Punctual Hilbert schemes for Kleinian singularities as quiver varieties
Alastair Craw, Søren Gammelgaard, Ádám Gyenge, Balázs Szendrői
aa r X i v : . [ m a t h . AG ] N ov PUNCTUAL HILBERT SCHEMES FOR KLEINIAN SINGULARITIESAS QUIVER VARIETIES
ALASTAIR CRAW, SØREN GAMMELGAARD, ´AD ´AM GYENGE, AND BAL ´AZS SZENDR ˝OI
Abstract.
For a finite subgroup Γ ⊂ SL(2 , C ) and n ≥
1, we construct the (reduced scheme underlyingthe) Hilbert scheme of n points on the Kleinian singularity C / Γ as a Nakajima quiver variety for the framedMcKay quiver of Γ, taken at a specific non-generic stability parameter. We deduce that this Hilbert schemeis irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. Moregenerally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKayquiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these newalgebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices ofstability parameter.
Contents
1. Introduction 12. Variation of GIT quotient for quiver varieties 33. Cornering the preprojective algebra 74. Identifying the posets for the coarse and fine moduli problems 115. Punctual Hilbert schemes for Kleinian singularities 12Appendix A. Bounding the dimension vectors of θ J -stable modules 15References 221. Introduction
Let Γ ⊂ SL(2 , C ) be a finite subgroup. One can associate various Hilbert schemes to the action of Γ onthe affine plane C and the Kleinian singularity C / Γ. For N := | Γ | and any natural number n , the actionof Γ on C induces an action of Γ on the Hilbert scheme Hilb [ nN ] ( C ) of nN points on the affine plane.The scheme n Γ- Hilb( C ), parametrising Γ-invariant ideals I in C [ x, y ] such that the quotient C [ x, y ] /I isisomorphic to the direct sum of n copies of the regular representation of Γ, is a union of components of thefixed point set of the Γ-action on Hilb [ nN ] ( C ). It is thus nonsingular and quasi-projective. One may alsoconsider the Hilbert scheme of n points Hilb [ n ] ( C / Γ) on the singular surface C / Γ, parametrising ideals inthe invariant ring C [ x, y ] Γ that have codimension n . This Hilbert scheme is quasi-projective, and in thisintroduction we endow it with the reduced scheme structure.These two kinds of Hilbert schemes are related by the morphism n Γ- Hilb( C ) −→ Hilb [ n ] ( C / Γ) (1.1) ending a Γ-invariant ideal I in C [ x, y ] to the ideal I ∩ C [ x, y ] Γ ; this set-theoretic map is indeed a morphismof schemes by Brion [Bri13, 3.4]. By composing with the Hilbert-Chow morphism of the surface C / Γ, wesee that (1.1) is in fact a morphism of schemes over the affine scheme Sym n ( C / Γ).Until recently, not much was known about the schemes Hilb [ n ] ( C / Γ) for n >
1. Gyenge, N´emethiand Szendr˝oi [GNS18] computed the generating function of their Euler characteristics for Γ of type A andD (the cyclic and dihedral cases), giving an answer with modular properties. Zheng [Zhe17] proved thatHilb [ n ] ( C / Γ) is always irreducible, and gave a homological characterisation of its smooth points through adetailed analysis of Cohen-Macaulay modules over C / Γ. Yamagishi [Yam17] studied symplectic resolutionsof the Hilbert squares Hilb [2] ( C / Γ), and described completely the central fibres of these resolutions, fromwhich he deduced that Hilb [2] ( C / Γ) admits a unique symplectic resolution.The aim of our paper is to study the spaces appearing in (1.1), and all possible ways in which themorphism from (1.1) can be decomposed, using quiver-theoretic techniques in a uniform way. The startingpoint is the McKay correspondence, which associates a quiver (oriented graph) to the subgroup Γ ⊂ SL(2 , C ).Representation spaces of a framed variant of the McKay quiver, each depending on a stability parameter, wereintroduced in Kronheimer and Nakajima [KN90] and studied further by Nakajima [Nak94]. Subsequently, forany n ≥ C ± in the spaceof stability parameters for which the corresponding representation space M θ is isomorphic to the punctualHilbert scheme Hilb [ n ] ( S ) of the minimal resolution S of C / Γ for θ ∈ C − , and to the scheme n Γ- Hilb( C )from (1.1) for θ ∈ C + , respectively. Much more recently, Bellamy and Craw [BC18] gave a completedescription of the wall-and-chamber structure on the space of stability parameters in this situation, andidentified a simplicial cone F containing C ± that is isomorphic as a fan to the movable cone of n Γ- Hilb( C )for n >
1; in particular, chambers in this simplicial cone correspond one-to-one with projective, symplecticresolutions of Sym n ( C / Γ) (see Figure 1 below for an example).The main result of our paper reconstructs the morphism from (1.1) by variation of GIT quotient. Explicitly,we vary a generic stability parameter θ ∈ C + to a parameter θ in a particular extremal ray of the closureof C + ; the induced morphism M θ → M θ coincides with the morphism (1.1). As a corollary, we obtain thefollowing result. Theorem 1.1.
Let Γ ⊂ SL(2 , C ) be a finite subgroup and let n ≥ . The (reduced) Hilbert scheme Hilb [ n ] ( C / Γ) red is an irreducible, normal scheme with symplectic, hence rational Gorenstein, singularities.Furthermore, it admits a unique projective, symplectic resolution given by (1.1) . We reiterate that irreducibility is due originally to Zheng [Zhe17]. The existence of a nowhere-vanishing2 n -form in the type A case, which follows from having symplectic singularities, was shown in the same paper[Zhe17, Theorem D], while the existence and uniqueness of the symplectic resolution for n = 2 is due toYamagishi [Yam17].Our main tool is to furnish Hilb [ n ] ( C / Γ) with a quiver-theoretic interpretation as a fine quiver modulispace by the process of cornering [CIK18]. More generally, we provide a fine moduli space description ofthe quiver varieties M θ for all non-generic stability parameters that lie in the closure of the cone C + . Ourmethods give conceptual proofs of the geometric properties of Hilb [ n ] ( C / Γ) listed in Theorem 1.1, andallow us to obtain all possible projective factorisations of the morphism (1.1) by universal properties of theresulting fine moduli spaces. Our proofs avoid case-by-case analysis, with the exception of a bound on the imension vector for quiver representations that are stable with respect to a non-generic stability condition;our long case-by-case argument for this statement is given in Appendix A.Quiver varieties with degenerate stability conditions identical to ours were considered before by Nakajimain [Nak09]. We hope to return to the relationship between the results of ibid. , and those of [GNS18] and thecurrent paper, in later work. Acknowledgements.
S.G. is supported by an Aker Scholarship. ´A.Gy. and B.Sz. are supported by EPSRCgrant EP/R045038/1. We thank Gwyn Bellamy, Ben Davison and Hiraku Nakajima for helpful discussions.
Notation.
