Deformations of Toric Singularities

Klaus Altmann

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

In the present paper we investigate the deformation theory of toric singularities (i.e. of those singularities occurring in toric varieties). The best results are obtained for isolated toric Gorenstein singularities: We describe their mini-versal (or semi-universal) deformation in several ways (by a combinatorial construction, by equations, and by interpreting its irreducible components). See section (1.2) for a more detailed survey.

My investigations on this subject started during a one-year stay at the university of Kaiserslautern (1990/91). I would like to thank Duco van Straten, Gert-Martin Greuel, and Theo de Jong for turning my interest toward deformation theory and for many fruitful discussions and hints (and the DFG for financial support).

At the same time I was introduced by Kurt Behnke, Jan Christophersen, and Jan Stevens to the latest developments of deforming two-dimensional quotient singularities. This led to the definition of so-called toric deformations and my first attempts of classifying them. Then, during a short stay (in 1992) at Cornell University, discussions with Bernd Sturmfels helped very much to form the idea that seems to be the key to describing toric deformations: Those deformations are closely related to splittings of certain polyhedra into Minkowski sums. I am very greatful to all these people for their support.

Finally, I found the ultimative construction of the versal deformation for isolated, toric Gorenstein singularities during a one-year stay at M.I.T. during the academic year 1993/94. I would like to thank Richard Stanley for giving me the opportunity (and the DAAD for giving me the money) to spend a year at this very stimulating place.

This thesis consists mainly of revised versions of several manuscripts and papers. We have tried to bring the notations into a common standard, and, during the last year, we have added several new sections. Roughly we can give the following description:

• Sections (3.1)-(3.3) and (4.1)-(4.3) consist of the papers [Al 3] and [Al 5].

• Most parts of (3.4), (3.5), and (4.4) are new.

• Chapter 5 together with (2.2), (2.3) form the paper [Al 6].

• Chapter 6 is completely new.

• Chapter 7 (except (7.4.4), which is new) is contained in [Al 4].

Finally, I want to thank Herbert Kurke for constant support and encouragement.

Contents

1 Introduction 4 1.1 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 What is this paper about? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Convex cones, polytopes, and toric varieties 11 2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The Minkowski scheme of a lattice polytope . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Proof of the previous statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 T^{1} and T^{2} for affine toric varieties 24 3.1 The modules T^{1} and T^{2} in general . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 The first T^{i}-formulas for toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Proof of the T^{i}-formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 An alternative to the complex L(E^{R})_{•} . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 First applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 The cup product 51 4.1 The cup product in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 The cup product for toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Proof of the cup product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Alternative cup product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Toric Gorenstein singularities 64 5.1 I Q-Gorenstein cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 The tautological cone over C(Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 A flat family over M̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.4 The Kodaira-Spencer map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 The obstruction map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.6 The components of the reduced versal family . . . . . . . . . . . . . . . . . . . . . 83 5.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Deformations of negative degree 90 6.1 Negative degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.2 The tautological cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.3 A flat family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.4 The Kodaira-Spencer map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5 The obstruction map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.6 Regular functions as variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7 Toric deformations 98

7.1 Toric deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.2 Constructing homogeneous, toric regular sequences . . . . . . . . . . . . . . . . . . 101 7.3 Proof of the previous theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.4 The Kodaira-Spencer map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Bibliography 115

Chapter 1

Introduction

1.1 Deformations

(1.1.1) In the very beginning, deformation theory appeared as the investigation of how complex structures may vary on a fixed compact, smooth manifold. In his famous paper “Theorie der abelschen Funktionen” (1857; cf. [Rm]) Riemann already mentioned the 3g −3 moduli determining the complex structure of an algebraic curve (“Riemann surface”) of genus g ≥ 2.

Looking for an analogeous description in higher dimensions, Kodaira and Spencer started to develop the machinery of deformation theory (cf. [KoSp]). Because of the misery that (beginning in dimension two) descent moduli spaces do not always exist, they used a modified, weaker concept: The versal (or semi-universal) deformation f : X → S of a manifold Y = X_{0} = f^{−1}(0 ∈ S) “contains” all possible deformations; but even the minimal one no longer provides a one-to-one correspondence between fibers and complex structures. Under the assumption that H^{2}(X_{0}, θ_{X_{0}}) = 0 (i.e. no obstructions occur) they have shown that a miniversal deformation always exists. Later, allowing singular spaces for the base, Kuranishi (cf. [Ku]) completed their work and got rid of the H^{2}-assumption. The tangent space of the versal base space S equals H^{1}(X_{0}, θ_{X_{0}}) (i.e. locally S ⊆ H^{1}(X_{0}, θ_{X_{0}})), and the dual vector space H^{2}(X_{0}, θ_{X_{0}})^{∗} provides the (non-linear) equations.

See [Kod] for a detailed description of these results, including the ups and downs during the process of developing this theory.

(1.1.2) In a similar manner, we may regard deformations of germs of analytic spaces (or equivalently, deformations of local, analytic I C-algebras). If Y = (Y, 0) is such a germ (often called “singularity”, since smooth germs are not the interesting ones), we define the following functor:

Def_{Y} ((T, 0)) := {g : Z → T (flat), together with g^{−1}(0) →^{∼} Y } /isomorphisms .

Definition: A deformation f : X → S (i.e. f ∈ Def_{Y} (S)) is called versal, if

(i) every other deformation g : Z → T of Y may be induced via base change from some analytic map T → S, and

(ii) this map may be prescribed on closed, analytic subgerms T′ ⊆ T.

Moreover, f : X → S is called mini-versal, if

(iii) not T → S, but at least its differential map T_{T,0} → T_{S,0} is uniquely determined. (Equivalently: T_{S,0} → Def_{Y} ( I C[ε]/ε^{2}) is an isomorphism.)

It is easy to see that the mini-versal deformation (if it exists) is uniquely determined. Y being a complete intersection, this is a family over a smooth base space obtained by certain perturbations of the defining equations. However, as soon as we leave this class of singularities, the structure of the versal family or even the base space will be more complicated. In (5.7.2) we will present a (three-dimensional) example of an almost rigid singularity admitting a fat point as base space of its versal deformation. Since the whole deformation theory of a singularity is encoded in this construction, the mini-versal deformation (or just its base space) contains important information about the given germ Y. For instance, it may be regarded a source of numerical invariants.

(1.1.3) Good references for facts about deformation theory are Artin’s Lecture notes [Art 2], the large introduction in Palamodov’s paper [Pa], or Steven’s thesis [St 2]. Nevertheless, we would like to mention some important points:

General facts. In [Sch 1] Schlessinger has investigated (in a more general setting) the restriction of functors such as Def_{Y} to the category of local Artinian algebras over I C. He got explicit conditions for the existence of a formal, formally versal object which are easy to check. In case of our deformation functor most of them are met automatically; it remains to have the vector space Def_{Y} ( I C[ε]/ε^{2}) be finite-dimensional, which will be true at least for isolated singularities. Moreover, Schlessinger’s construction leads, on the one hand, to an explicit method of constructing the versal deformation up to some order via power series Ansatz and, on the other hand, to a very useful criterion of versality. The latter one has been explicitly described and used in [Arn] and [JS 4]; for details see (3.1).

As before in the deformation theory of compact analytic spaces, it was easier to prove the existence of an analytic, mini-versal deformations under the assumption that no obstructions occur. Tjurina did so in [Tju] for normal, isolated singularities (assuming Ext_{Y}^{2} (Ω_{Y}^{1} , O_{Y}) = 0). Later, in 1972, Grauert treated the general case in [Gra].

Tangent space and obstructions. It is not difficult to show that the vector spaces T_{Y}^{1} and T_{Y}^{2} (in the form as they are used in (3.1)) equal Def_{Y} ( I C[ε]/ε^{2}) or contain the obstructions for lifting deformations to a higher order, respectively. However, much more effort is necessary to obtain these vector spaces as the first two groups of a general cohomology theory for (commutative) algebras R → A. The crucial step is to define the so-called cotangent complex C_{A/R}^{·} of A-modules. Then, one obtains the functors T^{i} and T_{i} via

T_{i}(A/R; M) := H_{i}(C_{A/R}^{·} ⊗_{A} M) and T^{i}(A/R; M) := H^{i}(Hom_{A}(C_{A/R}^{·}, M)) .

(M is an arbitrary A-module, and the relations to the above mentioned T_{Y}^{i} are T_{Y}^{i} = T^{i}(O_{Y,0}/ I C; O_{Y,0}).) The advantages of this more general point of view are the following:

(1) In the explicit formulas of (3.1), T_{Y}^{1} and T_{Y}^{2} seem to depend on the special embedding of Y into a smooth space I C^{w+1} and on some other things. Hence, it is a better way to show at once that the complex C_{A/R}^{·} is defined (up to homotopy) in a natural way.

(2) We get the notion of relative T^{i}’s (instead just those for I C → O_{Y,0}).

(3) We obtain formulas for the behavior under base change and localization.

(4) There are two long exact sequences, either for changing the A-module M or the algebras (starting with ring homomorphisms R → A → B). Since T_{0}(A/R; M) = Ω_{A/R}^{1} ⊗_{A} M and T^{0}(A/R; M) = Der_{R}(A, M), the latter one extends the usual exact sequences for Kähler differentials or their duals (cf. [Ma]).

(5) There is a spectral sequence governing the relations between the T^{i}’s and the T_{i}’s. (Moreover, if the previous definition is extended to the level of sheaves, we also obtain a local-global spectral sequence.)

(6) The complex Hom_{A}(C_{A/R}^{·}, A)) has the additional structure of a commutative graded algebra which provides the same structure on ⊕_{i} T^{i}(A/R; A) extending the Lie algebra structure on Der_{R}(A, A). In particular, we obtain the so-called cup product T^{1} × T^{1} → T^{2} (cf. (4.1) for a description in terms of the T_{Y}^{i} ’s). Moreover, Massey products may be defined.

A frequently used reference is the paper [LiS] of Lichtenbaum-Schlessinger who defined the cotangent complex up to the second position (obtaining the right T^{2}’s). For the whole complex see [An 2], [Ld], or [Qu] (with a categorial approach) or [Pa] which gives a rather down-to-earth description of the cotangent complex via the so-called Tjurina-resolvent. See §9 in [Qu] for a comparison of both definitions. A very readable summary of the properties one has to know (but without proofs) is the first section in [BeC].

Particular investigations. After the general framework is settled, it remains to ask for the shape of the versal deformation for given singularities or for certain classes of them - for instance for general theorems about the smoothness or about the number and dimension of irreducible components of the base space. What does it mean that there might be several components or that the base space might be non-reduced? The first example showing that the situation is not always as boring as in the complete intersection case was given by Pinkham (cf. [Pi 1]):

Example: Let Y ⊆ I C^{5} be the cone over the rational normal curve of degree four. Y is I Q-Gorenstein, and it is given by the equations

rank ( \begin{matrix}y_{0} & y_{1} & y_{2} & y_{3} \\ y_{1} & y_{2} & y_{3} & y_{4} \end{matrix} ) ≤ 1 .

Destroying the symmetry by introducing three parameters \underline{t} = (t_{1}, t_{2}, t_{3}) induces a flat family Y_{t} → I C^{3} defined by

rank ( \begin{matrix}y_{0} & y_{1} + t_{1} & y_{2} + t_{2} & y_{3} + t_{3} \\ y_{1} & y_{2} & y_{3} & y_{4} \end{matrix} ) ≤ 1 .

On the other hand, Y could be defined by the equations

rank ( \begin{matrix}y_{0} & y_{1} & y_{2} \\ y_{1} & y_{2} & y_{3} \\ y_{2} & y_{3} & y_{4} \end{matrix} ) ≤ 1 ,

too. They provide a one-parameter deformation Y_{s} → I C via

rank ( \begin{matrix}y_{0} & y_{1} & y_{2} \\ y_{1} & y_{2} + s & y_{3} \\ y_{2} & y_{3} & y_{4} \end{matrix} ) ≤ 1 .

The versal deformation of Y equals the union of these two families. Its base space is the union of a hyperplane and a line in I C^{4}. (In particular, it is not possible to find any flat family over a smooth parameter space containing both deformations Y_{t} → I C^{3} and Y_{s} → I C.)

We would like to give some examples of particular investigations. For a detailed survey of some of these developments we refer to the thesis [St 2].

• In [Sch 2] Schlessinger shows that isolated quotient singularities of dimension at least three are rigid.

• De Jong and van Straten investigated normal surface singularities via projecting them onto non-isolated hypersurfaces in I C^{3}. They compared the deformation theories of the former with a suitable adaption of that of the latter. As an application, they were able to determine the versal deformation of rational quadruple points (cf. [JS 1], [JS 2], [JS 3]).

• Using their own projection method, de Jong and van Straten have developed in [JS 5] the so-called picture method. It describes the versal deformation of two-dimensional sandwiched singularities (i.e. modifications of smooth germs) in terms of deformations of “pictures”, i.e. of certain incidences between points and algebraic curves (e.g. lines).

• Which deformations of a singularity admit a simultaneous resolution? See [Wa 1], [Lf], or [Al 1] for investigations concerning this question. In case of rational surface singularities, Artin has shown in [Art 1] that those parameters form their own component of the versal base space.

• In [KoSh] Kollár and Shephard-Barron applied the theory of three-dimensional varieties to obtain for two-dimensional cyclic quotient singularities a one-to-one correspondence between so-called P-resolutions on the one hand, and components of the versal base space on the other hand. (This generalizes the concept of the Artin component mentioned above.) Using this approach and the results of [Ri], it was Arndt ([Arn]), Christophersen ([Ch 3]), and Stevens ([St 1]) who were able to give an explicit description of the components.

• Many investigations have been done for cones over projective varieties (e.g. projective curves). For computations of the dimension of T_{Y}^{1} see [Pi 1], [Mu], [Sch 3], and [Wa 5].

1.2 What is this paper about?

(1.2.1) The starting point was the following observation Christophersen made while investigating the versal deformation of two-dimensional cyclic quotient singularities (i.e. two-dimensional, affine toric varieties): The total spaces over components of the versal base space are again toric. This fact raises the following questions:

(i) Does this statement hold for higher-dimensional toric varieties as well?

(ii) Assume that we are given a polyhedral cone σ describing an affine toric variety Y. Is it possible to describe the total spaces over versal components just by combinatorial methods? (Given σ, what do the cones describing the total spaces look like?)

(iii) What does the whole versal deformation of an affine toric variety look like? This question not only touches questions about number and dimension of irreducible components; the base space might even be non-reduced. Moreover, the goal is not just to get equations (as you can get by using computer algebra systems), but to obtain a combinatorial description displaying patterns.

(iv) What does the deformation theory of hypersurfaces (or complete intersections) in affine toric varieties look like?

In this thesis, we deal with the first three questions; we definitely do not say anything about (iv). A key for all investigations is the notion of a Minkowski summand of some polyhedron (arising as affine cross cut of our cone σ). It occurs in the description of infinitesimal deformations (vector space T^{1}), of genuine one-parameter deformations, as well as in the construction of the versal deformation for toric Gorenstein singularities. Our results are the following:

• We describe the infinitesimal deformation theory of affine toric varieties by calculating T^{1}, T^{2}, and the cup product T^{1} × T^{1} → T^{2} (see chapters 3 and 4).

• Those genuine deformations with toric total spaces (called “toric deformations”) are investigated and described in chapter 7 (approaching question (ii)). Via the Kodaira-Spencer map, they are always contained in the “negative” part ⊕_{R∈σ^{∨}∩M}T_{Y}^{1} (−R) of T_{Y}^{1} . In particular, we obtain that (i) is not true: As soon as there are non-negative infinitesimal deformations (for instance, if Y is the cone over the weighted projective space IP (1, 2, 3)), they cannot be toric.

• If Y is smooth in codimension two, then we can do even better: For each fixed R ∈ σ^{∨} ∩ M we can describe the “versal deformation of degree −R” (see chapter 6). However, besides the non-negative degrees, another problem remains: It is not clear how to put these homogeneous, versal pieces together to obtain the whole versal deformation. This problem does not occur if T_{Y}^{1} is concentrated in one single degree (as is the case for isolated, three-dimensional Gorenstein singularities treated in chapter 5). In particular, for those toric varieties the above question (iii) is answered.

Now we give a more detailed guide about how this paper is organized:

(1.2.2) In chapter 2 we try to put the reader in the mood for polytopes, Minkowski summands, and toric varieties. In particular, sections (2.1) and (2.4) are used to remind the reader of basic notions and to fix notation.

On the other hand, in (2.2) (right before starting to talk about toric varieties), we present a construction of an affine scheme M̄ assigned to a lattice polytope Q. It reflects the several possibilities of splitting Q into Minkowski summands: Those decompositions involving only lattice summands (similar to a non-unique prime factorization) are encoded as irreducible components of M̄_{red} (cf. Theorem (2.2.5)); the “remaining” ones cause a non-reduced structure of M̄. Proofs are contained in section (2.3).

(1.2.3) In chapter 3 we begin to investigate deformation theory of toric varieties by doing the job which must always be done first: We “compute” the vector spaces T_{Y}^{1} and T_{Y}^{2} . Computing means presenting combinatorial formulas relating these spaces to the given datum, i.e. the cone σ representing the affine toric variety Y = Y_{σ} we want to deform.

There are two different ways to present those formulas (connected by an exact sequence mentioned in (3.4.1)). The first version (Theorem (3.2.4)) is close to the language of equations and syzygies of Y_{σ}; it is very useful if we want keep control of the actual perturbations (cf. Theorem (3.2.6) and (3.2.7)). On the other hand, Theorems (3.4.1) and (3.4.3) are closer to the concept of Minkowski summands, which will be eventually the main tool to describe versal deformations. The latter formula is my favourite one, even if it looks a little bit ugly at first sight. However, if one excludes singularities which are not smooth in codimension two, then it reduces to Theorem (3.4.5).

Chapter 3 ends with applications of the T_{Y}^{i} -formulas obtained so far. We describe vanishing results for certain degrees of T_{Y}^{1} (cf. (3.5.2), (3.5.3)), and we treat the case of three-dimensional isolated singularities separately (cf. (3.5.4), (3.5.5)).

(1.2.4) Chapter 4 is the most technical one. Here we use our T_{Y}^{i} -formulas to describe the cup-product T_{Y}^{1} × T_{Y}^{1} → T_{Y}^{2} . Again, this can be done in the two different languages mentioned above (see Theorem (4.2.2) for the first version and Proposition (4.4.3) for the second one).

Knowing the cup product (in particular in its second version) gives the information about the versal base space up to the second order. Even if not needed for the proof of versality in chapters 5 and 7, it was the formula in Theorem (4.4.4) which gave us the first idea what the actual equations of the versal base space could look like. Hence, the calculation of (4.4.5) might serve in the same way to guess versal equations for the cases yet to be solved.

On the other hand, if Q is a lattice polytope with primitive edges, then we have defined in (2.2.2) the vector space V (Q)/\underline{1}. It can be interpreted in the following two different ways:

(i) V (Q)/\underline{1} equals T^{1} of the affine, toric Gorenstein singularity Y_{Cone(Q)} (cf. Corollary (5.1.2)), and, on the other hand,

(ii) V (Q)/\underline{1} equals PicY_{Σ(Q)}/O(1), if Y_{Σ(Q)} denotes the projective toric variety induced by Q in the dual sense (see (2.4.3)).

Comparing with the results of Batyrev ([Bat]), the spirit of mirror symmetry seems to be involved here. Hence, it might be interesting whether the relation between (i) and (ii) extends to T^{2} and higher Chow groups, respectively. Moreover, if it does somehow, we could use our knowledge of the cup product to compare it with the intersection pairing of divisors.

(1.2.5) In chapter 5 we investigate toric Gorenstein singularities Y which are smooth in codimension two. They may be described via cones over lattice polytopes Q (with primitive edges) embedded into height one.

We start in (5.1) with drawing from chapter 3 all possible information about T^{1} and T^{2} applied to this particular case. The main observation is that T_{Y}^{1} , if finite-dimensional, is concentrated in the single degree −R^{∗} (with [R^{∗} = 1] being the affine hyperplane containing Q). Since we have still problems combining deformations arising in different degrees (e.g. constructed as in chapters 6 or 7), it is exactly that fact which makes it possible to construct the whole versal deformation for Y.

Sections (5.2) and (5.3) describe that construction. Once we have the knowledge that deformations of Y are related in some way to Minkowski summands (cf. (1.2.7)), the following idea arises quite naturally: Use the “moduli space” of Minkowski summands (the cone C(Q) defined in (2.2.2)) and its “universal family” (the tautological cone C̃(Q) defined in (5.2.2)) to build up the first stage toward the versal deformation - just by applying the functor of transfering cones into affine toric varieties. The actual versal deformation is not toric at all (neither the base, nor the total space), but it emerges from our construction with just a little push. It turns out that the base space is exactly the M̄(Q) as constructed in the very beginning (cf. section (2.2) and Theorem (5.3.1)).

Unfortunely, being a natural construction is not sufficient for being versal, so we have to present a proof for this fact. The proof is divided into computing the Kodaira-Spencer-map as well as the obstruction map (cf. sections (5.4) and (5.5), respectively).

We conclude this chapter by giving an explicit description of the components of the versal base space and its total spaces (cf. section (5.6)) and with several examples presented in section (5.7). (The example of the hexagon Q_{6} defined in section (2.2) - yielding the cone over the Del Pezzo surface of degree 6 - accompanies the whole construction anyway.)

(1.2.6) Still assuming that Y is smooth in codimension two, in some sense the method of chapter 5 goes through also for non-Gorenstein varieties: Restricting ourselves to some fixed degree R ≥ 0 (i.e. R ∈ σ^{∨} ∩ M), we describe in chapter 6 the adjustments to be made for obtaining the “versal deformation in degree −R”. If (as it was in the Gorenstein case) T^{1} is concentrated in one single so-called negative degree, then this equals the actual versal deformation. On the other hand, even if T^{1} consists of negative degrees only, we are not (yet) able to combine pieces sitting in different degrees together. See (6.6) for a conjecture concerning this question.

(1.2.7) Finally, in chapter 7 we present the whole problem from a different point of view. Historically it was my first approach toward deforming toric varieties: Hoping (from Christophersen’s results in [Ch 3]) that total spaces over all or at least some components of the versal base space are toric again, we have created the corresponding notion of so-called toric deformations (cf. (7.1.1)) and have tried to classify them. The result is Theorem (7.2.5) which says roughly that toric deformations arise from dividing certain polytopes into Minkowski summands and putting them into parallel planes. This has been the starting point of the rather direct approach described in the earlier chapters 5 and 6.

However, chapter 7 has been included not only for nostalgic reasons - it is not just a part of the results obtained in the chapters 5 and 6: The approach of chapter 7 includes the more difficult case of singularities that might be singular in codimension two (such as two-dimensional cyclic quotient singularities or cones over singular surfaces). As we have already seen in Theorem (3.4.3), the description of T^{1} of those singularities does not only involve the Minkowski summands (encoded in V (Q)), but also some additional parameters (contained in W (Q)). Now, the calculations of chapter 7 (including the description of the Kodaira-Spencer map in section (7.4), cf. (7.4.4)) show how they fit into the game.

On the other hand, Proposition (7.1.3) and Theorem (7.2.5) show that our several constructions involving Minkowski summands of affine cross cuts of σ have no chance to cover more than the case of negative deformations (i.e. R ∈ σ^{∨} ∩ M). If T^{1} involves pieces from non-negative degrees also (e.g. Y equals the Gorenstein cone over the weighted projective space IP (1, 2, 3)), then we need completely new methods for constructing the versal deformation or just some genuine deformations (i.e. over reduced parameter spaces) in those degrees. Studying those deformations for non-isolated three-dimensional toric Gorenstein singularities would be important for a better understanding of three-dimensional flips (cf. [Re 2]).

Convex cones, polytopes, and toric varieties

2.1 Basic notions

(2.1.1) We are going to introduce briefly the notions we need from convex geometry. It should be considered a good opportunity to fix notations, on the one hand, and to bring readers from algebraic geometry in the mood for cones and polytopes, on the other hand. Standard references for the details are [Grü] or the new book [Zi].

Convex cones: Throughout the paper we use the word cone for rational, convex, polyhedral cones. If N, M are two mutually dual, free Abelian groups of finite rank, then a cone σ ⊆ N_{IR} := N ⊗_{Z Z} IR can be given either by its fundamental generators

σ = ⟨a^{1},…, a^{M}⟩ := ∑_{j=1}^{M} IR_{≥0} · a^{j} (a^{1},…, a^{M} ∈ N)

or by finitely many inequalities

σ = {a ∈ N_{IR} | ⟨a, r^{i}⟩ ≥ 0 ; i = 1,…, K} (r^{1},…, r^{K} ∈ M) .

The elements a^{j} ∈ N and r^{i} ∈ M can be normalized by asking for primitive ones, i.e. not being proper multiples. (I hope the reader will not be confused by abuse of notation: M, N might stand for free Abelian groups as well as for some natural numbers. Moreover, we use the symbol ⟨…⟩ for both the pairing between the mutually dual lattices N, M and for indicating the generators of cones; “<” also denotes the face relation.) The concept of duality interchanges both representations: The cone dual to σ is defined as

σ^{∨} := {r ∈ M_{IR} | ⟨a, r⟩ ≥ 0 for all a ∈ σ} .

It has r^{1},…, r^{K} ∈ M as fundamental generators, or it can be given by the inequalities provided by a^{1},…, a^{M} ∈ N.

Polytopes and polyhedra: Let (A A, IL) be a finite-dimensional real vector (or affine) space with a lattice. (Rational) polyhedra in (A A, IL) are given as intersections of finitely many rational half spaces. If additionally compact, they will be called polytopes.

Definition: For two polyhedra Q′, Q′′ ⊆ A A we define their Minkowski sum as the polyhedron Q′ + Q′′ := {p′ + p′′ | p′ ∈ Q′, p′′ ∈ Q′′}. Obviously, this notion also makes sense for translation classes of polyhedra (which is important, if A A is just an affine space).

It is not difficult to see that the faces of Q′ + Q′′ equal the Minkowski sums of the corresponding faces (defined by the same hyperplane in A A) of Q′ and Q′′. In particular, up to dilatation, the set of edges of Q′ + Q′′ equals the union of the corresponding sets for Q′ and Q′′, respectively.

Example: Let us introduce the following example, which will be continued through the paper: For Q we take the plane 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); hexagon Q_{6}]

Our polytope Q_{6} splits into

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

(2.1.2) Every polyhedron Q ⊆ A A is decomposable into the Minkowski sum Q = Q^{c} + Q^{∞} of a (compact) polytope Q^{c} and the so-called cone of unbounded directions Q^{∞}. The latter one is uniquely determined by Q, the compact summand is not. However, we can take for Q^{c} the minimal one - given as the convex hull of the vertices of Q itself. (If Q was already compact, then Q^{c} = Q and Q^{∞} = 0.)

Cross cuts: Let us turn to the following special setting - it serves as an example at the moment, but it reflects a rather typical situation in the sequel.

Let σ ⊆ N_{IR} be a cone and fix some primitive element R ∈ M. Then A A := [R = 1] := {a ∈ N_{IR} | ⟨a, R⟩ = 1} ⊆ N_{IR} is an affine space with lattice IL := [R = 1] ∩ N. We define the cross cut of σ in degree R as the polyhedron Q := σ ∩ [R = 1] ⊆ A A. It has the cone of unbounded directions Q^{∞} = σ ∩ [R = 0] ⊆ N_{IR}.

On the other hand, we obtain the compact part Q^{c} of Q by describing its vertices. Obviously, they correspond exactly to those fundamental generators a^{j} of σ meeting ⟨a^{j}, R⟩ ≥ 1 - the actual vertices equal ā^{j} = a^{j}/⟨a^{j}, R⟩.

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

(2.1.3) One of the most frequently used notions will be that of Minkowski summands of a given polyhedron Q ⊆ A A. Of course, a Minkowski summand Q′ of Q should be at least a summand in the usual sense (i.e. there has to be a Q′′ such that Q = Q′ + Q′′). However, since Q = Q′ + Q^{∞} for every Q^{c} ⊆ Q′ ⊆ Q, this might not be enough; we would like to avoid additional face structure of Q′ (not “coming” from Q). We take the following definition from [Sm]:

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

That means, a Minkowski summand has always the same cone of unbounded directions and, up to dilatation, the same compact edges as the original polyhedron.

Remark: Sometimes in the literature, the notion “Minkowski summand” has a slightly different meaning. Speaking with our definition, it has been used to describe Minkowski summands not of Q, but of sufficiently high multiples of Q. Those summands form a semigroup with cancellation property, i.e. it can be embedded into a group. Elements of that group can be represented as formal differences between two Minkowski summands - and we will call them generalized Minkowski summands (of multiples of Q).

(2.1.4) Finally, we would like to make some remarks about fans. They are a basic tool for describing non-affine toric varieties (gluing of cones translates directly into gluing affine pieces). Though we are dealing with the local case, i.e. with cones instead of fans, there is an interesting relation between fans and Minkowski summands we should mention. For further information and details we refer to arbitrary textbooks introducing toric varieties (as mentioned in section (2.4)).

Definition: A (finite) fan Σ is a finite collection of cones in N_{IR} such that

(1) for every cone σ ∈ Σ each of its faces belongs to Σ, and

(2) the intersection σ ∩ τ of two cones σ, τ ∈ Σ is always a common face of both of them.

The set |Σ| := ∪_{σ∈Σ}σ is called the support of Σ.

Let Q ⊆ M_{IR} be a rational polyhedron. By a ↦ min⟨a, Q⟩ we obtain a piecewise linear function defined on the cone (Q^{∞})^{∨} ⊆ N_{IR}. Its maximal domains of linearity generate a fan Σ(Q) supported on (Q^{∞})^{∨} (the inner normal fan of Q). If dim Q = dim M_{IR}, then each cone of Σ(Q) admits an apex (in 0). Now, we have the following well known fact:

Proposition: Let Q, Q′ ⊆ M_{IR} be rational polyhedra. Then, Q′ is a Minkowski summand of some positive multiple of Q, if and only if Σ(Q) is (a possibly trivial) subdivision of Σ(Q′) (i.e., if |Σ(Q)| = |Σ(Q′)|, and every cone of Σ(Q) is contained in some cone of Σ(Q′)).

Example: Looking at the hexagon Q_{6} from (2.1.1), we see that Σ(Q_{6}) is a plane fan built from six descent cones. The inner normal fans of the triangle summands of Q_{6} can be obtained by deleting every other ray; the fans assigned to the three one-dimensional summands consist of two half spaces each.

2.2 The Minkowski scheme of a lattice polytope

(2.2.1) Let Q ⊆ A A be a lattice polytope, i.e. the vertices are contained in IL. We will assume that the edges do not contain any interior lattice points (cf. (2.3.6)), hence, after choosing orientations they are given by primitive vectors d^{1},…, d^{N} ∈ IL.

Definition: For every two-dimensional face (“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 to 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: We consider our hexagon Q_{6}.

[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

\begin{matrix}d^{1} = (1, 0); & d^{2} = (1, 1); & d^{3} = (0, 1); \\ d^{4} = (−1, 0); & d^{5} = (−1, −1); & d^{6} = (0, −1). \end{matrix}

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

(2.2.2) We define the vector space V (Q) ⊆ 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 a rational, polyhedral cone in V, and its points correspond to the Minkowski summands of positive multiples of Q: Given a point (t_{1},…, t_{N}) ∈ C(Q), the corresponding polytope Q\underline{t} is built by using the edges t_{i} · d^{i} instead of the plain d^{i} contained in Q (cf. (5.2.1)). For a particular Minkowski summand Q′ of a positive multiple of Q we will denote the corresponding point in the cone by ϱ(Q′) ∈ C(Q). (By the way, the vector space V (Q) consists exactly of the generalized Minkowski summands mentioned in (2.1.3).)

Example: 1) ϱ(t · Q) = (t,…, t) ∈ C(Q) ⊆ V ⊆ IR^{N}.

2) We look at the two different possibilities shown in (2.1.1) to decompose Q_{6} into a Minkowski sum. Applying ϱ, these two equations become

\begin{aligned}(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) . \end{aligned}

(2.2.3) 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 A A, the g_{ε,k}(\underline{t}) turn into regular polynomials; for each pair (ε, k) we will get two linearly independent ones. We obtain an ideal

J := (g_{ε,k} | ε < Q is a 2-face, 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} mentioned in (2.2.1) 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, just finitely many equations are sufficient to generate the ideal J - but we can even give an effective criterion to see which equations can be dropped:

Proposition: Let ε < Q be a 2-face. Then, ε is contained in a two-dimensional subspace of A A, and this vector space comes with a natural lattice (the restriction of the big lattice IL). 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. (2.3.3).

Corollary: If Q is contained in n linearly independent strips (defined by pairs of parallel hyperplanes) of lattice-thickness ≤ k_{0}, then all polynomials g_{ε,k} with k > k_{0} are superfluous.

Example: Obviously, Q_{6} is contained in at least three strips of thickness 2. Hence, J is generated in 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}) .

(2.2.4) 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: (compare with Remark (5.2.5))

(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 ∈ A A^{∗}\}.

(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. (2.3.4).

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}.

(2.2.5) We want to describe the structure of the underlying reduced spaces of M or M̄.

First, we mention the following trivial observations concerning the cone C(Q):

(i) Minkowski summands Q′ of Q itself (instead of a positive multiple of Q) are characterized by the property ϱ(Q) − ϱ(Q′) ∈ IR_{≥0}^{N}, i.e. all components of ϱ(Q′) have to be contained in the interval [0, 1].

(ii) For a Minkowski summand Q′ (of some positive multiple of Q) the property of being a lattice polytope is equivalent to the fact that ϱ(Q′) ∈ ZZ^{N}.

Now, let Q = R_{0} + … + R_{n} be a decomposition of Q into a Minkowski sum of n + 1 lattice polytopes. Then, the N-tuples ϱ(R_{0}),…, ϱ(R_{n}) consist of numbers 0 and 1 only, and they sum up to (1,…, 1). In particular, the (n + 1)-plane I C · ϱ(R_{0}) + … + I C · ϱ(R_{n}) ⊆ I C^{N} (or its n-dimensional image via ℓ) is contained in M (in M̄, respectively).

Remark:

(1) That (n + 1)-plane (or its image via ℓ) is given by the linear equations t_{i} − t_{j} = 0 (if d^{i}, d^{j} belong to a common summand R_{k}).

(2) Refinements of Minkowski decompositions (they form a partial ordered set) correspond to inclusions of the associated planes.

Theorem: M_{red} and M̄_{red} equal the union of those flats corresponding to maximal Minkowski decompositions of Q into lattice summands.

Proof: cf. (2.3.5).

Example: M(Q_{6}) and M̄(Q_{6}) are already reduced schemes (for non-reduced examples see section 5.7). 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)}.

mentioned in (2.2.2).

(2.2.6)  M̄ (or M = ℓ^{−1}(M̄)) reflect the possibilities of constructing Minkowski decomposi-

tions 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.

Remark: The ideals defining M and M̄ are homogeneous. Hence, there are projective versions of these schemes, too.

2.3 Proof of the previous statements

(2.3.1) Using vectors c ∈ IL^{∗} (or selected c ∈ A A^{∗}) 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 section (2.2) can be reduced to much simpler lemmas, which we will present here.

Then, all those 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.

(2.3.2) For the whole section (2.3) we use the following notations:

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

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}).

(2.3.3) 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 the g_{1},…, g_{k_{0}}. (This implies Proposition (2.2.3).)

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

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

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

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

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

which proves the lemma. □

(2.3.4) Lemma:

(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.

(This implies Theorem (2.2.4).)

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) The polynomial rings I C[\underline{t}] and I C[t_{1}, \underline{t} − t_{1}] are equal, i.e. each polynomial q(\underline{t}) can uniquely be written as

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

Moreover, 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.

Let us determine the components of the polynomial p - we will start with our special case again:

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). Now, there are 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

σ_{k}(\underline{X}) − σ_{k}(\underline{Y}) = \begin{matrix}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}) \end{matrix}

= ∑_{v=1}^{k−1} q_{v}(\underline{X}, \underline{Y}) · g_{v}(\underline{X}, \underline{Y}) + c_{k} · g_{k}(\underline{X}, \underline{Y})

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. □

(2.3.5) Lemma: Let \underline{ξ} = (ξ_{1},…, ξ_{N}) ∈ I C^{N} be a point such that g_{k}(\underline{ξ}) = 0 for each k ≥ 1. Then, for every fixed ξ ∈ I C, we have ∑_{ξ_{i}=ξ} d_{i} = 0. (This implies Theorem (2.2.5).)

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

(2.3.6) The polytope Q was assumed to have primitive edges only. Actually, we never needed this fact neither in the previous lemmata nor in their proofs. It is only important to translate these results into the language of Minkowski summands used in section (2.2).

Dropping this condition, similar constructions are possible. However, by declaring some or just all lattice points contained in edges of Q to be additional, artificial vertices of Q, several possibilities arise with equal rights. The two extremal cases (add either no or all possible generalized vertices) seem to be the most interesting ones.

Remark:

(1) For a natural number g ∈ IN, the polytopes Q (with some fixed set of possibly artificial vertices) and g·Q (with the corresponding set of vertices) induce the same Minkowski scheme M̄.

(2) Let Q_{1} ⊆ Q_{2} be the same polytopes with different sets of generalized vertices. Then, M̄_{1} is a closed subscheme of M̄_{2}. It is defined by identifying the variables associated to those generalized edges of Q_{2} that are contained in the same generalized edge of Q_{1}.

Conjecture: Let Q be a lattice polytope such that each extremal Minkowski summand of Q is a lattice polytope, too. Then, using all generalized vertices of Q, the affine schemes M and M̄ are reduced.

In particular, if Q is an arbitrary lattice polytope (with primitive edges), then M̄_{Q} could be embedded into some reduced M̄_{g·Q}. The non-reduced structure of M̄_{Q} would arise as a germ of components visible in M̄_{g·Q} only.

2.4 Toric Varieties

(2.4.1) Most books concerned with toric varieties define them as normal varieties Y containing some torus ( I C^{∗})^{dim Y} as a dense, open subset such that its group structure may be extended to an action on the whole Y. Then, starting with the definitions M := character group, and N := set of 1-parameter subgroups of the torus, those varieties are completely described in terms of fans in N_{IR}. Standard references for an introduction into that subject are the survey article [Da] of Danilov, the first chapter of [Ke], or the books [Od] and [Fu 2].

Although very special, toric varieties are useful as examples or first test objects for general theories - many theorems in algebraic geometry have their combinatorial counterpart in the world of cones and polytopes. (For instance, in [Re 1] the main ideas of Mori theory can be seen very clearly.) Since we are mainly interested in the deformation theory of (singular) germs of varieties (which is well understood for complete intersections), toric singularities become interesting for us by the following reason: The combinatorial description makes it possible to handle a huge amount (compared with the codimension) of equations in a rather efficient way.

