Natural G-Constellation Families

Timothy Logvinenko

Received: April 22, 2007 Revised: December 9, 2008

Communicated by Peter Schneider

Abstract. Let G be a finite subgroup of GL_{n}(ℂ). G-constellations are a scheme-theoretic generalization of orbits of G in ℂ^{n}. We study flat families of G-constellations parametrised by an arbitrary resolution of the quotient space ℂ^{n}/G. We develop a geometrical naturality criterion for such families, and show that, for an abelian G, the number of equivalence classes of these natural families is finite.

The main intended application is the derived McKay correspondence.

2000 Mathematics Subject Classification: Primary 14J17; Secondary 14J10, 14D20, 14J40.

0 Introduction

Let G ⊆ GL_{n}(ℂ) be a finite subgroup. In this paper we classify flat families of G-constellations parametrised by a given resolution Y of the singular quotient space X = ℂ^{n}/G.

\xymatrix{Y \ar[dr]^{π} &  & ℂ^{n} \ar[dl]_{q} \\  & X}

A G-constellation is a scheme-theoretical generalization of a set-theoretical orbit of G in ℂ^{n}. They first arose in the context of moduli space constructions of crepant resolutions of X. Interpreting G-constellations in terms of representations of the McKay quiver of G, it is possible to use the methods of [Kin94] to consruct via GIT fine moduli spaces of stable G-constellations. The main irreducible component of such a moduli space turns out to be a projective crepant resolution of X. By varying the stability parameter θ it is possible to obtain different resolutions M_{θ}. In case of n = 3 and G abelian, it is possible to obtain all projective crepant resolutions in this way [CI04]. For further details see [Cra01], [CI04], [CMT07a], [CMT07b]. The formal definition of a G-constellation is:

Definition 0.1 ([Cra01]). A G-constellation is a G-equivariant coherent sheaf F whose global sections Γ(ℂ^{n}, F), as a representation of G, are isomorphic to the regular representation.

Families of G-constellations also occur naturally as objects defining Fourier-Mukai transforms (cf. [BKR01],[CI04], [BO95] and [Bri99]) which give a category equivalence D(Y) ⟶^{∼} D^{G}(ℂ^{n}) between the bounded derived categories of coherent sheaves on Y and of G-equivariant coherent sheaves on ℂ^{n}, respectively. This equivalence is known as the derived McKay correspondence (cf. [Rei97], [BKR01], [Kaw05], [Kal08]). It is the derived category interpretation of the classical McKay correspondence between the representation theory of G and the geometry of crepant resolutions of ℂ^{n}/G. It was conjectured by Reid in [Rei97] to hold for any finite subgroup G of SL_{n}(ℂ) and any crepant resolution Y of ℂ^{n}/G. In this paper we take an arbitrary resolution Y → ℂ^{n}/G and prove that it can support only a finite number (up to a twist by a line bundle) of flat families of Gconstellations. We give a complete classification of these families which allows one to explicitly compute them. For the precise statement of the classification see the end of this introduction. A motivation for this study is the fact that if a flat family of G-constellations on a crepant resolution Y of ℂ^{n}/G is sufficiently orthogonal, then it defines an equivalence D(Y) → D^{G}(ℂ^{n}) ([Log08], Theorem 1.1), i.e. the derived McKay correspondence conjecture holds for Y. For an example of a specific application of this see [Log08], §4, where the first known example of a derived McKay correspondence for a non-projective crepant resolution is explicitly constructed. This paper is laid out as follows. At the outset we allow Y to be any normal scheme birational to the quotient space X and first of all we move from the category Coh^{G}(ℂ^{n}) to the equivalent category Mod^{fg}-R ⋊ G of the finitelygenerated modules for the cross product algebra R ⋊ G, where R denotes the coordinate ring ℂ[x_{1}, …, x_{n}] of ℂ^{n}. This makes a family of G-constellations into a vector bundle on Y. In Section 1 we develop a geometrical naturality criterion for such families: mimicking the moduli spaces M_{θ} of θ-stable Gconstellations and their tautological families, we demand for a G-constellation parametrised in a family F by a point p ∈ Y to be supported precisely on the G-orbit corresponding to the point π(p) in the quotient space X. In other words, the support of the corresponding sheaf on Y × ℂ^{n} must lie within the fibre product Y ×_{X} ℂ^{n}. We call the families which satisfy this condition gnatfamilies (short for a geometrically natural) and demonstrate (Proposition 1.5) that they enjoy a number of other natural properties, including being equivalent (locally isomorphic) to the natural family π^{∗}q_{∗}O_{ℂ^{n}} on the open set of Y which lies over the free orbits in X. In this natural family a G-constellation which lies over a free orbit is the unique G-constellation supported on that orbit - its reduced subscheme structure. Thus, in a sense, gnat-families can be viewed as flat deformations of free orbits of G. Another property which characterises gnat-families is that it is possible to embed them into K(ℂ^{n}), considered as a constant sheaf on Y. This leads us to study G-equivariant locally free sub-O_{Y} -modules of K(ℂ^{n}). In Section 2, we study the rank one case. A G-invariant invertible sub-O_{Y} -module of K(ℂ^{n}) is just a Cartier divisor, and we define G- Car(Y), a group of G-Cartier divisors on Y, as a natural extension of the group of Cartier divisors which fits into a short exact sequence

1 → Car(Y) → G- Car(Y) ⟶^{ρ} G^{∨} → 1

where G^{∨} is the group of 1-dimensional irreducible representations of G. We then define ℚ-valued valuations of these G-Cartier divisors at prime Weil divisors of Y and define G-Div Y, the group of G-Weil divisors of Y, as a torsionfree subgroup of ℚ-Weil divisors which fits into a following exact sequence:

\xymatrix{1 \ar[r] & Car Y \ar@{^{(}->}[d]_{_{val_{K}}} \ar[r] & G- Car Y \ar[d]_{_{val_{KG}}} \ar[r]^{ρ} & G^{∨} \ar@{->>}[d]_{_{val_{G∨}}} \ar[r] & 1 \\ 0 \ar[r] & Div Y \ar[r] & G- Div Y \ar[r] & val_{G^{∨}}(G^{∨}) \ar[r] & 0}

We then show that the three vertical maps in this diagram, val_{K}, the ordinary ℤ-valued valuation of Cartier divisors, val_{K_{G}}, the ℚ-valued valuation of G-Cartier divisors, and their quotient val_{G^{∨}}, a ℚ/ℤ-valued valuation of G^{∨}, are all isomorphisms when Y is smooth and proper over X. Then, in Section 3, we observe that when our group G is abelian all its irreducible representations are of rank 1, so any gnat-family splits into invertible G-eigensheaves. Thus G-Weil divisors are all that we need to classify it after an embedding into K(ℂ^{n}). We further show that, since any gnat-family F embedded into K(ℂ^{n}) must be closed under the natural action of R on the latter, all the G-eigensheaves into which F decomposes must be, in a certain sense, close to each other inside K(ℂ^{n}). Up to a twist by a line bundle, this leaves only a finite number of possibilities for the corresponding G-Weil divisors. Thus, surprisingly, the number of equivalence classes of gnat-families on any Y is finite. Our main result (Theorem 4.1) is:

