Deformations of affine torus varieties

Klaus Altmann^{∗}

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

November 13, 1996

Contents

1 Introduction 1

2 Relative complete intersections in the category of affine torus varieties 3

3 Proof of the theorem (2.4) 7

4 Cones with simplicial faces only 14

5 The hexagon singularities 30

1 Introduction

(1.1) Based on the paper [KSB] of Kollár/ Shepherd-Barron, many investigations of the base space of the versal deformation of a two dimensional cyclic quotient singularity were done in the last years: In [Ar], J. Arndt gave an algorithm for computing the equations of the base space; in [St] and [Ch], J. Stevens and J.A. Christophersen described the components of the reduced base space in a more qualitative way using continued fractions.

Cyclic quotient singularities are exactly those singularities that appear as two dimensional affine torus varieties:

∗partially supported by a DFG grant

X(n, q) (quotient of the ZZ/nZZ-action ξ ↦ ( \begin{matrix}ξ & 0 \\ 0 & ξ^{q} \end{matrix} ) on I C^{2}) can be

built as an affine torus variety by the cone

σ := < (1, 0); (−q, n) > ⊆ N := ZZ^{2}. (For the notion of an affine torus variety: cf. [Ke].)

On the other hand, in [Ch] (CDV; §2) Christophersen made the following observation: Deforming such a cyclic quotient singularity, the total spaces over the components of the reduced base space are (after a finite base change) torus varieties, too.

(1.2) Having this in mind, the following problem seems to be very interesting: Classify (or describe in combinatorical terms) all possible fibre product diagrammes

\begin{aligned}& i \\ Y & ↪ & X \\ ↓ & ⊗ & ↓ & f \\ \{0\} & ↪ & S \end{aligned}

with: X, Y = affine torus varieties with closed orbits 0 ∈ Y, 0 ∈ X, respectively; 0 ∈ Y is an isolated singularity (0 ∈ X need not to be isolated); i = morphism of torus varieties with i(0) = 0; f is a flat map.

In particular, i maps the torus T_{Y} of Y in the torus T_{X} of X, and i : T_{Y} ↪ T_{X} is induced by a surjection of lattices:

i^{∗} : M ⟶ ⟶ M/L

(M ≅ ZZ^{n}, T_{X} = Spec I C[M];

L ⊆ M is an m − dimensional sublattice with M/L torsion free, T_{Y} = Spec I C[M/L]).

If X is given by the cone σ ⊆ N ⊗ IR (N ≅ ZZ^{n} is the dual lattice of M), the dual cone σ^{∨}⊆ M ⊗ IR is squeezed to a cone σ̄^{∨} ⊆ M/L ⊗ IR. (In the dual language: σ̄ = σ ∩ L^{⊥} in N ⊗ IR). σ and σ̄ are top dimensional cones (in N ⊗ IR and L^{⊥}, respectively) that do not contain any linear subspace.

(1.3) In §2, all such deformations were completely described in terms of the cone σ and the sublattice L ⊆ M. The corresponding theorem is proved in §3.

In §4, this theorem is used to give an explicit list of all possible pairs [σ, L] (in a more or less coarse sense) for the special case that all proper faces τ < σ are simplicial, i.e. X\{0} contains only cyclic quotient singularities of arbitrary dimensions.

§5 contains specified computations of the sporadic cases belonging to the previous paragraph. These computations can be used as examples for the main theorem (2.4) also.

(1.4) In [Al], we have proved a formula to compute the vector space T^{1} of the affine toric variety Yσ̄. The spirit of this formula is very closed to the problem of splitting some affine cuts of the cone σ̄ into a Minkowski sum of two polyhedra. In a forthcoming paper, we will use the result (2.4) to prove that 1-parameter-- deformations, indeed, correspond to special Minkowski sums (such that both summands do not admit any common direction of faces, i.e. the summands are extremal in some sense). The total spaces can be constructed by taking the convex hull of a suitable configuration of both summands.

The next task will be to generalize this point of view to deformations with parameter spaces of higher dimension (more than two Minkowski summands have to be involved). Then, it will be clear, under which conditions two 1-parameter-- deformations belong to the same irreducible component of the versal base space.

(1.5) Acknowledgements: Most of this paper was worked out during a one year stay at the university of Kaiserslautern (paid for by the DFG project “normal surface singularities”). It is a pleasure for me to thank the members of the working group “Algebraic geometry” for their hospitality and warm atmosphere; in particular, I wish to thank G.-M. Greuel and D. van Straten for many helpful discussions and hints. Finally, I wish to thank R. Schultz for helping me to imagine some 3-dimensional convex bodies, Ms. Dobers and Ms. Bitsch for typing (with LaTEX) and correcting the manuscript and T. de Jong for letting me use his (meanwhile slightly damaged) car.

2 Relative complete intersections in the category of affine torus varieties

(2.1) With the notations of the previous chapter, a deformation diagramme \begin{matrix}Y & ↪^{i} & X \\ ↓ & ⊗ & ↓ & f \\ \{0\} & ↪ & S \end{matrix} of (1.2) would induce a deformation of the corresponding torus T_{Y}: \begin{matrix}T_{Y} & ↪ & T_{X} \\ ↓ & ⊗ & ↓ & f \\ \{0\} & ↪ & S & . \end{matrix}

T_{Y} and T_{X} are smooth. Hence, T_{X} would split locally in a product T_{X} ≅ T_{Y} × S, and the germ (S, 0) would have to be smooth also.

(2.2) Let a deformation as in (1.2) be given.

Proposition: Let I := Ker (I C[σ^{∨} ∩M] → I C[σ̄^{∨} ∩ M/L]) be the ideal that defines

Y ⊆ X. Then, I can be generated by m polynoms of the form x^{r} − x^{s} ∈ I C[σ^{∨} ∩M]

(r, s ∈σ^{∨} ∩M; r − s ∈ L).

In particular, every deformation X of Y described above can be realized by a flat map f : X → I C^{m}, which makes Y a relative complete intersection in X.

Proof: Step 1: On the level of the local rings we obtain the following diagramme:

O_{X,0} ⟶ ⟶^{i^{∗}}  O_{Y,0}
↑flat ⊗  ↑
O_{S,0} ⟶ ⟶ O_{S,0}/m_{S,0} = I C.

Therefore, I · O_{X,0} = m_{S,0} · O_{X,0} is generated by m elements g_{1},…, g_{m} ((S, 0) is smooth), and by the Nakayama Lemma we can choose these generators among the elements of the form x^{r} − x^{s} (r, s ∈ σ^{∨} ∩M; r − s ∈ L).

Step 2: Let Ĩ := (g_{1},…, g_{m}) ⊆ I C[σ^{∨} ∩M]. Then, Ĩ ⊆ I are ideals in I C[σ^{∨} ∩M] meeting the following properties:

(i) Ĩ, I are homogeneous with respect to the M/L-grading;

(ii) Ĩ = I in the local ring O_{X,0}.

Now, we want to show Ĩ = I. For this let g ∈ I be an arbitrary M/L-homogeneous element. By (ii) there exists an h ∈ I C[σ^{∨} ∩M] with

h · g ∈ Ĩ and
h ∉ m_{0} := ⨁_{{r∈σ^{∨}∩M \atop r≠0}} I C · x^{r} (i.e. h contains a nonvanishing constant term);

by (i) we can assume in addition h to be M/L-homogeneous.

Finally, the closed embedding i : Y ↪ X sends 0 to 0 (cf. (1.2)), and this property implies L∩ σ^{∨}= {0}. Hence, the M/L-homogeneous element h has to be a constant.

□

Remark: The m vectors r − s that correspond to the generators of I are free generators of the m-dimensional lattice L.

(2.3) Now, our task (cf. (1.2)) is reduced to describe all relative complete intersections in the category of affine torus varieties. For doing this, we will change our point of view slightly:

Let M, N be n-dimensional dual lattices as before; let σ =< a^{1},…, a^{N} > be a top dimensional, rational polyhedral cone in N ⊗ IR that does not contain any linear subspace. (a^{1},…, a^{N} ∈ N denote the fundamental generators of σ. They are primitive elements of the lattice, i.e. the coordinates of each a^{j} have the greatest common divisor 1.)

For r_{0}^{1},…, r_{0}^{m}; r_{1}^{1},…, r_{1}^{m} ∈σ^{∨} ∩M (1 ≤ m ≤ n − 2) we define L := ∑_{i=1}^{m} ZZ · (r_{0}^{i} − r_{1}^{i}) ⊆

M. Suppose that L is an m-dimensional sublattice of M with M/L torsion free and

L∩ σ^{∨}= {0}.

The last condition implies that the ”squeezed” cone σ̄ := σ ∩ L^{⊥} is top dimensional in L^{⊥} ⊆ N ⊗ IR; we obtain the affine torus varieties X := Spec I C[σ^{∨} ∩M] and Y := Spec I C[σ̄^{∨} ∩ M/L] together with a map i : Y → X.

Remark: The canonical homomorphism I C[σ^{∨} ∩M] → I C[σ̄^{∨} ∩ M/L] need not be surjective (i.e. i would not be a closed embedding) in general. Look at the following example:

M := ZZ^{2}, σ^{∨}:=< (1, 0); (0, 1) >⊆ IR^{2};
r_{0}^{1} := (2, 0), r_{1}^{1} := (0, 3).

Then, L = ZZ · (2, −3), M splits into M = L ⊕ ZZ · (1, −1), and the canonical sur-

jection \begin{matrix}M & ⟶ ⟶ & M/L ≅ ZZ \\ (1, 0) & ↦ & 3 \\ (0, 1) & ↦ & 2. \end{matrix} is given by

Restricting this map to σ^{∨} ∩M, 1 ∈ ZZ will not be contained in the image of

σ^{∨} ∩M → σ̄^{∨} ∩ ZZ (= IN).

(The reason for having such cases is that the image of the semigroup σ^{∨} ∩M need not be saturated.)

(2.4) Theorem: Y has an isolated singularity in 0 ∈ Y, i is a closed embedding, and the ideal I := Ker (I C[σ^{∨} ∩M] → I C[σ̄^{∨} ∩ M/L]) is generated by the elements

x^{r_{0}^{1}} − x^{r_{1}^{1}},…, x^{r_{0}^{m}} − x^{r_{1}^{m}} ∈ I C[σ^{∨} ∩M] (i.e. the situation of (1.2) with S = I C^{m}), iff for all proper faces τ < σ one of the following conditions is fulfilled (with a suitable order of the r^{i}’s and a^{j}’s and with a possible permutation of r_{0}^{•} and r_{1}^{•}):

(I) ∃i ∈ {1,…, m} : r_{0}^{i} ∈ τ^{⊥}; r_{1}^{i} ∉ τ^{⊥} , or

(II) τ =< a^{1},…, a^{k} > is smooth (i.e. generated by a part of a ZZ-basis of N), and there exists an ℓ (0 ≤ ℓ ≤ k − 1) with

• < a^{j}, r_{0}^{i} >= δ_{ij} (i = 1,…, ℓ; j = 1,…, k)

• < a^{j}, r_{1}^{i} >= 0 (1 ≤ j ≤ i ≤ ℓ)

• < a^{j}, r_{v}^{i} >= 0 (i = ℓ + 1,…, m; j = 1,…, k; v = 0, 1).

