The punctual Hilbert schemes for the curve singularities of types $E_6$ and $E_8$

TL;DR

This paper analyzes the structure of punctual Hilbert schemes for $E_6$ and $E_8$ curve singularities, introducing algorithms and algebraic tools to decompose these schemes into affine cells.

Contribution

It provides a comprehensive study of all degrees of punctual Hilbert schemes for $E_6$ and $E_8$ singularities, using computational and algebraic methods.

Findings

Decomposition of punctual Hilbert schemes into affine cells.

Algorithms for analyzing scheme structures.

Application of Gr"obner basis theory and Jacobian results.

Abstract

Pfister and Steenbrink studied punctual Hilbert schemes for irreducible curve singularities. In particular, they investigated the structure of special punctual Hilbert schemes for certain monomial curve singularities. In this paper, we study the punctual Hilbert schemes of all degrees for the curve singularities of types $E_6$ and $E_8$. For our analysis, we introduce computational algorithms to decompose a punctual Hilbert schemes into affine cells. We also use the theory of Gr\"obner basis and known results about the compactified Jacobian of singular curves to prove our main theorems.