Arithmetic geometry (= numbertheory + algebraic geometry) is notorious for being an intricate subject with a long and steep learning curve. The student must become familiar with important but difficult results, many of which are only expounded upon in the primary literature. This has been the case for the geometric theory of arithmetic surfaces, as well as for the theory of stable reduction of algebraic curves, as it appears for instance in the seminal paper of Deligne and Mumford on the irreducibility of the moduli space of curves. The book under review fills these unfortunate gaps by providing a thorough and systematic presentation of these topics, preceded by an equally thorough exposition of the concepts, results, and techniques required in order to understand them.

The book has two parts. The first presents the modern (post-Grothendieck) approach to algebraic geometry, centered around the concepts of schemes and morphisms. The second part covers the birational geometry and intersection theory of surfaces, as well as the reduction theory of curves, culminating in the Artin-Winters proof of Deligne and Mumford's theorem. As should be the case for a project that started as lecture notes, the text is very much written with the student in mind, as evidenced by the large number of examples and by the many exercises that conclude each section.

Being well-written and practically self-contained, this book is very well suited for graduate students wishing to get a firm grasp on the foundations of the subject before moving on to more specialized topics. The only prerequisite is an introductory course in commutative algebra; more advanced commutative algebra topics are discussed in the first chapter of the book, as well as scattered throughout the text wherever they are needed. Despite its great expository qualities, the book does require a high level of mathematical sophistication that makes it generally unsuitable for undergraduate students.

Alex Ghitza is assistant professor of mathematics at Colby College.