The conditions (I) and (II) of the theorem may look a little bit technical, but they are very strong and easy to handle. As we will see in §4, these two conditions can be used to restrict the choice of possible cones σ to a great extent. Moreover, a coarser (but much nicer) version of the above theorem is sufficient for taking the first big steps to classifying squeezable cones:

(2.5) Notations:

1) Let R := {(r_{0}^{1})^{⊥},…, (r_{0}^{m})^{⊥}; (r_{1}^{1})^{⊥},…, (r_{1}^{m})^{⊥}}, i.e. we regard the elements r_{ν}^{i} as faces of σ (written as “r_{ν}^{i} ∈ R” or “r_{ν}^{i} ∉ R”). In general, these faces will be of dimension n − 1. We obtain #R ≤ 2m ≤ 2n − 4.

2) A top dimensional face τ < σ that does not belong to R will be called a “good face”.

Theorem: If Y has an isolated singularity in 0 ∈ Y, i is a closed embedding, and the ideal I := Ker (I C[σ^{∨} ∩M] → I C[σ̄^{∨} ∩ M/L]) is generated by the elements

x^{r_{0}^{1}} − x^{r_{1}^{1}},…, x^{r_{0}^{m}} − x^{r_{1}^{m}} ∈ I C[σ^{∨} ∩M] (i.e. the situation of (1.2) with S = I C^{m}), then, each good (top dimensional) face τ < σ admits the following properties:

(i) It is simplicial (even smooth).

(ii) With a suitable order of the r^{i}’s and with a possible permutation of r_{0}^{•} and r_{1}^{•} there holds:

• r_{0}^{1},…, r_{0}^{m} intersect τ in m different (n−2)-dimensional faces. ( r_{0}^{1},…, r_{0}^{m} were called the R-neighbours of τ.)

• For i = 1,…, m the set (r_{0}^{1} ∩… ∩ r_{0}^{i}) ∪ r_{1}^{i} contains all vertices of τ.

Remark: If r is an R-neighbour of τ, then either r is an (n − 2)-dimensional edge of τ, or r is a top dimensional face of σ that is adjacent to τ. In the first case, r can only be an R-neighbour for at most two different good faces of σ.

3 Proof of the theorem (2.4)

(3.1) With the notations of (2.3) let Ỹ ⊆ X be the zero set of the ideal (x^{r_{0}^{1}} − x^{r_{1}^{1}},…, x^{r_{0}^{m}} − x^{r_{1}^{m}}) ⊆ I C[σ^{∨} ∩M]. Then, the map i : Y → X factorizes through

Y → Ỹ ↪^{closed} X.

Lemma: For an arbitrary proper face τ < σ let orb τ := Spec I C[τ^{⊥} ∩ M] be the corresponding orbit; denote by 1_{τ} ∈ orb τ the unit of this torus (defined by sending all monomials to 1 ∈ I C). Now, the following conditions are equivalent:

(1) 1_{τ} ∈ Ỹ;

(2) orb τ ∩ Ỹ ≠ ∅;

(3) ∀i = 1,…, m : [r_{0}^{i} ∈ τ^{⊥} ⇔ r_{1}^{i} ∈ τ^{⊥}].

Proof: (2) ⟹ (3): The existence of a common point of orb τ and Ỹ yields a ring homomorphism φ : I C[σ^{∨} ∩M] ⟶ ⟶ I C with the following properties:

(i) r ∉ τ^{⊥} ⟺ φ(x^{r}) = 0 (φ factorizes through I C[σ^{∨} ∩M] ↪ I C[τ^{∨} ∩ M] ⟶ ⟶ I C[τ^{⊥} ∩ M]. Hence, φ maps x^{r} to 0 for r ∉ τ^{⊥} and to a unit otherwise.)

(ii) φ(x^{r_{0}^{i}} − x^{r_{1}^{i}}) = 0 for i = 1,…, m.

Let τ ∈ {1,…, m}. Then, we obtain φ(x^{r_{0}^{i}}) = φ(x^{r_{1}^{i}}), hence

φ(x^{r_{0}^{i}}) = 0 iff φ(x^{r_{1}^{i}}) = 0.

By (i), this equivalence implies the condition (3).

(3) ⟹ (1): The ideal of 1_{τ} ∈ X is generated (as a I C-vectorspace) by

• x^{s} (for s ∈σ^{∨} ∩M; s ∉ τ^{⊥}) and

• x^{s} − 1 (for s ∈ (σ^{∨} ∩τ^{⊥}) ∩ M).

Hence, this ideal contains the ideal of Ỹ ⊆ X. □

Corollary: Let (Ỹ , 0) be an isolated singularity. Then, for singular, proper faces τ < σ (i.e. X is not smooth in orb τ) the condition (I) of the theorem is fulfilled.

Proof: Failing (I) is equivalent to the condition (3) of the previous lemma, i.e. 1_{τ} ∈ Ỹ. Now, τ ≠ σ implies 1_{τ} ≠ 0, and 1_{τ} ∈ Ỹ has to be a smooth point. Hence, in 1_{τ} the torus variety X would be non-singular too. □

(3.2) Lemma: Let (Ỹ , 0) be an isolated singularity, let τ < σ be a proper face that fails the condition (I) of the theorem. (In particular, τ =< a^{1},…, a^{k} > is a smooth face.) Then, with a suitable order of the r^{i}’s and a^{j}’s and with a possible permutation of r_{0}^{•} and r_{1}^{•}, there exists an ℓ (0 ≤ ℓ ≤ k) with

• < a^{j}, r_{0}^{i} >= δ_{ij} (i = 1,…, ℓ; j = 1,…, k)

• < a^{j}, r_{ν}^{i} >= 0 (i = ℓ + 1,…, m; j = 1,…, k; ν = 0, 1).

Proof: Step 1: Completing the fundamental generators a^{1},…, a^{k} of τ to a ZZ-basis a^{1},…, a^{k}; b^{k+1},…, b^{n} of the lattice N, we obtain coordinates that split into two blocks:

r ∈ M can be written as r = (r̄, r^{=}) (r̄ ∈ ZZ^{k}; r^{=}∈ ZZ^{n−k}) with (r̄)_{j} =< a^{j}, r >,

and in the same way (x̄, x^{=}) = (x_{1},…, x_{n}) denote the coordinates of I C^{k} × (I C^{∗})^{n−k} ≅ Spec[τ^{∨} ∩ M] ↪^{open} X. (orb τ ⊆ Spec I C[τ^{∨} ∩ M] is then given by the equation x̄ = 0, and 1_{τ} ∈ orb τ corresponds to the point (¯0, ¯1).)

Now, Ỹ contains 1_{τ}, and we can compute the partial derivatives of the equations g_{i} := x^{r_{0}^{i}} − x^{r_{1}^{i}} in this point. Let ℓ ∈ IN (0 ≤ ℓ ≤ m) such that (after a permutation of the indices i ∈ {1,…, m}, if necessary)

r_{0}^{i}, r_{1}^{i} ∉ τ^{⊥} (i.e. r̄_{0}^{i}, r̄_{1}^{i} ≠ ¯0) for i = 1,…, ℓ;
r_{0}^{i}, r_{1}^{i} ∈ τ^{⊥} (i.e. r̄_{0}^{i} = r̄_{1}^{i} = ¯0) for i = ℓ + 1,…, m.

Then, we obtain

\frac{∂g_{i}}{∂x_{j}} (1_{τ}) = δ_{r̄_{0}^{i},e_{j}} − δ_{r̄_{1}^{i},e_{j}}  (i = 1,…, m; j = 1,…, k; \{e_{1},…, e_{k}\} denotes the canonical basis of ZZ^{k});

\frac{∂g_{i}}{∂x_{j}} (1_{τ}) = 0 (i = 1,…, ℓ; j = k + 1,…, n);
\frac{∂g_{i}}{∂x_{j}} (1_{τ}) = (r_{0}^{i} − r_{1}^{i})_{j} =< b^{j}, r_{0}^{i} − r_{1}^{i} > (i = ℓ + 1,…, m; j = k + 1,…, n).

Introducing the following notation

R̄_{•}^{i} := \{ \begin{matrix}r̄_{•}^{i} ∈ ZZ^{k} & (if r̄_{•}^{i} is a unit vector) \\ ¯0 ∈ ZZ^{k} & (otherwise) \end{matrix} ,

we can write down our result in a shorter way:

\frac{∂g}{∂x}(1_{τ}) = ( \begin{matrix}R̄_{0}^{i} − R̄_{1}^{i} & & 0 \\ & r_{0}^{i} − r_{1}^{i} \end{matrix}  ) \begin{aligned}i & = & \{ ⋮_{ℓ}^{1} \\ i & = & \{ ⋮_{m}^{ℓ+1} \end{aligned} .

\overbrace{1,…, k}  \overbrace{k + 1,…, n}

Step 2: After a possible changing of the order of the indices i and j and after a possible permutation of R̄_{0}^{•} and R̄_{1}^{•}, we obtain R̄_{0}^{i} = e_{i} for i = 1,…, ℓ (≤ k).

Since 1_{τ} is a smooth point of Ỹ, the Jacobian matrix \frac{∂g}{∂x}(1_{τ}) has the maximal rank m. Hence, rank(R̄_{0}^{i} − R̄_{1}^{i})_{i=1,...,ℓ} = ℓ, and this implies ℓ ≤ k. Now, by selecting ℓ suitable columns of (R̄_{0}^{i} − R̄_{1}^{i})_{i=1,...,ℓ} we obtain a new (ℓ × ℓ)- matrix A with det A ≠ 0 and all non-vanishing entries equal to ±1. It would be enough to prove that there exist ℓ such entries that admit a position of ℓ castles that cannot beat each other in a ℓ × ℓ chess game. But this follows easily by induction from the Laplacian Entwicklungssatz:

Let A = ( \begin{matrix}a_{11} & … & a_{1ℓ} \\ ⋮ & & ⋮ \\ a_{ℓ1} & … & a_{ℓℓ} \end{matrix} ); then ∑_{j=1}^{ℓ}(−1)^{j+1}a_{1j} det A_{1j} = det A ≠ 0,

and there exists an index j with a_{1j} ≠ 0 and det A_{1j} ≠ 0. □

Remark: For τ to meet the condition (II) of the theorem, only the proof of < a^{j}, r_{1}^{i} >= 0 (1 ≤ j ≤ i ≤ ℓ) is left. This will be done in (3.4).

(3.3) Let Y = Ỹ, i.e. the ideal I = Ker (I C[σ^{∨} ∩M] → I C[σ̄^{∨} ∩ M/L]) is gener-

ated by the elements g_{i} = x^{r_{0}^{i}} − x^{r_{1}^{i}} (i = 1,…, m). For any index set Λ ⊆ {1,…, m} we denote by L_{Λ} the corresponding sublattice

L_{Λ} := ⨁_{i∈Λ} ZZ(r_{0}^{i} − r_{1}^{i}) ⊆ L of M.

