The Hilbert schemes of points on surfaces with rational double point singularities

0.4 MB·

by  Xudong Zheng

THE HILBERT SCHEMES OF POINTS ON SURFACES WITH RATIONAL DOUBLE
POINT SINGULARITIES
XUDONGZHENG
7 Abstract. WeprovethattheHilbertschemeofpointsonanormalquasi-projectivesurfacewithat
1 worstrationaldoublepointsingularitiesisirreducible.
0
2

n TheprimarypurposeofthisarticleistoexplorethegeometryoftheHilbertschemeofpoints
a
Hilbd(X) for a surface X with certain isolated singularities. We confirm the irreducibility of
J
0 Hilbd(X)forXwithonlyrationaldoublepoints.
1 Hilbertschemesofpointsonquasi-projectivecomplexvarieties,especiallysmoothcurvesand
surfaces have been studied extensively in recent years. One of the common themes underlying

G] the existing literature is that studying the Hilbert schemes Hilbd(X) for varying d all at once
canrevealstructureaboutthevarietyX,suchassomeenumerativegeometricinformation. The
A
smoothnessofHilbd(X)oftenallowsinvestigationoftheirfurtherstructures. Ontheotherhand,
.
h the complexity of the singularities of Hilbd(X) if X is three-dimensional or higher has been one
t
a of the obstructions to generalizing results on smooth curves and surfaces. In the case that X
m is singular, the geometry of the Hilbert schemes become more complicated. It is necessary to
[ understandtheHilbertschemesofpointsofasingularvarietysincesingularvarietiesarenatural

degenerations of smooth varieties in families. For a singular curve C, a naturally related object
1
v is the compactified Jacobian J(C) of C, which admits the Abel-Jocobi map from Hilbd(C), and
5 Hilbd(C) is irreducible if and only if locally any point of C has embedding dimension at most 2
3 (locallyplanarsingularities) ([AIK,R80]). AgeneralizedMacdonald’sformula isappliedinthe
4
contextoftheuniversalHilbertschemesofpointsofafamilyofcurvesleadingtoaproofofGöttsche’s
2
conjectureonthenumberofnodalcurvesonsurfaces([G98,T12,S12]). Indimension2,ifasurface
0
. X has a single isolated cone singularity over a smooth curve of degree at least 5, then Hilbd(X)
1
0 isreducibleforsufficientlylargenumberd([MP13]). Yetitisanopenquestionwhetherthereis
7 asingularsurfacewhoseHilbertschemesofpointsareirreducibleforanyd > 0,particularlyfor
1 surfaceswithmildsingularities.
:
v Rationaldoublepointsonsurfacesinsomesenseareamongthesimplestsurfacesingularities
i withacompleteclassification. Aswewillseethestudyofreflexivemodulesandtheirdeformations
X
is essential in the proof of the main theorem, whereas we take advantage of the following facts
r
aboutrationaldoublepoints. First,therearefinitelymanyisomorphismclassesofindecomposable
a
reflexive modules; and second, every module admits a free resolution over the local ring of the
singularity via a matrix factorization. However, the irreducibility of the Hilbert scheme is not
valid for the affine cone over a twisted cubic curve, casting doubts on generalizing our results
to other quotient singularities. For a general rational surface singularity other than a rational
double point, the Hilbert scheme will become reducible if d is sufficiently large. We will leave
the investigation of the Hilbert schemes on surfaces with minimally elliptic singularities as a
separatedproject.
Date:January11,2017.
Keywordsandphrases. Hilbertschemeofpoints,smoothability,maximalCohen-Macaulaymodules,rationaldouble
pointsurfacesingularities.
1
2 XUDONGZHENG
TheoremA(Theorem3.3). SupposeXisaquasi-projectivenormalsurfacewithatworstrationaldouble
pointsingularities. LetHilbd(X)betheHilbertschemeoflengthdsubschemesofX. Thenforanypositive
integerd,Hilbd(X)isirreducibleofdimension2d.
We collect some geometric consequences of the irreducibility of Hilbd(X). First of all, one
canproveFogarty’ssmoothnesstheoremforHilbd(X)onasmoothsurfaceusingthetheoremof
Hilbert-Burch on free resolutions of ideal sheaves of zero-dimensional subschemes (see [FGA,
Chap. 7]). Inourcase,weshowasimilarsmoothnesscriterion:
TheoremB(Theorem4.1). SupposeZbealengthdsubschemeofthesurfaceXwithonlyADEsingu-
larities. ThenHilbd(X)issmoothat[Z]ifandonlyifI hasfinitehomologicaldimensionoverX.
