Pith. sign in

REVIEW 3 major objections 7 minor 43 references

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

T0 review · 3 major / 7 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves an exact identity relating well-spaced enumerative counts of elliptic curves in toric threefolds to logarithmic Gromov-Witten invariants, with two explicit tropical correction terms.

desk verdict A genuinely new comparison formula between well-spaced and logarithmic counts in genus one, but the proof has a load-bearing unproved displacement claim and several fixable gaps; worth refereeing but not accepting yet. read the letter →

arxiv 2608.05486 v1 pith:3P7OWL5P submitted 2026-08-06 math.AG

Sae Koyama This is my paper · ORCID
classification math.AG MSC 14N3514T9014H10
keywords ellipticcurvestoricthreefoldslogarithmicGromov-Witteninvariantstropicalwell-spacednessGetzler-Pandharipanderelationdoubleramificationcyclesfloordiagrams
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 aims to establish a precise relationship between two ways of counting genus-one curves in a smooth toric threefold: the enumerative 'well-spaced' count, an integer, and the logarithmic Gromov–Witten invariant, a virtual count defined via logarithmic stable maps. The main theorem says that the well-spaced count equals the logarithmic invariant plus two correction terms built from tropical curves: one term counting non-rigid well-spaced curves, and one term involving squared areas attached to genus-zero tropical curves. A sympathetic reader would care because it gives the first explicit logarithmic analogue of the Getzler–Pandharipande relation, and it makes the difference between virtual and enumerative counts computable by tropical methods. As a consequence, the paper also shows that for projective three-space with its toric boundary, the logarithmic virtual invariants eventually become strictly smaller than ordinary Gromov–Witten invariants.

What carries the argument

The central object is the identity W = L + E1 + E0/24, proved by decomposing both sides into sums over tropical types and comparing the multiplicities M_γ w_γ (well-spaced) and M_γ $m^{{vir}}$_γ (logarithmic). The comparison is organized by the excess dimension e(γ) of the tropical type: the differences for e(γ)=1 and 2 produce the E1 term, while the e(γ)=3 case produces the E0 term through a double ramification cycle integral on M_{1,3} that evaluates to −1/24 times the squared area of the parallelogram spanned by the two edge directions at the relevant vertex.

What would settle it

For the tropical type of Example 2.3.5 (two four-valent vertices joined by two edges, with the cycle contained in a plane), compute the intersection product of the image of ∏_v M_v with the diagonal in (∏_e D_e)^2 using Minkowski weights and generic displacement; if the intersection number is nonzero, the virtual multiplicity does not vanish, contradicting Lemma 4.3.1 and requiring an extra term in Theorem A.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem A: for a smooth toric threefold X and curve class β, with conditions given by a lines and b points, the well-spaced invariant W_{X,β}(φ) equals L_{X,β}(φ) + E1 + (1/24)E0, where E1 is the number of non-rigid well-spaced tropical curves with multiplicity M_γ w_γ, and E0 is the sum, over genus-zero tropical curves through the same conditions, of ∑_{v∈V(γ)} |u_1 ∧ u_2|^2 M_γ, with u_1, u_2 the directions of any two edges leaving vertex v. This identity is obtained by decomposing both invariants into contributions indexed by tropical types and comparing type by type according to the excess dimension e(γ) of the deformation space. For e(γ)=0 the two counts agree; for e(γ)=1 and for e(γ)=2 with the cycle in a plane, the well-spaced contribution survives while the logarithmic virtual contribution vanishes; for e(γ)=3 the well-spaced contribution vanishes and the logarithmic contribution is evaluated by a double ramification cycle computation to be −(1/24)|u_1∧u_2|^2 M_γ. The paper presents this as a logarithmic analogue of the Getzler–Pandharipande formula for elliptic curves in $P^{3}$.

Load-bearing premise

The proof hinges on the unproved claim in Lemma 4.3.1 that a certain tropical intersection--the product of vertex moduli spaces against the diagonal of edge divisors--can be generically displaced to be empty, which is what makes all excess-dimension-1 logarithmic contributions vanish.

