{"id":"22b46925-00d8-4859-a4a4-f6b35b375416","arxiv_id":"2607.18593","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Smooth Schubert varieties in rational homogeneous spaces are homologically rigid when marked roots are long; for subdiagram Schubert varieties the paper gives the full rigid/non-rigid list and Schur rigidity except projective-space fiber bundles.","lead":"This paper proves that all smooth Schubert varieties in rational homogeneous spaces are homologically rigid when the Dynkin diagram is marked only at long roots, and gives the complete list of rigid and non-rigid cases for subdiagram Schubert varieties. It also proves Schur rigidity for such varieties in the long-root case unless they fiber over projective space.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.10's induction does not control rigidity of Picard-one bases X_K(v), and its ρ=1 base case misuses Theorem 2.6; Theorem 1.2's proof is incomplete as written.","rationale":"The Reader's weakest assumption was the completeness of the exceptional-pair classification imported from Hong-Mok and Hong-Kwon (Proposition 3.4). That is indeed load-bearing for Theorem 1.3 and for the no-exceptionality claims in Theorem 1.2. However, a second-pass reading of Proposition 3.10 shows a more immediate internal gap: even granting Proposition 3.4, the induction proof does not justify rigidity of the intermediate bases X_K(v). The proof's base case for ρ=1 appears to require a Billey-Postnikov decomposition that Theorem 2.6 only guarantees for Picard number at least 2, and the ρ≥2 step uses rigidity of the base X_K(v) without it being part of the tower hypothesis. Since Theorem 1.2 is proved by invoking Proposition 3.10, this gap affects the central claim. It is plausible that the long-root hypothesis repairs the gap, because then every Picard-one base is automatically non-exceptional by Theorem 1.1 and a secondary induction on ambient Picard number can be written out. But the paper does not provide that repair, so the verdict should be conditional: the main theorem is likely true, but the proof as written needs an additional argument or a restricted statement of Proposition 3.10. I therefore adjust the Reader's ACCEPT to CONDITIONAL, with the concrete audit above as the test that would settle whether the gap is formal or merely expository.","tokens_in":27137,"tokens_out":31913,"duration_ms":345863,"concrete_test":"Run the proof of Proposition 3.10 on Example 3.13 with K'=J∪{α_{k-1}}: list every pair appearing in the tower, including the base X_{K'}(v), and check whether the 'base case ρ=1' applies to it. Then, under the long-root hypothesis of Theorem 1.2, audit each BP step used in the induction: verify explicitly that each Picard-one base X_K(v) is non-exceptional by Theorem 1.1 when G/P^K has Picard number one, and supply the secondary induction when it has higher Picard number. If this audit fails at any step, Theorem 1.2 is not established by the given proof.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Proposition 3.10 assumes only that the central fibers of the Billey-Postnikov tower are non-exceptional. The induction step for ρ≥2 then requires moving the image family N_t back to X_K(v) by the 'base case ρ=1'; this is a rigidity statement for the Picard-one Schubert variety X_K(v) in G/P^K, but the hypothesis says nothing about X_K(v), which is a base rather than a fiber. The ρ=1 base case is also unjustified: it invokes Theorem 2.6 to produce a root α with a Billey-Postnikov decomposition, but the existence part of Theorem 2.6 ('Moreover') requires the Schubert variety to have Picard number at least 2. For ρ=1 the argument would amount to proving every Picard-one smooth Schubert variety is rigid, contradicting the exceptional pairs listed in Proposition 3.4. Example 3.13 shows the gap concretely: the chosen good tower has non-exceptional fibers, but its base X_{K'}(v) is itself an exceptional pair from Proposition 3.4(2), so the step 'h(t)·N_t = X_K(v)' is not justified. In the long-root setting of Theorem 1.2 this particular obstruction may disappear because Theorem 1.1 makes every Picard-one base non-exceptional, but the paper never states or verifies such a base condition. Thus the proof of the central theorem is incomplete as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies homological and Schur rigidity for smooth Schubert varieties in rational homogeneous spaces of higher Picard number. The main result, Theorem 1.2, asserts that if all marked roots are long — in particular for G of type ADE — then every smooth Schubert variety is homologically rigid. Theorem 1.3 gives a complete list, for Schubert varieties of subdiagram type, of the cases where homological rigidity fails. Theorem 1.7 extends this to Schur rigidity under a long-root condition