Galois lines for a canonical curve of genus 4, II: 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 zero. For a line $l$, 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 referred to as a cyclic line if the extension $k(C)/{\pi }_l^{\ast }(k({\mathbb{P}}^1))$ is cyclic. A line $l\subset {\mathbb{P}}^3$ is said to be skew if $C\cap l=\varnothing$. We prove that the number of skew cyclic lines is equal to $0,1,3$ or $9$. We determine curves that have nine skew cyclic lines.