Euler characteristics of Hilbert schemes of points on simple surface singularities

TL;DR

This paper investigates the topology of Hilbert schemes of points on orbifold surfaces with singularities of type A or D, providing explicit formulas for their Euler characteristics and a combinatorial decomposition into affine strata.

Contribution

It introduces a new combinatorial decomposition of the Hilbert scheme into affine space strata and computes their Euler characteristics using affine Lie algebra representations, extending known results to type D.

Findings

Explicit formula for Euler characteristics in type A

New results for type D singularities

Conjecture for type E extension

Abstract

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.