On the locus of smooth plane curves with a fixed automorphism group

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 non-trivial group) is isomorphic to a subgroup of Aut(δ), the full automorphism group of δ, and let Mg(G) be the subset of curves δ such that G ∼ = Aut(δ). Now, for an integer d ≥ 4, let M P l 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 M P l g (G) := M P l g ∩ Mg(G) and M P l g (G) := Mg(G) ∩ M P l g . In this paper, we study some aspects of the irreducibility of M P l 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 M P l g (G). In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus M P l g (G) is represented by a single "normal form". Henn, in [12], and in [14], observed that M P l 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 M P l g (Z/(d − 1)Z) is not ES-Irreducible for any odd integer d ≥ 5, and the number of its irreducible components is at least two. Furthermore, we conclude the previous result when d = 6 for the locus M P l 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.