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We give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura variety of \({{\mathcal {A}}}_{g-1}\), contained in the Prym locus. First we give such a bound for a germ passing through a Prym variety of a k-gonal curve in terms of the gonality k. Then we deduce a bound only depending on the genus g.

A bound on the dimension of a totally geodesic submanifold in the Prym locus