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Algebraic curves and Riemann surfaces. (English) Zbl 0820.14022
Graduate Studies in Mathematics. 5. Providence, RI: AMS, American Mathematical Society (AMS). xxi, 390 p. (1995).
The basic objects of algebraic geometry, namely systems of polynomial equations and their sets of solutions, have always been of natural ubiquity and particular interest in various kinds of mathematical modelling. In this sense, algebraic geometry may be considered as being one of the oldest branches of mathematics at all. The first approach to the subject, characterized by concrete geometric constructions in affine or projective space, on the one hand, and by using the methods of complex function theory in the simplest case of one-dimensional objects (curves), on the other hand, flourished during the late 19th and the early 20th century. Due to this historical development, the theories of complex algebraic curves and compact Riemann surfaces comprise the most evolved part of algebraic geometry, arguably even the topic of greatest beauty and depth within the whole subject.
After the rigorous refoundation of algebraic geometry in the second half of the 20th century, which was essentially carried out by O. Zariski, A. Weil, J.-P. Serre, A. Grothendieck, M. Artin, D. Mumford, and many others, and which was conceptually based on the new framework of abstract algebra, homological algebra, sheaf theory, and cohomology theory, the subject has been transformed – possibly more than anything else in mathematics – into a uniquely multifarious, highly advanced and methodically rather involved complex of notions, tools, and deep-going results.
These developments in algebraic geometry, over the last fifty years, have not only changed its footing and its framework in a radical way, but also brought forth a fascinating new inflorescence of the subject itself and, simultaneously, an enormously higher level of interrelations with other areas of pure and applied mathematics. The ideas, principles, methods, and results of modern algebraic geometry have become indispensible, yet sometimes still considered as being sophisticated ingredients of the today’s research in algebra, number theory, complex analysis, and differential geometry, on the one hand, and in many applied disciplines such as partial differential equations, mathematical physics, coding theory, optimization, etc., on the other hand.
All this means that the present-day mathematician, be he an active researcher or a student, will be well-advised to acquire some basic knowledge of the methods of modern algebraic geometry. However, being in the position of finding tools of remarkable power at his disposal, the beginner in algebraic geometry empirically starts with a didactic problem, and so does the teacher of the subject: what is the most efficient way to go about learning or, respectively, teaching algebraic geometry at its present state? Should one begin with classical projective algebraic varieties, just to develop a good sense of the basic objects studied in a more down-to-earth manner, or should one deal with the modern approach à la Grothendieck, early and in full strength, in order to develop quickly the level of knowledge needed for active research in the field? The teacher, in particular, has to come to grips with the problem of how to handle the various technical prerequisites for actual algebraic geometry, without discouraging the student by tedious preliminaries before reaching the proper topic, but continuously keeping him motivated, and still providing him with some substantial parts of the rich framework of current algebraic geometry, in as much beauty and depth as possible, so that he becomes able to grasp and to appreciate the insights (and their significance) offered by it.
During the last twenty years, several authors have written introductory textbooks on modern algebraic geometry, from different didactic viewpoints and, by means of various approaches to the subject, of varying degrees of generality and depth. One of the most natural and, in the meantime, most well-tried approaches to algebraic geometry is to introduce the subject by its oldest, most well-known and perhaps most fascinating examples: the complex algebraic curves. For the beginner, this approach has many advantages. It allows to keep the prerequisites from commutative algebra, categorical algebra, and homological algebra at a minimum, already illustrates the interplay between algebraic geometry and complex analysis in full beauty and depth, makes it possible to convey the language of modern algebraic geometry to the student in an as yet enlightening way, and it already leads to very deep, generally important and utmost fascinating results. The fact that complex algebraic curves (or compact Riemann surfaces), together with their classification theory, have recently gained significant importance in theoretical physics (e.g., soliton equations, conformal quantum field theories, etc.), too, emphasizes the independent role of this approach within the textbook literature on algebraic geometry, expecially with regard to a wider public.
The present book provides another introduction to complex algebraic geometry via this well-established approach through curves and Riemann surfaces. The text grew out of lecture notes for courses which the author has taught several times during the last ten years. Now, in its evolved and fully ripe form, the text impressively reflects his apparently outstanding teaching skills as well as his admirable ability for combining great expertise in the field with masterly aptitude for representation and didactical sensibility.