Theorem (Classification of gnat-families). Let G be a finite abelian subgroup of GL_{n}(ℂ), X the quotient of ℂ^{n} by the action of G and Y a resolution of X. Then isomorphism classes of gnat-families on Y are in 1-to-1 correspondence with linear equivalence classes of G-divisor sets {D_{χ}}_{χ∈G^{∨}}, each D_{χ} a χ-Weil divisor, which satisfy the inequalities

D_{χ} + (f) − D_{χρ(f)} ≥ 0 ∀ χ ∈ G^{∨}, G-homogeneous f ∈ R

Here ρ(f) ∈ G^{∨} is the homogeneous weight of f. Such a divisor set {D_{χ}} corresponds then to a gnat-family ⨁ L(−D_{χ}). This correspondence descends to a 1-to-1 correspondence between equivalence classes of gnat-families and sets {D_{χ}} as above and with D_{χ_{0}} = 0, where χ_{0} is the trivial character. Furthermore, each divisor D_{χ} in such a set satisfies

M_{χ} ≥ D_{χ} ≥ −M_{χ^{−1}}

where {M_{χ}} is a fixed divisor set defined by

M_{χ} = ∑_{P}( min_{f∈R_{χ}} v_{P} (f))P

As a consequence, the number of equivalence classes of gnat-families on Y is finite.

Acknowledgements: The author would like to express his gratitude to Alastair Craw, Akira Ishii and Dmitry Kaledin for useful discussions on the subject and to Alastair King for the motivation, the discussions and the support. This paper was completed during the author’s stay at RIMS, Kyoto, and one would like to thank everyone at the institute for their hospitality.

1 gnat-Families

1.1 Families of G-Constellations

Let G be a finite abelian group and let V_{giv} be an n-dimensional faithful representation of G. We identify the symmetric algebra S(V_{giv}^{∨}) with the coordinate ring R of ℂ^{n} via a choice of such an isomorphism that the induced action of G on ℂ^{n} is diagonal. The (left) action of G on V_{giv} induces a (left) action of G on R, where we adopt the convention that

g.f(v) = f(g^{−1}.v)  ∀ g ∈ G, f ∈ R, v ∈ V_{giv},  (1.1)

When we consider the induced scheme morphisms g : ℂ^{n} → ℂ^{n} and the induced sheaf morphisms g : O_{ℂ^{n}} → g_{∗}^{−1}O_{ℂ_{n}}, the convention above ensures that for any point x ∈ ℂ^{n} and any function f in the stalk O_{ℂ^{n},x} at x, the function g.f is, naturally, an element of the stalk O_{ℂ^{n},g.x} at g.x Corresponding to the inclusion R^{G} ⊂ R of the subring of G-invariant functions we have the quotient map q : ℂ^{n} → X, where X = Spec R^{G} is the quotient space. This space is generally singular. We first wish to establish a notion of a family of G-Constellations parametrised by an arbitrary scheme.

Definition 1.1 ([CI04]). A G-constellation is a G-equivariant coherent sheaf F on ℂ^{n} such that H^{0}(F) is isomorphic, as a ℂ[G]-module, to the regular representation V_{reg}.

We would like for a family of G-constellations to be a locally free sheaf on Y, whose restriction to any point of Y would give us the respective G-constellation. We’d like this restriction to be a finite-dimensional vector-space, and for this purpose, it would be better to consider, instead of the whole G-constellation F, just its space of global sections Γ(F). It is a vector space with G and R actions, satisfying

g.(f.v) = (g.f).(g.v)  (1.2)

On the other hand, for any vector space V with G and R actions satisfying (1.2), we can define maps g : Ṽ → g_{∗}^{−1}Ṽ to give the sheaf Ṽ = V ⊗_{R} O_{ℂ^{n}} a G-equivariant structure. It is convenient to view such vector spaces as modules for the following non-commutative algebra:

Definition 1.2. A cross-product algebra R ⋊ G is an algebra, which has the vector space structure of R ⊗_{ℂ} ℂ[G] and the product defined by setting, for all g_{1}, g_{2} ∈ G and f_{1}, f_{2} ∈ R,

(f_{1} ⊗ g_{1}) × (f_{2} ⊗ g_{2}) = (f_{1}(g_{1}.f_{2})) ⊗ (g_{1}g_{2})  (1.3)

This is not a pure formalism - R ⋊ G is one of the non-commutative crepant resolutions of ℂ^{n}/G, a certain class of non-commutative algebras introduced by Michel van den Bergh in [dB02] as an analogue of a commutative crepant resolution for an arbitrary non-quotient Gorenstein singularity. For threedimensional terminal singularities, van den Bergh shows ([dB02], Theorem 6.3.1) that if a non-commutative crepant resolution Q exists, then it is possible to construct commutative crepant resolutions as moduli spaces of certain stable Q-modules. Functors Γ(•) and ˜• = (•) ⊗_{R} O_{ℂ^{n}} give an equivalence (compare to [Har77], p. 113, Corollary 5.5) between the categories of quasi-coherent G-equivariant sheaves on ℂ^{n} and of R ⋊ G-modules. G-constellations then correspond to R ⋊ G-modules, whose underlying G-representation is V_{reg}. As an abuse of notation, we shall use the term ‘G-constellation’ to refer to both the equivariant sheaf and the corresponding R ⋊ G-module.

Definition 1.3. A family of G-constellations parametrised by a

scheme S is a sheaf F of (R ⋊ G) ⊗_{ℂ} O_{S}-modules, locally free as an O_{S}- module, and such that for any point ι_{p} : Spec ℂ ↪ S, its fiber F_{|p} = ι_{p}^{∗}F is a G-constellation. We shall say that two families F and F′ are equivalent if they are locally isomorphic as (R ⋊ G) ⊗_{ℂ} O_{S}-modules.

1.2 gnat-Families

Let Y be a normal scheme and π : Y → X be a birational map.

