# The moduli space of quasistable spin curves

Alex Abreu, Marco Pacini, and Danny Taboada

###### Abstract.

We study a compactification of the moduli space of theta characteristics,
giving a modular interpretation of the geometric points and describing the
boundary stratification. This space is different from the moduli space of spin
curves. The modular description and the boundary stratification of the new
compactification are encoded by a tropical moduli space. We show that this
tropical moduli space is a refinement of the moduli space of spin tropical
curves. We describe explicitly the induced decomposition of its cones.

The second author was supported by CNPq-PQ, 301671/2019-2. The third author
was supported by Capes.

MSC (2010): 14T05, 14H10, 14H40

Key words: spin curve, spin tropical curve, moduli spaces, refinement.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Tropical moduli spaces
  4. 4 Root-graphs and their categories
  5. 5 Moduli of tropical theta characteristics
  6. 6 Moduli of spin quasistable roots
  7. 7 Moduli of quasistable spin curves

##  1\. Introduction

###  1.1. Background and motivation

The goal of this paper is the study of a compactification, denoted
$\overline{\mathcal{Q}}_{g,n}$, of the moduli space $\mathcal{S}_{g,n}$ of
theta characteristics on $n$-pointed smooth curves. A distinguished
compatification, $\overline{\mathcal{S}}_{g,n}$, of $\mathcal{S}_{g,n}$ over
the moduli space $\overline{\mathcal{M}}_{g,n}$ of $n$-pointed stable curves
was introduced by Cornalba in [17]. The stack $\overline{\mathcal{S}}_{g,n}$
is finite over $\overline{\mathcal{M}}_{g,n}$ and parametrizes _spin curves_ ,
which are certain pairs of nodal curve and invertible sheaves on it. The
construction was later generalized in [9] for higher roots of any sheaf. There
are at least two other different approaches to this compactification problem.
The first uses torsion-free rank-1 sheaves on stable curves and is introduced
by Jarvis (see [21] and [22]). The second involves the use of twisted curves
(nodal curves with a stacky structure at the nodes) and is employed by
Abramovich and Jarvis in [1] and by Chiodo in [15]. All these
compactifications of $\mathcal{S}_{g,n}$ are isomorphic to
$\overline{\mathcal{S}}_{g,n}$.

We define $\overline{\mathcal{Q}}_{g,n}$ as the closure of $\mathcal{S}_{g,n}$
in the Esteves universal compactified Jacobian of quasistable torsion-free
rank-1 sheaves on stable curves. We call $\overline{\mathcal{Q}}_{g,n}$ the
moduli space of _quasistable spin curves_. We give a modular description of
the geometric points of $\overline{\mathcal{Q}}_{g,n}$ and study its boundary
stratification. The upshot is that $\overline{\mathcal{Q}}_{g,n}$ is a
compactification of $\mathcal{S}_{g,n}$ which is not isomorphic to
$\overline{\mathcal{S}}_{g,n}$. In a way, this is a natural reflection of the
existence of two distinguished and different universal Jacobians: Caporaso
universal Picard variety, constructed in [7], and the Esteves universal
compactified Jacobian, introduced by Melo in [24] bulding upon the work of
Esteves [18]. These compactified Jacobians are different. For instance, using
the terminology in [8], the first one, in general, is not of Néron type, while
the second one is always of Néron type. Caporaso compactified Jacobian
contains $\overline{\mathcal{S}}_{g,n}$ (at least set-theoretically), so it is
natural to expect a difference between $\overline{\mathcal{Q}}_{g,n}$ and
$\overline{\mathcal{S}}_{g,n}$.

We achieve our results by resorting to the tropical counterpart of the
problem. In the last few years, tropical geometry became a powerful tool to
encode moduli problems and describe the boundary stratifications of
compactified moduli spaces. The tropical setting often helps in understanding
which combinatorial data are relevant to describe the strata. The associated
tropical moduli space is a generalized cone complex which can be regarded as a
“dual” of the algebro-geometric space. Usually, we have a reverse-inclusion
bijection between the strata of the moduli space and the cells of the cone
complex switching dimension and codimension.

This phenomenon appears in the study of $\overline{\mathcal{M}}_{g,n}$, the
moduli space of $n$-pointed stable curves, and of its tropical counterpart
$M_{g,n}^{\textnormal{trop}}$, the moduli space of $n$-pointed tropical
curves. In fact, a deep connection between $\overline{\mathcal{M}}_{g,n}$ and
$M_{g,n}^{\textnormal{trop}}$ was revealed in [5], where Abramovich, Caporaso,
and Payne showed that $\overline{M}_{g,n}^{\textnormal{trop}}$ is identified
with the Berkovich skeleton of $\overline{\mathcal{M}}_{g,n}$. In [11],
Caporaso, Melo, and the second author, did a similar analysis for the moduli
space of spin curves $\overline{\mathcal{S}}_{g,n}$. It is worth mentioning
that the new insight of the boundary stratification given by tropical geometry
was recently used in [13] and [14] to give strong results about the rational
cohomology of $\mathcal{M}_{g}$ and the $S_{n}$-equivariant Euler
characteristic of the top weight cohomology of $\mathcal{M}_{g,n}$.

We proceed in a similar way: we define the relevant tropical moduli space and
we use it to describe the geometric points and the boundary stratification of
$\overline{\mathcal{Q}}_{g,n}$.

###  1.2. Outline of the results

Let us give more details about our results. We start with the tropical
setting. In [2] and [4] two universal tropical Jacobians have been introduced.
The first one, denoted $J^{\textnormal{trop}}_{\mu,g,n}$, parametrizes
$(\mu,p_{0})$-quasistable divisors on tropical curves of genus $g$. The second
one, denoted $P^{\textnormal{trop}}_{\mu,g,n}$, parametrizes $\mu$-polystable
divisors on tropical curves of genus $g$. In both cases, $\mu$ denotes a
universal polarization. The two universal tropical Jacobians are different. In
fact, in Theorem 3.7, we show that $J^{\textnormal{trop}}_{\mu,g,n}$ is a
refinement of $P^{\textnormal{trop}}_{\mu,g,n}$, and we describe the induced
decomposition of the cones of $P^{\textnormal{trop}}_{\mu,g,n}$. This
generalizes to the universal setting a similar result for the Jacobian of a
tropical curve, proved in [4, Proposition 3.7] and [16, Corollay 5.10].

Motivated by the algebro-geometric picture, we can consider the locus inside
the tropical Jacobian parametrizing square roots of the canonical divisor.
This leads to a notion of _tropical theta characteristics_ of a tropical
curve, first studied by Zharkov in [25]. A moduli space,
$T_{g,n}^{\textnormal{trop}}$, of theta characteristics on $n$-pointed
tropical curves of genus $g$, is constructed in [12, Section 4], based on the
existence of certain representatives of theta characteristics that are well-
behaved under specializations. In Proposition 5.18 we show that in the
equivalence class of a theta characteristic there is exactly one
$\mu$-polystable divisor, where $\mu$ is the canonical polarization of degree
$g-1$. It is possible to write explicitly this distinguished representative
(see equation (20)). This gives rise to an explicit injection
$T_{g,n}^{\textnormal{trop}}\hookrightarrow P^{\textnormal{trop}}_{\mu,g,n}$.

However $T^{\textnormal{trop}}_{g,n}$ is not the tropical counterpart of the
moduli space $\overline{\mathcal{S}}_{g,n}$ of spin curves. In fact,
$T^{\textnormal{trop}}_{g,n}$ is connected, since all tropical theta
characteristic admits a specialization to the unique theta characteristic on
the tropical curve consisting of one vertex, while the stack
$\overline{\mathcal{S}}_{g,n}$ is not (it has two connected components,
corresponding to the parity of the space of sections of a spin curve). The
right tropical counterpart of $\overline{\mathcal{S}}_{g,n}$ is the space
$S^{\textnormal{trop}}_{g,n}$ of spin tropical curves constructed in [11],
parametrizing tropical theta characteristics together with the datum of a sign
function. We have a forgetful map $S^{\textnormal{trop}}_{g,n}\rightarrow
T^{\textnormal{trop}}_{g,n}$, and hence a chain of maps

$S^{\textnormal{trop}}_{g,n}\rightarrow
T^{\textnormal{trop}}_{g,n}\hookrightarrow P^{\textnormal{trop}}_{\mu,g,n}.$

The idea is to pull-back the refinement
$J^{\textnormal{trop}}_{\mu,g,n}\rightarrow P^{\textnormal{trop}}_{\mu,g,n}$
of $P^{\textnormal{trop}}_{\mu,g,n}$ to $S^{\textnormal{trop}}_{g,n}$ via the
chain of maps above. We look for a generalized cone complex,
$Q^{\textnormal{trop}}_{g,n}$, whose associated poset could encode the data of
the induced refinement of $S^{\textnormal{trop}}_{g,n}$. This amounts to
impose the right condition on the poset associated to
$P^{\textnormal{trop}}_{\mu,g,n}$. The last one is the poset of triples
$(\Gamma,\mathcal{E},D)$ where $\Gamma$ is a stable graph of genus $g$ with
$n$ legs, $\mathcal{E}$ is a subset of the set of edges of $\Gamma$, and $D$
is a $\mu$-polystable divisor on the graph $\Gamma^{\mathcal{E}}$ obtained
from $\Gamma$ by inserting a vertex in the interior of any edge
$e\in\mathcal{E}$. The key condition is:

(1) $2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}},$

where $\operatorname{Div}(\phi)$ is the divisor associated to some acyclic
flow $\phi$ on $\Gamma^{\mathcal{E}}$ and $K_{\Gamma^{\mathcal{E}}}$ is the
canonical divisor of $\Gamma^{\mathcal{E}}$. The flow $\phi$ is uniquely
determined if $D$ is $\mu$-semistable and plays an important role for us. We
refer to Theorem 4.7 for more properties of the flow $\phi$ (for example,
$|\phi(e)|\leq 1$ for every oriented edge $e$ of $\Gamma^{\mathcal{E}}$).
Using this terminology, the poset underlying $T^{\textnormal{trop}}_{g,n}$ is
given by the triples $(\Gamma,\mathcal{E},D)$, where $D$ is $\mu$-polystable
and satisfies equation (1). We can also reinterpret the poset underlying
$S^{\textnormal{trop}}_{g,n}$ simply adding a sign function to these triples.
The poset associated to $Q^{\textnormal{trop}}_{g,n}$ is formed by tuples
$(\Gamma,\mathcal{E},D,s)$ where $D$ is $\mu$-quasistable and satisfies
equation (1), and $s$ is a sign function. An important step in the
construction of the generalized cone complex $Q^{\textnormal{trop}}_{g,n}$ is
the explicit description of the equations of its cones. These are obtained by
intersecting the cone $\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{\geq 0}$ with the
linear subspace:

$\Lambda_{(\Gamma,\mathcal{E},D,s)}\colon\sum_{e\in\gamma}\phi(e)x_{e}=0$

where $\gamma$ runs over the oriented cycles of $\Gamma^{\mathcal{E}}$ and
$x_{e}$ is the coordinates of $\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$
corresponding to an edge $e$ of $\Gamma^{\mathcal{E}}$ (see equation (11)).
The following result is Theorem 6.9.

######  Theorem A.

We have a refinement $Q_{g,n}^{\textnormal{trop}}\rightarrow
S_{g,n}^{\textnormal{trop}}$ of generalized cone complexes. There is an
explicit description of the associated decomposition of the cones of
$S_{g,n}^{\textnormal{trop}}$, and of their interiors.

Next, we come back to algebraic geometry. The moduli space,
$\mathcal{S}_{g,n}$, of theta characteristics on $n$-pointed smooth curves of
genus $g$ naturally sits inside $\overline{\mathcal{J}}_{\mu,g,n}$, the
Esteves universal compactified Jacobian. As we already mentioned, we define
$\overline{\mathcal{Q}}_{g,n}$ as the closure of $\mathcal{S}_{g,n}$ in
$\overline{\mathcal{J}}_{\mu,g,n}$. For our purposes, it is convenient to
interpret $\overline{\mathcal{J}}_{\mu,g,n}$ as the moduli space of pair
$(X,L)$, where $X$ is a quasistable curve and $L$ is a
$(\mu,p_{0})$-quasistable invertible sheaf on $X$ with degree $-1$ on
exceptional components (see Proposition 7.1). The tropical setting suggests a
modular interpretation of $\overline{\mathcal{Q}}_{g,n}$. In fact, given a
triple $(\Gamma,\mathcal{E},D)$ satisfying equation (1), we can consider the
subgraph $P_{(\mathcal{E},D)}$ of $\Gamma^{\mathcal{E}}$ over which $\phi$ is
zero. If $X$ is a nodal curve with dual graph $\Gamma^{\mathcal{E}}$, we can
consider the subcurves of $X$ corresponding to the connected component of
$P_{(\mathcal{E},D)}$, which we call the _$\phi$ -subcurves_ of $X$. The
following result is Theorem 7.10.

######  Theorem B.

Let $(X,L)$ be a pair parametrized by a geometric point of
$\overline{\mathcal{J}}_{\mu,g,n}$. The following conditions are equivalent:

  * (1)

The point parametrizing $(X,L)$ is contained in
$\overline{\mathcal{Q}}_{g,n}$.

  * (2)

The combinatorial type $(\Gamma,\mathcal{E},D)$ of $(X,L)$ satisfies equation
(1) and for every $\phi$-subcurve $Z$ of $X$ with corresponding subgraph
$P_{Z}$ of $\Gamma^{\mathcal{E}}$, we have an isomorphism:

$\Big{(}L|_{Z}\otimes\mathcal{O}_{Z}\Big{(}-\sum p_{e}\Big{)}\Big{)}^{\otimes
2}\cong\omega_{Z}$

where $p_{e}$ runs over the nodes of $X$ corresponding to the oriented edges
$e$ of $\Gamma^{\mathcal{E}}$ with target contained in $P_{Z}$ and source not
contained in $P_{Z}$, such that $\phi(e)=-1$.

Finally we study the boundary stratification of the open immersion
$\mathcal{S}_{g,n}\hookrightarrow\overline{\mathcal{Q}}_{g,n}$. We define
$\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}$ as the locus of points of
$\overline{\mathcal{Q}}_{g,n}$ whose combinatorial type is a fixed tuple
$(\Gamma,\mathcal{E},D,s)$.

######  Theorem C.

The following decomposition:

$\overline{\mathcal{Q}}_{g,n}=\coprod_{(\Gamma,\mathcal{E},D,s)}\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}.$

is a stratification of $\overline{\mathcal{Q}}_{g,n}$ indexed by the poset
underlying $Q^{\textnormal{trop}}_{g,n}$.

We have an explicit formula for the codimension of the strata
$\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}$. This allows us to show that the
natural forgetful map
$\overline{\mathcal{Q}}_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n}$ is
finite exactly over the locus of tree-like curves. In particular,
$\overline{\mathcal{Q}}_{g,n}$ is not isomorphic to
$\overline{\mathcal{S}}_{g,n}$. We believe that Theorem A should help in the
construction of an explicit blowup
$\overline{\mathcal{Q}}_{g,n}\rightarrow\overline{\mathcal{S}}_{g,n}$, locally
given by a toric morphism induced by the decomposition of cones associated to
the refinement $Q^{\textnormal{trop}}_{g,n}\rightarrow
S^{\textnormal{trop}}_{g,n}$. We plan to investigate this problem in a
different paper. We can also give a combinatorial characterization of the
codimension 1 strata $\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}$: there are more
boundary divisors in $\overline{\mathcal{Q}}_{g,n}$ that in
$\overline{\mathcal{S}}_{g,n}$. Finally, we have a reverse-inclusion bijection
between the strata of $\overline{\mathcal{Q}}_{g,n}$ and the cells of
$Q_{g,n}^{\textnormal{trop}}$ switching dimension and codimension, proving, as
expected, an interesting “duality” between $Q^{\textnormal{trop}}_{g,n}$ and
$\overline{\mathcal{Q}}_{g,n}$. We refer to Theorem 7.17 for the precise
statement of the results.

##  2\. Preliminaries

###  2.1. Graphs

We denote by $\Gamma$ a weighted graph (or, simply, a _graph_) with legs,
where $V(\Gamma)$ and $E(\Gamma)$ are the sets of vertices and edges, $w\colon
V(\Gamma)\rightarrow\mathbb{Z}_{\geq 0}$ is the weight function, and
$L_{\Gamma}(v)$ (or, simply, $L(v)$) is the number of legs incident to $v\in
V(\Gamma)$.

Let $\Gamma$ be a graph. For every subset $\mathcal{E}\subset E(\Gamma)$, we
let $\deg_{\mathcal{E}}(v)$ be the number of edges in $\mathcal{E}$ incident
to $v$ (with loops counting twice). We set $\deg(v)=\deg_{E(\Gamma)}(v)$.
Given subsets $V,W\subset V(\Gamma)$, we let $E(V,W)\subset E(\Gamma)$ be the
set of edges with one end-vertex in $V$ and the other in $W$. If
$\Gamma_{1},\Gamma_{2}$ are subgraphs of $\Gamma$, we abuse notation writing
$E(\Gamma_{1},\Gamma_{2})=E(V(\Gamma_{1}),V(\Gamma_{2}))$. We set
$E(V)=E(V,V)$. For every subset $V\subset V(\Gamma)$, we let $\Gamma(V)$ be
the subgraph of $\Gamma$ whose set of vertices is $V(\Gamma)$ and with set of
edges $E(V)$. For $V\subset V(\Gamma)$ we set $V^{c}=V(\Gamma)\setminus V$.

We denote by $\textnormal{Aut}(\Gamma)$ be the group of automorphism of
$\Gamma$. We also let $b_{0}(\Gamma)$ be the number of connected components of
$\Gamma$ and we set $b_{1}(\Gamma)=\\#E(\Gamma)-\\#V(\Gamma)+b_{0}(\Gamma)$.
The _genus_ of $\Gamma$ is defined as $g(\Gamma)=\sum_{v\in
V(\Gamma)}w(v)+b_{1}(\Gamma)$. We say that $\Gamma$ is _stable_ if
$2w(v)-2+\deg_{\Gamma}(v)+L_{\Gamma}(v)>0$ for every $v\in V(\Gamma)$. Given a
subset $\mathcal{E}\subset E(\Gamma)$, we let $\Gamma_{\mathcal{E}}$ be the
graph obtained by removing $\mathcal{E}$, so that $\Gamma$ and
$\Gamma_{\mathcal{E}}$ have the same vertices and the same legs. We say that
$\mathcal{E}$ is _non-disconnecting_ if $\Gamma_{\mathcal{E}}$ is connected.
We denote by $\Gamma_{\mathcal{E},\operatorname{leg}}$ the graph obtained by
replacing every edge $e\in\mathcal{E}$ with end-vertices $v_{1},v_{2}$ by a
pair of legs, the first incident to $v_{1}$ and the second to $v_{2}$. A
_cycle_ of $\Gamma$ is a connected subgraph of $\Gamma$ all of whose vertices
$v$ satisfy $\deg_{\Gamma}(v)=2$. We will often identify a cycle and its set
of edges. A _cyclic subgraph_ of $\Gamma$ is a (possibly empty) union of
cycles of $\Gamma$ with no common edges.

We define the set $\overrightarrow{E}(\Gamma)=E_{1}\sqcup E_{2}$, whose
elements are called _oriented edges_ , where $E_{i}=E(\Gamma)$. We consider
the function $u\colon\overrightarrow{E}(\Gamma)\rightarrow E(\Gamma)$ such
that $u(e)=e$, and the functions
$s,t\colon\overrightarrow{E}(\Gamma)\rightarrow V(\Gamma)$ (the _source and
target_), such that

  * (1)

$\\{s(e),t(e)\\}$ is the set of vertices incident to $u(e)$,
$\forall\;e\in\overrightarrow{E}(\Gamma)$;

  * (2)

$s(e_{1})=t(e_{2})$ for oriented edges $e_{1}\in E_{1}$, $e_{2}\in E_{2}$ such
that $u(e_{1})=u(e_{2})\in E(\Gamma)$.

For every subset $\mathcal{E}\subset E(\Gamma)$, we let
$\overrightarrow{\mathcal{E}}=u^{-1}(\mathcal{E})\subset\overrightarrow{E}(\Gamma)$.

