Prym maps of cyclic coverings of hyperelliptic curves

TL;DR

The paper studies Prym maps for etale cyclic coverings of hyperelliptic curves, establishing global injectivity for even degrees d≥6 that are not odd prime powers and generic injectivity for odd degrees d≥9, with injectivity when d has multiple prime divisors. A dihedral group action on Prym varieties is developed to reconstruct coverings from Prym data, with detailed analyses for both even and odd degrees and a reduction to prime‑degree cases. These methods yield a complete picture of Prym map fibers in genus 2 and unify several existing results across degree types. The work advances Prym theory for higher degree cyclic coverings by providing a constructive approach to recover coverings from their Prym varieties and clarifying the boundary between injectivity and generic injectivity.

Abstract

We prove that the Prym map corresponding to étale cyclic coverings of hyperelliptic curves is injective whenever the degree of the covering $d \geq 6$ is not a power of an odd prime. For other degrees $d\geq 9$, 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.

Theorems & Definitions (34)

• Theorem 1.1: Theorem \ref{['maineven']}
• Theorem 1.2: Theorem \ref{['mainodd']} and Theorem \ref{['mainodd2primes']}
• Proposition 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1