2.1 Graphs, divisors and flows

We begin with a short review of some essential notions from the theory of graphs. The notions are well-known, but we include them to avoid excessive referencing and to fix notation. We will follow the definitions of graphs and their morphisms from [Reference Melo, Molcho, Ulirsch and VivianiMMUV22], although we will substitute oriented edges in the place of half-edges in our presentation. Let $\Gamma $ be a graph. We write $V(\Gamma )$ for the set of vertices of $\Gamma $ , $\mathcal {E}(\Gamma )$ for the set of oriented edges, $E(\Gamma )$ for the set of edges and $L(\Gamma )$ for the set of legs. A genus function on $\Gamma $ is a function

$$\begin{align*}h_\Gamma: V(\Gamma) \to \mathbb{N}. \end{align*}$$

The genus of $\Gamma $ is the natural number

$$\begin{align*}g = g(\Gamma) := \sum_{v \in V(\Gamma)} h_\Gamma(v) + \dim H_1(\Gamma). \end{align*}$$

When speaking about the genus of a graph, it is to be understood that a genus function has been specified. However, unless explicitly mentioned, our results do not require fixing the genus and apply equally well to graphs with or without a genus function. A marking on $\Gamma $ is a bijection between $\{1,\cdots ,n\}$ and $L(\Gamma )$ . A graph with a marking is called an n-marked graph.

Of course, $\mathsf {Div}(\Gamma )$ is nothing but the free abelian group $\mathbb {Z}^{V(\Gamma )}$ on $V(\Gamma )$ , but we insist on the notation for clarity.

Example 2.2.1. Let $\Gamma $ be an n-marked genus g graph. Let $(a_1,\cdots ,a_n) \in \mathbb {Z}^n$ , $k \in \mathbb {Z}$ . Let

$$\begin{align*}A = \sum_{v \in V(\Gamma)} k(2h_\Gamma(v)-2 + \text{val}(v))v + \sum_{i=1}^n a_iv_i, \end{align*}$$

where $\text {val}(v)$ is the valence of v – the number of oriented edges $\vec {e}$ with $\mathfrak {r}(\vec {e})=v$ plus the number of legs on v – and $v_i$ is the vertex that contains the i-th leg. Then A is a divisor on $\Gamma $ . When the $a_i$ are all $0$ , A is called the log canonical divisor. When the $a_i$ are all $-1$ , it is called the canonical divisor.

There is an evident two-to-one cover $\mathcal {E}(\Gamma ) \to E(\Gamma )$ from the set of oriented edges to the set of edges, and every choice of orientation $\vec {\Gamma }$ on $\Gamma $ gives a section $E(\Gamma ) \to \mathcal {E}(\Gamma )$ , whose image we denote by $E(\vec {\Gamma })$ .

Definition 2.4. A flow on $\Gamma $ is a function

$$ \begin{align*}s:\mathcal{E}(\Gamma) \to \mathbb{Z} \end{align*} $$

satisfying . We write $\mathsf {Flow}(\Gamma )$ for the group of flows on $\Gamma $ .

Of course, as above, $\mathsf {Flow}(\Gamma )$ can be identified with $\mathbb {Z}^{E(\vec {\Gamma })}$ after choosing an orientation on $\Gamma $ .

Any flow on s on $\Gamma $ determines a divisor $\mathsf {div}(s)$ on $\Gamma $ by setting

$$ \begin{align*}\mathsf{ord}_v(s) = \sum_{\{\vec{e} : \mathfrak{t}(\vec{e}) = v\}} s(\vec{e}) \end{align*} $$

and

$$ \begin{align*}\mathsf{div}(s) = \sum_{v \in V(\Gamma)} (\mathsf{ord}_v(s))v. \end{align*} $$

Here, the notation $\{\vec {e}: \mathfrak {t}(\vec {e})=v\}$ means the set of oriented edges whose terminal point is v. This procedure gives a homomorphism

$$ \begin{align*}\mathsf{div}:\mathsf{Flow}(\Gamma) \to \mathsf{Div}(\Gamma). \end{align*} $$

Definition 2.5. Let $\Gamma $ be a graph. A basic subdivision of $\Gamma $ is a graph $\Gamma '$ obtained from $\Gamma $ by replacing an oriented edge $\vec {e}$ Footnote ⁵ with two oriented edges $\vec {e_1},\vec {e_2}$ and a new vertex u, and setting

$$ \begin{align*} \mathfrak{r}(\vec{e_1}) = \mathfrak{r}(\vec{e})\\ \mathfrak{t}(\vec{e_1}) = \mathfrak{r}(\vec{e_2}) = u \\ \mathfrak{t}(\vec{e_2}) = \mathfrak{t}(\vec{e}). \end{align*} $$

There is an evident map $\Gamma ' \to \Gamma $ which sends $e_1,e_2,u$ to e and all other vertices and edges to themselves.

A subdivision of $\Gamma $ is a composition of basic subdivisions.

Let $\Gamma ' \to \Gamma $ be a subdivision. The additional vertices on $\Gamma '$ are called exceptional. A refinement of $\Gamma '$ is a further subdivision $\Gamma " \to \Gamma '$ .

Suppose $\Gamma ' \to \Gamma $ is a subdivision, and s is a flow on $\Gamma '$ . It is often desirable to find the minimal subdivision of $\Gamma $ on which s can be defined.

Definition 2.6. We say that $\Gamma '$ is minimal with respect to s if

$$ \begin{align*}\mathsf{ord}_v(s) \neq 0 \end{align*} $$

on all exceptional vertices v of $\Gamma '$ .

Proof. Define $\Gamma _s$ as the subdivision of $\Gamma $ obtained by keeping only the exceptional vertices of $\Gamma '$ on which

$$ \begin{align*}\mathsf{ord}_v(s) \neq 0. \end{align*} $$

Since the slope of s changes on the vertices v, any subdivision that supports s must refine $\Gamma _s$ , whereby the uniqueness of $\Gamma _s$ follows.

2.8 Equidimensionality

Let $\Gamma $ be a graph, and s a flow on $\Gamma $ . Then s defines a partial orientation on $\Gamma $ , by orienting the edges so that $s(\vec {e})>0$ . The orientation is partial, as it is not defined on edges with $s(e) = 0$ . We call such edges contracted. The flow s defines an honest orientation on the graph $\overline {\Gamma }$ obtained from $\Gamma $ by contracting the contracted edges.

An acyclic flow s defines a partial order on the vertices of $\Gamma $ : the order is generated by the relation

$$ \begin{align*}v < w \end{align*} $$

if $v,w$ are the endpoints of an oriented edge $\vec {e}$ from v to w in the orientation determined by s. Endpoints of contracted edges are not comparable to one another.

The flow s lifts to any subdivision $\Gamma ' \to \Gamma $ , by declaring $s(\vec {e}_i)=s(\vec {e})$ whenever $\vec {e} \in \mathcal {E}(\Gamma )$ is subdivided into $\vec {e}_1,\cdots ,\vec {e_n}$ in $\Gamma '$ , and an ordering extending s on $\Gamma $ lifts to an ordering extending s on $\Gamma '$ uniquely. We can rephrase the notion of ordering in the following convoluted way, which nevertheless will be meaningful in the next section:

We can then consider maps $\Gamma \to X$ . Morphisms for us take cells into cells (i.e. vertices or edges into vertices or edges). A morphism $\phi : \Gamma \to X$ also defines a partial order on $V(\Gamma )$ , by declaring $v<w$ if $\phi (v) = v_i$ , $\phi (w) = v_{j}$ , with $i < j$ . Vertices that map to the same vertex of X are incomparable with one another, and we do not define the order on vertices that map into edges of X. The following class of morphisms is then special:

Thus, an equidimensional morphism $\Gamma \to X$ defines a total order on the vertices of noncontracted edges (i.e., those edges which map to an edge instead of a vertex) of $\Gamma $ .

