Computing resolutions of quotient singularities
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
arXiv:1603.00071v1 [math.AG] 29 Feb 2016
MARIA DONTEN-BURY AND SIMON KEICHER Abstract. Let G ⊆ GL(n) be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution X → Cn /G, which is based just on the geometry of the singularity Cn /G, without further knowledge of its resolutions. As an application, we determine the Cox rings of resolutions X → C3 /G for all G ⊆ GL(3) with the aforementioned property and of order |G| ≤ 12. We also provide examples in dimension 4.
1. Introduction We consider quotient spaces Cn /G = Spec C[x1 , . . . , xn ]G , where G is a finite group acting linearly on Cn . By Noether’s theorem the ring of invariants of G is finitely generated, hence such quotients are (complex) algebraic varieties. They are usually singular and in such a case are called quotient singularities. In their construction, geometry meets finite group theory and there have been many attempts of extending this relation to resolutions of Cn /G. An example, probably the most important one, of describing the geometric structure of crepant (which in this case means that the canonical divisor is linearly trivial) resolutions of Cn /G in terms of algebraic properties of the group G is the McKay correspondence; for an introduction see, e.g., [21]. Though proved in several cases (see section 3.2), in general it reveals how much there is still to learn about this class of singularities and their resolutions. In this paper, we study quotient singularities X0 := Cn /G and their resolutions in terms of the Cox ring M Γ (X0 , O(D)) , R (X0 ) = Cl(X)
see [1] for details on the construction. Cox rings have already been successfully used by various authors to study resolutions of quotient singularities, in particular for symplectic quotients in [8, 6], described by a generating set in a simpler ring, and via an algorithmic approach based on toric ambient modifications in [11]. In this article we generalize the methods used in [11]. While finishing a draft of this paper, we also found out about Yamagishi’s work [25], which extends [8, 6]. However, our approach is different from [25] and our methods are not restricted to the class of crepant resolutions. In particular, as explained below, we do not try to construct the generators of the Cox ring directly from the group structure data, but we obtain it via toric ambient modifications, chosen in an intermediate step based on tropical geometry. Our first contribution, presented in Section 2, is an an algorithm to compute and verify a candidate for the Cox ring of a resolution X → X0 without requiring further knowledge of X0 . More precisely, Algorithm 2.3 computes R(X0 ) and Algorithm 2.8 2010 Mathematics Subject Classification. 14E15, 14Q10, 14Q15, 14C20, 14L24. This work was completed while the first author held a Dahlem Research School Postdoctoral Fellowship at Freie Universit¨ at Berlin. The first author was partially supported by a Polish National Science Center project 2013/11/D/ST1/02580 and the second author was supported by Proyecto FONDECYT Postdoctorado N. 3160016. 1
2
M. DONTEN-BURY AND S. KEICHER
then computes a candidate for the Cox ring of a resolution X → X0 and verifies the result. Here, the main tool are toric ambient modifications as in [10, 12]: we embed X0 into an affine toric variety Z0 and compute a resolution Z → Z0 . The proper transform X → X0 then is the desired candidate for a resolution which we can verify algorithmically. Note that the choice of the toric resolution involves a tropical step; this is in the spirit of Tevelev and Teissier [22, 23]. Algorithm 2.8 is a variant of the algorithm [18] given for Mori dream spaces, which, in turn, is based on [1, 14] where the algorithm has been shown to work in the setting of complete rational complexity one T -varieties. See also [13] for the case of affine C∗ -surfaces. Our algorithms are implemented in the Open Source computer algebra system Singular [5]. Our second contribution concerns resolutions of three-dimensional quotient singularities; it is presented in Section 3. We first classify in Proposition 3.2 the finite, non-abelian groups G ⊆ GL(3) with |G| ≤ 12 and without pseudo-reflections (see Remark 2.2). When G is abelian, the quotient singularity is toric; since the algorithm of constructing a (toric) resolution and the structure of the Cox ring is known in this case, we do not consider it. Using our algorithms from Section 2, we first present the Cox rings of all singularities C3 /G on the list, see Proposition 3.3 and then their resolutions, see Theorem 3.5. Then we discuss certain properties of the obtained resolutions in order to understand what can be expected from the output of the algorithm in general. In particular, we check whether the resolutions for subgroups of SL(3) on the list are crepant. Finally, in Section 4, we emphasize that the method is not restricted to the 3dimensional setting by providing two 4-dimensional examples. Note that in dimension 4 less is known about resolutions of quotient singularities, hence computational experiments are even more valuable than in dimension 3. We provide the Cox rings of X0 = C4 /G for chosen representations of G = D8 and G = Q8 and compute the Cox rings of modifications X → X0 . For one of the cases, we are able to retrieve a crepant resolution that was also found in [6], see Proposition 4.5. Acknowledgement: We would like to thank J¨ urgen Hausen for several helpful discussions. 2. Algorithms In this section, we describe algorithms to compute the Cox ring of a quotient singularity X0 := Cn /G for a finite group G and the Cox ring of a resolution X → X0 . The latter algorithm is a variant of [1, 18]. It produces candidates using a tropical step and verifies them. We work with representations of G, usually assuming that they are faithful. In this case we will often identify the representation with its image in GL(n), i.e., consider G as a matrix group. Definition 2.1. A matrix group G ⊆ GL(n) is small if it does not have pseudoreflections; these are A ∈ G ⊆ GL(n) of finite order such that the subspace of fixed points (Cn )A is a hyperplane. We also say that a representation of an abstract group G is small, if its image in GL(n) is small. Remark 2.2. Note that the assumption on the representation of G not having pseudo-reflections is not restrictive: let H ⊆ G be the normal subgroup generated by all pseudo-reflections. Then Cn /G ∼ = (Cn /H) (G/H) ∼ = Cn (G/H) since Cn /H is smooth by the Chevalley-Shephard-Todd theorem. In particular, the singularity for G is the same as the one for the smaller group G/H.
The degree map of the Cox ring R(X0 ) is a homomorphism Zs → Cl(X0 ) that sends the i-th canonical basis vector ei ∈ Zs to deg(Ti ). We call the matrix with columns deg(T1 ), . . . , deg(Ts ) degree matrix.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
3
Algorithm 2.3 (Cox ring of X0 ). Input: a small group G ⊆ GL(n). • Compute generators f1 , . . . , fs for the invariant ring C[S1 , . . . , Sn ][G,G] . • Then G′ := G/[G, G] acts on the C-vector space V := linC (f1 , . . . , fs ). Compute matrices Mi such that, as a group, G′ = hM1 , . . . , Mk i ⊆ GL(s). • Compute a basis g1 , . . . , gs ∈ V of eigenvectors for V with respect to all the Mi . • In C[S1 , . . . , Sn , T1 , . . . , Ts ], consider the ideal J := hTi − gi ; 1 ≤ i ≤ si. Compute the elimination ideal J ′ := J ∩ C[T1 , . . . , Ts ] and the saturation I0 := J ′ : (T1 · · · Ts )∞ . • Define the matrix Q0 with the i-th column (χi1 , . . . , χik ) where χij is the eigenvalue of gi with respect to Mj . Output: I0 ⊆ C[T1 , . . . , Ts ] and Q0 . Then C[T1 , . . . , Ts ]/I0 is a presentation of R(Cn /G) in terms of generators and relations and Q0 is the degree matrix. Proof. This proof is a generalization of the one given in [11, Proposition 3.1]: By a result by Arzhantsev and Ga˘ıfullin [2], the Cox ring R(Cn /G) is isomorphic to the invariant ring C[S1 , . . . , Sn ][G,G] . Then the gi are X(G′ )-homogeneous prime generators for the invariant ring. The desired presentation C[T1 , . . . , Ts ]/I0 is obtained by considering C[T1 , . . . , Tr ] → C[S1 , . . . , Sn ][G,G] ,
Ti 7→ gi
and computing the ideal I0 of relations among the generators, e.g., by computing the listed elimination ideal. Example 2.4. We consider the faithful representation of the quaternion group G := Q8 with 8 elements whose image G ⊆ GL(2) is generated by 0 −i i 0 0 −i , −i 0 ,
where i ∈ C denotes the imaginary unit. There are no pseudo-reflections since no matrix in G has 1 as eigenvalue. We now apply the steps of Algorithm 2.3. Generators for the invariant ring C[S1 , S2 ][G,G] , where [G, G] = h−idi, are f2 = S12 ,
f 1 = S1 S2 ,
f3 = S22 .
Moreover, the group G′ = G/[G, G] is isomorphic to (Z/2Z)2 and acts on the linear hull V := linC (f1 , f2 , f3 ); we work with the description −1 0 0 1 0 0 0 0 1 0 −1 0 ⊆ GL(3). , G′ ∼ = 0
0
−1
0
1
0
The vector space V then has the G′ -invariant subspaces linC (f1 ) and linC (f2 , f3 ). We have a basis (g1 , g2 , g3 ) for V consisting of the Eigenvectors g1 := f1 ,
g2 := f2 + f3 ,
g3 := −f2 + f3 ∈ V
where the weights wi := (χi1 , χi2 ) ∈ (Z/2Z)2 ∼ = G′ are w1 = (0, 1), w2 = (1, 0) and w3 = (1, 1). The degree map is 0 1 1 3 2 Q0 : Z → (Z/2Z) , ei 7→ · ei 1
0
1
In the ring C[S1 , S2 , T1 , T2 , T3 ], we consider the ideal J := hT1 − g1 , . . . , T3 − g3 i and compute its elimination ideal J ′ := J ∩ C[T1 , T2 , T3 ] as well as the saturation I0 := J ′ : (T1 T2 T3 )∞ . We arrive at the G′ -graded Cox ring with degree matrix Q: R(C2 /G) ∼ = C[T1 , T2 , T3 ]/I0 ,
I0 = h4T12 − T22 + T32 i.
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M. DONTEN-BURY AND S. KEICHER
We recall some theory for the resolution algorithm and fix the setting. Assume we know a presentation R(X0 ) = C[T1 , . . . , Ts ]/I0 and its degree matrix Q0 . Consider the affine variety X 0 := Spec(R(X0 )) ⊆ Cs . We fix a gale dual matrix P0 for Q0 , i.e., a matrix fitting into the dual exact sequences (1)
/ Zs
0
/L
0o
Cl(X0 ) o
Q0
Zs o
P0
/N
P0∗
ker(Q0 ) o
0
where P ∗ denotes the transpose of P ; see, e.g., [18] for its computation. Let us shortly recall the construction of the tropical variety Trop(X0 ). Construction 2.5. Let the setting be as above. Setting T := C∗ , we obtain the tropical varieties \ Trop X 0 ∩ Ts := Trop(f ) ⊆ Qs , f ∈I0
Trop(X0 ) := P Trop X 0 ∩ Ts
s
⊆ N
where Trop(f ) ⊆ Q is the support of the codimension-one skeleton of the normal fan over the Newton polytope of the polynomial f ∈ C[T1 , . . . , Ts ]. It is possible to establish a fan structure on Trop(X0 ), see, e.g., [4, 20]. Construction 2.6. See [1, Section III] and [10]. Let X be a normal, irreducible A2 maximal variety with Γ(X, O∗ ) = C∗ such that both K := Cl(X) and R := R(X) are finitely generated. Fix a system of pairwise non-associated, K-prime generators F := (f1 , . . . , fm ) for R, the degree matrix Q and a gale dual matrix P . Then X is the good quotient p of an open subset b ⊆ X := Spec(R) ⊆ Cr X
b can be described by the action of the characteristic quasitorus Spec(C[K]), and X combinatorially by a set Φ of pairwise overlapping polyhedral cones in K ⊗Z Q. The triple (R, F, Φ) already determines X up to isomorphism. Moreover, a face γ0 Qr≥0 is a relevant F-face if Y p fi ∈ / hfi ; ei ∈ / γ0 i ei ∈γ0
and Q(γ0 ) ∈ Φ holds. Write cov(X) for the set of minimal relevant F-faces. Then one defines the canonical ambient toric variety as the toric variety Z ⊇ X whose fan Σ has the maximal cones Σmax = {P (γ0∗ ); γ0 ∈ cov(X)} where γ0∗ (Qr≥0 )∨ is the dual face. Further note that we have a pairwise disjoint decomposition [ X(γ0 ), X(γ0 ) := p {z ∈ X; zi 6= 0 ⇔ ei ∈ γ0 } , X = γ0 ∈cov(X)
By the decomposition from Construction 2.6, we can test smoothness of X by testing each X(γ0 ) with the following straight-forward test; it was also used in [11]. Write T := C∗ . Algorithm 2.7 (Smoothness test). Input: the Cox ring R = C[T1 , . . . , Tr ]/I with degree matrix Q, of a normal, irreducible A2 -maximal variety X, a relevant F-face γ0 Qr≥0 . Let P be a gale dual (n × r) matrix for Q. • If σ := P (γ0∗ ) is not smooth, return false.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
5
• Find A ∈ GL(n, Z) such that Avij = ej holds for all rays Q≥0 · vij of σ, vij primitive. • Let Iγ0 ⊆ C[T1 , . . . , Tr ] be the ideal obtained from I by setting Ti = 0 ⇔ ei ∈ / γ0 . • With the toric morphisms α : Tn → Tn and p : Tr → Tn corresponding to A and P , compute the pushforward and pullback Jγ0 := α∗ p∗ Iγ0 ⊆ C[Y1 , . . . , Yn ]. • Let jγ0 ⊆ C[Y1 , . . . , Yn ] be the jacobian ideal, i.e., the ideal generated by all (n − dim(Jγ0 ) × (n − dim(Jγ0 )) minors of the Jacobian of Jγ0 . • Return true if 1 ∈ (Jγ0 + jγ0 ) : (Ym+1 · · · Yn )∞ holds in C[Y1 , . . . , Yn ]. Return false otherwise. Output: true if X(γ0 ) is smooth, false otherwise. Proof. The first steps produce equations for X(γ0 ) in O(Cm × Tn−m ) ∼ = C[σ ∨ ∩ n Z ]. The second to last step computes the ideal of the singular locus V (Jγ0 )sing of V (Jγ0 ) ⊆ Cn . The last step checks whether V (Jγ0 )sing ∩ Cm × Tn−m is empty. The following algorithm computes a candidate for the Cox ring R(X) of a resolution X → X0 and verifies it; it is close to [18, Algorithm 2.4.8] which in turn is based on [1, Theorem III.4.4.9] where the complete, complexity-one-case is treated. See also [14, Chapter 3] and consider [13, Section 3] for the case of affine C∗ -surfaces. The idea is to embed X0 ⊆ Z0 into its ambient affine toric variety Z0 = Cn /G′ , perform a toric resolution Z → Z0 and consider the proper transform X → X0 . The step Z → Z0 involves the tropical variety. Algorithm 2.8 (Cox ring of a resolution X → X0 = Cn /G). Input: a small group G ⊆ GL(n). • Compute the Cox ring R(X0 ) = C[T1 , . . . , Ts ]/I0 of X0 with Algorithm 2.3. Determine Q0 and P0 as in (1). • Compute σ := P (Qn≥0 ) ⊆ Qn . It determines the ambient affine toric variety Z0 = Cn /G′ where G′ := G/[G, G]. • Compute a fan Υ with support Trop(X0 ). • Form the fan Σ := {σ} ∩ Υ by cone-wise intersection. • Resolve the toric singularities by stellar subdivisions of Σ at primitive vectors v1 , . . . , vm . • Write π : Z → Z0 for the toric resolution. Compute a gale dual Q out of the enlarged matrix P := [P0 , v1 , . . . , vm ] as in (1). • Compute the Cox ring of the proper transform X → X0 under π using [12, Algorithm 3.6] with the ’verify’-option: – Compute the ideal I ′ := p∗ (p0 )∗ I0 ⊆ C[T1 , . . . , Tr ], i.e., the pullback of the pushforward under the toric maps p, p0 corresponding to P , P0 . – Compute the saturation I := I ′ : (T1 · · · Tr )∞ . – Verify that all Ti define prime elements in R := C[T1 , . . . , Tr ]/I. – Verify that dim(I) − dim(hTi , Tj i + I) ≥ 2 for all i 6= j. – Verify that R is normal. • Verify smoothness of the variety X determined by (R, Σ): – For each τ ∈ Σmax , do: ∗ determine γ0 Qr≥0 with P (γ0∗ ) = τ . ∗ Verify that X(γ0 ) is smooth with Algorithm 2.7. Output: R and Q. If the verifications of the last two steps were successful, then R is the Cox ring of X of a resolution X → X0 and Q is the degree matrix of R. Lemma 2.9. In Algorithm 2.8, suppose R is the Cox ring of X. With the fan Σ, the minimal relevant (T 1 , . . . , T r )-faces are cov(X) = {γ0∗ Qr≥0 ; P (γ0∗ ) ∈ Σmax }.
