Tropical geometry in torus bundles

Emily Dodwell

REVIEW 3 major objections 4 minor 13 references

Tropical geometry in torus bundles

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that tropical curves in nontrivial torus bundles obey a balancing condition corrected by first Chern classes: $\sum_\rho \langle e_i, v_\rho\rangle \omega_\rho = c_1(L_i)\cdot \pi_*[Z]$.

desk verdict A genuine, short result: the balancing condition for tropical curves acquires a Chern-class correction in non-trivial torus bundles; worth refereeing, but the theorem's field hypothesis needs to be stated honestly. read the letter →

arxiv 2509.02569 v1 pith:7CF7IYJV submitted 2025-08-19 math.AG

Emily Dodwell This is my paper · ORCID

classification math.AG MSC 14T0514M2514C17

keywords tropicalgeometrybalancingconditiontorusbundlestoricvarietyChernclassescompactificationintersectiontheory

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

Tropical geometry traditionally records subvarieties of algebraic tori as weighted polyhedral complexes whose edge weights satisfy a zero-sum balancing condition. This paper moves the setting to torus bundles, families of algebraic tori varying over a base, and proves that for a curve in such a bundle the balancing condition is no longer zero. Instead, the weighted sum of primitive edge vectors equals a vector of Chern-class pairings, $\sum_\rho \langle e_i, v_\rho\rangle \omega_\rho = c_1(L_i)\cdot \pi_*[Z]$ for each coordinate. When the bundle is trivial the correction vanishes and the classical zero-sum condition returns; when the bundle is nontrivial, the deviation is exactly the first Chern classes of the line bundles defining the torus bundle. That matters because it gives an elementary, intersection-theoretic route to balancing conditions in settings beyond tori, including logarithmic curves.

What carries the argument

The machinery is geometric tropicalization via toric variety bundles, together with intersection-theoretic weights. A toric variety bundle is a fiber bundle whose fibers are toric varieties, obtained by compactifying each fiber of the torus bundle; its horizontal divisors $D_\rho$ correspond to the rays of the toric fan. For a curve $Z$, the weight $\omega_\rho$ on a ray is the intersection multiplicity of the closure of $Z$ with $D_\rho$. The key algebraic input is a divisor relation on the toric variety bundle, stated as Lemma 4.2.3: in the codimension-one Chow group, $\sum_\rho \langle e_i, v_\rho\rangle\, D_\rho = \pi^* c_1(L_i)$ for each $i$. Intersecting this relation with the curve class $[Z]$ and applying the projection formula turns the divisor relations into the $n$ numerical equations of Theorem 4.3.1, which is precisely the modified balancing condition.

What would settle it

On the Hirzebruch surface $\mathbb{P}(\mathcal{O}(k)\oplus\mathcal{O})$ over $\mathbb{P}^1$, take a curve $Z$ and compute its intersection multiplicities $\omega_1=Z\cdot D_1$ and $\omega_2=Z\cdot D_2$ with the two horizontal boundary divisors using classical intersection theory; the theorem predicts $\omega_1-\omega_2 = k\,\deg(\pi_*[Z])$. A curve for which this identity fails would show that the divisor relation in Lemma 4.2.3 or the weight definition is incorrect.

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Extended reading notes

Core claim

The paper's central discovery is that the balancing condition for tropicalizations of curves extends from algebraic tori to torus bundles, but with an extra term controlled by the geometry of the bundle. For a curve $Z$ in a torus bundle $X$ over a smooth projective base $B$, where $X$ is built from line bundles $L_1,\dots,L_n$ by deleting zero sections and fibering them over $B$, the tropicalization is at most a one-dimensional weighted polyhedral complex. The main theorem, Theorem 4.3.1, says the weights $\omega_\rho$ assigned to the rays satisfy $$\sum_\rho \langle e_i, v_\rho\rangle\,\omega_\rho = c_1(L_i)\cdot \pi_*[Z]$$ for every basis vector $e_i$ of the cocharacter lattice, where $v_\rho$ is the primitive vector along the ray and $\beta=\pi_*[Z]\in H_2(B)$. The usual balancing condition $\sum_\rho \langle e_i,v_\rho\rangle\omega_\rho=0$ is therefore a special case, valid exactly when the Chern-class terms vanish on the pushed-forward curve class. Non-triviality of the bundle is detected precisely by this failure, and the failure is captured by first Chern classes. The argument shows that the weights are intersection multiplicities with the horizontal toric boundary divisors, and the divisor relations on the toric variety bundle convert the classical zero-sum relation into the modified identity.