An _oriented cycle_ of $\Gamma$ is a sequence of oriented edges
$e_{1},\dots,e_{n}$ such that $t(e_{n})=s(e_{1})$ and $t(e_{i})=s(e_{i+1})$
for every $i=1,\dots,n-1$ and whose associated set of edges forms a cycle of
$\Gamma$.

A flow on a graph $\Gamma$ is a function
$\phi\colon\overrightarrow{E}(\Gamma)\rightarrow\mathbb{Z}$ such that
$\phi(e_{1})+\phi(e_{2})=0$ for different
$e_{1},e_{2}\in\overrightarrow{E}(\Gamma)$ such that $u(e_{1})=u(e_{2})\in
E(\Gamma)$. Notice that $\phi(e_{1})=0$ if and only if $\phi(e_{2})=0$, so,
sometimes, we will abuse notation writing $\phi(e)=0$, where
$e=u(e_{1})=u(e_{2})\in E(\Gamma)$. If $v\colon
E(\Gamma)\rightarrow\overrightarrow{E}(\Gamma)$ is a section of $u$, we will
often define $\phi$ just defining it over the image of $v$.

Given a flow $\phi$ on a graph $\Gamma$ and vertices $v_{1},v_{2}\in
V(\Gamma)$, a _$\phi$ -path_ from $v_{1}$ to $v_{2}$ is a sequence
$e_{1},\dots,e_{n}\in\overrightarrow{E}(\Gamma)$ such that $s(e_{1})=v_{1}$,
$t(e_{n})=v_{2}$, $t(e_{i})=s(e_{i+1})$ for $i=1,\dots,n-1$ and
$\phi(e_{i})\geq 0$ for $i=1,\dots,n$. A $\phi$-path is called _positive_
(_null_) if $\phi(e_{i})>0$ (if $\phi(e_{i})=0$) for every $i$. A _path_ is a
$\phi$-path for $\phi=0$. A _$\phi$ -cycle_ is a $\phi$-path whose edges form
a cycle of $\Gamma$; it is called _positive_ (null) if it is so as a
$\phi$-path. A _$\phi$ -loop_ is a $\phi$-cycle consisting of a single
oriented edge. A flow $\phi$ is _acyclic_ if there are no non-null
$\phi$-cycles in $\Gamma$. If $\phi$ is acyclic and $\phi(e)\neq 0$ for every
$e\in\overrightarrow{E}(\Gamma)$, then there is at least a _source_ $v$ and a
_sink_ $v^{\prime}$ of $\phi$, that is, $\phi(e)>0$ if $s(e)=v$ and
$\phi(e)>0$ if $t(e)=v^{\prime}$.

A _$\phi$ -cut_ of $\Gamma$ is a subset $E(W_{1},W_{2})\subset E(\Gamma)$ for
some disjoint subsets $W_{1},W_{2}\subset V(\Gamma)$ such that for every
$v_{1}\in W_{1}$ and $v_{2}\in W_{2}$ there is no $\phi$-path from $v_{1}$ to
$v_{2}$ and from $v_{2}$ to $v_{1}$ whose edges are contained in
$\overrightarrow{E}(\Gamma)\setminus\overrightarrow{E}(W_{1},W_{2})$.

We define the graph $\Gamma/\mathcal{E}$ obtained by the contraction of the
edges of $\mathcal{E}$. Notice that there is a natural surjection
$V(\Gamma)\rightarrow V(\Gamma/\mathcal{E})$ and an identification
$E(\Gamma/\mathcal{E})=E(\Gamma)\setminus\mathcal{E}$. A graph $\Gamma$
_specializes_ to a graph $\Gamma^{\prime}$, and we write
$\iota\colon\Gamma\rightarrow\Gamma^{\prime}$, if there is an isomorphism
$\Gamma^{\prime}\cong\Gamma/\mathcal{E}$. A specialization
$\Gamma\rightarrow\Gamma^{\prime}$ is equipped with a surjective map
$\iota\colon V(\Gamma)\rightarrow V(\Gamma^{\prime})$ and an injective map
$E(\Gamma^{\prime})\rightarrow E(\Gamma)$.

Let $\Gamma$ be a graph. A _divisor_ $D$ on $\Gamma$ is a function $D\colon
V(\Gamma)\rightarrow\mathbb{Z}$. We often denote a divisor by $D=\sum_{v\in
V(\Gamma)}D(v)v$. The degree of $D$ is the integer $\deg D=\sum_{v\in
V(\Gamma)}D(v)$. We denote by $\operatorname{Div}(\Gamma)$ the group of
divisors of $\Gamma$. A _polarization_ $\mu$ on $\Gamma$ is a function
$\mu\colon V(\Gamma)\rightarrow\mathbb{R}$ such that $\sum_{v\in
V(\Gamma)}\mu(v)$ is an integer, called the _degree_ of the polarization.
Given a flow $\phi$ on $\Gamma$, the _divisor of $\phi$_ is defined as
$\operatorname{Div}(\phi)=\sum_{e\in\overrightarrow{E}(\Gamma)}\phi(e)t(e)$.
For a vertex $v\in V(\Gamma)$, we define the acyclic flow $\phi_{v}$ such that
for $e\in\overrightarrow{E}(\Gamma)$ we have

$\phi_{v}(e)=\begin{cases}\begin{array}[]{cl}1&\text{ if }s(e)=v\neq t(e)\\\
-1&\text{ if }t(e)=v\neq s(e)\\\ 0&\text{ otherwise}.\end{array}\end{cases}$

We set $\operatorname{Div}(v):=\operatorname{Div}(\phi_{v})$. A _principal
divisor_ on $\Gamma$ is a divisor of type $\sum_{v\in
V(\Gamma)}a_{v}\operatorname{Div}(v)$, for $a_{v}\in\mathbb{Z}$. Of course,
this divisor is equal to $\operatorname{Div}(\phi)$, where $\phi$ is the
acyclic flow $\phi=\sum_{v\in V(\Gamma)}a_{v}\phi_{v}$.

Given a specialization $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ of
graphs, a divisor $D$ and a polarization $\mu$ on $\Gamma$, we define the
divisor and polarization $\iota_{*}(D)$ and $\iota_{*}(\mu)$ on
$\Gamma^{\prime}$ taking every $v^{\prime}\in V(\Gamma^{\prime})$ to

$\iota_{*}(D)(v^{\prime})=\sum_{v\in\iota^{-1}(v^{\prime})}D(v)\;\text{ and
}\;\iota_{*}(\mu)(v^{\prime})=\sum_{v\in\iota^{-1}(v^{\prime})}\mu(v).$

Given a flow $\phi$ on $\Gamma$, we let $\iota_{*}\phi$ be the flow on
$\Gamma^{\prime}$ such that $\iota_{*}\phi(e)=\phi(e)$, $\forall\;e\in
E(\Gamma^{\prime})\subset E(\Gamma)$.

Let $D$ and $\mu$ be a divisor and a polarization on $\Gamma$ of degree $d$.
For every subset $W\subset V(\Gamma)$, we set
$\delta_{\Gamma,W}:=\\#E(W,W^{c})$ (or, simply, $\delta_{W}$) and
$\mu(W)=\sum_{v\in W}\mu(v)$, and we define:

$\beta_{D}(W):=\deg D|_{W}-\mu(W)+\frac{\delta_{\Gamma,W}}{2}.$

We say that the divisor $D$ on a connected graph $\Gamma$ is _$\mu$ -stable_
(respectively, _$\mu$ -semistable_), if $\beta_{D}(W)>0$ (respectively,
$\beta_{D}(W)\geq 0$) for every $W\subsetneq V(\Gamma)$. Moreover, $D$ is
_$(v_{0},\mu)$ -quasistable_, for some $v_{0}\in V(\Gamma)$, if
$\beta_{D}(W)\geq 0$ for every $W\subsetneq V(\Gamma)$, with strict inequality
if $v_{0}\in W$.

If $\Gamma$ is a non-connected graph and $\mu$ is a degree-$d$ polarization on
$\Gamma$, the notion of $\mu$-semistability naturally extends for divisors on
$\Gamma$ (see [4, Remark 3.1]). We say that a divisor $D$ of degree $d$ on
$\Gamma$ is $\mu$-_stable_ if it is $\mu$-semistable and
$D|_{\Gamma^{\prime}}$ is $\mu|_{\Gamma^{\prime}}$-stable for every connected
component $\Gamma^{\prime}$ of $\Gamma$.

######  Proposition 2.1.

Let $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ be a specialization of
graphs. Let $D$ and $\mu$ be a divisor and a polarization of degree $d$ on
$\Gamma$. We have $\beta_{\iota_{*}(D)}(W)=\beta_{D}(\iota^{-1}(W))$ for every
subset $W\subset V(\Gamma^{\prime})$. In particular, if $D$ is $\mu$-stable,
$\mu$-semistable, $(v_{0},\mu)$-quasistable for some $v_{0}\in V(\Gamma)$,
then $\iota_{*}(D)$ is $\iota_{*}(\mu)$-stable, $\iota_{*}(\mu)$-semistable,
$(\iota(v_{0}),\iota_{*}(\mu))$-quasistable, respectively.

###### Proof.

For every subset $W\subset V(\Gamma^{\prime})$ we have

$\displaystyle\beta_{\iota_{*}(D)}(W)$
$\displaystyle=\deg(\iota_{*}(D)|_{W})-\iota_{*}(\mu)(W)+\frac{\delta_{W}}{2}$
$\displaystyle=\deg(D|_{\iota^{-1}(W)})-\mu(\iota^{-1}(W))+\frac{\delta_{\iota^{-1}(W)}}{2}$
$\displaystyle=\beta_{D}(\iota^{-1}(W)).$

The result follows. ∎

Given a graph $\Gamma$ and a subset $\mathcal{E}\subset E(\Gamma)$, we let
$\Gamma^{\mathcal{E}}$ be the graph obtained by inserting exactly one vertex
in the interior of every edge $e\in\mathcal{E}$. Notice that there is a
natural inclusion $V(\Gamma)\subset V(\Gamma^{\mathcal{E}})$. We call
_exceptional_ a vertex in $V(\Gamma^{\mathcal{E}})\setminus V(\Gamma)$. An
edge which is incident to an exceptional vertex in the interior of an edge
$e\in E(\Gamma)$ is called _exceptional_ and we say that it is an edge _over_
$e$. A _pseudo-divisor_ on $\Gamma$ is a pair $(\mathcal{E},D)$, where
$\mathcal{E}\subset E(\Gamma)$ and $D$ is a divisor on $\Gamma^{\mathcal{E}}$
such that $D(v)=-1$ for every exceptional vertex $v$.

More generally, a _refinement_ of a graph $\Gamma$ is a graph
$\Gamma^{\prime}$ obtained by inserting a certain number of vertices in the
interior of each edge of $\Gamma$. We have a natural injective map
$\operatorname{Div}(\Gamma)\rightarrow\operatorname{Div}(\Gamma^{\prime})$
induced by the inclusion $V(\Gamma)\subset V(\Gamma^{\prime})$.

We say that a pseudo-divisor $(\mathcal{E},D)$ on $\Gamma$ is _simple_ if
$\Gamma_{\mathcal{E}}$ is connected. We say that $(\mathcal{E},D)$ is
$\mu$-stable, $\mu$-semistable, $(v_{0},\mu)$-quasistable for some $v_{0}\in
V(\Gamma)$, if so is $D$ on $\Gamma^{\mathcal{E}}$ with respect to the
polarization $\mu^{\mathcal{E}}$ on $\Gamma^{\mathcal{E}}$ extending $\mu$
such that $\mu^{\mathcal{E}}(v)=0$ for every exceptional vertex $v\in
V(\Gamma^{\mathcal{E}})\setminus V(\Gamma)$.

Let $D$ be a divisor and $\mu$ a polarization on a graph $\Gamma$. For a
subset $\mathcal{E}\subset E(\Gamma)$, we let $D_{\mathcal{E}}$ and
$\mu_{\mathcal{E}}$ be the divisor and the polarization on
$\Gamma_{\mathcal{E}}$ such that $D_{\mathcal{E}}(v)=D(v)$ and
$\mu_{\mathcal{E}}(v)=\mu(v)+\frac{1}{2}\deg_{\mathcal{E}}(v)$. For every
subset $W\subset V(\Gamma_{\mathcal{E}})=V(\Gamma)$, we set

$\beta_{\mathcal{E},D}(W)=\deg(D|_{W})-\mu_{\mathcal{E}}(W)+\frac{\delta_{\Gamma_{\mathcal{E}},W}}{2}.$

We say that $(\mathcal{E},D)$ is $\mu$-polystable if
$\beta_{\mathcal{E},D}(V)\geq 0$ for every subset $V\subset V(\Gamma)$, with
strict inequality if $E(V,V^{c})\not\subset\mathcal{E}$. Equivalently,
$(\mathcal{E},D)$ is $\mu$-polystable if $D_{\mathcal{E}}$ is
$\mu_{\mathcal{E}}$-stable on $\Gamma_{\mathcal{E}}$.

Given a specialization $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ and a
subset $\mathcal{E}\subset E(\Gamma)$ we have a specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow\Gamma^{\prime\mathcal{E}^{\prime}}$,
where $\mathcal{E}^{\prime}=\mathcal{E}\cap E(\Gamma^{\prime})$; we say that
$\widehat{\iota}$ is _induced_ by $\iota$. If $(\mathcal{E},D)$ is a pseudo-
divisor on $\Gamma$, we define the pseudo-divisor
$\iota_{*}(\mathcal{E},D)=(\mathcal{E}^{\prime},\widehat{\iota}_{*}(D))$ on
$\Gamma^{\prime}$.

Given a specialization of graphs $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$
and subsets $\mathcal{E}\subset E(\Gamma)$, $\mathcal{E}^{\prime}\subset
E(\Gamma^{\prime})$, we say that a specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$
is $\iota$-compatible, if the diagrams:

$\textstyle{V(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{V(\Gamma^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V(\Gamma^{\mathcal{E}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\iota}}$$\textstyle{V({\Gamma^{\prime}}^{\mathcal{E}^{\prime}})}$
$\textstyle{E(\Gamma^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{E(\Gamma)}$$\textstyle{E({\Gamma^{\prime}}^{\mathcal{E}^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\iota}}$$\textstyle{E(\Gamma^{\mathcal{E}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

are commutative. If $\iota$ is the identity, we simply say that
$\widehat{\iota}$ is compatible.

Given pseudo-divisors $(\mathcal{E},D)$ on $\Gamma$ and
$(\mathcal{E}^{\prime},D^{\prime})$ on $\Gamma^{\prime}$, we say that the
triple $(\Gamma,\mathcal{E},D)$ _specializes_ to the triple
$(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$, and we write
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$,
if there is a specialization $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$
such that $\mathcal{E}^{\prime}\subset\mathcal{E}\cap E(\Gamma^{\prime})$, and
a $\iota$-compatible specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow\Gamma^{\mathcal{E}^{\prime}}$
such that $\widehat{\iota}_{*}(D)=D^{\prime}$. When $\Gamma=\Gamma^{\prime}$,
we will omit $\Gamma$ in the notation
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.

Notice that a specialization of triple
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is determined by the associated specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$.
Indeed, the unique specialization
$\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ for which $\widehat{\iota}$ is
$\iota$-compatible is the one contracting the edges $e\in E(\Gamma)$ such that
all the edges of $\Gamma^{\mathcal{E}}$ over $e$ are contracted by
$\widehat{\iota}$.

######  Remark 2.2.

Given a $\mu$-semistable pseudo-divisor $(\mathcal{E},D)$ on a graph $\Gamma$,
there is a unique minimal $\mu$-polystable pseudo-divisor on $\Gamma$, denoted
$\operatorname{pol}(\mathcal{E},D)$, such that
$\operatorname{pol}(\mathcal{E},D)\geq(\mathcal{E},D)$ (see [4, Proposition
5.4]).

###  2.2. Tropical curves

An $n$-pointed tropical curve is a pair $X=(\Gamma,\ell)$ where $\Gamma$ is a
graph with $n$ legs and $\ell\colon E(\Gamma)\rightarrow\mathbb{R}_{>0}$ is a
function (the _length function_). The graph $\Gamma$ is called a _model_ of
$X$. We will consider tropical curves up to isometries. Given a point $p\in
X$, we define $\deg_{X}(p)=\deg_{\Gamma}(p)$, $w_{X}(p)=w_{\Gamma}(p)$ and
$L_{X}(p)=L_{\Gamma}(p)$ if $p\in V(\Gamma)$, otherwise we set
$\deg_{X}(p)=2$, $w_{X}(p)=0$ and $L_{X}(p)=0$. The _genus_ of $X$ is
$g=\sum_{v\in X}w_{X}(v)+b_{1}(\Gamma)$. We say that $X$ is _stable_ if, for
every $p\in X$ such that $\deg_{X}(p)\leq 1$, we have
$\deg_{X}(p)+2w_{X}(p)+L_{X}(p)\geq 3$.

Let $X$ be a tropical curve. A _divisor_ on $X$ is a function
$\mathcal{D}\colon X\rightarrow\mathbb{Z}$ such that $\mathcal{D}(p)\neq 0$
only for a finite set of points $p\in X$ (called the _support_ of
$\mathcal{D}$ and written $\operatorname{supp}(\mathcal{D})$). We often denote
a divisor by $\mathcal{D}=\sum_{p\in X}\mathcal{D}(p)p$. The _degree_ of
$\mathcal{D}$ is the integer $\sum_{p\in X}\mathcal{D}(v)$. The _canonical
divisor_ of $X$ is the divisor $K_{X}=\sum_{p\in X}(w_{X}(p)-2+\deg_{X}(p))p$.
The degree of $K_{X}$ is $2g-2$. We will implicitly consider a fixed model
$\Gamma$ of $X$ such that the support of $K_{X}$ is contained in $V(\Gamma)$.

A divisor $\mathcal{D}$ on $X$ is _$\Gamma$ -unitary_ (or simply _unitary_),
if for every $e\in E(\Gamma)$ we have $\mathcal{D}(p)=0$ for each point $p\in
e^{\circ}$, except for at most one point $p_{0}\in e^{\circ}$ for which
$\mathcal{D}(p_{0})=-1$. Given a $\Gamma$-unitary divisor $\mathcal{D}$ on
$X$, the _combinatorial type_ of $\mathcal{D}$ is the pseudo-divisor
$(\mathcal{E},D)$ on $\Gamma$ such that

$\mathcal{E}=\\{e\in E(\Gamma)|D(p)=-1,\text{ for some }p\in e^{\circ}\\},$

while $D(v)=\mathcal{D}(v)$, for every $v\in V(\Gamma)$, and $D(v)=-1$,
$\forall$ exceptional vertex $v\in V(\Gamma^{\mathcal{E}})$.

One can introduce stability conditions on divisors on a tropical curve (see
[4, Section 5] and [4, Section 5]). It is easy to see that a quasistable or
polystable divisor is always unitary. In this paper we use the equivalent
definition that a unitary divisor $\mathcal{D}$ on $X$ is _$\mu$ -stable_,
_$\mu$ -semistable_, _$(p_{0},\mu)$ -quasistable_ for a point $p_{0}\in
V(\Gamma)$, _$\mu$ -polystable_ if so is its combinatorial type
$(\mathcal{E},D)$, respectively (see [2, Proposition 5.3] and [4, Definition
5.8]).

A _rational function_ on $X$ is a piecewise linear continuous function
$f\colon X\rightarrow\mathbb{R}$ with integer slopes. Given a point $p\in X$,
we let $\operatorname{ord}_{p}(f)$ be the sum of the slopes of $f$ outgoing
$p$. A _principal divisor_ on $X$ is a divisor
$\operatorname{Div}(f)=\sum_{p\in X}\operatorname{ord}_{p}(f)p$, for some
rational function $f$ on $X$. Given divisors $D$ and $D^{\prime}$ on $X$, we
say that $D$ is equivalent to $D^{\prime}$, written $D\sim D^{\prime}$, if
$D-D^{\prime}$ is a principal divisor. Given a rational function $f$ on $X$
and a model $\Gamma$ such that the support of $\operatorname{Div}(f)$ is
contained in $V(\Gamma)$, we define a flow $\phi_{f}$ on $\Gamma$ such that
$\phi_{f}(e)$ is the slope of $f$ on $e$, for every
$e\in\overrightarrow{E}(\Gamma)$.

