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arxiv: 2112.06361 · v3 · pith:XLB3GDIHnew · submitted 2021-12-13 · 🧮 math.AG

Logarithmic resolution via multi-weighted blow-ups

Dan Abramovich , Ming Hao Quek This is my paper

Pith reviewed 2026-05-24 12:48 UTC · model grok-4.3

classification 🧮 math.AG
keywords resolution of singularitiesmulti-weighted blow-upsArtin stackslogarithmic resolutioncharacteristic zerosimple normal crossingsproper transformsHironaka resolution
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The pith

Multi-weighted blow-ups on smooth Artin stacks resolve singularities of a reduced subscheme in a smooth scheme over a characteristic-zero field by turning its singular locus into a simple normal crossing divisor.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces multi-weighted blow-ups to construct an explicit algorithm for functorial logarithmic resolution of singularities in characteristic zero. For a singular reduced closed subscheme X inside a smooth scheme Y, it builds a sequence of these blow-ups where each ambient space remains a smooth Artin stack. At every step the center is the worst singular locus of the current proper transform, and the operation produces an immediate improvement. The process terminates when the singular locus of X has become a simple normal crossing divisor on the final proper transform. A sympathetic reader would care because the construction supplies a systematic, step-by-step method that guarantees progress without relying on abstract existence results.

Core claim

For a singular, reduced closed subscheme X of a smooth scheme Y over a field of characteristic zero, the singularities of X are resolved by taking proper transforms X_i ⊂ Y_i along a sequence of multi-weighted blow-ups Y_N → Y_{N-1} → ⋯ → Y_0 = Y which satisfies: (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 immediate improvement; (iii) the singular locus of X is transformed into a simple normal crossing divisor on X_N.

What carries the argument

Multi-weighted blow-ups, a blow-up operation performed on smooth Artin stacks that generalizes ordinary weighted blow-ups and is chosen so each step improves the singularities of the proper transform.

If this is right

  • The resolution algorithm is functorial with respect to the input data.
  • Every intermediate space in the sequence is a smooth Artin stack whose exceptional locus has simple normal crossings.
  • Each blow-up step produces a strict improvement in the singularities of the proper transform of X.
  • The final proper transform X_N has its entire singular locus contained in a simple normal crossing divisor.
  • The centers are always determined by the worst singularities present at that stage.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same centers might be usable in other resolution problems where ordinary blow-ups stall.
  • Because the construction works on stacks, it may adapt to situations with finite group actions or quotient singularities.
  • An implementation could yield explicit equations for the centers at each step, making the resolution computable for concrete examples.
  • The method indicates that allowing stacky structure is essential for obtaining immediate improvement at every step.

Load-bearing premise

Multi-weighted blow-ups exist and can always be centered at the current worst singular locus so that the proper transform shows immediate, verifiable improvement while the ambient stack stays smooth with simple normal crossing exceptional locus.

What would settle it

A specific reduced singular subscheme X inside a smooth Y over a characteristic-zero field for which every possible finite sequence of multi-weighted blow-ups centered at successively worst singular loci either fails to keep the ambient spaces smooth Artin stacks or fails to turn the singular locus into a simple normal crossing divisor.

read the original 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 $X_N$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript introduces the notion of multi-weighted blow-ups and applies them to construct an explicit functorial algorithm for logarithmic resolution of singularities in characteristic zero. For a singular reduced closed subscheme X of a smooth scheme Y over a field of char 0, it produces a finite sequence of multi-weighted blow-ups Y_N → ⋯ → Y_0 = Y such that each Y_i is a smooth Artin stack with simple normal crossings exceptional locus, each step blows up the worst singular locus of the proper transform X_i and obtains immediate improvement on X_{i+1}, and the final singular locus of X becomes a simple normal crossing divisor on X_N.

Significance. If the stated properties of multi-weighted blow-ups and the immediate-improvement claim are verified, the construction would supply a concrete, stack-theoretic algorithm for Hironaka-style logarithmic resolution that is functorial and terminates after finitely many steps with an explicit improvement criterion at each stage. This could be of interest for both theoretical understanding of resolution and potential computational implementations in algebraic geometry.

minor comments (1)
  1. The abstract and claim summary do not specify the precise definition of a multi-weighted blow-up or the ordering used to identify the 'worst singular locus'; these should be stated explicitly in §2 or §3 with reference to the relevant equations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our work and for noting its potential significance for both theoretical and computational aspects of resolution of singularities. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces multi-weighted blow-ups as a new notion and then constructs an explicit sequence of such blow-ups to achieve logarithmic resolution, with the three listed properties verified directly on the proper transforms. No equations, parameters, or results are defined in terms of each other by construction, no fitted inputs are relabeled as predictions, and no load-bearing steps rely on self-citations. The derivation is a self-contained constructive algorithm in characteristic-zero algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The construction rests on the new notion of multi-weighted blow-ups together with the background existence of resolution in characteristic zero; no numerical parameters are visible.

axioms (1)
  • domain assumption Hironaka's theorem guarantees resolution of singularities exists in characteristic zero
    The paper works in the setting where such resolution is known to exist and seeks an explicit algorithmic version.
invented entities (1)
  • multi-weighted blow-up no independent evidence
    purpose: Generalized blow-up operation used to build the resolution sequence
    New technical device introduced in the paper to achieve the stated properties (i)–(iii).

pith-pipeline@v0.9.0 · 5704 in / 1445 out tokens · 79441 ms · 2026-05-24T12:48:01.438299+00:00 · methodology

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Reference graph

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