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Genus one correspondence between tropical and algebraic curves

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Pith's one-line read This paper proves that enumerative counts of algebraic elliptic curves in toric varieties equal counts of well-spaced tropical curves weighted by explicit lattice-point multiplicities, extending the correspondence theorem from genus 0 to ge

desk verdict A serious, credible genus-one tropical-algebraic correspondence in arbitrary toric dimension, with one load-bearing stabilizer step that needs a real proof before the weight formula is fully trusted. read the letter →

arxiv 2607.06426 v2 pith:TG6EWCZU submitted 2026-07-07 math.AG

Alessio Cela , Sae Koyama This is my paper · ORCID
classification math.AG MSC 14N1014N3514T0514H10
keywords tropicalgeometryenumerativegenusonetoricvarietieslogarithmicmapswell-spacedcurvescurvecountinglatticepoints
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

Counting algebraic curves of positive genus is hard; counting tropical ones is combinatorial. This paper proves that for genus one the two counts coincide: for any smooth projective toric variety, the genuinely enumerative number of algebraic elliptic curves through prescribed incidence conditions equals the number of well-spaced tropical genus-one curves through the corresponding tropical conditions, each weighted by an explicit combinatorial multiplicity — essentially the number of interior lattice points of a region attached to the tropical curve's unique cycle, divided by its automorphism count. This completes the genus-1 generalization of a correspondence that previously existed only in genus 0, in arbitrary dimension, where the realizability question for tropical curves had blocked exact tropical enumeration. The paper also gives a practical criterion for when a genus-one tropical curve admits an algebraic lift.

What carries the argument

The load-bearing objects are the moduli space W_Γ(X) of well-spaced genus-one logarithmic stable maps and its tropical analogue W_Γ(Σ(X)), a generalized cone complex whose maximal cones are tropical curve types. Well-spacedness — a condition on how the curve leaves each hyperplane containing its cycle — is exactly what makes a genus-one tropical curve realizable. The proof runs through the tropicalization map, Minkowski weights (balanced integer functions on cones computing toric intersections) on X^n × M_{1,1}, and a lifting criterion (Theorem 5.4) converting logarithmic enhancements into conditions on meromorphic functions at the nodes. The multiplicities count solutions of a monomial syst

What would settle it

Compute both sides of the correspondence for a case with a high-valence vertex on the cycle and compare: the paper's own example — degree-3 elliptic curves in P^3 through 5 points and meeting 1 line, which must number zero — is a sharp test, as is a direct linear-algebra check of Proposition 1.9's basis statement for the three maximal cone shapes the paper lists in dimension 3.

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Extended reading notes

Core claim

The paper's central claim (Theorems A, B, D) is an equality of enumerative counts: for a smooth projective toric variety X, the algebraic number of genus-one curves of a fixed class through general subvarieties equals the count of well-spaced tropical genus-one curves through the corresponding conditions, each weighted by the number of interior lattice points of a region P(σ') built from the direction vectors of edges leaving the curve's unique cycle, divided by the automorphism order (just the reciprocal when no cycle vertex is at least 4-valent). The fixed-j version holds too. A lifting criterion (Theorem 5.4) determines when a tropical curve is algebraic, resolving genus-one realizability

Load-bearing premise

The load-bearing premise is the basis statement quoted in Proposition 1.9 — that the non-cycle edge directions around the tropical curve's cycle project to an R-basis of the quotient H_1/H_0 — because the counting lemma behind the lattice-point multiplicities rests on it directly, and the proof also leans on a stabilizer comparison asserted without full expansion in §7.3, Claim (ii).

Editorial extensions

If this is right

  • Genus-one enumerative invariants of any toric variety become finitely computable from purely combinatorial data: list the well-spaced tropical curves, count interior lattice points, divide by automorphism orders.
  • Because the correspondence is genuinely enumerative rather than virtual, it provides a check on virtual-count formulas; the paper exhibits one published formula for elliptic curves in P^3 and shows it fails its simplest consistency check.
  • The fixed-j version gives tropical control over the complex structure of the elliptic curve itself: the length of the tropical cycle is the tropical j-invariant, so both position and j-invariant are read off the tropical data.
  • On toric surfaces, the new weights differ from earlier ad hoc tropical multiplicities in specific low-deficiency configurations, while the final fixed-j count for plane curves reproduces the known algebraic value.
  • The lifting criterion (Theorem 5.4) resolves the genus-one realizability problem in arbitrary dimension, converting a qualitative existence question into an exact checkable system of equations at the special points of the curve.

