Counting 3-uple Veronese surfaces

Anand Deopurkar; Anand Patel

REVIEW 3 major objections 5 minor 9 references

Counting 3-uple Veronese surfaces

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Thirteen general points in $\mathbb{P}^9$ determine exactly 4,246 3-uple Veronese surfaces.

desk verdict A new 4246 count for 3-Veronese surfaces through 13 points, with a transparent but partly hand-verified localization proof that deserves peer review and a careful check of the local multiplicity 4. read the letter →

arxiv 2411.14232 v1 pith:MFLG4YNC submitted 2024-11-21 math.AG

Anand Deopurkar , Anand Patel This is my paper · ORCID

classification math.AG MSC 14N1014C1714N15

keywords 3-upleVeronesesurfacesenumerativegeometrycompletetrianglessingulartriadsassociationandGaletransformAtiyah-BottlocalizationHilbertschemeofpointsintersectiontheory

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

This paper settles the next open case of a classical enumerative problem: it proves that 13 general points in $\mathbb{P}^9$ lie on exactly 4,246 Veronese surfaces, the 3-uple embeddings of $\mathbb{P}^2$. The result extends the classical 2-uple case, where 9 points in $\mathbb{P}^5$ determine 4 such surfaces. The proof avoids counting surfaces directly. Instead, it establishes a bijection between Veronese surfaces through 13 points and certain planar objects called singular triads, then builds new geometric spaces to count those triads exactly. The final number is obtained by equivariant localization after correcting an excess contribution that would otherwise give 57,728.

What carries the argument

The load-bearing object is the space of complete triangles CT, defined as the closure of the graph of the map sending three non-collinear points in $\mathbb{P}^2$ to the three lines they span. CT is a smooth 6-dimensional modification of the Hilbert scheme $\mathrm{Hilb}^3\,\mathbb{P}^2$, resolving the failure of the squared ideal of a length-3 scheme to remain of length 9. Over CT the sheaf of quintic forms singular along the triangle becomes a genuine vector bundle. The paper then passes to SQP, the Grassmannian bundle of 2-dimensional subspaces, or pencils, in the space $V_5$ of quintics through the squared triangle. The cycle $\mathrm{Bpt}(p)$ of pencils whose base scheme contains a point $p$ splits as $[\mathrm{Dom}(p)]+4[\mathrm{Inc}(p)]+[\mathrm{Lin}(p)]$, and the multiplicity 4 is the key transversality input. Bott localization over the 31 fixed points of a torus action computes the 13-fold intersection and yields 4246.

What would settle it

An independent verification could run a numerical homotopy-continuation solve of the 13 incidence equations on the parameter space for a random configuration of points; if the number of isolated solutions differs from 4246, the theorem is false. Alternatively, making the omitted local calculation in Lemma 4.11 fully explicit and finding a multiplicity other than 4 would change the final integral.

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Extended reading notes

Core claim

The paper's central claim is the exact enumerative constant $\nu_{3,2}=4246$: for a general choice of 13 points in $\mathbb{P}^9$, the number of 3-uple Veronese surfaces containing them is 4246. This is established through a chain of identifications. Theorem 2.3 provides a bijection between such surfaces and singular triads in the plane, triples of non-collinear points where the relevant quintic curves are singular. Theorem 4.2 expresses the count as the integral of $[\mathrm{Dom}(p)]^{13}$ over a smooth 26-dimensional space SQP of singular quintic pencils. The integral is evaluated by Atiyah-Bott localization, after subtracting an excess cycle whose multiplicities are fixed by Theorem 4.12.

Load-bearing premise

Everything rests on a local transversality and multiplicity statement saying that two divisors meet with multiplicity exactly 4 along the incidence component; the paper explicitly omits part of that local calculation.

