# Balancing properties of tropical moduli maps

Karl Christ, Xiang He, and Ilya Tyomkin  Department of Mathematics
University of Texas at Austin
Austin
TX
78712<EMAIL_ADDRESS>Yau Mathematical Sciences Center,
Shuangqing Complex Building, Tsinghua University, Haidian District, Beijing,
China, 100084<EMAIL_ADDRESS>Department of Mathematics
Ben-Gurion University of the Negev
P.O.Box 653
Be’er Sheva
84105
Israel<EMAIL_ADDRESS>
###### Abstract.

Given a family of parameterized algebraic curves over a strictly semistable
pair, we show that the simultaneous tropicalization of the curves in the
family forms a family of parameterized tropical curves over the skeleton of
the strictly semistable pair. We show that the induced tropical moduli map
satisfies a certain balancing condition, which allows us to describe
properties of its image and deduce a new liftability criterion.

KC was partially supported by NSF FRG grant DMS–2053261 and a Minerva Short-
Term Research Grant. XH is supported by the National Key R$\&$D Program of
China 2022YFA1007100, the NSFC grant 12301057, and the ERC Consolidator Grant
770922 - BirNonArchGeom.

##  1\. Introduction

In this paper, we study certain properties of tropicalizations of families of
parameterized curves over a valued field $K$. Given a family of smooth curves
$f\colon{\mathcal{C}}\to U$ together with a map ${\mathcal{C}}\to X$ to a
toric variety $X$, we construct a family of tropical curves
$\Gamma_{\Lambda}\to\Lambda$ together with a map
$h\colon\Gamma_{\Lambda}\to{\mathbb{R}}^{n}$. This generalizes the
construction for $1$-dimensional families in our previous paper [CHT23] and
relies on results of de Jong [dJ96] on the existence of semistable models and
of Gubler, Rabinoff, and Werner [GRW16] on skeletons of such models.

While other versions of tropicalizations of families of curves have been
established in the literature, see for example [ACGS20, CCUW20, Ran22, CGM22],
we propose here to focus on properties of the induced moduli map. Namely, the
family $\Gamma_{\Lambda}\to\Lambda$ induces a tropical moduli map

$\alpha\colon\Lambda\to M_{g,n,\nabla}^{\mathrm{trop}}$

to the moduli space $M_{g,n,\nabla}^{\mathrm{trop}}$ of tropical parameterized
curves of genus $g$, degree $\nabla$, and with $n$ marked points. As usual, it
is given by sending a point in $\Lambda$ to the isomorphism class of its fiber
in the tropical family. One could expect such a tropical moduli map to be a
tropical map itself, i.e., to satisfy a balancing condition. Our main results
give first evidence for this expectation.

Before we can state these results, recall that
$M_{g,n,\nabla}^{\mathrm{trop}}$ is a (generalized) polyhedral complex with
polyhedra $M_{[\Theta]}$ parameterizing tropical curves of fixed combinatorial
type $\Theta$. Two classes of combinatorial types are of particular interest
for our purposes:

First, weightless and $3$-valent types. The corresponding strata in
$M_{g,n,\nabla}^{\mathrm{trop}}$ are maximal with respect to inclusion in the
closure. Thus one may think of them as parameterizing tropically general
curves.

Second, weightless and almost $3$-valent types, where we allow for a single
$4$-valent vertex. A fixed weightless and almost $3$-valent stratum is
contained in the closure of at most three other strata: the $3$ ways to
resolve the $4$-valent vertex into a pair of adjacent $3$-valent vertices,
each giving rise to a potential adjacent stratum some of which might get
identified by an automorphism of the combinatorial type. Notice that the slope
of the resolving edge is determined uniquely by the harmonicity condition, and
each of these strata is weightless and $3$-valent.

Here is an informal version of our main result; see Theorem 3.1 for a precise
statement:

######  Main Theorem.

Let $\alpha\colon\Lambda\to M_{g,n,\nabla}^{\mathrm{trop}}$ be as above
induced by the tropicalization of a family of parameterized curves. Then,

  1. (1)

The map $\alpha$ is (quasi-)harmonic on the interior of weightless and
3-valent strata.

  2. (2)

The map $\alpha$ is either (quasi-)harmonic or locally combinatorially
surjective on the interior of weightless and almost 3-valent strata.

More precisely, the map $\alpha$ is harmonic on codimension $1$ strata of
$\Lambda$ and quasi-harmonic on strata of higher codimension. Here, quasi-
harmonic means that a positive linear combination of the images of the
primitive normal vectors to a stratum is contained in the image of the
stratum; harmonic requires that the coefficients in the linear combination can
be chosen to be $1$. Locally combinatorially surjective, on the other hand,
means that the image of $\alpha$ contains points in each of the weightless and
3-valent strata adjacent to the weightless and almost 3-valent stratum. See
Definitions 2.4 and 2.9.

######  Remark 1.1.

In [CHT22] we also considered types $\Theta$ that are not weightless or almost
$3$-valent. Namely, ones that contain either a single vertex of weight $1$ or
two vertices of valence $4$ adjacent to a flattened cycle. We established the
corresponding liftability results for such strata in [CHT22, Lemmas 4.5 and
4.6] in the case of a $1$-dimensional family of parameterized curves. Using
the arguments we present in the current article, one can also generalize these
results to the setting of higher dimensional bases.

The Main Theorem allows us, in particular, to establish the following
corollary, which we again state informally and refer to Corollary 3.2 for a
precise statement:

######  Corollary.

With assumptions as in the Main Theorem we have:

  1. (1)

If $\alpha$ is locally surjective at a point in the interior of a weightless
and 3-valent stratum, it is surjective onto the closure of that stratum.

  2. (2)

If $M_{[\Theta]}$ is a weightless and almost 3-valent stratum and $\alpha$ is
surjective onto one of its adjacent weightless and 3-valent strata, it is
surjective also onto all other such adjacent strata.

We view this result as a new kind of liftability/realizability criterion for
parameterized tropical curves. To illustrate this, let us explain how we will
use the corollary in [CHT24]. In _loc. cit._ we apply the corollary to the
universal family ${\mathcal{C}}\to V$ over an irreducible component $V$ of a
Severi variety of a toric surface $X$. Given a general point $[C]\in V$,
parameterizing a curve of geometric genus $g$ on $X$, we can cut out an
$N$-dimensional neighborhood of $[C]$ in $V$ by varying $N$ general point
constraints, where $N$ is the dimension of $V$. Choosing the points general
enough ensures that the combinatorial type $\Theta$ of the tropicalization
$\mathrm{trop}(C)$ of $C$ is weightless and 3-valent, and the tropicalization
of the $N$ points cuts out an $N$-dimensional neighborhood of
$\mathrm{trop}(C)$ in $M_{[\Theta]}$. As it turns out, $\dim M_{[\Theta]}=N$,
and hence part (1) of the corollary applies. Using part (2), it follows that
the tropicalization of the universal family over $V$ contains curves in the
closure of any weightless and 3-valent stratum that can be reached from
$M_{[\Theta]}$ by passing through weightless and (almost) 3-valent strata.

Thus, the question of whether a given tropical curve is the tropicalization of
a curve in ${\mathcal{C}}\to V$ is reduced to a purely combinatorial question,
reminiscent of establishing connectivity in codimension $1$. Consequently, we
show for a large class of toric surfaces that each component $V$ contains a
point representing a curve with a certain tropicalization. In a second step,
we show by deformation-theoretic arguments that curves with such a
tropicalization can be contained in at most one irreducible component $V$,
which allows us to conclude the irreducibility of Severi varieties for these
toric surfaces.

##  2\. Notation and terminology

Throughout this paper, $F$ is a complete discretely valued field with an
algebraically closed residue field, and $K$ is its algebraic closure. We
denote the valuation by $\nu\colon K\rightarrow\mathbb{R}\cup\\{\infty\\}$,
the ring of integers by $K^{0}$, its maximal ideal by $K^{00}$, and the
residue field by $\widetilde{K}$, and similarly for $F$.

For a subset $P\subset{\mathbb{R}}^{n}$ we denote by ${\rm Lin}(P)$ the linear
subspace acting simply transitively on the affine hull of $P$. If $P$ is a
polyhedron with rational slopes, then ${\rm Lin}(P)$ admits a natural integral
structure induced from ${\mathbb{R}}^{n}$, i.e., the lattice of integral
vectors. By an integral structure on $P$, we mean the integral structure on
${\rm Lin}(P)$.

###  2.1. Polyhedral complexes

######  Definition 2.1.

An (abstract) polyhedral complex with integral structure is a connected
topological space $\Lambda$ together with a finite set $\mathcal{F}(\Lambda)$
of closed subsets, called faces of $\Lambda$, and a map $f_{W}\colon
W\rightarrow N_{W}\otimes{\mathbb{R}}$ for every $W\in\mathcal{F}(\Lambda)$
such that

  1. (1)

For every $W\in\mathcal{F}(\Lambda)$, $N_{W}$ is a lattice of finite rank;

  2. (2)

The map $f_{W}$ is a homeomorphism from $W$ to a polyhedron with rational
slopes of dimension ${\rm rank}(N_{W})$. By abuse of language, we will address
$W$ itself as a polyhedron and speak about its faces, interior, dimension,
integral structure, etc.;

  3. (3)

$\Lambda=\bigsqcup_{W\in\mathcal{F}(\Lambda)}W^{\circ}$, where $W^{\circ}$
denotes the interior of $W$;

  4. (4)

For any $W,W^{\prime}\in\mathcal{F}(\Lambda)$, the intersection $W\cap
W^{\prime}$ is a union of faces of $W$ and of $W^{\prime}$. And each face in
the intersection belongs to $\mathcal{F}(\Lambda)$;

  5. (5)

If $W\in\mathcal{F}(\Lambda)$ and $W^{\prime}\in\mathcal{F}(\Lambda)$ is a
face of $W$, then the integral structure on $W^{\prime}$ is the restriction of
the integral structure on $W$, i.e., the linear part of $f_{W}\circ
f_{W^{\prime}}^{-1}$ takes $N_{W^{\prime}}$ to a saturated sublattice in
$N_{W}$.

We say that $\Lambda$ is rational if each $f_{W}(W)$ has rational vertices in
$N_{W}\otimes{\mathbb{R}}$. All polyhedral complexes considered in this paper
will be assumed to have integral structures, and we will call them polyhedral
complexes for short.

######  Remark 2.2.

Axiom (2) implies that each $W\in\mathcal{F}(\Lambda)$ has a structure of a
polyhedron with rational slopes, axioms (3) and (4) mean that $\Lambda$ is
“glued” from polyhedra along their faces, and axiom (5) ensures that the
integral structures on the polyhedra are compatible with the gluing.