\xymatrix{Y \ar[dr]^{π} &  & ℂ^{n} \ar[dl]_{q} \\  & X}

We wish to refine the definition (1.3) above and develop a notion of a geometrically natural family of G-constellations parametrised by Y. Any free G-orbit supports a unique G-cluster Z ⊂ ℂ^{n}: the reduced induced closed subscheme structure. Let U be an open subset of Y such that π(U) consists of free orbits of G and consider the sheaf π^{∗}q_{∗}O_{ℂ^{n}} restricted to U. It has a natural (R ⋊ G)-module structure induced from O_{ℂ^{n}}. It is locally free as an O_{U} module, since the quotient map q is flat wherever G acts freely. Its fiber at a point p ∈ Y is Γ(O_{Z}), where Z is the G-cluster corresponding to the free orbit q^{−1}π(p). Thus π^{∗}q_{∗}O_{ℂ^{n}} is a natural family of G-constellations, indeed of G-clusters, on U ⊂ Y. Its fiber at the generic point of Y is K(ℂ^{n}). The Normal Basis Theorem from Galois theory ([Gar86], Theorem 19.6) gives an isomorphism from K(ℂ^{n}) to the generic fiber of any G-constellation family on Y, which we can write as K(Y) ⊗_{ℂ} V_{reg}, but this isomorphism is only K(Y) and G, but not necessarily R, equivariant. On the other hand, for any G-constellation in a sense of G-equivariant sheaf, we can consider its support in ℂ^{n}. For instance, in the natural family π^{∗}q_{∗}O_{ℂ^{n}} discussed above the support of the G-constellation parametrised by a point p ∈ U is precisely the G-orbit q^{−1}π(p). This turns out to be the criterion we seek and we shall show that any family satisfying it is generically equivalent to the natural one.

Definition 1.4. A gnat-family F (short for geometrically natural family) is a family of G-constellations parametrised by Y such that for any p ∈ Y

q (Supp_{ℂ^{n}} F_{|p}) = π(p)  (1.4)

Proposition 1.5. Let Y be a normal scheme and π : Y → X be a birational map. Let F be a family of G-constellations on Y. Then the following are equivalent:

1. On any U ⊂ Y, such that πU consists of free orbits, F is equivalent to π^{∗}q_{∗}O_{ℂ^{n}}.

2. There exists an (R ⋊ G) ⊗_{ℂ} K(Y)-module isomorphism:

F_{|p_{Y}} ⟶^{∼} (π^{∗}q_{∗}O_{ℂ^{n}})_{p_{Y}}

where p_{Y} is the generic point of Y.

3. There exists an (R ⋊ G) ⊗_{ℂ} O_{Y} -module embedding

F ↪ K(ℂ^{n})

where K(ℂ^{n}) is viewed as a constant sheaf on Y and given a O_{Y} -module structure via the birational map π : Y → X.

4. F is a gnat-family.

5. The action of (R ⋊ G) ⊗_{ℂ} O_{Y} on F descends to the action of (R ⋊ G) ⊗_{R^{G}} O_{Y}, where R^{G}-module structure on O_{Y} is induced by the map π : Y → X.

Proof. 1 ⇒ 2 is restricting any of the local isomorphisms to the stalk at the generic point p_{Y} of Y. 2 ⇒ 3: the embedding is given by the natural map F ↪ F ⊗ K(Y). As Y is irreducible and F is locally free, F ⊗ K(Y) is isomorphic to F_{p_{Y}}, and hence to K(ℂ^{n}). 3 ⇒ 5 is immediate by inspecting the natural R ⋊ G⊗_{ℂ} O_{Y} -module structure on K(ℂ^{n}). 5 ⇒ 4 is also immediate, as the descent of the action of R ⋊ G⊗_{ℂ} O_{Y} to that of R ⋊ G⊗_{R^{G}} O_{Y} implies that for any p ∈ Y we have 𝔪_{π(p)} ⊂ Ann_{R} F_{|p}, where 𝔪_{π(p)} ⊂ R^{G} is the maximal ideal of π(p). Therefore 𝔪_{π(p)} = (Ann_{R} F_{|p})^{G}, which is equivalent to (1.4). 4 ⇒ 5: Consider the following composition of algebra morphisms:

R ⋊ G ⊗_{ℂ} O_{Y} ⟶^{α} End_{O_{Y}} (F) ⟶^{β_{p}} End_{ℂ}(F_{|p})

where α is the action map of R ⋊ G ⊗_{ℂ} O_{Y} on F and β_{p} is restriction to the fiber at a point p ∈ Y. To show that α filters through R ⋊ G ⊗_{R^{G}} O_{Y} it suffices to show that for any f ∈ R^{G} we have f⊗1−1⊗f ∈ ker(α). From (1.4) we have 𝔪_{π(p)} = (Ann_{R} F_{|p})^{G}, and therefore

β_{p}α((f − f(p)) ⊗ 1) = 0

Observe that β_{p}α(f(p) ⊗ 1) = f(p) 1_{End_{ℂ} F_{|p}} = β_{p}α(1 ⊗ f), and therefore

β_{p}α(f ⊗ 1 − 1 ⊗ f) = 0  (1.5)

As End_{O_{Y}} F is locally free, (1.5) holding ∀ p ∈ Y implies α(f ⊗ 1 − 1 ⊗ f) = 0, as required. 5 ⇒ 1: We have the R ⋊ G ⊗_{R^{G}} O_{Y} -action on F:

R ⋊ G ⊗_{R^{G}} O_{Y} ⟶^{α} End_{O_{Y}} (F)

LHS is isomorphic to π^{∗} End_{O_{X}} (q_{∗}O_{ℂ^{n}}). Over U, since q is flat over π(U), LHS is further isomorphic to End_{O_{U}} (π^{∗}q_{∗}O_{ℂ_{n}}). Thus we have:

End_{O_{U}} (π^{∗}q_{∗}O_{ℂ_{n}}) ⟶^{α′} End_{O_{U}} (F)  (1.6)

This map (1.6) is an O_{U}-algebra homomorphism of (split) Azumaya algebras over U of the same rank. By a general result on Azumaya algebras any such is an isomorphism (see [ACvdE05], Theorem 5.3, for full generality, but the original result in [AG60], Corollary 3.4 will also suffice here). Now Skolem-Noether theorem for Azumaya algebras ([Mil80], IV, §2, Proposition 2.3) implies that locally α′ must be induced by isomorphisms π^{∗}q_{∗}O_{ℂ^{n}} ⟶^{∼} F.

2 G-Cartier and G-Weil divisors

If F is a gnat-family, by Proposition 1.5 we can embed it into K(ℂ^{n}). We need, therefore, to study G-subsheaves of K(ℂ^{n}) which are locally free on Y. In this section we treat the rank 1 case, i.e. the invertible sheaves. Now, on an arbitrary scheme S, an invertible sheaf together with its embedding into K(S) defines a unique Cartier divisor on S. But here we embed not into K(Y) but into its Galois extension K(ℂ^{n}). Recall that we identify K(Y) with K(ℂ^{n})^{G} via the birational map Y ⟶^{π} X. We therefore seek to extend the familiar construction of Cartier divisors to accommodate for this fact.

2.1 G-Cartier divisors

We write G^{∨} for Hom(G, ℂ^{∗}), the group of irreducible representations of G of rank 1.

