By Edoardo Ballico, Fabrizio Catanese, Ciro Ciliberto

M. Andreatta,E.Ballico,J.Wisniewski: Projective manifolds containing huge linear subspaces; - F.Bardelli: Algebraic cohomology sessions 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 common moduli; - B. van Geemen: Projective versions of Picard modular kinds; - J.Kollar,Y.Miyaoka,S.Mori: Rational curves on Fano kinds; - R. Salvati Manni: Modular sorts of the fourth measure; A. Vistoli: Equivariant Grothendieck teams and equivariant Chow teams; - Trento examples; Open difficulties

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Moreover ~ ( X ) - ~ B is flat over B if B is a smooth curve, the general fibre of it: X - ~ B is smooth and the special fibre is a curve of the type s h o w n in Fig. i. Proof. 3 in [EH1], w h e r e the existence of the scheme ~ ( X ) - ~ B is proved. Let B be a smooth curve and let q be a closed point of B such that the fibre Xq of it: X - , B is of compact type with n nodes. W e m a y assume that for all points q' of B-{q}, the fibre Xq. is smooth. It is proved in [EH1] that, after having shrinked B, there is a scheme F'-~B with the following properties.

Amer. Journal of Math. 50 (1928),8792. Ciro Ciliberto Dipartimento di Matematica Urdversit£ di Roma Via Orazio Raimondo 00173 Roma Italy Gerard van der Geer Mathematisch Instituut Universiteit van Amsterdam Plantage Muidergracht 24 1018 WB Amsterdam The Netherlands 4 On t h e e n d o m o r p h i s m s of Jac(W~(C)) w h e n p : 1 and C has g e n e r a l m o d u l i Ciro Ciliberto, Joe Harris and Montserrat Teixidor i Bigas (*) 1 . - Introduction Let C be a smooth connected curve of genus g_~i over C and let, as usual, W~(C) be the subscheme of Picd(c) parametrizing all isomorphism classes of line bundles of degree d whose space of sections has dimension at least r and let G~(C) be the scheme parametrizing all linear series of degree d and dimension r on C.

D. - G dr (C,P,a) is connected if p'>0. Moreover, if C is general in its moduli space and P is general on C, then G d (C,P,a) is smooth of dimension p" at any point corresponding to a complete linear series ILl. r Proof. Let us prove the second part of the proposition. Assume the m a p s ~i are surjective for all i=0 ..... r. Then it is easy to see that dim(TL(G:(C,P,a)))=p'. On the other hand one has dlm(Gd(C,P,a))_> P . Hence if all the m a p s ~i are surjective, then G d (C,P,a) is smooth of dimension p' at ILl.