Sarkisov program for algebraically integrable and threefold foliations

Jihao Liu; Yanze Wang; Yifei Chen

REVIEW 2 major objections 4 minor 41 references

Sarkisov program for algebraically integrable and threefold foliations

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that any two Mori fiber spaces obtained from the MMP of an lc algebraically integrable adjoint foliated structure are connected by a finite sequence of Sarkisov links, and derives the Sarkisov program for algebraically…

desk verdict Genuinely new Sarkisov theorems for foliations, but the load-bearing floor is the unvetted companion MMP preprints; referee it, and make the dependencies the focus. read the letter →

arxiv 2505.15115 v1 pith:62SU32T6 submitted 2025-05-21 math.AG

Yifei Chen , Jihao Liu , Yanze Wang This is my paper · ORCID

classification math.AG MSC 14E3037F75

keywords SarkisovprogramalgebraicallyintegrablefoliationadjointfoliatedstructureMorifiberspaceminimalmodelthreefoldkltsingularitybirationalgeometry

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

Birational geometry routinely produces many different Mori fiber spaces from the same variety, and the Sarkisov program asks whether these outputs are connected by elementary birational steps. This paper answers that question for algebraically integrable foliations: any two Mori fiber spaces obtained by running the foliated minimal model program can be joined by a finite chain of Sarkisov links. The proof works in the more general setting of adjoint foliated structures, which package a foliation with a boundary divisor and a nef moduli part, and it yields the same connectivity for rank-one foliations on klt threefolds and for foliated log smooth foliations in dimension at most three. If correct, it gives foliations a well-defined deformation-and-factorization skeleton, the same kind of structure that underpins the study of birational automorphism groups in the classical case.

What carries the argument

The load-bearing objects are adjoint foliated structures A = (X,F,B,M,t), a foliation F on a normal variety X with a boundary divisor B, a nef b-divisor M, and a real parameter t in [0,1], whose canonical divisor is K_A = tK_F + (1-t)K_X + B + M_X. The argument is carried by a double-scaling induction with two parameters (l_i, r_i) that interpolate between the two given Mori fiber spaces; each step constructs a Sarkisov link either by extracting a divisor of negative discrepancy and running a K-trivial MMP (Type I or II links) or by contracting a K-trivial extremal face (Type III or IV links). Termination is obtained from the finiteness of weak lc models for adjoint foliated structures together with monotone decreases in discrepancy along the chain.

What would settle it

Construct an lc algebraically integrable adjoint foliated structure on a Q-factorial klt variety with two K_A-Mori fiber spaces that cannot be connected by any finite sequence of Sarkisov links; equivalently, exhibit a counterexample to the finiteness of weak lc models for such structures.

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Extended reading notes

Core claim

The central result is Theorem 1.3: the Sarkisov program holds for lc (log canonical) algebraically integrable adjoint foliated structures on Q-factorial klt varieties. Concretely, if A_W/U is such a structure on a Q-factorial klt variety W, then any two Mori fiber spaces f: X→Z and f': X'→Z' obtained from A by running the K_A-MMP over U are connected by a finite sequence of Sarkisov links over U. The proof converts a K_F-MMP for a foliation into an MMP for a klt adjoint foliated structure by pulling in an ample divisor, so the problem is moved from the category of lc foliations on klt varieties to the more rigid category of klt adjoint foliated structures. From Theorem 1.3 the paper derives the Sarkisov program for lc algebraically integrable foliations on Q-factorial klt varieties, for rank-one lc foliations on Q-factorial projective klt threefolds, and, together with the known rank-two threefold case, for foliated log smooth foliations in dimension at most three.

Load-bearing premise

The whole proof depends on the companion minimal model program for adjoint foliated structures: specifically, the existence of Q-factorial terminalizations and the finiteness of weak lc models are imported as black boxes, so any gap in that foundation would undermine the Sarkisov program proved here.

Editorial extensions

If this is right

Any two Mori fiber spaces from the MMP of an lc algebraically integrable foliation on a Q-factorial klt variety are Sarkisov-linked (Theorem 1.4).
The Sarkisov program holds for rank-one lc foliations on Q-factorial projective klt threefolds (Theorem 1.1).
With the existing rank-two threefold case, the Sarkisov program holds for all foliated log smooth foliations in dimension at most three (Theorem 1.2).
Every birational map between such Mori fiber spaces factors into finitely many links of the four classical types, so the foliated MMP outputs are organized by a finite-dimensional graph.

Reading between the lines

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

A natural next step is that the same factorization controls the birational automorphism group of a foliated Mori fiber space, mirroring the classical applications to Cremona groups.
If the underlying MMP for adjoint foliated structures advances to broader classes, the same double-scaling mechanism would automatically extend the Sarkisov program to higher dimensions and higher-rank foliations.
The relative formulation over arbitrary bases U suggests that an equivariant version of the foliated Sarkisov program should hold whenever the ambient MMP is equivariant.