Let π : X → Y be a projective morphism of schemes over an affine base Y . For a globallygenerated line bundle L on X , write | L | := Proj Y L k ≥ H ( X, L k ) for the (relative) linear series of L , and ϕ | L | : X → | L | for the induced morphism over Y .2. Variation of GIT quotient for quiver varieties
Let Γ ⊂ SL(2 , C ) be a finite subgroup. Let V denote its given two-dimensional representation, definedby this inclusion. Write ρ , ρ , . . . , ρ r for the irreducible representations of Γ, with ρ the trivial one. The McKay graph of Γ has vertex set { , , . . . , r } where vertex i corresponds to the representation ρ i of Γ, andthere are dim Hom Γ ( ρ j , ρ i ⊗ V ) edges between vertices i and j . By the McKay correspondence [McK80], theMcKay graph is an extended Dynkin diagram of ADE type. Add a framing vertex ∞ , together with anedge between vertices ∞ and 0, and let Q be the set of pairs consisting of an edge in this graph and anorientation of the edge. If a is an edge with orientation, we write a ∗ for the same edge with the oppositeorientation. The framed McKay quiver Q has vertex set Q = {∞ , , , . . . , r } and arrow set Q , where foreach oriented edge a ∈ Q we write t( a ) , h( a ) for the tail and head of a respectively.Let C Q denote the path algebra of Q . For i ∈ Q , let e i ∈ C Q denote the idempotent correspondingto the trivial path at vertex i . Let ǫ : Q → {± } be any map such that ǫ ( a ) = ǫ ( a ∗ ) for all a ∈ Q . Thepreprojective algebra Π is the quotient of C Q by the ideal generated by the relation X a ∈ Q ǫ ( a ) aa ∗ . Equivalently, multiplying both sides of this relation by the idempotent e i shows that Π can be presented asthe quotient of C Q by the ideal X h( a )= i ǫ ( a ) aa ∗ | i ∈ Q . (2.1)The preprojective algebra Π does not depend on the choice of the map ǫ [CBH98, Lemma 2.2]. Let R (Γ)denote the representation ring of Γ. Introduce a formal symbol ρ ∞ so that { ρ i | i ∈ Q } provides a Z -basisfor Z Q ∼ = Z ⊕ R (Γ) considered as Z -modules.For a natural number n ≥ v := ( v i ) i ∈ Q := ρ ∞ + X i ≥ n dim( ρ i ) ρ i ∈ Z Q . The group G ( v ) := C × × Q ≤ i ≤ r GL (cid:0) n dim( ρ i ) , C (cid:1) acts on the space Rep( Q, v ) := L a ∈ Q Hom (cid:0) C v t( a ) , C v h( a ) (cid:1) of representations of the quiver Q of dimension vector v by conjugation. The diagonal scalar subgroup actstrivially, and the action of the quotient G := G ( v ) / C × induces a moment map µ : Rep( Q, v ) → g ∗ such that closed point lies in µ − (0) if and only if the corresponding C Q -module satisfies the relations (2.1) of thepreprojective algebra Π. If we write Θ v = { θ : Z Q → Q | θ ( v ) = 0 } , then each character of G is χ θ : G → C × for some integer-valued θ ∈ Θ v , where χ θ ( g ) = Q i ∈ Q det( g i ) − θ i for g ∈ G ( v ).Given a stability parameter θ ∈ Θ v , recall that a Π-module M is θ -stable (respectively semistable) if θ (dim M ) = 0 and for every proper, nonzero submodule N ⊂ M , we have θ (dim N ) > θ (dim N ) ≥ θ -semistable Π-modules M, M ′ are said to be S -equivalent, if they admit filtrations0 = M ⊂ M ⊂ · · · ⊂ M s = M and 0 = M ′ ⊂ M ′ ⊂ · · · ⊂ M ′ s = M ′ such that each M i and each M ′ j is θ -semistable, and s M i =1 M i /M i − ∼ = s M i =1 M ′ i /M ′ i − . Every S -equivalence class has a representative unique up to isomorphism that is a direct sum of θ -stablemodules, the so-called polystable module.Given θ ∈ Θ v , the quiver variety M θ := ( µ − (0) // θ G ) red is the categorical quotient of the locus of χ θ -semistable points of µ − (0) by the action of G . It is the coarsemoduli space of S-equivalence classes of θ -semistable Π-modules of dimension vector v . As indicated, weconsider these GIT quotients with their reduced scheme structure everywhere below. Lemma 2.1.
For all θ ∈ Θ v , the scheme M θ is irreducible and normal, with symplectic singularities.Proof. See Bellamy and Schedler [BS16, Therorem 1.2, Proposition 3.21]. (cid:3)
The set of stability conditions Θ v admits a preorder ≥ , where θ ≥ θ ′ iff every θ -semistable Π-module is θ ′ -semistable. It is well known [DH98, Tha96] that we obtain a wall-and-chamber structure on Θ v , where θ, θ ′ ∈ Θ v lie in the relative interior of the same cone if and only if both θ ≥ θ ′ and θ ′ ≥ θ hold in thispreorder, in which case M θ ∼ = M θ ′ . The interiors of the top-dimensional cones in Θ v are GIT chambers ,while the codimension-one faces of the closure of each GIT chamber are
GIT walls . We say that θ ∈ Θ v is generic with respect to v , if it lies in some GIT chamber; equivalently, θ is generic if every θ -semistableΠ-module is θ -stable. Since v is indivisible, King [Kin94, Proposition 5.3] proves that for generic θ ∈ Θ v ,the quiver variety M θ is the fine moduli space of isomorphism classes of θ -stable Π-modules of dimensionvector v . In this case, the universal family on M θ is a tautological locally-free sheaf R := M i ∈ Q R i together with a C -algebra homomorphism φ : Π → End( R ), where R ∞ is the trivial bundle on M θ and whererank( R i ) = n dim( ρ i ) for i ≥ M θ was investigated recently by the first author withBellamy [BC18]. The following result records a surjectivity statement that will be useful later on. emma 2.2. Let θ, θ ′ ∈ Θ v satisfy θ ≥ θ ′ . Then the morphism π : M θ → M θ ′ obtained by variation of GITquotient is a surjective, projective and birational morphism of varieties over Sym n ( C / Γ) .Proof. If θ is generic and θ ′ = 0, then the morphism M θ → M ∼ = Sym n ( C / Γ) is a projective symplecticresolution [BC18, Theorem 4.5] and the result holds. For the general case, combining [BS16, Lemma 3.22]and [BC18, Lemma 4.4], we get dim M θ = 2 n . This holds for any θ ∈ Θ v , so dim M θ ′ = 2 n . The morphism π : M θ → M θ ′ is projective, so the image Z := π ( M θ ) is closed in M θ ′ . Deform θ if necessary to a genericparameter η such that η ≥ θ . Then the resolution M η → M ∼ = Sym n ( C / Γ) factors through π by variationof GIT quotient, so dim( Z ) = 2 n and hence π is birational onto its image. It follows that Z is an irreduciblecomponent of M θ ′ . However, M θ ′ is irreducible [BS16, Proposition 3.21]; so π is surjective. (cid:3) The GIT wall-and-chamber structure on Θ v was computed explicitly in [BC18, Theorem 4.6]. In thispaper, we focus on the distinguished GIT chamber C + := (cid:8) θ ∈ Θ v | θ ( ρ i ) > i ≥ (cid:9) . (2.2)It is well known that the quiver variety M θ for θ ∈ C + admits a description as an equivariant Hilbert scheme.Recall from the Introduction that n Γ- Hilb( C ) is the scheme parametrising Γ-invariant ideals I ⊳ C [ x, y ]with quotient isomorphic as a representation of Γ to the direct sum of n copies of the regular representationof Γ. Theorem 2.3 ([VV99, Wan99, Kuz07]) . Let Γ n := Γ n ⋊ S n ⊂ Sp(2 n, C ) denote the wreath product of Γ with the symmetric group S n . For θ ∈ C + , there is a commutative diagram n Γ - Hilb( C ) M θ C n / Γ n ∼ = Sym n ( C / Γ) M ∼ π ∼ in which the horizontal arrows are isomorphisms and the vertical arrows are symplectic resolutions. We now study partial resolutions of Sym n ( C / Γ) through which the resolution from Theorem 2.3 factors.The main result of [BC18, Theorem 1.2] implies that for n >
1, the nef cone of n Γ- Hilb( C ) over Sym n ( C / Γ)is isomorphic to the closure C + of the chamber from (2.2). For n = 1, the relation between these two conesis described in [BC18, Proposition 7.11] (see Remark 5.5 for more on the case n = 1). In any case, for n ≥ C + and any GIT parameterfrom the relative interior of that face; then perform variation of GIT quotient as the parameter moves to theorigin in Θ v .Every face of C + is of the form σ J := (cid:8) θ ∈ C + | θ ( ρ j ) > j ∈ J (cid:9) for some (possibly empty) subset J ⊆ { , , . . . , r } . The parameter θ J ∈ C + defined by setting θ J ( ρ i ) = − P j ∈ J n dim( ρ j ) for i = ∞ i ∈ J i ∈ { , , . . . , r } \ J lies in the relative interior of the face σ J . To simplify notation, in the case J = { } we occasionally denote θ := θ { } . roposition 2.4. The face poset of the cone C + can be identified with the poset on the set of quiver varieties M θ J for subsets J ⊆ { , , . . . , r } , where edges in the Hasse diagram of the poset are realised by the surjective,projective and birational morphisms π J,J ′ : M θ J → M θ J ′ .Proof. This is standard for variation of GIT quotient apart from surjectivity and birationality of each π J,J ′ .This was established in Lemma 2.2. (cid:3) Remark 2.5.
When J ′ = ∅ and J = { , . . . , r } , the morphism M θ J → M θ J ′ is the resolution n Γ-Hilb( C ) → Sym n ( C / Γ) from Theorem 2.3. The statement of Proposition 2.4 implies that the paths in the Hasse diagramof the face poset of C + from the unique maximal element to the unique minimal element provide all possibleways in which this resolution can be decomposed via primitive morphisms [Wil92]. Example 2.6.
Consider the case Γ ∼ = µ , corresponding to Dynkin type A , and n = 3. Figure 1 showsa transverse slice of the GIT wall-and-chamber structure inside a specific closed cone F in the space Θ v ofstability parameters. According to [BC18, Theorem 1.2], this decomposition of the cone is isomorphic as afan to the closure of the movable cone of this particular n Γ- Hilb( C ), with its natural subdivision into nefcones of birational models. The open subcone C + corresponds to the ample cone of n Γ- Hilb( C ) itself. InSection 5 we focus on the distinguished ray h θ i in the boundary of F . C + h θ i Figure 1.
