The class group of a minimal model of a quotient singularity

Let $V$ be a finite-dimensional vector space over the complex numbers and let $G\leq \operatorname{SL}(V)$ be a finite group. We describe the class group of a minimal model (that is, $\mathbb Q$-factorial terminalization) of the linear quotient $V/G$. We prove that such a class group is completely controlled by the junior elements contained in $G$.


Introduction
Let V be a finite-dimensional vector space over C and let G ≤ GL(V ) be a finite group.By the classical theorem of Chevalley-Serre-Shephard-Todd [Ser68, Théorème 1'], the linear quotient of V by G is smooth if and only if G is generated by reflections, that is, elements g ∈ G with rk(g − id V ) = 1.If V /G is singular, we also refer to this affine variety as a quotient singularity.Regarding the class group of Weil divisors Cl(V /G) of V /G, there is a nowadays well-known theorem by Benson [Ben93] which says that Cl(V /G) ∼ = Hom(G/K, C × ), where K ≤ G is the subgroup generated by the reflections contained in G. Restricting to subgroups G ≤ SL(V ), we will show that the class group of a certain partial resolution of V /G can be described in a similar way.
The inclusion G ≤ SL(V ) implies that G cannot contain any reflections and V /G is a singular variety.Further, by [BCHM10], there is a Q-factorial terminalization -or minimal model -φ : X → V /G, that is, a crepant, partial resolution of singularities of V /G with at most terminal singularities; see below for the precise definition.Q-factorial terminalizations of quotient singularities are well studied and, starting with the famous McKay correspondence [McK80], it is often observed that properties of φ : X → V /G are controlled only by the action of G on V , see for example the survey [Rei02].This leads to the expectation that questions regarding the birational geometry of V /G should be answered by only considering G, see Reid' In this note, we give further evidence of this phenomenon by describing the class group Cl(X) of X.By a version of the McKay correspondence due to Ito and Reid [IR96], the rank of the free part of this finitely generated abelian group coincides with the number of conjugacy classes of junior elements in G; we recall the definition of these distinguished elements of G below.In the literature, one further finds a sufficient condition and a full characterization of the freeness of Cl(X) by Donten-Bury-Wiśniewski [DW17, Lemma 2.11] and Yamagishi [Yam18,Proposition 4.14], respectively.
We study the torsion part Cl(X) tors of Cl(X) and obtain a theorem, which reads similar to Benson's result on Cl(V /G).
Theorem 1.1 (= Theorem 5.1).Let G ≤ SL(V ) be a finite group and let H ≤ G be the subgroup generated by the junior elements contained in G. Let φ : X → V /G be a Q-factorial terminalization of V /G.Then we have a canonical isomorphism of abelian groups Cl(X) tors ∼ = Hom(G/H, C × ) , which is induced by the push-forward map φ * : Cl(X) → Cl(V /G).
Combining our result with [IR96] gives a complete description of the class group.
Corollary 1.2 (= Corollary 5.2).With the assumptions in Theorem 1.1, we have where m is the number of conjugacy classes of junior elements in G.
Remark 1.3.We emphasize that by Corollary 1.2, the class group of a Q-factorial terminalization is completely controlled by the group G itself.This agrees well with the mentioned principle of Reid [Rei02, Principle 1.1].
Remark 1.4.We make a further philosophical observation.Recall that by the theorem of Chevalley-Serre-Shephard-Todd, the variety V /G is smooth if and only if G ≤ GL(V ) is generated by reflections.This is mirrored by a theorem of Yamagishi [Yam18, Theorem 1.1] which generalizes a result of Verbitsky [Ver00, Theorem 1.1] and says that, if a Q-factorial terminalization X → V /G is smooth, then G ≤ SL(V ) must be generated by junior elements.We feel that Theorem 1.1 mirrors Benson's theorem (Theorem 2.1) in the same way.In both cases the geometry of the linear quotient V /G is controlled by the reflections contained in G and the junior elements control the geometry of the Q-factorial terminalization X → V /G.Still, it appears that this picture is far from complete.The theorem of Verbitsky and Yamagishi on the smoothness of X is not an equivalence and the freeness of the class group also depends in a somewhat convoluted way on the junior elements, see Remark 6.1.
