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arxiv: 2604.05553 · v2 · submitted 2026-04-07 · 🧮 math.AG

Minimal-Degree Foliations on Cominuscule Grassmannians

Vladimiro Benedetti (LJAD) , Crislaine Kuster , Alan Muniz (UFPE) This is my paper

Pith reviewed 2026-05-10 19:01 UTC · model grok-4.3

classification 🧮 math.AG
keywords cominuscule Grassmanniansfoliationsminimal degreecotangent bundleHermitian symmetric spacespencils of hyperplanescodimension-one foliationsexterior powers
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The pith

Any codimension-one foliation of degree zero on a cominuscule Grassmannian is a pencil of hyperplanes.

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

The paper computes the smallest integer l(p) such that the bundle of p-forms on the cotangent sheaf of a cominuscule Grassmannian, twisted by l(p), admits global sections. This threshold directly implies that every codimension-one foliation of degree zero must arise as a pencil of hyperplanes. The same calculation determines the structure of codimension-one foliations of degree one and produces explicit families of minimal-degree foliations in higher codimensions on Grassmannians, Lagrangian Grassmannians, Spinor varieties, and the Cayley plane.

Core claim

By determining the minimal l(p) for which H^0(Ω^p_X(l(p))) is nonzero on a cominuscule Grassmannian X, any codimension-one foliation of degree zero is necessarily a pencil of hyperplanes. The computation likewise fixes the possible form of degree-one foliations in codimension one and supplies concrete families of high-codimension foliations that attain the minimal degree on the classical cominuscule spaces.

What carries the argument

The threshold function l(p), the smallest integer making the p-th exterior power of the cotangent bundle have nonzero global sections; this value sets the lowest possible degree for foliations generated by integrable sections of the corresponding twisted bundle.

If this is right

  • Every codimension-one foliation of degree zero must be a pencil of hyperplanes.
  • Codimension-one foliations of degree one are limited to those arising from sections at the next computed twist level.
  • Explicit minimal-degree foliations of arbitrary codimension exist on Grassmannians, Lagrangian Grassmannians, Spinor varieties, and the Cayley plane.

Where Pith is reading between the lines

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

  • The method of locating the first nonzero sections of twisted exterior powers could classify foliations on other homogeneous spaces beyond the cominuscule case.
  • These minimal-degree examples provide concrete test objects for studying rigidity or deformation questions for foliations on projective homogeneous varieties.
  • One could verify whether l(p) coincides with known formulas for the regularity of the cotangent sheaf on these spaces.

Load-bearing premise

That the degree of any foliation is captured exactly by the minimal twist at which a generating section of the twisted exterior power appears.

What would settle it

A single explicit example of a codimension-one foliation of degree zero on any cominuscule Grassmannian that is not a pencil of hyperplanes would disprove the classification.

read the original abstract

Given $X$ a cominuscule Grassmannian (or irreducible Hermitian symmetric space) and an integer $p,$ we compute the minimum $l(p)$ such that $H^0 (\Omega^p_X (l(p)))$ is not 0. This allows us to conclude that any codimension-one foliation of degree zero on a cominuscule Grassmannian is a pencil of hyperplanes, improving a result of the first and third authors with D. Faenzi. We also deduce the structure of codimension-one foliations of degree one. Finally, we provide families of examples of high codimensional foliations of minimal degree on classical Grassmannians, Lagrangian Grassmannians, Spinor varieties, and the Cayley plane.

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

2 major / 3 minor

Summary. The paper computes the minimal l(p) such that H^0(Ω^p_X(l(p))) ≠ 0 for cominuscule Grassmannians X (irreducible Hermitian symmetric spaces). This threshold is used to classify codimension-one foliations of degree zero, proving they are pencils of hyperplanes (improving a prior result with Faenzi), to describe the degree-one case, and to construct families of minimal-degree examples in higher codimensions on classical Grassmannians, Lagrangian Grassmannians, Spinor varieties, and the Cayley plane.

Significance. The explicit computation of the minimal l(p) via representation theory and vanishing theorems on homogeneous spaces yields a clean classification for low-degree codimension-one foliations and supplies concrete high-codimension examples. The parameter-free nature of the threshold and the direct link to the integrability condition for foliations strengthen the geometric conclusions.

major comments (2)
  1. §4, the proof that degree-zero foliations are pencils of hyperplanes: the argument relies on the vanishing H^0(Ω^1_X(l)) = 0 for l < l(1) together with the integrability condition; it is not immediately clear how the correspondence between nonzero sections and foliations is made bijective when the section is not necessarily integrable a priori.
  2. §5, the families of high-codimension examples on the Cayley plane: the claimed minimal degree is asserted by comparing the computed l(p) against the degree of the constructed foliations, but the explicit check that these examples achieve equality (rather than merely being bounded by l(p)) is only sketched.
minor comments (3)
  1. Notation for the twisted cotangent bundle Ω^p_X(l) should be introduced once in §2 with a reminder that l is the twisting by O_X(l) in the usual Plücker embedding.
  2. The statement of the main classification theorem (presumably Theorem 1.1 or 4.1) would benefit from an explicit list of the cominuscule Grassmannians to which it applies.
  3. A short table summarizing the values of l(p) for small p across the classical cases would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: §4, the proof that degree-zero foliations are pencils of hyperplanes: the argument relies on the vanishing H^0(Ω^1_X(l)) = 0 for l < l(1) together with the integrability condition; it is not immediately clear how the correspondence between nonzero sections and foliations is made bijective when the section is not necessarily integrable a priori.

    Authors: We thank the referee for this observation on the exposition. The standard correspondence identifies codimension-one foliations of degree d with integrable sections of Ω^1_X(d+1). Our argument uses the vanishing for l < l(1) to rule out lower-degree foliations and then identifies the minimal-degree sections with the known pencils of hyperplanes. To make the bijection fully explicit, we will revise §4 by adding a short paragraph that recalls the correspondence and confirms that the sections spanning H^0(Ω^1_X(l(1))) satisfy the integrability condition, using the representation-theoretic description already developed in the paper. revision: yes

  2. Referee: §5, the families of high-codimension examples on the Cayley plane: the claimed minimal degree is asserted by comparing the computed l(p) against the degree of the constructed foliations, but the explicit check that these examples achieve equality (rather than merely being bounded by l(p)) is only sketched.

    Authors: We appreciate this comment. In §5 the constructions on the Cayley plane are given explicitly via the geometry of E_6, and their degrees are compared with the computed l(p). We agree that the verification that equality is attained can be presented more explicitly. In the revised manuscript we will expand the relevant paragraph to include a direct computation of the degree of each constructed foliation, confirming that it equals the minimal l(p) for the corresponding p. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation computes the minimal l(p) such that H^0(Ω^p_X(l(p))) is nonzero via representation theory and vanishing theorems on homogeneous spaces, then applies this threshold to bound foliation degrees and classify the degree-zero case as pencils of hyperplanes. This is a direct, parameter-free cohomological calculation on cominuscule Grassmannians that does not reduce to fitted inputs, self-definitions, or load-bearing self-citations; the cited prior result of the authors is improved by the new explicit computation rather than presupposed. The integrability check for foliations follows from the cohomology statement without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work relies on standard axioms of algebraic geometry (cohomology on homogeneous spaces, properties of cominuscule varieties) with no free parameters or invented entities identifiable from the abstract.

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