{"id":"167eb155-e141-437f-8faa-9a499b244804","arxiv_id":"2411.13481","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For ADE types with an extremal or minuscule vertex, each opposite Schubert variety on one arm of the graph G_k has defining ideal given by a residual intersection of the linked variety on the other arm.","lead":"The authors show that the defining ideals of certain opposite Schubert varieties in ADE flag varieties can be written as residual intersections of two linked Schubert varieties, following a uniform graph pattern. This connects Schubert varieties to known constructions of free resolutions in commutative algebra and extends earlier results by Huneke and by Kustin and Ulrich to exceptional types.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2(2)–(3) relies on an unproved vanishing-coordinate statement for the intermediate Schubert varieties X_{y_l}, X_{z_m}; the §4.1 Levi/crystal argument only treats the endpoint nodes y1,z1.","rationale":"The reader's weakest assumption identifies the asserted Demazure/crystal identification for j = y_1,z_1. Our stress test sharpens this: the gap propagates to all intermediate nodes y_l,z_m through the sentence in §4 that asserts their vanishing coordinates. That assertion is needed for parts (2) and (3) of Theorem 1.2 and is not derived from part (1). The theorem is supported by explicit type A, E6, and E7 computations, and the Levi-branching strategy is plausible, but as written the uniform proof is incomplete at this step. We therefore keep the CONDITIONAL verdict rather than moving to reject or unverified. Agreement is partial because the reader focused on the endpoint nodes whereas the load-bearing issue is that the same kind of identification is silently used for every arm node in the residual intersection formulas.","tokens_in":17721,"tokens_out":9395,"duration_ms":105192,"concrete_test":"Independently compute the kernel of the restriction map res_{y_l}: V(ω_k)^* → V(y_l ω_k)^{opp,*} for a non-type-A configuration covered by Theorem 1.2, e.g. D_5/P_5 or E_6/P_6, using the standard monomial basis from Lakshmibai–Littelmann [LL03, Theorem 16]. Then check whether the dual Plücker coordinates lying in this kernel and in the coordinate set of the graph G_k are exactly {p_∅,...,p_{y_{l-1}}} for every l; if an extra coordinate appears, the equality I(X_{y_l}) = (p_∅,...,p_{y_{l-1}}):I(X_{z_1}) fails, and if no extra coordinate appears for all l and all ADE configurations, the missing lemma is confirmed and Theorem 1.2(2)–(3) follows.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2(2) reduces, via Remark 2.3, to the statement that the intersection I(X_{z_1}) ∩ I(X_{y_l}) is generated by p_∅,...,p_{y_{l-1}}. The only justification in §4 is the sentence: \"Out of these, only p*_∅,...,p*_u,...p*_{y_1}...p*_{y_{l-1}} vanish on X_{y_l}.\" That sentence is not a consequence of part (1), which concerns only X_{y_1} and X_{z_1}; it is exactly the missing vanishing description for every intermediate node y_l, and symmetrically z_m. The uniform argument for part (1) considers the Levi subalgebra g_j only for j = y_1,z_1, identifies one irreducible component of res^g_{g_j} V(ω_k), and concludes that the vanishing coordinates on X_j are the opposite arm nodes. Even accepting this for y_1,z_1, it is not written out for y_l or z_m, where the relevant Demazure module is generated by p_{y_l}, not by p_{y_1}. Section 6 shows the delicacy: for non-minuscule starting vertices, extra non-extremal Plücker coordinates appear in the vanishing ideal (e.g. §6.1.2), so the extremal-coordinate description is not automatic from the embedding of G_k into the crystal. Since the colon formula in part (2) holds only if the common vanishing coordinates are exactly p_∅,...,p_{y_{l-1}}, this missing lemma is load-bearing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a uniform pattern, for simply-laced Dynkin diagrams D and extremal or minuscule vertices k, under which the defining ideals of certain opposite Schubert varieties in G/P_k are residual intersections of two geometrically linked opposite Schubert varieties. Theorem 1.2 states that the defining ideals of the endpoint varieties X_{y_1}, X_{z_1} are cut out by extremal Plücker coordinates on the arms of a T-shaped graph G_k, and that the intermediate varieties X_{y_l}, X_{z_m} arise as residual intersections, extending the Huneke and Kustin-Ulrich examples to ADE types. The proof is representation-theoretic and uniform, with case-by-case Macaulay2 computations in the exceptional minuscule cases and explicit formulas