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

Ashima Bansal; Indranil Biswas; Souradeep Majumder

REVIEW 5 major objections 4 minor 1 cited by

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

T0 review · 5 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For a parabolic vector bundle on a curve, every positive cone of its projectivization is generated by two explicit divisor classes, and parabolic semistability is exactly the equality of the dual pseudoeffective and nef cones.

desk verdict A natural but under-polished extension of Miyaoka–Fulger to parabolic bundles; the semistable case is promising, while Theorem 5.5's statement is inconsistent and its proof is only a sketch. read the letter →

arxiv 2506.17594 v2 pith:QEAAY2DQ submitted 2025-06-21 math.AG

Ashima Bansal , Indranil Biswas , Souradeep Majumder This is my paper · ORCID

classification math.AG MSC 14H6014L2414F05

keywords parabolicvectorbundleprojectivizationnefconepseudoeffectiveHarder-Narasimhanfiltrationorbifoldsemistabilitypositivecones

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

This paper establishes that the positivity of a parabolic vector bundle over a smooth complex projective curve is fully encoded in the cones of its projectivization. The projectivization of a parabolic bundle need not be smooth, and for singular varieties the numerical groups of cycles and their duals must be treated separately; the paper computes both sides. It shows that in every intermediate codimension the pseudoeffective and nef cones are generated by exactly two explicit classes built from the parabolic Harder-Narasimhan slopes. As an application, it proves that a parabolic bundle is parabolic semistable if and only if every pseudoeffective cone of its projectivization coincides with the corresponding nef cone. This turns a condition on all parabolic subbundles into a finite, explicit equality of cones.

What carries the argument

The load-bearing mechanism is the orbifold correspondence. A parabolic bundle $E^*$ with rational weights is induced by a unique orbifold bundle $\widetilde E$ on a finite Galois cover $Y\to X$, and the parabolic projectivization is the quotient $\mathbb{P}(E^*)\cong \mathbb{P}(\widetilde E)/\Gamma$. Since $\mathbb{P}(\widetilde E)$ is a smooth projective bundle, its nef and pseudoeffective cones are known from the smooth theory. The paper proves explicit pushforward identities (Lemma 5.1) relating Chern monomials on the cover to Chern monomials on the quotient, and uses pushforward results for pseudoeffective cones to transfer the two generators down to $\mathbb{P}(E^*)$. The Harder-Narasimhan filtration enters through its graded pieces, whose slopes and degrees feed into the constants $\nu_k$.

What would settle it

For a fixed parabolic bundle, Theorem 5.5 gives an explicit predicted boundary class for every cone. One could take a rank-two parabolic bundle on $\mathbb{P}^1$ with one parabolic point and weights $0$ and $1/2$, compute the effective cone of curves of $\mathbb{P}(E^*)$ directly by intersecting the fiber and a section against $\xi$ and $L$, and compare the boundary slope with $\nu_1=(\mu_1-d)N(E^*)$. A mismatch, or an unstable example in which $\operatorname{Eff}^1=\operatorname{Nef}^1$, would refute the central claim.

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Extended reading notes

Core claim

The central claim, Theorem 5.5, is that for a parabolic vector bundle $E^*$ of rank $r$ on a curve, with rational parabolic weights, the pseudoeffective cone $\operatorname{Eff}_k(\mathbb{P}(E^*))$ is simplicial for each $k=1,\ldots,r-1$, spanned by $c_1(\xi)^{r-k} + \nu_k c_1(\xi)^{r-k-1}c_1(L)$ and $c_1(\xi)^{r-k-1}c_1(L)$, where $\xi$ is the tautological line bundle class, $L$ is the class of a fiber over a point outside the parabolic divisor, and $\nu_k$ is a constant determined by the Harder-Narasimhan filtration: for the appropriate Harder-Narasimhan stage $s$ and position $j$, $\nu_k=(j\mu_s-\mathbf d_{s-1})N(E^*)$. The dual cones $\operatorname{Nef}^k$ and $\operatorname{Eff}^k$ are then computed as dual cones, and Theorem 6.3 identifies parabolic semistability precisely with the equality $\operatorname{Eff}^k(\mathbb{P}(E^*))=\operatorname{Nef}^k(\mathbb{P}(E^*))$ for all $k$.

Load-bearing premise

The orbifold correspondence is used as the bridge to the smooth case, and that correspondence is stated only when all parabolic weights are rational; the cone formulas are therefore proved only for rational parabolic weights, and the irrational-weight case is not covered.

