LONDONMATHEMATICALSOCIETYSTUDENTTEXTS ManagingEditor:ProfessorD.Benson, DepartmentofMathematics,UniversityofAberdeen,UK 48 Settheory,ANDRÁSHAJNAL&PETERHAMBURGER.TranslatedbyATTILAMATE 49 AnintroductiontoK-theoryforC*-algebras,M.RØRDAM,F.LARSEN&N.J.LAUSTSEN 50 Abriefguidetoalgebraicnumbertheory,H.P.F.SWINNERTON-DYER 51 Stepsincommutativealgebra:Secondedition,R.Y.SHARP 52 FiniteMarkovchainsandalgorithmicapplications,OLLEHÄGGSTRÖM 53 Theprimenumbertheorem,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, PETERSARNAK&ALAINVALETTE 56 Logic,inductionandsets,THOMASFORSTER 57 IntroductiontoBanachalgebras,operatorsandharmonicanalysis,GARTHDALESetal. 58 Computationalalgebraicgeometry,HALSCHENCK 59 Frobeniusalgebrasand2-Dtopologicalquantumfieldtheories,JOACHIMKOCK 60 Linearoperatorsandlinearsystems,JONATHANR.PARTINGTON 61 AnintroductiontononcommutativeNoetherianrings:Secondedition,K.R.GOODEARL& R.B.WARFIELD,JR 62 Topicsfromone-dimensionaldynamics,KARENM.BRUCKS&HENKBRUIN 63 Singularpointsofplanecurves,C.T.C.WALL 64 AshortcourseonBanachspacetheory,N.L.CAROTHERS 65 ElementsoftherepresentationtheoryofassociativealgebrasI,IBRAHIMASSEM,DANIEL SIMSON&ANDRZEJSKOWRON´SKI 66 An introduction to sieve methods and their applications, ALINA CARMEN COJOCARU & M.RAMMURTY 67 Ellipticfunctions,J.V.ARMITAGE&W.F.EBERLEIN 68 Hyperbolicgeometryfromalocalviewpoint,LINDAKEEN&NIKOLALAKIC 69 LecturesonKählergeometry,ANDREIMOROIANU 70 Dependencelogic,JOUKUVÄÄNÄNEN 71 ElementsoftherepresentationtheoryofassociativealgebrasII,DANIELSIMSON&ANDRZEJ SKOWRON´SKI 72 Elements of the representation theory of associative algebras III, DANIEL SIMSON & ANDRZEJSKOWRON´SKI 73 Groups,graphsandtrees,JOHNMEIER 74 RepresentationtheoremsinHardyspaces,JAVADMASHREGHI 75 Anintroductiontothetheoryofgraphspectra,DRAGOŠCVETKOVIC´,PETERROWLINSON &SLOBODANSIMIC´ 76 NumbertheoryinthespiritofLiouville,KENNETHS.WILLIAMS 77 Lecturesonprofinitetopicsingrouptheory,BENJAMINKLOPSCH,NIKOLAYNIKOLOV& CHRISTOPHERVOLL 78 Cliffordalgebras:Anintroduction,D.J.H.GARLING 79 Introduction to compact Riemann surfaces and dessins d’enfants, ERNESTO GIRONDO & GABINOGONZÁLEZ-DIEZ 80 TheRiemannhypothesisforfunctionfields,MACHIELVANFRANKENHUIJSEN 81 Numbertheory,Fourieranalysisandgeometricdiscrepancy,GIANCARLOTRAVAGLINI 82 Finitegeometryandcombinatorialapplications,SIMEONBALL 83 Thegeometryofcelestialmechanics,HANSJÖRGGEIGES 84 Randomgraphs,geometryandasymptoticstructure,MICHAELKRIVELEVICHetal 85 Fourieranalysis:PartI–Theory,ADRIANCONSTANTIN 86 Dispersivepartialdifferentialequations,M.BURAKERDOG˘AN&NIKOLAOSTZIRAKIS 87 Riemannsurfacesandalgebraiccurves,R.CAVALIERI&E.MILES Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:04:06, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252 Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:04:06, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252 LondonMathematicalSocietyStudentTexts87 Riemann Surfaces and Algebraic Curves A First Course in Hurwitz Theory RENZO CAVALIERI ColoradoStateUniversity ERIC MILES ColoradoMesaUniversity Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:04:06, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252 One(cid:29)Liberty(cid:29)Plaza,(cid:29)20th(cid:29)Floor,NewYorkNY10006,USA CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107149243 (cid:2)c RenzoCavalieriandEricMiles2016 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2016 PrintedintheUnitedStatesofAmericabySheridanBooks,Inc. AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-Publicationdata Names:Cavalieri,Renzo,1976–|Miles,Eric(EricW.) Title:Riemannsurfacesandalgebraiccurves:afirstcoursein Hurwitztheory/RenzoCavalieri,ColoradoStateUniversity, EricMiles,ColoradoMesaUniversity. Description:NewYorkNY:CambridgeUniversityPress,[2017]| Series:LondonMathematicalSocietystudenttexts;87 Identifiers:LCCN2016025911|ISBN9781107149243 Subjects:LCSH:Riemannsurfaces|Curves,Algebraic.