Birational geometry of symplectic quotient singularities

For a finite subgroup Γ⊂SL(2,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \subset \mathrm {SL}(2,\mathbb {C})$$\end{document} and for n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document}, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity C2/Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^2/\Gamma $$\end{document}. It is well known that X:=Hilb[n](S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X:={{\,\mathrm{{\mathrm {Hilb}}}\,}}^{[n]}(S)$$\end{document} is a projective, crepant resolution of the symplectic singularity C2n/Γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^{2n}/\Gamma _n$$\end{document}, where Γn=Γ≀Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _n=\Gamma \wr \mathfrak {S}_n$$\end{document} is the wreath product. We prove that every projective, crepant resolution of C2n/Γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^{2n}/\Gamma _n$$\end{document} can be realised as the fine moduli space of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-stable Π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document}-modules for a fixed dimension vector, where Π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document} is the framed preprojective algebra of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} and θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-stability conditions to birational transformations of X over C2n/Γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^{2n}/\Gamma _n$$\end{document}. As a corollary, we describe completely the ample and movable cones of X over C2n/Γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^{2n}/\Gamma _n$$\end{document}, and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.


Introduction
For a finite subgroup Γ ⊂ SL(2, C), let S → C 2 /Γ denote the minimal resolution of the corresponding Kleinian singularity. The well-known paper by Kronheimer [44] realises S as a hyperkähler quotient, describes the ample cone of S as a Weyl chamber of the root system of type ADE associated to Γ by the McKay correspondence, and constructs the simultaneous resolution of the semi-universal deformation of C 2 /Γ. In the present paper we provide a natural generalisation of these results to higher dimensions by studying symplectic resolutions of the quotient singularity C 2n /Γ n for any n ≥ 1, where Γ n = Γ ≀ S n is the wreath product. We prove that every projective crepant resolution of C 2n /Γ n can be realised as a Nakajima quiver variety, generalising the description of the Hilbert scheme X := Hilb [n] (S) of n-points on S by Kuznetsov [45] (established independently by both Haiman and Nakajima). We also obtain a complete understanding of the birational geometry of X over C 2n /Γ n by describing explicitly the movable cone of X over C 2n /Γ n in terms of an extended Catalan hyperplane arrangement determined by the ADE root system associated to Γ.
Finally, we construct, using quiver GIT, the simultaneous resolution of the universal Poisson deformation of C 2n /Γ n .
1.1. Quiver varieties and the linearisation map. For n ≥ 1 and for a finite subgroup Γ ⊂ SL(2, C), the symmetric group S n acts on the direct product Γ n by permuting the factors, and the wreath product is defined to be the semidirect product Γ n := Γ n ⋊ S n ⊂ Sp(2n, C). Throughout the introduction, we assume n > 1 and Γ is non-trivial (see Section 4.4, Remark 7.8 and Proposition 7.11 for these degenerate cases). It is well-known that the Hilbert scheme of n points on S provides a symplectic resolution f : X := Hilb [n] (S) −→ Y := C 2n /Γ n of the corresponding quotient singularity. In particular, f is a projective, crepant resolution of singularities.
In order to study the birational geometry of X over Y , we first recall that X can be constructed by GIT as a quiver variety. Consider the affine Dynkin graph associated to Γ by McKay [55], where the vertex set is by definition the set of irreducible representations of Γ. Define the dimension vector v := nδ and the framing vector w = ρ 0 , where δ and ρ 0 are the regular and trivial representations of Γ respectively.
If we write R(Γ) for the representation ring of Γ, then our interest lies in studying the quiver varieties M θ := M θ (v, w), where the GIT stability parameter θ can be regarded as an element in the rational vector space Θ := Hom(R(Γ), Q); equivalently, M θ can be regarded as a moduli space of θ-semistable Π-modules, where Π is the framed preprojective algebra of Γ (see section 3.1 for details). A result of Kuznetsov [45] (also due to Haiman [34] and Nakajima [57]) determines an open GIT chamber C − in Θ and a commutative diagram X = Hilb [n] for any θ ∈ C − , where the horizontal arrows are isomorphisms and where the right-hand symplectic resolution is obtained by variation of GIT quotient.
Our first main result calculates explicitly the GIT chamber decomposition of Θ and describes the geometry of the quiver varieties M θ whenever θ lies in a chamber. To state the result, write Φ + for the set of positive roots in the ADE root system of finite type associated to Γ by the McKay correspondence, and for γ ∈ R(Γ) we write γ ⊥ := {θ ∈ Θ | θ(γ) = 0} for the dual hyperplane.
In particular, for any such θ ∈ Θ, there is a birational map ψ θ : X M θ over Y that is an isomorphism in codimension-one.
As a result, taking the proper transform along ψ θ enables us to identify canonically the Néron-Severi spaces of X and M θ for any generic θ, so we may regard the ample cone of M θ as lying in N 1 (X/Y ).
In order to understand the birational geometry of X, we exploit the link between wall-crossing for stability parameters and birational transformations of quiver varieties provided by the linearisation map arising from the GIT construction of M θ . To define this map, let C be any GIT chamber in Θ and fix θ ∈ C. The quiver variety M θ comes equipped with a tautological locally-free sheaf R := i∈I R i , where the summand R i has rank v i for each vertex i ∈ I in the framed affine Dynkin graph (see section 3.4). The linearisation map for the chamber C is the Q-linear map L C : Θ −→ N 1 (X/Y ) defined by sending η = (η i ) i∈I to the class of the line bundle L C (η) = i∈I det(R i ) ⊗ηi . Theorem 1.1 is a key ingredient in enabling us to prove that L C is a linear isomorphism that identifies the chamber C with the ample cone Amp(M θ /Y ) for θ ∈ C. In order to understand how the linearisation maps L C are related as we cross a wall between adjacent chambers in Θ, we focus our attention initially on chambers contained in the simplicial cone F := δ, ρ 1 , . . . , ρ r ∨ := {θ ∈ Θ | θ(δ) ≥ 0, θ(ρ i ) ≥ 0 for 1 ≤ i ≤ r}. (1.1) Note that F is a union of the closures of GIT chambers by Theorem 1.1.
Theorem 1.2. The linearisation maps L C for chambers C in the cone F glue to define an isomorphism of rational vector spaces that identifies the GIT wall-and-chamber structure of F with the decomposition of Mov(X/Y ) into Mori chambers. In particular, for any generic θ ∈ F , the moduli space M θ is the birational model of X determined by the line bundle L F (θ).
This result provides information only about the quiver varieties M θ for generic parameters θ in the cone F (see Theorem 1.7 for a stronger statement), but this is all that we require to establish the following result: Corollary 1.3. For n ≥ 1 and a finite subgroup Γ ⊂ SL(2, C), suppose that X ′ → C 2n /Γ n is a projective, crepant resolution. Then there exists a generic stability parameter θ such that X ′ ∼ = M θ .
Analogous statements appear in the literature for certain classes of singularities in dimension three: for Gorenstein affine toric threefolds, see Craw-Ishii [16], Ishii-Ueda [36]; and for compound du Val singularities, see Wemyss [68]. In higher dimensions, the analogous result for nilpotent orbit closures is due to Fu [28].

