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

We prove that the ramified Prym map $mathcal P_{g, r}$ which sends a covering $pi:Dlongrightarrow C$ ramified in $r$ points to the Prym variety $P(pi):=text{Ker}(text{Nm}_{pi})$ is an embedding for all $rge 6$ 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.