On the Locus of Smooth Plane Curves with a Fixed Automorphism Group

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), the full automorphism group of δ, and let M̃g(G) be the subset of curves δ such that G ∼= Aut(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 such 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 . In this paper, we study some aspects of the irreducibility of M̃Pl g (G) and its interrelation with the existence of “normal forms”, i.e. non-singular plane equations (depending on a set of parameters) such that a specialization of the parameters gives a certain non-singular plane model associated to the elements of M̃Pl g (G). In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus M̃Pl g (G) is represented by a single “normal form”. Henn, in [11], and Komiya-Kuribayashi, in [13], observed that M̃Pl 3 (G) is ES-Irreducible. In this paper we prove that this phenomena does not occur for any odd d > 4. More precisely, let Z/mZ be the cyclic group of order m, we prove that, for any odd integer d ≥ 5, ̃ MPl g (Z/(d− 1)Z) is not ES-Irreducible and the number of the irreducible components of such loci is at least two. Furthermore, we conclude the previous result when d = 6 for the locus ̃ MPl 10 (Z/3Z). Lastly, we prove the analogy of these statements when K is any algebraically closed field of positive characteristic p such that p > (d− 1)(d− 2) + 1.