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arxiv: 2602.14266 · v2 · submitted 2026-02-15 · 🧮 math.AG

Resolution Except for the Normal-Crossing Locus and Galois actions

Jaros{\l}aw W{\l}odarczyk This is my paper

Pith reviewed 2026-05-15 21:49 UTC · model grok-4.3

classification 🧮 math.AG MSC 14E15
keywords resolution of singularitiesnormal crossingsweighted blow-upsGalois actionsDeligne-Mumford stackscharacteristic zeroalgebraic geometry
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The pith

Weighted blow-ups resolve singularities in characteristic zero while strictly preserving the normal crossings locus.

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

The paper builds a canonical and functorial resolution procedure that uses weighted blow-ups to eliminate singularities outside the normal crossings locus without altering that locus. It works for both reduced and non-reduced normal crossings singularities and ends with a normal crossings Deligne-Mumford stack. The construction rests on two geometric facts: the openness of the normal crossings locus and the topological rigidity of certain maximal admissible weighted centers, which are proved by viewing normal crossings singularities as quotients of simple normal crossings singularities by finite Galois groups that permute branches.

Core claim

In characteristic zero, we construct a canonical, functorial resolution algorithm by weighted blow-ups that strictly preserves the normal crossings (nc) locus, effectively answering Kollar's problem. Operating in full generality, our approach handles both reduced and non-reduced nc singularities alongside the simple normal crossings (snc) exceptional divisor setup, terminating with a normal crossings Deligne-Mumford stack. The resolution is governed by the openness of the nc locus and the topological rigidity of canonical maximal admissible weighted centers, established via a direct Galois-theoretic analysis of splitting forms.

What carries the argument

The Center Theorem, asserting topological rigidity of canonical maximal admissible weighted centers, obtained by Galois-theoretic analysis of nc singularities viewed as quotients of etale-local snc singularities by finite groups permuting branches.

If this is right

  • The procedure yields a canonical compactification of nc Deligne-Mumford stacks.
  • It supplies a functorial nc-preserving resolution for subvarieties and stacks.
  • The algorithm terminates with a normal crossings Deligne-Mumford stack structure.

Where Pith is reading between the lines

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

  • The Galois-quotient perspective may allow similar weighted-blow-up resolutions for other quotient singularities that appear in moduli problems.
  • Preserving the nc locus exactly could simplify inductive arguments on log smooth pairs in birational geometry.
  • Stack structures arising at the end suggest the method produces objects already adapted to stack-theoretic compactifications.

Load-bearing premise

The topological rigidity of the canonical maximal admissible weighted centers holds via the Galois splitting analysis.

What would settle it

A concrete normal crossings singularity in characteristic zero for which the algorithm's chosen weighted center either fails to reduce the singularity or moves the normal crossings locus would falsify the claim.

read the original abstract

In characteristic zero, we construct a canonical, functorial resolution algorithm by weighted blow-ups that strictly preserves the normal crossings (nc) locus, effectively answering Kollar's problem. Operating in full generality, our approach handles both reduced and non-reduced nc singularities alongside the simple normal crossings (snc) exceptional divisor setup, terminating with a normal crossings Deligne-Mumford stack. The resolution is governed by two fundamental geometric properties: the openness of the nc locus and the topological rigidity of canonical maximal admissible weighted centers (the Center Theorem). We establish these via a direct Galois-theoretic analysis of splitting forms. By viewing general nc singularities as quotients of 'etale-local snc singularities by finite Galois groups permuting their branches, we reveal the intrinsic necessity of weighted blow-ups and stack structures. Building on the weighted blow-up framework of Abramovich, Temkin, and Wlodarczyk and our logarithmic refinement, this structural mechanism yields a robust resolution algorithm. Applications include a canonical compactification of nc Deligne-Mumford stacks and a functorial nc-preserving resolution of subvarieties and stacks.

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

2 major / 2 minor

Summary. The manuscript constructs a canonical, functorial resolution algorithm in characteristic zero for both reduced and non-reduced normal crossings (nc) singularities via weighted blow-ups. The algorithm strictly preserves the nc locus, terminates with a normal crossings Deligne-Mumford stack, and is governed by the openness of the nc locus together with the Center Theorem on topological rigidity of canonical maximal admissible weighted centers. The Center Theorem is established by Galois-theoretic analysis of splitting forms, viewing general nc singularities as quotients of étale-local snc singularities by finite groups permuting branches, building on the weighted blow-up framework of Abramovich-Temkin-Włodarczyk and a logarithmic refinement.

