Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory

195 Pages·2016·3.306 MB·English

by  Renzo Cavalieri, Eric Miles

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LondonMathematicalSocietyStudentTexts87
Riemann Surfaces and Algebraic Curves
A First Course in Hurwitz Theory
RENZO CAVALIERI
ColoradoStateUniversity
ERIC MILES
ColoradoMesaUniversity
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(cid:13)c RenzoCavalieriandEricMiles2016
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Firstpublished2016
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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
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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
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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
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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
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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
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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,
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