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Foliated and Mather-Jacobian discrepancies via tangential arcs

T0 review · 1 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Tangential arcs on non-resonant foliations reduce foliated discrepancies to ordinary log-pair calculations on branches and conductors.

desk verdict Solid, carefully scoped arc-space reduction for non-resonant logarithmic foliations; the confinement theorem and tangential EMY transfer are real and usable inside the stated sector. read the letter →

arxiv 2607.01809 v2 pith:JPUQC7NN submitted 2026-07-02 math.AG math.CVmath.DS

Maurício Corrêa This is my paper · ORCID
classification math.AGmath.CVmath.DS MSC 14E3014B0532S6514E18
keywords foliateddiscrepanciestangentialarcsseparatrix–conductorsystemEin–Mustaţă–YasudaMather–Jacobiandiscrepancylogarithmicfoliationsminimallogadjunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that for logarithmic simple co-rank one foliations on threefolds, when residues satisfy a positive non-resonance condition, reduced infinite arcs that stay tangent to the foliation and centre on a prescribed invariant divisor cannot leave that divisor. Those arcs are therefore exactly the ordinary arcs on the divisor, which can be presented by a finite normalised system of branches and their pairwise conductors. Foliated adjunction turns the branches and conductors into ordinary log pairs; the classical Ein–Mustaţă–Yasuda codimension formula on those pairs then becomes a tangential codimension formula that recovers a new tangential discrepancy. On adapted toroidal invariant divisors read on branches, this tangential discrepancy coincides with the usual foliated log discrepancy of the foliated MMP. The same identification yields toroidal 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 on the image separatrix system. A sympathetic reader cares because the construction supplies the first arc-space calculus that computes foliated discrepancies by ordinary discrepancy calculations on a finite collection of surfaces and curves.

What carries the argument

The reduced tangential arc-confinement theorem (and its coequaliser presentation by the normalised separatrix–conductor system). Non-resonance forces every reduced tangential formal arc centred on the invariant divisor to factor through that divisor; the resulting coequaliser of branch and conductor arc spaces carries all subsequent codimension and discrepancy calculations.

What would settle it

In a pure logarithmic model whose residues satisfy a positive resonance relation, exhibit an explicit reduced formal arc that is tangent yet whose generic point leaves the invariant divisor; if such arcs exist, confinement (and therefore the codimension formula) fails.

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Extended reading notes

Core claim

In the non-resonant logarithmic simple adapted setting, the reduced tangential arc sector of a co-rank one foliation on a threefold is identified with the ordinary reduced arc space of a fixed invariant normal-crossing separatrix divisor, and is therefore presented by the coequaliser of arcs on the normalised branches and conductors. Foliated adjunction transfers the discrepancy calculus to ordinary log pairs on those strata; applying the Ein–Mustaţă–Yasuda theorem there yields that the tangential logarithmic codimension of every adapted toroidal tangential divisorial cylinder equals q times the corresponding tangential discrepancy, which for branch data agrees with the usual foliated discre

Load-bearing premise

The residues of the logarithmic form never form a positive integer relation; if they do, reduced tangential arcs can escape the prescribed invariant divisor and the whole branch–conductor reduction collapses.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 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 simple adapted setting, reduced tangential arcs centred on the tangential locus are confined to the invariant divisor (Theorem 4.1 / Theorem 1.2). The reduced tangential sector is then presented by the seminormal branch–conductor system (coequaliser of reduced arc functors). Foliated adjunction transfers the discrepancy calculus to ordinary log pairs on the normalised branches and conductors; the Ein–Mustaţă–Yasuda theorem on those strata yields a tangential codimension formula (Corollary 1.4 / Theorem 12.3) identifying logarithmic codimensions of adapted toroidal tangential divisorial cylinders with the tangential discrepancies a_tan. For adapted toroidal invariant divisors read on normalised branches one has a_tan = A_F (usual foliated log discrepancy). Applications include toroidal tangential inversion of adjunction, branch–conductor descriptions of non-lc/non-klt loci, a cylinder criterion for tangential log canonicity, lower semicontinuity of tmld_tor, and a relative Mather–Jacobian refinement on the canonical image separatrix system.

