One parameter families containing three dimensional toric Gorenstein singularities

Klaus Altmann

Contents

1 Introduction 1

2 Visualizing T^{1} 3

3 Genuine deformations 9

4 Three dimensional toric Gorenstein singularities 14

5 Work in progress and open problems 24

1 Introduction

1.1

Let σ be a rational, polyhedral cone and Y = Y_{σ} the affine toric variety it defines, which is normal but possibly singular; we study the deformation theory of Y. The infinitesimal deformation space T_{Y}^{1} is a multigraded vector space, with homogeneous pieces determined by the combinatorial formulas of [Al 1].

If Y_{σ} has only an isolated Gorenstein singularity, we can say even more (cf. [Al 2], [Al 3]): T^{1} is concentrated in a single multidegree, and the corresponding homogeneous piece has an elementary geometric description in terms of Minkowski summands of a certain lattice polytope. We even obtain the entire versal deformation of Y_{σ} (cf. [Al 4]).

1.2

Our first aim in the present paper is to interpret geometrically the formula for T^{1} for arbitrary toric singularities in every multidegree. We again do this in terms of Minkowski summands of certain polyhedra; however, these polyhedra are not necessarily compact any longer, and their vertices are not necessarily lattice points (cf. 2.6).

In [Al 2] we studied so-called toric deformations. These only exist in negative multidegrees (that is, with multidegrees in −σ^{∨}). They are genuine deformations with smooth parameter space, and are characterized by the fact that their total space is again toric. Now, armed with our new description of T_{Y}^{1}, we describe the Kodaira–Spencer map in these terms in Theorem 3.3. Moreover, in 3.5, we extend the construction of genuine deformations to nonnegative degrees using a partial modification of our singularity Y_{σ}. Although the total space is no longer toric, we can still describe it and its Kodaira– Spencer map combinatorially.

1.3

After that, we focus on three dimensional Gorenstein toric singularities. As already mentioned, everything is known in the case of isolated singularities. However, as soon as Y_{σ} has a one dimensional singular locus (necessarily of transversal type A_{k}), the situation changes drastically: T_{Y}^{1} spreads in general into infinitely many multidegrees. In 4.3, using our geometric description of the graded pieces of T_{Y}^{1}, we detect all the nontrivial ones, and determine their dimension (which is usually 1). The easiest example of this kind is the cone over the weighted projective plane ℙ(1, 2, 3) (cf. 4.4 and 4.8).

At present it seems impossible to describe the entire versal deformation, which is an infinite dimensional space. However, the infinitesimal deformations corresponding to the one dimensional homogeneous pieces of T_{Y}^{1} are unobstructed, and we lift them in 4.5 to genuine one parameter families. Since the corresponding multidegrees are in general nonnegative, this can be done using the construction introduced in 3.5. Our treatment of the cone over ℙ(1, 2, 3) continues in this spirit in 4.8.

These one parameter families can be thought of as a kind of skeleton, from which one hopes eventually to build the entire versal deformation. The most important open questions are the following:

1. which sets of one parameter families belong to a common irreducible component of the base space?

2. how can those families be combined to find a general fiber of this component (a smoothing of Y_{σ})?

Answers to these questions would provide important information about three dimensional flips.

2 Visualizing T^{1}

2.1 Notation

As usual in toric geometry (see [Oda] for a detailed introduction), N and M denote dual lattices (that is, finitely generated, free Abelian groups) with a perfect pairing ⟨ , ⟩ : N × M → ℤ, and N_{ℝ}, M_{ℝ} the corresponding ℝ-vector spaces obtained by extension of scalars. Let σ ⊂ N_{ℝ} be the polyhedral cone with vertex at 0 spanned by fundamental generators a^{1}, …, a^{M} ∈ N. We assume that the a^{i} are primitive, that is, not proper multiples of other elements of N. We write σ = ⟨a^{1}, …, a^{M}⟩.

The dual cone σ^{∨} := {r ∈ M_{ℝ} | ⟨σ, r⟩ ≥ 0} is given by the inequalities corresponding to a^{1}, …, a^{M}. Intersecting σ^{∨} with the lattice M yields a finitely generated semigroup σ^{∨} ∩ M. We denote by E ⊂ σ^{∨} ∩ M its minimal set of generators, and call it the Hilbert basis. Then the affine toric variety Y_{σ} := Spec ℂ[σ^{∨}∩M] ⊂ ℂ^{E} has defining equations corresponding to the linear dependence relations among elements of E.

2.2

Most of the rings and modules relevant to the study of Y_{σ} are M-graded (or multigraded); this applies to the modules T_{Y}^{i}, that describe infinitesimal deformations of Y_{σ} and obstructions to extending them. Let R ∈ M. In [Al 1] and [Al 3], we defined the subsets

E_{j}^{R} := {r ∈ E | ⟨a^{j}, r⟩ < ⟨a^{j}, R⟩} ⊂ E

for j = 1, …, M. These provide the main tool for building a complex Span(E^{R})_{•} of free Abelian groups, with

Span(E^{R})_{−k} := ⨁_{τ a face of σ with dim τ = k} Span(E_{τ}^{R}),

where

E_{τ}^{R} := ⋂_{a^{j}∈τ} E_{j}^{R} for faces τ < σ, but E_{0}^{R} := ⋃_{j=1}^{N} E_{j}^{R} ,

and where the differentials are the obvious maps.

Theorem 2.1 (cf. [Al 1], [Al 3]) The homogeneous piece of T_{Y}^{1} in degree −R is given by

T_{Y}^{1}(−R) = H^{1}( Span(E^{R})∗_{•} ⊗_{ℤ} ℂ).

If we assume in addition that Y_{σ} is smooth in codimension two, the same thing holds for T_{Y}^{2}:

T_{Y}^{2}(−R) = H^{2}( Span(E^{R})∗_{•} ⊗_{ℤ} ℂ).

In particular, to calculate T_{Y}^{1}(−R), we need the vector spaces Span_{ℂ} E_{j}^{R} and Span_{ℂ} E_{jk}^{R}, where a^{j}, a^{k} span a two dimensional face of σ. The first of these is easily determined:

Span_{ℂ} E_{j}^{R} = \begin{cases}0 & if ⟨a^{j}, R⟩ ≤ 0, \\ (a^{j})^{⊥} & if ⟨a^{j}, R⟩ = 1, \\ M_{ℂ} & if ⟨a^{j}, R⟩ ≥ 2. \end{cases}

The second is always contained in (Span_{ℂ} E_{j}^{R}) ∩ (Span_{ℂ} E_{k}^{R}) as a subspace of codimension ≤ 2. As the following example shows, its actual size reflects the infinitesimal deformations of the two dimensional cyclic quotient singularity corresponding to the plane cone spanned by a^{j}, a^{k}. (These singularities are exactly the transversal types of the singularities of Y_{σ} in codimension two.)

Example 2.2 Write Y_{n,q} for the two dimensional quotient of ℂ^{2} by the action of ℤ/nℤ via ( \begin{matrix}ξ & 0 \\ 0 & ξ^{q} \end{matrix} ), where ξ is a primitive nth root of unity and q is coprime to n. Then Y_{n,q} is the toric variety given by the cone σ = ⟨(1, 0); (−q, n)⟩ ⊂ ℝ^{2}. The set E ⊂ σ^{∨}∩ℤ^{2} consists of the lattice points r^{0}, …, r^{w} along the compact faces of the boundary of the convex hull conv ((σ^{∨} \ {0}) ∩ ℤ^{2}). There are integers a_{v} ≥ 2 such that r^{v−1} + r^{v+1} = a_{v}r^{v} for v = 1, …, w − 1. These may be obtained by expanding n/(n − q) as a negative continued fraction (cf. [Oda], §1.6).

Assume w ≥ 2 and let a^{1} = (1, 0) and a^{2} = (−q, n). Then only two sets E_{1}^{R} and E_{2}^{R} are involved, and the previous theorem states that

T_{Y}^{1}(−R) = (\frac{Span_{ℂ} E_{1}^{R} ∩ Span_{ℂ} E_{2}^{R}}{Span_{ℂ}(E_{1}^{R} ∩ E_{2}^{R})} )^{∗} .

The only multidegrees R ∈ ℤ^{2} contributing to T_{Y}^{1} are as follows:

(i) R = r^{1} (and the case R = r^{w−1} is similar): then Span_{ℂ} E_{1}^{R} = (a^{1})^{⊥}, Span_{ℂ} E_{2}^{R} = ℂ^{2} (or (a^{2})^{⊥} if w = 2), and Span_{ℂ} E_{12}^{R} = 0. Hence, dim T^{1}(−R) = 1 (or 0 if w = 2).

(ii) R = r^{v} with 2 ≤ v ≤ w − 2: Span_{ℂ} E_{1}^{R} = Span_{ℂ} E_{2}^{R} = ℂ^{2}, and Span_{ℂ} E_{12}^{R} = 0. Thus dim T^{1}(−R) = 2.

(iii) R = p · r^{v} with 1 ≤ v ≤ w − 1, and 2 ≤ p < a_{v} for w ≥ 3; (or v = 1 = w − 1 with 2 ≤ p ≤ a_{1} for w = 2): then Span_{ℂ} E_{1}^{R} = Span_{ℂ} E_{2}^{R} = ℂ^{2}, and Span_{ℂ} E_{12}^{R} = ℂ · R. In particular, dim T^{1}(−R) = 1.

2.3 Minkowski summands

Definition 2.3 We define the Minkowski sum of two polyhedra Q′, Q′′ ⊂ ℝ^{n} to be the polyhedron Q′ + Q′′ := {p′ + p′′ | p′ ∈ Q′, p′′ ∈ Q′′}. Obviously, this notion also makes sense for translation classes of polyhedra in arbitrary affine spaces.

[figure: =; +; =; +; +]

Every polyhedron Q decomposes as the Minkowski sum Q = Q^{c} + Q^{∞} of a (compact) polytope Q^{c} and the cone of unbounded directions Q^{∞}. The latter is uniquely determined by Q, whereas the compact summand is not. However, we can take Q^{c} to be the minimal one – given as the convex hull of the vertices of Q itself. If Q was already compact, then Q^{c} = Q and Q^{∞} = 0.

