HILBERT SCHEMES OF ABELIAN GROUP ORBITS

IKU NAKAMURA

To the memory of Professor Nobuo Sasakura

Abstract. We construct a crepant resolution of the quotient variety A^{3}/G in the canonical manner when G is an abelian subgroup of SL(3, k).

0. Introduction

Let k be any algebraically closed field and G a finite subgroup of SL(3, k) of order prime to the characteristic of k. The group G acts on the affine 3-space A^{3} and the quotient space A^{3}/G is a normal Gorenstein variety with trivial canonical sheaf.

No natural choice of a crepant smooth resolution of A^{3}/G seems to be known in the general theories such as the theory of minimal models in birational geometry and the theory of torus embeddings. In this article we give a crepant smooth resolution of A^{3}/G canonical in some sense when G is abelian. The G-Hilbert scheme (or The G-orbit Hilbert scheme) Hilb^{G} := Hilb^{G}(A^{3}) introduced in [IN96] is in fact such a resolution. The G-Hilbert scheme is roughly speaking a largest connected subset of the set consisting of all G-invariant zero-dimensional subschemes of A^{3} of length n := |G|, the order of G. The zero-dimensional subschemes of A^{3} of length n are something which substitute for n-points in A^{3}, therefore those Ginvariant zero-dimensional subschemes of length n are substitutes for Gorbits in A^{3} of length n. Hence Hilb^{G} will be a substitute of A^{3}/G, the space of G-orbits. We will prove that for G abelian Hilb^{G} is a smooth torus embedding associated to a certain fan in R^{3} with apices junior elements of G (Theorem 4.2). As a corollary to it, we will see

Theorem 0.1. For any abelian subgroup G of SL(3, k) of order prime to the characteristic of k, Hilb^{G}(A^{3}) is a crepant smooth resolution of A^{3}/G.

Theorem 0.1 might sound a little unexpected because Hilb^{n}(A^{3}) is known to be very singular. We conjecture that the same is true for any finite subgroup G of SL(3, k) if the order of G is prime to the characteristic of k. The first half of the present article is devoted to describing Hilb^{G} as a toric variety in arbitrary dimension. In the second half of it we will prove that if G is an abelian subgroup of SL(3, k), the fan associated with Hilb^{G} is nonsingular, that is, Hilb^{G} is smooth. Some examples will be given in Section 5 and Section 6. See also [Reid97] and Ito-Nakajima [INkjm98].

Date: 25 September 1998.

1991 Mathematics Subject Classification. Algebraic Geometry, Primary 14M25; Secondary 14L30, 14C99.

Key words and phrases. Crepant resolution, G-orbit, Hilbert scheme, McKay correspondence, Toric variety, Quotient singularity.

Research was supported in part by the Grant-in-aid (No. 08404001) for Scientific Research, the Ministry of Education.

Acknowledgement. We would like to thank Professors M. Reid, H. Nakajima and Y. Ito for discussions during the preparation of the article. The present article was completed during the author’s stay in 1998 at Max-Planck Institut für Mathematik in Bonn. We would like to thank all the people in the institute for their hospitality.

1. Lattices and G-graphs

1.1. Abelian subgroups. Let G be a finite abelian subgroup of GL(r, k) of order prime to the characteristic of k and n the order of G. Without loss of generality we may assume that any element of G is a diagonal matrix.

Let μ_{n} be the group of n-th roots of unity and G_{m} the group of units in k. We choose and fix a primitive n-th root ζ of unity. Given a set F we denote its cardinality by |F |. We denote by Z_{0} (resp. R_{0}) the set of all nonnegative integers (resp. nonnegative real numbers).

Definition 1.2. Let N_{0} := ⊕_{i=1}^{r}Ze_{i} be a free Z module with e_{i} a Z-basis, M_{0} := Hom_{Z}(N_{0}, Z) = ⊕_{i=1}^{r}Zf_{i} and Δ := ∑_{i=1}^{r} R_{0}e_{i} where f_{i}(e_{j}) = δ_{ij}. In what follows we assume that any element of the group G is a diagonal r × r-matrix, therefore we denote it by an r-vector consisting of r diagonal coefficients. Given a nontrivial element g of G, we write g = (ζ^{a_{1}}, ⋯, ζ^{a_{r}}) for some 0 < a_{i} < n and identify it with an r-vector (a_{1}/n)e_{1} + ⋯ + (a_{r}/n)e_{r} ∈ N_{0} ⊗_{Z} Q. We define N := N_{0} +∑_{g∈G} Zg and M = Hom_{Z}(N, Z). We note N/N_{0} ≃ G and M_{0}/M ≃ G^{∨}. In fact, we see that e^{2α(g)πi} ∈ μ_{n} is well-defined for α ∈ M_{0}/M and g ∈ G, which gives a nondegenerate pairing N/N_{0} × M_{0}/M → μ_{n}. This pairing enables us to identify G^{∨} with M_{0}/M. We call this isomorphism weight and denote it by wt : M_{0}/M → G^{∨}. We denote a ≡ b for a, b ∈ M_{0} if wt(a) = wt(b), as is the same, a − b ∈ M.

1.3. The semigroup S(I). Let M_{0}^{0} (resp. M_{0}^{+}) be the semigroup generated by 0 and f_{i} (resp. f_{i}) (1 ≤ i ≤ r).

Since by the semigroup homomorphism f_{i} ↦ x_{i} there is an obvious bijection between the set of subsemigroups of M_{0}^{0} and the set of monomial subsemigroups in the polynomial ring k[A^{r}] = k[x_{1}, ⋯, x_{r}], we will often identify them if no confusion is possible.

Suppose that an ideal I of k[A^{r}] is generated by monomials. The monomial generators of I, in an obvious manner, give rise to a subsemigroup S(I) of k[A^{r}]. Let S(I)^{⊥} be the set of monomials in x_{i} which do not belong to S(I). Then we easily see that 1 ∉ S(I), fS(I) ⊂ S(I) for any monomial f.

From a certain reason we will explain in Subsection 2.1, we are particularly interested in those I satisfying the condition (ii) in Definition 1.4. Content-Length: 68561 X-Lines: 2130 Status: RO

Definition 1.4. For a subsemigroup S of M_{0}^{+} we define a subset S^{⊥} of M_{0}^{0} by S^{⊥} := M_{0}^{0} \ S. The subset S^{⊥} of M_{0}^{0} is called a G-graph if the following conditions are true:

(i) S + M_{0}^{+} ⊂ S, or as is equivalent, if a + b ∈ S^{⊥} for some a, b ∈ M_{0}^{0}, then a ∈ S^{⊥} and b ∈ S^{⊥},

(ii) wt : S^{⊥} → G^{∨} is a bijection where wt(α)(g) = α(g) (α ∈ S(I)^{⊥}, g ∈ G).

We denote S^{⊥} by Γ(S^{⊥}) or simply by Γ. We note 0 ∈ Γ(S^{⊥}) because S ⊂ M_{0}^{+}. If G ⊂ SL(r, k), then f_{1} + f_{2} + ⋯ + f_{r} ∉ Γ(S^{⊥}). Therefore Γ(S^{⊥}) consists of at most r (r − 1) -dimensional graphs Γ_{k} := Γ(S^{⊥}) ∩ (∑_{i≠k} Rf_{i}).

Definition 1.5. Let Γ be a G-graph. We define the weight map wt_{Γ} : M_{0}^{0} → Γ as follows. For any u ∈ M_{0}^{0} there exists a unique element v ∈ Γ such that wt(v) ≡ wt(u). We denote v by wt_{Γ}(u). Then we define a cone σ(Γ), a subsemigroup S(Γ) of M and a Z-submodule M(Γ) of M by

σ(Γ) := {α ∈ N_{0} ⊗_{Z} R; α(v − wt_{Γ}(v))) ≥ 0, ∀v ∈ M_{0}^{0}},
S(Γ) := ∑_{v∈M_{0}^{0}} Z_{0}(v − wt_{Γ}(v)) (⊂ M),

M(Γ) := ∑_{v∈M_{0}^{0}} Z(v − wt_{Γ}(v)) (⊂ M).

Lemma 1.6. For any finite abelian subgroup G of GL(r, k), there exists a G-graph.

Proof. If G is cyclic, then let g = (⋯, 1/n) be a generator of G. Let Γ be a G-graph with Γ = {xf_{r}; x ∈ [0, n − 1]}. Then it is clear that Γ is a G-graph. If G is not cyclic, then we prove Lemma by the induction on r. We may assume there exists a g ∈ G such that g = (⋯, 1/ℓ) is of the maximal order ℓ in G. Then there exists a subgroup H of G ∩ GL(r − 1, k) such that G/H (≃ Z/ℓZ) is generated by the image of g. By the induction hypothesis there exists an H-graph Γ′ ⊂ ⊕_{i=1}^{r−1}Zf_{i}. Let Γ := Γ′ × ([0, ℓ − 1]f_{r}). Then it is clear that Γ is a G-graph.

Lemma 1.7. Let Γ be a G-graph. Then S(Γ) is a finitely generated semigroup and M(Γ) = M.

Proof. Though it seems clear, we give a proof. First we note that M(Γ) ⊂ M and M ∩ M_{0}^{0} ⊂ M(Γ) because wt_{Γ}(v) = 0 for v ∈ M ∩ M_{0}^{0}. Since M ∩ M_{0}^{0} generates M as a group, M ⊂ M(Γ). It follows M(Γ) = M. There exists a finite subset A of M_{0}^{0} such that M_{0}^{0} is generated by A and M ∩ M_{0}^{0} as a semigroup. Any w ∈ M_{0}^{0} is written as w = v + a (∃a ∈ A, v ∈ M ∩ M_{0}^{0}).