######  Remark 2.3.

By [2, Theorem 5.6] and [4, Theorem 5.9] given a divisor $\mathcal{D}$ on $X$
there is a $(p_{0},\mu)$-quasistable divisor $\mathcal{D}^{\prime}$ and a
$\mu$-polystable divisor $\mathcal{D}^{\prime\prime}$ on $X$ such that
$\mathcal{D}$, $\mathcal{D}^{\prime}$, $\mathcal{D}^{\prime\prime}$ are
equivalent; the divisor $\mathcal{D}^{\prime}$ is uniquely determined.

##  3\. Tropical moduli spaces

In this section we recall the tropical moduli space we will deal with. First
of all we need to recall the notion of generalized cone complex.

Given a finite set $S\subset\mathbb{R}^{n}$, we let

$\operatorname{cone}(S):=\Big{\\{}\sum_{s\in
S}\lambda_{s}s|\lambda_{s}\in\mathbb{R}_{\geq 0}\Big{\\}}.$

A _polyhedral cone_ is a subset $\sigma\subset\mathbb{R}^{n}$ such that
$\sigma=\operatorname{cone}(S)$ for some finite set $S\subset\mathbb{R}^{n}$.
A cone $\sigma=\operatorname{cone}(S)$ is _rational_ if
$S\subset\mathbb{Z}^{n}$. Every polyhedral cone $\sigma\subset\mathbb{R}^{n}$
is the intersection of finitely many closed half spaces of $\mathbb{R}^{n}$.
We denote by $\dim(\sigma)$ the _dimension_ of $\sigma$, that is, the
dimension of the minimal linear subspace of $\mathbb{R}^{n}$ containing
$\sigma$. The relative interior $\sigma^{\circ}$ of a cone $\sigma$ is the
interior of $\sigma$ inside this minimal linear subspace. A _face_ of a cone
$\sigma$ is either $\sigma$ or the intersection of $\sigma$ with some linear
subspace $H\subset\mathbb{R}^{n}$ of codimension one such that $\sigma$ is
contained in one of the two closed half-spaces determined by $H$. A face of
$\sigma$ is also a polyhedral cone. A _facet_ is a face of codimension $1$. We
will simply use the terminology _cone_ to mean _rational polyhedral cone_.

A morphism $f\colon\sigma\rightarrow\sigma^{\prime}$ between cones
$\sigma\subset\mathbb{R}^{n}$ and $\sigma^{\prime}\subset\mathbb{R}^{m}$ is
the restriction to $\sigma$ of an integral linear transformation
$T\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ such that
$T(\sigma)\subset\sigma^{\prime}$. We say that $f$ is an isomorphism if there
exists an inverse morphism of cones
$f^{-1}\colon\sigma^{\prime}\rightarrow\sigma$. A morphism
$f\colon\sigma\rightarrow\sigma^{\prime}$ is called a _face morphism_ if $f$
is an isomorphism between $\sigma$ and a (not necessarily proper) face of
$\sigma^{\prime}$.

A _generalized cone complex_ $\Sigma$ is the colimit (as a topological space)
of a finite diagram of cones, denoted $D$, with face morphisms. A generalized
cone complex is written as
$\Sigma=\underset{\longrightarrow}{\lim}\,\sigma_{x}$, where
$x\rightarrow\sigma_{x}$ is a contravariant functor from a category
$\mathbf{C}$ to the category of rational polyhedral cones. The group
$\textnormal{Aut}(x)$ acts on $\sigma_{x}$ as a subgroup of automorphisms of
$\sigma_{x}$. Notice that the set $\mathcal{P}$ of isomorphism classes of
objects of $\mathbf{C}$ is a poset.

A _morphism_ $\Sigma\rightarrow\Sigma^{\prime}$ between generalized cone
complexes $\Sigma$ and $\Sigma^{\prime}$ with diagrams of cones $D$ and
$D^{\prime}$ is a continuous map of topological spaces such that for every
cone $\sigma\in D$ there exists a cone $\sigma^{\prime}\in D^{\prime}$ such
that the induced map $\sigma\rightarrow\Sigma^{\prime}$ factors through a cone
morphism $\sigma\rightarrow\sigma^{\prime}$. We say that a morphism of
generalized cone complexes $\Sigma\rightarrow\Sigma^{\prime}$ is a
_refinement_ if it is a bijection.

######  Definition 3.1.

Let $\mathcal{M}$ be a Deligne-Mumford stack and $\mathcal{P}$ be a poset. A
_stratification of $\mathcal{M}$ indexed by $\mathcal{P}$_ is a decomposition
$\mathcal{M}=\coprod_{x\in\mathcal{P}}\mathcal{M}_{x}$ of $\mathcal{M}$ such
that

  1. (1)

$\mathcal{M}_{x}$ is an irreducible substack of $\mathcal{M}$;

  2. (2)

the closure $\overline{\mathcal{M}}_{x}$ of $\mathcal{M}_{x}$ has a
decomposition $\overline{\mathcal{M}}_{x}=\coprod_{x^{\prime}\geq
x}\mathcal{M}_{x^{\prime}}$.

We call $\mathcal{M}_{x}$ a _stratum_ of the stratification.

###  3.1. The moduli space of tropical curves

Generalized cone complexes are often used to parametrize tropical objects. The
prototype example is the moduli space of pointed tropical curve
$M_{g,n}^{\textnormal{trop}}$. The moduli space $M_{g,n}^{\textnormal{trop}}$,
first constructed in [6], has the structure of a generalized cone complex (see
[5]).

We let $\mathbf{G}_{g,n}$ be the category whose objects are stable graphs of
genus $g$ with $n$ legs and whose arrows are specialization of graphs. We let
$\mathcal{G}_{g,n}=\mathbf{G}_{g,n}/\sim$ be the poset whose elements are
isomorphism classes of graphs in $\mathbf{G}_{g,n}$.

Given a graph $\Gamma$ we define the cone
$\tau_{\Gamma}:=\mathbb{R}^{E(\Gamma)}_{\geq 0}$ and its open subcone
$\tau_{\Gamma}^{\circ}:=\mathbb{R}^{E(\Gamma)}_{>0}$. Notice that given a
specialization of graphs $\Gamma\rightarrow\Gamma^{\prime}$, then we have a
natural face morphism $\tau_{\Gamma^{\prime}}\rightarrow\tau_{\Gamma}$ whose
image is the face of $\tau_{\Gamma}$ where the coordinates corresponding to
$E(\Gamma)\setminus E(\Gamma^{\prime})$ vanish. We define the generalized cone
complex:

$M_{g,n}^{\textnormal{trop}}=\underset{\longrightarrow}{\lim}\;\tau_{\Gamma}=\coprod_{[\Gamma]}\tau^{\circ}_{\Gamma}/\textnormal{Aut}(\Gamma)$

where the limit is taken over the category $\mathbf{G}_{g,n}$ and the union is
taken over all equivalence classes $[\Gamma]$ in $\mathcal{G}_{g,n}$. The
points in $M_{g,n}^{\textnormal{trop}}$ are in bijective correspondence with
(isomorphism classes of) tropical curves of genus $g$.

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of $n$-pointed stable
curves, and $\mathcal{M}_{g,n}$ be the locus corresponding to smooth curves.
Given an $n$-pointed stable curve $X$, we let $\Gamma_{X}$ be the usual dual
graph of $X$, whose vertices and edges are the components and the nodes of
$X$, respectively. For every graph $\Gamma$ with $n$ legs, we define the locus
$\mathcal{M}_{\Gamma}$ in $\overline{\mathcal{M}}_{g,n}$ given by:

$\mathcal{M}_{\Gamma}:=\\{[X]\in\overline{\mathcal{M}}_{g,n}\big{|}\Gamma_{X}\cong\Gamma\\}.$

We have a stratification
$\overline{\mathcal{M}}_{g,n}=\sqcup_{\Gamma\in\mathcal{G}_{g,n}}\mathcal{M}_{\Gamma}$
indexed by $\mathcal{G}_{g,n}$. Recall that $\mathcal{M}_{\Gamma}$ is
irreducible thanks to the existence of a natural étale finite map
$\widetilde{\mathcal{M}}_{\Gamma}\rightarrow\mathcal{M}_{\Gamma}$, where

(2) $\widetilde{\mathcal{M}}_{\Gamma}=\prod_{v\in
V(\Gamma)}\mathcal{M}_{w_{\Gamma}(v),\deg_{\Gamma}(v)+L_{\Gamma}(v)}.$

###  3.2. Universal tropical Jacobians

A _universal genus- $g$ polarization $\mu$ (of degree $d$)_ is the datum of a
polarization $\mu_{\Gamma}$ (of degree $d$) for every stable genus-$g$ graph
with $n$ legs such that $\iota_{*}(\mu_{\Gamma})=\mu_{\Gamma^{\prime}}$, for
every specialization of graphs $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$.
The _universal genus- $g$ canonical polarization of degree $d$_ takes every
graph $\Gamma$ and a vertex $v\in V(\Gamma)$ to:

$\mu_{\Gamma}(v):=\frac{d(2w_{\Gamma}(v)-2+\deg_{\Gamma}(v))}{2g-2}=\frac{d}{2g-2}K_{\Gamma}(v).$

Let $\mu$ be a universal genus-$g$ polarization. We denote by
$\mathbf{QD}_{\mu,g,n}$ the category whose objects are triples
$(\Gamma,\mathcal{E},D)$, where $\Gamma$ is a stable graph of genus $g$ with
$n$ legs, and $(\mathcal{E},D)$ is a $(v_{0},\mu)$-quasistable pseudo-divisor
on $\Gamma$ (the first leg is incident to $v_{0}$), and whose arrows are given
by specialization of triples. We let
$\mathcal{QD}_{\mu,g,n}=\mathbf{QD}_{\mu,g,n}/\sim$ be the poset of the
isomorphism classes of triples $(\Gamma,\mathcal{E},D)$.

Given a graph $\Gamma$ and a $v_{0}$-quasistable pseudo-divisor
$(\mathcal{E},D)$ on $\Gamma$, we define

$\tau_{(\Gamma,\mathcal{E},D)}:=\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{\geq
0}\quad\text{and}\quad\tau^{\circ}_{(\Gamma,\mathcal{E},D)}:=\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{>0}.$

Notice that, given a specialization
$i\colon(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$,
we have a natural face morphism
$\tau_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\rightarrow\tau_{(\Gamma,\mathcal{E},D)}$.

######  Definition 3.2.

Let $\mu$ be a universal genus-$g$ polarization. The _universal tropical
Jacobian of $\mu$-quasistable divisors_ is the generalized cone complex:

$J^{\textnormal{trop}}_{\mu,g,n}:=\underset{\longrightarrow}{\lim}\;\tau_{(\Gamma,\mathcal{E},D)}=\coprod_{[\Gamma,\mathcal{E},D]}\tau^{\circ}_{(\Gamma,\mathcal{E},D)}/\textnormal{Aut}(\Gamma,\mathcal{E},D),$

where the limit is taken over the category $\mathbf{QD}_{\mu,g,n}$ and the
union is taken over all equivalence classes $[\Gamma,\mathcal{E},S]$ in
$\mathcal{QD}_{\mu,g,n}$ (see [2, Definition 5.11]).

By [2, Proposition 5.12] the cone complex $J^{\textnormal{trop}}_{\mu,g,n}$
parametrizes equivalence classes $(X,\mathcal{D})$, where X is a stable
$n$-pointed tropical curve of genus $g$ and $\mathcal{D}$ is a
$(p_{0},\mu)$-quasistable divisor on $X$. By [2, Theorem 5.14] there is a
morphism of generalized cone complexes

$\pi^{\textnormal{trop}}_{J}\colon J_{\mu,g,n}^{\textnormal{trop}}\rightarrow
M_{g,n}^{\textnormal{trop}}$

taking a class $(X,\mathcal{D})$ to the tropical curve $X$. Indeed, we have
the linear map:

(3)
$\mathcal{S}_{\mathcal{E}}\colon\mathbb{R}^{E(\Gamma^{\mathcal{E}})}\rightarrow\mathbb{R}^{E(\Gamma)}$

taking $(x_{e})_{e\in E(\Gamma^{\mathcal{E}})}$ to $(y_{e})_{e\in E(\Gamma)}$,
where $y_{e}=x_{e_{1}}+x_{e_{2}}$ if $e_{1},e_{2}\in E(\Gamma^{\mathcal{E}})$
are over $e$, and $y_{e}=x_{e}$, otherwise. This map induces a morphism of
cones $\tau_{(\Gamma,\mathcal{E},D)}\rightarrow\tau_{\Gamma}$ factoring the
natural composition

$\tau_{(\Gamma,\mathcal{E},D)}\rightarrow
J_{\mu,g,n}^{\textnormal{trop}}\stackrel{{\scriptstyle\pi^{\textnormal{trop}}_{J}}}{{\rightarrow}}M_{g,n}^{\textnormal{trop}}.$

Consider a graph $\Gamma$ and a subset $\mathcal{E}\subset E(\Gamma)$. For
each subset $V\subset V(\Gamma)$ such that $E(V,V^{c})\subset\mathcal{E}$, we
define $\alpha_{V}\colon E(\Gamma^{\mathcal{E}})\rightarrow\\{0,\pm 1\\}$ as
$\alpha_{V}(e)=1$ (respectively, $-1$), if $e$ is incident to $V$
(respectively, to $V^{c}$) and lies over an edge in $E(V,V^{c})$. Otherwise,
we define $\alpha_{V}(e)=0$. Denote by $u_{e}$ the vectors of the canonical
basis of $\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$, for $e\in
E(\Gamma^{\mathcal{E}})$. We define the vector in
$\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$:

(4) $u_{V}=\sum_{e\in E(\Gamma^{\mathcal{E}})}\alpha_{V}(e)u_{e}.$

We let $\mathcal{L}_{\mathcal{E}}\subset\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$
be the subspace generated by all vectors $u_{V}$, as $V$ runs through all
subsets of $V(\Gamma)$ such that $E(V,V^{c})\subset\mathcal{E}$. Notice that
$\dim\mathcal{L}_{\mathcal{E}}=b_{0}(\Gamma_{\mathcal{E}})-1$. Of course, if
$(\mathcal{E},D)$ is simple, then $\mathcal{L}_{\mathcal{E}}=0$. We also
denote by

(5)
$\mathcal{T}_{\mathcal{E}}\colon\mathbb{R}^{E(\Gamma^{\mathcal{E}})}\rightarrow\mathbb{R}^{E(\Gamma^{\mathcal{E}})}/\mathcal{L}_{\mathcal{E}}$

the quotient linear map. For a pseudo-divisor $(\mathcal{E},D)$ on $\Gamma$,
we define the cone:

$\sigma_{(\Gamma,\mathcal{E},D)}=\mathcal{T}_{\mathcal{E}}(\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{\geq
0})\subset\mathbb{R}^{E(\Gamma^{\mathcal{E}})}/\mathcal{L}_{\mathcal{E}}.$

######  Remark 3.3.

If
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is a specialization, then
$\mathcal{L}_{\mathcal{E}^{\prime}}=\mathcal{L}_{\mathcal{E}}\cap\mathbb{R}^{E(\Gamma^{\mathcal{E}^{\prime}})}$,
and hence we have a natural inclusion
$\sigma_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\subset\sigma_{(\Gamma,\mathcal{E},D)}$
which is a face morphism. By [4, Proposition 6.10], every face of
$\sigma_{(\Gamma,\mathcal{E},D)}$ arises in this way.

If $\mu$ is a genus-$g$ universal polarization, we denote by
$\mathbf{PSD}_{\mu,g,n}$ the category whose objects are triples
$(\Gamma,\mathcal{E},D)$ where $\Gamma$ is a genus-$g$ stable graph with $n$
legs and $(\mathcal{E},D)$ is a $\mu$-polystable pseudo-divisor on $\Gamma$,
and morphisms are given by specializations of triples. As usual, we let
$\mathcal{PSD}_{\mu,g,n}$ be the poset $\mathbf{PSD}_{\mu,g,n}/\sim$, where
$\sim$ means isomorphism.

######  Definition 3.4.

Let $\mu$ be a genus-$g$ universal polarization. The _universal tropical
Jacobian of $\mu$-polystable divisors_ is the generalized cone complex:

$P^{\textnormal{trop}}_{\mu,g,n}=\underset{\longrightarrow}{\lim}\;\sigma_{(\Gamma,\mathcal{E},D)}=\coprod_{[\Gamma,\mathcal{E},D]}\sigma^{0}_{(\Gamma,\mathcal{E},D)}/\textnormal{Aut}(\Gamma,\mathcal{E},D),$

where the limit is taken over the category $\mathbf{PSD}_{\mu,g,n}$ and the
union is taken over all equivalence classes $[\Gamma,\mathcal{E},D]$ in
$\mathcal{PSD}_{\mu,g,n}$.

######  Proposition 3.5.

The generalized cone complex $P^{\textnormal{trop}}_{\mu,g,n}$ parametrizes
equivalence classes of pairs $(X,\mathcal{D})$, where $X$ is a stable
$n$-pointed tropical curve of genus $g$ and $\mathcal{D}$ is a
$\mu$-polystable divisor on $X$, and two pairs $(X,\mathcal{D})$ and
$(X^{\prime},\mathcal{D}^{\prime})$ are equivalent if there is an isomorphism
$i\colon X\rightarrow X^{\prime}$ such that $i_{*}(\mathcal{D})$ is linearly
equivalent to $\mathcal{D}^{\prime}$.

###### Proof.

The argument is the same as in [4, Proposition 6.14]. ∎

By Proposition 3.5, we can define a natural forgetful morphism of cone
complexes

$\pi^{\textnormal{trop}}_{P}\colon P_{\mu,g,n}^{\textnormal{trop}}\rightarrow
M_{g,n}^{\textnormal{trop}}$

taking the class of a pair $(X,\mathcal{D})$ to the class of the tropical
curve $X$. Indeed, the map of cones
$\tau_{(\Gamma,\mathcal{E},D)}\rightarrow\tau_{\Gamma}$ induced by equation
(3) factors through a map of cones
$\sigma_{(\Gamma,\mathcal{E},D)}\rightarrow\tau_{\Gamma}$ and the last map
factors the natural composition:

$\sigma_{(\Gamma,\mathcal{E},D)}\rightarrow
P_{\mu,g,n}^{\textnormal{trop}}\stackrel{{\scriptstyle\pi^{\textnormal{trop}}_{P}}}{{\rightarrow}}M_{g,n}^{\textnormal{trop}}.$

We can also define the map

$\rho^{\textnormal{trop}}_{P}\colon J^{\textnormal{trop}}_{\mu,g,n}\rightarrow
P^{\textnormal{trop}}_{\mu,g,n}$

taking the point parametrizing the class of a pair $(X,\mathcal{D})$ to the
point parametrizing the class of the pair $(X,\mathcal{D}^{\prime})$, where
$\mathcal{D}^{\prime}$ is a polystable divisor linearly equivalent to
$\mathcal{D}$ (the existence follows by [4, Theorem 5.9]).

The map $\rho^{\textnormal{trop}}_{P}$ is not only a morphism of generalized
cone complexes: it is a refinement. Before giving the precise statement, we
need an easy lemma.

######  Lemma 3.6.

Let
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
be a specialization of triples. Let
$\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ be the induced specialization of
graphs. If $(\mathcal{E},D)$ is $\mu$-polystable on $\Gamma$ and
$(\mathcal{E}^{\prime},D^{\prime})$ is $(v_{0},\iota_{*}(\mu))$-quasistable on
$\Gamma^{\prime}$, then there is a specialization of triples
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime}))$
factoring
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.