######  Definition 2.3.

A map $f\colon\Lambda\rightarrow\Lambda^{\prime}$ is called piecewise integral
affine if for each face $W\in\mathcal{F}(\Lambda)$ there is a face
$W^{\prime}\in\mathcal{F}(\Lambda^{\prime})$ such that $f(W)\subseteq
W^{\prime}$ and $f|_{W}\colon W\to W^{\prime}$ is integral affine.

For a face $W\in\mathcal{F}(\Lambda)$, let $\\{W_{i}\\}_{i}$ be the set of
faces in $\mathcal{F}(\Lambda)$ containing $W$ as a codimension-one face.
Then, for each $i$, the lattice $N_{W}$ is naturally identified with a
saturated sublattice of $N_{W_{i}}$ of corank one. Let $\vec{e}_{i}+N_{W}$ be
the generator of $N_{W_{i}}/N_{W}$ for which $f_{W_{i}}(W_{i})\subseteq
f_{W_{i}}(W)+\mathbb{R}_{\geq 0}\vec{e}_{i}$, and denote by ${\rm Star}(W)$
the collection $\\{\vec{e}_{i}+N_{W}\\}$. We will often consider a small open
neighborhood of $W^{\circ}$ in $W^{\circ}\cup\bigcup_{i}W_{i}^{\circ}$. To
distinguish this set from ${\rm Star}(W)$, we denote it by ${\rm
Star}(W^{\circ})$.

######  Definition 2.4.

Let $\Lambda$ be a polyhedral complex and $W\in\mathcal{F}(\Lambda)$ a face. A
piecewise integral affine map $f\colon\Lambda\rightarrow{\mathbb{R}}^{n}$ is
called quasi-harmonic at $W$ if

$\sum_{\vec{e}+N_{W}\in{\rm Star}(W)}a_{\vec{e}}\cdot\frac{\partial
f}{\partial\vec{e}}\in{\rm Lin}(f(W))$

for some positive integers $a_{\vec{e}}$. It is called harmonic at $W$ if one
can choose all $a_{\vec{e}}=1$. Finally, the map $f$ is called
(quasi-)harmonic if it is (quasi-)harmonic at all codimension-one faces.

######  Remark 2.5.

Plainly, being (quasi-)harmonic at $W$ depends only on the restriction of the
map $f$ to ${\rm Star}(W^{\circ})$.

###  2.2. Tropical curves

####  2.2.1. Abstract and parameterized tropical curves

The graphs in this paper are finite graphs having vertices, edges, and half-
edges, called legs. We denote the set of vertices of a graph $\mathbb{G}$ by
$V({\mathbb{G}})$, edges by $E({\mathbb{G}})$, and legs by $L({\mathbb{G}})$.
We set $\overline{E}(\mathbb{G}):=E({\mathbb{G}})\cup L({\mathbb{G}})$. By a
tropical curve $\Gamma$ we mean a finite graph $\mathbb{G}$ with ordered legs
equipped with a length function $\ell\colon
E(\mathbb{G})\rightarrow\mathbb{R}_{>0}$ and a weight (or genus) function
$g\colon V(\mathbb{G})\rightarrow\mathbb{Z}_{\geq 0}$. We say that a curve is
weightless if the weight function is identically zero. For any leg $l\in
L({\mathbb{G}})$ we set $\ell(l):=\infty$.

For $e\in\overline{E}({\mathbb{G}})$, we use $\vec{e}$ to indicate a choice of
orientation on $e$. If $e\in L({\mathbb{G}})$ is a leg, then it will always be
oriented away from the vertex. Bounded edges will be considered with both
possible orientations. We can always view tropical curves as polyhedral
complexes by dividing the loops with the middle point, identifying the edges
of ${\mathbb{G}}$ with bounded closed intervals of the corresponding lengths,
and identifying the legs of ${\mathbb{G}}$ with semi-bounded closed intervals.
This way, ${\rm Star}(v)$ is in one-to-one correspondence with the collection
of oriented edges and legs having $v$ as their tail. In particular, ${\rm
Star}(v)$ contains two vectors/edges for every loop based at $v$. The number
of elements in ${\rm Star}(v)$ is called the valence of $v$. The genus of
$\Gamma$ is defined to be
$g(\Gamma)=g({\mathbb{G}}):=1-\chi(\mathbb{G})+\sum_{v\in V(\mathbb{G})}g(v)$,
where $\chi({\mathbb{G}}):=b_{0}({\mathbb{G}})-b_{1}({\mathbb{G}})$ denotes
the Euler characteristic of ${\mathbb{G}}$. Finally, a tropical curve $\Gamma$
is called stable if $|{\rm Star}(v)|+2g(v)\geq 3$ for every vertex $v\in
V({\mathbb{G}})$.

Let $N$ be a lattice and set $N_{\mathbb{R}}:=N\otimes{\mathbb{R}}$. In the
sequel, we will always let $N$ be the lattice of cocharacters of a toric
variety. A parameterized tropical curve is a harmonic piecewise integral
affine map $h\colon\Gamma\rightarrow N_{\mathbb{R}}$ from a tropical curve
$\Gamma$ (considered as a polyhedral complex) to $N_{\mathbb{R}}$. The
combinatorial type $\Theta$ of a parameterized tropical curve is defined to be
the weighted underlying graph $\mathbb{G}$ with ordered legs equipped with the
collection of slopes $\frac{\partial h}{\partial\vec{e}}$ for
$e\in\overline{E}({\mathbb{G}})$. We denote the group of automorphisms of a
combinatorial type $\Theta$ by $\mathrm{Aut}(\Theta)$, and the isomorphism
class of $\Theta$ by $[\Theta]$. We define the extended degree
$\overline{\nabla}$ of a parameterized tropical curve to be the sequence of
slopes $\left(\frac{\partial h}{\partial\vec{l}}\right)_{l\in
L({\mathbb{G}})}$, and the degree $\nabla$ the subsequence
$\left(\frac{\partial h}{\partial\vec{l}_{i}}\right)$ of non-zero slopes.

####  2.2.2. Moduli of parameterized tropical curves

We denote by $M_{g,n,\nabla}^{\mathrm{trop}}$ the moduli space of
parameterized stable tropical curves of genus $g$, degree $\nabla$, and with
$n$ contracted legs. We always assume that the first $n$ legs
$l_{1},\dotsc,l_{n}$ are the contracted ones. The space
$M_{g,n,\nabla}^{\mathrm{trop}}$ is a generalized polyhedral complex with
integral structure in the sense of [ACP15]. It is stratified by subsets
$M_{[\Theta]}$ indexed by combinatorial types $\Theta$ with the fixed
invariants $g,\nabla,$ and $n$. If ${\mathbb{G}}$ is the underlying graph of a
combinatorial type $\Theta$, then
$M_{[\Theta]}=M_{\Theta}/\mathrm{Aut}(\Theta)$, where $M_{\Theta}$ is the
interior of a polyhedron $\overline{M}_{\Theta}$ in
${\mathbb{R}}^{|E({\mathbb{G}})|}\times N_{\mathbb{R}}^{|V({\mathbb{G}})|}$
and parameterizes tropical curves $h\colon\Gamma\to N_{{\mathbb{R}}}$ of type
$\Theta$. The $e$-coordinate for an edge $e\in E({\mathbb{G}})$ is the length
$\ell(e)$ of $e$; the $v$-coordinate for a vertex $v\in V({\mathbb{G}})$ is
the coordinate of $h(v)\in N_{\mathbb{R}}$.

We call the stratum $M_{\Theta}$ weightless and $3$-valent if so is the type
$\Theta$, and we call it weightless and almost $3$-valent if the type $\Theta$
is weightless and $3$-valent except for a unique $4$-valent vertex. If
$M_{\Theta}$ is weightless and almost $3$-valent it is contained in the
closure of at most $3$ strata, all of which are weightless and $3$-valent.
Furthermore, it has codimension one in each of them.

######  Remark 2.6.

The nice strata (resp. simple walls) that are used in [CHT23] are particular
cases of weightless and $3$-valent (resp. almost $3$-valent) strata, where the
strata are in addition required to be regular, i.e., of the expected
dimension.

####  2.2.3. Families of parameterized tropical curves over polyhedral
complexes

In this subsection, we generalize the notion of one-parameter families of
parameterized tropical curves in [CHT23, §3.1.3] to arbitrary dimension. Let
$\Lambda$ be a polyhedral complex. Consider a datum (†) consisting of the
following:

  * •

an extended degree $\overline{\nabla}$;

  * •

a combinatorial type $\Theta_{W}=\left({\mathbb{G}}_{W},\left(\frac{\partial
h}{\partial\vec{\gamma}}\right)\right)$ of extended degree $\overline{\nabla}$
for each $W\in\mathcal{F}(\Lambda)$;

  * •

an integral affine function $\ell_{W}(\gamma,\cdot)\colon
W\to{\mathbb{R}}_{\geq 0}$ for each $W\in\mathcal{F}(\Lambda)$ and $\gamma\in
E({\mathbb{G}}_{W})$;

  * •

an integral affine function $h_{W}(u,\cdot):W\to N_{\mathbb{R}}$ for each
$W\in\mathcal{F}(\Lambda)$ and $u\in V({\mathbb{G}}_{W})$;

  * •

a weighted contraction
$\phi_{W^{\prime},W}\colon\mathbb{G}_{W^{\prime}}\to\mathbb{G}_{W}$ preserving
the order of the legs for each pair of faces $W,W^{\prime}$ such that $W$ is a
face of $W^{\prime}$.

For any $W\in\mathcal{F}(\Lambda)$ and $q\in W^{\circ}$, set
$\Gamma_{q}:=(\mathbb{G}_{W},\ell_{q})$, where $\ell_{q}\colon
E({\mathbb{G}}_{W})\to{\mathbb{R}}_{\geq 0}$ is the function defined by
$\ell_{q}(\gamma):=\ell_{W}(\gamma,q)$. Denote by
$h_{q}\colon\Gamma_{q}\rightarrow N_{\mathbb{R}}$ the unique piecewise affine
map for which $h_{q}(u)=h_{W}(u,q)$ for all $u\in V({\mathbb{G}}_{W})$ and
$\left(\frac{\partial h}{\partial\vec{l}_{i}}\right)=\overline{\nabla}$.

