Automorphism Groups of Non-singular Plane Curves of Degree 5

Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite non-trivial group) is isomorphic to a subgroup of Aut(δ) and let M̃g(G) be the subset of curves δ such that G ∼= Aut(δ) where Aut(δ) is the full automorphism group of δ. Now, for an integer d ≥ 4, let MPl g be the subset of Mg representing smooth, genus g plane curves of degree d (in this case, g = (d−1)(d−2)/2) and consider the sets MPl g (G) := MPl g ∩Mg(G) and M̃Pl g (G) := M̃g(G) ∩MPl g . Henn in [7] and Komiya-Kuribayashi in [10], listed the groups G for which M̃Pl 3 (G) is non-empty. In this paper, we determine the loci M̃Pl 6 (G), corresponding to non-singular degree 5 projective plane curves, which are non-empty. Also, we present the analogy results of Henn for quartic curves concerning non-singular plane model equations associated to these loci. Similar arguments can be applied to deal with higher degrees.