JBrownVLakshmibaithemaingoalofthispaperistobringoutthisrelationship - PDF document

1 J.Brown&V.Lakshmibaithemaingoalofthispap
J.Brown&V.Lakshmibaithemaingoalofthispaperistobringoutthisrelationship.OurrstmainresultisadescriptionofthesingularlocusofaG-Htoricvarietyintermsofthefacesoftheassociatedpolyhedralcone;inparticular,wegiveaproofoftheconjectureof[9]usingonlythecombinatoricsofthepolyhedralconeassociatedtothetoricvarietyFurthermore,weprove(Theorem6.19)thatthesingularlocusofispureandofcodimension3inandthatthegenericsingu-laritiesareofconetype(moreprecisely,thesingularitytypeisthesameasthatatthevertexoftheconeoverthequadricsurface0inwealsodeterminethetangentconetoatagenericsingularity,whichturnsouttobeatoricvariety(bya“genericsingularity”wemeanapointsuchthattheclosureofthetorusorbitthroughisanirreduciblecomponentofSing,thesingularlocusofWefurtherobtain(seeCorollary7.3andTheorem7.11)aninterpretationofthemultiplicitiesatsomeofthesingularitiesascertainCata-lannumberswhenistheBruhatposetofSchubertvarietiesintheGrassmannianof2-planesinWealsopresentaproductformula(Theorem7.17).Itturnsoutthattheconjectureof[9]doesnotextendtoageneralSection9.2foracounterexample).However,in[4]weprovedtheconjectureof[9]forotherminusculeposets.ThispapercontainsmoreresultsfortheGrassmann–Hibitoricvarietiesthatcannotbededucedfromtheresultsof[4];forexample,themultiplicityformulasprovidedinSections7and8ofth

2 ispaperarenotdiscussedin[4].Thesingular-
ispaperarenotdiscussedin[4].Thesingular-itiesoftheHibitoricvarietywerealsostudiedbyWagnerin[18],whereitwasshownthatallHibitoricvarietieshaveasingularlocusofcodimensionthispaper,wegointomuchmoredetailaboutthesingularitiesofaG-Htoricvariety.Thebalanceofthepaperisorganizedasfollows.InSections1and2werecallgeneralitiesontoricvarietiesanddistributivelattices,respectively.InSection3,weintroducetheHibitoricvarietyandrecallsomeresultsfrom[9;14]onInSection4,werecallresultsfrom[14]onthepolyhedralconeassociatedinSection5,weintroducetheGrassmann–Hibitoricvariety.InSec-tion6weproveourrstmainresult,givingthedescriptionofSingintermsoffacesoftheconeassociatedtowealsopresentourresultsonthetan-gentconesanddeducethemultiplicitiesattheassociatedpoints.InSection7wepresentcertainproductformulasfor,whereistheBruhatposetofSchu-bertvarietiesintheGrassmannianof2-planesinInSection8,wepresentaformulaforthemultiplicityattheunique-xedpointoftheBruhatposetofSchubertvarietiesintheGrassmannianof-planesinInSection9wepresentacounterexampletoshowthattheconjectureof[9]doesnotextendtoageneralinthissection,wealsopresenttwoconjecturesthatconcernextendingthemultiplicityformulasofSections7and8.1.GeneralitiesonToricVarietiesOurmainobjectofstudyisacertainafnetoricvariety,sointhissectionwerecall

3 somebasicdenitionsonafnetoricvarieties
somebasicdenitionsonafnetoricvarieties.Letbean SingularLociofGrassmann–HibiToricVarieties-dimensionaltorus.Letbetheafne-space(i.e.,-tuplesofelementsofthe
1.1[7;12].Anequivariantafneembeddingofatorusisanafnevarietyasanopensubsetandequippedwithaextendingtheactiongivenbymultiplication.Ifinisnormal,theniscalledanafnetoricvariety.1.2.TheConeAssociatedtoaToricVariety.Letbethecharactergroup,andletbethe-dualof,andrecall[7;12]thatthereexistsastronglyconvexrationalpolyhedralconesuchthatthat]
[],whereisthesubsemigrouptheconeindualtoNotethatisanitelygeneratedsubsemigroupin1.3.OrbitDecompositioninAfÞneToricVarieties.WeshalldenoteWemaysuppose,withoutlossofgenerality,thatsothatthedimensionofequalsdim(By“dimensionof”wemeanthevec-torspacedimensionofthespanof1.4.Aisaconvexpolyhedralsubconeofoftheformforsome,anditisdenotedNotethatitselfisconsid-eredaface.Wehavethatisaprincipalopensubsetofnamely,Eachfacedeterminesa(closed)point:thepointcorrespondingtothemaximalidealinin]
[]givenbythekernelofof]
,wherewehave1if0otherwise.1.5.Asapointinmaybeidentiedwiththe-tuplewith1atthethplaceifisinandwith0otherwise.(Here,denotestheweightofthe-weightvector—theclassofof]1.6.OrbitDecomposition.Letdenotethe-orbitinhavethefollowingorbitdecompositionin ,dimSee[

