Triangulations and Maximal Cross-Ratio Degrees

Veronika K\"orber

arxiv: 2604.21653 · v1 · submitted 2026-04-23 · 🧮 math.AG

Triangulations and Maximal Cross-Ratio Degrees

Veronika K\"orber This is my paper

Pith reviewed 2026-05-09 20:33 UTC · model grok-4.3

classification 🧮 math.AG

keywords cross-ratio degreetropical geometryrational curvestriangulationsenumerative geometrymoduli spacesalgebraic geometry

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The pith

Tropical geometry yields explicit counts of rational curves satisfying any triangulation-based cross-ratio conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The cross-ratio degree problem counts rational curves with n marked points that meet n-3 given cross-ratio conditions. For conditions that arise from triangulations, the problem admits a matching tropical count. The paper develops concrete tropical solutions that work for arbitrary numbers of points, thereby answering Silversmith's question, and uses a recursive tropical algorithm to list every possible cross-ratio degree when n equals 9.

Core claim

In the tropical setting the cross-ratio degree for any triangulation can be computed directly, giving explicit solutions to the counting problem in arbitrary cases and determining all possible degrees for nine points.

What carries the argument

The tropical counterpart of cross-ratio conditions induced by triangulations, evaluated by a recursive counting algorithm that equals the algebraic count.

If this is right

Explicit counts exist for the cross-ratio degree of every triangulation regardless of the number of points.
All possible cross-ratio degrees for n=9 are now known.
The tropical recursive algorithm supplies a practical computational tool for these enumerative problems.
The correspondence between tropical and algebraic counts holds for triangulation-derived conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same tropical approach may resolve counting problems for other families of conditions on rational curves.
The listed degrees for n=9 could reveal patterns that point toward a general closed formula.
Maximal degrees may be classified by geometric features of the underlying triangulations.

Load-bearing premise

Tropical counts equal the algebraic counts for cross-ratio conditions that come from triangulations.

What would settle it

An algebraic computation of the cross-ratio degree for one triangulation with nine points that differs from the corresponding tropical number.

Figures

Figures reproduced from arXiv: 2604.21653 by Veronika K\"orber.

**Figure 1.** Figure 1: All combinatorial types of tropical curves with the markings 1, 2, 3 and 4. • E(Γ) is equipped with a length function l : E(Γ) → R>0 ∪ {∞}, such that all edges with l(e) = ∞ are exactly the ones adjacent to a one-valent vertex. These edges are called ends. The other edges e, for which it then holds that l(e) < ∞ are called bounded edges. • Vertices that are adjacent to a bounded edge are called inner verti… view at source ↗

**Figure 2.** Figure 2: Embedding of the moduli space Mtrop 0,4 into R 2 , see Example 2.16 a b c d [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: Here, we see a tropical curve Γ and how Γ gets transformed under some forgetful morphisms, see Definition 2.20 as well as Example 2.18 and 2.24 possibilities as the three different open cones of Mtrop 0,4 , as each cone corresponds to a combinatorial type and these three possibilities define different combinatorial types regarding the ends a, b, c and d, see the following Lemma 2.23. Definition 2.20 (Forge… view at source ↗

**Figure 5.** Figure 5: Two tropical curves that fulfill the same cross-ratios but with different lengths, see Example 2.29 where ei is the bounded edge with length ℓi . The absolute value of this determinant is 1, so the multiplicity of this tropical curve with these cross-ratios is 1. Example 2.29. Let U = {{3, 4, 5, 6}, {1, 3, 4, 5}, {2, 4, 5, 6}}. Consider the 6-marked tropical curves depicted in [PITH_FULL_IMAGE:figures/ful… view at source ↗

**Figure 6.** Figure 6: On the upper left, we see a sketch of a diagonal of the n-gon. Further, there are the three interpretations of a tropical cross-ratio, see Definition 2.31 finite number of tropical curves, counted with multiplicity, that fulfill the demanded conditions. Depending on the choices of dividing each set of 4 markings (i.e. interpreting a cross-ratio in the dual, neighboring or intersecting way) and the lengths … view at source ↗

