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

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

######  Proposition 5.22.

The generalized cone complex $R^{\textnormal{trop}}_{g,n}$ 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
$p_{0}$-quasistable root of $X$, and two pairs $(X,\mathcal{D})$ and
$(X^{\prime},\mathcal{D}^{\prime})$ are equivalent if there is an isomorphism
$\iota\colon X\rightarrow X^{\prime}$ such that
$\iota_{*}(\mathcal{D})=\mathcal{D}^{\prime}$.

###### Proof.

We can repeat verbatim the proof of Proposition 5.17, replacing the words
“polystable” with “$p_{0}$-quasistable”, $\kappa_{(\Gamma,\mathcal{E},D)}$
with $\lambda_{(\Gamma,\mathcal{E},D)}$, and $\mathbf{PSR}_{g,n}$ with
$\mathbf{QR}_{g,n}$. ∎

By [2, Proposition 5.12] and Proposition 5.22, we have an inclusion

$\psi_{R}^{\textnormal{trop}}\colon R_{g,n}^{\textnormal{trop}}\rightarrow
J_{\mu,g,n}^{\textnormal{trop}}$

which is a map of generalized cone complexes. Indeed, for every
$v_{0}$-pseudo-root, the inclusion
$\lambda_{(\Gamma,\mathcal{E},D)}\subset\mathbb{R}^{E(\Gamma^{\mathcal{E}})}_{\geq
0}\cong\tau_{(\Gamma,\mathcal{E},D)}$ factors the composition

$\lambda_{(\Gamma,\mathcal{E},D)}\rightarrow
R_{g,n}^{\textnormal{trop}}\stackrel{{\scriptstyle\psi_{R}^{\textnormal{trop}}}}{{\longrightarrow}}J_{\mu,g,n}^{\textnormal{trop}}.$

Moreover, the composition

$R_{g,n}^{\textnormal{trop}}\stackrel{{\scriptstyle\psi_{R}^{\textnormal{trop}}}}{{\longrightarrow}}J_{\mu,g,n}^{\textnormal{trop}}\stackrel{{\scriptstyle\rho_{P}^{\textnormal{trop}}}}{{\longrightarrow}}P_{\mu,g,n}^{\textnormal{trop}}.$

factors through the inclusion $\psi_{T}^{\textnormal{trop}}\colon
T_{g,n}^{\textnormal{trop}}\hookrightarrow P_{\mu,g,n}^{\textnormal{trop}}$,
because, if $\mathcal{D}$ is a $p_{0}$-quasistable root on a tropical curve
$X$, then a polystable divisor $\mathcal{D}^{\prime}$ on $X$ equivalent to
$\mathcal{D}$ is clearly a polystable root of $X$. The induced map

$\rho_{T}^{\textnormal{trop}}\colon R_{g,n}^{\textnormal{trop}}\rightarrow
T_{g,n}^{\textnormal{trop}}$

is a morphism of generalized cone complexes. Indeed, for every
$v_{0}$-quasistable pseudo-divisor $(\mathcal{E},D)$ on a graph $\Gamma$, we
have that Proposition 4.33 together with Remarks 5.6 and 5.7 give rise to an
inclusion
$\lambda_{(\Gamma,\mathcal{E},D)}=\kappa_{(\Gamma,\mathcal{E},D)}\subset\kappa_{(\Gamma,\operatorname{pol}(\mathcal{E},D))}$
which factors the composition

$\lambda_{(\Gamma,\mathcal{E},D)}\rightarrow
R_{g,n}^{\textnormal{trop}}\stackrel{{\scriptstyle\rho_{T}^{\textnormal{trop}}}}{{\longrightarrow}}T_{g,n}^{\textnormal{trop}}.$

######  Lemma 5.23.

Let
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$
be a specialization of semistable root-graphs. If $(\mathcal{E},D)$ is
polystable and $(\mathcal{E}^{\prime},D^{\prime})$ is $v_{0}$-quasistable,
then there exists a specialization of semistable roots-graphs
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime}))$
factoring
$(\Gamma,\mathcal{E},D)\geq(\Gamma,\mathcal{E}^{\prime},D^{\prime})$.

###### Proof.

We let $\iota\colon\Gamma\rightarrow\Gamma^{\prime}$ and
$\widehat{\iota}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$
be the specializations associated to the specialization
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.
We set
$(\widehat{\mathcal{E}},\widehat{D})=\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})$.
By Proposition 4.24, we know that there is a specialization of semistable
root-graphs
$(\Gamma^{\prime},\widehat{\mathcal{E}},\widehat{D})\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.
We let
$\widehat{\iota}^{\prime}\colon{\Gamma^{\prime}}^{\widehat{\mathcal{E}}}\rightarrow{\Gamma^{\prime}}^{\mathcal{E}^{\prime}}$
be the associated specialization. By Lemma 3.6 we have a specialization of
triples
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\widehat{\mathcal{E}},\widehat{D})$
factoring
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.
We let
$\widehat{j}\colon\Gamma^{\mathcal{E}}\rightarrow{\Gamma^{\prime}}^{\widehat{\mathcal{E}}}$
be the induced specialization. By Corollary 4.20, we need only to prove that
$\widehat{j}$ is induced by $\iota$. Consider an edge $e\in\mathcal{E}$ and
let $e_{1},e_{2}$ be the exceptional edges of $\Gamma^{\mathcal{E}}$ over $e$.
We have to prove that, if $e_{1}$ is contracted by $\widehat{j}$, then $e_{2}$
is contracted by $\widehat{j}$ as well. If $e_{1}$ was contracted by
$\widehat{j}$ and $e_{2}$ was not, then
$\phi_{(\widehat{\mathcal{E}},\widehat{D})}(e_{2})=0$ by Lemma 4.16. Using
that
$\widehat{\iota}_{*}(\phi_{(\mathcal{E},D)})=\phi_{(\mathcal{E}^{\prime},D^{\prime})}=\widehat{\iota}^{\prime}_{*}(\phi_{(\widehat{\mathcal{E}},\widehat{D})})$,
we would have:

$|\phi_{(\mathcal{E},D)}(e_{2})|=|\phi_{(\mathcal{E}^{\prime},D^{\prime})}(e_{2})|=|\phi_{(\widehat{\mathcal{E}},\widehat{D})}(e_{2})|=0.$

But we would get a contradiction to Lemma 4.16, since $e_{2}$ is an
exceptional edge of $\Gamma^{\mathcal{E}}$. ∎

######  Proposition 5.24.

The morphism of generalized cone complexes $\rho_{T}^{\textnormal{trop}}\colon
R_{g,n}^{\textnormal{trop}}\rightarrow T_{g,n}^{\textnormal{trop}}$ is a
refinement. For every polystable pseudo-root $(\Gamma,\mathcal{E},D)$, we have
decompositions:

$\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}=\underset{(\mathcal{E},D)=\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})}{\coprod_{(\mathcal{E}^{\prime},D^{\prime})\in\mathbf{QR}_{g,n}}}\lambda^{\circ}_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}\quad\text{
and
}\quad\kappa_{(\Gamma,\mathcal{E},D)}=\underset{(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}{\coprod_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})\in\mathbf{QR}_{g,n}}}\lambda^{\circ}_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})}.$

where in the last formula $\geq$ means specialization of semistable root-
graphs.

###### Proof.

Notice the morphism of cone complexes $\rho_{T}^{\textnormal{trop}}$ is
bijective because in an equivalence class of a polystable divisor
$\mathcal{D}$ on a tropical curve $X$ there is exactly one
$(v_{0},\mu)$-quasistable representative; such a representative is a root of
$X$ if and only if $\mathcal{D}$ is a root of $X$. Thus
$\rho_{T}^{\textnormal{trop}}$ is a refinement.

The stated decompositions of $\kappa^{\circ}_{(\Gamma,\mathcal{E},D)}$ and
$\kappa_{(\Gamma,\mathcal{E},D)}$ can be proved exactly as in Theorem 3.7,
where we use the following computation for the former:

$\lambda^{\circ}_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}=\kappa^{\circ}_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}=\kappa_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}\cap\sigma^{\circ}_{(\Gamma,\mathcal{E}^{\prime},D^{\prime})}\subset\kappa_{(\Gamma,\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime}))}\cap\sigma^{\circ}_{(\Gamma,\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime}))}=\kappa^{\circ}_{(\Gamma,\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime}))}$

(recall Remark 5.7), and we use Proposition 4.24 and Lemma 5.23 for the
latter. ∎

##  6\. Moduli of spin quasistable roots

Given a spin semistable root-graph $(\Gamma,\mathcal{E},D,s)$, we define the
cones

(21)
$\lambda_{(\Gamma,\mathcal{E},D,s)}:=\lambda_{(\Gamma,\mathcal{E},D)}\quad\text{
and }\quad\kappa_{(\Gamma,\mathcal{E},D,s)}:=\kappa_{(\Gamma,\mathcal{E},D)}$

and their interiors:

$\lambda^{\circ}_{(\Gamma,\mathcal{E},D,s)}:=\lambda^{\circ}_{(\Gamma,\mathcal{E},D)}\quad\text{
and
}\quad\kappa_{(\Gamma,\mathcal{E},D,s)}^{\circ}=\kappa_{(\Gamma,\mathcal{E},D)}^{\circ}.$

If
$(\Gamma,\mathcal{E},D,s)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})$
is a specialization of spin polystable root-graphs, using Lemma 5.14 we get a
face morphism
$\kappa_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})}\subset\kappa_{(\Gamma,\mathcal{E},D,s)}$
and every face of $\kappa_{(\Gamma,\mathcal{E},D,s)}$ arises in this way.

######  Definition 6.1.

We define the generalized cone complex, called the _moduli space of spin
tropical curves_ , as

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

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

Recall that if $\mathcal{D}$ is a unitary semistable root of a tropical curve
$X$ with model $\Gamma_{X}$, then its combinatorial type is a pseudo-root of
$\Gamma_{X}$ (see Lemma 5.11).

######  Proposition 6.2.

The generalized cone complex $S_{g,n}^{\textnormal{trop}}$ parametrizes
classes of triples $(X,\mathcal{D},s)$, where $X$ is a stable $n$-pointed
tropical curve of genus $g$, together with a polystable root $\mathcal{D}$ of
$X$ with a sign function $s$ on the combinatorial type $(\mathcal{E},D)$ of
$\mathcal{D}$, where two triples $(X,\mathcal{D},s)$ and
$(X^{\prime},\mathcal{D}^{\prime},s^{\prime})$ are equivalent if
$(X,\mathcal{D})$ and $(X^{\prime},\mathcal{D}^{\prime})$ are equivalent and
$\widehat{\iota}_{*}(s)=s^{\prime}$, where
$\widehat{\iota}\colon\Gamma_{X}^{\mathcal{E}}\rightarrow\Gamma_{X^{\prime}}^{\mathcal{E}^{\prime}}$
is the induced isomorphism.

###### Proof.

The argument is similar to the one used to prove Proposition 5.17 and is left
to the reader. ∎

######  Remark 6.3.

The moduli space $S^{\textnormal{trop}}_{g,n}$ is isomorphic (as a generalized
cone complex) to the moduli space of tropical spin curves introduced in [11],
which parametrizes classes of triples $(X,P,s)$ where $X$ is a stable
$n$-pointed tropical curve of genus $g$, together with a cyclic subgraph $P$
of $\Gamma_{X}$ and a sign function $s\colon
V(\Gamma_{X}/P)\rightarrow\mathbb{Z}/2\mathbb{Z}$. The isomorphism takes the
class of a triples $(X,P,s)$ to $(X,\mathcal{D}_{P},s)$ (we use Remark 4.18
and Proposition 5.18). Indeed, Lemma 5.14 guarantees that the given bijection
is a morphism of generalized cone complexes.

By Propositions 5.17 and 6.2 we obtain a forgetful morphism of generalized
cone complexes

$\phi_{S}^{\textnormal{trop}}\colon S_{g,n}^{\textnormal{trop}}\rightarrow
T_{g,n}^{\textnormal{trop}}$

taking the point parametrizing the class of a triples $(X,\mathcal{D},s)$ to
the point of $T_{g,n}^{\textnormal{trop}}$ parametrizing the class of the pair
$(X,\mathcal{D})$. We also have the morphism of cone complexes

$\pi^{\textnormal{trop}}_{S}=\pi^{\textnormal{trop}}_{T}\circ\phi^{\textnormal{trop}}_{S}\colon
S_{g,n}^{\textnormal{trop}}\rightarrow M_{g,n}^{\textnormal{trop}}$

taking the point parametrizing the class of $(X,\mathcal{D},s)$ to the point
parametrizing $X$.

###  6.1. Refining the moduli space of spin tropical curves

Recall that if $(\Gamma,\mathcal{E},D,s)$ is simple (for example,
$v_{0}$-quasistable), then
$\lambda_{(\Gamma,\mathcal{E},D,s)}=\kappa_{(\Gamma,\mathcal{E},D,s)}$ (see
Remark 5.6). By Lemma 5.2, if
$(\Gamma,\mathcal{E},D,s)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})$
is a specialization of spin simple semistable root-graphs, then we have a face
inclusion
$\lambda_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})}\rightarrow\lambda_{(\Gamma,\mathcal{E},D,s)}$,
and every face of $\lambda_{(\Gamma,\mathcal{E},D,s)}$ arises in this way.

