$\mathscr{S}_{{\mathrm{can}}}:=\left\\{\mathfrak{p}_{{\boldsymbol{\tau}}}\,\Big{|}\,\begin{subarray}{c}\hbox{${\boldsymbol{\tau}}$
an isomorphism class of decorated wall}\\\ \hbox{type with total curve class
lying in $Q\setminus I$}\end{subarray}\right\\}.$

We note here we do not exclude walls with $W_{{\boldsymbol{\tau}}}=0$, so this
wall structure may include an infinite set of trivial walls, i.e., with
attached function $1$. So technically this is not a wall structure, but if we
remove all trivial walls, it becomes a finite set as in Proposition 3.11.
However, for bookkeeping purposes, it will prove useful to include these
trivial walls.

To see that this gives a wall structure equivalent (in the sense of Definition
3.5) to the previous definition, fix a wall type $\tau$ and a curve class
$A\in Q\setminus I$ such that $W_{\tau,A}\not=0$; this data will give one wall
in the earlier definition of $\mathscr{S}_{{\mathrm{can}}}$. With
$\beta=(\\{u_{\tau}\\},A)$ the associated class of punctured map, we have the
canonical morphism
$\mathfrak{M}(\mathcal{X},\tau)\rightarrow\mathfrak{M}_{\tau}(\mathcal{X},\beta)$,
which is surjective and finite of degree $|\operatorname{Aut}(\tau)|$.
Further, we have a Cartesian diagram in all categories [ACGS2, Prop. 5.19]

$\textstyle{\coprod_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}\mathscr{M}(X,{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}}$$\scriptstyle{\varepsilon_{\tau}}$$\textstyle{\mathscr{M}(X,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{\mathfrak{M}(\mathcal{X},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\mathfrak{M}(\mathcal{X},\beta)}$

Here ${\bf A}$ runs over all decorations of $\tau$ with total curve class $A$.
As $j$ is a map of degree $|\operatorname{Aut}(\tau)|$ onto
$\mathfrak{M}_{\tau}(\mathcal{X},\beta)$, we see that

(3.14) $\displaystyle
W_{\tau,A}={\deg\varepsilon^{!}[\mathfrak{M}_{\tau}(\mathcal{X},\beta)]}={\deg\varepsilon^{!}j_{*}[\mathfrak{M}(\mathcal{X},\tau)]\over|\operatorname{Aut}(\tau)|}={\deg
j^{\prime}_{*}\varepsilon_{\tau}^{!}[\mathfrak{M}(\mathcal{X},\tau)]\over|\operatorname{Aut}(\tau)|}=\sum_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}{\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}(\tau)|}.$

Here the third equality holds by push-pull of [Mn, Thm. 4.1], and the last
summation runs over all choices of decorations ${\bf A}$ of $\tau$ with total
curve class $A$. Note however that $\operatorname{Aut}(\tau)$ acts on the set
of all choices of decoration ${\bf A}$, with orbits of this action giving
isomorphism classes of decorated types ${\boldsymbol{\tau}}$. The stabilizer
of a given ${\boldsymbol{\tau}}$ is the subgroup
$\operatorname{Aut}({\boldsymbol{\tau}})\subseteq\operatorname{Aut}(\tau)$,
and hence the orbit containing ${\boldsymbol{\tau}}$ is of size
$|\operatorname{Aut}(\tau)|/|\operatorname{Aut}({\boldsymbol{\tau}})|$. Thus
the final expression of (3.14) can be now expressed as a sum over isomorphism
classes of decorations ${\boldsymbol{\tau}}$ of $\tau$ with total class $A$:

(3.15)
$W_{\tau,A}=\sum_{{\boldsymbol{\tau}}}{\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}(\tau)|}\cdot{|\operatorname{Aut}(\tau)|\over|\operatorname{Aut}({\boldsymbol{\tau}})|}=\sum_{{\boldsymbol{\tau}}}{\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}({\boldsymbol{\tau}})|}.$

From this, we see the collection of walls in the new definition of
$\mathscr{S}_{{\mathrm{can}}}$ coming from all ${\boldsymbol{\tau}}$ with
underlying type $\tau$ and total curve class $A$ is equivalent to the
corresponding single wall in the original definition of
$\mathscr{S}_{{\mathrm{can}}}$.

######  Example 3.14.

The construction of $\mathscr{S}_{{\mathrm{can}}}$ agrees (up to equivalence)
with the construction of the canonical scattering diagram of [GHK] when $\dim
X=2$ and $K_{X}+D=0$, so that $B=\Sigma(X)$. In other words, the construction
given here is a generalization of the construction of [GHK].

To see this, we analyze decorated wall types ${\boldsymbol{\tau}}$ in
dimension two. Since we require $\dim\tau=\dim X-2$, in fact such types will
be rigid. Thus there is a unique tropical map $h:G\rightarrow B$ of type
$\tau$, and all vertices of $G$ are mapped to $0\in B$; otherwise, rescaling
$B$ provides a non-trivial deformation of $h$. Further, we require $\dim
h(\tau_{\mathrm{out}})=1$. So $h(\tau_{\mathrm{out}})$ is a ray in $B$ with
endpoint $0$. Hence there is no choice but for $u_{\tau}\in
B(\mathbb{Z})\setminus\\{0\\}$ and $h(\tau_{\mathrm{out}})=\mathbb{R}_{\geq
0}u_{\tau}$. Further, any edge of $G$ contracted by $h$ has length a free
parameter of the tropical curve, and hence again by rigidity, there are no
edges. Thus the only possibility for ${\boldsymbol{\tau}}$ is that $G$ has one
vertex, with an attached curve class $A$, and one leg, $L_{\mathrm{out}}$,
with ${\bf u}(L_{\mathrm{out}})=u_{\tau}\in B(\mathbb{Z})$. Note further that
$k_{\tau}$ is then just the index of $u_{\tau}$, i.e., the degree of
divisibility of $u_{\tau}$ in $B(\mathbb{Z})$. Of course
$|\operatorname{Aut}({\boldsymbol{\tau}})|=1$.

To make a comparison with the setup of [GHK], it is then convenient, for $u\in
B(\mathbb{Z})$ primitive, to define

$W_{A,u,k}:=W_{{\boldsymbol{\tau}}},$

where ${\boldsymbol{\tau}}$ is the decorated wall type as described above with
curve class $A$ and $u_{\tau}=ku$. In [GHK], only one wall for each ray
$\mathbb{R}_{\geq 0}u$ of rational slope occurs, and hence it is then more
useful to write the canonical wall structure in the equivalent form

$\mathscr{S}^{\prime}_{{\mathrm{can}}}:=\left\\{\Big{(}\mathbb{R}_{\geq
0}u,\exp\Big{(}\sum_{A,k}kW_{A,u,k}t^{A}z^{-ku}\Big{)}\Big{)}\,\Big{|}\,\hbox{$u\in
B(\mathbb{Z})\setminus\\{0\\}$ primitive}\right\\}.$

Here the sum is over all positive integers $k$ and all curve classes $A$.
However, if we write $u=aD_{i}^{*}+bD_{j}^{*}$ for some non-negative $a,b$
with $a+b>0$, then by the balancing condition as expressed in [GS7, Cor.
1.14], we may in fact sum over only those curve classes with $A\cdot D_{i}=a$,
$A\cdot D_{j}=b$, $A\cdot D_{k}=0$ for $k\not=i,j$.

In [GHK], a wall function is associated to each ray
$\mathfrak{d}=\mathbb{R}_{\geq 0}u$ of rational slope using invariants defined
as follows. Assuming $\mathfrak{d}$ does not coincide with a ray of
$\mathscr{P}$, one performs a toric (log étale) blow-up
$\pi:\widetilde{X}\rightarrow X$ by refining $(B,\mathscr{P})$ along the given
ray $\mathfrak{d}$. Otherwise, if $\mathfrak{d}$ coincides with a ray of
$\mathscr{P}$, we may take $\pi$ to be the identity. Let $\widetilde{A}$ be a
curve class on $\widetilde{X}$ with the following properties, depending on
whether or not $\pi$ is the identity. If $\pi$ is not the identity, then we
require that $\widetilde{A}$ has trivial intersection number with all boundary
components of $\widetilde{X}$ except for the exceptional divisor $E$ of $\pi$.
If $\pi$ is the identity, let $E$ be the component of $D$ corresponding to the
ray $\mathfrak{d}$. We then require that $\widetilde{A}$ has zero intersection
number with each component of $D$ except for $E$. Then [GHK], following [GPS,
§4], defines a number $N_{\widetilde{A}}$. This number is defined as a
_relative_ Gromov-Witten invariant for the non-compact pair
$(\widetilde{X}^{\circ},E^{\circ})$ where
$\widetilde{X}^{\circ}\subseteq\widetilde{X}$ is obtained by removing the
closure of $\pi^{-1}(D)\setminus E$ from $\widetilde{X}$, and
$E^{\circ}=E\cap\widetilde{X}^{\circ}$. This relative invariant counts
rational curves of class $\widetilde{A}$ with one marked point, with contact
order $k_{\widetilde{A}}:=\widetilde{A}\cdot E$ with $E^{\circ}$.

