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Let C be a locally planar curve. Its versal deformation admits astratification by the genera of the fibres. The strata are singular; we showthat their multiplicities at the central point are determined by the Eulernumbers of the Hilbert schemes of points on C. These Euler numbers have madetwo prior appearances. First, in certain simple cases, they control thecontribution of C to the Pandharipande-Thomas curve counting invariants ofthree-folds. In this context, our result identifies the strata multiplicitiesas the local contributions to the Gopakumar-Vafa BPS invariants. Second, when Cis smooth away from a unique singular point, a special case of a conjecture ofOblomkov and Shende identifies the Euler numbers of the Hilbert schemes withthe "U(infinity)" invariant of the link of the singularity. We make contactwith combinatorial ideas of Jaeger, and suggest an approach to the conjecture.

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation