### Algebraic Varieties by G. Kempf

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During this booklet, Professor Kempf provides an advent to the speculation of algebraic kinds from a sheaf theoretic viewpoint. through taking this view he's capable of supply a fresh and lucid account of the topic, with a purpose to be simply obtainable to all rookies to algebraic types.

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If ~e prove the claim then 1rpn* is a proper closed subset of p*. Clearly its complement is the open subset of the theorem. ·. ny x in X the fiber 'iT- (x) . d'1m X 1 Now the fiber 'iT-l(x) is the a such that - · h as di mens1on n (*) =O and d lz =O. X 1s smooth, dim Tz ~ dim X. Then 'iT-l(x) has the required dimens1on. 1 Introduction to curves A curve is an irreducible separated one dimensional variety. A convenient way to use this assumption is that a point e of curve C is determined by the local ring Oc,c in k( C).

The ring k(C(X)] is graded. Let M be a graded k[C(X))-module. We want to define a sheaf M X. We have the old sheaf M on C(X). Let = Mlc(X)-o· 1 Let U be an open subset oí X. Then M(7r- U) is graded. By definition M(U) =· (M(11'- 1 U))degree o· Explicitly if f is a homogeneous element of k(C(X)] then Mio(/) = M(l)-:;;ree 0 where D(/) =Spec k[C(X))(J) degree o· Thus Mis quasi-coherent on X and it is coherent ií M is finitely generated. 1. shea11e1 on X have the form M. s a finitely generated graded k[C(X))-module.

Thus it is enough to prove the 82 Smooth. isma proposition for X = X done by induction. ¡- 1 (U) Bertini'a theorem which we may assume to have been O 83 subsets V o( Y with the obvious restriction. i. A e L. A(V) = ve V (integral closure in Lo{ Ov, ). A of Oy-algebras on a variety Y. Ox where f: X-+ Y is an affine morphism. A is quasi-coh~rent. A(V) 1s finitely generated and has no nilpotent. 1. A is a coherent Oy-module. A)-. e normalization o/ Y in L. Proof. The problem is local on Y. ffine. A(Y) = integral closure in L o{ k[Y).

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