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Given a smooth curve on a smooth surface, the Hilbert scheme of the surfaceis stratified according to the length of the intersection with the curve. Thestrata are highly singular. We show that this stratification admits a naturallog-resolution, namely the stratified blowup. As a consequence, the inducedPoisson structure on the Hilbert scheme of a Poisson surface has unobstructeddeformations.

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