Reading between the lines

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

  • Because the multiplicity is a count of interior lattice points of a region built from the cycle's edge directions, the formula may extend beyond toric targets to any target admitting a suitable toric degeneration; one could test this by computing both sides for a simple non-toric surface.
  • The parallel structure of the fixed-j and unfixed-j counts suggests a tropical analogue of the string equation at genus one: summing the lattice-point multiplicities over cycle lengths should reproduce the unfixed count, a relation that could be checked symbolically for P^2 even before any new theory is built.
  • A computational stress-test of Proposition 1.9 would localize the exact scope of the theorem: verify by linear algebra, for each maximal cone shape in dimensions 2 and 3, that the non-cycle directions form a basis of H_1/H_0; a failure in some exotic cone would pinpoint precisely where the purely tropical multiplicity needs correction.
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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

3 major / 6 minor

Summary. The paper claims a complete genus-one correspondence between enumerative algebraic curve counts and weighted tropical curve counts for smooth projective toric varieties over C. For effective curve classes and general subvarieties Z_i, the algebraic number of smooth genus-one maps with prescribed incidences and with either free or fixed j-invariant is asserted to equal a tropical count of well-spaced genus-one tropical curves, weighted by explicit combinatorial multiplicities (Theorem A). The proof strategy is to use the Ranganathan–Santos-Parker–Wise moduli stack W_Γ(X) of well-spaced genus-one logarithmic maps, establish a Chow-theoretic correspondence via Minkowski weights and logarithmic blow-ups (Theorem B, Proposition 1.17), and then compute the tropical weights combinatorially in terms of lattice-point counts in an explicit polytope P(σ') (Theorem D), with a separate result relating saturation indices, lattice indices, loop multiplicities, and automorphism groups (Theorem C). The paper also gives low-dimensional specializations and comparisons with earlier ad hoc multiplicities of Kerber–Markwig and Len–Ranganathan.

Significance. If correct, the paper is a major advance: it provides the first enumerative (rather than virtual) genus-one tropical correspondence in arbitrary dimension, extending the Nishinou–Siebert genus-zero theorem. The weights are genuinely combinatorial and computable, and the paper gives nontrivial consistency checks against previous surface calculations and Pandharipande's fixed-j counts. The authors use substantial logarithmic-geometric machinery correctly in broad outline, and the paper is rich in explicit formulas and examples. The main caveat is that two load-bearing steps — the stabilizer comparison in §7.3 and the use of Proposition 1.9 in Lemma 8.21 — are not fully established in the text. These are fixable in principle, and the surrounding framework is credible, so the appropriate assessment is major revision rather than rejection.

major comments (3)
  1. [§7.3, proof of Theorem B, Claim (ii) and Equation (45)] Claim (ii) is the step where the proof passes from the sum over cones of the logarithmic blow-up \tilde W_Γ(X) to the sum over cones of W_Γ(X). It asserts that the pushforward of [V(\tilde σ)] is [V(σ)], i.e. that the stabilizers of the corresponding points coincide. The proof consists of an asserted Cartesian diagram, an invocation of Claim (i), and the sentence 'BT_{\tilde σ}→BT_σ is an isomorphism'. This is not enough: the diagram is not fully specified, and the map on tori is not justified in this setting. Footnote 7 explicitly warns that in the analogous situation X=[A^2/μ_2] the stabilizer comparison fails when a cone does not map injectively to the base fan. If Claim (ii) fails, the coefficient [V(\tilde σ)] would be a rational multiple (|Aut(f_σ)|/|Aut(f_{\tilde σ})|) of [V(σ)], and the weight w(σ) in Theorem B would not match the tropical multiplicity in Theorem D. This is inter
  2. [Proposition 1.9 and Lemma 8.21] The computation of n(σ) as the number of interior lattice points of P(σ) in Lemma 8.21 depends crucially on the assertion that the projected non-cycle edge directions {\bar u_{j,k}} form an R-basis of H_1/H_0. This is Proposition 1.9, quoted without proof from [Tor14, Proposition 3.2.18(b)]. The matrix M in Lemma 8.21 is square and invertible precisely because of this basis statement; if the quoted result does not apply to every maximal well-spaced cone in the present generality, the identity 'number of solutions = i(P(σ))' fails, and with it Theorem D and the 'purely tropical' form of the multiplicity. Since this is a central combinatorial input rather than a side remark, please either include a self-contained proof adapted to the present setting or state explicitly why the cited thesis result applies to all cones that occur.
  3. [§8.3.3 and proof of Theorem D] The proof of Theorem D relies on the claim that after solving Equation (59), the possible automorphisms of the resulting basic fine maps are fully understood. The text says in the proof of Theorem 8.9 that 'the proof of Theorem 8.22 explicitly lists all the basic fine maps ... and their automorphisms', but Theorem 8.22 as stated only lists the number n(σ), not the automorphism groups. The subsequent proof of Theorem D asserts that nontrivial automorphisms occur only in cases (2a), (3a), and (3b) and are μ_2. This is plausible but is not expanded in detail. Since the weighted factor n_Aut(σ') in Equation (6) divides by |Aut(f_i^fine)|, and since the comparison tables with [KM09] and [LR18] are sensitive to such factors, I would like an explicit verification that the exceptional solutions identified in the proof of Theorem 8.22 are the only ones with nontrivial automorphisms, and that the
minor comments (6)
  1. [§1.5] Typo: 'low dimenisonal' should be 'low dimensional'.
  2. [§8.2, Example 8.10] Typo: 'are are 0, ∞' should be 'are 0, ∞'.
  3. [Equation (56)] The product expression for ρ appears to have an index mismatch: the notation ρ=(ρ_{i,j})_{i=2,...,e}^{j=2,...,h_i} is not consistent with the preceding decomposition into factors G^{h_i-1}_m. Please clarify.
  4. [Lemma 8.16, proof] The sentence 'k′ = h_i, k′ = h_i' contains a duplicated bound; one of the two occurrences should presumably be a different index.
  5. [§7.2] The statement that 'both the point class and the unit class on M_{1,1} are pulled back from P^1' would benefit from a one-line justification, since the map M_{1,1} → P^1 is not an isomorphism at the stacky points.
  6. [Acknowledgments] 'Tubigen' should be 'Tübingen'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the combinatorial multiplicities are derived from a genuine lifting criterion and an explicit solution count, not defined to match the algebraic answer.

