{"id":"4c723947-3826-4f8c-a1f7-32607aa22675","arxiv_id":"2509.03747","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In Grassmannians, dimension 3 and codimension 3 classes are realizable by irreducible subvarieties exactly when b²≥ac, and in G(2,n) classes are realizable over Q exactly when their coefficients form a log-concave sequence with no internal zeros.","lead":"This paper determines which cohomology classes of complex Grassmannians can be represented by irreducible subvarieties, giving complete answers in dimensions and codimensions 2 and 3 and for the Grassmannians G(2,n) and G(3,6). The results connect a classical problem in algebraic geometry to log-concavity and convex geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hodge index steps in Theorems 7.1 and 8.3 require the intersected classes to be limits of irreducible surfaces; effectiveness alone is insufficient, and the paper only justifies this in Theorem 6.1.","rationale":"The reader identified the pseudoeffectiveness step as the weakest assumption. I agree that Theorems 7.1 and 8.3 omit the justification that the Hodge index hypothesis holds, but I sharpen the issue: the required property is not mere pseudoeffectiveness but that the class is a limit of irreducible surfaces. Effective reducible surfaces can violate the Hodge index inequality, so the proof needs a Bertini-type or Debarre-type irreducibility argument. This is load-bearing because all three necessity proofs reduce to the same determinant inequality, and Theorems 7.1 and 8.3 are central to the paper's classification claims. The sufficiency constructions are explicit and credible, and Theorem 6.1 does state the needed limit property, so the issue is localized and potentially fixable. I therefore recommend keeping the conditional verdict rather than accepting outright.","tokens_in":25615,"tokens_out":38519,"duration_ms":394909,"concrete_test":"Settle the missing hypothesis for Theorem 7.1: for a nontrivial irreducible sixfold Y in G(3,6), take its proper transform X = f^{-1}(Y) in G(2,5)×G(3,6), and compute X ∩ (gΣ_{2,2} × G(3,6)) for a general translate gΣ_{2,2} of the Schubert surface in G(2,5). Use Macaulay2/Schubert2 with Y obtained by Theorem 4.2 from a smooth surface class ax^2+bxy+cy^2 with a,b,c>0, so the class has all coefficients positive. If the intersection is reducible and its Gram matrix with D=σ_1⊗1, E=1⊗σ_1 has positive determinant, the Hodge index argument in Theorem 7.1 is invalid as stated and needs an explicit limit-of-irreducible-surfaces argument. If the intersection is always irreducible (or the class is a limit of irreducible surfaces), the concern is purely a missing sentence.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The necessity directions of Theorems 6.1, 7.1, and 8.3 all use the Hodge index theorem to force b^2 >= ac. Theorem 6.1 explicitly states that [Z]·x is a limit of irreducible surfaces. Theorems 7.1 and 8.3 instead apply Hodge index directly to σ_{2,2}⊗1·[f^{-1}Y] and x^2·[f^{-1}Y]. Even granting that these classes are effective, effectiveness is not enough: the Hodge index inequality D^2E^2 <= (D·E)^2 can fail for a reducible surface. For example, in P^2 × P^2, let S = ({p}×P^2) ∪ (P^2×{q}), D = pr_1^*O(1), E = pr_2^*O(1). Then D^2·S = 1, E^2·S = 1, D·E·S = 0, so the Gram matrix has determinant 1 > 0, even though S is connected and D,E are nef. The paper does not show that the specific classes in Theorems 7.1 and 8.3 are limits of irreducible surfaces, so the Hodge index step is not justified as written. If the actual intersections are reducible in the way of the example, the necessity proof collapses; if they are limits of irreducible surfaces, the gap is only expository.