A variation on a conjecture of Faber and Fulton

Abstract

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 the analogue of the F-conjecture for GIT configurations of points on \(\mathbb{P}^1\), that is we show that every extremal ray of the Mori cone of effective curves on 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. On the way to this result 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.