Streamlining resolution of singularities with weighted blow-ups

TL;DR

This paper extends a graphical approach to resolution of singularities using weighted blow-ups from plane curves to higher-dimensional varieties, significantly reducing complexity in characteristic zero.

Contribution

It generalizes the Newton graph method for singularity resolution to arbitrary dimensions, simplifying the process and lowering complexity compared to previous methods.

Findings

Extended graphical approach to higher dimensions

Achieved factorial reduction in complexity

Provided self-contained constructions and proofs

Abstract

In 2019, Abramovich--Temkin--Wlodarczyk and McQuillan used weighted blow-ups to obtain very fast and functorial algorithms for resolution of singularities in characteristic zero. Recently, Abramovich--Quek--Schober simplified the construction of the centre of blow-up introduced by Abramovich--Temkin--Wlodarczyk in the case of plane curves by using the Newton graph of the defining function. Their work follows the line of Schober's previous polyhedral analysis of the Bierstone--Milman invariant. In this paper, we extend their graphical approach to varieties of arbitrary (co)dimension in characteristic zero. This yields a factorial reduction in complexity in comparison with Abramovich--Temkin--Wlodarczyk, as previously achieved by Wlodarczyk. Our approach builds on the formalism of weighted blow-ups via filtrations of ideals developed and used by Loizides--Meinrenken, Quek--Rydh and Wlodarczyk, and on its interplay with systems of parameters. All constructions and proofs -- including that of resolution of varieties by Deligne--Mumford stacks -- are self-contained.