  We show that the conditions imposed on a second order linear differential
equation with rational coefficients on the complex line by requiring it to have
regular singularities with fixed exponents at the points of a finite set $P$
and apparent singularities at a finite set $Q$ (disjoint from $P$) determine a
linear system of maximal rank. In addition, we show that certain auxiliary
parameters can also be fixed. This enables us to conclude that the family of
such differential equations is of the expected dimension and to define a
birational map between an open subset of the moduli space of logarithmic
connections with fixed logarithmic points and regular semi-simple residues and
the Hilbert scheme of points on a quasi-projective surface.
