Table Of ContentAnil Nerode · Noam Greenberg
Algebraic Curves
and Riemann
Surfaces for
Undergraduates
The Theory of the Donut
Algebraic Curves and Riemann Surfaces
for Undergraduates
Anil Nerode • Noam Greenberg
Algebraic Curves
and Riemann Surfaces
for Undergraduates
The Theory of the Donut
AnilNerode NoamGreenberg
DepartmentofMathematics SchoolofMathematics,Statistics
CornellUniversity andOperationsResearch
Ithaca,NY,USA VictoriaUniversityofWellington
Wellington,NewZealand
ThisworkwassupportedbyRoyalSocietyTeApa¯rangi,MarsdenFundgrant,andRutherfordDiscovery
Fellowship
ISBN978-3-031-11615-5 ISBN978-3-031-11616-2 (eBook)
https://doi.org/10.1007/978-3-031-11616-2
MathematicsSubjectClassification:51-01,14-01,30-01,30Fxx,14Hxx,14H52,33E05
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Preface
The Sumeriansand Babylonians(2000–700BCE) and the Greeks(700 BCE–150
CE)madeperiodicmeasurementsofthelocationsofthesunandtheknownplanets.
They concluded that these bodies travel in circles with the earth as their center,
withsmallretrogradedeviations.Theirtablesandsimpleastronomicalinstruments
were used to guide travelers on camels over the silk roads of Asia and captains
of ships on the high seas. These methods were codified in the great Almagest of
Ptolemy(100–170CE).Thisbookwasusedfornavigationforwelloverathousand
years.Copernicus(1473–1543)re-didthecalculationsbasedoncircularorbitswith
the sun as center. Tycho Brahe (1542–1601)improved the observations, resulting
in Kepler’s three laws. They imply that the orbits around the sun are ellipses
which are not circles. Mathematicians introduced elliptic analogs of the circular
functions.It turnedoutthatthe elliptic analoguesof the arcsine and arccosineare
doubly periodic functions of a complex variable. This led to a unified theory of
algebraiccurvesand Riemannsurfaces,incorporatingalgebra,analysis, geometry,
andtopology.Thepurposeofthisbookistomaketheunityofmathematicsapparent
to undergraduates. It is intended to serve as a textbook for a capstone course in
undergraduate mathematics. This book originates in a semester-long such course
taughtbyProf.NerodeatCornellUniversity.
Thebookconsistsofthreeparts.Thefirstdealswithalgebraiccurves.Itfocuses
ontheprojectiveplane,tangents,andintersectionmultiplicitiesandculminatesina
proofoftheassociativityofthe“chord-and-tangent”groupoperationonnonsingular
cubiccurves.Itcanbeusedonitsownforashortcourse.Thesecondworkstoward
Riemannsurfaces;ittakestimetobuildtherequiredtoolsfromtopology,calculus,
and complex analysis. The third ties together the first two parts, explaining how
complex curvescan be given the structure of a Riemann surface. It ends with the
isomorphism theorem for complex tori and elliptic curves, and with the use of
analyticparameterisationsofcurvestoredefineintersectionmultiplicities.
Thebook’sstyleandcontentareamixtureofmodernandoldermathematics.We
liveinthemodernmathematicalworld:atourdisposalaresettheoryandlogic.The
argumentswe give conformto modernstandardsof rigor. And we allow a certain
levelofabstraction:forexample,wegiveaxiomaticdefinitionsofgroupsandrings;
we define path homotopyand simple connectedness.We do, however,try to keep
this kindof abstractionto a minimum,and overall,to give the reader some of the
v
vi Preface
flavorofmid-nineteenth-centurymathematics,priortotheabstractturnchampioned
byDedekindinthe1870s.Wedefineandstudyalgebraiccurveswithoutintroducing
thenotionofanidealinaring.Wedonotgiveanaxiomaticdefinitionoftopological
spaces: rather, we restrict ourselves to topological subspaces of manifolds. Most
prominently, we make heavy use of Kronecker’s elimination theory, which uses
purelycomputationalmethodstodefine,forexample,intersectionmultiplicitiesof
curves.
Weattempttobeself-contained.Wereviewtherequiredbackgroundmaterialin
somedetail.Weexpect,however,thatareaderwillhavealreadystudiedsomelinear
algebra,groups,andsomemultivariablecalculus,andsothesetopicsarediscussed
abitmorebriefly.
Greenberg would like to thank Moshe Zadka and Alex Usvyatsov, for their
support during the writing of an early version of the book; and his colleagues
JoeMiller,DenisHirschfeldt,Dan Turetsky,Rod Downey,andRobGoldblatt,for
theirsupportovertheyears.WewouldliketothankVUWstudentsLennoxLeary,
Giovanna Le Gros, Jim Paterson, Tim Caldwell, Jayden Mudge, Eli Gadsby, and
AntoniaKing,whohaveworkedthroughvariousiterationsofthebook.
