The fibers of the ramified Prym map

TL;DR

This paper investigates the fibers of the ramified Prym map, providing geometric descriptions in specific cases and presenting an explicit example of a totally geodesic curve within a fiber.

Contribution

It offers a detailed geometric analysis of the generic fibers of the ramified Prym map in six special cases and constructs an explicit example of a totally geodesic curve as a Prym fiber component.

Findings

Geometric description of generic fibers in six special cases.

Explicit example of a totally geodesic curve as a Prym fiber component.

Analysis of the dimension discrepancy between source and target spaces.

Abstract

We study the ramified Prym map $\mathcal P_{g,r} \longrightarrow \mathcal A_{g-1+\frac r2}^{\delta}$ 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 ${\mathcal P}_{1,2}$.