Quot schemes for Kleinian orbifolds

For a finite subgroup $\Gamma\subset {\mathrm{SL}}(2,\mathbb{C})$, we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold $[\mathbb{C}^2/\Gamma]$. We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of $\Gamma$, taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our previous work on the Hilbert scheme of points on $\mathbb{C}^2/\Gamma$; we present arguments that completely bypass the ADE classification.


Introduction
Let Γ ⊂ SL(2, C) be a finite subgroup and let Hilb n C 2 /Γ denote the Hilbert scheme of n points on the singular surface C 2 /Γ, parametrising ideals of colength n in the coordinate ring C[x, y] Γ of C 2 /Γ. In an earlier paper [7], we proved that (the reduced scheme underlying) this scheme is irreducible, normal, and admits a unique projective symplectic resolution π n : nΓ-Hilb C 2 −→ Hilb n C 2 /Γ red , where nΓ-Hilb C 2 is a certain component of the Γ-fixed locus Hilb C 2 Γ on the Hilbert scheme of points on C 2 itself. It was well known that nΓ-Hilb C 2 is a quiver variety M θ (1, nδ) associated to the framed McKay quiver of Γ, where θ is generic in a GIT chamber C + nδ and where the dimension vector on the McKay quiver is taken as a multiple of a fixed vector δ. In [7, Theorem 1.1] we reconstructed the morphism π n by variation of GIT quotient for quiver varieties; specifically, we identified a parameter θ 0 in the boundary of the chamber C + nδ such that π n is the morphism from M θ (1, nδ) to M θ 0 (1, nδ). This allowed us to interpret by variation of GIT quotient all possible ways in which π n can be factored as a sequence of primitive contractions.
To prove these results in [7], we introduced a family of algebras A I obtained by 'cornering', each indexed by a non-empty subset I of the set of irreducible representations of Γ. We showed This paper is a contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Gottsche.html that the fine moduli spaces of modules M A I (1, nδ I ) for these algebras with very specific dimension vectors are isomorphic to the quiver varieties mentioned above. Moreover, for I consisting of the trivial representation only, we recovered the space Hilb n C 2 /Γ .
In this paper, we have three main goals. First, we aim to understand better the moduli spaces M A I , constructing them as a kind of Quot scheme that is both natural and geometric. Second, with applications in mind, we consider arbitrary dimension vectors, not just multiples of some set of fixed dimension vectors as above. One interpretation of our results is a new fine moduli space structure, as well as a (noncommutative) geometric interpretation, of a large class of Nakajima quiver varieties for certain non-generic stability parameters. Our third and final goal is to provide proofs that completely bypass any case-by-case analysis of ADE diagrams.
To state our main result, let r denote the number of nontrivial irreducible representations of Γ. For any non-empty subset I ⊆ {0, 1, . . . , r}, let M θ I (1, v) denote the Nakajima quiver variety associated to the affine ADE graph for some dimension vector (1, v), where θ I is a specific stability condition determined by our choice of I; see (5.1) for the definition. 2) when I is a singleton corresponding to a one-dimensional representation of Γ, the orbifold Quot scheme is isomorphic to a classical Quot scheme for C 2 /Γ; 3) the orbifold Quot scheme is non-empty if and only if the quiver variety M θ I (1, v) is nonempty for some vector v ∈ N r+1 satisfying v i = n i for all i ∈ I, in which case, after changing the values of v k for k ∈ I if necessary, we have where on the left we take Quot n I I C 2 /Γ with the reduced scheme structure.
In particular, when it is non-empty, Quot n I I C 2 /Γ red is irreducible, normal, and it has symplectic, hence rational Gorenstein, singularities. Moreover, it admits at least one projective symplectic resolution. Remark 1.2. Fix the dimension vector v = nδ that we refer to in our opening paragraph.  (1, v) lies in an extremal ray of the movable cone described in [3], so it admits a unique projective symplectic resolution. In fact, our arguments here reprove the main result of [7] without having to resort to any case-by-case analysis of ADE diagrams.
2. For I = {1, . . . , r}, the stability condition θ I lies in the relative interior of a wall of the chamber C + nδ . It follows from [3, Theorem 1.2] that the polarising ample bundle on M θ I (1, v) lies in the interior of the movable cone, so M θ I (1, v) admits more than one (in fact, precisely two) projective crepant resolutions.
Thus, while the projective symplectic resolution is unique in [7, Theorem 1.1], it is not possible to assert uniqueness in Theorem 1.1 above.
While some of our arguments follow [7] closely, we introduce Quot schemes in a general noncommutative context, establishing a basic representability result in Proposition 3.1 that may be of broader interest. Furthermore, while the morphism induced by varying a generic stability condition θ to a special one θ I is always surjective for the dimension vector v = nδ relevant to the case Hilb n C 2 /Γ discussed above, the same is not true in general. Instead, one has to correct v to obtain a surjective morphism of quiver varieties.
The main application of this paper will be in studying generating functions attached to these moduli spaces, following [12,13]. In forthcoming work, various generating functions of Euler numbers of Hilbert and Quot schemes for ADE singularities and orbifolds will be studied, demonstrating specialisation phenomena also on the level of these generating series. A poset of Quot schemes corresponding to the poset of subsets of {0, . . . , r}, generalising that of [7, Section 5], will play an important role in that study. As in [12,13], it will be essential that we work with arbitrary dimension vectors, not just multiples of δ.
The structure of the paper is as follows. In Section 2 we collect the necessary preliminaries on semi-invariants of finite group actions and the McKay correspondence. In Section 3, we define Quot schemes for Kleinian orbifolds and discuss some of their properties. In Section 4, we introduce the algebras A I and the associated fine moduli spaces, and we identify these moduli spaces with Quot schemes for Kleinian orbifolds. In Section 5, we obtain a resolution of singularities for the quiver varieties of interest. Finally, in Section 6 we establish the key isomorphism from Theorem 1.1(3).
Conventions. We work throughout over C. In particular, all schemes are C-schemes and all tensor products are taken over C unless otherwise indicated. We often use the following partial order on dimension vectors: for any n ∈ N and for u = ( We adopt the sign convention of King [15] for θ-stability, so a module M over an algebra is θ-stable (resp. θ-semistable) if θ(M ) = 0 and every nonzero proper submodule N M satisfies θ(N ) > 0 (resp. θ(N ) ≥ 0).