in type A and type D.","tokens_in":18064,"tokens_out":7890,"duration_ms":84198,"significance":"If the proof can be completed, the result is significant: it would place the Huneke and Kustin-Ulrich residual intersection examples into one uniform Schubert-theoretic framework valid for all ADE types, and it would give a uniform explanation of previously sporadic codimension-three and higher residual intersections. The paper also contains useful explicit material: the type A minor formulas in §5.1, the Pfaffian identities in §5.2, the Macaulay2 verifications in §5.3 for E6/P1 and E7/P7, and the geometric approach to the Kustin-Ulrich resolutions in §5.2.4. These computations are a strength. However, the uniform proof of Theorem 1.2 is not yet fully established because the proof of parts (2) and (3) relies on an unproved vanishing statement for intermediate Schubert varieties, and the proof of part (1) compresses the key crystal and Levi identification into assertions.","major_comments":[{"comment":"The reduction to Remark 2.3 requires proving that I(X_{z_1}) ∩ I(X_{y_l}) is generated by p_∅,...,p_{y_{l-1}}. The proof asserts only that 'Out of these, only p*_∅,...,p*_u,...,p*_{y_1},...,p*_{y_{l-1}} vanish on X_{y_l}' and gives no argument for this vanishing statement. Part (1), which is invoked at this point, describes only the endpoint varieties X_{y_1} and X_{z_1}; it does not describe the defining ideal of X_{y_l} for l ≥ 2. Since the colon formula in part (2) holds exactly when the common vanishing coordinates are p_∅,...,p_{y_{l-1}}, this missing lemma is load-bearing. The same issue applies symmetrically to X_{z_m} in part (3). The material in Section 6 (e.g., §6.1.2 and §6.2.2) shows that non-extremal Plücker coordinates can appear in defining ideals of non-minuscule Schubert varieties, so the extremal-coordinate description cannot be treated as automatic from the embedding of G_k into the crystal graph.","section":"§4, proof of Theorem 1.2(2)–(3)"},{"comment":"The proof of the endpoint case is compressed to unproved identifications: the opposite Demazure module generated by p_{y_1} (resp. p_{z_1}) is identified with the module generated by the Lie subalgebra generated by b together with the Levi subalgebra g_j, and the irreducible component of res^g_{g_j} V(ω_k) corresponding to the opposite arm is asserted without derivation. The sentence explaining that g_j is a product of two type A algebras and that the component is an irreducible sl(r,C)-crystal does not by itself identify the component or rule out additional vanishing coordinates. A precise proof or a complete reference is needed before Theorem 1.2(1) is established. The authors' own remark that the identification fails when k is not extremal or minuscule, and their alternative offer of a computer-aided case-by-case proof, underscore that this is a substantive point rather than a formality.","section":"§4.1, proof of Theorem 1.2(1)"}],"minor_comments":[{"comment":"In the E7/P7 paragraph the notation is inconsistent: f_3 is set equal to p6* after f_2 was already assigned p6*, and f_7 and y_1234 are both assigned p1*. Please correct the labels so that the Macaulay2 computations are reproducible.","section":"§5.3.2"},{"comment":"The sentence 'We have k = xx_{c-2}' appears to contain a typo; it should presumably read k = x_{c-2}.","section":"§4.1"},{"comment":"The condition 1/(c-1) + 1/(d+1) + 1/(t+1) ≥ 1 is stated for the integers c,d,t but is not used afterward and appears inconsistent with a finite Y-shaped diagram; please clarify its role or omit it.","section":"Introduction and Figure 2"},{"comment":"The list of data for E6/P1 ends with 'z_2 =' and no value is given; please complete the definition.","section":"§5.3.1"},{"comment":"The complexes in Proposition 5.6 are presented without explicit differentials and without a precise statement of which parts are quoted from [KU92] and which are new; since the differentials are the main content of a free resolution, please indicate the source more carefully.","section":"§5.2.4"},{"comment":"The sentence 'Now, by Theorem 3.3 we know that I(X_{z_1}) ... is generated in degree 1 by the dual Plücker coordinates ...' misattributes the statement: the degree-one generation of the defining ideal of a single opposite Schubert variety is part (1) together with Theorem 3.2, not Theorem 3.3, which concerns unions.","section":"§4, proof of Theorem 1.2(2)–(3)"}],"recommendation":"major_revision","confidential_remarks":"The central idea is attractive and the explicit computations are a useful contribution, so I would