Negative Deformations of Toric Singularities that are Smooth in Codimension Two

Abstract

Abstract. Given a cone σ ⊆ N R with smooth two-dimensional faces and, moreover, an element R ∈ σ ∨ ∩ M of the dual lattice, we describe the part of the versal deformation of the associated toric variety TV(σ) that is built from the deformation parameters of multidegree R. The base space is (the germ of) an affine schemeM that reflects certain possibilities of splitting Q := σ ∩ [R = 1] into Minkowski summands.

Polyhedra, lattice structures, and extensions of semigroups

“…\end{equation*}$$In particular, in contrast to $${\eta }$$ and $${\widetilde{\eta }}$$, the functions $${\eta _{\mathbb {Z}}}$$ and $${\widetilde{\eta }_{\mathbb {Z}}}$$ are no longer piecewise linear. (ii)The pair $$[{c},{\widetilde{\eta }_{\mathbb {Z}}}({c})]$$ is a quite natural lifting of $$[{c},{\eta _{\mathbb {Z}}}({c})]$$ from $${M}\times \mathbb {N}$$ to $${M}\times {\mathcal {T}^*}({P})$$. However, even when asking for some positivity, it might be not the only lifting — see [3, 3.7] for an example.…”

“…Finally, the sub‐semigroup $$\operatorname{span}_{\mathbb {N}}\lbrace t_{ij}\rbrace \subseteq {C}({P})^{\scriptscriptstyle \vee }$$ of (ii) encodes the versal deformation itself, which is a much finer information than just its linear ambient space $$T^1_X(-{R})$$. Remark In the Gorenstein case, that is, when $${P}$$ was a lattice polytope, then the smoothness of $$X$$ in codimension two could be easily expressed by the primitivity of its lattice edges. In [3], we already got rid of the Gorenstein assumption, but we heavily depended on the assumption of smoothness in codimension two. In the non‐Gorenstein case, this condition can still be expressed in the combinatorial language.…”

“…Our theorem is a full generalization of [3,4], in the sense that no restrictions are imposed on the polyhedron. We tried to keep our notation and terminology as close as possible to those two papers.…”

On deformations of affine Gorenstein toric varieties

“…More precisely, we construct a miniversal deformation in degrees −kR * for k ∈ N, see Definition 3.1. Besides [1] there were also the papers [4] and [6] where a miniversal deformation of an affine toric variety in a single primitive lattice degree was constructed under some assumptions. For an affine Gorenstein toric variety X all three papers can only describe the miniversal deformation in the degree −R * under additional assumption that X is smooth in codimension 2.…”

“…The next two main sections are Section 4 and Section 6 where we show that the Kodaira-Spencer map of this deformation family is bijective and that the obstruction map is injective, respectively. The computations are different from the ones appearing in [1], [4] and [6] since we are considering all degrees −kR * at the same time. We conclude the paper by showing that the components of the reduced miniversal deformation correspond to maximal Minkowski decompositions of P in Section 7.…”