To compare $N_{\widetilde{A}}$ with the type of number considered in this
paper, note that the data of $\widetilde{A}$ and the contact order
$k_{\widetilde{A}}$ also specifies a type $\widetilde{\beta}$ of logarithmic
map to the pair $(\widetilde{X},\pi^{-1}(D))$, and we may compare the
logarithmic Gromov-Witten invariant
$\deg[\mathscr{M}(\widetilde{X},\widetilde{\beta})]^{\mathrm{virt}}$ with
$N_{\widetilde{A}}$. In fact, it follows from [AMW] that these two numbers
will agree provided that every stable log map in the moduli space
$\mathscr{M}(\widetilde{X},\widetilde{\beta})$ factors through
$\widetilde{X}^{\circ}$. However, it is an elementary exercise in tropical
geometry to show that if given a stable log map
$f:C/W\rightarrow\widetilde{X}$ of type $\widetilde{\beta}$ with $W$ a log
point, any tropical map in the corresponding family of tropical maps has image
$\mathfrak{d}$. This may be proved using an argument similar to the argument
given in Lemma 2.5, but a similar tropical argument in two dimensions has
already appeared in the proof of [Bo1, Lem. 12], which may also be viewed as a
tropical interpretation of [GPS, Prop. 4.2]. On the other hand, any stable log
map of type $\widetilde{\beta}$ which does not factor through
$\widetilde{X}^{\circ}$ will necessarily have a tropicalization whose image
does not coincide with $\mathfrak{d}$, as follows immediately from the
construction of tropicalization.

Finally, let $A=\pi_{*}\widetilde{A}$. Note that in the case $\pi$ is not the
identity, the curve class $\widetilde{A}$ is uniquely determined by $A$ and
the constraint that $\widetilde{A}$ has zero intersection number with
components of the strict transform of $D$. Then if $u$ is the primitive
generator of $\mathfrak{d}\cap B(\mathbb{Z})$, we may interpret
$k_{\widetilde{A}}u\in B(\mathbb{Z})$ as a contact order, and the class $A$
and the contact order $k_{\widetilde{A}}u$ determines a type of log map
$\beta$ to $X$. It then follows from the birational invariance of log Gromov-
Witten theory of [AW, Thm. 1.1.1] that
$\deg[\mathscr{M}(\widetilde{X},\widetilde{\beta})]^{\mathrm{virt}}=\deg[\mathscr{M}(X,\beta)]^{\mathrm{virt}}$.
In particular, $N_{\widetilde{A}}=W_{A,u,k_{\widetilde{A}}}$.

Conversely, given the data of $A$, $u$ and $k$ giving a wall type $\tau$ with
$\mathscr{M}(X,\tau)$ non-empty, we obtain via the above blow-up procedure a
curve class $\widetilde{A}$. Thus we have a one-to-one correspondence between
the set of data contributing to $\mathscr{S}^{\prime}_{{\mathrm{can}}}$ and
the data contributing to the canonical scattering diagram of [GHK]. Further,
we have just observed an equality of the invariants. The equivalence of
$\mathscr{S}^{\prime}_{{\mathrm{can}}}$ with the canonical scattering diagram
of [GHK] now follows by inspection of the definition of the canonical
scattering diagram in [GHK].

######  Example 3.15.

Returning to Examples 1.4 and 1.11, we first introduce notation for the
relevant curve classes. Let $e_{i}$ be the class of a fibre of
$\pi|_{E_{i}}:E_{i}\rightarrow Z_{i}$. Let $f$ be the class of a curve of the
form $\\{a\\}\times\mathbb{P}^{1}\times\\{b\\}$ disjoint from the centers
$Z_{1},Z_{2}$.

One can show that the only decorated wall types ${\boldsymbol{\tau}}$ with
$W_{{\boldsymbol{\tau}}}\not=0$ are of the following 5 types. In each case,
the underlying graph $G$ of ${\boldsymbol{\tau}}$ has one vertex $v$, no
edges, and of course a unique leg $L_{\mathrm{out}}$.

  1. (1)

For some $k>0$, we have $u_{\tau}=kD_{2,0}^{*}$,
${\boldsymbol{\sigma}}(L_{\mathrm{out}})=\mathfrak{p}_{1}:=\mathbb{R}_{\geq
0}D_{2,0}^{*}+\mathbb{R}_{\geq 0}D_{1,0}^{*}$, and ${\bf A}(v)=ke_{1}$.

  2. (2)

For some $k>0$, we have $u_{\tau}=kD_{2,0}^{*}$,
${\boldsymbol{\sigma}}(L_{\mathrm{out}})=\mathfrak{p}_{2}:=\mathbb{R}_{\geq
0}D_{2,0}^{*}+\mathbb{R}_{\geq 0}D_{1,\infty}^{*}$, and ${\bf A}(v)=ke_{1}$.

  3. (3)

For some $k>0$, we have $u_{\tau}=k(E_{2}^{*}-D_{1,\infty}^{*})$,
${\boldsymbol{\sigma}}(L_{\mathrm{out}})=\mathfrak{p}_{3}:=\mathbb{R}_{\geq}E_{2}^{*}+\mathbb{R}_{\geq
0}D_{1,\infty}^{*}$, and ${\bf A}(v)=k(f-e_{1}-e_{2})$.

  4. (4)

For some $k>0$, we have $u_{\tau}=kD_{2,\infty}^{*}$,
${\boldsymbol{\sigma}}(L_{\mathrm{out}})=\mathfrak{p}_{4}:=\mathbb{R}_{\geq
0}E_{2}^{*}+\mathbb{R}_{\geq 0}D_{2,\infty}^{*}$, and ${\bf A}(v)=k(f-e_{1})$.

  5. (5)

For some $k>0$, we have $u_{\tau}=kD_{2,\infty}^{*}$,
${\boldsymbol{\sigma}}(L_{\mathrm{out}})=\mathfrak{p}_{5}:=\mathbb{R}_{\geq
0}D_{1,0}^{*}+\mathbb{R}_{\geq 0}D_{2,\infty}^{*}$, and ${\bf
A}(v)=k(f-e_{1})$.

In all cases,
$h(\tau_{\mathrm{out}})={\boldsymbol{\sigma}}(L_{\mathrm{out}})$, so walls
only have five possible supports. Further, in every case, $k_{\tau}=k$ and
$W_{{\boldsymbol{\tau}}}=(-1)^{k-1}/k^{2}$. After passing to an equivalent
scattering diagram with only one wall with a given support, we see, for
example, that we have a wall

$\left(\mathfrak{p}_{1},\exp\Big{(}\sum_{k>0}k{(-1)^{k-1}\over
k^{2}}t^{ke_{1}}z^{-kD_{2,0}^{*}}\Big{)}\right)=\left(\mathfrak{p}_{1},1+t^{e_{1}}z^{-D_{2,0}^{*}}\right).$

Similarly, we have four other walls:

$\left(\mathfrak{p}_{2},1+t^{e_{1}}z^{-D_{2,0}^{*}}\right),\left(\mathfrak{p}_{3},1+t^{f-e_{1}-e_{2}}z^{D_{1,\infty}^{*}-E_{2}^{*}}\right),\left(\mathfrak{p}_{4},1+t^{f-e_{1}}z^{-D_{2,\infty}^{*}}\right),\left(\mathfrak{p}_{5},1+t^{f-e_{1}}z^{-D_{2,\infty}^{*}}\right).$

Together, these walls cover the affine plane contained in $B$ which is the
union of all two-dimensional cones of $\mathscr{P}$ not containing
$D_{3,0}^{*}$ or $D_{3,\infty}^{*}$. We omit a derivation of these results.
Showing (1)–(5) are the only possibilities is not difficult using [GS7, Cor.
1.14] to encode balancing requirements for punctured maps. A direct
calculation of $W_{{\boldsymbol{\tau}}}$ has not been carried out, but
presumably these multiple cover calculations can be carried out as in the two-
dimensional case of [GPS, Prop. 5.2]. Instead, the above formulas for the wall
functions are proved in a more general context in [AG]. In fact, the formulas
for these wall functions follows from the consistency of
$\mathscr{S}_{{\mathrm{can}}}$ proved in the present paper.

The walls $\mathfrak{p}_{i}$ for $i\not=3$ arise from zero-dimensional strata
in one-dimensional moduli spaces of ordinary (non-punctured) stable log maps.
For example, the inclusion of every fibre of $\pi|_{E_{1}}:E_{1}\rightarrow
Z_{1}$ into $X$ may be viewed as a stable log map. The walls
$\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ capture the curves in this family
which are degenerate with respect to the log structure on $X$, i.e., fall into
$D_{1,0}$ or $D_{1,\infty}$. For $k>1$, we count multiple covers of these
fibres.

The same holds for $\mathfrak{p}_{4}$ and $\mathfrak{p}_{5}$, with the curves
in question being, in general, strict transforms of curves of the form
$\\{a\\}\times\mathbb{P}^{1}\times\\{1\\}\subseteq\overline{X}$. Note as these
curves intersect $Z_{1}$ at one point, the class of the strict transform is
indeed $f-e_{1}$. However, as $a\rightarrow\infty$, this curve degenerates to
a union $C=C_{1}\cup C_{2}$ of two irreducible components, of class
$f-e_{1}-e_{2}$ (the strict transform $C_{1}$ of
$\\{\infty\\}\times\mathbb{P}^{1}\times\\{1\\}$) and of class $e_{2}$ (the
curve $C_{2}=\pi^{-1}(\infty,\infty,1)$). It is this degenerate curve and its
multiple covers which contribute to $\mathfrak{p}_{4}$. Meanwhile the strict
transform of $\\{0\\}\times\mathbb{P}^{1}\times\\{1\\}$ and its multiple
covers contribute to the wall $\mathfrak{p}_{5}$.

