The versal Deformation of an isolated toric Gorenstein Singularity

TL;DR

The paper develops a combinatorial framework for the versal deformation of affine toric Gorenstein singularities $Y$ arising from a lattice polytope $Q$, via the Minkowski scheme $M(Q)$ that encodes Minkowski decompositions. It introduces the tautological cone $ ilde{C}(Q)$ and builds a flat family over a base $ar{{\cal M}}$ whose special fiber is $Y$, and analyzes the Kodaira-Spencer and obstruction maps to prove versality in the isolated case, while giving a precise description of $T^1_Y$ and $T^2_Y$ in terms of polygonal data. The reduced versal base decomposes into flats corresponding to maximal Minkowski decompositions, yielding explicit components and their total spaces, often realized as cones over products of projective varieties. Through detailed examples (including Del Pezzo cones and the polygons $Q_4$, $Q_6$, and $Q_8$), the work demonstrates how the deformation theory of toric singularities is governed by the combinatorics of Minkowski decompositions, enabling a constructive, combinatorial approach to versal deformations.

Abstract

Given a lattice polytope Q in R^n, we define an affine scheme M(Q) that reflects the possibilities of splitting Q into a Minkowski sum. On the other hand, Q induces a toric Gorenstein singularity Y, and we construct a flat family over M(Q) with Y as special fiber. In case Y has an isolated singularity only, this family is versal. (This revised version contains the proof now.)