The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and their Jacobians (Lecture Notes in Mathematics, 1358)

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 learn i) 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. 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 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 graduates or mathematicians in other fields wishing to quickly learn aboutalgebraic geometry. This new edition includes an appendix that gives an overview of the theory of curves, their moduli spaces and their Jacobians -- one of the most exciting fields within algebraic geometry. The Book Under Review Is A Reprint Of Mumford's Famous Harvard Lecture Notes, Widely Used By The Few Past Generations Of Algebraic Geometers. Springer-verlag Has Done The Mathematical Community A Service By Making These Notes Available Once Again.... The Informal Style And Frequency Of Examples Make The Book An Excellent Text. (mathematical Reviews) Contents: Varieties -- Preschemes -- Local Properties Of Schemes. David Mumford. The basic object of study in algebraic geometry is an arbitrary prescheme.