Equisingular Deformations of Plane Curve and of Sandwiched Singularities

TL;DR

This paper investigates the relationship between equisingular deformations of plane curve singularities and their associated sandwiched singularities, providing formulas for deformation dimensions and methods for computing equisingularity ideals.

Contribution

It establishes an equivalence between deformations of a curve and its decorated version for large decorations, and offers a way to compute equisingularity ideals from minimal resolutions.

Findings

Equisingular deformation functors of $C$ and $(C,l)$ are equivalent for large $l$.

Derived a formula for the dimension of the equisingular stratum.

Provided a method to compute the equisingularity ideal from minimal resolution.

Abstract

Let $C$ be an isolated plane curve singularity, and $(C,l)$ be a decorated curve. In this article we compare the equisingular deformations of $C$ and the sandwiched singularity $X(C,l)$. We will prove that for $l \gg 0$ the functor of equisingular deformations of $C$ and $(C,l)$ are equivalent. From this we deduce a proof of a formula for the dimension of the equisingular stratum. Furthermore we will show how compute the equisingularity ideal of the curve singularity $C$, given the minimal (good) resolution of $C$.