not reject the paper. However, the uniform proof of Theorem 1.2 is not complete as written: the proof of parts (2) and (3) needs a vanishing statement for the intermediate Schubert varieties X_{y_l} and X_{z_m}, and the proof of part (1) needs a fuller justification of the crystal-Levi identification. These gaps are likely repairable within the scope of the paper, perhaps using standard monomial theory or a more detailed argument from [LL03], but they are load-bearing for the main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about arXiv:2411.13481. First, the main theorem is genuinely new: a uniform ADE statement that packages the Huneke determinantal and Kustin–Ulrich Pfaffian residual intersections as Schubert varieties, and adds E6/P1 and E7/P7 minuscule cases plus non-minuscule examples. That is useful for liaison theory and free resolutions. Second, the proof has a real gap: parts (2) and (3) do not follow from part (1) as written, because part (1) only identifies the vanishing coordinates for the endpoint Schubert varieties X_{y1} and X_{z1}. The proof of (2) needs the same description for every intermediate X_{y_l}—specifically that among the degree-one generators of I(X_{z1}), exactly p_∅,...,p_{y_{l-1}} vanish on X_{y_l}. That is asserted in one sentence in Section 4 and not derived. The §4.1 Levi/crystal argument treats only j = y1, z1, where the relevant component is an sl(r) crystal of highest weight ω1 or ωr; for y_l with l > 1 the analogous statement is not written out. Section 6 shows why this matters: for non-minuscule starting vertices, extra non-extremal Plücker coordinates do appear, so the extremal-coordinate description is not automatic.\n\nCredit where due: the paper is honest about this state of affairs. The abstract says a computer-aided case-by-case proof is possible, and Section 5 supplies explicit Macaulay2 checks for E6 and E7. The reductions via Remark 2.3 and Theorem 3.3 are coherent, and the explicit type A and D computations match known results. The self-citation to [FTW23] is appropriate for terminology and exceptional-type context. The main theorem is not circular; parts (2) and (3) are derived, and part (1) is attempted from crystal/Levi data. I do not see invented entities or parameter fitting.\n\nThe soft spots are proportionate: one load-bearing gap, plus typos (e.g., 'A_n' vs 'An', a garbled sentence in §4.1) and no standalone code. None of these destroy the result—the E6/E7 computations are persuasive—but the uniform proof needs either a proper lemma for intermediate nodes or an explicit statement that it is a conjecture with case-by-case verification.\n\nWho is this for? Commutative algebraists working on liaison and residual intersections, and people who care about equations of Schubert varieties. It deserves a serious referee: the statement is important enough, and the explicit computations are concrete evidence. My recommendation: send to peer review, but the referee should ask for the gap to be closed or clearly re-scoped.","headline":"A genuinely new uniform ADE statement with persuasive exceptional-type computations, but the proof of the residual intersection formula for intermediate Schubert varieties has a gap that needs fixing.","tokens_in":18626,"tokens_out":5025,"would_cite":true,"duration_ms":45496,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C99","13H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every simply-laced Dynkin diagram and every extremal or minuscule vertex, the defining ideals of a chain of opposite Schubert varieties are residual intersections computed by one colon-ideal formula from a linked Schubert variety.","keywords":["residual intersections","Schubert varieties","opposite Schubert varieties","Plücker coordinates","linkage","minuscule representations","crystal graphs","free resolutions"],"falsifier":"Take the least-explicit ADE case, for instance $E_8$ with an extremal vertex $k$, and compute the ideal $I(X_{y_1})$ in the coordinate ring of $G/P_k$; if it contains any Plücker coordinate beyond the extremal coordinates on the left arm of $G_k$, then the degree-one generation asserted in Theorem 1.2(1) fails, and the colon formula $I(X_{y_l})=(p_\\emptyset,\\ldots,p_{y_{l-1}}):I(X_{z_1})$ cannot hold. The paper's own non-minuscule examples show such extra coordinates do occur once the extremal or minuscule hypothesis is dropped, so the computation is a genuine test rather than a formality.","tokens_in":17504,"feed_emoji":"📐","tokens_out":13655,"duration_ms":138753,"temperature":0.7,"pith_summary":"The paper aims to prove one