1.2.
Wall-crossing and Ext-graphs. To prove Theorem 1.2, we fix a reference chamber C in F and study how the quiver varieties M θ and, where necessary, the tautological bundles R i , change as we cross walls. In fact, the reference chamber need not be the chamber C − defining X = Hilb [n] (S), so all of our results are independent of the paper [45] cited above.
For each wall of a chamber C in F , we study the morphism f : M θ → M θ0 obtained by varying a parameter θ ∈ C to a parameter θ 0 that is generic in the wall. The idea is to understand fétale locally by providing a relative version of theétale local description of M θ by Bellamy-Schedler [7], which in turn builds on work of Crawley-Boevey [21], Nakajima [56] and Kronheimer [44]. The key statement is the following result that is valid more generally for the Nakajima quiver variety M θ (v, w) associated to any graph, any choice of dimension and framing vectors, and any stability condition; see section 3.2 for the relevant definitions and a more precise statement. Returning to the proof of Theorem 1.2, our description of the hyperplane arrangement in Theorem 1.1, provides enough information to compute explicitly the Ext-graph associated to any closed point on the quiver variety M θ0 , where θ 0 is generic in any GIT wall that lies in F . The graphs that arise are quite simple, including for example, the disjoint union of a collection of graphs each comprising one vertex and one edge loop. As such, it is possible to recognise the morphisms M ̺ (m, n) → M 0 (m, n) from Theorem 1.4 that appear in theétale local description of the contractions induced by each wall. In this way, we show for every wall in the interior of F separating chambers C and C ′ in F , that the birational map induced by crossing the wall is a flop, and moreover, that the line bundles det(R i ) on M θ are each the proper transform along ϕ of the corresponding line bundle det(R ′ i ) on M θ ′ . It then follows from the definition that the linearisations maps L C and L C ′ of the chambers on either side of the wall agree. Repeating this argument across all chambers in F determines the linear isomorphism L F from Theorem 1.2 whose restriction to any chamber C in F identifies C with the cone Amp(M θ /Y ) for any θ ∈ C. In addition, we demonstrate that the morphism f : M θ → M θ0 induced by moving a GIT parameter from a chamber C of F into a boundary wall of F is necessarily a divisorial contraction; this includes the wall δ ⊥ ∩ F of the chamber C − which induces the Hilbert-Chow morphism Hilb [n] (S) → Sym n (S).
1.3. The movable cone. The hyperplanes in the arrangement A from Theorem 1.1 that pass through the interior of the cone F can be computed explicitly, so the decomposition of the movable cone Mov(X/Y ) into Mori chambers can be obtained easily from Theorem 1.2: Theorem 1.5. The division of the movable cone Mov(X/Y ) = L F (F ) into Mori chambers is determined by the images under the isomorphism L F of the hyperplanes (mδ − α) ⊥ for all 0 < m < n and α ∈ Φ + .
We have that Amp(X/Y ) = L F (C − ); more generally, the chambers in this decomposition are precisely the ample cones of the projective, crepant resolutions of Y .
This generalises the result of Andreatta-Wiśniewski [1, Theorem 1.1] in the case when n = 2 and Φ is of type A r (see Example 6.7), and provides an answer to the question of Fu [29,Problem 1].
It turns out that an affine slice of the movable cone admits a purely combinatorial description. The affine hyperplane Λ := {θ ∈ Θ v | θ(δ) = 1} lies parallel to the supporting hyperplane δ ⊥ of F . In particular, the slice F ∩ Λ determines completely the wall-and-chamber decomposition of F , so the image of this slice under L F determines completely the Mori chamber decomposition of Mov(X/Y ): Corollary 1.6. The intersection of Mov(X/Y ) with the affine hyperplane {L F (θ) | θ(δ) = 1} is isomorphic to the decomposition of the fundamental chamber of the (n − 1)-extended Catalan hyperplane arrangement associated to Φ.
This result provides a geometric realisation of the extended Catalan hyperplane arrangement that was introduced originally by Postnikov-Stanley [64] and studied further by Athanasiadis [3].
Our description of the movable cone also provides new proofs for several results from the literature: • Corollary 1.6 implies that the number of non-isomorphic projective crepant resolutions of C 2n /Γ n is • Define an action of W on N 1 (X/Y ) by setting s δ to be reflection in L F (δ ⊥ ) and s i to be reflection • We provide a purely quiver-theoretic proof of the fact that X is a relative Mori Dream Space over Y (see Corollary 6.5). This is a special case of [1, Theorem 3.2] when n = 2, and follows from work of Namikawa [63, Lemma 1, Lemma 6] when n > 2.
It is perhaps worth making a philosophical remark about the final point. Any (relative) Mori Dream Space has a finitely generated Cox ring, and in our situation this ring was described by generators and relations by Donten-Bury-Grab [24, Section 5] when n = 2 and Φ is of type A 1 . While we do not make use of the Cox ring in this paper, the fact that our Corollary 1.3 reconstructs all small birational models of X by GIT as quiver varieties for the affine Dynkin graph of Γ suggests that the preprojective algebra Π should be thought of as a kind of 'noncommutative Cox ring' for each of the varieties Hilb [n] (S) with n ≥ 1.
1.4. Strong version via the Namikawa Weyl group. We now explain how to understand the quiver varieties M θ for parameters θ that lie beyond the simplicial cone F . For 1 ≤ i ≤ r, write s i : Θ → Θ for the reflection in the hyperplane ρ ⊥ i , and write s δ : Θ → Θ for reflection in the hyperplane δ ⊥ . The Namikawa Weyl group is the group W := s δ , s 1 , . . . , s r generated by these reflections. We prove (see Proposition 2.2) that the action of W permutes the set of GIT chambers in Θ, and that the simplicial cone F introduced in (1.1) above is a fundamental domain for the action of W on Θ. The next result provides a stronger version of Theorem 1.2: Theorem 1.7.
(i) Under the identification of the Néron-Severi spaces induced by the birational maps from Theorem 1.1, the linearisation maps L C glue to a piecewise-linear, continuous map (ii) The map L is invariant with respect to the action of W on Θ, i.e. L(θ) = L(wθ) for all w ∈ W and θ ∈ Θ. In particular, the image of L is the movable cone Mov(X/Y ).
(iii) The map L is compatible with the chamber decomposition of Θ and the Mori chamber decomposition of Mov(X/Y ), in the sense that for any chamber C ⊂ Θ v and for any θ ∈ C, the moduli space M θ is the birational model of X determined by the line bundle L(θ).
(iv) For each chamber C ⊂ Θ, the map L| C : The following result was anticipated by Losev [48].
Once we prove that L is invariant under the action of W as in Theorem 1.7(i), then parts (ii)-(iv) follow from Theorem 1.2. To achieve this, for each reflection s 1 , . . . , s r , we study the corresponding Nakajima reflection functor and its effect on the tautological bundles of M θ . The case of the reflection s δ has to be treated separately by studying the isomorphism from M θ to M −ι(θ) , where ι is either the identity or is induced by an order two symmetry of the McKay graph of Γ (see section 7.2).
The work of Bezrukavnikov-Kaledin [9] shows that every symplectic resolution X ′ → C 2n /Γ n possesses a collection of tilting bundles (called Procesi bundles) that induce derived equivalences between the derived category of coherent sheaves on X ′ and the derived category of Γ n -equivariant coherent sheaves on C 2n .
Remarkably, these Procesi bundles were classified completely by Losev [48], at least when X ′ is a quiver variety M θ (v, w). Moreover, he confirmed the first half of a conjecture of Haiman [35,Conjecture 7.2.13], that there is a unique Procesi bundle on each M θ (v, w) whose Γ n−1 -invariant summand is the tautological bundle R θ (v, w). Corollaries 1.3 and 1.8 now imply the following: Corollary 1.9. Let X ′ → C 2n /Γ n be a projective, symplectic resolution, and let C ⊂ F be the chamber satisfying L(C) = Amp(X ′ /Y ). For every normalised Procesi bundle P on X ′ , there exists a unique w ∈ W such that the Γ n−1 -invariant part of P is the tautological bundle R w(θ) on M w(θ) ∼ = X ′ for θ ∈ C. Moreover, every normalised Procesi bundle on X ′ arises in this way. In particular, there is a bijection between elements of W and the normalised Procesi bundles on each projective crepant resolution of C 2n /Γ n .
In addition, confirmation of the second half of Haiman's Γ n -constellation conjecture, when combined with Corollary 1.3, would imply every projective crepant resolution X ′ of C 2n /Γ n is a fine moduli space of stable modules over the skew group algebra C[V ×n ] ⋊ Γ n . It would then follow from Bayer-Craw-Zhang [4,Section 7], together with the derived equivalence of [9], that every such X ′ can be realised as a moduli space of Bridgeland-stable objects in the derived category of coherent sheaves on X.
1.5. The universal Poisson deformation. Kronheimer's realisation of the minimal resolution of the Kleinian singularity as a morphism of quiver varieties led to a new construction of the semiuniversal deformation of C 2 /Γ and its simultaneous resolution. In higher dimensions, the semi-universal deformation does not behave well; instead the natural object to consider is the universal graded Poisson deformation, as defined by Ginzburg-Kaledin [31] and Namikawa [62].
It was shown by Kaledin-Verbitsky [41] and Losev [47] that each symplectic resolution M θ of the quotient singularity Y admits a universal graded Poisson deformation Namikawa [62] showed that Y also admits a universal graded Poisson deformations Y → h/W . On the other hand, by taking the preimage under the moment map of a point λ ∈ h, one gets a graded Poisson deformation M θ → h for any θ ∈ Θ. The morphism f θ : M θ → M 0 obtained by variation of GIT quotient extends to a projective morphism f θ : M θ → M 0 . Theorem 1.10. Let C ⊆ Θ be a chamber, and let θ ∈ C.
(i) There exists a unique w ∈ W and graded Poisson isomorphism M θ ∼ → X such that the diagram commutes. Thus, the flat family M θ → h is the universal graded Poisson deformation of M θ .
(ii) The morphism f θ : M θ → M 0 is a crepant resolution, and an isomorphism in codimension one.
The situation is summarised in the following commutative diagram where the lower right rectangle is shown to be Cartesian. the stability space Θ studied here is isomorphic to the rational Picard group of X rather than the rational Grothendieck group. Put another way, the framed McKay quiver has too few vertices to be the quiver encoding a tilting bundle on X when n > 1. In particular, the quiver varieties M θ (v, w) that we study here cannot be realised directly as moduli spaces of Bridgeland-stable objects in the derived category of coherent sheaves on X in the manner described in [4,Section 7].
Recently, McGerty-Nevins [54] have shown that Kirwan surjectivity holds for quiver varieties. That is, for each chamber C, with θ ∈ C, the natural map K C : Our map L C fits naturally into a commutative diagram is the space of characters of g. As noted above, it follows easily from Theorem 1.1 that L C is an isomorphism. The two vertical maps are also isomorphisms, so the map K C : is an isomorphism too in our case. Therefore, our main results demonstrate precisely the extent to which the linear map K C extends across chambers to give a linear map on a union of chambers.
Since this paper was written, Theorem 1.5 enabled Gammelgaard, Gyenge, Szendrői and the second author [15] to construct the Hilbert scheme of n points on C 2 /Γ as a quiver variety M θ for a particular nongeneric parameter θ. Nakajima [60]  We also thank the anonymous referee for many helpful comments and corrections.

