基本信息
•【作 者】：J.Harris編
•【又/譯名】：Algebraic Geometry A First Course
•【叢編項】：Graduate Texts in Mathematics
•【裝幀項】：簡裝本 大32開 / 328
•【出版項】：世界圖書出版公司(此信息作廢)/ 2000-6-1
•【ISBN號】：9787506247146 / 7506247143
圖書簡介
This book is based on one-semester courses given at Harvard in 1984， at Brown in 1985， and at Harvard in 1988. It is intended to be， as the title suggests， a first introduction to the subject. Even so， a few words are in order about the purposes of the book. Algebraic geometry has developed tremendously over the last century. During the 19th century， the subject was practiced on a relatively concrete， down-to-earth level; the main objects of study were projective varieties， and the techniques for the most part were grounded in geometric constructions. This approach flourished during the middle of the century and reached its culmination in the work of the Italian school around the end of the 19th and the beginning of the 20th centuries.Ultimately， the subject was pushed beyond the limits of its foundations： by the end of its period the Italian school had progressed to the point where the language and techniques of the subject could no longer serve to express or carry out the ideas of its best practitioners.本書為英文版。
本書目錄
Preface
Acknowledgments
UsingThisBook
PARTI:EXAMPLESOFVARIETIESANDMAPS
LECTURE1
AffineandProjectiveVarieties
ANoteAboutOurField
AffineSpaceandAffineVarieties
ProjectiveSpaceandProjectiveVarieties
LinearSpaces
FiniteSets
Hypersurfaces
AnalyticSubvarietiesandSubmanifolds
TheTwistedCubic
RationalNormalCurves
DeterminantalRepresentationoftheRationalNormalCurve
AnotherParametrizationoftheRationalNormalCurve
TheFamilyofPlaneConics
ASyntheticConstructionoftheRationalNormalCurve
OtherRationalCurves
VarietiesDefinedoverSubfieldsofK
ANoteonDimension,Smoothness,andDegree
LECTURE2
RegularFunctionsandMaps
TheZariskiTopology
RegularFunctionsonanAftineVariety
ProjectiveVarieties
RegularMaps
TheVeroneseMap
DeterminantalRepresentationofVeroneseVarieties
SubvarietiesofVeroneseVarieties
TheSegreMaps
SubvarietiesofSegreVarieties
ProductsofVarieties
Graphs
FiberProducts
CombinationsofVeroneseandSegreMaps
LECTURE3
Cones,Projections,andMoreAboutProducts
Cones
Quadrics
Projections
MoreCones
MoreProjections
ConstructibleSets
LECTURE4
FamiliesandParameterSpaces
FamiliesofVarieties
TheUniversalHyperplane
TheUniversalHyperplaneSection
ParameterSpacesofHypersurfaces
UniversalFamiliesofHypersurfaces
AFamilyofLines
LECTURE5
IdealsofVarieties,IrreducibleDecomposition,andtheNullstellensatz
GeneratingIdeals
IdealsofProjectiveVarieties
IrreducibleVarietiesandIrreducibleDecomposition
GeneralObjects
GeneralProjections
GeneralTwistedCubics
DoublePointLoci
ALittleAlgebra
RestatementsandCorollaries
LECTURE6
GrassmanniansandRelatedVarieties
Grassmannians
SubvarietiesofGrassmannians
TheGrassmannianG(1,3)
AnAnalogoftheVeroneseMap
IncidenceCorrespondences
VarietiesofIncidentPlanes
TheJoinofTwoVarieties
FanoVarieties
LECTURE7
RationalFunctionsandRationalMaps
RationalFunctions
RationalMaps
GraphsofRationalMaps
BirationalIsomorphism
TheQuadricSurface
Hypersurfaces
DegreeofaRationalMap
Blow-Ups
BlowingUpPoints
BlowingUpSubvarieties
TheQuadricSurfaceAgain
TheCubicScrollinP4
Unirationality
LECTURE8
MoreExamples
TheJoinofTwoVarieties
TheSecantPlaneMap
SecantVarieties
TrisecantLines,etc.