Three algebras arising from the McKay graph
Let Γ ⊂ GL(V ) be a finite subgroup of the general linear group for an m-dimensional C-vector space V . Choosing a basis of V , we can write the symmetric algebra of the dual vector space V * as a polynomial ring R = C[x 1 , . . . , x m ]. The group Γ acts dually on R: for g ∈ Γ and f ∈ R, List the irreducible representations of Γ as {ρ 0 , ρ 1 , . . . , ρ r }, where ρ 0 is the trivial representation. The decomposition of R as an R Γ -module can be written [11,Proposition 4 From now on, let Γ ⊂ SL(2, C) be a finite subgroup acting as above on R = C[x, y]. The McKay graph of Γ has vertex set {0, 1, . . . , r} corresponding to the irreducible representations of Γ, where for each 0 ≤ i, j ≤ r, we have dim Hom Γ (ρ j , ρ i ⊗ V ) edges joining vertices i and j.
McKay [18] observed that this graph is an affine Dynkin diagram of ADE type. We frame this graph by introducing an additional vertex, denoted ∞, together with an edge joining the vertices ∞ and 0; we refer to the resulting graph as the framed McKay graph of Γ.
We now associate a doubled quiver to each graph. For this, consider the set of pairs Q 1 comprising an edge in the framed McKay graph and an orientation of the edge. For each a ∈ Q 1 , we write t(a), h(a) for the vertices at the tail and head respectively of the oriented edge, and we write a * for the same edge with the opposite orientation. Define the framed McKay quiver of Γ, denoted Q, to be the quiver with vertex set Q 0 = {∞, 0, . . . , r} and arrow set Q 1 . The (unframed) McKay quiver, denoted Q Γ , is the complete subquiver of Q on the vertex set {0, 1, . . . , r}. Using these quivers we define the following three algebras.
First, let CQ denote the path algebra of the framed McKay quiver Q. For i ∈ Q 0 , let e i ∈ CQ denote the idempotent corresponding to the trivial path at vertex i. Let : Q 1 → {±1} be any map such that (a) = (a * ) for all a ∈ Q 1 . The preprojective algebra of Q, denoted Π, is defined as the quotient of CQ by the ideal The preprojective algebra Π does not depend on the choice of the map . For each i ∈ Q 0 , we simply write e i ∈ Π for the image of the corresponding vertex idempotent. Second, in the framed McKay quiver Q, let b * ∈ Q 1 be the unique arrow with head at vertex ∞. Define a C-algebra as the quotient of the preprojective algebra Π by the two-sided ideal generated by the class of b * : Equivalently, if we define a quiver Q * to have vertex set Q * 0 = {∞, 0, 1, . . . , r} and arrow set Q * 1 = Q 1 \ {b * }, then A is the quotient of the path algebra CQ * by the ideal of relations Third, let CQ Γ denote the path algebra of the (unframed) McKay quiver Q Γ . Since CQ Γ is a subalgebra of CQ, we use the same symbol e i ∈ CQ Γ for the idempotent corresponding to the trivial path at vertex i of Q Γ . The preprojective algebra of Q Γ , denoted B, is defined as the quotient of CQ Γ by an ideal defined similarly to that from (2.1) for the quiver Q Γ . Remark 2.1. For each vertex 0 ≤ i ≤ r we denote by e i the class in A of the idempotent at the vertex i. Note that B is not a subalgebra of Π. On the other hand, B is isomorphic to the subalgebra ⊕ 0≤i,j≤r e j Ae i of A due to [7,Lemma 3.2], and moreover as complex vector spaces. In particular, the classes e i , 0 ≤ i ≤ r are also contained in B.

