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 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),...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2016 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:240652 |
| Acceso en línea: | https://ddd.uab.cat/record/240652 https://dx.doi.org/urn:doi:10.1142/S0218196716500168 |
| Access Level: | acceso abierto |
| Palabra clave: | Non-singular curves Plane models Automorphism groups Moduli spaces |
| Sumario: | 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 MPlg be the subset of Mg representing smooth, genus g curves that admit a non-singular plane model of degree d (in this case, g = (d-1)(d-2)/2) and consider the sets MPlg(G):= MPlg∩Mg(G) and MPlg(G)˜:= Mg(G)˜∩MPlg. In this paper we first determine, for an arbitrary but a fixed degree d, an algorithm to list the possible values m for which MPlg(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∈MPlg 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∈MPlg(G)˜ and also we determine the locus MPlg(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. |
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