Finally, $\mathfrak{p}_{3}$ arises not from a family of curves with one marked
point of positive contact order with the boundary, but from the punctured log
map whose image is $C_{1}$. This involves a negative contact order with the
divisor $D_{1,\infty}$, which contains $C_{1}$. This curve is rigid, and if we
did not include a wall for this curve, we would not get a consistent
scattering diagram. This shows how it is essential, unlike in the case that
$\dim X=2$, to take into account punctured curves rather than just marked
curves, as without $\mathfrak{p}_{3}$, $\mathscr{S}_{{\mathrm{can}}}$ would
not be consistent.

######  Construction 3.16 (The torus action).

As with the canonical wall structure in two dimensions in [GHK, §5], there is
a natural torus action on the mirror family constructed using
$\mathscr{S}_{{\mathrm{can}}}$ in all dimensions. We refer the reader to [GHS,
§4.4] for the general setup for torus actions in the context of families of
varieties built via wall structures, and do not review the notation here.

In this case, there is an action of a torus with character lattice

$\Gamma:=\operatorname{Div}_{D}(X),$

i.e., the free abelian group generated by boundary divisors. In the language
of [GHS, §4.4], the base ring $A$ is taken here to be the ground field
$\Bbbk$. To specify the torus action, we must specify the maps $\delta_{Q}$
and $\delta_{B}$ of the diagram [GHS, (4.8)]. Note that each irreducible
component $D_{i}$ of $D$ defines an $\mathbb{R}$-valued PL function on $B$;
indeed, by construction
$B\subseteq\operatorname{Div}_{D}(X)^{*}_{\mathbb{R}}$, and the linear
functional $D_{i}$ on this vector space restricts to a piecewise linear
function on $B$. We continue to denote this PL function as $D_{i}$. We may
then define

$\delta_{Q}:Q\longrightarrow\Gamma,\quad\quad\delta_{Q}(A)=\sum_{i}(D_{i}\cdot
A)D_{i}$

for $A\in Q$, and

$\delta_{B}:\operatorname{PL}(B)^{*}\longrightarrow\Gamma,\quad\quad\delta_{B}(\beta)=\sum_{i}\beta(D_{i})D_{i}$

for $\beta\in\operatorname{PL}(B)^{*}$. Commutativity of [GHS, (4.8)] then
follows by direct computation. Indeed, note that in our current context,
$Q_{0}$ is the free monoid with generators $e_{\rho}$,
$e_{\rho}\in\mathscr{P}^{[n-1]}$, and $h:Q_{0}\rightarrow Q$ is defined by
$h(e_{\rho})=[X_{\rho}]$, the kink of our choice of MPL function $\varphi$
along $\rho$. Then

$\delta_{Q}(h(e_{\rho}))=\sum_{i}(D_{i}\cdot X_{\rho})D_{i}.$

On the other hand, $g:Q_{0}\rightarrow\operatorname{PL}(B)^{*}$ takes
$e_{\rho}$ to the linear functional on $\operatorname{PL}(B)$ taking
$\psi\in\operatorname{PL}(B)$ to the kink of $\psi$ along $\rho$. It is easy
to check from toric geometry that the kink of the PL function induced by
$D_{i}$ along $\rho$ is $D_{i}\cdot X_{\rho}$ (noting that if $D_{i}$ is not
good, it induces the zero function on $B$). The claimed equality
$\delta_{Q}\circ h=\delta_{B}\circ g$ then follows.

Thus, by [GHS, Thm. 4.17], it follows there is an induced
$\operatorname{Spec}\Bbbk[\Gamma]$-action on the flat family
$\check{\mathfrak{X}}\rightarrow\operatorname{Spec}(\Bbbk[Q]/I)$ constructed
from the wall structure $\mathscr{S}_{{\mathrm{can}}}$ provided that
$\mathscr{S}_{{\mathrm{can}}}$ is homogeneous in the sense of [GHS, Def.
4.16]. For this, it is enough to check that if ${\boldsymbol{\tau}}$ is a
decorated wall type with $W_{{\boldsymbol{\tau}}}\not=0$ and total curve class
$A$, then $\deg_{\Gamma}t^{A}z^{-u_{\tau}}=0$. Here
$\deg_{\Gamma}t^{A}=\delta_{Q}(A)=\sum_{i}(D_{i}\cdot A)D_{i}$. If we write
$u_{\tau}=\sum u_{i}D_{i}^{*}$ as a tangent vector to
${\boldsymbol{\sigma}}(L_{\mathrm{out}})$ then $\deg_{\Gamma}(z^{-u_{\tau}})$
is computed from [GHS, (4.9)] as $-\sum_{i}u_{i}D_{i}$. It then follows from
[GS7, Cor. 1.14] that necessarily $u_{i}=D_{i}\cdot A$, and hence
$\deg_{\Gamma}t^{A}z^{-u_{\tau}}=0$ as desired.

####  3.2.3. The relative case

######  Construction 3.17.

Suppose we are in the situation of Proposition 1.19. It is useful (see [AG])
to compare the wall structures for $(X,D)$ and $(X_{s},D_{s})$, which we do as
follows.

Recall from that proposition that after extending the affine structure on $B$
across the interior of cells $\omega\in\mathscr{P}^{[n-2]}_{\partial}$,
$\partial B$ may be viewed as the polyhedral affine pseudomanifold
corresponding to the pair $(X_{s},D_{s})$. As such, write $(\partial B)_{0}$
for the complement in $\partial B$ of the union of cones of
$\mathscr{P}^{[n-3]}_{\partial}$. Let $\Lambda_{\partial B}$ denote the sheaf
on $(\partial B)_{0}$ of integral tangent vectors. We may view
$\Lambda_{\partial B}$ as a subsheaf of $\Lambda|_{\partial B}$, where the
latter sheaf is now extended across the interiors of cells
$\omega\in\mathscr{P}^{[n-2]}_{\partial}$.

Further, Proposition 1.19, (3) then implies that if $\mathcal{P}_{\partial B}$
is defined using the MPL function $\varphi|_{\partial B}$ on $\partial B$, we
obtain a natural inclusion $\mathcal{P}_{\partial
B}\subseteq\mathcal{P}|_{\partial B}$. Again, the latter sheaf is viewed as
extending across the interiors of $\omega\in\mathscr{P}^{[n-2]}_{\partial}$.
Note that $\mathcal{P}_{\partial B}$ consists of those sections $m$ of
$\mathcal{P}|_{\partial B}$ such that $\bar{m}$ is tangent to $\partial B$.

On the other hand, we also have the MPL function $\varphi_{(X_{s},D_{s})}$
taking values in $H_{2}(X_{s})\otimes_{\mathbb{Z}}\mathbb{R}$. This leads to a
sheaf $\mathcal{P}_{(X_{s},D_{s})}$ on $(\partial B)_{0}$, and the map
$\iota:H_{2}(X_{s})\rightarrow H_{2}(X)$ then induces a map

$\iota_{*}:\mathcal{P}_{(X_{s},D_{s})}\rightarrow\mathcal{P}_{\partial B}.$

This allows us to define a map, for $x\in(\partial B)_{0}$,

$\iota_{*}:\Bbbk[\mathcal{P}_{(X_{s},D_{s}),x}^{+}]\rightarrow\Bbbk[\mathcal{P}_{\partial
B,x}^{+}],$

where $\mathcal{P}_{\partial B,x}^{+}:=\mathcal{P}_{\partial
B,x}\cap\mathcal{P}^{+}_{x}$. Finally, we may define

$\iota(\mathscr{S}_{{\mathrm{can}},s}):=\big{\\{}(\mathfrak{p},\iota_{*}(f_{\mathfrak{p}}))\,|\,(\mathfrak{p},f_{\mathfrak{p}})\in\mathscr{S}_{(X_{s},D_{s})}\big{\\}}.$

Like $\mathscr{S}_{{\mathrm{can}},s}$, this is a wall structure on $\partial
B$.

On the other hand, we may define the _asymptotic wall structure_ of the wall
structure $\mathscr{S}_{{\mathrm{can}}}$ on $B$ by

(3.16)
$\mathscr{S}^{\mathrm{as}}_{{\mathrm{can}}}:=\\{(\mathfrak{p}\cap\partial
B,f_{\mathfrak{p}})\,|\,\hbox{$(\mathfrak{p},f_{\mathfrak{p}})\in\mathscr{S}$
with $\dim\mathfrak{p}\cap\partial B=n-2$}\\}.$

We note that for each such wall,
$f_{\mathfrak{p}}\in\Bbbk[\mathcal{P}^{+}_{\partial
B,x}]\subseteq\Bbbk[\mathcal{P}^{+}_{x}]$. Indeed, this follows from
Proposition 3.7, which implies that for a wall type $\tau$ for $X$, $u_{\tau}$
is tangent to fibres of $g_{\mathrm{trop}}$, and in particular tangent to
$\partial B$. Hence $\mathscr{S}^{\mathrm{as}}_{{\mathrm{can}}}$ may be viewed
as a wall structure on $\partial B$.

