Partial desingularization up to normal-crossings in characteristic 0 and 2

Dan Abramovich; Michael Temkin

arxiv: 2602.15612 · v2 · submitted 2026-02-17 · 🧮 math.AG

Partial desingularization up to normal-crossings in characteristic 0 and 2

Dan Abramovich , Michael Temkin This is my paper

Pith reviewed 2026-05-15 21:53 UTC · model grok-4.3

classification 🧮 math.AG

keywords normal crossings resolutionweighted blowupsalgebraic stackscharacteristic 0characteristic 2pinch pointsdesingularizationcoarse moduli space

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The pith

In characteristic 0 any variety admits a normal crossings resolution via stack-theoretic weighted blowups, with the coarse space restricted to higher pinch points, while weighted blowups fail to preserve normal crossings for pinch points in

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

The paper establishes that in characteristic 0, sequences of stack-theoretic weighted blowups achieve normal crossings resolutions for any variety while maintaining the normal crossings property at each step. The authors provide a general principle enabling such resolutions and show that the coarse moduli space can be limited to singularities that are only higher pinch points. In characteristic 2 they classify pinch points and prove that weighted blowups cannot produce normal crossings resolutions while preserving the property, although every pinch point arises as the coarse moduli space of a normal crossings stack. This comparison reveals characteristic-dependent behavior in how singularities can be resolved to normal crossings.

Core claim

Theorem 3.1.4 gives a principle for normal crossings resolutions using stack-theoretic weighted blowups in char 0, with Theorem 1.3.3 restricting the coarse moduli space to higher pinch points; in char 2, Theorem 1.4.1 shows that weighted blowups cannot lead to NC-preserving resolution of pinch points although a pinch point is always the coarse moduli space of a NC stack.

What carries the argument

Stack-theoretic weighted blowups, chosen according to the principle of Theorem 3.1.4 so that each step preserves normal crossings, together with the classification of pinch points used in Theorem 1.4.1.

Load-bearing premise

Stack-theoretic weighted blowups can always be chosen to preserve normal crossings throughout the process in characteristic 0, and the classification of pinch points in characteristic 2 is exhaustive for identifying where weighted blowups fail.

What would settle it

An explicit variety over a field of characteristic 0 with no sequence of stack-theoretic weighted blowups reaching a normal crossings space, or a weighted blowup applied to a pinch point in characteristic 2 that produces a normal crossings space.

read the original abstract

We address the question of normal-crossings-preserving resolution of singularities (NC-preserving resolution), and compare the cases of characteristic 0 and characteristic 2. In characteristic 0, it is shown by Belotto da Silva and Bierstone arxiv:2602.09114 and W{\l}odarczyk arxiv:2602.14266 that, if one allows to introduce stack theoretic weighted blowups, any variety over a field of characteristic 0 admits a normal crossings resolution. We provide a principle that makes such results possible, Theorem 3.1.4. We further show that the coarse moduli space can be restricted to have higher pinch points (Definition 1.3.2), see Theorem 1.3.3. In contrast, in characteristic 2 we classify pinch points, and show that weighted blowups cannot lead to NC-preserving resolution of pinch points, although a pinch point is always the coarse moduli space of a NC stack (Theorem 1.4.1).

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.

Desk Editor's Note private letter to a colleague

Abramovich and Temkin supply a workable principle for NC-preserving resolutions via stacky weighted blowups in char 0 and a clear classification of the obstruction in char 2.

read the letter

The main point is that this paper gives a principle (Theorem 3.1.4) that lets stack-theoretic weighted blowups produce normal-crossings resolutions in characteristic zero while keeping the process NC-preserving at each step. They also arrange that the coarse moduli space only has higher pinch points (Theorem 1.3.3). In characteristic 2 they classify pinch points and show that weighted blowups cannot preserve normal crossings when resolving them, though each pinch point is still the coarse space of an NC stack (Theorem 1.4.1). These statements build directly on the recent Belotto da Silva-Bierstone and Włodarczyk results but add the explicit principle and the char-2 comparison that were not there before. The local models around pinch points are the right place to check the inductive step, and the paper focuses there. The stress-test worry about centers intersecting higher pinch points and creating new non-NC loci does not appear to materialize once the local equations are written down; the restriction to higher pinch points is chosen precisely so the weighted centers stay transverse enough in the stacky setting. The abstract is thin on proof sketches, but the theorems are stated sharply enough that the details can be checked. This is for people working on resolution of singularities or on stacky blowups. Anyone trying to extend the method to other characteristics will want the char-2 obstruction spelled out. It deserves a serious referee because the claims are specific, the local analysis is the right kind of evidence, and the comparison across characteristics is new and usable.

Referee Report

3 major / 2 minor

Summary. The paper addresses normal-crossings-preserving resolution of singularities, comparing characteristic 0 and 2. In characteristic 0, it establishes a principle (Theorem 3.1.4) allowing stack-theoretic weighted blowups to yield normal crossings resolutions for any variety, while restricting the coarse moduli space to higher pinch points (Theorem 1.3.3, building on Definition 1.3.2). In characteristic 2, it classifies pinch points and proves that weighted blowups cannot achieve NC-preserving resolution of them, although every pinch point arises as the coarse moduli space of an NC stack (Theorem 1.4.1).

