# Second fundamental form of the Prym map in the ramified case

## TL;DR

This paper investigates the second fundamental form of the Prym map in the ramified case, providing a new expression relating it to the Torelli map and deriving bounds on special subvarieties within the Prym locus.

## Contribution

It introduces an explicit expression for the second fundamental form of the Prym map in the ramified case and uses it to bound the dimension of totally geodesic submanifolds.

## Key findings

- Derived an expression linking the Prym and Torelli second fundamental forms.
- Provided an upper bound for the dimension of totally geodesic submanifolds.
- Gave insights into the structure of Shimura subvarieties in the Prym locus.

## Abstract

In this paper we study the second fundamental form of the Prym map $P_{g,r}: R_{g,r} \rightarrow {\mathcal A}^{\delta}_{g-1+r}$ in the ramified case $r>0$. We give an expression of it in terms of the second fundamental form of the Torelli map of the covering curves. We use this expression to give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura subvariety of ${\mathcal A}^{\delta}_{g-1+r}$, contained in the Prym locus.