We then have:

######  Proposition 3.18.

Suppose that all good divisors contained in $g^{-1}(0)$ have multiplicity one
in $g^{-1}(0)$. Then the wall structures
$\iota(\mathscr{S}_{{\mathrm{can}},s})$ and
$\mathscr{S}^{\mathrm{as}}_{{\mathrm{can}}}$ are equivalent.

###### Proof.

Note $\iota(\mathscr{S}_{{\mathrm{can}},s})$ is equal to the canonical wall
structure for $(X_{s},D_{s})$ constructed using the curve class group
$H_{2}(X)$ rather than $H_{2}(X_{s})$. In the proof, we generally use
${\boldsymbol{\tau}}^{\prime}$ to denote a decorated wall type for $(X,D)$ and
${\boldsymbol{\tau}}$ for a decorated wall type for $(X_{s},D_{s})$, with
curve class decoration ${\bf A}$ taking values in $H_{2}(X)$.

Given a decorated wall type ${\boldsymbol{\tau}}^{\prime}$, the corresponding
cone $\tau^{\prime}$ comes with a structure map
$p:\tau^{\prime}\rightarrow\mathbb{R}_{\geq 0}$ coming from the fact the type
is defined over $S$. Note $p$ is surjective as $u_{\tau}$ is tangent to fibres
of $g_{\mathrm{trop}}$ by Proposition 3.7 and
$h_{\tau^{\prime}}(\tau^{\prime}_{\mathrm{out}})\not\subseteq\partial B$ by
Definition 3.6,(3). Then $p^{-1}(0)$ is a proper face of $\tau^{\prime}$, and
hence corresponds to a contraction morphism
$\phi:{\boldsymbol{\tau}}^{\prime}\rightarrow{\boldsymbol{\tau}}$ of decorated
types. The type $\tau$ is realizable provided that the length of
$L_{\mathrm{out}}\in L(G^{\prime})$ isn’t zero on the face $p^{-1}(0)$. Thus
$\tau$ realizable and $\dim\tau=n-3$ is equivalent to
$\dim\mathfrak{p}_{{\boldsymbol{\tau}}^{\prime}}\cap\partial B=n-2$. From this
it is immediate that $\tau$ is a wall type for $(X_{s},D_{s})$ if and only if
$\dim\mathfrak{p}_{{\boldsymbol{\tau}}^{\prime}}\cap\partial B=n-2$.

Consider the sets

$\displaystyle S^{\prime}:={}$
$\displaystyle\\{{\boldsymbol{\tau}}^{\prime}\,|\,\hbox{${\boldsymbol{\tau}}^{\prime}$
is a decorated wall type for $X$ with
$\dim\mathfrak{p}_{{\boldsymbol{\tau}}^{\prime}}\cap\partial B=n-2$}\\},$
$\displaystyle S:={}$
$\displaystyle\\{{\boldsymbol{\tau}}\,|\,\hbox{${\boldsymbol{\tau}}$ is a
decorated wall type for $X_{s}$}\\}.$

The discussion of the previous paragraph has constructed a map
$\Phi:S^{\prime}\rightarrow S$. To show the desired equivalence of wall
structures, it is enough to show that for ${\boldsymbol{\tau}}\in S$ we have

(3.17)
$k_{\tau}W_{{\boldsymbol{\tau}}}=\sum_{{\boldsymbol{\tau}}^{\prime}\in\Phi^{-1}({\boldsymbol{\tau}})}k_{\tau^{\prime}}W_{{\boldsymbol{\tau}}^{\prime}}.$

To do so, we use the decomposition setup of [ACGS2, Thm. 5.21, Def. 5.22, Thm.
5.23]. Note that for $0\not=s\in S$, $s$ is a trivial log point, so in fact
$\mathscr{M}(X_{s},{\boldsymbol{\tau}})=\mathscr{M}(X_{s}/s,{\boldsymbol{\tau}})$.
Thus in particular we have

$W_{{\boldsymbol{\tau}}}={\deg[\mathscr{M}(X_{s}/s,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}({\boldsymbol{\tau}})|}={\deg[\mathscr{M}(X_{0}/0,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}({\boldsymbol{\tau}})|},$

the first equality by definition and the second equality by [ACGS2, Thm.
5.23,(1)]. Further, by [ACGS2, Thm. 5.23,(2)], we have

(3.18)
$\deg[\mathscr{M}(X_{0}/0,{\boldsymbol{\tau}})]^{\mathrm{virt}}=\sum_{{\boldsymbol{\tau}}^{\prime}}{m_{{\boldsymbol{\tau}}^{\prime}}\over|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime}/{\boldsymbol{\tau}})|}\deg[\mathscr{M}(X_{0}/0,{\boldsymbol{\tau}}^{\prime})]^{\mathrm{virt}},$

where the sum is over codimension one degenerations
${\boldsymbol{\tau}}^{\prime}$ of ${\boldsymbol{\tau}}$ in the sense of
[ACGS2, Def. 5.22,(2)], and $m_{{\boldsymbol{\tau}}^{\prime}}$ is the order of
the cokernel of $p_{*}:\Lambda_{\tau^{\prime}}\rightarrow\mathbb{Z}$. It is
immediate from that definition that if
$\mathscr{M}(X_{0}/0,{\boldsymbol{\tau}}^{\prime})\not=\emptyset$ then such a
${\boldsymbol{\tau}}^{\prime}$ is a decorated wall type for $X$. (We need to
assume non-emptiness to guarantee that $\tau^{\prime}$ is balanced.) It also
follows from Proposition 3.7 and the proof of Lemma 3.9 that
$\mathscr{M}(X_{0}/0,{\boldsymbol{\tau}}^{\prime})=\mathscr{M}(X/S,{\boldsymbol{\tau}}^{\prime})=\mathscr{M}(X,{\boldsymbol{\tau}}^{\prime})$.
Thus we may rewrite (3.18) as

$k_{\tau}{\deg[\mathscr{M}(X_{0}/0,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}({\boldsymbol{\tau}})|}=\sum_{{\boldsymbol{\tau}}^{\prime}}{m_{{\boldsymbol{\tau}}^{\prime}}k_{\tau}\over|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime}/{\boldsymbol{\tau}})||\operatorname{Aut}({\boldsymbol{\tau}})|}\deg[\mathscr{M}(X,{\boldsymbol{\tau}}^{\prime})]^{\mathrm{virt}}.$

Certainly,
$|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime}/{\boldsymbol{\tau}})||\operatorname{Aut}({\boldsymbol{\tau}})|=|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime})|$.
Further,
$\Lambda_{\tau^{\prime}_{\mathrm{out}}}=\Lambda_{\tau_{\mathrm{out}}}\oplus\mathbb{Z}$
and
$p_{*}(\Lambda_{\tau^{\prime}_{\mathrm{out}}})=m_{\tau^{\prime}}\mathbb{Z}$.
Next
$\Lambda_{{\boldsymbol{\sigma}}^{\prime}(L_{\mathrm{out}})}=\Lambda_{{\boldsymbol{\sigma}}(L_{\mathrm{out}})}\oplus\mathbb{Z}$,
with the induced map
$g_{\mathrm{trop},*}:\Lambda_{{\boldsymbol{\sigma}}^{\prime}(L_{\mathrm{out}})}\rightarrow\mathbb{Z}$
being surjective because of the assumption that $g^{-1}(0)$ is reduced. Thus
an elementary diagram chase gives a short exact sequence

$0\rightarrow\operatorname{coker}(\Lambda_{\tau_{\mathrm{out}}}\rightarrow\Lambda_{{\boldsymbol{\sigma}}(L_{\mathrm{out}})})\rightarrow\operatorname{coker}(\Lambda_{\tau^{\prime}_{\mathrm{out}}}\rightarrow\Lambda_{{\boldsymbol{\sigma}}^{\prime}(L_{\mathrm{out}})})\rightarrow\mathbb{Z}/m_{\tau^{\prime}}\mathbb{Z}\rightarrow
0,$

so $m_{\tau^{\prime}}k_{\tau}=k_{\tau^{\prime}}$, giving (3.17). ∎

###  3.3. Logarithmic broken lines and theta functions

Wall types defined in the previous subsection allowed us to define the
canonical wall structure. Consistency of a wall structure is defined partly
via _broken lines_ (reviewed in Definition 4.2). A key point of our proof of
consistency is a correspondence result that associates counts of broken lines
to certain punctured invariants. In this subsection, we shall define these
punctured invariants. We proceed quite analogously with the notion of wall
types.

######  Definition 3.19.

A _(non-trivial) broken line type_ is a type
$\tau=(G,{\boldsymbol{\sigma}},{\mathbf{u}})$ of tropical map to $\Sigma(X)$
such that:

  1. (1)

$G$ is a genus zero graph with $L(G)=\\{L_{\mathrm{in}},L_{\mathrm{out}}\\}$
with ${\boldsymbol{\sigma}}(L_{\mathrm{out}})\in\mathscr{P}$ and

$u_{\tau}:={\bf u}(L_{\mathrm{out}})\not=0,\quad p_{\tau}:={\bf
u}(L_{\mathrm{in}})\in{\boldsymbol{\sigma}}(L_{\mathrm{in}})\setminus\\{0\\}$

(so that $L_{\mathrm{in}}$ represents a marked rather than punctured point).