###### Proof.

We have specializations
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\iota_{*}(\mathcal{E},D))\rightarrow(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.
By [4, Lemma 5.2] the pseudo-divisor $\iota_{*}(\mathcal{E},D)$ is
$\iota_{*}(\mu)$-polystable on $\Gamma^{\prime}$. By the minimal property of
$\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})$ in Remark 2.2, we have a
specialization
$\iota_{*}(\mathcal{E},D)\geq\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})$,
concluding the proof. ∎

######  Theorem 3.7.

The map $\rho^{\textnormal{trop}}_{P}\colon
J^{\textnormal{trop}}_{\mu,g,n}\rightarrow P^{\textnormal{trop}}_{\mu,g,n}$ is
a refinement of generalized cone complexes. For every $\mu$-polystable pseudo-
divisor $(\mathcal{E},D)$ on a stable graph $\Gamma$ with $n$ legs, we have
decompositions

$\sigma^{\circ}_{(\Gamma,\mathcal{E},D)}=\underset{(\mathcal{E},D)=\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})}{\coprod_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})\in\mathbf{QD}_{\mu,g,n}}}\tau^{\circ}_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}\quad\text{
and
}\quad\sigma_{(\Gamma,\mathcal{E},D)}=\underset{(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}{\coprod_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})\in\mathbf{QD}_{\mu,g,n}}}\tau^{\circ}_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}.$

###### Proof.

First of all, if $\mathcal{D}$ is a $\mu$-polystable divisor on a tropical
curve $X$ equivalent to a $(p_{0},\mu)$-quasistable divisor
$\mathcal{D}^{\prime}$ on $X$, then the combinatorial type of $\mathcal{D}$ is
equal to $\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})$, where
$(\mathcal{E}^{\prime},D^{\prime})$ is the combinatorial type of
$\mathcal{D}^{\prime}$ (see the proof of [4, Theorem 5.9 (1)]). Thus we can
describe locally the map $\rho^{\textnormal{trop}}_{P}$ as follows. Let
$(\mathcal{E}^{\prime},D^{\prime})$ be a $(v_{0},\mu)$-quasistable pseudo-
divisor on a stable graph $\Gamma$ with $n$ legs, and set
$(\mathcal{E},D)=\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})$. Since
$(\mathcal{E}^{\prime},D^{\prime})$ is simple, we have that
$\tau_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}=\sigma_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}$.
Since $(\mathcal{E},D)\geq(\mathcal{E}^{\prime},D^{\prime})$, by Remark 3.3 we
have an inclusion
$\tau_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}=\sigma_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}\hookrightarrow\sigma_{(\Gamma,\mathcal{E},D)}$
which factors the composition

$\tau_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}=\sigma_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}\rightarrow
J_{\mu,g,n}^{\textnormal{trop}}\stackrel{{\scriptstyle\rho^{\textnormal{trop}}_{P}}}{{\rightarrow}}P_{\mu,g,n}^{\textnormal{trop}}.$

Then $\rho^{\textnormal{trop}}_{P}$ is a morphism of generalized cone
complexes. The morphism $\rho^{\textnormal{trop}}_{P}$ is also bijective by
Remark 2.3, hence it is a refinement of generalized cone complexes.

Denote $\mathbf{PSD}:=\mathbf{PSD}_{\mu,g,n}$ and
$\mathbf{QD}:=\mathbf{QD}_{\mu,g,n}$. Given
$(\Gamma,\mathcal{E}^{\prime},D^{\prime})\in\mathbf{QD}$, by [4, Proposition
6.9] we have
$\tau^{\circ}_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}=\sigma^{\circ}_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}\subset\sigma^{\circ}_{(\Gamma,\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime}))}$.
Thus, the decomposition of $\sigma^{\circ}_{(\Gamma,\mathcal{E},D)}$ is as
stated. Consider $(\Gamma,\mathcal{E},D)\in\mathbf{PSD}$. By the definition of
$P_{\mu,g,n}^{\textnormal{trop}}$ and the decomposition just proved, we have:

$\displaystyle\sigma_{(\Gamma,\mathcal{E},D)}$
$\displaystyle=\underset{(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime\prime},D^{\prime\prime})}{\coprod_{(\Gamma^{\prime},\mathcal{E}^{\prime\prime},D^{\prime\prime})\in\mathbf{PSD}}}\sigma^{\circ}_{(\Gamma^{\prime},\mathcal{E}^{\prime\prime},D^{\prime\prime})}$
$\displaystyle=\underset{(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime\prime},D^{\prime\prime})}{\coprod_{(\Gamma^{\prime},\mathcal{E}^{\prime\prime},D^{\prime\prime})\in\mathbf{PSD}}}\;\;\underset{(\mathcal{E}^{\prime\prime},D^{\prime\prime})=\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})}{\coprod_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})\in\mathbf{QD}}}\tau^{\circ}_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}.$

By Lemma 3.6 and by the existence of a specialization
$\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})\geq(\mathcal{E}^{\prime},D^{\prime})$,
the decomposition of $\sigma_{(\Gamma,\mathcal{E},D)}$ is as stated. ∎

##  4\. Root-graphs and their categories

In this section we introduce the notion of pseudo-root and root-graph, and
study stability conditions on these objects. We will define the category of
root-graphs, which will be important in defining later some tropical moduli
spaces.

###  4.1. Pseudo-roots on graphs

Let $g$ be a nonnegative integer. From now on, $\mu$ will be the canonical
polarization of degree $g-1$, defined over a graph $\Gamma$ as the
polarization taking $v\in V(\Gamma)$ to

$\mu(v)=\frac{2w(v)-2+\deg(v)}{2}.$

All the stability conditions will be taken with respect to $\mu$.

######  Definition 4.1.

Let $\Gamma$ be a graph. A _pseudo-root_ on $\Gamma$ is a pseudo-divisor
$(\mathcal{E},D)$ on $\Gamma$ such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$, for some acyclic flow
$\phi$ on $\Gamma^{\mathcal{E}}$. A pseudo-root $(\mathcal{E},D)$ is,
respectively, _stable_ , _semistable_ , _polystable_ , _$v_{0}$ -quasistable_
for some vertex $v_{0}$ of $\Gamma$, if so is the pseudo-divisor
$(\mathcal{E},D)$. (Notice that a pseudo-root has degree $g-1$, where $g$ is
the genus of its underlying graph.)

We will see in Proposition 4.7 that $\phi$ is uniquely determined if
$(\mathcal{E},D)$ is semistable. Given a pseudo root $(\mathcal{E},D)$ on a
graph $\Gamma$ with associated acyclic flow $\phi$, we define

(6) $P_{(\mathcal{E},D)}=\\{e\in E(\Gamma^{\mathcal{E}})|\phi(e)=0\\}.$

######  Remark 4.2.

Let $(\mathcal{E},D)$ be a pseudo-root on a graph $\Gamma$ and $\phi$ be an
acyclic flow on $\Gamma^{\mathcal{E}}$ such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$. Consider the
specialization
$\iota\colon\Gamma^{\mathcal{E}}\rightarrow\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)}$.
Since $\iota_{*}$ is linear on divisors, we have
$2\iota_{*}(D)+\operatorname{Div}(\iota_{*}(\phi))=K_{\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)}}$.
Notice that $\iota_{*}(\phi)$ is acyclic by definition. Then
$(\emptyset,\iota_{*}(D))$ is a pseudo-root on
$\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)}$.

######  Definition 4.3.

A _root-graph_ is a triple $(\Gamma,\mathcal{E},D)$, where $\Gamma$ is a graph
and $(\mathcal{E},D)$ is a pseudo-root on $\Gamma$. A root-graph is _stable_ ,
_semistable_ , _polystable_ , _$v_{0}$ -quasistable_ for some vertex $v_{0}$
of $\Gamma$, if so is its underlying pseudo-root.

######  Lemma 4.4.

Let $(\Gamma,\mathcal{E},D)$ be a root-graph. Let $\phi$ be an acyclic flow on
$\Gamma^{\mathcal{E}}$ such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$. The pseudo-divisor
$(\mathcal{E},D)$ is stable if and only if $\phi=0$.

###### Proof.

If $(\mathcal{E},D)$ is stable, then $\mathcal{E}=\emptyset$. By Proposition
2.1 and Remark 4.2, we can assume $\phi(e)\neq 0$,
$\forall\;e\in\overrightarrow{E}(\Gamma)$. Assume that
$E(\Gamma)\neq\emptyset$ and let $v\in V(\Gamma)$ be a sink of $\phi$. Thus
$\operatorname{Div}(\phi)(v)\geq\deg_{\Gamma}(v)$. The relation
$2D(v)+\operatorname{Div}(\phi)(v)=K_{\Gamma}(v)$ gives us

$D(v)\leq-\frac{\deg_{\Gamma}(v)}{2}+w_{\Gamma}(v)-1+\frac{\deg_{\Gamma}(v)}{2}=\mu(v)-\frac{\deg_{\Gamma}(v)}{2}.$

Hence $\beta_{D}(v)\leq 0$, and $D$ is not $\mu$-stable, a contradiction.

Conversely, if $\phi=0$, then for every proper subset $W\subset
V(\Gamma^{\mathcal{E}})$ we have

$\beta_{D}(W)=\deg D|_{W}-\mu(W)+\frac{\delta_{W}}{2}=\frac{\deg
K_{\Gamma^{\mathcal{E}}}}{2}-\mu(W)+\frac{\delta_{W}}{2}=\frac{\delta_{W}}{2}.$

Hence $\beta_{D}(W)>0$ and $(\mathcal{E},D)$ is stable. ∎

######  Lemma 4.5.

Let $(\Gamma,\mathcal{E},D)$ be a semistable root graph. Let $\phi$ be an
acyclic flow on $\Gamma^{\mathcal{E}}$ such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$. Then $|\phi(e)|\leq 1$
for every $e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$.

###### Proof.

By Proposition 2.1 and Remark 4.2, we can assume $\phi(e)\neq 0$, for every
$e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$. By contradiction, suppose that
$\phi(e_{0})\geq 2$ for some
$e_{0}\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$. We let $V$ be the set of
vertices $v\in V(\Gamma^{\mathcal{E}})$ for which there is a positive
$\phi$-path from $s(e_{0})$ to $v$. Then $t(e_{0})\in V$ and, since $\phi$ is
acyclic, we have $s(e_{0})\in V^{c}$, so $V$ is a proper nonempty subset of
$V(\Gamma)$. By the definition of $V$, we have $\phi(e)\geq 1$ for every
$e\in\overrightarrow{E}(V,V^{c})$ with $t(e)\in V$. Hence:

(7) $\deg(\operatorname{Div}(\phi)|_{V})=\phi(e_{0})+\underset{t(e)\in
V}{\sum_{e\in\overrightarrow{E}(V,V^{c})\setminus\\{e_{0}\\}}}\phi(e)>\\#E(V,V^{c})=\delta_{\Gamma^{\mathcal{E}},V}.$

On the other hand, we have

$\deg(\operatorname{Div}(\phi)|_{V})=\deg(K_{\Gamma^{\mathcal{E}}}|_{V})-2\deg(D|_{V})=2\mu(V)-2\deg(D|_{V})\leq\delta_{\Gamma^{\mathcal{E}},V},$

where we use that $\beta_{D}(V)\geq 0$ by the $\mu$-semistability of $D$, and
this contradicts equation (7). ∎

######  Lemma 4.6.

Let $(\Gamma,\mathcal{E},D)$ be a semistable root-graph. Let $\phi$ be an
acyclic flow on $\Gamma^{\mathcal{E}}$ such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$. Given a subset
$V\subset V(\Gamma^{\mathcal{E}})$, we have $\beta_{D}(V)=0$ if and only if
$\phi(e)=1$ for every $e\in\overrightarrow{E}(V,V^{c})$ such that $t(e)\in V$.

###### Proof.

First of all, we have

(8)
$\deg(\operatorname{Div}(\phi)|_{V})=\deg(K_{\Gamma^{\mathcal{E}}}|_{V})-2\deg
D|_{V}=2\mu(V)-2\deg D|_{V}.$

Thanks to equation (8), the condition $\beta_{D}(V)=\deg
D|_{V}-\mu(V)+\frac{\delta_{\Gamma^{\mathcal{E}},V}}{2}=0$ is equivalent to
the condition
$\deg(\operatorname{Div}(\phi)|_{V})=\delta_{\Gamma^{\mathcal{E}},V}$. We are
done by Lemma 4.5. ∎

######  Theorem 4.7.

Let $(\Gamma,\mathcal{E},D)$ be a semistable root-graph. Let $\phi$ be an
acyclic flow on $\Gamma^{\mathcal{E}}$ such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$. Then

  1. (1)

$\phi(e)=1$ for every $e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$ such that
$t(e)$ is an exceptional vertex of $\Gamma^{\mathcal{E}}$;

  2. (2)

$\phi$ is the unique acyclic flow such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$.

###### Proof.

Let us prove item (1). Let $e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$ such
that $t(e)$ is an exceptional vertex of $\Gamma^{\mathcal{E}}$. The relation
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$ implies that
$\operatorname{Div}(\phi)(t(e))=2$. Since, by Lemma 4.5, we have $\phi(e)\leq
1$, we deduce that $\phi(e)=1$.

Let us prove item (2). Notice that $(\Gamma^{\mathcal{E}},\emptyset,D)$ is a
semistable root-graph. So assume $\mathcal{E}=\emptyset$. We proceed by
induction on $\\#E(\Gamma)$, being the statement clear for $\\#E(\Gamma)=0$.
If $(\emptyset,D)$ is stable, then we are done by Lemma 4.4. So we can assume
that there is a proper subset $V\subset V(\Gamma)$ such that $\beta_{D}(V)=0$.
By Lemma 4.6 we know that $\phi(e)=1$ for every
$e\in\overrightarrow{E}(V,V^{c})$ such that $t(e)\in V$. Set
$\Gamma_{1}=\Gamma(V)$ and $\Gamma_{2}=\Gamma(V^{c})$. Define:

(9) $D_{1}:=D|_{\Gamma_{1}}\quad\text{ and }\quad
D_{2}:=D|_{\Gamma_{2}}-\sum_{v\in V^{c}}\deg_{E(V,V^{c})}(v)v.$

We claim that $(\Gamma_{i},\emptyset,D_{i})$ is a semistable root-graph for
$i=1,2$. Indeed, consider the flow $\phi_{i}:=\phi|_{\Gamma_{i}}$ on
$\Gamma_{i}$, for $i\in\\{1,2\\}$. First of all, we have:

$\displaystyle 2D_{1}$ $\displaystyle=2D|_{\Gamma_{1}}$
$\displaystyle=K_{\Gamma}|_{\Gamma_{1}}-\operatorname{Div}(\phi)|_{\Gamma_{1}}$
$\displaystyle=K_{\Gamma_{1}}+\sum_{v\in
V}\deg_{E(V,V^{c})}(v)v-\operatorname{Div}(\phi_{1})-\underset{t(e)\in
V}{\sum_{e\in\overrightarrow{E}(V,V^{c})}}\phi(e)t(e)$
$\displaystyle=K_{\Gamma_{1}}-\operatorname{Div}(\phi_{1})$

and

$\displaystyle 2D_{2}$ $\displaystyle=2D|_{\Gamma_{2}}-2\sum_{v\in
V^{c}}\deg_{E(V,V^{c})}(v)v$
$\displaystyle=K_{\Gamma}|_{\Gamma_{2}}-\operatorname{Div}(\phi)|_{\Gamma_{2}}-2\sum_{v\in
V^{c}}\deg_{E(V,V^{c})}(v)v$ $\displaystyle=K_{\Gamma_{2}}-\sum_{v\in
V^{c}}\deg_{E(V,V^{c})}(v)v-\operatorname{Div}(\phi_{2})-\underset{t(e)\in
V^{c}}{\sum_{e\in\overrightarrow{E}(V,V^{c})}}\phi(e)t(e)$
$\displaystyle=K_{\Gamma_{2}}-\operatorname{Div}(\phi_{2}).$

We now check the semistability of $(\Gamma_{i},\emptyset,D_{i})$. Let
$\mu_{i}$ be the canonical polarization of degree $g_{\Gamma_{i}}-1$ on
$\Gamma_{i}$, which is the polarization taking a vertex $v\in V(\Gamma_{i})$
to

$\mu_{i}(v)=\mu(v)-\frac{\deg_{E(V,V^{c})}(v)}{2}.$

For every subset $W\subset V(\Gamma_{1})$, we have

$\displaystyle\beta_{D_{1}}(W)$
$\displaystyle=\deg(D_{1}|_{W})-\mu_{1}(W)+\frac{\delta_{\Gamma_{1},W}}{2}$
$\displaystyle=\deg(D|_{W})-\mu(W)+\frac{\\#E(V^{c},W)}{2}+\frac{\delta_{\Gamma,W}-\\#E(V^{c},W)}{2}$
$\displaystyle=\beta_{D}(W)\geq 0,$

where the last inequality follows by the $\mu$-semistability of $D$ on
$\Gamma$. So the root-graph $(\Gamma_{1},\emptyset,D_{1})$ is
$\mu_{1}$-semistable. For every subset $W\subset V(\Gamma_{2})$, we have

$\displaystyle\beta_{D_{2}}(W)$
$\displaystyle=\deg(D_{2}|_{W})-\mu_{2}(W)+\frac{\delta_{\Gamma_{2},W}}{2}$
$\displaystyle=\deg(D|_{W})-\\#E(V,W)-\mu(W)+\frac{\\#E(V,W)}{2}+\frac{\delta_{\Gamma,W}-\\#E(V,W)}{2}$
$\displaystyle=\beta_{D}(W)-\\#E(V,W)$ $\displaystyle=\beta_{D}(V\cup
W)-\beta_{D}(V)+\\#E(V,W)-\\#E(V,W)$ $\displaystyle=\beta_{D}(V\cup W)\geq 0,$

where we use [2, Lemma 4.1] and the $\mu$-semistability of $D$ on $\Gamma$. So
the root-graph $(\Gamma_{2},\emptyset,D_{2})$ is $\mu_{2}$-semistable. This
concludes the proof of the claim.

By the induction hypothesis, $\phi_{1}$ and $\phi_{2}$ are uniquely
determined, so is $\phi$. ∎

######  Notation 4.8.

Given a semistable root-graph $(\Gamma,\mathcal{E},D)$, we denote by
$\phi_{(\mathcal{E},D)}$ the unique acyclic flow on $\Gamma^{\mathcal{E}}$
such that
$2D+\operatorname{Div}(\phi_{(\mathcal{E},D)})=K_{\Gamma^{\mathcal{E}}}$. In
Figure 1 we see the possibilities for the value $\phi_{(\mathcal{E},D)}(e)$
for $e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$.

$0$$1$$1$$1$exceptional vertex Figure 1. Possible values for the flow
$\phi_{(\mathcal{E},D)}$.

######  Remark 4.9.

Since $\phi(e^{\prime})\neq 0$ for every $e^{\prime}$ lying over an edge
$e\in\overrightarrow{\mathcal{E}}$, the graph $P_{(\mathcal{E},D)}$ defined in
equation (6) can be seen as a subgraph of either one of the following graphs:
$\Gamma_{\mathcal{E}}$, $\Gamma$, $\Gamma^{\mathcal{E}}$.

We denote by
$\overline{P}_{(\mathcal{E},D)}=\Gamma^{\mathcal{E}}_{\mathcal{F},\operatorname{leg}}$,
where $\mathcal{F}=E(\Gamma^{\mathcal{E}})\setminus P_{(\mathcal{E},D)}$. The
set of connected components of $\overline{P}_{(\mathcal{E},D)}$ are in natural
bijection with the set $V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$.

######  Example 4.10.

Let $\Gamma$ be the graph with two vertices $v_{0}$ and $v_{1}$ of weight
zero, connected by two edges $e_{0},e_{1}$. Let $\mathcal{E}=\\{e_{0}\\}$ and
$D=v_{0}-w_{0}\in\operatorname{Div}(\Gamma^{\mathcal{E}})$, where $w_{0}$ is
the exceptional vertex of $e_{0}$. Then $(\mathcal{E},D)$ is a
$v_{0}$-quasistable pseudo-root (hence semistable), and
$\phi_{(\mathcal{E},D)}$ has value $1$ over every
$e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$ oriented as in Figure 2. In
this case, $P_{(\mathcal{E},D)}=0$, and $\overline{P}_{(\mathcal{E},D)}$
consists of the disjoint union of the vertices $v_{0},v_{1},w_{0}$, with two
legs attached to each vertex and no edge.