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies which integral cohomology classes of Grassmannians G(k,n) can be represented by irreducible subvarieties. The main results are: (i) a complete classification in dimension 2 and codimension 2 (Theorem 5.1, 5.2); (ii) a classification in dimension 3 and codimension 3 for 3 ≤ k ≤ n−k, giving the condition a,b,c ≥ 0 and b² ≥ ac, with explicit boundary exceptions for G(3,6) and G(2,n) (Theorems 6.1, 7.1, 6.2, 7.2); (iii) a classification of all classes realizable over Q in G(2,n), namely log-concave sequences with no internal zeros (Theorem 9.1); and (iv) additional results for G(3,6) and G(4,8) (Theorems 8.2, 8.3, 9.2). The sufficiency proofs use incidence correspondences and birational cone constructions from products of projective spaces, together with known realizability results of Huh, Hong, and Coskun–Robles. The necessity proofs use the Hodge index theorem to derive the determinant inequality b² ≥ ac. Stabilization results in Section 3 reduce many statements to small Grassmannians.","tokens_in":25983,"tokens_out":16194,"duration_ms":177855,"significance":"If the results are correct, they represent substantial progress on a natural realizability problem for Grassmannian cohomology classes. The paper gives explicit, practically checkable criteria in several nontrivial dimensions and codimensions, and it identifies a clean obstruction from the Hodge-Riemann relations. The stabilization theorems (Corollaries 3.7 and 3.9) are useful and reduce the classification to a finite set of cases. The constructions via birational image of Grassmannian bundles over products of projective spaces are elegant and produce irreducibility in a wide range of cases. The reliance on previously published theorems of Huh, Hong, and Coskun–Robles is appropriate; I see no circularity in the main classification. The main weakness is that several Hodge-index necessity arguments are not fully justified as written, as detailed below.","major_comments":[{"comment":"The necessity argument applies the Hodge index theorem to the class S = σ_{2,2} ⊗ 1 · [f^{-1}(Y)]. This is only valid if S is a limit of irreducible surface classes (or otherwise satisfies the weak Lorentzian property). The text does not show this, and effectiveness alone is insufficient. For example, in P² × P², the reducible surface S = ({p}×P²) ∪ (P²×{q}) has D²·S = E²·S = 1 and D·E·S = 0 for D = pr₁*O(1), E = pr₂*O(1), so the Hodge index determinant is positive. The proof in Theorem 6.1 explicitly states that [Z]·x is a limit of irreducible surfaces, but Theorem 7.1 has no such justification for σ_{2,2}⊗1·[f^{-1}Y]. Since this step is the only proof of the inequality b² ≥ ac in the codimension-3 classification, it is load-bearing. Please add a proof, a reference, or an equivalent argument showing that S is a limit of irreducible surfaces (or replace the Hodge-index step with another","section":"Theorem 7.1 (and Section 7, paragraph after the pullback formulas)"},{"comment":"The same issue appears in the proof of Theorem 8.3: after forming x²·[f^{-1}(Y)], the text says 'The Hodge index inequality...' without explaining why this class is a limit of irreducible surfaces. Here the situation is more easily repaired than in Theorem 7.1, because x is the pullback of the hyperplane class from P⁴ and x² is a complete intersection of two basepoint-free divisors; intersecting an irreducible variety with general members of such a linear system yields an irreducible surface by Bertini. As written, however, the necessary justification is omitted. Please state it explicitly.","section":"Theorem 8.3"},{"comment":"The proof of Theorem 9.1 also applies the Hodge index theorem to the class x^{n−4}·[f^{-1}(Y)] without stating why this is a limit of irreducible surfaces (or, if the dimension is not 2, why the Hodge-Riemann form for the relevant cycle class is weakly Lorentzian). The same concern as in Theorems 7.1 and 8.3 applies. Since Theorem 9.1 is one of the main classification results, this step needs a clear justification. If the dimension count forces this class to be a surface for the relevant m, the argument can likely be repaired by selecting n−4 general hyperplanes and invoking Bertini, but the text should say so.","section":"Theorem 9.1 (inductive step)"}],"minor_comments":[{"comment":"In the last paragraph, the text reads 'By [Ho05], bσ_{3,1,1} is realizable...' but the class under discussion is bσ_{2,2}. Either the notation should be corrected or the intended Schubert class should be clarified.","section":"Theorem 9.2, proof"},{"comment":"The statement of part (2) does not specify