######  Definition 2.7.

Let $\Lambda$ be a polyhedral complex. We say that a datum (†) is a family of
parameterized tropical curves over $\Lambda$ if the following compatibilities
hold for all pairs $W,W^{\prime}\in\Lambda$ such that $W$ is a face of
$W^{\prime}$, and all $q\in W^{\circ}$:

  1. (1)

$(\Gamma_{q},h_{q})$ is a parameterized tropical curve of combinatorial type
$\Theta_{W}$;

  2. (2)

$\ell_{W}(\phi_{W^{\prime},W}(\gamma),q)=\ell_{W^{\prime}}(\gamma,q)$ for all
$\gamma\in E({\mathbb{G}}_{W^{\prime}})$;

  3. (3)

$h_{W}(\phi_{W^{\prime},W}(u),q)=h_{W^{\prime}}(u,q)$ for all $u\in
V({\mathbb{G}}_{W^{\prime}})$.

A family of parameterized tropical curves over $\Lambda$ will be denoted by
$h\colon\Gamma_{\Lambda}\to N_{\mathbb{R}}$ or $(\Gamma_{\Lambda},h)$. The
tropical curve $(\Gamma_{q},h_{q})$ will be referred to as the fiber of
$(\Gamma_{\Lambda},h)$ over $q\in\Lambda$.

######  Definition 2.8.

Let $\Lambda$ be a polyhedral complex, and $\alpha\colon\Lambda\to
M_{g,n,\nabla}^{\mathrm{trop}}$ a continuous map. We say that $\alpha$ is
piecewise integral affine if for any $W\in\Lambda$ the restriction
$\alpha|_{W}$ lifts to an integral affine map $W\to\overline{M}_{\Theta}$ for
some combinatorial type $\Theta$.

Let $\nabla$ be the degree associated to the extended degree
$\overline{\nabla}$ by removing the zero slopes. Any family of parameterized
tropical curves $h\colon\Gamma_{\Lambda}\to N_{\mathbb{R}}$ induces a
piecewise integral affine map $\alpha\colon\Lambda\to
M_{g,n,\nabla}^{\mathrm{trop}}$ by sending $q\in\Lambda$ to the point
parameterizing the isomorphism class of the fiber $h_{q}\colon\Gamma_{q}\to
N_{\mathbb{R}}$. Furthermore, $\alpha$ lifts to an integral affine map from
the interior of each face of $\Lambda$ to the corresponding stratum
$M_{\Theta}$. The following definition is a generalization of [CHT23,
Definition 3.4].

######  Definition 2.9.

Let $\Lambda$ be a polyhedral complex, $\alpha\colon\Lambda\to
M_{g,n,\nabla}^{\mathrm{trop}}$ a piecewise integral affine map, $W$ a face of
$\Lambda$, and $\Theta$ a combinatorial type such that
$\alpha(W^{\circ})\subset M_{[\Theta]}$.

  1. (1)

Suppose $\alpha({\rm Star}(W^{\circ}))\subset M_{[\Theta]}$. We say that
$\alpha$ is (quasi-)harmonic at $W$ if $\alpha_{|_{{\rm Star}(W^{\circ})}}$
lifts to a (quasi-)harmonic map ${\rm Star}(W^{\circ})\to M_{\Theta}$.

  2. (2)

Suppose $\alpha({\rm Star}(W^{\circ}))\nsubseteq M_{[\Theta]}$. We say that
$\alpha$ is locally combinatorially surjective at $W$ if for any combinatorial
type $\Theta^{\prime}$ with an inclusion
$\overline{M}_{\Theta}\hookrightarrow\overline{M}_{\Theta^{\prime}}$, we have
$\alpha({\rm Star}(W^{\circ}))\cap M_{[\Theta^{\prime}]}\neq\emptyset$.

###  2.3. Families of curves

By a _family of curves_ , we mean a flat, projective morphism of finite
presentation and relative dimension one. By a collection of _marked points_ on
a family of curves, we mean a collection of disjoint sections contained in the
smooth locus of the family. A family of curves with marked points is
_prestable_ if its geometric fibers have at-worst-nodal singularities. It is
called _(semi-)stable_ if so are its geometric fibers. A prestable curve with
marked points defined over a field is called _split_ if the irreducible
components of its normalization are geometrically irreducible and smooth and
the preimages of the nodes in the normalization are defined over the ground
field. A family of prestable curves with marked points is called split if all
of its fibers are so; cf. [dJ96, § 2.22]. If $U\subset Z$ is open, and
$({\mathcal{C}},\sigma_{\bullet})$ is a family of curves with marked points
over $U$, then by a _model_ of $({\mathcal{C}},\sigma_{\bullet})$ over $Z$ we
mean a family of curves with marked points over $Z$, whose restriction to $U$
is $({\mathcal{C}},\sigma_{\bullet})$; and similarly, for prestable and semi-
stable models.

###  2.4. Toric varieties and parameterized curves

For a toric variety $X$, we denote the lattice of characters by $M$, of
cocharacters by $N$, and the monomial functions by $x^{m}$ for $m\in M$. A
_parameterized curve_ in a toric variety $X$ is a smooth projective curve with
marked points $(C,\sigma_{\bullet})$ together with a map $f\colon C\to X$ such
that $f(C)$ does not intersect orbits of codimension greater than one, and the
image of $C\setminus\left(\bigcup_{i}\sigma_{i}\right)$ under $f$ is contained
in the dense orbit of $X$.

A _family of parameterized curves_ $f\colon{\mathcal{C}}\rightarrow X$ over
$K$ consists of the following data:

  1. (1)

a family of smooth marked curves $({\mathcal{C}}\to B,\sigma_{\bullet})$ over
a base $B$, and

  2. (2)

a map $f\colon{\mathcal{C}}\rightarrow X$, such that for any geometric point
$p\in B$ the restriction ${\mathcal{C}}_{p}\to X$ is a parameterized curve.

###  2.5. Tropicalization of parameterized curves

Let $f\colon C\to X$ be a parameterized curve, and $C^{0}\to{\rm Spec}(K^{0})$
a prestable model. Denote by $\widetilde{C}$ the fiber of $C^{0}$ over the
closed point of ${\rm Spec}(K^{0})$. The _tropicalization_ $\mathrm{trop}(C)$
of $C$ with respect to the model $C^{0}$ is the tropical curve
$\Gamma=(\mathbb{G},\ell)$ defined as follows: The underlying graph
${\mathbb{G}}$ is the dual graph of the central fiber $\widetilde{C}$, i.e.,
the vertices of $\mathbb{G}$ correspond to irreducible components of
$\widetilde{C}$, the edges – to nodes, the legs – to marked points, and the
natural incidence relation holds. For a vertex $v$ of $\mathbb{G}$, its weight
is defined to be the geometric genus of the corresponding component
$\widetilde{C}_{v}$ of the reduction $\widetilde{C}$. As for the length
function, if $e\in E({\mathbb{G}})$ is the edge corresponding to a node
$z\in\widetilde{C}$, then $\ell(e)$ is defined to be the valuation of
$\lambda$, where $\lambda\in K^{00}$ is such that étale locally at $z$, the
total space of $C^{0}$ is given by $xy=\lambda$. Although $\lambda$ depends on
the étale neighborhood, its valuation does not, and hence the length function
is well-defined. Finally, notice that the order on the set of marked points
induces an order on the set of legs of $\Gamma$.

Next, we explain how to construct the parameterization
$h\colon\mathrm{trop}(C)\to N_{\mathbb{R}}$. Let $\widetilde{C}_{v}$ be an
irreducible component of $\widetilde{C}$. Then, for any $m\in M$, the pullback
$f^{*}(x^{m})$ of the monomial $x^{m}$ is a non-zero rational function on
$C^{0}$ since the preimage of the dense orbit is dense in $C$. Thus, there is
a $\lambda_{m}\in K^{\times}$, unique up to an element invertible in $K^{0}$,
such that $\lambda_{m}f^{*}(x^{m})$ is an invertible function at the generic
point of $\widetilde{C}_{v}$. The function $h(v)$, associating to $m\in M$ the
valuation $\nu(\lambda_{m})$, is clearly linear, and hence $h(v)\in
N_{\mathbb{R}}$. The parameterization $h\colon\mathrm{trop}(C)\to
N_{\mathbb{R}}$ is defined to be the unique piecewise integral affine function
with values $h(v)$ at the vertices of $\mathrm{trop}(C)$, whose slopes along
the legs satisfy the following: for any leg $l$ and $m\in M$ we have
$\frac{\partial
h}{\partial\vec{l}}(m)=-\mathrm{ord}_{\sigma_{i}}f^{*}(x^{m})$, where
$\sigma_{i}$ is the marked point corresponding to $l$. Then
$h\colon\mathrm{trop}(C)\to N_{\mathbb{R}}$ is a parameterized tropical curve,
by [Tyo12, Lemma 2.23]. It is called the _tropicalization_ of $f\colon C\to X$
with respect to the model $C^{0}$.

Plainly, the tropical curve $\mathrm{trop}(C)$ is independent of the
parameterization and depends only on $C^{0}$. If the family $C\to X$ is stable
and $C^{0}$ is the stable model, then the corresponding tropicalization is
called simply the tropicalization of $C$ (resp. of $f\colon C\to X$).

######  Remark 2.10.

Let $l$ be the leg corresponding to a marked point $\sigma_{i}$. If
$\sigma_{i}$ is mapped to the boundary divisor $D$, then, by definition,
$f(\sigma_{i})$ belongs to a unique component of $D$, and in particular, to
the smooth locus of $D$. Thus, the direction of the slope $\frac{\partial
h}{\partial\vec{l}}$ is completely determined by the irreducible component of
$D$ containing the image of $\sigma_{i}$. Furthermore, the integral length of
$\frac{\partial h}{\partial\vec{l}}$ is the multiplicity of $\sigma_{i}$ in
$f^{*}D$. Note also that $\frac{\partial h}{\partial\vec{l}}=0$ if and only if
$f(\sigma_{i})$ is contained in the dense orbit of $X$.

###  2.6. Strictly semistable pairs and their tropicalization

To tropicalize families, we will first tropicalize the base of the family. To
do so, we shall use higher-dimensional tropicalizations. Different versions of
such tropicalizations have appeared in the literature; see, for example,
[MN15, ACGS20, CCUW20, Ran22, CGM22]. Here we will follow the construction by
Gubler, Rabinoff, and Werner [GRW16]. Let us recall their construction.

