In the preface of Miles Reid’s book Undergraduate Algebraic Geometry, written in 1988, he writes that “there are several good recent textbooks on algebraic geometry at the graduate level, but not (to my knowledge) any designed for an undergraduate course” and therefore he is presenting the notes from his course at Warwick as a “self-contained introductory textbook.” When Reid wrote these words in 1988, it may have been the case that there was a lack of good elementary algebraic geometry texts, but this is much less true today. Books by Kirwan, Fulton, Miranda, Hassett, and Kendig (among others) all come to mind as books accessible to undergraduates that give tastes of different flavors of algebraic geometry. Still, even 25 years later, Reid’s book is one of the best books on the subject and one of the first books I would hand to a student looking to learn about this deep and exciting field of mathematics.
Reid opens his book with a “woffle,” which is “intended as a cultural introduction.” In these pages, he introduces the concept of a variety as “(roughly) a locus defined by polynomial equations” and discusses some questions from number theory, topology and other areas of mathematics where it would be useful to understand varieties. He then discusses both the strengths and drawbacks of the algebraic approach to studying them. He finally gives an overview of the rest of the book, which is divided into three chapters.
The first of these chapters is entitled “Playing With Plane Curves.” It begins with a section dedicated to plane conics, in which Reid gives friendly introductions to the projective plane, Bezout’s Theorem, and a number of other topics related to conic curves. The next section is dedicated to cubic curves and includes a discussion of why there is no rational parametrization as well as the group law on cubic curves and Pascal’s mystic hexagon theorem. A third section (which Reid prefers to call an appendix) takes a more topological point of view, introducing the idea of the genus of a curve and giving brief discussions of topics such as the Mordell-Weil Theorem.
The second chapter of Reid’s book is entitled “The Category Of Affine Varieties,” and (as its name suggests) gets a bit more technical. In particular, Reid gives a crash course in concepts from commutative algebra so that he can discuss topics such as the Hilbert Basis Theorem, the Zariski Topology of algebraic sets, and the Nullstellensatz. Despite the level of technical detail Reid must reach, he keeps the exposition light and gives many examples along the way in order to keep the book accessible. He then embraces the notion that one of the best ways to understand a variety is to understand the functions on that variety, an idea which he develops in the fourth section for affine varieties and in the fifth section for projective varieties. This leads to a discussion of singular and nonsingular points as well as tangent spaces and dimensions. There is then a section devoted to the famous theorem that a nonsingular cubic surface defined over an algebraically closed field contains exactly 27 lines.
As its title suggests, Undergraduate Algebraic Geometry is indeed a book for undergraduates. There are many places where Reid chooses to wave hands and point the reader at harder books, and a more advanced mathematician is likely to point out several places where Reid does not discuss things in the most general or “correct” manner. For that reason, while I would recommend this book to a student who is new to the area, the first seven sections of this book will not be all that useful to people who are already familiar with the field.
That said, the final section of Reid’s book is worth reading even if you are an expert in algebraic geometry. In it, Reid gives a “history and sociology of the modern subject” in which he does not hold back from sharing some opinions about trends in the field through the 20th century such as how a “whole generation of students (mainly French) got themselves brainwashed into the foolish belief that a problem that can’t be dressed up in high-powered abstract formalism is unworthy of study.” He also gives some further thoughts to instructors who wish to use these notes as a text, and while his thoughts on computation may show their age they is well worth reading. He concludes the book with some self-professed “name-dropping.”
As should be clear, Reid has written the book in a very informal tone despite the notoriously technical nature of some of the material. The style of the book will not be for everyone’s tastes, and it is definitely not an exhaustive treatment of algebraic geometry. That said, Reid is a fine expositor who relies heavily on concrete examples and has many exercises for reader’s to try on their own. Undergraduate Algebraic Geometry remains one of the clearest introductions to the field, and is a book that belongs in every library.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College. His mathematical interests include number theory, Galois theory, and algebraic geometry though his first introduction to the field did not come from Reid’s book. He can be reached at firstname.lastname@example.org.