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

Bolognesi, Michele; Massarenti, Alex

doi:10.2140/ant.2021.15.515

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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

DOI: 10.2140/ant.2021.15.515

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
${(ℙ^{1})}^{n}$ // $PGL (2)$ , 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 ${(ℙ^{1})}^{n}$ // $PGL (2)$ in its natural embedding, and its automorphism group.

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Keywords

Mori dream spaces, moduli of curves

Mathematical Subject Classification 2010

Primary: 14D22, 14H10, 14H37

Secondary: 14N05, 14N10, 14N20

Milestones

Received: 20 March 2020

Revised: 4 July 2020

Accepted: 21 August 2020

Published: 7 April 2021

Authors

Michele Bolognesi
IMAG - Université de Montpellier Montpellier France

Alex Massarenti
Dipartimento di Matematica e Informatica Università di Ferrara Ferrara Italy

Vol. 15, No. 2, 2021