Editorial extensions

If this is right

The number $\nu_{3,2}=4246$ becomes a proved constant, joining the classical $\nu_{2,2}=4$ as the only known nontrivial values in the Veronese counting problem.
Theorem 2.3 gives a new, self-contained correspondence between Veronese surfaces through points and planar singular triads, providing a route that does not rely on a special elliptic curve as in the 2-uple case.
The excess formula $[\mathrm{Bpt}(p)]=[\mathrm{Dom}(p)]+4[\mathrm{Inc}(p)]+[\mathrm{Lin}(p)]$ shows exactly why a naive Porteous computation gives the wrong value 57,728, and how to correct it.
The smooth space CT, constructed directly as an unordered moduli space of complete triangles, resolves the non-flatness of squaring ideals and may serve as a foundation for similar counting problems.
The full computation is reproducible from the published fixed-point weight tables and the included code, so the integer 4246 is checkable line by line.

Reading between the lines

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

The same three-step pattern—correspondence to planar objects, resolution by complete triangles, and localization on a Grassmannian bundle—could plausibly be iterated for higher Veronese folds, once the analogous complete spaces are constructed.
The appearance of association as a composite of a Cremona transformation with a Veronese embedding, which the authors observe in all known cases, suggests a general machine for translating Veronese problems into planar counting problems; making that machine precise is a natural next step.
The excess multiplicity 4 is a new enumerative invariant of this planar incidence problem, and testing whether it persists for other degree-5 interpolation problems would isolate where the true difficulty of the count lies.
If the weight tables remain valid in characteristic 2, the same localization computation should produce a modified count, possibly mirroring how the classical 2-uple count drops from 4 to 2 in that characteristic.

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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

The paper claims nu_{3,2}=4246, the first genuinely new case of Coble's Veronese counting problem since nu_{2,2}=4. That claim is new and important, and the proof architecture is serious: a correspondence to singular triads, a new unordered space of complete triangles, an excess-free Grassmannian bundle, and Atiyah-Bott localization. The included Sage code is real evidence; it reproduces both the wrong 57728 and the final 4246, with sanity checks like H^6=15, Inc^3=1, and vanishing of Lin^3. For a computation this heavy, shipping working code is a genuine step up.

What the paper does well: the construction of CT as a PGL(3)-equivariant resolution of the squaring map's failure, the re-proof of the singular-triad correspondence, and the careful decomposition of Bpt(p) into Dom, Inc, Lin are all coherent and well-motivated. The exposition is clear, and the authors are honest about where calculations are deferred. The fixed-point lists and weight tables are organized and reproducible from the code.

The soft spots are real but concentrated. The load-bearing multiplicity 4 in Theorem 4.12 rests on an omitted local calculation in Lemma 4.11 and a 2-dimensional slice where two divisors are asserted to have ordinary nodes with distinct tangent cones. If that multiplicity is 2 or 6, the final count changes. The fixed-point weight lists in Propositions 3.25 and 3.26 are only partially checked in the text; the reader is told the rest is easy. These are not red flags by themselves, but here the entire conclusion depends on them. I cannot certify the count from the text alone, and neither can the reader.

Proportionately: the architecture is convincing, the final number is consistent with all internal checks, and I found no circularity or fitted parameters. The main unresolved issue is verification of the omitted local geometry, either by a reimplementation or by filling in the calculations. This is not a fatal flaw; it is a paper that should go to a serious referee, and the referee should be asked to scrutinize Theorem 4.12 and the weight tables.

Who is this for? Enumerative algebraic geometers, especially those working with localization and Hilbert schemes. It deserves peer review with the expectation of revision. I would bring it to a reading group and cite it once the count is independently verified.

Referee Report

3 major / 5 minor

Summary. The paper claims a new enumerative constant: 13 general points in P^9 determine exactly 4246 3-uple Veronese surfaces (Theorem 1.2). The proof strategy is to translate this by Coble association and Goppa's lemma into the count of singular triads in P^2, then to replace Hilb^3 P^2 by a new 6-dimensional 'space of complete triangles' CT, and then to move to a 26-dimensional Grassmannian bundle SQP of singular quintic pencils. The count is expressed as the degree of a class [Dom(p)]^13 on SQP, and this degree is evaluated by Atiyah-Bott localization, with the arithmetic carried out in the appended Sage code. The final expression is Eq. (24), and the paper also includes a number of checks and open questions.

