By David Mumford

From the studies: "Although numerous textbooks on glossy algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the crimson ebook of sorts and schemes, now as prior to essentially the most first-class and profound primers of contemporary algebraic geometry. either books are only precise classics!" Zentralblatt

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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.