Wall-and-chamber structure inside the cone F for Γ ∼ = µ and n = 3We conclude this section with a lemma that identifies the key geometric fact that makes the chamber C + special; our argument depends crucially on this observation. For θ ∈ C + and for any θ ′ ∈ Θ v , we considerthe line bundle L C + ( θ ′ ) := N ≤ i ≤ r det( R i ) θ ′ i on M θ ; the line bundle L J := L C + ( θ J ) will play a special rolein particular. Lemma 2.7.
Let θ ∈ C + . Then (i) for each θ ′ ∈ C + , the line bundle L C + ( θ ′ ) on M θ is globally generated; (ii) for any J ⊆ { , . . . , r } , after multiplying θ J by a positive integer if necessary, the morphism to thelinear series of L J decomposes as the composition of π J and a closed immersion: M θ | L J | . M θ J π J ϕ | LJ | (2.3) roof. Since θ ∈ C + , the tautological bundles R i on the quiver variety M θ are globally generated for i ∈ Q by [CIK18, Corollary 2.4]. Hence L C + ( θ ′ ) is globally generated because θ ′ i ≥ ≤ i ≤ r . In particular,since θ J ∈ C + , the rational map ϕ | L J | is defined everywhere. The line bundle L J induces the morphism π J : M θ → M θ J ⊂ | L J | by [BC18, Theorem 1.2], where we take a positive multiple of θ J if necessary toensure that the polarising ample bundle on M θ J is very ample. This proves the result. (cid:3) Remark 2.8.
We choose a sufficiently high multiple of θ (and the same high multiple of each θ J ) to ensurethat the polarising ample line bundle on M θ J is very ample for every subset J ⊆ { , . . . , r } .3. Cornering the preprojective algebra
In general, the quiver variety M θ J is the coarse moduli space for S-equivalence classes of θ J -semistableΠ-modules of dimension vector v . However, in the special case J = { , . . . , r } it is the fine moduli space ofisomorphism classes of θ J -stable Π-modules. We now introduce an alternative, fine moduli space constructionfor each M θ J by defining an algebra Π J obtained from Π by the process of ‘cornering’.For any subset J ⊆ { , , . . . , r } , define the idempotent e J := e ∞ + P j ∈ J e j and consider the subalgebraΠ J := e J Π e J of Π spanned over C by paths in Q whose tail and head both lie in the set {∞} ∪ J . The process of passingfrom Π to Π J is called cornering ; see [CIK18, Remark 3.1]. Then v J := ρ ∞ + X j ∈ J n dim( ρ j ) ρ j ∈ Z ⊕ Z J is a dimension vector for Π J -modules, and we consider the stability condition η J : Z ⊕ Z J → Q given by η J ( ρ i ) = ( − P j ∈ J n dim( ρ j ) for i = ∞ i ∈ J It follows directly from the definition that a Π J -module N of dimension vector v J is η J -stable if and only ifthere exists a surjective Π J -module homomorphism Π J e ∞ → N .The vector v J is indivisible and η J is a generic stability condition for Π J -modules, so the construction ofKing [Kin94, Proposition 5.3] defines the fine moduli space M (Π J ) of η J -stable Π J -modules of dimensionvector v J . Let T J := L i ∈{∞}∪ J T i denote the tautological bundle on M (Π J ), where T ∞ is the trivial bundleand T j has rank n dim( ρ j ) for j ∈ J . The line bundle L J := O j ∈ J det( T j )is the polarising ample bundle on M (Π J ) given by the GIT construction. Lemma 3.1.
Let θ ∈ C + , and let J ⊆ { , . . . , r } be any subset. There is a universal morphism τ J : M θ → M (Π J ) (3.1) satisfying τ ∗ J ( T i ) ∼ = R i for i ∈ {∞} ∪ J .Proof. In light of the universal property of M (Π J ), it suffices to show that the locally-free sheaf R J := M i ∈{∞}∪ J R i f rank 1 + P j ∈ J n dim( ρ j ) on the quiver variety M θ is a flat family of η J -stable Π J -modules of dimensionvector v J . Multiplying the tautological C -algebra homomorphism φ : Π → End( R ) on the left and right bythe idempotent e J determines a C -algebra homomorphism Π J → End( R J ) which makes R J into a flat familyof Π J -modules of dimension vector v J . To establish stability, write L i ∈ I R i,y for the fibre of R over a closedpoint y ∈ M θ . The fact that L i ∈ I R i,y is θ -stable is equivalent to the existence of a surjective Π-modulehomomorphism Π e ∞ → L i ∈ I R i,y . Applying e J on the left produces a surjective Π J -module homomorphismΠ J e ∞ → L i ∈{∞}∪ J R i,y which in turn is equivalent to η J -stability of the fibre L i ∈{∞}∪ J R i,y of R J over y ∈ M θ . In particular, R J is a flat family of η J -stable Π J -modules of dimension vector v J . (cid:3) Remarks 3.2. (1) An alternative proof of Lemma 3.1 uses the fact that the tautological bundles R i on M θ are globally generated for i ∈ I by [CIK18, Corollary 2.4], in which case one can adapt the proofof [CIK18, Proposition 2.3] to deduce that R J is a flat family of η J -stable Π J -modules of dimensionvector v J . In particular, global generation is the key feature in Lemma 3.1, just as in the proof ofLemma 2.7. This is not a coincidence; see Theorem 3.7.(2) Building on Remark 2.8, we now take an even higher multiple of θ if necessary (and the same highmultiple of each η J and each θ J ) to ensure that the polarising ample line bundles on M (Π J ) and on M θ J are very ample for all relevant J ⊆ { , . . . , r } .From now on in this section, we assume that J = ∅ ; see Remarks 3.8 (3). Lemma 3.3.
Let θ ∈ C + and assume J ⊆ { , . . . , r } is nonempty. There is a commutative diagram M θ M θ J M (Π J ) | L J | |L J | π J τ J ϕ | LJ | ϕ |L J | ψ (3.2) of schemes over Sym n ( C / Γ) , where ψ is an isomorphism.Proof. The commutative triangle on the left of (3.2) was constructed in Lemma 2.7. For the quadrilateralon the right, our choice of η J ensures that the polarising line bundle L J on M (Π J ) is very ample, so themorphism ϕ |L J | is well-defined. Since pullback commutes with tensor operations on the T i , the isomorphisms τ ∗ J ( T i ) ∼ = R i for i ∈ J imply that L J = τ ∗ J ( L J ). If O |L J | (1) denotes the polarising ample bundle on |L J | ,then ( ϕ |L J | ◦ τ J ) ∗ ( O |L J | (1)) = τ ∗ J ( L J ) = L J = ϕ ∗| L J | (cid:0) O | L J | (1) (cid:1) (3.3)on M θ . The morphism to a complete linear series is unique up to an automorphism of the linear series, sothere is an isomorphism ψ : | L J | → |L J | such that ϕ |L J | ◦ τ J = ψ ◦ ϕ | L J | as required.It remains to show that (3.2) is a diagram of schemes over Sym n ( C / Γ). The Leray spectral sequence forthe resolution π : M θ → M ∼ = Sym n ( C / Γ) gives H ( O M θ ) ∼ = H ( O M ) ∼ = ( C [ V ] Γ ) S n because Sym n ( C / Γ)has rational singularities. It follows that π = ϕ |O M θ | , i.e. π is the structure morphism of M θ as a varietyover Sym n ( C / Γ). Repeating the argument from (3.3), with the roles of L J , L J and O |L J | (1) played insteadby the trivial bundles on M θ , M (Π J ) and Sym n ( C / Γ) respectively, shows that M (Π J ) is a scheme overSym n ( C / Γ). It follows that (3.2) is a diagram of schemes over Sym n ( C / Γ). (cid:3) ur goal for the rest of this section is to add a morphism ι J : M θ J → M (Π J ) to diagram (3.2) and toshow that ι J is an isomorphism on the underlying reduced schemes. Consider the functorsΠ -mod Π J -mod j ∗ j ! defined by j ∗ ( − ) := e J Π ⊗ Π ( − ) and j ! ( − ) := Π e J ⊗ Π J ( − ). These are two of the six functors in a recollementof the module category Π -mod [FP04]. In particular, j ∗ is exact, j ∗ j ! is the identity functor, and for anyΠ J -module N , the Π-module j ! ( N ) provides the maximal extension by Π / (Π e J Π)-modules; see [CIK18,Equation (3.4)].
Lemma 3.4.