4 7;12]fordetails. denesanorder-reversing
7;12]fordetails. denesanorder-reversingbijectionbetweenfacesof-orbitclosuresin1.7[7,Sec.3.1].Foraface ]
[] J.Brown&V.Lakshmibai2.FiniteDistributiveLatticesWeshallstudyaspecialclassoftoricvarieties—namely,thetoricvarietiesassoci-atedtodistributivelattices.Weshallrstcollectsomedenitionsonnitepartiallyorderedsets.Apartiallyorderedsetisalsocalledaposet.2.1.Aniteposetiscalledifithasbothauniquemaxi-malandauniqueminimalelement,denoted1and0respectively.Atotallyorderediscalledachain,andthenumber#1iscalledtheofthechain.Aboundedposetissaidtobegradedranked)ifallmaximalchainshavethesamelength.Ifisgraded,thenthelengthofamaximalchainincalledtherank2.2.Letbeagradedposet.For,thegradediscalledtheintervalfromandisdenotedbyby!, ]DeÞnition2.3.Letbeagradedposet,andletorderedpairiscalledacover(andwealsosaythatcoversif[2.4.Aisapartiallyorderedsetsuchthat,foreverypairofelements,thereexistelements,called(respec-tively)theandthe,denedby:,andif,andifItiseasytocheckthattheoperationsarecommutativeandassociative.2.5.Givenalattice,asubsetiscalledaimpliesthat2.6.Alatticeiscalleddistributiveifthefollowingidentitieshold:(ii)"2.7.Anelementofalatticeiscalledjoin-irreducibleirreducible)ifimplieseitherThesetofjoin-irreducibleelementsofisdenotedbyThefollowinglemmaiseasily

5 checked.beanitedistributivelattice.Then
checked.beanitedistributivelattice.Thenthereexistsatmostonecoveroftheform2.9.Asubsetofaposetiscalledanif,forall SingularLociofGrassmann–HibiToricVarieties2.10[3,Chap.Sec.3].beanitedistributivelatticewithminimalelementandletwiththeinducedpartialorderofisisomorphictothelatticeofidealsofbymeansofthelatticeFor,letdenotetheidealcorrespondingtoundertheisomorphisminTheorem2.10.2.11.AsaconsequenceofTheorem2.10,wehavethateverynitedis-tributivelatticeisgraded.3.TheVarietyThroughoutthefollowingsections,letbeanitedistributivelattice.Considerthepolynomialalgebraalgebra,
],andletbetheidealgeneratedbyThenoneknows[10]thatthat,
] $isanormaldomain;inparticular,wehavethatisaprimeideal.Letbetheafnevarietyofthezeroesin(hereisanafnenormalvarietydenedbybinomials;ontheotherhand,by[6],abinomialprimeidealisatoricideal(here,“toricideal”isinthesenseof[17]).HenceisatoricvarietyfortheactionbyasuitabletorusInthesequel,weshalldenotedenote,
] $Also,forweshalldenotetheimageof3.1.ThevarietywillbecalledaHibitoricvariety.3.2.Anextensivestudyofappearedrstin[10].Wehavethatdim3.3[14].Thedimensionofisequaltowhichisalsoequaltothecardinalityofthesetofelementsinamaximalchainof3.4.Foranitedistributivelattice,wecallthecardinalityof,denoteddimisasublatticeof,thentheisdenedasd

6 im3.5[18].Asublatticeiscalledanembeddeds
im3.5[18].AsublatticeiscalledanembeddedsublatticeofGivenasublattice,considerthevarietyandthecanonicalembed-3.6[9,Prop.5.16].isasubvarietyofifandonlyifisanembeddedsublatticeof J.Brown&V.Lakshmibai3.7.MultiplicityofattheOrigin.Letbea-gradedandnitely-algebra,)denotetheHilbertfunction,)andletdenotetheHilbertpolynomialofRecallthat:that: ]degdimProj,say;andtheleadingcoefcientofisoftheform3.8.Thenumberiscalledthedegreeofthegradedring,orthedegreeofProjThedegreeofof
]isequaltothenumberofmaximalchainsProof.beasbefore.Webeginbyputtingamonomialorderonon,
]Considerthereversepartialorderonandextendittoatotalorder,denoted,onthevariablesWenowtakethemonomialorderdenedasfollows.For,wesayifandonlyifeither+*andthereexists,+suchthatFrom[9]wehavenon-comparableisaGröbnerbasisforforthismonomialorder.Hence,lettingbetheidealgeneratedbyinitialtermsofelementsof,wehavethatisageneratingsetLetusdenotedenote
]byy,
] $by&By[5,Sec.15.8]wehaveaatdegenerationofSpectoSpec&Hence,thedegreeofthedegreeofbeasubsetofsuchthatNotethatthusachainoflength1inWehavee-1,,-*],where#runsoverallchainsofanylengthin&Notethatforsufcientlylarge,theleadingtermappearsinthesummationonlyofmaximalcardinalityTheresultfollowsfromthis. Nextwerecallmult,themultiplicity

