The Hilbert scheme of 11 points in A3 is irreducible TheodosiosDouvropoulos,JoachimJelisiejew,BerntIvarUtstølNødland,and ZachTeitler 7 1 0 2 n a J 1 1 Abstract We provethattheHilbertschemeof11pointsona smooththreefoldis ] irreducible.Inthecourseoftheproof,wepresentseveralknownandnewtechniques G forproducingcurvesontheHilbertscheme. A . h at 1 Introduction m [ Let X be a smooth connected quasi-projective variety. The Hilbert scheme of d 1 pointsinX istheschemeparametrizingfinitesubschemesofX ofdegreed.There v areampleintroductoryreadingsonthe Hilbertschemeofpointsavailable,includ- 9 ing[16,17,19,26,30,37,38]. 8 The Hilbert scheme of points is quasi-projective (projective iff X is) and con- 0 nected[16,17,23].Moreover,Fogarty[17]provedthatfordimX≤2itissmoothof 3 0 dimensiond·(dimX).Forhigher-dimensionalX,muchlessisknown.Thequestions . ofirreducibilityoftheHilbertschemeofpointsisespeciallyinteresting,becauseit 1 ensuresthatallfiniteschemesarelimitsofreducedones;see[4]foranapplication. 0 7 This question is local and only dependson the dimensionof X: the answer for n- 1 dimensional X will be the same as for An, see [1, p. 4] or [10, Lemma 2.2]. We : v i X TheodosiosDouvropoulos r SchoolofMathematics,UniversityofMinnesota,Minneapolis,e-mail:[email protected] a JoachimJelisiejew Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland e-mail: [email protected] BerntIvarUtstølNødland DepartmentofMathematics,UniversityofOslo,Norway.e-mail:[email protected] ZachTeitler BoiseStateUniversity,DepartmentofMathematics,1910UniversityDrive,Boise,Idaho83725- 1555,USA.e-mail:[email protected] 1 2 T.Douvropoulos,J.Jelisiejew,B.I.U.Nødland,Z.Teitler denotetheHilbertschemeofd pointsinAn byHd.Ourmotivatingquestionisthe n following: Forwhichpairs(n,d)istheHilbertschemeHd irreducible? n By Fogarty’sresults, all Hd are irreducible.Mazzola [36] provedirreducibility of 2 Hd forallnandd≤7.Iarrobino[27,28]showedthatforeveryn≥3andd≥78 n the scheme Hd is reducible. Emsalem and Iarrobino proved that Hd is reducible n n ford≥8andn≥4,see[29,Section2.2,p.158]andalso[8].BorgesdosSantos, Henni,andJardim[2]showedthatH9 andH10 areirreduciblebycomparingthem 3 3 with appropriatespacesofcommutingmatricesandusing theresults ofSˇivic [40, Theorems 26, 32]. Thus, the reducibility of Hd was unknown only for the values n n=3and11≤d≤77.Hereweimprovethelowerbound. Theorem1.1.TheHilbertschemeof11pointsinasmoothirreduciblethreefoldis irreducibleofdimension33. WeproveTheorem1.1inSection4.WereviewbackgroundinformationinSec- tion2.InSection3wegiveanoverviewofstrategy,gathergeneralresultsthatwill beusedintheproofoftheabovetheorem,anddemonstratehowtouseMacaulay2 [21]forsomecomputations. In Section 5 we discuss a special class of subschemes, which appeared in the earliestexampleofreducibleHd,duetoIarrobino[27].Namely,letmbetheideal 3 of the origin of A3. Fix d and