Smooth plane curves with a unique outer Galois point and their automorphism groups

TL;DR

This paper classifies smooth plane curves with a unique outer Galois point, detailing their equations and automorphism groups, especially focusing on non-cyclic automorphism groups over characteristic zero fields.

Contribution

It provides a complete classification of such curves, characterizing their equations and automorphism groups for each non-cyclic reduced automorphism group case.

Findings

Curves are cyclic coverings of the projective line due to the Galois point.

Full automorphism group structures are determined for each case.

Explicit equations are provided for each automorphism group type.

Abstract

We consider smooth plane curves $\mathcal{X}$ of degree $d\geq4$, defined over an algebraically closed field of characteristic $0$, that possess a unique outer Galois point. This geometric condition forces the curve to be a cyclic covering of the projective line, and ensures that its automorphism group fits into a specific theoretical framework. For each possible non-cyclic reduced automorphism group $\operatorname{Aut}_{\operatorname{red}}(\mathcal{X})$, we fully characterize the defining equation of $\mathcal{X}$ and the precise structure of its full automorphism group $\operatorname{Aut}(\mathcal{X})$. This comprehensive analysis not only identifies the exact form of the equation for each automorphism type but also establishes the detailed criteria under which these scenarios can occur, thereby offering a complete classification of defining equations for smooth plane curves with a unique outer Galois point and a non-cyclic reduced automorphism group.