Projective plane curves whose automorphism groups are simple and primitive

TL;DR

This paper characterizes when complex projective plane curves are invariant under certain simple, primitive automorphism groups, providing conditions on the degree for the existence of nonsingular curves.

Contribution

It establishes necessary and sufficient conditions on the degree for the existence of nonsingular curves invariant under specific simple primitive groups in PGL(3,C).

Findings

Conditions on degree d for nonsingular invariant curves.

Analysis of automorphism groups A6, A5, PSL(2,7).

Extension to integral curves.

Abstract

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $PGL(3, \mathbb{C})$, which is isomorphic to $A_{6}$, $A_{5}$ or $PSL(2, \mathbb{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.