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Residual Intersections and Schubert Varieties

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.13481 v1 pith:KKQV4VIC submitted 2024-11-20 math.AG math.ACmath.RT

Sara Angela Filippini , Xianglong Ni , Jacinta Torres , Jerzy Weyman This is my paper · ORCID
classification math.AGmath.ACmath.RT MSC 13C9913H10
keywords residualintersectionsSchubertvarietiesoppositePlückercoordinateslinkageminusculerepresentationscrystalgraphsfreeresolutions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $\mathfrak{g}_j$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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].

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

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.

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 (2)
  1. [§4, proof of Theorem 1.2(2)–(3)] 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.
  2. [§4.1, proof of Theorem 1.2(1)] 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.
minor comments (6)
  1. [§5.3.2] 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.
  2. [§4.1] The sentence 'We have k = xx_{c-2}' appears to contain a typo; it should presumably read k = x_{c-2}.
  3. [Introduction and Figure 2] 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.
  4. [§5.3.1] The list of data for E6/P1 ends with 'z_2 =' and no value is given; please complete the definition.
  5. [§5.2.4] 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.
  6. [§4, proof of Theorem 1.2(2)–(3)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the residual intersection formulas are derived from independent Schubert-variety and crystal data; the main proof gap is an unproved vanishing statement, not a circular one.

full rationale

The paper's central claim, Theorem 1.2, is not taken as an input. The Schubert varieties X_{y_l}, X_{z_m} are defined independently via Bruhat closures in G/P_k, and the residual intersection formulas are derived using Remark 2.3 together with Theorem 3.3 and an identification of the relevant opposite Demazure modules from crystal/Levi data. No fitted parameter is renamed as a prediction, and no defining equation of the Schubert varieties is chosen to be the colon formula itself. The proof of parts (2) and (3) does rely on an asserted vanishing-coordinate statement: 'Out of these, only p*_∅,...,p*_u,...p*_{y_1}...p*_{y_{l-1}} vanish on X_{y_l}', which is not derived in the text and is exactly the load-bearing missing lemma for intermediate nodes. That is a correctness gap or an omitted proof, not circularity, because the Schubert variety X_{y_l} is not defined by that vanishing set. The self-citations, chiefly [FTW23] in Section 5.3, are not load-bearing: the exceptional-type E6/E7 residual intersection identities are re-verified by explicit inline Macaulay2 computations, and the uniform argument in Section 4 does not depend on those citations. Thus the derivation chain is not equivalent to its own inputs, and no significant circularity is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numbers are fitted. The proof rests on standard theorems (Krull altitude, linkage, Lakshmibai-Littelmann, crystal graphs, Borel-Weil-Bott) and on the explicit ADE setup of an extremal or minuscule vertex. There are no invented entities.

assumptions (7)
  • standard math Krull's altitude theorem and standard linkage facts, including the definition and basic properties of residual intersections.
    Used in the proof of Theorem 1.2 to bound ht(J) by the minimal number of generators of the union ideal and to recognize an ideal quotient as a residual intersection (Remark 2.3).
  • domain assumption Lakshmibai-Littelmann theorem [LL03, Theorem 16]: opposite Schubert varieties and their unions are defined scheme-theoretically by linear Plücker coordinates together with the quadrics of G/P.
    Cited as Theorem 3.2 and used to identify generators of the ideals I(X_{y1}) and I(X_{z1}).
  • standard math Standard monomial theory for Schubert varieties, as in [LL03].
    Used in the proof of Theorem 3.3 to show that the ideal of a union of two opposite Schubert varieties is generated by its degree one elements.
  • standard math Crystal graph theory and Levi branching for representations of simply-laced Lie algebras, as in [KN94].
    Used in the proof of Theorem 1.2(1) to identify the embedded graph G_k inside B(ω_k) and to locate the coordinates vanishing on X_{y1} and X_{z1}.
  • standard math Borel-Weil-Bott theorem and the geometric syzygy method.
    Used in Proposition 5.6 to construct free resolutions over Sym(Λ^2 V* ⊕ U ⊗ V*).
  • standard math Kustin-Ulrich Theorem 10.2 on residual intersections of Pfaffian ideals.
    Used in Section 5.2.3 and 5.2.4 to connect the Schubert variety ideals to the Kustin-Ulrich complexes.
  • domain assumption Setup restriction: D is simply-laced, k is an extremal or minuscule vertex, and the coordinates are extremal Plücker coordinates arranged as the T-shaped graph T_{c-1,d+1,t+1} with 1/(c-1)+1/(d+1)+1/(t+1) ≥ 1.
    The domain of the theorem. This is stated in the introduction and not derived.

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Cite this review

Pith. "Pith review of Residual Intersections and Schubert Varieties." pith.science (2026). https://pith.science/paper/KKQV4VIC

@misc{pith2026241113481,
  author       = {Pith},
  title        = {Pith review of: Residual Intersections and Schubert Varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKQV4VIC}},
  note         = {Machine review of arXiv:2411.13481}
}
read the original abstract

Inspired by the work of Ulrich and Huneke-Ulrich, we describe a pattern to show that the ideals of certain opposite embedded Schubert varieties defined by this pattern arise by taking residual intersections of two geometrically linked opposite Schubert varieties. This pattern is uniform for the ADE types. Some of the free resolutions of the Schubert varieties in question are important for the structure of finite free resolutions. Our proof is representation theoretical and uniform for our pattern, however it is possible to derive our results using case-by-case analysis and the aid of a computer.

Figures

Figures reproduced from arXiv: 2411.13481 by the authors.

Figure 1
Figure 1. The graph Gk. xc−2 . . . x1 u y1 . . . yd z1 . . . zt [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The graph Tc−1,d+1,t+1. We will label the vertices of Gk as defined by the picture in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Bruhat graph for W/WP2 for A5 Proposition 5.2. Let j < n and I = (pin : 1 ≤ i < n) Kj = (pin : j ≤ i < n) Ij = (pst : j ≤ s < t ≤ n). Then Kj : I = Ij . Proof. The Pl¨ucker relation pstpin = psiptn − psnptn implies Kj : I ⊇ Ij . The same relation implies that a given dual Pl¨ucker coordinate pst belongs to the colon ideal Kj : I if and only if pst ∈ Ij . 5.2. Residual intersections of Pfaffian ideals and Schubert Va… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Crystal graph for the half-spin representation [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The graph G6 for E6. R = QQ[y_0,y_12,y_13,y_14,y_15,y_23,y_24,y_25,y_34,y_35,y_45,y_1234,y_1235,y_1245,y_1345,y_2345] zb_1 = y_15*y_1234 - y_14*y_1235 +y_13*y_1245 - y_12*y_1345 zb_2 = y_25*y_1234 - y_24*y_1235 +y_23*y_1245 - y_12*y_2345 zb_3 = y_35*y_1234 -y_34*y_1235…
Figure 6
Figure 6. Figure 6: The graph G7 for E7 R = QQ[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11,x_12,x_13,x_14, x_15,x_16,x_17,x_18,x_19, x_20,x_21,x_22,x_23,x_24,x_25,x_26,x_27] Qe7e6 = x_13*x_14*x_15-x_11*x_15*x_16-x_12*x_13*x_17+x_10*x_16*x_17+x_9*x_15*x_18-x_8*x_17*x_18+ x_11*x_12*x_19-x…

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