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 P1 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.P1/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.P1/n== PGL.2/ in its natural embedding, and its automorphism group.