|Geometry,Algebraic. Classification:LCCQA333.C382017|DDC515/.93–dc23 LCrecordavailableathttps://lccn.loc.gov/2016025911 ISBN978-1-107-14924-3Hardback ISBN978-1-316-60352-9Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Titlepage:TopologyTree,byKrisBarz,2015.http://krisbarz.squarespace.com Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:04:06, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252 Contents Introduction pageix 1 FromComplexAnalysistoRiemannSurfaces 1 1.1 Differentiability 2 1.2 Integration 4 1.3 Cauchy’sIntegralFormulaandConsequences 7 1.4 InverseFunctions 9 2 IntroductiontoManifolds 14 2.1 GeneralDefinitionofaManifold 15 2.2 BasicExamples 17 2.3 ProjectiveSpaces 19 2.4 CompactSurfaces 23 2.5 ManifoldsasLevelSets 28 3 RiemannSurfaces 32 3.1 ExamplesofRiemannSurfaces 33 3.2 CompactRiemannSurfaces 37 4 MapsofRiemannSurfaces 47 4.1 HolomorphicMapsofRiemannSurfaces 47 4.2 LocalStructureofMaps 50 4.3 MapsofCompactRiemannSurfaces 54 4.4 TheRiemann–HurwitzFormula 56 4.5 ExamplesofMapsofCompactRiemannSurfaces 59 5 LoopsandLifts 63 5.1 Homotopy 64 5.2 TheFundamentalGroup 67 5.3 CoveringSpaces 75 v Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:03:12, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252 vi Contents 6 CountingMaps 80 6.1 HurwitzNumbers 81 6.2 Riemann’sExistenceTheorem 84 6.3 HyperellipticCovers 86 7 CountingMonodromyRepresentations 90 7.1 FromMapstoMonodromy 91 7.2 FromMonodromytoMaps 96 7.3 MonodromyRepresentationsandHurwitzNumbers 100 7.4 ExamplesandComputations 102 7.5 DegenerationFormulas 106 8 RepresentationTheoryofS 111 d 8.1 TheGroupRingandtheGroupAlgebra 111 8.2 Representations 113 8.3 Characters 116 8.4 TheClassAlgebra 118 9 HurwitzNumbersandZ(S ) 120 d 9.1 Genus0 120 9.2 GenusandCommutators 122 9.3 BurnsideFormula 124 10 TheHurwitzPotential 128 10.1 GeneratingFunctions 128 10.2 ConnectedHurwitzNumbers 133 10.3 Cut-and-Join 137 AppendixA HurwitzTheoryinPositiveCharacteristic 143 A.1 Introduction 143 A.2 AlgebraicCurvesinPositiveCharacteristic 144 A.3 SmoothAlgebraicCurves 145 A.4 TheGenusandtheRiemann–HurwitzFormula 146 A.5 TheAffineLineIsNotSimplyConnectedOverk 149 AppendixB TropicalHurwitzNumbers 151 B.1 TropicalGeometry:WhereDoesItComeFrom? 151 B.2 AxiomaticTropicalGeometry 152 B.3 TropicalCovers 153 B.4 TropicalCoversasShadows–MoviesofMonodromy Representations 156 B.5 BendingandBreaking 157 Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:03:12, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252 Contents vii AppendixC HurwitzSpaces 161 C.1 ModuliSpaces 161 C.2 HurwitzSpaces 163 C.3 ApplicationsofHurwitzSpaces 164 AppendixD Does Physics Have Anything to Say About HurwitzNumbers? 169 D.1 PhysicalMathematics 169 D.2 HurwitzNumbersandStringTheory 171 Bibliography 179 Index 181 Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:03:12, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252 Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:03:12, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252 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 intendstopresentHurwitztheorytoanundergraduateaudience,payingspecial 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 materialinthebook. HurwitztheoryistheenumerativestudyofanalyticfunctionsbetweenRie- mann Surfaces – complex compact manifolds of dimension one. A Hurwitz numbercountsthenumberofsuchfunctionswhentheappropriatesetofdis- creteinvariantsisfixed.Thishasitsorigininthe1800sintheworkofRiemann, whofirsthadtheinsightthatmulti-valuedinversesofcomplexanalyticfunc- tionscanbenaturallyseenasfunctionsdefinedonadomainwhichislocally, butnotglobally,identifiablewiththecomplexplane:i.e.aRiemannSurface. StudyinganalyticfunctionsdefinedonRiemannSurfacesleadstothegeom- etry