Editorial extensions

If this is right

For every parabolic bundle $E^*$ of rank $r$, the numerical groups and positive cones in every codimension are two-dimensional and simplicial, so a single constant $\nu_k$ per codimension describes the whole positivity structure.
The generators are explicit from the Harder-Narasimhan filtration, so the nef and pseudoeffective cones are computable once the parabolic slopes are known.
A parabolic bundle is semistable exactly when $\operatorname{Eff}^k(\mathbb{P}(E^*))=\operatorname{Nef}^k(\mathbb{P}(E^*))$ for every $k$, giving a finite numerical semistability test that avoids checking all parabolic subbundles.
Because the cone generators are effective, the pseudoeffective cone is actually the effective cone generated by those two classes, not just a closure.

Reading between the lines

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

Since the constants $\nu_k$ vary continuously with the parabolic slopes and weights, the formulas could probably be extended from rational to real weights by a limiting argument; the paper leaves this open.
A natural extension, consistent with the announced scope, is to apply the same pushforward identities to the product of two Galois covers to compute the cones for the fiber product of two parabolic projective bundles.
The semistability criterion could be used algorithmically: checking the cone equality requires only the Harder-Narasimhan slopes and the two generators, which is numerically cheaper in principle than verifying slope inequalities for all parabolic subbundles.

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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

Dear colleague,

Here is my read of arXiv:2506.17594. The paper extends Miyaoka's and Fulger's cone theorems to projectivizations of parabolic bundles over a curve, using Biswas's orbifold correspondence. The semistable case (Theorems 4.3 and 5.2) looks right, and the method is natural: pull back to a smooth orbifold cover, apply the known smooth results, push forward. The application to a semistability criterion (Theorem 6.3) is a nice idea.

But the paper as written is not in good shape. The abstract promises a computation for the fiber product of two parabolic projective bundles; there is no such result in the body. The title says products over a curve, but the body only treats one projectivization. That needs fixing.

The bigger problem is Theorem 5.5, the main result for unstable bundles. Its proof is one sentence: 'immediately by induction' with l=1,2 cases illustrated. The induction step depends on Lemma 5.4, which defines g_k = ep_* ∘ f_k ∘ ep_*^{-1} without proving that ep_* is an isomorphism of Eff cones for the possibly singular quotient. The cited results may give surjectivity, but you need injectivity too, and the paper does not supply it.

Worse, Theorem 5.5's statement is internally inconsistent. The clause ν_{r_{s-1}+j} = ν_{r-r_{s-1}-j} := (j μ_s - d_{s-1})N(E*) forces ν_k = ν_{r-k}. But in the semistable case, Theorem 5.2 gives ν_k = -(r-k) μ N(E*) and ν_{r-k} = -k μ N(E*). For a rank-3 semistable bundle with μ ≠ 0, these differ. The printed formula cannot be right; it must be a typo, but the asymmetry between ν_k and ν_{r-k} is exactly what makes Corollary 5.9 work. So the statement needs correction and, more importantly, a real proof.

Lemma 5.1 also has questionable equalities in the pushforward formulas; the factors of |Γ| and N(E*) do not line up in simple degree checks. I would want that verified.

One limitation the paper does state is the rationality of parabolic weights; the orbifold correspondence needs it. Fine, but it means the results do not cover parabolic bundles with irrational weights.

Overall: the core idea is sound and the semistable formulas are plausible and likely correct. The general case is not proved as written, and the presentation needs serious cleanup. I would send it to a referee who knows the orbifold correspondence and the Fulger–Lehmann pushforward results, but I would expect major revision.

My call: accept for peer review, but flag the missing proof and the inconsistent statement.

Best

Referee Report

5 major / 4 minor

Summary. The paper studies the positive cones attached to the projectivization P(E*) of a parabolic vector bundle E* on a smooth complex projective curve X, using the Biswas correspondence between parabolic bundles and orbifold bundles. It computes the Néron–Severi group and the nef cone of P(E*), describes generators for the higher nef and pseudoeffective cones Eff_k(P(E*)) and their duals, and derives a criterion for parabolic semistability in terms of equality of Eff^k and Nef^k. The abstract and title also announce results for fiber products of two parabolic projective bundles, but the body of the manuscript contains no such results.

