Moduli of curves on Enriques surfaces

Abstract

We compute the number of moduli of all irreducible components of the moduli space of smooth curves on Enriques surfaces. In most cases, the moduli maps to the moduli space of Prym curves are generically injective or dominant. Exceptional behavior is related to existence of Enriques–Fano threefolds and to curves with nodal Prym-canonical model.