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

2019/01/13 by Howard Nuer, Nuer, Howard, Kōta Yoshioka +1 · 3 citations
Mathematics · #14D20 (primary) #14E30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1901.04848

openalex publication_date 2019/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space M_σ(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 M_τ(v) and M_σ(v) are birational. As a consequence, we show that for primitive v of odd rank M_σ(v) is birational to a Hilbert scheme of points. Similarly, in even rank we show that M_σ(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 θ_v,σ:v^⊥\toPic(M_σ(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).

Cited by

• EPW varieties as moduli spaces on ordinary GM surfaces and special GM threefolds
• Nef cones of Hilbert schemes of points on some K3 surfaces
• Preservation of stability under the Fourier-Mukai transform whose kernel is the Poincare line bundle

Related

• Birational geometry of moduli spaces of stable objects on Enriques\n surfaces
• Stable sheaves on bielliptic surfaces: from the classical to the modern
• The birational geometry of moduli space of sheaves on the projective plane
• Projectivity and Birational Geometry of Bridgeland moduli spaces
• Bridgeland-Stable Moduli Spaces for K-Trivial Surfaces