{"id":"5e8908d0-a653-4cb4-8656-ed1cdfbe68b4","arxiv_id":"2608.09663","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Rank one foliations with canonical singularities have unique minimal models; co-rank one threefolds admit D-flops in klt and F-dlt settings, while new examples show flops and canonical models can fail.","lead":"This paper proves that minimal models of rank one foliations are unique, and shows when flops exist for co-rank one foliations on threefolds. It also builds examples where flops and canonical models fail, revealing new pathologies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 relies on an unrestricted cone theorem for rank one foliations, [CS25a, Thm 4.8], whose hypotheses and dimension range are not stated; if it is not available in full generality, the uniqueness proof is unsupported.","rationale":"The reader identified the unstated hypotheses of the cone theorem as the weakest point, and I agree. Theorem 1.1 is the central claim, and its proof hinges on the existence of a (K_F1 + D1)-negative curve C that is either F1-invariant or a non-lc locus of the pair; without the cone theorem in the required generality, the final contradiction collapses. I examined other parts of the proof: the jet-Bertini theorem (Theorem 3.3) appears sound, the reduction to small map follows standard negativity-lemma arguments, and the final negativity-lemma computation around H2·C < 0 is clear. The only extraneous step that worries me is the use of the cone theorem without quoting its hypotheses. This is not a demonstrated error, but it is exactly the kind of omitted hypothesis that can invalidate a theorem when the citation only covers a special case. Therefore the verdict should remain CONDITIONAL: the paper is likely correct, but the authors should state the cited theorem and either verify its hypotheses in this setting or restrict the statement. No change to the reader's verdict is needed.","tokens_in":114,"tokens_out":8898,"duration_ms":139862,"concrete_test":"Locate [CS25a, Theorem 4.8] (Cascini–Spicer, 'Foliation adjunction', Math. Ann. 2025) and check its precise statement. Determine whether it applies to a rank one foliation F1 with canonical singularities on an arbitrary-dimensional normal projective Q-factorial variety Y1, with D1 a Q-divisor such that (F1, D1) is only log canonical on Exc φ, not necessarily globally. If the theorem has a dimension bound or extra hypotheses, test the final paragraph of the proof of Theorem 1.1 under those conditions: does the produced curve C still exist in the setting of Theorem 1.1? If not, the proof of uniqueness in arbitrary dimension fails; the paper would need to either prove the needed cone theorem or restrict Theorem 1.1 accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 (Section 3, final paragraph) invokes the foliated cone theorem, cited as [CS25a, Theorem 4.8], to assert that because K_F1 + D1 is not nef, there exists a curve C with (K_F1 + D1)·C < 0 and either (F1, D1) is not log canonical at the general point of C or C is F1-invariant. This is the pivotal step: it produces the invariant curve C contained in Exc φ that contradicts the earlier claim that Exc φ contains no log canonical centre. However, the paper never states the hypotheses or dimension range of [CS25a, Theorem 4.8]. If that theorem is only proved for threefolds, or requires (F1, D1) to be log canonical, or imposes F-dlt/terminal conditions, or requires an algebraic integrability assumption, then the argument does not prove the claimed uniqueness in arbitrary dimension. In particular, the pair (F1, D1) is only shown to be log canonical at points of Exc φ; it may not satisfy a global log-canonicity hypothesis, and the curve C is only known a posteriori to lie in Exc φ. The gap is a verification gap rather than an internal contradiction, but it is load-bearing and must be closed by stating the cited theorem fully.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies birational rigidity and flops for foliations. The main result (Theorem 1.1) asserts that if F is a rank one foliation with canonical singularities on a normal projective Q-factorial variety and K_F is pseudo-effective, then any two outputs of a K_F-MMP are isomorphic, not merely flop-connected. For co-rank one foliations on threefolds, the paper proves existence of D-flops both in the F-dlt setting under hypotheses involving separatrices (Theorem 1.2) and in the klt setting (Theorem 1.3). The final section constructs two pathologies: a rank one canonical foliation with a flopping contraction that admits no D-flop (Theorem 5.2), and a rank one canonical foliation on a smooth projective threefold with K_H nef and big but no canonical model in the category of algebraic spaces (Theorem 5.3). The proofs use foliated jets, a Bertini-type avoidance statement, the foliated cone theorem, Camacho-Sad indices, formal separatrix geometry, and Hilbert modular surface examples.","tokens_in":18278,"tokens_out":10219,"duration_ms":96208,"significance":"If correct, the paper makes a substantial contribution: Theorem 1.1 is a strong uniqueness statement that goes beyond the classical Kawamata flop-connectedness result, and Theorems 5.2 and 5.3 identify genuinely new failures of the MMP for