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. Cover LNM 1358 Title Page Preface to the Second Edition Preface to the First Edition Table of Contents I. Varieties §1. Some algebra §2. Irreducible algebraic sets §3. Definition of a morphism §4. Sheaves and affine varieties §5. Definition of prevarieties and morphisms §6. Products and the Hausdorff axiom §7. Dimension §8. The fibres of a morphism §9. Complete varieties §10. Complex varieties II. Preschemes §1. Spec (R) §2. The category of preschemes §3. Varieties and preschemes §4. Fields of definition §5. Closed subpreschemcs §G. The functor of points of a prescheme §7. Proper morphisms and finite morphisms §8. Specialization III. Local Properties of Schemes §1. Quasi-coherent modules §2. Coherent modules §3. Tangent cones §4. Non-singularity and differentials §5. Etale morphisms §6. Uniformizing parameters §7. Non-singularity and the UFD property §8. Normal varieties and normalization §9. Zariski's Main Theorem §10. Flat and smooth morphisms Appendix: Curves and Their Jacobians I. What is a Curve and How Explicitly Can We Describe Them II. The Moduli Space of Curves: Definition, Coordinatization, and Some Properties III. How Jacobians and Theta Functions Arise IV. The Torelli Theorem and the Schottky Problem Survey of Work on the Schottky Problem up to 1996 by Enrico Arbarello References: The Red Book of Varieties and Schemes Guide to the Literature and References: Curves and Their Jacobians Supplementary Bibliography on the Schottky Problem by Enrico Arbarello 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) The basic object of study in algebraic geometry is an arbitrary prescheme.