Lemma:

1) Let Λ ⊆ {1,…, m}; let p ∈ L \ L_{Λ}, s ∈ M be elements with s, s − p ∈σ^{∨}. Then there exists an index i ∈ {1,…, m} \ Λ such that [(s − r_{ν}^{i}) + L_{Λ}]∩ σ^{∨}≠ ∅ (for some ν ∈ {0, 1}).

2) Given a face τ =< a^{1},…, a^{k} > < σ, we denote by Λ_{τ} the index set Λ_{τ} := {i | r_{0}^{i}, r_{1}^{i} ∈ τ^{⊥}}. Now, for p, s as before there exist i ∈ {1,…, m} \ Λ and ν ∈ {0, 1} such that < a^{j}, s − r_{ν}^{i} > ≥ 0 for all j = 1,…, k.

Proof: Step 1: We define a graph with M as the set of vertices as follows:

s, t ∈ M will be connected by an edge iff there exists an index i ∈ {1,…, m} such that either

s − r_{0}^{i} = t − r_{1}^{i} ∈σ^{∨} or s − r_{1}^{i} = t − r_{0}^{i} ∈σ^{∨} .

Now, let elements p ∈ L, s ∈ M be given such that s, s − p ∈σ^{∨}; x^{s} − x^{s−p} ∈ I yields an equation

(∗) x^{s} − x^{s−p} = ∑_{μ} c_{μ}x^{t_{μ}}(x^{r_{0}^{i(μ)}} − x^{r_{1}^{i(μ)}}) (c_{μ} ∈ I C; t_{μ} ∈σ^{∨} ∩M).

The exponents t_{μ} + r_{0}^{i(μ)} and t_{μ} + r_{1}^{i(μ)} are connected by an edge in the above graph M. Therefore, by deleting all terms that admit exponents not contained in the connected component of s ∈ M, the equation (∗) keeps its form at the right-hand side. Only two different cases can arise:

(i) x^{s} − x^{s−p} = ∑_{μ}′ c_{μ}x^{t_{μ}}(x^{r_{0}^{i(μ)}} − x^{r_{1}^{i(μ)}}) or

(ii) x^{s} = ∑_{μ}′ c_{μ}x^{t_{μ}}(x^{r_{0}^{i(μ)}} − x^{r_{1}^{i(μ)}}) (“∑′” means a part of “∑”).

The latter case would imply x^{s} ∈ I, i.e. xs̄ = 0 in the ring I C[M/L]. Hence, only the first case is possible, and we obtain that s and s − p are contained in a common connected component of M.

Step 2: Any path from s to s − p provides a sequence s = s_{0},…, s_{N} = s − p with either

s_{μ−1} − r_{0}^{i(μ)} = s_{μ} − r_{1}^{i(μ)} ∈σ^{∨} ∩M, or
s_{μ−1} − r_{1}^{i(μ)} = s_{μ} − r_{0}^{i(μ)} ∈σ^{∨} ∩M (μ = 1,…, N).

The element p = s_{0} − s_{N} = ∑_{μ=1}^{N} ±(r_{0}^{i(μ)} − r_{1}^{i(μ)}) was presumed not to be contained

in the sublattice L_{Λ} ⊆ L. Hence, at least one of the indices i(μ) does not belong to Λ.

Assume μ^{∗} to be the smallest index meeting this property, then we obtain

s_{0} − s_{μ^{∗}−1} = ∑_{μ=1}^{μ^{∗}−1} ±(r_{0}^{i(μ)} − r_{1}^{i(μ)}) ∈ L_{Λ} and
∃ν ∈ {0, 1} :  (s − r_{ν}^{i}) − (s_{0} − s_{μ^{∗}−1}) = s_{μ^{∗}−1} − r_{ν}^{i} ∈σ^{∨} ∩M (i := i(μ^{∗}) ∉ Λ).

Finally, the second part of the lemma is a direct consequence from the first part. □

Corollary: Let τ =<a^{1},…, a^{k} > < σ be a smooth face; denote Λ_{τ} := {i | r_{0}^{i}, r_{1}^{i} ∈ τ^{⊥}} as before. Then, for each p ∈ L \ L_{Λ} there exist i ∈ {1,…, m} \ Λ and ν ∈ {0, 1} such that

< a^{j}, r_{ν}^{i} > ≤ max{< a^{j}, p >; 0} for all j = 1,…, k.

Proof: For a given p ∈ L we construct the following elements of the lattice M:

1) Since τ is a smooth cone (i.e. {a^{1},…, a^{k}} is a part of a ZZ-basis of the lattice N), there exists an s^{1} ∈ M such that

< a^{j}, s^{1} >= max{< a^{j}, p >; 0} for all j = 1,…, k.

2) Let s^{2} ∈ [int(σ^{∨} ∩τ^{⊥})] ∩ M. Then, we obtain

< a^{j}, s^{2} > = 0 (j = 1,…, k) and
< a^{μ}, s^{2} > > 0 for all remaining fundamental generators
a^{μ} of the cone σ.

3) Now, s := s^{1} + g · s^{2} (g ∈ IN; g ≫ 0) admits the following properties:

s, s − p ∈ σ^{∨}  and
< a^{j}, s > = max{< a^{j}, p >; 0} (j = 1,…, k).

Applying the previous lemma to τ, p and s completes the proof. □

(3.4) Lemma: Let Y = Ỹ have an isolated singularity in 0 (that means a combination of the presumptions of (3.2) and (3.3)). Let τ =< a^{1},…, a^{k} > < σ be a proper face that fails the conditions (I) of the theorem. Then, additionally to the claim of Lemma (3.2), we can achieve < a^{j}, r_{1}^{i} >= 0 for 1 ≤ j ≤ i ≤ ℓ.

Proof: In (3.2) we have shown

< a^{j}, r_{0}^{i} >= δ_{ij} (i = 1,…, ℓ (≤ k); j = 1,…, k) and
< a^{j}, r_{ν}^{i} >= 0 (i = ℓ + 1,…, m; j = 1,…, k; ν = 0, 1);

in particular Λ_{τ} = {ℓ + 1,…, m}. Now, we prove the above lemma by induction; let ℓ′ ≤ ℓ such that

< a^{j}, r_{1}^{i} >= 0 for 1 ≤ j ≤ i ≤ ℓ; j ≤ ℓ′ − 1.

Claim: There exists an index j (ℓ′ ≤ j ≤ ℓ) such that < a^{j}, r_{1}^{i} >= 0 for each i ≥ ℓ′.

Proof: Let this not be the case, i.e. < a^{j}, ∑_{i≥ℓ′} r_{1}^{i} > ≥ 1 for each j = ℓ′,…, ℓ.

Then, we define p := ∑_{i≥ℓ′}(r_{0}^{i} − r_{1}^{i}) ∈ L \ L_{Λ}, which admits the following prop-

erties:

1) < a^{j}, p >= 0 for j = 1,…, ℓ′ − 1 (since < a^{j}, r_{0}^{i} >=< a^{j}, r_{1}^{i} >= 0 for j ≤ ℓ′ − 1, i ≥ ℓ′);

2) < a^{j}, p > ≤ 0 for j = ℓ′,… ℓ

(since < a^{j}, ∑_{i≥ℓ′} r_{0}^{i} >= 1, < a^{j}, ∑_{i≥ℓ′} r_{1}^{i} >≥ 1 for ℓ′ ≤ j ≤ ℓ);

3) < a^{j}, p > ≤ 0 for j ≥ ℓ + 1 (since < a^{j}, r_{0}^{i} >= 0 for j ≥ ℓ + 1).

Therefore, max{< a^{j}, p >; 0} = 0 for j = 1,…, k, and we can apply the corollary of (3.3): There exist i ∈ {1,…, ℓ} and ν ∈ {0, 1} such that < a^{j}, r_{v}^{i} > ≤ 0 for all j = 1,…, k. By τ failing the condition (I) of the theorem this implies r_{0}^{i} ∈ τ^{⊥}, and we obtain a contradiction.

Permuting the indices ℓ′ and j in both sets {a^{1},…, a^{k}} and {r^{1},…, r^{ℓ}} the induction hypothesis together with the above claim imply

< a^{j}, r_{1}^{i} >= 0 for 1 ≤ j ≤ i ≤ ℓ; j ≤ ℓ′.

The statements of lemma (3.2) also stay valid during this permutation; hence, the lemma is proven. □

Remark:

1) The lemma implies that ℓ has to be smaller than k.

2) The lemmata (3.2) and (3.4) show that meeting the properties (I) or (II) in the theorem (2.4) is a necessary condition (for Y = Ỹ having an isolated singularity). It remains to prove the opposite direction of this theorem.

(3.5) Assume that for each proper face τ =< a^{1},…, a^{k} > < σ at least one of the conditions (I) or (II) is fulfilled.

Lemma: Let τ < σ be a proper face such that orbτ∩Ỹ ≠ ∅. Then, Spec I C[τ^{∨} ∩M] ∩ Ỹ is irreducible, smooth of dimension n − m, and contains the torus Spec I C[M/L] as an open, dense subset.

Proof: orb τ ∩ Ỹ ≠ ∅ implies that τ fails the condition (I) (cf. lemma (3.1)), hence, condition (II) must be fulfilled. We use the language of the proof of lemma (3.2): U_{τ} := Spec I C[τ^{∨} ∩M] ≅ I C^{k} × (I C^{∗})^{n−k} is an open, dense subset of X; the equations of Ỹ in U_{τ} have the following form:

x_{1}· x^{=}^{r^{=}^{1}^{0}} = x̄^{r̄_{1}^{1}}· x^{=}^{r^{=}{1 \atop 1}} (the right-hand side does not contain x_{1})
x_{2}· x^{=}^{r^{=}{2 \atop 0}} = x̄^{r̄_{1}^{2}}· x^{=}^{r^{=}{2 \atop 1}} (the right-hand side does not contain x_{1}, x_{2})
⋮  ⋮  ⋮
x_{ℓ}· x^{=}^{r^{=}{ℓ \atop 0}} = x̄^{r̄_{1}^{ℓ}}· x^{=}^{r^{=}{ℓ \atop 1}} (the right-hand side does not contain x_{1},…, x_{ℓ})
and x^{=}^{r^{=}{i \atop 0}} = x^{=}^{r^{=}{i \atop 1}} for i = ℓ + 1,…, m.

The last m − ℓ equalities define a subtorus (I C^{∗})^{(n−m)−(k−ℓ)} ⊆ (I C^{∗})^{n−k}. The variables x_{ℓ+1},…, x_{k} (belonging to x̄) can free range in a space I C^{k−ℓ}, but the variables x_{1},…, x_{ℓ} are completely determined by the first ℓ equations. Hence, U_{τ} ∩ Ỹ ≅ I C^{k−ℓ} × (I C^{∗})^{(n−m)−(k−ℓ)}; the torus Spec I C[M/L] corresponds to the open, dense subset (I C^{∗})^{k−ℓ} × (I C^{∗})^{(n−m)−(k−ℓ)} = (I C^{∗})^{n−m}. □

Corollary: Ỹ \ {0} is irreducible and smooth of dimension n − m. It contains Spec I C[M/L] as an open, dense subset, and the torus action T × X → X induces an action of the small torus

Spec I C[M/L] × (Ỹ \ {0}) → (Ỹ \ {0}).

