The paper studies fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov’s description we are able to prove that the only such morphisms are forgetful maps. This allows us to show that the automorphism group of ${\overline{M}}_{0,n}$ is the permutation group on $n$ elements as soon as $n\ge 5$.

These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel${}^{\text{'}}$fand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.