All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions

Dimitrios I. Dais, Martin Henk and Günter M. Ziegler

D.I.D.: Mathematisches Institut der Universität Bonn, Beringstraße 4, D-53115 Bonn, Germany

                                   dais@mpim-bonn.mpg.de

M.H.: Konrad-Zuse-Zentrum (ZIB), Takustraße 7, D-14195 Berlin-Dahlem, Germany

                                        henk@zib.de

G.M.Z.: TU-Berlin, FB Mathematik, MA 6-1, Straße des 17. Juni 136, D-10623 Berlin, Germany

                                 ziegler@math.tu-berlin.de

Abstract

For Gorenstein quotient spaces ℂ^{d}/G, a direct generalization of the classical McKay correspondence in dimensions d ≥ 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstein abelian quotient singularities, which are embeddable as complete intersections of hypersurfaces in an affine space, have torus-equivariant projective crepant resolutions in all dimensions. We use techniques from toric and discrete geometry.

1. Introduction

Up to isomorphism, the underlying spaces ℂ^{d}/G, d ≥ 2, of Gorenstein quotient singularities can be always realized by finite subgroups G of SL(d, ℂ) acting linearly on ℂ^{d}. For d = 2, these Gorenstein quotient spaces are embeddable in ℂ^{3} as hypersurfaces of A-D-E type. In 1979, McKay [18] observed a remarkable connection between the representation theory of the finite subgroups of SL(2, ℂ) and the Dynkin diagrams of certain irreducible root systems. This was the starting-point for Gonzalez-Sprinberg and Verdier [8], and Knörrer [15] to construct a purely geometric, direct correspondence

Ψ : Irr^{0} (G) → Exc (f)

“of McKay-type” between the set Irr^{0} (G) of non-trivial irreducible representations of G, and the set Exc(f) of exceptional prime divisors of the minimal desingularization f : X → ℂ^{2}/G, (or, equivalently, between Irr^{0} (G) and the members of the natural basis of the cohomology ring H^{∗} (X, ℚ)). The bijection Ψ induces, in fact, an isomorphism between the graph of Irr^{0} (G) and the dual (resolution-) graph w.r.t. f, i.e., the product of the images of two distinct elements of Irr^{0} (G) under Ψ is mapped onto the exceptional prime divisor corresponding to the “right” graph-vertex. (For various applications of the geometry of Kleinian singularities and more recent results, including also “quiver-theoretic” methods, we refer to Slodowy [28]).

There are several difficulties (not only of group-theoretic nature) to generalize (even partially) McKay-type correspondences in higher dimensions. Already for d = 3, many things change drastically. For instance, the minimal embedding dimension of ℂ^{3}/G is in many cases very high, and its singular locus is only rarely a singleton. In addition, crepant desingularizations (i.e., the high-dimensional analogues of the above f, cf. [22, 24]) are unique only up to “isomorphism in codimension 1”, and there are lots of examples of non-projective f’s. Nevertheless, Markushevich [17], Ito [10, 11, 12] and Roan [26, 27] proved case-by-case the existence of crepant resolutions of ℂ^{3}/G, for all possible finite subgroups G of SL(3, ℂ), by making use of Blichfeldt’s classification results, and Ito & Reid [13] established a canonical one-to-one correspondence between the conjugacy classes of the Tate twist G (−1) with age = 1 and the crepant discrete valuations on ℂ^{d}/G for any d ≥ 2. It is worth mentioning that the existence of terminal Gorenstein quotient singularities for d ≥ 4 (cf. [19, 20]) means automatically that not all Gorenstein quotient singularities can be resolved by crepant birational morphisms. On the other hand, Batyrev [2] and Kontsevich [16] recently announced proofs of the invariance of all cohomology dimensions of the overlying spaces of all “full” crepant desingularizations for arbitrary d ≥ 2. (For the case of abelian acting groups, see [3, 5.4]). For these reasons, to proceed the investigation of these quotient spaces in dimensions d ≥ 4 one has either to work with at most partial crepant morphisms (and lose the above quoted cohomology dimension invariance) or to determine special classes of quotients for which this classical type of McKay-correspondence makes sense. As there are only very few known examples to follow the latter approach to the problem (cf. Hirzebruch-Höfer [9, p. 257] and Roan [26, § 5]), Reid asked about conditions on G’s which would guarantee the existence of the required “full” crepant (preferably projective) resolutions of the corresponding quotient spaces (see [25], [13, § 4.5]). We believe that one significant class of Gorenstein quotient spaces which enjoys this property is that of complete intersections (“c.i.’s”).

Conjecture 1.1. For all finite subgroups G of SL(d, ℂ), for which ℂ^{d}/G is minimally embeddable as complete intersection (in an affine space ℂ^{r}, r ≥ d+1), the quotient space ℂ^{d}/G admits crepant, projective desingularizations for all d ≥ 2.

In this paper, we prove the following:

Theorem 1.2. Conjecture 1.1 is true for all abelian finite groups G ⊂ SL(d, ℂ) (for which ℂ^{d}/G is a “c.i.”).

For other, non-c.i. abelian quotient spaces having such desingularizations, we refer to [5]. Our proof makes use of toric geometry and Watanabe’s classification result [32], and is motivated by the so-called “nice polyhedral subdivisions” of Knudsen & Mumford [14, Ch. III]. More precisely, the paper is organized as follows: In § 2 we introduce those notions of the theory of toric varieties which will enable a convenient combinatorial characterization of the inner structure of ℂ^{d}/G. Useful tools and arguments coming from convex geometry and the theory of triangulations of polyhedral complexes are presented in § 3. In section 4 we explain why the existence of projective, crepant resolutions can be deduced from the existence of b.c.b.-triangulations of the junior simplex. The method of how one passes from “special data” (parametrizing the defining groups of c.i.-quotient spaces) to “Watanabe forests” and to their lattice-geometric realization via “Watanabe simplices” is described explicitly in § 5. Section 6 provides the inductive, constructive procedure leading to the desired “nice” b.c.b.-triangulations of all Watanabe simplices. Finally, in § 7 we give concrete formulae for the computation of the dimensions of the cohomology groups of the spaces desingularizing ℂ^{d}/G’s (which are “c.i”) by any crepant morphism.

Concerning the general terminology, we always work with normal complex varieties, i.e., with normal, integral, separated schemes defined over ℂ. A partial desingularization of such an X is a proper birational morphism f : Y → X on X, in which Y is assumed to be normal. f : Y → X is called a full (or overall) desingularization of X if, in addition, we have Sing(Y) = ∅, i.e., if the overlying space Y is smooth. For brevity’s sake, the word desingularization of X (or the phrase resolution of singularities of X) is sometimes used as synonymous with the phrase “full (or overall) desingularization of X”. When we refer to partial desingularizations, we mention it explicitly. (By the word singularity we intimate either a singular point or the germ of a singular point, but the meaning will be in each case clear from the context.)

A partial desingularization f : Y → X of a ℚ-Gorenstein complex variety X with global index j is called non-discrepant or simply crepant if ω_{X}^{[j]} ≅ f_{∗} (ω_{Y}^{⊗j}), or, in other words, if the (up to rational equivalence uniquely determined) difference jK_{Y} − f^{∗} (jK_{X}) contains exceptional prime divisors which have vanishing multiplicities. (ω_{X}, K_{X} and ω_{Y} , K_{Y} denote the dualizing sheaves and the canonical divisors of X and Y respectively). Furthermore, f : Y → X is projective if Y admits an f-ample Cartier divisor.

2. A brief toric glossary

We recall some basic facts from the theory of toric varieties and fix the notation which will be used in the sequel. For details the reader is referred to the standard textbooks [6] [7] [14] [21].

(a) For a set A of vectors of ℝ^{d}, the linear hull, the positive hull, the affine hull and the convex hull of A is

lin (A) = {∑_{i=1}^{k} μ_{i}x_{i} | x_{i} ∈ A, μ_{i} ∈ ℝ, k ∈ ℕ} ,

pos (A) = {∑_{i=1}^{k} μ_{i}x_{i} | x_{i} ∈ A, μ_{i} ∈ ℝ, μ_{i} ≥ 0, k ∈ ℕ} ,

aff (A) = {∑_{i=1}^{k} μ_{i}x_{i} | x_{i} ∈ A, μ_{i} ∈ ℝ, ∑_{i=1}^{k} μ_{i} = 1, k ∈ ℕ} , and

conv (A) = {∑_{i=1}^{k} μ_{i}x_{i} | x_{i} ∈ A, μ_{i} ∈ ℝ_{≥0}, ∑_{i=1}^{k} μ_{i} = 1, k ∈ ℕ} ,

respectively. Moreover, we define the integral affine hull of A as

aff_{ℤ} (A) = {∑_{i=1}^{k} μ_{i}x_{i} | x_{i} ∈ A, μ_{i} ∈ ℤ, ∑_{i=1}^{k} μ_{i} = 1, k ∈ ℕ} .

For a set A ⊂ ℝ^{d} the dimension of its affine hull is denoted by dim(A).

(b) Let N ≅ ℤ^{d} be a free ℤ-module of rank d ≥ 1. N can be regarded as a lattice in N_{ℝ} := N ⊗_{ℤ} ℝ ≅ ℝ^{d}. (We shall represent the elements of N by column vectors). If {n_{1}, …, n_{d}} is a ℤ-basis of N, then

det (N) := |det (n_{1}, …, n_{d})|

is the lattice determinant. An n ∈ N is primitive if conv({0, n}) ∩ N contains no other lattice points except 0 and n.

Let N ≅ ℤ^{d} be a lattice as above, M := Hom_{ℤ} (N, ℤ) its dual, N_{ℝ}, M_{ℝ} their real scalar extensions, and ⟨., .⟩ : N_{ℝ} × M_{ℝ} → ℝ the natural ℝ-bilinear pairing. A subset σ of N_{ℝ} is a strongly convex polyhedral cone (s.c.p. cone, for short), if there exist n_{1}, …, n_{k} ∈ N_{ℝ}, such that σ = pos({n_{1}, …, n_{k}}), and σ ∩ (−σ) = {0}. Its dimension dim(σ) is defined to be the dimension of the ℝ-vector space lin(σ) and its relative interior int(σ) is the usual topological interior of it, considered as a subset of lin(σ). The dual cone of an s.c.p. cone σ is defined by

σ^{∨} := {x ∈ M_{ℝ} | ⟨x, y⟩ ≥ 0, for all y ∈ σ }

and satisfies: σ^{∨} + (−σ^{∨}) = M_{ℝ} and dim(σ^{∨}) = d. A subset τ of a s.c.p. cone σ is called a face of σ (notation: τ ≺ σ), if τ = {y ∈ σ | ⟨m_{0}, y⟩ = 0}, for some m_{0} ∈ σ^{∨}. A s.c.p. cone σ is simplicial (resp. rational) if σ = pos({n_{1}, …, n_{k}}), where the vectors n_{1}, …, n_{k} are ℝ-linearly independent (resp. if n_{1}, …, n_{k} ∈ N_{ℚ}). If σ ⊂ N_{ℝ} is a rational s.c.p. cone, then σ is “pointed” and the subsemigroup σ ∩ N of N is a monoid having the origin 0 as its neutral element. Using its dual M ∩ σ^{∨}, one constructs a finitely generated, normal ℂ-subalgebra ℂ [M ∩ σ^{∨}] of ℂ [M] and a d-dimensional affine complex variety

U_{σ} := Max-Spec (ℂ [M ∩ σ^{∨}]) .

(c) For N ≅ ℤ^{d} we define an algebraic torus T_{N} ≅ (ℂ^{∗})^{d} by T_{N} := Hom_{ℤ} (M, ℂ^{∗}) = N ⊗_{ℤ} ℂ^{∗}. Every m ∈ M assigns a character e (m) : T_{N} → ℂ^{∗} with e (m) (t) := t (m), for all t ∈ T_{N}. We have

e (m + m′) = e (m) · e (m′) , for m, m′ ∈ M, and e (0) = 1.

Moreover, for each n ∈ N we define an 1-parameter subgroup of T_{N} by

γ_{n} : ℂ^{∗} → T_{N} with γ_{n} (λ) (m) := λ^{⟨m,n⟩}, for λ ∈ ℂ^{∗}, m ∈ M,

(γ_{n+n′} = γ_{n}∘ γ_{n′}, for n, n′ ∈ N). We can therefore identify M with the character group of T_{N} and N with the group of 1-parameter subgroups of T_{N}. Furthermore, U_{σ} (as above) can be identified with the set of semigroup homomorphisms:

U_{σ} = {u : M ∩ σ^{∨} → ℂ^{∗} | u (0) = 1, u (m + m′) = u (m) · u (m′) , for all m, m′ ∈ M ∩ σ^{∨} } ,

where e (m) (u) := u (m), for all m ∈ M ∩ σ^{∨} and for all u ∈ U_{σ}.

(d) A fan Δ w.r.t. N ≅ ℤ^{d} is a finite collection of rational s.c.p. cones in N_{ℝ} such that:

(i) any face τ of σ ∈ Δ belongs to Δ, and (ii) for σ_{1}, σ_{2} ∈ Δ, the intersection σ_{1} ∩ σ_{2} is a face of both σ_{1} and σ_{2}. The union |Δ| := ⋃ {σ | σ ∈ Δ} is called the support of Δ. Furthermore, we define

Δ (i) := {σ ∈ Δ | dim (σ) = i} , for 0 ≤ i ≤ d.

If ϱ ∈ Δ (1), then there exists a unique primitive vector n (ϱ) ∈ N ∩ ϱ with ϱ = ℝ_{≥0} n (ϱ) and each cone σ ∈ Δ can be therefore written as

σ = ∑_{\begin{matrix}ϱ∈Δ(1) \\ ϱ≺σ \end{matrix}} ℝ_{≥0} n (ϱ) .