######  Definition 2.11.

([GRW16, Definition 3.1]) A strictly semistable pair $(B^{0},H^{0})$ over
$K^{0}$ consists of an irreducible proper flat scheme $B^{0}$ over $K^{0}$ and
a sum $H^{0}=\sum H_{k}$ of distinct effective Cartier divisors $H_{k}$ on
$B^{0}$ such that $B^{0}$ is covered by open subsets $U$ which admit an étale
morphism

$\varphi\colon U\rightarrow\mathrm{Spec}(K^{0}[x_{0},\dotsc,x_{d}]/\langle
x_{0}\cdot\ldots\cdot x_{a}-\lambda\rangle)$ (2.1)

for some $a\leq d$ and $\lambda\in K^{00}$. Moreover, we assume that each
$H_{k}$ has irreducible support and $H_{k}\cap U$ is either empty or defined
by $\varphi^{*}(x_{j})$ for some $a+1\leq j\leq d$. Similarly, define a
strictly semistable pair over $F^{0}$ by replacing $K^{0}$ with $F^{0}$.

For a strictly semistable pair $(B^{0},H^{0})$, denote by $\widetilde{B}$ the
special fiber of $B^{0}$ with irreducible components $\widetilde{B}_{j}$. Set

$D\coloneqq\widetilde{B}\cup(\bigcup_{k}H_{k})\subset B^{0}$ (2.2)

with irreducible components $\\{D_{i}\\}_{i\in
I}=\\{H_{k}\\}_{k}\cup\\{\widetilde{B}_{j}\\}_{j}$. Then the
$\widetilde{B}_{j}$’s are called the _vertical components_ of $D$ and the
$H_{k}$’s the _horizontal components_. The divisor $D$ admits a stratification
by the irreducible components of $\bigcap_{i\in
J}D_{i}\backslash\big{(}\bigcup_{i\not\in J}D_{i}\big{)}$ for any $J\subseteq
I$. There is a natural partial order on the set of strata of $D$ given by
$S\leq T$ if and only if $S\subseteq\overline{T}$. Let
$\mathrm{Str}(B^{0},H^{0})$ denote the set of strata with vertical support,
namely, those contained in $\widetilde{B}$. Then for any
$S\in\mathrm{Str}(B^{0},H^{0})$, the closure $\overline{S}$ is smooth over
$\widetilde{K}$, cf. [GRW16, §3.15]. In particular, each component of the
reduction $\widetilde{B}_{j}$ is smooth over $\widetilde{K}$. Notice that our
notation differs slightly from the one in loc. cit., where the set of strata
$\mathrm{Str}(B^{0},H^{0})$ is denoted by ${\rm str}(\widetilde{B},H^{0})$.

We can now describe the skeleton of a strictly semistable pair $(B^{0},H^{0})$
following [GRW16, §4]. Notice, however, that in loc. cit. the construction is
described for a formal strictly semistable pair obtained from $(B^{0},H^{0})$
by a formal completion with respect to a non-zero element of $K^{00}$. To a
stratum $S\in\mathrm{Str}(B^{0},H^{0})$, one associates a polyhedron
$\Delta_{S}$ as follows. Pick a sufficiently small open neighborhood $U$ of a
general point of $S$ such that there exists an étale morphism $\varphi$ as in
(2.1) and such that for any stratum $T$ the following holds: $S\leq T$ if and
only if $T\cap U\neq\emptyset$. Let $\\{H_{i}\\}_{1\leq i\leq b}$ be the
horizontal divisors that intersect $U$, where $0\leq b\leq d-a$. We may assume
that $H_{i}$ is defined by $\varphi^{*}(x_{a+i})$, and that $S$ is given by
$\varphi^{*}(x_{0})=\cdots=\varphi^{*}(x_{a+b})=0$. Set
$\Delta_{S}:=\Delta(a,\lambda)\times\mathbb{R}^{b}_{\geq 0}$, where

$\Delta(a,\lambda)=\\{y=(y_{0},\dotsc,y_{a})\in\mathbb{R}_{\geq
0}^{a+1}|y_{0}+\cdots+y_{a}=\nu(\lambda)\\},$

and notice that it is strictly convex, i.e., contains no lines. Following
[GRW16], we call $\nu(\lambda)$ the length of $S$.

For $U$ and $\Delta_{S}$ as above, consider the locus $U_{S}\subset U\setminus
D$ of $K$-points $p$, whose specialization belongs to the union of strata
$\cup_{S\leq T}T$. There is a natural tropicalization map $\tau_{U}\colon
U_{S}\rightarrow\Delta_{S}$ given by

$p\mapsto\left(\nu\left(\varphi^{*}(x_{0})(p)\right),\dotsc,\nu\left(\varphi^{*}(x_{a+b})(p)\right)\right).$

Then $\Delta_{S}$ is independent of the choice of $U$ and $\varphi$, up to
reordering the coordinates, and the tropicalization map is also independent of
the choice of $\varphi$, and compatible on the intersections for different
choices of $U$; cf. [GRW16, §4.3].

If $S\leq T$ are strata, then up to reordering of the coordinates, there
exists $a^{\prime}\leq a$ and $b^{\prime}\leq b$, such that $T$ is given
locally by
$\varphi^{*}(x_{0})=\cdots=\varphi^{*}(x_{a^{\prime}})=\varphi^{*}(x_{a+1})=\cdots=\varphi^{*}(x_{a+b^{\prime}})=0.$
Therefore, $\Delta_{T}$ can be identified naturally with a face of
$\Delta_{S}$. Moreover, the tropicalization maps agree on $U_{T}$. Since
$B^{0}$ is proper over ${\rm Spec}(K^{0})$, any $K$-point specializes to one
of the strata in $\mathrm{Str}(B^{0},H^{0})$, and therefore, the
tropicalization maps glue to a well-defined tropicalization map $\tau$ from
the set of $K$-points of $B^{0}\setminus D$ to the limit
$S(B^{0},H^{0}):=\bigcup_{S\in\mathrm{Str}(B^{0},H^{0})}\Delta_{S}$; cf.
[GRW16, §4.6, §4.9]. In the sequel, e.g., in Theorem 3.1, we will abuse the
notation and denote the tropicalization map $\tau$ by $\mathrm{trop}$.

######  Definition 2.12.

The polyhedral complex
$S(B^{0},H^{0}):=\bigcup_{S\in\mathrm{Str}(B^{0},H^{0})}\Delta_{S}$ is called
the skeleton of the strictly semistable pair $(B^{0},H^{0})$.

The skeleton is naturally a rational polyhedral complex with an integral
structure. Indeed, any face $\Delta_{S}$ admits a natural embedding
$\Delta_{S}=\Delta(a,\lambda)\times\mathbb{R}^{b}_{\geq
0}\rightarrow{\mathbb{R}}^{a+1+b}$, which induces the integral structure on
$\Delta_{S}$. By construction, these integral structures are compatible for
different faces of the skeleton.

##  3\. Main theorem

In this section, we fix a pair of dual lattices $M$ and $N$, and a toric
variety $X$ with lattice of monomials $M$. Let
$\overline{{\mathcal{M}}}_{g,n}$ be the algebraic stack over $\mathbb{Z}$
classifying stable $n$-pointed curves of genus $g$. Then
$\overline{{\mathcal{M}}}_{g,n}$ admits a finite surjective morphism from a
projective scheme $\overline{M}$, over which there is a universal family of
curves, cf. [dJ96, §2.24].

######  Theorem 3.1.

Let $(B^{0},H^{0})$ be a strictly semistable pair over $K^{0}$ and
$({\mathcal{C}}^{0}\rightarrow B^{0},\sigma^{0}_{\bullet})$ a split family of
stable marked curves over $B^{0}$. Let $B$, $H$ and ${\mathcal{C}}$ be the
generic fibers of $B^{0}$, $H^{0}$, and ${\mathcal{C}}^{0}$, respectively.
Suppose ${\mathcal{C}}^{0}$ is obtained as a pullback of the universal family
over $\overline{M}$ and is smooth over $B^{\prime}:=B\backslash H$. Suppose
furthermore that $f\colon{\mathcal{C}}|_{B^{\prime}}\rightarrow X$ is a family
of parameterized curves over $B^{\prime}$. Then there exists a family of
parameterized tropical curves $h\colon\Gamma_{\Lambda}\to N_{\mathbb{R}}$ over
$\Lambda:=S(B^{0},H^{0})$ such that

  1. (1)

For any $\eta\in B^{\prime}(K)$, the fiber of $(\Gamma_{\Lambda},h)$ over
$\mathrm{trop}(\eta)$ is the tropicalization of
$f\colon{\mathcal{C}}_{\eta}\to X$;

  2. (2)

For any face $W\in{\mathcal{F}}(\Lambda)$ of codimension 1 (resp. of
codimension at least 2) such that the tropical curves over the interior of $W$
are weightless and $3$-valent except for at most one $4$-valent vertex, the
induced map $\alpha\colon\Lambda\to M^{\mathrm{trop}}_{g,n,\nabla}$ is either
harmonic (resp. quasi-harmonic) or locally combinatorially surjective along
$W$.

The following corollary is useful in applications, see, e.g., [CHT24].

######  Corollary 3.2.

Let $X$ be a toric variety and $f\colon{\mathcal{C}}\rightarrow X$ a family of
parameterized curves defined over the field $K$. Assume that the base
$B^{\prime}$ of the family is quasi-projective, and consider the set of
tropicalizations

$\Sigma:=\left\\{\left[\mathrm{trop}(f_{\eta},C_{\eta})\right]\;:\;\eta\in
B^{\prime}(K)\right\\}\subseteq M^{\mathrm{trop}}_{g,n,\nabla}.$

Let $M_{[\Theta]}\subset M^{\mathrm{trop}}_{g,n,\nabla}$ be a stratum. Then,

  1. (1)

The closure $\overline{\Sigma}\subseteq M^{\mathrm{trop}}_{g,n,\nabla}$ is the
image of a rational polyhedral complex $\Lambda$ under a piecewise integral
affine map;

  2. (2)

If $M_{[\Theta]}$ is weightless and $3$-valent and $\dim(\Sigma\cap
M_{[\Theta]})=\dim M_{[\Theta]}$, then
$\overline{M}_{[\Theta]}\subseteq\overline{\Sigma}$;

  3. (3)

If $M_{[\Theta]}$ is weightless and almost $3$-valent and $\overline{\Sigma}$
contains one of its adjoint weightless and $3$-valent strata, then
$\overline{\Sigma}$ contains all other such strata as well.