Proof: All these properties can be checked locally, and Ỹ \ {0} is covered by the open subsets U_{τ} ∩ Ỹ (taking all proper faces τ < σ such that orb τ ∩ Ỹ ≠ ∅). □

Therefore, Ỹ itself is irreducible of dimension n − m, and 0 ∈ Ỹ is an isolated singularity. As Ỹ is a relative complete intersection in the Cohen-Macaulay variety X, Ỹ has to be Cohen-Macaulay, too. Hence, Ỹ is reduced and normal (non-singular in codimension 1 since m ≤ n − 2).

Spec I C[M/L] ⊆ Ỹ is an open, dense subset, and the group law of the torus extends to an action on Ỹ (which is the restriction of the action of T_{X} on X). Therefore, Ỹ must be an affine torus variety, too, i.e. we obtain Y = Ỹ. This completes the proof of the theorem.

4 Cones with simplicial faces only

(4.1) We use the notations of (2.3). Let σ ⊆ N_{IR} be a cone such that for all good faces τ < σ the conditions (i) and (ii) of the theorem (2.5) are fulfilled. This means a very strong restriction to the form of σ, and we want to determine all abstract cones (i.e. their structure is reduced to the incidence relation between all faces) that can appear in this way.

Remark: Each n-dimensional σ is a cone over a compact, convex polyhedron P_{σ} of dimension n − 1. This polyhedron yields the same incidence relations as σ; it can be recovered by intersection of σ with a suitable affine hypersurface in N_{IR}. Sometimes (in particular for n = 4) it is useful to imagine this polyhedron P_{σ} instead of σ itself. According to this, the fundamental generators of σ were often called vertices also.

(4.2) For the rest of this chapter we make the following assumption:

Let σ be a cone such that all proper faces are simplicial ones. That means the corresponding toric variety X = Spec I C[σ^{∨} ∩M] contains at most cyclic quotient singularities outside 0 ∈ X.

Definition: For top dimensional faces τ_{1}, τ_{2} < σ we define

ϱ(τ_{1}, τ_{2}) := (n − 1) − dim(τ_{1} ∩ τ_{2})
= #{vertices of τ_{1} that do not belong to τ_{2}}
= #{vertices of τ_{2} that do not belong to τ_{1}}.

Lemma:

1) 0 ≤ ϱ(τ_{1}, τ_{2}) ≤ n − 1.

2) ϱ(τ_{1}, τ_{2}) = 0 ⟺ τ_{1} = τ_{2}; ϱ(τ_{1}, τ_{2}) = 1 ⟺ τ_{1} and τ_{2} are adjacent.

3) If τ_{1} and τ_{2} are connected by a path meeting only k (n − 2)-dimensional faces of σ but no faces of smaller dimension, then ϱ(τ_{1}, τ_{2}) ≤ k.

4) Let K = {τ_{i}} be a set of top dimensional faces such that any two faces of K are adjacent to each other. Then, if a face τ < σ is adjacent to two faces of K, τ is adjacent to all faces of K.

5) Let {τ_{1},…, τ_{k}} and {φ_{1},…, φ_{ℓ}} (k, ℓ ≥ 2) be two sets of top dimensional faces such that ϱ(τ_{i}, φ_{j}) = 1 for all pairs [i, j]. Then there are only two possible cases:

a) k = ℓ = 2; ϱ(τ_{1}, τ_{2}) = ϱ(φ_{1}, φ_{2}) = 2, or

b) ϱ(τ_{i}, τ_{j}) = 1, ϱ(φ_{i}, φ_{j}) = 1 for all possible pairs [i, j].

(In particular, if σ is not a simplex, (b) implies k + ℓ ≤ n − 1 !)

The proof is easy and will be omitted here.

(4.3) We start our investigations by regarding (as a first case) the following class of cones σ:

Assume that there exist (top dimensional) faces τ_{L} and τ_{R} such that ϱ(τ_{L}, τ_{R}) ≥ 3.

Then the sets of all faces that are adjacent to τ_{L} and τ_{R}, respectively, have no common elements. The cone σ must contain at least 2n various faces:

(n−1) (n−1)

(Each fat point represents a face; the edges between two of these fat points mean that both faces are adjacent ones.)

τ_{L} τ_{R}

The faces τ_{L} and τ_{R} have the following form:

τ_{L} =< A, B, C, τ_{L} ∩ τ_{R},… >; τ_{R} =< P, Q, R, τ_{L} ∩ τ_{R},… >

(A, B, C, P, Q, R ∈ σ are vertices, and

the points “…” can be omitted, iff ϱ(τ_{L}, τ_{R}) = 3).

For any vertex X ∈ τ_{•} we denote by (X) the adjacent face of τ_{•} that does not contain X. Using this language, the faces (A), (B), (C) are the only possible adjacencies of τ_{L} for ϱ(., τ_{R}) ≤ 2 (analog (P), (Q), (R) on the right-hand side).

Therefore, we define the following sets:

B_{L} := {τ_{L}; adjacencies of τ_{L} that contain A, B and C};
B_{R} := {τ_{R}; adjacencies of τ_{R} that contain P, Q and R}.

[figure: τ_{L}; τ_{R}; (A); (B); (C); (P); (Q); (R); B_{L}; B_{R}]

Claim: B_{L} and B_{R} admit the following properties:

1) #B_{L} = #B_{R} = n − 3.

2) There exists a bijection Φ : B_{L} \ {τ_{L}} →^{∼} B_{R} \ {τ_{R}} such that

(i) ϱ(τ, Φ(τ)) ≥ 2 for each τ ∈ B_{L} \ {τ_{L}}, and

(ii) ϱ(., .) ≥ 3 for the remaining pairs of B_{L} × B_{R}.

(In particular, ϱ(τ_{L}, B_{R}) ≥ 3 and ϱ(B_{L}, τ_{R}) ≥ 3.)

Proof: If ϱ(τ_{L}, τ_{R}) ≥ 4, then there exists at most one pair [τ_{1}, τ_{2}] ∈ B_{L} × B_{R} such that ϱ(τ_{1}, τ_{2}) ≤ 2. For ϱ(τ_{L}, τ_{R}) = 3, we can define Φ as Φ : (S) ↦ (S) for vertices S ∈ τ_{L} ∩ τ_{R}: Let τ_{1} ∈ B_{L}, let τ_{2} = (S) ∈ B_{R} \ {τ_{R}} (S is a vertex of τ_{L} ∩ τ_{R}). Then, A, B, C ∈ τ_{1}, but at most one of these vertices belongs to τ_{R}. Hence, ϱ(τ_{1}, τ_{2}) ≤ 2 implies that S cannot be contained in the face τ_{1}, too, i.e. τ_{1} = (S). □

(4.4) Proposition: Let σ be a cone such that all proper faces are simplicial ones. Assume that there exist (top dimensional) faces τ_{L} and τ_{R} such that ϱ(τ_{L}, τ_{R}) ≥ 3. Then, for 2 ≤ m ≤ n − 3, σ cannot meet the condition of theorem (2.5).

Proof: Otherwise we would be in the situation of (4.1) - (4.3). Then, we distinguish between two cases:

1st case: B_{L} ⊆ R At most 2m−(n−3) ≤ m elements of R can sit at the right-hand side of our picture in (4.3). Hence, the remaining (at least three) faces have to be good ones that admit all the same set R \B_{L} of R-neighbours. This fact contradicts to (5) of lemma (4.2).

2nd case: τ_{L}, τ_{R} are good faces.

(After renaming we can assume that τ_{L} is a good face. Because the first case cannot happen even for the right-hand side, there must be another good face τ ∈ B_{R}. Since ϱ(τ_{L}, τ) ≥ 3 we can rename the faces once more to obtain τ = τ_{R}.)

Case (2.1): B_{L} \ {τ_{L}} ⊆ R; B_{R} \ {τ_{R}} ⊆ R. Then, each good face at the left-hand side admits at least one R-neighbour at the right-hand side. (Otherwise, there would be a good face τ adjacent to τ_{L} such that τ and τ_{L} admit the same set of R-neighbours. Again by the lemma (4.2)(5), the faces τ, τ_{L} and all these R-neighbours would be adjacent to each other, and so would all other left-hand faces admitting at least one left-hand R-neighbour (cf. lemma (4.2)(4)). Hence, there had to exist a good face adjacent to τ_{L} that admits right-hand Rneighbours only, but this would imply m ≤ 1.)

Only m = n − 3 is still possible, and we can assume the following situation:

(A), (P) ∈ R, but (B), (C), (Q), (R) are good faces;
ϱ((A), (Q)) = ϱ((A), (R)) = ϱ((B), (P)) = ϱ((C), (P)) = 1.

This implies P ∈ (A); P ∉ (B), (C), hence, ϱ((A), (B)), ϱ((A), (C)) ≥ 2. Therefore, (B) and (C) have B_{L} \ {τ_{L}} as common set of R-neighbours, i.e. (B) and (C) have to be adjacent. We can apply lemma (4.2) again: ϱ(τ_{L}, (B)) = ϱ(τ_{L}, (C)) = ϱ((B), (C)) = ϱ((B), (P)) = ϱ((C), (P)) = 1 would imply ϱ(τ_{L}, (P)) = 1 but this means a contradiction.

Case (2.2): There are good faces τ_{L}, τ ∈ B_{L}; τ_{R} ∈ B_{R}. From τ_{L} and τ having the same R-neighbours we can derive the existence of a lefthand good face ψ that admits the same R-neighbours as τ_{R} (cf. the beginning of case (2.1)). In particular, m = 2 and these two right-hand R-neighbours have to be outside B_{R}.

Hence, B_{R} consists of good faces only. We have found n − 2 faces (B_{R} and ψ) admitting the same two R-neighbours, which is a contradiction because of lemma (4.2). □

(4.5) As a next step we will investigate the case 2 ≤ m = n − 2. In this situation, the property (ii) (cf. theorem (2.5)) of good faces implies

(ii)’ There exist not only m = n − 2 but m + 1 = n − 1 R-neighbours.

(ii)” If #R = 2m = 2n − 4, then there is an element s ∈ R (different from the n − 1 R-neighbours) that intersects the good face τ in an (n − 3)-dimensional edge, i.e. ϱ(τ, s) ≤ 2.

Proposition: Let σ be a cone such that all proper faces are simplicial ones. Assume that there exist (top dimensional) faces τ_{L} and τ_{R} such that ϱ(τ_{L}, τ_{R}) ≥ 3. Then, for 2 ≤ m = n − 2, σ cannot meet the condition of theorem (2.5).

Proof: As in the proof of the last proposition, we assume that σ would meet this condition. Obviously, τ_{L} and τ_{R} cannot be good faces simultaneously (each of them had to have n − 1 neighbours from a set with at most 2n − 4 elements). Hence, by the same arguments as in the proof of the previous proposition, we may assume B_{L} ⊆ R.