Notice that the mentioned theorem of Verbitsky and Yamagishi together with our result implies that if X is smooth, then the class group is free.This result was already shown in [DW17, Lemma 2.11] using different methods; a stronger statement appears in [Yam18, Proposition 4.14], see also Corollary 5.4.
To prove Theorem 1.1, we study the Cox ring R(V /G) which is graded by Cl(V /G).This directly extends the work of [Yam18].The Cox ring R(V /G) was introduced by Cox [Cox95] in the context of toric varieties and transferred to a more general setting of birational geometry by Hu and Keel [HK00].
After fixing the notation and recalling fundamental results in Section 2, we present a correspondence of homogeneous elements in R(V /G) and effective divisors on V /G in Section 3. We do not claim originality for the results in these sections, but only unify the different sources and transfer them to our setting.In Section 4, we analyse the grading of the Cox ring R(V /G) by Cl(V /G) in more detail and finally use this in Section 5 to prove our theorem.We close with some small examples in Section 6.

Preliminaries
Let V be a finite-dimensional vector space over C and let G ≤ GL(V ) be a finite group.Note that we will restrict to subgroups of SL(V ) after a short general discussion.Write V /G := Spec C[V ] G for the linear quotient, where C[V ] G is the invariant ring of G.The variety V /G is normal and Q-factorial, see [Ben93].
2.1.The class group of V /G.Recall that by a reflection (or pseudo-reflection), we mean an element g ∈ GL(V ) with rk(g − id V ) = 1.The class group of V /G was first described by Benson [Ben93, Theorem 3.9.2]building on work of Nakajima [Nak82, Theorem 2.11].
Theorem 2.1 (Benson).Let G ≤ GL(V ) be a finite group and let K ≤ G be the subgroup generated by the reflections contained in G. Then there is an isomorphism With K as in the theorem, let Ab(G/K) := (G/K)/[G/K, G/K] be the abelianization of G/K and write Ab(G/K) ∨ for the group of irreducible (hence linear) characters of this group.By elementary character theory [BKZ18, Theorem I.9.5], we have Hom and we hence often write Ab(G/K) ∨ for the class group of V /G.
Definition 2.2 (Q-factorial terminalization).Let Y be a normal Q-factorial variety.A Q-factorial terminalization of Y is a projective birational morphism φ : X → Y such that X is a normal Q-factorial variety with terminal singularities and φ is crepant.
In our context, a Q-factorial terminalization X → V /G is often referred to as minimal model, see for example [IR96].However, the usage of this terminology is not entirely uniform, which is why we decided to use the more unwieldy term 'Q-factorial terminalization'.
We have the following special case of the deep result achieved in [BCHM10].[IR96].We recall some notation from [IR96].
For the following definition, let g ∈ GL(V ) be of finite order r and fix a primitive r-th root of unity ζ r .Then there are integers 0 ≤ a i < r, such that the eigenvalues of g are given by the powers ζ a1 r , . . ., ζ an r , where dim V = n.Definition 2.4 (Age and junior elements).We call the number age(g the age of g.Elements of age 1 are called junior. By construction, the number age(g) is an integer if g ∈ SL(V ) and the junior elements in SL(V ) are hence the non-trivial elements of minimal age 1.The age is by definition invariant under conjugacy and we refer to the conjugacy classes of a group G ≤ SL(V ) consisting (only) of junior elements as junior conjugacy classes.
Remark 2.5.The age as defined above depends on the choice of the root of unity ζ r , although this is hidden in the notation.See [IR96, p. 224, Remark 3] for an example demonstrating this.However, the subgroup generated by the junior elements, which is relevant for Theorem 1.1, is independent of any choices.We give a short argument for this in Appendix A, see Lemma A.4. Definition 2.6 (Monomial valuation).For non-negative integers a 1 , . . ., a n ∈ Z ≥0 with gcd(a 1 , . . ., a n ) = 1, we construct a discrete valuation v : We call v a monomial valuation.
This construction indeed gives a well-defined discrete valuation, see [Kal02, Definition 2.1].
