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arxiv: 2604.08100 · v1 · submitted 2026-04-09 · 🧮 math.AG · math.CV

Rank one foliations on toroidal varieties

Calum Spicer , Luca Tasin This is my paper

Pith reviewed 2026-05-10 17:52 UTC · model grok-4.3

classification 🧮 math.AG math.CV
keywords rank one foliationslog canonical pairsbirational geometrytoroidal varietieslog homogeneous varietiescanonical classfoliation singularities
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The pith

For log canonical pairs with globally generated twisted logarithmic tangent sheaf, rank one foliations admit a divisor making the canonical classes equivalent while preserving log canonicity

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the canonical class of a rank one foliation can be realized through a log canonical pair on the ambient variety after adjustment by a fixed divisor. Given a log canonical pair (X, B) and Cartier divisor D making T_X(-log B) ⊗ O(D) locally free and globally generated, for any log canonical rank one foliation F there exists Gamma with (X, Gamma) log canonical and K_X + Gamma ~ K_F + D. This matters for a reader because it supplies a direct link between foliation invariants and the geometry of pairs, allowing birational questions about foliations to be reduced to standard pair techniques. The authors then apply the result to obtain several statements on the birational geometry of such foliations when the ambient space is log homogeneous.

Core claim

Consider a log canonical pair (X, B) such that there is a Cartier divisor D for which T_X(-log B) ⊗ O(D) is locally free and globally generated. Let F be a log canonical foliation of rank 1 on X. We prove that there exists a divisor Γ such that (X, Γ) is log canonical and K_X + Γ ∼ K_F + D. We then apply this result to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.

What carries the argument

The divisor Γ that makes (X, Γ) log canonical while satisfying the linear equivalence K_X + Γ ~ K_F + D, enabled by the global generation of the twisted logarithmic tangent sheaf.

If this is right

  • The result is applied to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.
  • The canonical class equivalence allows the geometry of the foliation to be compared directly with that of a log canonical pair on the same space.
  • Birational properties of the foliation can be read off from the adjusted pair (X, Γ).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction may extend to toroidal varieties by specializing the log homogeneous case to explicit torus actions.
  • One could check the result on low-dimensional examples such as weighted projective spaces to see explicit forms of Γ.
  • The same generation hypothesis might be used to relate higher-rank foliations to log canonical pairs in future work.

Load-bearing premise

The twisted logarithmic tangent sheaf must be locally free and globally generated, and the foliation must be log canonical of rank one.

What would settle it

A concrete counterexample consisting of a log canonical pair (X, B), Cartier divisor D satisfying the sheaf condition, and log canonical rank one foliation F for which no divisor Γ exists with (X, Γ) log canonical and K_X + Γ ∼ K_F + D.

read the original abstract

Consider a log canonical pair $(X,B)$ such that there is a Cartier divisor $D$ for which $T_X(-\log B) \otimes \mathcal O(D)$ is locally free and globally generated. Let $\mathcal F$ be a log canonical foliation of rank 1 on $X$. We prove that there exists a divisor $\Gamma$ such that $(X, \Gamma)$ is log canonical and $K_X + \Gamma \sim K_{\mathcal F} + D$. We then apply this result to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that given a log canonical pair (X, B) with a Cartier divisor D such that T_X(-log B) ⊗ O(D) is locally free and globally generated, and a log canonical rank-1 foliation F on X, there exists a divisor Γ making (X, Γ) log canonical with K_X + Γ ∼ K_F + D. This existence result is then applied to derive several statements on the birational geometry of rank-1 log canonical foliations on log homogeneous varieties.

Significance. If the central existence result holds, it supplies a concrete construction of a log canonical divisor tied to the canonical class of the foliation via global sections of the twisted logarithmic tangent sheaf. This appears to be a useful technical device in the logarithmic category for toroidal varieties, with direct implications for the minimal model program and classification questions for rank-1 foliations. The applications to log homogeneous varieties constitute a natural follow-up once the main theorem is granted.

minor comments (2)
  1. [§1] §1 (Introduction): the statement of the main theorem could be restated with explicit reference to the global-generation hypothesis on T_X(-log B) ⊗ O(D) to make the logical dependence clearer before the applications are listed.
  2. The notation for the foliation canonical divisor K_F is introduced without a preliminary definition or reference to the standard definition in the literature on foliations; adding a short sentence or citation would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of the utility of the central existence result in the logarithmic category, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; standard existence result from hypotheses

full rationale

The central claim is a conditional existence theorem: given a log canonical pair (X, B) with T_X(-log B) ⊗ O(D) locally free and globally generated, and a log canonical rank-1 foliation F, there exists Γ such that (X, Γ) is log canonical and K_X + Γ ∼ K_F + D. This follows directly from the stated assumptions via standard constructions in the logarithmic category (global sections of the twisted tangent sheaf producing the divisor). No step reduces by the paper's own equations to a fitted parameter, self-referential definition, or self-citation chain. The subsequent applications to birational geometry of rank-1 foliations on log homogeneous varieties are conditional on this independently derived existence and do not create circularity. The derivation is self-contained against external benchmarks in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard background notions of log canonical pairs, foliations, and linear equivalence of divisors in algebraic geometry; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard properties of log canonical pairs and rank-1 foliations in algebraic geometry hold.
    Invoked implicitly to state the hypotheses and conclusion.

pith-pipeline@v0.9.0 · 5392 in / 1292 out tokens · 32302 ms · 2026-05-10T17:52:53.428451+00:00 · methodology

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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The minimal volume of stable surfaces of rank one

    math.AG 2026-05 unverdicted novelty 7.0

    The minimal volume of rank-one stable surfaces is determined and achieved uniquely by one surface up to isomorphism, resolving the Alexeev-Liu conjecture.

  2. The minimal volume of stable surfaces of rank one

    math.AG 2026-05 unverdicted novelty 7.0

    The minimal volume of stable surfaces of rank one is determined with uniqueness up to isomorphism, resolving a conjecture of Alexeev and the second author.

  3. Birational boundedness of stable families

    math.AG 2026-04 unverdicted novelty 6.0

    Algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded, which implies birational boundedness for stable families of maximal variation.

Reference graph

Works this paper leans on

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