The Fermat curve x^n+y^n+z^n: the most symmetric non-singular algebraic plane curve

TL;DR

This paper proves that for algebraic curves of degree d>=8 over complex numbers, the Fermat curve uniquely maximizes symmetry among non-singular plane curves, extending known results for degrees up to 7.

Contribution

It extends the classification of maximally symmetric non-singular plane algebraic curves to all degrees d>=8, showing the Fermat curve's uniqueness beyond previously known degrees.

Findings

Fermat curve is the unique maximally symmetric non-singular curve for d>=8.

The result generalizes known classifications for degrees up to 7.

The characterization holds over algebraically closed fields of characteristic zero.

Abstract

A projective non-singular plane algebraic curve of degree d<=4 is called maximally symmetric if it attains the maximum order of the automorphism groups for complex non-singular plane algebraic curves of degree d. For d<=7, all such curves are known. Up to projectivities, they are the Fermat curve for d=5,7, see \cite{kmp1,kmp2}, the Klein quartic for d=4, see \cite{har}, and the Wiman sextic for d=6, see \cite{doi}. In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every d>=8 showing that the Fermat curve is the unique maximally symmetric non-singular curve of degree d with d>=8, up to projectivity. For d=11,13,17,19, this characterization of the Fermat curve has already been obtained, see \cite{kmp2}.