Undergraduate Algebraic Geometry

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ISBN: 9780521356626 | 0521356628
Cover: Paperback
Copyright: 1/27/1989

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Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. With the minimum of prerequisites, Dr. Reid introduces the reader to the basic concepts of algebraic geometry, including: plane conics, cubics and the group law, affine and projective varieties, and nonsingularity and dimension. He stresses the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book contains numerous examples and exercises illustrating the theory.

Woffle

(8)

Reasons for studying algebraic geometry, the `subset' problem; different categories of geometry, need for commutative algebra, partially defined function; character of the author

Prerequisites, relations with other courses, list of books

Playing with plane curves

(39)

Plane conics

(18)

General familiarity with P2 and homogeneous coordinates, relations of A2 to P2; parametrisation, every smooth conic C ⊂ P2 is ≅ P1

Easy cases of Bezout's theorem: line ∩ curve of degree d = d points, conic ∩ curve of degree d = 2d points; linear system of conics through P1,..Pn

Cubics and the group law

(21)

The curve (y2 = x(x-1)(x-λ)) has no rational parametrisation

Linear systems Sd(P1,..Pn); pencil of cubics through 8 points `in general position'; group law on cubic; Pascal's mystic hexagon

Appendix to Chapter I

Curves and their genus

(5)

Topology of nonsingular plane cubics over C; informal discussion of the genus of a curve; topology, differential geometry, moduli, number theory, Mordell-Weil-Faltings

The category of affine varieties

(31)

Affine varieties and the Nullstellensatz

(18)

Noetherian rings, Hilbert Basis Theorem; correspondences V and I, irreducible algebraic sets, Zariski topology, statement of Nullstellensatz; irreducible hypersurface

Noether normalisation and proof of Nullstellensatz; reduction to a hypersurface

Functions on varieties

(13)

Coordinate ring and polynomial maps; morphisms and isomorphisms; affine varieties

Rational function field and rational maps; dominant rational maps, and composing rational maps; standard open sets; addition law on elliptic curve is a morphism

Applications

(50)

Projective varieties and birational equivalence

(15)

Motivation: there are varieties strictly bigger than any affine variety; homogeneous V-I correspondences; projective versus affine

Examples: quadric surfaces; Veronese surface

Birational equivalence, rational varieties; every variety is birational to a hypersurface; products

Tangent space and nonsingularity, dimension

(8)

Motivation: implicit function theorem, varieties and manifolds

Definition of affine tangent space; nonsingular points are dense

Tangent space and m/m2, tangent space is intrinsic; dimension of X = tr degk k(X)

Resolution of singularities by blow-ups

The 27 lines on a cubic surface

102

(12)

Lines on a nonsingular cubic surface S

Proof of the existence of a line by elimination; polar form

The 5 pairs of lines meeting a given line

S is rational

The classical configuration of 27 lines

The Hessian

A case when all the lines are rational

Final comments

114

(15)

History and sociology

Choice of topics, highbrow remarks and technical notes

Substitute for preface; acknowledgements and name-dropping

Index

129

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Undergraduate Algebraic Geometry

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Table of Contents

Supplemental Materials