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We study the moduli space of logarithmic connections of rank 2 on the Riemannsphere minus n points with fixed spectral data. There are two naturalLagrangian maps: one towards apparent singularities of the associated fuchsianscalar equation, and another one towards moduli of parabolic bundles. We showthat these are transversal and dual to each other. In case n=5, we recover thebeautiful geometry of Del Pezzo surfaces of degree 4.

Lagrangian fibration in duality on moduli space of rank two logarithmic connections over the projective line