The right-hand side contains at most (2n − 4) − (n − 3) = n − 1 faces of R, i.e. at least one face τ is left to be good. On the other hand, more than or equal to two good faces at the right-hand side would admit the same set R \ B_{L} of R-neighbours, and this would be a contradiction because of lemma (4.2). Hence, the right-hand side consists exactly of τ and its n−1 R-neighbours. This fact implies τ = τ_{R}, and we obtain the contradiction by missing the announced element s ∈ R for the good face τ_{R} (cf. (ii)” at the beginning of (4.5)). □

(4.6) In the case m = 1 there are two positive examples. The set R = {r_{0}, r_{1}} consists of two top dimensional faces that have no common vertex:

(I) The octahedron (m = 1; n = 4)

[figure: r_{0}; r_{1}; (cf. (5.1)).; ^{r_{0}  r_{1}  P_{σ}}]

(II) The tent (m = 1; n = 4)

[figure: r_{0}; r_{1}; (cf. (5.2)).; ^{r_{0}  r_{1} P_{σ}}]

Proposition: Let σ be a cone such that all proper faces are simplicial ones. Assume that there exist (top dimensional) faces τ_{L} and τ_{R} such that ϱ(τ_{L}, τ_{R}) ≥ 3. Then, for 1 = m ≤ n − 2, the above two examples are the only ones such that σ can meet the condition of theorem (2.5).

Proof: Each face of σ has to be adjacent either to r_{0} or r_{1}. In particular, σ consists of exactly 2n faces, and r_{0}, r_{1} are two of them. By lemma (4.2)(1) we obtain n ≥ 4.

Claim: τ_{R} is the only face of σ such that ϱ(τ_{L}, .) ≥ 3. (Let τ < σ such that ϱ(τ_{L}, τ) ≥ 3. Then, the set of all top dimensional faces of σ can be written as a disjoint union in two different ways:

{τ_{L} and its neighbours} ⊥⊥ {τ_{R} and its neighbours} = {faces of σ} =
= {τ_{L} and its neighbours} ⊥⊥ {τ and its neighbours}.

Hence, {τ_{L} and its neighbours} = {τ and its neighbours}, and this implies τ_{L} = τ.)

In particular, the set B_{R} consists of the single element τ_{R} only. That means n = 4, and we obtain the following picture:

[figure: τ_{L}=<A,B,C>; τ_{R}=<P,Q,R>; (C); (B); (A); (R); (Q); (P)]

The two examples described above arise from the following two cases:

(I) The sets {(A), (B), (C)} and {(P), (Q), (R)} both do not contain any pair of adjacent faces.

(II) ϱ((A), (B)) = 1. □

(4.7) Lemma: Let σ be an n-dimensional cone (n ≥ 4) such that all proper faces are simplicial ones. Then, the following three statements are equivalent:

(1) ϱ(., .) ≤ 2;

(2) σ admits at most n + 1 vertices;

(3) σ^{∨} is equal to the cone over Δ^{k} × Δ^{n−k−1} for a suitable k = 0,…, [\frac{n−1}{2}]. (Δ^{i} denotes the i-dimensional, compact simplex.)

Remark: The equivalence of (2) and (3) is, of course, well known (cf. [Gr]).

Proof of the Lemma: Assume ϱ(., .) ≤ 2. We will deal with the compact polyhedron P_{σ} instead of σ (cf. (4.1)) and have to introduce some notations: choose a top dimensional face I that will be regarded as basic face of P_{σ}. The vertices of I will be represented by the integers from 1 to n − 1: I =< 1,…, n − 1 >. Let A = {a^{1},…, a^{N}} be the set of the remaining vertices (not contained in I) of the polyhedron P_{σ}.

Then, the faces of P_{σ} split into only two different types: type 1: [i, a(i)] :=< 1,…, î,…, n − 1; a(i) ∈ A >,

and a : {1,…, n − 1} → A becomes a map which is constant iff σ is equal to a simplex.

type 2: [i, j; a, b] :=< 1,…, î,…, ĵ,…, n − 1; a, b ∈ A >, but the elements a, b ∈ A are not uniquely determined by giving the pair [i, j]. It is even possible that there does not exist any face of this type for given i, j.

For each pair i, j ∈ I we define a non-oriented graph G(i, j) with its set of vertices contained in A as follows:

\overline{ab} is an edge in G(i, j), iff [i, j; a, b] is a face of σ (a, b ∈ A).

We obtain the following properties:

(i) If a(i) = a(j), then there is no face of the form [i, j; a(i), .].

(ii) If a(i) ≠ a(j), then there exists exactly one element b ∈ A such that [i, j; a(i), b] is a face of σ.

(iii) Let a ≠ a(i), a(j). Then, the number of faces [i, j; a, .] of σ is either 0 or 2.

Consequences for the graph G(i, j): G(i, j) is the disjoint union of

a) a path from a(i) to a(j) that does not contain any loop (in particular, it is empty if a(i) = a(j)) and

b) possibly some loops that do not contain a(i), a(j).

(iv) Let i, j, k ∈ I; a, b ∈ A. Then, the number of σ-faces among [i, j; a, b], [j, k; a, b] and [i, k; a, b] is either 0 or 2. [That means that the edge \overline{ab} appears in either 0 or 2 of the graphs G(i, j), G(j, k), or G(i, k).]

(v) Let i, j, k ∈ I be three different elements. Then, for each face [i, j; a, b], either a = a(k) or b = a(k) has to be true. [The graph G(i, j) is star shaped with a(k) as its centre.]

(vi) Let i, j, k ∈ I be three different elements. If [i, j; a, b] and [j, k; c, d] are faces of σ, then the sets {a, b} and {c, d} have at least one common element. [Any two edges of G(i, j) and G(j, k), respectively, have at least one common vertex.]

(vii) Let i, j, k, ℓ ∈ I be four different elements. If [i, j; a, b] and [k, ℓ; c, d] are faces of σ, then the sets {a, b} and {c, d} are equal. [If G(i, j), G(k, ℓ) are not empty, both graphs coincide and consist of a single edge only.] (viii) If σ is not a simplex, then for each element a ∈ A there exists a face of the

form [., .; a, .].

[Each element of the set A appears as a vertex in some graph G(i, j).]

The proofs of (i) - (viii) are easy; they result from three basic ideas: regard a suitable (n − 2)-dimensional face of σ (i.e. a (n − 3)-dimensional face of P_{σ}) and use the fact that it is always the intersection of exactly two top dimensional faces ((i) - (iv)).

(v) - (vii): If the claim would not be true, the corresponding faces would differ in more than two vertices (regard [k, a(k)] in (v)). In (viii), the dual statement is trivial: Let P ∈ σ be a vertex that is not contained in a top dimensional face F. Then, there exists another top dimensional face F̃ not containing P such that F and F̃ have contact of codimension 1.

Now, we can use the properties (i) - (iii) and (v) to obtain the following description of the graphs G(i, j):

1) a(i) = a(j) iff G(i, j) = ∅.

2) If a(i) ≠ a(j), then there are two possibilities:

a) G(i, j) = \{edge \overline{a(i)a(j)}\} and a(k) ∈ \{a(i), a(j)\} for each k ≠ i, j, or

b) G(i, j) = \{edges \overline{a(i)b}, \overline{a(j)b}\} and a(k) = b for each k ≠ i, j.

In particular, if σ is not a simplex, N = #A can be equal to 2 or 3 only; we will consider these two cases separately:

Case N = 3: Each non-empty graph G(i, j) has to be of the form that is described in (b). Hence, the map a : {1,…, n − 1} → A is surjective, and we can assume the following situation:

n = 4; A = {a, b, c};
a(1) = a, a(2) = b, a(3) = c;
G(1, 2) = \{\overline{ac}, \overline{bc}\}, G(1, 3) = \{\overline{ab}, \overline{bc}\}, G(2, 3) = \{\overline{ab}, \overline{ac}\}.

On the other hand, such a polyhedron P_{σ} cannot exist by the Euler polyhedron formulae (notice the equation 2·#(edges) = 3·#(faces) because of P_{σ} having simplicial faces only).

Case N = 2: We change our notation slightly: Let A = \{A_{0}, B_{0}\} and I = \{A_{1},…, A_{k}; B_{1},…, B_{n−k−1}\} (1 ≤ k ≤ [\frac{n−1}{2}] ) such that a(A_{i}) = A_{0} and a(B_{j}) = B_{0}.

Then, we obtain the following graphs:

G(A_{i}, A_{j}) = G(B_{i}, B_{j}) = ∅;
G(A_{i}, B_{j}) = \{edge \overline{A_{0}B_{0}}\}.

In particular, P_{σ} consists of the top dimensional faces

S_{ij} :=< A_{0},…, Â_{i},…, A_{k}; B_{0},…, B̂_{j},…, B_{n−k−1} >
(i = 0,…, k; j = 0,…, n − k − 1)

only, i.e. P_{σ} is the polyhedron that is dual to Δ^{k} × Δ^{n−k−1}. □

(4.8) We are looking for cones σ that are able to meet the condition of theorem (2.5). By the previous lemma, only the investigation of the polyhedra P_{σ} = (Δ^{k} × Δ^{n−k−1})^{∨} (k = 0,…, [\frac{n−1}{2}]) is left. (The case n = 3 (and m = 1) is integrated, since only triangles and quadrangles are candidates for squeezing.)

Cones over (Δ^{k} × Δ^{n−k−1})^{∨} have the following form:

σ = < a^{0},…, a^{k}; b^{0},…, b^{n−k−1} >;
(ν, μ) := < a^{0},…, â^{ν},…, a^{k}; b^{0},…, b̂^{μ},…, b^{n−k−1} > are the top dimensional
faces of σ (ν = 0,…, k; μ = 0,…, n − k − 1).

These faces are illustrated as vertices of a lattice:

[figure: k; k-1; 2; 1; 0; ⋮; 0; 1; 2; 3; n-k-1; …; ^{...}]

Then, ϱ((ν, μ); (ν̃, μ̃)) = 1 means that the points (ν, μ) and (ν̃, μ̃) sit in a common line (either row or column).

Lemma: Let σ meet the condition of theorem (2.5). If r denotes the cardinality of R = {(r_{0}^{1})^{⊥},…, (r_{0}^{m})^{⊥}; (r_{1}^{1})^{⊥},…, (r_{1}^{m})^{⊥}} (cf. (2.5)), then the following inequality is true:

(n + m + 1)r ≥ (k + 1)(n − k)m + max \{r, \frac{r^{2}}{n + 1}\} + max \{r, \frac{r^{2}}{n − k} \}.

(If R does not contain top dimensional faces only, then correct the definition of r as follows: replace all faces of dimension less or equal than n − 2 by one of the enclosed (n − 1)-dimensional ones.)

Proof: We may assume that R consists of top dimensional faces only. (Otherwise, the changes suggested above have to be done, but the conditions (i) and (ii) of (2.5) still keep true.) Now, we look at the “face lattice” drawn above and count the number of R-faces sitting in each row and column, respectively:

s_{ν} := #(R ∩ νth line)  (ν = 0,…, k);
t_{μ} := #(R ∩ μth column) (μ = 0,…, n − k − 1).

These two numbers admit the following properties:

1) ∑_{ν=0}^{k} s_{ν} = r; ∑_{μ=0}^{n−k−1} t_{μ} = r.

2) If (ν, μ) ∉ R, then s_{ν} + t_{μ} ≥ m (by (ii) of (2.5)).