A wall-and-chamber structure
We begin by providing an elementary description of a wall-and-chamber decomposition of a rational vector space Θ, together with an action of the Namikawa Weyl group on Θ. This section is purely combinatorics and uses no machinery from Geometric Invariant Theory (GIT).
2.1. The chamber decomposition. Let Γ ⊂ SL(2, C) be a finite subgroup and let n ≥ 1 be an integer. Let V denote the given 2-dimensional representation of Γ and list the irreducible representations as ρ 0 , . . . , ρ r , where ρ 0 is the trivial representation.
The McKay graph is the affine Dynkin diagram associated to Γ by the McKay correspondence; explicitly, the vertex set is Irr(Γ), and there are dim Hom Γ (ρ i , ρ j ⊗ V ) edges between vertices ρ i and ρ j . Since V is self-dual, this is symmetric in ρ i and ρ j . Let A Γ denote the adjacency matrix of this graph. McKay [55] observed that if we define C Γ := 2Id − A Γ and equip the integral representation ring Zρ i with the symmetric bilinear form given by (α, β) Γ := α t C Γ β, then we obtain the root lattice of an affine root system Φ aff of type ADE in which the McKay graph is the Dynkin diagram, the irreducible representations {ρ 0 , . . . , ρ r } provide a system of simple roots, and the regular representation is the minimal imaginary root. The corresponding root system of finite type Φ ⊂ Φ aff is the intersection of Φ aff with the integer span of the nontrivial irreducible representations. Let Φ + denote the set of positive roots.
Given v ∈ R(Γ), we let v ⊥ := {θ ∈ Θ | θ(v) = 0} denote the dual hyperplane; and given v 1 , . . . , v m ∈ R(Γ), Consider the hyperplane arrangement in Θ given by A chamber in Θ is the intersection with Θ of a connected component of the locus and we let Θ reg denote the union of all chambers in Θ. The closure of each chamber defines a top-dimensional cone in a chamber complex in Θ determined by A, and the codimension-one faces of these top-dimensional cones are called walls in Θ. Each wall is contained in a unique hyperplane from A.
is a chamber of Θ. Indeed, since C − is cut out by specifying a strict inequality for each hyperplane in A, the claim follows provided that we show that it's non-empty. Let h = 0≤i≤r δ i be the Coxeter number of Φ. If we set θ i = 1 for i ≥ 1 and θ 0 = 1 2n − h + 1, then θ(mδ) = m 2n and θ(α) ≥ 1 for all α ∈ Φ + . This shows that θ ∈ C − as required.
(2) The interior C + of the closed cone is also a chamber in Θ, because if we set θ i = 1 for i ≥ 0, then θ ∈ C + and the statement follows by the same logic as in part (1). In fact, C + can be described more simply as the interior of the cone where β is the highest positive root, we have that the closure of C + is contained in (2.4). For the opposite inclusion, mδ > α for 1 ≤ m < n, so each of δ, α, mδ ± α can be expressed as a positive sum of ρ 0 , . . . , ρ r . It follows that the inequalities defining C + in (2.3) can be deduced from the inequalities θ i ≥ 0 for 0 ≤ i ≤ r which characterise the cone (2.4).
2.2. The Namikawa Weyl group. We now introduce an action of a finite group on Θ. For 1 ≤ i ≤ r, let for any 0 ≤ j ≤ r, if we write c i,j for the (i, j)-th entry of the Cartan matrix C Γ , then In addition, consider the involution s δ : R(Γ) → R(Γ) defined by sending δ to −δ and fixing ρ 1 , . . . , ρ r ; explicitly, We caution the reader that the subspace in R(Γ) orthogonal (with respect to (−, −) Γ ) to ρ differs from ρ ⊥ = {θ ∈ Θ | θ(ρ) = 0}, which is a hyperplane in the dual space Θ.
We are primarily interested in the dual action on Θ. For 1 ≤ i ≤ r, we use the same notation s i : Θ → Θ for the linear map defined for θ ∈ Θ and 0 ≤ j ≤ r by setting s i (θ)(ρ j ) = θ(s −1 i (ρ j )) = θ(s i (ρ j )), where we use the fact that s i is an involution. It follows for θ ∈ Θ and 1 ≤ i ≤ r that s i (θ) ∈ Θ has components Similarly, the dual map s δ : Θ → Θ sends θ to the vector s δ (θ) with components Now define the Namikawa Weyl group to be the group W := s δ , s 1 , . . . , s r generated by these reflections of Θ.
Proposition 2.2. The Namikawa Weyl group W satisfies the following properties: where W Γ is the Weyl group of the root system Φ; (ii) the simplicial cone in Θ defined by we see that the cone F is a fundamental domain for the action of W . For (iii), the action of the generator s δ fixes δ ⊥ and exchanges (mδ + α) ⊥ with (mδ − α) ⊥ for all 0 ≤ m < n and α ∈ Φ + , while the simple reflections s 1 , . . . , s r permute the hyperplanes in A. It follows that W permutes the chambers.
2.3. Counting chambers. The supporting hyperplanes of F lie in A, so F ∩ Θ reg is a union of chambers.
Our next goal is to count the number of chambers in F . We begin with a useful lemma. Proof. We give two proofs. For the first, we claim that F is equal to the cone in Θ generated by the closures of the cones C ± from Example 2.1. Indeed, the cone F is generated by the vectors f 0 , . . . , f r ∈ Θ where f 0 := (1, 0, . . . , 0) and, for 1 ≤ i ≤ r, the vector f i satisfies We have that f 0 ∈ C + and f 1 , . . . , f r ∈ C − , so F lies in the cone generated by C − ∪ C + . For the opposite inclusion, we have C + ⊂ F by (2.4), while the inequalities θ(α) > 0 for α ∈ Φ + defining C − include θ(ρ i ) > 0 for 1 ≤ i ≤ r and hence C − ⊂ F . This proves the claim. It follows that the walls passing through the interior of F are those for which the corresponding defining inequality changes from > to < or vice-versa when we compare C ± . The result follows by comparing the lists from (2.2) and (2.3).
The second approach is more explicit. The hyperplanes δ ⊥ and α ⊥ for α ∈ Φ + support the facets of F and can be discarded. Notice that if θ is in the interior of F then θ(δ), θ(α) > 0 implies that θ(mδ + α) > 0, so the hyperplanes (mδ + α) ⊥ for α ∈ Φ + and 0 < m < n do not intersect the interior of F . Hence it suffices to show that (mδ − α) ⊥ intersects the interior of F for all 0 < m < n and α ∈ Φ + . Let θ i = 1 for 1 ≤ i ≤ r.
Theorem 2.4. The cone F contains precisely chambers, where r is the rank, h is the Coxeter number and d 1 , . . . , d r are the degrees of the basic polynomial invariants of W Γ .
Proof. Every chamber in F intersects the affine hyperplane Λ := {θ ∈ Θ | θ(δ) = 1} in an open region of dimension r, so it suffices to count the number of these regions. The hyperplanes ρ ⊥ i for 1 ≤ i ≤ r intersect Λ to give a system of coordinate hyperplanes in Λ ∼ = Q r with origin at f 0 = (1, 0, . . . , 0) ∈ Θ, and Λ ∩ F is the positive orthant Q r ≥0 . More generally, the intersection of Λ with the hyperplanes from Lemma 2.3 and the hyperplanes ρ ⊥ i for 1 ≤ i ≤ r defines the following collection of affine hyperplanes in Λ: this is the (n − 1)-extended Catalan hyperplane arrangement of Φ from [3], or one of the truncated Φ aff -affine arrangements from [64]. (1) The proof of Theorem 2.4 shows that the unbounded regions in the (n − 1)-extended Catalan hyperplane arrangement of Φ are precisely the intersection with the affine hyperplane Λ of those chambers in F whose closure touches the facet δ ⊥ of F .
(2) Proposition 2.2 and Theorem 2.4 together imply that there are Example 2.6. For n = 4 and Φ of type A 2 , Figure 1 illustrates in two ways the decomposition of the cone F into chambers: Figure 1(a) shows all 22 regions in the fundamental chamber of the 3-extended Catalan hyperplane arrangement of Φ in the affine plane Λ parallel to δ ⊥ that was introduced in the proof of Theorem 2.4; Figure 1(b) shows the height-one slice of F and its division into 22 chambers. In each case, we indicate where the chambers C ± lie. The seven unbounded regions in Figure 1(a) correspond to the seven chambers in Figure 1(b) whose closure touches the facet of F in δ ⊥ . Notice that three chambers in F are not the interior of a simplicial cone.

3.Étale local normal form for quiver varieties
Modelled on Crawley-Boevey'sétale local description of affine quiver varieties, we give anétale local normal form for the morphism between quiver varieties defined by variation of GIT quotient. Pullback allows us to identify the tautological bundles on the quiver variety with the tautological bundles on the normal form. The results of this section hold for arbitrary quiver varieties.
The group G(v) := r k=0 GL(V k ) acts naturally on the space This action of G(v) is Hamiltonian for the natural symplectic structure on M(v, w) and, after identifying the via the trace pairing, the corresponding moment map µ : ). Though one can talk about arbitrary stability conditions in this context, as was done in [59], it is easier in our case to apply the trick of Crawley-Boevey [19] and reduce to the case where each W k = 0 by introducing a framing vertex.
The set H associated to the graph can be thought of as the arrow set of a quiver. We frame this quiver by adding an additional vertex ∞, as well as w i arrows from vertex ∞ to vertex i and another w i arrows of representations of Q of dimension vector v in such a way that the G(v)-action on M(v, w) corresponds to the action of the group G(v) := i∈I GL(v i ) /C × on Rep(Q, v) by conjugation and, moreover, that the above map µ corresponds to the moment map µ induced by this G(v)-action on Rep(Q, v). If we write . For θ ∈ Θ v , after replacing θ by a positive multiple if necessary, the (Nakajima) quiver variety associated to θ is the categorical quotient slice of the coordinate ring of the affine variety µ −1 (0). Note that C × acts on M(v, w) by scaling, and this action descends to an action on M θ (v, w). is said to be θ-polystable if it is a direct sum of θ-stable Π-modules. King [42] proved that a Π-module M of dimension vector v is θ-semistable (resp. θ-stable) if and only if the corresponding point of µ −1 (0) is χ θsemistable (resp. χ θ -stable) in the sense of GIT. In fact [42, Propositions 3.2,5.2] establishes that the quiver The geometry of the quiver varieties M θ (v, w) may change as we vary the stability parameter θ ∈ Θ v . We work of Dolgachev-Hu [23] and Thaddeus [65] implies that there is a wall-and-chamber structure on the cone of effective stability parameters in Θ v , where two generic parameters θ, θ ′ ∈ Θ v lie in the same (open) GIT chamber if and only if the notions of θ-stability and θ ′ -stability coincide, in which case The GIT walls of a GIT chamber are the codimension-one faces of the closure of the chamber.
Remark 3.1. Note that a priori, the locus of generic stability parameters could be empty.