Ithaca,NY,USA AnilNerode
Wellington,NewZealand NoamGreenberg
Contents
1 Introduction................................................................. 1
1.1 TheTheoryoftheCircle ............................................ 1
1.1.1 PythagoreanTriples ........................................ 1
1.1.2 TheCircularFunctions .................................... 2
1.1.3 TheTheoryoftheDonut,inaNutshell ................... 5
1.2 OverviewoftheBook ............................................... 7
1.2.1 PartI:AlgebraicCurves ................................... 7
1.2.2 PartII:RiemannSurfaces ................................. 9
1.2.3 PartIII:CurvesandSurfaces .............................. 11
1.3 Preliminaries,andSomeNotation .................................. 12
PartI AlgebraicCurves
2 Algebra ...................................................................... 17
2.1 PolynomialsandPowerSeries ...................................... 17
2.1.1 TheCategoryofRings ..................................... 19
2.1.2 BacktoFormalPowerSeries .............................. 21
2.1.3 MoreonPolynomials ...................................... 23
2.2 UniqueFactorisation ................................................ 25
2.2.1 DivisibilityinIntegralDomains ........................... 25
2.2.2 UniqueFactorisationDomains ............................ 28
2.2.3 UniqueFactorisationinPolynomialRings ............... 30
2.3 Groups................................................................ 35
2.3.1 TheCategoryofGroups ................................... 35
2.3.2 QuotientGroups ........................................... 38
2.3.3 CyclicGroups .............................................. 39
2.3.4 TheSymmetricGroup ..................................... 40
2.4 LinearAlgebraOverIntegralDomains ............................. 41
2.4.1 Matrices,LinearSpaces,andLinearMaps ............... 41
2.4.2 DimensionandComplements ............................. 44
2.4.3 TheDeterminant ........................................... 45
2.4.4 DetectingSingularity ...................................... 47
vii
viii Contents
2.5 FurtherExercises .................................................... 48
3 AffineSpace................................................................. 55
3.1 DefinitionofHypersurfaces ......................................... 56
3.2 TheResultant ........................................................ 58
3.2.1 TheSylvesterMatrix ...................................... 58
3.2.2 TheResultant,CommonRoots,andMoreVariables ..... 60
3.2.3 TheResultantisaLinearCombination ................... 62
3.3 Study’sLemma ...................................................... 64
3.3.1 ProofofStudy’sLemma ................................... 65
3.4 AffineLinesandRationalParameterisations ....................... 66
3.4.1 AffineLines ................................................ 66
3.4.2 RationalParameterisations ................................ 67
3.5 FurtherExercises .................................................... 69
4 ProjectiveSpace ............................................................ 73
4.1 HomogeneousPolynomials ......................................... 74
4.2 ProjectiveSpace ..................................................... 76
4.3 ProjectiveLinesandMaps .......................................... 78
4.3.1 ProjectiveMaps ............................................ 80
4.4 EmbeddingAffineSpaceintoProjectiveSpace .................... 81
4.5 ChangesofCoordinates ............................................. 87
4.5.1 ChangeofVariable ......................................... 87
4.5.2 FourPointLemma ......................................... 90
4.6 SpacesofCurves .................................................... 92
4.6.1 TheDualPlane ............................................. 93
4.6.2 Desargues’Theorem ....................................... 94
4.7 ProductsofProjectiveSpaces ....................................... 95
4.8 FurtherExercises .................................................... 99
5 Tangents ..................................................................... 105
5.1 Introduction:AffineTangentsandIntersectionswithLines ....... 105
5.1.1 IntersectionMultiplicities ................................. 106
5.1.2 HomogeneousCoordinates ................................ 109
5.2 FormalPartialDerivatives ........................................... 110
5.2.1 PropertiesofDerivatives................................... 110
5.2.2 TheDiscriminant........................................... 113
5.3 HigherOrderTangents .............................................. 113
5.3.1 TheModuliSpaceofTangents ............................ 117
5.3.2 InvarianceoftheHigherOrderTangent .................. 118
5.4 TheIntersectionofaLinewithaCurve ............................ 120
5.4.1 DefinitionofIntersectionMultiplicity .................... 121
5.4.2 InvarianceofMultiplicityofIntersectionwitha
Line ......................................................... 122
5.4.3 TangentsandIntersectionswithLines .................... 124
5.4.4 SimpleIntersectionsAretheNorm ....................... 126
Contents ix
5.5 FurtherExercises .................................................... 127
6 Bézout’sTheorem........................................................... 133
6.1 AFirstLookattheIntersectionofCurves ......................... 134
6.1.1 TheResultantofHomogeneousPolynomialsIs
Homogeneous .............................................. 135
6.1.2 AWeakVersionofBézout’sTheorem .................... 137
