THE VERSAL DEFORMATION SPACE OF A REFLEXIVE MODULE ON A RATIONAL CONE

Pure Mathematics

ISBN 82–553–1404–0 No. 33 ISSN 0806–2439 October 2003

THE VERSAL DEFORMATION SPACE OF A REFLEXIVE MODULE ON A RATIONAL CONE

TROND STØLEN GUSTAVSEN AND RUNAR ILE

Abstract. By an approach based on results of A. Ishii, we describe the versal deformation space of any reflexive module on the cone over the rational normal curve of degreem. To each component a resolution is given as the total space of a vector bundle on a Grassmannian. The vector bundle is a sum of copies of the cotangent bundle, the canonical sub-bundle, the dual of the canonical quotient bundle, and the trivial line bundle. Via an embedding in a trivial bundle, we obtain the components by projection. In particular we give equations for the minimal stratum in the Chern class filtration of the versal deformation space.

We obtain a combinatorial description of the local deformation relation and a classification of the components. In particular we give a formula for the number of components.

1. Introduction

The versal deformation space is in general highly singular and difficult to de- scribe. Some explicit results have been given, e.g. for surface singularities [7, 23, 1], and for torsion free sheaves on singular curves [21, 22]. The aim of this article is to describe the versal deformation space of any (not necessarily indecomposable) reflexive module M on the cone over the rational normal curve of degreem.

AssumingX is a rational surface singularity, A. Ishii proves in [16, 4.9] an inter- esting theorem giving a filtration of the versal deformation spaceRfor deformations of a reflexive moduleM onX, which, in the caseXis a rational double point, is the stratification with respect to isomorphism classes of modules. More precisely; let π:Xe →X be a minimal resolution, then a reflexive moduleM onX corresponds to a full sheaf Mf on Xe. For each d ∈ PicXe, Ishii defines a functor of families (parametrised by arbitrary schemes overRred) of semi-full sheavesE onXe with an isomorphism. The functor is represented by a regular schemeF^dwhich is projective over R_red. Asdvaries, a finite stratification`S^d of R_red is obtained such that if the fibre of the versal family at t ∈ R is the reflexive module N, then t ∈ S^d if and only if the full sheaf Ne has Chern class d. Moreover; S^d is regular for all d.

In particular; each component in the reduced versal deformation space is given as the closure of an S^d. By the McKay correspondence this gives the stratification by isomorphism classes if X is a rational double point. In the latter case Ishii also gives an explicit example of anF^d; assume c1(Mf) is minus the fundamental cycle, thenF⁰ is the minimal resolution ofX∼=R_red, [16, 5.3]. If X is a rational double point, Ishii describes the closure of the minimal stratum in terms of resolutions, in [16, 5.6], and in particular obtains the local deformation relation [16, 5.5].

In the case X is the cone over the rational normal curve of degreem, there are m isomorphism classes of rank one reflexive modules and any reflexive moduleM is a direct sum of these. We findF^d for allM and alld. In Theorem 1 an intrinsic

2000Mathematics Subject Classification. Primary 14B12, 14D20; Secondary 13C14, 14J17.

Acknowledgement. The authors are grateful for partial financial support from NorFA through the research network NORDAG and from RCN’s Strategic University Program in Pure Mathem- atics at the Dept. of Mathematics, University of Oslo (No 154077/420).

1

description is given: F^d(ask-scheme) is the total space of a vector bundle of relative extensions Ext¹

X×A/Ae (EA,EA) on a Grassmannian A, where (A,EA) represents a functor of embeddings of semi-full sheavesE with c1(E) =d, see Proposition 1.

In Theorem 2 we calculateExt¹

X×A/Ae (EA,EA) as a sum of copies of the cotangent bundle, the canonical sub-bundle, the dual of the canonical quotient bundle, and the trivial line bundle onA. The number of copies is given by the dimension of certain cohomology groups associated to a sub-sheaf of Mf. Remark that this strengthens and generalises the description of the “generic” minimal stratum in Ishii’s [16, 5.6ii]. Theorem 2 also gives an embedding of the vector bundle in the trivial vector bundle Ext¹_X(M, M)×A and the map to Ris obtained as the composition of the embedding with the projection to Ext¹_X(M, M). From the equations in Corollary 1 defining the embedding, an explicit expression for the image R^d ofF^d in R for all the minimal strata is obtained in Corollary 2;R^dis the cone over a Segre embedding times an incidence variety times an affine space intersected with certain hyperplanes and quadratic hypersurfaces. In Corollary 3 we give an ideal Id of minors which givesT^dby blowing upR^d. It givesT^das the strict transform in resolutions of rank singularities and we observe how the Chern class filtration of the versal deformation space is related to the rank filtration.

From Theorem 1 a combinatorial description of the local deformation relation is obtained in Lemma 3. In contrast to the rational double point case, there are many non-trivial Chern class preserving deformations, they give smooth strata in Rred. In Theorem 3 the components of the reduced versal deformation space are classified and a formula for the number of components is given. The components correspond to the geometrically rigid modules, and they are listed in Corollary 4. Further observations concerning the local deformation relation are given in Corollaries 4–8.

Three elementary examples are found in the final section.

The study of reflexive modules on rational surface singularities may be traced back to the 1960’s. D. Mumford in characteristic zero [18] and J. Lipman in char- acteristic p >0 [17] proved that a surface singularityX is rational [3] if and only ifX has finitely many isomorphism classes of rank one reflexive modules. Later J.

