Plane Non-singular Curves with an Element of “large” Order in Its 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) 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 . In this note we first determine, for an arbitrary but a fixed degree d, an algorithm to list the possible values m for which MPl g (Z/m) is non-empty where Z/m denotes the cyclic group of order m. In particular, we prove that m should divide one of the integers: d− 1, d, d2 − 3d+3, (d− 1)2, d(d− 2) or d(d− 1). Secondly, consider a curve δ ∈ MPl g with g = (d− 1)(d− 2)/2 such that Aut(δ) has an element of “very large” order, in the sense that this element is of order d2 − 3d + 3, (d − 1)2, d(d − 2) or d(d − 1). Then we investigate the groups G for which δ ∈ M̃Pl g (G) and also we determine the locus M̃Pl g (G) in these situations. Moreover, we work with the same question when Aut(δ) has an element of “large” order ld, l(d− 1) or l(d− 2) with l ≥ 2 an integer.