The set Sk^{1} (σ) := {n (ϱ) | ϱ ∈ Δ (1) , ϱ ≺ σ } is called the (pure) first skeleton of σ. The toric variety X (N, Δ) associated to a fan Δ w.r.t. the lattice N is by definition the identification space

X (N, Δ) := ( ⋃_{σ∈Δ} U_{σ}) / ∼

with U_{σ_{1}} ∋ u_{1} ∼ u_{2} ∈ U_{σ_{2}} if and only if there is a τ ∈ Δ, such that τ ≺ σ_{1} ∩ σ_{2} and u_{1} = u_{2} within U_{τ}. As a complex variety, X (N, Δ) turns out to be irreducible, normal, Cohen-Macaulay and to have at most rational singularities (cf. [7, p. 76] and [21, Thm. 1.4, p. 7 and Cor. 3.9, p. 125]). X (N, Δ) admits a canonical T_{N}-action which extends the group multiplication of T_{N} = U_{{0}}:

T_{N} × X (N, Δ) ∋ (t, u) ⟼ t · u ∈ X (N, Δ)  (2.1)

where, for u ∈ U_{σ}, (t · u) (m) := t (m) · u (m), for all m ∈ M ∩ σ^{∨}. The orbits w.r.t. the action (2.1) are parametrized by the set of all the cones belonging to Δ. For a τ ∈ Δ, we denote by orb(τ) (resp. by V (τ)) the orbit (resp. the closure of the orbit) which is associated to τ. The spaces orb(τ) and V (τ) have the following properties (cf. [7, pp. 52-55], [21, § 1.3]):

(i) V (τ) = ∐ {orb (σ) | σ ∈ Δ, τ ≺ σ } and orb(τ) = V (τ) ∖ ⋃ {V (σ) | τ ⪵ σ } . (ii) V (τ) = X (N (τ) , Star (τ)) is itself a toric variety w.r.t.

N (τ) := N/N_{τ} , N_{τ} := N ∩ lin (τ) , Star (τ) := \{\overline{σ} | σ ∈ Δ, τ ≺ σ \} ,

where \overline{σ} := (σ + (N_{τ})_{ℝ}) / (N_{τ})_{ℝ} denotes the image of σ in N (τ)_{ℝ} = N_{ℝ} / (N_{τ})_{ℝ} . (iii) For τ ∈ Δ, V (τ) has a natural affine open covering {U_{σ} (τ) | τ ≺ σ } consisting of “intermediate” spaces

U_{τ} (τ) = orb (τ) ↪ U_{σ} (τ) ↪ U_{σ}

being defined by: U_{σ} (τ) := Max-Spec(ℂ [\overline{σ}^{∨} ∩ M (τ)]), with M (τ) denoting the dual of N (τ). It should be pointed out, that every T_{N}-invariant subvariety of U_{σ} has the form U_{σ} (τ) and that dim(U_{σ} (τ)) = dim(σ) − dim(τ) .

(e) Let X (N, Δ) be a d-dimensional (not necessarily compact) toric variety. A function ψ : |Δ| → ℝ is an (integral) Δ-support function if ψ (N ∩ |Δ|) ⊂ ℤ and ψ|_{σ} is linear for all σ ∈ Δ. This means that for all σ ∈ Δ there has to be some m_{σ} ∈ M such that

ψ(x) = ⟨m_{σ}, x⟩ for all x ∈ σ.

We denote by SF(N, Δ) the additive group of all Δ-support functions. A ψ ∈ SF(N, Δ) is strictly upper convex if for every σ ∈ Δ the m_{σ} can be chosen such that ψ(x) ≤ ⟨m_{σ}, x⟩, with equality if and only if x ∈ σ. Let

SUCSF (N, Δ) := {ψ ∈ SF (N, Δ) | ψ strictly upper convex} .

The group T_{N}-CDiv(X (N, Δ)) of T_{N}-invariant Cartier divisors on X (N, Δ) has the ℤ-basis {ℤ V (ϱ) | ϱ ∈ Δ (1)} (cf. [21, p. 68]).

Theorem 2.1. The relationship between the Δ-support functions and the T_{N}-invariant Cartier divisors on X (N, Δ) is given by the following bijections:

ψ ∈  SF (N, Δ)  ⊃  SUCSF (N, Δ)
↕  ↕ 1:1  ↕ 1:1
D_{ψ} ∈ T_{N}-CDiv (X (N, Δ)) = ⨁_{ϱ∈Δ(1)} ℤ V (ϱ) ⊃ \{ \begin{matrix}ample divisors \\ on X (N, Δ) \end{matrix} \}

(2.2)

with

D_{h} := − ∑_{ϱ∈Δ(1)} ψ (n (ϱ)) V (ϱ) .

Proof. For the first bijection see [21, Prop. 2.1, p. 68-69]. The proof of the second bijection of (2.2), in the case in which X (N, Δ) is compact, is given in [21, Cor. 2.14, p. 83]. The general case is treated in [14, Ch. I, Thm. 9, p. 42, and Thm. 13, p. 48]. _{□}

Corollary 2.2. Let X (N, Δ) be a toric variety (resp. a compact toric variety). Then X (N, Δ) is quasiprojective (resp. projective) if and only if SUCSF(N, Δ) ≠ ∅.

(f) A map of fans ϖ : (N′, Δ′) → (N, Δ) is a ℤ-linear homomorphism ϖ : N′ → N whose extension ϖ_{ℝ} : N′_{ℝ} → N_{ℝ} satisfies the property

for all σ′ ∈ Δ′ there is some σ ∈ Δ such that ϖ_{ℝ}(σ′) ⊆ σ.

Every map of fans ϖ : (N′, Δ′) → (N, Δ) induces an holomorphic map

ϖ_{∗} : X (N′, Δ′) → X (N, Δ)

which is equivariant w.r.t. the actions of T_{N′} and T_{N} on the toric varieties X (N′, Δ′), X (N, Δ).

Theorem 2.3. If ϖ : (N′, Δ′) → (N, Δ) is a map of fans, ϖ_{∗} is proper if and only if ϖ^{−1} (|Δ|) = |Δ′| . In particular, if N = N′ and Δ′ is a refinement of Δ, i.e., if each cone of Δ is a union of cones of Δ′, then id_{∗} : X (N, Δ′) → X (N, Δ) is proper and birational.

Proof. See [21, Thm. 1.15, p. 20, and Cor. 1.18, p. 23]. _{□}

(g) Let N ≅ ℤ^{d} be a lattice of rank d and σ ⊂ N_{ℝ} a simplicial, rational s.c.p. cone of dimension k ≤ d. σ can be obviously written as σ = ϱ_{1} + ⋯ + ϱ_{k}, for distinct 1-dimensional cones ϱ_{1}, …, ϱ_{k}. We denote by

Par (σ) := {x ∈ (N_{σ})_{ℝ} | x = ∑_{j=1}^{k} ε_{j} n(ϱ_{j}), with 0 ≤ ε_{j} < 1 for 1 ≤ j ≤ k}

the fundamental (half-open) parallelotope which is associated to σ. The multiplicity mult(σ, N) of σ with respect to N is defined as

mult (σ, N) := # (Par (σ) ∩ N_{σ}) = Vol (Par (σ) ; N_{σ}) ,

where

Vol (Par (σ) ; N_{σ}) := Vol (Par (σ)) ; det (N_{σ})

is the relative volume of Par(σ) w.r.t. N_{σ}. The affine toric variety U_{σ} is smooth if and only if mult(σ, N) = 1.

Theorem 2.4. Every toric variety X (N, Δ) admits a T_{N}-equivariant desingularization

f = id_{∗} : X (N, Δ′) → X (N, Δ)

by a suitable refinement Δ′ of Δ.

In fact, one can always make Δ simplicial without introducing additional rays. (This step relies on Carathéodory’s theorem, cf. [21, p. 175]). In a second step, this new simplicial Δ will be subdivided further into subcones of strictly smaller multiplicities than those of the cones of the starting-point. (Since for each σ ∈ Δ, mult(σ, N) is a volume, mult(σ̂, N) < mult(σ, N) for every simplicial subcone σ̂ of σ). Δ′ is constructed after finitely many subdivisions of this kind (cf. [14, pp. 31-35]).

(h) For the germ (U_{σ}, orb (σ)) of an affine d-dimensional toric variety w.r.t. a singular point orb(σ), (with dim(σ) = d = dim(N)), we define the splitting codimension splcod(orb (σ) ; U_{σ}) of orb(σ) in U_{σ} as

splcod (orb (σ) ; U_{σ}) := max \{κ ∈ \{2, …, d\} | \begin{matrix}U_{σ} ≅ U_{σ}′ × ℂ^{d−κ}, dim (σ′) = κ, \\ and Sing (U_{σ}′) ≠ ∅ \end{matrix} \} .

(For orb(σ) regular we can formally define splcod(orb (σ) ; U_{σ}) = 0.) If splcod(orb (σ) ; U_{σ}) = d, then orb(σ) will be called an msc-singularity, i.e., a singularity having the maximum splitting codimension. An msc-singularity (U_{σ}, orb (σ)) is absolutely unbreakable if U_{σ} cannot be isomorphic even to the product of (at least two) singular affine toric varieties.

(i) For any abelian finite G ⊂ GL(d, ℂ), d ≥ 2, of order l ≥ 2, which is small (i.e. which has no pseudoreflections), ℂ^{d}/G is singular, with singular locus Sing(ℂ^{d}/G) containing at least the image [0] of the origin under the canonical quotient-map ℂ^{d} → ℂ^{d}/G. If we fix a decomposition

G ≅ (ℤ/q_{1}ℤ) × ⋯ × (ℤ/q_{κ}ℤ)

into cyclic groups, take primitive q_{i}th roots of unity ζ_{q_{i}}, and choose eigencoordinates z_{1}, …, z_{d}, then the action of G on ℂ^{d} is determined by

(ℤ/q_{i}ℤ) × ℂ^{d} ∋ (g_{i}, (z_{1}, …, z_{d})) ⟼ diag (ζ_{q_{i}}^{α_{i,1}}z_{1}, …, ζ_{q_{i}}^{α_{i,d}}z_{d}) ∈ ℂ^{d}, 1 ≤ i ≤ κ,

for generators g_{i} = diag(ζ_{q_{i}}^{α_{i,1}}, …, ζ_{q_{i}}^{α_{i,d}}), where the weights (α_{i,1}, …, α_{i,d}) are unique up to only the usual conjugacy relations. Using the above toric glossary, we may identify ℂ^{d}/G with the affine toric variety U_{σ_{0}} = Max-Spec(ℂ [σ_{0}^{∨} ∩ M_{G}]) being associated to the positive orthant σ_{0} = pos({e_{1}, …, e_{d}}) with respect to the lattice of weights

N_{G} = ℤ^{d} + ∑_{i=1}^{κ} ℤ (\frac{1}{q_{i}} (α_{i,1}, …, α_{i,d})^{⊺}) with det(N_{G}) = \frac{1}{l},

where M_{G} denote the dual of N_{G}, which can be viewed as the lattice parametrizing all Ginvariant Laurent monomials. Formally, we identify [0] with orb(σ_{0}) and U_{σ_{0}} with the toric variety X (N_{G}, Δ_{G}) w.r.t. the fan Δ_{G} consisting of σ_{0} itself, together with all its faces. In these terms, and since σ_{0} is a simplicial s.c.p. cone, the singular locus of X (N_{G}, Δ_{G}) can be written as the union

Sing (X (N_{G}, Δ_{G})) = orb (σ_{0}) ∪ (⋃ {U_{σ_{0}} (τ) | τ ⪵ σ_{0}, int(Par(τ)) ∩ N_{G} ≠ ∅})

For 0 ≤ i ≤ d, Δ_{G} (i) has \binom{d}{i} simplicial cones. In particular, Δ_{G} (0) = \{0\}, and for 1 ≤ i ≤ d,

Δ_{G} (i) = {σ_{0} (ν_{1}, …, ν_{i}) := pos({e_{ν_{1}}, …, e_{ν_{i}}}) | 1 ≤ ν_{1} < ν_{2} < ⋯ < ν_{i} ≤ d} .

For any cone σ_{0} (ν_{1}, …, ν_{i}) ∈ Δ_{G} (i), U_{σ_{0}} (σ_{0} (ν_{1}, …, ν_{i})) is nothing but

({z = (z_{1}, …, z_{d}) ∈ ℂ^{d} | z_{ν_{1}} = ⋯ = z_{ν_{i}} = 0})/G ,

i.e., a (d − i)-dimensional coordinate-subspace of ℂ^{d} divided by the inherited G-action.

Proposition 2.5. For an abelian G (as above) the following conditions are equivalent :

(i) X (N_{G}, Δ_{G}) = U_{σ_{0}} = ℂ^{d}/G is Gorenstein, (ii) G ⊂ SL(d, ℂ), (iii) ∑_{j=1}^{d} α_{i,j} ≡ 0 mod q_{i}, for 1 ≤ i ≤ κ,

(iv) ⟨(1, 1, …, 1), n⟩ ≥ 1, for all n ∈ σ_{0} ∩ (N_{G} ∖ {0}),

(v) (U_{σ_{0}}, orb (σ_{0})) is a canonical singularity of index 1.

Proof. (i) ⇔ (ii) follows from [31] and (i) ⇔ (v) from [22, 23]. All the other implications can be checked easily. _{□}

From now on, let X (N_{G}, Δ_{G}) be Gorenstein. The cone σ_{0} = ℝ_{≥0} 𝔰_{G} is supported by the so-called junior lattice simplex 𝔰_{G} = conv({e_{1}, …, e_{d}}) (w.r.t. N_{G}; cf. [13] [3]). Note that up to 0 there is no other lattice point of σ_{0} ∩ N_{G} lying “under” the affine hyperplane of ℝ^{d} containing 𝔰_{G}. Moreover, the lattice points representing the l − 1 non-trivial group elements are exactly those belonging to the intersection of a dilation λ 𝔰_{G} of 𝔰_{G} with Par (σ_{0}), for some integer λ, 1 ≤ λ ≤ d − 1. For c.i.-U_{σ_{0}}’s, our purpose is to construct crepant, projective (full) desingularizations

f : X (N_{G}, Δ̂_{G}) → X (N_{G}, Δ_{G}) = U_{σ_{0}}

which are T_{N_{G}}-equivariant (with T_{N_{G}} = (ℂ^{∗})^{d} /G denoting the algebraic torus embedded in U_{σ_{0}}), and have overlying (quasiprojective) spaces X (N_{G}, Δ̂_{G}) associated to fans Δ̂_{G} which refine appropriately Δ_{G}.

3. On b.c.b.-triangulations of lattice polytopes

In this section we introduce “b.c.b.-triangulations” and study their behaviour with respect to joins and dilations. (We shall mostly use the standard terminology from the theory of polyhedral complexes and polytopes, cf. [30] [33]).

(a) By vert(S) we denote the set of vertices of a polyhedral complex S. By a triangulation T of a polyhedral complex S we mean a geometric simplicial subdivision of S with vert(S) ⊂ vert(T ). If T is d-dimensional, then by T (i) we denote the set of i-dimensional simplices of T . A polytope P will be frequently identified with the polyhedral complex consisting of P itself together with all its faces.

