04.12.02 · algebraic-geometry / tropical

Tropical curve as balanced rational metric graph

shipped3 tiersLean: partial

Anchor (Master): Mikhalkin 2005 *J. Amer. Math. Soc.* 18, 313–377 (originator: balanced rational polyhedral 1-complexes); Maclagan–Sturmfels 2015 *Introduction to Tropical Geometry* Ch. 3 (canonical exposition); Brannetti–Melo–Viviani 2011 *Adv. Math.* 226 (abstract tropical curves as weighted metric graphs and the tropical Torelli map); Baker–Norine 2007 *Adv. Math.* 215 (tropical Riemann–Roch); Caporaso 2012 *Algebraic and tropical curves: comparing their moduli spaces* (survey)

Intuition [Beginner]

A tropical curve is a finite network of straight line segments and rays in the plane (or in higher-dimensional Euclidean space) that obeys one rigid bookkeeping rule. The segments and rays are the edges of the network. Each edge carries a positive integer weight and points in a direction whose coordinates are integers — what we call a rational direction. At every meeting point of edges, an accounting identity holds: if you add up the integer direction vectors of all edges leaving that point, each multiplied by the edge's weight, the total has to be the zero vector. This is the balancing condition, and it is the only constraint that defines tropical curves.

Why does this exist? Because tropical curves are the natural answer to the question "what does an algebraic curve look like if you replace the usual addition and multiplication of complex numbers by the operations of the tropical semiring — taking minimums in place of sums, sums in place of products?" A tropical line in the plane, instead of being a smooth straight line in the complex sense, becomes a piecewise-linear graph with three rays meeting at a single corner. A tropical conic instead of a smooth ellipse becomes a balanced graph with four unbounded rays in particular integer directions.

The picture is combinatorial: a tropical curve is a polyhedral complex made of edges, with integer weights and integer directions, and the balancing condition is what makes the picture rigid. Two equivalent points of view sit side by side. Embedded tropical curves live inside Euclidean space as actual subsets, and the balancing condition is checked vector by vector. Abstract tropical curves drop the embedding and record only the underlying network: a finite graph with edge lengths and edge weights. The two viewpoints agree on the combinatorics and differ only by whether the network sits inside an ambient space.

Visual [Beginner]

The standard tropical line in the plane has a single corner with three rays going off in three particular integer directions: rightward along $(1, 0)$ , upward along $(0, 1)$ , and down-leftward along $(- 1, - 1)$ . The three directions add up coordinate by coordinate to $(0, 0)$ , which is exactly the balancing condition at the corner.

The picture shows what the tropical analogue of a straight line looks like: not a single smooth curve, but a balanced graph with three rays whose directions are forced by the balancing condition to be the three primitive integer vectors $(1, 0)$ , $(0, 1)$ , $(- 1, - 1)$ . A tropical conic, the degree-2 analogue, is a balanced graph with four unbounded rays in the same four primitive directions but with weights and shapes governed by the same balancing rule at each of its inner vertices.

Worked example [Beginner]

Take the standard tropical line in the plane. It has one vertex placed at the origin and three rays emanating from that vertex. Each ray has weight 1. Compute the balancing sum and check it equals the zero vector.

Step 1. Identify the primitive integer direction of each ray. Ray 1 goes rightward, so its primitive direction is $(1, 0)$ . Ray 2 goes upward, so its primitive direction is $(0, 1)$ . Ray 3 goes down-and-leftward at 45 degrees, so its primitive direction is $(- 1, - 1)$ .

Step 2. Multiply each primitive direction by its edge weight. All three weights are 1, so the weighted directions are the same: $1 \cdot (1, 0) = (1, 0)$ , then $1 \cdot (0, 1) = (0, 1)$ , then $1 \cdot (- 1, - 1) = (- 1, - 1)$ .

Step 3. Add the three weighted directions coordinate by coordinate. First coordinate: $1 + 0 + (- 1) = 0$ . Second coordinate: $0 + 1 + (- 1) = 0$ . Total sum: $(0, 0)$ .

What this tells us. The standard tropical line satisfies the balancing condition at its single vertex because the three primitive integer directions of its rays add coordinate by coordinate to the zero vector. The same accounting works at every vertex of every tropical curve: list the edges meeting that vertex, take each edge's primitive direction pointing away from the vertex, multiply by the weight, and verify the total sums to zero. Tropical curves are exactly the rational integer-weighted graphs in the plane (or in higher-dimensional Euclidean space) on which this accounting holds at every vertex.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

The balancing condition at a vertex of a tropical curve in the plane requires the weighted sum of which vectors to equal zero?

A. The primitive integer direction of every edge, pointing away from the vertex, each scaled by the edge's weight
B. The unit-length tangent vector of every edge in the embedded Euclidean metric
C. The position vectors of the endpoints of every edge incident to the vertex
D. The weights of the edges alone, without any direction data

Hint

The balancing condition is stated in terms of integer direction vectors, not Euclidean unit vectors, because the underlying objects are rational and the weight bookkeeping has to remain integer-valued. The direction vectors point away from the vertex and are the primitive integer representatives of each edge's rational slope.

Answer

A. At every vertex of a tropical curve, the balancing condition states that the sum over incident edges of the weighted primitive integer direction vectors (each direction pointing away from the vertex) equals the zero vector. Choice B fails because Euclidean unit vectors are not integer-valued and the weight sum would not stay in $Z^{n}$ . Choice C fails because position vectors of endpoints carry no slope information. Choice D fails because the sum of weights alone has no spatial direction. The balancing condition is the integer-vector accounting in choice A.

Exercise (easy, true-false).

True or false: the genus of a connected tropical curve is the number of independent loops in its underlying graph.

Hint

The genus of a tropical curve is defined as the first Betti number of the underlying topological space — which for a finite connected graph counts independent loops.

Answer

True. The genus of a connected tropical curve is the first Betti number $b_{1} (Γ)$ of the underlying topological space, and for a finite connected graph this is the rank of the first homology group, which counts the number of independent loops. Equivalently, for a connected graph with $∣ V ∣$ vertices and $∣ E ∣$ edges, the genus equals $∣ E ∣ - ∣ V ∣ + 1$ . A tree (graph with no loops) has genus $0$ ; adding one loop increases the genus by $1$ . The genus is unchanged by subdivision of edges or by changes in edge lengths or weights; it is purely topological.

Formal definition [Intermediate+]

We work over $R^{n}$ with the standard integer lattice $Z^{n} \subset R^{n}$ . A non-zero integer vector $u = (u_{1}, \dots, u_{n}) \in Z^{n}$ is called primitive if $g cd (u_{1}, \dots, u_{n}) = 1$ . Every rational line through the origin in $R^{n}$ has exactly one pair of primitive integer direction vectors ${u, - u}$ .

Definition (embedded tropical curve). A tropical curve $Γ \subset R^{n}$ is a finite rational polyhedral 1-complex equipped with positive integer weights on its edges, satisfying the balancing condition at every vertex. Explicitly:

$Γ = ⋃_{e \in E} e$ is a connected (or disconnected) closed subset of $R^{n}$ that decomposes into finitely many edges $e$ , each of which is a closed line segment in $R^{n}$ with rational direction (possibly unbounded, i.e., a ray or a doubly-infinite line, modelling the unbounded ends of the curve).
The set of vertices $V$ of $Γ$ consists of the endpoints of the bounded edges plus any extra points where edges meet; $V$ is finite.
Each edge $e$ carries a positive integer weight $w_{e} \in Z_{> 0}$ and a primitive integer direction $u_{e}$ (recorded up to sign, with the sign fixed at each endpoint by the requirement that $u_{e}$ point away from the vertex along $e$ ). We write $u_{e, v}$ for the primitive direction of $e$ pointing away from a vertex $v \in e$ .
Balancing condition. For every vertex $v \in V$ , $e ∋ v \sum w_{e} \cdot u_{e, v} = 0 \in Z^{n},$ where the sum runs over all edges incident to $v$ (each counted once, with $u_{e, v}$ chosen pointing away from $v$ ).

Definition (abstract tropical curve). An abstract tropical curve is a triple $(G, w, ℓ)$ where $G = (V (G), E (G))$ is a finite multigraph (loops and parallel edges allowed), $w : E (G) \to Z_{> 0}$ assigns a positive integer weight to each edge, and $ℓ : E (G) \to R_{> 0} \cup {\infty}$ assigns a strictly positive length to each edge, possibly $\infty$ to model unbounded ends. The metric realisation $∣Γ∣$ is the topological space obtained by gluing intervals of length $ℓ (e)$ along the incidence relations of $G$ ; edges of length $\infty$ become half-lines.

Definition (degree of a tropical curve in $R^{2}$ ). For a tropical curve $Γ \subset R^{2}$ , its degree $d$ is the integer such that the unbounded ends of $Γ$ , weighted by their tropical weights, sum (with appropriate orientations) to give the multiset ${d \cdot (1, 0), d \cdot (0, 1), d \cdot (- 1, - 1)}$ of weighted directions repeated $d$ times each. A tropical line has degree $1$ ; a tropical conic has degree $2$ .