We note that stability is not an absolute notion, but it depends on the original graph $\Gamma $ on which s is defined.

Suppose $\Gamma ' \to X$ is an equidimensional lift of s. Then $\Gamma ' \to X$ defines an ordering $\kappa $ extending s, as it orders the vertices of $\Gamma '$ that lie on the noncontracted edges of $\Gamma '$ (i.e., those on which $s \neq 0$ , which are precisely the subdivisions of the noncontracted edges of $\Gamma $ ), and since $V(\Gamma ) \subset V(\Gamma ')$ , also the vertices of $\Gamma $ which lie on the noncontracted edges of $\Gamma $ . Conversely, given an ordering $\kappa $ extending s, we can define a combinatorial line $X_\kappa $ by taking one vertex $v_i$ for each vertex of a noncontracted edge of $\Gamma $ according to the order determined by $\kappa $ . This defines an evident function

$$ \begin{align*}\phi: \Gamma \to X_\kappa, \end{align*} $$

extending s. This is, however, not a morphism: edges of $\Gamma $ can map into unions of cells of $X_\kappa $ . There is a minimal subdivision $\Gamma _\kappa $ which turns $\Gamma \to X_\kappa $ into a morphism by adjoining the preimages of the vertices $\phi ^{-1}(v_i)$ to the noncontracted edges of $\Gamma _\kappa $ . The following lemma then follows:

Lemma 2.14. Orderings extending s are equivalent to stable equidimensional lifts of s, under the correspondence

$$ \begin{align*}\kappa \leftrightarrow \phi: \Gamma_\kappa \to X_\kappa. \end{align*} $$

2.15 Numerical stability conditions

Let $\Gamma $ be a graph.

In other words, the subdivision $\Gamma ' \to \Gamma $ introduces at most one exceptional vertex on each edge of $\Gamma $ .

Finally, we recall that a stability condition $\theta $ on $\Gamma $ is simply a function

$$ \begin{align*}\theta: V(\Gamma) \to \mathbb{R}. \end{align*} $$

It is nondegenerate or generic if, for every $S \subset V(\Gamma )$ , we have

$$ \begin{align*}\theta(S) \pm \frac{E(S,S^c)}{2} \notin \mathbb{Z}, \end{align*} $$

where $\theta (S) = \sum _{v \in S} \theta (v)$ and $E(S,S^c)$ is the number of edges between S and its complement.

A stability condition determines a list of semistable divisors on $\Gamma $ : those D for which

$$ \begin{align*}\theta(S) - \frac{E(S,S^c)}{2} \le D(S) \le \theta(S) + \frac{E(S,S^c)}{2} \end{align*} $$

for all $S \subset V(\Gamma )$ . The divisor is stable if the inequalities are strict. Thus, a stability condition is nondegenerate if and only if all semistable divisors are stable.

A stability condition on $\Gamma $ lifts canonically to a quasi-stable model $\Gamma ' \to \Gamma $ by declaring its value on exceptional vertices to be $0$ .

Definition 2.18. Let $\theta $ be a stability condition on $\Gamma $ , and $\Gamma ' \to \Gamma $ a quasi-stable model. We call an admissible divisor D on $\Gamma '\ \theta $ -semistable if for every subgraph $S \subset \Gamma '$ , we have

$$ \begin{align*}\theta(S) - \frac{E(S,S^c)}{2} \le D(S) \le \theta(S) + \frac{E(S,S^c)}{2}. \end{align*} $$

We note that if $\theta $ is generic, the inequalities above are strict for every divisor supported on $V(\Gamma ) \subset V(\Gamma ')$ . However, equality can hold for divisors that have support on exceptional vertices.Footnote ⁶

We thus arrive at the key combinatorial notions of this paper. Let A denote a fixed divisor on $\Gamma $ , and $\theta $ a stability condition.

Definition 2.19. A $\theta $ -flow balancing A (or $\theta $ -flow for short) consists of a quasi-stable model $\Gamma ' \to \Gamma $ , a $\theta $ -semistable divisor D and an acyclic flow s with

$$ \begin{align*}\mathsf{div}(s) = A - D. \end{align*} $$

2.21 Specialization

The data discussed above specializes with respect to edge contractions. Namely, if $\overline {\Gamma }$ is obtained from $\Gamma $ by contracting some edges, and $\phi : \Gamma \to \overline {\Gamma }$ denotes the contraction map,

1. (1) Divisors on $\Gamma $ specialize to divisors on $\overline {\Gamma }$ by $D \to \overline {D}$

   $$ \begin{align*}\overline{D}(v) = \sum_{w \in \phi^{-1}(v) \cap V(\Gamma)} D(w). \end{align*} $$

2. (2) Stability conditions specialize exactly analogously as $\overline {\theta }(v) = \sum \theta (w)$ .

3. (3) Flows specialize by $s \to \overline {s}$ , with

   $$ \begin{align*}\overline{s}(\vec{e}) = s(\vec{e}) \end{align*} $$

   under the natural inclusion $\mathcal {E}(\overline {\Gamma }) \subset \mathcal {E}(\Gamma )$ .

4. (4) The genus function $h_{\overline {\Gamma }}$ of $\overline {\Gamma }$ is defined by

   $$\begin{align*}h_{\overline{\Gamma}}(v) = g_{\Gamma}(\phi^{-1}(v)) = \sum_{w \in V(\Gamma) \cap \phi^{-1}(v)} h_\Gamma(w) + \dim H_1(\phi^{-1}(v)). \end{align*}$$

All notions discussed, starting with subdivisions and culminating with $\theta $ -stable equidimensional flows, specialize under these definitions.

Example 2.21.1. Let $\Gamma $ be an n-marked graph with a genus function $h_\Gamma $ . Let $(a_1,\cdots ,a_n) \in \mathbb {Z}^n$ , $k \in \mathbb {Z}$ and

$$\begin{align*}A = \sum_{v \in V(\Gamma)} k(2h_\Gamma(v)-2 + \text{val}(v))v + \sum_{i=1}^n a_iv_i \end{align*}$$

the divisor of example 2.2.1. Then A specializes to the analogous divisor

$$\begin{align*}\sum_{v \in V(\overline{\Gamma})} k(2h_{\overline{\Gamma}}(v)-2 +\text{val}(v))v + \sum_{i=1}^na_i\overline{v}_i \end{align*}$$

under any contraction $\Gamma \to \overline {\Gamma }$ , where $\overline {v}_i$ is the vertex of $\overline {\Gamma }$ that contains the i-th leg.

3.4 Subdivisions of tropical curves

Let $\Gamma $ be a tropical curve. A subdivision of $\Gamma $ is a tropical curve $\Gamma '$ metrized by M, such that the underlying graph of $\Gamma '$ is a subdivision of the underlying graph of $\Gamma $ and such that the lengths of the edges $e' \in E(\Gamma ')$ that subdivide an edge $e \in E(\Gamma )$ add up to the length of e: if $\phi :E(\Gamma ') \to E(\Gamma )$ is the induced map of edges, we must have

$$ \begin{align*}\sum_{e' \in \phi^{-1}(e)} \ell_{e'} = \ell_e. \end{align*} $$

Subdivisions $\Gamma '$ come with an evident map $\Gamma ' \to \Gamma $ . The collection of vertices in $\Gamma '$ that are not in $\Gamma $ are called exceptional vertices. A refinenent of $\Gamma ' \to \Gamma $ is a further subdivision $\Gamma " \to \Gamma '$ .

Suppose $\Gamma ' \to \Gamma $ is a subdivision, and $\alpha $ is a piecewise linear function on $\Gamma '$ . It is often desirable to find the minimal subdivision of $\Gamma $ on which $\alpha $ can be defined.