Herzog [14], H. Esnault [10] and M. Auslander [6] proved that a rational surface singularity is a quotient singularity if and only if there is a finite number of indecom- posable reflexive modules onX. As shown in [11, 2], the intersection of the Chern class of the full sheaf with the exceptional divisor, sets up a correspondence between the set of isomorphism classes of non-trivial indecomposable reflexive modules and the components of the exceptional in the caseX is a rational double point. This is the McKay correspondence. The Chern character (i.e. rank and Chern class) does not determine the corresponding reflexive module for general quotient singularities, cf. [10]. In [25] J. Wunram determined the full sheaves for cyclic quotient singu- larities, and following [10] he gave in [26] a cohomological criterion on a full sheaf such that a generalised McKay correspondence may be set up for the corresponding sub-class of indecomposable reflexive modules.

2. Preliminaries

In this section we introduce notation which is fixed and cite standard results which will be used freely throughout the article. Let X be a surface singularity, i.e. X = SpecOX where OX is the Henselisation of a local, normal, essentially finitely generated k-algebra of dimension 2 over an algebraically closed field k of any characteristic. There exists a minimal resolutionπ:Xe →X of the singularity in all characteristics, [17, 4.1], andX is arational surface singularity if R¹π∗O

Xe = 0, [3]. Remark that Hⁱ(X,e F) = 0 for any coherent sheaf F and i > 2 by the

Theorem on Formal Functions, cf. [13, 11.1]. In this article X will be the affine cone over the m-uple embedding of P¹k in P^mk, i.e. OX = k[u^m, u^m−1v, . . . , v^m]^h where “h” denotes Henselisation. The exceptional divisor C=Xe×XSpeck⊆Xe is therefore isomorphic to P¹k. There is an intersection theory onXe, see [18, 3, 17], and C∼ −mD where D is any curve intersecting C transversally in one point, in particularC²=−m. Moreover; by [17] we have PicXe ∼=Zgenerated byD.

A finitely generated O_X-module M is called reflexive if the canonical map to its double O_X-dual, M → M^∨∨, is an isomorphism. Since X is 2-dimensional and normal, a reflexive module is the same as a maximal Cohen-Macaulay module.

In particular; M restricted to the regular locus U ⊆X is locally free. Let Mf= π^∗M/torsion and more generally, ifM_S is anS-flat family of reflexive modules on X for a k-scheme S, then MfS is the image of the canonical map from π_S^∗MS to its double O

X×Se -dual where πS : X×Se → X×S is the pullback ofπ. Following [10], Mfis called a full sheaf, and M = H⁰(X, π∗Mf). Moreover; a sheaf E on Xe is shown to be full if and only if E is locally free, generated by global sections and R¹π_∗E^ω = 0, where E^ω := Hom

Xe(E, ω

Xe). In particular M = H⁰(X, π_∗E) is a reflexive OX-module with Mf= E since they are generated by the same global sections. For our particular X we have rank one reflexive OX-modules Mi = (uⁱ, uⁱ⁻¹v, uⁱ⁻²v², . . . , vⁱ) for 06i6m−1 withMf_i ∼=O

Xe(iD). Since the group H = coker(Z−→^·C PicXe)∼=Z/mZby [17] classifies the rank one reflexive modules, the following lemma implies that theMi are the only indecomposables.

Lemma 1. IfM is a reflexive module onX,thenM is isomorphic to a direct sum of rank one reflexive modules.

Proof. It is sufficient to show thatMfis a direct sum of line bundles. More generally we show that a vector bundleE onXe is isomorphic to a direct sum of line bundles.

Let i be maximal such that H⁰(E⊗OC(−i)) 6= 0. From the exact sequence 0 → E(−C−iD)→ E(−iD)→ E⊗OC(−i)→0 we get H⁰(E(−iD)) = 0⇒H⁰(E(−C− iD)) = 0, twisting the sequence several times byO(−C) gives H⁰(E(−nC−iD)) = 0 for all n>0 which is impossible sinceO(−C) is very ample relative toX. Hence we have a non-zero section s∈ H⁰(E(−iD)) which defines a short exact sequence of locally free sheaves

0→ O(iD)−→ E −→ F →^s 0

by the maximality ofi. The lemma follows by induction on the rank since one from the maximality of igets Ext¹

Xe(F,O(iD)) = 0.

Remark 1. If chark = 0, then X is the cyclic quotient singularity defined by the action of the cyclic group G=hζ·idk²i onA², whereζ is a primitivem^th root of unity in k, see [19]. Moreover; there is a correspondence between the irreducible representations of G and the indecomposable reflexive OX-modules where Mi =

k[u, v]⊗kξi^G

if the irreducible representationξi is given byζ·id_k2 7→ζⁱ for 06 i6m−1, see [25].

Note that by adjunction c1(ω

Xe)·C=−C²−2 =m−2 thusω

Xe =O

Xe((m−2)D) and since π_∗ω

Xe = ωX for all rational surface singularities, we getωX = M_m−2. We say that a locally free sheaf E on Xe is semi-full (over a reflexive module M) if R¹π_∗E = 0 and H⁰(X, π_∗E) ∼= M. By [16, 1.8] there are natural embeddings Mf⊆ E ⊆Mg^ωω where M^ω = Hom_X(M, ωX). We have Mf_i^ω =O

Xe((m−2−i)D) for 06i6m−1 and H⁰(X, π∗(E^ω)) =M^ω [10], hence we get M_i^ω=Mm−2−i for

06i6m−2 andM_m−1^ω =M_m−1. We obtain

Mg_i^ωω= (

Mfi if 06i6m−2 OXe(−D) if i=m−1.

