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We study the cohomology of Jacobians and Hilbert schemes of points on reducedand locally planar curves, which are however allowed to be singular andreducible. We show that the cohomologies of all Hilbert schemes of allsubcurves are encoded in the cohomologies of the fine compactified Jacobians ofconnected subcurves, via the perverse Leray filtration.

A support theorem for Hilbert schemes of planar curves, II