Algebraic Geometry. I: Complex projective varieties.
type: Item
instance of: scholarly article
title: Algebraic Geometry. I: Complex projective varieties.
zbMATH Open document ID: 0821.14001
published in: Classics in Mathematics
publication date: 1995-05-23T00:00:00Z
review text: In the 20th century, algebraic geometry has undergone several revolutionary changes with respect to its conceptual foundations, technical framework, and intertwining with other branches of mathematics. Accordingly the way it is taught has gone through distinct phases. The theory of algebraic schemes, together with its full-blown machinery of sheaves and their cohomology, being for now the ultimate stage of this evolution process in algebraic geometry, had created -- around 1960 -- the urgent demand for new textbooks reflecting these developments and (henceforth) various facets of algebraic geometry. The famous volumes ``Éléments de géométrie algébrique'' as a series in Publ. Math., Inst. Hautes Étud. Sci. (1960-1967) by \textit{A. Grothendieck} and \textit{J. Dieudonné} were entirely written in the new language of schemes, without being linked up with the classical roots, and the so far existing textbooks just dealed with classical methods. It was \textit{David Mumford}, who at first started the project of writing a textbook on algebraic geometry in its new setting. His mimeographed Harvard notes ``Introduction to algebraic geometry: Preliminary version of the first three chapters'' (bound in red) were distributed in the mid 1960's, and they were intended as the first stage of a forthcoming, more inclusive textbook. For some years, these mimeographed notes represented the almost only, however utmost convenient and abundant source for non-experts to get acquainted with the basic new concepts and ideas of modern algebraic geometry. Their timeless utility, in this regard, becomes apparent from the fact that two reprints of them have appeared, since 1988, as a proper book under the title ``The red book of varieties and schemes'' [cf. Lect. Notes Math. 1358 (1988; Zbl 0658.14001)]. In the process of exending his Harvard notes to a comprehensive textbook, the author's teaching experiences led him to the didactic conclusion that it would be better to split the book into tw…
Mathematics Subject Classification ID: 14-01
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14A10
Mathematics Subject Classification ID: 14Hxx
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14Jxx
zbMATH DE Number: 758271
zbMATH Keywords: affine varieties
zbMATH Keywords: projective varieties
zbMATH Keywords: correspondences
zbMATH Keywords: linear systems
zbMATH Keywords: algebraic curves
zbMATH Keywords: algebraic surfaces
zbMATH Keywords: birational algebraic geometry
reviewed by: Werner Kleinert
MaRDI profile type: Publication
author: David Mumford
description: scientific article; zbMATH DE number 758271