Projectivity and Birational Geometry of Bridgeland Moduli spaces on an Enriques Surface

Abstract

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 Macrì'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: 1) We obtain a region in the ample cone of the moduli space of Gieseker-stable sheaves which works for all unnodal Enriques surfaces. 2) 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 $\varphi$-function. 3) We recover some classical results on linear systems on Enriques surfaces and obtain some new ones about $n$-very ample line bundles.

52 pages, 1 figure. Fixed a mistake about primitive vectors of even rank and added some more explanation to the proof of Corollary 13.2. Comments Welcome! arXiv admin note: text overlap with arXiv:1203.4613 by other authors

References in corpus (5)

• Bridgeland-Stable Moduli Spaces for K-Trivial Surfaces
• Birational Geometry of Singular Moduli Spaces of O'Grady Type
• A Note on the existence of stable vector bundles on Enriques surfaces
• Singularities and Kodaira dimension of moduli scheme of stable sheaves on Enriques surfaces
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• MMP Via Wall-crossing for Moduli Spaces of Stable Sheaves on an Enriques surface
• Nef divisors for moduli spaces of complexes with compact support
• On upper bounds of Manin type
• A note on stable sheaves on Enriques surfaces II
• Gauss-Prym maps on Enriques surfaces
• Stable sheaves on bielliptic surfaces: from the classical to the modern
• A local compactification of the Bridgeland stability manifold
• On the birational geometry of Hilbert schemes of points and Severi divisors
• Wall-Crossing implies Brill-Noether. Applications of stability conditions on surfaces
• On two families of Enriques categories over K3 surfaces