A polyhedron Q′ is called a Minkowski summand of Q if there is a Q′′ such that Q = Q′ + Q′′ and if, additionally, (Q′)^{∞} = Q^{∞}.

In particular, a Minkowski summand always has the same cone of unbounded directions as the original polyhedron, whereas its compact edges are a dilation of those of the original polyhedron (the dilation factor 0 is allowed).

2.4 Setup

Consider the cone σ ⊂ N_{ℝ} and fix some element R ∈ M. Then

𝔸(R) := [R = 1] = {a ∈ N_{ℝ} | ⟨a, R⟩ = 1} ⊂ N_{ℝ}

is an affine space; provided that R is primitive, it comes with a lattice 𝕃(R) := [R = 1]∩N. The corresponding vector space is 𝔸_{0}(R) := [R = 0]; this always has the lattice 𝕃_{0}(R) := [R = 0] ∩ N. We define the crosssection of σ in degree R to be the polyhedron

Q(R) := σ ∩ [R = 1] ⊂ 𝔸(R)

(here Q stands for Querschnitt). This has cone of unbounded directions Q(R)^{∞} = σ ∩ 𝔸_{0}(R) ⊂ N_{ℝ}. The compact part Q(R)^{c} is generated by its vertices \overline{a}^{j} := a^{j}/ ⟨a^{j}, R⟩ for j satisfying ⟨a^{j}, R⟩ ≥ 1. A trivial but nevertheless important observation is the following: the vertex \overline{a}^{j} is a lattice point (that is, \overline{a}^{j} ∈ 𝕃(R)), if and only if ⟨a^{j}, R⟩ = 1.

Fundamental generators of σ contained in R^{⊥} can still be “seen” as edges in Q(R)^{∞}, but those with ⟨•, R⟩ < 0 are “invisible” in Q(R). In particular, we can recover the cone σ from Q(R) if and only if R ∈ σ^{∨}.

2.5

Write d^{1}, …, d^{N} ∈ R^{⊥} ⊂ N_{ℝ} for the compact edges of Q(R). As in [Al 4], §2, for each compact 2-face ε < Q(R) we define its sign vector \underline{ε} ∈ \{0, ±1\}^{N} to be

ε_{i} := \begin{cases}±1 & if d^{i} is an edge of ε, \\ 0 & otherwise, \end{cases}

where the signs are chosen so that the oriented edges ε_{i} · d^{i} fit into a cycle along the boundary of ε. This determines \underline{ε} up to sign (and either choice will do). In particular, ∑_{i} ε_{i}d^{i} = 0.

Definition 2.4 For each R ∈ M we define the vector spaces

V (R) := {(t_{1}, …, t_{N}) | ∑_{i} t_{i}ε_{i}d^{i} = 0 for every compact 2-face ε < Q(R)}
W (R) := ℝ^{#{vertices of Q(R) not in N}}.

The cone C(R) := V (R) ∩ ℝ_{≥0}^{N} measures the dilation of each compact edge, and therefore parametrizes exactly the Minkowski summands of positive multiples of Q(R). For this reason, we call elements of V (R) generalized Minkowski summands; they may have edges of negative length. (See [Al 4], Lemma 2.2 for a discussion of the compact case.) The vector space W (R) provides coordinates s_{j} for each vertex \overline{a}^{j} ∈ Q(R) \ N, that is, ⟨a^{j}, R⟩ ≥ 2.

2.6

Each compact edge d^{jk} = \overline{\overline{a}^{j}\overline{a}^{k}} gives rise to a set of equations G_{jk} relating elements (\underline{t}, \underline{s}) ∈ V (R) ⊕ W (R). These^{1} sets are of one of the following three types:

(0) G_{jk} = ∅;

(1) G_{jk} = {s_{j} − s_{k} = 0} whenever both coordinates exist in W (R);

(2) G_{jk} = {t_{jk} − s_{j} = 0, t_{jk} − s_{k} = 0}, omitting equations that do not make sense.

1The edge corresponds to a codim 2 singular locus of Y_{σ}, whose transversal type is the cyclic quotient singularity \frac{1}{n}(1, q) described in Example 2.2. As one can see at the end of 2.7, the choice of G_{jk} is essentially governed by the position of the monomial relative to the classification of Example 2.2 into cases (i), (ii), (iii).

Restricting V (R) ⊕ W (R) to the (at most) three coordinates t_{jk}, s_{j}, s_{k}, the actual choice of G_{jk} is made such that these equations yield a subspace of dimension 1+dim T_{⟨a^{j},a^{k}⟩}^{1}(−R). Notice that the dimension of T^{1}(−R) for the

two dimensional quotient singularity assigned to the plane cone ⟨a^{j}, a^{k}⟩ can be obtained from Example 2.2.

Theorem 2.5 The infinitesimal deformations of Y_{σ} in degree −R equal

T_{Y}^{1}(−R) = \{(\underline{t}, \underline{s}) ∈ V_{ℂ}(R) ⊕ W_{ℂ}(R) | \begin{matrix}(\underline{t}, \underline{s}) satisfies the \\ equations G_{jk} of 2.6 \end{matrix} \} /ℂ · (\underline{1}, \underline{1}).

The vector space V (R) (encoding Minkowski summands) is, in a sense, the main tool to describe infinitesimal deformations. Depending on which of the above Types (0)–(2) the G_{jk} belong to, the elements of W (R) either provide additional parameters, or introduce conditions that exclude Minkowski summands not of some prescribed type.

If Y is smooth in codimension two, then G_{jk} is always of Type (2). In particular, the variables \underline{s} are completely determined by the \underline{t}, and we obtain:

Corollary 2.6 If Y is smooth in codimension two, then T_{Y}^{1}(−R) is contained in V_{ℂ}(R)/ℂ · (\underline{1}). It is made up of those \underline{t} such that t_{jk} = t_{kl} whenever d^{jk}, d^{kl} are compact edges with a common non-lattice vertex \overline{a}^{k} of Q(R). Thus T_{Y}^{1}(−R) equals the set of equivalence classes of those Minkowski summands of ℝ_{≥0} · Q(R) that preserve the stars of non-lattice vertices of Q(R) up to homothety.

2.7

Proof of Theorem 2.5, Step 1 From Theorem 2.1, we know that T_{Y}^{1}(−R) equals the complexified cohomology of the complex

N_{ℝ} → ⨁_{j} (Span_{ℝ} E_{j}^{R})^{∗} → ⨁_{⟨a^{j},a^{k}⟩<σ} (Span_{ℝ} E_{jk}^{R})^{∗} .

According to 2.2, an element of ⨁_{j} (Span_{ℝ} E_{j}^{R})^{∗} can be represented by a family of elements

\begin{cases}b^{j} ∈ N_{ℝ} & if ⟨a^{j}, R⟩ ≥ 2, \\ b^{j} ∈ N_{ℝ}/ℝ · a^{j} & if ⟨a^{j}, R⟩ = 1. \end{cases}

Dividing by the image of N_{ℝ} means shifting this family by a common vector b ∈ N_{ℝ}. On the other hand, the family {b^{j}} must map to 0 in the complex; that is, for each compact edge \overline{\overline{a}^{j}, \overline{a}^{k}} < Q the functions b^{j} and b^{k} must be equal on Span_{ℝ} E_{jk}^{R}. Since

(a^{j}, a^{k})^{⊥} ⊂ Span_{ℝ} E_{jk}^{R} ⊂ (Span_{ℝ} E_{j}^{R}) ∩ (Span_{ℝ} E_{k}^{R}),

we immediately obtain the necessary condition b^{j} −b^{k} ∈ ℝa^{j} +ℝa^{k}. However, the actual behavior of Span_{ℝ} E_{jk}^{R} will require a closer look in Step 3 below.

Step 2 We introduce new “coordinates”:

\overline{b}^{j} := b^{j} − ⟨b^{j}, R⟩ \overline{a}^{j} ∈ R^{⊥}, which is defined even in the case ⟨a^{j}, R⟩ = 1;

s_{j} := − ⟨b^{j}, R⟩ for j meeting ⟨a^{j}, R⟩ ≥ 2 (inducing an element of W (R)).

The shift of the b^{j} by an element b ∈ N_{ℝ} (that is, (b^{j})′ = b^{j} + b) appears in these new coordinates as

(\overline{b}^{j})′ = (b^{j})′ − ⟨(b^{j})′, R⟩ \overline{a}^{j} = b^{j} + b − ⟨b^{j}, R⟩ \overline{a}^{j} − ⟨b, R⟩ \overline{a}^{j}
= \overline{b}^{j} + b − ⟨b, R⟩ \overline{a}^{j},
s′_{j} = − ⟨(b^{j})′, R⟩ = s_{j} − ⟨b, R⟩ .

In particular, an element b ∈ R^{⊥} does not change the s_{j}, but shifts the points \overline{b}^{j} inside the hyperplane R^{⊥}. Hence, the set of the \overline{b}^{j} should be considered modulo translation inside R^{⊥} only. On the other hand, the condition b^{j} −b^{k} ∈ ℝa^{j} + ℝa^{k} changes into \overline{b}^{j} − \overline{b}^{k} ∈ ℝ\overline{a}^{j} + ℝ\overline{a}^{k} or even \overline{b}^{j} − \overline{b}^{k} ∈ ℝ(\overline{a}^{j} − \overline{a}^{k}) (consider the values of R). Hence, the \overline{b}^{j} form the vertices of a Minkowski summand of Q(R), or at least a generalized Minkowski summand. Modulo translation, this summand is completely described by the dilation factors t_{jk} obtained from

\overline{b}^{j} − \overline{b}^{k} = t_{jk} · (\overline{a}^{j} − \overline{a}^{k}).

Now, the remaining part of the action of b ∈ N_{ℝ} comes down to an action of ⟨b, R⟩ ∈ ℝ only:

t′_{jk} = t_{jk} − ⟨b, R⟩ , and
s′_{j} = s_{j} − ⟨b, R⟩ , as we already know.

Up to now, we have found that T_{Y}^{1}(−R) ⊂ V_{ℂ}(R) ⊕ W_{ℂ}(R)/(\underline{1}, \underline{1}).

