS is a fully faithful abelian subcategory of MHS 00 • Note that similar categories of infinite dimensional mixed Hodge structures already appeared in the literature, see Hain [20] and Morgan [31). , whose objects and morphisms are obtained by formally add to MHS (small) filtered colimits of objects in MHS with colimit morphisms. Consider the case A = Q. In this case, in the category MHS 00 we have infinite products of those families of objects {H; }; EI such that the induced families of filtrations {W;};EI and {F;};EI are finite.

15) is equivalent to dim(Y)- l :::: dim(X) -ca(X). It follows in particular that Nr1x is (d - I)-ample, with d := dim(Y). Therefore Y is G2 in X by theorem l in [BS]. 5). 7), (i). 16) ([BS]). x of X x X is G3 in X x X. x in X x X is just the tangent bundle Tx of X. According to Goldstein [Go], ca(X) > 1. In particular, Tx is (d- I)-ample, with d := dim(X). x is G2 in X x X. 7), (i)), since X x X is a rational projective homogeneous space. 7), (i), is a lot easier than the argument given in [BS] (which was based on Faltings' connectivity theorem for rational homogeneous spaces, see [Fl ]).