Prym maps of cyclic coverings of hyperelliptic curves

We prove that the Prym map corresponding to étale cyclic coverings of hyperelliptic curves is injective whenever the degree of the covering $d\ge 6$ is not a power of an odd prime. For other degrees $d\ge 9$, we show that the Prym map is generically injective. In particular, we complete the study of Prym maps of étale cyclic coverings of genus 2 curves.