and an exclusion of fiber bundles over projective space. The proofs are built on the Billey–Postnikov decomposition of smooth Schubert varieties and on the known Picard-number-one rigidity theorems of Hong–Mok and Hong–Kwon, with an induction on Picard number.","tokens_in":27506,"tokens_out":16427,"duration_ms":179080,"significance":"If the main theorem is correct, it is a substantial extension of the rigidity theory of Schubert varieties from Picard-number-one homogeneous spaces to arbitrary Picard numbers, and the classification in Theorem 1.3 is a valuable complete answer for the subdiagram case. The paper explicitly builds on external classifications and does not appear to be circular; the main ingredients are cited from Hong–Mok, Hong–Kwon, and Richmond–Slofstra. However, the proof of the central induction, Proposition 3.10, has a load-bearing gap concerning Picard-one bases of the Billey–Postnikov tower. The gap is likely repairable in the long-root setting of Theorem 1.2, but as written the proof of the main theorem is incomplete.","major_comments":[{"comment":"The proof is incomplete. In the induction step, after producing the family N_t = π(M_t), the proof asserts: “From the base case ρ=1 of the induction, there exists h(t)·N_t = X_K(v)”. This is a rigidity statement for the Picard-one Schubert variety X_K(v) in G/P_K. The hypothesis of Proposition 3.10 only excludes exceptional pairs among the fibers of the tower; it says nothing about the base X_K(v). The ρ=1 base case is not proved: the argument invokes Theorem 2.6 for the existence of α, but that existence statement is stated only for Picard number at least 2, and a Picard-one Schubert variety appearing as a base need not be rigid. Example 3.13 is a concrete instance: after taking K' = J∪{α_{k-1}}, the base X_{K'}(v) is an exceptional pair of Proposition 3.4(2) (type C_n, marked root α_k), hence is not rigid in the Picard-one sense. Thus the step moving N_t back to X_K(v) is unjustified.","section":"Section 3.2, Proposition 3.10"},{"comment":"The proof of Theorem 1.2 checks only that the fibers in the Billey–Postnikov tower are non-exceptional. To run the induction in Proposition 3.10, one must also know that every Picard-one Schubert variety that occurs as a base of the tower is rigid in its ambient space. In the long-root setting this follows from Theorem 1.1, but the paper never states or verifies this base condition. Since Theorem 1.1 does supply the missing rigidity when all marked roots are long, I believe the statement of Theorem 1.2 is likely correct, but the proof as written is not.","section":"End of Section 3.2, proof of Theorem 1.2"},{"comment":"The induction hypothesis is applied to the fiber X_J(u) in P_K/P_J, but the hypotheses of Proposition 3.10 are stated for the original tower associated with X_J(w). One must justify that the restricted tower for X_J(u) again satisfies the no-exceptional-pairs condition. This is probably true by restricting the original tower to the fiber, but it is not stated, and the proof depends on it.","section":"Section 3.2, induction on the fiber X_J(u)"}],"minor_comments":[{"comment":"There are numerous typographical errors: “homogenous” for “homogeneous”, “subdigram” for “subdiagram”, “clousure”, “veriety”, “denot”, “irreucible”, and “the the” in Example 2.8. These should be corrected.","section":"Throughout"},{"comment":"The notation R_1 and R_2 is potentially confusing because the root system is also denoted R. Consider using different letters, e.g. A_1 and A_2, or explicitly saying R_1, R_2 are subsets of the set of simple roots.","section":"Theorem 1.3"},{"comment":"The notation F• for a flag is informal; define it, e.g., F• = (F_1, F_2).","section":"Section 2.6, Example 2.8"},{"comment":"The proofs of Lemmas 3.15–3.17 refer to marked diagrams and product structures; these arguments are very terse. In particular, the transition from “exceptional pairs appear in every Billey–Postnikov decomposition” to “rigidity holds” in Lemma 3.16 needs more detail, even if it is repairable.","section":"Section 3.3"},{"comment":"Figure 1, “Billey-Postnikov decomposition,” appears to be missing from the text; there is only a caption. Either include the figure or delete the caption.","section":"Figure 1 and Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the main results are plausibly correct and significant. The major issue is the proof of Proposition 3.10, which is the engine for Theorem 1.2 and the subdiagram classification; the missing base-rigidity condition needs to be added and verified. In the long-root setting the repair seems straightforward using Theorem 1.1, so I do not recommend rejection. I also suggest that the authors re-examine the case-by-case arguments in Section 3.3 once the induction is fixed, since those arguments rely on Proposition 3.10."