######  Remark 6.4.

It is worth mentioning that Lemma-Definition 5.3 and Corollary 5.4 can be
rephrased and extended to spin semistable root-graphs and spin
$v_{0}$-quasistable root-graphs.

######  Definition 6.5.

We define the generalized cone complex, called the _moduli space of spin
quasistable roots_ , as

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

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

######  Remark 6.6.

By Remark 4.29, $Q^{\textnormal{trop}}_{g,n}$ has two connected components
$(Q^{\textnormal{trop}}_{g,n})^{+}$ and $(Q^{\textnormal{trop}}_{g,n})^{-}$
corresponding, respectively, to the posets $\mathcal{SPQ}^{+}_{g,n}$ and
$\mathcal{SPQ}^{-}_{g,n}$.

######  Proposition 6.7.

The generalized cone complex $Q_{g,n}^{\textnormal{trop}}$ parametrizes
classes of triples $(X,\mathcal{D},s)$, where $X$ is a stable $n$-pointed
tropical curve of genus $g$, together with a $p_{0}$-quasistable root
$\mathcal{D}$ of $X$ with a sign function $s$ on the combinatorial type
$(\mathcal{E},D)$ of $\mathcal{D}$, where two triples $(X,\mathcal{D},s)$ and
$(X^{\prime},\mathcal{D}^{\prime},s^{\prime})$ are equivalent if
$(X,\mathcal{D})$ and $(X^{\prime},\mathcal{D}^{\prime})$ are equivalent and
$\widehat{\iota}_{*}(s)=s^{\prime}$, where
$\widehat{\iota}\colon\Gamma_{X}^{\mathcal{E}}\rightarrow\Gamma_{X^{\prime}}^{\mathcal{E}^{\prime}}$
is the induced isomorphism.

###### Proof.

The argument is similar to the one used to prove Proposition 5.17 and is left
to the reader. ∎

By Propositions 5.22, 6.2, and 6.7, we obtain a commutative diagram of
morphisms of generalized cone complexes:

$\textstyle{Q_{g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{Q}^{\textnormal{trop}}}$$\scriptstyle{\rho_{S}^{\textnormal{trop}}}$$\textstyle{R_{g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{T}^{\textnormal{trop}}}$$\textstyle{S_{g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{S}^{\textnormal{trop}}}$$\textstyle{T_{g,n}^{\textnormal{trop}}}$

Let us describe the new morphisms appearing in the diagram. Consider a point
$q$ of $Q^{\textnormal{trop}}_{g,n}$ parametrizing the class of a triple
$(X,\mathcal{D},s)$. The map $\phi_{Q}^{\textnormal{trop}}$ takes $q$ to the
point parametrizing the class of the pair $(X,\mathcal{D})$. This is a
morphism of cone complexes by the definition of the cone
$\lambda_{(\Gamma,\mathcal{E},D,s)}$ in equation (21).

If $\mathcal{D}^{\prime}$ is a polystable divisor on $X$ equivalent to
$\mathcal{D}$ and $(\mathcal{E},D)$ is the combinatorial type of
$\mathcal{D}$, then the combinatorial type of $\mathcal{D}^{\prime}$ is
$\operatorname{pol}(\mathcal{E},D)$. The map $\phi_{S}^{\textnormal{trop}}$
takes $q$ to the point parametrizing the class
$(X,\mathcal{D}^{\prime},\operatorname{pol}(s))$. By the definition of the
cone $\kappa_{(\Gamma,\mathcal{E},D,s)}$ in equation (21) it is easy to see
that $\phi_{S}^{\textnormal{trop}}$ is a morphism of generalized cone
complexes.

######  Lemma 6.8.

Let
$(\Gamma,\mathcal{E},D,s)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})$
be a specialization of spin semistable root-graphs. If $(\mathcal{E},D,s)$ is
polystable and $(\mathcal{E}^{\prime},D^{\prime},s^{\prime})$ is
$v_{0}$-quasistable, then there exists a specialization of spin semistable
root-graphs
$(\Gamma,\mathcal{E},D,s)\geq(\Gamma^{\prime},\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime},s^{\prime}))$
factoring
$(\Gamma,\mathcal{E},D,s)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})$.

###### Proof.

Set
$(\widehat{\mathcal{E}},\widehat{D}):=\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})$.
By Lemma 5.23 we have a specialization of semistable root-graphs
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\widehat{\mathcal{E}},\widehat{D})$
factoring the specialization of semistable root-graphs
$(\Gamma,\mathcal{E},D)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})$.
By Proposition 4.24 we have an identification
$P_{(\mathcal{E}^{\prime},D^{\prime})}=P_{(\widehat{\mathcal{E}},\widehat{D})}$.
Hence we get a commutative diagram of specializations:

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

where $\widetilde{\iota}$ and $\widetilde{\iota}^{\prime}$ are the
specializations of Lemma 4.27. Hence
$\widetilde{\iota}^{\prime}_{*}(s)=\widetilde{\iota}_{*}(s)=s^{\prime}=\operatorname{pol}(s^{\prime})$,
and we are done. ∎

######  Theorem 6.9.

The morphism of generalized cone complexes $\rho_{S}^{\textnormal{trop}}\colon
Q_{g,n}^{\textnormal{trop}}\rightarrow S_{g,n}^{\textnormal{trop}}$ is a
refinement. For every spin polystable pseudo-root $(\Gamma,\mathcal{E},D,s)$,
we have decompositions:

$\kappa^{\circ}_{(\Gamma,\mathcal{E},D,s)}=\underset{(\mathcal{E},D,s)=\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime},s^{\prime})}{\coprod_{(\mathcal{E}^{\prime},D^{\prime},s^{\prime})\in\mathbf{SPQ}_{g,n}}}\lambda^{\circ}_{(\Gamma,\mathcal{E}^{\prime},D^{\prime},s^{\prime})}\quad\text{
and
}\quad\kappa_{(\Gamma,\mathcal{E},D,s)}=\underset{(\Gamma,\mathcal{E},D,s)\geq(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})}{\coprod_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})\in\mathbf{SPQ}_{g,n}}}\lambda^{\circ}_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})},$

where in the last formula $\geq$ means specialization of spin semistable
pseudo-roots.

###### Proof.

We know that the map $\rho_{T}^{\textnormal{trop}}$ is a bijection. Given a
sign $s$ on a polystable pseudo-root $(\mathcal{E},D)$ on a graph $\Gamma$ and
a $v_{0}$-quasistable pseudo-root $(\mathcal{E}^{\prime},D^{\prime})$ on
$\Gamma$ such that
$\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime})=(\mathcal{E},D)$, we have
that $s^{\prime}=s$ is the unique sign on $(\mathcal{E}^{\prime},D^{\prime})$
such that $\operatorname{pol}(s^{\prime})=s$ (we use that, by Proposition
4.24, $P_{(\mathcal{E},D)}=P_{(\mathcal{E}^{\prime},D^{\prime})}$). It follows
that $\rho^{\textnormal{trop}}_{S}$ is bijective as well, hence it is a
refinement.

For the stated decompositions, we can argue as in the proof of Proposition
5.24, using Proposition 4.33 and Lemma 6.8 for the second decomposition. ∎

We sum up the morphisms of generalized cone complexes constructed so far in
the following commutative diagram:

$\textstyle{Q_{g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{Q}^{\textnormal{trop}}}$$\scriptstyle{\rho_{S}^{\textnormal{trop}}}$$\textstyle{R_{g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{T}^{\textnormal{trop}}}$$\scriptstyle{\psi_{R}^{\textnormal{trop}}}$$\textstyle{J_{\mu,g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{P}^{\textnormal{trop}}}$$\scriptstyle{\pi_{J}^{\textnormal{trop}}}$$\textstyle{S_{g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{S}^{\textnormal{trop}}}$$\textstyle{T_{g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\textnormal{trop}}_{T}}$$\textstyle{P_{\mu,g,n}^{\textnormal{trop}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{P}^{\textnormal{trop}}}$$\textstyle{M_{g,n}^{\textnormal{trop}}}$

In the diagram, the $\rho^{\textnormal{trop}}$ maps are refinements, the
$\pi^{\textnormal{trop}}$ and the $\phi^{\textnormal{trop}}$ maps are
forgetful maps, and the $\psi^{\textnormal{trop}}$ maps are injections. We
conclude the section with an example illustrating the decompositions given by
Theorem 6.9.

######  Example 6.10.

Consider a graph $\Gamma$ with two vertices connected by two edges. Set
$V(\Gamma)=\\{v_{0},v_{1}\\}$ and $E(\Gamma)=\\{e_{0},e_{1}\\}$. Consider the
polystable pseudo-root $(\mathcal{E},D)$ on $\Gamma$, where
$\mathcal{E}=E(\Gamma)$ and

$D=v_{0}+v_{1}-w_{0}-w_{1}.$

Here $w_{i}\in V(\Gamma^{\mathcal{E}})$ is the vertex in the interior of
$e_{i}$. Then $\phi_{(\mathcal{E},D)}$ has value 1 over every
$e\in\overrightarrow{E}(\Gamma^{\mathcal{E}})$ oriented as in Figure 3 and
$P_{(\mathcal{E},D)}$ has no edges. Consider the sign function
$s:V(\Gamma/P_{(\mathcal{E},D)})\rightarrow\mathbb{Z}/2\mathbb{Z}$ which has
value 0 on every $v\in V(\Gamma/P_{(\mathcal{E},D)})=V(\Gamma)$.

$v_{0}$$v_{1}$$w_{0}$$w_{1}$$y$$x$$w$$z$ Figure 3. The refinement of $\Gamma$.

We have the injection
$\kappa_{(\Gamma,\mathcal{E},D,s)}=\kappa_{(\Gamma,\mathcal{E},D)}=\mathcal{T}_{\mathcal{E}}(\lambda_{(\Gamma,\mathcal{E},D)})\subset\mathbb{R}^{4}/{\mathcal{L}_{\mathcal{E}}}$,
where $\mathcal{L}_{\mathcal{E}}$ is the one-dimensional linear space spanned
by the vector with coordinates $(1,1,-1,-1)$. By Lemma 5.14, we have
identifications $\mathbb{R}^{2}_{\geq
0}\stackrel{{\scriptstyle\delta_{\mathcal{E}}}}{{\cong}}\Delta_{\mathcal{E}}\stackrel{{\scriptstyle\mathcal{T}_{\mathcal{E}}}}{{\cong}}\kappa_{(\Gamma,\mathcal{E},D,s)}\subset\mathbb{R}^{2}$.
We get a commutative diagram:

$\textstyle{\mathbb{R}_{\geq
0}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\delta_{\mathcal{E}}}$$\textstyle{\Delta_{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{T}_{\mathcal{E}}}$$\textstyle{\kappa_{(\Gamma,\mathcal{E},D,s)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{R}^{2}_{\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\mathcal{E}}}$$\textstyle{\mathbb{R}^{4}_{\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{T}_{\mathcal{E}}}$$\textstyle{\sigma_{(\Gamma,\mathcal{E},D)}}$

where $\alpha$ is the inclusion in equation (15). We let $(u,v)$ be the
coordinates of $\mathbb{R}^{2}$ and $(x,y,z,w)$ the coordinates of
$\mathbb{R}^{4}$. The $v_{0}$-quasistable spin pseudo-roots
$(\mathcal{E}^{\prime},D^{\prime},s^{\prime})$ of $\Gamma$ such that
$\operatorname{pol}(\mathcal{E}^{\prime},D^{\prime},s^{\prime})=(\mathcal{E},D,s)$
are:

$\begin{array}[]{ll}(\mathcal{E}_{1},D_{1},s)=(\\{e_{1}\\},v_{0}-w_{0},s),\;\;(\mathcal{E}_{2},D_{2},s)=(\\{e_{2}\\},v_{0}-w_{1},s),\;\;(\mathcal{E}_{3},D_{3},s)=(\emptyset,v_{0}-v_{1},s).\end{array}$

In Figure 4 we draw a picture of the underlying pseudo-roots.

$v_{0}$$v_{1}$$w_{0}$$y$$x$$z$$v_{0}$$v_{1}$$x$$w_{1}$$y$$w$$v_{0}$$v_{1}$$x$$y$
Figure 4. The pseudo-roots
$(\mathcal{E}_{1},D_{1})$,$(\mathcal{E}_{2},D_{2})$,
$(\mathcal{E}_{3},D_{3})$.