(b) A triangulation T of a polyhedral d-complex S is coherent (or regular) if there exists a strictly upper convex T -support function ψ : |T | → ℝ, i.e., a real function on the underlying space |T | for which

ψ(t x + (1 − t)y) ≥ t ψ(x) + (1 − t)ψ (y) for all x, y ∈ |T | and t ∈ ℝ ∩ [0, 1] ,

and ψ |_{s} ≠ ψ |_{s′} for any two maximal simplices s, s′, s ≠ s′, of T . The set of all strictly upper convex T -support functions is denoted by SUCSF(T ).

Lemma 3.1 (Patching Lemma). Let P ⊂ ℝ^{d} be a d-polytope, T = {s_{i} | i ∈ I } (with I a finite set) a coherent triangulation of P, and T_{i} = {s_{i,j} | j ∈ J_{i} } (J_{i} finite, for all i ∈ I) a coherent triangulation of s_{i}, for all i ∈ I. If ψ_{i} : |T_{i}| → ℝ denote strictly upper convex T_{i}-support functions, such that

ψ_{i} |_{s_{i}∩s_{i′}} = ψ_{i′} |_{s_{i}∩s_{i′}}

for all (i, i′) ∈ I × I, then

T̃ := {all the simplices s_{i,j} | j ∈ J_{i}, i ∈ I}

forms a coherent triangulation of the initial polytope P.

The above ψ_{i}’s can be canonically “patched together” to construct an element ψ of SUCSF(T̃); see [14, Cor. 1.12, p. 115] or [4, Lemma 3.2.2].

(c) A triangulation T of a d-dimensional simplex s (or, more generally, of any pure d-dimensional simplicial complex S) is balanced if its graph can be (d + 1)-coloured, i.e., if there is a function

φ : vert (T ) → {0, 1, 2, …, d}

such that any two adjacent vertices receive different values (“colours”) under φ. If T is a balanced triangulation of a d-polytope, then all facets of T receive all the colours, and the colouring function φ is unique (up to permutation of the colours).

Example 3.2. The first two 2-dimensional triangulations (A) and (B) of Figure 1 are balanced, whereas (C) and (D) are not.

[figure: 0; 1; 2; 0; 2; 0; 1; 1; 2; 0; (A); 0; 0; 2; 1; 1; (B); (C); (D)]

Figure 1

(d) Let N be a d-dimensional lattice. A lattice polytope (w.r.t. N) is a polytope in N_{ℝ}≅ ℝ^{d} all of whose vertices belong to N. If {n_{0}, n_{1}, …, n_{k}} is a set of affinely independent lattice points (k ≤ d), s the k-simplex s = conv({n_{0}, n_{1}, …, n_{k}}), and N_{s} := lin({n_{1} − n_{0}, …, n_{k} − n_{0}}) ∩ N, then s is basic if it satisfies any one of the following equivalent conditions:

(i) aff_{ℤ}(n_{0}, …, n_{k}) = N ∩ aff(s), (ii) {n_{1} − n_{0}, n_{2} − n_{0}, …, n_{k} − n_{0}} is a ℤ-basis of N_{s}, (iii) {n_{1} − n_{0}, n_{2} − n_{0}, …, n_{k} − n_{0}} is a part of a ℤ-basis of N,

(iv) s has relative volume Vol(s; N_{s}) = Vol (s) ; det (N_{s}) = \frac{1}{k!} (w.r.t. N_{s}).

A lattice triangulation T of a lattice polytope P (w.r.t. N ≅ ℤ^{d}) is a triangulation of P with vertices in N, and it is called basic if all its simplices are basic. If T is a basic triangulation of P w.r.t. N, then in particular we have

aff_{ℤ}(vert((T)) = N ∩ aff(P).

Affine maps Φ : N_{ℝ} → N_{ℝ} will be called affine integral transformations (w.r.t N) if they satisfy N = Φ(N), that is, if they have the form Φ (x) = Mx + β, with M ∈ GL_{ℤ} (N) ≅ GL(d, ℤ) and β ∈ N (i.e., if they are composed of N-unimodular transformations and N-integral translations).

Definition 3.3. From now on, b.c.b.-triangulation is used as an abbreviation for a basic, coherent, balanced triangulation of a lattice polytope.

Lemma 3.4 (Preservation of the “b.c.b.-property” under affine transformations). Let T be a lattice triangulation of a lattice polytope P ⊂ N_{ℝ} (w.r.t. N ≅ ℤ^{d}) and let Φ : N_{ℝ} → N_{ℝ} be a (not necessarily integral) regular affine transformation. If T is a b.c.b.- triangulation w.r.t. N, then its image Φ (T ) := {Φ (s) | s ∈ T } under Φ is again a b.c.b.- triangulation of the transformed lattice d-polytope Φ (P) w.r.t. the lattice Φ(N).

(e) If T_{1} (resp. T_{2}) is a triangulation of a d_{1}-polytope P_{1} (resp. of a d_{2}-polytope P_{2}) in ℝ^{d} such that dim(P_{1} ∪ P_{2}) = dim(P_{1}) + dim(P_{2}) + 1, that is, such that aff(P_{1}) and aff(P_{2}) are skew affine subspaces of N_{ℝ}, then the join T_{1} ∗ T_{2} of T_{1} and T_{2} is defined by

T_{1} ∗ T_{2} = {conv (s_{1} ∪ s_{2}) | s_{1} ∈ T_{1}, s_{2} ∈ T_{2} } .

It is a triangulation of the usual join P = conv(P_{1} ∪ P_{2}) of P_{1} with P_{2}, which is a polytope of dimension d_{1} + d_{2} + 1 (cf. [30, Ex. 7, p. 136] and [33, Ex. 9.9, p. 323]).

Theorem 3.5 (Preservation of the “b.c.b.-property” for joins). Let T_{1}, T_{2} be triangulations of polytopes P_{1}, P_{2} ⊂ ℝ^{d} of dimensions d_{1} resp. d_{2} in skew affine subspaces of N_{ℝ}, i.e., satisfying dim(P_{1} ∪ P_{2}) = d_{1} + d_{2} + 1.

If T_{1}, T_{2} are b.c.b. triangulations of P_{1}, P_{2} w.r.t. N, then T_{1} ∗ T_{2} is a b.c.b.-triangulation of the (d_{1} + d_{2} + 1)-polytope P = conv(P_{1} ∪ P_{2}) w.r.t. N if and only if

aff_{ℤ}( vert(T_{1}) ∪ vert(T_{2}) ) = N ∩ aff(P_{1} ∪ P_{2}).

Proof. By (e) it is obvious that T_{1} ∗ T_{2} is a lattice triangulation of P w.r.t. the lattice N.

(i) Let s be a (d_{1} + d_{2} + 1)-simplex of T_{1} ∗ T_{2}. By definition it can be written as

s = conv({n_{0}^{(1)}, …, n_{d_{1}}^{(1)}, n_{0}^{(2)}, …, n_{d_{2}}^{(2)}}),

where conv(n_{0}^{(i)}, …, n_{d_{i}}^{(i)}) ∈ T_{i}(d_{i}) is a basic simplex w.r.t. N ∩aff(P_{i}), for i = 1, 2. In particular, we have aff_{ℤ}(n_{0}^{(1)}, …, n_{d_{1}}^{(1)}, n_{0}^{(2)}, …, n_{d_{2}}^{(2)}) = aff_{ℤ}(T_{1} ∗ T_{2}). Thus, s is basic w.r.t. N ∩ aff(P_{1} ∪ P_{2}) if and only if

aff_{ℤ}( vert(T_{1}) ∪ vert(T_{2}) ) = N ∩ aff(P_{1} ∪ P_{2}).

(ii) If ψ_{1} ∈ SUCSF(T_{1}), ψ_{2} ∈ SUCSF(T_{2}), then the function Ψ defined by

Ψ (t x + (1 − t) y) := t ψ_{1} (x) + (1 − t) ψ_{2} (y) , for all x ∈ P_{1}, y ∈ P_{2}, and t ∈ ℝ ∩ [0, 1] ,

belongs to SUCSF(T_{1} ∗ T_{2}). (iii) If T_{1} and T_{2} are both balanced and have dimensions d_{1} and d_{2}, with colouring functions

φ_{1} : vert (T_{1}) → {0, 1, 2, …, d_{1}} and φ_{2} : vert (T_{2}) → {0, 1, 2, …, d_{2}} ,

then φ_{1} ∗ φ_{2} : vert(T_{1} ∗ T_{2}) = vert(T_{1}) ∪ vert(T_{2}) → {0, 1, …, d_{1} + d_{2} + 1} defined by

(φ_{1} ∗ φ_{2}) (x) := \{ \begin{matrix}φ_{1} (x) , & if x ∈ vert (T_{1}) \\ φ_{2} (x) + d_{1} + 1, & if x ∈ vert (T_{2}) \end{matrix}

is a colouring function for T_{1} ∗ T_{2}. _{□} (f) Let now T be a triangulation of d-polytope P ⊂ ℝ^{d}. If λ P = {λ x |x ∈ P } is the λ-times dilated P, i.e., the image of P under the dilation map d_{λ} : ℝ^{d} ∋ x ↦ λ x ∈ ℝ^{d}, where λ ∈ ℤ, λ ≥ 1, then λ T will denote the corresponding triangulation of λ P under the same dilation map, i.e., λ T = {λ s | s ∈ T }.

Lemma 3.6 (Preservation of the “c.b.-property” for dilations). Let T denote a triangulation of a d-polytope P ⊂ ℝ^{d}, and λ ≥ 1 an integer.

(i) If T is coherent, then λ T is coherent. (ii) If T is balanced, then λ T is balanced. (iii) If P is a lattice polytope w.r.t. a lattice N ⊂ ℝ^{d}, and T a lattice triangulation, then λ T is a lattice triangulation too w.r.t. the same lattice.

Proof. For (i), if ψ ∈ SUCSF(T ), then ψ ∘ (d_{λ})^{−1} ∈ SUCSF(λ T ). For (ii), if φ is a colouring function for T , then φ ∘ (d_{λ})^{−1} serves as a colouring function for λ T . (iii) is obvious because λ is an integer. _{□}

Remark 3.7. If T is a basic triangulation of a lattice d-polytope P ⊂ ℝ^{d}, then for λ ≥ 2, the dilation λ T is non-basic (see Figure 2). Nevertheless, as we shall see below in Prop. 6.1, any dilation of a b.c.b. triangulation admits a b.c.b. triangulation.

[figure: T; 3T]

Figure 2

4. Torus-equivariant, crepant, projective resolutions

Reverting to (singular) Gorenstein abelian quotient spaces

X (N_{G}, Δ_{G}) = U_{σ_{0}} = ℂ^{d}/G, G ⊂ SL (d, ℂ) , d ≥ 2,

we explain how the desired desingularizations can be constructed by means of triangulations. Let f = id_{∗} : X(N_{G}, Δ̂_{G}) → X (N_{G}, Δ_{G}) be an arbitrary T_{N_{G}}-equivariant desingularization of X (N_{G}, Δ_{G}) (as in Thm. 2.4). There are one-to-one correspondences:

ϱ  ∈  (Δ̂_{G} (1) ∖ {ℝ_{≥0} e_{1}, …, ℝ_{≥0} e_{d}})
↕  ↕
n (ϱ)  ∈ (⋃ {Sk^{1} (σ) | σ ∈ Δ̂_{G}} ∖ {e_{1}, …, e_{d}})
↕  ↕
D_{n(ϱ)} := V (ϱ) = V (ℝ_{≥0} n (ϱ)) ∈  \{ \begin{matrix}exceptional prime divisors \\ with respect to f \end{matrix} \}

(4.1)

Furthermore, D_{e_{i}} := V (ℝ_{≥0} e_{i}), in X(N_{G}, Δ̂_{G}), corresponds to the strict transform of

({z = (z_{1}, …, z_{d}) ∈ ℂ^{d} | z_{i} = 0}) /G

with respect to f, for 1 ≤ i ≤ d.

Proposition 4.1. A T_{N_{G}}-equivariant partial desingularization f : X (N_{G}, Δ̂_{G}) → U_{σ_{0}} (de-

fined by any fan Δ̂_{G} refining Δ_{G} w.r.t. N_{G}) is crepant if and only if

⋃ {Sk^{1} (σ) | σ ∈ Δ̂_{G}} ⊂ {x = (x_{1}, …, x_{d})^{⊺} ∈ (N_{G})_{ℝ} | ∑_{i=1}^{d} x_{i} = 1} .  (4.2)

Proof. Let m_{1}, …, m_{d} be ℝ-linearly independent vectors of M_{G}. The dualizing sheaf ω_{T_{N_{G}}} of T_{N_{G}} is generated by the rational differential form

ϕ_{T_{N_{G}}} := \frac{±det(M_{G})}{det( ⨁_{1≤i≤d} ℤm_{i})} · \frac{du_{1}}{u_{1}} ∧ ⋯ ∧ \frac{du_{d}}{u_{d}} ,

where u_{1} = e (m_{1}) , …, u_{d} = e (m_{d}) ( ϕ_{T_{N_{G}}} is independent of the specific choice of m_{1}, …, m_{d}). Moreover, for the dualizing sheaf ω_{X(N_{G},Δ_{G})} of X (N_{G}, Δ_{G}) we have

ω_{X(N_{G},Δ_{G})} = O_{X(N_{G},Δ_{G})} (K_{X(N_{G},Δ_{G})}) = ℂ [M_{G} ∩ int (σ_{0}^{∨})] · ϕ_{T_{N_{G}}}

(cf. [22, Lemma 3.3, p. 293]). On the other hand, conditions (i)-(v) of Prop. 2.5 are equivalent to the triviality of ω_{X(N_{G},Δ_{G})}, as one can easily verify by using Ishida’s criteria [21, p. 126]. This means, in particular, that the semigroup ideal M_{G}∩ int(σ_{0}^{∨}) of the semigroup ring M_{G} ∩ σ_{0}^{∨} is principal. In fact, it is generated by the element (1, 1, …, 1, 1) and ℂ [M_{G} ∩ int (σ_{0}^{∨})] by e ((1, 1, …, 1, 1)), where ord_{V (ℝ_{≥0}e_{i})}(((1, …, 1)) · ϕ_{T_{N_{G}}} ) = 1 for 1 ≤ i ≤ d.

f is crepant if and only if the difference

K_{X(N_{G},Δ̂_{G})} − f^{∗} (K_{X(N_{G},Δ_{G})})

between the canonical divisor of X (N_{G}, Δ̂_{G}) and the pull-back of the canonical (trivial) divisor of X (N_{G}, Δ_{G}) vanishes. In this case, K_{X(N_{G},Δ̂_{G})} is trivial too. Let

X(N_{G}, Δ̂_{G}) = ( ∐_{σ∈Δ̂_{G}} Û_{σ})/ ∼,

σ′ ∈ Δ̂_{G} (d) an arbitrary maximal cone of Δ̂_{G}, ϱ a ray from (Δ̂_{G} (1) ∖ {ℝ_{≥0} e_{1}, …, ℝ_{≥0} e_{d}}) with ϱ ≺ σ′, and D_{n(ϱ)} the corresponding exceptional prime divisor (as in (4.1)). Since

H^{0} (Û_{σ}′; ω_{X(N_{G},Δ̂_{G})}) = ℂ [M_{G} ∩ int ((σ′)^{∨})] · ϕ_{T_{N_{G}}} ,

by the same argument as above, if ωÛ_{_{σ}′} is trivial, then it is generated by e ((1, 1, …, 1, 1))·ϕ_{T_{N_{G}}}. The discrepancy of D_{n(ϱ)} with respect to f|Û_{_{σ}′} is equal to ord_{D_{n(ϱ)}}(e((1, …, 1)) · ϕ_{T_{N_{G}}} ) − 1,

where this vanishing order of (e ((1, …, 1)) · ϕ_{T_{N_{G}}} ) along D_{n(ϱ)} equals

ord_{D_{n(ϱ)}}(e((1, …, 1)) · ϕ_{T_{N_{G}}} ) = ⟨(1, …, 1), n(ϱ)⟩ ,

by [7, lemma of p. 61] and the definition of ϕ_{T_{N_{G}}}. Hence f is crepant if and only if the condition (4.2) is satisfied. _{□} Corollary 4.2. All T_{N_{G}}-equivariant partial crepant desingularizations of U_{σ_{0}} = ℂ^{d}/G are of the form

f_{T} : X (N_{G}, Δ̂_{G} (T )) → U_{σ_{0}}  (4.3)

for some fan Δ̂_{G} (T ) = {σ_{s}, s ∈ T }, where σ_{s} := {γx ∈ (N_{G})_{ℝ} | x ∈ s, γ ∈ ℝ_{≥0} } and T denotes a lattice triangulation of the junior simplex 𝔰_{G}.

