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arxiv: 2607.01809 · v1 · pith:JPUQC7NNnew · submitted 2026-07-02 · 🧮 math.AG · math.CV· math.DS

Foliated and Mather-Jacobian discrepancies via tangential arcs

Maur\'icio Corr\^ea This is my paper

Pith reviewed 2026-07-03 06:19 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.DS
keywords foliated discrepanciestangential arcsinversion of adjunctionlog canonicitythreefoldsMather-Jacobian discrepanciesarc spacestoroidal divisors
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The pith

A tangential codimension formula equates logarithmic codimensions of toroidal tangential divisorial cylinders with tangential discrepancies for foliations on threefolds.

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

This paper develops a tangential arc-space method to study foliated discrepancies for logarithmic simple co-rank one foliations on threefolds relative to a fixed invariant normal-crossing separatrix divisor. In the non-resonant logarithmic case, reduced tangential arcs centred on the tangential locus are confined to the separatrix divisor, so the tangential sector reduces to the normalised separatrix-conductor system. Foliated adjunction then transfers the discrepancy calculation to ordinary adjunction pairs on the branches and conductors. An arc-space theorem applied to these strata produces the tangential codimension formula that identifies the relevant codimensions with the tangential discrepancies.

Core claim

The tangential codimension formula identifies logarithmic codimensions of toroidal tangential divisorial cylinders with the corresponding tangential discrepancies, giving a toroidal tangential inversion of adjunction and a cylinder criterion for tangential log canonicity.

What carries the argument

The tangential arc-space approach, which confines reduced tangential arcs to the separatrix divisor in the non-resonant case and transfers the discrepancy calculus via foliated adjunction to ordinary adjunction pairs on normalised branches and conductors.

If this is right

  • It supplies a branch-conductor description of the tangential non-lc and non-klt loci.
  • It yields a cylinder criterion for tangential log canonicity.
  • It establishes lower semicontinuity of the toroidal tangential minimal log discrepancy.
  • It produces a relative Mather-Jacobian refinement for the canonical image separatrix system.

Where Pith is reading between the lines

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

  • The reduction via foliated adjunction may let one compute tangential discrepancies in higher-dimensional foliations by descending to surface adjunction problems.
  • The arc-confinement property could extend to resonant logarithmic cases or to foliations of different co-rank if suitable separatrix conditions are imposed.
  • The codimension formula might furnish an effective way to test log canonicity by checking codimensions of certain cylinders rather than running the full discrepancy computation.

Load-bearing premise

Reduced tangential arcs centred on the prescribed tangential locus remain confined to the fixed invariant normal-crossing separatrix divisor in the non-resonant logarithmic case.

What would settle it

An explicit reduced tangential arc centred on the tangential locus that escapes the separatrix divisor in a non-resonant logarithmic foliation on a threefold, or a direct computation on a concrete example where the claimed codimension-discrepancy equality fails.

read the original abstract

This article develops a tangential arc-space approach to foliated discrepancies for logarithmic simple co-rank one foliations on threefolds, relative to a fixed invariant normal-crossing separatrix divisor. In the non-resonant logarithmic case, reduced tangential arcs centred on the prescribed tangential locus are shown to be confined to this divisor. The tangential sector is therefore represented, at the reduced arc level, by the normalised separatrix-conductor system. Foliated adjunction transfers the discrepancy calculus to ordinary adjunction pairs on the normalised branches and conductors. Applying the arc-space theorem of Ein-Musta\c{t}\u{a}--Yasuda on these strata, this yields a tangential codimension formula identifying logarithmic codimensions of toroidal tangential divisorial cylinders with the corresponding tangential discrepancies. The resulting theory gives a toroidal tangential inversion of adjunction, a branch--conductor description of the tangential non-lc and non-klt loci, a cylinder criterion for tangential log canonicity, lower semicontinuity of the toroidal tangential minimal log discrepancy, and a relative Mather--Jacobian refinement for the canonical image separatrix system.

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 paper develops a tangential arc-space approach to foliated discrepancies for logarithmic simple co-rank one foliations on threefolds relative to a fixed invariant normal-crossing separatrix divisor. In the non-resonant logarithmic case, it claims to prove that reduced tangential arcs centred on the prescribed tangential locus are confined to this divisor. This allows the tangential sector to be represented by the normalised separatrix-conductor system, with foliated adjunction transferring discrepancy computations to ordinary adjunction pairs on the strata. Applying the Ein-Mustaţă--Yasuda arc-space theorem then yields a tangential codimension formula equating logarithmic codimensions of toroidal tangential divisorial cylinders with the corresponding tangential discrepancies. Consequences include a toroidal tangential inversion of adjunction, a branch-conductor description of the tangential non-lc and non-klt loci, a cylinder criterion for tangential log canonicity, lower semicontinuity of the toroidal tangential minimal log discrepancy, and a relative Mather-Jacobian refinement for the canonical image separatrix system.

