Plane non-singular curves with an element of "large" order in its automorphism group
Badr, Eslam (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Bars Cortina, Francesc (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Data:	2016
Resum:	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.
Ajuts:	Ministerio de Economía y Competitividad MTM2013-40680-P
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Llengua:	Anglès
Document:	Article ; recerca ; Versió sotmesa a revisió
Matèria:	Non-singular curves ; Plane models ; Automorphism groups ; Moduli spaces
Publicat a:	International Journal of Algebra and Computation, Vol. 26, Issue 2 (March 2016) , p. 399-433, ISSN 0218-1967

DOI: 10.1142/S0218196716500168

Preprint 25 p, 374.7 KB

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