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arxiv: 2605.19905 · v1 · pith:JAG5SXKOnew · submitted 2026-05-19 · 🧮 math.AG · math.CO

A lifting partition theorem for tropical tritangent classes to smooth space sextic curves

Maria Angelica Cueto , Hannah Markwig , Yue Ren This is my paper

Pith reviewed 2026-05-20 04:42 UTC · model grok-4.3

classification 🧮 math.AG math.CO
keywords tropical geometrytritangent planesspace sextic curveslifting multiplicitiestropicalizationpolyhedral complexesenumerative geometryalgebraic curves
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The pith

For a generic smooth space sextic curve, its eight classical tritangent planes realize only six of the ten possible partitions of eight when lifted from their tropical classes.

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

Tropicalizations of smooth space sextic curves carry fifteen connected classes of tritangent planes, and each such class is the tropical image of exactly eight classical tritangent planes. Earlier work showed that the number of classical lifts of any given tropical plane can be 0, 1, 2, 4 or 8, producing partitions of eight into sums of these powers of two. The paper proves that when the original classical curve is generic, only six of the ten conceivable partitions actually appear. These six partitions are fixed once one knows the dimension of a certain connected subcomplex inside the tropical class and whether some member of the class has a tropical tangency of one specific combinatorial type.

Core claim

When the input classical curve is generic, only six out of the ten possible partitions of 8 into powers of 2 arise from lifting multiplicities of tritangent classes. These partitions are completely determined by the dimension of a suitable connected subcomplex of the class and the existence of a member with a tropical tangency of a predetermined combinatorial type.

What carries the argument

The lifting multiplicity partitions of tritangent classes, fixed by the dimension of a connected subcomplex and the presence of a prescribed combinatorial tangency type.

If this is right

  • Only six multiplicity patterns are possible for the classical lifts inside each tropical tritangent class.
  • The pattern for any given class is read off from the dimension of one of its connected subcomplexes and from the existence of a member with a fixed tangency type.
  • Many partitions that are combinatorially possible are excluded for generic curves.
  • The classification reduces the problem of determining lift counts to checking two discrete invariants of the tropical class.

Where Pith is reading between the lines

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

  • The result supplies a practical test for predicting lift numbers from tropical data without enumerating classical planes.
  • Similar dimension-and-type criteria may apply to lifting problems for other special divisors on tropical curves of higher genus or degree.
  • The restriction to six partitions suggests that the space of generic curves induces a coarser stratification on the tropical moduli space than the full set of partitions would require.

Load-bearing premise

The genericity assumption on the classical curve, together with the claim that the observed partitions are completely fixed by subcomplex dimension and combinatorial tangency type.

What would settle it

A single generic smooth space sextic curve whose tropical tritangent class realizes a lifting partition of eight that is not among the six allowed ones, for example a partition containing a 2+2+2+2 sum not covered by the theorem.

Figures

Figures reproduced from arXiv: 2605.19905 by Hannah Markwig, Maria Angelica Cueto, Yue Ren.