Significance. If the proof is correct, this is the first new number in the Veronese counting problem since Coble's 1922 result, and it is a substantial piece of enumerative geometry. The construction of CT directly on the Hilbert scheme of length-3 subschemes is original and likely to be useful beyond this example. The paper is unusually transparent in providing the complete Sage transcript, and the localization arithmetic is internally checked against known Chern-class evaluations; the final number 4246 never enters as an input, so there is no visible circularity. The main caveat is that a small number of local geometric computations are load-bearing and are deferred or asserted rather than fully shown.

major comments (3)

[§4.3, Lemma 4.11] The proof of Lemma 4.11, which establishes transversality on Lin(p)^† and hence fixes the coefficient of Lin(p) in Theorem 4.12, contains the explicit sentence 'A local calculation (omitted)' and then asserts that the three membership conditions of (14) 'are met' without demonstrating them. Since Lemma 4.11 is used directly in the cycle decomposition (15), and since Eq. (15) is used in Eq. (24) to obtain 4246, this is a load-bearing omitted calculation. Please supply the complete coordinate computation, or a formal verification of the three equalities φ_{f,Pi}(f') = φ_{g,Pi}(g') for i = E,F,C, together with the stated nonvanishing f'(D) = 0 and g'(D) ≠ 0.
[§4.3, Theorem 4.12] The multiplicity 4 at Inc(p)^† is derived in the proof from the assertions that the two local equations of B_1(p) and B_2(p) on a general 2-dimensional étale slice have ordinary nodes at E and that their tangent cones share no line. The latter is justified only by the phrase 'as can easily be checked in local coordinates' earlier in §4.3 and by the generic-slice hypotheses near the displayed polynomial expansions. This is the most load-bearing point of the paper: if the local intersection multiplicity were 2 or 6 rather than 4, the class equality (15) would change and Eq. (24) would no longer give 4246. The Sage code verifies the localization arithmetic once the coefficient 4 is assumed, but it does not verify that coefficient. Please provide a complete local calculation, including the explicit tangent cones and the proof that they are distinct, or an independent formal verification.
[§3.7, Propositions 3.25 and 3.26] The Atiyah-Bott computation uses the full six-orbit fixed-point weight tables for both the vector bundle E and the tangent bundle of CT, but the proofs verify only one representative for E (case (2)) and two representatives for T (cases (3) and (6)), with the remaining cases delegated to 'the reader can then check that no new complications arise'. The included Sage transcript recapitulates the asserted tables but does not derive them. Since the final localization sum (11) and the integral (24) depend on every entry of these tables, please provide a systematic derivation for all six orbit types, or a machine-checkable verification of the tables themselves.

minor comments (5)

[§4.3, Eq. (14)] The description of the tangent space of the frame bundle E^† in display (14) also relies on an asserted local-coordinate check regarding the Hessian H_f and the induced node deformation. Please include this check explicitly or relegate the formula to a lemma with proof, since it is used throughout the later transversality arguments.
[§3.7, Proposition 3.27] The notation O[3], O(1)[3], O(2)[3], and O(3)[3] for the pulled-back tautological rank-3 bundles is easy to confuse with powers of line bundles; please rename these bundles, for example O_i^{(3)}, and define the notation before the proposition.
[§7, Sage code] The final computations are only present as commented-out print statements. Please include a short transcript or output block showing Wrong(a=45,b=3,c=10) = 57728 and Answer(a=-20,b=9,c=7) = 4246, so that a reader can verify the reported numbers without rerunning the code.
[§2.2, Theorem 2.3] The proof of Theorem 2.3 says that 'a simple dimension count, which we omit, shows that A and B are finite sets'. Since the bijection is used to pass from Veronese surfaces to singular triads, please include the one-paragraph dimension count for completeness.
[§4.4, Theorem 4.25] The phrase 'Bézout's theorem' in the final line of the proof is used loosely for an intersection-theoretic identification on a 6-dimensional variety; consider replacing it with a more precise reference to the intersection product and the fact that all components have the expected dimension.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the count 4246 never enters as an input; the one self-citation to [LP19] for the correspondence is re-proved and non-load-bearing.

