Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve

Abstract

A projective nonsingular plane algebraic curve of degree \(d\ge 4\) is called maximally symmetric if it attains the maximum order of the automorphism groups for complex nonsingular plane algebraic curves of degree \(d\). For \(d\le 7\), all such curves are known. Up to projectivities, they are the Fermat curve for \(d=5,7\); see Kaneta et al. (RIMS Kokyuroku 1109:182–191, 1999) and Kaneta et al. (Geom. Dedic. 85:317–334, 2001), the Klein quartic for \(d=4\), see Hartshorne (Algebraic Geometry. Springer, New York, 1977), and the Wiman sextic for \(d=6\); see Doi et al. (Osaka J. Math. 37:667–687, 2000). 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 \(d\ge 8\) showing that the Fermat curve is the unique maximally symmetric nonsingular curve of degree \(d\) with \(d\ge 8\), up to projectivity. For \(d=11,13,17,19\), this characterization of the Fermat curve has already been obtained; see Kaneta et al. (Geom. Dedic. 85:317–334, 2001).