Morita equivalence
The skew group algebra S := R * Γ = C[x, y] * Γ of the finite subgroup Γ ⊂ SL(2, C) contains the group algebra CΓ as a subalgebra. For 0 ≤ i ≤ r, choose an idempotent f i ∈ CΓ such that CΓf i ∼ = ρ i .
Proof . As in [9, Lemma 1.1], S has a basis given by all elements of the form x j y k g, where j, k ≥ 0 and g ∈ Γ. As f 0 is invariant under multiplication by any g ∈ Γ, the subspace Sf 0 has a basis given by elements of the form x j y k f 0 for j, k ≥ 0. This gives the required isomorphism of vector spaces which is readily seen to be an isomorphism of S-R Γ -bimodules.
To recall the key result, define f = f 0 + · · · + f r .
Proposition 2.3. The skew group algebra S and the preprojective algebra B are Morita equivalent via an isomorphism f Sf ∼ = B; explicitly, the equivalence on left modules is Proof . The Morita equivalence is a bi-product of the proof of [23, Proposition 2.13], and may have been first stated explicitly in [9, Theorem 0.1]. The algebra isomorphism f Sf ∼ = B that in particular sends f i to e i for 0 ≤ i ≤ r is described in [9,Theorem 3.4]. The description of S as End R 0 (R) follows from [1], spelled out in [17,Theorem 5.12], while the description of B follows by applying the isomorphism f Sf ∼ = B; see for example Buchweitz [6].
Proof . Apply f i to both sides of f Sf while simultaneously applying e i to both sides of B; the algebra isomorphism f Sf ∼ = B then restricts to the algebra isomorphism f i Sf i ∼ = e i Be i . For the second statement, apply e i and e 0 to the left and right respectively of the isomorphism We finally look at how the Morita equivalence Φ acts on dimension vectors.
Proof . Applying Lemma 2.2 shows that Φ sends the polynomial ring R ∼ = Sf 0 to f Sf 0 = (f Sf )f 0 = Be 0 . It follows that for any S-submodule J ⊆ R, the quotient R/J is an Smodule whose image under Φ is Be 0 /f J. Finally, Φ induces a Z-linear isomorphism between the Grothendieck groups of the categories of finite-dimensional left modules over S and B which we identify with the representation ring Rep(Γ) and with Z r+1 respectively.

Quot schemes for modules over associative algebras
Let H denote an arbitrary finitely generated, not necessarily commutative C-algebra, and let M be a finitely generated left H-module. For a scheme X, we denote by sending a scheme X to the set of isomorphism classes of left H X -modules Z equipped with a surjective H X -module homomorphism M X → Z such that Z, when considered as an O Xmodule, is locally free of rank n.
The following result may be known to experts, though we could not find it in the literature. Proof . As H is finitely generated as a C-algebra, we can fix a surjection H + → H with H + a free noncommutative C-algebra on a finite number of generators. Then M is also a left H +module. Define J := ker H + → H . Consider a left H + -module Z that is a quotient of M via a surjective H + -module morphism M → Z.
Since any element of the two-sided ideal J of H + acts trivially on M , and so also on quotients of M , Z is automatically an H-module. Conversely, any quotient of M as an H-module is automatically an H + -module. It follows that there is a canonical isomorphism of functors Q n H (M ) = Q n H + (M ). Thus, we may assume that H is a finitely generated and free C-algebra. Fix a left H-module surjection ψ : H r → M for some r. Write elements of ker ψ in the form 1≤k≤r w k e k , where the e k ∈ H r are the standard module generators and w k ∈ H. For any quotient module q : M → Z, composition with ψ presents Z as a quotientq : H r → Z, such that ker ψ ⊆ kerq. The equations 1≤k≤r w kq (e k ) = 0 give a closed condition on [q] ∈ Quot n H (H r ): the vanishing of a collection of vectors in the vector space underlying Z. When trivialising the universal sheaf on some open cover of Q n H (H r ), these closed conditions glue together, and we see that this construction realises as a closed subfunctor.
To conclude, it suffices to show that Q n H (H r ) is representable for a free noncommutative C-algebra H. This follows from [2, Theorem 2.5]; while that paper discusses the case when H is freely generated by 3 elements, the proof generalises to finitely many generators.

