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