Definition 3.5. We say that $\Gamma '$ is minimal with respect to $\alpha $ if

$$\begin{align*}\mathsf{div}(\alpha)(v) \neq 0 \end{align*}$$

on all exceptional vertices v of $\Gamma '$ .

Proof. Define $\Gamma _\alpha $ as the subdivision of $\Gamma $ obtained by keeping only the exceptional vertices of $\Gamma '$ on which

$$ \begin{align*}\mathsf{div}(\alpha)(v) \neq 0. \end{align*} $$

Since the slope of $\alpha $ changes on the vertices v, any subdivision that supports $\alpha $ must refine $\Gamma _\alpha $ , whereby the uniqueness of $\Gamma _\alpha $ follows.

3.7 Equidimensional piecewise linear functions and twists

Any integral monoid $M \subset M^{\text {gp}}$ can be regarded as a partial order on $M^{\text {gp}}$ : for $x,y \in M^{\text {gp}}$ , we declare $x \le y$ if $y - x \in M$ . Thus, a piecewise linear function $\alpha $ on $\Gamma $ comes with a partial ordering of its values $\alpha (v) \in M^{\text {gp}}$ . This partial order is evidently compatible with the orientation on $\Gamma $ induced by the underlying twist of $\alpha $ .

Remark 3.9. The condition that the values $\alpha (v)$ of the function $\alpha $ at the vertices v are totally ordered can be equivalently phrased in terms of the underlying flow s of $\alpha $ . Since for any $v,w$ we have that

$$\begin{align*}\alpha(w) = \alpha(v) + \sum_{\vec{e} \in P_{v \to w}} s(\vec{e})\ell_e \end{align*}$$

for any oriented path $P_{v \to w}$ , to say that $\alpha (w)$ comes after $\alpha (v)$ in a total order for $\alpha $ is to say that

$$\begin{align*}\sum_{\vec{e} \in P_{v \to w}} s(\vec{e})\ell_e \in M \subset M^{\text{gp}} \end{align*}$$

for any oriented path $P_{v \to w}$ , or, equivalently, that the evaluation

$$\begin{align*}\sum_{\vec{e} \in P_{v \to w}} s(\vec{e})\ell_e(x) \ge 0 \end{align*}$$

for any $x \in \sigma = M^\vee $ .

Borrowing ideas from the theory of semistable reduction, we make the following definition.

While on its face the definition of equidimensionality seems dependent on the values of $\alpha $ , the definition is, in fact, invariant under translation of the values $\alpha (v)$ by any common element $x \in M^{\text {gp}}$ . Thus, the definition descends to twists, and it makes sense to talk about equidimensional twists.

The definition of equidimensional piecewise linear function can perhaps be clarified by introducing its image, which is a one dimensional tropical target, also referred to as a tropical line.

We spell out the meaning of the definition: a tropical line is, thus, as a polyhedral complex, simply a combinatorial line X with a length assignment $\ell _{f_i} \in M-0$ for each of its edges. But to say that this polyhedral complex is a polyhedral complex structure on $\mathbb {R}$ over M means that it furthermore comes with a chosen piecewise linear embedding

$$ \begin{align*}\iota_X: X \subset \mathbb{R}. \end{align*} $$

This data is very similar to the definition of a piecewise linear function on a tropical curve: an element

$$ \begin{align*}\iota_X(v_i) := \gamma_i \in M^{\text{gp}} \end{align*} $$

for each vertex $v \in V(X)$ , such that

$$ \begin{align*}\gamma_{i+1}- \gamma_{i} = \ell_{f_i}. \end{align*} $$

In particular, the ordering on the vertices has been chosen so that $\gamma _{i+1}>\gamma _i$ by assumption, which means that the piecewise linear structure is compatible with the canonical orientation $E(\vec {X})$ of definition 2.11 on X. We will also consider the trivial polyhedral decomposition as an allowable tropical line, where $\mathbb {R}$ is considered as a single cell. In that case, we will simply write $\mathbb {R}$ for the tropical line.

Let $\Gamma $ be a tropical curve and X a tropical line metrized by M. By definition, a map of polyhedral complexes $\Gamma \to X$ is a piecewise linear map that respects the cell structure of the polyhedral decomposition: each cell (that is, vertex or edge) of $\Gamma $ maps into a cell of X (rather than a union of more than one cells). In particular, a piecewise linear function $\alpha $ on $\Gamma $ can be tautologically thought of as a map

$$ \begin{align*}\Gamma \to \mathbb{R} \end{align*} $$

to the trivial tropical line. The piecewise linear function $\alpha $ may or may not factor through X:

Since $\iota _X$ is a monomorphism, the arrow $\beta $ , if it exists, is unique.

As tropical curves and tropical lines are particularly simple examples of polyhedral complexes, the meaning of equidimensionality of a map

$$ \begin{align*}\beta: \Gamma \to X \end{align*} $$

is very simple to describe: it says that each vertex $v \in \Gamma $ must map to a vertex in X.

Proof. Suppose e is an edge of $\Gamma $ with endpoints $v,w$ , and $\alpha (w)> \alpha (v)$ . Suppose u is a vertex of $\Gamma $ with $\alpha (v) \le \alpha (u) \le \alpha (w)$ . If $\alpha $ factors through X, then $\beta (v)$ and $\beta (w)$ must be consecutive vertices $v_{i},v_{i+1}$ of X (otherwise, the edge e would map to a union of cells). But to say that $\beta $ is equidimensional is to say that $\beta (u)$ must be a vertex of X, and hence one of $v_{i},v_{i+1}$ . So either $\alpha (u)=\alpha (v)$ or $\alpha (u)=\alpha (w)$ . Conversely, given an equidimensional function $\alpha $ , we build X by taking one vertex $v_i$ for each distinct value in { $\alpha (v): v \in V(\Gamma )\}$ and define

$$ \begin{align*}\iota_X(v_i) = \alpha(v) \end{align*} $$

to be the corresponding value.

Remark 3.14. Note that while the definition of equidimensionality requires that vertices of $\Gamma $ must map to vertices of X, edges can map either onto edges or vertices of X. Edges that map onto vertices of X are precisely those on which the slope of $\alpha $ is $0$ (i.e., the contracted edges). We point out that in order for a given $\alpha $ to factor through X (i.e., in order for $\alpha = \iota _X \circ \beta $ to hold), strict constraints must hold between the lengths of the edges e of $\Gamma $ and the edges f of X. Suppose $\vec {f} \in E(\vec {X})$ is a canonically oriented edge of X (c.f. definition 2.11) between vertices $v_i$ and $v_{i+1}$ , and suppose an edge e of $\Gamma $ maps to f. Then, orienting e so that $s(\vec {e})> 0$ , we must have that $\beta $ maps the initial point $\mathfrak {r}(\vec {e}) = v$ to $v_i$ and the terminal point $\mathfrak {t}(\vec {e})=w$ to $v_{i+1}$ , and furthermore,

$$\begin{align*}s(\vec{e})\ell_e = \alpha(w) - \alpha(v) = \iota_X(v_{i+1})-\iota_X(v_i) = \ell_f. \end{align*}$$

Thus, in order for a given acyclic flow s to lift to an equidimensional twist with underlying combinatorial type $\Gamma \to X$ , the system of equations

$$\begin{align*}\text{ } s(\vec{e})\ell_e = \ell_f \text{ for all }\vec{e} \text{ that map to } \vec{f}, \vec{f} \in E(\vec{X}) \end{align*}$$

needs to be satisfied. We note that this system of equations is stronger than the equations $\left \langle \gamma , s \right \rangle = 0$ required for the flow to lift to a twist: since X is contractible, the image of any loop $\gamma \in H_1(\Gamma )$ in X is an oriented path $P=\sum \vec {f}$ where each edge appears an equal number of times with opposite orientations. But then

$$\begin{align*}\left \langle s, \gamma \right \rangle = \sum_{\vec{e} \in \gamma} s(\vec{e})\ell_e = \sum_{\vec{f} \in P} o(\vec{f}) \ell_f = 0, \end{align*}$$

where the $o(\vec {f})=1$ if $\vec {f}$ is canonically oriented and $-1$ otherwise.

