Berlin: Springer-Verlag. xviii, 303 p. DM 68.00; öS 530.40; sFr. 68.00 (1994).
The present book, published in two volumes, is the English translation of the second revised and expanded edition of the author’s famous introductory textbook on algebraic geometry. The Russian edition appeared in 1988 (Zbl 0675.14001), and rewardingly has been translated (and slightly commented on by M. Reid. The translator has attempted – with the author’s permission – to put the text into the language used by the present generation of English-speaking algebraic geometers and, moreover, added some footnotes concerning terminology or further references. In this English translation, the two volumes have now a common index and list of references, in contrast to the Russian original edition.
As for the present revised and expanded edition of the author’s textbook itself, a few remarks might be appropriate. The very first edition (in Russian) appeared in 1972 (Zbl 0258.14001). At that time, this textbook was the first and only one which built bridges between the geometric intuition, the classical origins and achievements of algebraic geometry, the modern concepts and methods, and the complex-analytic aspects in algebraic geometry. The English translation of this unique textbook was published in 1974 under the title “Basic algebraic geometry” (Zbl 0284.14001). In the meantime, it has become one of the most valuable, recommended and used textbooks on algebraic geometry, together with the subsequent standard texts by R. Hartshorne, D. Mumford, Ph. A. Griffiths and J. Harris, and others. The special feature of the author’s book, in comparison to the others, has always been provided by the fact that it really conveys the many different aspects of modern algebraic geometry, without particularly focusing on any special approach, and without assuming any advanced prerequisites such as commutative algebra, differential geometry, functions of several complex variables, etc. In this sense, it has proved an extremely useful addition to the other (here and there) more thorough-going textbooks, in particular for beginners, and – simultaneously – a highly recommendable introduction to them and to the current research literature. Besides, and in any case, the author’s book is still a lovely and fascinating invitation to algebraic geometry.
Now, in the second edition, he maintains his ground-rules and the tried arrangement of the original text. That means, he has left the aims, the character, and the chapters basically intact. However, taking into account the rapid development and the various interconnections of algebraic geometry during the past two decades, he has added – in an organic manner – some important topics of current interest as well as some further motivating and instructive examples.
The first volume of the new edition corresponds to chapters I–IV of the first edition. The material is enhanced by the additional treatment of various examples of concrete algebraic varieties such as plane cubic curves, cubic surfaces, grassmannians and determinantal varieties. The study of singularities of algebraic varieties and maps is remarkably deepened and applied to the current topic of degenerations in algebraic families of varieties, in particular to degenerations of quadrics and elliptic curves. Also, the Bertini theorems are now included, as is a discussion of normal singularities of algebraic surfaces. Furthermore, some arithmetic aspects are worked into the text, for example: the zeta function for algebraic varieties over a finite ground field, a version of the Riemann conjecture for elliptic curves, and other applications.
Finally, in order to keep the text as self-contained as possible, the author has added an appendix entitled “Algebraic supplements”, in which he compiles the basic algebraic facts utilized in the text.
The many instructive exercises (of various degrees of difficulty) and the updated bibliography have been adjusted to the reworked material, some inaccuracies in the original text have been removed, and several proofs of theorems have been ameliorated. Thus the first part of the author’s well-tested textbook has undergone an evident enrichment in divers regards. The author has managed, with his inimitable masterly skill, to organically insert more concrete, advanced and topical material, to lucidly present the interrelations, the complexity, and the vividness of algebraic geometry in a comprehensible way, and to make his already outstanding textbook even more useful for both learning and teaching.
[See also the following review].