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

TL;DR

This paper computes the effective divisor cone of the Hilbert scheme of points in the plane, linking stable bundles, Bridgeland stability, and geometric properties of point configurations.

Contribution

It generalizes Gaeta's theorem, relates effective divisors to stable bundles and Bridgeland stability, and confirms a conjecture connecting Mori and Bridgeland walls.

Findings

Effective cone of the Hilbert scheme is computed.

Stable bundles satisfy natural interpolation conditions.

The results align with the conjectured Mori-Bridgeland wall correspondence.

Abstract

We compute the cone of effective divisors on the Hilbert scheme of 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 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 which predicts a correspondence between Mori and Bridgeland walls for the Hilbert scheme.