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M. DONTEN-BURY AND S. KEICHER
Proof. By a result of Tevelev [22, Lemma 2.2] and the definition of the tropical variety Trop(X), we obtain γ0 Qr≥0 relevant (T 1 , . . . , T r )-face
P (γ0∗ )◦ ∩ Trop(X) 6= ∅ and P (γ0∗ ) ∈ Σ.
⇔
As |Σ| = Trop(X), the first condition on the right-hand-side can be dropped. Considering only minimal relevant (T 1 , . . . , T r )-faces means to replace the last condition by requiring P (γ0∗ ) ∈ Σmax . Proof of Algorithm 2.8. By construction and [12, Algorithm 3.6], X → X0 is a modification and R the Cox ring of X; note that [12, Algorithm 3.6] directly translates to the affine setting. By the smoothness verification step and Lemma 2.9, X is smooth. Example 2.10. Consider the Cox ring R(X0 ) = C[T1 , T2 , T3 ]/I0 computed in Example 2.4 with its degree-map Q : Z3 → (Z/2Z)2 . We apply the steps of Algorithm 2.8 to X0 . The canonical ambient toric variety is the affine toric variety Z0 = Z0 (σ) with σ ⊆ Q3 being the polyhedral cone spanned by (1, 0, 0), (1, 2, 0), (1, 0, 2). The following picture shows the steps for the toric resolution. (1, 2, 0)
(1, 2, 0)
v3 v1 v4 (0, 0, 0)
(0, 0, 0)
(1, 0, 0) (1, 0, 2)
v2 (1, 0, 0) (1, 0, 2)
Σ and new rays Q≥0 · vi (red)
σ (gray) and Σ (blue)
The two-dimensional fan Σ = {σ}∩Trop(X0 ) has already the additional ray Q≥0 ·v1 where v1 := (3, 2, 2). The fan can be resolved by insertion of the further rays through v2 := (2, 1, 2),
v3 := (2, 2, 1),
v4 := (2, 1, 1).
This yields a toric resolution Z → Z0 which then induces a resolution X → X0 as the proper transform of Z → Z0 ; its Z4 -graded Cox ring R(X) and degree matrix are R(X) = C[T1 , . . . , T7 ]/h4T12 T7 − T6 T22 + T32 T5 i, # " −1 0 0 0
−1 −1 0 −1
−1 0 −1 −1
1 1 1 0
0 −1 1 1
0 1 −1 1
0 −1 −1 −1
.
One verifies that X is smooth. As X is a surface, we can verify minimality of the b Ti ): they are resolution by inspecting the self intersection numbers of the V (X; −1, −1, −1, −2, −2, −2, −2. We will use Algorithms 2.3 and 2.8 in Section 3 to compute resolutions of quotient singularities.
Remark 2.11 (Implementation). We have started an implementation of Algorithms 2.3 and 2.8 in the library quotsingcox.lib for the Open Source computer algebra system Singular [5]. It will be made available at [7]. 3. Resolutions of 3-dimensional quotient singularities 3.1. Cox rings. In this section, we compute Cox rings of resolutions of quotient singularities C3 /G where G is a group of order at most 12 and discuss the results. As explained in Remark 2.2, we will consider only faithful representations of G without pseudo-reflections.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
7
Notation 3.1. In the following, we denote by S3 the permutation group of three elements, by D2n the dihedral group of 2n elements, by Q8 the Quaternion group and by A4 the alternating group of four elements. Moreover, we write BD3 for the binary dihedral group, i.e., the abstract group with 12 elements
BD3 = a, x; a2n = 1, x2 = an , x−1 ax = a−1 . Proposition 3.2. Up to conjugacy in GL(3), the 3-dimensional faithful small representations of non-abelian groups up to order 12 are as follows. No.
G
gen.s in GL(3)
1
S3
0 1 0
−1 −1 0
0 0 1
,
−1 0 0
2
D8
0 1 0
−1 0 0
0 0 1
,
Q8
i 0 0
0 −i 0
0 0 1
,
Q8
i 0 0
0 −i 0
0 0 −1
D10
ζ5 0 0
D12
−ζ3 0 0
A4
0 1 0
BD3
2 −ζ3 0 0
BD3
2 −ζ3 0 0
BD3
2 −ζ3 0 0
3
4
5
6
7
8
9
10
0
0 0 1
4 ζ5
0
0
0 0 1
−ζ3 0
0 0 1
1 0 0
0
−ζ3 0
0 −ζ3 0
0 −ζ3 0
0 0 1
0 0 −1
0 0 1
.
1 0 0
0 −1 0
0 0 −1
.
0 1 0
−1 0 0
0 0 1
0 1 0
0 1 0
,
,
0 0 −1
,
,
1 1 0
−1 0 0
,
,
0 1 0
,
1 0 0
0 −i 0
−i 0 0
0 −i 0
.
.
0 0 −1
0 0 1
0 −i 0
0 0 −1
0 −1 0
.
0 0 1
−1 0 0
1 0 0
−i 0 0
−i 0 0
.
.
0 0 1
.
0 0 i
.
0 0 −1
.
In the table, i ∈ C denotes the imaginary unit, and ζk ∈ C is a primitive k-th root of unity. All listed representations are reducible except for the A4 -case. Proof. To classify these representations we use the library of groups of small order, which is a part of GAP [9]. We do the following steps: (i) view the character tables to get dimensions of irreducible representations,
8
M. DONTEN-BURY AND S. KEICHER
(ii) find all irreducible representations: for a group of small order and a given character one can either locate a representation in the literature or construct easily, (iii) combine irreducible representations to get all faithful 3-dimensional representations of a given group (in particular, direct sums of 1-dimensional representations are not allowed since they are not faithful representations – the image is an abelian group), (iv) eliminate representations with pseudo-reflections (see Remark 2.2), (v) check whether obtained representations have different groups as images (if representations differ just by a permutation of conjugacy classes in G, they give the same quotient); for any two representations of the same group on the list above the set of matrix traces are different, hence this condition is satisfied. Proposition 3.3. In the setting of Proposition 3.2, the Cox rings of X0 := C3 /G are as listed the following table. No.
G
Cl(X0 )
1
S3
Z/2Z
2
D8
Z/2Z × Z/2Z
3
Q8
4
degree matrix
R(X0 )
C[T1 , . . . , T4 ]/I with I gen. by 4T33 − T22 − 27T42
1
1
0
0
h
1 1
1 0
0 1
0 0
i
C[T1 , . . . , T4 ]/I with I gen. by 4T12 + T22 − T42
Z/2Z × Z/2Z
h
1 1
1 0
0 1
0 0
i
C[T1 , . . . , T4 ]/I with I gen. by T12 − T22 + 4T32
Q8
Z/2Z × Z/2Z
h
1 1
1 0
1 0
0 1
i
C[T1 , . . . , T4 ]/I with I gen. by T12 − T32 + 4T42
5
D10
Z/2Z
1
1
0
0
6
D12
Z/2Z × Z/2Z
1 1
1 0
0 1
0 0
7
A4
Z/3Z
0
0
2
1
8
BD3
Z/4Z
3
1
2
0
9
BD3
Z/4Z
3
1
1
2
10
BD3
Z/4Z
3
1
2
2
h
i
C[T1 , . . . , T4 ]/I with I gen. by 4T35 + T22 − T42
C[T1 , . . . , T4 ]/I with I gen. by 4T43 + T12 − T22
C[T1 , . . . , T4 ]/I with I gen. by T13 + T33 − 3T1 T3 T4 + T43 − 27T22 C[T1 , . . . , T4 ]/I with I gen. by 4T33 + T12 − T22
C[T1 , . . . , T4 ]/I with I gen. by 4T43 + T12 − T32
C[T1 , . . . , T4 ]/I with I gen. by 4T43 + T12 − T22
Proof. The listed Cox rings are obtained by applying Algorithm 2.3 to the list of representations from Proposition 3.2 using our implementation 2.11.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
9
Remark 3.4. Note that several rings on the list (without grading considered) are isomorphic. This is because the ring structure of the Cox ring of Cn /G is just the invariant ring C[x1 , . . . , xn ][G,G] and derived subgroups in cases 2, 3, 4 (D8 and Q8 representations) and also 6, 8, 9, 10 (D12 and BD3 representations) are conjugate subgroups of GL(3). Theorem 3.5. In the setting of Proposition 3.3, a resolution X → X0 of the quotient X0 = C3 /G has the following Cox ring, respectively. No.
1
2
3
4
5
6
7
8
9
G
Cl(X)
S3
Z4
D8
Z4
Q8
Z4
Q8
Z4
D10
D12
A4
BD3
BD3
degree matrix
" "
Z5
Z5
Z8
0 0 0 1
"
#
−1 −1 −1 0 1 0 0 −2 −1 −1 0 0 2 0 1 0 −1 0 0 −2 2 2 0 0 0 0 −3 1
−1 0 −1 −1 1 0 −2 −1 −1 −1 0 2 1 0 0 −1 0 −2 2 1 0 0 0 −3
0 0 0 1
0 0 2 1
0 0 0 1
0 −5 −2 −5 1 0 0 0 −1 −1 0 0 0 2 0 0 1 0 −1 −1 0 −2 1 0 2 0 −1 −2 0 −4 0 1 2 0 −1 −3 0 −5 0 0
−3 −3 0 −2 2 0 0 −1 0 −1 0 0 2 0 1 1 0 0 −1 0 1 2 1 0 1 −1 −1 0 2 2 0 1 −2 0 0
0 0 −1 −1 −1 −2 −2
−3
#
0 0 0 0 1
0 0 0 0 0 0 1 0
0 2 00 0 0 −3 2 0 0 0 2 −2 1 0 0 5 −5 0 1 0 4 −4 0 0
0 0 0 0 1
C[T1 , . . . , T9 ]/I with I gen. by 4T35 T73 T8 + T22 T6 − T42 T9
0 000 0 0 −2 −1 0 3 0 0 0 0 0 1 0 0 −2 1 0 0 0 −1 0 0 0 −2 0 1 0 0 −1 2 0 0 −4 0 0 1 0 −2 1 0 0 −3 0 0 0 1 −2 1 0 0 −4 0 0 0 0 −2 2 0 0 −5 0 0 0 0
−2 2 −2 −3 −3
C[T1 , . . . , T8 ]/I with I gen. by −T12 T6 − 4T42 T7 + T32 T8
0 0 0 0 1
−3 3 −2 −5 −4
C[T1 , . . . , T8 ]/I with I gen. by −4T12 T6 + T42 T7 − T22 T8
C[T1 , . . . , T8 ]/I with I gen. by T12 T6 + 4T32 T7 − T22 T8
0 0 0 1 0
−3 2 −1 −2 −2
C[T1 , . . . , T8 ]/I with I gen. by 4T33 T7 − T22 T6 − 27T42 T8
#
0 0 0 1
−2 −3 −2 −2 1
Z5
Z8
1 00 0 20 0 −2 1 0 −3 0
−1 −1 0 −1 1 0 0 −1 0 −1 0 0 2 0 −1 −1 0 −2 0 0 2 1 0 0 1 0 −1 −1
"
0 −3 −2 −3 −1 −1 0 0 1 0 −1 −1 1 0 −1 −2
R(X)
#
C[T1 , . . . , T9 ]/I with I gen. by 4T43 T72 T9 + T12 T6 − T22 T8
0 0 0 0 0 0 0 1
C[T1 , . . . , T9 ]/I with I gen. by 4T33 T72 T9 + T12 T6 − T22 T8
0 −3 −2 1 0 0 0 0 0 0 −1 −1 −1 −2 0 2 0 0 0 0 0 −1 0 0 2 0 −3 2 0 0 0 0 3 1 0 −3 0 5 −5 1 0 0 0 2 1 0 −3 0 4 −4 0 1 0 0 2 0 0 −3 0 4 −4 0 0 1 0 3 1 0 −4 0 6 −6 0 0 0 1 4 2 0 −6 0 9 −9 0 0 0 0
C[T1 , . . . , T12 ]/I with I gen. by T33 T62 T7 T82 T11 + T43 T6 T72 T92 T12 2 T2 −3T1 T3 T4 T6 T7 T8 T9 T11 T12 + T13 T8 T9 T11 12 −27T22 T10
0 0 0 0 0 0 0 1
C[T1 , . . . , T12 ]/I with I gen. by 4T43 T62 T9 T10 + T12 T7 T12 − T32 T8
10
M. DONTEN-BURY AND S. KEICHER
10 BD3
Z5
−3 −3 0 −2 2 0 0 0 1 1 0 0 −1 1 0 0 2 3 −1 2 −3 0 2 0 −2 −5 2 −3 5 0 −5 1 −2 −4 2 −3 4 0 −4 0
0 0 0 0 1
C[T1 , . . . , T9 ]/I with I gen. by 4T43 T62 T9 + T12 T7 − T22 T8
Proof. The Cox rings of the X0 = C3 /G have been presented in Proposition 3.3. We then obtain the Cox rings of a resolution X → X0 using our implementation 2.11 of Algorithm 2.8. The verifications of the primality of the variables Ti are done directly computationally, the smoothness tests are done with Algorithm 2.7 as implemented in MDSpackage [11]. The normality follows from Serre’s criterion, see, e.g., [19, 6.2]. Remark 3.6. Using these methods and its implementation, one can directly go to higher group orders |G|. Since the number of different representations to consider grows quickly for isomorphism types of G with higher order, we end the table at order 12. 3.2. Properties of quotients and their resolutions. We discuss certain geometric properties of the quotient singularities and their resolutions corresponding to the Cox rings from Theorem 3.5. At first, we turn to torus actions on quotient spaces: we consider the relation between the form of the Cox ring and the existence of a torus action on Cn /G. Proposition 3.7. Let V be an affine space which is a direct sum of n representations of a given group G. Then V admits an action of (C∗ )n which commutes with the action of G. Proof. Let (xi,1 . . . , xi,ni ) be the coordinates of the i-th representation (of dimension di ). The following action commutes with the action of G. (C∗ )n × V → V,
=
(t1 , . . . , tn ) · (x1,1 . . . , x1,d1 , . . . , xn,1 . . . , xn,dn ) (t1 x1,1 . . . , t1 x1,d1 , . . . , tn xn,1 . . . , tn xn,dn )