Z
WealsoobtainsomegeneralizationsonresultsontheaffineplaneduetoHaiman. Thesmooth
quasi-projectivevarietyHilbd(A2)hastrivialcanonicalbundle,whereas:
TheoremC(Theorem4.2). SupposeX=Spec(C[x,y,z]/(cid:104)xz−yn+1(cid:105))istheaffinesurfacewithonlyA
n
singularityattheorigin. ThenHilbd(X)hasanowherevanishing2d-form.
HaimanalsoshowsthatHilbd(A2)isisomorphictotheblow-upofthesymmetricproductA2(d)
alongsomeideal,wherethisblow-upisidentifiedwiththeHilbert-Chowmorphism. Weobtain
analogousstatementfortheaffinequadricconeQ=Spec(C[x,y,z]/(cid:104)xz−y2(cid:105)).
TheoremD(Theorem4.5). TheHilbertschemeHilbd(Q)togetherwiththeHilbert-Chowmorphismis
isomorphictotheblow-upofthed-thsymmetricproductQ(d) alongtheidealconsistingofproductsoftwo
S -alternatingfunctionsonQ.
d
Thestudyofdeformationsofclosedsubschemesthataresupportedataclosedpointinavariety
reliesheavilyontheformalanalyticpropertiesofaneighborhoodofthepoint. Wenotethatthisis
allowableforourpurposeaszero-dimensionalschemesarealgebraicinanyeventandembedded
deformationofclosedsubschemesisformalinessence.
The paper is organized as follows. In section 1, we fix terminologies on Hilbert schemes
of points and review the structure of reflexive modules on ADE surface singularities. Section
2 contains the results on deformations of reflexive modules, which, as syzygy modules of zero-
dimensionalsubschemes,willgiverisetodeformationsofthecorrespondingsubschemes. Section
3 contains the main theorem on the irreducibility of the Hilbert scheme on surfaces with ADE
singularities (Theorem 3.3). Section 4 collects a few geometric applications of the irreducibility
ofHilbd(X). Inparticular,ifXistheaffinequadricconein3-space,theHilbert-Chowmorphism
hasanalternativeinterpretationanalogoustothecaseoftheaffineplane. Inthelastsection,we
providesomeexamplesofreducibleHilbertschemes,inparticular,ofatleast8pointsonthecone
overatwistedcubiccurve.
Acknowledgement. The results on the A singularities case have appeared as parts of the
n
author’sPh.D.ThesisattheUniversityofIllinoisatChicago. Itisapleasuretoacknowledgethe
overwhelminginfluenceofProfessorsLawrenceEinandKevinTuckeronthiswork. Hethanks
ProfessorsFrank-OlafSchreyerandRobinHartshorneforinspiringconversationscontributingto
thepaper. HeisgratefultoProfessorsHailongDao, TrondGustavsen, AkiraIshii, andMichael
Wemyssforansweringquestionsrelatedtothepaperinemailcorrespondence.
1. Preliminaryandmotivations
Throughoutthisarticle,weworkoverthefieldofcomplexnumberswithinthecategoryC−Sch
oflocallyNoetherianC-schemes. FiberproductsareoverSpec(C)ifnotspecified.
HILBERTSCHEMESOFPOINTSONSINGULARSURFACES 3
1.1. Definitions and background. Suppose X is a (quasi)-projective scheme and P(x) ∈ Q[x] a
numericalpolynomial.
The Hilbert functor on X (with Hilbert polynomial P) HilbP is the contravariant functor from
X
thecategoryC−SchtothecategorySetofsets,whichassignstoeachschemeUthesetofclosed
U-flat subschemes Z ⊂ U×X that have Hilbert polynimal P; and for a morphism φ : U → V ∈
MorC−Sch(U,V),HilbX(φ)istheinducedmapofsetsviafiberproduct. TheHilbertschemeofXwith
Hilbert polynomial P, denoted by HilbP(X), is the quasi-projective scheme which represents the
functorHilbP. Inparticular,ifP(x)=d∈Nistheconstantpolynomial,theschemeHilbd(X)isthe
X
HilbertschemeofdpointsofX. ThereistheuniversalfamilyH(cid:103)ilbd(X)→−π Hilbd(X).
