Versality in toric geometry

TL;DR

This paper investigates the deformation theory of affine toric varieties, constructing specific maximal deformations with prescribed tangent spaces using polyhedral methods, advancing understanding of their versal deformations.

Contribution

It constructs the homogeneous part of the versal deformation for affine toric varieties of certain degrees, using polyhedral cuts and length assumptions on edges.

Findings

Constructed the homogeneous part of the versal deformation for degree -R.

Provided explicit methods under length assumptions on polyhedral edges.

Enhanced understanding of deformation spaces of affine toric singularities.

Abstract

We study deformations of affine toric varieties. The entire deformation theory of these singularities is encoded by the so-called versal deformation. The main goal of our paper is to construct the homogeneous part of some degree -R of this, i.e. a maximal deformation with prescribed tangent space T^1(-R) for a given character R. To this aim we use the polyhedron obtained by cutting the rational cone defining the affine singularity with the hyperplane defined by [R=1]. Under some length assumptions on the edges of this polyhedron, we provide the versal deformation for primitive degrees R.