Numerical Reduction and Sharp Thresholds for Adjoint Singularities of Foliated Surfaces

Shi Xu

REVIEW 2 major objections 4 minor 14 references

Numerical Reduction and Sharp Thresholds for Adjoint Singularities of Foliated Surfaces

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For foliated surface singularities, every ε-adjoint log canonical singularity is foliated log canonical when 0<ε<1/5, every ε-adjoint canonical singularity is foliated lc and surface klt when 0<ε<1/4, and both bounds are sharp.

desk verdict A genuinely new classification with the right-looking thresholds, but the completeness argument has a real gap around Proposition 4.14 that a referee should force them to close. read the letter →

arxiv 2512.20744 v3 pith:H4NYT3M6 submitted 2025-12-23 math.AG

Shi Xu This is my paper · ORCID

classification math.AG MSC 14J2914B0532S6532M25

keywords Foliationsadjointdivisorslogcanonicalsingularitiesfoliatedsurfacesminimalmodelprogramnegativedefiniteconfigurationssingularitythresholds

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 studies what happens to foliated surface singularities when the foliation's canonical class is perturbed by a small multiple of the ambient surface's canonical class, forming the adjoint divisor K_F + εK_X. It proves a complete classification of ε-adjoint log canonical singularities for 0<ε<1/3, showing that the first stability threshold is ε=1/5: below it, every ε-adjoint log canonical singularity is already foliated log canonical and the underlying surface singularity is log canonical, while at ε=1/5 a new boundary configuration appears. Imposing the stronger ε-adjoint canonical condition moves the threshold to ε=1/4: below it, the singularity is foliated log canonical and the surface is klt. Both thresholds are sharp and realized by explicit examples. If correct, the results give a stability range for the adjoint minimal model program on surfaces and determine the interpolated lc threshold gap constant τ(2)=1/6.

What carries the argument

The engine is a numerical reduction procedure for negative definite exceptional configurations. On the minimal resolution, the ε-adjoint log canonical condition becomes a system of linear inequalities for discrepancy coefficients, expressible as the condition that the exceptional divisor E is a K≥0-graph for K = K_F + εK_X with 0<ε<1/3. The procedure identifies components with K·C<0, starts special K>0-chains from them (F-chains and related linear configurations of F-invariant and non-invariant curves), peels off these chains by subtracting their intersection-theoretic projections M(K,Θ), and then analyzes the residual divisor K₁ on the remaining components. Finite combinatorial checks—using

What would settle it

Solve the integer systems in Lemma 4.12, namely (4.6)–(4.8) for nonnegative integers (x,l) with 0<ε<1/3, and Lemma 4.13, (4.10)–(4.13) for (x,k,l,y). If any solution beyond the listed pairs (2,0),(1,2),(0,3),(0,4) and quadruples (2,0,0,0),(1,2,0,0),(0,2,0,1),(0,2,1,0) exists, the classification misses a configuration and the sharpness claims fail. Concretely, a reader could attempt to construct an exceptional divisor with an extra F-chain of type (B) attached to the boundary curve C in Type (III) that still satisfies K·C≤0 for some ϵ<1/5.

Watch

Extended reading notes

Core claim

The central claim is a numerical classification of negative definite exceptional configurations. Passing to the minimal resolution, ε-adjoint log canonical singularities correspond exactly to configurations whose exceptional divisor is a K≥0-graph for K = K_F + εK_X, and the paper determines all such graphs for 0<ε<1/3. The classification separates into foliated canonical configurations, foliated lc non-canonical configurations, and a single boundary type: a smooth rational F-invariant curve with Z(F,C)=3 connecting two F-chains of type (2,3), possibly with an extra chain of (−2)-F-curves. This boundary type is not foliated log canonical and becomes ε-adjoint log canonical precisely at ε=1/5

Load-bearing premise

The whole classification depends on the claim that the short lists of possible exceptional-curve configurations produced by finite numerical inspections—especially the 'straightforward inspection' steps in Lemmas 4.12 and 4.13 and the combinatorial analysis in Proposition 4.6—are complete; if any admissible configuration was missed, the thresholds 1/5 and 1/4 could shift or acquire companions.

