Geometry of syzygies of sheaves on $\mathbb{P}^2$ via interpolation and Bridgeland stability

TL;DR

This paper explores the relationship between syzygies of sheaves on the projective plane and the geometry of their moduli spaces, revealing how minimal free resolutions influence the structure of effective divisors.

Contribution

It introduces a novel connection between minimal free resolutions of semi-stable sheaves and extremal rays of the effective cone, linking syzygies to wall-crossing phenomena.

Findings

Minimal free resolutions contain subcomplexes determining extremal rays.

Syzygies influence the structure of the effective cone of moduli spaces.

New computations of movable cones and Mori decompositions using syzygies.

Abstract

We show that the minimal free resolution of a general semi-stable sheaf $U$ on $\mathbb{P}^2$ contains a subcomplex that determines an extremal ray of the cone of effective divisors of its moduli space. We provide evidence that this is part of a general phenomenon in which minimal free resolutions, for distinct Betti tables, contain subcomplexes depending on wall-crossing. From this viewpoint, we provide new computations of the movable cones and Mori decompositions of some moduli spaces of sheaves using syzygies.