Torsion of differentials on toric varieties

Klaus Altmann^{∗}

Institut für reine Mathematik, Humboldt-Universität zu Berlin Ziegelstr. 13a, D-10099 Berlin, Germany. E-mail: altmann@mathematik.hu-berlin.de

Abstract

We introduce an invariant for semigroups with cancellation property. When the semigroup equals the set of lattice points in a rational, polyhedral cone, then this invariant describes the torsion of the differential sheaf on the associated toric variety. Finally, as an example, we present the case of two-dimensional cones (corresponding to two-dimensional cyclic quotient singularities).

1 An invariant for semigroups

(1.1) Let S be a commutative semigroup with 0 and cancellation property (i.e. a + s = b + s implies a = b for a, b, s ∈ S). In particular, S can be embedded into a group, and the notion −a for a ∈ S makes sense. Assume that (inside this group) S ∩ (−S) = {0}; then via

a ≥ b :⟺ a − b ∈ S ,

S turns also into a partially ordered set.

For each ℓ ∈ S we will define a certain abelian group T_{ℓ}. Their direct sum T := ⊕_{ℓ∈S}T_{ℓ} or the T_{ℓ}’s itself should be considered an invariant of the original semigroup. If S is finitely generated, then so will the T_{ℓ}’s. In case S equals the set of lattice points of some rational, polyhedral cone (i.e. generated by a finite set of rays through rational points) in IR^{n}, we will show that T ⊗ I C coincides with the torsion of the differential sheaf Ω_{Y}^{1} of the affine toric variety Y = Spec I C[S] ( I C[S] denotes the semigroup algebra of S over I C).

∗This paper was written at M.I.T. and supported by a DAAD-fellowship.

(1.2) For any subset P ⊆ S we denote by L(P) the abelian group of integral relations between elements of P, i.e.

L(P) := ker (ZZ^{⊕P} → span_{ZZ}P ⊆ S − S) .

For each q ∈ L(P) we define its support as

supp q := {s ∈ P | q_{s} ≠ 0};

it is a finite subset of P. Then, every q ∈ L(P) is uniquely representable as a difference q = q^{+} − q^{−} with q^{+}, q^{−} ∈ IN^{⊕P} and supp q^{+} ∩ supp q^{−} = ∅. In particular,

∑_{s∈S} q_{s}^{+} · s = ∑_{s∈S} q_{s}^{−} · s ∈ span_{IN}P ⊆ S ,

and we denote this element by q̄ (the “bar-value” of q).

Definition: For every ℓ ∈ S let S_{ℓ} := {s ∈ S | s ≤ ℓ}. Then, we define

T_{ℓ} := L(S^{ℓ})/⟨q ∈ L(S_{ℓ}) | q̄ ≤ ℓ⟩ = ⟨q ∈ L(S) | supp q ≤ ℓ⟩/⟨q ∈ L(S) | q̄ ≤ ℓ⟩ .

(⟨q ∈ L(S_{ℓ}) | q̄ ≤ ℓ⟩ means the subgroup generated by those q’s, and “supp q ≤ ℓ” is just an abbreviation for “s ≤ ℓ for all s ∈ supp q”.)

(1.3) Proposition: Let E ⊆ S be a subset that generates the semigroup S. Denoting E_{ℓ} := E ∩ S_{ℓ} = {s ∈ E | s ≤ ℓ}, we obtain for ℓ ∈ S

T_{ℓ} = L(E^{ℓ})/⟨q ∈ L(E_{ℓ}) | q̄ ≤ ℓ⟩ = ⟨q ∈ L(E) | supp q ≤ ℓ⟩/⟨q ∈ L(E) | q̄ ≤ ℓ⟩ .

Proof: We regard the canonical map

φ : L(E^{ℓ})/⟨q ∈ L(E_{ℓ}) | q̄ ≤ ℓ⟩ ⟶ L(S^{ℓ})/⟨q ∈ L(S_{ℓ}) | q̄ ≤ ℓ⟩ .

