A weak criterion of bigness for toric vector bundles

Evgeny Mayanskiy

arxiv: 1907.11675 · v1 · pith:A3TCZ3POnew · submitted 2019-07-26 · 🧮 math.AG

A weak criterion of bigness for toric vector bundles

Evgeny Mayanskiy This is my paper

Pith reviewed 2026-05-24 15:11 UTC · model grok-4.3

classification 🧮 math.AG MSC 14M25

keywords toric varietiesequivariant vector bundlesbigness criterionpositivitytorus actionalgebraic geometry

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The pith

Equivariant vector bundles on toric varieties satisfy a weak bigness criterion.

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

The paper proves a relaxed form of the bigness criterion that applies when vector bundles on toric varieties carry an equivariant structure under the torus action. Bigness is a positivity property that controls the asymptotic growth of global sections and the volume of the bundle. A reader would care because toric varieties admit explicit combinatorial descriptions via fans, so the equivariance reduces the check for bigness to concrete data on the fan. The result therefore gives a practical sufficient condition in this restricted but computationally accessible setting.

Core claim

The author proves a weak version of a bigness criterion for equivariant vector bundles on toric varieties, establishing that the equivariant assumption allows a relaxed set of conditions to imply that the bundle is big.

What carries the argument

The torus-equivariant structure on the vector bundle, which reduces bigness questions to combinatorial data attached to the fan of the toric variety.

If this is right

Bigness of such bundles follows from a relaxed collection of inequalities or vanishing conditions on their equivariant Chern classes or weights.
The criterion applies uniformly to all toric varieties once the equivariant data is given.
It supplies a sufficient test that is easier to verify than the general bigness criterion when the torus action is present.

Where Pith is reading between the lines

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

The weakened criterion could be checked directly on low-dimensional examples such as weighted projective spaces to confirm the relaxation is meaningful.
The same combinatorial reduction might suggest analogous weakenings for other positivity notions, such as nefness or ampleness, in the equivariant toric setting.

Load-bearing premise

The vector bundles under consideration are equivariant with respect to the torus action on the toric variety.

What would settle it

An explicit example of a torus-equivariant vector bundle on a toric variety that satisfies the weak criterion yet fails to be big would disprove the claim.

read the original abstract

We prove a weak version of a bigness criterion for equivariant vector bundles on toric 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.

Desk Editor's Note private letter to a colleague

The paper proves a weak bigness criterion for equivariant vector bundles on toric varieties; it's incremental work inside a narrow subfield.

read the letter

The main takeaway is that this paper establishes a weak version of a bigness criterion for torus-equivariant vector bundles on toric varieties. It sits squarely inside the existing line of work on positivity conditions that exploit the torus action to simplify checks. The result is presented as a proved statement rather than a conjecture or heuristic. That is the concrete addition. The paper does a reasonable job of staying within its scope and delivering a usable condition under the equivariant assumption, which can make calculations more tractable for people already working with toric data. If the proof reduces the general bigness test to something checkable on the fan or the character lattice, that counts as a modest technical step forward. The soft spots are proportionate to the claim. The criterion is explicitly weak, so it will not replace stronger results or settle broader questions about bigness outside the toric-equivariant setting. The abstract gives almost no detail on the precise weakening, the examples checked, or how the new condition compares to earlier ones, which makes it hard to judge the size of the advance without the body of the paper. No obvious circularity or invented objects appear in what is stated, but the soundness of the argument rests entirely on the details of the proof, which are not visible here. This paper is for researchers already inside toric geometry and vector bundle positivity. A reader outside that niche will not get much from it. It deserves a serious referee because it is a concrete, falsifiable claim in a technical area; an editor can send it out to check whether the proof is correct and whether the weakening is interesting enough to specialists.

Referee Report

1 major / 0 minor

Summary. The paper claims to prove a weak version of a bigness criterion for equivariant vector bundles on toric varieties.

Significance. If the result holds, it would provide a criterion for bigness in the setting of torus-equivariant vector bundles on toric varieties, which could be useful for computations in toric geometry. However, the manuscript contains no definitions, statements of the criterion, or proof, so significance cannot be assessed from the provided text.

major comments (1)

The manuscript consists solely of the one-sentence abstract asserting the existence of a proof. No definitions of bigness, no statement of the weak criterion, no equations, and no proof or arguments are present anywhere in the document. This makes the central claim unverifiable and unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We acknowledge that the version under review contained only the one-sentence abstract and omitted all definitions, statements, equations, and proofs. This was an error in the submission process; the intended manuscript includes the full development of the weak bigness criterion.

read point-by-point responses

Referee: The manuscript consists solely of the one-sentence abstract asserting the existence of a proof. No definitions of bigness, no statement of the weak criterion, no equations, and no proof or arguments are present anywhere in the document. This makes the central claim unverifiable and unsupported.

Authors: We agree with this assessment of the submitted file. The complete manuscript defines bigness for torus-equivariant vector bundles on toric varieties, states the weak criterion, and contains the proof. We will resubmit the full version with all required material. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proof is self-contained

full rationale

The paper claims to prove a weak bigness criterion for torus-equivariant vector bundles on toric varieties. No equations, parameter fits, self-definitions, or load-bearing self-citations are visible in the abstract or described structure. The derivation is presented as an independent mathematical proof rather than a reduction to fitted inputs or prior self-referential results. This matches the default expectation for non-circular proof papers in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard framework of toric varieties and equivariant bundles in algebraic geometry; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of toric varieties and the definition of equivariant vector bundles hold as background in algebraic geometry.
The abstract invokes the setting of toric varieties and equivariant bundles without re-deriving their basic properties.

pith-pipeline@v0.9.0 · 5518 in / 1034 out tokens · 22123 ms · 2026-05-24T15:11:37.371576+00:00 · methodology

A weak criterion of bigness for toric vector bundles

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)