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Professor D. Benson, Department of Mathematics, University of Aberdeen, UK LONDON MATHEMATICAL SOCIETY STUDENT TEXTS Managing Editor: Professor D. Benson, Department of Mathematics, University of Aberdeen, UK 48 Set theory, ANDRÁS HAJNAL & PETER HAMBURGER. Translated by ATTILA MATE 49 An introduction to K-theory for C*-algebras, M. RØRDAM, F. LARSEN & N. J. LAUSTSEN 50 A brief guide to algebraic number theory, H. P. F. SWINNERTON-DYER 51 Steps in commutative algebra: Second edition, R. Y. SHARP 52 Finite Markov chains and algorithmic applications, OLLE HÄGGSTRÖM 53 The prime number theorem, G. J. O. JAMESON 54 Topics in graph automorphisms and reconstruction, JOSEF LAURI & RAFFAELE SCAPELLATO 55 Elementary number theory, group theory and Ramanujan graphs, GIULIANA DAVIDOFF, PETER SARNAK & ALAIN VALETTE 56 Logic, induction and sets, THOMAS FORSTER 57 Introduction to Banach algebras, operators and harmonic analysis, GARTH DALES et al. 58 Computational algebraic geometry, HAL SCHENCK 59 Frobenius algebras and 2-D topological quantum field theories, JOACHIM KOCK 60 Linear operators and linear systems, JONATHAN R. PARTINGTON 61 An introduction to noncommutative Noetherian rings: Second edition, K. R. GOODEARL & R. B. WARFIELD, JR 62 Topics from one-dimensional dynamics, KAREN M. BRUCKS & HENK BRUIN 63 Singular points of plane curves, C. T. C. WALL 64 A short course on Banach space theory, N. L. CAROTHERS 65 Elements of the representation theory of associative algebras I, IBRAHIM ASSEM, DANIEL SIMSON & ANDRZEJ SKOWRONSKI´ 66 An introduction to sieve methods and their applications, ALINA CARMEN COJOCARU & M. RAM MURTY 67 Elliptic functions, J. V. ARMITAGE & W. F. EBERLEIN 68 Hyperbolic geometry from a local viewpoint, LINDA KEEN & NIKOLA LAKIC 69 Lectures on Kähler geometry, ANDREI MOROIANU 70 Dependence logic, JOUKU VÄÄNÄNEN 71 Elements of the representation theory of associative algebras II, DANIEL SIMSON & ANDRZEJ SKOWRONSKI´ 72 Elements of the representation theory of associative algebras III, DANIEL SIMSON & ANDRZEJ SKOWRONSKI´ 73 Groups, graphs and trees, JOHN MEIER 74 Representation theorems in Hardy spaces, JAVAD MASHREGHI 75 An introduction to the theory of graph spectra, DRAGOŠ CVETKOVIC,´ PETER ROWLINSON & SLOBODAN SIMIC´ 76 Number theory in the spirit of Liouville, KENNETH S. WILLIAMS 77 Lectures on profinite topics in group theory, BENJAMIN KLOPSCH, NIKOLAY NIKOLOV & CHRISTOPHER VOLL 78 Clifford algebras: An introduction, D. J. H. GARLING 79 Introduction to compact Riemann surfaces and dessins d’enfants, ERNESTO GIRONDO & GABINO GONZÁLEZ-DIEZ 80 The Riemann hypothesis for function fields, MACHIEL VAN FRANKENHUIJSEN 81 Number theory, Fourier analysis and geometric discrepancy, GIANCARLO TRAVAGLINI 82 Finite geometry and combinatorial applications, SIMEON BALL 83 The geometry of celestial mechanics, HANSJÖRG GEIGES 84 Random graphs, geometry and asymptotic structure, MICHAEL KRIVELEVICH et al 85 Fourier analysis: Part I – Theory, ADRIAN CONSTANTIN 86 Dispersive partial differential equations, M. BURAK ERDOGAN˘ & NIKOLAOS TZIRAKIS 87 Riemann surfaces and algebraic curves, R. CAVALIERI & E. MILES 7D647#&'&"(975"[email protected]%#5$91&)'079).%79'0"!61C'9@1AE"7(""#9D1%3!"76"8?"'$"B87"#A"8B8K8!"@B$2'%A"A7"A0'"#5$91&)'"79'"A'95@"7("B@' 7D647#&'&"(975"[email protected]%#5$91&)'079).%79'0"!61C'9@1AE"7(""#9D1%3!"76"8?"'$"B87"#A"8B8K8!"@B$2'%A"A7"A0'"#5$91&)'"79'"A'95@"7("B@' London Mathematical Society Student Texts 87 Riemann Surfaces and Algebraic Curves A First Course in Hurwitz Theory RENZO CAVALIERI Colorado State University ERICMILES Colorado Mesa University 7D647#&'&"(975"[email protected]%#5$91&)'079).%79'0"!61C'9@1AE"7(""#9D1%3!"76"8?"'$"B87"#A"8B8K8!"@B$2'%A"A7"A0'"#5$91&)'"79'"A'95@"7("B@' One Liberty Plaza, 20th Floor, New York NY 10006, USA Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107149243 c Renzo Cavalieri and Eric Miles 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United States of America by Sheridan Books, Inc. A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication data Names: Cavalieri, Renzo, 1976– | Miles, Eric (Eric W.) Title: Riemann surfaces and algebraic curves : a first course in Hurwitz theory / Renzo Cavalieri, Colorado State University, Eric Miles, Colorado Mesa University. Description: New York NY : Cambridge University Press, [2017] | Series: London Mathematical Society student texts ; 87 Identifiers: LCCN 2016025911 | ISBN 9781107149243 Subjects: LCSH: Riemann surfaces | Curves, Algebraic. | Geometry, Algebraic. Classification: LCC QA333 .C38 2017 | DDC 515/.93–dc23 LC record available at https://lccn.loc.gov/2016025911 ISBN 978-1-107-14924-3 Hardback ISBN 978-1-316-60352-9 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Title page: Topology Tree, by Kris Barz, 2015. http://krisbarz.squarespace.com 7D647#&'&"(975"[email protected]%#5$91&)'079).