full rationale

Circularity pass: no equation in the derivation takes 4246 as an input. The Atiyah-Bott localization data are computed from explicit fixed-point ideals and bundle definitions in Section 3.7, and the final evaluation in Section 7 is a rational-function calculation in the torus weights a, b, c. Theorem 2.3, the correspondence previously obtained by the authors in [LP19], is given a full proof in Section 2.2 via Goppa's lemma and Cremona transformations, so the self-citation is not load-bearing. The intermediate value 57728 is explicitly identified as the wrong, excess-laden answer in Section 4.1 and is discarded; the corrected integral uses Theorem 4.12, whose multiplicity 4 is derived from a local node-and-tangent-cone analysis, not from fitting the final number. The sanity checks in Section 7 (for example H^6 = 15, H^4*Inc = 3, c3(O(2)[3])^2 = 4, c3(O(3)[3])^2 = 84, and Inc^3 = 1) are independent enumerative or Chern-class evaluations used only to validate the localization arithmetic; they do not pin down 4246. The omitted local calculation in Lemma 4.11 and the phrase 'as can easily be checked in local coordinates' in Section 4.3 are genuine gaps that could affect the coefficient 4 and hence the final count, but they are correctness risks, not circularity: no passage assumes the desired result as part of the derivation. The only mild self-citation is the origin of the correspondence in [LP19], and because the proof is reproduced, the central claim remains independent. Score 2 reflects that minor self-citation rather than any fitted-input or definitional circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 2 invented entities

No fitted constants are used: the torus weights a,b,c are arbitrary generic integers and the computation is invariant under their choice. The central claim rests on standard localization, Porteous, and Gorenstein duality background, plus the newly constructed CT and SQP spaces and a set of hand-computed weight tables and excess multiplicities that are not independently checked.

assumptions (7)

standard math Gorenstein duality and association/Gale transform supply the bijection in Theorem 2.3.
Used in Section 2 to define associated tuples and the singular-triad correspondence; accepted background in algebraic geometry.
standard math Goppa's lemma for non-special line bundles on smooth curves (Lemma 2.2).
Invoked in Section 2.2 to identify associated subspaces; a classical tool.
standard math Atiyah-Bott localization formula for Gm actions with finite fixed points.
Used in Sections 3.6 and 5 to reduce integrals over CT and SQP to fixed point sums.
standard math Porteous formula for degeneracy loci.
Used in Section 3.6 and Lemma 5.2 to express degeneracy cycles; the first application produced the wrong number 57728 and had to be corrected.
domain assumption Orbit classification of length-3 subschemes of P2 and the behavior of squaring ideals on Hilb3 P2.
Relied on in Section 3 to define CT and to prove Proposition 3.12; the classification is standard and checked orbit by orbit.
ad hoc to paper Correctness and completeness of the 31 fixed point weight tables for E and T (Propositions 3.25 and 3.26).
These hand-computed tables are direct inputs to the localization sum; only two representative cases are shown and no independent derivation is provided.
ad hoc to paper The excess decomposition [Bpt(p)] = [Dom(p)] + 4[Inc(p)] + [Lin(p)] with the computed multiplicities.
The count passes through this identity in Section 5; Lemma 4.11 contains an omitted local calculation and the key multiplicity 4 is only sketched.

invented entities (2)

Space of complete triangles (CT)
purpose: A PGL(3)-equivariant modification of Hilb3 P2 that resolves the non-flatness of squared ideals and hosts the rank-9 vector bundle E.
CT is introduced for this proof; its geometry is studied internally, but it makes no falsifiable prediction outside the paper.
Space of singular quintic pencils (SQP)
purpose: A Grassmannian bundle Gr(2,V5) over CT used to eliminate excess in the degeneracy calculation.
SQP is an auxiliary construction internal to the proof, with no independent empirical handle.