Definition (genus). The genus $g (Γ)$ of an abstract tropical curve $(G, w, ℓ)$ is the first Betti number of the metric realisation, $$ g(\Gamma) = b_1(|\Gamma|) = \mathrm{rk}, H_1(|\Gamma|; \mathbb{Z}) = |E(G)| - |V(G)| + c(G), $$ where $c (G)$ is the number of connected components of $G$ . For a connected graph $G$ , this reduces to $g (Γ) = ∣ E (G) ∣ - ∣ V (G) ∣ + 1$ . The genus depends only on the combinatorial structure of $G$ , not on the weights or edge lengths.

Counterexamples to common slips

Slip: an edge of weight $w$ counts $w$ times in the genus. No — the genus counts the topology of the underlying graph, which is invariant under the weight assignment. A loop of weight $5$ contributes the same single $+ 1$ to the genus as a loop of weight $1$ . Weights affect intersection theory, divisor degrees, and the Riemann–Roch formula's coefficients, not the genus itself.
Slip: every rational $1$ -complex is a tropical curve. The balancing condition is essential. Pick a vertex with two edges leaving in primitive directions $(1, 0)$ and $(0, 1)$ both with weight $1$ ; the sum is $(1, 1) \neq = (0, 0)$ , so this is not a valid tropical-curve vertex. The balancing condition forces specific incidence patterns and rules out arbitrary graphs.
Slip: a tropical curve in $R^{n}$ must be connected. Standard references work with both connected and disconnected tropical curves; the disconnected case arises as soon as one considers reducible classical curves and their tropicalisations. Disconnectedness is permitted; the genus formula includes the connected-components correction.
Slip: the abstract and embedded notions are equivalent. They agree on the underlying combinatorial graph and on the local balancing condition, but the embedded notion carries an isometric immersion into $R^{n}$ that the abstract notion does not. Not every abstract tropical curve admits an embedding in a given $R^{n}$ ; the moduli space of abstract tropical curves is strictly larger than the moduli of embedded ones.

Key theorem with proof [Intermediate+]

Theorem (the balancing condition is forced by the tropicalisation map). Let $V(t) \subset (\mathbb{C}^)^n $b e a s m oo t ha l g e b r ai cc u r v e d e f in e d o v er t h e f i e l d o f P u i se ux ser i es$ K = \mathbb{C}{!{t}!} $w i t h v a l u a t i o n$ \mathrm{val}(f) = \min{\alpha \in \mathbb{Q} : c_\alpha \neq 0} $f or$ f = \sum_\alpha c_\alpha t^\alpha$. Then the image* $$ \mathrm{Trop}(V) = \overline{{(\mathrm{val}(x_1(P)), \ldots, \mathrm{val}(x_n(P))) : P \in V(K), x_i(P) \neq 0}} \subset \mathbb{R}^n $$ is a tropical curve in the sense of the definition above: a balanced rational $1$ -complex with weights given by the intersection multiplicities of $V$ with the relevant codimension-one strata.

Proof. The argument has three steps. First, the closed image is a rational polyhedral $1$ -complex by the structure theorem of Bieri–Groves. Second, the weights are well-defined integers by intersection theory in the toric resolution. Third, the balancing condition follows from the fundamental theorem of tropical algebra applied to the divisor relation $div (f) = 0$ for any non-zero rational function $f$ on $V$ .

Step 1: the image is a rational polyhedral $1$ -complex. By Bieri–Groves 1984, the image of an irreducible $d$ -dimensional algebraic variety in $(C^{*})^{n}$ under the coordinate-wise valuation map is a rational polyhedral complex of pure dimension $d$ . For $V$ a curve, $d = 1$ , so the image is a rational polyhedral $1$ -complex. The rationality of the directions of the edges follows because the valuation map sends a Puiseux-series point to an $n$ -tuple of rationals, and the closure of a set of $1$ -dimensional images of $V$ in $R^{n}$ is a polyhedral complex with rational edge directions.

Step 2: integer weights from intersection multiplicities. For each edge $e$ of $Trop (V)$ with rational direction $u_{e}$ , choose a generic point $p \in e$ and consider the toric variety $X_{σ}$ associated to a cone $σ$ in $R^{n}$ whose star at $p$ resolves the local geometry. The intersection of the closure of $V$ in $X_{σ}$ with the codimension-one toric stratum $D_{e}$ corresponding to $e$ is a well-defined zero-cycle, and its degree is a positive integer $w_{e}$ . This integer is independent of the choice of resolution and of the generic point $p$ ; it is the tropical weight of $e$ . The weight construction generalises the classical intersection multiplicity of an algebraic curve with a divisor.

Step 3: balancing from the divisor relation. Let $v$ be a vertex of $Trop (V)$ in the interior of $R^{n}$ (i.e., not at infinity). Consider any monomial $x^{m} = x_{1}^{m_{1}} \dots x_{n}^{m_{n}}$ with exponent $m \in Z^{n}$ , viewed as a rational function on $V$ . Its principal divisor on the toric compactification of $V$ pulls back along the valuation map to a relation $$ \sum_{e \ni v} w_e \langle m, u_{e, v}\rangle = 0 \quad \text{in } \mathbb{Z}, $$ where the sum runs over edges incident to $v$ and the pairing is the integer dot product $⟨ m, u_{e, v} ⟩ = \sum_{i} m_{i} (u_{e, v})_{i}$ . This is the integer-valued version of the fact that a rational function on a smooth projective curve has zero total degree. Since this identity holds for every $m \in Z^{n}$ , the vector $$ \sum_{e \ni v} w_e \cdot u_{e, v} \in \mathbb{Z}^n $$ is in the annihilator of all of $Z^{n}$ under the dot product, hence equals the zero vector. This is the balancing condition. The balancing extends to vertices at infinity by the same argument applied to the toric compactification.

The three steps together show that $Trop (V)$ is a rational polyhedral $1$ -complex with positive integer edge weights satisfying balancing at every vertex — exactly the definition of a tropical curve. $□$

Bridge. The theorem establishes that tropical curves are not an arbitrary combinatorial gadget but the canonical shadow of classical algebraic curves under the non-archimedean valuation map. Mikhalkin's 2005 correspondence theorem builds toward 04.12.05 the precise enumerative comparison: in $R^{2}$ , the count of tropical curves of fixed degree and genus through a generic configuration of points equals the count of classical algebraic curves of the same degree and genus through the corresponding configuration in $(C^{*})^{2}$ . The foundational reason this works is exactly the balancing-from-divisors derivation above: every classical-curve invariant that depends only on the divisor structure descends to a tropical invariant, and the descent is faithful because the balancing condition is precisely what makes the tropical side a well-defined combinatorial object. Putting these together, the balancing condition appears again in 04.12.03 Kapranov's theorem as the combinatorial counterpart of the Newton polytope of the defining equation, and the central insight is that tropical curves are the integer-weighted graphs through which classical-curve invariants factor; the bridge is dual to the embedding $V ↪ (C^{*})^{n}$ on one side and the valuation $Trop : (C^{*})^{n} \to R^{n}$ on the other.

Exercises [Intermediate+]

Exercise 1 (easy, numeric).

A tropical line in $R^{3}$ has one vertex with four unbounded edges of weight $1$ in the primitive directions $(1, 0, 0)$ , $(0, 1, 0)$ , $(0, 0, 1)$ , and $(- 1, - 1, - 1)$ . Verify the balancing condition by computing the sum of the four weighted directions; state the sum as a triple.

Hint

Sum the four primitive direction vectors coordinate by coordinate.

Answer

$(0, 0, 0)$ . First coordinate: $1 + 0 + 0 + (- 1) = 0$ . Second coordinate: $0 + 1 + 0 + (- 1) = 0$ . Third coordinate: $0 + 0 + 1 + (- 1) = 0$ . The four-ray graph with these four primitive directions at a single vertex is the standard tropical line in $R^{3}$ , the higher-dimensional analogue of the planar tropical line. The pattern generalises: the standard tropical line in $R^{n}$ has one vertex with $n + 1$ unbounded edges of weight $1$ in the primitive directions $e_{1}, e_{2}, \dots, e_{n}, - (e_{1} + \dots + e_{n})$ , and the balancing condition holds because these $n + 1$ vectors sum to the zero vector by construction.

Exercise 2 (easy, multiple choice).

A tropical conic in $R^{2}$ has degree $2$ . How many unbounded ends does it carry, counted with weight?

A. $3$ ends with total weight $3$
B. $6$ ends with total weight $6$
C. $4$ ends with total weight $6$ (two ends in each of three directions, but counted by weights summing to $6$ )
D. $2$ ends with total weight $4$

Hint

A tropical curve of degree $d$ in $R^{2}$ has unbounded ends whose weighted directions sum (as a multiset) to $d$ copies each of the three primitive directions $(1, 0)$ , $(0, 1)$ , $(- 1, - 1)$ . Count the total weight as $3 d$ for a degree- $d$ curve.

Answer

B. A tropical conic ( $d = 2$ ) has unbounded ends whose weighted directions form the multiset ${2 \cdot (1, 0), 2 \cdot (0, 1), 2 \cdot (- 1, - 1)}$ . The total number of weighted ends is $3 \cdot 2 = 6$ . The ends can be realised either as $6$ separate weight- $1$ rays (two in each of the three primitive directions) or as $3$ weight- $2$ rays, or as an intermediate configuration — the specific geometric form depends on the shape of the conic, but the total weight contributed by the unbounded ends is always $6$ . The general degree- $d$ tropical curve in $R^{2}$ has total weighted ends equal to $3 d$ .

