At its heart, algebraic geometry is nothing more than a dictionary. Many theorems in the area simply involve taking properties of polynomial equations and translating them into properties of the geometric objects they describe. Other theorems take properties of well defined geometric objects and translate them into properties of their defining equations. Some of the complicated theorems involve bouncing back and forth between the two languages for a while.
Many books on algebraic geometry spend their energy describing high-tech machines (such as sheaves, schemes, and cohomology theories) that have been developed in order to make the area as powerful as possible. But Ideals, Varieties, and Algorithms by David Cox, John Little, and Donal O'Shea takes a very concrete (some might even say naïve) approach to the subject; it puts the focus squarely on the dictionary relating the algebra and the geometry. This is not to say that there is no beautiful or hard mathematics in the book; on the contrary, the recently released third edition of the book contains quite a bit of the breadth and depth one finds in the areas of algebraic geometry and commutative algebra. However, the book is designed to have little in the way of prerequisites (although there is an appendix recapping some highlights from algebra, such as the definitions and basic results on groups, rings, and fields) and the authors start by introducing the relationship between polynomials, ideals, and affine varieties. Later chapters build on this to cover many of the topics one would expect from an algebraic geometry text such as Hilbert's Nullstellensatz, Zariski closure, Bezout's Theorem, and notions of dimension.
In recent years the computer has found itself increasingly at the center of mathematics research, and some of the best uses of computers in pure mathematics come from algebraic geometry. Much of Cox, Little, and O'Shea's book is dedicated towards the computational sides of algebraic geometry such as Groebner Bases and Elimination Theory. The authors believe that understanding these approaches to various problems in algebraic geometry will make the subject more concrete and help to develop the reader's intuition. The book is sprinkled with just enough pseudocode and descriptions of algorithms that a reader who is interested in the computations would be able to start implementing the algorithms (in fact, there are several appendices dedicated to further descriptions of computer algebra systems and programming), but these are not essential and readers who want to focus instead on the theory will not be distracted or forced to the computer.
One of the things that makes this book stand out from other books about algebraic geometry is the chapter on applications of algebraic geometry to robotic movement and automatic theorem proving. The authors give short but elegant introductions to these two areas, and describe how many basic problems in them can be related to problems in algebraic geometry. Another chapter discusses the invariants of finite groups, and the relationship between the generators of invariant rings and the underlying geometry. A final section worth noting gives a number of suggestions for computer projects and research papers that would be extremely useful to motivated students and their teachers.
For fans of the earlier editions of the book, we should note that the authors have not made any large substantive changes. The biggest change is in the appendices on computer algebra systems, and implementations of the algorithms in the book. The authors maintain a website containing Maple and Mathematica packages relevant to the book, as well as other notes and errata. The latter is particularly important, as, despite the authors' offer of cash rewards for typos, there were still quite a few small mistakes sprinkled throughout the third edition. However, this annoyance is more than made up for by the many good things about this book. In a review of an earlier edition of this book appearing in The American Mathematical Monthly, wrote that "Ideals, Varieties, and Algorithms offers the heart and soul of modern commutative algebra and algebraic geometry." This reviewer couldn't agree more.
Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include Algebraic Geometry, Number Theory, and Cryptography. He can be reached at email@example.com.