Galois lines for a canonical curve of genus 4, III: non-cyclic Galois lines

TL;DR

This paper classifies non-cyclic Galois lines for genus 4 canonical curves in projective 3-space, establishing upper bounds on the number of lines with specific Galois groups.

Contribution

It provides bounds on the number of $S_3$-lines and $K_4$-lines for genus 4 canonical curves, expanding understanding of Galois lines in algebraic geometry.

Findings

Number of $S_3$-lines is at most 10.

Number of $K_4$-lines is at most 15.

Classifies non-cyclic Galois lines for genus 4 curves.

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.\ $15$).