Computation of the vector space T^{1} for affine toric varieties

Klaus Altmann

Fachbereich Mathematik der Humboldt-Universität zu Berlin PSF 1297, D(O)-1086 Berlin, Germany

Abstract

We compute the vector space T^{1} of first order infinitesimal deformations for affine toric varieties X_{σ}. The result is given by two different formulas which use combinatorial data of the given cone σ only. One of these formulas suggests that there is a relation between deformations of affine toric varieties on the one hand and splitting of certain polyhedra into a Minkowski sum on the other hand.

As an application, the result is used to investigate when three-dimensional affine toric varieties are rigid.

Contents

1 Introduction 1

2 The first T^{1}-formula for affine toric varieties 4

3 Proof of Theorem (2.3) 8

4 The second T^{1}-formula 14

5 Applications and Examples 19

1 Introduction

(1.1) Let X be an algebraic variety (affine, complete or a germ) over I C. Then, the corresponding deformation functor is defined as

Def_{X}((T, 0)) := {X̃ → T (flat) together with an isomorphism

between the special fibre and X} (modulo equivalence).

If X admits only isolated singularities (i.e. X is a smooth manifold except in finitely many points), there exists the semiuniversal deformation [X → S] ∈ Def_{X}(S) that is characterized by the following property:

Given an arbitrary deformation [X̃ → T] ∈ Def_{X}(T), there exists a map T → S (uniquely determined on the level of tangent spaces) such that this deformation is induced by the semiuniversal one. That means, up to

equivalence, we have a fibre product diagramme \begin{aligned}X̃ & → & X \\ ↓ & & ↓ \\ T & → & S \end{aligned} compatible

with the isomorphisms between the special fibres and X. Now, the idea is to obtain information about X by describing the base space S of its semiuniversal deformation. Numbers such as the dimension, the embedding dimension or the number of irreducible components of S at 0 provide interesting numerical invariants of X.

(1.2) Let us denote the ring I C[t]/t^{2} by the symbol I C[ε] (i.e. the relation ε^{2} = 0 is included). Then, the non-reduced space Spec I C[ε] is a fat point containing not only information about the point but also about the tangent direction starting from this point. Therefore, the tangent-space T_{S,0} of S in 0 can be computed as

T_{S,0} = {Spec I C[ε] → (S, 0)} = Def_{X}(Spec I C[ε]).

This vector space is an important invariant of the space X. It is denoted by T_{X}^{1}, and its dimension is equal to the embedding dimension of S. For defining T_{X}^{1} := Def_{X}(Spec I C[ε]) it is not necessary that X admits only isolated singularities, and we will drop this assumption from now on. (But then, T_{X}^{1} may be infinite-dimensional.)

(1.3) There are two extremal cases for X: If X is a “local variety” (i.e. a germ or an affine variety), then the whole interest is concentrated on the singular points of X (T_{X}^{1} = 0 for smooth X). One may compute T_{X}^{1} using the well-developed machinery of cotangential complexes of X which provides T_{X}^{1} as a certain cohomology group (cf. André, Quillen,[LSch] or [KPR], ch. 5).

In the global case, even the case of a smooth varity X is not without interest: Deformations of X are always locally trivial, and computing T_{X}^{1} means to study how these local pieces are patched together. The result is T_{X}^{1} = H^{1}(X, Θ_{X}) (Θ_{X} denotes the tangent sheaf on X).

In general, both extremal cases are combined by using the corresponding local-global spectral sequence for T_{X}^{1}.

(1.4) In this paper, we investigate the case of an affine variety X. Let us mention three “down to earth”-methods to compute T_{X}^{1}. We always use the following notations:

• X = Spec A, i.e. A = Γ(X, O_{X}) is the ring of regular functions;

• X ↪ I C^{n} is given by the corresponding surjection I C[x_{1},…, x_{n}] ⟶ ⟶ A;

• I ⊆ I C[x_{1},…, x_{n}] denotes the ideal of the embedding X ↪ I C^{n}.

(1.4.1) There are exact sequences of O_{X}-sheaves

0 → Θ_{X} → Θ_{I C^{n}} ⊗ O_{X} → N_{X| I C^{n}} → T_{X}^{1} → 0,

or of A-modules

0 → Der_{I C}(A, A) ⟶ A^{n} ⟶ Hom_{A}(I/I^{2}, A) → T_{X}^{1} → 0
D  ↦ (D(x_{1}),...,D(x_{n}))
e_{i}  ↦ [f ∈ I ↦ \frac{∂f}{∂x_{i}}]  , respectively.

(Θ_{X} = Der_{I C}(A, A) denotes the tangent sheaf, i.e. the derivations from A to A;

Θ_{I C^{n}} = ⨁_{i=1}^{n} I C[x_{1},…, x_{n}] · \frac{∂}{∂x_{i}} ≅ I C[x_{1},…, x_{n}]^{n};

N_{X| I C^{n}} = Hom_{A}(I/I^{2}, A) denotes the normal sheaf of X ↪ I C^{n}.) (cf. [KPR], ch. 5)

Example: If X = [f = 0] is a hypersurface, then Hom_{A}(f /f^{2}, A) ≅ A (φ ↦ φ(f)), and T_{X}^{1} can be given as

T_{X}^{1} = A /(\frac{∂f}{∂x_{1}} ,…, \frac{∂f}{∂x_{n}} ) = I C[x^{1},…, x^{n}] /(f, \frac{∂f}{∂x_{1}} ,…, \frac{∂f}{∂x_{n}} ) .

(1.4.2) Let A be a normal ring. Then, T_{X}^{1} is computable as T_{X}^{1} = Ext_{A}^{1}(Ω_{A| I C}^{1}, A) (Ω_{A| I C}^{1} = sheaf of Kähler differentials, i.e. Θ_{X} = Hom(Ω_{A| I C}^{1}, A)). (cf. [KPR], ch. 5)

(1.4.3) Let X = Spec A be a normal surface singularity with the only singular point 0 ∈ X. Then, T_{X}^{1} is the kernel of the map

H^{1}(X\0, Θ_{X\0}) → H^{1}(X\0, Θ_{I C^{n}| _{X\0}}).

(cf. [Sch], §1)

Remark: The embedding T_{X}^{1} ⊆ H^{1}(X\0, Θ_{X\0}) corresponds to the restriction map Def_{X} → Def_{X\0}.

(1.5) The aim of this paper is to compute T_{X}^{1} for affine toric varieties by using the method (1.4.2). The result is given in the Theorems (2.3) and (4.4). (In [LS] a very abstract theory that deals with cohomology groups of semigroup algebras was developed. Nevertheless, direct formulas for computing T_{X}^{1} were obtained in the cyclic quotient case (dim X = 2) only.)

Acknowledgements: I should like to thank Duco van Straten who pointed out to me a mistake resulting from using the “wrong differentials”. I am also grateful to Bernd Sturmfels and the MSI for inviting me to Cornell. During very interesting and stimulating discussions, the idea of the relation between deformations of toric varieties and the decompositions of certain polyhedra into Minkowski sums was formed. Finally, I wish to thank the referee for many useful comments and Ms. Dobers and Ms. Kerger for typing (with LaTEX) and correcting the manuscript .