Dividing the two inclusions Mf ⊆ E ⊆ Mg^ωω with Mf give inclusions of sheaves 0 ⊆ E ⊆ OC(−1)^r on C, hence if E is semi-full, with π_∗E ∼= M = Lm−1

i=0 M_iⁿⁱ, then (by the proof of Lemma 1) E ∼= FL

G where F = Li=m−2

i=0 O(iD)ⁿⁱ and G=O(−D)^sLO((m−1)D)^r−s(r=n_m−1), this notation will be fixed.

Let Hens_k be the category of local, Henselian k-algebras O_S with residue field k. The deformation functor Def_M : Hens_k → Sets associates to O_S the set of equivalence classes of deformations of M to OS. A deformation (or flat lifting) of M to OS is an (OX⊗kOS)^h-module MS, flat as OS-module together with an (OX⊗kOS)^h-linear map π : M_S → M with π⊗OSk : M_S⊗OSk −^'→ M. Two deformations are equivalent if they are isomorphic over M. Maps are induced by tensorisation. If the module is of finite type over an algebraic ring, i.e. the Henselisation of a k-algebra essentially of finite type, such that the locus whereM is not free is of finite length, then, using [5] and [9, Thm. 3], it is shown in [24] and in [16] that there exists aversal family (R, MR) for Def_M where in particularRis algebraic. We fix such a versal family where we assume that the Zariski tangent space is of minimal dimension at the central point and put XR = SpecO^h_X×R. Moreover; since Def_M is a functor locally of finite presentation, there exists agerm representing (R, MR), i.e. an affine k-pointed k-scheme R^ft of finite type and an O_Rft-flat family of reflexive modules M_Rft, finitely generated as O_X×Rft-module, such that the Henselisation at thek-point gives (R, MR).

Definition 1. If M and N are two reflexive modules on a surface singularity, let Loc(N) be the set ofk-pointst∈R^ft(k) such that the pullbackMtofM_Rft totis isomorphic to N. ThenM locally deforms to N, denoted M 99KN, if the Zariski closure Loc(N) strictly contains the central k-point t0 corresponding to M. If, possibly after restricting to a Zariski open set in R^ft containingt₀, the pullback of M_Rft to Loc(N)r{t0}is non-empty and only containsN as k-fibres, then Loc(N) is called an absolute minimal stratum ofR^ft and the local deformation ofM to N is called minimal.

It follows that the relation99Kis independent of choice of germ, and by openness of versality [16, 2.13] it follows that the local deformation relation istransitive.

In [16] Ishii introduces a sub-functor Def_M⁰ ⊆Def_M of deformations such that the induced deformation of the determinant bundle ofMS restricted to the regular locusU is trivial. Ishii shows that there is a versal family (R⁰, MR⁰) for Def_M⁰ and that R⁰ ∼=Rred.

Let d∈ PicXe, then the Ishii functor of semi-full sheaves with isomorphism is defined as follows:

Definition 2 (A. Ishii [16, 4.2]). Let the functor

(1) F_M^d

R0 :Sch_R⁰ →Sets for any (ψS :S→R⁰) inSchR⁰be given as the setF_M^d

R0(ψS:S→R⁰) of equivalence classes of pairs (ES, ϕS) where

i) ES is a locally free sheaf onXeS =Xe×XXS,

ii) R¹π∗Et= 0 and c1(Et) =dfor all pullbacksEtofES tok-pointst∈S(k), iii) ϕS :π_S∗ES

−'→ψ^∗_SMR⁰ onXS =XR⁰×R⁰S.

Two pairs (ES, ϕS) and (E_S⁰, ϕ⁰_S) are equivalent if there is an isomorphismτ :ES

−'→ E_S⁰ such thatπS∗(τ) = (ϕ⁰_S)⁻¹ϕS.

Ishii’s main theorem [16, 4.9] states that the functor F_M^d

R0 is represented by a R⁰-schemeψ_Fd :F^d=F_M^d

R0 →R⁰ which is projective overR⁰, regular, non-empty for a finite set of Chern classesd, and their images{R^d}inR⁰constitutes a filtration of R⁰. There is an isomorphism between the locus S^d in R⁰ of reflexive modules with first Chern class equal to d and the open set in F_M^d

R0 corresponding to full sheaves.

We may take the structure map ψS into the functor and define T_M^d

R0(S) for a k-scheme S as equivalence classes of tuples (ES, ψ_S :S →R⁰, ϕ_S) where (ES, ϕ_S) defines an element inF_M^d

R0(ψ_S :S→R⁰). One can check thatT_M^d

R0 is representable if and only ifF_M^d

R0 is representable. We extendT_M^d

R0 by allowing the range ofψ_S to be the the non-reducedRand define T^d =T_M^d_R as equivalence classes of triples (ES, ψS : S →R, ϕS) with (ES, ϕS) as in Definition 2 withR substituting R⁰. A priori T_M^d

R0 ⊆ T_M^d

R, but forX a rational cone we show in Theorem 1 thatT_M^d

R is represented by a regular scheme T^d, henceψ_Td :T^d →R factors throughR⁰ and F^d=T^d ask-schemes.

By “module” we will usually mean “reflexive module”. As a convention the first projection will usually be denoted p, like in p : X×Se →Xe, and the second projectionq.

3. Representing the Ishii functor Theorem 1 states that the representing space forT_M^d

R is given as the total space of a vector bundle of relative extensions of a locally free sheafE_Awith itself over a Grassmannian A. LetA=A^d_M be the sub-functor of T^d of tuples (E_S, ψ_S :S → R, ϕS) whereψS factorises through Speck, i.e. is trivial. In Proposition 1 we show that A^d_M is represented byA=A^d_M with a universal locally free sheafEA onXe×A and in Proposition 2 we give a natural embedding of the sheaf of relative extensions ofEA with itself into the trivial pullback of Ext¹_X(M, M) toA.

