One of the great pleasures of being a number theorist these days is the unavoidability of algebraic geometry. My own schooling was heavily oriented toward modular forms: in the late 1970s and early 1980s it couldn’t be otherwise. At that time, at least where I was (UCLA, UCSD), the focus fell on such things as representation theory and trace formulas, Hecke algebras, oldforms and newforms, and connections with elliptic curves. Yes, although what is now called arithmetic geometry was already beginning to knock on the door, Wiles’ work was still over a decade in the future, and the floodgates hadn’t really been fully opened yet. The algebraic geometers had their seminars, and the number theorists had theirs, and the overlap was rather sporadic. Also, the algebraic geometry was primarily geometry (Mumford’s geometric invariant theory was a major theme at UCLA, for example), while the number theory was primarily arithmetic (what a bizarre way to put it!); in any case, there was, for instance, no mention of sheaf theory in my number theory courses.
Before long a profound infusion of algebraic-geometric methods occurred, becoming universal after the stunning proof of Fermat’s Last Theorem as a consequence of settling the Shimura-Taniyama-Weil Conjecture. The remarkably fecund interplay between modular forms, Galois representations, and elliptic curves ascended to center stage as an exemplar of a new distribution of emphases both in algebraic number theory and algebraic geometry. Accordingly what is now called arithmetic geometry quickly reached the prominence it so properly enjoys.
From a broader (or deeper) standpoint, the handwriting had already been on the wall for many years with Weil’s visionary work, Grothendieck’s development of the architecture that would support the most dazzling and spectacular constructs, the work of Deligne on the Weil conjectures, and Faltings’s treatment of the Mordell Conjecture. So, in a true sense, the work done by Wiles should be regarded as the climactic culminating chapter of a long story, with the excitement building every step of the way. But to say that it is the final chapter is of course ridiculous: the plot continues to thicken every day.
Naturally, this parochial view of algebraic geometry as a tool for number theory is at the same time disingenuous: the subject is alive and well and indeed is flourishing. Hand in glove with this is the fact that books on algebraic geometry are plentiful, the most prominent still probably being Hartshorne’s Algebraic Geometry.
Thus, now more than ever, the market for algebraic geometry is strong: every aspiring mathematician (except maybe the most exclusive of logicians) needs to learn a sizable chunk of it. But it is not the case that one can approach this beautiful subject without first preparing the way, and this means in particular a thorough study of commutative algebra. In my own case that meant working through Atiyah-MacDonald, but other classics abound, such as the classic two-volume source by Zariski and Samuel. It cannot be denied, however, that these are reasonably austere texts, the former book’s terseness notwithstanding (do all the exercises!), and after dealing with them algebraic geometry itself is still ahead.
I had a marvelous time, in due course, working from MacDonald’s old book Algebraic Geometry, an Introduction to Schemes and parts of Serre’s famous paper Faisceaux Algébriques Cohérents even before starting in properly on Hartshorne, and I am very happy with this trajectory. However, there is in judo a pithy Japanese phrase that describes how things should be done, and I certainly didn’t do things that way: the phrase is seiryoku zenyo (maximally efficient use of energy), and my patchwork approach was not optimal. There are far smoother ways to get off the ground with this material, and the book under review, by Ernst Kunz, would certainly aid in this regard. It is filled with material that in my own meanderings in algebraic geometry I came across in half a dozen different sources, pitched at different levels of austerity. It would have been so much easier to have it all together in one place, done this well.
The book’s arrangement is wonderful and effective: varieties à dimension à regular and rational functions + localization à the local-global principle (which in my youth was still called the Hasse Principle, at least in algebraic number theory) à the number of equations cutting out a variety à regular and singular points à projective resolutions. The book is not very “sheafy” and not at all “schemey,” there’s nothing cohomological in it, and Grothendieck only makes a brief appearance on p.210, as an adjective for the now ubiquitous group he defined in the context of isomorphism classes of modules over a ring. Indeed this introduction of the Grothendieck group occurs in the setting of problem 3 of the set of exercises following section VII.2, “Homological characterization of regular rings and local complete intersections,” and illustrates one of the pedagogical virtues of the book: there are a huge number of exercises, and they are carefully placed and carefully designed. The usual rules are in place: the serious student should go at them with guns blazing.
Kunz says in the Foreword to the book that “[its] center of gravity lies more in commutative algebra than in algebraic geometry. For a continued study of algebraic geometry I recommend one of the excellent books that have recently appeared …” He wrote this comment in 1978, of course, and his bibliography on algebraic geometry as such (cf. pp.224–225) includes in addition to Hartshorne (1977), Shafarevich’s eminently readable Basic Algebraic Geometry (1974), and Griffiths-Harris’ Principles of Algebraic Geometry (1978). — as the song goes, “Ah, yes, I remember it well …” And it is still the case that these three books serve spectacularly in the cause to get a student into the beautiful and important subject of algebraic geometry more deeply, with seiryoku zenyo. Kunz’ book is a wonderful springboard for all this.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.