$v_{0}$$v_{1}$$w_{0}$ Figure 2. The orientation on the graph
$\Gamma^{\mathcal{E}}$.

######  Definition 4.11.

Given a semistable pseudo-root $(\Gamma,\mathcal{E},D)$, the
_$\phi_{(\mathcal{E},D)}$ -graphs_ of $\Gamma^{\mathcal{E}}$ are the connected
components $P_{u}$ of $\overline{P}_{(\mathcal{E},D)}$, for $u\in
V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$.

######  Remark 4.12.

Let $(\Gamma,\mathcal{E},D)$ be a semistable root-graph. Consider a vertex
$u\in V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$. Using that
$2D+\operatorname{Div}(\phi_{(\mathcal{E},D)})=K_{\Gamma^{\mathcal{E}}}$, we
deduce that

$2D|_{P_{u}}+\sum_{w\neq u}\underset{t(e)\in
V(P_{u})}{\sum_{e\in\overrightarrow{E}(P_{u},P_{w})}}(\phi_{(\mathcal{E},D)}(e)-1)t(e)=K_{P_{v}}.$

######  Definition 4.13.

We say that a semistable root-graph $(\Gamma,\mathcal{E},D)$ _specializes_ to
a semistable root-graph $(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$,
and we write
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$,
if there is a specialization of triples
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
with associated specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$
such that
$\widehat{\iota}_{*}(\phi_{(\mathcal{E},D)})=\phi_{(\mathcal{E}^{\prime},D^{\prime})}$.

We consider the category $\mathbf{SSR}_{g,n}$ whose objects are semistable
roots-graphs $(\Gamma,\mathcal{E},D)$, where $\Gamma$ is a stable genus-$g$
graph with $n$ legs, and the arrows are given by the specializations of
semistable roots-graphs. We let $\mathcal{SSR}_{g,n}$ be the poset
$\mathbf{SSR}_{g,n}/\sim$, where the relation $\sim$ is isomorphism.

######  Remark 4.14.

Not every semistable pseudo divisor $(\mathcal{E},D)$ on a graph $\Gamma$ is
such that $(\Gamma,\mathcal{E},D)$ is an object of $\mathbf{SSR}_{g,n}$. Let
$\Gamma$ be the graph with two vertices $v_{0}$ and $v_{1}$ of weight zero,
and four edges connecting them. Then $(\emptyset,2v_{0})$ is a
$v_{0}$-quasistable pseudo-divisor (and hence it is semistable), but there is
no acyclic flow $\phi$ on $\Gamma$ such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma}$.

######  Remark 4.15.

It is not true that every specialization
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
of triples is a morphism in $\mathbf{SSR}_{g,n}$, for objects
$(\Gamma,\mathcal{E},D)$ and
$(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$ in $\mathbf{SSR}_{g,n}$.
Let $\Gamma$ and $(\mathcal{E},D)$ be the graph and the pseudo-divisor on it
as in Example 4.10. The pseudo-divisor $(\mathcal{E}^{\prime},D^{\prime})$ on
$\Gamma$, where $\mathcal{E}^{\prime}=\emptyset$ and $D^{\prime}=0$, is a
$v_{0}$-quasistable pseudo-root, with
$\phi_{(\mathcal{E}^{\prime},D^{\prime})}=0$. We have a specialization
$\iota\colon(\Gamma,\mathcal{E},D)\geq(\Gamma,\mathcal{E}^{\prime},D^{\prime})$
but, if
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$
is the associated specialization, then $\widehat{\iota}_{*}(\phi)\neq 0$: this
specialization is not a morphism in $\mathbf{SSR}_{g,n}$.

We denote by $\mathbf{SR}_{g,n}$, $\mathbf{PSR}_{g,n}$ and
$\mathbf{QR}_{g,n}$, respectively, the subcategory of $\mathbf{SSR}_{g,n}$ of
simple semistable, polystable and $v_{0}$-quasistable root-graphs. We let
$\mathcal{SR}_{g,n}$, $\mathcal{PSR}_{g,n}$ and $\mathcal{QR}_{g,n}$ be
respectively the associated posets. Notice that $\mathbf{PSR}_{g,n}$ is a
subcategory of $\mathbf{PSD}_{g,n}$, while $\mathbf{QR}_{g,n}$ is a sub-
category of both $\mathbf{QD}_{g,n}$ and of $\mathbf{SR}_{g,n}$ (by [2,
Proposition 4.6]).

###  4.2. Polystable root-graphs

In this section we analyze more closely polystable root-graphs. We show that
the category of polystable pseudo-roots can be identified with the one of
cyclic subgraphs.

######  Lemma 4.16.

Let $(\mathcal{E},D)$ be a polystable pseudo-root on a graph $\Gamma$. We have
$\phi_{(\mathcal{E},D)}(e)=0$ if and only if
$e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$ is a non-exceptional oriented
edge. In particular, $\operatorname{Div}(\phi_{(\mathcal{E},D)})$ is a
principal divisor on $\Gamma^{\mathcal{E}}$, with
$\operatorname{Div}(\phi_{(\mathcal{E},D)})=-\sum_{e\in\mathcal{E}}\operatorname{Div}(w_{e})$,
where $w_{e}$ is the exceptional vertex in the interior of $e$.

###### Proof.

Since $(\mathcal{E},D)$ is polystable, we have that the divisor
$D_{\mathcal{E}}$ is $\mu_{\mathcal{E}}$-stable on $\Gamma_{\mathcal{E}}$.
Thus $D_{1}=D_{\mathcal{E}}-\sum_{v\in
V(\Gamma^{\mathcal{E}})}\text{deg}_{\mathcal{E}}(v)$ is $\mu_{1}$-stable,
where $\mu_{1}$ is the canonical divisor of degree
$g_{\Gamma^{\mathcal{E}}}-1$ of $\Gamma^{\mathcal{E}}$. Set
$\phi=\phi_{(\mathcal{E},D)}$. By Theorem 4.7 (1), we have
$2D_{1}+\operatorname{Div}(\phi|_{\Gamma_{\mathcal{E}}})=K_{\Gamma_{\mathcal{E}}}$,
then by Lemma 4.4 we have that $\phi|_{\Gamma_{\mathcal{E}}}=0$, that is,
$\phi(e)=0$ for every non-exceptional oriented edge
$e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$. We can conclude using Theorem
4.7 (1). ∎

As in [12, Section 4], we let $\operatorname{{\bf CYC}}_{g,n}$ be the category
whose objects are pairs $(\Gamma,P)$, where $\Gamma$ is a stable genus-$g$
graph with $n$ legs and $P$ is a cyclic subgraph of $\Gamma$, and whose arrows
$(\Gamma,P)\rightarrow(\Gamma^{\prime},P^{\prime})$ are given by a
specialization $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ of graphs with
$n$ legs such that $\iota_{*}(P)=P^{\prime}$.

For every graph $\Gamma$ and a cyclic subgraph $P$ of $\Gamma$, we denote by
$(\mathcal{E}_{P},D_{P})$ the pseudo-divisor on $\Gamma$ such that
$\mathcal{E}_{P}=E(\Gamma)\setminus P$ and

(10) $D_{P}=\sum_{v\in
V(\Gamma)}\left(\deg_{\Gamma}(v)-\frac{\deg_{P}(v)}{2}-1+w_{\Gamma}(v)\right)v-\sum_{e\in\mathcal{E}_{P}}w_{e}$

where $w_{e}$ is the exceptional vertex inside $e$.

######  Proposition 4.17.

Let $\Gamma$ be a graph. The pseudo-divisors $(\mathcal{E}_{P},D_{P})$, for
$P$ a cyclic subgraph of $\Gamma$, are all the polystable pseudo-roots on
$\Gamma$. In particular, $\Gamma$ admits exactly $2^{b_{1}(\Gamma)}$
polystable pseudo-roots.

###### Proof.

Let $\phi$ be the acyclic flow associated to the principal divisor
$-\sum_{e\in\mathcal{E}_{P}}\operatorname{Div}(w_{e})$. We have:

$2D_{P}+\operatorname{Div}(\phi)=\sum_{v\in
V(\Gamma)}\left(2\deg_{\Gamma}(v)-\deg_{P}(v)-2+2w_{\Gamma}(v)-\deg_{\mathcal{E}_{P}}(v)\right)v=K_{\Gamma^{\mathcal{E}}}.$

Moreover, since $\mu$ is the canonical polarization, we have:

$\mu_{\mathcal{E}_{P}}(v)=\mu(v)+\frac{\deg_{\mathcal{E}_{P}}(v)}{2}=(D_{P})_{\mathcal{E}_{P}}(v)$

for every $v\in V(\Gamma)$. Thus the divisor $(D_{P})_{\mathcal{E}_{P}}$ is
$\mu_{\mathcal{E}_{P}}$-stable on $\Gamma_{\mathcal{E}_{P}}$, so
$(\mathcal{E}_{P},D_{P})$ is polystable. We deduce that
$(\mathcal{E}_{P},D_{P})$ is a polystable pseudo-root of $\Gamma$.

Conversely, let $(\mathcal{E},D)$ be a polystable pseudo-divisor on $\Gamma$.
By Lemma 4.16:

$\displaystyle 2D$ $\displaystyle=$ $\displaystyle
K_{\Gamma^{\mathcal{E}}}+\sum_{e\in\mathcal{E}}\operatorname{Div}(w_{e})$
$\displaystyle=$ $\displaystyle\sum_{v\in
V(\Gamma)}(\deg_{\Gamma}(v)-2+2w_{\Gamma}(v)+\deg_{\mathcal{E}}(v))v-2\sum_{e\in\mathcal{E}}w_{e}$
$\displaystyle=$ $\displaystyle\sum_{v\in
V(\Gamma)}(2\deg_{\Gamma}(v)-\deg_{E(\Gamma)\setminus\mathcal{E}}(v)-2+2w_{\Gamma}(v))v-2\sum_{e\in\mathcal{E}}w_{e}.$

If we set $P:=E(\Gamma)\setminus\mathcal{E}$, we see that $\deg_{P}(v)$ is
even for every $v\in V(\Gamma)$, that is $P$ is a cyclic subgraph of $\Gamma$,
and $(\mathcal{E},D)=(\mathcal{E}_{P},D_{P})$, as wanted. ∎

######  Remark 4.18.

Let $(\mathcal{E},D)$ be a polystable pseudo-root on a graph $\Gamma$. If
$(\mathcal{E},D)=(\mathcal{E}_{P},D_{P})$ for some cyclic subgraph $P$ of
$\Gamma$, then $P=P_{(\mathcal{E},D)}$ by Lemma 4.16.

######  Proposition 4.19.

Let $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ be a specialization of
graphs. Given a cyclic subgraph $P$ of $\Gamma$, we have
$\iota_{*}(\mathcal{E}_{P},D_{P})=(\mathcal{E}_{\iota_{*}(P)},D_{\iota_{*}(P)})$
and
$\iota_{*}(\phi_{(\mathcal{E}_{P},D_{P})})=\phi_{(\mathcal{E}_{\iota_{*}(P)},D_{\iota_{*}(P)})}$.

###### Proof.

First of all, notice that
$\iota_{*}(\mathcal{E}_{P})=\mathcal{E}_{\iota_{*}(P)}$. We now show that
$\iota_{*}(D_{P})=D_{\iota_{*}(P)}$. We can assume that $\iota$ is the
contraction of just one edge $e\in E(\Gamma)$. Let $v_{1}$ and $v_{2}$ be the
(possibly coinciding) vertices of $\Gamma$ incident to $e$ and let $v_{0}$ be
the vertex of $\Gamma^{\prime}$ to which $e$ contracts. Moreover, we have that
$\iota_{*}(D_{P})(v)=D_{\iota_{*}P}(v)$ for every $v\in
V({\Gamma^{\prime}}^{\mathcal{E}_{\iota_{*}(P)}})$ with $v\neq v_{0}$. If $e$
is not a loop, then

$\displaystyle\iota_{*}(D_{P})(v_{0})$ $\displaystyle=$
$\displaystyle\deg_{\Gamma}(v_{1})+\deg_{\Gamma}(v_{2})-\frac{\deg_{P}(v_{1})}{2}-\frac{\deg_{P}(v_{2})}{2}-2+$
$\displaystyle+w_{\Gamma}(v_{1})+w_{\Gamma}(v_{2})-[e\in\mathcal{E}]$
$\displaystyle=$
$\displaystyle\deg_{\Gamma^{\prime}}(v_{0})-\left(\frac{\deg_{P}(v_{1})}{2}+\frac{\deg_{P}(v_{2})}{2}+[e\in\mathcal{E}]\right)+w_{\Gamma^{\prime}}(v_{0})$
$\displaystyle=$
$\displaystyle\deg_{\Gamma^{\prime}}(v_{0})-\left(\frac{\deg_{\iota_{*}(P)}(v_{0})}{2}+1\right)+w_{\Gamma^{\prime}}(v_{0})$
$\displaystyle=$ $\displaystyle D_{\iota_{*}(P)}(v_{0}).$

On the other hand, if $e$ is a loop, we set $\overline{v}:=v_{1}=v_{2}$. Then

$\displaystyle\iota_{*}(D_{P})(v_{0})$ $\displaystyle=$
$\displaystyle\deg_{\Gamma}(\overline{v})-\frac{\deg_{P}(\overline{v})}{2}-1+w_{\Gamma}(\overline{v})-[e\in\mathcal{E}]$
$\displaystyle=$
$\displaystyle\deg_{\Gamma^{\prime}}(v_{0})-\left(\frac{\deg_{P}(\overline{v})}{2}+[e\in\mathcal{E}]\right)+\left(1+w_{\Gamma}(\overline{v})\right)$
$\displaystyle=$
$\displaystyle\deg_{\Gamma^{\prime}}(v_{0})-\left(\frac{\deg_{\iota_{*}(P)}(v_{0})}{2}+1\right)+w_{\Gamma^{\prime}}(v_{0})$
$\displaystyle=$ $\displaystyle D_{\iota_{*}(P)}(v_{0}).$

Summing up, we get
$\iota_{*}(\mathcal{E}_{P},D_{P})=(\mathcal{E}_{\iota_{*}(P)},D_{\iota_{*}(P)})$.
By Lemma 4.16, we also have
$\iota_{*}(\phi_{(\mathcal{E}_{P},D_{P})})=\phi_{(\mathcal{E}_{\iota_{*}(P)},D_{\iota_{*}(P)})}$,
and we are done. ∎

######  Corollary 4.20.

Let $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ be a specialization of
graphs. If $(\Gamma,\mathcal{E},D)$ is a polystable root-graph, then
$(\Gamma^{\prime},\iota_{*}(\mathcal{E},D))$ is a polystable root-graph, and
we have a specialization of polystable root-graphs
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\iota_{*}(\mathcal{E},D))$.

###### Proof.

The result readily follows from Propositions 4.17 and 4.19. ∎

######  Proposition 4.21.

Let
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
be a specialization of polystable root-graphs with associated specializations
$\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ and
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$.
Then $\mathcal{E}^{\prime}=\mathcal{E}\cap E(\Gamma^{\prime})$, so in
particular $\widehat{\iota}$ is the specialization induced by $\iota$.

###### Proof.

Let $e_{1},e_{2}$ be different edges of $\Gamma^{\mathcal{E}}$ incident to the
same exceptional vertex of $\Gamma^{\mathcal{E}}$. If $\widehat{\iota}$
contracts $e_{1}$, then $\widehat{\iota}$ contracts $e_{2}$ as well, otherwise
$e_{2}$ would be a non-exceptional edge of
${\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$ and hence
$\phi_{(\mathcal{E}^{\prime},D^{\prime})}(e_{2})=0$ by Lemma 4.16. But
$\phi_{(\mathcal{E}^{\prime},D^{\prime})}(e_{2})=\widehat{\iota}_{*}(\phi_{(\mathcal{E},D)})(e_{2})=1$
by Theorem 4.7, a contradiction. ∎

######  Corollary 4.22.

The categories $\mathbf{PSR}_{g,n}$ and $\operatorname{{\bf CYC}}_{g,n}$ are
canonically isomorphic.

###### Proof.

This is a consequence of Propositions 4.17, 4.19, 4.21. ∎

######  Lemma 4.23.

Let $(\mathcal{E},D)$ be a semistable pseudo-root on a graph $\Gamma$. Assume
that $V\subset V(\Gamma)$ is a subset with $\beta_{\mathcal{E},D}(V)=0$ and
$E(V,V^{c})\not\subset\mathcal{E}$. Then there is a unique semistable pseudo-
root $(\mathcal{E}\cup E(V,V^{c}),D_{1})$ on $\Gamma$ admitting a
specialization $(\Gamma,\mathcal{E}\cup
E(V,V^{c}),D_{1})\geq(\Gamma,\mathcal{E},D)$ of semistable root-graphs.
Moreover, we have $P_{(\mathcal{E}\cup
E(V,V^{c}),D_{1})}=P_{(\mathcal{E},D)}$.

###### Proof.

By [4, Proposition 5.3] the pseudo-divisor $(\mathcal{E}\cup
E(V,V^{c}),D_{1})$ on $\Gamma$, where

$D_{1}(v)=\begin{cases}D(v)+\text{deg}_{E(V,V^{c})\setminus\mathcal{E}}(v)&\text{
if $v\in V$,}\\\ D(v)&\text{ if $v\notin V$,}\\\ -1&\text{ if $v$ is
exceptional,}\end{cases}$

is the unique semistable pseudo-divisor on $\Gamma$ admitting a specialization
of pseudo-divisors $(\mathcal{E}\cup
E(V,V^{c}),D_{1})\rightarrow(\mathcal{E},D)$. The associated specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}\cup
E(V,V^{c})}\rightarrow\Gamma^{\mathcal{E}}$ gives rise to a natural inclusion
$E(\Gamma^{\mathcal{E}})\subset E(\Gamma^{\mathcal{E}\cup E(V,V^{c})})$. We
define the flow $\phi_{1}$ on $\Gamma^{\mathcal{E}\cup E(V,V^{c})}$ such that
$(\phi_{1})|_{\overrightarrow{E}(\Gamma^{\mathcal{E}})}=\phi_{(\mathcal{E},D)}$
and $\phi_{1}(e)=1$ for every $e\in\overrightarrow{E}(\Gamma^{\mathcal{E}\cup
E(V,V^{c})})\setminus\overrightarrow{E}(\Gamma^{\mathcal{E}})$ such that
$t(e)$ is an exceptional vertex. We define

$\widetilde{V}=V\cup\\{w_{e}|e\in\mathcal{E}\cap(E(V)\cup E(V,V^{c}))\\},$

where $w_{e}$ is the exceptional vertex inside $e\in\mathcal{E}$. It is easy
to check that $\beta_{D}(\widetilde{V})=\beta_{\mathcal{E},D}(V)=0$. So, by
Lemma 4.6, we have that $\phi_{(\mathcal{E},D)}(e)=1$ for every
$e\in\overrightarrow{E}(V,V^{c})\setminus\overrightarrow{\mathcal{E}}\subset\overrightarrow{E}(\widetilde{V},\widetilde{V}^{c})$
such that $t(e)\in V$. We deduce that
$\widehat{\iota}_{*}(\phi_{1})=\phi_{(\mathcal{E},D)}$.

Notice that $\phi_{1}$ is an acyclic flow, since $\phi_{(\mathcal{E},D)}$ is
acyclic and $\widehat{\iota}^{S}_{*}(P)$ is a non-null
$\phi_{(\mathcal{E},D)}$-cycle on $\Gamma^{\mathcal{E}}$ for every non-null
$\phi_{1}$-cycle $P$ on $\Gamma^{\mathcal{E}\cup E(V,V^{c})}$.