uniform statement behind a family of commutative-algebra examples: for a simply-laced Dynkin diagram $D$, an extremal or minuscule vertex $k$, and the T-shaped graph $G_k$ built from $D$, the defining ideal of the opposite Schubert variety $X_{y_l}$ at the $l$-th node of one arm is the residual intersection $I(X_{y_l})=(p_\\emptyset,\\ldots,p_{y_{l-1}}):I(X_{z_1})$, and the symmetric formula holds on the other arm. Here the $p_j$ are extremal Plücker coordinates attached to the nodes of $G_k$, and $I(X_{z_1})$ is the ideal of the first node on the opposite arm. These equalities turn a geometric link between two Schubert varieties into an algebraic colon computation, and they reproduce the classical maximal-minors and Pfaffian residual intersections as special cases. The interest is that a single representation-theoretic pattern, rather than a case-by-case list, describes all these defining ideals in the ADE types.","feed_headline":"One recipe gives Schubert ideals in all ADE types","feed_subtitle":"Opposite Schubert variety ideals all come from one colon formula, tying Pfaffian and minor examples together","key_machinery":"The T-shaped graph $G_k$ is the organizing object: its nodes are extremal Plücker coordinates $p_j$ in $V(\\omega_k)^*$, with $p_\\emptyset$ at the bottom and the two arms starting at $y_1,z_1$. The graph embeds in the crystal graph $B(\\omega_k)$ of the fundamental representation, so it records which coordinates vanish on which Schubert variety. The algebraic workhorse is the residual-intersection observation of Remark 2.3: if $I$ and $K$ are prime ideals with $I\\not\\subset K$ and $A=I\\cap K$, then $K=A:I$, and $K$ is an $s$-residual intersection whenever the generator count satisfies $\\mu(A)=s$. Theorems 3.2 and 3.3 supply the degree-one generation of ideals of opposite Schubert varieties and their unions, reducing the proof to checking that the common vanishing coordinates are exactly the nodes before the target on the opposite arm. The paper's uniform representation-theoretic input is the assertion that for $j=y_1,z_1$ the opposite Demazure module generated by $p_j$ is the module generated by the Borel subalgebra together with the Levi subalgebra $\\mathfrak{g}_j$.","core_discovery":"Theorem 1.2 is the central claim. If $D$ is simply-laced and $k$ is extremal or minuscule, with $c$ the codimension of $X_{y_1}$ and $d,t$ the arm lengths of $G_k$, then for every $1\\le l\\le d$ and $1\\le m\\le t$ the scheme-theoretic defining ideals satisfy $I(X_{y_l})=(p_\\emptyset,\\ldots,p_{y_{l-1}}):I(X_{z_1})$ and $I(X_{z_m})=(p_\\emptyset,\\ldots,p_{z_{m-1}}):I(X_{y_1})$. The proof uses the fact that the union $X_{z_1}\\cup X_{y_l}$ has its ideal generated in degree one by exactly the coordinates $p_\\emptyset,\\ldots,p_{y_{l-1}}$, so Remark 2.3 applies: with $I=I(X_{z_1})$, $K=I(X_{y_l})$, and $A=(p_\\emptyset,\\ldots,p_{y_{l-1}})$, the ideal $K$ is an $(l+c)$-residual intersection of $I$. Part (1) of the theorem, which identifies the degree-one generators of $I(X_{y_1})$ and $I(X_{z_1})$ as the extremal Plücker coordinates on the opposite arms, is the representation-theoretic step; it is carried by an identification of opposite Demazure modules with modules generated by a Borel subalgebra together with a Levi subalgebra, and the authors point to a computer-aided case-by-case proof as an alternative where that identification is delicate. In the minuscule cases the paper writes the equations explicitly as minors and Pfaffians, and in the codimension-three case the resulting free resolutions are those of the complexes of [KU92].","pith_inferences":["A natural extension is to minuscule weights outside the simply-laced case: for symplectic and orthogonal Grassmannians the Bruhat graph still coincides with the crystal graph, so the same T-shaped pattern may hold, but the Levi subalgebra argument would need replacement because the Levi pieces are no longer both of type A.","The paper's non-minuscule examples suggest that when $k$ is not extremal or minuscule, the extra vanishing coordinates are duals of additional irreducible components in the branching decomposition, so a generalized residual intersection formula with more generators in the ideal $A$ might still exist; this is an inference, not a claim of the paper.","If the uniform identification in Section 4.1 fails in some larger ADE type, the authors' computer-aided alternative would still certify the formula case by case; the result would then be a collection of true identities without a uniform explanation, and the open question would be whether