Klein coverings over hyperelliptic genus 3 curves

TL;DR

This paper characterizes the moduli space of étale Klein coverings over hyperelliptic genus 3 curves and proves the Prym map's injectivity and generic finiteness, advancing understanding of these coverings' geometry.

Contribution

It provides a detailed description of the moduli space of étale Klein coverings of hyperelliptic genus 3 curves and establishes key properties of the Prym map.

Findings

Prym map on each component is injective

Prym map of étale Klein coverings of genus 3 is generically finite

Characterization of the moduli space of Klein coverings

Abstract

We characterize the moduli space of \'etale Klein coverings (i.e. Galois with deck group $\mathbb{Z}_2^2$) of hyperelliptic curves of genus 3. We prove that the Prym map on each component is injective. As an application, we show that the Prym map of \'etale Klein coverings of genus 3 curves is generically finite.