Birational geometry of moduli spaces of stable objects on Enriques surfaces

Abstract

Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes and that under an extra assumption every minimal model can be described as a moduli space. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.

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

“…During the writing phase of this project, the authors became aware of the recent preprint [8]. In that paper, the author proves part (1) of Theorem 1.1 as well as Theorems 1.2 and 1.3 for generic Enriques surfaces (that is, when Picp r…”

“…Xq " ̟ ˚PicpX q) and for Mukai vectors v such that ̟ ˚v is primitive (see Theorem 4.4, Proposition 4.6, and Proposition 4.7, respectively, in [8] for the precise results). The approach is entirely different from ours and is again based on hyperkähler geometry, specifically the concept of a constant cycle subvariety of a hyperkähler manifold.…”

“…In the case of[8, Proposition 4.6], the author actually does not need to assume that X is generic, but instead uses a clever deformation argument with relative moduli spaces of stable sheaves to prove his analogue of Theorem 1.2 only for moduli of Gieseker stable sheaves. This is another benefit of our method; although, as the author points out, one can obtain the same generality as in Theorem 1.2 using the construction of relative moduli spaces of Bridgeland stable objects in the forthcoming article[5] …”