Let S be the symmetric group in d elements, and let Xd = X × ··· × X be the d-fold fiber
d
product of X. The group S acts on Xd by permuting the factors. The d-th symmetric product of
d
XistheprojectiveschemeofS -quotientofXd,denotedbyX(d). Explicitly,fixinganembedding
d
X(cid:44)→P(E),bygeometricinvarianttheoryonecanequiptheset-theoreticalquotientXd/S witha
d
quasi-projectiveschemestructure. TheclosedpointsofX(d) aretheeffective0-cyclesoflengthd,
henceX(d) isregardedasaChowvarietyof0-cycles. ThesymmetricproductX(d) coincideswith
Rydh’s scheme of divided powers Γd(X) as over C. Over an arbitrary field k, for an affine scheme
Y = Spec(A)theaffineschemeofdividedpowersofY isthespectrumofthealgebraofdivided
powers Γd(A), which is not necessarily isomorphic to the d-th symmetric algebra Symd(A) (see
k
[R08]fordetails).
TheHilbert-Chowmorphismisthefollowingmorphism,denotedbyh:Hilbd(X)→X(d). Suppose
Z ⊂ X is a length d scheme supported at r distinct points z ,...,z ∈ X, giving rise to a closed
1 r
point [Z] ∈ Hilbd(X). Then h([Z]) = (cid:80)ri=1mizi, where mi = dimkOZ,zi. This description of the
Hilbert-Chow morphism is only on the set of closed points. The target X(d) should adapt the
reducedschemestructure. Foraprecisedefinitionsee[FGA,Chap. 7].
If X is a non-singular connected curve, then h is an isomorphism between two non-singular
varieties. Inparticular,thecohomologyringofX(d) isexpressedintermsofthecohomologyring
ofthecurveXbyMacdonald’sformula. IfXisanon-singularconnectedsurface,thenHilbd(X)is
non-singularofdimension2dandisacrepantresolutionofsingularitiesofX(d)viah([F68]).
Let I ⊂ O be an ideal sheaf which defines a zero-dimensional subscheme Z of X of length
X
d=l(Z),i.e.,dimCOX/I=d. WesaythattheidealIhascolengthd. TheHilbertfunctionofIisthe
integral-valued vector h = (h0,h1,...) where hi = dimCH0(X,OX(i))−dimCH0(X,I(i)) for i ≥ 0.
Localizing at a closed point x ∈ X, the punctual Hilbert scheme Hilbd(X,x) is the reduced closed
subschemeofHilbd(X)parameterizinglengthdsubschemesofXsupportedatx. Ifx∈Xisanon-
singularpointonasurface,thenHilbd(X,x)isirreducibleofdimensiond−1([B77,I72,ES98,EL99]).
When X = P2, the schemes Hilbd(P2) and Hilbd(P2,x) are stratified into affine spaces whose
numbers of affine cells of each dimension compute the ranks of their Borel-Moore homology
groups([B77,ES87,ES88,G88]). TheBettinumbersofHilbd(X),ChowgroupsandtheChowmo-
tivewithQ-coefficientswerecomputedforanarbitrarysmoothprojectivesurface([G90,deCM]).
IfXisaK3surface,Hilbd(X)isahyperKählermanifold([B83,F83]). WriteH (cid:66) H∗(Hilbd(X),Q)
d (cid:76)
fortherationalcoefficientcohomologyoftheHilbertschemeofpointsandH(cid:66) H ,thenH
n≥0 d
isanirreduciblemoduleoveraHeisenbergalgebra([N97,G96]). Inthecaseoftheaffineplane,
thenaturaltorus-actiononA2 extendstoHilbd(A2). Thehighercohomologygroupsoftwistsof
thetautologicallinebundlesonHilbd(A2,0)vanish. ThereisanAtiyah-BottformulafortheEuler
characteristic of the twists of the tautological line bundle on Hilbd(A2,0) which has seen some
combinatorial applications including the Macdonald positivity conjecture and the n! conjecture
([H98,H01]).
4 XUDONGZHENG
1.2. Maximal Cohen-Macaulay modules. In this subsection we review some well-known facts
about maximal Cohen-Macaulay modules over ADE singularities. The standard references are
[GV83,AV85,E85,A86].
Definition1.1. Let(B,m,k)beaNoetherianlocalring,andMafinitelygeneratedB-module. The
B-moduleMissaidtobemaximalCohen-Macaulay,orMCM,ifMsatisfiesdepth(M) = dimB. M
issaidtobereflexive,ifM (cid:27) M∗∗,wherethefunctor(•)∗ isthedualHom (•,B)onthecategoryof
B
B-modules.
Example1.2. LetBbeanytwo-dimensionalnormaldomainandI⊂Banidealoffinitecolength.
ThenthefirstsyzygymoduleofIisMCMoverBandisreflexive.