Proposition 1. Let d= c1(fM) +sC with 0 6 s6 r where r is the multiplicity of Mm−1 in M. Then the functor A^d_M is represented by the Grassmannian A = Grass(s, r).

Proof. There is a natural isomorphism α : A −^'→ B valid for all rational surface singularities where B(S) is defined as the set of equivalence classes of embeddings ιS : ES ,→ p^∗Mg^ωω where ES is a locally free coherent sheaf with c1(Et) = d, R¹π_∗Et= 0 andπ_∗ιt:π_∗Et

−'→M for allt∈S(k) and withιS ∼ι⁰_S if imιS= imι⁰_S. It is not obvious thatBis a functor, i.e. whether a pullback ofιS will be an injective map, this is however a consequence of the following argument. Given (ES, ϕS) in A(S) the inclusionιS =α(ES, ϕS) is the following composition

(2) ES

−'→(E_S^ω)^ω,→(π_S^∗πS∗E_S^ω)^ω

∼= πS∗(E_S^ωe)ω (gc⁻¹)^ω

−−−−→

∼= (πS∗(ES)^ωe)ω ϕf^ω_S^ω

−−−→∼=

p^^∗M^ω

ω∼=p^∗Mg^ωω

where all maps are canonical exceptϕf^ω_S^ω. The first map is an isomorphism sinceE_S is locally free. For the second one the cokernel of the natural map π_S^∗πS∗E_S^ω→ E_S^ω has support on C×S, dualising in p^∗ω

Xe hence gives an inclusion. The next iso- morphism follows similarly. The canonical isomorphism π_∗ω

Xe →ωX induces the isomorphism c :π_S∗(E_S^ω)→(πS∗ES)^ω. The two last isomorphisms are clear. The construction of ιS from ϕS is clearly functorial, hence α is well defined: An iso- morphismθ:ES → E_S⁰ compatible withϕS andϕ⁰_S givesϕf^ω_S^ω=gϕ^0ω_S^ω◦ (πS∗(θ)^ωe)ω

and imιS = imι⁰_S.

The inverse β:B → A toαis given by

(ι:E_S ,→p^∗Mg^ωω)7→(E_S, π_S∗(ι) :π_S∗(E_S)→π_S∗p^∗Mg^ωω=p^∗M), ι_∗=πS∗(ι) is an isomorphism by definition ofB. In particular;Bis a functor.

Now the general idea is to map ιS : ES ,→ p^∗Mg^ωω in B(S) to the induced embeddingι_S :E_S =E_S/ι⁽⁻¹⁾_S (p^∗Mf),→p^∗(gM^ωω/Mf) with support onC×S so that one is left to study a Quot-functor on the exceptional fibre. Hereι⁽⁻¹⁾_S is defined as follows. Pulling the inverse (ι∗)⁻¹ back to π_S^∗(ι⁻¹_∗ ) :p^∗π^∗M −^'→π_S^∗πS∗ES followed by the canonicalπ_S^∗π_S∗ES → ES induces a mapι⁽⁻¹⁾_S :p^∗Mf→ ES, functorial inϕS, the composition ιS◦ι⁽⁻¹⁾_S : p^∗Mf→p^∗Mg^ωω is the natural inclusion, [15, 1.8], and henceι⁽⁻¹⁾_S is an inclusion. In our case there is a canonical splittingES =p^∗F ⊕GS

and ES = GS/ι⁽⁻¹⁾_S (p^∗O

Xe((m−1)D)^r). Since ι⁽⁻¹⁾_S is functorial in S, the short exact sequence 0→p^∗Mf ^ι

(−1)

−−−→ ES S → ES →0 is natural for pullbacks of S, hence E_S is S-flat. It follows that the inclusion ι_S : E_S ,→ p^∗O_C(−1)^r is natural for pullbacks of S, hence Et is a locally free sheaf for t ∈ S(k). From the inclusion ιS we have H⁰(C,Et) = 0, and since Et is semi-full 0 = H¹(X,e Et) H¹(C,Et), henceEt∼=OC(−1)^s0. We claim thats⁰ =s. Ifρ= rkMfwe chooseρ−1 generic sections in H⁰(X,e Mf)−^'→H⁰(X,e Et) which define inclusions of the trivial sheafO^ρ−1

Xe

in Mfand in Et to obtain representatives of the first Chern class as the cokernel, see [2]. There is a commutative diagram of sheaves on Xe with two horizontal and two vertical short exact sequences from which the claim follows:

(3) 0 //O^ρ−1

Xe //

Mf_

ι⁽⁻¹⁾_t

//O

Xe(c₁(Mf))

_ //0

0 //O^ρ−1

Xe //Et

//O

Xe(c1(fM) +sC)

//0

Et

' //O_sC(c₁(fM) +sC)

By Cohomology and Base Change [13, III.12.11], twisting by 1 and pushing down to S gives an embedding of a locally free sheaf of ranks; q_∗ι_S(1) :q_∗ES(1),→ O^r_S, hence an element in Quot^s_kr(S).

For the inverse, let 0→ V → O^r_S −→ O^τ _S^r/V →0 be an element in Quot^s_kr(S). Let GS = kerη⊗τ;

0→ GS g_S

−→p^∗O

Xe(−D)⊗q^∗O_S^r −^η⊗τ−−→p^∗OC(−1)⊗q^∗O^r_S/V →0 where η is the quotient map in the short exact sequence

(4) 0→ O

Xe(−C−D)−→ O

Xe(−D)−→ O^η C(−1)→0.

