Non-singular plane curves with an element of ``large'' order in its automorphism group (Q2799127)

Let \(M_g\) denote the moduli space of nonsingular, projective, geometrically irreducible genus \(g\) curves, and \(G\) a group. In this paper the authors investigate the following subsets of \(M_g\): (1) \(M_g(G)\) the set of elements \(\delta\in M_g\) such that \(G\) is isomorphic to a subgroup of \(\text{Aut}(\delta)\), (2) \(\tilde{M_g(G)}\) the set of elements \(\delta\in M_g\) such that \(G\) is isomorphic to \(\text{Aut}(\delta)\), (3) For \(d\geq 4\), \(M_g^{Pl}\) the set of elements of \(M_g\) that admits a nonsingular model of degree \(d\), (4) \(M_g^{Pl}(G):=M_g^{Pl}\cap M_g(G)\), (5) \(\tilde{M_g^{Pl}(G)}:=M_g^{Pl}\cap \tilde{M_g(G)}\).NEWLINENEWLINEFor a fixed \(d\), they first state an algorithm to decide whether \(M_g^{Pl}(\mathbb Z_m)\neq\emptyset\). In this case \(m\) should divide either \(d-1\), \(d\), \(d^2-3d+3\), \((d-1)^2\), \(d(d-2)\), or \(d(d-1\).NEWLINENEWLINELet \(\delta\in M_g^{Pl}\), \(g=(d-1)(d-2)/2\), such that \(\text{Aut}(\delta)\) has an element of order either \(d^2-3d+3\), \((d-1)^2\), \(d(d-2)\), \(d(d-1)\), \(\ell d\), \(\ell (d-1)\), or \(\ell(d-2)\) with \(\ell\geq 2\) an integer. Secondly, they investigate the groups \(G\) such that \(\delta\in \tilde{M_g^{Pl}}(G)\) and the corresponding locus.