Lagrangian Fibrations in Duality on Moduli Spaces of Rank 2 Logarithmic Connections Over the Projective Line

We study the moduli space of logarithmic connections of rank 2 on $\mathbb P^1_\mathbb C$ minus n points with fixed spectral data. There are two natural Lagrangian maps: one toward apparent singularities of the associated Fuchsian scalar equation and another one toward moduli of parabolic bundles. We show that these are transversal and dual to each other. In case n=5, we recover the beautiful geometry of Del Pezzo surfaces of degree 4.