Editorial extensions

If this is right

  • The formula provides a practical tropical algorithm for computing logarithmic Gromov–Witten invariants of elliptic curves in any toric threefold: enumerate well-spaced tropical curves, compute M_γ and w_γ, and add the modified genus-zero contribution E0 via floor diagrams.
  • For X = P^3 with its toric boundary, combining Theorem A with the Getzler–Pandharipande relation yields (d!)^4 GW^1_{a,b} ≥ L_{a,b}, with strict inequality for all line conditions once d ≥ 3 and for all point conditions once d ≥ 4.
  • The correction E1 has geometric meaning as curves whose circuit components map into a plane, and the paper shows how to compute it via floor diagrams in examples, including the full degree-4 case (a,b)=(2,7).
  • The equality holds for every smooth toric threefold and any curve class, independent of the explicit tropical multiplicities computed in prior work, so it can serve as a consistency check for independent calculations of either invariant.

Reading between the lines

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

  • If Theorem A holds, the same type-by-type comparison might extend to higher genus curves in toric threefolds, with correction terms expressed through higher double ramification cycles and more elaborate vertex multiplicities; the genus-one case is the first non-trivial instance.
  • The strict inequality for P^3 suggests that for other Fano toric threefolds the logarithmic invariants relative to the toric boundary will eventually fall below the ordinary Gromov–Witten invariants, with the threshold determined by the smallest degree at which a marked floor diagram develops a floor of divergence 3 or a weight-2 elevator.
  • The unproved displacement claim in Lemma 4.3.1 could be verified or refuted by a direct tropical intersection computation; if it fails, the equality W = L + E1 + E0/24 would need an additional term for planar excess-dimension-1 types.
  • The modified floor diagrams used to compute E0 may be adaptable to refined or descendent invariants in genus one, since the vertex multiplicity |u_1∧u_2|^2 resembles refined tropical multiplicities.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper studies the enumerative geometry of genus-one curves in smooth toric threefolds. It compares the well-spaced enumerative invariants W_{X,β}(φ) defined by Cela–Koyama with the logarithmic Gromov–Witten invariants L_{X,β}(φ). The main result, Theorem A, states that W_{X,β}(φ) = L_{X,β}(φ) + E_1 + (1/24)E_0, where E_1 counts non-rigid well-spaced tropical curves with multiplicity M_γ · w_γ and E_0 is an explicit weighted sum over genus-zero tropical curves. The proof strategy is to decompose the logarithmic invariant into tropical contributions via a decomposition theorem, then compare each contribution with the well-spaced tropical correspondence, establishing vanishing or equality in excess dimensions 0, 1, 2, and computing the excess-dimension-3 contribution by a double ramification cycle calculation. Applications include Corollary B, a strict inequality between logarithmic and ordinary Gromov–Witten invariants for P^3 at sufficiently large degree, and several worked examples using floor diagrams.

Significance. If the proof can be completed, Theorem A is a substantial result: it gives a logarithmic analogue of the Getzler–Pandharipande relation and provides an explicit, parameter-free comparison between virtual and enumerative invariants in a higher-dimensional setting. The correction terms E_0 and E_1 have concrete tropical and geometric interpretations, and the paper includes computable examples and a low-degree analysis supporting Corollary B. The use of existing double ramification cycle computations (including a computer-algebra check via admcycles) is a strength. However, the proof is not complete as written: several load-bearing steps are asserted rather than proved, and one of them, the vanishing argument in Lemma 4.3.1, appears to have a genuine logical gap. The result is likely to be of interest to researchers in logarithmic Gromov–Witten theory, tropical geometry, and enumerative geometry, provided those gaps are addressed.

