{"id":"a82fa2bb-b9cb-45cd-b034-72b54aac5dbe","arxiv_id":"2506.17594","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The nef and pseudoeffective cones of the projectivization of a parabolic vector bundle over a curve are determined by the parabolic Harder-Narasimhan slopes, yielding a semistability criterion.","lead":"This paper computes the positivity cones (nef and pseudoeffective) for the projectivization of a parabolic vector bundle over a complex curve, using a correspondence with orbifold bundles. It also gives a cone-theoretic criterion for parabolic semistability.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.5 is not proved: the HN induction is sketched only for l=1,2, and the printed equality ν_k=ν_{r−k} is false for semistable rank-3 bundles.","rationale":"The reader's verdict of CONDITIONAL is well justified, but the primary obstacle is not (only) the rationality of weights. The most load-bearing defect is that the main theorem's proof is absent for l≥3 and its printed statement contains a false equality ν_k=ν_{r−k}, which fails in the simplest nontrivial semistable case. This is a correctness risk, not merely a presentation issue. The rational-weight restriction is real and should be stated in the theorems, but it is a scope limitation; the unproved induction and the garbled ν formula directly undermine the central claim and its application Theorem 6.3. A corrected version that states the theorem with rational weights, removes the false equality, and supplies the full induction would likely be valid, hence CONDITIONAL rather than REJECT. I partially agree with the reader: the rational weights were identified as the weakest assumption, but I find the incomplete/garbled proof of Theorem 5.5 more load-bearing.","tokens_in":15914,"tokens_out":46452,"duration_ms":444721,"concrete_test":"Recompute ν_1 and ν_2 for a semistable rank-3 parabolic bundle with slope µ ≠ 0, once from the displayed formula in Theorem 5.5 (which forces ν_1 = ν_2) and once from the expansion of (ξ − µN(E*)L)^{r−k} used in Theorem 5.2. If the two values differ, the equality clause in Theorem 5.5 is false; then repair the statement by deleting '=ν_{(r−r_{s−1}−j)}' and separately verify the l=3 induction step, applying Lemma 5.4 twice to check that the recursive pushforward reproduces the claimed generator with coefficient (jµ_3 − d_2)N(E*).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim, Theorem 5.5, asserts the generators of Eff_k(P(E*)) for every rank r and every Harder–Narasimhan step. Its proof is 'immediately by induction' and only the cases l=1 and l=2 are written out. The induction step would need Lemma 5.4, whose proof defines g_k = ep_*∘f_k∘ep_*^{-1}; this requires ep_* to be an isomorphism of Eff cones for the possibly singular quotient P(E*) ← P(\\tilde E), but the paper only cites surjectivity-type results, never proves the inverse map. Even before that, the statement itself is internally inconsistent: it reads ν_{r_{s-1}+j} = ν_{(r−r_{s−1}−j)} := (jµ_s − d_{s−1})N(E*), i.e. ν_k = ν_{r−k}. For a semistable rank-3 bundle (l=1, µ_1=µ, d=3µ, N=N(E*)), the cone formula of Theorem 5.2 gives Eff_1 generated by (ξ−µN L)^2, so ν_1 = −2µN, and Eff_2 generated by (ξ−µN L)^1, so ν_2 = −µN. These are unequal for µ≠0. Thus, as printed, Theorem 5.5 cannot be correct; the equality clause must be a typo, but the intended asymmetry between ν_k and ν_{r−k} is exactly what controls the duality between Eff_k and Eff^{r−k} (Corollary 5.9). The paper therefore needs a corrected statement and a complete proof of the induction, not just the two illustrative cases.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16248,"tokens_out":11078,"duration_ms":114478,"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":[{"comment":"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":"Section 5, Theorem 5.5"},{"comment":"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":"Section 5, Lemma 5.1 and proof"},{"comment":"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.","section":"Section 5, proof of Theorem 5.5"},{"comment":"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.","section":"Title and Abstract"},{"comment":"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.","section":"Sections 3–6, rationality assumption"}],"minor_comments":[{"comment":"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":"Section 5, proof of Theorem 5.2"},{"comment":"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 \tilde r_i and \tilde d_i, would prevent confusion in the formula for ν_k.","section":"Section 5, Theorem 5.5 notation"},{"comment":"Reference [6] is dated 2007, but the cited Fulger–Lehmann paper in Algebraic Geometry 4 was published in 2017; please correct the year.","section":"References"},{"comment":"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.","section":"Proposition 4.1 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and contains a plausible and potentially useful approach, but the main theorem is not yet proved as printed. The false equality in Theorem 5.5 and the inconsistency in Lemma 5.1 are likely repairable, and the missing induction step may be fillable, but the authors need to supply a complete argument, not just the l=1,2 cases. The absence of the advertised fiber-product results is also a substantive mismatch between the abstract and the content."