Therefore

w − wt_{Γ}(w) = v + a − wt_{Γ}(a)

because wt_{Γ} v = 0. Since M ∩ M_{0}^{0} is finitely generated, S(Γ) is finitely generated.

Lemma 1.8. Let Γ be a G-graph and A a finite subset of M_{0}^{0} \ Γ such that M_{0}^{0} \ Γ = A + M_{0}^{0}. Then S(Γ) is generated by v − wt_{Γ}(v) (v ∈ A) as a semigroup.

Proof. Let λ(v) := v − wt_{Γ}(v) and we denote by S^{∗}(λ(A)) the semigroup generated by λ(A). First we see nf_{i} ∈ σ^{∨}(Γ) ∩ M where n = |G| because wt_{Γ}(nf_{i}) = 0. Hence σ(Γ) ⊂ Δ. Next we notice that σ(Γ) is a g-dimensional cone, that is, there is a solution α ∈ Int(σ(Γ)) to the strict inequalities

α(λ(v)) > 0 (∀v ∈ M_{0}^{0} \ Γ). (1)

We choose and fix such an α ∈ Int(σ(Γ)). Then since σ(Γ) ⊂ Δ, we see that α(v) > 0 for any nonzero v ∈ M_{0}^{0} and the subset {v ∈ M_{0}^{0}; α(v) < c} is finite for any constant c.

Now we prove our Lemma by the induction on the value α(z) (z ∈ M_{0}^{0}\Γ). If z ∈ A, then λ(z) ∈ S^{∗}(λ(A)). Let z ∈ M_{0}^{0}. By the assumption there exist v ∈ A and b ∈ M_{0}^{0} such that z = v + b. We see

λ(z) = λ(v) + λ(wt_{Γ}(v) + b). (2)

We easily see α(z) = α(v) + α(b) > α(wt_{Γ}(v) + b). By the induction hypothesis λ(wt_{Γ}(v) + b) ∈ S^{∗}(λ(A)). This completes the proof.

Definition 1.9. For any G-graph Γ we define

U_{σ(Γ)} := Spec k[σ^{∨}(Γ) ∩ M],
V (Γ) := Spec k[S(Γ)],
I(Γ) := (w^{v}; v ∉ Γ) ⊂ k[A^{r}],
I^{vers}(Γ) := (w^{v} − s^{v−wt_{Γ}(v)}w^{wt_{Γ}(v)}; v ∈ M_{0}^{0})

⊂ k[S(Γ)] ⊗_{k} k[A^{r}] ⊂ k[σ^{∨}(Γ) ∩ M] ⊗_{k} k[A^{r}]

where s^{v} (v ∈ M) is the torus coordinate of the toric varieties U_{σ(Γ)} and V (Γ). Let Z(Γ) (resp. Z^{vers}(Γ)) be a G-invariant subscheme of A^{r} (resp. a G-invariant subscheme of A_{V (Γ)}^{r}) defined by the ideal I(Γ) (resp. I^{vers}(Γ)).

2. The G-orbit Hilbert scheme

2.1. The G-orbit Hilbert scheme. Let G be a finite subgroup of GL(r, k) and n = |G|. Let Hilb^{n}(A^{3}) be the Hilbert scheme of n points in A^{r} and S^{n}(A^{r}) the n-th symmetric product of A^{r}. We have a natural morphism π_{n} from Hilb^{n}(A^{r}) onto S^{n}(A^{r}). Since π_{n} is G-equivariant, we have a natural morphism between their G-fixed point sets. We see easily that the quotient variety A^{r}/G is one of the irreducible components of the G-fixed point set S^{n}(A^{r})^{G−inv} with reduced structure. Then the G-orbit Hilbert scheme Hilb^{G} := Hilb^{G}(A^{r}) is by definition the unique irreducible component, endowed with reduced structure, of the G-fixed point set of Hilb^{n}(A^{r}) which dominates A^{r}/G by the map π_{n}. Let π be the natural morphism from Hilb^{G}(A^{r}) onto A^{r}/G. We call a zero-dimensional subscheme (resp. a G-invariant subscheme) Z of A^{r} of length n a cluster (resp. a G-cluster). A G-cluster Z is smoothable if and only if Z ∈ Hilb^{G}(A^{r}). Let I be the ideal defining the (G-invariant) smoothable cluster. Then we write I ∈ Hilb^{n}(A^{r}) resp. I ∈ Hilb^{G}(A^{r}) since there seems to be no fear of confusion.

For instance, if r = 3 and G is generated by g = (1/6, 1/2, 3/6), then (x^{6}, y, z), (x^{2}, y^{3}, z), and (x^{2}, xy, y^{2}, z^{2}) are examples of ideals generated by monomials in Hilb^{G}(A^{3}). See Table 2.

By [IN98, Lemma 9.4] if I ∈ Hilb^{G}(A^{r}), then the quotient module k[A^{r}]/I is isomorphic to the group algebra k[G] (the regular representation of G) as a G-module. In other words, dim(k[A^{r}]/I)^{ρ} = 1 (∀ρ ∈ G^{∨}) where (k[A^{r}]/I)^{ρ} is the ρ-eigensubspace of k[A^{r}]/I for any character ρ ∈ G^{∨}. This is just the condition (ii) in Definition 1.4.

Definition 2.2. Let Hilb_{norm}^{G}(A^{r}) be the normalization of Hilb^{G}(A^{r}). Since Hilb_{norm}^{G}(A^{r}) is a normal torus embedding, it is covered with finitely many toric charts. To each chart there corresponds a cone of N ⊗ Q. Since Hilb^{|G|}(A^{r}) is projective over S^{|G|}(A^{r}), so is Hilb^{G}(A^{r}) over A^{r}/G. Therefore there exists a fan {σ} with support Δ and a covering of Hilb^{G}(A^{r}) consisting of toric charts U_{σ} with dim σ = r, each of which has a unique zero-dimensional toric stratum corresponding to a G-cluster Z_{σ} ∈ Hilb^{G}(A^{r}) defined by a monomial G-ideal I_{σ} of k[A^{r}]. There exists a G-graph Γ such that Z_{σ} = Z(Γ), I_{σ} = I(Γ), equivalently σ = σ(Γ).

Lemma 2.3. Assume that Γ is a G-graph. Let Z^{vers} := Z^{vers}(Γ) and V := V (Γ) in Definition 1.9. Let Z_{s}^{vers} be a closed fibre of Z^{vers} over s ∈ V (k). Then

(i) Z^{vers} is V -flat and Z_{s}^{vers} ∈ Hilb^{G} for any s ∈ V (k). In particular, Z(Γ) ∈ Hilb^{G}.

(ii) Γ is a k-basis of k[A^{r}]/I^{vers}(Γ) ⊗ k(s) for any s ∈ V (k). (iii) At any point s ∈ V (k) Z^{vers} is versal for flat G-equivariant defor-

mations of Z_{s}^{vers} with generic support, that is, whose support is not contained in A^{r} \ G_{m}^{r}.

Proof. The scheme Z as well as V admits a torus action w^{v} ↦ λ^{v}w^{v}, s^{v−wt_{Γ}(v)} ↦ λ^{v−wt_{Γ}(v)}s^{v−wt_{Γ}(v)} (∀v ∈ M_{0}^{0}, λ ∈ G_{m}^{r}). For s ∈ G_{m}^{r}(k), the number of solutions of the system of the equations

w^{v} = s^{v−wt_{Γ}(v)}w^{wt_{Γ}(v)} (∀v ∈ M_{0}^{0}) (3)

is equal to |Γ| = |G| by Lemma 1.7. Hence Z is flat over G_{m}^{r} (⊂ V). Let B be the subscheme of V consisting of all s ∈ V such that dim_{k} k[A^{r}]/I^{vers}(Γ) ⊗ k(s) ≥ |G| + 1. The subset B is invariant under λ ∈ G_{m}^{r}, whence if B ≠ ∅, then B contains s = 0, the unique 0-stratum of V because B is closed. However dim_{k} k[A^{r}]/I(Γ) = |G| because Γ is a G-graph. It follows that B = ∅, Z is V -flat. In particular, Z_{s} is smooth for s ∈ G_{m}^{r} and Z_{s} ∈ Hilb^{G} for any s ∈ V (k). This proves (i).

Let B′ = {s ∈ V; Γ is not a k(s)-basis of k[A^{r}]/I^{vers}(Γ) ⊗ k(s)}. Then B′ is a closed G_{m}^{r}-invariant subscheme of V. It is easy to see that 0 ∉ B′. This shows B′ = ∅, which proves (ii).

