  We study necessary and sufficient conditions for a meromorphic quadratic
differential with prescribed poles to be the Schwarzian derivative of a
rational map. We give geometric interpretations of these conditions. We also
study the pole-dependency of these Schwarzian derivatives. We show that, in the
cubic case, the analytic dependency fails precisely when the poles are
displaced at the vertices of a regular ideal tetrahedron of the hyperbolic
3-ball.
