  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 $n$.