7 ofanalgebraicvarietyatapointbethetangent
ofanalgebraicvarietyatapointbethetangentconeatthatis,..ThenthemultiplicityofisdenedtobedegProj.deg.Thus,usingthenotationfromSection3.7,weobtain),themultiplicityofSpec)attheorigin. SingularLociofGrassmann–HibiToricVarietiesThefollowingresultisadirectconsequenceofTheorem3.9.3.10.Themultiplicityofattheoriginisequaltothenumberofmaximalchainsin4.ConeandDualConeof,withbasisbethe-dualof,withbasisdualtoWedenotethetorusactingonthetoricvariety,andweidentifywiththecharactergroupThus,for(undertheidenticationof,welet,DenotebythelatticeofidealsofFor,setbetheconesuchthat,andletbetheconethatisdualto,sothatthat
]equalsthesemigroupalgebraalgebra]From[10;14,Prop.4.6]wehavethefollowingstatement.4.1.Thesemigroupisgeneratedby#bethesetofmaximalelementsintheposet1#notethesetofallcoversintheposet(i.e.,"2"intheposet,andthereisnootherelementsuchthat"22"Foracover1#,denote4.2[14,Prop.4.7].Theconeisgeneratedby#1#AnalysisofFacesofWeshallconcernourselvesjustwiththeclosedpointsinSointhesequel,byapointinweshallmeanaclosedpoint.beafaceofbethedistinguishedpointofwiththeassociatedmaximalidealbeingthekernelofthemapmap]
,,1if0otherwise.Then,forapoint(identiedwithapointinanddenotingthcoordinateof,wehave1if0otherwise.Nowlet J

8 .Brown&V.LakshmibaiTheBijection(cf.[14])
.Brown&V.LakshmibaiTheBijection(cf.[14]).Wehaveabijectionfacesofembeddedsublatticesof4.5[14,Prop.4.11].beafaceof 45.TheDistributiveLatticeandtheGrassmann–HibiToricVarietyWenowturnourfocustoaparticulardistributivelattice—namely,ThepartialorderisgivenbyFor,wedenotethethentryin -5.1.Itisawell-knownfact(seee.g.[13])thatthepartiallyorderedsetisisomorphictotheposetdeterminedbythesetofSchubertvarietiesintheGrassmannianof-dimensionalsubspacesinan-dimensionalspace,wheretheSchubertvarietiesarepartiallyorderedbyinclusion.From[15,Sec.3]wegetisadistributivelattice.5.2.SomereadersmayprefertoworkwiththelatticeofYoungdia-gramsthattintoarectanglewithrowsandcolumns,whichwewilldenotebyInthiscaseonemaygofromusingthefollow-ingbijection:Inthenextlemma,byasegmentweshallmeanasetconsistingofconsecutiveintegers.5.3[9].Forthefollowingstatementshold.Theelementisjoin-irreducibleifandonlyifeitherisasegmentweshallcalltheseelementsTypeIconsistsoftwodisjointsegmentsstartingwith1TypeII(ii)Theelementismeet-irreducibleifandonlyifeitherisasegmentorconsistsoftwodisjointsegmentsendingwith(iii)Theelementisjoin-irreducibleandmeet-irreducibleifandonlyifeitherisasegmentorconsistsoftwodisjointsegmentsstartingwith1andendingwith5.4.ThejoinirreducibleelementsofarethoseYoungdiagramsthatare

9 rectangles(i.e.,thenonzerorowsallhavethe
rectangles(i.e.,thenonzerorowsallhavethesamelength). SingularLociofGrassmann–HibiToricVarieties5.5.Weshalldenote$byjustandwillrefertoitasaGrassmann–Hibitoricvariety,oraG-Htoricvarietyforshort.6.SingularFacesoftheG-HToricVarietyrepresentthedistributivelatticeFromLemma5.3wehavethattheele-mentsofareoftwotypes,TypeIandTypeII.Sincethegeneratorsoftheconearedeterminedby(Proposition4.2),wewilloftenconsiderasapartiallyorderedsetwiththepartialorderinducedNoticethathasonemaximalelement,whichisalsothemaximalelementofhasoneminimalelement,whichisalsotheminimalelementofForeachelementthereareatmosttwocoversoftheformForexample,ifthenwehave,form-ingthetwocoversof(if1,thenorifisofTypeI,thenhasonlyonecover.Thefollowinglemmaisacorollaryof[15,Prop.3.2].6.1.Thepartiallyorderedset#$isadistributivelattice.6.2.Asalattice,lookslikeatessellationofdiamondsintheshapeofarectanglewithsidesoflength1andForexample,letisthefollowinglattice.167156127145126134125124123 AsinSection4,letbetheconeassociatedto6.3.For11and11,let J.Brown&V.Lakshmibaiai!-, -]Remark6.4.(i)By[9,Lemma11.5]wehavethatisanembeddedsublattice.(ii)Forforsome1and1thuseverydiamondinhasaasitsminimalelement.6.5.Letbethefaceofforwhich6.6.Afaceisa)faceifisasingular(resp.nonsingular)pointofOurrstresultisthatisasingularf

