Dan Abramovich ; Ming Hao Quek - Logarithmic resolution via multi-weighted blow-ups

epiga:9793 - Épijournal de Géométrie Algébrique, October 31, 2024, Volume 8 - https://doi.org/10.46298/epiga.2024.9793
Logarithmic resolution via multi-weighted blow-upsArticle

Authors: Dan Abramovich ; Ming Hao Quek

    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 $X_N$.


    Volume: Volume 8
    Published on: October 31, 2024
    Accepted on: April 23, 2024
    Submitted on: July 15, 2022
    Keywords: Mathematics - Algebraic Geometry,14E15 (Primary), 14A20, 14M25 (Secondary)

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