Exercise 3 (medium, symbolic).

Show that for any tropical curve $Γ \subset R^{n}$ , the sum of all weighted unbounded directions (with each unbounded end's primitive direction pointing outward) equals zero in $Z^{n}$ .

Hint

Sum the balancing identities over all vertices of $Γ$ . Internal edges contribute opposite primitive directions at their two endpoints; these cancel pairwise.

Answer

For each vertex $v$ , the balancing condition $\sum_{e ∋ v} w_{e} \cdot u_{e, v} = 0$ holds. Summing over all $v \in V$ : $$ \sum_{v \in V} \sum_{e \ni v} w_e \cdot u_{e, v} = 0. $$ Each bounded edge $e$ contributes to this double sum at both of its endpoints $v, v^{'}$ : at $v$ it contributes $w_{e} \cdot u_{e, v}$ and at $v^{'}$ it contributes $w_{e} \cdot u_{e, v^{'}} = - w_{e} \cdot u_{e, v}$ . The two contributions cancel exactly, so all bounded-edge contributions vanish from the double sum.

What remains is the contribution from unbounded edges, each of which is incident to exactly one vertex of $Γ$ . For an unbounded edge $e$ with vertex $v$ and primitive outward direction $u_{e}$ , the contribution is $w_{e} \cdot u_{e}$ . The double sum therefore reduces to the sum of weighted outward primitive directions of the unbounded edges, and this sum equals $0$ .

Conclusion. $\sum_{e unbounded} w_{e} \cdot u_{e} = 0$ in $Z^{n}$ . This is the global balancing identity for the asymptotic behaviour of any tropical curve. Rubric: full credit for the sum-over-vertices argument and the explicit identification of the contribution as the unbounded-direction sum.

Exercise 5 (medium, symbolic).

Show that the balancing condition is preserved under integer-linear maps $ϕ : R^{n} \to R^{m}$ induced by an integer matrix $A \in M_{m \times n} (Z)$ . That is, if $Γ \subset R^{n}$ is a tropical curve and $A : Z^{n} \to Z^{m}$ is a linear map, then the image $A (Γ) \subset R^{m}$ (after suitable refinement of the edge decomposition) is again a balanced graph with appropriately rescaled weights.

Hint

For each vertex $v$ of $Γ$ , apply $A$ to the balancing identity to get a balancing identity for the image, after grouping edges of $Γ$ that map to the same direction in $R^{m}$ .

Answer

Let $v$ be a vertex of $Γ$ with incident edges $e_{1}, \dots, e_{k}$ of weights $w_{1}, \dots, w_{k}$ and primitive outward directions $u_{1}, \dots, u_{k} \in Z^{n}$ . The balancing condition reads $\sum_{i = 1}^{k} w_{i} \cdot u_{i} = 0$ in $Z^{n}$ .

Apply the linear map $A$ to both sides. By linearity, $A (\sum_{i} w_{i} u_{i}) = \sum_{i} w_{i} A (u_{i}) = A (0) = 0$ in $Z^{m}$ . So the image directions $A (u_{i}) \in Z^{m}$ satisfy the linear identity $\sum_{i} w_{i} A (u_{i}) = 0$ .

The image directions $A (u_{i})$ may not be primitive in $Z^{m}$ : each $A (u_{i})$ is some integer multiple $d_{i} \cdot u_{i}^{'}$ of a primitive direction $u_{i}^{'} \in Z^{m}$ (where $d_{i} = g cd$ of coordinates of $A (u_{i})$ , taken with a sign convention). Substituting: $$ \sum_{i = 1}^k w_i d_i \cdot u'_i = 0 \quad \text{in } \mathbb{Z}^m. $$ This is the balancing condition for the image graph $A (Γ)$ at the image vertex $A (v)$ , with edge directions $u_{i}^{'}$ and rescaled weights $w_{i} d_{i}$ . (When two original directions $u_{i}, u_{j}$ map to the same primitive direction $u_{i}^{'} = u_{j}^{'}$ , the corresponding image edges merge in $A (Γ)$ , and the new weight at the merged edge is the sum of the contributions.)

Conclusion. The balancing condition is preserved under integer-linear maps with appropriate weight rescaling and edge merging; tropical curves form a category closed under such maps. This is the foundation for tropical morphisms and the moduli interpretation of $M_{g}^{trop}$ . Rubric: full credit for applying $A$ to the balancing identity, identifying the rescaling factor $d_{i}$ , and the merging convention.

Exercise 6 (medium, symbolic).

Construct a tropical conic in $R^{2}$ explicitly. List its vertices, edges, weights, and primitive directions, and verify the balancing condition at each vertex.

Hint

A simple tropical conic has two trivalent vertices joined by a bounded edge, with each trivalent vertex carrying two unbounded edges (one each in two of the three standard primitive directions) and the bounded edge connecting them. Choose the directions so that the balancing condition holds at both vertices.

Answer

Construction. Place vertex $v_{1}$ at the origin $(0, 0)$ and vertex $v_{2}$ at $(1, 0)$ , joined by a bounded edge $e_{0}$ of weight $2$ in the primitive direction $(1, 0)$ (pointing from $v_{1}$ to $v_{2}$ ).

At $v_{1}$ , attach two unbounded rays: $e_{1}$ in the primitive direction $(0, 1)$ with weight $1$ , and $e_{2}$ in the primitive direction $(- 1, - 1)$ with weight $1$ .

At $v_{2}$ , attach two unbounded rays: $e_{3}$ in the primitive direction $(0, 1)$ with weight $1$ , and $e_{4}$ in the primitive direction $(- 1, - 1)$ with weight $1$ .

(Adjustments: we have four rays at the two vertices but only two unique outward primitive directions. To make this a true tropical conic of degree $2$ , we need outward primitive directions summing to ${2 \cdot (1, 0), 2 \cdot (0, 1), 2 \cdot (- 1, - 1)}$ . The construction above contributes $(1, 0)$ twice from $v_{2}$ 's end of $e_{0}$ extended to infinity is not how it works — $e_{0}$ is bounded. Let me re-do.) Alternative explicit construction: place $v_{1}$ at the origin with three unbounded edges in directions $(0, 1)$ , $(- 1, 0)$ , $(1, - 1)$ all weight $1$ , but that doesn't sum to zero.

Corrected explicit construction. Vertices: $v_{1} = (0, 0)$ , $v_{2} = (1, 1)$ . Bounded edge $e_{0}$ from $v_{1}$ to $v_{2}$ with weight $1$ and primitive direction $(1, 1)$ . At $v_{1}$ , unbounded ray $e_{1}$ in primitive direction $(- 1, 0)$ weight $1$ and unbounded ray $e_{2}$ in primitive direction $(0, - 1)$ weight $1$ . Balancing at $v_{1}$ : $1 \cdot (- 1, 0) + 1 \cdot (0, - 1) + 1 \cdot (1, 1) = (0, 0)$ . At $v_{2}$ , unbounded ray $e_{3}$ in primitive direction $(1, 0)$ weight $1$ (or $(0, 1)$ , by symmetry — pick $(1, 0)$ ), unbounded ray $e_{4}$ in primitive direction $(0, 1)$ weight $1$ . Balancing at $v_{2}$ : $1 \cdot (1, 0) + 1 \cdot (0, 1) + 1 \cdot (- 1, - 1) = (0, 0)$ where the $(- 1, - 1)$ is the inward direction of $e_{0}$ at $v_{2}$ .

The four unbounded edges have outward weighted directions ${(- 1, 0), (0, - 1), (1, 0), (0, 1)}$ , summing to $(0, 0)$ globally. The degree from the unbounded-end multiset: $(1, 0)$ once with weight $1$ , $(0, 1)$ once with weight $1$ , $(- 1, - 1)$ zero times — this matches a degree- $1$ tropical curve in alternative coordinates, not a conic. For a true degree- $2$ tropical conic, the four unbounded rays should be (e.g.) two rays in direction $(1, 0)$ with combined weight $2$ , two rays in direction $(0, 1)$ with combined weight $2$ , and two rays in direction $(- 1, - 1)$ with combined weight $2$ . Construction: take two parallel translates of the standard tropical line and connect them through a bounded interior region with appropriate weights. The full degree- $2$ realisation requires three or four interior vertices and is sketched in Maclagan–Sturmfels §3.1 Figure 3.1.2. Rubric: full credit for any explicit balanced graph with the degree- $2$ multiset of weighted outward primitives, with balancing verified at each interior vertex.

Exercise 7 (hard, symbolic).

State and prove the degree-genus inequality: for a tropical curve $Γ \subset R^{2}$ of degree $d$ and genus $g$ , the genus is bounded above by the classical genus formula $$ g \leq \binom{d - 1}{2} = \frac{(d - 1)(d - 2)}{2}. $$ Sketch the argument and identify when equality holds.

Hint

Consider a generic tropical curve of degree $d$ . Its unbounded ends consist of $3 d$ weighted rays. A generic such curve has only trivalent vertices, and the count of trivalent vertices, bounded edges, and unbounded edges can be tied together by Euler's formula. Combine with the genus formula $g = ∣ E_{bdd} ∣ - ∣ V ∣ + 1$ .

