Non-singular plane curves with an element of "large" order in its automorphism group

© 2016 World Scientific Publishing Company. 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 Mg(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 MgPl be the subset of Mg representing smooth, genus g curves that admit a non-singular plane model of degree d (in this case, g = 1/2(d - 1)(d - 2)) and consider the sets MgPl(G):= MgPl ⊂ Mg(G) and MgPl(G):= Mg(G) ⊂ MgPl. In this paper we first determine, for an arbitrary but a fixed degree d, an algorithm to list the possible values m for which MgPl(Z/mZ) is non-empty, where Z/mZ 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 MgPl with g = 1/2(d - 1)(d - 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 MgPl(G) and also we determine the locus MgPl(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.