Given a point $(x,y,z,0)\in\lambda^{\circ}_{(\mathcal{E}_{1},D_{1},s)}$, we
have $x-z-y=0$. Hence the points
$(x,y,z,0)\in\lambda^{\circ}_{(\mathcal{E},D,s)}$ and
$(\frac{x+z}{2},\frac{y}{2},\frac{x+z}{2},\frac{y}{2})$ get identified with
the same point
$[\frac{x+z}{2},\frac{y}{2},\frac{x+z}{2},\frac{y}{2}]\in\kappa_{(\Gamma,\mathcal{E},D)}\subset\mathbb{R}^{4}/\mathcal{L}_{\mathcal{E}}$.
Under the identification
$\kappa_{(\Gamma,\mathcal{E},D)}\cong\mathbb{R}^{2}_{\geq 0}$, the point
$[\frac{x+z}{2},\frac{y}{2},\frac{x+z}{2},\frac{y}{2}]$ corresponds to the
point $(u,v)=(\frac{x+z}{2},\frac{y}{2})\in\mathbb{R}^{2}_{\geq 0}$. Thus we
get $u>v$, since $y=x-z$. It is easy to see that for every $u>v>0$, the point
$(u,v)$ corresponds to the point
$(u+v,2v,u-v,0)\in\lambda^{\circ}_{(\mathcal{E}_{1},D_{1},s)}$. A similar
argument can be done for $(\mathcal{E}_{2},D_{2},s)$ and
$(\mathcal{E}_{3},D_{3},s)$. Hence

$\kappa^{\circ}_{1}:=\kappa^{\circ}_{(\Gamma,\mathcal{E}_{1},D_{1},s)}\cong\\{(u,v)\in\mathbb{R}^{2}_{>0}\;|\;u>v\\}$
$\kappa^{\circ}_{2}:=\kappa^{\circ}_{(\Gamma,\mathcal{E}_{2},D_{2},s)}\cong\\{(u,v)\in\mathbb{R}^{2}_{>0}\;|\;u<v\\}$
$\kappa^{\circ}_{3}:=\kappa^{\circ}_{(\Gamma,\mathcal{E}_{3},D_{3},s)}\cong\\{(u,v)\in\mathbb{R}^{2}_{>0}\;|\;u=v\\}.$

The decomposition in Theorem 6.9 of $\kappa^{\circ}_{(\mathcal{E},D,s)}$ is
$\kappa^{\circ}_{(\Gamma,\mathcal{E},D,s)}=\kappa^{\circ}_{1}\sqcup\kappa^{\circ}_{2}\sqcup\kappa^{\circ}_{3}$.

Next, for $i\in\\{1,2\\}$, consider the specialization
$\iota_{i}\colon\Gamma\to\Gamma_{i}:=\Gamma/\\{e_{3-i}\\}$. There is just one
$v_{0}$-quasistable spin pseudo-root on $\Gamma_{i}$, which is
$(\mathcal{E}_{i},D^{\prime}_{i},s^{\prime}_{i})=(\\{e_{i}\\},v_{0}-w_{i-1},\iota_{i*}(s))$.
Consider the specialization
$\iota_{3}\colon\Gamma\to\Gamma_{3}:=\Gamma/\\{e_{1},e_{2}\\}$. There is just
one spin pseudo-divisor on $\Gamma_{3}$, which is
$(\mathcal{E}_{3},D^{\prime}_{3},s^{\prime}_{3})=(\emptyset,0,\iota_{3*}(s))$.
We draw a picture of these psedo-roots in Figure 5.

$v_{0}$$w_{0}$$x$$z$$v_{0}$$w_{1}$$w$$y$$v_{0}$ Figure 5. The pseudo-roots on
$\Gamma_{1},\Gamma_{2},\Gamma_{3}$.

The decomposition of
$\kappa_{(\Gamma,\mathcal{E},D,s)}\cong\mathbb{R}^{2}_{\geq 0}$ is as
prescribed by Theorem 6.9, since:

$\displaystyle\kappa^{\prime\circ}_{1}:=\kappa^{\circ}_{(\Gamma_{1},\mathcal{E}^{\prime}_{1},D^{\prime}_{1},s^{\prime}_{1})}$
$\displaystyle\cong\\{(u,0);u\in\mathbb{R}_{>0}\\}$
$\displaystyle\kappa^{\prime\circ}_{2}:=\kappa^{\circ}_{(\Gamma_{2},\mathcal{E}^{\prime}_{2},D^{\prime}_{2},s^{\prime}_{2})}$
$\displaystyle\cong\\{(0,v);v\in\mathbb{R}_{>0}\\}$
$\displaystyle\kappa^{\prime\circ}_{3}:=\kappa^{\circ}_{(\Gamma_{3},\mathcal{E}^{\prime}_{3},D^{\prime}_{3},s^{\prime}_{3})}$
$\displaystyle\cong\\{(0,0)\\}.$

The construction is illustrated in Figure 6, where we draw a section of the
relevant cones.

$\sigma_{(\Gamma,\mathcal{E},D)}$$\kappa_{(\Gamma,\mathcal{E},D,s)}$$\kappa^{\prime\circ}_{2}$$\kappa^{\prime\circ}_{1}$$\kappa^{\circ}_{1}$$\kappa^{\circ}_{2}$$\kappa^{\circ}_{3}$$\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}$$x$$w$$z$$y$$\tau_{(\Gamma,\mathcal{E}_{1},D_{1})}$$\tau_{(\Gamma,\mathcal{E}_{2},D_{2})}$

$\tau_{(\Gamma,\mathcal{E}_{3},D_{3})}$

Figure 6. The decomposition of $k_{(\Gamma,\mathcal{E},D,s)}$.

##  7\. Moduli of quasistable spin curves

###  7.1. Esteves universal Jacobian

A _(pointed) curve_ will always mean a reduced connected projective scheme of
dimension $1$ defined over an algebraically closed field $k$, whose
singularities are nodes, together with a tuple of smooth points. We consider
curves of genus $g\geq 2$. The dual graph of a curve $X$ is denoted
$\Gamma_{X}$; we let $p_{e}$ be the node of $X$ corresponding to an edge $e\in
E(\Gamma)$.

A _chain of rational components_ is a curve which is a union of smooth
connected rational components $E_{1},\dots E_{n}$ such that $E_{i}\cap
E_{j}=\emptyset$ for $|i-j|>1$ and $\\#(E_{i}\cap E_{i+1})=1$ for
$i=1,\dots,n-1$. A _semistable modification_ $X$ of a curve $C$ is a curve $X$
which is the union of the normalization $C_{\Delta}$ of $C$ at a subset
$\Delta\subset C$ of nodes and chains $E_{p}=\cup_{1\leq i\leq n_{p}}E_{p,i}$
of rational components (called _exceptional components_), for $p\in\Delta$,
such that $\\#(E_{p}\cap C_{\Delta})=2$ and $\\#(E_{p,1}\cap
C_{\Delta})=\\#(E_{p,n_{p}}\cap C_{\Delta})=1$. A _quasistable modification_
is a semistable modification such that $n_{p}=1$ for every $p\in\Delta$. There
is a map $X\rightarrow C$ contracting the exceptional components.

Recall that a pointed curve $X$ is _stable_ if so is its dual graph. We say
that $X$ is _quasistable_ if it is a quasistable modification of a stable
curve. Given a curve $X$, we denote by $\operatorname{st}(X)$ the _stable
model_ of $X$, obtained by contracting all exceptional components of $X$.
Similarly, we can define the stable model
$\operatorname{st}(\mathcal{X})\rightarrow S$ for a family
$\mathcal{X}\rightarrow S$ of nodal curves.

Let $I$ be a torsion-free rank-$1$ sheaf on a nodal curve $X$. Consider a
smooth point $p_{0}\in X$ and a polarization $\mu$ on the dual graph $\Gamma$
of $X$. For every subcurve $Y$ of $X$, we set $\mu(Y)=\sum_{v\in
V_{Y}}\mu(v)$, where $V_{Y}\subset V(\Gamma)$ is the set of vertices
corresponding to the components of $X$ contained in $Y$, and
$\delta_{Y}:=\delta_{V_{Y}}=|Y\cap\overline{X\setminus Y}|$. Let $I_{Y}$ be
the restriction of $I$ to $Y$ modulo torsion. We define:

$\beta_{Y}(I)=\deg I_{Y}-\mu(Y)+\frac{1}{2}\delta_{Y}.$

We say that $I$ is _$(p_{0},\mu)$ -quasistable_ if $\beta_{Y}(I)\geq 0$ for
every proper subcurve $Y$ where the inequality is strict if $p_{0}\in Y$. The
definition extends to families: given a family
$\pi\colon\mathcal{X}\rightarrow S$ of nodal curves with a section $\sigma$ of
$\pi$ through its smooth locus and a polarization $\mu$ on $\mathcal{X}/S$, we
say that a sheaf $\mathcal{I}$ on $\mathcal{X}/S$ is _$(\sigma,\mu)$
-quasistable_ if the restriction of $\mathcal{I}$ to every fiber $\pi^{-1}(s)$
is $(\sigma(s),\mu|_{\pi^{-1}(s)})$-quasistable for every geometric point
$s\in S$.

The construction also extends to the universal setting, giving rise to a
universal compactified Jacobian $\overline{\mathcal{J}}_{\mu,g,n}$. This is a
proper and separated Deligne-Mumford stack over $\overline{\mathcal{M}}_{g,n}$
whose sections over a $k$-scheme $S$ are:

$\overline{\mathcal{J}}_{\mu,g,n}(S)=\frac{\left\\{(\pi\colon\mathcal{X}\rightarrow
S,\mathcal{I})\big{|}\begin{array}[]{l}\pi\text{ is a family of stable
$n$-pointed genus-$g$ curves}\\\ \text{with sections $(\sigma_{i})$ and
$\mathcal{I}$ is a $(\sigma_{1},\mu)$-quasistable}\\\ \text{torsion-free
rank-1 sheaf on $\mathcal{X}/S$}\end{array}\right\\}}{\sim}$

where $(\pi\colon\mathcal{X}\rightarrow
S,\mathcal{I})\sim(\pi^{\prime}\colon\mathcal{X}^{\prime}\rightarrow
S,\mathcal{I}^{\prime})$ if there is an $S$-isomorphism
$f\colon\mathcal{X}\to\mathcal{X}^{\prime}$ and an invertible sheaf
$\mathcal{L}$ on $S$, such that $\sigma^{\prime}_{i}=f\circ\sigma_{i}$ are the
sections of $\pi^{\prime}$ and $\mathcal{I}\cong
f^{*}\mathcal{I}^{\prime}\otimes\pi^{*}\mathcal{L}$. The morphisms between two
objects $(\pi\colon\mathcal{X}\rightarrow S,\mathcal{I})$ and
$(\pi^{\prime}\colon\mathcal{X}^{\prime}\rightarrow
S^{\prime},\mathcal{I}^{\prime})$ are given by Cartesian diagrams:

$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathcal{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{S^{\prime}}$

such that there is an isomorphism
$f^{*}\mathcal{I}^{\prime}\cong\mathcal{I}\otimes\pi^{*}(\mathcal{L})$ for
some invertible sheaf $\mathcal{L}$ on $S$, and
$f\circ\sigma_{i}=\sigma^{\prime}_{i}\circ h$. We refer to [24, Theorems A and
B] and [23, Corollary 4.4 and Remark 4.6] for more details.

We now give another useful presentation of the stack
$\overline{\mathcal{J}}_{\mu,g,n}$. Consider the category
$\overline{\mathcal{P}}_{\mu,g,n}$ fibered in grupoids over the category of
$k$-schemes, whose sections over a $k$-scheme $S$ are:

$\overline{\mathcal{P}}_{\mu,g,n}(S)=\frac{\left\\{(\pi\colon\mathcal{X}\rightarrow
S,\mathcal{L})\big{|}\begin{array}[]{l}\pi\text{ is a family of $n$-pointed
genus-$g$ quasistable}\\\ \text{curves with sections $(\sigma_{i})$ and
$\mathcal{L}$ is a $(\sigma_{1},\mu)$-}\\\ \text{quasistable invertible sheaf
on $\mathcal{X}/S$ with}\\\ \text{degree $-1$ on exceptional
components}\end{array}\right\\}}{\sim}$

where the equivalence relation and the morphisms between objects are the ones
introduced above.

######  Proposition 7.1.

There is an isomorphism
$\overline{\mathcal{J}}_{\mu,g,n}\cong\overline{\mathcal{P}}_{\mu,g,n}$.

###### Proof.

Given $(\pi\colon\mathcal{X}\rightarrow
S,\mathcal{L})\in\overline{\mathcal{P}}_{\mu,g,n}(S)$, we take
$(\operatorname{st}(\pi)\colon\operatorname{st}(\mathcal{X})\rightarrow
S,\psi_{*}(\mathcal{L}))$, where
$\psi\colon\mathcal{X}\rightarrow\operatorname{st}(\mathcal{X})$ is the
contraction morphism and the sections of $\operatorname{st}(\pi)$ are given by
composing the sections of $\pi$ with $\psi$. Conversely, given
$(\pi\colon\mathcal{X}\rightarrow
S,\mathcal{I})\in\overline{\mathcal{J}}_{\mu,g,n}(S)$, we define
$\mathbb{P}_{\mathcal{X}}(\mathcal{I}^{\vee}):=\operatorname{Proj}(\operatorname{Sym}(\mathcal{I}^{\vee}))$,
with its natural structure morphism
$\psi\colon\mathbb{P}_{\mathcal{X}}(\mathcal{I}^{\vee})\rightarrow\mathcal{X}$,
and we take
$(\pi\circ\psi\colon\mathbb{P}_{\mathcal{X}}(\mathcal{I}^{\vee})\rightarrow
S,\mathcal{O}_{\mathbb{P}_{\mathcal{X}}(\mathcal{I}^{\vee})}(-1))$, where the
sections of $\pi\circ\psi$ are given by pulling-back the sections of $\pi$ via
$\psi$. The fact that we have established an isomorphism between
$\overline{\mathcal{J}}_{\mu,g,n}$ and $\overline{\mathcal{P}}_{\mu,g,n}$
follows from [20, Theorems 3.1, 4.1] and a small modification of [20,
Propositions 5.4, 5.5]. ∎

From now on, we will identify $\overline{\mathcal{J}}_{\mu,g,n}$ and
$\overline{\mathcal{P}}_{\mu,g,n}$. For every pair $(X,L)$, where $X$ is a
pointed quasistable curve of genus $g$ and $L$ is a $(\sigma,\mu)$-quasistable
invertible sheaf on $X$ with degree $-1$ on exceptional components, we let
$[X,L]$ be the geometric point of $\overline{\mathcal{J}}_{\mu,g,n}$
parametrizing $(X,L)$.