3.2.
A local normal form. We begin by describing anétale local form for M θ (v, w) based on Luna's slice theorem, generalising [21,Section 4]. Let A be the adjacency matrix of the framed (doubled) quiver Q, i.e.
Then A is a symmetric matrix and we define a symmetric bilinear form on Z I by setting Let p be the quadratic form on Z I defined by setting Let θ ∈ Θ v , and choose θ 0 ∈ Θ v that lies in the boundary of the closure of the GIT chamber containing θ (the stability condition θ 0 should not be confused with the component of θ corresponding to the vertex 0; in all that follows, we never refer to the latter). Then there is a projective morphism induced by variation of GIT quotient. In this generality, f need not be birational. where n i = −(β (∞) , β (i) ). Finally, define the exponent ̺ ∈ Hom(Z k+1 , Q) for a rational character of G(m) by M 0 (m, n) × C 2ℓ of x and 0 respectively, together with a projective morphism ξ : Z → Z and a closed point z ∈ Z, forming a diagram where both squares are Cartesian, all horizontal maps areétale, and where p(z) = x, q(z) = 0.
Theorem 3.2 implies: Passing to the formal neighbourhood of x in V , and the formal neighbourhood of f −1 (x) in f −1 (V ), we deduce, as was explained in [8, Section 2.1.6], that there is a commutative diagram of formal schemes: . As in loc. cit., we let µ y denote the composition of µ with the quotient map g(v) * → g(m) * and let L be a G(m)-stable complement to g(m) in g(v). As in loc. cit., we define ν : where ω is the G-invariant symplectic form on M(v, w). For c ∈ C, g ∈ g(m) and l ∈ L we calculate The following two results are each a relative version of [7, Theorem 3.3].
Lemma 3.5. There exists a G(m)-saturated affine open subset U 0 of 0 ∈ C such that: (iii) these maps induce a Cartesian diagram with both horizontal mapsétale.
. We now apply [7, Equation (4), Lemma 3.9] to obtain a G(m)-saturated affine open neighbourhood U 0 of 0 in C such that (i) holds.
Then h is ̺-semi-invariant, and hence c ∈ U ̺ . For the opposite inclusion, the tensor product of the counit and the identity map gives a G(m) Frobenius reciprocity [38,Proposition 3.4] gives This also implies that that diagram is Cartesian, with the horizontal maps beingétale and G(v)-equivariant. Taking the GIT quotient gives the Cartesian diagram (3.7), so (iii) holds as required.
Let µ m denote the moment map for the action of G(m) on M(m, n). It is explained in [21, §4] that C ∩ (g(v) · y) ⊥ can be identified, as a G(m)-module, with representations of a certain doubled quiver.
This doubled quiver is precisely the framed doubled quiver associated to the Ext-graph described in section 3.2, except that we have neglected to include the p(β (∞) ) = ℓ loops at vertex ∞ in our Ext-graph. Since the dimension vector m of the framed doubled quiver satisfies m ∞ = 1, there is a factor of C 2ℓ in the representation space, corresponding to the value of the endomorphisms at the loops, on which G(m) acts trivially. That is, we can identify C ∩ (g(v) · y) ⊥ = M(m, n) × C 2ℓ as G(m)-modules, where G(m) acts trivially on C 2ℓ , in such a way that µ m is identified with the restriction of µ y to C ∩ (g(v) · y) ⊥ .   Recall that Q is a doubled quiver with framing vertex ∞, and v denotes the dimension vector for Q associated to a dimension vector v for Q {∞}. Since v is primitive, King [42,Proposition 5.3] proves that In this case, the universal family on M θ (v, w) is a tautological locally-free sheaf together with a C-algebra homomorphism Π → End(R). Explicitly, for i ∈ I we write F i for the representation of G(v) obtained by pulling back the vectorial representation from the ith factor of G(v). Then R i is the locally-free sheaf associated to the vector bundle When we wish to emphasise the dependence of R i on the dimension vectors, we write R i (v, w). Since R is defined only up to tensor product by an invertible sheaf, we normalise by fixing R ∞ to be the trivial bundle.
As in section 3.2, choosing a closed point x ∈ M θ0 (v, w), where θ 0 lies in the closure of the GIT chamber containing θ, determines k ≥ 0 and dimension vectors β (0) , . . . , β (k) ∈ Z I , dimension vectors m, n for the Ext-graph of x, and a stability condition ̺ ∈ Hom(Z k+1 , Q), which determine the quiver variety M ̺ (m, n).
Recall now the statement of Theorem 3.2, and specifically, diagram (3.4): Proposition 3.8. The quiver variety M ̺ (m, n) carries a tautological locally-free sheaf j R j (m, n), where j ranges over the set {∞, 0, . . . , k}. Moreover, for each i ∈ I \ {∞}, there is an isomorphism of bundles on Z ′ given by Proof. The vector m determines a dimension vector m for the framed (doubled) quiver satisfying m ∞ = 1, so m is primitive. In light of [42,Proposition 5.3], it remains to show that ̺ is generic in order to prove the first statement.
We claim that if θ is generic with respect to v then ̺ is generic with respect to m. Our argument is based The fact that r is strictly polystable means that G(m) r ′ , and hence G(v) y ′ , is non-trivial. This in turn implies that x ′ is strictly polystable, contradicting the assumption that θ is generic for v.
For the second statement, the locally-free sheaf p * here the first isomorphism follows from the proof of Lemma 3.5, and the second is a consequence of Luna's slice theorem [49]. n). Thus, the result follows from the fact that This latter decomposition is simply the fact that x corresponds to the θ 0 -polystable representation This completes the proof.
3.5. A stratification. There are two natural ways of defining a finite stratification of the quiver variety.
The first is Luna's stratification, coming from the fact that it is a GIT quotient by a reductive group. The second is the stratification into symplectic leaves, which is finite since the quiver variety has symplectic singularities [7, Theorem 1.2]. It is known that these two stratifications agree [7,Proposition 3.6]. In this section, we recall the definition of these strata. In order to do so, we first recall Crawley-Boevey's canonical decomposition of a dimension vector.
As explained in [39], associated to the Cartan matrix C of the framed (doubled) quiver Q = (I, Q 1 ) is a (3.10) In general, it is very difficult to compute Σ θ , but notice that if θ(β) = 0 for all roots β < α, then α ∈ Σ θ .
where γ (i) ∈ Σ θ for 0 ≤ i ≤ ℓ, such that any other decomposition of v as a sum of elements from Σ θ is a refinement of the sum (3.11); this is the canonical decomposition of v with respect to θ.
The strata of M θ (v, w) are labelled by the "representation types" of v, which we now recall. A represen- where β (i) ∈ Σ θ , n i ∈ Z >0 and k i=0 n i β (i) = v. The stratum labelled by the representation type τ is: Remark 3.10. By [7,Corollary 3.25], each stratum is a connected locally-closed, smooth subvariety of M θ (v, w). Since the dimension of the locus of θ-stable points in M θ (v, w) equals 2p(v), Theorem 3.9(i) implies that the stratum M θ (v, w) τ is non-empty if and only if β (i) = β (j) when β (i) and β (j) are real roots.
The stratification is finite, and See [7, §3.5] and references therein for more information.

Quiver varieties for the framed McKay quiver
We now specialise to the case where the quiver varieties M θ (v, w) are constructed from the affine Dynkin As explained in section 3.5, associated to the quiver Q is the root system R ⊂ Z I , and R + = N I ∩ R the set of positive roots. We can recover the affine root system Φ aff associated to Γ in section 2.1 as follows.
The proof requires two preliminary results, the first of which is an application of the Frenkel-Kac theorem.
where m = p(γ). Conversely, any vector of this form lies in R + .
Proof. Let V (ω 0 ) be the vacuum module (of level one) for the Kac-Moody algebra with root system R. The Frenkel-Kac Theorem [39, Lemma 12.6] says that w is a weight of V (ω 0 ) if and only if for some ν ∈ i≥1 Zρ i and m ≥ 0. Then, the statement follows from [59, Lemma 2.14].
Lemma 4.4. For θ ∈ Θ v , we have v ∈ R + θ and p(v) = n. In particular, if n > 1 then v is an anisotropic root.

4.2.
Characterising smooth quiver varieties. The following description of the affine quotient corresponding to the stability parameter θ = 0 is well-known.

Lemma 4.5.
There is an isomorphism of algebraic varieties M 0 ∼ = C 2n /Γ n that is also an isomorphism of Poisson varieties (up to rescaling the bracket by some t ∈ C × ).
Proof. This follows from results of Crawley-Boevey [20]; see also Kuznetsov [45,Proposition 33]. Indeed, Proposition 4.2 shows that the canonical decomposition of v in Σ 0 is ρ ∞ + δ + · · · + δ, where δ appears n To state the main result of this section, identify the space Θ v of GIT stability parameters with Θ via the projection away from the θ ∞ coordinate. We obtain from (2.1) a hyperplane arrangement in Θ v given by where now each γ ∈ Z I = Z ⊕ R(Γ) defines the dual hyperplane γ ⊥ := {θ ∈ Θ v | θ(γ) = 0}. As before, a chamber in Θ v is the intersection with Θ v of a connected component of the locus and we let Θ reg v denote the union of all chambers in Θ v . A wall is a codimension-one face of the closure of a chamber. A vector w ∈ Σ θ is said to be minimal (with respect to θ) if it does not admit a proper decomposition w = β (0) + · · · + β (k) with k > 0 and β (i) ∈ Σ θ .
Theorem 4.6. For n > 1, the following conditions on a stability parameter θ ∈ Θ v are equivalent: (i) the variety M θ is smooth; (ii) the canonical decomposition of v with respect to θ is of the form σ (0) + · · · + σ (ℓ) where each σ (i) ∈ Σ θ is minimal and a given imaginary root can appear at most once as a summand; (iii) the parameter θ lies in Θ reg v ; (iv) the parameter θ is generic, i.e. every θ-semistable Π-module of dimension vector v is θ-stable.
For the equivalence of (iii) and (iv), the previous paragraph shows that if θ lies on any of the hyperplanes in A v , then either v admits a canonical decomposition with more than one term (i.e., the generic polystable Πmodule of dimension vector v is not θ-stable), or v admits a proper decomposition. Grouping like terms, such a decomposition defines a representation type τ consisting of more than one term. This implies that there are θ-semistable representations of dimension v that are not θ-stable. Therefore (iv) implies (iii). Conversely, if θ does not lie on a hyperplane in A v , then the previous paragraph shows that the only decomposition of v as a sum of elements from Σ θ is v itself. Therefore every θ-semistable representation is θ-stable. (1) Theorem 4.6 implies that the wall-and-chamber structure on Θ v determined by A v coincides with the wall-and-chamber structure arising from the GIT construction of the quiver varieties M θ for θ ∈ Θ v . In particular, all results of Section 2 hold for the GIT decomposition of Θ v , so the walls and chambers introduced there are GIT walls and GIT chambers.
(2) It follows from Proposition 4.2 that whenever the equivalent conditions of Theorem 4.6 hold, the canonical decomposition of v is v, and moreover v is minimal.

