# Euler characteristics of Hilbert schemes of points on simple surface singularities

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## Abstract

We study the geometry and topology of Hilbert schemes of points on the orbifold surface Open image in new window , respectively the singular quotient surface Open image in new window , where Open image in new window is a finite subgroup 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*.

## Keywords

Hilbert scheme Singularities Euler characteristic Generating series Young wall## Mathematics Subject Classification

14N10 05E10## Notes

### Acknowledgements

The authors would like to thank Gwyn Bellamy, Alastair Craw, Eugene Gorsky, Ian Grojnowski, Kevin McGerty, Iain Gordon, Tomas Nevins and Tamás Szamuely for helpful comments and discussions.

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