We would like to give a brief sketch of the subject of toric varieties here. We start with the affine ones, then we will mention more global (and in particular projective) toric varieties. For that purpose, our point of view will be the following: Consider toric varieties just a special class of algebraic varieties defined by many equations, but encoded in some nice and easy combinatorics. For details and proofs we refer to the books mentioned above.

(2.4.2) Affine toric varieties. Let M, N be two mutually dual, free Abelian groups of finite rank (as in (2.1.1)).

Definition: If σ ⊆ N_{IR} is a cone with apex, then we define by Y_{σ} := Spec I C[σ^{∨} ∩ M] its associated, affine toric variety. ( I C[σ^{∨} ∩ M] denotes the semigroup ring - obtained by regarding elements r ∈ σ^{∨} ∩ M as exponents of some “abstract symbol” x.) Let σ_{1} ⊆ N_{IR}^{1}, σ_{2} ⊆ N_{IR}^{2} be two cones. Then, a ZZ-linear maps f : N^{1} → N^{2} such that f(σ_{1}) ⊆ σ_{2} induces an algebraic morphism f : Y_{1} → Y_{2} in an obvious way. Those maps will be regarded the morphisms in the category of affine, toric varieties.

The semigroup σ^{∨} ∩M is generated by the finite set E of its irreducible elements. (E is often called the Hilbert basis of that semigroup.) Assigning to each element r ∈ E a variable z_{r}, our affine toric variety Y_{σ} can be embedded into I C^{E}. It is defined by the binomial equations obtained by “raising” linear dependencies between the r’s into the exponents of the z_{r}’s. To give an example, the relation r + 2s = 3t + u turns into z_{r} z_{s}^{2} = z_{t}^{3} z_{u}.

Examples:

(1) The cone σ := IR_{≥0}^{k} ⊆ IR^{k} (i.e. N := ZZ^{k}) yields σ^{∨} ∩ M = IN^{k}, hence Y_{σ} = I C^{k}.

(2) Let E ⊆ σ^{∨} ∩ M be the Hilbert basis for an arbitrary cone σ ⊆ N_{IR}. Then, assigning each a ∈ N the E-tuple (⟨a, r⟩)_{r∈E} ∈ ZZ^{E}, defines a ZZ-linear map N → ZZ^{E} putting σ into IR_{≥0}^{E}. At the level of toric varieties, this yields exactly the embedding Y_{σ} ↪ I C^{E} described above.

(3) Let n, q ∈ IN be relatively prime numbers. With (α, β) ↦ (−q α + β, n α), we obtain a ZZ-linear map f : ZZ^{2} → ZZ^{2} sending IR_{≥0}^{2} onto σ := ⟨(1, 0), (−q, n)⟩ ⊆ IR^{2}. (By the way, any two-dimensional cone can be written that way.) At the toric level this means we have a morphism π : I C^{2} → Y_{σ}. The dual cone equals σ^{∨} = ⟨[0, 1], [n, q]⟩, and we obtain for f^{∗}(σ^{∨} ∩ ZZ^{2}) the semigroup

f^{∗}(σ^{∨} ∩ ZZ^{2}) = IN^{2} ∩ {(r_{1}, r_{2}) ∈ ZZ^{2} | r_{1} + q r_{2} ≡ 0 mod n} .

Hence, the affine coordinate ring I C[σ^{∨} ∩ ZZ^{2}] of Y_{σ} equals the subring of I C[z_{1}, z_{2}] consisting

of polynomials invariant under ( \begin{matrix}ξ & 0 \\ 0 & ξ^{q} \end{matrix} ) (with ξ being a primitive n^{th} root of unity). In

particular, Y_{σ} is a cyclic quotient singularity, and π is the quotient map. For n = 4, q = 1 we obtain the example mentioned in (1.1.3).

We have seen that almost all two-dimensional cones yield singular toric varieties. This reflects the general situation - smooth, affine toric varieties are boring: If σ is a top-dimensional cone, then Y_{σ} is smooth, if and only if σ is a simplex generated by a ZZ-basis of N (i.e. the determinant of its fundamental generators has to be ±1). If this is the case, then Y_{σ} is isomorphic to the affine space.

Remark: If σ_{1} ⊆ N_{IR}^{1}, σ_{2} ⊆ N_{IR}^{2}, then σ_{1} × σ_{2} is a cone in N_{IR}^{1} ⊕ N_{IR}^{2}. Moreover, we have Y_{σ_{1}×σ_{2}} = Y_{σ_{1}} × Y_{σ_{2}}. In particular, if τ ⊆ N_{IR} is not top-dimensional, then τ = τ × {0} ⊆ span_{IR}(τ) × complement. That means Y_{τ} = Y_{τ⊆span τ} × ( I C^{∗})^{codim τ}, and we see that the previous assumption about dim σ was not essential.

Finally, we would like to remark that special cones yield special toric varieties: If Q ⊆ (A A, IL) is a lattice polytope, then we can define the cone σ := IR_{≥0} ·(Q, 1) in N_{IR} := A A⊕IR (with N := IL⊕ZZ). It is exactly the toric varieties defined by those cones which are Gorenstein (cf. (5.1.1)). We will treat those cones and varieties in chapter 5 - their deformation theory can be described in terms of Minkowski summands of Q.

(2.4.3) As already mentioned, morphisms between affine toric varieties arise from ZZ-linear maps f : N^{1} → N^{2} such that f(σ_{1}) ⊆ σ_{2}. A very important special case is that of f : N → N being the identical map and σ_{1} being a face of σ_{2}. If r ∈ σ2^{∨} ∩ M is actually cutting out that face (i.e. σ_{1} = σ_{2} ∩ r^{⊥}), then I C[σ2^{∨} ∩ M] equals the localization of I C[σ1^{∨} ∩ M] by the element x^{r}. In particular, the induced map Y_{σ_{1}} → Y_{σ_{2}} is an open embedding (identifying the first variety with the open subset [x^{r} ≠ 0] ⊆ Y_{σ_{1}}). Moreover, every open embedding in our category arises that way.

Definition: If Σ is a fan in N_{IR}, then the toric variety Y_{Σ} is obtained by gluing together the affine pieces Y_{σ} (σ ∈ Σ) along common faces of Σ-cones. A map between toric varieties Y_{Σ^{1}}, Y_{Σ^{2}} is given by a ZZ-linear f : N^{1} → N^{2} such that for each σ_{1} ∈ Σ^{1} there is some σ_{2} ∈ Σ^{2} meeting f(σ_{1}) ⊆ σ_{2}.

Proposition: The morphism Y_{Σ^{1}} → Y_{Σ^{2}} induced from f : N^{1} → N^{2} is proper, if and only if |Σ^{1}| = f^{−1}(|Σ^{2}|). In particular, a toric variety Y_{Σ} is complete, if and only if |Σ| = N_{IR}. Projectivity is equivalent to Σ being the inner normal fan of some polytope in M_{IR} (cf. the upcoming (2.4.4)).

This criterion makes it easy to construct three-dimensional toric varieties that are complete, but not projective.

On the other hand, proper birational morphisms onto a toric variety Y_{Σ} correspond to polyhedral subdivisions of Σ. Hence, by Proposition (2.1.4), looking for projective contractions of a given projective toric variety (assigned to some polytope P) means looking for Minkowski summands (of P).

Examples:

(1) The k-dimensional fan spanned by the canonical basis vectors e^{1},…, e^{k} of ZZ^{k} and −e (with e := e^{1} + … + e^{k}) defines the projective space IP^{k}.

(2) The subdivision of the smooth cone σ = IR_{≥0}^{k} = ⟨e^{1},…, e^{k}⟩ into the union of σ_{i} := ⟨e^{1},…, ê^{i},…, e^{k}, e⟩ (i = 1,…, k) describes the blowing up of the origin in the affine kspace.

(3) If σ = ⟨a^{1},…, a^{M}⟩ is an arbitrary cone (with apex), then the normalized blowing up of Y_{σ} in the closed orbit is given by the subdivision of σ into the union of σ_{r} := {a ∈ σ | ⟨a, r⟩ ≤ ⟨a, E⟩} (and r is runing through the Hilbert basis E of σ^{∨} ∩ M).

(4) Let σ be the three-dimensional cone over the unit square (contained in height one). Y_{σ} equals the cone over the Segre embedding of IP^{1} × IP^{1} into IP^{3}, i.e. the three-dimensional node. Now, the two possibilities of drawing diagonals into that given square correspond to the two small resolutions of Y_{σ}.

In general, every fan can be subdivided into a “smooth” fan. That means, every toric variety admits a toric desingularization.

Invertible sheaves on Y_{Σ}: Trivializing it on the torus, an invertible sheaf L is given by monomial generators x^{r(σ)} on Y_{σ} for each cone σ ∈ Σ. In particular, Γ(Y_{σ}, L) = ⊕_{r∈r(σ)+(σ^{∨}∩M)} I C · x^{r}.

We define the polyhedron Δ(L) := ∩_{σ∈Σ}(r(σ) + σ^{∨}) ⊆ M_{IR}. If Σ is complete, then Δ(L) is bounded (i.e. a polytope). By definition, the global sections of L form the vector space Γ(Y_{Σ}, L) = ⊕_{r∈M∩Δ(L)} I C · x^{r} which will be finite-dimensional, if Δ(L) is a polytope. Moreover, L is generated by global sections, if and only if r(σ) ∈ Δ(L) for every cone σ ∈ Σ. If this is the case, then the r(σ) form the vertices of that polyhedron, and Σ is a subdivision of the inner normal fan Σ(Δ(L)).

In particular, if Σ = Σ(P) for some polytope P (i.e. if Y_{Σ} is projective), then globally generated invertible sheaves correspond to lattice polytopes that are Minkowski summands of some multiple of P. Hence, Pic Y_{Σ} = V (P), and C(P) coincides with the ample cone (cf. (2.2.2) for the definition of V and C).

(2.4.4) Finally, we would like to make some additional remarks about projective toric varieties. Let P ⊆ (A A, IL) be a lattice polytope. Then, we can associate to P the polarized projective variety (X_{P} , O_{X}(P)) in three different ways:

(1) Denote by Σ := Σ(P) the inner normal fan (contained in A A^{∗}) of the polytope P. We define X_{P} := X_{Σ}, and O_{X}(P) is just the ample invertible sheaf associated to P itself. In particular, h^{0}(X_{P} , O_{X}(P)) = #(P ∩ IL), and the corresponding linear system defines a map X_{P} → IP^{#(P ∩IL)−1}.

(2) The homogeneous coordinates of IP^{#(P ∩IL)−1} are in a one-to-one correspondence with the lattice points of P. We translate the affine relations between them (analogously to (2.4.2), by writing the relations multiplicatively) into homogeneous, monomial equations on IP^{#(P ∩IL)−1}. They define a projective subvariety X′_{P} ⊆ IP^{#(P ∩IL)−1} which turns out to be the scheme theoretical image of X_{P} under the previously described map. In the other way around, we can obtain X_{P} from X′_{P} just by taking the normalization and O_{X}(P) by pulling back O(1).

(3) Embedding P via P ⊆ A A ≅ A A × {1} ↪ A A × IR =: M_{IR} in the next dimension (in a dual way as we did with Q at the end of (2.4.2)), we obtain rational, polyhedral cones σ^{∨} := IR_{≥0}·P ⊆ M_{IR} and σ := σ^{∨∨} ⊆ N_{IR}. The latter one contains a^{∗} := (\underline{0}, 1) as an interior point; the equation [a^{∗} = 1] cuts out P from σ^{∨} again. Moreover, a^{∗} can be used to make the ring I C[σ^{∨} ∩ M] ZZ-graded (deg x^{r} := ⟨a^{∗}, r⟩), and we obtain X_{P} = Proj I C[σ^{∨} ∩ M]. (In fact, it is not difficult to see that the homogeneous localizations by monomials yield those semigroup rings induced by the fan Σ.)

On the other hand, we can use the cone σ to define an affine toric variety Y_{σ} = Spec I C[σ^{∨} ∩ M]. Since I C[σ^{∨} ∩ M] = ⊕_{i≥0}H^{0}(X_{P} , O_{X}(P)^{i}) (as graded rings), Y equals the cone over (X_{P} , O_{X}(P)). Let us elucidate this fact from the combinatorial view point: The open subset Y \ {0} ⊆ Y is given by the fan ∂σ contained in N_{IR}, and the projection π : Y \ {0} → X_{P} corresponds to the ZZ-linear map π : N ⟶ ⟶ N/ZZ · a^{∗} = A A^{∗} (sending proper faces of σ onto Σ-cones).

Claim: π maps the facets of σ isomorphically (including the lattice structure) onto the topdimensional cones of Σ. In particular, Y \ {0} is locally isomorphic to X_{P} × I C^{∗}.

Proof: Considering just the real structure of the cones, there is no problem at all. On the other hand, let (for some facet τ < σ) a ∈ τ ⊆ N_{IR} such that π(a) belongs to the lattice (π(a) ∈ π(τ) ∩ N/ZZa^{∗}). The special shape of σ implies that τ is given as τ = σ ∩ r^{⊥} for some vertex r ∈ P. In particular, ⟨a^{∗}, r⟩ = 1, and, if π(a) is represented by some b ∈ N, then b − ⟨b, r⟩a^{∗} ∈ π^{−1}(π(τ)) ∩ r^{⊥} = τ. Hence, a = b − ⟨b, r⟩a^{∗} is an element of N. □

T^{1} and T^{2} for affine toric varieties

3.1 The modules T^{1} and T^{2} in general

(3.1.1) For an affine scheme Y = Spec A, there are two important A-modules, T_{Y}^{1} and T_{Y}^{2} carrying information about its deformation theory: T_{Y}^{1} describes the infinitesimal deformations, and T_{Y}^{2} contains the obstructions for extending deformations of Y to larger base spaces (cf. [KPR], [LiS], or [Pa]; see also (1.1.3) in this thesis). In case Y admits a versal deformation, T_{Y}^{1} is the tangent space of the versal base space S. Moreover, if J denotes the ideal defining (S, 0) as a closed subscheme of the germ (T_{Y}^{1} , 0), the module

(J/m_{T^{1},0} J )^{∗} can be canonically embedded into T_{Y}^{2} , i.e. (T_{Y}^{2} )^{∗}-elements induce the equations

defining S in T_{Y}^{1} .

The vector spaces T_{Y}^{i} come with a cup product T_{Y}^{1} × T_{Y}^{1} → T_{Y}^{2} . The associated quadratic form T_{Y}^{1} → T_{Y}^{2} describes the quadratic part of the elements of J, i.e. it can be used to get a better approximation of the versal base space S as it is its tangent space (see section (4.1)).

(3.1.2) If Y is given as a closed subscheme of some I C^{w+1}, then the standard exact sequence of differentials can be completed on the right hand side yielding

0 → Θ_{Y} → Θ_{I C^{w+1}} ⊗ O_{Y} → N_{Y | I C^{w+1}} → T_{Y}^{1} → 0.

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

Θ_{I C^{w+1}} = ⨁_{v=0}^{w} I C[\underline{z}] · \frac{∂}{∂z_{v}} ≅ I C[z_{0},…, z_{w}]^{w+1}; N_{Y | I C^{w+1}} denotes the normal sheaf of Y ↪ I C^{w+1}.)

Let Y ⊆ I C^{w+1} be given by equations f_{1},…, f_{m}, i.e. its ring of regular functions equals

A = P/I with P = I C[z_{0},…, z_{w}] ; I = (f_{1},…, f_{m}) .

Then, using d : I/I^{2} → A^{w+1} (d(f_{i}) := (\frac{∂f_{i}}{∂z_{0}} ,… \frac{∂f_{i}}{∂z_{w}})), the above sequence means that the vector space T_{Y}^{1} equals

T_{Y}^{1} = Hom_{A}(I/I^{2}, A)/ ; Hom_{A}(A^{w+1}, A) .

Remark: (additional T^{1}-formulas)

(i) Let A be a reduced ring. Then, T_{Y}^{1} is also computable as T_{Y}^{1} = Ext_{A}^{1}(Ω_{A| I C}^{1}, A) (Ω_{A| I C}^{1} = sheaf of Kähler differentials, i.e. Θ_{X} = Hom(Ω_{A| I C}^{1}, A); cf. [KPR], ch. 5). In fact, we will use this formula for our computation in section (3.3).

3.1. THE MODULES T^{1} AND T^{2} IN GENERAL 25

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

H^{1}(Y \0, Θ_{Y \0}) → H^{1}(Y \0, Θ_{I C^{w+1}| _{Y \0}})

(cf. [Sch 2], §1). The embedding T_{Y}^{1} ⊆ H^{1}(Y \0, Θ_{Y \0}) corresponds to the restriction map Def_{Y} → Def_{Y \0}.

(3.1.3) Let R ⊆ P^{m} denote the P -module of relations between the equations f_{1},…, f_{m}. If {e^{1},…, e^{m}} denotes the canonical basis of P^{m}, then R contains the so-called Koszul relations R_{0} := ⟨f_{i} e^{j} − f_{j} e^{i}⟩ as a submodule. Now, T_{Y}^{2} can be obtained as

T_{Y}^{2} = Hom_{P} (R/R_{0}, A)/ ; Hom_{P} (P^{m}, A) .

Remark: Let A be a normal ring. Then, analogeously to T_{Y}^{1} , the vector space T_{Y}^{2} is computable as T_{Y}^{2} = Ext_{A}^{2}(Ω_{A| I C}^{1}, A). (We may use the same reference [KPR]; though a slightly different notion of T_{Y}^{2} has been used there, both vector spaces coincide for normal singularities.)

(3.1.4) Finally, we construct the so-called obstruction map. It detects all possible infinitesimal extensions of a flat family to another one defined over some larger base space. We follow the explanation given in §4 of [JS 4].

Let 0 → W → R′ →^{π} R → 0 be a small extension of a local I C-algebra R, i.e. W · m_{R′} = 0, and W is a finite-dimensional I C-vector space. Moreover, assume that we are given a flat R-algebra A_{R} = R[\underline{z}]/(F_{1},…, F_{m}) deforming A; in particular, the equations F_{i}(\underline{z}) specialize to f_{i}(\underline{z}) via R ⟶ ⟶ I C. Then, we may construct an element λ(A_{R}) ∈ T_{Y}^{2} ⊗_{I C} W in the following way:

(i) For any relation s ∈ R ⊆ I C[\underline{z}]^{m} we choose a lifting S′ ∈ R′[\underline{z}]^{m} such that ∑_{i} S′_{i}(\underline{z})·F_{i}(\underline{z}) = 0 in R[\underline{z}].

(ii) Take any F′_{i}(\underline{z}) ∈ R′[\underline{z}] lifting the equations F_{i}. Then, ∑_{i} S′_{i}(\underline{z}) · F′_{i}(\underline{z}) ∈ P ⊗_{I C} W ⊆ R′[\underline{z}], and we define λ(A_{R}; s) as its image in A ⊗_{I C} W.

It is not hard to check that λ(A_{R}) ∈ T_{Y}^{2} ⊗_{I C} W does not depend on the choices being made in the previous construction. Moreover, we obtain

Proposition:

(1) λ(A_{R}) = 0, if and only if the deformation A_{R} lifts to one over R′.

(2) The corresponding map λ(A_{R}) : W^{∗} → T_{Y}^{2} (“obstruction map”) is injective, if and only if A_{R} is not liftable to any deformation over a ring between R and R′ (and different from R).

Applied to the following situation, we obtain a very useful criterion of versality:

Theorem: ([Arn]) Let R = I C\{\underline{t}\}/J and R′ = I C\{\underline{t}\}/(\underline{t})J. Then, A_{R} is the mini-versal deformation of A, if and only if the Kodaira-Spencer-map is an isomorphism and the obstruction map is injective.

26 CHAPTER 3. T^{1} AND T^{2} FOR AFFINE TORIC VARIETIES

3.2 The first T^{i}-formulas for toric varieties

(3.2.1) As far as I know, the deformation theory of affine toric varieties (e.g. calculation of its infinitesimal deformations) is hardly mentioned in the literature. There are just the papers [IsOd] and [LdSl]. The first one does not deal with toric singularities directly, but with varieties defined by monomial equations. The latter paper builds up a cohomology theory for semigroups; via a spectral sequence argument the main result is that this theory describes exactly the algebra cohomology (in particular T^{1}, T^{2}) of the corresponding toric variety. However, an explicit description of the vector spaces T^{i} is lacking except for the case of two-dimensional quotient singularities. We will describe T^{1} and T^{2} in such a way that it can be used for further investigations (such as describing Kodaira-Spencer or obstruction maps of certain deformations).

(3.2.2) We start with fixing (or recalling from (2.4.2)) the usual notations when dealing with affine toric varieties (cf. [Ke] or [Od]):

• Let M, N be mutually dual free Abelian groups, we denote by M_{IR}, N_{IR} the associated real vector spaces obtained by base change with IR.

• Let σ = ⟨a^{1},…, a^{M}⟩ ⊆ N_{IR} be a rational, polyhedral cone with apex in 0 - given by its fundamental generators. σ^{∨} := {r ∈ M_{IR} | ⟨σ, r⟩ ≥ 0} ⊆ M_{IR} is called the dual cone. It induces a partial order on the lattice M via [ a ≥ b iff a − b ∈ σ^{∨}].

• A := I C[σ^{∨} ∩ M] denotes the semigroup algebra. It is the ring of regular functions of the toric variety Y = Spec A associated to σ. The ring A itself as well as most of its important modules (such as T^{1} and T^{2}) admit an M-(multi)grading. It is this grading which will make computations possible.

• Denote by E ⊂ σ^{∨} ∩M the minimal set of generators of this semigroup (”the Hilbert basis”). E equals the set of all indecomposable elements of σ^{∨} ∩M. In particular, there is a surjection of semigroups π : IN^{E} ⟶ ⟶ σ^{∨}∩M, and this fact translates into a closed embedding Y ↪ I C^{E}. To make the notations coherent with section (3.1), assume that E = {r^{0},…, r^{w}} consists of w + 1 elements (inducing coordinates z_{0},…, z_{w}).

(3.2.3) To a fixed degree R ∈ M we associate “thick facets” K_{j}^{R} of the dual cone σ^{∨} defined by

K_{j}^{R} := {r ∈ σ^{∨} ∩ M | ⟨a^{j}, r⟩ < ⟨a^{j}, R⟩} (j = 1,…, M).

3.2. THE FIRST T^{I}-FORMULAS FOR TORIC VARIETIES 27

[figure: K_{1}^{R}; K_{2}^{R}; K_{1}^{R}; R; R; 0; 0; K_{1}^{R} ∩ K_{2}^{R}; K_{2}^{R} = ∅; or]

Lemma: (properties of the sets K_{j}^{R})

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

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

(3) ∪_{j}K_{j}^{R} = (σ^{∨} ∩ M) \ (R + σ^{∨}).

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

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

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

are equivalent.

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

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

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

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

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

Hence,

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

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

Mostly, we will need two special cases of the last part of the previous lemma:

Corollary: Let r_{1},…, r_{m} ∈ σ^{∨} ∩ M.

(a) For j = 1,…, M we obtain that

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

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

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

Intersecting those sets K_{j}^{R} with E ⊆ σ^{∨}∩M, we obtain the basic objects for describing the modules T_{Y}^{i}:

E_{j}^{R} := K_{j}^{R} ∩ E = {r ∈ E | ⟨a^{j}, r⟩ < ⟨a^{j}, R⟩} ,
E_{0}^{R} := ⋃_{j=1}^{N} E_{j}^{R} , and
E_{τ}^{R} := ⋂_{a^{j}∈τ} E_{j}^{R} for faces τ < σ .

We obtain a complex L(E^{R})_{•} of free Abelian groups via

L(E^{R})_{−k} := ⨁ ; \begin{matrix}τ < σ \\ dim τ = k \end{matrix} L(E_{τ}^{R})

with the usual differentials. (L(…) denotes the free Abelian group of integral, linear dependencies.) The most interesting part (k ≤ 2) can be written explicitly as

L(E^{R})_{•} : ⋯ → ⊕_{⟨a^{j},a^{k}⟩<σ}L(E_{j}^{R} ∩ E_{k}^{R}) ⟶ ⊕_{j}L(E_{j}^{R}) ⟶ L(E_{0}^{R}) → 0 .

(3.2.4) Theorem:

(1) T_{Y}^{1} (−R) = H^{0} (L(E^{R})_{•}^{∗} ⊗_{Z Z} I C)

(2) T_{Y}^{2} (−R) ⊇ H^{1} (L(E^{R})_{•}^{∗} ⊗_{Z Z} I C)

(3) Moreover, if Y is smooth in codimension 2 (i.e. if the 2-faces ⟨a^{j}, a^{k}⟩ < σ are spanned by a part of a ZZ-basis of the lattice N), then

T_{Y}^{2} (−R) = H^{1} (L(E^{R})_{•}^{∗} ⊗_{Z Z} I C) .

(4) Module structure: If x^{s} ∈ A (i.e. s ∈ σ^{∨}∩M), then E_{j}^{R−s} ⊆ E_{j}^{R}, hence L(E^{R})_{•}^{∗} ⊆ L(E^{R−s})_{•}^{∗}. The induced map in cohomology corresponds to the multiplication with x^{s} in the A-modules T_{Y}^{1} and T_{Y}^{2} , respectively.

The proof is contained in section (3.3).

Remark: The assumption made in (3) cannot be dropped: Taking for Y a two-dimensional cyclic quotient singularity given by some two-dimensional cone σ, then there are only two different sets E_{1}^{R} and E_{2}^{R} (for each R ∈ M). In particular, H^{1}(L(E^{R})_{•}^{∗} ⊗_{Z Z} I C) = 0.

(3.2.5) Nevertheless, we can use two-dimensional cyclic quotient singularities (i.e. twodimensional affine toric varieties, cf. example (3) in (2.4.2)) as an object to demonstrate how the T^{1}-formula of the previous theorem works.

If Y (n, q) denotes the quotient of I C^{2} by the ZZ/nZZ-action ξ ↦ \binom{ξ 0}{0 ξ^{q}}, then Y (n, q) is given by the cone σ = ⟨(1, 0); (−q, n)⟩ ⊆ IR^{2}. We develop \frac{n}{n−q} into a (negative) continued fraction

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

Then E ⊆ σ^{∨} ∩ ZZ^{2} is given as the set E = {r^{0},…, r^{w}} ⊆ ⟨[0, 1], [n, q]⟩ ∩ ZZ^{2} with

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

• r^{v−1} + r^{v+1} = a_{v} · r^{v} (v = 1,…, w − 1) (cf. [Ri] or [Od]).

[figure: …; [0, 0]; r^{0} = [0, 1]; r^{1}; r^{2}; r^{w} = [n, q]; σ^{∨} ∩ ZZ^{2}]

Now, there are four different cases for the multidegree R ∈ M ≅ ZZ^{2}. (We assume w ≥ 2, i.e. Y is not smooth; let a^{1} = (1, 0) and a^{2} = (−q, n).):

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

theorem yields T_{Y}^{1} (−R) ≅ \{ \begin{matrix}I C & for & w ≥ 3 \\ 0 & for & w = 2. \end{matrix}

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

(iii) R = p · r^{v} (1 ≤ v ≤ w − 1, 2 ≤ p < a_{v} for w ≥ 3; or v = 1 = w − 1, 2 ≤ p ≤ a_{1} for w = 2): We obtain E_{1}^{R} = {r^{0},…, r^{v}} and E_{2}^{R} = {r^{v},…, r^{w}}, and the theorem yields T_{Y}^{1} (−R) ≅ I C.

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

(3.2.6) We want to describe the isomorphisms connecting the general T^{i}-formulas of (3.1.2) and (3.1.3) with the toric ones given in (3.2.4).

Y ↪ I C^{w+1} is given by the equations

f_{ab} := \underline{z}^{a} − \underline{z}^{b} (a, b ∈ IN^{w+1} with π(a) = π(b) in σ^{∨} ∩ M),

and it is easier to deal with this infinite set of equations (which generates the ideal I as a I C-vector space) instead of selecting a finite number of them in some non-canonical way. In particular, for m of (3.1.2) and (3.1.3) we take

m := {(a, b) ∈ IN^{w+1} × IN^{w+1} | π(a) = π(b)} .

The general T^{i}-formulas mentioned in (3.1.2) and (3.1.3) remain true.

Theorem: For a fixed element R ∈ M let φ : L(E)_{I C} → I C induce some element of

(^{L(ER 0)/}∑_{j} L(E_{j}^{R}))^{∗} ⊗_{Z Z} I C ≅ T_{Y}^{1} (−R) (cf. Theorem (3.2.4)(1)).

Then, the A-linear map

I/I^{2} ⟶ A
\underline{z}^{a} − \underline{z}^{b} ↦ φ(a − b) · x^{π(a)−R}

provides the same element via the formula (3.1.2).

Again, the proof is contained in (3.3) (see the end of (3.3.5)).

Remark: A simple, but nevertheless important check shows that the map (\underline{z}^{a} − \underline{z}^{b}) ↦ φ(a − b) · x^{π(a)−R} goes into A, indeed: Assume π(a) − R ∉ σ^{∨}. Then, there exists an index j such that ⟨a^{j}, π(a) − R⟩ < 0. Denoting by ”supp q” (for a q ∈ IR^{E}) the set of those r ∈ E providing a non-vanishing entry q_{r}, we obtain

supp (a − b) ⊆ supp a ∪ supp b ⊆ E_{j}^{R} ,

i.e. φ(a − b) = 0.

(3.2.7) The P -module R ⊆ P^{m} is generated by relations of two different types:

r(a, b; c) := e^{a+c, b+c} − \underline{z}^{c} e^{a,b} (a, b, c ∈ IN^{w+1}; π(a) = π(b)) and
s(a, b, c) := e^{b,c} − e^{a,c} + e^{a,b} (a, b, c ∈ IN^{w+1}; π(a) = π(b) = π(c)) .

(e^{•,•} denote the standard basis vectors of P^{m}.)

Theorem: For a fixed element R ∈ M let ψ_{j} : L(E_{j}^{R})_{I C} → I C be linear maps such that ψ_{j} = ψ_{k} on L(E_{j}^{R} ∩ E_{k}^{R})_{I C} = L(E_{⟨a^{j},a^{k}⟩}^{R})_{I C} for each two-dimensional face ⟨a^{j}, a^{k}⟩ < σ. In particular, they induce some element of

(\begin{matrix}Ker & (⊕_{j}L(E_{j}^{R}) ⟶ L(E_{0}^{R})) \\ im & (⊕_{⟨a^{j},a^{k}⟩<σ}L(E_{j}^{R} ∩ E_{k}^{R}) → ⊕_{j}L(E_{j}^{R})) \end{matrix})^{∗} ⊗_{Z Z} I C ⊆ T_{Y}^{2} (−R) (cf. (3.2.4)(2)).

3.3. PROOF OF THE T^{I}-FORMULAS 31

Then, the P -linear map

R/R_{0} ⟶ A
r(a, b; c) ↦ \{ \begin{matrix}ψ_{j}(a − b) x^{π(a+c)−R} & for π(a) ∈ K_{j}^{R}; π(a + c) ≥ R \\ 0 & for π(a) ≥ R or π(a + c) ∈ ⋃_{j} K_{j}^{R} \end{matrix}
s(a, b, c) ↦ 0

is correct defined, and via the formula (3.1.3) it induces the same element of T_{Y}^{2} .

For the proof we refer to (3.3). Nevertheless, we check the correctness of the definition of the P -linear map R/R_{0} → A instantly:

(i) If π(a) is contained in two different sets K_{j}^{R} and K_{k}^{R}, then the two fundamental generators a^{j} and a^{k} can be connected by a sequence a^{j_{0}},…, a^{j_{p}}, such that

• a^{j_{0}} = a^{j}, a^{j_{p}} = a^{k},

• a^{j_{v−1}} and a^{j_{v}} are the edges of some 2-face of σ (v = 1,…, p), and

• π(a) ∈ K_{j_{v}}^{R} for v = 0,…, p.

Hence, supp (a − b) ⊆ E_{j_{v−1}}^{R} ∩ E_{j_{v}}^{R} (v = 1,…, p), and we obtain

ψ_{j}(a − b) = ψ_{j_{1}}(a − b) = … = ψ_{j_{p−1}}(a − b) = ψ_{k}(a − b) .

(ii) There are three types of P -linear relations between the generators r(…) and s(…) of R:

0 = \underline{z}^{d} r(a, b; c) − r(a, b; c + d) + r(a + c, b + c; d) ,
0 = r(b, c; d) − r(a, c; d) + r(a, b; d) − s(a + d, b + d, c + d) + \underline{z}^{d} s(a, b, c) ,
0 = s(b, c, d) − s(a, c, d) + s(a, b, d) − s(a, b, c) .

Our map respects them all.

(iii) Finally, the typical element (\underline{z}^{a} − \underline{z}^{b})e^{cd} − (\underline{z}^{c} − \underline{z}^{d})e^{ab} ∈ R_{0} equals

−r(c, d; a) + r(c, d; b) + r(a, b; c) − r(a, b; d) − s(a + c, b + c, a + d) + s(b + c, a + d, b + d) .

It will be sent to 0, too.

3.3 Proof of the T^{i}-formulas

(3.3.1) We will use the sheaf Ω_{Y}^{1} = Ω_{A| I C}^{1} of Kähler differentials (and the Ext-formulas mentioned in remarks (3.1.2) and (3.1.3)) for computing the modules T_{Y}^{i}.

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

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

After killing torsion of Ω_{Y}^{1} , this gives an embedding: M ⊗_{Z Z} I C[M] can be identified with the module of Kähler differentials on the torus ( I C^{∗})^{rank M} ≅ Spec I C[M], and the previous map corresponds to the restriction of differentials from Y onto the open subset ( I C^{∗})^{rank M} ⊆ Y. Since Ω_{( I C^{∗})^{rank M}}^{1} is just a localization of Ω_{Y}^{1} , we obtain that the above map has exactly tors (Ω_{Y}^{1} ) as its kernel.

In particular, we prefer to use the module Ω1 Y / ; tors (Ω_{Y}^{1} ) ⊆ M ⊗_{Z Z} I C[M] instead of the original

Ω_{Y}^{1} . Since tors (Ω_{Y}^{1} ) is concentrated at the singular locus of Y, the maps

α_{i} : Ext_{A}^{i} ( Ω1 Y / ; tors (Ω_{Y}^{1} ) , A) ↪ Ext_{A}^{i} (Ω_{Y}^{1} , A) ≅ T_{Y}^{i} (i = 1, 2)

are injective. Moreover, they are isomorphisms for

• i = 1, since Y is normal, and for

• i = 2, if Y is smooth in codimension 2.

(3.3.2) We will build a special M-graded, A-free resolution

E ⟶^{d_{E}} D ⟶^{d_{D}} C ⟶^{d_{C}} B ⟶^{d_{B}} Ω1 Y / ; tors (Ω_{Y}^{1} ) → 0

to compute the groups Ext^{1} and Ext^{2}. Denoting supp q := {r ∈ E | q_{r} ≠ 0} (q ∈ L(E)),

L^{2}(E) := L(L(E)) , L^{3}(E) := L(L^{2}(E)), and

supp^{2}ξ := ⋃_{q∈supp ξ} supp q (ξ ∈ L^{2}(E)), supp^{3}ω := ⋃_{ξ∈supp ω} supp^{2}ξ (ω ∈ L^{3}(E)) ,

the resolution can be constructed as follows:

(B) Starting with

B := ⊕_{r∈E} A · B(r) , d_{B} : B(r) ↦ d x^{r} , and deg B(r) := r ,

our first task is to determine the kernel of d_{B} : B → M ⊗_{Z Z} I C[M] (B(r) ↦ r ⊗ x^{r}). Defining deg(s ⊗ x^{r}) := r, the map d_{B} indeed respects the grading. Hence, it suffices to look for the M-homogeneous elements of the kernel. They are generated (as a I C-vector space) by

C(q; ℓ) := ∑_{r∈E} q_{r} · x^{ℓ−r}B(r) (q ∈ L(E) ⊆ ZZ^{E}; ℓ ≥ supp q) .

(In general, for a relation q ∈ L(E), there is no smallest element ℓ ≥ supp q. Hence, it is more naturally to regard all of those elements ℓ (as well as all relations q ∈ L(E)) instead of some finite, but non-canonical selection of generators. This leads us to consider the I C-linear basis of ker d_{B} instead of some A-linear generating system.)

(C) The previous formula suggests

C := ⊕_{\begin{matrix}q∈L(E) \\ ℓ≥supp q \end{matrix}}A · C(q; ℓ) , deg C(q; ℓ) := ℓ

and says, how to define d_{C}. (Caution: Our policy of regarding all elements q and ℓ causes free A-modules of infinite rank.) Now, there are two kinds of homogeneous generators of ker d_{C}:

(i) D(q; ℓ, η) := C(q; η) − x^{η−ℓ} · C(q; ℓ) (q ∈ L(E) , η ≥ ℓ ≥ supp q) compare the different possibilities of choosing lattice points dominating supp q, and

(ii) D(ξ; η) := ∑_{q∈L(E)} ξ_{q} · C(q; η) (ξ ∈ L^{2}(E) , η ≥ supp^{2}ξ) is of the usual type.

(D) As usual, we define

D := (⊕_{\begin{matrix}q∈L(E) \\ η≥ℓ≥supp q \end{matrix}}A · D(q; ℓ, η)) ⊕ (⊕_{\begin{matrix}ξ∈L^{2}(E) \\ η≥supp^{2}ξ \end{matrix}}A · D(ξ; η)) , deg D(…; η) := η ,

and the the map d_{D} is induced by the formulas (C)(i) and (C)(ii). Its kernel is generated by

(i) E(q; ℓ, η, μ) := D(q; η, μ) − D(q; ℓ, μ) + x^{μ−η} · D(q; ℓ, η)

(q ∈ L(E) , μ ≥ η ≥ ℓ ≥ supp q),

(ii) E(ξ; η, μ) := D(ξ; μ) − x^{μ−η} · D(ξ; η) − ∑_{q∈L(E)} ξ_{q} · D(q; η, μ)

(ξ ∈ L^{2}(E) , μ ≥ η ≥ supp^{2}ξ),

(iii) E(ω; μ) := ∑_{ξ∈L^{2}(E)} ω_{ξ} · D(ξ; μ) (w ∈ L^{3}(E) , μ ≥ supp^{3}w).

(E) Finally, it is clear, how to define E and the map d_{E}. Moreover, the generators E(…; μ) of E are defined to have degree μ ∈ M.

