Hilbert scheme of points on cyclic quotient singularities of type (p,1)

TL;DR

This paper studies the generating series of Euler characteristics for Hilbert schemes of points on cyclic quotient singularities of type (p,1), connecting combinatorics to p-fountains and expressing the series via a two-variable generating function.

Contribution

It introduces a novel combinatorial approach using p-fountains to analyze the generating series of Euler characteristics on these singularities.

Findings

Derived a new representation of the generating series as a coefficient in a two-variable series.

Linked the combinatorics of p-fountains to the geometry of Hilbert schemes.

Provided explicit formulas for the generating series in terms of combinatorial objects.

Abstract

In this note we investigate the generating series of the Euler characteristics of Hilbert scheme of points on cyclic quotient singularities of type (p,1). We link the appearing combinatorics to p-fountains, a generalization of the notion of fountain of coins. We obtain a representation of the generating series as coefficient of a two variable generating series.