{"id":"a8e895f6-3fb5-41cc-8381-4c1eae4b3ca3","arxiv_id":"2509.17857","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The quantum cohomology of the smooth Schubert divisor X_{w0 s_{n-1}} in Fl_n is presented, with a quantum Chevalley formula and quantum Schubert polynomials identical to those of Fl_n.","lead":"This paper computes the quantum cohomology ring and quantum multiplication rules for a smooth hypersurface inside the complete flag variety Fl_n, called a Schubert divisor. It is one of the first systematic treatments of quantum Schubert calculus for a non-homogeneous Schubert variety.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 4.11's vanishing proof relies on an unproved codimension-one statement about the image of \\hat\\pi; without it, Theorems 1.3 and 1.6 lack support for the exceptional degrees.","rationale":"I read the paper in good faith. The algebraic framework in Sections 3 and 5 is coherent: the Givental/Kim presentation for Fl_n is standard, and the transition-equation proof of Theorem 1.6 is formally sound once the quantum Monk-Chevalley formula is available. The main risk is exactly where the reader located it: the proof of Proposition 4.11. That proposition is the only place the exceptional degree classes d = alpha_{in} + alpha_{n-1,n} are shown to produce no quantum contributions, and the argument contains an unproved dimension-drop assertion. The rest of the vanishing argument in Proposition 4.10 is a standard curve-neighborhood count and is not the weak point. My proposed test directly verifies the disputed dimension count in the first nontrivial case; it would either support the proof's codimension-one claim or show that the proof as written cannot be trusted. Since this concern does not move the reader's conditional verdict, I mark the recommended verdict as UNCHANGED: conditional acceptance with a request to supply a rigorous proof or a cited reference for the dimension-drop step in Prop. 4.11 is appropriate.","tokens_in":28189,"tokens_out":24679,"duration_ms":190579,"concrete_test":"Check the dimension claim in the smallest nontrivial case: n=4, i=2, d=(0,1,2), P=F\\ell_{1,2;4}, Y={F_1 subset V_2}, e=pi_*d. Compute dim im(\\hat\\pi) by analyzing the incidence I={(g,p): g in M_{0,2}(P,e), p in im g cap Y}: over a liftable g, lifts are parameterized by a choice of p in Y on g(C) together with the choice of a degree-one rational curve in the P^1-fiber over p attached at that point (including node position and automorphisms). If dim im(\\hat\\pi) = dim M_{0,2}(P,e), the asserted 'reduces one more dimension' inequality in Prop. 4.11 is false and the proof fails; if dim im(\\hat\\pi) = dim M_{0,2}(P,e)-1, the codimension-one claim is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The vanishing of all Gromov-Witten invariants for degrees d with d_{n-1} >= 2 is load-bearing for the quantum Monk-Chevalley formula (Theorem 1.3) and hence for Theorem 1.6. Proposition 4.10 handles all such degrees except d = alpha_{in} + alpha_{n-1,n}; Proposition 4.11 is the only place those exceptional degrees are killed. In case i < n-1, the proof asserts that the induced morphism \\hat\\pi: M_{0,2}(X,d) -> M_{0,2}(P, pi_*d) 'cannot be surjective, since any stable map in the image has an extra constraint that C intersects with Y,' and immediately concludes dim im(\\hat\\pi) <= dim M_{0,2}(P, pi_*d) - 1. This step is not proved. Non-surjectivity alone gives a proper closed image only if the target is irreducible, and the statement that 'C intersects Y' is a codimension-one condition is not established; Y has codimension 2 in P, so this requires a real incidence-count argument. The subsequent dimension estimate, and the equality case via [BCMP13, Lemma 3.8], both depend on this one-codimension drop. If the image of \\hat\\pi has full dimension, the contradiction disappears and no vanishing is obtained for these degrees, leaving the truncation in Theorem 4.2 incomplete. The theorem may still be true by another argument, but as written the proof has a load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops quantum Schubert calculus for the smooth Schubert divisor X = X_{w_0 s_{n-1}} of the complete flag variety Fl_n, defined by the condition F^1 ⊂ V^{n-1}. The main results are: a Borel-type presentation of QH^*(X) as a quotient by the deformed elementary symmetric relations (Theorem 1.2); a quantum Monk-Chevalley formula for products with divisor classes (Theorem 1.3); the statement that the Fomin-Gelfand-Postnikov quantum Schubert polynomials represent the pullback Schubert classes of X under this presentation (Theorem 1.6); and an explicit quantum Lefschetz-type ring