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_σ(v) of $σ$-semistable objects of class v for a generic stability condition $σ$ on an arbitrary Enriques surface X. In particular, we show that for any other generic stability condition $τ$, the two moduli spaces M_τ(v) and M_σ(v) are birational. As a consequence, we show that for primitive v of odd rank M_σ(v) is birational to a Hilbert scheme of points. Similarly, in even rank we show that M_σ(v) is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when (v)=1. As an added bonus of our work, we prove that the Donaldson-Mukai map θv,σ:v^⊥(M_σ(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).