Smoothing of surface singularities via equivariant smoothing of lci covers

TL;DR

This paper investigates how equivariant smoothing of locally complete intersection covers can be used to smooth certain surface singularities, providing classifications for specific types like elliptic and cusp singularities.

Contribution

It offers new results on smoothing surface singularities through equivariant smoothing of LCI covers and classifies cases for specific singularity types.

Findings

Classification of smoothing scenarios for simple elliptic and cusp singularities

Identification of conditions where equivariant smoothing induces surface singularity smoothing

Extension of smoothing techniques to cyclic quotient singularities

Abstract

We provide some results of the smoothing of surface singularities by Looijenga-Wahl and study smoothing of isolated surface singularities induced by equivariant smoothing of locally complete intersection ($\lci$) singularities. We classify the situation where the smoothing of a simple elliptic singularity, a cusp singularity or its cyclic quotient is induced by the equivariant smoothing of the $\lci$ covers.