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

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.