Pith. Sign in

REVIEW

Mirror Counts of Spectral Curves

PaperPDF from arXiv

data files

ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Tom Graber, Mingyuan Hu, Eric Zaslow

Symplectic Geometry · Submitted 6 Sep 2026 · 78 references

Pith's verdict: Rulings are a genuinely new combinatorial handle on curve counts, with strong independent checks, but Theorem 3.1 as stated outruns the dimer-consistency input the proof actually relies on.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 10, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →
Challenge this review Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID2609.06821 (version 1)
Pith IDHRR665RQ, Pith's permanent record of this paper and its review
CitationEric Zaslow, Mingyuan Hu, Tom Graber. "Mirror Counts of Spectral Curves." arXiv:2609.06821 (2026). https://pith.science/paper/HRR665RQ
Cite this reviewBibTeX for Pith's review of this paper
Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Rulings of constructible sheaves count rational nodal curves on toric surfaces, unifying six enumerative methods.

The reading

The paper proves that for a convex lattice polygon defining a toric surface, the number of rational nodal curves with fixed toric boundary intersection can be computed in six different ways and that all six counts coincide. The six methods are: the classical Severi variety, logarithmic Gromov-Witten theory, tropical geometry, the Euler characteristic of coherent-sheaf moduli, the Euler characteristic of constructible-sheaf moduli, and a novel combinatorial object called a 'ruling'. The authors establish the two new equalities involving rulings and constructible sheaves, thereby unifying the entire set. If correct, this result provides a purely combinatorial handle on an important enumerative invariant and suggests deeper connections to refined invariants and higher-genus curve counting via a conjecture relating rulings to Block-Göttsche polynomials.

What carries the argument

The central object is a bipartite graph Γ embedded in the 2-torus, whose zigzag paths form a Legendrian link Λ_Γ. The Goncharov-Kenyon integrable system on Γ, combined with mirror symmetry via the coherent-constructible correspondence, identifies the moduli space of pure one-dimensional coherent sheaves on the toric surface (the relative Jacobian of the linear system) with a moduli space of constructible sheaves on the torus with singular support in Λ_Γ. The ruling decomposition of this constructible-sheaf moduli space—analogous to the Severi stratification by genus—provides the bridge: each ruling ρ corresponds to a stratum of dimension 2g(ρ) + r(ρ), and rational rulings (g=0) give the count of nodal rational curves.

Extended reading notes

Core claim

For any convex lattice polygon Δ, the rational nodal curve count N_Δ equals the number of rational rulings of the Legendrian link Λ_Γ associated to a dimer model on the torus. This is proved by first showing that the moduli space of constructible sheaves microlocally supported on Λ_Γ has a ruling stratification, each stratum isomorphic to (C^*)^{2g} × C^r, where g is the genus of the ruling surface and r the number of returns. Rational rulings correspond to genus-zero strata, and additivity of the Euler characteristic gives N_Δ = number of rational rulings. The equality of N_Δ with all other definitions (tropical, log Gromov-Witten, coherent-sheaf, Severi) then follows from previously known equivalences assembled in the literature, together with the coherent-constructible correspondence for toric stacks established in earlier work.

Load-bearing premise

The dimer model Γ must satisfy a consistency condition (Section 2.7) so that its zigzag Legendrian Λ_Γ is isotopic to the FLTZ skeleton Λ_Σ(Δ)°; the paper does not prove that every convex lattice polygon admits such a consistent dimer model, and the calculations in Section 7 assume one exists for the chosen polygon.

Editorial extensions

If this is right

  • The ruling count provides a purely combinatorial method to compute N_Δ for any polygon admitting a consistent dimer model, bypassing algebraic equations or Gromov-Witten theory.
  • If Conjecture 6.22 holds, the ruling polynomial R_Δ(z) equals the refined tropical invariant N_Δ(q) of Block–Göttsche, meaning rulings encode higher-genus log Gromov–Witten invariants via Bousseau’s theorem.
  • The ruling decomposition yields an explicit formula for the Hirzebruch genus χ_{-y} of the moduli space of constructible sheaves, connecting to motivic invariants of moduli spaces.
  • For triangles with primitive edges, N_Δ = 2·Area(Δ) (Proposition 6.21), providing a fast computable formula for an infinite family of examples.