Step 3 Actually, the elements b^{j} and b^{k} must coincide on Span_{ℝ} E_{jk}^{R}, which may be larger than just (a^{j}, a^{k})^{⊥}. To measure the difference, consider the quotient space Span_{ℝ} E_{jk}^{R}/(a^{j}, a^{k})^{⊥} contained in the two dimensional vector space M_{ℝ}/(a^{j}, a^{k})^{⊥} = Span_{ℝ}(a^{j}, a^{k})^{∗}. Since this quotient space coincides with the set Span_{ℝ} E\overline{R} ; jk corresponding to the two dimensional cone ⟨a^{j}, a^{k}⟩ ⊂ Span_{ℝ}(a^{j}, a^{k}), where \overline{R} denotes the image of R in Span_{ℝ}(a^{j}, a^{k})^{∗}, we may assume that σ = ⟨a^{1}, a^{2}⟩ (that is, j = 1, k = 2) represents a two dimensional cyclic quotient singularity. In particular, we only need to discuss the three cases (i)–(iii) of Example 2.2:

In (i) and (ii) we have Span_{ℝ} E_{12}^{R} = 0, that is, no additional equation is needed. This means G_{12} = ∅ is of Type (0) (see 2.6). On the other hand, if T_{Y}^{1} = 0, then the vector space ℝ_{(t_{12},s_{1},s_{2})}^{3}/ℝ·(\underline{1}) has to be killed by identifying the three variables t_{12}, s_{1} and s_{2}; we obtain Type (2).

Case (iii) provides Span_{ℝ} E_{12}^{R} = ℝ · R. Hence, as an additional condition we obtain that b^{1} and b^{2} have to be equal on R. By definition of s_{j}, this means that s_{1} = s_{2}, and G_{12} must be of Type (1). □

3 Genuine deformations

3.1

In [Al 2] we studied the so-called toric deformations in a given multidegree −R ∈ M. These are genuine deformations, in the sense that they are defined over smooth parameter spaces; they are characterized by the fact that the total space together with the embedding of the special fiber still belongs to the toric category. Despite the fact they look so special, it seems that toric deformations cover a big part of the versal deformation of Y_{σ}. They only exist in negative degrees (that is, R ∈ σ^{∨} ∩ M), but here they form a kind of skeleton. If Y_{σ} is an isolated toric Gorenstein singularity, then toric deformations even provide all irreducible components of the versal deformation (cf. [Al 4]).

After briefly recalling the idea of this construction, we show how the new formula for T_{Y}^{1} of Theorem 2.1 can be used to describe the Kodaira–Spencer map of toric deformations. We follow this by the study of nonnegative degrees: if R ∉ σ^{∨} ∩ M, then we are still able to construct genuine deformations of Y_{σ}; however, these are no longer toric.

3.2

Let R ∈ σ^{∨} ∩ M. Then, as in [Al 2], §3, an m-parameter toric deformation of Y_{σ} in degree −R corresponds to a splitting of Q(R) as a Minkowski sum

Q(R) = Q_{0} + Q_{1} + ⋯ + Q_{m}

satisfying the following conditions:

(i) Q_{0} ⊂ 𝔸(R) and Q_{1}, …, Q_{m} ∈ 𝔸_{0}(R) are polyhedra with common cone of unbounded directions Q(R)^{∞}.

(ii) Each supporting hyperplane t of Q(R) defines a face F (Q_{i}, t) of each of the polyhedra Q_{0}, Q_{1}, …, Q_{m}, and the Minkowski sum of these faces equals F (Q(R), t). With at most one exception (which can depend on t), these faces contain lattice vertices (points of N).

Remark 3.1 In [Al 2] we distinguished between the case of primitive and nonprimitive elements R ∈ M: if R is a multiple of some element of M, then 𝔸(R) does not contain lattice points at all. In particular, condition (ii) just means that Q_{1}, …, Q_{m} must be lattice polyhedra.

On the other hand, for primitive R, the (m + 1) summands Q_{i} appear on an equal footing and may be put into the same space 𝔸(R). Their Minkowski sum must then be interpreted inside this affine space.

Given a Minkowski decomposition, how to obtain the corresponding toric deformation?

Defining Ñ := N ⊕ℤ^{m} (and M̃:= M ⊕ℤ^{m}), we have to embed the summands as (Q_{0}, 0), (Q_{1}, e^{1}), …, (Q_{m}, e^{m}) into the vector space Ñ_{ℝ}; here {e^{1}, …, e^{m}} denotes the standard basis of ℤ^{m}. Together with (Q(R)^{∞}, 0), these polyhedra generate a cone σ̃ ⊂ Ñ containing σ via the inclusion

N ↪ Ñ defined by a ↦ (a; ⟨a, R⟩ , …, ⟨a, R⟩).

Actually, σ equals σ̃ ∩ N_{ℝ}, and we obtain an inclusion Y_{σ} ↪ Xσ̃ between the associated toric varieties.

On the other hand, [R, 0]: Ñ → ℤ and pr_{ℤ^{m}}: Ñ → ℤ^{m} induce regular functions f: Xσ̃ → ℂ and (f^{1}, …, f^{m}): Xσ̃ → ℂ^{m}, respectively. The resulting map (f^{1} − f, …, f^{m} − f): Xσ̃ → ℂ^{m} is flat and has Y_{σ} ↪ Xσ̃ as special fiber.

3.3

Let R ∈ σ^{∨} ∩ M and Q(R) = Q_{0} + ⋯ + Q_{m} be a decomposition satisfying conditions (i) and (ii) of 3.2. Denote by (\overline{a}^{j})_{i} the vertex of Q_{i} induced from \overline{a}^{j} ∈ Q(R), so that \overline{a}^{j} = (\overline{a}^{j})_{0} + ⋯ + (\overline{a}^{j})_{m}.

Theorem 3.2 The Kodaira–Spencer map of the corresponding toric deformation Xσ̃ → ℂ^{m} is the map

ϱ: ℂ^{m} = T_{ℂ^{m},0} ⟶ T_{Y}^{1}(−R) ⊂ V_{ℂ}(R) ⊕ W_{ℂ}(R)/ℂ · (\underline{1}, \underline{1})

sending e^{i} to the pair [Q_{i}, \underline{s}^{i}] ∈ V (R) ⊕ W (R) (for i = 1, …, m) with

s_{j}^{i} := \begin{cases}0 & if the vertex (\overline{a}^{j})_{i} of Q_{i} belongs to the lattice N, \\ 1 & if (\overline{a}^{j})_{i} is not a lattice point. \end{cases}

Remark 3.3 Setting e^{0} := −(e^{1} +⋯+e^{m}), we obtain ϱ(e^{0}) = [Q_{0}, \underline{s}^{0}] where \underline{s}^{0} is defined in the same way as \underline{s}^{i} above.

3.4 Proof of Theorem 3.2

We to derive the formula for the Kodaira–Spencer map from the more technical result of [Al 2], Theorem 5.3. If we use in addition [Al 3], Theorem 6.1, this theorem describes ϱ(e^{i}) ∈ T_{Y}^{1}(−R) = H^{1}( Span_{ℂ}(E^{R})∗_{•}) as follows:

Let E = {r^{0}, …, r^{w}} ⊂ σ^{∨} ∩ M. Its elements may be lifted via M̃ ↠ M to r̃^{v} ∈ σ̃^{∨} ∩M̃ (for v = 0, …, w); denote their ith entry of the ℤ^{m}-part by η_{i}^{v}. Then, given elements v^{j} ∈ Span E_{j}^{R}, we may represent them as v^{j} = ∑_{v} q_{v}^{j}r^{v} with q^{j} ∈ ℤ^{E_{j}^{R}}, and ϱ(e^{i}) sends v^{j} to the integer − ∑_{v} q_{v}^{j}η_{i}^{v}. Using the notation of 2.7 for ϱ(e^{i}), this means that b^{j} sends elements r^{v} ∈ E_{j}^{R} onto −η_{i}^{v} ∈ ℤ.

By construction of σ̃, we have the inequalities

⟨((\overline{a}^{j})_{0}, 0), r̃^{v}⟩ ≥ 0 and ⟨((\overline{a}^{j})_{i}, e^{i}), r̃^{v}⟩ ≥ 0 for i = 1, …, m,

which sum up to ⟨\overline{a}^{j}, r^{v}⟩ = ⟨(\overline{a}^{j}, \underline{1}), r̃^{v}⟩ ≥ 0. On the other hand, r^{v} ∈ E_{j}^{R} is equivalent to ⟨\overline{a}^{j}, r^{v}⟩ < 1. Hence, for i = 0, …, m, whenever (\overline{a}^{j})_{i} ∈ Q_{i} belongs to the lattice, the corresponding inequality becomes an equality. With at most one exception, this must always happen. Hence for i = 1, …, m,

⟨(\overline{a}^{j})_{i}, r^{v}⟩ + η_{i}^{v} = \begin{cases}0 & if (\overline{a}^{j})_{i} ∈ N, \\ ⟨\overline{a}^{j}, r^{v}⟩ & if (\overline{a}^{j})_{i} ∉ N, \end{cases}

meaning that b^{j} = (\overline{a}^{j})_{i} or b^{j} = (\overline{a}^{j})_{i} − \overline{a}^{j} respectively. By definition of \overline{b}^{j} and s_{j} given in 2.7, we are done. □

3.5

We now treat the case of nonnegative degrees; let R ∈ M \ σ^{∨}. It often happens that the easiest way to solve a problem is to change the question until there is no problem left. We can do this here by changing our cone σ into some τ^{R} such that the degree −R becomes negative. We define

τ := τ^{R} := σ ∩ [R ≥ 0], that is, τ^{∨} = σ^{∨} + ℝ_{≥0} · R.