2 The first T^{1}-formula for affine toric varieties

(2.1) For the notion of toric varieties cf. [Ke], [Oda] or [Da]. We only want to fix the following notations: Let σ ⊆ IR^{n} be a rational, polyhedral cone with vertex in 0; let σ̌ := {r ∈ IR^{n}| ⟨a, r⟩ ≥ 0 for all a ∈ σ} be the dual cone. The corresponding lattices are denoted by N ⊆ IR^{n} and M ⊆ IR^{n}, respectively. Then, the affine toric variety X_{σ} is defined as

X_{σ} := Spec I C[σ̌ ∩ M].

Remark: As generally known, the affine toric varieties are exactly those affine, normal varieties admitting a (I C^{∗})^{n}-action with an open, dense orbit.

Example: The two-dimensional cyclic quotient singularities are exactly the twodimensional affine toric varieties. If X(n, q) denotes the quotient by the ZZ/nZZ-action ξ ↦ \binom{ξ 0}{0 ξ^{q}}, then X(n, q) is given by the cone σ = ⟨(1, 0); (−q, n)⟩.

We will use these singularities as an example to illustrate how the general formula for T^{1} works.

(2.2) Let E be the minimal set that generates the semigroup σ̌ ∩ M. That means, E is finite and consists exactly of the irreducible elements of σ̌ ∩ M:

E = {r ∈ σ̌ ∩ M | 1) r ≠ 0 and
2) r = r^{1} + r^{2} (r^{i} ∈ σ̌ ∩ M) always implies
r^{1} = 0 or r^{2} = 0}.

Example: In case of X = X(n, q) we can develop \frac{n}{n−q} into a continued fraction

\begin{matrix}\frac{n}{n − q} = a_{1} − \frac{1}{a_{2} − \frac{1}{a_{3}}} \\ & . & . & . \\ & & & & −\frac{1}{a_{r}} \end{matrix} (a_{i} ≥ 2).

Then, E is given as the set E = {w^{0},…, w^{r+1}} with elements w^{i} ∈ ZZ^{2} and

• w^{0} = [0, 1], w^{1} = [1, 1], w^{r+1} = [n, q];

• w^{i−1} + w^{i+1} = a_{i} · w^{i} (i = 1,…, r). (cf. [Ch])

[figure: …; [0, 0]; w^{0} = [0, 1]; w^{1}; w^{2}; w^{r+1} = [n, q]; σ̌ ∩ M]

(2.3) T_{X}^{1} is a I C[σ̌ ∩ M]-module which also admits an M-grading. Therefore, we choose and fix an arbitrary element R ∈ M – the corresponding task will be to compute the (always finite dimensional) homogeneous piece T_{X}^{1}(−R). If σ ⊆ IR^{n} is given by its fundamental generators σ = ⟨a^{1},…, a^{N}⟩ (i.e. a^{i} are the edges of σ, and the dual cone is given by σ̌ = {r ∈ IR^{n}| ⟨a^{i}, r⟩ ≥ 0 for i = 1,…, N}), then we define the following sets:

K_{i} := {r ∈ σ̌ ∩ M | (0 ≤)⟨a^{i}, r⟩ < ⟨a^{i}, R⟩} (i = 1,… N);
E_{i} := E ∩ K_{i} (i = 1,…, N);
E′ := ⋃_{i=1}^{N} E_{i}.

Attention: Though it is not visible in the notation: The sets K_{i}, E_{i} and E′ depend on the special choice of the lattice point R ∈ M!

0  0 R  R K_{1}  K_{2}  K_{1} K_{1} ∩ K_{2}  or  K_{2} = ∅

Theorem:

T_{X}^{1}(−R) = (L(E′)_{/∑_{i=1}^{N} L(E_{i})} )^{∗} ⊗_{IR} I C.

(• If F ⊆ IR^{n} is an arbitrary subset, we denote by L(F) the vector space of all linear dependences between elements of F.

• Take the canonical embeddings L(E_{i}) ↪ L(E′).)

(2.4) Let us explain this theorem for cyclic quotient singularities X = X(n, q). There are four different cases for the multidegree R ∈ M ≅ ZZ^{2}:

(i) R = w^{1} (or analogously R = w^{r}) We obtain E_{1} = {w^{0}} and E_{2} = {w^{2},…, w^{r+1}}, and the theorem yields

T^{1}(−R) ≅ \{ \begin{matrix}I C & for & r ≥ 2 \\ 0 & for & r = 1. \end{matrix}

(ii) R = w^{i} (2 ≤ i ≤ r − 1) We obtain E_{1} = {w^{0},…, w^{i−1}} and E_{2} = {w^{i+1},…, w^{r+1}}, and the theorem yields T^{1}(−R) ≅ I C^{2}.

(iii) R = ℓ · w^{i} (1 ≤ i ≤ r; 2 ≤ ℓ < a_{i}) We obtain E_{1} = {w^{0},…, w^{i}} and E_{2} = {w^{i},…, w^{r+1}}, and the theorem yields T^{1}(−R) ≅ I C.

(iv) For the remaining R ∈ M, either E_{1} ⊆ E_{2} or E_{2} ⊆ E_{1} or #(E_{1} ∩ E_{2}) ≥ 2. In all these cases the theorem yields T^{1}(−R) = 0.

(2.5) In [Sm] Smilansky has proved the following result concerning the decomposition of a polyhedron P into a Minkowski sum: If P^{∗} denotes the dual polyhedron (vertices of P^{∗} correspond to facets of P), and if F^{∗} < P^{∗} runs through all faces that arise as dual ones of proper, compact faces of P, then

λ(P) := dim (D(P^{∗}) /∑ D(F^{∗}) ) + 1

(D(…) := vector space of the affine dependences between the vertices of the argument)

is equal to the dimension of the summand set of P. In particular, P is decomposable into a non-trivial Minkowski sum if and only if λ(P) = 1.

As Bernd Sturmfels pointed out to me, the λ(P)-formula looks very similar to the T_{X}^{1}-formula written down in Theorem (2.3). This reflects the fact that there is a correspondence between the decomposition of certain polyhedra into a Minkowski sum and the deformation of affine toric varieties. (This has already been proved for one parameter deformations and will appear in a forthcoming paper.)

Example: (cf. (5.4) of [Al]) Let H ⊆ IR^{2} be the hexagon H := conv{(0, 0); (1, 0); (2, 1); (2, 2); (1, 2); (0, 1)}; putting H into the affine hyperplane [z = 1] ⊆ IR^{3}, we obtain a cone σ := Cone(H) ⊆ IR^{3}. X_{σ} is a three-dimensional singularity (cone over a Del Pezzo surface) with dim_{I C} T_{X}^{1} = 3. The whole vector space is concentrated in the single degree R_{0} = [0, 0, 1] that defines the [z = 1]-plane containing H:

σ = ⟨a^{1}, a^{2},…, a^{6}⟩; σ̌ = ⟨r^{1}, r^{2},…, r^{6}⟩.

