Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures

This is the authors' abstract. We don't add key points for this paper.

Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations.