The Geometry of Schemes

This text is intended to fill the gap between texts on classical algebraic geometry and the full-blown accounts of the theory of schemes. The text focuses on interesting examples, with a minimum of machinery, to show what is happening in the field. Included is a large number of exercises, spread throughout the text. The prerequisites for reading this book are modest: a little commutative algebra and an acquaintance with algebraic varieties.

Table of Contents

I Basic Definitions

1.1 Affine Schemes

1.2 Schemes in General

1.3 Relative Schemes

1.4 The Functor of Points

II Examples

II.I Reduced Schemes over Algebraically Closed Fields

II .2 Reduced Schemes over Non-Algebraically Closed Fields

II .3 Nonreduced Schemes

II .4 Arithmetic Schemes

Ill Projective Schemes

Ill. I Attributes of Morphisms

III .2 Proj of a Graded Ring

III .3 Invariants of Projective Schemes

IV Classical Constructions

IV.1 Flexes of Plane Curves

IV.2 Blow-ups

IV.3 Fano schemes

IV.4 Forms

V Local Constructions

VI Schemes and Functors

Review

“A great subject and expert authors!”

Nieuw Archief voor Wiskunde,June 2001

“Both Eisenbud and Harris are experienced and compelling educators of modern mathematics. This book is strongly recommended to anyone who would like to know what schemes are all about.”

Newsletter of the New Zealand Mathematical Society, No. 82, August 2001