Let $X\rightarrow C$ be a quasistable modification of a curve $C$ and let $L$
be an invertible sheaf on $X$ with degree $-1$ on exceptional components. We
define the _combinatorial type_ $(\Gamma_{C},\mathcal{E}_{L},D_{L})$, where
$(\mathcal{E}_{L},D_{L})$ is the pseudo-divisor on the dual graph $\Gamma_{C}$
of $C$ such that $\mathcal{E}_{L}\subset E(\Gamma_{C})$ is the set of nodes of
$C$ to which the exceptional components of $X$ contract and $D_{L}$ is given
by the multidegree of $L$. One can see that $L$ is $(p_{0},\mu)$-quasistable
if and only if its combinatorial type is $(v_{0},\mu)$-quasistable, where
$v_{0}\in V(\Gamma)$ is the vertex corresponding to the component of $X$
containing $p_{0}$. We let:

(22)
$\tau\colon\overline{\mathcal{J}}_{\mu,g,n}\rightarrow\mathcal{QD}_{\mu,g,n}$

be the function taking $[X,L]$ to
$(\Gamma_{\operatorname{st}(X)},\mathcal{E}_{L},D_{L})$.

By [2, Proposition 6.4], the moduli space $\overline{\mathcal{J}}_{\mu,g,n}$
admits a stratification indexed by $\mathcal{QD}_{\mu,g,n}$ whose strata
$\mathcal{J}_{(\Gamma,\mathcal{E},D)}$ are given by the fibers of $\tau$,
i.e., for every $(\Gamma,\mathcal{E},D)\in\mathcal{QD}_{\mu,g,n}$,

$\mathcal{J}_{(\Gamma,\mathcal{E},D)}=\\{[X,L]\big{|}(\Gamma_{\operatorname{st}(X)},\mathcal{E}_{L},D_{L})\cong(\Gamma,\mathcal{E},D)\\}.$

Next, consider a quasistable modification $X\rightarrow C$ of a curve $C$ and
an invertible sheaf $L$ on $X$ with degree $-1$ on exceptional components.
Assume that the combinatorial type of $(X,L)$ is a semistable root-graph. Let
$\\{P_{u}\\}_{u\in V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})}$ be the set of
$\phi_{(\mathcal{E},D)}$-graphs of $\Gamma^{\mathcal{E}}$ (see Definition
4.11).

######  Definition 7.2.

The _$\phi_{(\mathcal{E},D)}$ -curves_ of $X$ are the subcurves $Z_{u}$ of $X$
whose dual graph corresponds to $P_{u}$, for $u\in
V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$. Notice that any exceptional
component of $X$ is a $\phi_{(\mathcal{E},D)}$-curve.

Notice that the two different $\phi_{(\mathcal{E},D)}$-subcurves intersect
each other at a (possibly empty) subset of nodes of $X$, and we have
$X=\cup_{u\in V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})}Z_{u}$.

######  Notation 7.3.

Given $u\in V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$ we denote by
$\mathcal{N}_{u}$ the subset of vertices $u^{\prime}\in
V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$ such that
$\phi_{(\mathcal{E},D)}(e)=-1$ for some (and hence all) edges
$e\in\overrightarrow{E}(P_{u^{\prime}},P_{u})$ such that $t(e)\in V(P_{u})$.

Hence for every $u\in V(\Gamma^{\mathcal{E}}/P_{(\mathcal{E},D)})$ we get

(23) $\mathcal{O}_{Z_{u}}\Big{(}\underset{t(e)\in
V(P_{u})}{\sum_{e\in\overrightarrow{E}(P_{u}^{c},P_{u})}}\phi_{(\mathcal{E},D)}(e)p_{e}\underset{t(e)\in
V(P_{u})}{-\sum_{e\in\overrightarrow{E}(P_{u}^{c},P_{u})}}p_{e}\Big{)}=\mathcal{O}_{Z_{u}}\Big{(}-2\sum_{u^{\prime}\in\mathcal{N}_{u}}\;\sum_{e\in
E(P_{u^{\prime}},P_{u})}p_{e}\Big{)}.$

######  Definition 7.4.

A _semistable spin curve_ is a pair $(X,L)$, where $X$ is a quasistable curve,
and $L$ is an invertible sheaf on $X$ such that:

  * (1)

the restriction of $L$ to any exceptional component has degree $-1$.

  * (2)

the combinatorial type of $(X,L)$ is a semistable root-graph.

  * (3)

for every $\phi_{(\mathcal{E}_{L},D_{L})}$-curve $Z_{u}$ of $X$ we have an
isomorphism $\widetilde{L}_{u}^{\otimes 2}\cong\omega_{Z_{u}}$, where

(24)
$\widetilde{L}_{u}:=L|_{Z_{u}}\otimes\mathcal{O}_{Z_{u}}\Big{(}-\sum_{u^{\prime}\in\mathcal{N}_{u}}\sum_{e\in
E(P_{u^{\prime}},P_{u})}p_{e}\Big{)}.$

We say that $(X,L)$ is _polystable_ and _$p_{0}$ -quasistable_, respectively,
if so is its combinatorial type.

######  Definition 7.5.

Let $(X,L)$ be a semistable spin curve and $(\Gamma,\mathcal{E},D)$ be its
combinatorial type. We define the sign function $s_{L}\colon
V(\Gamma/P_{(\mathcal{E},D)})\rightarrow\mathbb{Z}/2\mathbb{Z}$ taking $u\in
V(\Gamma/P_{(\mathcal{E},D)})$ to $s_{L}(u)$, the parity of
$h^{0}(Z_{u},\widetilde{L}_{u})$. The semistable spin root-graph
$(\Gamma,\mathcal{E},D,s_{L})$ is called the _spin combinatorial type_ of
$(X,L)$. We say that $(X,L)$ is _even_ or _odd_ if so is
$(\Gamma,\mathcal{E},D,s_{L})$.

Notice that $s_{L}$ is indeed a sign function: if the weight $w(u)$ is zero
for some $u\in V(\Gamma/P_{(\mathcal{E},D)})$, then by Corollary 4.25 we have
that $Z_{u}\cong\mathbb{P}^{1}$ and
$\widetilde{L}_{u}\cong\mathcal{O}_{\mathbb{P}^{1}}(-1)$, hence $s_{L}(u)=0$.

###  7.2. Spin curves and their moduli

The moduli space of spin curves $\overline{\mathcal{S}}_{g,n}$ compactifies
the moduli space $\mathcal{S}_{g,n}$ of isomorphism classes of pairs $(X,L)$,
where $X$ is a $n$-pointed smooth curve of genus $g$ and $L$ is an invertible
sheaf on $X$ such that $L^{\otimes 2}\cong\omega_{X}$. The moduli space
$\overline{\mathcal{S}}_{g,n}$ is finite of degree $2^{2g}$ over
$\overline{\mathcal{M}}_{g,n}$, hence $\overline{\mathcal{S}}_{g,n}$ has
dimension $3g-3+n$.

We use a different, but equivalent, modular description of the moduli space
$\overline{\mathcal{S}}_{g,n}$, which better suits our purposes. The closed
points of $\overline{\mathcal{S}}_{g,n}$ are isomorphism classes of pairs
$(X,L)$, where $X$ is a $n$-pointed quasistable curve and $L$ is an invertible
sheaf with degree $-1$ on exceptional components of $X$ and $L|_{Y}^{\otimes
2}(-2\sum_{E\in\operatorname{Ex}(X)}\sum_{p\in Y\cap E}p)\cong\omega_{Y}$,
where $Y$ is the complement of the union of the exceptional components of $X$.
Notice that $Y$ could be non-conneted. The space
$\overline{\mathcal{S}}_{g,n}$ has two connected components,
$\overline{\mathcal{S}}^{+}_{g,n}$ and $\overline{\mathcal{S}}^{-}_{g,n}$,
corresponding to even and odd spin curves.

######  Remark 7.6.

If $(X,L)$ is a pair parametrized by a point of
$\overline{\mathcal{S}}_{g,n}$, then $(X,L)$ is a polystable spin curve. In
fact, by [11, Theorem 3.4.3], the moduli space $\overline{\mathcal{S}}_{g,n}$
admits a stratification indexed by the poset $\mathcal{SP}_{g,n}$. Recall that
$\mathcal{SP}_{g,n}$ is isomorphic to the poset of isomorphism classes of
triples $(\Gamma,P,s)$, for a graph $\Gamma$, a cyclic subgraph $P$ of
$\Gamma$, and a sign function $s\colon
V(\Gamma/P)\rightarrow\mathbb{Z}/2\mathbb{Z}$ (see Remark 4.30). The strata
are defined, for every triple $(\Gamma,P,s)$, as:

$\mathcal{S}_{(\Gamma,P,s)}=\\{[X,L]\in\overline{\mathcal{S}}_{g,n}\big{|}(\Gamma_{\operatorname{st}(X)},\mathcal{E}_{L},D_{L},s_{L})\cong(\Gamma,\mathcal{E}_{P},D_{P},s)\\}.$

The moduli space $\mathcal{S}_{g,n}$ of theta characteristics on $n$-pointed
smooth curves of genus $g$ naturally sits inside
$\overline{\mathcal{J}}_{\mu,g,n}$. We define $\overline{\mathcal{Q}}_{g,n}$
as the closure of $\mathcal{S}_{g,n}$ inside
$\overline{\mathcal{J}}_{\mu,g,n}$. By definition, we have inclusions over
$\overline{\mathcal{M}}_{g,n}$:

$\overline{\mathcal{S}}_{g,n}\supset\mathcal{S}_{g,n}\subset\overline{\mathcal{Q}}_{g,n}\subset\overline{\mathcal{J}}_{\mu,g,n}.$

In particular, the dimension of $\overline{\mathcal{Q}}_{g,n}$ is $3g-3+n$.
Moreover, $\overline{\mathcal{Q}}_{g,n}$ is a Deligne-Mumford separated stack,
since so is $\overline{\mathcal{J}}_{\mu,g,n}$.

Our next goal is to give a modular interpretation of the space
$\overline{\mathcal{Q}}_{g,n}$ using root graphs. Before, we need to prove an
easy result.

######  Lemma 7.7.

A flow $\phi$ on a graph is acyclic if and only if there are positive integers
$h_{e}$, for each $e\in E(\Gamma)$ such that

$\sum_{e\in\gamma}\phi(e)h_{e}=0$

for every oriented cycle $\gamma$ of $\Gamma$. In particular, if
$\widehat{\Gamma}$ is the refinement of $\Gamma$ obtained by inserting
$h_{e}-1$ vertices over the edge $e$, then $\operatorname{Div}(\phi)$ is
principal as a divisor on $\widehat{\Gamma}$.

###### Proof.

Consider the vector space $\mathbb{R}^{E(\Gamma)}$ with coordinates
$\\{x_{e}\\}_{e\in E(\Gamma)}$. We define the linear subspace $\Lambda$ of
$\mathbb{R}^{E(\Gamma)}$ given by the equations:

(25) $\sum_{e\in\gamma}\phi(e)x_{e}=0,$

where $\gamma$ runs over the set of oriented cycles of $\Gamma$. There is a
vector $(h_{e})_{e\in E(\Gamma)}\in\Lambda\cap\mathbb{Z}_{>0}^{E(\Gamma)}$ if
and only if $\operatorname{Div}(\phi)$ is principal on the refinement of
$\Gamma$ obtained by inserting exactly $h_{e}-1$ vertices in the interior of
each edge $e\in E(\Gamma)$. It is clear that
$\Lambda\cap\mathbb{Z}_{>0}^{E(\Gamma)}$ is empty if $\phi$ is not acyclic.

If $\phi$ is acyclic and nontrivial, then $|V(\Gamma)|\geq 2$. The space
$\Lambda$ is given by intersecting at most $b_{1}(\Gamma)$ independent linear
hyperspaces in a space of dimension $|E(\Gamma)|$, hence
$\dim\Lambda\geq|V(\Gamma)|-1\geq 1$. Since $\phi$ is acyclic, for every
oriented cycle $\gamma$ for which (25) is not a trivial equation, there are
two nonempty subset $\gamma^{+},\gamma^{-}\subset\gamma$ such that
$\phi(e)\neq 0$ for $e\in\gamma^{+}\cup\gamma^{-}$, so equation (25) reads as:

$\sum_{e\in\gamma^{+}}|\phi(e)|x_{e}=\sum_{e\in\gamma^{-}}|\phi(e)|x_{e}.$

We deduce that $\Lambda\cap\mathbb{Z}_{>0}^{E(\Gamma)}$ is not empty. ∎

######  Definition 7.8.

A _one-parameter deformation_ of a curve $X$ is a family of curves
$\pi\colon\mathcal{X}\rightarrow B=\operatorname{Spec}(k[[t]])$ whose special
fiber is $X$. A _twister_ of $\mathcal{X}$ is an invertible sheaf of
$\mathcal{X}$ associated to an effective Cartier divisor of $\mathcal{X}$
supported on the special fiber. A _smoothing of $X$_ is a one-parameter
deformation with smooth generic fiber.

