Tropical and algebraic elliptic plane curves with fixed j-invariant

Alessio Cela; Sae Koyama

REVIEW 2 major objections 5 minor 45 references

Tropical and algebraic elliptic plane curves with fixed j-invariant

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A tropical counting argument proves the classical formula for elliptic plane curves with fixed j-invariant.

desk verdict A tropical proof of Pandharipande's formula with a real internal gap in the large-j enumeration and a heavy reliance on an unreviewed companion correspondence. read the letter →

arxiv 2608.13081 v1 pith:242AIO26 submitted 2026-08-13 math.AG

Alessio Cela , Sae Koyama This is my paper · ORCID

classification math.AG MSC 14N1014H5214T9014N35

keywords tropicalgeometryellipticcurvesj-invariantenumerativecorrespondencetheoremwell-spacedMikhalkinmultiplicityplane

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 proves a tropical formula for the number $E_{d,j}$ of degree-$d$ elliptic plane curves through $3d-1$ general points with a fixed $j$-invariant. Using the authors' genus-one tropical correspondence theorem, it shows $E_{d,j}=E^{\mathrm{trop}}_{d,j}=\binom{d-1}{2}N_d$, where $N_d$ is the number of rational plane curves of degree $d$ through $3d-1$ general points. The point is that the tropical count is now computed with multiplicities that genuinely reflect algebraic lifts, repairing an earlier ad hoc weighting that had no evident geometric meaning. The result recovers the classical algebraic enumeration as a corollary of tropical reasoning, and it exhibits a closed expression for $E^{\mathrm{trop}}_{d,j}$ in terms of triangles in Newton subdivisions of rational tropical curves.

What carries the argument

The load-bearing object is the moduli space $W_{\Gamma_d}(\mathbb{R}^2)$ of well-spaced parametrized tropical genus-one curves of degree $d$ in $\mathbb{R}^2$, together with the correspondence-theoretic multiplicities $m(\sigma)$ attached to its maximal cones. Well-spacedness is a genericity condition on how branches attach to the core cycle; deficiency measures whether the cycle image spans $\mathbb{R}^2$, a line, or a point. The multiplicities $m(\sigma)$ come from an automorphism-weighted count of algebraic lifts and simplify here to the loop multiplicity in deficiency 0, to entries involving $\gcd(a,b)$ and determinant factors in deficiency 1, and to interior-lattice-point counts of dual polygons in deficiency 2. These weights, multiplied by the index $\det_{\mathrm{ev}\times j}(\sigma)$ of the tropical evaluation-and-$j$-invariant map, give the contribution of each cone. The evaluation determinants convert tropical edge-length coordinates into counts of algebraic lifts, and the whole sum is matched to the genus-zero count via the tropical correspondence for rational curves and Pick's theorem.

What would settle it

For degree $d=3$, the theorem predicts $E^{\mathrm{trop}}_{3,j}=12$ for any general point configuration and general $j$. Carrying out the weighted tropical enumeration over $W_{\Gamma_3}(\mathbb{R}^2)$ explicitly and obtaining any number other than 12 would falsify the claimed equality.

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Extended reading notes

Core claim

The central claim is that for general interpolation points and general $j\notin\{0,1728\}$, the enumerative count $E_{d,j}$ of degree-$d$ elliptic plane curves with fixed $j$-invariant equals the weighted tropical count over maximal cones of the moduli space of well-spaced tropical genus-one curves in $\mathbb{R}^2$, and that both equal $\binom{d-1}{2}N_d$. The tropical count is computed by specializing the correspondence theorem to $\mathbb{P}^2$: for large $j$, the contributing curves have a contracted bounded cycle edge, and summing their multiplicities over rational tails gives $\binom{d-1}{2}\det\mathrm{ev}(\sigma')$ for every rational tropical curve $\sigma'$; for small $j$, the contributing cycle types are classified by deficiency $0,1,2$, where deficiency measures the codimension of the span of the cycle image, and the resulting sum over triangles of the Newton subdivision yields $2\operatorname{Area}(T)^2-\tfrac12$ per triangle. Thus the classical formula is derived tropically with algebraically meaningful weights, and the same computation shows the Gromov–Witten count $E^{\mathrm{vir}}_{d,j}=d^2N_d$ differs from the enumerative one.