Since a simplex s ∈ T is basic if and only if mult(σ_{s}) = 1, i.e., if and only if the toric variety U_{σ_{s}} is smooth, we have:

Lemma 4.3. f_{T} : X (N_{G}, Δ̂_{G} (T )) → U_{σ_{0}} is a crepant (full) desingularization if and only if T is basic.

Example 4.4. Consider the 3-dim. Gorenstein cyclic quotient singularity of type \frac{1}{7}(3, 3, 1) (in the sense of [24, § 4.2]). The junior simplex 𝔰_{G} = conv({e_{1}, e_{2}, e_{3}}) contains three additional inner lattice points, namely n_{1} = \frac{1}{7} (3, 3, 1)^{⊺}, n_{2} = \frac{1}{7} (2, 2, 3)^{⊺}, n_{3} = \frac{1}{7} (1, 1, 5)^{⊺}. The unique basic triangulation T of 𝔰_{G} is drawn in Figure 3. The three exceptional prime divisors on X (N_{G}, Δ̂_{G} (T )) are isomorphic to

D_{n_{1}} ≅ ℙ_{ℂ}^{2} , D_{n_{2}} ≅ 𝔽_{3}, and D_{n_{3}} ≅ 𝔽_{5},

respectively, where 𝔽_{j} := ℙ (O_{ℙ_{ℂ}^{1}} ⊕ O_{ℙ_{ℂ}^{1}} (j)) , j ≥ 0, denote the Hirzebruch-surfaces.

f_{T} e_{1}  e_{2} e_{3}  e_{1}  e_{2} e_{3} n_{1} n_{2} n_{3}

Figure 3

Coming back to (4.3), we see that there is a bijection

SUCSF (N_{G}, Δ̂_{G}(T )) ∋ ψ ↦ ψ|_{T} ∈ SUCSF(T )

(with inverse defined by taking the extension ψ̃ of any ψ ∈ SUCSF(T ), so that ψ̃ (γx) = ψ (x), for all x ∈ s, s ∈ T (d), and γ ∈ ℝ_{≥0}). Therefore, by 2.1, 2.2, we obtain:

Proposition 4.5. f_{T} is a projective crepant morphism if and only if T is coherent.

Hence, the existence of a projective, crepant (full) desingularization of U_{σ_{0}} = ℂ^{d}/G is equivalent to the existence of a basic, coherent triangulation of the junior simplex 𝔰_{G}. Our use of balanced complexes is a requirement that is not needed from the toric point of view; however, it is our main technical tool for “gluing” triangulations together. In particular, it is not clear whether every dilation of a basic triangulation has a basic refinement; the same is proved for balanced basic triangulations, and for b.c.b. triangulations, below.

5. From special data to Watanabe simplices

In 1980 Watanabe [32, 1.7, 2.1 and 2.8] classified all abelian quotient c.i.-singularities in terms of “special data” that encode complete descriptions of the corresponding polynomial rings and acting groups. Although his method for proving this theorem is purely algebraic, he also indicated how could one represent these “special data” by certain graphs (which are actually forests, and will be henceforth called Watanabe forests). As we shall see in the present section, this graph-theoretic approach to the Classification Theorem 5.3 yields a very useful latticegeometric interpretation. More precisely, we work out the following one-to-one correspondences:

\{ \begin{matrix}special non-trivial \\ data & 𝔻 = & (𝔇, w) \\ (up to isomorphisms ≃ ) \end{matrix} \} ↔  \{ \begin{matrix}abelian quotient c.i.-singularities \\ (ℂ^{d}/G, [0]) & ≅ & (ℂ^{d}/G_{𝔻}, [0]) \\ (up to analytic isomorphisms & ≅ & ) \end{matrix} \}
↕  ↕

\{ non-trivial Watanabe forests W_{𝔻} (up to graph-theoretic isomorphisms ≅_{wg} ) \} ↔ \{ \begin{matrix}Gorenstein, simplicial, singular, toric \\ varieties X (N_{G}, Δ_{G}) = U_{σ_{0}} for which \\ 𝔰_{G} is a Watanabe simplex w.r.t. N_{G} \\ (up to affine integral transformations) \end{matrix} \}

Let us first formulate Watanabe’s result.

Definition 5.1. Let d ≥ 2 be an integer. A special datum 𝔻 = (𝔇, w) (w.r.t. d) is a pair consisting of a set of non-empty subsets of {1, 2, …, d} (i.e. 𝔇 ⊆ 2^{{1,2,...,d}} ∖ {∅}), together with a “weight-function” w : 𝔇 → ℕ, such that:

(i) For each i ∈ {1, 2, …, d} we have {i} ∈ 𝔇. (ii) For every pair of index-subsets J, J′ ∈ 𝔇, either J ⊆ J′, or J′ ⊆ J, or J ∩ J′ = ∅. (iii) If J is a maximal element with respect to the inclusion relation “⊆”, then w (J) = 1. (iv) If J, J′ ∈ 𝔇 and J ⫋ J′, then w (J) > w (J′) and w (J′) |w (J).

(v) For J_{1}, J_{2}, J ∈ 𝔇, with J_{1} ⊏ J, J_{2} ⊏ J, we have w (J_{1}) = w (J_{2}). For this the binary cover relation “⊏” on the sets of 𝔇 is defined by:

J ⊏ J′ ⟺_{def} (J ⫋ J′ and ∄J′′, J′′ ∈ 𝔇 : J ⫋ J′′ ⫋ J′) .

𝔻 = (𝔇, w) is non-trivial if there is at least one subset J ∈ 𝔇 with cardinality # (J) ≥ 2.

Definition 5.2. Let d ≥ 2 be an integer, and let 𝔻 = (𝔇, w) be a special datum w.r.t. d. We define the polynomial ring R_{𝔻} and the group G_{𝔻} by

R_{𝔻} := ℂ [𝔵_{J} | J ∈ 𝔇] with 𝔵_{J} := ( ∏_{i∈J} 𝔵_{i})^{w(J)}, and

G_{𝔻} := ⟨\{diag(1, …, 1, \underbrace{ζ_{w}}_{i-th pos.}, 1, …, 1, \underbrace{ζ_{w}^{−1}}_{j-th pos.}, 1, …, 1) | \begin{matrix}i ∈ J_{1}, j ∈ J_{2} \\ for all J_{1}, J_{2}, J ∈ 𝔇 \\ with J_{1} ⊏ J, J_{2} ⊏ J \\ and w = w (J_{1}) = w (J_{2}) \end{matrix} \}⟩

Here ζ_{w} denotes a primitive wth root of unity, and the diagonal matrices generating G_{𝔻} are (d × d)-matrices in SL(d, ℂ).

Theorem 5.3 (Watanabe’s Classification Theorem). Let d ≥ 2 be an integer and G be a finite, abelian subgroup of SL(d, ℂ) of order ≥ 2. The following conditions are equivalent:

(i) The quotient space ℂ^{d}/G is minimally embeddable as a complete intersection of hypersurfaces in an affine (complex) space. (ii) There is a special datum 𝔻 = (𝔇, w) (w.r.t. d), such that

ℂ^{d}/G ≅ Max-Spec (R_{𝔻})

and G is conjugate to G_{𝔻} (within SL(d, ℂ)).

In other words,

(ℂ^{d}/G, [0]) ≅ (ℂ^{d}/G_{𝔻}, [0]) ,

i.e., up to an analytic isomorphism, all (germs of ) abelian quotient c.i.-singularities are parameterized by the set of non-trivial special data 𝔻 = (𝔇, w) (w.r.t. d).

Remark 5.4. Let 𝔻 = (𝔇, w) be a special datum (w.r.t. d) and let J_{♦}^{[1]},…, J_{♦}^{[κ]} denote the maximal elements of 𝔇 (w.r.t. “⊆”). By the properties (i) and (ii) in 5.1, we have:

(i) ⋃_{i=1}^{κ} J_{♦}^{[i]} = {1, 2, …, d} (set-theoretically). (ii) For any J ∈ 𝔇 that is not a singleton (#(J) ≥ 2), there is a set-theoretic partition:

J = ⋃^{•} {J′ ∈ 𝔇 | J′ ⊏ J } .

Remark 5.5. The elements of J’s determine the positions of the roots of unity within the diagonals of the matrices generating G_{𝔻}. As there are many choices for the elements of J’s leading to the same matrices (up to permutations of their entries), and since Theorem 5.3 gives the classification up to isomorphism, we shall define “isomorphisms” between special data in order to work only with convenient representatives of the corresponding equivalence-classes. For a special datum 𝔻 = (𝔇, w) (w.r.t. d) let

𝔇 (p) := {J ∈ 𝔇 | # (J) = p} , 1 ≤ p ≤ d,

denote the subset of 𝔇 consisting of all index-sets of fixed cardinality p.

Two special data 𝔻 = (𝔇, w) and 𝔻′ = (𝔇′, w′) (w.r.t. d) are isomorphic (and we denote this isomorphism by 𝔻 ≃ 𝔻′) if there exists a bijection θ : {1, …, d} → {1, …, d} for which (i) J ∈ 𝔇 ⟺ Θ(J) ∈ 𝔇′ (i.e., θ induces a bijection Θ : 𝔇(p) ↔ 𝔇′(p) for 1 ≤ p ≤ d), and (ii) w′ (Θ (J)) = w (J), for all J ∈ 𝔇. It is easy to verify the following equivalence-implications:

𝔻 ≃ 𝔻′ ⟺ ( \begin{matrix}G_{𝔻} and G_{𝔻′} belong to the same \\ conjugacy class (within SL & (d, ℂ) & ) \end{matrix} ) ⟺ (ℂ^{d}/G_{𝔻}, [0]) ≅ (ℂ^{d}/G_{𝔻′}, [0])

Convention A. From now on, as representatives 𝔻 = (𝔇, w) of the equivalence-classes from

({all special data (w.r.t. d)} / ≃) ,

we shall consider (without loss of generality) only 𝔇’s all of whose index subsets J have the form J = {ν, ν + 1, …, ξ − 1, ξ}, 1 ≤ ν ≤ ξ ≤ d, i.e., contiguous segments of {1, 2, …, d}. Convention B. We refer to the minimum and maximum elements of J = {ν, ν + 1, …, ξ − 1, ξ} with the notation ν = ν_{J} and ξ = ξ_{J}.

Definition 5.6. Let d ≥ 2 be an integer, 𝔻 = (𝔇, w) a special datum w.r.t. d, and

J_{0} = {ν_{0}, ν_{0} + 1, …, ξ_{0} − 1, ξ_{0}} ∈ 𝔇

a fixed index-set. We define the subdatum 𝔻_{J_{0}}= (𝔇_{J_{0}}, w_{J_{0}}) of 𝔻 relative to J_{0} by

𝔇_{J_{0}} := \{J ∈ 𝔇 | J ⊆ J_{0} \} , and w_{J_{0}} (J) := \frac{w (J)}{w (J_{0})}, for all J ∈ 𝔇_{J_{0}}.

The subdatum 𝔻_{J_{0}}= (𝔇_{J_{0}}, w_{J_{0}}) can be viewed as an “autonomous” special datum 𝔻′= (𝔇′, w′) w.r.t. ξ_{0} − ν_{0} + 1 via the bijection

ϑ_{0} : 𝔇_{J_{0}} → 𝔇′ = {1, 2, …, ξ_{0} − ν_{0} + 1}

defined by

ϑ_{0} (ν_{0}) = 1, ϑ_{0} (ν_{0} + 1) = 2, …,ϑ_{0} (ξ_{0}) = ξ_{0} − ν_{0} + 1,

(where w′ (J) := w_{J_{0}} (ϑ_{0}^{−1} (J)), for all J ∈ 𝔇′). Convention C. From now on we shall use a subdatum of a special datum 𝔻 relative to some J_{0} in both “roles” without referring explicitly to the identification map ϑ_{0}. (In the first case we shall emphasize the induced embedding R_{𝔻_{J_{0}}} ↪ R_{𝔻} or G_{𝔻_{J_{0}}} ⊂ G_{𝔻}; in the latter, its own right to enjoy properties (i)-(v) of 5.1).

Next we recall some definitions concerning trees. We shall mostly use standard terminology. As usual, a graph is determined by the set of its vertices and the set of its edges. A vertex v is a neighbour of another vertex v′, if v and v′ are adjacent in the considered graph. A graph is connected if there is a path between any two vertices of it. A cycle in a graph is a simple path from a vertex to itself. By a weighted graph is meant a graph with weights assigned to its vertices. Two weighted graphs are isomorphic to each other (denoted by ≅_{wg} ) if there exists a bijection between the sets of their vertices preserving both adjacency and weights.

