Galois lines for a canonical curve of genus 4, III: non-cyclic Galois 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 $l \subset \mathbb{P}^3$, we consider the projection $\pi_l: C \to \mathbb{P}^1$ from $l$ and the induced extension of function fields $\pi_l^*: k(\mathbb{P}^1)\hookrightarrow k(C)$. A line $l$ is called an \emph{$S_3$-line} (resp. a \emph{$K_4$-line}) if the extension $k(C)/\pi_l^*(k(\mathbb{P}^1))$ is Galois and its Galois group is isomorphic to the symmetric group $S_3$ on three letters (resp. the Klein four-group $K_4$). We prove that the number of $S_3$-lines (resp.\ $K_4$-lines) is at most $10$ (resp.\ $15$).