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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 main result here is real: Theorem 1.1 settles the rank-one threefold case of the Sarkisov program, which was open because Bertini-type theorems fail, and Theorem 1.3 gives a general statement for algebraically integrable adjoint foliated structures. The reduction from lc foliations to klt adjoint structures via the parameter t is a genuine technical step, and the log and adjoint versions are a sensible way to package the work. The paper is well written and honest about what it assumes.

The soft spot is exactly where the stress-test note lands. Lemma 3.14, which powers termination, is one sentence: the map is a weak lc model, so Theorem 2.20 applies. Theorem 2.20's proof is deferral to [CHLMSSX25, Theorems 2.5.2 and 2.2.3], and Definition-Theorem 2.15 (Q-factorial terminalization) also relies on [CHLMSSX25, Theorem 2.2.3]. Lemma 2.17, which constructs the common terminal model for the two Mori fiber spaces, uses that same terminalization. So the factorization algorithm is coherent, but the entire proof stands on two companion preprints from the same group that are not yet peer-reviewed. This is not a hidden circularity, and the paper is upfront about the citations. But it is a genuine fragility: if the finiteness statement in [CHLMSSX25] has a missing hypothesis (say, a bigness condition that fails along the path (l_i,r_i)), then Lemma 3.14 fails and so does Theorem 1.3. The stress-test's conditional is fair.

What the paper does well is the internal engine: the Sarkisov links are constructed cleanly, the discrepancy comparisons in Lemmas 3.7 and 3.11 are standard but carefully executed, and the termination argument is reduced to finiteness of weak lc models in a way that is easy to audit once the black boxes are accepted. The citation pattern is appropriate; the self-citations are to the MMP foundations on which the work explicitly builds.

This paper is for experts in the MMP for foliations, and it deserves a serious referee. A competent referee will need to read the two companion papers alongside it, and the editor should say so. My recommendation is to send it to peer review, with a note that the main risk is external dependence, not internal error.

Referee Report

2 major / 4 minor

Summary. The paper develops a Sarkisov program for algebraically integrable adjoint foliated structures on Q-factorial klt varieties (Theorem 1.3), and derives from it the Sarkisov program for rank-one lc foliations on Q-factorial klt threefolds (Theorem 1.1), for lc algebraically integrable foliations on Q-factorial klt varieties (Theorem 1.4), and for foliated log smooth foliations in dimension at most three (Theorem 1.2). The proof adapts the classical Hacon--McKernan and Liu strategies to adjoint foliated structures, running a Sarkisov algorithm (Section 3) whose termination is proved by a discrepancy-monotonicity argument, with the key finiteness input (Lemma 3.14) supplied by the companion preprint [CHLMSSX25].

Significance. If the companion MMP results are valid, this paper establishes a substantial and natural extension of the Sarkisov program to foliations, unifying the classical case (t=0, F=T_X) and the rank-one threefold case. The reduction from potentially non-klt lc foliations to klt adjoint foliated structures is a useful conceptual innovation, and the paper gives a coherent, structured proof that should have lasting value for the birational geometry of foliations. The paper is clearly written and carefully organizes the many technical hypotheses, and it explicitly identifies the main external dependencies.

major comments (2)

[Lemma 3.14] The one-sentence proof of Lemma 3.14 invokes Theorem 2.20, but Theorem 2.20 applies only to a fixed rational polytope C in the space of structures, with every A in C klt and B+M_X big over U. The proof does not identify such a polytope containing the structures AW(li,ri) produced along the Sarkisov algorithm, nor does it check that the convex hull of these points satisfies the klt condition. Since termination (Theorem 3.18) and hence Theorem 1.3 depend directly on Lemma 3.14, this is a load-bearing gap. The argument must either exhibit a uniform polytope and verify all hypotheses, or give a separate finiteness argument for the specific sequence of weak lc models.
[Definition-Theorem 2.15 and Theorem 2.20] Both the construction of the common Q-factorial terminal model AW (Definition-Theorem 2.15, via [CHLMSSX25, Theorem 2.2.3]) and the finiteness of weak lc models (Theorem 2.20, via [CHLMSSX25, Theorems 2.5.2 and 2.2.3]) are entirely deferred to a companion preprint that is not peer-reviewed and whose authors overlap with those of this paper. The main theorem is therefore conditional on those external results, but the abstract and introduction state the theorems unconditionally. The authors should either include proofs of these statements, or explicitly and prominently state that the main theorems assume [CHLMSSX25] (and indicate whether that preprint has been accepted or posted in final form).

minor comments (4)

[Abstract] The word 'aformentioned' is a typo; it should be 'aforementioned'.
[References] In the reference [BM97], the title has 'Sarkisov proram'; it should be 'Sarkisov program'.
[References] In reference [HM13], the author name is typeset as 'M cKernan' with an extra space; the standard spelling is 'McKernan'.
[Definition 3.1] The commutative diagrams for the four types of Sarkisov links are hard to read in the current LaTeX rendering; in the final journal version, they should be typeset as proper commutative diagrams so that the maps p, q, alpha, beta and the contractions are clearly displayed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is derived from prior MMP results for adjoint foliated structures, not assumed.

