Interpolation, Bridgeland stability and monomial schemes in the plane

Abstract

Given a zero-dimensional scheme Z, the higher-rank interpolation problem asks for the classification of slopes μ such that there exists a vector bundle E of slope μ satisfying Hi(E⊗IZ)=0 for all i. In this paper, we solve this problem for all zero-dimensional monomial schemes in P2. As a corollary, we obtain detailed information on the stable base loci of Brill–Noether divisors on the Hilbert scheme of points on P2. We prove the correspondence between walls in the Bridgeland stability manifold and walls in the Mori chamber decomposition of the effective cone conjectured in [2] 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.