Michele Bolognesi, Alex Massarenti

In this paper, we study the geometry of GIT configurations of $n$ ordered points on ${\mathbb{P}}^{1}$ both from the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient

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