Let R be a complete local domain with R/m ≃ k and K the quotient field, m the maximal ideal of R. Choose and fix s_{0} ∈ V (k). Let X be an R-flat (embedded) G-equivariant deformation of Z_{s_{0}} with generic support, in other words, the support of X_{K} is not contained in A_{K}^{r} \ G_{m,K}^{r}. Let J be a G-invariant ideal of Γ(O_{A_{R}^{r}}) defining X. Since X is flat and X_{0} = Z_{s_{0}}, Γ is a free R-basis of Γ(O_{X}) ≃ R ⊗ k[A^{r}]/J. Therefore for any w^{v} there exist a unique γ ∈ Γ and c ∈ R such that w^{v} = c · γ mod J. In other words, w^{v} − cw^{wt_{Γ}(v)} ∈ J. Moreover J is generated by the elements of this form because Γ(O_{X}) ≃ R ⊗ k[A^{r}]/J. Hence we see that

(4)  J = (w^{v} − q_{v}w^{wt_{Γ}(v)} (∀v ∈ M_{0}^{0} \ Γ))

for some q_{v} ∈ R. Suppose that we are given v_{j}, u_{i} ∈ M_{0}^{0} such that

∑_{j}(v_{j} − wt_{Γ}(v_{j})) = ∑_{i}(u_{i} − wt_{Γ}(u_{i})). (5)

Then we prove

∏_{j} q_{v_{j}} = ∏_{i} q_{u_{i}}. (6)

In fact, let a := ∑_{j} wt_{Γ}(v_{j}) and b := ∑_{i} wt_{Γ}(u_{i}). Then

w^{a+b} ∏_{j} q_{v_{j}} = w^{b} ∏_{j} w^{v_{j}} = w^{a} ∏_{i} w^{u_{j}} = w^{a+b} ∏_{i} q_{u_{j}} mod J.

Since X is a generic-support-deformation, the relation (6) follows. This proves (iii).

Corollary 2.4. Z(Γ) is smoothable for any G-graph Γ. The map I ↦ Γ(S(I)^{⊥}) is a bijection between the set of I ∈ Hilb^{G}(A^{r}) generated by monomials and the set of G-graphs.

See also [INkjm98] for the case where G ⊂ SL(3, k).

Lemma 2.5. Let Γ be a G-graph and σ(Γ) the same as before. Let τ be a one-codimensional face of σ(Γ). Then there exists v^{∗} := v_{−} − v_{+} ∈ M indivisible in M by any positive integer ≥ 2 such that v_{−} ∈ M_{0}^{0}, v_{+} = wt_{Γ}(v_{−}) ∈ Γ and v_{±} has no common factors in M_{0}^{0} ^{1}, τ = σ(Γ) ∩ (v^{∗})^{⊥} and σ(Γ) ⊂ {v_{−} ≥ v_{+}}, v^{∗} is a generator of the lattice τ^{⊥} ∩ M.

1This means that there is no nonzero u ∈ M_{0}^{0} such that v_{±} − u ∈ M_{0}^{0}

Proof. Since τ is one codimensional in σ(Γ), there exists v^{∗} := v_{−} − v_{+} ∈ M such that v_{+} = wt_{Γ}(v_{−}) and τ = σ(Γ) ∩ {v^{∗}}^{⊥}, σ(Γ) ⊂ {v_{−} ≥ v_{+}} for some v_{±} ∈ M_{0}. We note that if v^{∗} := v_{−} − v_{+} ∈ M such that v_{−} ∈ M_{0}^{0}, v_{+} = wt_{Γ}(v_{−}) ∈ Γ and if v_{±} has a common factor u, then v_{+}−u ∈ Γ because Γ is a G-graph. Similarly v_{−} − u ∈ M_{0}^{0} and wt_{Γ}(v_{−} − u) = v_{+} − u. Hence without loss of generality we may assume that v_{±} has no factors in common. Moreover we may assume that v_{−} − v_{+} is primitive in M, that is, v_{−} − v_{+} is divisible in M by no positive integer ≥ 2. In fact, suppose that there exist e_{±} ∈ M_{0}^{0} with no common factors and a positive integer ℓ such that v_{−} − v_{+} = ℓ(e_{−} − e_{+}) and e_{−} − e_{+} ∈ M. It follows that v_{−} + ℓe_{+} = v_{+} + ℓe_{−} and v_{±} = ℓe_{±}. Since Γ is a G-graph, e_{+} ∈ Γ whence e_{−} ∉ Γ and wt_{Γ}(e_{−}) = e_{+} because e_{−} − e_{+} ∈ M. Moreover τ = σ ∩ {e_{−} = e_{+}}, σ ⊂ {e_{−} ≥ e_{+}}. Thus we may assume v_{−} − v_{+} is divisible in M by no positive integer ≥ 2. This also shows that v^{∗} is a generator of the lattice τ^{⊥} ∩ M. This completes the proof of Lemma.

Definition 2.6. Let Γ be a G-graph, v^{∗}, v_{+} and v_{−} as in Lemma 2.5. Suppose v_{±} ≠ 0. A G-igsaw transform Γ′ of Γ is then defined to be

G_{ig}(v_{+}, v_{−})(v) = v + c_{max}^{+}(v)(v_{−} − v_{+}) ∈ Γ′, (7)

Γ′ := G_{ig}(v_{+}, v_{−})(Γ) (8)

where c_{max}^{+}(v) := max{c ∈ Z; v − cv_{+} ∈ M_{0}^{0}}.

We will see in Lemma 2.8 that Γ′ is also a G-graph adjacent to Γ in the sense that σ(Γ) and σ(Γ′) are adjacent, i.e. have a one-codimensional face in common.

2.7. One parameter family of clusters. Let 𝔪 and 𝔫 be respectively the maximal ideal of k[A^{r}] at the origin, and the ideal of k[A^{r}] generated by all G-invariant monomials vanishing at the origin.

Let I ∈ Hilb^{G}(A^{r}) and {I(q); q ∈ A^{1}} a k[q]-flat G-equivariant deformation of I. Suppose I(q) ⊂ 𝔪. Then 𝔫 ⊂ I(q) because k[A^{r}]/I(q) ≃ k[G], hence the G-invariant part of k[A^{r}]/I(q) is spanned by constant functions. Then dim_{k} k[A^{q}]/I(q) is constant so that it determines a natural morphism ϕ : A^{1} → Gr into the Grassmann variety Gr := Gr(V, n) of n-codimensional subspaces of V where V := k[A^{r}]/𝔫, n = |G|. Let B := ϕ(A^{1}), B̄ the closure of B and ϕ(∞) := B̄ \ B. Since ϕ(∞) ∈ Gr, it determines a unique ideal I_{ϕ(∞)} of k[A^{r}] such that ϕ(∞) = I_{ϕ(∞)}/𝔫 ∈ Gr.

Lemma 2.8. Let Γ be a G-graph, τ and v^{∗} := v_{−} − v_{+} the same as in Lemma 2.5. Suppose v_{±} ≠ 0. Define a deformation {I(Γ, v^{∗})(q)}_{q∈A^{1}} of I(Γ) in the v^{∗}-direction by

(9) I(Γ, v^{∗})(q) = (w^{v_{−}} − qw^{v_{+}}, w^{v} (v ∈ M_{0}^{0} \ Γ, v − wt_{Γ}(v) ∉ τ^{⊥})).

Then I(Γ, v^{∗})(∞) := (lim_{q→∞} I(Γ, v^{∗})(q)/𝔫) + 𝔫 = I(G_{ig}(v_{+}, v_{−})(Γ)). In particular, G_{ig}(v_{+}, v_{−})(Γ) is a G-graph.

Proof. First we prove I(Γ, v^{∗})(0) = I(Γ). Let σ = σ(Γ) and J(Γ) := I(Γ, v^{∗})(0). Let s^{v} (v ∈ M) be the torus coordinate of the torus embedding U_{σ} := Spec k[σ^{∨} ∩ M]. Let q = s^{v_{−}−v_{+}}. It is clear that J(Γ) ⊂ I(Γ). Conversely let v ∈ M_{0}^{0} \ Γ. If v − wt_{Γ}(v) ∉ τ^{⊥}, then w^{v} ∈ J(Γ). Assume v − wt_{Γ}(v) ∈ τ^{⊥}. It follows that there exists n ≥ 1 such that v − wt_{Γ}(v) = n(v_{−} − v_{+}). Hence v − nv_{−} ∈ M_{0}^{0} and wt_{Γ}(v) − nv_{+} ∈ M_{0}^{0}. Hence w^{v} = w^{nv_{−}}w^{v−nv_{−}} ∈ w^{nv_{−}}k[A^{r}] ⊂ J(Γ). Therefore I(Γ) = J(Γ).

Let Γ′ := G_{ig}(v_{+}, v_{−})(Γ). Then we see

M_{0}^{0} \ Γ′ = {v ∉ Γ, v − wt_{Γ}(v) ∉ τ^{⊥}} ∪ {v ∈ Γ; v − v_{+} ∈ M_{0}^{0}}. (10)

We can easily prove that I(Γ, v^{∗})(q) is coflat over A_{v^{∗}}^{1} ≃ Spec k[q] and that Γ is a k-basis of k[A^{3}]/I(Γ, v^{∗})(q). Since I(Γ, v^{∗})(q) ⊂ 𝔪, this implies that 𝔫 ⊂ I(Γ, v^{∗})(q). Therefore we can define I(Γ, v^{∗})(∞) := lim_{q→∞}(Γ, v^{∗})(q) as in Subsection 2.7. Hence 𝔫 ⊂ I(Γ, v^{∗})(∞) Since I(Γ, v^{∗})(∞) is G_{m}^{r}- invariant, there is a G-graph Γ′′ such that I(Γ, v^{∗})(∞) = I(Γ′′). We prove Γ′ = Γ′′. Since |Γ′| = |Γ′′| = |G|, it suffices to prove Γ′′ ⊂ Γ′. If v ∉ Γ and v − wt_{Γ}(v) ∉ τ^{⊥}, then w^{v} ∈ I(Γ, v^{∗})(q) for any q, hence w^{v} ∈ I(Γ, v^{∗})(∞). If v ∈ Γ and v − v_{+} ∈ M_{0}^{0}, then w^{v} = w^{v−v_{+}}w^{v_{+}} ∈ I(Γ, v^{∗})(∞). Hence by (10) (M_{0}^{0} \ Γ′) ⊂ (M_{0}^{0} \ Γ′′). Hence Γ′′ ⊂ Γ′. It follows that I(Γ, v^{∗})(∞) = I(G_{ig}(v_{+}, v_{−})(Γ)). This completes the proof.