10 ace.Toprovethis,westartbydeterminingaset
ace.Toprovethis,westartbydeterminingasetofgeneratorsfor6.7.Letusdenoteby8(orsimplythesetofgeneratorsfor,asdescribedinProposition4.2.Foraface,dene83(Here,isasinSection4.4.)Then8givesasetofgeneratorsforDeterminationof8ItwillaidourproofbelowtoobserveafewfactsaboutthegeneratorsofFirstofall,isnotageneratorforany,sinceforall11and11andsinceisnonzeroonSimilarly,forthecoverisnotageneratorforanySecond,foranycover,ifisnotagener-atorofThus,indeterminingelementsof8weneedonlybeconcernedwithelementssuchthatthat!-, -]Theelementsofof!-, -]arefor0for0NotethatInthenexttheoremweprovethat8consistsofpreciselyfourelements,formingadiamondinthedistributivelatticeasthesmallestelement.8wherearedenedintheproof.Proof.Wedividetheproofintotwocases,1andCase1.1and1Herewehave SingularLociofGrassmann–HibiToricVarietiesAsdiscussedpreviously,wendthatiscoveredinWehavethatbothareintheintervalal!-, -]LetbethejoinofinthelatticeNotethatarecoversinWerstobservethat,,if(ThisfollowsbecauseimplythatandhenceClaim(i):arebothin8Weshallprovetheclaim(theproofforissimilar).Toprovethatisin8weneedtoshowthattheredoesnotexistansuchthatButthisfollowsfrom(6.1)(whichimpliesthat,for,ifClaim(ii):arein8TheproofissimilartothatofClaim(i).Againweshowtheresultfor(theproofforissi

11 mi-lar).Wemustdemonstratethattheredoesno
mi-lar).WemustdemonstratethattheredoesnotexistanbutAgainthisfollowsfrom(6.1)(notethatimpliesinparticularthatClaim(iii):8Inthecaseunderconsideration,since1itfollowsthattheonlyelementsofof!-, -]areofthefollowingforms:for0for1thuswehavecoversoffor0andoftypefor0Observethat,andInClaims(i)and(ii)wehaveshownthatthecovers,andyieldele-mentsof8AlsonotethatistheonlycoverofHence,itonlyremainstoshowthat8for2andthat8for1Foreachofthesecovers,weshallexhibitansuchthatisnonzeroonthecoverunderconsideration.for2isnonzeroonfor21andonfor2For,notethatisnonzeroThiscompletestheproofofCase1.Case2.Nowlet21and1Wehave J.Brown&V.LakshmibaiAsinCase1,welookforcoversofTheyaretobethejoinofinthelatticeClaim(iv):arein8Werstob-servethat,,ifForsupposethatnowtogetherimplythatandthusforsome11forClearly,nosuchexistsandthus(6.2)follows.By(6.2)wehavethat,ifissuchthat,thenClaim(iv)follows.Claim(v):8AsinClaim(iii),wewillshowthatallothercoversinoftheformform!-, -],arenotin8AsinSection6.8,alloftheelementsofof!-, -]arefor0for0(notethatWewillexaminecoversoftheseelements;noticethat,andfor1,andletFirstwewanttoshowthatthecoversdonotyieldelementsin8Observethat8isnonzero;thus8,andwemayrestrictourattentiontofor2Now,ontheinterval21,wehavethefollowingfacts:Facts(1),(3),a

12 nd(4)holdforthecaseitisjustaseparatechec
nd(4)holdforthecaseitisjustaseparatecheck.Hence,for2(resp.2wehavethatisnonzeroon SingularLociofGrassmann–HibiToricVarietiesNext,wemustconcernourselveswithcoversinvolvingfor1TocompleteClaim(v),wemustshowthatthecoversdonotyieldelementsof8Notethatandthusdoesyieldanelementof8isnonzero;wecanthereforerestrictourattentionto,Ontheinterval2,wehavethefollowingfacts:Therefore,ontheinterval2(resp.2,wehavethatisnonzeroonThiscompletesClaim(v),Case2,andtheproofofTheorem6.9. 6.10.ThefacecorrespondstothefollowingdiamondinThisdiamondisaposetofrank2.6.11.Thefacehasdimension3.Proof.Wehave,asetofgeneratorsforWecanseethatasubsetofthreeofthesegeneratorsislinearlyindependent.Thus,ifthefourthgeneratorcanbeputintermsoftherstthree,theresultfollows.No-ticethat OurnexttheoremisanimmediateconsequenceofTheorem6.9andLemma6.11.6.12.Wehaveanidenticationoftheafnepieceincor-respondingtothefacewiththeproductwhereistheconeoverthequadricsurfaceWenowprovetwolemmasthatholdforageneraltoricvariety.6.13.beanafnetoricvarietywithastheassociatedcone.isanonsingularvarietyifandonlyifitisnonsingularatthedistin-guishedpointProof.Onlytheimplicationrequiresaproof.Letthenbeasmoothpoint.Letusassume(ifpossible)thatSingWehavethefollowingfacts. J.Brown&V.LakshmibaiSingisaclosed-stablesubsetof foreve

13 ryface(seeSection1.6);inparticular, fors
ryface(seeSection1.6);inparticular, forsomefacesuchthatisasingularpoint(suchaexistsbecause,byourassump-tion,Singisnonempty).Wethusobtainthat,acontradiction.Henceourassumptioniswrongandtheresultfollows. 6.14.beafaceofaconvexpolyhedralconeisasmoothpointofifandonlyifisasmoothpointof—thatis,ifandonlyifisgeneratedbyapartofabasisofwhereisthe-dualofthecharactergroupofthetorusProof.WehavethatisaprincipalopensubsetofisnonsingularifandonlyifisnonsingularatByLemma6.13,isnonsingularatifandonlyifisanonsingularvariety;butby[7,Sec.2.1],thisistrueifandonlyifisgeneratedbyapartofabasisof Wenowreturntothecasewhereistheconvexpolyhedralconeassociated$6.15.Thenthefollowingstatementshold.(ii)WehaveanidenticationofasinTheorem6.12furthermore,isatoricvariety.(iii)ThesingularityatisofthesametypeasthatatthevertexoftheconeoverthequadricsurfaceInparticular,Proof.Assertion(i)followsfromLemma6.13,Lemma6.14,andTheorem6.12.isopenin,wemayidentify,whichinturncoincideswithisofconetype,whereisidentiedwiththeori-gin).Assertion(ii)followsfromthisinviewofTheorem6.12andgiventhatisatoricvariety.Assertion(iii)isimmediatefrom(ii). Next,wewillshowthatthefacescontainingsomearetheonlysingularfaces.Werstprovesomepreparatorylemmas.6.16.isinthesetofgeneratorsofasde-scribedinProposition4.2isin8cf.