φ is surjective: Suppose a relation q ∈ L(S_{ℓ}) involves s^{1},…, s^{N} ∈ S_{ℓ} (i.e. supp q = {s^{1},…, s^{N}}). Each s^{v} can be represented as a sum of elements from E providing a special relation q(s^{v}) with \overline{q(s^{v})} = s^{v} ≤ ℓ. Then, if q equals the relation [∑_{v} q_{v} · s^{v} = 0], we obtain q − ∑_{v} q_{v} · q(s^{v}) ∈ L(E_{ℓ}). (By the way, q̃:= q − ∑_{v} q_{v} · q(s^{v}) can be regarded as similar to q, but each occurrence of s^{v} is replaced by an appropriate sum of elements from E. In particular, the bar-value of q̃ is less or equal than that of q.)

φ is injective: Let q ∈ L(E_{ℓ}) such that φ(q) = 0, i.e. it splits into a sum q = ∑_{i} q^{i} with q^{i} ∈ L(S_{ℓ}), q̄^{i} ≤ ℓ. Treating each of the summands as we did q previously, we obtain

q^{i} = q̃^{i} + ∑_{v} q_{v}^{i} · q(s^{v}) with q̃^{i} ∈ L(E_{ℓ}), \overline{q̃^{i}} ≤ ℓ, and s^{v} ∈ S_{ℓ}.

Hence, up to relations from ⟨q ∈ L(E_{ℓ}) | q̄ ≤ ℓ⟩, we may assume that

q = ∑_{s^{v}∈S_{ℓ}\E_{ℓ}} g_{v} · q(s^{v}) ∈ L(S_{ℓ}) ⊆ ZZ^{S_{ℓ}}

for some integer coefficients g_{v}. However, since q is actually contained in L(E_{ℓ}) ⊆ ZZ^{E_{ℓ}}, evaluating this equation at each of the s^{v} ∈ S_{ℓ} \ E_{ℓ} yields always g_{v} = 0. □

Corollary: If S is a finitely generated semigroup, then the T_{ℓ} are finitely generated abelian groups.

(1.4) We conclude this general section with a trivial, but important lemma.

Lemma: Let ℓ ∈ S, and define S′ ⊆ S as the semigroup generated by S_{ℓ}. The relation “≤_{S′}” associated to S′ is not the restriction of that of S; in general, “≤_{S′}” is stricter than “≤_{S}”. However,

(i) (S′)_{ℓ} = S_{ℓ}, and

(ii) the abelian groups T_{ℓ} coincide for S and S′.

Proof: For (i) assume that s ∈ S fulfills s ≤_{S} ℓ. By definition, there is an t ∈ S such that s + t = ℓ. In particular, s, t ∈ S_{ℓ} ⊆ S′, and we obtain s ≤_{S′} ℓ. The same trick works for (ii): If q ∈ L(S_{ℓ}) is a relation with q̄ ≤_{S} ℓ, then there exists a t ∈ S such that q̄ + t = ℓ. In particular, t ∈ S_{ℓ} ⊆ S′, hence q̄ ≤_{S′} ℓ. □

Hence, to compute T_{ℓ}, we may always change the semigroup and assume that E = E_{ℓ}.

2 Rational, polyhedral cones

(2.1) For this section we assume that S ⊆ ZZ^{n} equals the set of lattice points of some rational, polyhedral cone in IR^{n}. In particular, it defines an affine toric variety Y := Spec I C[S]. (For general facts about toric varieties see [Da] or [Od]).

Definition: Let Y = Spec B be an arbitrary affine scheme over the complex numbers, i.e. let B be an arbitrary (commutative) I C-algebra. Then, the B-module of Kähler differentials Ω_{Y}^{1} together with the canonical differential d : B → Ω_{Y}^{1} are defined by the following universal property:

(i) d is additive, kills the constants, and it fulfills the rule

d(fg) = f d(g) + g d(f) for f, g ∈ B.

(ii) For each B-module Q all maps d′ : B → Q meeting the properties similar to (i) factorize through d by a uniquely determined B-linear map Ω_{Y}^{1} → Q.

Indeed, the Kähler differentials Ω_{Y}^{1} do exist, and they are uniquely determined by the previous definition. For a proof of these facts and basic properties (for instance the two fundamental exact sequences) see Chapter 10 of [Ma].