A graph having no cycle is acyclic. A tree is a connected acyclic graph. (A trivial tree is a tree consisting of only one vertex). An arbitrary acyclic graph is called a forest. (So all connected components of a forest are trees). A leaf is a vertex of degree at most 1, i.e., a vertex being contained in at most one edge. A rooted tree distinguishes one vertex as its root. If v is a non-root vertex of a rooted tree, then its parent is the neighbour of v on the path connecting v with its root. The children of v are its other neighbours. In this case, the leaves are exactly the vertices without children.

A rooted plane tree is a rooted tree which is embedded in the plane, i.e., a rooted tree which is endowed with a left-to-right ordering specified for the children of each vertex. (By a plane forest we mean a forest having only rooted plane trees as connected components).

Definition 5.7. Let d be an integer ≥ 2 and 𝔻 = (𝔇, w) be a special datum w.r.t. d. The Watanabe forest associated to 𝔻 is defined to be a weighted plane forest W_{𝔻} whose vertices are in one-to-one correspondence with the elements of 𝔇 and whose edges are to be drawn as follows: If J, J′ are two elements of 𝔇, then the corresponding vertices of W_{𝔻}, say v_{J}, v_{J′}, should be joinable by an edge of W_{𝔻} if and only if either J ⊏ J′ or J′ ⊏ J. If W_{𝔻} happens to be a tree, then we shall speak of a Watanabe tree (being associated to 𝔻). For a J ∈ 𝔇, we associate the weight w (v_{J}) := w (J) to v_{J}. Moreover, for J, J′ ∈ 𝔇, with J ⊏ J′, using Watanabe’s original upside-down convention, we shall join v_{J} and v_{J′} in such a way, that v_{J′} lies over v_{J}. It is therefore useful to regard the edge connecting v_{J} with v_{J′} as “directed” and denote it by \overrightarrow{v_{J}, v_{J′}}. (For typographical reasons, in our figures we shall denote the vertices of the forest W_{𝔻} by v_{J}, w(J) enclosing also their weights.)

Remark 5.8. (i) By 5.1 (i) we have ⋃^{•} {J ∈ 𝔇 | v_{J} leaf} = {1, 2, …, d} (set-theoretically). (ii) By definition,

𝔻 ≃ 𝔻′ ⟺ W_{𝔻} ≅_{wg} W_{𝔻′}

(iii) Let 𝔻 = (𝔇, w) be a special datum and assume that its Watanabe forest W_{𝔻} has connected components

W_{𝔻^{[1]}}, W_{𝔻^{[2]}}, …, W_{𝔻^{[κ−1]}}, W_{𝔻^{^{[}^{κ}^{]}}}

whose vertices are in one-to-one correspondence with the elements of

𝔇^{[1]}, 𝔇^{[2]}, …, 𝔇^{[κ−1]}, 𝔇^{[κ]}

for a partition 𝔇 = ⋃_{1≤i≤κ}^{•} 𝔇^{[i]} of 𝔇. Then each component W_{𝔻^{[i]}} has root v_{J_{♦}^{[i]}}, with J_{♦}^{[i]}

as in 5.4, and 𝔻^{[i]} := (𝔇^{[i]}, w |_{𝔇^{[i]}} ) is to be identified with the subdatum 𝔻_{J_{♦}^{[i]}} (in the sense of

5.6), for 1 ≤ i ≤ κ. Furthermore, the group G_{𝔻} splits into the direct product

G_{𝔻} = G_{𝔻_{1}} × G_{𝔻_{2}} × ⋯ × G_{𝔻_{κ−1}} × G_{𝔻_{κ}}.

Remark 5.9. Let 𝔻 = (𝔇, w) be a special datum (w.r.t. an integer d ≥ 2). Then we have:

(i) (ℂ^{d}/G_{𝔻}, [0]) ≅ (U_{σ_{0}}, orb (σ_{0})) is a singularity if and only if W_{𝔻} contains at least one nontrivial tree. In this case,

splcod (orb (σ_{0}) ; U_{σ_{0}}) = d − # {all trivial trees of W_{𝔻}} .

(ii) (ℂ^{d}/G_{𝔻}, [0]) is a d-dimensional msc-singularity if and only if all connected components W_{𝔻^{[i]}} of W_{𝔻} are non-trivial trees. In this case, for d ≥ 3, all G_{𝔻}’s are abelian, non-cyclic groups. (iii) (ℂ^{d}/G_{𝔻}, [0]) is a d-dimensional absolutely unbreakable msc-singularity if and only if W_{𝔻} itself is a non-trivial Watanabe tree. If, in addition, this singularity is a hypersurface-singularity, then Theorem 5.3 can be simplified as follows:

Proposition 5.10. Let G be a finite abelian subgroup of SL(d, ℂ), d ≥ 2, of order l ≥ 2. Then (X (N_{G}, Δ_{G}) , orb (σ_{0})) is an absolutely unbreakable hypersurface-msc-singularity if and only if G is conjugate (within SL(d, ℂ)) to a group of the form

G (d; k) := ⟨\{diag (1, 1, …, 1, \underbrace{ζ_{k}}_{i-th position}, \underbrace{ζ_{k}^{−1}}_{(i+1)-position}, 1, …, 1, 1) | 1 ≤ i ≤ d − 1\}⟩ ,

for a k ≥ 2. In this case, l = k^{d−1}, and one has the following analytic germ isomorphisms

(X (N_{G}, Δ_{G}) , orb (σ_{0})) ≅ (ℂ^{d}/G (d; k) , [0_{ℂ^{d}}])

and

(ℂ^{d}/G (d; k) , [0_{ℂ^{d}}]) ≅ ({(z_{0}, .., z_{d}) ∈ ℂ^{d+1} | z_{0}^{k} = ∏_{i=1}^{d} z_{i} } , 0_{ℂ^{d+1}}) .  (5.1)

In particular, G (d; k) = G_{𝔻} ≅ (ℤ/kℤ)^{d−1}, where 𝔻 = (𝔇, w) denotes the special datum with

𝔇 = {{1} , {2} , …, {d} , {1, 2, …, d}} , w ({j}) = k, 1 ≤ j ≤ d, and w ({1, 2, …, d}) = 1.

Its associated Watanabe tree is given in Figure 4.

v_{{1}}, k  v_{{2}}, k ⋯  ⋯ v_{{d−1}}, k  v_{{d}}, k [figure: v_{{1,...,d}}, 1]

Figure 4

Definition 5.11. We shall call the above singularities (5.1) (d; k)-hypersurface-singularities, or simply (d; k)-hypersurfaces.

Remark 5.12. (i) Using the projection map

{(z_{0}, z_{1}, …, z_{d}) ∈ ℂ^{d+1} | z_{0}^{k} = ∏_{i=1}^{d} z_{i} } ∋ (z_{0}, z_{1}, …, z_{d}) ⟼ (z_{1}, …, z_{d}) ∈ ℂ^{d}

one may regard (ℂ^{d}/G (d; k) , [0_{ℂ^{d}}]) as the total space of a k-sheeted covering of (ℂ^{d}, 0_{ℂ^{d}}) having the union of all coordinate hyperplanes of ℂ^{d} as its branching locus. (ii) Removing the assumption for (X (N_{G}, Δ_{G}) , orb (σ_{0})) to be absolutely unbreakable, we obtain a direct product of such hypersurface-singularities.

To present a fairly short proof of our main Theorem 1.2 (by avoiding to work simultaneously with forests and triangulations), we introduce a new term, under the name “Watanabe simplex.”

Definition 5.13. Let d ≥ 0 be an integer and N a d-dimensional lattice in N_{ℝ} ≅ ℝ^{d}. The Watanabe simplices w.r.t. N are the lattice simplices s satisfying

aff_{ℤ}(s) = aff(s) ∩ N

that arise in the following manner:

(i) Every 0-dimensional lattice simplex s = {n}, n ∈ N, is a Watanabe simplex. (ii) A lattice simplex s ⊂ N_{ℝ} of dimension d′ ≥ 1 is a Watanabe simplex if and only if

• either s = s_{1} ∗ s_{2}, where s_{1}, s_{2} are Watanabe simplices of dimensions d_{1}, d_{2} ≥ 0 with d′ = d_{1} + d_{2} + 1, , with respect to sublattices N_{1} ⊂ aff(s_{1}), N_{2} ⊂ aff(s_{2}) of N, such that N = aff_{ℤ}(N_{1} ∪ N_{2}),

• or s is a lattice translate of some dilation λ s′, where λ ≥ 2 is an integer, and s′ is a d′-dimensional Watanabe simplex w.r.t. N. (These conditions are mutually exclusive; with this definition every affine integral transformation that preserves N also preserves the Watanabe simplices of N.)

Example 5.14. The 2-dimensional lattice simplices (A) and (B) (w.r.t. ℤ^{2}) of Figure 5 are Watanabe simplices, whereas (C) and (D) are not Watanabe simplices.

[figure: (A); (B); (C); (D)]

Figure 5

We are now going to prove the following:

Theorem 5.15 (Reduction-Theorem). Let d ≥ 2 be an integer, 𝔻 = (𝔇, w) a non-trivial special datum (w.r.t. d), W_{𝔻} the associated Watanabe forest, and (ℂ^{d}/G_{𝔻}, [0]) the corresponding abelian quotient c.i.-singularity. If we identify the underlying space ℂ^{d}/G_{𝔻} with the toric variety X (N_{G_{𝔻}}, Δ_{G_{𝔻}}) = U_{σ_{0}} (as in § 2, (i)), then the junior simplex 𝔰_{G_{𝔻}} is a (non-basic) Watanabe simplex; and conversely, every (non-basic) Watanabe simplex w.r.t. an N ≅ ℤ^{d} is (up to an affine integral transformation) the junior simplex corresponding to some abelian quotient c.i.-singularity.

The proof will be done in four steps. We begin with the case in which W_{𝔻} is a Watanabe tree.

• First step. Let W_{𝔻} denote a non-trivial Watanabe tree. At first we explain how the labeled weights naturally lead to “free parameters”. By 5.4 (i), J_{♦} = {1, 2, …, d} is the maximal element of 𝔇 (w.r.t. inclusion “⊆”). For any J ∈ 𝔇, J ⫋ J_{♦}, there exists a unique chain of oversets

J_{0} = J ⊏ J_{1} ⊏ J_{2} ⊏ ⋯ ⊏ J_{ρ−1} ⊏ J_{ρ} = J_{♦} (with ρ ≥ 1)

which, in fact, corresponds to the directed path

[\overrightarrow{v_{J_{0}}, v_{J_{1}}}, \overrightarrow{v_{J_{1}}, v_{J_{2}}}, …, \overrightarrow{v_{J_{ρ−2}}, v_{J_{ρ−1}}}, \overrightarrow{v_{J_{ρ−1}}, v_{J_{ρ}}}]

connecting v_{J} with the root v_{J_{♦}} of W_{𝔻}. By 5.1 (iii)-(iv), the weights of the vertices of W_{𝔻} can be written in the form

w (v_{J}) = w (J) = \{ \begin{matrix}1 & if J = J_{♦} \\ ∏_{i=0}^{ρ−1} k_{J_{i},J_{i+1}} & otherwise \end{matrix}

for integers k_{J_{0},J_{1}}, k_{J_{1},J_{2}}, …, k_{J_{ρ−1},J_{ρ}} (which are ≥ 2), assigned to the edges

\overrightarrow{v_{J_{0}}, v_{J_{1}}}, \overrightarrow{v_{J_{1}}, v_{J_{2}}}, …, \overrightarrow{v_{J_{ρ−2}}, v_{J_{ρ−1}}}, \overrightarrow{v_{J_{ρ−1}}, v_{J_{ρ}}} .

Definition 5.16. By the above procedure we assign an integer k_{J,J′} ≥ 2 to every edge \overrightarrow{v_{J}, v_{J′}} of W_{𝔻} (J ⊏ J′). These integers will be called the free parameters of 𝔻 (or of W_{𝔻}). For fixed J′, the value k_{J,J′} is the same for all J’s for which J ⊏ J′ (see 5.1 (v)). Thus for k_{J,J′} we can use the alternative notation

k_{J,J′} =: k_{ν_{J′},ξ_{J′}}

which indicates the “free parameter source” (with ν_{J′} and ξ_{J′} as defined in 5.5). For example, the free parameter k of (d; k)-hypersurfaces equals k_{1,d}.

Example 5.17. Up to an analytic isomorphism, all four-dimensional absolutely unbreakable, msc-, quotient c.i.-singularities are either (4; k)-hypersurfaces or singularities corresponding to exactly one of the four special data 𝔻, whose associated Watanabe trees W_{𝔻} and free parameters k... are depicted in Figures 6a)-6d).

[figure: v_{{1}}, k_{1,4} · k_{1,3}; v_{{2}}, k_{1,4} · k_{1,3}; v_{{3}}, k_{1,4} · k_{1,3}; k_{1,3}; k_{1,3}; k_{1,3}; v_{{1,2,3}}, k_{1,4}; v_{{4}}, k_{1,4}; k_{1,4}; k_{1,4}; v_{{1,2,3,4}}, 1]

Figure 6a)

[figure: v_{{1}}, k_{1,4} · k_{1,2}; v_{{2}}, k_{1,4} · k_{1,2}; k_{1,2}; k_{1,2}; v_{{1,2}}, k_{1,4}; v_{{3}}, k_{1,4}; v_{{4}}, k_{1,4}; k_{1,4}; k_{1,4}; k_{1,4}; v_{{1,2,3,4}}, 1]

Figure 6b)

[figure: v_{{1}}, k_{1,4} · k_{1,2}; v_{{2}}, k_{1,4} · k_{1,2}; k_{1,2}; k_{1,2}; v_{{1,2}}, k_{1,4}; v_{{3}}, k_{1,4} · k_{3,4}; v_{{4}}, k_{1,4} · k_{3,4}; k_{3,4}; k_{3,4}; v_{{3,4}}, k_{1,4}; k_{1,4}; k_{1,4}; v_{{1,2,3,4}}, 1]

Figure 6c)

[figure: v_{{1}}, k_{1,4} · k_{1,3} · k_{1,2}; v_{{2}}, k_{1,4} · k_{1,3} · k_{1,2}; k_{1,2}; k_{1,2}; v_{{1,2}}, k_{1,4} · k_{1,3}; v_{{3}}, k_{1,4} · k_{1,3}; k_{1,3}; k_{1,3}; v_{{1,2,3}}, k_{1,4}; v_{{4}}, k_{1,4}; k_{1,4}; k_{1,4}; v_{{1,2,3,4}}, 1]

Figure 6d)

• Second step. Let W_{𝔻} be, as before, a non-trivial Watanabe tree associated to a special datum 𝔻 = (𝔇, w) w.r.t. a d ≥ 2. Keeping conventions A- C (of 5.5, 5.6) in mind, we fix an enumeration, say J_{1}, …, J_{p}, of all members of {J ∈ 𝔇 | J ⊏ J_{♦}}, so that

J_{i} = {ν_{i}, ν_{i} + 1, …, ξ_{i} − 1, ξ_{i}} for 1 ≤ i ≤ p,

where

ν_{1} = 1 ≤ ξ_{1}, ν_{i} = ξ_{i−1} + 1, for 2 ≤ i ≤ p and ν_{p} ≤ ξ_{p} = d,

and denote by W_{1}, …, W_{p} the subtrees of W_{𝔻} which begin with v_{J_{1}}, …, v_{J_{p}} as in Figure 7.

