The Red Book of Varieties and Schemes

David Mumford (auth.)354063293X, 9783540632931

Mumford’s famous Red Book gives a simple readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra. It is aimed at graduate students or mathematicians in other fields wishing to learn quickly what algebraic geometry is all about. This new edition also includes an overview of the theory of curves, their moduli spaces and their Jacobians, one of the most exciting fields within algebraic geometry. The book is aimed at graduate students and professors seeking to learni) the concept of “scheme” as part of their study of algebraic geometry and ii) an overview of moduli problems for curves and of the use of theta functions to study these.

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Table of contents :
Front Matter….Pages N2-V
Front Matter….Pages 1-1
Some algebra….Pages 2-7
Irreducible algebraic sets….Pages 7-15
Definition of a morphism: I….Pages 15-24
Sheaves and affine varieties….Pages 24-35
Definition of prevarieties and morphism….Pages 35-45
Products and the Hausdorff Axiom….Pages 46-55
Dimension….Pages 56-67
The fibres of a morphism….Pages 67-75
Complete varieties….Pages 75-80
Complex varieties….Pages 80-89
Front Matter….Pages 91-92
Spec (R)….Pages 93-108
The category of preschemes….Pages 108-121
Varieties are preschemes….Pages 121-131
Fields of definition….Pages 131-142
Closed subpreschemes….Pages 143-155
The functor of points of a prescheme….Pages 155-167
Proper morphisms and finite morphisms….Pages 168-176
Specialization….Pages 177-189
Front Matter….Pages 191-191
Quasi-coherent modules….Pages 193-205
Coherent modules….Pages 205-215
Front Matter….Pages 191-191
Tangent cones….Pages 215-228
Non-singularity and differentials….Pages 228-242
Étale morphisms….Pages 242-254
Uniformizing parameters….Pages 254-259
Non-singularity and the UFD property….Pages 259-271
Normal varieties and normalization….Pages 272-286
Zariski’s Main Theorem….Pages 286-295
Flat and smooth morphisms….Pages 295-308
Back Matter….Pages 309-315