*Geometry of Algebraic Curves*, Volume I, by Enrico Arbarello, Maurizio Cornalba, Phillip A. Griffiths, Joseph D. Harris
*Geometry of Algebraic Curves*, Volume II, by Enrico Arbarello, Maurizio Cornalba, Phillip A. Griffiths

The theory of complex algebraic curves has a long and distinguished history that reached a summit at the end of the 19^{th} century with the Abel-Jacobi and Riemann-Roch theorems. For the last few years of the 19^{th} century and the first decades of the 20^{th}, Clebsch, Noether, Castelnuovo, Enriques and Severi and many other geometers with various degrees of rigor were able to bring the methods of algebra into the realm, freeing it from some specifically complex-analytic methods.

The summits of the late 19^{th} century are now the starting points to begin studying the geometry of curves, and the first volume of the encyclopedic treatise under review mentions, but leaves to the exercises, the two theorems in a preliminary chapter (Riemann-Roch in page 7, Abel-Jacobi in page 18). That chapter then focuses on the interplay between the theta divisor on the Jacobian of a curve and the geometric properties of the curve itself, for example to obtain a first instance of Torelli’s theorem that a curve can be recovered from its Jacobian and its theta divisor. This sets the tone for the first volume: Its aim is to study a fixed algebraic curve by studying special divisors or linear series on it. The methods are algebro-geometric, rather than complex-analytic or field-theoretic (as in algebraic function fields in one variable).

As can be gathered from the description of the preliminary chapter, this is not an introductory textbook on the theory of algebraic curves. It is a comprehensive account of the deepest results of the geometry of algebraic curves that were obtained in the second half of the 20^{th} century using some of the more advanced techniques of abstract algebraic geometry, e.g., Grothendieck’s version of the Riemann-Roch theorem.

After two preliminary chapters, the main part of the first volume starts in Chapter 3 with the introduction of special divisors on smooth curves. The authors prove results such as Clifford’s theorem, Castelnuovo bound on the genus of a curve, and M. Noether’s theorem on the normality of canonically embedded curves. Chapter 4 introduces the varieties of special divisors and linear series on a curve and in Chapter 5 the main theorems of Brill-Noether theory are discussed and illustrated with some low-genus examples, with proofs deferred to chapter 7. Chapter 6 caps all the theory developed in the previous chapters by focusing first on the geometry of Riemann’s theta divisor, giving Riemann’s geometric interpretation of the multiplicities of the singular points of the theta divisor and then proving George Kemp’s beautiful generalization of Riemann’s singularity theorem as a consequence of the formal theory developed in previous chapters. The next sections give Andreotti’s proof of Torelli’s theorem and Andreotti and Mayer approach to the Schottky problem of giving equations for the locus of the Jacobians inside the Siegel upper half-plane. Two appendices complement this chapter, the first on norm maps, Weil pairings and theta characteristics and the second on the main properties of Prym varieties (abelian varieties associated to unramified double covers of curves). Chapter 7 gives the proofs of the main results of Brill-Noether theory, and chapter 8 is an introduction to the enumerative geometry of curves, now with methods coming from abstract algebraic geometry, recalling first the necessary results.

I should mention that at the end of every chapter there bibliographical notes that guide the reader to the original literature and further developments and sets of exercises that complement the theory, sometimes taking the interested reader to important results not included on the main text or give alternate proofs of results proved or just formulated in the corresponding chapter.

The first volume became an immediate standard reference for researchers and students working on the geometry of algebraic curves when it appeared in 1985. But we were kept waiting for the announced second volume, noticing that Springer Verlag had reserved volume 268 of the *Grundlehren* series for it.

We waited for a quarter of a century for the second volume, but it was worthwhile. As announced in 1985, now the focus is not on an individual curve but on families of curves. Some of the goals sketched on the introduction of the first volume have (of course) been reformulated due to the spectacular advances in moduli theory in the last 25 years.

The most important goal of this second volume is the classification, up to isomorphism, of smooth algebraic curves of a given genus. As with all classification problems, the first step is to give a precise formulation in functorial terms. Then the question is whether the corresponding functor is representable. If it is, we call the representing scheme (or algebraic space or more generally, algebraic stack) a *fine moduli* *space*. Weakening the representability condition we have *coarse* *moduli spaces*.

The moduli space that it is interesting in this context parametrizes smooth algebraic curves of a given genus. For smooth complex curves of genus 1, isomorphism classes correspond to lattices in the complex plane and bases for these lattices can be chosen of the form 1, τ with τ in the upper half-plane. Thus the upper half plane is the moduli space for isomorphism classes of smooth complex curves of genus 1. For curves of genus g > 1 there is a moduli stack M_{g}^{ }that classifies smooth projective curves of genus g > 1. The compactification of M_{g}^{ }is obtained by adding isomorphism classes of stable nodal curves as boundary points. There are some variants of this construction, for instance, by marking some points on the curves.

The first four chapters of volume 2, numbered 9 to 14, give the construction of the compactified moduli space of smooth curves of genus g > 1 with *n* marked points. Chapter 9 reviews the necessary facts about Hilbert schemes. Nodal curves are studied in Chapter 10, deformation theory in Chapter 11, with special emphasis on deformation of stable nodal curves. The construction of the corresponding moduli space is given in Chapter 12, first as an analytic space, and finally as a Deligne-Mumford stack. The fact that this moduli space is a scheme is proved in Chapter 14.

Then the book takes a turn back from the algebraic realm to the analytic one. Chapter 15 gives an introduction to Teichmüller theory that is needed for the discussion of smooth Galois covers of moduli spaces in Chapter 16 and which is used in Chapter 17 for the theory of cycles on moduli spaces. A combinatorial study of the moduli space M_{g,n} is given in Chapter 18, paving the way for the formulation and proof of Witten’s conjecture in the next two chapters, with Chapter 20 devoted entirely to Kontsevitch’s proof of Witten’s conjecture. The last chapter retakes one of the main topics of the first volume, namely Brill-Noether theory but now for smooth curves moving with moduli.

As the reader may gather from the sketchy summary given above, this a deep scholarly monograph, with the two volumes collecting in one place the most important results and developments in the theory of algebraic curves. As with volume 1, it is a safe bet to say that this second volume will become the standard reference for researchers and students working on the algebraic geometry of curves. With almost 700 items in the rich 42-page bibliography, bibliographical notes at the end of every chapter to guide the reader and sets of (guided) exercises as in the first volume, this second volume is an interactive resource for everyone seriously interested on this beautiful part of algebraic geometry. We owe the authors a heartfelt thank you for writing such a rich, beautiful and full treatise.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.