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Birational geometry of moduli spaces of configurations of points on the line
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-04-07, DOI: 10.2140/ant.2021.15.515
Michele Bolognesi, Alex Massarenti

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.