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1、ALGEBRAICGEOMETRYJ.S.MILNEAbstract.ThesearethenotesforMath631,taughtattheUniversityofMichigan,Fall1993.Theyareavailableatwww.math.lsa.umich.edu/∼jmilne/Pleasesendcommentsandcorrectionstomeatjmilne@umich.edu.v2.01(August24,1996).Firstversionontheweb.v3.01(June13,
2、1998).Added5sections(25pages)andanindex.MinorchangestoSections0–8.ContentsIntroduction20.AlgorithmsforPolynomials41.AlgebraicSets142.AffineAlgebraicVarieties303.AlgebraicVarieties444.LocalStudy:TangentPlanes,TangentCones,Singularities595.ProjectiveVarietiesandComp
3、leteVarieties806.FiniteMaps1017.DimensionTheory1098.RegularMapsandTheirFibres.1179.AlgebraicGeometryoveranArbitraryField13110.DivisorsandIntersectionTheory13711.CoherentSheaves;InvertibleSheaves.14312.Differentials14913.AlgebraicVarietiesovertheComplexNumbers1511
4、4.FurtherReading153Index156c1996,1998J.S.Milne.Youmaymakeonecopyofthesenotesforyourownpersonaluse.12J.S.MILNEIntroductionJustasthestartingpointoflinearalgebraisthestudyofthesolutionsofsystemsoflinearequations,naijXj=di,i=1,...,m,(*)j=1thestartingpointforalgebr
5、aicgeometryisthestudyofthesolutionsofsystemsofpolynomialequations,fi(X1,...,Xn)=0,i=1,...,m,fi∈k[X1,...,Xn].Noteimmediatelyonedifferencebetweenlinearequationsandpolynomialequations:theoremsforlinearequationsdon’tdependonwhichfieldkyouareworkingover,1butthoseforpol
6、ynomialequationsdependonwhetherornotkisalgebraicallyclosedand(toalesserextent)whetherkhascharacteristiczero.SinceIintendtoemphasizethegeometryinthiscourse,wewillworkoveralgebraicallyclosedfieldsforthemajorpartofthecourse.Abetterdescriptionofalgebraicgeometryistha
7、titisthestudyofpolynomialfunc-tionsandthespacesonwhichtheyaredefined(algebraicvarieties),justastopologyisthestudyofcontinuousfunctionsandthespacesonwhichtheyaredefined(topo-logicalspaces),differentialgeometry(=advancedcalculus)thestudyofdifferentiablefunctionsandthe
8、spacesonwhichtheyaredefined(differentiablemanifolds),andcomplexanalysisthestudyofholomorphicfunctionsandthespacesonwhichtheyaredefined(Riemannsurfacesandcomplexmanifolds