Next, we already observed that $\phi_{(\mathcal{E},D)}(e)=1$ for every
$e\in\overrightarrow{E}(V,V^{c})\setminus\overrightarrow{\mathcal{E}}$ such
that $t(e)\in V$. Then for every $v\in V(\Gamma^{\mathcal{E}\cup E(V,V^{c})})$
we have:

$\operatorname{Div}(\phi_{1})(v)=\begin{cases}\operatorname{Div}(\phi)(v)-2\text{deg}_{E(V,V^{c})\setminus\mathcal{E}}(v)&\text{
if $v\in V$,}\\\ \operatorname{Div}(\phi)(v)&\text{ if $v\notin V$,}\\\
2&\text{ if $v$ is exceptional}\end{cases}$

and hence

$2D_{1}(v)+\operatorname{Div}(\phi_{1})(v)=\begin{cases}2D(v)+\operatorname{Div}(\phi)(v)&\text{
if $v\in V(\Gamma)$,}\\\ 0&\text{ if $v$ is exceptional.}\end{cases}$

Since $2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$, it follows that
$2D_{1}+\operatorname{Div}(\phi_{1})=K_{\Gamma^{\mathcal{E}\cup E(V,V^{c})}}$,
and hence $(\mathcal{E}\cup E(V,V^{c}),D_{1})$ is a semistable pseudo-root on
$\Gamma$. By construction, we have a specialization of semistable root-graphs
$(\Gamma,\mathcal{E}\cup E(V,V^{c}),D_{1})\geq(\Gamma,\mathcal{E},D)$ and
$P_{(\mathcal{E}\cup E(V,V^{c}),D_{1})}=P_{(\mathcal{E},D)}$. ∎

######  Proposition 4.24.

Let $(\Gamma,\mathcal{E},D)$ be a semistable root-graph. Then:

  * (1)

$(\Gamma,\operatorname{pol}(\mathcal{E},D))$ is a polystable root-graph; it is
the unique minimal polystable root-graph admitting a specialization of
semistable root-graphs to $(\Gamma,\mathcal{E},D)$;

  * (2)

$P_{(\mathcal{E},D)}=P_{\operatorname{pol}(\mathcal{E},D)}$.

###### Proof.

A semistable pseudo-root $(\mathcal{E},D)$ on $\Gamma$ is polystable if and
only if $\beta_{\mathcal{E},D}(V)>0$ for every subset $V\subset V(\Gamma)$
such that $E(V,V^{c})\not\subset\mathcal{E}$. Therefore, by Lemma 4.23, we can
construct a sequence of specializations of semistable root-graphs

$(\Gamma,\mathcal{E}_{k},D_{k})>(\Gamma,\mathcal{E}_{k-1},D_{k-1})>\dots>(\Gamma,\mathcal{E}_{0},D_{0})=(\Gamma,\mathcal{E},D),$

where $(\mathcal{E}_{k},D_{k})$ is polystable and $(\mathcal{E}_{j},D_{j})$ is
not polystable for $j<k$. The sequence of specializations above is the same as
in [4, Equation 21]. By [4, Proposition 5.4] we have that
$(\mathcal{E}_{k},D_{k})=\operatorname{pol}(\mathcal{E},D)$, which proves (1).
By Lemma 4.23 we also have $P_{(\mathcal{E}_{i},D_{i})}=P_{(\mathcal{E},D)}$
for every $i\in\\{1,\dots,k\\}$, which proves (2). ∎

######  Corollary 4.25.

If $(\Gamma,\mathcal{E},D)$ is a semistable root-graph, then
$P_{(\mathcal{E},D)}$ is a cyclic subgraph of $\Gamma$. In particular no
connected component of $\overline{P}_{(\mathcal{E},D)}$ is a tree with at
least one edge.

###### Proof.

The result follows from Remark 4.18 and Proposition 4.24. ∎

###  4.3. Spin pseudo-roots

In this section we consider an additional structure to root-graphs. This is a
sign function which, as we shall see later, is the combinatorial counterpart
for the parity of the number of sections of a sheaf on curves.

######  Definition 4.26.

Let $\Gamma$ be a graph. A _spin semistable pseudo-root_ on $\Gamma$ is a
triple $(\mathcal{E},D,s)$, where $(\mathcal{E},D)$ is a semistable pseudo-
root on $\Gamma$ and $s\colon
V(\Gamma/P_{(\mathcal{E},D)})\rightarrow\mathbb{Z}/2\mathbb{Z}$ is a function,
called _sign function on $(\Gamma,\mathcal{E},D)$_, such that $s(v)=0$ for
every vertex $v\in V(\Gamma/P_{(\mathcal{E},D)})$ of weight
$w_{\Gamma/P_{(\mathcal{E},D)}}(v)=0$. We say that $(\mathcal{E},D,s)$ is
_polystable_ (respectively, _$v_{0}$ -quasistable for some $v_{0}\in
V(\Gamma)$_) if so is $(\mathcal{E},D)$. A _spin semistable_ (_polystable,
$v_{0}$-quasistable_) _root-graph_ is a tuple $(\Gamma,\mathcal{E},D,s)$,
where $\Gamma$ is a graph and $(\mathcal{E},D,s)$ is a spin semistable
(polystable, $v_{0}$-quasistable) pseudo-root on $\Gamma$. We say that
$(\Gamma,\mathcal{E},D,s)$ is _even_ or _odd_ , respectively, if so is
$\sum_{v\in V(\Gamma/P_{(\mathcal{E},D)})}s(v)$.

######  Lemma 4.27.

If
$\iota\colon(\Gamma,\mathcal{E},D)\rightarrow(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is a specialization of semistable root-graphs, then
$P_{(\mathcal{E}^{\prime},D^{\prime})}=\iota_{*}(P_{(\mathcal{E},D)})$. In
particular, we have an induced specialization
$\widetilde{\iota}\colon\Gamma/P_{(\mathcal{E},D)}\rightarrow\Gamma^{\prime}/P_{(\mathcal{E}^{\prime},D^{\prime})}$
giving rise to a commutative diagram

$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\Gamma^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma/P_{(\mathcal{E},D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\iota}}$$\textstyle{\Gamma^{\prime}/P_{(\mathcal{E}^{\prime},D^{\prime})}}$

###### Proof.

The proof is left to the reader. ∎

######  Definition 4.28.

A spin semistable root-graph $(\Gamma,\mathcal{E},D,s)$ _specializes_ to a
spin semistable root-graph
$(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})$, and we write
$(\Gamma,\mathcal{E},D,s)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})$,
if there is a specialization of semistable root-graphs
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
such that $s^{\prime}=\widetilde{\iota}_{*}(s)$, where
$\widetilde{\iota}\colon\Gamma/P_{(\mathcal{E},D)}\rightarrow{\Gamma^{\prime}}/P_{(\mathcal{E}^{\prime},D^{\prime})}$
is the induced specialization (see Lemma 4.27) and $\widetilde{\iota}_{*}(s)$
is the sign function on $(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
taking $u^{\prime}\in
V(\Gamma^{\prime}/P_{(\mathcal{E}^{\prime},D^{\prime})})$ to

$\widetilde{\iota}_{*}(s)(u^{\prime})=\underset{u\in\widetilde{\iota}^{-1}(u^{\prime})}{\sum_{u\in
V(\Gamma/P_{(\mathcal{E},D)})}}s(u).$

We consider the category $\mathbf{SS}_{g,n}$ whose objects are spin semistable
root-graphs, where the arrows are given by the specializations of spin
semistable root-graphs.

We consider the sub-categories $\mathbf{SP}_{g,n}$ and $\mathbf{SPQ}_{g,n}$ of
$\mathbf{SS}_{g,n}$ whose objects are, respectively, spin polystable and spin
$v_{0}$-quasistable root-graphs. We let $\mathcal{SP}_{g,n}$ and
$\mathcal{SPQ}_{g,n}$ be, respectively, the sub-posets
$\mathbf{SP}_{g,n}/\sim$ and $\mathbf{SPQ}_{g,n}/\sim$.

######  Remark 4.29.

The parity of a spin semistable pseudo-root is preserved under
specializations. Moreover, every even (respectively, odd) spin root-graph
admits a specialization to $(\Gamma,\emptyset,D,s)$, where $\Gamma$ is the
graph with one vertex $v$ and no edges, $D=(g-1)v$ and $s(v)=0$ (respectively,
$s(v)=1$). Hence the poset $\mathcal{SPQ}_{g,n}$ has exactly two connected
components $\mathcal{SPQ}^{+}_{g,n}$ and $\mathcal{SPQ}^{-}_{g,n}$
corresponding to even and odd spin root-grahs.

######  Remark 4.30.

Using Corollary 4.22, it is easy to check that the category of spin graphs of
genus $g$ and $n$ legs, as defined in [11], is canonically isomorphic to the
category $\mathbf{SP}_{g,n}$. The isomorphism sends a spin graph
$(\Gamma,P,s)$, where $\Gamma$ is a graph, $P$ is a cyclic subgraph of
$\Gamma$ and $s\colon V(\Gamma/P)\rightarrow\mathbb{Z}/2\mathbb{Z}$ is a sign
function, to $(\Gamma,D_{P},\mathcal{E}_{P},s)$.

######  Remark 4.31.

Let $(\Gamma,\emptyset,D,s)$ be a spin semistable root graph such that
$\beta_{D}(V)=0$ for a proper subset $V\subset V(\Gamma)$. Set
$\Gamma_{1}=\Gamma(V)$, $\Gamma_{2}=\Gamma(V^{c})$. Define $D_{1}^{V}:=D_{1}$
and $D_{2}^{V}:=D_{2}$ as in equation (9). By the claim in the proof of
Theorem 4.7 we have that $(\Gamma_{i},\emptyset,D_{i}^{V})$ is a semistable
root-graph for $i=1,2$. By Lemma 4.6 we have $\phi_{(\mathcal{E},D)}(e)=1$,
$\forall\;e\in\overrightarrow{E}(\Gamma_{1},\Gamma_{2})$ such that $t(e)\in
V$. Then $P_{(\emptyset,D_{i}^{V})}=P_{(\emptyset,D)}\cap\Gamma_{i}$, so there
is an inclusion
$\Gamma_{i}/P_{(\emptyset,D_{i}^{V})}\subset\Gamma/P_{(\emptyset,D)}$. We
define the spin semistable root-graph
$(\Gamma_{i},\emptyset,D_{i}^{V},s_{i}^{V})$, where $s_{i}^{V}$ be the
restriction of $s$ to $V(\Gamma_{i}/P_{(\emptyset,D_{i}^{V})})$.

Assume that
$\iota\colon(\Gamma^{\prime},\emptyset,D^{\prime},s^{\prime})\rightarrow(\Gamma,\emptyset,D,s)$
is a specialization of spin semistable root-graphs and set
$V^{\prime}:=\iota^{-1}(V)\subset V(\Gamma^{\prime})$. By Proposition 2.1 we
have $\beta_{D^{\prime}}(V^{\prime})=\beta_{D}(V)=0$. Set
$\Gamma^{\prime}_{1}=\Gamma(V^{\prime})$ and
$\Gamma^{\prime}_{2}=\Gamma({V^{\prime}}^{c})$. By construction, there is a
specialization
$(\Gamma^{\prime}_{i},\emptyset,D_{i}^{V^{\prime}},s_{i}^{V^{\prime}})\geq(\Gamma_{i},\emptyset,D_{i}^{V},s_{i}^{V})$
of spin semistable root-graphs for $i=1,2$.

######  Definition 4.32.

Given a spin semistable pseudo-root $(\mathcal{E},D,s)$ on a graph $\Gamma$,
we define $\operatorname{pol}(\mathcal{E},D,s)$ as the spin polystable pseudo-
root $(\operatorname{pol}(\mathcal{E},D),\operatorname{pol}(s))$ on $\Gamma$,
where $\operatorname{pol}(s)=s$ (view as a sign function on
$(\Gamma,\operatorname{pol}(\mathcal{E},D))$ thanks to the identification
$P_{(\mathcal{E},D)}=P_{\operatorname{pol}(\mathcal{E},D)}$, see Proposition
4.24 (2)).

######  Proposition 4.33.

Let $(\Gamma,\mathcal{E},D,s)$ be a spin semistable root-graph. We have that
$(\Gamma,\operatorname{pol}(\mathcal{E},D,s))$ is the unique minimal spin
polystable root-graph admitting a specialization of spin semistable root-
graphs to $(\Gamma,\mathcal{E},D,s)$.

###### Proof.

By Proposition 4.24 we have that $(\Gamma,\operatorname{pol}(\mathcal{E},D))$
is the minimal polystable root-graph admitting a specialization to
$(\Gamma,\mathcal{E},D)$ and, since
$P_{(\mathcal{E},D)}=P_{\operatorname{pol}(\mathcal{E},D)}$, the sign function
$\operatorname{pol}(s)=s$ is the unique sign function for which there is a
specialization of spin semistable root-graphs
$(\Gamma,\operatorname{pol}(\mathcal{E},D,s))\geq(\Gamma,\mathcal{E},D,s)$. ∎

The transformation $\operatorname{pol}$ gives rise to a commutative diagram:

$\textstyle{\mathbf{SPQ}_{g,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pol}}$$\textstyle{\mathbf{QR}_{g,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pol}}$$\textstyle{\mathbf{QD}_{\mu,g,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pol}}$$\textstyle{\mathbf{SP}_{g,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{PSR}_{g,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{PSD}_{\mu,g,n}}$

##  5\. Moduli of tropical theta characteristics

In this section we introduce and study the moduli space of tropical theta
characteristics. We describe an important refinement of this space.

###  5.1. Cones associated to semistable pseudo-roots

First of all, we need to introduce the cone associated to a semistable root-
graph.

Consider a semistable root-graph $(\Gamma,\mathcal{E},D)$. Consider the vector
space $\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$ with coordinates $x_{e}$ for
$e\in E(\Gamma^{\mathcal{E}})$. For every oriented edge
$e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$, we will abuse notation writing
$x_{e}$ for the coordinates associated to the unoriented edge associated to
$e$. We define
$\lambda_{(\Gamma,\mathcal{E},D)}\subset\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$
as the cone obtained by intersecting
$\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{\geq 0}$ with the linear subspace
$\Lambda_{(\Gamma,\mathcal{E},D)}$ defined by the equations:

(11)
$\Lambda_{(\Gamma,\mathcal{E},D)}:\;\;\sum_{e\in\gamma}\phi_{(\mathcal{E},D)}(e)x_{e}=0$

where $\gamma$ runs over the set of oriented cycles of $\Gamma^{\mathcal{E}}$.

######  Remark 5.1.

Notice that if $(\mathcal{E},D)$ is semistable, then
$\mathcal{L}_{\mathcal{E}}\subset\Lambda_{(\Gamma,\mathcal{E},D)}$.

We let $\lambda^{\circ}_{(\Gamma,\mathcal{E},D)}$ be the interior of
$\lambda_{(\Gamma,\mathcal{E},D)}$, that is,
$\lambda^{\circ}_{(\Gamma,\mathcal{E},D)}=\lambda_{(\Gamma,\mathcal{E},D)}\cap\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{>0}$.
We also set

(12)
$b_{0}(\mathcal{E},D):=b_{0}(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})\quad\;\text{
and }\;\quad
b_{1}(\mathcal{E},D):=b_{1}(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)}).$

Notice that the number of independent equations in (11) is
$b_{1}(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$, which is equal to
$b_{1}(\mathcal{E},D)-b_{0}(\mathcal{E},D)+1+\\#\mathcal{E}$. Hence

(13)
$\dim\lambda_{(\Gamma,\mathcal{E},D)}=\\#E(\Gamma)-b_{1}(\mathcal{E},D)+b_{0}(\mathcal{E},D)-1.$

We define

$\Sigma_{(\Gamma,\mathcal{E},D)}=\\{\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}|(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})\in\mathbf{SSR}_{g,n}\text{
and
}(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})\\}.$

######  Theorem 5.2.

Let $(\Gamma,\mathcal{E},D)$ be a semistable pseudo-root.

  * (1)

If
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is a specialization of semistable root-graphs, then we have a face morphism
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\subset\lambda_{(\Gamma,\mathcal{E},D)}$.

  * (2)

Every face of $\lambda_{(\Gamma,\mathcal{E},D)}$ is equal to a cone
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}$ for some
specialization of semistable root-graphs
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.

  * (3)

We have a decomposition

(14)
$\lambda_{(\Gamma,\mathcal{E},D)}=\underset{(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}{\coprod_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})\in{\bf
SSR}_{g,n}}}\lambda^{\circ}_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}.$

  * (4)

The set $\Sigma_{(\Gamma,\mathcal{E},D)}$ is a fan whose support is
$\lambda_{(\Gamma,\mathcal{E},D)}$.

###### Proof.

We prove (1). If
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is an isomorphism, the result is clear. Otherwise, consider the hyperplane
$H\subset\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$ given in the coordinates
$(x_{e})_{e\in E(\Gamma^{\mathcal{E}})}$ of
$\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$ by the equation

$H:\sum_{e\in E(\Gamma^{\mathcal{E}})\setminus
E({\Gamma^{\prime}}^{\mathcal{E}^{\prime}})}x_{e}=0.$

Notice that $\lambda_{(\Gamma,\mathcal{E},D)}$ in contained in one of the two
half spaces determined by $H$. Moreover, the equations defining the
intersection $\lambda_{(\Gamma,\mathcal{E},D)}\cap H$ are obtained by putting
$x_{e}=0$ in equation (11), $\forall e\in E(\Gamma^{\mathcal{E}})\setminus
E({\Gamma^{\prime}}^{\mathcal{E}^{\prime}})$. These equations are the
equations defining
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}$, hence
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}=\lambda_{(\Gamma,\mathcal{E},D)}\cap
H.$

Let us prove (3). By (1), for every
$(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$ as in the right hand side
of equation (14) we have
$\lambda^{\circ}_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\subset\lambda_{(\Gamma,\mathcal{E},D)}$.
Let us show that
$\lambda^{\circ}_{(\Gamma_{1},\mathcal{E}_{1},D_{1})}\cap\lambda^{\circ}_{(\Gamma_{2},\mathcal{E}_{2},D_{2})}=\emptyset$
for distinct $(\Gamma_{1},\mathcal{E}_{1},D_{1})$,
$(\Gamma_{2},\mathcal{E}_{2},D_{2})$ as in the right hand side of equation
(14). This is true if
$(\Gamma_{1},\mathcal{E}_{1})\neq(\Gamma_{2},\mathcal{E}_{2})$, because in
this case
$\mathbb{R}^{E({\Gamma_{1}^{\mathcal{E}_{1}}})}_{>0}\cap\mathbb{R}^{E(\Gamma_{2}^{\mathcal{E}_{2}})}_{>0}=\emptyset$
(as subspaces of $\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$) and
$\lambda^{\circ}_{(\Gamma_{i},\mathcal{E}_{i},D_{i})}\subset\mathbb{R}^{E(\Gamma_{i}^{\mathcal{E}_{i}})}_{>0}$
for $i=1,2$. If $\Gamma^{\prime}:=\Gamma_{1}=\Gamma_{2}$ and
$\mathcal{E}^{\prime}:=\mathcal{E}_{1}=\mathcal{E}_{2}$, then we can assume
that $(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D_{1})$
and $(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D_{2})$
have the same associated specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$,
otherwise
$\mathbb{R}^{E({\Gamma_{1}^{\mathcal{E}_{1}}})}_{>0}\cap\mathbb{R}^{E(\Gamma_{2}^{\mathcal{E}_{2}})}_{>0}=\emptyset$
and we argue as before. Hence $D_{1}=\widehat{\iota}_{*}(D)=D_{2}$, a
contradiction. So the right hand side of equation (14) is contained in the
left hand side.

On the other hand, let $(x_{e})_{e\in
E(\Gamma^{\mathcal{E}})}\in\lambda_{(\Gamma,\mathcal{E},D)}$. Let
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
be the specialization of triples determined by the specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$
contracting the edges $e$ of $E(\Gamma^{\mathcal{E}})$ for which $x_{e}=0$ (so
$D^{\prime}=\widehat{\iota}_{*}(D)$). Since
$2D+\operatorname{Div}(\phi_{(\mathcal{E},D)})=K_{\Gamma^{\mathcal{E}}}$, we
have
$2D^{\prime}+\operatorname{Div}(\widehat{\iota}_{*}(\phi_{(\mathcal{E},D)}))=K_{{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}}$
and by equation (11) the flow $\widehat{\iota}_{*}(\phi_{(\mathcal{E},D)})$ is
acyclic. Hence $(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$ is a
semistable root-graph and $(x_{e})_{e\in
E({\Gamma^{\prime}}^{\mathcal{E}^{\prime}})}\in\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}^{\circ}$.
This concludes the proof of (3).

Notice that (2) and (4) are consequences of (3) and (1), hence we are done. ∎

Recall the $\phi_{(\mathcal{E},D)}$-graphs $P_{u}$ of $\Gamma^{\mathcal{E}}$,
for $u\in V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$ (see Definition 4.11).

######  Lemma-Definition 5.3.

Let
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
be a specialization of triples where $(\Gamma,\mathcal{E},D)$ is a semistable
root-graph. Assume that the associated specialization
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$
satisfies one of the following conditions:

  * (1)

$\widehat{\iota}$ is the contraction of an edge $e\in
P_{(\mathcal{E},D)}\subset E(\Gamma)$.

  * (2)

$\widehat{\iota}$ is the contraction of $E(P_{u},P_{u^{\prime}})$ for distinct
vertices $u,u^{\prime}\in V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$ such
that $E(P_{u},P_{u^{\prime}})$ is a $\phi_{(\mathcal{E},D)}$-cut of
$\Gamma^{\mathcal{E}}$.

Then
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is a specialization of semistable root-graphs. We call
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
elementary (of type $1$ or $2$, respectively, if (1) or (2) holds).

###### Proof.

Since
$2D+\operatorname{Div}(\phi_{(\mathcal{E},D)})=K_{\Gamma^{\mathcal{E}}}$, we
have
$2D^{\prime}+\operatorname{Div}(\widehat{\iota}_{*}(\phi_{(\mathcal{E},D)}))=K_{{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}}$.
So, by Proposition 2.1 we are left to show that
$\widehat{\iota}_{*}\phi_{(\mathcal{E},D)}$ is acyclic. This is clear if (1)
holds, since $\phi_{(\mathcal{E},D)}(e)=0$ for $e\in P_{(\mathcal{E},D)}$. If
(2) holds, then $\widehat{\iota}_{*}\phi_{(\mathcal{E},D)}$ is acyclic by the
definition of $\phi_{(\mathcal{E},D)}$-cut. ∎

######  Corollary 5.4.

The following properties hold.

  * (1)

For a semistable pseudo-root $(\mathcal{E},D)$ on a graph $\Gamma$, a cone
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\in\Sigma_{(\Gamma,\mathcal{E},D)}$
is a facet of $\lambda_{(\Gamma,\mathcal{E},D)}$ if and only if the associated
specialization
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is elementary.

  * (2)

Every specialization of semistable pseudo-roots can be written (up to
isomorphisms) as a chain of elementary specializations.

  * (3)

An elementary specialization of semistable root-graphs can not be factored as
a non-trivial chain of specializations of semistable root-graphs.

###### Proof.

Consider an elementary specialization
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.
If
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is of type 1, then $\\#E(\Gamma^{\prime})=\\#E(\Gamma)-1$ and
$\dim\Lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}=\dim\Lambda_{(\Gamma,\mathcal{E},D)}$.
If
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is of type 2, then
$\\#E(\Gamma^{\prime})=\\#E(\Gamma)-\\#E(P_{u},P_{u^{\prime}})$ and

$\dim\Lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}=\dim\Lambda_{(\Gamma,\mathcal{E},D)}-\\#E(P_{v},P_{v^{\prime}})+1.$

Hence, in both cases,
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}$ is a facet of
$\lambda_{(\Gamma,\mathcal{E},D)}$. Conversely, assume that
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}$ is a facet of
$\lambda_{(\Gamma,\mathcal{E},D)}$ and let
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
and
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow\Gamma^{\prime\mathcal{E}^{\prime}}$
be the associated specializations. In particular,
$\widehat{\iota}_{*}(\phi_{(\mathcal{E},D)})$ is acyclic. If
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is not elementary, by what we have already shown we have that there are
distinct vertices $u,u^{\prime}\in
V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$ and a proper non-empty set
$F\subset E(P_{u},P_{u^{\prime}})$ such that $\widehat{\iota}$ contracts
$E(P_{u},P_{u^{\prime}})\setminus F$ and does not contract $F$. Thus
$\widehat{\iota}$ sends each oriented edge in $\overrightarrow{F}$ to a
positive $\widehat{\iota}_{*}(\phi_{(\mathcal{E},D)})$-loop in
$(\Gamma^{\prime})^{\mathcal{E}^{\prime}}$, which is a contradiction. This
proves (1). Notice that (2) and (3) are consequences of (1) and Theorem 5.2. ∎

######  Remark 5.5.

If $(\Gamma,\mathcal{E},D)$ is $v_{0}$-quasistable, then the statements of
Theorem 5.2 and Corollary 5.4 hold if we replace the word _semistable_ with
_$v_{0}$ -quasistable_ (specializations preserves semistability and
quasistability).

Recall the definition of the map $\mathcal{T}_{\mathcal{E}}$ in equation (5).
For a semistable root-graph $(\Gamma,\mathcal{E},D)$, we define the cone
$\kappa_{(\Gamma,\mathcal{E},D)}$ and its interior
$\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}$ as:

$\kappa_{(\Gamma,\mathcal{E},D)}=\mathcal{T}_{\mathcal{E}}(\lambda_{(\Gamma,\mathcal{E},D)})\quad\text{
and
}\quad\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}=\mathcal{\mathcal{T}}_{\mathcal{E}}(\lambda_{(\Gamma,\mathcal{E},D)}\cap\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{>0}).$

By construction, we have

(15)
$\kappa_{(\Gamma,\mathcal{E},D)}\subset\sigma_{(\Gamma,\mathcal{E},D)}\subset\mathbb{R}^{E(\Gamma^{\mathcal{E}})}/\mathcal{L}_{\mathcal{E}}.$

######  Remark 5.6.

Notice that if $(\mathcal{E},D)$ is simple, then we have
$\mathcal{L}_{\mathcal{E}}=0$ in equation (5), and hence
$\lambda_{(\Gamma,\mathcal{E},D)}=\kappa_{(\Gamma,\mathcal{E},D)}$.

######  Remark 5.7.

If
$(\Gamma,\mathcal{E},D)\rightarrow(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is a specialization of semistable root-graphs, then using Remark 3.3 and
Theorem 5.2, we deduce that

$\kappa_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}=\kappa_{(\Gamma,\mathcal{E},D)}\cap\sigma_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}=\kappa_{(\Gamma,\mathcal{E},D)}\cap\mathbb{R}^{E(\Gamma^{\mathcal{E}^{\prime}})}/\mathcal{L}_{\mathcal{E}^{\prime}},$

and hence we get an inclusion
$\kappa_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\subset\kappa_{(\Gamma,\mathcal{E},D)}$.

######  Remark 5.8.

As we shall see in Proposition 5.24, the inclusion
$\kappa_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\subset\kappa_{(\Gamma,\mathcal{E},D)}$
might fail to be a face morphism when $(\mathcal{E},D)$ is polystable and
$(\mathcal{E}^{\prime},D^{\prime})$ is $v_{0}$-quasistable. Nevertheless, in
Lemma 5.14, we will see that it is a face morphism if $(\mathcal{E},D)$ and
$(\mathcal{E}^{\prime},D^{\prime})$ are polystable pseudo-roots.

######  Proposition 5.9.

We have
$\dim\kappa_{(\Gamma,\mathcal{E},D)}=\\#E(\Gamma)-b_{1}(\mathcal{E},D)$, for
every semistable root-graph $(\Gamma,\mathcal{E},D)$.

###### Proof.

By (13) and the following computation:

$\dim\mathcal{L}_{\mathcal{E}}=b_{0}(\Gamma_{\mathcal{E}})-1=b_{0}(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})-1=b_{0}(\mathcal{E},D)-1,$

we need only to check that
$\mathcal{L}_{\mathcal{E}}\subset\Lambda_{(\Gamma,\mathcal{E},D)}$, which
easily follows by Theorem 4.7. ∎

######  Definition 5.10.

Let $X$ be a tropical curve. A _root_ of $X$ is a divisor $\mathcal{D}$ on $X$
such that the divisor $2\mathcal{D}$ is linearly equivalent to the canonical
divisor $K_{X}$. A root is _unitary, semistable, polystable,
$v_{0}$-quasistable_ for some point $p_{0}$ of $X$, if so is $\mathcal{D}$ as
a divisor on $X$.

######  Lemma 5.11.

Let $\mathcal{D}$ be a unitary semistable root of a tropical curve $X$ with
model $\Gamma$. Then the combinatorial type $(\mathcal{E},D)$ of $\mathcal{D}$
is a pseudo-root on $\Gamma$.

###### Proof.

There is a rational function $f$ on $X$ such that
$2\mathcal{D}+\operatorname{Div}(f)=K_{X}$. Since $\mathcal{D}$ is unitary, we
have $\operatorname{supp}(\mathcal{D})\subset V(\Gamma^{\mathcal{E}})$. But
$\operatorname{supp}(K_{X})\subset V(\Gamma)$, hence
$\operatorname{supp}(\operatorname{Div}(f))\subset V(\Gamma^{\mathcal{E}})$.
Then $\operatorname{Div}(\phi_{f})$ is a divisor on $\Gamma^{\mathcal{E}}$,
and we have $2D+\operatorname{Div}(\phi_{f})=K_{\Gamma^{\mathcal{E}}}$. Since
$\phi_{f}$ is the flow of a rational function, it is acyclic, and we are done.
∎

######  Proposition 5.12.

Let $(\Gamma,\mathcal{E},D)$ be a semistable root-graph. The open cone
$\kappa^{0}_{(\Gamma,\mathcal{E},D)}$ parametrizes classes of pairs
$(X,\mathcal{D})$, where $X$ is a $n$-pointed tropical curve with model
$\Gamma$ and $\mathcal{D}$ is a semistable unitary root of $X$ with
combinatorial type $(\mathcal{E},D)$, and $(X,\mathcal{D})$ is equivalent to
$(X^{\prime},\mathcal{D}^{\prime})$ if $X=X^{\prime}$ and $\mathcal{D}$ is
linearly equivalent to $\mathcal{D}^{\prime}$.

###### Proof.

By [4, Proposition 4.6], the open cone
$\sigma^{\circ}_{(\Gamma,\mathcal{E},D)}$ parametrizes equivalence classes of
pairs $(X,D)$, where $X$ is a tropical curve with model $\Gamma$ and
$\mathcal{D}$ is a unitary divisor with combinatorial type $(\mathcal{E},D)$
(hence $\mathcal{D}$ is semistable as a divisor on $X$), and two pairs
$(X,\mathcal{D})$ and $(X^{\prime},D^{\prime})$ are equivalent if
$X=X^{\prime}$ and the divisors $\mathcal{D}$ and $\mathcal{D}^{\prime}$ are
linearly equivalent.

Consider a point $(x_{e})_{e\in
E(\Gamma^{\mathcal{E}})}\in\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{>0}$ whose
class in $\sigma^{\circ}_{(\Gamma,\mathcal{E},D)}$ corresponds to a pair
$(X,\mathcal{D})$. A model of $X$ is $\Gamma^{\mathcal{E}}$, with length
function taking an edge $e\in E(\Gamma^{\mathcal{E}})$ to $x_{e}$. If
$\mathcal{D}$ is a root of $X$, then there is a rational function $f$ on $X$
such that $2\mathcal{D}+\operatorname{Div}(f)=K_{X}$. Thus
$\operatorname{Div}(\phi_{f})$ is a divisor on $\Gamma^{\mathcal{E}}$ such
that $2D+\operatorname{Div}(\phi_{f})=K_{\Gamma^{\mathcal{E}}}$. Hence
$\phi_{f}=\phi_{(\mathcal{E},D)}$ (by Theorem 4.7) and
$\phi_{(\mathcal{E},D)}$ satisfies equations (11) because it is the flow
associated to a rational function on $X$. This implies that $(x_{e})_{e\in
E(\Gamma^{\mathcal{E}})}$ is a point of
$\lambda^{\circ}_{(\Gamma,\mathcal{E},D)}$, so its class belongs to
$\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}$. Conversely, if $(x_{e})_{e\in
E(\Gamma^{\mathcal{E}})}$ is a point of
$\lambda^{\circ}_{(\Gamma,\mathcal{E},D)}$, then equations (11) implies the
existence of a rational function $f$ on $X$ such that
$\phi_{f}=\phi_{(\mathcal{E},D)}$. Thus
$2D+\operatorname{Div}(\phi_{f})=K_{\Gamma^{\mathcal{E}}}$, which implies that
$2\mathcal{D}+\operatorname{Div}(f)=K_{X}$, i.e., that $\mathcal{D}$ is a root
of $X$. ∎

######  Remark 5.13.

A similar argument can be used to prove the statement of Proposition 5.12
where we replace the word _semistable_ either with _polystable_ , or with
_$v_{0}$ -quasistable_.

###  5.2. The moduli space of tropical theta characteristics

Let $\Gamma$ be a graph. Given a subset $\mathcal{E}\subset E(\Gamma)$, we
consider the linear injective map

(16)
$\delta_{\mathcal{E}}\colon\mathbb{R}^{E(\Gamma)}\rightarrow\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$

taking $(x_{e})_{e\in E(\Gamma)}$ to $(x_{e^{\prime}})_{e^{\prime}\in
E(\Gamma^{\mathcal{E}})}$, where $x_{e^{\prime}}=x_{e}$ if the edge
$e^{\prime}\in E(\Gamma^{\mathcal{E}})$ is over the edge $e\in E(\Gamma)$.
Denote by
$\Delta_{\mathcal{E}}:=\delta_{\mathcal{E}}(\mathbb{R}^{E(\Gamma)}_{\geq 0})$,
so that we have

$\Delta_{\mathcal{E}}=\\{(x_{e})_{e\in
E(\Gamma^{\mathcal{E}})}|x_{e_{1}}=x_{e_{2}}\geq 0,\forall e_{1},e_{2}\text{
over the same edge
}e\in\mathcal{E}\\}\subset\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{\geq 0}.$

######  Lemma 5.14.

Let $(\Gamma,\mathcal{E},D)$ be a polystable root-graph.

  * (1)

We have isomorphisms:

$\mathbb{R}^{E(\Gamma)}_{\geq
0}\stackrel{{\scriptstyle\underset{\simeq}{\delta_{\mathcal{E}}}}}{{\rightarrow}}\Delta_{\mathcal{E}}\stackrel{{\scriptstyle\underset{\simeq}{\mathcal{T}_{\mathcal{E}}}}}{{\rightarrow}}\kappa_{(\Gamma,\mathcal{E},D)}\subset\mathbb{R}^{E(\Gamma^{\mathcal{E}})}/\mathcal{L}_{\mathcal{E}};$

  * (2)

If
$(\Gamma,\mathcal{E},D)\rightarrow(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is a specialization of polystable root-graphs, then we have a face morphism
$\kappa_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\rightarrow\kappa_{(\Gamma,\mathcal{E},D)}$.
Conversely, every face of $\kappa_{(\Gamma,\mathcal{E},D)}$ arises in this
way.

###### Proof.

We show (1). It is clear that
$\delta_{\mathcal{E}}\colon\mathbb{R}^{E(\Gamma)}_{\geq
0}\rightarrow\Delta_{\mathcal{E}}$ is an isomorphism. By Lemma 4.16 we have
that $\Delta_{\mathcal{E}}\subset\lambda_{(\Gamma,\mathcal{E},D)}$ and, by
equation (4), we have $\Delta_{\mathcal{E}}\cap\mathcal{L}_{\mathcal{E}}=0$.
Therefore, by restricting $\mathcal{T}_{\mathcal{E}}$ to
$\Delta_{\mathcal{E}}$, we obtain an injective linear map

(17)
$\mathcal{T}_{\mathcal{E}}\colon\Delta_{\mathcal{E}}\rightarrow\kappa_{(\Gamma,\mathcal{E},D)}.$

We show that the map in equation (17) is surjective. Assume first that
$(\mathcal{E},D)$ is simple (recall Remark 5.6). Fix an spanning tree $T$ of
$\Gamma$ not intersecting $\mathcal{E}$. Hence, for each
$e^{\prime}\in\overrightarrow{\mathcal{E}}$, there exists a unique oriented
cycle $\gamma_{e^{\prime}}$ on $\Gamma$ such that
$E(\gamma_{e^{\prime}})\setminus T=\\{e^{\prime}\\}$. Then, seeing
$\gamma_{e^{\prime}}$ as an oriented cycle on $\Gamma^{\mathcal{E}}$, from the
equation:

$\sum_{e\in\gamma_{e^{\prime}}}\phi_{(\mathcal{E},D)}(e)x_{e}=0,$

we have that $x_{e_{1}}=x_{e_{2}}$ for $e_{1},e_{2}$ over the associated
unoriented edge to $e^{\prime}\in\overrightarrow{\mathcal{E}}$. So, in this
case, we also have
$\kappa_{(\Gamma,\mathcal{E},D)}=\lambda_{(\Gamma,\mathcal{E},D)}\subset\Delta_{\mathcal{E}}$,
concluding the proof of (1) when $(\mathcal{E},D)$ is simple.

Next, assume that $(\mathcal{E},D)$ is not simple. It suffices to prove that
for every vector $v\in\lambda_{(\Gamma,\mathcal{E},D)}$ there is a vector
$w\in\mathcal{L}_{\mathcal{E}}$ such that $v+w\in\Delta_{\mathcal{E}}$. We
argue by induction on $\\#\mathcal{E}$. Recall that $u_{e}$ denotes the vector
of the canonical basis of $\mathbb{R}^{E(\Gamma^{\mathcal{E}})}$ corresponding
to $e\in E(\Gamma^{\mathcal{E}})$. If $\\#\mathcal{E}=1$, we have that
$\mathcal{L}_{\mathcal{E}}$ is generated by $u_{e_{1}}-u_{e_{2}}$, where
$e_{1}$ and $e_{2}$ are the edges over the unique edge in $\mathcal{E}$. In
this case, we conclude observing that, given a vector
$v\in\lambda_{(\Gamma,\mathcal{E},D)}$ with coordinates $(x_{e})_{e\in
E(\Gamma^{\mathcal{E}})}$,

$v+\frac{x_{e_{2}}-x_{e_{1}}}{2}(u_{e_{1}}-u_{e_{2}})\in\Delta_{\mathcal{E}}.$

If $\\#\mathcal{E}>1$, we can choose $\mathcal{E}^{\prime}\subset\mathcal{E}$
such that $\mathcal{E}^{\prime}$ is a nonempty minimal disconnecting subset of
edges of $\Gamma$. Hence $\Gamma_{\mathcal{E}^{\prime}}$ has two connected
components, which we denote $\Gamma_{1}$ and $\Gamma_{2}$. Fix
$i\in\\{1,2\\}$. For a vector $v\in\lambda_{(\Gamma,\mathcal{E},D)}$ with
coordinates $(x_{e})_{e\in E(\Gamma^{\mathcal{E}})}$, we let
$v_{i}:=\pi_{i}(v)\in\mathbb{R}^{E(\Gamma_{i}^{\mathcal{E}_{i}})}$, where
$\mathcal{E}_{i}=E(\Gamma_{i})\cap\mathcal{E}$ and $\pi_{i}$ is the natural
projection
$\pi_{i}\colon\mathbb{R}^{E(\Gamma^{\mathcal{E}})}\rightarrow\mathbb{R}^{E(\Gamma_{i}^{\mathcal{E}_{i}})}$.
Since $v$ satisfies equations (11) for cycles in $\Gamma$, we see that $v_{i}$
satisfies equations (11) for cycles in $\Gamma_{i}$, for $i=1,2$. Moreover,
set:

$P_{i}=E(\Gamma_{i})\setminus\mathcal{E}_{i}\quad\text{ and }\quad
D_{i}=D|_{\Gamma_{i}}-\sum_{v\in
V(\Gamma_{i})}\deg_{\mathcal{E}^{\prime}}(v)v.$

Then $P_{i}$ is a cyclic subgraph, because $P=E(\Gamma)/\mathcal{E}$ is cyclic
and $P=P_{1}\sqcup P_{2}$, and, by equation (10), we have
$(\mathcal{E}_{i},D_{i})=(\mathcal{E}_{P_{i}},D_{P_{i}})$. By Proposition
4.17, we have that $(\mathcal{E}_{i},D_{i})$ is a polystable root on
$\Gamma_{i}$ , and hence
$v_{i}\in\lambda_{(\Gamma_{i},\mathcal{E}_{i},D_{i})}$.

