We define a natural compactification of an arrangement complement in a ball quotient. We show that when this complement has a moduli space interpretation, then this compactification is often one that appears naturally by means of geometric invariant theory. We illustrate this with the moduli spaces of smooth quartic curves and rational elliptic surfaces.
"Compactifications defined by arrangements, I: The ball quotient case." Duke Math. J. 118 (1) 151 - 187, 15 May 2003. https://doi.org/10.1215/S0012-7094-03-11816-5