###### Proof.

After replacing $B^{\prime}$ with a finite covering, we may assume that the
family of marked curves over $B^{\prime}$ is a pullback of the universal
family over the projective scheme $\overline{M}$. Plainly, it suffices to
prove the corollary for each irreducible component of $B^{\prime}$.
Furthermore, we may replace $B^{\prime}$ with $B^{\prime}_{\rm red}$, and
hence we will assume that $B^{\prime}$ is integral, i.e., a quasi-projective
variety. Thus, there is a projective variety $B$ containing $B^{\prime}$ as a
dense open subset. After replacing $B$ with the closure of the graph
$B^{\prime}\to\overline{M}\times B$, we may assume that the family of marked
curves over $B^{\prime}$ extends to $B$, and the latter family is also a
pullback of the universal family over $\overline{M}$. Set $H:=B\setminus
B^{\prime}$.

By [Sta20, Tag 0C2S], $B$ admits a flat projective model $B^{0}$ over $K^{0}$.
Let $\overline{M}^{0}$ be the trivial model of $\overline{M}$ over $K^{0}$.
After replacing $B^{0}$ with the schematic image of the graph
$B\to\overline{M}^{0}\times_{{\rm Spec}(K^{0})}B^{0}$, which is also flat over
$K^{0}$ by loc. cit., we may assume that the family of marked curves extends
to $B^{0}$. By [dJ96, Lemma 5.3], there exist a projective alteration
$\varphi_{1}\colon B_{1}^{0}\to B^{0}$ such that the pullback of
${\mathcal{C}}$ to $B_{1}^{0}$ is split over $B_{1}^{0}$. Replacing
$B_{1}^{0}$ with an irreducible component that dominates $B^{0}$ if necessary,
we may assume that $B_{1}^{0}$ is integral. Since the loci $\Sigma$ associated
to $B^{\prime}$ and to $\varphi_{1}^{-1}(B^{\prime})$ coincide, we may assume
that $B_{1}^{0}=B^{0}$ and hence the family ${\mathcal{C}}$ is split over
$B^{0}$. Finally, by [dJ96, Theorem 6.5], there exists an alteration
$\varphi_{2}\colon B_{2}^{0}\to B^{0}$ such that $B_{2}^{0}$ is projective
over $K^{0}$, the generic fiber of $B_{2}^{0}\to{\rm Spec}(K^{0})$ is
irreducible, and the pair $\left(B_{2}^{0},\overline{\varphi_{2}^{-1}(H)}_{\rm
red}\right)$ is strictly semistable. Here $\overline{\varphi_{2}^{-1}(H)}_{\rm
red}$ is the closure of $\varphi_{2}^{-1}(H)_{\rm red}$ in $B_{2}^{0}$. As
before, we may assume that $B_{2}^{0}=B^{0}$.

Denote by $H^{0}$ the union of the horizontal components of $H\cup\tilde{B}$.
Then $(B^{0},H^{0})$ is a strictly semistable pair, and the family of marked
curves extends to $B^{0}$ by pulling back the universal family over
$\overline{M}$. Therefore, we may assume that we are given all the data
$B^{0},H^{0},{\mathcal{C}}^{0},\sigma^{0}_{\bullet},B^{\prime}$ satisfying the
assumptions of Theorem 3.1. Consider the tropicalization
$h\colon\Gamma_{\Lambda}\to N_{\mathbb{R}}$ and the induced map
$\alpha\colon\Lambda\to M^{\mathrm{trop}}_{g,n,\nabla}$ provided by Theorem
3.1.

(1) Since $\alpha$ is piecewise integral affine, it follows that
$\alpha(\Lambda)\subseteq M^{\mathrm{trop}}_{g,n,\nabla}$ is closed. Now, by
Theorem 3.1 (1), $\Sigma\subset\alpha(\Lambda)$, and therefore
$\overline{\Sigma}\subset\alpha(\Lambda)$. Furthermore, $\alpha^{-1}(\Sigma)$
is dense in $\Lambda$. Thus, $\Sigma$ is dense in $\alpha(\Lambda)$, and hence
$\overline{\Sigma}=\alpha(\Lambda)$ as asserted.

(2) Set $k:=\dim M_{[\Theta]}$, and assume to the contrary that
$\overline{M}_{[\Theta]}\nsubseteq\alpha(\Lambda)$. By (1),
$\alpha(\Lambda)\subseteq M^{\mathrm{trop}}_{g,n,\nabla}$ is closed, and
therefore $M_{[\Theta]}\nsubseteq\alpha(\Lambda)$. Furthermore, there is a
face $W\in{\mathcal{F}}(\Lambda)$ such that $\alpha(W^{\circ})$ has dimension
$k-1$, $\alpha\left({\rm Star}(W^{\circ})\right)$ has dimension $k$, and
$\alpha(W^{\circ})$ belongs to the relative boundary of $\alpha(\Lambda)$ in
$M_{[\Theta]}$.

After replacing $W$ with one of its faces, we may assume that $\dim(W)=k-1$.
Indeed, since $\dim(\alpha(W^{\circ}))=k-1$, the dimension of $W$ is at least
$k-1$. If it is greater than $k-1$, then the kernel of the differential
$d\alpha$ along $W$ is non-trivial. But $\Lambda$ is the skeleton of a
strictly semistable pair, and therefore $W$ contains no lines. Thus,
$\alpha(W)=\alpha(\partial W)$, and by induction, we can find a face of $W$ of
dimension $k-1$ satisfying the requirements.

By construction, $\alpha\left({\rm Star}(W^{\circ})\right)$ belongs to a half-
space supported by $\alpha(W^{\circ})$ and the differential $d\alpha$ is not
identically zero on ${\rm Star}(W^{\circ})$. Therefore $\alpha$ is not quasi-
harmonic along $W$ contradicting Theorem 3.1 (2).

(3) Let $M_{[\Theta^{\prime}]}$ be a weightless and $3$-valent stratum
adjacent to $M_{[\Theta]}$ that is contained in $\overline{\Sigma}$, and
$M_{[\Theta^{\prime\prime}]}$ be another weightless and $3$-valent stratum
adjacent to $M_{[\Theta]}$. Then there exists a face
$W^{\prime}\in\mathcal{F}(\Lambda)$ and a face $W$ of $W^{\prime}$ of
codimension one such that $\alpha\left((W^{\prime})^{\circ}\right)\subseteq
M_{[\Theta^{\prime}]}$ has dimension $\dim M_{[\Theta^{\prime}]}$, and
$\alpha(W^{\circ})\subseteq M_{[\Theta]}$. Thus, $\alpha$ is locally
combinatorially surjective along $W$ by Theorem 3.1 (3). In particular, there
exists a face $W^{\prime\prime}\in\mathcal{F}(\Lambda)$ containing $W$ as a
codimension-one face such that
$\alpha\left((W^{\prime\prime})^{\circ}\right)\subseteq
M_{[\Theta^{\prime\prime}]}$, and hence

$\dim\alpha\left(W^{\prime\prime}\right)\geq\dim\alpha(W)+1=\dim\left(M_{[\Theta^{\prime\prime}]}\right)$

since $\alpha|_{W^{\prime\prime}}$ is an affine map and
$\alpha(W)\subsetneq\alpha(W^{\prime\prime})$. Therefore,
$\overline{M}_{[\Theta^{\prime\prime}]}\subseteq\overline{\Sigma}$ by (2), and
we are done. ∎

######  Remark 3.3.

The above results, while applicable to the more general case of weightless and
(almost) $3$-valent strata, are conceived with a view towards their
application in [CHT24]. There we will use simple walls and nice strata of
[CHT23], which are weightless and (almost) $3$-valent strata, which in
addition are regular, i.e., of the expected dimension. Two weightless and
(almost) $3$-valent, but not regular, strata were studied already in [CHT22]:

A non-regular, weightless and almost $3$-valent type $\Theta$ is given by a
contracted loop adjacent to the $4$-valent vertex, and no other contracted
edges. In this case, the stratum $M_{[\Theta]}$ has dimension one more than
expected. While Corollary 3.2 (2) does not apply in this case, since the
stratum is not $3$-valent, the claim still holds: since $\alpha$ is not
locally combinatorially surjective at $M_{[\Theta]}$ by the proof of [CHT22,
Lemma 4.6], it is (quasi-)harmonic by Theorem 3.1. This in turn is the only
statement we used to deduce Corollary 3.2 (2).

A non-regular, weightless and $3$-valent type $\Theta$ on the other hand
appears when $\Theta$ is $3$-valent but contains a single flattened cycle as
in [CHT22, Definition 3.7]. In this case, we again have dimension one more
than expected, and $M_{[\Theta]}$ contains curves that are not realizable
[CHT22, Proposition 3.8]. In particular, the condition of local surjectivity
of $\alpha$ in Corollary 3.2 (2) can never be satisfied. However, one can
describe the realizable sublocus in $M_{[\Theta]}$ explicitly, and
(quasi-)harmonicity of $\alpha$ again ensures that if $\alpha$ is locally
surjective at a point in this locus, it is surjective onto the whole
realizable locus in $M_{[\Theta]}$.

The rest of the section is devoted to the proof of Theorem 3.1. Since $K$ is
the algebraic closure of a complete discretely valued field $F$, any
$K$-scheme of finite type is defined over a finite extension $F^{\prime}$ of
$F$, which is also a complete discretely valued field since so is $F$. Thus,
we may view a $K$-scheme of finite type as the base change of a scheme over a
finite extension of $F$. In the proofs below, we will work over $F^{\prime}$
in order to have a well-behaved total space of the families we consider. To
ease the notation, we will assume that all points we are interested in are
defined already over $F$ and integral models – over $F^{0}$. For the rest of
the proof, we fix a uniformizer $\pi\in F^{00}$, and assume without loss of
generality that $\nu(\pi)=1$. In particular, we may assume that $\eta$ and $f$
are defined over $F$ and $({\mathcal{C}}^{0}\rightarrow
B^{0},\sigma^{0}_{\bullet})$ over $F^{0}$.

Throughout the proof, we will use the following notation: $\widetilde{B}$ and
$\widetilde{\mathcal{C}}$ will denote the special fibers,
$s:=\mathrm{red}(\eta)\in\widetilde{B}$ the reduction of $\eta$, and
$\widetilde{\mathcal{C}}_{s}$ the corresponding fiber of ${\mathcal{C}}^{0}\to
B^{0}$. We set $D_{B^{0}}:=\widetilde{B}\cup H$ and
$D_{{\mathcal{C}}^{0}}:=\widetilde{\mathcal{C}}\cup{\mathcal{C}}_{H}\cup\left(\bigcup_{j}\sigma_{j}^{0}\right)$,
where ${\mathcal{C}}_{H}$ is the restriction of ${\mathcal{C}}$ to $H$. Then,
the pullback of any monomial function $f^{*}(x^{m})$ is regular and invertible
on ${\mathcal{C}}^{0}\setminus D_{{\mathcal{C}}^{0}}$ by the assumptions of
the theorem. Recall that $D_{B^{0}}$ has a stratification, and we denote by
$\mathrm{Str}(B^{0},H^{0})$ the set of strata with vertical support.