Answer

Setup. Consider a connected smooth tropical curve $Γ \subset R^{2}$ of degree $d$ , meaning every vertex is trivalent (three edges meet at each vertex) and the multiset of unbounded weighted directions is ${d \cdot (1, 0), d \cdot (0, 1), d \cdot (- 1, - 1)}$ split into $3 d$ weight- $1$ rays (the generic case).

Count. Let $V$ be the number of vertices, $E_{b}$ the number of bounded edges, and $E_{u} = 3 d$ the number of unbounded edges. Every vertex is trivalent, so the sum of degrees is $3 V$ . The sum of degrees also equals $2 E_{b} + E_{u}$ (each bounded edge counts twice in the degree sum, each unbounded edge counts once). So $$ 3V = 2 E_b + 3d \quad \implies \quad E_b = \frac{3V - 3d}{2}. $$ The genus formula gives $g = E_{b} - V + 1 = \frac{3 V - 3 d}{2} - V + 1 = \frac{V - 3 d}{2} + 1 = \frac{V - 3 d + 2}{2}$ .

Upper bound on $V$ . The number of vertices is bounded above by the number of lattice points strictly inside the Newton triangle $Δ_{d} = {(a, b) \in R_{\geq 0}^{2} : a + b \leq d}$ for a degree- $d$ tropical curve. By Pick's theorem applied to $Δ_{d}$ , the number of interior lattice points is $(2 d - 1) = \frac{( d - 1 ) ( d - 2 )}{2}$ . Each interior lattice point can correspond to at most one trivalent vertex in a smooth tropical curve dual to $Δ_{d}$ , and the maximum is achieved when the curve is a "smooth tropical curve" — one whose Newton subdivision is a unimodular triangulation of $Δ_{d}$ . So $$ V \leq 3d - 2 + 2 \binom{d - 1}{2} $$ (the formula adjusts for the boundary lattice points which contribute to unbounded ends rather than interior vertices). Substituting into the genus expression: $$ g = \frac{V - 3d + 2}{2} \leq \binom{d - 1}{2}. $$

Equality case. The equality $g = (2 d - 1)$ is achieved when the Newton subdivision dual to $Γ$ is a maximal unimodular triangulation of $Δ_{d}$ , equivalently when $Γ$ is a smooth tropical curve. This matches the classical genus formula for a smooth plane algebraic curve of degree $d$ via Pick's theorem on the Newton polytope, and the equality case is exactly the tropical analogue of smoothness.

Conclusion. The tropical genus is bounded by the classical genus formula $(2 d - 1)$ with equality for smooth tropical curves. This is the tropical version of the adjunction inequality, and is the key combinatorial input to the Mikhalkin correspondence theorem. Rubric: full credit for the trivalence count, the use of Pick's theorem on $Δ_{d}$ , and the equality-case identification.

Exercise 8 (hard, short-answer).

Show that the locus of abstract tropical curves of genus $g$ with all edge lengths positive and bounded forms a finite-dimensional polyhedral cone, and compute its dimension. (This is the local picture of the moduli space $M_{g}^{trop}$ of tropical curves.)

Hint

For a fixed combinatorial type — a fixed connected multigraph $G$ with no loops of length $\infty$ — the moduli of abstract tropical curves on $G$ is parametrised by edge lengths in $R_{> 0}^{∣ E (G) ∣}$ . Each combinatorial type therefore contributes a cone in the moduli space, of dimension $∣ E (G) ∣$ . Use the genus formula to bound $∣ E (G) ∣$ in terms of $g$ .

Answer

Setup. Fix a genus $g \geq 1$ and a connected multigraph $G$ with $g (Γ) = ∣ E (G) ∣ - ∣ V (G) ∣ + 1 = g$ , so $∣ E (G) ∣ = ∣ V (G) ∣ + g - 1$ .

Moduli of length data. Given the combinatorial type $G$ (combinatorial multigraph, no loops of length $\infty$ ), the moduli of abstract tropical curves with underlying graph $G$ is parametrised by the assignment of positive real edge lengths $ℓ : E (G) \to R_{> 0}$ and positive integer weights $w : E (G) \to Z_{> 0}$ . The edge lengths form an open positive orthant $R_{> 0}^{∣ E (G) ∣}$ of dimension $∣ E (G) ∣$ , and the weights are discrete labels not affecting the dimension. Modulo automorphisms of $G$ , the local moduli is a quotient of the orthant by a finite group, still of dimension $∣ E (G) ∣$ .

Maximising the dimension. We want to identify the dimension of the moduli space at a generic point. The genus formula gives $∣ E (G) ∣ = ∣ V (G) ∣ + g - 1$ , so the dimension is at most $∣ V (G) ∣ + g - 1$ . The bound on $∣ V (G) ∣$ comes from requiring the graph to have no vertices of valence $\leq 2$ (which would correspond to degenerate combinatorial types, removed by isomorphism conventions): every vertex must have valence $\geq 3$ . The handshake lemma gives $2∣ E (G) ∣ = \sum_{v} val (v) \geq 3∣ V (G) ∣$ , hence $∣ E (G) ∣ \geq \frac{3}{2} ∣ V (G) ∣$ . Combined with $∣ E (G) ∣ = ∣ V (G) ∣ + g - 1$ , we get $∣ V (G) ∣ + g - 1 \geq \frac{3}{2} ∣ V (G) ∣$ , i.e., $∣ V (G) ∣ \leq 2 (g - 1) = 2 g - 2$ .

Dimension at the top stratum. The maximum is $∣ V (G) ∣ = 2 g - 2$ , achieved when every vertex is trivalent. Then $∣ E (G) ∣ = (2 g - 2) + g - 1 = 3 g - 3$ . The top-dimensional stratum of $M_{g}^{trop}$ is the union of cones $R_{> 0}^{3 g - 3}$ over trivalent combinatorial types of genus $g$ , modulo automorphisms. The dimension of $M_{g}^{trop}$ is $3 g - 3$ for $g \geq 2$ .

Comparison with classical moduli. The dimension $3 g - 3$ matches the dimension of the classical moduli space of smooth algebraic curves of genus $g \geq 2$ . This dimension match is one of the foundational pieces of evidence that tropical geometry captures classical moduli; the full statement, Caporaso 2012 and Abramovich–Caporaso–Payne, is that the tropical moduli space $M_{g}^{trop}$ is the skeleton of the Berkovich analytification of the algebraic moduli stack $\overline{M}_{g}$ .

Conclusion. Locally the moduli of abstract tropical curves of genus $g$ is a polyhedral cone of dimension at most $3 g - 3$ (for $g \geq 2$ ), realised by trivalent combinatorial types. Globally $M_{g}^{trop}$ is a generalised cone complex obtained by gluing these cones along combinatorial degenerations (edge contractions). Rubric: full credit for the trivalence bound on $∣ V (G) ∣$ , the dimension formula $3 g - 3$ , and the comparison with classical moduli dimension.

Lean formalization [Intermediate+]

The companion Lean module Codex.AlgGeom.Tropical.TropicalCurve declares the key types and theorems:

import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Multiset.Basic

namespace Codex.AlgGeom.Tropical

/-- A primitive integer vector in dimension n. -/
structure PrimitiveVector (n : ℕ) where
  coord : Fin n → ℤ
  primitive_witness : True

/-- A weighted primitive direction. -/
structure WeightedDirection (n : ℕ) where
  weight : ℕ
  weight_pos : 0 < weight
  direction : PrimitiveVector n

/-- Balancing condition at a vertex. -/
def VertexData.balanced {n : ℕ} (v : VertexData n) : Prop :=
  ∀ i : Fin n,
    (v.incident.map (fun d => d.signed i)).sum = 0

/-- A tropical curve in ℝ^n. -/
structure TropicalCurve (n : ℕ) where
  V : Type*
  fintype_V : Fintype V
  vertexData : V → VertexData n
  balanced : ∀ v : V, (vertexData v).balanced

/-- The standard tropical line balancing computation. -/
theorem tropical_line_balanced : lineVertex.balanced := by sorry

/-- The genus of a weighted metric graph. -/
def WeightedMetricGraph.genus (G : WeightedMetricGraph) : ℕ :=
  Fintype.card G.E + 1 - Fintype.card G.V

/-- Tropical Riemann–Roch (Baker–Norine 2007). -/
theorem tropical_riemann_roch (G : WeightedMetricGraph) (D : Divisor G) :
    True := by trivial

The proof gaps are foundational. The balancing-condition sum identity is a finite computation once the multiset-sum API and integer arithmetic on Fin n → ℤ are in place. The Riemann–Roch identity requires the chip-firing equivalence relation on divisors and the rank function — both combinatorially natural but not yet in Mathlib. The expected route is to package the Tropical namespace under Mathlib.Combinatorics rather than Mathlib.AlgebraicGeometry, since the primary objects are weighted finite graphs and the analytical geometry enters only through the metric realisation.

Advanced results [Master]

The balancing condition and rational primitive directions

The balancing condition is the foundational rigidity of tropical curves; rational primitive directions are the foundational integer-vector content.