Definition 2.1. We shall say that a rational function f ∈ K(ℂ^{n}) is

G-homogeneous of weight χ ∈ G^{∨} if

g.f = χ(g^{−1})f ∀ g ∈ G  (2.1)

We shall denote by K_{χ}(ℂ^{n}) the subset of K(ℂ^{n}) of homogeneous elements of a specific weight χ and by K_{G}(ℂ^{n}) the subset of K(ℂ^{n}) of all the G-homogeneous elements. We shall use R_{χ} and R_{G} to mean R ∩ K_{χ}(ℂ^{n}) and R ∩ K_{G}(ℂ^{n}) respectively.

NB: The choice of a sign is dictated by wanting f ∈ R to be homogeneous of weight χ ∈ G^{∨} if f(g.v) = χ(g)f(v) for all g ∈ G and v ∈ ℂ^{n}. The invertible elements of K_{G}(ℂ^{n}) form a multiplicative group which we shall denote by K_{G}^{∗}(ℂ^{n}). We have a short exact sequence:

1 → K^{∗}(Y) → K_{G}^{∗}(ℂ^{n}) ⟶^{ρ} G^{∨} → 1  (2.2)

The following replicates, almost word-for-word, the definition of a Cartier divisor in [Har77], pp. 140-141.

Definition 2.2. A group of G-Cartier divisors on Y, denoted by G-Car(Y) is the group of global sections of the sheaf of multiplicative groups K_{G}^{∗}(ℂ^{n})/O_{Y}^{∗}, i.e. the quotient of the constant sheaf K_{G}^{∗}(ℂ^{n}) on Y by the sheaf O_{Y}^{∗} of invertible regular functions.

Observe that (2.2) gives a well-defined short exact sequence:

1 → Car(Y) → G- Car(Y) ⟶^{ρ} G^{∨} → 1  (2.3)

Given a G-Cartier divisor, we call its image χ ∈ G^{∨} under ρ the weight of the divisor and say, further, that the divisor is χ-Cartier. A G-Cartier divisor can be specified by a choice of an open cover {U_{i}} of Y and functions {f_{i}} ⊆ K_{G}^{∗}(ℂ^{n}) such that f_{i}/f_{j} ∈ Γ(U_{i} ∩ U_{j}, O_{Y}^{∗}). In such case, the weight of the divisor is the weight of any one of f_{i}. As with ordinary Cartier divisors, we say that a G-Cartier divisor is principal if it lies in the image of the natural map K_{G}^{∗}(ℂ^{n}) → K_{G}^{∗}(ℂ^{n})/O_{Y}^{∗} and call two divisors linearly equivalent if their difference is principal. Consider now a χ-Cartier divisor D on Y specified by a collection {(U_{i}, f_{i})} where U_{i} form an open cover of Y and f_{i} ∈ K_{χ}^{∗}(ℂ^{n}). We define an invertible sheaf L(D) on Y as the sub-O_{Y} -module of K(ℂ^{n}) generated by f_{i}^{−1} on U_{i}. Observe that G acts on L(D), the action being the restriction of the one on K(ℂ^{n}), and that it acts on every section by the character χ.

Proposition 2.3. The map D → L(D) gives an isomorphism between G-Car Y and the group of invertible G-subsheaves of K(ℂ^{n}). Furthermore, it descends to an isomorphism of the group G- Cl of G-Cartier divisors up to linear equivalence and the group G- Pic of invertible G-sheaves on Y.

Proof. A standard argument in [Har77], Proposition 6.13, shows everything claimed, apart from the fact we can embed any invertible G-sheaf L, with G acting by some χ ∈ G^{∨}, as a sub-O_{Y} -module into K(ℂ^{n}). Given such L, we consider the sheaf L ⊗_{O_{Y}} K(Y). On every open set U_{i} where L is trivial, it is G-equivariantly isomorphic to the constant sheaf K_{χ}(ℂ^{n}). On an irreducible scheme a sheaf constant on an open cover is constant itself, so as Y is irreducible we have L ⊗_{O_{Y}} K(Y) ≃ K_{χ}(ℂ^{n}) and a particular choice of this isomorphism gives the necessary embedding as

L → L ⊗_{O_{Y}} K(Y) ≃ K_{χ}(ℂ^{n}) ⊂ K(ℂ^{n})

2.2 Homogeneous valuations

We now aim to develop a matching notion of G-Weil divisors. Recall that the homomorphism from ordinary Cartier to ordinary Weil divisors is defined in terms of valuations of rational functions at prime Weil divisors of Y. Valuations at prime divisors of Y define a unique group homomorphism val_{K} from K^{∗}(Y) to Div Y, the group of Weil divisors. Looking at the short exact sequence (2.2), we see that val_{K} must extend uniquely to a homomorphism val_{K_{G}} from K_{G}^{∗}(ℂ^{n}) to ℚ- Div Y, as G^{∨} is finite and ℚ is injective. We further obtain a quotient homomorphism val_{G^{∨}} from G^{∨} to ℚ/ℤ- Div Y. Explicitly, we set:

Definition 2.4. Let P be a prime Weil divisor on Y. For any f ∈ K_{G}^{∗}(ℂ^{n}), observe that f^{|G|} is necessarily of trivial weight and hence lies in K(Y). We define valuation of f at P to be

v_{P} (f) = \frac{1}{|G|}v_{P} (f^{|G|}) ∈ ℚ  (2.4)

where v_{P} (f^{|G|}) is the ordinary valuation in the local ring of P. For any χ ∈ G^{∨}, observe that for any f, f′ homogeneous of weight χ their ratio f/f′ is of trivial character and therefore has integer valuation. We define

valuation of χ at P to be

v_{P} (χ) = frac(v_{P} (f)) ∈ ℚ/ℤ  (2.5)

where f is any homogeneous function of weight χ and frac(-) denotes the fractional part.

It can be readily verified that val_{K_{G}} = ∑ v_{P} (-)P and val_{G^{∨}} = ∑ v_{P} (-)P. Furthermore, the short exact sequence (2.3) becomes a commutative diagram:

\xymatrix{1 \ar[r] & Car Y \ar[d]_{_{val_{K}}} \ar[r] & G- Car Y \ar[d]_{_{val_{K_{G}}}} \ar[r]^{ρ} & G^{∨} \ar[d]_{_{val_{G^{∨}}}} \ar[r] & 1 \\ 0 \ar[r] & Div Y \ar[r] & ℚ- Div Y \ar[r] & ℚ/ℤ− Div Y \ar[r] & 0}  (2.6)

2.3 G-Weil divisors

Aiming to have a short exact sequence similar to (2.3), we now define the group G-Div Y of G-Weil divisors to be the subgroup of ℚ-Div Y, which consists of the pre-images of val_{G^{∨}}(G^{∨}) ⊂ ℚ/ℤ-Div Y.