Let $S\in\mathrm{Str}(B^{0},H^{0})$ be the stratum containing $s$, and $s\in
U\subset B^{0}$ an open neighborhood admitting an étale morphism
$\varphi\colon
U\rightarrow\mathrm{Spec}\left(F^{0}[x_{0},\dotsc,x_{d}]/\langle x_{0}\cdots
x_{a}-\lambda\rangle\right)$ as in Definition 2.11, such that $S\cap U$ is
given by $\varphi^{*}(x_{0})=\cdots=\varphi^{*}(x_{a+b})=0$ for some $0\leq
b\leq d-a$. Then the tropicalization

$\mathrm{trop}(\eta)=\left(\nu\left(\varphi^{*}(x_{0})(\eta)\right),\dotsc,\nu\left(\varphi^{*}(x_{a+b})(\eta)\right)\right)$

is contained in the interior of
$\Delta_{S}=\Delta(a,\lambda)\times\mathbb{R}^{b}_{\geq 0}$. Without loss of
generality we may assume that $\lambda=\pi^{l}$, where $l=\nu(\lambda)$ is the
length of $S$. We denote by $\widetilde{\mathcal{C}}_{S}$ and
$\widetilde{\mathcal{C}}_{\overline{S}}$ the restriction of
$\widetilde{\mathcal{C}}$ to $S$ and $\overline{S}$, respectively. Similarly,
for an irreducible component $D_{i}\subseteq D_{B^{0}}$, given locally by
$\varphi^{*}(x_{i})=0$ for some $0\leq i\leq a+b$, we denote by
${\mathcal{C}}_{D_{i}}$ the restriction of ${\mathcal{C}}^{0}$ to $D_{i}$.
Finally, we denote by $\psi$ the map from the preimage of $U$ in
${\mathcal{C}}^{0}$ to $\mathrm{Spec}\left(F^{0}[x_{0},\dotsc,x_{d}]/\langle
x_{0}\cdots x_{a}-\pi^{l}\rangle\right)$.

###  3.1. Proof of part (1) Theorem 3.1

We need to specify a datum (†) as in Section 2.2.3 that satisfies the
compatibility conditions of Definition 2.7. Since the tropicalizations of
$K$-points give rise only to rational points of $\Lambda$, we will work with
those and extend the family by linearity to the non-rational points in the
very end. To ease the notation, for a face
$W=\Delta_{T}\in{\mathcal{F}}(\Lambda)$ corresponding to a stratum
$T\in\mathrm{Str}(B^{0},H^{0})$, set $\Theta_{T}:=\Theta_{W}$,
${\mathbb{G}}_{T}:={\mathbb{G}}_{W}$, $\ell_{T}:=\ell_{W}$, and
$h_{T}:=h_{W}$. Similarly, if
$W^{\prime}=\Delta_{T^{\prime}}\in{\mathcal{F}}(\Lambda)$ is such that $W$ is
a face of $W^{\prime}$, i.e., $T^{\prime}\leq T$, then set
$\phi_{T^{\prime},T}:=\phi_{W^{\prime},W}$.

#### Step 1: The tropicalization $\mathrm{trop}({\mathcal{C}}_{\eta})$ depends
only on $\mathrm{trop}(\eta)\in\Lambda$, the underlying graph of
$\mathrm{trop}({\mathcal{C}}_{\eta})$ depends only on $S$, and the length of
each edge of $\mathrm{trop}({\mathcal{C}}_{\eta})$ is an integral affine
function over $\Delta_{S}^{\circ}$ which extends to $\Delta_{S}$ in a
compatible way.

By definition, the underlying graph of $\mathrm{trop}({\mathcal{C}}_{\eta})$
is the dual graph $\mathbb{G}_{s}$ of $\widetilde{\mathcal{C}}_{s}$. We first
show that $\mathbb{G}_{s}$ only depends on the stratum $S$. Let $z$ be a node
of $\widetilde{\mathcal{C}}_{s}$. Since ${\mathcal{C}}^{0}$ is pulled back
from $\overline{M}$, there exist an étale neighborhood $V$ of $s$ in $B^{0}$
and a function $g_{z}\in\mathcal{O}_{V}(V)$ vanishing at $s$ such that the
family ${\mathcal{C}}^{0}\times_{B^{0}}V$ is given by $xy=g_{z}$ étale locally
near $z$. After shrinking $V$, we may assume that the latter is true in a
neighborhood of any node of $\widetilde{\mathcal{C}}_{s}$, and that the
morphism $V\to B^{0}$ factors through $U\subseteq B^{0}$. By assumption,
${\mathcal{C}}$ is smooth over $B\backslash H$, hence each $g_{z}$ is
invertible on the complement of $D_{B^{0}}$. However, $V$ is normal, and
$g_{z}(s)=0$ for all nodes $z$, hence each $g_{z}$ must vanish along some
component of $D_{B^{0}}\cap V$ that intersects $S$, and therefore contains
$V\cap S$. Thus, all the $g_{z}$’s vanish identically along $S$. This implies
that the dual graph $\mathbb{G}_{s}$ is étale locally constant along $S$.
Moreover, since $\widetilde{\mathcal{C}}_{S}\rightarrow S$ is split, the
identification of $\mathbb{G}_{s}$ with the dual graph of the fiber over the
generic point of $S$ is canonical. Therefore, $\mathbb{G}_{s}$ only depends on
$S$. Set ${\mathbb{G}}_{S}:={\mathbb{G}}_{s}$. Then for any $S\leq
T\in\mathrm{Str}(B^{0},H^{0})$ there is a natural contraction
$\phi_{S,T}\colon{\mathbb{G}}_{S}\rightarrow{\mathbb{G}}_{T}$ obtained by the
specialization of the curve over the generic point of $T$ to the curve over
the generic point of $S$.

Pick a node $z\in\widetilde{{\mathcal{C}}}_{s}$. We claim that there exist
integers $n_{i}$ and $n_{\pi}$ such that

$g_{z}\cdot\pi^{-n_{\pi}}\cdot\prod_{i=0}^{a+b}\varphi^{*}(x_{i})^{-n_{i}}$
(3.1)

is regular and invertible along $S$. Here, by abuse of notation, we denote by
$\varphi^{*}(x_{i})$ the pullback of the monomial $x_{i}$ to $V$ rather than
$U$. Consider the log-structure on ${\rm Spec}(F^{0})$ given by the
uniformizer $\pi$, the monomial log-structure on
$\mathrm{Spec}(F^{0}[x_{0},\dotsc,x_{d}]/\langle x_{0}\cdot\ldots\cdot
x_{a}-\pi^{l}\rangle)$ given by $\pi$ and the monomials $x_{i}$’s, and the
monomial log-structure on $V$ given by the pullback of the monomial log-
structure on $\mathrm{Spec}(F^{0}[x_{0},\dotsc,x_{d}]/\langle
x_{0}\cdot\ldots\cdot x_{a}-\pi^{l}\rangle)$. Then

$\mathrm{Spec}(F^{0}[x_{0},\dotsc,x_{d}]/\langle x_{0}\cdot\ldots\cdot
x_{a}-\pi^{l}\rangle)\to{\rm Spec}(F^{0})$

is log-smooth by [Kat94, §8.1] and
$V\to\mathrm{Spec}(F^{0}[x_{0},\dotsc,x_{d}]/\langle x_{0}\cdot\ldots\cdot
x_{a}-\pi^{l}\rangle)$ log-étale by [Kat89, Proposition 3.8]. Since ${\rm
Spec}(F^{0})$ is log-regular, so is $V$ by [Kat94, §8.2]. Now, since $V$ is
log-regular, the monomial log-structure is given by the pushforward of the
sheaf of invertible functions on the complement of $D_{B^{0}}$ by [Kat94,
Theorem 11.6]. Finally, since $g_{z}$ is invertible away from $D_{B^{0}}$, it
follows that the function $g_{z}$ is a monomial in $\pi$ and
$\varphi^{*}(x_{i})$’s up-to an invertible function as asserted. Proofs of
similar claims in slightly different settings can be found in [Wło22, Lemma
4.8.2] and [Gub07, Proposition 2.11].

Next, we compute the lengths of the edges of ${\mathbb{G}}_{s}$ in
$\mathrm{trop}({\mathcal{C}}_{\eta})$. By (3.1), the length of $\gamma\in
E(\mathbb{G}_{s})$ corresponding to $z$ is given by

$\ell(\gamma)=\nu(g_{z}(\eta))=n_{\pi}+\sum_{i=0}^{a+b}n_{i}\cdot\nu\left(\varphi^{*}(x_{i})(\eta)\right)=\sum_{i=0}^{a+b}k_{i}\cdot\nu\left(\varphi^{*}(x_{i})(\eta)\right),$

where $k_{i}:=n_{i}+\frac{1}{l}n_{\pi}=\frac{1}{l}\mathrm{ord}_{D_{i}}(g_{z})$
if $i\leq a$ and $k_{i}:=n_{i}=\mathrm{ord}_{D_{i}}(g_{z})$ otherwise. We
claim that the $k_{i}$’s are independent of the choice of $s\in S$. Indeed,
for another $s^{\prime}\in S$, the corresponding $g_{z}$ and $g^{\prime}_{z}$
differ by an invertible function on the intersection of the corresponding
étale neighborhoods of $s$ and $s^{\prime}$, and hence also at the generic
points of the $D_{i}$’s for $0\leq i\leq a+b$. Thus,
$\mathrm{ord}_{D_{i}}(g_{z})=\mathrm{ord}_{D_{i}}(g^{\prime}_{z})$ and
therefore $k_{i}=k^{\prime}_{i}$ as claimed. Furthermore, we see that the
length function $\ell(\gamma)$ depends only on the tropicalization
$q:=\mathrm{trop}(\eta)=\left(\nu\left(\varphi^{*}(x_{0})(\eta)\right),\dotsc,\nu\left(\varphi^{*}(x_{a+b})(\eta)\right)\right)$
rather than the point $\eta$ itself.