3) ∑_{(ν,μ)∈R} s_{ν} = ∑_{ν=0}^{k}(#{(ν, .)/(ν, .) ∈ R} · s_{ν}) = ∑_{ν=0}^{k} s_{ν}^{2};

∑_{(ν,μ)∈R} t_{μ} = ∑_{μ=0}^{n−k−1} t_{μ}^{2}.

Hence, we obtain

(n + 1)r = (n − k) ∑_{ν=0}^{k} s_{ν} + (k + 1) ∑_{μ=0}^{n−k−1} t_{μ} = ∑_{(ν,μ)}(s_{ν} + t_{μ}) =
= ∑_{(ν,μ)∉R}(s_{ν} + t_{μ}) + ∑_{(ν,μ)∈R}(s_{ν} + t_{μ})

≥ [(k + 1)(n − k) − r]m + ∑_{ν=0}^{k} s_{ν}^{2} + ∑_{μ=0}^{n−k−1} t_{μ}^{2}.

Finally, ∑_{ν} s_{ν}^{2} (and ∑_{μ} t_{μ}^{2}) can be estimated in two different ways:

a) s_{ν}^{2} ≥ s_{ν} provides ∑_{ν} s_{ν}^{2} ≥ ∑_{ν} s_{ν} = r, and

b) \sqrt{\frac{1}{k+1} ∑_{ν} s_{ν}^{2}} ≥ \frac{1}{k+1} ∑_{ν} s_{ν} implies ∑_{ν} s_{ν}^{2} ≥ \frac{r^{2}}{k+1}. □

Remark:

1) The inequality of the lemma could be improved by finding a better estimation

for ( ∑_{ν} s_{ν}^{2} + ∑_{μ} t_{μ}^{2}).

2) If (n, k, m, r) satisfies not only the inequality but also the sharp version (n + m + 1)r = (k + 1)(n − k)m + \frac{r^{2}}{k+1} + \frac{r^{2}}{n−k}, then the values of s_{0},…, s_{k} and t_{0},…, t_{n−k−1} have to be equal, respectively. In particular, r has to be divisible by k + 1 and n − k.

3) In the case m = n − 2 we are able to use (ii)’ of (4.5) instead of (ii) of (2.5). In particular, m can be replaced by m+1, and we obtain the following inequality:

2nr ≥ (k + 1)(n − k)(n − 1) + max \{r, \frac{r^{2}}{k + 1}\} + max \{r, \frac{r^{2}}{n − k} \}.

Corollary: Only few possiblities for σ remain:

a) k ≤ 1, or

b) k = 2; n = 5, 6, 7, 8, or

c) k = 3; n = 7.

Proof: Denote q := (k + 1)(n − k). Then, k ≥ 2 implies q > r, and we obtain the following inequalities:

(n + 1)r ≥ (q − r)m + \frac{r^{2}}{k + 1} + \frac{r^{2}}{n − k} ≥ (q − r)\frac{r}{2} + \frac{r^{2}}{n + 1} + \frac{r^{2}}{n − k} ,
2q(n + 1)r ≥ q(q − r)r + 2((n − k) + (k + 1))r^{2},
2q(n + 1) ≥ q(q − r) + 2(n + 1)r,
2(n + 1)(q − r) ≥ q(q − r), and this means
2(n + 1) ≥ q.

Therefore, the inequalities q ≥ 3(n − 2) (for k ≥ 2) and q ≥ 4(n − 3) (for k ≥ 3) provide the restrictions n ≤ 8 and n ≤ 7, respectively. □

(4.9) The investigation of the remaining cases has to be more careful now. We still assume that R consists of top dimensional faces only and recall the essential part of theorem (2.5) (applied to the special case P_{σ} = (Δ^{k} × Δ^{n−k−1})^{∨}):

1) R consists of m pairs of faces of σ (but these pairs need not to be disjoint ones).

2) Given a good face τ < σ (i.e. τ ∉ R) it is possible to introduce an order of both, the set of pairs in R on the one hand, and the elements of each pair on the other hand: R consists of (r_{0}^{1}, r_{1}^{1}),…, (r_{0}^{m}, r_{1}^{m}). These orders (that depend on the choice of τ) admit the following properties:

a) The faces r_{0}^{1},…, r_{0}^{m} correspond to m distinct fat points sitting in the same line (row or column) as τ (cf. the picture at the beginning of (4.8)).

b) For i = 1,…, m, the line that contains r_{0}^{i} but not τ does not pass through any of the fat points r_{1}^{i}, r_{1}^{i+1},…, r_{1}^{m}.

Case k ≥ 2: 13 quadruples (n, k, m, r) (such that k ≤ [\frac{n−1}{2}], 1 ≤ m ≤ n − 2 and m + 1 ≤ r ≤ 2m) are left by the inequality of the previous lemma ((2) and (3) of the added remark included):

(5, 2, 1, 2); (5, 2, 2, 3);  • • •
(5, 2, 2, 4); (5, 2, 3, 6) • • • (P_{σ} = (Δ^{2} × Δ^{2})^{∨})
• • •
(6, 2, 1, 2); (6, 2, 2, 4);  • • • •
(6, 2, 3, 6) • • • • (P_{σ} = (Δ^{2} × Δ^{3})^{∨})
• • • •
(7, 2, 2, 4); (7, 2, 3, 6); • • • • •
(7, 2, 4, 8)  • • • • • (P_{σ} = (Δ^{2} × Δ^{4})^{∨})
• • • • •
(8, 2, 3, 6)  • • • • • •
• • • • • • (P_{σ} = (Δ^{2} × Δ^{5})^{∨})
• • • • • •
(7, 3, 1, 2); (7, 3, 2, 4)  • • • •
• • • •  (P_{σ} = (Δ^{3} × Δ^{3})^{∨}).
• • • •
• • • •

After checking the above conditions (1), (2a) and (2b) in each of these 13 cases, the only quadruple that survives is (5, 2, 2, 3). Hence, we are able to continue our positive list started in (4.6):

(III) The super triangle P_{σ} = (Δ^{2} × Δ^{2})^{∨} (n = 2, n = 5)

[figure]

r_{0}^{1}

r_{1}^{1}=r_{0}^{2}

r_{1}^{2}

Squeezing of the corresponding cone σ^{∨} means the contraction of the triangle that is spanned by the three R-elements to one point (cf. (5.3)).

Case k = 1: P_{σ} = (Δ^{1} × Δ^{n−2})^{∨} admits the following lattice of top dimensional faces:

• • • … • • • • … •

_{n−1 columns}

Applying the conditions (1), (2a) and (2b) to this special situation, we obtain:

- There exist good faces in each of the two lines.

- Each line contains exactly one element of each R-pair. We number these pairs and denote their elements by r^{i} and s^{i} (sitting in the upper or lower line, respectively).

- r = 2m.

(If this would not be the case, we would be able to assume r^{1} = r^{2}. Then, the condition (2a) would imply that there had to be an arrangement of faces of

the form ( \begin{matrix}x & r^{1} = r^{2} & x \\ s^{1} & ? & s^{2} \end{matrix} ) (“x” denotes a good face), but this contradicts

(2b).)

- If we would draw edges between both elements of each R-pair on the one hand and between any two elements of R that are contained in the same column on the other hand, then no loops are allowed to arise.

(That means, constellations as [figure: s^{2}; s^{3}; s^{1}; r^{1}; r^{2}; r^{3}] or \begin{matrix}r^{1} \\ ∥ \\ s^{1} \end{matrix} are forbidden.)

These conditions lead to the following class of squeezable cones:

(IV)_{P_{1},...,P_{ℓ}} The double simplex

Pσ^{∨} = (B_{p_{1}},…, B_{p_{ℓ}}) is built up from blocks of the form

B_{p} := ([figure: s^{1}; s^{2}; s^{3}; s^{p}; x; x; r^{1}; r^{2}; r^{p−1}; r^{p}; …]) (p ≥ 0; “x” denotes a good face).

Therefore, ∑_{i=1}^{ℓ} p_{i} = m and m + ℓ = ∑_{i=1}^{ℓ}(p_{i} + 1) = n − 1; also we can asume

that 0 ≤ p_{1} ≤ ⋯ ≤ p_{ℓ} is true.

(4.10) We still regard the polyhedron P_{σ} = (Δ^{k} × Δ^{n−k−1})^{∨}. But now, we want to omit the restriction that R is allowed to contain top dimensional faces only.

If K ∈ R is an (n − 2)-dimensional face of a squeezable cone σ (and we also will assume that it is the only member of R being not of dimension n − 1), then, in (4.8) and (4.9) K has been replaced by one of the two top dimensional faces containing this edge: R changes to R′, and (σ; R′) remains squeezable (at least in the sense of (2.5)). Let r^{i} ∈ R′ be the face that arises from K ∈ R. Then, (σ, R′) must belong to one of the two classes (III) or (IV)_{p_{1},...,p_{ℓ}}, and the following additional properties have to be satisfied:

- Replacing r^{i} by the other top dimensional face that contains K, the result still has to belong to the classes (III) or (IV), respectively.

- There exists almost one good face τ of (σ; R′) that requires r^{i} to be its Rneighbour of the ith pair. Then, K has to be the edge connecting both faces, τ and r^{i}.

- Let α be a good face such that K has to play the role of r_{1}^{i} in (σ; R). Then, in (σ; R′) not only r^{i} but also τ satisfies the condition for r_{1}^{i} described in (2b) of (4.9).

If R contains more than one (n − 2)-dimensional face we can proceed inductively: Step by step (in an arbitrary order) these faces can be changed into top dimensional ones - the whole system always keeps squeezable.

Having all this in mind, the following two additional classes of squeezable cones will be obtained:

(IV)_{p,q}^{1} The double simplex (with a single edge)

Pσ^{∨} = ( \begin{matrix}x \\ s^{1} \end{matrix} | B_{p} | B_{q}), and r^{1} ∈ R equals the (n − 2)-dimensional edge

between the two upper good faces of B_{p} and B_{q}, respectively. (p, q ≥ 0; p + q = m − 1 = n − 4. We also assume p ≤ q.)

(IV)_{p,q}^{2} The double simplex (with two edges)

Pσ^{∨} = ( \begin{matrix}x \\ s^{1} \end{matrix} | B_{p} | B_{q} | \begin{matrix}r^{m} \\ x \end{matrix} ), and r^{1} ∈ R (s^{m} ∈ R) equals the (n − 2)-

dimensional edge between the two upper (lower) good faces of B_{p} and B_{q}, respectively. (p, q ≥ 0; p + q = m − 2 = n − 5. We also assume p ≤ q.)

Remark: If R contains an edge ℓ of dimension less than n − 2, then the condition of theorem (2.5) cannot be satisfied: Otherwise we could replace ℓ by an enclosed edge of dimension n − 2, and the whole set up would “factorize” through (IV)_{p,q}^{1}. But, in this class, the edge r^{1} is needed to be the R-neighbour of two good faces, i.e. it can not be replaced by a smaller one.

(4.11) In the last two sections we have regarded the polyhedra P_{σ} = (Δ^{k} × Δ^{n−k−1})^{∨} with k ≥ 1. Therefore, only one class of cones remains to be mentioned:

(V) The simplex

σ < a^{1},…, a^{n} >; 1 ≤ m ≤ n − 2.