Definition 2.5. We say that a ℚ-Weil divisor ∑ q_{P} P on Y is a G-Weil divisor if there exists χ ∈ G^{∨} such that

frac(q_{P}) = v_{P} (χ)  for all prime Weil P  (2.7)

We call a G-Weil divisor principal if it is an image of a single function f ∈ K_{G}^{∗}(ℂ^{n}) under val_{K^{G}}, call two G-Weil divisors linearly equivalent if their difference is principal and call a divisor ∑ q_{i}D_{i} effective if all q_{i} ≥ 0. We now have a following commutative diagram:

\xymatrix{1 \ar[r] & Car Y \ar@{^{(}->}[d]_{_{val_{K}}} \ar[r] & G- Car Y \ar[d]_{_{val_{K_{G}}}} \ar[r]^{ρ} & G^{∨} \ar@{->>}[d]_{_{val_{G^{∨}}}} \ar[r] & 1 \\ 0 \ar[r] & Div Y \ar[r] & G- Div Y \ar[r] & val_{G^{∨}}(G^{∨}) \ar[r] & 0}  (2.8)

A warning: for general Y, even a smooth one, G-Cartier and G-Weil divisors may not be very well behaved. For an example let Y be the smooth locus of X. It can be shown, that while val_{K} is an isomorphism, val_{K_{G}} is not even injective as G-Car Y has torsion. And val_{G^{∨}} is the zero map, thus G-Div Y is just Div Y.

Proposition 2.6. If Y is smooth and proper over X, then val_{K}, val_{K_{G}} and val_{G^{∨}} in (2.8) are isomorphisms.

Proof. If Y is smooth, or at least locally factorial, val_{K} is well-known to be an isomorphism ([Har77], Proposition 6.11). It therefore suffices to show that val_{G^{∨}} is injective and hence an isomorphism. As diagram (2.8) commutes, val_{K_{G}} will then also have to be an isomorphism. Fix χ ∈ G^{∨}. Let Y_{χ} denote the normalisation of Y ×_{X} (ℂ^{n}/ ker χ). It is a Galois covering of Y whose Galois group is χ(G). By Zariski-Nagata’s purity of the branch locus theorem ([Zar58], Proposition 2), the ramification locus of Y_{χ} → Y is either empty or of pure codimension one. As Y is smooth, Y_{χ} → Y being finite and unramified would make it an étale cover. Which is impossible, since a resolution of a quotient singularity is well known to be simply-connected (see, for instance, [Ver00], Theorem 4.1). Thus, we can assume there exists a ramification divisor P ⊂ Y_{χ}. Let Q be its image in Y. Let Ram(P) be the subgroup of G which fixes P pointwise. Then n_{ram} = |Ram(P)/ ker χ| is the ramification index of P. We can take ordinary integer valuations of K_{χ}^{∗}(ℂ^{n}) on prime divisors of Y_{χ} as K_{χ}^{∗}(ℂ^{n}) ⊂ K(ℂ^{n})^{ker χ}. It is easy to see that for any f ∈ K_{χ}^{∗}(ℂ^{n})

v_{Q}(f) = \frac{1}{nram} v_{P} (f)  (2.9)

where LHS is a rational valuation in sense of Definition 2.4. If v_{Q}(χ) = 0, then v_{Q}(K_{χ}^{∗}(ℂ^{n})) ⊂ ℤ. Then necessarily v_{Q}(K_{χ}^{∗}(ℂ^{n})) = ℤ, as K_{χ}^{∗}(ℂ^{n}) is a coset of K(Y) in K_{G}^{∗}(ℂ^{n}). In particular, there would exist f_{χ} ∈ K_{χ}^{∗}(ℂ^{n}), such that v_{Q}(f_{χ}) = 0, i.e. f_{χ} is a unit in O_{Y_{χ},P}. Which is impossible: any g ∈ Ram(P) fixes P pointwise, in particular f − g.f ∈ 𝔪_{Y,P} for any f ∈ O_{Y,P}. As Ram(P)/ ker χ is non-trivial we can choose g such that χ(g) ≠ 1 and then f_{χ} = \frac{1}{1−χ(g)}(f_{χ} − g.f_{χ}) must lie in 𝔪_{Y,P}. This finishes the proof. For abelian G, this all can be seen very explicitly by exploiting the toric structure of the singularity: even though we do not assume the resolution Y to be toric, it has been proven by Bouvier ([Bou98], Theorem 1.1) and by Ishii and Kollár ([KI03], Corollary 3.17, in a more general context of Nash problem) that every essential divisor over X (i.e. a divisor which must appear on every resolution) is toric. The set of essential toric divisors is well understood - it can be identified with the Hilbert basis of the positive octant of the toric lattice of weights, and then with a subset of Ext^{1}(G^{∨}, ℤ) = Hom(G^{∨}, ℚ/ℤ). This correspondence sends each divisor precisely to the valuation of G^{∨} at it, see [Log04], Section 4.3 for more detail.

We also show that, away from a finite number of prime divisors on Y, all G-Weil divisors are ordinary Weil.

Proposition 2.7. Unless a prime divisor P ⊂ Y is exceptional or its image in X is a branch divisor of ℂ^{n} → X, the valuation v_{P} : G^{∨} → ℚ/ℤ is the zero-map.

Proof. If P is not exceptional, let Q be its image in X. The valuations at P and Q are the same, so it suffices to prove the statement about v_{Q}. Let P′ be any divisor in ℂ^{n} which lies above Q. As in Proposition 2.6, for any f ∈ K_{G}^{∗}(ℂ^{n}) we have v_{Q}(f) = \frac{1}{nram} v_{P′}(f) where nram is the ramification index of P′. Unless Q is a branch divisor, nram = 1 and v_{Q} = v_{P′}. Which makes v_{Q} integer-valued on K_{G}^{∗}(ℂ^{n}) and makes the quotient homomorphism G^{∨} → ℚ/ℤ the zero map.

3 Classification of gnat-families

3.1 Reductor Sets

From now on, in addition to assuming that G is a finite group acting faithfully on V_{giv}, we also assume that G is abelian. We further assume that Y is smooth and π : Y → X is proper. Let F be a gnat-family on Y. Write the decomposition of F into G-eigensheaves as ⨁_{χ∈G^{∨}} F_{χ}. By Proposition 1.5 we can embed F into K(ℂ^{n}) and, as was demonstrated in Proposition 2.3, the image of each F_{χ} defines a χ-Cartier divisor. Hence F ≃ ⨁_{χ} L(−D_{χ}) for some set {D_{χ}}_{χ∈G^{∨}} of G-Weil divisors.

