By Yang K.

A textbook for second-year graduate scholars who're acquainted with algebraic topology, functionality concept, and uncomplicated differential geometry. the gathering of seminar notes constitutes an creation to complicated algebraic geometry, concentrating on its transcendental element.

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**Extra info for Complex Algebraic Geometry: An Introduction to Curves and Surfaces**

**Example text**

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.