Algebraic Geometry Seminar: "Compactifications of moduli of points and lines in the plane"

Abstract: Projective duality identifies the moduli space B_n parametrizing configurations of n general points in projective 2-space with X(3,n), parametrizing configurations of n general lines in the dual projective plane. When considering degenerations of such objects, it is interesting to compare different compactifications of the above moduli spaces. In this work, we consider Gerritzen-Piwek's compactification \overline{B}_n and Kapranov's Chow quotient compactification \overline{X}(3,n) and we show they have isomorphic normalizations. We prove that \overline{B}_n does not admit a modular interpretation claimed by Gerritzen and Piwek, namely a family of n-pointed central fibers of Mustafin joins associated with one-parameter degenerations of n points in the plane. We construct the correct compactification of B_n which admits such a family, and we describe it for n=5,6. This is joint work in progress with Jenia Tevelev.

Host: Matt Kerr and Patricio Gallardo