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The generic element of the moduli space of logarithmic connections withparabolic points on holomorphic vector bundle over the Riemann sphere can berepresented by a Fuchsian equation with some singularities and some apparentsingularities. We analyze the case of rank $3$ vector bundle which leads tothird order Fuchsian equation. We find coordinates on an open subset of themoduli space and we construct a non-trivial part of the moduli space by blowingup along a variety in a special case.

The locus of the representation of logarithmic connections by Fuchsian equations