This beautiful book is an exposition, starting from scratch, of the theory of algebraic curves (over $ C$) and their Jacobians with emphasis on explicit realizations of curves and moduli questions. Most of the material is presented without proof; some special topics (either new or lesser known results) are sketched. The book is a slightly expanded version of four Ziwet lectures given by the author at the University of Michigan in the Fall of 1974; hence the (pleasantly) informal style. As the author explains in the introduction, the book is designed for any reader having the basic standard training in topology, complex analysis and algebra. No special knowledge of commutative algebra or the foundations of algebraic geometry is required. In the author's words: "Because of time constraints I had to avoid digressions on any foundational topics and to rely on the standard definitions and intuitions of mathematicians in general. This is not always simple in algebraic geometry since its foundational systems have tended to be more abstract and apparently more idiosyncratic than in other fields such as differential or analytic geometry, and have therefore not become widely known to non-specialists. My idea was to get around this problem by imitating history: i.e., by introducing all the characters simultaneously in their complex analytic and algebraic forms."
The book is divided into four lectures: the first two on curves, the last two on Jacobians. Lecture 3 (basic material on Jacobians and theta functions) is parallel to Lecture 1 (description of curves); lecture 4 (the relation between the moduli of Abelian varieties and the moduli of curves via the Jacobian, i.e., on the Torelli theorem and the Schottky problem) is parallel to Lecture 2 (the moduli of curves). In each lecture one salient result is presented in more detail than the rest. This provides an agreeable break in the exposition.

It should be noted that the book is a photocopy of a typescript. It would be nice if a second edition (perhaps augmented, especially in the bibliography) could be published in a more appealing format.

It is difficult to summarize the book, which is itself a summary of over one hundred years of mathematics. It is immediately apparent in the first lecture that the author has gone to great lengths to be completely explicit and down to earth, and this is one of the nicest aspects of the book. He guides us through the "zoo" of curves of low genus, mentioning briefly but clearly their special features: e.g., in genus 1 the Weierstrass $\wp$-function, addition on the plane cubic, metrics and the Gel\cprime fond-Schneider theorem; in genus 3 the distinction between the curves that can be represented as plane quartics and the hyperelliptic ones, etc. This is followed by a discussion of the problem of representing the general curve of genus $g$ as a plane curve with ordinary singularities; this leads naturally to the question of putting all curves of genus $g$ in one family, thus preparing Lecture 2. The lecture concludes with a detailed account of work of K. Petri on the canonically embedded curve.

The second lecture discusses the moduli spaces $\scr M_{g,n}$ and the corresponding Teichmüller spaces $\scr T_{g,n}$. As in the first lecture the author begins with a discussion of some low genus cases, especially $\scr M_{1,1}$ and $\scr T_{1,1}$ which are of course well known to analysts. Various methods for coordinatizing $\scr M_{g,n}$ are given. The compactification of $\scr M_{g,n}$ by stable curves is described in down-to-earth terms. The last topic, developed at greater length, is the various elliptic and hyperbolic of $\scr M_{g,n}$ and $\scr T_{g,n}$, in particular the Arakelov-Par\v sin-Manin-Grauert rigidity theorem, a proof of which is sketched in the appendix.

The third lecture introduces the Jacobian. In order to tie in with the first lecture, many low genus examples are given: in particular how the map from the symmetric product of the curve to its Jacobian reflects the geometry of the curve. This culminates in Riemann's theorem on the singularities of the image of the symmetric product in the Jacobian, and G. Kempf's generalization. The last section of the lecture introduces the theta function and discusses three problems concerning Jacobians: projective embeddings; the relation of the group structure to the function theory (theta-nulls); the pullback of functions on the Jacobian to the curve and prime forms.

The last chapter is substantially more difficult than the others, and is probably of more use to the specialist to get an overview of the situation, than to the novice. It is devoted to two central problems in the theory of curves: the Torelli theorem and the Schottky problem. Four different approaches to these questions are presented extremely concisely and elegantly. Here are the titles of the sections and the names that can be attached to each: (1) reducibility of $\Theta\cap\Theta_a$, where $\Theta$ is the theta divisor and $a$ its translate by $\Theta$ (Weil, Matsusaka, Andreotti and Martens) (this section is particularly illuminating); (2) $\Theta$ of translation type (Lie, Wirtinger, Saint-Donat); (3) singularities of $\Theta$ (Andreotti and Mayer); (4) Prym varieties (Schottky, Mumford).

Reviewed by Henry C. Pinkham