{"id":"231a02c3-242f-473e-9944-4f5c62ebf579","arxiv_id":"2501.00470","paper_version":7,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For foliated algebraic surfaces, the paper computes the Zariski decomposition of adjoint divisors K_F+D and gives explicit very-ampleness bounds for their canonical models.","lead":"This mathematics paper studies algebraic surfaces carrying a foliation, and asks how the adjoint divisor K_F+D splits and when large multiples of it become very ample. It gives an exact description of the negative part of that splitting and explicit bounds, yielding an effective answer to a boundedness question for foliated surfaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2's Zariski decomposition rests on applying the separatrix theorem to curves in N(D) when F is only log canonical for ε>0; the required reducedness of singularities on those curves is not established, so the assertion h≥1 in the proof may fail.","rationale":"The reader's weakest assumption is exactly the place where Theorem 4.2 is least secure: the separatrix theorem is invoked for a foliation that is only log canonical when ε>0, and its reducedness hypothesis is not checked. This is the single most load-bearing concern because the entire classification of N(D) as a sum over maximal (D,F)-chains, and hence the canonical model conclusions, depends on the inequalities (4.5)–(4.6) and on the disjointness of chains, both of which rely on the separatrix theorem. If the h≥1 bound fails, negative part components outside the (D,F)-chains could exist, and the description of the canonical model singularities in Theorem 5.1 and Corollaries 5.3–5.4 would not follow. The paper contains genuine independent support for other parts of the argument: the use of Tan's effective generation theorem and the explicit volume estimates are concrete and checkable, and the proofs of Proposition 6.4 and Corollary 6.8 are detailed. Those parts are not the Achilles heel; the gap is localized to the separatrix invocation. I do not assert that Theorem 4.2 is false, only that the given proof has a missing verification. The proposed concrete test—either a rederivation of h≥1 using log canonical singularity classification or an explicit check of the two-curve configuration—would settle whether the concern lands. Since the reader already issued a CONDITIONAL verdict, and this concern reinforces that condition rather than overturning the paper, the verdict should remain unchanged.","tokens_in":29879,"tokens_out":10406,"duration_ms":114961,"concrete_test":"Give a proof of the assertion 'h≥1' in Theorem 4.2 that avoids Theorem 2.10 when F is log canonical, or exhibit a counterexample: construct a log canonical foliation F with ε<1/4 and two (-2)-curves Γ and C meeting transversely at one point, where Γ is a maximal (D,F)-chain, C has no singularities other than the node, D·C=0, and K_F·C=-1; compute W·C=K_F·C+M(D,Θ)·C+D·C and check whether W·C<0, which would contradict (4.2). Equivalently, verify from the classification of log canonical surface foliation singularities (e.g., [Chen23, Theorem 1.1]) whether every F-invariant contractible curve in the negative part must have a singularity outside the nodes of the (D,F)-chains; if the classification permits a saddle-node on C, recompute the inequalities in Lemma 5.13 case (F) with that singularity type to see if the contradiction still holds for all ε<1/4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 4.2, after choosing an F-invariant curve C with W·C<0, the authors write 'By the separatrix theorem (cf. Theorem 2.10), we have h≥1', where h counts singularities of F on C outside the intersection points p_i with the maximal (D,F)-chains Θ_i. Theorem 2.10 requires all singularities of F on the contracted tree to be reduced and the dual graph to be a tree. When ε>0, F is only log canonical (Remark 4.3(1)), and log canonical foliation singularities include non-reduced saddle-nodes. The authors do not prove that C, or the tree C∪(∪Θ_i), has only reduced singularities. The subsequent inequality (4.5)–(4.6) uses h(F,C)≥1 to rule out configurations such as a (-2)-curve C meeting a single maximal (D,F)-chain of a (-2)-curve Γ transversely, with no other singularities on C; such a configuration would give W·C<0 and violate the asserted decomposition if it existed. Thus the separatrix theorem is load-bearing for both the disjointness of maximal (D,F)-chains and the h≥1 bound, and its hypotheses are not verified in the log canonical regime. The same gap propagates into Theorem 5.1 and Corollaries 5.3–5.4, since Lemma 5.13 relies on the same h≥1 estimates.