  2. (2)

$\tau$ is realizable and balanced.

  3. (3)

Let $h:\Gamma(G,\ell)\rightarrow\Sigma(X)$ be the corresponding universal
family of tropical maps, and let $\tau_{\mathrm{out}}\in\Gamma(G,\ell)$ be the
cone corresponding to $L_{\mathrm{out}}$. Then $\dim\tau=n-1$ and $\dim
h(\tau_{\mathrm{out}})=n$.

We also consider the possibility of a _trivial broken line type_ $\tau$, which
is not an actual type: the underlying graph $G$ consists of just one leg
$L_{\mathrm{in}}=L_{\mathrm{out}}$ and no vertices, with the convention that
$u_{\tau}=-p_{\tau}$. Note this does not correspond to an actual punctured
curve.888Such a broken line type will correspond, as discussed in §4, to a
broken line contained entirely in ${\boldsymbol{\sigma}}(L_{\mathrm{in}})$ and
which does not bend.

A _decorated broken line type_ is a decorated type
${\boldsymbol{\tau}}=(\tau,\mathbf{A})$ with $\tau$ a broken line type. In the
trivial case, as there are no vertices, this does not involve any extra
information, and in this case, the total curve class is taken to be $0$.

A _degenerate broken line type_ is a type $\tau$ which satisfies conditions
(1) and (2) above and instead of (3),

  * $(3^{\prime})$

$\dim\tau=n-2$ and $\dim h(\tau_{\mathrm{out}})=n-1$.

######  Lemma 3.20.

Let ${\boldsymbol{\tau}}$ be a non-trivial decorated broken line type. Then
$\mathscr{M}(X,{\boldsymbol{\tau}})$ is proper over $\operatorname{Spec}\Bbbk$
and carries a virtual fundamental class of dimension zero.

###### Proof.

The argument is essentially identical to that of the proof of Lemma 3.9, and
we leave it to the reader to make the necessary modifications, except for one
point. In that proof, we appealed to Proposition 3.7 to argue that in the
relative case, for each $v\in V(G)$, we have
$\Sigma(g):h(\tau_{v})\rightarrow\mathbb{R}_{\geq 0}$ surjective. Crucially, a
broken line type is not in general a type of tropical map to $X/S$, so a
different argument is necessary. However, note that by the assumption that
$\tau$ is a balanced type, for any $s\in\operatorname{Int}(\tau)$,
$\Sigma(g)\circ h_{s}:G\rightarrow\mathbb{R}_{\geq 0}$ is a balanced tropical
map. Necessarily
$\Sigma(g)_{*}(\mathbf{u}(L_{\mathrm{in}}))=\Sigma(g)(p_{\tau})\in\mathbb{R}_{\geq
0}$, and hence by balancing, $\Sigma(g)_{*}(u_{\tau})\in\mathbb{R}_{\leq 0}$.
This implies, again by balancing, that if $v_{\mathrm{in}}$,
$v_{\mathrm{out}}$ are the vertices adjacent to $L_{\mathrm{in}}$,
$L_{\mathrm{out}}$ respectively, then $a=\Sigma(g)\circ
h_{s}(v_{\mathrm{out}})\leq\Sigma(g)\circ h_{s}(v_{\mathrm{in}})=b$, and for
any other vertex $v\in V(G)$, $\Sigma(g)\circ h_{s}(v)\in[a,b]$. Further,
since $\dim h(\tau_{\mathrm{out}})=n$, necessarily
$h(\tau_{\mathrm{out}})\not\subseteq\partial B$, and thus
$\Sigma(g):h(\tau_{\mathrm{out}})\rightarrow\mathbb{R}_{\geq 0}$ is
surjective, and so
$\Sigma(g):h(\tau_{v_{\mathrm{out}}})\rightarrow\mathbb{R}_{\geq 0}$ is also
surjective. Putting this together, we see that
$\Sigma(g):h(\tau_{v})\rightarrow\mathbb{R}_{\geq 0}$ is surjective for any
vertex $v\in V(G)$. This is sufficient to complete the argument of the proof
of Proposition 3.9 for properness of the moduli spaces involved. ∎

We now define, for ${\boldsymbol{\tau}}$ a non-trivial decorated broken line
type,

(3.19)
$N_{{\boldsymbol{\tau}}}:={\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}({\boldsymbol{\tau}})|}.$

We also have a map
$h_{*}:\Lambda_{\tau_{\mathrm{out}}}\rightarrow\Lambda_{{\boldsymbol{\sigma}}(L_{\mathrm{out}})}$,
necessarily of finite index, and define

(3.20)
$k_{\tau}:=|\operatorname{coker}h_{*}|=|\Lambda_{{\boldsymbol{\sigma}}(L_{\mathrm{out}})}/h_{*}(\Lambda_{\tau_{\mathrm{out}}})|.$

For a trivial decorated broken line type ${\boldsymbol{\tau}}$, we set

$N_{{\boldsymbol{\tau}}}:=1,\quad k_{{\boldsymbol{\tau}}}:=1.$

######  Definition 3.21.

Fix $p\in B(\mathbb{Z})\setminus\\{0\\}$,
$\sigma\in\mathscr{P}_{{\operatorname{max}}}$,
$x\in\operatorname{Int}(\sigma)$ not contained in any rationally defined
hyperplane in $\sigma$. We then define

$\vartheta^{\log}_{p}(x)=\sum_{{\boldsymbol{\tau}}}k_{\tau}N_{{\boldsymbol{\tau}}}t^{A}z^{-u_{\tau}}\in\Bbbk[\mathcal{P}^{+}_{x}]/I_{x}$

where the sum is over all isomorphism classes of decorated broken line types
${\boldsymbol{\tau}}$ with $p_{\tau}=p$, $x\in h(\tau_{\mathrm{out}})$. Here
$A$ is the total curve class of ${\boldsymbol{\tau}}$, and we require $A\in
Q\setminus I$.

######  Lemma 3.22.

For $p\in B(\mathbb{Z})\setminus\\{0\\}$, the number of decorated broken line
types ${\boldsymbol{\tau}}$ with $p_{\tau}=p$, total curve class in
$Q\setminus I$, and $N_{{\boldsymbol{\tau}}}\not=0$ is finite. In particular,
the sum defining $\vartheta^{\log}_{p}(x)$ is finite.

###### Proof.

As $Q\setminus I$ is finite, there are only a finite number of total curve
classes. Further, there are only a finite number of possibilities for
${\boldsymbol{\sigma}}(L_{\mathrm{out}})$. For each curve class $A$ and choice
of ${\boldsymbol{\sigma}}(L_{\mathrm{out}})$, it follows from [GS7, Cor. 1.14]
that $u_{\tau}$ is determined by $p$ if $\mathscr{M}(X,{\boldsymbol{\tau}})$
is non-empty. If $\beta$ is the punctured curve class given by contact orders
$p$ and $u_{\tau}$ and curve class $A$, then $\mathscr{M}(X,\beta)$ is of
finite type, and there are only a finite number of decorated types
${\boldsymbol{\tau}}$ appearing as the types of tropicalizations of punctured
maps in $\mathscr{M}(X,\beta)$. Thus there are only a finite number of
possible choices of ${\boldsymbol{\tau}}$ with given total curve class $A$ and
${\boldsymbol{\sigma}}(L_{\mathrm{out}})$ satisfying
$\mathbf{u}(L_{\mathrm{in}})=p$ and $\mathbf{u}(L_{\mathrm{out}})=u_{\tau}$. ∎

##  4\. A correspondence theorem for broken lines

###  4.1. Broken lines and statement of the correspondence theorem

We first recall the definition of _broken lines_ [GHS, Def. 3.3] for a wall
structure $\mathscr{S}$ on $B$. We continue to work with the specific choice
of $(B,\mathscr{P})$ derived from $(X,D)$, a monoid $Q\subset H_{2}(X)$ and
monoid ideal $I\subset Q$ with $Q\setminus I$ finite as usual.