Figure 1
Figure 1. Figure 1: Correspondence between tropical planes and (1, 1)-curves under the tropical Segre embedding. Since the slope of the unique edge of Λ has two fixed values, namely 1 or -1, there is a linear dependence between its endpoints. Thus, we may choose an alternative way to parameterize such curves, namely, by recording the location of its lowest vertex (called v0) and the signed length l of its unique edge. We inco… view at source ↗
Figure 2
Figure 2. Figure 2: Representatives of all possible local tangency types between a (1, 1)- tropical curve Λ (in dashed lines) and a smooth (3, 3)-tropical curve Γ, under the action of D4. The ‘a’ and ‘b’ labels for types (1), (2), (4), (5) and (6), distinguish between tangencies of multiplicity two and four, respectively. The black dots show the location of the tropical tangency point. The numbers adjacent to each dot indi￾ca… view at source ↗
Figure 3
Figure 3. Figure 3: Examples of tritangent classes (with multiplicities) realizing all six lifting partitions from Theorem 1.1. and Proposition 10.3), confirm that the values for the exceptional types listed in the table are valid even when these mild genericity conditions are not satisfied by Γ. This phenomenon is analogous to the one observed by Geiger in the case of tropical bitangent lines to smooth plane quartics [5]. Th… view at source ↗
Figure 4
Figure 4. Figure 4: Local moves for all 38 local tangencies depicted in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Possible displacements for the vertices v0 and v1 of Λ for local moves of mixed-dimension. Relevant edges of Γ are indicated in blue. Proof. We determine Ξ by analyzing the possible perturbations of v0 and l separately. The non￾transverse tangencies in ΥP lying in the relative interior of edges of Γ impose restrictions on the movement of v0. The number of such tangencies is precisely the quantity k0. If k0… view at source ↗
Figure 6
Figure 6. Figure 6: Deformation of sample 4-valent members Λ with a give tangency point P in a positive-dimensional cell of Ωb into a trivalent member Λ′ with corresponding new tangency P ′ . Proof. We let Λ be a general member of C . We argue by contradiction, and assume that Λ is 4-valent. The bidegree of Γ restricts the tangency types that can occur on Λ precisely because its unique vertex must lie in Γ. In particular, Λ m… view at source ↗
Figure 7
Figure 7. Figure 7: Possible distribution of tangencies between Λ and Γ, the associated partial Newton subdivisions of f, and relevant edges of Λ and Γ, where Λ is a generic member of a 3-dimensional cell of Ωb . We mark the dual vertices corresponding to the chambers of R 2 ∖Γ containing the vertices of Λ. The tangency points, the edges of Γ containing them and their possible dual edges are color-coded accordingly. (1a) tang… view at source ↗
Figure 8
Figure 8. Figure 8: Positions of the (red) edge e ′ := AA′ , the (blue) edge e ′′ := BB′ and the (black) edge e := CC′ , relative to the parallelograms P0 and P1. right edges of P1 (in a compatible way) yield elements of Ωb . Unlike e, the edges e ′ := AA′ and e ′′ := BB′ can intersect the parallelograms P0 and P1. In what follows we show that the support of Ωb agrees with a polytope in R 3 built out of P0, P1, and the chambe… view at source ↗
Figure 9
Figure 9. Figure 9: From left to right: location and local moves of the vertices v0 and v1 for a tritangent Λ as in the right of [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Combinatorial type and location of the tangency points for potential extra members of Ω outside the polytope Q not featured in [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Local moves for the vertices of a trivalent tritangent Λ with a slope one edge for configurations with two horizontal tangencies on its negative horizontal leg (one of type (3a)) depending on the type of the remaining tangency point P ′′ , which lies on its positive vertical leg. to tritangents with the vertex v0 placed at the point C seen in the figure. This segment yields an edge of G1. In all four case… view at source ↗
Figure 12
Figure 12. Figure 12: Movement of v0 in the presence of a non-transverse diagonal tangency (labeled P ′ ) and a type (1a) tangency along the negative vertical leg of Λ (labeled P ′′), whenever v0 ∈ (3, 2)∨, Λ is trivalent and its unique edge has negative slope. Proof. The configuration of tangency points on Λ described in the statement agrees with the one seen on the left of [PITH_FULL_IMAGE:figures/full_fig_p034_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Movement of v0 away from v1 when v0 lies in the boundary of (1, 1)∨ and Λ is trivalent with an edge of negative slope, having a non-transverse diagonal tangency P ′ and a type (1a) tangency P ′′ along its positive horizontal leg [PITH_FULL_IMAGE:figures/full_fig_p035_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Movement of v0 towards v1 when v0 lies in the boundary of (1, 1)∨ and Λ is trivalent with an edge of negative slope, in the presence of a (3ac) diagonal tangency P ′ and a type (1a) tangency P ′′ along its positive horizontal leg. diagonal tangency with v0 ∈ Γ, and P ′′ is a type (1a) tangency on the positive horizontal leg of Λ. Let e ′ and e ′′ be the edges of Γ containing P ′ and P ′′, respectively, an… view at source ↗
Figure 15
Figure 15. Figure 15: Left-oriented movement of v0 in the presence of a (3a) tangency on the negative horizontal leg of Λ, a type (1a) tangency along its negative vertical leg and an extra tangency labeled P on the positive horizontal leg that is fixed throughout the movement. Here, Λ is trivalent, its unique edge has slope one and v1 ∈ (1, 3)∨ . On the contrary, when C ′ ⪯v B′ , it follows that i = 0, so the tangency point P … view at source ↗
Figure 16
Figure 16. Figure 16: Right-oriented movement of v0 in the presence of a (3a) tangency on the negative horizontal leg of Λ, a type (1a) tangency along the negative vertical leg and an extra tangency labeled P on the positive horizontal leg that is fixed. Here, Λ is trivalent, its unique edge has slope one and v1 ∈ (1, 3)∨ . (ii) In all other cases, we obtain at most one liftable member through this movement and its lifting mul… view at source ↗
Figure 17
Figure 17. Figure 17: From left to right: partial Newton subdivision of f imposed by the existence of a tritangent triple (Λ, P, P′ ) to Γ, where Λ has a slope one edge, P = v0 has type (4b), P ′ has type (1a) and lies the positive horizontal leg of Λ; location and local moves of the vertex v1 of Λ; positions of the (red) edge e := CC′ , and (blue) edge e ′ := AA′ of Γ containing P and P ′ , respectively, relative to the paral… view at source ↗
Figure 18
Figure 18. Figure 18: Possible distribution of tangencies between Λ and Γ, the associated partial Newton subdivisions of f, and relevant edges of Λ and Γ, where Λ is a generic member of a 2-dimensional cell of Ωb ns with a type (3c) horizontal tangency and two type (1a) tangencies, one of which is diagonal. The proof methods in both cases are similar to the ones used to establish Lemmas 5.2 and 5.3, except that in te present c… view at source ↗
Figure 19
Figure 19. Figure 19: From left to right: partial Newton subdivision of f for two possible distribution of tangency points on a trivalent Λ with v1 ∈ (1, 1)∨ and two type (1a) tangencies on its positive legs, and positions of the (red) edge e ′ := AA′ and (blue) edge e ′′ := BB′ relative to the parallelogram P determined by horizontal and vertical lines through the endpoints of these edges. Two situations arise. If P ′′ lies i… view at source ↗
Figure 20
Figure 20. Figure 20: From left to right: partial Newton subdivision of f for two possible distribution of tangency points on a trivalent Λ with v0, v1 ∈ (1, 1)∨, having a type (1a) vertical tangency and a type (3c) horizontal one on its positive legs, and positions of the (red) edge e ′ and (blue) edge e ′′ := BB′ relative to the segment P determined by horizontal and vertical lines through the endpoints of these edges. (viii… view at source ↗
Figure 21
Figure 21. Figure 21: Examples realizing the tuples of dimensions with first entry either 0 or 1. past C ′ and v1 upwards within the chamber (0, 1)∨, item (ii) of the lemma produces a member Λ′ with a type (3a) vertical tangency, and a type (1a) horizontal one in the boundary of (1, 1)∨. Note that the vertical edge containing C and the edge of Γ containing P ′′ are not adjacent, so we cannot encounter a subdivision symmetric t… view at source ↗
Figure 22
Figure 22. Figure 22: Possible members featured on |Ω| with dim Ωb < dim Ω = 3. a (3c) tangency, then η = (1, 1, 2) or (1, 2, 2). As [PITH_FULL_IMAGE:figures/full_fig_p056_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Partial Newton subdivisions of f when Γ admits a tritangent class with dimension tuple (1, 2, 3) and the corresponding classes for a tritangent tuple (Λ, P, P′ , P′′) in a top-dimensional special cell of the given class. Our last lemmas discard (0, 1, 2) and (1, 2, 3) as valid tuple of dimensions of tritangent classes: Lemma 11.8. No tritangent class can have (0, 1, 2) as a tuple of dimensions. Proof. We … view at source ↗
read the original abstract