%79'0"!61C'9@1AE"7(""#9D1%3!"76"8?"'$"B87"#A"8B8K8!"@B$2'%A"A7"A0'"#5$91&)'"79'"A'95@"7("B@' Contents Introduction page ix 1 From Complex Analysis to Riemann Surfaces 1 1.1 Differentiability 2 1.2 Integration 4 1.3 Cauchy’s Integral Formula and Consequences 7 1.4 Inverse Functions 9 2 Introduction to Manifolds 14 2.1 General Definition of a Manifold 15 2.2 Basic Examples 17 2.3 Projective Spaces 19 2.4 Compact Surfaces 23 2.5 Manifolds as Level Sets 28 3 Riemann Surfaces 32 3.1 Examples of Riemann Surfaces 33 3.2 Compact Riemann Surfaces 37 4 Maps of Riemann Surfaces 47 4.1 Holomorphic Maps of Riemann Surfaces 47 4.2 Local Structure of Maps 50 4.3 Maps of Compact Riemann Surfaces 54 4.4 The Riemann–Hurwitz Formula 56 4.5 Examples of Maps of Compact Riemann Surfaces 59 5 Loops and Lifts 63 5.1 Homotopy 64 5.2 The Fundamental Group 67 5.3 Covering Spaces 75 v 6C536"%&%"'864")@@79.CCC0$"4#80%(&068(.$68&0" 50B&890@D"6'"!"8C0$2!"65"8?"&#"B87""@"8B8?7B!"9A#1&$@"@6"@)&""4#80%(&"68&"@&849"6'"A9& vi Contents 6 Counting Maps 80 6.1 Hurwitz Numbers 81 6.2 Riemann’s Existence Theorem 84 6.3 Hyperelliptic Covers 86 7 Counting Monodromy Representations 90 7.1 From Maps to Monodromy 91 7.2 From Monodromy to Maps 96 7.3 Monodromy Representations and Hurwitz Numbers 100 7.4 Examples and Computations 102 7.5 Degeneration Formulas 106 8 Representation Theory of Sd 111 8.1 The Group Ring and the Group Algebra 111 8.2 Representations 113 8.3 Characters 116 8.4 The Class Algebra 118 9 Hurwitz Numbers and Z(Sd ) 120 9.1 Genus 0 120 9.2 Genus and Commutators 122 9.3 Burnside Formula 124 10 The Hurwitz Potential 128 10.1 Generating Functions 128 10.2 Connected Hurwitz Numbers 133 10.3 Cut-and-Join 137 Appendix A Hurwitz Theory in Positive Characteristic 143 A.1 Introduction 143 A.2 Algebraic Curves in Positive Characteristic 144 A.3 Smooth Algebraic Curves 145 A.4 The Genus and the Riemann–Hurwitz Formula 146 A.5 The Affine Line Is Not Simply Connected Over k 149 Appendix B Tropical Hurwitz Numbers 151 B.1 Tropical Geometry: Where Does It Come From? 151 B.2 Axiomatic Tropical Geometry 152 B.3 Tropical Covers 153 B.4 Tropical Covers as Shadows – Movies of Monodromy Representations 156 B.5 Bending and Breaking 157 6C536"%&%"'864")@@79.CCC0$"4#80%(&068(.$68&0" 50B&890@D"6'"!"8C0$2!"65"8?"&#"B87""@"8B8?7B!"9A#1&$@"@6"@)&""4#80%(&"68&"@&849"6'"A9& Contents vii Appendix C Hurwitz Spaces 161 C.1 Moduli Spaces 161 C.2 Hurwitz Spaces 163 C.3 Applications of Hurwitz Spaces 164 Appendix D Does Physics Have Anything to Say About Hurwitz Numbers? 169 D.1 Physical Mathematics 169 D.2 Hurwitz Numbers and String Theory 171 Bibliography 179 Index 181 6C536"%&%"'864")@@79.CCC0$"4#80%(&068(.$68&0" 50B&890@D"6'"!"8C0$2!"65"8?"&#"B87""@"8B8?7B!"9A#1&$@"@6"@)&""4#80%(&"68&"@&849"6'"A9& 6C536"%&%"'864")@@79.CCC0$"4#80%(&068(.$68&0" 50B&890@D"6'"!"8C0$2!"65"8?"&#"B87""@"8B8?7B!"9A#1&$@"@6"@)&""4#80%(&"68&"@&849"6'"A9& Introduction Hurwitz theory is a beautiful algebro-geometric theory that studies maps of Riemann Surfaces. Despite being (relatively) unsophisticated, it is typically unapproachable at the undergraduate level because it ties together several branches of mathematics that are commonly treated separately. This book intends to present Hurwitz theory to an undergraduate audience, paying special attention to the connections between algebra, geometry and complex analy- sis that it brings about. We illustrate this point by giving an overview of the material in the book. Hurwitz theory is the enumerative study of analytic functions between Rie- mann Surfaces – complex compact manifolds of dimension one. A Hurwitz number counts the number of such functions when the appropriate set of dis- crete invariants is fixed. This has its origin in the 1800s in the work of Riemann, who first had the insight that multi-valued inverses of complex analytic func- tions can be naturally seen as functions defined on a domain which is locally, but not globally, identifiable with the complex plane: i.e. a Riemann Surface.
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