(3.3.3) Combining the two exact sequences

R/I R ⟶ A^{m} ⟶I/I^{2} → 0 and I/I^{2} ⟶ Ω_{I C^{w+1}}^{1} ⊗ A ⟶ Ω_{Y}^{1} → 0 ,

we get a complex (not exact at the place of A^{m}) involving Ω_{Y}^{1} . We will compare in the following

commutative diagram this complex with the previous resolution of Ω1 Y / ; tors (Ω_{Y}^{1} ) :

[figure: I/I^{2}; R/I R; A^{m}; Ω_{I C^{w+1}}^{1} ⊗A; Ω_{Y}^{1}; 0; im d_{D}; E; D; C; B; Ω1 Y/ ; tors (Ω_{Y}^{1} ); 0; d; p_{D}; p_{C}; p_{B} ∼; d_{E}; d_{D}; d_{C}]

Let us explain the three labeled vertical maps:

(B) p_{B} : dz_{r} ↦ B(r) is an isomorphism between two free A-modules of rank w + 1.

(C) p_{C} : e^{ab} ↦ C(a − b; π(a)). In particular, the image of this map is spanned by those C(q, ℓ) meeting ℓ ≥ q̄:= π(q^{+}) (cf. (4.2.1)) which is stronger than just ℓ ≥ supp q.

(D) Finally, p_{D} arises as pull back of p_{C} to R/I R. It can be described by r(a, b; c) ↦ D(a − b; π(a), π(a + c)) and s(a, b, c) ↦ D(ξ; π(a)) (ξ denotes the relation ξ = [(b − c) − (a − c) + (a − b) = 0]).

Remark: Starting with the typical R_{0}-element mentioned in (3.2.7)(iii), the previous description of the map p_{D} yields 0 (even in D).

(3.3.4) By (3.3.1) we get the A-modules T_{Y}^{i} by computing the cohomology of the complex dual to that of (3.3.2).

Denote by G one of the capital letters B, C, D, or E. Then, an element ψ of the dual free module G^{∗} = (⨁_{G} I C[σ^{∨} ∩ M] · G)^{∗} can be described by giving elements g(x) ∈ I C[σ^{∨} ∩ M] to be the images

of the generators G (g stands for b, c, d, or e, respectively).

For ψ to be homogeneous of degree −R ∈ M, g(x) has to be a monomial of degree

deg g(x) = −R + deg G.

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

g ≠ 0 implies − R + deg G ≥ 0 (i.e. − R + deg G ∈ σ^{∨}).

(3.3.5) For computing T_{Y}^{1} (−R), the interesting part of the dualized complex (3.3.2)^{∗} in degree −R equals the complex of infinite-dimensional I C-vector spaces

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

with coordinates \underline{b}, \underline{c}, and \underline{d}, respectively:

B_{−R}^{∗} = \{\underline{b(r)} | b(r) = 0 for r − R ∉ σ^{∨}\} ⊆ I C^{E}
C_{−R}^{∗} = \{\underline{c(q; ℓ)} | c(q; ℓ) = 0 for ℓ − R ∉ σ^{∨}\} ⊆ I C^{#\{C(q,ℓ)\}}
D_{−R}^{∗} = \{[\underline{d(q; ℓ, η)}, \underline{d(ξ; η)}] | \begin{matrix}d(q; ℓ, η) & = & 0 & for & η − R ∉ σ^{∨}, and \\ d(ξ; η) & = & 0 & for & η − R ∉ σ^{∨}\} . \end{matrix}

(The parameters run through the following sets: r ∈ E for \underline{b(r)}; q ∈ L(E), ℓ ≥ supp q for \underline{c(q; ℓ)}; q ∈ L(E), η ≥ ℓ ≥ supp q for \underline{d(q; ℓ, η)}; and ξ ∈ L^{2}(E), η ≥ supp^{2}ξ for \underline{d(ξ; η)}.)

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

c(q; ℓ)  = ∑_{r∈E} q_{r} · b(r)  and
d(q; ℓ, η) = c(q; η) − c(q; ℓ),
d(ξ; η) = ∑_{q∈L(E)} ξ_{q} · c(q; η) .

Hence, we are able to compute ker d_{D}^{∗} and im d_{C}^{∗}:

ker d_{D}^{∗} = \{\underline{c(q, ℓ)} | \begin{matrix}• c(q, ℓ) = 0 if ℓ ≱ R \\ • c(q, ℓ) does not depend on ℓ \\ • ∑_{q∈L(E)} ξ_{q} · c(q, ℓ) = 0 for ξ ∈ L^{2}(E) \} \end{matrix}

= \{\underline{c(q)} | \begin{matrix}• c(q) = 0 if ∃ℓ ≥ supp q : ℓ ≱ R \\ • ∑_{q∈L(E)} ξ_{q} · c(q) = 0 for ξ ∈ L^{2}(E) \} \end{matrix}
= {c ∈ L(E)_{I C}^{∗} | c(q) = 0 if ∃ℓ ≥ supp q : ℓ ≱ R}

and

im d_{C}^{∗} = \{c ∈ L(E)_{I C}^{∗} | ∃ \underline{b(r)} : \begin{matrix}• b(r) = 0 if r ≱ R \\ • c(q) = ∑_{r∈E} q_{r} · b(r) \} . \end{matrix}

Involving the sets K_{j}^{R} introduced in (3.2.3) and using part (a) of the corollary contained in that section, we can reformulate the description of ker d_{D}^{∗}:

ker d_{D}^{∗} = {c ∈ L(E)_{I C}^{∗} | c(q) = 0 if ∃j, ∃ℓ ≥ supp q : ℓ ∈ K_{j}^{R}}
= {c ∈ L(E)_{I C}^{∗} | c(q) = 0 if ∃j : supp q ⊆ K_{j}^{R}}
= ker [L(E)_{I C}^{∗} → ⊕_{j}L(E_{j}^{R})_{I C}^{∗}] .

Moreover, we have

\begin{aligned}im d_{C}^{∗} & = & im [ I C^{E\E_{0}^{R}} ↪ I C^{E} ⟶ ⟶ L(E)_{I C}^{∗}] \\ & = & ker [L(E)_{I C}^{∗} ⟶ ⟶ L(E_{0}^{R})_{I C}^{∗}] \end{aligned}

(dualize the exact sequence 0 → L(E_{0}^{R}) → L(E) → I C^{E\E_{0}^{R}} to get the latter equality). Hence,

T^{1}(−R) = ker d∗ D/im d_{C}^{∗} = ker [L(E_{0}^{R})_{I C}^{∗} → ⊕_{j}L(E_{j}^{R})_{I C}^{∗}]
= (^{L(ER 0)/} ∑_{j} L(E_{j}^{R}))^{∗} ⊗_{Z Z} I C ,

and the first part of Theorem (3.2.4) is proved.

To compare this T^{1}-formula with that involving Hom(I/I^{2}, A) (i.e. to prove Theorem (3.2.6)), we regard the commutative diagram constructed in (3.3.3): Given a linear map φ : L(E)_{I C} → I C, we obtain (through the recent identifications) an element of C_{−R}^{∗} - it maps C(q; ℓ) onto φ(q) · x^{ℓ−R}. Pulling via p_{C}^{∗} this homomorphism back to A^{m}, it sends the basis vector e^{ab} to φ(a − b) · x^{π(a)−R}. On the other hand, it is the same e^{ab} that stands for the binomial equation \underline{z}^{a} − \underline{z}^{b} ∈ I. Hence, Theorem (3.2.6) is proved.

(3.3.6) Proving the formula for T^{2}(−R) requires to consider the vector space

E_{−R}^{∗} = \{[\underline{e(q; ℓ, η, μ)}, \underline{e(ξ; η, μ)}, \underline{e(ω; μ)}] | each coordinate vanishes for μ − R ∉ σ^{∨}\}

(additionally to C_{−R}^{∗} and D_{−R}^{∗} mentioned in (3.3.5)). The differential d_{E}^{∗} is given by

\begin{aligned}e(q; ℓ, η, μ) & = & d(q; η, μ) − d(q; ℓ, μ) + d(q; ℓ, η), \\ e(ξ; η, μ) & = & d(ξ; μ) − d(ξ; η) − ∑_{q∈F} ξ_{q} · d(q; η, μ), \\ e(ω; μ) & = & ∑_{ξ∈G} ω_{ξ} · d(ξ; μ), \end{aligned}

and we obtain

ker d_{E}^{∗} = \{[\underline{d(q; ℓ, η)}; \underline{d(ξ; η)}] | \begin{matrix}d(q; ℓ, η) = d(ξ; η) = 0 for η − R ∉ σ^{∨}, \\ d(q; ℓ, μ) = d(q; ℓ, η) + d(q; η, μ) (μ ≥ η ≥ ℓ ≥ supp q), \\ d(ξ; μ) = d(ξ; η) + ∑_{q} ξ_{q} · d(q; η, μ) (μ ≥ η ≥ supp^{2}ξ), \\ ∑_{ξ∈L^{2}(E)} ω_{ξ} d(ξ; μ) = 0 for ω ∈ L^{3}(E) with μ ≥ supp^{3}ω \}, \end{matrix}

im d_{D}^{∗} = \{[\underline{d(q; ℓ, η)}; \underline{d(ξ; η)}] | ∃c(q, ℓ) : \begin{matrix}c(q, ℓ) = 0 for ℓ − R ∉ σ^{∨}, \\ d(q; ℓ, η) = c(q; η) − c(q; ℓ), \\ d(ξ; η) = ∑_{q∈L(E)} ξ_{q} · c(q; η) \}. \end{matrix}

Again, these two vector spaces can be simplified through involving the sets K_{j}^{R} (or E_{j}^{R}). Let us define

W := \{[\underline{x_{j}(q)}_{(q∈L(E_{j}^{R}))}] | \begin{matrix}x_{j}(q) = x_{k}(q) for • ⟨a^{j}, a^{k}⟩ < σ is a 2-face and • q ∈ L(E_{j}^{R} ∩ E_{k}^{R}) , \\ ξ ∈ L^{2}(E_{j}^{R}) implies ∑_{q} ξ_{q} · x_{j}(q) = 0 \} and \end{matrix}

W′ := \{[\underline{x(q)}_{(q∈∪_{j}L(E_{j}^{R}))}] | ξ ∈ L(⋃_{j} L(E_{j}^{R})) implies ∑_{q} ξ_{q} · x(q) = 0 \}.

Lemma: The linear map W → ker d_{E}^{∗} defined by

d(q; ℓ, η) := \{ \begin{matrix}x_{j}(q) & for & ℓ ∈ K_{j}^{R}, η ≥ R \\ 0 & for & ℓ ≥ R or η ∈ ⋃_{j} K_{j}^{R}; \end{matrix}
d(ξ; η) := 0

induces injective maps

W / ; W′ ↪ ker d_{E}^{∗}/ ; im d_{D}^{∗} ↪ T^{2}(−R) .

If Y is smooth in codimension 2, both of them will be isomorphisms. (We already have known that for the latter map, cf. (3.3.1).)

Proof: 1) The map W → ker d_{E}^{∗} is correct defined: On the one hand, an argument as used in (3.2.7)(i) shows that ℓ ∈ K_{j}^{R} ∩ K_{k}^{R} would imply x_{j}(q) = x_{k}(q). On the other hand, the image of [x_{j}(q)]_{q∈L(E_{j}^{R})} meets all conditions in the description of ker d_{E}^{∗}.

2) W′ maps to im d_{D}^{∗} (take c(q, ℓ) := x(q) for ℓ ≥ R and c(q, ℓ) := 0 otherwise).

3) The map between the two factor spaces is injective: Assume for [x_{j}(q)]_{q∈L(E_{j}^{R})} that there exist elements c(q, ℓ), such that

c(q; ℓ) = 0 for ℓ ∈ ⋃_{j} K_{j}^{R} ,
x_{j}(q) = c(q; η) − c(q; ℓ) for η ≥ ℓ, ℓ ∈ K_{j}^{R}, η ≥ R ,
0 = c(q; η) − c(q; ℓ) for η ≥ ℓ and [ℓ ≥ R or η ∈ ⋃_{j} K_{j}^{R}] , and
0 = ∑_{q} ξ_{q} · c(q; η) for η ≥ supp^{2}ξ .

In particular, x_{j}(q) do not depend on j, and these elements have the property

∑_{q} ξ_{q} · x_{•}(q) = 0 for ξ ∈ L(⋃_{j} L(E_{j}^{R})).

4) If Y is smooth in codimension 2, the map is surjective : Given an element [d(q; ℓ, η), d(ξ; η)] ∈ ker d_{E}^{∗}, there exist complex numbers c(q; η) such that:

(i) d(ξ; η) = ∑_{q} ξ_{q} · c(q; η) ,

(ii) c(q; η) = 0 for η ∉ R + σ^{∨} (i.e. η ∈ ⋃_{j} K_{j}^{R}) .

(Do this separately for each η and distinguish between the cases η ∈ R + σ^{∨} and η ∉ R + σ^{∨}.) In particular, [c(q; η) − c(q; ℓ), d(ξ; η)] ∈ im d_{D}^{∗}. Hence, up to now, we have seen that we may assume d(ξ; η) = 0.

Let us choose some sufficiently high degree ℓ^{∗} ≥ E. Then,

x_{j}(q) := d(q; ℓ, η) − d(q; ℓ^{∗}, η)

(with ℓ ∈ K_{j}^{R}, ℓ ≥ supp q (cf. Corollary (3.2.3)(a)), and η ≥ ℓ, ℓ^{∗}, R) defines some preimage:

(i) It is independent from the choice of η: Using a different η′ generates the difference d(q; η, η′)− d(q; η, η′).

(ii) It is independent from ℓ ∈ K_{j}^{R}: Choosing another ℓ′ ∈ K_{j}^{R} with ℓ′ ≥ ℓ would add the summand d(q; ℓ, ℓ′) which is 0; for the general case use Corollary (3.2.3)(a).

(iii) If ⟨a^{j}, a^{k}⟩ < σ is a 2-face with supp q ⊆ L(E_{j}^{R})∩L(E_{k}^{R}), then by Corollary (3.2.3)(b) we may choose an ℓ ∈ K_{j}^{R} ∩ K_{k}^{R} achieving x_{j}(q) = x_{k}(q).

3.4. AN ALTERNATIVE TO THE COMPLEX L(E^{R})_{•} 37

(iv) For ξ ∈ L^{2}(E_{j}^{R}) we have

∑_{q} ξ_{q} · d(q; ℓ, η) = ∑_{q} ξ_{q} · d(q; ℓ^{∗}, η) = 0 ,

and this gives the corresponding relation for the x_{j}(q)′s.

(v) Finally, if we apply to [\underline{x_{j}(q)}] ∈ W the linear map W → ker d_{E}^{∗}, the result differs from [d(q; ℓ, η), 0] ∈ ker d_{E}^{∗} by the im d_{D}^{∗}-element built from

c(q; ℓ) := \{ \begin{matrix}d(q; ℓ, η) − d(q; ℓ^{∗}, η) & if ℓ ≥ R \\ 0 & otherwise . \end{matrix}

□

(3.3.7) Now, it is easy to complete the proofs for Theorem (3.2.4) (part 2 and 3) and Theorem (3.2.7):

First, for a tuple [\underline{x_{j}(q)}]_{q∈L(E_{j}^{R})}, the condition

ξ ∈ L^{2}(E_{j}^{R}) implies ∑_{q} ξ_{q} · x_{j}(q) = 0

is equivalent to the fact that the coordinates x_{j}(q) are induced by elements x_{j} ∈ L(E_{j}^{R})_{I C}^{∗}. The other condition for elements of W just says that for 2-faces ⟨a^{j}, a^{k}⟩ < σ there is x_{j} = x_{k} on L(E_{j}^{R} ∩ E_{k}^{R})_{I C} = L(E_{j}^{R})_{I C} ∩ L(E_{k}^{R})_{I C}. In particular, we obtain

W = ker ( ⊕_{j} L(E_{j}^{R})_{I C}^{∗} → ⊕_{⟨a^{j},a^{k}⟩<σ}L(E_{j}^{R} ∩ E_{k}^{R})_{I C}^{∗}) .

In the same way we get

W′ = (∑_{j} L(E_{j}^{R})_{I C} )^{∗} ,

and our T^{2}-formula is proven.

Moreover, if ψ_{j} : L(E_{j}^{R})_{I C} → I C are linear maps defining an element of W, then they induce the following A-linear map on D (even on im d_{D}):

D(q; ℓ, η) ↦ \{ \begin{matrix}ψ_{j}(q) · x^{η−R} & for & ℓ ∈ K_{j}^{R}, η ≥ R \\ 0 & for & ℓ ≥ R or η ∈ ⋃_{j} K_{j}^{R} \end{matrix}
D(ξ; η) ↦ 0 .

Now, looking at the diagram of (3.3.3), this translates exactly into the claim of Theorem (3.2.7).

3.4 An alternative to the complex L(E^{R})_{•}

(3.4.1) Let R ∈ M be fixed for the whole section. The complex L(E^{R})_{•} introduced in (3.2.3) fits naturally into the exact sequence

0 → L(E^{R})_{•} ⟶ (ZZ^{E^{R}})_{•} ⟶ span(E^{R})_{•} → 0

of complexes built in the same way as L(E^{R})_{•}, i.e.

(ZZ^{E^{R}})_{−k} := ⊕\begin{matrix}τ<σ \\ dim τ=k \end{matrix}ZZ^{E_{τ}^{R}}  and  span(E^{R})_{−k} := ⊕\begin{matrix}τ<σ \\ dim τ=k \end{matrix}span(E_{τ}^{R}) .

Lemma: The complex (ZZ^{E^{R}})_{•} is exact.

Proof: The complex (ZZ^{E^{R}})_{•} can be decomposed into a direct sum

(ZZ^{E^{R}})_{•} = ⨁_{r∈E}(ZZ^{E^{R}})(r)_{•}

showing the contribution of each r ∈ E. The complexes occurring as summands are defined as

(ZZ^{E^{R}})(r)_{−k} := ⊕\begin{matrix}τ<σ \\ dim τ=k \end{matrix}\{ \begin{matrix}ZZ = ZZ^{\{r\}} & for r ∈ E_{τ}^{R} \\ 0 & otherwise \end{matrix} \}

= ZZ^{#{τ | dim τ=k; r∈E_{τ}^{R}}} .

Denote by H^{+} the halfspace

H^{+} := {a ∈ N_{IR} | ⟨a, r⟩ < ⟨a, R⟩} ⊆ N_{IR}.

Then, for τ ≠ 0, the fact that r ∈ E_{τ}^{R} is equivalent to τ \ {0} ⊆ H^{+}. On the other hand, r ∈ E_{0}^{R} corresponds to the condition σ ∩ H^{+} ≠ ∅. In particular, (ZZ^{E^{R}})(r)_{•}, shifted by one place, equals the complex for computing the reduced homology of the topological space ∪{τ | τ \ {0} ⊆ H^{+}} ⊆ σ cut by some affine hyperplane. Since this space is contractable, the complex is exact. □

As a corollary, we obtain

Theorem: The complexes L(E^{R})_{•}^{∗} and span(E^{R})_{•}^{∗}[1] are quasiisomorphic. In particular, under the usual assumptions (cf. Theorem (3.2.4)), we obtain

T_{Y}^{i} (−R) = H^{i} (span(E^{R})_{•}^{∗} ⊗_{Z Z} I C) .

For later use, we mention explicitly how to change from the L(…)- to the span-language at the T^{1}-level: Starting with q ∈ L(E_{0}^{R}) (i.e. ∑_{v} q_{v}r^{v} = 0), we split it into q = ∑_{j} q^{j} (q^{j} ∈ ZZ^{E_{j}^{R}}) and take v^{j} := − ∑_{v} q_{v}^{j}r^{v} for the elements of span E_{j}^{R}. Obviously, they sum up to 0 ∈ M. (The strange sign in the definition of v^{j} is, of course, not necessary - it just makes some upcoming formulas nicer.)

Example: In the case of two-dimensional cyclic quotient singularities (cf. (3.2.5)), the formula for T_{Y}^{1} (−R) comes down to

T_{Y}^{1} (−R) = (^{(span^{I C}ER 1) ∩ (span^{I C}ER 2)/}span_{I C}(E_{1}^{R} ∩ E_{2}^{R}))^{∗} .

The several possibilities for R (as mentioned in (3.2.5)) translate into

(i) span_{I C}E_{1}^{R} = (a^{1})^{⊥}, span_{I C}E_{2}^{R} = I C^{2} (or (a^{2})^{⊥}, if w = 2), and span_{I C}E_{12}^{R} = 0.

(ii) span_{I C}E_{1}^{R} = span_{I C}E_{2}^{R} = I C^{2} , and span_{I C}E_{12}^{R} = 0.

(iii) span_{I C}E_{1}^{R} = span_{I C}E_{2}^{R} = I C^{2} , and span_{I C}E_{12}^{R} = I C · R.

(iv) span_{I C}E_{12}^{R} = span_{I C}E_{1}^{R} (or span_{I C}E_{12}^{R} = span_{I C}E_{2}^{R}).

Remark: The reason that makes the recent formula easier to handle than the old one is the simpler description of span_{I C}E_{j}^{R} (compared with L(E_{j}^{R})_{I C}):

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

Moreover, if Y is smooth in codimension two, then span_{I C}E_{jk}^{R} = (span_{I C}E_{j}^{R}) ∩ (span_{I C}E_{k}^{R}) holds for 2-faces ⟨a^{j}, a^{k}⟩ < σ.

(3.4.2) We are going to use the previous theorem to develop an alternative description of T_{Y}^{1} (−R) which relates infinitesimal deformations of Y (of degree −R) to Minkowski summands of the polyhedron Q := σ ∩ [R = 1]. (It neither needs to be compact, nor its vertices to be lattice points.) If Y is smooth in codimension two, then this relation will be quite straight.

Remark: For ⟨a^{j}, R⟩ ≥ 1, we define ā^{j} := a^{j}/⟨a^{j}, R⟩. Those points form exactly the set of vertices of Q. Moreover, ⟨a^{j}, R⟩ = 1, if and only if ā^{j} is a lattice point (i.e. ā^{j} ∈ N).

Nevertheless, even in this more general situation, we are able to keep using the definition of C(Q) ⊆ V (Q) (cf. (2.2.2)) - we just have to deal with the compact faces only: Denoting by d^{1},…, d^{N} ∈ R^{⊥} ⊆ N_{IR} the compact edges of Q, we can assign to each compact 2-face ε < Q its sign vector \underline{ε} ∈ \{0, ±1\}^{N} in the usual way (cf. (2.2.1)). Then,

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

Again, the points of the cone C(Q) := V (Q) ∩ IR_{≥0}^{N} parametrize exactly the Minkowski summands (in the sense of Definition (2.1.3)) of positive multiples of Q. Moreover, we need the additional vector space

W (Q) := IR^{#{vertices not belonging to N}}

with coordinates s_{j} for each vertex ā^{j} ∈ Q \ N (i.e. ⟨a^{j}, R⟩ ≥ 2).

(3.4.3) Theorem: For each compact edge d^{jk} = \overline{ā^{j}ā^{k}} there is a subspace U_{jk} ⊆ V (Q)^{∗} ⊕ W (Q)^{∗} belonging to one of the following three (mutually overlapping) types:

(0) U_{jk} = 0,

(1) U_{jk} is generated by s_{j} − s_{k} (drop those elements, i.e. set U_{jk} = 0, if not both coordinates exist), or

(2) U_{jk} is generated by t_{jk} − s_{j}, t_{jk} − s_{k} (again, drop generators that do not make sense).

(The actual type of U_{jk} depends on the edge \overline{ā^{j}ā^{k}} only and will be discussed in the sequel. Anyway, if Y is smooth in codimension two, then type (2) applies for each edge.)

Using that notation, we have

T_{Y}^{1} (−R) = ker [V_{I C}(Q) ⊕ W_{I C}(Q) ⟶ ⟶ (∑_{d^{jk}} U_{jk})^{∗}]/ I C · (1,…, 1; 1,…, 1) .

In some sense, the vector space V (Q) (encoding the Minkowski summands) can be considered the main tool to describe infinitesimal deformations. The elements of W (Q) can - depending on the type of the U_{jk}’s - either be additional parameters, or they imply dependencies between elements of V (Q) (excluding Minkowski summands not fitting in a certain type).

Now, we start describing how the type of U_{jk} depends on the shape of the particular edge \overline{ā^{j}ā^{k}} (only being important in case Y has two-codimensional singularities). The easiest way to do so seems to be the following one: Replace Q by the line segment \overline{ā^{j}ā^{k}} and take for U_{jk} exactly that type yielding the right T^{1}(−R)- dimension for the two-dimensional cyclic quotient singularity induced by the cone < a^{j}, a^{k} > (cf. (i)-(iv) in (3.2.5) or (3.4.1)). If one prefers a more explicit description, then denote by g^{jk} the affine line (in N_{IR}) containing the edge \overline{ā^{j}ā^{k}} of Q and distinguish between the following two cases:

(a) The line g^{jk} contains lattice points (equivalently: R is primitive, when regarded as an element of M /(a^{j}, a^{k})^{⊥}):

(a.1) If the edge \overline{ā^{j}ā^{k}} contains an interior lattice point, then we assign to it the space U_{jk} of type (0).

(a.2) Otherwise, of type (2).

(b) The line g^{jk} contains no lattice points at all:

Denote by p ∈ IN \ {0} the smallest number such that lattice points are contained in p · g^{jk} (i.e. R ≡ p · R^{0} (mod (a^{j}, a^{k})^{⊥}) for some R^{0} ∈ M being primitive in M /(a^{j}, a^{k})^{⊥}). In particular, p ≥ 2 and p|⟨a^{j}, R⟩; ⟨a^{k}, R⟩. Hence, both coordinates s_{j} and s_{k} are well defined. Now, in any case, we assign to our edge at least the equation s_{j} = s_{k} - that means, type (0) will be excluded for U_{jk}. Actually, we assign to \overline{ā^{j}ā^{k}} the subspace U_{jk} of

(b.1) type (1), if the closed line segment p · \overline{ā^{j}ā^{k}} contains at least p interior lattice points (or at least p − 1 ones, if both bounds p ā^{j} and p ā^{k} belong to N themselves),

(b.2) type (2), otherwise. (This case implies that d^{jk} has (lattice-) length smaller than one.)

Remark:

• If R ∈ M is not primitive, then neither it is in the factors M /(a^{j}, a^{k})^{⊥}. In particular, we are always in case (b), and all coordinates s_{j} have to be equal on T_{Y}^{1} (−R). Setting them all zero (we can do this mod I C · (\underline{1}, \underline{1})), we can consider T_{Y}^{1} (−R) a subspace of V_{I C}(Q). The condition for a (complexified) Minkowski summand belonging to T_{Y}^{1} (−R) is the following: Whenever an edge d^{jk} of Q is so “short” fitting into case (b.2), then it has to be erased in the Minkowski summand.

• If Y is smooth in codimension two, then for each edge d^{jk} the elements a^{j}, a^{k} form a part of a ZZ-basis for the lattice N. In particular, the line segment p · \overline{ā^{j}ā^{k}} does not contain any interior lattice point. Hence, it fits into case (a.2) or (b.2) - always inducing U_{jk} of type (2).

(3.4.4) Proof (of the previous theorem): Step 1: From Theorem (3.4.1) we know that T_{Y}^{1} (−R) equals the complexification of the cohomology of the complex

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

According to the remark of (3.4.1), we can represent an element of ⊕_{j} (span_{IR}E_{j}^{R})^{∗} by a family of • b^{j} ∈ N_{IR}, if ⟨a^{j}, R⟩ ≥ 2 and

• b^{j} ∈ N^{IR}/IR · a^{j}, if ⟨a^{j}, R⟩ = 1.

Dividing by the image of N_{IR} means to shift this family by common vectors b ∈ N_{IR}. More interesting is the condition for families \{b^{j}\} mapping onto 0: For each compact edge \overline{ā^{j}, ā^{k}} < Q the functions b^{j} and b^{k} have to be equal on span_{IR}E_{jk}^{R}. Since

(a^{j}, a^{k})^{⊥} ⊆ span_{IR}E_{jk}^{R} ⊆ (span_{IR}E_{j}^{R}) ∩ (span_{IR}E_{k}^{R}) ,

we obtain as a necessary condition b^{j} − b^{k} ∈ IRa^{j} + IRa^{k} immediately. However, in case the cone ⟨a^{j}, a^{k}⟩ is singular, the actual behavior of span_{IR}E_{jk}^{R} requires a closer look.

Step 2: We introduce new “coordinates”:

• b̄^{j} := b^{j} − ⟨b^{j}, R⟩ā^{j} ∈ R^{⊥} (being well defined even in the case ⟨a^{j}, R⟩ = 1);

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

The shift of the b^{j}’s by an element b ∈ N_{IR} (i.e. (b^{j})′ = b^{j} + b) appears in the new coordinates as follows:

(b̄^{j})′ = (b^{j})′ − ⟨(b^{j})′, R⟩ā^{j} = b^{j} + b − ⟨b^{j}, R⟩ā^{j} − ⟨b, R⟩ā^{j}
= b̄^{j} + b − ⟨b, R⟩ā^{j} ,
s′_{j} = −⟨(b^{j})′, R⟩ = s_{j} − ⟨b, R⟩ .

In particular, an element b ∈ R^{⊥} does not change the s_{j}’s but shifts the b̄^{j}’s inside the hyperplane R^{⊥}. Hence, the set of the b̄^{j}’s should be considered modulo translation (inside R^{⊥}) only.

On the other hand, the condition b^{j} − b^{k} ∈ IRa^{j} + IRa^{k} changes into b̄^{j} − b̄^{k} ∈ IRā^{j} + IRā^{k} or even b̄^{j} − b̄^{k} ∈ IR(ā^{j} − ā^{k}) (look at the values of R). Hence, the b̄^{j}’s form the vertices of an (at least generalized) Minkowski summand of Q. Modulo translation, this summand is completely described by the dilatation factors defined by

b̄^{j} − b̄^{k} = t_{jk} · (ā^{j} − ā^{k}) .

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

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

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

Step 3: Actually, the elements b^{j} and b^{k} have to be equal on span_{IR}E_{jk}^{R} which may be a larger space than just (a^{j}, a^{k})^{⊥}. To measure the difference we consider the factor span_{IR}E_{jk}^{R}/(a^{j}, a^{k})^{⊥}

contained in the two-dimensional vector space M^{IR}/(a^{j}, a^{k})^{⊥} = span_{IR}(a^{j}, a^{k})^{∗}. Since this factor

coincides with the set span_{IR}ER̄ ; jk assigned to the two-dimensional cone ⟨a^{j}, a^{k}⟩ ⊆ span_{IR}(a^{j}, a^{k}) (R̄ denotes the image of R in span_{IR}(a^{j}, a^{k})^{∗}), we may assume that σ = ⟨a^{1}, a^{2}⟩ (i.e. j = 1, k = 2) represents a two-dimensional cyclic quotient singularity. In particular, we have just to discuss the four cases (i)-(iv) from the example contained in (3.2.5) and (3.4.1):

Lemma: Let σ = ⟨a^{1}, a^{2}⟩ be a two-dimensional cone as in (3.2.5).

(1) Let R ∈ int σ^{∨}∩M be primitive. Then, R is indecomposable (i.e. an element of {r^{1},…, r^{w−1}}), if and only if the line segment \overline{ā^{1}ā^{2}} = σ ∩ [R = 1] contains an interior lattice point.

(2) Assume R = r^{v} (v ∈ {1,…, w − 1}). Then, the number a_{v} (defined by r^{v−1} + r^{v+1} = a_{v} · r^{v}) is one bigger than the number of interior lattice points of \overline{ā^{1}ā^{2}}.

Proof: Both statements are contained in the explanations of [Od], §1.6. Nevertheless, we present the short proof here directly:

(1) If R = r^{v}, then {r^{v}, r^{v+1}} is a ZZ-basis of M, and the equations ⟨a^{∗}, r^{v}⟩ = ⟨a^{∗}, r^{v+1}⟩ = 1 define an interior lattice point a^{∗} of σ. For the reverse implication, assume that a^{∗} ∈ int σ ∩ N ∩ [R = 1]. Then, a decomposition R = R_{1} + R_{2} inside σ^{∨} ∩ M would (w.l.o.g.) imply that ⟨a^{∗}, R_{1}⟩ = 0 - contradicting the fact that a^{∗} belongs to the interior of σ.

(2) An interior lattice point a^{∗} of \overline{ā^{1}ā^{2}} is characterized by the conditions

⟨a^{∗}, r^{v}⟩ = 1 ; ⟨a^{∗}, r^{v+1}⟩ ∈ ZZ; ⟨a^{∗}, r^{v+1}⟩ ≥ 1 ; ⟨a^{∗}, r^{v−1}⟩ ≥ 1 .

Since r^{v−1} + r^{v+1} = a_{v} · r^{v}, those points are parametrized by integers g (= ⟨a^{∗}, r^{v+1}⟩) meeting 1 ≤ g ≤ a_{v} − 1. □

Now, let us start with case (a), i.e. we assume that R is a primitive element of M. By the first part of the previous lemma we know that (a.1) is a synonymous for R ∈ {r^{1},…, r^{w−1}}. Those degrees were discussed in (i) and (ii) of (3.2.5) and (3.4.1) - we have obtained span_{IR}E_{12}^{R} = 0. The lack of any additional equations for the b^{j}’s means that U_{12} has to be zero, i.e. of type (0). The remaining primitive degrees yield T_{Y}^{1} (−R) = 0 or, equivalently,

span_{IR}E_{12}^{R} = (span_{IR}E_{1}^{R}) ∩ (span_{IR}E_{2}^{R}) .

Hence, at least after a suitable lifting of b^{j} to N_{IR}, we can use the equation b^{1} = b^{2} in N_{IR}:

b̄^{1} = b^{1} − ⟨b^{1}, R⟩ā^{1} = b^{2} − ⟨b^{1}, R⟩ā^{1} = b̄^{2} − ⟨b^{1}, R⟩ā^{1} + ⟨b^{2}, R⟩ā^{2}

yields

t_{12} · (ā^{1} − ā^{2}) = b̄^{1} − b̄^{2} = −⟨b^{1}, R⟩ā^{1} + ⟨b^{2}, R⟩ā^{2} .

In particular, as far as defined, the variables s_{j} (j = 1, 2) equal t_{12}. This is encoded in the vector space U_{12} of type (2).

For case (b), we assume that R = p · R^{0} with some primitive R^{0} ∈ M. Using the first part of the previous lemma again, we see that condition (b.1) implies R^{0} ∈ {r^{1},…, r^{w−1}}. Moreover, it is the second part of that lemma telling us that (b.1) is actually equivalent to (iii) from (3.2.5) and (3.4.1). Hence, assuming (b.1), we obtain span_{IR}E_{12}^{R} = IR · R as the space where b^{1} and b^{2} have to be equal on. That means s_{1} = s_{2}, and we need U_{12} of type (1). On the other hand, the class (b.2) implies T_{Y}^{1} (−R) = 0 again. Similarly to the case (a.1), we have to choose type (2) for U_{12}.

Theorem (3.4.3) (including the characterization of the vector spaces U_{jk}) is proven. □

(3.4.5) Finally, we assume that Y is smooth in codimension two. Then, U_{jk} is always of the (maximal) type (2), and the additional coordinates s_{j} of W (Q) are completely determined by those of V (Q) (but still enforcing some dependencies between them). In particular, as a corollary from (3.4.3), we obtain

Theorem: If Y is smooth in codimension two, then T_{Y}^{1} (−R) ⊆ V^{I C}(Q)/ I C · (\underline{1}) is given by the equations

t_{jk} = t_{kl} (if d^{jk}, d^{kl} are compact edges with a common non-lattice vertex ā^{k}).

That means, infinitesimal deformations of Y are in one-to-one correspondence to those equivalence classes of Minkowski summands of Q keeping the stars of non-lattice vertices in Q unchanged up to homothety.

(3.4.6) Still assuming Y to be smooth in codimension two, we try to illuminate the behavior of T_{Y}^{2} (−R). Let us start with the following commutative diagram built from the first terms of the span-complex defined in (3.4.1):

[figure: ⊕_{ε<σ}( ∩_{a^{j}∈ε} span E_{j}^{R})^{∗}; ⊕_{a^{j}∈σ}(span E_{j}^{R})^{∗}; ⊕_{⟨a^{j},a^{k}⟩<σ}(span E_{jk}^{R})^{∗}; ⊕_{ε<σ}(span E_{ε}^{R})^{∗}]

(ε < σ stands for three-dimensional faces of σ.) The vertical map is surjective; it would be an isomorphism, if Y was smooth in codimension three. According to Theorem (3.4.1), T_{Y}^{2} (−R) equals the cohomology of the (complexified) bottom row. In particular, there is a map

Ψ : T_{Y}^{2} (−R) → ⨁_{ε<σ} (∩_{a^{j}∈ε}span E_{j}^{R}/span E_{ε}^{R})^{∗} ⊆ ⨁_{ε<σ} ( ∩_{a^{j}∈ε} span_{I C}E_{j}^{R})^{∗} .

“evaluating” this vector space.

Proposition: If R ∈ M is always different from 1 on the fundamental generators a^{j} of σ (e.g. equals a proper multiple of some other degree), or if dim σ = 3, then Ψ is injective.

Proof: All we have to show is exactness of the top row in the previous diagram. Since this is obvious in the three-dimensional case, let us assume ⟨a^{j}, R⟩ ≠ 1 for j = 1,…, M. Then, the Abelian groups span E_{j}^{R} involved in that complex are either M or 0, depending on ⟨a^{j}, R⟩ > 0 or not. Similarly to the proof of Lemma (3.4.1) this means that our complex measures the reduced homology (⊗_{Z Z}M) of the topological space ∪{τ | τ \ {0} ⊆ [R > 0]} ⊆ σ cut by some affine hyperplane. □

Corollary: If R does not yield 1 on the a^{j}’s, then Ψ provides an embedding

Ψ : T_{Y}^{2} (−R) ↪ N_{I C}^{{ε<σ | R>0 on ε}} .

Moreover, its image in the ε-th summand is killed by span_{I C}E_{ε}^{R}.

3.5 First applications

(3.5.1) For an R ∈ M we denote by τ(R) the smallest face of σ containing all generators a^{j} meeting ⟨a^{j}, R⟩ ≥ 1. (In particular, ⟨a^{k}, R⟩ ≤ 0 for a^{k} ∈ σ \ τ.) If Y (R) stands for the affine variety assigned to τ(R) (considered as a top-dimensional cone in span_{IR}τ(R) = τ(R) − τ(R)), then Y (R) × ( I C^{∗})^{dim σ−dim τ(R)} is an open subset of Y.

Proposition: Let R̄ be the image of R in (span_{IR}τ(R))^{∗} = M/τ(R)^{⊥}. Then, the infinitesimal

deformations of Y in degree −R are caused by those of Y (R), i.e. T_{Y}^{1} (−R) = T_{Y (R)}^{1}(−R̄).