Now let R be any 2-dimensional rational double point singularity (i.e., the completion Rˆ is
a two-dimensional normal local domain with singularity one of the types A ,D ,E ,E or E ),
n n 6 7 8
then R is a hypersurface ring, in particular, Gorenstein. What follows about free resolutions of
finitelygeneratedR-modulesholdstrueforanyabstracthypersurfaceinthesenseof[E80,Remark
6.2] (the maximal ideal of R is minimally generated by dimR+1 elements, and the zero ideal
of R is analytically unmixed, or alternatively, Rˆ is the quotient of a regular local ring modulo a
principalideal. MinimalfreeR-resolutionofanyfinitelygeneratedR-module,ifnotfinite,willbe
periodicofperiod2afteratmost3steps([E80,Theorem6.1]). Aperiodicresolutionofperiod2is
necessarilygivenbyamatrixfactorizationassociatedtothehypersurface.
Example1.3. InthecaseofthequadricconeR=C[x,y,z]/(cid:104)xz−y2(cid:105)andQ=Spec(R)thereisonly
onenon-freeindecomposableMCMR-moduleuptoisomorphismwhichisgivenbythematrix
factorizationsM=M(φ,ψ),where
(cid:34) (cid:35) (cid:34) (cid:35)
x −y z y
φ= −y z , and ψ= y x ,
andthisMCMmoduleisgivenbythecokernelofeitheroneofthetwomatricesabove. Wedenote
the cokernel of φ by P. Note that P has rank 1 corresponding to a Weil divisor. Geometrically,
PisisomorphictothefractionalidealdefiningareducedlineLthroughthesingularpointonQ,
which can be understood as the first Chern class of P in Cl(R). The projective line L¯ as a Weil
divisorintheprojectiveclosureQ¯ isnotCartier,but2L¯ isCartier,i.e.,[P]is2-torsioninCl(R). The
φ ψ ψ φ
minimalR-freeresolutionofPisF•(P) : ··· → R⊕2 →− R⊕2 −→ ... −→ R⊕2 →− R⊕2 → 0. Truncating
thesequencewehaveashortexactsequence
(1.1) 0→P→R⊕2 →P→0.
Thissequenceisaspecialcasethatwillbestudiedindetail.
Ingeneral,theclassificationofindecomposablereflexivemodulesover2-dimensionalrational
doublepointsingularitiesarefullyunderstood([AV85,Theorem1.11]). Evenmoregenerally,for
anytwo-dimensionalquotientsingularitygivenbyafinitegroupG ≤ GL(2,C)thereisaone-to-
one correspondence between the finite set of isomorphism classes of indecomposable reflexive
modulesandthatofisomorphismclassesofirreduciblerepresentationsofthefinitegroupG. In
the case that G ≤ SL(2,C), there is a third finite set which yields a bijection with the preceding
two,namelytheverticesofthedualgraphoftheintersectionpairingoftheexceptionaldivisorin
theminimalresolutionofsingularity,theDynkindiagram. ThisisreferredasthegeometricMcKay
correspondence. AlltheindecomposablereflexiveR-modulescanbeexhaustedintheconcreteway
as follows: regarding the 2-dimensional regular ring C[u,v] as a regular representation of the
finitegroupGovertheringofinvariantsR=C[u,v]G,thenC[u,v]decomposesintothedirectsum
ofirreduciblerepresentationsofGoverRwitheachirreduciblerepresentationappearingexactly
once,thenthesearepreciselyalloftheindecomposablereflexiveR-modules.
HILBERTSCHEMESOFPOINTSONSINGULARSURFACES 5
Example1.4. ExplicitlyintheA singularitycase,wecanexpressRasR=C[un+1,uv,vn+1]. Then
n
anynontrivialindecomposablereflexiveR-modulehastheformM = R(cid:104)uavb | a,b ∈ N,a+b ≡ i(
i
mod n+1)(cid:105)fori=1,...,n. Ontheotherhand,asanirreduciblerepresentationofG,eachmodule
M asasubmoduleofC[u,v]isisomorphictoafractionalidealinC(u,v),M (cid:27) R(cid:104)1,ui/vn+1−i(cid:105)for
i i
i=1,...,n.
2. DeformationsofMCMmodules
Suppose (X,p) is the germ of a rational double point. In this section, we prove the following
resultsinthecontextofanarbitrary2-dimensionalrationaldoublepoints: (1)giventhesyzygy
module M = Ω(I ) of a zero-dimensional subscheme Z on X, a deformation of M induces a flat
Z
embedded deformation of Z in X (Proposition 2.3); (2) closed subschemes of finite projective
dimensiononXissmoothable(Theorem2.4). Bothstatementscontributetothecoreoftheproof
ofthemaintheoremontheirreducibilityoftheHilbertschemes(Theorem3.3). And(3),ifthesocle
dimensionsoc(O )ofazero-dimensionalsubschemeZ⊂Xsatisfiestheequalitye(I )−soc(O )=1
Z Z Z
thenZhasfiniteprojectivedimension. Wealsoincludesomeexplicitstudiesoftheextensionsof
reflexivemodules.