Reading between the lines

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

  • Because the construction requires a consistent dimer model, the class of polygons for which all six counts are known to agree may be narrower than the class of all convex lattice polygons. Future work could determine whether every polygon admits such a model or whether the count N_Δ can be defined via rulings even when no dimer model exists.
  • The visual similarity between rulings and coamoebas of singular curves (Appendix B) suggests a possible direct interpretation of the ruling surface as the set of critical points of a mirror map, which could lead to a direct geometric construction of the ruling from the original curve family.
  • The gonality gap condition (Conjecture 6.25) mirrors the conifold gap in topological string theory, hinting that ruling polynomials might satisfy modularity properties or recursion relations analogous to those in the Gromov-Witten theory of local Calabi-Yau threefolds.
Original abstract

Given a convex lattice polygon $\Delta\subset \mathbb R^2$, let $N_\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\Delta$ on the toric variety $\mathbb P_\Delta$ defined by $\Delta$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\Delta$. We perform calculations of $N_\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-G\"ottsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,200 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.32 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.85 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Given a convex lattice polygon $\Delta\subset \mathbb R^2$, let $N_\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\Delta$ on the toric variety $\mathbb P_\Delta$ defined by $\Delta$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\Delta$. We perform calculations of $N_\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-G\"ottsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Given a convex lattice polygon $\Delta\subset \mathbb R^2$, let $N_\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\Delta$ on the toric variety $\mathbb P_\Delta$ defined by $\Delta$, having fixed intersection with the toric boundary. background 0.99 0.13 0.04 asserted 0.54
s1 We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. result 0.67 0.93 0.31 asserted 1.00
s2 In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\Delta$. contribution 0.53 0.94 0.53 asserted 1.00
s3 We perform calculations of $N_\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. method 0.61 0.95 0.13 asserted 1.00
s4 Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. result 0.90 0.86 0.07 asserted 1.00
s5 We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-G\"ottsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau. interpretation 0.53 0.16 0.15 speculative 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 2. 1: Zigzag paths and conjugate surface near an edge (left) and for a torus graph Γ (right) with ΛΓ = ΛΣ(∆)◦ where ∆ = □ and P∆ = P 1 × P 1 . Then SΓ is homeomorphic to a thickening of Γ with a twist in each ribbon edge. The intersection form ⟨ , ⟩ on SΓ defines the Poisson structure on Loc1(SΓ) ∼= Loc1(Γ) ∼= Hom(H1(Γ; Z); C ∗ ) ∼= (C ∗ ) b1(Γ): let a, b ∈ H1(Γ; Z) and let z a , zb ∈ O(Loc1(Γ)). Then {z a , zb} = ⟨a, b⟩z a+b . The boundary curves of SΓ are precisely the zigzags γ, and these span the kernel of the intersection form, so z γ are the Casimirs of the Poisson struct
Figure 5. 2: Portion of a path P ⊂ T (left, in blue) and its deformation (right). Proof. We first fix (i, j), 1 ≤ i ≤ s and 1 ≤ j ≤ di . Step 1) First note that while H is not connected, the product of permutation groups G = Sdi−1× Q k̸=i Sdk acts transitively on the set of connected components, and the invari￾ants are unchanged under the action of G composed with a simultaneous relabeling of endpoints. So we can prove invariance by proving invariance under continuous paths h : I → H, i.e. N trop,h(0) ∆ (q) = Ntrop,h(1)(q). Step 2) Next we note that the number and degrees of T are bounded by…
Figure 6. 2: An example of a ruling (middle) and the skeleton of the corre￾sponding ruling surface (right). The Newton polygon is drawn on the left. 6.3. From Sheaves to Rulings. We explain how to get a ruling from a sheaf F ∈ Sh(T,Λ)η 0 . First recall a standard fact in microlocal sheaf theory: Proposition 6.10 ([Gui23, Corollary 4.4.3]). A constructible sheaf E ∈ Shb (R) admits a unique decomposition: (6.5) E = M α CIα [dα] where {Iα ⊂ R}α∈A is a locally finite family of intervals (open, closed, or half-open), and dα are finite integers. Moreover, let e : R → S 1 be the covering map. A
Figure 7. 4: Left: dimer model (blue) and zigzag curves (black) for primitive edge triangle (right), when k = 2 and n = 3. As for tropical curves, recall that the point conditions on the boundary determine the location of the rays of the tropical curves. Specifically, for each marked point pi in the domain curve C, the incidence condition f(pi) ∈ Dρ determines via the Log map an equivalence class in NR/(Rρ) labeling the ith external ray of the tropical curve, with direction ρ. Tropical Curve Counting. Unit-edges triangles have b = 3, so Lemma 7.1 requires that all tropical curves have just one…

Showing 4 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.06821 by the authors.

References

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.