######  Definition 4.1.

Let $\gamma:(a,b)\rightarrow B_{0}$ be a path with $t\in(a,b)$ such that
$\gamma(t)\in|\mathscr{S}|\setminus\operatorname{Sing}(\mathscr{S})$. Let
$n\in\check{\Lambda}_{\gamma(t)}$ be a primitive cotangent vector annihilating
the tangent space to $|\mathscr{S}|$ at $\gamma(t)$. Thinking of $n$ as
defining a linear function locally near $\gamma(t)$ which is zero along
$|\mathscr{S}|$, we assume further that $n$ is positive on
$\gamma(t-\epsilon,t)$ and negative on $\gamma(t,t+\epsilon)$ for some
$\epsilon>0$. Suppose $\gamma(t-\epsilon)\in\sigma$,
$\gamma(t+\epsilon)\in\sigma^{\prime}$ with
$\sigma,\sigma^{\prime}\in\mathscr{P}_{{\operatorname{max}}}$. Given an
expression $az^{m}$ with $a\in\Bbbk[Q]/I$, $m\in\Lambda_{\sigma}$, $\langle
m,n\rangle>0$, we define a _result of transport along $\gamma$_ of $az^{m}$ to
be $a_{i}z^{m_{i}}$ chosen as follows. Write

$f_{\gamma(t)}^{\langle
n,m\rangle}\cdot\mathfrak{t}_{\sigma,\sigma^{\prime}}(az^{m})=\sum_{i}a_{i}z^{m_{i}}$

inside $(\Bbbk[Q]/I)[\Lambda_{\sigma^{\prime}}]$, with the
$m_{i}\in\Lambda_{\sigma^{\prime}}$ mutually distinct. We then take
$a_{i}z^{m_{i}}$ to be one of the terms in this sum. Here $\mathfrak{t}$ is as
defined in (3.7) and $f_{\gamma(t)}$ is as defined in (3.8), which may be
viewed via transport as an element of
$(\Bbbk[Q]/I)[\Lambda_{\sigma^{\prime}}]$.

######  Definition 4.2.

A _broken line_ for a wall structure $\mathscr{S}$ on $(B,\mathscr{P})$ is a
proper continuous map

$\beta:(-\infty,0]\rightarrow B_{0}$

with image disjoint from $\operatorname{Sing}(\mathscr{S})$, along with a
sequence $-\infty=t_{0}<t_{1}<\cdots<t_{r}=0$ for some $r\geq 1$ with
$\beta(t_{i})\in|\mathscr{S}|$ for $1\leq i\leq r-1$, and for each
$i=1,\ldots,r$ an expression $a_{i}z^{m_{i}}$ with $a_{i}\in\Bbbk[Q]/I$,
$m_{i}\in\Lambda_{\beta(t)}$ for any $t\in(t_{i-1},t_{i})$, subject to the
following conditions:

  1. (1)

$\beta|_{(t_{i-1},t_{i})}$ is a non-constant affine map with image disjoint
from $|\mathscr{S}|$, hence contained in the interior of some
$\sigma\in\mathscr{P}_{{\operatorname{max}}}$, and $\beta^{\prime}(t)=-m_{i}$
for all $t\in(t_{i-1},t_{i})$.

  2. (2)

For each $i=1,\ldots,r-1$ the expression $a_{i+1}z^{m_{i+1}}$ is a result of
transport of $a_{i}z^{m_{i}}$ along $\beta|_{(t_{i-1},t_{i+1})}$.

  3. (3)

$a_{1}=1$.

The _asymptotic monomial_ of $\beta$ is $m_{1}$. Note that since
$\beta((-\infty,t_{1}])$ is an affine half-line contained in a cone
$\sigma\in\mathscr{P}$, we may identify $m_{1}$ with an integral point of
$\sigma$, and hence view $m_{1}\in B(\mathbb{Z})$.

Given a broken line $\beta$, we write $a_{\beta}=a_{r}$, $m_{\beta}=m_{r}$.

######  Remark 4.3.

This only differs in one way from [GHS, Def. 3.3], namely the requirement that
$a_{1}=1$. In [GHS], broken lines satisfying this additional condition are
called _normalized_ broken lines. As we never use any other type of broken
line here, we include this in the definition.

Broken lines define theta functions:

######  Definition 4.4.

Fix $p\in B(\mathbb{Z})\setminus\\{0\\}$,
$\sigma\in\mathscr{P}_{{\operatorname{max}}}$,
$x\in\operatorname{Int}(\sigma)$ not contained in any rationally defined
hyperplane in $\sigma$. We then define

$\vartheta_{p}(x)=\sum_{\beta}a_{\beta}z^{m_{\beta}}\in\Bbbk[\mathcal{P}_{x}^{+}]/I_{x},$

where the sum is over all broken lines $\beta$ with asymptotic monomial $p$
and $\beta(0)=x$.

We note this sum is finite, e.g., by [GHS, Lem. 3.7].

The main result of this section may now be stated as:

######  Theorem 4.5.

For $p\in B(\mathbb{Z})\setminus\\{0\\}$,
$\sigma\in\mathscr{P}_{{\operatorname{max}}}$,
$x\in\operatorname{Int}(\sigma)$ not contained in any rationally defined
hyperplane, we have

$\vartheta_{p}(x)=\vartheta^{\log}_{p}(x).$

Before proving this theorem, we first recast it as a tropical correspondence
theorem between broken lines and punctured log curves. To do so, we need to
deal with a certain amount of book-keeping. To this end, we first refine the
notion of broken line. This refinement keeps track of which wall types cause
the bending of the broken line.

######  Definition 4.6.

A _decorated broken line_ for $\mathscr{S}_{{\mathrm{can}}}$ is a proper
continuous map

$\beta:(-\infty,0]\rightarrow B_{0}$

with image disjoint from $\operatorname{Sing}(\mathscr{S}_{{\mathrm{can}}})$,
along with a sequence $-\infty=t_{0}<t_{1}<\cdots<t_{r}=0$ for some $r\geq 1$
with $\beta(t_{i})\in|\mathscr{S}_{{\mathrm{can}}}|$ for $1\leq i\leq r-1$,
along with additional data:

  * (i)

For each $i=1,\ldots,r$ an expression $a_{i}z^{m_{i}}$ with
$a_{i}\in\Bbbk[Q]/I$, $m_{i}\in\Lambda_{\beta(t)}$ for any
$t\in(t_{i-1},t_{i})$.

  * (ii)

For each $i=1,\ldots,r-1$, let
$\mathscr{S}_{i}:=\\{\mathfrak{p}_{{\boldsymbol{\tau}}}\in\mathscr{S}_{{\mathrm{can}}}\,|\,\beta(t_{i})\in\mathfrak{p}_{{\boldsymbol{\tau}}}\\}$.
Then we are given in addition a function
$\mu_{i}:\mathscr{S}_{i}\rightarrow\mathbb{N}$.

This data is subject to the following conditions:

  1. (1)

$\beta|_{(t_{i-1},t_{i})}$ is a non-constant affine map, and
$\beta^{\prime}(t)=-m_{i}$ for all $t\in(t_{i-1},t_{i})$.

  2. (2)

Define the _support_ of $\mu_{i}$ to be the subset of $\mathscr{S}_{i}$ on
which $\mu_{i}$ takes non-zero value. Then the support of $\mu_{i}$ must be
finite, and non-empty if $\beta(t_{i})$ lies in the interior of a maximal cone
of $\mathscr{P}$. Further, if $\beta(t)\in\rho\in\mathscr{P}^{[n-1]}$, then
$t=t_{i}$ for some $i$.

  3. (3)

For each $i=1,\ldots,r-1$, if $\beta(t_{i}-\epsilon)\in\sigma$,
$\beta(t_{i}+\epsilon)\in\sigma^{\prime}$, we have

(4.1)
$a_{i+1}z^{m_{i+1}}=\mathfrak{t}_{\sigma,\sigma^{\prime}}(a_{i}z^{m_{i}})\prod_{\mathfrak{p}_{{\boldsymbol{\tau}}}\in\mathscr{S}_{i}}{(\langle
n_{i},m_{i}\rangle
k_{\tau}W_{{\boldsymbol{\tau}}}t^{A}z^{-u_{\tau}})^{\mu_{i}(\mathfrak{p}_{{\boldsymbol{\tau}}})}\over\mu_{i}(\mathfrak{p}_{{\boldsymbol{\tau}}})!}.$

Here $n_{i}\in\check{\Lambda}_{\beta(t_{i})}$ is primitive, vanishing on the
tangent space to $|\mathscr{S}_{{\mathrm{can}}}|$ at $\beta(t_{i})$, and
positive on $m_{i}$.

  4. (4)

$a_{1}=1$.

######  Remark 4.7.

There are a couple of points of distinction between the above definition and
Definition 4.2. First, it is possible that
$\beta(t)\in|\mathscr{S}_{{\mathrm{can}}}|$ with $t\not=t_{i}$ for any $i$
provided $\beta(t)$ lies in the interior of a maximal cell. Second, the data
of the $\mu_{i}$ completely determines the change of monomial. To explain the
expression (4.1), note that by the description (3.13) of
$\mathscr{S}_{{\mathrm{can}}}$,

$f_{\beta(t_{i})}^{\langle
n_{i},m_{i}\rangle}=\exp\left(\sum_{\mathfrak{p}_{{\boldsymbol{\tau}}}\in\mathscr{S}_{i}}\langle
n_{i},m_{i}\rangle
k_{\tau}W_{{\boldsymbol{\tau}}}t^{A}z^{-u_{\tau}}\right)=\sum_{\mu_{i}}\prod_{\mathfrak{p}_{{\boldsymbol{\tau}}}\in\mathscr{S}_{i}}{(\langle
n_{i},m_{i}\rangle
k_{\tau}W_{{\boldsymbol{\tau}}}t^{A}z^{-u_{\tau}})^{\mu_{i}(\mathfrak{p}_{{\boldsymbol{\tau}}})}\over\mu_{i}(\mathfrak{p}_{{\boldsymbol{\tau}}})!}$

where the sum is over all functions
$\mu_{i}:\mathscr{S}_{i}\rightarrow\mathbb{N}$ with finite support. Thus (4.1)
arises from one of the summands, and if $\beta(t_{i})$ falls in the interior
of a maximal cell, we do not allow the trivial summand $1$ (with
$\mu_{i}\equiv 0$). This would correspond, in Definition 4.2, to a situation
where $a_{i}z^{m_{i}}=a_{i+1}z^{m_{i+1}}$.