The first proposition motivates the entire investigation of subschemes of finite homological
dimension.
Proposition2.1. SupposeasubschemeZhasfiniteprojectivedimensiononthesurfaceX,thentheZariski
tangentspaceofHilbd(X)at[Z]hasdimension2d.
Proof. The proof of smoothness of the Hilbert scheme of points on a smooth surface works ver-
batim. We write I for the ideal sheaf of Z in O . Under the assumption of the finiteness
Z X
of the homological dimension of Z, by the Auslander-Buchsbaum formula and the theorem of
Hilbert-Burch-Schaps,aminimalfreeresolutionofI ofZtakestheform:
Z
0→O⊕r →O⊕r+1 →I →0
X X Z
forsomepositiveintegerr. WecancomputethatExt2(I ,O )=0andExt1(I ,O )(cid:27)Ext1(I ,O )(cid:27)
X Z Z X Z Z X Z X
ω ,whichimplythatχ(O ,O )=0. HencetheZariskitangentspaceT Hilbd(X)(cid:27)Hom (I ,O )(cid:27)
Z Z Z [Z] X Z Z
Ext1(O ,O )hasdimension2d. (cid:3)
X Z Z
Remark 2.2. The preceding proposition does not necessarily prove that the closed point [Z] in
Hilbd(X)isasmoothpoint,asitisunclearwhetherHilbd(X)locallyat[Z]hasdimensionsmaller
than2d.
The key construction is given in the following proposition, which produces a family of sub-
schemesfromafamilyofreflexivemodules. WewriteX0 =X\{p}andX=X×A1fortheisotrivial
1-parameterfamilyofXwithtwoprojectionsπX :X→XandπA1 :X→A1.
Proposition2.3. LetM beafamilyofreflexiveO -modulesofconstantrankroverA1. SupposeM is
t X 0
the syzygy module of the ideal I of a zero-dimensional subscheme Z of X with Supp(Z ) = {p}. Then
0 0 0
thereexistsanopenneighborhoodUof0∈A1suchthatforanyt∈UthecorrespondingmoduleM isthe
t
syzygymoduleoftheidealI ofazero-dimensionalsubschemeZ ofXwiththesamelengthasZ .
t t 0
Proof. Byassumption,thereisashortexactsequenceofR-modules:
φ
0→M −→0 Or+1 →I →0.
0 X 0
WeconsiderthefollowingsequenceonX:
φ
0→F →− F →Q(cid:66)Coker(φ)→0,
1 0
whichsatisfiesthefollowingproperties:
6 XUDONGZHENG
1. F isafreeO [t]-moduleofrankr+1;
0 X
2. F isareflexiveO [t]-moduleofrankrwithdepth (F )=3foranyclosedpointq∈X;
1 X q 1
3. therestrictionofφtotheclosedfiberX over0∈A1isφ .
0 0
Therestoftheproofwillbeaseriesofrestrictionsintheaffinelinetoanopenneighborhoodof0
suchthatineachfiberthemoduleQrestrictstoanidealofasubschemeofX = Xwithconstant
t
lengthasZ .
0
To begin with, note that the support of the torsion part of Q in A1 is disjoint from 0. We let
U ⊂ A1 bethecomplementofthetorsionpartofQandletX = X×U betherestrictedfamily.
1 1 1
NowwecanassumethatQisarank1torsionfreesheafonX suchthatQ| (cid:27)I .
The double dual Q∗∗ of Q with respect to R[t] is a rank 1 r1eflexive R[t]-mX0odu0le. Hence there
existsaWeildivisorDofX1withQ∗∗ (cid:27)OX1(D). SinceXisatrivialfamilyofXovertheaffineline,
wehaveisomorphismoftheirfirstChowgroups,whichsurjectsontotheChowgroupofX :
1
CH1(X)(cid:27)CH1(X)→CH1(X )→0.
1
Therefore there exists a Weil divisor D0 of X0 such that OX1(D) (cid:27) π∗XOX(D0). Restricting this
identificationtoX0weseethatD0ishomologousto0,soitiswithD. SowehaveQ∗∗ (cid:27)OX1.
WecanwriteQ = I astheidealsheafofaclosedsubschemeW ofX . ThissubschemeW is
W 1
notnecessarilyequidimensionalandhencenotfiniteflatoverU . WewillfurtherrestrictinU to
1 1
whereW becomefiniteflatoverthebaseofdegreeequaltothelengthofZ . FirstletW bethe
0 1
union of the 2-dimensional components of W (possibly empty). We see that W is disjoint from
1
X .
