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

## TL;DR

This paper investigates the Prym map for degree 4 cyclic covers of hyperelliptic curves, proving injectivity for genus g ≥ 3 and describing fibers for genus 2, along with new descriptions of the moduli space.

## Contribution

It establishes the injectivity of the Prym map on a specific moduli component for genus g ≥ 3 and characterizes fibers for genus 2, providing new descriptions of the moduli space.

## Key findings

- Prym map is injective for g ≥ 3.
- Fibers for g=2 are mostly projective lines minus 8 points.
- New equations and descriptions of hyperelliptic covers.

## Abstract

In this paper, we study the Prym map associated to degree 4 \'etale cyclic covers of genus $g$ hyperelliptic curves restricted to the irreducible component $\mathcal{RH}_g[4]^{hyp}$ of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for $g \geq 3$ the Prym map is injective on $\mathcal{RH}_g[4]^{hyp}$. In the case $g=2$ (where $\mathcal{RH}_2[4]^{hyp} = \mathcal{RH}_2[4]$) 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 $\mathcal{RH}_g[4]^{hyp}$ in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers.