As promised by the title, this book is an elementary introduction to Algebraic Geometry. Of course, one has to make clear what "elementary" means. We will get into that later.
Algebraic Geometry is, roughly speaking, the study of the set of solutions of systems of polynomial equations in several variables. One usually start with an algebraically closed field (the complex numbers is a favorite) and with questions such as existence of solutions, dimension of the solution set and one moves to more elaborate questions such as the topology of the solution set, expressed in terms of cohomology groups and so on. Although over the complex numbers some of these issues can be tackled analytically, the subject handles them algebraically, so as to mantain the validity of the results over more general fields and also to obtain results valid only for algebraic (as opposed to, say, analytic) varieties. Non-algebraically closed fields can also be considered and give the subject an arithmetic flavor. The book covers the first few of these aspects but not the latter ones.
The book starts by defining affine and projective varieties and maps between them, both rational maps and morphisms. Then the notions of smooth points and dimension are discussed. These are general chapters, covering the basic definitions, with the main result being the equivalence of the various definitions of dimension.
The book then changes tone and moves to discussion of some special classes of varieties. First a chapter on cubic curves, followed by one on cubic surfaces and an introduction the the general theory of curves. I liked the discussion of cubic surfaces, which included a detailed proof of the classical result that a smooth cubic surface contains exactly twenty seven lines.
The book balances theory and examples well and the exercises are well-chosen to further illustrate the basic concepts. All in all, the book does an excellent job of explaining what Algebraic Geometry is about, what are the basic results and it invites the reader to continue exploring the subject.
The blurb and the preface state that the book is aimed at undergraduates and beginning graduate students. Perhaps in Germany or Harvard. At most American universities this would be a bit of a stretch, although the missing prerequisites are not many and it's a pity the author hasn't included a chapter covering them. A short chapter on polynomial rings and Noetherian rings would have sufficed. The author needs (and uses without warning) the equivalence of the various definitions of Noetherian ring and the Hilbert basis theorem. This is standard fare in a graduate algebra class but not in an undergraduate one.
I wouldn't recommend the book for an undergraduate course but I would definitely recommend it as reading material to a bright undergraduate who has taken a basic course on rings and fields and has read about Noetherian rings. It is certainly suitable for a one-semester graduate course, for students who have taken a graduate algebra class. Mathematicians from other areas will also enjoy the book if they get a quick refresher on its prerequisites first.
The book reminds me of more old-fashioned books on Algebraic Geometry, such as Introduction to Algebraic Geometry by Semple and Roth, but updated to our modern standards of rigor and shorter attention span.
José Felipe Voloch is Professor of Mathematics at the University of Texas at Austin. His main interests are in Number Theory and Algebraic Geometry and applications to Coding Theory and Cryptography.