Klaus Hulek (Leibniz University Hannover) - Configuration spaces of points and the universe of Deligne-Mostow varieties

There are several approaches to studying configuration spaces of points on curves. Examples are the Deligne-Mumford and Hassett spaces or, in the case of points on a projective line, a straightforward GIT construction. Deligne and Mostow further investigated the situation where moduli spaces of weighted points on the projective line have ball quotient models. This leads to a finite set of configuration spaces of ordered or partially ordered points on P^1. In all of these cases, the GIT compactification and the Baily-Borel compactification of the ball quotient are isomorphic. These spaces have natural (partial) desingularizations, namely the Kirwan blow-up and the toroidal compactification. It follows from results of Gallardo-Kerr-Schaffler and Kiem-Moon that, in the case of ordered configurations, the Deligne-Mostow map lifts to an  isomorphism of these compactifications. This can fail for only partially ordered configurations. The main result of this talk is to explain a criterion exactly when these two compactifications coincide. This is also related to the Log Minimal Model Program. This is joint work with Yota Maeda.