Automorphism groups of smooth plane curves

Abstract

The author classifies finite groups acting on smooth plane curves of degree at least four. Furthermore, he gives some upper bounds for the order of automorphism groups of smooth plane curves and determines the exceptional cases in terms of defining equations. This paper also contains a simple proof of the uniqueness of smooth plane curves with the full automorphism group of maximum order for each degree.

On the automorphism group of non-singular plane curves fixing the degree

“…Theorem 7 (Harui). (see [11] §2) Let G be a subgroup of Aut(C). Then G satisfies one of the following statements:…”

“…It follows by Mitchell [16] §5, that Aut( C) should fix a point, a line or a triangle. If Aut( C) fixes a triangle and neither a line nor a point is leaved invariant then, by Harui [11] §5, C is a descendant of the Fermat curve F 5 or the Klein curve K 5 but this is impossible because 4 ∤ |Aut(F 5 )|(= 150) and 4 ∤ |Aut(K 5 )|(= 39). Therefore, Aut( C) should fix a line and a point off that line.…”

“…The motivation of this work is that we did not expect M P l g (G) to be ES-Irreducible in general. In order to construct counter examples where M P l g (G) is not ES-Irreducible: we need first, a group G such that there exist at least two injective representations ρ i : G ֒→ P GL 3 (K) with i = 1, 2 which are not conjugate (i.e there is no transformation P ∈ P GL 3 (K) with P ρ 1 (G)P −1 = ρ 2 (G), more details are included in §2) and for the zoo of groups that could appear for non-singular plane curves [11], we consider G a cyclic group of order m. Secondly, one needs to prove the existence of non-singular plane curves with automorphism group conjugate to ρ i (G) for each i = 1, 2.…”

“…Abstract. This note is devoted, after the result of Harui [11], to solve the following questions for non-singular plane curve of degree d over an algebraically closed field K of zero characteristic.…”

“…The results of Harui in [11] simplify the problem to a certain list of finite groups that one should work with. These groups can be classified up to conjugation as; cyclic groups, an element of Ext 1 (−, −) of a cyclic group by a dihedral group, an alternating group A 4 (resp.…”