Editorial extensions

If this is right

For 0<ε<1/5, every ε-adjoint log canonical singularity is both foliated log canonical and an lc surface singularity, so small adjoint perturbations cannot create new non-lc foliated behavior.
For 0<ε<1/4, every ε-adjoint canonical singularity is foliated log canonical and the underlying surface singularity is klt, giving a log-to-canonical stability interval.
The adjoint minimal model program for foliated surfaces runs for all rational ε in (0,1/4), producing models whose surfaces have klt singularities and foliations have lc singularities, and ample canonical classes when K_F+εK_X is big.
The interpolated lc threshold t₀ satisfies t₀=1 or t₀≤5/6, and the dimension-two gap constant for the 1-gap conjecture is τ(2)=1/6, with sharpness shown by the boundary configuration at ε=1/5.
Both thresholds are optimal: at ε=1/5 a boundary configuration enters the admissible region, and at ε=1/4 a configuration that is ε-adjoint canonical but not foliated canonical exists, so no larger uniform interval is possible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The numerical reduction procedure appears to be a general mechanism, not a one-off classification: the paper itself notes that minor parameter adjustments recover the classical classifications of log canonical surface and foliated singularities, suggesting the same framework could classify other adjoint families.
The specific values 1/5 and 1/4 are not arbitrary constants but are forced by the determinants 2 and 3 of the F-chains in the boundary Type (III) configurations, so one might expect analogous walls in higher dimensions to be governed by small determinant chains rather than by global dimension constants.
A testable extension would be to run the same reduction on non-minimal resolutions or on singularities with boundary divisors, checking whether the walls 1/5 and 1/4 persist or shift when extra marked curves are allowed.
The sharpness examples suggest that the failure of foliated lc at the walls is caused by very specific local models; one could look for whether these models are the only obstructions to extending the stability intervals in a relative or logarithmic setting.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper gives a systematic classification of ε-adjoint singularities on foliated surfaces for 0<ε<1/3 and extracts sharp walls at ε=1/5 and ε=1/4. If the classification is complete, this is the right result: stability of adjoint lc on (0,1/5), canonical-to-lc stability on (0,1/4), and the dimension-two 1-gap constant τ(2)=1/6. That is a genuine step beyond SS23 and Vas25, which only handle smooth and klt base points.

What is good: the reduction to discrepancy inequalities and K≥0-graphs is coherent; Lemmas 4.2 and 4.3 give a clean list of K-negative curves; the special K>0-chains in Proposition 4.6 are structurally useful. The explicit boundary examples in Section 6 are concrete and I see no issue with them. The sign slips flagged in Lemmas 5.2 and 5.4 are real but minor: the displayed inequalities are off as printed, yet the intended positivity still follows from the surrounding bounds, so they do not sink the argument.

Soft spots: the paper leans on several “straightforward inspection” checks that are not shown. Lemmas 4.12 and 4.13 reduce to small integer inequalities, so a referee can verify those in an afternoon. The worrying spot is Proposition 4.14: after subtracting maximal special chains, the residual K₁=0 case is dismissed in a single sentence (“ruled out by the Separatrix Theorem and assumption (∗)”). The stress-test note is right that no residual-configuration analysis is actually presented. The theorem’s exhaustiveness — and hence both thresholds — depends on that assembly step. A second fragility is the repeated invocation of the Separatrix Theorem on tree-shaped subgraphs before the full configuration is known; the hypotheses may hold here, but the paper does not verify them case by case. If a missed residual configuration exists, the walls could shift or acquire companions, so any referee should demand the missing case analysis be written out.

Bottom line: this deserves serious refereeing, and with the Proposition 4.14 gap filled I would expect it to stand. Anyone working on foliated MMP or lc thresholds should read it. Send it to a competent referee rather than desk-rejecting.

Referee Report

2 major / 4 minor

Summary. The paper studies the adjoint divisors K_F + εK_X for foliated surfaces and classifies ε-adjoint log canonical singularities for 0 < ε < 1/3 by a numerical reduction of negative definite exceptional configurations. The main results are: (i) for 0 < ε < 1/5 every ε-adjoint lc singularity is foliated lc and surface lc, with sharpness at 1/5 via a boundary configuration not foliated lc; (ii) imposing ε-adjoint canonicity gives stability for 0 < ε < 1/4 and a classification in Theorem 5.7; (iii) applications to the adjoint minimal model program and to interpolated lc thresholds, including τ(2)=1/6. The proof reduces the adjoint lc condition to nonnegativity of solutions of the intersection equations, then proposes a finite assembly classification via special chains, residual divisors, and several finite combinatorial checks.

Significance. If the classification is complete, the paper establishes sharp and explicit stability thresholds for adjoint singularities of foliated surfaces, improving the range of the adjoint MMP from ε∈(0,1/5) to ε∈(0,1/4) and giving a precise 1-gap constant τ(2)=1/6. The numerical reduction method is natural and the deterministic computations behind the walls 1/5 and 1/4 are coherent. The claimed sharpness is supported by concrete examples (Example 6.3). The argument does not appear circular: the classification of which configurations are foliated lc uses the independent results of Chen and Alexeev/Kollár–Mori. However, the exhaustiveness of the finite assembly is asserted in several places rather than demonstrated, and this is load-bearing for the central classification.

major comments (2)

