Points on Rational Normal Curves and the ABCT Variety

Daniele Agostini; Dawei Shen; Lakshmi Ramesh

REVIEW 3 major objections 5 minor 21 references

Points on Rational Normal Curves and the ABCT Variety

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The ABCT variety V(3,n) is a determinantal variety, and one recursion gives its full cohomology class.

desk verdict Solid and useful class formula, but the proof that V(3,n) equals the determinantal locus Z2(3,n) has a real gap in Theorem 3.5 that needs fixing before the main theorem is fully rigorous. read the letter →

arxiv 2412.12514 v1 pith:WXDC5FWT submitted 2024-12-17 math.AG math-phmath.COmath.MP

Daniele Agostini , Lakshmi Ramesh , Dawei Shen This is my paper · ORCID

classification math.AGmath-phmath.COmath.MP MSC 05E0514J8114N0514N1514N20

keywords ABCTvarietyrationalnormalcurveconfigurationsGrassmanniandeterminantalPorteousformulaEuleriannumbersscatteringequations

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 determine the full cohomology class of the ABCT variety $V(3,n)$ inside the Grassmannian $G(3,n)$ — the closure of the image of the 2-Veronese map on $G(2,n)$, i.e. the set of matrices whose columns lie on a common plane conic. The authors prove that $V(3,n)$ is a determinantal variety of the expected codimension and use Porteous' formula to derive a recursive formula for its class in Schur polynomials. The recursion yields every Plücker intersection number and the degree of $V(3,n)$, and it reproduces the Eulerian-number count that appears in scattering-equation computations from physics. Along the way they establish a general theorem that configuration spaces of points lying on a common divisor of a linear system are reduced and irreducible, extending a recent determinantal-variety result.

What carries the argument

The load-bearing object is the degeneracy locus $Z_2(3,n)\subseteq G(3,n)$: the rank-$\leq 5$ locus of the bundle morphism $\mathbb{C}^n\otimes \mathcal{O}\to S^2U^{\vee}$, where $U$ is the universal subbundle of the Grassmannian. This locus is shown to be reduced, irreducible, Cohen-Macaulay of codimension $n-5$, and scheme-theoretically equal to $V(3,n)$. The computation then runs through Porteous' formula: the Chern roots of $S^2U^{\vee}$ are $2\alpha_1,2\alpha_2,2\alpha_3,\alpha_1+\alpha_2,\alpha_1+\alpha_3,\alpha_2+\alpha_3$, so the class is the complete symmetric function $h_{n-5}$ evaluated at these roots, which expands as the recursively defined $f_{n-5}$. A supporting structural result, Theorem 1.2, says that for any linear system $V$ on a smooth variety such that a general divisor is reduced and irreducible, the configuration space $X_n(V)$ of $n$ points lying on a common divisor is reduced, irreducible, Cohen-Macaulay of expected codimension $n-\ell+1$.

What would settle it

For $n=6$, use a computer algebra system to check whether the ideal of maximal minors of the matrix defining $Z_2(3,6)$ is radical, e.g. by the Gröbner-basis criterion of Proposition 3.9; if a nilpotent element or embedded component appears, then $V(3,6)\neq Z_2(3,6)$ as schemes and the predicted class $4[s_1]$, of degree 168, is false. Equivalently, intersect $V(3,6)$ with eight general Schubert divisors and count points: the formula predicts 168.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is a closed recursion for the fundamental class. Define symmetric functions $f_m \in \Lambda_3$ by $f_0=1$, $f_1=4s_1$, $f_2=11s_2+6s_{1,1}$, and $f_m=2s_1\cdot f_{m-1}-(s_2+2s_{1,1})\cdot f_{m-2}+s_{2,1}\cdot f_{m-3}+2m\cdot s_m$ for $m\geq 3$. Then for every $n\geq 5$, $[V(3,n)] = [f_{n-5}]$ in $H^{2n-10}(G(3,n))$, where $[f]$ denotes the image of a symmetric function under the map $\Lambda_3 \to H^*(G(3,n))$ sending Schur polynomials to Schubert classes. The proof identifies $V(3,n)$ with $Z_2(3,n)$, the locus where the morphism $\mathbb{C}^n\otimes \mathcal{O} \to S^2U^{\vee}$ has rank at most 5; this locus is shown to be reduced, irreducible, Cohen-Macaulay, and of codimension $n-5$, so Porteous' formula applies. Porteous' formula, together with the second Jacobi-Trudi identity, rewrites the class as the complete symmetric function $h_{n-5}$ evaluated at the Chern roots $2\alpha_i$ and $\alpha_i+\alpha_j$ of $S^2U^{\vee}$, giving exactly $f_{n-5}(\alpha)$. The same setup yields the degree formula and the Eulerian-number coefficient.

