Birational geometry of moduli spaces of configurations of points on the line

In this paper we study the geometry of GIT configurations of $n$ ordered points on ${\mathbb{P}}^1$ both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient $({\mathbb{P}}^1{)}^n\mathrm{/}\mathrm{/}PGL(2)$, taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of $({\mathbb{P}}^1{)}^n\mathrm{/}\mathrm{/}PGL(2)$ in its natural embedding, and its group of automorphisms.