The Generic Isogeny Decomposition of the Prym Variety of a Cyclic Branched Covering

Abstract

Let $f:S^{\prime }⟶S$ be a cyclic branched covering of smooth projective surfaces over $\mathbb{C}$ whose branch locus $\Delta \subset S$ is a smooth ample divisor. Pick a very ample complete linear system $|\mathcal{H}|$ on $S$, such that the polarized surface $(S,|\mathcal{H}|)$ is not a scroll nor has rational hyperplane sections. For the general member $[C]\in |\mathcal{H}|$ consider the ${\mu }_n$-equivariant isogeny decomposition of the Prym variety $Prym(C^{\prime }/C)$ of the induced covering $f:C^{\prime }=f^{-1}(C)⟶C$:\[Prym(C'/C)\sim\prod_{d|n,\ d\neq1}\mathcal{P}_{d}(C'/C).\] We show that for the very general member $[C]\in |\mathcal{H}|$ the isogeny component ${\mathcal{P}}_d(C^{\prime }/C)$ is ${\mu }_d$-simple with $En{d}_{{\mu }_d}({\mathcal{P}}_d(C^{\prime }/C))\cong \mathbb{Z}[{\zeta }_d]$. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map ${\mathcal{P}}_d(C^{\prime }/C)\subset Jac(C^{\prime })⟶Alb(S^{\prime })$.

arXiv comment13 pages, corrected a typo in the paper's title, comments are welcome