Definition 3.1. Let {D_{χ}}_{χ∈G^{∨}} be a set of G-Weil divisors on Y. We call it a reductor set if each D_{χ} is a χ-Weil divisor and ⊕L(−D_{χ}) is a gnat-family on Y. We call a reductor set normalised if D_{χ_{0}} = 0. We say that two reductor sets {D_{χ}} and {D′_{χ}} are linearly equivalent if there exists f ∈ K(Y) such that D_{χ} − D′_{χ} = Div f for all χ ∈ G^{∨}.

Lemma 3.2. Let {D_{χ}} and {D′_{χ}} be two reductor sets. Any (R ⋊ G) ⊗ O_{Y} - module morphism ϕ : ⨁ L(−D_{χ}) → ⨁ L(−D′_{χ}) is necessarily a multiplication inside K(ℂ^{n}) by some f ∈ K(Y).

Proof. Because of G-equivariance ϕ decomposes as ⨁_{χ∈G^{∨}} ϕ_{χ} with ϕ_{χ} a morphism L(−D_{χ}) → L(−D′_{χ}). Each ϕ_{χ} is a morphism of invertible sub-O_{Y} - modules of K(ℂ^{n}) and so is necessarily a multiplication by some f_{χ} ∈ K(Y): consider the induced map O_{Y} → L(−D_{χ} + D′_{χ}) and take f_{χ} to be the image of 1 under this map. It remains to show that all f_{χ} are equal. Fix any χ ∈ G^{∨} and consider any Ghomogeneous m ∈ R of weight χ. Take any s ∈ L(−D_{χ_{0}}). Then ms ∈ L(−D_{χ}) and by R-equivariance of ϕ

ϕ_{χ}(ms) = mϕ_{χ_{0}}(s) = f_{χ_{0}}ms  (3.1)

and hence f_{χ} = f_{χ_{0}} for all χ ∈ G^{∨}.

Corollary 3.3. Isomorphism classes of gnat-families on Y are in 1-to-1 correspondence with linear equivalence classes of reductor sets.

Proof. If in the proof of Lemma 3.2 each ϕ_{χ} is an isomorphism, then f, by construction, globally generates each L(−D′_{χ} + D_{χ}). Thus D_{χ} − D′_{χ} = Div(f).

Proposition 3.4. Let {D_{χ}} and {D′_{χ}} be two reductor sets. Then ⨁ L(−D_{χ}) and ⨁ L(−D′_{χ}) are equivalent (locally isomorphic) if and only if there exists a Weil divisor N such that D_{χ} − D′_{χ} = N for all χ ∈ G^{∨}.

Proof. The ‘if’ direction is immediate. Conversely, if the families are equivalent, then by applying Lemma 3.2 to each local isomorphism, we obtain the data {U_{i}, f_{i}}, where U_{i} are an open cover of Y and on each U_{i} multiplication by f_{i} is an isomorphism ⨁ L(−D_{χ}) ⟶^{∼} ⨁ L(−D′_{χ}). One can readily check that such {U_{i}, f_{i}} must define a Cartier divisor and that the corresponding Weil divisor is the requisite divisor N.

Corollary 3.5. In each equivalence classes of gnat-families there is precisely one family whose reductor set is normalised.

3.2 Reductor Condition

We now investigate when is a set {D_{χ}} of G-divisors a reductor set. This issue is the issue of ⨁ L(−D_{χ}) actually being (R ⋊ G) ⊗ O_{Y} -module. By definition it is a sub-O_{Y} -module of K(ℂ^{n}), but there is no a priori reason for it to also be closed under the natural R ⋊ G-action on K(ℂ^{n}). If it is closed, it can be checked that it trivially satisfies all the other requirements in Proposition 1.5, item 3, which makes it a gnat-family. We further observe that ⨁ L(−D_{χ}) is always closed under the action of G, so it all boils down to the closure under the action of R. Recall, that we write R_{G} for R ∩ K_{G}^{∗}(ℂ^{n}), the G-homogeneous regular polynomials, and R_{χ} for R ∩ K_{χ}^{∗}(ℂ^{n}), the G-homogeneous regular polynomials of weight χ ∈ G^{∨}.

Proposition 3.6 (Reductor Condition). Let {D_{χ}}_{χ∈G^{∨}} be a set with each D_{χ} a χ-Weil divisor. Then it is a reductor set if and only if, for any f ∈ R_{G}, the divisor

D_{χ} + (f) − D_{χρ(f)} ≥ 0  (3.2)

i.e. it is effective.

Remarks:

1. If we choose a G-eigenbasis of V_{giv}, then its dual basis, a set of basic monomials x_{1}, …, x_{n}, generates R_{G} as a semi-group. As condition (3.2) is multiplicative on f, it is sufficient to check it only for f being one of x_{i}. This leaves us with a finite number of inequalities to check.

2. Numerically, if we write each D_{χ} as ∑ q_{χ,P} P, inequalities (3.2) subdivide into independent sets of inequalities

q_{χ,P} + v_{P} (f) − q_{χρ(f),P} ≥ 0  ∀ χ ∈ G^{∨}  (3.3)

a set for each prime divisor P on Y. This shows that a gnat-family can be specified independently at each prime divisor of Y: we can construct reductor sets {D_{χ}} by independently choosing for each prime divisor P any of the sets of numbers {q_{χ,P} }_{χ∈G^{∨}} which satisfy (3.3).

3. There is an interesting link here with the work of Craw, Maclagan and Thomas in [CMT07a] which bears further investigation. In a toric context, they have rediscovered these inequalities as dual, in a certain sense, to the defining equations of the coherent component Y_{θ} of the moduli space M_{θ} of θ-semistable G-constellations. They then use them to compute the distinguished θ-semistable G-constellations parametrised by torus orbits of Y_{θ}. In particular, their Theorem 7.2 allows them to explicitly write down the tautological gnat-family on Y_{θ} and suggests that, up to a reflection, it is the gnat family which minimizes θ.{D_{χ}}. We shall see an example of that for the case of Y_{θ} = Hilb^{G} in our Proposition 3.17.

Proof. Take an open cover U_{i} on which all L(−D_{χ}) are trivialised and write g_{χ,i} for the generator of L(−D_{χ}) on U_{i}. As R is a direct sum of its G-homogeneous parts, it is sufficient to check the closure under the action of just the homogeneous functions. Thus it suffices to establish that for each f ∈ R_{G}, each U_{i} and each χ ∈ G^{∨}

fg_{χ,i} ∈ O_{Y} (U_{i})g_{χρ(f),i}

On the other hand, with the notation above, G-Cartier divisor D_{χ}+(f)−D_{χρ(f)} is given on U_{i} by \frac{fg_{χ,i}}{g_{χρ(f),i}} and it being effective is equivalent to

\frac{fg_{χ,i}}{g_{χρ(f),i}} ∈ O_{Y} (U_{i})

for all U_{i}’s. The result follows.