Since Tor^X×S_i^e (p^∗OC(−1)⊗q^∗O^r_S/V,−) = 0 for i >2,GS is a coherent locally free OX×Se -sheaf. Together with the trivial embedding of p^∗F, gS gives the element ι_S :p^∗F ⊕GS ,→p^∗Mg^ωωin B(S). One checks that this gives an inverse.

Let

(5) ι= (id⊕g) :EA=p^∗F ⊕GA,→p^∗F ⊕p^∗O

Xe(−D)^r=p^∗Mg^ωω

be the universal embedded locally free sheaf onXe×AwhereAis the Grassmannian representing A. Letτ :O_A^r → Qbe the universal quotient map on the Grassman- nian.

Lemma 2. There is a natural short exact sequence of locally free sheaves onXe×A (6) 0→p^∗O

Xe(−D)⊗q^∗S −→ GA−→p^∗O

Xe(−C−D)⊗q^∗Q →0 which is locally split on A.

Proof. The short exact sequence

(7) 0→ GA

−g

→p^∗O

Xe(−D)⊗q^∗O^r_A−^η⊗τ−−→p^∗OC(−1)⊗q^∗Q →0 defines GA. Pull the canonical short exact sequence

(8) 0→p^∗O

Xe(−D)⊗q^∗S −→p^∗O

Xe(−D)⊗q^∗O^r_A−→p^∗O

Xe(−D)⊗q^∗Q →0 back along the inclusionp^∗O

Xe(−C−D)⊗q^∗Q →p^∗O

Xe(−D)⊗q^∗Q. Theng is the induced inclusion GA → p^∗O

Xe(−D)⊗q^∗O^r_A, and the short exact sequence (6) is obtained. A local splitting of the universal inclusionS ,→ O_A^r gives a local splitting of (8) which induces the local splitting of p^∗O

Xe(−D)⊗q^∗S,→ GA. Assume Mis a quasi-coherent sheaf onX×S. Lete Ext^∗

X×S/Se (M,−) denote the derived functor ofq_∗Hom

X×Se (M,−) :Mod

X×Se →ModS, [16, 5.4]. There is a first quadrant cohomological spectral sequence

(9) E^ij₂ = Rⁱq_∗Ext^j

X×Se (M,−)⇒ Ext^∗

X×S/Se (M,−). IfMis locally free the spectral sequence degenerates to

(10) Rⁱq_∗Hom

X×Se (M,−) =Extⁱ

X×S/Se (M,−).

Proposition 2. There is a natural injective homomorphism ofO_A-sheaves Ext¹

X×A/Ae (EA,EA),→Ext¹_X(M, M)⊗kOA.

Proof. The universal inclusionι :EA ,→p^∗Mg^ωω and the induced inclusionι⁽⁻¹⁾: p^∗M ,f→ EA(cf. the proof of Proposition 1) give two maps

(11) Ext¹

X×A/Ae (EA,EA) ^(ι

(−1))^∗

−−−−−→ Ext¹

X×A/Ae (p^∗M ,f EA)−→ Ext^{ι∗ 1}

X×A/Ae (p^∗M , pf ^∗Mg^ωω). In Theorem 2 we prove in a more precise statement that this composition is an inclusion. By (10)

Ext¹

X×A/Ae (p^∗M , pf ^∗Mg^ωω)∼= R¹q∗Hom

X×Ae (p^∗M , pf ^∗Mg^ωω)

∼= R¹q_∗p^∗Hom

Xe(M ,fMg^ωω)∼= Ext¹

Xe(M ,fMg^ωω)⊗kOA.

The proposition then follows from composing with the following isomorphism, valid for all rational surface singularities

(12) π_∗: Ext¹

Xe(fM ,Mg^ωω)−^'→Ext¹_X(M, M).

The latter is proved as in [15, 3.5] with ^∨ changed to^ω. Set E = Spec Sym_k(Ext¹_X(M, M)^∗) and let E^h be the Henselisation of E at the origin. There is an embedding of R into E^h which is not canonical; two em- beddings differ by an automorphism of E^h which induces the identity on the Za- riski tangent space. To accommodate this inconvenience we consider the functor R^d = R^d_MR which is defined to be the image ofT^d in Hom_Sch

k(−, R) under the map (ES, ψS, ϕS)7→ψS.

Theorem 1. Let (R, M_R)be the versal family for the reflexive module M on the coneX over the rational normal curve of degreem,and letd= c₁(fM)+sC∈PicXe with 06 s6r where r is the multiplicity of M_m−1 in M. If EA is the universal sheaf on Xe×A from (5), then the functorT_M^d

R is represented by thek-scheme T^d=SpecSym_OA Ext¹

X×A/Ae (EA,EA)^∨

×EE^h

where the map to E is induced by the inclusion in Proposition 2 and where the Grassmannian A= Grass(s, r)represents the functorA^d_M.

Moreover; letR^d be the image ofT^d inE^hand let Ψ^d :T^d→R^d be the induced map. ThenR^d representsR^d_M

R andT_M^d

R → R^d_M

R is induced by Ψ^d.

Proof. LetT andR^din the following argument be theT^dand theR^d“without the h” unless pullback toE^h is called for. The proof has two parts.

(1) Construction of an element (ET, ψT, ϕT) inT^d(T).

(2) Show that this element is universal forT^d.

