Logarithmic resolution via multi-weighted blow-ups

TL;DR

This paper introduces multi-weighted blow-ups to develop an explicit, efficient algorithm for functorial logarithmic resolution of singularities in characteristic zero, improving singularity handling.

Contribution

The paper systematically constructs a new class of blow-ups and an algorithm for resolving singularities functorially in characteristic zero.

Findings

Develops the concept of multi-weighted blow-ups.

Provides an explicit algorithm for logarithmic resolution.

Ensures the resolution process improves singularities at each step.

Abstract

We first introduce and study the notion of multi-weighted blow-ups, which is later used to systematically construct an explicit yet efficient algorithm for functorial logarithmic resolution in characteristic zero, in the sense of Hironaka. Specifically, for a singular, reduced closed subscheme $X$ of a smooth scheme $Y$ over a field of characteristic zero, we resolve the singularities of $X$ by taking proper transforms $X_i \subset Y_i$ along a sequence of multi-weighted blow-ups $Y_N \to Y_{N-1} \to \dotsb \to Y_0 = Y$ which satisfies the following properties: (i) the $Y_i$ are smooth Artin stacks with simple normal crossing exceptional loci; (ii) at each step we always blow up the worst singular locus of $X_i$, and witness on $X_{i+1}$ an immediate improvement in singularities; (iii) and finally, the singular locus of $X$ is transformed into a simple normal crossing divisor on $X_N$.