Galois lines for normal elliptic space curves, II

TL;DR

This paper classifies Galois lines for linearly normal elliptic space curves in projective 3-space, revealing specific configurations and limitations on Galois points for genus one plane quartic curves.

Contribution

It explicitly determines the Galois lines and their arrangements for elliptic space curves, including special cases based on the j-invariant, and deduces properties of Galois points on genus one quartic curves.

Findings

Elliptic curves have exactly six V_4-lines.

When j(C)=1, there are eight Z_4-lines in addition.

Each plane quartic of genus one has at most one Galois point.

Abstract

For each linearly normal elliptic curve $C$ in $\mathbb P^3$, we determine Galois lines and their arrangement. The results are as follows: the curve $C$ has just six $V_4$-lines and in case $j(C)=1$, it has eight $Z_4$-lines in addition. The $V_4$-lines form the edges of a tetrahedron, in case $j(C)=1$, for each vertex of the tetrahedron, there exist just two $Z_4$-lines passing through it. We obtain as a corollary that each plane quartic curve of genus one does not have more than one Galois point.