foliations. The arguments are mostly detailed and self-contained, and the main results are not simply repackaged earlier theorems; the examples are explicit and falsifiable. However, the two points flagged below involve load-bearing citations whose hypotheses are not verified, so the significance can only be assessed once those gaps are closed.","major_comments":[{"comment":"The pivotal step in the proof of Theorem 1.1 uses [CS25a, Theorem 4.8] to produce a curve C with (K_F1+D1).C<0 that is either F1-invariant or a non-lc centre, but the paper never states the hypotheses or dimension range of this cone theorem. The pair (F1,D1) is only shown to be log canonical at points of Excφ; no global F-dlt or log-canonicity hypothesis is stated, and the same theorem is invoked in Section 5.3 to infer that K_H is nef from the absence of invariant rational curves. If the cited theorem requires e.g. threefolds, a global F-dlt condition, or algebraic integrability, then Theorem 1.1 in arbitrary dimension and Theorem 5.3 are unsupported. Please state the theorem in full and verify that (F1,D1) and H satisfy its hypotheses, or prove the needed cone statement here.","section":"Section 3, proof of Theorem 1.1; Section 5.3"},{"comment":"The Camacho-Sad formula is applied on a possibly singular Q-factorial klt threefold in the proof of Theorem 1.3, after reducing to a small Q-factorialisation, but [Per22, Theorem 3.9] is stated for smooth complex varieties. Remark 4.1 says the proof 'easily adapts' to quotient singularities in codimension two; since this formula is used to conclude S_i|S_i ≡f 0, which is the step that makes S f-nef and hence controls the perturbation of Γ, the adaptation is load-bearing. Please provide a proof or a precise statement with hypotheses.","section":"Remark 4.1 and proof of Theorem 1.3"},{"comment":"Theorem 3.3 is stated only for log canonical centres, but the proof of Theorem 1.1 uses it to assert that a general ample divisor contains no F1-invariant subvariety outside Sing+F1 and hence that H1 is not F2-invariant. Either every invariant subvariety outside Sing+F is an lc centre, in which case this should be stated and proved, or Theorem 3.3 should be strengthened to the class of invariant subvarieties; the local argument in the proof appears to establish the stronger statement, so this is likely easy to fix.","section":"Theorem 3.3 and proof of Theorem 1.1"}],"minor_comments":[{"comment":"The arXiv identifiers in [CM24] and [JV23] are formatted inconsistently ('arXiv.2410.05178', 'arXiv.2305.19728'); they should be 'arXiv:2410.05178' and 'arXiv:2305.19728'.","section":"References"},{"comment":"The abbreviation 'e.g.l.' is used without definition; please spell out 'elliptic Gorenstein leaf' at first use.","section":"Section 5.1"},{"comment":"In the definition of a flopping contraction, condition (2) says 'Exc f is one-dimensional', and the later argument works with an irreducible curve C, but the notation C:=Exc f in Theorem 1.2 would be cleaner if it explicitly said C is an irreducible component and that the contraction is assumed to have irreducible exceptional locus, as condition (1) does.","section":"Section 2.1"},{"comment":"The final sentence of the construction says that no morphism f:X0→Σ with dim Σ≤1 makes N^*_{X0/P} f-ample, and uses this to conclude that no birational morphism c:P→Z can contract the null locus. The implication would be clearer if the role of the normal bundle in the contraction criterion were stated explicitly.","section":"Section 5.3"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the unverified use of the authors' own cone theorem [CS25a, Theorem 4.8] in the proof of the flagship Theorem 1.1 and again in Theorem 5.3. This is likely fixable by stating the theorem and checking hypotheses, but it is load-bearing and must be settled before publication. The Camacho-Sad adaptation in Remark 4.1 also needs concrete detail. I see no indication of circularity or of overlap with prior work that would undermine novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper before you referee anything else in foliated MMP. It proves that rank one foliations with canonical singularities have unique minimal models, not merely flop-connected, and it explains why flops are pathological for rank one but exist for co-rank one on threefolds.\n\nThe uniqueness theorem (Thm 1.1) is genuinely new and sharply contrasts with Kawamata. The technical engine is a Bertini-type theorem via foliated jets (Thm 3.3) that keeps general ample divisors clear of non-singular log canonical centres; the jet construction is worked out cleanly. The D-flop existence results for co-rank one (Thms 1.2, 1.3) are substantial, and the counterexamples (Thms 5.2, 5.3) are detailed and credible, showing a nef-big canonical class without a canonical model even as algebraic space.