ItisnoweasytoseethatlinearlyequivalentdivisorsgiverisetoisomorphiclinebundlesandcanberecapturedfromthosebyconsideringzerosofmeromorphicsectionsIftheyaree ectivethentheycanberecapturedaszeroesofho ID: 340107
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LinebundlesAlinebundleLoverXisabration:L!Xwhosebersarelines(C)andsuchthatthebrationislocallytrivial.Sowhatdoesthatmean?WeshouldconsiderXtobecoveredwithopensubsetsUioverwhichistrivial.Wecanthenwrite�1(Ui)=UiC.Thoseneedtobegluedtogether,andhowdoweexpressglueingdata?Wecanthinkoflocalco-ordinatesas(zi;t)for�1(Ui)whereziissotospeakthelocalberco-ordinate.NowifwehavetwoopencoversintersectinginUij=Ui\Ujweshouldhavezi= ij(t)zjwhere ij(t)6=0anddenedontheintersectionUij.Itisnaturaltothinkof ijasholomorphicfunctionsifwewantLtobeaholomorphicbundle.Nowthose ijarereferredtoasthetransitionfunctions,andtheycannotbearbitrarilychosen.IfUijk=Ui\Uj\Ukisnon-emptywehavemanyidenticationsoverpointsinUijkandtheyhavetobecompatible.Whatisthecompatibilitycondition?Itiseasilyseentobegivenby ik= ij jkwherewemakethenaturalconventionthat ij=1= jiandhence ii=1.Nowthosetransitionfunctionsarenotuniquelydeterminedbythelinebun-dle,afterallwecane ectchangesoftheberco-ordinates.Chosingnon-zeroholomorphicfunctionsfionUiwecanintroducenewco-ordinatesz0i=fiziandthenewtransitionfunctions 0ijwillbetransformedaccordingly 0ij=fi=fj ij.Itisclearthatif ijsatisesthecompatibilityconditionsodoes 0ij.Inpartic-ularifwecanndfunctionsfisuchthat ij=fi=fjthenwecanusethosetogetaglobaltrivializationofL,namelytowriteLasCX.DigressiononCzechcohomologyByaCzechcochainismeantfunctionstakingvaluesinsomesheafFandde-pendingonintersectionsofopensetsofanopencovering.Iwillnotgiveaformaldenitionofasheaf,itiseasyenough,butyoushouldthinkaboutitasfunctionsdenedonopensubsetsofatopologicalsetX.Ifthefunctionsareconstant,wethinkofthesheafasoneoflocallyconstantfunctions.Bya0-cochainwecanthinkofasfunctionsfUdenedonanopensetUinthecovering,whilea1-cochainisgivenbyfunctionsfUVdenedonU\Vwhilea2-cochainissimilarlygivenbyfunctionsfUVWdenedonintersectionsU\V\W.Nowwecandeneaboundaryoperator@denedasfollows(@f)UV=fU�fVand(@f)UVW=fUV�fUW+fVWandsoon.Whatdoesitmeanthat@f=0fora0-chain?SimplythatfU=fVonU\V.InotherwordsthatwecanndafunctionFgloballydenedsuchthatFU=fUwesaythatthelocaldatapatchesuptoagloballydenedfunction.Whatdoesitmeanthat@f=0fora1-chain?NamelythatfUV=fUW+fWV(NotethatfXY=�fYX).Thisisalmostthesamethingasthecompatibilityconditionfortransitionfunctions,exceptthatitiswrittenadditivelyinsteadofmultiplicatively.Furthermorea1boundaryisoftheformfu�fvwhichistheadditiveinterpretationofthemultiplicativepresentationoftransitionfunctionswhicharetrivial.2 Itisnoweasytoseethatlinearlyequivalentdivisorsgiverisetoisomorphicline-bundles,andcanberecapturedfromthosebyconsideringzerosofmero-morphicsections.Iftheyaree