{"id":"82192975-3448-4e08-849f-04f09340f232","arxiv_id":"2512.20744","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every ε-adjoint log canonical singularity of a foliated surface is foliated log canonical for 0<ε<1/5, and every ε-adjoint canonical singularity is foliated lc and surface klt for 0<ε<1/4; explicit examples show 1/5 and 1/4 are sharp.","lead":"This paper classifies all ε-adjoint singularity types for foliations on complex surfaces and fixes the first sharp stability thresholds at ε = 1/5 and ε = 1/4. It tells birational geometers how far the adjoint perturbation K_F + ε K_X can be pushed before log canonical or canonical behavior changes, with explicit examples showing both walls are optimal.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness of Prop 4.14's residual K1=0 classification is asserted rather than shown; a missed configuration would move the 1/5 and 1/4 walls.","rationale":"The reader's CONDITIONAL verdict is justified: the classification is coherent and the numerical thresholds follow from the listed configurations, but the proof's completeness rests on finite checks that are asserted rather than exhibited. I focused on Prop 4.14's residual K1=0 step because it is more global than the integer inspections in Lemmas 4.12–4.13: it claims that every connected graph assembled from zero-contribution building blocks is ruled out by the Separatrix Theorem, without presenting the case analysis. This is the exact point where an omitted admissible configuration would undermine the sharpness of 1/5 and 1/4. I do not see an actual counterexample, and the sign slips noted by the reader do not affect the main thresholds, so the appropriate verdict remains CONDITIONAL rather than ACCEPT or REJECT.","tokens_in":28433,"tokens_out":16235,"duration_ms":161352,"concrete_test":"Re-derive the final paragraph of Proposition 4.14: enumerate all connected negative definite configurations made from special K>0-chains of types (A)–(D) plus the components in Lemma 4.12(2)–(3) and Lemma 4.13(3)–(5), with K1·C=0 on every component, and prove by an explicit induction on the dual graph that each violates assumption (∗) or the Separatrix Theorem. If any configuration survives, test it numerically via the discrepancy system (4.1); this would either exhibit a counterexample to Theorem 4.15 or close the gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the exhaustiveness of Theorem 4.15/5.7. The most load-bearing step is not the sign slips in §5 but the unshown finite residual analysis in Proposition 4.14. After subtracting all maximal special K>0-chains, the proof asserts: if no component has K1·C<0, then E is a (K1)=0-graph combining only the listed building blocks, and 'any such E is ruled out by the Separatrix Theorem and assumption (∗)' — with no case analysis. Completeness therefore rests on an unstated graph-theoretic argument. The same pattern appears in Proposition 4.6 ('straightforward combinatorial analysis') and in Lemmas 4.12/4.13 ('straightforward inspection' of (4.6)–(4.8), (4.10)–(4.13)). Those integer enumerations are small and checkable, but the global K1=0 assembly is less transparent. A missing residual configuration would be ε-adjoint lc for ε<1/3 and could enter before ε=1/5, breaking both stability propositions and the claimed sharpness.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28670,"tokens_out":10568,"duration_ms":101907,"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":[{"comment":"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.","section":"§4.4.3, Proposition 4.14"},{"comment":"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.","section":"§4.3, Proposition 4.6 and §4.4.2, Proposition 4.10"}],"minor_comments":[{"comment":"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.","section":"§5, Lemma 5.2"},{"comment":"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.","section":"§4.4.3, Lemmas 4.12 and 4.13"},{"comment":"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.","section":"Theorem 4.15 / Proposition 4.14"},{"comment":"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.","section":"§5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper has a clear and plausible central argument, and the two threshold values 1/5 and 1/4 appear well motivated by the determinant computations. My main concern is purely about proof completeness: the classification's final step—the exclusion of all residual (K_1)=0 graphs—is asserted rather than proved, and the same pattern recurs in several 'straightforward' combinatorial steps. These are fixable in revision, so I do not recommend rejection, but the manuscript is not ready until the missing case analyses are supplied or replaced by a machine-checked enumeration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSoft 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.\n\nBottom 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.","headline":"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.","tokens_in":29217,"tokens_out":2194,"would_cite":true,"duration_ms":24766,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J29","14B05","32S65","32M25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Foliations","adjoint divisors","log canonical singularities","foliated surfaces","minimal model program","negative definite configurations","singularity thresholds"],"falsifier":"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.","tokens_in":28267,"feed_emoji":"🌀","tokens_out":3727,"duration_ms":39708,"temperature":0.7,"pith_summary":"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.","feed_headline":"1/5 and 1/4: the sharp walls of adjoint lc and canonical stability","feed_subtitle":"A numerical reduction classifies all ε-adjoint foliated singularities and shows the adjoint MMP holds on (0,1/4).","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Sharp adjoint thresholds at 1/5 and 1/4 for foliated surfaces","Numerical reduction classifies adjoint lc singularities: walls at 1/5, 1/4","epsilon-adjoint walls: sharp at 1/5 and 1/4","Peeling method yields sharp 1/5, 1/4 thresholds for adjoint lc","Exact adjoint lc and canonical boundaries: 1/5 and 1/4"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sharp adjoint thresholds at 1/5 and 1/4 for foliated surfaces","Numerical reduction classifies adjoint lc singularities: walls at 1/5, 1/4","epsilon-adjoint walls: sharp at 1/5 and 1/4","Peeling method yields sharp 1/5, 1/4 thresholds for adjoint lc","Exact adjoint lc and canonical boundaries: 1/5 and 1/4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000929,"raw_usage":{"total_tokens":3851,"prompt_tokens":812,"completion_tokens":3039,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":2916}},"tokens_in":556,"tokens_out":3039,"duration_ms":19168,"temperature":1.0,"reasoning_tokens":2916,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T14:20:01.407862+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}