major comments (3)
  1. [Section 4.3, Lemma 4.3.1] The proof of Lemma 4.3.1 contains the key assertion that the tropical image of ×_v M_v in (×_e D_e)^2 "can be generically displaced so that it does not meet the tropical image of △" because each vertex image lies in A = {z_1 = z_2}. This does not follow: the diagonal △ is itself contained in A, so containment in A gives no information about whether the image can be translated off the diagonal. The allowed displacements must come from deformations of the vertex moduli spaces, and in the minimal two-vertex, two-edge model of Example 2.3.5 the gluing leaves exactly a one-parameter family in the common z-direction; a one-sided translation avoiding △ is not induced by a deformation of the source. This step is load-bearing, as it is used to kill all excess-dimension-1 virtual contributions in Sections 4.3 and 4.4. If the displacement claim is false, the right-hand side of Theorem A would acquire additional correction terms. The proof also mentions that the excess bundle is trivial but does not connect this to the claimed vanishing through the excess intersection formula. A complete proof requires either a rigorous excess-intersection computation or a precise deformation argument showing that the intersection actually vanishes.
  2. [Section 2.4.2, Proposition 2.4.6] The decomposition theorem is stated and 'proved' in four terse steps, but two of the steps are not justified. Step 1 asserts the existence of a subdivision such that the induced morphism of Artin fans is flat and that (X^n)† can be taken to be a smooth toric variety; this is nontrivial and no reference is provided. Step 4 concludes that c_vir(σ) = sum_γ m_γ m_vir_γ from the description of the relevant piecewise polynomial, but the identification of the multiplicity M_γ with the determinant of the tropical evaluation map is not derived. Since Proposition 2.4.6 is the bridge between the logarithmic invariant L_{X,β} and the tropical sum that underlies Theorem A, this needs either a detailed proof or a precise citation to a version with a complete argument.
  3. [Section 4.5, Lemmas 4.5.1 and 4.5.2] The derivation of the E_0 correction term depends on Lemma 4.5.1, which is proved by invoking a factorization theorem from [33] without stating it precisely. The claim that, after an appropriate logarithmic modification, the strict transform of TC_1(a,b,0) is the product of the strict transforms of TC_1(a,b) and DR_1(0), and that pushing down gives TC_1(a,b)DR_1(0), is not justified. In particular, the assertion that the third vector being 0 'imposes no further subdivisions' does not by itself imply the product structure of strict transforms. This lemma is used to compute the coefficient -1/24 in Lemma 4.5.2, so the gap directly affects the stated form of E_0 in Theorem A. The authors should either provide the precise statement from [33] with a proof of the needed case, or supply a self-contained argument.
minor comments (7)
  1. [Section 1.1, Theorem A] The index ranges in the statement of Theorem A are inconsistent: it should read φ_1, ..., φ_a ∈ H^4(X) and φ_{a+1}, ..., φ_{a+b} ∈ H^6(X), rather than φ_1, ..., φ_{a+1} ∈ H^4(X) and φ_a, ..., φ_{a+b} ∈ H^6(X).
  2. [Definition 2.1.4] The definition says 'A tropical map from a tropical curve Γ to Σ_X is a tropical map from R^r ...' which should presumably be 'a tropical map from Γ to R^r'.
  3. [Section 2.4.2, Step 1] There is a typo: 'subidivison' should be 'subdivision'.
  4. [Definition 3.3.5(3)(a)] The text 'Let i(P) be the Then the number of interior lattice points ...' is incomplete; i(P) should be defined as the number of interior lattice points, and the sentence should be completed.
  5. [Section 4.5, Lemma 4.5.2] In the proof of Lemma 4.5.2, the phrase 'Applying admcycles5, we obtain Then ∫ ...' contains a stray 'Then' and should be rephrased for readability.
  6. [Section 5.4] The displayed formula L_{2,7} = W_{2,7} - (1/24)E_0 = (4!)^4 - 191/6 is dimensionally inconsistent with the preceding computations, which give W_{2,7} = 4(4!)^4 and E_0 = 860(4!)^4; the correct value is L_{2,7} = (-191/6)(4!)^4, or the displayed identity should be for L_{2,7}/(4!)^4.
  7. [Section 5.3, Proposition 5.3.1 and Figure 13] Proposition 5.3.1 claims positivity 'precisely for the (a,b) given in Figure 13', but the figure only shows a shaded region in the (a,d)-plane and the proof does not give the exact inequalities; the region should be described explicitly or given as a table.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A compares independently defined invariants and the proof derives the relation rather than assuming it.

