The fibers of the ramified Prym map

Abstract

We study the ramified Prym map $\mathscr{P}_{g, r} \rightarrow \mathscr{A}_{g-1+\frac{r}{2}}^\delta$ which assigns to a ramified double cover of a smooth irreducible curve of genus $g$ ramified in $r$ points the Prym variety of the covering. We focus on the six cases where the dimension of the source is strictly greater than the dimension of the target giving a geometric description of the generic fiber. We also give an explicit example of a totally geodesic curve which is an irreducible component of a fiber of the Prym map $\mathscr{P}_{1,2}$.