Corollary 3.8. A quotient of C3 by a direct sum of a 2-dimensional and a 1dimensional representation is a T -variety of complexity one. Remark 3.9. All representations in the table in Proposition 3.2 except the case of A4 are direct sums of two irreducible representations, as it can be easily seen from the generating matrices. In particular, all varieties X0 in Proposition 3.3 except possibly for the A4 case are T -varieties of complexity one. Thus, the only 3-dimensional quotient singularities which are possibly not T varieties of complexity one correspond to irreducible representations. The representation of A4 in Proposition 3.2 is irreducible and we expect that there is no (C∗ )2 action. Note that, in spite of the (C∗ )2 -action from Remark 3.9 for the other cases, we still did not know a priori that Algorithm 2.8 would compute the Cox ring of a resolution: to our knowledge, only for complete T -varieties [1, Theorem III.4.4.9] and for affine C∗ -surfaces [13, Section 3] this is known. However, it seems natural that, as in the cases of complete T -varieties and C∗ -surfaces, the single relations of the Cox rings of the singularity and its resolution are in all cases except in the A4 -case of trinomial shape. Remark 3.10. By the previous discussing, it would be interesting to continue the list of results in Theorem 3.5 with irreducible, 3-dimensional representations. However, the three smallest cases of such representations (one of order 21 and two of order 27) are at the moment computationally out of reach on our machines.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
11
We now turn to properties of the resolutions found in Theorem 3.5. Recall that a resolution of singularities π : X → X0 is called crepant if KX = π ∗ KX0 . For surface quotient singularities for G ⊆ SL(2), i.e., du Val singularities, crepant resolutions are the minimal ones: the special fiber is a tree of smooth rational curves dual to a Dynkin diagram An , Dn , E6 , E7 or E8 . The relation between the structure of G (its conjugacy classes or irreducible representations) and the shape of the diagram of the resolution of C/G was noticed by McKay. The postulated relation between the geometry of crepant resolutions of (Gorenstein) quotient singularities and the structure of the group is called the McKay correspondence. It has been studied in several special cases and in different formulations. In particular, it is proved in dimension 3, see e.g. [15], for symplectic singularities in dimension 4, see [16], and a weak version for any G ⊂ SL(n) (the equality of the dimension of cohomology space and the number of conjugacy classes of G) is due to Batyrev [3]. It is a natural question to ask how good the resolutions obtained using Algorithm 2.8 are and what properties to expect them to have. In particular, we would like to know whether they are crepant for G ⊂ SL(n). To test this property for the 3-dimensional results in Theorem 3.5 we can use the McKay correspondence. Remark 3.11. Among groups listed in Proposition 3.2 the cases 1, 2, 3, 5, 6, 7 and 8 are in SL(3). Proposition 3.12. The Cox rings of resolutions of singularities in cases 1 (S3 ), 5 (D10 ) and 7 (A4 ), given in Theorem 3.5, are not the Cox rings of crepant resolutions. The resolutions in cases 2 (D8 ), 3 (Q8 ), 6 (D12 ) and 8 (BD3 ) are crepant. Proof. By [8, Lemma 2.11], the class group Cl(X) of a resolution π : X → Cn /G of a quotient singularity is a free group and its rank m is equal to the number of irreducible components of the exceptional divisor. By the McKay correspondence in dimension n = 3 we know that m is the number of conjugacy classes of junior elements in G where an element g ∈ G is junior if n 1X ak age(g) = 1 where age(g) := r k=1
2πiak r
with ak coming from the exponents of eigenvalues e of g, see e.g. [21]. Thus the conjugacy classes of junior elements can be determined with simple computations (e.g., using GAP, [9]), and the results are as follows: case number junior classes
1 2
2 4
3 4
5 3
6 5
7 3
8 5
Comparing them with the rank of Cl(X) given in Theorem 3.5 we obtain that the resolutions in cases 1, 5 and 7 have too many components of the exceptional divisor to be crepant. We sketch the argument that in the remaining cases obtained resolutions are crepant. The exceptional divisors which have to be present in the crepant resolution come from the minimal resolutions of transversal du Val singularities along lines in C3 fixed by an element g of age 1, possibly divided by the action of the normalizer of g in G. It follows from the McKay correspondence that the number of such divisors is the same as the number of conjugacy classes of junior elements in G. Hence the resolutions with the rank of the Picard group equal to the number of junior classes are crepant. Note that in dimension 3 there is one Cox ring corresponding to all crepant resolutions, because they are all birationally equivalent and flops preserves smoothness in dimension 3.
12
M. DONTEN-BURY AND S. KEICHER
Remark 3.13. We can describe the geometry of all crepant resolutions for groups 2, 3, 6 and 8 of Proposition 3.2 by computing GIT quotients of the spectrum of their Cox rings. To determine the GIT fan describing the variation of the quotients, we use [17]. case number max. GIT-cones within Mov(X)
2 9
3 1
6 16
8 1
Actually, the cases 3 and 11 are products of a finite subgroup of SL(2) and a trivial group, so the resolutions will be just products of minimal resolutions of du Val singularities by C. Hence, there is just one crepant resolution, i.e., just one chamber in the GIT fan restricted to the cone of movable divisor classes Mov(X). 4. Two 4-dimensional examples In this section, we present two 4-dimensional examples. In dimension 4 much less is known about the resolutions of quotient singularities. In particular, crepant resolutions do not always exist, and the McKay correspondence has been proved just for the symplectic case (see [16]). Hence it is a very appropriate setting for computational experiments with constructing resolutions via Cox rings. Moreover, it is interesting also from the point of view of the Cox ring theory: while in dimension 3 all the examples we treat can be defined with a single relation, almost always trinomial, here the ring structure will be more complicated. The first of the examples, the representation of D8 , has also been treated in [6] where the Cox ring of symplectic resolutions was constructed. Proposition 4.1. Consider the 4-dimensional quotient singularities C4 /G for the two small groups G ⊆ GL(4) G
gen.s in GL(4)
D8
"
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
Q8
"
i 0 0 0
0 −i 0 0
0 0 i 0
0 0 0 −i
# #
,
"
,
"
1 0 0 0
0 −1 0 0
0 1 0 0
−1 0 0 0
0 0 1 0
0 0 0 1
0 0 0 −1
0 0 −1 0
# #
.
.
where i ∈ C is the imaginary unit. The Cox ring of the quotient space X0 := Cn /G, the degree matrix and the class group of X0 are as follows. G D8
Cl(X0 ), degree matrix and R(X0 ) Z/2Z × Z/2Z
h
1 1
1 1
1 1
C[T1 , . . . , T10 ]/I with I gen. by T5 T8 + T4 T9 − 2T6 T10 2 , T72 − T8 T9 + T10 T5 T7 + T2 T9 − 2T3 T10 , 2T3 T7 − T6 T9 + T5 T10 , 2T1 T7 + T6 T8 − T4 T10 , T2 T5 − 2T3 T6 + T7 T9 , T3 T4 − T1 T5 − T7 T10 , 4T32 + T52 − T92 , 2 , 4T1 T3 + T62 − T10 2T1 T2 + T4 T6 − T8 T10 ,
1 0
1 0
1 0
0 1
0 0
0 0
0 0
i
T3 T8 + T1 T9 − T2 T10 , T6 T7 + 2T1 T9 − T2 T10 , T4 T7 − T2 T8 + 2T1 T10 , T2 T7 − T4 T9 + T6 T10 , 2 , T4 T5 − T62 + T8 T9 − T10 2T1 T5 − T2 T6 + T7 T10 , T2 T4 − 2T1 T6 − T7 T8 , 2T2 T3 + T5 T6 − T9 T10 , T22 + T62 − T8 T9 , 4T12 + T42 − T82
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
Q8
Z/2Z × Z/2Z
h
1 1
1 1
1 1
1 0
1 0
1 0
0 1
0 1
0 1
0 0
C[T1 , . . . , T10 ]/I with I gen. by 2 , T82 − 4T7 T9 − T10 T5 T8 − 2T6 T9 + T2 T10 , T2 T8 − 2T3 T9 + T5 T10 , T5 T7 − T4 T9 + T3 T10 , T2 T7 − T1 T9 + T6 T10 , T3 T5 − T2 T6 − 2T9 T10 , T3 T4 − T1 T6 + 2T7 T10 , T32 − T62 + 4T7 T9 , T1 T3 − T4 T6 + 2T7 T8 , 2 , T1 T2 − T4 T5 + 4T7 T9 + 2T10
13
i
T6 T8 − 2T4 T9 + T3 T10 , T3 T8 − 2T1 T9 + T6 T10 , 2T6 T7 − T4 T8 + T1 T10 , 2T3 T7 − T1 T8 + T4 T10 , 2 , T4 T5 − T62 − T10 T1 T5 − T3 T6 − T8 T10 , T2 T4 − T3 T6 + T8 T10 , T2 T3 − T5 T6 + 2T8 T9 , T22 − T52 + 4T92 , T12 − T42 + 4T72
Proof. This is again an application of Algorithm 2.3.
We now intend to compute the Cox ring of a resolution for the both cases X0 = C4 /D8 and X0 = C4 /Q8 presented in the previous proposition with Algorithm 2.8. As the Cox rings turn out to big for all smoothness tests to be computationally feasible, we therefore present Cox rings of modifications X → X0 (smoothness is not fully verified) in the following two examples. However, in Proposition 4.1 we present a resolution for the D8 -case thereby retrieving a result of Grab and the first author [6]. Example 4.2 (Case D8 ). In the case of G = D8 in Proposition 4.1, applying Algorithm 2.8 to X0 = C4 /G, we obtain a modification X → X0 with Cl(X) = Z16 and the Cox ring R(X) = C[T1 , . . . , T26 ]/I with generators for I and the degree matrix given by T6 T7 T17 − T1 T8 + (4i)T3 T9 , T2 T7 T11 − T4 T8 + 2T5 T9 , 2 2 2T9 T10 T18 T20 + T1 T2 T11 − T4 T6 T17 , T8 T10 T18 T20 + (2i)T2 T3 T11 − T5 T6 T17 , 2 2 T7 T10 T18 T20 + (2i)T3 T4 − T1 T5 , T2 T6 T16 T25 T26 + (−2i)T3 T4 − T1 T5 , 2 2 2 T6 T10 T14 T22 T23 − T4 T8 − 2T5 T9 , 2T3 T10 T14 T22 T23 + (i)T2 T8 T16 T25 T26 + (−i)T5 T7 T17 , 2 2 2 2 T25 T26 − T4 T7 T17 , T5 T10 T15 T19 T21 − T6 T8 T16 T25 T26 + (2i)T3 T7 T11 , T1 T10 T14 T22 T23 − 2T2 T9 T16 2 2 2 T4 T10 T15 T19 T21 − 2T6 T9 T16 T25 T26 − T1 T7 T11 , T2 T10 T15 T19 T21 − T1 T8 + (−4i)T3 T9 , 2 2 2 T2 T5 T15 T19 T21 + (−2i)T3 T6 T14 T22 T23 − T7 T8 T18 T20 , 2 2 2 T2 T4 T15 T19 T21 − T1 T6 T14 T22 T23 + 2T7 T9 T18 T20 , 2 2 2 2 2 T2 T11 T15 T19 T21 − T6 T14 T17 T22 T23 + 4T8 T9 T18 T20 , 2 2 2 T25 T26 T25 T26 − 2T1 T5 T8 T9 T16 T1 T4 T82 T16 T25 T26 + (−4i)T3 T4 T8 T9 T16 2 T25 T26 + (−2i)T3 T4 T72 T11 T17 − T1 T5 T72 T11 T17 , +(8i)T3 T5 T92 T16 2 2 2 2 2 T4 T7 T8 T17 T18 T20 + 2T5 T7 T9 T17 T18 T20 + (2i)T1 T3 T4 T8 T14 T22 T23 2 2 2 T22 T23 + (4i)T1 T3 T5 T9 T14 T22 T23 , T22 T23 + 8T32 T4 T9 T14 −T12 T5 T8 T14 2 2 2 2 4T92 T16 T18 T20 T25 T26 − T42 T15 T17 T19 T21 + T12 T11 T14 T22 T23 , 2 2 2 2 2 2 2T8 T9 T16 T18 T20 T25 T26 + T7 T11 T17 T18 T20 − T4 T5 T15 T17 T19 T21 + (2i)T1 T3 T11 T14 T22 T23 , 2 2 2 2 T82 T16 T18 T20 T25 T26 − T52 T15 T17 T19 T21 − 4T32 T11 T14 T22 T23 , 2 2 2 2 2 T10 T14 T18 T20 T22 T23 + T2 T11 T16 T25 T26 − 2T4 T5 T17 , 2 T2 T2 T T T 2 2 T10 15 18 19 20 21 − T6 T16 T17 T25 T26 + (4i)T1 T3 T11 , 2 2 2 2 2 T6 T14 T16 T22 T23 T25 T26 + T72 T11 T18 T20 − 2T4 T5 T15 T19 T21 , 2 2 2 2 T22 T23 , T20 + (−4i)T1 T3 T14 T16 T19 T21 T25 T26 − T72 T17 T18 T22 T15 2 2 2 2 2 T10 T14 T15 T19 T21 T22 T23 − 4T8 T9 T16 T25 T26 − T7 T11 T17 , 2 2 2 2 2 2 T1 T8 T14 T16 T22 T23 T25 T26 + (−8i)T1 T3 T8 T9 T14 T16 T22 T23 T25 T26 2 2 2 2 2 2 −16T32 T92 T14 T16 T22 T23 T25 T26 + T74 T11 T17 T18 T20 − 2T4 T5 T72 T15 T17 T19 T21 , 2 2 2 2 T42 T82 T15 T16 T19 T21 T25 T26 − 4T4 T5 T8 T9 T15 T16 T19 T21 T25 T26 2 T2 T T T T 4 2 2 2 2 2 +4T52 T92 T15 16 19 21 25 26 − T7 T11 T17 T18 T20 + (−4i)T1 T3 T7 T11 T14 T22 T23 , 4 2 2 2 2 2 T74 T11 T17 T18 T20 − 2T4 T5 T72 T15 T17 T18 T19 T20 T21 + (4i)T1 T3 T72 T11 T14 T18 T20 T22 T23 2 2 2 2 2 2 T19 T21 T22 T23 T15 T19 T21 T22 T23 + T12 T52 T14 T15 T15 T19 T21 T22 T23 + (−4i)T1 T3 T4 T5 T14 −4T32 T42 T14
14
M. DONTEN-BURY AND S. KEICHER
−1
1 −1 −1 0 0 1 1 0 0 −1 −1 −2 −1 −1 −2
−1 0 0 0 −1 −1 1 1 −1 0 −1 −1 −1 −1 −2 −2
−1 −1 1 −1 0 0 1 1 −1 0 0 −2 −1 −1 −2 −1
0 0 −2 0 −1 0 −1 0 0 0 −1 1 0 0 1 0
0 −2 0 0 −1 0 −1 0 −1 0 0 0 1 0 0 1
