The versal deformation of an isolated toric Gorenstein singularity

Klaus Altmann^{∗}

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

Abstract

Given a lattice polytope Q ⊆ IR^{n}, we define an affine scheme M̄ that reflects the possibilities of splitting Q into a Minkowski sum. Denoting by Y the toric Gorenstein singularity induced by Q, we construct a flat family over M̄ with Y as special fiber. In case Y has an isolated singularity, this family is versal.

1 Introduction

(1.1) The whole deformation theory of an isolated singularity is encoded in its so-called versal deformation. For complete intersection singularities this is a family over a smooth base space obtained by certain perturbations of the defining equations. As soon as we leave this class of singularities, the structure of the family, and sometimes even the base space, will be more complicated. It is well known that the base space may consist of several components or may be non-reduced. In (9.2) we will present a (three-dimensional) example of a singularity admitting a fat point as base space of its versal deformation.

For two-dimensional cyclic quotient singularities (coinciding with the two-dimensional affine toric varieties), the computations of Arndt, Christophersen, Kollár/ Shepherd-Barron, Riemenschneider, and Stevens provide a description of the versal family - in particular, the number and dimension of the components of the reduced base (these components are smooth) are computed. Christophersen observed that the total spaces over these components are toric varieties again (cf. [Ch]). This suggests that the entire deformation theory of affine toric varieties might remain inside this category. It should be a challenge to find the versal deformation, its base space, or the total spaces over the components by purely combinatorial methods.

(1.2) Affine toric varieties are constructed from rational, polyhedral cones σ ⊆ IR^{n+1}: One takes the dual cone

σ^{∨} := {r ∈ (IR^{n+1})^{∗} | ⟨a, r⟩ ≥ 0 for each a ∈ σ} ,

and Y_{σ} is defined as the spectrum of the semigroup algebra I C[σ^{∨} ∩ (ZZ^{n+1})^{∗}]. In particular, equations of Y_{σ} are induced from affine relations between lattice points of σ^{∨} ⊆ (IR^{n+1})^{∗}. (In the following we will no longer differentiate between IR^{n+1} and its dual space; however, for vectors, we will try to use parentheses and brackets for primal and dual ones, respectively.) See [Ke] or [Od] for an introduction into the subject of toric varieties. For investigating versal deformation spaces, Gorenstein singularities could serve as the first class to study beyond complete intersections. Ishida gave a nice description of this class inside toric varieties (cf. [Ish], Theorem 7.7): Y_{σ} is Gorenstein if and only if σ equals the cone over some lattice polytope Q ⊆ IR^{n}, i.e. its vertices are lattice points embedded into height one. Therefore, our point of view will be the following: Given a lattice polytope Q ⊆ IR^{n}, we want to study the deformation theory of the affine, toric variety Y_{σ} with σ := Cone(Q) ⊆ IR^{n+1}. Examples of these singularities are Del Pezzo surfaces of degree ≥ 6 (cf. (9.1)).

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

(1.3) The main tool to describe our results is the notion of Minkowski sums:

Definition: For two polytopes P, P′ ⊆ IR^{n} we define their Minkowski sum as the polytope P +P′ := {p+p′ | p ∈ P, p′ ∈ P′}. Obviously, this notion also makes sense for translation classes of polytopes.

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

(See (9.3) for another illustration of this notion.) Each Minkowski summand of a given polytope Q ⊆ IR^{n} (or some multiple of Q) contains, up to the length, the same edges as Q itself. This fact enables us to handle the “moduli space” C(Q) of Minkowski summands which is a polyhedral cone (cf. (2.2)). Attaching each Minkowski summand at the point that represents it in C(Q) yields the so-called tautological cone C̃(Q) together with a projection onto C(Q). Its construction is very similar to that of a universal bundle, and indeed, applying the functor that makes toric varieties from cones will provide the main step toward constructing the versal base space of Y_{σ} (cf. §4).

(1.4) For a given lattice polytope Q ⊆ IR^{n} with primitive edges, i.e. they do not contain any interior lattice points, we begin in §2 with describing an affine scheme M̄ which seems to be interesting independently from the toric or deformation context. It describes the possibilities of splitting Q into Minkowski summands. The underlying reduced space is an arrangement of planes corresponding to those Minkowski decompositions involving summands that are lattice polytopes themselves. Since all the proofs are based on quite the same method, we have collected them in a separate section. Each theorem of §2 can be translated into an easier language and corresponds to a certain lemma of §3. In §4 we study the tautological cone C̃(Q). This leads in §5 to the construction of a flat family over M̄ with Y_{σ} (σ = Cone(Q)) as special fiber. Note that for Y_{σ} the assumption of Q having primitive edges means smoothness in codimension two. Computing the Kodaira-Spencer map (in §6) as well as the obstruction map (in §7) shows that for isolated singularities the family is versal (nevertheless trivial for dim Q ≥ 3, cf. (6.3)). Its components are described in §8. In the general case, the Kodaira-Spencer map is an isomorphism onto the homogeneous part of T_{Y}^{1} with the most interesting multidegree (cf. Theorem (6.2)), and the obstruction map is still injective (cf. Theorem (7.2)). Throughout the paper, an example accompanies the general theory. Further examples can be found in §9.

(1.5) Acknowledgements: I am very grateful to Duco van Straten and Theo de Jong for constant encouragement and valuable hints. This paper was written during a one-year stay at MIT. I would like to thank Richard Stanley and all the other people who made it possible for me to work at this very interesting and stimulating place.

2 The Minkowski scheme of a lattice polytope

(2.1) Let Q ⊆ IR^{n} be a lattice polytope, i.e. the vertices are contained in ZZ^{n}. We will always assume that the edges do not contain any interior lattice points. Hence, after choosing orientations they are given by primitive vectors d^{1},…, d^{N} ∈ ZZ^{n}.

Definition: For every 2-face ε < Q we define its sign vector \underline{ε} = (ε_{1},…, ε_{N}) ∈ \{0, ±1\}^{N} by

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

such 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 we choose one of both possibilities. In particular, ∑_{i} ε_{i}d^{i} = 0.

Example: Let us introduce the following example, which will be continued throughout the paper: For Q we take the hexagon

Q_{6} := Conv {(0, 0), (1, 0), (2, 1), (2, 2), (1, 2), (0, 1)} ⊆ IR^{2}.

[figure: (0, 0); (1, 0); (2, 1); (2, 2); (1, 2); (0, 1); d^{1}; hexagon Q_{6}]

Starting with d^{1} := \overline{(0, 0)(1, 0)}, the anticlockwise oriented edges are denoted by d^{1},…, d^{6}. As vectors they equal

d^{1} = (1, 0) ; d^{2} = (1, 1) ; d^{3} = (0, 1) ; d^{4} = (−1, 0) ; d^{5} = (−1, −1) ; d^{6} = (0, −1) .

Q_{6} is 2-dimensional, hence, it is its own unique 2-face ε = Q. For \underline{Q} we take \underline{Q} = (1,…, 1).

(2.2) We define the vector space V ⊆ IR^{N} by

V := V (Q) := {(t_{1},…, t_{N}) | ∑_{i} t_{i} ε_{i} d^{i} = 0 for every 2-face ε < Q}.

Then, C(Q) := V ∩ IR_{≥0}^{N} is obviously a rational, polyhedral cone in V.

Lemma: The points of C(Q) correspond to the Minkowski summands of positive multiples of Q.

Proof: For an element (t_{1},…, t_{N}) ∈ C(Q), the corresponding summand Q\underline{t} is built by the edges t_{i} · d^{i} (i = 1,…, N) as follows: Assume that 0 ∈ IR^{n} coincides with some vertex of the lattice polytope Q. Then, each vertex a of Q can be reached from 0 by some walk along the edges of Q. We obtain

a = ∑_{i=1}^{N} λ_{i} d^{i} for some \underline{λ} = (λ_{1},…, λ_{N}) , λ_{i} ∈ ZZ.

Now, given an element \underline{t} ∈ C(Q), we may define the corresponding vertex a\underline{t} by

a\underline{t} := ∑_{i=1}^{N} t_{i} λ_{i} d^{i}.

The linear equations defining V = span C(Q) ensure that this definition does not depend on the particular path from 0 to a through the 1-skeleton of Q. The polytope Q\underline{t} is defined as the convex hull of all the a\underline{t}. Finally, it is clear that all Minkowski summands arise in this way. □

For a particular Minkowski summand Q′ we will denote the corresponding point in the cone by ϱ(Q′) ∈ C(Q).

Example: 1) Applying ϱ, the two splittings of Q_{6} drawn in (1.3) become

(1, 1, 1, 1, 1, 1) = (1, 0, 1, 0, 1, 0) + (0, 1, 0, 1, 0, 1) = (1, 0, 0, 1, 0, 0) + (0, 0, 1, 0, 0, 1) + (0, 1, 0, 0, 1, 0) .

2) In any case we have ϱ(t · Q) = (t,…, t) ∈ C(Q) ⊆ V ⊆ IR^{N}.

(2.3) For each 2-face ε < Q and for each integer k ≥ 1 we define the (vector valued) polynomial

g_{ε,k}(\underline{t}) := ∑_{i=1}^{N} t_{i}^{k} ε_{i} d^{i} .

Using coordinates of IR^{n}, the g_{ε,k}(\underline{t}) become regular polynomials; for each pair (ε, k) we will get two linearly independent ones. We obtain an ideal

J := (g_{ε,k} | ε < Q, k ≥ 1) ⊆ I C[t_{1},…, t_{N}]

which defines an affine closed subscheme

M := Spec I C[\underline{t}]/ ; J ⊆ V_{I C} ⊆ I C^{N}.

Example: For our hexagon Q_{6} introduced in (2.1) we obtain

J = (t_{1}^{k} + t_{2}^{k} − t_{4}^{k} − t_{5}^{k}, t_{2}^{k} + t_{3}^{k} − t_{5}^{k} − t_{6}^{k} | k ≥ 1) .

Of course, finitely many polynomials are sufficient to generate the ideal J - but we can even give an effective criterion to see which equations may be dropped:

Proposition: Let ε < Q be a 2-face. Then ε is contained in a two-dimensional subspace of IR^{n}, and this vector space comes with a natural lattice (the restriction of the big lattice ZZ^{n}). If ε is contained in two different strips defined by pairs of parallel lines of lattice-distance ≤ k_{0} each, then the equations g_{ε,k} (k > k_{0}) are contained in the ideal generated by g_{ε,1},…, g_{ε,k_{0}}.

Proof: cf. (3.3).

Example: Obviously, Q_{6} is contained in at least three different strips of thickness 2. Hence, J is generated by polynomials of degree ≤ 2:

J = (t_{1} + t_{2} − t_{4} − t_{5}, t_{2} + t_{3} − t_{5} − t_{6}, t_{1}^{2} + t_{2}^{2} − t_{4}^{2} − t_{5}^{2}, t_{2}^{2} + t_{3}^{2} − t_{5}^{2} − t_{6}^{2}) .

(2.4) Denote by ℓ the canonical projection

ℓ : I C^{N} ⟶ ⟶ I C^{N}/ ; I C · (1,…, 1) = I C^{N}/ ; I C · ϱ(Q) .

On the level of regular functions this corresponds to the inclusion I C[t_{i} − t_{j} | 1 ≤ i, j ≤ N] ⊆ I C[t_{1},…, t_{N}].

Theorem: (See also Remark (4.4))

(1) J is generated by polynomials from I C[t_{i} − t_{j}], i.e. M = ℓ^{−1}(M̄) for some affine closed

subscheme M̄ ⊆ V^{I C}/ ; I C · ϱ(Q) ⊆ I C^{N}/ ; I C · ϱ(Q) . M̄ is defined by the ideal J ∩ I C[t_{i} − t_{j}].

(2) J ⊆ I C[t_{1},…, t_{N}] is the smallest ideal that meets property (1) and, on the other hand, contains the “toric” equations

∏_{i=1}^{N} t_{i}^{d_{i}^{+}} − ∏_{i=1}^{N} t_{i}^{d_{i}^{−}} with

\underline{d} ∈ ZZ^{N} ∩ span \{[⟨ε_{1}d^{1}, c⟩,…, ⟨ε_{N}d^{N}, c⟩] | ε < Q 2-face, c ∈ IR^{n}\}. (For an integer h we denote

h^{+} := \{ \begin{matrix}h & if h ≥ 0 \\ 0 & otherwise \end{matrix} ; h^{−} := \{ \begin{matrix}0 & if h ≥ 0 \\ −h & otherwise \end{matrix} .)

Proof: cf. (3.4).

Example: Toric equations for Q_{6} are for instance t_{1} t_{2} − t_{4} t_{5}, t_{2} t_{3} − t_{5} t_{6}, and t_{1} t_{6} − t_{3} t_{4}.

(2.5) We want to describe the structure of the underlying reduced spaces of M or M̄. Let Q = R_{0} + … + R_{m} be a decomposition of Q into a Minkowski sum of m + 1 lattice polytopes. Then, the N-tuples ϱ(R_{0}),…, ϱ(R_{m}) have entries 0 and 1 only, and they sum up to (1,…, 1). In particular, the (m + 1)-plane I C · ϱ(R_{0}) + … + I C · ϱ(R_{m}) ⊆ I C^{N} is contained in M. It is given by the linear equations t_{i} − t_{j} = 0 (if d^{i}, d^{j} belong to a common summand R_{v}). Refinements of Minkowski decompositions (they form a partially ordered set) correspond to inclusions of the associated planes.

Theorem: M_{red} equals the union of those flats corresponding to maximal Minkowski decompositions of Q into lattice summands. M̄_{red} consists of their images via ℓ.

Proof: cf. (3.5).

Example: M(Q_{6}) and M̄(Q_{6}) are reduced schemes (for non-reduced examples cf. §9). Let us study them directly:

• The linear equations allow the following substitution:

\begin{aligned}t & := & t_{1} \\ s_{1} & := & t_{1} − t_{3} \\ s_{2} & := & t_{4} − t_{2} \\ s_{3} & := & t_{1} − t_{4} \end{aligned}  \begin{aligned}t_{1} & = & t \\ t_{2} & = & t − s_{2} − s_{3} \\ t_{3} & = & t − s_{1} \\ t_{4} & = & t − s_{3} \\ t_{5} & = & t − s_{2} \\ t_{6} & = & t − s_{1} − s_{3} . \end{aligned}

• The two quadratic equations transform into s_{1} s_{3} = s_{2} s_{3} = 0.

In particular, M̄ is the union of a line and a 2-plane - corresponding to the Minkowski decompositions

Q_{6} = Conv {(0, 0), (1, 0), (1, 1)} + Conv {(0, 0), (0, 1), (1, 1)} and
Q_{6} = Conv {(0, 0), (1, 0)} + Conv {(0, 0), (0, 1)} + Conv {(0, 0), (1, 1)}

already mentioned in (2.2) and depicted in (1.3).

(2.6)  M̄ (or M = ℓ^{−1}(M̄)) reflects the possibilities of Minkowski decompositions of Q:

• The underlying reduced space encodes the decompositions of Q into lattice summands.

• Extremal decompositions into rational summands are hidden in the scheme structure of M̄. Its tangent space in 0 (the smallest affine space containing M̄) equals V^{I C}/ ; I C · ϱ(Q) - it is the vector space arising from the cone C(Q) of Minkowski summands by killing the summands homothetic to Q.

Therefore, we will call M̄ the (affine) Minkowski scheme of Q.

