On the Prym map of degree 4 cyclic covers of hyperelliptic curves

2025/08/28 by Anatoli Shatsila, Shatsila, Anatoli · 1 citation
Engineering · Mathematics · #14H30 #14H40 #14H45 #14H55 #14K12 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2508.20838

openalex publication_date 2025/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus g hyperelliptic curves restricted to the irreducible component RH_g[4]^hyp of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for g ≥ 3 the Prym map is injective on RH_g[4]^hyp. In the case g=2 (where RH₂[4]^hyp = RH₂[4]) we prove that non-empty fibers of the Prym map, apart from two exceptional fibers, are isomorphic to the projective line without 8 points. Moreover, we obtain a new description of the space RH_g[4]^hyp in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers.

Citations

• Prym maps of cyclic coverings of hyperelliptic curves
• Simplicity of some Jacobians with many automorphisms
• Hyperelliptic genus 3 curves with involutions and a Prym map
• Cyclic Coverings of genus 2 curves of Sophie Germain type
• Klein coverings of genus 2 curves
• Variétés de Prym associées aux revêtements n-cycliques d'une courbe hyperelliptique

Cited by

• Prym maps of cyclic coverings of hyperelliptic curves

Related

• Prym maps of cyclic coverings of hyperelliptic curves
• Prym varieties of étale covers of hyperelliptic curves
• The fibres of the Prym map of étale cyclic coverings of degree 7
• Pryms of ℤ₃×ℤ₃ coverings of genus 2 curves
• Prym varieties of cyclic coverings