2012/11/18 by Enrico Arbarello, Arbarello, Enrico, Giulia Saccà +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #math.AG
paper · pdf · doi:10.48550/arxiv.1211.4268
46 pages. Some sections rearranged, one section added, a few minor fixes
openalex publication_date 2012/11/18 · arxiv created 2013/05/26 · arxiv updated 2013/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given an Enriques surface T, its universal K3 cover f: S→ T, and a genus g linear system |C| on T, we construct the relative Prym variety PH=\Prymv, H(\D/\CC), where \CC→ |C| and \D→ |f^*C| are the universal families, v is the Mukai vector (0,[D], 2-2g) and H is a polarization on S. The relative Prym variety is a (2g-2)-dimensional possibly singular variety, whose smooth locus is endowed with a hyperkähler structure. This variety is constructed as the closure of the fixed locus of a symplectic birational involution defined on the moduli space Mv,H(S). There is a natural Lagrangian fibration η: PH → |C|, that makes the regular locus of PH into an integrable system whose general fiber is a (g-1)-dimensional (principally polarized) Prym variety, which in most cases is not the Jacobian of a curve. We prove that if |C| is a hyperelliptic linear system, then PH admits a symplectic resolution which is birational to a hyperkähler manifold of K3[g-1]-type, while if |C| is not hyperelliptic, then PH admits no symplectic resolution. We also prove that any resolution of PH is simply connected and, when g is odd, any resolution of PH has h2,0-Hodge number equal to one.