Abelian covers and the second fundamental form

Abstract We give some conditions on a family of abelian covers of P¹ P 1 of genus g curves, that ensure that the family yields a subvariety of Ag 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 g > M, then the family yields a subvariety of Ag A g which is not totally geodesic. We prove then analogous results for families of abelian covers of Cₜ P¹ = Cₜ/G C ~ t → P 1 = C ~ t / G ~ with an abelian Galois group G G ~ of even order, proving that under some conditions, if G σ ∈ G ~ is an involution, the family of Pryms associated with the covers Cₜ Cₜ= Cₜ/ C ~ t → C t = C ~ t / ⟨ σ ⟩ yields a subvariety of A₏^ A p δ which is not totally geodesic. As a consequence, we show that if G= (Z/N Z) ᵐ G ~ = (Z / N Z) m with N even, and σ is an involution in G G ~, there exists an integer M (N) which only depends on N such that, if g= g (Cₜ) > M (N) g ~ = g (C ~ t) > M (N), then the subvariety of the Prym locus in { A}^ ₏ A p δ induced by any such family is not totally geodesic (hence it is not Shimura).