######  Lemma 7.9.

Let $\pi\colon\mathcal{X}\rightarrow B$ be a one-parameter deformation of a
curve $X$ with generic fiber $X_{\eta}$. Let $\Gamma$ and $\Gamma_{\eta}$ be
the dual graphs of $X$ and $X_{\eta}$. For each edge $e\in E(\Gamma)\setminus
E(\Gamma_{\eta})$ we assume that the local étale equation of $\mathcal{X}$ at
the node of $X$ corresponding to $e$ is $xy=t^{h_{e}}$, where $x,y$ are local
étale coordinates and $h_{e}$ is a positive integer. Let $T$ be a twister of
$\mathcal{X}$ and $D_{T}$ be the divisor on $\Gamma$ given by the multidegree
of $T|_{X}$. Then $D_{T}=\operatorname{Div}(\phi)$ for some acyclic flow
$\phi$ on $\Gamma$ such that $\phi(e)=0$ for every
$e\in\overrightarrow{E}(\Gamma_{\eta})\subset\overrightarrow{E}(\Gamma)$.
Moreover, if $\widehat{\Gamma}$ is the refinement of $\Gamma$ obtained by
inserting $h_{e}-1$ vertices on each edge $e\in E(\Gamma)\setminus
E(\Gamma_{\eta})$, then $D_{T}$, seen as a divisor on $\widehat{\Gamma}$, is
principal.

###### Proof.

Denote by $(X_{v})_{v\in V(\Gamma)}$ the components of $X$. Since $T$ is
supported on $X$, we can write $T=\mathcal{O}_{\mathcal{X}}(\sum_{v\in
V(\Gamma)}\alpha_{v}X_{v})$, for some $\alpha_{v}\in\mathbb{Z}$.

We claim that $\alpha_{s(e)}=\alpha_{t(e)}$ for every
$e\in\overrightarrow{E}(\Gamma_{\eta})\subset\overrightarrow{E}(\Gamma)$. The
completion of the local ring of $\mathcal{X}$ at the node $p_{e}$
corresponding to $e$ is

$\widehat{\mathcal{O}}_{\mathcal{X},p_{e}}\cong R:=\frac{k[[x,y,t]]}{(xy)},$

where $x,y$ are local étale coordinates. Assume that $f$ is the local equation
of $T$. In particular, $f$ is not a divisor of $0$ in $R$. Since $T$ is
supported on the special fiber, we have that $(t)\subset\sqrt{(f)}$ and hence
$t^{m}\in(f)$, which means that $t^{m}=f\overline{f}$ for some
$\overline{f}\in R$, so there exists an integer $\alpha$ such that
$f=t^{\alpha}u$ with $u$ invertible. This proves that
$\alpha_{s(e)}=\alpha_{t(e)}=\alpha$.

Next, for a node $p_{e}$ of $X$, with
$e\in\overrightarrow{E}(\Gamma)\setminus\overrightarrow{E}(\Gamma_{\eta})$, we
let $xy=t^{h_{e}}$ be the local equation of $\mathcal{X}$ at $p_{e}$, for some
positive integer $h_{e}$, where $x,y$ are local étale coordinates. Since $T$
is Cartier, there is an integer $a_{e}$ such that
$\alpha_{t(e)}-\alpha_{s(e)}=h_{e}a_{e}$. Notice $a_{e}+a_{\overline{e}}=0$
for $e$ and $\overline{e}$ with the same associated edge. This means that

(26)
$T|_{X_{v}}\cong\mathcal{O}_{X_{v}}(\sum_{\begin{subarray}{c}e\in\overrightarrow{E}(\Gamma)\setminus\overrightarrow{E}(\Gamma_{\eta})\\\
s(e)=v\end{subarray}}a_{e}p_{e}).$

Define the flow $\phi$ on $\Gamma$ such that $\phi(e)=-a_{e}$ for every
$e\in\overrightarrow{E}(\Gamma)\setminus\overrightarrow{E}(\Gamma_{\eta})$ and
$\phi(e)=0$ for every $e\in\overrightarrow{E}(\Gamma_{\eta})$. Then we have
$D_{T}=\operatorname{Div}(\phi)$.

We show that $\phi$ is acyclic. If $e_{1},\dots,e_{n}$ was a no-null
$\phi$-cycle, with $v_{i}:=s(e_{i})=t(e_{i-1})$, then

(27) $0=\alpha_{v_{1}}-\alpha_{v_{n}}+\sum_{1\leq i\leq
n-1}(\alpha_{v_{i+1}}-\alpha_{v_{i}})=\sum_{1\leq i\leq
n}h_{e_{i}}a_{e_{i}}=-\sum_{1\leq i\leq n}h_{e_{i}}\phi(e_{i})$

where $\phi(e_{i})\geq 0$ and $\phi(e_{i})$ is not always zero, which is a
contradiction. Hence $\phi$ is acyclic.

The last statement follows from Equation (27) and Lemma 7.7. ∎

######  Theorem 7.10.

Let $X$ be a quasistable $n$-pointed curve and $L$ be an invertible sheaf on
$X$. The following conditions are equivalent

  * (1)

$[X,L]$ is a geometric point of $\overline{\mathcal{Q}}_{g,n}$.

  * (2)

$(X,L)$ is a $p_{0}$-quasistable spin curve.

###### Proof.

Assume that $[X,L]$ is a geometric point of $\overline{\mathcal{Q}}_{g,n}$.
Set $(\Gamma,\mathcal{E},D):=\tau([X,L])\in\mathcal{QD}_{\mu,g,n}$. There is a
smoothing $\pi\colon\mathcal{X}\rightarrow B$ of $X$ and an invertible sheaf
$\mathcal{L}$ on $\mathcal{X}/B$ such that $\mathcal{L}|_{X}\cong L$ and
$\mathcal{L}|_{X_{\eta}}^{\otimes 2}\cong\omega_{X_{\eta}}$, where $X_{\eta}$
is the generic fiber of $\pi$. There is a twister $T$ of $\mathcal{X}$ such
that

(28) $\mathcal{L}^{\otimes 2}\otimes T\cong\omega_{\mathcal{X}}.$

By Lemma 7.9, the multidegree of the invertible sheaf $T$ is a principal
divisor $D_{T}$ on $\Gamma^{\mathcal{E}}$ and
$D_{T}=\operatorname{Div}(\phi)$, for an acyclic flow $\phi$ on
$\widehat{\Gamma}$. By (28), we deduce the relation
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$. Hence
$(\Gamma,\mathcal{E},D)$ is a root-graph. By Remark 4.12 and equations (26),
(23), (28) we have an isomorphism as in equation (24), for every
$\phi_{(\mathcal{E},D)}$-curve $Z_{v}$ of $X$.

Next assume that $(X,L)$ is a $v_{0}$-quasistable spin curve. Let
$(\Gamma,\mathcal{E},D)$ be the combinatorial type of $(X,L)$. By Lemma 7.7
there is a refinement $\widehat{\Gamma}$ of $\Gamma$ such that
$\operatorname{Div}(\phi_{(\mathcal{E},D)})$ is principal as a divisor on
$\widehat{\Gamma}$. Let $Y$ be the nodal curve having $\widehat{\Gamma}$ as
dual graph such that $\operatorname{st}(Y)=X$, and $\pi_{Y}\colon Y\rightarrow
X$ be the induced contraction map. By equations (23) and (24), the multidegree
of $\pi_{Y}^{*}(\omega_{X}\otimes L^{-2})$ is equal to
$\operatorname{Div}(\phi_{(\mathcal{E},D)})$ as a divisor on
$\widehat{\Gamma}$. Therefore, by [19, Corollary 6.9] there is a smoothing
$\pi\colon\mathcal{Y}\rightarrow B$ of $Y$ and a twister $T$ of $\mathcal{Y}$
such that $\pi_{Y}^{*}(\omega_{X}\otimes L^{-2})\cong T|_{Y}$. Hence by [9,
Remark 3.0.6] we get an invertible sheaf $\mathcal{M}$ on $\mathcal{Y}/B$ such
that $\mathcal{M}|_{Y}\cong\pi_{Y}^{*}(L)$ endowed with an isomorphism
$\mathcal{M}^{\otimes 2}\otimes T\cong\omega_{\mathcal{Y}/B}$. If
$\pi\colon\mathcal{Y}\rightarrow\mathcal{X}$ is the contraction of the
exceptional components of $Y$ over $X$, then $(\pi_{*}\mathcal{M})^{\otimes
2}\otimes\pi_{*}(T)\cong\omega_{\mathcal{X}/B}$ and
$\pi_{*}\mathcal{M}|_{X}\cong L$. Hence $[X,L]$ is a geometric point of
$\overline{\mathcal{Q}}_{g,n}$ as $(X,L)$ is the limit of a family of theta
characteristics on smooth curves. ∎

The previous result justifies the following definition.

######  Definition 7.11.

We call $\overline{\mathcal{Q}}_{g,n}$ the _moduli space of quasistable spin
curves_.

###  7.3. The stratification of $\overline{\mathcal{Q}}_{g,n}$

Next, consider a $v_{0}$-quasistable spin root-graph
$(\Gamma,\mathcal{E},D,s)$. We let $\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}$ be
the locus in $\overline{\mathcal{Q}}_{g,n}$ defined as:

$\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}=\\{[X,L]\big{|}(\Gamma_{\operatorname{st}(X)},\mathcal{E}_{L},D_{L},s_{L})\cong(\Gamma,\mathcal{E},D,s)\\}.$

We have an inclusion
$\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}\subset\mathcal{J}_{(\Gamma,\mathcal{E},D)}$.
Recall the definition of $\widetilde{\mathcal{M}}_{\Gamma}$ in equation (2).
Let $\mathcal{C}_{\Gamma}\rightarrow\mathcal{M}_{\Gamma}$ be the universal
family over $\mathcal{M}_{\Gamma}$ and set
$\widetilde{\mathcal{C}}_{\Gamma}:=\mathcal{C}_{\Gamma}\times_{\mathcal{M}_{\Gamma}}\widetilde{\mathcal{M}}_{\Gamma}$.
Let $\widetilde{\mathcal{C}}_{\Gamma,\mathcal{E}}$ be the partial
normalization of $\widetilde{\mathcal{C}}_{\Gamma}$ at the nodes corresponding
to the edges of $\mathcal{E}$. Let
$\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$ be the Jacobian over
$\widetilde{\mathcal{M}}_{\Gamma}$ parametrizing invertible sheaves of
multidegree $D_{\mathcal{E}}$ over the family
$\widetilde{\mathcal{C}}_{\Gamma,\mathcal{E}}\to\widetilde{\mathcal{M}}_{\Gamma}$.
Define
$\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}\subset\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$
by the following fiber diagram

${\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}}$${\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}}$${\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}}$${\mathcal{J}_{(\Gamma,\mathcal{E},D)}}$

Recall the definition of $b_{1}(\mathcal{E},D)$ in equation (12).

######  Lemma 7.12.

Every irreducible component of
$\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}$ has dimension at least
$\dim\mathcal{M}_{\Gamma}+b_{1}(\mathcal{E},D)$.

###### Proof.

We have that $\widetilde{\mathcal{C}}_{\Gamma,\mathcal{E}}$ can be written as
a union $\bigcup_{u\in
V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})}{\mathcal{Z}}_{u}$, where
$\mathcal{Z}_{u}\rightarrow\widetilde{\mathcal{M}}_{\Gamma}$ is a family whose
fibers have dual graph isomorphic to the connected component $P_{u}$ of
$\overline{P}_{(\mathcal{E},D)}$ corresponding to $u$. Let
$\sigma_{\mathcal{N}_{u}}$ be the set of sections of
$\mathcal{Z}_{u}\rightarrow\widetilde{\mathcal{M}}_{\Gamma}$ whose
corresponding legs of $P_{u}$ is obtained from an edge $e\in
E(\Gamma^{\mathcal{E}})$ connecting $u$ and a vertex
$u^{\prime}\in\mathcal{N}_{u}$ (see Notation 7.3). Let $\mathcal{L}$ be the
universal invertible sheaf over the family

$\widetilde{\mathcal{C}}_{\Gamma,\mathcal{E}}\times_{\widetilde{\mathcal{M}}_{\Gamma}}\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}\rightarrow\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}.$

Set
$\widetilde{\mathcal{Z}}_{u}:=\mathcal{Z}_{u}\times_{\widetilde{\mathcal{M}}_{\Gamma}}\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$.
Let $\mathcal{L}_{u}$ be the restriction of $\mathcal{L}$ to the family
$f_{u}\colon\widetilde{\mathcal{Z}}_{u}\rightarrow\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$,
and $\widetilde{\sigma}_{\mathcal{N}_{u}}$ be the set of sections of $f_{u}$
obtained as the pull-back of the sections in $\sigma_{\mathcal{N}_{u}}$. We
define

$\widetilde{\mathcal{L}}_{u}:=\mathcal{L}_{u}\otimes\mathcal{O}_{\widetilde{\mathcal{Z}}_{u}}(-\sum_{\sigma_{i}\in\sigma_{\widetilde{\mathcal{N}}_{u}}}\sigma_{i}(\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)})).$

Finally, let $\mathcal{Q}_{u}$ be the locus of points $p$ of
$\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$ where
$\widetilde{\mathcal{L}}_{u}^{\otimes
2}|_{f_{u}^{-1}(p)}\cong\omega_{f_{u}^{-1}(p)}$.