3.3 Canonical family

We have not yet given any evidence of any gnat-families actually existing on an arbitrary resolution Y of X.

Proposition 3.7 (Canonical family). Let Y be a resolution of X = ℂ^{n}/G. Define the set {C_{χ}}_{χ∈G^{∨}} of G-Weil divisors by

C_{χ} = ∑ v(P, χ)P

where P runs over all prime Weil divisors on Y and v(P, χ) are the numbers introduced in Definition 2.4 (lifted to [0, 1) ⊂ ℚ). Then {C_{χ}}_{χ∈G^{∨}} is a reductor set. We call the corresponding family the canonical gnat-family on Y.

Proof. We must show that {C_{χ}} satisfies the inequalities (3.2). Choose any χ ∈ G^{∨}, any f ∈ R_{G} and any prime divisor P on Y. Observe that 0 ≤ v_{P} (χ), v_{P} (χρ(f)) < 1 by definition, while v_{P} (f) ≥ 0 since f^{|G|} is regular on all of Y. So we must have

v_{P} (χ) + v_{P} (f) − v_{P} (χρ(f)) > −1

As the above expression must be integer-valued, we further have

v_{P} (χ) + v_{P} (f) − v_{P} (χρ(f)) ≥ 0

as required.

This family has a following geometrical description:

Proposition 3.8. On any resolution Y, the canonical family is isomorphic to the pushdown to Y of the structure sheaf N of the normalisation of the reduced fiber product Y ×_{X} ℂ^{n}.

Proof. First we construct a (R ⋊ G)⊗O_{Y} -module embedding of N into K(ℂ^{n}). Let α be the map O_{Y} ⊗_{R^{G}} R → K(ℂ^{n}) which sends a ⊗ b to ab. It is R ⋊ G ⊗ O_{Y} -equivariant. If we show that ker α is the nilradical of O_{Y} ⊗_{R^{G}} R, then N can be identified with the integral closure of the image of α in K(ℂ^{n}). Due to G-equivariance α decomposes as ⨁_{χ∈G^{∨}} α_{χ} with each α_{χ} a morphism O_{Y} ⊗_{R^{G}} R_{χ} → K_{χ}(ℂ^{n}). Observe that (O_{Y} ⊗_{R^{G}} R_{χ})^{|G|} ⊂ O_{Y} ⊗_{R^{G}} R_{χ_{0}} = O_{Y} as a product of |G| homogeneous functions is invariant. Hence (ker α_{χ})^{|G|} ⊂ ker α_{χ_{0}} = 0 as required. Write ⨁_{χ∈G^{∨}} N_{χ} for the decomposition of N into G-eigensheaves. Fix a point p ∈ Y and observe that f ∈ K_{χ}(ℂ^{n}) is integral over the local ring N_{p} if and only if f^{|G|} ∈ (N_{χ_{0}})_{p} = O_{Y,p}. Therefore

(N_{χ})_{p} = {f ∈ K_{χ}(ℂ^{n}) | G-Weil divisor Div(f) is effective at p}

In particular, the generator c_{χ} of C_{χ} at p lies in (N_{χ})_{p}. Observe further that for any f ∈ (N_{χ})_{p} the Weil divisor Div(f) − C_{χ} is effective at p as the coefficients of C_{χ} are just the fractional parts of those of Div(f) and the latter is effective. Therefore c_{χ} generates (N_{χ})_{p} as O_{Y,p}-module, giving N_{χ} = L(−C_{χ}) as required.

3.4 Symmetries

Having demonstrated that the set of equivalence classes of gnat-families is always non-empty, we now establish two types of symmetries which this set possesses. It is worth noting that from the description of the symmetries of the chambers in the parameter space of the stability conditions for G-constellations described in [CI04], Section 2.5, it follows that all the symmetries described below take the subset of gnat-families on Y consisting of universal families of stable G-constellations into itself.

Proposition 3.9 (Character Shift). Let {D_{χ}} be a normalised reductor set. Then for any χ in G^{∨}

D′_{χλ} = D_{χ} − D_{λ^{−1}}  (3.4)

is also a normalised reductor set. We call it the χ-shift of {D_{χ}}.

Proof. Writing out the reductor condition (3.2) for the new divisor set {D′_{χ}} we get:

(D_{χ} − D_{λ^{−1}}) + (m) − (D_{χρ(m)} − D_{λ^{−1}}) ≥ 0

Cancelling out D_{λ}^{−1}, we obtain precisely the reductor condition for the original set {D_{χ}}. And since

D′_{χ_{0}} = D′_{λ^{−1}λ} = D_{λ^{−1}} − D_{λ^{−1}} = 0

we see that the new reductor set is normalised.

NB: Observe, that for a reductor set {D_{χ}} and for any χ-Weil divisor N, the set {D_{χ} + N} is linearly equivalent to the χ-shift of {D_{χ}}.

Proposition 3.10 (Reflection). Let {D_{χ}} be a normalised reductor set. Then the set {−D_{χ^{−1}}} is also a normalised reductor set, which we call the

reflection of {D_{χ}}.

Proof. We need to show that

−D_{χ^{−1}} + (m) − (−D_{χ^{−1}ρ(m)^{−1}}) ≥ 0

Rearranging we get

D_{χ^{−1}ρ(m)^{−1}} + (m) − D_{χ^{−1}ρ(m)^{−1}ρ(m)} ≥ 0

which is one of the reductor equations the original set {D_{χ}} must satisfy. As D′_{χ_{0}} = −D_{χ_{0}} = 0, the new set is normalised.

3.5 Maximal shift family and finiteness

We now examine the individual line bundles L(−D_{χ}) in a gnat-family and show that the reductor condition imposes a restriction on how far apart from each other they can be.

Lemma 3.11. Let {D_{χ}} be a reductor set. Write each D_{χ} as ∑ q_{χ,P} P, where P ranges over all the prime Weil divisors on Y. For any χ_{1}, χ_{2} ∈ G^{∨} and for any prime Weil divisor P, we necessarily have

min_{f∈R_{χ_{1}/χ_{2}}} v_{P} (f) ≥ q_{χ_{1},P} − q_{χ_{2},P} ≥ − min_{f∈R_{χ_{2}/χ_{1}}} v_{P} (f)  (3.5)

Proof. Both inequalities follow directly from the reductor condition (3.2): the right inequality by setting χ = χ_{1} ∈ G^{∨}, ρ(f) = \frac{χ_{2}}{χ_{1}} and letting f vary within R_{ρ(f)}; the left inequality by setting χ = χ_{2} and ρ(f) = \frac{χ_{1}}{χ_{2}} .

This suggests the following definition:

Definition 3.12. For each character χ ∈ G^{∨}, we define the maximal shift

χ-divisor M_{χ} to be

M_{χ} = ∑_{P}( min_{f∈R_{χ}} v_{P} (f))P  (3.6)

where P ranges over all prime Weil divisors on Y.