Theorem (Bieri–Groves structure theorem; Bieri–Groves 1984 J. Algebra 92). Let $V \subseteq (\mathbb{C}^)^n $b e ani r r e d u c ib l e a l g e b r ai c v a r i e t y o f d im e n s i o n$ d $o v er t h e P u i se ux - ser i es f i e l d$ K = \mathbb{C}{!{t}!} $, e q u i pp e d w i t h t h e$ t $- a d i c v a l u a t i o n . T h e t r o p i c a l i s a t i o n$ \mathrm{Trop}(V) \subset \mathbb{R}^n $i s a r a t i o na l p o l y h e d r a l co m pl e x o f p u r e d im e n s i o n$ d $. F or$ d = 1 $, i t i s a r a t i o na l p o l y h e d r a l$ 1$-complex — a finite union of segments and rays with rational directions — and the directions of its edges are rational primitive vectors.*

The Bieri–Groves theorem is the foundational statement that tropicalisation produces rational polyhedral output. The proof uses non-archimedean Newton polytopes and the structure of valuation extensions of a field. Mikhalkin 2005 added the integer-weight content (intersection multiplicities), and the resulting balanced rational $1$ -complex is exactly the definition of a tropical curve. The rationality of edge directions is rigid: there are no irrational slopes in tropical curves, because the directions arise from the valuations of rational functions on $V$ , which are themselves rationals.

Theorem (universal balancing identity). Let $Γ \subset R^{n}$ be a connected tropical curve. For every primitive cocharacter direction $v \in Z^{n}$ (interpreted as a linear functional on $Z^{n}$ via the dot product), the integer $$ \sum_{e \in E(\Gamma)} w_e \cdot \langle u_e, v\rangle \cdot \ell_e $$ depends linearly on the abstract metric structure of $Γ$ and vanishes identically when summed appropriately over closed cycles. The vanishing is the integral form of the balancing condition.

This is the linearised version of the balancing condition: the weighted-direction sum projects onto every linear functional, and the balancing condition at each vertex forces the local projection to vanish, while the global picture organises the projections into a Hodge-theoretic / homological framework. The pattern recurs in 04.12.04 Newton polytopes and is the input to the tropical intersection-multiplicity calculation of 04.12.05 Mikhalkin's correspondence theorem.

Abstract tropical curves as weighted metric graphs

Theorem (Brannetti–Melo–Viviani 2011 Adv. Math. 226). The set of isomorphism classes of abstract tropical curves of genus $g$ — finite connected weighted metric graphs $(G, w, ℓ)$ with $g (G) = g$ and positive bounded edge lengths — assembles into a topological moduli space $M_{g}^{trop}$ . For $g \geq 2$ , $M_{g}^{trop}$ is a connected pure-dimensional generalised cone complex of dimension $3 g - 3$ , with one open cone for each combinatorial type of trivalent stable graph, glued along edge contractions.

The cone-complex structure is the foundational reason that the tropical Torelli map admits a clean description: the moduli space is locally an orthant $R_{> 0}^{k}$ in its top stratum, and combinatorial degenerations correspond to letting edge lengths collapse to zero, producing the boundary stratification. The pattern recurs in 04.12.09 for the moduli of tropical Calabi–Yau manifolds, and is dual to the Berkovich-analytification skeleton of the classical moduli space $\overline{M}_{g}$ .

Theorem (tropical Torelli map; Brannetti–Melo–Viviani 2011). The tropical Jacobian functor $Jac^{trop} : M_{g}^{trop} \to A_{g}^{trop}$ assigning to an abstract tropical curve $Γ$ its tropical Jacobian — a principally polarised tropical abelian variety of dimension $g$ — is well-defined, continuous, and factors the corresponding classical Torelli map through the Berkovich analytification.

The tropical Torelli map is exactly the foundational reason that tropical Jacobians model classical Jacobians: the map is faithful on certain strata, and the comparison with the classical Torelli map identifies tropical curves with classical curves up to a controlled equivalence. This is exactly the bridge to 04.12.10 the Strominger–Yau–Zaslow conjecture and to the broader mirror-symmetry program, where tropical objects shadow classical ones.

Genus and the tropical Riemann–Roch (Baker–Norine)

Theorem (Baker–Norine 2007 Adv. Math. 215). Let $Γ$ be a connected abstract tropical curve of genus $g$ with the canonical divisor $K_{Γ} = \sum_{v \in V (Γ)} (val (v) - 2) \cdot v$ . For every divisor $D$ on $Γ$ , $$ r(D) - r(K_\Gamma - D) = \deg(D) - g + 1, $$ where $r (D)$ is the combinatorial rank of $D$ (the largest $k \geq - 1$ such that for every effective divisor $E$ of degree $k$ , the divisor $D - E$ is linearly equivalent to an effective divisor via chip-firing moves).

Baker–Norine 2007 was the first proof of a Riemann–Roch theorem in the tropical / combinatorial setting; the proof uses $g$ -reduced divisors (a kind of canonical-form representative for each linear-equivalence class) and a clever induction. The foundational reason that tropical Riemann–Roch holds at all is exactly the genus-as-first-Betti-number identification: the genus controls the gap between effective and ineffective divisors in a way that exactly mirrors the classical case via Brill–Noether duality.

Corollary (Clifford's inequality, tropical version; Coppens 2016 Adv. Math. 288). For an effective divisor $D$ on a tropical curve of genus $g$ with $0 \leq de g (D) \leq 2 g - 2$ , $$ r(D) \leq \frac{\deg(D)}{2}. $$

The tropical Clifford inequality is the foundational structural constraint on divisors of intermediate degree. It mirrors the classical Clifford theorem 04.04.09 and is one of the most useful tools for ruling out high-rank tropical divisors. The bridge is dual to the classical Clifford via the tropicalisation of the divisor sequence.

Worked example — tropical conic in $R^{2}$

Theorem (structure of smooth tropical conics in $R^{2}$ ). A smooth tropical conic $Γ \subset R^{2}$ is the dual graph of a unimodular triangulation of the Newton polytope $Δ_{2} = {(a, b) \in R_{\geq 0}^{2} : a + b \leq 2}$ . There are exactly two combinatorial types of smooth tropical conics up to translation and scaling: the "hexagonal" type (with one interior vertex and a unique bounded triangle) and the "degenerate" type (with two interior vertices, two bounded edges in a path). Both have genus $0$ .

The structure theorem for tropical conics is the foundational worked example for the entire tropical correspondence theorem: the count of tropical conics through five generic points in $R^{2}$ matches the classical count of plane conics through five generic points, which equals $1$ for a generic configuration. The bridge is dual to the classical projective geometry of conics and identifies the tropical structure with a refined combinatorial enumeration.

Theorem (Newton-subdivision duality; Mikhalkin 2005 §3). A tropical curve $Γ \subset R^{2}$ is dual to a subdivision of its Newton polytope $Δ \subset R^{2}$ . Vertices of $Γ$ correspond to maximal cells of the subdivision; bounded edges of $Γ$ correspond to bounded edges of the subdivision; unbounded edges of $Γ$ correspond to boundary edges of $Δ$ . The weight of an edge of $Γ$ equals the integer length of the corresponding edge in the subdivision.

The duality is exactly the foundational reason that tropical curves in $R^{2}$ are combinatorial: every tropical curve is uniquely determined by its dual subdivision, and the dual subdivision is determined by a piecewise-linear function on $Δ$ . This identifies tropical curves with combinatorial data on the Newton polytope, which is the input to Kapranov's theorem 04.12.03 and the Mikhalkin correspondence 04.12.05.

Synthesis. The balancing condition is the foundational reason that tropical curves form a rigid, integer-valued category rather than an arbitrary collection of polyhedral $1$ -complexes. The central insight is that the balancing condition is forced by classical-curve divisor identities under the valuation map, so tropical curves are exactly the integer-vector shadows of algebraic curves; this identifies the integer-weighted balanced graphs with a faithful combinatorial proxy for algebraic curves in $(C^{*})^{n}$ .

Putting these together with the abstract-tropical-curve viewpoint, the moduli space $M_{g}^{trop}$ is a generalised cone complex of dimension $3 g - 3$ that appears again in 04.12.09 as the base of the Gross–Siebert reconstruction and is dual to the Berkovich skeleton of the classical $\overline{M}_{g}$ . The pattern recurs across the tropical-mirror-symmetry program: the moduli of tropical objects shadows the moduli of classical objects, the tropical invariants match the classical invariants, and the bridge is the valuation map combined with the balancing-from-divisors derivation.

The genus is the first Betti number of the underlying topological space, and the tropical Riemann–Roch theorem of Baker–Norine is exactly the foundational reason that the genus controls the rank-degree gap in a way that mirrors the classical Riemann–Roch theorem. This generalises in two directions. To higher dimensions, tropical varieties are balanced rational polyhedral $d$ -complexes, and tropical intersection theory mirrors classical intersection theory; the pattern recurs in 04.12.08 tropical manifolds and dual intersection complexes. To the level of moduli, the tropical Torelli map identifies tropical Jacobians with the Berkovich skeleton of classical Jacobians, building toward 04.12.10 the Strominger–Yau–Zaslow conjecture and the broader connection between tropical and mirror-symmetric geometry. In all directions, the central insight is the same: tropical objects are the combinatorial integer-valued shadows of classical algebraic-geometric objects, and the bridge is faithful enough to transport enumerative invariants.

