This classic is one of the cornerstones of modern algebraic geometry. At the same time, it is entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals.

Reviews

"It is not often that one reviews a text written 53 years ago, updated 37 years ago, and still as relevant today as it was in its previous incarnations ... This book has played an important role in establishing the mathematical foundations of Algebraic Geometry and in providing its accepted language. Although there have been very significant subsequent ideological shifts in the subject, this book is just as fresh today as it was when it first appeared."

-- Monatshefte für Mathematik

Table of Contents

• Algebraic preliminaries
• Algebraic theory of specializations
• Analytic theory of specializations
• The geometric language
• Intersection-multiplicities (special case)
• General intersection-theory
• The geometry on abstract varieties
• The calculus of cycles
• Divisors and linear systems
• Comments and discussions
• Appendix I. Normal varieties and normalization
• Appendix II. Characterization of the $i$-symbol by its properties
• Appendix III. Varieties over topological fields
• Index of definitions