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Streamlining resolution of singularities with weighted blow-ups

Maxim Jean-Louis Brais

TL;DR

Building on the weighted blow-up framework for resolution of singularities in characteristic zero, the paper generalizes the Newton-graph approach to arbitrary codimensions and achieves factorial reductions in complexity relative to prior constructions. It introduces two complementary methods for constructing the associated centre: a parameter-based local method and a global method via derivatives of ideals, both yielding the maximal quasi-homogeneous approximation that drives the resolution step. The analysis leverages the degeneration to the weighted normal cone and a $\mathbb{G}_m$-action to prove that the invariant decreases on the blow-up, establishing functoriality and compatibility with smooth morphisms. The globalisation results and invariance properties enable non-embedded resolution and point toward broader applications, including logarithmic settings and foliations, enhancing the practicality and scope of resolution techniques.

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.

Streamlining resolution of singularities with weighted blow-ups

TL;DR

Building on the weighted blow-up framework for resolution of singularities in characteristic zero, the paper generalizes the Newton-graph approach to arbitrary codimensions and achieves factorial reductions in complexity relative to prior constructions. It introduces two complementary methods for constructing the associated centre: a parameter-based local method and a global method via derivatives of ideals, both yielding the maximal quasi-homogeneous approximation that drives the resolution step. The analysis leverages the degeneration to the weighted normal cone and a -action to prove that the invariant decreases on the blow-up, establishing functoriality and compatibility with smooth morphisms. The globalisation results and invariance properties enable non-embedded resolution and point toward broader applications, including logarithmic settings and foliations, enhancing the practicality and scope of resolution techniques.

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.

Paper Structure

This paper contains 27 sections, 45 theorems, 158 equations, 5 figures.

Table of Contents

  1. Our formalism: weightings, smooth centres, and marked centres
  2. The associated centre as the optimal quasi-homogeneous approximation
  3. Blowing up the optimal quasi-homogeneous approximation improves the singularities
  4. Method 1: constructing the associated centre in parameters
  5. Method 2: constructing the associated centre with derivatives of ideals
  6. Overview of the paper
  7. Further applications
  8. Weightings and marked centres
  9. Definitions and first properties
  10. Valuation and conditions for compatible parameters
  11. Differential operators
  12. Marked centres
  13. Weighted blow-ups and resolution
  14. Weighted blow-ups
  15. What is needed for resolution?
  16. ...and 12 more sections

Key Result

Proposition 1

Let $\mathcal{F}_\bullet$ be a weighting with weights $w_1,\dots,w_k$. Then, for all $j\geq 0$, $\mathcal{F}_j/\mathcal{F}_{j+1}$ is a locally free $\mathcal{O}_Z$-module of rank equal to the number of multi-indices $\beta$ of length $k$ such that $\sum\beta_i w_i=j$.

Figures (5)

  • Figure 1: Illustration of $H(a_1,a_2)\leq\mathbf{Newt}(\mathbf x)$
  • Figure 2: Graphical construction of the associated centre for \ref{['ex: newt curve']}
  • Figure 3: $\Xi(\alpha)$ is the $v_2$-intercept of $L(\alpha)$
  • Figure 4: Illustration of \ref{['thm: num']}
  • Figure 5: Two illustrations of the inequality \ref{['eq: leq max']} when $\mathbf{v}(\gamma)$ lies under $M$

Theorems & Definitions (121)

  • Definition 1: Parameters
  • Remark 1
  • Definition 2: Weighting
  • Remark 2
  • Remark 3
  • Example 1
  • Example 2
  • Example 3
  • Proposition 1
  • proof
  • ...and 111 more