By M. Andreatta, E. Ballico, J. Wiśniewski (auth.), Edoardo Ballico, Fabrizio Catanese, Ciro Ciliberto (eds.)

M. Andreatta,E.Ballico,J.Wisniewski: Projective manifolds containing huge linear subspaces; - F.Bardelli: Algebraic cohomology periods on a few specialthreefolds; - Ch.Birkenhake,H.Lange: Norm-endomorphisms of abelian subvarieties; - C.Ciliberto,G.van der Geer: at the jacobian of ahyperplane part of a floor; - C.Ciliberto,H.Harris,M.Teixidor i Bigas: at the endomorphisms of Jac (W1d(C)) while p=1 and C has basic moduli; - B. van Geemen: Projective versions of Picard modular forms; - J.Kollar,Y.Miyaoka,S.Mori: Rational curves on Fano kinds; - R. Salvati Manni: Modular types of the fourth measure; A. Vistoli: Equivariant Grothendieck teams and equivariant Chow teams; - Trento examples; Open difficulties

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Additional info for Classification of Irregular Varieties: Minimal Models and Abelian Varieties. Proceedings of a Conference held in Trento, Italy, 17–21 December, 1990

Example text

As the fibres of ~}~(X)~B h a v e dimension p, one has dim(~}~(X))=p+dim(B). Hence dim(F')=dim(B)+p+nr(r+l) and F' is flat. 3ince all fibres of F ' ~ ( X ) have the s a m e dimension, each component of ~ ( X ) surjects onto B, hence ~ ( X ) ~ B x is flat. In fact the associated points of ~dr (), which correspond to the irreducible components of ~ ( X ) Macaulay, m a p to the generic point of B . d. 7) yields another proof of the connectedness of ~rd(g). F (ii) The assertion about the t r a n s v e r s a l i t y of the c o m p o n e n t s of Gd(Co) appearing in fig.

__ __~-" ~- E 1 E2 A~- p i Fig. i Consider a one-dimensional family of curves whose general m e m b e r is nonsingular and whose special m e m b e r is C o . This paragraph is devoted to recall s o m e facts from [EHI] and [EH2] about the limit G~(C o) of G~(C) as C goes to C o. In w h a t follows w e will only deal with the case r= I, j0=I; anyhow, since there is no gain in working in this particular case, w e will consider here the m o r e general situation r_~l, p = l . Recall that a (crude) limit linear series g~ on C O is a (g+l)-ple ~=(8,~i .....

We collect some well-known results needed later. Let C be a complete irreducible non-singular curve and let J a c ( C ) be its jacobian. Consider the surface C x C. Its second cohomology H2(C x C) with Z or Q -coefficients splits by the K/inneth formula. If F1, F2 denote the classes of a vertical and a horizontal fibre in H 2 then this gives H2(C x C)/(F1,F2) ~- H'(C) ® Hi(C). On the other hand we have canonically H~(C) ~- H l ( j a c ( C ) ) . The polarization J a c ( C ) ~ Jac(C)* induces an identification of H 1 with ( H 1)* hence of ( H 1)* ® H i with E n d ( g ~ (Jac(C))) and we have then E n d ° ( H 1(Jac(C), Q)) fl H °'° = E n d ° ( J a c ( C ) ) , where E n d ° stands for End ®z Q and H °'° for the endomorphisms preserving the Hodge structure.

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