By the induction hypothesis, there exists
$w_{i}\in\mathcal{L}_{\mathcal{E}_{i}}$ such that
$v_{i}+w_{i}\in\Delta_{\mathcal{E}_{i}}$. Notice that for each $V\subset
V(\Gamma_{i})$ such that $E(\Gamma_{i})\cap E(V,V^{c})\subset\mathcal{E}_{i}$,
we have that $E(V,V^{c})\subset\mathcal{E}_{i}\cup\mathcal{E}^{\prime}$. Then
each vector in $\mathcal{L}_{\mathcal{E}_{i}}$ is the projection in
$\mathbb{R}^{E(\Gamma_{i}^{\mathcal{E}_{i}})}$ of a vector in
$\mathcal{L}_{\mathcal{E}}\cap\mathbb{R}^{E(\Gamma^{\mathcal{E}})\setminus
E(\Gamma_{{}_{3-i}}^{{}^{\mathcal{E}_{3-i}}})}$. Let $u_{i}$ be a vector in
$\mathcal{L}_{\mathcal{E}}\cap\mathbb{R}^{E(\Gamma^{\mathcal{E}})\setminus
E(\Gamma_{{}_{3-i}}^{{}^{\mathcal{E}_{3-i}}})}$ such that its projection to
$\mathbb{R}^{E(\Gamma_{i}^{\mathcal{E}_{i}})}$ is $w_{i}$. Define the vector
$u:=v+u_{1}+u_{2}$. Then $u$ projects to $v_{i}+w_{i}$ in
$\mathbb{R}^{E(\Gamma_{i}^{\mathcal{E}_{i}})}$ via $\pi_{i}$, since
$\pi_{i}(u_{3-i})=0$. So the coordinates $(y_{e})_{e\in
E(\Gamma^{\mathcal{E}})}$ of $u$, for any edge
$e\in\mathcal{E}\setminus\mathcal{E}^{\prime}$, satisfy

(18) $y_{e_{1}}=y_{e_{2}},$

where $e_{1}$, $e_{2}$ are over $e$. Set $V_{1}=V(\Gamma_{1})$. We have that
$E(V_{1},V_{1}^{c})=\mathcal{E}^{\prime}$ and
$u_{V_{1}}\in\mathbb{R}^{E(\Gamma^{\mathcal{E}})\setminus(E(\Gamma_{1}^{\mathcal{E}_{1}})\cup
E(\Gamma_{2}^{\mathcal{E}_{2}}))}$ (recall Equation (4)). If
$\\#\mathcal{E}^{\prime}=1$, with $e_{1}$ and $e_{2}$ the edges over the
unique edge $e\in\mathcal{E}^{\prime}$, then $u_{V_{1}}=u_{e_{1}}-u_{e_{2}}$
and hence we are done since:

$u+\frac{x_{e_{2}}-x_{e_{1}}}{2}u_{V_{1}}\in\Delta_{\mathcal{E}}.$

We are left with the case $\\#\mathcal{E}^{\prime}\geq 2$. For every two
different edges $e^{\prime},e^{\prime\prime}\in\mathcal{E}^{\prime}$, we can
choose an oriented cycle $\gamma_{e^{\prime},e^{\prime\prime}}$ of $\Gamma$
such that
$E(\gamma_{e^{\prime},e^{\prime\prime}})\cap\mathcal{E}^{\prime}=\\{e^{\prime},e^{\prime\prime}\\}$
(recall that $\mathcal{E}^{\prime}$ is minimal and hence $V_{1}$ and
$V_{1}^{c}$ induce connected subgraphs). Since
$\mathcal{L}_{\mathcal{E}}\subset\Lambda_{(\Gamma,\mathcal{E},\mathcal{D})}$
by Remark 5.1, we have that the vector $u\in\Lambda_{(\Gamma,\mathcal{E},D)}$
(recall Equation (11)). Then:

$\sum_{e\in\gamma_{e^{\prime},e^{\prime\prime}}}\phi_{(\mathcal{E},D)}(e)y_{e}=0,$

(where we see $\gamma_{e^{\prime},e^{\prime\prime}}$ as an oriented cycle in
$\Gamma^{\mathcal{E}}$) and so, if $e^{\prime}_{1},e^{\prime}_{2}$ are over
$e^{\prime}$, and $e^{\prime\prime}_{1},e^{\prime\prime}_{2}$ over
$e^{\prime\prime}$, with $e^{\prime}_{1},e^{\prime\prime}_{1}$ incident to
$V(\Gamma_{1})$, we get:

$0=\sum_{e\in\gamma_{e^{\prime},e^{\prime\prime}}}\phi_{(\mathcal{E},D)}(e)y_{e}=y_{e^{\prime}_{1}}-y_{e^{\prime}_{2}}-(y_{e^{\prime\prime}_{1}}-y_{e^{\prime\prime}_{2}})+\sum_{e\in\gamma_{e^{\prime},e^{\prime\prime}}\setminus\mathcal{E}^{\prime}}\phi_{(\mathcal{E},D)}(e)y_{e}.$

By Equation (18) and Lemma 4.16 we have that
$\sum_{e\in\gamma_{e^{\prime},e^{\prime\prime}}\setminus\mathcal{E}^{\prime}}\phi_{(\mathcal{E},D)}(e)y_{e}=0$
and hence

(19)
$y_{e^{\prime}_{1}}-y_{e^{\prime}_{2}}=y_{e^{\prime\prime}_{1}}-y_{e^{\prime\prime}_{2}}.$

Fix an edge $e^{\prime}\in\mathcal{E}^{\prime}$ and let
$e^{\prime}_{1},e^{\prime}_{2}$ be the different edges of
$\Gamma^{\mathcal{E}}$ over $e^{\prime}$, with $e^{\prime}_{1}$ incident to
$V(\Gamma_{1})$. The vector
$u+\frac{y_{e^{\prime}_{2}}-y_{e^{\prime}_{1}}}{2}u_{V_{1}}$, with coordinates
$(z_{e})_{e\in E(\Gamma^{\mathcal{E}})}$, satisfies
$z_{e_{1}}=y_{e_{1}}=y_{e_{2}}=z_{e_{2}}$ for every $e_{1},e_{2}$ over an edge
$e\in\mathcal{E}\setminus\mathcal{E}^{\prime}$ and

$z_{e^{\prime\prime}_{1}}=y_{e^{\prime\prime}_{1}}+\frac{y_{e^{\prime}_{2}}-y_{e^{\prime}_{1}}}{2}=y_{e^{\prime\prime}_{2}}-\frac{y_{e^{\prime}_{2}}-y_{e^{\prime}_{1}}}{2}=z_{e^{\prime\prime}_{2}}$

for every edge $e^{\prime\prime}\in\mathcal{E}^{\prime}$ (where in the second
equality we used (19)).

Moreover, for every edge $e\in\mathcal{E}$ we have that
$2z_{e_{1}}=2z_{e_{2}}=z_{e_{1}}+z_{e_{2}}=x_{e_{1}}+x_{e_{2}}$, where
$e_{1},e_{2}$ are the edges over $e$. Then the vector
$v+u_{1}+u_{2}+\frac{y_{e^{\prime}_{2}}-y_{e^{\prime}_{1}}}{2}u_{V_{1}}$ has
non-negative coordinates, so it is contained in $\Delta_{\mathcal{E}}$. This
concludes the proof of (1).

Let
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
be a specialization of polystable root-graphs. If
$\Gamma\rightarrow\Gamma^{\prime}$ is the induced specialization, then we have
a natural face morphism $\mathbb{R}^{E(\Gamma^{\prime})}_{\geq
0}\rightarrow\mathbb{R}^{E(\Gamma)}_{\geq 0}$ which, by the first part of the
proof can be identified with a face morphism
$\kappa_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\rightarrow\kappa_{(\Gamma,\mathcal{E},D)}$.
It is clear that all the faces arises in this way, since every face of
$\mathbb{R}^{E(\Gamma)}_{\geq 0}$ is of type
$\mathbb{R}^{E(\Gamma^{\prime})}_{\geq 0}$, for some graph $\Gamma^{\prime}$
and some specialization $\Gamma\rightarrow\Gamma^{\prime}$. ∎

We are now in a position to introduce the following generalized cone complex.

######  Definition 5.15.

We define the generalized cone complex, called the _moduli space of tropical
theta characteristics_ , as

$T^{\textnormal{trop}}_{g,n}:=\underset{\longrightarrow}{\lim}\;\kappa_{(\Gamma,\mathcal{E},D)}=\coprod_{[\Gamma,\mathcal{E},D]}\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}/\textnormal{Aut}(\Gamma,\mathcal{E},D),$

where the limit is taken over the category $\mathbf{PSR}_{g,n}$ and the union
is taken over all equivalence classes $[\Gamma,\mathcal{E},D]$ in
$\mathcal{PSR}_{g,n}$.

######  Remark 5.16.

It is worth noting that although $\mathbb{R}^{E(\Gamma)}_{\geq 0}$ is
isomorphic to $\kappa_{(\Gamma,\mathcal{E},D)}$ we take quotient via the group
$\textnormal{Aut}(\Gamma,\mathcal{E},D)$ which is a (possibly proper) subgroup
of $\textnormal{Aut}(\Gamma)$

######  Proposition 5.17.

The generalized cone complex $T^{\textnormal{trop}}_{g,n}$ parametrizes
equivalence classes of pairs $(X,\mathcal{D})$, where $X$ is a stable
$n$-pointed tropical curve of genus $g$, and $\mathcal{D}$ is a polystable
root of $X$, and where $(X,\mathcal{D})$ is equivalent to
$(X^{\prime},\mathcal{D}^{\prime})$ if there is an isomorphism $\iota\colon
X\rightarrow X^{\prime}$ such that $\iota_{*}(\mathcal{D})$ is linearly
equivalent to $\mathcal{D}^{\prime}$.

###### Proof.

Let $X$ be a $n$-pointed stable tropical curve of genus $g$ and let
$\Gamma:=\Gamma_{X}$ be the genus-$g$ stable graph with $n$ legs which is a
model of $X$. By Lemma 5.11, a polystable root divisor $\mathcal{D}$ of $X$
has combinatorial type equal to a polystable pseudo-root $(\mathcal{E},D)$ on
$\Gamma$. Therefore, by Lemma 5.12, we see that $(X,\mathcal{D})$ corresponds
to a point in $\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}$, and hence to a point
in $T^{\textnormal{trop}}_{g,n}$. If $(X,\mathcal{D})$ and
$(X^{\prime},\mathcal{D}^{\prime})$ are equivalent, then there is an
isomorphism $\iota\colon X\to X^{\prime}$ such that $\iota_{*}(\mathcal{D})$
is equivalent to $\mathcal{D}^{\prime}$. Consider the three points
$p_{1},p_{2},p_{3}$ in $\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}$ corresponding
to $(X,\mathcal{D})$, $(X^{\prime},\iota_{*}(\mathcal{D}))$,
$(X^{\prime},\mathcal{D}^{\prime})$. The points $p_{1}$ and $p_{2}$ will get
identified in
$\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}/\textnormal{Aut}(\Gamma,\mathcal{E},D)$,
while $p_{2}$ and $p_{3}$ are equal in
$\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}$. Hence, $p_{1}$ and $p_{3}$ will
correspond to the same point in

$\kappa_{\Gamma,\mathcal{E},D}^{\circ}/\textnormal{Aut}(\Gamma,\mathcal{E},D)\subset
T_{g,n}^{\textnormal{trop}}.$

If $(X,\mathcal{D})$ and $(X^{\prime},\mathcal{D}^{\prime})$ correspond to the
same point contained in some cell
$\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}/\textnormal{Aut}(\Gamma,\mathcal{E},D)$
of $T^{\textnormal{trop}}_{g,n}$, then there is an isomorphism
$\Gamma=\Gamma_{X}\stackrel{{\scriptstyle\iota}}{{\cong}}\Gamma_{X^{\prime}}$
such that $\iota_{*}(\mathcal{E},D)=(\mathcal{E}^{\prime},D^{\prime})$.
Moreover, it follows that the metrics of $X$ and $X^{\prime}$ are equal, hence
$\iota$ induces an isomorphism of tropical curves $\iota\colon X\to
X^{\prime}$. This implies that $(X^{\prime},\mathcal{D}^{\prime})$ and
$(X^{\prime},\iota_{*}(\mathcal{D}))$ are the same point in
$\kappa_{(\Gamma,\mathcal{E},D)}^{\circ}$. By Lemma 5.12, the divisors
$\mathcal{D}^{\prime}$ and $\iota_{*}(\mathcal{D})$ are equivalent, so
$(X,\mathcal{D})$ and $(X^{\prime},\mathcal{D}^{\prime})$ are equivalent, as
required.

Conversely, given a triple $(\Gamma,\mathcal{E},D)$ in $\mathbf{PSR}_{g,n}$,
using again Lemma 5.12 we see that every point in
$\kappa_{(\Gamma,\mathcal{E},D)}^{\circ}$ corresponds to a pair
$(X,\mathcal{D})$, where $X$ is a tropical curve and $\mathcal{D}$ is a
polystable root of $X$. It is easy to see that we have established a one-to-
one correspondence between the set of points of $T_{g,n}^{\textnormal{trop}}$
and the set of equivalence classes of pairs $(X,\mathcal{D})$. ∎

Given a tropical curve $X$ with underlying graph $\Gamma$, and a cyclic
subgraph $P$ of $\Gamma$, we let $\mathcal{E}_{P}:=E(\Gamma)\setminus E(P)$,
and we define the divisor $\mathcal{D}_{P}$ on $X$ as:

(20) $\mathcal{D}_{P}=\sum_{v\in
V(\Gamma)}\left(\deg_{X}(v)-\frac{\deg_{P}(v)}{2}-1+w_{X}(v)\right)v-\sum_{e\in\mathcal{E}_{P}}p_{e},$

where $p_{e}$ is the mid-point of the edge $e$. Notice that the combinatorial
type of $\mathcal{D}_{P}$ is $(\mathcal{E}_{P},D_{P})$, hence
$\mathcal{D}_{P}$ is a polystable divisor on $X$ by Lemma 4.17.

Recall that a theta characteristic on a tropical curve is the class
$[\mathcal{D}]$ of a divisor $\mathcal{D}$ such that $2\mathcal{D}$ is
equivalent to $K_{X}$.

######  Proposition 5.18.

Let $X$ be a tropical curve with underlying graph $\Gamma$. Then, the divisor
$\mathcal{D}_{P}$ is a polystable root of $X$ for every cyclic subgraph $P$ of
$\Gamma$. Conversely, every theta characteristic on $X$ is the class of a
divisro $\mathcal{D}_{P}$, for a unique cyclic subgraph $P$.

###### Proof.

Consider a cyclic subgraph $P$ of $\Gamma$. Given $e\in\mathcal{E}_{P}$
incident to the vertices $v_{1},v_{2}$ of $\Gamma$, we let $f_{e}$ be the
rational function on $X$ with slope $1$ over the segments $[v_{1},p_{e}]$ and
$[v_{2},p_{e}]$, and slope $0$ everywhere else. We have

$\mathcal{D}_{P}+\sum_{e\in\mathcal{E}_{P}}\operatorname{Div}(f_{e})=\sum_{v\in
V(\Gamma)}\left(\frac{\deg_{P}(v)}{2}-1+w_{X}(v)\right)v+\sum_{e\in\mathcal{E}_{P}}p_{e}.$

Thus, by Theorem [12, Proposition 3.6], we deduce that every theta
characteristic has exactly one representative equal to $\mathcal{D}_{P}$. ∎

The following corollary is an immediate consequence of Propositions 5.17 and
5.18, and justifies the name “moduli space of theta characteristics” given to
$T_{g,n}^{\textnormal{trop}}$.

######  Corollary 5.19.

The generalized cone complex $T^{\textnormal{trop}}_{g,n}$ parametrizes
equivalence classes of pairs $(X,[\mathcal{D}])$ where $X$ is a stable
$n$-pointed tropical curve of genus $g$ and $[\mathcal{D}]$ is a theta
characteristic on $X$, and where $(X_{1},[\mathcal{D}_{1}])$ is equivalent to
$(X_{2},[\mathcal{D}_{2}])$ if there is an isomorphism $\iota\colon
X_{1}\rightarrow X_{2}$ such that
$[\iota_{*}(\mathcal{D}_{1})]=[\mathcal{D}_{2}]$.

By Propositions 3.5 and 5.17 it is clear that there is an inclusion

$\psi^{\textnormal{trop}}_{T}\colon T^{\textnormal{trop}}_{g,n}\hookrightarrow
P^{\textnormal{trop}}_{\mu,g,n},$

which is a morphism of generalized cone complexes. Indeed, for every genus-$g$
stable graph $\Gamma$ with $n$ legs and a polystable pseudo-root
$(\mathcal{E},D)$ on $\Gamma$, the inclusion
$\kappa_{(\Gamma,\mathcal{E},D)}\subset\sigma_{(\Gamma,\mathcal{E},D)}$ in
equation (15) factors the composition:

$\kappa_{(\Gamma,\mathcal{E},D)}\rightarrow
T_{g,n}^{\textnormal{trop}}\stackrel{{\scriptstyle\psi^{\textnormal{trop}}_{T}}}{{\longrightarrow}}P_{\mu,g,n}^{\textnormal{trop}}.$

We can view $T_{g,n}^{\textnormal{trop}}$ as a generalized sub cone complex of
$P_{\mu,g,n}^{\textnormal{trop}}$. The restriction of the forgetful morphism
$\pi_{P}^{\textnormal{trop}}\colon P_{\mu,g,n}^{\textnormal{trop}}\rightarrow
M_{g,n}^{\textnormal{trop}}$ to $T_{g,n}^{\textnormal{trop}}$ gives rise to a
forgetful morphism of generalized cone complexes

$\pi^{\textnormal{trop}}_{T}\colon T_{g,n}^{\textnormal{trop}}\rightarrow
M_{g,n}^{\textnormal{trop}}$

taking the point parametrizing the class of a pair $(X,\mathcal{D})$, for a
tropical curve $X$ and a polystable square root $\mathcal{D}$ of $X$ to the
point parametrizing the class of $X$.

###  5.3. Refining the moduli space $T_{g,n}^{\textnormal{trop}}$

We construct a refinement of the moduli space $T_{g,n}^{\textnormal{trop}}$ of
polystable tropical theta characteristics using $v_{0}$-quasistability, as we
did for the universal tropical Jacobian $P_{\mu,g,n}^{\textnormal{trop}}$.

Recall that, if $(\Gamma,\mathcal{E},D)$ is a simple semistable graph, then
$\kappa_{(\Gamma,\mathcal{E},D)}=\lambda_{(\Gamma,\mathcal{E},D)}$ (see Remark
5.6). By Lemma 5.2, if
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
is a specialization of simple semistable root-graphs, then we have a face
inclusion
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}\rightarrow\lambda_{(\Gamma,\mathcal{E},D)}$,
and every face of $\lambda_{(\Gamma,\mathcal{E},D)}$ arises in this way.

######  Definition 5.20.

We define the generalized cone complex, called the _moduli space of simple
semistable roots_ , as

$\Theta^{\textnormal{trop}}_{g,n}:=\underset{\longrightarrow}{\lim}\;\lambda_{(\Gamma,\mathcal{E},D)}=\coprod_{[\Gamma,\mathcal{E},D]}\lambda^{\circ}_{(\Gamma,\mathcal{E},D)}/\textnormal{Aut}(\Gamma,\mathcal{E},D),$

where the limit is taken over the sub-category of $\mathbf{SR}_{g,n}$ and the
union is taken over all equivalence classes $[\Gamma,\mathcal{E},D]$ in the
poset $\mathcal{SR}_{g,n}$.

Recall that a quasistable pseudo-divisor is simple.

######  Definition 5.21.

The _moduli space of quasistable roots_ is the sub-cone complex
$R_{g,n}^{\textnormal{trop}}\subset\Theta^{\textnormal{trop}}_{g,n}$ obtained
by restricting $\mathbf{SR}_{g,n}$ to the sub-category $\mathbf{QR}_{g,n}$,
that is,