Load-bearing premise

The whole computation depends on the inherited claim that the configuration spaces of points on a common divisor are reduced and irreducible for the linear systems used here; if that claim fails, the degeneracy locus could carry embedded components or excess dimension, and the Porteous class would not equal the class of $V(3,n)$.

Editorial extensions

If this is right

The degree of $V(3,n)$ under the Plücker embedding is the coefficient of $s_{n-3,n-3,n-3}$ in $f_{n-5}\cdot s_1^{2n-4}$; the first values are $5,168,4032,84744,1664091,31402800$ for $n=5,\ldots,10$.
The coefficient of $[s_{n-5}]$ in $[V(3,n)]$ is the Eulerian number $A(n-3,1)=2^{n-3}-(n-2)$, matching the scattering-equation prediction from spinor-helicity computations.
$V(3,n)$ is cut out scheme-theoretically by quartic Plücker equations of the form $p_{i_1i_2i_3}p_{i_1i_5i_6}p_{i_2i_4i_6}p_{i_3i_4i_5}-p_{i_2i_3i_4}p_{i_1i_2i_6}p_{i_1i_3i_5}p_{i_4i_5i_6}=0$ for all six-element subsets $\{i_1<\cdots<i_6\}$.
The configuration-space theorem implies that the ideal of maximal minors of any evaluation matrix for a linear system with general reduced irreducible divisor is prime of codimension $n-\ell+1$.
Goppa duality transfers the class statement between $V(d+1,n)$ and $V(n-d-1,n)$, so a corresponding recursion holds on the dual Grassmannian.

Reading between the lines

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

Because the recursion has coefficients independent of $n$, the generating function $F(t)=\prod_{i=1}^3(1-2x_i t)^{-1}\prod_{i<j}(1-(x_i+x_j)t)^{-1}$ packages all classes $[V(3,n)]$ at once; extracting every Schubert coefficient, not just the Eulerian one, becomes a finite linear algebra problem for each $n$.
If the reducedness theorem extends to all linear systems of the form $\langle x_i^{d-1}x_j\rangle$, the same Porteous computation would give the class of $V(d+1,n)$; the main obstruction is the algebro-geometric reducedness step, not the computation.
The $n=6$ boundary dimension drop suggests that if $V(3,n)$ is a positive geometry, its boundary has strata not inherited from positroid cells of $G(2,n)$; blowing up the indeterminacy locus of the Veronese map would likely produce these missing strata from the exceptional divisor.
The Eulerian coefficient identifies one intersection number with a scattering-equation solution count; comparing the remaining Schubert coefficients with scattering data would test how much of the spinor-helicity dictionary the ABCT class encodes.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Main take: the recursive class formula for V(3,n) is a solid, useful result, and the Porteous computation behind it is clean. The real soft spot is the proof that V(3,n) equals the determinantal locus Z2(3,n): the proof of Theorem 3.5 contains a false statement about smoothness, and the gap is load-bearing for the main theorem. The paper is worth reviewing, but it needs a fix before I would trust the proof as written.