Theorem: In the toric situation the I C[S]-module of Kähler differentials Ω_{Y}^{1} and its torsion submodule tors (Ω_{Y}^{1} ) are ZZ^{n}-graded. For ℓ ∈ S we have

tors (Ω_{Y}^{1} )(ℓ) = T_{ℓ} ⊗_{ZZ} I C ,

and tors (Ω_{Y}^{1} ) vanishes in the remaining degrees. Moreover, if s ∈ S, then the canonical map T_{ℓ} → T_{ℓ+s} describes the multiplication with x^{s} ∈ I C[S].

(2.2) Proof: Step 1: Let E = {s^{0},…, s^{w}} be a generating set of the semigroup S; in particular, we have a surjective map π : IN^{w+1} ⟶ ⟶ S , a ↦ ∑_{i} a_{i} s^{i}. Let m := {(a, b) | a, b ∈ IN^{w+1}; π(a) = π(b) in S}. Then, Y can be regarded as the closed subset of I C^{w+1} defined by the ideal I = (\underline{z}^{a} − \underline{z}^{b} | (a, b) ∈ m) ⊆ I C[z_{0},…, z_{w}]. We obtain the standard exact sequence (cf. Theorem 58 in [Ma])

I/I^{2} ⟶^{d} Ω_{I C^{w+1}}^{1} ⊗_{I C[z_{0},...,z_{w}]} I C[S] ⟶ Ω_{Y}^{1} ⟶ 0 ,

inducing

I C[S]^{⊕m} ⟶^{α} I C[S]^{w+1} ⟶ Ω_{Y}^{1} ⟶ 0

via composing d with the canonical surjection I C[S]^{⊕m} ⟶ ⟶ I/I^{2}.

The maps in the latter exact sequence could be described as follows: Denote by {e^{ab}} and {e^{i}} the standard bases of I C[S]^{⊕m} and I C[S]^{w+1}, respectively; and for an s ∈ S denote by x^{s} ∈ I C[S] the corresponding element in the semigroup algebra, e.g. x^{s^{i}} ≡ z_{i} (mod I). Then, the image of e^{ab} in I/I^{2} is the equation \underline{z}^{a} − \underline{z}^{b}, and this maps onto d(\underline{z}^{a} − \underline{z}^{b}) = ∑_{i=0}^{w}(a_{i} − b_{i}) x^{π(a)−s^{i}} d z_{i} ∈ Ω_{I C^{w+1}}^{1} ⊗ I C[S]. In particular,

α(e^{ab}) = ∑_{i=0}^{w}(a_{i} − b_{i}) · x^{π(a)−s^{i}} · e^{i} .

(At the first glance, α might not always map into I C[S]^{w+1}. However, if π(a)−s^{i} ∉ S, then this would imply a_{i} = b_{i} = 0.) Finally, e^{i} ∈ I C[S]^{w+1} maps onto d x^{s^{i}} ∈ Ω_{Y}^{1} by the second map.

Step 2: The I C-linear map I C[S] → ZZ^{n} ⊗_{ZZ} I C[ZZ^{n}] , x^{s} ↦ s ⊗ x^{s} is a I C-derivation: It kills the constants, and for s, t ∈ S we have x^{s}(t ⊗ x^{t}) + x^{t}(s ⊗ x^{s}) = (s + t) ⊗ x^{s+t}. Hence, by definition of the Kähler differentials, we obtain a I C[S]-linear map

Ω_{Y}^{1} ⟶ ZZ^{n} ⊗_{ZZ} I C[ZZ^{n}] , d x^{s} ↦ s ⊗ x^{s} .

On the other hand, ZZ^{n} ⊗_{ZZ} I C[ZZ^{n}] can be identified with the module of Kähler differentials on the torus ( I C^{∗})^{n} = Spec I C[ZZ^{n}], and the previous map corresponds to the restriction of differentials from Y onto the open subset ( I C^{∗})^{n} ⊆ Y. Since Ω_{( I C^{∗})^{n}}^{1} is just a localisation of Ω_{Y}^{1} , we obtain that this map has exactly tors (Ω_{Y}^{1} ) as its kernel. Putting this fact together with the information from the first step, we obtain the sequence of I C[S]-modules

(∗) \begin{matrix}I C[S]^{⊕m} & ⟶^{α} & I C[S]^{w+1} & ⟶^{β} & ZZ^{n} ⊗_{ZZ} I C[ZZ^{n}] \\ & & e^{i} & ↦ & s^{i} ⊗ x^{s^{i}} \end{matrix}

and, moreover, tors (Ω_{Y}^{1} ) = ker β/im α.