Lemma 3.13. The G-Weil divisor set {M_{χ}} is a normalised reductor set. We call the corresponding family the maximal shift gnat-family on Y.

Proof. We need to show that for any f ∈ R_{G} and any prime divisor P

v_{P} (m_{χ}) + v_{P} (f) − v_{P} (m_{χρ(f)}) ≥ 0

where m_{χ} and m_{χρ(f)} are chosen to achieve the minimality in (3.6). Observe that m_{χ}f is also a G-homogeneous element of R, therefore by the minimality of v_{P} (m_{χρ(f)}) we have

v_{P} (m_{χ}f) ≥ v_{P} (m_{χρ(f)})

as required. To establish that M_{χ_{0}} = 0 we observe that for any G-homogeneous f ∈ R we have v_{P} (f) ≥ 0 on any prime Weil divisor P as f^{|}G| is globally regular. Moreover for f in R_{χ_{0}} = R^{G} this lower bound is achieved by f = 1.

Observe that with Lemma 3.13 we have established another gnat-family which always exists on any resolution Y. While sometimes it coincides with the canonical family, generally the two are distinct.

Proposition 3.14 (Maximal Shifts). Let {D_{χ}} be a normalised reductor set. Then for any χ ∈ G^{∨}

M_{χ} ≥ D_{χ} ≥ −M_{χ^{−1}}  (3.7)

Moreover both the bounds are achieved.

Proof. To establish that (3.7) holds set χ_{2} = χ_{0} in Lemma 3.11. Lemma 3.13 shows that the bounds are achieved.

Proposition 3.15. If the coefficient of a maximal shift divisor M_{χ} at a prime divisor P ⊂ Y is non-zero, then either P is an exceptional divisor or the image of P in X is a branch divisor of ℂ^{n} ⟶^{q} X.

Proof. Let P be a prime divisor on X which is not a branch divisor of q. Let χ ∈ G^{∨}. By the defining formula (3.6) it suffices to find f ∈ R_{χ} such that v_{P} (f) = 0. As R is a PID, there exist t_{1}, …, t_{k} ∈ R such that (t_{1}), …, (t_{k}) are all the distinct prime divisors lying over P in ℂ^{n}. Observe that the product t_{1} … t_{n} must be G-homogeneous. Since P is not a branch divisor, there exists u ∈ R such that t_{1} … t_{k}u is invariant and u ∉ (t_{i}) for all i. Then u′ = u^{|G|−1} is a Ghomogeneous function of the same weight as t_{1} … t_{k} and v_{P} (u′) = 0. Now take any f ∈ R_{χ} and consider its factorization into irreducibles. G-homogeneity of f implies that all t_{i} occur with the same power k. Now replacing (t_{1} … t_{n})^{k} in the factorization by (u′)^{k} we obtain an element of R_{χ} whose valuation at P is zero.

Corollary 3.16. The number of equivalence classes of gnat-families on Y is finite.

Proof. Let {D_{χ}} be a normalised reductor set. Coefficients of D_{χ} at prime divisors P of Y have fixed fractional parts (Definition 2.5), are bound above and below (Proposition 3.14) and are zero at all but finite number of P (Proposition 3.15). This leaves only a finite number of possibilities.

For one particular resolution Y the family provided by the maximal shift divisors has a nice geometrical description.

Proposition 3.17. Let Y = Hilb^{G} ℂ^{n}, the coherent component of the moduli space of G-clusters in ℂ^{n}. If Y is smooth, then ⨁ L(−M_{χ}) is the universal family F of G-clusters parametrised by Y, up to the usual equivalence of families.

Proof. Firstly F is a gnat-family, as over any set U ⊂ X such that G acts freely on q^{−1}(U) we have F|_{U} ≃ π^{∗}q_{∗}O_{ℂ^{n}}|_{U}. Write F as ⊕L(−D_{χ}) for some reductor set {D_{χ}}. Take an open cover {U_{i}} of Y and consider the generators {f_{χ,i}} of D_{χ} on each U_{i}. Working up to equivalence, we can consider {D_{χ}} to be normalised and so f_{χ_{0},i} = 1 for all U_{i}. Now any G-cluster Z is given by some invariant ideal I ⊂ R and so the corresponding G-constellation H^{0}(O_{Z}) is given by R/I. In particular note that R/I is generated by R-action on the generator of χ_{0}-eigenspace. Therefore any f_{χ,i} is generated from f_{χ_{0},i} = 1 by R-action, which means that all f_{χ,i} lie in R. But this means that for any prime Weil divisor P on Y we have

v_{P} (f_{χ,i}) ≥ min_{f∈R_{χ}} v_{P} (f)

and therefore D_{χ} ≥ M_{χ}. Now Proposition 3.14 forces the equality.

4 Conclusion

We summarise the results achieved in the following theorem:

Theorem 4.1 (Classification of gnat-families). Let G be a finite abelian subgroup of GL_{n}(ℂ), X the quotient of ℂ^{n} by the action of G, Y nonsingular and π : Y → X a proper birational map. Then isomorphism classes of gnat-families on Y are in 1-to-1 correspondence with linear equivalence classes of G-divisor sets {D_{χ}}_{χ∈G^{∨}}, each D_{χ} a χ-Weil divisor, which satisfy the inequalities

D_{χ} + (f) − D_{χρ(f)} ≥ 0 ∀ χ ∈ G^{∨}, G-homogeneous f ∈ R

Such a divisor set {D_{χ}} corresponds then to a gnat-family ⨁ L(−D_{χ}). This correspondence descends to a 1-to-1 correspondence between equivalence classes of gnat-families and sets {D_{χ}} as above and with D_{χ_{0}} = 0. Furthermore, each divisor D_{χ} in such a set satisfies inequality

M_{χ} ≥ D_{χ} ≥ −M_{χ^{−1}}

where {M_{χ}} is a fixed divisor set defined by

M_{χ} = ∑_{P}( min_{f∈R_{χ}} v_{P} (f))P

As a consequence, the number of equivalence classes of gnat-families is finite.

Proof. Corollary 3.3 establishes the correspondence between isomorphism classes of gnat-families and linear equivalence classes of reductor sets. Proposition 3.6 gives description of reductor sets as the divisor sets satisfying the reductor condition inequalities. Corollary 3.5 gives the correspondence on the level of equivalence classes of gnat-families and normalised reductor sets. Proposition 3.14 establishes the bounds on the set of all normalised reductor sets and Corollary 3.16 uses it to show that the set of all normalised reductor sets is finite.

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Timothy Logvinenko Department of Mathematical Sciences University of Liverpool Peach Street Liverpool, L69 7ZL, UK T.Logvinenko@liverpool.ac.uk 824

Documenta Mathematica 13 (2008)