Then, the semigroup σ̌ ∩ M is generated by {R_{0}; r^{1},…, r^{6}}, and we obtain E_{i} = {r^{i}, r^{i+1}} (i = 1,…, 6; r^{7} := r^{1}). These six sets consist of linear independent vectors only, i.e.

T_{X}^{1}(−R_{0}) = L(r^{1}, r^{2},…, r^{6})^{∗} ⊗_{IR} I C
= I C^{6} _{/{(⟨a, r^{1}⟩,…, ⟨a, r^{6}⟩) / a ∈ N_{IR}}}
= Div Xσ̌/Cart Xσ̌ ⊗_{ZZ} I C.

(Div Xσ̌ and Cart Xσ̌ denote the sets of Weil and Cartier (i.e. principal) divisors, respectively. Divisors are determined by their so-called order functions (cf. [Ke]), hence, by giving six integers as values of r^{1},…, r^{6}.)

The base space of the semi-universal deformation consists of two irreducible components of dimension 1 and 2, respectively.

These two components correspond to the following Minkowski decompositions of the hexagon H:

(i) H = conv{(0, 0); (1, 1); (0, 1)} + conv{(0, 0); (1, 0); (1, 1)} and

(ii) H = conv{(0, 0); (1, 0)} + conv{(0, 0); (0, 1)} + conv{(0, 0); (1, 1)}.

3 Proof of Theorem (2.3)

(3.1) Let E ⊆ σ̌ ∩ M be an arbitrary finite set that generates σ̌ ∩ M as a semigroup (minimality is not necessary); let K_{i} and E_{i} (for a fixed degree R ∈ M) as defined in (2.3). As mentioned in (1.4.2) we have T_{X}^{1} = Ext_{I C[σ̌∩M]}^{1}(Ω_{X}^{1}, I C[σ̌ ∩ M]), and the I C[σ̌ ∩ M]- module Ω_{X}^{1} can be computed modulo its torsion by the following map (cf. [Da]):

Ω_{X}^{1} ⟶^{Φ} V ⊗_{I C} I C[σ̌ ∩ M] (V := M ⊗_{ZZ} I C)
dx^{r} ⟼ r ⊗ x^{r}.

That means, Q := Im Φ = Ω1 X/ ; tors(Ω_{X}^{1}) is the submodule of V ⊗_{I C} I C[σ̌ ∩ M] that is

generated by the elements B(r) := r ⊗ x^{r} (r ∈ E). On the other hand, Φ is injective (i.e. Q = Ω_{X}^{1}) outside a closed subset of X of codimension 2 (X_{σ} is normal), hence

T_{X}^{1} = Ext_{I C[σ̌∩M]}^{1}(Q, I C[σ̌ ∩ M]).

(3.2) We want to build a I C[σ̌ ∩ M]-free resolution of the module Q; hence, our first step has to be to determine the kernel of the surjection

B := ⨁_{r∈E} I C[σ̌ ∩ M] · B(r) ⟶ Q.

As we are in a graded situation, it will suffice to regard the M-homogeneous pieces of the kernel. They consist of the following elements:

If q ∈ L(E) ⊆ IR^{E} is a relation within E, i.e. ∑_{r∈E} q_{r} · r = 0, then we

define

C(q; ℓ) := ∑_{r∈E} q_{r} · x^{ℓ−r}B(r)

(for those ℓ ∈ M such that ℓ ≥ r for all r with q_{r} ≠ 0).

(The partial ordering “≥” on M is defined with respect to the cone σ̌. That means ℓ ≥ r :⇔ ℓ − r ∈ σ̌.)

By restricting to the set F ⊆ IR^{E} of minimal relations q (up to a constant factor they are uniquely determined by the set of elements r ∈ E that are involved) we obtain a finite set of generators of the kernel. (For each q ∈ F only finitely many elements ℓ are needed.) Attention: Since we are looking for generators of a certain I C[σ̌ ∩ M]-module, it is not enough for F to consist of a linear basis of L(E) only.

(3.3) Analogously we determine the kernel of the map

C := ⨁_{q∈F;ℓ} I C[σ̌ ∩ M] · C(q; ℓ) → ⨁_{r∈E} I C[σ̌ ∩ M] · B(r) = B.

Now, there are two kinds of homogeneous generators:

(i) D(q; ℓ_{1}, ℓ_{2}; η) := x^{η−ℓ_{1}}C(q; ℓ_{1}) − x^{η−ℓ_{2}}C(q; ℓ_{2}) (for η ≥ ℓ_{1}; ℓ_{2})

(ii) If ξ ∈ IR^{F} is a (minimal) relation within F, i.e. ∑_{q∈F} ξ_{q} · q = 0, then we define

D(ξ; \underline{ℓ}; η) := ∑_{q∈F} ξ_{q} · x^{η−ℓ_{q}} · C(q, ℓ_{q})

(for those η ∈ M such that η ≥ ℓ_{q} for all q with ξ_{q} ≠ 0).

(3.4) By defining B, C and D to be homogeneous of the right degree (it is always visible in the last argument: r, ℓ and η, respectively), we obtain an M-graded complex of free I C[σ̌ ∩ M]-modules

D  ⟶^{d_{C}}  C  ⟶^{d_{B}}  B
∥  ∥  ∥
⨁_{D} I C[σ̌ ∩ M] · D ⟶^{d_{C}} ⨁_{C} I C[σ̌ ∩ M] · C ⟶^{d_{B}} ⨁_{B} I C[σ̌ ∩ M] · B,

which is a resolution of Q. Now, we can compute T_{X}^{1} as the homology of the dualized complex.

(3.5) Denote by G one of the capital letters B, C or D. Then, an element φ of the dual free module (⨁_{G} I C[σ̌ ∩ M] · G)^{∗} can be described by considering elements

g(x) ∈ I C[σ̌ ∩ M] to be the images of the generators G (g stands for b, c, or d, respectively).

For φ to be homogeneous of degree −R ∈ M, g(x) will have to be a monomial of degree

deg g(x) = −R + deg G.

In particular, the corresponding complex coefficient g ∈ I C (i.e. g(x) = g · x^{−R+deg G}) admits the property that

g ≠ 0 implies − R + deg G ≥ 0 (i.e. − R + deg G ∈ σ̌).

Therefore, the homogeneous part of degree −R of the dualized complex (3.4)* is equal to the complex

B_{−R}^{∗} ⟶^{d_{B}^{∗}} C_{−R}^{∗} ⟶^{d_{C}^{∗}} D_{−R}^{∗}

of I C-vector spaces with coordinates \underline{b}, \underline{c}, and \underline{d}, respectively:

B_{−R}^{∗} = \{\underline{b(r)} ∈ I C^{E}| b(r) = 0 for r − R ∉ σ̌\}
C_{−R}^{∗} = \{\underline{c(q; ℓ)}_{q∈F,ℓ}| c(q; ℓ) = 0 for ℓ − R ∉ σ̌\}
D_{−R}^{∗} = \{[\underline{d(q; ℓ_{1}, ℓ_{2}; η)}, \underline{d(ξ; \underline{ℓ}; η)}] | \begin{matrix}d(q; ℓ_{1}, ℓ_{2}; η) & = & 0 & for & η − R ∉ σ̌, and \\ d(ξ; \underline{ℓ}; η) & = & 0 & for & η − R ∉ σ̌\}. \end{matrix}

The differentials d_{B}^{∗} and d_{C}^{∗} can be described by

c(q; ℓ)  = ∑_{r∈E} q_{r} · b(r)  and
d(q; ℓ_{1}, ℓ_{2}; η) = c(q; ℓ_{1}) − c(q; ℓ_{2}),
d(ξ; \underline{ℓ}; η)  = ∑_{q∈F} ξ_{q} · c(q, ℓ_{q}).