Significance. If the main formulas are correct, the paper gives a useful parabolic analogue of Fulger's computation of effective cones on projective bundles over curves, and the semistability criterion in Theorem 6.3 is an attractive statement. The semistable case is plausibly derived from Miyaoka's theorem through a finite pullback, and the advertised cone generators are explicit and computable from the parabolic Harder–Narasimhan data. The paper would be a meaningful contribution to the positivity theory of parabolic bundles. However, the central general statement and some of the auxiliary lemmas currently contain false equalities or incomplete proofs, so the main result is not yet established in the form presented.

major comments (5)

[Section 5, Theorem 5.5] The displayed definition of the coefficients ν_k is internally inconsistent. For l=1, the theorem gives r_0=0 and d_0=d, and the equality asserts ν_j = ν_{r-j} = (j μ_1 - d)N(E*) for all j. But the l=1 computation in the proof, together with Theorem 5.2, gives ν_k = (k μ_1 - d)N(E*) = -(r-k) μ_1 N(E*). For a semistable rank-3 bundle with μ_1 ≠ 0 this yields ν_1 = -2 μ_1 N(E*) and ν_2 = -μ_1 N(E*), which are unequal. Thus the printed equality ν_{r_{s-1}+j} = ν_{(r-r_{s-1}-j)} is false, and the later Corollary 5.9 relies on the asymmetry between ν_k and ν_{r-k}. The statement of Theorem 5.5 must be corrected before the main result can be evaluated.
[Section 5, Lemma 5.1 and proof] The proof of Lemma 5.1 contains the equality c1(L̃) = |Γ| c1(L̃), which can hold only if c1(L̃) = 0. This is not a harmless typo: the surrounding argument confuses the class of the single fiber eπ^*O_Y(y) with the pullback class (p∘eπ)^*O_X(x), whose numerical class is |Γ| times the single-fiber class. The same ambiguity appears in Section 4 in the identities ep^*L = L̃ and ep^*(L') = L̃'. Since Theorem 5.2 and later results depend on Lemma 5.1, the lemma and the numerical relations in Section 3 need a clean statement with one consistent convention for L̃.
[Section 5, proof of Theorem 5.5] The proof of Theorem 5.5 is not an induction: only the cases l=1 and l=2 are written, and the text says the rest follows 'immediately by induction' without giving the induction step. Lemma 5.4 provides an isomorphism of entire Eff cones, but it does not compute the images of the specific generators in the general block, and the map g_k is defined using ep_*^{-1} on Eff cones, whose existence as an isomorphism is only cited from [5] and [6]. The general generator formula for all Harder–Narasimhan blocks is therefore not proved as written.
[Title and Abstract] The title and abstract advertise results on the fiber product of two parabolic projective bundles over a curve, but the manuscript contains no such theorem, section, or computation; all results concern a single projectivization P(E*). Either the promised product results should be added, or the title and abstract should be revised to describe only the projectivization of one parabolic bundle.
[Sections 3–6, rationality assumption] The orbifold correspondence used throughout is available only for rational parabolic weights, and Section 3 explicitly states 'fixed rational parabolic weights'. However, the main theorems (Theorems 4.3, 5.2, 5.5, and 6.3) are stated for an arbitrary parabolic vector bundle without repeating this hypothesis. As stated, these theorems do not cover parabolic bundles with irrational weights. The rational-weight hypothesis must be included in every main statement, or a separate argument must be given for irrational weights.

minor comments (4)

[Section 5, proof of Theorem 5.2] The equality (c1(ξ̃) - μ c1(L̃))^{r-k} = (c1(ξ̃) - μ̃ c1(L̃))^{r-k} is unexplained: if c1(L̃) is the single-fiber class then μ̃ = |Γ| μ makes the equality false, while if c1(L̃) is the pullback class then the notation should be fixed consistently.
[Section 5, Theorem 5.5 notation] The notation r_i and d_i is overloaded: the same symbols are used for the rank and degree of the quotient Q_i^* and for the cumulative rank and degree of (E/E_i)^*. Using different letters, such as ilde r_i and ilde d_i, would prevent confusion in the formula for ν_k.
[References] Reference [6] is dated 2007, but the cited Fulger–Lehmann paper in Algebraic Geometry 4 was published in 2017; please correct the year.
[Proposition 4.1 proof] In the proof of injectivity of ep^*, the phrase 'surjectivity of ep' should specify that ep is a surjective finite morphism, and the constant α in the projection-formula argument should be identified as the degree of the finite map; the argument is correct in spirit but the wording should be tightened.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the parabolic cone theorems are obtained by pushing forward independent external cone results on smooth orbifold projectivizations, not by assuming the target cones.