Proof: The polyhedra Q := σ ∩ [R = 1] ⊆ N_{IR} and Q̄ := τ ∩ [R̄ = 1] ⊆ span_{IR}τ(R) ⊆ N_{IR} are built from the same vertices (ā^{j} with ⟨a^{j}, R⟩ ≥ 1) but different cones of unbounded directions (σ ∩ R^{⊥} for Q and τ ∩ R̄^{⊥} = τ ∩ R^{⊥} for Q̄). In particular, the compact faces of Q and Q̄ are the same, and we obtain

V (Q) = V (Q̄) , W (Q) = W (Q̄) , and U_{jk}(Q) = U_{jk}(Q̄) . □

Corollary: If T_{Y}^{1} is finite-dimensional, then for all proper faces τ < σ the toric variety Y_{τ} associated to the cone τ is rigid.

Proof: Let R̄ ∈ M/τ^{⊥} (represented by some R ∈ M) such that T_{Y_{τ}}^{1} (−R̄) ≠ 0. If r ∈ σ^{∨} ∩ M defines the face τ (i.e. τ = σ ∩ r^{⊥}), then R − k · r still represents R̄, and, moreover, τ(R − k · r) ⊆ τ for k ≫ 0. In particular, T_{Y}^{1} (−[R − k · r]) = T_{Y_{τ}}^{1} (−R̄) ≠ 0 for those (infinitely many) k. □

Since none of the two-dimensional cyclic quotient singularities are rigid, we finally obtain: At least three-dimensional affine toric varieties have an infinite-dimensional T^{1} unless they are smooth in codimension two.

(3.5.2) Assume that Y is smooth in codimension two.

Proposition: T_{Y}^{1} (−R) ≠ 0 implies that {a^{j} | ⟨a^{j}, R⟩ ≤ 1} is not contained in any proper face of σ.

Proof: Assume that {a^{j} | ⟨a^{j}, R⟩ ≤ 1} ⊆ τ. We choose some vertex ā ∈ Q not coming from τ (in particular, not being contained in the lattice N); from (3.4.5) we know that the same dilatation factor, say t, has to be assigned to all edges containing ā. Now, we consider the remaining compact edges of Q. Those which are not contained in τ can be connected via a path along edges through non-τ-vertices with ā. In particular, they obtain the same dilatation factor t. On the other hand, if we are given a compact edge d contained in τ, then there is at least one two-dimensional compact face ε not contained in τ but containing this given edge d. Then, we already know that the dilatation factors of its edges (except that of d) equal t. However, using the equation for ε in the definition of V (Q) (cf. (3.4.2)), this has to hold for d, too. □

Corollary: There are just finitely many R ≥ 0 such that T_{Y}^{1} (−R) ≠ 0.

Proof: There are just finitely many different choices for selecting elements a^{j} to yield ⟨•, R⟩ = 0 or 1. Fix on of those choices and denote

H^{0} := {a^{j} | ⟨a^{j}, R⟩ = 0}; H^{1} := {a^{j} | ⟨a^{j}, R⟩ = 1} .

We already know that H^{0} ∪ H^{1} may not be contained in any proper face of σ. We distinguish two cases:

(a) span_{IR}(H^{0} ∪ H^{1}) = N_{IR}: Then, R (if there is any) is uniquely determined by H^{0} and H^{1}.

(b) span_{IR}(H^{0} ∪ H^{1}) =: H ⊆ N_{IR}: Then, H contains interior points of σ - let b be one of them. Since the value of R on b is determined by H^{0} and H^{1}, we can show that the value of R on the fundamental generators a^{j} is bounded: On the one hand, we have ⟨a^{j}, R⟩ ≥ 0. On the other hand, to each a^{j} we can associate the line through a^{j} and b. Moreover, we can choose some point (a^{j})′ ∈ σ sitting at this line - but at the other side of b as a^{j} does. Then, the condition ⟨(a^{j})′, R⟩ ≥ 0 gives an upper bound for ⟨a^{j}, R⟩. □

Remark: Degrees R ∈ M such that R ≥ 0 are, at least in some sense, easier to handle. In section (4.4) it is exactly that class making a nicer description of the cup product possible. Moreover, only for R ∈ σ^{∨} ∩ M the cone σ can be reconstructed from the polyhedron Q (via σ = \overline{IR · Q}). The latter is the main reason for the construction of the versal deformation working well in those degrees (cf. chapter 6). On the other hand, in dimension three, the modules T_{Y}^{2} (−k R) (k ≥ 1) vanish unless R is contained in int σ^{∨} (cf. the upcoming section (3.5.4)). Hence, deformations of those degrees are unobstructed.

(3.5.3) Proposition: We still assume that Y is smooth in codimension two. Let R ∈ int σ^{∨} ∩ M. If every 2-face of Q := σ ∩ [R = 1] contains at most three vertices belonging to the lattice, then T_{Y}^{1} (−R) = 0.

Proof: Since R is an interior point of σ^{∨}, the polyhedron Q is compact. Now, for every 2-face ε < Q, the assumption of containing at most three lattice points as vertices means that there are in fact only three different dilatation factors left in ε. On the other hand, triangles having mutually equal angles are homothetical. It is that simple fact (encoded in the ε-equation of V (Q)) telling that even those three dilatation factors have to be equal. □

Corollary: Let Y be smooth in codimension two. If the three-dimensional faces of σ are simplicial cones (i.e. built from three fundamental generators), then T_{Y}^{1} (−R) = 0 for each R ∈ int σ^{∨}∩M.

However, it is not always true that those cones (even if smooth in codimension three) provide rigid singularities:

Example: Let σ = ⟨a, b, x, y, z⟩ be the (four-dimensional) cone over some double tetrahedron.

[figure: a; b; z; x; y; \begin{aligned}a & = & (1, 0, 0, 0) \\ b & = & (0, 1, 0, 0) \\ y & = & (0, 0, 1, 0) \\ x & = & (1, 1, 1, −1) \\ z & = & (1, 1, 1, 1) \end{aligned}]

(The two partial cones generated by a, b, y, x and a, b, y, z, respectively, are smooth. Hence, so are the facets of σ.) Let R = [0, 0, 1, 0]. Then, the compact part of Q just consists of two edges (containing the latticevertices x̄, ȳ, z̄).

[figure: +; Q^{∞}; 0; b; a; x̄; z̄; ȳ; Q =]

In particular, dim T_{Y}^{1} (−R) = 1. (Moreover, for me, it seems to be that all examples for non-rigid, four-dimensional, isolated toric singularities are more or less of that type.)

(3.5.4) Three-dimensional toric varieties with isolated singularities: Let σ = ⟨a^{1},…, a^{M}⟩ be a three-dimensional cone with smooth facets, i.e. every pair {a^{j}, a^{j+1}} (j = 1,…, M; M + 1 := 1) can be completed to some ZZ-basis of the lattice N. As usual, fixing some degree R ∈ M provides the polygon Q := σ ∩ [R = 1] with its cone of infinite directions Q^{∞}. Now, we construct a second polygon Q̄ by cutting all compact corners containing non-lattice vertices:

Q̄ := conv {ā^{j} ∈ Q | ⟨a^{j}, R⟩ = 1, or ā^{j} is contained in some non-compact edge} + Q^{∞} ⊆ Q .

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

Two examples for Q̄ ⊆ Q

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

Proposition: Infinitesimal deformations of Y in degree −R correspond to (complexified) Minkowski summands of Q̄ (modulo homothety). Actually,

dim T_{Y}^{1} (−R) = \{ \begin{matrix}#\{ā^{j} | ā^{j} ∈ N, i.e. ⟨a^{j}, R⟩ = 1\} − 3 & if R > 0 \\ #\{ā^{j} | ā^{j} ∈ N, not contained in a non-compact edge\} & if R ≯ 0 . \end{matrix}

For the obstructions in degree −R, we obtain

T_{Y}^{2} (−R)^{∗} = ∩_{j=1}^{M} (span_{I C}E_{j}^{R})/ ; span_{I C}(∩_{j=1}^{M} E_{j}^{R}) ,

which is a subquotient of M_{I C}. In particular, T_{Y}^{2} (−R) is at most three-dimensional and vanishes unless R > 0.

Proof: The first part is a direct consequence of Theorem (3.4.5). For the T^{2}-formula we could either look at (3.4.6), or we take Theorem (3.4.1) and use the obvious fact

ker [⊕_{j}span E_{j j+1}^{R} → ⊕_{j}span E_{j}^{R}] = ∩_{j=1}^{M} span E_{j}^{R} . □

Remark:

(1) Assume that those edges of Q̄ containing interior lattice points admit lattice points as vertices, too. Then, the cones σ = \overline{IR_{≥0} · Q} and σ̄ := \overline{IR_{≥0} · Q̄} define affine toric varieties with the same infinitesimal deformations in degree −R.

(2) It is good to know obstructions in proper multiples of some fixed degree. For R ∈ M and k ≥ 2 we obtain

T_{Y}^{2} (−kR)^{∗} = \{ \begin{matrix}M^{I C} /span_{I C}(∩_{j=1}^{M} E_{j}^{kR}) & if R > 0 \\ 0 & if R ≯ 0 . \end{matrix}

Moreover, we know that R ∈ span_{I C}(∩_{j=1}^{M} E_{j}^{kR}) i.e. T_{Y}^{2} (−kR) can be at most two-dimensional.

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

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

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

• α_{1}a^{1} + α_{3}a^{3} = α_{2}a^{2} + α_{4}a^{4}

• There exists an R ∈ M such that ⟨a^{j}, R⟩ = g_{j}, if and only if α_{1}g_{1} + α_{3}g_{3} = α_{2}g_{2} + α_{4}g_{4}. (The fundamental generators a^{j} ∈ ZZ^{3} are in the most general position that is possible when (1) holds.) We denote those R by R ∼ [g_{1}, g_{2}, g_{3}, g_{4}].

• Y has an isolated singularity (i.e. is smooth in codimension two), if and only if gcd(α_{j}, α_{j+1}) = 1 (j = 1,…, 4, α_{5} := α_{1}).

For the behavior of T_{Y}^{1} , there are three mutually excluding cases:

(i) α_{1} ∈ α_{2} + α_{4} + INα_{2} + INα_{4} + INα_{3} or α_{3} ∈ α_{2} + α_{4} + INα_{2} + INα_{4} + INα_{1} (implying α_{1} ≥ α_{2} + α_{4} or α_{3} ≥ α_{2} + α_{4}, respectively): Then, for every R ∈ M (and there is at least one) shaped as R ∼ [1, ≥ 1, ≤ 0, ≥ 1] or R ∼ [≤ 0, ≥ 1, 1, ≥ 1], we have dim T_{Y}^{1} (−R) = 1.

(ii) Analogously, there is the case α_{2} ∈ α_{1} + α_{3} + INα_{1} + INα_{3} + INα_{4} or α_{4} ∈ α_{1} + α_{3} + INα_{1} + INα_{3} + INα_{2}.

(iii) Finally, we assume α_{1} + α_{3} = α_{2} + α_{4}. This condition is equivalent to Y being Gorenstein (cf. (5.1.1)) and implies that the whole space T_{Y}^{1} will be one-dimensional (concentrated in degree R ∼ [1, 1, 1, 1]).

Y will be rigid, if the numbers α_{1},…, α_{4} do not satisfy any of the previous three conditions.

(3.5.5) Cones over smooth, projective, toric surfaces: Let P ⊆ IR^{2} be a lattice polygon defining some projective, toric surface X_{P} (cf. (2.4.4)). That surface will be smooth, if and only if the vectors along adjacent edges form ZZ-bases of ZZ^{2}. (Equivalently: The lattice parallelograms at the corners of P have to have volume one each.)

Polygon P with parallelograms of volume one drawn at some corners

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

We use the notation of (2.4.4): σ ⊂ N_{IR} := IR^{3} is the dual of the cone over P (embedded in an affine plane of height one in IR^{3}); Y = Y_{σ} is the cone (in the sense of algebraic geometry) over X_{P}. Since Y is a cone, its affine coordinate ring as well as modules such as T_{Y}^{1} admit a natural ZZgrading. It is induced from our more frequently used M-(multi)grading via ⟨a^{∗}, •⟩.

Proposition: Except P is (as a lattice polytope) isomorphic to the unit square (yielding Y = [xy − zw = 0] ⊆ I C^{3}), we have

(T_{Y}^{1} )_{−} := ⊕_{k≥1} T_{Y}^{1} (−k) = T_{Y}^{1} (−1) = ⊕_{R∈P} T_{Y}^{1} (−R) .

Proof: Step 1: In the direct sum (T_{Y}^{1} )_{−} = ⊕_{⟨a^{∗},R⟩≥1} T_{Y}^{1} (−R) there occur only elements R ∈ σ^{∨} ∩ M: Assume that T_{Y}^{1} (−R) ≠ 0 and ⟨a^{∗}, R⟩ ≥ 1. The first condition implies that there is an a^{j} ∈ σ meeting

⟨a^{j}, R⟩ = 1 , ⟨a^{j−1}, R⟩, ⟨a^{j+1}, R⟩ ≥ 1 ;

in particular, R is non-negative on the cone σ′ := ⟨a^{j−1}, a^{j}, a^{j+1}, a^{∗} − a^{j}⟩ ⊆ N_{IR}. Hence, it remains to show that σ ⊆ σ′:

Let a ∈ σ ∩ N but a ∉ σ′. Since a cannot be contained in the σ-edges emerging from a^{j}, the sum a + a^{j} has to be even an interior point of σ, i.e. a − (a^{∗} − a^{j}) = (a + a^{j}) − a^{∗} ∈ σ ∩ N. Continuing this process, there are just two different possibilities: Case 1: a−k(a^{∗} −a^{j}) ∈ σ∩N for every k ∈ IN - but this would imply that a^{j} −a^{∗} = −(a^{∗} −a^{j}) ∈ σ (and a^{j} would not longer be a fundamental generator of σ). Case 2: There is a k ∈ IN such that a−k(a^{∗}−a^{j}) ∈ σ′. Hence, a = [a−k(a^{∗}−a^{j})]+k(a^{∗}−a^{j}) ∈ σ′.

Step 2: ⟨a^{∗}, R⟩ ≥ 2 (and R ∈ σ^{∨} ∩ M) implies T_{Y}^{1} (−R) = 0: Assuming that ⟨a^{∗}, R⟩ ≥ 2, the point R̄ := R/⟨a^{∗}, R⟩ is contained in the polytope P, and each equation ⟨a^{j}, R⟩ = 1 implies that ⟨a^{j}, R̄⟩ = 1/⟨a^{∗}, R⟩ ≤ 1/2. The strips S_{j} := {R̄ | ⟨a^{∗}, R̄⟩ = 1 ; 0 ≤ ⟨a^{j}, R̄⟩ ≤ 1/2} along the edges (a^{j})^{⊥} of P intersect the adjacent edges in the first half on the way to the next lattice point.

[figure: a^{j}; a^{j+1}; a^{j−1}; Polygon P; S_{j}]

In particular, neither there could be ⟨a^{j}, R⟩ = 1 (i.e. R̄ ∈ S_{j}) and R̄ ∈ (a^{k})^{⊥} at the same time for non-adjacent a^{j}, a^{k}, nor (up to the exception made in the proposition) R̄ could be contained in four different strips S_{•}. Hence, T_{Y}^{1} (−R) = 0. □

Which of the cones over smooth, projective, toric surfaces admit non-trivial, negative deformations? We fix some lattice point R ∈ P and ask for conditions for T_{Y}^{1} (−R) ≠ 0:

(i) Assume that R ∈ ∂P: Then, there has to be some a^{j} such that ⟨a^{j}, R⟩ = 1 and ⟨a^{j−1}, R⟩, ⟨a^{j+1}, R⟩ ≥ 1. Hence, R is as well a boundary point of P as an interior point of the (in the picture dashed) line connecting the two lattice points on ∂P next to the edge (a^{j})^{⊥}.

[figure: a^{j}; a^{j+1}; a^{j−1}; Polygon P]

In particular, this dashed line has to be an edge of P by itself, i.e. P is contained in a strip of (lattice-) thickness one. P is a quadrangle with (after choosing suitable coordinates) vertices [0, 0], [a, 0], [b, 1], [0, 1] (a, b ∈ ZZ; 1 ≤ a ≤ b). It does not contain any interior lattice points, but each lattice point R (except the four vertices) on ∂P provides a one-dimensional T_{Y}^{1} (−R). Y is a scroll given by the cone σ = ⟨(1, 0, 0), (0, 1, 0), (−1, b − a, a), (0, −1, 1)⟩ (i.e. α_{1} = α_{3} = 1, α_{2} = b, α_{4} = a in the language of Example (3.5.4)) or by the equations

rank ( \begin{matrix}y_{0} & y_{1} & … & y_{a−1} & z_{0} & … & z_{b−1} \\ y_{1} & y_{2} & … & y_{a} & z_{1} & … & z_{b} \end{matrix} ) ≤ 1 .

Except for a = b = 1 (yielding [xy − zw = 0] ⊆ I C^{3} again), we obtain dim(T_{Y}^{1} )_{−} = a + b − 2. Moreover, it is possible to write down the negative part of the versal deformation (with parameters s_{v}, t_{μ}) directly:

rank ( \begin{matrix}y_{0} & y_{1} + s_{1} & … & y_{a−1} + s_{a−1} & z_{0} & z_{1} + t_{1} & … & z_{b−1} + t_{b−1} \\ y_{1} & y_{2} & … & y_{a} & z_{1} & z_{2} & … & z_{b} \end{matrix} ) ≤ 1 .

(ii) Assume that R is an interior lattice point of P: In a similar way as in (i), we obtain first that P has to be contained in strip of (lattice-) thickness two. Actually, for P and R ∈ int P there are the following possibilities:

(ii.1) We have four special polygons containing exactly one interior lattice point R^{∗} each. In either case, Y is Gorenstein and T_{Y}^{1} = (T_{Y}^{1} )_{−} is concentrated in degree −R^{∗} (of dimension 1, 1, 2, and 3, respectively).

[figure: Polygon P′_{4}; Polygon P_{4}; Polygon P_{5}; Polygon P_{6}]

The versal base space of toric Gorenstein singularities will be constructed in chapter 5. P′_{4} corresponds to (IP^{1}×IP^{1}, O(2, 2)), and the polygons P_{4}, P_{5}, and P_{6} induce Del Pezzo surfaces of degree 8, 7, and 6, respectively; for a detailed discussion of the latter three examples we refer to (5.7.1) (also showing the duals Q_{i} of P_{i}).

(ii.2) Moreover, there are also two series of polygons: pentagons with vertices [1, 2], [0, 1], [0, 0], [a, 0], and [a + 2b, 2] and hexagons with the two vertices [a + b, 1] and [a + 2b − 1, 2] instead of [a + 2b, 2] (with a, b ∈ ZZ; a, b ≥ 1).

[figure: [0, 0]; [0, 1]; [1, 2]; [a + 2b − 1, 2]; [a + 2b, 2]; [a + b, 1]; [a, 0]]

Except for a = b = 1, the vector space (T_{Y}^{1} )_{−} is one-dimensional in the pentagon case (concentrated in degree −[1, 1]) and two-dimensional in the hexagon case (concentrated in the degrees −[1, 1] and −[a + b − 1, 1]). For a = b = 1, we obtain our polygons P_{5} and P_{6} from (ii.1) again.

The cup product

4.1 The cup product in general

(4.1.1) Using the notation of section (3.1), it is well known (cf. [Ld], (5.1.5)) that the cup product T^{1} × T^{1} → T^{2} can be defined in the following way:

(i) Starting with an φ ∈ Hom_{A}(I/I^{2}, A), we lift the images of the f_{i} obtaining elements φ̃(f_{i}) ∈ P.

(ii) Given a relation r ∈ R, the linear combination ∑_{i} r_{i} φ̃(f_{i}) vanishes in A, i.e. it is contained in the ideal I ⊆ P. Denote by λ(φ) ∈ P^{m} any set of coefficients such that

∑_{i} r_{i} φ̃(f_{i}) + ∑_{i} λ_{i}(φ) f_{i} = 0 in P.

(Of course, λ depends on r also.)

(iii) If φ, ψ ∈ Hom_{A}(I/I^{2}, A) represent two elements of T_{Y}^{1} , then we define for each relation r ∈ R

(φ ∪ ψ)(r) := ∑_{i} λ_{i}(φ) ψ(f_{i}) + ∑_{i} φ(f_{i}) λ_{i}(ψ) ∈ A .

Remark: The definition of the cup product does not depend on the choices we made:

(a) Choosing a λ′(φ) instead of λ(φ) yields λ′(φ)−λ(φ) ∈ R, i.e. in A we obtain the same result.

(b) Let φ̃′(f_{i}) be different liftings to P. Then, the difference φ̃′(f_{i}) − φ̃(f_{i}) is contained in I, i.e. it can be written as some linear combination

φ̃′(f_{i}) − φ̃(f_{i}) = ∑_{j} t_{ij} f_{j} .

Hence,

∑_{i} r_{i} φ̃′(f_{i}) = ∑_{i} r_{i} φ̃(f_{i}) + ∑_{i,j} t_{ij} r_{i} f_{j} ,

and we can define λ′_{j}(φ) := λ_{j}(φ) − ∑_{i} t_{ij} r_{i} (corresponding to φ̃′ instead of φ̃). Then, we obtain for the cup product

(φ ∪ ψ)′(r) − (φ ∪ ψ)(r) = − ∑_{i} r_{i} · (∑_{j} t_{ij} ψ(f_{j})) ,

but this expression comes from some map P^{m} → A.

(4.1.2) The associated to the cup product quadratic form T^{1} → T^{2} can also be obtained in the following way: Considering the exact sequence

0 → I C ⟶^{·ε^{2}} I C[ε]/ε^{3} ⟶ I C[ε]/ε^{2} → 0 ,

we may ask for lifting infinitesimal first-order deformations ξ (elements of T^{1}) to the second order. Hence, using the construction of (3.1.4) (W := I C), each ξ ∈ T^{1} is assigned the obstruction λ(ξ) ∈ T^{2} governing this construction. It is not difficult to see that λ(ξ) = ξ ∪ ξ.

As a corollary, we obtain that the associated to the cup product quadratic form λ : T^{1} → T^{2} describes the equations of the versal base space up to the second order.

4.2 The cup product for toric varieties

(4.2.1) Turning to the particular case of toric varieties (use the notation of (3.2)), the cup product T_{Y}^{1} × T_{Y}^{1} → T_{Y}^{2} respects the M-grading. It splits into pieces

T_{Y}^{1} (−R) × T_{Y}^{1} (−S) ⟶ T_{Y}^{2} (−R − S) (R, S ∈ M) .

To describe these maps in our combinatorial language, we choose some set-theoretical section Φ : M → ZZ^{w+1} of the ZZ-linear map

π : ZZ^{w+1} ⟶ M
a ↦ ∑_{v} a_{v} r^{v}

with the additional property Φ(σ^{∨} ∩ M) ⊆ IN^{w+1}.

Let q ∈ L(E) ⊆ ZZ^{w+1} be an integral relation between the generators of the semigroup σ^{∨} ∩ M. We introduce the following notations:

• q^{+}, q^{−} ∈ IN^{w+1} denote the positive and the negative part of q, respectively. (With other words: q = q^{+} − q^{−} and ∑_{v} q_{v}^{−} q_{v}^{+} = 0.)

• q̄:= π(q^{+}) = ∑_{v} q_{v}^{+} r^{v} = ∑_{v} q_{v}^{−} r^{v} = π(q^{−}) ∈ M.

• If φ, ψ : L(E) → ZZ are linear maps and R, S ∈ M, then we define

t_{φ,ψ,R,S}(q) := φ(q) · ψ (Φ(q̄ − R) + Φ(R) − q^{−}) + ψ(q) · φ (Φ(q̄ − S) + Φ(S) − q^{+}) ,

or, more generally, for (α, β) ∈ m

t_{φ,ψ,R,S}(α, β) := \begin{matrix}φ(α − β) · ψ (Φ(π(α) − R) + Φ(R) − β) + \\ +ψ(α − β) · φ (Φ(π(α) − S) + Φ(S) − α) . \end{matrix}

(Both expressions are related via t(q) = t(q^{+}, q^{−}).)

Assume that φ and ψ vanish on ∑_{j} L(E_{j}^{R}) and ∑_{j} L(E_{j}^{S}), respectively. In particular, they represent elements from T_{Y}^{1} (−R) and T_{Y}^{1} (−S) via Theorem (3.2.4).

Lemma: Let α, β, γ ∈ IN^{E} with π(α) = π(β) = π(γ).

(1) t(α, β) = −t(β, α); in particular t(−q) = −t(q) for q ∈ L(E).

(2) t(α, β) = t(α − β) as long as π(α) ∈ ⋃_{j} K_{j}^{R+S}. (Do not mix up that property with t(q) = t(q^{+}, q^{−}) being valid without further assumptions.)

(3) t(β, γ) − t(α, γ) + t(α, β) = 0.

Proof: (1) t(α, β) + t(β, α) = φ(α − β) ψ(α − β) + ψ(α − β) φ(β − α) = 0.

(2) It is enough to show that t(α + r, β + r) = t(α, β) for r ∈ IN^{E}, π(α + r) ∈ ⋃_{j} K_{j}^{R+S}. The difference of these two terms equals

\begin{matrix}t(α + r, β + r) − t(α, β) = φ(α − β) · ψ[Φ(π(α + r) − R) − Φ(π(α) − R) − r]+ \\ +ψ(α − β) · φ[Φ(π(α + r) − S) − Φ(π(α) − S) − r] , \end{matrix}

and both summands vanish: For instance, on the one hand, π(α) ∈ ⋃_{j} K_{j}^{R} would cause φ(α − β) = 0, and, on the other hand, π(α) − R ≥ 0 and π(r + Φ(π(α) − R)) ∈ ⋃_{j} K_{j}^{S} imply ψ[Φ(π(α + r) − R) − Φ(π(α) − R) − r] = 0.

(3) By extending φ and ψ to linear maps I C^{E} → I C, we obtain

t(α, β) = \begin{matrix}[φ(α − β) ψ (Φ(π(α) − R) + Φ(R)) + ψ(α − β) φ (Φ(π(α) − S) + Φ(S))]+ \\ +[φ(β) ψ(β) − φ(α) ψ(α)]. \end{matrix}

Now, since π(α) = π(β) = π(γ), both types of summands add up to 0 separately in the expression t(β, γ) − t(α, γ) + t(α, β). □

Remark: The previous lemma does not imply that t(q) is ZZ-linear in q. The assumption for π(α) made in (2) is really essential.

(4.2.2) Theorem: Assume that Y is smooth in codimension 2. Let R, S ∈ M, and let φ, ψ : L(E)_{I C} → I C be linear maps vanishing on ∑_{j} L(E_{j}^{R})_{I C} and ∑_{j} L(E_{j}^{S})_{I C}, respectively. In particular, they define elements φ ∈ T_{Y}^{1} (−R), ψ ∈ T_{Y}^{1} (−S) (which involves a slight abuse of notations).

(1) Each integral relation q ∈ L(E_{j}^{R+S}) admits a decomposition q = ∑_{l} q^{l} such that q^{l} ∈ L(E_{j}^{R+S}) and, moreover, ⟨a^{j}, q̄^{l}⟩ < ⟨a^{j}, R + S⟩ (i.e. q̄^{l} ∈ K_{j}^{R+S}).

(2) ∑_{l} q^{l} = 0 (with q̄^{l} ∈ K_{j}^{R+S}) implies ∑_{l} t(q^{l}) = 0. In particular, the decomposition of (1) induces a well defined element t_{φ,ψ,R,S}^{j} ∈ L(E_{j}^{R+S})^{∗} via t^{j}(q) := ∑_{l} t(q^{l}).

(3) For adjacent a^{j}, a^{k} the relations q ∈ L(E_{j}^{R+S} ∩ E_{k}^{R+S}) admit decompositions q = ∑_{l} q^{l} that work for both j and k. In particular, t^{j}(q) = t^{k}(q).

(4) According to Theorem (3.2.4), the family (t_{φ,ψ,R,S}^{j}) ∈ ⊕_{j}L(E_{j}^{R+S})^{∗} defines an element of T_{Y}^{2} (−R − S). It equals the cup product φ ∪ ψ.

The proof of the theorem is contained in (4.3).

Remark: Replacing all the terms Φ(•) in the t’s of the previous formula for φ ∪ ψ by arbitrary liftings from M to ZZ^{w+1}, the result in T_{Y}^{2} (−R − S) will be unchanged as long as we obey the following two rules:

(i) Use always (for all q, q^{l}, and j) the same liftings of R and S to ZZ^{w+1}.

(ii) Elements of σ^{∨} ∩ M always have to be lifted to IN^{w+1}.

Proof: Replacing Φ(R) by Φ(R) + d (d ∈ L(E)) at each occurrence changes all maps (φ ∪ ψ)_{j} = t_{φ,ψ,R,S}^{j} by the summand ψ(d)·φ(•). However, this additional linear map comes from L(E)^{∗}, hence it is trivial on Ker (⊕_{j}L(E_{j}^{R+S}) → L(E_{0}^{R+S}) ⊆ L(E)).

Let us look at the terms Φ(q̄− R) in t(q) now: Unless q̄ ≥ R, the factor φ(q) vanishes (cf. Remark (3.2.6)). On the other hand, the expression t(q) is never used for those relations q satisfying q̄ ≥ R + S (cf. conditions for the q^{l}’s). Hence, we may assume that

(q̄ − R) ≥ 0 and, moreover, (q̄ − R) ∈ ⋃_{j} K_{j}^{S} .

Now, each two liftings of (q̄ − R) to IN^{w+1} differ by an element of Ker ψ only (apply the method of Remark (3.2.6) again), in particular, they cause the same result for t(q). □

4.3 Proof of the cup product formula

(4.3.1) Let us fix an element j ∈ {1,…, M}. Since σ^{∨} ∩ M contains elements r with ⟨a^{j}, r⟩ = 1, some of them must be contained in the generating set E, too. We choose one of these elements and call it r(j). Now, to each r ∈ E we associate some relation p^{j}(r) ∈ L(E) by

p^{j}(r) := e^{r} − ⟨a^{j}, r⟩ · e^{r(j)} + [suitable element of ZZ^{E∩(a^{j})^{⊥}}] .

The two essential properties of these special relations are

(i) ⟨a^{j}, \overline{p^{j}(r)}⟩ = ⟨a^{j}, r⟩ (see (4.2.1) for the definition of q̄ for a relation q), and

(ii) if q ∈ L(E) is any relation, then q and ∑_{r∈E} q_{r} · p^{j}(r) differ by some element of L(E ∩ (a^{j})^{⊥}) only.

In particular, this proves (1). For (3), let a^{j}, a^{k} be two adjacent edges of σ. We adapt the construction of the elementary relations p^{j}(r); instead of the r(j)’s, we will use elements r(j, k) ∈ E characterized by the property

⟨a^{j}, r(j, k)⟩ = 1 , ⟨a^{k}, r(j, k)⟩ = 0 .

(Those elements exist, since Y is assumed to be smooth in codimension 2.) Now, we define

p^{jk}(r) := e^{r} − ⟨a^{j}, r⟩ · e^{r(j,k)} − ⟨a^{k}, r⟩ · e^{r(k,j)} + [suitable element of ZZ^{E∩(a^{j})^{⊥}∩(a^{k})^{⊥}}] .

These special p(r)’s meet the usual properties (i) and (ii) - but for the two different indices j and k at the same time. In particular, if q ∈ L(E) is any relation, then q and ∑_{r∈E} q_{r} · p^{jk}(r) differ by some element of L(E ∩ (a^{j})^{⊥} ∩ (a^{k})^{⊥}) only.

(4.3.2) Claim: Let q^{l} ∈ L(E) be relations such that ∑_{l}⟨a^{j}, q̄^{l}⟩ < ⟨a^{j}, R + S⟩. Then, ∑_{l} t(q^{l}) = t(∑_{l} q^{l}).

(Proof: We can restrict ourselves to the case of two summands, q^{1} and q^{2}. Then, by Lemma (4.2.1),

t(q^{1}) + t(q^{2}) = t ((q^{1})^{+}, (q^{1})^{−}) + t ((q^{2})^{+}, (q^{2})^{−})
= t ((q^{1})^{+} + (q^{2})^{+}, (q^{1})^{−} + (q^{2})^{+}) + t ((q^{2})^{+} + (q^{1})^{−}, (q^{2})^{−} + (q^{1})^{−})
= t ((q^{1})^{+} + (q^{2})^{+}, (q^{2})^{−} + (q^{1})^{−})
= t(q^{1} + q^{2}) .  □)

In particular, if ∑_{l} q^{l} = 0 (with q̄^{l} ∈ K_{j}^{R+S}), then this applies for the special decompositions

q^{l} = ∑_{r} q_{r}^{l} · p^{j}(r) + q^{0,l} (q^{0,l} ∈ L(E ∩ (a^{j})^{⊥}))

of the summands q^{l} themselves. We obtain

∑_{q_{r}^{l}>0} q_{r}^{l} · t (p^{j}(r)) + t(q^{0,l}) = t ( ∑_{q_{r}^{l}>0} q_{r}^{l} p^{j}(r) + q^{0,l}) =: t(q^{1,l})

and

∑_{q_{r}^{l}<0} q_{r}^{l} · t (p^{j}(r)) = t ( ∑_{q_{r}^{l}<0} q_{r}^{l} p^{j}(r)) =: t(q^{2,l}) .

Up to elements of E ∩ (a^{j})^{⊥}, the relations q^{1,l} and q^{2,l} are connected by the common

(q^{1,l})^{−} = −q_{r(j)}^{1,l} · e^{r(j)} = ⟨a^{j}, q̄^{l}⟩ · e^{r(j)} = q_{r(j)}^{2,l} · e^{r(j)} = (q^{2,l})^{+} .

Hence, Lemma (4.2.1) yields

∑_{r} q_{r}^{l} · t (p^{j}(r)) + t(q^{0,l}) = t(q^{1,l}) + t(q^{2,l}) = t (q^{1,l} + q^{2,l}) = t(q^{l}) ,

and we conclude

∑_{l} t(q^{l}) = ∑_{l} (∑_{r} q_{r}^{l} · t (p^{j}(r)) + t(q^{0,l}))

= ∑_{r} (∑_{l} q_{r}^{l}) t (p^{j}(r)) + t (∑_{l} q^{0,l}) (cf. previous claim)

= 0 + t (∑_{l} q^{l} − ∑_{l,r} q_{r}^{l} p^{j}(r))
= 0 .

(4.3.3) It remains to relate the family (t_{φ,ψ,R,S}^{j}) to the cup product φ ∪ ψ, i.e. to show (4). Fix an R ∈ M, and let φ ∈ L(E)_{I C}^{∗} induce some element (also denoted by φ) of T_{Y}^{1} (−R). Using the notations of (4.1.1) we can take

φ̃(f_{αβ}) := φ(α − β) · \underline{z}^{Φ(π(α)−R)}

for the auxiliary P -elements needed to compute the λ(φ)’s (cf. Theorem (3.2.6)).

Now, we have to distinguish between the two several types of relations generating the P -module R ⊆ P^{m} (cf. (3.2.7)):

(r) Regarding the relation r(a, b; c) we obtain

∑_{(α,β)∈m} r(a, b; c)_{αβ} · φ̃(f_{αβ}) = φ̃(f_{a+c,b+c}) − \underline{z}^{c} φ̃(f_{ab})

= φ(a − b) · (\underline{z}^{Φ(π(a+c)−R)} − \underline{z}^{c+Φ(π(a)−R)})
= φ(a − b) · f_{Φ(π(a+c)−R), c+Φ(π(a)−R)} .

In particular,

λ_{αβ}^{r(a,b;c)}(φ) = \{ \begin{matrix}φ(a − b) & for [α, β] = [c + Φ(π(a) − R), Φ(π(a + c) − R)] \\ 0 & otherwise . \end{matrix}

(s) The corresponding result for the relation s(a, b, c) is much nicer:

∑_{(α,β)∈m} s(a, b, c)_{αβ} · φ̃(f_{αβ}) = φ̃(f_{bc}) − φ̃(f_{ac}) + φ̃(f_{ab})
= [φ(b − c) − φ(a − c) + φ(a − b)] · \underline{z}^{Φ(π(a)−R)}
= 0 .

In particular, λ^{s(a,b,c)}(φ) = 0.

Now, let R, S, φ, and ψ as in the assumption of Theorem (4.2.2). Using formula (4.1.1)(iii), our previous computations yield (φ ∪ ψ)(s(a, b, c)) = 0 and

(φ ∪ ψ)(r(a, b; c)) = ∑_{α,β} λ_{αβ}^{r(a,b;c)}(φ) · ψ(f_{αβ}) + ∑_{α,β} φ(f_{αβ}) · λ_{αβ}^{r(a,b;c)}(ψ)

= \begin{matrix}φ(a − b) · ψ (c + Φ(π(a) − R) − Φ(π(a + c) − R)) · x^{π(c+Φ(π(a)−R))−S}+ \\ +ψ(a − b) · φ (c + Φ(π(a) − S) − Φ(π(a + c) − S)) · x^{π(c+Φ(π(a)−S))−R} \end{matrix}

= \begin{matrix}[φ(a − b) · ψ (c + Φ(π(a) − R) − Φ(π(a + c) − R)) + \\ +ψ(a − b) · φ (c + Φ(π(a) − S) − Φ(π(a + c) − S))] · x^{π(a+c)−R−S} . \end{matrix}

Unless π(a + c) ≥ R + S, both summand in the brackets will vanish. (Use the same argument as done in the proof of the second part of Lemma (4.2.1).)

(4.3.4) To apply Theorem (3.2.7) we would like to remove the argument c from the big coefficient of x^{π(a+c)−R−S}. This will be done by adding a suitable coboundary T.

Define the P -linear map T ∈ Hom(P^{m}, A) by

T : e^{αβ} ↦ \{ \begin{matrix}t(α, β) x^{π(α)−R−S} & for π(α) ≥ R + S \\ 0 & otherwise . \end{matrix}

Pulling back T to R ⊆ P^{m} yields (in case of π(a + c) ≥ R + S)

T (r(a, b; c)) = \{ \begin{matrix}[t(a + c, b + c) − t(a, b)] · x^{π(a+c)−R−S} & for π(a) ≥ R + S \\ t(a + c, b + c) · x^{π(a+c)−R−S} & otherwise \end{matrix}
= \{ \begin{matrix}−(φ ∪ ψ)(r(a, b; c)) & for π(a) ≥ R + S \\ t(a, b) x^{π(a+c)−R−S} − (φ ∪ ψ)(r(a, b; c)) & otherwise \end{matrix}

and T (s(a, b, c)) = 0 (by (2) of the previous lemma).