Corollary 2.9. Let Γ be a G-graph and Γ′ := G_{ig}(v_{+}, v_{−})(Γ). Let v^{∗} := v_{−} − v_{+}, v_{+} ∈ Γ, v_{−} ∈ Γ′ and h := s^{v_{−}−v_{+}} the same as before. Then k[S(Γ)][1/h] ≃ k[S(Γ′)][h].

Proof. For any v ∈ M_{0}^{0}, there exists a unique integer n ≥ 0 such that wt_{Γ′}(v) = wt_{Γ}(v)+n(v_{−}−v_{+}), that is, v−wt_{Γ}(v) = v−wt_{Γ′}(v)+n(v_{−}−v_{+}). In particular, v_{+} − wt_{Γ′}(v_{+}) = v_{+} − v_{−}. Hence we have

(11)  w^{v−wt_{Γ}(v)} = w^{v−wt_{Γ′}(v)}w^{n(v_{−}−v_{+})}
(12)  w^{v_{−}−v_{+}} = (w^{(v_{+}−v_{−})})^{−1}.

This proves the isomorphism.

Remark 2.10. Thus by Corollary 2.9 we can glue together all V (Γ) (Γ ∈ Graph(G)) to obtain an irreducible variety W (G) := T_{M}(Graph(G)). We see also that there is a zero-dimensional (universal) subscheme Z^{vers} over W (G) such that Z^{vers} ×_{W (G)} V (Γ) ≃ Z^{vers}(Γ). The torus embedding T_{M}(Fan(G)) is the normalization of W (G). It is not known whether V (Γ) as well as Hilb^{G} is normal.

Theorem 2.11. Let G be a finite abelian subgroup of GL(r, k) whose order is prime to the characteristic of k. Let Graph(G) be the set of all G-graphs and Fan(G) the set of all σ(Γ) with Γ ∈ Graph(G). Then

(i) Fan(G) is a finite fan with its support Δ, (ii) Fan(G) is obtained from a G-graph by G-igsaw tranformations, (iii) T_{M}(Graph(G)) ≃ Hilb^{G}(A^{r}) and it is projective over A^{r}/G, (iv) T_{M}(Fan(G)) is the normalization of Hilb^{G}(A^{r}), projective over A^{r}/G.

Proof. First we notice that A^{r}/G ≃ Spec k[Δ^{∨} ∩ M], the latter of which we

denote by T_{M}(Δ) where Δ^{∨} is the dual cone of Δ in M_{R} := M ⊗ R.

Now we prove (iii) and (iv). In fact, let W (G) = T_{M}(Graph(G)) and X the normalization of it. By the universality of Hilb(A^{r}) the subscheme Z^{vers} gives rise to a morphism h : W (G) → Hilb^{G} := Hilb^{G}(A^{r}). We prove that h is an isomorphism. Let H be the normalization of Hilb^{G}. Since Hilb^{G} hence H admits a T_{M}-action, H is a normal torus embedding. Hence we have a sequence of morphisms of normal torus embeddings

X →^{h̃} H → A^{r}/G

where h̃ is the morphism induced from h : X → Hilb^{G}. If h̃ is not an isomorphism, some toric 1-strata will be contracted by h̃, hence by h because X is finite over W (G). This contradicts Lemma 2.3 (ii). Hence X ≃ H, whence W (G) is finite and generically bijective over Hilb^{G}. Since W (G) is locally isomorphic to Hilb^{G} by Lemma 2.3 (ii), W (G) ≃ Hilb^{G}. Hence W (G) is projective over A^{r}/G. The normalization T_{M}(Fan(G)) of W (G) is also projective over A^{r}/G because it is finite over W (G).

Next we prove (i) and (ii). Since T_{M}(Fan(G)) ≃ Hilb^{G}(A^{r}) is projective over A^{r}/G ≃ T_{M}(Δ), the support of Fan(G) is exactly Δ. Let B := nM_{0}^{+} + M_{0}^{0}. Since w^{v} ∈ I(Γ) (∀v ∈ B) for any G-graph Γ, Γ is contained in the fixed finite subset M_{0}^{0} \ B of M_{0}^{0}. Therefore there are only finitely many G-graphs. Hence Fan(G) is a finite fan with its support Δ. For any G-graph Γ, the subscheme Z(Γ) is smoothable by Lemma 2.3. Hence by Lemmas 2.5 and 2.8 and the connectivity of Δ any G-graph is obtained from a given G-graph by G-igsaw transformations.

3. G-graphs in dimension three

In what follows we study Hilb^{G}(A^{3}) when G ⊂ SL(3, k).

3.1. G-admissible fans. Let n = |G| and g = (a/n, b/n, c/n) ∈ G such that 1 ≤ a, b, c ≤ n − 1. Then the sum (a + b + c)/n is equal to either 1 or 2. If it is equal to 1 (resp. 2), then we call g junior (resp. senior). Let Jun(G) be the set of all junior elements in G, and Jun^{∗}(G) := {e_{1}, e_{2}, e_{3}} ∪ Jun(G).

We notice A^{3}/G ≃ T_{M}(Δ), a torus embedding. We subdivide Δ into simplicies with apices in Jun^{∗}(G) in an arbitrary manner to get a fan F_{AN}. F_{AN} is called G-admissible if any 3-dimensional cone τ ∈ F_{AN} is generated by three vertices v_{1}, v_{2}, v_{3} ∈ Jun^{∗}(G) such that τ ∩ Δ ∩ N = {v_{1}, v_{2}, v_{3}}. By [Roan94], the torus embedding T_{M}(F_{AN}) is a crepant smooth resolution of A^{3}/G if F_{AN} is G-admissible. Any crepant smooth resolution of A^{3}/G is obtained this way.

The purpose of this section is to prove that given an abelian subgroup G of SL(3, k) the fan Fan(G) is G-admissible in the above sense and that Hilb^{G} ≃ T_{M}(Fan(G)).

Lemma 3.2. Assume G ⊂ SL(3, k). Let Γ be a G-graph and p_{k} = max{u ∈ Z; uf_{k} ∈ Γ} (k ∈ Z/3Z). Then there exist unique nonnegative integers a_{i}, b_{i} such that

(p_{i} + 1)f_{i} ≡ a_{i+1}f_{i+1} + b_{i−1}f_{i−1} ∈ Γ.

Proof. We first prove that (p_{1} + 1)f_{1} ≡ jf_{2} + kf_{3} ∈ Γ for some j, k. In fact, by the definition of p_{1} we have (p_{1} + 1)f_{1} ∉ Γ. Then by Definition 1.4 there exists a unique if_{1} + jf_{2} + kf_{3} ∈ Γ such that (p_{1} + 1)f_{1} ≡ if_{1} + jf_{2} + kf_{3} ∈ Γ. In particular, (p_{1} + 1)f_{1} ≠ if_{1} + jf_{2} + kf_{3}. If i ≥ 1, then (i − 1)f_{1} + jf_{2} + kf_{3} ∈ Γ by Definition 1.4, which contradicts Definition 1.4 because p_{1}f_{1} ≡ (i − 1)f_{1} + jf_{2} + kf_{3} ∈ Γ and p_{1}f_{1} ∈ Γ. Therefore i = 0.

Lemma 3.3. (Unique Valley Lemma) Suppose r ≥ 0, s ≥ 0. Assume

rf_{k+1} + sf_{k−1} ∈ Γ, (r + 1)f_{k+1} + sf_{k−1} ∈ Γ,
rf_{k+1} + (s + 1)f_{k−1} ∈ Γ, (r + 1)f_{k+1} + (s + 1)f_{k−1} ∉ Γ.

Then r = a_{k+1}, s = b_{k−1} and (p_{k} + 1)f_{k} ≡ rf_{k+1} + sf_{k−1}.

Proof. Let k = 1 for simplicity. By Definition 1.4 there exists af_{1} + bf_{2} + cf_{3} ∈ Γ such that (r + 1)f_{2} + (s + 1)f_{3} ≡ af_{1} + bf_{2} + cf_{3}. Since (r + 1)f_{2} + (s + 1)f_{3} ∉ Γ, we have (r + 1)f_{2} + (s + 1)f_{3} ≠ af_{1} + bf_{2} + cf_{3}. If b ≥ 1, then rf_{2} + (s + 1)f_{3} ≡ af_{1} + (b − 1)f_{2} + cf_{3} ∈ Γ, which contradicts Definition 1.4 because rf_{2} + (s + 1)f_{3} ∈ Γ. Hence b = 0. Similarly c = 0. It follows that (r + 1)f_{2} + (s + 1)f_{3} ≡ af_{1} ∈ Γ. Since f_{1} + f_{2} + f_{3} ≡ 0, we have rf_{2} + sf_{3} ≡ (a + 1)f_{1}, where rf_{2} + sf_{3} ∈ Γ. It follows from Definition 1.4 that (a + 1)f_{1} ∉ Γ. Therefore a = p_{1}.