14 Denition6.7forsomewhereProof.isequaltos
Denition6.7forsomewhereProof.isequaltosome,thenmustbeoneofthetwocoversofandwearedonebyTheorem6.9.Sowewillassumethatisajoinirreducibleofoneofthefollowingtwoforms.Case1:forsome,andisacoverinhasonlyonecoverin,whichmustbeisanelementof8,asshowninTheorem6.9. SingularLociofGrassmann–HibiToricVarietiesCase2:notethat72.8Thenwehave,andisacoverinAlso,wemusthave,andisanelementof8byCase1ofTheorem6.9. WenowreturntothecaseofaGrassmann–Hibitoricvariety.6.17.beafacesuchthatisnotcontainedinanyThentheassociatedfaceisnonsingulari.e.,ifafacedoesnotcontainanyoneisnonsingularProof.ByLemma6.14,fortobenonsingularitmustbegeneratedbypartofabasisforisgeneratedbyasubset8,fortobesingularitsgeneratorswouldhavetobelinearlydependent.(Generallythisisnotenoughtoprovethatafaceissingularornonsingular,butsinceallgeneratorsinhaveco-efcientsequalto1,anylinearlyindependentsetwillserveaspartofabasisforissingular;thenthereissomesubsetoftheelementsof8suchthat0,withcoefcientsnonzeroforatleastone-Recallthattheelementsofcanberepresentedasallthelinesegmentsinthewiththeexceptionof(seediagraminRemark6.2).Therefore,thelinearlydependentgeneratorsofmustrepresenta“loop”oflinesegmentsinThisloopwillhaveatleastonebottomcorner,leftcorner,topcorner,andrightcorner.ChoosesomeparticularByThe

15 orem6.9,8Thesefourgeneratorsarereprese
orem6.9,8ThesefourgeneratorsarerepresentedbythefoursidesofadiamondinThus,byhypothesis,thegeneratorsofrepresentaloopinthatdoesnottraverseallfoursidesofthediamondrepresentingallfourgen-eratorsofByhypothesis,isnotcontainedinanyfor11and1hencetheremustbeatleastoneelementofintheinterval[[!-, -]WehaveasdenedintheproofofTheorem6.9.Basedonhowcompareswithboth,wecaneliminatecertainelementsof8Therearefourpossibilities;welistallfour,aswellasthecorrespondinggeneratorsin8thatarenotin8(i.e.,those8suchthat38888Therefore,itisimpossibletohavecontainedin8Thisistrueforanyandso,inviewofLemma6.16,our“loop”inthatrepresentedthegeneratorsofcannothavealeftcornerorarightcorner.Thusitisreallynotpossibletohavealoopatall;hencethegenera-torsofarelinearlyindependent,andtheresultfollows. J.Brown&V.Lakshmibai6.18.TheG-HtoricvarietyissmoothalongtheorbitandonlyifthefacedoesnotcontainanyCombiningthiscorollarywithTheorem6.15andLemma6.11yieldsourrstmaintheoremasfollows.6.19.Thenthefollowingstatementshold.(i)Sing wheretheunionistakenoveralltheasinTheorem6.9(ii)Singispureandofcodimension3inandthegenericsingulari-tiesareofconetypemoreprecisely,thesingularitytypeisthesameasthatatthevertexoftheconeoverthequadricsurface(iii)Forisatoricvarietyand6.20.Theorem6.19t

16 husprovestheconjectureof[9]usingjustthec
husprovestheconjectureof[9]usingjustthecom-binatoricsoftheconeassociatedtothetoricvariety(forastatementoftheconjectureof[9],seeRemark9.1).Further,itgivesadescriptionofSingpurelyintermsofthefacesoftheconeassociatedto7.MultiplicitiesofSingularFacesofInthissectionwetake,determinethemultiplicityof$forcertainofthesingularfacesof,anddeduceaproductformula.Forwehavedenedandthecorrespondingfacefor11andhence,forweneedonlyconsiderfor1Forexample,thefollowingdiagramistheposetofjoinirreduciblesforinsideeachdiamondbecausethefoursegmentssurroundingitrepresentthefourgeneratorsoftheface.5,61,64,51,53,41,42,31,31,2 Inordertogofromthejoinirreduciblesof,wesimplyadd1,76,7totheposetabove,formingWewillseethatthismakesthecalculationofthemultiplicitiesofsingularfacesofmucheasier.Inthesequel,weshalldenotethesetofjoinirreduciblesofalso,asintheprevioussections,willdenotethepolyhedralconecorrespondingto SingularLociofGrassmann–HibiToricVarietiesisnowofconetype(i.e.,thevanishingidealishomogeneous),wehaveacanonicalidenticationof(thetangentconeHence,byTheorem3.10,multequalsthenum-berofmaximalchainsinSowebeginbycountingthenumberofmaximalchainsinAswemovethroughachainfrom1,2,atanypointwehaveatmosttwopossibilitiesforthenextpoint,Foreachcoverinourchain,weassignavalue:foracoveroftype-

