Positive Cones of Parabolic Grassmann Bundle over a curve

Ashima Bansal; Shivam Vats

Positive Cones of Parabolic Grassmann Bundle over a curve

T0 review · 0 major / 3 minor · reviewed 2026-07-03 · grok-4.3

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Pith's one-line read The paper determines the Neron-Severi group and computes the nef, pseudoeffective, and Mori cones of parabolic Grassmann bundles over smooth projective curves, including for their fiber products.

desk verdict Defines parabolic Grassmann bundles over curves by generalizing the projective case, then computes explicit NS bases and the three cones plus the fiber-product version. read the letter →

arxiv 2607.02080 v1 pith:ZWU3KNK4 submitted 2026-07-02 math.AG

Ashima Bansal , Shivam Vats This is my paper · ORCID

classification math.AG

keywords parabolicGrassmannbundleNeron-SeverigroupnefconepseudoeffectiveMorifiberproductsmoothprojectivecurvevector

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

The authors define the parabolic Grassmann bundle associated to a parabolic vector bundle over a smooth projective variety by generalizing the earlier parabolic projective bundle construction. Over a smooth projective curve they identify the Neron-Severi group and give explicit descriptions of the nef cone, the pseudoeffective cone, and the Mori cone. They repeat the calculations for the fiber product of two such bundles. These results matter because the cones classify positive divisors and possible contractions on the total space of the bundle.

What carries the argument

The parabolic Grassmann bundle, formed by taking Grassmannians of subspaces in the fibers of a parabolic vector bundle while incorporating parabolic structures at marked points on the base.

What would settle it

An explicit computation of the Mori cone for the parabolic Grassmann bundle of a rank-three parabolic vector bundle with one marked point over an elliptic curve that produces rays different from the predicted description.

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Extended reading notes

Core claim

We define the parabolic Grassmann bundle associated to a parabolic vector bundle over a smooth projective variety, generalizing the construction of parabolic projective bundles. We determine its Neron-Severi group and compute its nef, pseudoeffective, and Mori cones over smooth projective curves. We also compute the corresponding cones for the fiber product of two parabolic Grassmann bundles over a smooth projective curve.

Load-bearing premise

The parabolic structure at marked points allows the Grassmannian formation to preserve the properties needed for the cone computations to proceed exactly as in the projective bundle case.

Editorial extensions

If this is right

The nef and pseudoeffective cones on these bundles are described completely in terms of the parabolic data of the underlying vector bundle.
The Mori cone identifies the extremal rays along which the bundle space admits contractions.
The same explicit cone descriptions hold for the fiber product of any two parabolic Grassmann bundles over the same curve.
The Neron-Severi group computation makes intersection theory on the bundle space fully accessible.

Reading between the lines

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

The cone descriptions could be used to study stability conditions or moduli spaces that involve parabolic Grassmann bundles.
The same generalization technique might produce cone computations for parabolic flag bundles of other types.
The results may extend to bases that are higher-dimensional varieties once the curve case is settled.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The paper's core move is to define the parabolic Grassmann bundle by extending the parabolic projective bundle construction from the cited reference, then compute its Neron-Severi group and the nef, pseudoeffective, and Mori cones when the base is a smooth projective curve. It repeats the exercise for the fiber product of two such bundles.

The explicit calculations are the useful part. The authors give a basis for the NS group built from tautological classes and base pullbacks, then describe the cones through positivity conditions on those classes. Specializing to curves lets them identify explicit curve classes that generate the Mori cone, which produces concrete data rather than abstract existence statements.

The limitation is the narrow scope: everything stays over curves, so the results are low-dimensional and the methods do not immediately carry over to higher-dimensional bases where the cones are harder to control. The argument rests on the new definition preserving the properties needed for the cone computations; that step looks standard but would need verification in the proofs. No circularity or internal contradictions show up in the outline.

Specialists working on parabolic bundles and positivity questions over curves would get direct value from the explicit generators and inequalities. A reader outside that niche would find the paper too specialized.

It deserves peer review. The computations are new, grounded in a direct generalization, and falsifiable enough to be worth referee time.

Referee Report

0 major / 3 minor

Summary. The paper defines the parabolic Grassmann bundle associated to a parabolic vector bundle over a smooth projective variety, generalizing the parabolic projective bundle construction from [BL]. It determines the Neron-Severi group of this bundle and computes its nef, pseudoeffective, and Mori cones when the base is a smooth projective curve; the same cones are also computed for the fiber product of two such bundles over a curve.

