Hilbert schemes of points on some classes of surface singularities

TL;DR

This paper investigates the geometry of Hilbert schemes of points on certain surface singularities, providing explicit formulas for their Euler characteristics and revealing connections to affine Lie algebras and modular properties.

Contribution

It offers a new combinatorial decomposition of Hilbert schemes for type A and D singularities, extending known results and proposing conjectures for type E, with implications for representation theory.

Findings

Explicit formula for Euler characteristics in type A and D

Decomposition of Hilbert schemes into affine space strata

Connections to affine Lie algebra representations and modular properties

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 for type A, and are new for type D. The crystal basis theory of the fundamental representation of the affine Lie algebra corresponding to the surface singularity (via the McKay correspondence) plays an important role in our approach. The result gives a generalization of G\"ottsche's formula and has interesting modular properties related to the S-duality conjecture. The moduli space of torsion free sheaves on surfaces are higher rank analogs of the Hilbert schemes. In type A our results reveal their Euler characteristic generating function as well. Another very interesting class of normal surface singularities is the so-called cyclic quotient singularities of type (p,1). As an outlook we also obtain some results about the associated generating functions.