Set $y_{i}:=\nu\left(\varphi^{*}(x_{i})(\eta)\right)$. Then,
$q=(y_{0},\dotsc,y_{a+b})$ and the length function

$\ell_{S}(\gamma,q)=n_{\pi}+\sum_{i=0}^{a+b}n_{i}y_{i}=\sum_{i=0}^{a+b}k_{i}y_{i}$

is an integral affine function on $\Delta_{S}^{\circ}$ that extends naturally
to $\Delta_{S}$. It remains to show that $\ell_{S}(\gamma,q)$ satisfies the
compatibility condition of Definition 2.7 (2). Note that $g_{z}$ is defined
over an étale open subset of $B^{0}$ which intersects all strata
$T\in\mathrm{Str}(B^{0},H^{0})$ satisfying $S\leq T$. Let $T$ be the stratum
given locally by

$\varphi^{*}(x_{0})=\cdots=\varphi^{*}(x_{a^{\prime}})=0=\varphi^{*}(x_{a+1})=\cdots=\varphi^{*}(x_{a+b^{\prime}}),$

for some $a^{\prime}\leq a$ and $b^{\prime}\leq b$. Then $\Delta_{T}$ is the
face of $\Delta_{S}$ given by

$y_{a^{\prime}+1}=\cdots=y_{a}=y_{a+b^{\prime}+1}=\cdots=y_{a+b}=0.$

If the node $z$ is smoothed out in the family of curves over $T$, then $g_{z}$
is invertible at the generic point of $D_{i}$ with $0\leq i\leq a^{\prime}$ or
$a+1\leq i\leq a+b^{\prime}$, and the edge $\gamma$ gets contracted to a
vertex in ${\mathbb{G}}_{T}$. Hence $k_{i}=0$ for $0\leq i\leq a^{\prime}$ or
$a+1\leq i\leq a+b^{\prime}$ and the compatibility holds:

$\ell_{T}(\phi_{S,T}(\gamma),q)=0=\ell_{S}(\gamma,q),\ \mathrm{for\ all}\
q\in\Delta_{T}.$

Assume now that $z$ is the specialization of a node of the curve over the
generic point of $T$. Then $\gamma$ is not contracted by $\phi_{S,T}$. In this
case, again for any $s^{\prime}\in T$, the corresponding $g^{\prime}_{z}$ and
$g_{z}$ differ by a function invertible at the generic points of all $D_{i}$
that contain $T$, namely, with $0\leq i\leq a^{\prime}$ or $a+1\leq i\leq
a+b^{\prime}$. Therefore,
$\mathrm{ord}_{D_{i}}(g_{z})=\mathrm{ord}_{D_{i}}(g^{\prime}_{z})$ for those
$D_{i}$’s, and hence the restriction of $\ell_{S}(\gamma,\cdot)$ to
$\Delta_{T}$ coincides with $\ell_{T}(\phi_{S,T}(\gamma),\cdot)$ as needed.

#### Step 2: The parameterization
$h_{\eta}\colon\mathrm{trop}({\mathcal{C}}_{\eta})\rightarrow N_{\mathbb{R}}$
depends only on $\mathrm{trop}(\eta)$, and varies integral affine linearly on
$\Delta_{S}^{\circ}$ with constant combinatorial type. It also extends to
$\Delta_{S}$ in a compatible way.

Since each section $\sigma^{0}_{\bullet}$ of the family
${\mathcal{C}}\rightarrow B^{0}$ is mapped to a unique toric divisor or the
dense orbit in $X$, the slopes of the legs of
$\mathrm{trop}({\mathcal{C}}_{\eta})$ are independent of $\eta$ by Remark
2.10. Therefore, the extended degree of
$h_{\eta}\colon\mathrm{trop}({\mathcal{C}}_{\eta})\rightarrow N_{\mathbb{R}}$
is independent of $\eta$ and will be denoted by $\overline{\nabla}$.

Recall that the closure $\overline{S}$ of $S$ is a component of the
intersection of $D_{0},\dotsc,D_{a+b}$. Let $u$ be a vertex of
${\mathbb{G}}_{S}$. Denote by $\widetilde{\mathcal{C}}_{u}$ the component of
$\widetilde{\mathcal{C}}_{S}$ corresponding to $u$ and by
$\widetilde{\mathcal{C}}_{s,u}$ the component of $\widetilde{\mathcal{C}}_{s}$
corresponding to $u$. As in Step 1, let
$q=(y_{0},\dotsc,y_{a+b}):=\mathrm{trop}(\eta)$. Let
$\widetilde{\mathcal{C}}_{u}^{\prime}\subset\widetilde{\mathcal{C}}_{u}$ be
the open subset of non-special points. Then in a neighborhood of
$\widetilde{\mathcal{C}}_{u}^{\prime}$, the toroidal structure on
$\mathscr{C}^{0}$ is pulled back from the one on $B^{0}$. Therefore, as in
Step 1, for any $m\in M$, there exist integers $n_{i}$ for $0\leq i\leq a+b$
and $n_{\pi}$ such that

$f^{*}(x^{m})\cdot\pi^{-n_{\pi}}\cdot\prod_{0\leq i\leq
a+b}\psi^{*}(x_{i})^{-n_{i}}$

is regular and invertible along $\widetilde{\mathcal{C}}_{u}^{\prime}$. Hence,
by definition, the inner product of $h_{\eta}(u)\in N_{\mathbb{R}}$ with $m$
is given by

$-\mathrm{ord}_{\widetilde{\mathcal{C}}_{s,u}}\left(f^{*}(x^{m})|_{{\mathcal{C}}_{\overline{\eta}}}\right)=-n_{\pi}-\sum_{i=0}^{a+b}n_{i}\cdot\nu\left(\varphi^{*}(x_{i})(\eta)\right)=-n_{\pi}-\sum_{i=0}^{a+b}n_{i}y_{i},$

which only depends on $q=\mathrm{trop}(\eta).$ Let us write
$h_{S}(u,q)=h_{\eta}(u)$, then $h_{S}(u,\cdot)$ is an integral affine function
on $\Delta_{S}$. By a similar argument as in Step 1, we see that this function
satisfies the compatibility of Definition 2.7 (3).

As the parameterization $h_{\eta}$ varies continuously with respect to
$q=\mathrm{trop}(\eta)$, while the set of combinatorial types with underlying
graph ${\mathbb{G}}_{S}$ is a discrete set, we see that the combinatorial type
of $h_{\eta}$ is constant, which we denote by $\Theta_{S}$.

#### Step 3: The conclusion of the proof of (1).

In the first two steps, we have constructed the data of a family of
parameterized topical curve over $\Lambda$ \- the extended degree, the
combinatorial type for each face of $\Lambda$, the weighted contractions, and
the integral affine functions $\ell_{S}$ and $h_{S}$. Furthermore, we have
verified the compatibility conditions of Definition 2.7, and have seen that
the resulting family of parameterized topical curves over $\Lambda$ satisfies
assertion (1).

### Proof of Part (2) of Theorem 3.1

Denote by $\chi\colon B^{0}\rightarrow\overline{M}$ the map inducing the
family ${\mathcal{C}}^{0}$. Let $S\in\mathrm{Str}(B^{0},H^{0})$ be the stratum
corresponding to $W$. Then the closure $\overline{S}$ of $S$ is a proper and
smooth variety. Denote by $\Theta$ the combinatorial type of
$\alpha(W^{\circ})$. There are two cases to consider:

Case 1: $\chi$ contracts $\overline{S}$. In this case we will prove that
$\alpha$ is harmonic along $W$ if $W$ has codimension one and quasi-harmonic
along $W$ otherwise. Set $p:=\chi(\overline{S})$ and let $C_{p}$ be the fiber
of the universal family over $p$. Since ${\mathcal{C}}^{0}$ is a pullback of
the universal family along $\chi$, it follows that
$\widetilde{\mathcal{C}}_{\overline{S}}=\overline{S}\times C_{p}$.
Furthermore, ${\mathbb{G}}_{S}$ is the dual graph of $C_{p}$, and $\alpha$
maps ${\rm Star}(W^{\circ})$ to $M_{[\Theta]}$, which lifts to a map to
$M_{\Theta}\subset{\mathbb{R}}^{|E({\mathbb{G}}_{S})|}\times
N_{\mathbb{R}}^{|V({\mathbb{G}}_{S})|}$. By abuse of notation, the lifting is
also denoted by $\alpha$.

Recall that each point $s\in S$ admits an open neighborhood $s\in U\subset
B^{0}$ and an étale morphism $\varphi\colon
U\rightarrow\mathrm{Spec}\left(F^{0}[x_{0},\dotsc,x_{d}]/\langle x_{0}\cdots
x_{a}-\pi^{l}\rangle\right)$ such that $S\cap U$ is given by
$\varphi^{*}(x_{0})=\cdots=\varphi^{*}(x_{a+b})=0$ for some $0\leq b\leq d-a$,
and hence $W=\Delta(a,\pi^{l})\times\mathbb{R}^{b}_{\geq 0}$. Let
$v_{0},\dotsc,v_{a}$ be the vertices of $W$. For $a+1\leq i\leq a+b$, let
$\vec{e}_{i}$ be the standard basis of $\mathbb{R}^{b}$ and set
$v_{i}=v_{0}+\vec{e}_{i}$. Then ${\rm Lin}(\alpha(W))$ is generated by
$\alpha(v_{i})-\alpha(v_{0})$ for $1\leq i\leq a+b$.

Let $\\{S_{i}\\}$, where $a+b+1\leq i\leq a+b+r+t$, be the set of strata of
codimension one in $\overline{S}$. Each $S_{i}$ corresponds to a face
$W_{i}\in\mathcal{F}(\Lambda)$ containing $W$ as a codimension-one face.
Suppose $\overline{S}_{i}=\overline{S}\cap D_{i}$ for some component $D_{i}$
of $D_{B^{0}}$. We may further assume that $D_{i}$ is vertical for $a+b+1\leq
i\leq a+b+r$, and horizontal for $a+b+r+1\leq i\leq a+b+r+t$. Note that the
$D_{i}$’s are uniquely determined by the $S_{i}$’s but may not necessarily be
distinct. In the sequel, we will mainly consider the following components of
$D_{B^{0}}$, which have non-empty intersection with $\overline{S}$: the
components $D_{0},\dotsc,D_{a+b}$, which are distinct and contain
$\overline{S}$; and the components $D_{a+b+1},\dotsc,D_{a+b+r+t}$ constructed
above.