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies adjoint foliated structures K_F+D on algebraic surfaces, where D is either εK_X with ε∈[0,1/4] or an effective non-invariant boundary Δ. The central claim is that, under the absence of certain (−1)-curves and pseudo-effectivity of K_F+D, the negative part N(D) of the Zariski decomposition is exactly the sum of M(D,Θ_i) over all maximal (D,F)-chains (Theorem 4.2), with ⌊N(D)⌋=0. From this, the authors derive a classification of the exceptional curves contracted to the canonical model (Theorem 5.1), describe the resulting singularities as cyclic or dihedral quotient singularities (Corollaries 5.3–5.4), and obtain effective very ampleness statements for multiples of K_G+εK_Y (Theorem 6.7, Corollary 1.2). They also state variants for canonical foliations and for a non-invariant curve with zero tangency (Theorems 4.4, 4.9, 5.8) and use the results to give an effective answer to a boundedness problem of Hacon and Langer. The proof of the main theorem relies on Camacho–Sad, the Hodge index theorem, and the separatrix theorem, and the effective bounds are expressed explicitly in terms of i(F), ε, and the volume.","tokens_in":30172,"tokens_out":11355,"duration_ms":119072,"significance":"If the main theorem is correct, the paper is a substantial contribution: it gives a complete description of the negative part of the Zariski decomposition for adjoint divisors K_F+εK_X on foliated surfaces, a classification of canonical-model singularities in the log canonical range, and fully explicit effective constants for very ampleness, improving and complementing work of McQuillan, Chen, Spicer–Svaldi, Tan, and Hacon–Langer. The use of the nonnegative index h_p(F,C) and the reduction of the problem to maximal (D,F)-chains are natural and promising. The effective bound in Corollary 6.8 depending only on ε, i(F), and the volume is a genuine, falsifiable statement. However, the proof of the central Zariski decomposition contains a gap concerning the hypotheses of the separatrix theorem, and this gap propagates into the canonical-model classification and the effective results. The paper is not formalized or machine-checked, and several variant theorems are delegated with 'Proof similar to...'.","major_comments":[{"comment":"The invocation of the separatrix theorem is not justified. The proof asserts 'By the separatrix theorem (cf. Theorem 2.10), we have h ≥ 1' for an F-invariant curve C with W·C<0. Theorem 2.10 requires all singularities of F on the contracted curve to be reduced. In the range ε>0, the foliation is only log canonical (Remark 4.3(1) states this explicitly), and the paper does not prove that the singularities on C∪(∪Θ_i) are reduced; log canonical foliation singularities include non-reduced saddle-nodes. Moreover, even granting reducedness, the conclusion of the separatrix theorem does not by itself produce a singularity on C outside the intersection points p_i: at a transverse intersection p_i, the two invariant branches C and Θ_i already provide separatrices, so the theorem's conclusion can be satisfied with h=0. The estimate h(F,C)≥1 is then used to derive (4.6), to exclude the case k=1 in Case 1, and to rule out the configurations (n1,n2)=(2,2) and (2,3). Without this estimate, the inequality 0>W·C is not contradicted in the stated setting, so the equality N(D)=Σ_iM(D,Θ_i) is not established as written.","section":"§4.1, proof of Theorem 4.2, around equations (4.5)–(4.6)"},{"comment":"The classification of F-invariant curves in Null(P(D)) and the resulting description of canonical-model singularities inherit the same gap. In the proof of Lemma 5.13, the line h(F,C)=KF·C+2−2g(C)−k ≥ 1 is again used without verifying the reducedness hypotheses needed for the separatrix theorem. Cases (E)–(G) and the inequalities leading