Load-bearing premise

The balancing formula rests on a cited divisor relation in the Chow group of the toric variety bundle, and the proof only covers conical tropical curves, so the statement is established only when both the divisor relation and the conical assumption hold.

Editorial extensions

If this is right

For a trivial torus bundle, or more generally whenever $c_1(L_i)\cdot \pi_*[Z]=0$ for all $i$, the modified condition coincides with the ordinary zero-sum balancing condition, so the classical theorem is recovered as the vanishing case.
On a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$ with $L=\mathcal{O}(k)$, the two horizontal divisors satisfy $D_1-D_2 = \pi^* c_1(\mathcal{O}(k))$, giving the explicit balancing equation $\omega_1-\omega_2 = k\,\deg(\pi_*[Z])$; here $k$ is the obstruction to the zero-sum rule.
For a simple normal crossings pair, the embedding into a toric variety bundle makes this formula a concrete balancing condition for tropicalizations of logarithmic curves, since the pullback of the horizontal toric boundary is the boundary divisor of the pair.
Because the tropicalization is independent of the chosen compactifying toric variety bundle, the theorem gives an invariant constraint: the weights must satisfy the Chern-class equations for any smooth complete fan used to compactify the bundle.
The formula constrains which weighted polyhedral complexes can appear as tropicalizations: for a fixed curve class $\beta$, the vector $(c_1(L_1)\cdot\beta,\dots,c_1(L_n)\cdot\beta)$ must be realized as the weighted sum of primitive ray vectors.

Reading between the lines

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

For non-Archimedean curves, where tropicalizations contain bounded edges and interior vertices, the same divisor-relation argument should apply vertex-by-vertex: each vertex should satisfy the same Chern-class-modified balancing equation, but this requires treating bounded-edge contributions explicitly and is not established in the paper.
The theorem suggests an enumerative correspondence for torus bundles: counting weighted tropical curves of a given degree should recover algebraic curve counts only after the Chern-class correction is inserted, since the correction measures the nontriviality of the bundle along the curve class.
In higher dimensions, the same horizontal-divisor relations should yield analogous constraints on the weights of codimension-one faces of tropicalizations of subvarieties of torus bundles, using Chow classes of toric variety bundles; the paper proves only the curve case.
Read as an existence obstruction, the formula says a proposed tropical curve with prescribed weights and degree $\beta$ is realizable only if the Chern-class vector lies in the lattice generated by the primitive ray vectors, so nontrivial bundles restrict tropical degrees in a concrete, testable way.

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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

Dear [Name],

The short version: this is a real result, not a repackaging. The paper proves a balancing condition for tropicalizations of curves in torus bundles: instead of the usual zero-sum condition, you get sum_rho omega_rho = c1(L_i)·beta. That is new, and it is exactly the right statement. The proof is a clean intersection-theoretic argument: the divisor relations on a toric variety bundle (from Sankaran-Uma) give a relation among the horizontal divisors, intersect with the curve, and apply the projection formula. The examples, especially the Hirzebruch surface case, make the phenomenon concrete. The paper is well-written and the logic in the conical case is sound.

The main soft spot is scope, and it is real. Theorem 4.3.1 is stated with no field hypothesis, but the proof only covers the conical, trivially-valued (e.g. complex) setting. The tropicalization in Definition 3.1.2 is a fan-like complex with only rays and a single vertex; non-Archimedean curves have bounded edges and interior vertices, and Section 1.2 explicitly defers that to future work. So the theorem as written overreaches its proof. This is fixable: either restrict the statement to curves over C (or a trivially valued field), or actually handle bounded edges. As it stands, a reader could reasonably expect the theorem to apply to Puiseux-coefficient curves and be misled.

