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

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