It is then clear that we have a refined description

$\vartheta_{p}(x)=\sum_{\beta}a_{\beta}z^{m_{\beta}}\in\Bbbk[\mathcal{P}^{+}_{x}]/I_{x}$

as a sum over decorated broken lines with asymptotic monomial $p\in
B(\mathbb{Z})$ and $\beta(0)=x$. Each term in the summation of Definition 4.4
is now split up into a sum over a number of decorated broken lines.

######  Construction 4.8.

We may now associate to a decorated broken line $\beta$ an isomorphism class
of a decorated broken line type
${\boldsymbol{\tau}}_{\beta}=(G_{\beta},{\boldsymbol{\sigma}}_{\beta},{\bf
u}_{\beta},{\bf A}_{\beta})$. This is done as follows.

First, we adopt the following notation. For each
$\mathfrak{p}_{{\boldsymbol{\tau}}}\in\mathscr{S}_{i}$, write
$(G_{{\boldsymbol{\tau}}},{\boldsymbol{\sigma}}_{{\boldsymbol{\tau}}},{\bf
u}_{{\boldsymbol{\tau}}})$ and ${\bf A}_{{\boldsymbol{\tau}}}$ for the data
giving the decorated type ${\boldsymbol{\tau}}$. Let
$L_{\mathrm{out},{\boldsymbol{\tau}}}\in L(G_{{\boldsymbol{\tau}}})$ be the
unique leg, adjacent to a unique $v_{\mathrm{out},{\boldsymbol{\tau}}}\in
V(G_{{\boldsymbol{\tau}}})$.

The graph $G_{\beta}$ has a spine comprising of the legs $L_{\mathrm{in}}$ and
$L_{\mathrm{out}}$ and the edges $E_{1},\ldots,E_{r-2}$ and their vertices
$v_{1},\ldots,v_{r-1}$, with the vertices of $E_{i}$ being $v_{i}$, $v_{i+1}$
and the vertices of $L_{\mathrm{in}}$ and $L_{\mathrm{out}}$ being $v_{1}$ and
$v_{r-1}$ respectively. At the vertex $v_{i}$ we glue
$\mu_{i}(\mathfrak{p}_{{\boldsymbol{\tau}}})$ copies of
$G_{{\boldsymbol{\tau}}}$ to $v_{i}$ along the unique leg of
$G_{{\boldsymbol{\tau}}}$.

Next, we define ${\boldsymbol{\sigma}}_{\beta}$ as follows. We define
${\boldsymbol{\sigma}}_{\beta}(v_{i})$ or
${\boldsymbol{\sigma}}_{\beta}(E_{i})$ to be the minimal cone of $\mathscr{P}$
containing $\beta(t_{i})$ or $\beta((t_{i},t_{i+1}))$ respectively, and
${\boldsymbol{\sigma}}_{\beta}(L_{\mathrm{in}})$ and
${\boldsymbol{\sigma}}_{\beta}(L_{\mathrm{out}})$ to be the minimal cones
containing $\beta((-\infty,t_{1}))$ or $\beta((t_{r-1},t_{r}))$ respectively.
Further, for each $i$, $\mathfrak{p}_{{\boldsymbol{\tau}}}\in\mathscr{S}_{i}$,
we may view each of the $\mu_{i}(\mathfrak{p}_{{\boldsymbol{\tau}}})$ copies
of $G_{{\boldsymbol{\tau}}}$ as a subgraph of $G_{\beta}$, and we take
${\boldsymbol{\sigma}}_{\beta}$ to agree with
${\boldsymbol{\sigma}}_{{\boldsymbol{\tau}}}$ on each copy of
$G_{{\boldsymbol{\tau}}}$.

To define $\mathbf{u}_{\beta}$, we again take it to agree with
$\mathbf{u}_{{\boldsymbol{\tau}}}$ on any copy of $G_{{\boldsymbol{\tau}}}$,
and take $\mathbf{u}_{\beta}(E_{i})=-m_{i+1}$ with $E_{i}$ oriented from
$v_{i}$ to $v_{i+1}$. We also set $\mathbf{u}_{\beta}(L_{\mathrm{in}})=m_{1}$,
$\mathbf{u}_{\beta}(L_{\mathrm{out}})=-m_{r}$ (using the standard convention
that legs are oriented away from their vertex).

This defines the global type $\tau_{\beta}$. For the decoration ${\bf
A}_{\beta}$, it similarly agrees with ${\bf A}_{{\boldsymbol{\tau}}}$ on each
copy of $G_{{\boldsymbol{\tau}}}$. If $v=v_{i}$, we take
${\mathbf{A}}_{\beta}(v_{i})=0$ if $\beta(t_{i})$ lies in the interior of a
maximal cell of $\mathscr{P}$, but if
$\beta(t_{i})\in\operatorname{Int}(\rho)$ with $\rho\in\mathscr{P}^{[n-1]}$,
then we take ${\mathbf{A}}_{\beta}(v_{i})=\langle
n_{i},m_{i}\rangle[X_{\rho}]\in H_{2}(X)$.

######  Lemma 4.9.

Let $\beta$ be a decorated broken line with
$x=\beta(0)\in\operatorname{Int}(\sigma)$,
$\sigma\in\mathscr{P}_{{\operatorname{max}}}$, $x$ not contained in any
rationally defined hyperplane. Then ${\boldsymbol{\tau}}_{\beta}$ is a
decorated broken line type.

###### Proof.

Condition (1) of Definition 3.19 is immediate.

For condition (2), we first show $\tau_{\beta}$ is realizable, i.e., there is
a tropical map $h:G_{\beta}\rightarrow\Sigma(X)$ of type $\tau_{\beta}$. On
the spine $G^{\prime}_{\beta}$ of $G_{\beta}$, consisting of vertices
$v_{1},\ldots,v_{r-1}$, edges $E_{1},\ldots,E_{r-2}$ and legs
$L_{\mathrm{in}},L_{\mathrm{out}}$, we define $h$ to agree with $\beta$,
extending $\beta$ linearly on $L_{\mathrm{out}}$ if necessary. For each $i$
and for each $\mathfrak{p}_{{\boldsymbol{\tau}}}\in\mathscr{S}_{i}$,
${\boldsymbol{\tau}}$ is a wall type, and hence is realizable with an
$n-2$-dimensional universal family with output leg tracing out the wall
$\mathfrak{p}_{{\boldsymbol{\tau}}}$. Further,
$\beta(t_{i})\in\operatorname{Int}(\mathfrak{p}_{{\boldsymbol{\tau}}})$. Thus
there is a unique tropical map
$h_{{\boldsymbol{\tau}}}:G_{{\boldsymbol{\tau}}}\rightarrow\Sigma(X)$ of type
$\tau$ with $\beta(t_{i})\in
h_{{\boldsymbol{\tau}}}(L_{\mathrm{out},{\boldsymbol{\tau}}})$. Thus we may
define $h$ restricted to any copy of $G_{{\boldsymbol{\tau}}}$ in $G_{\beta}$
to be given by $h_{{\boldsymbol{\tau}}}$, appropriately truncating the edge
$L_{\mathrm{out},{\boldsymbol{\tau}}}$ as necessary. This shows realizability.

Next, we observe that $\tau_{\beta}$ is balanced. Because this is true for
each wall type $\tau$ appearing in $\tau_{\beta}$, this only needs to be
checked at the vertices $v_{i}$, but such balancing follows from (4.1).
Indeed, by considering exponents, that equation tells us that

$m_{i+1}=m_{i}-\sum_{\mathfrak{p}_{{\boldsymbol{\tau}}}\in\mathscr{S}_{i}}\mu_{i}(\mathfrak{p}_{{\boldsymbol{\tau}}})u_{\tau},$

which is the balancing condition at $v_{i}$ given the specified contact orders
in Construction 4.8 of edges adjacent to $v_{i}$. In the relative case, we
also need to verify that $\Sigma(g)\circ h$ is balanced as a map to
$\Sigma(S)=\mathbb{R}_{\geq 0}$. However, this again holds at all vertices of
$G_{\beta}$ except for the $v_{i}$’s because the same is assumed true of the
wall types $\tau$. At a vertex $v_{i}$, the result follows from the balancing
in $B$ and the fact (Lemma 1.15) that $g_{\mathrm{trop}}$ is an affine
submersion.