Remark: As in the previous cases it remains to classify the possible constellations of R-elements in this class also. But, because of the absence of sufficient good faces, to carry out this task seems to be very stupid and will be omitted here.

(4.12) In this chapter we have proved the following statement: let i : Yσ̄ ↪ X_{σ} be a closed embedding of two affine torus varieties that is a relative complete intersection (cf. chapter 2). If, moreover, Yσ̄ \ {0} is smooth and X_{σ} \ {0} admits at most cyclic quotient singularities (i.e. the cone σ consists of simplicial faces only), then for σ together with the equations x^{r_{0}^{i}} −x^{r_{1}^{i}} (i = 1,…, m; r_{ν}^{i} ∈σ^{∨} ∩M) there are only the possibilities (I) - (V) described above.

In the next chapter we will complete our work for the classes (I) - (III): all squeezable cones will be classified not only as abstract cones, but up to automorphisms of the lattice N. Moreover, we will characterize those (3-dimensional) isolated toric singularities that can appear in this way. The classes (IV) and (V) consist of the simplest cones one can think about - but there are many different cases for the arrangement of the R-neighbours belonging to each good face. We omit further investigations here (cf. chapter 6 for the special cases m = n − 2; n = 3, 4), but it seems to be quite interesting to keep these cases in mind: for instance, the whole series of cones over scrolls (given by the determinantic equations

det ( \begin{matrix}x_{10} & x_{11} & & x_{1 k_{1}−1} & x_{20} & x_{21} & & x_{2 k_{2}−1} & & x_{ℓ k_{ℓ}−1} \\ & & … & & & & … & & … \\ x_{11} & x_{12} & & x_{1 k_{1}} & x_{21} & x_{22} & & x_{2 k_{2}} & & x_{ℓ k_{ℓ}} \end{matrix} ) = 0 )

is hidden in class (IV).

5 The hexagon singularities

The classes (I) - (III) of squeezable cones (cf. (4.6) and (4.9)) provide 3-dimensional toric singularities that are given by cones over a hexagon. Now, by using theorem (2.4) we will describe these cases in detail:

(5.1) Class (I): The octahedron

A  a  b B Y r_{1} r_{0}  P_{σ} X

< A, B, X > is a good face. In particular, it is smooth, and we can assume

A = (1, 0, 0, 0); B = (1, 1, 0, 0); X = (1, 0, 1, 0).

Hence, < A, r_{0} >=< X, r_{0} >= 0 and < B, r_{0} >= 1 imply r_{0} = [0, 1, 0, ∗], and we can choose our coordinates of N in such a way that r_{0} = [0, 1, 0, 0].

Now, by the properties of good faces or simply by incidence relations only, we obtain the following equations:

< a, r_{0} > = < B, r_{1} > = < b, r_{1} > = < Y, r_{1} > = 0;
< b, r_{0} > = < Y, r_{0} > = < A, r_{1} > = < a, r_{1} > = < X, r_{1} > = 1.

In particular, that means r_{1} = [1, −1, 0, ∗]. We can use the remaining feasibilities of choosing the coordinates well to kill the last component of r_{1}; then, we obtain the result

\begin{matrix}\begin{matrix}A = (1, 0, 0, 0) \\ a = (1, 0, a_{3}, a_{4}) \end{matrix} & X = (1, 0, 1, 0); & r_{0} = [0, 1, 0, 0] \\ \begin{matrix}B = (1, 1, 0, 0) \\ b = (1, 1, b_{3}, b_{4}) \end{matrix} & Y = (1, 1, Y_{3}, Y_{4}); & r_{1} = [1, −1, 0.0]. \end{matrix}

On the other hand, for P_{σ} to be a convex body of the form drawn above, some 4 × 4 determinants have to be positive. This provides conditions for our parameters a_{3}, a_{4}, b_{3}, b_{4}, Y_{3}, Y_{4} ∈ ZZ:

(5.1.1) \begin{matrix}a_{4}, b_{4}, Y_{4} > 0, \\ | \begin{matrix}a_{3} & a_{4} \\ Y_{3} & Y_{4} \end{matrix} | , | \begin{matrix}b_{3} & b_{4} \\ Y_{3} & Y_{4} \end{matrix} | > 0, \\ | \begin{matrix}b_{3} − Y_{3} & b_{4} − Y_{4} \\ a_{3} & a_{4} \end{matrix} | , | \begin{matrix}b_{3} − Y_{3} & b_{4} − Y_{4} \\ a_{3} − 1 & a_{4} \end{matrix} | , | \begin{matrix}b_{3} & b_{4} \\ a_{3} − 1 & a_{4} \end{matrix} | > 0. \end{matrix}

The smoothness of the good faces (only < A, B, X > was already used) yields

(5.1.2) \begin{matrix}gcd(a_{3}, a_{4}) = gcd(a_{3} − 1, a_{4}) = gcd(b_{3}, b_{4}) = 1; \\ gcd(Y_{3}, Y_{4}) = gcd(b_{3} − Y_{3}, b_{4} − Y_{4}) = 1. \end{matrix}

Remark:

1) The vertices of σ together sit inside the affine hyperplane [a_{1} = 1] of N ⊗ IR. That means that the corresponding toric variety X_{σ} is Gorenstein.

2) X_{σ} has an isolated singularity in 0, iff a_{4} = | \begin{matrix}b_{3} & b_{4} \\ Y_{3} & Y_{4} \end{matrix} | = 1.

(Then, this even implies all gcd-conditions.)

Now, we are ready to determine the special fibre Yσ̄ of the map (x^{r_{0}} −x^{r_{1}}) : X_{σ} → I C^{1}. We compute the “squeezed” cone σ̄ = σ ∩ [r_{0} − r_{1}]^{⊥} : r_{0} −r_{1} = [−1, 2, 0, 0], i.e. the linear subspace [r_{0} −r_{1}]^{⊥} ⊆ N is given by the equation a_{1} = 2a_{2}. The points of intersection of this subspace with the six edges \overline{AB}, \overline{BX}, \overline{Xb}, \overline{ba}, \overline{aY} and \overline{Y A}, respectively, are

(2, 1, 0, 0); (2, 1, 1, 0); (2, 1; b_{3} + 1, b_{4}); (2, 1, a_{3} + b_{3}, a_{4} + b_{4});
(2, 1, a_{3} + Y_{3}, a_{4} + Y_{4}), and (2, 1, Y_{3}, Y_{4}).

Taking (2, 1, 0, 0); (0, 0, 1, 0); (0, 0, 0, 1) as a ZZ-basis of [r_{0} − r_{1}]^{⊥}, we can omit the first component “2” in each point. Then, \overline{σ} can be seen to be the cone over the hexagon

H_{I} :=< (0, 0); (1, 0); (b_{3} + 1, b_{4}); (a_{3} + b_{3}, a_{4} + b_{4}); (a_{3} + Y_{3}, a_{4} + Y_{4}); (Y_{3}, Y_{4}) >

that sits in the affine hyperplane [ā_{1} = 1] (ā_{1}, ā_{2}, ā_{3} denote the coordinates of [r_{0} − r_{1}]^{⊥}).

[figure: (1,0); Y⃗; a⃗; b⃗−Y⃗; a⃗−(1,0); b⃗; P_{0}=(0,0); P_{1}=(1,0); P_{2}=(b_{3}+1,b_{4}); P_{3}=(a_{3}+b_{3},a_{4}+b_{4}); P_{4}=(a_{3}+Y_{3},a_{4}+Y_{4}); P_{5}=(Y_{3},Y_{4})] Remark:

3) Yσ̄ is Gorenstein, too.

4) The conditions of (5.1.1) express exactly the fact that H_{I} is a convex, nondegenerate hexagon of the form drawn above.

5) The conditions of (5.1.2) express that the (oriented) edges of H_{I} consist of primitive vectors only. But this fact is equivalent to (Yσ̄, 0) having at most an isolated singularity.

6) P_{0} + P_{2} + P_{4} = P_{1} + P_{3} + P_{5}, i.e. the centres of both triangles, ΔP_{0}P_{2}P_{4} and ΔP_{1}P_{3}P_{5}, coincide. This condition is not only necessary, but also sufficient (up to ZZ-isomorphisms) for H_{I} to have those special vertices as mentioned above.

Therefore, it is exactly the isolated Gorenstein hexagon singularities Y with P_{0} + P_{2} + P_{4} = P_{1} + P_{3} + P_{5} that admit toric deformations of type (I). The corresponding total space X_{σ} is uniquely determined by Y.

[figure: P_{σ}^{∨} and P ∨ ; σ̄]

(The view of the dual situation illustrates the notion of squeezing: The hexagon results by pressing the cube into dimension 2.)

(5.2) Class (II): The tent

P_{σ} A B  X  a  b  Y r_{1} r_{0}

We proceed as in the last section; for shortness we will only list the results in the order of arising:

(i) The good face < A, B, X > together with a good choice of coordinates yields A = (1, 0, 0, 0); B = (1, 1, 0, 0); X = (s, 0, 1, 0) and r_{0} = [0, 1, 0, 0]. (s ∈ ZZ is a parameter which keeps for us the possibility to change the first component of X if it is necessary.)

(ii) The equations

< a, r_{0} > = < B, r_{1} > = < b, r_{1} > = < y, r_{1} > = 0;
< b, r_{0} > = < A, r_{1} > = < a, r_{1} > = 1

provide r_{1} = [1, −1, 0, 0] (the last two components vanish after improving the coordinates again - the parameter s of (i) is fixed now). We obtain the following result:

\begin{matrix}\begin{matrix}A = (1, 0, 0, 0) \\ a = (1, 0, a_{2}, a_{4}) \end{matrix} & X = (s, 0, 1, 0); & r_{0} = [0, 1, 0, 0] \\ \begin{matrix}B = (1, 1, 0, 0) \\ b = (1, 1, b_{3}, b_{4}) \end{matrix} & Y = (t, t, Y_{3}, Y_{4}); & r_{1} = [1, −1, 0, 0]. \end{matrix}

(iii) Conditions for the parameters s, t, a_{3}, a_{4}, b_{3}, b_{4}, Y_{3}, Y_{4} ∈ ZZ:

(5.2.1) \begin{matrix}s, t, a_{4}, b_{4}, Y_{4} > 0; \\ | \begin{matrix}a_{3} & a_{4} \\ b_{3} & b_{4} \end{matrix} | , | \begin{matrix}b_{3} & b_{4} \\ Y_{3} & Y_{4} \end{matrix} | , | \begin{matrix}a_{3} & a_{4} \\ Y_{3} & Y_{4} \end{matrix} | > 0; \\ s | \begin{matrix}a_{3} & a_{4} \\ Y_{3} & Y_{4} \end{matrix} | > Y_{4}; t | \begin{matrix}a_{3} & a_{4} \\ b_{3} & b_{4} \end{matrix} | > | \begin{matrix}a_{3} & a_{4} \\ Y_{3} & Y_{4} \end{matrix} |; tb_{4} > Y_{4}. \end{matrix}

(5.2.2) \begin{matrix}gcd(a_{3}, a_{4}) = gcd(b_{3}, b_{4}) = gcd(Y_{3}, Y_{4}) = 1; \\ gcd(sa_{3} − 1, a_{4}) = gcd(Y_{3} − tb_{3}, Y_{4} − tb_{4}) = 1. \end{matrix}

Remark:

1) X_{σ} is Gorenstein, iff X_{σ} is I Q-Gorenstein, iff s = t = 1.

