Vol. 12, No. 4, 2018

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A generic slice of the moduli space of line arrangements

Kenneth Ascher and Patricio Gallardo

Vol. 12 (2018), No. 4, 751–778
Abstract

We study the compactification of the locus parametrizing lines having a fixed intersection with a given line, inside the moduli space of line arrangements in the projective plane constructed for weight one by Hacking, Keel and Tevelev and for general weights by Alexeev. We show that this space is smooth, with normal crossing boundary, and that it has a morphism to the moduli space of marked rational curves which can be understood as a natural continuation of the blow up construction of Kapranov. In addition, we prove that our space is isomorphic to a closed subvariety inside a nonreductive Chow quotient.

Keywords
moduli spaces, hyperplane arrangements, chow quotient, wonderful compactifications, stable pairs, birational geometry, minimal model program
Mathematical Subject Classification 2010
Primary: 14J10
Secondary: 14D20
Milestones
Received: 13 March 2016
Revised: 11 August 2017
Accepted: 17 March 2018
Published: 11 July 2018
Authors
Kenneth Ascher
Mathematics Department
Massachusetts Institute of Technology
Cambridge, MA
United States
Patricio Gallardo
Department of Mathematics
Washington University in St. Louis
St. Louis, MO
United States