The Universe of Deligne-Mostow Varieties

Deligne and Mostow investigated period maps on the configuration spaces $M_{0,n}$ of $n$ ordered points on $\mathbb{P}^1$. The images of these maps are open subsets of certain ball quotients. Moreover, they extend to isomorphisms between GIT-quotients and the Baily-Borel compactifications. Building on a theorem of Gallardo, Kerr and Schaffler, the period maps lift to isomorphisms between two natural compactifications, namely the Kirwan blow-up and the toroidal compactification. In this paper, we look at the more general situation where we also allow unordered or partially ordered $n$-tuples. Our main result is an easily verifiable criterion that, in this broader setting, determines when the Deligne-Mostow period maps still lift to isomorphisms between the Kirwan blow-up and the toroidal compactification. We further investigate a partial ordering among Deligne-Mostow varieties, which reduces this problem to considering minimal or maximal Deligne-Mostow varieties with respect to this partial ordering. As a byproduct, we prove that, in general, Kirwan's resolution pair is not a log canonical log minimal model and not log $K$-equivalent to the unique toroidal compactification.