Algebraic Geometry: An Introduction

Daniel Perrin (auth.)978-2-7598-0048-3

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject and assumes only the standard background of undergraduate algebra. It is developed from a masters course given at the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.

The book starts with easily-formulated problems with non-trivial solutions – for example, Bézout’s theorem and the problem of rational curves – and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. The treatment uses as little commutative algebra as possible by quoting without proof (or proving only in special cases) theorems whose proof is not necessary in practice, the priority being to develop an understanding of the phenomena rather than a mastery of the technique. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.

Table of contents :
Front Matter….Pages I-XX
Affine algebraic sets….Pages 9-24
Projective algebraic sets….Pages 25-36
Sheaves and varieties….Pages 37-68
Dimension….Pages 69-86
Tangent spaces and singular points….Pages 87-99
Bézout’s theorem….Pages 101-112
Sheaf cohomology….Pages 113-129
Arithmetic genus of curves and the weak Riemann-Roch theorem….Pages 131-148
Rational maps, geometric genus and rational curves….Pages 149-171
Liaison of space curves….Pages 173-195
Back Matter….Pages 199-263