## Journal of Differential Geometry

### Relative Prym varieties associated to the double cover of an Enriques surface

#### 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.

#### Article information

Source
J. Differential Geom., Volume 100, Number 2 (2015), 191-250.

Dates
First available in Project Euclid: 4 May 2015

https://projecteuclid.org/euclid.jdg/1430744121

Digital Object Identifier
doi:10.4310/jdg/1430744121

Mathematical Reviews number (MathSciNet)
MR3343832

Zentralblatt MATH identifier
1362.14035

#### Citation

Arbarello, E.; Saccà, G.; Ferretti, A. Relative Prym varieties associated to the double cover of an Enriques surface. J. Differential Geom. 100 (2015), no. 2, 191--250. doi:10.4310/jdg/1430744121. https://projecteuclid.org/euclid.jdg/1430744121