a deeper mechanism produces them."],"forward_implications":["For every ADE type, all Schubert varieties sitting on the two arms of $G_k$ have their defining ideals given by one colon formula, so the whole chain is controlled by a single linked pair $X_{y_1}$, $X_{z_1}$.","The classical examples are special cases: in type A the formula is the residual intersection of maximal minors, and in the codimension-three spinor case it is the Pfaffian residual intersection behind the complexes of [KU92].","Because the formulas are scheme-theoretic and the union ideals are linear, the minimal free resolutions of these Schubert ideals can be studied uniformly; in the minuscule cases the paper writes explicit Betti tables and resolutions via vector-bundle pushdowns.","The linkage behind the formulas means the dual of the mapping cone of the resolution of one Schubert variety gives a resolution of the other, up to shift, so liaison theory transfers resolutions across the T-shaped graph.","For $c=3$ the paper's varieties coincide with the family for which the complexes of [KU92] were built, giving a Schubert-geometric interpretation of those resolutions."],"supporting_citations":[{"why":"Supplies the linkage framework and the initial theorem that the two first-arm Schubert varieties are linked by a regular sequence.","marker":"[Ulr90]"},{"why":"The motivating source for residual intersections and the pattern the paper generalizes.","marker":"[HU88]"},{"why":"Defines the Ulrich pairs of opposite Schubert varieties and provides the exceptional-type computations and notation used throughout.","marker":"[FTW23]"},{"why":"Provides the codimension-three residual intersection complexes that Theorem 1.2 recovers as Schubert varieties, and the theorem used in the Pfaffian proof.","marker":"[KU92]"},{"why":"Gives Theorems 3.2 and 3.3 on degree-one generation of ideals of Schubert varieties and their unions, the key step in the colon equality.","marker":"[LL03]"},{"why":"Contains the original maximal-minors residual intersection recovered in the type A case and the argument for one inclusion.","marker":"[Hun83]"},{"why":"Supplies the geometric method for computing syzygies used to describe free resolutions in the minuscule cases.","marker":"[Wey03]"},{"why":"Connects Schubert varieties with finite free resolutions of length three, the context for the c=3 case.","marker":"[SW21]"},{"why":"Provides the crystal graph description used to identify the T-shaped graph inside the crystal of the fundamental representation.","marker":"[KN94]"}],"fun_headline_variants":["One colon formula for Schubert ideals in ADE types","Schubert ideals from two linked varieties, one pattern","ADE Schubert ideals: all from residual intersections","Uniform residual intersections yield Schubert ideals","Schubert ideals via linked pairs: ADE uniform"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the set of Plücker coordinates vanishing on the first Schubert varieties $X_{y_1}$ and $X_{z_1}$ is exactly the set of extremal coordinates on the opposite arm of the T-shaped graph; the paper asserts this identification in Section 4.1 rather than deriving it, and notes that it fails when $k$ is not extremal or minuscule.","fun_headline_variants_meta":{"raw":{"variants":["One colon formula for Schubert ideals in ADE types","Schubert ideals from two linked varieties, one pattern","ADE Schubert ideals: all from residual intersections","Uniform residual intersections yield Schubert ideals","Schubert ideals via linked pairs: ADE uniform"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000672,"raw_usage":{"total_tokens":3097,"prompt_tokens":1016,"completion_tokens":2081,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":2006}},"tokens_in":632,"tokens_out":2081,"duration_ms":14908,"temperature":1.0,"reasoning_tokens":2006,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:23:00.488550+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the least-explicit ADE case, for instance $E_8$ with an extremal vertex $k$, and compute the ideal $I(X_{y_1})$ in the coordinate ring of $G/P_k$; if it contains any Plücker coordinate beyond the extremal coordinates on the left arm of $G_k$, then the degree-one generation asserted in Theorem 1.2(1) fails, and the colon formula $I(X_{y_l})=(p_\\emptyset,\\ldots,p_{y_{l-1}}):I(X_{z_1})$ cannot hold. The paper's own non-minuscule examples show such extra coordinates do occur once the extremal or minuscule hypothesis is dropped, so the computation is a genuine test rather than a formality.","supporting_citations":[],"review_version":1}