631 - Introduction to Algebraic Geometry Igor V Dolgachev...

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Unformatted text preview: Introduction to Algebraic Geometry Igor V. Dolgachev April 30, 2010 ii Contents 1 Systems of algebraic equations 1 2 Affine algebraic sets 7 3 Morphisms of affine algebraic varieties 13 4 Irreducible algebraic sets and rational functions 21 5 Projective algebraic varieties 31 6 B´ ezout theorem and a group law on a plane cubic curve 45 7 Morphisms of projective algebraic varieties 57 8 Quasi-projective algebraic sets 69 9 The image of a projective algebraic set 77 10 Finite regular maps 83 11 Dimension 93 12 Lines on hypersurfaces 105 13 Tangent space 117 14 Local parameters 131 15 Projective embeddings 147 iii iv CONTENTS 16 Blowing up and resolution of singularities 159 17 Riemann-Roch Theorem 175 Index 191 Lecture 1 Systems of algebraic equations The main objects of study in algebraic geometry are systems of algebraic equa-tions and their sets of solutions. Let k be a field and k [ T 1 , . . . , T n ] = k [ T ] be the algebra of polynomials in n variables over k . A system of algebraic equations over k is an expression { F = 0 } F ∈ S , where S is a subset of k [ T ] . We shall often identify it with the subset S . Let K be a field extension of k . A solution of S in K is a vector ( x 1 , . . . , x n ) ∈ K n such that, for all F ∈ S, F ( x 1 , . . . , x n ) = 0 . Let Sol ( S ; K ) denote the set of solutions of S in K . Letting K vary, we get different sets of solutions, each a subset of K n . For example, let S = { F ( T 1 , T 2 ) = 0 } be a system consisting of one equation in two variables. Then Sol ( S ; Q ) is a subset of Q 2 and its study belongs to number theory. For example one of the most beautiful results of the theory is the Mordell Theorem (until very recently the Mordell Conjecture) which gives conditions for finiteness of the set Sol ( S ; Q ) . Sol ( S ; R ) is a subset of R 2 studied in topology and analysis. It is a union of a finite set and an algebraic curve, or the whole R 2 , or empty. Sol ( S ; C ) is a Riemann surface or its degeneration studied in complex analysis and topology. 1 2 LECTURE 1. SYSTEMS OF ALGEBRAIC EQUATIONS All these sets are different incarnations of the same object, an affine algebraic variety over k studied in algebraic geometry. One can generalize the notion of a solution of a system of equations by allowing K to be any commutative k-algebra. Recall that this means that K is a commutative unitary ring equipped with a structure of vector space over k so that the multiplication law in K is a bilinear map K × K → K . The map k → K defined by sending a ∈ k to a · 1 is an isomorphism from k to a subfield of K isomorphic to k so we can and we will identify k with a subfield of K . The solution sets Sol ( S ; K ) are related to each other in the following way....
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This note was uploaded on 02/24/2012 for the course MATH 632 taught by Professor Igordolgache during the Fall '08 term at University of Michigan-Dearborn.

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631 - Introduction to Algebraic Geometry Igor V Dolgachev...

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