2) X_{σ} has an isolated singularity in 0, iff a = | \begin{matrix}b_{3} & b_{4} \\ Y_{3} & Y_{4} \end{matrix} | = 1.

(Then, this even implies all gcd-conditions.)

(iv) The 3-dimensional cone σ̄ = σ ∩ [r_{0} − r_{1}]^{⊥} is spanned by the vertices

(1, 0, 0); (s, 1, 0); (1, a_{3}, a_{4}); (1, a_{3} + b_{3}, a_{4} + b_{4}); (1, b_{3} + b_{4}); (t, Y_{3}, Y_{4}),

which arise as points of intersection of [r_{0} − r_{1}]^{⊥} with the edges \overline{AB}, \overline{BX}, \overline{Ba}, \overline{ab}, \overline{bA} and \overline{Y A}, respectively. That means, σ̄ is the cone over the hexagon

H_{II} :=< (0, 0); \frac{1}{s}(1, 0); (a_{3}, a_{4}); (a_{3} + b_{3}, a_{4} + b_{4}); (b_{3}, b_{4}); \frac{1}{t}(Y_{3}, Y_{4}) >

that sits in the affine hyperplane [ā_{1} = 1]. Attention: The points \frac{1}{s}(1, 0) and \frac{1}{t}(Y_{3}, Y_{4}) belong to the lattice of [r_{0} − r_{1}]^{⊥}, iff s = t = 1!

[figure: a⃗; b⃗; P_{0}=(0,0); P_{1}= \frac{1}{s} (1,0); P_{5}= \frac{1}{t} (Y_{3},Y_{4}); P_{4}=(b_{3},b_{4}); P_{3}=(a_{3}+b_{3},a_{4}+b_{4}); P_{2}=(a_{3},a_{4})] Remark:

3) Yσ̄ is Gorenstein, iff it is I Q-Gorenstein, iff s = t = 1.

4) The conditions of (5.2.1) express exactly the fact that H_{II} is a convex, nondegenerate hexagon of the form drawn above.

5) The conditions of (5.2.2) express that (Yσ̄, 0) has, at most, an isolated singularity.

6) P_{0} + P_{3} = P_{2} + P_{4}, i.e. the vertices P_{0}, P_{2}, P_{3}, P_{4} form a parallelogram inside H_{II}. This condition is not only necessary, but also sufficient (up to ZZ-isomorpisms) for H_{II} to have those special vertices as mentioned above.

Therefore, the isolated singularities Y that admit toric deformations of type (II) are exactly those hexagon singularities such that P_{0}, P_{2}, P_{3}, P_{4} form an (isolated) Gorenstein parallelogram singularity. The corresponding total space X_{σ} is uniquely determined by Y.

(5.3) Class (III): The super triangle

[figure: r_{0}^{1}; r_{1}^{1}=r_{0}^{2}; r_{1}^{2}; (a^{2}); (a^{1}); (a^{0}); (b^{0}); (b^{1}); (b^{2}); Pσ^{∨} = Δ^{2} × Δ^{2}]

The cone σ =< a^{0}, a^{1}, a^{2}; b^{0}, b^{1}, b^{2} > admits 9 top dimensional faces (ν, μ) :=< a^{0},…, â^{ν},… a^{2}; b^{0},…, b̂^{μ},…, b^{2} >; each face is represented by a fat point in the (3 × 3)-lattice drawn above.

In particular,

r_{0}^{1} = (0, 0) = < a^{1}, a^{2}, b^{1}, b^{2} >,
r := r_{1}^{1} = r_{0}^{2} = (1, 1) = < a^{0}, a^{2}, b^{0}, b^{2} >, and
r_{1}^{2} = (2, 2) = < a^{0}, a^{1}, b^{0}, b^{1} > .

Now, we can proceed as usual (kill the last components of r_{0}^{1}, r and r_{1}^{2}, respectively, by choosing the coordinates of N well):

(i) The property for (0, 2) =< a^{1}, a^{2}, b^{0}, b^{1} > to be a good face provides

a^{1} = (1, 1, 0, 0, 0); a^{2} = (1, 0, 1, 0, 0); b^{0} = (1, 0, 0, 0, 0); b^{1} = (1, 1, 0, 1, 0)

and r_{0}^{1} = [1, −1, −1, 0, 0]; r_{1}^{2} = [0, 0, 1, 0, 0].

(ii) Again, there are many equations:

< b^{2}, r_{0}^{1} > = < a^{0}, r > = < a^{2}, r > = < b^{0}, r > = < b^{2}, r > = < a^{0}, r_{1}^{2} > = 0;
< a^{0}, r_{0}^{1} > = < a^{1}, r > = < b^{1}, r > = < b^{2}, r_{1}^{2} > = 1.

Hence, we obtain the following result:

\begin{matrix}a^{0} = (1, 0, 0; a_{4}, a_{5}) \\ a^{1} = (1, 1, 0; 0, 0) \\ a^{2} = (1, 0, 1; 0, 0) \\ b^{0} = (1, 0, 0; 0, 0) \\ b^{1} = (1, 1, 0; 1, 0) \\ b^{2} = (1, 0, 1; b_{4}, b_{5}) \end{matrix}  \begin{matrix}r_{0}^{1} = [1, −1, −1, 0, 0] \\ r = [0, 1, 0, 0, 0] \\ r_{1}^{2} = [0, 0, 1, 0, 0]. \end{matrix}

(iii) Conditions for the parameters a_{4}, a_{5}, b_{4}, b_{5}:

(5.3.1) a_{5}, b_{5}, | \begin{matrix}a_{4} & a_{5} \\ b_{4} & b_{5} \end{matrix} | > 0.

(5.3.2) gcd(a_{4}, a_{5}) = gcd(b_{4}, b_{5}) = 1.

Remark:

1) X_{σ} is Gorenstein.

2) X_{σ} has an isolated singularity in 0, iff a_{4} = b_{4} + 1 and a_{5} = b_{5} = 1.

(iv) The linear subspace L^{⊥} = [r_{0}^{1} −r]^{⊥} ∩[r −r_{1}^{2}]^{⊥} ⊆ N is given by the equations a_{1} = 3a_{3}; a_{2} = a_{3}. To determine σ̄ = σ ∩ L^{⊥} it is sufficient to compute the intersection with each of the 3-dimensional faces of σ only. Then, exactly the six faces of the form < .^{0}, .^{1}, .^{2} > give a contribution, and we obtain σ̄ to be the cone over

H_{III} :=< (0, 0); (1, 0); (a_{4} + 1, a_{5}); (a_{4} + b_{4} + 1, a_{5}, b_{5}); (a_{4} + b_{4}, a_{5} + b_{5}); (b_{4}, b_{5}) >

that sits in the affine hyperplane [ā_{1} = 1] ⊆ L^{⊥}.

[figure: b⃗; (1,0); −a⃗; P_{0}=(0,0); P_{1}=(1,0); P_{4}=(a_{4}+b_{4},a_{5}+b_{5}); P_{3}=(a_{4}+b_{4}+1,a_{5}+b_{5}); P_{5}=(b_{4},b_{5}); P_{2}=(a_{4}+1,a_{5})]

Remark:

3) Yσ̄ is Gorenstein.

4) The conditions of (5.3.1) mean that H_{III} is a convex, non-degenerate hexagon of the form drawn above.

5) The condition of (5.3.2) expresses that (Yσ̄, 0) has at most an isolated singularity.

6) P_{0} + P_{3} = P_{1} + P_{4} = P_{2} + P_{5}, i.e. opposite edges are parallel ones which have the same length too. This condition is not only necessary, but also sufficient (up to ZZ-isomorphism) for H_{III} to have those special vertices as mentioned above.

Therefore, the isolated singularities Y that admit toric deformations of type (III) are exactly those Gorenstein hexagon singularities such that the hexagon consists of three parallelograms as shown in the picture above. The corresponding total space X_{σ} is uniquely determined by Y.

(5.4) Which isolated hexagon singularities admit toric deformations of more than one type? Obviously, the type (II) is not compatible with any of the two remaining types. On the other hand, the hexagons H belonging to both classes, (I) and (III) are easy to describe: they consist of six copies of a fixed triangle, and these copies are arranged in such a way that any two adjacent ones form a parallelogram.

[figure: (1,0); a⃗; (0,0); (1,0); (a_{4}−1,a_{5}); (2a_{4}−1,2a_{5}); (2a_{4},2a_{5}); (a_{4}+1,a_{5}); {a_{5} > 0; \atop gcd(a_{4}, a_{5}) = gcd(a_{4} − 1, a_{5}) = 1.}]

Remark: The total spaces of both toric deformations have isolated singularities, iff a_{5} = 1. If this is the case, then the coordinates can be improved once more to obtain a_{4} = 1 also. In particular, the corresponding toric variety Yσ̄ is uniquely determined; the two total spaces are isomorphic to the cones over IP^{1} × IP^{1} × IP^{1} or IP^{2} × IP^{2}, respectively.

x_{1}∼(0,0) (1,0)∼x_{2}  (2,1)∼y_{3} (2,2)∼y_{1} y_{2}∼(1,2) x_{3}∼(0,1)  (1,1)∼z

Yσ̄ has embedding dimension 7 and can be given by 9 equations:

x_{1}y_{1} = x_{2}y_{2} = x_{3}y_{3} = z^{2}

x_{1}z = x_{2}x_{3}; x_{2}z = x_{1}y_{3}; x_{3}z = x_{1}y_{2};
y_{1}z = y_{2}y_{3}; y_{2}z = x_{3}y_{1}; y_{3}z = x_{2}y_{1}.

In fact, as J. Stevens pointed out to me, this singularity is the cone over the Del Pezzo surface. Moreover, by explicit computations with the computer programme “Macaulay”, Duco van Straten has shown that the above two total spaces represent, indeed, exactly the two components of the reduced versal base space of Yσ̄.

References

[Al] Altmann, K.: Computation of the vector space T^{1} for affine toric varieties. (appears as preprint of the MSI (Cornell University, NY))

[Ar] Arndt, J.: Verselle Deformationen zyklischer Quotientensingularitäten. Dissertation, Universität Hamburg, 1988

[Ch] Christophersen, J.A.: Obstruction spaces for rational singularities and deformations of cyclic quotients. Thesis, University of Oslo, 1989/1990

[Gr] Grünbaum, B.: Convex Polytopes.

Wiley 1967

[Ke] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings I. Lect. Notes Math., vol. 339. Berlin-Heidelberg-New York: Springer 1973

[KSB] Kollár J., Shepherd-Barron, N.I.: Threefolds and deformations of surface singularities. Invent. math. 91 (1988), 299-338

[St] Stevens, J.: On the versal deformation of cyclic quotient singularities. Preprint, Mathematisches Seminar der Universität Hamburg