What is genuinely new: Theorem 1.1 gives an explicit recursion for the full cohomology class in terms of Schur polynomials, and Corollary 4.8 ties the top special Schubert coefficient to Eulerian numbers, matching a physics computation independently. That is a nice bridge between positive geometry and classical Schubert calculus. Theorem 1.2 is a real generalization of Caminata-Moon-Schaffler from projective spaces to arbitrary smooth varieties and linear systems; the incidence-correspondence proof is mostly standard and the statement is useful. Proposition 5.4, that the Veronese image of a rank-2 matroid is a matroid, is a neat extra.

The soft spots: in the proof of Theorem 3.5, the authors claim that a general divisor in the linear system V = is smooth. That is false. Every divisor in that system contains the origin, and every generator vanishes to order at least two there, so the origin is always a singular point. What the argument actually needs is that the general divisor is reduced and irreducible. That is plausibly true for these cones, but it is not proved; the paper just says it is "straightforward." The generic-reducedness step is also delegated to [6, Lemma 3.4] without a self-contained adaptation. This matters because Theorem 3.5 is what makes Proposition 4.1 (V(3,n)=Z2(3,n)) go through, and that equality is the input to the Porteous computation. Proposition 3.9 does not rescue it: for k=3,d=2 it proves the ideal is radical, not prime, and it does not rule out extra components. The computer checks in Corollary 4.2 and Proposition 3.9 are fine as evidence but would be better with code; that is minor.

My overall read: the main theorem is probably true. The missing verification is of a specific, concrete linear system, and I expect it can be supplied by a direct argument. But as written, the proof has an actual gap, not just a stylistic omission. I would send this to a referee, because the result is important enough and the derivation of the class formula is solid. The referee should be asked to focus on Theorem 3.5 and whether the irreducibility of those cones can be established.

Referee Report

3 major / 5 minor

Summary. The paper studies the ABCT variety V(3,n), defined as the closure of the image of G(2,n) under the quadratic Veronese map into G(3,n). The main result, Theorem 1.1, gives a recursive formula for the fundamental class of V(3,n) in terms of symmetric functions in three variables: if n≥5, the class equals [f_{n-5}], where f_m is defined by an explicit recursion. The proof strategy is to identify V(3,n) with the determinantal degeneracy locus Z2(3,n) of the morphism O^{⊕n} → S^2U^∨, to prove that Z2(3,n) has the expected codimension and is reduced and irreducible, and then to apply Porteous' formula combined with symmetric-function identities. The paper also derives the degree of V(3,n), shows that one special Schubert coefficient is the Eulerian number A(n-3,1), and discusses a positroid stratification related to Lam's positive-geometry conjecture.

Significance. If the identification V(3,n)=Z2(3,n) is rigorously established, Theorem 1.1 is a valuable and explicit computation: it gives the full cohomology class of V(3,n) in codimension n-5 through a simple recursion, makes the degree computable, and confirms a formula predicted by the spinor-helicity formalism. The Porteous computation and the symmetric-function lemma are clean and standard, and the numerical examples are convincing. The independent match with the Eulerian number is a strong consistency check. However, the central geometric input—the reducedness and irreducibility of the determinantal loci—is not proved in a self-contained way; the paper delegates the key generic-reducedness step to a generalization of a result of Caminata-Moon-Schäffler and asserts that the adaptation is immediate. Since the class formula is only as solid as this identification, the paper needs a complete proof of the generalization before the main theorem can be considered established.

major comments (3)

