Enrique Arrondo Introduction to projective varieties

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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