Algebraic diffeomorphisms of real rational surface

arXiv:0804.3846v1[math.AG]24Apr2008ALGEBRAICDIFFEOMORPHISMSOFREALRATIONALSURFACESANDWEIGHTEDBLOW-UPSINGULARITIESJOHANNESHUISMANANDFR´ED´ERICMANGOLTEAbstract.LetXbearealrationalsurfacehavingonlyweightedblow-upsingularities.DenotebyDiffalg(X)thegroupofalgebraicautomor-phismsoralgebraicdiffeomorphismsofXintoitself.Letnbeanaturalintegerandlete=[e1,...,eℓ]beapartitionofn.DenotebyXethesetofℓ-tuples(P1,...,Pℓ)ofdistinctnonsingularinfinitelynearpointsofXoforders(e1,...,eℓ).WeshowthatthegroupDiffalg(X)actstransi-tivelyonXe.Thisstatementgeneralizesearlierworkwherethecaseofthetrivialpartitione=[1,...,1]wastreatedunderthesupplementaryconditionthatXisnonsingular.Asanapplicationweclassifyrationalrealsurfaceshavingonlyweight-edblow-upsingularities.MSC2000:14P25,14E07Keywords:realalgebraicsurface,rationalsurface,geometricallyrationalsurface,weightedblow-upsingularity,algebraicdiffeomor-phism,algebraicautomorphism,transitiveaction1.IntroductionLetXbeanonsingularcompactconnectedrealalgebraicsurface,i.e.XisanonsingularcompactconnectedrealalgebraicsubsetofsomeRmofdimension2.RecallthatXisrationalifthefieldofrationalfunctionsR(X)ofXisapurelytranscendentfieldextensionofRoftranscendencedegree2.Moregeometrically,XisrationaliftherearenonemptyZariskiopensubsetsUandVofR2andX,respectively,suchthatthereisanisomorphismofrealalgebraicvarieties—inthesenseof[BCR98]—betweenUandV.Looselyspeaking,XisrationalifanonemptyZariskiopensubsetofXadmitsarationalparametrizationbyanonemptyZariskiopensubsetofR2.AtypicalexampleofarationalcompactconnectedrealalgebraicsurfaceistheunitsphereS2inR3.Arationalparametrizationinthatcaseistheinversestereographicprojection.Ithasrecentlybeenshownthatanyrationalnonsingularcompactcon-nectedrealalgebraicsurfaceisisomorphiceithertotherealalgebraictorusS1×S1,ortoarealalgebraicsurfaceobtainedfromtherealalgebraicsphereS2byblowingupafinitenumberofpoints[BH07,HM08].Inthesequel,itwillbeconvenienttoidentifytherealalgebraicsurfaceXwiththeaffineschemeSpecR(X),whereR(X)denotestheR-algebraofallalgebraic—alsocalledregular—functionsonX[BCR98].Areal-valuedfunc-tionfonXisalgebraiciftherearerealpolynomialspandqinx1,...,xmTheresearchofthesecondnamedauthorwaspartiallysupportedbytheANRgrant”JCLAMA”ofthefrench”AgenceNationaledelaRecherche”.HebenefittedalsofromthehospitalityoftheUniversityofPrincetonwhenpreparingthefinalversionofthearticle.1WEIGHTEDBLOW-UPSINGULARITIESONREALRATIONALSURFACES2suchthatqdoesnotvanishonXandsuchthatf=p/qonX.ThealgebraR(X)isthelocalizationofthecoordinateringR[x1,...,xm]/IwithrespecttothemultiplicativesystemofallpolynomialsthatdonotvanishonX,whereIdenotesthevanishingidealofX.ItisthesubringofR(X)ofallrationalfunctionsonXthatdonothaveanypolesonX.Thankstotheaboveconvention,wecandefineaninfinitelynearpointofXtobeaclosedsubschemePofXthatisisomorphictoSpecR[x]/(xe),forsomenonzeronaturalintegere.WecallethelengthortheorderofP.AninfinitelynearpointofXisalsocalledaninfinitesimalarconX.Aninfinitelynearpointoflength1isanordinarypointofX.Aninfinitelynearpointoflength2isapair(P,L),wherePisapointofXandLis1-dimensionalsubspaceofthetangentspaceTPXofXatP.Equivalently,aninfinitelynearpointPofXoflength2isapointoftheexceptionaldivisoroftherealalgebraicsurfaceobtainedbyblowingupXinanordinarypoint.Byinduction,aninfinitelynearpointofXoflengtheisaninfinitelynearpointoflengthe−1ontheexceptionaldivisorEofablow-upofX(cf.[Mu,p.171]).LetPbeaninfinitelynearpointofX.ThereducedschemePredisanordinarypointofX.LetPandQbeinfinitelynearpointsofX.WesaythatPandQaredistinctifthepointsPredandQredofXaredifferent.Letnbeanaturalintegerandlete=[e1,...,eℓ]beapartitionofn,whereℓissomenaturalinteger.DenotebyXethesetofℓ-tuples(P1,...,Pℓ)ofdistinctinfinitelynearpointsP1,...,PℓofXoforderse1,...,eℓ,respec-tively.RecallthatanalgebraicdiffeomorphismoralgebraicautomorphismofXisabijectivemapffromXintoitselfsuchthatallcoordinatefunctionsoffandf−1arealgebraicfunctionsonX[HM08].DenotebyDiffalg(X)thegroupofalgebraicdiffeomorphismsofXintoitself.Equivalently,Diffalg(X)isthegroupofR-algebraautomorphismsofR(X).OnehasanaturalactionofDiffalg(X)onXe.Oneofthemainresultsofthepaperisthefollowing.Theorem1.1.LetXbeanonsingularrationalcompactconnectedrealalge-braicsurface.Letnbeanaturalintegerandlete=[e1,...,eℓ]beapartitionofn,forsomenaturalintegerℓ.ThenthegroupDiffalg(X)actstransitivelyonXe.Roughlyspeaking,Theorem1.1statesthatthegroupDiffalg(X)actsℓ-transitivelyoninfinitelynearpointsofX,foranyℓ.Thestatementgener-alizesearlierworkwheren-transitivitywasprovedforordinarypointsonly,i.e.,incaseofthetrivialpartitione=[1,...,1](cf.[HM08]).ThestatementofTheorem1.1motivatesthefollowingquestion.Question1.2.LetXbeanonsingularrationalcompactconnectedrealal-gebraicsurface.IsthesubsetDiffalg(X)ofalgebraicdiffeomorphismsofXdenseinthesetDiff(X)ofalldiffeomorphismsofX?Equivalently,cananydiffeomorphismofXbeapproximatedbyalgebraicdiffeomorphisms?Theproblemofapproximatingsmoothmapsbetweenrealalgebraicva-rietiesbyalgebraicmapshasbeenstudiedbynumerousauthors[BK87a,BK87b,BKS97,Ku99,JK03,JM04,Ma06].WEIGHTEDBLOW-UPSINGULARITIESONREALRATIONALSURFACES3