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Geometry of Schemes

ISBN-10: 0387986375
ISBN-13: 9780387986371
Edition: 2000
List price: $49.95 Buy it from $46.21
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Description: The theory of schemes is the foundation for algebraic geometry proposed and elaborated by Alexander Grothendieck and his co-workers. It has allowed major progress in classical areas of algebraic geometry such as invariant theory and the moduli of  More...

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Book details

List price: $49.95
Copyright year: 2000
Publisher: Springer
Publication date: 11/29/2001
Binding: Paperback
Pages: 300
Size: 6.25" wide x 9.50" long x 0.75" tall
Weight: 1.386
Language: English

The theory of schemes is the foundation for algebraic geometry proposed and elaborated by Alexander Grothendieck and his co-workers. It has allowed major progress in classical areas of algebraic geometry such as invariant theory and the moduli of curves. It integrates algebraic number theory with algebraic geometry, fulfilling the dreams of earlier generations of number theorists. This integration has led to proofs of some of the major conjectures in number theory (Deligne's proof of the Weil Conjectures, Faltings' proof of the Mordell Conjecture).This book is intended to bridge the chasm between a first course in classical algebraic geometry and a technical treatise on schemes. It focuses on examples, and strives to show "what is going on" behind the definitions. There are many exercises to test and extend the reader's understanding. The prerequisites are modest: a little commutative algebra and an acquaintance with algebraic varieties, roughly at the level of a one-semester course. The book aims to show schemes in relation to other geometric ideas, such as the theory of manifolds. Some familiarity with these ideas is helpful, though not required.

Introduction
Basic Definitions
Affine Schemes
Schemes in General
Relative Schemes
The Functor of Points
Examples
Reduced Schemes over Algebraically Closed Fields
Reduced Schemes over Non-Algebraically Closed Fields
Nonreduced Schemes
Arithmetic Schemes
Projective Schemes
Attributes of Morphisms
Proj of a Graded Ring
Invariants of Projective Schemes
Classical Constructions
Flexes of Plane Curves
Blow-ups
Fano Schemes
Forms
Local Constructions
Images
Resultants
Singular Schemes and Discriminants
Dual Curves
Double Point Loci
Schemes and Functors
The Functor of Points
Characterization of a Space by its Functor of Points
References
Index

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