General problem of finding affine pavings of Hilbert schemes of points

Determine affine pavings for the punctual Hilbert schemes Hilb^n(C,0) of points, beyond the currently known cases, in order to facilitate the computation of their homology.

Background

Computing the homology of Hilbn(C,0) often relies on the existence of affine pavings. Several special cases are known—e.g., Hilbn(C2), certain singular irreducible curves, and generic curve singularities with specified Puiseux exponents.

However, the existence and construction of affine pavings in general remain largely unresolved, posing a broad and fundamental challenge for the study of Hilbert schemes of points on singular (especially non-reduced) curves.

References

We list a few known affine pavings, and in general the problem of finding affine pavings of the Hilbert scheme is wide open.

— Hilbert scheme of points on non-reduced nodal curves  (2604.03111 - Luan, 3 Apr 2026) in Subsection: Affine paving of the Hilbert scheme (Introduction)