Let N be an η J -stable Π J -module of dimension vector v J . The Π -module j ! ( N ) is θ J -semistable.Proof. Since N is η J -stable, there is a surjective Π J -module homomorphism Π J e ∞ → N . The proof of[CIK18, Lemma 3.6] applies verbatim to construct a surjective Π-module homomorphism Π e ∞ → j ! ( N ) and,moreover, to show that the finite dimensional Π-module j ! ( N ) satisfies dim i j ! ( N ) = dim i N for i ∈ {∞} ∪ J .Recall that θ J ( ρ i ) = 0 for i
6∈ {∞} ∪ J , so θ J (cid:0) j ! ( N ) (cid:1) = θ J X i ∈{∞}∪ J dim i ( j ! ( N )) ρ i = η J X i ∈{∞}∪ J dim i ( N ) ρ i = η J ( N ) = 0 . Now let M ⊂ j ! ( N ) be a proper submodule. If dim ∞ M = 1, then surjectivity of the map Π e ∞ → j ! ( N ) gives M = j ! ( N ) which is absurd, so dim ∞ M = 0. But θ J ( ρ i ) ≥ i = ∞ , so θ J ( M ) ≥ (cid:3) Lemma 3.5.
Let N be an η J -stable Π J -module of dimension vector v J . Then there exists a θ J -semistable Π -module M such that j ∗ M ∼ = N and dim i M ≤ n dim( ρ i ) for all i
6∈ {∞} ∪ J .Proof. By Lemma 3.4, j ! ( N ) is θ J -semistable. If dim i j ! ( N ) ≤ n dim( ρ i ) for i
6∈ {∞} ∪ J , then we cansimply set M := j ! ( N ), as j ∗ j ! is the identity. Otherwise, consider the θ J -polystable module L λ M λ thatis S -equivalent to j ! ( N ). Let M λ ∞ denote the unique summand satisfying dim ∞ M λ ∞ = 1. Since M λ ∞ is by construction a θ J -stable Π-module, it follows that dim i M λ ∞ = n dim( ρ i ) for all i ∈ J , and hencedim i M λ = 0 for all λ = λ ∞ and all i ∈ {∞} ∪ J . For each index λ and for all i ∈ {∞} ∪ J , we havedim i j ∗ M λ = dim e i (cid:0) e J Π ⊗ Π ( M λ ) (cid:1) = dim e i Π ⊗ Π M λ = dim i M λ . It follows that dim i j ∗ M λ = 0 for all λ = λ ∞ and i ∈ {∞} ∪ J , and hence j ∗ M λ = 0 for λ = λ ∞ .We claim that j ∗ M λ ∞ is isomorphic to N . Indeed, the Π-module j ! ( N ) is θ J -semistable by Lemma 3.4,and the θ J -stable Π-modules M λ are by construction the factors in the composition series of j ! ( N ) in thecategory of θ J -semistable Π-modules. It follows from exactness of j ∗ that the Π J -modules j ∗ M λ are thefactors in the composition series of j ∗ j ! ( N ) ∼ = N in the category of η J -semistable Π J -modules. But j ∗ M λ = 0for λ = λ ∞ , so the only nonzero factor of the composition series is j ∗ M λ ∞ . It follows that j ∗ M λ ∞ ∼ = N ,because the factor j ∗ M λ ∞ can only appear once in the composition series.As a result, the θ J -stable Π-module M λ ∞ satisfies j ∗ M λ ∞ ∼ = N and dim i M λ ∞ = n dim( ρ i ) for all i ∈ J .Therefore M λ ∞ is the required Π J -module as long as dim i M λ ∞ ≤ n dim( ρ i ) for i
6∈ {∞} ∪ J . We establishthis key inequality in Appendix A. (cid:3) Remark 3.6.
The modules M λ for λ = λ ∞ in the proof of Lemma 3.5 are in fact all 1-dimensional vertexsimples. To see this, note that removing any nonempty set of vertices and their incident edges from an xtended Dynkin diagram gives a diagram in which every connected component is Dynkin of finite type.Thus removing the vertices {∞} ∪ J and all incident edges from the framed extended diagram leaves us witha collection of Dynkin diagrams of finite type. Choose λ = λ ∞ . Since dim j M λ = 0 for all j ∈ {∞} ∪ J , M λ isa simple module of the preprojective algebra of a quiver of finite type. But such modules are one-dimensionalby [ST11, Lemma 2.2]. Theorem 3.7.
For any nonempty J ⊆ { , . . . , r } , there is a commutative diagram of morphisms M θ M θ J M (Π J ) , π J τ J ι J (3.4) where ι J is an isomorphism of the underlying reduced schemes. In particular, M (Π J ) is irreducible, and itsunderlying reduced scheme is normal and has symplectic singularities.Proof. Let σ J : M θ J → |L J | be the composition of the isomorphism ψ of Lemma 3.3 with the closed immersion M θ J ֒ → | L J | from diagram (3.2). Since σ J is a closed immersion, it identifies M θ J with Im( σ J ). Surjectivityof π J and commutativity of diagram (3.2) then imply that M θ J is isomorphic to the subscheme Im( σ J ◦ π J ) =Im( ϕ |L J | ◦ τ J ) of |L J | . Since L J is the polarising very ample line bundle on the GIT quotient M (Π J ), theclosed immersion ϕ |L J | induces an isomorphism λ J : Im( ϕ |L J | ) → M (Π J ). The morphism ι J := λ J ◦ σ J : M θ J → M (Π J )is therefore a closed immersion. Note that ι J ◦ π J = λ J ◦ σ J ◦ π J = λ J ◦ ϕ |L J | ◦ τ J = τ J , so diagram (3.4) commutes. In order to prove that ι J is an isomorphism of the underlying reduced schemes,it suffices to show that ι J is surjective on closed points.Consider a closed point [ N ] ∈ M (Π J ), where N is an η J -stable Π J -module of dimension vector v J . Let M be the θ J -semistable Π-module from Lemma 3.5. For i
6∈ {∞} ∪ J , define m i := n dim( ρ i ) − dim i M ≥ S i := C e i denote the vertex simple Π-module at vertex i ∈ Q . The Π-module M := M ⊕ M i ∈{ ,...,r }\ J S ⊕ m i i is θ J -semistable of dimension vector v by construction, and it satisfies j ∗ ( M ) = j ∗ ( M ) = N . Write [ M ] ∈ M θ J for the corresponding closed point, and let f M be any θ -stable Π-module of dimension vector v such thatthe closed point [ f M ] ∈ M θ satisfies π J ([ f M ]) = [ M ] ∈ M θ J . Then j ∗ ( f M ) = j ∗ ( M ) = N , hence τ J ([ f M ]) = [ N ],and commutativity of diagram (3.4) gives that ι J ([ M ]) = ( ι J ◦ π J ) (cid:0) [ f M ] (cid:1) = τ J (cid:0) [ f M ] (cid:1) = [ N ] , so ι J is indeed surjective. The final statement of Theorem 3.7 follows from Lemma 2.1 and Lemma 3.3. (cid:3) Remarks 3.8. (1) If J = { , . . . , r } , then the stability parameter θ J lies in the boundary of the GITchamber C + , so M θ J does not admit a universal family of θ J -semistable Π-modules of dimensionvector v . However, the fine moduli space M (Π J ) does carry a universal family T J of η J -stableΠ J -modules of dimension vector v J , and hence under the isomorphism of Theorem 3.7, the bundle ι ∗ J ( T J ) on M θ J pulls back along π J to the summand L i ∈{∞}∪ J R i of the tautological bundle on M θ .
2) In the course of the proof of Theorem 3.7, we deduce directly that τ J is surjective on closed points.(3) For J = ∅ , we have M θ J ∼ = Sym n ( C / Γ). However, M (Π J ) is an affine scheme that does not dependon n , so M θ J = M (Π J ) when J = ∅ .4. Identifying the posets for the coarse and fine moduli problems
We now establish that the morphisms ι J : M θ J → M (Π J ) from Theorem 3.7 are compatible with themorphisms π J,J ′ : M θ J → M θ J ′ that feature in the poset introduced in Proposition 2.4. Lemma 4.1.