(3.6) To compute the homology we start describing the kernel V of the second differential. Setting \underline{d} = 0, the first observation we make is that c(q) := c(q; ℓ) does not depend on ℓ:

Ṽ := I C^{F} = {(…, c(q), ⋯)| q ∈ F }.

Now, we can interpret the remaining conditions “d(ξ; \underline{ℓ}; η) = 0” and “c(q; ℓ) = 0 for ℓ − R ∉ σ̌” to obtain

V = \{\underline{c(q)} ∈ Ṽ | • The complex numbers c(q) inherit all linear

dependences from the original vectors q ∈ F. (Equivalently: The linear equations

∑_{r∈E} q_{r} · x_{r} = c(q) (q ∈ F) admit a common

solution \underline{x} = (…, x_{r},…) ∈ I C^{E}.)

• The existence of an ℓ ∈ M with • ℓ ≥ r for q_{r} ≠ 0 and

• ℓ − R ∉ σ̌ implies c(q) = 0.}.

The image W of the first differential is easy to describe:

W̃ := I C^{E} = {(…, b(r),…)| r ∈ E}
W = \{\underline{b(r)} ∈ W̃ | b(r) = 0 for r − R ∉ σ̌\}.

Finally, the embedding W ↪ V is given by the formula

c(q) = ∑_{r∈E} q_{r} · b(r),

and we obtain T_{X}^{1}(−R) =V/W.

(3.7) Now, it is time for involving the sets K_{i}. By definition

⋃_{i} K_{i} = (σ̌ ∩ M) \ (R + (σ̌ ∩ M)),

i.e. for r occurring in the definition of W, the condition “r − R ∉ σ̌” is equivalent to “∃ i : r ∈ K_{i}”. For reformulating the conditions in the definition of V , too, we have to look closer at the sets K_{i}:

Lemma: (properties of the sets K_{i})

1) ⟨a^{i}, R⟩ ≤ 0 iff K_{i} = ∅; ⟨a^{i}, R⟩ = 1 iff K_{i} = [a^{i}-face of σ̌] ∩ M; ⟨a^{i}, R⟩ ≥ 2 iff K_{i} contains an interior point of σ̌.

2) K_{i} is closed for descent, i.e. r ∈ K_{i} and 0 ≤ s ≤ r imply s ∈ K_{i}.

3) Let ⟨a^{1},…, a^{k}⟩ < σ be a smooth face of σ (i.e. {a^{1},…, a^{k}} is a part of a ZZ-basis of the lattice N). Then, for elements r_{1},…, r_{m} ∈ σ̌ ∩ M the conditions

(i) r_{1},…, r_{m} ∈ K_{1} ∩… ∩ K_{k} and

(ii) ∃ ℓ ≥ r_{1},…, r_{m} : ℓ ∈ K_{1} ∩… ∩ K_{k}

are equivalent.

Proof: The only non-trivial part is the third one. It will be sufficient to prove it for m = 2. Let r_{1}, r_{2} ∈ K_{1} ∩… ∩ K_{k} be given. Step 1: Since ⟨a^{1},…, a^{k}⟩ < σ is a smooth face, there exist elements s^{1},…, s^{k} ∈ σ̌ ∩ M such that

⟨a^{i}, s^{j}⟩ = δ_{ij} (1 ≤ i, j ≤ k).

Step 2: We can assume that ⟨a^{i}, r_{1}⟩ = ⟨a^{i}, r_{2}⟩ for i = 1,…, k. (In fact, let ⟨a^{i}, r_{1}⟩−⟨a^{i}, r_{2}⟩ = g_{i} ≥ 0. Then, r_{2} can be corrected by r_{2} := r_{2} +g_{i} ·s^{i}.)

Step 3: Let s ∈ M be an interior point of the dual face ⟨a^{1},…, a^{k}⟩^{∗} = (a^{1})^{⊥} ∩… ∩ (a^{k})^{⊥} ∩ σ̌ of the cone σ̌, i.e.

⟨a^{i}, s⟩ = 0 for i = 1,…, k and
⟨a^{v}, s⟩ > 0 for v ∉ {1,…, k}.

Hence,

⟨a^{i}, r_{1} − r_{2} + g · s⟩ = 0 for i = 1,…, k and
⟨a^{v}, r_{1} − r_{2} + g · s⟩ ≥ 0 for v ∉ {1,…, k} and g ≫ 0,

and we can define ℓ := r_{1} + g · s = r_{2} + (r_{1} − r_{2} + g · s). □

We will need two special cases of the third part of the previous lemma:

Corollary: Let σ = ⟨a^{1},…, a^{N}⟩; r_{1},…, r_{m} ∈ σ̌ ∩ M.

a) For i = 1,…, N we obtain that

r_{1},…, r_{m} ∈ K_{i} iff ∃ ℓ ≥ r_{1},…, r_{m} : ℓ ∈ K_{i}.

b) Let X_{σ} be smooth in codimension 2; let ⟨a^{i}, a^{j}⟩ < σ be a two-dimensional face of the cone σ. Then, we obtain

r_{1},…, r_{m} ∈ K_{i} ∩ K_{j} iff ∃ ℓ ≥ r_{1},…, r_{m} : ℓ ∈ K_{i} ∩ K_{j}.

Now, we can use (a) to reformulate the second condition contained in the definition of the vector space V (cf. (3.6)). Altogether we obtain the following situation:

V = \{\underline{c(q)} | • The linear equations ∑_{r∈E} q_{r} · x_{r} = c(q) (q ∈ F)

admit a common solution \underline{x}.

• For each q the condition {r| q_{r} ≠ 0} ⊆ E_{i}

(for some i) implies c(q) = 0.};

W = \{\underline{b(r)} | b(r) = 0 for r ∈ E′\};

T_{X}^{1}(−R) =V/W (W ⊆ V is embedded by c(q) = ∑_{r∈E} q_{r} · b(r)).

(3.8) Let Φ be the following map:

Φ : V ⟶ L_{I C}(E)^{∗}
\underline{c(q)} ⟼ [Q = ∑_{q∈F} ξ_{q} · q ↦ ∑ ξ_{q} · c(q)] .

(The first condition contained in the definition of V guarantees that Φ is defined correctly. It does not depend on the special representation of Q as a linear combination of elements q ∈ F.)

