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

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