Crepant resolutions.
We now investigate how the quiver varieties M θ are related to each other. where the wreath product Γ n := S n ≀ Γ = S n ⋉ Γ n acts on (C 2 ) n ∼ = C 2n in the natural way. That f is an example of one of the morphisms from Proposition 4.8 is due to Kuznetsov [45]; the same observation was made independently by Haiman [34] and Nakajima [57]. To state the result, Remark 4.7 enables us to regard the chamber C − from Example 2.1(1) as a GIT chamber in Θ v .    Proof. The proof is identical to the proof of Theorem 4.6, except in this case when θ ∞ = 0, the minimal imaginary root δ only appears with multiplicity one in the canonical decomposition. Therefore, a parameter θ on the hyperplane δ ⊥ , which is not contained in any α ⊥ for α ∈ Φ, is such that M θ is smooth. Every point of M θ corresponds to a strictly polystable representation even though the moduli space is smooth.
In particular, when n = 1 we recover the chamber structure for the unframed McKay quiver moduli construction due to Kronheimer [44] (see also Cassens-Slodowy [14]) in the hyperplane δ ⊥ . Indeed, it is well-known that the chamber decomposition in this case is the Weyl chamber decomposition for Φ. bracket. We now explain how to identify this family, when θ is generic, with the universal graded Poisson deformation of M θ whose existence was shown by Kaledin-Verbitsky [41] in the ungraded case, and adapted by Losev [47] to the graded case.
There is a commutative diagram where f θ : M θ → M 0 is the morphism given by variation of GIT.
Proof. The fact that the morphism is projective and Poisson follows directly from the definition of variation of GIT. To see that it is birational it suffices to note that there is a finite collection of hyperplanes in h such that f θ is an isomorphism away from those hyperplanes. Indeed, let S denote the complement to all hyperplanes α ⊥ for α < v a positive root; these define proper hyperplanes in h because v is indivisible. If λ ∈ S then the only positive root α ≤ v in the set Σ λ is v itself. By [19,Theorem 1.2], this implies that every representation of the deformed preprojective algebra Π λ of dimension v is simple, and hence θ-stable.
(i) The morphism σ 0 is flat, and M 0 is irreducible, normal, and Cohen-Macaulay.
(ii) If θ ∈ Θ reg v , then σ θ is flat and M θ is smooth and connected.
Proof. For (i), the locus µ −1 (h) is cut out by n 2 |Γ| − r equations in the affine space M(v, w) of dimension 2n(n|Γ| + 1), so it is a complete intersection of dimension n 2 |Γ| + 2n + r by [ normality, it suffices to show that M 0 satisfies (R 1 ). Let U be the set of points x in M 0 such that x is a regular point in the fiber σ −1 0 (σ 0 (x)). Then [53,Theorem 23.7] says that U is contained in the smooth locus of M 0 . Thus, we need to show that the complement C to U in M 0 has codimension at least two. The image of C in h is contained in the finitely many hyperplanes described in the proof of Lemma 4.13 since the set S described there satisfies σ −1 0 (S) ⊂ U . Since C ∩ σ −1 0 (λ) has codimension at least one in σ −1 0 (λ) (in fact it has codimension at least two since each fiber σ −1 0 (λ) is normal), codim M0 C ≥ codim h σ 0 (C) + 1 ≥ 2. For (ii), an argument similar to that for σ 0 above shows that σ θ is also flat. As for the variety M θ , all maps appearing in diagram (4.6) are equivariant for the natural scaling action of C × described in section 3.1, if we make C × act on h with weight one. This implies that the singular locus of M θ is a C × -stable closed subvariety. In particular, if it is non-empty then it would intersect the zero fiber σ −1 θ (0) = M θ non-trivially. However, since σ θ is flat and M θ , h are smooth, [53,Theorem 23.7] says that this intersection is trivial.
Thus, M θ is smooth. Similarly, each connected component of M θ is a closed C × -stable subvariety, and hence must intersect M θ non-trivially. But the latter is irreducible, so M θ is connected. The degeneracy locus of φ θ is closed and C × -stable. In particular, if it were non-empty it would intersect the fiber σ −1 θ (0) = M θ non-trivially. However, if i : M θ ֒→ M θ is the embedding of the zero fibre, then i * φ θ : Ω 1 M θ → T M θ is an isomorphism since the Poisson bracket restricted to M θ is non-degenerate. Therefore, we deduce that φ θ is an isomorphism. Thus, there is a relative symplectic form ω Mθ /h on M θ , dual to the Poisson bracket. Since h is obviously smooth, this implies that σ θ is smooth and, if ω h is a nowhere vanishing top form on h, then ω n M θ /h ∧ σ * θ ω h is a nowhere vanishing top form on M θ . In particular, the canonical divisor K M θ is trivial.
Similarly, on M 0 , we have φ 0 : Ω 1 M0/h → T M0/h 2 . Let D ⊂ M 0 be its degeneracy locus. This is a closed, Then, by Chevalley's Theorem, each D ≤i is closed and C × -stable. We know that dim D ∩ σ −1 0 (0) ≤ 2n − 2 since the map Ω 1 M0 → T M0 is an isomorphism on the smooth locus of M 0 . Therefore, we deduce D = D ≤2n−2 since the C × -action contracts every λ ∈ h to 0. This implies that dim D ≤ (2n − 2) + dim h = dim M 0 − 2, i.e. there exists an open subset U ⊂ M 0 whose complement D has codimension at least two, such that φ 0 is an isomorphism over U . We deduce that there exists a relative symplectic form ω M0/h on the open set U . Moreover, the fact that f θ is Poisson implies that f * θ ω M 0 /h = ω M θ /h . Again, the top form ω n M0/h ∧ σ * 0 ω h trivialises the canonical divisor K M 0 over U , and we deduce by normality that K M 0 = 0, i.e. M 0 is Gorenstein. Thus, f * θ K M 0 = K M θ , i.e. f θ is crepant.
Finally, we check that f θ is an isomorphism in codimension one. Suppose otherwise, so f θ contracts a divisor E in M θ . Note that E is necessarily stable under the action of C × . The map σ θ is proper, therefore σ θ (E) is closed in h. Since it lies in the complement to the set S defined in the proof of Lemma 4.13, it is a proper subset of h. Therefore, there must exist a fiber of σ θ | E : E → h of dimension at least 2n. Arguing as above, we must in fact have dim( is generically an isomorphism. In particular, a generic fiber is zero dimensional. This is a contradiction.
In the proof of Proposition 4.15, we also established the following result: Given any conic symplectic singularity Y that admits a symplectic resolution X → Y , Namikawa [62] showed that there exist universal graded Poisson deformations Y → h/W and X → h of Y and X respectively.
Write q : h → h/W for the quotient map. Proof. This follows from the main results of [6] and [63]. To see this, first identify h with the subset where the large rectangle at the front of the diagram is Cartesian.

Variation of GIT quotient in the cone F
We now describe the geometry of the quiver varieties M θ under variation of GIT quotient as the stability parameter θ passes from a chamber of F into a wall of the chamber. We also study the geometry of M θ and track what happens to the tautological bundles as θ crosses any wall that passes through the interior of F .

Classification of walls.
Consider the wall-and-chamber structure on the space Θ v of GIT stability conditions, as in equation (4.4). We now classify the walls into three families. We say that a wall is: (i) an imaginary boundary wall if it is contained in δ ⊥ ; (ii) a real boundary wall if it is contained in α ⊥ for some α ∈ Φ + ; and (iii) a real internal wall otherwise. In particular, f is a divisorial contraction. The statement of Corollary 5.2 below is therefore not new.
Nevertheless, we present the next result to provide an independent proof of this fact and, furthermore, to illustrate our approach via the Ext-graph as described in section 3.2.
In particular, v − (k + l)γ must be a positive root. Lemma 4.3 implies that i.e. 0 ≤ k + l ≤ N . Moreover, γ is a real root, so p(γ) = 0 and In particular, this applies to v (in the case k = 0) and shows that every representation type of v is of the form (1, v − kγ; k, γ). The symplectic leaves of M θ0 are in bijection with these representation types, with the type (1, v − kγ; k, γ) corresponding to But γ is real, so up to isomorphism there is only one θ 0 -stable representation of dimension γ. Therefore L k is isomorphic to the fine moduli space of θ 0 -stable Π-modules of dimension vector v − kγ. The fact that v − kγ ∈ Σ θ0 implies that L k is non-empty of dimension 2p(v − kγ) = 2(n − k(k + m)).