Notation 2.7.Let g ∈ SL(V ) of finite order r be a junior element with respect to the primitive r-th root of unity ζ r .That is, the eigenvalues of g are ζ a1 r , . . ., ζ an r with 1 r n i=1 a i = 1.For d := gcd(a 1 , . . ., a n ), we obtain 1 d as the age of g with respect to the root of unity ζ d r .As age(g) is an integer, we conclude d = 1 and we can therefore define a monomial valuation The construction of v g again depends on the choice of a root of unity, see Remark 2.5 above.The valuation v g is stable under conjugacy of g and we can hence associate valuations to conjugacy classes in G without needing to specify a particular representative.
Theorem 2.8 (Ito-Reid, 'McKay correspondence').Let G ≤ SL(V ) be a finite group and let X → V /G be a Q-factorial terminalization.Then there is a one-toone correspondence between the junior conjugacy classes of G and the irreducible exceptional divisors on X.
More precisely, if E is a divisor corresponding to a conjugacy class of a junior element g ∈ G of order r in this way, then v E = 1 r v g , where v E is the valuation of E and we identify C(X) = C(V ) G via the birational morphism X → V /G.See [IR96, Section 2.8] for a proof.
Let m ∈ Z ≥0 be the number of junior conjugacy classes in G. Using [Har77, Proposition II.6.5 (c)], we have an exact sequence of abelian groups where E i are the irreducible exceptional divisors on X.As Cl(V /G) is finitely generated, this implies that Cl(X) is finitely generated as well.
Notation 2.9.We write Cl(X) tors ≤ Cl(X) for the torsion subgroup of Cl(X) and Cl(X) free for the corresponding factor group, that is, Cl(X) free = Cl(X)/ Cl(X) tors .
The sequence in (1) can be extended to a short exact sequence by Grab [Gra19].
Proposition 2.10 (Grab).There is a short exact sequence of abelian groups where E i ∈ Div(X) are the irreducible components of the exceptional divisor of φ and φ * : Cl(X) → Cl(V /G) is the induced push-forward map.
Remark 2.11.As Cl(V /G) is a torsion group, we can deduce with Theorem 2.8 and Proposition 2.10 that Cl(X) free ∼ = Z m for the free part of Cl(X), where m ∈ Z ≥0 is the number of junior conjugacy classes in G.

Cox rings.
To understand the torsion part Cl(X) tors of Cl(X) we use the These properties are in particular fulfilled for linear quotients V /G and we may hence speak about the Cox ring R(V /G).
We summarize a result on the Cox ring of R(V /G) by Arzhantsev and Gaǐfullin [AG10].Recall that Cl(V /G) ∼ = Ab(G) ∨ as G ≤ SL(V ) cannot contain any reflections.There is an action of Ab(G) on the ring C[V ] [G,G] induced by the action of G.This action induces a grading by Ab(G) ∨ by setting the graded component of a character χ ∈ Ab(G) ∨ to be where we write γ.f for the action of Theorem 2.12 (Arzhantsev-Gaǐfullin).Let G ≤ SL(V ) be a finite group.Then there is an See [AG10, Theorem 3.1] for a proof.

Correspondence of effective divisors and homogeneous elements
To be able to deduce information about Cl(X), we use the connection between effective divisors and canonical sections in the Cox ring R(V /G) of V /G.We recall this correspondence and adapt it to our setting.Remark 3.2.Working with the ring R(V /G) brings two subtle problems.First of all, homogeneous elements f ∈ R(V /G) are only residue classes of elements of the function field C(V ) G as Cl(V /G) is a torsion group.We hence cannot immediately identify such elements f with a function in C(V ) G .However, for a divisor D ∈ Div(V /G) we have an isomorphism by [ADHL15, Lemma 1.4.3.4].That means, once we fixed a representative of the degree of a homogeneous element f ∈ R(V /G) we can uniquely lift f to an element of C(V ) G .
The second problem comes from the fact that we make heavy use of the graded isomorphism Ψ : ] as in Theorem 2.12 to the extent that one might forget that the isomorphism is not an identity.This is in particular important when we work with a valuation v : C(V ) → Z.We can only use v on elements of G] and cannot apply v to elements of R(V /G) in a well-defined way without choosing a system of representatives for the class group.For D ∈ Div(V /G), we have an isomorphism of vector spaces ψD : by setting ψD := Ψ • ψ D .Notice that for the trivial divisor, this gives an identity as we have , where 1 denotes the trivial character.