0 −1 0 0 0 2 0 0 0 0 0 0 0 0 0 −1 0 −2 0 −1 −2 1 0 0 0 0 0 0 0 0 −1 0 0 −2 −1 −2 0 1 0 0 0 0 0 0 0 −1 0 0 0 −1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 0 −1 0 −1 −1 0 0 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −2 0 0 0 0 0 2 0 0 0 0 0 −1 −1 −1 −2 0 0 0 0 0 0 1 0 0 0 0 −1 0 −1 0 0 0 0 0 0 −1 0 1 0 0 0 −1 −1 −1 −1 0 0 0 0 0 −1 0 0 1 0 0 0 −1 −1 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 −1 1 0 0 0 0 0 −2 0 0 0 0 0 0 −1 −1 1 0 0 0 0 0 −2 0 0 0 0 0 −1 −1 −1 0 0 0 0 0 0 −2 0 0 0 0 0 −2 −1 −1 1 0 0 0 0 0 −3 0 0 0 0 0 −1 −2 −1 1 0 0 0 0 0 −3 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
All Ti are prime and the codimension test succeeds. For the normality check, recall that by [1, Lemma IV.1.2.7], if R is an integral domain, and f ∈ R is prime, then R is normal if Rf is. We use this lemma repeatedly by localizing with respect to certain variables Ti until we reach a polynomial ring. We expect X → X0 to be a resolution but cannot show this at the moment as some smoothness tests were infeasible on our machines. Example 4.3 (Case Q8 ). In the case of G = Q8 in Proposition 4.1, applying Algorithm 2.8 to X0 = C4 /G, we obtain a modification X → X0 with Cl(X) = Z18 and the Cox ring R(X) = C[T1 , . . . , T28 ]/I with generators for I and the degree matrix given by 2 T 2 T6 T10 T17 21 + T2 T7 − T1 T9 , 2T8 T10 T18 T20 + T2 T4 − T1 T5 , 2 2 2 2 T3 T10 T11 T19 + T5 T7 − T4 T9 , T2 T3 T11 T19 + 2T8 T9 T18 T20 − T5 T6 T17 T21 , 2 2 2 2 2 T1 T3 T11 T19 + 2T7 T8 T18 T20 − T4 T6 T17 T21 , T3 T6 T16 T25 T28 + T8 T10 T18 T20 − T1 T5 , 2 2 2 2 2 T6 T8 T14 T24 T27 + T3 T10 T11 T19 − 2T4 T9 , T2 T8 T14 T24 T27 − 2T3 T9 T16 T25 T28 + T5 T10 T17 T21 , 2 2 2 2 2 2 T1 T8 T14 T24 T27 − 2T3 T7 T16 T25 T28 − T4 T10 T17 T21 , T5 T8 T15 T23 T26 − 2T6 T9 T16 T25 T28 + T2 T10 T11 T19 , 2 2 2 2 2 T4 T8 T15 T23 T26 − 2T6 T7 T16 T25 T28 − T1 T10 T11 T19 , T3 T8 T15 T23 T26 + T6 T10 T17 T21 − 2T1 T9 , 2 2 2 2 2 2 T20 , T24 T27 + 2T7 T10 T18 T23 T26 − T1 T6 T14 T20 , T3 T4 T15 T24 T27 − 2T9 T10 T18 T23 T26 − T2 T6 T14 T3 T5 T15 2 2 2 2 2 T82 T14 T18 T20 T24 T27 + T32 T11 T16 T19 T25 T28 − T4 T5 T17 T21 , 2 2 2 2 2 2 T52 T15 T17 T21 T23 T26 − T22 T11 T14 T19 T24 T27 − 4T92 T16 T18 T20 T25 T28 , 2 2 2 2 2 2 T20 T25 T28 , T18 T19 T24 T27 − 4T72 T16 T14 T21 T23 T26 − T12 T11 T17 T42 T15 2 2 2 2 2 2 2 T8 T15 T18 T20 T23 T26 − T6 T16 T17 T21 T25 T28 + T1 T2 T11 T19 , 2 2 2 2 2 T20 , T21 T24 T27 + 4T7 T9 T18 T17 T19 T23 T26 − T62 T14 T15 T32 T11 2 2 2 2 2 2 T62 T14 T16 T24 T25 T27 T28 + T10 T11 T18 T19 T20 − T4 T5 T15 T23 T26 , 2 2 2 2 2 2 T32 T15 T16 T23 T25 T26 T28 − T10 T17 T18 T20 T21 − T1 T2 T14 T24 T27 , 2 2 2 2 2 2 T82 T14 T15 T23 T24 T26 T27 − T10 T11 T17 T19 T21 − 4T7 T9 T16 T25 T28 , 2 2 2 2 2 2 T2 T4 T10 T11 T17 T19 T21 + T1 T5 T10 T11 T17 T19 T21 2 2 T25 T28 T25 T28 + 2T2 T4 T7 T9 T16 −2T2 T5 T72 T16 2 2 +2T1 T5 T7 T9 T16 T25 T28 − 2T1 T4 T92 T16 T25 T28 , 2 2 2 2 2 2 T21 T23 T26 T17 T19 T20 T21 − T4 T5 T15 T18 T17 T11 2T10 2 T2 T T T 2 2 +T1 T2 T11 14 19 24 27 + 4T7 T9 T16 T18 T20 T25 T28 , 2 2 4 2 2 2 2 2 T4 T5 T10 T15 T17 T21 T23 T26 + T1 T2 T10 T11 T14 T17 T19 T21 T24 T27 2 2 2 2 2 2 T16 T24 T25 T27 T28 T18 T20 T21 T25 T28 − 2T22 T72 T14 T17 +4T7 T9 T10 T16 2 2 2 2 2 2 +4T1 T2 T7 T9 T14 T16 T24 T25 T27 T28 − 2T1 T9 T14 T16 T24 T25 T27 T28 , 2 2 2 2 2 2 2 2 2T4 T5 T10 T15 T17 T18 T20 T21 T23 T26 − 2T1 T2 T10 T11 T14 T18 T19 T20 T24 T27 2 2 4 2 2 2 +8T7 T9 T10 T16 T18 T20 T25 T28 − T22 T42 T14 T15 T23 T24 T26 T27 2 2 2 2 2 2 +2T1 T2 T4 T5 T14 T15 T23 T24 T26 T27 − T1 T5 T14 T15 T23 T24 T26 T27 , 2 T2 T2 T2 T T T T 2 4 2 2 T4 T5 T10 11 15 17 19 21 23 26 + T1 T2 T10 T11 T14 T19 T24 T27 2 2 2 2 2 2 T23 T25 T26 T28 T16 T19 T20 T25 T28 − 2T52 T72 T15 T11 T16 T18 −4T7 T9 T10 2 2 2 2 T16 T23 T25 T26 T28 +4T4 T5 T7 T9 T15 T16 T23 T25 T26 T28 − 2T42 T92 T15
−1
−1 −1 0 0 0 0 0 0 −2 −1 0 −2 −1 0 −1 −2 0 −1 −2 0 −1 −2 −1 −1 −1 0 0 0 −1 0 −1 0 0 −1 −1 −1 0 0 −1 0 0 −1 0 0 −1 −1 0 0 0 0 −1 −1 −1 0 0 0 0 0 0 0 0 −1 −1 −2 −2 −2 −1 −1 −1 −1 −1 1 1 1 0 0 0 −1 −1 2 2 2 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 −1 0 1 0 2 1 2 1 0 0 1 0 2 1 1 1 0 0 1 1 1 2 1 −1 0 0 −1 −1 2 3 3 0 1 0 −1 −1 2 3 2 0 1 0 −1 0
0 −2 0 0 0 −1 0 −1 −1 −1 0 0 0 0 0 0 0 0
−1 −1 −1 0 0 0 −1 −1 −2 0 1 1 1 2 2 1 2 2
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 −2 2 0 0 0 0 0 −3 1 1 0 0 0 0 −2 0 0 1 0 0 0 −1 −1 0 0 1 0 0 −3 0 0 0 0 1 0 −2 −1 0 0 0 0 1 −3 1 0 0 0 0 0 −5 2 0 0 0 0 0 −4 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
.
As in Example 4.2 all Ti are prime, the codimension test is successful and we obtain normality of R by repeated by localizations. We expect X → X0 to be a resolution but not all smoothness tests were feasible on our machines.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
15
Remark 4.4. The groups considered in the two previous examples are symplectic. The first one, D8 , is conjugate to the group considered in [6, Section 5]. In particular, in this case the symplectic, i.e., crepant, resolutions exists, and their Cox ring is given in [6, Proposition 5.6]. Comparing it with the Cox ring of the resolution of C4 /D8 in Proposition 4.1, we see that our resolution is not crepant. The second example, Q8 , is not generated by symplectic reflections, hence by [24] there is no crepant resolution. It turns out, however, that after some changes in Algorithm 2.8, inspired by the results on the Cox ring of crepant resolutions for a group with similar structure [8, 11], we can obtain the Cox ring of the crepant resolution for D8 . Proposition 4.5. In the case of G = D8 in Proposition 4.1, applying Algorithm 2.8 to X0 = C4 /G, we obtain a resolution X → X0 with Cl(X) = Z2 and the Cox ring is R(X) = C[T1 , . . . , T12 ]/I where generators for I and the degree matrix are given by T5 T8 + T4 T9 − 2T6 T10 , T2 T5 − 2T3 T6 + T7 T9 , 2T3 T4 − T2 T6 − T7 T10 , T72 T12 − T22 + 4T1 T3 , T5 T7 T12 + T2 T9 − 2T3 T10 , 2 , T4 T5 T12 − T62 T12 + T8 T9 − T10 2T3 T7 T11 − T6 T9 + T5 T10 , 2T1 T7 T11 + T6 T8 − T4 T10 , 2T2 T3 T11 + T5 T6 T12 − T9 T10 , 2T1 T2 T11 + T4 T6 T12 − T8 T10 ,
h
−1 1
−1 1
−1 1
0 −1
0 −1
T3 T8 + T1 T9 − T2 T10 , 2T1 T5 − T2 T6 + T7 T10 , T2 T4 − 2T1 T6 − T7 T8 , T6 T7 T12 + 2T1 T9 − T2 T10 , T4 T7 T12 − T2 T8 + 2T1 T10 , T72 T11 + T4 T5 − T62 , T2 T7 T11 − T4 T9 + T6 T10 , 4T32 T11 + T52 T12 − T92 , 2 , 4T1 T3 T11 + T4 T5 T12 + T8 T9 − 2T10 4T12 T11 + T42 T12 − T82 0 −1
−1 0
0 0
0 0
0 0
2 −2
0 2
i
.
After a suitable linear change of coordinates one sees that this ring is isomorphic to the Cox ring of symplectic resolutions of considered representation of D8 computed in [6]. Proof. This is an application of Algorithm 2.3 where instead of forming Σ = {σ} ∩ Υ, we perform the following steps as in [11, Section 3]: There are three minimal elements σ1 , σ2 , σ3 of the set consisting of all P (γ0∗ ) such that γ0 is an F-face and P (γ0∗ ) is singular. We form the cone-wise intersections Σi = {σi } ∩ Υ and resolve the fans: Σ′i → Σi . Denoting by primit(Ξ(1) ) the primitive generators of the rays of a fan or cone Ξ, we have 3 [
i=1
(1) ′(1) = {v1 , v2 }. \ primit σi primit Σi
The remaining steps of Algorithm 2.8 deliver the result. All verifications succeed; as in Example 4.2, we repeatedly apply [1, Lemma IV.1.2.7] to obtain normality of the ring. References [1] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface. Cox rings, volume 144 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2014. [2] I. V. Arzhantsev and S. A. Ga˘ıfullin. Cox rings, semigroups, and automorphisms of affine varieties. Mat. Sb., 201(1):3–24, 2010. [3] V. V. Batyrev. Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc. (JEMS), 1(1):5–33, 1999. [4] T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas. Computing tropical varieties. J. Symbolic Comput., 42(1-2):54–73, 2007. [5] W. Decker, G.-M. Greuel, G. Pfister, and H. Sch¨ onemann. Singular 4-0-3 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2016. [6] M. Donten-Bury and M. Grab. Cox rings of some symplectic resolutions of quotient singularities. 2015. Preprint. arXiv:1504.07463.
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[7] M. Donten-Bury and S. Keicher. quotsingcox.lib – a Singular library to compute Cox rings of quotient singularities, 2016. www.mathematik.uni-tuebingen.de/˜ keicher/quotsingcox/. [8] M. Donten-Bury and J. A. Wi´sniewski. On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32. 2014. Preprint. arXiv:1409.4204. [9] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.7.5, 2014. [10] J. Hausen. Cox rings and combinatorics. II. Mosc. Math. J., 8(4):711–757, 847, 2008. [11] J. Hausen and S. Keicher. A software package for Mori dream spaces. LMS J. Comput. Math., 18(1):647–659, 2015. [12] J. Hausen, S. Keicher, and A. Laface. Computing Cox rings. Math. Comp., 85(297):467–502, 2016. [13] J. Hausen and M. Wrobel. Non-complete rational T-varieties of complexity one. 2015. Preprint. arXiv:1512.08930. [14] E. Huggenberger. Fano Varieties with Torus Action of Complexity One. PhD thesis, Universit¨ at T¨ ubingen, Wilhelmstr. 32, 72074 T¨ ubingen, 2013. [15] Y. Ito and M. Reid. The McKay correspondence for finite subgroups of SL(3, C). In Higherdimensional complex varieties (Trento, 1994), pages 221–240. de Gruyter, Berlin, 1996. [16] D. Kaledin. McKay correspondence for symplectic quotient singularities. Invent. Math., 148(1):151–175, 2002. [17] S. Keicher. Computing the GIT-fan. Internat. J. Algebra Comput., 22(7):1250064, 11, 2012. [18] S. Keicher. Algorithms for Mori Dream Spaces. PhD thesis, Universit¨ at T¨ ubingen, 2014. [19] H. Kraft. Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. [20] D. Maclagan and B. Sturmfels. Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015. [21] M. Reid. La correspondance de McKay. Ast´ erisque, (276):53–72, 2002. S´ eminaire Bourbaki, Vol. 1999/2000. [22] J. Tevelev. Compactifications of subvarieties of tori. Amer. J. Math., 129(4):1087–1104, 2007. [23] J. Tevelev. On a question of B. Teissier. Collect. Math., 65(1):61–66, 2014. [24] M. Verbitsky. Holomorphic symplectic geometry and orbifold singularities. Asian J. Math., 4(3):553–563, 2000. [25] R. Yamagishi. On smoothness of minimal models of quotient singularities by finite subgroups of SLn (C). 2016. Preprint. arXiv:1602.01572. University of Warsaw, Institute of Mathematics, Banacha 2, 02-097 Warszawa, Poland ¨ t Berlin, Mathematisches Institut, Arnimallee 3, 14195 Berlin, Ger& Freie Universita many E-mail address: [email protected] Departamento de Matematica, Facultad de Ciencias Fisicas y Matematicas, Universi´ n, Casilla 160-C, Concepcio ´ n, Chile dad de Concepcio E-mail address: [email protected]
arXiv:1603.00071v1 [math.AG] 29 Feb 2016
MARIA DONTEN-BURY AND SIMON KEICHER Abstract. Let G ⊆ GL(n) be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution X → Cn /G, which is based just on the geometry of the singularity Cn /G, without further knowledge of its resolutions. As an application, we determine the Cox rings of resolutions X → C3 /G for all G ⊆ GL(3) with the aforementioned property and of order |G| ≤ 12. We also provide examples in dimension 4.