ective,thentheycanberecapturedaszeroesofholomorphicsections.KeyPoint.Thecrucialdataaregivenbytransitionfunctions.Thosecanbemanipulatedveryeasily,muchmoreeasilythanindividualsectionsandcorrespondingdivisors.Inherentinthemisthe'movingproperty'andassuchtheywillgiveaneasyproofofBezoutaswewillseelater.FunctorialpropertiesofLine-bundlesIfwehaveamap:X!Ywecanliftanyline-bundleLofYtoonedenoted(L)aboveX.Thecorrespondingline-bundleiscalledthepull-backandisalmosttriviallydenedbyconsideringitstransitionfunctions( ij):= ij()bycomposition.Ifisaninclusion,thenwesimplyrestrictthetransitionfunctionsfromYtoX.MapsassociatedtolinearsystemsIfwehavetwosectionss0;s1tothesameline-bundlethenwehaveseenthats1=s0isawell-denedmeromorphicfunction,andhenceamapintoP1.Wecanalsorepresentitasthemapx7!(s0(x);s1(x).Moregenerally,letVbealinearsubspaceofH0(D).Suchaspaceisreferredtoasalinearsystem.Ifwechooseabasiss0;s1;:::snforitwegetamapofXintoPnintheobviousway.Nowweshouldbecareful.Ifallsi(x)=0wesaythatxbelongstothebaselocusofthelinearsystem,anditisonlyoutsidethebaselocusthatwehaveawell-denedmap.Wecanmakethemapsintoprojectivespaceco-ordinatefree,providedthatwemapintotheirduals.Infactifxdoesnotbelongtothebase-locus,thesectionsvanishingatxformahyperplane.Givenamapp:X!Pnassociatedtoaline-bundleL.WecanthenconsiderthehyperplanesHofPnandlookatp(H)thosearereferredtoasthehyperplanesections,andtheycorrespondtodivisorsofsectionsofLandallsuchoccurinthisway.Ifweconsideranarbitrarylinearsystemi.e.alinearsubspaceofthecom-pletelinearsystem,themapontothatcanbethoughtofasaprojectionfromthecompletePn.Pencils,whichwealreadyhaveencountered,aresimply1-dimensionallinearsystems.TheygiverisetobrationsoverP12-dimensionalsystemsarecallednetsclassically,while3-dimensionalarereferredtoaswebs.4 1-formsandvectoreldsontheRiemannsphereNowbythechainrule@F=@z=(@F=@w)(dw=dz)thusavectoreldgivenbys0(z)@ @z=s1(w)@ @wwillsatisfys0(z)(dw=dz)=s1(w)ors0(z)=�z2s1(w).Thustherewillbeplentyofholomorphicvectoreldsonthesphere,infacttheyaregivenbythebinaryquadrics.Thedualcaseof1-formsisdi erent.Clearlydw=dw dzdzandhenceifwehaves0(z)dz=s1(w)dwweobtains0(z)=�z�2s1(z)hencetherearenoglobal1-formsonthesphere,exceptthetrivial.WehavethusthefundamentalTheorem:ThedegreeofthecanonicallinebundleofCP1is�2MapsoflinearsystemsonCP1ThemapCP1!CP2givenbyallbinaryquadricshasasanimageacurveofdegree2i.e.aconic.Andconverselyallconicsoccurinthiswayaswehavealreadynoticed.ThemapCP1!CP3givenbyallbinarycubicshasasanimageacurveofdegree3.ItisbiholomorphictoCP1andisthusasocalledrationalcurve,referredtoasthetwistedcubic.IfweconsiderthelinearsystemofcubicspassingthroughaxedpointonCP1thisformsanetandhencegivesaprojectionofthetwistedcubicontotheplane.Theimagewillbeaconic,aswecanwritethecubicsasquadricsmul-tipliedwithaconstantlinearfactorcuttingoutthexedpoint.GeometricallyitmeansthatforanypointponthetwistedcubicCthatcurveiscontainedinaquadriccone.Giventwoquadricconescorrespondingtothepointsp;qtheirintersectionwillconsistofthetwistedcubicalongwiththelinepassingthroughpandq.Thetwistedcubicisnotacompleteintersection,youneedthreequadricstocutitout.Givenanysuchthreequadricstheywillnotonlycutitoutbutalsogenerateallquadricswhichcontainthetwistedcubic,whichconstituteanetwiththetwistedcubicasabaselocus.6