Rational curves on the compactified moduli space of rational curves with marked points

We investigate the possible homological classes of rational curves on the moduli space $X_n=\overline{{\mathcal{M}}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to certain polyhedral cones which we describe explicitly. Not all homology classes in the cone correspond to rational curves, but all classes sufficiently far from the walls of the cone do. The proof for $X_5$ is explicit, the proof for $X_6$ was carried out using computers.