full rationale

Theorem A relates the logarithmic virtual invariant L_{X,beta}(phi), defined as an integral over the virtual class of the logarithmic stable maps space M_Lambda(X) in Section 2.3, to the well-spaced count W_{X,beta}(phi), defined as an integral over the moduli space W_Lambda(X) in Section 3.3 and identified with a tropical sum via [14, Theorem B]. The correction terms E1 and E0 are defined directly as tropical sums, in Sections 3.3 and 4.5 respectively, and are not fitted parameters or renamed versions of L. The proof of Theorem A proceeds by comparing per-tropical-type contributions (Table 1) and by establishing vanishing and matching statements in Lemmas 4.3.1, 4.3.2, 4.4.6, 4.5.2, and 4.5.5; the target identity is not used as an input. Although several cited results, especially [14], [30], [31], [33], and [39], are authored by the author or their advisor, these are external theorems with their own stated assumptions and do not assume the formula being proved. The paper even emphasizes that Theorem A is independent of the explicit multiplicities found in [14]. The unproved generic-displacement assertion in Lemma 4.3.1 is a possible correctness gap in the proof of the vanishing of the excess contribution, but it is not a circular reduction: the claimed vanishing is argued from geometry and Minkowski-weight intersection, not derived from the equality W = L + E1 + E0/24. Self-citation alone does not constitute circularity, and no equation in the paper reduces to another by definition or by fitted constants.

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

The paper introduces no new geometric objects, particles, or fitted constants. The correction terms E0 and E1 are derived quantities, not independent inputs. The main imported assumptions are the existence and properties of the well-spaced moduli space, the gluing and rigidification theorems, the tropical correspondence theorem from the author's companion paper, and a factorization statement for higher double ramification cycles. The ledger is dominated by domain assumptions from previous work by the same group.

assumptions (7)
  • domain assumption The tropical correspondence theorem for well-spaced curves W = sum M_gamma w_gamma (Theorem 3.3.4) holds for all toric threefolds and the conditions considered.
    Stated as Theorem B of [14] by the same author; Theorem A of this paper relies on it for the enumerative side of the comparison but disclaims dependence on the explicit weights.
  • domain assumption The well-spaced, radially-aligned moduli space W_Lambda(X) exists, is logarithmically smooth, and gives well-spaced invariants (Theorem 3.2.4, [39, Theorem B]).
    Imported from prior work [39]; its properties are used in Definition 3.3.1 and in identifying the boundary strata.
  • domain assumption The gluing formula (Theorem 2.5.2, [31, Theorem 8.3.2]) and the rigidification results [30] apply to the expanded target moduli M_Lambda as used in Lemma 4.3.1 and Section 4.5.
    Section 2.5 and Sections 4.3 and 4.5 invoke these external results without proof; the vanishing of virtual multiplicities in Lemmas 4.3.1 and 4.5.2 depends on them.
  • domain assumption Higher double ramification cycles factorize after a logarithmic modification in the sense of [33], used in Lemma 4.5.1.
    Lemma 4.5.1 is a key computation for the E0 coefficient and is proved by citing [33] for a factorization statement; no proof is given in this paper.
  • ad hoc to paper The tropical image vanishing argument in Lemma 4.3.1: the tropical image of the product of vertex moduli can be generically displaced from the tropical diagonal, so the fiber product intersection vanishes.
    This displacement step is stated without a detailed proof in Lemma 4.3.1 and is load-bearing for the vanishing of the excess-dimension-1 logarithmic contribution.
  • domain assumption General position assumptions (Definitions 4.1.1 to 4.1.5) on point and line conditions exclude finitely many positive-codimension loci, so all tropical curves that appear have the stated rigidity and excess dimensions.
    The classification of contributing tropical types (Propositions 4.4.1 and 4.5.3) depends on these generality conditions; the paper notes they are generic but does not parameter-count every case.
  • standard math DR1(0) = -lambda1 on M_{1,n} and the Janda-Pandharipande-Pixton-Zvonkine double ramification formula hold as cited.
    Used in Lemma 4.5.1 and the lambda1-integral factorization; standard results in the double ramification cycle literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Elliptic curve counting in toric threefolds: virtual, enumerative, and tropical." pith.science (2026). https://pith.science/paper/3P7OWL5P

