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

Abstract

We compute the cone of effective divisors on the Hilbert scheme of n points in the projective plane. We show 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 [ABCH] which predicts a correspondence between Mori and Bridgeland walls for the Hilbert scheme.

The ample cone of moduli spaces of sheaves on the plane

“…We now recall basic facts concerning Bridgeland stability conditions on P 2 developed in [ABCH13, CH14] and [Hui15].…”

Mori cones of holomorphic symplectic varieties of K3 type

“…The case of punctual Hilbert schemes of P 2 and del Pezzo surfaces was investigated by Arcara, Bertram, Coskun, and Huizenga [ABCH13,Hui12,BC13,CH13]. The effective cone on (P 2 )…”

The effective cone of the moduli space of sheaves on the plane

“…Theorem 4.8 ( [H,Theorem 3.2]). Let γ ∈ E ′′ = E ∩ (0, 1 2 ) be an exceptional slope, and put α = par L (γ) and β = par R (γ).…”