1. Introduction We consider quotient spaces Cn /G = Spec C[x1 , . . . , xn ]G , where G is a finite group acting linearly on Cn . By Noether’s theorem the ring of invariants of G is finitely generated, hence such quotients are (complex) algebraic varieties. They are usually singular and in such a case are called quotient singularities. In their construction, geometry meets finite group theory and there have been many attempts of extending this relation to resolutions of Cn /G. An example, probably the most important one, of describing the geometric structure of crepant (which in this case means that the canonical divisor is linearly trivial) resolutions of Cn /G in terms of algebraic properties of the group G is the McKay correspondence; for an introduction see, e.g., [21]. Though proved in several cases (see section 3.2), in general it reveals how much there is still to learn about this class of singularities and their resolutions. In this paper, we study quotient singularities X0 := Cn /G and their resolutions in terms of the Cox ring M Γ (X0 , O(D)) , R (X0 ) = Cl(X)
see [1] for details on the construction. Cox rings have already been successfully used by various authors to study resolutions of quotient singularities, in particular for symplectic quotients in [8, 6], described by a generating set in a simpler ring, and via an algorithmic approach based on toric ambient modifications in [11]. In this article we generalize the methods used in [11]. While finishing a draft of this paper, we also found out about Yamagishi’s work [25], which extends [8, 6]. However, our approach is different from [25] and our methods are not restricted to the class of crepant resolutions. In particular, as explained below, we do not try to construct the generators of the Cox ring directly from the group structure data, but we obtain it via toric ambient modifications, chosen in an intermediate step based on tropical geometry. Our first contribution, presented in Section 2, is an an algorithm to compute and verify a candidate for the Cox ring of a resolution X → X0 without requiring further knowledge of X0 . More precisely, Algorithm 2.3 computes R(X0 ) and Algorithm 2.8 2010 Mathematics Subject Classification. 14E15, 14Q10, 14Q15, 14C20, 14L24. This work was completed while the first author held a Dahlem Research School Postdoctoral Fellowship at Freie Universit¨ at Berlin. The first author was partially supported by a Polish National Science Center project 2013/11/D/ST1/02580 and the second author was supported by Proyecto FONDECYT Postdoctorado N. 3160016. 1
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M. DONTEN-BURY AND S. KEICHER
then computes a candidate for the Cox ring of a resolution X → X0 and verifies the result. Here, the main tool are toric ambient modifications as in [10, 12]: we embed X0 into an affine toric variety Z0 and compute a resolution Z → Z0 . The proper transform X → X0 then is the desired candidate for a resolution which we can verify algorithmically. Note that the choice of the toric resolution involves a tropical step; this is in the spirit of Tevelev and Teissier [22, 23]. Algorithm 2.8 is a variant of the algorithm [18] given for Mori dream spaces, which, in turn, is based on [1, 14] where the algorithm has been shown to work in the setting of complete rational complexity one T -varieties. See also [13] for the case of affine C∗ -surfaces. Our algorithms are implemented in the Open Source computer algebra system Singular [5]. Our second contribution concerns resolutions of three-dimensional quotient singularities; it is presented in Section 3. We first classify in Proposition 3.2 the finite, non-abelian groups G ⊆ GL(3) with |G| ≤ 12 and without pseudo-reflections (see Remark 2.2). When G is abelian, the quotient singularity is toric; since the algorithm of constructing a (toric) resolution and the structure of the Cox ring is known in this case, we do not consider it. Using our algorithms from Section 2, we first present the Cox rings of all singularities C3 /G on the list, see Proposition 3.3 and then their resolutions, see Theorem 3.5. Then we discuss certain properties of the obtained resolutions in order to understand what can be expected from the output of the algorithm in general. In particular, we check whether the resolutions for subgroups of SL(3) on the list are crepant. Finally, in Section 4, we emphasize that the method is not restricted to the 3dimensional setting by providing two 4-dimensional examples. Note that in dimension 4 less is known about resolutions of quotient singularities, hence computational experiments are even more valuable than in dimension 3. We provide the Cox rings of X0 = C4 /G for chosen representations of G = D8 and G = Q8 and compute the Cox rings of modifications X → X0 . For one of the cases, we are able to retrieve a crepant resolution that was also found in [6], see Proposition 4.5. Acknowledgement: We would like to thank J¨ urgen Hausen for several helpful discussions. 2. Algorithms In this section, we describe algorithms to compute the Cox ring of a quotient singularity X0 := Cn /G for a finite group G and the Cox ring of a resolution X → X0 . The latter algorithm is a variant of [1, 18]. It produces candidates using a tropical step and verifies them. We work with representations of G, usually assuming that they are faithful. In this case we will often identify the representation with its image in GL(n), i.e., consider G as a matrix group. Definition 2.1. A matrix group G ⊆ GL(n) is small if it does not have pseudoreflections; these are A ∈ G ⊆ GL(n) of finite order such that the subspace of fixed points (Cn )A is a hyperplane. We also say that a representation of an abstract group G is small, if its image in GL(n) is small. Remark 2.2. Note that the assumption on the representation of G not having pseudo-reflections is not restrictive: let H ⊆ G be the normal subgroup generated by all pseudo-reflections. Then Cn /G ∼ = (Cn /H) (G/H) ∼ = Cn (G/H) since Cn /H is smooth by the Chevalley-Shephard-Todd theorem. In particular, the singularity for G is the same as the one for the smaller group G/H.
The degree map of the Cox ring R(X0 ) is a homomorphism Zs → Cl(X0 ) that sends the i-th canonical basis vector ei ∈ Zs to deg(Ti ). We call the matrix with columns deg(T1 ), . . . , deg(Ts ) degree matrix.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
3
Algorithm 2.3 (Cox ring of X0 ). Input: a small group G ⊆ GL(n). • Compute generators f1 , . . . , fs for the invariant ring C[S1 , . . . , Sn ][G,G] . • Then G′ := G/[G, G] acts on the C-vector space V := linC (f1 , . . . , fs ). Compute matrices Mi such that, as a group, G′ = hM1 , . . . , Mk i ⊆ GL(s). • Compute a basis g1 , . . . , gs ∈ V of eigenvectors for V with respect to all the Mi . • In C[S1 , . . . , Sn , T1 , . . . , Ts ], consider the ideal J := hTi − gi ; 1 ≤ i ≤ si. Compute the elimination ideal J ′ := J ∩ C[T1 , . . . , Ts ] and the saturation I0 := J ′ : (T1 · · · Ts )∞ . • Define the matrix Q0 with the i-th column (χi1 , . . . , χik ) where χij is the eigenvalue of gi with respect to Mj . Output: I0 ⊆ C[T1 , . . . , Ts ] and Q0 . Then C[T1 , . . . , Ts ]/I0 is a presentation of R(Cn /G) in terms of generators and relations and Q0 is the degree matrix. Proof. This proof is a generalization of the one given in [11, Proposition 3.1]: By a result by Arzhantsev and Ga˘ıfullin [2], the Cox ring R(Cn /G) is isomorphic to the invariant ring C[S1 , . . . , Sn ][G,G] . Then the gi are X(G′ )-homogeneous prime generators for the invariant ring. The desired presentation C[T1 , . . . , Ts ]/I0 is obtained by considering C[T1 , . . . , Tr ] → C[S1 , . . . , Sn ][G,G] ,
Ti 7→ gi
and computing the ideal I0 of relations among the generators, e.g., by computing the listed elimination ideal. Example 2.4. We consider the faithful representation of the quaternion group G := Q8 with 8 elements whose image G ⊆ GL(2) is generated by 0 −i i 0 0 −i , −i 0 ,
where i ∈ C denotes the imaginary unit. There are no pseudo-reflections since no matrix in G has 1 as eigenvalue. We now apply the steps of Algorithm 2.3. Generators for the invariant ring C[S1 , S2 ][G,G] , where [G, G] = h−idi, are f2 = S12 ,
f 1 = S1 S2 ,
f3 = S22 .
Moreover, the group G′ = G/[G, G] is isomorphic to (Z/2Z)2 and acts on the linear hull V := linC (f1 , f2 , f3 ); we work with the description −1 0 0 1 0 0 0 0 1 0 −1 0 ⊆ GL(3). , G′ ∼ = 0
0
−1
0
1
0
The vector space V then has the G′ -invariant subspaces linC (f1 ) and linC (f2 , f3 ). We have a basis (g1 , g2 , g3 ) for V consisting of the Eigenvectors g1 := f1 ,
g2 := f2 + f3 ,
g3 := −f2 + f3 ∈ V
where the weights wi := (χi1 , χi2 ) ∈ (Z/2Z)2 ∼ = G′ are w1 = (0, 1), w2 = (1, 0) and w3 = (1, 1). The degree map is 0 1 1 3 2 Q0 : Z → (Z/2Z) , ei 7→ · ei 1
0
1
In the ring C[S1 , S2 , T1 , T2 , T3 ], we consider the ideal J := hT1 − g1 , . . . , T3 − g3 i and compute its elimination ideal J ′ := J ∩ C[T1 , T2 , T3 ] as well as the saturation I0 := J ′ : (T1 T2 T3 )∞ . We arrive at the G′ -graded Cox ring with degree matrix Q: R(C2 /G) ∼ = C[T1 , T2 , T3 ]/I0 ,
I0 = h4T12 − T22 + T32 i.
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M. DONTEN-BURY AND S. KEICHER
We recall some theory for the resolution algorithm and fix the setting. Assume we know a presentation R(X0 ) = C[T1 , . . . , Ts ]/I0 and its degree matrix Q0 . Consider the affine variety X 0 := Spec(R(X0 )) ⊆ Cs . We fix a gale dual matrix P0 for Q0 , i.e., a matrix fitting into the dual exact sequences (1)
/ Zs
0
/L
0o
Cl(X0 ) o
Q0
Zs o
P0
/N
P0∗
ker(Q0 ) o
0
where P ∗ denotes the transpose of P ; see, e.g., [18] for its computation. Let us shortly recall the construction of the tropical variety Trop(X0 ). Construction 2.5. Let the setting be as above. Setting T := C∗ , we obtain the tropical varieties \ Trop X 0 ∩ Ts := Trop(f ) ⊆ Qs , f ∈I0
Trop(X0 ) := P Trop X 0 ∩ Ts
s
⊆ N
where Trop(f ) ⊆ Q is the support of the codimension-one skeleton of the normal fan over the Newton polytope of the polynomial f ∈ C[T1 , . . . , Ts ]. It is possible to establish a fan structure on Trop(X0 ), see, e.g., [4, 20]. Construction 2.6. See [1, Section III] and [10]. Let X be a normal, irreducible A2 maximal variety with Γ(X, O∗ ) = C∗ such that both K := Cl(X) and R := R(X) are finitely generated. Fix a system of pairwise non-associated, K-prime generators F := (f1 , . . . , fm ) for R, the degree matrix Q and a gale dual matrix P . Then X is the good quotient p of an open subset b ⊆ X := Spec(R) ⊆ Cr X
b can be described by the action of the characteristic quasitorus Spec(C[K]), and X combinatorially by a set Φ of pairwise overlapping polyhedral cones in K ⊗Z Q. The triple (R, F, Φ) already determines X up to isomorphism. Moreover, a face γ0 Qr≥0 is a relevant F-face if Y p fi ∈ / hfi ; ei ∈ / γ0 i ei ∈γ0
and Q(γ0 ) ∈ Φ holds. Write cov(X) for the set of minimal relevant F-faces. Then one defines the canonical ambient toric variety as the toric variety Z ⊇ X whose fan Σ has the maximal cones Σmax = {P (γ0∗ ); γ0 ∈ cov(X)} where γ0∗ (Qr≥0 )∨ is the dual face. Further note that we have a pairwise disjoint decomposition [ X(γ0 ), X(γ0 ) := p {z ∈ X; zi 6= 0 ⇔ ei ∈ γ0 } , X = γ0 ∈cov(X)
By the decomposition from Construction 2.6, we can test smoothness of X by testing each X(γ0 ) with the following straight-forward test; it was also used in [11]. Write T := C∗ . Algorithm 2.7 (Smoothness test). Input: the Cox ring R = C[T1 , . . . , Tr ]/I with degree matrix Q, of a normal, irreducible A2 -maximal variety X, a relevant F-face γ0 Qr≥0 . Let P be a gale dual (n × r) matrix for Q. • If σ := P (γ0∗ ) is not smooth, return false.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
5
• Find A ∈ GL(n, Z) such that Avij = ej holds for all rays Q≥0 · vij of σ, vij primitive. • Let Iγ0 ⊆ C[T1 , . . . , Tr ] be the ideal obtained from I by setting Ti = 0 ⇔ ei ∈ / γ0 . • With the toric morphisms α : Tn → Tn and p : Tr → Tn corresponding to A and P , compute the pushforward and pullback Jγ0 := α∗ p∗ Iγ0 ⊆ C[Y1 , . . . , Yn ]. • Let jγ0 ⊆ C[Y1 , . . . , Yn ] be the jacobian ideal, i.e., the ideal generated by all (n − dim(Jγ0 ) × (n − dim(Jγ0 )) minors of the Jacobian of Jγ0 . • Return true if 1 ∈ (Jγ0 + jγ0 ) : (Ym+1 · · · Yn )∞ holds in C[Y1 , . . . , Yn ]. Return false otherwise. Output: true if X(γ0 ) is smooth, false otherwise. Proof. The first steps produce equations for X(γ0 ) in O(Cm × Tn−m ) ∼ = C[σ ∨ ∩ n Z ]. The second to last step computes the ideal of the singular locus V (Jγ0 )sing of V (Jγ0 ) ⊆ Cn . The last step checks whether V (Jγ0 )sing ∩ Cm × Tn−m is empty. The following algorithm computes a candidate for the Cox ring R(X) of a resolution X → X0 and verifies it; it is close to [18, Algorithm 2.4.8] which in turn is based on [1, Theorem III.4.4.9] where the complete, complexity-one-case is treated. See also [14, Chapter 3] and consider [13, Section 3] for the case of affine C∗ -surfaces. The idea is to embed X0 ⊆ Z0 into its ambient affine toric variety Z0 = Cn /G′ , perform a toric resolution Z → Z0 and consider the proper transform X → X0 . The step Z → Z0 involves the tropical variety. Algorithm 2.8 (Cox ring of a resolution X → X0 = Cn /G). Input: a small group G ⊆ GL(n). • Compute the Cox ring R(X0 ) = C[T1 , . . . , Ts ]/I0 of X0 with Algorithm 2.3. Determine Q0 and P0 as in (1). • Compute σ := P (Qn≥0 ) ⊆ Qn . It determines the ambient affine toric variety Z0 = Cn /G′ where G′ := G/[G, G]. • Compute a fan Υ with support Trop(X0 ). • Form the fan Σ := {σ} ∩ Υ by cone-wise intersection. • Resolve the toric singularities by stellar subdivisions of Σ at primitive vectors v1 , . . . , vm . • Write π : Z → Z0 for the toric resolution. Compute a gale dual Q out of the enlarged matrix P := [P0 , v1 , . . . , vm ] as in (1). • Compute the Cox ring of the proper transform X → X0 under π using [12, Algorithm 3.6] with the ’verify’-option: – Compute the ideal I ′ := p∗ (p0 )∗ I0 ⊆ C[T1 , . . . , Tr ], i.e., the pullback of the pushforward under the toric maps p, p0 corresponding to P , P0 . – Compute the saturation I := I ′ : (T1 · · · Tr )∞ . – Verify that all Ti define prime elements in R := C[T1 , . . . , Tr ]/I. – Verify that dim(I) − dim(hTi , Tj i + I) ≥ 2 for all i 6= j. – Verify that R is normal. • Verify smoothness of the variety X determined by (R, Σ): – For each τ ∈ Σmax , do: ∗ determine γ0 Qr≥0 with P (γ0∗ ) = τ . ∗ Verify that X(γ0 ) is smooth with Algorithm 2.7. Output: R and Q. If the verifications of the last two steps were successful, then R is the Cox ring of X of a resolution X → X0 and Q is the degree matrix of R. Lemma 2.9. In Algorithm 2.8, suppose R is the Cox ring of X. With the fan Σ, the minimal relevant (T 1 , . . . , T r )-faces are cov(X) = {γ0∗ Qr≥0 ; P (γ0∗ ) ∈ Σmax }.