On the other hand, T yields a trivial element of T_{Y}^{2} (−R − S), i.e. inside this group we may replace φ ∪ ψ by (φ ∪ ψ) + T to obtain

(φ ∪ ψ)(r(a, b; c)) = \{ \begin{matrix}t(a, b) · x^{π(a+c)−R−S} & for π(a) ∈ ⋃_{j} K_{j}^{R+S}; π(a + c) ≥ R + S \\ 0 & otherwise , \end{matrix}
(φ ∪ ψ)(s(a, b, c)) = 0 .

Having Theorem (3.2.7) in mind, this formula for φ ∪ ψ is exactly what we were looking for: Given an r(a, b; c) with π(a) ∈ K_{j}^{R+S}, let us compute t_{φ,ψ,R,S}^{j}(q := a − b). We do not need to split q = a − b into a sum q = ∑_{l} q^{l} - the element q itself already satisfies the condition

⟨a^{j}, q̄⟩ ≤ ⟨a^{j}, π(a)⟩ < ⟨a^{j}, R + S⟩.

In particular, t_{φ,ψ,R,S}^{j}(a − b) = t_{φ,ψ,R,S}(a − b) = t_{φ,ψ,R,S}(a, b), and we are done.

4.4 Alternative cup product formula

(4.4.1) We are going to describe the cup product using the description of T_{Y}^{1} and T_{Y}^{2} given in section (3.4). Unfortunately, this has been possible only for degrees R, S ≥ 0.

Lemma:

(1) If S ≥ 0 (i.e. S ∈ σ^{∨} ∩ M), then t(q) = 0 unless q̄ ≥ R.

(2) If R, S ≥ 0 and ⟨a^{j}, R + S⟩ ≥ 1, then t vanishes on L(E ∩ (a^{j})^{⊥}).

Proof: (1) As used many times, the property q̄ ∈ ⋃_{j} E_{j}^{R} implies φ(q) = 0. Now, we can distinguish between two cases: Case 1: q̄ ∈ ⋃_{j} E_{j}^{S}. We obtain ψ(q) = 0, in particular, both summands of t(q) vanish. Case 2: q̄ ≥ S. Then, q̄− S, S ∈ σ^{∨} ∩ M, and Φ lifts these elements to IN^{w+1}. Now, the condition q̄ ∈ ⋃_{j} E_{j}^{R} implies that φ (Φ(q̄ − S) + Φ(S) − q^{+}) = 0.

(2) is a direct consequence of the first part: Since ⟨a^{j}, R + S⟩ ≥ 1, elements q ∈ L(E ∩ (a^{j})^{⊥}) cannot meet both q̄ ≥ R and q̄ ≥ S. □

We will assume R, S ≥ 0 (and Y being smooth in codimension two) for the whole section.

Remark: In the special case of R ≥ S ≥ 0 we can choose liftings Φ(R) ≥ Φ(S) ≥ 0 in IN^{w+1}. In case of q̄ ≥ R we may assume that q^{+} ≥ Φ(R) is true in IN^{w+1} (the general q’s are differences of those ones). Then, t can be computed as the product t(q) = φ(q) ψ(q).

Proof: We can choose Φ(q̄− R) := q^{+} − Φ(R) and Φ(q̄− S) := q^{+} − Φ(S). Then, the claim follows straight forward. □

(4.4.2) Assume that we start with elements of T_{Y}^{1} (−R) and T_{Y}^{1} (−S) that are given (via Theorem (3.4.1)) by linear maps φ^{j} : span E_{j}^{R} → ZZ and ψ^{j} : span E_{j}^{S} → ZZ (j = 1,…, M), respectively. (In particular, φ^{j} = φ^{k} and ψ^{j} = ψ^{k} for two-dimensional faces ⟨a^{j}, a^{k}⟩ < σ.)

We define φ : ZZ^{E} → ZZ as zero on ZZ^{E\E_{0}^{R}} and as φ(e^{r}) := φ^{j}(r) for r ∈ E_{j}^{R}. Following the remark right after Theorem (3.4.1), the restriction of −φ to L(E_{0}^{R}) ⊆ ZZ^{E} equals (via Theorem (3.2.4)) the same element of T_{Y}^{1} (−R) as we have started with. (Of course, the same arguments apply to ψ^{j} and ψ.) By abuse of notation we will sometimes write r for e^{r} ∈ ZZ^{E} (e.g. φ(r) for φ(e^{r})). However, keep always in mind that the arguments of φ^{j} are elements of span E_{j}^{R} ⊆ M, but the arguments of φ come from ZZ^{E}.

We need the notions r(j), r(j, k), p^{j}(r), and p^{jk}(r) introduced in (4.3.1). Moreover, we define

ℓ(j, k) := e^{r(j,k)} − e^{r(j)} + [suitable element of ZZ^{E∩(a^{j})^{⊥}}] ⊆ L(E) .

Remark:

(i) p^{jk}(r) = p^{kj}(r) and

(ii) p^{j}(r) − p^{jk}(r) ≡ ⟨a^{j}, r⟩ · ℓ(j, k) (mod L(E ∩ (a^{j})^{⊥})).

Then, we can describe the cup product φ ∪ ψ ∈ T_{Y}^{2} (−R − S) of the two given infinitesimal deformations as a certain map (φ ∪ ψ) : ⊕_{⟨a^{j},a^{k}⟩<σ}span E_{jk}^{R+S} → ZZ:

Proposition: Let ⟨a^{j}, a^{k}⟩ < σ be a two-dimensional face. Then, the map (φ ∪ ψ)_{jk} can be given as (φ ∪ ψ)_{jk} = t[ℓ(k, j)] · a^{k} − t[ℓ(j, k)] · a^{j} ∈ N. (The data of φ and ψ have been involved in the definition of t.) Moreover, that particular (φ ∪ ψ)_{jk} does not depend from the special choice of elements r(j, k) or L(E ∩ (a^{j})^{⊥})-summands in the definition of ℓ(j, k). On the other hand, changing r(j) means changing (φ ∪ ψ)_{jk} by a coboundary (coming from ⊕_{j}(span E_{j}^{R+S})^{∗}).

Proof: Of course, we will use Theorem (4.2.2). Assume that we are given elements q^{jk} ∈ span E_{jk}^{R+S} (for two-faces ⟨a^{j}, a^{k}⟩ < σ) representing a class q in H^{2} (span(E^{R})_{•}). Then, its image in ⊕_{j}L(E_{j}^{R+S}) can be described as follows (cf. (3.4.1)): Represent q^{jk} as q^{jk} = ∑_{r∈E} q_{r}^{jk} · r with

q_{•}^{jk} ∈ ZZ^{E_{jk}^{R+S}} and take

q_{•}^{j} := ∑_{a^{k}adjacent to a^{j}} sgn ⟨j, k⟩ · q_{•}^{jk} ∈ L(E_{j}^{R+S}) ⊆ ZZ^{E} .

(The sign sgn ⟨j, k⟩ measures the difference between two orientations of ⟨a^{j}, a^{k}⟩: On the one hand, each face of σ is oriented somehow from the very beginning, and, on the other hand, considering a^{j} as the “first” vertex of ⟨a^{j}, a^{k}⟩ yields a second one.)

According to Theorem (4.2.2) we obtain

(φ ∪ ψ)(q) = ∑_{j=1}^{M} t^{j}(q_{•}^{j}) = ∑_{j=1}^{M} [∑_{r∈E} q_{r}^{j} · t[p^{j}(r)] + t[L(E ∩ (a^{j})^{⊥})-element]] .

Since E_{j}^{R+S} = ∅ (i.e. the j-th summand vanishes) unless ⟨a^{j}, R + S⟩ ≥ 1, we may apply Lemma (4.4.1)(2) killing the second t-summands. Hence,

(φ ∪ ψ)(q) = ∑_{j=1}^{M} ∑_{r∈E} ∑_{a^{k}adj. to a^{j}} sgn ⟨j, k⟩ · q_{r}^{jk} · t[p^{j}(r)]
= ∑_{⟨a^{j},a^{k}⟩<σ} ∑_{r∈E} q_{r}^{jk} · (t[p^{k}(r)] − t[p^{j}(r)]) .

On the other hand, if q_{r}^{jk} ≠ 0 (i.e. ⟨a^{j}, r⟩ < ⟨a^{j}, R + S⟩), then by Theorem (4.2.2)(2) and part (ii) of the previous remark we know t[p^{j}(r)] = t[p^{jk}(r)] + ⟨a^{j}, r⟩ · t[ℓ(j, k)]. Doing the same thing with t[p^{k}(r)] and using p^{jk} = p^{kj}, we obtain

t[p^{k}(r)] − t[p^{j}(r)] = ⟨a^{k}, r⟩ · t[ℓ(k, j)] − ⟨a^{j}, r⟩ · t[ℓ(j, k)] .

Finally, we investigate the influence of the choices being made in the definition of ℓ(j, k):

1) ℓ(j, k) has been defined up to relations from L(E ∩(a^{j})^{⊥}) only. However, if ⟨a^{j}, R+S⟩ ≥ 2, then those summands have no influence to t[ℓ(j, k)] (cf. Theorem (4.2.2)(2) and Lemma (4.4.1)(2)). On the other hand, if ⟨a^{j}, R + S⟩ ≤ 1, then E_{jk}^{R+S} ⊆ (a^{j})^{⊥}, i.e. the coefficient of a^{j} in (φ ∪ ψ)_{jk} has no meaning, anyway.

2) Let r′(j, k) ∈ E be another element meeting ⟨a^{j}, r′(j, k)⟩ = 1 and ⟨a^{k}, r′(j, k)⟩ = 0. Then, there is a relation

d(j, k) := e^{r′(j,k)} − e^{r(j,k)} + [suitable element of ZZ^{E∩(a^{j},a^{k})^{⊥}}] ⊆ L(E ∩ (a^{k})^{⊥})

providing ℓ′(j, k) = ℓ(j, k) + d(j, k). In particular, at least in the case ⟨a^{j}, R + S⟩ ≥ 2, we have t[ℓ′(j, k)] = t[ℓ(j, k)] + t[d(j, k)] and t[d(j, k)] = 0.

3) Similarly, changing r(j) into r′(j) yields a relation d(j) = e^{r′(j)} − e^{r(j)} + … ∈ L(E), and we obtain t[ℓ′(j, k)] = t[ℓ(j, k)] − t[d(j)]. □

(4.4.3) We are going to replace the rather strange elements t[ℓ(j, k)]. In case of ⟨a^{j}, R⟩ = ⟨a^{k}, R⟩ = ⟨a^{j}, S⟩ = ⟨a^{k}, S⟩ = 1, we define (writing R, S ∈ IN^{E} for Φ(R) and Φ(S), respectively)

φ^{jk} := φ(S) − φ^{j}[r(k, j)] − φ^{k}[r(j, k)] + φ^{j or k}[r(j, k) + r(k, j) − S]
= φ[S − r(k, j)] − φ^{k}[S − r(k, j)]
= φ[S − r(j, k)] − φ^{j}[S − r(j, k)]

and, similarly,

ψ^{jk} := ψ(R) − ψ^{j}[r(k, j)] − ψ^{k}[r(j, k)] + ψ^{j or k}[r(j, k) + r(k, j) − R] .

(The latter two descriptions of φ^{jk} show that this number (and analogously ψ^{jk}) does not depend on the particular choices of r(j, k) and r(k, j).)

Proposition: The elements (φ ∪ ψ)_{jk} ∈ (span E_{jk}^{R+S})^{∗} vanish unless either

(1) ⟨a^{j}, R⟩ = ⟨a^{k}, R⟩ = ⟨a^{j}, S⟩ = ⟨a^{k}, S⟩ = 1 (yielding (φ ∪ ψ)_{jk} = φ^{jk} · ψ^{jk} · (a^{j} − a^{k})) or

(2) R = S ∈ E and

(i) ⟨a^{j}, R⟩ = 1, ⟨a^{k}, R⟩ ≥ 2 (yielding (φ ∪ ψ)_{jk} = φ^{k}(R) · ψ^{k}(R) · a^{j}) or

(ii) ⟨a^{j}, R⟩ ≥ 2, ⟨a^{k}, R⟩ = 1 (yielding (φ ∪ ψ)_{jk} = −φ^{j}(R) · ψ^{j}(R) · a^{k}).

Remark: If R = S ∉ E (in particular, R has to be decomposable in σ^{∨} ∩ M) and ⟨a^{j}, R⟩ = 1, ⟨a^{k}, R⟩ ≥ 2, then the previous proposition implies that (φ ∪ ψ)_{jk} = 0. On the other hand, all calculations remain valid after enlarging E (actually, we have never used minimality of E). In particular, we may turn R into an “artificial” element of E yielding a possibly different value for (φ ∪ ψ)_{jk}. Looking like a contradiction, this fact can be explained as follows: First, one has to be careful anyway, since with E also the functions φ and ψ will change. On the other hand, there is a coboundary “connecting” the results of those two different approaches.

Proof (of the proposition): Step 1: t[ℓ(j, k)] = 0 unless ⟨a^{j}, R⟩ = ⟨a^{j}, S⟩ = 1: Using Lemma (4.4.1)(1), we know that t[ℓ(j, k)] ≠ 0 implies \overline{ℓ(j, k)} ≥ R, S. In particular, ⟨a^{j}, R⟩, ⟨a^{j}, S⟩ ≤ 1, and we have already mentioned that in case of ⟨a^{j}, R + S⟩ ≤ 1 (i.e. E_{jk}^{R+S} ⊆ (a^{j})^{⊥}) the coefficient t[ℓ(j, k)] of a^{j} does not matter.

Step 2: Assume ⟨a^{j}, R⟩ = ⟨a^{j}, S⟩ = 1. Then (using a special choice for r(j)), we obtain

t[ℓ(j, k)] = (φ[S − r(j, k)] − φ^{j}[S − r(j, k)]) · (ψ[r(j, k) − R] − ψ^{j}[r(j, k) − R]) .

(In particular, we are done in the case of ⟨a^{j}, R⟩ = ⟨a^{k}, R⟩ = ⟨a^{j}, S⟩ = ⟨a^{k}, S⟩ = 1.) We may choose r(j) := R, even on the IN^{E}-level (r(j) := S would yield the same result). Then, we apply the general definition of t in (4.2.1) to the relation ℓ(j, k) = e^{r(j,k)} − e^{R} + s^{+}(j, k) − s^{−}(j, k) (with s^{+}, s^{−} ∈ IN^{E∩(a^{j})^{⊥}}). The first summand in t vanishes, since the argument of ψ equals Φ([R + s^{−}] − R) + Φ(R) − e^{R} − s^{−} = 0 (Φ(R) and e^{R} are just synonyms, and for Φ(s^{−}) we may choose s^{−} itself by Remark (4.2.2)). For the second t-summand, we consider

\begin{aligned}ψ(r(j, k) − R + s^{+}(j, k) − s^{−}(j, k)) & = & ψ(r(j, k) − R) + ψ^{j}(s^{+}(j, k) − s^{−}(j, k)) \\ & = & ψ(r(j, k) − R) − ψ^{j}(r(j, k) − R) \end{aligned}

and

\begin{aligned}φ(Φ[ℓ̄− S] + S − [r(j, k) + s^{+}(j, k)]) & = & φ(S − r(j, k)) + φ^{j}(Φ[ℓ̄− S] − s^{+}(j, k)) \\ & = & φ(S − r(j, k)) − φ^{j}(S − r(j, k)) . \end{aligned}

Step 3: t[ℓ(j, k)] = 0 unless ⟨a^{k}, R⟩ = ⟨a^{k}, S⟩: Assume ⟨a^{k}, R⟩ > ⟨a^{k}, S⟩. We will use that S − r(j, k) − ⟨a^{k}, S⟩ · r(k, j) ∈ (a^{j}, a^{k})^{⊥} and call this element s(j, k). Hence,

φ(S − r(j, k)) = φ^{k}(S − r(j, k)) (since ⟨a^{k}, R⟩ > ⟨a^{k}, S⟩)
= φ^{k}(⟨a^{k}, S⟩ · r(k, j) + s(j, k))
= φ^{j}(⟨a^{k}, S⟩ · r(k, j) + s(j, k)) (\begin{matrix}if ⟨a^{k}, R⟩ ≥ 2, then φ^{j} = φ^{k} on (a^{j})^{⊥} ⊆ E_{jk}^{R}; \\ if ⟨a^{k}, R⟩ = 1, then ⟨a^{k}, S⟩ = 0) \end{matrix}
= φ^{j}(S − r(j, k)) .

Step 4: Assume that ⟨a^{j}, R⟩ = ⟨a^{j}, S⟩ = 1 and ⟨a^{k}, R⟩ = ⟨a^{k}, S⟩ ≥ 2. We obtain φ^{k}(S − r(j, k)) = φ^{j}(S − r(j, k)) as in the previous step and continue

φ(S − r(j, k)) − φ^{j}(S − r(j, k)) = φ(S − r(j, k)) − φ^{k}(S − r(j, k)) = φ(S) − φ^{k}(S) .

Hence,

t[ℓ(j, k)] = −(φ(S) − φ^{k}(S)) · (ψ(R) − ψ^{k}(R)) .

Now, we distinguish between the following cases:

(i) Assume R ≠ S. In particular, there must be some l ∈ σ such that w.l.o.g. ⟨a^{l}, R⟩ > ⟨a^{l}, S⟩ (we may even assume a^{l} being adjacent to a^{k}). Hence, φ(S) = φ^{l}(S) = φ^{k}(S) (the latter equality follows from S ∈ (span E_{k}^{R}) ∩ (span E_{l}^{R})) implying t[ℓ(j, k)] = 0.

(ii) Assume R = S is not contained in E. Hence, R = R_{1} + R_{2} splits even on the IN^{E}-level. In particular, ⟨a^{k}, R_{1}⟩ ≤ ⟨a^{k}, R⟩ and moreover, if they are equal, there is an adjacent a^{l} such that ⟨a^{l}, R_{1}⟩ < ⟨a^{l}, R⟩. Similarly to (i), this means that φ(R_{1}) = φ^{k}(R_{1}), and we obtain φ(R) = φ(R_{1}) + φ(R_{2}) = φ^{k}(R_{1}) + φ^{k}(R_{2}) = φ^{k}(R).

(iii) Assume R = S is contained in E. Obviously, this implies φ(R) = ψ(R) = 0. □

(4.4.4) Assume R = S (≥ 0). We are computing the cup product T_{Y}^{1} (−R) × T_{Y}^{1} (−R) → T_{Y}^{2} (−2R). According to the special shape of the degree 2R (a proper multiple of an element of M), we may use the embedding Ψ : T_{Y}^{2} (−2R) ↪ N_{I C}^{{ε<σ | R>0 on ε}} (cf. (3.4.6)) to display our result. For a three-dimensional face ε = ⟨a^{1},…, a^{m}⟩ < σ (with R > 0 on ε) we have (φ ∪ ψ)_{ε} := Ψ_{ε}(φ ∪ ψ) = ∑_{j=1}^{m}(φ ∪ ψ)_{j j+1} (with m + 1 := 1).

On the other hand, we would like to deal with elements of T_{Y}^{1} (−R) as Minkowski summands of Q := σ ∩[R = 1]. Recalling the recipe of (3.4.4) and (3.4.5), the collection of maps φ^{j} : span E_{j}^{R} → ZZ corresponds to a Minkowski summand \underline{t} ∈ V_{I C}(Q)/\underline{1} in the following way:

• t_{jk} are the dilatation factors for the edges, i.e. if the Minkowski summand is given by its vertices b̄^{j} ∈ R^{⊥}, then b̄^{j} − b̄^{k} = t_{jk} · (ā^{j} − ā^{k}) (with ā^{j} = a^{j}/⟨a^{j}, R⟩).

• If ⟨a^{j}, R⟩ ≥ 2, then t_{jk} does not depend on k and will be denoted by s_{j}, too. (Set s_{j} := 0, if ⟨a^{j}, R⟩ = 1.)

• With b^{j} = b̄^{j} − s_{j} ā^{j} (hence s_{j} = −⟨b^{j}, R⟩), we have φ^{j} = ⟨b^{j}, •⟩.

(Take t′_{jk}, s′_{j}, b^{′j}, b̄^{′j} for the data corresponding to ψ.)

Theorem: Let \underline{t}, \underline{t}′ ∈ T_{Y}^{1} (−R) ⊆ V_{I C}(Q)/\underline{1}. Then, if ε = conv\{ā^{1},…, ā^{m}\} < Q is a compact, two-dimensional face, the ε-component of the cup product is given by

(\underline{t} ∪ \underline{t}′)_{ε} := Ψ_{ε}(\underline{t} ∪ \underline{t}′) = ∑_{j=1}^{m} t_{j j+1} t′_{j j+1} · (ā^{j} − ā^{j+1}) ∈ R^{⊥} ⊆ N_{I C} .

In particular, focusing on the degree-(−R)-part of the versal deformation of Y_{σ}, the quadratic equations look exactly (up to the new exponents 2) like the linear ones (defining V_{I C}(Q) as a subspace of I C^{N}, cf. (3.4.2)).

Proof: We have just to translate the result obtained in the previous proposition into the right language. Let us assume R ∈ E.

First, if ⟨a^{j}, R⟩ = ⟨a^{k}, R⟩ = 1, then

φ^{jk} = φ(R) − φ^{j}[r(k, j)] − φ^{k}[r(j, k)] + φ^{j or k}[r(j, k) + r(k, j) − R]
= 0 − ⟨b^{j}, r(k, j)⟩ − ⟨b^{k}, r(j, k)⟩ + ⟨b^{j}, r(j, k) + r(k, j) − R⟩
= ⟨b^{j} − b^{k}, r(j, k)⟩ − ⟨b^{j}, R⟩
= t_{jk} · ⟨a^{j} − a^{k}, r(j, k)⟩ − 0 = t_{jk} .

Hence, (φ ∪ ψ)_{j k} = t_{jk} t′_{jk} · (a^{j} − a^{k}).

On the other hand, if for instance ⟨a^{j}, R⟩ = 1, ⟨a^{k}, R⟩ ≥ 2, then

(φ ∪ ψ)_{j k} = ⟨b^{k}, R⟩ · ⟨b^{′k}, R⟩ · a^{j} = s_{k} · s′_{k} · a^{j}

(and (φ ∪ ψ)_{j k} = −s_{j} s′_{j} · a^{k} if ⟨a^{j}, R⟩ ≥ 2, ⟨a^{k}, R⟩ = 1).

Since edges with ⟨a^{j}, R⟩, ⟨a^{k}, R⟩ ≥ 2 yield (φ ∪ ψ)_{j k} = 0, we are done by taking the sum over all edges of ε < Q. □

(4.4.5) Assume R ≠ S (and put both into E). In that case, Ψ does not need to be injective anymore. Nevertheless, it pays to analyze what Ψ is doing with the cup product of elements from T_{Y}^{1} (−R) and T_{Y}^{1} (−S). We might find at least some of the quadratic equations for the versal base space with mixed degrees.

Definition: a^{j} ∈ σ will be called a 1-vertex, if ⟨a^{j}, R⟩ = ⟨a^{j}, S⟩ = 1. Moreover, a 1-edge is a two-dimensional face ⟨a^{j}, a^{k}⟩ < σ connecting two 1-vertices a^{j} and a^{k}. (It has been only the 1-edges contributing a non-trivial (φ ∪ ψ)_{jk}.)

Lemma: Let \underline{t} ∈ T_{Y}^{1} (−R) ⊆ V_{I C}(Q)/\underline{1}. Then, for each 1-vertex a^{j} ∈ σ, there is a value t_{j} ∈ I C such that

(1) If a^{j}, a^{k} ∈ σ are 1-vertices that can be connected within [R = S] ⊆ σ by some chain of edges (i.e. 2-faces of σ), then t_{j} = t_{k}. In particular, this holds for 1-edges ⟨a^{j}, a^{k}⟩ < σ.

(2) If ⟨a^{j}, a^{k}⟩ < σ is a 1-edge, then φ^{jk} = t_{jk} − t_{j} = t_{jk} − t_{k}.

(3) Moreover, in case of R ≰ S, we have t_{j} = t_{jl} for all vertices a^{l} ∈ [R ≥ S] ⊆ σ adjacent to a^{j}, but not forming a 1-edge with a^{j}.

Proof: Let us first assume that R ≰ S. In particular, there is at least one “neighbor” a^{l} of a^{j} being contained in [R > S] ⊆ σ, and we define t_{j} := t_{jl}.

To prove (3), we take another a^{l′} (even ⟨a^{l′}, R⟩ = ⟨a^{l′}, S⟩ will be allowed, if ⟨a^{l′}, R⟩ ≥ 2). Using an induction argument, we may assume that a^{j}, a^{l}, and a^{l′} are contained in a common threedimensional face ε ⊆ [R ≥ S] ⊆ σ. Then, we have R > 0 on ε, and R has value 1 in at most three different vertices including a^{j}. (Indeed, in [R ≥ S] the fact ⟨a^{•}, R⟩ = 1 implies ⟨a^{•}, S⟩ = 0 or a^{•} ∈ [R = S].) Hence, the linear equations defining V (Q) (cf. (3.4.2)) imply that all dilatation factors for edges in ε are equal.

To prove (1) in the case R ≰ S (i.e. we may use (3)), it is sufficient to deal with 1-edges ⟨a^{j}, a^{k}⟩ < σ. Again, we choose some three-dimensional face ε < σ containing ⟨a^{j}, a^{k}⟩ (and the remaining vertices from [R > S]) and look for vertices with R = 1: If there are at most three of them, then we are done as in the previous discussion (implying not only t_{l} = t_{k}, but also t_{j} = t_{jk}). On the other hand, if ⟨a^{j}, R⟩ = ⟨a^{k}, R⟩ = ⟨a^{l}, R⟩ = ⟨a^{l′}, R⟩ = 1, then ⟨a^{l}, S⟩ = ⟨a^{l′}, S⟩ = 0, and it is the V (Q)-equation for ε evaluated in S bringing the result.

Now, we consider the case R ≤ S. Defining t_{j} := φ^{j}(S), we would like to prove (1): We may assume that on the edges (in [R = S] ⊆ σ) between a^{j} and a^{k} there are no further 1-vertices. In particular, all dilatation factors are equal, and we call them t_{jk}. Then,

t_{j} − t_{k} = φ^{j}(S) − φ^{k}(S) = ⟨b^{j} − b^{k}, S⟩ = t_{jk} · ⟨a^{j} − a^{k}, S⟩ = 0 .

Finally, one obtains (2) for both cases (R ≰ S and R ≤ S) by straight computation (like in the proof of Theorem (4.4.4)). □

Theorem: Let \underline{t} ∈ T_{Y}^{1} (−R) and \underline{t}′ ∈ T_{Y}^{1} (−S). Then, if ε = ⟨a^{1},…, a^{m}⟩ < σ is a three-dimensional face, the Ψ_{ε}-image of the cup product is given by

(1) Ψ_{ε}(\underline{t} ∪ \underline{t}′) = ∑_{j=1}^{m} t_{j j+1} t′_{j j+1} · (ā^{j} − ā^{j+1}) ∈ (R, S)^{⊥} ⊆ N_{I C}, if ε is contained in [R = S]. (In particular, this result fits into (4.4.4).)

(2) Ψ_{ε}(\underline{t} ∪ \underline{t}′) = 0, otherwise.

Proof: In the first case (ε ⊆ [R = S]) we use Proposition (4.4.3) and the second part of the previous lemma to obtain

Ψ_{ε}(\underline{t} ∪ \underline{t}′) = ∑_{1-edges of ε}(t_{jk} − t_{j}) · (t′_{jk} − t′_{j}) · (ā^{j} − ā^{k}) .

Moreover, part (1) of the previous lemma tells that t_{j} does not depend on j - let us call it t_{ε} (and do the same thing for the t′-factor). On the other hand, since R ≠ S, we may assume (w.l.o.g.) that R ≰ S. In particular, t_{ε} equals all dilatation factors of non-1-edges of ε (by (3) of the same lemma). Hence,

Ψ_{ε}(\underline{t} ∪ \underline{t}′) = ∑_{j=1}^{m} (t_{j j+1} − t_{ε}) · (t′_{j j+1} − t′_{ε}) · (ā^{j} − ā^{j+1})
= \begin{matrix}∑_{j=1}^{m} t_{j j+1} t′_{j j+1} (ā^{j} − ā^{j+1}) − t_{ε} · ∑_{j=1}^{m} t′_{j j+1} (ā^{j} − ā^{j+1}) − \\ − t′_{ε} · ∑_{j=1}^{m} t_{j j+1} (ā^{j} − ā^{j+1}) + t_{ε} t′_{ε} · ∑_{j=1}^{m} (ā^{j} − ā^{j+1}) \end{matrix}

= ∑_{j=1}^{m} t_{j j+1} t′_{j j+1} (ā^{j} − ā^{j+1}) − 0 − 0 + 0 .

Finally, let us assume R ≥ S, R ≠ S on ε. Then, there is at most one 1-edge ⟨a^{j}, a^{k}⟩ in ε, and we know Ψ_{ε}(\underline{t} ∪ \underline{t}′) = (t_{jk} − t_{j}) · (t′_{jk} − t′_{j}) · (ā^{j} − ā^{k}). On the other hand, similarly to the proof of the previous lemma, there are at most three different dilatation factors for edges in ε: t_{jk} for ⟨a^{j}, a^{k}⟩, some t for a possible edge being contained in [S = 0], and t_{j} for the remaining ones. In particular, t_{jk} = t = t_{j}. □

Corollary: If dim σ = 3, then we have a complete description of the cup product T_{Y}^{1} (−R) × T_{Y}^{1} (−S) → T_{Y}^{2} (−R − S) (R, S ≥ 0):

\underline{t} ∪ \underline{t}′ = \{ \begin{matrix}∑_{j=1}^{m} t_{j j+1} t′_{j j+1} (ā^{j} − ā^{j+1}) ∈ R^{⊥} ⊆ N_{I C} (if R = S) \\ 0 (if R ≠ S) . \end{matrix}

Proof: On the one hand, σ is its only 3-face. On the other hand, we know that Ψ = Ψ_{σ} is always injective in the three-dimensional case. □

Toric Gorenstein singularities

5.1 I Q-Gorenstein cones

(5.1.1) The aim of this chapter is to apply the previous results to the special case of toric ( I Q-) Gorenstein singularities. On the one hand, this notion is the next one if we are looking for a wider class than that of complete intersections (which yields no interesting deformation theory). On the other hand, the property “ I Q-Gorenstein” admits a very clear description in the language of toric varieties and convex cones. (As far as I know, this was first mentioned by Ishida, [Ish], Theorem 7.7.)

In general, the dualizing sheaf ω on a Cohen-Macaulay variety can be obtained as

(i) ω_{Z} := Ω_{Z}^{dim Z} (sheaf of the highest differential forms), if Z is smooth, and

(ii) π_{∗}ω_{Y} := \underline{Hom}_{O_{Z}} (π_{∗}O_{Y} , ω_{Z}) for a flat, finite map π : Y → Z.

Definition: A variety Y is called ( I Q-) Gorenstein if (the reflexive hull of some tensor power of) ω_{Y} is an invertible sheaf on Y.

Since toric varieties are normal, the dualizing sheaf can be obtained as the push forward of the canonical sheaf on its smooth part. Hence, in our special situation, ω_{Y} equals the T (orus)-invariant, complete fractional ideal that is given by the order function mapping each fundamental generator onto 1 ∈ ZZ (cf. Theorem I/9 in [Ke]). In particular, we obtain the following

Fact: Let Y = Spec I C[σ^{∨} ∩M] be an affine toric variety given by a cone σ = ⟨a^{1},…, a^{M}⟩. Then, Y is I Q-Gorenstein, if and only if there is a primitive element R^{∗} ∈ M and a natural number g ∈ IN such that

⟨a^{j}, R^{∗}⟩ = g for each j = 1,…, M.

Y is Gorenstein, if and only if additionally g = 1.

In particular, toric I Q-Gorenstein singularities are obtained by putting a lattice polytope Q ⊆ A A into the affine hyperplane A A×{g} ⊆ A A×IR =: N_{IR} and defining σ := Cone(Q). Then, the canonical degree R^{∗} equals [\underline{0}, 1].

(5.1.2) The crucial point making I Q-Gorenstein cones easier to handle is the following: Affine conditions for fundamental generators a^{j} (such as “⟨a^{j}, R⟩ = c”) can be translated into homogeneous ones (as “⟨a^{j}, g R − c R^{∗}⟩ = 0”).

5.1. I Q-GORENSTEIN CONES 65

Lemma: Let σ be a I Q-Gorenstein cone which is smooth in codimension two. For each R ∈ M, the existence of some a^{j} meeting ⟨a^{j}, R⟩ ≥ 2 implies T_{Y}^{1} (−R) = 0.

Proof: We consider the set τ := {a^{j} | ⟨a^{j}, R⟩ ≤ 1} (which does not need to represent any face of σ). It is built from exactly those fundamental generators of σ that are contained in the half space [⟨•, g R − R^{∗}⟩ ≤ 0]. Now, we can proceed along the lines of the proof of Proposition (3.5.2). □

Let us fix some R such that T_{Y}^{1} (−R) ≠ 0. Using the notation of (3.5.1), our previous lemma implies that R yields exactly 1 on each fundamental generator of τ(R) (and, by definition of τ(R), non-positive values outside). There are just two different cases:

• τ(R) = σ: That fact is equivalent to R = R^{∗}, g = 1 (i.e. Y is Gorenstein and R = R^{∗} is the particular degree cutting out the lattice polytope Q). Moreover, T_{Y}^{1} (−R^{∗}) = V_{I C}(Q)/ I C · \underline{1}.

• τ(R) ≠ σ: Then, T_{Y}^{1} (−R) = V_{I C}(τ(R) ∩ [R = 1])/ I C · \underline{1} (the classes of Minkowski summands of a proper face of Q). Since (3.5.1), the whole T_{Y}^{1} is infinite-dimensional.

Corollary: Let σ be an at least three-dimensional I Q-Gorenstein cone. Assume that T_{Y}^{1} is finitedimensional (for instance, if Y has an isolated singularity). Then,

(1) Y is rigid unless it is Gorenstein.

(2) If Y is Gorenstein (g = 1), then T_{Y}^{1} = T_{Y}^{1} (−R^{∗}) = V_{I C}(Q)/ I C · \underline{1}.

(5.1.3) Let Q ⊆ A A be a lattice polytope; via affine embedding into height one, it provides a Gorenstein cone σ = IR_{≥0} · Q ⊆ A A × IR = N_{IR}.

Proposition:

(1) Y is smooth in codimension two, if and only if the edges of Q do not contain interior lattice points (i.e. they are represented by primitive elements d^{i} ∈ N ∩ R^{⊥}).

(2) Assume, moreover, dim Y ≥ 4 (i.e. dim Q ≥ 3). If dim T_{Y}^{1} < ∞, then every 2-face of Q has to be a triangle (with three primitive edges).

(3) If every 2-face of Q is a triangle with three primitive edges (for instance, if Y is smooth in codimension three), then Y will be rigid.

Proof: The first part is straight forward. The second part follows from (3.5.1) - each polygon with at least four edges admits non-trivial Minkowski summands. For the last part, we start with the homogeneous part T_{Y}^{1} (−R^{∗}) - it vanishes since (3.5.3). If R ∈ M is an arbitrary degree, we use (3.5.1) telling that T_{Y}^{1} (−R) = T_{Y (R)}^{1}(−R̄). Moreover, since (3.5.2) and (5.1.2), Y (R) is a Gorenstein singularity of smaller dimension, and R̄ is the corresponding canonical degree (called R^{∗} for Y). Hence, we have T_{Y (R)}^{1}(−R̄) = 0 again. □

In particular, if we were interested in isolated Gorenstein singularities, then the three-dimensional case would be the only interesting one. However, this case should be considered an important example for the upcoming investigations.

Remark: Just to be fashionable, we want to mention the following fact: As explained in (2.4.4), we may assign to Q also the projective variety X_{Σ(Q)}. Then, Corollary (5.1.2) says that T_{Y}^{1} = Pic(X_{Σ(Q)})/O_{X}(1), i.e. some kind of mirror symmetry seems to be involved.

5.2 The tautological cone over C(Q)

(5.2.1) Let Q ⊆ A A be a lattice polytope with primitive edges. We are going to investigate the deformation theory of Y = Y_{σ} in detail. In (2.2.2) 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). However, defining Q\underline{t} as a particular polytope inside its translation class requires a closer look:

Assume that 0 ∈ A A coincides with some vertex of the lattice polytope Q. Then, each vertex a of Q can be reached from there 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 can define the corresponding vertex a\underline{t} (and finally the polytope Q\underline{t} as the convex hull of all of them) by

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

(The linear equations defining V (Q) = span C(Q) ensure that this definition does not depend on the particular path from 0 to a through the 1-skeleton of Q.)

In the whole chapter 5 we use the notation of section (2.2).

(5.2.2) Definition: The tautological cone C̃(Q) ⊆ A A × V =: Ñ_{IR} ⊆ A A × IR^{N} is defined as

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

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

• a^{j} is a vertex of Q and

• \underline{t}^{l} is 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 a vertex a ∈ Q and \underline{t}, \underline{t}′ ∈ C(Q).)

With σ = Cone(Q) ⊆ A A × IR (putting Q into the hyperplane (t = 1)), we obtain a fiber product diagram of rational polyhedral cones:

[σ ⊆ A A × IR] ↪^{i} [C̃(Q) ⊆ A A × V]
↓↓pr_{IR}  ↓↓pr_{V}
IR_{≥0}  ↪^{·ϱ(Q)} [C(Q) ⊆ V]

(The vertical maps are projections onto the IR- and the V -component, respectively. The inclusion i is given by (t · a; t) ↦ (t · a; t,…, t).)

(5.2.3) The three cones σ = Cone(Q) ⊆ A A × IR, C̃(Q) ⊆ A A × V, and C(Q) ⊆ V define affine toric varieties called Y, X, and S, respectively. Using the abbreviations

M := IL^{∗} × ZZ, M_{IR} := A A^{∗} × IR, M̃ := IL^{∗} × V_{Z Z}^{∗}, M̃_{IR} := A A^{∗} × V^{∗} ,

5.2. THE TAUTOLOGICAL CONE OVER C(Q) 67

the corresponding rings of regular functions are A(Y) = I C[σ^{∨} ∩ M], A(X) = I C[C̃(Q)^{∨} ∩ M̃], and A(S) = I C[C(Q)^{∨} ∩ V_{Z Z}^{∗}]. These varieties come with the following maps:

(i) The diagram of (5.2.2) induces a fiber product diagram

\begin{matrix}Y & ↪^{i} & X \\ ↓ & & ↓π \\ I C & ↪ & S . \end{matrix}

Both horizontal maps are closed embeddings. (These claims will be checked in (5.2.5) and (5.2.8)(1).)

(ii) C(Q) = V ∩ IR_{≥0}^{N} is contained in IR_{≥0}^{N}, and this inclusion provides a morphism p : S → I C^{N} inducing functions t_{1},…, t_{N} on S. The composition I C ↪ S →^{p} I C^{N} sends t to (t,…, t).