@misc{pith2026260805486,
  author       = {Pith},
  title        = {Pith review of: Elliptic curve counting in toric threefolds: virtual, enumerative, and tropical},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3P7OWL5P}},
  note         = {Machine review of arXiv:2608.05486}
}
abstract

We study the enumerative geometry of elliptic curves in toric threefolds. We consider enumerative integer invariants, called well-spaced counts, which can be studied using well-spaced genus-one tropical curves in $\mathbb{R}^3$. By comparing this with the logarithmic degeneration formula, we obtain an explicit relationship between logarithmic virtual invariants and these geometric invariants. The result is a logarithmic analogue of a formula of Getzler--Pandharipande for elliptic curves in $\mathbb{P}^3$. As an application, we show that the virtual logarithmic invariants for $\mathbb{P}^3$ with respect to its toric boundary are strictly less than the ordinary Gromov--Witten invariants once the degree is sufficiently large. Several examples are included.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 40 canonical work pages

  1. [33]

    MOLCHO ANDD

    S. MOLCHO ANDD. RANGANATHAN,A case study of intersections on blowups of the moduli of curves, Alg. Number Th., 18 (2024), pp. 1767–1816. 26

    work page 2024

  2. [14]

    Genus one correspondence between tropical and algebraic curves

    A. CELA ANDS. KOYAMA,Genus one correspondence between tropical and algebraic curves, arXiv preprint arXiv:2607.06426, (2026). 1, 2, 5, 11, 12, 18, 19

    work page Pith review arXiv 2026

  3. [1]

    ABRAMOVICH ANDQ

    D. ABRAMOVICH ANDQ. CHEN,Stable logarithmic maps to Deligne-Faltings pairs II, Asian J. Math., 18 (2014), pp. 465 – 488. 1

    work page 2014

  4. [2]

    ABRAMOVICH, Q

    D. ABRAMOVICH, Q. CHEN, M. GROSS,ANDB. SIEBERT,Decomposition of degenerate Gromov–Witten invariants, Compos. Math., 156 (2020). 1, 2, 3, 5, 8, 11, 12, 14

    work page 2020

  5. [3]

    ,Punctured logarithmic maps, Mem. Eur. Math. Soc., (2024). 5

    work page 2024

  6. [4]

    BATTISTELLA, N

    L. BATTISTELLA, N. NABIJOU,ANDD. RANGANATHAN,Curve counting in genus one: Elliptic singularities and relative geometry, Algebr. Geom., (2021), pp. 637–679. 5 ELLIPTIC CURVE COUNTING IN TORIC THREEFOLDS: VIRTUAL, ENUMERATIVE, AND TROPICAL 35 2 1 1 µ=2, n=1 2 1 1 µ=1, n=2 FIGURE19. Marked floor diagrams of genus1for(a, b) = (2, 7)with non-zero multiplicity

    work page 2021

  7. [5]

    BERTRAND, E

    B. BERTRAND, E. BRUGALL ´E,ANDG. MIKHALKIN,Genus 0 characteristic numbers of the tropical projective plane, Compos. Math., 150 (2014), pp. 46–104. 5

    work page 2014

  8. [6]

    BLECHMAN ANDE

    L. BLECHMAN ANDE. SHUSTIN,Refined descendant invariants of toric surfaces, Discrete & Computational Geometry, 62 (2019), pp. 180–208. 5

    work page 2019

Show all 43 references
  1. [7]

    BOUSSEAU,Refined floor diagrams from higher genera and lambda classes, Selecta Math., 27 (2021), p

    P. BOUSSEAU,Refined floor diagrams from higher genera and lambda classes, Selecta Math., 27 (2021), p. 43. 5

    2021

  2. [8]

    BRUGALL ´E,Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces, Adv

    E. BRUGALL ´E,Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces, Adv. Math., 279 (2015), pp. 438–500. 5

    2015

  3. [9]