of oriented topological surfaces, which Riemann Surfaces are. The local behavioroffunctionsrevealsahighdegreeofstructure:analyticfunctionsare ramifiedcoverings;thatis,coveringsexceptatadiscretesetofpointswherea phenomenoncalledramificationoccurs. Ramifiedcoveringsnaturallygiverisetomonodromyrepresentations,which arehomomorphismsfromthefundamentalgroupofthepuncturedtargetsur- facetoasymmetricgroup.Theramificationatthepreimagesofapointbinthe baseiscapturedbythecycletypeofthepermutationassociatedwithasmall loopwindingaroundthepointb. Thecountofallsuchrepresentationscanbeidentifiedwithacoefficientof aspecificproductofvectorsintheclassalgebraofthesymmetricgroup:with a vector space which has a basis indexed by conjugacy classes. Elements of ix Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:02:04, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252.001 x Introduction thisbasisaregivenbyformalsumsofallpermutationsinthesameconjugacy class. A commutative multiplication is then defined by extending the group operationofthesymmetricgroupbybilinearity. Theclassalgebraisknowntobesemisimple:itadmitsabasiswithrespect to which multiplication is idempotent. Computing the product above in the semisimple basis yields closed formulas for Hurwitz numbers in terms of charactersofthesymmetricgroup. Tosummarize,thecountofanalyticfunctionswastranslatedtoageometric count of topological covers, then to an algebraic count of group homomor- phisms,andfinallyreducedtoarepresentationtheoreticcomputation. Inadifferentdirection,RiemannSurfacescanbedegeneratedtonodalsur- faces by shrinking loops. These nodal surfaces look like “smaller” Riemann Surfacesgluedatpoints,andsodegenerationcreatesinfinitefamiliesofrecur- sive relations among Hurwitz numbers. We conclude the book by showing that when Hurwitz numbers are encoded as coefficients of a formal power series(ageneratingfunctioncalledtheHurwitzpotential),someoftheserecur- sionstranslateintopartialdifferentialequationsthataresolvedbytheHurwitz potential. Whether this summary makes perfect sense or no sense at all depends on the background of the reader. In any case, we hope that at least two things are apparent: first, that keywords from several different undergraduate courseshavebeenused;andsecond,thatnoexceptionallysophisticatedterm appeared. Thisbookarisesfromtwoexperimentalundergraduatecoursesthatthefirst authortaughtatColoradoStateUniversityin2014and2015.Thecourseswere offeredasafollow-uptoclassesintopologyanddifferentialgeometry;amain goal was to depart from the structure of a traditional course and offer the students a mode of approaching the study of mathematics closer to that of a researcherfacinganewproblem. At a school like Colorado State University, most advanced math majors have typically taken semester-long courses in some of the areas mentioned in the above synopsis, and typically have not taken all those courses. There issomeanalogywiththesituationthatmathematicalresearchersareinwhen they tackle an open problem. First of all, translation and reformulation of a problemisoftenaveryimportanttoolinmathematicalresearch.Problemsthat aretoodifficultwhenstudiedinacertainwaymaybecomeapproachablewhen thepointofviewischanged.Whenmathematicalresearcherstranslateaques- tion in order to find ways to solve it, they are often taken into mathematical areas out of their comfort zone. And they don’t have the opportunity to take a semester-long course, or to read a whole book on each topic that they use, Downloaded from https:/www.cambridge.org/core. University of Warwick, on 03 Feb 2017 at 02:02:04, subject to the Cambridge Core terms of use , available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316569252.001