Tropical trigonal curves

This paper establishes the equivalence between the existence of a specific divisor and a non-degenerate harmonic morphism on 3-edge connected tropical curves to define their moduli spaces, ultimately proving that the dimension of the moduli space of tropical trigonal curves matches that of their algebraic counterparts.

The Gist

Imagine you are an architect designing a city, but instead of buildings and roads, you are working with rubber bands and strings. In the world of mathematics, these rubber band structures are called metric graphs (or "tropical curves"). They are like skeletons of shapes that can stretch, shrink, and wiggle, but they must stay connected.

The paper you are asking about, written by Margarida Melo and Angelina Zheng, is a guidebook for understanding a specific type of these rubber band cities: the ones that can be "folded" in a very specific way.

Here is the story of the paper, broken down into simple concepts.

1. The Big Question: How "Foldable" is Your Shape?

In the world of algebraic geometry (the study of smooth curves), mathematicians ask: "Can I map this complex shape onto a simple line (like a straight road) without tearing it?"

• If you can map it using 2 paths, it's called Hyperelliptic (like a double-decker bus).
• If you need 3 paths, it's called Trigonal (like a three-lane highway).

The authors are interested in the Trigonal case. They want to know: What does a "3-lane highway" rubber band city look like?

2. The Two Ways to Describe a "3-Lane" City

The paper solves a puzzle by showing that there are two different ways to describe these special shapes, and surprisingly, they mean the exact same thing (under certain conditions).

Description A: The "Divisor" Method (The Treasure Map)
Imagine you place 3 special "treasure markers" (divisors) on your rubber band city. If you can move these markers around freely (using a mathematical rule called "linear equivalence") and still keep them balanced, your city is "Trigonal."

• Analogy: It's like having 3 coins on a table. If you can slide them around the table in any direction without them falling off, the table has a special property.

Description B: The "Harmonic Morphism" Method (The Folding Map)
Imagine you have a complex, knotted rubber band shape. Can you fold it down onto a simple straight line (a tree) so that:

1. Every point on the line is covered by exactly 3 strands of your rubber band?
2. The folding is "harmonious" (no weird stretching or tearing)?

• Analogy: Think of a 3-strand braid. If you pull the braid tight, it becomes a single straight line. The paper asks: "Can every 3-lane city be unbraided into a straight line?"

The Big Discovery:
For a long time, mathematicians knew these two descriptions were the same for simple shapes (2-lane cities). But for 3-lane cities, it was a mystery.
Melo and Zheng proved that yes, they are the same! If you have 3 treasure markers that can move freely, you can always fold the shape into a 3-strand braid leading to a straight line.

3. The "Tropical Modification" (The Magic Glue)

Here is the tricky part. Sometimes, the rubber band city has "loops" (like a figure-8). If you try to fold a figure-8 directly onto a straight line, it might get stuck or break the rules.

The authors introduce a concept called Tropical Modification.

• Analogy: Imagine you have a tangled knot. You can't untie it directly. But if you glue a small extra string (a tree) onto a specific point of the knot, suddenly the whole thing becomes easy to fold into a straight line.
• The paper proves that even if your shape has loops, you can always "glue on" these extra strings to make the folding work. This is the tropical version of adding "rational tails" in classical geometry.

4. The "3-Ladder" (The Perfect Shape)

The authors then ask: "What is the most complex, 'maximal' version of this 3-lane city?"
They invent a shape called a 3-Ladder.

• Analogy: Imagine a standard ladder. Now, imagine you have three identical ladders standing side-by-side. You connect the rungs of these three ladders together in a specific pattern.
• This "3-Ladder" is the ultimate 3-lane city. It's the most complex shape that still fits the rules.

They prove that every 3-lane city is just a "shrunken" or "contracted" version of one of these 3-Ladders. If you take a 3-Ladder and squash some of its rungs together, you get any other 3-lane city you can imagine.

5. Why Does This Matter? (The Dimension Match)

In mathematics, we often count the "degrees of freedom" or the dimension of a space.