consider the ideals ms ⊂I ⊂ms+1 such thatV(I) hasdegreed;thensisuniquelydetermined.Callsuchidealsverycompressed and denotebyHmax,d theirfamily.LetRd denotetheclosureinHd oftheopensetof 3 3 smoothsubschemes.ThecomponentRd iscalledthesmoothablecomponent.Ithas 3 dimension3d.Theresultof[27]isthatford≥96wehavedimHmax,d ≥3d and, thus,ageneralverycompressedidealdoesnotlieinthesmoothablecomponent.We showthatford≤95thefamilyHmax,d isinfactcontainedinRd. 3 Proposition1.2.ThefamilyHd,max ofverycompressedidealsiscontainedinthe smoothablecomponentifandonlyifd≤95. The key points of the proof are the use of smoothings by degenerating to initial idealsandaMacaulay2calculation,seeSection5. We now explain our approachto the proof of Theorem1.1. We build uponthe strategyof[8].Asexplainedthere,questionsaboutsmoothabilityofaspecifiedideal IareeasilyreducedtothecasewhereIislocalandhasfullembeddingdimension3. TherearefifteenpossibleHilbertfunctionsofI,seeTable1.ForeachHilbertfunc- tionh,theschemeHh parameterizeslocalidealswithfixedHilbertfunctionhand 3 the standard graded Hilbert scheme Hh parameterizes homogeneousideals with 3 fixed Hilbert function h. We apply three differentstrategies to show that for each Hilbertfunctionhinourlist,wehaveHh⊂R11. 3 3 First, for some cases the knowledge about the Hilbert function of an ideal I is enoughtoproduceadeformation(viarayfamiliesintroducedin[9])whosespecial fiberisIandgeneralfiberisreducible.ByLemma1.4,suchanIissmoothable,see Section4.1. TheHilbertschemeof11pointsinA3isirreducible 3 1. (1,3,1,1,1,1,1,1,1) §4.1 6. (1,3,5,1,1) §4.1 11. (1,3,2,2,2,1) §4.4 2. (1,3,2,1,1,1,1,1) §4.1 7. (1,3,3,4) §4.2 12. (1,3,3,2,2) §4.5 3. (1,3,2,2,1,1,1) §4.1 8. (1,3,4,3) §4.2 13. (1,3,3,3,1) §4.6 4. (1,3,3,1,1,1,1) §4.1 9. (1,3,5,2) §4.2 14. (1,3,3,2,1,1) §4.7 5. (1,3,4,1,1,1) §4.1 10. (1,3,4,2,1) §4.3 15. (1,3,6,1) §4.8 Table1 HilbertfunctionshinH11andthecorrespondingsections. 3 Second, most of the schemes Hh contain smooth points of the Hilbert scheme 3 which lie in the smoothable component R11. Such points are called smooth and 3 smoothablepoints;examplesincludepointscorrespondingtoGorensteinalgebras, see[10,Corollary2.6]. Lemma1.3.IfZ⊆H11isanirreduciblesetthatcontainsasmoothandsmoothable 3 point,thenwehaveZ⊆R11. 3 Proof. ThelocusofsmoothandsmoothablepointsisopenandcontainedinR11,so 3 theintersectionZ∩R11 containsanopensubsetofZ.Then,thesubsetZ∩R11⊂Z 3 3 isdenseandclosed,soitisequaltoZ. ⊓⊔ Toapplytheabovelemma,wewriteHh asaunionofirreduciblesetsZ andshow 3 that each Z contains a smooth and smoothable point. To find the sets Z we may takeadvantageofthemorphismp h:H3h→H3htakinganidealI toitsinitialideal, see[8].Weemploythefollowing3-stepstrategy: 1. DecomposeHhintoirreduciblestrata. 3 2. Usingthemorphismp h:H3h→H3h,decomposeH3hintoirreduciblestrata. 