5.4.
The real boundary walls. Fix a positive real root α ∈ Φ + . The walls of F contained in α ⊥ are the real boundary walls. Assume that θ is chosen in a chamber C ⊂ F with θ 0 ∈ C ∩ α ⊥ a generic point on the wall α ⊥ . Assume that x ∈ L k for some k ≥ 1. The representation type of v labelling this leaf is (1, v − kα; k, α).
Note first that ℓ := p(v − kα) = n − k 2 . Also, the Ext-graph has one vertex 0, with p(α) = 0 loops at this vertex. We have m = (k) and the fact that −(v − kα, α) = 2k implies that n = (2k). Finally, we have ̺(∞) = −θ(kα) < 0 and ̺(0) = θ(α) > 0. If we fix a vector space Λ of dimension 2k, then an explicit calculation of quiver varieties shows that there is a commutative diagram in which the horizontal arrows are isomorphisms and the vertical arrows are symplectic resolutions; here, is the nilpotent orbit of rank k matrices whose square is zero, and G(k, Λ) is the Grassmanian of k-planes in Λ. It now follows from Theorem 3.2 that,étale locally at x, the morphism f : M θ → M θ0 can be identified with with x mapped to 0 ∈ C 2(n−k 2 ) × O. We claim that the exceptional locus of this morphism is an irreducible divisor. It suffices to assume that n = k 2 . Let Exp be the exceptional locus. If π : T * G(k, Λ) → G(k, Λ) is projection to the zero section, then we claim that π| Exp is a Zariski locally trivial fibre bundle, with each fibre π −1 (V ) ∩ Exp an irreducible hypersurface in π −1 (V ). Fixing a basis of V and extending this to a basis of Λ, we can identify the space π −1 (V ) ∩ Exp with the space of all k × k matrices B of rank < k (note that dim V = k and dim Λ = 2k). This is the zero set of the determinant, and hence a hypersurface. Being a special case of a determinantal variety, it is well-known to be irreducible. The claim follows and we deduce that the exceptional locus of f is irreducible. We note that this argument shows that the exceptional divisor is singular. Finally, if x belongs to the open subset L 1 of Z, then (5.1) becomes C 2n−2 ×T * P 1 → C 2n−2 ×C 2 /Z 2 which is a divisorial contraction.

5.5.
Internal walls. In order to state the main result for real internal walls, we recall the definition of a Mukai flop of type A due to Namikawa [61].
Let Λ be a finite dimensional vector space and choose an integer 1 ≤ k < dim Λ. Let O ⊂ gl(Λ) be the nilpotent orbit of rank k matrices whose square is zero, and let G(k, Λ) be the Grassmanian of k-planes in Λ. Provided 2k = dim Λ, the symplectic singularity O admits two different symplectic resolutions such that the diagram is a flop [61,Lemma 5.4]; that is, g ± are small projective resolutions, and if L + is a g + -ample line bundle on T * G(k, Λ) then the proper transform L − is such that L −1 − is g − -ample. In particular, ψ is not regular. One can obtain the above diagram by variation of GIT quotient for a certain quiver variety: begin with the graph with one vertex 0 and no arrows; take vectors v = (k) and w = (dim Λ); and calculate that The (real) interior walls are precisely those that are contained in γ ⊥ for γ = mδ − α, where α ∈ Φ + and 0 < m < n. There are uniquely defined chambers C, C ′ in F such that the wall is C ∩ C ′ . Choose θ ∈ C, θ ′ ∈ C ′ and let θ 0 be a generic point of this wall; fix conventions so that θ(γ) > 0 and θ ′ (γ) < 0. Let N be the largest positive integer such that N (N + m) ≤ n. Proof. First we note by Lemma 5.3 that M θ0 = N k=0 L k admits a finite stratification by smooth locally closed subvarieties, where the codimension of L k is 2k(k + m). Choose x ∈ L k ⊂ M θ0 . The representation type of v labelled by x is (1, β (∞) ; k, β (0) ), where β (∞) = v − kγ and β (0) = γ, so the Ext-graph has only one vertex in this case. Since p(γ) = 0, there are no loops at this vertex. The dimension vector m equals (k) and n = (m + 2k) because − β (∞) , β (0) = −(v − kγ, γ) = m + 2k. We have ℓ = p β (∞) = p(v − kγ) = n−k(m+k). Finally, the stability condition ̺ is given by ̺ 0 = θ(γ) and ̺ ∞ = θ(v−kγ). Similarly, ̺ ′ 0 = θ ′ (γ) and ̺ ′ ∞ = θ ′ (v − kγ). In particular, ̺ ∞ > 0 and ̺ ′ ∞ < 0. Thus, if we choose a vector space Λ of dimension m + 2k, it follows thatétale locally, the morphism f θ : M θ → M θ0 is isomorphic, in a neighbourhood of x, to the morphism C 2ℓ ×T * G(k, Λ) → C 2ℓ ×O in a neighbourhood of 0. Similarly, the morphism f θ ′ : is isomorphic, in a neighbourhood of x, to the morphism C 2ℓ × T * G(k, Λ * ) → C 2ℓ × O in a neighbourhood of 0. Since the maps g ± in diagram (5.2) are small contractions, the result follows.
Remark 5.6. We prove in Corollary 6.3 below that the rational map ϕ from Theorem 5.5 is a flop. In fact, using quite different techniques it is possible to show that these maps are 'stratified Mukai flops of type A' in the sense of Fu [29]. We will come back to this in future work.
The following proposition will be important later.
Proposition 5.7. Let Z ⊂ M θ denote the unstable locus for the wall containing θ 0 , i.e. Z is the set of points consisting of θ-stable Π-modules that are θ ′ -unstable. Similarly for Z ′ ⊂ M θ ′ . Then: (ii) under the identification from part (i), the restriction of the tautological bundle R i to M θ Z is identified with the restriction of the tautological bundle R ′ i to M θ ′ Z ′ ; and (iii) the closed subschemes Z and Z ′ have codimension at least m + 1 in M θ and M θ ′ respectively.
For the opposite inclusion, let x ∈ M θ0 M s θ0 and apply Lemma 5.3 when k = 0 to see that x corresponds to the polystable representation M ∞ ⊕ M ⊕k 0 , where dim M 0 = γ and dim M ∞ = v − kγ, for some k > 0. Since θ(γ) > 0, and θ ′ (γ) < 0, any point y ∈ f −1 θ (x) must correspond to a θ-stable representation N fitting into a sequence Any such point is clearly θ ′ -unstable, giving f −1 θ (M θ0 M s θ0 ) ⊆ Z which proves the claim. The isomorphism M θ Z ∼ = M θ ′ Z ′ from part (i) is given by the restriction of f −1 θ ′ • f θ to M θ Z. Part (ii) follows from part (i) because both M θ Z and M θ ′ Z ′ parametrise precisely those Π-modules of dimension vector v that are simultaneously θ-stable and θ ′ -stable. For part (iii), the codimension of Z is the maximum codimension of the locally closed subsets f −1 θ (L k ) for k > 0. Since f θ is semi-small by Theorem A.1, this number is at least m + 1 by Lemma 5.3.