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real extension of the Hong-Mok/Hong-Kwon rigidity results to higher Picard numbers. The long-root homological rigidity theorem and the subdiagram classification look right, and the Schur rigidity theorem is a solid addition. The Billey-Postnikov induction is a sensible new tool for this problem. But the paper has a genuine presentation gap in Proposition 3.10 and a wrong example (3.13) that a referee needs to catch.\n\nWhat is new: Theorem 1.2 for long-root marks, the complete list in Theorem 1.3, and the Schur rigidity theorem 1.7. These are not just corollaries of the Picard-one results; the induction over BP towers is doing real work. The reliance on Richmond-Slofstra for the existence of BP decompositions is explicit and appropriate.\n\nWhere the soft spots are. First, the base case ρ=1 of Proposition 3.10 cites Theorem 2.6 for the existence of a root α with a BP decomposition, but Theorem 2.6's existence statement requires Picard number at least two. The tower assumption should provide that root, so the argument is repairable, but as written it is a mis-citation.\n\nMore important: in the induction step, after moving the image family N_t back to X_K(v), the proof invokes the ρ=1 base case to control X_K(v). The hypothesis only mentions fibers. The natural reading of 'all pairs in the fibers of the iterated fiber bundle' does cover X_K(v), because in the next step of the tower X_K(v) is itself a fiber (when its Picard number is one). But the paper never states this, and Example 3.13 actively confuses the issue: it claims a tower with K'=J∪{α_{k-1}} has no exceptional pair, but the base X_{K'}(v) is an exceptional pair of type C_n by Proposition 3.4(2). That base becomes the fiber in the second step, so the tower does not satisfy the hypothesis. The example should be corrected, and the proof should spell out why the base is controlled.\n\nFor Theorem 1.2 specifically, the long-root hypothesis saves the day: every Picard-one base is non-exceptional by Theorem 1.1. So I think the main results are right. But the paper needs a careful revision of Proposition 3.10 and Example 3.13 before I'd trust it as written.\n\nBottom line: yes, send it to peer review. It deserves a serious referee, and the referee should be asked to focus on the induction logic and the case-by-case B_n/C_n/F_4 arguments, which are terse.","headline":"Solid extension of the rigidity program, but Proposition 3.10 has a real gap in how it treats Picard-one bases, and Example 3.13 is wrong as written.","tokens_in":27949,"tokens_out":21853,"would_cite":true,"duration_ms":211711,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","32G10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that smooth Schubert varieties in rational homogeneous spaces are homologically rigid whenever all marked Dynkin roots are long roots, covering all ADE types, and supplies a complete rigidity classification for subdiagram-t","keywords":["Schubert varieties","homological rigidity","Schur rigidity","rational homogeneous spaces","Billey–Postnikov decomposition","marked Dynkin diagrams","exceptional pairs","subdiagram type"],"falsifier":"Exhibit a smooth Schubert variety in a type A or D rational homogeneous space of Picard number at least two together with a subvariety in the same homology class that is not a group translate. The paper's Theorem 1.2 predicts that no such pair exists.","tokens_in":27052,"feed_emoji":"📐","tokens_out":7833,"duration_ms":85035,"temperature":0.7,"pith_summary":"Schubert varieties are the basic cycle classes of rational homogeneous spaces—flag varieties and their relatives—and the question is whether a Schubert variety is the only subvariety in its homology class up to the action of the automorphism group ('homological rigidity'), and whether multiples of its class force sums of translates ('Schur rigidity'). The paper proves that in any rational homogeneous space whose marked Dynkin roots are all long roots, every smooth Schubert variety is homologically rigid; because all roots are long in types A, D, and E, this settles the simply-laced cases completely. For Schubert varieties of subdiagram type, it also gives the complete list of non-rigid cases, all of which involve short marked roots in types B, C, or F. In the long-root subdiagram setting, Schur rigidity holds unless the variety is a fiber bundle over projective space. The method is an induction along the Billey-Postnikov decomposition, a tower of fiber bundles reducing the problem to single-marked-root cases.","feed_headline":"Smooth Schubert varieties in ADE spaces are homologically rigid","feed_subtitle":"Long roots force any subvariety in the same homology class to be a translate—and the rare exceptions are classified.","key_machinery":"The