3. ForeachstratumofHh,findasmoothpointoftheHilbertschemewhichliesin 3 thesmoothablecomponentandconcludethatthewholestratumliesthere. Insteps1and2,weuseMacaulay’sinversesystems,seeSection2.Inthesim- plest cases, we find thatthere is a bijection betweenirreduciblestrata of Hh and 3 Hh,butthisisnotalwaystrue,seeforexampleSection4.5. 3 For step 3 we introduce cleavable ideals. An ideal is said to be cleavable (or limit-reducible)ifitcanbedeformedtoanidealwhosesupportconsistsofatleast twopoints. Lemma1.4.AcleavableidealI∈H11issmoothable. 3 Proof. Let I be a one-parameter flat family of ideals with I =I and for t 6=0, t 0 I supported at more than one point. Each irreducible componentof I has length t t strictlylessthan11,soitissmoothable.Hence,theidealIisalsosmoothable. ⊓⊔ To show that an ideal I is cleavable, we construct a family over Speck[t] whose generalfiberisreducibleandcheckthatitisflat,seeSection3.1. Third, there is a case where both previous methods do not apply. This is the caseh=(1,3,6,1),seeProposition4.22.ThestratumHhdoesnotseemtocontain 3 4 T.Douvropoulos,J.Jelisiejew,B.I.U.Nødland,Z.Teitler smoothpoints.However,thestratumisirreducibleandwe candescribewhatgen- eralpointslooklike.We buildadeformationshowingthatsuchgeneralpointsare smoothable,hence,byirreducibility,theentirestratumhastobesmoothable. Weworkoveranalgebraicallyclosedfieldkofcharacteristiczero. 2 Prerequisites Hilbert schemes and smoothability. The Hilbert scheme Hd parameterizessub- n schemesofAn ofdimensionzeroanddegreed. Moreformally,Hd representsthe n functorwhichassignstoeachk-schemeX thesetofsubschemesofAn×X which are flat over X and for which all fibers are finite of degree d, see [26, Chapter 1]. Equivalently,lettingT =k[a ,a ,...,a ],theschemeHd parameterizesidealsIfor 1 2 n n whichT/I isavectorspaceofdimensiond.Inotherwords,Hd alsorepresentsthe n functorwhichassignstoeachk-algebraAthesetofidealsI inT⊗Asuchthatthe quotientsT⊗A/IarelocallyfreeA-modulesofrankd. The Zariski tangent space to Hd at the point representing I is the T-module n Hom(I,T/I), see [26, Theorem 1.1]. Using Macaulay2 [21], we can compute the dimensionofthistangentspace.Westressthatapointissmoothifandonlyifthe pointlies on only one irreduciblecomponentof the scheme and the dimension of thetangentspaceatthatpointequalsthedimensionofthecomponentofthescheme containingthepoint.Thedimensionofthetangentspaceincreasesatsingularpoints. On Hd, there is a distinguished componentcorrespondingto smooth schemes. n Indeed,aslightlyperturbedtupleofdclosedpointsinAnisjustanothersuchtuple. Thus, the set of tuples of points is open in the Hilbert scheme and their closure is a component.It is called the smoothable component of Hd and denoted by Rd. n n Clearly, Rd is generically smooth of dimension nd. Since Hd is smooth, we have n 2 Rd =Hd. 2 2 ApointofRd issaidtobesmoothable.Thus,anidealIissmoothableifandonly n ifitcanbedeformedtoanidealofd