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

TL;DR

This paper studies the decomposition of Prym varieties associated with cyclic branched covers of surfaces, showing that for general curves, the components are simple and have endomorphism rings isomorphic to cyclotomic integers.

Contribution

It establishes the generic isogeny decomposition of Prym varieties for cyclic branched covers and proves the simplicity and endomorphism structure of the components.

Findings

The Prym variety decomposes into components indexed by divisors of n.

Each component is μ_d-simple with endomorphism ring isomorphic to olds cyclotomic integers.

Results hold for very general members of a linear system on the surface.

Abstract

Let $f\colon S'\longrightarrow S$ 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 $Prym(C'/C)$ of the induced covering $f\colon C'= f^{-1}(C)\longrightarrow C$:\[Prym(C'/C)\sim\prod_{d|n,\ d\neq1}\mathcal{P}_{d}(C'/C).\] We show that for the very general member $[C]\in|\mathcal{H}|$ the isogeny component $\mathcal{P}_{d}(C'/C)$ is $\mu_{d}$-simple with $End_{\mu_{d}}(\mathcal{P}_{d}(C'/C))\cong\mathbb{Z}[\zeta_{d}]$. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map $\mathcal{P}_{d}(C'/C)\subset Jac(C')\longrightarrow Alb(S')$.