This book, with its eleven authors, grew out of the Park City Mathematics Institute’s Undergraduate Faculty Program on Algebraic and Analytic Geometry, which was held in the summer of 2008 in Park City, Utah, USA. Generally, the Park City Mathematics Institute (PCMI) holds annual three-week summer programs for researchers and postdoctoral scholars, graduate and undergraduate students, mathematics teachers, mathematics education researchers, and undergraduate faculty. One of PCMI’s main goals is to foster the interaction between research and ecucation in mathematics, thereby bringing modern concepts in mathematics to the attention of educators, on the one hand, and involving professional mathematicians in education, on the other. Each summer a different topic is chosen as the focus of the PCMI’s program, where the basics of algebraic geometry were part of the summer program in 2008. 0n that occasion, T. Garrity led a group of mathematicians, with the main goal being for the participants to be able to teach undergraduate algebraic geometry at their own college or university. The outcome of this rewarding undertaking is the text on algebraic geometry under review, which is largely based on the reader’s active work through solving many exercises. Actually, the text consists of a series of concrete problems in algebraic geometry, supplemented by some related definitions, clarifying explanations, and a few fundamental theorems.
As for the precise contents, the book comprises six chapters, each of which contains several sections. Chapter 1 is concerned with conics in the complex projective plane, their projective transformations, and their classification up to algebraic isomorphisms. Also, degenerate conics, tangents, singular points, and the notion of duality in the projective plane are both defined and exemplified in this first, mainly motivating chapter. Cubic curves and elliptic curves are introduced in Chapter 2, including their inflection points, the group law on cubics, normal forms, cross-ratios and the \(j\)-invariant, complex tori, and the classification of elliptic curves up to isomorphism.
Chapter 3 explores curves of higher degree in the projective plane. In this context, the topology of such curves, their intersection theory up to Bézout’s theorem, their rings of regular functions and function fields, divisors, the Riemann-Roch Theorem for plane curves, and the technique of blowing-up for singular curves are developed through respective explanations and instructive exercises. Chapter 4 uses abstract algebra to describe the geometry of affine varieties. The basic topics touched upon in this more algebraic part include Hilbert’s Nullstellensatz, algebraic sets and ideals, coordinate rings and function fields, the Zariski topology, local rings, tangent spaces, the concept of dimension, singular points, morphisms, rational maps, and products of affine varieties. In Chapter 5, this work is extended to the study of algebraic varieties in projective \(n\)-space, culminating in the study of rational and birational maps between projective varieties. Finally, Chapter 6 sketches the beginnings of sheaf theory and cohomology in the context of algebraic geometry, there by recasting the previous study of divisors and the statement of the Riemann-Roch Theorem into the language of invertible sheaves and their Čech cohomology. While the first three chapters are appropriate for students who have taken courses in multivariable calculus in linear algebra, the following three chapters require a first course in abstract algebra as basic background knowledge.
As the present text is essentially a problem book, with some explanations and necessary information provided along the way, its study certainly requires additional reading in basic algebraic geometry. The authors have given a large number of references in this regard, both in the preface and in the rich bibliography at the end of the book, which will help the reader work successfully with the current, primarily example-driven introduction to the subject. This seems all the more important in view of the fact that (unfortunately) no solutions to the exercises are given anywhere in the book, and that therefore the reader’s possibilities for self-control are beyond the content of the present primer. On the other hand, the text depicts the ideas, concepts, and methods of algebraic geometry in very original and instructive a manner for non-specialists. No doubt, this book is an excellent invitation to algebraic geometry and its vast contemporary standard texts.