Kalinin Effectivity and Wonderful Compactifications

TL;DR

This paper reviews Kalinin effectivity, explores its properties in wonderful compactifications, and demonstrates its applications in effective spaces, including Deligne-Mumford spaces and Smith-Thom maximality.

Contribution

It introduces methods for constructing effective spaces, proves Kalinin effectivity for certain compactifications, and applies this to real rational curves and Hilbert squares.

Findings

Wonderful compactifications are Kalinin effective.

Deligne-Mumford space of real rational curves is effective.

Kalinin effectivity aids in studying Smith-Thom maximality.

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.