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

作者：Badr Eslam^*; Bars Francesc^*

来源：International Journal of Algebra and Computation, 2016, 26(2): 399-433.

DOI：10.1142/S0218196716500168

• Non-singular curves
• plane models
• automorphism groups
• moduli spaces

Let M-g be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by M-g(G) the subset of M-g of curves delta such that G (as a finite nontrivial group) is isomorphic to a subgroup of Aut(delta) and let <(Mg(G))over tilde> be the subset of curves delta such that G (similar or equal to) Aut(delta), where Aut(delta) is the full automorphism group of delta. Now, for an integer d >= 4, let M-g(Pl) be the subset of M-g 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 M-g(Pl) (G) := M-g(Pl) boolean AND Mg(G) and (MPl g) over tilde (G) := <(Mg(G))over tilde>Mg(G) boolean AND M-g(Pl).
In this paper we first determine, for an arbitrary but a fixed degree d, an algorithm to list the possible values m for which M-g(Pl) (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, d(2) - 3d + 3, (d - 1)(2), d(d - 2) or d(d - 1). Secondly, consider a curve delta is an element of M-g(Pl) with g = 1/2 (d - 1)(d - 2) such that Aut(delta) 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 delta is an element of M-g(Pl) (G) and also we determine the locus <(MPl g (G))over tilde> in these situations. Moreover, we work with the same question when Aut(delta) has an element of "large" order: ld, l(d - 1) or l(d - 2) with l >= 2 an integer.

• 出版日期2016-3