17 ,weassignforacoveroftype-,weassig
,weassignforacoveroftype-,weassignAmaximalchaincontains23latticepoints,soeverychaincanbeuniquelyrepresentedbya-tupleof1sand1s;letusdenotethistuplebyForanysuch,itisclearthat1and1occurprecisely2times.Also,wecanseethat1andthat,forany14,if1sthan1sthenwehavearrivedatapoint2-,whichisnotalatticepoint.Thus,wemusthave0forevery17.2[16,Cor.6.2.3].TheCatalannumber countsthenumberofsequencesof1sand1swith1,2,7.3.ThemultiplicityofisequaltotheCatalannumber 7.4.multNextweshalldeterminemultofblocktype(seeDenition7.7tofollow).Letbeafaceofsuchthattheassociated(embeddedisoftheformform1,2272,73,
1,]$1%$2(say),where$1[1,2]andand72,73,
1,]for13and0Weshallnowdetermine8(cf.Section6.7).Letdenotetheintervalal1,2,72,73]in J.Brown&V.Lakshmibai7.5.Withasjustdescribed,wehavethat8acoverinProof.Clearly,(theelementin8correspondingtotheuniquemaxi-malelementisnotin8,sinceLetusdenoteClaim1:ForacoverforallclaimfollowsinviewofthefollowingfactsforacoverandhenceforallandhenceforallClaim2:ForacovernotcontainedinthereexistsansuchthatNotethatacoverinisoneofthefollowingthreeTypeI:TypeII:-TypeIII:--LetnowbeacovernotcontainedinisofTypeI,then,whereeitherwehaveisofTypeII,then-,whereeither2or,wehaveisofType--,whereei

18 ther1orwehaveTherequiredresultnowfollows
ther1orwehaveTherequiredresultnowfollowsfromClaims1and2. 7.6.WithasinLemma7.5,wehave%%7.7.WedeneafaceasinLemma7.5asa-block(i.e.,isaunionofconsecutive7.8.Notethataunionoffacesneednotbeaface.7.9.TheHibiVarietyForaninteger3,letdenotethedistribu-tivelattice1,2WedenetobetheHibivarietyasso-ciatedtoNote(cf.Proposition4.2)thattheconeassociatedtohasa SingularLociofGrassmann–HibiToricVarietiessetofgeneratorsconsistingof,whereisacoverinthesub-lattice[1,3]of(thesetofjoinirreduciblesofInviewofTheorem3.10wehavedenotestheorigin.7.10.beafaceofthatisa“-block”ofasinDenition7.7WehaveanidenticationoftheopenafnepieceofcorrespondingtowhereProof.InviewofSection1.6andProposition4.5,wehave4elementsinamaximalchaininFromthisitisclearthatcodimNext,inviewofLemma7.5andSection7.9,weobtainanidenticationofinthetheorem). 7.11.beasinTheorem7.10.Wehaveanidenticationofwhereisatoricvariety.(ii)mult Proof.isopenin,wemayidentify,whichinturncoincideswith(becauseisofconetype,whereisidentiedwiththeorigin).Assertion(i)followsfromthisinviewofTheorem(andthefactisatoricvariety).Assertion(ii)followsfrom(i)andCorollary7.3. 7.12.AProductFormula.Herewegiveaproductformulaformultisaunionofpairwisenonintersectingandnonconsecutive-blocks(see7.15).beafaceofsuchthattheassociated(embeddedsublattic

19 e)isofthethe1,221712,1713,2,
e)isofthethe1,221712,1713,2,21]%[2722,2723,
1,]#1%#2%#3(say),where17112andwherewhere1,221712,1713,2,21],#3[2722,2723,
1,]Considerthefollowingsublatticesin(thesetofjoinirreduciblesin J.Brown&V.Lakshmibai7.13.Withasbefore,wehave8isacoverinProof.WeproceedasintheproofofLemma7.5,whereisnotin8Letusdenote:Foranycover,weclearlyhaveandhenceandhenceforallThusweobtainthat0forall(7.1)Next,ifisacoverin,thenandhenceforallisacoverin,thenandhenceforallNotethatisthesmallest(resp.largest)elementin0forall(7.2)Together,(7.1)and(7.2)implytheinclusion“”.Weshallprovetheinclusion”byshowingthat,ifacoverisnotcontainedin,thenthereexistssuchthatThisproofrunsalonglinessimilartotheproofofLemma7.5.Letthenbeacoverinnotcontainedinisconvenienttointroducethefollowingsublatticesinin1,221,1713,21,22],&[1,2723,
1,]Wedistinguishthreecasesasfollows.Case1:isoftypeIcf.proofofLemma7.5(i)Ifiscontainedin,thenWeletNotethat(ii)Ifiscontainedin,then2(resp.WeletNotethatCase2:isoftypeII—say-Then32,oraccordinglyasiscontainedin,orWeletNotethataccordinglyasiscontainedin,andCase3:isoftypeIII—say--(i)Ifiscontainedin,thenWelet-Notethat(ii)Ifiscontainedin,then1(resp.Welet-Notethataccordinglyasiscontainedin,and S