For $a+b+1\leq i\leq a+b+r$, let $l_{i}$ be the length of $S_{i}$ and
$v^{\prime}_{i}$ be the vertex of $W_{i}$ not contained in $W$. In fact,
$l_{i}=l$ unless $a=0$. Let
$\vec{e}_{i}=\frac{1}{l_{i}}(v^{\prime}_{i}-v_{0})$ be the primitive integral
vector parallel to $v^{\prime}_{i}-v_{0}$, and set $v_{i}:=v_{0}+\vec{e}_{i}$.
For $a+b+r+1\leq i\leq a+b+r+t$ we have $W_{i}=W\times{\mathbb{R}}_{\geq 0}$.
In this case, set $v_{i}:=v_{0}+\vec{e}_{i}$, where $\vec{e}_{i}$ is the unit
normal vector to $W$ in $W_{i}$. Then $W_{i}$ is contained in
$W+{\mathbb{R}}_{\geq 0}\vec{e}_{i}$ for $a+b+1\leq i\leq a+b+r+t$.
Consequently, we have ${\rm Star}(W)=\\{\vec{e}_{i}+N_{W}\\}_{a+b+1\leq i\leq
a+b+r+t}$ and
$\frac{\partial\alpha}{\partial\vec{e}_{i}}=\alpha(v_{i})-\alpha(v_{0})$.
Thus, to prove quasi-harmonicity (resp. harmonicity), it remains to show that

$\sum_{i=a+b+1}^{a+b+r+t}a_{i}\left(\alpha(v_{i})-\alpha(v_{0})\right)\in{\rm
Lin}(\alpha(W))$ (3.2)

for some positive integers $a_{i}$ (resp. $a_{i}=1$).

Since $\overline{S}$ is étale locally isomorphic to an affine space and the
$\overline{S}_{i}$’s correspond to the coordinate hyperplanes, we can pick a
(possibly reducible) curve $C$ in $\overline{S}$ that has positive
intersection numbers with each $\overline{S}_{i}$ and none of whose components
is contained in $\cup\overline{S}_{i}$. In particular, when $\dim S=1$
(equivalently, $W$ has codimension one in $\Lambda$), we set
$C:=\overline{S}$. Set $l_{i}:=l$ for $i\leq a$, and $l_{i}:=1$ for $a+1\leq
i\leq a+b$ and for $i>a+b+r$. Then (3.2) would follow from the following
equality

$\sum_{i=a+b+1}^{a+b+r+t}\overline{S}_{i}\cdot C\
\big{(}\alpha(v_{i})-\alpha(v_{0})\big{)}=-\sum_{i=0}^{a+b}l_{i}D_{i}\cdot C\
\big{(}\alpha(v_{i})-\alpha(v_{0})\big{)},$ (3.3)

where the intersection product on the left is in $\overline{S}$, while the one
on the right is in $B^{0}$. Note that when $\dim S=1$ we have
$\overline{S}_{i}\cdot C=1$ by construction.

We now prove (3.3) coordinatewise. Let $\gamma\in E({\mathbb{G}}_{S})$ be an
edge corresponding to a node $z\in C_{p}$. Let $u,u^{\prime}\in
V({\mathbb{G}}_{S})$ be the vertices adjacent to $\gamma$, let $C_{u}$ and
$C_{u^{\prime}}$ be the corresponding components of $C_{p}$, and
$\widetilde{{\mathcal{C}}}_{u},\widetilde{{\mathcal{C}}}_{u^{\prime}},Z$ the
pullbacks of $C_{u},C_{u^{\prime}},z$ to
$\widetilde{{\mathcal{C}}}_{\overline{S}}$, respectively. The universal curve
over $\overline{M}$ is given étale locally at $z$ by $xy=m_{z}$, where $m_{z}$
is defined on an étale neighborhood of $p$, and vanishes at $p$. Thus,
${\mathcal{C}}^{0}$ is given by $xy=g_{Z}:=\chi^{*}(m_{z})$ in an étale
neighborhood of $Z$. Notice that we constructed a function defined in a
neighborhood of the whole family of nodes $Z$, which globalizes the local
construction of Step 1 in the particular case we consider here.

According to Step 1, the $\gamma$-coordinate of $\alpha(v_{i})-\alpha(v_{0})$
is equal to $\frac{1}{l_{i}}\left({\rm ord}_{D_{i}}(g_{Z})-{\rm
ord}_{D_{0}}(g_{Z})\right)$ when $D_{i}$ is vertical, namely $0\leq i\leq a$
or $a+b+1\leq i\leq a+b+r$; and equal to ${\rm ord}_{D_{i}}(g_{Z})$ otherwise.
Set $k_{i}:=\frac{1}{l_{i}}{\rm ord}_{D_{i}}(g_{Z})$.

Then the $\gamma$-coordinate of the left-hand side of (3.3) is given by

$\sum_{i=a+b+1}^{a+b+r+t}k_{i}\overline{S}_{i}\cdot
C-\sum_{i=a+b+1}^{a+b+r}k_{0}\frac{l}{l_{i}}\overline{S}_{i}\cdot C;$

while on the right-hand side it is given by

$-\sum_{i=0}^{a+b}l_{i}k_{i}D_{i}\cdot C+\sum_{i=0}^{a}l_{0}k_{0}D_{i}\cdot
C=-\sum_{i=0}^{a+b}l_{i}k_{i}D_{i}\cdot C+\sum_{i=0}^{a}lk_{0}D_{i}\cdot C.$

Therefore, (3.3) is equivalent to

$\sum_{i=0}^{a+b}l_{i}k_{i}D_{i}\cdot
C+\sum_{i=a+b+1}^{a+b+r+t}k_{i}\overline{S}_{i}\cdot
C=lk_{0}\Big{(}\sum_{i=0}^{a}D_{i}\cdot
C+\sum_{i=a+b+1}^{a+b+r}\frac{1}{l_{i}}\overline{S}_{i}\cdot C\Big{)},$ (3.4)

which holds true since both sides vanish. Indeed, since $\overline{S}_{i}\cdot
C=l_{i}D_{i}\cdot C$ for all $i>a+b$, the left-hand side of (3.4) is
$\mathrm{div}(g_{Z})\cdot C$, which vanishes since $C$ is complete, while the
right-hand side is $k_{0}l\widetilde{B}\cdot C=0$ since $\widetilde{B}$ is a
principal divisor of $B^{0}$ defined by $\pi$.

Next we verify (3.3) for the $u$-coordinates. Pick $u\in V({\mathbb{G}}_{S})$.
For $1\leq i\leq a+b+r+t$, let ${\mathcal{C}}_{i}$ be the irreducible
component of ${\mathcal{C}}_{D_{i}}$ that contains some fiber of
$\widetilde{\mathcal{C}}_{u}$. Denote $k^{\prime}_{i}=\frac{1}{l_{i}}{\rm
ord}_{{\mathcal{C}}_{i}}f^{*}(x^{m})$. In this case, by taking inner product
of both sides of (3.3) with an arbitrary $m\in M$, a similar computation as
above shows that (3.3) is equivalent to

$\sum_{i=0}^{a+b}l_{i}k^{\prime}_{i}D_{i}\cdot
C+\sum_{i=a+b+1}^{a+b+r+t}k^{\prime}_{i}\overline{S}_{i}\cdot
C=lk^{\prime}_{0}\Big{(}\sum_{i=0}^{a}D_{i}\cdot
C+\sum_{i=a+b+1}^{a+b+r}\frac{1}{l_{i}}\overline{S}_{i}\cdot C\Big{)}.$

Take a section $\widetilde{\sigma}_{S}$ of
$\widetilde{\mathcal{C}}_{u}\rightarrow\overline{S}$ that is disjoint with any
section $\sigma^{0}_{\bullet}$ of ${\mathcal{C}}^{0}$ and the singular locus
of ${\mathcal{C}}^{0}$, and let $\widetilde{\sigma}_{C}$ (resp.
$\widetilde{\sigma}_{i}$) be the restriction of $\widetilde{\sigma}_{S}$ on
$C$ (resp. $\overline{S}_{i}$). By projection formula, we have $D_{i}\cdot
C={\mathcal{C}}_{i}\cdot\widetilde{\sigma}_{C}$. Then (3.3) is equivalent to

$\sum_{i=0}^{a+b}l_{i}k^{\prime}_{i}\mathscr{C}_{i}\cdot\widetilde{\sigma}_{C}+\sum_{i=a+b+1}^{a+b+r+t}k^{\prime}_{i}\widetilde{\sigma}_{i}\cdot\widetilde{\sigma}_{C}=lk^{\prime}_{0}\Big{(}\sum_{i=0}^{a}D_{i}\cdot
C+\sum_{i=a+b+1}^{a+b+r}\frac{1}{l_{i}}\overline{S}_{i}\cdot C\Big{)},$

and again, the latter equality holds true since both sides vanish. Indeed, the
left-hand side is now the intersection
$\widetilde{\sigma}_{C}\cdot\mathrm{div}f^{*}(x^{m})$ and the right-hand side
is $lk^{\prime}_{0}\widetilde{B}\cdot C$. This concludes the proof of (3.3)
and of Case 1.

Case 2: $\chi$ does not contract $\overline{S}$. By the assumptions of the
theorem, the graph ${\mathbb{G}}_{S}$ is weightless and $3$-valent except for
at most one $4$-valent vertex. Since rational curves with three marked points
have no moduli, it follows that ${\mathbb{G}}_{S}$ has a $4$-valent vertex,
which we denote by $u\in{\mathbb{G}}_{S}$. In this case we will show that the
map $\alpha$ is locally combinatorially surjective at $W$. Consider
$\widetilde{\mathcal{C}}_{u}$ as above. By construction,
$\widetilde{\mathcal{C}}_{u}\to\overline{S}$ is a family of rational curves
and since $\widetilde{\mathcal{C}}\to\widetilde{B}$ is split, the four marked
points in each fiber define sections of the family. Let
$\xi\colon\overline{S}\to\overline{M}_{0,4}\simeq{\mathbb{P}}^{1}$ be the
induced map. Since $\chi$ does not contract $\overline{S}$, it follows that
$\xi$ is not constant, and hence surjective. Thus, the preimage of each
boundary point of $\overline{M}_{0,4}$ in $\overline{S}$ is the union of
closures of some codimension-one strata $S^{\prime}$. Such strata correspond
to faces $W^{\prime}\in\mathcal{F}(\Lambda)$ containing $W$ as a codimension-
one face. Since $\xi$ is surjective, it follows that for each weightless and
$3$-valent types $\Theta^{\prime}$ corresponding to one of the three possible
resolutions of the $4$-valent vertex $u$, there exists a face $W^{\prime}$ as
above for which $\alpha(W^{\prime})\cap M_{[\Theta^{\prime}]}\neq\emptyset$.
Therefore $\alpha$ is locally combinatorially surjective along $W$ as claimed.
∎

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