Given a finite morphism between smooth projective curves one can canonically associate to it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map. It is a classical result that the Prym map is generically injective for étale double coverings over curves of genus at least 7.
In this talk I will show the global injectivity of the Prym map for ramified double coverings over curves of genus g ≥ 1 and ramified in at least 6 points. This is a joint work with J.C. Naranjo.
I will finish with an overview on what is known for the degree of the Prym map for ramified cyclic coverings of degree d ≥ 2.