Resolution Except for the Normal-Crossing Locus and Galois actions

TL;DR

The paper presents a canonical, functorial resolution algorithm in characteristic zero that preserves the normal crossings locus using weighted blow-ups, addressing a longstanding problem and extending to stacks.

Contribution

It introduces a new resolution method that handles both reduced and non-reduced nc singularities with stack structures, based on Galois-theoretic analysis and weighted blow-ups.

Findings

Constructs a resolution algorithm that preserves the nc locus.

Handles non-reduced and stack-based singularities.

Provides applications to stack compactification and resolution of subvarieties.

Abstract

In characteristic zero, we construct a canonical, functorial resolution algorithm by weighted blow-ups that strictly preserves the normal crossings (nc) locus, effectively answering Kollar's problem. Operating in full generality, our approach handles both reduced and non-reduced nc singularities alongside the simple normal crossings (snc) exceptional divisor setup, terminating with a normal crossings Deligne-Mumford stack. The resolution is governed by two fundamental geometric properties: the openness of the nc locus and the topological rigidity of canonical maximal admissible weighted centers (the Center Theorem). We establish these via a direct Galois-theoretic analysis of splitting forms. By viewing general nc singularities as quotients of 'etale-local snc singularities by finite Galois groups permuting their branches, we reveal the intrinsic necessity of weighted blow-ups and stack structures. Building on the weighted blow-up framework of Abramovich, Temkin, and Wlodarczyk and our logarithmic refinement, this structural mechanism yields a robust resolution algorithm. Applications include a canonical compactification of nc Deligne-Mumford stacks and a functorial nc-preserving resolution of subvarieties and stacks.