1. There is a covering ofAby open affines{V_i}such thatExt¹

X×A/Ae (E_A,E_A)_|Vi is a freeOV_i =OA(Vi)-sheaf for alliby Lemma 2 and (10). LetEV_i:=E_A|Vi then

Γ(Vi,Ext¹

X×A/Ae (EA,EA))∼= H¹(X×Ve i,End

X×Ve i(EVi)) =: H¹

with a Bi := OA(Vi)-basis {η⁽ⁱ⁾₁ , . . . , ηn⁽ⁱ⁾}. There is a covering{U0, U1} of Xe by affine open sub-schemes. LetU01=U0∩U1, thenEV_i is defined by a transition map θ ∈ Γ(U01×Vi,End

X×Ae (EA)) (e.g. by Lemma 2 since PicXe is generated by (any) D). DefineEi on

T_i= Spec Sym_B

i (H¹)^∨

=B_i[t₁, . . . , t_n] by extending the transition map to eθ=θ+Pn

j=1η_j⁽ⁱ⁾t_j wheret_j =t⁽ⁱ⁾_j in (H¹)^∨is Bi-dual toη_j⁽ⁱ⁾. Remark thatθe mod (t1, . . . , tn)² gives the universal lifting of EV_i

toBi[t1, . . . , tn]/(t1, . . . , tn)². In fact theEiglue together on the full scheme giving ET, sinceθeis independent of choice of basis, which follows since thetj have degree one in the Bi-linear grading ofTi, and this grading is preserved by localisation of Bi. By Lemma 2, after pullback toE^h, R¹π∗(ET⊗O_Tk(t)) = 0 for allt∈T(k).

In order to findψT :T →RandϕT :πT∗ET

−'→ψ^∗_TMRwe first show thatαe_∗ET

is a locally free sheaf, henceH-flat, whereαe:X×Te →Xe×H is the pullback ofρ: T →H= Spec Sym_kH⁰(T,OT). OnUi,C is defined byvi, pushout of (6) restric- ted toU_ialong the isomorphismp^∗O_Ui(−D)⊗q^∗S−−−→^·vi⊗1 p^∗O_Ui(−C−D)⊗q^∗Sgives the exact sequence 0 →p^∗O_Ui(−C−D)⊗q^∗S → G_A⁰ →p^∗O_Ui(−C−D)⊗q^∗Q →0 with G_A⁰ ∼=GA. The natural maps

Ext¹_Ui×A(OUi(−D)Q,OUi(−D)S)∼=OUi⊗kExt¹_A(Q,S)→ Ext¹_U

i×A(OUi(−C−D)Q,OUi(−D)S)∼=OUi(C)⊗kExt¹_A(Q,S)−−−→^·vi⊗1 Ext¹_Ui×A(OU_i(−C−D)Q,OU_i(−C−D)S)∼=OU_i⊗kExt¹_A(Q,S)

takes the canonical sequence on A tensor O_Ui(−D) to (6) and further on to the canonical sequence onAtensorO_Ui(−C−D), thereforeG_A|Ui×A∼=O_Ui(−D)O^r_A and henceE_A|Ui×A is free. Thus the tensor product pre-sheafE_A|Ui×A⊗_OAOT is a sheaf andET|Ui×T =E_A|Ui×A⊗_OAOT. Therefore

αe_∗ET|Ui×H(U_i×H)∼=E_A|Ui×A(U_i×A)⊗_OA(A)OT(T)∼=OUi(U_i)^rk(E)⊗kOH. Remark that H is of finite type since T → E is a projective morphism which follows from Propositions 1, 2. Since αe_∗ET is locally free, we have that π_H∗αe_∗ET

is an OH-flat family of reflexive modules onX by [15, 3.4], with central fibre M.

After Henselisation there is, by versality of (R, M_R), a map γ : H → R and an isomorphism f : (πHα)e ∗ET

−'→(id×γ)^∗MR. We have a commutative diagram

Xe×T ^αe //

π_T

Xe×H

π_H

X×T ^α //X×H Let f : α^∗α_∗(πT)_∗ET

−'→ ((id×γ)α)^∗MR be the pullback of f by α to X×T. We claim that the canonical map θ : α^∗α_∗(πT)_∗ET → πT∗ET is an isomorphism and we put ϕT = f θ⁻¹. For the surjectivity of θ, there is a natural map u : (πT∗ET)⊗_OTOA→π_A∗(ET⊗_OTOA)∼=M⊗kOA. To show thatuis an isomorphism we consider the sheafified ˇCech complex 0 → ET → C⁰ → C¹ → 0 on Xe×T, applying πT∗ gives a short exact sequence since R¹πT∗ET = 0 by Cohomology and Base Change [13, III.12.11]. Applying −⊗OTOA leaves the sequence exact since π_T∗C¹ is flat as OT-module. Since we also have (α^∗α_∗(π_T)_∗ET)⊗OTOA ∼= M⊗kOA, θ⊗OTOA is an isomorphism and cokerθ = 0. Injectivity follows from (kerθ)⊗OTOA ∼= Tor^O₁^T(OA, π_T∗ET) = 0 ([15, 3.4]). With ψ_T = γ◦ρ, we get a T-pointξ= [(ET, ψT, ϕT)]∈ T^d(T).

