# Moduli of curves on Enriques surfaces

## TL;DR

This paper calculates the moduli of smooth curves on Enriques surfaces, revealing cases where the moduli maps are injective or dominant, and exploring special behaviors linked to Enriques--Fano threefolds and nodal Prym-canonical models.

## Contribution

It provides a comprehensive computation of the moduli of curves on Enriques surfaces and analyzes the nature of the associated moduli maps, including exceptional cases.

## Key findings

- Most moduli maps are either generically injective or dominant.
- Exceptional cases relate to Enriques--Fano threefolds and nodal Prym-canonical models.
- The work advances understanding of the geometry 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 behaviour is related to existence of Enriques--Fano threefolds and to curves with nodal Prym-canonical model.