One-Parameter Toric Deformations of Cyclic Quotient Singularities

TL;DR

This paper classifies all one-parameter toric deformations of two-dimensional cyclic quotient singularities using Minkowski decompositions, providing explicit equations, descriptions of fibers, and resolutions.

Contribution

It offers a complete classification of one-parameter toric deformations for cyclic quotient singularities, including explicit equations and resolution methods.

Findings

Classification of all such deformations via Minkowski decompositions

Explicit equations for each deformation

Descriptions of singularities and resolutions in general fibers

Abstract

In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions. In particular, we describe to which components each such deformation maps, show how to induce each deformation from a versal family, give explicit equations for each deformation, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.