• For algebraic curves (smooth, real-world shapes), the space of all 3-lane cities has a specific size (dimension).
• For tropical curves (rubber band shapes), the authors built a "moduli space" (a map of all possible 3-lane rubber band cities).

The Punchline:
They calculated the size of their rubber band map and found it exactly matches the size of the map for the real-world algebraic curves.

• Why this is cool: It means the "skeleton" (tropical) version captures all the essential complexity of the "flesh and blood" (algebraic) version. The rubber band model is a perfect, simplified mirror of the complex reality.

Summary

Melo and Zheng wrote a guidebook that says:

1. Two definitions match: A shape is "3-lane" if it has movable markers OR if it can be folded into a 3-strand braid.
2. Loops are fixable: If the shape is knotted, just glue on a little extra string (tropical modification) to make it foldable.
3. The Master Shape: All these shapes are just variations of a "3-Ladder."
4. Perfect Match: The map of these rubber band shapes is the exact same size as the map of the real mathematical curves.

This work helps mathematicians understand the "boundary" of complex shapes by studying their simpler, rubber-band skeletons. It's like understanding the structure of a complex building by studying its wireframe model.

Technical Summary

Here is a detailed technical summary of the paper "Tropical Trigonal Curves" by Margarida Melo and Angelina Zheng.

1. Problem Statement and Motivation

The paper addresses the relationship between two distinct definitions of gonality for metric graphs (tropical curves) in the context of trigonality (gonality $d=3$).

• Context: In algebraic geometry, a curve is $d$-gonal if it admits a degree $d$ map to $\mathbb{P}^1$. In tropical geometry, there are two analogous definitions:
  1. Divisorial Gonality: The existence of a divisor of degree $d$ and rank at least 1 (analogous to the existence of a $g^1_d$).
  2. Geometric Gonality: The existence of a non-degenerate harmonic morphism of degree $d$ from the graph (or a tropical modification of it) to a tree.
• The Gap: While these two definitions are known to be equivalent for hyperelliptic curves ($d=2$) and for combinatorial graphs with $d=3$ (under 3-edge connectivity), they diverge for general metric graphs when $d=3$. Specifically, a metric graph can be divisorially trigonal without admitting a non-degenerate harmonic morphism of degree 3 to a tree directly.
• Objective: The authors aim to:
  1. Establish the precise equivalence between divisorial trigonality and geometric trigonality for 3-edge connected metric graphs, utilizing tropical modifications.
  2. Construct the moduli spaces of 3-edge connected tropical trigonal curves and covers.
  3. Prove that the dimension of this tropical moduli space matches the dimension of the classical algebraic moduli space of trigonal curves.

2. Methodology

The authors employ a combination of combinatorial graph theory, divisor theory on metric graphs, and tropical geometry techniques.

• Tropical Modifications: A key tool is the concept of a tropical modification, where trees are glued to a metric graph at specific points. This allows the authors to "fix" metric obstructions that prevent a direct harmonic morphism.
• Dhar's Burning Algorithm: Used extensively to analyze the rank of divisors and to prove non-existence results (e.g., proving 3-edge connected graphs cannot be hyperelliptic).
• Canonical Models: The analysis focuses on the canonical loopless model ($G^-$) of a metric graph. The authors distinguish between the loopless case and the case with loops, showing that loops require specific tropical modifications (adding leaves) to satisfy the harmonic condition.
• Admissible Representatives: For a divisor $D$ of rank 1, the authors construct "admissible representatives" (divisors supported on vertices and specific edge points) to build the combinatorial structure of the morphism.
• Moduli Construction: The moduli spaces are constructed as generalized cone complexes (colimits of rational polyhedral cones). The cones are defined by length constraints imposed by the harmonic morphisms and their equivalence relations.

