Synopsis
In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of a one semester of complex variable! theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.
Review
"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 ... 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 ... A wealth of concrete examples ... 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 ... Altogether, the present book is a masterly written, irresistible invitation to complex algebraic geometry and its generalization to the rich theory of algebraic schemes ... 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." ---- Zentralblatt MATH
"Has a perspective and charm that makes it an excellent addition to the survey literature on the subject ... a leisurely and well-presented introduction to algebraic geometry through the study of algebraic curves over the complex numbers ... contains an abundance of examples and problems and develops the basic notions ... thoroughly and carefully ... excellent for self-study by beginners in the field ... repays examination by anyone interested in the field for some interesting insights and for a number of excellent ideas about the development and presentation of the material ... a charming book ... [recommended] both to those advanced undergraduates who have an interest in this area and to any graduate students who wish to learn more about this important and lively area of mathematics ... both beginners and experts as well will find a number of fascinating topics that do not normally appear in introductory texts." ---- Bulletin of the AMS
"The author takes great care in explaining how analytic concepts and algebraic concepts agree, and there is also a fine discussion of monodromy ... on the whole, this is a welcome addition to the texts in this area." ---- Mathematical Reviews
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