Remark 3.15. The definition of equidimensionality may seem convoluted from the vantage of tropical curves metrized by monoids, but it is natural from the dual point of view of cone complexes and semistable reduction. We find the dual point of view more intuitive, but we stick with the tropical perspective as it is more ubiquitous in the literature. Namely, a tropical curve $\Gamma $ metrized by M is equivalent data to a map of cone complexes (with integral structure)

$$ \begin{align*}\Sigma_C \to \Sigma_S := M^{\vee}. \end{align*} $$

Plainly, one builds $\Sigma _C$ out of $\Gamma $ as a fibration over $\Sigma _S$ . The fiber over $x \in \Sigma _S$ is obtained by attaching a vertex $v_x$ for each $v \in V(\Gamma )$ and an edge of length $\ell _e(x) \in \mathbb {R}_{\ge 0}$ for each $e \in E(\Gamma )$ . Then, a piecewise linear function $\alpha $ on $\Gamma $ corresponds to a piecewise linear map $\tilde {\alpha }$

A tropical line X corresponds to a subdivision $\Sigma _X \to \Sigma _S \times \mathbb {R} \to \Sigma _S$ so that the composed map $\Sigma _X \to \Sigma _S$ is equidimensional (maps cells onto cells) and furthermore sends integral structures onto integral structures, and an equidimensional piecewise linear function that factors through X corresponds to a factorization of $\tilde {\alpha }$ through $\Sigma _X$ that sends cells of $\Sigma _C$ onto cells of $\Sigma _X$ . The name equidimensional comes from the fact that maps of fans which send cones onto cones are the ones that induce equidimensional maps of toric varieties.

Suppose $\alpha $ is a piecewise linear function or twist on $\Gamma $ , and $\Gamma ' \to \Gamma $ is a subdivision. Then $\alpha $ lifts to a piecewise linear function on $\Gamma '$ . Suppose $\alpha $ is equidimensional on $\Gamma '$ . We say that $\Gamma '$ is a minimal subdivision on which $\alpha $ is equidimensional if the following stability condition holds:

$(\star )$ For every exceptional vertex v of $\Gamma '$ , there exists a nonexceptional vertex w of $\Gamma $ such that $\alpha (v) = \alpha (w)$ .

We remark that the minimal $\Gamma '$ on which $\alpha $ is equidimensional is, in general, finer than the minimal model $\Gamma _\alpha $ on which $\alpha $ is defined. Furthermore, the model is unique if it exists. This is very similar to the combinatorial analogues in 2.14, but the tropical picture deviates here: a model where $\alpha $ becomes equidimensional may not exist, as the required combinatorial subdivisions may not lift with respect to the metric structure. Experts will recognize here that one can always find a minimal equidimensional model, but only after altering the base monoid M. However, one can say the following with relative ease:

Definition 3.17. Suppose $\alpha $ is a piecewise linear function on $\Gamma $ , and that $\alpha $ lifts to an equidimensional function

$$ \begin{align*}\Gamma' \to X \end{align*} $$

on some subdivision of $\Gamma $ . We call the lift stable if $\Gamma ' \to X$ is the minimal such lift (i.e., satisfies $(\star )$ ). We write

$$ \begin{align*}\alpha: \mathsf{Eq}_{\Gamma}(\alpha) \to X \end{align*} $$

for the minimal lift.

4.1 Cone complexes and cone stacks

Since we want to build tropical moduli spaces parametrizing our objects – tropical curves with various types of piecewise linear functions on them – and our objects come with automorphisms, the notion of a cone complex is inadequate. Rather, we will use the formalism of cone stacks, developed in [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20]. We recall only the very basics of the theory that we will use from loc. cit and refer the interested reader there for a comprehensive treatment.

A face morphism or face map of cones $\tau \to \sigma $ is an isomorphism of $\tau $ with a face of $\sigma $ . Note that in the definition we consider $\sigma $ itself as a face, so arbitrary isomorphisms $\tau \to \sigma $ are considered face morphisms as well. A covering of a cone complex $\Sigma $ by face morphisms is a collection of face maps $\tau \to \Sigma $ that is jointly surjective. Such coverings generate a Grothendieck topology on the category of cone complexes. A cone stack is a stack on this site, with the (essentially straightforward) appropriate notion of algebraicity. Since face maps are, however, so simple, the notion of cone stack is, in fact, equivalent to the following much simpler definition [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20][Proposition 2.19]:

Definition 4.2. A cone stack is a category fibered in groupoids

$$\begin{align*}\Sigma \to \textbf{RPC}^f,\end{align*}$$

where $\textbf {RPC}^f$ is the full subcategory of $\textbf {RPC}$ with face maps as morphisms.

Most frequently, cone stacks natually arise as colimits. More precisely, suppose $\mathcal {D}$ is a category, and $F: \mathcal {D} \to \textbf {RPC}^f$ is a functor. Concretely, this amounts to the data of cones $\sigma _{\alpha }$ for $\alpha \in \text {Ob}(\mathcal {D})$ , and face morphisms $u_{\alpha \to \beta }: \sigma _\alpha \to \sigma _\beta $ for any morphism $\alpha \to \beta $ in $\mathcal {D}$ . Then one constructs a cone stack

$$\begin{align*}\varinjlim_{\mathcal{D}} \sigma_{\alpha} \to \textbf{RPC}^f \end{align*}$$

by taking the fiber over $\sigma $ to be the groupoid with objects

$$\begin{align*}\phi \in \coprod_{\alpha \in \text{Ob}(\mathcal{D})} \text{Hom}_{\textbf{RPC}}(\sigma,\sigma_{\alpha}) \end{align*}$$

(the notation here means all $\textbf {RPC}$ morphisms, not just face morphisms) and with an isomorphism from $\phi : \sigma \to \sigma _\alpha $ to $\psi :\sigma \to \sigma _\beta $ if there exists $\alpha \to \beta $ in $\mathcal {D}$ such that $u_{\alpha \to \beta } \circ \phi = \psi $ . We note that when all morphisms $u_{\alpha \to \beta }$ appearing in $F(\mathcal {D})$ are proper face morphisms (i.e., face morphisms to a proper face), the groupoid above is equivalent to a set. This way, one recovers the notion of generalized cone complexes of [Reference Abramovich, Caporaso and PayneACP15] or cone spaces of [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20]. Moreover, when there exists at most one morphism between any two $\sigma _\alpha \to \sigma _\beta $ , one gets an ordinary polyhedral complex. The following table, while simply an analogy, is perhaps helpful:

Example 4.3.1. Fix a genus g and a number of markings n. There is a cone stack $\mathcal {M}_{g,n}^{\text {trop}} \to \mathbf {RPC}^f$ which over a cone $\sigma $ parametrizes genus g, n-marked curves $\Gamma $ metrized by $M = \sigma ^\vee $ , defined by

$$\begin{align*}\mathcal{M}_{g,n}^{\text{trop}}(\sigma) = \{(\Gamma,\ell_e:E(\Gamma) \to M-0): g(\Gamma)=g, \{1,\cdots,n\} \cong L(\Gamma) \}. \end{align*}$$

For further details on the CFG structure, the interested reader is referred to [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20].

