MMP Via Wall-crossing for Moduli Spaces of Stable Sheaves on an Enriques surface

We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space $M_\sigma(\mathbf{v})$ of $\sigma$-semistable objects of class $\mathbf{v}$ for a generic stability condition $\sigma$ on an arbitrary Enriques surface $X$. In particular, we show that for any other generic stability condition $\tau$, the two moduli spaces $M_\tau(\mathbf{v})$ and $M_\sigma(\mathbf{v})$ are birational. As a consequence, we show that for primitive $\mathbf{v}$ of odd rank $M_\sigma(\mathbf{v})$ is birational to a Hilbert scheme of points. Similarly, in even rank we show that $M_\sigma(\mathbf{v})$ is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when $\ell(\mathbf{v})=1$. As an added bonus of our work, we prove that the Donaldson-Mukai map $\theta_{\mathbf{v},\sigma}:\mathbf{v}^\perp\to\mathrm{Pic}(M_\sigma(\mathbf{v}))$ is an isomorphism for these classes. Finally, we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on $X$ that are uniruled (and thus not K-trivial).