Straight Equisingular Deformations and Punctual Hilbert Schemes

TL;DR

This paper investigates straight equisingular deformations of plane curve singularities, describing their seminuniversal deformation and linking the base space to punctual Hilbert schemes, offering new insights into their structure.

Contribution

It introduces the concept of straight equisingular deformations, describes their seminuniversal deformation, and connects it to punctual Hilbert schemes, providing a novel perspective on singularity deformations.

Findings

Semiuniversal straight equisingular deformation described by an ideal containing the Tjurina ideal.

Base space of deformation appears as a fiber of a morphism from the μ-constant stratum.

Potential new insights into the structure of equisingular deformations of plane curve singularities.

Abstract

We study "straight equisingular deformations", a linear subfunctor of all equisingular deformations and describe their seminuniversal deformation by an ideal containing the fixed Tjurina ideal. Moreover, we show that the base space of the seminuniversal straight equisingular deformation appears as the fibre of a morphism from the {\mu}-constant stratum onto a punctual Hilbert scheme parametrizing certain zero-dimensional schemes concentrated in the singular point. Although equisingular deformations of plane curve singularities are very well understood, we believe that this aspect may give a new insight in their inner structure.