distinctpoints.Thismeansthatonecanbuild aone-parameterflatfamilyofschemesoveradiscretevaluationringforwhichthe generalmemberconsistsofd distinctpointsandthespecialfiberisT/I,see[6,8] fordetails.Inparticular,adisjointunionofsmoothableschemesissmoothableand alimitofsmoothableschemesissmoothable. Hilbert functions. In analyzing the Hilbert scheme Hd, it is useful to use work n withaninvariantthatrefinesthedegreed.Therearetwoclosely-relatednotionsof Hilbertfunction: • ForagradedT-moduleM,itsHilbertfunctionisdefinedbyh(i)=dim(M).In i particular,givenahomogeneousidealI⊂T,weconsidertheHilbertfunctionof thequotientringT/I. • ForafilteredT-moduleMwithdescendingfiltrationM=M ⊇M ⊇M ⊇···, 0 1 2 the Hilbert function h is defined by h(i)=dim(M/M ). In particular, if the i i+1 schemeassociatedto anidealI ⊂T issupportedata point,thenT/I isa local TheHilbertschemeof11pointsinA3isirreducible 5 ring(A,m),andtheHilbertfunctionhwithrespecttothefiltrationbypowersof misdefinedtobeh(i)=dim(mi/mi+1). IfI ishomogeneousandT/I islocal, thetwonotionscoincide. Wewritehasavector(h(0),h(1),...),trimmingitafterthelastpositiveentry. LetA=T/I whereT =k[a ,a ,...,a ]isapolynomialringwithitsstandard 1 2 n gradingandI isahomogeneousideal.AssumethatI containsnolinearforms.We callsuchanalgebrastandardgraded. Macaulay’sboundisanupperboundforthegrowthofHilbertfunctionsofstan- dardgradedalgebras,definedasfollows.First, forpositiveintegersh andd, there existuniquelydeterminedintegersd ≥1andkd >kd−1>···>kd ≥d suchthat h= kd + kd−1 +···+ kd . (cid:18)d(cid:19) (cid:18)d−1(cid:19) (cid:18)d (cid:19) This expression is called the d-binomial expansion of h and denoted h . The (d) d-binomialexpansionofhcanbefoundgreedily:letk bethegreatestintegersuch d that kd ≤h, then find the (d−1)-binomial expansion of h− kd . Now hhdi is d d defin(cid:0)ed(cid:1)asfollows.Ifh = kd + kd−1 +···+ kd thenwedefi(cid:0)ne(cid:1) (d) d d−1 d (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) hhdi:= kd+1 +···+ kd +1 . (cid:18)d+1(cid:19) (cid:18)d +1(cid:19) Example2.1.Wehave5 = 3 + 2 ,so5h2i= 4 + 3 =7.Similarly,wehave (2) 2 1 3 2 4 = 3 + 1 ,so4h2i= 4 (cid:0)+(cid:1)2 (cid:0)=(cid:1)5. (cid:0) (cid:1) (cid:0) (cid:1) (2) 2 1 3 2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Example2.2.Ifh≤dthenwehaveh = d + d−1 +···+ d−h+1 andhhdi=h. (d) d d−1 d−h+1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Theorem2.3(Macaulay’sbound,[34]or[3,Theorem4.2.10]).LetAbeastan- dard graded k-algebra with Hilbert function h. For every non-negativeinteger d, wehaveh(d+1)≤h(d)hdi. Corollary2.4.Let A be a standard graded k-algebra with Hilbert function h. If d≥0issuchthath(d)≤d,thenwehaveh(d)≥h(d+1)≥h(d+2)≥···. OncetheMacaulayboundisattainedthenitwillalso beattainedforallhigher degreesprovidedthatnonewgeneratorsoftheidealappear: Theorem2.5(Gotzmann’s Persistence Theorem, [20] or [3, Theorem 