Step 3: Defining deg(e^{ab}) := π(a) = π(b) ∈ S for I C[S]^{⊕m}, deg(e^{i}) := s^{i} ∈ S for I C[S]^{w+1}, and deg(s ⊗ x^{t}) := t ∈ ZZ^{n} for ZZ^{n} ⊗_{ZZ} I C[ZZ^{n}], the sequence (∗) turns out to be ZZ^{n}-graded. In particular, to calculate ker β/im α, we can deal with each degree ℓ ∈ S separately (degrees from ZZ^{n} \ S do not appear).

Claim: T_{ℓ} ⊗_{ZZ} I C → (ker β/im α)_{ℓ} , q ↦ ∑_{s^{i}∈E} q_{i} · x^{ℓ−s^{i}} · e^{i} is an isomorphism of I C-vector spaces. Proof: For q ∈ L(E_{ℓ}), the sum ∑_{s^{i}∈E} q_{i} · x^{ℓ−s^{i}} · e^{i} defines an element of ( I C[S]^{w+1})_{ℓ}. Applying β, we obtain

β(∑_{s^{i}∈E} q_{i} · x^{ℓ−s^{i}} · e^{i}) = ∑_{s^{i}∈E} q_{i} · x^{ℓ−s^{i}} · (s^{i} ⊗ x^{s^{i}})
= (∑_{s^{i}∈E} q_{i} s^{i}) ⊗ x^{ℓ}
= 0 .

Obviously, each element of (ker β)_{ℓ} can be obtained that way: If ∑_{s^{i}∈E} q_{i}(x) · e^{i} ∈ I C[S]^{w+1} is of degree ℓ, we obtain q_{i}(x) = q_{i} x^{ℓ−s^{i}} (q_{i} ∈ I C with q_{i} = 0 unless ℓ ≥ s_{i}). Moreover, we have just seen that β kills this element if and only if ∑_{s^{i}∈E} q_{i} s^{i} = 0, i.e. it comes from a relation q ∈ L(E_{ℓ}). On the other hand, q ∈ L(E_{ℓ}) with q̄ ≤ ℓ can be written as q = a − b (a, b ∈ IN^{w+1}; π(a) = π(b) ≤ ℓ). Then,

∑_{s^{i}∈E}(a_{i} − b_{i}) · x^{ℓ−s^{i}} · e^{i} = x^{ℓ−π(a)} · α(e^{ab}) ,

and those elements generate (im α)_{ℓ}. □

(2.3) Remark: We never used the fact that Y = Spec I C[S] is a normal variety. Hence, Theorem (2.1) remains true, if S ⊆ ZZ^{n} is just supposed to be a finitely generated semigroup with S ∩ (−S) = {0}. In particular, over IR, S generates a rational, polyhedral cone σ ⊆ IR^{n}, but there may exist finitely many points of σ ∩ ZZ^{n} not belonging to S.

Example: If S ⊆ IN, then Y is a curve singularity. For instance, S := IN \ {1} is generated by the integers 2 and 3, and Y ⊆ I C^{2} is the cusp given by the equation y^{2} − x^{3} = 0. Let us compute the torsion of Ω_{Y}^{1} in this case: With E = {s^{0} := 2, s^{1} := 3} we have the only relation q := [3s^{0} − 2s^{1} = 0]. Since q ∈ S_{5}, on the one hand, but q̄ = 6, on the other hand, we obtain T = T_{5} = ZZ · q, i.e. tors (Ω_{Y}^{1} ) is a one-dimensional vector space concentrated in the singular point.

3 Two-dimensional, cyclic quotient singularities

(3.1) Two-dimensional, cyclic quotient singularities coincide with the twodimensional, affine toric varieties; they were first investigated by Riemenschneider, [Ri]. We want to compute our invariant for this special case.

Let S be the set of lattice points in the two-dimensional cone spanned by some primitive vectors s^{0}, s^{w} ∈ ZZ^{2}. Then, following §1.6 in [Od], E = {s^{0},…, s^{w}} is built from all lattice points of the compact part of the boundary of conv (S \ {0}); in particular, ∂(conv (S \ {0})) consists of w primitive edges (containing no interior lattice point). Every pair of adjacent elements s^{i−1}, s^{i} (i = 1,…, w) provides a ZZ-basis for the lattice ZZ^{2}, and there are relations

s^{i−1} + s^{i+1} = a_{i} · s^{i} (a_{i} ∈ IN, a_{i} ≥ 2; i = 1,…, w − 1) .

