Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface

We construct moduli spaces of semistable objects on an Enriques surface for generic Bridgeland stability condition and prove their projectivity. We further generalize classical results about moduli spaces of semistable sheaves on an Enriques surface to their Bridgeland counterparts. Using Bayer and Macri`'s construction of a natural nef divisor varying with the stability condition, we begin a systematic exploration of the relation between wall-crossing on the Bridgeland stability manifold and the minimal model program for these moduli spaces. We give three applications of our machinery to obtain new information about the classical moduli spaces of Gieseker stable sheaves. We obtain a region in the ample cone of the moduli space of Gieseker stable sheaves over Enriques surfaces. We determine the nef cone of the Hilbert scheme of n points on an unnodal Enriques surface in terms of the classical geometry of its half-pencils and the Cossec-Dolgachev ;φ-function. We recover some classical results on linear systems on an unnodal Enriques surfaces and obtain some new ones about n-very ample line bundles.