Quot schemes for Kleinian orbifolds
Let B be the preprojective algebra of the unframed McKay quiver and R = C[x, y] as in Section 2.1. Recall that R 0 = R Γ , and that for each For any subset I ⊆ {0, . . . , r}, consider the idempotent e I := i∈I e i in B, and define the C-algebra comprising linear combinations of classes of paths in Q Γ whose tails and heads lie in the set I. The process of passing from B to B I is called cornering; see [8, Remark 3.1]. Then is naturally a finitely generated left B I -module. For a given dimension vector n I = (n i ) ∈ N I , consider the contravariant functor Sch op → Sets sending a scheme X to the set of isomorphism classes of left B I,X -modules Z equipped with a surjective B I,X -module homomorphism such that each submodule e i Z, for i ∈ I, when considered as an O X -module, is locally free of rank n i . Note that we can recover Z from these submodules via The next result provides the link between this functor and that introduced in Section 3.1.
into open and closed subfunctors. In particular, each functor Q n I B I R I is represented by a scheme Quot n I I C 2 /Γ of finite type over C.
Proof . The dimension vector n I = (n i ) ∈ N I is locally constant in a flat family of B I -modules of fixed rank n, with each of the entries being lower semicontinuous by the Schur lemma. So we have a decomposition (3.2) into open and closed subfunctors. As R I is finitely generated as a left B I -module, the functor on the left hand side of (3.2) is represented by a scheme of finite type over C by Proposition 3.1. The last claim then follows.
When the index set is a singleton I = {i}, we often simply write Quot n i i C 2 /Γ for Quot 3 for a discussion of this special case. As is common with Hilbert schemes in other contexts, we consider collections of our Quot schemes for all possible dimension vectors. Define the orbifold Quot scheme for C 2 /Γ and for the index set I to be where n I = (n i ) i∈I ∈ N I as before. Again, when I = {i} is a singleton, we simply write Denote by Hilb C 2 /Γ the Hilbert scheme of Γ-invariant finite colength subschemes of C 2 . As explained in more detail in [13, Section 1.1], this space decomposes as a disjoint union of quasi-projective varieties which respects the decompositions on both sides into pieces indexed by v ∈ N r+1 .
Proof . This follows from the Morita equivalence between the algebras S and B from Proposition 2.3, and its amplification Corollary 2.5, together with the fact that a finite colength left

Quot schemes for Kleinian singularities
In this section, we discuss the relation of the construction from Section 3.2 to the more classical (commutative) version of the Quot scheme, in the case when I = {i} is a singleton. Note that in this case, R i ∼ = e i Be 0 is both a left B i -module and a right R 0 ∼ = e 0 Be 0 -module.
Let Quot i C 2 /Γ be the scheme parameterising finite codimension R Γ -submodules of R i or, equivalently, finite colength subsheaves of the coherent sheaf on the Kleinian singularity C 2 /Γ corresponding to R i . Again, this space decomposes as  ι(0) C 2 /Γ represents the functor that sends a scheme X to the set of isomorphism classes of right B 0,Xmodule quotients R ι(0),X → Z which are locally free of rank n ι(0) over O X . The claim follows.   Recall that A = Π/(b * ) is the quotient of the preprojective algebra of the framed McKay quiver of Γ by the two sided ideal (b * ), where b * is the arrow from 0 to ∞. Let I ⊆ {0, . . . , r} be a non-empty subset; we work under the assumption that I = ∅ for the rest of the paper. Define the idempotent e I := e ∞ + i∈I e i ∈ A. The subalgebra A I := e I Ae I (4.1) of A comprises linear combinations of classes of paths in Q whose tails and heads lie in the set {∞} ∪ I; this is also an instance of cornering [8, Remark 3.1]. Let n I := (n i ) i∈I = i∈I n i ρ i ∈ N I . Then (1, n I ) := ρ ∞ + n I ∈ N ⊕ N I is a dimension vector for A I -modules, and we consider the stability condition η I : Z ⊕ Z I → Q given by where T ∞ is the trivial bundle and T i has rank n i for i ∈ I. After multiplying η I by a positive integer m ≥ 1 if necessary, we may assume that the polarising ample line bundle L I := i∈I det(T i ) m on M A I (1, n I ) given by the GIT construction is very ample.