to them depend on this estimate. Consequently, Theorem 5.1, Corollaries 5.3–5.4, and Theorem 1.1(2) are not supported for ε>0. A necessary repair is either to prove that every F-invariant curve C with W·C<0 and C∪Supp(N(D)) negative definite has only reduced singularities on the relevant tree, or to supply a log canonical analogue of Theorem 2.10; without one of these, the range ε∈(0,1/4) is not justified.","section":"Lemma 5.13 and Theorem 5.1"},{"comment":"The effective boundedness and very ampleness results depend on the same unproved claim. In the proof of Theorem 4.9, the claim that 3i(Δ,F)P(Δ)+K_X is nef uses Lemma 5.13 for curves with P(Δ)·C=0, and Corollary 4.10 and Corollary 6.8 rely on the consequences of that claim. Thus the affirmative answer to the Hacon–Langer boundedness question in the form stated is not fully proven. If the missing reducedness argument is supplied for Theorem 4.2, the authors should also verify that it propagates through Lemma 5.13 and Theorem 4.9 without further hypotheses.","section":"Theorem 4.9 and Corollary 4.10"}],"minor_comments":[{"comment":"There are numerous typos and inconsistencies in notation: 'maxiaml' in Theorem 4.2 and its proof, 'Therem 2.10' in §4.1, 'Esc(σ)' for 'Exc(σ)' in §5, 'F oliations' in §2.1, and an erroneous repeated index in the direct sum in the proof of Proposition 6.4 after equation (6.8). These should be corrected.","section":"Throughout"},{"comment":"Both theorems are stated with 'Proof similar to...' and no indication of the modifications needed for their different coefficient ranges (ε≤1/2 and ε≤1 in Theorem 4.4, and the additional case (5) in Theorem 5.8). Since these are nontrivial generalizations, a proof sketch or a precise reduction to the proved cases should be included.","section":"Theorem 4.4 and Theorem 5.8"},{"comment":"The singularity assumption on F is not stated in Definition 4.1 or Theorem 4.2, but Remark 4.3(1) says the theorem applies to canonical singularities for ε=0 and log canonical singularities for ε∈(0,1/4]. This assumption should be made explicit in the theorem statement, especially because the proof of Theorem 4.2 invokes Theorem 2.10 where the singularity type is essential.","section":"Definition 4.1 and Remark 4.3(1)"},{"comment":"The quantities l(q_i) and a(q_i) used in the computation of K_{X'} are not defined before use, and the verification that the resulting model is not log canonical is sketched rather than shown. The example would be easier to check if the blow-up sequence and the relevant discrepancies were written out.","section":"Example 5.6"},{"comment":"The sentence 'In particular, in this case, if m_p(C)≥2, then p is a dicritical singularity with a local generator v=x∂/∂x+λy∂/∂y for λ∈Q+' seems to conflate saddle-nodes with non-degenerate nodes; a saddle-node with a strong separatrix should not be given by a linear vector field with two nonzero eigenvalues. Please clarify the intended statement.","section":"Lemma 2.4(2)"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the separatrix theorem is real and lands exactly on the load-bearing step of Theorem 4.2. The proof does not verify the reducedness hypotheses, and the inference h≥1 is not a direct consequence of the theorem even in the reduced case. Since the same estimate drives Lemma 5.13 and the effective results, the central claim is currently unproven. I do not see a concrete numerical contradiction in the rest of the paper, and the authors' strategy is plausible; the issue may be fixable by adding a log canonical separatrix-type argument or by restricting the main theorem to cases where reducedness can be checked. If the gap is repaired, the paper would be a strong contribution. The 'Proof similar to' theorems and the typos are secondary and can be handled in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Jun, here's my take on Lu-Wu-Xu's Canonical Models of Adjoint Foliated Structures on Surfaces.\n\nThe genuinely new thing is the explicit Zariski decomposition for K_F+Δ+εK_X with ε<1/4, expressed in terms of maximal (D,F)-chains, and the resulting effective very-ampleness where the multiplier depends only on ε, the index i(F), and the volume. That is a real advance over Spicer-Svaldi's ε<1/5 and the ad hoc cases in [LW24] and [LLTX23]. The paper does a good job of building the chain machinery, and the case analysis in Theorem 4.2 is quite thorough. The classification of quotient singularities in Theorem 5.1 and the connection to Hacon-Langer boundedness are genuinely useful.