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M. DONTEN-BURY AND S. KEICHER
Proof. By a result of Tevelev [22, Lemma 2.2] and the definition of the tropical variety Trop(X), we obtain γ0 Qr≥0 relevant (T 1 , . . . , T r )-face
P (γ0∗ )◦ ∩ Trop(X) 6= ∅ and P (γ0∗ ) ∈ Σ.
⇔
As |Σ| = Trop(X), the first condition on the right-hand-side can be dropped. Considering only minimal relevant (T 1 , . . . , T r )-faces means to replace the last condition by requiring P (γ0∗ ) ∈ Σmax . Proof of Algorithm 2.8. By construction and [12, Algorithm 3.6], X → X0 is a modification and R the Cox ring of X; note that [12, Algorithm 3.6] directly translates to the affine setting. By the smoothness verification step and Lemma 2.9, X is smooth. Example 2.10. Consider the Cox ring R(X0 ) = C[T1 , T2 , T3 ]/I0 computed in Example 2.4 with its degree-map Q : Z3 → (Z/2Z)2 . We apply the steps of Algorithm 2.8 to X0 . The canonical ambient toric variety is the affine toric variety Z0 = Z0 (σ) with σ ⊆ Q3 being the polyhedral cone spanned by (1, 0, 0), (1, 2, 0), (1, 0, 2). The following picture shows the steps for the toric resolution. (1, 2, 0)
(1, 2, 0)
v3 v1 v4 (0, 0, 0)
(0, 0, 0)
(1, 0, 0) (1, 0, 2)
v2 (1, 0, 0) (1, 0, 2)
Σ and new rays Q≥0 · vi (red)
σ (gray) and Σ (blue)
The two-dimensional fan Σ = {σ}∩Trop(X0 ) has already the additional ray Q≥0 ·v1 where v1 := (3, 2, 2). The fan can be resolved by insertion of the further rays through v2 := (2, 1, 2),
v3 := (2, 2, 1),
v4 := (2, 1, 1).
This yields a toric resolution Z → Z0 which then induces a resolution X → X0 as the proper transform of Z → Z0 ; its Z4 -graded Cox ring R(X) and degree matrix are R(X) = C[T1 , . . . , T7 ]/h4T12 T7 − T6 T22 + T32 T5 i, # " −1 0 0 0
−1 −1 0 −1
−1 0 −1 −1
1 1 1 0
0 −1 1 1
0 1 −1 1
0 −1 −1 −1
.
One verifies that X is smooth. As X is a surface, we can verify minimality of the b Ti ): they are resolution by inspecting the self intersection numbers of the V (X; −1, −1, −1, −2, −2, −2, −2. We will use Algorithms 2.3 and 2.8 in Section 3 to compute resolutions of quotient singularities.
Remark 2.11 (Implementation). We have started an implementation of Algorithms 2.3 and 2.8 in the library quotsingcox.lib for the Open Source computer algebra system Singular [5]. It will be made available at [7]. 3. Resolutions of 3-dimensional quotient singularities 3.1. Cox rings. In this section, we compute Cox rings of resolutions of quotient singularities C3 /G where G is a group of order at most 12 and discuss the results. As explained in Remark 2.2, we will consider only faithful representations of G without pseudo-reflections.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
7
Notation 3.1. In the following, we denote by S3 the permutation group of three elements, by D2n the dihedral group of 2n elements, by Q8 the Quaternion group and by A4 the alternating group of four elements. Moreover, we write BD3 for the binary dihedral group, i.e., the abstract group with 12 elements
BD3 = a, x; a2n = 1, x2 = an , x−1 ax = a−1 . Proposition 3.2. Up to conjugacy in GL(3), the 3-dimensional faithful small representations of non-abelian groups up to order 12 are as follows. No.
G
gen.s in GL(3)
1
S3
0 1 0
−1 −1 0
0 0 1
,
−1 0 0
2
D8
0 1 0
−1 0 0
0 0 1
,
Q8
i 0 0
0 −i 0
0 0 1
,
Q8
i 0 0
0 −i 0
0 0 −1
D10
ζ5 0 0
D12
−ζ3 0 0
A4
0 1 0
BD3
2 −ζ3 0 0
BD3
2 −ζ3 0 0
BD3
2 −ζ3 0 0
3
4
5
6
7
8
9
10
0
0 0 1
4 ζ5
0
0
0 0 1
−ζ3 0
0 0 1
1 0 0
0
−ζ3 0
0 −ζ3 0
0 −ζ3 0
0 0 1
0 0 −1
0 0 1
.
1 0 0
0 −1 0
0 0 −1
.
0 1 0
−1 0 0
0 0 1
0 1 0
0 1 0
,
,
0 0 −1
,
,
1 1 0
−1 0 0
,
,
0 1 0
,
1 0 0
0 −i 0
−i 0 0
0 −i 0
.
.
0 0 −1
0 0 1
0 −i 0
0 0 −1
0 −1 0
.
0 0 1
−1 0 0
1 0 0
−i 0 0
−i 0 0
.
.
0 0 1
.
0 0 i
.
0 0 −1
.
In the table, i ∈ C denotes the imaginary unit, and ζk ∈ C is a primitive k-th root of unity. All listed representations are reducible except for the A4 -case. Proof. To classify these representations we use the library of groups of small order, which is a part of GAP [9]. We do the following steps: (i) view the character tables to get dimensions of irreducible representations,
8
M. DONTEN-BURY AND S. KEICHER
(ii) find all irreducible representations: for a group of small order and a given character one can either locate a representation in the literature or construct easily, (iii) combine irreducible representations to get all faithful 3-dimensional representations of a given group (in particular, direct sums of 1-dimensional representations are not allowed since they are not faithful representations – the image is an abelian group), (iv) eliminate representations with pseudo-reflections (see Remark 2.2), (v) check whether obtained representations have different groups as images (if representations differ just by a permutation of conjugacy classes in G, they give the same quotient); for any two representations of the same group on the list above the set of matrix traces are different, hence this condition is satisfied. Proposition 3.3. In the setting of Proposition 3.2, the Cox rings of X0 := C3 /G are as listed the following table. No.
G
Cl(X0 )
1
S3
Z/2Z
2
D8
Z/2Z × Z/2Z
3
Q8
4
degree matrix
R(X0 )
C[T1 , . . . , T4 ]/I with I gen. by 4T33 − T22 − 27T42
1
1
0
0
h
1 1
1 0
0 1
0 0
i
C[T1 , . . . , T4 ]/I with I gen. by 4T12 + T22 − T42
Z/2Z × Z/2Z
h
1 1
1 0
0 1
0 0
i
C[T1 , . . . , T4 ]/I with I gen. by T12 − T22 + 4T32
Q8
Z/2Z × Z/2Z
h
1 1
1 0
1 0
0 1
i
C[T1 , . . . , T4 ]/I with I gen. by T12 − T32 + 4T42
5
D10
Z/2Z
1
1
0
0
6
D12
Z/2Z × Z/2Z
1 1
1 0
0 1
0 0
7
A4
Z/3Z
0
0
2
1
8
BD3
Z/4Z
3
1
2
0
9
BD3
Z/4Z
3
1
1
2
10
BD3
Z/4Z
3
1
2
2
h
i
C[T1 , . . . , T4 ]/I with I gen. by 4T35 + T22 − T42
C[T1 , . . . , T4 ]/I with I gen. by 4T43 + T12 − T22
C[T1 , . . . , T4 ]/I with I gen. by T13 + T33 − 3T1 T3 T4 + T43 − 27T22 C[T1 , . . . , T4 ]/I with I gen. by 4T33 + T12 − T22
C[T1 , . . . , T4 ]/I with I gen. by 4T43 + T12 − T32
C[T1 , . . . , T4 ]/I with I gen. by 4T43 + T12 − T22
Proof. The listed Cox rings are obtained by applying Algorithm 2.3 to the list of representations from Proposition 3.2 using our implementation 2.11.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
9
Remark 3.4. Note that several rings on the list (without grading considered) are isomorphic. This is because the ring structure of the Cox ring of Cn /G is just the invariant ring C[x1 , . . . , xn ][G,G] and derived subgroups in cases 2, 3, 4 (D8 and Q8 representations) and also 6, 8, 9, 10 (D12 and BD3 representations) are conjugate subgroups of GL(3). Theorem 3.5. In the setting of Proposition 3.3, a resolution X → X0 of the quotient X0 = C3 /G has the following Cox ring, respectively. No.
1
2
3
4
5
6
7
8
9
G
Cl(X)
S3
Z4
D8
Z4
Q8
Z4
Q8
Z4
D10
D12
A4
BD3
BD3
degree matrix
" "
Z5
Z5
Z8
0 0 0 1
"
#
−1 −1 −1 0 1 0 0 −2 −1 −1 0 0 2 0 1 0 −1 0 0 −2 2 2 0 0 0 0 −3 1
−1 0 −1 −1 1 0 −2 −1 −1 −1 0 2 1 0 0 −1 0 −2 2 1 0 0 0 −3
0 0 0 1
0 0 2 1
0 0 0 1
0 −5 −2 −5 1 0 0 0 −1 −1 0 0 0 2 0 0 1 0 −1 −1 0 −2 1 0 2 0 −1 −2 0 −4 0 1 2 0 −1 −3 0 −5 0 0
−3 −3 0 −2 2 0 0 −1 0 −1 0 0 2 0 1 1 0 0 −1 0 1 2 1 0 1 −1 −1 0 2 2 0 1 −2 0 0
0 0 −1 −1 −1 −2 −2
−3
#
0 0 0 0 1
0 0 0 0 0 0 1 0
0 2 00 0 0 −3 2 0 0 0 2 −2 1 0 0 5 −5 0 1 0 4 −4 0 0
0 0 0 0 1
C[T1 , . . . , T9 ]/I with I gen. by 4T35 T73 T8 + T22 T6 − T42 T9
0 000 0 0 −2 −1 0 3 0 0 0 0 0 1 0 0 −2 1 0 0 0 −1 0 0 0 −2 0 1 0 0 −1 2 0 0 −4 0 0 1 0 −2 1 0 0 −3 0 0 0 1 −2 1 0 0 −4 0 0 0 0 −2 2 0 0 −5 0 0 0 0
−2 2 −2 −3 −3
C[T1 , . . . , T8 ]/I with I gen. by −T12 T6 − 4T42 T7 + T32 T8
0 0 0 0 1
−3 3 −2 −5 −4
C[T1 , . . . , T8 ]/I with I gen. by −4T12 T6 + T42 T7 − T22 T8
C[T1 , . . . , T8 ]/I with I gen. by T12 T6 + 4T32 T7 − T22 T8
0 0 0 1 0
−3 2 −1 −2 −2
C[T1 , . . . , T8 ]/I with I gen. by 4T33 T7 − T22 T6 − 27T42 T8
#
0 0 0 1
−2 −3 −2 −2 1
Z5
Z8
1 00 0 20 0 −2 1 0 −3 0
−1 −1 0 −1 1 0 0 −1 0 −1 0 0 2 0 −1 −1 0 −2 0 0 2 1 0 0 1 0 −1 −1
"
0 −3 −2 −3 −1 −1 0 0 1 0 −1 −1 1 0 −1 −2
R(X)
#
C[T1 , . . . , T9 ]/I with I gen. by 4T43 T72 T9 + T12 T6 − T22 T8
0 0 0 0 0 0 0 1
C[T1 , . . . , T9 ]/I with I gen. by 4T33 T72 T9 + T12 T6 − T22 T8
0 −3 −2 1 0 0 0 0 0 0 −1 −1 −1 −2 0 2 0 0 0 0 0 −1 0 0 2 0 −3 2 0 0 0 0 3 1 0 −3 0 5 −5 1 0 0 0 2 1 0 −3 0 4 −4 0 1 0 0 2 0 0 −3 0 4 −4 0 0 1 0 3 1 0 −4 0 6 −6 0 0 0 1 4 2 0 −6 0 9 −9 0 0 0 0
C[T1 , . . . , T12 ]/I with I gen. by T33 T62 T7 T82 T11 + T43 T6 T72 T92 T12 2 T2 −3T1 T3 T4 T6 T7 T8 T9 T11 T12 + T13 T8 T9 T11 12 −27T22 T10
0 0 0 0 0 0 0 1
C[T1 , . . . , T12 ]/I with I gen. by 4T43 T62 T9 T10 + T12 T7 T12 − T32 T8
10
M. DONTEN-BURY AND S. KEICHER
10 BD3
Z5
−3 −3 0 −2 2 0 0 0 1 1 0 0 −1 1 0 0 2 3 −1 2 −3 0 2 0 −2 −5 2 −3 5 0 −5 1 −2 −4 2 −3 4 0 −4 0
0 0 0 0 1
C[T1 , . . . , T9 ]/I with I gen. by 4T43 T62 T9 + T12 T7 − T22 T8
Proof. The Cox rings of the X0 = C3 /G have been presented in Proposition 3.3. We then obtain the Cox rings of a resolution X → X0 using our implementation 2.11 of Algorithm 2.8. The verifications of the primality of the variables Ti are done directly computationally, the smoothness tests are done with Algorithm 2.7 as implemented in MDSpackage [11]. The normality follows from Serre’s criterion, see, e.g., [19, 6.2]. Remark 3.6. Using these methods and its implementation, one can directly go to higher group orders |G|. Since the number of different representations to consider grows quickly for isomorphism types of G with higher order, we end the table at order 12. 3.2. Properties of quotients and their resolutions. We discuss certain geometric properties of the quotient singularities and their resolutions corresponding to the Cox rings from Theorem 3.5. At first, we turn to torus actions on quotient spaces: we consider the relation between the form of the Cox ring and the existence of a torus action on Cn /G. Proposition 3.7. Let V be an affine space which is a direct sum of n representations of a given group G. Then V admits an action of (C∗ )n which commutes with the action of G. Proof. Let (xi,1 . . . , xi,ni ) be the coordinates of the i-th representation (of dimension di ). The following action commutes with the action of G. (C∗ )n × V → V,
=
(t1 , . . . , tn ) · (x1,1 . . . , x1,d1 , . . . , xn,1 . . . , xn,dn ) (t1 x1,1 . . . , t1 x1,d1 , . . . , tn xn,1 . . . , tn xn,dn )