0
LetX (cid:66) X \W . RestrictingonX wecanassumethatW isaone-dimensionalscheme(not
2 1 1 2
necessarilyreducedorirreducible). WehaveapresentationofI :
W
φ
0→F →− F →I →0.
1 0 W
Bycondition3onthedepthofF above,weconcludethatWisaCohen-Macaulaycurve. Moreover,
1
therecouldbecomponentsofWwhicharecontractedbyπA1. Theseverticalcomponentsarenot
contractedtotheclosedpoint0. Byfurtherrestrictingtoapossiblysmalleropenneighborhood
U ⊂U of0wecanassumethatWdoesnotcontainanyverticalcomponentsandhenceflatover
2 1
U .
2
Writeψ:W →U2fortherestrictionoftheprojectionπA1 fromX2. Notethatψ−1(0)red ={(p,0)},
and that both X and U are affine schemes. We can apply Zariski’s main theorem to obtain a
2 2
factorization of ψ through ψ˜ : Z → U and we can find an open neighborhood U of (p,0) in W
2
suchthattheinducedmorphismU →Zisanopenimmersion. (cid:3)
Theorem2.4. Suppose(X,p)isthegermofarationaldoublepointsurfacesingularitywithX=Spec(R),
theclosedpointp∈Xbeingtheonlysingularpoint. SupposeZ isaclosedsubschemeofXsupportedatp
0
withidealI . WeassumethatI hasalength2freeresolutiononX. ThenZissmoothable.
0 0
Proof. Theproofisstreamlinedbyaconcreteconstructionofaflatfamilysmoothingthesubscheme
Z. WriteX =X\{p}andX=X×A1fortheisotrivial1-parameterfamilyofXwithtwoprojections
0
πX :X→XandπA1 :X→A1. SupposeI0hasaminimalR-freeresolutionoftheform
φ
(2.1) 0→R⊕r−1 −→0 R⊕r →I →0,
0
whereφ isar×(r−1)matrixwithallentries f inthemaximalidealmofR. Nowweconstruct
0 ij
athreeshortexactsequenceofR[t]-modulesasintheproofoftheprecedingpropositionwhich
restrictsto(2.1):
φ
(2.2) 0→R[t]⊕r−1 →− R[t]⊕r →Q→0.
BythepreviousconstructionwecanassumethatQ=I forsomeclosedsubschemeWofXfinite
W
andflatoveranopenneighborhoodof0∈A1.
HILBERTSCHEMESOFPOINTSONSINGULARSURFACES 7
Nowweneedtodeform(2.2)sothattherestrictiontoageneralfiberisthefreeresolutionof
(cid:34) (cid:35)
anotherideal. Wedefinea1-parameterfamilyofmatrices: φ˜ (cid:66)φ+tI(cid:48) ,whereI(cid:48) = Ir−1 .
r−1 r−1 0
Note that for t = 0 we have φ˜ = φ , and for any t (cid:44) 0, the top (r−1)×(r−1)-minor of φ˜ is
0 0
invertiblesinceitsdeterminantistr−1. Consequently,thecokernelofφ˜ fort (cid:44) 0istheidealofa
t
closedsubschemeofXoflengthdwhichdoesnothavepinitssupport. (cid:3)
Remark2.5. See[Y04,Theorem2.2,3.2]forresultsondegenerationsofmodulesinamoregeneral
setting,and[Y04,Remark2.5]forastatementonspecializationsofanextensionclass.
Nextweprovidewithaclassofexamplesofsubschemesoffinitehomologicaldimension,for
whichtheassumptionisclearinthecasewherethesurfaceissmooth([EL99,Lemma2]).
Proposition2.6. SupposethesyzygymoduleΩ(I )ofI isgivenbyaminimalpresentation
Z Z
0→Ω(I )→Rr+3 →I →0,
Z Z
for some integer r ≥ 0, i.e., Ω(IZ) has free rank zero. Suppose dimCExt1R(IZ,R) = r+2, then IZ has
homologicaldimension1overR.
Proof. Supposeη1,...,ηr+2formasetofgeneratorsofExt1R(IZ,R). Thenonecanformtheextension
η(cid:66)(η1,...,ηr+2)forsomeR-moduleM(IZ):
φ
η:0→Rr+2 →− M(I )→I →0.
Z Z
ApplyingHom (•,R)toη,wehave
R
0→R→(M(I ))∗ →Rr+2→−γ Ext1(I ,R)→Ext1(M(I ),R)→0.
