Tools for the computation of families of coverings. (English) Zbl 1016.14012

Couveignes, Jean-Marc

Völklein, Helmut (ed.) et al., Aspects of Galois theory. Papers from the conference on Galois theory, Gainesville, FL, USA, October 14-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 256, 38-65 (1999).

The author is interested by methods to compute algebraic models for coverings of the projective line (over \(\mathbb{C})\) with a prescribed ramification data. Such coverings are described by moduli spaces, the Hurwitz spaces. The specification of the ramification data implies a lot of calculations, to compute the coefficients of the equations defining the coverings, and, unless for small cases, it is not possible to make the calculations, to solve the nonlinear equations giving the coefficients. J.-M. Couveignes and L. Granboulan [in: The Grothendieck theory of dessins d’enfants, Lond. Math. Soc. Lect. Notes Ser. 200, 79-113 (1994; Zbl 0835.14010)] have given a today famous example of computation of a covering of \(\mathbb{P}^1_\mathbb{C}\) with 4 branch points and with a big monodromy group (Mathieu group of degree 24). The arguments and the different steps of this coverings are explained, and the paper contains also a list of other effective methods.
Note that the pages of the paper are not well numbered, three of them have to be changed following the cycle (51, 52, 53): page 51 is in fact page 52, this one being page 53 and this last one being page 51.
For the entire collection see [Zbl 0941.00014].

Reviewer: Marc Reversat (Toulouse)

MSC:

14H30	Coverings of curves, fundamental group
14E22	Ramification problems in algebraic geometry
14E20	Coverings in algebraic geometry
12F12	Inverse Galois theory

Keywords:

coverings of the projective line; Hurwitz spaces; ramification

Citations:

Zbl 0835.14010