Corollary 3.8. A quotient of C3 by a direct sum of a 2-dimensional and a 1dimensional representation is a T -variety of complexity one. Remark 3.9. All representations in the table in Proposition 3.2 except the case of A4 are direct sums of two irreducible representations, as it can be easily seen from the generating matrices. In particular, all varieties X0 in Proposition 3.3 except possibly for the A4 case are T -varieties of complexity one. Thus, the only 3-dimensional quotient singularities which are possibly not T varieties of complexity one correspond to irreducible representations. The representation of A4 in Proposition 3.2 is irreducible and we expect that there is no (C∗ )2 action. Note that, in spite of the (C∗ )2 -action from Remark 3.9 for the other cases, we still did not know a priori that Algorithm 2.8 would compute the Cox ring of a resolution: to our knowledge, only for complete T -varieties [1, Theorem III.4.4.9] and for affine C∗ -surfaces [13, Section 3] this is known. However, it seems natural that, as in the cases of complete T -varieties and C∗ -surfaces, the single relations of the Cox rings of the singularity and its resolution are in all cases except in the A4 -case of trinomial shape. Remark 3.10. By the previous discussing, it would be interesting to continue the list of results in Theorem 3.5 with irreducible, 3-dimensional representations. However, the three smallest cases of such representations (one of order 21 and two of order 27) are at the moment computationally out of reach on our machines.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
11
We now turn to properties of the resolutions found in Theorem 3.5. Recall that a resolution of singularities π : X → X0 is called crepant if KX = π ∗ KX0 . For surface quotient singularities for G ⊆ SL(2), i.e., du Val singularities, crepant resolutions are the minimal ones: the special fiber is a tree of smooth rational curves dual to a Dynkin diagram An , Dn , E6 , E7 or E8 . The relation between the structure of G (its conjugacy classes or irreducible representations) and the shape of the diagram of the resolution of C/G was noticed by McKay. The postulated relation between the geometry of crepant resolutions of (Gorenstein) quotient singularities and the structure of the group is called the McKay correspondence. It has been studied in several special cases and in different formulations. In particular, it is proved in dimension 3, see e.g. [15], for symplectic singularities in dimension 4, see [16], and a weak version for any G ⊂ SL(n) (the equality of the dimension of cohomology space and the number of conjugacy classes of G) is due to Batyrev [3]. It is a natural question to ask how good the resolutions obtained using Algorithm 2.8 are and what properties to expect them to have. In particular, we would like to know whether they are crepant for G ⊂ SL(n). To test this property for the 3-dimensional results in Theorem 3.5 we can use the McKay correspondence. Remark 3.11. Among groups listed in Proposition 3.2 the cases 1, 2, 3, 5, 6, 7 and 8 are in SL(3). Proposition 3.12. The Cox rings of resolutions of singularities in cases 1 (S3 ), 5 (D10 ) and 7 (A4 ), given in Theorem 3.5, are not the Cox rings of crepant resolutions. The resolutions in cases 2 (D8 ), 3 (Q8 ), 6 (D12 ) and 8 (BD3 ) are crepant. Proof. By [8, Lemma 2.11], the class group Cl(X) of a resolution π : X → Cn /G of a quotient singularity is a free group and its rank m is equal to the number of irreducible components of the exceptional divisor. By the McKay correspondence in dimension n = 3 we know that m is the number of conjugacy classes of junior elements in G where an element g ∈ G is junior if n 1X ak age(g) = 1 where age(g) := r k=1
2πiak r
with ak coming from the exponents of eigenvalues e of g, see e.g. [21]. Thus the conjugacy classes of junior elements can be determined with simple computations (e.g., using GAP, [9]), and the results are as follows: case number junior classes
1 2
2 4
3 4
5 3
6 5
7 3
8 5
Comparing them with the rank of Cl(X) given in Theorem 3.5 we obtain that the resolutions in cases 1, 5 and 7 have too many components of the exceptional divisor to be crepant. We sketch the argument that in the remaining cases obtained resolutions are crepant. The exceptional divisors which have to be present in the crepant resolution come from the minimal resolutions of transversal du Val singularities along lines in C3 fixed by an element g of age 1, possibly divided by the action of the normalizer of g in G. It follows from the McKay correspondence that the number of such divisors is the same as the number of conjugacy classes of junior elements in G. Hence the resolutions with the rank of the Picard group equal to the number of junior classes are crepant. Note that in dimension 3 there is one Cox ring corresponding to all crepant resolutions, because they are all birationally equivalent and flops preserves smoothness in dimension 3.
12
M. DONTEN-BURY AND S. KEICHER
Remark 3.13. We can describe the geometry of all crepant resolutions for groups 2, 3, 6 and 8 of Proposition 3.2 by computing GIT quotients of the spectrum of their Cox rings. To determine the GIT fan describing the variation of the quotients, we use [17]. case number max. GIT-cones within Mov(X)
2 9
3 1
6 16
8 1
Actually, the cases 3 and 11 are products of a finite subgroup of SL(2) and a trivial group, so the resolutions will be just products of minimal resolutions of du Val singularities by C. Hence, there is just one crepant resolution, i.e., just one chamber in the GIT fan restricted to the cone of movable divisor classes Mov(X). 4. Two 4-dimensional examples In this section, we present two 4-dimensional examples. In dimension 4 much less is known about the resolutions of quotient singularities. In particular, crepant resolutions do not always exist, and the McKay correspondence has been proved just for the symplectic case (see [16]). Hence it is a very appropriate setting for computational experiments with constructing resolutions via Cox rings. Moreover, it is interesting also from the point of view of the Cox ring theory: while in dimension 3 all the examples we treat can be defined with a single relation, almost always trinomial, here the ring structure will be more complicated. The first of the examples, the representation of D8 , has also been treated in [6] where the Cox ring of symplectic resolutions was constructed. Proposition 4.1. Consider the 4-dimensional quotient singularities C4 /G for the two small groups G ⊆ GL(4) G
gen.s in GL(4)
D8
"
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
Q8
"
i 0 0 0
0 −i 0 0
0 0 i 0
0 0 0 −i
# #
,
"
,
"
1 0 0 0
0 −1 0 0
0 1 0 0
−1 0 0 0
0 0 1 0
0 0 0 1
0 0 0 −1
0 0 −1 0
# #
.
.
where i ∈ C is the imaginary unit. The Cox ring of the quotient space X0 := Cn /G, the degree matrix and the class group of X0 are as follows. G D8
Cl(X0 ), degree matrix and R(X0 ) Z/2Z × Z/2Z
h
1 1
1 1
1 1
C[T1 , . . . , T10 ]/I with I gen. by T5 T8 + T4 T9 − 2T6 T10 2 , T72 − T8 T9 + T10 T5 T7 + T2 T9 − 2T3 T10 , 2T3 T7 − T6 T9 + T5 T10 , 2T1 T7 + T6 T8 − T4 T10 , T2 T5 − 2T3 T6 + T7 T9 , T3 T4 − T1 T5 − T7 T10 , 4T32 + T52 − T92 , 2 , 4T1 T3 + T62 − T10 2T1 T2 + T4 T6 − T8 T10 ,
1 0
1 0
1 0
0 1
0 0
0 0
0 0
i
T3 T8 + T1 T9 − T2 T10 , T6 T7 + 2T1 T9 − T2 T10 , T4 T7 − T2 T8 + 2T1 T10 , T2 T7 − T4 T9 + T6 T10 , 2 , T4 T5 − T62 + T8 T9 − T10 2T1 T5 − T2 T6 + T7 T10 , T2 T4 − 2T1 T6 − T7 T8 , 2T2 T3 + T5 T6 − T9 T10 , T22 + T62 − T8 T9 , 4T12 + T42 − T82
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
Q8
Z/2Z × Z/2Z
h
1 1
1 1
1 1
1 0
1 0
1 0
0 1
0 1
0 1
0 0
C[T1 , . . . , T10 ]/I with I gen. by 2 , T82 − 4T7 T9 − T10 T5 T8 − 2T6 T9 + T2 T10 , T2 T8 − 2T3 T9 + T5 T10 , T5 T7 − T4 T9 + T3 T10 , T2 T7 − T1 T9 + T6 T10 , T3 T5 − T2 T6 − 2T9 T10 , T3 T4 − T1 T6 + 2T7 T10 , T32 − T62 + 4T7 T9 , T1 T3 − T4 T6 + 2T7 T8 , 2 , T1 T2 − T4 T5 + 4T7 T9 + 2T10
13
i
T6 T8 − 2T4 T9 + T3 T10 , T3 T8 − 2T1 T9 + T6 T10 , 2T6 T7 − T4 T8 + T1 T10 , 2T3 T7 − T1 T8 + T4 T10 , 2 , T4 T5 − T62 − T10 T1 T5 − T3 T6 − T8 T10 , T2 T4 − T3 T6 + T8 T10 , T2 T3 − T5 T6 + 2T8 T9 , T22 − T52 + 4T92 , T12 − T42 + 4T72
Proof. This is again an application of Algorithm 2.3.
We now intend to compute the Cox ring of a resolution for the both cases X0 = C4 /D8 and X0 = C4 /Q8 presented in the previous proposition with Algorithm 2.8. As the Cox rings turn out to big for all smoothness tests to be computationally feasible, we therefore present Cox rings of modifications X → X0 (smoothness is not fully verified) in the following two examples. However, in Proposition 4.1 we present a resolution for the D8 -case thereby retrieving a result of Grab and the first author [6]. Example 4.2 (Case D8 ). In the case of G = D8 in Proposition 4.1, applying Algorithm 2.8 to X0 = C4 /G, we obtain a modification X → X0 with Cl(X) = Z16 and the Cox ring R(X) = C[T1 , . . . , T26 ]/I with generators for I and the degree matrix given by T6 T7 T17 − T1 T8 + (4i)T3 T9 , T2 T7 T11 − T4 T8 + 2T5 T9 , 2 2 2T9 T10 T18 T20 + T1 T2 T11 − T4 T6 T17 , T8 T10 T18 T20 + (2i)T2 T3 T11 − T5 T6 T17 , 2 2 T7 T10 T18 T20 + (2i)T3 T4 − T1 T5 , T2 T6 T16 T25 T26 + (−2i)T3 T4 − T1 T5 , 2 2 2 T6 T10 T14 T22 T23 − T4 T8 − 2T5 T9 , 2T3 T10 T14 T22 T23 + (i)T2 T8 T16 T25 T26 + (−i)T5 T7 T17 , 2 2 2 2 T25 T26 − T4 T7 T17 , T5 T10 T15 T19 T21 − T6 T8 T16 T25 T26 + (2i)T3 T7 T11 , T1 T10 T14 T22 T23 − 2T2 T9 T16 2 2 2 T4 T10 T15 T19 T21 − 2T6 T9 T16 T25 T26 − T1 T7 T11 , T2 T10 T15 T19 T21 − T1 T8 + (−4i)T3 T9 , 2 2 2 T2 T5 T15 T19 T21 + (−2i)T3 T6 T14 T22 T23 − T7 T8 T18 T20 , 2 2 2 T2 T4 T15 T19 T21 − T1 T6 T14 T22 T23 + 2T7 T9 T18 T20 , 2 2 2 2 2 T2 T11 T15 T19 T21 − T6 T14 T17 T22 T23 + 4T8 T9 T18 T20 , 2 2 2 T25 T26 T25 T26 − 2T1 T5 T8 T9 T16 T1 T4 T82 T16 T25 T26 + (−4i)T3 T4 T8 T9 T16 2 T25 T26 + (−2i)T3 T4 T72 T11 T17 − T1 T5 T72 T11 T17 , +(8i)T3 T5 T92 T16 2 2 2 2 2 T4 T7 T8 T17 T18 T20 + 2T5 T7 T9 T17 T18 T20 + (2i)T1 T3 T4 T8 T14 T22 T23 2 2 2 T22 T23 + (4i)T1 T3 T5 T9 T14 T22 T23 , T22 T23 + 8T32 T4 T9 T14 −T12 T5 T8 T14 2 2 2 2 4T92 T16 T18 T20 T25 T26 − T42 T15 T17 T19 T21 + T12 T11 T14 T22 T23 , 2 2 2 2 2 2 2T8 T9 T16 T18 T20 T25 T26 + T7 T11 T17 T18 T20 − T4 T5 T15 T17 T19 T21 + (2i)T1 T3 T11 T14 T22 T23 , 2 2 2 2 T82 T16 T18 T20 T25 T26 − T52 T15 T17 T19 T21 − 4T32 T11 T14 T22 T23 , 2 2 2 2 2 T10 T14 T18 T20 T22 T23 + T2 T11 T16 T25 T26 − 2T4 T5 T17 , 2 T2 T2 T T T 2 2 T10 15 18 19 20 21 − T6 T16 T17 T25 T26 + (4i)T1 T3 T11 , 2 2 2 2 2 T6 T14 T16 T22 T23 T25 T26 + T72 T11 T18 T20 − 2T4 T5 T15 T19 T21 , 2 2 2 2 T22 T23 , T20 + (−4i)T1 T3 T14 T16 T19 T21 T25 T26 − T72 T17 T18 T22 T15 2 2 2 2 2 T10 T14 T15 T19 T21 T22 T23 − 4T8 T9 T16 T25 T26 − T7 T11 T17 , 2 2 2 2 2 2 T1 T8 T14 T16 T22 T23 T25 T26 + (−8i)T1 T3 T8 T9 T14 T16 T22 T23 T25 T26 2 2 2 2 2 2 −16T32 T92 T14 T16 T22 T23 T25 T26 + T74 T11 T17 T18 T20 − 2T4 T5 T72 T15 T17 T19 T21 , 2 2 2 2 T42 T82 T15 T16 T19 T21 T25 T26 − 4T4 T5 T8 T9 T15 T16 T19 T21 T25 T26 2 T2 T T T T 4 2 2 2 2 2 +4T52 T92 T15 16 19 21 25 26 − T7 T11 T17 T18 T20 + (−4i)T1 T3 T7 T11 T14 T22 T23 , 4 2 2 2 2 2 T74 T11 T17 T18 T20 − 2T4 T5 T72 T15 T17 T18 T19 T20 T21 + (4i)T1 T3 T72 T11 T14 T18 T20 T22 T23 2 2 2 2 2 2 T19 T21 T22 T23 T15 T19 T21 T22 T23 + T12 T52 T14 T15 T15 T19 T21 T22 T23 + (−4i)T1 T3 T4 T5 T14 −4T32 T42 T14
14
M. DONTEN-BURY AND S. KEICHER
−1
1 −1 −1 0 0 1 1 0 0 −1 −1 −2 −1 −1 −2
