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

摘要

A projective nonsingular plane algebraic curve of degree d ≥ 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 ≤ 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 ≥ 8 showing that the Fermat curve is the unique maximally symmetric nonsingular 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 Kaneta et al. (Geom. Dedic. 85:317–334, 2001).