Klein coverings over hyperelliptic genus 3 curves

TL;DR

This work characterizes the moduli of étale Klein coverings of hyperelliptic genus 3 curves by distinguishing isotropic and non-isotropic Klein subgroups of 2-torsion, revealing four distinct cases. For each case, the authors analyze the Prym variety, its polarization, and the isotypical decomposition to recover the covering from the Prym data, establishing Prym-map injectivity on each irreducible component. They then show that, globally, the Klein Prym maps on genus 3 are generically finite, with degree one on the hyperelliptic-type loci I.2, II.1, and II.2, and provide a fibered-product framework to reconstruct coverings from Prym data. These results yield a concrete route to understanding the Prym map’s fibers and demonstrate injectivity and finite-ness properties across the four Klein-covering types, with implications for the geometry of corresponding abelian varieties.

Abstract

We characterize the moduli space of étale 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 étale Klein coverings of genus 3 curves is generically finite.

Theorems & Definitions (62)

• Theorem 1
• Theorem 2
• Lemma 1.1
• Definition 1.2
• Lemma 1.3
• proof
• Lemma 1.4
• proof
• Lemma 1.5
• proof