3. Key Contributions and Results

A. Equivalence of Gonality Definitions (The Main Theorem)

The central result is Theorem 1.1, which establishes the equivalence for 3-edge connected metric graphs $\Gamma$:

• Condition A: $\Gamma$ is divisorially trigonal (i.e., $W^1_3(\Gamma) \neq \emptyset$).
• Condition B: There exists a tropical modification $\tilde{\Gamma}$ of $\Gamma$ and a non-degenerate harmonic morphism of degree 3 from $\tilde{\Gamma}$ to a metric tree.

Nuances:

• If $\Gamma$ is loopless, no modification is needed; $\Gamma$ itself admits the morphism (Theorem 3.3).
• If $\Gamma$ has loops, a tropical modification is required. Specifically, for each loop, a leaf is attached at a point determined by the divisor $D$ to ensure the morphism remains harmonic and of degree 3 (Theorem 3.15).
• Counter-example: The paper provides examples showing that without the 3-edge connectivity assumption, or without allowing tropical modifications, the equivalence fails.

B. Construction of Moduli Spaces

The authors define two moduli spaces:

1. $H^{trop,(3)}_{g,3}$: The moduli space of 3-edge connected tropical trigonal covers (pairs of graph and morphism).
2. $T^{trop,(3)}_g$: The moduli space of 3-edge connected tropical trigonal curves (the image of the covers under the forgetful map).

These spaces are constructed as colimits of rational polyhedral cones $\tau_{(G,w,\phi)}$, where the cones are defined by length constraints derived from the harmonic morphism $\phi$.

C. Dimension and Maximal Cells

• Maximal Cells: The authors identify the maximal cells of the moduli space $T^{trop,(3)}_g$. These correspond to graphs constructed from trees via a specific process called "3-ladders" (Definition 27). A 3-ladder is formed by taking three copies of a tree $T$ and connecting them with vertical edges according to the valence of vertices in $T$.
• Dimension Formula:
  • For genus $g=3$, the dimension is 6.
  • For genus $g \geq 4$, the dimension is $2g + 1$.
  • This matches exactly the dimension of the classical algebraic moduli space of trigonal curves ($M^1_{g,3}$).
• Connectivity: The moduli space $T^{trop,(3)}_g$ is proven to be connected through codimension 1 (Corollary 4.12), meaning any two maximal cells can be connected by a sequence of codimension 1 faces.

D. Relation to Admissible Covers

The authors show that their tropical trigonal covers correspond to tropical admissible covers (as defined by Cavalieri, Markwig, and Ranganathan). They prove that for 3-edge connected trigonal types, the tropical modification required to make the morphism harmonic satisfies the local Riemann-Hurwitz equality, confirming they are indeed tropical admissible covers.

4. Significance

• Bridging Algebraic and Tropical Geometry: The paper successfully translates the geometry of the boundary of the trigonal locus in the algebraic moduli space $M_g$ into the tropical setting. It confirms that the tropical moduli space of trigonal curves has the expected dimension, validating the tropicalization approach for studying the topology of $M_g$.
• Resolution of the $d=3$ Discrepancy: It resolves the ambiguity regarding the equivalence of divisorial and geometric gonality for $d=3$ by introducing the necessary role of tropical modifications, a concept less critical for $d=2$.
• Topological Implications: The results suggest that studying the topology of the 3-edge connected tropical trigonal locus is sufficient to understand the rational cohomology of the algebraic trigonal locus, as the "bad" loci (bridges, cut vertices) are contractible. This provides a combinatorial framework for computing cohomology of $M_g$ via graph complexes, extending the work of Chan, Galatius, and Payne on the top-weight cohomology.
• Generalization: While focused on $d=3$, the methodology (using tropical modifications to fix metric obstructions) offers a blueprint for studying higher gonality cases where the equivalence between divisorial and geometric definitions is not yet fully understood.

In summary, this paper provides a rigorous combinatorial and geometric foundation for the moduli of tropical trigonal curves, proving that they faithfully model the dimension and connectivity properties of their algebraic counterparts, provided one accounts for tropical modifications in the presence of loops.