, let f ∈ C(V ) G be the rational function mapping to f via the isomorphism determined by D as in Remark 3.2.We associate to f an effective divisor The construction of a [D]-divisor is not limited to our setting; see [ADHL15, Construction 1.5.2.1] for more details and the general case.We point out that G] is in general not an element of C(V ) G , that is, there is no meaning in writing div(f ).
The [D]-divisor behaves well with respect to the multiplication of elements.
Proposition 3.5.Let E ∈ Div(V /G) be an effective divisor.There exist a class The element f is unique up to constants; it is called a canonical section of E.
Using the correspondence between effective divisors and homogeneous elements one can derive a precise description of the image of the strict transform of an effective divisor D ∈ Div(V /G) in the free group Cl(X) free .The general idea of this argument appeared to our knowledge first in [DW17, Lemma 3.22].We require a bit of notation.
Recall that by Theorem 2.8 we have a one-to-one correspondence between the junior conjugacy classes of G and the irreducible components of the exceptional divisor of φ : X → V /G.Let {g 1 , . . ., g m } ∈ G be a minimal set of representatives of the junior conjugacy classes corresponding to exceptional prime divisors E 1 , . . ., E m ∈ Div(X).For each i ∈ {1, . . ., m}, write v i for the monomial valuation on C(V ) defined by g i and recall from Theorem 2.8 that we have v Ei = 1 ri v i , where v Ei is the divisorial valuation of E i and r i the order of g i .
The following also appears in [Gra19, Proposition 4.1.9].We present the argument from [Yam18, Lemma 4.3] for completeness.Denote the canonical projection by ρ : Cl(X) → Cl(X) free .Proposition 3.6.Let D ≥ 0 be an effective divisor on V /G and let be a canonical section.Write D := φ −1 * (D) for the strict transform of D via φ.Then we have the equality in Cl(X) free .
Proof.As f is homogeneous with respect to the action of Ab(G), there is r , where the first equality is by Lemma 3.4, the second by the independence of choice of representative and the third is by the fact that Hence, we have the equality of classes We may finally cancel r in the free group Cl(X) free giving Remark 3.7.Notice that the proof of Proposition 3.6 in fact computes the degree of a preimage of the canonical section f under a graded surjective morphism R(X) → R(V /G) induced by φ.See [ADHL15, Proposition 4.1.3.1] for the construction of this morphism between the Cox rings.

A digression on gradings
To understand the group Cl(X), we first have to get a better understanding of the grading of C[V ] [G,G] by Ab(G) ∨ .Unfortunately, there are a few subtle details involved, turning this into quite a technical discussion.
Again, let g 1 , . . ., g m ∈ G be representatives of the junior conjugacy classes corresponding to the exceptional divisors E 1 , . . ., E m ∈ Div(X) of φ and write v 1 , . . ., v m for the monomial valuations corresponding to the g i .
At first, fix i ∈ {1, . . ., m}.Let the eigenvalues of g i be given by ζ In particular, there is a graded morphism given by the identity on the rings and by the projection Z → Z/r i Z on the grading groups.
Write g i .ffor the linear action of g i on f ∈ C[V ].Observe that for every 1 ≤ i ≤ m we have an action of G] as required.Hence the grading by As the actions of the elements g 1 , . . ., g m on C[V ] [G,G] commute, we can consider all the induced gradings at the same time and hence obtain a grading by the group The g i do not commute with each other in general, so we cannot decompose their actions on C[V ] into a common eigenbasis.Hence, we cannot put the above gradings together to obtain a grading by as there are in general no polynomials which are homogeneous with respect to all gradings at the same time.
