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

TL;DR

This paper studies a stratification of Hilbert schemes of smooth surfaces based on intersection length with a curve, resolving its singularities and applying results to Poisson structures to show their deformations are unobstructed.

Contribution

It introduces a natural log-resolution for the stratification of Hilbert schemes and demonstrates unobstructed deformations of Poisson structures on these schemes.

Findings

Stratification admits a natural log-resolution via stratified blowup.

Poisson structures on Hilbert schemes of Poisson surfaces have unobstructed deformations.

Abstract

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.