We will use the map π : X → S to construct the versal deformation of Y.

(5.2.4) 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. Let us start with the dual cone of σ:

To each non-trivial c ∈ IL^{∗} we associate a vertex a(c) of Q and an integer η_{0}(c) meeting the properties

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

For c = 0 we define a(0) := 0 ∈ IL and η_{0}(0) := 0 ∈ ZZ.

Remark:

(1) 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.

(2) Since 0 ∈ Q, the integers η_{0}(c) are non-negative.

By the definition of η_{0}, we have

∂σ^{∨} ∩ M = {[c, η_{0}(c)] | c ∈ IL^{∗}} .

Moreover, if c^{1},…, c^{w} ∈ IL^{∗} \ 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

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

form the minimal generator set E for σ^{∨} ∩ M as a semigroup. Among them are all pairs [c, η_{0}(c)] describing facets (i.e. top dimensional faces) of Q.

E has always been used to obtain a closed embedding Y ↪ I C^{w+1}. To emphasize the special role of the element [\underline{0}, 1] ∈ E, we will denote the corresponding coordinate of I C^{w+1} by t rather than the usual z_{0} (keeping z_{1},…, z_{w} unchanged).

Example: We continue our example Q_{6} from section (2.2). 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

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

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 .

(5.2.5) 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^{∗}].

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. (5.3.2)). 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 ∈ A A^{∗}} .

Remark: Using our new notations, we can reformulate Theorem (2.2.4) now: M ⊆ I C^{N} is the

largest closed subscheme that is contained in S̄ and additionally comes from I C^{N}/ ; I C · ϱ(Q) via ℓ.

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 coordinate 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 got a first contribution to proof the claims made in (5.2.3)(i).

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

Definition: For c ∈ IL^{∗} choose some path from 0 ∈ Q to a(c) ∈ Q through the 1-skeleton of Q and let \underline{λ}^{c} = (λ_{1}^{c},…, λ_{N}^{c}) ∈ ZZ^{N} be the vector counting how often (and in which direction) we went through each particular edge (cf. (5.2.1)). 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 special choice of the 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 ∈ IL^{∗} we have \underline{η}(c) ≥ 0 (in the sense of Definition (5.2.5)).

(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 ∈ IL^{∗}.

Proof: (ii) a(c) is a vertex of Q providing 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

∑_{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 .

(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 ∈ A A^{∗}. Then, η_{0}(c) ∈ IR and \underline{η}(c) ∈ V^{∗} do not need to be contained in the lattices anymore. The previous lemma will keep valid (even for g_{v} ∈ IR_{≥0} in (iii)), if the relation “≥ 0” is replaced by the weaker version “∈ C(Q)^{∨}”.

(5.2.7) Proposition:

(1) C̃(Q)^{∨} = \{[c, \underline{η}] ∈ A A^{∗} × 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})] and C(Q)^{∨} ∩ V_{Z Z}^{∗} (embedded as [0, C(Q)^{∨}]) generate the semigroup C̃(Q)^{∨} ∩ M̃. (For recalling the definition of the c^{1},…, c^{w}, cf. (5.2.4).)

Proof: (1) Let [c, \underline{η}] ∈ A A^{∗} × 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 a [c, \underline{η}] ∈ C̃(Q)^{∨} it is possible to choose IR^{N}-representatives for \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) then.

(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 ∈ IL^{∗}). 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

\begin{aligned}[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)] . \end{aligned}

□

(5.2.8) Finally, we will take a short look at the geometrical situation reached at this point.

5.3. A FLAT FAMILY OVER M̄ 71

(1) The linear map

\begin{matrix}C̃(Q)^{∨} ∩ M̃ & ⟶ ⟶ & σ^{∨} ∩ M \\ [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, the map Y ↪ X is a closed embedding. Moreover, comparing with the similar statement concerning C(Q)^{∨} and IN at the end of (5.2.5), we obtain that the diagram of (5.2.3)(i) is a fiber product diagram, indeed.

(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 (5.2.4). Moreover, for v = 1,…, w, Z_{v} is the only monomial function lifting z_{v} from Y to X.

We have obtained the following commutative diagram:

Y ↪ I C^{w} × I C = I C^{w} × I C
↓ ⊗ ↓ ↓ Δ
X ↪ I C^{w} × S ⟶^{p} I C^{w} × I C^{N}
↓  ↓
S  ⟶^{p}  I C^{N}  ⟶^{ℓ} I C^{N−1} .

5.3 A flat family over M̄

(5.3.1) 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 (cf. section (2.2)) 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 section (5.3).

(5.3.2) 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 those 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 (5.2.7)(3)), the same arguments as for S and S̄ apply. Hence, Part (1) of the previous theorem is proved.

(5.3.3) 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^{∨} ∩ M, respectively, the equations of Y ⊆ I C^{w+1} (cf. (3.2.6)) are shaped as

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 them to the following elements of A(S̄)[Z_{1},…, Z_{w}] (described by using 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 - choose one of the possibilities. (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 (5.2.6), 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 defines an element of A(S̄)[Z_{1},…, Z_{w}], indeed.

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: Mapping F into A(X) = ⊕_{[c,\underline{η}]} I Cx^{[c,\underline{η}]} ([c, \underline{η}] runs through all elements of C̃(Q)^{∨} ∩ (IL^{∗} ×

V_{Z Z}^{∗}); Z_{v} ↦ x^{[c^{v},\underline{η}(c^{v})]}, t_{i} ↦ x^{[0,e_{i}]}) yields

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). □

(5.3.4) 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.

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

Theorem: 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 to 0 ∈ M̄ of the equations defining X̃ ⊆ I C^{w+1} ×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.

Proof: According to [Ma], (20.C), Theorem 49, flatness of π̃ in 0 ∈ M̄ is equivalent to the vanishing of Tor_{1}^{O_{M̄,0}}((π̃_{∗}OX̃)_{0}, I C) ( I C becomes an O_{M̄,0}-module via evaluating in 0 ∈ M̄). Using the embedding X̃ ↪ I C^{w+1} × M̄ (together with the defining equations and linear relations between them) we obtain an O_{M̄,0}[Z_{0},…, Z_{w}]-free (hence O_{M̄,0}-free) resolution of (π̃_{∗}OX̃)_{0} up to the second term. Now, the condition that relations between Y -equations lift to those between X̃-equations is equivalent to the fact that our (partial) resolution remains exact under ⊗_{O_{M̄,0}} I C. □

For our special situation take X̃ := X̄ ×S̄ M (and M̄ := M̄, Y := Y); in (5.3.3) 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 (5.3.1), we have just to take the linear relations between the f_{(a,b,α,β)}’s and lift them to relations between the F_{(a,b,α,β)}’s.

According to the special shape of our generator set E, there are three types of relations between the f_{(a,b,α,β)}’s (in (3.2.7), type (i) was called s(…); the types (ii) and (iii) will fit into the r(…)- relations):

(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_{(q,p+r,ξ,0)} in (5.5.3)).

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

The proof of Theorem (5.3.1) is complete.

(5.3.6) 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 (5.3.3) 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 (2.2.3), they describe a family contained

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

5.4 The Kodaira-Spencer map

(5.4.1) To each vertex a^{j} ∈ Q (or equally named fundamental generator a^{j} := (a^{j}, 1) ∈ σ) and each element R ∈ M we have associated the subset

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

For the special degree R^{∗} = [\underline{0}, 1] ∈ M (corresponding to the affine hyperplane containing Q) the associated subsets of E = \{[c^{1}, η_{0}(c^{1})],…, [c^{w}, η_{0}(c^{w})], [\underline{0}, 1]\} equal

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

In (5.2.6), for each c ∈ IL^{∗}, 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(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(E ∩ ∂σ^{∨}).) Moreover, if q comes from one of the submodules L(E_{j}^{R^{∗}}) ⊆ L(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 .

(5.4.2) Theorem: The Kodaira-Spencer map of the family X̄ ×S̄ M → M̄ of section (5.3) 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, composing this map with the isomorphism T_{Y}^{1} (−R^{∗}) →^{∼} V_{I C}/\underline{1} provided in (3.4.3) (see also (5.1.2)) yields the identity.

Proof: Using the same symbol J for the ideal J ⊆ I C[t_{1},…, t_{N}] as well as for the intersection J ∩ I C[t_{i} − t_{j} | 1 ≤ i, j ≤ N] (cf. (2.2.4)), 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) (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 notations of (5.3.3), in A\underline{t}^{0} it holds

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

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}))) .

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

On the other hand, we use Theorem (3.2.6): Fixing R^{∗} ∈ M, it is the element of L_{I C}(E ∩ ∂σ^{∨})^{∗} given by q ↦ ∑_{i,v} t_{i}^{0} q_{v} η_{i}(c^{v}) that corresponds to the infinitesimal deformation of T_{Y}^{1} (−R^{∗}) defined by the map

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]} .

Finally, we have to combine Φ with the isomorphism (L(E_{0}^{R^{∗}})/ ∑_{j} L(E_{j}^{R^{∗}}))^{∗} →^{∼} V (Q)/\underline{1} pro-

vided in section (3.4). In (3.4.1), we have related relations q ∈ L(E_{0}^{R^{∗}}) to elements (v^{j}) ∈ ker (⊕_{j} span E_{j}^{R^{∗}} → M) via q = ∑_{j} q^{j} (q^{j} ∈ ZZ^{E_{j}}) and v^{j} = − ∑_{v} q_{v}^{j} r^{v}. Hence, similarly to the calculations preceding the theorem, we obtain

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

Using the language of (3.4.4), this means that the Kodaira-Spencer map assigns to \underline{t} ∈ V (Q)/\underline{1} the elements b^{j} = (aj ; \underline{t}, 0) ∈ N (j = 1,…, M) - just forming the vertices of the Minkowski summand Q\underline{t} in (R^{∗})^{⊥}. □

(5.4.3) Corollary: If Y is a three-dimensional, isolated Gorenstein singularity, then the Kodaira-Spencer map T_{0}M̄ → T_{Y}^{1} is an isomorphism. (Of course, this is right in higher dimensions, too. However, in (5.1.3) we have seen that those singularities are rigid anyway.)

Proof: From section (5.1) we know that T_{Y}^{1} (−R^{∗}) = T_{Y}^{1} . □

5.5 The obstruction map

(5.5.1) In this section we build up the obstruction map as it was described in (3.1.4). 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 comes as the kernel in 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. (5.3.5)), 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 (5.4.2) we have proved that the Kodaira-Spencer map is an isomorphism (at least onto the homogeneous piece T_{Y}^{1} (−R^{∗})). By Theorem (3.1.4), this fact combined with injectivity of the obstruction map implies versality. □

The remaining part of section (5.5) contains the proof of the previous theorem.

(5.5.2) We have to improve the notations of the sections (5.2) and (5.3). Since M̄ ⊆ S̄ ⊆ I C^{N}, we were able to use the toric equations (cf. (2.2.4)) during computations modulo J. In particular, the exponents \underline{η} ∈ ZZ^{N} of \underline{t} needed be known modulo V^{⊥} only; 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 with refining the definitions of (5.2.6):

(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 ∈ IL^{∗} 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 ∈ IL^{∗} 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 (5.2.6). 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 (5.3.3). Let a, b ∈ IN^{w}, α, β ∈ IN such that ([a, α], [b, β]) ∈ m ⊆ IN^{w+1} × IN^{w+1}, i.e.

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}^{\underline{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)}) .

(5.5.3) We have to discuss the same three types of relations as we did in (5.3.5). Since there is only one single element c ∈ IL involved in the relations (i) and (ii), computing 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 notations:

• 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{q} := \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 (5.3.5) 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_{(q,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}^{q} · (\underline{t}^{αe_{1}+∑_{v}(a_{v}+r_{v}−q_{v})\underline{η̃}(c^{v})} − \underline{t}^{βe_{1}+∑_{v}(b_{v}+r_{v}−q_{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}^{q} \underline{t}^{∑_{v}(p_{v}+r_{v}−q_{v})\underline{η̃}(c^{v})} − \underline{Z}^{p+r}) \end{matrix}

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

As in (5.3.5)(iii), we can see that λ(s) vanishes modulo J (or even in A(S̄)) - just identify \underline{η} and \underline{η̃}.

(5.5.4) In (5.5.1) we 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]

(k_{0} := ∑_{i} A_{i}; c_{k} are the constants occurred in (2.3.4)). In particular, the coefficients from W_{k} vanish for k > k_{0}.

Proof: First, we remark that it is allowed to assume that A = \underline{d}^{+}, B = \underline{d}^{−}, i.e. \underline{t}^{A} − \underline{t}^{B} = p\underline{d}(\underline{t}) (cf. (2.3.2)). (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 (2.3.4) 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 (2.3.3) 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}^{q} · 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 the a- and the b-summand of the λ(s)-formula of (5.5.3). For the first one we obtain

\underline{d}^{(a)} = \begin{matrix}[αe_{1} + ∑_{v}(a_{v} + r_{v} − q_{v}) ˜ \underline{η}(c^{v})] − \\ − [αe_{1} + ∑_{v}(p_{v} + r_{v} − q_{v}) ˜ \underline{η}(c^{v}) + ∑_{v}(a_{v} − p_{v}) \underline{η}(c^{v})] \end{matrix}

= ∑_{v}(a_{v} − p_{v}) · (\underline{η̃}(c^{v}) − \underline{η}(c^{v}))  and

k_{0} = ∑_{i} (αe_{1} + ∑_{v}(a_{v} + r_{v} − q_{v}) \underline{η̃}(c^{v}))_{i}
= α + ∑_{v}(a_{v} + r_{v} − q_{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}

□

(5.5.5) Now, we try to approach the obstruction map λ from the opposite direction. Using the description of T_{Y}^{2} given in (3.2.4) we construct an element of T_{Y}^{2} ⊗_{I C} W, that afterwards will turn out to equal λ.

For ρ ∈ ZZ^{N} induced from some path along the edges of Q, we will denote

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

the vector showing the behavior of c ∈ IL^{∗} passing each particular edge. If, moreover, ρ comes from a closed path, then \underline{d}(ρ, c) will be 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, •) : & A A^{∗} & ⟶ & V^{⊥} & ⟶ & W_{k} \\ & c & & ↦ & & g_{\underline{d}(ρ,c),k}(\underline{t} − t_{1}) . \end{aligned}

Remark:

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

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

(2) Let c ∈ IL^{∗} (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: The reason for (1) is the fact that the elements \underline{d}(ε, c) (ε < Q 2-face; c ∈ IL^{∗}) generate V^{⊥} as a vector space. For the proof of (2), we consider \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 (5.5.4) we obtain g_{\underline{d},k}(\underline{t} − t_{1}) ∈ J̃ by Lemma (2.3.3). □

Considering the sets E_{j} for the degrees R := k R^{∗}, k ≥ 2, we obtain

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

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^{j}, a^{l}⟩ < Q be an edge of the polyhedron Q. Then, on L(E_{j}^{kR^{∗}} ∩ E_{l}^{kR^{∗}}) = L(E_{j}^{kR^{∗}}) ∩ L(E_{l}^{kR^{∗}}), the maps ψ_{j}^{(k)} and ψ_{l}^{(k)} coincide. In particular (cf. Theorem (3.2.4)), the ψ_{j}^{(k)}’s induce a linear map ψ^{(k)} : T_{Y}^{2} (−kR^{∗})^{∗} → W_{k}.

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

\begin{aligned}ψ_{j}^{(k)}(q) − ψ_{l}^{(k)}(q) & = & ∑_{v} q_{v} · g^{(k)} (\underline{λ}(a^{j}) + \underline{μ}^{v}(a^{j}) − \underline{λ}(a^{l}) + \underline{μ}^{v}(a^{l}), c^{v}) \\ & = & \begin{matrix}g^{(k)} (\underline{λ}(a^{j}) − \underline{λ}(a^{l}) + ρ^{jl}, ∑_{v} q_{v} c^{v}) + \\ + ∑_{v} q_{v} · g^{(k)} (\underline{μ}^{v}(a^{j}) − \underline{μ}^{v}(a^{l}) − ρ^{jl}, 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 remark: If q_{v} ≠ 0, then the assumption about q implies the inequalities

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

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

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

Proof: Using Theorem (3.2.7), 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. (5.3.5)) 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 onto 0 (in particular, if there is not any j meeting the desired condition).

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^{∗}} ; 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} ; \begin{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{matrix}

(and mapping onto 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^{∗}) ⊗ W_{k} (still representing ψ^{(k)}), but improves 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, those relations map onto 0;

• for relations of type (iii) (i.e. γ = 0) we can compare the previous formula with the result obtained in Corollary (5.5.4): The coefficients coincide, and the monomial \underline{z}^{q} 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). □

(5.5.7) 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}A A^{∗} ⊗_{IR} I C ⟶ ⟶ T_{Y}^{2} (−kR^{∗})^{∗} yielding ψ^{(k)} ∘ p^{k} = ∑_{ε<Q} g^{(k)}(\underline{ε}, •). Then, the result follows from the first part of Remark (5.5.5).

The degree kR^{∗} (k ≥ 2) is always different from 1 on the fundamental generators of σ. In particular, it meets the assumption made in Corollary (3.4.6) yielding an injection Ψ : T_{Y}^{2} (−kR^{∗}) ↪ ⊕_{ε<Q}N_{I C}. Moreover, since R^{∗} = [\underline{0}, 1] ∈ E_{j}^{kR^{∗}}, the image of Ψ is killed by R^{∗}, and we can use

Ψ^{∗} : ⊕_{ε<Q}M_{I C} ⟶ ⟶ ⊕_{ε<Q}M_{I C}/I C · R^{∗} = ⊕_{ε<Q}A A_{I C}^{∗} ⟶ ⟶ T_{Y}^{2} (−kR^{∗})^{∗}

for the auxiliary map p^{k}. Taking a closer look at that construction, 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}] ∈ M (and, as just mentioned, only the c ∈ IL^{∗} 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^{∗}}. (This corresponds to the lifting from span(E^{R})_{•} to (ZZ^{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}ZZ^{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 (5.5.5) 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 Remark (5.5.5)(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 (5.5.1) is proven.

5.6 The components of the reduced versal family

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

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

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

(i) Define the cone

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

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

(ii) The projection A A × IR^{n+1} ⟶ ⟶ IR^{n+1} provides n + 1 regular functions on X_{0}, i.e. we obtain a

map X_{0} → I C^{n+1}. Composing this morphism with ℓ : I C^{n+1} ⟶ ⟶ I C^{n+1}/I C · (1,…, 1) yields the flat map X_{0} → M̄_{0}.

We will use (5.6.2) and (5.6.3) to prove the theorem.

(5.6.2) We already know (cf. (2.2.5)) that both the space

M̄_{0} = I C^{n+1} /I C · (1,…, 1) ⊆ I C^{N}/I C · (1,…, 1)

and its pullback M_{0} ⊆ I C^{N} are given by the equations t_{i} − t_{j} = 0 (if d^{i}, d^{j} belong to some common summand R_{k} of Q).

There is a chain of inclusions M_{0} ⊆ M ⊆ S̄ ⊆ I C^{N}, and the map M_{0} ↪ S̄ factorizes through an embedding M_{0} ↪ S. It is given by the surjection of I C-algebras

\begin{matrix}I C[C(Q)^{∨} ∩ V_{Z Z}^{∗}] & ⟶ ⟶ & I C[T_{0},…, T_{n}] \\ t_{i} & ↦ & T_{k} with d^{i} ∈ R_{k} \end{matrix}

coming from the linear map

\begin{matrix}IR^{n+1} & ↪ & V & ↪ & IR^{N} . \\ e^{k} & ↦ & ∑_{d^{i}∈R_{k}} e^{i} . \end{matrix}

The matrix of IR^{n+1} ↪ IR^{N} equals the incidence matrix between edges and Minkowski summands of Q, and the space M_{0} = I C^{n+1} corresponds to the cone IR_{≥0}^{n+1} = IR^{n+1} ∩ C(Q).

(5.6.3) The family X_{0} → M̄_{0} arises from X̄ ×S̄ M → M̄ via base change M̄_{0} ↪ M̄. We obtain

\begin{aligned}X_{0} & = & (X̄ ×S̄ M) ×M̄ M̄_{0} & = & X̄ ×S̄ (M ×M̄ M̄_{0}) & = & X̄ ×S̄ M_{0} \\ & = & X̄ ×S̄ (S ×_{S} M_{0}) & = & (X̄ ×S̄ S) ×_{S} M_{0} & = & X ×_{S} M_{0} . \end{aligned}

Hence, with σ̃ as defined is the theorem, it remains to show that

I C[C̃(Q)^{∨} ∩ (IL^{∗} × V_{Z Z}^{∗})] ⟶^{ψ}  I C[σ̃^{∨} ∩ (IL^{∗} × ZZ^{n+1})]
⋃|  ⋃|
I C[C(Q)^{∨} ∩ V_{Z Z}^{∗}]  ⟶ ⟶ I C[IN^{n+1}] = I C[T_{0},…, T_{n}]

is a tensor product diagram:

(i) σ̃ is the preimage of IR_{≥0}^{n+1} ⊆ C(Q) via the projection C̃(Q) ⟶ ⟶ C(Q). In particular, σ̃ ⊆ C̃(Q) causes a surjective map ψ_{IR} : C̃(Q)^{∨} ⟶ ⟶ σ̃^{∨}.

(ii) To show surjectivity at the level of lattices (i.e. surjectivity of ψ) we start with an element [c, \underline{η}] ∈ C̃(Q)^{∨} and suppose its image ψ_{IR}([c, \underline{η}]) to be contained in σ̃^{∨} ∩ (IL^{∗} × ZZ^{n+1}). In particular, c ∈ IL^{∗}, and we obtain [c, \underline{η}(c)] ∈ C̃(Q)^{∨} ∩ (IL^{∗} × V_{Z Z}^{∗}) implying that

[0, \underline{η} − \underline{η}(c)] = [c, \underline{η}] − [c, \underline{η}(c)] ∈ [0, C(Q)^{∨}] ⊆ C̃(Q)^{∨}

maps to an element of IN^{n+1} ⊆ σ̃^{∨} ∩ (IL^{∗} × ZZ^{n+1}). On the other hand, surjectivity of C(Q)^{∨} ∩ V_{Z Z}^{∗} ⟶ ⟶ IN^{n+1} causes that this element can be reached by some [0, \underline{μ}] ∈ [0, C(Q)^{∨} ∩ V_{Z Z}^{∗}], too. Hence, [c, \underline{η}(c) + \underline{μ}] ∈ C̃(Q)^{∨} ∩ (IL^{∗} × V_{Z Z}^{∗}) is a lattice-preimage of ψ_{IR}([c, \underline{η}]).

(iii) The same methods applies for showing that ker ψ is generated by the same elements as ker( I C[C(Q)^{∨} ∩ V_{Z Z}^{∗}] → I C[IN^{n+1}]): If [c^{1}, \underline{η}^{1}], [c^{2}, \underline{η}^{2}] ∈ C̃(Q)^{∨} ∩ (IL^{∗} × V_{Z Z}^{∗}) have the same image in I C[σ̃^{∨} ∩ (IL^{∗} × ZZ^{n+1})] (i.e.

x^{[c^{1},\underline{η}^{1}]} − x^{[c^{2},\underline{η}^{2}]} ∈ ker ψ), then c^{1} = c^{2}, and the elements

\underline{μ}^{1} := \underline{η}^{1} − \underline{η}(c^{1}), \underline{μ}^{2} := \underline{η}^{2} − \underline{η}(c^{2}) ∈ C(Q)^{∨} ∩ V_{Z Z}^{∗}

have the same image in I C[IN^{n+1}]. In particular, x^{\underline{μ}^{1}} − x^{\underline{μ}^{2}} ∈ ker( I C[C(Q)^{∨} ∩ V_{Z Z}^{∗}] → I C[IN^{n+1}]), and

x^{[c^{1},\underline{η}^{1}]} − x^{[c^{2},\underline{η}^{2}]} = x^{[c^{1},\underline{η}(c^{1})]} · (x^{\underline{μ}^{1}} − x^{\underline{μ}^{2}}) .

(5.6.4) Example: At the end of (2.2.5) 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

\begin{matrix}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)⟩ . \end{matrix}

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.

5.7 Examples

(5.7.1) 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} survive. (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. (5.7.2).)

(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 has been spread in the paper.

(5.7.2) We will use the polygon Q_{4} := Conv {(0, 0), (1, 0), (2, 2), (0, 1)} of (5.7.1)(iv) for a more detailed demonstration 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 information to obtain the versal family of Y_{4} = Spec I C[Cone(Q_{4})^{∨} ∩ ZZ^{3}]:

– 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 do 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 (5.2.6) and (5.3.3):

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

[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].

[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 sum 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 equations:

\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

\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}^{5} = [0, 2, 0, 0] = [0, 0, 1, 1],
\underline{η}^{6} = [0, 0, 0, 2], \underline{η}^{7} = [0, 0, 0, 1], \underline{η}^{8} = [0, 0, 0, 0].

– Now, we can lift our 20 equations 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:

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}.

(5.7.3) 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 into 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) .

Deformations of negative degree

6.1 Negative degrees

(6.1.1) The construction of chapter 5 works in a more general case: Let σ ⊆ N_{IR} be a cone providing an affine toric variety that is smooth in codimension two. Then, for a fixed R ∈ σ^{∨} ∩ M, we will obtain the “versal deformation of degree −R”, i.e. that part of the entire versal deformation which remains after setting zero all T_{Y}^{1} -variables of degree different from −R. Those degrees comming from −(σ^{∨} ∩ M) we will call (strictly) negative degrees. In case that T_{Y}^{1} is concentrated in one single negative degree −R (as is for three-dimensional, isolated toric Gorenstein singularities with R = R^{∗}), the versal deformation of degree −R is just versal.

(6.1.2) To obtain the whole versal deformation of Y, two further steps would be necessary:

(1) Non-negative degrees have to be considered, too. As already discussed in (3.5.2) and (3.5.3), the negative degrees are special in some sense - for instance, the cone σ can be reconstructed from the cross cut Q := σ ∩ [R = 1].

(2) If there are two different R, S ∈ M (or even R, S ∈ σ^{∨} ∩ M), then it is not clear how to put the homogeneous pieces together getting a family containing both. (It works in many examples, and a possible idea might roughly speaking be the following: After proceeding in degree −R analogously to the Gorenstein case, lift S ∈ σ^{∨} ∩ M to an element of C̃(Q)^{∨} ∩ (IL^{∗} × V_{Z Z}^{∗}) and try to deform the singularity given by the cone C̃(Q) in that lifted degree. However, one of the problems arising that way is: We cannot stay inside the class of singularities which are smooth in codimension two.)

(6.1.3) The generalization from the Gorenstein case is rather straight. Nevertheless, we have preferred to present that special case first: Some of the notations have become simpler, making clearer the idea of the construction.

To avoid boring repetitions in the upcoming sections, we will proceed as follows: We will just mention how notions or statements have to be changed to fit into the general case. Whenever a calculation or a proof remain essentially the same, we will omit it (or just emphasize the adjustments to be made). We will even use the same numbering as in the previous chapter.

6.2 The tautological cone

(6.2.1) Let σ ⊆ N_{IR} be a cone such that the two-dimensional faces ⟨a^{j}, a^{k}⟩ < σ are smooth (i.e. {a^{j}, a^{k}} could be extended to a ZZ-basis of N). Let R ∈ σ^{∨} ∩ M be a primitive element which will be fixed through the whole chapter. (According to Remark (3.4.3), non-primitive R’s would yield T_{Y}^{1} (−R) = 0 anyway.) The affine hyperplane A A := [⟨•, R⟩ = 1] ⊆ N_{IR} containing the lattice points IL := A A ∩ N has (lattice-) distance one from the origin. We define Q := σ ∩ [R = 1] - it is a polyhedron in A A with Q^{∞} := σ ∩ R^{⊥} as its cone of unbounded directions. We fix one of the IL-vertices of Q to be the origin 0 ∈ A A - if there are not any vertices contained in the lattice at all, T_{Y}^{1} (−R) will vanish again.

According to the general description of T_{Y}^{1} (−R) given in Theorem (3.4.5), we have to replace the vector space V (Q) (which for general polyhedra Q was defined in (3.4.2)) by its subspace

V′(Q) := \{(t_{1},…, t_{N}) | \begin{matrix}• ∑_{i} t_{i} ε_{i} d^{i} = 0 for every compact 2-face ε < Q \\ • t_{i} = t_{j} if d^{i}, d^{j} are compact edges containing a common non- lattice vertex of Q \}. \end{matrix}

Of course, we also will use C′(Q) := V′(Q) ∩ IR_{≥0}^{N} instead of C(Q) ⊆ V (Q). For \underline{t} ∈ C′(Q), the corresponding Minkowski summand Q\underline{t} is defined as

Q\underline{t} = conv \{ā\underline{t} | ā ∈ Q is a vertex\} + Q^{∞} .

Nevertheless, just to simplify notations, we are going to use the usual symbols V (Q) and C(Q) for the new objects V′(Q) and C′(Q), respectively.

(6.2.2) Since Q does not need to be compact (if R ∈ ∂σ^{∨}), we have to take the closure of IR_{≥0} · Q ⊆ N_{IR} to get the cone σ back. Similarly, we define

C̃(Q) := \overline{\{(a, \underline{t}) | \underline{t} ∈ C(Q); a ∈ Q\underline{t}\}} = \{(a, \underline{t}) | \underline{t} ∈ C(Q); a ∈ Q\underline{t}\} + (Q^{∞}, 0) .

It is generated by the usual pairs (ā_{\underline{t}^{l}}^{j}, \underline{t}^{l}) (cf. (5.2.2)) plus the additional elements (a^{k}, 0) with a^{k} ∈ Q^{∞} = σ ∩ R^{⊥}.

(6.2.3) The diagrams of cones (given in (5.2.2)) and the corresponding affine toric varieties (cf. (5.2.3)) remain unchanged. However, the latter one might fail to be a fiber product diagram, as it will be explained in (6.2.8).

(6.2.4) To describe the dual cone σ^{∨} we will still use the vertices a(c) and scalars η_{0}(c) as we did in (5.2.4). However,

• just to emphasize that in the non-Gorenstein case they might be different from the corresponding fundamental generators a(c) ∈ σ, we prefer to call those vertices ā(c) ∈ Q from now on (i.e. ā(c) = a(c)/⟨a(c), R⟩), and

• they are only defined for c ∈ (Q^{∞})^{∨} ⊆ A A^{∗}, and - even for c ∈ IL^{∗} ∩ (Q^{∞})^{∨} - it will be no longer true that the η_{0}(c)’s have to be integers; we just know η_{0}(c) ∈ IR_{≥0}.

Denote by η_{0}^{∗}(c) the smallest integer greater or equal than η_{0}(c). Then,

σ^{∨} = \{[c, η_{0}(c)] | c ∈ (Q^{∞})^{∨}\} + IR_{≥0} · [\underline{0}, 1]

and

σ^{∨} ∩ M = \{[c, η_{0}^{∗}(c)] | c ∈ IL^{∗} ∩ (Q^{∞})^{∨}\} + IN · [\underline{0}, 1] .

([\underline{0}, 1] equals the element R ∈ M fixed in the beginning.) In particular, analogously to (5.2.4), we can choose the generating set E ⊆ σ^{∨} ∩ M as some

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

(It just might happen that [\underline{0}, 1] splits inside σ^{∨} ∩ M. Then, the generating set E will not be a minimal one.)

(6.2.5) According to the changes of V = V (Q) in (6.2.1), we have to adapt the descriptions of V^{⊥} (yielding the equations of S̄ ⊆ I C^{N}) and V^{∗}:

V^{⊥} = span \begin{cases}• [⟨ε_{1}d^{1}, c⟩,…, ⟨ε_{N}d^{N}, c⟩] | ε < Q is a compact 2-face, c ∈ A A^{∗}; \\ • [0,…, 1_{i},…, −1_{j},…, 0] if d^{i}, d^{j} contain a common non-lattice vertex of Q \} . \end{cases}

To deal with the dual space V^{∗}, the following point of view might be useful: In the previous chapter we have described its elements by using the surjection IR^{N} ⟶ ⟶ V^{∗}. In particular, an element \underline{η} ∈ V^{∗} was given by coordinates η_{i} corresponding to the edges d^{i} of Q. Now, caused by the additional equations t_{i} − t_{j}, we may collect the edges into several “connected components”: Two edges are said to be directly connected, if they contain a common non-lattice vertex of Q. For each component built that way, not the single coordinates but just their sum is important to know.

(6.2.6) For c ∈ (Q^{∞})^{∨} ⊆ A A^{∗} we take the usual definition of \underline{η}(c) ∈ C(Q)^{∨} ⊆ V^{∗} via paths \underline{λ}^{c} ∈ ZZ^{N}. Except for statements concerning the lattice (in the sense of Remark (5.2.6)), Lemma (5.2.6) remains true in the general case. On the other hand, if c ∈ IL^{∗} ∩ (Q^{∞})^{∨}, it is the equation ∑_{i} η_{i}(c) = η_{0}(c) showing that we cannot expect \underline{η}(c) belonging to V_{Z Z}^{∗}.

Definition:

(1) Let ā ∈ Q be a vertex not contained in the lattice IL. Then, we denote by e[ā] ∈ V_{Z Z}^{∗} the element represented by [0,…, 1_{i},…, 0] ∈ ZZ^{N} for some compact edge d^{i} containing ā. (In the language of (6.2.5), e[ā] yields 1 at the “connected component” containing ā, and 0 at the remaining components.)

(2) \underline{η}^{∗}(c) := \underline{η}(c) + (η_{0}^{∗}(c) − η_{0}(c)) · e[ā(c)] ∈ V^{∗}. (If ā(c) ∈ IL, then η_{0}^{∗}(c) − η_{0}(c) = 0 implies that we do not need e[ā(c)] in that case.)

Lemma: Let c ∈ IL^{∗} ∩ (Q^{∞})^{∨}. Then,

(i) \underline{η}^{∗}(c) ∈ V_{Z Z}^{∗}. This element equals \underline{η}(c) if and only if ⟨ā(c), c⟩ ∈ ZZ (in particular, if ā(c) ∈ IL).

(ii) \underline{η}^{∗}(c) ≥ 0 in the sense that \underline{η}^{∗}(c) ∈ im [IN^{N} → V_{Z Z}^{∗}] ⊆ C(Q)^{∨} ∩ V_{Z Z}^{∗}.

(iii) For c^{v} ∈ IL^{∗} ∩ (Q^{∞})^{∨} and g_{v} ∈ IN we have ∑_{v} g_{v} \underline{η}^{∗}(c^{v}) ≥ \underline{η}^{∗}(∑_{v} g_{v} c^{v}).

(iv) ∑_{i=1}^{N} η_{i}^{∗}(c) = η_{0}^{∗}(c).

Proof: The last part is a direct consequence of both the corresponding statement for \underline{η}(c) and the definition of \underline{η}^{∗}(c). Now, for c ∈ IL^{∗} ∩ (Q^{∞})^{∨}, we will show that \underline{η}^{∗}(c) ∈ V^{∗} can be represented by a vector of IR^{N} having just non-negative integers as coordinates. (However, according to (6.2.5), we may consider elements \underline{η} ∈ IR^{N} in the factor IR^{N}/span\{t_{i} − t_{j} | d^{i}, d^{j} connected\}. In particular, we can speak about coordinates of \underline{η} assigned to certain “components” of the edge set, e.g. assigned to certain non-lattice vertices of Q.) This will imply (i) and (ii) of the lemma.

Proceeding as in the proof of Lemma (5.2.6), we choose some path along the edges of Q passing ā^{0} = 0,…, ā^{p} = ā(c) and decreasing the value of c at each step. It provides some vector \underline{λ}^{c} ∈ ZZ^{N} yielding \underline{η}(c) via η_{i}(c) := −λ_{i}^{c} ⟨d^{i}, c⟩. Denote by ā^{j_{0}},…, ā^{j_{q}} ({j_{0},…, j_{q}} ⊆ {0,…, p}) the IL-vertices on the path. Then, for v = 1,…, q, the edges between ā^{j_{v−1}} and ā^{j_{v}} (say d^{i_{1}},…, d^{i_{k}}) belong to the same “component”. In particular, not the single η_{i_{1}}^{∗}(c),…, η_{i_{k}}^{∗}(c) but their sums have to be considered:

∑_{μ=1}^{k} η_{i_{μ}}^{∗}(c) = ∑_{μ=1}^{k} η_{i_{μ}}(c) = ⟨− ∑_{μ=1}^{k} λ_{i_{μ}}^{c}d^{i_{μ}}, c⟩ = ⟨ā^{j_{v−1}} − ā^{j_{v}}, c⟩ ∈ IN .

If ā(c) belongs to the lattice IL, then we are done. Otherwise, there might be at most one noninteger coordinate (assigned to ā(c) ∉ IL) in \underline{η}^{∗}(c). However, this cannot be the case, since the sum taken over all coordinates of \underline{η}^{∗}(c) yields the integer η_{0}^{∗}(c).

To prove (iii) we start as in (5.2.6). Then, inside the vector space IR^{N}/span{t_{i}−t_{j} | d^{i}, d^{j} connected}, we obtain the (componentwise) inequality

∑_{v} g_{v} \underline{η}^{∗}(c^{v}) ≥ ∑_{v} g_{v} \underline{η}(c^{v}) ≥ \underline{η} (∑_{v} g_{v} c^{v}) .

On the other hand, \underline{η} (∑_{v} g_{v} c^{v}) and \underline{η}^{∗} (∑_{v} g_{v} c^{v}) might differ in at most one coordinate (assigned to ā(∑_{v} g_{v} c^{v})). If so, then by definition of \underline{η}^{∗} the latter one equals the smallest integer not smaller than the first one. Hence, we are done, since the left hand side of our inequality involves integers only. □

(6.2.7) Proposition:

(1) C̃(Q)^{∨} = \{[c, \underline{η}] ∈ (Q^{∞})^{∨} × V^{∗} ⊆ A A^{∗} × V^{∗} | \underline{η} − \underline{η}(c) ∈ C(Q)^{∨}\}

(2) In particular, [c, \underline{η}(c)] ∈ C̃(Q)^{∨}; it is the only preimage of [c, η_{0}(c)] ∈ σ^{∨} via the surjection i^{∨} : C̃(Q)^{∨} ⟶ ⟶ σ^{∨}. Moreover, for c ∈ IL^{∗} ∩ (Q^{∞})^{∨}, it holds [c, \underline{η}^{∗}(c)] ∈ C̃(Q)^{∨} ∩ M̃. These elements are liftings of [c, η_{0}^{∗}(c)] ∈ σ^{∨} ∩ M - but, in general, they are not the only ones (even to C̃(Q)^{∨} ∩ M̃).

