The Topology of Equivariant Hilbert Schemes

TL;DR

This paper investigates the topological properties of equivariant Hilbert schemes associated with finite abelian groups acting on the plane, revealing periodic and quasipolynomial behaviors of invariants related to the group's order.

Contribution

It establishes the periodicity and quasipolynomial nature of topological invariants of equivariant Hilbert schemes for abelian groups, using combinatorial and geometric methods.

Findings

Topological invariants are periodic in the order of the group G.

Invariants exhibit quasipolynomial behavior across certain families of abelian subgroups.

The Bialynicki-Birula decomposition links topology to combinatorics of partitions.

Abstract

For $G$ a finite group acting linearly on $\mathbb{A}^2$, the equivariant Hilbert scheme $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ is a natural resolution of singularities of $\operatorname{Sym}^r(\mathbb{A}^2/G)$. In this paper we study the topology of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ for abelian $G$ and how it depends on the group $G$. We prove that the topological invariants of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ are periodic or quasipolynomial in the order of the group $G$ as $G$ varies over certain families of abelian subgroups of $GL_2$. This is done by using the Bialynicki-Birula decomposition to compute topological invariants in terms of the combinatorics of a certain set of partitions.