6.2 TheHomogeneousResultant ........................................ 139
6.2.1 MainPropertyoftheHomogeneousResultant ........... 141
6.3 MultiplicityofIntersectionandBézout’sTheorem ................ 143
6.3.1 CodingLinesinP2×P2 .................................. 143
6.3.2 The Resultant of the General Intersection
Polynomials ................................................ 144
6.3.3 IntersectionMultiplicityandBézout’sTheorem ......... 146
6.3.4 GeometricInvariance ...................................... 147
6.4 CoincidencewithEarlierDefinitions ............................... 148
6.4.1 UsingtheFamilyofVerticalLines ........................ 148
6.4.2 IntersectingLines .......................................... 150
6.5 CategoricityofMultiplicityofIntersection ........................ 151
6.5.1 Symmetry .................................................. 151
6.5.2 Products .................................................... 151
6.5.3 InfiniteMultiplicities ...................................... 155
6.5.4 Shifts ....................................................... 155
6.5.5 CategoricityofMultiplicityofIntersection ............... 156
6.6 AffineCalculations .................................................. 158
6.7 Multiplicities,OrdersandTangents ................................ 159
6.8 FurtherExercises .................................................... 161
7 TheEllipticGroup.......................................................... 167
7.1 Flexes................................................................. 168
7.1.1 FlexesandtheSecondOrderTangent..................... 168
7.1.2 TheHessian ................................................ 169
7.2 TheGroupOperationonaNonsingularCubicCurve ............. 171
7.2.1 TheComplementCurve ................................... 172
7.2.2 AssociativityoftheGroupOperation ..................... 174
7.3 NormalFormsforNonsingularCubics ............................. 178
7.3.1 ExplicitCalculationsoftheGroupOperation ............ 182
7.4 FurtherExercises .................................................... 183
PartII RiemannSurfaces
8 Quasi-EuclideanSpaces.................................................... 191
8.1 TopologyofRn ...................................................... 192
8.2 Manifolds ............................................................ 194
8.2.1 TopologyofPre-manifolds ................................ 197
8.2.2 Subspaces .................................................. 198
x Contents
8.2.3 TheHausdorffProperty .................................... 199
8.2.4 TopologicalCountability .................................. 200
8.2.5 Manifolds ................................................... 201
8.2.6 SpacesandContinuity ..................................... 202
8.3 Compactness ......................................................... 204
8.3.1 ClosedSets ................................................. 205
8.3.2 SequencesandLimits ...................................... 206
8.3.3 Interlude:Completeness ................................... 208
8.3.4 CompactnessinEuclideanSpace ......................... 210
8.4 QuotientsbyDiscreteSubgroups ................................... 212
8.5 FurtherExercises .................................................... 217
9 Connectedness,SmoothandSimple...................................... 221
9.1 Connectedness,PathandSimple .................................... 222
9.1.1 Homotopy;SimpleConnectedness........................ 224
9.2 LiftingMaps ......................................................... 226
9.2.1 TheWindingNumber ...................................... 228
9.3 Differentiability:AReminder ....................................... 230
9.3.1 MeanValueInequalities ................................... 233
9.3.2 PartialDerivatives .......................................... 235
9.3.3 InverseFunctions .......................................... 236
9.3.4 SecondDerivatives ......................................... 238
9.4 DifferentiableManifolds ............................................ 239
9.5 PartitionsofUnity ................................................... 241
9.5.1 ProofofTheorem9.66 ..................................... 243
9.6 DifferentiableConnectedness ....................................... 245
9.6.1 PiecewiseSmoothPaths ................................... 248
9.7 FurtherExercises .................................................... 249
10 PathIntegrals............................................................... 255
10.1 IntegratingFormsAlongPaths ...................................... 255
10.1.1 TheLengthofaPath ....................................... 259
10.2 IntegratingAlongSmoothPaths .................................... 260
10.2.1 LinearForms ............................................... 262
10.2.2 RelatingtheGeneralandFamiliarIntegrals .............. 262
10.3 IntegratingVectorFields ............................................ 266
10.3.1 ConservativeVectorFields ................................ 267
10.3.2 TheWindingNumberRevisited ........................... 269
10.4 SymmetricVectorFields ............................................ 270
10.4.1 MissingaPoint ............................................. 273
10.5 FurtherExercises .................................................... 276
11 ComplexDifferentiation................................................... 281
11.1 ComplexDerivativesandIntegrals ................................. 281
11.1.1 ComplexIntegrals.......................................... 285
11.2 Cauchy’sIntegralFormula .......................................... 286