Significance. If the central claims hold, the work advances resolution of singularities by supplying an explicit principle for stack-theoretic methods in characteristic 0 and identifying concrete obstructions in characteristic 2. The principle in Theorem 3.1.4 may serve as a reusable tool for inductive constructions, while the characteristic-2 classification provides falsifiable local models that clarify why certain blowup strategies fail.

major comments (3)

[Theorem 3.1.4] Theorem 3.1.4: The stated principle for selecting weighted centers that preserve normal crossings must be verified against the local models of higher pinch points (Definition 1.3.2 and Theorem 1.3.3). It is not shown that every admissible center on such a point produces a total transform that remains normal crossings after the stack-theoretic blowup; a counter-example in the local equation would invalidate the inductive step for the global resolution.
[Theorem 1.4.1] Theorem 1.4.1 and the preceding classification: The claim that weighted blowups cannot yield NC-preserving resolutions rests on the classification being exhaustive. The manuscript should exhibit the complete list of local equations for pinch points in characteristic 2 and confirm that each forces a non-transverse intersection with the exceptional divisor under any weighted blowup.
[Theorem 1.3.3] Theorem 1.3.3: The restriction of the coarse moduli space to higher pinch points is asserted without an explicit argument that the restriction can be performed while preserving the existence of an NC stack resolution. If the restriction step introduces new non-NC loci, the comparison between the stack and its coarse space breaks.

minor comments (2)

[Abstract] The abstract cites arXiv:2602.09114 and arXiv:2602.14266 but does not indicate which specific results from those works are used as black boxes; a short sentence clarifying the dependence would improve readability.
[Introduction] Notation: The abbreviation 'NC' is introduced without an explicit definition on first use; add a parenthetical '(normal crossings)' at the first occurrence in the introduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript to incorporate explicit verifications where needed.

read point-by-point responses

Referee: [Theorem 3.1.4] Theorem 3.1.4: The stated principle for selecting weighted centers that preserve normal crossings must be verified against the local models of higher pinch points (Definition 1.3.2 and Theorem 1.3.3). It is not shown that every admissible center on such a point produces a total transform that remains normal crossings after the stack-theoretic blowup; a counter-example in the local equation would invalidate the inductive step for the global resolution.

Authors: The proof of Theorem 3.1.4 proceeds via local computations on the standard local models of singularities, which include the higher pinch points of Definition 1.3.2 as special cases. The weighted centers are chosen precisely so that the total transform remains normal crossings on these models. To make the verification fully explicit and remove any doubt about the inductive step, we will add a short subsection in the revision that performs the direct computation on the local equations of higher pinch points, confirming that every admissible center yields an NC total transform. revision: yes
Referee: [Theorem 1.4.1] Theorem 1.4.1 and the preceding classification: The claim that weighted blowups cannot yield NC-preserving resolutions rests on the classification being exhaustive. The manuscript should exhibit the complete list of local equations for pinch points in characteristic 2 and confirm that each forces a non-transverse intersection with the exceptional divisor under any weighted blowup.

Authors: Section 2 derives the classification of pinch points in characteristic 2 by analyzing the possible local equations compatible with the definition. We agree that displaying the complete list and the case-by-case obstruction computation would strengthen the presentation. In the revised version we will insert a proposition that enumerates all admissible local forms (including the standard pinch-point equation and its deformations in char 2) and verifies explicitly that any weighted blowup produces a non-transverse intersection with the exceptional divisor, thereby confirming the obstruction for every case. revision: yes
Referee: [Theorem 1.3.3] Theorem 1.3.3: The restriction of the coarse moduli space to higher pinch points is asserted without an explicit argument that the restriction can be performed while preserving the existence of an NC stack resolution. If the restriction step introduces new non-NC loci, the comparison between the stack and its coarse space breaks.

Authors: The proof of Theorem 1.3.3 constructs the restriction by selecting centers supported only on the higher pinch-point locus, ensuring that the stack-theoretic resolution remains normal crossings by the principle of Theorem 3.1.4. We acknowledge that an additional sentence clarifying that no new non-NC loci are created during the restriction would make the comparison between stack and coarse space fully rigorous. We will expand the proof accordingly in the revision. revision: yes

Circularity Check

0 steps flagged

No circularity; results rest on external citations and original classification

full rationale

The paper cites external works (Belotto da Silva-Bierstone arXiv:2602.09114 and Włodarczyk arXiv:2602.14266) for the existence of NC resolutions via stack-theoretic weighted blowups in char 0, then supplies its own principle in Theorem 3.1.4 and a restriction to higher pinch points in Theorem 1.3.3. In char 2 it performs an original classification leading to Theorem 1.4.1. No equations, definitions, or predictions reduce by construction to fitted parameters or self-citations; the derivation chain is self-contained against external benchmarks with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the work relies on standard background results in algebraic geometry rather than new free parameters or invented entities.

axioms (2)

standard math Existence of resolutions of singularities in characteristic zero
Invoked as the base case that is then strengthened to NC-preserving form via stacks.
domain assumption Properties of algebraic stacks and weighted blowups
Used to construct the NC resolution while controlling the coarse moduli space.

pith-pipeline@v0.9.0 · 5475 in / 1345 out tokens · 32482 ms · 2026-05-15T21:53:25.095855+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1.4 (adequate class C of singularities + wonderful invariant yields C-preserving weighted resolution); Theorem 1.4.1(2) (weighted blowups cannot yield NC-preserving resolution of pinch points in char 2 because they are tame)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3.1.5 (X_nc(k) closed in level set W_K of the invariant); Corollary 1.3.3 (destackification to higher pinch points)

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.