The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Abstract

Let f: S′⟶ S be a cyclic branched covering of smooth projective surfaces over C whose branch locus Δ ⊂ S is a smooth ample divisor. Pick a very ample complete linear system | H| on S, such that the polarized surface (S, | H|) is not a scroll nor has rational hyperplane sections. For the general member [C] ∈ | H| consider the μn-equivariant isogeny decomposition of the Prym variety Prym(C′/C) of the induced covering f: C′: = f- 1(C) ⟶ C: Prym(C′/C)∼∏d|n,d≠1Pd(C′/C).We show that for the very general member [C] ∈ | H| the isogeny component Pd(C′/ C) is μd-simple with Endμd(Pd(C′/C))≅Z[ζd]. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map Pd(C′/C)⊂Jac(C′)⟶Alb(S′). © 2022, The Author(s).