homomorphism from QH^*(Fl_n) to QH^*(X) (Theorem 1.7). The presentation is derived from Givental's J-function and the quantum Lefschetz theorem, the quantum Chevalley formula from a mix of low-degree moduli computations and curve-neighborhood vanishing, and the Schubert-polynomial statement from a transition-equation induction.","tokens_in":28541,"tokens_out":6381,"duration_ms":50617,"significance":"If the proofs are fully valid, the paper would provide the first complete quantum Schubert calculus for a smooth Schubert divisor of the complete flag variety: a ring presentation, a quantum Chevalley rule, and quantum Giambelli-type polynomials. The results are concrete and falsifiable, and the techniques — Givental's J-function, the quantum Lefschetz theorem, curve neighborhoods, and transition equations — are standard and well matched to the problem. A notable strength is that the ring presentation is derived geometrically rather than fitted, and Theorem 1.3 gives an explicit correction term whose geometric origin is explained. The paper also spells out a surprising failure of functoriality for the naive pullback and a corrected quantum Lefschetz map. However, a central vanishing statement in Section 4 is not proved at the required level of detail, and this gap is load-bearing for the main quantum Chevalley formula.","major_comments":[{"comment":"The proof of the vanishing for d = α_{in} + α_{n-1,n} hinges on the assertion that the induced map \\hat\\pi : M_{0,2}(X,d) → M_{0,2}(P,π_*d) 'cannot be surjective, since any stable map in the image has an extra constraint that C intersects with Y', and on the resulting one-codimension drop dim im(\\hat\\pi) ≤ dim M_{0,2}(P,π_*d) - 1. This step is not proved. Non-surjectivity alone does not give a proper closed image unless the target is irreducible, and the statement that 'C intersects Y' is a codimension-one condition is not established; Y has codimension 2 in P, so a genuine incidence-count argument is needed. The subsequent dimension estimate for ev_*[M_{0,2}(X,d)]^{vir}, and the equality case via [BCMP13, Lemma 3.8], both depend on this one-codimension drop. If the image of \\hat\\pi has full dimension, the contradiction disappears and no vanishing is obtained for these exceptional degree","section":"§4.2.2, Prop. 4.11, case i < n-1"},{"comment":"Even if the asserted non-surjectivity is granted, the final dimension contradiction assumes that the equality cases can be identified precisely: that ev_P restricts to a finite-degree morphism Z_1 → Z_2, that Z_1 is a codimension-one component of im(\\hat\\pi), and that Z_2 is a component of im(ev_P) contained in Y×Y. The citation to 'the proof of [BCMP13, Lemma 3.8]' is used to describe the fiber of the projection of Z_2 to Y_1 as the curve neighborhood Γ^P_{π_*d}(y), of dimension ℓ(z^P_{π_*d}) = ℓ(t_{in}) - 1. This identification requires that the relevant components of the moduli space and the evaluation images are ordinary cycles of the asserted dimension, which is not justified independently of the missing codimension-one statement. Because the proof of Proposition 4.11 is the only support for the 'specific degrees' vanishing, this should be made into a complete geometric argument.","section":"§4.2.2, Prop. 4.11, equality case"}],"minor_comments":[{"comment":"There are several typographical errors: 'gezus-0' should be 'genus-0', 'tiangular' should be 'triangular', 'polynimals' should be 'polynomials', and 'contract' should be 'contrast'.","section":"§1, Introduction"},{"comment":"In the displayed computation, the notation appears as Fℓ3: 'Z_{[M_{0,3}(Fℓ3, ι∗d′)]}' should presumably be Fℓ_n. Please correct the subscript to avoid confusion.","section":"§4.1.2, Proof of Prop. 4.5"},{"comment":"The phrase 'C intersects with Y' is ambiguous: it should say explicitly that the image f(C) meets the subvariety Y (or the appropriate incidence condition on the stable map), and not the domain curve as an abstract curve.","section":"§4.2.2, Prop. 4.11"},{"comment":"The notation 'deg q_j' in Equation (2.13) is not formally defined; it is clear from context that it means the degree in the quantum grading, but a short sentence would help.","section":"§2.3"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The paper is squarely within the scope of the journal and the central results are plausible and well-motivated. The only blocking issue is the unproved geometric assertion in Proposition 4.11, which is load-bearing for Theorems 1.3 and 1.6; I expect this can be repaired with a precise incidence argument, but it is not a purely cosmetic gap. The references to the authors' own in-preparation work [LRY] are used only as motivation and do not create circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this is the first paper to work out a quantum Schubert calculus for a smooth Schubert divisor of a flag variety, and most of it is solid. Second, the proof of the key vanishing result (Prop 4.11) has a hand-wavy step that is load-bearing; I would want that fixed before fully trusting Theorems 1.3 and 1.6.