Linearisation map to the movable cone
We now construct an isomorphism L C : Θ v → N 1 (X/Y ) of rational vector spaces for each chamber C in Θ v . For any two chambers C, C ′ in the simplicial cone F , it is shown that the isomorphisms L C and L C ′ are equal. This allows us to describe explicitly the chamber structure of the movable cone of X over Y . deg L| ℓ = deg L ′ | ℓ for every proper curve ℓ ⊂ X. Given a line bundle L on X, we use the same notation L ∈ N 1 (X/Y ) for the corresponding numerical class.
We now introduce several cones in N 1 (X/Y ). The (relative) movable cone Mov(X/Y ) ⊂ N 1 (X/Y ) is the closure of the convex cone generated by the divisor classes D for which the linear system |mD| has no fixed component for m ≫ 0. The (relative) nef cone Nef(X/Y ) ⊂ N 1 (X/Y ), is the closed cone of divisor classes D satisfying D · ℓ ≥ 0 for every curve ℓ contracted by f , and the (relative) ample cone Amp(X/Y ) is the interior of Nef(X/Y ). Note that Suppose now that f ′ : X ′ → Y is another projective, crepant resolution. Since X and X ′ are birational minimal models over Y [43,Corollary 3.54], there is a commutative diagram where the birational map ψ : X X ′ is an isomorphism in codimension-one. Taking the proper transform along ψ enables us to identify canonically the vector space N 1 (X ′ /Y ) with N 1 (X/Y ), and the movable cone Mov(X ′ /Y ) with Mov(X/Y ). The ample and nef cones of X ′ over Y , however, depend on curves in X ′ , but by taking the proper transform along ψ we may nevertheless identify them with the cones ψ * Amp(X ′ /Y ) and ψ * Nef(X ′ /Y ) respectively in Mov(X/Y ).
We now turn our attention to the quiver varieties for the framed McKay quiver. For any chamber C ⊂ Θ v and any θ ∈ C, diagram (6.1) specialises to the commutative diagram where f θ : M θ → Y is the symplectic resolution obtained from Proposition 4.8 and Lemma 4.5, and where the birational map ψ θ : X M θ is an isomorphism in codimension-one. We may therefore identify N 1 (M θ /Y ) with N 1 (X/Y ) and Mov(M θ /Y ) with Mov(X/Y ) by taking the proper transform along ψ θ . We also identify the ample and nef cones of M θ over Y with the cones ψ * θ (Amp(M θ /Y )) and ψ * θ (Nef(M θ /Y )) respectively in Mov(X/Y ). All of these cones are of top dimension in N 1 (X/Y ) because M θ is projective over Y .
6.2. The linearisation map. Let C ⊂ Θ v be any chamber. Recall that for θ ∈ C, the quiver variety M θ carries a tautological locally free sheaf R := i∈I R i that depends on the choice of chamber C, where R ∞ ∼ = O M θ and where R i has rank n dim(ρ i ) for i ∈ I \ {∞}. Define a Q-linear map as follows: for integer-valued maps η : Z I → Z, set and define L C in general by extending linearly over Q. The arguments that follow depend only on the choice of η up to a positive multiple, so we may assume without loss of generality that η takes only integer values.
The line bundle L C (θ) descends from the trivial bundle O µ −1 (0) linearised by the character χ θ , so L C (θ) is the ample line bundle O M θ (1) obtained from the GIT construction of M θ . More generally, for any stability parameter η ∈ C, we have that where g : M θ → M η is the morphism obtained by variation of GIT quotient. hence N 1 (X) ∼ = H 2 (X, Q). Since Y is affine, the dimension of N 1 (X/Y ) is also equal to r + 1. In particular, Θ v and N 1 (X/Y ) have the same dimension.
Suppose for a contradiction that L C does not have full rank. We claim that the image of C under L C equals that of the boundary ∂C. Indeed, for the non-obvious inclusion, suppose there exists ℓ ∈ L C (C) \ L C (∂C).
The intersection of the affine subspace L −1 C (ℓ) with C is a polyhedron P that contains some point of C, so P has positive dimension. The boundary of P is L −1 C (ℓ) ∩ ∂C, but this is empty by assumption, so P is an affine subspace of positive dimension. In particular, C contains an affine subspace of positive dimension, and hence it contains a vector subspace of positive dimension. But this is a contradiction, because the explicit hyperplane arrangement A v from (4.3) allows no room for the closure C of any chamber to contain a nonzero subspace. Therefore, if L C does not have full rank, then L C (C) = L C (∂C) as claimed. If we can deduce from this that L C (∂C) is contained in the boundary of the nef cone, then we obtain a contradiction because L C (C) is contained in the interior of the nef cone. Thus, to prove that L C is an isomorphism, we need only prove that L C (ζ) is nef but not ample for ζ ∈ ∂C. Suppose otherwise. After replacing ζ by a positive multiple if necessary, L C (ζ) is very ample and (6.3) implies that g : M θ → M ζ is a closed immersion.
Since M θ and M ζ have the same dimension, g is an isomorphism. This is absurd because M θ is smooth whereas M ζ is singular by Theorem 4.6, so L C (ζ) lies in the boundary of Nef(M θ /Y ) and hence L C is an isomorphism.
Since L C is a linear isomorphism, it identifies C with the interior of a polyhedral cone of full dimension in Amp(M θ /Y ). Moreover, we proved above that L C sends the boundary of C into the boundary of the nef cone. In particular, the supporting hyperplanes of the closure of the cone L C (C) must lie in the boundary of the nef cone. This implies L C (C) = Amp(M θ /Y ) and hence L C (C) = Nef(M θ /Y ).
Remark 6.2. The proof of Proposition 6.1 shows that the rank of the Picard group of X = Hilb [n] (S) is equal to 1 + rk(Pic(S)). If S were projective, this would follow from the main result of Fogarty [27]. Corollary 6.3. Let C, C ′ be adjacent chambers in F , and let θ 0 be generic in the separating wall C ∩ C ′ .
Then for θ ∈ C and θ ′ ∈ C ′ , the diagram involving the maps from Theorem 5.5 is a flop.
Proof. In light of Theorem 5.5, it remains to prove that the proper transform of the f θ -ample bundle L C (θ) is f θ ′ -antiample. Proposition 5.7 shows that det(R i ) is the proper transform along ϕ of det(R ′ i ) for all i ∈ I. In particular, the linearisation maps for the chambers C and C ′ agree, i.e.
It follows that the proper transform of L C (θ) is L C ′ (θ).
The ample bundle (6.3). It is possible to choose θ ∈ C and θ ′ ∈ C ′ such that θ 0 = 1 2 (θ + θ ′ ), giving Proposition 6.1 implies that the set of curve classes contracted by f θ ′ is non-empty. Since L C ′ (θ ′ ) and f * θ ′ (L 0 ) have positive and zero degree respectively on all such curves, it follows that has positive degree on all such curves, so its inverse L C ′ (θ) is f θ ′ -antiample.
Proof of Theorem 1.2. Choose any chamber C in F and define L F (θ) := L C (θ) for θ ∈ Θ v . To see that L F is well-defined, independent of the choice of C, let C ′ be a chamber in F that lies adjacent to C. A key fact, established in equation (6.5), is that the linearisation maps L C and L C ′ agree. Repeating this successively for all internal walls in F shows that L F is well-defined. Each L C is an isomorphism of vector spaces by Proposition 6.1, hence so is L F .
For each chamber C in F , Proposition 6.1 shows that the restriction L F | C = L C identifies C with the ample cone Amp(M θ /Y ) for θ ∈ C. In particular, the restriction of L F to the interior of (2) that Mov(X/Y ) is a simplicial cone. This recovers a special case of [1,Theorem 4.1].
In addition, Theorem 1.2 leads to a new, purely quiver-theoretic proof of the following result that is due originally to Andreatta-Wiśniewski [1, Theorem 3.2] for n = 2, and to Namikawa [63, Lemma 1, Lemma 6] for n > 2. Recall that a divisor class D is semi-ample if kD is basepoint-free for some k ≥ 1. (ii) there are only finitely many small birational models X = X 0 , X 1 , . . . , X k of X over Y , and Nef(X i /Y ), (6.6) where each cone in this description is generated by finitely many semi-ample line bundles.
Proof. The closure C of each chamber C in Θ v is finitely generated because the hyperplane arrangement A v is finite, hence so is the cone Nef(M θ /Y ) for any θ ∈ C by Proposition 6. Example 6.7. Let n = 2 and suppose that Φ is of type A r . Let e 0 , . . . e r denote the standard basis of Q r+1 .
with the cone e 0 , e 1 , . . . , e r ∨ . Moreover, if we write α i,j := ρ i + ρ i+1 + · · · + ρ j for 1 ≤ i < j ≤ r, then the normal vector δ − α i,j to each hyperplane in A v passing through F is identified with Therefore the Mori chamber decomposition of Mov(X/Y ) from Theorem 1.5 coincides with the decomposition of the cone e 0 , e 1 , . . . , e r ∨ obtained by cutting with the hyperplanes e 0 − (e i + · · ·+ e j ) ⊥ for 1 ≤ i < j ≤ r.
of Mov(X/Y ) is obtained by applying the isomorphism L F to the slice of F shown in Figure 1(a); and similarly for the transverse slice of Mov(X/Y ) shown in Figure 1(b).