Definition 3.4. We call 𝔳_{k} := a_{k+1}f_{k+1} + b_{k−1}f_{k−1} (k ∈ Z/3Z) a valley of Γ if the condition of Lemma 3.2 is true. We also call 𝔰_{k} := p_{k}f_{k} a summit of Γ. The vector rf_{k+1} + sf_{k−1} of Γ is called a geometric valley of Γ if the conditions in Lemma 3.3 are true.

Lemma 3.3 is true even if r = 0 or if s = 0. By Lemma 3.3 Γ has at most a unique geometric valley on Γ_{k} := Γ ∩ (Rf_{i} + Rf_{j}) (i, j ≠ k). It gives a geometric way of computing a_{k} and b_{k}.

3.5. Classification of G-graphs. By Lemma 3.3 any G-graph Γ is one of the following:

(i) a linear graph, (ii) a planar graph with no geometric valleys, (iii) a planar graph with a geometric valley, (iv) three planar graphs with no geometric valleys,

(v) a planar graph with a geometric valley and two planar graph with no geometric valleys, (vi) two planar graphs, each with a geometric valley and a planar graph

with no geometric valley, (vii) three planar graphs, each with a geometric valley.

Definition 3.6. Let Γ be a G-graph. Let 𝔳_{i} := a_{i+1}f_{i+1} + b_{i−1}f_{i−1} be the valleys of Γ (i ∈ Z/3Z). Then we define

v_{i} := (p_{i} + 1)f_{i} − 𝔳_{i},
w_{i} := f_{1} + f_{2} + f_{3} − v_{i},
v_{c} = w_{c} := f_{1} + f_{2} + f_{3}.

A G-graph Γ is called v-crepant (resp. w-crepant) if v_{1} + v_{2} + v_{3} = f_{1} + f_{2} + f_{3} (resp. w_{1} + w_{2} + w_{3} = f_{1} + f_{2} + f_{3}).

We call Γ crepant if Γ is either v-crepant or w-crepant. We remark that no G-graph Γ is v-crepant and w-crepant.

Lemma 3.7. Let Γ be a G-graph. Then S(Γ) = σ^{∨}(Γ) ∩ M. If Γ is vcrepant (resp. w-crepant), then S(Γ) = Z_{0}v_{1} + Z_{0}v_{2} + Z_{0}v_{3} (resp. S(Γ) = Z_{0}w_{1} + Z_{0}w_{2} + Z_{0}w_{3}). In particular, S(Γ) = σ^{∨}(Γ) ∩ M.

Proof. By Lemma 1.8 and Subsection 3.5, S(Γ) is generated by v_{i}, w_{i} and v_{c} (i = 1, 2, 3). If Γ is v-crepant, then v_{c} = v_{1} +v_{2} +v_{3} and w_{i} = v_{c} −v_{i}. Hence S(Γ) = Z_{0}v_{1} + Z_{0}v_{2} + Z_{0}v_{3}. Since M = M(Γ), we have S(Γ) = σ^{∨}(Γ) ∩ M. The rest is similar.

Notation 3.8. Let Γ be a G-graph and v_{i} the valleys of Γ as in Lemma 3.2. For a = (a_{1}, a_{2}, a_{3}), b = (b_{1}, b_{2}, b_{3}) we denote Γ by Γ = Γ(a, b) (resp. Γ^{∗}(a, b)) if Γ is v-crepant (resp. w-crepant). For the same a and b we also define

∥a∥ = a_{1} + a_{2} + a_{3}, a ∗ b = a_{1}b_{2} + a_{2}b_{3} + a_{3}b_{1},
d(a, b) = 1 + ∥a∥ + ∥b∥ + a ∗ a + a ∗ b + b ∗ b.

Lemma 3.9. Let Γ be a G-graph.

(i) If Γ = Γ(a, b), then w_{i} = v_{i−1} + v_{i+1}, p_{i} = a_{i} + b_{i} and |Γ| = d(a, b), (ii) If Γ = Γ^{∗}(a, b), then v_{i} = w_{i−1} + w_{i+1}, p_{i} = a_{i} + b_{i} + 1 and |Γ| =

3 + ∥a∥ + ∥b∥ + d(a, b).

Proof. Clear.

3.10. Crepant G-graphs. Any v-crepant Γ in the first six classes is viewed as one of the special cases of crepant graphs in (vii). In fact, Γ_{k} := Γ∩{x_{k} = 0} has no geometric valley if and only if a_{k−1}b_{k+1} = 0. Therefore if at least one of a_{i} or b_{j} equals zero, (vii) is reduced to the cases (i)-(vi). For instance, the case (i) is obtained by setting a = (a_{1}, 0, 0) and b = (b_{1}, 0, 0) with a_{1} > 0, b_{1} > 0, in which case Γ = Γ(a, b) = {if_{1}; i ∈ [0, a_{1} + b_{1}]}. It is v-crepant but cannot be w-crepant. Since various proofs and computations are carried out more or less in the same manner as (vii), we mainly discuss the case (vii) in what follows. In the case (vii) Γ is given by

Γ = \begin{cases}if_{k} + jf_{k+1}; & (i, j) ∈ [0, p_{k}] × [0, b_{k+1}] or \\ (k ∈ Z/3Z) & (i, j) ∈ [0, a_{k}] × [b_{k+1} + 1, p_{k+1}] \end{cases}\}

where p_{k} = a_{k} + b_{k}.

Let Γ = Γ^{∗}(a, b) be a w-crepant G-graph. Then p_{i} = a_{i} + b_{i} + 1 ≥ 1 by Lemma 3.9. Therefore (i)-(iii) is impossible.

Let us consider the case (iv). Then by Lemma 3.9 we have

|Γ| = 1 + ∥p∥ + p ∗ p = 4 + 2∥p∥ + a ∗ a + a ∗ b + b ∗ a + b ∗ b.

Hence 3 + ∥a∥ + ∥b∥ + b ∗ a = 0, which is impossible. Similarly we derive a contradiction in the cases (v) and (vi). Thus there are no w-crepant G-graphs in (iv)-(vi). However there do exist infinitely many w-crepant Ggraphs in (vii). If Γ is w-crepant with a_{k} = b_{k} = 0, then Γ = {0, f_{1}, f_{2}, f_{3}}, and G is an abelian group ≃ Z/2Z ⊕ Z/2Z of order 4 consisting of diagonal matrices with entries ±1 and determinant one. See also Corollary 2.4.

The following Lemmas follow from Definition 2.6 readily.

Lemma 3.11. Let Γ = Γ(a, b) be a v-crepant G-graph, and v_{1} = v_{−} − v_{+}, v_{−} = (p_{1} + 1)f_{1} and v_{+} = a_{2}f_{2} + b_{3}f_{3}. Then the G-igsaw transformation G_{ig}(v_{−}, v_{+})(Γ) of Γ is a crepant G-graph adjacent to Γ and it is given by

(13) G_{ig}(v_{+}, v_{−})(Γ) = \begin{cases}Γ^{∗}(a − f_{2}, b − f_{3}) & if a_{2} ≥ 1, b_{3} ≥ 1 \\ Γ(a′, b′) & if a_{2} ≥ 1, b_{3} = 0 \\ Γ(a′′, b′′) & if a_{2} = 0, b_{3} ≥ 1 \end{cases}

where

a′ = (([p_{2}/a_{2}] − 1)(p_{1} + 1) + a_{1}, (1 + [p_{2}/a_{2}])a_{2} − p_{2} − 1, 0),
b′ = (p_{1} + 1, p_{2} − [p_{2}/a_{2}]a_{2}, a_{3}),
a′′ = (p_{1} + 1, b_{2}, p_{3} − [p_{3}/b_{3}]b_{3}),
b′′ = (([p_{3}/b_{3}] − 1)(p_{1} + 1) + b_{1}, 0, (1 + [p_{3}/b_{3}])a_{3} − p_{3} − 1).

Proof. Let v_{i} be as before and Γ′ = G_{ig}(v_{+}, v_{−})(Γ). Then by Lemma 2.8, Γ′ is a G-graph. Therefore we have v′_{i} := v_{i}(Γ′) and w′_{i} := w_{i}(Γ′) by Definition 3.6. The G-graph Γ′ is given as above. In fact, suppose a_{2} ≥ 1, b_{3} ≥ 1. Then we see that p_{1}(Γ′) = p_{1}(Γ)+1, a_{2}(Γ′) = a_{2}(Γ)−1 and b_{3}(Γ′) = b_{3}(Γ)−1 and for the other invariants a_{i}(Γ′) = a_{i}(Γ) (i = 1, 3), b_{j}(Γ′) = b_{j}(Γ) (j = 1, 2) and p_{k}(Γ′) = p_{k}(Γ) (k = 2, 3). Therefore p_{2}(Γ′) = a_{2}(Γ′) + b_{2}(Γ′), Γ′ = Γ^{∗}(a−f_{2}, b−f_{3}) and w_{i}(Γ′) = (1−δ_{i1})(f_{1} +f_{2} +f_{3})−v_{i}. In particular, w′_{1} = −v_{1}, w′_{1} + w′_{2} + w′_{3} = f_{1} + f_{2} + f_{3}. Hence Γ′ is w-crepant.

If a_{2} ≥ 1, b_{3} = 0, then v′_{1} = (k + 1)v_{1} + v_{2}, v′_{2} = −v_{1}, v′_{3} = v_{3} − (k − 1)v_{1} where k = [p_{2}/a_{2}]. Hence v′_{1} + v′_{2} + v′_{3} = f_{1} + f_{2} + f_{3}. This proves that Γ′ is v-crepant. The rest is similar.