full rationale

The central derivation of Theorem 1.3 takes the minimal model program for lc algebraically integrable adjoint foliated structures as input. Theorem 2.20 (finiteness of weak lc models) is proved here by reducing to [CHLMSSX25, Theorem 2.5.2], and Lemma 3.14 invokes it to bound the possible birational maps. This is a conditional use of an external theorem, not a definitional or fitted circularity: the cited finiteness theorem concerns existence of minimal models and does not include the Sarkisov program among its assumptions. Similarly, Definition-Theorem 2.15 uses [CHLMSSX25, Theorem 2.2.3] to extract finitely many non-klt places. Authorship overlap with the companion papers is not itself circularity, and no equation in the paper reduces by construction to an input. The termination proof is contingent on the companion finiteness results, but conditionality on unverified external theorems is a correctness risk, not circularity. No fitted parameter is relabelled as a prediction, and no known result is renamed. Therefore the paper receives score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central proof introduces no fitted parameters or invented entities; it is a pure mathematical derivation. The relevant 'input' is the prior MMP theory, listed as axioms.

assumptions (6)

domain assumption Existence and termination of MMP for lc algebraically integrable adjoint foliated structures, including cone theorem, contraction theorem, flips, and MMP with scaling (CHLMSSX24).
Used as the core engine in Section 3: Lemma 2.19, Construction 3.6, etc. The current paper does not prove these statements; they are taken from the companion paper arXiv:2408.14258.
domain assumption Finiteness of weak lc models for klt adjoint foliated structures with big boundary (CHLMSSX25, Theorem 2.5.2).
Used in the proof of Theorem 2.20 (Finiteness of weak lc models), which is then used in Lemma 3.14 and Theorem 3.18 for termination.
domain assumption Q-factorial terminalization for klt algebraically integrable sub-adjoint foliated structures (CHLMSSX25, Theorem 2.2.3).
Used in Definition-Theorem 2.15 and Lemma 2.17 to construct terminal models; also used in Lemma 3.8.
domain assumption Any K_F-MMP for an lc foliation can be perturbed to an MMP for a klt adjoint foliated structure (CHLMSSX25, Lemma 3.29).
Stated as the crucial observation in the introduction and invoked in the proof of Theorem 1.3 (Section 4).
domain assumption Algebraic integrability of rank one foliations on projective klt threefolds with non-pseudo-effective canonical class (Campana-Paun [CP19], Liu-Luo-Meng [LLM23, Theorem 3.1]).
Used in the proof of Theorem 1.1 to reduce to the algebraically integrable case (Section 4, last paragraph).
domain assumption Sarkisov program for Q-factorial F-dlt foliations of rank two on threefolds (Mascharak [Mas24, Theorem 1.1]).
Used in the proof of Theorem 1.2 for foliations of rank two in dimension three.

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Pith. "Pith review of Sarkisov program for algebraically integrable and threefold foliations." pith.science (2026). https://pith.science/paper/62SU32T6

@misc{pith2026250515115,
  author       = {Pith},
  title        = {Pith review of: Sarkisov program for algebraically integrable and threefold foliations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/62SU32T6}},
  note         = {Machine review of arXiv:2505.15115}
}

abstract

By applying the theory of the minimal model program for adjoint foliated structures, we establish the Sarkisov program for algebraically integrable foliations on klt varieties: any two Mori fiber spaces of such structure are connected by a sequence of Sarkisov links. Combining with a result of R. Mascharak, we establish the Sarkisov program for foliations in dimension at most $3$ with mild singularities. Log version and adjoint foliated version of the aformentioned Sarkisov programs are also established.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Ambro, P

F. Ambro, P. Cascini, V. V. Shokurov, and C. Spicer, Positivity of the moduli part, arXiv:2111.00423

arXiv

[2] [2]

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F. Bernasconi, A. Fanelli, J. Schneider, and S. Zimmermann, Explicit Sarkisov program for regular surfaces over arbitrary fields and applications, arXiv:2404.03281

arXiv

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J. Blanc, S. Lamy, and S. Zimmermann, Quotients of higher dimensional Cremona groups, Acta Math. 226 (2021), no. 2, 211--318

work page 2021

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Birkar, Existence of log canonical flips and a special LMMP, Publ

C. Birkar, Existence of log canonical flips and a special LMMP, Publ. Math. IHÉS 115 (2012), 325--368

work page 2012

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Birkar, P

C. Birkar, P. Cascini, C. D. Hacon, and J. M c Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405--468

2010

[6] [6]

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Bruno and K

A. Bruno and K. Matsuki, Log Sarkisov proram, Int. J.Math. 8 (1997), no. 4, 451--494

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