Abelian covers and the second fundamental form

Abstract

We give some conditions on a family of abelian covers of \({\mathbb P}^1\) of genus g curves, that ensure that the family yields a subvariety of \({\mathsf A}_g\) which is not totally geodesic, hence it is not Shimura. As a consequence, we show that for any abelian group G, there exists an integer M which only depends on G such that if \(g >M\), then the family yields a subvariety of \({\mathsf A}_g\) which is not totally geodesic. We prove then analogous results for families of abelian covers of \({\tilde{C}}_t \rightarrow {\mathbb P}^1 = {\tilde{C}}_t/{\tilde{G}}\) with an abelian Galois group \({\tilde{G}}\) of even order, proving that under some conditions, if \(\sigma \in {\tilde{G}}\) is an involution, the family of Pryms associated with the covers \({\tilde{C}}_t \rightarrow C_t= {\tilde{C}}_t/\langle \sigma \rangle \) yields a subvariety of \({\mathsf A}_{p}^{\delta }\) which is not totally geodesic. As a consequence, we show that if \({\tilde{G}}=(\mathbb Z/N\mathbb Z)^m\) with N even, and \(\sigma \) is an involution in \({\tilde{G}}\), there exists an integer M(N) which only depends on N such that, if \({\tilde{g}}= g({\tilde{C}}_t) > M(N)\), then the subvariety of the Prym locus in \({{\mathsf A}}^{\delta }_{p}\) induced by any such family is not totally geodesic (hence it is not Shimura).