3 Proofs of the statements of §2

(3.1) Using vectors c ∈ ZZ^{N} (or certain c ∈ IR^{N}) we can evaluate the edges d^{1},…, d^{N} to get integers

d_{1} := ⟨ε_{1}d^{1}, c⟩,…, d_{N} := ⟨ε_{N}d^{N}, c⟩

for every given 2-face ε < Q. Doing so, the statements of §2 can be reduced to much simpler lemmas which will be presented here. Then, all these lemmas are proved using the following recipe:

(i) Assume d_{i} = ±1 - then the lemmas reduce to well known facts concerning symmetric functions.

(ii) Move to the general case by specialization of variables.

(3.2) For the whole §3 we use the following notation:

Let d_{1},…, d_{N} ∈ ZZ such that d_{1},…, d_{M} ≥ 0, d_{M+1},…, d_{N} ≤ 0, and ∑_{i=1}^{N} d_{i} = 0 .

\begin{aligned}g_{k}(\underline{t}) & := & g_{\underline{d},k}(\underline{t}) & := & ∑_{i=1}^{N} d_{i} t_{i}^{k} , \\ p(\underline{t}) & := & p\underline{d}(\underline{t}) & := & t_{1}^{d_{1}} ·. . . · t_{M}^{d_{M}} − t_{M+1}^{d_{M+1}} ·. . . · t_{N}^{d_{N}} . \end{aligned}

Denote by σ_{k} and s_{k} the k-th elementary symmetric polynomial and the sum of the k-th powers of a given set of variables, respectively.

Remark: For 1 ≤ i, j ≤ M or M + 1 ≤ i, j ≤ N, identifying the two variables t_{i} and t_{j} (i.e. switching from I C[\underline{t}] to I C[\underline{t}]/t_{i} − t_{j}) yields the following situation:

• t_{i}, t_{j} are replaced by a common new variable t̃ (i.e. N is replaced by N − 1),

• d_{i}, d_{j} are replaced by d̃:= d_{i} + d_{j}, but

• g_{k}(\underline{t}), p(\underline{t}) keep their shapes in the new set up.

In particular, the general situation can always be obtained via factorization from the special case d_{1} = … = d_{M} = 1; d_{M+1} = … = d_{N} = −1 (and N = 2M). Renaming t_{i} = x_{i}, t_{M+i} = y_{i} (i ≤ M) it looks like

\begin{aligned}g_{k}(\underline{x}, \underline{y}) & = & (∑_{i=1}^{M} x_{i}^{k}) − (∑_{i=1}^{M} y_{i}^{k}) = s_{k}(\underline{x}) − s_{k}(\underline{y}) , \\ p(\underline{x}, \underline{y}) & = & (x_{1} ·. . . · x_{M}) − (y_{1} ·. . . · y_{M}) = σ_{M}(\underline{x}) − σ_{M}(\underline{y}). \end{aligned}

(3.3) Lemma: If k_{0} := ∑_{i=1}^{M} d_{i} = − ∑_{i=M+1}^{N} d_{i}, then the polynomials g_{k} (k > k_{0}) are I C[\underline{t}]- linear combinations of g_{1},…, g_{k_{0}}. (This implies Proposition (2.3).)

Proof: As previously discussed, we may regard the special case d_{i} = ±1. In particular, k_{0} = M. Now, for an arbitrary k (> M), the expression s_{k}(\underline{x}) is a polynomial in s_{1}(\underline{x}),…, s_{M}(\underline{x}), say

s_{k}(\underline{x}) = P_{k} (s_{1}(\underline{x}),…, s_{M}(\underline{x})) .

Then,

g_{k}(\underline{x}, \underline{y}) = s_{k}(\underline{x}) − s_{k}(\underline{y}) = P_{k} (s_{1}(\underline{x}),…, s_{M}(\underline{x})) − P_{k} (s_{1}(\underline{y}),…, s_{M}(\underline{y})) ,

and for each monomial s_{1}^{r_{1}} s_{2}^{r_{2}} … s_{M}^{r_{M}} occurring in P_{k}, we have

s_{1}(\underline{x})^{r_{1}} ·. . . · s_{M}(\underline{x})^{r_{M}} − s_{1}(\underline{y})^{r_{1}} ·. . . · s_{M}(\underline{y})^{r_{M}} =

= ∑_{v=1}^{M} ∑_{i=1}^{r_{v}}[s_{v}(\underline{x}) − s_{v}(\underline{y})] · s_{1}(\underline{x})^{r_{1}} … s_{v−1}(\underline{x})^{r_{v−1}} s_{v}(\underline{x})^{i−1} · s_{v}(\underline{y})^{r_{v}−i} s_{v+1}(\underline{y})^{r_{v+1}} … s_{M}(\underline{y})^{r_{M}}

= ∑_{v=1}^{M} g_{v}(\underline{x}, \underline{y}) · ( ∑_{i=1}^{r_{v}} s_{1}(\underline{x})^{r_{1}} … s_{v−1}(\underline{x})^{r_{v−1}} s_{v}(\underline{x})^{i−1} · s_{v}(\underline{y})^{r_{v}−i} s_{v+1}(\underline{y})^{r_{v+1}} … s_{M}(\underline{y})^{r_{M}} ) ,

proving the lemma. □

(3.4) Lemma: (implying Theorem (2.4))

(1) The ideal J := (g_{k} | k ≥ 1) ⊆ I C[t_{1},…, t_{N}] is generated by polynomials in t_{i} − t_{1} (i = 2,…, N) only.

(2) J is the smallest ideal generated by polynomials in t_{i} − t_{1}, which additionally contains p.

Proof: (1) Replacing t_{i} by t_{i} − t_{1} as arguments in g_{k} yields

g_{k}(t_{1} − t_{1},…, t_{N} − t_{1}) = ∑_{i=1}^{N} d_{i} (t_{i} − t_{1})^{k} = ∑_{i=1}^{N} d_{i} · (∑_{v=0}^{k}(−1)^{v} t_{1}^{v} t_{i}^{k−v})

= ∑_{v=0}^{k}(−1)^{v} t_{1}^{v} · (∑_{i=1}^{N} d_{i} t_{i}^{k−v}) = ∑_{v=0}^{k}(−1)^{v} t_{1}^{v} g_{k−v}(\underline{t}).

In particular, (g_{k}(\underline{t}) | k ≥ 1) and (g_{k}(\underline{t} − t_{1}) | k ≥ 1) are the same ideals in I C[\underline{t}].

(2) Each polynomial q(\underline{t}) can be written uniquely as

q(\underline{t}) = ∑_{v≥0} q_{v}(t_{2} − t_{1},…, t_{N} − t_{1}) · t_{1}^{v}.

If J ⊆ I C[\underline{t}] is an ideal generated by polynomials in \underline{t} − t_{1} only, then for each q(\underline{t}) ∈ J the components q_{v} are automatically contained in J, too. Hence, we should look for the components of the polynomial p. In the polynomial ring I C[\underline{X}, \underline{Y} , T] we know that

p(T + \underline{X}, T + \underline{Y}) = (T + X_{1}) ·. . . · (T + X_{M}) − (T + Y_{1}) ·. . . · (T + Y_{M})

has σ_{k}(\underline{X})−σ_{k}(\underline{Y}) as coefficient of T^{M−k} (k = 1,…, M). On the other hand, there is a polynomial P_{k} and a non-vanishing rational number c_{k} (not depending on M) such that

σ_{k}(\underline{X}) = P_{k}(s_{1}(\underline{X}),…, s_{k−1}(\underline{X})) + c_{k} · s_{k}(\underline{X}).

As in the proof of the previous lemma we obtain

\begin{aligned}σ_{k}(\underline{X}) − σ_{k}(\underline{Y}) & = & P_{k}(s_{1}(\underline{X}),…, s_{k−1}(\underline{X})) − P_{k}(s_{1}(\underline{Y}),…, s_{k−1}(\underline{Y})) + c_{k} · s_{k}(\underline{X}) − c_{k} · s_{k}(\underline{Y}) \\ & = & ∑_{v=1}^{k−1} q_{v}(\underline{X}, \underline{Y}) · g_{v}(\underline{X}, \underline{Y}) + c_{k} · g_{k}(\underline{X}, \underline{Y}) \end{aligned}

for some coefficients q_{v}. Specialization (first by T ↦ x_{1}, X_{i} ↦ x_{i} − x_{1}, Y_{i} ↦ y_{i} − x_{1}, then followed by the usual one) shows that the ideal generated by the components p_{v}(\underline{t} − t_{1}) of p equals J. □

(3.5) Lemma: Let \underline{c} = (c_{1},…, c_{N}) ∈ I C^{N} be a point such that g_{k}(\underline{c}) = 0 for each k ≥ 1. Then, for every fixed c ∈ I C, we have ∑_{c_{i}=c} d_{i} = 0. (This implies Theorem (2.5).)

Proof: The equations ∑_{i=1}^{N} d_{i} c_{i}^{k} = 0 present 0 as a linear combination of the vectors (c_{i}, c_{i}^{2}, c_{i}^{3},…). On the other hand, the Vandermonde tells us that this linear combination has to be a trivial one, i.e. the sum of the coefficients d_{i} belonging to equal variables vanishes. □

4 The tautological cone over C(Q)

(4.1) In (2.2) we have introduced the cone C(Q) of Minkowski summands of IR_{≥0} · Q. For an element (t_{1},…, t_{N}) ∈ C(Q) the corresponding summand Q\underline{t} was built by the edges t_{i} · d^{i} (i = 1,…, N). Now, we paste the summands at the points they are assigned to:

Definition: The tautological cone C̃(Q) ⊆ IR^{n} × V ⊆ IR^{n+N} is defined as

C̃(Q) := \{(a, \underline{t}) | \underline{t} ∈ C(Q); a ∈ Q\underline{t}\}.

It comes with a natural projection C̃(Q) → C(Q).

C̃(Q) is (as C(Q)) a rational, polyhedral cone. It is generated by the pairs (a_{\underline{t}^{j}}^{i}, \underline{t}^{j}) with

• a^{i} a vertex of Q and

• \underline{t}^{j} a fundamental generator of C(Q).

This follows from the simple rule (a_{\underline{t}+\underline{t}′}, \underline{t}+\underline{t}′) = (a\underline{t}, \underline{t})+(a\underline{t}′, \underline{t}′) for vertices a ∈ Q and \underline{t}, \underline{t}′ ∈ C(Q). Defining σ := Cone(Q) ⊆ IR^{n+1} by putting Q into the hyperplane (t = 1), we obtain a fiber product diagram of rational polyhedral cones:

[figure: [C̃(Q) ⊆ IR^{n} × V]; [C(Q) ⊆ V]; IR_{≥0}^{N}; [σ ⊆ IR^{n+1}]; IR_{≥0}; pr_{V}; i; pr_{n+1}; ·ϱ(Q); diagonal]

(The horizontal maps are projections onto the V and the (n + 1)-th component, respectively. The inclusion i is given by (t · a; t) ↦ (t · a; t,…, t).)

(4.2) Assigning toric varieties to polyhedral cones is functorial, i.e. we can proceed so with the whole diagram. We obtain affine toric varieties Y, X, and S with coordinate rings A(Y) = I C[σ^{∨} ∩ ZZ^{n+1}], A(X) = I C[C̃(Q)^{∨} ∩ (ZZ^{n} × V_{Z Z}^{∗})], and A(S) = I C[C(Q)^{∨} ∩ V_{Z Z}^{∗}], respectively. (Recall that Y = Y_{σ} is the toric Gorenstein singularity we want to deform.) The varieties are arranged in the following commutative diagram:

[figure: X_{C̃(Q)}; S_{C(Q)}; I C^{N}; I C^{N}/(1,…, 1); Y_{σ}; I C; {0}; π; p; ℓ; i; diagonal]

In (4.4) and (4.7) we will see that Y ↪ X is the pull back of the closed embedding I C ↪ S. Notice that p : S → I C^{N} defines functions t_{1},…, t_{N} on S.

(4.3) To study the toric varieties Y, X, and S it is important to understand the dual cones of σ, C̃(Q), and C(Q), respectively. Beginning with the dual cone of σ, to each non-trivial c ∈ ZZ^{n} we associate a vertex a(c) of Q and a non-negative integer η_{0}(c) meeting the properties

⟨Q, −c⟩ ≤ η_{0}(c) and ⟨a(c), −c⟩ = η_{0}(c).

With respect to Q, c ≠ 0 is the inner normal vector of the affine supporting hyperplane ⟨•, −c⟩ = η_{0}(c) through a(c). In particular, η_{0}(c) is uniquely determined, while a(c) is not. For c = 0 we define a(0) := 0 ∈ IR^{n} and η_{0}(0) := 0 ∈ ZZ. Recall that the dual cone of σ is defined as σ^{∨} := {r ∈ IR^{n+1} | ⟨σ, r⟩ ≥ 0}. Hence, by the definition of η_{0}, we have

∂σ^{∨} ∩ ZZ^{n+1} = {[c, η_{0}(c)] | c ∈ ZZ^{n}} .

Moreover, if c^{1},…, c^{w} ∈ ZZ^{n} \ 0 are those elements producing irreducible pairs [c, η_{0}(c)] (i.e. not allowing any non-trivial lattice decomposition [c, η_{0}(c)] = [c′, η_{0}(c′)] + [c′′, η_{0}(c′′)]), then the elements

[c^{1}, η_{0}(c^{1})],…, [c^{w}, η_{0}(c^{w})], [\underline{0}, 1]

form the minimal generator set for σ^{∨} ∩ ZZ^{n+1} as a semigroup. Among them are all pairs [c, η_{0}(c)] corresponding to facets (i.e. top dimensional faces) of Q. We obtain a closed embedding Y ↪ I C^{w+1}. The coordinate functions of I C^{w+1} will be denoted by z_{1},…, z_{w}, t corresponding to [c^{1}, η_{0}(c^{1})],…, [c^{w}, η_{0}(c^{w})], [\underline{0}, 1], respectively.

Example: We continue our example Q_{6} from §2. Here, the facets of Q_{6} equal its edges d^{1},…, d^{6}, and they are sufficient for producing all irreducible pairs [c^{1}, η_{0}(c^{1})],…, [c^{6}, η_{0}(c^{6})]. We have

c^{1} = [0, 1], c^{2} = [−1, 1], c^{3} = [−1, 0], c^{4} = [0, −1], c^{5} = [1, −1], c^{6} = [1, 0] .

The corresponding vertices are (for instance)

a(c^{6}) = a(c^{1}) = (0, 0), a(c^{2}) = a(c^{3}) = (2, 1), a(c^{4}) = a(c^{5}) = (1, 2),

and we obtain

η_{0}(c^{1}) = 0, η_{0}(c^{2}) = 1, η_{0}(c^{3}) = 2, η_{0}(c^{4}) = 2, η_{0}(c^{5}) = 1, η_{0}(c^{6}) = 0 .

(4.4) Thinking of C(Q) as a cone in IR^{N} instead of V allows dualizing the equation C(Q) = IR_{≥0}^{N} ∩ V to get C(Q)^{∨} = IR_{≥0}^{N} + V^{⊥}. Hence, for C(Q) as a cone in V we obtain

C(Q)^{∨} = IR_{≥0}^{N} + V^{⊥}_{/V^{⊥}} = im [IR_{≥0}^{N} ⟶ V^{∗}].

(As with IR^{n}, we do not use different notation for IR^{N} and its dual space. Let e_{1},…, e_{N} be the canonical basis of the latter one.) The surjection IR_{≥0}^{N} ⟶ ⟶ C(Q)^{∨} induces a map IN^{N} ⟶ C(Q)^{∨} ∩ V_{Z Z}^{∗}, which does not need to be surjective at all. This leads to the following definition:

Definition: On V_{Z Z}^{∗} we introduce a partial ordering “⪰” by

\underline{η} ⪰ \underline{η}′ ⟺ \underline{η} − \underline{η}′ ∈ im [IN^{N} → V_{Z Z}^{∗}] ⊆ C(Q)^{∨} ∩ V_{Z Z}^{∗}.

On the geometric level, the non-saturated semigroup im [IN^{N} → V_{Z Z}^{∗}] ⊆ C(Q)^{∨} ∩ V_{Z Z}^{∗} corresponds to the scheme theoretical image S̄ of p : S → I C^{N}, and S → S̄ is its normalization (cf. (5.2)). The equations of S̄ ⊆ I C^{N} are collected in the kernel of

I C[t_{1},…, t_{N}] = I C[IN^{N}] ⟶^{φ} I C[C(Q)^{∨} ∩ V_{Z Z}^{∗}] ⊆ I C[V_{Z Z}^{∗}],

and it is easy to see that

ker φ = (∏_{i=1}^{N} t_{i}^{d_{i}^{+}} − ∏_{i=1}^{N} t_{i}^{d_{i}^{−}} | \underline{d} ∈ ZZ^{N} ∩ V^{⊥})  with
V^{⊥} = span {[⟨ε_{1}d^{1}, c⟩,…, ⟨ε_{N}d^{N}, c⟩] | ε < Q is a 2-face, c ∈ IR^{n}} .

Remark: Using our new notation, we can reformulate Theorem (2.4) as: M ⊆ I C^{N} is the largest closed subscheme that is contained in S̄ and, additionally, comes from I C^{N}/ϱ(Q) via ℓ^{−1}.

On the other hand, dualizing the embedding IR_{≥0} ↪ C(Q) yields

\begin{matrix}C(Q)^{∨} ∩ V_{Z Z}^{∗} & ⟶ ⟶ & IN \\ \underline{η} & ↦ & ∑_{i} η_{i} \end{matrix}

at the level of semigroups. This map is surjective, even after restricting to the subset im [IN^{N} → V_{Z Z}^{∗}]: All vectors e_{i} corresponding to the functions t_{i} map onto 1 ∈ IN. Geometrically this means that both maps I C → S and I C → S̄ are closed embeddings, and the corresponding ideals are (x\underline{η} − x^{\underline{η}′} | \underline{η}, \underline{η}′ ∈ C(Q) ∩ V_{Z Z}^{∗} with ∑_{i} η_{i} = ∑_{i} η′_{i} ) and (t_{i} − t_{j} | 1 ≤ i, j ≤ N), respectively. In particular, we have made a first step towards proving the claims made in (4.2).

(4.5) In the next two sections we take a closer look at the dualized cone C̃(Q)^{∨}.

Definition: For c ∈ ZZ^{n} let \underline{λ}^{c} = (λ_{1}^{c},…, λ_{N}^{c}) ∈ ZZ^{N} describe some path from 0 ∈ Q to a(c) ∈ Q through the 1-skeleton of Q (similar to that in (2.2)). Then,

\underline{η}(c) := [−λ_{1}^{c}⟨d^{1}, c⟩,…, −λ_{N}^{c}⟨d^{N}, c⟩] ∈ ZZ^{N}

defines an element \underline{η}(c) ∈ V_{Z Z}^{∗} not depending on the choice of the particular path \underline{λ}^{c}.

(Let \underline{λ̃}^{c} be a different path from 0 to a(c). It will differ from \underline{λ}^{c} by some linear combination ∑_{ε<Q} g_{ε} \underline{ε} (g_{ε} ∈ ZZ for 2-faces ε < Q) only. In particular, λ̃_{i}^{c}⟨d^{i}, c⟩ − λ_{i}^{c}⟨d^{i}, c⟩ = ∑_{ε<Q} g_{ε}⟨ε_{i} d^{i}, c⟩, and we obtain \underline{η}(c)λ̃ − \underline{η}(c)_{λ} ∈ V^{⊥}.)

Lemma: (i) \underline{η}(0) = 0 ∈ V_{Z Z}^{∗}.

(ii) For all c ∈ ZZ^{n} we have \underline{η}(c) ⪰ 0 (in the sense of Definition (4.4)).

(iii) \underline{η} is convex: ∑_{v} g_{v} \underline{η}(c^{v}) ⪰ \underline{η}(∑_{v} g_{v} c^{v}) for natural numbers g_{v} ∈ IN.

(iv) ∑_{i=1}^{N} η_{i}(c) = η_{0}(c) for arbitrary c ∈ ZZ^{n}.

Proof: (ii) a(c) is a vertex of Q providing the minimal value of the linear function ⟨•, c⟩. In particular, we can choose a path \underline{λ}^{c} from 0 ∈ Q to a(c) such that this function decreases in each step, i.e. λ_{i}^{c}⟨d^{i}, c⟩ ≤ 0 (i = 1,…, N).

(iii) We define the following paths through the 1-skeleton of Q:

• \underline{λ} := path from 0 ∈ Q to a(∑_{v} g_{v} c^{v}) ∈ Q,

• \underline{μ}^{v} := path from a(∑_{v} g_{v} c^{v}) ∈ Q to a(c^{v}) ∈ Q such that μ_{i}^{v}⟨d^{i}, c^{v}⟩ ≤ 0 for each i = 1,…, N.

Then, \underline{λ}^{v} := \underline{λ} + \underline{μ}^{v} is a path from 0 ∈ Q to a(c^{v}), and for i = 1,…, N we obtain

\begin{aligned}∑_{v} g_{v} η_{i}(c^{v}) − η_{i}( ∑_{v} g_{v} c^{v}) & = & − ∑_{v} g_{v} (λ_{i} + μ_{i}^{v}) ⟨d^{i}, c^{v}⟩ + λ_{i}⟨d^{i}, ∑_{v} g_{v} c^{v}⟩ \\ & = & − ∑_{v} g_{v} μ_{i}^{v} ⟨d^{i}, c^{v}⟩ ≥ 0 . \end{aligned}

(iv) By definition of \underline{λ}^{c} we have ∑_{i=1}^{N} λ_{i}^{c} d^{i} = a(c). In particular,

∑_{i=1}^{N} η_{i}(c) = − ∑_{i=1}^{N}⟨λ_{i}^{c} d^{i}, c⟩ = −⟨a(c), c⟩ = η_{0}(c).

□

Example: In our hexagon Q_{6} we choose the following paths from (0, 0) to the vertices a(c^{1}),…, a(c^{6}), respectively:

\underline{λ}^{6} = \underline{λ}^{1} := \underline{0}, \underline{λ}^{2} = \underline{λ}^{3} := [1, 1, 0, 0, 0, 0], \underline{λ}^{4} = \underline{λ}^{5} := [1, 1, 1, 1, 0, 0] .

They provide

\begin{aligned}\underline{η}(c^{1}) & = & [0, 0, 0, 0, 0, 0] , & \underline{η}(c^{2}) & = & [1, 0, 0, 0, 0, 0] , & \underline{η}(c^{3}) & = & [1, 1, 0, 0, 0, 0] , \\ \underline{η}(c^{4}) & = & [0, 1, 1, 0, 0, 0] , & \underline{η}(c^{5}) & = & [−1, 0, 1, 1, 0, 0] , & \underline{η}(c^{6}) & = & [0, 0, 0, 0, 0, 0] . \end{aligned}

Since [1, 0, −1, −1, 0, 1] = [⟨d^{1}, [1, −1]⟩,…, ⟨d^{6}, [1, −1]⟩] ∈ V^{⊥}, the vector \underline{η}(c^{5}) can be transformed into [0, 0, 0, 0, 0, 1].

Remark: The definitions of a(c), η_{0}(c), and \underline{η}(c) also make sense for general c ∈ IR^{n}. However, η_{0}(c) ∈ IR and \underline{η}(c) ∈ V^{∗} no longer need to be contained in the lattices. The previous lemma will remain valid (even for g_{v} ∈ IR_{≥0} in (iii)), if the relation “⪰ 0” is replaced by the weaker version “∈ C(Q)^{∨}”.

(4.6) Proposition: (1) C̃(Q)^{∨} = \{[c, \underline{η}] ∈ IR^{n} × V^{∗} | \underline{η} − \underline{η}(c) ∈ C(Q)^{∨}\}

(2) In particular, [c, \underline{η}(c)] ∈ C̃(Q)^{∨}, and moreover, it is the only preimage of [c, η_{0}(c)] ∈ σ^{∨} via the surjection i^{∨} : C̃(Q)^{∨} ⟶ ⟶ σ^{∨}.

(3) [c^{1}, \underline{η}(c^{1})],…, [c^{w}, \underline{η}(c^{w})] together with C(Q)^{∨} ∩ V_{Z Z}^{∗} (embedded as [0, C(Q)^{∨}]) generate the semigroup C̃(Q)^{∨} ∩ (ZZ^{n} × V_{Z Z}^{∗}). (For the definition of c^{1},…, c^{w}, cf. (4.3).)

Proof: (1) Let [c, \underline{η}] ∈ IR^{n} × V^{∗} be given; if some representative of \underline{η} in IR^{N} is needed, then it will be denoted by the same name. We have the following equivalences:

[c, \underline{η}] ∈ C̃(Q)^{∨} ⟺ ⟨(Q\underline{t}, \underline{t}), [c, \underline{η}]⟩ ≥ 0 for each \underline{t} ∈ C(Q)
⟺ ⟨Q\underline{t}, c⟩ + ⟨\underline{t}, \underline{η}⟩ ≥ 0 for each \underline{t} ∈ C(Q)
⟺ ⟨a(c)\underline{t}, c⟩ + ⟨\underline{t}, \underline{η}⟩ ≥ 0 for each \underline{t} ∈ C(Q).

Using some path \underline{λ}^{c} we obtain:

[c, \underline{η}] ∈ C̃(Q)^{∨} ⟺ ∑_{i=1}^{N} t_{i} λ_{i}^{c}⟨d^{i}, c⟩ + ⟨\underline{t}, \underline{η}⟩ ≥ 0 for each \underline{t} ∈ C(Q)

⟺ ∑_{i=1}^{N} t_{i} · (λ_{i}^{c} ⟨d^{i}, c⟩ + η_{i}) ≥ 0 for each \underline{t} ∈ C(Q)
⟺ [λ_{1}^{c} ⟨d^{1}, c⟩ + η_{1},…, λ_{N}^{c} ⟨d^{N}, c⟩ + η_{N}] ∈ C(Q)^{∨}.

(2) By part (1) we know that for each [c, \underline{η}] ∈ C̃(Q)^{∨} it is possible to choose IR^{N}-representatives of \underline{η}, \underline{η}(c) such that η_{i} ≥ η_{i}(c) for i = 1,…, N. On the other hand, the two equalities ∑_{i} η_{i}(c) = η_{0}(c) (cf. (iv) of the previous lemma) and ∑_{i} η_{i} = η_{0}(c) (corresponding to the fact [c, \underline{η}] ↦ [c, η_{0}(c)]) imply \underline{η} = \underline{η}(c).

(3) Let [c, \underline{η}] ∈ C̃(Q)^{∨}. Then, [c, η_{0}(c)] is representable as a non-negative linear combination [c, η_{0}(c)] = ∑_{v=1}^{w} p_{v} [c^{v}, η_{0}(c^{v})] (p_{v} ∈ IN if c ∈ ZZ^{n}). Since both elements [c, \underline{η}(c)] and ∑_{v} p_{v}[c^{v}, \underline{η}(c^{v})] are preimages of [c, η_{0}(c)] via i^{∨}, they must be equal by (2), and we obtain

[c, \underline{η}] = [c, \underline{η}(c)] + [0, \underline{η} − \underline{η}(c)] = ∑_{v} p_{v} [c^{v}, \underline{η}(c^{v})] + [0, \underline{η} − \underline{η}(c)] .

□

(4.7) Finally, we will take a short look at the geometrical situation reached at this point. The linear map

\begin{matrix}C̃(Q)^{∨} ∩ (ZZ^{n} × V_{Z Z}^{∗}) & ⟶ ⟶ & σ^{∨} ∩ ZZ^{n+1} \\ [c, \underline{η}] & ↦ & [c, ∑_{i} η_{i}] \end{matrix}

is surjective ([c, \underline{η}(c)] ↦ [c, η_{0}(c)]; [0, e_{i}] ↦ [0, 1]). Since x^{[c,\underline{η}]} − x^{[c,\underline{η}′]} = x^{[c,\underline{η}(c)]} · (x^{[0,\underline{η}−\underline{η}(c)]} − x^{[0,\underline{η}′−\underline{η}(c)]}), the kernel of the corresponding homomorphism between the semigroup algebras equals the ideal

(x^{[0,\underline{η}]} − x^{[0,\underline{η}′]} | ∑_{i} η_{i} = ∑_{i} η′_{i}).

In particular, Y ↪ X is a closed embedding. Moreover, looking at the similar statement concerning C(Q)^{∨} and IN at the end of (4.4), we see that this map equals the pull back of I C ↪ S as claimed in (4.2). The elements [c^{1}, \underline{η}(c^{1})],…, [c^{w}, \underline{η}(c^{w})] ∈ C̃(Q) induce some regular functions Z_{1},…, Z_{w} on X. They define a closed embedding X ↪ I C^{w} × S lifting the embedding Y ↪ I C^{w+1} of (4.3).

X ↪ I C^{w} × S
↑  ↑
Y ↪ I C^{w} × I C

Moreover, for i = 1,…, N, Z_{i} is the only monomial function lifting z_{i} from Y to X.

5 A flat family over M̄

(5.1) Theorem: Denote by X̄ and S̄ the scheme theoretical images of X and S in I C^{w} × I C^{N} and I C^{N}, respectively. Then,

(1) X → X̄ and S → S̄ are the normalization maps.

(2) π : X → S induces a map π̄ : X̄ → S̄, and π can be recovered from π̄ via base change S → S̄.

(3) Restricting to M ⊆ S̄ and composing with ℓ turns π̄ into a family X̄ ×S̄ M ⟶^{π̄} M ⟶^{ℓ} M̄. It is flat in 0 ∈ M̄ ⊆ I C^{N−1}, and the special fiber equals Y.

The proof of this theorem will fill §5.

(5.2) The ring of regular functions A(S̄) is given as the image of the map I C[t_{1},…, t_{N}] → A(S). Since ZZ^{N} ⟶ ⟶ V_{Z Z}^{∗} is surjective, the rings A(S̄) ⊆ A(S) ⊆ I C[V_{Z Z}^{∗}] have the same field of fractions. On the other hand, while t-monomials with negative exponents might be involved in A(S), the surjectivity of IR_{≥0}^{N} ⟶ ⟶ C(Q)^{∨} tells us that sufficiently high powers of these monomials always come from A(S̄). In particular, A(S) is normal over A(S̄). A(X̄) is given as the image A(X̄) = im ( I C[Z_{1},…, Z_{w}, t_{1},…, t_{N}] → A(X)). Since A(X) is generated by Z_{1},…, Z_{w} over its subring A(S) (cf. Proposition (4.6)(3)), the same arguments as for S and S̄ apply. Hence, Part (1) of the previous theorem is proved.

(5.3) Recalling that z_{1},…, z_{w}, t ∈ A(Y) stand for the monomials with exponents [c^{1}, η_{0}(c^{1})], …,[c^{w}, η_{0}(c^{w})], [0, 1] ∈ σ^{∨} ∩ ZZ^{n+1}, respectively, we obtain the following equations describing Y as a subset of I C^{w+1}:

f_{(a,b,α,β)}(\underline{z}, t) := t^{α} ∏_{v=1}^{w} z_{v}^{a_{v}} − t^{β} ∏_{v=1}^{w} z_{v}^{b_{v}}
with \begin{matrix}a, b ∈ IN^{w} : ∑_{v} a_{v} c^{v} = ∑_{v} b_{v} c^{v} and \\ α, β ∈ IN : ∑_{v} a_{v} η_{0}(c^{v}) + α = ∑_{v} b_{v} η_{0}(c^{v}) + β . \end{matrix}

Defining c := ∑_{v} a_{v} c^{v} = ∑_{v} b_{v} c^{v} we can lift f_{(a,b,α,β)} to the following element of A(S̄)[Z_{1},…, Z_{w}] (described via the map I C[Z_{1},…, Z_{w}, t_{1},…, t_{N}] ⟶ ⟶ A(S̄)[Z_{1},…, Z_{w}]):

F_{(a,b,α,β)}(\underline{Z}, \underline{t}) := f_{(a,b,α,β)}(\underline{Z}, t_{1}) − \underline{Z}^{[c,\underline{η}(c)]} · (\underline{t}^{αe_{1}+∑_{v} a_{v} \underline{η}(c^{v})} − \underline{t}^{βe_{1}+∑_{v} b_{v} \underline{η}(c^{v})}) · \underline{t}^{−\underline{η}(c)} .

Remark: (1) The symbol \underline{Z}^{[c,\underline{η}(c)]} means ∏_{v=1}^{w} Z_{v}^{p_{v}} with natural numbers p_{v} ∈ IN such that [c, \underline{η}(c)] = ∑_{v} p_{v} [c^{v}, \underline{η}(c^{v})] or equivalently [c, η_{0}(c)] = ∑_{v} p_{v} [c^{v}, η_{0}(c^{v})]. This condition does not determine the coefficients p_{v} uniquely. Any choice satisfying the equation will do. Choosing other coefficients q_{v} with the same property yields Z_{1}^{p_{1}} · . . . · Z_{w}^{p_{w}} − Z_{1}^{q_{1}} · . . . · Z_{w}^{q_{w}} = F_{(p,q,0,0)}(\underline{Z}, \underline{t}) = f_{(p,q,0,0)}(\underline{Z}, t), anyway.

(2) By part (iii) of Lemma (4.5), we have ∑_{v} a_{v}\underline{η}(c^{v}), ∑_{v} b_{v}\underline{η}(c^{v}) ⪰ \underline{η}(c). In particular, representatives of the \underline{η}’s can be chosen such that all t-exponents occurring in monomials of F are non-negative, i.e. F indeed defines an element of A(S̄)[Z_{1},…, Z_{w}].

Lemma: The polynomials F_{(a,b,α,β)} generate ker (A(S̄)[\underline{Z}] → A(X)), i.e. they can be used as equations for X̄ ⊆ I C^{w} × S̄.

Proof: Recall first that the map from A(S̄)[\underline{Z}] to A(X) = ⊕_{[c,\underline{η}]} I Cx^{[c,\underline{η}]} , where [c, \underline{η}] runs through

all elements of C̃(Q)^{∨} ∩ (ZZ^{n} × V_{Z Z}^{∗}), sends Z_{v} ↦ x^{[c^{v},\underline{η}(c^{v})]} and t_{i} ↦ x^{[0,e_{i}]}. Hence,

F_{(a,b,α,β)} = \begin{matrix}(t_{1}^{α} ∏_{v} Z_{v}^{a_{v}} − \underline{Z}^{[c,\underline{η}(c)]} \underline{t}^{αe_{1}+∑_{v} a_{v} \underline{η}(c^{v})−\underline{η}(c)}) − \\ − (t_{1}^{β} ∏_{v} Z_{v}^{b_{v}} − \underline{Z}^{[c,\underline{η}(c)]} \underline{t}^{βe_{1}+∑_{v} b_{v} \underline{η}(c^{v})−\underline{η}(c)}) \end{matrix}

↦ \begin{matrix}(x^{α[0,e_{1}]+∑_{v} a_{v}[c^{v},\underline{η}(c^{v})]} − x^{[c,\underline{η}(c)]+α[0,e_{1}]+∑_{v} a_{v}[0,\underline{η}(c^{v})]−[0,\underline{η}(c)]}) − \\ − (x^{β[0,e_{1}]+∑_{v} b_{v}[c^{v},\underline{η}(c^{v})]} − x^{[c,\underline{η}(c)]+β[0,e_{1}]+∑_{v} b_{v}[0,\underline{η}(c^{v})]−[0,\underline{η}(c)]}) \end{matrix}

= 0 − 0 = 0 .

On the other hand, ker (A(S̄)[\underline{Z}] → A(X)) is obviously generated by the binomials

\underline{t}\underline{η} Z_{1}^{a_{1}} ·. . . · Z_{w}^{a_{w}} − \underline{t}\underline{μ} Z_{1}^{b_{1}} ·. . . · Z_{w}^{b_{w}} such that

\begin{matrix}∑_{v} a_{v}[c^{v}, \underline{η}(c^{v})] + [0, \underline{η}] = ∑_{v} b_{v}[c^{v}, \underline{η}(c^{v})] + [0, \underline{μ}] , \\ i.e. & \begin{matrix}• & c := ∑_{v} a_{v} c^{v} = ∑_{v} b_{v} c^{v} \\ • & ∑_{v} a_{v} \underline{η}(c^{v}) + \underline{η} = ∑_{v} b_{v} \underline{η}(c^{v}) + \underline{μ} . \end{matrix} \end{matrix}

However,

\underline{t}\underline{η} \underline{Z}^{a} − \underline{t}\underline{μ} \underline{Z}^{b} = \begin{matrix}\underline{t}\underline{η} · (∏_{v} Z_{v}^{a_{v}} − \underline{Z}^{[c,\underline{η}(c)]} \underline{t}^{∑_{v} a_{v} \underline{η}(c^{v})−\underline{η}(c)}) − \\ −\underline{t}\underline{μ} · (∏_{v} Z_{v}^{b_{v}} − \underline{Z}^{[c,\underline{η}(c)]} \underline{t}^{∑_{v} b_{v} \underline{η}(c^{v})−\underline{η}(c)}) \end{matrix}

= \underline{t}\underline{η} · F_{(a,p,0,α)} − \underline{t}\underline{μ} · F_{(b,p,0,β)}

with p ∈ IN^{w} such that ∑_{v} p_{v}[c^{v}, \underline{η}(c^{v})] = [c, \underline{η}(c)], α = ∑_{v} a_{v}η_{0}(c^{v})−η_{0}(c), and β = ∑_{v} b_{v}η_{0}(c^{v})− η_{0}(c). □

(5.4) Using exponents \underline{η}, \underline{μ} ∈ ZZ^{N} (instead of IN^{N}), the binomials \underline{t}\underline{η} \underline{Z}^{a} − \underline{t}\underline{μ} \underline{Z}^{b} generate the kernel of the map

A(S)[\underline{Z}] = A(S̄)[\underline{Z}] ⊗_{A(S̄)} A(S) ⟶ ⟶ A(X̄) ⊗_{A(S̄)} A(S) ⟶ ⟶ A(X) .

Since \underline{Z}^{a} ⊗ \underline{t}\underline{η} − \underline{Z}^{b} ⊗ \underline{t}\underline{μ} = \underline{Z}^{[c,\underline{η}(c)]} ⊗ (\underline{t}^{∑_{v} a_{v} \underline{η}(c^{v})−\underline{η}(c)+\underline{η}} − \underline{t}^{∑_{v} b_{v} \underline{η}(c^{v})−\underline{η}(c)+\underline{μ}}) = 0 in A(X̄) ⊗_{A(S̄)}

A(S), this implies that the surjection A(X̄) ⊗_{A(S̄)} A(S) ⟶ ⟶ A(X) is injective, too. In particular, part (2) of our theorem is proved.

(5.5) We are going to use the following well known criterion of flatness:

Theorem: ([Ma], (20.C), Theorem 49) Let π̃ : X̃ ↪ I C^{w+1} × M̄ ⟶ ⟶ M̄ be a map with special fiber Y = π̃^{−1}(0); in particular, Y ⊆ I C^{w+1} is defined by the restrictions of the equations defining X̃ ⊆ I C^{w+1}×M̄ to 0 ∈ M̄. Then, π̃ is flat if and only if each linear relation between the (restricted) equations for Y lifts to some linear relation between the original equations for X̃.

For our special situation take X̃ := X̄ ×S̄ M (and M̄ := M̄, Y := Y); in (5.3) we have seen how the equations defining Y ↪ I C^{w} × I C can be lifted to those defining X̄ ↪ I C^{w} × S̄, hence X̄ ×S̄ M ↪ I C^{w} × M →^{∼} I C^{w} × I C × M̄. In particular, to show (3) of Theorem (5.1), we just have to determine the linear relations between the f_{(a,b,α,β)}’s and lift them to relations between the F_{(a,b,α,β)}’s. There are three types of relations between the f_{(a,b,α,β)}’s:

(i) f_{(a,r,α,γ)} + f_{(r,b,γ,β)} = f_{(a,b,α,β)}

with \begin{matrix}• & ∑_{v} a_{v}c^{v} = ∑_{v} r_{v}c^{v} = ∑ b_{v}c^{v} and \\ • & ∑_{v} a_{v}η_{0}(c^{v}) + α = ∑_{v} r_{v}η_{0}(c^{v}) + γ = ∑_{v} b_{v}η_{0}(c^{v}) + β . \end{matrix}

For this relation, the same equation between the F ’s is true.

(ii) t · f_{(a,b,α,β)} = f_{(a,b,α+1,β+1)} lifts to t_{1} · F_{(a,b,α,β)} = F_{(a,b,α+1,β+1)}.

(iii) \underline{z}^{r} · f_{(a,b,α,β)} = f_{(a+r,b+r,α,β)}.

With c := ∑_{v} a_{v}c^{v} = ∑_{v} b_{v}c^{v}, c̃:= c + ∑_{v} r_{v}c^{v} we obtain

\underline{Z}^{r} · F_{(a,b,α,β)} − F_{(a+r,b+r,α,β)} =

\begin{matrix}= \underline{Z}^{[c̃,\underline{η}(c̃)]} · (\underline{t}^{αe_{1}+∑_{v} a_{v} \underline{η}(c^{v})+∑_{v} r_{v} \underline{η}(c^{v})} − \underline{t}^{βe_{1}+∑_{v} b_{v} \underline{η}(c^{v})+∑_{v} r_{v} \underline{η}(c^{v})}) · \underline{t}^{−\underline{η}(c̃)}− \\ −\underline{Z}^{[c,\underline{η}(c)]} \underline{Z}^{r} · (\underline{t}^{αe_{1}+∑_{v} a_{v} \underline{η}(c^{v})} − \underline{t}^{βe_{1}+∑_{v} b_{v} \underline{η}(c^{v})}) · \underline{t}^{−\underline{η}(c)} \end{matrix}

\begin{matrix}= (\underline{t}^{αe_{1}+∑_{v} a_{v} \underline{η}(c^{v})−\underline{η}(c)} − \underline{t}^{βe_{1}+∑_{v} b_{v} \underline{η}(c^{v})−\underline{η}(c)}) · \\ (\underline{t}^{\underline{η}(c)+∑_{v} r_{v} \underline{η}(c^{v})−\underline{η}(c̃)}\underline{Z}^{[c̃,\underline{η}(c̃)]} − \underline{Z}^{[c,\underline{η}(c)]}\underline{Z}^{r}) . \end{matrix}

Now, the inequalities

∑_{v} a_{v}\underline{η}(c^{v}), ∑_{v} b_{v}\underline{η}(c^{v}) ⪰ \underline{η}(c) and \underline{η}(c) + ∑_{v} r_{v}\underline{η}(c^{v}) − \underline{η}(c̃) ⪰ 0

imply that

– the first factor is contained in the ideal defining 0 ∈ M̄, and

– the second factor is an equation of X̄ ⊆ I C^{w} × S̄ (called F_{(p̃,p+r,ξ,0)} in (7.4)).

In particular, we have found a lift for the third relation, too.

The proof of Theorem (5.1) is complete.

(5.6) Example: The singularity Y_{6} induced by the hexagon Q_{6} equals the cone over the Del Pezzo surface of degree 6 obtained by blowing up three points of (IP^{2}, O(3)). As a closed subset of I C^{7}, it is given by the following 9 equations:

\begin{matrix}f_{(e_{1},e_{6}+e_{2},1,0)} = z_{1} t − z_{6} z_{2}, & f_{(e_{2},e_{1}+e_{3},1,0)} = z_{2} t − z_{1} z_{3}, & f_{(e_{3},e_{2}+e_{4},1,0)} = z_{3} t − z_{2} z_{4}, \\ f_{(e_{4},e_{3}+e_{5},1,0)} = z_{4} t − z_{3} z_{5}, & f_{(e_{5},e_{4}+e_{6},1,0)} = z_{5} t − z_{4} z_{6}, & f_{(e_{6},e_{5}+e_{1},1,0)} = z_{6} t − z_{5} z_{1}, \\ f_{(\underline{0},e_{1}+e_{4},2,0)} = t^{2} − z_{1} z_{4}, & f_{(\underline{0},e_{2}+e_{5},2,0)} = t^{2} − z_{2} z_{5}, & f_{(\underline{0},e_{3}+e_{6},2,0)} = t^{2} − z_{3} z_{6} . \end{matrix}

Then, the construction described in (5.3) yields the liftings

\begin{aligned}F_{(e_{1},e_{6}+e_{2},1,0)} & = & (Z_{1} t_{1} − Z_{6} Z_{2}) − Z_{1}(t_{1} − t_{1}) & = & Z_{1} t_{1} − Z_{6} Z_{2}, \\ F_{(e_{2},e_{1}+e_{3},1,0)} & = & (Z_{2} t_{1} − Z_{1} Z_{3}) − Z_{2}(t_{1}^{2} − t_{1} t_{2}) t_{1}^{−1} & = & Z_{2} t_{2} − Z_{1} Z_{3}, \\ F_{(e_{3},e_{2}+e_{4},1,0)} & = & (Z_{3} t_{1} − Z_{2} Z_{4}) − Z_{3}(t_{1}^{2}t_{2} − t_{1} t_{2} t_{3}) t_{1}^{−1}t_{2}^{−1} & = & Z_{3} t_{3} − Z_{2} Z_{4}, \\ F_{(e_{4},e_{3}+e_{5},1,0)} & = & (Z_{4} t_{1} − Z_{3} Z_{5}) − Z_{4}(t_{1} t_{2} t_{3} − t_{2} t_{3} t_{4}) t_{2}^{−1}t_{3}^{−1} & = & Z_{4} t_{4} − Z_{3} Z_{5}, \\ F_{(e_{5},e_{4}+e_{6},1,0)} & = & (Z_{5} t_{1} − Z_{4} Z_{6}) − Z_{5}(t_{1} t_{6} − t_{2} t_{3}) t_{6}^{−1} & = & Z_{5} t_{5} − Z_{4} Z_{6}, \\ F_{(e_{6},e_{5}+e_{1},1,0)} & = & (Z_{6} t_{1} − Z_{5} Z_{1}) − Z_{6}(t_{1} − t_{6}) & = & Z_{6} t_{6} − Z_{5} Z_{1}, \\ F_{(\underline{0},e_{1}+e_{4},2,0)} & = & (t_{1}^{2} − Z_{1} Z_{4}) − (t_{1}^{2} − t_{2} t_{3}) = t_{2} t_{3} − Z_{1} Z_{4} & = & t_{5} t_{6} − Z_{1} Z_{4}, \\ F_{(\underline{0},e_{2}+e_{5},2,0)} & = & (t_{1}^{2} − Z_{2} Z_{5}) − (t_{1}^{2} − t_{3} t_{4}) & = & t_{3} t_{4} − Z_{2} Z_{5}, \\ F_{(\underline{0},e_{3}+e_{6},2,0)} & = & (t_{1}^{2} − Z_{3} Z_{6}) − (t_{1}^{2} − t_{1} t_{2}) & = & t_{1} t_{2} − Z_{3} Z_{6} . \end{aligned}

Together with the four equations mentioned at the end of (2.3), they describe a family contained

in I C^{6} × I C^{6} \underrightarrow{pr_{2}} I C^{6}/ I C · (1,…, 1).