\n\nWhat is genuinely new is the Borel-type presentation of QH*(X) via Givental's J-function and the quantum Lefschetz theorem, the quantum Monk-Chevalley formula with its surprising minus sign, and the identification of quantum Schubert polynomials with the FGP polynomials. The transition-equation machinery in Section 5 is elegant: it reduces Theorem 1.6 to a small part of the quantum Chevalley formula. The paper is also honest about where the predicted presentation came from [LRY], which is fine.\n\nThe soft spot is Proposition 4.11. For d = alpha_{in} + alpha_{n-1,n} with i < n-1, the authors need to show all Gromov-Witten invariants vanish. They assert that the map hat(pi): M_{0,2}(X,d) -> M_{0,2}(P, pi_*d) cannot be surjective because stable maps in the image have an extra constraint that the curve intersects Y. That assertion is not proved. Non-surjectivity alone does not give the required one-dimensional drop unless the target is irreducible and the condition is of codimension one; Y has codimension 2 in P, so an incidence-count argument is needed. The subsequent dimension estimate and the use of [BCMP13, Lemma 3.8] both depend on that drop. If the image has full dimension, the contradiction disappears and no vanishing is obtained for these degrees, leaving the truncation in Theorem 4.2 incomplete. The claim may well be true, but as written it is a gap.\n\nMinor point: Theorem 1.7 cites [GI] for the quantum Lefschetz interpretation, but the proof of the theorem is self-contained, so that is not a real issue.\n\nWho is this for? People working on quantum cohomology of Schubert varieties and flag varieties. It is a solid contribution that opens a new subfield. I would send it to a serious referee. My own verdict: conditional, pending a rigorous proof of Prop 4.11.","headline":"First real quantum Schubert calculus for a smooth Schubert divisor of Fl_n; the paper deserves refereeing, but the vanishing proof for exceptional degrees has a load-bearing gap that needs closing.","tokens_in":29068,"tokens_out":4122,"would_cite":true,"duration_ms":187158,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14M15","05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper gives the first complete quantum Schubert calculus for a smooth Schubert divisor of the complete flag variety, including a ring presentation, a quantum Chevalley formula, and matching quantum Schubert polynomials.","keywords":["quantum Schubert calculus","Schubert divisors","quantum cohomology","complete flag variety","Gromov-Witten invariants","Borel presentation","quantum Monk-Chevalley formula","quantum Schubert polynomials"],"falsifier":"For $n=4$, compute the quantum product $\\xi_{s_1} \\star \\xi_{s_2}$ in the ring of Theorem 1.2; if the coefficient of $q_1 q_2 q_3^2$ (degree $(1,1,2)$) is nonzero, the geometric vanishing of Proposition 4.11 fails, and the quantum Chevalley formula would need correction.","tokens_in":28071,"feed_emoji":"🧮","tokens_out":11126,"duration_ms":74608,"temperature":0.7,"texified_at":"2026-08-05T20:30:53.846912+00:00","pith_summary":"The paper builds a quantum Schubert calculus for the smooth Schubert divisor $X$ of the complete flag variety $\\operatorname{Fl}_n$ defined by the incidence $F_1 \\subset V_{n-1}$. It proves a Borel-type presentation for the quantum cohomology ring $QH^*(X,\\mathbb{Z})$ using deformed elementary symmetric polynomials, derives a quantum Monk–Chevalley formula that describes all products of Schubert divisor classes, and shows that the quantum Schubert polynomials of $X$ are exactly those of $\\operatorname{Fl}_n$. A quantum Lefschetz homomorphism from $QH^*(\\operatorname{Fl}_n)$ to $QH^*(X)$ is also constructed. If correct, this is the first complete quantum Schubert calculus for a smooth non-homogeneous Schubert variety.