We claim that the codimension of $\mathcal{Q}_{u}$ in
$\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$ is at most the genus
$g(P_{u})$ of $P_{u}$. Indeed, define $\mathcal{J}_{u}$ as the relative
Jacobian of $f_{u}$ parametrizing invertible sheaves of multidegree equal to
the multidegree of $\omega_{f_{u}}$. We have that
$\mathcal{J}_{u}\to\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$ has two
sections induced by $\omega_{f_{u}}$ and by
$\widetilde{\mathcal{L}}_{u}^{\otimes 2}$, and $\mathcal{Q}_{u}$ is set-
theoretically the locus where these sections agree. Since the relative
dimension of
$\mathcal{J}_{u}\to\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$ is
$g(P_{u})$ and $\mathcal{J}_{u}$ is smooth, each irreducible component of
$\mathcal{Q}_{u}$ has codimension at most $g(P_{u})$.

Let $\mathcal{Q}_{u,s(u)}$ be the locus in $\mathcal{Q}_{u}$ where
$h^{0}(X,\widetilde{\mathcal{L}}_{v}|_{X})\equiv s(u)\mod(2)$ for every fiber
$X$ of $f_{u}$. By deformation invariance of the parity of
$h^{0}(X,\widetilde{\mathcal{L}}_{u}|_{X})$, each irreducible component of
$\mathcal{Q}_{u,s(u)}$ is an irreducible component of $\mathcal{Q}_{u}$. Hence
each irreducible component of $\mathcal{Q}_{u,s(u)}$ has codimension at most
$g(P_{u})$ in $\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$. By Theorem
7.10 we have $\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}=\bigcap_{u\in
V(\Gamma_{\mathcal{E}}/P_{\mathcal{E},D})}\mathcal{Q}_{u,s(u)}$ (set-
theoretically). Then, by the claim,
$\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}$ has codimension at most
$\sum_{u\in V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})}g(P_{u})$ in
$\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}$. Thus:

$\displaystyle\dim\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}$
$\displaystyle\geq\dim\widetilde{\mathcal{J}}_{(\Gamma,\mathcal{E},D)}-\sum_{u\in
V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})}g(P_{u})$
$\displaystyle=\dim(\mathcal{M}_{\Gamma})+g(\Gamma_{\mathcal{E}})-\sum_{u\in
V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})}g(P_{u})$
$\displaystyle=\dim(\mathcal{M}_{\Gamma})+g(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})-\sum_{u\in
V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})}g(P_{u})$
$\displaystyle=\dim(\mathcal{M}_{\Gamma})+b_{1}(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})$
$\displaystyle=\dim(\mathcal{M}_{\Gamma})+b_{1}(\mathcal{E},D),$

which concludes the proof. ∎

######  Proposition 7.13.

Let $(\Gamma,\mathcal{E},D,s)$ be a $v_{0}$-quasistable spin root-graph. Given
$u\in V(\Gamma/P_{(\mathcal{E},D)})$, let $P_{u}$ be the associated connected
components of $\overline{P}_{(\mathcal{E},D)}$ and set $s_{u}=s(u)$. Then
there are morphisms

$\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}\stackrel{{\scriptstyle\rho}}{{\rightarrow}}\prod_{u\in
V(\Gamma/P_{(\mathcal{E},D)})}\mathcal{S}_{(P_{u},P_{u},s_{u})}\stackrel{{\scriptstyle\theta}}{{\rightarrow}}\mathcal{S}_{(\Gamma,P_{(\mathcal{E},D)},s)}.$

where $\theta$ is étale and the fibers of $\rho$ are isomorphic to
$(k^{*})^{b_{1}(\mathcal{E},D)}$. In particular
$\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}$ is irreducible of codimension
$\\#E(\Gamma)-b_{1}(\mathcal{E},D)$ in $\overline{\mathcal{Q}}_{g,n}$.

###### Proof.

The map $\theta$ is the natural étale morphism defined in the proof of [11,
Proposition 3.6.3], where we use that
$P_{\operatorname{pol}(\Gamma,\mathcal{E},D)}=P_{(\mathcal{E},D)}$ by
Proposition 4.24 and $\operatorname{pol}(s)=s$ by Definition 4.32.

Next, consider a $k$-scheme $S$. A section $(\pi\colon\mathcal{X}\rightarrow
S,\mathcal{L})\in\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}(S)$ gives
rise to a family $\pi_{u}\colon\mathcal{Z}_{u}\rightarrow S$ of curves whose
fiber have dual graph isomorphic to $P_{u}$. Let $\sigma_{\mathcal{N}_{u}}$ be
the set of sections of $\pi_{u}$ whose corresponding legs of $P_{u}$ is
obtained from an edge $e\in E(\Gamma^{\mathcal{E}})$ connecting $u$ and a
vertex $u^{\prime}\in\mathcal{N}_{u}$ (see Notation 7.3). The map $\rho$ in
the statement takes $(\mathcal{X}\rightarrow S,\sigma,\mathcal{L})$ to

$\rho(\mathcal{X}\rightarrow S,\mathcal{L})=\prod_{u\in
V(\Gamma/P_{(\mathcal{E},D)})}(\pi_{u}\colon\mathcal{Z}_{u}\rightarrow
S,\mathcal{L}|_{\mathcal{Z}_{u}}\otimes\mathcal{O}_{\mathcal{Z}_{u}}\Big{(}-\sum_{\sigma_{i}\in\sigma_{\mathcal{N}_{u}}}\sigma_{i}(S))\Big{)}.$

Indeed, by Theorem 7.10, we have that $\rho(\mathcal{X}\rightarrow
S,\mathcal{L})\in\prod_{u\in
V(\Gamma/P_{(\mathcal{E},D)})}\mathcal{S}_{(P_{u},P_{u},s_{u})}$.

The fiber of $\rho$ passing through a geometric point $[X,L]$ of
$\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}$ is given by all possible
gluing of $L|_{\mathcal{Z}_{u}}$ to an invertible sheaf on $X$, where gluings
at exceptional components of $X$ do not count for the dimension of the fiber.
Hence the fibers of $\rho$ are as claimed in the statement.

By [11, Theorem 4.2.4] we have that $\mathcal{S}_{(P_{u},P_{u},s_{u})}$ is
irreducible. Hence, from what we have shown so far, we get that there is a
unique irreducible component of
$\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}$ that dominates
$\prod_{v\in V(\Gamma/P_{(\mathcal{E},D)})}\mathcal{S}_{(P_{u},P_{u},s_{u})}$.
The last product has dimension equal to $\dim\mathcal{M}_{\Gamma}$, since it
is étale over $\mathcal{S}_{(\Gamma,P_{(\mathcal{E},D),s})}$, which is étale
over $\mathcal{M}_{\Gamma}$. Therefore, by Lemma 7.12, we see that
$\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}$ is irreducible and its
dimension is $\dim\mathcal{M}_{\Gamma}+b_{1}(\mathcal{E},D)$. Since
$\dim\overline{\mathcal{Q}}_{g,n}=3g-3+n$, we can conclude just observing that
there is a finite (in fact étale) map
$\widetilde{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}\rightarrow\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}$.
∎

######  Proposition 7.14.

Let $\pi\colon\mathcal{X}\to B$ be a one-parameter deformation of a curve
$X^{\prime}$, with generic fiber $X$, and $\mathcal{L}$ an invertible sheaf on
$\mathcal{X}/B$ such that $(X,\mathcal{L}|_{X})$ and
$(X^{\prime},\mathcal{L}|_{X^{\prime}})$ are, respectively, semistable spin
curves of spin combinatorial types $(\Gamma,\mathcal{E},D,s)$ and
$(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})$. Then there is
a specialization
$(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})\geq(\Gamma,\mathcal{E},D,s)$
of spin semistable root-graphs.

###### Proof.

We have a specialization of triples
$\iota\colon(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime})\geq(\Gamma,\mathcal{E},D)$
(see [10, Proposition 4.4.2]). Adding a suitable number of sections, we can
make $\pi$ a family of $n$-pointed stable curves. Hence we can assume
$\mathcal{E}=\mathcal{E}^{\prime}=\emptyset$.

_Case 1_. Assume that $(\Gamma,\emptyset,D)$ is stable. We have
$\mathcal{L}|_{X}^{2}=\omega_{X}$, and hence
$\omega_{\mathcal{X}}\otimes\mathcal{L}^{-2}\cong T$ for some twister $T$ of
$\mathcal{X}$. Let $D_{T}$ be the divisor on $\Gamma^{\prime}$ given by the
multidegree of $T|_{X^{\prime}}$. By Lemma 7.9 we have
$D_{T}=\operatorname{Div}(\phi)$, for some acyclic flow $\phi$ on
$\Gamma^{\prime}$ such that $\phi(e)=0$ for every
$e\in\overrightarrow{E}(\Gamma)\subset\overrightarrow{E}(\Gamma^{\prime})$.
Since $2D^{\prime}+\operatorname{Div}(\phi)=K_{\Gamma^{\prime}}$, we deduce
that $\phi=\phi_{(\emptyset,D^{\prime})}$ (we use Theorem 4.7). We have
$\iota_{*}(\phi_{(\emptyset,D^{\prime})})=\iota_{*}(\phi)=0=\phi_{(\emptyset,D)}$,
which means that
$(\Gamma^{\prime},\emptyset,D^{\prime})\geq(\Gamma,\emptyset,D)$ is a
specialization of root-graphs.

Set
$(\widetilde{\mathcal{E}}^{\prime},\widetilde{D}^{\prime})=\operatorname{pol}(\emptyset,D^{\prime})$.
To conclude, we show the existence of a specialization of polystable root-
graphs
$(\Gamma^{\prime},\widetilde{\mathcal{E}}^{\prime},\widetilde{D}^{\prime},\operatorname{pol}(s^{\prime}))\geq(\Gamma,\emptyset,D,s)$.
Indeed, if
$\widetilde{\iota}\colon\Gamma^{\prime}/P_{(\widetilde{\mathcal{E}}^{\prime},\widetilde{D}^{\prime})}\rightarrow\Gamma/P_{(\emptyset,D)}$
is the associated specialization, then
$\widetilde{\iota}_{*}(s^{\prime})=\widetilde{\iota}_{*}(\operatorname{pol}(s^{\prime}))=s$
(we use the identification
$\Gamma^{\prime}/P_{(\emptyset,D^{\prime})}\cong\Gamma^{\prime}/P_{(\widetilde{\mathcal{E}}^{\prime},\widetilde{D}^{\prime})}$).

For every edge $e\in E(\Gamma^{\prime})\setminus E(\Gamma)$ let $k_{e}$ be the
positive integer such that the local étale equation of $\mathcal{X}$ at the
node of $X^{\prime}$ corresponding to $e$ is $xy=t^{k_{e}}$, where $x,y$ are
local étale coordinates. Let $\widehat{\Gamma}^{\prime}$ be the refinement of
$\Gamma^{\prime}$ obtained by inserting $k_{e}-1$ vertices in the interior of
each $e\in E(\Gamma^{\prime})\setminus E(\Gamma)$. Notice that
$\phi_{(\emptyset,D^{\prime})}$ induces a natural acyclic flow on
$\widetilde{\Gamma}^{\prime}$, which we continue to denote by
$\phi_{(\emptyset,D^{\prime})}$. By Lemma 7.9, we have that
$\operatorname{Div}(\phi_{(\emptyset,D^{\prime})})$ is a principal divisor on
$\widehat{\Gamma}^{\prime}$, i.e.,
$\operatorname{Div}(\phi_{(\emptyset,D^{\prime})})=\sum_{v\in
V(\widehat{\Gamma}^{\prime})}a_{v}\operatorname{Div}(v)$ for
$a_{v}\in\mathbb{Z}$. Notice that
$\phi(\widehat{e})=a_{s(\widehat{e})}-a_{t(\widehat{e})}$, for every
$\widehat{e}\in\overrightarrow{E}(\widehat{\Gamma}^{\prime})$ and
$k_{e}\phi(e)=a_{s(e)}-a_{t(e)}$ for every
$e\in\overrightarrow{E}(\Gamma^{\prime})$. Recall that $|\phi(e)|\leq 1$ for
every $e\in\overrightarrow{E}(\widehat{\Gamma}^{\prime})$, by Theorem 4.7. Let
$\widetilde{\Gamma}^{\prime}$ be the refinement of $\Gamma^{\prime}$ obtained
from $\widehat{\Gamma}^{\prime}$ by inserting one vertex in the interior of
any edge $e\in E(\widehat{\Gamma}^{\prime})$. For every
$e\in\overrightarrow{E}(\Gamma^{\prime})$ we let
$f_{0},\dots,f_{2k_{e}-1}\in\overrightarrow{E}(\widetilde{\Gamma}^{\prime})$
be the oriented edges over $e$, and we set $v_{e,i}:=t(f_{i-1})=s(f_{i})$, for
every $i=1,\dots,2k_{e}-1$. If $\phi(e)=-1$, we set

$a_{e,i}:=\begin{cases}a_{s(e)}+i&\text{ for }i=1,\dots,k_{e}-1\\\
a_{t(e)}&\text{ for }i=k_{e},\dots,2k_{e}-1.\end{cases}$

If $\phi(e)=0$, we set $a_{e,i}:=a_{s(e)}=a_{t(e)}$. For each vertex
$\widetilde{v}\in V(\widetilde{\Gamma}^{\prime})$ we define $b_{v}$ as

$b_{\widetilde{v}}:=\begin{cases}a_{e,i}&\text{ if $\widetilde{v}=v_{e,i}$ for
some $e\in\overrightarrow{E}(\Gamma^{\prime})$ and $i=1,\ldots,2k_{e}-1$,}\\\
a_{v}&\text{ if $\widetilde{v}=v\in V(\Gamma^{\prime})\subset
V(\widetilde{\Gamma}^{\prime})$}.\end{cases}$

We note that $b_{\widetilde{v}}$ is well defined. In Figure 7 we illustrate
the construction for $\phi(e)=-1$.

