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

Given an Enriques surface $T$, its universal K3 cover $f:S\to T$, and a genus $g$ linear system $\mathrm{\mid }C\mathrm{\mid }$ on $T$, we construct the relative Prym variety $P_H={\text{Prym}}_{v,H}(\text{D}\mathrm{/}\text{CC})$, where $\text{CC}\to \mathrm{\mid }C\mathrm{\mid }$ and $\text{D}\to \mathrm{\mid }{f}^{\ast }C\mathrm{\mid }$ 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 hyperk"ahler 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 $\eta :P_H\to \mathrm{\mid }C\mathrm{\mid }$, 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 $\mathrm{\mid }C\mathrm{\mid }$ is a hyperelliptic linear system, then $P_H$ admits a symplectic resolution which is birational to a hyperk"ahler manifold of K3${}^{[g-1]}$-type, while if $\mathrm{\mid }C\mathrm{\mid }$ 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.