Macdonald formula for curves with planar singularities

We generalize Macdonald’s formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes essential use of Ngô’s support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we give a cohomological interpretation of the numerator of the Hilbert-zeta function of curves with planar singularities.