Significance. If the reduction chain holds, the paper supplies a clean arc-space dictionary for a restricted but natural class of foliated singularities (logarithmic simple non-resonant, adapted toroidal). The confinement argument is elementary and explicit in formal power series; the subsequent transfer of ordinary EMY / Mather–Jacobian formulas to the branch–conductor strata is carefully functorial under adapted blow-ups and crepant under common refinements. The comparison a_tan = A_F for branch data, the lower-semicontinuity statement for tmld_tor, and the relative MJ refinement are concrete contributions that sit usefully between Carter’s jet schemes of foliations and the foliated MMP of Cascini–Spicer–Svaldi. The theory is deliberately narrow and the manuscript flags its own limits (non-resonance, reduced infinite arcs, image-system MJ); that honesty is a strength rather than a defect.

major comments (1)
  1. The central load-bearing hypothesis is positive non-resonance of the logarithmic residues (Definition 2.1). Theorem 4.1 / Theorem 1.2 and the whole branch–conductor presentation rest on it; Proposition 4.3 correctly shows sharpness. The manuscript already restricts every global statement to the non-resonant logarithmic simple adapted toroidal sector and never claims the resonant or second simple type. No further major technical gap is visible in the reduction chain (confinement → coequaliser of reduced arcs → crepant adjunction → ordinary EMY on strata). The comparison a_tan = A_F is likewise correctly restricted to branch data (Corollary 1.5 / Corollary 11.3). I therefore raise no load-bearing objection that would require a major revision of the argument.
minor comments (5)
  1. The introduction and abstract correctly emphasise that only reduced infinite arcs are used, yet several later passages still allude to Carter jet schemes without always repeating the reduced-infinite restriction. A single clarifying sentence at the start of §4 would help the reader.
  2. Notation for the two discrepancy conventions (α(E;G,Δ_W) versus A_F) is introduced in Notation 3.2 but reappears with slight variations later; a short table or a consistent choice of symbol throughout would improve readability.
  3. The Mather–Jacobian section (§16) is long and partly independent of the main EMY transfer. A brief roadmap at the beginning of that section, or a clearer separation of the relative versus intrinsic statements, would help.
  4. Examples 19.1–19.7 are useful local models; a short pointer in the introduction to which example illustrates coefficient-one cancellation versus transverse defect would make them easier to locate.
  5. Minor typographical inconsistencies appear in the arXiv version (e.g., spacing around “Ein–Mustaţă–Yasuda”, occasional missing accents). These are easily cleaned.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: tangential discrepancies and codimensions are ordinary EMY/MJ quantities after an independently proved geometric reduction (non-resonant confinement + branch–conductor coequaliser + crepant adjunction).

full rationale

The derivation chain is: (i) positive non-resonance implies reduced tangential arcs are confined to D_inv (Thm 4.1, proved by leading-term residue calculation on the logarithmic generator, with sharpness shown separately in Prop 4.3); (ii) the reduced arc functor of the SNC divisor is presented by the coequaliser of normalised branch and conductor arc spaces (Lem 4.8, Prop 4.9); (iii) normalised foliated adjunction produces ordinary log pairs (V, B_V) on those strata, crepant under adapted blow-ups (Thm 8.2, Prop 8.3); (iv) adapted toroidal tangential cylinders are identified with ordinary maximal divisorial sets N_q(F) on those pairs (Thm 9.2, Def 12.1); (v) ordinary EMY (or MJ) on (V, B_V) therefore yields lcodim_tan(N_tan_q(E)) = q a(F; V, B_V). The quantity a_tan is defined to be that ordinary discrepancy (Def 9.3), so the codimension formula (Thm 12.3 / Cor 1.4) is immediate once the geometric identification is established; it is not a fitted or self-referential prediction. The comparison a_tan = A_F for branch data (Cor 11.3) is a separate local coefficient-matching computation (Prop 10.1, Lem 11.2) against the foliated MMP convention, not an input. All background citations (Cano, Cascini–Spicer, Spicer–Svaldi, EMY, de Fernex–Docampo, etc.) are external; there is no load-bearing self-citation, uniqueness import, or ansatz smuggling. The theory is deliberately restricted to the non-resonant logarithmic simple adapted toroidal sector, which the paper itself flags. No step reduces the central claim to its own definition or to a fitted parameter.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No free parameters or fitted constants. The theory rests on standard algebraic geometry (arc spaces, log discrepancies, seminormal schemes) plus domain assumptions from the foliated MMP (simple non-dicritical singularities, foliated adjunction) and paper-specific restrictions (logarithmic non-resonant adapted charts, fixed toroidal category generated by coordinate stratum blow-ups, reduced infinite arcs only). Invented entities are definitional constructions (tangential discrepancy, tmld_tor, image separatrix system), not postulated physical objects; independent evidence is internal consistency and comparison with A_F on branches.

