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arxiv: 2301.08706 · v2 · submitted 2023-01-20 · 🧮 math.DG

A series of Nash resolutions of a singular foliation

Ruben Louis This is my paper

Pith reviewed 2026-05-24 10:26 UTC · model grok-4.3

classification 🧮 math.DG
keywords singular foliationNash modificationblowupDebord foliationLie infinity-algebroidprojective singular foliationresolution
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The pith

Any singular foliation becomes a Debord foliation after one Nash blowup of its universal Lie infinity-algebroid.

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

The paper constructs a series of blowups of singular foliations by repeatedly applying the Nash modification to their universal Lie infinity-algebroid. This process recovers earlier blowups due to Sertöz at the first step and Mohsen at the second. The central result shows that a single application of this modification turns any singular foliation into a Debord foliation, which is a projective singular foliation. Sympathetic readers would care because this offers a uniform resolution method that works for all such structures. It unifies previous constructions under one framework.

Core claim

By applying the Nash modification to the universal Lie ∞-algebroid of a singular foliation, one obtains a sequence of blowups where after exactly one step the foliation satisfies the Debord property, meaning it is projective.

What carries the argument

The Nash modification of the universal Lie ∞-algebroid, which generates the blowups and enforces the projective property after one iteration.

If this is right

  • Recovers the blowup introduced by Sinan Sertöz for i=0.
  • Recovers the notion due to Omar Mohsen for i=1.
  • Any singular foliation becomes a Debord foliation after one blowup.
  • Examples of these resolutions can be constructed explicitly.

Where Pith is reading between the lines

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

  • Further iterations might provide higher-order resolutions or invariants for classifying foliations.
  • This approach could be compared to resolution techniques in algebraic geometry for testing generality.
  • Applying the method to concrete examples like specific singular foliations on manifolds could verify the projective property directly.

Load-bearing premise

The Nash modification applied to the universal Lie ∞-algebroid always produces a blowup that is a Debord foliation after exactly one application.

What would settle it

A counterexample singular foliation where the structure after one Nash modification on its universal Lie ∞-algebroid remains non-Debord.

read the original abstract

We construct a series of blowups $(\widetilde M_i,\pi_i)_{i\in \mathbb N_0}$ of a singular foliation by applying to the universal Lie $\infty$-algebroid of a singular foliation the so-called Nash modification. For $i=0$, we recover a blowup introduced Sinan Sert\"oz, and for $i=1$, we recover a notion due to Omar Mohsen. One of the important features is that any singular foliation becomes a Debord foliation (= projective singular foliation) after one blowup. Examples are also given.

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 series of blowups (M̃_i, π_i)_{i∈ℕ₀} of an arbitrary singular foliation by iteratively applying the Nash modification to its universal Lie ∞-algebroid. The principal claim is that the resulting structure is a Debord (projective) singular foliation after exactly one step; the i=0 and i=1 cases recover the constructions of Sertöz and Mohsen, respectively, and illustrative examples are supplied.

Significance. If the one-step resolution to a Debord foliation holds, the work supplies a canonical, uniform procedure that converts any singular foliation into a projective one, thereby extending the reach of techniques available for regular or projective foliations and unifying earlier ad-hoc blowups within the framework of the universal Lie ∞-algebroid.

major comments (2)
  1. [Main theorem on the one-step Debord property] The central claim that the Nash modification yields a Debord foliation after precisely one iteration is load-bearing; the manuscript must supply an explicit verification (e.g., in the section containing the main theorem) that the image of the modified anchor map is a vector subbundle of the tangent bundle, rather than merely a sheaf of modules.
  2. [Construction of the modified universal Lie ∞-algebroid] The preservation of the foliation axioms (in particular, the Lie ∞-structure and the involutivity condition) under the Nash modification is invoked but not derived in detail; a step-by-step check that the modified bracket remains a derivation of the correct degree is required to confirm that the output remains a singular foliation.
minor comments (2)
  1. The recursive definition of the sequence (M̃_i, π_i) for i ≥ 2 should be stated explicitly, including how the universal Lie ∞-algebroid is updated at each step.
  2. Notation for the Nash modification and the resulting anchor map should be introduced with a short table or diagram contrasting the original and modified structures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the two major comments below.

read point-by-point responses

  1. Referee: [Main theorem on the one-step Debord property] The central claim that the Nash modification yields a Debord foliation after precisely one iteration is load-bearing; the manuscript must supply an explicit verification (e.g., in the section containing the main theorem) that the image of the modified anchor map is a vector subbundle of the tangent bundle, rather than merely a sheaf of modules.

    Authors: We agree that an explicit verification is required. In the revised manuscript we will insert, in the section stating the main theorem, a direct argument showing that the image of the modified anchor is a vector subbundle of the tangent bundle. revision: yes

  2. Referee: [Construction of the modified universal Lie ∞-algebroid] The preservation of the foliation axioms (in particular, the Lie ∞-structure and the involutivity condition) under the Nash modification is invoked but not derived in detail; a step-by-step check that the modified bracket remains a derivation of the correct degree is required to confirm that the output remains a singular foliation.

    Authors: We agree that a more detailed verification is needed. The revised version will contain an explicit, step-by-step check that the modified bracket is a derivation of the appropriate degree and that the Lie ∞-structure and involutivity are preserved, thereby confirming that the output remains a singular foliation. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a sequence of blowups by applying the Nash modification to the universal Lie ∞-algebroid of a given singular foliation, recovering known constructions for i=0 (Sertöz) and i=1 (Mohsen). The central claim—that the result is Debord/projective after exactly one step—follows directly from the stated properties of these established inputs rather than from any internal fit, self-definition, or load-bearing self-citation. No equations or steps in the abstract reduce the claimed resolution to a renamed input or a parameter fitted to the target quantity itself. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the pre-existing notions of singular foliation, universal Lie ∞-algebroid, Nash modification, and Debord foliation; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence and properties of the universal Lie ∞-algebroid associated to any singular foliation
    Invoked as the object to which the Nash modification is applied in the construction.

pith-pipeline@v0.9.0 · 5611 in / 1297 out tokens · 26105 ms · 2026-05-24T10:26:29.708448+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On Nash resolution of (singular) Lie algebroids

    math.DG 2024-04 unverdicted novelty 6.0

    Defines Nash blow-up Nash(A) for Lie algebroids yielding short exact sequence 0 to K to Nash(A) to D to 0 with K Lie algebra bundle and D having dense injective anchor, plus extension to singular subalgebroids.

  2. On longitudinal differential operators and Nash blowups

    math.DG 2025-09 unverdicted novelty 5.0

    Links Helffer-Nourrigat cone of singular foliations to Nash algebroids and characterizes longitudinally elliptic operators via symplectic leaves of holonomy Lie algebroids.

Reference graph

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