Let H ≤ G be the subgroup of G generated by the junior elements contained in G.In general, the representatives g 1 , . . ., g m do not suffice to generate H. Let and notice that this group is generated by the residue classes g 1 , . . ., g m modulo [G, G].This gives a map which is surjective onto H.This surjection corresponds to an embedding of character groups a projection of characters Ab(G) ∨ → H ∨ by restriction.We conclude: Lemma 4.2.The gradings on C[V ] [G,G] coming from the actions of the groups Ab(G), H and ⟨g 1 ⟩ × • • • × ⟨g m ⟩ are compatible in the sense that there is a graded morphism We state for later reference: Proof.For the first statement, we note that the image of [G, G] under the projection and an application of the isomorphism theorems gives the claim.The second statement follows directly as ∨ is a contravariant functor.□ The following three lemmas are key ingredients for our theorem on Cl(X).G] be Ab(G) ∨ -homogeneous.For every index i ∈ {1, . . ., m}, we have G] be Ab(G) ∨ -homogeneous.Fix an i ∈ {1, . . ., m}.Then f is deg i -homogeneous by Lemma 4.2.By Lemma 4.1, there exist deg i -and deg i -homogeneous elements f i,j ∈ C[V ] such that f = j f i,j and deg i (f i,j ) < deg i (f i,j ′ ) whenever j < j ′ .In particular, we have deg i (f i,1 ) = v i (f ) and deg i (f i,1 ) = deg i (f ).Hence, we conclude G] be Ab(G) ∨ -homogeneous.We have Proof.This is saying that the relative invariants with respect to the linear characters of Ab(G) on C[V ] [G,G] are non-empty which holds by [Nak82, Lemma 2.1].□ Notice that the lemma also implies that we can find an effective divisor in any class of divisors in Cl(V /G).

The class group
We are now prepared for our theorem.
Theorem 5.1.Let G ≤ SL(V ) be a finite group and let H ≤ G be the subgroup generated by the junior elements contained in G. Let φ : X → V /G be a Q-factorial terminalization of V /G.Then we have a canonical isomorphism of abelian groups Cl(X) tors ∼ = Ab(G/H) ∨ = Hom(G/H, C × ) , which is induced by the push-forward map φ * : Cl(X) → Cl(V /G).
Proof.For ease of notation, we identify Cl(V /G) with Ab(G) ∨ via Theorem 2.1 and use both groups synonymously.Notice that Ab(G/H) ∨ is the subgroup of Ab(G) ∨ consisting of those characters which take value 1 on every junior element.We claim that restricting φ * to Cl(X) tors induces a bijection onto Ab(G/H) ∨ .
We first show that we indeed have φ * (Cl(X) tors ) ⊆ Ab(G/H) ∨ .Let D ∈ Div(X) be a divisor on X.By Lemma 4.6, there is with a i ∈ Z and where E 1 , . . ., E m ∈ Div(X) are the irreducible components of the exceptional divisor of φ.As before let ρ : Cl(X) → Cl(X) free := Cl(X)/ Cl(X) tors be the canonical projection.Applying ρ on both sides of (2) and using Proposition 3.6 yields We now prove that ψ is bijective.For injectivity, notice that the morphism of groups is injective.This follows from the exactness of the sequence in Proposition 2.10 noticing that the group m i=1 ZE i embeds into Cl(X) free , see also [Gra19,Lemma 4.1.4].The injectivity of ϑ implies the injectivity of ψ: if we have ψ( Now let χ ∈ Hom(G/H, C × ) be a character, which we identify with a class of divisors [D] ∈ Cl(V /G).By Lemma 4.6, there exists 0 and we may assume without loss of generality that D ∈ Div(V /G) is effective and f is the canonical section of D as in Proposition 3.5.By the assumption on χ, we have Corollary 5.2.Let G ≤ SL(V ) be a finite group and let H ≤ G be the subgroup generated by the junior elements contained in G. Let φ : X → V /G be a Q-factorial terminalization of V /G.Then we have where m is the number of junior conjugacy classes in G. Further, the canonical embedding ι : Proof.The first part follows directly from the mentioned theorems.For the second part, we combine Proposition 2. We obtain [Yam18, Proposition 4.14] as a further corollary.
Corollary 5.4 (Yamagishi).Let G ≤ SL(V ) be a finite group and let φ : X → V /G be a Q-factorial terminalization of V /G.Then the class group Cl(X) is free if and only if G is generated by the junior elements contained in G together with [G, G].

Examples and closing remarks
Remark 6.1.Note that in Corollary 5.4 we cannot drop the part 'together with [G, G]' for the equivalence, that is, there are groups which are not generated by junior elements such that Cl(X) is free.For example, let I ≤ SL 2 (C) be the binary icosahedral group [LT09, Theorem 5.14] and set G := {diag(g, g) | g ∈ I} ≤ SL 4 (C).The abelianization Ab(I) = {1} is trivial, so the same is true for Ab(G).However, every non-trivial element in I is of age 1, hence all non-trivial elements of G are of age 2 and G does not contain any junior elements.Hence, the class group of a Q-factorial terminalization of C 4 /G is trivial and therefore free.For an example of a non-trivially free class group, one considers the direct product of G with a group generated by junior elements.