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere 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.\n\nBut 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.\n\nThe 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.\n\nWorse, 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.\n\nLemma 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.\n\nOne 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.\n\nOverall: 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.\n\nMy call: accept for peer review, but flag the missing proof and the inconsistent statement.\n\nBest","headline":"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.","tokens_in":16834,"tokens_out":11189,"would_cite":false,"duration_ms":99683,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H60","14L24","14F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["parabolic vector bundle","projectivization","nef cone","pseudoeffective cone","Harder-Narasimhan filtration","orbifold bundle","semistability","positive cones"],"falsifier":"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.","tokens_in":15693,"feed_emoji":"📐","tokens_out":13285,"duration_ms":122830,"temperature":0.7,"pith_summary":"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.","feed_headline":"Semistable parabolic bundles: all positive cones agree","feed_subtitle":"Explicit generators for every nef and pseudoeffective cone turn semistability into a cone equality.","key_machinery":"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 $\\nu_k$.","core_discovery":"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 $k$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the parabolic-bundle/orbifold-bundle correspondence and the equivalence of semistability, which lets the argument pass to a smooth Galois cover.","marker":"[2]"},{"why":"Constructs the projectivization of a parabolic bundle and its tautological line bundle, the object whose cones are computed.","marker":"[4]"},{"why":"Supplies the generator description for pseudoeffective cones of effective cycles on a smooth projective bundle over a curve, applied to the orbifold bundle.","marker":"[7]"},{"why":"Gives the nef cone of an ordinary projective bundle over a curve, the model boundary case used in the parabolic nef cone computation.","marker":"[14]"},{"why":"Provides the pushforward results for pseudoeffective cones used to transfer cone generators from the cover to the quotient.","marker":"[5]"},{"why":"Provides the dual-cycle formalism and the behavior of dual pseudoeffective and nef cones under pushforward, used in the dual statements.","marker":"[6]"},{"why":"Supplies the covering lemma used to produce the Galois cover on which the orbifold bundle lives; this is where the rational-weight assumption enters.","marker":"[10]"}],"fun_headline_variants":["Parabolic semistability: all positive cones coincide","Explicit generators for higher nef and pseudoeffective cones","Cone equality criterion for parabolic vector bundle stability","Simplicial cones reveal parabolic stability via cone equality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Parabolic semistability: all positive cones coincide","Explicit generators for higher nef and pseudoeffective cones","Cone equality criterion for parabolic vector bundle stability","Simplicial cones reveal parabolic stability via cone equality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000362,"raw_usage":{"total_tokens":1929,"prompt_tokens":898,"completion_tokens":1031,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":966}},"tokens_in":514,"tokens_out":1031,"duration_ms":10893,"temperature":1.0,"reasoning_tokens":966,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:08:12.951058+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Biswas, Parabolic bundles as orbifold bundles,Duke Math","cited_arxiv_id":null,"evidence_quote":"Establishes the parabolic-bundle/orbifold-bundle correspondence and the equivalence of semistability, which lets the argument pass to a smooth Galois cover."},{"cited_title":"Biswas and F","cited_arxiv_id":null,"evidence_quote":"Constructs the projectivization of a parabolic bundle and its tautological line bundle, the object whose cones are computed."},{"cited_title":"Fulger, The cones of effective cycles on projective bundles over curves,Math","cited_arxiv_id":null,"evidence_quote":"Supplies the generator description for pseudoeffective cones of effective cycles on a smooth projective bundle over a curve, applied to the orbifold bundle."},{"cited_title":"Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic Geometry, Sendai, 1985,Adv","cited_arxiv_id":null,"evidence_quote":"Gives the nef cone of an ordinary projective bundle over a curve, the model boundary case used in the parabolic nef cone computation."},{"cited_title":"Fulger and B","cited_arxiv_id":null,"evidence_quote":"Provides the pushforward results for pseudoeffective cones used to transfer cone generators from the cover to the quotient."},{"cited_title":"Fulger and B","cited_arxiv_id":null,"evidence_quote":"Provides the dual-cycle formalism and the behavior of dual pseudoeffective and nef cones under pushforward, used in the dual statements."},{"cited_title":"Kawamata, K","cited_arxiv_id":null,"evidence_quote":"Supplies the covering lemma used to produce the Galois cover on which the orbifold bundle lives; this is where the rational-weight assumption enters."}],"review_version":2}