Fibrations of K3-surfaces and Belyi-maps

Abstract

In a paper by Miranda and Persson [MP89], the authors study semi-stable elliptic fibrations over P1 of K3-surfaces with 6 singular fibres. In their paper the authors give a list of possible fiber types for such fibrations. It turns out that there are 112 cases. The corresponding J-invariant is a so-called Belyi-function. More particularly, J is a rational function of degree 24, it ramifies of order 3 in every point above 0, it ramifies of order 2 in every point above 1, and the only other ramification occurs above infinity. To every such map we can associate a so-called ’dessin d’enfant’ (a name coined by Grothendieck) which in its turn uniquely determines the Belyi map. If f : C → P1 is a Belyi map, the dessin is the inverse image under f of the real segment [0,1]. Several papers, e.g. [Ir03],[Schu04], [TY04], have been devoted to the calculation of some of the rational J-invariants for the Miranda-Persson list. It turns out that explicit calculations quickly become too cumbersome (even for a computer) if one is not careful enough. The goal of this paper is to compute all J-invariants corresponding to the Miranda-Persson list. We use a trick which enables us to reduce the calculation to the solution of a set of three polynomial equations in three unknowns (see Section 7 for details).