\n\nThe main soft spot is not the results but a citation gap. In the proof of Thm 1.1, the existence of an F1-invariant curve C in Exc phi comes from the cone theorem [CS25a, Thm 4.8]. The hypotheses and dimension range of that theorem are never stated. If it is only proved in dimension three or under extra conditions, the arbitrary-dimension uniqueness claim is unsupported. This is a verification gap, not an internal contradiction, but it is load-bearing. Similarly, Remark 4.1 extends the Camacho-Sad formula to quotient singularities in codim two by saying the proof 'easily adapts,' citing [DO22]—that adaptation matters for Thm 1.3 and should be written out or precisely cited. The paper also relies heavily on the authors' own prior work; that is fine if the cited results are published and independent, but a referee needs those hypotheses explicitly.\n\nNone of this makes me doubt the main conclusions; the arguments are plausible and detailed, and the examples are self-checking. But the gap is real.\n\nThis paper is for anyone working in foliated birational geometry or MMP. It deserves a serious referee, and with the cone theorem hypothesis stated, I expect the theorem to stand. I would cite the uniqueness theorem.\n\nMy recommendation: send it to a good referee, and make the authors clarify the hypotheses of the cone theorem and the Camacho-Sad adaptation. The paper is a solid contribution and, conditional on those clarifications, publishable.","headline":"A strong, important paper: uniqueness of rank-one foliated minimal models is new and convincing, but the proof leans on an unstated cone-theorem hypothesis that needs to be pinned down.","tokens_in":18892,"tokens_out":2976,"would_cite":true,"duration_ms":26202,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30","14E05","37F75"],"pacs":[],"model":"deepseek-v4-flash","headline":"Minimal models of rank-one foliations are unique, not just flop-connected.","keywords":["rank one foliations","foliated minimal models","flops","co-rank one foliations","canonical singularities","foliated jets","basepoint-free theorem","MMP"],"falsifier":"Verify the dimension and hypothesis range of the cone theorem for rank one foliations invoked in the proof of Theorem 1.1: if it fails or has unstated restrictions in dimension at least 4, the uniqueness statement is unsupported. Concretely, try to construct a rank one foliation with canonical singularities on a smooth projective fourfold, with pseudo-effective K_F, that has two distinct K_F-MMP outputs; any such example would disprove Theorem 1.1.","tokens_in":17815,"feed_emoji":"🍃","tokens_out":6332,"duration_ms":51471,"temperature":0.7,"pith_summary":"This paper shows that, unlike the classical minimal model program, rank one foliations have unique minimal models: for a rank one foliation with canonical singularities on a normal projective Q-factorial variety whose canonical class is pseudo-effective, any two outputs of the K_F-MMP are isomorphic. The argument introduces a Bertini-type statement for foliated jets, ensuring a general ample divisor avoids invariant subvarieties. In the co-rank one case on threefolds, the paper proves existence of D-flops under F-dlt and klt hypotheses, using separatrices and the Camacho-Sad formula. It also constructs rank one examples where a flopping contraction admits no D-flop, and where a nef and big canonical divisor admits no canonical model even in the category of algebraic spaces.","feed_headline":"Minimal models of rank-one foliations are unique","feed_subtitle":"For canonical rank-one foliations with pseudo-effective canonical class, any two MMP outputs are isomorphic.","key_machinery":"The key object is the vector bundle of foliated m-jets J_F^m L attached to a line bundle L along a rank one foliation F, with the natural jet map j_F^m. Theorem 3.3, a Bertini-type statement, shows that after replacing an ample line bundle L by a multiple, a general section avoids every log canonical centre of F not contained in Sing^+F; the proof uses the fact that the foliated n-jet of a general section is nowhere vanishing on the regular loci. This avoidance statement is what forces the exceptional loci of the two MMP outputs to be empty. For the flop existence results, the machinery is the classification of transverse types of co-rank one singularities (logarithmic or saddle-node), the Camacho-Sad formula for self-intersection of separatrices, and a reduction to the construction of a klt ample model via [BCHM10].","core_discovery":"The central discovery is that the birational rigidity of rank one foliations forces uniqueness of minimal models as outputs of the MMP: Theorem 1.1 states that if F is a rank one foliation with canonical singularities on a normal projective Q-factorial variety X with K_F pseudo-effective, then any two outputs f_i: X --> Y_i of a K_F-MMP are isomorphic over X. This is stronger than the classical Kawamata result, where minimal models are only flop-connected. The proof shows the birational map between outputs is an isomorphism in codimension one and then uses a cone theorem (cited from [CS25a]) to rule out any remaining exceptional locus. For co-rank one foliations on threefolds, Theorems 1.2 and 1.3 establish existence of D-flops in the F-dlt and klt settings, respectively. The examples of Section 5 exhibit pathologies: a flopping contraction for a rank one foliation with canonical singularities that has no D-flop (Theorem 5.2), and a smooth projective threefold with a rank one foliation of canonical type whose canonical divisor is nef and big yet the null locus cannot be contracted in algebraic spaces (Theorem 5.3).","pith_inferences":["If