the value of n. From the context it appears to be n = 5; please state this explicitly.","section":"Theorem 6.2(2)"},{"comment":"The theorem states that the coefficients are nonnegative rational numbers, but the induction in the proof repeatedly says 'nonnegative integers.' The statements should be harmonized.","section":"Theorem 9.1 and its proof"},{"comment":"There are several typographical and grammatical issues, e.g., 'all the classes that can be realizable' in the abstract, 'Propsition' in Theorem 7.2, and 'prooof' in Theorem 8.3. A careful proofreading pass is recommended.","section":"Throughout"},{"comment":"The use of the same notation mσ_r and mσ_{1^r} for exceptions in both codimension-r and dimension-r statements is confusing, especially with the convention that σ_λ := σ_{λ^c} in dimension statements. Please clarify the notation in these two propositions.","section":"Propositions 3.6 and 3.8"}],"recommendation":"major_revision","confidential_remarks":"The main technical gap is the unjustified Hodge-index step in Theorems 7.1 and 9.1 (and, to a lesser extent, 8.3). If the authors can supply a proof that the relevant intersection classes are limits of irreducible surfaces, or replace those steps with a valid weak-Lorentzian argument, the paper is likely acceptable. The scope and choice of examples are otherwise appropriate for the journal. I do not see a circularity problem; the external inputs are prior published results with independent standing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a worthwhile paper. It settles the realization problem for dimension/codimension 2 and 3 classes in Grassmannians, and gives the log-concavity characterization for G(2,n) over Q. The stabilization results (Cor 3.7, 3.9) look correct and useful. The constructions via cones over products of projective spaces are clean, and the paper honestly credits HHMWW25 for the dimension 2 overlap.\n\nThe main body of the proof is the necessity: Hodge index forces b^2 >= ac in Theorems 6.1, 7.1, 8.3, and 9.1. That step is where I have a real reservation. Hodge index applies to classes of irreducible surfaces, or limits of those. In Theorem 6.1 the paper says explicitly that [Z]·x is a limit of irreducible surfaces. In Theorems 7.1 and 8.3 it doesn't, it just applies Hodge index to σ_{2,2}⊗1·[f^{-1}Y] and x^2·[f^{-1}Y]. Effectiveness alone isn't enough — a reducible surface can have a positive intersection matrix (the P^2×P^2 example in the stress-test is correct). So the necessity direction is under-justified as written. I think it's fixable: in Theorem 8.3 the class is a complete intersection of divisors, so Bertini should give irreducible surfaces for general choices, and in Theorem 9.1 the same is likely true. But Theorem 7.1 uses σ_{2,2}⊗1, which is not a divisor, and it's not obvious without an argument that the class is a limit of irreducible surfaces. The authors should either prove that or find another way to get the Hodge index inequality.\n\nThere are also minor errors: in Theorem 9.2 proof, 'bσ_{3,1,1}' should be 'bσ_{2,2}'; in the proof of Theorem 9.1, the induction base cites Theorem 7.2 when it should cite Theorem 6.2. Nothing that affects the main results.\n\nThe paper does deserve a serious referee. The classifications are important, the constructions are explicit, and the gap, while real, looks repairable. I'd recommend conditional acceptance: ask the authors to justify the Hodge index step in Theorems 7.1, 8.3, and 9.1, and fix the small typos.","headline":"Main classifications look right, but the Hodge index necessity has a gap in Theorems 7.1 and 8.3 that needs an explicit limit-of-surfaces argument.","tokens_in":26452,"tokens_out":6677,"would_cite":true,"duration_ms":68591,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14C25","14C17"],"pacs":[],"model":"deepseek-v4-flash","headline":"A quadratic inequality b²≥ac exactly characterizes which dimension-3 and codimension-3 cohomology classes of a Grassmannian are realizable by irreducible subvarieties, with a short list of boundary exceptions.","keywords":["Grassmannian","Schubert classes","irreducible subvarieties","realizable classes","Hodge index