For nonempty subsets J ′ ⊂ J ⊂ { , , . . . , r } , there is a commutative diagram M θ J M (Π J ) M θ J ′ M (Π J ′ ) ι J π J,J ′ τ J,J ′ ι J ′ (4.1) in which the horizontal arrows are isomorphisms on the underlying reduced schemes and the vertical arrowsare surjective, projective, birational morphisms.Proof. The subbundle L i ∈{∞}∪ J ′ T i of the tautological bundle T J on M (Π J ) is a flat family of η J ′ -stableΠ J ′ -modules of dimension vector v J ′ , so there is a universal morphism τ J,J ′ : M (Π J ) −→ M (Π J ′ )satisfying τ ∗ J,J ′ ( T ′ i ) = T i for i ∈ {∞} ∪ J ′ , where L i ∈{∞}∪ J ′ T ′ i is the tautological bundle on M (Π J ′ ). Now (cid:0) τ J,J ′ ◦ τ J (cid:1) ∗ ( T ′ i ) = τ ∗ J ( T i ) = R i = τ ∗ J ′ ( T ′ i )for all i ∈ {∞} ∪ J ′ , and since this property characterises the morphism τ J ′ , we have a commutative diagram M θ M (Π J ) M (Π J ′ ) . τ J τ J τ J,J ′ (4.2)Proposition 2.4 gives a similar commutative diagram expressing the identity π J,J ′ ◦ π J = π J ′ for morphismsbetween quiver varieties, while Theorem 3.7 establishes the identities ι J ◦ π J = τ J and ι J ′ ◦ π J ′ = τ J ′ . Takentogether, these identities show that the maps in all four triangles in the following pyramid diagram commute: M θ M θ J M (Π J ) M θ J ′ M (Π J ′ ) . π J τ J ι J π J,J ′ τ J,J ′ ι J ′ π J ′ τ J ′ (4.3)To show that the morphisms around the pyramid’s square base commute, choose for any closed point x ∈ M θ J a lift y ∈ π − J ( x ) ⊂ M θ . Commutativity of the triangles in the diagram gives (cid:0) ι J ′ ◦ π J,J ′ (cid:1) ( x ) = ι J ′ ( π J ′ ( y ) (cid:1) = τ J ′ ( y ) = τ J,J ′ ( τ J ( y ) (cid:1) = (cid:0) τ J,J ′ ◦ ι J (cid:1) ( x ) , and since x ∈ M θ was arbitrary and π J is surjective, we have that ι J ′ ◦ π J,J ′ = τ J,J ′ ◦ ι J as required. (cid:3) e deduce the following. Proposition 4.2.
The face poset of the cone C + can be identified with the poset on the set of fine modulispaces M (Π J ) for nonempty subsets J ⊆ { , . . . , r } together with C n / Γ n , where edges in the Hasse diagramof the poset indicating inequalities M (Π J ) > M (Π J ′ ) and M (Π J ) > C n / Γ n are realised by the universalmorphisms τ J,J ′ and the structure morphisms ϕ |O M (Π J ) | respectively. Punctual Hilbert schemes for Kleinian singularities
In this section, we specialise to the case J = { } and study the algebra Π J , before establishing the linkbetween the fine moduli space M (Π J ) and the Hilbert scheme of n points on C / Γ. It will be convenient towrite dimension vectors of Π J -modules as pairs ( v ∞ , v ) in this case.Like the algebra Π, the algebra Π J can also be presented as a quiver algebra with relations. The relationsappear to be fairly complicated, but for J = { } there is a simple presentation of its quotient algebra Π J / ( b ∗ ),where b ∗ is the class of a particular arrow in Q . This will turn out to be sufficient for our purposes. To spellthis out, write b for the generator corresponding to the arrow going from ∞ to 0 in the path algebra C Q of the framed McKay quiver, and b ∗ for the opposite arrow from 0 to ∞ . Through slight abuse of notation,we use the same symbols for the respective images of these elements in the preprojective algebra Π and itssubalgebra Π J .On the other hand, define a new quiver Q ′ with vertex set Q ′ = {∞ , } and arrow set Q ′ comprising onearrow α from ∞ to 0, and loops α , α , α at vertex 0 as shown in Figure 2. ∞ α α α α Figure 2.
The quiver Q ′ used in the presentation of a quotient of Π J for J = { } .To state the key result, recall that the quotient singularity C / Γ is famously a hypersurface in C , with theΓ-invariant subring C [ x, y ] Γ having three minimal generators z , z , z satisfying one relation f ( z , z , z ) = 0. Lemma 5.1.
For J = { } , let b ∗ ∈ Π J denote the class of the arrow in Q from to ∞ . The algebra Π J / ( b ∗ ) is isomorphic to the quotient of C Q ′ by the two-sided ideal K = (cid:0) f ( α , α , α ) , α α − α α , α α − α α , α α − α α (cid:1) , (5.1) where f ∈ C [ z , z , z ] is the defining equation of the hypersurface C / Γ ⊆ Spec C [ z , z , z ] .Proof. The unframed McKay quiver Q Γ is the complete subquiver of Q on the vertex set { , , . . . , r } . WriteΠ Γ for the corresponding preprojective algebra. The natural epimorphism C Q → C Q Γ that kills any pathin Q touching vertex ∞ induces a short exact sequence0 −→ ( bb ∗ ) −→ e Π e φ −→ e (Π Γ ) e −→ . Recall from [CBH98, Theorem 0.1] the isomorphism e (Π Γ ) e ∼ = C [ x, y ] Γ . We may therefore associate toeach of the three minimal C -algebra generators of C [ x, y ] Γ an element β i ∈ e (Π Γ ) e for 1 ≤ i ≤
3. For each i , choose a linear combination of cycles in C Q Γ mapping to β i in e (Π Γ ) e . The same linear combinationof cycles can be thought of as an element of C Q , and its image e β i ∈ e Π e ⊂ Π J defines a lift of β i withrespect to φ .For each i , let b i be the image of e β i in Π J / ( b ∗ ). Mapping β i to b i and mapping e to the class of thetrivial path at vertex 0 in Π J / ( b ∗ ) defines a map t in the following commutative diagram, where both rowsare exact; a simple diagram chase shows that t is indeed well-defined.0 ( bb ∗ ) e Π e e (Π Γ ) e
00 ( b ∗ ) Π J Π J / ( b ∗ ) 0 φ tp (5.2)In particular, the elements b i ∈ Π J / ( b ∗ ) commute. Furthermore, since f ( β , β , β ) = 0, we have0 = t ( f ( β , β , β )) = f (cid:0) t ( β ) , t ( β ) , t ( β ) (cid:1) = f ( b , b , b ) . We next show that t is injective. Let γ ∈ ker( t ). For any element e γ ∈ e Π e such that φ ( e γ ) = γ , thismeans that e γ ∈ ( b ∗ ) as an element of Π J . However, since ( b ∗ ) ∩ e Π e = ( bb ∗ ), it must be the case that e γ ∈ ( bb ∗ ). Therefore γ = 0, and t is injective.Since Π J is a quotient of the algebra e J C Qe J , it is generated by e ∞ , b, b ∗ and the class of every cycle in Q starting and ending at the vertex 0. Inside this, the subalgebra generated by classes of cycles starting andending at 0 equals e Π e . By commutativity of diagram (5.2), the image of this subalgebra under p equalsthe image Im( t ) of t . It is then clear that Π J / ( b ∗ ) is generated as a C -algebra by e ∞ , b and Im( t ) ≃ C [ x, y ] Γ ,in other words by the elements e ∞ , b, e , b , b , b .Now we define a map ψ : C Q ′ /K −→ Π J / ( b ∗ )by sending the classes of the trivial paths at vertices ∞ and 0 to e ∞ and e respectively, and by setting ψ ( α ) = b and ψ ( α i ) = b i for 1 ≤ i ≤
3. The above discussion and the definition of K shows that ψ is awell-defined surjective homomorphism. To see that ψ is injective, we note that it maps the C -subalgebraΛ ⊂ C Q ′ /K generated by ( e , α , α , α ) bijectively onto Im( t ). In addition, ψ is injective when restrictedto the two-sided ideal ( e ∞ , α ). Finally, we must show that if ζ = 0 is an element of ( e ∞ , α ), then ψ ( ζ ) doesnot lie in Im( t ). Since e ∞ · e = 0, it suffices to consider such a ζ of the form ξα for some nonzero ξ ∈ Λ.Therefore ψ ( ζ ) = cb for some c ∈ Im( t ). But b does not start at 0, so cb Im( t ). It follows that no nonzerolinear combination of an element of the ideal ( e ∞ , α ) with an element of Λ can be mapped to 0 by ψ . Butevery element of C Q ′ /K is of this form, so ψ is injective. Thus ψ is an isomorphism as required. (cid:3) Note that a Π J -module M for which b ∗ acts as 0 is the same thing as a Π J / ( b ∗ )-module, and thereforethe same as a C Q ′ /K -module. Proposition 5.2.