6 The Kodaira-Spencer map

(6.1) Denote by E ⊆ σ^{∨} ∩ ZZ^{n+1} the minimal generating set

E := \{[c^{1}, η_{0}(c^{1})],…, [c^{w}, η_{0}(c^{w})], [\underline{0}, 1]\}

mentioned in (4.3). To each vertex a^{j} ∈ Q (or identically named fundamental generator a^{j} := (a^{j}, 1) ∈ σ) and each element R ∈ ZZ^{n+1} we associate the subset

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

Theorem: (cf. [Al 1]) The vector space T_{Y}^{1} of infinitesimal deformations of Y is ZZ^{n+1}-graded, and in degree −R it equals

T_{Y}^{1} (−R) = ( ^{L_{I C} (∪_{j}E_{j}^{R})/}∑_{j} L_{I C}(E_{j}^{R}))^{∗} ,

where L(…) denotes the vector space of linear relations.

(6.2) There is a special degree R^{∗} = [\underline{0}, 1] ∈ ZZ^{n+1} corresponding to the affine hyperplane containing Q. The associated subsets of E equal

E_{j}^{R^{∗}} = E ∩ (a^{j})^{⊥} = {[c^{v}, η_{0}(c^{v})] | ⟨a^{j}, −c^{v}⟩ = η_{0}(c^{v})} .

In (4.5), for each c ∈ ZZ^{n}, we have defined the linear form \underline{η}(c) ∈ V_{Z Z}^{∗}. Restricted to the cone C(Q), it maps \underline{t} to Max⟨Q\underline{t}, −c⟩ = ⟨a(c)\underline{t}, −c⟩. This induces the following bilinear map:

\begin{aligned}Φ : & V^{Z Z}/(1,…, 1) & × & L_{Z Z}(E ∩ ∂σ^{∨}) & ⟶ & ZZ \\ & \underline{t} & , & q & ↦ & ∑_{v,i} t_{i} q_{v} η_{i}(c^{v}) . \end{aligned}

(Indeed, for \underline{t} := \underline{1} we obtain ∑_{v,i} q_{v} η_{i}(c^{v}) = ∑_{v} q_{v} η_{0}(c^{v}) = 0 since q ∈ L_{Z Z}(E ∩ ∂σ^{∨}).) Moreover, if q comes from one of the submodules L_{Z Z}(E_{j}^{R^{∗}}) ⊆ L_{Z Z}(E ∩ ∂σ^{∨}), we obtain

Φ(\underline{t}, q) = ∑_{v} q_{v} · Max⟨Q\underline{t}, −c^{v}⟩ = ∑_{v} q_{v} · ⟨aj ; \underline{t}, −c^{v}⟩
= ⟨aj ; \underline{t}, − ∑_{v} q_{v}c^{v}⟩ = 0 .

Theorem: The Kodaira-Spencer map of the family X̄ ×S̄ M → M̄ of §5 equals the map

T_{0}M̄ =V^{I C}/(1,…, 1) ⟶ ( ^{L^{I C}(E ∩ ∂σ^{∨})/}∑_{j} L_{I C}(E_{j}^{R^{∗}}))^{∗} = T_{Y}^{1} (−R^{∗})

induced by the previous pairing. Moreover, this map is an isomorphism.

Proof: Using the same symbol J for the ideal J ⊆ I C[t_{1},…, t_{N}] and the intersection J ∩ I C[t_{i} − t_{j} | 1 ≤ i, j ≤ N] (cf. (2.4)), our family corresponds to the flat I C[t_{i} − t_{j}]/J -module I C[\underline{Z}, \underline{t}]/(J , F_{•}(\underline{Z}, \underline{t})). Now, we fix a non-trivial tangent vector \underline{t}^{0} ∈ V_{I C}. Via t_{i} ↦ t + t_{i}^{0} ε it induces the infinitesimal family given by the flat I C[ε]/ε^{2}-module

A\underline{t}^{0} := I C[\underline{z}, t, ε]/(ε^{2}, F_{•}(\underline{z}, t + \underline{t}^{0} ε)) .

To obtain the associated A(Y)-linear map I/I^{2} → A(Y), where I := (f_{•}(\underline{z}, t)) denotes the ideal of Y in I C^{w+1}, we have to compute the images of f_{•}(\underline{z}, t) in ε A(Y) ⊆ A\underline{t}^{0} and divide them by ε: Using the notation of (5.3), in A\underline{t}^{0}

0 = F_{(a,b,α,β)}(\underline{z}, t + \underline{t}^{0} ε)
= \begin{matrix}f_{(a,b,α,β)}(\underline{z}, t + t_{1}^{0} ε)− \\ −\underline{z}^{[c,\underline{η}(c)]} · ((t + \underline{t}^{0} ε)^{αe_{1}+∑_{v} a_{v} \underline{η}(c^{v})−\underline{η}(c)} − (t + \underline{t}^{0} ε)^{βe_{1}+∑_{v} b_{v} \underline{η}(c^{v})−\underline{η}(c)}) . \end{matrix}

The relation ε^{2} = 0 yields

f_{(a,b,α,β)}(\underline{z}, t + t_{1}^{0}ε) = f_{(a,b,α,β)}(\underline{z}, t) + ε · (α t^{α−1} t_{1}^{0} \underline{z}^{a} − β t^{β−1} t_{1}^{0} \underline{z}^{b}) ,

and similarly we can expand the other terms. Eventually, we obtain

\begin{aligned}f_{(a,b,α,β)}(\underline{z}, t) & = & \begin{matrix}−ε t_{1}^{0} (α t^{α−1} \underline{z}^{a} − β t^{β−1} \underline{z}^{b}) + ε \underline{z}^{[c,\underline{η}(c)]} t^{α+∑_{v} a_{v}η_{0}(c^{v})−η_{0}(c)−1}· \\ · [t_{1}^{0} (α − β) + ∑_{i} t_{i}^{0} (∑_{v}(a_{v} − b_{v})η_{i}(c^{v}))] \end{matrix} \\ & = & ε · x^{∑_{v} a_{v}[c^{v},η_{0}(c^{v})]+[0,α−1]} · ( ∑_{i} t_{i}^{0} ( ∑_{v}(a_{v} − b_{v})η_{i}(c^{v}))) . \end{aligned}

(In ε A(Y) we were able to replace the variables t and z_{i} by x^{[\underline{0},1]} and x^{[c^{v},η_{0}(c^{v})]}, respectively.)

On the other hand, we use Theorem (3.4) of [Al 3]: Fixing R^{∗} ∈ ZZ^{n+1}, the element of L_{I C}(E∩∂σ^{∨})^{∗} given by q ↦ ∑_{i,v} t_{i}^{0} q_{v} η_{i}(c^{v}) corresponds to the infinitesimal deformation of T_{Y}^{1} (−R^{∗}) defined by the map

\begin{matrix}I_{/I^{2}} & ⟶ & A(Y) \\ t^{α} \underline{z}^{a} − t^{β} \underline{z}^{b} & ↦ & ( ∑_{i,v} t_{i}^{0} (a_{v} − b_{v})η_{i}(c^{v})) · x^{∑_{v} a_{v}[c^{v},η_{0}(c^{v})]+[0,α−1]} . \end{matrix}

□

(6.3) To discuss the meaning of the homogeneous part T_{Y}^{1} (−R^{∗}) inside the whole vector space T_{Y}^{1} , we have to look at the results of [Al 2], (6.5): If dim T_{Y}^{1} < ∞ (for instance, if Y has an isolated singularity), then

(1) T_{Y}^{1} = T_{Y}^{1} (−R^{∗}), but

(2) T_{Y}^{1} = 0 for dim Y ≥ 4.

In particular, the interesting cases arise from 2-dimensional lattice polygons Q with primitive edges only. The corresponding 3-dimensional toric varieties Y have an isolated singularity, and the Kodaira-Spencer map T_{0}M̄ → T_{Y}^{1} is an isomorphism.

If T_{Y}^{1} has infinite dimension, then this comes from the existence of infinitely many non-trivial homogeneous pieces T_{Y}^{1} (−R). Whenever ⟨a^{j}, R⟩ ≤ 1 holds for all vertices a^{j} ∈ Q, we have

T_{Y}^{1} (−R) = V_{I C}(conv{a^{j} | ⟨a^{j}, R⟩ = 1}) ,

i.e. T_{Y}^{1} (−R) equals the vector space of Minkowski summands of some face of Q, whereas T_{Y}^{1} (−R) = 0 for all other R ∈ ZZ^{n+1}. In particular, T_{Y}^{1} (−R^{∗}) is a typical, but nevertheless extremal and perhaps the most interesting part of T_{Y}^{1} .

7 The obstruction map

(7.1) Dealing with obstructions in the deformation theory of Y involves the A(Y)-module T_{Y}^{2} . Usually, it is defined in the following way:

Let m := \{([a, α], [b, β]) ∈ IN^{w+1} × IN^{w+1} | \begin{matrix}∑_{v} a_{v} c^{v} = ∑_{v} b_{v} c^{v}; \\ ∑_{v} a_{v} η_{0}(c^{v}) + α = ∑_{v} b_{v} η_{0}(c^{v}) + β\} \end{matrix}

denote the set parametrizing the equations f_{(a,b,α,β)} generating the ideal I ⊆ I C[\underline{z}, t] of Y. Then,

R := ker ( I C[\underline{z}, t]^{m} ⟶ ⟶ I)

is the module of linear relations between these equations; it contains the submodule R_{0} of the so-called Koszul relations.

Definition: T_{Y}^{2} := Hom (R/R_{0}, A(Y))/ ; Hom ( I C[\underline{z}, t]^{m}, A(Y)) .

Now, we have a similar theorem for T_{Y}^{2} as we had in (6.1) for T_{Y}^{1} ; in particular, we use the notation introduced there.

Theorem: (cf. [Al 3]) The vector space T_{Y}^{2} is ZZ^{n+1}-graded, and in degree −R it equals

T_{Y}^{2} (−R) = (\begin{matrix}ker & (⊕_{j}L_{I C}(E_{j}^{R}) ⟶ L_{I C}(E)) \\ im & (⊕_{⟨a^{i},a^{j}⟩<Q}L_{I C}(E_{i}^{R} ∩ E_{j}^{R}) → ⊕_{i}L_{I C}(E_{i}^{R})) \end{matrix})^{∗} .

(7.2) In this section we build up the so-called obstruction map. It detects all possible infinitesimal extensions of our family over M̄ to a flat family over some larger base space. We follow the explanation given in §4 of [JS]. As before,

J = (g_{ε,k}(\underline{t} − t_{1}) | ε < Q, k ≥ 1) = (g_{\underline{d},k}(\underline{t} − t_{1}) | \underline{d} ∈ V^{⊥} ∩ ZZ^{N}, k ≥ 1) ⊆ I C[t_{i} − t_{j}]

denotes the homogeneous ideal of the base space M̄. Let

J̃ := (t_{i} − t_{j})_{i,j} · J + J_{1} · I C[t_{i} − t_{j}] ⊆ I C[t_{i} − t_{j} | 1 ≤ i, j ≤ N] .

Then, W := J/ ; J̃ is a finite-dimensional, ZZ-graded vector space (W = ⊕_{k≥2}W_{k}, and W_{k} is generated by the polynomials g_{\underline{d},k}(\underline{t} − t_{1})). It is the kernel of the exact sequence

0 → W ⟶ I C[t^{i} − t^{j}]/ ; J̃ ⟶ I C[t^{i} − t^{j}]/ ; J → 0 .

Identifying t with t_{1} and \underline{z} with \underline{Z}, the tensor product with I C[\underline{z}, t] (over I C) yields the important, exact sequence

0 → W ⊗_{I C} I C[\underline{z}, t] ⟶ I C[\underline{Z}, \underline{t}]/ ; J̃ · I C[\underline{Z}, \underline{t}] ⟶ I C[\underline{Z}, \underline{t}]/ ; J · I C[\underline{Z}, \underline{t}] → 0 .

Now, let s be any relation with coefficients in I C[\underline{z}, t] between the equations f_{(a,b,α,β)}, i.e.

∑ s_{(a,b,α,β)} f_{(a,b,α,β)} = 0 in I C[\underline{z}, t] .

By flatness of our family (cf. (5.5)), the components of s can be lifted to I C[\underline{Z}, \underline{t}] obtaining an s̃ such that

λ(s) := ∑ s̃_{(a,b,α,β)} F_{(a,b,α,β)} ↦ 0 in I C[\underline{Z}, \underline{t}]/ ; J · I C[\underline{Z}, \underline{t}] .

In particular, each relation s ∈ R induces some element λ(s) ∈ W ⊗_{I C} I C[\underline{z}, t], which is well defined after the additional projection to W ⊗_{I C} A(Y). This procedure describes a certain element λ ∈ T_{Y}^{2} ⊗_{I C} W = Hom(W^{∗}, T_{Y}^{2} ) called the obstruction map.

Theorem: The obstruction map λ : W^{∗} → T_{Y}^{2} is injective.

Corollary: If dim T_{Y}^{1} < ∞, our family equals the versal deformation of Y. In general, we could say that it is “versal in degree −R^{∗}”.

Proof: In (6.2) we have proved that the Kodaira-Spencer map is an isomorphism (at least onto the homogeneous piece T_{Y}^{1} (−R^{∗})). By a criterion also described in [JS], this fact combined with injectivity of the obstruction map implies versality. □

The remaining part of §7 contains the proof of the previous theorem.

(7.3) We have to improve the notation of §4 and §5. Since M̄ ⊆ S̄ ⊆ I C^{N}, we were able to use the toric equations (cf. (2.4)) during computations modulo J. In particular, the exponents \underline{η} ∈ ZZ^{N} of \underline{t} only needed to be known modulo V^{⊥}; it was enough to define \underline{η}(c) as elements of V_{Z Z}^{∗}. However, to compute the obstruction map, we have to deal with the smaller ideal J̃ ⊆ J. Let us start by refining the definitions of (4.5):

(i) For each vertex a ∈ Q we choose the following paths through the 1-skeleton of Q:

• \underline{λ}(a) := path from 0 ∈ Q to a ∈ Q .

• \underline{μ}^{v}(a) := path from a ∈ Q to a(c^{v}) ∈ Q such that μ_{i}^{v}(a)⟨d^{i}, c^{v}⟩ ≤ 0 for each i = 1,…, N.

• \underline{λ}^{v}(a) := \underline{λ}(a) + \underline{μ}^{v}(a) is then a path from 0 ∈ Q to a(c^{v}), which depends on a.

(ii) For each c ∈ ZZ^{n} we use the vertex a(c) to define

\underline{η}^{c}(c) := [−λ_{1}(a(c))⟨d^{1}, c⟩,…, −λ_{N}(a(c))⟨d^{N}, c⟩] ∈ ZZ^{N}

and

\underline{η}^{c}(c^{v}) := [−λ_{1}^{v}(a(c))⟨d^{1}, c^{v}⟩,…, −λ_{N}^{v}(a(c))⟨d^{N}, c^{v}⟩] ∈ ZZ^{N} .