theorem","log-concave sequences","cohomology classes","algebraic cycles"],"falsifier":"An irreducible subvariety of G(3,6) with class σ₃+σ₁,₁,₁ would violate b²≥ac (here b=0, ac=1) and disprove the classification. Conversely, if no such subvariety exists, one can still check the proof's key step directly by computing whether the class σ₂,₂⊗1·[f^{-1}Y] lies in the closure of the effective cone for an arbitrary irreducible sixfold Y—a concrete effective-cone membership calculation in the cohomology of G(2,5)×G(3,6).","tokens_in":25523,"feed_emoji":"🧮","tokens_out":8328,"duration_ms":82856,"temperature":0.7,"pith_summary":"This paper asks which cohomology classes of a Grassmannian G(k,n)—the space of k-dimensional linear subspaces of an n-dimensional vector space—can be represented by a single irreducible algebraic subvariety. In dimensions 2 and 3 and codimensions 2 and 3, it gives a complete answer: apart from a few boundary cases, the only condition beyond nonnegativity of the Schubert coefficients is a quadratic inequality, b²≥ac, between the three coefficients of the class. For the Grassmannians G(2,n), it classifies all classes that are realizable over the rationals: the coefficient sequence must be log-concave with no internal zeros. The result matters because it turns a hard geometric existence question into a checkable algebraic inequality, and it shows exactly where the Hodge index theorem—rather than merely numerical effectiveness—starts to govern which cycles are irreducible.","feed_headline":"One inequality decides which 3-cycles in Grassmannians are irreducible","feed_subtitle":"Dimension-3 and codimension-3 classes in G(k,n) are realizable exactly when b²≥ac, with a few boundary exceptions.","key_machinery":"The argument is carried by two mechanisms. First, an incidence-correspondence and cone construction (Propositions 4.1–4.6) builds irreducible subvarieties of G(k,n) from irreducible subvarieties of products of projective spaces or smaller Grassmannians, and computes their Schubert classes exactly; this is what realizes every class with b²≥ac. Second, the Hodge index theorem applied to classes like [Y]·x or σ₂,₂⊗1·[f^{-1}Y] forces the determinant inequality ac−b²≤0 on any realizable class, which is what makes the classification sharp. The boundary cases are settled by the known classification of multi-rigid Schubert classes, which says which positive multiples of a single Schubert class can b","core_discovery":"The central claim is a complete classification in the first nontrivial ranges. Write a dimension-3 or codimension-3 class in G(k,n) as ν=aσ₃+bσ₂,₁+cσ₁,₁,₁, with σλ the Schubert classes forming the standard basis of the cohomology ring. The paper proves that for 3≤k≤n−k, such a class is the class of an irreducible subvariety over Z if and only if a,b,c≥0 and b²≥ac, with precisely listed exceptions: when k=3 and n=6 the only pure classes that survive are σ₃ and σ₁,₁,₁ themselves, and when k=3 and n>6 the class σ₁,₁,₁ must still occur with coefficient 1 while σ₃ may occur with any positive coefficient. The necessity of b²≥ac is derived from the Hodge index theorem applied to limits of surfaces","pith_inferences":["One can test the paper's implicit pseudoeffectiveness step directly: in the two cases where it is not proved, compute whether the classes σ₂,₂⊗1·[f^{-1}Y] and x²·[f^{-1}Y] lie in the closure of the effective cone for every realizable Y; if a counterexample appears, the necessity direction would need a different proof even though the theorem may survive.","The G(2,n) log-concavity result suggests reading 'realizable over Q' as a convexity phenomenon; one could probe whether the same holds for the other cominuscule Grassmannians, where Schubert classes are indexed by other strict partitions.","The boundary rigidities vanish as soon as one passes to a larger Grassmannian: multiples of σ₁,₁,₁ that are not realizable in G(3,6) become realizable in G(3,n) or G(k,n) for k>3. This suggests that rigidity is a small-dimensional phenomenon and that the stabilizing Grassmannian is the right arena for the classification."],"forward_implications":["For every Grassmannian with 3≤k≤n−k, the dimension-3 