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1、AlgebraicGeometryAProblemSolvingApproachParkCityMathematicsInstitute2008UndergraduateFacultyProgramProjectLeadTomGarrityWilliamsCollegeAuthorsRichardBelshoffJunalynNavarra-MadsenMissouriStateUniversityTexasWoman’sUniversityLynetteBoosPedroPoitevin1TrinityC
2、ollegeSalemStateCollegeRyanBrownShawnRobinsonGeorgiaCollege&StateUniversityUniversityofMaine,PresqueIsleJimDrouihletBrianSnyderMinnesotaStateUniversity2LakeSuperiorStateUniversityCarlLienertCarynWernerFortLewisCollegeAlleghenyCollegeDavidMurphyHillsdaleCol
3、lege1AlsoProvidenceCollege2JimpassedawayJanuary23,2009ContentsPrefacev0.1.Algebraicgeometryv0.2.Overviewvi0.3.Problembookvii0.4.Historyofbookviii0.5.Anasideonnotationviii0.6.ThanksixChapter1.Conics11.1.ConicsovertheReals11.2.ChangesofCoordinates101.3.Conic
4、sovertheComplexNumbers201.4.TheComplexProjectivePlane?2291.5.ProjectiveChangeofCoordinates371.6.TheComplexProjectiveLine?1391.7.Ellipses,Hyperbolas,andParabolasasSpheres461.8.DegenerateConics-Crossinglinesanddoublelines.511.9.TangentsandSingularPoints551.1
5、0.Conicsvialinearalgebra641.11.Duality72Chapter2.CubicCurvesandEllipticCurves792.1.Cubicsin?2792.2.In?ectionPoints852.3.GroupLaw1112.4.Normalformsofcubics1242.5.TheGroupLawforaSmoothCubicinCanonicalForm1482.6.CubicsasTori1562.7.Cross-Ratiosandthej-Invarian
6、t1592.8.CrossRatio:AProjectiveInvariant1702.9.Thej-Invariant1762.10.Torusas?/Λ179iiiAlgebraicGeometry:AProblemSolvingApproach2.11.Mapping?/ΛtoaCubic188Chapter3.HigherDegreeCurves1953.1.HigherDegreePolynomialsandCurves1953.2.HigherDegreeCurvesasSurfaces1973
7、.3.B′ezout’sTheorem2053.4.RegularFunctionsandFunctionFields2303.5.TheRiemann-RochTheorem2413.6.SingularitiesandBlowingUp283Chapter4.A?neVarieties3054.1.ZeroSetsofPolynomials3054.2.AlgebraicSets3074.3.ZeroSetsviaV(I)3084.4.FunctionsonZeroSetsandtheCoordinat
8、eRing3104.5.HilbertBasisTheorem3114.6.HilbertNullstellensatz3134.7.VarietyasIrreducible:PrimeIdeals3154.8.Subvarieties3174.9.FunctionFields3194.10.PointsasMaximalIdeals3204.11.TheZariskiTopology3214.12.Points
