On the structure of the fiber cone of ideals with

arXiv:math/0603042v1[math.AC]2Mar2006ONTHESTRUCTUREOFTHEFIBERCONEOFIDEALSWITHANALYTICSPREADONETERESACORTADELLASANDSANTIAGOZARZUELAAbstract.Foragivenalocalring(A,m),westudythefiberconeofidealsinAwithanalyticspreadone.Inthiscase,thefiberconehasastructureasamoduleoveritsNoethernormalizationwhichisapolynomialringinonevariableovertheresiduefield.Onemaythenapplythestructuretheoremformodulesoveraprincipaldomaintogetacompletedescriptionofthefiberconeasamodule.Weanalyzethisstructureinordertostudyandcharacterizeintermsoftheidealitselfthearithmeticalpropertiesandothernumericalinvariantsofthefiberconeasmultiplicity,reductionnumberorCastelnuovo-Mumfordregularity.1.IntroductionLet(A,m)beaNoeherianlocalringandletIbeanidealofA.ThefiberconeofI(orthespecialfiberoftheReesalgebraA[It])istheringF(I)=Mn≥0In/mIn=A[It]⊗AA/m.ItsKrulldimensioniscalledtheanalyticspreadofIandwewilldenoteitbyl(I).AnidealJ⊆IiscalledareductionofIifthereexistsanintegernsuchthatIn+1=JIn.Phrasedotherwise,JisareductionofIifA[Jt]֒→A[It]isafinitemorphismofgradedalgebras.Equivalently,itisknownthatJisareductionofIifandonlyifIisintegraloverJ.AreductionJofIisaminimalreductionifJisminimalwithrespecttoinclusionamongreductionsofI.ByNorthcottandRees[32]minimalreductionsalwaysexist.LetJbeareductionofIandassumeinadditionthattheresiduefieldofAisinfinite.Then,JisaminimalreductionofIif,andonlyif,Jisminimallygeneratedbyl(I)elementsif,andonlyif,JisgeneratedbyafamilyofanalyticallyindependentelementsinI.Therefore,givenJaminimalreductionofI,theringF(J)isisomorphictoapolynomialringinl(I)variablesoverA/mandtheequalitiesmIn∩Jn=mJnaresatisfiedforalln.Thatis,thegradedmorphismF(J)֒→F(I)isaNoethernormalization.Fora∈I,wewilldenotebya0theclassofainI/mI.MinimalreductionsalsoprovidehomogeneoussystemsofparametersofF(I).Concretely,iftheresiduefieldofAisinfinite,afamilyofelementsa1,...,al∈Iisaminimalsetofgenerators2000MathematicsSubjectClassification.Primary13A30;Secondary13H10,13H15,13A02.BothauthorssupportedbyMTM2004-01850(Spain).12TERESACORTADELLASANDSANTIAGOZARZUELAofaminimalreductionofIifandonlyifa01,...,a0lisahomogeneoussystemofparametersofF(I).Assumenowthattheresiduefieldisinfiniteandl(I)=1.IfJ=(a)isaminimalreductionofI,thenF(J)isisomorphictoapolynomialringinonevariableoverA/mandF(I)isagradedfinitemoduleoverF(J).SowemayapplythestructuretheoremoffinitelygeneratedgradedmodulesoveraprincipalidealgradeddomaintogetasetofinvariantsdescribingtheprecisestructureofF(I)asF(J)-module.Ourpurposeinthispaperistoanalyzeindetailtheinformationprovidedbythissetofinvariantsinordertostudythepropertiesoffiberconesofdimensionone.Inparticular,theCohen-Macaulay,GorensteinorBuchsbaumproperties,andothernumericalinformationsuchasCastelnuovo-Mumfordregularity,multiplicity,Hilbertfuntion,reductionnumberorpostulationnumber.Aswewillsee,althoughthestructureofF(I)asF(J)-moduleislessrichthanthestructureofF(I)asF(J)-algebra,itsufficesinthiscasetocharacterizealltheabovepropertiesintermsoftheidealitself.ThefiberconeofanidealIisoneofthesocalledblowupalgebrasofIanditsProjrepresentsthefiberofthemaximalidealmbytheblowupofAwithcenterI.Moreover,itprovidesinterestinginformationabouttheidealitself:TheHilbertfunctionofthefiberconedescribestheminimalnumberofgeneratorsofthepowersofIand,whentheresiduefieldisinfinite,itsdimensioncoincideswiththeminimalnumberofgeneratorsofanyminimalreductionofI.Forthemaximalidealitself,thefiberconecoincideswiththeassociatedgradedring,andsointhisparticularsituationithasbeenextensivelystudied,thecaseofanalyticspreadonebeingthetangentconesofcurvesingularities.Butforageneralideal,thepropertiesofthefiberconearemuchlessknown.Nevertheless,inrecentyearssomeefforthasbeendonebyseveralauthorsinordertounderstanditsbehaviour.Withrespecttothearithmeticalpropertiesofthefibercone,oneofthefirstknownresultswasgivenbyHunekeandSally[27]whoprovedthat,ifAisCohen-Macaulay,thefiberconeofanym-primaryidealofreductionnumberoneisCohen-Macaulay.ThisresultwaslaterextendedbyK.Shah[35,36]toequimultipleidealsofreductionnumberone,givingalsosomeconditionsfortheCohen-Macaulaynessofthefiberconeofequimultipleidealsofreductionnumbertwo.SubsequentresultsbyCortadellasandZarzuela[6,7],D’Cruz,RaghavanandVerma[12],andD’CruzandVerma[13]completedtheresultsofShahformoregeneralfamiliesofideals.Also,thefiberconeofthedefiningidealofamonomialcurveinP3lyingonaquadricwasproventobeCohen-MacaulaybyMoralesandSimis[31].ThisresultwaslaterextendedbyP.Gimenez[16]andBarileandMorales[1]tothedefiningidealofaprojectivemonomialvarietyofcodimensiontwo.Ontheotherhand,motivatedbyworkofR.H¨ubl[22],H¨ublandHuneke[23]studiedtheCohen-Macaulaypropertyofthefiberconeofspecialidealsinconnec-tionwiththetheoryofevolutionsintroducedbyEisenbudandMazur[15],whichisrelatedtoA.Wiles’sworkonFermat’sLastTheorem[41].H¨ublandSwanson[24]havealsomadesomeconcretecomputationsonfiberconesinthiscontext.Morerecentworkconcerningthepropertiesofthefibercones(multiplicity,Hilbertfunction,Cohen-Macaulayness,Gorensteiness,depth...)hasbeendonebyCorso,Ghezzi,PoliniandUlrich[4],Corso,PoliniandVasconcelos[3],T.Cortadellas[5],D’CruzandPuthepur