4.3.3]). Let A=T/I be a standardgradedalgebrawith Hilbert functionh. Ifd ≥0 is an integersuchthath(d+1)=h(d)hdiandIisgeneratedindegrees≤d,thenwehave h(k+1)=h(k)hki forallk≥d. Apolarityandinversesystems. Akeytoolintheanalysisoffiniteschemesisthe technique of Macaulay’s inverse systems, also known as apolarity. General refer- encesinclude[15,18],[30,Section1.3,Chapter5],[39]. LetS=k[x ,x ,...,x ] andT =k[a ,a ,...,a ] be polynomialringswith the 1 2 n 1 2 n standardgrading.Whenn≤3,weinsteadusevariablesx,y,zanda ,b ,g .Wewrite 6 T.Douvropoulos,J.Jelisiejew,B.I.U.Nødland,Z.Teitler S for d S ,andsimilarlyT .ThepolynomialringT actsonSbylettinga ≤d k=0 k ≤d i actaspaLrtialdifferentiationbyxi.Thisiscalledtheapolarityaction.Wedenotethis actionby ,sothata F = ¶ F forF ∈S.ThisgivesbilinearmapsT ×S →S foralld,e.Inparticuliar,fore¶axcihdthepairingT ×S →S =kisapderfecetpairien−gd. d d 0 Definition2.6.ForanysubsetJ⊂Stheapolarideal,orannihilatingidealJ⊥⊂T istheidealofelementsQ ∈T suchthatQ F =0forallF∈J.ForF∈Swewrite F⊥for({F})⊥. WhenJisspannedbyhomogeneouselements,theapolaridealishomogeneous. When J consists of a single element F, then the ideal F⊥ is Gorenstein, see [12, Section21.2]. Example2.7.IfF =xa1xa2···xan,thenweclaimthatF⊥=(a a1+1,...,a an+1).In- 1 2 n 1 n deed, it is easy to see that each a ai+1 ∈ F⊥. Conversely, if Q ∈ T has a term i a 1b1a 2b2···a nbn with each bi ≤ai, then the apolar pairing of this term with F is a monomialthatdeterminesthe b, meaningthatitcannotbecancelledbythe other i termsofQ .Hence,ifQ ∈F⊥,theneachtermofQ mustlieintheindicatedideal. ThelinearmapT →SgivenbyQ 7→Q F providesasimpleapproachtocom- putingF⊥. The apolaridealF⊥ is the kernelof this map.We can computeJ⊥ by intersectingtheidealsF⊥ forF inJ.IfJ isak-vectorspace,thenitissufficientto considerF inabasisforJ. Example2.8.ForF =x3+yz,wehaveF⊥=(a 3−6bg ,ab ,ag ,b 2,g 2). Example2.9.ForF =x2y+y2z,wehaveF⊥=(g 2,ag ,a 2−bg ,b 3,ab 2). Definition2.10.AMacaulayinversesystem,orsimplyinversesystem,isaT-sub- moduleofS.Thatis,aninversesystemisak-vectorsubspaceJ⊆Swhichisclosed underdifferentiation:ifF ∈J,thenallofthederivativesa F,...,a F lieinJ. 1 n The inverse system generated by a subset f ,...,f of S is hf ,f ,...,f i = 1 s 1 2 s Tf +Tf +···+Tf , that is, the vector space spanned by the f together with 1 2 s i allhigherpartialderivatives.Clearly,wehavehf ,...,f i⊥= s hfi⊥= s f⊥. 1 s i=1 i i=1 i AninversesystemishomogeneousifitisgeneratedbyhomogTeneouselemTents. Remark2.11.ThemappingJ7→J⊥ sendsfinite-dimensionalinversesystemstolo- calidealssupportedattheorigin,thatis, m-primaryidealswheremistheidealof theorigin.Themappingisone-to-one,sinceJ maybecomputedfromJ⊥ similarly to the discussion above. In fact it is a bijection, as shown by Macaulay [35], or seeforexample[15,Corollaire2].WhenI isalocalideal,wewillwriteI⊥ forits