The cone τ defines an affine toric variety Y_{τ}. Since τ ⊂ σ, it comes with a map g: Y_{τ} → Y_{σ}; in other words, Y_{τ} is an open part of a modification of Y_{σ}. The important observation is

\begin{aligned}τ ∩ [R = 0] & = & σ ∩ [R = 0] & = & Q(R)^{∞}, and \\ τ ∩ [R = 1] & = & σ ∩ [R = 1] & = & Q(R), \end{aligned}

implying T_{Y_{τ}}^{1} (−R) = T_{Y_{σ}}^{1}(−R) by Theorem 2.6. Moreover, even the genuine toric deformations Xτ̃ → ℂ^{m} of Y_{τ} carry over to m-parameter (nontoric) deformations X → ℂ^{m} of Y_{σ}:

Theorem 3.4 Each Minkowski decomposition Q(R) = Q_{0} + Q_{1} + ⋯ + Q_{m} satisfying (i) and (ii) of 3.2 provides an m-parameter deformation X → ℂ^{m} of Y_{σ}. Via some birational map g̃: Xτ̃ → X, it is compatible with the toric deformation Xτ̃ → ℂ^{m} of Y_{τ} presented in 3.2.

Y_{τ}  \overline{g^{}}Y_{σ}

[figure]

[figure: Xτ̃; g̃; X; Zσ̃]

[figure]

ℂ^{m}^{}

The total space X is no longer toric, but it sits via birational maps between Xτ̃ and some affine toric variety Zσ̃ that still contains Y_{σ} as a closed subset.

3.6 Proof

We first construct Ñ, M̃ and τ̃ ⊂ Ñ_{ℝ} by the recipe of 3.2. In particular, N ⊂ Ñ, and the projection π: M̃ → M sends [r; g_{1}, …, g_{m}] onto r+(∑_{i} g_{i})R. Defining σ̃ := τ̃ + σ (hence σ̃^{∨} = τ̃^{∨} ∩ π^{−1}(σ^{∨})), we obtain the commutative diagram

[figure: ℂ[τ̃^{∨} ∩ M̃]; ℂ[σ̃^{∨} ∩ M̃]; ℂ[τ^{∨} ∩ M]; π; ℂ[σ^{∨} ∩ M]; π]

with surjective vertical maps. The canonical elements e_{1}, …, e_{m} ∈ ℤ^{m} ⊂ M̃ together with [R; 0] ∈ M̃ are preimages of R ∈ M. Hence, the corresponding monomials x^{e_{1}}, …, x^{e_{m}}, x^{[R,0]} in the semigroup algebra ℂ[τ̃^{∨} ∩ M̃] (these were called f^{1}, …, f^{m}, f in 3.2) map onto x^{R} ∈ ℂ[τ^{∨} ∩ M], which is not regular on Y_{σ}. We define Zσ̃ as the affine toric variety corresponding to σ̃, and X as

X := Spec B, where B := ℂ[σ̃^{∨} ∩ M̃][f^{1} − f, …, f^{m} − f] ⊂ ℂ[τ̃^{∨} ∩ M̃].

This means that X is obtained from Xτ̃ by eliminating all the variables except those lifted from Y_{σ} and the deformation parameters themselves. By construction of B, the vertical algebra homomorphisms π induce a surjection B ↠ ℂ[σ^{∨} ∩ M].

Lemma 3.5 Any element of ℂ[τ̃^{∨} ∩ M̃] can be written in a unique way as a sum

∑_{(v_{1},...,v_{m})∈ℕ^{m}} c_{v_{1},...,v_{m}} · (f^{1} − f)^{v_{1}} ⋯ (f^{m} − f)^{v_{m}},

with c_{v_{1},...,v_{m}} ∈ ℂ[τ̃^{∨} ∩ M̃] such that s − e_{i} ∉ τ̃^{∨} (for i = 1, …, m) for any of its monomial terms x^{s}. Moreover, these sums belong to the subalgebra B if and only if their coefficients c_{v_{1},...,v_{m}} do so.

Proof (a) Existence: Let s − e_{i} ∈ τ̃^{∨} for some s, i. Then, with s′ := s − e_{i} + [R, 0] we obtain

x^{s} = x^{s′} + x^{s−e_{i}}(x^{e_{i}} − x^{[R,0]}) = x^{s′} + x^{s−e_{i}}(f^{i} − f).

Since e_{i} = 1 and [R, 0] = 0 if evaluated on (Q_{i}, e^{i}) ⊂ τ̃, this process eventually stops.

(b) Membership of B: For the previous reduction step, we have to show that if s ∈ ℂ[σ̃^{∨} ∩ M̃], then the same holds for s′ and s − e_{i}. Since π(s′) = π(s) ∈ σ^{∨}, this is clear for s′. It remains to check that π(s − e_{i}) ∈ σ^{∨}. Let a ∈ σ be an arbitrary test element; we distinguish two cases:

Case 1: ⟨a, R⟩ ≥ 0. Then a belongs to the subcone τ, and π(s − e_{i}) ∈ τ^{∨} yields ⟨a, π(s − e_{i})⟩ ≥ 0.

Case 2: ⟨a, R⟩ ≤ 0. This implies

⟨a, π(s − e_{i})⟩ = ⟨a, s⟩ − ⟨a, R⟩ ≥ ⟨a, s⟩ ≥ 0.

(c) Uniqueness: Let p := ∑ c_{v_{1},...,v_{m}} · (f^{1} − f)^{v_{1}} … (f^{m} − f)^{v_{m}} (satisfying the above conditions) be equal to 0 in ℂ[τ̃^{∨} ∩ M̃]. Using the projection π: M̃ → M makes everything M-graded. Since the factors (f^{i} − f) are homogeneous (of degree R), we may assume that the same holds for p, hence also for its coefficients c_{v_{1},...,v_{m}}.

Claim 3.6 These coefficients are just monomials.

Indeed, if s, s′ ∈ τ̃^{∨} had the same image under π, we could assume that some e_{i}-coordinate of s′ is smaller than that of s. Hence, s − e_{i} would still be equal to s on (Q_{0}, 0) and on any (Q_{j}, e^{j}) (for j ≠ i), but even greater than or equal to s′ on (Q_{i}, e^{i}). This would imply s − e_{i} ∈ τ̃^{∨}, contradicting our assumption for p.

Say c_{v_{1},...,v_{m}} = λ_{v_{1},...,v_{m}}x^{•}; we use the projection M̃ → ℤ^{m} to take p into the ring ℂ[ℤ^{m}] = ℂ[y_{1}^{±1}, …, y_{m}^{±1}]. The elements x^{•}, f^{i}, f map onto y^{•}, y_{i}, and 1, respectively. Hence, p turns into

\overline{p} = ∑_{(v_{1},...,v_{m})∈ℕ^{m}} λ_{v_{1},...,v_{m}} · y^{•} · (y_{1} − 1)^{v_{1}} ⋯ (y_{m} − 1)^{v_{m}}.

By induction through ℕ^{m}, we obtain that vanishing of \overline{p} implies the vanishing of its coefficients: replace y_{i}−1 by z_{i}, and take partial derivatives. This proves the claim.

Now, we can easily see that X → ℂ^{m} is flat and has Y_{σ} as special fiber: Lemma 3.5 means that for k = 0, …, m, we have inclusions

B/(f^{1} − f, …, f^{k} − f) ↪ ℂ[τ̃^{∨} ∩ M̃]/(f^{1} − f, …, f^{k} − f).

The values k < m yield that (f^{1} − f, …, f^{m} − f) forms a regular sequence even in the subring B, meaning that X → ℂ^{m} is flat. With k = m we obtain that the surjective map B/(f^{1} − f, …, f^{m} − f) → ℂ[σ^{∨} ∩ M] is also injective. □

4 Three dimensional toric Gorenstein singularities

4.1

By [Ish], Theorem 7.7, toric Gorenstein singularities always arise from the following construction: assume given a lattice polytope P ⊂ ℝ^{n}. We embed the whole space (containing P) into the “height one” slice of N_{ℝ} := ℝ^{n} ⊕ ℝ, and consider the cone σ generated by P; write M_{ℝ} := (ℝ^{n})^{∗} ⊕ ℝ for the dual space and N, M for the natural lattices. Our polytope P may be recovered from σ as

P = Q(R^{∗}) ⊂ 𝔸(R^{∗}) with R^{∗} := [\underline{0}, 1] ∈ M.

[figure: P; cone(P)]

The fundamental generators a^{1}, …, a^{M} ∈ 𝕃(R^{∗}) of σ coincide with the vertices of P. (This involves a slight abuse of notation; we use the same symbol a^{j} for both a^{j} ∈ ℤ^{n} and (a^{j}, 1) ∈ M.)

If \overline{a^{j}a^{k}} is an edge of P, we denote by ℓ(j, k) ∈ ℤ its “length” induced from the lattice ℤ^{n} ⊂ ℝ^{n}. Every edge provides a codimension two singular stratum of Y_{σ} with transversal type A_{ℓ(j,k)−1}. In particular, Y_{σ} is smooth in codimension two if and only if every edge of P is primitive, that is, of length ℓ = 1.

4.2

As usual, we fix some element R ∈ M. We know the vector spaces V (R) and W (R) from 2.5; we introduce the subspace

V′(R) := \{\underline{t} ∈ V (R) | t_{jk} ≠ 0 ⇒ 1 ≤ ⟨a^{j}, R⟩ = ⟨a^{k}, R⟩ ≤ ℓ(j, k)\} ⊂ V (R).

This represents the Minkowski summands of Q(R) that contract to a point any compact edge not satisfying ⟨a^{j}, R⟩ = ⟨a^{k}, R⟩ ≤ ℓ(j, k).

Theorem 4.1 For T_{Y}^{1}(−R), we distinguish two different types of R ∈ M:

(i) If R ≤ 1 on P (or equivalently ⟨a^{j}, R⟩ ≤ 1 for j = 1, …, M), then T_{Y}^{1}(−R) = V_{ℂ}(R)/(\underline{1}). Moreover, concerning Minkowski summands, we may replace the polyhedron Q(R) by its compact part P ∩ [R = 1] (a face of P).

(ii) If R does not satisfy the previous condition, then T_{Y}^{1}(−R) = V′(R).

Proof The first case follows from Theorem 2.6 simply because W (R) = 0. For (ii), we assume that there are vertices a^{j} contained in the affine halfspace [R ≥ 2]. These vertices can be connected to one another inside this halfspace via paths along edges of P.