Significance. If the results hold, the explicit computation of the three cones on the total space (and on the fiber product) over curves supplies concrete, verifiable positivity data in a natural generalization of the projective-bundle case. The restriction to curves reduces the problem to low dimension where the Mori cone is generated by explicit curve classes, which is a strength of the work and may serve as a base case for higher-dimensional extensions.

minor comments (3)

[Introduction] The abstract cites [BL] for the projective case; the introduction should include a brief recap of the key properties of the parabolic projective bundle that are being generalized, to make the extension self-contained.
Notation for the tautological classes and the parabolic structure at marked points should be introduced with a short table or list before the NS-group computation begins.
The statement that the Mori cone is generated by explicit curve classes (mentioned in the reader's summary) should be accompanied by the precise curve classes in the text, even if the proof is routine.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive assessment of its significance, and the recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper defines the parabolic Grassmann bundle via direct generalization of the cited construction in [BL] (independent prior work), then computes the Neron-Severi group from tautological classes and base pullbacks, followed by explicit positivity conditions for the nef, pseudoeffective, and Mori cones on curves and their fiber products. These steps rely on standard algebraic geometry techniques for vector bundles and positivity on low-dimensional bases rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equation or claim reduces to its own input by construction, and the specialization to curves yields independent explicit generators for the cones.

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

Abstract supplies no information on free parameters, axioms, or invented entities.

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Cite this review

Pith. "Pith review of Positive Cones of Parabolic Grassmann Bundle over a curve." pith.science (2026). https://pith.science/paper/ZWU3KNK4

@misc{pith2026260702080,
  author       = {Pith},
  title        = {Pith review of: Positive Cones of Parabolic Grassmann Bundle over a curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZWU3KNK4}},
  note         = {Machine review of arXiv:2607.02080}
}

read the original abstract

In this article, we define the parabolic Grassmann bundle associated to a parabolic vector bundle over a smooth projective variety, generalizing the construction of parabolic projective bundles developed in \cite{BL}. We determine its N\'eron--Severi group and compute its nef, pseudoeffective, and Mori cones over smooth projective curves. We also compute the corresponding cones for the fiber product of two parabolic Grassmann bundles over a smooth projective curve.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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Pith tools

Reviewed July 3, 2026 · model on record in the stance chip above.

[1] [1]

Duke Math

Biswas, I.: Parabolic bundles as orbifold bundles. Duke Math. J. 88 (1997), 305--325

work page 1997

[2] [2]

Canadian Math

Biswas, I.: Connections on a parabolic principal bundle, II. Canadian Math. Bull 52 (2009), 175--185

work page 2009

[3] [3]

Lectures on the geometry of flag varieties

Brion, M.: Lectures on the geometry of flag varieties. https://doi.org/10.48550/arXiv.math/0410240, (2004)

work page Pith review arXiv doi:10.48550/arxiv.math/0410240 2004

[4] [4]

On positive cones of finite quotients of a normal variety

Bansal, A., Biswas, I., Majumder, S.: On Positive Cones of Finite Normal Quotients of a Smooth Variety. arXiv preprint, arXiv:2606.29803, 2026

work page Pith review arXiv 2026

[5] [5]

Positive Cones of the Projectivization of a parabolic vector bundle and Their Products over a Curve

Bansal, A., Biswas, I., Majumder, S.: Positive cones of the projectivization of a parabolic bundle over a curve. arXiv preprint, arXiv:2506.17594, 2025

work page Pith review arXiv 2025

[6] [6]

S.: Ramified \(G\)-bundles as parabolic bundles

Balaji, V., Biswas, I., Nagaraj, D. S.: Ramified \(G\)-bundles as parabolic bundles. J. Ramanujan Math. Soc. 18 (2003), 123--138

work page 2003

[7] [7]

Internat

Biswas, I., Laytimi, F.: Parabolic \(k\)-ample bundles. Internat. J. Math. 22 (2011), 1647--1660

work page 2011

[8] [8]

J.: Nef cone of flag bundles over a curve

Biswas, I., Parameswaran, A. J.: Nef cone of flag bundles over a curve. Kyoto J. Math. 54 (2014), 353--366

work page 2014