inversesystem. RecallthatHhandHhconsistofallhomogeneousandlocalideals,respectively, n n with Hilbert function h. On the other hand Hd, consists of all zero-dimensional n schemesoflengthd inAn,notonlylocalonesoronessupportedattheorigin. Proposition2.12([18,RemarkafterProposition2.5]).IfJ isahomogeneousin- versesystemthen,Jisisomorphicasagradedk-vectorspacetoT/J⊥. TheHilbertschemeof11pointsinA3isirreducible 7 Proposition2.13([15,Proposition2(a)]).Forafinite-dimensionalinversesystem J,wehavedimkJ=dimkT/J⊥. Proof. LetdbelargeenoughsothatJ⊆S .ItfollowsthatthemapT →T/J⊥is ≤d ≤d surjective.Hence,bothofthedimensionsareequaltothecodimensionofJ⊥∩T ≤d inT . ⊓⊔ ≤d Remark2.14.For an inverse system J, for each integer k, J denotes the vector ≤k spaceofpolynomialsofdegreeatmostk inJ.Theseformanincreasingfiltration, J ⊆J ⊆···.TheinversesystemJisafilteredT-module,soitsHilbertfunction ≤0 ≤1 h is given by h(k)=dimJ≤k−dimJ≤k−1 for each k and (cid:229) h(i)=dimkJ. If J is homogeneous,thenh(k)=dimJ . k Proposition2.15([30,Lemma2.12]).Let f ∈Sbeahomogeneousformofdegree d.IfhistheHilbertfunctionoftheinversesystemhfi,thenh=(h(0),...,h(d))is symmetric:h(i)=h(d−i)foralli. Proposition2.16([7]).Supposethat f ∈Sisahomogeneousformofdegreed.Let hbetheHilbertfunctionofhfi.Ifh(d−1)=k,thatish=(...,k,1),thenthereare independentlinearfunctionsℓ ,ℓ ,...,ℓ ∈S andahomogeneousformgsuchthat 1 2 k 1 f =g(ℓ ,ℓ ,...,ℓ ).Equivalently,thereisalinearchangeofcoordinatessothat f 1 2 k dependsonlyonthevariablesx ,...,x ,notonx ,...,x . 1 k k+1 n Remark2.17.Using the above proposition, one can show that if hfi has Hilbert function(...,1,1),then f =ℓd forsomelinearfunctionℓandhfihasHilbertfunc- tion(1,1,...,1,1).Ifh(d−2)=h(d−1)=2,theneither f =ℓd+mdor f =ℓd−1m forsomeindependentlinearfunctionsℓ,m∈S ,andeitherwayhfihasHilbertfunc- 1 tion(1,2,2,...,2,2,1).Forproofseeforexample[30,Theorem1.44]:intheirno- tation,s=2,and f⊥hasaquadraticgenerator,whichuptoachangeofcoordinates iseitherab orb 2. Dealing with nonhomogeneous inverse systems is much harder than working with homogeneousones. Fortunately,each inversesystem J has an associated ho- mogeneousinversesystemlead(J). Definition2.18.The leading form of a polynomial is its highest degree homoge- neouspart.Thismaynotbea monomial.ForaninversesystemJ ⊂S,the inverse systemofleadingformsofJ,denotedlead(J),isthevectorsubspaceofSspanned byleadingformsofalltheelementsofJ. Forexample,theinversesystemhx3+y2i=span{x3+y2,x2,x,y,1}has lead(hx3+y2i)=span{x3,x2,x,y,1}=hx3,yi. ThereisatightconnectionbetweenasystemJandlead(J). Proposition2.19.TheHilbertfunctionsofJ andlead(J)areequal. 