4.4 Stacks of twists

We make the simplifying assumption that all graphs that appear have at least one leg.

Definition 4.5. We define cone stacks

$$ \begin{align*}\Sigma_{\textbf{Div}}, \Sigma_{\textbf{Ord}}, \Sigma_{\textbf{Rub}} \to \textbf{RPC}^f \end{align*} $$

by setting, for a cone $\sigma $ with dual monoid $M = \sigma ^\vee $

$$ \begin{align*} \Sigma_{\textbf{Div}}(\sigma) = \{\Gamma, \alpha \in \mathsf{PL}(\Gamma)\} \\ \Sigma_{\textbf{Ord}}(\sigma) = \{\Gamma, \alpha \in \mathsf{PL}(\Gamma) \text{ which is totally ordered} \}\\ \Sigma_{\textbf{Rub}}(\sigma) = \{\Gamma, \alpha \in \mathsf{PL}(\Gamma), \mathsf{Eq}_\Gamma(\alpha) \to X \}, \end{align*} $$

where

• • $\Gamma $ is metrized by M.

• • $\alpha $ vanishes on the vertex of $\Gamma $ containing the first leg.

Isomorphisms are isomorphism of graphs that respects the functions, orderings and equidimensional lifts.

Let $\textbf {CombDiv}$ be the category whose objects consist of a graph $\Gamma $ and an acyclic flow s. A map $(\Gamma ,s) \to (\overline {\Gamma },\overline {s})$ in $\textbf {CombDiv}$ is given by a map $f: \Gamma \to \overline {\Gamma }$ , where $\overline {\Gamma }$ is an edge contraction of $\Gamma $ , such that $\overline {s} \circ f = s$ (in particular, automorphisms respecting the flow are allowed). Similarly, we define $\textbf {CombOrd}$ to consist of pairs $(\Gamma ,s)$ and a total ordering $\kappa $ on $\Gamma $ extending s, and $\textbf {CombRub}$ to consist of a pair $(\Gamma ,s)$ and a stable equidimensional lift $\Gamma ' \to X$ of s (2.13).

For each $(\Gamma ,s) \in \textbf {CombDiv}$ , define a cone

$$ \begin{align*}\sigma_{(\Gamma,s)} \subset \mathbb{R}_{\ge 0}^{E(\Gamma)} \end{align*} $$

consisting of the $\ell $ that satisfy the equations

$$ \begin{align*}\left \langle s,\gamma \right \rangle_{\ell} =0 \end{align*} $$

for all $\gamma \in H_1(\Gamma )$ . The pairing here is the intersection pairing of Section 3. By definition, the pairing requires a length on each edge of $\Gamma $ . The subscript $\ell $ here means that on the point $\ell = (\ell _{e})_{e \in E(\Gamma )}$ of $\mathbb {R}_{\ge 0}^{E(\Gamma )}$ , we give the tautological length $\ell _e$ to the edge e. Under a morphism $(\Gamma ,s) \to (\overline {\Gamma },\overline {s})$ , we get a face morphism

$$ \begin{align*}\sigma_{(\overline{\Gamma},\overline{s})} \to \sigma_{(\Gamma,s)} \end{align*} $$

and so we may glue the cones into a cone stack

$$ \begin{align*}\Sigma_{\textbf{Div}}' : = \varinjlim_{(\Gamma,s)} \sigma_{(\Gamma,s)}. \end{align*} $$

Similarly, as in the discussion in remark 3.14, for a triple $(\Gamma ,s,\Gamma ' \to X)$ in $\textbf {CombRub}$ , we take the cone

$$ \begin{align*}\sigma_{(\Gamma,s,\Gamma' \to X)} \subset \mathbb{R}_{\ge 0}^{E(\Gamma')} \times \mathbb{R}_{\ge 0}^{E(X)} \end{align*} $$

consisting of the $(\ell _{e},\ell _f)_{e \in E(\Gamma '),f \in E(X)}$ that satisfy

$$\begin{align*}s(\vec{e})\ell_e = \ell_f \end{align*}$$

whenever $\vec {e} \in \mathcal {E}(\Gamma ')$ maps to $\vec {f}$ in $E(\vec {X}) \subset \mathcal {E}(X)$ , and glue to a cone stack

$$ \begin{align*}\Sigma_{\textbf{Rub}}':=\varinjlim_{(\Gamma,s,\Gamma \to X)} \sigma_{(\Gamma,s,\Gamma' \to X)}. \end{align*} $$

Finally, as in remark 3.9, for $(\Gamma ,s,\kappa ) \in \mathbf {CombOrd}$ , we take the cone

$$ \begin{align*}\sigma_{(\Gamma,s,\kappa)} \subset \mathbb{R}_{\ge 0}^{E(\Gamma)} \end{align*} $$

defined by the equations

$$ \begin{align*}\left \langle s, \gamma \right \rangle_\ell = 0 \end{align*} $$

for $\gamma \in H_1(\Gamma )$ , and the additional condition: for any two vertices $v,w$ such that w comes later than v in the ordering $\kappa $ , and an oriented path $P_{v \to w}$ from v to w, we keep the lengths $\ell =(\ell _e)_{e \in E(\Gamma )} \subset \mathbb {R}_{\ge 0}^{E(\Gamma )}$ for which additionally,

$$\begin{align*}\sum_{\vec{e} \in P_{v \to w}} s(\vec{e})\ell_e \ge 0. \end{align*}$$

Footnote ⁸ Equivalently, we can fix a minimal vertex $v_0$ for $\kappa $ , and an oriented path $P_{v}$ from $v_0$ to v for each v, and keep the lengths $\ell $ such that

$$\begin{align*}\sum_{\vec{e} \in P_{v} } s(\vec{e})\ell_e \le \sum_{\vec{e} \in P_{w}} s(\vec{e})\ell_e \end{align*}$$

whenever $v \le w$ in the ordering $\kappa $ (the advantage of doing so is that one has to consider fewer paths: one for each vertex v instead of one for each pair of vertices $v,w$ ). We glue these cones to a cone stack

$$ \begin{align*}\Sigma_{\textbf{Ord}}' = \varinjlim_{(\Gamma,s,\kappa)} \sigma_{(\Gamma,s,\kappa)}. \end{align*} $$

Proof. Let M be a monoid, with $M^\vee = \sigma $ . An element $\Sigma _{\textbf {Div}}(M)$ is a tropical curve $\Gamma $ metrized by M, together with a piecewise linear function $\alpha $ vanishing on the vertex containing the first leg. This data defines a map

$$ \begin{align*} \sigma \to \mathbb{R}_{\ge 0}^{E(\Gamma)} \\ x \to (\ell_e(x)) \end{align*} $$

and an underlying flow $s_\alpha :=s$ . But the flow is a twist, and so

$$ \begin{align*}\left \langle s, \gamma \right \rangle \in M^{\text{gp}} \end{align*} $$

is $0$ . So the $\ell _e(x)$ map into $\sigma _{(\Gamma ,s)}$ . Conversely, a map $f: \sigma \to \sigma _{(\Gamma ,s)}$ defines a metric on $\Gamma $ , by taking $\ell _e \in M$ to be the composition

of f with the e-th projection. The acyclic flow s is a twist on $\Gamma $ , since the lengths have been chosen so that the equations $\left \langle s,\gamma \right \rangle = 0$ are satisfied. The twist only lifts to a piecewise linear function on $\Gamma $ up to translation by an element of $M^{\text {gp}}$ , but it lifts uniquely if we assume that its value is $0$ on the vertex containing the first leg. This shows

$$ \begin{align*}\Sigma_{\textbf{Div}} \cong \Sigma_{\textbf{Div}}'. \end{align*} $$

The proofs for $\textbf {Ord},\textbf {Rub}$ are similar, using the fact that the defining equations of the cones $\sigma _{(\Gamma ,s,\kappa )}$ and $\sigma _{(\Gamma ,s,\Gamma ' \to \Gamma )}$ are, according to remarks 3.9 and 3.14, precisely the conditions necessary for s to lift to a totally ordered or equidimensional twist, respectively.

