Classics in Mathematics. Berlin: Springer-Verlag. x, 186 p. DM 59.00; öS 460.20; sFr. 57.00 (1995).
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 A. Grothendieck and 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 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 two volumes, thereby starting with complex projective varieties (in volume I), and proceeding with schemes and their cohomology (in volume II). – In 1976, the author published the first volume under the title “Algebraic geometry. I: Complex projective varieties” (1976; Zbl 0356.14002; corrected second edition 1980; Zbl 0456.14001), where the corrections concerned the wiping out of some misprints, inconsistent notations, and other slight inaccuracies.
The book under review is an unchanged reprint of this corrected second edition from 1980. Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford’s “Volume I” is, together with its predecessor “The red book of varieties and schemes”, now as before, one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!
As to the intended volume II of the book under review, the author planned to publish it in collaboration with D. Eisenbud and J. Harris. This would have been based on existing but unpublished notes of the author (partially revised by S. Lang), but then the author and his co-authors came to the conclusion that such a second volume was not really what was needed anymore, because R. Hartshorne’s famous book “Algebraic geometry” (1977; Zbl 0367.14001) already covered a good part of the material they had planned to include. Instead, D. Eisenbud and J. Harris published what they felt is needed more: a brief introduction to schemes [cf. D. Eisenbud and J. Harris, “Schemes: The language of modern algebraic geometry” (1992; Zbl 0745.14002)]. Their booklet may be regarded as a bridge between D. Mumford’s thorough classic (under review) and the now existing several textbooks on “scheme- theoretic” algebraic geometry, including Hartshorne’s book as well as Mumford’s other classic, the “Red book of varieties and schemes”.