Lemma 3.12. Let Γ = Γ^{∗}(a, b) be a w-crepant G-graph and w_{1} = w_{−}−w_{+}, w_{−} = (a_{2} + 1)f_{2} + (b_{3} + 1)f_{3} and w_{+} = p_{1}f_{1}. Then G_{ig}(w_{+}, w_{−})(Γ)) = Γ(a + f_{2}, b + f_{3}). In particular, G_{ig}(v_{+}, v_{−})(Γ) is crepant and adjacent to Γ.

Proof. This case is the converse to the first case in Lemma 3.11.

Lemma 3.13. Any G-graph is crepant if G ⊂ SL(3, k).

Proof. Suppose that there is at least a crepant G-graph Γ. Then any Ggraph adjacent to Γ is obtained by a G-igsaw transformation from Γ by Theorem 2.11 and it is crepant by Lemmas 3.11 and 3.12. Thus we can find adjacent crepant G-graphs successively so that we can eventually cover the whole cone Δ with σ(Γ) for crepant G-graphs Γ. It remains to find a crepant G-graph.

If G is cyclic, then let g = (r/n, s/n, 1/n) be a generator with 1 ≤ r ≤ s ≤ n − 1 and 1 + r + s = n. Let Γ be a linear graph in Subsection 3.5 (i) with Γ = {xf_{3}; x ∈ [0, n − 1]}. Then it is easy to check that Γ is a crepant G-graph. We see Γ = Γ(a, b) where a_{i} = b_{i} = 0 (i = 1, 2), a_{3} = s and b_{3} = r.

Next consider an abelian group G with two generators. Let m be the maximal order of elements in G. Then without loss of generality we may assume that G is a group of order ℓm generated by g := (r/m, s/m, 1/m) and f := (−1/ℓ, 1/ℓ, 0) for some 2 ≤ ℓ ≤ m. Let Γ = {pf_{2} + qf_{3}; p ∈ [0, ℓ − 1], q ∈ [0, m − 1]}, a planar G-graph in Subsection 3.5 (ii) with no valleys. Let a_{1} = b_{1} = b_{2} = 0, a_{2} = ℓ − 1, a_{3} the unique integer such that 0 ≤ a_{3} ≤ m − 1, a_{3} = ℓs mod m, and b_{3} := m − 1 − a_{3} (≥ 0). Let a = (a_{i}), b = (b_{i}). Then it turns out that Γ = Γ(a, b) and Γ is v-crepant. This proves the existence of crepant G-graphs. This completes the proof of Lemma.

Table 1. G-graphs

type v/w  conditions  geom. valleys
(i)  v a_{k} = a_{k+1} = b_{k} = b_{k+1} = 0  none
(ii) v  a_{k} = b_{k} = a_{k−1}b_{k+1} = 0  none
(iii) v  a_{k} = b_{k} = 0  𝔳_{k}
(iv) v  a_{2}b_{1} = a_{3}b_{2} = a_{1}b_{3} = 0  none
(v) v  a_{k}b_{k−1} = a_{k+1}b_{k} = 0  𝔳_{k}
(vi) v  a_{k+1}b_{k} = 0  𝔳_{k},𝔳_{k+1}
(vii) v  a_{k} ≥ 1, b_{k} ≥ 1  𝔳_{k} (k = 1, 2, 3)
(vii) w  a_{k} ≥ 0, b_{k} ≥ 0  𝔳_{k} (k = 1, 2, 3)

3.14. Generators of σ(Γ). By Definition 1.5 σ(Γ) is given explicitly. Let

(14)  𝔞(a, b) = (\begin{matrix}F (a_{2}, a_{3}, b_{2}, b_{3}) \\ F (a_{3}, a_{1}, b_{3}, b_{1}) \\ F (a_{1}, a_{2}, b_{1}, b_{2}) \end{matrix})

where a = (a_{1}, a_{2}, a_{3}), b = (b_{1}, b_{2}, b_{3}) and F (x, y, u, v) = (xy + xv + uv)/|Γ|.

It turns out that σ(Γ) is generated by three junior elements of G via the isomorphism G ≃ N/N_{0}. Let Γ = Γ(a, b) and

A = A(Γ) = (\begin{matrix}a_{1} + b_{1} + 1 & −a_{2} & −b_{3} \\ −b_{1} & a_{2} + b_{2} + 1 & −a_{3} \\ −a_{1} & −b_{2} & a_{3} + b_{3} + 1 \end{matrix}) . (15)

Then we have

(16)  A^{−1} := (𝔞_{1}(Γ), 𝔞_{2}(Γ), 𝔞_{3}(Γ))
:= (𝔞(a + f_{2}, b + f_{3}), 𝔞(a + f_{3}, b + f_{1}), 𝔞(a + f_{1}, b + f_{2})).

The column vectors 𝔞_{k}(Γ) are junior elements of G via the isomorphism G ≃ N/N_{0} because (1, 1, 1)A = (1, 1, 1) by the v-crepancy of Γ. Next take Γ = Γ^{∗}(a, b) and set

A^{∗} = A^{∗}(Γ) = (\begin{matrix}−a_{1} − b_{1} − 1 & a_{2} + 1 & b_{3} + 1 \\ b_{1} + 1 & −a_{2} − b_{2} − 1 & a_{3} + 1 \\ a_{1} + 1 & b_{2} + 1 & −a_{3} − b_{3} − 1 \end{matrix}) (17)

and (A^{∗})^{−1} = (𝔞_{1}^{∗}(Γ), 𝔞_{2}^{∗}(Γ), 𝔞_{3}^{∗}(Γ)). The vectors 𝔞_{k}^{∗}(Γ) are junior elements of G via the isomorphism G ≃ N/N_{0} where

𝔞_{1}^{∗}(Γ) = 𝔞(a + f_{3} + f_{1}, b + f_{1} + f_{2}),
(18)  𝔞_{2}^{∗}(Γ) = 𝔞(a + f_{1} + f_{2}, b + f_{2} + f_{3}),
𝔞_{3}^{∗}(Γ) = 𝔞(a + f_{2} + f_{3}, b + f_{3} + f_{1}).

Thus σ(Γ(a, b)) is generated by 𝔞_{k}(Γ(a, b)) (k ∈ Z/3Z), while σ(Γ^{∗}(a, b)) is generated by 𝔞_{k}^{∗}(Γ^{∗}(a, b)) (k ∈ Z/3Z).

4. A crepant smooth resolution

Lemma 4.1. Let Graph(G) be the set of all G-graphs. Let Fan(G) be the set of all σ(Γ) such that Γ ∈ Graph(G). Then Fan(G) is a finite fan with Δ its support, which consists of exactly |G| 3-dimensional G-cones and their faces. The associated torus embedding T_{M}(Fan(G)) is a crepant smooth resolution of T_{M}(Δ) ≃ A^{3}/G.

Proof. By Lemma 3.7 T_{M}(Fan(G)) is smooth. It remains to prove crepancy. Let Γ ∈ Graph(G) and let u_{i} be a v-basis or a w-basis of M. Then X := T_{M}(Fan(G)) has a globally well-defined rational 3-form

ω_{rat} := ∏_{i=1}^{3} dw^{u_{i}}/w^{u_{i}} = |G| ∏_{i=1}^{3} dw^{f_{i}}/w^{f_{i}}. (19)

By the crepancy of Γ ∈ Graph(G), w^{f_{1}+f_{2}+f_{3}}ω_{rat} is a nonvanishing regular 3-form on X because char. k is prime to n = |Γ|, whence the dualizing sheaf of X is trivial. This proves Lemma.

Theorem 4.2. T_{M}(Fan(G)) ≃ Hilb^{G}(A^{3}).

Proof. By Theorem 2.11 and Lemma 3.7 we see Hilb^{G} ≃ T_{M}(Graph(G)) ≃ T_{M}(Fan(G)).

Remark 4.3. The crepancy of G-graphs is interpreted almost as smoothness of versal generic-support-deformation spaces of Z(Γ) when G ⊂ SL(3, k).

With the notation in Subsection 3.1 A^{3} ≃ Spec k[Δ^{∨}∩M_{0}] ≃ Spec k[x, y, z] where x,y and z are the torus coordinates corresponding to the characters of M_{0} f_{1}, f_{2} and f_{3} respectively. We also denote k[x, y, z] by k[A^{3}].

Let Γ be a G-graph and Z(Γ) the associated G-cluster.

The generic-support G-equivariant deformation theory of the cluster Z(Γ) is more or less the same for any crepant G-graph Γ. The cases (i)-(vi) are obtained by specialization of the case (vii)_{v}. Now suppose that Γ := Γ(a, b) is a v-crepant G-graph in the class (vii). The ideal I(Γ) defining Z(Γ) is given by

I(Γ) := (\begin{matrix}x^{p_{1}+1}, y^{p_{2}+1}, z^{p_{3}+1}, xyz \\ y^{a_{2}+1}z^{b_{3}+1}, z^{a_{3}+1}x^{b_{1}+1}, x^{a_{1}+1}y^{b_{2}+1} \end{matrix}) .

Let A^{v} := k[s_{1}, s_{2}, s_{3}] and Z^{v} a G-invariant subscheme of A_{A^{v}}^{3} defined by the ideal

I^{v}(Γ)(s) := (\begin{matrix}x^{p_{1}+1} − s_{1}y^{a_{2}}z^{b_{3}}, y^{p_{2}+1} − s_{2}z^{a_{3}}x^{b_{1}}, z^{p_{3}+1} − s_{3}x^{a_{1}}y^{b_{2}} \\ y^{a_{2}+1}z^{b_{3}+1} − s_{2}s_{3}x^{p_{1}}, z^{a_{3}+1}x^{b_{1}+1} − s_{3}s_{1}y^{p_{2}} \\ x^{a_{1}+1}y^{b_{2}+1} − s_{1}s_{2}z^{p_{3}}, xyz − s_{1}s_{2}s_{3} \end{matrix}) .