There are evident maps $\Sigma _{\textbf {Rub}} \to \Sigma _{\textbf {Ord}} \to \Sigma _{\textbf {Div}}$ obtained by forgetting the additional structure at each step.

Proof. We first look at $\Sigma _{\textbf {Ord}} \to \Sigma _{\textbf {Div}}$ . It is clear that the map is a monomorphism, so it suffices to show that the map is bijective on $\mathbb {N}$ points. Given a tropical curve $\Gamma $ with integer lengths, and an $\alpha : \Gamma \to \mathbb {R}$ , the values of $\alpha $ are elements of $\mathbb {Z}$ , and so automatically totally ordered.

For the map $\Sigma _{\textbf {Rub}} \to \Sigma _{\textbf {Ord}}$ , it suffices to show that the map is bijective on $\mathbb {Q}_{\ge 0}$ points. We start with an $\alpha $ and an ordering $\kappa $ of its values and build a tropical line X by taking one vertex $v_i$ for each distinct value $\alpha (v)$ , with the ordering of $\kappa $ . We define an embedding $\iota _X: X \to \mathbb {R}$ by

$$ \begin{align*}\iota_X(v_i) = \text{ value of corresponding } \alpha(v). \end{align*} $$

There is a map of topological spaces $\Gamma \to X$ , but it does not respect cell structures, as interior points of edges of $\Gamma $ map to vertices of X. We refine $\Gamma $ to $\Gamma '$ obtained by subdividing along the preimages of the vertices of X. This gives a stable equidimensional PL function on $\alpha $ .

Remark 4.9. The argument in the proof essentially shows how the finite index inclusion in $\Sigma _{\textbf {Rub}} \to \Sigma _{\textbf {Ord}}$ arises. When subdividing $\Gamma $ to $\Gamma '$ , the new points may have rational coordinates; for instance, if an edge e from v to w has slope $s(e)$ , and $\alpha (u)$ is an intermediate value between $\alpha (v)$ and $\alpha (w)$ , the function $\alpha $ hits $\alpha (u)$ at the point

$$ \begin{align*}\frac{\alpha(u)-\alpha(v)}{s(e)} \end{align*} $$

of e, and thus e needs to be subdivided there. This point is in $M_{\mathbb {Q}}^{\text {gp}}$ , but not necessarily in $M^{\text {gp}}$ .

Proof. It suffices to check that the cones $\sigma _{(\Gamma ,s,\Gamma ' \to X)}$ of $\Sigma _{\textbf {Rub}}$ are isomorphic to $\mathbb {N}^k$ for some k. But the equations

$$ \begin{align*}s(\vec{e})\ell_e = \ell_f \end{align*} $$

for $\vec {e}$ mapping to $\vec {f} \in E(\vec {X})$ show that the coordinates of the noncontracted edges in $E(\Gamma )$ are redundant, and the cone is, in fact, isomorphic to

$$ \begin{align*}\mathbb{N}^{E(X)} \times \mathbb{N}^{E^c(\Gamma)}, \end{align*} $$

where $E^c(\Gamma )$ denotes the set of contracted edges of $\Gamma $ . The fact that $\Sigma _{\textbf {Ord}}$ is simplicial follows from the fact that its cones are isomorphic to those of $\Sigma _{\textbf {Rub}}$ after tensoring with $\mathbb {Q}$ .

In fact, more can be said. From the isomorphism of the real points of a cone in $\Sigma _{\textbf {Ord}}$ with $\mathbb {R}_{\ge 0}^{E(X)} \times \mathbb {R}_{\ge 0}^{E^c(\Gamma )}$ , it follows that the rays (i.e., the one dimensional cones) in $\Sigma _{\mathbf {Ord}}$ are precisely those that parametrize maps $\alpha $ consisting of

• • Either a single contracted edge.

• • Or, maps without contracted components to a target with exactly one edge.

In either case, the map $\alpha $ automatically lifts to an equidimensional one. Thus, the sublattice structure of $\Sigma _{\textbf {Rub}}$ agrees with that of $\Sigma _{\textbf {Ord}}$ on rays. Since $\Sigma _{\textbf {Ord}}$ is simplicial, we obtain the following:

Thus, $\Sigma _{\textbf {Ord}}$ is in a certain sense a coarse moduli space of $\Sigma _{\textbf {Rub}}$ : see, for example, [Reference Gillam and MolchoGM15, Subsection 3.2].

4.12 Carving out small subcomplexes

The complexes $\Sigma _{\textbf {Div}}, \Sigma _{\textbf {Rub}}, \Sigma _{\textbf {Ord}}$ are very large: they are indexed by additional data over the category of all graphs, which is itself notoriously large. When algebraizing, as we will in the next section, the resulting schemes have infinitely many connected components and are highly nonseparated. Here, we want to impose several increasingly stringent conditions that carve out subcomplexes which algebraize to much more pleasant spaces.

First, we can, as usual, restrict our attention to genus g, n-marked graphs. We may then write

$$ \begin{align*}\Sigma_{\textbf{Div}_n} \end{align*} $$

which decomposes

$$ \begin{align*}\Sigma_{\textbf{Div}} = \coprod_{n \ge 1} \Sigma_{\textbf{Div}_{n}}. \end{align*} $$

Similar descriptions are available for the order and rubber versions.

Given an n-marked graph $\Gamma $ , we can consider $n+1$ -marked graphs $\Gamma _{c}$ , one for every cell c of $\Gamma $ (vertex or edge), obtained by attaching a leg $l_{n+1}$ on c when it is a vertex, and a bivalent vertex with a single leg $l_{n+1}$ on c when it is an edge. A map $\Gamma _{c} \to \Gamma $ is obtained by deleting the latter vertex. There is a map

$$ \begin{align*}\mathbb{R}_{\ge 0}^{E(\Gamma_c)} \to \mathbb{R}_{\ge 0}^{E(\Gamma)} \end{align*} $$

which is an isomorphism when $c=v$ is a vertex, and which is the fiber product

when e is an edge. Given a piecewise linear function $\alpha $ on $\Gamma $ , with underlying flow $s=s_\alpha $ , it lifts canonically to a piecewise linear function on $\Gamma _c$ by not changing the slopes if e has been subdivided. The union of the cones

$$\begin{align*}\sigma_{(\Gamma,s,n+1)} \subset \Sigma_{\mathbf{Div}_{n+1}} \end{align*}$$

forms a subcomplex with a forgetful map to $\sigma _{(\Gamma ,s,n)}$ .

Lemma 4.13. The universal family of $\Sigma _{\mathbf {Div}_{n}}$ restricts to

$$ \begin{align*}\varinjlim_{c \in V(\Gamma) \cup E(\Gamma)} \sigma_{(\Gamma_c,s,n+1)} \end{align*} $$

over $\sigma _{(\Gamma ,s,n)}$ .

Proof. Let $\mathcal {C} \to \Sigma _{\textbf {Div}_n}$ be the universal family, and $\sigma $ a cone. A map $\sigma \to \mathcal {C}$ is a map $\sigma \to \Sigma _{\textbf {Div}_n}$ , together with a section of

$$ \begin{align*}\mathcal{C} \times_{\Sigma_{\mathbf{Div}_n}} \sigma \to \sigma. \end{align*} $$

In other words, it consists of a tropical curve $\Gamma $ metrized by $M = \sigma ^\vee $ , a piecewise linear function $\alpha $ and a section of $\Gamma $ . The section is a point of $\Gamma $ , which is either a vertex or lies on an edge. Sections that land in e are in bijection with subdivisions $\Gamma '$ of $\Gamma $ whose underlying graph is $\Gamma _e$ , and are thus parametrized by the choice of lengths of the two pieces of e determined by the section. Since $\alpha $ lifts canonically to $\Gamma '$ by not altering the slopes, the triples $(\Gamma ,\alpha ,\text {section through } e)$ are in bijection with maps

$$ \begin{align*}\sigma \to \sigma_{(\Gamma_e,s,n+1)} \end{align*} $$

as claimed.