(3.8.1) Im Φ ⊆ L_{I C}(E_{i})^{⊥} for i = 1,…, N : Let \underline{c(q)} ∈ V and Q ∈ L_{I C}(E_{i}). Then, since F ⊆ L_{I C}(E) is more than a linear basis (cf. (3.2)), Q can be written as a linear combination of elements q ∈ F ∩ L_{I C}(E_{i}). We obtain

⟨Φ(\underline{c(q)}), Q⟩ = ⟨Φ(\underline{c(q)}), ∑_{q∈F ∩L(E_{i})} ξ_{q} · q⟩
= ∑_{q∈F ∩L(E_{i})} ξ_{q} · c(q) = ∑_{q} ξ_{q} · 0 = 0.

(3.8.2) Φ : V → ⋂_{i=1}^{N} L_{I C}(E_{i})^{⊥} (⊆ L_{I C}(E)^{∗}) is surjective: Let φ : L_{I C}(E) → I C be a linear map such that φ| L_{I C}(E_{i}) = 0 for i = 1,…, N. Then, c(q) := φ(q) define an element of V that maps onto φ.

(3.8.3) Finally, the restriction to E′ ⊆ E provides the surjective maps

Φ′ : V ⟶ ⟶^{Φ} (L^{I C}(E) /∑_{i} L_{I C}(E_{i}) )^{∗} ⟶ ⟶ (L^{I C}(E′) /∑_{i} L_{I C}(E_{i}) )^{∗} .

It is clear that Ker Φ′ contains the vector space W. Let us look at the reversed inclusion: Given an element \underline{c(q)} ∈ Ker Φ′ we obtain the following property:

If Q = ∑_{q∈F} ξ_{q} · q fulfils Q_{r} = 0 for r ∉ E′, then ∑_{q∈F} ξ_{q} · c(q) = 0.

But this means that the equations ∑_{r∈E\E′} q_{r} · x_{r} = c(q) (q ∈ F) admit a common

solution \underline{x}, i.e. \underline{c(q)} is contained in W.

Now, the proof of Theorem (2.3) is complete.

4 The second T^{1}-formula

(4.1) As in (2.3) we fix the degree R ∈ M. We are looking for a different T_{X}^{1}(−R)-formula that does not involve the semigroup structure of σ̌ ∩ M (by using E) but only the structure of the real cone σ̌. Let us start with defining those vector spaces that play the crucial role in this formula:

V_{i} := span_{IR}(E_{i}) = \{ \begin{matrix}0  for & ⟨a^{i}, R⟩ ≤ 0 \\ [a^{i} = 0] ⊆ M_{IR} for & ⟨a^{i}, R⟩ = 1 \\ M_{IR} = IR^{n}  for & ⟨a^{i}, R⟩ ≥ 2 \end{matrix}
(i = 1,…, N);
V_{ij} := span_{IR}(E_{i} ∩ E_{j}) ⊆ V_{i} ∩ V_{j}  (i, j = 1,…, N).

Remark: It is not necessary to have any information about the lattice points inside σ̌ (or about E ⊆ σ̌ ∩ M) for defining the vector spaces V_{i}. However, this statement is not true for the vector spaces V_{ij}.

(4.2) Proposition:

(i) Let π : V_{1} ⊕ … ⊕ V_{N} ⟶ V_{1} + … + V_{N} be the canonical map, and let V_{ij} ⊆ V_{i} ∩ V_{j} be embedded as subspaces of Ker π ⊆ V_{1} ⊕… ⊕ V_{N}. Then,

T_{X}^{1}(−R) = (Ker π /∑_{i,j} V_{ij} )^{∗} ⊗_{IR} I C.

(ii) For computing ∑_{i,j} V_{ij} it is possible to restrict the sum to those pairs (i, j) such that ⟨a^{i}, a^{j}⟩ < σ is a two-dimensional face (“edge”) of σ.

Proof: (i) We will use Theorem (2.3), hence, the first step of the proof is to define a map

ψ : L(E′) → Ker π/∑_{i,j} V_{ij} .

For q ∈ L(E′) ⊆ IR^{E′} there exists a (not unique) decomposition

q = ∑_{i=1}^{N} q^{i} with q^{i} ∈ IR^{E_{i}} ⊆ IR^{E′}.

Then, v_{i} := ∑_{r∈E′} q_{r}^{i} · r ∈ V_{i} are vectors with

∑_{i=1}^{N} v_{i} = ∑_{i=1}^{N} ∑_{r∈E′} q_{r}^{i} · r = ∑_{r∈E′} ( ∑_{i=1}^{N} q_{r}^{i}) · r = ∑_{r∈E′} q_{r} · r = 0,

and we define

ψ(q) := (v_{1},…, v_{N}).

To investigate the behaviour of the vectors v_{1},…, v_{N} by choosing different decom-

positions q = ∑_{i=1}^{N} q^{i} = ∑_{i=1}^{N} q̃^{i}, we use induction on the number of pairs (i, r) such that

q_{r}^{i} ≠ q̃_{r}^{i}: Let q_{r}^{i}, q̃_{r}^{i} be different numbers (λ := q_{r}^{i} − q̃_{r}^{i}). Then, there exists an index j ≠ i such that q_{r}^{j} and q̃_{r}^{j} are different too, and we can define another decomposition (q̄^{1},…, q̄^{N}) by

q̄_{r}^{i} := q̃_{r}^{i} + λ = q_{r}^{i},
q̄_{r}^{j} := q̃_{r}^{j} − λ, and
q̄_{•}^{•} := q̃_{•}^{•} for the remaining positions.

On the one hand, (q̄^{1},…, q̄^{N}) sits “between” (q^{1},…, q^{N}) and (q̃^{1},…, q̃^{N}), and on the other hand, the difference ψ(q = ∑ q̄^{i}) − ψ(q = ∑ q̃^{i}) is equal to the tuple

\begin{aligned}(0,…, 0, & λ · r & , 0,…, 0, & −λ · r & , 0,…, 0) \\ & [i] & & [j] \end{aligned} . But this element corresponds to

λ · r ∈ V_{ij} via the embedding mentioned in the proposition. In particular, ψ is defined correctly.

Since ∑_{i} L(E_{i}) ⊆ Ker ψ we obtain a map

ψ : L(E′)/∑_{i} L(E_{i}) ⟶ Ker π/∑_{i,j} V_{ij} .

To construct the inverse of ψ, let (v_{1},…, v_{N}) ∈ Ker π be given. Each v_{i} ∈ V_{i} = span(E_{i}) can be written as

v_{i} = ∑_{r∈E_{i}} q_{r}^{i} · r (for some coefficients q_{r}^{i}),

hence, q := ∑_{i=1}^{N} q^{i} ∈ L(E′) is a pre-image of (v_{1},…, v_{N}). (Different decompositions

of v_{i} yield elements q and q′ which are equal modulo ∑_{i} L(E_{i}).)