A second, minor issue: the paper uses intersections D_rho · [Z] and pushforward pi_*[Z] without stating that Z is proper/complete. For the intersection numbers to be genuine degrees, Z should be a complete curve. This is a small missing hypothesis, easily added.

I would not count the reliance on Sankaran-Uma and Ulirsch against the paper. The contribution is the formulation and the observation that the Chern class correction appears; the proof being short is a feature, not a flaw. The citation pattern is honest and the paper credits the sources appropriately.

Who is this for? Tropical and logarithmic geometers, and anyone working on compactifications of subvarieties of torus bundles. It's a useful reference and a good starting point for the non-Archimedean extension. I'd send it to a serious referee. It is not a major reorganization of the field, but it is a correct, clean result that deserves to be in the literature after the scope issue is fixed.

Referee Report

3 major / 4 minor

Summary. The paper proposes a tropicalization for subvarieties of a torus bundle X over a smooth projective base B, where X is the product of the total spaces of line bundles L_1,...,L_n with their zero sections removed. The author defines geometric tropicalization in this setting by compactifying X to a toric variety bundle Y and assigning a weight to each ray of the tropicalization as the intersection multiplicity of the curve Z with the corresponding horizontal toric divisor D_rho. The main result, Theorem 4.3.1, states that for a curve Z in X, the weighted sum of primitive integral directions satisfies sum_rho <e_i,v_rho> omega_rho = c_1(L_i) * pi_*[Z] for each i, replacing the classical zero-sum balancing condition by a Chern-class-modified relation when the bundle is nontrivial. The proof is a direct application of a divisor relation for toric variety bundles, Lemma 4.2.3, combined with the projection formula.

Significance. If the result holds in the stated generality, it is a meaningful contribution: it gives an elementary, intersection-theoretic derivation of a balancing-type condition for curves in torus bundles and explains exactly how nontriviality of the bundle deforms the classical balancing condition. The main theorem is clean, the proof is short, and the Hirzebruch-surface example in Section 4.1 correctly illustrates the phenomenon. The paper also makes a plausible compatibility claim with Ulirsch's logarithmic tropicalization. There are no fitted parameters or circular definitions: the right-hand side of the balancing formula is a genuine Chern class evaluation. The principal weakness is that the theorem is stated more broadly than the proof actually covers, and the key divisor relation is imported from a citation without the required hypotheses being checked in this setting.

major comments (3)

[Section 4.3 / Theorem 4.3.1 and Section 1.2] Theorem 4.3.1 is stated without any field hypothesis, but the proof and Definitions 3.1.2-3.1.3 only treat the conical tropicalization coming from a compactifying fan, where every edge is an unbounded ray from the origin. For a curve over a non-Archimedean field such as the Puiseux series field, the standard valuation tropicalization defined in Section 3.2 contains bounded edges and interior vertices; the paper assigns no weight to bounded edges and proves no local balancing equation at those vertices. Since Section 1.2 explicitly defers non-Archimedean tropicalization to future work, the theorem must either be restricted to the trivially valued (conical) case or the statement and proof must be extended to handle bounded edges and interior vertices.
[Lemma 4.2.3] Lemma 4.2.3 is the sole load-bearing input for the proof of Theorem 4.3.1, but its proof is only a citation to Sankaran-Uma [4, Theorem 1.2]. The author should state the precise hypotheses under which the relation sum_rho <e_i,v_rho> D_rho = pi^* c_1(L_i) holds in the relevant Chow group, and should verify that these hypotheses are satisfied by the toric variety bundles Y constructed in Construction 3.0.3 for arbitrary smooth complete fans Sigma. As written, a reader cannot check whether the cited theorem applies to this construction without consulting a different paper and supplying compatibility arguments herself.
[Definitions 3.1.3 and proof of Theorem 4.3.1] The weight omega_rho is introduced as an intersection multiplicity of Z with D_rho, but Z is a subvariety of the open torus bundle X while D_rho lies in the toric boundary Y\X. The intersection number D_rho * [Z] in the proof therefore requires an explicit convention that Z means the closure \overline{Z} in a chosen compactifying toric bundle Y meeting the toric boundary transversely. The independence of omega_rho and of the balancing equation from the choice of Y should also be stated; without this convention the formula is not well defined.

