There is no doubt that modern algebraic geometry is a difficult subject for the student. First, the student needs to have a good understanding of basic (analytic) geometry and, at least, some basic knowledge of topology. Most algebraic geometry courses begin with some flavor of commutative algebra, combined with the basic theory of algebraic and projective sets, leading to Hilbert's Nullstellensatz (aka Hilbert's zeros theorem). The commutative algebra, can be more or less difficult, depending on the depth of the treatment. However, the real difficulties lie ahead in the concepts of sheaves, schemes and sheaf cohomology, combining commutative algebra and topology with new definitions which may seem dramatically contrived to the student if these concepts are not properly justified and explained through numerous examples.
Even though there are standard advanced algebraic geometry texts available in the market (e.g. Hartshorne's Algebraic Geometry or Mumford's The Red Book of Varieties and Schemes), there is (as far as I am aware) no standard basic text on the subject — perhaps Shafarevich's Basic Algebraic Geometry comes closest. This lack of a standard reference represents an opportunity for authors.
The book under review, Algebraic Geometry, by Daniel Perrin, is an introductory text on modern algebraic geometry. It is aimed to be the text for a first basic course for graduate students. The principle of the book is to use some easy-to-state problems (such as Bézout's theorem on intersections of curves, or the question of whether a curve has a rational parametrization) to motivate the theory. The author avoids commutative algebra as much as possible by simply stating the theorems that he needs along the way. This allows him to keep the emphasis on the main subject: geometry. Some may argue that these theorems in commutative algebra are essential to algebraic geometry, but it is undoubtedly true that including a detailed treatment of commutative algebra, with proofs, adds a large amount of material to an already heavy load. Perrin also avoids using the language of schemes (except for finite schemes in the context of intersection multiplicities).
Perrin covers affine algebraic sets (including the Nullstellensatz) and projective sets; sheaves are introduced very early on (page 37) together with varieties, structure sheafs and sheaves of modules. There follow treatments of the concept of dimension; tangent spaces and algebraic singularities; Bézout's theorem; sheaf cohomology; arithmetic genus of curves and the weak Riemann-Roch theorem; rational maps, geometric genus, rational curves and blow-ups; and the concept of scheme-theoretic complete intersection of space curves. Each chapter is complete, with a good set of exercises (two or three pages per chapter) and there are several additional exercises in the appendix, of a lengthier and more involved nature (which would make good projects for the student).
The appendix also contains a summary of useful results from commutative algebra and a brief introduction to the language of schemes. Moreover, Perrin has also included a number of exams that he used in his own courses.
So, will this book help the student overcome the difficulty of the subject? I think so. As far as I can tell, the book is a very solid introduction to the subject and I believe the contents of the book would make for a very nice graduate course. The book is very nicely written (and very nicely translated into English too).
What I liked the most about this book is that Perrin has included many, many remarks aimed to explain and deconstruct definitions and theorems. I believe these remarks will be very valuable to the reader in order to gain the very much needed intuition for the theory. My only wish is that the author had included as many examples as remarks. There are some examples, but I do not find them to be sufficient in number. A few more concrete examples in each section could make this book the standard basic text that we are still missing.
Álvaro Lozano-Robledo is H. C. Wang instructor at Cornell University.
Foreword.- Notation.- Introduction.- Affine algebraic sets.- Projective algebraic sets.- Sheaves and varieties.- Dimension.- Tangent spaces and singular points.- Bézout’s theorem.- Sheaf cohomology.- Arithmetic genus of curves.- Rational maps and geometric genus.- Liaison of space curves.- Appendices: Summary of useful results from algebra.- Schemes.- Problems.- References.- Index.- Index of notation.