Finiteness and injectivity of Prym maps for cyclic coverings

Abstract

The structure of the Prym map for coverings of degrees $d\geq 3$ is mostly unknown. Only recently, under mild numerical assumptions, a generic injectivity of the Prym maps for étale cyclic coverings of hyperelliptic curves of prime degrees has been shown. In the paper, we prove that the Prym maps is generically injective for all remaining degrees (i.e. composite numbers $d\geq 6$) and we prove global injectivity if $d$ is not a power of an odd prime. In particular, we complete the study of Prym maps of étale cyclic coverings of genus 2 curves. As an application, we fully characterise for which $d$ the Prym map of cyclic coverings of degree $d$ of genus $3$ curves is generically finite and we conjecture that it is injective.