This book is by far much more than just another text on algebraic curves, among several others, for it offers many new and unique features. Apart from the two main aims for the book, namely to keep the prerequisites to a bare minimum while still treating the major theorems rigorously, on the one hand, and to begin to convey to the reader some of the language (and methods) of contemporary algebraic geometry, on the other hand, one prominent feature is provided by the fact that the analytic viewpoint (Riemann surfaces) and the algebraic aspect (projective curves) are discussed in a well-balanced fashion. This method is not very common; the majority of books on this subject discusses curves and Riemann surfaces either exclusively, in each case, or at least separately as part of distinct general theories. The author’s strategy is to start with the analytic theory (of compact Riemann surfaces), to discuss then the projective curves as the main examples, to develop subsequently the algebro-geometric theory of curves, culminating in an algebraic proof of the Riemann-Roch theorem, to return then to the analytic framework with respect to Abel’s theorem, and to repeat the progression to the algebraic category again when sheaves and cohomology are introduced. Line bundles and their cohomology, Picard groups, Jacobians, coverings, and first- order deformations of projective curves (with respect to the Zariski topology) represent then the algebro-geometric highlights of the second half of the text. A wealth of concrete examples, as they are barely found in any other text, at least not in such a variety and detail, and many carefully selected exercises and hints, in particular those for further reading, enhance the rich theoretical material developed in the course of the exposition, very much so to the benefit of the reader. Another advantage of this excellent text is provided by the pleasant and vivid manner of writing, by which the author has tried to ease the reader’s effort at understanding abstract concepts, general principles, deeper interrelations, and strategies of proof.
As to the contents of the book, the text consists of eleven chapters. The first three chapters discuss the basic theory of compact Riemann surfaces. Chapter I gives the definition of a Riemann surface and its genus, the typical examples, and a first treatment of affine and projective curves in this context. – Chapter II deals with holomorphic and meromorphic functions on Riemann surfaces, holomorphic maps between them, and the basic global properties of holomorphic maps and meromorphic functions, concluding with the Riemann-Hurwitz formula. – Chapter III is devoted to further examples of Riemann surfaces such as complex tori, hyperelliptic curves, nodal plane curves, cyclic coverings of the projective line, etc. This chapter also includes discussions of group actions on Riemann surfaces, quotient surfaces, ramification behavior under quotient maps, automorphism groups and Hurwitz’s bound for their orders, the concept of monodromy and monodromy representations, lots of examples again and, concludingly, the basic notions of the algebraic geometry of projective varieties, with special emphasis on projective curves and their projections. – Chapter IV treats the integration theory on Riemann surfaces. The framework of holomorphic and meromorphic differential forms is introduced to derive the residue theorem in a very direct (analytic) way, and the topological concepts of homotopy, the fundamental group, and the first homology group of a Riemann surface are explained at the end of this chapter.