full rationale

The paper's derivation chain is not circular. The parabolic cone computations are reduced to the smooth orbifold projectivization P(\tilde E) via the Biswas orbifold correspondence ([2], Section 2.5) and the quotient construction of P(E*) ([4], Section 2.6); both are prior constructions with independent proofs and do not presuppose the nef or pseudoeffective cones of P(E*). The cone descriptions for P(\tilde E) are quoted from Fulger ([7]) and Fulger--Lehmann ([5], [6]), which are external results about ordinary projective bundles over smooth curves, not about the parabolic bundles whose cones are being derived. The push-forward step uses intersection formulas (Lemma 5.1) and known behaviour of effective cones under finite morphisms; no parameter is fitted to the cones that are then 'predicted'. The semistability criterion (Theorem 6.3) is a consequence of the computed cone equalities and of the strict inclusion for unstable bundles imported from [7, Lemma 3.2]; it is not used as an input. The self-citations [2] and [4] are load-bearing constructions, but they are not unverified assertions of the target results, so they do not constitute circularity. The rational-weight restriction stated in Section 3 and the sketchy induction in Theorem 5.5 (only l=1,2 written out) are completeness or correctness issues, not self-referential reasoning.

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

No free parameters are fitted: the parabolic weights and N(E*) are inputs from the parabolic structure, not ad hoc constants. The paper relies on the orbifold correspondence and on standard cone theorems for ordinary projective bundles. No new entities are postulated.

assumptions (6)

domain assumption All parabolic weights are rational numbers
The inverse orbifold correspondence in Section 2.5 requires all parabolic weights to be rational; the paper fixes rational weights in Section 3.
domain assumption Orbifold correspondence between parabolic bundles and orbifold bundles (Biswas 1997)
The paper uses the equivalence from [2] as the bridge to compute cones; see Section 2.5 and Propositions 2.6 and 2.7.
standard math Fulger's description of effective cones on ordinary projective bundles over curves
Eff_k(P(eE)) formulas in Theorems 5.2 and 5.5 are taken from [7].
standard math Miyaoka's nef cone theorem for projective bundles over curves
Used in Theorem 4.3 to identify Nef(P(eE)).
standard math Fulger-Lehmann isomorphism results for pushforwards of pseudoeffective cones under finite morphisms
Used to transfer Eff_k(P(eE)) to Eff_k(P(E*)) in Theorems 5.2 and 5.5 and Lemma 5.4.
standard math Kawamata covering lemma
Produces the Galois cover Y to X and the orbifold bundle eE; cited in Section 2.5.

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Pith. "Pith review of Positive Cones of the Projectivization of a parabolic vector bundle and Their Products over a Curve." pith.science (2026). https://pith.science/paper/QEAAY2DQ

@misc{pith2026250617594,
  author       = {Pith},
  title        = {Pith review of: Positive Cones of the Projectivization of a parabolic vector bundle and Their Products over a Curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QEAAY2DQ}},
  note         = {Machine review of arXiv:2506.17594}
}

read the original abstract

We compute the positive cones of the projectivization of a parabolic vector bundle and the fiber product of two parabolic projective bundles over a smooth complex projective curve. Specifically, we determine their N\'eron--Severi groups and compute their nef and pseudoeffective cones. Moreover, for the projectivization of a parabolic vector bundle, we explicitly describe the generators of the higher nef and pseudoeffective cones. As an application, we obtain a necessary and sufficient criterion for the semistability of a parabolic vector bundle.

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

Positive Cones of Parabolic Grassmann Bundle over a curve
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Defines parabolic Grassmann bundles and computes their Neron-Severi group and positivity cones over curves.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [5]

Fulger and B

M. Fulger and B. Lehmann, Morphisms and faces of pseudo-effective cones,Proc. Lond. Math. Soc. 112(2016), 651–676

work page 2016

[2] [6]

Fulger and B

M. Fulger and B. Lehmann, Positive cones of dual cycle classes,Alg. Geom.4(2007), 1–28

work page 2007

[3] [1]

Balaji, I

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

work page 2003

[4] [2]

Biswas, Parabolic bundles as orbifold bundles,Duke Math

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

work page 1997

[5] [3]

Biswas, Connections on a parabolic principal bundle,Canadian Jour

I. Biswas, Connections on a parabolic principal bundle,Canadian Jour. Math.58(2006), 262–281

work page 2006

[6] [4]

Biswas and F

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

work page 2011

[7] [7]

Fulger, The cones of effective cycles on projective bundles over curves,Math

M. Fulger, The cones of effective cycles on projective bundles over curves,Math. Zeit.269(2011), 449–459

work page 2011

[8] [8]

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W. Fulton,Intersection theory, Second edition, Springer-Verlag, Berlin, 1998

work page 1998