Relative MMP without Q-factoriality

TL;DR

This paper explores the minimal model program for non-Q-factorial varieties, demonstrating that many steps are simpler than expected and extending key theorems to broader classes of singularities.

Contribution

It shows that all flips are 1-complemented in non-Q-factorial cases and removes the Q-factoriality assumption from several important theorems.

Findings

All flips are 1-complemented in the studied cases.

Simplification of the minimal model program steps for non-Q-factorial varieties.

Extension of theorems to log terminal singularities without Q-factoriality.

Abstract

We consider the minimal model program for varieties that are not Q-factorial. We show that, in many cases, its steps are simpler than expected. In particular, all flips are 1-complemented. The main applications are to log terminal singularities, removing the earlier Q-factoriality assumption from several theorems of Hacon--Witaszek and de~Fernex--Koll\'ar--Xu. Version 2: many small changes.