Homology of Hilbert schemes of points on a locally planar curve

Abstract

Let $C$ be a proper, integral, locally planar curve, and consider its Hilbert schemes of points $C^{[}n]$. We define four creation/annihilation operators acting on the rational homology groups of these Hilbert schemes and show that the operators satisfy the relations of a Weyl algebra. The action of this algebra is similar to that defined by Grojnowski and Nakajima for a smooth surface. As a corollary, we compute the cohomology of $C^{[}n]$ in terms of the cohomology of the compactified Jacobian of $C$ together with an auxiliary grading on the latter. This recovers and slightly strenghtens a formula recently obtained in a different way by Maulik and Yun and independently Migliorini and Shende.