(iii) For each c ∈ ZZ^{n} we fix a representation c = ∑_{v} p_{v}^{c} c^{v} (p_{v}^{c} ∈ IN) such that η_{0}(c) = ∑_{v} p_{v}^{c} η_{0}(c^{v}). (That means, c is represented only by those generators c^{v} that define faces of Q containing the face defined by c itself.)

Remark: Let a ∈ IN^{w}. Denoting c := ∑_{v} a_{v}c^{v} we obtain ∑_{v} a_{v} \underline{η}^{c}(c^{v}) − \underline{η}^{c}(c) ∈ IN^{N} by arguments as in Lemma (4.5). Moreover, for the special representation c = ∑_{v} p_{v}^{c}c^{v}, the equation ∑_{v} p_{v}^{c} \underline{η}^{c}(c^{v}) = \underline{η}^{c}(c) is true.

Now, we improve the definition of the polynomials F_{•}(\underline{Z}, \underline{t}) given in (5.3). Let a, b ∈ IN^{w}, α, β ∈ IN such that

c := ∑_{v} a_{v} c^{v} = ∑_{v} b_{v} c^{v} and ∑_{v} a_{v} η_{0}(c^{v}) + α = ∑_{v} b_{v} η_{0}(c^{v}) + β .

Then,

F_{(a,b,α,β)}(\underline{Z}, \underline{t}) := f_{(a,b,α,β)}(\underline{Z}, t_{1}) − \underline{Z}^{p^{c}} · (\underline{t}^{αe_{1}+∑_{v} a_{v} \underline{η}^{c}(c^{v})−\underline{η}^{c}(c)} − \underline{t}^{βe_{1}+∑_{v} b_{v} \underline{η}^{c}(c^{v})−\underline{η}^{c}(c)}) .

(7.4) We have to discuss the same three types of relations as we did in (5.5). Since there is only one single element c ∈ ZZ^{n} involved in the relations (i) and (ii), calculating modulo J̃ instead of J makes no difference in these cases - we always obtain λ(s) = 0. Let us regard the relation s := [\underline{z}^{r} · f_{(a,b,α,β)} − f_{(a+r,b+r,α,β)} = 0] (r ∈ IN^{w}). We will use the following notation:

• c := ∑_{v} a_{v} c^{v} = ∑_{v} b_{v} c^{v}; \underline{p} := \underline{p}^{c}; \underline{η} := \underline{η}^{c};

• c̃:= ∑_{v}(a_{v} + r_{v}) c^{v} = ∑_{v}(b_{v} + r_{v}) c^{v} = ∑_{v}(p_{v} + r_{v}) c^{v}; \underline{p̃}:= \underline{p}c̃; ˜ \underline{η} := \underline{η}c̃;

• ξ := ∑_{i} ((∑_{v}(p_{v} + r_{v})η̃_{i}(c^{v})) − η̃_{i}(c̃)) = ∑_{v}(p_{v} + r_{v})η_{0}(c^{v}) − η_{0}(c̃) .

Using the same lifting of s to s̃ as in (5.5) yields

λ(s) = \begin{matrix}\underline{Z}^{r} · F_{(a,b,α,β)} − F_{(a+r,b+r,α,β)} − \\ − (\underline{t}^{αe_{1}+∑_{v} a_{v} \underline{η}(c^{v})−\underline{η}(c)} − \underline{t}^{βe_{1}+∑_{v} b_{v} \underline{η}(c^{v})−\underline{η}(c)}) · F_{(p̃,p+r,ξ,0)} \end{matrix}

= \begin{matrix}−\underline{Z}^{p+r} · (\underline{t}^{αe_{1}+∑_{v}(a_{v}−p_{v})\underline{η}(c^{v})} − \underline{t}^{βe_{1}+∑_{v}(b_{v}−p_{v})\underline{η}(c^{v})}) + \\ + \underline{Z}p̃ · (\underline{t}^{αe_{1}+∑_{v}(a_{v}+r_{v}−p̃_{v})\underline{η̃}(c^{v})} − \underline{t}^{βe_{1}+∑_{v}(b_{v}+r_{v}−p̃_{v})\underline{η̃}(c^{v})}) − \\ − (\underline{t}^{αe_{1}+∑_{v}(a_{v}−p_{v})\underline{η}(c^{v})} − \underline{t}^{βe_{1}+∑_{v}(b_{v}−p_{v})\underline{η}(c^{v})}) · (\underline{Z}p̃ \underline{t}^{∑_{v}(p_{v}+r_{v}−p̃_{v})\underline{η̃}(c^{v})} − \underline{Z}^{p+r}) \end{matrix}

= \begin{matrix}\underline{Z}p̃ · (\underline{t}^{αe_{1}+∑_{v}(a_{v}+r_{v}−p̃_{v})\underline{η̃}(c^{v})} − \underline{t}^{αe_{1}+∑_{v}(p_{v}+r_{v}−p̃_{v})\underline{η̃}(c^{v})+∑_{v}(a_{v}−p_{v})\underline{η}(c^{v})}) − \\ −\underline{Z}p̃ · (\underline{t}^{βe_{1}+∑_{v}(b_{v}+r_{v}−p̃_{v})\underline{η̃}(c^{v})} − \underline{t}^{βe_{1}+∑_{v}(p_{v}+r_{v}−p̃_{v})\underline{η̃}(c^{v})+∑_{v}(b_{v}−p_{v})\underline{η}(c^{v})}) . \end{matrix}

As in (5.5)(iii), we can see that λ(s) vanishes modulo J (or even in A(S̄)) merely by identifying \underline{η} and \underline{η̃}.

(7.5) In (7.2) we have already mentioned the isomorphism

W ⊗_{I C} I C[\underline{z}, t] ⟶^{∼} J · I C[\underline{Z}, \underline{t}]/ ; J̃ · I C[\underline{Z}, \underline{t}]

obtained by identifying t with t_{1} and \underline{z} with \underline{Z}. Now, with λ(s), we have obtained an element of the right hand side, which has to be interpreted as an element of W ⊗_{I C} I C[\underline{z}, t].

Lemma: Let A, B ∈ IN^{N} such that \underline{d} := A − B ∈ V^{⊥}, i.e. \underline{t}^{A} − \underline{t}^{B} ∈ J · I C[\underline{Z}, \underline{t}]. Then, via the previously mentioned isomorphism, \underline{t}^{A} − \underline{t}^{B} corresponds to the element

∑_{k≥1} c_{k} · g_{\underline{d},k}(\underline{t} − t_{1}) · t^{k_{0}−k} ∈ W ⊗_{I C} I C[\underline{z}, t] ,

where k_{0} := ∑_{i} A_{i}, and c_{k} are the constants occurred in (3.4). In particular, the coefficients from W_{k} vanish for k > k_{0}.

Proof: First, we remark that we may assume that A = \underline{d}^{+}, B = \underline{d}^{−}, i.e. \underline{t}^{A} − \underline{t}^{B} = p\underline{d}(\underline{t}) (cf. (3.2)). Otherwise we could write this binomial as

\underline{t}^{A} − \underline{t}^{B} = \underline{t}^{C} · (\underline{t}^{\underline{d}^{+}} − \underline{t}^{\underline{d}^{−}}) (C ∈ IN^{N}),

and since

\underline{t}^{C} = (t_{1} + [\underline{t} − t_{1}])^{C} ≡ t_{1}^{∑_{i} C_{i}} (mod (t_{i} − t_{j})) ,

we would obtain

\underline{t}^{A} − \underline{t}^{B} ≡ t_{1}^{∑_{i} C_{i}} · (\underline{t}^{\underline{d}^{+}} − \underline{t}^{\underline{d}^{−}}) (mod J̃) .

In (3.4) we have seen that

p\underline{d}(\underline{t}) = ∑_{k=1}^{k_{0}} t_{1}^{k_{0}−k} · (∑_{v=1}^{k−1} q_{v,k}(\underline{t} − t_{1}) · g_{\underline{d},v}(\underline{t} − t_{1}) + c_{k} · g_{\underline{d},k}(\underline{t} − t_{1}))

with k_{0} := ∑_{i} d_{i}^{+}. Since q_{v,k}(\underline{t} − t_{1}) ∈ (t_{i} − t_{j}) · I C[t_{i} − t_{j}], this implies

p\underline{d}(\underline{t}) ≡ ∑_{k=1}^{k_{0}} t_{1}^{k_{0}−k} · c_{k} · g_{\underline{d},k}(\underline{t} − t_{1}) (mod J̃) .

On the other hand, for k > k_{0}, Lemma (3.3) tells us that g_{\underline{d},k}(\underline{t} − t_{1}) is a I C[t_{i} − t_{j}]-linear combination of the elements g_{\underline{d},1}(\underline{t} − t_{1}),…, g_{\underline{d},k_{0}}(\underline{t} − t_{1}). Then, the degree k part of the corresponding equation shows g_{\underline{d},k}(\underline{t} − t_{1}) ∈ J̃. □

Corollary: Transferred to W ⊗_{I C} I C[\underline{z}, t], the element λ(s) equals

∑_{k≥1} c_{k} · g_{\underline{d},k}(\underline{t} − t_{1}) · \underline{z}p̃ · t^{k_{0}−k} with \begin{aligned}\underline{d} & := & ∑_{v}(a_{v} − b_{v}) · (\underline{η̃}(c^{v}) − \underline{η}(c^{v})) , \\ k_{0} & := & α + ∑_{v}(a_{v} + r_{v}) η_{0}(c^{v}) − η_{0}(c̃) . \end{aligned}

The coefficients vanish for k > k_{0}.

Proof: We apply the previous lemma to both summands of the λ(s)-formula of (7.4). For the first one we obtain

\underline{d}^{a} = [αe_{1} + ∑_{v}(a_{v} + r_{v} − p̃_{v}) ˜ \underline{η}(c^{v})] − [αe_{1} + ∑_{v}(p_{v} + r_{v} − p̃_{v}) ˜ \underline{η}(c^{v}) + ∑_{v}(a_{v} − p_{v}) \underline{η}(c^{v})]
= ∑_{v}(a_{v} − p_{v}) · (\underline{η̃}(c^{v}) − \underline{η}(c^{v}))  and
k_{0} = ∑_{i} (αe_{1} + ∑_{v}(a_{v} + r_{v} − p̃_{v}) \underline{η̃}(c^{v}))_{i}
= α + ∑_{v}(a_{v} + r_{v} − p̃_{v}) η_{0}(c^{v}) = α + ∑_{v}(a_{v} + r_{v}) η_{0}(c^{v}) − η_{0}(c̃) .

k_{0} has the same value for both the a- and b-summand, and

\begin{aligned}\underline{d} = \underline{d}^{a} − \underline{d}^{b} & = & ∑_{v}(a_{v} − p_{v}) · (\underline{η̃}(c^{v}) − \underline{η}(c^{v})) − ∑_{v}(b_{v} − p_{v}) · (\underline{η̃}(c^{v}) − \underline{η}(c^{v})) \\ & = & ∑_{v}(a_{v} − b_{v}) · (\underline{η̃}(c^{v}) − \underline{η}(c^{v})) . \end{aligned}

□

(7.6) Now, we try to approach the obstruction map λ from the opposite direction. Using the description of T_{Y}^{2} given in (7.1), we construct an element of T_{Y}^{2} ⊗_{I C} W that afterwards will turn out to equal λ.

For a path ϱ ∈ ZZ^{N} along the edges of Q, we denote by

\underline{d}(ϱ, c) := [⟨ϱ_{1} d^{1}, c⟩,…, ⟨ϱ_{N} d^{N}, c⟩] ∈ ZZ^{N}

the vector measuring the behavior of c ∈ ZZ^{n} passing each particular edge. If, moreover, ϱ comes from a closed path, \underline{d}(ϱ, c) is also contained in V^{⊥}. On the other hand, for each k ≥ 1, we can use the \underline{d}’s from V^{⊥} to get elements g_{\underline{d},k}(\underline{t} − t_{1}) ∈ W_{k} generating this vector space. Composing both procedures we obtain, for each closed path ϱ ∈ ZZ^{N}, a map

\begin{aligned}g^{(k)}(ϱ, •) : & IR^{n} & ⟶ & V^{⊥} & ⟶ & W_{k} \\ & c & & ↦ & & g_{\underline{d}(ϱ,c),k}(\underline{t} − t_{1}) . \end{aligned}

Lemma:

(1) Taking the sum over all 2-faces we get a surjective map

∑_{ε<Q} g^{(k)}(\underline{ε}, •) : ⊕_{ε<Q} I C^{n} ⟶ ⟶ W_{k} .

(2) Let c ∈ ZZ^{n} (having integer coordinates is very important here). If ϱ^{1}, ϱ^{2} ∈ ZZ^{N} are two paths each connecting vertices a, b ∈ Q such that

• |⟨a, c⟩ − ⟨b, c⟩| ≤ k − 1 and

• c is monotone along both paths, i.e. ⟨ϱ_{i}^{1} d^{i}, c⟩; ⟨ϱ_{i}^{2} d^{i}, c⟩ ≥ 0 for i = 1,…, N,

then ϱ^{1} − ϱ^{2} ∈ ZZ^{N} will be a closed path yielding g^{(k)}(ϱ^{1} − ϱ^{2}, c) = 0 in W_{k}.

Proof: (1) is a consequence of the fact that the elements \underline{d}(ε, c) (ε < Q 2-face; c ∈ ZZ^{n}) generate V^{⊥} as a vector space. For the proof of (2), we look at \underline{d} := \underline{d}(ϱ^{1} − ϱ^{2}, c). Since d_{i} = ⟨ϱ_{i}^{1} d^{i}, c⟩ − ⟨ϱ_{i}^{2} d^{i}, c⟩ is the difference of two non-negative integers, we obtain d_{i}^{+} ≤ ⟨ϱ_{i}^{1} d^{i}, c⟩. Hence,

∑_{i} d_{i}^{+} ≤ ∑_{i}⟨ϱ_{i}^{1} d^{i}, c⟩ = ⟨b, c⟩ − ⟨a, c⟩ ≤ k − 1 ,

and as in (7.5) we obtain g_{\underline{d},k}(\underline{t} − t_{1}) ∈ J̃ by Lemma (3.3). □

(7.7) Using the notation introduced in (6.1) we obtain for R := k R^{∗}, k ≥ 2

E_{j}^{kR^{∗}} = {[c^{v}, η_{0}(c^{v})] | ⟨a^{j}, c^{v}⟩ + η_{0}(c^{v}) ≤ k − 1} ∪ {R^{∗}} ⊆ σ^{∨} ∩ ZZ^{n+1} .

Then, we can define the following linear maps :

\begin{aligned}ψ_{j}^{(k)} : & L(E_{j}^{kR^{∗}}) & ⟶ & W_{k} \\ & q & ↦ & ∑_{v} q_{v} · g^{(k)} (\underline{λ}(a^{j}) + \underline{μ}^{v}(a^{j}) − \underline{λ}(a(c^{v})), c^{v}) . \end{aligned}

(The q-coordinate corresponding to R^{∗} ∈ E_{j}^{kR^{∗}} is not used in the definition of ψ_{j}^{(k)}.)

Lemma: Let ⟨a^{i}, a^{j}⟩ < Q be an edge of the polyhedron Q. Then, on L(E_{i}^{kR^{∗}} ∩ E_{j}^{kR^{∗}}) = L(E_{i}^{kR^{∗}}) ∩ L(E_{j}^{kR^{∗}}), the maps ψ_{i}^{(k)} and ψ_{j}^{(k)} coincide. In particular (cf. Theorem (7.1)), the ψ_{j}^{(k)}’s induce a linear map ψ^{(k)} : T_{Y}^{2} (−kR^{∗})^{∗} → W_{k}.