This is the versal deformation given in Lemma 2.3. Similarly for a wcrepant G-graph Γ := Γ^{∗}(a, b) the versal deformation of Z^{w} := Z(Γ) is given by

I^{w}(Γ)(t) := (\begin{matrix}x^{p_{1}+1} − t_{2}t_{3}y^{a_{2}}z^{b_{3}}, y^{p_{2}+1} − t_{3}t_{1}z^{a_{3}}x^{b_{1}}, z^{p_{3}+1} − t_{1}t_{2}x^{a_{1}}y^{b_{2}} \\ y^{a_{2}+1}z^{b_{3}+1} − t_{1}x^{p_{1}}, z^{a_{3}+1}x^{b_{1}+1} − t_{2}y^{p_{2}} \\ x^{a_{1}+1}y^{b_{2}+1} − t_{3}z^{p_{3}}, xyz − t_{1}t_{2}t_{3} \end{matrix}) .

Thus in each case the distinguched term xyz − s_{1}s_{2}s_{3} or xyz − t_{1}t_{2}t_{3} is present in the ideal defining a versal deformation, which implies crepancy of Hilb^{G}(A^{3}).

By Lemma 3.7 U_{σ(Γ)} := Spec k[σ^{∨}(Γ) ∩ M] turns out to be

U_{σ(Γ(a,b))} : = Spec k[w^{v_{1}}, w^{v_{2}}, w^{v_{3}}] ≃ Spec k[s_{1}, s_{2}, s_{3}],
U_{σ(Γ^{∗}(a,b))} : = Spec k[w^{w_{1}}, w^{w_{2}}, w^{w_{3}}] ≃ Spec k[t_{1}, t_{2}, t_{3}].

See Definition 3.6 for the notation.

4.4. The exceptional set. Let π : Hilb^{G} → A^{3}/G be the natural morphism and E(G) the execeptional set of π. Any irreducible component of E(G) is by Theorem 4.2 a smooth torus embedding corresponding to a junior element g ∈ G, which we denote by E(g). Hence E(G) = ∪_{g∈Jun(G)} E(g). Let Graph(G, g) be the set of Γ ∈ Graph(G) with g ∈ σ(Γ), and U_{g}(σ(Γ)) := Spec k[σ^{∨}(Γ)∩g^{⊥}∩M]. Then E(g) is covered with U_{g}(σ(Γ)) (Γ ∈ Graph(G, g)).

Definition 4.5. Let g_{i} ∈ Jun(G) (i = 1, 2, 3) be distinct elements. Then E(g_{1}) ∩ E(g_{2}) is a smooth rational curve if and only if Graph(G, g_{1}) ∩ Graph(G, g_{2}) ≠ ∅ if and only if σ(g_{1}, g_{2}) := R_{0} g_{1} + R_{0} g_{2} ∈ Sk^{2} Fan(G). Similarly ∩_{i=1,2,3} E(g_{i}) is a point if and only if ∩_{i=1,2,3} Graph(G, g_{i}) ≠ ∅ if and only if σ(g_{1}, g_{2}, g_{3}) := ∑_{i=1,2,3} R_{0} g_{i} ∈ Sk^{3} Fan(G). In this case |∩_{i=1,2,3} Graph(G, g_{i})| = 1. Thus E(G) is stratified into 0,1,2-strata labeled by Sk^{p} Fan(G) (p = 3, 2, 1) respectively. Let τ ∈ Sk^{p} Fan(G) and E^{0}(τ) a (3 − p)-stratum of E(G) corresponding to τ and E(τ) the closure of E^{0}(τ). This means that E^{0}(σ) ∩ E^{0}(τ) = ∅ for σ ≠ τ and E(σ) ⊂ E(τ) if τ ≺ σ (τ, σ ∈ Fan(G)), where E^{0}(τ) = Spec k[τ^{⊥} ∩ M] ≃ G_{m}^{3−p}.

Definition 4.6. For a subset A of M_{0} and τ ∈ Fan(G) we define

Graph(G, τ) := ∩_{g∈τ∩Jun(G)} Graph(G, g),
Γ(τ) := ∩_{Γ∈Graph(G,τ)} Γ, Γ(τ, A) := ∩_{A⊂Γ∈Graph(G,τ)} Γ.

Now one easily sees

Lemma 4.7. Let Γ = Γ(a − f_{2}, b − f_{3}) with a_{i}, b_{j} ≥ 1 and g := 𝔞_{1}(Γ) = 𝔞(a, b) with the notation in Subsection 3.14. Then

Graph(G, g) = \begin{cases}Γ^{∗}(a − f_{i} − f_{i+1}, b − f_{i+1} − f_{i−1}) \\ Γ(a − f_{i+1}, b − f_{i+2}) (i ∈ Z/3Z) \end{cases}\} .

Corollary 4.8. Let the G-graph Γ and the junior element g be as above. Then E(g) is a regular hexagon – P^{2} blown up at three distinct points.

Remark 4.9. As in [IN98] there is a certain (rather complicated) correspondence between the set of irreducible exceptional subvarieties and the set of minimal G-submodules generating I ∈ Hilb^{G}. This would be a kind of phenomenon generalizing [McKay80] from the view point of representations of G. However it seems difficult to understand this correspondence only from irreducible decompositions of tensor products. We omit it. A generalization of McKay correspondence in dimension three analogous to [GSV83] is given in [INkjm98].

5. Examples in dimension 3

5.1. The case G = G_{1,2,3}. We will explain the theory in the previous sections by the examples in smaller values. Let G be a cyclic group of order 6, g = (1, 2, 3)/6 a generator of G. We denote G = G_{1,2,3} when necessary. Figure 1 plots all the junior elements for G = G_{1,2,3}. Any Γ ∈ Graph(G) is one of the following table (Table 2). Def_{Γ} is the set of toric parameters for deforming the G-cluster Z(Γ). Let us explain the theory in this case in detail.

First we start with the ideal I(Γ_{1}) = (x^{6}, y, z). It is clear dim k[A^{3}]/I(Γ_{1}) = 6, and that k[A^{3}]/I(Γ_{1}) is spanned by Γ_{1}. We can deform Z(Γ_{1}) into smooth G-subschemes by (x^{6} − s, y, z), whence Z(Γ_{1}) ∈ Hilb^{G}. The versal deformation of Z(Γ_{1}) is given by the ideal

(x^{6} − s, y − tx^{2}, z − ux^{3}, x^{4}y − st, x^{3}z − us, xyz − stu)

where x^{2} and x^{3} are the unique elements of Γ_{1} such that wt(y) = wt(x^{2}), wt(z) = wt(x^{3}). Def_{Γ_{1}} is now understood as 3 free parameters s, t, u hence may be understood as x^{6}, yx^{−2} and zx^{−3}. In this case there are two directions yx^{−2} and zx^{−3} of deformations which give rise to monomial G-ideals (i.e. G-ideals generated by monomials). This is written in Figure 2 as b−2a and c − 3a. The graph Γ_{2} is obtained from Γ_{1} by replacing 2a in any vector of Γ_{1} by b repeatedly. In general, in the process of taking the limit of clusters in the direction v_{−} − v_{+} the second G-graph is obtained from the first G-graph by replacing v_{+} by v_{−} repeatedly.

For instance v_{+} = a + b and v_{−} = c in the process from Γ_{2} to Γ_{4}. Hence a + b ∈ Γ_{2} (resp. a + 2b) is transformed into c (resp. b + c) so that Γ_{2} is changed into Γ_{4} under (G_{1,2,3})_{ig}(a + b, c). Figure 2 lists the possible solutions to the jigsaw puzzle (the G-igsaw puzzle !) between G-graphs when G = G_{1,2,3}.

Table 2. G_{1,2,3}-graphs/cones
No.  Γ  I(Γ)
1 {1, x, x^{2}, x^{3}, x^{4}, x^{5}}  (x^{6}, y, z)
2 {1, x, y, xy, y^{2}, xy^{2}}  (x^{2}, y^{3}, z)
3 {1, y, z, yz, y^{2}, y^{2}z}  (x, y^{3}, z^{2})
4  {1, x, y, z, yz, y^{2}} (xy, y^{2}z, zx, x^{2}, y^{3}, z^{2})
5 {1, x, y, z, yz, zx} (x^{2}, xy, y^{2}, z^{2})
6 {1, x, x^{2}, z, xz, x^{2}z}  (x^{3}, y, z^{2})
No.  Def_{Γ}  σ(Γ)
1  x^{6}, y/x^{2}, z/x^{3}  (e_{2}, e_{3}, g)
2  x^{2}/y, y^{3}, z/xy  (e_{3}, g, g^{3})
3  x/y^{2}z, y^{3}, z^{2}  (e_{1}, g^{3}, g^{4})
4 xy/z, y^{2}z/x, zx/y^{2}  (g, g^{3}, g^{4})
5  x^{2}/y, y^{2}/zx, z^{2}  (g, g^{2}, g^{4})
6  x^{3}/z, y/x^{2}, z^{2}  (e_{2}, g, g^{2})

5.2. Jigsaw puzzles. For a given G-graph the possible solutions to the jigsaw puzzle are clear at first sight if the graph is complicated enough, as we see in the case of G_{1,5,31} below. See Figure 3. Let Γ = Γ(g^{10}, g^{11}, g^{17}). Then the G-graph Γ is v-crepant. There are exactly three jigsaw puzzle solutions only in the v_{k}-directions where v_{k} is among 4a − 2b − c, 5c − 2a − b and 4b − a − 3c. For instance, the adjacent G-graph of Γ in the (5c − 2a − b)- direction is obtained by moving the hook at 2a+b consisting of 4 small black dots, each surrounded by a large white circle, into the position 5c as in the lower-left of the Figure 3. On the other hand there are exactly three jigsaw puzzle solutions only in the w_{k}-directions for any w-crepant Γ, for instance, Γ = Γ(g^{4}, g^{10}, g^{11}). See also Tbale 3 for another example of G-graphs/cones.