20 ingularLociofGrassmann–HibiToricVarietie
ingularLociofGrassmann–HibiToricVarietiesAsanimmediateconsequenceofLemma7.13andCorollary7.6,wehavethefol-lowingresult.7.14.beasinLemma7.13.where%%%%7.15.WerefertoapairoffacesasinCorollary7.14asintersecting-blocks.7.16.wherearetwononintersecting-blocksseeCorollaryWehaveanidenticationoftheopenafnepieceofcorrespondingtowhereTheproofissimilartothatofTheorem7.11(usingLemma7.13).Ournexttheoremfollowsasanimmediateconsequence.wherearetwononintersecting-blocks.Wehaveanidenticationofwhereinparticular,isatoricvariety.(ii)multTheproofissimilartothatofTheorem7.11(usingTheorem7.16).7.18.Itisclearthatwecanextendthismultiplicativepropertyto%%,aunionofpairwisenonintersecting,nonconsecutive8.AMultiplicityFormulaforInthissectionwegiveaformulaformultByTheorem3.10,multequalsthenumberofmaximalchainsinWeshallprovideanexplicitformulaforthenumberofmaximalchainsinObservethatthenumberofchainsin1,2,isthesameasthenumberofchains0,0,hence,foranyinthechain,NowsetForany,let—thatis,thenumberofstandardYoungtableauxofshape(cf.[16]).8.1[16,Prop.7.10.3].beapartitionofThenthenumbercountsthelatticepathswhere fromtheori-witheachstepacoordinatevector,andstayingwithintheregionorcone J.Brown&V.LakshmibaiThus,forasdescribedin(8.1),thenumberofmaximalchainsinisequalAnexp

21 licitdescriptionofisgivenin[16,Cor.7.21.
licitdescriptionofisgivenin[16,Cor.7.21.5]. ThestatementofthepropositionreferstoasaboxintheYoungdiagramofandtoasthe“hooklength”ofThehooklengthiseasilydenedasthenumberofboxestotherightandbelowof,includingLetustake,forexample,3,3,3,andtheYoungdiagramofwithhooklengthsgivenintheircorrespondingboxesisasfollows. 5 4 3 4 3 2 3 2 1 Therefore,/!9! Infact,inthescenarioourderivedpartition(givenby(8.1))willalwaysbearectangle,andwecandeduceaformulaforthatdoesnotrequiretheYoungtableau.Thetopleftboxofwillalwayshavehooklengththeboxdirectlybelowit(andtheboxdirectlytotherightofit)willhaveForanyboxof,theboxbelowandtheboxtotherightwillhavehooklength1lessthanthatoftheboxwithwhichwestarted.Sincetheposets,andareisomorphic,wemayassumethatThuswearriveatthefollowingstatement.Themultiplicityofisequalto 9.ConjecturesInthissection,wegivetwoconjecturesonthemultiplicityatasingularpoint.WealsomentionaresultrelevanttothispaperonSingtheBruhatposetofSchubertvarietiesinanyminuscule Thegeneratingset8ofafaceconsistsofforcertaincovers(assumingthat,sothatisnotin88terminesasubset==suchthat8consistsofallthe SingularLociofGrassmann–HibiToricVarietiescoversinthe==wouldbethediamondgiveninRe-mark6.10.InSection7.12,ifapairofnonconsecutiveand-blocks,t

22 hen===1.Themultiplicityformulafor
hen===1.ThemultiplicityformulaforinTheorem7.17extendstoNamely,letbetheconvexpolyhedralconeassociatedtoandletbefacesofsuchthatThen,if==isempty,wehavemultTheorem7.11impliesthatmultifbothblocksofthesamelength;inparticular,==areisomorphic.Guidedbythisphenomenon,wemakethefollowingconjecture.2.ForafaceofanyHibitoricvariety,multisdeter-minedbytheposet=Bythiswemeanthatifaresuchthat==areisomorphicposets,thenthemultiplicitiesofatthepointsthesame.9.1.TowardgeneralizingTheorem6.19tootherHibivarieties,wewillrstexplainhowthelatticepointswerechosen.Letbetwoincomparablemeetandjoinirreduciblesinsay,InviewofTheorems6.15and6.17,wehavethefollowingstatement.isasmoothpointifandonlyif,foreverypairofjoinandmeetirreducibles,thereisann:,:]suchthatcoordinateofisnonzero.Infact,thisisthecontentoftheconjectureof[9,Sec.11].Theseresultssuggestthatwelookatsuchpairsofjoin–meetirreduciblesinotherdistributivelatticesandexpectthecomponentsofthesingularlocusoftheassoci-atedHibitoricvarietytobegivenbyTheorem6.19(i)forthecaseofHowever,thisisnottrueingeneral,asthefollowingcounterexampleshows..Letbetheinterval[1,3,42,5,6],asublattice 235 136 234



