Topology of the punctual Hilbert schemes of plane curve singularities with a single Puiseux pair

TL;DR

This paper investigates the topology of punctual Hilbert schemes associated with plane curve singularities having a single Puiseux pair, providing computational insights into their Euler and Betti numbers.

Contribution

It offers a computational perspective on the topology of punctual Hilbert schemes, extending previous theoretical results to explicit calculations.

Findings

Euler numbers of punctual Hilbert schemes computed

Betti numbers of these schemes determined

Affine cell decompositions confirmed for the schemes

Abstract

Piontkowski proved the existence of affine cell decompositions of Jacobian factors of plane curve singularities with a single Puiseux pair. He also provided a combinatorial description of the Euler numbers and Betti numbers of these Jacobian factors. Following his results, Oblomkov, Rasmussen, and Shende demonstrated the existence of affine cell decompositions of punctual Hilbert schemes for the same type of singularity. In the present paper, we revisit their theorem from a computational perspective and describe the Euler numbers and Betti numbers of the punctual Hilbert schemes.