Constructing of relative Prym varieties associated to a linear system on an Enriques surface

In a joint work with Giulia Saccà and Andrea Ferretti, we construct relative Prym varieties associated to a linear system on an Enriques surface. These are singular symplectic varieties whose smooth locus is a completely integrable Hamiltonian system. We describe these varieties in the hyperelliptic and non-hyperelliptic case. We show that their non-singular model is simply connected and possesses a unique holomorphic 2-form, up to scalars. The analysis of singularities in the hyperelliptic case leads to Nakajima's quiver varieties.