Example 6.2.As a 'reality check', let G ≤ SL(V ) be a group which does not contain any junior elements.Then age(g) > 1 for every non-trivial g ∈ G, so V /G has terminal singularities by s 'principle of the McKay correspondence' [Rei02, Principle 1.1].
Notation 3.1.For a divisor D ∈ Div(V /G), we write χ [D] ∈ Ab(G) ∨ for the character corresponding to the class [D] ∈ Cl(V /G) under the isomorphism in Theorem 2.1.
10 and the first part to obtain Ab(G) ∨ ∼ = coker(ι) ⊕ Ab(G/H) ∨ and then the claim follows by Lemma 4.3.□ Remark 5.3.As the isomorphism in Theorem 5.1 is induced by φ * , we can see the sequence in Proposition 2.10 as the direct sum of the short exact sequences 0 [Kol13, Theorem 3.21].Hence, V /G is a Q-factorial terminalization of itself and Corollary 5.2 gives Cl(V /G) = Ab(G) ∨ as in Theorem 2.1.Example 6.3.For a non-trivial example, we consider the groupG := diag(−1, −1, −ζ 3 , −ζ 23 ) ≤ SL 4 (C) of order 6, where ζ 3 is a primitive third root of unity.As G does not contain any reflections, we have Cl(C 4 /G) ∼ = Z/6Z.Let g ∈ G be an element of order r ∈ Z >0 .There are 0 ≤ a i < r such that the eigenvalues of g are given by ζ e(G)a i r.Then the eigenvalues of φ(g) are given by (Let ζ, η ∈ C × be primitive e(G)-th roots of unity and writeH ζ ≤ G, respectively H η ≤ G,for the subgroup of G generated by the ζ-junior elements, respectively the η-junior elements.Then we have H ζ = H η .Proof.Let φ : G → G be the bijection in Lemma A.3, so taking powers by some a ∈ Z >0 .If g ∈ H ζ is a ζ-junior element, then φ(g) = g a ∈ H η is an η-junior element.Further, we clearly have g a ∈ H ζ , hence H η ⊆ H ζ and an analogous argument gives the reverse inclusion.□ -th root of unity ζ ri and integers 0 ≤ a i,j < r i , where r i is the order of g i in G and n = dim V .This induces a Z-grading deg i on C[x 1 , . . ., x n ] by putting deg i (x j ) := a i,j .For a polynomial f ∈ C[x 1 , . . ., x n ], the valuation v i (f ) is then the degree of the homogeneous component of f of minimal degree with respect to deg i .Notice that in this construction, we consider V in an eigenbasis of g which we denote by deg i .Write (C[V ], Z/r i Z, deg i ) for the ring C[V ] graded by Z/r i Z via deg i .We directly obtain: Lemma 4.1.With the above notation, if f ∈ C[V ] is deg i -homogeneous, then f is deg i -homogeneous as well and we have i giving rise to the isomorphism C[V ] ∼ = C[x 1 , . .., x n ].However, the grading deg i is well-defined on C[V ] for any basis of V , although the variables of the polynomial ring are in general not homogeneous.As we endow the same ring with gradings by different groups, we use the non-standard notation (C[V ], Z, deg i ) for the ring C[V ] graded by Z via deg i .The group ⟨g i ⟩ acts on C[V ] and hence induces a grading by ⟨g i ⟩ ∨ ∼ = Z/r i Z, m} if and only if f ∈ C[V ] H , where H ≤ G is the subgroup generated by the junior elements contained in G.Proof.By Lemma 4.4, we havev i (f ) ≡ deg i (f ) mod r i for every i.Therefore, r i | v i (f ) is equivalent to deg i (f ) =0 for every i.Equivalently, every g i acts trivially on f .Since f is furthermore [G, G]-invariant, we conclude that this is the case if and only if every junior element in G leaves f invariant and hence f ∈ C[V ] H . □