The topology of equivariant Hilbert schemes

For G a finite group acting linearly on A(2), the equivariant Hilbert scheme Hilb'[A(2)/G] is a natural resolution of singularities of Sym'(A(2)/G). In this paper, we study the topology of Hilb'[A(2)/G] for abelian G and how it depends on the group G. We prove that the topological invariants of Hilb'[A(2)/G] are periodic or quasipolynomial in the order of the group Gas 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.