[figure] [figure] [figure] [figure]

s^{w}

s^{0}

s^{1}

s^{2}

s^{3}

s^{4}

This is a rough draft of S; in fact it cannot exactly be shaped as shown in the figure.

Remark: (cf. Lemma 1.20 in [Od]) The coefficients a_{i} can be obtained from the continued fraction

\frac{n}{n − q} = a_{1} − \frac{1}{a_{2} − \frac{1}{a_{3}−...}}

with n := det(s^{0}, s^{w}) and q ∈ {1,…, n − 1} such that n|(qs^{0} + s^{w}).

(3.2) Since a_{i} ≥ 2 (i = 1,…, w − 1), the sets E_{ℓ} ⊆ E are “connected” (i.e. they are shaped as E_{ℓ} = {s^{p}, s^{p+1},…, s^{p+k}} for some p, k) for every ℓ ∈ S. Hence, to compute T_{ℓ}, we may assume that E_{ℓ} = {s^{0},…, s^{w}} (cf. Lemma (1.4)).

Lemma: Let ℓ ∈ S such that E_{ℓ} = {s^{0},…, s^{w}}. Then,

(i) ℓ ≥ s^{0} + s^{w}, and

(ii) T_{ℓ} = 0 unless ℓ = s^{0} + s^{w}.

Proof: (i) The vectors s^{0} and s^{w} form a IR-basis of IR^{2}, and, for an s ∈ ZZ^{2}, the condition “s ∈ S” is equivalent to having non-negative coordinates only. Hence, the inequalities ℓ ≥ s^{0}, s^{w} imply that both coordinates of ℓ are not smaller than 1, i.e. ℓ − s^{0} − s^{w} has non-negative coordinates.

(ii) Let ℓ > s^{0} + s^{w} be given; we will show that T_{ℓ} = 0. We distinguish two cases:

Case 1: ℓ − (s^{0} + s^{w}) ≥ s^{0} (or similarly ℓ − (s^{0} + s^{w}) ≥ s^{w}). Then, since s^{i} ≤ s^{0} + s^{w}, we know that s^{0} + s^{i} ≤ 2s^{0} + s^{w} ≤ ℓ for i = 2,…, w. Summing up the equations s^{j−1} + s^{j+1} = a_{j} s^{j} (1 ≤ j ≤ i − 1) provides relations q^{i}

(q^{i}) s^{0} + s^{i} = (s^{1} + s^{i−1}) + ∑_{j=1}^{i−1}(a_{j} − 2) s^{j} (i = 2,…, w)

having exactly s^{0} + s^{i} ≤ ℓ as their bar-values. In particular, they yield zero in T_{ℓ}. On the other hand, if we are given an arbitrary relation q ∈ L(E), then we can use q^{w},…, q^{2} to eliminate s^{w},…, s^{2} step by step from the support of q (without changing its value in T_{ℓ}). Since s^{0} and s^{1} are linearly independent, q has to be trivial then.

Case 2: Not Case 1; in particular, ℓ − (s^{0} + s^{w}) ≥ s^{i} for some i = 1,…, w − 1. Then, we have ℓ − (s^{0} + s^{w}) = ∑_{j=1}^{w−1} g_{j} s^{j} with non-negative integers g_{j} and, moreover, g_{i} ≥ 1. (There are no summands involving s^{0} or s^{w}, because this would fit in the first case.) Again, we start with an arbitrary q ∈ L(E). First, we use the relation s^{0} + s^{w} = (s^{1} + s^{w−1}) + ∑_{j=1}^{w−1}(a_{j} − 2) s^{j} (bar-value s^{0} + s^{w} ≤ ℓ) to eliminate s^{0} (if i = 1) or s^{w} (if i ≥ 2) from supp q. If w = 2, then we are already done. Otherwise, we use the similar relations expressing s^{i} + s^{w} (if i = 1) or s^{0} + s^{i} (if i ≥ 2) by the generators in between to eliminate s^{w} or s^{0}, respectively. The result is a relation q with supp q ⊆ {s^{1},…, s^{w−1}}. Finally, we know that

ℓ = (s^{0} + s^{w}) + ∑_{j=1}^{w−1} g_{j} s^{j} = (s^{1} + s^{w−1}) + ∑_{j=1}^{w−1}(g_{j} + a_{j} − 2) s^{j} .

