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**Methods of algebraic geometry. Volume I. Book I: Algebraic preliminaries. Book II: Projective space.**
*(English)*
Zbl 0796.14001

Cambridge Mathematical Library. Cambridge: Cambridge University Press. viii, 440 p. (1994).

[The first edition of this volume appeared in 1947 and as reprint in 1968; cf. Zbl 0157.275).]

In the history of algebraic geometry only a few textbooks on this subject have played such a dominating role as the author’s classic work did for about a quarter of a century. After the first flourishing period of algebraic geometry, which lasted from the middle of the last century to the 1930’s, and was mainly influenced by the extensive work of the Italian and the German school, it became painfully evident that algebraic geometry urgently needed a rigorous and solid foundation. Such a substantial base for algebraic geometry was laid by B. L. van der Waerden, A. Weil, O. Zariski, and others, essentially in the 1930’s and 1940’s. The basic ingredient for this solid foundation of algebraic geometry was the thorough interweaving of the apparatus of modern algebra, which made it possible to model the existing, almost purely geometric framework in a rigorous manner. B. L. van der Waerden’s “Einführung in die algebraische Geometrie” (1939; Zbl 0021.25001) and A. Weil’s famous “Foundations of algebraic geometry” (New York 1946; 2nd edition 1962; Zbl 0168.187) were the two most important textbooks, at that time, which reflected these new developments. However, both books, especially A. Weil’s work, emphasized the new abstract algebraic viewpoint, and did not incorporate the rich and beautiful geometric theory that had been built up in the past. Thus there was certainly a need for a comprehensive textbook on algebraic geometry, which would provide a systematic and unified account of the foundations and methods of algebraic geometry as they existed after World War II.

The authors made it their mission to write such a textbook. Their book “Methods of algebraic geometry” appeared in three volumes, published successively within the period 1947-1954, and represented the first comprehensive text combining the new algebraic-structural framework with the classic geometric-intuitive achievements of algebraic geometry. Until the mid 1960’s, after the ultimate revolution in algebraic geometry by Serre’s and Grothendieck’s sheaf-theoretic and cohomological approach had taken place and found its reflection in henceforth modern treatises and textbooks, the authors’ three volumes represented the most complete and consulted textbook on contemporary algebraic geometry. However, even after these tremendous developments, the work maintained its significance, since then as a valuable linking connection between Grothendieck’s renewal of algebraic geometry and its classic preliminary stages, as an extremely rich source of concrete geometric background materials and examples for the new abstract concepts, and as a still useful guide for the beginner’s first steps in algebraic geometry.

Also in these days, this particular character of the authors’ classic work is unalteredly vivid. It is and remains one of the standard texts and reference books in algebraic geometry, and an everlasting milestone of its historical development in this century. Therefore it is a gratifying fact that the three volumes of this outstanding textbook are again available, to the benefit of new generations of students and researchers in algebraic geometry.

The present volume I is a reprint of the original edition which appeared in 1947. It consists of two parts (book I and book II) and provides the purely algebraic preliminaries (book I) and the classical geometry of projective spaces (book II). The latter part contains many classical constructions and examples, which are difficult to find somewhere else, at least in such a systematic and condensed form.

In the history of algebraic geometry only a few textbooks on this subject have played such a dominating role as the author’s classic work did for about a quarter of a century. After the first flourishing period of algebraic geometry, which lasted from the middle of the last century to the 1930’s, and was mainly influenced by the extensive work of the Italian and the German school, it became painfully evident that algebraic geometry urgently needed a rigorous and solid foundation. Such a substantial base for algebraic geometry was laid by B. L. van der Waerden, A. Weil, O. Zariski, and others, essentially in the 1930’s and 1940’s. The basic ingredient for this solid foundation of algebraic geometry was the thorough interweaving of the apparatus of modern algebra, which made it possible to model the existing, almost purely geometric framework in a rigorous manner. B. L. van der Waerden’s “Einführung in die algebraische Geometrie” (1939; Zbl 0021.25001) and A. Weil’s famous “Foundations of algebraic geometry” (New York 1946; 2nd edition 1962; Zbl 0168.187) were the two most important textbooks, at that time, which reflected these new developments. However, both books, especially A. Weil’s work, emphasized the new abstract algebraic viewpoint, and did not incorporate the rich and beautiful geometric theory that had been built up in the past. Thus there was certainly a need for a comprehensive textbook on algebraic geometry, which would provide a systematic and unified account of the foundations and methods of algebraic geometry as they existed after World War II.

The authors made it their mission to write such a textbook. Their book “Methods of algebraic geometry” appeared in three volumes, published successively within the period 1947-1954, and represented the first comprehensive text combining the new algebraic-structural framework with the classic geometric-intuitive achievements of algebraic geometry. Until the mid 1960’s, after the ultimate revolution in algebraic geometry by Serre’s and Grothendieck’s sheaf-theoretic and cohomological approach had taken place and found its reflection in henceforth modern treatises and textbooks, the authors’ three volumes represented the most complete and consulted textbook on contemporary algebraic geometry. However, even after these tremendous developments, the work maintained its significance, since then as a valuable linking connection between Grothendieck’s renewal of algebraic geometry and its classic preliminary stages, as an extremely rich source of concrete geometric background materials and examples for the new abstract concepts, and as a still useful guide for the beginner’s first steps in algebraic geometry.

Also in these days, this particular character of the authors’ classic work is unalteredly vivid. It is and remains one of the standard texts and reference books in algebraic geometry, and an everlasting milestone of its historical development in this century. Therefore it is a gratifying fact that the three volumes of this outstanding textbook are again available, to the benefit of new generations of students and researchers in algebraic geometry.

The present volume I is a reprint of the original edition which appeared in 1947. It consists of two parts (book I and book II) and provides the purely algebraic preliminaries (book I) and the classical geometry of projective spaces (book II). The latter part contains many classical constructions and examples, which are difficult to find somewhere else, at least in such a systematic and condensed form.

Reviewer: W.Kleinert (Berlin)