Reflection functors and the Namikawa Weyl group
Our results thus far, and specifically Theorem 1.2, give an understanding of the quiver varieties M θ for generic parameters θ that lie in the simplicial cone F . We now use the fact that F is a fundamental domain for the action of the Namikawa Weyl group W to study the moduli spaces M θ for any generic θ ∈ Θ v . 7.1. Reflection functors. Reflection functors were introduced by Nakajima [58], but were also studied independently by Lusztig [50], Crawley-Boevey-Holland [22], and Maffei [51].
Let C ⊂ Θ v be a chamber and let θ ∈ C. Recall that for 1 ≤ i ≤ r, we write s i : Θ v → Θ v for reflection in the hyperplane ρ ⊥ i . By Proposition 2.2(iii), the action of the Namikawa Weyl group W permutes the chambers in Θ v , so s i (C) is a chamber containing the parameter s i (θ). The following result is a special case of a general result of Losev [47,Lemma 6.4.2]. Recall that for j ∈ I, we let R j denote the corresponding tautological bundle on the fine moduli space M θ . For simplicity, we write R ′ j for the corresponding tautological bundle on M si(θ) .
for all η ∈ Θ v . In particular, the linearisation maps for C and s i (C) fit into a commutative diagram Proof. For simplicity, we follow the set-up described in [51, §3]. We let Z : [51, §3]; note that θ is denoted m there. We may assume without loss of generality that θ i > 0.
Then there are explicit isomorphisms Moreover, statements in and preceding [51,Definition 27] imply that for indices j = i there is an isomorphism of bundles on Z, where the sum is over arrows a ∈ H; here we use the isomorphisms p * (R tl(a) ) ∼ = (p ′ ) * (R ′ tl(a) ) for arrows with hd(a) = i and the fact that the McKay quiver Q has no loops. Taking determinants, we Since S i := p • (p ′ ) −1 , the left hand side of equation (7.1) is therefore Recall from equation (2.5) that s i (η) j = η j − c i,j η i for any η ∈ Θ v , so the right hand side of (7.1) is where J = {j ∈ I | j = i, j = tl(a) for some a ∈ Q 1 with hd(a) = i}. This equals (7.2) as required. It now follows directly from the definition that S * i (L C (η)) ∼ = L si(C) (s i (η)) for all η ∈ Θ v . The final statement of the lemma follows by considering this isomorphism for η ∈ C and using the equalities L C (C) = Amp(M θ /Y ) from Proposition 6.1 for the chambers C and s i (C).
The orbit of the cone F under the action of the subgroup of W generated by the reflections s 1 , . . . , s r covers the half-space {θ ∈ Θ v | θ(δ) ≥ 0}. By combining the results from this section with Theorem 1.2, it follows that we now understand the moduli spaces M θ and their tautological bundles for all generic parameters θ that lie in this halfspace. 7.2. Crossing the hyperplane δ ⊥ . Ideally, we would like to reflect at the vertex 0 as well, but this is not possible since v is not fixed by s 0 . In order to study the moduli M θ for θ in the half-space {θ ∈ Θ v | θ(δ) < 0}, we instead use the fact that the preprojective algebra is isomorphic to its dual.
Let ι denote the involution on Irr(Γ) given by ρ → ρ * . Since V is self-dual, this is a graph automorphism.
It is described as follows: if Γ is of type A n (n > 1), D n (n odd) or E 6 , then ι is the involution of Irr(Γ) induced by the order 2 symmetry of the McKay graph described in [10,item (XI) in Planche I-VII]; otherwise, Γ is of type A 1 , D n (n even), E 7 , E 8 and ι is the identity. Let H aff ⊂ H denote the set of oriented edges of the unframed affine Dynkin graph. Then ι is uniquely defined on H aff by the rule tl(ι(a)) = ι(hd(a)), hd(ι(a)) = ι(tl(a)).
Let A := C[V ] ⋊ Γ be the skew-group algebra. We define an analogous anti-involution ν : Proof. The proof of this lemma is rather lengthy since we must show that the isomorphism in the "Key Lemma" [22,Lemma 3.2] can be made compatible with the anti-involutions. First, we note that the case where Γ is of type A can be checked explicitly by hand, so we assume that Γ is of type D or E. In this case, the definition of ι implies that there is no arrow a in H aff such that a : i → ι(i) for some vertex i, and that ι(a) = a for all a. In particular, we can choose an orientation Ω aff ⊂ H aff of the McKay graph so that ǫ(ι(a)) = −ǫ(a) for all arrows a.
Next, as in [22], we let C 1 = V ⊗ C CΓ be the Γ-bimodule with diagonal left action and right action on the right factor only. Let C 2 = C 1 ⊗ Γ C 1 . Then, if there is an arrow in H aff from i to j, and zero otherwise. The "Key Lemma" of [22], rephrased as [22,Lemma 3.3], exhibits a basis θ a of f hd(a) C 1 f tl(a) such that in C 2 , where x, y is the standard basis of V and where δ i = dim C ρ i . Now, Lemma 7.4 and the fact that ν is an anti-involution implies that ν(θ a ) = m a θ ι(a) for some m a ∈ C with m ι(a) = m −1 a . Next, Since there is at most one arrow from i to j in H aff , we deduce that either and hence m a * = m −1 a . The fact that ι(a) = a for all a implies that either (i) a * = ι(a) (when ι(tl(a)) = tl(a) and ι(hd(a)) = hd(a)), in which case we let t a be a square root of m a and set t ι(a) = t −1 a ; or (ii) a, a * , ι(a), ι(a) * are all distinct, in which case let t a again be a square root of m a and set t a = t ι(a) * = t −1 a * = t −1 ι(a) . Finally, define θ a := 1 ta θ a . Then and θ a θ a * = θ a θ a * . Combining this with the proof of [22,Theorem 3.4], we obtain a well-defined isomorphism Π aff → f Af sending ρ i to f i and a to θ a that is compatible with the anti-involutions ι and ν| f Af .
Let Π aff -mod and A-mod denote the categories of finite dimensional left Π aff -modules and A-modules Proof. Let M = (V i , ψ a | i ∈ I, a ∈ H) be a point in µ −1 (0). Then D(M ) = (V * i , ψ T ι(a) ). We now define ι : as required. We now compute what happens to the tautological bundles under D. For i ∈ I, write R i and R ′ i for the corresponding tautological bundles on M θ and M −ι(θ) respectively.
Remark 7.8. In fact all of the results of this section are valid even when Γ is trivial if we set ι to be the identity. This gives rise to an isomorphism D : M −θ → M θ over the base Sym n (C 2 ). In this case, there are only two GIT chambers and it is well known that M θ is the Hilbert scheme of n-points on C 2 .
7.3. The main results. We can now prove the main result which gives an understanding of the moduli spaces M θ for any generic θ ∈ Θ v . It is convenient to first establish a compatibility result for the linearisation maps associated to the chambers C and w(C) for any w ∈ W ; here, w(C) is a chamber by Proposition 2.2(iii).
Lemma 7.9. For any chamber C ⊂ Θ v , we have that L w(C) (w(θ)) = L C (θ) for all w ∈ W and θ ∈ C.
Proof. The birational map ψ θ : X M θ of schemes over M 0 from section 6.1 is unique; it's determined by the linear system |L C− (ℓθ)| for sufficiently large ℓ, where C − is the unique chamber in F satisfying X ∼ = M θ for θ ∈ C − (see Theorem 4.9). For clarity, in the course of this proof we choose not to suppress making reference to these maps when identifying nef cones and line bundles, so in fact our goal is to prove that for all w ∈ W and θ ∈ C. since w 0 (β) = −β. By definition, s δ w 0 also acts on Θ v as multiplication by −ι. Since W is generated by {s 1 , . . . , s r } and s δ w 0 , it suffices to check that equation (7.3) holds for w = s i for 1 ≤ i ≤ r, and for s δ w 0 .
We are finally in a position to prove the strong form of our main result.
Proof of Theorem 1.7. For (i), define L by setting L(θ) := L C (θ) for θ ∈ Θ v , where C is any chamber satisfying θ ∈ C. To see that L is well-defined, we need only show that for adjacent chambers C, C ′ ⊂ Θ v , the maps L C , L C ′ agree on the separating wall, i.e. that L C (θ 0 ) = L C ′ (θ 0 ) for all θ 0 ∈ C ∩ C ′ (we do not assume θ 0 is generic in the wall). Since F is a fundamental domain for W , we may assume without loss of generality by Lemma 7.9 that C ⊂ F . There are three cases to consider. First, if C ′ also lies in F , then the proof of Theorem 1.2 shows that L C (η) = L C ′ (η) for all η ∈ Θ v . Second, if the wall separating C from C ′ is a real boundary wall of F , then it is contained in a hyperplane ρ ⊥ i for some 1 ≤ i ≤ r. In this case, C ′ = s i (C) and θ 0 = s i (θ 0 ) for all θ 0 ∈ C ∩ C ′ , giving L C ′ (θ 0 ) = L si(C) (s i θ) = L C (θ), as required. Finally, if the wall separating C from C ′ is an imaginary boundary wall, then C ′ = s δ (C) and, as above, we deduce that L C ′ (θ 0 ) = L C (θ 0 ). Therefore L is a well-defined, piecewise-linear map.
For (ii), let C be any chamber such that θ ∈ C. Then w(θ) ∈ w(C) and hence Lemma 7.9 gives L(w(θ)) = L w(C) (w(θ)) = L C (θ) = L(θ) which establishes W -invariance. We have L| F = L F by construction, so L(F ) = Mov(X/Y ) by Theorem 1.2, and since F is a fundamental domain for W , the W -invariance of L now implies that L(Θ v ) = Mov(X/Y ).
For (iii), observe first that L| F is compatible with the chamber decompositions of Θ v and Mov(X/Y ) by Theorem 1.2. Part (ii) now implies that L| w(F ) is compatible with the chamber decompositions for each w ∈ W , and the statement of part (iii) follows from Proposition 2.2(ii). Part (iv) follows from Proposition 6.1, because the restriction of L to C equals L C . Corollary 7.10. Let C, C ′ ⊂ Θ v be chambers and let θ ∈ C, θ ′ ∈ C ′ . Then M θ ∼ = M θ ′ as schemes over Y if and only if there exists w ∈ W such that w(C) = C ′ .
For completeness, we now present the analogues of Proposition 6.1 and Theorem 1.7 in the degenerate case when n = 1. Note that X ∼ = S and Y ∼ = C 2 /Γ. Proposition 7.11. Let n = 1.
(ii) These maps glue to give a piecewise-linear, continuous map L : Θ v −→ N 1 S/(C 2 /Γ)) that is invariant with respect to the action of W on Θ v , and whose image is Nef(S/(C 2 /Γ)).
If a ∈ Q 1 is the unique arrow with tail at ρ ∞ , then the relation a * a = 0 in Π ensures that any θ-stable point (V i , ψ a | i ∈ I, a ∈ Q 1 ) in µ −1 (0) satisfies ψ a = 0 and ψ a * = 0. Therefore, we have a nowhere-zero morphism R ρ∞ → R ρ0 which is necessarily an isomorphism. It follows that κ := (−1, 1, 0, . . . , 0) lies in the kernel of L C . Gonzalez-Sprinberg-Verdier [32] implies that the line bundles det(R i ) for 1 ≤ i ≤ r provide an integral basis of N 1 (S/(C 2 /Γ)), so L C is surjective and hence κ spans the kernel of L C . It remains to note that the image of C in Θ v / κ ∼ = δ ⊥ is a Weyl chamber in the decomposition associated to Φ, so the final statement from (i) follows from Kronheimer [44] and Lemma 4.12. The case where C lies in the half-space {θ ∈ Θ v | θ(δ) > 0} is similar. For part (ii), the proof of Theorem 1.7 carries over verbatim, bearing in mind that F is the closure of a unique chamber since n = 1.

Appendix A. Variation of GIT for quiver varieties
In the appendix we describe the properties of quiver varieties under variation of GIT that are required in the main body of the article. We adopt once again the notation and assumptions of section 3.1, so that (v, w) are a pair of dimension vectors for a fixed graph with vertex set {0, 1, . . . , r} and v is the dimension vector of the corresponding framed doubled quiver Q = (I, Q 1 ) for I = {∞, 0, 1, . . . , r}.
Theorem A.1. Let θ ≥ θ 0 ∈ Θ v such that M θ = ∅. There exists a unique representation type τ such that: (i) the morphism f : M θ → M θ0 obtained by variation of GIT quotient satisfies Im f = M θ0,τ ; and (ii) the resulting morphism f : M θ → M θ0,τ is birational and semi-small.
Remark A.2. The above theorem was shown by Nakajima [56,59] in the case where M θ is smooth. We give a different proof that does not rely on the topological arguments of loc. cit.
A.2. Proof. The difficult part of Theorem A.1 is to show that f is semi-small. We reduce this statement to the following, whose proof relies on a result of Bozec. Recall that a root α is anisotropic if p(α) > 1. Abusing notation, let us write Ω for a quiver whose double Ω equals the framed doubled quiver Q. Let N be the number of loops in Ω, so that Q has 2N loops and dim M 0,0 = 2N (one has the freedom to assign to each loop in Q a scalar). Let L ⊂ Ω 1 be the set of loops in Ω, and let Λ ⊂ Rep(Q, v) denote the closed subset of seminilpotent representations, as defined by Bozec [11]. Then Λ is isotropic by [11, Lemma 1.2], so dim Λ ≤ 1 2 dim Rep(Q, v). We define Λ 0 = {x ∈ Λ | Tr(a * ) = 0, ∀ a ∈ L}.
Next, write v = dα, where d ∈ Z >0 and α an indivisible dimension vector. We recall from [7, Theorem 2.2] that α is an anisotropic root, and so too is every multiple mα of it. Our assumption that θ is sufficiently general is introduced to ensure that θ(β) = 0 for all positive roots β ≤ v that are not a multiple of α. It then follows from [7, Theorem 2.2] that either (a) α ∈ Σ θ , or (b) every proper multiple of α belongs to Σ θ but α itself does not. Then the only representation types of v are ν = (n 0 , ν 0 α; . . . , ; n k , ν k α), Inequality (A.1), applied to each v = ν i α, gives for all 0 ≤ i ≤ k. Combining this with (A.2) gives Since dim M 0 (mα) 0 = 2N is independent of m, and since i dim M θ (ν i α) = dim M θ,ν ≤ dim M θ , we have which implies the statement of the proposition.