Full proof set [Master]

Proposition (balancing is forced by tropicalisation), restated. Given in the Intermediate-tier section: every tropical curve $Γ \subset R^{n}$ arising as $Trop (V)$ for an algebraic curve $V \subset (C^{*})^{n}$ over $K = C {{t}}$ satisfies the balancing condition at every vertex, with the proof reducing to the integer-valued zero-degree identity for principal divisors on the toric compactification of $V$ . $□$

Proposition (global balancing identity). For any tropical curve $Γ \subset R^{n}$ , $\sum_{e \in E (Γ) unbounded} w_{e} \cdot u_{e} = 0$ in $Z^{n}$ , where $u_{e}$ is the primitive outward direction of the unbounded edge $e$ .

Proof. The argument is the sum-over-vertices computation of Exercise 3. Sum the local balancing identities $\sum_{e ∋ v} w_{e} \cdot u_{e, v} = 0$ over all vertices $v \in V$ . Each bounded edge $e$ contributes the term $w_{e} \cdot u_{e, v}$ at one endpoint $v$ and the term $w_{e} \cdot u_{e, v^{'}} = - w_{e} \cdot u_{e, v}$ at the other endpoint $v^{'}$ ; the two contributions cancel. Each unbounded edge $e$ is incident to exactly one vertex and contributes $w_{e} \cdot u_{e}$ to the global sum. The local balancing identities therefore aggregate to $\sum_{e unbounded} w_{e} \cdot u_{e} = 0$ . $□$

Proposition (genus invariance). The genus of an abstract tropical curve $(G, w, ℓ)$ depends only on the combinatorial graph $G$ and not on the weights $w$ or edge lengths $ℓ$ .

Proof. The genus formula $g (Γ) = ∣ E (G) ∣ - ∣ V (G) ∣ + c (G)$ involves only the cardinalities of $V (G)$ and $E (G)$ and the number of connected components $c (G)$ , all combinatorial invariants of $G$ . The weight function $w$ and length function $ℓ$ play no role. The first Betti number identification $g (Γ) = b_{1} (∣Γ∣)$ follows because the metric realisation $∣Γ∣$ has the same homotopy type as the underlying combinatorial graph $G$ (each edge is contractible to a point, preserving homotopy type), and $b_{1} (G) = ∣ E (G) ∣ - ∣ V (G) ∣ + c (G)$ by the standard formula for the first Betti number of a finite graph. $□$

Proposition (Newton-subdivision duality), restated. A tropical curve $Γ \subset R^{2}$ is dual to a subdivision of its Newton polytope $Δ \subset R^{2}$ , with vertices of $Γ$ in bijection with maximal cells of the subdivision and bounded edges of $Γ$ in bijection with bounded edges of the subdivision.

Proof. (Sketch.) For each unbounded direction $u$ in the multiset of unbounded ends of $Γ$ , the corresponding boundary edge of $Δ$ has integer length equal to the total weight of unbounded edges in direction $u$ . The bounded edges of $Γ$ , viewed as a polyhedral complex in $R^{2}$ , partition the plane into bounded regions. Each bounded region of $R^{2} ∖ Γ$ corresponds to a vertex of the Newton subdivision; each bounded edge of $Γ$ corresponds to an edge of the subdivision; each vertex of $Γ$ corresponds to a maximal cell of the subdivision. The duality is established by the Legendre-transform construction: given a piecewise-linear concave function on $Δ$ , its corner locus is a tropical curve and its domain of linearity is a maximal cell of the subdivision. Conversely, every tropical curve in $R^{2}$ arises this way from a unique (up to additive constant) piecewise-linear concave function. The weight of an edge of $Γ$ equals the integer length of the dual edge in $Δ$ (this is the integer-length convention of Mikhalkin 2005 §2). $□$

Proposition (tropical Riemann–Roch identity; Baker–Norine 2007). For any divisor $D$ on a connected abstract tropical curve $Γ$ of genus $g$ , $r (D) - r (K_{Γ} - D) = de g (D) - g + 1$ .

Proof. (Outline of the Baker–Norine argument.) The proof has three components. (i) Define the chip-firing equivalence relation $\sim$ on divisors: $D \sim D^{'}$ if $D^{'} = D - L (f)$ for some integer-valued function $f$ on vertices, where $L$ is the graph Laplacian. The Picard group $Pic (Γ) = Div (Γ) / \sim$ is a finitely generated abelian group. (ii) Define $g$ -reduced divisors: a divisor $D$ is $g$ -reduced at a vertex $v_{0}$ if for every non-empty subset $A \subseteq V ∖ {v_{0}}$ , firing the chip-firing move at $A$ would produce negative coefficients in $D$ . Every linear-equivalence class contains a unique $g$ -reduced divisor with respect to a fixed base vertex $v_{0}$ . (iii) Verify Riemann–Roch by induction on the rank: for $de g (D) < 0$ both sides are clear ( $r (D) = - 1$ by convention), for $de g (D) \geq 2 g - 1$ the divisor $D$ has rank $de g (D) - g$ by an analogue of the classical Riemann–Roch, and the intermediate cases are handled by the induction step using the $g$ -reduced representative. The proof identifies the rank function with a combinatorial winning condition in a chip-firing game, and the genus controls the gap between effective and ineffective configurations. $□$

Proposition (tropical Clifford inequality), restated. For an effective divisor $D$ on a tropical curve $Γ$ of genus $g$ with $0 \leq de g (D) \leq 2 g - 2$ , $r (D) \leq de g (D) /2$ .

Proof. (Sketch.) The proof mirrors the classical Clifford theorem via Brill–Noether duality. Set $E = K_{Γ} - D$ ; both $D$ and $E$ are effective for the relevant degree range (when $r (D), r (E) \geq 0$ ), with degrees summing to $2 g - 2$ . Apply Riemann–Roch: $r (D) - r (E) = de g (D) - g + 1$ , so $r (D) + r (E) = (r (D) - r (E)) + 2 r (E) = de g (D) - g + 1 + 2 r (E) \geq de g (D) - g + 1$ . Combined with the inequality $r (D) + r (E) \leq g - 1$ for effective $D, E$ with $D + E$ in the canonical class (the tropical version of the Brill–Noether inequality), we get $2 r (D) \leq r (D) + r (E) + (r (D) - r (E)) = r (D) + r (E) + (de g (D) - g + 1) \leq (g - 1) + (de g (D) - g + 1) = de g (D)$ , hence $r (D) \leq de g (D) /2$ . $□$

Proposition (smooth tropical conic structure), restated. A smooth tropical conic $Γ \subset R^{2}$ is dual to a unimodular triangulation of $Δ_{2}$ ; there are two combinatorial types, the "hexagonal" type (with one interior vertex in the triangulation $Δ_{2}$ ) and the "two-vertex" type (no interior vertex but a substantive bounded edge); both have genus $0$ .

Proof. The Newton polytope $Δ_{2} = {(a, b) \in R_{\geq 0}^{2} : a + b \leq 2}$ is a right triangle with vertices at $(0, 0)$ , $(2, 0)$ , $(0, 2)$ . Its lattice points are six: the three vertices, the midpoints $(1, 0)$ , $(0, 1)$ , $(1, 1)$ . A unimodular triangulation uses all six lattice points as vertices (the dimension count: $6$ lattice points partition into $4$ unimodular triangles by Pick's theorem, since the area is $2$ and each unimodular triangle has area $1/2$ ). There is essentially one such triangulation up to symmetry (the "fine subdivision"), which dualises to a tropical conic with one trivalent interior vertex and six unbounded edges in three direction-pairs. The alternative non-unimodular subdivisions of $Δ_{2}$ (using only some of the lattice points) dualise to "non-smooth" tropical conics with reduced combinatorial type. The genus in all cases is $0$ , since a smooth tropical curve of degree $d$ has genus $(2 d - 1)$ by Exercise 7, and $(2 1) = 0$ for $d = 2$ . $□$

Connections [Master]

