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 $M_{\sigma }(\mathbf{v})$ 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).

84 pages, 3 figures, Comments Welcome. arXiv admin note: text overlap with arXiv:1301.6968 by other authors

References in corpus (2)

• On some moduli of complexes on K3 surfaces
• Some remarks on Bridgeland stability conditions on K3 and Enriques surfaces