Automorphism Groups of Nonsingular Plane Curves of Degree 5

Abstract

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 nontrivial group) is isomorphic to a subgroup of Aut(δ), and let 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 be the subset of Mg representing smooth, genus g, plane curves of degree d, i.e. smooth curves that admits a plane non-singular model of degree d, (in this case, g = (d − 1)(d − 2)/2), and consider the sets and .Henn in [7] and Komiya and Kuribayashi in [10], listed the groups G for which is nonempty. In this article, we determine the loci , corresponding to nonsingular degree 5 projective plane curves, which are nonempty. Also, we present the analogy of Henn's results for quartic curves concerning nonsingular plane model equations associated to these loci (see Table 2 for more details). Similar arguments can be applied to deal with higher degrees.