Hence, the fact T_{ℓ} = 0 for the cone spanned by s^{1} and s^{w−1} (induction by w) tells us that L(s^{1},…, s^{w−1}) is spanned by relations with bar-value not greater than ℓ. In particular, our q can be reduced to zero. □

(3.3) Lemma: Assume ℓ = s^{0} + s^{w} (including E_{ℓ} = {s^{0},…, s^{w}}). Then,

T_{ℓ} ≅ \{ \begin{matrix}0 & for w ≤ 2 \\ ZZ^{[(w−1)/2]} & for a_{1} = … = a_{w−1} = 2 \\ ZZ & otherwise . \end{matrix}

(The second case (a_{1} = … = a_{w−1} = 2) means that the points s^{0},…, s^{w} are sitting on an affine line; the corresponding cyclic quotient singularity equals the cone over the rational normal curve of degree w.)

Proof: The case w ≤ 2 is obvious. Assuming a_{1} = … = a_{w−1} = 2, the entire collection of relations with bar-value not greater than ℓ is given by the [w/2] equations

s^{0} + s^{w} = s^{1} + s^{w−1} = … = s^{[w/2]} + s^{w−[w/2]} ,

and we obtain (w − 1) − [w/2] = [(w − 1)/2] for the rank of T_{ℓ}. Finally, if w ≥ 3 and not a_{1} = … = a_{w−1} = 2, then

ℓ = s^{0} + s^{w} = s^{1} + s^{w−1} + ∑_{i=1}^{w−1}(a_{i} − 2) s^{i} >_{⟨s^{1},s^{w−1}⟩} s^{1} + s^{w−1}

shows (cf. Lemma (3.2)) that L(s^{1},…, s^{w−1}) ⊆ ⟨q ∈ L(E) | q̄ ≤ ℓ⟩. On the other hand, the only relations involving s^{0} or s^{w} and having a bar-value ≤ ℓ are those representing s^{0} + s^{w} as linear combination of s^{1},…, s^{w−1}. (Indeed, if for instance s^{0} + ∑_{i=0}^{w} g_{i} s^{i} = ∑_{i=1}^{w} h_{i} s^{i} is such a relation (g_{i}, h_{i} ≥ 0; g_{i} h_{i} = 0), then s^{0} + ∑_{i=0}^{w} g_{i} s^{i} ≤ ℓ = s^{0} + s^{w} implies ∑_{i} g_{i} s^{i} ≤ s^{w}. Hence g_{w} = 0 or 1, and g_{i} = 0 for i ≠ w. Moreover, since s^{0} is not representable by other generators, this implies that s^{0} + s^{w} forms the left hand side of the relation - and the right hand side has to be built from s^{1},…, s^{w−1} then.) □

(3.4) We are gathering our results and obtain the following description of the invariants T_{ℓ} for a two-dimensional cyclic quotient singularity:

Theorem: Let S and a_{1},…, a_{w−1} as in (3.1); define a_{0} := a_{w} := 3.

(1) Let s^{p}, s^{p+k} ∈ E (0 ≤ p < p + k ≤ w) be elements such that

(i) k ≥ 3, and

(ii) at least one of the numbers a_{p},…, a_{p+k} is greater than two.

Then, ℓ := s^{p} + s^{p+k} uniquely determines p and k, and we have

T_{ℓ} ≅ \{ \begin{matrix}ZZ^{[(k−1)/2]} & for a_{p+1} = … = a_{p+k−1} = 2 \\ ZZ & otherwise . \end{matrix}

The abelian group T_{ℓ} vanishes in the remaining degrees.

(2) For T = ⊕_{ℓ}T_{ℓ} we obtain T ≅ ZZ^{(w−1)(w−2)/2}. In particular,

dim (tors (Ω_{Y}^{1} )) = \binom{w − 1}{2} .

(w + 1 equals the embedding dimension of the cyclic quotient singularity Y).

