  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 $(\mathbb{P}^1)^n//PGL(2)$, 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
$(\mathbb{P}^1)^n//PGL(2)$ in its natural embedding, and its group of
automorphisms.
