Minimal free resolutions of sheaves on the projective plane and the stable base locus decomposition of their moduli spaces

TL;DR

This paper integrates minimal free resolutions into the birational geometry of moduli spaces of sheaves on the projective plane, linking them to Bridgeland stability and effective cone computations, and proposes a program for stable base locus decomposition.

Contribution

It introduces a novel approach connecting minimal free resolutions with Bridgeland destabilizing objects to analyze the birational geometry of sheaf moduli spaces.

Findings

Reconstruction of Bridgeland destabilizing objects from free resolutions.

Computation of the effective cone of $M(\xi)$ using free resolutions.

Proposal of a program for stable base locus decomposition of $P^{2[n]}$.

Abstract

The purpose of this paper is to incorporate minimal free resolutions into the study of the birational geometry of the moduli space of coherent sheaves on the plane with character $\xi$, denoted by $M(\xi)$. We show that it is possible to recover the relevant Bridgeland destabilizing object from the minimal free resolution in order to compute the effective cone $\mathrm{Eff}(M(\xi))$ and conjecture a full relationship between Bridgeland destabilizing objects and minimal free resolutions. Moreover, we also prove that minimal free resolutions, paired with interpolation for vector bundles, yield the movable cone of the Hilbert scheme of $n$ points on the plane $\mathbb{P}^{2[n]}$, for certain values of $n$. We propose a program that computes the full stable base locus decomposition of $\mathbb{P}^{2[n]}$ based on free resolutions and interpolation. We show that this programs yields correct answers for small values of $n$.