Proof: For (1) assume that we are given some ℓ ∈ S. Then, E_{ℓ} is shaped as E_{ℓ} = {s^{p},…, s^{p+k}}, and by Lemma (3.2) we know that T_{ℓ} = 0 unless ℓ = s^{p} + s^{p+k}. On the other hand, if ℓ = s^{p} + s^{p+k}, then Lemma (3.3) tells us about T_{ℓ}. The only thing being left is asking the other way around: What is the condition for an ℓ := s^{p} + s^{p+k} to yield exactly E_{ℓ} = {s^{p},…, s^{p+k}}? Obviously, E_{ℓ} does always contain {s^{p},…, s^{p+k}}, and we show that it is exactly the condition a_{p} = … = a_{p+k} = 2 saying that both sets are not equal: Assume E_{ℓ} = {s^{p−i},…, s^{p},…, s^{p+k},…, s^{p+k+j}} (w.l.o.g. i ≥ j ≥ 0, i ≠ 0), then we obtain a chain of inequalities

s^{p} + s^{p+k} = ℓ ≥ s^{p−i} + s^{p+k+j} ≥ s^{p−i+1} + s^{p+k+j−1} ≥ … ≥ s^{p−i+j} + s^{p+k} .

If i > j, this would imply that two different elements of E (s^{p} and s^{p−i+j}) would be comparable in S. Hence, i = j, and all signs in the previous chain turn into equalities implying a_{p−i+1} = … = a_{p} = … = a_{p+k} = … = a_{p+k+i−1} = 2. The reversed direction is easy; the equalities a_{p} = … = a_{p+k} = 2 imply ℓ = s^{p−1} + s^{p+k+1}, hence s^{p−1}, s^{p+k+1} ∈ E_{ℓ}.

To prove the second part of the theorem, we have to count dimensions. Assume that the compact part of the boundary ∂(conv (S \ {0})) consists of m edges, each containing w_{i} − 1 (i = 1,…, m) interior lattice points. In particular, ∑_{i=1}^{m} w_{i} = w.

Then, we have

(a) \binom{w+1}{2} possibilities of choosing two different points s^{p} and s^{p+k} from E;

(b) 2w − 1 of those pairs with 1 ≤ k ≤ 2;

(c) \binom{w_{i}+1}{2} possibilities of choosing two different points s^{p} and s^{p+k} from the i-th edge;

(d) 2w_{i} − 1 of those pairs with 1 ≤ k ≤ 2.

Hence, we obtain

\binom{w + 1}{2} − (2w − 1) − ∑_{i=1}^{m} \binom{w_{i} + 1}{2} + ∑_{i=1}^{m}(2w_{i} − 1) = \binom{w − 1}{2} − ∑_{i=1}^{m} \binom{w_{i} − 1}{2}

possibilities of choosing pairs s^{p}, s^{p+k} ∈ E with k ≥ 3 and such that at least one of the numbers a_{p+1},…, a_{p+k−1} is greater than two. Those pairs yield T_{ℓ} = ZZ.

On the other hand, let {s^{q},…, s^{q+w_{i}}} form the i-th edge. Then, its only pairs (s^{p}, s^{p+k}) meeting the assumption that at least one of the numbers a_{p},…, a_{p+k} is greater than two are (s^{q}, s^{q+k}) and (s^{q+w_{i}−k}, s^{q+w_{i}}) with 1 ≤ k ≤ w_{i}. Each of them providing a contribution of ZZ^{[(k−1)/2]} (which is automatically zero if k = 1, 2), we obtain for the entire i-th edge

2 · ∑_{k=3}^{w_{i}−1} [\frac{k − 1}{2}] + 1 · [\frac{w_{i} − 1}{2}] = \binom{w_{i} − 1}{2}

dimensions for T. □

Acknowledgement: I would like to thank the referee for many useful hints and suggestions.

References

[Da] Danilov, V.I.: The Geometry of Toric Varieties. Russian Math. Surveys 33/2 (1978), 97-154.

[Ma] Matsumura, H.: Commutative Algebra. W.A.Benjamin, New York 1970.

[Ri] Riemenschneider, O.: Deformationen von Quotientensingularitäten (nach zyklischen Gruppen). Math. Ann. 209 (1974), 211-248.

[Od] Oda, T.: Convex bodies and algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3/15), Springer-Verlag, 1988.