⋯ W_{1}· · · · · · v_{J_{1}}, k_{1,d}  ⋯ W_{2}· · · · · · v_{J_{2}}, k_{1,d}  ⋯  · · · · · · W_{p−1}· · · · · · v_{J_{p−1}}, k_{1,d}  ⋯ W_{p}· · · · · · v_{J_{p}}, k_{1,d} [figure: k_{1,d}; k_{1,d}; k_{1,d}; k_{1,d}; v_{{1,...,d}}, 1] Figure 7

To the subtrees W_{1}, …, W_{p} we assign the “autonomous” Watanabe trees W_{𝔻_{1}}, …, W_{𝔻_{p}}, induced by the subdata 𝔻_{i} = (𝔇_{i}, w_{i}) of 𝔻, where

𝔇_{i} := 𝔇_{J_{i}}, w_{i} := w_{J_{i}}, for 1 ≤ i ≤ p,

(in the notation introduced in 5.6). W_{𝔻_{i}} is derived from W_{𝔻} after the killing of the parent v_{J_{♦}} of v_{J_{i}} (w.r.t. W_{𝔻}), the christening of v_{J_{i}} as its root, and the adoption of the appropriate weights. In other words, the children of the root of W_{𝔻} become autonomous roots (and parents).

• In the arguments of the next four lemmas we shall use induction on the cardinality number d of J_{♦}, i.e., on the dimension of the singularity. (That all assertions are correct for d = 2 can be checked easily, and will be therefore omitted). Our “induction hypothesis” is that all assertions (formulated below) are true for all Watanabe trees having roots corresponding to maximal index-sets of cardinality < d, and, in particular, true for W_{𝔻_{1}}, …, W_{𝔻_{p}}. (We do not exclude the possibility that some of the W_{𝔻_{i}}’s are trivial trees.)

Lemma 5.18. (i) If we define

𝔜_{𝔻} := ⋃_{J∈𝔇} \{diag(1, …, 1, \underbrace{ζ_{w(J′)}}_{ν_{J} -th pos.}, 1, …, 1, \underbrace{ζ_{w(J′)}^{−1}}_{ν_{J′}-th pos.}, 1, …, 1) | \begin{matrix}for all J′ ∈ 𝔇 \\ with J′ ⊏ J \\ and ν_{J} < ν_{J′} \end{matrix} \} ,

then G_{𝔻} = ⟨𝔜_{𝔻}⟩, and 𝔜_{𝔻} is a minimal generating system for G. (ii) Identifying ℂ^{d}/G_{𝔻} with X (N_{G_{𝔻}}, Δ_{G_{𝔻}}) = U_{σ_{0}} as in §2(i), we obtain

N_{G_{𝔻}} = ∑_{j=1}^{d} ℤ e_{j} + ℤ 𝔑_{𝔻},

where {e_{1}, …, e_{d}} are the unit vectors of (N_{G_{𝔻}})_{ℝ} ≅ ℝ^{d}, and

𝔑_{𝔻} := ⋃_{J∈𝔇} \{\frac{1}{w (J′)} (e_{ν_{J}} − e_{ν_{J′}} ) | \begin{matrix}for all J′ ∈ 𝔇 \\ with J′ ⊏ J \\ and ν_{J} < ν_{J′} \end{matrix} \} .

(iii) Furthermore, # (𝔜_{𝔻}) = # (𝔑_{𝔻}) = d − 1 .

Proof. (i) Obviously, each element of the generating system given in the definition 5.2 of G_{𝔻} can be written as product of elements of 𝔜_{𝔻}, and any group generated by a proper subset of 𝔜_{𝔻} is a proper subgroup of G_{𝔻}. (ii) This follows from the usual representation of the group elements of 𝔜_{𝔻} by lattice points in N_{G_{𝔻}}. (iii) By induction hypothesis, # (𝔜_{𝔻_{i}}) = # (𝔑_{𝔻_{i}}) equals ν_{i} − ξ_{i}, for all i, 1 ≤ i ≤ p. Hence,

# (𝔜_{𝔻}) = # (𝔑_{𝔻}) = ∑_{i=1}^{p} (ν_{i} − ξ_{i}) + (p − 1) = ∑_{i=1}^{p} ν_{i} − ∑_{i=2}^{p} ξ_{i} + (p − 2)

= ∑_{i=1}^{p} ν_{i} − ∑_{i=2}^{p} (ν_{i−1} + 1) + (p − 2) = d − 1,

and the proof is completed. _{□}

In the following we denote by {n_{2}^{(𝔻)}, n_{3}^{(𝔻)}, …, n_{d}^{(𝔻)}} an enumeration of the d − 1 elements of 𝔑_{𝔻} such that

n_{ν_{i}}^{(𝔻)} = \frac{1}{k_{1,d}} (e_{1} − e_{ν_{i}}) for 2 ≤ i ≤ p.  (5.2)

Furthermore, we assume that we have chosen an enumeration of the elements {n_{ν_{i}+1}^{(𝔻_{i})}, …, n_{ξ_{i}}^{(𝔻_{i})}} of 𝔑_{𝔻_{i}} such that

n_{ν_{i}+1}^{(𝔻)} = \frac{1}{k_{1,d}} n_{ν_{i}+1}^{(𝔻_{i})}, n_{ν_{i}+2}^{(𝔻)} = \frac{1}{k_{1,d}} n_{ν_{i}+2}^{(𝔻_{i})}, …, n_{ξ_{i}}^{(𝔻)} = \frac{1}{k_{1,d}} n_{ξ_{i}}^{(𝔻_{i})}, for 1 ≤ i ≤ p.

Lemma 5.19.

(i) {e_{1}, n_{2}^{(𝔻)}, n_{3}^{(𝔻)}, …, n_{d}^{(𝔻)}} is a ℤ-basis of the lattice N_{G_{𝔻}}.

(ii) {n_{2}^{(𝔻)}, n_{3}^{(𝔻)}, …, n_{d}^{(𝔻)}} forms a ℤ-basis of N_{G_{𝔻}} ∩ lin ({e_{1} − e_{2}, e_{1} − e_{3}, …, e_{1} − e_{d}}) .

Proof. (i) By definition of N_{G_{𝔻}} it suffices to show that ℤ^{d} ⊂ ℤ{e_{1}, n_{2}^{(𝔻)}, n_{3}^{(𝔻)}, …, n_{d}^{(𝔻)}}. By induction hypothesis, we may assume

ℤ e_{ν_{i}} + ℤ e_{ν_{i}+1} + ⋯ + ℤ e_{ξ_{i}} ⊂ ℤ {e_{ν_{i}}, n_{ν_{i}+1}^{(𝔻_{i})}, …, n_{ξ_{i}}^{(𝔻_{i})}} , for 1 ≤ i ≤ p.

Since k_{1,d} ∈ ℤ we also have

ℤ e_{ν_{i}} + ℤ e_{ν_{i}+1} + ⋯ + ℤ e_{ξ_{i}} ⊂ ℤ {e_{1}, n_{ν_{i}}^{(𝔻)}, n_{ν_{i+1}}^{(𝔻)}, …, n_{ξ_{i}}^{(𝔻)}}

and thus

⋃_{i=2}^{p} {e_{1}, n_{ν_{i}}^{(𝔻)}, n_{ν_{i}+1}^{(𝔻)}, …, n_{ξ_{i}}^{(𝔻)}} = {e_{1}, n_{2}^{(𝔻)}, n_{3}^{(𝔻)}, …, n_{d}^{(𝔻)}}

is indeed a ℤ-basis of ℤ^{d}. (ii) By (i), it suffices to prove that each vector n_{i}^{(𝔻)}, 2 ≤ i ≤ d, belongs to the subspace lin({e_{1} − e_{2}, …, e_{1} − e_{d}}). By (5.2) this is true for all n_{ν_{i}}^{(𝔻)}, 2 ≤ i ≤ p. By induction hypothesis, we may assume that

n_{ν_{i}+j}^{(𝔻_{i})} ∈ lin ({e_{ν_{i}} − e_{ν_{i}+1}, …, e_{ν_{i}} − e_{ξ_{i}}}) , 1 ≤ i ≤ p, 1 ≤ j ≤ ξ_{i} − ν_{i},

and hence that

n_{ν_{i}+j}^{(𝔻)} ∈ lin ({e_{ν_{i}} − e_{ν_{i}+1}, …, e_{ν_{i}} − e_{ξ_{i}}})

too. Since e_{ν_{i}} − e_{ν_{i}+j} = (e_{1} − e_{ν_{i}+j}) − (e_{1} − e_{ν_{i}}), we are done. _{□}

• Third step. Since the lattice N_{G_{𝔻}} is “skew” and d-dimensional, it is not so convenient to work directly with it. For this reason, we shall perform two affine transformations (cf. Lemma 3.4), such that N_{G_{𝔻}} ∩ aff(𝔰_{G_{𝔻}}) = N_{G_{𝔻}} ∩ aff(e_{1}, …, e_{d}) becomes the standard lattice ∑_{i=2}^{d} ℤe_{i} ≅ ℤ^{d−1} ⊂ ℝ^{d}.

Define first Φ_{1}^{𝔻} : (N_{G_{𝔻}})_{ℝ} → (N_{G_{𝔻}})_{ℝ} by

Φ_{1}^{𝔻} (x) := e_{1} − x .

Obviously,

Φ_{1}^{𝔻} (𝔰_{G_{𝔻}}) = conv ({0, e_{1} − e_{2}, e_{1} − e_{3}, …, e_{1} − e_{d}}) ,

and Φ_{1}^{𝔻}(N_{G_{𝔻}}) = N_{G_{𝔻}}. After that define the affine (linear) transformation (cf. Lemma 5.19) Φ_{2}^{𝔻} : (N_{G_{𝔻}})_{ℝ} → (N_{G_{𝔻}})_{ℝ} by setting Φ_{2}^{𝔻}(e_{1}) = e_{1} and

Φ_{2}^{𝔻} (\frac{1}{w (J′)} (e_{ν_{J}} − e_{ν_{J′}} )) := e_{ν_{J′}} , for all elements \frac{1}{w (J′)} (e_{ν_{J}} − e_{ν_{J′}} ) of 𝔑_{𝔻} .

Now let Φ^{𝔻} : (N_{G_{𝔻}})_{ℝ} → (N_{G_{𝔻}})_{ℝ} be the decomposition Φ^{𝔻} := Φ_{2}^{𝔻} ∘ Φ_{1}^{𝔻} and let

𝔰̃_{G_{𝔻}} := Φ^{𝔻}(𝔰_{G_{𝔻}}) and Ñ_{G_{𝔻}}^{𝔰} := Φ^{𝔻} (N_{G_{𝔻}} ∩ aff(𝔰_{G_{𝔻}})) .

Observe that Ñ_{G_{𝔻}}^{𝔰} = ∑_{i=2}^{d} ℤe_{i} ≅ ℤ^{d−1} ⊂ ℝ^{d}. To provide a convenient description of the vertices of Φ^{𝔻}(𝔰_{G_{𝔻}}) incorporating our inductive argumentation, we denote the vertex set of 𝔰̃_{G_{𝔻_{i}}} = Φ^{𝔻_{i}}(𝔰_{G_{𝔻_{i}}}) ⊂ ∑_{j=ν_{i}+1}^{ξ_{i}} ℤe_{j} corresponding to the tree W_{𝔻_{i}} by

vert \binom{}{𝔰_{G_{𝔻_{i}}}} = \{0, 𝔶_{ν_{i}+1}^{(𝔻_{i})}, 𝔶_{ν_{i}+2}^{(𝔻_{i})}, …, 𝔶_{ξ_{i}}^{(𝔻_{i})}\} ,

in such a way that

𝔶_{ν_{i}+j}^{(𝔻_{i})} := Φ^{𝔻_{i}} (e_{ν_{i}+j}) = Φ_{2}^{𝔻_{i}} (e_{ν_{i}} − e_{ν_{i}+j}) , 1 ≤ i ≤ p, 1 ≤ j ≤ ν_{i} − ξ_{i}.

Analogously, we label the vertices in vert(𝔰̃_{G_{𝔻}}) = {0, 𝔶_{2}^{(𝔻)}, 𝔶_{3}^{(𝔻)}, …, 𝔶_{d}^{(𝔻)}} with

𝔶_{j+1}^{(𝔻)} := Φ^{𝔻} (e_{j+1}) = Φ_{2}^{𝔻} (e_{1} − e_{j+1}) for 1 ≤ j ≤ d − 1.

Lemma 5.20. Using the notation introduced above,

𝔶_{ν_{i}}^{(𝔻)} = k_{1,d} · e_{ν_{i}}, 2 ≤ i ≤ p,  (5.3)
𝔶_{ν_{1}+j}^{(𝔻)} = k_{1,d} · 𝔶_{ν_{1}+j}^{(𝔻_{1})}, 1 ≤ j ≤ ξ_{1} − ν_{1},  (5.4)
𝔶_{ν_{i}+j}^{(𝔻)} = k_{1,d} · (𝔶_{ν_{i}+j}^{(𝔻_{i})} + e_{ν_{i}}) , 2 ≤ i ≤ p, 1 ≤ j ≤ ξ_{j} − ν_{j}.  (5.5)

Proof. (5.3) follows from (5.2) because

𝔶_{ν_{i}}^{(𝔻)} = Φ_{2}^{𝔻} (e_{1} − e_{ν_{i}}) = k_{1,d} · Φ_{2}^{𝔻} (n_{ν_{i}}^{(𝔻)}) = k_{1,d} · e_{ν_{i}} .

