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The red book of varieties and schemes. (English) Zbl 0658.14001
Lecture Notes in Mathematics, 1358. Berlin etc.: Springer-Verlag. v, 309 p. DM 50.00 (1988).
From the author’s preface: “These notes originated in several classes that I taught in the mid 60’s to introduce graduate students to algebraic geometry. I had intended to write a book, entitled “Introduction to Algebraic Geometry”, based on these courses and, as a first step, began writing class notes. The Harvard mathematics department typed them up and distributed them. They were called “Introduction to Algebraic Geometry: Preliminary version of the first three chapters” and were bound in red. The intent was to write a much more inclusive book. But as the years progressed, my ideas of what to include in this book changed. The book became two volumes, and eventually, with almost no overlap with these notes, the first volume appeared in 1976, entitled “Algebraic Geometry. I: Complex projective varieties” (1976; Zbl 0356.14002; second edition 1980). The present plan is to publish shortly the second volume, entitled “Algebraic Geometry. II: Schemes and cohomology”, in collaboration with David Eisenbud and Joe Harris.
D. Gieseker and several others have, however, convinced me to let reprint the original notes, on the grounds that they serve a quite distinct purpose. These old notes had been intended only to explain in a quick and informal way what varieties and schemes are, and give a few key examples illustrating their simplest properties. The hope was to make the basic objects of algebraic geometry as familiar to the reader as the basic objects of differential geometry and topology.
This volume is a reprint of the old notes without change, except that the title has been changed to clarify their aim.
It may be of some interest to recall how hard it was for algebraic geometers to find a satisfactory language. At the time these notes were written, the “foundations” of the subject had been described in at least half a dozen different mathematical “languages”.
Then Grothendieck came along and turned a confused world of researchers upside down, overwhelming them with the new terminology of schemes as well as with a huge production of new and very exciting results. These notes attempted to show something that was still very controversial at that time: that schemes really were the most natural language for algebraic geometry.”
The language of schemes is now fully accepted as the correct one for the problems of algebraic geometry. Therefore the book is welcome as an introduction to this topic and serves very well as such in spite of the long time since it was first published.
The contents covers: Varieties (chapter I); preschemes (chapter II); local properties of schemes (chapter III).
Reviewer: G.-E.Winkler

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14A10 Varieties and morphisms
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14A15 Schemes and morphisms