\n\nThe soft spot is exactly where you'd expect: the proof of Theorem 4.2 uses the separatrix theorem to assert that an F-invariant curve C in the negative part has at least one singularity off the chains (h≥1). The theorem as stated requires reduced singularities on the contracted tree. When ε>0 the foliation is only log canonical, and log canonical singularities need not be reduced in the relevant sense. The authors do not verify that C∪(∪Θ_i) satisfies the hypotheses. Moreover, even if the tree is reduced, the separatrix theorem only gives a separatrix through some point of the tree not contained in it, which could be one of the intersection points with a chain, not an extra singularity on C. So the h≥1 assertion doesn't follow as written. This is load-bearing for the inequality (4.5)-(4.6) and hence for the whole classification of the negative part and the canonical model. It propagates into Theorem 5.1 and the corollaries.\n\nI also note that Theorems 4.4 and 5.8 are proved 'similarly,' which is unsatisfying given that Theorem 4.4 widens the ε range and 5.8 handles the tangency-zero case. And the paper leans on unpublished preprints ([LLTX23], [Tan24], [Liu19]), which makes checking harder. The typos in figures and notation are minor.\n\nIs the main idea right? Probably. The (D,F)-chain formalism is natural, and the gap looks fixable: one might prove a variant of the separatrix theorem for irreducible curves in the negative part under log canonical assumptions, or handle the h=0 case separately with a numerical argument. The paper deserves a serious referee, not a desk rejection. If I were editing, I'd send it to someone who knows foliation singularities and the Camacho-Sad formula, and ask them to focus on Theorem 4.2's use of Theorem 2.10.\n\nWho should read it: anyone working on effective boundedness of foliated surfaces. It's not ready as is, but it's worth engaging with.","headline":"New explicit Zariski decomposition for adjoint foliated divisors with ε<1/4, but the proof of the key step relies on an unverified application of the separatrix theorem.","tokens_in":30738,"tokens_out":6680,"would_cite":true,"duration_ms":61659,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C21","32S65","37F75"],"pacs":[],"model":"deepseek-v4-flash","headline":"For foliated surfaces, the negative part of the Zariski decomposition of $K_{\\mathcal{F}}+D$ is a disjoint union of maximal $(D,\\mathcal{F})$-chains, and this makes the canonical model's adjoint divisor admit explicit very ample multiples.","keywords":["foliation","adjoint divisor","canonical model","Zariski decomposition","F-chain","boundedness","effective very ampleness","foliated surface"],"falsifier":"Look for a log minimal foliated surface satisfying the hypotheses of Theorem 4.2 with $\\epsilon\\in(0,1/4)$ for which the Zariski negative part $N(\\epsilon)$ has an irreducible component not contained in any maximal $(\\epsilon K_X,\\mathcal{F})$-chain, or for which some connected component of $\\mathrm{Exc}(\\sigma)$ has a dual graph not among the listed types. Because the authors prove the bound $\\epsilon<1/4$ is sharp, a second test is to build a global example at $\\epsilon=1/4$ where the canonical model's foliation is not log canonical, mirroring their local Example 5.6.","tokens_in":29668,"feed_emoji":"🌀","tokens_out":12122,"duration_ms":111636,"temperature":0.7,"pith_summary":"This paper establishes a structural description of adjoint foliated divisors $K_{\\mathcal{F}}+D$ on algebraic surfaces when $D$ is either $\\epsilon K_X$ with $0<\\epsilon<1/4$ or a boundary divisor whose components are not $\\mathcal{F}$-invariant. The Zariski negative part of such a divisor is always a disjoint union of maximal $(D,\\mathcal{F})$-chains, and on the canonical model the ambient surface has only cyclic, dihedral, or cusp quotient singularities, with cusps excluded in the $\\epsilon K_X$ case. When $K_{\\mathcal{F}}+D$ is big, $K_{\\mathcal{G}}+D_Y$ on the canonical model is ample, and an explicit multiple of it is very ample, with the multiple depending only on $\\epsilon$, the index of the positive part, and the volume. This gives an effective bound for a boundedness problem for foliated surfaces of general type.","feed_headline":"Chain classification governs canonical models of foliated surfaces","feed_subtitle":"The negative part of K_F + εK_X splits into maximal F-chains, giving explicit very ample multiples.","key_machinery":"The load-bearing object is the $(D,\\mathcal{F})$-chain: a chain of $\\mathcal{F}$-invariant rational curves whose first curve has Gomez-Mont--Seade--Verjovsky index $1$ and whose later curves have index $0$, together with a $\\mathbb{Q}$-divisor $M(D,\\Theta)$ supported on the chain and defined by $M(D,\\Theta)\\cdot\\Gamma_i=-(K_{\\mathcal{F}}+D)\\cdot\\Gamma_i$. The chains are exactly what the negative part of the Zariski decomposition is made of, and $M(D,\\Theta)$ is the contribution of each chain. The proof also relies on the separatrix theorem, which guarantees an extra separatrix through every negative-definite tree of reduced invariant curves with normal crossings; this is the mechanism that forces the negative part to split into disjoint maximal chains and rules out curves outside them. Finally, a surface positivity estimate (nefness of $3A+K_X$ with $A=i(D,\\mathcal{F})P(D)$) converts the chain classification into explicit very ampleness through a criterion for multiple linear systems.","core_discovery":"The central claim is Theorem 4.2: for $D=\\Delta+\\epsilon K_X$ with $\\epsilon\\in[0,1/4]$ and no exceptional $(-1)$-curves of the listed types, if $K_{\\mathcal{F}}+D$ is pseudo-effective then its Zariski decomposition has negative part $N(D)=\\sum_i M(D,\\Theta_i)$, summed over all maximal $(D,\\mathcal{F})$-chains, and $\\lfloor N(D)\\rfloor=0$. The same chain language controls the canonical model: Theorem 5.1 lists the possible dual graphs for a connected component of the contracted locus, and Corollary 5.4 gives that $\\mathcal{G}$ has at most log canonical singularities and $Y$ only rational quotient singularities when $\\epsilon>0$. Consequently, when $K_{\\mathcal{F}}+\\epsilon K_X$ is big, $K_{\\mathcal{G}}+\\epsilon K_Y$ is ample and $i(\\epsilon K_X,\\mathcal{F})\\cdot(\\alpha(\\epsilon K_X,\\mathcal{F})+3)\\cdot(K_{\\mathcal{G}}+\\epsilon K_Y)$ is very ample; under the additional hypothesis that $K_{\\mathcal{F}}$ is pseudo-effective, a multiple depending only on $\\epsilon$, $i(\\mathcal{F})$, and $\\operatorname{Vol}(K_{\\mathcal{F}}+\\epsilon K_X)$ is very ample.","pith_inferences":["Editorial extension: the same chain argument should constrain higher-dimensional adjoint foliated pairs only after replacing the string combinatorics by a more general skeleton of invariant subvarieties; the sharp threshold $\\epsilon<1/4$ suggests the coefficient $1/4$ will reappear as a log canonical threshold in that setting.","Editorial extension: the dual graphs in Theorem 5.1 determine the local analytic types of the quotient singularities, so one could extract explicit local equations and monodromy data from the graphs even though the paper does not write them down.","Editorial extension: a testable computational check is whether, for small values of the volume, the multiplier $n$ in Corollary 1.2 is actually optimal; this would require computing the invariant $\\alpha$ for explicit foliated surfaces and comparing the vanishing threshold with the base-point-free threshold."],"forward_implications":["For $D=\\epsilon K_X$ with $0<\\epsilon<1/4$, the negative part of $K_{\\mathcal{F}}+\\epsilon K_X$ is a disjoint union of maximal $(\\epsilon K_X,\\mathcal{F})$-chains with $\\lfloor N(\\epsilon)\\rfloor=0$, so the Zariski decomposition is explicitly computable from the foliation's invariant curves.","On the canonical model, $K_{\\mathcal{G}}+\\epsilon K_Y$ is ample whenever $K_{\\mathcal{F}}+\\epsilon K_X$ is big, and the divisor $i(\\epsilon K_X,\\mathcal{F})(\\alpha(\\epsilon K_X,\\mathcal{F})+3)(K_{\\mathcal{G}}+\\epsilon K_Y)$ is very ample.","If $K_{\\mathcal{F}}$ is pseudo-effective, there is a uniform integer $n$ depending only