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ID:39833679
大小:1.24 MB
页数:140页
时间:2019-07-12
《Enrique Arrondo Introduction to projective varieties》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
1、IntroductiontoprojectivevarietiesbyEnriqueArrondo(*)VersionofSeptember5,2007Thisisstillprobablyfarfrombeingafinalversion,especiallysinceIhadnotimeyettocompletethesecondpart(whichissofarnotwellconnectedwiththefirstone).Anyway,anykindofcommentsareverywelcome,inparticularthoseco
2、ncerningthegeneralstructure,orsuggestionsforshorterand/orcorrectproofs.0.Algebraicbackground1.Projectivesetsandtheirideals;WeakNullstellensatz2.Irreduciblecomponents3.Hilbertpolynomial.Nullstellensatz4.Gradedmodules;resolutionsandprimarydecomposition5.Dimension,degreeandari
3、thmeticgenus6.Productofvarieties7.Regularmaps8.Propertiesofmorphisms9.Resolutionsanddimension10.Ruledvarieties11.Tangentspacesandcones;smoothness12.Transversality13.Parameterspaces14.Affinevarietiesvsprojectivevarieties;sheaves15.Thelocalringatapoint16.Introductiontoaffineandpr
4、ojectiveschemes17.Vectorbundles18.Coherentsheaves19.Schemes(*)DepartamentodeAlgebra,FacultaddeCienciasMatem´´aticas,UniversidadCom-plutensedeMadrid,28040Madrid,Spain,EnriqueArrondo@mat.ucm.es1Thescopeofthesenotesistopresentasoftandpracticalintroductiontoalgebraicgeometry,i.
5、e.withveryfewalgebraicrequirementsbutarrivingsoontodeepresultsandconcreteexamplesthatcanbeobtained“byhand”.ThenotesarebasedonsomebasicPhDcourses(Milan1998andFlorence2000)andasummercourse(Perugia1998)thatItaught.Idecidedtoproducethesenoteswhilepreparingnewsimilarcourses(Mila
6、nandPerugia2001).Myapproachconsistsofavoidingallthealgebraicpreliminariesthatastandardal-gebraicgeometrycourseusesforaffinevarietiesandthusstartdirectlywithprojectivevarieties(whicharethevarietiesthathavegoodproperties).ThemaintechniqueIuseistheHilbertpolynomial,fromwhichitis
7、possibletorigorouslyandintuitivelyintroducealltheinvariantsofaprojectivevariety(dimension,degreeandarithmeticgenus).ItisalsopossibletoeasilyprovetheprojectiveNullstellensatz(fromwhichthestandardaffineNullstellensatzcaninfactbeobtained).Thepricetopayforthisshortcutisthattheway
8、toproducetheimportantresults(themostimportantoneforpracticalpurposesisthetheoremab
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