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 component to 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 types of Picard modular forms; - J.Kollar,Y.Miyaoka,S.Mori: Rational curves on Fano types; - 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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8) L e m m a . We have Horn(If(C, S), Alb(S)) = (0). P r o o f . If not, then K ( C , S) maps with finite kernel to AIb(S). Since we have E n d ( I f ( C , S)) = Z this implies by (3) that Jac(C) is rigid. But then S is a scroll or C is of genus zero, cf. 6, contrary to our assumption. [] 4. R e f e r e n c e s . : Geometry of algebraic curves I. Grundlehren der math. Wiss. 267. Springer Verlag1985. : Compact complex surfaces. F,rgebnisse der Mat12. 4. Springer Verlag, 1984. 40 [C] H. Clemens, J.

Therefore we assume that we have an endomorphism 77 of K~ defined over C(A). Using the curves P r we find that Dy - Dv, lies in the image of Pic(S) ~ Pic(C). Therefore we find that (13 - 7)(Y - Y') + T c ( y - y') e Image of Pic(S) --+ Pic(C). 3) that e lies in Z C End(K(C, S)) and this now proves that E n d ( K ( C , S)) = Z. The following lemma then finishes the proof of the Theorem. 8) L e m m a . We have Horn(If(C, S), Alb(S)) = (0). P r o o f . If not, then K ( C , S) maps with finite kernel to AIb(S).

A r) at P. - TL(G~(C,P,a)) is isomorphic to Ker(~o). r F Proof. By the completeness assumption on gd, Gd(C,P,a) is isomorphic, around r r the point [LI, to its image W~(C,P,a) in the jacobian of C. Hence T =TL(Gd(C,P,a)) is the same as the tangent space to the scheme %q~r (C,P,a) at the point corresponding to L. Recall that the deformations of the line bundle L correspond to the elements of HI(C,0c) in the following way. Take an affine covering {U h} of C over which L trivializes. Let {fhk} be the corresponding family of transition functions.