(ii) Let an element r ∈ E_{i} ∩ E_{j} ⊆ K_{i} ∩ K_{j} be given. If we define the following open half space

H^{+} := {a ∈ IR^{n}| ⟨a, R − r⟩ > 0},

then a^{i} and a^{j} are contained in H^{+}. Hence, there exists a chain of fundamental generators of σ a^{i} = a^{i_{0}},…, a^{i_{k}} = a^{j} such that

• a^{i_{v}} ∈ H^{+} (i.e. r ∈ E_{i_{v}}) and

• ⟨a^{i_{v}}, a^{i_{v+1}}⟩ < σ is always an edge. □

Example: In the case of two-dimensional cyclic quotients X = X(n, q) we obtain T_{X}^{1}(−R) = (V^{1} ∩ V^{2}/V_{12})^{∗} ⊗_{IR} I C.

To give an explicit description of the vector spaces V_{1}, V_{2}, and V_{12}, we use the terminology of (2.4):

(i) V_{1} = IR · w^{0}; V_{2} = \{ \begin{matrix}IR^{2} for & r ≥ 2 \\ IR · w^{r+1}for & r = 1 \end{matrix} ; V_{12} = 0

(ii) V_{1} = V_{2} = IR^{2}; V_{12} = 0

(iii) V_{1} = V_{2} = IR^{2}; V_{12} = IR · w^{i}

(iv) V_{1} = V_{12} or V_{2} = V_{12} or V_{12} = IR^{2}.

(4.3) Let τ = ⟨a^{1},…, a^{k}⟩ < σ be an arbitrary face of σ. Then, the corresponding lattices are denoted by

N_{τ} ⊆ N  and
M_{τ} = M /τ^{⊥}.

As usual, we fix a degree R ∈ M ⟶ ⟶^{π} M_{τ}.

Lemma:

1) E^{0} := E ∩ τ^{⊥} generates (σ̌ ∩ τ^{⊥}) ∩ M as a semigroup. In particular, span(E^{0}) = τ^{⊥}.

2) As soon as they are not empty, the sets E_{i} (i = 1,…, k) contain E^{0}.

3) Let r ∈ E. Then, π(r) splits into a sum of elements of E^{(τ)}.

4) Each element r̄ ∈ E^{(τ)} can be lifted to E.

5) Let r ∈ σ̌ ∩ M. Then, π(r) ∈ τ̌ ∩ M_{τ}, and for i = 1,…, k, the conditions “r ∈ K_{i}” and “π(r) ∈ K_{i}^{(τ)}” are equivalent. In particular, (3) and (4) will remain valid if we replace E and E^{(τ)} by E_{i} (or E_{i} ∩ E_{j}) and E_{i}^{(τ)} (or E_{i}^{(τ)} ∩ E_{j}^{(τ)}), respectively.

Proof: Most of the claims are quite clear - we only want to prove (4):

Let r ∈ M be an arbitrary lift of r̄ to the lattice M. By r := r + g · s (g ≫ 0; s ∈ int(τ^{⊥} ∩ σ̌)) we may asume that r ∈ σ̌ ∩ M, hence, r can be split into a sum

r = r_{1} + … + r_{m} (r_{j} ∈ E).

On the M_{τ}-level this means that we have found a decomposition

r̄ = π(r) = π(r_{1}) + … + π(r_{m}).

Then, since r̄ ∈ E^{(τ)} is irreducible, there has to exist a summand r_{j} such that r̄ = π(r_{j}), i.e. r_{j} ∈ E is the desired lift of r̄. (The remaining summands r_{v} are contained in E^{0}.) □

Corollary: Let i, j ∈ {1,…, k}. Then, if V_{i}, V_{j} ≠ 0 (i.e. ⟨a^{i}, R⟩, ⟨a^{j}, R⟩ ≥ 1), we will obtain

V_{i} = π^{−1}(V_{i}^{(τ)}) and
V_{ij} = π^{−1}(V_{ij}^{(τ)}).

(4.4) Proposition: Let T_{X_{σ}}^{1} be a finite-dimensional vector space. Then, all proper faces τ < σ are rigid (i.e. T_{X_{τ}}^{1} = 0).

Proof: X_{σ} contains X_{τ} × (I C^{∗})^{dim σ−dim τ} as an open, dense subset, hence, the claim of the proposition is not very surprising. Nevertheless, we will give a proof in terms of our T^{1}-formula; we use the terminology of (4.3). Let R ∈ M such that T_{X_{τ}}^{1} (−R̄) ≠ 0 (R̄ := π(R)). Then, for

R_{g} := R − g · s (g ≫ 0; s ∈ int(τ^{⊥} ∩ σ̌))

we obtain the following properties:

• V_{j} = 0 if j ∉ {1,…, k};
• V^{1} ⊕… ⊕ V^{k} /∑ V_{ij}  ⟶ ⟶^{ψ}  V_{1} + … + V_{k}
↓ ↓

V{(τ) \atop 1} ⊕… ⊕ V{(τ) \atop k} /∑ V_{ij}^{(τ)} ⟶ ⟶^{ψ^{(τ)}} V_{1}^{(τ)} + … + V_{k}^{(τ)}

is a commutative diagramme involving surjections only.

If p = (p_{1},…, p_{k}) ∈V^{1} ⊕… ⊕ V^{k} /∑ V_{ij} is an arbitrary pre-image of a non-trivial

element of T_{X_{τ}}^{1} (−R̄) = Ker ψ^{(τ)}, then ψ(p) will be contained in τ^{⊥}. Hence, we can modify p by p′ := (p_{1}−ψ(p), p_{2},…, p_{k}) to obtain a non-trivial element of T_{X_{σ}}^{1}(−R_{g}).

That means that T_{X_{σ}}^{1} would have non-trivial homogeneous summands in infinitely many degrees R_{g} (g ≫ 0), i.e. dim_{I C} T_{X_{σ}}^{1} = ∞. □

Theorem: Let dim X_{σ} ≥ 3.

1) If T_{X}^{1} is a finite-dimensional vector space, X will be smooth in codimension 2.

2) If this is the case (i.e. if all “edges” of σ are smooth), we will obtain the formula

T_{X}^{1}(−R) = Ker [V^{1} ⊕… ⊕ V^{N} / ; ∑_{{a^{i},a^{j}}<σ} V_{i} ∩ V_{j} ⟶ ⟶ V_{1} + … + V_{N} ]^{∗} .

Proof: (1) We apply the previous proposition to the two-dimensional faces {a^{i}, a^{j}} < σ. The corresponding toric varieties X_{τ} are two-dimensional cyclic quotient singularities, hence, rigidity implies smoothness.

(2) Looking at proposition (4.2), the only point that remains to be checked is V_{ij} = V_{i} ∩ V_{j} ({a^{i}, a^{j}} < σ being a face provided). By Corollary (4.3) this question can be investigated on the face being fixed instead of σ. X_{{a^{i},a^{j}}} was assumed to be smooth, hence, V_{ij} = V_{i} ∩ V_{j} becomes trivial at this level (cf. the example in (4.2)). □

5 Applications and Examples

(5.1) Let σ = ⟨a^{1},…, a^{n}⟩ be an n-dimensional simplicial cone that is smooth in codimension 2. Then, each pair of “vertices” yields an “edge” ⟨a^{i}, a^{j}⟩ < σ, hence,

T_{X_{σ}}^{1}(−R) = Ker [V^{1} ⊕… ⊕ V^{n} /∑_{i,j} V_{i} ∩ V_{j} ⟶ ⟶ V_{1} + … + V_{n}]^{∗} .

