Hartshorne Algebraic Geometry Solutions

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1、AlgebraicGeometryBy:RobinHartshorneSolutionsSolutionsbyJoeCutroneandNickMarshburn1Foreword:Thisisourattempttoputacollectionofpartiallycompletedsolutionsscatteredontheweballinoneplace.ThisstartedasourpersonalcollectionofsolutionswhilereadingHartshorne.Wewerestuck(andarestill)onseveralprobl

2、ems,whichledtoourwebsearchwherewefoundsomeextremelycleversolutionsby[SAM]and[BLOG]amongothers.Somesolutionsinthis.pdfarealltheirsandjustrepeatedhereforconvenience.Inotherplacestheauthorsmadecorrectionsorclarications.Duecredithastriedtobeproperlygivenineachcase.Ifyoulookontheirwebsites(li

3、stedinthereferences)andcomparesolutions,itshouldbeobviouswhenweusedtheirideasifnotexplicitlystated.Whilemostsolutionsaredone,theyarenottypedatthistime.Iamtryingtobeonpacewithonesolutionaday(...whichrarelyhappens),soIwillupdatethisfrequently.Checkbackfromtimetotimeforupdates.AsIamusingthis

4、reallyasalearningtoolformyself,pleaserespondwithcommentsorcorrections.Aswithanymathpostedanywhere,readatyourownrisk!21Chapter1:Varieties1.1AneVarieties1.(a)LetYbetheplanecurvedenedbyy=x2.ItscoordinateringA(Y)isthenk[x;y]=(yx2)=k[x;x2]=k[x].(b)A(Z)=k[x;y]=(xy1)=k[x;1],whichistheloca

5、lizationofk[x]atxx.Anyhomomorphismofk-algebras':k[x;1]!k[x]mustmapxxintok,sincexisinvertible.Then'isclearlynotsurjective,soinparticular,notanisomorphism.(c)Letf(x;y)2k[x;y]beanirreduciblequadratic.Theprojectiveclosureisdenedbyz2f(x;y):=F(x;y;z).Intersectingthisvarietyzzwiththehyperplanea

6、tinnityz=0givesahomogeneouspolynomialF(x;y;0)intwovariableswhichsplitsintotwolinearfactors.IfFhasadoubleroot,thevarietyintersectsthehyperplaneatonlyonepoint.SinceanynonsingularcurveinP2isisomorphictoP1,Z(F)n1=P1n1=A1.SoZ(f)=A1.IfFhastwodistinctroots,sayp;q,thentheoriginalcurveisP1minus

7、2points,whichisthesameasA1minusonepoint,callitp.Changecoordinatestosetp=0sothatthecoordinateringisk[x;1].x2.YisisomorphictoA1viathemapt7!(t;t2;t3),withinversemapbeingtherstprojection.SoYisananevarietyofdimension1.ThisalsoshowsthatA(Y)isisomorphictoapolynomialringi