The following chapters V to VIII are much more advanced and form the technical heart of the book. Chapter V provides a detailed account on divisors on compact Riemann surfaces. This chapter includes the basic constructions of and via divisors, canonical divisors, linear equivalence of divisors, the degree of smooth projective plane curves, Bezout’s theorem and Plücker’s formula for them, spaces of meromorphic functions and meromorphic 1-forms associated to a divisor, Riemann’s bound for the dimension of these spaces, linear systems of divisors, holomorphic maps to projective space defined by base point-free linear systems, and embedding criteria for such maps. The entire material is amply illustrated by instructive and important examples, among which are complex tori, rational normal curves, and elliptic normal curves. – Chapter VI turns towards complex algebraic curves. Here they are defined in an analytic way, namely as those compact Riemann surfaces whose field of meromorphic functions separates points and tangents. The culminating points in this chapter are the Riemann-Roch theorem and Serre’s duality theorem for complex algebraic curves. The proofs presented here are based on the interpretation of the involved cohomology groups as obstruction spaces to solving Mittag-Leffler problems with respect to Laurent tail divisors. This very concrete approach avoids, for the time being, any sheaves and their cohomology, but gives a strong motivation for their explicit introduction later on. – Chapter VII is devoted to various applications of the Riemann-Roch theorem within the theory of algebraic curves. It provides a particularly beautiful treatment of the great classic theorems like Clifford’s theorem, the structure of the canonical map, Riemann’s parameter count for curves of genus $$g$$, and Castelnuovo’s bound on the genus of curves in projective space. In addition, the author offers a large and thorough discussion of the classification theory for curves of low genus, inflection points and Weierstrass points, flexes and bitangents, the monodromy of the hyperplane divisor, and even some outlook to Castelnuovo curves in projective space. The whole chapter is extremely enlightening and stimulating, and it is mainly here that the reader will not only grasp the full depth and beauty of complex curve theory, but moreover feel the urge to learn more about curves (from the more advanced and more algebraically oriented monographs), and perhaps also more about higher-dimensional algebraic varieties. – Chapter VIII is entitled “Abel’s theorem”. The author explains the first homology group of a compact Riemann surface, period matrices, Jacobians, the Abel- Jacobi map, and Riemann’s bilinear relations. Abel’s theorem is then proved by applying an algebraic proof of the residue theorem (using trace operations), and the end of this chapter is devoted to elliptic curves in the light of Abel’s theorem.
The last three chapters are dealing with the sheaf-theoretic framework in complex curve theory. Chapter IX introduces sheaves and their Čech cohomology in the analytic set-up, i.e., by using the classical topology of Riemann surfaces. The main examples are here the standard sheaves of holomorphic and meromorphic functions and forms, their cohomology groups and sequences, and the isomorphism theorems of De Rham and Dolbeault for Riemann surfaces. – Chapter X discusses the algebraic counterpart, that is algebraic sheaves and their cohomology with respect to the Zariski topology of complex algebraic curves. The concluding Chapter XI takes up the theory of line bundles on complex algebraic curves. It starts with invertible sheaves, their relation to divisors, and the Picard groups they define, turns then to algebraic line bundles and their equivalence to invertible sheaves, discusses subsequently the various interpretations of the Picard group of an algebraic curve in this context, and finishes with explaining why the first cohomology group $$H^ 1 (X, {\mathcal O}^*_ X)$$ of a curve is useful to classify locally trivial objects (like coverings, extensions of invertible sheaves, first-order deformations of complex structures, etc.) in general. Riemann’s count of the number of moduli of algebraic curves of genus $$g$$ is here repeated, and that as a reprise showing the power of deformation-theoretic and cohomological methods.
Altogether, the present book is a masterly written, irresistible invitation to complex algebraic geometry and its generalization to the rich theory of algebraic schemes. Although starting from scratch, and assuming an absolute minimum of prerequisites, it presents many of the most beautiful parts of curve theory in a brilliant manner. As to the spirit and the strategy, the text comes closest to the standard work of Ph. Griffiths and J. Harris [“Principles of algebraic geometry” (1978; Zbl 0408.14001)], which treats complex algebraic geometry in general and would perhaps be the best reference for subsequent reading, and also to the surely most complete work on complex curve theory by E. Arbarello, M. Cornalba, Ph. A. Griffiths and J. Harris [“Geometry of algebraic curves”, Volume I (1985; Zbl 0559.14017); Volume II (to appear)], which is particularly recommended for the further study of the moduli theory of curves. The reader who is more interested in the algebraic context could continue, certainly without any problem, with one of the great standard books such as R. Hartshorne’s “Algebraic geometry” (1977; Zbl 0367.14001), D. Mumford’s “The red book of varieties and schemes” (1988; Zbl 0658.14001) or I. R. Shafarevich’s “Basic algebraic geometry”, Vol. 1 and 2 (1994; Zbl 0797.14001 and 14002); he will find himself well- prepared after having studied the present text. – The author has added, at the end of each chapter, detailed suggestions for further reading concerning the various single topics touched upon in the text.
The present book is perfectly suited for graduate students, partly even for senior undergraduate students, for self-teaching non-experts, and also – as an extraordinarily inspiring source and reference book – for teachers and researchers.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 30Fxx Riemann surfaces 14H40 Jacobians, Prym varieties 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14C20 Divisors, linear systems, invertible sheaves