The analogous result does not hold for $\Sigma _{\textbf {Ord}}$ . The reason is that an ordering $\kappa $ does not lift canonically to an ordering on the universal curve. All that can be said is that the universal curve of $\Sigma _{\textbf {Ord}}$ is the pullback of the universal curve of $\Sigma _{\textbf {Div}}$ .

The result is also false for $\Sigma _{\textbf {Rub}}$ , but a curious intermediate statement can be obtained. Points of $\Sigma _{\textbf {Rub}}$ have more structure, contained in the map $\Gamma ' \to X$ . Suppose $\kappa $ is the induced ordering. The section of the universal curve in particular factors through some cell $\Gamma ^{\prime }_c$ of $\Gamma '$ now, and thus the ordering $\kappa $ lifts canonically to $\Gamma ^{\prime }_c$ : points in any cell of $\Gamma '$ are in a unique order relative to $\alpha $ . It follows that the universal curve of $\Sigma _{\textbf {Rub}_n}$ factors through $\Sigma _{\textbf {Ord}_{n+1}} \to \Sigma _{\textbf {Div}_{n+1}}$ . However, it does not necessarily factor through $\Sigma _{\textbf {Rub}_{n+1}}$ . The reason is that while $\alpha $ and the total ordering lift canonically to $\Gamma ^{\prime }_c$ , the equidimensional lift

$$ \begin{align*}\Gamma' \to X \end{align*} $$

does not. The induced map

$$ \begin{align*}\Gamma^{\prime}_{c} \to X \end{align*} $$

is no longer equidimensional. Further subdivision of X and consequently of $\Gamma ^{\prime }_c$ is required, which may require extracting additional roots, as in 4.9. Nevertheless, the argument suffices to show the following:

We now continue our carving mission much more aggressively. As usual, we can fix the genus g of the graph, along with its n markings. This way, we obtain substacks

$$ \begin{align*}\Sigma_{?_{g,n}}. \end{align*} $$

Next, we fix a universal stability condition $\theta $ [Reference Kass and PaganiKP19]. This means a stability condition for the universal family $\mathcal {C}_{g,n}^{\text {trop}} \to \mathcal {M}_{g,n}^{\text {trop}}$ of stable tropical curves: a numerical stability condition in the sense of Subsection 2.15 for each stable graph $\Gamma $ of genus g with n legs, which are compatible with respect to all contractions of edges and automorphisms. We restrict to stable graphs here for simplicity but, in fact, a similar procedure even works for the cone stack of all genus g, n-marked tropical curves $\mathfrak {M}_{g,n}^{\text {trop}}$ .

Let A be the divisor of example 2.2.1 associated to a vector $(a_1,\cdots ,a_n) \in \mathbb {Z}^n$ , $k \in \mathbb {Z}$ , which is compatible with contractions according to example 2.21.1.

Figure 1 Above, a point of $\Sigma _{\textbf {Ord}}$ , representing a piecewise linear function $\alpha $ with $\alpha (v)<\alpha (u)<\alpha (w)$ , and a lift to its universal family. The relation between $\alpha (u)$ and $\alpha (v_{n+1})$ on the universal family is undetermined. Below, an analogous point of $\Sigma _{\textbf {Rub}}$ . Here, the relation $\alpha (v_{n+1})<\alpha (u)=\alpha (u')$ is forced.

We write $\Sigma _{\mathbf {Div}_{g,A}^\theta } $ for the subcomplex of $\Sigma _{\mathbf {Div}_{g,n}}$ consisting of the cones $\sigma _{(\Gamma ,s)}$ such that the graph $\Gamma $ is quasi-stable, and $D = A - \mathsf {div}(s)$ is $\theta $ -stable. In other words, $(\Gamma ,s)$ is a $\theta $ -flow relative to the stabilization $\Gamma ^{\mathsf {st}}$ . We write

$$ \begin{align*}\Sigma_{\mathbf{Ord}_{g,A}^\theta} \end{align*} $$

for the subcomplex consisting of $(\Gamma ,s,\kappa )$ with $\Gamma $ quasi-stable and $D = A - \mathsf {div}(s)\ \theta $ -stable, and

$$ \begin{align*}\Sigma_{\mathbf{Rub}_{g,A}^\theta} \end{align*} $$

for the subcomplex consisting of $(\Gamma ,s,\Gamma ' \to X)$ with $\Gamma $ quasi-stable, $D = A - \mathsf {div}(s)\ \theta $ -stable and $\Gamma ' \to X$ a stable equidimensional lift. In other words, $(\Gamma ,s,\Gamma ' \to X)$ is a $\theta $ -stable equidimensional flow relative to the stable graph $\Gamma ^{\mathsf {st}}$ .

The space $\Sigma _{\mathbf {Div}^\theta _{g,A}}$ has a stabilization map to $\mathcal {M}_{g,n}^{\text {trop}}$ . It is shown in theorem [Reference Holmes, Molcho, Pandharipande, Pixton and SchmittHMP+22, Theorem 23] that for nondegenerate $\theta $ , this map factors isomorphically through a subdivision. We then get the following:

Corollary 4.17. Let $\theta $ be a nondegenerate stability condition. The map

$$ \begin{align*}\Sigma_{\mathbf{Rub}_{g,A}^\theta} \to \mathcal{M}_{g,n}^{\text{trop}}\end{align*} $$

factors isomorphically through the composition of a subdivision and a finite index sublattice inclusion.

5.1 Tropicalization and tropical operations

In this paper, we use the language of logarithmic geometry as our main means to access algebro-geometric problems via combinatorial tools. We rapidly lay out our conventions. For a thorough treatment of logarithmic geometry, we refer the interested reader to [Reference KatoKat89] and [Reference OgusOgu18]. For a shallower treatment more adapted to our needs here, we refer to [Reference Holmes, Molcho, Pandharipande, Pixton and SchmittHMP+22, Reference Molcho and RanganathanMR21].

A log scheme (or log DM stack) is a pair $(S,M_S)$ consisting of a scheme (or DM stack) S with a log structure $M_S$ (i.e., a sheaf of monoids on the étale site of S with a homomorphism of monoids)

$$\begin{align*}\epsilon: M_S \to \mathcal{O}_S \end{align*}$$

to the structure sheaf of S with its multiplicative monoid structure, such that $\epsilon ^{-1}(\mathcal {O}_S^*) \cong \mathcal {O}_S^*$ . As is common, we drop $M_S$ from the notation; however, when speaking about a log scheme S, it it to be understood that a log structure $M_S$ is present.

The quotient

$$\begin{align*}\overline{M}_S := M_S/\mathcal{O}_S^* \end{align*}$$

is a constructible sheaf on S, called the characteristic monoid. The characteristic monoid stratifies S; the strata are the connected components of the loci of S on which $\overline {M}_S$ is locally constant. As with our conventions on monoids, we will assume throughout that $\overline {M}_S$ is a sheaf of finitely generated and saturated monoids.

