Negative deformations of toric singularities that are smooth in codimension two

TL;DR

This paper investigates specific negative deformations of toric singularities that are smooth in codimension two, focusing on the structure of their versal deformations linked to lattice points and Minkowski decompositions.

Contribution

It provides a detailed description of the versal deformation space of toric varieties with certain smoothness properties, connecting deformation parameters to Minkowski summands of associated polyhedra.

Findings

Describes the part of the versal deformation space related to a lattice point R.

Connects the deformation parameters to Minkowski decompositions of polyhedron Q.

Characterizes the base space as an affine scheme reflecting splitting possibilities.

Abstract

Given a polyhedral cone sigma with smooth two-dimensional faces and, moreover, a lattice point R in the dual cone of sigma, we describe the part of the versal deformation of the associated toric variety TV(sigma) that is built from the deformation parameters of multidegree R. Let Q the polyhedron obtained by intersecting sigma with the hyperplane R=1. Then the base space is (the germ of) an affine scheme that reflects certain possibilities of splitting Q into Minkowski summands.