Hilbert schemes of points on fold-like curves and their combinatorics

TL;DR

This paper studies the structure and singularities of Hilbert schemes of points on curves with n-fold singularities, revealing complex combinatorial patterns that influence their geometric properties.

Contribution

It provides a detailed description of the irreducible components and singularities of these Hilbert schemes, highlighting new combinatorial insights.

Findings

Identified the number and structure of irreducible components.

Analyzed the singularities and their combinatorial patterns.

Revealed rich geometric and combinatorial structures.

Abstract

We investigate the Hilbert scheme of points on curves with n-fold singularities, that is curves that look locally around their singular points as the axis in an affine space. We describe the structure and number of its irreducible components, and provide a detailed analysis of their singularities, revealing rich combinatorial patterns governing its geometry.