8 T.Douvropoulos,J.Jelisiejew,B.I.U.Nødland,Z.Teitler Proof (sketch). Let f ,f ,...,f be a vector space basis for lead(J) consisting of 1 2 s homogeneouselementsandlet g ,g ,...,g ∈J with lead(g)= f. Onecan show 1 2 s i i theg areabasisforJ.ExpressingtheHilbertfunctionsofJandlead(J)intermsof i theg and f givestheresult. ⊓⊔ i i Theinitialformorlowestdegreeformofapolynomialg isitslowestdegreeho- i mogeneouspart.TheinitialidealofanidealK,denotedin(K),istheidealgenerated bytheinitialformsofallelementsofK. Proposition2.20([15,Proposition3]).LetJbeafinite-dimensionalinversesystem with ideal J⊥ =I. We have lead(J)⊥ =in(I). In other words, T/lead(J)⊥ is the associatedgradedalgebraofT/J⊥. Proof. IfQ ∈in(I),thenQ =in(Y ),forsomeY ∈I.ToseethatQ ∈lead(J)⊥,let F=lead(G)forG∈J.ItfollowsthatQ F isthehighestdegreepartofY G=0, soitiszero.Thisshowsthatin(I)⊆lead(J)⊥.Wehave dimkJ=dimklead(J)=dimkT/lead(J)⊥ ≤dimkT/in(I)=dimkT/I=dimkJ, where the first equality is by Proposition2.19 and the last is by Proposition 2.13. Thiscompletestheproof. ⊓⊔ Remark2.21.ByProposition2.19andProposition2.20,theHilbertfunctionofan inversesystem J is also the Hilbertfunctionof a standardgradedalgebra,namely the associated gradedalgebra of T/J⊥. Hence, Macaulay’sand Gotzmann’stheo- remsapplytothesefunctions.ThisenablesustoprovethattheonlypossibleHilbert functionshoflocalidealsI=J⊥ inH11withfullembeddingdimension3,equiva- 3 lentlyh(1)=3,aretheoneslistedinTable1.Sinceh(2)≤6,weneedtoconsider everypossiblevalueforh(2),1≤h(2)≤6.Also,(cid:229) h(i)=dimkT/I=11.Finally, ifh(i)≤2foranyi≥2,thenhisnonincreasingfromtheithsteponward,byCorol- lary2.4.ItistheneasytolistthepossibleHilbertfunctionsandtocheckthatallof themareinTable1. Proposition2.22([15,§C.2]).LetF(t)={f (t),f (t),...,f (t)}⊂S[[t]]beacol- 1 2 s lectionofpolynomialsinS[[t]],whichwe regardaspolynomialsinSwhosecoeffi- cientsarecontinuousfunctionsofaparametert inaneighborhoodof0.Thefam- ilyofapolarideals{F(t)⊥}satisfieslim F(t)⊥⊆F(0)⊥.Iftheinversesystems t→0 hF(t)ihavethesameHilbertfunctionforallt,thenwehavelim F(t)⊥=F(0)⊥ t→0 and{F(t)⊥}isaflatfamily. Proof. IfQ ∈lim F(t)⊥,writeQ =Q (0)=lim Q (t)whereQ (t)∈F(t)⊥for t→0 t→0 t6=0.Foreacht6=0wethenhavethatQ (t) f(t)=0,fori=1,...,s.Bycontinuity, i we also have that Q (0) f(0)=0. This shows Q ∈F(0)⊥ and lim F(t)⊥ ⊆ i t→0 F(0)⊥. The equality of Hilbert functions implies equality of dimensions, so the idealsareequal. ⊓⊔ TheHilbertschemeof11pointsinA3isirreducible 9 Definition2.23.When J =hf (t),f (t),...,f (t)i is a parametrized family of in- t 1 2 s versesystemsgeneratedbypolynomials f whosecoefficientsarecontinuousfunc- i tionsoft,wewillsaylim J =J ifandonlyiflim J⊥=J⊥. t→0 t 0 t→0 t 0 Example2.24.ConsiderthefamiliesW ={hℓd,mdi|ℓ,m∈S , independent}and 1 1 W ={hℓd,ℓd−1mi|ℓ,m∈S , independent}.Sincethelimit 2 1 (ℓ+tm)d−ℓd lim =ℓd−1m, t→0 dt wehave,byProposition2.22,that (ℓ+tm)d−ℓd limhℓd,(ℓ+tm)di=lim ℓd, =hℓd,ℓd−1mi. t→0 t→0(cid:28) dt (cid:29) ThisisbecauseeveryinversesystemineachfamilyhasHilbertfunction(1,2,...,2). ThisimpliesthatW isintheclosureofW intheZariskitopology. 