Load-bearing premise

The argument rests on the genus-one correspondence theorem taken from the companion paper, which asserts that the algebraic count equals the weighted tropical sum; if that theorem's multiplicities or determinant factors were wrong, the tropical computation would not equal the algebraic count.

Editorial extensions

If this is right

For every degree $d$, the enumerative count of genus-one plane curves with fixed general $j$-invariant equals $\binom{d-1}{2}N_d$, so the classical formula is confirmed by tropical geometry.
The tropical count can be computed by summing, over rational tropical curves through the $3d-1$ points, the expression $\sum_T (2\operatorname{Area}(T)^2-\tfrac12)\operatorname{mult}(C)$ over triangles in the Newton subdivision.
The genus-one enumerative count differs from the genus-one Gromov–Witten count, which is $d^2N_d$; the two invariants must be distinguished in applications.
The earlier tropical formula is recovered, but now with multiplicities that reflect algebraic lifts, so the tropical count is a genuine enumerative invariant rather than an ad hoc weighting.
The large-$j$ computation shows all contributing curves have a contracted cycle edge, which ties the genus-one enumeration directly to the genus-zero tropical counts.

Reading between the lines

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

The same specialization of the correspondence theorem could be applied to other toric surfaces; a discrepancy with earlier cone complexes suggests some fixed-$j$ elliptic counts on toric surfaces may need re-examination.
Formula (2) expresses $N_d$ purely in terms of local data (triangle areas) of rational tropical curves; one could try to turn it into a new recursion or closed form for $N_d$.
Since both the small-$j$ and large-$j$ computations give the same expression, the two extreme computations could serve as a combinatorial proof that the count is independent of $j$ and of the point configuration.

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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

If you read one thing in this paper, make it the proof of Proposition 5.3. There is a concrete mistake there: the identity Σ_e (w(e)−1) = Σ_T (b(T)−3)/2 only holds if every bounded edge of the tropical curve C′ is adjacent to two triangles. Simple rational tropical curves have crossings, and those produce parallelograms in the Newton subdivision. For an edge shared by a triangle and a parallelogram, the left side gets w(e)−1 but the right side only (w(e)−1)/2; for an edge between two parallelograms, the right side gets 0. The later algebra reintroduces only half of the parallelogram boundary contribution, so the claimed total is not proven. The gap is in the large-j case, which is half of the proof of Theorem 1.1. I would ask the authors to check this with a concrete degree-3 example before trusting the count.

That said, the paper has real content. It is an alternative derivation of Pandharipande's formula using the algebraically grounded multiplicities from the authors' companion paper [CK26], rather than the ad hoc weights in [KM09]. The small-j case (Proposition 5.9 and Lemmas 5.10–5.12) appears much more self-contained, and the comparison with the Gromov-Witten count (Theorem 1.2) is a nice addition. The exposition is clear and the tables comparing [CK26], [KM09], and [LR18] are genuinely useful.

The other soft spot is the black-box reliance on [CK26, Theorem 4.11]. The correspondence theorem is the bridge between the algebraic count and the tropical sum; if that theorem has any issue, everything downstream moves with it. This is not a fatal objection—papers cite companion work all the time—but for a paper whose entire purpose is to validate a new correspondence, not having the theorem in the same package makes it hard to assess. There are also several steps delegated to [KM09] with 'same as' arguments, and a parity typo in Lemma 5.11 (the last line should presumably say 'w(e) even').

Who is this for? People working on tropical enumerative geometry and log geometry will want to see it, mainly because the well-spaced correspondence is important. But as it stands, the main theorem is not fully proven. I'd send it to a serious referee—the work is honest and the gap might be repairable—but I would not let it pass without having the authors fix Proposition 5.3 and provide the details of [CK26].

Referee Report

2 major / 5 minor

Summary. The paper claims a tropical proof of Pandharipande's formula equating the enumerative count E_{d,j} of degree-d elliptic plane curves through 3d-1 general points with fixed general j-invariant to binom(d-1,2) N_d, where N_d is the Kontsevich count of rational plane curves. The proof works in the moduli space of well-spaced tropical curves: Section 5.1 treats the large-j limit by removing a contracted cycle edge, Section 5.2 treats the small-j limit by degenerating the cycle, and both sections reduce the elliptic counts to explicit weighted sums over rational tropical curves. The paper also proves that the virtual Gromov-Witten count is d^2 N_d (Theorem 1.2) and contrasts it with the enumerative count. The main input is the genus-one tropical correspondence theorem of the authors' companion paper [CK26], which is quoted as Theorem 4.11.