Z R Z R Z
By construction, γ : Rr+2 → Ext1(I ,R) is surjective. Hence Ext1(M(I ),R) = 0. Then M(I ) is
R Z R Z Z
maximalCohen-Macaulay. Infact,HerzogandKühl([HK])showthatM(I )(cid:27)Ω(I )∗⊕R.
Z Z
The r+2 components of the map φ : Rr+2 → M(I ) projecting to Ω(I )∗ defines a map φ˜ :
Z Z
Ω(I )→Rr+2ofrankr+2. Onecantakethepush-outdiagram
Z
The bottom row of the diagram is denoted by (∗). Now the two extensions η and (∗) differ
possibly by an automorphism of Rr+2 since φ˜ has full rank. Comparing η with (∗), we get the
followingdiagram
Bytheshort-fivelemma,themiddleverticalmapisanisomorphism: M(I )(cid:27)M(cid:48). Combining
Z
thiswiththepreviousdiagramwehave
8 XUDONGZHENG
Sinceτhasfullrank,bytheSnakeLemmaweseethatker(τ) = 0. Nowsupposethecokernal
of the left two vertical maps τ and ρ is K. Then K is an R-module of finite length. Applying
Hom (•,R)tothelefttwocolumnswegetExt1(Ω(I ),R)(cid:27)Ext2(K,R). ButsinceΩ(I )ismaximal
R Z Z
Cohen-Macaulay,thisext-groupiszero. ByMatlisduality,K=0. HenceΩ(I )isaR-free. (cid:3)
Z
The rest of the section is devoted to the study of deformations of reflexive modules which
appearassyzygiesofzero-dimensionalsubschemesonthesurface. Roughlyspeaking,ifthereis
ashortexactsequenceofreflexivemodules
0→M→M(cid:48)(cid:48) →M(cid:48) →0
thenthedirectsumM⊕M(cid:48)canbeconsideredasaspecializationofM(cid:48)(cid:48)bytrivializingtheextension.
Ontheotherhand,forafixedpairofmodulesMandM(cid:48),theC-vectorspaceExt1(M(cid:48),M)isstratified
R
into finitely many strata according to the isomorphism classes of the middle term, and there is
a “generic extension”, i.e., a dense subset of Ext1(M(cid:48),M) which all have the same middle term,
R
denotedbyE(M(cid:48),M). Consequently,alloftheextensionscanbethoughtasspecializationsofthe
genericextension.
To proceed, we first compute the dimensions of the ext-groups of pairs of reflexive modules.
Suchcomputationsappearin[G80]andinabroadercontextin[IM]. SupposeM ,...,M arethe
1 n
distinct isomorphism classes of indecomposable reflexive R-modules. For each M there are at
i
leasttwonaturalshortexactsequencesendingwithM: oneisthepresentationafterchoosinga
i
minimalsetofgeneratorsofM:
i
(2.3) 0→M∗ →R⊕2rk(Mi) →M →0
i i
∗
whereM istheR-dualofM. TheotheristheAuslander-Reitensequence,whichhasbothendings
i i
M:
i
(2.4) 0→M →E(M)→M →0,
i i i
where E(M) is the direct sum of all the indecomposable modules corresponding to the vertices
i
adjacent to the one for M in the extended Dynkin diagram. For the reader’s convenience,
i
we briefly reproduce the construction of both cases below. For a rigorous exposition on the
sequence (2.4) in an appropriate generality, see [W88]. For the coincidence between the McKay
quiverassociatedthethefinitegroupG ≤ SL (C)andtheAuslander-Reitenquiveronthesetof
2
indecomposablereflexivemoduleswereferthereadertoAuslander’soriginalpaper([A86]).
AnextendedDynkindiagramisobtainedfromtheusualDynkindiagrambyaddingonevertex
correspondingtothetrivialmoduleRattheappropriateplace. Inthediagramsbelow,welabel
the vertices in the A case 1,...,n from the left to the right, and in the D case 1,...,n from the
n n
lefttotheright(thelasttwoverticesn−1andnarenotdistinguished). ForA ,oneputstheextra
n
vertexontopofthechainofnexistingverticesandclosetheloopbyaddingedgesfromvertices
1andntothis(n+1)-stvertex. ForD ,the(n+1)-stvertexisaddedtovertex2onthelefttailso
n
thatthepicturewouldbecomesymmetric. ForE theextravertexisplacedandconnectedbyan
6
edgeontopofvertex4;forE andE theextravertexisconnectedtovertex1(labelledbyacircle).