JoinsofCorrespondingPoints
RationalNormalScrolls
Higher-DimensionalScrolls
MoreIncidenceCorrespondences
FlagManifolds
MoreJoinsandIntersections
QuadricsofRank4
RationalNormalScrollsII
LECTURE9
DeterminantalVarieties
GenericDeterminantalVarieties
SegreVarieties
SecantVarietiesofSegreVarieties
LinearDeterminantalVarietiesinGeneral
RationalNormalCurves
SecantVarietiestoRationalNormalCurves
RationalNormalScrollsIII
RationalNormalScrollsIV
MoreGeneralDeterminantalVarieties
SymmetricandSkew-SymmetricDeterminantalVarieties
FanoVarietiesofDeterminantalVarieties
LECTURE10
AlgebraicGroups
TheGeneralLinearGroupGLnK
TheOrthogonalGroupSOnK
TheSymplecticGroupSp2nK
GroupActions
PGLn+1KactsonPn
PGL2KActsonP2
PGL2KActsonP3
PGL2KActsonPn
PGL3KActsonP5
PGL3KActsonP9
POnKActsonPn-1(automorphismsoftheGrassmannian)
PGLnKActsonP(∧kKn)
Quotients
QuotientsofAffineVarietiesbyFiniteGroups
QuotientsofAffineSpace
SymmetricProducts
QuotientsofProjectiveVarietiesbyFiniteGroups
WeightedProjectiveSpaces
PARTII:ATTRIBUTESOFVARIETIES
LECTURE11
DefinitionsofDimensionandElementaryExamples
Hypersurfaces
CompleteIntersections
ImmediateExamples
TheUniversalk-Plane
VarietiesofIncidentPlanes
SecantVarieties
SecantVarietiesinGeneral
JoinsofVarieties
FlagManifolds
(Some)SchubertVarieties
LECTURE12
MoreDimensionComputations
DeterminantalVarieties
FanoVarieties
ParameterSpacesofTwistedCubics
TwistedCubics
TwistedCubicsonaGeneralSurface
CompleteIntersections
CurvesofType(a,b)onaQuadric
DeterminantalVarieties
GroupActions
GL(V)ActsonSymdVand∧kV
PGLn+1KActson(Pn)landG(k,n)l
LECTURE13
HilbertPolynomials
HilbertFunctionsandPolynomials
HilbertFunctionoftheRationalNormalCurve
HilbertFunctionoftheVeroneseVariety
HilbertPolynomialsofCurves
Syzygies
ThreePointsinP2
FourPointsinP2
CompleteIntersections:KoszulComplexes
LECTURE14
SmoothnessandTangentSpaces
TheZariskiTangentSpacetoaVariety
ALocalCriterionforIsomorphism
ProjectiveTangentSpaces
DeterminantalVarieties
LECTURE15
GaussMaps,TangentialandDualVarieties
ANoteAboutCharacteristic
GaussMaps
TangentialVarieties
TheVarietyofTangentLines
JoinsofIntersectingVarieties
TheLocusofBitangentLines
DualVarieties
LECTURE16
TangentSpacestoGrassmannians
TangentSpacestoGrassmannians
TangentSpacestoIncidenceCorrespondences
VarietiesofIncidentPlanes
TheVarietyofSecantLines
VarietiesSweptoutbyLinearSpaces
TheResolutionoftheGenericDeterminantalVariety
TangentSpacestoDualVarieties
TangentSpacestoFangVarieties
LECTURE17
FurtherTopicsInvolvingSmoothnessandTangentSpaces
GaussMapsonCurves
OsculatingPlanesandAssociatedMaps
TheSecondFundamentalForm
TheLocusofTangentLinestoaVariety
Bertini'sTheorem
Blow-ups,NashBlow-ups,andtheResolutionofSingularities
SubadditivityofCodimensionsofIntersections
LECTURE18
Degree
Bezout'sTheorem
TheRationalNormalCurves
MoreExamplesofDegrees
VeroneseVarieties
SegreVarieties
DegreesofConesandProjections
JoinsofVarieties
UnirationalityofCubicHypersurfaces
LECTURE19
FurtherExamplesandApplicationsofDegree
MultidegreeofaSubvarietyofaProduct
ProjectiveDegreeofaMap
JoinsofCorrespondingPoints
VarietiesofMinimalDegree
DegreesofDeterminantalVarieties
DegreesofVarietiesSweptoutbyLinearSpaces
DegreesofSomeGrassmannians
Harnack'sTheorem
LECTURE20
SingularPointsandTangentCones
TangentCones
TangentConestoDeterminantalVarieties
Multiplicity
ExamplesofSingularities
ResolutionofSingularitiesforCurves
LECTURE21
ParameterSpacesandModuliSpaces
ParameterSpaces
ChowVarietSes
HilbertVarieties
CurvesofDegree2
ModuliSpaces
PlaneCubics
LECTURE22
Quadrics
GeneralitiesaboutQuadrics
TangentSpacestoQuadrics
PlaneConics
QuadricSurfaces
QuadficsinPn
LinearSpacesonQuadrics
LinesonQuadrics
PlanesonFour-DimensionalQuadrics
FanoVarietiesofQuadricsinGeneral
FamiliesofQuadrics
TheVarietyofQuadricsinP1
TheVarietyofQuadricsinP2
CompleteConics
QuadricsinPn
PencilsofQuadrics
HintsforSelectedExercises
References
Index
