基础代数几何（第1卷）

BOOK1.VarietiesinProjectiveSpace
ChapterI.BasicNotions
1.AlgebraicCurvesinthePlane
1.1.PlaneCurves
1.2.RationalCurves
1.3.RelationwithFieldTheory
1.4.RationalMaps
1.5.SingularandNonsingularPoints
1.6.TheProjectivePlane
Exercisesto1
2.ClosedSubsetsofAffineSpace
2.1.DefinitionofClosedSubsets
2.2.RegularFunctionsonaClosedSubset
2.3.RegularMaps
Exercisesto2
3.RationalFunctions
3.1.IrreducibleAlgebraicSubsets
3.2.RationalFunctions
3.3.RationalMaps
Exercisesto3
4.QuasiprojectiveVarieties
4.1.ClosedSubsetsofProjectiveSpace
4.2.RegularFunctions
4.3.RationalFunctions
4.4.ExamplesofRegularMaps
Exercisesto4
5.ProductsandMapsofQuasiprojectiveVarieties
5.1.Products
5.2.TheImageofaProjectiveVarietyisClosed
5.3.FiniteMaps
5.4.NoetherNormalisation
Exercisesto5
6.Dimension
6.1.DefinitionofDimension
6.2.DimensionofIntersectionwithaHypersurface
6.3.TheTheoremontheDimensionofFibres
6.4.LinesonSurfaces
Exercisesto6
ChapterII.LocalProperties
1.SingularandNonsingularPoints
1.1.TheLocalRingofaPoint
1.2.TheTangentSpace
1.3.IntrinsicNatureoftheTangentSpace
1.4.SingularPoints
1.5.TheTangentCone
Exercisesto1
2.PowerSeriesExpansions
2.1.LocalParametersataPoint
2.2.PowerSeriesExpansions
2.3.VarietiesovertheRealsandtheComplexes
Exercisesto2
3.PropertiesofNonsingularPoints
3.1.CodimensioniSubvarieties
3.2.NonsingularSubvarieties
Exercisesto3
4.TheStructureofBirationalMaps
4.1.BlowupinProjectiveSpace
4.2.LocalBlowup
4.3.BehaviourofaSubvarietyunderaBlowup
4.4.ExceptionalSubvarieties
4.5.IsomorphismandBirationalEquivalence
Exercisesto4
5.NormalVarieties
5.1.NormalVarieties
5.2.NormalisationofanAffineVariety
5.3.NormalisationofaCurve
5.4.ProjectiveEmbeddingofNonsingularVarieties
Exercisesto5
6.SingularitiesofaMap
6.1.Irreducibility
6.2.Nonsingularity
6.3.Ramification
6.4.Examples
Exercisesto6
ChapterIII.DivisorsandDifferentialForms
1.Divisors
1.1.TheDivisorofaFunction
1.2.LocallyPrincipalDivisors
1.3.MovingtheSupportofaDivisorawayfromaPoint
1.4.DivisorsandRationalMaps
1.5.TheLinearSystemofaDivisor
1.6.PencilofConicsoverp1
Exercisesto1
2.DivisorsonCurves
2.1.TheDegreeofaDivisoronaCurve
2.2.Bezout'sTheoremonaCurve
2.3.TheDimensionofaDivisor
Exercisesto2
3.ThePlaneCubic
3.1.TheClassGroup
3.2.TheGroupLaw
3.3.Maps
3.4.Applications
3.5.AlgebraicallyNonclosedField
Exercisesto3
4.AlgebraicGroups
4.1.AlgebraicGroups
4:2.QuotientGroupsandChevalley'sTheorem
4.3.AbelianVarieties
4.4.ThePicardVariety
Exercisesto4
5.DifferentialForms
5.1.RegularDifferential1-forms
5.2.AlgebraicDefinitionoftheModuleofDifferentials
5.3.Differentialp-forms
5.4.RationalDifferentialForms
Exercisesto5
6.ExamplesandApplicationsofDifferentialForms
6.1.BehaviourUnderMaps
6.2.InvariantDifferentialFormsonaGroup
6.3.TheCanonicalClass
6:4.Hypersurfaces
6.5.HyperellipticCurves
6.6.TheRiemann-RochTheoremforCurves
6.7.ProjectiveEmbeddingofaSurface
Exercisesto6
ChapterIV.IntersectionNumbers
1.DefinitionandBasicProperties
1.1.DefinitionofIntersectionNumber
1.2.Additivity
1.3.InvarianceUnderLinearEquivalence
1.4.TheGeneralDefinitionofIntersectionNumber
Exercisesto1
2.ApplicationsofIntersectionNumbers
2.1.Bezout'sTheoreminProjectiveandMultiprojective
Space
2.2.VarietiesovertheReals
2.3.TheGenusofaNonsingularCurveonaSurface
2.4.TheRiemann-RochInequalityonaSurface
2.5.TheNonsingularCubicSurface
2.6.TheRingofCycleClasses
Exercisesto2
3.BirationalMapsofSurfaces
3.1.BlowupsofSurfaces
3.2.SomeIntersectionNumbers
3.3.ResolutionofIndeterminacy
3.4.FactorisationasaChainofBlowups
3.5.RemarksandExamples
Exercisesto3
4.Singularities
4.1.SingularPointsofaCurve
4.2.SurfaceSingularities
4.3.DuValSingularities
4.4.DegenerationofCurves
Exercisesto4
AlgebraicAppendix
1.LinearandBilinearAlgebra
2.Polynomials
