Galois lines for a canonical curve of genus 4, I: Non-skew cyclic lines

Abstract

Let $C\subset {\mathbb{P}}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic $0$. For a line $l\subset {\mathbb{P}}^3$, we consider the projection ${\pi }_l:C\to {\mathbb{P}}^1$ with center $l$ and the extension of the function fields ${\pi }_l^{\ast }:k({\mathbb{P}}^1)\hookrightarrow k(C)$. A line $l$ is assumed to be cyclic for $C$, if the extension $k(C)/{\pi }_l^{\ast }(k({\mathbb{P}}^1))$ is cyclic. A line $l$ is assumed to be non-skew, if $C\cap l\ne \varnothing$, i.e., $\mathrm{deg}{\pi }_l<\mathrm{deg}C=6$. We investigate the number of non-skew cyclic lines for $C$. As main results, we explicitly give the equation of $C$ in the particular case in which $C$ has two cyclic trigonal morphisms; we prove that the number of cyclic lines with $\mathrm{deg}{\pi }_l=4$ is at most $1$, and the number of cyclic lines with $\mathrm{deg}{\pi }_l=5$ is at most $1$.