For the subset J = { } , there is a morphism of schemes ω n : Hilb [ n ] ( C / Γ) → M (Π J ) over Sym n ( C / Γ) , which is an isomorphism of the underlying reduced subschemes.Proof. We begin by constructing the morphism of schemes ω n . Let T denote the tautological rank n bundleon Hilb [ n ] ( C / Γ), and write O for the trivial bundle. In light of the universal property of M (Π J ), it suffices o show that O ⊕ T is a flat family of η J -stable Π J -modules of dimension vector v J = (1 , n ) on Hilb [ n ] ( C / Γ).Since
O ⊕ T is a flat family of C -vector spaces, it suffices to study the fibres over closed points. A pointof Hilb [ n ] ( C / Γ) corresponds to a codimension n ideal I ⊳ C [ x, y ] Γ ∼ = C [ z , z , z ] / ( f ). The quotient vectorspace C [ x, y ] Γ /I is of dimension n , it carries the action of commuting arrows α , α , α satisfying the relation f , and has a distinguished generator [1] ∈ C [ x, y ] Γ /I , which can be thought of as the image of a map w froma one-dimensional vector space. Lemma 5.1 now shows that we get the data of a Π J -module of dimensionvector (1 , n ) for which the map w ∗ representing the class b ∗ ∈ Π J is the zero map. This module is moreovercyclic with generator at vertex ∞ , so it is η J -stable as required. Since the bundle O ⊕ T inducing ω n has O as a summand, and since the trivial bundle on any scheme induces the structure morphism, we see that ω n commutes with the structure morphisms to Sym n ( C / Γ).Next we claim that conversely, for any closed point of M (Π J ) for J = { } , the corresponding η J -stableΠ J -representation N has w ∗ = 0; here w ∗ is the map representing the class b ∗ ∈ Π J . To show this, considerthe morphism τ J : M θ → M (Π J ) constructed in Lemma 3.1, which is surjective by Theorem 3.7. Lift therepresentation N to a θ -stable representation ˜ N of Π. From the description of the morphism τ J in Lemma 3.1it is clear that the map w ∗ for the representation N between the vector spaces at vertices 0 and ∞ is simplythe restriction of the same map in the representation ˜ N . On the other hand, this latter representation canequivalently be thought of, using the isomorphism M θ ∼ = n Γ- Hilb( C ), as a Γ-invariant ideal I of C [ x, y ]. Inthis language, the restriction to the 0 vertex is the Γ-invariant part of the quotient C [ x, y ] /I , and it is wellknown that the induced map w ∗ to the one-dimensional vector space at the vertex ∞ vanishes in this case.Finally, we construct an inverse to ω n on M (Π J ) red . Given a finitely generated C -algebra A , a Spec A -valued point of M (Π J ) is given by the data of two finite flat A -modules M ∞ and M of ranks 1 and n respectively (computed over the localisation of A at any prime ideal, over which finite flat modules are free),and A -module homomorphisms( w, w ∗ ) ∈ Hom A ( M ∞ , M ) ⊕ Hom A ( M , M ∞ ) , { w , . . . , w i ′ } ⊂ End A ( M )for some integer i ′ , satisfying the relations defining Π J . Moreover, for every maximal ideal of A , restrictingto the corresponding closed point of Spec A gives an η J -stable Π J -module. By the argument of the previousparagraph, we must have that the arrow w ∗ becomes zero when restricted to all closed points of Spec A . If A is reduced, this implies w ∗ = 0. Using Lemma 5.1 again, we can now reverse the construction of the firstparagraph, and get a Spec A -valued point of Hilb [ n ] ( C / Γ) for reduced rings A . We thus obtain a map ofschemes M (Π J ) red → Hilb [ n ] ( C / Γ) which is by construction the inverse of ω n when we restrict to reducedclosed subschemes on both sides. (cid:3) We deduce Theorem 1.1 announced in the Introduction.
Corollary 5.3.
For any n ≥ , the reduced scheme underlying Hilb [ n ] ( C / Γ) is isomorphic to the quivervariety M θ for the parameter θ = ( − n, , , . . . , (compare Figure 1). In particular, Hilb [ n ] ( C / Γ) red is a normal, irreducible scheme over C n / Γ n with symplectic singularities that admits a unique projectivesymplectic resolution, namely the morphism n Γ - Hilb( C ) → Hilb [ n ] ( C / Γ) red that sends an ideal I in C [ x, y ] to the ideal I ∩ C [ x, y ] Γ . roof. The first statement follows from Theorem 3.7 and Proposition 5.2, while the geometric properties ofHilb [ n ] ( C / Γ) red are all inherited from its manifestation as M θ via Lemma 2.1.Next we prove the statement about the resolution. In the notation of [BC18, Theorem 1.2], the extremalray ρ ⊥ ∩ · · · ∩ ρ ⊥ r of the cone F that contains θ = ( − n, , , . . . ,
0) lies in the closure of precisely one chamber,namely the chamber C + . Under the isomorphism L F from ibid. , it follows that there is exactly one projectivesymplectic resolution of M θ , namely the fine moduli space M θ for θ ∈ C + . By Theorem 2.3, this resolutionis indeed M θ ∼ = n Γ- Hilb( C ).The last statement of the Corollary was essentially already demonstrated in the proof of Proposition 5.2above. Indeed, there we noted that the map τ J , in the language of ideals, takes a Γ-invariant ideal I of C [ x, y ], and restricts the corresponding representation to the 0 vertex as the Γ-invariant part of the quotient C [ x, y ] /I . Now we conclude using the evident isomorphism ( C [ x, y ] /I ) Γ ∼ = C [ x, y ] Γ / C [ x, y ] Γ ∩ I . (cid:3) Remark 5.4. (1) Irreducibility of Hilb [ n ] ( C / Γ) was first established by Zheng [Zhe17] through thestudy of maximal Cohen–Macaulay modules on Kleinian singularities using a case-by-case analysisfollowing the ADE classification.(2) Uniqueness of the symplectic resolution of Hilb [ n ] ( C / Γ) was previously known in the special case n = 2 by the work of Yamagishi [Yam17, Proposition 2.10].(3) Our approach does not shed light on whether Hilb [ n ] ( C / Γ) is reduced in its natural scheme structure,coming from its moduli space interpretation.
Remark 5.5.
For n = 1, the statement of Theorem 1.1 is well known because Hilb [1] ( C / Γ) ∼ = C / Γ, whilethe statement of Theorem 3.7 is a framed version of [CIK18, Theorem 1.2] for Γ ⊂ SL(2 , C ). Nevertheless,the approach of the current paper is valid for n = 1 and shows in particular that M θ J ∼ = C / Γ for J = { } .In fact, this result follows from [BC18, Proposition 7.11]. Indeed, ibid . constructs a surjective linear map L C + : Θ v → N ( S/ ( C / Γ)) with kernel equal to the subspace spanned by ( − , , , . . . , L C + ( C + )is the ample cone of S over C / Γ. Since θ J = ( − , , , . . . ,
0) for J = { } and n = 1, it follows that M θ J ∼ = C / Γ in that case. In addition, this explicit description of the kernel of L C + for n = 1 shows thatthe morphisms π J,J ′ and τ J,J ′ from Propositions 2.4 and 4.2 are isomorphisms if and only if J ′ \ J = { } . Appendix A. Bounding the dimension vectors of θ J -stable modules A.1.
The key statement.
We use the term ‘diagram’ to mean ‘framed extended Dynkin diagram’, anduse the notation A i , D i , E i for the framed extended versions of these Dynkin diagrams. A module M of thepreprojective algebra Π of the appropriate type naturally determines a representation V of the correspondingquiver Q that satisfies the preprojective relations; we will call these simply ’quiver representations’ below.The notion of θ J -stability for M defines a notion of θ J -stability for V .For i ∈ Q = {∞ , , , . . . , r } we write v i := dim i V , and for 0 ≤ i ≤ r we write δ i := dim( ρ i ), so that theregular representation δ = P ≤ i ≤ r δ i ρ i coincides with the minimal imaginary root of the affine Lie algebraassociated to the extended Dynkin diagram.The goal of this appendix is to prove the following result, which we require in the proof of Lemma 3.5. Proposition A.1.
Let J ⊆ { , , . . . , r } be a nonempty subset. Assume that V is a θ J -stable quiver rep-resentation with v ∞ = 1 and v i = nδ i for i ∈ J and some fixed natural number n . Then v j ≤ nδ j for j J ∪ {∞} . e argue by contradiction, splitting the proof into several parts. The basic idea is as follows. First, ifthe inequality v i ≤ nδ i holds for a vertex i but not its neighbour j , we deduce a basic inequality (A.1) (seeLemma A.3(1)). We then show that this inequality can be ‘pushed along’ the branches of the diagram (seeLemma A.3(2)). If the diagram branches at a trivalent vertex, then we push the inequality further along atleast one branch (see Lemma A.4). This leads either to a contradiction or to strong constraints on dim V .Together, these results settle several cases of the proposition. In particular, Lemma A.7 suffices to proveevery case with 0 ∈ J , except if the diagram is A or D . This is all that is required to prove the case ofprimary interest, namely when J = { } . Some remaining cases require arguments directly depending on thediagram structure.Our main tool for deriving a contradiction is the following estimate, the proof of which is inspired by aresult of Crawley-Boevey [CB01, Lemma 7.2]. This inequality is the only consequence of θ J -stability thatwe use in the subsequent numerical argument. Proposition A.2.
Let V be a θ J -stable quiver representation. If i J , then v i ≤ P { a ∈ Q | h( a )= i } v t( a ) .Proof. Define V ⊕ := M a ∈ Q , h( a )= i V t( a ) . The maps in V determined by arrows with tail and head at vertex i combine to define maps f : V i → V ⊕ and g : V ⊕ → V i satisfying g ◦ f = 0. If ker( f ) = 0, then V admits a nonzero subrepresentation W suchthat W i = ker( f ) and W j = 0 for j = i . But then W corresponds to a Π-submodule of M supported atvertex i . This submodule would be θ J -semistable, which contradicts the θ J -stability of V . Thus f is injective.Similarly, if Im( g ) ( V i , then V admits a subrepresentation U such that U i = Im( g ) and U j = V j for j = i .Then U is θ J -semistable, which again contradicts the θ J -stability of V . So g is surjective. It follows that thecomplex 0 −→ V i f −→ V ⊕ g −→ V i −→ V ⊕ , so dim V ⊕ ≥ V i . (cid:3) Proof of Proposition A.1.
From now on, assume that V is a θ J -stable quiver representation with v ∞ = 1and v i = nδ i for i ∈ J . We split the proof into several sections for better readibility.A.2. Proof in the case ∈ J , excluding types A and D . The inequality (A.1) below is the basicinequality that we will ‘push around’ the diagram.
Lemma A.3. (1) Let i, i − be adjacent vertices of the diagram. If v i > nδ i and v i − ≤ nδ i − , then δ i − v i > δ i v i − . (A.1) (2) Suppose the vertex i J is bivalent, and neither of its neightbours is ∞ :. . . i − i i +1 . . . (A.2) Then δ i − v i > δ i v i − implies δ i v i +1 > δ i +1 v i . If in addition v i > nδ i , then v i +1 > nδ i +1 . roof. (1) is immediate. Since i and ∞ are not neighbours, 2 δ i = δ i − + δ i +1 holds. (2) follows by combiningthis equality withthe assumed inequality δ i − v i > δ i v i − and 2 v i ≤ v i − + v i +1 coming from Proposition A.2. The laststatement is again immediate. (cid:3) Lemma A.4.
Suppose that the diagram has a trivalent vertex i J , not adjacent to the vertex ∞ :. . . i − i j . . . k ... (A.3) and assume that δ i − v i > δ i v i − .(1) At least one of the inequalities δ j v i < δ i v j and δ k v i < δ i v k must hold.Suppose now that v i > nδ i , that δ j v i < δ i v j holds, and furthermore that the branch starting at j does notbranch further. Then(2) the branch starting at j does not contain any vertices in J , and(3) the same branch must terminate at the framing vertex ∞ , and in this case δ i v j = δ j v i + 1 . Remark A.5.
The only framed extended Dynkin diagrams where a trivalent vertex is adjacent to theframing vertex are of type A i for i >
1. We handle the case of such a vertex not being in J in Lemma A.8. Proof.
For (1), combining 2 δ i = δ i − + δ j + δ k with 2 v i ≤ v i − + v k + v j and δ i − v i > δ i v i − leads to δ j v i + δ k v i < δ i v j + δ i v k which implies the result. For (2) and (3), we denote the vertices as. . . i j . . . j + l − j + l ... (A.4)if the branch does not contain the framing vertex, or. . . i j . . . j + l − j + l ∞ ... (A.5)if it does. To simplify notation, we take j − i in the following argument. One of the following mustoccur. • The branch contains another vertex in J . Suppose that j ′ is the node with smallest index on thebranch such that j ′ = i and j ′ ∈ J . Lemma A.3(2) gives δ j ′ − v j ′ > δ j ′ v j ′ − and v j ′ > nδ j ′ ,contradicting j ′ ∈ J . • The branch contains no vertices in J ∪ ∞ . Repeated applications of Lemma A.3(2) show that δ j + l − v j + l > δ j + l v j + l − . However, since 2 δ j + l = δ j + l − , this implies 2 v j + l > v j + l − , contradictingProposition A.2. The branch contains no vertices of J , and terminates at ∞ . We prove a slightly stronger statement, namely that for any vertex m = ∞ on the branch, wehave δ m − v m = δ m v m − + 1. We proceed by induction on the number of edges that lie between ∞ and m . For the base case m = j + l , note that δ j + l − v j + l > δ j + l v j + l − implies 2 v j + l > v j + l − .However, since 2 v j + l ≤ v j + l − + 1 by Proposition A.2, we must have 2 v j + l = v j + l − + 1. If there ismore than one edge between ∞ and m , then the induction hypothesis gives δ m v m +1 = δ m +1 v m + 1.Combining this with 2 v m ≤ v m − + v m +1 from Lemma A.2 and 2 δ m = δ m +1 + δ m − shows that δ m − v m ≤ δ m v m − + 1. Lemma A.3(2) gives δ m − v m > δ m v m − and the result follows.This concludes the proof. (cid:3) Lemma A.6.
Suppose that in the chain of bivalent vertices. . . i i +1 . . . i + k . . . (A.6) we have both i, i + k ∈ J . Then there is no j ∈ [ i, i + k ] such that v j > nδ j .Proof. Without loss of generality, we can assume that no vertices between i and i + k lie in J . Now let t be the smallest integer such that v i + t > nδ i + t . Thus δ i + t − v i + t > δ i + t v i + t − . Repeated applications ofLemma A.3(2) lead to v i + k > nδ i + k , a contradiction. (cid:3) We move on to completing the proof of Proposition A.1 in the case 0 ∈ J . The key is that in this case,removing J and all incident edges creates a component consisting only of the node ∞ . Lemma A.7.
Let i ∈ Q \ {∞} be any vertex such that every path in the diagram from ∞ to i passes throughan element of J . Assume in addition that the diagram is not of type D . Then v i ≤ nδ i .Proof. We introduce some notation. Given a path γ in our diagram, we define d γ ( a, b ) := 1 + { vertices on γ between a and b } . Assume that the statement does not hold, i.e. v i > nδ i . Let γ be a path in the diagram from ∞ to i that doesnot touch a given vertex more than once, and let j be the vertex in J on γ for which d γ ( ∞ , j ) is maximal.There must be a pair k , k of adjacent vertices along γ such that(1) d γ ( ∞ , j ) ≤ d γ ( ∞ , k ) = d γ ( ∞ , k ) − d γ ( ∞ , k ) ≤ d γ ( ∞ , i ); and(2) v k ≤ nδ k and v k > nδ k .It follows that δ k v k < δ k v k . We now use the above lemmas to ‘push’ this inequality away from ∞ untilwe reach a contradiction. Formally, we apply Lemma A.3(2) to the pair k , k . Reapplying the same lemma,we either reach a contradiction, or a node a of valency 3. Ignoring for now the branch containing k , observethat if either of the other branches contains an additional branch then Lemma A.4(3) gives a contradiction.Otherwise, there must be a branch starting at a that reaches another branching point of valency at most3. Let b be the vertex on this branch that lies adjacent to a . Lemma A.4 gives δ b v a < δ a v b . ApplyingLemma A.3(2) repeatedly to pairs of adjacent vertices along the branch starting at b enables us to push thisbasic inequality to the second branching point, where Lemma A.4(2) gives a contradiction. (cid:3) Proof of Proposition A.1 in the case ∈ J except for types A and D . If our diagram is not of type A or D , then all vertices have valency at most 3 and there is no double edge. Also, the framing vertex ∞ is djacent only to the vertex labelled 0. Thus, if 0 ∈ J , any path from ∞ to any vertex that does not lie in J must pass through an element of J . Hence Lemma A.7 proves Proposition A.1 in this case. (cid:3) A.3.
Proof for types A and D .Lemma A.8. Proposition A.1 holds for A and D .Proof. For type A , we have the diagram ∞ == (A.7)where the symbol == indicates that there are two edges in the diagram. If J = { , } there is nothing toprove, so we take J to be either { } or { } . A straightforward adaptation of Proposition A.2 shows that if J = { } , we must have 2 v ≤ v , so v ≤ n . Similarly, if J = { } , we obtain 2 v ≤ v + 1 = 2 n + 1, giving v ≤ n .For type D , the diagram is: ∞ . (A.8)Consider first the case of 0 ∈ J . If v > nδ = 2 n , we get from Proposition A.2 that 2 v i ≤ v for i ∈ { , , } .This implies that 4 v ≤ v + 2 v + 2 v + 2 v ≤ n + 3 v , contradicting that v > n . On the other hand, if v > nδ = n , Proposition A.2 implies that v > n . Thesame argument as before leads to a contradiction. By symmetry, the same argument applies if v i > n for i = 3 or i = 4.If v > nδ = n , and 2 ∈ J , Lemma A.4 immediately gives a contradiction.So suppose without loss of generality that 1 ∈ J . Then any other vertex i with v i > nδ i will, by Lemma A.4or Proposition A.2 give that v > n . The same lemmas show that4 v ≤ v + 2 v + 2 v + 2 v ≤ n + 3 v + 1 (A.9)and thus v ≤ n + 1. So v = 2 n + 1, but then Proposition A.2 gives that v = v = v = n . Plugging thisinto (A.9) gives 6 n + 3 = 3 v ≤ n + 1, a contradiction. (cid:3) A.4.
Proof in the general case.
We next handle the cases where 0 J . For this, we need to considereach diagram type individually. Lemma A.9.
