Interpolation, Bridgeland stability and monomial schemes in the plane

TL;DR

This paper solves the higher-rank interpolation problem for all zero-dimensional monomial schemes in the projective plane, linking Bridgeland stability, effective cone walls, and cohomology computations.

Contribution

It provides a complete classification of slopes for vector bundles orthogonal to monomial schemes and establishes a correspondence between Bridgeland and Mori chamber walls.

Findings

Classification of slopes for monomial schemes

Correspondence between Bridgeland and Mori walls

New cohomology-friendly resolution for ideal sheaves

Abstract

Given a zero-dimensional scheme Z, the higher-rank interpolation problem asks for the classification of slopes of vector bundles cohomologically orthogonal to the ideal sheaf of Z. In this paper, we solve this problem for all zero-dimensional monomial schemes in the projective plane. As a corollary, we obtain detailed information on the stable base loci of divisors on the Hilbert scheme of points on the plane. We prove the correspondence between walls in the Bridgeland stability manifold and walls in the Mori chamber decomposition of the effective cone conjectured by Arcara-Bertram-Coskun-Huizenga for monomial schemes. We determine the Harder-Narasimhan filtration of ideal sheaves of monomial schemes for suitable Bridgeland stability conditions and, as a consequence, obtain a new resolution better suited for cohomology computations than other standard resolutions such as the minimal free resolution.