Translator Disclaimer
June 2015 Relative Prym varieties associated to the double cover of an Enriques surface
E. Arbarello, G. Saccà, A. Ferretti
J. Differential Geom. 100(2): 191-250 (June 2015). DOI: 10.4310/jdg/1430744121

Abstract

Given an Enriques surface $T$, its universal $\mathrm{K3}$ cover $f : S \to T$, and a genus $g$ linear system $\lvert C \rvert$ on $T$, we construct the relative Prym variety $P_H = \textrm{Prym}_{v, H} (\mathcal{D/C})$, where $\mathcal{C} \to \lvert C \rvert$ and $\mathcal{D} \to \lvert f*C \rvert$ 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 $M_{v,H} (S)$. There is a natural Lagrangian fibration $\eta : P_H \to \lvert C \rvert$ that makes the regular locus of $P_H$ 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 $\lvert C \rvert$ is a hyperelliptic linear system, then $P_H$ admits a symplectic resolution which is birational to a hyperkähler manifold of $K3^{[g-1]}$-type, while if $\lvert C \rvert$ is not hyperelliptic, then $P_H$ admits no symplectic resolution. We also prove that any resolution of $P_H$ is simply connected and, when $g$ is odd, any resolution of $P_H$ has $h^{2,0}$-Hodge number equal to one.

Citation

Download Citation

E. Arbarello. G. Saccà. A. Ferretti. "Relative Prym varieties associated to the double cover of an Enriques surface." J. Differential Geom. 100 (2) 191 - 250, June 2015. https://doi.org/10.4310/jdg/1430744121

Information

Published: June 2015
First available in Project Euclid: 4 May 2015

zbMATH: 1362.14035
MathSciNet: MR3343832
Digital Object Identifier: 10.4310/jdg/1430744121

Rights: Copyright © 2015 Lehigh University

JOURNAL ARTICLE
60 PAGES


SHARE
Vol.100 • No. 2 • June 2015
Back to Top