    BRUGALL ´E ANDG

    E. BRUGALL ´E ANDG. MIKHALKIN,Enumeration of curves via floor diagrams, Comptes Rendus Mathematique, 345 (2007), pp. 329–334. 27, 30, 31

    2007

  4. [10]

    ,Floor decompositions of tropical curves: the planar case, Proceedings of the G ¨okova Geometry-Topology confer- ence, (2009), pp. 64–90. 1, 3, 27

    2009

  5. [11]

    CAROCCI ANDN

    F. CAROCCI ANDN. NABIJOU,Rubber tori in the boundary of expanded stable maps, J. Lond. Math. Soc., 109 (2024), p. e12874. 25, 26

    2024

  6. [12]

    CAVALIERI, H

    R. CAVALIERI, H. MARKWIG,ANDD. RANGANATHAN,Tropical and Logarithmic Methods in Enumerative Geometry, 01 2023. 1, 12, 23

    2023

  7. [13]

    CAVALIERI, H

    R. CAVALIERI, H. MARKWIG,ANDJ. SCHMITT,k-leaky double Hurwitz descendants, in Forum Math. Sigma, vol. 13, Cambridge University Press, 2025, p. e60. 5

    2025

  8. [15]

    CHEUNG, L

    M.-W. CHEUNG, L. FANTINI, J. PARK,ANDM. ULIRSCH,Faithful realizability of tropical curves, Int. Math. Res. Not., (2015), p. rnv269. 4

    2015

  9. [16]

    DELECROIX, J

    V. DELECROIX, J. SCHMITT,ANDJ.VANZELM,admcycles-a sage package for calculations in the tautological ring of the moduli space of stable curves, JSAG, 11 (2022), pp. 89–112. 26, 27

    2022

  10. [17]

    DING ANDJ

    Y. DING ANDJ. HU,Refined floor diagrams relative to a conic and Caporaso–Harris type formula, arXiv preprint arXiv:2509.06004, (2025). 5

    2025

  11. [18]

    FULTON ANDB

    W. FULTON ANDB. STURMFELS,Intersection theory on toric varieties, Topology, 36 (1997), pp. 335–353. 11, 12, 13

    1997

  12. [19]

    GETZLER,Intersection theory onM 1,4 and elliptic Gromov–Witten invariants, J

    E. GETZLER,Intersection theory onM 1,4 and elliptic Gromov–Witten invariants, J. Am. Math. Soc., 10 (1997), pp. 973–

    1997

  13. [20]

    GRABER, J

    T. GRABER, J. KOCK,ANDR. PANDHARIPANDE,Descendant invariants and characteristic numbers, Am. J. Math., 124 (2002), pp. 611–647. 5

    2002

  14. [21]

    GROSS,Remarks on gluing punctured logarithmic maps, arXiv preprint arXiv:2306.02661, (2023)

    M. GROSS,Remarks on gluing punctured logarithmic maps, arXiv preprint arXiv:2306.02661, (2023). 14

    2023 arXiv

  15. [22]

    GROSS ANDB

    M. GROSS ANDB. SIEBERT,Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc., 26 (2013), pp. 451–510. 1, 10 36 SAE KOYAMA

    2013

  16. [23]

    HOLMES, S

    D. HOLMES, S. MOLCHO, R. PANDHARIPANDE, A. PIXTON,ANDJ. SCHMITT,Logarithmic double ramification cycles, Invent. Math., (2025), pp. 1–87. 1, 3, 5, 26

    2025

  17. [24]

    JANDA, R

    F. JANDA, R. PANDHARIPANDE, A. PIXTON,ANDD. ZVONKINE,Double ramification cycles on the moduli spaces of curves, Publications math´ematiques de l’IH ´ES, 125 (2017), pp. 221–266. 5, 26

    2017

  18. [25]

    KANNAN ANDT

    S. KANNAN ANDT. D. SONG,The dual complex ofM 1,n(Pr, d)via the geometry of the Vakil–Zinger moduli space, arXiv preprint arXiv:2411.03518, (2024). 17

    2024 arXiv

  19. [26]

