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A Guide to Plane Algebraic Curves

Mathematical Association of America
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I have admired Keith Kendig’s expository skills for about 35 years now, ever since I picked up, as a graduate student, his Graduate Texts in Mathematics volume (#44) on algebraic geometry, which I thought at the time was the clearest and most accessible available entrée into the subject. Unfortunately, in the decades since I first read that book, I have not had a great deal of time to spend on algebraic geometry, and so, while I once knew, for example, what the Riemann-Roch theorem said and meant, I had long since forgotten that. If, a month ago, somebody had asked me to state the theorem, I could only have mumbled something about “divisors” and hope I would not be asked just what a divisor was.

What a pleasure, therefore, for me to, after all these years, once again be reminded by Kendig of the joys of algebraic geometry, this time at a much more expository level than in his earlier book — exactly what I, as a mathematically-trained person with no plans to do research in the area but with a desire to at least have an overview of the subject, needed.

The goal of this book (as with other books in the “MAA Guides” series) is to provide a quick introduction to an area of mathematics, without getting mired down in a lot of technicalities. To this end, the book focuses on plane algebraic curves (zero sets of polynomials in two variables) rather than discussing algebraic geometry in full generality. The emphasis in this book is on understanding what the ideas mean, so instead of lots of proofs and details we instead have lots of examples and pictures, which I think is a splendid way to either be exposed to this highly visual subject for the first time, or to be reminded of what you once knew and have since forgotten.

Because proofs are either sketched or (more commonly) omitted altogether, and because there are no exercises, this Guide will probably not find use as a primary course textbook, but would make an excellent supplementary text for students as well as a source of information for mathematicians wanting to know more about this subject. People who want or need to know full details can of course then consult the more sophisticated texts, and will undoubtedly find that since they already have at least an intuitive understanding of the concepts, the gory details have become somewhat more digestible.

The book begins in chapter 1 with a bestiary of real algebraic curves of all shapes and sizes, along with a nicely motivated introduction to the concept of the resultant and a discussion of various “tricks of the trade.” As just noted, the emphasis here is on developing intuition for what these objects of study look like and at least some sense of a few techniques used to study them.

The next two chapters broaden the scope of the inquiry, first by introducing the idea of points at infinity and the real projective plane (chapter 2) and then by extending this discussion to affine and projective planes over the complex numbers (chapter 3). The motivation for broadening our horizons in this way is to answer a question posed by the author on the bottom of page 29: given all these different kinds of algebraic curves, “where are the nice theorems?” One of these “nice theorems” is Bézout’s theorem on the number of points of intersection (in the complex projective plane) of two plane curves; chapter 3 culminates in a ten-page discussion of this result, including a statement of several equivalent formulations of the theorem, some illustrations, a sketch of a proof, and some applications, including a proof of Pascal’s theorem in projective geometry.

In the next chapter, the author discusses topological ideas in the context of the complex projective plane. The first major result is a well-motivated proof that complex projective curves are path-connected. Although Kendig strives to make this chapter as self-contained as possible, it is clear that some prior background in complex analysis will considerably facilitate understanding of this material. (The preface does list basic complex analysis as one of the prerequisites for the book, along with the “rudiments of coffee cup and donut topology” and some understanding of basic ring and field theory.) A proof is sketched that a smooth irreducible algebraic curve in the projective complex plane is an oriented two-manifold and that the genus of such a curve of degree n is given by the formula g = (n–1)(n–2)/2. Although the author does define “oriented two-manifold,” he assumes some prior familiarity with the notion of genus and with Euler’s famous formula V – E + F = 2 – 2g.

Chapter five, a long chapter comprising more than a quarter of the text, discusses the notion of singularities of curves, starting with several different but equivalent definitions of singularity, some geometric, some algebraic. There follow illustrations of the connections between singularities and the genus of a curve. The chapter culminates with the construction of the function field of a curve, on which, to use the author’s phrase, one can “do math on a curve,” by which he means “doing elementary complex analysis in one variable on that curve.” As always, the emphasis is on extended discussions of interesting examples, illustrating results that are stated but not generally proved. (References to proofs are, however, given.)

The final and most mathematically demanding chapter of the book addresses the basic equivalence of three concepts: algebraic curves in the complex projective plane, compact Riemann surfaces, and algebraic extensions of the field of complex rational functions in a single variable (i.e., extensions of the field of complex numbers of transcendence degree 1). Although these terms are defined in the text, it would, as in chapter 4, certainly be helpful for a reader to have a good background in complex analysis in order to get maximum benefit from this chapter, which also talks at some length about elliptic curves (defined here as projective nonsingular curves of genus 1). The Riemann-Roch theorem is stated, as are a number of other theorems, and as usual references are given to the proofs, with this book providing detailed discussion of examples and useful motivation.

There are, of course, other textbooks that attempt to make algebraic geometry accessible through a discussion of algebraic curves, but none (to my knowledge) are quite like this one. Robert Bix’s Conics and Cubics: A Concrete Introduction to Algebraic Curves, for example, is intended for a much less sophisticated audience; it mostly deals with algebraic curves of degree 2 or 3, largely eschews the use of algebra, analysis and topology in favor of direct calculations, and does not discuss topics like the Riemann-Roch theorem at all; it is probably accessible to undergraduates in a way that this book is not. Frances Kirwan’s Complex Algebraic Curves is another very nice introduction to algebraic curves, but one that seems to be pitched a bit above this book in terms of difficulty and formality, as is the book Plane Algebraic Curves by Kunz. The books by Kunz and Kirwan are two of the sources the author gives for details of the proofs of the theorems stated here; others are listed in the three-page bibliography to this book.

In summary, this is a very nicely written, interesting and useful little book which anybody teaching or studying algebraic curves will benefit from looking through.

Mark Hunacek ( teaches mathematics at Iowa State University.

Date Received: 
Monday, January 23, 2012
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Keith Kendig
Dolciani Mathematical Expositions 46/MAA Guides 7
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Mark Hunacek
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1. A Gallery of Algebraic Curves
2. Points at Infinity
3. From Real to Complex
4. Topology of Algebraic Curves in P2 (C)
5. Singularities
6. The Big Three: C, K, S

Publish Book: 
Modify Date: 
Thursday, May 31, 2012