Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles

We compute the cone of effective divisors on the Hilbert scheme of n points in the projective plane. We show that the sections of many stable vector bundles satisfy a natural interpolation condition and that these bundles always give rise to the edge of the effective cone of the Hilbert scheme. To do this, we give a generalization of Gaeta's theorem on the resolution of the ideal sheaf of a general collection of n points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that ideal sheaves of collections of points are destabilized by exceptional bundles. By studying the Bridgeland stability of exceptional bundles, we also show that our computation of the effective cone of the Hilbert scheme is consistent with a conjecture in a 2013 work by Arcara, Bertram, Coskun and the author which predicts a correspondence between Mori and Bridgeland walls for the Hilbert scheme.