The two dimensional cyclic quotient singularities corresponding to the edges \overline{a^{j}a^{k}} of P are themselves Gorenstein. In the language of Example 2.2, this means that w = 2, and we obtain

dim T_{⟨a^{j},a^{k}⟩}^{1}(−R) = \begin{cases}1 & if ⟨a^{j}, R⟩ = ⟨a^{k}, R⟩ = 2, …, ℓ(j, k) \\ & (case (iii) in 2.2), \\ 0 & otherwise. \end{cases}

In particular, T_{⟨a^{j},a^{k}⟩}^{1}(−R) cannot be two dimensional, and (in the notation

of 2.6) the equations s_{j} −s_{k} = 0 belong to G_{jk} whenever ⟨a^{j}, R⟩ , ⟨a^{k}, R⟩ ≥ 2. This means that for elements of

T_{Y}^{1} ⊂ (V_{ℂ}(R) ⊕ W_{ℂ}(R))/ℂ · (\underline{1}, \underline{1}),

all entries of the W_{ℂ}(R)-part must be equal, or even zero, after dividing by ℂ · (\underline{1}, \underline{1}). Moreover, if not both ⟨a^{j}, R⟩ and ⟨a^{k}, R⟩ equal one, the vanishing of T_{⟨a^{j},a^{k}⟩}^{1}(−R) implies that G_{jk} also contains the equation t_{jk} − s_{•} = 0. □

Corollary 4.2 For toric Gorenstein singularities, the condition 3.2, (ii) that allow us to build genuine deformations simplifies: Q_{1}, …, Q_{m} just have to be lattice polyhedra.

Proof If R ≤ 1 on P, then Q(R) is itself a lattice polyhedron. Hence, condition (ii) automatically comes down to this simpler form.

In the second case, T_{Y}^{1}(−R) involves some W (R) part. On the one hand, via the Kodaira–Spencer map, it indicates which vertices of which polyhedron Q_{i} belong to the lattice. On the other, we have observed in the previous proof that the entries of W (R) are all equal. This exactly implies our claim. □

4.3

To treat three dimensional toric Gorenstein singularities, we now focus on plane lattice polygons P ⊂ ℝ^{2}. The vertices a^{1}, …, a^{M} are arranged in a cycle. For j ∈ ℤ/Mℤ, we write d^{j} := a^{j+1} − a^{j} ∈ 𝕃_{0}(R^{∗}) for the edge from a^{j} to a^{j+1} (see 2.4), and ℓ(j) := ℓ(j, j + 1) for its length.

Let s^{1}, …, s^{M} be the fundamental generators of the dual cone σ^{∨}, labelled so that σ ∩ (s^{j})^{⊥} equals the face spanned by a^{j}, a^{j+1} ∈ σ. In particular, skipping the last coordinate of s^{j} yields the (primitive) inner normal vector at the edge d^{j} of P.

Remark 4.3 For the convenience of those who prefer to live in M rather than N, we show how to see the integers ℓ(j) in the dual world: choose a fundamental generator s^{j} and denote by r, r′ ∈ M the elements of the Hilbert basis closest to s^{j} along the two adjacent faces of σ^{∨}, respectively (see Figure 1 in 4.7). Then, {R^{∗}, s^{j}} together with either r or r′ form a basis of the lattice M, and (r + r′) − ℓ(j)R^{∗} is a multiple of s^{j}.

In the very special case of plane lattice polygons (that is, three dimensional toric Gorenstein singularities), we can describe T_{Y}^{1} and the genuine deformations (for fixed R ∈ M) explicitly. First, we can easily spot the degrees carrying infinitesimal deformations:

Theorem 4.4 In general (with the exceptions (4–5)), T_{Y}^{1}(−R) is only nontrivial in the following cases:

(1) R = R^{∗} with dim T_{Y}^{1}(−R) = M − 3;

(2) R = qR^{∗} (for q ≥ 2) with dim T_{Y}^{1}(−R) = max{0, #{j | q ≤ ℓ(j)} − 2};

(3) R = qR^{∗} − ps^{j} with 2 ≤ q ≤ ℓ(j) and p ∈ ℤ sufficiently large such that R ∉ int(σ^{∨}). In this case, T_{Y}^{1}(−R) is one dimensional.

Additional degrees exist only in the following two (overlapping) exceptional cases:

(4) P contains a pair of parallel edges d^{j}, d^{k}, both longer than every other edge. Then dim T_{Y}^{1}(−qR^{∗}) = 1 for q in the range

max{ℓ(l) | l ≠ j, k} < q ≤ min{ℓ(j), ℓ(k)}.

(5) P contains a pair of parallel edges d^{j}, d^{k} with distance d (d := ⟨a^{j}, s^{k}⟩ = ⟨a^{k}, s^{j}⟩). If ℓ(k) > d ≥ max{ℓ(l) | l ≠ j, k}, then dim T_{Y}^{1}(−R) = 1 for R = qR^{∗} + ps^{j} with 1 ≤ q ≤ ℓ(j) and 1 ≤ p ≤ (ℓ(k) − q)/d.

The cases (1), (2), (4), and (5) yield at most finitely many (negative) degrees in T_{Y}^{1}. Type (3) consists of ℓ(j) − 1 infinite series corresponding to any vertex a^{j} ∈ P; these series contain only nonnegative degrees, except possibly for the leading elements (R might sit on ∂σ^{∨}).

Proof These assertions are straightforward consequences of Theorem 4.1, so that the following brief remark should be sufficient: the condition ⟨a^{j}, R⟩ = ⟨a^{j+1}, R⟩ means d^{j} ∈ R^{⊥}. Moreover, if R ∉ ℤ · R^{∗}, there is at most one edge (or a pair of parallel edges) having this property. □

4.4

Example 4.5 A typical example of a nonisolated, three dimensional toric Gorenstein singularity is the cone over the weighted projective space ℙ(1, 2, 3). We use it to illustrate our calculations of T^{1} as well as the forthcoming construction of genuine one parameter families. P has the vertices (−1, −1), (2, −1), (−1, 1), so that σ is the cone generated by

a^{1} = (−1, −1; 1), a^{2} = (2, −1; 1), a^{3} = (−1, 1; 1).

Since our singularity is a cone over a projective variety, σ^{∨} also appears as a cone over some lattice polygon. In this example, σ and σ^{∨} happen to be isomorphic. We obtain

σ^{∨} = ⟨s^{1}, s^{2}, s^{3}⟩ with s^{1} = [0, 1; 1], s^{2} = [−2, −3; 1], s^{3} = [1, 0; 1].

The Hilbert basis E ⊂ σ^{∨} ∩ ℤ^{3} consists of these three fundamental generators together with

R^{∗} = [0, 0; 1], v^{1} = [−1, −2; 1], v^{2} = [0, −1; 1], w = [−1, −1; 1].

[figure: a^{1}; a^{2}; a^{3}; s^{2}; s^{3}; s^{1}; w; v^{1}; v^{2}; R^{∗}; σ = cone(P); σ^{∨}]

In particular, Y_{σ} has embedding dimension 7. The edges of P have length ℓ(1) = 3, ℓ(2) = 1, and ℓ(3) = 2. Thus Y_{σ} has one dimensional singularities of transversal type A_{2} and A_{1}. According to Theorem 4.4, Y_{σ} admits only infinitesimal deformations of type (3). Their degrees come in three series:

(α) 2R^{∗} − p_{α}s^{3} with p_{α} ≥ 1. Even the leading element R^{α} = [−1, 0, 1] is not contained in σ^{∨}.

(β) 2R^{∗}−p_{β}s^{1} with p_{β} ≥ 1. The leading element equals R^{β} = v^{2} = [0, −1, 1] and sits on the boundary of σ^{∨}.

(γ) 3R^{∗} − p_{γ}s^{1} with p_{γ} ≥ 2. The leading element is R^{γ} = [0, −2, 1] ∉ σ^{∨}.

[figure: σ^{∨} ⊂ M_{ℝ}; R^{α}; R^{β}; R^{γ}]

4.5

Each degree belonging to type (3) (that is, R = qR^{∗} − ps^{j} with 2 ≤ q ≤ ℓ(j)) provides an infinitesimal deformation. To show that they are unobstructed by describing how they lift to genuine one parameter deformations should be no problem: just split the polygon Q(R) into a Minkowski sum satisfying conditions (i) and (ii) of 3.2, then construct τ̃, σ̃ and (f^{1} − f) as in 3.2 and 3.5.

However, we prefer to present the result for our special case all at once using new coordinates. Let P ⊂ 𝔸(R^{∗}) = ℝ^{2} × {1} ⊂ ℝ^{3} = N_{ℝ} be a lattice polygon as in 4.3, and R = qR^{∗} − ps^{j} as just mentioned. Then σ, τ ⊂ N_{ℝ} are the cones over P and P ∩ [R ≥ 0] respectively, and the one parameter family in degree −R is obtained as follows:

Proposition 4.6 The cone τ̃ ⊂ N_{ℝ} ⊕ ℝ = ℝ^{4} is generated by the elements:

(i) (a, 0) − ⟨a, R⟩ (\underline{0}, 1) as a ∈ P ∩ [R ≥ 0] runs through the vertices from the R^{⊥}-line to a^{j},

(ii) (a, 0) − ⟨a, R⟩ (d^{j}/ℓ(j), 1) as a ∈ P ∩ [R ≥ 0] runs from a^{j+1} back up to the line R^{⊥}, and

(iii) (\underline{0}, 1) and (d^{j}/ℓ(j), 1).

The vector space N_{ℝ} containing σ sits in N_{ℝ} ⊕ ℝ as N_{ℝ} × {0}. Via this embedding, we have as usual σ̃ = τ̃ + σ. The monomials f and f^{1} are given by their exponents [R, 0] and [R, 1] ∈ M ⊕ ℤ respectively.

Geometrically, one can think of τ̃ as generated by the interval I with vertices as in (iii) and by the polygon P′ obtained as follows: “squeeze” P ∩ [R ≥ 0] along R^{⊥} by a cone with base q/ℓ(j) · \overline{a^{j}a^{j+1}} and some vertex on the line R^{⊥}; take − ⟨•, R⟩ as an additional, fourth coordinate. Then, [R^{∗}, 0] is still 1 on P′ and equals 0 on I. Moreover, [R, 0] vanishes on I and on the R^{⊥}-edge of P′; [R, 1] vanishes on the whole of P′.