(3) [c^{1}, \underline{η}^{∗}(c^{1})],…, [c^{w}, \underline{η}^{∗}(c^{w})] and C(Q)^{∨} ∩V_{Z Z}^{∗} (embedded as [0, C(Q)^{∨}]) generate the semigroup Γ := \{[c, \underline{η}] ∈ (IL^{∗} ∩ (Q^{∞})^{∨}) × V_{Z Z}^{∗} | \underline{η} − \underline{η}^{∗}(c) ∈ C(Q)^{∨}\} ⊆ C̃(Q)^{∨} ∩ M̃. (C̃(Q)^{∨} ∩ M̃ is the saturation of that subsemigroup.)

Proof: We just say some words to the third part. First, the condition “\underline{η} − \underline{η}^{∗}(c) ∈ C(Q)^{∨}” indeed describes a semigroup; this is a consequence of (iii) of Lemma (6.2.6). On the other hand, let [c, \underline{η}^{∗}(c)] be given. Using some representation [c, η_{0}^{∗}(c)] = ∑_{v=1}^{w} p_{v} [c^{v}, η_{0}^{∗}(c^{v})] (p_{v} ∈ IN), we obtain by the same lemma

∑_{v} p_{v} \underline{η}^{∗}(c^{v}) − \underline{η}^{∗}(c) = ∑_{v} p_{v} \underline{η}^{∗}(c^{v}) − \underline{η}^{∗}(∑_{v} p_{v} c^{v}) ∈ C(Q)^{∨} (or even ≥ 0).

Since, at the same time, the sum taken over all coordinates of that difference vanishes, the whole difference has to be zero. □

Example: We present an example showing that there might be liftings of [c, η_{0}^{∗}(c)] different from [c, \underline{η}^{∗}(c)] (i.e. Γ is a proper subsemigroup of C̃(Q)^{∨} ∩ M̃). Let σ ⊆ IR^{3} be generated from

a^{0} := (0, 0; 1) , a^{1} := (2, −1; 1) , a^{2} := (1, 1; 1) , a^{3} := (0, 1; 1) , and a := (7, 0; 4) .

We consider the degree R = [0, 0; 1]. Then, Q is a pentagon with four vertices a^{0},…, a^{3} ∈ IL and the non-lattice vertex ā = (7/4, 0) (if a^{0} ∈ Q coincides with the origin).

[figure: a^{0}; a^{3}; a^{2}; a^{1}; ā; t_{01}; t_{12}; t_{12}; t_{23}; t_{03}] The equations for V (Q) are
t_{01} · (2, −1) + t_{12} · [(−1/4, 1) + (−3/4, 1)] + t_{23} · (−1, 0) + t_{03} · (−1, 0) = (0, 0) ,

i.e. t_{23} = 2 t_{01} − t_{12} and t_{03} = 2 t_{12} − t_{01}. Hence, V (Q) = IR^{2} (with coordinates t_{01} and t_{12}), and the cone C(Q) ⊆ V (Q) is given by the inequalities t_{12} ≤ 2 t_{01} ≤ 4 t_{12}.

Now, let c := [−2, −1]. The vertex with minimal c-value is ā(c) = ā = (7/4, 0), hence η_{0}(c) = 7/2 and η_{0}^{∗}(c) = 4. Choosing the path from a^{0} to ā via a^{1}, we obtain \underline{η}(c) = 3 t_{01} + 1/2 t_{12} and \underline{η}^{∗}(c) = 3 t_{01} + t_{12}. On the other hand, \underline{η}′ := 4 t_{01} is another lifting of η_{0}^{∗}(c) (belonging to C̃(Q)^{∨} since \underline{η}′ − \underline{η}(c) = t_{01} − 1/2 t_{12} ∈ C(Q)^{∨}).

(6.2.8) The statements of (5.2.8) remain true, if the semigroup C̃(Q)^{∨} ∩ M̃ will be replaced by its (non-saturated) subgroup Γ mentioned in the last part of Proposition (6.2.7) (and X by X′ := Spec Γ, respectively). Then, X′ equals the scheme theoretical image of the map X → I C^{w} × S induced by the elements [c^{1}, \underline{η}^{∗}(c^{1})],…, [c^{w}, \underline{η}^{∗}(c^{w})] ∈ Γ. And, in return, X is the normalization of X′.

6.3 A flat family

(6.3.1) Theorem (5.3.1) remains valid with just one exception: X̄ ⊆ I C^{w} × I C^{N} is the scheme theoretical image of X as well as of X′. On the other hand, if we start with π̄ : X̄ → S̄, then base change via S → S̄ brings back X′ → S instead of X → S. The spaces M ⊆ V (Q) and M̄ ⊆ V (Q)/\underline{1} are still defined by the equations g_{ε,k}(t) (cf. section (2.2)) assigned to compact two-faces ε < Q. (However, as it was explained in (6.2.1), we are using a modified version of the vector space V (Q) through the whole chapter.)

(6.3.2) Replace X by X′.

(6.3.3) Replace η_{0} and \underline{η} by η_{0}^{∗} and \underline{η}^{∗}, respectively.

(6.3.4) Replacing A(X) by A(X′), we obtain an isomorphism A(X̄) ⊗_{A(S̄)} A(S) ⟶^{∼} A(X′), and these rings are contained in A(X).

(6.3.5) There are no additional changes necessary.

6.4 The Kodaira-Spencer map

(6.4.1) The condition for an element [c, η_{0}^{∗}(c)] to be contained in E_{j}^{R} looks different in the non-Gorenstein case:

E_{j}^{R} = {[c^{v}, η_{0}^{∗}(c^{v})] | ⟨ā^{j}, −c^{v}⟩ > η_{0}^{∗}(c^{v}) − 1} .

(In particular, [c, η_{0}^{∗}(c)] is always contained in E_{ā(c)}^{R}.) This leads to the following description of \underline{η}^{∗}(c) generalizing its definition from (6.2.6):

Lemma: Assume that [c, η_{0}^{∗}(c)] is contained in E_{j}^{R}. Then,

\underline{η}^{∗}(c) = ⟨ā_{•}^{j}, −c⟩ + (η_{0}^{∗}(c) + ⟨ā^{j}, c⟩) · e[ā^{j}] .

(ā_{•}^{j} denotes the map assigning \underline{t} ∈ V (Q) the vertex āj ; \underline{t} of the (generalized) Minkowski summand Q\underline{t}.)

Proof: If ā^{j} ∈ IL, then the condition ⟨ā^{j}, −c⟩ > η_{0}^{∗}(c) − 1 is equivalent to ⟨ā^{j}, −c⟩ = η_{0}(c) = η_{0}^{∗}(c). Hence, the second summand in our formula vanishes, and we are done. On the other hand, if ā^{j} ∉ IL, then there is not any lattice point contained in the strip ⟨ā^{j}, c⟩ ≥ ⟨•, c⟩ > ⟨ā(c), c⟩. In particular, every edge on the path from ā^{j} to ā(c) (decreasing the c-value at each step) belongs to the “component” induced by ā^{j} (cf. (6.2.5)). Now, our formula follows from the definition of \underline{η}^{∗}(c). □

Similarly to (5.4.1) we consider the bilinear map

\begin{aligned}Φ : & V^{Z Z}/(1,…, 1) & × & L(∪_{j}E_{j}^{R}) & ⟶ & ZZ \\ & \underline{t} & , & q & ↦ & ∑_{v,i} t_{i} q_{v} η_{i}^{∗}(c^{v}) , \end{aligned}

and we obtain Φ(\underline{t}, q) = 0 for q ∈ L(E_{j}^{R}) again. (Indeed,

Φ(\underline{t}, q) = ∑_{v} q_{v} ⟨\underline{t}, \underline{η}^{∗}(c^{v})⟩
= ∑_{v} q_{v} · (⟨āj ; \underline{t}, −c^{v}⟩ + (η_{0}^{∗}(c^{v}) − ⟨ā^{j}, −c^{v}⟩) · t_{}_{ā^{j}})
= ⟨āj ; \underline{t}, − ∑_{v} q_{v}c^{v}⟩ + (∑_{v} q_{v} η_{0}^{∗}(c^{v}) − ⟨ā^{j}, ∑_{v} q_{v}c^{v}⟩) · t_{}_{ā^{j}} = 0 .)

(6.4.2) Theorem (5.4.2) (stating that Φ, on the one hand, describes the Kodaira-Spencer map and, on the other hand, yields exactly the isomorphism built in section (3.4)) remains true in the non-Gorenstein case. Even the proof of the first part transfers completely (with the usual replacements). To prove the second statement we start as in (3.4.1): Take a q ∈ L(E_{0}^{R}), decompose it into q = ∑_{j} q^{j} (q^{j} ∈ ZZ^{E_{j}}) and consider v^{j} := − ∑_{v} q_{v}^{j} [c^{v}, η_{0}^{∗}(c^{v})]. Now,

Φ(\underline{t}, q) = ∑_{j} Φ(\underline{t}, q^{j}) = ∑_{j,v} q_{v}^{j} · ⟨\underline{t}, \underline{η}^{∗}(c^{v})⟩
= ∑_{j,v}⟨āj ; \underline{t}, −c^{v}⟩ + ∑_{j,v} q_{v}^{j} · [η_{0}^{∗}(c^{v}) + ⟨ā^{j}, c^{v}⟩] · t_{}_{ā^{j}}
= ∑_{j}⟨(āj ; \underline{t}, 0), v^{j}⟩ − ⟨(ā^{j}, 1), v^{j}⟩ · t_{}_{ā^{j}} .

Hence, in the sense of (3.4.1), our Φ(\underline{t}, •) ∈ L(E_{0}^{R})^{∗} corresponds to the elements b^{j} := (āj ; \underline{t}, 0) − (ā^{j}, 1) t_{}_{ā^{j}} ∈ N_{IR} defining some map ⊕_{j} span_{IR}E_{j}^{R} → IR. On the other hand, it is exactly those elements b^{j} bringing back the original \underline{t} ∈ V (Q) via the construction of (3.4.4): Recalling that ā^{j} is just an abbreviation for (ā^{j}, 1), we obtain b̄^{j} = (āj ; \underline{t}, 0) and s_{j} = t_{}_{ā^{j}}.

In particular, the Kodaira-Spencer map T_{M̄,0} → T_{Y}^{1} (−R) is always an isomorphism (even in the non-Gorenstein case).

6.5 The obstruction map

(6.5.1) According to the description of V^{⊥} in (6.2.5), the generators g_{\underline{d},k}(\underline{t} − t_{1}) (\underline{d} ∈ V^{⊥} ∩ ZZ^{N}, k ≥ 1) of J split into the following two types:

• g_{ε,k}(\underline{t} − t_{1}) for compact two-faces ε < Q and

• t_{i} − t_{j}, if the edges d^{i}, d^{j} contain a common non-lattice vertex of Q.

We can get rid of the latter type by the usual recipe: Do not consider coordinates t_{i} corresponding to single edges, but to entire “components”. This point of view is related to the following fact: t_{i} − t_{j} ∈ J_{1} ⊆ J̃, i.e. t_{i} − t_{j} vanishes in W = J /J̃ anyway.

Theorem (5.5.1) still holds - but the corollary now just states that we have found the versal deformation of degree −R.

(6.5.2) We use the definitions of \underline{η}^{c}(c) and \underline{η}^{c}(c^{v}) given in (5.5.2)(ii). Additionally, we need (similarly to (6.2.6)(2))

\underline{η}^{∗c}(c) := \underline{η}^{c}(c) + [η_{0}^{∗}(c) − η_{0}(c)] · e[ā(c)] and \underline{η}^{∗c}(c^{v}) := \underline{η}^{c}(c^{v}) + [η_{0}^{∗}(c^{v}) − η_{0}(c^{v})] · e[ā(c^{v})] .

The coefficients p_{v}^{c} ∈ IN introduced in (iii) are now defined via [c, η_{0}^{∗}(c)] = ∑_{v} p_{v}^{c} [c^{v}, η_{0}^{∗}(c^{v})] for c ∈ (Q^{∞})^{∨} ∩ IL^{∗}. Similarly, add stars to every η_{0} or \underline{η} occurring in the sequel.

(6.5.3) Add stars to the η’s.

(6.5.4) The lemma remains valid. In the corollary, it should be k_{0} = α+∑_{v}(a_{v} +r_{v}) η_{0}^{∗}(c^{v})− η^{∗}(c̃) now. On the other hand, it holds

\underline{d} = ∑_{v}(a_{v} − b_{v}) · (\underline{η̃}(c^{v}) − \underline{η}(c^{v})) = ∑_{v}(a_{v} − b_{v}) · (\underline{η̃}^{∗}(c^{v}) − \underline{η}^{∗}(c^{v})) ,

since the e[ā(c^{v})]-terms kill each other.

(6.5.5) For c ∈ IL^{∗} and paths ρ ∈ ZZ^{N} it is not in general true that \underline{d}(ρ, c) = [⟨ρ_{1} d^{1}, c⟩,…, ⟨ρ_{N} d^{N}, c⟩] ∈ IR^{N} is contained in ZZ^{N}. However, if ρ governs the walk between two lattice vertices, and if \underline{d}(ρ, c) will be regarded modulo t_{i} − t_{j} (if d^{i}, d^{j} contain a common non-lattice vertex), then it is. (Proceed as in the proof of Lemma (6.2.6).) In particular, the property \underline{d}(ρ, c) ∈ ZZ^{N} still holds for closed paths.

Looking at Remark (5.5.5), we just take the sum in (1) over all compact two-faces ε < Q. The map remains surjective, since the generators g_{\underline{d},k} of type t_{i} − t_{j} vanish in W. Part (2) remains true for arbitrary vertices ā, b̄ ∈ Q, even for |⟨ā, c⟩ − ⟨b̄, c⟩| < k.

Now, we use the sets

E_{j}^{kR} = {[c^{v}, η_{0}^{∗}(c^{v})] | ⟨ā^{j}, c^{v}⟩ + η_{0}^{∗}(c^{v}) < k} ∪ {R} (k ≥ 2)

again. The maps ψ_{j}^{(k)} : L(E_{j}^{kR}) → W_{k} are defined as in the Gorenstein case. Even the lemma remains valid, i.e. we obtain a linear map ψ^{(k)} : T_{Y}^{2} (−kR) → W_{k}.

(6.5.6) Just make the usual changes. It may be helpful to keep in mind that, for instance, the statements ∑_{v} a_{v} η_{0}^{∗}(c^{v}) − η_{0}(c) + (α − k) ≥ 0 and ∑_{v} a_{v} η_{0}^{∗}(c^{v}) − η_{0}^{∗}(c) + (α − k) ≥ 0 are equivalent (since a_{v}, α, k ∈ IN).

(6.5.7) Finally, one shows surjectivity of ψ^{(k)} by ψ^{(k)} ∘ p^{k} = ∑_{ε<Q} g^{(k)}(\underline{ε}, •) just as in the Gorenstein case.

6.6 Regular functions as variables

Finally, we give a conjecture how the equations of the versal base space could look like, if all negative degrees are involved at the same time. The idea is to replace ordinary variables for the base space by variable Laurent series: A former variable in T^{1}(−R) occurs as the (variable) x^{R}- coefficient of some of these Laurent series. (Actually, since just negative degrees will be involved, our variables will already be contained in I C[σ^{∨} ∩ M], i.e. equal (variable) regular functions on Y.)

Definition-Proposition: Let σ = ⟨a^{1},…, a^{M}⟩. If D := ∑_{j} \overline{orb(a^{j})} denotes the negative of the canonical divisor on Y, then the map

\begin{matrix}N & ⟶ & θ_{Y} (−log D) \\ a & ↦ & ∂_{a} := [x^{r} ↦ ⟨a, r⟩ · x^{r}] \end{matrix}

induces an isomorphism N ⊗_{Z Z} O_{Y} →^{∼} θ_{Y} (−log D).

Now, we introduce for every two-dimensional face ⟨a^{j}, a^{k}⟩ < σ some variable f_{jk}(\underline{x}) ∈ I C[σ^{∨} ∩ M].

Conjecture: The equations for the negative part of the versal base space of Y are

• [∂_{a^{k}} − id](f_{jk} − f_{kl}) = 0, whenever ⟨a^{j}, a^{k}⟩, ⟨a^{k}, a^{l}⟩ < σ are two-dimensional faces, and

• ∑_{v=1}^{k} f_{a^{j_{v−1}} ,a^{jv}}^{q} (\underline{x})·[a^{j_{v−1}}−a^{j_{v}}] = 0 (q ≥ 1) for every three-dimensional face ε = ⟨a^{j_{1}},…, a^{j_{k}}⟩ < σ (with j_{0} := j_{k}).

(In the previous sections we have proved, that these are indeed the right equations, if restricted to each degree −R (R ∈ σ^{∨} ∩ M) separately.)

Toric deformations

7.1 Toric deformations

(7.1.1) From chapter 5 we know that (at least for isolated toric Gorenstein singularities) the total spaces over irreducible components of the versal base space are toric again. Hence, it pays to turn the whole situation upside down: Even if Y is not smooth in codimension two, we ask for deformations of Y with toric total space.

Definition: A deformation of Y, i.e. a flat map f : X → S with an isomorphism Y →^{∼} f^{−1}(0) (0 ∈ S), is said to be toric, if

(i) X is an affine, toric variety,

(ii) i : Y →^{∼} f^{−1}(0) ↪ X is a morphism in the category of toric varieties, i.e. it induces an algebraic group homomorphism T_{Y} ↪ T_{X} between the embedded tori which makes i equivariant, and

(iii) i sends the closed T_{Y} -orbit in Y isomorphically onto the closed T_{X}-orbit in X.

Remark: The rather technical condition (iii) could be replaced by the weaker (and perhaps more natural) one that asks for mapping the closed T_{Y} -orbit in Y into the closed T_{X}-orbit in X. This would not essentially change the notion of a toric deformation, but the actual version of (iii) makes the theory more convenient. In most applications the closed orbits under the torus action are points, anyway. Since we are actually interested in the germs of Y, X, and S only, the condition (iii) arises quite naturally then.

Example: Both families Y_{t} → I C^{3} and Y_{s} → I C of (1.1.3) are toric deformations of the cone over the rational normal curve of degree four.

(7.1.2) For X and Y as before we introduce the following notation:

• Let M̃, Ñ be mutually dual lattices; let σ̃ be a rational polyhedral cone in Ñ_{IR} := Ñ⊗_{Z Z}IR that does not contain any linear subspace. These objects are used to build X = Spec I C[σ̃^{∨} ∩ M̃].

• Analogously, we use M, N, and σ to obtain Y = Spec I C[σ^{∨} ∩ M].

Let Y and f : X → S meet the conditions (i) and (ii) of Definition (7.1.1). Then, the equivariant closed embedding i : Y ↪ X corresponds to an embedding

i : N ↪ Ñ

on the level of lattices. The dual map i^{∗} induces a surjection

i^{∗} : σ̃^{∨} ∩ M̃ ⟶ ⟶ σ^{∨} ∩ M

even between the semigroups. The kernel L := ker i^{∗} ⊆ M̃ is a sublattice (say of dimension n), and we obtain

N = Ñ ∩ L^{⊥} ; σ = σ̃ ∩ L^{⊥} ⊆ N_{IR} .

Lemma: The condition (iii) of Definition (7.1.1) is equivalent to σ̃^{∨} ∩ L_{IR} = {0}.

Proof: The condition that i maps the closed orbit of Y into the closed orbit of X can be written as

i^{∗}(σ̃^{∨} \ σ̃^{⊥}) ⊆ σ^{∨} \ σ^{⊥}.

Moreover, to obtain equality of both orbits, the map i^{∗} has to induce a bijection σ̃^{⊥}∩M̃ ⟶^{∼} σ^{⊥}∩M, which is equivalent to the injectivity of i^{∗} on σ̃^{⊥}. Now, these two conditions can be translated into

σ̃^{∨} ∩ (L + σ̃^{⊥}) = σ̃^{⊥} (i.e. σ̃^{∨} ∩ L ⊆ σ̃^{⊥}) and σ̃^{⊥} ∩ L = {0},

which is equivalent to σ̃^{∨} ∩ L = {0}. □

(7.1.3) Proposition: Let f : X → S be a toric deformation of Y. Then, the germ (S, 0) is smooth, and the ideal I := ker ( I C[σ̃^{∨} ∩ M̃] → I C[σ^{∨} ∩ M]) defining Y ⊆ X can be generated by n

binomials x^{u^{1}} − x^{v^{1}},…, x^{u^{n}} − x^{v^{n}} ∈ I C[σ̃^{∨} ∩ M̃] (u^{i}, v^{i} ∈ σ̃^{∨} ∩ M̃; u^{i} − v^{i} ∈ L). In particular, they form a binomial regular sequence, and Y is a relative complete intersection in X.

Proof: Step 1: A deformation diagram

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

induces a deformation of the corresponding torus T_{Y}:

T_{Y} ↪ T_{X}
↓ ⊗ ↓ f
{0} ↪ S .

T_{Y} and T_{X} are smooth. Hence, T_{X} splits locally into a product T_{X} ≅ T_{Y} × S, and the germ (S, 0) has to be smooth, too.

Step 2: On the level of local rings (at the general points of the closed orbits and at the special point of S, respectively) we obtain the following diagram:

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 n elements g_{1},…, g_{n} ((S, 0) is smooth), and by the Nakayama lemma we can choose these generators among the elements of the form x^{u} − x^{v} (u, v ∈ σ̃^{∨} ∩ M̃; u − v ∈ L).

Step 3: Let Ĩ := (g_{1},…, g_{n}) ⊆ I C[σ̃^{∨} ∩ M̃]. Then, Ĩ ⊆ I are ideals in I C[σ̃^{∨} ∩ M̃] meeting the following conditions:

(i) Ĩ and I are homogeneous with respect to the M-grading;

(ii) Ĩ = I in the local ring O_{X,0}.

We want to show Ĩ = I. For that purpose, let g ∈ I be an arbitrary M-homogeneous element. By (ii) there exists an h ∈ I C[σ̃^{∨} ∩ M̃] with h · g ∈ Ĩ and

h ∉ m_{0} := ⨁_{{u∈σ̃^{∨}∩M̃ \atop u∉σ̃^{⊥}}} I C · x^{u}

(i.e. h contains a term of I C[σ̃^{⊥} ∩ M̃]); by (i) we can additionally assume h to be M-homogeneous. Since σ̃^{∨} ∩ L_{IR} = {0}, this implies that h is a monomial of I C[σ̃^{⊥} ∩ M̃], i.e. h is invertible. □

Remark: The n vectors u^{1} − v^{1},…, u^{n} − v^{n} built from the exponents of the binomial regular sequence generating I are free generators of the sublattice L ⊆ M̃.

Proof: Let L′ := span_{Z Z}(u^{1} − v^{1},…, u^{n} − v^{n}) ⊆ L. In particular, the ideal I is homogeneous under the M̃/L′-grading of I C[σ̃^{∨} ∩ M̃]. Now, for each l ∈ L there are u, v ∈ σ̃^{∨} ∩ M̃ such that l = u − v. The monomials x^{u} and x^{v} map onto equal functions in Γ(Y, O_{Y}), hence x^{u} − x^{v} ∈ I. Since I does not contain monomials at all, x^{u} − x^{v} has to be M̃/L′-homogeneous itself, i.e. u − v ∈ L′. □

Definition: Those binomial regular sequences defining a relative complete intersection between affine toric varieties satisfying (7.1.1)(ii) and (iii) (as in the previous proposition) are said to be toric regular sequences. (They give rise to a so-called format of Y; cf. [Rö].)

(7.1.4) Christophersen [Ch 2] developed the notion of relative deformations.

Definition: If Y ↪ X is a relative complete intersection (i.e. given by a regular sequence in X), then a relative deformation (of Y in X) over S is given by a commutative diagram

Ỹ ↪ X × S
↓ f  ↓ pr_{S}
S =  S

(f is flat) and an isomorphism Y →^{∼} f^{−1}(0) compatible with the embeddings into X. Two relative deformations are said to be equivalent, if they are so as abstract deformations (without regarding the embeddings).

Relative deformations can be obtained by arbitrary perturbations of the regular sequence. In particular, they form a smooth subfunctor Def_{Y ↪X} of the deformation functor Def_{Y}. If Y admits a versal deformation with base space S_{Y}, then the versal relative deformation of Y ↪ X equals the subfamily over some smooth subscheme S_{Y ↪X} ⊆ S_{Y}. Christophersen has worked out that S_{Y ↪X} should be a good candidate for components of the reduced base space (S_{Y})_{red}.

(7.1.5) Let f : X → S be a toric deformation of Y. Then, the diagram

X ↪^{(id,f)} X × S
↓ f  ↓ pr_{S}
S =  S

and Proposition (7.1.3) show that f : X → S is a relative deformation for the embedding Y ↪ X given by the toric regular sequence g := (x^{u^{1}} − x^{v^{1}},…, x^{u^{n}} − x^{v^{n}}). In particular, it is induced by some morphism S → S_{Y ↪X}. If, moreover, f equals the restriction of the versal family of Y to some component S of (S_{Y})_{red}, then S and S_{Y ↪X} are equal as germs. On the other hand, the regular sequence g itself provides a special toric deformation by regarding the (flat) map g : X → I C^{n}. f and g need not be equivalent in general. However, similar to the case of f, the deformation g is induced by some map I C^{n} → S_{Y ↪X}, i.e. S and I C^{n} map at least into the same component of (S_{Y})_{red}. If, moreover, S = S_{Y ↪X} (for instance, if f is a component of the versal deformation), then I C^{n} → S_{Y ↪X} = S is an isomorphism, i.e. f = g.

7.2 Constructing homogeneous, toric regular sequences

(7.2.1) The previous section provides motivation for looking for those pairs (Y, X) of affine toric varieties such that Y is a relative complete intersection in X, given by a toric regular sequence g. One possibility for doing so is fixing the “big” space X and searching for those binomial regular sequences yielding a toric variety as zero set. This was the principal approach in [Al 2]. However, starting with Y and looking for toric X to map into, is a completely different story. In this section we will solve this problem, if the sequences are additionally assumed to be homogeneous:

Definition: If g = (x^{u^{1}} −x^{v^{1}},…, x^{u^{n}} −x^{v^{n}}) is a toric regular sequence defining Y ↪ X, then the common images ū^{i} ∈ M of u^{i}, v^{i} ∈ M̃ are called the degrees of g. The sequence g (and its associated toric deformation) is said to be homogeneous of degree R, if R = ū^{1} = … = ū^{n}. (Obviously, if m = 1, then this will always be the case.)

Lemma: Up to ZZ-linear transformations, a homogeneous, toric regular sequence g has the shape g = (x^{u^{1}} − x^{u^{0}},…, x^{u^{n}} − x^{u^{0}}), i.e. u^{0} = v^{1} = … = v^{n}.

Proof: Choose an arbitrary u^{0} ∈ {u^{1},…, u^{n}, v^{1},…, v^{n}}. Then, since u^{i} − u^{0}, v^{i} − u^{0} ∈ L, the 2n binomials x^{u^{i}} − x^{u^{0}}, x^{v^{i}} − x^{u^{0}} (i = 1,…, n) are contained in I. Moreover, since x^{u^{i}} − x^{v^{i}} = (x^{u^{i}} − x^{u^{0}}) − (x^{v^{i}} − x^{u^{0}}), they generate this ideal, and we can choose n among them still doing so (cf. the proof of Proposition (7.1.3)). It remains to show that this changing of generators could be done using a ZZ-linear transformation only. We regard the following, more general situation: Let x^{u} − x^{v} be contained in an ideal I generated by binomials x^{u^{1}} − x^{v^{1}},…, x^{u^{n}} − x^{v^{n}}, and assume that all exponents u, v, u^{i}, v^{i} ∈ M̃ map onto a single R ∈ M. Using the natural M-grading, all these binomials are homogeneous of degree R. Hence, representing x^{u} − x^{v} as a I C[σ̃^{∨} ∩ M̃]-linear combination of the generators of I can be done by using homogeneous coefficients of degree 0 ∈ M only. Since σ̃^{∨} ∩ L = {0}, they have to be constants. Moreover, it is obvious that they can be taken even from ZZ then. □

Remark: If g = (x^{u^{1}} − x^{u^{0}},…, x^{u^{n}} − x^{u^{0}}), then

L = ∑_{i=1}^{n} ZZ · (u^{i} − u^{0}) = ∑_{i,j=0}^{n} ZZ · (u^{i} − u^{j}) = ker (deg : ⨁_{i=0}^{n} ZZ · u^{i} ⟶ ⟶ ZZ)

(deg(u^{i}) := 1). The elements u^{0},…, u^{n} are linearly independent in M̃_{IR}.

(7.2.2) Now, we approach our main issue. We will produce homogeneous toric regular sequences (yielding pairs (Y, X)) from so-called deformation data.

Definition: Let (A A, IL) be a pair of a real vector space A A and a lattice IL ⊆ A A. A deformation datum of size n is a tuple (R_{0},…, R_{n}; C; p) admitting the following properties:

(i) C ⊆ A A is a rational polyhedral cone with apex in 0 ∈ A A, and p ≥ 1 is a natural number.

(ii) R_{0},…, R_{n} ⊆ A A are rational polyhedra with C as their common cone of unbounded directions, i.e. R_{i} = R̄_{i} + C for suitable compact polytopes R̄_{i}, i = 0,…, n. (It is possible to choose these polytopes in a canonical way by taking the convex hull of the vertices of R_{i}.)

(7.2.3) For a given polyhedron P ⊆ A A and a non-trivial linear form t ∈ (P^{∞})^{∨} ⊆ A A^{∗} (i.e. t ≠ 0, t bounded below on P) we denote by F (P, t) the face of P that is defined by t being minimal on it.

Definition: A deformation datum (R_{0},…, R_{n}; C; p) is said to be admissible, if it meets the following conditions:

Case 1: p = 1. For each t ∈ C^{∨} ⊆ A A^{∗} at least n of the n + 1 faces F (R_{i}, t) of R_{i} (i = 0,…, n) contain lattice points.

Case 2: p ≥ 2. R_{1},…, R_{n} are lattice polyhedra, i.e. they admit only lattice points as vertices.

The following alternative description will be useful:

Lemma: A deformation datum (R_{0},…, R_{n}; C; p) is admissible, if and only if for each t ∈ IL^{∗}∩C^{∨} the values of t on at least n of the n + 1 faces F (R_{i}, t) of R_{i} (i = 0,…, n) (or exactly on the faces F (R_{1}, t),…, F (R_{n}, t), if p ≥ 2) are integers.

Proof: Linear forms t ∈ IL^{∗} ∩ C^{∨} yield integers as values on lattice points. Hence, admissible deformation data always admit the property described in the lemma. To obtain the opposite implication, we proceed in three steps:

Step 1: Perturbing the linear form t ∈ C^{∨} slightly (inside the cone C^{∨}), the corresponding faces F (R_{i}, t) of R_{i} will at most be replaced by smaller ones. Hence, it will be sufficient to regard only those t such that each F (R_{i}, t) is a vertex of R_{i}. Moreover, if convenient, it will be possible to replace t by suitable linear forms t′ ∈ C^{∨} close to t again; they provide the same vertex as t.

Step 2: Claim: Let b^{0}, b^{1} ∈ A A \ IL, and let t ∈ C^{∨} \ {0}. Then, there exists a linear form t′ ∈ (IL^{∗} ∩ C^{∨}) \ {0} such that

• t′/∥t′∥ and t/∥t∥ are arbitrarily close to each other, and

• ⟨b^{0}, t′⟩, ⟨b^{1}, t′⟩ ∉ ZZ.

Proof: First we try to meet the latter condition. Choose t^{0}, t^{1} ∈ IL^{∗} having no integer value on b^{0}, b^{1}, respectively. If there exists a t^{j} among them such that both ⟨b^{0}, t^{j}⟩, ⟨b^{1}, t^{j}⟩ ∉ ZZ, then take t′ := t^{j}. Otherwise, we know that ⟨b^{0}, t^{1}⟩, ⟨b^{1}, t^{0}⟩ ∈ ZZ and ⟨b^{0}, t^{0}⟩, ⟨b^{1}, t^{1}⟩ ∉ ZZ. Hence, t′ := t^{1} +t^{2} has the desired property. Now, we have to improve our linear form t′ to obtain the additional property t′ ∈ C^{∨}. If s ∈ IL^{∗} ∩ (int C^{∨}), and if N ∈ IN \ {0} such that N · b^{0}, N · b^{1} are contained in the lattice IL, then ⟨b^{0}, N · s⟩, ⟨b^{1}, N · s⟩ ∈ ZZ. Each element of IL^{∗} can be put into the cone C^{∨} by adding a sufficiently large multiple of s. Hence, we substitute t′ := t′ + (kN) · s (with k ≫ 0). Finally, we can add sufficiently large multiples of t (which are contained in the lattice and yield integers as values on b^{0}, b^{1}) to t′. This operation ensures that the directions t′/∥t′∥ and t/∥t∥ are arbitrarily close to each other.

Step 3: If a deformation datum (R_{0},…, R_{n}; C; p) is not admissible, then there exists an element t ∈ C^{∨} such that two of the faces F (R_{i}, t) (i = 0,…, n) equal vertices (denoted by b^{0}, b^{1}) that are not contained in the lattice (Case p ≥ 2: one of the faces F (R_{i}, t) (i = 1,…, n), denoted by b^{0} = b^{1}). Let t′ be a linear form as constructed in the second step; suppose that t and t′ define the same vertices of R_{0},…, R_{n} (including b^{0} and b^{1}). Then, t′ violates the conditions of the lemma. □

(7.2.4) To a given deformation datum (R_{0},…, R_{n}; C; p) we associate the following objects:

(7.2.4.1) Define the polyhedron Q to be the Minkowski sum

Q := R_{0} + … + R_{n} = C + (R̄_{0} + … + R̄_{n}) ⊆ A A.

We embed the whole space as an affine hyperplane into a higher-dimensional space:

• N_{IR} := A A × IR is a vector space containing the lattice N := IL × ZZ (M_{IR} := N_{IR}^{∗}, M := N^{∗});

• ψ_{1} : A A ↪ N_{IR} ; a ↦ (a, p^{−1}).

In particular, ψ_{1}(Q) is a polyhedron in N_{IR}. Denoting by

ψ : A A ↪ N_{IR} ; a ↦ (a, 0)

the associated linear embedding, we can define Y := Spec I C[σ^{∨} ∩ M] as the affine toric variety that is given by the cone

σ := \overline{IR_{≥0} · ψ_{1}(Q)} = ψ(C) ∪ IR_{≥0} · ψ_{1}(Q) .

(7.2.4.2) To define σ̃ and X, we put the polyhedra R_{0},…, R_{n} into parallel affine planes of a vector space that is large enough.

• Ñ_{IR} := A A × IR^{n+1}, Ñ := IL × ZZ^{n+1}; M̃_{IR} := Ñ_{IR}^{∗}, M̃ := Ñ^{∗}. Denote by Φ : Ñ_{IR} ⟶ IR^{n+1} the projection onto the second factor.

• ϕ_{i} : A A ↪ Ñ_{IR}; a ↦ \{ \begin{matrix}(a, p^{−1}e^{0}) & for i = 0 \\ (a, e^{i}) & for i = 1,…, n \end{matrix}

(e^{0},…, e^{n} denotes the standard basis of ZZ^{n+1}).

On the homogeneous level, the affine maps ϕ_{0},…, ϕ_{n} correspond to the trivial embedding ϕ : A A ↪ Ñ_{IR} (a ↦ (a, 0)).

Now, we denote by P the convex hull

P := conv (⋃_{i=0}^{n} ϕ_{i}(R_{i})) = ϕ(C) + conv (⋃_{i=0}^{n} ϕ_{i}(R̄_{i})) ⊆ Ñ_{IR}

and define X := Spec I C[σ̃^{∨} ∩ M̃] as the affine toric variety given by the cone

σ̃ := \overline{IR_{≥0} · P} = ϕ(C) ∪ IR_{≥0} · P.

(7.2.4.3) If pr_{i} : IR^{n+1} → IR denotes the projection onto the i-th factor, we can define linear maps u^{0},…, u^{n} : Ñ → ZZ by

u^{i} := \{ \begin{matrix}p · (pr_{0} ∘ Φ) & for i = 0 \\ pr_{i} ∘ Φ & for i = 1,…, n. \end{matrix}

By construction, these maps correspond to elements u^{i} ∈ σ̃^{∨} ∩ M̃.

(7.2.4.4) N can be considered a sublattice of Ñ via the inclusion map

N ↪ Ñ , (a; 1) ↦ (a; 1, p,…, p).

This embedding admits the following properties:

(i) N = Ñ ∩ ⋂_{i,j}(u^{i} − u^{j})^{⊥} = Ñ ∩ ⋂_{i=1}^{n}(u^{i} − u^{0})^{⊥}

(ii) σ = σ̃ ∩ N_{IR}.

In particular, we obtain a map Y → X which sends Y into the special fiber of the morphism X → I C^{n} defined by the regular functions x^{u^{1}} − x^{u^{0}},…, x^{u^{n}} − x^{u^{0}} ∈ I C[σ̃^{∨} ∩ M̃].

(7.2.5) Theorem: Starting with an admissible deformation datum (R_{0},…, R_{n}; C; p), the previous construction provides a pair (Y, X) of affine toric varieties such that Y ⊆ X is given by a homogeneous toric regular sequence x^{u^{1}} − x^{u^{0}},…, x^{u^{n}} − x^{u^{0}}. (Looking at (7.2.4.4), Y equals the special fiber of X → I C^{n}.) Moreover, all those pairs (Y, X) arise that way.

The proof is contained in section 7.3.

Remark: Up to isomorphisms, the construction (7.2.4) will yield the same result, if the polyhedra R_{i} from the deformation datum (R_{0},…, R_{n}; C; p) are shifted via vectors from IL (for i ≥ 1) or even from p^{−1}IL (for i = 0).

(7.2.6) Eventually, let us switch to the natural viewpoint that an affine toric variety Y = Spec I C[σ^{∨} ∩ M] is given, and we are asking for its (homogeneous) toric deformations. Fixing some degree R ∈ σ^{∨} ∩ M corresponds to the choice of an affine cross cut Q of the cone σ ⊆ N_{IR}. Then, the previous theorem tells us that homogeneous toric deformations of Y arise from certain subdivisions of Q into a Minkowski sum. More precisely, we have to proceed as follows:

(i) Define the vector space A A_{0} := {a ∈ N_{IR}| ⟨a, R⟩ = 0}, which contains the lattice IL_{0} := A A_{0} ∩N.

(ii) Let p be the greatest common divisor of the coordinates of R, i.e. p^{−1}R is a primitive element of M.

(iii) Define the affine space A A := {a ∈ N_{IR}| ⟨a, R⟩ = 1}. Fixing some point 0 ∈ A A ∩ p^{−1}N, we obtain a sublattice via IL := 0 + IL_{0}. (In case p = 1, IL equals A A ∩ N.) Moreover, we can use the point 0 to identify (A A, IL) with the pair (A A_{0}, IL_{0}) providing a linear structure which was assumed in (7.2.2).