Algebraic torus and character/cocharacter lattices 04.11.01. The ambient space $R^{n}$ in which embedded tropical curves live is the real tangent space $N_{R} = N \otimes_{Z} R$ of the algebraic torus $T = (C^{*})^{n}$ , with $N$ the cocharacter lattice from 04.11.01. The integer dot product $⟨ u, v ⟩$ on $Z^{n}$ that appears in the balancing-from-divisors derivation is exactly the natural pairing $M \times N \to Z$ of 04.11.01. The lattice formalism developed there is the foundational reason tropical curves are integer-valued objects: every directional datum of a tropical curve is a primitive vector in the cocharacter lattice $N$ , and every weight is an integer arising from intersection theory in the toric compactification of $T$ .
Tropical semiring and tropical polynomial 04.12.01. The defining equation of a tropical curve in $R^{n}$ is a tropical polynomial — a piecewise-linear concave function obtained by replacing classical $+$ and $\times$ with tropical $min$ and $+$ — and the curve is the corner locus where the function fails to be smooth. The pattern recurs in the dual-subdivision construction of the Newton polytope: the tropical polynomial determines the subdivision of the Newton polytope, which dualises to the embedded tropical curve. The prerequisite tropical-semiring framework supplies the algebraic operations under which the geometry of 04.12.02 is closed.
Kapranov's theorem 04.12.03. The downstream specialisation identifies the tropicalisation $Trop (V (f))$ of a hypersurface defined by a Laurent polynomial $f \in K [x_{1}^{\pm 1}, \dots, x_{n}^{\pm 1}]$ with the corner locus of the tropicalisation $trop (f) : R^{n} \to R$ of the polynomial. For curves ( $f$ in two variables), Kapranov's theorem gives the explicit construction of every embedded tropical curve from a Laurent polynomial, complementing the abstract characterisation of 04.12.02 via balancing.
Mikhalkin correspondence theorem 04.12.05. The chapter-closing synthesis of 04.12.05 proves that the count of tropical curves of fixed degree and genus through generic point configurations in $R^{2}$ equals the count of classical complex algebraic curves of the same degree and genus through corresponding configurations in $(C^{*})^{2}$ . The balancing-condition machinery of 04.12.02 is the foundational rigidity that makes this correspondence well-defined: tropical curves form a discrete combinatorial set that can be counted, and the balancing constraint exactly matches the divisor-class constraint on the classical side.
Riemann–Roch theorem for curves 04.04.01. The tropical Riemann–Roch theorem of Baker–Norine 2007 — $r (D) - r (K_{Γ} - D) = de g (D) - g + 1$ — is the exact combinatorial analogue of the classical Riemann–Roch theorem on a smooth algebraic curve. The classical and tropical statements correspond term-by-term under the dictionary $h^{0} (D) \leftrightarrow r (D) + 1$ , $de g \leftrightarrow de g$ , $g \leftrightarrow g$ , $K_{C} \leftrightarrow K_{Γ}$ . The bridge is the specialisation map from divisors on a degenerating family of algebraic curves to divisors on the tropical curve of the special fibre.
Hurwitz formula 04.04.02. The classical Hurwitz formula extends to tropical morphisms of tropical curves: for a finite cover $π : Γ \to Γ^{'}$ of degree $d$ between connected tropical curves of genera $g, g^{'}$ , $2 g - 2 = d (2 g^{'} - 2) + de g R_{π}$ , where $R_{π}$ is the tropical ramification divisor recording the local "branching" of the cover at vertices where the local degree exceeds the lifted slope index. The tropical Hurwitz formula was developed in work of Caporaso 2014 and Cavalieri–Markwig–Ranganathan; its proof mirrors the classical Hurwitz derivation but takes place entirely on the combinatorial graph.
Newton polytope and non-archimedean amoeba 04.12.04. The Newton polytope $Δ$ of a Laurent polynomial $f$ defining an algebraic curve carries a piecewise-linear concave function (the Legendre transform of the tropicalisation $trop (f)$ ), and the dual subdivision of $Δ$ is in bijection with the tropical curve $Trop (V (f))$ via the Newton-subdivision duality of this unit's Master theorem set. The non-archimedean amoeba — the image of $V$ under the absolute value with respect to a non-archimedean valuation — has the tropical curve as its boundary at infinity. The downstream unit 04.12.04 elaborates the Newton-polytope formalism that this unit's Master-tier duality theorem opens.
Algebraic curves 04.04.03. Elliptic curves and higher-genus algebraic curves have tropical counterparts: a tropical elliptic curve is a connected weighted metric graph of genus $1$ (a circle with finitely many edges, or any homotopy-equivalent graph), and a tropical curve of genus $g \geq 2$ has $3 g - 3$ moduli (matching the classical dimension of $M_{g}$ ). The Jacobian of a classical elliptic curve has a tropical analogue $Jac^{trop} (Γ) ≅ R / Z$ (a tropical $1$ -dimensional torus) computed combinatorially from the loop. The classical-to-tropical specialisation of curves is one of the foundational bridges in the program of Mikhalkin–Zharkov 2008 and Caporaso 2012.
Berkovich analytification [pending]. The classical-to-tropical comparison is mediated by the Berkovich analytification $V^{an}$ of an algebraic curve $V$ over a non-archimedean field: the underlying topological space $∣ V^{an} ∣$ deformation-retracts onto a finite metric graph $Γ$ , which is the tropical curve associated to a chosen semistable model of $V$ . This identifies tropical curves with skeletons of Berkovich analytifications, and the moduli space $M_{g}^{trop}$ with the skeleton of the Berkovich analytification of $\overline{M}_{g}$ . The skeleton viewpoint is the modern lens through which the entire tropical-curve formalism is now organised.
Nishinou-Siebert correspondence 04.12.06. The further downstream unit generalises the curve-level correspondence to higher-dimensional toric targets: tropical curves on the dual intersection complex $B (Ξ)$ of a toric degeneration lift to log stable maps via the Nishinou-Siebert toric-degeneration mechanism. The balancing-and-multiplicity infrastructure of the present unit specialises in their setup to balanced rational metric graphs on the dual intersection complex, with the Nishinou-Siebert vertex-multiplicity formula reducing at trivalent vertices to a determinant of primitive edge directions — the same lattice-multiplicity datum carried by the tropical curves of this unit. The bridge between the surface case and the higher-dimensional case is the replacement of the ambient $R^{2}$ by the polyhedral cone complex $B (Ξ)$ while retaining the balancing condition on the embedded metric graph.
Toric degeneration of a Calabi-Yau variety 04.12.07. The downstream unit places tropical curves on the dual intersection complex $B (P)$ of a toric degeneration of a Calabi-Yau variety: the balanced metric graphs of the present unit are the wall-crossing data on the dual intersection complex feeding the Gross-Siebert reconstruction. The balancing condition holds at every vertex when read against the integral affine structure on $B (P)$ , with edge weights given by integer multiplicities. The present unit supplies the abstract combinatorial framework; [04.12.07] supplies the Calabi-Yau-degeneration ambient on which the framework is enumerated.
Dual intersection complex; tropical manifold $B$ 04.12.08. The downstream unit identifies the dual intersection complex $(B, P)$ of a Calabi-Yau toric degeneration as the higher-dimensional ambient that replaces $R^{n}$ : tropical curves on $(B, P)$ are balanced rational metric graphs in the precise sense of the present unit, with the balancing condition read against the integral affine structure of $B$ . The dual-intersection-complex unit is the polyhedral-manifold home in which the present unit's balanced-metric-graph formalism is enumerated for mirror-symmetry purposes.
Gross-Siebert reconstruction theorem 04.12.09. The downstream reconstruction theorem assembles the smooth mirror Calabi-Yau from balanced rational metric graphs on the dual intersection complex via the scattering / wall-crossing algorithm: the genus- $0$ tropical-curve counts on $(B, P)$ supply the order-by-order coefficients of the slab and wall functions, and these counts are exactly enumerations of the balanced metric graphs of the present unit. The cone-complex structure of the moduli of tropical curves recurs in [04.12.09] as the polyhedral base of the reconstruction; the balanced-metric-graph framework here is the combinatorial primitive on which the reconstruction is built.
Strominger-Yau-Zaslow conjecture 04.12.10. The downstream SYZ-conjecture unit identifies the balanced rational metric graphs of the present unit as the combinatorial object enumerated on the SYZ base in the tropical-curve interpretation of mirror symmetry: balanced metric graphs on the integral affine SYZ base $B$ are the tropical curves that compute Gromov-Witten invariants of the Calabi-Yau via the SYZ identification, with edge weights given by integer multiplicities. The present unit's balancing-and-multiplicity framework is the discrete combinatorial substrate of the SYZ enumeration.
Slab function and structure of a tropical manifold 04.12.11. The downstream slab-function unit takes the walls of the structure $S$ as sub-pieces of tropical hypersurfaces in $B$ whose top cells carry the lattice multiplicities of the present unit: each wall function is built from a tropical-curve fragment with the balancing condition holding at every interior vertex. The broken lines through the structure are tropical-curve fragments with prescribed boundary conditions, and the present unit's balanced-metric-graph framework is the foundational polyhedral object of which slabs and walls are polynomial-coefficient enrichments.
Theta function of a polarised tropical manifold 04.12.12. The downstream theta-function unit constructs canonical basis sections via broken-line enumeration on the polarised tropical manifold; broken lines are the unmarked-endpoint analogue of the balanced metric graphs of the present unit, with bending compatibility at wall crossings replacing the balancing condition at internal vertices. The two structures are dual via the broken-line / tropical-disk correspondence (Mandel 2019): broken lines correspond to tropical disks with one boundary on the smooth locus of $B$ , and the balanced-metric-graph framework here is the closed-tropical-curve counterpart.

Historical & philosophical context [Master]

The tropical-curve formalism arose at the confluence of two independent developments in the late twentieth century. The first is the non-archimedean / valuative side, opened by Robert Bieri and Joseph Groves in The geometry of the set of characters induced by valuations (J. reine angew. Math., 1984) ^{[Bieri1984Groves]}, where the tropicalisation $Trop (V)$ of a subvariety of an algebraic torus is shown to be a rational polyhedral complex. The second is the combinatorial / polyhedral side, developed by Mikhail Kapranov, Maxim Kontsevich, and others in the 1990s as a tool for enumerative geometry of toric varieties. The two strands were unified and given their canonical form by Grigory Mikhalkin in Enumerative tropical algebraic geometry in $R^{2}$ (Journal of the American Mathematical Society, 2005) ^{[Mikhalkin2005]}, where the definition of tropical curves as balanced rational polyhedral $1$ -complexes with positive integer weights was crystallised and the foundational correspondence theorem with classical enumerative invariants was proved.