central mechanism is the Billey-Postnikov decomposition: a smooth Schubert variety with Picard number at least two is an iterated Zariski-locally trivial fiber bundle whose fibers are smooth Schubert varieties in single-marked-root rational homogeneous spaces. The proof pushes rigidity down this tower. At each step the key object is an 'exceptional pair'—a single-marked-root Schubert variety admitting a local deformation not coming from the group action—and the induction succeeds exactly when no fiber contains one.","core_discovery":"The paper establishes that on a rational homogeneous space X=G/P, represented by a marked Dynkin diagram, a smooth Schubert variety X_0 is homologically rigid whenever all marked roots are long roots: any subvariety with the same homology class must be g·X_0 for some automorphism g. Since all roots are long in types A, D, and E, every smooth Schubert variety in those spaces is homologically rigid, with no homogeneity assumption on X_0. For Schubert varieties of subdiagram type, the paper gives a complete classification: homological rigidity holds except for explicit configurations in F_4, B_n, and C_n that all involve short marked roots. It further proves Schur rigidity in the long-root subd","pith_inferences":["The same induction would likely extend to singular Schubert varieties if a Billey-Postnikov tower exists for them; smoothness enters through the exceptional-pair classification rather than through the induction itself.","The non-rigid examples in types B_n and C_n are products containing a linear factor with a larger automorphism group; this suggests that in short-root cases the general failure mechanism is again linear-subspace deformations, so a full short-root classification might reduce to tracking maximal linear subspaces.","Theorem 1.7's 'not a fiber bundle over projective space' condition is probably sharp in the long-root subdiagram setting: the proof uses it exactly at the Schur step, so any projective-bundle case should admit explicit counterexamples to multiple-class rigidity.","Because the proof is combinatorial once the Billey-Postnikov tower is known, the rigidity of any given smooth Schubert variety can in principle be verified by a finite check on Dynkin subdiagrams, independent of the ambient group."],"forward_implications":["In all type A, D, E rational homogeneous spaces, homological rigidity holds for every smooth Schubert variety without requiring homogeneity.","The complete exception list for subdiagram-type Schubert varieties identifies exactly which F_4, B_n, and C_n configurations can deform, so rigidity can be read off the marked Dynkin diagram.","In long-root cases, Schur rigidity fails only in the presence of a fiber-bundle structure over projective space; outside that geometric obstruction, multiple homology classes have only the obvious sum-of-translates representatives.","The Billey-Postnikov criterion gives a local-to-global test: checking rigidity of any smooth Schubert variety reduces to checking finitely many single-marked-root fibers.","The results cover non-homogeneous smooth Schubert varieties, not just group orbits, extending the scope of known rigidity statements."],"fun_headline_variants":["Long roots force homological rigidity in Schubert varieties","ADE Schubert varieties: homological rigidity guaranteed","Smooth Schubert varieties rigid in long-root spaces","Homological rigidity: long roots seal Schubert varieties","Smooth Schubert varieties are rigid for long roots"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof rests on a previously compiled list of all exceptional pairs in single-marked-root spaces; if that list is incomplete, the induction over the fiber-bundle tower could miss a non-rigid Schubert variety.","fun_headline_variants_meta":{"raw":{"variants":["Long roots force homological rigidity in Schubert varieties","ADE Schubert varieties: homological rigidity guaranteed","Smooth Schubert varieties rigid in long-root spaces","Homological rigidity: long roots seal Schubert varieties","Smooth Schubert varieties are rigid for long roots"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002021,"raw_usage":{"total_tokens":7756,"prompt_tokens":826,"completion_tokens":6930,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":6855}},"tokens_in":570,"tokens_out":6930,"duration_ms":45535,"temperature":1.0,"reasoning_tokens":6855,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T14:56:29.219033+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a smooth Schubert variety in a type A or D rational homogeneous space of Picard number at least two together with a subvariety in the same homology class that is not a group translate. The paper's Theorem 1.2 predicts that no such pair exists.","supporting_citations":[],"review_version":1}