Elementary Algebraic Geometry

This book and I go back a long way. It was first published, as volume 44 in Springer’s Graduate Texts in Mathematics series, back in 1977, when I was a graduate student about one year away from finishing up. Like most students in that position, I was narrowly focused on my dissertation topic, and feeling somewhat nostalgic for the good old days when I had the time to spend reading about interesting-sounding areas of mathematics just for fun. One day, while in the university bookstore, I noticed Kendig’s book, which looked, at first glance, to be quite readable. Not having learned as much algebraic geometry as much as I would have liked, I purchased it on impulse, and a year or so later, freed from the constraints of having to think about my thesis all the time, actually spent a few months reading it. My initial reaction to it proved correct: I thought at the time that it was beautifully written, very intuitive and insightful, and quite accessible.

In fact, it was primarily my very favorable impression of this book that led me, years later and now a contributor to this column, to request Kendig’s much more recent A Guide to Plane Algebraic Curves to review; that book, too, proved to be a very enjoyable read, enabling me to get reacquainted, quickly and painlessly, with many of the topics that I had learned from the original text but which had faded from memory over the years.

And now we have a Dover reprint of Kendig’s original text. It is identified on the cover as a second edition, but there don’t appear to be many significant differences in textual content from the first: the table of contents is unchanged, and in a (purely unscientific) random selection and comparison of a dozen or so pages, I found only one difference between the first and second editions: page 26 notes the solution to Fermat’s Last Theorem. The figures do look a little different, however; the author states in the preface that these were done with the help of Adobe Illustrator, which, like the proof of Fermat’s Last Theorem, did not exist when the first edition was published.

Re-reading the text after all these years reminded me why I liked it so much originally. The book has held up remarkably well, and remains a model for clear, insightful, reader-friendly exposition. I do think, however, that the back-cover blurb that this book is suitable for “advanced undergraduates” is overly optimistic. Although some of the material in the book is certainly accessible to strong, well-prepared undergraduates, the optimal audience, it seems to me, would be early graduate students who are looking to learn the rudiments of the subject, perhaps as a prelude to more sophisticated, scheme-theoretic, accounts. This is in line with the author’s own description, in the Preface, of the people for whom the book is designed. A year’s worth of abstract algebra, perhaps at the level of two good undergraduate courses, is a prerequisite for the book, but a specialized background in commutative algebra is not; the basics of that subject are introduced later, as needed. Some prior familiarity with basic topological terminology is also helpful (the phrase “Hausdorff space”, for example, appears on page 18), and at times some previous exposure to complex analysis would also be useful.

The text begins with an introductory and example-based chapter on plane curves, designed to give the reader some intuition about what they are and what they look like. These examples motivate the need to consider curves over an algebraically closed field and to consider projective rather than just affine spaces. Even in this introductory chapter, we see a theme that will dominate the book: an attempt to use lots of figures, concrete examples, and geometric reasoning to help illustrate many of the underlying ideas of algebraic geometry. A closing section also talks about curves over the field of rational numbers and illustrates how these concepts, depending on the ground field, can have a strong number-theoretic feel to them.

The ideas that were informally introduced in the first chapter are then revisited and made more precise in the second. Projective space is defined precisely, and a number of theorems about curves, particularly of a topological nature (example: a curve is connected) are given rigorous proofs.

In the first two chapters, Kendig tries to keep things simple by (for the most part) sticking to plane curves (i.e., zero sets of a single nonconstant polynomial with only two variables, or three if you “homogenize” the polynomial by working in the projective plane). Of course, quite a lot of algebraic geometry is concerned not just with curves but with the more general concept of varieties, solution sets to a collection of polynomials. Varieties (affine and projective) are the subject of Chapter IV, a chapter that is largely concerned with the concepts of dimension and intersection multiplicity, culminating in a statement and proof of Bezout’s theorem; by way of preparation for this, Chapter III first offers up a nice discussion of the topics in commutative algebra that are particularly useful for algebraic geometry. Statements and proofs are given of results ranging from the basic to the fairly sophisticated: the Hilbert basis theorem, correspondence between ideals and varieties, Nullstellensatz, Noether Normalization Lemma, etc.

The fifth, final and most difficult chapter (one that pretty much requires at least some prior background in basic complex function theory to get much out of) investigates the notion of a variety (mostly, in this chapter, a curve) as a space on which one does mathematics. After some more commutative algebra (valuation rings, local rings), the author discusses the function field of a curve and explains how one can develop a theory of functions defined on an irreducible nonsingular curve in the complex projective plane that parallels classical complex analysis (which takes place on the Riemann sphere, or the complex projective line). There are Riemann surfaces lurking around here, but the author, in contrast to his more recent Guide, stops short of introducing that terminology, presumably to keep the exposition as elementary as possible. The chapter culminates in a proof of the Riemann-Roch theorem. Although Kendig strives to make this chapter as accessible as possible, there is no point kidding ourselves that this material is easy, and as a result this chapter is, I think, a level of magnitude more difficult than the rest of the book; I am not, however, aware of any source that makes this material more transparent.

And speaking of other sources: in the years since the first edition of this book was published, there have appeared several other books that aim to make algebraic geometry comprehensible to a relatively less-than-advanced audience. Two examples that spring to mind are Ideals, Varieties and Algorithms by Cox, Little and O’Shea (CLO), and Undergraduate Algebraic Geometry by Miles Reid. Kendig’s book, however, is substantially different than either of these: unlike CLO, it does not stress the computational point of view. It is three times as long and and more fleshed-out than Reid’s book, which does not, for example, cover the Riemann-Roch theorem. In addition, Kendig’s text, although having substantial topical overlap with Algebraic Geometry: A Problem-Solving Approach by Garrity et al, is also much different from that text, which is a problem book in the “Moore method” tradition, leaving proofs for the readers; the book under review, by contrast, though relying heavily on motivation and concrete examples, does not shy away from giving rigorous proofs.

In short: this was a very good book 40 years ago, and it has held up well over the years. If anything, its current availability as an inexpensive (about $17 on amazon.com, as I write this) Dover paperback makes it even more attractive now, and I commend it highly to anybody who would like to know something about algebraic geometry, or who plans to teach it.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.