135  134



 J.Brown&V.LakshmibaiNoticethathasonlyonepairofjoin–meetirreduc

23 ibles,2,3,41,5,6andthusthecorrespondingi
ibles,2,3,41,5,6andthusthecorrespondinginterval[]istheentirelattice.Therefore,ifourresult(Theorem6.19(i))onthesingularlocusofG-HtoricvarietiesweretogeneralizetootherHibitoricvarieties,thenanyproperfacewouldbenonsingu-lar.Thisfollowsbecauseanyfacemustcorrespondtoanembeddedsublattice,andnaturallythissublatticewillintersecttheinterval,whichisjustButthisisnottrue!Forexample,letbethefaceofsuchthat1,5,6145156136156135145135136134135isasetofgeneratorsforClearly,isnotgeneratedbythesubsetofabasis,soisasingularface(seeLemma6.14).Nevertheless,Theorem6.19holdsforminusculelatticesasdescribednext.Letbesemisimple,andletbeamaximalparabolicsubgroupwithastheasso-ciatedfundamentalweight.LetbetheWeylgroupofThentheSchubertvarietiesinareindexedby8 8minuscule,whichwemeanthattheweightsinthefundamentalrepresentationassociatedtoformoneorbitundertheWeylgroup.ItisknownthattheBruhatposet8 8oftheSchubertvarietiesinisadistributivelattice;see[11]fordetails.9.3.Wecall8 8minusculelatticeHibitoricvariety.9.4.AnyGrassmann–HibitoricvarietyisalsoaBruhat–Hibitoricvariety.Now,foraminusculelatticeasinDenition9.3,considerapairofin-comparablejoin–meetirreducibleelements.Ithasrecentlybeenshown[4]thataBruhat–Hibitoricvarietyissmoothatafaceofifandonlyif,foreachincomparablepairofjoin–meetirreduc

24 iblesin,thereexistsatleastoneone%,
iblesin,thereexistsatleastoneone%,%]suchthat?isnonzero.References[1]V.V.Batyrev,I.Ciocan-Fontanine,B.Kim,andD.vanStraten,ConifoldtransitionsandmirrorsymmetryforCalabi–YaucompleteintersectionsinGrassmannians,NuclearPhys.B514(1998),640–666. Mirrorsymmetryandtoricdegenerationsofpartialagmanifolds,Math.184(2000),1–39.[3]G.Birkhoff,Latticetheory,3rded.,Amer.Math.Soc.Colloq.Publ.,Amer.Math.Soc.,Providence,RI,1967.[4]J.BrownandV.Lakshmibai,SingularlociofBruhat–Hibitoricvarieties,J.Algebra319(2008),4759–4779.[5]D.Eisenbud,Commutativealgebrawithaviewtowardalgebraicgeometry,TextsinMath.,150,Springer-Verlag,Berlin,1995.[6]D.EisenbudandB.Sturmfels,Binomialideals,DukeMath.J.84(1996),1–45. SingularLociofGrassmann–HibiToricVarieties[7]W.Fulton,Introductiontotoricvarieties,Ann.ofMath.Stud.,PrincetonUniv.Press,Princeton,NJ,1993.[8]N.GonciuleaandV.Lakshmibai,DegenerationsofagandSchubertvarietiestotoricvarieties,Transform.Groups1(1996),215–248. Schubertvarieties,toricvarietiesandladderdeterminantalvarieties,Ann.Inst.Fourier(Grenoble)47(1997),1013–1[10]T.Hibi,Distributivelattices,afnesemigrouprings,andalgebraswithstraighteningCommutativealgebraandcombinatorics(Kyoto,1985),Adv.Stud.PureMath.,11,pp.93–109,Elsevier,Amsterdam,1987.[11]H.Hiller,GeometryofCoxetergro

25 ups,Res.NotesMath.,54,Pitman,Boston,1982
ups,Res.NotesMath.,54,Pitman,Boston,1982.[12]G.Kempfetal.,Toroidalembeddings,LectureNotesinMath.,339,Springer-Verlag,Berlin,1973.[13]V.LakshmibaiandN.Gonciulea,Flagvarieties,Hermann,Paris,2001.[14]V.LakshmibaiandH.Mukherjee,SingularlociofHibitoricvarieties,J.RamanujanMath.Soc.(submitted).[15]R.A.Proctor,Bruhatlattices,planepartitiongeneratingfunctions,andminusculerepresentations,EuropeanJ.Combin.5(1984),331–350.[16]R.P.Stanley,Enumerativecombinatorics,vol.2,CambridgeStud.Adv.Math.,62,CambridgeUniv.Press,Cambridge,1999.[17]B.Sturmfels,Gröbnerbasesandconvexpolytopes,Univ.LectureSer.,8,Amer.Math.Soc.,Providence,RI,1996.[18]D.G.Wagner,SingularitiesoftoricvarietiesassociatedwithnitedistributiveJ.AlgebraicCombin.5(1996),149–165.J.BrownDepartmentofMathematicsNortheasternUniversityBoston,MA02115brown.justin1@neu.eduV.LakshmibaiDepartmentofMathematicsNortheasternUniversityBoston,MA02115 MichiganMath.J.59(2010),243–267SingularLociofGrassmann–HibiToricVarietiesJ.Brown&V.LakshmibaiIntroductiondenotethebaseeld,whichweassumetobealgebraicallyclosedandofar-bitrarycharacteristic.Givenadistributivelattice,letdenotetheafnevari-etyin ReceivedNovember10,2008.RevisionreceivedMarch25,2009.ThesecondauthorwaspartiallysupportedbyNSFGrantDMS-0652386andNortheasternUniversityRSD