Kalinin Effectivity and Wonderful Compactifications

Abstract

We review the definition and main properties of Kalinin effectivity and describe methods for constructing effective spaces together with several examples. We analyze the Kalinin effectivity of wonderful compactifications and prove that the wonderful compactifications of hyperplane arrangements and of configuration spaces associated to Kalinin effective compact complex manifolds are themselves Kalinin effective. As an application, we show that the Deligne-Mumford space of real rational curves with marked points is effective. Finally, we apply Kalinin effectivity to study Smith-Thom maximality for Hilbert squares.