Bielliptic curves of genus 3 in the hyperelliptic moduli

TL;DR

This paper characterizes the intersection of bielliptic genus 3 curves with hyperelliptic moduli, providing explicit equations, parametrizations, and models over their fields of moduli, enhancing understanding of their algebraic and automorphism structures.

Contribution

It determines the explicit equation and birational parametrization of the intersection space of bielliptic genus 3 curves with hyperelliptic moduli, including automorphism group analysis and models over fields of moduli.

Findings

The intersection space is an irreducible, 3-dimensional rational algebraic variety.

Explicit equations in terms of $Gl(2, k)$-invariants are provided.

Models over the field of moduli are constructed for curves with automorphism group size greater than 4.

Abstract

In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $\S$ is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the $Gl(2, k)$-invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization of $\S$. We also compute all possible subloci of curves for all possible automorphism group $G$. Moreover, for every rational moduli point $\p \in \S$, such that $| \Aut (\p) | > 4$, we give explicitly a rational model of the corresponding curve over its field of moduli in terms of the $Gl(2, k)$-invariants.