{"id":"ce2f14e4-bbc8-4076-acd3-4c113c6abe3f","arxiv_id":"2412.12514","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives a recursive formula for the cohomology class of the ABCT variety V(3,n) inside the Grassmannian G(3,n), using a determinantal realization and Porteous' formula, and matches special coefficients to Eulerian numbers.","lead":"This paper computes the full cohomology class of the ABCT variety V(3,n), a space of configurations of n points on a conic, and finds a recursive formula for it in terms of symmetric functions. It also proves a general theorem that configuration spaces of points lying on a common divisor of a smooth variety are reduced and irreducible, and it confirms a known Eulerian-number coefficient from the Cachazo-He-Yuan scattering formulas.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The class formula depends on the identification V(3,n)=Z2(3,n), which rests on the delegated generic-reducedness argument in Theorem 1.2; the paper's claimed smoothness of the general divisor in P(V) is literally false, so the gap is real but likely repairable.","rationale":"The reader's diagnosis is essentially correct: the unproved-in-full generic reducedness of the configuration space is the soft spot in the reduction of the class computation. I add a sharper observation: the paper's own smoothness assertion in the proof of Theorem 3.5 is false for the affine linear system it uses, because the origin is a common singular point of every divisor in that system. However, Theorem 1.2 only requires reducedness and irreducibility of the general divisor, and for the specific system in question those properties are quite plausible and are supported indirectly by the later Eulerian-number check and by the radicality statement in Proposition 3.9 for k=3,d=2. The Porteous algebra itself, the symmetric-function recursion, and the degree and Eulerian-number corollaries are internally consistent and match known physics results. Thus I do not see a demonstrated mathematical failure, but the paper is not fully self-contained at the critical point, so the CONDITIONAL verdict should stand. My agreement with the reader is partial because the precise failure mode is not quite the smoothness of a general hypersurface; it is the unverified reducedness of the configuration space under weaker hypotheses, plus a false overstatement in the proof that should be corrected.","tokens_in":16757,"tokens_out":55670,"duration_ms":557757,"concrete_test":"For k=3, d=2, and n=8, compute in Macaulay2 or OSCAR the primary decomposition of the ideal of maximal minors of the matrix (14) in the standard affine chart of G(3,8). Verify that the ideal is prime of height 3 with no embedded components. Since this chart is dense in G(3,8), a single prime component of height 3 proves that Z2(3,8) is reduced and irreducible there, directly supporting Theorem 3.5 and Proposition 4.1 for this case; an extra component or embedded prime would show the delegated reducedness step is unreliable. If the computation passes, repeat the same primary-decomposition check for n=9 to confirm the pattern is not an artifact of small n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a Porteous computation once V(3,n)=Z2(3,n) is accepted, so the load-bearing step is Theorem 3.5: Z2(3,n) is reduced, irreducible, and of expected codimension. Its proof applies Theorem 1.2 to the linear system V of non-pure degree-d monomials, and the generic-reducedness part is delegated to [6, Lemma 3.4] without a self-contained verification. This is not merely cosmetic: the proof of Theorem 3.5 asserts that a general hypersurface in P(V) is smooth, but for every such hypersurface the origin of C^k lies on the divisor and all defining monomials have degree at least 2, so the gradient vanishes at the origin. Thus the stated smoothness hypothesis is false. The theorem only needs the general divisor to be reduced and irreducible, which is plausible for these cones, but the paper does not supply the detailed verification. If the delegated argument fails for this linear system, Z2(3,n) could have embedded components or excess dimension, and the Porteous class would no longer be the class of V(3,n). The independent algebraic check in Proposition 3.9 only proves radicality of the ideal for k=3,d=2, not primality or the absence of extra components, so it does not fully substitute for the missing generic-reducedness step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17022,"tokens_out":7093,"duration_ms":66262,"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":[{"comment":"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":"Section 3, proof of Theorem 3.5"},{"comment":"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":"Section 4, Proposition 4.1"},{"comment":"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.","section":"Section 4, Proposition 4.1"}],"minor_comments":[{"comment":"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":"Corollary 4.2"},{"comment":"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.","section":"Section 5"},{"comment":"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":"Remark 4.9 and Proposition 4.10"},{"comment":"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":"Section 2, first paragraph"},{"comment":"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.","section":"Section 2, first paragraph"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a genuinely attractive computation, and I do not see a reason to believe the main formula is false. However, the central identification V(3,n)=Z2(3,n) is not yet proved, because Theorem 1.2 is delegated to [6] and the smoothness claim in Theorem 3.5 is false. These are repairable in principle, but the authors must supply either a complete proof of the generalized configuration theorem or a different argument establishing reducedness and irreducibility of Z2(3,n). I would not recommend rejection, but the current version is not ready for publication as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nMy 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.","headline":"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.","tokens_in":17567,"tokens_out":5740,"would_cite":true,"duration_ms":50714,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","14J81","14N05","14N15","14N20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The ABCT variety V(3,n) is a determinantal variety, and one recursion gives its full cohomology class.","keywords":["ABCT variety","rational normal curve","configurations","Grassmannian","determinantal variety","Porteous formula","Eulerian numbers","scattering equations"],"falsifier":"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.","tokens_in":16546,"feed_emoji":"📐","tokens_out":18135,"duration_ms":133406,"temperature":0.7,"pith_summary":"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.","feed_headline":"One recursion computes the ABCT variety's full class","feed_subtitle":"It yields every Plücker intersection number and the degree, matching scattering-equation counts.","key_machinery":"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 $n-\\ell+1$.","core_discovery":"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.","pith_inferences":["Because the recursion has coefficients independent of $n$, the generating function $F(t)=\\prod_{i=1}^3(1-2x_i t)^{-1}\\prod_{i