assumptions (6)
  • standard math Ein–Mustaţă–Yasuda theorem: for a log smooth (or log-resolved) pair, logarithmic codimension of a maximal divisorial cylinder N_q(F) equals q·a(F;V,B_V).
    Invoked in Theorem 12.3 / Corollary 1.4 after reduction to branch–conductor pairs; cited as [11].
  • domain assumption Foliated adjunction for invariant divisors of co-rank-one foliated pairs supplies a different Diff_S(Δ) so that ν*((K_G+Δ_W)|S) ~_Q K_{S^ν}+Θ_S with coefficient-one invariant traces.
    Used in Theorem 8.2 and throughout the discrepancy transfer; cited Cascini–Spicer / Spicer–Svaldi [8,14].
  • domain assumption Cano reduction: codimension-one foliations on smooth threefolds admit resolutions to simple singularities by blow-ups in the singular locus.
    Background for the logarithmic simple adapted setting (Remark 2.4); the paper works only in the non-resonant logarithmic part of simple singularities.
  • ad hoc to paper Positive non-resonance: a·λ ≠ 0 for all nonzero a in Z^r_≥0 in every logarithmic simple adapted chart.
    Definition 2.1; required for arc confinement (Theorem 4.1). The paper proves sharpness when resonance occurs (Proposition 4.3).
  • ad hoc to paper All global statements are made inside the fixed adapted toroidal category Adm(X,F,Δ;W_0) generated by coordinate stratum blow-ups from a chosen logarithmic adapted model.
    Definition 3.1 and Theorem 3.10; model independence is only conditional on common adapted refinements and the same image separatrix system.
  • ad hoc to paper Discrepancy calculations use only reduced infinite tangential arcs, not equality of finite Carter tangent jet schemes.
    Section 4 and Example 4.5; finite-level branch decompositions fail even for xy=0.
invented entities (2)
  • Tangential discrepancy a_tan and toroidal tangential mld tmld_tor
    purpose: Measure singularities of the foliated pair via ordinary log discrepancies on normalised branch–conductor adjunction pairs after arc confinement.
    Defined in Definitions 9.3 and 12.4; compared with A_F only for adapted toroidal invariant branch data (Corollary 1.5). Independent evidence is the EMY codimension match and the branch comparison lemma, not external experiment.
  • Canonical image separatrix system S^sn_X and relative tangential Mather–Jacobian discrepancy a_tan_MJ
    purpose: Handle possibly non-algebraic formal separatrices by working with the algebraic image of the adapted toroidal category and applying ordinary MJ theory downstairs.
    Construction 16.1 and Definition 16.3; intrinsic comparison requires Assumption 16.9 (algebraic separatrix hypothesis).

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Pith. "Pith review of Foliated and Mather-Jacobian discrepancies via tangential arcs." pith.science (2026). https://pith.science/paper/JPUQC7NN

@misc{pith2026260701809,
  author       = {Pith},
  title        = {Pith review of: Foliated and Mather-Jacobian discrepancies via tangential arcs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JPUQC7NN}},
  note         = {Machine review of arXiv:2607.01809}
}
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 confined to this divisor. The tangential sector is therefore presented, at the reduced arc level, by the normalised separatrix-conductor system. Foliated adjunction transfers the discrepancy calculus to ordinary log pairs obtained by adjunction on the normalised branches and conductors. The arc-space theorem of Ein-Musta\c{t}\u{a}-Yasuda, applied on these strata, then gives a tangential codimension formula identifying logarithmic codimensions of toroidal tangential divisorial cylinders with the corresponding tangential discrepancies. For toroidal invariant divisors read on branches, this tangential discrepancy agrees with the usual foliated discrepancy. The resulting theory gives 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.

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Works this paper leans on

15 extracted references · 2 linked inside Pith

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