7 8
HILBERTSCHEMESOFPOINTSONSINGULARSURFACES 9
DynkindiagramsofADEsingularities
(ineachcasesofD andE nodesofthesamecolorcorrespondtomodules
n n
withthesamedeterminantinthelocalclassgroupofthesingularity)
Toexplainsequence(2.3),wehavethefollowing
Lemma2.7. TheR-dualofeachfiniterankreflexivemoduleislistedasbelow:
A . M (cid:27)M∗ fori=1,...n.
n i n+1−i
10 XUDONGZHENG
Dn. Mi (cid:27) M∗i fori = 1,...,n−2. Ifniseven,thenMn−1 andMn arealsoself-dual;ifnisodd,then
theyaredualtoeachother.
E . In E , M (cid:27) M∗,M (cid:27) M∗, and M and M are self-dual. In E and E every indecomposable
n 6 1 6 2 5 4 3 7 8
moduleisself-dual.
LetΩ(cid:102) bethereflexivehullofthesheafofholomorphic1-formsontheopensurfaceX extended
X 0
toX. ThenΩ(cid:102) isareflexiveO -module. ThereisthefundamentalsequenceonX:
X X
(2.5) 0→ω →Ω(cid:102) →O →C→0.
X X X
Suppose M is a non-trivial indecomposable reflexive module on X, then the Auslander-Reiten
sequence ending with M can be obtained by tensoring the sequence (2.5) with M and taking
reflexivehulls,resultingwithasequenceoftheform
0→τ(M)→(Ω(cid:102) ⊗M)∗∗ →M→0,
X
whereτ(M) = (ω ⊗M)∗∗ iscalledtheAuslander-ReitentransposeofM. Inthecurrentcase, since
X
thesurfaceisGorensteinτ(M)(cid:27)M,andthemiddletermwasdenotedbyE(M)in(2.4).
However,thereareinfactmanymoreextensionsofreflexivemodules. Acombinatorialwayto
findthedimensionsofExt1(M,M )foranypairofmodulesM andM isdeveloppedbyIyama
R i j i j
andWemyss([IM,Theorem4.4,4.5]). Inthisarticlewecomputesomeexamplesfollowing([IM]).
Example 2.8. (An) In the An case, for any integers 1 ≤ a ≤ b ≤ n we have dimCExt1R(Mb,Ma) =
dimCExt1R(Ma,Mb)=max{a,n+1−b}.
Example2.9. (D6)SaywewanttocomputedimCExt1R(M2,M1). Wewillproduceachainoftuples
oftheindicesontheDynkindiagram. Westartwiththetuple(2)indicatingtheextensionsshould
endatM . Thenexttupleisthecollectionofallindicesadjacentto2,namely(1,3),andwewrite
2
(2)(cid:55)→(1,3). Thenexttupleisthecollectionofalladjacentverticesof(1,3)countingmultiplicities
subtracting the previous one, hence it is (2,2,4)−(2) = (2,4) and we write (2) (cid:55)→ (1,3) (cid:55)→ (2,4).
Thischainkeepsgoing:
(2)(cid:55)→(1,3)(cid:55)→(2,4)(cid:55)→(3,5,6)(cid:55)→(4,4)(cid:55)→(3,5,6)(cid:55)→(2,4)(cid:55)→(1,3)(cid:55)→(2)(cid:55)→0.
Eventually the sequence will stop at 0 and we count how many 1’s showing up in this entire
sequence. Therearetwo1’s,sodimExt1(M ,M )=2. ThismethodworksforanyADEsingulari-
R 2 1
ties.
Example2.10. (D4)ThesamecalculationasforD6saysthatdimCExt1R(Mi,Mi)=2foranyi=1,3,4.
Knowingthedimensionsoftheextensionsforeachpairofreflexivemodulesgivesapreliminary
estimationforthedeformationsofthemodules,andweneedtofindthestratificationwithineach
Ext-group and to determine the most general extensions. For a fixed singularity and a fix pair
ofreflexivemodulesM andM onit,wedeclareapartialorderonExt1(M,M ): twoextensions
i j R i j
satisfyη>τifτisaspecializationofη.
Lemma2.11. IneachcomplexvectorspaceExt1(M,M )thereisauniquemaximalelement,corresponding
R i j
tothe“genericextension”.
Proof. Thisshouldbeageneralhomologicalstatement. Supposetothecontrarythattherearetwo
non-isomorphicextensionsη,τ ∈ Ext1(M,M )suchthateachhasanopenneighborhoodU and
R i j 1
U in Ext1(M,M ) in its usual Euclidean topology. Then we connect η and τ by a line segment
2 R i j
andconstructahomotopyfromonetotheother. ThislinesegmentwillintersectwithU andU
1 2
bothonanopensubset. Thisisimpossiblesinceaone-dimensionalsubspaceofExt1(M,M )can
R i j
haveonlyonegenericclass. (cid:3)