If V_{1},…, V_{m} denote the non-trivial vector spaces (equal to [a^{i} = 0] ⊆ IR^{n} or IR^{n}), we will obtain the important equality

V_{i} ∩ ( ∑_{j≥i+1} V_{j}) = ∑_{j≥i+1}(V_{i} ∩ V_{j}) (i = 1,…, m − 1).

This means that, by induction, all elements of Ker[V_{1} ⊕… ⊕ V_{m} ⟶ ⟶ V_{1} + … + V_{m}] can be put into the vector space ∑_{i,j} V_{i} ∩ V_{j}.

Hence, T_{X_{σ}}^{1}(−R) = 0 for each degree R ∈ M, i.e X_{σ} is rigid.

Remark: Simplicial cones σ yield n-dimensional cyclic quotient singularities. Therefore, the rigidity of these toric varieties is a special case of the well-known result of Schlessiger (cf. [Sch]) concerning arbitrary isolated quotient singularities in dimension ≥ 3.

(5.2) Let σ = ⟨a^{0},…, a^{m}; b^{0},…, b^{n}⟩ (m, n ≥ 1) be an (m+n+1)-dimensional cone such that the fundamental generators admit the relation

∑_{i=0}^{m} λ_{i}a^{i} = ∑_{j=0}^{n} μ_{j}b^{j} (λ_{i}, μ_{j} > 0).

Then, the dual cone σ̌ is the cone over the product of two simplices Δ^{m} × Δ^{n}, i.e. the fundamental generators can be listed as pairs (i, j) (i = 0,…, m; j = 0,…, n). The relation between σ and σ̌ is given by

⟨a^{p}, (i, j)⟩ = 0 iff p ≠ i and
⟨b^{q}, (i, j)⟩ = 0 iff q ≠ j.

Assume that X_{σ} is smooth in codimension 2.

Case 1: Let m, n ≥ 2. Then, each pair of “vertices” yields an “edge” of σ, hence,

T_{X_{σ}}^{1}(−R) = Ker [^{(⨁_{i} V_{i})⊕(⨁_{j} W_{j})} /_{∑(V_{i_{1}}∩V_{i_{2}})+∑(V_{i}∩W_{j})+∑(W_{j_{1}}∩W_{j_{2}})} ⟶ ⟶ (∑_{i} V_{i}) + (∑_{j} W_{j})]^{∗} .

As in (5.1) we will regard only those vector space V_{i}, W_{j} that are non-trivial. Our task will be to put them into a new, linear order P_{1},…, P_{N} such that

P_{μ} ∩ ( ∑_{ν≥μ+1} P_{ν}) = ∑_{ν≥μ+1}(P_{μ} ∩ P_{ν}).

Well, this is very easy: For N ≥ 3, the only condition is to put two vector spaces of the same type (V or W) to the last two places P_{N−1} and P_{N}. The case N ≤ 2 is trivial, anyway. Therefore, T_{X_{σ}}^{1}(−R) = 0 for each degree R ∈ M, i.e. X_{σ} is rigid.

Case 2: Let m ≥ 2; n = 1 (σ is the cone over an (m + 1)-dimensional double simplex; σ̌ is the cone over a prism). Then, {b^{0}, b^{1}} is the only pair of vertices that does not correspond to a (twodimensional) face of σ. We can proceed in the usual way (to reorder the non-trivial V_{i}, W_{j} into a chain P_{1},…, P_{N}), but we have to achieve the stronger formula

P_{μ} ∩ ( ∑_{ν≥μ+1} P_{ν}) = ∑_{{ν≥μ+1 \atop \{Pμ,Pν \}≠\{W_{0},W_{1}\}}}(P_{μ} ∩ P_{ν}).

If there are at least two non-vanishing vector spaces V_{i}, V_{j}, this will be possible (put them to the last two places P_{N−1}, P_{N}). If there is at least one vector space V_{i} that is equal to IR^{m+2}, we will also succeed (put V_{i} to the place P_{N}). Hence, only two cases are left to be a candidat for T^{1}(−R) ≠ 0:

a) V_{0} = … = V_{m} = 0; W_{0}, W_{1} ≠ 0. This situation implies

⟨a^{i}, R⟩ ≤ 0 (i = 0,…, m) and
⟨b^{j}, R⟩ ≥ 1 (j = 0, 1),

i.e. we obtain a contradiction to the assumption ∑_{i=0}^{m} λ_{i}a^{i} = ∑_{j=0}^{1} μ_{j}b^{j} (λ_{i}, μ_{j} > 0).

b) V_{i_{0}} = [a^{i_{0}} = 0] ⊆ IR^{n}; V_{i} = 0 (i ≠ i_{0}); W_{0}, W_{1} ≠ 0. The two vector spaces V_{i_{0}} ∩W_{0}; V_{i_{0}} ∩W_{1} ⊆ V_{i_{0}} ⊕W_{0} ⊕W_{1} have 0 as the intersection, hence, to compute T_{X_{σ}}^{1}(−R) it will be suffice to count the dimensions:

dim_{I C} T_{X_{σ}}^{1}(−R) = dim_{IR}(V_{i_{0}} ⊕ W_{0} ⊕ W_{1}) − dim_{IR}(V_{i_{0}} + W_{0} + W_{1})
− dim_{IR}(V_{i_{0}} ∩ W_{0}) − dim_{IR}(V_{i_{0}} ∩ W_{1})
= ([m + 1] + dim W_{0} + dim W_{1}) − (m + 2)
−(dim W_{0} − 1) − (dim W_{1} − 1)
= 1.

On the other hand, this situation occurs iff

⟨a^{i_{0}}, R⟩ = 1;
⟨a^{i}, R⟩ ≤ 0 (i = 0,…, m; i ≠ i_{0});
⟨b^{j}, R⟩ ≥ 1 (j = 0, 1).

In particular, if X_{σ} is not rigid, the following condition will be implied:

λ_{i_{0}} ≥ λ_{i_{0}} + ∑_{i≠i_{0}} λ_{i} · ⟨a^{i}, R⟩ = ∑_{i=0}^{m} λ_{i} · ⟨a^{i}, R⟩ = ∑_{j=0}^{1} μ_{j}⟨b^{j}, R⟩ ≥ ∑_{j=0}^{1} μ_{j},

i.e. λ_{i_{0}} ≥ μ_{0} + μ_{1}.

Case 3: m = n = 1 ( σ and σ̌ are cones over a quadrangle). This case is included in the investigation of the three-dimensional toric varieties (cf. (5.3)).

Remark: In a discussion with Ancus Röhr, we have observed that the class of cones σ = ⟨a^{0},…, a^{m}; b^{0},…, b^{n}⟩ considered in this section contains the general determinantal singularities (2-minors of a (m + 1) × (n + 1) matrix with general entries). They correspond to those cones σ such that

(i) λ_{0} = … = λ_{m} = μ_{0} = … = μ_{n} = 1 and

(ii) each system of m+n+1 vectors of {a^{0},…, a^{m}; b^{0},…, b^{n}} provides a ZZ-basis of the lattice N ≅ ZZ^{m+n+1}.

