Examples of Lagrangian fibrations with many singular fibres

Let S be a K3 surface with a generically finite morphism of degree 2 to the projective plane such that the branching divisor D has mild singularities. This morphism gives rise to a Lagrangian fibration of a 4-dimensional Hyperkähler manifold whose discriminant divisor depends on the singularities of D. will describe the singular fibres over generic points of the discriminant divisor. This is a report on joint work with A. Czaplinski, A. Krug and S. Rollenske.