MMP via wall-crossing for moduli spaces of stable sheaves on an Enriques surface

Abstract

We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space Mσ(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 Mτ(v) and Mσ(v) are birational. As a consequence, we show that for primitive v of odd rank Mσ(v) is birational to a Hilbert scheme of points. Similarly, in even rank we show that Mσ(v) is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when ℓ(v)=1. As an added bonus of our work, we prove that the Donaldson-Mukai map θv,σ:v⊥→Pic(Mσ(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).