    KATZ,Lifting tropical curves in space and linear systems on graphs, Adv

    E. KATZ,Lifting tropical curves in space and linear systems on graphs, Adv. Math., 230 (2012), pp. 853–875. 4

    2012

  20. [27]

    KENNEDY-HUNT, Q

    P. KENNEDY-HUNT, Q. SHAFI,ANDA. URUNDOLILKUMARAN,Tropical refined curve counting with descendants, Commun. Math. Phys., 405 (2024), p. 240. 5

    2024

  21. [28]

    KERBER ANDH

    M. KERBER ANDH. MARKWIG,Counting tropical elliptic plane curves with fixedj-invariant, Comment. Math. Helv., 84 (2009), pp. 387–427. 5

    2009

  22. [29]

    MANDEL ANDH

    T. MANDEL ANDH. RUDDAT,Descendant log gromov–witten invariants for toric varieties and tropical curves, Trans. Am. Math. Soc., 373 (2020), pp. 1109–1152. 5

    2020

  23. [30]

    MAULIK ANDD

    D. MAULIK ANDD. RANGANATHAN,Gromov–Witten theory, degenerations, and the tautological ring, arXiv preprint arXiv:2510.04779, (2025). 14

    2025

  24. [31]

    ,Logarithmic enumerative geometry for curves and sheaves, Camb. J. Math., 13 (2025), pp. 51–172. 3, 8, 15

    2025

  25. [32]

    MIKHALKIN,Enumerative tropical algebraic geometry inR 2, J

    G. MIKHALKIN,Enumerative tropical algebraic geometry inR 2, J. Am. Math. Soc., 18 (2005), pp. 313–377. 1, 4

    2005

  26. [34]

    NISHINOU ANDB

    T. NISHINOU ANDB. SIEBERT,Toric degenerations of toric varieties and tropical curves, (2006). 1

    2006

  27. [35]

    RANGANATHAN,Skeletons of stable maps II: superabundant geometries, Res

    D. RANGANATHAN,Skeletons of stable maps II: superabundant geometries, Res. Math. Sci., 4 (2017). 4

    2017

  28. [36]

    Geom., 9 (2022), pp

    ,Logarithmic Gromov–Witten theory with expansions, Algebr. Geom., 9 (2022), pp. 714–761. 2, 5, 9, 10, 13, 14, 15

    2022

  29. [37]

    RANGANATHAN ANDA

    D. RANGANATHAN ANDA. U. KUMARAN,Logarithmic Gromov–Witten theory and double ramification cycles, J. Reine Angew. Math., 2024 (2024), pp. 1–40. 1, 5

    2024

  30. [38]

    RANGANATHAN, K

    D. RANGANATHAN, K. SANTOS-PARKER,ANDJ. WISE,Moduli of stable maps in genus one and logarithmic geometry, I, Geom. Topol., 23 (2019), pp. 3315–3366. 4, 5

    2019

  31. [39]

    Number Th, 13 (2019), pp

    ,Moduli of stable maps in genus one and logarithmic geometry, II, Alg. Number Th, 13 (2019), pp. 1765–1805. 1, 4, 5, 17, 18

    2019

  32. [40]

    D. E. SPEYER,Parameterizing tropical curves. I: Curves of genus zero and one., Alg. Number Th., 8 (2014), pp. 963–998. 1, 4

    2014

  33. [41]

    ULIRSCH,Functorial tropicalization of logarithmic schemes: the case of constant coefficients, Proc

    M. ULIRSCH,Functorial tropicalization of logarithmic schemes: the case of constant coefficients, Proc. Lond. Math. Soc., 114 (2017), pp. 1081–1113. 8

    2017

  34. [42]

    VAKIL,The enumerative geometry of rational and elliptic curves in projective space, arXiv: Algebraic Geometry, (1997)

    R. VAKIL,The enumerative geometry of rational and elliptic curves in projective space, arXiv: Algebraic Geometry, (1997). 18

    1997

  35. [43]

    VAKIL ANDA

    R. VAKIL ANDA. ZINGER,A desingularization of the main component of the moduli space of genus-one stable maps into Pn, Geom. Topol., 12 (2008), pp. 1–95. 4

    2008

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.