Global Prym-Torelli for double coverings ramified in at least 6 points

Abstract

We prove that the ramified Prym map ${\mathcal{P}}_{g,r}$ which sends a covering $\pi :D⟶C$ ramified in $r$ points to the Prym variety $P(\pi ):=\text{Ker}({\text{Nm}}_{\pi })$ is an embedding for all $r\ge 6$ and for all $g(C)>0$. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that ${\mathcal{P}}_{g,2}$ and ${\mathcal{P}}_{g,4}$ have positive dimensional fibers.

Minor changes. Accepted for publication in Journal of Algebraic Geometry