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

Given a smooth curve on a smooth surface, the Hilbert scheme of 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.