The Quot scheme as a fine moduli space of modules
We now establish the link between the fine moduli space M A I (1, n I ) and the orbifold Quot scheme Quot n I I C 2 /Γ . For any non-empty subset I ⊆ {0, . . . , r}, consider the algebras A I and B I from (4.1) and (3.1) respectively. In [7, Proposition 3.3], we used the isomorphism B I ∼ = End R 0 i∈I R i to introduce quivers Q I and Q * I and a commutative diagram of C-algebra homomorphisms where the vertical maps are surjective and the horizontal maps are injective. In particular, we obtain quiver presentations B I ∼ = CQ I / ker(α I ) and A I ∼ = CQ * I / ker(β I ). The vertex set of Q * I is {∞} ∪ I, while the edge set comprises three kinds of arrows: loops at each vertex i ∈ I; arrows between pairs of distinct vertices in I; and arrows from vertex ∞ to vertices in I. Note that Q I is the complete subquiver of Q * I on the vertex set I.

Then there is an isomorphism
Proof . It follows from (2.2) that there is an isomorphism of vector spaces For the middle summand, consider the map of left B I -modules defined on representing paths in the framed McKay quiver Q by composing with the arrow b on the right. This map is by definition surjective. But it is also injective, as every relation in A involving the arrow b can be factored into the product of b and a relation in B. This establishes the isomorphism of vector spaces. To enhance this to an isomorphism of C-algebras, note that the algebra structure on A I ∼ = B I ⊕ e I Bb ⊕ Ce ∞ is given by Since c 2 is a scalar, it commutes with b = be ∞ , giving b 1 r 2 b + r 1 bc 2 e ∞ = (b 1 r 2 + r 1 c 2 )b in the middle term above. The isomorphism (4.3) allows us to drop the b from the right-hand side, leaving the second statement as required.

Variation of GIT quotient for quiver varieties
Let v := (v i ) 0≤i≤r ∈ N r+1 be a dimension vector for the McKay quiver Q Γ . Recall that we identify Z Q 0 ∼ = Zρ ∞ ⊕ Rep(Γ), so we obtain the dimension vector for the space of stability conditions for the the framed quiver. Following Nakajima [19,20], for every θ ∈ Θ v , the quiver variety M θ (1, v) is the coarse moduli space of S-equivalence classes of θ-semistable Π-modules of dimension vector (1, v). The GIT construction of M θ (1, v) is summarised for example in [7, Section 2]; see also [22,Section 2]. Note in particular that we give M θ (1, v) the reduced scheme structure.
The set of stability conditions Θ v admits a preorder ≥, and this determines a wall-andchamber structure, 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 this preorder, in which case M θ (1, v) ∼ = M θ (1, v). The interiors of the top-dimensional cones in Θ v are chambers, while the codimension-one faces of the closure of each chamber are walls. We say that θ ∈ Θ v is generic with respect to v, if it lies in some GIT chamber. The vector (1, v) is indivisible, so again, [15,Proposition 5.3] implies that for generic θ ∈ Θ v the quiver variety M θ (1, v) is the fine moduli space of isomorphism classes of θ-stable Π-modules of dimension vector (1, v). In this case, the universal family on M θ (1, v) is a tautological locally-free sheaf R := ⊕ i∈Q 0 R i together with a C-algebra homomorphism ϕ : Π → End(R), where R ∞ is the trivial bundle on M θ (1, v) and where rank(R i ) = v j for i ≥ 0. 1. For all θ ∈ Θ v , the scheme M θ (1, v) is irreducible and normal, with symplectic singularities.