[§4.4.3, Proposition 4.14] The residual (K_1)=0 case is not proved. After excluding the configurations of Lemmas 4.12(1) and 4.13(1)–(2), the proof says: 'Hence E is a (K_1)=0-graph... any such E is ruled out by the separatrix theorem and assumption (∗).' This is an assertion, not an argument. A missing residual configuration could be ε-adjoint lc for ε<1/3 and could enter before ε=1/5, shifting both claimed walls. Please supply the complete case analysis, or a verifiable enumeration of all connected graphs assembled from the listed building blocks that satisfy the numerical conditions and then show each is excluded.
[§4.3, Proposition 4.6 and §4.4.2, Proposition 4.10] The proof excludes 'any additional case' by the Separatrix Theorem after a 'straightforward combinatorial analysis'. Similar uses appear in Proposition 4.11. This is load-bearing because the hypotheses of Theorem 2.10 (negative definiteness, tree dual graph, reduced singularities) are exactly conditions being verified for the exceptional configurations. Please make these exclusion arguments explicit: in each case, state which separatrix is produced, why it cannot be contained in the exceptional divisor, and why this contradicts minimality or the admissible graph structure.

minor comments (4)

[§5, Lemma 5.2] In the second displayed estimate of the proof, 'K'·C ≥ K_F C = −C^2 ≥ 2' is incorrect for a bad tail. By Proposition 2.8, for a rational invariant curve with Z(F,C)=3 one has K_F·C = 1, not −C^2; and K'·C can be as small as 1 (when C^2=−2). The conclusion W·C>0 still follows from the corrected bound 1 − 1/2 − 1/3 = 1/6 > 0, so this is a local error, but the displayed inequality should be corrected.
[§4.4.3, Lemmas 4.12 and 4.13] The finite integer enumerations are described as 'straightforward inspection'. These are small and checkable, but since they are load-bearing for the classification, please include the actual verification, for example in an appendix or a table, rather than leaving the reader to reproduce them.
[Theorem 4.15 / Proposition 4.14] For the boundary subtypes (III-1-b), (III-1-c), (III-2-b), (III-2-c), the text proves they cannot occur for ε<1/5 but does not explicitly verify that at ε=1/5 the discrepancy vector (a_i) is nonnegative. Example 6.3 realizes only one subtype. Please add a direct check that all listed boundary subtypes satisfy the adjoint lc inequalities at ε=1/5.
[§5.1] The contraction of a 'maximal adjoint collection' of (K_F+εK_X)≥0-graphs is used to produce the MMP morphism. Please state briefly why such a maximal collection exists and why contracting all of them gives a morphism with the stated negativity property; this is likely standard but should be explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: thresholds follow from discrepancy inequalities; the only self-citation is contextual.

full rationale

The derivation is self-contained. In Section 4 the epsilon-adjoint log canonical condition is converted into the linear system (4.1), equivalently requiring E to be a K>=0-graph. The numerical reduction procedure (special K>0 chains, subtraction of M(K,Theta_i) and M(K,Xi_j) to form K_1, and residual classification in Propositions 4.11-4.14) operates only on intersection numbers and discrepancy quantities. Lemmas 4.2 and 4.3 express K.C in terms of Z(F,C), tang(F,C), and d(C), and Proposition 4.6 classifies special chains by determinant inequalities. The walls 1/5 and 1/4 arise as roots of these inequalities, for example in Lemma 4.13 where (n_1,n_2)=(2,3) is allowed only when epsilon>=1/5, and in Lemmas 5.2-5.3 where positivity of W.C excludes types at epsilon<1/4. The external classifications of Theorem 2.24 and Theorem 2.25 are used for comparison and for the final lc/klt conclusions, not as inputs to the threshold computation. The only self-citation, [LWX25], appears in Remarks 1.6 and 5.8 as context or as a supplementary effective-basis statement; it is not load-bearing for Theorem 4.15 or Theorem 5.7. The phrases 'straightforward inspection' and 'ruled out by the Separatrix Theorem' in Lemmas 4.12-4.13 and Proposition 4.14 mark abbreviated case checks and completeness risks, but they do not assume the target epsilon-adjoint classifications, so they are not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: ε is the variable under study and the thresholds 1/5 and 1/4 are derived from inequalities. The paper does not introduce new physical entities; its new words (F-chain, bad tail, boundary configuration) are definitions of subconfigurations. The ledger entries are the external classification theorems and the unproved exhaustiveness of internal finite case checks.