the cone theorem [CS25a, Theorem 4.8] holds in all dimensions, Theorem 1.1 settles the uniqueness question completely for canonical rank one foliations; a natural next step is to test whether the same rigidity holds for log canonical singularities or for pseudo-effective twisted canonical classes.","The non-existence of D-flops in Theorem 5.2 suggests that the universe of birational transformations allowed for rank one foliations may need to be enlarged, perhaps to include formal or derived contractions, to restore a flop operation.","The construction of the null-locus obstruction in Theorem 5.3 relies on non-abundant tautological foliations; one could test whether the same obstruction appears for other non-abundant foliations or in higher dimension.","The foliated-jet technique may yield a proof of abundance for rank one foliations in some cases, since the vanishing of foliated jets along invariant subvarieties is closely tied to the numeric properties of K_F."],"forward_implications":["For rank one foliations with canonical singularities and pseudo-effective canonical class, the K_F-MMP has a uniquely determined end product; no flop ambiguity remains.","Birational invariants of rank one foliated minimal models are well-defined, so, for instance, the Kodaira dimension or the canonical ring of any output is a birational invariant.","The foliated-jet Bertini theorem (Theorem 3.3) is a new tool that can be applied to other situations where invariant subvarieties must be avoided, such as the construction of sufficiently general boundaries in the foliated MMP.","For co-rank one threefolds, D-flops exist in the klt setting and in the F-dlt setting under the stated hypotheses, giving a working flop operation that parallels the classical case; in particular flopping curves in the F-dlt setting are smooth rational curves.","The counterexamples show that the classical basepoint-free theorem fails for rank one foliations even in dimension three, and that the failure is not an artifact of working with varieties rather than algebraic spaces."],"supporting_citations":[{"why":"Supplies the cone theorem for rank one foliations (Theorem 4.8) that produces the curve C with (K_{F_1}+D_1).C < 0 in the proof of Theorem 1.1.","marker":"[CS25a]"},{"why":"Provides the foundational definitions of foliated singularities, F-dlt, non-dicriticality, and the local lemmas (e.g., Lemma 3.5, Lemma 3.8, Corollary 3.20) used in the flop existence proofs.","marker":"[CS21]"},{"why":"Brings in the Camacho-Sad formula (Theorem 3.9) which is used to compute self-intersections of separatrices and to prove the existence of a proper separatrix in Theorem 1.2.","marker":"[Per22]"},{"why":"Gives the existence of klt ample models (Corollary 1.3.1) that is used to produce the D-ample model, i.e., the D-flop.","marker":"[BCHM10]"},{"why":"Provides Lemma 8.14, which yields that the pair (X, sum of separatrices) is log canonical and gives the nefness properties of K_X + S used in the flop arguments.","marker":"[Spi20]"},{"why":"Supplies the Hilbert modular surface examples and the notion of non-abundant tautological foliations, which are the basis for the pathology constructions in Section 5.","marker":"[McQ08]"},{"why":"Gives the contraction of a curve with anti-ample normal bundle to an algebraic space, used to build the flopping contraction in Theorem 5.2.","marker":"[Art70]"}],"fun_headline_variants":["Rank-one foliation MMP outputs are always isomorphic","Minimal models unique for canonical rank-one foliations","Foliation rigidity: no flops between minimal models","Rank-one foliations: MMP outputs cannot differ","No D-flop for some rank-one foliation contractions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Theorem 1.1 depends on a cone theorem for rank one foliations cited from prior work, whose hypotheses and dimension range are not stated in this paper; if that theorem has hidden assumptions (for instance holding only in dimension three), then the arbitrary-dimensional uniqueness claim is not established.","fun_headline_variants_meta":{"raw":{"variants":["Rank-one foliation MMP outputs are always isomorphic","Minimal models unique for canonical rank-one foliations","Foliation rigidity: no flops between minimal models","Rank-one foliations: MMP outputs cannot differ","No D-flop for some rank-one foliation contractions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00058,"raw_usage":{"total_tokens":2719,"prompt_tokens":919,"completion_tokens":1800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":1725}},"tokens_in":535,"tokens_out":1800,"duration_ms":15893,"temperature":1.0,"reasoning_tokens":1725,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:08:29.705606+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify the dimension and hypothesis range of the cone theorem for rank one foliations invoked in the proof of Theorem 1.1: if it fails or has unstated restrictions in dimension at least 4, the uniqueness statement is unsupported. Concretely, try to construct a rank one foliation with canonical singularities on a smooth projective fourfold, with pseudo-effective K_F, that has two distinct K_F-MMP outputs; any such example would disprove Theorem 1.1.","supporting_citations":[],"review_version":1}