$1$$1$$1$$1$$a_{s(e)}$$a_{t(e)}$$a_{s(e)}+1$$a_{s(e)}+3$$a_{s(e)}+2$$a_{s(e)}$$a_{s(e)}+1$$a_{s(e)}+2$$a_{s(e)}+3$$a_{t(e)}$$a_{t(e)}$$a_{t(e)}$$a_{t(e)}$$a_{t(e)}$
Figure 7. The case $\phi(e)=-1$.

Consider the diagram

(29)
$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}\times_{B}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t\rightarrow
t^{2}}$$\textstyle{B}$

where $\mathcal{Y}$ is the desingularization of $\mathcal{X}\times_{B}B$ at
the nodes of $X^{\prime}$ corresponding to edges in
$E(\Gamma^{\prime})\setminus E(\Gamma)$. Denote by
$\varphi\colon\mathcal{Y}\rightarrow\mathcal{X}$ the induced map. The dual
graph of the special fiber $Y$ of $\mathcal{Y}\rightarrow B$ is
$\widetilde{\Gamma^{\prime}}$. We let $Y_{\widetilde{v}}$ be the component of
$Y$ corresponding to the vertex $\widetilde{v}\in
V(\widetilde{\Gamma}^{\prime})$. Define the twister $\widetilde{T}$ on
$\mathcal{Y}$ as:

$\widetilde{T}:=\mathcal{O}_{\mathcal{Y}}\Big{(}\sum_{\widetilde{v}\in
V(\widetilde{\Gamma}^{\prime})}b_{\widetilde{v}}Y_{\widetilde{v}}\Big{)}.$

Let $\psi\colon\mathcal{Y}\rightarrow\widetilde{\mathcal{X}}$ be the
contraction of the exceptional components of the special fiber of
$\mathcal{Y}\rightarrow B$ over which
$\varphi^{*}(\mathcal{L})\otimes\widetilde{T}$ has degree zero. Set
$\widetilde{\mathcal{L}}:=\psi_{*}(\varphi^{*}(\mathcal{L})\otimes\widetilde{T})$.
Then $(\widetilde{\mathcal{X}},\widetilde{\mathcal{L}})$ is a family of spin
curves over $B$ whose generic fiber is isomorphic to $(X,\mathcal{L}|_{X})$
and the special fiber has spin combinatorial type
$(\Gamma^{\prime},\widetilde{\mathcal{E}}^{\prime},\widetilde{D}^{\prime},\operatorname{pol}(s^{\prime}))$.
By Remark 7.6 we get a specialization of polystable root-graphs
$(\Gamma^{\prime},\widetilde{\mathcal{E}}^{\prime},\widetilde{D}^{\prime},\operatorname{pol}(s^{\prime}))\geq(\Gamma,\emptyset,D,s)$,
as wanted.

_Case 2_. Assume that $(\Gamma,\emptyset,D)$ is not stable. We argue by
induction on $\\#E(\Gamma)$. There is a proper subset $V\subset V(\Gamma)$
such that $\beta_{D}(V)=0$. Set $V^{\prime}:=\iota^{-1}(V)$. Let
$(\Gamma_{i},\emptyset,D_{i}^{V},s_{i}^{V})$ and
$(\Gamma^{\prime}_{i},\emptyset,D_{i}^{V^{\prime}},s_{i}^{V^{\prime}})$ be the
spin semistable root-graphs as in Remark 4.31. An edge
$e\in\mathcal{E}(V,V^{c})$ gives rise to a section $\sigma_{e}\colon
B\rightarrow\mathcal{X}$ of $\pi$. We let
$\pi_{1},\pi_{2}\colon\mathcal{X}_{i}\rightarrow B$ be the two families of
curves obtained by taking the normalization of $\mathcal{X}$ at the union of
$\sigma_{e}(B)$, $\forall\;e\in E(V,V^{c})$. The family $\pi_{i}$ could fail
to be quasistable. Though, we can and will consider the family to be
quasistable simply adding a suitable number of sections to $\pi_{i}$. Set

$\mathcal{M}_{1}:=\mathcal{L}|_{\mathcal{X}_{1}}\text{ and
}\mathcal{M}_{2}:=\mathcal{L}|_{\mathcal{X}_{2}}\otimes\mathcal{O}_{\mathcal{X}_{2}}\Big{(}-\sum_{e\in
E(V,V^{c})}\sigma_{e}(B)\Big{)}.$

If $X_{i}$ and $X^{\prime}_{i}$ are the generic and special fiber of
$\pi_{i}$, respectively, then $(X_{i},\mathcal{M}_{i}|_{X_{i}})$ and
$(X^{\prime}_{i},\mathcal{M}_{i}|_{X^{\prime}_{i}})$ are semistable spin
curves of spin combinatorial type $(\Gamma_{i},\emptyset,D_{i}^{V},s_{i}^{V})$
and $(\Gamma^{\prime}_{i},\emptyset,D_{i}^{V^{\prime}},s_{i}^{V^{\prime}})$.
By the induction hypothesis, there is a specialization of semistable root-
graphs
$\iota_{i}\colon(\Gamma^{\prime}_{i},\emptyset,D_{i}^{V^{\prime}},s_{i}^{V^{\prime}})\geq(\Gamma_{i},\emptyset,D_{i}^{V},s_{i}^{V})$.
We argue that $\iota_{*}(\phi_{(\emptyset,D^{\prime})})=\phi_{(\emptyset,D)}$:
this is clear away from $E(V,V^{c})$ by the existence of the specialization
$\iota_{i}$, while
$\iota_{*}(\phi_{(\emptyset,D^{\prime})})(e)=\phi_{(\emptyset,D)}(e)=\pm 1$,
$\forall\;e\in\overrightarrow{E}(V,V^{c})=\overrightarrow{E}(V^{\prime},V^{\prime
c})$ by Theorem 4.7 (we use the natural inclusion $E(\Gamma)\subset
E(\Gamma^{\prime})$ to identify $\overrightarrow{E}(V,V^{c})$ with
$\overrightarrow{E}(V^{\prime},V^{\prime c})$, and that
$\beta_{D^{\prime}}(V^{\prime})=0$ by Proposition 2.1). If
$\widetilde{\iota}\colon\Gamma^{\prime}/P_{(\emptyset,D^{\prime})}\rightarrow\Gamma/P_{(\emptyset,D)}$
is the associated specialization, we also deduce that
$\widetilde{\iota}_{*}(s^{\prime})=s$, and we are done. ∎

######  Proposition 7.15.

Let $(X^{\prime},L^{\prime})$ be a semistable spin curves and
$(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})$ be its spin
combinatorial type. Let
$\iota\colon(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})\geq(\Gamma,\mathcal{E},D,s)$
be a specialization of spin semistable root-graphs. Then there is a one-
parameter deformation $\mathcal{X}\to B$ of $X^{\prime}$, with generic fiber
$X$, and an invertible sheaf $\mathcal{L}$ on $\mathcal{X}/B$ such that
$\mathcal{L}|_{X^{\prime}}\cong L^{\prime}$ and $(X,\mathcal{L}|_{X})$ is a
semistable spin curves with spin combinatorial type
$(\Gamma,\mathcal{E},D,s)$.

###### Proof.

Adding a suitable number of section, we can make $X^{\prime}$ a $n$-pointed
stable curve, hence we can assume
$\mathcal{E}=\mathcal{E}^{\prime}=\emptyset$. Let
$\widetilde{\iota}\colon\Gamma^{\prime}/P_{(\emptyset,D^{\prime})}\rightarrow\Gamma/P_{(\emptyset,D)}$
be the specializations associated to
$(\Gamma^{\prime},\emptyset,D^{\prime},s^{\prime})\geq(\Gamma,\emptyset,D,s)$.

_Case 1._ Assume that $(\Gamma,\emptyset,D)$ is stable. Then
$\phi_{(\emptyset,D)}=0$ by Lemma 4.4, hence $P_{(\emptyset,D)}=\Gamma$ and
$\\#V(\Gamma/P_{(\emptyset,D)})=1$. By Corollary 5.4, we can assume that
$\iota$ is elementary. If it is of type $1$, then
$\phi_{(\emptyset,D^{\prime})}=0$, so
$(\Gamma^{\prime},\emptyset,D^{\prime},s^{\prime})$ is stable by Lemma 4.4,
and we are done by Remark 7.6.

If the specialization $\iota$ is elementary of type $2$, then we can write
$V(\Gamma^{\prime})=V(P^{\prime}_{1})\cup V(P^{\prime}_{2})$, where $\iota$ is
the contraction of $E(P^{\prime}_{1},P^{\prime}_{2})$. Let
$\Gamma^{\prime}_{1},\dots,\Gamma^{\prime}_{n}$ be the connected components of
the subgraph of $\Gamma^{\prime}$ having $E(P^{\prime}_{1},P^{\prime}_{2})$ as
a set of edges, and $\Gamma^{\prime}_{n+1},\dots,\Gamma^{\prime}_{m}$ be the
vertices in $V(\Gamma^{\prime})\setminus\cup_{1\leq i\leq
n}V(\Gamma^{\prime}_{i})$. Let $X^{\prime}_{i}$ be the subcurve of
$X^{\prime}$ corresponding to $\Gamma^{\prime}_{i}$, for all
$i\in\\{1,\dots,m\\}$. For $i\in\\{1,\dots,n\\}$, the multidegree of
$\omega_{X^{\prime}_{i}}\otimes L^{\prime}|_{X^{\prime}_{i}}^{-2}$ is equal to
the restriction of $\operatorname{Div}(\phi_{(\emptyset,D^{\prime})})$ to
$\Gamma^{\prime}_{i}$, which is a principal divisor on $\Gamma^{\prime}_{i}$.
By [19, Corollary 6.9], there is a smoothing $\mathcal{X}_{i}\rightarrow B$ of
$X^{\prime}_{i}$ and a twister $T_{i}$ of $\mathcal{X}_{i}$ such that
$T_{i}|_{X^{\prime}_{i}}\cong\omega_{X^{\prime}_{i}}\otimes
L^{\prime}|_{X^{\prime}_{i}}^{-2}$. For $i\in\\{n+1,\dots,m\\}$, consider the
trivial family $\mathcal{X}_{i}=X^{\prime}_{i}\times B$. Every edge $e\in
E(\Gamma)$ gives rise to two sections $\sigma^{i}_{e},\sigma^{j}_{e}\colon
B\rightarrow\mathcal{X}_{i}$, for $i$ and $j$ such that $\Gamma^{\prime}_{i}$
and $\Gamma^{\prime}_{j}$ are incident to $e$ (seen as an edge of
$\Gamma^{\prime}$). We let $\pi\colon\mathcal{X}\rightarrow B$ be the family
obtained from $\sqcup\mathcal{X}_{i}$ by identifying $\sigma^{i}_{e}(b)$ and
$\sigma^{j}_{e}(b)$, $\forall\;b\in B$. The generic fiber $X$ of $\pi$ has
$\Gamma$ as a dual graph.

We cover $\mathcal{X}$ with two open subsets $U$ and $V$, where
$U\cong(\sqcup_{1\leq i\leq
n}\mathcal{X}_{i})\setminus(\sqcup_{i,e}\sigma^{i}_{e}(B))$ and $V$ is the
complement of the singular points of $X^{\prime}$ corresponding to edges $e\in
E(P^{\prime}_{1},P^{\prime}_{2})$. We let $T_{U}$ be the sheaf on $U$
restricting to $T_{i}|_{U\cap\mathcal{X}_{i}}$ over $U\cap\mathcal{X}_{i}$.
Since $\operatorname{Pic}(B)=0$, we have $T_{U}|_{U\cap
V}\cong\mathcal{O}_{U\cap V}$, and hence we can glue $T_{U}$ and
$\mathcal{O}_{V}$ along $U\cap V$ to get a sheaf $T$ on $\mathcal{X}$ such
that $T|_{X^{\prime}}\cong\omega_{X^{\prime}}\otimes L^{\prime-2}$ and $T$ is
trivial away from $X^{\prime}$. By [9, Remark 3.0.6] there is an invertible
sheaf $\mathcal{L}$ such that $\mathcal{L}|_{X^{\prime}}\cong L^{\prime}$ and
$\mathcal{L}^{\otimes 2}\cong\omega_{\mathcal{X}/B}\otimes T^{-1}$. Since
$\mathcal{L}|_{X}^{\otimes 2}\cong\omega_{X}$, we have
$\mathcal{E}_{\mathcal{L}|_{X}}=\emptyset$ and
$D_{\mathcal{L}|_{X}}=K_{\Gamma}/2=D$. By Proposition 7.14 we have a
specialization
$(\Gamma^{\prime},\emptyset,D^{\prime},s^{\prime})\rightarrow(\Gamma,\emptyset,D,s_{\mathcal{L}|_{X}})$
of semistable root-graphs, then
$s_{\mathcal{L}|_{X}}=\widetilde{\iota}_{*}(s^{\prime})=s$, and we are done.