The main example to keep in mind is a normal crossings pair – a smooth S with a normal crossings divisor D. This defines a log structure by setting, for an étale map $i: U \to S$ ,

$$\begin{align*}M_S(i:U \to S) = \{f \in \mathcal{O}_U: f|_{i^{-1}(S-D)} \in \mathcal{O}_U^*(i^{-1}(S-D))\}. \end{align*}$$

The characteristic monoid $\overline {M}_S$ is, in that case, at a point x of the étale site of S, equal to

$$\begin{align*}\overline{M}_{S,x} = \mathbb{N}^k, \end{align*}$$

where k is the number of distinct (in any sufficiently small étale neighborhood) branches of D that contain x.

The important point for us is that any cone stack, as in Section 4, can be given a functor of points in the category of log schemes: for a cone stack $\Sigma $ , one obtains a prestack

$$ \begin{align*}\Sigma \rightarrow \textbf{LogSch} \end{align*} $$

by defining

$$ \begin{align*}T \to \Sigma \rightsquigarrow \mathsf{Hom}(\overline{M}_T(T)^\vee, \Sigma). \end{align*} $$

The associated stack is representable by an algebraic stack with logarithmic structure, referred to as an Artin fan. Instead of introducing Artin fans here, we will simply understand morphisms from a log scheme to $\Sigma $ to mean maps to the associated stack.

In particular, under our definition, a tropicalization for S is not unique. For any log smooth DM stack S locally of finite type (for example, a normal crossings pair), one can construct a canonical tropicalization $\Sigma _S$ of S as follows. When S is sufficiently small – the technical term is atomic, meaning that S has a unique closed stratum and the restriction map $\overline {M}_S(S) \to \overline {M}_{S,x}$ is an isomorpism for any x in the closed stratum – it is easy to check that the rational polyhedral cone $\Sigma _S = \overline {M}_S(S)^{\vee }$ is a tropicalization. In general, we can find an étale cover U of S by atomic log schemes, and an étale cover V of $U \times _S U$ by atomic log schemes. Then, the coequalizer

considered as the cone stack associated to a colimit as in Section 4 is a tropicalization of S. It is straightforward to check that the composed map $U \to \Sigma _U \to \Sigma _S$ descends to a map $S \to \Sigma _S$ .

Let S be a log DM stack with a tropicalization $S \to \Sigma _S$ . Then, any map $\Sigma \to \Sigma _S$ induces a map

$$ \begin{align*}S \times_{\Sigma_S} \Sigma \to S. \end{align*} $$

The fiber product $S \times _{\Sigma _S} \Sigma $ is then an algebraic stack over S, and the map $S \times _{\Sigma _S} \Sigma \to \Sigma $ is a tropicalization of $S \times _{\Sigma _S} \Sigma $ . The importance of this observation for us is that, in this fashion, one can lift combinatorial operations one performs on $\Sigma _S$ to algebraic operations on S; often, they have clear geometric meaning. The ones we have encountered in Section 4 are:

• • Subdivisions $\Sigma \to \Sigma _S$ . These lift to log modifications $\widetilde {S} = S \times _{\Sigma _S} \Sigma \to S$ , which are proper, birational, surjective representable maps.

• • Roots $\Sigma _S' \to \Sigma _S$ , which are maps that replace the integral structure of $\Sigma _S$ with an integral structure coming from a finite index sublattice. These correspond to root stacks of S algebraically, which are proper, birational, nonrepresentable maps $S' \to S$ , which are bijective on geometric points – a generalization of roots along divisors of S. Details can be found in [Reference Borne and VistoliBV12] and [Reference Gillam and MolchoGM15].

• • Inclusion of a subcomplex $\Sigma \subset \Sigma _S$ . These lift to open inclusions $V \subset S$ .

The three operations above are, furthermore, logarithmically étale and monomorphisms in the category of log schemes!

5.3 Logarithmic curves

Let $C \to S$ be a logarithmic curve (see [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20, Section 7]). Applying the tropicalization construction of the previous subsection yields a map of cone stacks

$$ \begin{align*}\Sigma_C \to \Sigma_S. \end{align*} $$

Pulling back to S and unwinding the definitions, one arrives at the data of a family of tropical curvesFootnote ¹⁰ over the scheme S, referred to as the tropicalization of $C \to S$ in the literature:

• • For each point $x \in S$ , an underlying graph $\Gamma _x$ : the dual graph of $C_x$ .

• • A tropical curve structure on $\Gamma _x$ metrized by $\overline {M}_{S,x}$ : for each edge $e \in E(\Gamma _x)$ , a length $\ell _e \in \overline {M}_{S,x}$ . The length $\ell _e$ is the ‘smoothing parameter’ of the corresponding node q in $C_x$ : there is a unique element $\ell _e$ in $\overline {M}_{S,x}$ such that

  $$ \begin{align*}\overline{M}_{C,q} \cong \overline{M}_{S,x} \oplus_{\mathbb{N}} {\mathbb{N}}^2\end{align*} $$
  under the map ${\mathbb {N}} \to \overline {M}_{S,x}$ sending $1 \to \ell _e$ , and the diagonal ${\mathbb {N}} \to {\mathbb {N}}^2$ .

• • Compatibility with étale specializations: for each étale specialization $\zeta : y \rightsquigarrow x$ , a map $f_\zeta :\Gamma _x \to \Gamma _y$ compatible with the induced map $\overline {M}_{S,x} \to \overline {M}_{S,y}$ .

The geometric notions on tropical curves discussed in the previous section globalize to logarithmic curves. The globalization works the same way for all concepts, by working fiber by fiber and demanding compatibility with étale specializations: a notion A on a tropical curve globalizes to the analogous notion on a logarithmic curve as a system of $A_x$ on $\Gamma _x/\overline {M}_{S,x}$ for each $x \in S$ , compatible with étale specializations. For example, a piecewise linear function on $C \to S$ is a collection of piecewise linear functions

$$ \begin{align*}\alpha_x \in \mathsf{PL}(\Gamma_x) \end{align*} $$

which are compatible with the maps $\Gamma _x \to \Gamma _y$ for each étale specialization $y \rightsquigarrow x$ . It is ordered, equidimensional, stable, etc. if all $\alpha _x$ are. For divisors, we use the term tropical divisor to avoid confusion with the traditional algebro-geometric notion. Thus, a tropical divisor on $C/S$ is a collection of divisors on each $\Gamma _x$ compatible with the contractions that arise from étale specializations.

Example 5.3.1. Let $C \to S$ be a logarithmic curve with n markings $x_1,\cdots ,x_n$ , and let $(a_1,\cdots ,a_n)$ be a vector of integers (for instance, one adding up to $-k(2g-2+n)$ ). The multidegree of $\omega _{C/S}^{\log }(\sum a_ix_i)$ defines a tropical divisor on $C/S$ : for each $x \in S$ , it is the divisor on $\Gamma _x/\overline {M}_{S,x}$ given by

$$ \begin{align*}A= \sum_{v \in V(\Gamma_x)} k\deg \omega^{\log}_{C_x}(v)+ \sum a_iv_i \end{align*} $$

for $v_i$ the vertex containing the marking $x_i$ , and where $\deg \omega _{C_x}^{\log }(v)$ is the degree of $\omega _{C_x}^{\log }$ on the component of $C_x$ corresponding to v. The divisor A is then precisely the divisor of example 2.2.1 and so is compatible with étale specializations by example 2.21.1. We warn the reader that our terminology here clashes with that of the introduction: the tropical divisor A does require the vector of integers $(a_1,\cdots ,a_n)$ as input but encodes more information.

Subdivisions deserve a special mention. A subdivision of the tropicalization of $C \to S$ is a subdivision of the fibers $\Gamma _x/\overline {M}_{S,x}$ compatible with étale specializations. These subdivisions certainly give rise to logarithmic modifications $C' \to C$ . However, the log modifications that arise this way are special, as the induced map $C' \to S$ remains a logarithmic curve.