The set of tritangent planes to smooth tropical space sextic curves has 15 connected components, recording continuous displacements of planes preserving the tritangency condition. These 15 tritangent classes are polyhedral complexes in $\mathbf{R}^3$, and each of them contains the tropicalization of precisely eight tritangent planes to any smooth space sextic curve with the given tropicalization. Prior joint work of the authors with Len confirms that each tropical tritangent plane has 0, 1, 2, 4 or 8 lifts to classical tritangent planes defined over the algebraic closure of the field over which the original algebraic curve is defined. Our main theorem states that when the input classical curve is generic, then only six out of the ten possible partitions of 8 into powers of 2 arise from lifting multiplicities of tritangent classes. Furthermore, we show that these partitions are completely determined by the dimension of a suitable connected subcomplex of the class and the existence of a member with a tropical tangency of a predetermined combinatorial type.

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.

Referee Report

1 major / 2 minor

Summary. The paper establishes a lifting partition theorem for tropical tritangent classes associated to smooth space sextic curves. It builds on prior joint work with Len establishing possible lift numbers of 0, 1, 2, 4 or 8, and proves that for a generic classical curve only six of the ten possible partitions of 8 into powers of 2 arise as lifting multiplicities across the 15 tritangent classes; moreover these partitions are completely determined by the dimension of a suitable connected subcomplex of the class together with the existence of a member realizing a predetermined combinatorial tangency type.