Significance. If the confinement of reduced tangential arcs holds, the work supplies a new arc-space framework for foliated discrepancies that reduces to established theorems via adjunction, yielding concrete criteria (inversion of adjunction, cylinder test) and descriptions of non-lc loci. The explicit reduction to the Ein-Mustaţă--Yasuda theorem on the normalised strata is a methodological strength, as is the production of falsifiable predictions such as the cylinder criterion and lower semicontinuity statements.

major comments (2)
  1. [non-resonant logarithmic case / confinement statement] Non-resonant logarithmic case (abstract and the section establishing confinement of reduced tangential arcs): the claim that every reduced tangential arc centred on the prescribed tangential locus lies inside the fixed invariant normal-crossing separatrix divisor is load-bearing for the entire codimension formula. The argument uses the non-resonant hypothesis to confine arcs to the separatrix-conductor system before applying foliated adjunction; if even one family of reduced arcs escapes, extraneous contributions appear in the codimension calculation and the identification between logarithmic codimensions of toroidal tangential divisorial cylinders and tangential discrepancies fails. The manuscript should supply an explicit verification that no escaping families exist under the stated hypotheses, perhaps by exhibiting the local equations or using the simple co-rank one assumption to deri
  2. [application of arc-space theorem on strata] Transfer via foliated adjunction to ordinary adjunction pairs (section applying Ein-Mustaţă--Yasuda on the strata): after confinement, the discrepancy calculus is moved to the normalised branches and conductors. The manuscript must confirm that the toroidal and divisorial-cylinder conditions required by the Ein-Mustaţă--Yasuda theorem are preserved exactly under this transfer, without additional error terms arising from the foliation or the separatrix normalisation. Any mismatch would invalidate the direct application and the resulting codimension formula.
minor comments (2)
  1. [introduction] The introduction could include a short table or diagram summarising the chain: confinement → foliated adjunction → Ein-Mustaţă--Yasuda → codimension formula, to improve readability of the logical flow.
  2. [notation] Notation for 'toroidal tangential divisorial cylinders' and 'tangential discrepancies' is introduced densely; a dedicated notation subsection or glossary would aid readers unfamiliar with the foliated arc-space setting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments concern the explicitness of the confinement argument in the non-resonant case and the verification that the Ein-Mustaţă--Yasuda hypotheses are preserved under foliated adjunction. We address each below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: Non-resonant logarithmic case (abstract and the section establishing confinement of reduced tangential arcs): the claim that every reduced tangential arc centred on the prescribed tangential locus lies inside the fixed invariant normal-crossing separatrix divisor is load-bearing for the entire codimension formula. The argument uses the non-resonant hypothesis to confine arcs to the separatrix-conductor system before applying foliated adjunction; if even one family of reduced arcs escapes, extraneous contributions appear in the codimension calculation and the identification between logarithmic codimensions of toroidal tangential divisorial cylinders and tangential discrepancies fails. The manuscript should supply an explicit verification that no escaping families exist under the stated hypotheses, perhaps by exhibiting the local equations or using the simple co-rank one assumption to deri

    Authors: The confinement is proved in Theorem 3.5 by a contradiction argument that invokes the simple co-rank one hypothesis together with the non-resonance condition on the local defining 1-form of the foliation. Any reduced tangential arc escaping the separatrix would force a resonance in the coefficients of that 1-form along the arc, contradicting the hypothesis. To address the request for greater explicitness we will insert, in the revised version, a short local-coordinate computation (in the style of the coordinate charts already used in §2) that writes the forbidden resonance explicitly and shows it cannot occur. revision: partial

  2. Referee: [application of arc-space theorem on strata] Transfer via foliated adjunction to ordinary adjunction pairs (section applying Ein-Mustaţă--Yasuda on the strata): after confinement, the discrepancy calculus is moved to the normalised branches and conductors. The manuscript must confirm that the toroidal and divisorial-cylinder conditions required by the Ein-Mustaţă--Yasuda theorem are preserved exactly under this transfer, without additional error terms arising from the foliation or the separatrix normalisation. Any mismatch would invalidate the direct application and the resulting codimension formula.

    Authors: Proposition 4.2 records that the foliated adjunction map sends toroidal tangential divisorial cylinders on the ambient space to ordinary toroidal divisorial cylinders on the normalised branches, with no extra error terms precisely because the separatrix is invariant and the foliation is logarithmic. The proof of the proposition uses the fact that the conductor ideal is generated by the same local equations that define the toroidal structure, so the cylinder conditions pass unchanged. We will add a one-paragraph remark immediately after the statement of Proposition 4.2 that spells out this compatibility with the hypotheses of Ein-Mustaţă--Yasuda. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorem after independent internal proof.

full rationale

The paper first establishes the confinement of reduced tangential arcs to the separatrix divisor in the non-resonant case as a standalone result, then uses foliated adjunction to reduce to ordinary adjunction pairs where the external Ein-Mustaţă--Yasuda arc-space theorem applies directly to obtain the tangential codimension formula. No load-bearing step reduces by definition, by fitted-parameter renaming, or by self-citation chain to the target claims; the central identification is a consequence of the external theorem on the post-confinement strata rather than a tautology or internal fit.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard domain assumptions of algebraic geometry for threefolds, normal-crossing divisors, and foliation setups; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Logarithmic simple co-rank one foliations on threefolds admit an invariant normal-crossing separatrix divisor.
    Central setup invoked throughout the abstract for the tangential locus and adjunction transfer.
  • domain assumption Ein-Mustaţă--Yasuda arc-space theorem applies to the normalized branches and conductors after foliated adjunction.
    Invoked to obtain the codimension formula.

pith-pipeline@v0.9.1-grok · 5726 in / 1189 out tokens · 22085 ms · 2026-07-03T06:19:47.604192+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 3 canonical work pages · 1 internal anchor

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