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Foliated Minimal Models and Flops

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Pith's one-line read Minimal models of rank-one foliations are unique, not just flop-connected.

desk verdict 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. read the letter →

arxiv 2608.09663 v1 pith:IUO32B7B submitted 2026-08-10 math.AG math.DS

Paolo Cascini , Roktim Mascharak , Calum Spicer This is my paper · ORCID
classification math.AGmath.DS MSC 14E3014E0537F75
keywords rankonefoliationsfoliatedminimalmodelsflopsco-rankcanonicalsingularitiesjetsbasepoint-freetheoremMMP
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

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.

What carries the argument

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].

What would settle it

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.

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Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 4 minor

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.

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 (3)
  1. [Section 3, proof of Theorem 1.1; Section 5.3] 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.
  2. [Remark 4.1 and proof of Theorem 1.3] 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.
  3. [Theorem 3.3 and proof of Theorem 1.1] 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.
minor comments (4)
  1. [References] 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'.
  2. [Section 5.1] The abbreviation 'e.g.l.' is used without definition; please spell out 'elliptic Gorenstein leaf' at first use.
  3. [Section 2.1] 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.
  4. [Section 5.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are proved using cited published tools, and the self-citations are independent, not reductions to the paper's own conclusions.

full rationale

Walking the derivation chain: Theorem 1.1 is not obtained by restating an earlier result. The proof begins with a Bertini-type statement (Theorem 3.3) proved self-contained via foliated jets, then uses the foliated cone theorem [CS25a, Theorem 4.8] only at the final step to extract an invariant curve in Exc phi. That cone theorem is published prior work by the same authors, but it is used as a tool with independent content; the paper's contribution is the reduction of uniqueness to the existence of such a curve, not the cone theorem itself. Theorem 3.3 does not depend on Theorem 1.1. The flop existence results (Theorems 1.2 and 1.3) are constructive: they use separatrices, Camacho-Sad indices, and reduction to klt ample models, with no fitted parameters and no prediction that is forced by an earlier fit. The examples in Section 5 are explicit constructions. The main caveats are verification gaps, not circularity: (i) the hypotheses and dimension range of [CS25a, Theorem 4.8] are not stated, so Theorem 1.1 may be unsupported in full generality if that theorem is restricted; and (ii) Remark 4.1 extends the Camacho-Sad formula to quotient singularities by saying the proof 'easily adapts', an unproved extension that is load-bearing for Theorem 1.3. These are concerns about missing support, not self-referential reductions, and they do not make the derivation equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claims rest on several published theorems from the foliated MMP literature, many authored by the present authors. None of these background results include the target theorems as hypotheses; they are used as tools. The main burden is that some tools are invoked with unstated hypotheses (the cone theorem) or adapted to wider settings (Camacho-Sad formula on klt threefolds). No ad hoc entities or fitted parameters are introduced.

assumptions (8)
  • domain assumption Cone theorem for rank one foliations: if K_F is not nef, there exists a K_F-negative curve which is F-invariant or a log canonical centre (cf. [CS25a, Theorem 4.8]).
    Invoked in the proof of Theorem 1.1 to produce the curve C with (K_{F1}+D1).C < 0, and in Section 5.3 to conclude K_H is nef. The exact hypotheses of this theorem are not stated in the paper.
  • domain assumption MMP for co-rank one foliations on threefolds, including existence and properties of flips, F-dlt adjunction, and extension of formal separatrices (cf. [CS21], [Spi20]).
    Used throughout Sections 2 and 4 for F-dlt foliations, simple singularities, and separatrices along flopping curves. This is prior published work by the authors, used as a black box.
  • domain assumption Camacho-Sad formula relating the self-intersection of a separatrix to the Camacho-Sad index (cf. [Per22, Theorem 3.9]).
    Used in the proof of Theorem 1.2 to derive a contradiction in both logarithmic and saddle-node cases. In Remark 4.1 the authors state the formula adapts to quotient singularities in codimension two, citing [DO22].
  • standard math MMP for klt pairs and existence of ample models (BCHM) (cf. [BCHM10, Corollary 1.3.1]).
    Used at the end of Theorems 1.2 and 1.3 to produce the D-ample model after reducing to a klt pair on a formal neighbourhood. This is a standard heavy theorem in birational geometry.
  • standard math Negativity lemma (cf. [KM98, Lemma 3.39], [CS21, Lemma 2.7]).
    Used in Theorems 1.1 and 1.2 to compare pullbacks of nef divisors and to constrain the normal bundle of a contractible curve. A standard tool in MMP.
  • standard math Artin's contraction theorem for algebraic spaces and algebraization of formal moduli (cf. [Art70]).
    Used in Section 5.2 to contract E to a point in an algebraic space, and to relate existence of a D-flop to finite generation of a formal algebra.
  • domain assumption Inversion of adjunction and canonical singularity criteria for foliations (cf. [SSV25, Theorem 4.12], [MP13, Fact III.i.3]).
    Used in Section 5.2 to conclude that the constructed rank one foliation H on Y has canonical singularities from its restriction to the invariant surface S.
  • standard math Foliated jet construction of Pereira (cf. [Per01, Section 6.2]) together with Serre vanishing and global generation.
    Underpins Lemma 3.2 and the Bertini-type statement Theorem 3.3 for avoiding log canonical centres. The jet construction is published and used as a tool.

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Pith. "Pith review of Foliated Minimal Models and Flops." pith.science (2026). https://pith.science/paper/IUO32B7B

@misc{pith2026260809663,
  author       = {Pith},
  title        = {Pith review of: Foliated Minimal Models and Flops},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IUO32B7B}},
  note         = {Machine review of arXiv:2608.09663}
}
abstract

We study minimal models and flops for foliations. We show that if $\mathcal F$ is a rank one foliation with canonical singularities on a normal projective $\mathbb Q$-factorial variety and $K_{\mathcal F}$ is pseudo-effective, then any two outputs of the $K_{\mathcal F}$-MMP are isomorphic. For co-rank one foliations on threefolds, we prove existence results for $D$-flops in the klt setting and, under additional hypotheses, in the F-dlt setting. By contrast, we construct examples showing that rank one foliations display pathologies absent from the classical MMP: flopping contractions need not admit $D$-flops, and nef and big canonical divisors need not give rise to canonical models, even in the category of algebraic spaces.

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