full rationale

The paper's central claimed derivation is not circular. The tropical weight w(σ) is initially defined in Definition 4.16 in terms of the algebraic automorphism groups of logarithmic maps, and Theorem B is a projection/formula statement. The combinatorial content is concentrated in Theorem D, whose proof goes through Theorem 5.4 (an independent existence criterion for fs logarithmic enhancements) and Lemma 8.21, which establishes a bijection between solutions of the system (59) and interior lattice points of the region P(σ) of Construction 1.10. That region was defined from tropical direction vectors before any counting, so the equality i(P(σ))=#solutions is a proved identification, not a definitional shortcut. The automorphism contributions are enumerated explicitly in Theorem 8.22 and Theorem 8.9, including the case (Example 8.10) where the inclusion Aut/G ↪ Aut(σ) is strict; the result is not deduced by assuming the tropical automorphism group equals the algebraic one. The only self-citations, [CK] and [Koy], are announcements of future work and are not used in the proof. The load-bearing external citations ([RSPW19b], [Wis19], [ACGS20], [Tor14]) are independent prior results, not outputs of this paper. The basis property quoted from [Tor14, Proposition 3.2.18(b)] is external support; if it failed, the system in Lemma 8.21 would not be square, but that is a mathematical dependence, not a circularity. There is an internal step, Claim (ii) in §7.3, that is asserted with a sketched Cartesian-diagram argument and a footnote noting a nearby stacky example where the analogous stabilizer comparison fails when the cone map is not an inclusion; this is an abbreviated proof and a potential correctness gap, but it does not make the theorem equal to its input by construction. Accordingly, the circularity score is 0.

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

The central claim rests on standard toric/log-geometric machinery and on RSPW's moduli stack; no free parameters are fitted to data. The paper introduces no new speculative entities; the region P(σ) and n_Aut(σ) are derived combinatorial constructs, not independent postulates.