minor comments (4)

[Section 2.1, first paragraph] The sentence beginning 'For simplicity, we will first discuss tropicalization for a curve in C in (C*)^2' contains a typo and should read 'a curve C in (C*)^2'.
[References] Reference [2] is incorrectly described as a Journal of the American Mathematical Society article; Maclagan and Sturmfels's 'Introduction to Tropical Geometry' is a book in the Graduate Studies in Mathematics series published by the AMS (volume 161, 2015).
[Section 2.2.1] The notation H2(X_Sigma) mixes cohomology with cycle groups; using A_1(X_Sigma) consistently, as in Section 4.2, would make the divisor-relation argument easier to follow.
[Section 4.1] The asserted identity D_1 - D_2 = k*[fiber] in the Hirzebruch-surface example depends on the labeling of the two direct-sum factors O(k) and O; the author should specify how D_1 and D_2 are associated to these factors to fix the sign convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the balancing formula is an immediate application of an external divisor relation to intersection-defined weights.

full rationale

The main theorem (Theorem 4.3.1) is not circular. The weights omega_rho are set in Definition 3.1.3 as intersection multiplicities omega_rho = D_rho · [Z], and beta is defined as pi_*[Z]. The proof of Theorem 4.3.1 substitutes the external divisor relation from Lemma 4.2.3, sum_rho <ei, vrho> D_rho = pi^* c1(L_i), into this definition and applies the projection formula. The right-hand side c1(L_i)·beta is therefore a genuine Chern-class evaluation, not a quantity chosen to match the weighted left-hand sum. Lemma 4.2.3 itself is quoted from Sankaran-Uma, a published external source with no author overlap, so the load-bearing cohomological input is independent rather than a self-citation chain. The tropicalization definitions in Section 3 do not assume the balancing equation; they only specify which rays appear and assign weights by intersection. The caveats in the paper concern scope, not circularity: Section 1.2 defers non-Archimedean tropicalization with bounded edges, while Theorem 4.3.1 is stated without a field hypothesis and its proof only treats the conical/geometric (complex) setting. That is a missing-hypothesis or proof-gap concern, not an equivalence between input and output. There is no fitted parameter, no ansatz smuggled by citation, and no renaming of a known pattern as a new result. Score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The result introduces no free parameters and no new objects. It relies on two external theorems, Ulirsch's compactification and Sankaran-Uma's divisor relation, plus standard intersection theory. No fitted values or ad hoc constants appear.

assumptions (4)

domain assumption Ulirsch's compactification theorem: every subvariety of a torus bundle admits a toric variety bundle compactification whose closure meets the toric boundary transversely.
Section 3.1 uses this to ensure a toric variety bundle Y with transverse closure exists; this also transfers tropicalization to a union of cones in a fan. The theorem is cited from Ulirsch and not proved.
domain assumption Sankaran-Uma divisor relation: sum_rho <e_i, v_rho> D_rho = pi^* c1(L_i) in the codimension-one Chow group of a toric variety bundle.
Lemma 4.2.3 is the central input; the paper quotes [4, Theorem 1.2] and does not prove it. The balancing formula is exactly this relation intersected with [Z].
standard math The fan Sigma of the toric variety bundle Y is smooth and complete.
Section 4 assumes smoothness and completeness to identify primitive ray vectors and use the divisor relations in the required Chow group.
standard math The projection formula for proper pushforwards and pullbacks in intersection theory.
Used in the proof of Theorem 4.3.1 to move pi^* c1(L_i) across the intersection with [Z].