3.QuasilinearMaps
4.Invariants
5.Fields
6.CommutativeRings
7.UniqueFactorisation
8.IntegralElements
9.LengthofaModule
References
Index
BOOK2.SchemesandVarieties
ChapterV.Schemes
1.TheSpecofaRing
1.1.DefinitionofSpecA
1.2.PropertiesofPointsofSpecA
1.3.TheZariskiTopologyofSpecA
1.4.Irreducibility,Dimension
Exercisesto1
2.Sheaves
2.1.Presheaves
2.2.TheStructurePresheaf
2.3.Sheaves
2.4.StalksofaSheaf
Exercisesto2
3.Schemes
3.1.DefinitionofaScheme
3.2.GlueingSchemes
3.3.ClosedSubschemes
3.4.ReducedSchemesandNilpotents
3.5.FinitenessConditions
Exercisesto3
4.ProductsofSchemes
4.1.DefinitionofProduct
4.2.GroupSchemes
4.3.Separatedness
Exercisesto4
ChapterVI.Varieties
1.DefinitionsandExamples
1.1.Definitions
1.2.VectorBundles
1.3.VectorBundlesandSheaves
1.4.DivisorsandLineBundles
Exercisesto1
2.AbstractandQuasiprojectiveVarieties
2.1.Chow'sLemma
2.2.BlowupAlongaSubvariety
2.3.ExampleofNon-QuasiprojectiveVariety
2.4.CriterionsforProjectivity
Exercisesto2
3.CoherentSheaves
3.1.SheavesofOx-modules
3.2.CoherentSheaves
3.3.DevissageofCoherentSheaves
3.4.TheFinitenessTheorem
Exercisesto3
4.ClassificationofGeometricObjectsandUniversalSchemes
4.1.SchemesandFunctors
4.2.TheHilbertPolynomial
4.3.FlatFamilies
4.4.TheHilbertScheme
Exercisesto4
BOOK3.ComplexAlgebraicVarietiesandComplexManifolds
ChapterVII.TheTopologyofAlgebraicVarieties
1.TheComplexTopology
1.1.Definitions
1.2.AlgebraicVarietiesasDifferentiableManifolds;
Orientation
1.3.HomologyofNonsingularProjectiveVarieties
Exercisesto1
2.Connectedness
2.1.PreliminaryLemmas
2.2.TheFirstProofoftheMainTheorem
2.3.TheSecondProof
2.4.AnalyticLemmas
2.5.ConnectednessofFibres
Exercisesto2
3.TheTopologyofAlgebraicCurves
3.1.LocalStructureofMorphisms
3.2.TriangulationofCurves
3.3.TopologicalClassificationofCurves
3.4.CombinatorialClassificationofSurfaces
3.5.TheTopologyofSingularitiesofPlaneCurves
Exercisesto3
4.RealAlgebraicCurves
4.1.ComplexConjugation
4.2.ProofofHarnack'sTheorem
4.3.OvalsofRealCurves
Exercisesto4
ChapterVIII.ComplexManifolds
1.DefinitionsandExamples
1.1.Definition
1.2.QuotientSpaces
1.3.CommutativeAlgebraicGroupsasQuotientSpaces
1.4.ExamplesofCompactComplexManifoldsnot
IsomorphictoAlgebraicVarieties
1.5.ComplexSpaces
Exercisesto1
2.DivisorsandMeromorphicFunctions
2.1.Divisors
2.2.MeromorphicFunctions
2.3.TheStructureoftheFieldM(X)
Exercisesto2
3.AlgebraicVarietiesandComplexManifolds
3.1.ComparisonTheorems
3.2.ExampleofNonisomorphicAlgebraicVarietiesthat
AreIsomorphicasComplexManifolds
3.3.ExampleofaNonalgebraicCompactComplex
ManifoldwithMaximalNumberofIndependent
MeromorphicFunctions
3.4.TheClassificationofCompactComplexSurfaces
Exercisesto3
4.KahlerManifolds
4.1.KahlerMetric
4.2.Examples
4.3.OtherCharacterisationsofKahlerMetrics
4.4.ApplicationsofKahlerMetrics
4.5.HodgeTheory
Exercisesto4
ChapterIX.Uniformisation
1.TheUniversalCover
1.1.TheUniversalCoverofaComplexManifold
1.2.UniversalCoversofAlgebraicCurves
1.3.ProjectiveEmbeddingofQuotientSpaces
Exercisesto1
2.CurvesofParabolicType
2.1.Thetafunctions
2.2.ProjectiveEmbedding
2.3.EllipticFunctions,EllipticCurvesandElliptic
Integrals
Exercisesto2
3.CurvesofHyperbolicType
3.1.PoincareSeries
3.2.ProjectiveEmbedding
3.3.AlgebraicCurvesandAutomorphicFunctions
Exercisesto3
4.UniformisingHigherDimensionalVarieties
4.1.CompleteIntersectionsareSimplyConnected
4.2.ExampleofManifoldwithaGivenFiniteGroup
4.3.Remarks
Exercisesto4
HistoricalSketch
1.EllipticIntegrals
2.EllipticFunctions
3.AbelianIntegrals
4.RiemannSurfaces
5.TheInversionofAbelianIntegrals
6.TheGeometryofAlgebraicCurves
7.HigherDimensionalGeometry
8.TheAnalyticTheoryofComplexManifolds
9.AlgebraicVarietiesoverArbitraryFieldsandSchemes
References
ReferencesfortheHistoricalSketch
Index