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Cite this review

Pith. "Pith review of Counting 3-uple Veronese surfaces." pith.science (2026). https://pith.science/paper/MFLG4YNC

@misc{pith2026241114232,
  author       = {Pith},
  title        = {Pith review of: Counting 3-uple Veronese surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFLG4YNC}},
  note         = {Machine review of arXiv:2411.14232}
}

read the original abstract

This paper culminates in the count of the number of 3-Veronese surfaces passing through 13 general points. This follows the case of 2-Veronese surfaces discovered by Coble in the 1920's. One important element of the calculation is a direct construction of a space of "complete triangles." Our construction is different from the classical ordered constructions of Schubert, Collino and Fulton, as it occurs directly on the Hilbert scheme of length 3 subschemes of the plane. We transport the enumerative problem into a 26-dimensional Grassmannian bundle over our space of complete triangles, where we perform Atiyah-Bott localization. Several important questions arise, which we collect at the end of the paper.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

[1]

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Alberto Collino, William Fulton, et al. Intersection rings of spaces of triangles. colloque en l’honneur de pierre samuel (orsay, 1987). MEMOIRE DE LA SOCIETE MATHEMATIQUE DE FRANCE , 38:75--117, 1989

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Izzet Coskun. The enumerative geometry of del pezzo surfaces via degenerations. American Journal of Mathematics , 128(3):751--786, 2006

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Explicit computations in Hilb ^ 3 P^ 2

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The projective geometry of the gale transform

David Eisenbud and Sorin Popescu. The projective geometry of the gale transform. Journal of Algebra , 230(1):127--173, 2000

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Bott’s formula and enumerative geometry

Geir Ellingsrud and Stein Str mme. Bott’s formula and enumerative geometry. Journal of the American Mathematical Society , 9(1):175--193, 1996

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Functorial construction of le barz’s triangle space with applications

Sean Keel. Functorial construction of le barz’s triangle space with applications. Transactions of the American Mathematical Society , 335(1):213--229, 1993

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[9]

Interpolation problems: Del pezzo surfaces

Aaron Landesman and Anand Patel. Interpolation problems: Del pezzo surfaces. ANNALI SCUOLA NORMALE SUPERIORE-CLASSE DI SCIENZE , pages 1389--1428, 2019

2019

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Intersection rings of spaces of triangles

Alberto Collino, William Fulton, et al. Intersection rings of spaces of triangles. colloque en l’honneur de pierre samuel (orsay, 1987). MEMOIRE DE LA SOCIETE MATHEMATIQUE DE FRANCE , 38:75--117, 1989

work page 1987

[2] [2]

Arthur B. Coble. Associated sets of points. Transactions of the American Mathematical Society , 24:1--20, 1922

work page 1922

[3] [3]

Degenerations of surface scrolls and the gromov-witten invariants of grassmannians

Izzet Coskun. Degenerations of surface scrolls and the gromov-witten invariants of grassmannians. Journal of Algebraic Geometry , 15(2):223--284, 2006

work page 2006

[4] [4]

The enumerative geometry of del pezzo surfaces via degenerations

Izzet Coskun. The enumerative geometry of del pezzo surfaces via degenerations. American Journal of Mathematics , 128(3):751--786, 2006

work page 2006

[5] [5]

Explicit computations in Hilb ^ 3 P^ 2

Georges Elencwajg and Patrick Le Barz. Explicit computations in Hilb ^ 3 P^ 2 . In Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12--19, 1986 , pages 76--100. Springer, 2006

work page 1986

[6] [6]

The projective geometry of the gale transform

David Eisenbud and Sorin Popescu. The projective geometry of the gale transform. Journal of Algebra , 230(1):127--173, 2000

2000

[7] [7]

Bott’s formula and enumerative geometry

Geir Ellingsrud and Stein Str mme. Bott’s formula and enumerative geometry. Journal of the American Mathematical Society , 9(1):175--193, 1996

work page 1996

[8] [8]

Functorial construction of le barz’s triangle space with applications

Sean Keel. Functorial construction of le barz’s triangle space with applications. Transactions of the American Mathematical Society , 335(1):213--229, 1993

work page 1993