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Cite this review

Pith. "Pith review of Tropical geometry in torus bundles." pith.science (2026). https://pith.science/paper/7CF7IYJV

@misc{pith2026250902569,
  author       = {Pith},
  title        = {Pith review of: Tropical geometry in torus bundles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CF7IYJV}},
  note         = {Machine review of arXiv:2509.02569}
}

abstract

We formulate and prove an analogue of the balancing condition for the tropicalization of a curve in a torus bundle $X$. We find that the usual balancing condition fails when the bundle is non-trivial and that the failure is captured by the first Chern classes of the line bundles associated to $X$. We discuss a geometric perspective of tropicalization where the weights arise naturally from intersection theory. The relations between divisors on a toric variety bundle put constraints on these weights which leads to the balancing condition.

Figures

Figures reproduced from arXiv: 2509.02569 by the authors.

**Figure 1.** Tropicalization of C = {(z1, z2) : z 2 1 + z2 = 1}. We can also find the tropicalization using the tropical polynomial given by g(x1, x2) = max{2x1, x2, 0}. We see that the maximum is attained at least twice when 2x1 = x2 and x1, x2 ≥ 0, or when x1 = 0 and x2 ≤ 0, or when x2 = 0 and x1 ≤ 0. Note that this agrees with the above. In the case where all coefficients aij are in C, the tropicalization is conical and we ca… view at source ↗

**Figure 2.** Visualisation of Z meeting the toric boundary non-transversely and transversely [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** A visualisation of a torus bundle. Construction 3.0.3. A toric variety bundle Y over B is constructed as follows: • Begin with a torus bundle X as in Definition 3.0.1 and a fan Σ in R n . • For a trivialising open U ⊆ B, compactify X|U = U × (C ∗ ) n in U × XΣ, where XΣ is the toric variety associated to the fan Σ. • Glue to get a toric variety bundle Y over B. We see that the toric variety bundle Y locally looks li… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A visualisation of a toric variety bundle. 3.1. Geometric tropicalization for torus bundles. We will now construct tropicalization for a subvariety Z of a torus bundle X. Consider a toric variety bundle Y compactifying X with toric variety XΣ in each fiber. For each ra…

**Figure 5.** Figure 5: Examples of curves in a P 1 -bundle over P 1 . 3.2. Tropicalization for torus bundles via valuation. By setting B to be a point, Z is a subvariety of (C ∗ ) n and we recover the original setting. We can also define tropicalization for subvarieties of a torus bundle coo…

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Works this paper leans on

13 extracted references · 10 canonical work pages

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Pith tools

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

[1] [1]

Tropical algebraic geometry

A. Gathmann, Tropical Algebraic Geometry. arXiv:math/0601322 (2006)

work page Pith review arXiv 2006

[2] [2]

Maclagan, B

D. Maclagan, B. Sturmfels, Introduction to Tropical Geometry . Journal of the American Mathematical Society. Vol. 161, (2015)

work page 2015

[3] [3]

Ulirsch, Tropical Compactification in Log-regular Varieties

M. Ulirsch, Tropical Compactification in Log-regular Varieties. Math. Z. 280, (2015), pp.195–210

work page 2015

[4] [4]

Sankaran, V

P. Sankaran, V. Uma, Cohomology of Toric Bundles . Comment. Math. Helv. 78, (2003), pp.540–554

work page 2003

[5] [5]

Tevelev, Compactifications of Subvarieties of Tori

J. Tevelev, Compactifications of Subvarieties of Tori . American Journal of Mathematics, 129(4), (2007), pp.1087-1104

work page 2007

[6] [6]

Fulton, Intersection Theory

W. Fulton, Intersection Theory. Springer New York, (1998)

work page 1998

[7] [7]

Cools, J

F. Cools, J. Draisma, S. Payne, E. Robeva, A Tropical Proof of the Brill-Noether Theorem . Advances in Mathematics. 230, (2012), pp. 759-776

work page 2012

[8] [8]

Farkas, D

G. Farkas, D. Jensen, S. Payne, The Kodaira Dimensions of M22 and M23. arXiv:2005.00622v2 (2023)

arXiv 2023