On the other hand,

Φ_{2}^{𝔻} (e_{ν_{i}} − e_{ν_{i}+j}) = k_{1,d} · Φ_{2}^{𝔻_{i}} (e_{ν_{i}} − e_{ν_{i}+j}) = k_{1,d} · 𝔶_{ν_{i}+j}^{(𝔻_{i})}

and (5.4) is clear by setting i = 1 (ν_{1} = 1). Finally, to get (5.5) we write

Φ_{2}^{𝔻} (e_{1} − e_{ν_{i}+j}) = Φ_{2}^{𝔻} ((e_{1} − e_{ν_{i}}) + (e_{ν_{i}} − e_{ν_{i}+j}))
= Φ_{2}^{𝔻} (e_{1} − e_{ν_{i}}) + Φ_{2}^{𝔻} (e_{ν_{i}} − e_{ν_{i}+j}) = k_{1,d} · e_{ν_{i}} + k_{1,d} · 𝔶_{ν_{i}+j}^{(𝔻_{i})}

and we are done. _{□}

Lemma 5.21. The image 𝔰̃_{𝔻} of the junior simplex 𝔰_{G_{𝔻}} under Φ^{𝔻} satisfies the following:

(i)

∑_{j=2}^{d} ℤe_{j} = Ñ_{G_{𝔻}}^{𝔰} = aff_{ℤ} ({ \atop N_{G_{𝔻_{1}}}^{𝔰}} ∪ ({ \atop N_{G_{𝔻_{2}}}^{𝔰}} + e_{ν_{2}}) ∪ … ∪ ({ \atop N_{G_{𝔻p}}^{𝔰}} + e_{ν_{p}})) .

(ii) d − 1 = dim \binom{}{𝔰_{G_{𝔻}}} = dim \binom{}{𝔰_{G_{𝔻_{1}}}} + ∑_{i=2}^{p} dim (𝔰̃_{G_{𝔻_{i}}} + e_{ν_{i}}) + p − 1,

(iii) The simplex 𝔰̃_{G_{𝔻}} can be expressed as a dilation of p − 1 simplex-joins

𝔰̃_{G_{𝔻}} = k_{1,d} (𝔰̃_{G_{𝔻_{1}}} ∗ (𝔰̃_{G_{𝔻_{2}}} + e_{ν_{2}}) ∗ ⋯ ∗ (𝔰̃_{G_{𝔻p}} + e_{ν_{p}})) ⊂ ∑_{j=2}^{d} ℤe_{j} ≅ ℤ^{d−1}.

Proof. By construction we have { \atop N_{G_{𝔻_{i}}}^{𝔰}} = ∑_{j=ν_{i}+1}^{ξ_{i}} ℤe_{j} which shows (i). Obviously, dim(𝔰̃_{G_{𝔻_{i}}}) = ξ_{i}−ν_{i} and since the simplices 𝔰̃_{G_{𝔻_{1}}}, (𝔰̃_{G_{𝔻_{2}}} +e_{ν_{2}}), …, (𝔰̃_{G_{𝔻p}} +e_{ν_{p}}) are contained in complementary subspaces, (ii) and (iii) are immediate consequences of Lemma 5.20. _{□}

Example 5.22. If we consider the 7-dimensional absolutely unbreakable, msc-, quotient c.i.- singularity corresponding to the special datum 𝔻 whose Watanabe tree W_{𝔻} shown in Figure 8, then the image 𝔰̃_{𝔻} of the junior simplex 𝔰_{𝔻} under Φ^{𝔻} is the Watanabe simplex defined as convex hull of the seven vectors:

\begin{matrix}(0, 0, 0, 0, 0, 0, 0)^{⊺}, \\ (0, 0, k_{1,7}, 0, 0, 0, 0)^{⊺}, \\ (0, 0, 0, 0, k_{1,7}, k_{1,7}k_{4,7}k_{4,5}, 0)^{⊺}, \\ (0, 0, 0, k_{1,7}, 0, k_{1,7}k_{4,7}, k_{1,7}k_{4,7}k_{4,5})^{⊺}. \end{matrix}  \begin{matrix}(0, k_{1,7}k_{1,2}, 0, 0, 0, 0, 0)^{⊺}, \\ (0, 0, 0, k_{1,7}, 0, 0, 0)^{⊺}, \\ (0, 0, 0, k_{1,7}, 0, k_{1,7}k_{4,7}, 0)^{⊺}, \end{matrix}

[figure: k_{1,7}k_{1,2}^{v_{{1}}}; k_{1,7}k_{1,2}^{v_{{2}}}; k_{1,2}; k_{1,2}; k_{1,7}^{v_{{1,2}}}; k_{1,7}^{v_{{3}}}; k_{1,7}k_{4,7}k_{4,5}^{v_{{4}}}; k_{1,7}k_{4,7}k_{4,5}^{v_{{5}}}; k_{4,5}; k_{4,5}; k_{1,7}k_{4,7}^{v_{{4,5}}}; k_{1,7}k_{4,7}k_{6,7}^{v_{{6}}}; k_{1,7}k_{4,7}k_{6,7}^{v_{{7}}}; k_{6,7}; k_{6,7}; k_{1,7}k_{4,7}^{v_{{6,7}}}; k_{4,7}; k_{4,7}; k_{1,7}^{v_{{4,5,6,7}}}; k_{1,7}; k_{1,7}; k_{1,7}; 1^{v_{{1,2,3,4,5,6,7}}}]

Figure 8

• Fourth step. By Lemma 5.21 and by construction, it is now clear that, up to an affine integral transformation, 𝔰_{G_{𝔻}} is a Watanabe simplex whenever W_{𝔻} is a Watanabe tree. If W_{𝔻} is a Watanabe forest, then we recall the “superscripts in square brackets” from 5.8 (iii): Taking its connected components to be W_{𝔻^{[i]}}, 1 ≤ i ≤ κ, the group G_{𝔻} splits into the direct product of G_{𝔻^{[i]}}’s, and by arguments similar to those of the previous steps we can show that 𝔰_{G_{𝔻}} is again a transformed join of 𝔰_{G_{𝔻^{[1]}}} with the junior simplices 𝔰_{G_{𝔻^{[i]}}}, 2 ≤ i ≤ κ, corresponding to the other components, after translating by unit vectors. (We do not need an extra dilation in this case). Hence, the one direction of the Reduction Theorem 5.15 is completely proved. The proof of the converse statement is based on the “backtracking method”, and on the fact that any toric affine variety being associated to a rational s.c.p. cone supported by a (non-basic) simplex, is the underlying space of an abelian quotient singularity; it is therefore omitted.

Example 5.23. Up to isomorphism, the only four-dimensional abelian, msc-, quotient c.i.- singularity which “breaks” is that corresponding to the datum 𝔻 and Watanabe forest W_{𝔻} of Figure 9. (In fact, it splits into two Hirzebruch-Jung singularities of types A_{k_{1,2}−1} and A_{k_{3,4}−1}, respectively).

[figure: v_{{1}}, k_{1,2}; v_{{2}}, k_{1,2}; k_{1,2}; k_{1,2}; v_{{1,2}}, 1; v_{{3}}, k_{3,4}; v_{{4}}, k_{3,4}; k_{3,4}; k_{3,4}; v_{{3,4}}, 1]

Figure 9

The junior tetrahedron 𝔰_{G_{𝔻}} (which is the join of two 1-dimensional Watanabe simplices), is drawn in Figure 10, for the values of free parameters k_{1,2} = 3, k_{3,4} = 5.

[figure: •; e_{1}; •; e_{2}; •; e_{4}; •; e_{3}; •; •; •; •; •; •]

Figure 10

6. Final step: Basic, coherent triangulations of the junior simplex

In this section we give the proof of our Main Theorem 1.2. To obtain the desired basic, coherent triangulations of the junior simplices 𝔰_{G} of all abelian quotient c.i.-singularities, it suffices (by 3.4, 5.15) to construct b.c.b.-triangulations for all Watanabe simplices.

Proposition 6.1. All dilations λ s for integral λ ≥ 2, of a basic lattice d-simplex s in ℝ^{d}, have b.c.b.-triangulations.

Proof. Up to an affine integral transformation, we may assume that s equals s_{d} (w.r.t. ℤ^{d}), where

s_{d} := conv ({0, e_{1}, e_{1} + e_{2}, …, e_{1} + ⋯ + e_{d}}) = {x ∈ ℝ^{d} | 0 ≤ x_{d} ≤ x_{d−1} ≤ … ≤ x_{1} ≤ 1} .

The affine hyperplane arrangement ℍ_{d} (of type Ã_{d}) consisting of the union of hyperplanes

H_{i} (k) = {x ∈ ℝ^{d} | x_{i} = k } , for 1 ≤ i ≤ d, k ∈ ℤ

and

H_{i,j} (k) = {x ∈ ℝ^{d} | x_{i} − x_{j} = k } for 1 ≤ i < j ≤ d, k ∈ ℤ

is infinite, but only the hyperplanes

H_{i} (k) = {x ∈ ℝ^{d} | x_{i} = k } , for 1 ≤ i ≤ d, 1 ≤ k ≤ λ − 1

and

H_{i,j} (k) = {x ∈ ℝ^{d} | x_{i} − x_{j} = k } for 1 ≤ i < j ≤ d, 1 ≤ k ≤ λ − 1

intersect the interior of λ s_{d}. The hyperplanes H_{i} (k) , 1 ≤ i ≤ d, k ∈ ℤ, subdivide ℝ^{d} into unit cubes. Given such a cube

C (μ) := [0, 1]^{d} + μ (where μ = (μ_{1}, …, μ_{d}) ∈ ℤ^{d})

and fixing i, j with 1 ≤ i < j ≤ d, we see that there is only one of the hyperplanes H_{i,j} (k) intersecting the interior of C (μ), namely H_{i,j} (μ_{i} − μ_{j}). The hyperplanes

{H_{i,j} (μ_{i} − μ_{j}) | 1 ≤ i < j ≤ d}

provide the “usual” basic triangulation of C (μ) into d! basic subsimplices of the form

s (μ,θ) = μ + conv ({0, e_{θ(1)}, e_{θ(1)+θ(2)}, …, e_{θ(1)+θ(2)+···+θ(d)}}) , for all permutations θ ∈ 𝔖_{d}.

Thus, ℍ_{d} defines a basic triangulation 𝔗_{(d)} of the entire space ℝ^{d} (cf. [14, Ch. III]). This triangulation is also coherent because

ℝ^{d} ∋ x ⟼ \overline{ψ} (x) = − ∑_{0≤i<j≤d} \{ ∑_{0≤k≤x_{j}−x_{i}} H(x_{j}−x_{i}−k) + ∑_{x_{j}−x_{i}≤k≤0} H(k−x_{j}+x_{i})\} ∈ ℝ

with x_{0} := 0, using the Heaviside function H : ℝ → ℝ

H(x) := \{ \begin{matrix}x & if x ≥ 0, \\ 0 & otherwise, \end{matrix}

is a strictly upper convex function for this triangulation. Finally, it is balanced via the colouring function φ_{d}:

ℤ^{d} ∋ μ = (μ_{1}, …, μ_{d}) ↦^{φ_{d}} (μ_{1} + ⋯ + μ_{d}) mod (d + 1) ∈ {0, 1, …, d}

which (d + 1)-colours all the facets s (μ,θ) of 𝔗_{(d)}.

Since the bounding hyperplanes of λ s_{d} are contained in ℍ_{d}, the restriction 𝔗_{[d;λ]} := 𝔗_{(d)} |_{λ s_{d}} is a b.c.b.-triangulation of λ s_{d}. _{□}

Example 6.2. Figure 11 provides the “nice” basic triangulation 𝔗_{[2;4]} of 4 s_{2} inherited from the affine hyperplane arrangement ℍ_{2}.

[figure]

Figure 11

Corollary 6.3. Theorem 1.2 is true for all (d; k)-hypersurface-singularities.

Proof. In this case the junior simplex 𝔰_{G} equals (up to an affine integral transformation) k s, where

s = conv (\{\frac{1}{k} e_{1}, …, \frac{1}{k} e_{d}\})

w.r.t. N_{G(d;k)}. But s is obviously basic. _{□}

Remark 6.4. Corollary 6.3 generalizes Roan’s result in [26, § 5]. (Roan proved the existence of crepant desingularizations of (d; 2)-hypersurfaces by successively blowing up the singular parts of the branching locus of the corresponding double covering of ℂ^{d}; see Remark 5.12(i)).

By Theorem 5.15, and what we have already explained in § 4, our Main Theorem 1.2 is now a consequence of the following:

Theorem 6.5. Every Watanabe simplex has a b.c.b.-triangulation.