(iv) Let C := σ ∩ A A_{0} and Q := σ ∩ A A. Then, by Theorem (7.2.5), homogeneous regular sequences of degree R (in some larger affine toric variety X) correspond to admissible splittings of Q into a Minkowski sum Q = R_{0} + … + R_{n}.

Remark: If Y_{σ} is smooth in codimension two, then (according to Remark (3.4.3)) we may assume that p = 1. This case has been discussed in chapter 6, and we should convince ourselves that Theorem (7.2.5) fits into the results obtained there:

Of course, splitting Q into Q = R_{0} + … + R_{n} corresponds to a decomposition of \underline{1} ∈ C(Q) ⊆ V (Q) into a sum \underline{1} = \underline{t}^{0} +…+\underline{t}^{n} (inside C(Q)). However, the crucial point is the admissibility condition; it is now equivalent to the fact that the summands \underline{t}^{i} (i = 0,…, n) belong to the lattice of the smaller vector space V′(Q) ⊆ V (Q) defined in (6.2.1). Hence, using an adapted version of (5.6.1) to describe the reduced structure of the versal base space M̄ in degree −R (as constructed in chapter 6), we obtain:

The irreducible components of M̄_{red} indeed correspond to the maximal, admissible decompositions of Q into Minkowski summands.

7.3 Proof of the previous theorem

(7.3.1) Let an admissible deformation datum (R_{0},…, R_{n}; C; p) be given. We have to show that the associated data σ̃, σ and u^{0},…, u^{n} (cf. (7.2.4)) provide indeed a toric regular sequence.

(7.3.1.1) σ̃^{∨} ∩ M̃ ⟶ ⟶ σ^{∨} ∩ M is surjective: Let s ∈ M̃ such that

⟨(b^{0} + … + b^{n}; p^{−1}, 1,…, 1), s⟩ ≥ 0 \begin{matrix}for b^{i} ∈ R_{i}, i = 0,…, n \\ (i.e. b^{0} + … + b^{n} ∈ Q) ; \end{matrix}

it means that s maps onto an element s̄ ∈ σ^{∨} ∩M. We have to show that s̄ can be lifted to σ̃^{∨} ∩M̃. Projecting s to the A A^{∗}−component, we obtain an element t = s|_{A A}∈ IL^{∗} ∩C^{∨}. Since our deformation element is admissible, we may assume that, on the faces F (R_{1}, t),…, F (R_{n}, t), the linear form t provides integers only. Hence, even on the embedded polyhedra ϕ_{1}(R_{1}),…, ϕ_{n}(R_{n}) ⊆ Ñ_{IR}, the minimal value of s is contained in ZZ. Denote these values by k_{i} and suppose that they occur at points (b̃^{i}, e^{i}) ∈ ϕ_{i}(R_{i}) ⊆ Φ^{−1}(e^{i}) ⊆ Ñ_{IR} (i = 1,…, n). Modifying s by s := s − ∑_{i=1}^{n} k_{i} · u^{i} + (∑_{i=1}^{n} k_{i}) · u^{0}, we can assume that s is non-negative on ϕ_{1}(R_{1}),…, ϕ_{n}(R_{n}) ⊆ Ñ_{IR} and, moreover, ⟨(b̃^{i}, e^{i}), s⟩ = 0 for i = 1,…, n. Now, if b^{0} ∈ R_{0} (embedded as (b^{0}, p^{−1}e^{0}) ∈ ϕ_{0}(R_{0}) ⊆ Ñ_{IR}) is given, we obtain

⟨(b^{0}, p^{−1}e^{0}) , s⟩ = ⟨(b^{0} + ∑_{i=1}^{n} b̃^{i}, p^{−1}e^{0} + ∑_{i=1}^{n} e^{i}) , s⟩ ≥ 0.

Hence, s is contained in σ̃^{∨}.

(7.3.1.2) σ̃^{∨} ∩ L_{IR} = {0} (L_{IR} := ker(M̃_{IR} ⟶ ⟶ M_{IR})) : The cone σ̃ contains points from Φ^{−1}(e^{0}),…, Φ^{−1}(e^{n}). In particular, a non-trivial element of L_{IR} = ker (deg : ⨁_{i=0}^{n} IR · u^{i} ⟶ ⟶ IR) cannot be contained in σ̃^{∨}.

(7.3.1.3) We show that I = ker ( I C[σ̃^{∨} ∩ M̃] ⟶ ⟶ I C[σ^{∨} ∩ M]) is generated by x^{u^{1}} − x^{u^{0}},…,

x^{u^{n}} − x^{u^{0}}: Let x^{u} − x^{v} ∈ I be any binomial element of I, i.e. u, v ∈ σ̃^{∨} ∩ M̃ , u − v ∈ L. Hence,

u − v = ∑_{i=0}^{n} g_{i} · u^{i} (g_{i} ∈ ZZ, ∑_{i} g_{i} = 0) .

Defining

s := u − ∑_{i} g_{i}^{+} · u^{i} = v − ∑_{i} g_{i}^{−} · u^{i} ∈ M̃ (g_{i}^{+}, g_{i}^{−} ∈ IN, g_{i}^{+} − g_{i}^{−} = g_{i}, g_{i}^{+}g_{i}^{−} = 0) ,

this linear form will equal u or v, if it is restricted to the polyhedra ϕ_{i}(R_{i}) with g_{i} ≤ 0 or g_{i} ≥ 0, respectively. In particular, s is non-negative at P, i.e. s ∈ σ̃^{∨} ∩ M̃. Now, expressing x^{u} − x^{v} as a linear combination of the generators x^{u^{i}} − x^{u^{0}} is straightforward.

(7.3.2) It remains to prove that all homogeneous, toric regular sequences can be obtained from deformation data. Hence, in the remainder of section (7.3) we assume that we are given such a sequence x^{u^{1}} − x^{u^{0}},…, x^{u^{n}} − x^{u^{0}} defining a relative complete intersection Y ↪ X as in (7.1.1) and (7.1.2).

Lemma: Let u, v ∈ σ̃^{∨} ∩ M̃ such that u − v ∈ L \ {0}. Then, there are two different indices i, j ∈ {0,…, n} satisfying u − u^{i}, v − u^{j} ∈ σ̃^{∨}.

Proof: Since x^{u} − x^{v} ∈ I, there must be some equation

x^{u} − x^{v} = ∑_{μ} c_{μ} x^{t^{μ}} (x^{u^{i(μ)}} − x^{u^{0}}) (c_{μ} ∈ I C; t^{μ} ∈ σ̃^{∨} ∩ M̃).

In particular, both exponents u and v have to occur somewhere on the right hand side providing the existence of the desired u^{i}, u^{j}. Moreover, if i = j, then we could apply that procedure to u′ := u − u^{i} and v′ := v − u^{i} again. This recursion eventually stops. □

(7.3.3) Lemma: Denote by R ∈ M = M̃/L the common image of the elements u^{0},…, u^{n} via the surjection M̃ ⟶ ⟶ M. We obtain:

(1) R is not trivial on σ ⊆ N_{IR}.

(2) Let R = p · R′ (p ∈ IN, R′ ∈ M primitive). Then, without loss of generality, u^{0} is also divisible by p and can be written as u^{0} = p · ũ^{0}. Moreover, the elements ũ^{0}, u^{1},…, u^{n} equal a part of a ZZ-basis of the lattice M̃.

Proof: Step 1: If R were trivial on σ, not only R but also −R would belong to σ^{∨}. In particular, there would be an r ∈ σ̃^{∨} (lifting −R) such that

r + u^{i} ∈ σ̃^{∨} ∩ L = {0}, i.e. − u^{i} ∈ σ̃^{∨} for i = 0,…, n.

Hence, the linearly independent vectors u^{0} − u^{i} would have to be contained in σ̃^{∨} ∩ L = {0}, which is impossible.

Step 2: Since σ̃^{∨} ∩ M̃ ⟶ ⟶ σ^{∨} ∩ M is surjective, the element R′ ∈ σ^{∨} ∩ M can be lifted to some r ∈ σ̃^{∨} ∩ M̃. Then, p · r differs from u^{0},…, u^{n} by IL-elements only, and we can apply the previous lemma. There has to be an index i ∈ {0,…, n} such that p · r − u^{i} is contained in σ̃^{∨}. On the other hand, p·r −u^{i} is obviously contained in the lattice L, and we obtain p·r −u^{i} ∈ σ̃^{∨} ∩L = {0}. We may assume that i = 0.

Step 3: We will show that there are lattice elements b^{i} ∈ Ñ for i = 0,…, n such that

⟨b^{i}, u^{i}⟩ = \{ \begin{matrix}p & for i = 0 \\ 1 & for i = 1,…, n \end{matrix}  and
⟨b^{i}, u^{j}⟩ = 0 for j ≠ i.

Since, for a fixed i, (u^{i} − u^{j})_{j∈{0,...,n}\{i}} is a basis of the lattice L ⊆ M̃, it is possible to choose an element b^{i} ∈ Ñ such that ⟨b^{i}, u^{i} − u^{j}⟩ = 1, i.e.

⟨b^{i}, u^{i}⟩ = ⟨b^{i}, u^{j}⟩ + 1 (for j ≠ i).

R′ ∈ M is primitive, hence, there is a b ∈ N = L^{⊥} ∩ Ñ such that ⟨b, R′⟩ = 1. Therefore,

⟨b, u^{j}⟩ = p for j = 0,…, n.

Case 1: i ≥ 1. The equation ⟨b^{i}, u^{i}⟩ = ⟨b^{i}, u^{j}⟩ + 1 (j ≠ i) implies

⟨b^{i}, u^{j}⟩ + 1 = ⟨b^{i}, u^{i}⟩ = ⟨b^{i}, u^{0}⟩ + 1 = ⟨b^{i}, p · ũ^{0}⟩ + 1 ,

in particular, p|⟨b^{i}, u^{j}⟩ for j ≠ i. Hence, there is a k ∈ ZZ such that the improved b^{i} := b^{i} + k · b additionally yields ⟨b^{i}, u^{j}⟩ = 0 (j ≠ i).

Case 2: i = 0. Here, we have to use the modification b^{0} := p · b^{0} + k · b (with suitable k ∈ ZZ) to obtain the equations ⟨b^{0}, u^{j}⟩ = 0 (j ≥ 1). □

(7.3.4) Lemma: Let u ∈ M̃ be such that, for each index i = 0,…, n, it can be pushed into σ̃^{∨} without using u^{i}:

u + ∑_{j≠i} λ_{j}^{i} · u^{j} ∈ σ̃^{∨} (for some λ_{j}^{i} ∈ ZZ).

Then, u itself is contained in σ̃^{∨}.

Proof: We will proceed by induction on ∑_{j} λ_{j}^{i}. To do so we first have to modify the presumption of the lemma slightly: For u ∈ M̃ suppose that

u + ∑_{j} λ_{j}^{i} · u^{j} ∈ σ̃^{∨} with λ_{j}^{i} ∈ ZZ, λ_{i}^{i} ≤ 0, and ∑_{j} λ_{j}^{i} is constant in i.

(The latter fact can be obtained by increasing some of the coefficients λ_{j}^{i} with j ≠ i.) Now, on the one hand, the sum ∑_{j} λ_{j}^{i} is bounded below. (Look at the vector space M_{IR}: Subtracting R sufficiently often from a given point leads out of the cone σ^{∨}.) On the other hand, as soon as the elements u+∑_{j} λ_{j}^{i} ·u^{j} are not mutually equal (for different i), we can apply (7.3.2) to all these elements. We obtain u + ∑_{j}(λ_{j}^{i})′ · u^{j} ∈ σ̃^{∨} with (λ_{j}^{i})′ = λ_{j}^{i} or λ_{j}^{i} − 1 and ∑_{j=0}^{n}(λ_{j}^{i})′ = ∑_{j=0}^{n} λ_{j}^{i} − 1. Therefore, only the case that the coefficients λ_{j}^{i} do not depend on i is left. This implies λ_{j}^{i} = λ_{j}^{j} ≤ 0; hence, with u + ∑_{j} λ_{j}^{i} · u^{j}, u has to be contained in σ̃^{∨}, too. □

Corollary: We can replace the fact L ∩ σ̃^{∨} = {0} by an even stronger statement: If any linear combination ∑_{i=0}^{n} λ_{i} · u^{i} (with integer coefficients λ_{i} ∈ ZZ) is contained in σ̃^{∨} ∩ M̃, then each of the coefficients will be non-negative.

Proof: Assume that λ_{i} < 0. Then, the element u := λ_{i} · u^{i} admits the following properties:

u + ∑_{j≠i} λ_{j} · u^{j} ∈ σ̃^{∨} and u − λ_{i} · u^{i} ∈ σ̃^{∨}.

In particular, u fulfills the assumption of the previous lemma. We obtain −u^{i} ∈ σ̃^{∨}, but this cannot be true (cf. Step 1 in the proof of Lemma (7.3.3)). □

(7.3.5) Now, we start with the direct construction of the deformation datum that will induce the given toric regular sequence. Let

(A A, IL) := [(u^{0})^{⊥} ∩… ∩ (u^{n})^{⊥}, (u^{0})^{⊥} ∩… ∩ (u^{n})^{⊥} ∩ Ñ] and
C := (u^{0})^{⊥} ∩… ∩ (u^{n})^{⊥} ∩ σ̃ ⊆ A A.

Fixing some point in each of the sets

{a ∈ p^{−1}Ñ | ⟨a, u^{0}⟩ = 1, ⟨a, u^{j}⟩ = 0 for j ≠ 0} and
{a ∈ Ñ | ⟨a, u^{i}⟩ = 1, ⟨a, u^{j}⟩ = 0 for j ≠ i} (for i = 1,…, n),

we obtain n + 1 different affine embeddings A A ↪ Ñ_{IR} which induce isomorphisms

A A ⟶^{ϕ_{i}} {a ∈ Ñ_{IR} | ⟨a, u^{i}⟩ = 1, ⟨a, u^{j}⟩ = 0 for j ≠ i} (i ≥ 0)  and
IL ⟶^{ϕ_{i}} {a ∈ Ñ | ⟨a, u^{i}⟩ = 1, ⟨a, u^{j}⟩ = 0 for j ≠ i} (only for i ≥ 1 or p = 1).

Hence, with

R_{i} := ϕ_{i}^{−1} (σ̃ ∩ {a ∈ Ñ_{IR} | ⟨a, u^{i}⟩ = 1, ⟨a, u^{j}⟩ = 0 for j ≠ i})

we have got n + 1 polyhedra R_{0},…, R_{n} ⊆ A A admitting C as their cone of unbounded directions (i.e. R_{i} = C + compact set).

Lemma:

(1) The polyhedra R_{0},…, R_{n} are not empty.

(2) σ̃ = C ∪ IR_{≥0} · conv (⋃_{i=0}^{n} ϕ_{i}(R_{i})).

Proof: (1) We define some auxiliary cones in Ñ_{IR}:

C_{i} := (u^{0})^{⊥} ∩… ∩ ˆ (u^{i})^{⊥} ∩… ∩ (u^{n})^{⊥} ∩ σ̃
= C ∪ IR_{≥0} · ϕ_{i}(R_{i}) (for i = 0,…, n).

Then, the claim R_{i} ≠ ∅ is equivalent to the fact that the cone C is properly contained in C_{i}. We will switch to the dual level showing Ci^{∨} ⊂ C^{∨}: The dual cone C^{∨} ⊆ M̃_{IR} can be obtained as the pull back of C^{∨} regarded as a subset of M̃_{IR}/span_{IR}(u^{0},…, u^{n}). Moreover, the latter one equals the image of σ̃^{∨} via the canonical projection. Looking at Ci^{∨} in the same way (take span_{IR}(u^{0},…, û^{i},…, u^{n}) instead of span_{IR}(u^{0},…, u^{n})), we obtain

Ci^{∨} + IR · u^{i} = C^{∨}.

Hence, it suffices to check that −u^{i} ∉ Ci^{∨}: The opposite would mean that there is an r ∈ σ̃^{∨} such that r and −u^{i} differ by an element of span_{IR}(u^{0},…, û^{i},…, u^{n}) only, i.e.

r + u^{i} = ∑_{j≠i} λ_{j} · u^{j} in M̃_{IR}.

Enlarging the coefficients λ_{j}, we may assume that they are integers, but then the fact −u^{i} + ∑_{j≠i} λ_{j} · u^{j} ∈ σ̃^{∨} contradicts the previous corollary.

(2) The second part of the lemma is equivalent to the fact σ̃ = ∑_{i=0}^{n} C_{i} (or to σ̃^{∨} = ⋂_{i=0}^{n} Ci^{∨}). However, since

Ci^{∨} = σ̃^{∨} + ∑_{j≠i} IR · u^{j} (obtained in the first part of the proof),

this was already shown in Lemma (7.3.4). □

(7.3.6) We finish the proof of Theorem (7.2.5) by showing that the deformation datum just constructed is admissible:

Step 1: Let t ∈ IL^{∗} ∩ C^{∨}, and we assume (without loss of generality) that it has no integers as values on the faces F (R_{0}, t) and F (R_{1}, t) (or only on F (R_{1}, t) in case of p ≥ 2).

Let b^{0},…, b^{n} ∈ Ñ_{IR} be points of the faces F (R_{0}, t),…, F (R_{n}, t) in the embedded polyhedra ϕ_{0}(R_{0}),…, ϕ_{n}(R_{n}). Then, t ∈ IL^{∗} can be lifted to T ∈ M̃ such that

⟨b^{0}, T ⟩; ⟨b^{1}, T ⟩ ∉ ZZ (even in case p ≥ 2).

(For i ≥ 1 or p = 1, the value of t on F (R_{i}, t) equals ⟨b^{i}, T ⟩ in I Q/ZZ. In case i = 0, p ≥ 2, the value of ⟨b^{0}, T ⟩ is determined up to p^{−1}ZZ only. In particular, we can choose T in such a way that this value is not an integer.)

Step 2: Denote by q_{0}, q_{1} ∈ I Q the numbers q_{i} := ⟨b^{i}, T ⟩. Then, there is an integer k ≥ 1 such that [kq_{0} + kq_{1}] = [kq_{0}] + [kq_{1}] + 1. ([…] denotes the integral part, and k can always be obtained as one of the two possibilities k = 1 or k = (common denominator of q_{0}, q_{1}) − 1.) Therefore, we obtain

[⟨b^{0}, T ⟩ + ⟨b^{1}, T ⟩] = [⟨b^{0}, T ⟩] + [⟨b^{1}, T ⟩] + 1 ,

if t and T are replaced by kt and kT, respectively. (This operation does not change the faces F (R_{i}, t) of R_{i}.)

Step 3: Now, we modify our T by adding elements of span_{Z Z}(u^{0},…, u^{n}); it remains a lifting of t, and the values of ⟨b^{i}, T ⟩ do not change in I Q/ZZ. It is possible to obtain the following situation:

⟨b^{0}, T ⟩ ∈ [−1, 0) and ⟨b^{i}, T ⟩ ∈ [0, 1) (for i = 1,…, n),

i.e. T is not contained in the cone σ̃^{∨} and even cannot be put into σ̃^{∨} by adding vectors of the sublattice L ⊆ M̃ only. On the other hand, the result of the second step yields that

⟨∑_{i=0}^{n} b^{i}, T ⟩ ≥ 0.

This means that T is non-negative on the cone σ = C ∪ IR_{≥0} · (ϕ_{0}(R_{0}) + … ϕ_{n}(R_{n})). Hence, we have found an element of σ^{∨} ∩ M that cannot be lifted to σ̃^{∨} ∩ M̃. This is a contradiction to the fact that σ̃ and u^{0},…, u^{n} define a toric regular sequence.

7.4 The Kodaira-Spencer map

(7.4.1) Let (R_{0},…, R_{n}; C; p) be an admissible deformation datum (R_{i}, C ⊆ A A). Then, in (7.2.4) we have defined cones σ ⊆ σ̃ ⊆ Ñ_{IR} and elements u^{0},…, u^{n} ∈ σ̃^{∨} ∩ M̃ such that the map

Spec I C[σ̃^{∨} ∩ M̃] ⟶^{g} I C^{n} (defined by the regular functions x^{u^{1}} − x^{u^{0}},…, x^{u^{n}} − x^{u^{0}} ∈ I C[σ̃^{∨} ∩ M̃])

yields a deformation of the special fiber Y = Spec I C[σ^{∨} ∩ M]. Now, we will compute the Kodaira-Spencer map

ϱ : I C^{n} = T_{0} I C^{n} ⟶ T_{Y}^{1}

associated to this deformation.

(7.4.2) The cone σ was defined as the cone over the polyhedron Q = R_{0} +…+R_{n} embedded into the affine hyperplane {(•, p^{−1})} ⊆ A A × IR = N_{IR}. Hence, the elements of E can be written as

r^{v} = [c^{v}, η^{v}] with c^{v} ∈ IL^{∗} ∩ C^{∨}, η^{v} ∈ ZZ, and ⟨a, −p c^{v}⟩ ≤ η^{v} for a ∈ Q (since ⟨(a, p^{−1}), [c^{v}, η^{v}]⟩ ≥ 0).

Let {r̃^{0},…, r̃^{w}} be an arbitrary lift of E to σ̃^{∨} ∩ M̃ ⊆ M̃ = IL^{∗} × ZZ^{n+1}, i.e.

r̃^{v} = [c^{v}; η_{0}^{v},…, η_{n}^{v}] with η_{i}^{v} ∈ ZZ, η_{0}^{v} + p η_{1}^{v} + … + p η_{n}^{v} = η^{v} and ⟨a, −p c^{v}⟩ ≤ \{ \begin{matrix}η_{0}^{v} & for a ∈ R_{0} \\ p η_{i}^{v} & for a ∈ R_{i} (i ≥ 1). \end{matrix}

Remark: Those integers η_{i}^{v} exist, because the given deformation datum is admissible. In n out of the n + 1 possibilities i = 0,…, n the linear form c^{v} has integer value on F (R_{i}, c^{v}), and we can take this value for −η_{i}^{v}.

Theorem:

(i) The Kodaira-Spencer map sends the whole space I C^{n} into the homogeneous summand T_{Y}^{1} (−R). (The lattice point R ∈ σ^{∨} ∩ M was defined as either the common image [0, p] of u^{0},…, u^{n}, or by the fact that Q is contained in the affine hyperplane [R = 1] of N_{IR}.)

(ii) Using the T_{Y}^{1} -formula of (3.2.4), the Kodaira-Spencer map equals

ϱ : I C^{n} ⟶ (L(∪_{j=1}^{M}E_{j}^{R})/∑_{j=1}^{M} L(E_{j}^{R}) )^{∗} ⊗_{IR} I C

induced by the bilinear map IR^{n} × L(E) → IR that is described by the matrix

( \begin{matrix}η_{1}^{0} & … & η_{1}^{w} \\ ⋮ & & ⋮ \\ η_{n}^{0} & … & η_{n}^{w} \end{matrix} ) .

Remark: The previous theorem justifies the notions “degree” or “homogeneous” built up in (7.2.1). On the other hand, it shows the limits of the concept of toric deformations; only “strictly negative” degrees (i.e. −R with R ∈ σ^{∨} ∩ M) occur in the image of the Kodaira-Spencer map.

(7.4.3) Proof: Step 1: We show that the maps ϱ(e^{i}) : L(E) → IR (q = (q_{v})_{v=0,...,w} ↦ ∑_{v=0}^{w} η_{i}^{v} q_{v}) are trivial on each L(E_{j}^{R}). First, we notice that

E_{j}^{R} ≠ ∅ ⇔ a^{j} ∉ C ⇔ ⟨a^{j}, R⟩ > 0 ⇔ a^{j} corresponds to a vertex ā^{j} of Q.

Let a^{j} = ⟨a^{j}, R⟩ · (ā^{j}, p^{−1}) be one of the fundamental generators of σ that meets these conditions. Then, ā^{j} splits into a sum ā^{j} = (ā^{j})_{0} + … + (ā^{j})_{n}, and (ā^{j})_{i} ∈ R_{i} are vertices defined all by the same hyperplane t ∈ IL^{∗} ∩ C^{∨} as ā^{j} ∈ Q. Since our deformation datum is admissible, n of these vertices (say (ā^{j})_{1},…, (ā^{j})_{n}) have to be contained in the lattice IL. For an element q ∈ L(E) the property “q ∈ L(E_{j}^{R})” means that the components q_{v} are allowed to be non-trivial at most for ⟨(ā^{j}, p^{−1}), [c^{v}, η^{v}]⟩ < ⟨(ā^{j}, p^{−1}), R⟩. Since R = [0, p], this condition is equivalent to

⟨ā^{j}, −p c^{v}⟩ > η^{v} − p.

Restricting ourselves to those indices v, we obtain

\begin{aligned}η^{v} − p & < & ⟨ā^{j}, −p c^{v}⟩ = ⟨(ā^{j})_{0}, −p c^{v}⟩ + ⟨(ā^{j})_{1}, −p c^{v}⟩ + … + ⟨(ā^{j})_{n}, −p c^{v}⟩ \\ & ≤ & η_{0}^{v} + p η_{1}^{v} + … + p η_{n}^{v} = η^{v} . \end{aligned}

The numbers ⟨(ā^{j})_{i}, −p c^{v}⟩ (i ≥ 1) are contained in pZZ; hence

η_{i}^{v} = ⟨(ā^{j})_{i}, −c^{v}⟩ (for i = 1,…, n) and η_{0}^{v} = η^{v} + ∑_{i=1}^{n}⟨(ā^{j})_{i}, p c^{v}⟩.

Therefore, if q ∈ L(E_{j}^{R}), the equation ∑_{v} q_{v} · [c^{v}, η^{v}] = 0 implies ∑_{v} q_{v} η_{i}^{v} = 0 for i = 0,…, n.

Step 2: To compute the image of the i-th canonical unit vector e^{i} ∈ I C^{n} via the Kodaira-Spencer map, we may assume without loss of generality that i = 1 and consider the ring

Ã := I C[σ̃^{∨} ∩ M̃] /((x^{u^{1}} − x^{u^{0}})^{2}, x^{u^{2}} − x^{u^{0}},…, x^{u^{n}} − x^{u^{0}}) .

Via ε ↦ (x^{u^{1}} − x^{u^{0}}) this is a flat I C[ε]/ε^{2}-algebra and, moreover, A := Ã ⊗ I C[ε]/ε equals the ring I C[σ^{∨} ∩ M]. Spec Ã is the infinitesimal deformation of Y = Spec A obtained by restricting our toric deformation to the tangent vector e^{1} ∈ I C^{n} = T_{0} I C^{n}, and the corresponding class in T_{Y}^{1} is induced from the A-linear map ψ : I/I^{2} → A described by

ψ : P (z_{0},…, z_{w}) ↦ [\frac{P (x^{r̃^{0}},…, x^{r̃^{w}})}{x^{u^{1}} − x^{u^{0}}} (computed in Ã)] ∈ A.

We compute ψ for P (\underline{z}) = \underline{z}^{a} − \underline{z}^{b} (as already mentioned in (3.2.6), those binomials with (a, b) ∈ m ⊆ IN^{w+1} × IN^{w+1} generate the ideal I). Inside the semigroup σ̃^{∨} ∩ M̃ it is possible to write

∑_{v=0}^{w}(a_{v} − b_{v}) · r̃^{v} = ∑_{i=0}^{n} g_{i} · u^{i} (g_{i} ∈ ZZ, ∑_{i} g_{i} = 0).

Claim: ψ(\underline{z}^{a} − \underline{z}^{b}) = g_{1} · x^{(∑ a_{v}r^{v})−R} ∈ A.

Proof: Similar to (7.3.1.3) we know that

s := ∑_{v=0}^{w} a_{v} · r̃^{v} − ∑_{i=0}^{n} g_{i}^{+} · u^{i} = ∑_{v=0}^{w} b_{v} · r̃^{v} − ∑_{i=0}^{n} g_{i}^{−} · u^{i}

is contained in σ̃^{∨} ∩ M̃. Hence,

P (x^{r̃^{0}},…, x^{r̃^{w}}) = x^{∑ a_{v}r̃^{v}} − x^{∑ b_{v}r̃^{v}} = x^{s} · (x^{∑ g_{i}^{+}u^{i}} − x^{∑ g_{i}^{−}u^{i}})
= x^{s} · (∏_{i=0}^{n}(x^{u^{i}})^{g_{i}^{+}} − ∏_{i=0}^{n}(x^{u^{i}})^{g_{i}^{−}}) (in the ring I C[σ̃^{∨} ∩ M̃]).

Assuming (w.l.o.g.) g_{1} ≥ 0, we obtain in the ring Ã:

\begin{aligned}P (x^{r̃^{0}},…, x^{r̃^{w}}) & = & x^{s} · (x^{u^{0}})^{(∑ g_{i}^{+})−g_{1}^{+}} · ((x^{u^{1}})^{g_{1}^{+}} − (x^{u^{0}})^{g_{1}^{+}}) \\ & = & [x^{u^{1}} − x^{u^{0}}] · x^{s} · (x^{u^{0}})^{(∑ g_{i}^{+})−g_{1}^{+}} · (∑_{μ=1}^{g_{1}^{+}}(x^{u^{1}})^{g_{1}^{+}−μ}(x^{u^{0}})^{μ−1}) . \end{aligned}

In particular,

ψ(\underline{z}^{a} − \underline{z}^{b}) = xs̄ · (x^{R})^{(∑ g_{i}^{+})−g_{1}^{+}} · (g_{1} · (x^{R})^{g_{1}^{+}−1}) = g_{1} · x^{(∑ a_{v}r^{v})−R} ∈ A.

Step 3: To finish the proof, we have to involve the η’s. Since

∑_{v=0}^{w}(a_{v} − b_{v}) r̃^{v} = ∑_{v=0}^{w}(a_{v} − b_{v}) · [c^{v}; η_{0}^{v}, η_{1}^{v},…, η_{n}^{v}]

= (0; ∑_{v=0}^{w} η_{0}^{v} (a_{v} − b_{v}), ∑_{v=0}^{w} η_{1}^{v} (a_{v} − b_{v}),…, ∑_{v=0}^{w} η_{n}^{v} (a_{v} − b_{v}))

= p^{−1} (∑_{v=0}^{w} η_{0}^{v} (a_{v} − b_{v})) · u^{0} + ∑_{i=1}^{n} (∑_{v=0}^{w} η_{i}^{v} (a_{v} − b_{v})) · u^{i} ,

we obtain g_{1} = ∑_{v} η_{1}^{v} (a_{v} − b_{v}). Hence,

ψ(\underline{z}^{a} − \underline{z}^{b}) = ∑_{v} η_{1}^{v} (a_{v} − b_{v}) · x^{(∑ a_{v}r^{v})−R} = ϱ(e^{1})(a − b) · x^{(∑ a_{v}r^{v})−R} ,

and we are done by Theorem (3.2.6). □

(7.4.4) Finally, we want to present the Kodaira-Spencer map using the formula

T_{Y}^{1} (−R) = ker [V_{I C}(Q) ⊕ W_{I C}(Q) ⟶ ⟶ (∑_{d^{jk}} U_{jk})^{∗}]/ I C · (\underline{1}, \underline{1})

from (3.4.3). Already knowing the result in the Gorenstein case (cf. (5.4.2)) or even for arbitrary Y that are smooth in codimension two, it will be no surprise that ϱ yields the Minkowski summands R_{i} ∈ V (Q)/IR · \underline{1} as images of the standard vectors e^{i} ∈ I C^{n}. However, it might be interesting to see what the W (Q)-part looks like.

Theorem: The Kodaira-Spencer map

ϱ : I C^{n} ⟶ T_{Y}^{1} (−R) ⊆ V_{I C}(Q) ⊕ W_{I C}(Q)/ I C · (\underline{1}, \underline{1})

sends e^{i} onto the pair [R_{i}, \underline{s}^{i}] ∈ V (Q) ⊕ W (Q) (i = 1,…, n) with

s_{j}^{i} = \{ \begin{matrix}0 & if the vertex (ā^{j})_{i} of R_{i} belongs to the lattice \\ 1 & if (ā^{j})_{i} is not a lattice point \end{matrix} (if p = 1),

or \underline{s}^{i} := \underline{0} , if p ≥ 2.

Remark: Defining e^{0} := −(e^{1} + … + e^{n}), we obtain ϱ(e^{0}) = [R_{0}, \underline{s}^{0}] with \underline{s}^{0} similar to \underline{s}^{i} in the theorem (for p = 1), but \underline{s}^{0} = \underline{1} for p ≥ 2.

Proof: Fixing some i ∈ {1,…, n}, Theorem (7.4.2) tells that ϱ(e^{i}) maps a relation q ∈ L(E_{0}^{R}) onto ∑_{v} q_{v} η_{i}^{v}. As usual (cf. (3.4.1)), we can translate this fact into

ϱ(e^{i}) : (v^{j}) ∈ ⊕_{j}span E_{j}^{R} (with ∑_{j} v^{j} = 0) ↦ (q^{j}) ∈ ⊕_{j}ZZ^{E_{j}^{R}} (with ∑_{v} q_{v}^{j}r^{v} = −v^{j})
↦ ∑_{j} (∑_{v} q_{v}^{j} η_{i}^{v})

meaning that ϱ(e^{i}) is given by the linear maps b^{j} : span E_{j}^{R} → ZZ sending v^{j} onto ∑_{v} q_{v}^{j} η_{i}^{v}. On the other hand, by the first step of the proof in (7.4.3), we know that the condition r^{v} = [c^{v}, η^{v}] ∈ E_{j}^{R} implies

η_{i}^{v} = \{ \begin{matrix}−⟨(ā^{j})_{i}, c^{v}⟩ & if p ≥ 2 or (ā^{j})_{i} ∈ IL (p = 1) \\ ⟨ā^{j}, c^{v}⟩ − ⟨(ā^{j})_{i}, c^{v}⟩ + η^{v} & if p = 1 and (ā^{j})_{i} ∉ IL . \end{matrix}

In the first case, we obtain

b^{j}(v^{j}) = − ∑_{v} q_{v}^{j} ⟨(ā^{j})_{i}, c^{v}⟩ = ⟨(ā^{j})_{i}, − ∑_{v} q_{v}^{j}c^{v}⟩ = ⟨((ā^{j})_{i}, 0), v^{j}⟩ ,

i.e. b^{j} = ((ā^{j})_{i}, 0) ∈ N_{IR}. Hence, in the language of (3.4.4), we have s_{j} = −⟨b^{j}, R⟩ = 0, and the elements b̄^{j} = b^{j} form the vertices of the Minkowski summand R_{i}. Similarly, we obtain in the second case (p = 1, (ā^{j})_{i} ∉ IL)

b^{j} = −(ā^{j}, 0) + ((ā^{j})_{i}, 0) − (\underline{0}, 1) .

Hence, s_{j} = −⟨b^{j}, R⟩ = 1 and b̄^{j} = b^{j} + (ā^{j}, 1) = ((ā^{j})_{i}, 0). □

7.5 Examples

(7.5.1) See the examples (5.6.4) and (5.7.3).

(7.5.2) The cone over the rational normal curve of degree four (cf. Example (1.1.3)) is a two-dimensional cyclic quotient singularity. These singularities are toric, and this special one is given by the cone σ := ⟨(1, 0); (−1, 4)⟩ ⊆ IR^{2}. The dual cone equals σ^{∨} = ⟨[0, 1], [4, 1]⟩, T^{1} is four-dimensional, and the homogeneous pieces T^{1}(−R) are non-trivial only for R = [1, 1], [2, 1], [3, 1]. Now, we cut σ with the affine hyperplanes corresponding to these values:

(i) A A := [⟨•, [1, 1]⟩ = 1]; IL := A A ∩ ZZ^{2} (in particular p = 1) . To get an isomorphism (A A, IL) ≅ (IR, ZZ) we have to choose and fix an origin (contained in IL) and a ZZ-basis of IL:

(A A, IL) = [(0, 1) + IR · (−1, 1), (0, 1) + ZZ · (−1, 1)] .

In particular, Q := σ ∩ A A = [(1, 0), (−1/3, 4/3)] corresponds to [−1, 1/3] ⊆ IR, and this interval admits only one admissible decomposition into a Minkowski sum:

[−1, 1/3] = [−1, 0] + [0, 1/3] .

(ii) Analogously, we consider A A := [⟨•, [3, 1]⟩ = 1] = (0, 1) + IR · (−1, 3) (p = 1). Then, Q = [−1/3, 1] splits into [−1/3, 1] = [−1/3, 0] + [0, 1].

(iii) Let A A := [⟨•, [2, 1]⟩ = 1] = (0, 1) + IR · (−1, 2) (p = 1). Then, Q = [−1/2, 1/2] admits two different decompositions:

[−1/2, 1/2] = [−1/2, 0] + [0, 1/2] = {1/2} + [−1, 0] .

The decompositions (i), (ii) and the first one of (iii) provide the (three-dimensional) Artin component in the versal deformation of our singularity; the remaining decomposition of (iii) yields the other (one-dimensional) component. The corresponding families equal Y_{t} → I C^{3} and Y_{s} → I C from (1.1.3), respectively.

(7.5.3) Finally, we want to determine those two-dimensional cyclic quotient singularities that correspond to rational intervals of length one in IR. (Considering admissible Minkowski decompositions, those intervals seem to be very interesting; they admit the funny decomposition into [0, 1] and an interval of length zero.) For a given parameter p ∈ IN, each positive rational number can be written as a quotient x/(pd) (x, d ∈ IN with gcd(x, d) = 1). Then, the interval Q = [x/(pd), x/(pd) + 1] yields the cone σ = ⟨(x/(pd), 1/p), (x/(pd) + 1, 1/p)⟩ = ⟨(x, d), (x + pd, d)⟩.

(i) | \begin{matrix}x + pd & d \\ x & d \end{matrix} | = xd + pd^{2} − xd = pd^{2}.

This is the order of the cyclic group acting on I C^{2}.

(ii) We will regard x as an (invertible) element of ZZ/dZZ. Then, we obtain

(pdx^{−1} − 1) · (x + pd, d) + (x, d) = (p^{2}d^{2}x^{−1}, pd^{2}x^{−1}),

i.e. both components are divisible by pd^{2}.

Hence, Q corresponds to the cyclic quotient singularity X(pd^{2}, pdx^{−1} − 1) which equals A_{p−1} for d = 1 and which is called a T -singularity for d ≥ 2. Those singularities form the fundamental bricks for building P -resolutions (cf. §3 of [KoSh]). The canonical decomposition [x/(pd), x/(pd) + 1] = {x/(pd)} + [0, 1] corresponds to the oneparameter deformation presenting X(pd^{2}, pdx^{−1} − 1) as a hypersurface in a three-dimensional cyclic quotient singularity (of type (d; x, d − x)).

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