Wall-and-chamber structure
Define δ := 0≤i≤r dim(ρ i )ρ i ∈ Rep(Γ). For the dimension vector v := nδ, the GIT walland-chamber structure on Θ nδ is computed explicitly in [3,Theorem 4.6]. More generally, for arbitrary v ∈ N r+1 , it is possible to compute the wall-and-chamber structure on Θ v by applying recent work of Dehority [10,Proposition 4.8]. We do not need the wall-and-chamber structure here, but we do use the following distinguished region of the space of stability conditions: This open cone need not be a GIT chamber, though it is always contained in one (and it is a chamber in the special case v = nδ, see [ For v ∈ N r+1 and θ ∈ C + v , there is an isomorphism Proof . In the special case v = nδ, the statement is proved in [16,Theorem 4.6]. However, the argument given there applies for an arbitrary dimension vector v; note in particular that the auxiliary [16, Corollary 3.5] does not need any assumption on the dimension vector (and we have that Hilb v C 2 /Γ is non-empty if and only if M θ (1, v) is non-empty).
Every face of the closed cone C + v is of the form When v = nδ, these faces are contained in walls of the wall-and-chamber structure on Θ nδ , though this need not be the case for arbitrary v ∈ N r+1 . In any case, the parameter lies in the relative interior of the face σ I .