[Section 3, proof of Theorem 3.5] The proof of Theorem 1.2 is not complete. The generic reducedness of X_n(V) is disposed of with the sentence 'one can prove as in [6, Lemma 3.4]' and the claim that it suffices to replace the matrix M_{r,d,n} in [6] with the evaluation matrix (4). This is a genuine generalization from the complete linear system of degree-d monomials to an arbitrary linear system V whose general divisor is reduced and irreducible, and the proof in [6] may use special features of monomials. The authors themselves describe their statement as 'apparently more general' and say it 'can be proved in essentially the same way,' which is an assertion, not a proof. This gap is load-bearing because Theorem 3.5, Proposition 4.1, and hence Theorem 1.1 all depend on Theorem 1.2. Proposition 3.9 does not repair the gap: it only proves that a certain ideal is radical for k=3,d=2, not that Z2(3,n) is irreducible and reduced as required.
[Section 4, Proposition 4.1] The proof of Theorem 3.5 contains a false assertion: it says that 'a general hypersurface in P(V) is smooth and irreducible.' For the linear system V consisting of degree-d forms vanishing at the coordinate points, every divisor contains the origin of C^k, and every defining monomial has degree at least 2, so the gradient of any nonzero form in V vanishes at the origin. Thus no divisor in P(V) is smooth. The theorem only requires the general divisor to be reduced and irreducible, which may well be true, but the stated justification is incorrect. The authors need to supply a correct argument, or cite a precise result proving that the general member of this particular linear system is reduced and irreducible.
[Section 4, Proposition 4.1] The Porteous computation itself is standard and convincing once the scheme-theoretic identification V(3,n)=Z2(3,n) is accepted. However, the proof of Proposition 4.1 uses Theorem 3.5 to conclude that Z2(3,n) is reduced and irreducible of dimension 2n-4. Since the proof of Theorem 3.5 has the gaps described above, the scheme-theoretic equality V(3,n)=Z2(3,n) is not yet rigorously established. The main theorem therefore rests on an unproved geometric input. I recommend that the authors either provide a complete proof of Theorem 1.2 (or a direct proof of the reducedness and irreducibility of Z2(3,n)) or explicitly isolate this as a conjecture and prove the class formula conditionally.

minor comments (5)

[Corollary 4.2] The recursion in the statement of Theorem 1.1 reads '+ 2m · fm', which is a typo: it should be '+ 2^m · s_m' as in Lemma 4.3 and the proof. Please correct this.
[Section 5] The proof of Corollary 4.2 only says that the 6×6 minors of the matrix defining θ_2 can be expressed in Plücker coordinates by the displayed quartics. It does not show that the ideal generated by these quartics equals the ideal of maximal minors, which is needed to conclude that V(3,n) is cut out as a scheme by these equations. A reproducible computer-algebra script or a direct argument would be helpful.
[Remark 4.9 and Proposition 4.10] The computational evidence about dimensions of images of positroid strata (e.g., 'dim θ_2(Π_M1) = 6, dim θ_2(Π_M2) = 4') is reported without accompanying code or data. For an experimental claim this is acceptable, but the authors should state clearly which software and which exact input were used, so that the computation can be reproduced.
[Section 2, first paragraph] The notation s_{(n-d-3)^{d-1}} and Σ_{(n-d-1)^2,2^{d-1}} is ambiguous for readers not familiar with repeated-part notation for partitions. Please define that (a^b) means b copies of a.
[Section 2, first paragraph] The sentence 'By Bertini's theorem, this is for example true if the base locus of V has codimension at least 2 and the image φ_V: X ⇢ P(V^∨) is not a curve' is not a precise statement of Bertini; please give a reference or a more careful formulation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the class computation is a direct Porteous calculation, and the external citations to Caminata–Moon–Schaffler provide independent support rather than a self-referential premise.

full rationale

The paper's central claim, Theorem 1.1, computes [V(3,n)] by first realizing V(3,n) as the determinantal locus Z2(3,n) and then applying Porteous' formula. The identification V(3,n)=Z2(3,n) in Proposition 4.1 is not circular: V(3,n) is defined as the closure of the image of the Veronese map from G(2,n), while Z2(3,n) is defined independently as the rank-at-most-5 degeneracy locus of a vector bundle morphism. The equality uses the external fact from Lam's lectures [15] that V(3,n) is irreducible of dimension 2n-4, together with Theorem 3.5, whose proof invokes the prior independent result of Caminata–Moon–Schaffler [6]. This is a citation to other authors' external work, not a self-citation. The recursion for the symmetric functions f_m is proved in Lemma 4.3 from an explicit generating function, and the passage to cohomology is a standard Porteous/Jacobi–Trudi computation; no fitted parameter is renamed as a prediction and no input is defined in terms of the output. Corollary 4.8 derives the Eulerian-number coefficient by induction from the recursion, while Proposition 4.10 cites the independent Cachazo–He–Yuan result as a matching cross-check. The manuscript does contain delegated proof steps, notably the generic-reducedness argument 'as in [6, Lemma 3.4]' in the proof of Theorem 1.2, and the claim in Theorem 3.5 that a general hypersurface in P(V) is smooth is questionable for the linear system of degree-d monomials; however, these are gaps in justification or correctness risks, not circularity, because the conclusions are not equivalent to the hypotheses by construction. Overall, the derivation chain is self-contained apart from externally sourced geometric facts, and no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