Significance. If the Center Theorem holds with the claimed invariance, the result would resolve Kollar's problem on nc-preserving resolution and supply a canonical compactification of nc DM stacks together with functorial resolutions of subvarieties. The Galois-theoretic derivation of canonicity and the explicit handling of non-reduced cases constitute the main potential contribution; the work also supplies reproducible algorithmic structure once the weighted centers are fully specified.

major comments (2)
  1. [Center Theorem] Center Theorem (Galois analysis section): the argument that canonical maximal admissible weighted centers are topologically rigid must explicitly verify invariance under all étale-local splitting forms when multiplicities exceed 1; without this check the claim that the center is independent of presentation and remains maximal admissible for non-reduced nc singularities is not yet load-bearing.
  2. [Resolution algorithm] Resolution algorithm (main construction): the sequence of weighted blow-ups is asserted to strictly preserve the nc locus at every step, but no explicit verification or inductive step is supplied showing that the chosen center lies outside the nc locus while the exceptional divisor remains nc after the blow-up; this step is central to the functoriality claim.
minor comments (2)
  1. [Abstract] The abstract refers to 'our logarithmic refinement' without defining the precise modification to the Abramovich-Temkin-Włodarczyk weighted blow-up; a short paragraph contrasting the new refinement with the cited prior work would improve readability.
  2. Notation for admissible weighted centers and splitting forms is introduced late; early definitions or a table summarizing the Galois action on branches would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the arguments require more explicit verification. We will revise the manuscript to strengthen the exposition on both the Center Theorem and the resolution algorithm while preserving the overall structure and claims.

read point-by-point responses

  1. Referee: [Center Theorem] Center Theorem (Galois analysis section): the argument that canonical maximal admissible weighted centers are topologically rigid must explicitly verify invariance under all étale-local splitting forms when multiplicities exceed 1; without this check the claim that the center is independent of presentation and remains maximal admissible for non-reduced nc singularities is not yet load-bearing.

    Authors: We agree that the Galois-theoretic argument would benefit from an explicit verification of invariance for multiplicities greater than 1. In the revised manuscript we will insert a new subsection that enumerates the possible étale-local splitting forms with higher multiplicities, confirms that the canonical maximal admissible weighted center remains unchanged under these forms, and thereby establishes topological rigidity and independence of presentation for the non-reduced case. revision: yes

  2. Referee: [Resolution algorithm] Resolution algorithm (main construction): the sequence of weighted blow-ups is asserted to strictly preserve the nc locus at every step, but no explicit verification or inductive step is supplied showing that the chosen center lies outside the nc locus while the exceptional divisor remains nc after the blow-up; this step is central to the functoriality claim.

    Authors: We accept that an explicit inductive argument is needed. The revised version will contain a new lemma in the main construction section that proceeds by induction on the number of blow-ups: at each stage the chosen weighted center is shown to lie outside the nc locus (by the openness property and the maximality condition), and the blow-up is shown to produce an exceptional divisor that is normal crossings, thereby preserving the nc locus and ensuring functoriality. revision: yes

Circularity Check

0 steps flagged

Minor reliance on prior self-cited weighted blow-up framework; Center Theorem proved independently via Galois analysis

full rationale

The derivation chain rests on the openness of the nc locus and the Center Theorem (topological rigidity of canonical maximal admissible weighted centers). The paper states it establishes the Center Theorem 'via a direct Galois-theoretic analysis of splitting forms' by viewing nc singularities as quotients of étale-local snc singularities by finite Galois groups. This argument is presented as internal to the paper and independent of the resolution algorithm. The only self-reference is the citation to 'the weighted blow-up framework of Abramovich, Temkin, and Wlodarczyk and our logarithmic refinement,' which supplies the underlying machinery but does not reduce the new nc-locus preservation or canonicity claim to a fit, definition, or unverified self-citation chain. No equations or steps reduce by construction to inputs; the Galois analysis supplies the load-bearing independence.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The construction rests on the characteristic-zero hypothesis and the openness of the nc locus together with the Center Theorem; these are treated as established geometric facts rather than derived inside the paper.

axioms (3)
  • domain assumption Work in characteristic zero
    Resolution algorithm stated only for char 0 fields.
  • domain assumption Openness of the nc locus
    Invoked as one of the two fundamental geometric properties governing the algorithm.
  • domain assumption Topological rigidity of canonical maximal admissible weighted centers (Center Theorem)
    Central structural result proved via Galois analysis but treated as input for the resolution procedure.

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