and codimension-3 realizability problem is closed: a class either meets the inequality or it does not.","Rational realizability in G(2,n) is a convex-geometric condition—log-concavity with no internal zeros—so checking it is an algorithmically simple test.","Stabilization reduces all higher-dimension questions to a boundary Grassmannian: a codimension-r class is realizable over Q in G(k,n) iff it is realizable in G(r,2r), so classification problems for fixed r can be solved once and for all.","The same Hodge-index obstruction should reappear in every higher dimension; the paper's Question 1.7 makes precise the hope that Hodge-Riemann relations give all obstructions."],"supporting_citations":[{"why":"Defines the realizability-over-Z/Q terminology and independently proves the dimension-2/codimension-2 surface classification that this paper extends.","marker":"[HHMWW25]"},{"why":"Classifies which surface classes in P²×P² are realizable over Z; this is the input for the cone construction in the dimension-3 and codimension-3 classifications.","marker":"[Hu13]"},{"why":"Proves the log-concavity criterion for classes in products of projective spaces used in Theorem 9.1 to transfer to G(2,n).","marker":"[Hu12]"},{"why":"Classifies multi-rigid Schubert classes; determines which multiples of σ₃ and σ₁,₁,₁ are realizable and hence pins down the boundary exceptions.","marker":"[Ho05]"},{"why":"Classifies when positive multiples of Schubert classes are realizable; used to show pure multiples are realizable outside the rigid cases.","marker":"[CR13]"},{"why":"Supplies the fact that effective cycles in Grassmannians have nonnegative Schubert coefficient expansions, the starting necessary condition.","marker":"[Co18]"},{"why":"Connectivity theorem used in the reverse-inclusion and stabilization arguments to show an irreducible representative can be intersected with a Schubert variety to remain irreducible.","marker":"[De96]"},{"why":"Supplies the Littlewood–Richardson rule used for the pullback formulas and intersection-matrix computations in the Hodge-index steps.","marker":"[Co09]"}],"fun_headline_variants":["Cohomology classes that are realizable: full classification in low ranks","B²≥AC: the test for irreducible subvarieties in G(k,n)","All realizable classes in G(2,n) and G(3,6) now classified","Hodge index yields exact criterion for realizable Schubert classes","Grassmannians: which cycles come from irreducible subvarieties?"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the classes obtained by intersecting an irreducible representative with certain divisor classes are still limits of genuine surfaces; the paper proves this in the dimension-3 case but leaves it implicit elsewhere, and if such a class were not a limit of effective surfaces, the numerical inequality that rules out all other classes would not be forced.","fun_headline_variants_meta":{"raw":{"variants":["Cohomology classes that are realizable: full classification in low ranks","B²≥AC: the test for irreducible subvarieties in G(k,n)","All realizable classes in G(2,n) and G(3,6) now classified","Hodge index yields exact criterion for realizable Schubert classes","Grassmannians: which cycles come from irreducible subvarieties?"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000781,"raw_usage":{"total_tokens":3277,"prompt_tokens":725,"completion_tokens":2552,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":469,"completion_tokens_details":{"reasoning_tokens":2451}},"tokens_in":469,"tokens_out":2552,"duration_ms":16813,"temperature":1.0,"reasoning_tokens":2451,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:42:36.281386+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An irreducible subvariety of G(3,6) with class σ₃+σ₁,₁,₁ would violate b²≥ac (here b=0, ac=1) and disprove the classification. Conversely, if no such subvariety exists, one can still check the proof's key step directly by computing whether the class σ₂,₂⊗1·[f^{-1}Y] lies in the closure of the effective cone for an arbitrary irreducible sixfold Y—a concrete effective-cone membership calculation in the cohomology of G(2,5)×G(3,6).","supporting_citations":[],"review_version":1}