−1 0 0 0 −1 −1 1 1 −1 0 −1 −1 −1 −1 −2 −2
−1 −1 1 −1 0 0 1 1 −1 0 0 −2 −1 −1 −2 −1
0 0 −2 0 −1 0 −1 0 0 0 −1 1 0 0 1 0
0 −2 0 0 −1 0 −1 0 −1 0 0 0 1 0 0 1
0 −1 0 0 0 2 0 0 0 0 0 0 0 0 0 −1 0 −2 0 −1 −2 1 0 0 0 0 0 0 0 0 −1 0 0 −2 −1 −2 0 1 0 0 0 0 0 0 0 −1 0 0 0 −1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 0 −1 0 −1 −1 0 0 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −2 0 0 0 0 0 2 0 0 0 0 0 −1 −1 −1 −2 0 0 0 0 0 0 1 0 0 0 0 −1 0 −1 0 0 0 0 0 0 −1 0 1 0 0 0 −1 −1 −1 −1 0 0 0 0 0 −1 0 0 1 0 0 0 −1 −1 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 −1 1 0 0 0 0 0 −2 0 0 0 0 0 0 −1 −1 1 0 0 0 0 0 −2 0 0 0 0 0 −1 −1 −1 0 0 0 0 0 0 −2 0 0 0 0 0 −2 −1 −1 1 0 0 0 0 0 −3 0 0 0 0 0 −1 −2 −1 1 0 0 0 0 0 −3 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
All Ti are prime and the codimension test succeeds. For the normality check, recall that by [1, Lemma IV.1.2.7], if R is an integral domain, and f ∈ R is prime, then R is normal if Rf is. We use this lemma repeatedly by localizing with respect to certain variables Ti until we reach a polynomial ring. We expect X → X0 to be a resolution but cannot show this at the moment as some smoothness tests were infeasible on our machines. Example 4.3 (Case Q8 ). In the case of G = Q8 in Proposition 4.1, applying Algorithm 2.8 to X0 = C4 /G, we obtain a modification X → X0 with Cl(X) = Z18 and the Cox ring R(X) = C[T1 , . . . , T28 ]/I with generators for I and the degree matrix given by 2 T 2 T6 T10 T17 21 + T2 T7 − T1 T9 , 2T8 T10 T18 T20 + T2 T4 − T1 T5 , 2 2 2 2 T3 T10 T11 T19 + T5 T7 − T4 T9 , T2 T3 T11 T19 + 2T8 T9 T18 T20 − T5 T6 T17 T21 , 2 2 2 2 2 T1 T3 T11 T19 + 2T7 T8 T18 T20 − T4 T6 T17 T21 , T3 T6 T16 T25 T28 + T8 T10 T18 T20 − T1 T5 , 2 2 2 2 2 T6 T8 T14 T24 T27 + T3 T10 T11 T19 − 2T4 T9 , T2 T8 T14 T24 T27 − 2T3 T9 T16 T25 T28 + T5 T10 T17 T21 , 2 2 2 2 2 2 T1 T8 T14 T24 T27 − 2T3 T7 T16 T25 T28 − T4 T10 T17 T21 , T5 T8 T15 T23 T26 − 2T6 T9 T16 T25 T28 + T2 T10 T11 T19 , 2 2 2 2 2 T4 T8 T15 T23 T26 − 2T6 T7 T16 T25 T28 − T1 T10 T11 T19 , T3 T8 T15 T23 T26 + T6 T10 T17 T21 − 2T1 T9 , 2 2 2 2 2 2 T20 , T24 T27 + 2T7 T10 T18 T23 T26 − T1 T6 T14 T20 , T3 T4 T15 T24 T27 − 2T9 T10 T18 T23 T26 − T2 T6 T14 T3 T5 T15 2 2 2 2 2 T82 T14 T18 T20 T24 T27 + T32 T11 T16 T19 T25 T28 − T4 T5 T17 T21 , 2 2 2 2 2 2 T52 T15 T17 T21 T23 T26 − T22 T11 T14 T19 T24 T27 − 4T92 T16 T18 T20 T25 T28 , 2 2 2 2 2 2 T20 T25 T28 , T18 T19 T24 T27 − 4T72 T16 T14 T21 T23 T26 − T12 T11 T17 T42 T15 2 2 2 2 2 2 2 T8 T15 T18 T20 T23 T26 − T6 T16 T17 T21 T25 T28 + T1 T2 T11 T19 , 2 2 2 2 2 T20 , T21 T24 T27 + 4T7 T9 T18 T17 T19 T23 T26 − T62 T14 T15 T32 T11 2 2 2 2 2 2 T62 T14 T16 T24 T25 T27 T28 + T10 T11 T18 T19 T20 − T4 T5 T15 T23 T26 , 2 2 2 2 2 2 T32 T15 T16 T23 T25 T26 T28 − T10 T17 T18 T20 T21 − T1 T2 T14 T24 T27 , 2 2 2 2 2 2 T82 T14 T15 T23 T24 T26 T27 − T10 T11 T17 T19 T21 − 4T7 T9 T16 T25 T28 , 2 2 2 2 2 2 T2 T4 T10 T11 T17 T19 T21 + T1 T5 T10 T11 T17 T19 T21 2 2 T25 T28 T25 T28 + 2T2 T4 T7 T9 T16 −2T2 T5 T72 T16 2 2 +2T1 T5 T7 T9 T16 T25 T28 − 2T1 T4 T92 T16 T25 T28 , 2 2 2 2 2 2 T21 T23 T26 T17 T19 T20 T21 − T4 T5 T15 T18 T17 T11 2T10 2 T2 T T T 2 2 +T1 T2 T11 14 19 24 27 + 4T7 T9 T16 T18 T20 T25 T28 , 2 2 4 2 2 2 2 2 T4 T5 T10 T15 T17 T21 T23 T26 + T1 T2 T10 T11 T14 T17 T19 T21 T24 T27 2 2 2 2 2 2 T16 T24 T25 T27 T28 T18 T20 T21 T25 T28 − 2T22 T72 T14 T17 +4T7 T9 T10 T16 2 2 2 2 2 2 +4T1 T2 T7 T9 T14 T16 T24 T25 T27 T28 − 2T1 T9 T14 T16 T24 T25 T27 T28 , 2 2 2 2 2 2 2 2 2T4 T5 T10 T15 T17 T18 T20 T21 T23 T26 − 2T1 T2 T10 T11 T14 T18 T19 T20 T24 T27 2 2 4 2 2 2 +8T7 T9 T10 T16 T18 T20 T25 T28 − T22 T42 T14 T15 T23 T24 T26 T27 2 2 2 2 2 2 +2T1 T2 T4 T5 T14 T15 T23 T24 T26 T27 − T1 T5 T14 T15 T23 T24 T26 T27 , 2 T2 T2 T2 T T T T 2 4 2 2 T4 T5 T10 11 15 17 19 21 23 26 + T1 T2 T10 T11 T14 T19 T24 T27 2 2 2 2 2 2 T23 T25 T26 T28 T16 T19 T20 T25 T28 − 2T52 T72 T15 T11 T16 T18 −4T7 T9 T10 2 2 2 2 T16 T23 T25 T26 T28 +4T4 T5 T7 T9 T15 T16 T23 T25 T26 T28 − 2T42 T92 T15
−1
−1 −1 0 0 0 0 0 0 −2 −1 0 −2 −1 0 −1 −2 0 −1 −2 0 −1 −2 −1 −1 −1 0 0 0 −1 0 −1 0 0 −1 −1 −1 0 0 −1 0 0 −1 0 0 −1 −1 0 0 0 0 −1 −1 −1 0 0 0 0 0 0 0 0 −1 −1 −2 −2 −2 −1 −1 −1 −1 −1 1 1 1 0 0 0 −1 −1 2 2 2 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 −1 0 1 0 2 1 2 1 0 0 1 0 2 1 1 1 0 0 1 1 1 2 1 −1 0 0 −1 −1 2 3 3 0 1 0 −1 −1 2 3 2 0 1 0 −1 0
0 −2 0 0 0 −1 0 −1 −1 −1 0 0 0 0 0 0 0 0
−1 −1 −1 0 0 0 −1 −1 −2 0 1 1 1 2 2 1 2 2
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 −2 2 0 0 0 0 0 −3 1 1 0 0 0 0 −2 0 0 1 0 0 0 −1 −1 0 0 1 0 0 −3 0 0 0 0 1 0 −2 −1 0 0 0 0 1 −3 1 0 0 0 0 0 −5 2 0 0 0 0 0 −4 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
.
As in Example 4.2 all Ti are prime, the codimension test is successful and we obtain normality of R by repeated by localizations. We expect X → X0 to be a resolution but not all smoothness tests were feasible on our machines.
COMPUTING RESOLUTIONS OF QUOTIENT SINGULARITIES
15
Remark 4.4. The groups considered in the two previous examples are symplectic. The first one, D8 , is conjugate to the group considered in [6, Section 5]. In particular, in this case the symplectic, i.e., crepant, resolutions exists, and their Cox ring is given in [6, Proposition 5.6]. Comparing it with the Cox ring of the resolution of C4 /D8 in Proposition 4.1, we see that our resolution is not crepant. The second example, Q8 , is not generated by symplectic reflections, hence by [24] there is no crepant resolution. It turns out, however, that after some changes in Algorithm 2.8, inspired by the results on the Cox ring of crepant resolutions for a group with similar structure [8, 11], we can obtain the Cox ring of the crepant resolution for D8 . Proposition 4.5. In the case of G = D8 in Proposition 4.1, applying Algorithm 2.8 to X0 = C4 /G, we obtain a resolution X → X0 with Cl(X) = Z2 and the Cox ring is R(X) = C[T1 , . . . , T12 ]/I where generators for I and the degree matrix are given by T5 T8 + T4 T9 − 2T6 T10 , T2 T5 − 2T3 T6 + T7 T9 , 2T3 T4 − T2 T6 − T7 T10 , T72 T12 − T22 + 4T1 T3 , T5 T7 T12 + T2 T9 − 2T3 T10 , 2 , T4 T5 T12 − T62 T12 + T8 T9 − T10 2T3 T7 T11 − T6 T9 + T5 T10 , 2T1 T7 T11 + T6 T8 − T4 T10 , 2T2 T3 T11 + T5 T6 T12 − T9 T10 , 2T1 T2 T11 + T4 T6 T12 − T8 T10 ,
h
−1 1
−1 1
−1 1
0 −1
0 −1
T3 T8 + T1 T9 − T2 T10 , 2T1 T5 − T2 T6 + T7 T10 , T2 T4 − 2T1 T6 − T7 T8 , T6 T7 T12 + 2T1 T9 − T2 T10 , T4 T7 T12 − T2 T8 + 2T1 T10 , T72 T11 + T4 T5 − T62 , T2 T7 T11 − T4 T9 + T6 T10 , 4T32 T11 + T52 T12 − T92 , 2 , 4T1 T3 T11 + T4 T5 T12 + T8 T9 − 2T10 4T12 T11 + T42 T12 − T82 0 −1
−1 0
0 0
0 0
0 0
2 −2
0 2
i
.
After a suitable linear change of coordinates one sees that this ring is isomorphic to the Cox ring of symplectic resolutions of considered representation of D8 computed in [6]. Proof. This is an application of Algorithm 2.3 where instead of forming Σ = {σ} ∩ Υ, we perform the following steps as in [11, Section 3]: There are three minimal elements σ1 , σ2 , σ3 of the set consisting of all P (γ0∗ ) such that γ0 is an F-face and P (γ0∗ ) is singular. We form the cone-wise intersections Σi = {σi } ∩ Υ and resolve the fans: Σ′i → Σi . Denoting by primit(Ξ(1) ) the primitive generators of the rays of a fan or cone Ξ, we have 3 [
i=1
(1) ′(1) = {v1 , v2 }. \ primit σi primit Σi
The remaining steps of Algorithm 2.8 deliver the result. All verifications succeed; as in Example 4.2, we repeatedly apply [1, Lemma IV.1.2.7] to obtain normality of the ring. References [1] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface. Cox rings, volume 144 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2014. [2] I. V. Arzhantsev and S. A. Ga˘ıfullin. Cox rings, semigroups, and automorphisms of affine varieties. Mat. Sb., 201(1):3–24, 2010. [3] V. V. Batyrev. Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc. (JEMS), 1(1):5–33, 1999. [4] T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas. Computing tropical varieties. J. Symbolic Comput., 42(1-2):54–73, 2007. [5] W. Decker, G.-M. Greuel, G. Pfister, and H. Sch¨ onemann. Singular 4-0-3 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2016. [6] M. Donten-Bury and M. Grab. Cox rings of some symplectic resolutions of quotient singularities. 2015. Preprint. arXiv:1504.07463.
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[7] M. Donten-Bury and S. Keicher. quotsingcox.lib – a Singular library to compute Cox rings of quotient singularities, 2016. www.mathematik.uni-tuebingen.de/˜ keicher/quotsingcox/. [8] M. Donten-Bury and J. A. Wi´sniewski. On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32. 2014. Preprint. arXiv:1409.4204. [9] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.7.5, 2014. [10] J. Hausen. Cox rings and combinatorics. II. Mosc. Math. J., 8(4):711–757, 847, 2008. [11] J. Hausen and S. Keicher. A software package for Mori dream spaces. LMS J. Comput. Math., 18(1):647–659, 2015. [12] J. Hausen, S. Keicher, and A. Laface. Computing Cox rings. Math. Comp., 85(297):467–502, 2016. [13] J. Hausen and M. Wrobel. Non-complete rational T-varieties of complexity one. 2015. Preprint. arXiv:1512.08930. [14] E. Huggenberger. Fano Varieties with Torus Action of Complexity One. PhD thesis, Universit¨ at T¨ ubingen, Wilhelmstr. 32, 72074 T¨ ubingen, 2013. [15] Y. Ito and M. Reid. The McKay correspondence for finite subgroups of SL(3, C). In Higherdimensional complex varieties (Trento, 1994), pages 221–240. de Gruyter, Berlin, 1996. [16] D. Kaledin. McKay correspondence for symplectic quotient singularities. Invent. Math., 148(1):151–175, 2002. [17] S. Keicher. Computing the GIT-fan. Internat. J. Algebra Comput., 22(7):1250064, 11, 2012. [18] S. Keicher. Algorithms for Mori Dream Spaces. PhD thesis, Universit¨ at T¨ ubingen, 2014. [19] H. Kraft. Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. [20] D. Maclagan and B. Sturmfels. Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015. [21] M. Reid. La correspondance de McKay. Ast´ erisque, (276):53–72, 2002. S´ eminaire Bourbaki, Vol. 1999/2000. [22] J. Tevelev. Compactifications of subvarieties of tori. Amer. J. Math., 129(4):1087–1104, 2007. [23] J. Tevelev. On a question of B. Teissier. Collect. Math., 65(1):61–66, 2014. [24] M. Verbitsky. Holomorphic symplectic geometry and orbifold singularities. Asian J. Math., 4(3):553–563, 2000. [25] R. Yamagishi. On smoothness of minimal models of quotient singularities by finite subgroups of SLn (C). 2016. Preprint. arXiv:1602.01572. University of Warsaw, Institute of Mathematics, Banacha 2, 02-097 Warszawa, Poland ¨ t Berlin, Mathematisches Institut, Arnimallee 3, 14195 Berlin, Ger& Freie Universita many E-mail address: [email protected] Departamento de Matematica, Facultad de Ciencias Fisicas y Matematicas, Universi´ n, Casilla 160-C, Concepcio ´ n, Chile dad de Concepcio E-mail address: [email protected]
