Functorial desingularization over Q: boundaries and the embedded case

TL;DR

This paper proves functorial desingularization for noetherian quasi-excellent schemes over Q with boundaries, extending to stacks, formal schemes, and analytic spaces, and includes a semistable reduction theorem.

Contribution

It establishes the first functorial desingularization in characteristic zero for a broad class of schemes and related objects, with applications to stacks and analytic spaces.

Findings

Functorial desingularization for quasi-excellent schemes over Q.

Extension to stacks, formal schemes, and analytic spaces.

A semistable reduction theorem for formal schemes.

Abstract

Our main result establishes functorial desingularization of noetherian quasi-excellent schemes over $\bfQ$ with ordered boundaries. A functorial embedded desingularization of quasi-excellent schemes of characteristic zero is deduced. Furthermore, a standard simple argument extends these results to other categories, including in particular, (equivariant) embedded desingularization of the following objects of characteristic zero: qe algebraic stacks, qe schemes, qe formal schemes, complex and non-archimedean analytic spaces. We also obtain a semistable reduction theorem for formal schemes.