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

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