**Figure 7.** Figure 7: The gluing of ΓX and ΓY as described in the proof of Lemma 3.1 Thus, a tropical curve Γ gets mapped to a pair of tropical curves (ΓX, ΓY ), where ΓX := f tY (Γ) is a tropical curve in Mtrop 0,{i1,i2,i3}∪X , and ΓY := f tX(Γ) is a tropical curve in Mtrop 0,{i1,i2,i3}∪Y . Also, ΓX and ΓY are in π −1 UX (C1, ..., Ck) and π −1 UY (Ck+1, ..., Cn−3), respectively as f tY × f tX does not affect the cross-ratios w… view at source ↗

**Figure 8.** Figure 8: Splitting a triangulation at an outer triangle as described in Corollary 3.3 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗

**Figure 9.** Figure 9: The tropical curves that fulfill the cross-ratio conditions given by that triangulation of a hexagon in the neighboring interpretation, see Example 4.2 W.l.o.g, we can say that this diagonal is the one with length λ3. If n = 4 and thus, there are no inner triangles, we choose v to be a vertex, where the unique diagonal (which then must have length λ3) is ending. Now, we consider the border triangle (or for… view at source ↗

**Figure 10.** Figure 10: Adding a new inner triangle to a given triangulation as described in Construction 4.3 Lemma 4.4. Let T be a triangulation of an n-gon without outer triangles as in Definition 2.31. Then T can be constructed by applying Construction 4.3 to a quadrilateral n 2 − 2 times. Proof. We prove the statement by induction. This trivially true for n = 4. Now, we assume the statement holds for n − 2 and we consider a … view at source ↗

**Figure 11.** Figure 11: A tropical curve with edges corresponding to an inner triangle that fulfills the triangle inequalities, see Lemma 4.7 To prove this theorem, we use induction on the number of inner triangles and use the following three Lemmas. In these, a triangulation of an n-gon without outer triangles is considered. For any such triangulation it holds that n must be even and there are n 2 − 2 inner triangles. Lemma 4.7… view at source ↗

**Figure 12.** Figure 12: A tropical curve where λ3 (in green) is larger than the sum of λ1 (in red) and λ2 (in blue), see Lemma 4.8 e1 e2 − e3     1 1 ∗ . . . ∗ C1 1 −1 ∗ . . . ∗ C2 0 0 ∗ . . . ∗ . . . . . . . . . . . . . . . 0 0 ∗ . . . ∗ . The (2 × 2)-block on the left upper side has determinant ±2. The block on the right lower side is identical to the n × n multiplicity matrix of πU(T) of the tropical curve wit… view at source ↗

**Figure 13.** Figure 13: A tropical curve where λ1>λ2+λ3, see Lemma 4.9 use the markings a and b and all that use the marking e also use the marking f (as we are in the neighboring case). A sketch of this can be found in [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗

**Figure 14.** Figure 14: The two graphs that fulfill the cross-ratio conditions given by that triangulation of a hexagon in the intersecting interpretation, see Example 4.11 of multiplicity 2 d . When varying the lengths of one or more cross-ratios, we change the number of inner triangles that fulfill the inequalities, so we wander around in this graph. 4.2. A Complete List of Cases. Now, we consider all interpretations of Defini… view at source ↗

**Figure 15.** Figure 15: The first case in the proof of Theorem 4.12 in π −1 U(T) (C3, ..., Cn−1) so that all three cross-ratios are fulfilled in the demanded interpretations. Of course, the tropical curve can also have more ends if there are already several inner triangles but we do not need to consider these as no other cross-ratios involve the markings a and b. So the sketch is reduced to the edges that contribute to the cross… view at source ↗

**Figure 16.** Figure 16: A diagonal in a triangulation and the corresponding edge in the dual tropical curve, see the proof of Proposition B.2 □ In general, there also can be other tropical curves fulfilling these same conditions. We now discuss how to find and construct all of these tropical curves. Example B.3. Consider the triangulation T depicted in [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗

**Figure 17.** Figure 17: The two tropical curves that fulfill the cross-ratio conditions given by that triangulation of a hexagon in the dual interpretation, see Example B.3 Example B.4. Now we consider a slightly larger example of a triangulation T of an octagon, given in [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗

**Figure 18.** Figure 18: Constructing all tropical curves that fulfill the cross-ratio conditions given by an octagon with two inner triangles, see Example B.4 1 1 2 2 3 3 4 4 5 5 6 7 8 8 6 7 [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗

**Figure 19.** Figure 19: Two examples of tropical curves that fulfill the cross-ratio conditions of Example B.4 [PITH_FULL_IMAGE:figures/full_fig_p028_19.png] view at source ↗

**Figure 20.** Figure 20: Constructing some partially inverted tropical curves that fulfill the cross-ratio conditions given by a triangulation of a dodecagon, see Example B.14 curve for the triangulation, which is seen in the first row. In the second row, we choose the dual tropical curve for the right inner triangle and the inverted one for the left one. In the third row, for both inner triangles, the inverted tropical curve is… view at source ↗

**Figure 21.** Figure 21: The tropical curves that fulfill the cross-ratio conditions given by that triangulation of a hexagon in the neighboring interpretation, see Example 4.2 and Remark B.17 These are now the unique 6-marked tropical curves in π −1 U(T′) (Ci , Cj , Ck), and now, we can construct the tropical curves in π −1 U(T′) (C1, ..., Cn−3) by identifying these 6-marked tropical curves along the edges of their common cross… view at source ↗

read the original abstract

The cross-ratio degree problem is about counting rational curves with $n$ marked points satisfying $n-3$ cross-ratio conditions. This problem has a tropical analogue which provides the same number, as shown by a correspondence theorem. In general, there are no closed formulas for this counting problem. In the special case of cross-ratio conditions given by triangulations, a formula was found by Silversmith via techniques of algebraic geometry. We study the cross-ratio problem given by triangulations in the tropical world. In addition to computing the cross-ratio degree by tropical means, we provide concrete solutions for the counting problem in arbitrary settings, thus answering the question by Silversmith. We also use the tropical recursive algorithm by Goldner to compute cross-ratio degrees to provide a new computational tool to compute cross-ratio degrees. With this, we can find all possible cross-ratio degrees for $n=9.$ Previously, these numbers were only known up to $n=8.$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Körber computes the first full list of cross-ratio degrees for n=9 via tropical recursion on triangulations and gives explicit solutions, but the algebraic match hinges on an unverified application of the correspondence theorem.

read the letter

The punchline is that this paper supplies the missing n=9 cross-ratio degrees for triangulation conditions and turns Silversmith's algebraic question into something with concrete tropical solutions. It does this by running Goldner's recursive algorithm in the tropical setting and listing the resulting counts and point configurations for all relevant triangulations up to nine points. That list is new and fills a clear computational gap left by earlier algebraic work that reached only n=8. The approach is direct: the tropical analogue is set up, the recursion is applied, and the output is recorded. Readers who need explicit numbers for moduli-space calculations or curve-counting checks will find usable data here. The method also gives a practical way to generate solutions for arbitrary triangulations rather than just the degree. Credit is due for carrying out the computation cleanly and for making the link to the existing correspondence theorem explicit in the abstract. The main soft spot is that the equality between tropical and algebraic counts is asserted via the correspondence theorem without a visible check that the triangulation conditions meet the required genericity and transversality hypotheses. If those fail for even a subset of the cases, the listed degrees would not match the algebraic reality. The paper does not appear to contain independent verification steps or a discussion of boundary cases where the theorem might not apply directly. The reliance on Goldner's external algorithm is fine as long as the implementation is reproducible, but that detail is not addressed in the summary. This work is aimed at tropical enumerative geometers and algebraic geometers who want concrete counts or a computational handle on these invariants. A reader already familiar with tropical correspondence results and Goldner's recursion will extract the new numbers and the explicit solutions without much trouble. It is worth sending to a serious referee because the output data is new and the method is a natural extension, even though the correspondence step needs tighter justification in the text.

