#### Vol. 15, No. 2, 2021

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

### Michele Bolognesi and Alex Massarenti

Vol. 15 (2021), No. 2, 513–544
##### Abstract

In this paper, we study the geometry of GIT configurations of $n$ ordered points on ${ℙ}^{1}$ 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
${\left({ℙ}^{1}\right)}^{n}$// $PGL\left(2\right)$, 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 ${\left({ℙ}^{1}\right)}^{n}$ // $PGL\left(2\right)$ in its natural embedding, and its automorphism group.

##### Keywords
Mori dream spaces, moduli of curves
##### Mathematical Subject Classification 2010
Primary: 14D22, 14H10, 14H37
Secondary: 14N05, 14N10, 14N20