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**History of algebraic geometry. An outline of the history and development of algebraic geometry. Transl. from the French by Judith D. Sally.**
*(English)*
Zbl 0629.14001

The Wadsworth Mathematics Series. Monterey, California: Wadsworth Advanced Books & Software. A division of Wadsworth, Inc. XII, 186 p. $ 32.95 (1985).

This book is essentially the English translation of the first part of the author’s “Cours de géométrie algébrique” (Presses Univ. France 1974). However, the last chapter is considerably enlarged and updated. The author attempted (and succeeded) to show how the geometers have been led progressively to broaden their ideas, and also how the fundamental ideas of algebraic geometry appeared and developd time after time in different guises. The influence of analysis, topology, commutative algebra or homological algebra is clearly emphasized. The book has eight chapters and an introduction. The first chapter is very short and deals with the prehistory of algebraic geometry (ca. 400 BC-1630 AD). The second chapter (exploration, 1630-1795) is also short and begins with the discovery of cartesian coordinates by Descartes and independently by Fermat. The third chapter discusses the so-called “golden age of projective geometry” (1795-1850), when the geometers began to study mainly the geometry of complex projective space of dimensions 2 and 3, and later on introduced the n-dimensional projective geometry. In this period important notions (such as Plücker coordinates, birational transformations, quadratic transformations, etc.) have been introduced. The fourth chapter discusses the tremendous contribution of Riemann to the regeneration of algebraic geometry from the point of view of the theory of functions and their integrals. The fifth chapter’s title is “Development and chaos” (1866-1920). It shows how the extraordinary richness of Riemann’s work influenced the geometers in various directions such as: (1) the algebraic school (Kronnecker, Dedekind, Weber), (2) the geometric school and linear series (Roch, Clebsch, Brill, M. Noether, Smith, Cayley, Halphen, Zeuthen, Cremona, Bertini), (3) the transcendental theory of algebraic functions (M. Noether, Picard, Poincaré, Hurwitz), (4) the italian school (Castelnuovo, Enriques, Severi), etc. The sixth chapter deals with the new structures in algebraic geometry (1920-1950) such as Kähler manifolds (Kähler, Hodge), abstract algebraic geometry and commutative algebra (O. Zariski, A. Weil, W. Krull, Van der Waerden, etc.). The seventh chapter \((1950- \quad)\) discusses how the considerable advances in algebraic topology, differential topology and the theory of analytic spaces have completely renovated the foundations and the methods of algebraic geometry. The concept of scheme and Grothendieck topology allowed to Grothendieck to undertake a vast program whose objective was a generalization of algebraic geometry aiming to applications and to arithmetic as well.

The last chapter has been completely rewritten. It gives a panorama of the most significant accomplishments made in algebraic geometry as the result of the developments of its methods and techniques such as: construction of moduli spaces of various types of geometric objects and the geometric invariant theory, resolution of singularities of algebraic varieties of arbitrary dimension, the Riemann-Roch-Hirzebruch- Grothendieck theorem and the K-theory, results in enumerative geometry, the deformation theory of algebraic varieties and of their singularities, recent developments in Hodge theory, the theory of abelian varieties, the introduction of étale cohomology, the solution of the Weil diophantine conjectures (Deligne), the theory of curves and abelian varieties over function fields (culminating with Faltings’ solution of the Mordell conjecture), etc. The most important open problems of algebraic geometry are also discussed.

This book is an impressive work, bringing great services to the mathematical community.

The last chapter has been completely rewritten. It gives a panorama of the most significant accomplishments made in algebraic geometry as the result of the developments of its methods and techniques such as: construction of moduli spaces of various types of geometric objects and the geometric invariant theory, resolution of singularities of algebraic varieties of arbitrary dimension, the Riemann-Roch-Hirzebruch- Grothendieck theorem and the K-theory, results in enumerative geometry, the deformation theory of algebraic varieties and of their singularities, recent developments in Hodge theory, the theory of abelian varieties, the introduction of étale cohomology, the solution of the Weil diophantine conjectures (Deligne), the theory of curves and abelian varieties over function fields (culminating with Faltings’ solution of the Mordell conjecture), etc. The most important open problems of algebraic geometry are also discussed.

This book is an impressive work, bringing great services to the mathematical community.

Reviewer: L.Bădescu