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":4332,"prompt_tokens":772,"completion_tokens":3560,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":772,"completion_tokens_details":{"reasoning_tokens":2804}},"feed_headline":"First complete quantum Schubert calculus for a Schubert divisor","feed_subtitle":"Ring presentation and Chevalley formula give explicit quantum product rules for a smooth Schubert divisor.","key_machinery":"The key objects are the matrices $M_{\\operatorname{Fl}_n}$ and $M_X$, whose characteristic polynomials define the quantum relations $E^n_i$ and $\\hat{E}^n_i$; these quantize the elementary symmetric polynomials and give the Borel presentation. The proof of the quantum Chevalley formula uses the geometry of two-pointed stable maps $M_{0,2}(X,d)$, comparing it with $M_{0,2}(\\operatorname{Fl}_n,d)$. For degrees with $d_{n-1} \\ge 2$ the curve-neighborhood method is used to show that most Gromov–Witten invariants vanish; the remaining 'two-chain' degrees are handled by a dimension count on a related moduli space. The transition equation for quantum Schubert polynomials then drives an induction that establishes the polynomial identity.","core_discovery":"The central claim is that $QH^*(X,\\mathbb{Z})$ is a deformation of $H^*(X,\\mathbb{Z})$ with a canonical presentation as $\\mathbb{Z}[x_1,\\ldots,x_n,q_1,\\ldots,q_{n-1}]/(\\hat{E}^n_1,\\ldots,\\hat{E}^n_{n-1},E^{n-1}_{n-1})$, where the $\\hat{E}^n_i$ are quantizations of elementary symmetric polynomials built from a tridiagonal matrix. The quantum product is governed by a Monk–Chevalley formula whose terms come directly from Gromov–Witten invariants; a distinctive minus sign in the formula corrects for nodal stable curves. The paper shows that for all Bruhat-constrained permutations, the quantum Schubert class $\\xi_w$ is represented by the same quantum Schubert polynomial $S^q_w$ as its counterpart in $\\operatorname{Fl}_n$, even though the multiplication laws diff","pith_inferences":["The geometric vanishing argument for degrees with d_{n-1} ≥ 2 likely extends to other smooth Schubert divisors of flag varieties, though the dimension count may need refinement for more complicated curve neighborhoods.","The quantum Lefschetz homomorphism might be a special case of a general quantum hyperplane-section map for any smooth hypersurface in a flag variety, giving a systematic construction of quantum cohomology for such divisors.","The authors' note that the equivariant version follows immediately suggests that the equivariant quantum Monk–Chevalley formula could be tested against known equivariant quantum Schubert calculus for Fl_n, providing an independent check."],"forward_implications":["The ring presentation gives an explicit finite presentation of QH^*(X,Z), reducing quantum product computations to polynomial arithmetic.","The quantum Monk–Chevalley formula determines all quantum structure constants of X in principle, since the divisor classes generate QH^*(X) as a Z[q]-algebra.","The quantum Schubert polynomials for X coincide with those of Fl_n, so the known combinatorial and positivity properties of those polynomials apply directly.","The quantum Lefschetz homomorphism exhibits a functoriality of quantum cohomology under the inclusion ι: X → Fl_n, which is generally false for arbitrary subvarieties.","This provides the first complete quantum Schubert calculus for a smooth Schubert divisor, opening the door to similar results for other smooth Schubert varieties."],"fun_headline_variants":["Quantum cohomology ring for Schubert divisors presented","Chevalley formula and ring presentation for Schubert divisors","Quantum Schubert polynomials match full flag variety","First explicit quantum cohomology for a Schubert divisor"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof that all quantum corrections in degrees with $d_{n-1} \\ge 2$ vanish depends on the claim that a certain map between moduli spaces of stable maps is not surjective; if that claim fails, the quantum product would acquire extra high-degree terms.","fun_headline_variants_meta":{"raw":{"variants":["Quantum cohomology ring for Schubert divisors presented","Chevalley formula and ring presentation for Schubert divisors","Quantum Schubert polynomials match full flag variety","First explicit quantum cohomology for a Schubert divisor"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000775,"raw_usage":{"total_tokens":3208,"prompt_tokens":630,"completion_tokens":2578,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":374,"completion_tokens_details":{"reasoning_tokens":2515}},"tokens_in":374,"tokens_out":2578,"duration_ms":16722,"temperature":1.0,"reasoning_tokens":2515,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T15:49:00.146300+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $n=4$, compute the quantum product $\\xi_{s_1} \\star \\xi_{s_2}$ in the ring of Theorem 1.2; if the coefficient of $q_1 q_2 q_3^2$ (degree $(1,1,2)$) is nonzero, the geometric vanishing of Proposition 4.11 fails, and the quantum Chevalley formula would need correction.","supporting_citations":[],"review_version":1}