  In this paper we give a bound on the dimension of a totally geodesic
submanifold of the moduli space of polarised abelian varieties of a given
dimension, which is contained in the Prym locus of a (possibly) ramified double
cover. This improves the already known bounds. The idea is to adapt the
techniques introduced by the authors in collaboration with A. Ghigi and G. P.
Pirola for the Torelli map to the case of the Prym maps of (ramified) double
covers.

Algebraic Geometry