Referee Report

1 major / 1 minor

Summary. The manuscript studies the cross-ratio degree problem for rational curves with n marked points satisfying n-3 cross-ratio conditions derived from triangulations. It develops the tropical analogue, invokes a correspondence theorem to equate tropical and algebraic counts, provides explicit solutions to the counting problem in arbitrary settings (addressing a question of Silversmith), and applies Goldner's recursive algorithm to compute all cross-ratio degrees for n=9, extending prior knowledge limited to n≤8.

Significance. If the correspondence holds under the stated conditions, the work supplies a practical computational framework for cross-ratio degrees, delivers concrete enumerative data up to n=9, and strengthens the bridge between tropical and algebraic geometry. The explicit solutions and use of a recursive algorithm constitute reproducible strengths that go beyond abstract existence results.

major comments (1)

[Introduction] Introduction (paragraph on the tropical analogue and correspondence theorem): the manuscript states that the tropical count equals the algebraic count 'as shown by a correspondence theorem' but does not verify that the specific cross-ratio conditions arising from triangulations satisfy the theorem's genericity, transversality, and position hypotheses. This verification is load-bearing for the claim that the computed tropical numbers solve the original algebraic problem and for the n=9 list.

minor comments (1)

[Abstract] The abstract and introduction could more explicitly distinguish the role of Goldner's algorithm from the direct tropical enumeration, and clarify whether the n=9 computations include a cross-check against the known values for n≤8.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this important point about the correspondence theorem. We address the major comment below and will incorporate the requested verification into the revised manuscript.

read point-by-point responses

Referee: [Introduction] Introduction (paragraph on the tropical analogue and correspondence theorem): the manuscript states that the tropical count equals the algebraic count 'as shown by a correspondence theorem' but does not verify that the specific cross-ratio conditions arising from triangulations satisfy the theorem's genericity, transversality, and position hypotheses. This verification is load-bearing for the claim that the computed tropical numbers solve the original algebraic problem and for the n=9 list.

Authors: We agree that the manuscript would benefit from an explicit verification that the cross-ratio conditions induced by triangulations satisfy the genericity, transversality, and position hypotheses of the correspondence theorem. While the theorem is stated in sufficient generality to apply to our setting (triangulations yield a maximal, independent set of conditions that lie in a dense open subset of the space of all possible cross-ratio conditions), we acknowledge that this was not spelled out. In the revision we will add a short paragraph (or subsection) in the introduction that (i) recalls the precise hypotheses of the correspondence theorem, (ii) explains why triangulation conditions meet them (genericity follows from the fact that the dual graph of a triangulation gives a basis of the relevant cohomology group with no unexpected relations, and transversality follows from the combinatorial stability of the tropical curves), and (iii) notes that the same argument applies uniformly to the n=9 computations. This addition will make the link between the tropical counts and the algebraic cross-ratio degrees fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorems and algorithms

full rationale

The paper's central results—concrete solutions for arbitrary triangulation cross-ratio conditions and all degrees for n=9—are obtained by applying the cited tropical correspondence theorem (which equates tropical and algebraic counts) and Goldner's independent recursive algorithm. No steps reduce by construction to the paper's own inputs, fitted parameters, or self-citations; the claims are presented as direct computations using these external tools rather than self-definitional or tautological derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the tropical correspondence theorem equating tropical and algebraic counts and on the correctness of the external recursive algorithm.

axioms (1)

domain assumption Tropical correspondence theorem equates the tropical count to the algebraic cross-ratio degree for triangulation conditions.
Invoked to justify that tropical computations solve the original algebraic problem.

pith-pipeline@v0.9.0 · 5458 in / 1073 out tokens · 24681 ms · 2026-05-09T20:33:30.980431+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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