Topics in the Theory of Riemann Surfaces
The book's main concern is automorphisms of Riemann surfaces, giving a foundational treatment from the point of view of Galois coverings, and treating the problem of the largest automorphism group for a Riemann surface of a given genus. In addition, the e
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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Robert D. M. Accola
Topics in the Theory of Riemann Surfaces
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Author Robert D. M. Accola Department of Mathematics Brown University Providence, Rhode Island 029 I2, USA
Mathematics Subject Classification (1991): 30FIO; 14HI5
ISBN 3-540-58721 -7 Springer-Verlag Berlin Heidelberg New York CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera-ready out put by the author SPIN: 10130221 46/3140-543210 - Printed on acid-free paper
Preface These are lecture notes for a course given during the Fall of 1988 at Brown University. The students were assumed to have had a previous course on the theory of Riemann surfaces or algebraic curves. Chapter One of these notes gives a review of most of the basic material needed later, but few proofs are given, and the reader is assumed to have some previous acquaintance with the materiaL Besides giving the author's point of view and notational conventions, the introduction gives a few proofs (or rather demonstrations) intended to bridge the gap between the more analytic approach to the subject as exemplified by the books of Ahlfors-Sario [4], Farkas-Kra [11] or Forster [12] and the more algebraic approach as exemplified by Walker [26]. In particular the proof of the genus formulas for a plane curve based on the Riemann-Hurwitz formula and other materials in Section 1.3 were inspired by a seminar given in the early '60's at Brown by S. Lefschetz. Despite the recent appearance of several excellent books on the theory of compact Riemann surfaces (which is, of course, basically the same as the theory of algebraic curves over the complex numbers), most of the material in these notes does not appear in these books. The two main subjects treated here are exceptional points on Riemann surfaces (Weierstrass points, higher-order Weierstrass points) and automorphisms of Riemann surfaces. A foundational treatment of the theory of automorphisms from the viewpoint of Galois coverings of Riemann surfaces is given in Chapters Four and Five, following and expanding to some extent the treatment of Ahlfors-Sario [4] and Seifert-Threlfal1 [25]. The treatment is technically different from that of A. M. Macbeath [19] and his students, although fundamentally it is the same
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