Dream resolution and principalization II: excellent schemes

TL;DR

This paper extends dream principalization and resolution methods to all excellent schemes of characteristic zero, providing a detailed description of invariants and a non-embedded resolution approach using stack-theoretical modifications.

Contribution

It generalizes previous results to all excellent schemes of characteristic zero and introduces a non-embedded resolution method with explicit invariants.

Findings

Extended dream principalization to all excellent schemes of characteristic zero.

Described invariants of canonical centers.

Established non-embedded resolution via stack modifications.

Abstract

This is the second paper in a project on dream (or memoryless) principalization and resolution methods. It extends this theory from the case of schemes with enough derivations, which was established in [Tem25], to general excellent schemes of characteristic zero. So, similarly to McQuillan's approach developed in [McQ20], the approach of [ATW24] is now extended to the generality of all excellent schemes of characteristic zero. In addition, we precisely describe the set of invariants of canonical centers and establish the resolution in the non-embedded form, where one applies simple (stack-theoretical) modifications along subschemes of a special form that we call tubes. In the regular case these are precisely the subschemes corresponding to canonical centers.