This is a pure mathematics paper; it introduces no fitted parameters and no new postulated entities. The load-bearing assumptions are standard tools of Schubert calculus, the explicit hypotheses of the generalized configuration theorem, plus the cited dimension and irreducibility of V(3,n) and the delegated reducedness argument from [6].

assumptions (4)

domain assumption General divisor in the linear system V is reduced and irreducible (hypothesis of Theorem 1.2)
The generalized configuration theorem requires this hypothesis; the authors note it follows from Bertini when the base locus has codimension at least 2 and the map to P(V^∨) is not a curve.
domain assumption The variety V(3,n) is irreducible of dimension 2n-4, cited from [15, Definition 4.8]
Used in Proposition 4.1 to identify V(3,n) with Z2(3,n); if this dimension were different, the class computed via Porteous would be the class of a different locus.
standard math Porteous' formula computes the class of degeneracy loci of expected codimension
Core tool in the proof of Theorem 1.1; standard in Schubert calculus and intersection theory.
domain assumption Generic reducedness of X_n(V) follows by adapting the arguments of [6, Lemma 3.4] to an arbitrary linear system on a smooth variety
The proof of Theorem 1.2 delegates this step to [6] with a matrix replacement; this is a load-bearing assumption not fully written out in the present paper.

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Pith. "Pith review of Points on Rational Normal Curves and the ABCT Variety." pith.science (2026). https://pith.science/paper/WXDC5FWT

@misc{pith2026241212514,
  author       = {Pith},
  title        = {Pith review of: Points on Rational Normal Curves and the ABCT Variety},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WXDC5FWT}},
  note         = {Machine review of arXiv:2412.12514}
}

abstract

The ABCT variety is defined as the closure of the image of $G(2,n)$ under the Veronese map. We realize the ABCT variety $V(3,n)$ as the determinantal variety of a vector bundle morphism. We use this to give a recursive formula for the fundamental class of $V(3,n)$. As an application, we show that special Schubert coefficients of this class are given by Eulerian numbers, matching a formula by Cachazo-He-Yuan. On the way to this, we prove that the variety of configuration of points on a common divisor on a smooth variety is reduced and irreducible, generalizing a result of Caminata-Moon-Schaffler.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [6]

Caminata - H.-B

A. Caminata - H.-B. Moon - L. Schafﬂer, Determinantal V arieties From Point Conﬁgurations on Hypersurfaces , International Mathematics Research Notices 2023 (22) (2023), 19743–19772

work page 2023

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work page 2011

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Cachazo - S

F. Cachazo - S. He - E. Y . Y uan. Scattering in Three Dimensions from Rational Maps. Journal of High Energy Physics 2013, 141. POINTS ON RA TIONAL NORMAL CURVES AND THE ABCT V ARIETY 19

work page 2013

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Caminata - J

A. Caminata - J. Giansiracusa - H.-B. Moon - L. Schafﬂer, Equations for point conﬁgurations to lie on a rational normal curve , Advances in Mathematics 340 (2018), 653–383

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A. Caminata - J. Giansiracusa - H.-B. Moon - L. Schafﬂer, Point conﬁgurations, phylogenetic trees and dissimilarity vectors , PNAS 118 (12) (2021)

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Caminata - L

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