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Algebraic Geometry: A Problem Solving Approach

American Mathematical Society/Institute for Advanced Study
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Algebraic geometry is a notoriously difficult subject for a novice to get the hang of, and therefore any book that is intended to make this subject accessible to beginners deserves serious consideration. The book under review is one such, and is certainly one that should be looked at carefully by anybody contemplating teaching a course in the subject or wanting to learn it by self-study. There is a caveat, however: this is, as the title makes very clear, a problem book rather than a textbook, and therefore the extent to which this book will succeed for any particular reader depends on that reader’s willingness to buy into a “Moore method”-style program.

There is much to recommend here. For one thing, the organization of the material is superb. The book starts with elementary material that should be comprehensible to people with only a modest mathematics background and gradually works its way to considerably more sophisticated mathematics.

To be more specific: chapter 1 is on conic sections. Assuming, as far as I could tell, only the rudiments of matrix language and notation, the authors discuss conics over the real numbers and then over the complex numbers, and then introduce projective spaces. By the end of the chapter the reader has not only seen that the complex projective line is a sphere and that non-degenerate conics are equivalent to spheres, but has also seen connections between Pythagorean triples and the geometry of conics.

The next two chapters are also about curves, but of higher degree: chapter 2 takes the discussion from conics to cubics (including the group law on a smooth cubic) and chapter 3 looks at curves of higher degree (culminating in the Riemann-Roch theorem). The advertising material on the back cover of the book, repeating a statement made in the preface, says that these chapters are “appropriate for people who have taken multivariable calculus and linear algebra”, but by this point in the book that is an overly optimistic statement. Since groups, rings and fields are all mentioned here (for example: the group law on a cubic, the ring of regular functions, and function fields), at least some prior exposure to these ideas would be valuable. While the term “group” is defined from scratch, the authors do seem to expect some prior knowledge of rings and fields, at least to the extent, for example, that the reader would understand phrases like “field of characteristic p”. In addition, parts of chapter 3 require some background in complex analysis.

The coverage of these first three chapters is roughly comparable to that of A Guide to Plane Algebraic Curves by Keith Kendig, another (very good) book that eschews technical details in favor of intuition and examples (although in that book the reader is not expected to work through problems to get the details). These three chapters could themselves form the basis for a one-semester course.

At this point, however, the book under review continues beyond the point where Kendig’s book leaves off. Abstract algebra becomes absolutely indispensable in the next few chapters, which expand the notion of a curve to the more general concepts of affine (chapter 4) and projective (chapter 5) varieties. The former chapter is very much the longer of the two, because most of the topics in the latter can be reduced to the affine case. The affine chapter includes, for example, discussion of the Nullstellensatz, the Zariski topology (the basics of topology are developed from scratch), dimension and other fairly standard topics; the analogues of these topics are discussed in the projective chapter.

The final chapter of the book treats sheaves and cohomology, topics usually encountered only by graduate students. No attempt is made to provide an encyclopedic reference, but the authors do a good job of motivating the basic definitions and introductory ideas. For obvious reasons, the level of mathematical maturity needed for an appreciation of this chapter is fairly high; although the authors do state that the entire text should be “fair game” for anybody who has completed a course in abstract algebra, I wonder if somebody who has just been exposed to a mere semester of abstract algebra will really get a great deal out of this chapter.

Good features of the text include a nice annotated bibliography that comprises part of the preface, which not only lists a number of other texts (at both the undergraduate and graduate level) but also discusses them, and an introductory paragraph that appears at the beginning of each section, describing what the goal or goals of that particular section are. Clearly, the authors have kept the needs of a beginning reader firmly in mind throughout the book, a fact which is also evidenced by a uniformly clear writing style. (With regard to the bibliography, I should say I was surprised by the fact that Silverman’s graduate text Arithmetic of Elliptic Curves was mentioned but the more elementary book that he co-authored, Rational Points on Elliptic Curves, was not. However, no bibliography can list everything.)

The foregoing is the good news. The bad news — or, I should say, potentially bad news (depending on the tastes of the reader) — is that this is, as previously noted, a problem book. The body of the text here is limited to definitions and some motivational discussion, but the main results and details of examples generally appear in a series of exercises, answers to which are not available in the text (or anywhere else, as far as I can tell; the webpage for the book does not mention the availability of any instructor’s manual).

The upshot of all this is that, while most people reading this column have probably told a student at one time or another that a math book cannot be read like a novel, this book cannot even be read like a math book; it has to be worked through, which for some people will be a selling point but which for others may detract from the book’s usefulness. Certainly, using a book like this as a text requires not only a willingness to buy into this kind of approach, but also a skill set on the part of the instructor that even excellent lecturers may not have.

The problems themselves (and I confess to having done only a sample) appear to range in difficulty from the absolutely trivial through some standard results that may prove challenging for a student (on one or two occasions, the authors tell the students that it is OK to look up a solution), up to problems that the authors amusingly state are intended for “mathematicians on an airplane” — i.e., problems that are difficult enough to be interesting, but not so difficult that they could not be done even with the distractions of a long airplane flight. Problems of more-than-average difficulty are generally divided into smaller, more manageable, pieces; for example, the classification of maximal ideals in the polynomial ring Z[x] is spread out over seven exercises.

One mild complaint: the index can be improved. Terms like “Riemann-Roch theorem” and “Nullstellensatz”, for example, do not appear there, although these topics are covered in the text.

To summarize: this is a well-written, well-motivated account of algebraic geometry, from the very basic fundamentals to topics that are usually reserved for graduate students. If working through a series of problems to get at the details, or directing students to do the same, is not a problem for you, then by all means take a look at this book.

Mark Hunacek ( teaches mathematics at Iowa State University.

Date Received: 
Friday, February 8, 2013
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Student Mathematical Library 66
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Friday, February 8, 2013