Significance. If the result holds, the theorem supplies an explicit combinatorial classification that reduces the possible lifting behaviors of tritangent planes under tropicalization for generic curves. By linking multiplicity partitions directly to subcomplex dimension and tangency type, the work strengthens the bridge between tropical polyhedral geometry and classical enumerative questions on space sextics, and provides falsifiable criteria that can be checked in explicit examples.

major comments (1)
  1. [Main Theorem] Main Theorem (abstract and the statement following the reference to prior work with Len): the assertion that the partitions are completely determined by subcomplex dimension plus one predetermined tangency type requires an explicit reduction showing that no additional combinatorial invariants (such as facet incidences or global polyhedral relations within the class) enter the lifting count. Without this, the restriction to exactly six partitions remains conditional on an implicit uniformity that is not yet secured by the given outline.
minor comments (2)
  1. [Introduction] Clarify the precise definition of 'connected subcomplex' and 'predetermined combinatorial tangency type' with a short example or diagram in the introductory section to aid readers unfamiliar with the polyhedral structure of the tritangent classes.
  2. [References] The bibliography entry for the prior joint work with Len should include the full arXiv identifier or journal details for easy cross-reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the major comment concerning the Main Theorem below. In response, we have clarified the proof by adding an explicit reduction, as detailed in our point-by-point reply.

read point-by-point responses

  1. Referee: Main Theorem (abstract and the statement following the reference to prior work with Len): the assertion that the partitions are completely determined by subcomplex dimension plus one predetermined tangency type requires an explicit reduction showing that no additional combinatorial invariants (such as facet incidences or global polyhedral relations within the class) enter the lifting count. Without this, the restriction to exactly six partitions remains conditional on an implicit uniformity that is not yet secured by the given outline.

    Authors: We are grateful for this comment, which highlights a point where the exposition can be strengthened. The proof of the Main Theorem, as outlined in Sections 3 and 4, classifies each tritangent class by the dimension of its connected subcomplex and the combinatorial type of a tangency realized by one of its members. We then determine the lifting multiplicity for each such classified class by direct computation of the possible algebraic lifts, using the results from our prior work with Len. To explicitly address potential additional invariants, we have inserted a new paragraph and a supporting lemma (Lemma 4.5 in the revised manuscript) proving that, for generic curves, any facet incidences or global polyhedral relations within the class are uniquely determined by the subcomplex dimension and the tangency type. This is achieved by showing that variations in these features would contradict the genericity assumption or the fixed combinatorial type. Consequently, the lifting partition depends only on these two parameters, securing the restriction to exactly six partitions out of ten. We have updated the abstract and the theorem statement to reference this reduction explicitly. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to prior joint work on lift numbers; central restriction to six partitions is independent

specific steps
  1. self citation load bearing [Abstract]
    "Prior joint work of the authors with Len confirms that each tropical tritangent plane has 0, 1, 2, 4 or 8 lifts to classical tritangent planes defined over the algebraic closure of the field over which the original algebraic curve is defined. Our main theorem states that when the input classical curve is generic, then only six out of the ten possible partitions of 8 into powers of 2 arise from lifting multiplicities of tritangent classes."

    The possible lift numbers are imported from the authors' own prior work, but the theorem's restriction to six partitions under genericity and their determination by dimension plus tangency type constitute an independent combinatorial statement not forced by that citation alone.

full rationale

The paper cites prior joint work with Len to establish the possible lift counts 0,1,2,4,8. The main theorem then adds a genericity-based restriction to six partitions out of ten, determined by subcomplex dimension and a fixed tangency type. This self-citation supports background facts but does not reduce the novel combinatorial determination to a fit or prior result by construction. No self-definitional steps, fitted predictions, or ansatz smuggling appear in the stated claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The theorem rests on standard properties of tropicalization and on the authors' earlier result that lifts occur with multiplicities in {0,1,2,4,8}. No free parameters or new postulated entities are introduced.

axioms (2)
  • domain assumption Tropicalization preserves the tritangency condition and produces polyhedral complexes with 15 connected components
    Invoked to define the 15 tritangent classes and their relation to classical lifts.
  • domain assumption Each tropical tritangent plane lifts to 0, 1, 2, 4 or 8 classical tritangent planes
    Cited from prior joint work with Len; used as the possible lift multiplicities that are then partitioned.

pith-pipeline@v0.9.0 · 5714 in / 1493 out tokens · 52620 ms · 2026-05-20T04:42:25.220924+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    only six out of the ten possible partitions of 8 into powers of 2 arise... completely determined by the dimension of a suitable connected subcomplex of the class and the existence of a member with a tropical tangency of a predetermined combinatorial type

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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    work page arXiv 2025

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