[figure: s^{j}; a^{j}; a^{j+1}; R^{⊥}; P ⊂ 𝔸(R^{∗}); P ∩ [R ≥ 0]; \frac{q}{ℓ(j)} · \overline{a^{j}a^{j+1}}]

Figure 1: Parallel lines R^{⊥} and a^{j}a^{j+1} at distance q/p apart in 𝔸(R^{∗})

Proof We change coordinates. If g := gcd(p, q) denotes the “length” of R, then we can find an s ∈ M such that {s, R/g} forms a basis of M ∩ (d^{j})^{⊥}. Adding some r ∈ M with ⟨d^{j}/ℓ(j), r⟩ = 1 (the r of Remark 4.3 will do) yields a ℤ-basis for the whole lattice M. We consider the following commutative diagram:

[figure: N; \begin{matrix}(s, r, R/g) \\ ∼ \end{matrix}; ℤ^{3}; N ⊕ ℤ; (id, 0); \begin{matrix}([s, 0], [r, 0], [R/g, 0], [R, 1]) \\ ∼ \end{matrix}; ℤ^{3} ⊕ ℤ; (id, g · pr_{3})]

The left hand side contains the data relevant for our proposition. Carrying them over to the right yields:

• [0, 0, g] ∈ (ℤ^{3})^{∗} as the image of R;

• [0, 0, g, 0], [0, 0, 0, 1] ∈ (ℤ^{4})^{∗} as the images of [R, 0] and [R, 1] respectively;

• τ becomes a cone with affine crosssection

Q([0, 0, g]) =

conv (( ⟨a, s⟩ / ⟨a, R⟩; ⟨a, r⟩ / ⟨a, R⟩; 1/g) | a ∈ P ∩ [R ≥ 0]);

• I changes into the unit interval (Q_{1}, 1) reaching from (0, 0, 0, 1) to (0, 1, 0, 1); • finally, cone(P′) maps onto the cone spanned by the convex hull (Q_{0}, 0)

of the points (⟨a, s⟩ / ⟨a, R⟩; ⟨a, r⟩ / ⟨a, R⟩; 1/g; 0) for a ∈ P ∩ [R ≥ 0] on the a^{j}-side and ( ⟨a, s⟩ / ⟨a, R⟩; ⟨a, r⟩ / ⟨a, R⟩ − 1; 1/g; 0) for a on the a^{j+1}-side respectively.

Since Q([0, 0, g]) equals the Minkowski sum of the interval Q_{1} ⊂ 𝔸_{0}([0, 0, g]) and the polygon Q_{0} ⊂ 𝔸([0, 0, g]), we are done by 3.2. □

4.6

To see how the original equations of the singularity Y_{σ} are perturbed, it is useful to study first the dual cones τ̃^{∨} or σ̃^{∨} = τ̃^{∨} ∩ π^{−1}(σ^{∨}):

Proposition 4.7 If s ∈ σ^{∨} ∩ M, the element of M ⊕ ℤ given by

S := \begin{cases}[s, 0] & if ⟨d^{j}, s⟩ ≥ 0 \\ [s, − ⟨d^{j}/ℓ(j), s⟩] & if ⟨d^{j}, s⟩ ≤ 0 \end{cases}

is a lift of s to σ̃^{∨} ∩ (M ⊕ ℤ). (Notice that it does not depend on p, q, but only on j.) Moreover, if s^{v} runs through the edges of P ∩ [R ≥ 0], the elements S^{v} together with [R, 0] and [R, 1] form the fundamental generators of τ̃^{∨}.

Proof Since we know τ̃ from the previous proposition, the calculations are straightforward and we omit them. □

4.7

Recall from 2.1 that E denotes the minimal set generating the semigroup σ^{∨} ∩ M. Any s ∈ E has an associated variable z_{s}, and Y_{σ} ⊂ ℂ^{E} is given by binomial equations arising from linear relations among elements of E. Everything will be clear by considering an example: a linear relation such as s^{1} + 2s^{3} = s^{2} + s^{4} transforms into z_{1}z_{3}^{2} = z_{2}z_{4}.

The fact that σ defines a Gorenstein variety (that is, σ is a cone over a lattice polytope) implies that E consists only of R^{∗} and elements of ∂σ^{∨} including the fundamental generators s^{v}. If E ∩ ∂σ^{∨} is ordered clockwise, then any two adjacent elements together with R^{∗} form a ℤ-basis of the three dimensional lattice M.

In particular, any three consecutive elements of E ∩ ∂σ^{∨} provide a unique linear relation among them and R^{∗}. (We have already met this fact in Remark 4.3, where r, s^{j}, r′ were consecutive elements.) The resulting “boundary” equations do not generate the ideal of Y_{σ} ⊂ ℂ^{E}. Nevertheless, to describe a deformation of Y_{σ}, it is sufficient to know only how this set of equations is perturbed. Moreover, if one has to avoid boundary equations “overlapping” a certain spot on ∂σ^{∨}, then it will even be possible to drop up to two of them from the list.

[figure: …; ⋮; R^{∗}; R; r; r′; s^{j}; s^{j−1}; s^{j+1}; σ^{∨}; [d^{j} ≥ 0]; [d^{j} ≤ 0]]

Theorem 4.8 The one parameter deformation of Y_{σ} in degree −(qR^{∗} − ps^{j}) is completely determined by the following perturbations:

(i) (Boundary) equations involving only variables induced from [d^{j} ≥ 0] ⊂ σ^{∨} remain unchanged. The same statement holds for [d^{j} ≤ 0].

(ii) The boundary equation z_{r}z_{r′} − z_{R^{∗}}^{ℓ(j)}z_{s^{j}}^{k} = 0 corresponding to the triple {r, s^{j}, r′} is perturbed to (z_{r}z_{r′} − z_{R^{∗}}^{ℓ(j)}z_{s^{j}}^{k}) − tz_{R^{∗}}^{ℓ(j)−q}z_{s^{j}}^{k+p} = 0. Divide everything by z_{s^{j}}^{k} if k < 0.

Proof Restricted to either [d^{j} ≥ 0] or [d^{j} ≤ 0], the map s ↦ S lifting elements of E to σ̃ ∩ (M ⊕ ℤ) is linear. Hence, any linear relation remains true, and part (i) is proved.

For the second part, we consider the boundary relation r+r′ = ℓ(j)R^{∗}+ks^{j} with a suitable k ∈ ℤ. By Proposition 4.6, the summands involved lift to the elements [r, 0], [r′, 1], [R^{∗}, 0], and [s^{j}, 0] respectively. In particular, the relation breaks down and has to be replaced by

\begin{aligned}[r, 0] + [r′, 1] & = & [R, 1] + (ℓ(j) − q)[R^{∗}, 0] + (k + p)[s^{j}, 0], and \\ ℓ(j)[R^{∗}, 0] + k[s^{j}, 0] & = & [R, 0] + (ℓ(j) − q)[R^{∗}, 0] + (k + p)[s^{j}, 0]. \end{aligned}

The monomials corresponding to [R, 1] and [R, 0] are f^{1} and f respectively. They are not regular on the total space X, but their difference t := f^{1} − f is. Hence, the difference of the monomial versions of both equations yields the result.

Finally, we should remark that (i) and (ii) cover all boundary equations except those overlapping the intersection of ∂σ^{∨} with \overline{R^{∗}R}. □

4.8

We return to Example 4.4 and discuss the one parameter deformations in degree −R^{α}, −R^{β}, and −R^{γ} respectively:

Case α: R^{α} = [−1, 0, 1] = 2R^{∗} − s^{3} means j = 3, q = ℓ(3) = 2, and p = 1. Hence, the line R^{⊥} has distance q/p = 2 from its parallel through a^{3} and a^{1}. In particular, τ = ⟨a^{1}, c^{1}, c^{3}, a^{3}⟩ with c^{1} = (1, −1, 1) and c^{3} = (3, −1, 3).

[figure: R^{⊥}; c^{3}; a^{1}; c^{1}; a^{2}; a^{3}; τ ⊂ σ]

We construct the generators of τ̃ by the recipe of Proposition 4.6: a^{3} treated via (i) and a^{1} treated via (ii) yield the same element A := (−1, 1, 1, −2); from the R^{⊥}-line we obtain C^{1} := (1, −1, 1, 0) and C^{3} := (3, −1, 3, 0); finally (iii) provides X := (0, 0, 0, 1) and Y := (0, −1, 0, 1). Hence, τ̃ is the cone over the pyramid with plane base XY C^{1}C^{3} and A as top. (The relation between the vertices of the quadrangle equals 3C^{1} + 2X = C^{3} + 2Y.) Moreover, σ̃ equals σ̃ = τ̃ + ℝ_{≥0}a^{2} with a^{2} := (a^{2}, 0). Since A + 2X + 2a^{2} = C^{3} and A + 2Y + 2a^{2} = 3C^{1}, σ̃ is a simplex generated by A, X, Y, and a^{2}.

Denoting the variables assigned to s^{1}, s^{2}, s^{3}, R^{∗}, v^{1}, v^{2}, w ∈ E ⊂ σ^{∨} ∩ M by Z_{1}, Z_{2}, Z_{3}, U, V_{1}, V_{2}, W respectively, there are six boundary equations:

Z_{3}W Z_{1} − U^{3} = Z_{1}Z_{2} − W^{2} = 0,
W V_{1} − UZ_{2} = Z_{2}V_{2} − V_{1}^{2} = V_{1}Z_{3} − V_{2}^{2} = V_{2}Z_{1} − U^{2} = 0.

Only the latter four are covered by Theorem 4.8. They are perturbed to

W V_{1} − UZ_{2} = Z_{2}V_{2} − V_{1}^{2} = V_{1}Z_{3} − V_{2}^{2} = V_{2}Z_{1} − U^{2} − t_{α}Z_{3} = 0.

Case β: R^{β} = [0, −1, 1] = 2R^{∗} − s^{1} means j = 1, ℓ(1) = 3, q = 2, and p = 1. Hence, R^{⊥} still has distance 2, but now from the line a^{1}a^{2}.

[figure: R^{⊥}; a^{1}; a^{2}; a^{3}; τ = σ]

We obtain

τ̃ = ⟨(−1, −1, 1, −2); (0, −1, 1, −2); (−1, 1, 1, 0); (0, 0, 0, 1); (1, 0, 0, 1)⟩ .

The boundary equation Z_{3}W Z_{1} − U^{3} = 0 corresponds to Theorem 4.8, (ii); it is perturbed to Z_{3}W Z_{1} − U^{3} − t_{β}UZ_{1} = 0.

Case γ: R^{γ} = [0, −2, 1] = 3R^{∗} − 2s^{1} means j = 1, q = ℓ(1) = 3, and p = 2.

[figure: R^{⊥}; a^{1}; a^{2}; a^{3}; τ ⊂ σ]

Here we have

τ̃ = ⟨(−1, −1, 1, −3); (−2, 1, 2, 0); (−1, 2, 4, 0); (0, 0, 0, 1); (1, 0, 0, 1)⟩ ,

and the previous boundary equation provides Z_{3}W Z_{1} − U^{3} − t_{γ}Z_{1}^{2} = 0.

5 Work in progress and open problems

5.1

For a fixed j, denote by t_{q,p} the parameter corresponding to R = qR^{∗} − ps^{j} and take them as coefficients of formal power series t_{q}(z) := ∑_{p≫0} t_{q,p}z^{p} for 2 ≤ q ≤ ℓ(j).

Conjecture 5.1 The one parameter families corresponding to a fixed j fit into a common huge family defined over a smooth, infinite dimensional parameter space. As in Theorem 4.8, the boundary equations sitting completely inside [d^{j} ≥ 0] or [d^{j} ≤ 0] remain unchanged. The equation z_{r}z_{r′} −z_{R^{∗}}^{ℓ(j)}z_{s^{j}}^{k} = 0 turns into (z_{r}z_{r′} − z_{R^{∗}}^{ℓ(j)}z_{s^{j}}^{k}) − ∑_{2≤q≤ℓ(j)} t_{q}(z_{s^{j}})z_{R^{∗}}^{ℓ(j)−q}z_{s^{j}}^{k} = 0.

If this conjecture is true, then it should be possible to prove it directly by lifting equations and relations, without using Minkowski decompositions.

Equations

Perturbations

(0) Z_{3}W Z_{1} − U^{3} −(t_{α} + Z_{3}ξ_{α})UZ_{3} − (t_{β} + Z_{1}ξ_{β})UZ_{1} − (t_{γ} + Z_{1}ξ_{γ})Z_{1}^{2}
(1) Z_{1}Z_{2} − W^{2} −t_{α}(t_{α} + Z_{3}ξ_{α})ξ_{β} − t_{α}V_{1} − ξ_{α}ξ_{β}(V_{2} + t_{β})Z_{1}
−ξ_{α}(V_{2} + t_{β})^{2}
(2) W V_{1} − UZ_{2} −(t_{α} + Z_{3}ξ_{α})t_{α}ξ_{γ} − ξ_{α}(t_{γ} + Z_{1}ξ_{γ})(V_{2} + t_{β})
(3) Z_{2}V_{2} − V_{1}^{2}  −(t_{α} + Z_{3}ξ_{α})ξ_{β}V_{1} − t_{α}ξ_{γ}W − ξ_{α}ξ_{β}(t_{γ} + Z_{1}ξ_{γ})U
−ξ_{α}(t_{γ} + Z_{1}ξ_{γ})^{2}
(4) V_{1}Z_{3} − V_{2}^{2}  +(t_{α} + Z_{3}ξ_{α})ξ_{β}Z_{3} − (t_{β} + Z_{1}ξ_{β})V_{2} − (t_{γ} + Z_{1}ξ_{γ})U
(5) V_{2}Z_{1} − U^{2}  −(t_{α} + Z_{3}ξ_{α})Z_{3}
(6) V_{2}W − UV_{1} −(t_{α} + Z_{3}ξ_{α})(t_{γ} + Z_{1}ξ_{γ} + Uξ_{β})
(7) Z_{3}W − UV_{2} −(t_{β} + Z_{1}ξ_{β})U − (t_{γ} + Z_{1}ξ_{γ})Z_{1}
(8) Z_{2}Z_{3} − V_{1}V_{2} −t_{α}ξ_{γ}U − (t_{β} + Z_{1}ξ_{β})V_{1} − (t_{γ} + Z_{1}ξ_{γ})W
(9) Z_{1}V_{1} − UW −(t_{α} + Z_{3}ξ_{α})(V_{2} + t_{β})

Table 5.1: Deformation of the cone over ℙ(1, 2, 3). The base space is defined by t_{α}t_{β} = t_{α}t_{γ} = 0.

5.2

Jan Stevens has used the computer algebra system Macauly to compute the versal deformation of our example Cone(ℙ(1, 2, 3)). In the list of perturbed equations given in Table 5.1, we denote the deformation parameters by t_{α}, t_{β}, t_{γ}, ξ_{α}, ξ_{β} and ξ_{γ}. The first ones are the same as used in 4.8 covering R^{α}, R^{β}, R^{γ}, respectively. The latter three parameters are formal power series ξ_{α} = ξ_{α}(Z_{3}), ξ_{β} = ξ_{β}(Z_{1}), and ξ_{γ} = ξ_{γ}(Z_{1}); their coefficients, including that of Z^{0} are the actual deformation parameters. Up to shift of degrees, the notation is similar to that of the previous conjecture; here we have deg_{M} ξ_{α} = R^{α} − s^{3}, deg_{M} ξ_{β} = R^{β} − s^{1}, and deg_{M} ξ_{γ} = R^{γ} − s^{1}, respectively.

Equation (0) is not needed as a generator of the ideal of Y_{σ} ⊂ ℂ^{7}. It was put into the list since (0)–(5) are the boundary equations mentioned in 4.8. I’m afraid that the plus sign in (4) seems to be no mistake since, for instance, the original relation between equations (4), (6), and (7) lifts to U · 4 + Z_{3} · 6 − V_{2} · 7 − (t_{γ} + Z_{1}ξ_{γ}) · 5 = 0.

The versal base space is given by t_{α}t_{β} = t_{α}t_{γ} = 0, that is, it consists of two (infinite dimensional) smooth components. Restricting the deformation to the subspaces [t_{α} = ξ_{α} = 0] and [t_{β} = t_{γ} = ξ_{β} = ξ_{γ} = 0] yields two smooth families corresponding to j = 1 and j = 3, respectively. They equal exactly the deformations predicted in 5.1 and contain the 3 one parameter families shown in 4.8.

5.3

Let us focus on the smooth two parameter family involving only t_{α} and ξ_{β} := ξ_{β}(0), setting the remaining parameters to zero. From the above list, only the following perturbation terms survive:

\begin{matrix}(0) & (Z_{3}W Z_{1} − U^{3}) − t_{α}UZ_{3} − ξ_{β}UZ_{1}^{2} & (5) & (V_{2}Z_{1} − U^{2}) − t_{α}Z_{3} \\ (1) & (Z_{1}Z_{2} − W^{2}) − t_{α}^{2}ξ_{β} − t_{α}V_{1} & (6) & (V_{2}W − UV_{1}) − t_{α}ξ_{β}U \\ (2) & (W V_{1} − UZ_{2}) & (7) & (Z_{3}W − UV_{2}) − ξ_{β}Z_{1}U \\ (3) & (Z_{2}V_{2} − V_{1}^{2}) − t_{α}ξ_{β}V_{1} & (8) & (Z_{2}Z_{3} − V_{1}V_{2}) − ξ_{β}Z_{1}V_{1} \\ (4) & (V_{1}Z_{3} − V_{2}^{2}) + t_{α}ξ_{β}Z_{3} − ξ_{β}Z_{1}V_{2} & (9) & (Z_{1}V_{1} − UW ) − t_{α}V_{2} \end{matrix}

Equation (3) turns into a binomial by restricting to the subfamily [ξ_{γ} = 0] or [t_{α} = 0]. This makes it difficult to hope that our two parameter family has a total space which is at least close to being toric (for instance, by some partial modifications).

5.4

Nevertheless, we try to obtain those “mixed” deformations involving degrees −(qR^{∗} − ps^{j}) with more than one j by the methods developed so far. We begin with the t_{α}-family introduced in 4.8 and try to deform varieties close to its total space. We recall the relevant cones:

σ = ⟨a^{1}, a^{2}, a^{3}⟩ and τ = ⟨a^{1}, c^{1}, c^{3}, a^{3}⟩, with
\begin{matrix}a^{1} = (−1, −1, 1), \\ a^{2} = (2, −1, 1), \\ a^{3} = (−1, 1, 1) \end{matrix} and \begin{matrix}c^{1} = (1, −1, 1), \\ c^{3} = (3, −1, 3); \end{matrix}

σ̃ = ⟨A, X, Y, a^{2}⟩ and τ̃ = ⟨A, C^{1}, C^{3}, X, Y ⟩, with
\begin{matrix}A = (−1, 1, 1, −2), \\ C^{1} = (1, −1, 1, 0), \\ C^{3} = (3, −1, 3, 0) \end{matrix} and \begin{matrix}X = (0, 0, 0, 1), \\ Y = (0, −1, 0, 1), \\ a^{2} = (a^{2}, 0). \end{matrix}

The two dimensional faces (“edges”) of the simplicial cone σ̃ are smooth. Hence, by 2.6 the variety Zσ̃ is rigid.

That means we have to focus on infinitesimal deformations of Xτ̃. We begin at the level of special fibers with a quick view of T_{τ}^{1} in comparison with T_{σ}^{1}: again from 2.6, we see that the vector space T_{τ}^{1}(−R) is nontrivial only for R = [0, −1, 0] and for the four series

R_{λ}^{(1)} = R^{α} − λs^{3}; R_{λ}^{(2)} = R^{β} − λs^{1}; R_{λ}^{(3)} = v^{1} − λR^{α}; R_{λ}^{(4)} = s^{1} − λs^{2}

with λ ≥ 0.

