Abstract
In this paper we study the second fundamental form of the Prym map \(P_{g,r}: {\mathcal R}_{g,r} \rightarrow {\mathcal A}^{\delta }_{g-1+r}\) 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.
The first author was partially supported by MIUR PRIN 2015 “Geometry of Algebraic Varieties”. The second author was partially supported by MIUR PRIN 2015 “Moduli spaces and Lie theory”. The authors were also partially supported by GNSAGA of INdAM. AMS Subject classification: 14H10, 14K12.
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Colombo, E., Frediani, P. (2020). Second Fundamental Form of the Prym Map in the Ramified Case. In: Neumann, F., Schroll, S. (eds) Galois Covers, Grothendieck-Teichmüller Theory and Dessins d'Enfants. GGT-DE 2018. Springer Proceedings in Mathematics & Statistics, vol 330. Springer, Cham. https://doi.org/10.1007/978-3-030-51795-3_4
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