Vol. 11 (1977) REPORTS ON MATHEMATICAL PHYSICS No. 3

BOOK REVIEWS

DAVID MUMFORD (Professor of mathematics, Harvard chusetts 02138, USA)

Algebraic Geometry I, Complex Projective Varieties

University, Cambridge, Massa-

Grundlehren der mathematischen Wissenschaften 221. (A series of Comprehensive Studies in Mathematics)

Editors: S. S. Chern, J. L. Doob, J. Douglas, jr., A. Grothendieck, E. Heinz, F. Hirze- bruch, E. Hopf, S. MacLane, W. Magnus, M. M. Postnikov, F. K. Schmidt, W. Schmidt, D. S. Scott, K. Stein, J. Tits, B. L. van der Waerden.

Springer-Verlag, Berlin, Heidelberg, New York 1976. Managing Editors: B. Eckmann, J. K. Moser.

Conrenk: Introduction (p. VII), Chap. 1. Affine Varieties (p. l), Chap. 2. Projective Varieties (p. 20), Chap. 3. Structure of Correspondences (p. 40), Chap. 4. Chow’s Theorem (p. 59), Chap. 5. Degree of a Pro- jective Variety (p. 70), Chap. 6. Linear Systems (p. 96), Chap. 7. Curves and Their Genus (p. 127), Chap. 8. The Birational Geometry of Surfaces (p. 156).

The author, Professor David Mumford, is an outstanding mathematician and a known lecturer in the field of algebraic geometry. His great experience allows him to handle the matter from several different sides. Didactically preferably, this book, as the first of several, introduces only complex projective varieties. Regarding that difficulties of traditional geometry lie in, as Mumford himself says, “short cutting the fine detailes of all proofs and ignoring at times the time-consuming analysis of special cases”, the striving of the renown mathematicians in the field as Zariski, Weil and Grothendieck was to apply new tools as a commutative algebra. There arose a difficulty of language in which to talk of projective varieties over the complex numbers, for instance. The advantageous progress in this domain seemed to be the Grothendieck’s schemas.

Algebraic geometry rapidly developed in the last ten years provides a deeper under- standing of many geometrical problems as singularities and the theory of cycles on them, especially in three and more dimensions.

The reviewed book contains an effective and thorough study of the complex projective varieties with topological and analytical techniques. The presented theory is thoroughly investigated, yet given in a clear and comprehensive manner, provided with graphic illustra- tions. As a very valuable and extremely interesting approach, the book is highly recom- mended to mathematicians interested in algebraic geometry.

F. RYNKIE WICZ

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