In particular, we can see that the general determinantal singularities of dimension ≥ 4 are rigid.

(5.3) Three-dimensional affine toric varieties Let σ = ⟨a^{1},…, a^{N}⟩ be a three-dimensional cone that is smooth in codimension 2 (i.e. X_{σ} admits an isolated singularity). Since ⟨a^{i}, a^{i+1}⟩ < σ (i ∈ZZ/NZZ) are the edges of σ, we obtain

T_{X_{σ}}^{1}(−R) = Ker [V^{1} ⊕… ⊕ V^{N} /∑_{i}(V_{i} ∩ V_{i+1}) ⟶ ⟶ V_{1} + … + V_{N} ]^{∗} .

As in the second case of (5.2), we try to count dimensions to compute T^{1}. The sum ∑_{i}(V_{i} ∩ V_{i+1}) is not a direct one, but we can measure the difference by the exact sequence

0 → ⋂_{i} V_{i} → ⨁_{i}(V_{i} ∩ V_{i+1}) → ∑_{i}(V_{i} ∩ V_{i+1}) → 0.

Hence,

dim_{I C} T_{X_{σ}}^{1}(−R) = ∑_{i} dim V_{i} − ∑_{i} dim(V_{i} ∩ V_{i+1}) + dim ⋂_{i} V_{i} − dim(∑_{i} V_{i}).

To distinguish between some cases, let V_{1},…, V_{k} ≠ 0 and V_{k+1} = … = V_{N} = 0. (The vertices that admit trivial vector spaces form a connected chain.) Case 1: k ≤ 2. Then, T_{X_{σ}}^{1}(−R) = 0. Case 2: 3 ≤ k ≤ N − 1.

dim_{I C} T_{X_{σ}}^{1}(−R) = ∑_{i=1}^{k} dim V_{i} − ∑_{i=1}^{k−1}(dim V_{i} + dim V_{i+1} − 3) + 0 − 3
= 3(k − 2) − ∑_{i=2}^{k−1} dim V_{i}.

In particular, T_{X_{σ}}^{1}(−R) = 0 iff dim V_{i} = 3 for i = 2,…, k − 1. Case 3: k = N.

dim_{I C} T_{X_{σ}}^{1}(−R) = ∑_{i∈^{Z Z}/_{NZ Z}} dim V_{i} − ∑_{i∈^{Z Z}/_{NZ Z}}(dim V_{i} + dim V_{i+1} − 3) + dim ⋂_{i} V_{i} − 3
= 3(N − 1) − ∑_{i=1}^{N} dim V_{i} + dim ⋂_{i} V_{i} ;
dim ⋂_{i} V_{i} = \{ \begin{matrix}3 for & dim V_{1} = … = dim V_{N} = 3 \\ 2 for & #\{i| dim V_{i} = 2\} = 1 \\ 1 for & #\{i| dim V_{i} = 2\} = 2 \\ 0 for & #\{i| dim V_{i} = 2\} ≥ 3. \end{matrix}

In particular, T_{X_{σ}}^{1}(−R) = 0 iff #{i| dim V_{i} = 2} ≤ 3. Otherwise, we obtain dim_{I C} T_{X_{σ}}^{1}(−R) = 3(N − 1) − ∑_{i=1}^{N} dim V_{i}.

Summary:

1) T_{X_{σ}}^{1} consists of homogeneous pieces T_{X_{σ}}^{1}(−R) of two different types: (I) R ∉ int σ̌.

dim_{I C} T_{X_{σ}}^{1}(−R) = #{i ∈^{ZZ}/_{NZZ}| ⟨a^{i−1}, R⟩ ≥ 1; ⟨a^{i}, R⟩ = 1; ⟨a^{i+1}, R⟩ ≥ 1}.

(II) R ∈ int σ̌.

dim_{I C} T_{X_{σ}}^{1}(−R) = Max {0; #{i ∈^{ZZ}/_{NZZ}| ⟨a^{i}, R⟩ = 1} − 3} .

2) In particular, X_{σ} is not rigid iff there exists an R ∈ M such that

(i) \begin{aligned}∃i : & ⟨a^{i−1}, R⟩ & ≥ 1, \\ & ⟨a^{i}, R⟩ & = 1, \\ & ⟨a^{i+1}, R⟩ & ≥ 1, \\ ∃j: & ⟨a^{j}, R⟩ & ≤ 0, & or \end{aligned}

(ii) ∃i_{v} (v = 1, 2, 3, 4) : ⟨a^{i_{v}}, R⟩ = 1.

Example: Let (α_{1}, −α_{2}, α_{3}, −α_{4}) ∈ ZZ^{4} be a primitive lattice point with α_{i} > 0 (i = 1, 2, 3, 4). Then, ^{ZZ^{4}}/_{ZZ·(α_{1},−α_{2},α_{3},−α_{4})} ≅ ZZ^{3}, i.e. we obtain an exact sequence

0 ⟶ ZZ ⟶  ZZ^{4}  ⟶ ZZ^{3} ⟶ 0
1 ↦ (α_{1},−α_{2},α_{3},−α_{4})
e^{i}  ⟼ a^{i}  .

Now, the cone σ := ⟨a^{1}, a^{2}, a^{3}, a^{4}⟩ ⊆ IR^{3} admits the following properties:

1) α_{1}a^{1} + α_{3}a^{3} = α_{2}a^{2} + α_{4}a^{4} (the real relation between the fundamental generators).

2) There exists an R ∈ M such that ⟨a^{i}, R⟩ = g_{i}, iff α_{1}g_{1} + α_{3}g_{3} = α_{2}g_{2} + α_{4}g_{4}. (The fundamental generators a^{i} ∈ ZZ^{3} are in the most general position that is possible when (1) holds.)

Proof: Consider the dual of the above sequence!

We want to describe T_{X_{σ}}^{1} for this cone:

(i) X_{σ} admits infinitesimal deformations of type (I) iff

α_{1} ∈ α_{2} + α_{4} + INα_{2} + INα_{4} + INα_{3} or
α_{3} ∈ α_{2} + α_{4} + INα_{2} + INα_{4} + INα_{1} or
α_{2} ∈ α_{1} + α_{3} + INα_{1} + INα_{3} + INα_{4} or
α_{4} ∈ α_{1} + α_{3} + INα_{1} + INα_{3} + INα_{2}.

These conditions imply α_{1} ≥ α_{2} + α_{4}, α_{3} ≥ α_{2} + α_{4}, α_{2} ≥ α_{1} + α_{3}, and α_{4} ≥ α_{1} + α_{3}, respectively.

(ii) X_{σ} admits infinitesimal deformations of type (II), iff α_{1} + α_{3} = α_{2} + α_{4}. (If this condition is satisfied, the whole space T_{X_{σ}}^{1} will be one-dimensional.)

References

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