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Hardcover ISBN:  9780821802687 
Product Code:  GSM/5 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470411404 
Product Code:  GSM/5.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821802687 
eBook ISBN:  9781470411404 
Product Code:  GSM/5.B 
List Price:  $184.00 $141.50 
MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsGraduate Studies in MathematicsVolume: 5; 1995; 390 ppMSC: Primary 14;
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 RiemannRoch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a MittagLeffler 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 one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a secondsemester course in complex variables or a yearlong course in algebraic geometry.
ReadershipGraduate students studying complex variables and algebraic geometry.

Table of Contents

Chapters

Chapter I. Riemann surfaces: Basic definitions

Chapter II. Functions and maps

Chapter III. More examples of Riemann surfaces

Chapter IV. Integration on Riemann surfaces

Chapter V. Divisors and meromorphic functions

Chapter VI. Algebraic curves and the RiemannRoch theorem

Chapter VII. Applications of RiemannRoch

Chapter VIII. Abel’s Theorem

Chapter IX. Sheaves and Čech cohomology

Chapter X. Algebraic sheaves

Chapter XI. Invertible sheaves, line bundles, and $\check {H}^1$


Reviews

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 wellbalanced 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 selfteaching nonexperts, 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 wellpresented 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 selfstudy 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 
One of the best introductory textbooks on the theory of algebraic curves and Riemann surfaces … very well organized … plenty of examples … strongly recommend this book as a textbook for an introduction to algebraic curves and Riemann surfaces … One of my students said that this is one of a very few books in algebraic geometry that he can read and understand. The price of the book is very affordable.
Pawel Gajer, Johns Hopkins University 
The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory.
William Goldman, University of Maryland


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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 RiemannRoch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a MittagLeffler 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 one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a secondsemester course in complex variables or a yearlong course in algebraic geometry.
Graduate students studying complex variables and algebraic geometry.

Chapters

Chapter I. Riemann surfaces: Basic definitions

Chapter II. Functions and maps

Chapter III. More examples of Riemann surfaces

Chapter IV. Integration on Riemann surfaces

Chapter V. Divisors and meromorphic functions

Chapter VI. Algebraic curves and the RiemannRoch theorem

Chapter VII. Applications of RiemannRoch

Chapter VIII. Abel’s Theorem

Chapter IX. Sheaves and Čech cohomology

Chapter X. Algebraic sheaves

Chapter XI. Invertible sheaves, line bundles, and $\check {H}^1$

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 wellbalanced 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 selfteaching nonexperts, 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 wellpresented 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 selfstudy 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 
One of the best introductory textbooks on the theory of algebraic curves and Riemann surfaces … very well organized … plenty of examples … strongly recommend this book as a textbook for an introduction to algebraic curves and Riemann surfaces … One of my students said that this is one of a very few books in algebraic geometry that he can read and understand. The price of the book is very affordable.
Pawel Gajer, Johns Hopkins University 
The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory.
William Goldman, University of Maryland