By Kenji Ueno

Glossy algebraic geometry is outfitted upon basic notions: schemes and sheaves. the idea of schemes used to be defined in Algebraic Geometry 1: From Algebraic forms to Schemes, (see quantity 185 within the comparable sequence, Translations of Mathematical Monographs). within the current publication, Ueno turns to the idea of sheaves and their cohomology. Loosely talking, a sheaf is a manner of keeping an eye on neighborhood info outlined on a topological area, comparable to the neighborhood holomorphic features on a fancy manifold or the neighborhood sections of a vector package deal. to check schemes, it really is beneficial to check the sheaves outlined on them, specifically the coherent and quasicoherent sheaves. the first software in knowing sheaves is cohomology. for instance, in learning ampleness, it really is usually helpful to translate a estate of sheaves right into a assertion approximately its cohomology.

The textual content covers the $64000 issues of sheaf conception, together with different types of sheaves and the basic operations on them, resembling ...

coherent and quasicoherent sheaves. right and projective morphisms. direct and inverse pictures. Cech cohomology.

For the mathematician surprising with the language of schemes and sheaves, algebraic geometry can appear far-off. despite the fact that, Ueno makes the subject look ordinary via his concise variety and his insightful reasons. He explains why issues are performed this manner and supplementations his reasons with illuminating examples. hence, he's capable of make algebraic geometry very obtainable to a large viewers of non-specialists.

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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.

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