Proof. Trivially, every Watanabe 0-simplex has a b.c.b.-triangulation. If T_{1} (resp. T_{2}) denotes a b.c.b.-triangulation of a Watanabe d_{1}-simplex s_{1} (resp. of a Watanabe d_{2}-simplex s_{2}), then T_{1} ∗ T_{2} is a b.c.b.-triangulation of s_{1} ∗ s_{2} by Theorem 3.5. Hence, it suffices to show that for a given Watanabe simplex s, being equipped with a b.c.b.-triangulation, say T , having a colouring function φ : s ∩ ℤ^{d} → {0, 1, …, d} and a strictly upper convex function ψ : |s| → ℝ, the λ-times dilation of s possesses itself a b.c.b.-triangulation. Every facet F of T gives rise to a unique affine integral transformation Φ_{F} : s_{d} → F (s_{d} as before in the proof of 6.1) which respects the colourings relatively to s_{d}, i.e., which sends every vertex of s_{d} to the vertex of F that has same colour. Φ_{F} maps λ s_{d} onto λ F. Thus, the image

T̃_{λ F} := Φ_{F} (𝔗_{[d;λ]}) = λ 𝔗_{(d)} |_{λ F}

constitutes a triangulation of λ F (which is a b.c.b.-triangulation by 3.6, 3.4, and 6.1). The triangulations {T̃_{λ F} | F facets of T } fit together to give a basic triangulation T̂ of λ s. The

triangulation T̂ is also coherent by Patching Lemma 3.1. Finally, T̂ is balanced because φ̂ : |T̂| → ℝ (with φ̂ |_{F} = φ̂_{F}) defined by

λ F ∋ x ↦ φ̂_{F} (x) := φ_{d} ((Φ_{F})^{−1} (x)) ∈ ℝ

on all facets F of T , is a colouring map. _{□}

Example 6.6. Let T denote the unique b.c.b.-triangulation of the 2-dimensional Watanabe simplex

s = conv (\{( \begin{matrix}0 \\ 0 \end{matrix} ) , ( \begin{matrix}−1 \\ 1 \end{matrix} ) , ( \begin{matrix}−1 \\ −1 \end{matrix} )\}) (w.r.t. ℤ^{2})

with the facets F_{1} = conv(\{\binom{−1}{1}, \binom{−1}{0}, \binom{0}{0}\}) and F_{2} = conv(\{\binom{−1}{−1}, \binom{−1}{0}, \binom{0}{0}\}). Figure 12 illustrates how one “glues together”

T̃_{3F_{i}} = Φ_{F_{i}} (𝔗_{[2;3]}) = 3 𝔗_{(2)} |_{3 F_{i}} , i = 1, 2,

to obtain the b.c.b.-triangulation T̂. The affine integral transformations Φ_{F_{i}} are given by

Φ_{F_{1}} (x) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) x + ( \begin{matrix}−1 \\ −1 \end{matrix} ) , Φ_{F_{2}} (x) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) x + ( \begin{matrix}−1 \\ 1 \end{matrix} ) , for x ∈ ℝ^{2}.

s the basic simplex s the triangulation T  the entire gluing triangulation T̂ the “nice” triangulation 𝔗_{[2;3]} of 3𝔰_{2} F_{1} F_{2}  Φ_{F_{1}} Φ_{F_{2}}  Φ_{F_{1}} Φ_{F_{2}} Figure 12

7. On the computation of cohomology group dimensions

Let ℂ^{d}/G = U_{σ_{0}} = X (N_{G}, Δ_{G}) be again the underlying space of an abelian quotient c.i.- singularity and

f_{T} : X (N_{G}, Δ̂_{G} (T )) → X (N_{G}, Δ_{G})  (7.1)

any partial crepant desingularization induced by a lattice triangulation T of the junior simplex 𝔰_{G}. It is easy to verify that the central fiber F_{T} = (f_{T})^{−1} ([0]) of f_{T} is a strong deformation retract of the overlying space X (N_{G}, Δ̂_{G} (T )). Theorem 1.2 guarantees the existence of at least one b.c.b.-triangulation T of 𝔰_{G}, so that (7.1) is a projective, crepant, full desingularization. But even if we let T go through the entire class of all possible basic triangulations of 𝔰_{G}, and use [3, Thm. 5.4], [13, Cor. 1.5], we obtain the one-to-one McKay-type correspondence

\{elements of G \} ⟷^{1:1} \{ \begin{matrix}a basis of H^{∗} (F_{T} , ℚ) consisting \\ of classes of algebraic cycles \end{matrix} \}

and, in particular,

\{elements of G of age 1\} ⟷^{1:1} \{ \begin{matrix}exceptional prime \\ divisors w.r.t. f_{T} \end{matrix} \} ⟷^{1:1} \{ \begin{matrix}a basis of H^{2} (F_{T} , ℚ) \\ consisting of classes of \\ algebraic cycles \end{matrix} \}

where the elements of age 1 are those having lattice-point-representatives lying on 𝔰_{G}. In fact, only the even cohomology groups of F_{T} are non-trivial. To compute their dimensions we need some concepts from enumerative combinatorics.

For a lattice d-polytope P ⊂ N_{ℝ} w.r.t. an N ≅ ℤ^{d}, and κ a positive integer, let

Ehr_{N} (P, κ) = a_{0} (P) + a_{1} (P) κ + ⋯ + a_{d−1} (P) κ^{d−1} + a_{d} (P) κ^{d} ∈ ℚ [κ]

denote the Ehrhart polynomial of P (w.r.t. N), where Ehr_{N} (P, κ) := # (κ P ∩ N), and

Ehr_{N} (P; 𝔱) := 1 + ∑_{κ=1}^{∞} Ehr_{N} (P, κ) 𝔱^{κ} ∈ ℚ [[𝔱]]

the corresponding Ehrhart series. Writing Ehr_{N} (P; 𝔱) as

Ehr_{N} (P; 𝔱) = \frac{δ_{0} (P) + δ_{1} (P) 𝔱 + ⋯ + δ_{d−1} (P) 𝔱^{d−1} + δ_{d} (P) 𝔱^{d}}{(1 − 𝔱)^{d+1}}

we get the so-called δ-vector δ (P) = (δ_{0} (P) , δ_{1} (P) , …, δ_{d−1} (P) , δ_{d} (P)) of P.

Definition 7.1. For any integer d ≥ 0 we introduce the transfer a-δ-matrix M_{d} ∈ GL(d + 1, ℚ) (depending only on d) to be defined as

M_{d} := (ℜ_{i,j})_{0≤i,j≤d} with ℜ_{i,j} := \frac{1}{d!} \{∑_{p=i}^{d} \left[{d \atop p}\right] \binom{p}{i} (d − j)^{p−i}\}

where \left[{d \atop p}\right] denotes the Stirling number (of the first kind) of d over p.

The following lemma can be proved easily.

Lemma 7.2. For a lattice d-polytope P ⊂ N_{ℝ} w.r.t. an N ≅ ℤ^{d}, we have

(a_{0} (P) , a_{1} (P) , …, a_{d−1} (P) , a_{d} (P)) = (δ_{0} (P) , δ_{1} (P) , …, δ_{d−1} (P) , δ_{d} (P)) · (M_{d})^{⊺} .

Theorem 7.3. For any integer d ≥ 2, and for any crepant (full) desingularization (7.1) of an abelian quotient c.i.-space ℂ^{d}/G, the non-trivial cohomology dimensions of F_{T} (or of X (N_{G}, Δ̂_{G} (T )), for any basic T ), can be determined inductively (via the coefficients of Ehrhart polynomials) as follows :

(i) If 𝔰_{G} is the join of two Watanabe simplices s_{1}, s_{2}, of dimensions d_{1}, d_{2}, (d_{1} + d_{2} = d − 2), then

dim_{ℚ}H^{2i} (F_{T} , ℚ) =

= ∑_{\begin{matrix}0≤p,q≤d−1 \\ with p+q=i \end{matrix}} (([(M_{d−1})^{−1}]_{p+1} · ( a_{0} (s_{1}) ; a_{1} (s_{1}) ; ⋮ ; a_{d−1} (s_{1}) )) × ([(M_{d−1})^{−1}]_{q+1} · ( a_{0} (s_{2}) ; a_{1} (s_{2}) ; ⋮ ; a_{d−1} (s_{2}) )))

(7.2)

(ii) If 𝔰_{G} (up to an affine integral transformation) is the dilation λ s, λ ≥ 2, of a Watanabe simplex s, then

dim_{ℚ}H^{2i} (F_{T} , ℚ) = [(M_{d−1})^{−1}]_{i+1} · ( a_{0} (s) ; λ a_{1} (s) ; ⋮ ; λ^{d−1} a_{d−1} (s) )  (7.3)

for all i, 0 ≤ i ≤ d − 1, where for all ρ, 1 ≤ ρ ≤ d, [(M_{d−1})^{−1}]_{ρ} denotes the ρ-th row-vector

of (M_{d−1})^{−1}.

Proof. By [3, Thm. 4.4], dim_{ℚ}H^{2i} (F_{T} , ℚ) equals the i-th component of the δ-vector of 𝔰_{G}. That the computation can be done inductively follows from the converse statement in Theorem 5.15 and Theorem 1.2. In fact, for any G being conjugate to a G_{𝔻}, the cohomology dimensions can be read off from the free parameters of W_{𝔻}.

(i) Since T is basic, the δ-vector of 𝔰_{G} is equal to its h-vector (see Stanley [29, 2.5]). Hence, (7.2) follows from the formula which provides the h-vector of the join of two simplices. (ii) (7.3) is obvious because a_{j} (λ s) = λ^{j} a_{j} (s) for all j, 0 ≤ j ≤ d − 1. _{□}

Corollary 7.4. Let G be conjugate to G (d; k) (within SL(d, ℂ)). Then the non-trivial cohomology dimensions of any crepant resolution (7.1) of the (d; k)-hypersurfaces equals

dim_{ℚ}H^{2i} (F_{T} , ℚ) = [(M_{d−1})^{−1}]_{i+1} · ( 1 ⋮ \underbrace{\frac{k^{j}}{(d − 1)!} \{∑_{p=j}^{d−1} \left[{d − 1 \atop p}\right] \binom{p}{j} (d − 1)^{p−j}\}}_{entry of the column at (j+1)-position (0≤j≤d−1)} ⋮ \frac{k^{d−1}}{(d − 1)!} ) (7.4)

for 0 ≤ i ≤ d − 1. In particular, the Euler-Poincaré characteristic χ (F_{T}) of F_{T} equals

χ (F_{T}) = χ (X (N_{G(d;k)}, { \atop Δ_{G(d;k)}} (T ))) = |G (d; k)| = k^{d−1}  (7.5)

Proof. In this case (up to an affine integral transformation) 𝔰_{G} equals k s, where

s = conv (\{\frac{1}{k} e_{1}, …, \frac{1}{k} e_{d}\})

w.r.t. N_{G(d;k)}. Since

Ehr_{N_{G(d;k)}} (s, κ) = \binom{κ + d − 1}{κ} = \binom{κ + d − 1}{d − 1} = \frac{1}{(d − 1)!} ∑_{p=0}^{d−1} \left[{d − 1 \atop p}\right] (κ + d − 1)^{p} ,

or, alternatively, since

δ_{0} (s) = 1 and δ_{1} (s) = ⋯ = δ_{d−1} (s) = 0,

we get

a_{j} (s) = \{ \begin{matrix}(j + 1) & -entry of the \\ first column of \\ the matrix M_{d−1} \end{matrix} \} = \frac{1}{(d − 1)!} \{∑_{p=j}^{d−1} \left[{d − 1 \atop p}\right] \binom{p}{j} (d − 1)^{p−j}\}

for all j, 0 ≤ j ≤ d − 1, and (7.4) follows from (7.3). Finally, formula (7.5) follows from the equality χ (F_{T}) = (d − 1)! a_{d−1} (s) . _{□}

8. Comments and open problems.

(i) All partial, crepant, projective desingularizations (7.1) of (ℂ^{d}/G, [0]) = (U_{σ_{0}}, orb (σ_{0})) can be studied by means of the secondary polytope of the junior simplex 𝔰_{G}, whose vertices parametrize all coherent triangulations T of 𝔰_{G}. Passing from one vertex of this polytope to another, we perform a finite series of flops. It should be mentioned that for d ≥ 4, it is possible to start from a vertex corresponding to a basic T , and arrive at another maximal, but non-basic triangulation T′. In the language of “toric MMP” (see Reid [23]), our b.c.b.-triangulations lead to smooth minimal models. (Note that all morphisms f_{T} can be written, by [23, 0.2-0.3], as compositions of finite sequences of more elementary toric contraction-morphisms).

(ii) For any d ≥ 3, the general hyperplane section of the above (ℂ^{d}/G, [0]) through [0] is either a rational or an elliptic Gorenstein singularity (see [24, § 3.10]). Already for d = 3, both possibilities occur. For example, the general hyperplane section of the (3; k)-hypersurfaces is a compound Du Val singularity (of type cD_{4}) for k = 2, and an elliptic Gorenstein surface singularity whenever k ≥ 3. It might be interesting to investigate the class of Gorenstein elliptic singularities obtained by this procedure by exploiting the inductive character of Watanabe’s classification, essentially via the forests W_{𝔻}. (What is the relationship between these generalhyperplane-section singularities and the free parameters of the starting-point singularities for d ≥ 4 ?)

(iii) Could Theorem 1.2 be generalized for the underlying spaces of Gorenstein, toric, nonquotient, c.i.-singularities? More precisely, what would be the geometric analogue of joins and dilations describing the structure of lattice polytopes which support the Gorenstein cones in this case?

(iv) Theorem 1.2 has various applications to global geometrical constructions. For instance, every “well-stratified” Calabi-Yau variety which is locally a complete intersection, and has at most abelian quotient singularities, possesses global, crepant, full resolutions in all dimensions. (Nevertheless, to check the projectivity of these globally desingularizing morphisms one needs to apply the Nakai-Moishezon criterion, and this is only possible if one has some extra information available about the intrinsic geometry of the varieties being under consideration).

(v) A special class of such Calabi-Yau varieties, which is of particular interest, is that of compactified hypersurfaces \overline{Z_{f}} ⊂ Y_{P} being embedded in a toric variety Y_{P}, associated to a reflexive, simple lattice polytope P. Assuming that \overline{Z_{f}} is P -regular (in Batyrev’s sense [1]), and that the polar lattice polytope P ^{⋆} of P has only Watanabe simplices as faces of codimension ≥ 2, there exist always global, crepant, full desingularizations Ẑ_{f} → \overline{Z_{f}} of \overline{Z_{f}}. One method to construct at least one of them is to triangulate the faces of P ^{⋆} as in the present paper and then to join the single interior point of P ^{⋆} with them. For example, the mirror-partner \overline{Z_{g}} of a marginally deformed Fermat-hypersurface \overline{Z_{f}} (or of any smooth hypersurface) of degree d in ℙ_{ℂ}^{d−1} has global, crepant, full desingularizations in all dimensions. (In this particular case, one can, in addition, easily construct a globally projective desingularizing morphism). Hence, the so-called “string-theoretic” Hodge numbers of this \overline{Z_{g}} (cf. [1] [3]) are nothing but the usual Hodge-numbers of an always existing smooth, projective Ẑ_{g}; in particular, the corresponding monomial-divisor mirror-map provides the usual dualism between the polynomial first-order deformations of \overline{Z_{f}} within H^{d−3,1} (\overline{Z_{f}}) on the one hand, and the toric part of H^{1,1} \binom{}{Z_{g}}, on the other. (So there are concrete exceptional divisors, and there is no need here to work in the category of singular spaces). This motivates the formulation of a purely combinatorial problem: to classify all reflexive, simplicial polytopes, at least in dimension 5, having only Watanabe simplices as faces of codimension at least 2.

Acknowledgements. The first author benefited greatly from the stimulating mathematical atmosphere of the Max-Planck-Institute in Bonn, where he spent the last part of his graduate studies under the direction of Prof. F. Hirzebruch, towards the end of which many of the ideas contained in his contribution here were conceived in their first form. Subsequent visits were also in more than one way beneficial. He also would like to thank Prof. M. Reid for his interest in our project concerning the existence of crepant resolutions, to DFG for the support by an one-year-fellowship, and to Mathematics Institute of Bonn University for hospitality. The second and third author wish to acknowledge the long-term support of their research by a DFG Gerhard-Hess Forschungsförderpreis.

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