assumptions (5)

standard math Seidenberg's resolution theorem for foliated surfaces (Theorem 2.3)
Guarantees a minimal resolution with reduced singularities, the setting for all exceptional-divisor classifications.
domain assumption Separatrix theorem (Theorem 2.10, Brunella)
Invoked in Prop 4.6, Prop 4.10, Lemmas 4.12 and 4.13 to rule out adjacencies and to derive Z(F,C) ≥ k+1; requires negative definiteness, tree dual graph, and reduced singularities.
domain assumption Chen's classification of log canonical foliated surface singularities (Theorem 2.25)
External benchmark used to identify types (I)/(II) as foliated lc/canonical and type (III) as not foliated lc.
standard math Alexeev/Kollár–Mori classification of log canonical surface singularities (Theorem 2.24)
Used in Corollaries 4.16 and 5.9 to translate foliated-lc classifications into surface lc/klt statements.
ad hoc to paper Completeness of the finite numerical inspections
Prop 4.6 says 'straightforward combinatorial analysis'; Lemmas 4.12/4.13 say 'straightforward inspection' yields only listed solutions. The exhaustive nature of these lists is load-bearing for Theorem 4.15 and both thresholds.

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Pith. "Pith review of Numerical Reduction and Sharp Thresholds for Adjoint Singularities of Foliated Surfaces." pith.science (2026). https://pith.science/paper/H4NYT3M6

@misc{pith2026251220744,
  author       = {Pith},
  title        = {Pith review of: Numerical Reduction and Sharp Thresholds for Adjoint Singularities of Foliated Surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H4NYT3M6}},
  note         = {Machine review of arXiv:2512.20744}
}

abstract

Let \((X,\mathcal F)\) be a foliated surface over the complex numbers. We study the variation of \(\epsilon\)-adjoint singularities associated with the adjoint divisor \[ K_{\mathcal F}+\epsilon K_X,\qquad \epsilon>0. \] Using a numerical reduction procedure for negative definite exceptional configurations, we classify \(\epsilon\)-adjoint log canonical singularities for \(0<\epsilon<1/3\). The reduction detects negative vertices, peels off the special chains generated by them, and reduces the classification to the residual intersections left after peeling. In this form, the first stability threshold is \(\epsilon=1/5\): for \(0<\epsilon<1/5\), every \(\epsilon\)-adjoint log canonical singularity is foliated log canonical, while at \(\epsilon=1/5\) a new boundary configuration appears. Imposing the stronger \(\epsilon\)-adjoint canonical condition gives a second classification below \(\epsilon=1/4\). The same residual mechanism detects the wall \(1/4\), which gives the sharp canonical-to-log-canonical stability interval. Both thresholds are sharp and are realized by explicit examples. As an application, we describe the negative part of the Zariski decomposition of \(K_{\mathcal F}+\epsilon K_X\) below the wall \(1/4\), and obtain the corresponding stability range for the adjoint minimal model program.

Figures

Figures reproduced from arXiv: 2512.20744 by the authors.

**Figure 1.** F-chain with the first curve Γ1 (2) Z(F, C) = 1 (resp. Z(F, C) = 2). Lemma 2.17. If C is a (−1)-F-curve with C 2 = −1, then C is F-exceptional. Moreover, there exists exactly one singularity of F on C, which is non-degenerate and reduced. Lemma 2.18. Suppose C is a (−2)-F-curve with C 2 = −1. Assume that every singularity of F on C is reduced. If C is not F-exceptional, then at the contraction point p of C, the indu… view at source ↗

**Figure 2.** C is a bad tail Definition 2.20. An F-dihedral fork is a divisor F = Γ1 + Γ2 + · · · + Γr, where (1) Γ1 and Γ2 are (−1)-F-curves with self-intersection −2; (2) Γ3 is a bad tail, which is attached to a chain of (−2)-F-curves Γ4 +· · ·+Γr. (See [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** F-dihedral fork 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: An elliptic Gorenstein leaf Definition 2.21 (egl). An elliptic Gorenstein leaf is either a rational F-invariant curve with a single node, or a cycle of (−2)-F-curves. (See [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: F-star graph centered at C Remark 2.23. The configurations introduced above arise naturally in the birational geometry of foliated surfaces. They will serve as the basic configurations throughout this paper. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: F-star chain centered at C, where Θ1 and Θ2 may be empty. 2.8. Classification of lc surfaces singularities and lc foliated surface singularities. In this subsection, we recall the classification of log canonical surface singularities and log canonical foliated surface…

**Figure 7.** Figure 7: 34 [PITH_FULL_IMAGE:figures/full_fig_p034_7.png]

**Figure 8.** Figure 8: Denote by qi (resp. Ei) the successive blow-up points (resp. exceptional curves). In this case, l(q1) = l(q2) = 1, l(q3) = a(q3) = 2. In particular, all Ei are F ′ -invariant. A direct computation gives KF′ = σ ∗ (KF ) − E3, KX′ = σ ∗ (KX) + E¯ 1 + 2E¯ 2 + 4E3. Hence, …

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Works this paper leans on

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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