The fibers of the ramified Prym map

Abstract

We study the ramified Prym map ${\mathcal{P}}_{g,r}⟶{\mathcal{A}}_{g-1+\frac{r}{2}}^{\delta }$ 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 fibre. We also give an explicit example of a totally geodesic curve which is an irreducible component of a fibre of the Prym map ${\mathcal{P}}_{1,2}$.

Final version. To appear in Communications in Contemporary Mathematics

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