Proof: Let q ∈ L(E_{i}^{kR^{∗}} ∩ E_{j}^{kR^{∗}}), and denote by ϱ^{ij} ∈ ZZ^{N} the path consisting of the single edge running from a^{i} to a^{j}. Then,

\begin{aligned}ψ_{i}^{(k)}(q) − ψ_{j}^{(k)}(q) & = & ∑_{v} q_{v} · g^{(k)} (\underline{λ}(a^{i}) + \underline{μ}^{v}(a^{i}) − \underline{λ}(a^{j}) − \underline{μ}^{v}(a^{j}), c^{v}) \\ & = & \begin{matrix}g^{(k)} (\underline{λ}(a^{i}) − \underline{λ}(a^{j}) + ϱ^{ij}, ∑_{v} q_{v} c^{v}) + \\ + ∑_{v} q_{v} · g^{(k)} (\underline{μ}^{v}(a^{i}) − \underline{μ}^{v}(a^{j}) − ϱ^{ij}, c^{v}) , \end{matrix} \end{aligned}

and both summands vanish for several reasons. The first one is killed simply by the equality ∑_{v} q_{v} c^{v} = 0. For the second one we can use (2) of the previous lemma: If q_{v} ≠ 0, then the assumption about q implies the inequalities

0 ≤ ⟨a^{i}, c^{v}⟩ − ⟨a(c^{v}), c^{v}⟩; ⟨a^{j}, c^{v}⟩ − ⟨a(c^{v}), c^{v}⟩ ≤ k − 1 .

Hence, assuming w.l.o.g. ⟨a^{i}, c^{v}⟩ ≥ ⟨a^{j}, c^{v}⟩, we can take ϱ^{1} := −\underline{μ}^{v}(a^{j}) − ϱ^{ij} and ϱ^{2} := −\underline{μ}^{v}(a^{i}) to see that g^{(k)} (\underline{μ}^{v}(a^{i}) − \underline{μ}^{v}(a^{j}) − ϱ^{ij}, c^{v}) = 0. □

(7.8) Proposition: ∑_{k≥1} c_{k} ψ^{(k)} equals λ^{∗}, the adjoint of the obstruction map.

Proof: In Theorem (3.5) of [Al 3] we gave a dictionary between the two T^{2}-formulas mentioned in (7.1). Using this result we can find an element of Hom(R/R_{0}, W_{k} ⊗ A(Y)) representing ψ^{(k)} ∈ T_{Y}^{2} ⊗ W_{k}. It sends relations of type (i) (cf. (5.5)) to 0 and deals with relations of type (ii) and (iii) in the following way:

[\underline{z}^{r} t^{γ} · f_{(a,b,α,β)} − f_{(a+r,b+r,α+γ,β+γ)} = 0] ↦ ψ_{j}^{(k)}(a − b) · x^{∑_{v}(a_{v}+r_{v})[c^{v},η_{0}(c^{v})]+(α+γ−k)R^{∗}}

if

⟨(Q, 1), ∑_{v}(a_{v} + r_{v}) [c^{v}, η_{0}(c^{v})] + (α + γ − k)R^{∗}⟩ ≥ 0

and j is such that

⟨(a^{j}, 1), ∑_{v} a_{v} [c^{v}, η_{0}(c^{v})] + (α − k)R^{∗}⟩ < 0 ;

otherwise the relation is sent to 0 (in particular, if there is not any j meeting the desired property).

On Q, the linear forms c := ∑_{v} a_{v} c^{v} and c̃ = ∑_{v}(a_{v} + r_{v})c^{v} admit their minimal values at the vertices a(c) and a(c̃), respectively. Hence, we can transform the previous formula into

[\underline{z}^{r} t^{γ} · f_{(a,b,α,β)} − f_{(a+r,b+r,α+γ,β+γ)} = 0] ↦ ψ_{a(c)}^{(k)}(a − b) · x^{∑_{v}(a_{v}+r_{v})[c^{v},η_{0}(c^{v})]+(α+γ−k)R^{∗}} ; \begin{aligned}if & \begin{matrix}∑_{v}(a_{v} + r_{v})η_{0}(c^{v}) − η_{0}(c̃) + (α + γ − k) = \\ = ⟨(a(c̃), 1), ∑_{v}(a_{v} + r_{v}) [c^{v}, η_{0}(c^{v})] + (α + γ − k)R^{∗}⟩ ≥ 0 , \end{matrix} \\ & ∑_{v} a_{v} η_{0}(c^{v}) − η_{0}(c) + (α − k) = ⟨(a(c), 1), ∑_{v} a_{v} [c^{v}, η_{0}(c^{v})] + (α − k)R^{∗}⟩ < 0 \end{aligned}

(or mapping to 0 otherwise). Adding the coboundary h ∈ Hom ( I C[\underline{z}, t]^{m}, W_{k} ⊗ A(Y))

h_{(a,α),(b,β)} := \{ \begin{matrix}ψ_{a(c)}^{(k)}(a − b) · x^{∑_{v} a_{v}[c^{v},η_{0}(c^{v})]+(α−k)R^{∗}} & for ∑_{v} a_{v} η_{0}(c^{v}) − η_{0}(c) + α ≥ k , \\ 0 & otherwise \end{matrix}

does not change the class in T_{Y}^{2} (−kR^{∗}), which still represents ψ^{(k)}, but it does improve the representative from Hom(R/R_{0}, W_{k} ⊗ A(Y)). It still maps type-(i)-relations to 0, and moreover

[\underline{z}^{r} t^{γ} · f_{(a,b,α,β)} − f_{(a+r,b+r,α+γ,β+γ)} = 0] ↦

↦ \{ \begin{matrix}(ψ_{a(c)}^{(k)}(a − b) − ψ_{a(c̃)}^{(k)}(a − b)) · x^{∑_{v}(a_{v}+r_{v})[c^{v},η_{0}(c^{v})]+(α+γ−k)R^{∗}} & for k_{0} + γ ≥ k \\ 0 & otherwise \end{matrix}

with k_{0} = α + ∑_{v}(a_{v} + r_{v}) η_{0}(c^{v}) − η_{0}(c̃). By definition of ψ_{j}^{(k)} and g^{(k)} we obtain

ψ_{a(c)}^{(k)}(a − b) − ψ_{a(c̃)}^{(k)}(a − b) =

= ∑_{v}(a_{v} − b_{v}) · g^{(k)} (\underline{λ}(a(c)) + \underline{μ}^{v}(a(c)) − \underline{λ}(a(c̃)) − \underline{μ}^{v}(a(c̃)), c^{v}) = ∑_{v}(a_{v} − b_{v}) · g^{(k)} (\underline{λ}^{v}(a(c)) − \underline{λ}^{v}(a(c̃)), c^{v})

= g_{\underline{d}, k}(\underline{t} − t_{1}) with \begin{aligned}\underline{d} & = & ∑_{v}(a_{v} − b_{v}) · \underline{d} (\underline{λ}^{v}(a(c)) − \underline{λ}^{v}(a(c̃)), c^{v}) \\ & = & ∑_{v}(a_{v} − b_{v}) · (˜\underline{η}(c^{v}) − \underline{η}(c^{v})) , \end{aligned}

and this completes our proof. Indeed,

• for relations of type (ii) (i.e. r = 0; γ = 1) we know c = c̃, hence, these relations map onto 0;

• for relations of type (iii) (i.e. γ = 0) we compare the previous formula with the result obtained in Corollary (7.5) - the coefficients coincide, and the monomial \underline{z}p̃ t^{k_{0}−k} ∈ I C[\underline{z}, t] maps onto

x^{∑_{v}(a_{v}+r_{v})[c^{v},η_{0}(c^{v})]+(α+γ−k)R^{∗}} ∈ A(Y). □

(7.9) It remains to show that the summands ψ^{(k)} of λ^{∗} are indeed surjective maps from T_{Y}^{2} (−kR^{∗})^{∗} to W_{k}. We will do so by composing them with auxiliary surjective maps p^{k} : ⊕_{ε<Q} I C^{n} ⟶ ⟶ T_{Y}^{2} (−kR^{∗})^{∗} yielding ψ^{(k)} ∘ p^{k} = ∑_{ε<Q} g^{(k)}(\underline{ε}, •). Then, the result follows from the first part of Lemma (7.6).

In §6 of [Al 3] we used a short exact sequence of complexes called

0 → L_{I C}(E^{R})_{•} ⟶ ( I C^{E^{R}})_{•} ⟶ span_{I C}(E^{R})_{•} → 0

to obtain from Theorem (7.1) an isomorphism

T_{Y}^{2} (−R) ≅ (\frac{im [⊕_{ε<Q} I C^{n+1} → ⊕_{⟨a^{i},a^{j}⟩<Q} I C^{n+1}]}{im [⊕_{ε<Q}span_{I C} (∩_{a^{j}∈ε}E_{j}^{R}) → ⊕_{⟨a^{i},a^{j}⟩<Q} I C^{n+1}]})^{∗} .

Since R^{∗} = [\underline{0}, 1] ∈ E_{j}^{kR^{∗}} for k ≥ 2, the induced surjective map ⊕_{ε<Q} I C^{n+1} ⟶ ⟶ T_{Y}^{2} (−kR^{∗})^{∗} factorizes through ⊕_{ε<Q} I C^{n+1}/I C · R^{∗} = ⊕_{ε<Q} I C^{n} yielding the auxiliary map p^{k} just mentioned. Taking a closer look at the construction of [Al 3] §6, we can give an explicit description of p^{k}; eventually we will be able to compute ψ^{(k)} ∘ p^{k}.

Let us fix some 2-face ε < Q. Assume that d^{1},…, d^{M} are its counterclockwise oriented edges, i.e. the sign vector \underline{ε} looks like ε_{i} = 1 for i = 1,…, M and ε_{j} = 0 otherwise. Moreover, we denote the vertices of ε < Q by a^{1},…, a^{M} such that d^{i} runs from a^{i} to a^{i+1} (M + 1 := 1). Starting with a [c, η_{0}] ∈ I C^{n+1} (and, as just mentioned, only the c ∈ I C^{n} is essential) we have to proceed as follows:

(i) For i = 1,…, M we represent [c, η_{0}] as a linear combination of elements of E_{i}^{kR^{∗}} ∩ E_{i+1}^{kR^{∗}}, which corresponds to the lifting from span_{I C}(E^{R})_{•} to ( I C^{E^{R}})_{•}.

[c, η_{0}] = ∑_{v} q_{iv} [c^{v}, η_{0}(c^{v})] + q_{i} [\underline{0}, 1] ,

and q_{iv} ≠ 0 implies [c^{v}, η_{0}(c^{v})] ∈ E_{i}^{kR∗} ∩ E_{i+1}^{kR^{∗}}, i.e.

⟨a^{i}, c^{v}⟩ + η_{0}(c^{v}) ≤ k − 1 ; ⟨a^{i+1}, c^{v}⟩ + η_{0}(c^{v}) ≤ k − 1 .

(ii) We map the result to ⊕_{i=1}^{M} I C^{E_{i}^{kR^{∗}}} by taking successive differences, corresponding to the application of the differential in the complex ( I C^{E^{R}})_{•}. The result is automatically contained in ker (⊕_{i}L(E_{i}^{kR^{∗}}) → L(E)), and its i-th summand is the linear relation

∑_{v}(q_{i,v} − q_{i−1,v}) · [c^{v}, η_{0}(c^{v})] + (q_{i} − q_{i−1}) · [\underline{0}, 1] = 0 .

(iii) Finally, we apply ψ^{(k)} to obtain

ψ^{(k)}(p^{k}(c)) = ∑_{i=1}^{M} ∑_{v}(q_{i,v} − q_{i−1,v}) · g^{(k)} (\underline{λ}(a^{i}) − \underline{λ}(a(c^{v})) + \underline{μ}^{v}(a^{i}), c^{v})
= \begin{matrix}∑_{i,v} g^{(k)} (\underline{λ}(a^{i}) − \underline{λ}(a(c^{v})) + \underline{μ}^{v}(a^{i}), q_{i,v} c^{v}) − \\ − ∑_{i,v} g^{(k)} (\underline{λ}(a^{i+1}) − \underline{λ}(a(c^{v})) + \underline{μ}^{v}(a^{i+1}), q_{i,v} c^{v}) \end{matrix}

= ∑_{i,v} g^{(k)} (\underline{λ}(a^{i}) − \underline{λ}(a^{i+1}) + \underline{μ}^{v}(a^{i}) − \underline{μ}^{v}(a^{i+1}), q_{i,v} c^{v}) .

Similar to the proof of Lemma (7.7) we introduce the path ϱ^{i} consisting of the single edge d^{i} only. Then, if q_{iv} ≠ 0 and w.l.o.g. ⟨a^{i}, c^{v}⟩ ≥ ⟨a^{i+1}, c^{v}⟩, the pair of paths \underline{μ}^{v}(a^{i}) and \underline{μ}^{v}(a^{i+1}) + ϱ^{i} meets the assumption of Lemma (7.6)(2) (cf. (i)). Hence, we can proceed as follows:

ψ^{(k)}(p^{k}(c)) = \begin{matrix}∑_{i,v} g^{(k)} (\underline{λ}(a^{i}) − \underline{λ}(a^{i+1}) + ϱ^{i}, q_{iv} c^{v}) + \\ + ∑_{i,v} g^{(k)} (\underline{μ}^{v}(a^{i}) − \underline{μ}^{v}(a^{i+1}) − ϱ^{i}, q_{iv} c^{v}) \end{matrix}

= ∑_{i=1}^{M} g^{(k)} (\underline{λ}(a^{i}) − \underline{λ}(a^{i+1}) + ϱ^{i}, ∑_{v} q_{iv} c^{v})
= ∑_{i=1}^{M} g^{(k)} (\underline{λ}(a^{i}) − \underline{λ}(a^{i+1}) + ϱ^{i}, c)
= g^{(k)} (∑_{i=1}^{M} ϱ^{i}, c)
= g^{(k)}(\underline{ε}, c) .

Hence, Theorem (7.2) is proven.

8 The components of the reduced versal family

(8.1) The components of the reduced base space M̄_{red} correspond to maximal decompositions of Q into a Minkowski sum Q = R_{0} +…+R_{m} with lattice polytopes R_{k} ⊆ IR^{n} as summands. Intersections of components are obtained by the finest Minkowski decompositions of Q that are coarser than all the maximal ones involved.

Theorem: Fix such a Minkowski decomposition. Then, the corresponding component (or inter-

section of components) M̄_{0} is isomorphic to I C^{m+1}/I C · (1,…, 1), and the restriction X_{0} → I C^{m} of the versal family can be described as follows:

(i) Defining the cone

σ̃ := Cone ( ⋃_{k=0}^{m}(R_{k} × {e^{k}})) ⊆ IR^{n+m+1} ,

it contains σ = Cone (Q×{1}) ⊆ IR^{n+1} via the diagonal embedding IR^{n+1} ↪ IR^{n+m+1} ((a, 1) ↦ (a; 1,…, 1)). The inclusion σ ⊆ σ̃ induces a closed embedding of the affine toric varieties defined by these cones, giving Y ↪ X_{0}.

(ii) The projection IR^{n+m+1} ⟶ ⟶ IR^{m+1} provides m + 1 regular functions on X_{0}, i.e. we obtain a map X_{0} → I C^{m+1}. Composing this map with ℓ : I C^{m+1} ⟶ ⟶ I C^{m+1}/I C · (1,…, 1) yields the family.

