Abstract
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.
The authors would like to thank A. Oblomkov for bringing this problem to their attention and V. Shende for helpful discussions. They are also grateful to an anonymous referee for helpful suggestions. D.M. is supported by the Clay Research Fellowship. Z.Y. is partially supported by the NSF grant DMS-0969470.
© 2014 by De Gruyter