Resolution of singularities
Let v ∈ N r+1 be a dimension vector and I ⊆ {0, . . . , r}. As mentioned above, the quiver variety M θ I (1, v) is singular in general for θ I as defined in (5.1) above. Moreover, for θ ∈ C + v , the morphism obtained by variation of GIT need not be birational. It is nevertheless possible to find a resolution of M θ I (1, v) using a quiver variety if one modifies the dimension vector v: Let v ∈ N r+1 and let I ⊆ {0, . . . , r} be non-empty. Assume further that M θ I (1, v) is non-empty. There exists a dimension vector v ∈ N r+1 satisfying v i ≤ v i for 0 ≤ i ≤ r and v i = v i for i ∈ I, such that there is a projective resolution of singularities where the stability condition θ satisfies θ ∈ C + v .
Proof . Following Nakajima [19, Section 6] (cf. [3, Section 3.5]), there is a finite stratification by smooth locally-closed subvarieties, where the union is over conjugacy classes of subgroups of the reductive group applied during the GIT construction of M θ I (1, v), and where the stratum M θ I (1, v) γ consists of S-equivalence classes of semistable modules with polystable representative having stabiliser in the conjugacy class γ. Since M θ I (1, v) is irreducible by Lemma 5.1, there is a unique dense stratum and we fix a conjugacy class γ such that the dense open stratum Apply [22,Proposition 2.25] to obtain a dimension vector v ∈ N r+1 as in the statement of the lemma together with an isomorphism where M s θ I (1, v) is the fine moduli space of θ I -stable Π-modules of dimension vector (1, v), and Y ⊂ M 0 (0, v − v) is a locally-closed subvariety. Note that M θ I (1, v) γ is non-empty by assumption, hence so are M s θ I (1, v) and Y . We claim that Y is a closed point. Indeed, M 0 (0, v − v) parametrises 0-polystable representations, i.e., direct sums of simple representations. The framing is 0-dimensional, and since I is non-empty, there is at least one i such that v i = v i . It follows that the representations parameterised by Y are supported on a doubled quiver obtained from an affine ADE diagram with at least one vertex removed. Removing a vertex from an affine ADE diagram leaves a graph in which every connected component is a subgraph of an finite type ADE diagram. Therefore M 0 (0, v − v) parametrises direct sums of simple representations of preprojective algebras of doubled quivers associated to finite type ADE diagrams. A simple representation of any such quiver is one-dimensional [25, Lemma 2.2], so the only polystable representation of dimension vector Indeed, ifμ and µ denote the moment maps for the actions of Gṽ := 0≤i≤r GL(ṽ i ) and G v := 0≤i≤r GL(v i ) on the representation spaces of Π-modules of dimension vectors (1,ṽ) and (1, v) respectively, then the above assignment induces an inclusionμ −1 (0) → µ −1 (0) that is equivariant with respect to the actions of Gṽ and G v onμ −1 (0) and µ −1 (0) respectively. Any submodule of π I (V ) is W ⊕ W for submodules W ⊆ V and W ⊆ 0≤i≤r S ⊕(v i −ṽ i ) i , and we have θ I (W ⊕ W ) = θ I (W ) ≥ 0. Thus, the image of theθ-stable locus ofμ −1 (0) lies in the θ I -semistable locus of µ −1 (0), and this inclusion induces the morphism π I as claimed.
It remains to establish the properties of π I . Adapting the proof from [4,Lemma 2.4] shows that π I is a projective morphism, while [4,Theorem 1.15] gives that M θ (1, v) is non-singular as θ is generic. Our explicit description of π I shows that it factors via the morphism Recall that M θ I (1, v) is a coarse moduli space of θ I -polystable modules up to isomorphism [22,Proposition 2.9(2)], and that every θ I -polystable module has a unique summand that has dimension 1 at the vertex ∞.  (1, v), and let M ∞ be the unique summand that has dimension 1 at the vertex ∞. Then Proof . The morphism π I from Proposition 5.3 is surjective, because quiver varieties are irreducible and π I is both proper and birational. Therefore [M ] = [π I (V )] for someθ-stable Π-module V of dimension vector (1,ṽ). Each vertex simple S i arising as a summand of π I (V ) has dimension 0 at the vertex ∞, so we must have M ∞ ⊆ V . This implies the lemma.
Suppose now that M θ I (1, v) is non-empty, and apply Proposition 5.3 to obtainṽ ≤ v and a resolution of singularities π I : 2) for any I ⊆ {0, . . . , r}, after multiplying θ I by a positive integer if necessary, the morphism to the linear series of L I decomposes as the composition of the morphism π I from Proposition 5.3 and a closed immersion: Since θ ∈ C + v , the tautological bundles R i on the quiver variety Mθ(1,ṽ) are globally generated for i ∈ Q 0 by [8,Corollary 2.4]. Therefore, L C + v (θ ) is globally generated as θ i ≥ 0 for all 0 ≤ i ≤ r. In particular, since θ I ∈ C + v , the rational map ϕ |L I | is defined everywhere. For (2), after taking a positive multiple of θ I if necessary, we may assume that the polarising ample line bundle L I on M θ I (1, v) is very ample. Our explicit construction of π I from the proof of Proposition 5.3 implies that the G v -equivariant line bundle on µ −1 (0) determined by the character associated to θ I restricts under the inclusionμ −1 (0) → µ −1 (0) to the Gṽ-equivariant line bundle onμ −1 (0) determined by the character associated to θ I . Thus, after descent, we obtain π * I (L I ) = L I as required. 6 Quiver varieties and moduli spaces for cornered algebras 6.1 The key commutative diagram Once and for all, fix a non-empty I ⊆ {0, 1, . . . , r} and a dimension vector n I = (n i ) i∈I ∈ N I . Proposition 6.1. Let v = (v j ) 0≤j≤r ∈ N r+1 be any vector satisfying v i = n i for all i ∈ I. Suppose that M θ I (1, v) is non-empty, and define v as in Proposition 5.3. Then v satisfies v i = n i for i ∈ I, and there is a commutative diagram of schemes over C for any θ ∈ C + v , where π I is surjective and where ι I is a closed immersion. In particular, the fine moduli space M A I (1, n I ) is non-empty.
Proof . The fact that v i = n i for all i ∈ I is immediate from Proposition 5.3. We now construct a commutative diagram  (1, v). Now, just as in the proof of Lemma 5.5, we deduce ϕ |L I | = ϕ |L I | • τ I , so the right-hand triangle commutes and hence so does the diagram.
It remains to construct the closed immersion ι I . Commutativity of the diagram, combined with surjectivity of π I , shows that M θ I (1, v) is isomorphic to the closed subscheme Im(σ I ) = Im(σ I •π I ) = Im(ϕ I •τ I ) of |L I |. The closed immersion ϕ I induces an isomorphism λ I : Im(ϕ I ) → M A I (1, n I ) and hence we obtain a closed immersion Since M θ I (1, v) is non-empty by assumption, it follows that M A I (1, n I ) is non-empty.

Selecting a suitable dimension vector
We don't yet know whether the morphism τ I from the key commutative diagram from Proposition 6.1 is surjective. We now introduce a collection of dimension vectors that will allow us to establish surjectivity.
For any vector v ∈ N r+1 , it is convenient to write v ∞ := 1, so that the dimension vector (1, v) has components v i for all i ∈ Q 0 . Lemma 6.2. Let v = (v j ) 0≤j≤r ∈ N r+1 satisfy v i = n i for all i ∈ I. Let V be a θ I -stable A-module of dimension vector (1, v). Then 2v k ≤ {e∈Q 1 |t(e)=k} v h(e) for all k / ∈ {∞} ∪ I.
Proof . Fix k / ∈ {∞} ∪ I and define The maps in V determined by arrows with head and tail at vertex k combine to define maps f : V k → V ⊕ and g : V ⊕ → V k satisfying g • f = 0. The same proof as in [7,Proposition A.2] implies that the complex has nonzero homology only at V ⊕ , so dim V ⊕ ≥ 2 dim V k . This proves the claim.
For our chosen vector n I = (n i ) i∈I ∈ N I , define Recall also that Q 1 = {b * } ∪ Q * 1 , so the indexing set in the sum from (6.1) differs from the set {e ∈ Q 1 | t(e) = 0} only for k = 0. Example 6.3. If there exists n > 0 such that n i = n dim ρ i for all i ∈ I, then equation (6.1) shows that the vector (1, v) with v k = n dim(ρ k ) for all 0 ≤ k ≤ r lies in V(n I ). This class of dimension vectors appears in [7]. Lemma 6.4. For any v ∈ N r+1 satisfying v i = n i for all i ∈ I, there exists (1, v ) ∈ V(n I ) such that v − v ≥ 0.
Proof . Write K := {0, 1, . . . , r} \ I. Set v ∞ = 1 and v i = n i for i ∈ I. Our goal is to define v k ∈ N for each k ∈ K such that v k ≥ v k and the inequality 2v k ≥ {e∈Q 1 |t(e)=k} v h(e) (6.2) holds for all k ∈ K. For this, define a subset K ⊆ K as follows. If 0 ∈ I, then set K = ∅. Otherwise, we have 0 ∈ K. In this case, choose any shortest path γ in the McKay graph starting at vertex 0 and ending at some vertex i ∈ I. The set K of vertices through which γ passes satisfies K ⊆ K \{0}. Now, fix N 0 and for each k ∈ K , let d k denote the distance in the graph from 0 to k. For k ∈ K, define v k = N dim(ρ k ) − d k if k ∈ K , N dim(ρ k ) otherwise.
Since N 0, we have v k ≥ v k for all k ∈ K. Also, for all k ∈ K and i ∈ I, we have v k v i .
It remains to establish the inequality (6.2) for all k ∈ K. To do this, consider three cases, depending on whether k ∈ K and whether k = 0.
First, suppose k ∈ K . If there is a vertex j ∈ I that lies adjacent to k, then (6.2) follows by combining (6.1) with the inequality v k v j . Otherwise, precisely two vertices adjacent to k lie in {0} ∪ K , while every other vertex adjacent to k lies in K. Then so (6.2) holds because k = 0. Next, suppose k ∈ K \ K with k = 0. Then each vertex j adjacent to k lies in {0, 1, . . . , r} and satisfies N dim(ρ j ) ≥ v j . Combine these inequalities with (6.1) to obtain 2v k = 2N dim(ρ k ) = This establishes the inequality (6.2) since k = 0. Finally, the only remaining case is when k = 0 ∈ K. If there is a vertex j ∈ I that lies adjacent to 0, then (6.2) follows by combining (6.1) with the inequality v 0 v j and v ∞ = 1. Otherwise, we have 0 ∈ K and one vertex of K lies adjacent to 0, giving The quiver Q 1 has a unique arrow with tail at 0 and head at ∞, so (6.2) holds for k = 0 if 0 ∈ K. This completes the proof. is θ I -semistable of dimension vector (1, v ) by construction.

Quiver varieties as fine moduli spaces
As before, n I = (n i ) ∈ N I is a dimension vector and η I : Z ⊕ Z I → Q is the stability condition for A I -modules of dimension vector (1, n I ) defined in (4.2). Let v = (v j ) 0≤j≤r ∈ N r+1 be a dimension vector satisfying v i = n i for all i ∈ I, and let θ I ∈ Θ v be the stability condition for A-modules defined in (5.1). We now use Corollary 6.5 to strengthen the statement of Proposition 6.1. Before stating the desired result, recall the idempotent e I = e ∞ + i∈I e i ∈ A and the algebra A I = e I Ae I .

A -mod
A I -mod j * I j I! defined by j * Proof . If M A I (1, n I ) = ∅, then Corollary 6.5 gives a vector v ∈ N r+1 satisfying v i = n i for i ∈ I such that M θ I (1, v) = ∅. The converse is immediate from Proposition 6.1.
Finally, we prove the results announced in Section 1.
Proof of Theorem 1.1. The isomorphism in (1), including the case when either space is empty, follows from Proposition 4.2. The isomorphism in (2)