Algebraic curves and Riemann surfaces / Rick Miranda

... 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. .... (Zentralblatt)

Bibliogr. p. 371-376 Index