Automorphism group of plane curve computed by Galois points, II

TL;DR

This paper constructs examples of smooth plane curves with automorphism groups of order 60d using Galois points, expanding understanding of automorphism groups classified by degree.

Contribution

It provides explicit examples and determines the structure of automorphism groups for curves with order 60d, building on prior classification results.

Findings

Constructed smooth plane curves with automorphism group order 60d

Determined the structure of these automorphism groups

Applied Galois points method to achieve these results

Abstract

Recently, the first author classified finite groups obtained as automorphism groups of smooth plane curves of degree $d \ge 4$ into five types. He gave an upper bound of the order of the automorphism group for each types. For one of them, the type (a-ii), that is given by $\max \left\{ 2 d (d - 2), 60 d \right\}$. In this article, we shall construct typical examples of smooth plane curve $C$ by applying the method of Galois points, whose automorphism group has order $60 d$. In fact, we determine the structure of the automorphism group of those curves.