The dualizing sheaf on first-order deformations of toric surface singularities

TL;DR

This paper explicitly describes the infinitesimal deformations of cyclic quotient singularities on toric surfaces, revealing differences among various deformation notions and their impact on moduli space structures.

Contribution

It provides a detailed analysis of deformation conditions for cyclic quotient singularities, highlighting differences among Wahl, Kollár-Shepherd-Barron, and Viehweg notions.

Findings

Different deformation notions are not equivalent in many cases.

The KSB and Viehweg moduli spaces share the same reduced structure but differ infinitesimally.

Explicit descriptions of deformations for cyclic quotient singularities are provided.

Abstract

We explicitly describe infintesimal deformations of cyclic quotient singularities that satisfy one of the deformation conditions introduced by Wahl, Koll\'ar-Shepherd-Barron and Viehweg. The conclusion is that in many cases these three notions are different from each other. In particular, we see that while the KSB and the Viehweg versions of the moduli space of surfaces of general type have the same underlying reduced subscheme, their infinitesimal structures are different.