2. Restricting to the reduced locusR⁰⊆R (see comments at the end of Section 2), we assume T_M^d

R0 is represented by the regular scheme F^d which is projective over R⁰ [16, 4.9]. SinceT is regular, ψT factors through R⁰ andξ∈ T^d(T) induce a map f :T →F^dofR⁰-schemes. The central fibre ofT and ofF^d isA, and (after the pullback toE^h) allk-rational points ofT are contained inA. Lett= [(E, ϕ)]∈ A^d(k) =T^d(k) =A(k) and let T_t and A_t be the associated deformation functors ofT^d andA^d at t. There is a short exact sequence ofk-vector spaces

(13) 0→ At(k[ε])−→ Tt(k[ε])−→Def_E(k[ε])→0

which follows from [16, 4.11] since the composition is the trivial map: If ι :E ,→ Mg^ωω is the embedding (2) corresponding to ϕ, At(k[ε])→Def_E(k[ε]) is the com- position Hom

Xe(E/ι⁽⁻¹⁾(Mf),Mg^ωω/ι(E))−^'→Hom

Xe(E,Mg^ωω/ι(E))−→^δ Ext¹

Xe(E,E) of natural maps. The composition map factors through Ext¹

Xe(E/ι⁽⁻¹⁾(fM),E) and since Ext¹

Xe(E,E)→Ext¹

Xe(fM ,E) is injective by Proposition 2 (R¹q_∗commutes with tensor products), it is trivial. From (13) and the construction of T it follows that f induces an isomorphism on the Zariski tangent spaces and hence ˆO_F,f(t)−^'→Oˆ_{T ,t} for allt∈T(k), by [12, 17.6.3]T →F is ´etale and since it is bijective onk-rational points it is an isomorphism.

One may also prove representability for the a priori larger T_M^d

R directly. Pro- representability of Tt follows essentially as [20, 3.2], dim_kTt(k[ε])<∞is given by (13). The substantial part is to show that an isomorphismτ :E1→ E2of liftings of E toSinArtkcompatible withϕi:πS∗Ei

−'→ψ^∗_RMRis uniquely determined, which follows by induction on the length ofS: The set of liftings ofτ in a small extension S⁰ S is a torsor over End

Xe(E)⊗ker(S⁰S), but End

Xe(E) = End_X(M) and πS∗τ is uniquely determined asϕ⁻¹₂ ϕ1. Let [(ET⁰, ψT⁰, ϕT⁰)]∈ T^d(T⁰) and denote by A⁰ ⊆ T⁰ the closed fibre. By Proposition 1 there is a unique map A⁰ → A which hence defines T⁰(k)→T(k) as a set map. By working locally onT⁰ (in the

´

etale topology) and applying Artin’s Approximation Theorem [4, 2.2] one obtains a mapf :T⁰ →T of schemes overR, using the uniqueness-of-isomorphism argument above one shows that there is a unique isomorphismf^∗ET

−'→ ET⁰.

For the last part of Theorem 1 we show that (απ_T)_∗E_T is the pullback of an O_Rd-flatO_X×Rd-moduleM_Rd, hence γ:H →R may be chosen as a factorisation through R^d. Since αe∗ET|Ui×H is free and the transition matrix in fact is defined

overO_Ui×Rd by construction ofE_T,αe_∗E_T descends to a locally freeXe×R^d-module which is pushed down to an R^d-flat X×R^d-module M_Rd by [15, 3.4]. Remark that the various t⁽ⁱ⁾_j (in the construction ofET) are global sections in the image H⁰(OE×A)→H⁰(OT), hence are contained in O_Rd. We conclude that anyψT⁰ ∈

R^d_MR(T⁰) factorises throughR^d.

Remark 2. By versality of (R, MR) and the last part of Theorem 1 one obtainsR^d as a uniquely defined closed sub-scheme ofR.

4. Equations for the strata

Recall the map Ψ^d which is the embedding T^d ,→E^h×A followed by the pro- jection onto its image R^d in E^h. In Theorem 2 we calculate both sides of the inclusion in Proposition 2 and the inclusion itself and hence, by Theorem 1, obtain an explicit description of T^d and of the embedding in terms of canonical bundles and maps between them on the GrassmannianA= Grass(s, r). Explicit equations are given in Corollary 1, and in Corollary 2 we obtain equations for the minimal strata R^d ⊆E^h (s= 1). An ideal-sheafId onE^h which define Ψ^d:T^d→R^d as a blowing up is given in Corollary 3. With notation and assumptions as in Theorem 1 we have:

Theorem 2. Let EA = p^∗F ⊕ GA be the universal sheaf on Xe ×A in (5). The map Ext¹

X×A/Ae (E_A,E_A)→Ext¹_X(M, M)⊗_kO_Astated in Proposition 2 gives an em- bedding T^d ,→ E^h×A and is the direct sum of the following four embeddings of coherent sheaves on A:

1) Ext¹

X×A/Ae (p^∗F, p^∗F) = H¹EndF

⊗kOA

−=→H¹EndF

⊗kOA

2) Ext¹

X×A/Ae (p^∗F,GA) = H¹F^∨⊗OC(−1)

⊗kS −−−→^id⊗σ H¹F^∨⊗OC(−1)

⊗kO^r_A 3) Ext¹

X×A/Ae (G_A, p^∗F) = H¹F ⊗O_C(−m+ 1)

⊗_kQ^∨

id⊗τ^∨

−−−−→H¹F ⊗OC(−m+ 1)

⊗k(O_A^r)^∨ 4) Ext¹

X×A/Ae (GA,GA) = H¹OC(−m)

⊗kHom_A(Q,S)

id⊗(τ^∗, σ∗)

−−−−−−−→H¹OC(−m)

⊗kEnd_A(O^r_A) where 0 → S −→ O^{σ r}_A −→ Q →^τ 0 is the canonical sequence on the Grassmannian A. In particular; if F = 0 we get k^m−1×T_A^∨ ,→ k^m−1×End_A(O^r_A), where T_A^∨ = Hom_A(Q,S)is the cotangent bundle on A.