Significance. If the proof is completed, the paper would give a genuinely tropical, structurally meaningful derivation of a known algebraic enumeration, and it would clarify which multiplicities in earlier work by Kerber-Markwig were ad hoc and which are algebraically grounded. The computation of the virtual count E^vir_{d,j}=d^2 N_d is a useful addition, and the explicit formulas in Propositions 5.3 and 5.9 are interesting in their own right. However, the central claim is not self-contained: it relies on the unpublished companion theorem [CK26], and the proof of Proposition 5.3, which is the load-bearing step for the large-j enumeration, contains a combinatorial identity that appears to be false when the Newton subdivision has parallelograms. These issues must be resolved before the main theorem can be regarded as established.

major comments (2)

[§5.1, Proposition 5.3] The proof of Proposition 5.3 contains a load-bearing algebraic error. The displayed identity "Σ_{e∈E(C')}(w(e)-1) = Σ_T (b(T)-3)/2" is false when the Newton subdivision contains parallelograms: for an internal edge e adjacent to a triangle and a parallelogram, the left-hand side receives w(e)-1, while the right-hand side receives only (w(e)-1)/2 because the parallelogram is absent from the sum over triangles. Consequently the subsequent chain of equalities does not prove the claimed identity Σ_T i(T)+Σ_e(w(e)-1)+Σ_P Area(P) = binom(d-1,2). In fact, the right-hand side of the display in the proof is Σ_T i(T)+Σ_P i(P)+Σ_T (b(T)-3)/2+Σ_P (b(P)-4)/2+#P, which the proof correctly identifies with i(Δ_d); but the left-hand side that must be computed is larger by Σ_P(b(P)-4)/2. A concrete numerical test is the degree-3 case: a parallelogram dual to a crossing of a weight-2 edge and a weight-1 edge has (b(P)-4)/2 = 1, so the per-curve contribution would be 2 instead of the claimed binom(2,2)=1. Thus the large-j enumeration of Theorem 1.1 is not established as written. The authors should either correct the combinatorial identity, adjust the multiplicities in Lemmas 5.6 and 5.7, or show that the extra parallelogram term vanishes in the relevant enumerative setting.
[§4.3.2, Theorem 4.11] The proof of Theorem 1.1 relies entirely on Theorem 4.11, the genus-one correspondence theorem quoted from the companion preprint [CK26]. This theorem is the bridge that equates the algebraic enumerative count E_{d,j} with the weighted tropical sum over maximal cones of the well-spaced moduli space W_{Γ_d}(R^2), including the multiplicities m(σ) and the determinant factors det ev×j(σ). Since [CK26] is not included in the manuscript and the theorem is not proved here, the paper is not self-contained and any error or missing hypothesis in [CK26] would propagate directly into Theorem 1.1. The authors should provide the precise statement with all hypotheses, and ideally include a proof or make the companion preprint available for verification, before the main result can be independently checked.

minor comments (5)

[§4.2.4, Definition 4.8] The phrase "det ev×j(σ) is the lattice of the image of the map" is ambiguous; it should read "the index of the image lattice" or "the determinant of the image lattice," since a determinant is a number rather than a lattice.
[§5.2, Lemma 5.11] In the second half of the proof of Lemma 5.11, the sentence "A similar calculation gives the same result for w(e) odd" should presumably read "for w(e) even," because the odd case was already treated immediately before.
[§5.2, Proposition 5.8] In the deficiency-1 case, the exclusion of a marking on the cycle is justified only by a brief genericity remark ("one can in fact exclude this possibility as well"); a precise statement of the general-position condition on the points x_i that rules this out would strengthen the proof.
[§1 and throughout] The manuscript contains numerous typographical errors and infelicities, including "maxima lcone", "oone", "ellitpic", "adjcant", and "the the". A careful proofreading pass is needed before publication.
[§1] The statement that the Len-Ranganathan cone complex "may create some inaccuracies in certain curve counts in toric surfaces" is vague; please provide a concrete example or a precise reference for the claimed discrepancy.

