Deformations of an affine Gorenstein toric pair

TL;DR

This paper studies deformations of affine Gorenstein toric pairs, computing their tangent and obstruction spaces, and generalizes existing deformation constructions to non-isolated singularities, linking components to Minkowski decompositions.

Contribution

It extends Altmann's deformation theory from isolated to non-isolated Gorenstein toric singularities, providing explicit descriptions and classifications.

Findings

Computed tangent and obstruction spaces for deformations.

Constructed miniversal deformations in various degrees.

Linked deformation components to Minkowski decompositions.

Abstract

We consider deformations of a pair $(X,\partial X)$, where $X$ is an affine toric Gorenstein variety and $\partial X$ is its boundary. We compute the tangent and obstruction space for the corresponding deformation functor and for an admissible lattice degree $m$ we construct the miniversal deformation of $(X,\partial X)$ in degrees $-km$, for all $k\in \mathbb{N}$. This in particular generalizes Altmann's construction of the miniversal deformation of an isolated Gorenstein toric singularity to an arbitrary non-isolated Gorenstein toric singularity. Moreover, we show that the irreducible components of the reduced miniversal deformation are in one to one correspondence with maximal Minkowski decompositions of the polytope $P\cap (m=1)$, where $P$ is the lattice polytope defining $X$.