The theorem is a straight consequence of knowing the versal deformation. Hence, we omit the proof here.

(8.2) Example: At the end of (2.5) we presented two decompositions of Q_{6} into a Minkowski sum of lattice summands. Let us describe now the restrictions of the versal family to the associated components of M̄:

(i) Putting the two triangles R_{0}, R_{1} into two parallel planes contained in IR^{3} yields an octahedron as the convex hull of the whole configuration. Then, σ̃ is the (4-dimensional) cone over this octahedron

σ̃ = ⟨(0, 0; 1, 0), (1, 0; 1, 0), (1, 1; 1, 0), (0, 0; 0, 1), (0, 1; 0, 1), (1, 1; 0, 1)⟩ .

(ii) Looking at the second decomposition, we have to put three line segments R_{0}, R_{1}, R_{2} on three parallel 2-planes in general position inside the affine space IR^{4}. Taking the convex hull of this configuration yields a 4-dimensional polytope that is dual to (triangle)×(triangle). Again, σ̃ is the (5-dimensional) cone over this polytope

σ̃ = ⟨(0, 0; 1, 0, 0), (1, 0; 1, 0, 0), (0, 0; 0, 1, 0), (0, 1; 0, 1, 0), (0, 0; 0, 0, 1), (1, 1; 0, 0, 1)⟩ .

The total spaces over the components arise as the toric varieties defined by σ̃. In our example, they equal the cones over IP^{1} × IP^{1} × IP^{1} and IP^{2} × IP^{2}, respectively.

9 Further examples

(9.1) Three examples of toric Gorenstein singularities arise as cones over the Del Pezzo surfaces obtained by blowing up (IP^{2}, O(3)) in one, two, or three points, respectively. They correspond to the following polygons:

[figure: Polygon Q_{4}; Polygon Q_{5}; Polygon Q_{6}]

Let us discuss these three examples:

(iv) The edges equal

d^{1} = (1, 0), d^{2} = (1, 2), d^{3} = (−2, −1), d^{4} = (0, −1) ,

and they imply the following equations of the versal base space as closed subscheme of I C^{4}/I C · (1, 1, 1, 1):

t_{1} + t_{2} = 2t_{3}, t_{3} + t_{4} = 2t_{2}, t_{1}^{2} + t_{2}^{2} = 2t_{3}^{2}, t_{3}^{2} + t_{4}^{2} = 2t_{2}^{2} .

Using the two linear equations, only two coordinates t := t_{1}, ε := t_{1} − t_{3} are sufficient. (We get the t_{i}’s back by t_{1} = t, t_{2} = t − 2ε, t_{3} = t − ε, t_{4} = t − 3ε.) Then, the two quadratic equations transform into 2ε^{2} = 0, i.e. the versal base space is a fat point. On the other hand, Q_{4} does not allow any splitting into a Minkowski sum involving lattice summands only. This reflects the triviality of the underlying reduced space. (Cf. (9.2).)

(v) The polygon Q_{5} allows the decomposition into the sum of a triangle and a line segment. In particular, the reduced base space of the versal deformation of Y_{5} has to be a line. We compute the true base space: d^{1} = (1, 1), d^{2} = (−1, 1), d^{3} = (−1, 0), d^{4} = (0, −1), d^{5} = (1, −1) yield the equations

t_{1} − t_{3} = t_{2} − t_{5} = t_{4} − t_{1} and t_{1}^{2} − t_{3}^{2} = t_{2}^{2} − t_{5}^{2} = t_{4}^{2} − t_{1}^{2} .

With t := t_{1}, s_{1} := t_{1} − t_{3}, s_{2} := t_{1} − t_{2} and t_{1} = t, t_{2} = t − s_{2}, t_{3} = t − s_{1}, t_{4} = t + s_{1}, t_{5} = t − s_{1} − s_{2}, they turn into

s_{1}^{2} = 2s_{1}s_{2} = 0 .

(vi) This example was spread throughout the paper.

(9.2) We will use the polygon Q_{4} := Conv {(0, 0), (1, 0), (2, 2), (0, 1)} of (9.1)(iv) for a more detailed demonstration of how the theory works. In particular, we will describe the versal family of Y_{4} over Spec I C[ε]/ε^{2}:

(1) The (t, ε)-coordinates of V correspond to the linear map

( \begin{matrix}1 & 0 \\ 1 & −2 \\ 1 & −1 \\ 1 & −3 \end{matrix} ) : IR^{2} ⟶^{∼} V ↪ IR^{4} .

We obtain

C(Q_{4}) = {(a, b) ∈ IR^{2} | a ≥ 0, a − 2b ≥ 0, a − b ≥ 0, a − 3b ≥ 0}
= {(a, b) ∈ IR^{2} | a ≥ 0, a − 3b ≥ 0}
= ⟨[1, 0], [1, −3]⟩^{∨} = ⟨(0, −1), (3, 1)⟩ ⊆ IR^{2} ,

and the map IN^{4} → C(Q_{4})^{∨} ∩ V_{Z Z}^{∗} sends e_{1}, e_{2}, e_{3}, e_{4} to [1, 0], [1, −2], [1, −1], [1, −3], respectively. In particular, this map is surjective, i.e. S_{4} = S̄_{4} and X_{4} = X̄_{4}.

(2) To compute the tautological cone C̃(Q_{4}), we need the Minkowski summands associated to the two fundamental generators of C(Q_{4}):

(Q_{4})_{(0,−1)} = Conv{(0, 0), (2, 4), (0, 3)}, (Q_{4})_{(3,1)} = Conv{(0, 0), (3, 0), (4, 2)} .

Hence,

C̃(Q_{4}) = ⟨(0, 0; 0, −1); (2, 4; 0, −1); (0, 3; 0, −1); (0, 0; 3, 1); (3, 0; 3, 1); (4, 2; 3, 1)⟩ .

(3) Now, we have all the information needed to obtain the versal family of Y_{4}:

– Restrict the family Spec I C[C̃(Q_{4})^{∨}∩ZZ^{4}] → Spec I C[C(Q_{4})^{∨}∩ZZ^{2}] ⊆ I C^{4} to the subspace I C^{2} ≃ V_{I C} ⊆ I C^{4}, i.e. use the (t, ε)-coordinates instead of (t_{1}, t_{2}, t_{3}, t_{4}).

– Compose the result with the projection I C^{2} ⟶ ⟶ I C^{1} ((t, ε) ↦ ε). That means we no longer regard t as a coordinate of the base space.

– Finally, we restrict our family to the closed subscheme defined by the equation ε^{2} = 0.

(4) To obtain equations, we could either take a closer look to the family constructed so far, or we can proceed more directly as described in (4.5) and (5.3):

– Computing the minimal generator set of the semigroup Cone (Q_{4})^{∨} ∩ ZZ^{3}, we get the elements [c^{v}; η_{0}(c^{v})]:

\begin{matrix}[c^{1}; η_{0}^{1}] = [0, 1; 0], & [c^{2}; η_{0}^{2}] = [−1, 1; 1], & [c^{3}; η_{0}^{3}] = [−2, 1; 2], & [c^{4}; η_{0}^{4}] = [−1, 0; 2], \\ [c^{5}; η_{0}^{5}] = [0, −1; 2], & [c^{6}; η_{0}^{6}] = [1, −2; 2], & [c^{7}; η_{0}^{7}] = [1, −1; 1], & [c^{8}; η_{0}^{8}] = [1, 0; 0]. \end{matrix}

[figure: Polygon Q4^{∨}; z_{3}; z_{4}; z_{5}; z_{6}; z_{7}; z_{8}; z_{1}; z_{2}; t]

Together with [0, 0; 1], they induce coordinates z_{1},…, z_{8}, t on Y_{4}, i.e. we have obtained an embedding Y_{4} ↪ I C^{9}. (The sums of the three components of the vectors are always 1. In the figure we have drawn the first two coordinates.)

– Y_{4} ⊆ I C^{9} is defined by the following 20 polynomials:

\begin{matrix}t^{2} − z_{4}z_{8}, & t^{2} − z_{1}z_{5}, & t^{2} − z_{2}z_{7}, & z_{1}t − z_{2}z_{8}, & z_{2}t − z_{3}z_{8}, \\ z_{2}t − z_{1}z_{4}, & z_{3}t − z_{2}z_{4}, & z_{4}t − z_{3}z_{7}, & z_{4}t − z_{2}z_{5}, & z_{5}t − z_{4}z_{7}, \\ z_{5}t − z_{2}z_{6}, & z_{6}t − z_{5}z_{7}, & z_{7}t − z_{5}z_{8}, & z_{7}t − z_{1}z_{6}, & z_{8}t − z_{1}z_{7}, \\ z_{1}z_{3} − z_{2}^{2}, & z_{3}z_{5} − z_{4}^{2}, & z_{4}z_{6} − z_{5}^{2}, & z_{6}z_{8} − z_{7}^{2}, & z_{3}z_{6} − z_{4}z_{5}. \end{matrix}

– Choosing paths from (0, 0) ∈ Q_{4} to the other vertices, we obtain the list

\begin{matrix}\underline{η}^{1} = [0, 0, 0, 0], & \underline{η}^{2} = [1, 0, 0, 0], & \underline{η}^{3} = [2, 0, 0, 0], & \underline{η}^{4} = [1, 1, 0, 0] = [0, 0, 2, 0], \\ \underline{η}^{6} = [0, 0, 0, 2], & \underline{η}^{7} = [0, 0, 0, 1], & \underline{η}^{8} = [0, 0, 0, 0], & \underline{η}^{5} = [0, 2, 0, 0] = [0, 0, 1, 1]. \end{matrix}

– Now, we can lift our 20 polynomials to the ring I C[Z_{1},…, Z_{8}, t_{1},…, t_{4}]:

\begin{matrix}t_{1}t_{2} − Z_{4}Z_{8}, & t_{2}^{2} − Z_{1}Z_{5}, & t_{1}t_{4} − Z_{2}Z_{7}, & Z_{1}t_{1} − Z_{2}Z_{8}, & Z_{2}t_{1} − Z_{3}Z_{8}, \\ Z_{2}t_{2} − Z_{1}Z_{4}, & Z_{3}t_{2} − Z_{2}Z_{4}, & Z_{4}t_{3} − Z_{3}Z_{7}, & Z_{4}t_{2} − Z_{2}Z_{5}, & Z_{5}t_{3} − Z_{4}Z_{7}, \\ Z_{5}t_{2} − Z_{2}Z_{6}, & Z_{6}t_{3} − Z_{5}Z_{7}, & Z_{7}t_{3} − Z_{5}Z_{8}, & Z_{7}t_{4} − Z_{1}Z_{6}, & Z_{8}t_{4} − Z_{1}Z_{7}, \\ Z_{1}Z_{3} − Z_{2}^{2}, & Z_{3}Z_{5} − Z_{4}^{2}, & Z_{4}Z_{6} − Z_{5}^{2}, & Z_{6}Z_{8} − Z_{7}^{2}, & Z_{3}Z_{6} − Z_{4}Z_{5}. \end{matrix}

– Finally, we restrict the family to the versal base space by switching to the (t, ε)- coordinates and obeying the equation ε^{2} = 0. Moreover, t is no longer a coordinate of the base space:

\begin{matrix}t(t − 2ε) − z_{4}z_{8}, & t(t − 4ε) − z_{1}z_{5}, & t(t − 3ε) − z_{2}z_{7}, & z_{1}t − z_{2}z_{8}, \\ z_{2}t − z_{3}z_{8}, & z_{2}(t − 2ε) − z_{1}z_{4}, & z_{3}(t − 2ε) − z_{2}z_{4}, & z_{4}(t − ε) − z_{3}z_{7}, \\ z_{4}(t − 2ε) − z_{2}z_{5}, & z_{5}(t − ε) − z_{4}z_{7}, & z_{5}(t − 2ε) − z_{2}z_{6}, & z_{6}(t − ε) − z_{5}z_{7}, \\ z_{7}(t − ε) − z_{5}z_{8}, & z_{7}(t − 3ε) − z_{1}z_{6}, & z_{8}(t − 3ε) − z_{1}z_{7}, & z_{1}z_{3} − z_{2}^{2}, \\ z_{3}z_{5} − z_{4}^{2}, & z_{4}z_{6} − z_{5}^{2}, & z_{6}z_{8} − z_{7}^{2}, & z_{3}z_{6} − z_{4}z_{5}. \end{matrix}

(9.3) At last we want to present an example involving more than only quadratic equations for the versal base space. Let Q_{8} be the “regular” lattice 8-gon; it is contained in two strips of lattice thickness 3.

[figure: Polygon Q_{8}]

Q_{8} admits three maximal Minkowski decompositions into a sum of lattice summands:

(i) Q_{8} = [figure] + [figure] + [figure] + [figure]

(ii) Q_{8} = [figure] + [figure] + [figure]

(iii) Q_{8} = [figure] + [figure] + [figure]

The decompositions (i), (ii) and (i), (iii) are refinements of the coarser decompositions

Q_{8} = [figure] + [figure] and Q_{8} = [figure] + [figure] , respectively. These facts translate directly into the geometry of the reduced base space of the versal deformation of Q_{8}:

• It is embedded in some affine space I C^{5} and equals the union of a 3-plane with two 2-planes (through 0 ∈ I C^{5}).

• The two 2-planes each have a common line with the 3-dimensional component. However, they intersect each other in 0 ∈ I C^{5} only.

On the other hand, we can write down the equations of the true versal base space (as a closed

subscheme of I C^{8}/I C · (1,…, 1)):

t_{1}^{k} + t_{2}^{k} + t_{8}^{k} = t_{4}^{k} + t_{5}^{k} + t_{6}^{k}, t_{2}^{k} + t_{3}^{k} + t_{4}^{k} = t_{6}^{k} + t_{7}^{k} + t_{8}^{k} (k = 1, 2, 3) .

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.: Obstructions in the deformation theory of toric singularities.

Preprint 61 “Europäisches Singularitätenprojekt”, Boston 1994; e-print alg-geom/9405008; to appear in J. Pure Appl. Algebra.

[Ar] Arndt, J.: Verselle Deformationen zyklischer Quotientensingularitäten.

Dissertation, Universität Hamburg, 1988.

[Ch] Christophersen, J.A.: On the Components and Discriminant of the Versal Base Space of Cyclic Quotient Singularities. In: Singularity Theory and its Applications, Warwick 1989, Part I: Geometric Aspects of Singularities, pp. 81-92, Springer-Verlag Berlin Heidelberg, 1991 (LNM 1462).

[Ish] Ishida, M.-N.: Torus embeddings and dualizing complexes.

Tôhoku Math. Journ. 32, 111-146 (1980).

[JS] Jong, T. de, Straten, D. van: On the deformation theory of rational surface singularities with reduced fundamental cycle. J. Algebraic Geometry 3, 117-172 (1994).

[Ke] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I.

Lecture Notes in Mathematics 339, Springer-Verlag, Berlin-Heidelberg-New York, 1973.

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

[Ma] Matsumura, H.: Commutative Algebra.

W.A. Benjamin, Inc., New York 1970.

[Od] Oda, T.: Convex bodies and algebraic geometry.

Ergebnisse der Mathematik und ihrer Grenzgebiete (3/15), Springer-Verlag, 1988.

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

[St] Stevens, J.: On the versal deformation of cyclic quotient singularities.

In: Singularity Theory and its Applications, Warwick 1989, Part I: Geometric Aspects of Singularities, pp. 302-319, Springer-Verlag Berlin Heidelberg, 1991 (LNM 1462).