Circularity Check

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No circular reduction: the tropical enumeration is computed from a cited correspondence theorem and independent lattice-point identities, not from the target formula.

full rationale

The derivation chain is: (1) import the genus-one correspondence Theorem 4.11 from the authors' prior work [CK26], asserting E_{d,j} = sum m(sigma) det(ev x j)(sigma); (2) compute that tropical weighted sum, separately for large and small j, in Sections 5.1 and 5.2; (3) identify the resulting expressions with binom(d-1,2) N_d using Mikhalkin's genus-zero correspondence and Kontsevich's count of rational curves. The target Pandharipande formula is never inserted as an input. The imported [CK26] theorem is a same-author citation, and it is load-bearing, but it is a general correspondence statement for elliptic curves in toric varieties, with multiplicities described as automorphism-weighted numbers of algebraic lifts (Remark 4.7), not as parameters fitted to reproduce the closed formula binom(d-1,2) N_d. The tropical computations themselves are independent lattice-point and determinant calculations (Lemmas 5.6, 5.7, 5.10-5.12), and the final identification with N_d uses external genus-zero results. The skeptically noted issue in Proposition 5.3, concerning the replacement of edge sums by triangle boundary sums when parallelograms occur, would be an internal correctness gap, not a circularity; a false identity or an omitted argument does not make the theorem's conclusion equivalent to its inputs by construction. No step in the paper reduces by definition or by fitted parameter to the claimed enumeration, so no significant circularity is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on three main external results: the Cela-Koyama correspondence theorem (a companion preprint by the same authors), the moduli theory of well-spaced stable maps from Ranganathan-Santos-Parker-Wise, and Mikhalkin's genus-zero correspondence theorem. None of these are proved in the present paper, and the first is not yet independently verified. No free parameters are fitted to data, and no new entities are postulated.

assumptions (5)

domain assumption The Cela-Koyama genus one correspondence theorem [CK26, Corollary 1.7] holds: for general points and general j, E_{d,j} = sum_{σ} m(σ) · det ev×j(σ) over maximal cones of W_{Γ_d}(R^2).
Stated as Theorem 4.11 and used as a black box; the present paper does not prove it. The main theorem reduces to this correspondence.
domain assumption The moduli stack W_{Γ_d}(P^2) of well-spaced stable maps is proper, logarithmically smooth, pure-dimensional, and its degree to (P^2)^{3d-1} × M_{1,1} equals the enumerative count E_{d,j} divided by (d!)^3.
Assumed from [RSPW19a, RSPW19b] and [CK26]; used in Section 3.2 to set up the correspondence.
domain assumption Mikhalkin's genus zero correspondence theorem equates N_d with the tropical count N_d^{trop}, and general-position tropical curves are simple.
Used in Section 4.3.1 and throughout Section 5; standard result from [Mik05].
standard math The combinatorial identity that the number of parallelograms in a Newton subdivision equals the number of interior lattice points that are vertices of the subdivision, for a genus zero source curve.
Used in the proof of Proposition 5.3; stated with a brief justification, not proved in detail.
domain assumption Kerber-Markwig determinant computations [KM09, Lemmas 4.10, 4.11, 7.1] are valid in the present setting where the multiplicities may differ but the determinant factors do not.
Used in Lemmas 5.6, 5.7, 5.10 and Proposition 5.3; these prior lemmas are cited rather than reproved.

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Cite this review

Pith. "Pith review of Tropical and algebraic elliptic plane curves with fixed j-invariant." pith.science (2026). https://pith.science/paper/242AIO26

@misc{pith2026260813081,
  author       = {Pith},
  title        = {Pith review of: Tropical and algebraic elliptic plane curves with fixed j-invariant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/242AIO26}},
  note         = {Machine review of arXiv:2608.13081}
}

abstract

We tropically enumerate planar well-spaced elliptic curves with a fixed $j$-invariant. Combined with the recent genus 1 correspondence theorem of Cela--Koyama, this yields an alternative proof of Pandharipande's algebraic enumeration.

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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Irene Schwarz , journal=. On the. 2018 , url=

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Gross, Mark and Siebert, Bernd , title =. Journal of the American Mathematical Society , volume =. 2013 , doi =

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Abramovich, Dan and Chen, Qile and Gross, Mark and Siebert, Bernd , title =. Compositio Mathematica , volume =. 2020 , doi =

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Ranganathan, Dhruv and Santos-Parker, Keli and Wise, Jonathan , title =. Geometry & Topology , volume =. 2019 , doi =

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