The classes of the quasihomogeneous Hilbert schemes of points on the plane

TL;DR

This paper provides a formula for the classes of irreducible components of certain quasi-homogeneous Hilbert schemes on the plane, offers a new geometric interpretation of the q,t-Catalan numbers, and explores their connection to nested Hilbert schemes.

Contribution

It introduces a formula for classes of components in quasi-homogeneous Hilbert schemes and links these to q,t-Catalan numbers and nested Hilbert schemes.

Findings

Derived a formula for classes in the Grothendieck ring

Provided a geometric interpretation of q,t-Catalan numbers

Explored connections to nested Hilbert schemes

Abstract

In this paper we give a formula for the classes (in the Grothendieck ring of complex quasi-projective varieties) of irreducible components of $(1,k)$-quasi-homogeneous Hilbert schemes of points on the plane. We find a new simple geometric interpretation of the $q,t$-Catalan numbers. Finally, we investigate a connection between $(1,k)$-quasi-homogeneous Hilbert schemes and homogeneous nested Hilbert schemes.