The abstract-tropical-curve viewpoint, in which the embedding into $R^{n}$ is forgotten and the underlying weighted metric graph is retained, was developed by Silvia Brannetti, Margarida Melo, and Filippo Viviani in On the tropical Torelli map (Advances in Mathematics, 2011) ^{[Brannetti2011]}. Their construction of the moduli space $M_{g}^{trop}$ as a generalised cone complex of dimension $3 g - 3$ for $g \geq 2$ established the modern foundation of tropical moduli theory; Lucia Caporaso's Algebraic and tropical curves: comparing their moduli spaces (Handbook of Moduli, 2012) ^{[Caporaso2012]} surveys the resulting comparison with classical moduli of algebraic curves, and Dan Abramovich, Caporaso, and Sam Payne 2015 The tropicalization of the moduli space of curves identified $M_{g}^{trop}$ with the skeleton of the Berkovich analytification of $\overline{M}_{g}$ .

The tropical Riemann–Roch theorem was proved by Matthew Baker and Serguei Norine in Riemann–Roch and Abel–Jacobi theory on a finite graph (Advances in Mathematics, 2007) ^{[Baker2007Norine]}, initially for the discrete (combinatorial) graph setting; the extension to metric tropical curves was carried out by Gathmann–Kerber 2008 and Mikhalkin–Zharkov 2008. The chip-firing equivalence relation underlying the proof has earlier roots in the work of Björner, Lovász, Shor, and Tardos on the abelian sandpile model in statistical mechanics (1990s) and was rediscovered in the tropical setting by Baker and Norine. The Brill–Noether theory of tropical curves, including the tropical Clifford inequality, was developed by Coppens, Caporaso, and others in the 2010s as part of the broader effort to transport classical moduli-of-divisors invariants to the tropical world. The story continues: tropical Hurwitz theory, tropical Jacobians, tropical theta functions (Mikhalkin–Zharkov 2008 ^{[MikhalkinZharkov2008]}), and the connection to the Gross–Siebert mirror-symmetry program 04.12.09 are each open chapters with active current research.

Bibliography [Master]

@article{Mikhalkin2005,
  author  = {Mikhalkin, Grigory},
  title   = {Enumerative tropical algebraic geometry in $\mathbb{R}^2$},
  journal = {Journal of the American Mathematical Society},
  volume  = {18},
  year    = {2005},
  pages   = {313--377}
}

@book{MaclaganSturmfels,
  author    = {Maclagan, Diane and Sturmfels, Bernd},
  title     = {Introduction to Tropical Geometry},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {161},
  year      = {2015}
}

@article{Brannetti2011,
  author  = {Brannetti, Silvia and Melo, Margarida and Viviani, Filippo},
  title   = {On the tropical Torelli map},
  journal = {Advances in Mathematics},
  volume  = {226},
  year    = {2011},
  pages   = {2546--2586}
}

@article{Baker2007Norine,
  author  = {Baker, Matthew and Norine, Serguei},
  title   = {Riemann--Roch and Abel--Jacobi theory on a finite graph},
  journal = {Advances in Mathematics},
  volume  = {215},
  year    = {2007},
  pages   = {766--788}
}

@article{Bieri1984Groves,
  author  = {Bieri, Robert and Groves, J. R. J.},
  title   = {The geometry of the set of characters induced by valuations},
  journal = {Journal f\"ur die reine und angewandte Mathematik},
  volume  = {347},
  year    = {1984},
  pages   = {168--195}
}

@article{Gathmann2006,
  author  = {Gathmann, Andreas},
  title   = {Tropical algebraic geometry},
  journal = {Jahresbericht der Deutschen Mathematiker-Vereinigung},
  volume  = {108},
  year    = {2006},
  pages   = {3--32}
}

@incollection{MikhalkinZharkov2008,
  author    = {Mikhalkin, Grigory and Zharkov, Ilia},
  title     = {Tropical curves, their Jacobians and theta functions},
  booktitle = {Curves and Abelian Varieties (Athens, GA, 2007)},
  publisher = {American Mathematical Society},
  series    = {Contemporary Mathematics},
  volume    = {465},
  year      = {2008},
  pages     = {203--230}
}

@incollection{Caporaso2012,
  author    = {Caporaso, Lucia},
  title     = {Algebraic and tropical curves: comparing their moduli spaces},
  booktitle = {Handbook of Moduli, Vol. I},
  publisher = {International Press},
  year      = {2012},
  pages     = {119--160}
}

@article{AbramovichCaporasoPayne,
  author  = {Abramovich, Dan and Caporaso, Lucia and Payne, Sam},
  title   = {The tropicalization of the moduli space of curves},
  journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
  volume  = {48},
  year    = {2015},
  pages   = {765--809}
}

@article{Coppens2016,
  author  = {Coppens, Marc},
  title   = {Clifford's theorem for tropical curves},
  journal = {Advances in Mathematics},
  volume  = {288},
  year    = {2016},
  pages   = {937--965}
}

Prerequisites

04.11.01

Tier anchors

beginner: Maclagan–Sturmfels *Introduction to Tropical Geometry* §3.1 informal — tropical lines and tropical conics in the plane as balanced graphs
intermediate: Mikhalkin 2005 *Enumerative tropical algebraic geometry in R^2*; Maclagan–Sturmfels §3.3–§3.4 (tropical curves, balancing, and the structure theorem)
master: Mikhalkin 2005 *J. Amer. Math. Soc.* 18, 313–377 (originator: balanced rational polyhedral 1-complexes); Maclagan–Sturmfels 2015 *Introduction to Tropical Geometry* Ch. 3 (canonical exposition); Brannetti–Melo–Viviani 2011 *Adv. Math.* 226 (abstract tropical curves as weighted metric graphs and the tropical Torelli map); Baker–Norine 2007 *Adv. Math.* 215 (tropical Riemann–Roch); Caporaso 2012 *Algebraic and tropical curves: comparing their moduli spaces* (survey)

References

TODO_REF
Mikhalkin 2005 — Enumerative tropical algebraic geometry in R^2 · J. Amer. Math. Soc. 18, 313–377; §2 (tropical curves as balanced graphs), §3 (correspondence theorem)
TODO_REF
Maclagan–Sturmfels — Introduction to Tropical Geometry · Graduate Studies in Mathematics 161, AMS 2015; Ch. 3 (tropical curves), §3.1–§3.4
TODO_REF
Brannetti–Melo–Viviani 2011 — On the tropical Torelli map · Adv. Math. 226, 2546–2586; §2 (abstract tropical curves), §3 (tropical Torelli map)
TODO_REF
Baker–Norine 2007 — Riemann–Roch and Abel–Jacobi theory on a finite graph · Adv. Math. 215, 766–788; Theorem 1.12 (tropical Riemann–Roch)
TODO_REF
Gathmann 2006 — Tropical algebraic geometry · Jahresber. Deutsch. Math.-Verein. 108, 3–32; survey of tropical curves and enumerative geometry
TODO_REF
Mikhalkin–Zharkov 2008 — Tropical curves, their Jacobians and theta functions · Curves and Abelian Varieties (Athens, GA 2007), Contemp. Math. 465, 203–230
TODO_REF
Caporaso 2012 — Algebraic and tropical curves: comparing their moduli spaces · Handbook of Moduli, Vol I, 119–160

Lean module

Codex.AlgGeom.Tropical.TropicalCurve

Mathlib gap

Mathlib has finite simple graphs (`SimpleGraph`), multisets, and
basic integer-lattice infrastructure, but the rational polyhedral
1-complex framework needed for embedded tropical curves is absent.
Specifically missing: (i) primitive integer direction vectors as a
named type with the gcd-equals-one predicate; (ii) weighted edge
data assembling a positive integer weight with a primitive
direction; (iii) the balancing condition at a vertex as a named
predicate on a multiset of weighted directions; (iv) the metric
graph realisation of an abstract weighted metric graph with edge
lengths in $\mathbb{R}_{> 0} \cup \{\infty\}$; (v) the genus as the
first Betti number $b_1(\Gamma) = |E| - |V| + 1$ of the underlying
topological space, with a proof that the genus is a topological
invariant; (vi) the tropical Picard group $\mathrm{Pic}(\Gamma) =
\mathrm{Div}(\Gamma) / \sim_{\text{chip-firing}}$ and the rank
function on divisors needed to state Baker–Norine Riemann–Roch.
A Mathlib-grade development of the tropical-curve formalism would
open downstream work on the tropical Jacobian, the moduli of
tropical curves $M_g^{\mathrm{trop}}$, and the correspondence with
classical algebraic curves via Berkovich analytification.

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 45m
master: 80m

Intuition [Beginner]

Visual [Beginner]

Worked example [Beginner]

Check your understanding [Beginner]

Formal definition [Intermediate+]

Counterexamples to common slips

Key theorem with proof [Intermediate+]

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Advanced results [Master]

The balancing condition and rational primitive directions

Abstract tropical curves as weighted metric graphs

Genus and the tropical Riemann–Roch (Baker–Norine)

Worked example — tropical conic in R2

Full proof set [Master]

Connections [Master]

Historical & philosophical context [Master]

Bibliography [Master]

Worked example — tropical conic in $R^{2}$