2 1 3 The Hilbertscheme of11points in3-space Inthissectionwe,useMacaulay2toperformsomecomputationsthatwillbeneeded lateronandgathersomegeneralmethodsapplicabletoseveralofthecases. 3.1 Macaulay2codeexamples TocheckifanidealIinT =k[a,b,c]issmoothwecanrunthefollowingcode.This isoneofthecaseswecheckintheproofofProposition4.16. i1 : T = QQ[a,b,c] i2 : I = ideal {b*c,a*b,aˆ2*c,aˆ3-cˆ2,bˆ5} i3 : (dim I, degree I, degree Hom(I,T/I)) o3 = (0, 11, 33) These computations show that we have a zero-dimensional scheme of degree 11 with tangent space dimension 33. If we now know that this is in the smoothable component, then it has to be a smooth point, since we know that the smoothable componenthasdimension3·11=33.Tocheckthatthispointisinthesmoothable component,weconstructadeformation.WeguessacandidateidealK,thencheck thatitsatisfiestheneededconditions. 10 T.Douvropoulos,J.Jelisiejew,B.I.U.Nødland,Z.Teitler i4 : R = T[t] i5 : K = ideal {b*c,a*b,aˆ2*c,aˆ3-cˆ2,bˆ5+t*bˆ4} i6 : assert (K:t == K) i7 : minimalPrimes K o7 = {ideal (c, a, t + b), ideal (c, b, a)} Here K is an ideal in k[a,b,c,t] whose special fiber (at t =0) is I. To check that this is a flat family over k[t], we appealto [25, PropositionIII.9.7]which implies that if the ideal (K :t) equals K, then the family is flat in a neighbourhoodof 0. Thegeneralfiberis supportedatthe two points(0,−t,0),(0,0,0).Thisshowsthe specialfiberI iscleavable,hence,byLemma1.4I isalsosmoothable. 3.2 Somegeneral methods InthissectionwecollectvariousresultswhichweuseinSection4. In our analysis of the irreducible componentsof some standard graded Hilbert scheme(andthefibersofp ),wewilloftenconsiderthesetofquadricgenerators h {q ,q ,...,q }ofahomogeneousidealI⊂T.Thefollowinglemmadescribesthe 1 2 k spaceofcubicshq ,q ,···,q i·T intheidealgeneratedbythesequadrics. 1 2 k 1 Lemma3.1.Let T =k[a ,a ,...,a ] be the polynomial ring in n variables. Let 1 2 n q ,...,q be linearly independent quadrics in T where 2 ≤ k ≤ n, and let I = 1 k (q ,...,q ).ThendimI ≥nk− k ,withequalityifandonlyiftheq shareacom- 1 k 3 2 i monlinearfactor,thatis,qi=ℓℓ(cid:0)if(cid:1)orsomelinearformsℓ,ℓ1,...,ℓk. Proof. Let h be the Hilbertfunctionof T/I. The 2-binomialexpansionof h(2) is givenbyh(2)= n+1 −k= n + n−k .Thus,h(3)≤h(2)h2i= n+1 + n−k+1 , 2 2 1 3 2 so (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) n+2 n+1 n−k+1 k dimI =dimT −h(3)≥ − − =nk− . 3 3 (cid:18) 3 (cid:19) (cid:18) 3 (cid:19) (cid:18) 2 (cid:19) (cid:18)2(cid:19) Supposethatequalityholds.Wewillshowthattheq sharealinearfactor.ByGotz- i mann’sPersistenceTheorem,seeTheorem2.5,theequalityh(3)=h(2)h2i implies thath(t+1)=h(t)hti forallt≥2,whichgivesbyinduction n+t−2 n−k+t−2 n+t−2 n−k+t−2 h(t)= + = + . (cid:18) t (cid:19) (cid:18) t−1 (cid:19) (cid:18) n−2 (cid:19) (cid:18) n−k−1 (cid:19) This shows thatthe projectiveschemeV ⊂Pn−1 defined by I has Hilbert polyno- mialofdegreen−2withleadingcoefficient1/(n−2)!.Bystandardpropertiesof