Table 3. G_{1,2,4}-graphs/cones

No.  Γ  I(Γ)
0 {1, x, y, z, xy, yz, zx} (x^{2}, y^{2}, z^{2}, xyz)
1 {1, y, z, z^{2}, z^{3}, yz, yz^{2}} (x, y^{2}, z^{4}, yz^{3})
2 {1, z, x, x^{2}, x^{3}, zx, zx^{2}} (y, z^{2}, x^{4}, zx^{3})
3 {1, x, y, y^{2}, y^{3}, xy, xy^{2}} (z, x^{2}, y^{4}, zx^{3})
4 {1, z, z^{2}, z^{3}, z^{4}, z^{5}, z^{6}}  (x, y, z^{7})
5 {1, y, y^{2}, y^{3}, y^{4}, y^{5}, y^{6}}  (x, y^{7}, z)
6 {1, x, x^{2}, x^{3}, x^{4}, x^{5}, x^{6}}  (x^{7}, y, z)
No.  Def_{Γ}  σ(Γ)
0  x^{2}/y, y^{2}/z, z^{2}/x  (g, g^{2}, g^{4})
1  x/z^{2}, z^{4}/y, y^{2}/z  (e_{1}, g^{2}, g^{4})
2  y/x^{2}, x^{4}/z, z^{2}/x  (e_{2}, g, g^{2})
3  z/y^{2}, y^{4}/x, x^{2}/y  (e_{3}, g, g^{4})
4  z^{7}, x/z^{2}, y/z^{4}  (e_{1}, e_{2}, g^{2})
5  y^{7}, z/y^{2}, x/y^{4}  (e_{3}, e_{1}, g^{4})
6  x^{7}, y/x^{2}, z/x^{4}  (e_{2}, e_{3}, g)

6. Miscellaneous remarks

6.1. A singular example in dimension four. There is an example of singular Hilb^{G}(A^{4}). Let G be an abelian subgroup of SL(4, k) consisting of all daigonal matrices with entries ±1, which we denote by G_{2^{3}}. We will prove that Hilb^{G}(A^{4}) is singular. The group G has order 8 with 6 junior elements and 12 monomial G-ideals. Among them there are essentially three different types of G-ideals whose G-graphs are Γ_{1}, Γ_{2} and Γ_{3} in the Table 4. Let U_{k} := Spec k[σ^{∨}(Γ_{k}) ∩ M]. We see that U_{k} (k = 1, 2) is smooth, while U_{3} is singular. Hence Hilb^{G}(A^{4}) is singular. The versal generic-support-G-equivariant deformation of Z(Γ_{k}) (k = 1, 2, 3) is given by the ideal

I(Γ_{1})(s) := (\begin{matrix}x − s_{1}yzw, y^{2} − s_{2}, z^{2} − s_{3}, w^{2} − s_{4} \\ xyzw − s_{1}s_{2}s_{3}s_{4} \end{matrix}) ,

I(Γ_{2})(t) := (\begin{matrix}x^{2} − t_{1}t_{2}t_{3}t_{4}, y^{2} − t_{1}t_{4}, z^{2} − t_{2}t_{4}, w^{2} − t_{3}t_{4} \\ xy − t_{1}zw, xz − t_{2}yw, xw − t_{3}yz \\ yzw − t_{4}x, xyzw − t_{1}t_{2}t_{3}t_{4}^{2} \end{matrix}) ,

I(Γ_{3})(u) := (\begin{matrix}x^{2} − u_{1}, y^{2} − u_{2}, z^{2} − u_{3}, w^{2} − u_{4} \\ yz − u_{5}xw, yw − u_{6}xz, zw − u_{7}xy \\ xyzw − u_{1}u_{2}u_{7} \end{matrix})

where

u_{2}u_{7} = u_{3}u_{6} = u_{4}u_{5}, u_{2}u_{3} = u_{1}u_{4}u_{5}^{2},
u_{2}u_{4} = u_{1}u_{3}u_{6}^{2}, u_{3}u_{4} = u_{1}u_{2}u_{7}^{2}.

Table 4. G_{2^{3}}-graphs/cones

\begin{matrix}No. & Γ & I(Γ) \\ 1 & \{1, y, z, w, yz, yw, zw, yzw\} & (x, y^{2}, z^{2}, w^{2}) \\ 2 & \{1, x, y, z, w, yz, yw, zw\} & (xy, xz, xw, x^{2}, y^{2}, z^{2}, w^{2}, yzw) \\ 3 & \{1, x, y, z, w, xy, xz, xw\} & (yz, yw, zw, x^{2}, y^{2}, z^{2}, w^{2}) \end{matrix}

6.2. The case of no crepant resolutions. There is an example of a smooth Hilb^{G}(A^{4}) which is not a crepant resolution of A^{4}/G. For instance let G be the subgroup of SL(4, k) of order two generated by minus the identity. We easily see that Hilb^{G}(A^{4}) is smooth, while it is well known that A^{4}/G has no crepant resolutions because it is terminal.

6.3. Abelian subgroups in GL. There is also an example of singular Hilb^{G} for a finite abelian subgroup G in GL(3, k) by [Reid97]. Very recently R. Kidoh [Kidoh98] gave a complete description of Hilb^{G}(A^{2}) for an abelian finite subgroup G of GL(2, k) by using two kinds of continued fractions. Her work essentially gives the G-graphs in our formulation explicitly.

6.4. Quot schemes. There are infinitely many algebro-geometric relatives of Hilbert schemes, that is, Grothedieck’s Quot schemes of coherent sheaves.

Conjecture 6.5. Suppose G ⊂ SL(3, k). Then any projective crepant resolution of A^{3}/G is isomorphic to Quot^{G}(F) for some G-invariant coherent sheaf F on A^{3}.

A similar conjecture was communicated to us by Nakajima in the quiver variety formulation.

References

[Kidoh98] R. Kidoh, Hilbert schemes and cyclic quotient surface singularities, Preprint (1998).

[GSV83] G. Gonzalez-Sprinberg, J. Verdier, Construction géométrique de la correspondence de McKay, Ann. scient. Éc. Norm. Sup. 16 (1983) 409–449.

[INkjm98] Y. Ito, H. Nakajima, McKay correspondence and Hilbert schemes in dimension three, Preprint (1998), to appear in Topology.

[IN96] Y. Ito, I. Nakamura, McKay correspondence and Hilbert schemes, Proc. Japan Acad. 72 (1996), 135–138.

[IN98] Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, to appear in the Proceedings of Algebraic Geometry Conference in Warwick 1996 (1998).

[IR94] Y. Ito, M. Reid, The McKay correspondence for finite subgroups of SL(3, C), in Higher Dimensional Complex Varieties (Trento, June 1994), M. Andreatta and others Eds., de Gruyter, 1996, 221–240.

[McKay80] J. McKay, Graphs, singularities, and finite group, Proc. Symp. Pure Math., AMS 37 (1980), 183–186.

[Roan94] S-S.Roan, On c_{1} = 0 resolution of quotient singularities, Intern. J. Math. 5 (1994), 523–536.

[Reid97] M. Reid, McKay correspondence, Preprint (1997).

Figure 1. FAN(G_{1,2,3})

[figure: e_{3}; e_{1}; g^{3}; σ_{2}; σ_{3}; e_{2}; g^{4}; σ_{5}; g^{2}; σ_{6}; g; σ_{1}; σ_{4}]

Figure 2. JIGSAW PUZZLE OF G_{1,2,3}-GRAPHS

[figure: Γ_{1}]

[figure: Γ_{2}; ⟶; a + b − c; ↙; 2a − b] [figure: Γ_{6}; ↘; 3a − c; ↙; 2a − b]

[figure: Γ_{5}]

[figure: Γ_{4}; ↖; a + c − 2b]

[figure: Γ_{3}; ↙; a − (2b + c)]

Department of Mathematics, Hokkaido University, Sapporo, 060, Japan

E-mail address: nakamura@@math.sci.hokudai.ac.jp

Figure 3. JIGSAW PUZZLE OF G_{1,5,31}-GRAPHS

Γ(g^{10}, g^{11}, g^{17}) a  3a b 3b c 4c  2a + b 2b + c  a + 3c [figure] [figure] \overline{}[figure] [figure] [figure] [figure] [figure] [figure]  ⟹ 4a − (2b + c) Γ(g^{4}, g^{10}, g^{11}) [figure] [figure]  4a [figure] [figure] [figure] [figure]

[figure: ⇓; 5c − (2a + b); ↘; ↘; 4b − (a + 3c); Γ(g^{11}, g^{17}, g^{18}); 5c; Γ(g^{10}, g^{16}, g^{17}); 4b]