[figure: s^{2}; s^{3}; s^{1}; s^{2}; s^{3}; s^{1}; σ^{∨}; τ^{∨}; (1); (2); (3); (4); (α); (β); (γ)]

T^{1}(−R) is always one dimensional, except that dim T_{τ}^{1}(−[0, −2, 0]) = 2; here the series (2) and (3) meet (with λ = 1). In particular, we have lost the γ-deformations in Y_{τ}. They are not compatible with the partial modification Y_{τ} → Y_{σ}.

As already mentioned in 4.8, τ̃ is the cone over the pyramid with plane base XY C^{1}C^{3} and A as its top. The edges of the triangular face AC^{1}C^{3} correspond to A_{1} singularities; the remaining edges are smooth. The Hilbert basis E of σ^{∨} ∩ M ⊂ τ^{∨} ∩ M lifts uniquely to τ̃^{∨} ∩ ℤ^{4}; together with [R^{α}, 0] and [R^{α}, 1] we obtain the Hilbert basis of the latter semigroup:

\begin{aligned}S^{1} & = & [0, 1, 1, 1] & V^{1} & = & [−1, −2, 1, 0] & R^{∗} & = & [0, 0, 1, 0] \\ S^{2} & = & [−2, −3, 1, 0] & V^{2} & = & [0, −1, 1, 0] & [R^{α}, 0] & = & [−1, 0, 1, 0] \\ S^{3} & = & [1, 0, 1, 0] & W & = & [−1, −1, 1, 0] & [R^{α}, 1] & = & [−1, 0, 1, 1] \end{aligned}

Using our formula of 2.6 for T^{1} again, we see that T 1 ; τ̃(−R) is at most one dimensional. Xτ̃ admits infinitesimal deformations in the following degrees:

(a) R_{λ}^{(a)} = [−1, −2, 1, −1] − λ[R^{α}, 1] for λ ≥ 1. Here, R equals 2 on the vertices A, C^{1}, C^{3}, and it is nonpositive otherwise. In particular, Q(R) = τ̃ ∩ [R = 1] is a triangle, and we obtain V (R) = W (R) = ℝ. Projecting ℤ^{4} → M maps this series onto R_{λ}^{(3)} (λ ≥ 1) which was mentioned above.

(b) R_{λ,μ}^{(b)} = [−1, −2, 1, 0] − λ[R^{α}, 0] − μ[R^{α}, 1] for λ, μ ≥ 0. The compact part of Q(R) equals the interval reaching from C^{1}/2 to C^{3}/2. Again, this series projects onto the third one for τ.

(c) R_{λ,μ}^{(c)} = [0, −1, 1, −1] − λS^{1} − μ[R^{α}, 1] for λ ≥ 2; μ ≥ 0.

Q(R) now has \overline{AC^{1}}/2 as compact part. Mapping the degrees into M yields the β-series R_{λ}^{(2)} with λ ≥ 2.

(d) R_{λ,μ}^{(d)} = [0, 1, 1, 0] − λS^{2} − μ[R^{α}, 1] for λ, μ ≥ 0.

This final series corresponds to the edge \overline{AC^{3}}, and it maps onto R_{λ}^{(4)} (λ ≥ 0).

5.5

From the previous section we know that there is no infinitesimal deformation either of Xτ̃ or Zσ̃ in any lift of degree −R_{1}^{(2)}. In particular, our attempt to reconstruct the (t_{α}, ξ_{β})-deformation of 5.3 has failed.

However, since we are just interested in mixing α- and β-deformations, we may try degree T := R_{2,0}^{(c)} = [0, −3, −1, −3] from the series (c) above. It is a lift of R_{2}^{(2)} ∈ M corresponding to the deformation parameter arising as the coefficient of Z in the formal power series ξ_{β}(Z). By abuse of notation, we will call this parameter ξ_{β} again. The corresponding family may be obtained from 5.3 via the substitution ξ_{β} := ξ_{β}Z_{1}.

Now we may construct the genuine deformation of Xτ̃ occurring in degree −T. The usual approach (splitting Q(R) into a Minkowski sum) yields the following result:

• On the partial modification of the new total space we have variables Z_{1}, Z_{2}, Z_{3}, U, V_{1}, V_{2}, W arising as lifts from σ^{∨}. Moreover, we call R_{0}, R_{1}, T_{0}, T_{1} the variables corresponding to the lifts of [R^{α}, 0], [R^{α}, 1] or to the two lifts of T, respectively. In particular, t_{α} = R_{1} −R_{0} and ξ_{β} = T_{1} −T_{0} are the deformation parameters.

• The modified total space corresponds to a cone

Δ = ⟨B^{1}, B^{2}, B^{3}, C^{3}, Y, Z⟩ ,

where B^{1}, B^{2}, B^{3} have arisen from the Minkowski splitting \overline{AC^{1}} = \{B^{1}\} + \overline{B^{2}B^{3}} inside the affine plane [T = 1]. The variables mentioned above correspond to the equally named points of Δ^{∨} ∩ ℤ^{5}. The Hilbert basis consists of {Z_{1}, Z_{2}, Z_{3}, V_{1}, V_{2}, R_{0}, R_{1}, T_{0}, T_{1}} and an additional element S (with U = S + Z_{1}, W = S + R_{1}). It might be useful to know Δ and its dual cone in detail. Therefore, we write down the following table indicating the pairing between the mutually dual cones:

Z_{1} Z_{2} Z_{3} U V_{1} V_{2} W R_{0} R_{1} T_{0} T_{1} S
B^{1} 0 0 0 1 0 0 1 0 0 2 0 1
B^{2} 0 0 0 0 0 0 0 1 0 0 1 0
B^{3} 0 1 1 0 1 1 0 0 0 0 1 0
C^{3} 2 0 6 3 2 4 1 0 0 0 0 1
Y 0 3 0 0 2 1 1 0 1 0 0 0
Z 2 0 0 3 0 0 3 6 2 0 0 0

• Proceeding as in the proof of Theorem 4.8, the ten original equations defining Y_{σ} (cf. the second column of Table 5.1) induce binomial equations for the modified total space:

(0) and (5) provide W Z_{1} − R_{1}U = U^{2} − Z_{1}^{3}T_{0} = R_{0}Z_{3} − Z_{1}^{3}T_{1} = V_{2}Z_{1} − R_{1}Z_{3} = 0;

(1) provides Z_{1}Z_{2} − R_{1}V_{1} = W^{2} − Z_{1}R_{1}^{2}T_{0} = R_{0}V_{1} − Z_{1}R_{1}^{2}T_{1} = 0;

(9) provides Z_{1}V_{1} − V_{2}R_{1} = UW − T_{0}R_{1}Z_{1}^{2} = V_{2}R_{0} − T_{1}R_{1}Z_{1}^{2} = 0; the equations (2), (3), (4), (6), (7), and (8) remain unchanged.

• To describe the deformation of Xτ̃ we have to eliminate T_{0} and T_{1}; just their difference ξ_{γ} = T_{1} − T_{0} may occur. There is nothing to do with equations (2), (3), (4), (6), (7), (8); for the remaining ones we obtain

\begin{matrix}(0; 5) & W Z_{1} − R_{1}U = U^{2} − R_{0}Z_{3} + ξ_{β}Z_{1}^{3} = V_{2}Z_{1} − R_{1}Z_{3} = 0 \\ (1) & Z_{1}Z_{2} − R_{1}V_{1} = W^{2} − R_{0}V_{1} + ξ_{β}R_{1}^{2}Z_{1} \\ (9) & Z_{1}V_{1} − V_{2}R_{1} = UW − V_{2}R_{0} + ξ_{β}R_{1}Z_{1}^{2}. \end{matrix}

• Finally, we should remember that we are eventually interested in deformations of X rather than Xτ̃. Hence, we have to do the same job again with t_{α} = R_{1} − R_{0}. The best result for equations (0), (1), (5), and (9) is

\begin{matrix}(0) & (Z_{3}W Z_{1} − U^{3}) − U(t_{α}Z_{3} + ξ_{β}Z_{1}^{3}) \\ (1) & (Z_{1}Z_{2} − W^{2}) − t_{α}V_{1} − ξ_{β}R_{1}^{2}Z_{1} \\ (5) & (V_{2}Z_{1} − U^{2}) − t_{α}Z_{3} − ξ_{β}Z_{1}^{3} \\ (9) & (Z_{1}V_{1} − UW) − t_{α}V_{2} − ξ_{β}R_{1}Z_{1}^{2}. \end{matrix}

In particular, we have not been able to eliminate R_{1} from equations (1) and (9). It indicates that our deformation of Xτ̃ does not blow down onto the X level.

5.6

We have used τ ⊂ σ to obtain a partial modification Y_{τ} → Y_{σ}. Deformations of Y_{σ} have been obtained by blowing down deformations of Y_{τ}.

Is it possible, or even better, to consider the whole modification Y_{Σ} → Y_{σ} instead of the open chart only? Σ is the fan consisting of the two cones τ = σ ∩ [R ≥ 0] and τ′ := σ ∩ [R ≤ 0]. Apart from the fact that this approach looks more natural, there might be another advantage: the lack of certain degrees providing infinitesimal deformations of Y_{τ} may be overcome by blowing down local trivial deformations of Y_{Σ}.

References

[Al 1] Altmann, K.: Computation of the vector space T^{1} for affine toric varieties. J. Pure Appl. Algebra 95 (1994), 239-259.

[Al 2] Altmann, K.: Minkowski sums and homogeneous deformations of toric varieties. Tôhoku Math. J. 47 (1995), 151-184.

[Al 3] Altmann, K.: Infinitesimal deformations and obstructions for toric singularities. J. Pure Appl. Algebra 119 (1997), 211-235.

[Al 4] Altmann, K.: The versal deformation of an isolated, toric Gorenstein singularity. Invent. math. 128 (1997), 443-479.

[Ish] Ishida, M.-N.: Torus embeddings and dualizing complexes. Tôhoku Math. J. 32, 111-146 (1980).

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

[Re] Reid, M.: Some flips. Talk at the conferences in Santa Cruz (July 1995) and Warwick (September 1995).

Klaus Altmann Institut für reine Mathematik, Humboldt-Universität zu Berlin Ziegelstr. 13A, D–10099 Berlin, Germany. e-mail: altmann@mathematik.hu-berlin.de