Proof. The map is induced by (ι⁽⁻¹⁾)^∗ andι_∗, see (11).

1.Ext¹

X×A/Ae (p^∗F, p^∗F)∼= R¹q_∗p^∗End

Xe(F)∼= H¹(End

Xe(F))⊗_kO_A by (10). The restrictions ofι⁽⁻¹⁾ andιare the identity maps.

2. The restriction of (ι⁽⁻¹⁾)^∗ is the identity while the restriction ofι∗ isg∗. We shall obtain a commutative diagram

R¹q_∗Hom

X×Ae (p^∗F,GA) ^g∗//

∼= α1

R¹q_∗Hom

X×Ae (p^∗F, p^∗O

Xe(−D)⊗q^∗O^r_A)

∼= α2

H¹ F^∨⊗OC(−1)

⊗kS ^id⊗σ //H¹ F^∨⊗OC(−1)

⊗kO_A^r

Apply Hom

X×Ae (p^∗F,−) to (7) and then apply q_∗ to the resulting short exact sequence. We get a short exact sequence of R¹q_∗-terms

0→R¹q_∗Hom

X×Ae (p^∗F,GA)→ H¹(X,e Hom

Xe(F,O

Xe(−D)))⊗kO^r_A−−−→^η∗⊗τ H¹(X,e Hom

Xe(F,OC(−1)))⊗kQ →0 since Hⁱ F^∨(−C−D)

= 0 fori >0, and hence we also get the isomorphism α1. Theα2is obtained analogously changing (7) to (4).

3. The restriction of ι_∗ is the identity map. The upper horizontal map in the following commutative diagram is by (10) the relevant inclusion.

R¹q_∗Hom

X×Ae (G_A, p^∗F) ^(ι

(−1))^∗//

∼= δ1

R¹q_∗Hom

X×Ae (p^∗O

Xe(−C−D)⊗q^∗O_A^r, p^∗F)

∼= δ2 R¹q∗Ext¹

X×Ae (p^∗OC(−1)⊗q^∗Q, p^∗F) ^τ∗ //R¹q∗Ext¹

X×Ae (p^∗OC(−1)⊗q^∗O^r_A, p^∗F) H¹ F ⊗OC(−m+ 1)

⊗kQ^{∨ id⊗τ}

∨ //

δ₃ ∼=

OO

H¹ F ⊗OC(−m+ 1)

⊗k(O_A^r)^∨

δ₄ ∼=

OO

Theδ1 is induced by the connecting map in the short exact sequence obtained by applyingHom

X×Ae (−, p^∗F) to (7). But Rⁱq_∗(p^∗F(D))∼= Hⁱ(F(D))⊗OA= 0 for all i > 0 henceδ1 is an isomorphism. Analogous arguing goes for δ2 using the short exact sequence (4) instead. Theδ3is the composition H¹ F ⊗OC(−m+1)

⊗kQ^∨∼= R¹q_∗ p^∗Hom

X×Ae (O

Xe(−C−D),F)⊗q^∗Q^∨ '

−→R¹q_∗Ext¹

X×Ae (p^∗OC(−1)⊗q^∗Q, p^∗F) obtained by considering connecting maps induced from (4), analogous for δ4.

4. Substitute GA for p^∗F in the previous diagram, analogous reasoning gives (ι⁽⁻¹⁾)^∗ as

R¹q_∗Hom

X×Ae (p^∗O

Xe(C−D)⊗q^∗Q,GA) ^τ

∗

−→R¹q_∗Hom

X×Ae (p^∗O

Xe(C−D)⊗q^∗O^r_A,GA) via the isomorphisms

R¹q_∗End

X×Ae (GA)−→^δ1 R¹q_∗Ext¹

X×Ae (p^∗OC(−1)⊗q^∗Q,GA)

δ₃

←−R¹q_∗Hom

X×Ae (p^∗O

Xe(C−D)⊗q^∗Q,GA). essentially because R¹q∗(p^∗O

Xe(D)⊗GA) = 0 which follows from (6). Apply Hom

X×Ae (p^∗O

Xe(−C−D)⊗q^∗τ ,−)

to (7) and obtain a map τ^∗ of short exact sequences. Applyingq_∗ and the projec- tion formula yields a map of short exact sequences in each cohomological degree essentially because Hⁱ(O

Xe) = 0 fori >0. Calculating the two last R¹q_∗-terms:

H¹ O

Xe(C)

⊗_kHom_A(Q,O^r_A) ^τ

∗ //

H¹(η(C−D))⊗τ∗

H¹ O

Xe(C)

⊗_kEnd_A(O^r_A)

H¹(η(C−D))⊗τ∗

H¹ OC(−m)

⊗kEnd_A(Q) ^τ∗ //H¹ OC(−m)

⊗kHom_A(O^r_A,Q) Again because Hⁱ(O

Xe) = 0 for i > 0, H¹(η(C−D)) is an isomorphism and the kernel of the vertical maps gives the relevant summand of (ι⁽⁻¹⁾)^∗:

H¹ OC(−m)

⊗kHom_A(Q,S)−→^τ∗ H¹ OC(−m)

⊗kHom_A(O_A^r,S). Forι_∗, calculating

Ext¹

X×A/Ae (O

Xe(−C−D)O_A^r,O

Xe(−D)O_A^r)∼= H¹ OC(−m)

⊗kEnd_A(O_A^r) gives the fourth summand ofι_∗which isg_∗=σ_∗: H¹(OC(−m))⊗kHom_A(O^r_A,S)→

H¹(OC(−m))⊗kEnd_A(O^r_A).