Monodromy of Elliptic Surfaces

Summary

We study monodromy groups of elliptic fibrations over the projective line and classify possible types for Jacobian elliptic K3 and rational surfaces.

In an algebraic family of elliptic fibrations the degree of j is bounded by the degree of the generic element. It follows that there is only a finite number of monodromy groups for each family. The number of subgroups of bounded index in SL(2, ℤ) grows superexponentially [8], similarly to the case of a free group (since SL(2, ℤ) contains a free subgroup of finite index). For T of large index, the number of M_Γ-represent at ions of the sphere 𝕊 ² is substantially smaller, however, still superexponential (see 3.5). Our goal is to introduce some combinatorial structure on the set of monodromy groups of elliptic fibrations which would help to answer some natural questions. Our original motivation was to describe the set of groups corresponding to rational or K3 elliptic surfaces, explain how to compute the dimensions of the spaces of moduli of surfaces in this class with given monodromy group etc. As a direct application of the methods developed in the present paper the authors and T. Petrov have obtained a proof of rationality and stable rationality of many classes of moduli spaces of elliptic fibrations with given monodromy, including all such moduli spaces of rational and elliptic K3 surfaces (see [4]). Our approach is based on a detailed study of the relation between special graphs on Riemann surfaces and subgroups of finite index in PSL(2, ℤ).