on $\\epsilon$, $i(\\mathcal{F})$, and $\\operatorname{Vol}(K_{\\mathcal{F}}+\\epsilon K_X)$ such that $n(K_{\\mathcal{G}}+\\epsilon K_Y)$ is very ample.","For foliated triples with $D=\\Delta$ of coefficients in $[0,1)$, the birational map given by $|m(K_{\\mathcal{G}}+\\Delta_Y)|$ is an isomorphism away from cusp singularities once $m\\ge i(\\Delta,\\mathcal{F})(\\alpha(\\Delta,\\mathcal{F})+3)$; when $\\Delta$ meets every elliptic Gorenstein leaf, $K_{\\mathcal{G}}+\\Delta_Y$ is ample.","For $m\\ge i(\\Delta,\\mathcal{F})(\\alpha(\\Delta,\\mathcal{F})+1)$ divisible by $i(\\Delta,\\mathcal{F})$, both $H^1$ and $H^2$ of $mP(\\Delta)$ vanish and $\\dim H^0(m(K_{\\mathcal{F}}+\\Delta))$ equals the explicit volume formula in Proposition 1.6."],"supporting_citations":[{"why":"Constructs the canonical model of a foliated surface, identifies cusp singularities with elliptic Gorenstein leaves, and supplies the results that Theorem 5.1 extends.","marker":"[McQ08]"},{"why":"Provides the index theorems, the separatrix theorem, and the F-chain formalism used throughout the proof of Theorem 4.2.","marker":"[Bru15]"},{"why":"Supplies Zariski decomposition and Weil-divisor index facts on normal surfaces, including the equality relating i(F) to the Q-Gorenstein index on the canonical model.","marker":"[Sak84]"},{"why":"Gives the effective criterion for multiple linear systems on surfaces that yields the explicit very ample multiples in Corollary 1.2.","marker":"[Tan04]"},{"why":"States the boundedness problem for foliated surfaces and shows the relevant multiplier depends on the Hilbert function, the target of Corollaries 1.2 and 1.7.","marker":"[HA21]"},{"why":"Establishes the D=0 version of the chain decomposition and plurigenera invariance, which the present theorems generalize.","marker":"[CF18]"},{"why":"Introduces the minimal partial du Val resolution and gives a prior boundedness result that Proposition 1.9 builds on.","marker":"[Chen21]"},{"why":"Provides the positivity lemma (3A+K_X nef) used to control P(Delta)+epsilon K_X in Theorem 4.9.","marker":"[PS19]"}],"fun_headline_variants":["Chain classification yields canonical models for foliated surfaces","Maximal F-chains determine canonical models of foliated surfaces","Effective boundedness via chain decomposition of adjoint foliations","Canonical models from Zariski decomposition of adjoint foliations","Boundedness for foliated surfaces solved via chain language"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification hinges on the separatrix theorem's assumptions—reduced singularities, normal crossings, and a negative-definite tree of invariant curves—which are not automatic for log canonical foliations when epsilon>0; without them the claimed description of the negative part and of the quotient singularities does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Chain classification yields canonical models for foliated surfaces","Maximal F-chains determine canonical models of foliated surfaces","Effective boundedness via chain decomposition of adjoint foliations","Canonical models from Zariski decomposition of adjoint foliations","Boundedness for foliated surfaces solved via chain language"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000153,"raw_usage":{"total_tokens":1174,"prompt_tokens":879,"completion_tokens":295,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":226}},"tokens_in":495,"tokens_out":295,"duration_ms":3251,"temperature":1.0,"reasoning_tokens":226,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:49:56.438144+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a log minimal foliated surface satisfying the hypotheses of Theorem 4.2 with $\\epsilon\\in(0,1/4)$ for which the Zariski negative part $N(\\epsilon)$ has an irreducible component not contained in any maximal $(\\epsilon K_X,\\mathcal{F})$-chain, or for which some connected component of $\\mathrm{Exc}(\\sigma)$ has a dual graph not among the listed types. Because the authors prove the bound $\\epsilon<1/4$ is sharp, a second test is to build a global example at $\\epsilon=1/4$ where the canonical model's foliation is not log canonical, mirroring their local Example 5.6.","supporting_citations":[],"review_version":1}