_Case 2._ Assume that $(\Gamma,\emptyset,D)$ is not stable. We argue by
induction on $\\#E(\Gamma)$. There is a proper subset $V\subset V(\Gamma)$
such that $\beta_{D}(V)=0$. Set $V^{\prime}:=\iota^{-1}(V)\subset
V(\Gamma^{\prime})$ and let
$(\Gamma^{\prime}_{i},\emptyset,D_{i}^{V^{\prime}},s_{i}^{V^{\prime}})\geq(\Gamma_{i},\emptyset,D_{i}^{V},s_{i}^{V})$
be the specialization of spin semistable root-graphs in Remark 4.31 for
$i=1,2$. Let $X^{\prime}_{i}$ be the subcurve of $X^{\prime}$ corresponding to
$\Gamma^{\prime}_{i}$, where we add a marked point in correspondence of any
$e\in E(\Gamma^{\prime}_{1},\Gamma^{\prime}_{2})$. Set
$L^{\prime}_{i}:=L^{\prime}|_{X^{\prime}_{i}}$. Define

$M^{\prime}_{1}:=L^{\prime}_{1}\;\text{ and
}\;M^{\prime}_{2}:=L^{\prime}_{2}\otimes\mathcal{O}_{X^{\prime}_{2}}\Big{(}-\sum_{e\in
E(\Gamma^{\prime}_{1},\Gamma^{\prime}_{2})}p_{e}\Big{)}$

By construction, $(X^{\prime}_{i},M^{\prime}_{i})$ is a semistable spin curve
of type
$(\Gamma^{\prime}_{i},\emptyset,D_{i}^{V^{\prime}},s_{i}^{V^{\prime}})$. Let
$\pi_{i}\colon\mathcal{X}_{i}\rightarrow B$ be the family obtained by the
induction hypothesis applied to the specialization
$(\Gamma^{\prime},\emptyset,D_{i}^{V^{\prime}},s_{i}^{V^{\prime}})\geq(\Gamma,\emptyset,D_{i}^{V},s_{i}^{V})$.
Every edge $e\in E(\Gamma_{1},\Gamma_{2})\subset\Gamma$ gives rise to a
section $\sigma^{i}_{e}$ of $\pi_{i}$. Let $\pi\colon\mathcal{X}\rightarrow B$
be the family obtained from $\mathcal{X}_{1}\sqcup\mathcal{X}_{2}$ by
identifying a point $\sigma^{1}_{e}(b)$ with $\sigma^{2}_{e}(b)$,
$\forall\;b\in B$. By construction, the generic fiber $X$ of $\pi$ has
$\Gamma$ as a dual graph and the special fiber of $\pi$ is $X^{\prime}$.

Set
$n:=\\#E(\Gamma_{1},\Gamma_{2})=\\#E(\Gamma^{\prime}_{1},\Gamma^{\prime}_{2})$.
There are exact sequences

$0\rightarrow\mathcal{O}^{*}_{\mathcal{X}}\rightarrow\nu_{*}\mathcal{O}^{*}_{\mathcal{X}_{1}\sqcup\mathcal{X}_{2}}\rightarrow\mathcal{K}\rightarrow
0\;\;\text{ and
}\;\;0\rightarrow\mathcal{O}_{X^{\prime}}^{*}\rightarrow\mathcal{O}^{*}_{X^{\prime}_{1}\sqcup
X^{\prime}_{2}}\rightarrow\prod_{e\in
E(\Gamma^{\prime}_{1},\Gamma^{\prime}_{2})}k^{*}_{p_{e}}\rightarrow 0$

where $\mathcal{K}$ is a principal bundle over $B$ with fiber $(k^{*})^{n}$.
The associated long exact sequences in cohomology gives rise to a commutative
diagram

$\textstyle{H^{0}(\mathcal{X},\mathcal{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{0}}$$\scriptstyle{\delta}$$\textstyle{\operatorname{Pic}(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\operatorname{Pic}(\mathcal{X}_{1})\times\operatorname{Pic}(\mathcal{X}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\rho_{1},\rho_{2})}$$\textstyle{0}$$\textstyle{(k^{*})^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Pic}(X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Pic}(X^{\prime}_{1})\times\operatorname{Pic}(X^{\prime}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\rho,\rho_{0},\rho_{1},\rho_{2}$ are the natural restriction maps.
Notice that $\mathcal{K}$ and $\prod_{e\in
E(\Gamma^{\prime}_{1},\Gamma^{\prime}_{2})}k^{*}_{p_{e}}$ are supported on the
image $\Sigma\subset\mathcal{X}$ of the sections $\sigma_{e}^{i}$. Moreover,
$\pi|_{\Sigma}$ is just a collection of isomorphisms. Then $\pi|_{\Sigma*}$ is
exact, and hence $\pi_{*}(\mathcal{K})\to(k^{*})^{n}$ is surjective. Since $B$
has only one closed point and only one open set containing this point, we get
a surjection $H^{0}(\mathcal{X},\mathcal{K})\rightarrow(k^{*})^{n}$ which we
can identify with $\rho_{0}$. Hence $\rho_{0}$ is surjective. It is clear that
$\rho,\rho_{1},\rho_{2}$ are surjective as well.

Let $\mathcal{L}\in\operatorname{Pic}(\mathcal{X})$ be an invertible sheaf
lifting
$(\mathcal{L}_{1},\mathcal{L}_{2})\in\operatorname{Pic}(\mathcal{X}_{1})\times\operatorname{Pic}(\mathcal{X}_{2})$,
where

$\mathcal{L}_{1}=\mathcal{M}_{1}\;\text{ and
}\;\mathcal{L}_{2}=\mathcal{M}_{2}\otimes\mathcal{O}_{\mathcal{X}_{2}}\Big{(}\sum_{e\in
E(\Gamma^{\prime}_{1},\Gamma^{\prime}_{2})}\sigma^{2}_{e}(B)\Big{)}.$

A simple diagram chasing shows the existence of an element $\alpha\in
H^{0}(\mathcal{X},\mathcal{K})$ such that
$\rho(\mathcal{L}\otimes\delta(\alpha))=L^{\prime}$. The family
$\pi\colon\mathcal{X}\rightarrow B$ and the sheaf
$\mathcal{L}\otimes\delta(\alpha)\in\operatorname{Pic}(\mathcal{X})$ are as
required by the statement. ∎

######  Corollary 7.16.

The following conditions are equivalent:

  * (1)

$\mathcal{Q}_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})}\cap\overline{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}\neq\emptyset$.

  * (2)

$(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})\geq(\Gamma,\mathcal{E},D,s)$
in the poset $\mathcal{SPQ}_{g,n}$.

  * (3)

$\mathcal{Q}_{(\Gamma^{\prime},\mathcal{E}^{\prime},D^{\prime},s^{\prime})}\subset\overline{\mathcal{Q}}_{(\Gamma,\mathcal{E},D,s)}$.

###### Proof.

The result readily follows from Propositions 7.14 and 7.15. ∎

We are ready to prove the main result of the section.

######  Theorem 7.17.

The following decomposition is a stratification of
$\overline{\mathcal{Q}}_{g,n}$ indexed by $\mathcal{SPQ}_{g,n}$:

(30)
$\overline{\mathcal{Q}}_{g,n}=\coprod_{(\Gamma,\mathcal{E},D,s)\in\mathcal{SPQ}_{g,n}}\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}.$

In particular, $\overline{\mathcal{Q}}_{g,n}$ has exactly two connected
components $\overline{\mathcal{Q}}^{+}_{g,n}$ and
$\overline{\mathcal{Q}}^{-}_{g,n}$ corresponding to even and odd quasistable
spin curves and:

  * (1)

the maps
$\overline{\mathcal{Q}}^{+}_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n}$ and
$\overline{\mathcal{Q}}^{-}_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n}$ are
finite exactly over the locus of tree-like curves, and hence
$\overline{\mathcal{Q}}_{g,n}^{+}$ and $\overline{\mathcal{Q}}_{g,n}^{-}$ are
not isomorphic, respectively, to $\overline{\mathcal{S}}_{g,n}^{+}$ and
$\overline{\mathcal{S}}_{g,n}^{-}$.

  * (2)

$\operatorname{codim}_{\overline{\mathcal{Q}}_{g,n}}\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}=\\#E(\Gamma)-b_{1}(\mathcal{E},D)$.

  * (3)

$\operatorname{codim}_{\overline{\mathcal{Q}}_{g,n}}\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}=1$
if and only if one of the following conditions hold:

    * (a)

$\\#V(\Gamma)=\\#E(\Gamma)=1$.

    * (b)

$\\#V(\Gamma)=2$, $\mathcal{E}=\emptyset$, and $P_{(\mathcal{E},D)}=0$.

  * (4)

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.

###### Proof.

By Proposition 7.13 and Corollary 7.16 we have that the decomposition in
equation (30) is a stratification indexed by $\mathcal{SPQ}_{g,n}$. The
statement about the connected components follows by Remark 6.6. By
Propositions 5.9 and 7.13 we have that (2) and (4) hold.

We prove (1). Assume that $\Gamma$ is not a tree-like graph. By upper
semicontituity, without loss of generality, we can assume that $V(\Gamma)=2$.
Let $e_{1},e_{2}$ be edges of $\Gamma$ that are not loops and set
$\mathcal{E}=E(\Gamma)\setminus\\{e_{1},e_{2}\\}$. Let $\phi$ be the acyclic
flow on $\Gamma$ such that, for $e\in\overrightarrow{E}(\Gamma)$, we have
$\phi(e)=1$ if either $t(e)$ is an exceptional vertex or $s(e)=v_{0}$. By
construction, we have that
$K_{\Gamma^{\mathcal{E}}}(v)-\operatorname{Div}(\phi)(v)$ is even for every
$v\in V(\Gamma^{\mathcal{E}})$, hence there is a divisor $D$ on
$\Gamma^{\mathcal{E}}$ such that
$2D+\operatorname{Div}(\phi)=K_{\Gamma^{\mathcal{E}}}$. It is easy to check
that $D$ is $v_{0}$-quasistable. Then $(\Gamma,\mathcal{E},D)$ is a
$v_{0}$-quasistable root-graph with $P_{(\mathcal{E},D)}=0$. If $H$ is the
subgraph of $\Gamma$ with $e_{1},e_{2}$ as edges, then we have
$b_{1}(\mathcal{E},D)=b_{1}(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})=b_{1}(H)=1$.
We can easily find an even and an odd sign functions on
$(\Gamma,\mathcal{E},D)$, which we denote by $s^{+}$ and $s^{-}$. So, by
Proposition 7.13 the maps
$\mathcal{Q}_{(\Gamma,\mathcal{E},D,s^{+})}\rightarrow\overline{\mathcal{M}}_{g,n}$
and
$\mathcal{Q}_{(\Gamma,\mathcal{E},D,s^{-})}\rightarrow\overline{\mathcal{M}}_{g,n}$
are not finite.

Conversely, let $\Gamma$ be a tree-like grah and $T$ be its spanning tree. Let
$(\Gamma,\mathcal{E},D,s)$ be a $v_{0}$-quasistable spin root-graph. Then
$\Gamma_{\mathcal{E}}$ is connected, hence $\mathcal{E}$ is formed by loops of
$\Gamma$. Since $\phi_{(\mathcal{E},D)}$ is acyclic and $P_{(\mathcal{E},D)}$
is cyclic, we have that $\mathcal{E}\cup E(P_{(\mathcal{E},D)})$ is the set of
loops of $\Gamma$. Hence
$b_{1}(\mathcal{E},D)=b_{1}(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})=b_{1}(T)=0$
and, by Proposition 7.13 the map
$\mathcal{Q}_{(\Gamma,\mathcal{E},D,s)}\rightarrow\overline{\mathcal{M}}_{g,n}$
is finite.

We prove (3). We have $\\#E(\Gamma)-b_{1}(\mathcal{E},D)=1$ if and only if
$\\#\mathcal{E}+\\#E(P_{(\mathcal{E},D)})+\\#V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})=2$,
which is satisfied if either (a) or (b) holds. Moreover,

  1. (i)

if $\\#V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})=1$, $\\#\mathcal{E}=1$ and
$P_{(\mathcal{E},D)}=0$, then (a) holds.

  2. (ii)

if $\\#V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})=1$, $\mathcal{E}=\emptyset$
and $\\#E(P_{(\mathcal{E},D)})=1$, then (a) holds.

  3. (iii)

if $\\#V(\Gamma_{\mathcal{E}}/P_{(\mathcal{E},D)})=2$, then
$\mathcal{E}=\emptyset$ and $P_{(\mathcal{E},D)}=0$, and hence (b) holds.

The proof is complete. ∎

## Acknowledgements

This is a part of the Ph.D. thesis of the third author supervised by the first
and second author. We thank Ethan Cotterill and Margarida Melo for some
discussions and for the positive comments on a preliminary version of the
paper.

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Alex Abreu, Marco Pacini, and Danny Taboada
Universidade Federal Fluminense, Rua Prof. M. W. de Freitas, S/N
Niterói, Rio de Janeiro, Brazil. 24210-201.