Lemma A.1 holds for any diagram of type A i with i > .Proof. We number the vertices as follows: r r − . . . ∞ . (A.10)Assume that some vertex k ′ = ∞ has v k ′ > nδ k ′ = n . ote that by Lemma A.7, we can assume that 0 J . So we consider a subdiagram. . . i . . . r . . . j . . . ∞ (A.11)where i, j (possibly equal) are the only vertices in J , with k ′ some vertex in this subdiagram. We can withoutloss of generality assume 0 ≤ k ′ < j . Then there are adjacent vertices k, k + 1 such that k ′ ≤ k, k + 1 ≤ j with v k > n ≥ v k +1 . Repeatedly applying Lemma A.3 gives v > v > n. (A.12)There must also be adjacent vertices l, l + 1 between i and 0 such that v l +1 > v l . In a similar way, this leadsto v > v r . Combining with (A.12), we deduce 2 v > v + v r + 1, contradicting Proposition A.2. (cid:3) Lemma A.10.
Lemma A.1 holds for diagrams of type D i , i > .Proof. We number the vertices as follows: ∞ . . . r − r − r (A.13)We show that three cases remaining from Lemma A.7 are also absurd. By the symmetry of the diagram,these are sufficient.(1) There is an i such that ≤ i ≤ r − , v i > nδ i = 2 n , and all j ∈ J have i < j . Let k be maximal amongthe vertices such that v k > nδ k . If k ≤ r −
2, we have δ k +1 v k > δ k v k +1 .If k = r −
1, we must have J = { r } , and by Lemma A.4 we get δ r − v r − > δ r − v r − . By symmetry,the case k = r also leads to δ r − v r − > δ r − v r − .Both cases lead to (by Lemma A.3) v > v , that is, v − ≥ v . Then Lemma A.4 gives 2 v = v + 1and 2 v ≤ v . Combining these with Proposition A.2 leads to4 v ≤ v + 2 v + 2 v ≤ v − , which is absurd.(2) v > nδ = n , and all j ∈ J have j > . This implies v > n . Let j be the least vertex such that v j ≤ nδ j . Applying Lemmas A.3 and A.4 to the vertices j − , j (or if j = r , the vertices r, r −
2) weagain find v > v . Then the conclusion of case (1) applies.(3) v > nδ , and all j ∈ J have j ≥ . If 2 ∈ J , we have v = 2 n , and then v > n leads to 2 v > n + 1,contradicting Proposition A.2. If 2 J we can again take j as the least vertex with v j ≤ nδ j and argueas in case (2).Hence all possibilities lead to a contradiction, and Lemma A.1 holds for diagrams of type D i with i > (cid:3) To conclude, we have to deal with diagrams of type E i . As the proof strategies for these are very similar,we only include the full argument for the E case. Lemma A.11.
Proposition A.1 holds for type E . roof. We number the vertices as follows: ∞ (A.14)This time, we split what remains after Lemma A.7 into four cases. Let k be the minimal vertex with v k > nδ k . The cases are:(1) k > and all j ∈ J have j < k , or k = 0 . By Lemma A.3 and Lemma A.4, we find that v > nδ = n .The same lemmas show that δ k +1 v k + 1 = δ k v k +1 . Let us temporarily use the designation 9 for thevertex marked 0. By Proposition A.2, we get2 δ k v k ≤ δ k v k +1 + δ k v k − ≤ δ k +1 v k + 1 + δ k nδ k − implying δ k − ( v k − nδ k ) ≤
1. But this contradicts v k > nδ k .(2) k = 4 and all j ∈ J have j < k : We have2 v ≤ v + v + v ≤ nδ + nδ + v = 7 n + v . Since we also have (Lemma A.4) 5 v + 1 = 6 v , this implies that 7 v − ≤ n . But since v > n , thisis impossible.(3) k = 2 and at least one of the vertices and are in J : By Proposition A.2, we must have v ≥ n + 1.By Lemma A.3 applied to the vertex chain 1 , ,
4, we find 6 v < v . Then Lemma A.4 shows that6 v = 5 v + 1. Now, if v ≤ nδ , the same lemma and Proposition A.2 imply12 v ≤ v + 6 v + 6 v ≤ v + 24 n + 1leading to 24 n +4 ≤ v ≤ n +1, a contradiction. So suppose that 1 ∈ J , and v > n . By Lemma A.3,we get 6 v < v . As above, we find12 v ≤ v + 6 v + 6 v ≤ v + 6 v + 1leading to 4 v ≤ v + 1. This implies that 4 v = 6 v + 1, which has no integer solutions. Hence wehave a contradiction.(4) k = 1 or k = 3 , and so J only consists of
2: Suppose that v = nδ = 3 n . Then, by Lemma A.4 andLemma A.3, we get v > n , say v = 4 n + t, t >
0. But then Proposition A.4 and Proposition A.2 give12 v ≤ v + 6 v + 6 v ≤ n + 4 v + 5 v + 1leading to 18 n + 3 t = 3 v ≤ n + 1, a contradiction.Thus Proposition A.1 holds for E . (cid:3) Analogous arguments apply for the diagrams of type E and E . The proof of Proposition A.1 is nowcomplete. (cid:3) eferences [BC18] G. Bellamy and A. Craw. Birational geometry of symplectic quotient singularities, 2018. arXiv:1811.09979 .[Bri13] M. Brion. Invariant Hilbert schemes. In Handbook of moduli. Vol. I , volume 24 of
Adv. Lect. Math. (ALM) , pages64–117. Int. Press, Somerville, MA, 2013.[BS16] G. Bellamy and T. Schedler. Symplectic resolutions of quiver varieties and character varieties, 2016. arXiv:1602.00164 .[CB01] W. Crawley-Boevey. Geometry of the moment map for representations of quivers.
Compositio Math. , 126(3):257–293,2001.[CBH98] W. Crawley-Boevey and M.P. Holland. Noncommutative deformations of Kleinian singularities.
Duke Math. J. ,92(3):605–635, 1998.[CIK18] A. Craw, Y. Ito, and J. Karmazyn. Multigraded linear series and recollement.
Math. Z. , 289(1-2):535–565, 2018.[DH98] I.V. Dolgachev and Y. Hu. Variation of geometric invariant theory quotients.
Inst. Hautes ´Etudes Sci. Publ. Math. ,(87):5–56, 1998. With an appendix by Nicolas Ressayre.[FP04] V. Franjou and T. Pirashvili. Comparison of abelian categories recollements.
Doc. Math. , 9:41–56, 2004.[GNS18] ´A. Gyenge, A. N´emethi, and B. Szendr˝oi. Euler characteristics of Hilbert schemes of points on simple surface singu-larities.
Eur. J. Math. , 4(2):439–524, 2018.[Kin94] A. D. King. Moduli of representations of finite-dimensional algebras.
Quart. J. Math. Oxford Ser. (2) , 45(180):515–530,1994.[KN90] P. Kronheimer and H. Nakajima. Yang-Mills instantons on ALE gravitational instantons.
Math. Ann. , 288(2):263–307,1990.[Kuz07] A. Kuznetsov. Quiver varieties and Hilbert schemes.
Mosc. Math. J. , 7(4):673–697, 767, 2007.[McK80] J. McKay. Graphs, singularities, and finite groups. In
The Santa Cruz Conference on Finite Groups (Univ. California,Santa Cruz, Calif., 1979) , volume 37 of
Proc. Sympos. Pure Math. , pages 183–186. Amer. Math. Soc., Providence,R.I., 1980.[Nak94] H. Nakajima. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras.
Duke Math. J. , 76(2):365–416,1994.[Nak09] H. Nakajima. Quiver varieties and branching.
SIGMA Symmetry Integrability Geom. Methods Appl. , 5:Paper 003, 37,2009.[ST11] A. Savage and P. Tingley. Quiver Grassmannians, quiver varieties and the preprojective algebra.
Pacific J. Math. ,251:393–429, 2011.[Tha96] M. Thaddeus. Geometric invariant theory and flips.
J. Amer. Math. Soc. , 9(3):691–723, 1996.[VV99] M. Varagnolo and E. Vasserot. On the K -theory of the cyclic quiver variety. Internat. Math. Res. Notices , (18):1005–1028, 1999.[Wan99] W. Wang. Hilbert schemes, wreath products, and the McKay correspondence, 1999. arXiv:math/9912104 .[Wil92] P.M.H. Wilson. The K¨ahler cone on Calabi-Yau threefolds.
Invent. Math. , 107(3):561–583, 1992.[Yam17] R. Yamagishi. Symplectic resolutions of the Hilbert squares of ADE surface singularities, 2017. arxiv:1709.05886 .[Zhe17] X. Zheng. The Hilbert schemes of points on surfaces with rational double point singularities, 2017. arxiv:1701.02435 . Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
E-mail address : [email protected] Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
E-mail address : [email protected] / [email protected] / [email protected]@maths.ox.ac.uk / [email protected] / [email protected]