assumptions (7)
  • domain assumption The moduli stack W_Γ(X) of well-spaced genus-one stable logarithmic maps is proper, logarithmically smooth, pure of dimension n+m+(r−dim Span(δ)), with tropicalization the cone complex W_Γ(Σ(X)).
    Introduced by RSPW19a,b; used throughout §§1–7 as the space whose intersection numbers are computed. Not reproven here.
  • domain assumption A genus-one tropical stable map admitting a minimal logarithmic map is realizable if and only if it is well-spaced ([RSPW19b, Theorem 4.4.8]).
    Used in §5 and in the proof of Proposition 8.13 to pass from existence of the log lift to realizability.
  • domain assumption The set {ū_{j,k}} of projections of non-cycle half-edge directions forms an R-basis for H_1/H_0 (Proposition 1.9, quoted from [Tor14, Proposition 3.2.18(b)]).
    Basis for the count of solutions in Lemma 8.21, hence for Theorem D's lattice-point multiplicity formula.
  • standard math Minkowski weights compute intersection products on smooth toric varieties ([FS97]).
    Used in the proof of Theorem B, §7.3, to convert algebraic Chow classes into sums over cones.
  • standard math The moduli stack M_{1,1} is isomorphic to P(4,6), with coarse space P^1, and the point/unit classes are pulled back from P^1.
    Used in §7.2 to replace the evaluation target Ev by the smooth toric variety X^n × P^1.
  • domain assumption For fs logarithmic stacks, Hom(T, A_Y) ≅ Hom_Cones(Σ(T), Σ(Y)) (Proposition 7.1, recast from ACGS20 Proposition 2.10).
    Proved in the paper but depends on the established theory of Artin fans and generalized cone complexes.
  • domain assumption Wise's theorem on the space of types ([Wis19, Theorem 1.1]): the space of logarithmic enhancements of a scheme map of a given type is controlled by a monomorphism into the type space with étale projection.
    Used in §8.1 to count the number of logarithmic enhancements of each fine map as the saturation index.

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Pith. "Pith review of Genus one correspondence between tropical and algebraic curves." pith.science (2026). https://pith.science/paper/TG6EWCZU

@misc{pith2026260706426,
  author       = {Pith},
  title        = {Pith review of: Genus one correspondence between tropical and algebraic curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TG6EWCZU}},
  note         = {Machine review of arXiv:2607.06426}
}
abstract

We show that the genuinely enumerative count of algebraic elliptic curves in any toric variety agrees with the count of the corresponding well-spaced tropical curves, weighted by explicit combinatorial multiplicities. This provides a complete genus-$1$ generalization of the celebrated Nishinou--Siebert correspondence theorem in genus $0$. The proof is algebro-geometric and relies on logarithmic deformation theory together with an explicit enumeration of logarithmic maps with fixed tropicalization.

Figures

Figures reproduced from arXiv: 2607.06426 by the authors.

Figure 1
Figure 1. General shape of the neighborhood of the cycle. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Dimension 1, deficiency 1. Contracted parts are shown dashed. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Dimension 2, deficiency 1. 3. Deficiency 2. We have the following cases (examples illustrated in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Dimension 2, deficiency 2. For dim X ≥ 3, there is no analogue of the dual polygon associated to a vertex of a tropical curve. On the other hand, the computation of n Aut(σ ′ ) exhibits a recursive structure. Indeed, the automorphism￾weighted factor n Aut(σ ′ ) depends…
Figure 5
Figure 5. Figure 5: Dimension 3, deficiency 3, neighborhood spans. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: A tropical curve mapping to the fan of P 2 . There are tropical evaluation and stabilization maps fitting into the following commutative diagram: MΓ(Σ(X)) Σ(X) n × Mtrop g,n MΓ((NX)R) (NX) n R × Mtrop g,n (10) The evaluation maps to Σ(X) and (NX)R are defined by sendin…
Figure 7
Figure 7. Figure 7: Example of radial alignment. The cycle is drawn as a large bullet, while the vertices outside the [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: A genus one tropical curve which does not have a logarithmic curve which tropicalizes to it. [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]
Figure 9
Figure 9. Figure 9: A parametrized tropical curve where the cycle consists of two vertices (depicted by large bullets) [PITH_FULL_IMAGE:figures/full_fig_p041_9.png]
Figure 9
Figure 9. Figure 9: A parametrized tropical curve where the cycle consists of two vertices (depicted by large bullets) [PITH_FULL_IMAGE:figures/full_fig_p043_9.png]
Figure 10
Figure 10. Figure 10: Cases in Theorem 8.22(2a), (3a) and (3b). [PITH_FULL_IMAGE:figures/full_fig_p051_10.png]
Figure 10
Figure 10. Figure 10: Cases in Theorem 8.22(2a), (3a) and (3b). [PITH_FULL_IMAGE:figures/full_fig_p053_10.png]

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Tropical and algebraic elliptic plane curves with fixed j-invariant

    math.AG 2026-08 conditional novelty 6.0 of 10

    A tropical proof of Pandharipande's formula for the number of degree-d elliptic plane curves with fixed j-invariant, using well-spaced tropical curves and corrected multiplicities.

  2. Elliptic curve counting in toric threefolds: virtual, enumerative, and tropical

    math.AG 2026-08 conditional novelty 6.0 of 10

    Well-spaced elliptic curve counts equal logarithmic Gromov-Witten invariants plus explicit genus-zero correction terms, yielding a logarithmic analogue of the Getzler-Pandharipande relation.

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