For condition (3), observe that, by elementary convex geometry, $\beta$ varies
in an $n$-dimensional family: each $\beta(t_{i})$ is constrained to live in an
$n-1$-dimensional subspace, and the location of $\beta(t_{1})$ and the $m_{i}$
completely determine the map $\beta$, up to the location of $\beta(0)$, which
gives an extra parameter. In particular, the dimension of the universal family
of tropical maps of type $\tau_{\beta}$ is then clearly $n-1$, and
$h(\tau_{\beta,\mathrm{out}})$ is $n$-dimensional. This shows $\tau_{\beta}$
is a broken line type. ∎

We note that the broken line type ${\boldsymbol{\tau}}_{\beta}$ had a very
specific form for the curve classes associated to the vertices $v_{i}$. We
codify this as follows:

######  Definition 4.10.

Let ${\boldsymbol{\tau}}=(G,{\boldsymbol{\sigma}},{\mathbf{u}},{\mathbf{A}})$
be a decorated broken line type with spine $G^{\prime}\subseteq G$ (see
Definition 2.4), the latter having vertices $v_{1},\ldots,v_{r-1}$ and edges
$E_{i}$ with vertices $v_{i},v_{i+1}$. By Lemma 2.5,
${\boldsymbol{\sigma}}(v_{i})\in\mathscr{P}^{[n]}\cup\mathscr{P}^{[n-1]}$. We
say ${\boldsymbol{\tau}}$ is _admissible_ if $\mathbf{A}(v_{i})=0$ whenever
${\boldsymbol{\sigma}}(v_{i})\in\mathscr{P}^{[n]}$, and
$\mathbf{A}(v_{i})=|\langle
n_{i},\mathbf{u}(E_{i})\rangle|[X_{{\boldsymbol{\sigma}}(v_{i})}]$ whenever
${\boldsymbol{\sigma}}(v_{i})\in\mathscr{P}^{[n-1]}$, where
$n_{i}\in\check{\Lambda}_{x}$ for
$x\in\operatorname{Int}({\boldsymbol{\sigma}}(v_{i}))$ is a choice of
primitive normal vector to ${\boldsymbol{\sigma}}(v_{i})$.

By construction, if $\beta$ is a decorated broken line, then
${\boldsymbol{\tau}}_{\beta}$ is admissible. The following is immediate from
Lemma 2.3:

######  Lemma 4.11.

Let ${\boldsymbol{\tau}}$ be a decorated broken line type. Then
$N_{{\boldsymbol{\tau}}}\not=0$ implies that ${\boldsymbol{\tau}}$ is
admissible.

We now reverse the procedure of Construction 4.8 and go from decorated broken
line types to decorated broken lines.

######  Construction 4.12.

Let ${\boldsymbol{\tau}}=(G,{\boldsymbol{\sigma}},{\bf u},{\bf A})$ be a
decorated broken line type, $h$ the universal tropical map parameterized by
the cone $\tau$, and $x\in\operatorname{Int}(h(\tau_{\mathrm{out}}))$. We may
construct a broken line $\beta_{{\boldsymbol{\tau}},x}$ as follows. There
exists a unique $s\in\operatorname{Int}(\tau)$ such that $x\in
h_{s}(L_{\mathrm{out}})$. Let $G^{\prime}$ be the spine of $G$. Label the
vertices of $G^{\prime}$ as $v_{1},\ldots,v_{r-1}$ with $L_{\mathrm{in}}$
adjacent to $v_{1}$, $L_{\mathrm{out}}$ adjacent to $v_{r-1}$, and edges
$E_{1},\ldots,E_{r-2}$, with $E_{i}$ having vertices $v_{i}$ and $v_{i+1}$.
Shrink $L_{\mathrm{out}}$ so that the non-vertex endpoint of
$L_{\mathrm{out}}$ maps to $x$ under $h_{s}$. We may then choose an
identification $\psi:\mathbb{R}_{\leq 0}\rightarrow G^{\prime}$ with the
following property. Let $t_{i}\in\mathbb{R}_{\leq 0}$ satisfy
$\psi(t_{i})=v_{i}$, $1\leq i\leq r-1$, take $t_{r}=0$, and set
$\beta_{{\boldsymbol{\tau}},x}=h_{s}|_{G^{\prime}}\circ\psi$. We require that
condition (1) of Definition 4.6 is satisfied for
$\beta_{{\boldsymbol{\tau}},x}$ with $m_{i}=-\mathbf{u}(E_{i})$, with $E_{i}$
oriented from $v_{i}$ to $v_{i+1}$. Note by construction that
$\beta_{{\boldsymbol{\tau}},x}(0)=x$.

We now specify the decorations, notably the functions $\mu_{i}$. Let
$G_{i1},\ldots,G_{is_{i}}$ be the closures of the connected components of
$G\setminus\\{v_{i}\\}$ not containing $L_{\mathrm{in}}$ or
$L_{\mathrm{out}}$. These are the graphs attached to $v_{i}$ but only
intersecting $G^{\prime}$ at $v_{i}$. We view the edge of $G_{ij}$ adjacent to
the vertex $v_{i}$ as a leg $L_{ij,\mathrm{out}}$ of $G_{ij}$, so that $v_{i}$
is not a vertex of $G_{ij}$. This gives rise to decorated types
${\boldsymbol{\tau}}_{i1},\ldots,{\boldsymbol{\tau}}_{is_{i}}$ by restriction
of ${\boldsymbol{\sigma}},\mathbf{u},\mathbf{A}$ to each graph $G_{ij}$.

We claim that each ${\boldsymbol{\tau}}_{ij}$ is a wall type. Indeed,
condition (1) of Definition 3.6 is immediate except for the statement that
$\mathbf{u}(L_{ij,\mathrm{out}})\not=0$, which we rule out below. For
condition (2), note ${\boldsymbol{\tau}}_{ij}$ is realizable and balanced
since ${\boldsymbol{\tau}}$ is realizable and balanced.

For condition (3), let $h_{ij}:\Gamma(G_{ij},\ell_{ij})\rightarrow\Sigma(X)$
be the universal tropical map of type ${\boldsymbol{\tau}}_{ij}$, defined over
the cone $\tau_{ij}$. First note that necessarily $h(\tau_{v_{i}})\subseteq
h_{ij}(\tau_{ij,\mathrm{out}})$, and by Lemma 2.5, (2), $\dim
h(\tau_{v_{i}})=n-1$. Thus $\dim h_{ij}(\tau_{ij,\mathrm{out}})\geq n-1$.
However by Lemma 2.5, (1), still assuming that
$\mathbf{u}(L_{ij,\mathrm{out}})\not=0$, $\dim
h_{ij}(\tau_{ij,\mathrm{out}})\leq n-1$ and thus this dimension is $n-1$, as
desired.

Suppose now that $\dim\tau_{ij}>n-2$. Thus for any fixed
$s^{\prime}\in\operatorname{Int}(\tau)$, there is a positive dimensional
subset $\omega\subseteq\tau_{ij}$ such that for $s^{\prime\prime}\in\omega$,
$h_{s^{\prime}}(v_{i})\in h_{ij,s^{\prime\prime}}(L_{ij,\mathrm{out}})$, where
$L_{ij,\mathrm{out}}$ is the unique leg of $G_{ij}$. Thus for each point $y\in
h(\tau_{v_{i}})$, there is a positive dimensional family of tropical maps of
type $\tau$ taking $v_{i}$ to $y$. Again since $\dim h(\tau_{v_{i}})=n-1$,
this shows $\dim\tau\geq n$, a contradiction.

We now eliminate the case that ${\bf u}(L_{ij,\mathrm{out}})=0$. Indeed, if
this is the case and $v$ is the unique vertex of $L_{ij,\mathrm{out}}$ in
$G_{ij}$, then $h_{ij}(\tau_{ij,v})=h_{ij}(\tau_{ij,L_{ij,\mathrm{out}}})$,
and this is again $n-1$-dimensional as above. Thus $\dim\tau_{ij}\geq n-1$. On
the other hand, if $E_{ij}$ is the edge of $G$ corresponding to the leg
$L_{ij,\mathrm{out}}$ of $G_{ij}$, we have ${\bf u}(E_{ij})={\bf
u}(L_{ij,\mathrm{out}})=0$, so the length of $E_{ij}$ is arbitrary, and thus
we obtain again that $\dim\tau\geq n$. Thus we conclude that
${\boldsymbol{\tau}}_{ij}$ is a wall type.

We may now define $\mu_{i}:\mathscr{S}_{i}\rightarrow\mathbb{N}$ by defining,
for $\mathfrak{p}_{{\boldsymbol{\tau}}^{\prime}}\in\mathscr{S}_{i}$,
$\mu_{i}(\mathfrak{p}_{{\boldsymbol{\tau}}^{\prime}})=|\\{j\,|\,{\boldsymbol{\tau}}_{ij}\simeq{\boldsymbol{\tau}}^{\prime}\\}|$.

Having defined the $\mu_{i}$, (4.1) defines the monomials $a_{i}z^{m_{i}}$.
There is one thing to check: as we have already defined $m_{i}$ in terms of
${\bf u}$, we need to check that (4.1) yields the same values for $m_{i}$.
However, this is immediate inductively by balancing at the vertices $v_{i}$.

Hence we have constructed a decorated broken line
$\beta_{{\boldsymbol{\tau}}}$.

Constructions 4.8 and 4.12 now immediately give:

######  Proposition 4.13.

Given $x\in B_{0}$ not contained in any rationally defined hyperplane and
$p\in B(\mathbb{Z})\setminus\\{0\\}$, there is a one-to-one correspondence

$\displaystyle\\{\beta\,|\,\hbox{ $\beta$ is a decorated broken line with
endpoint $x$ and asymptotic monomial $p$}\\}$ $\displaystyle\leftrightarrow$
$\displaystyle\left\\{{\boldsymbol{\tau}}\,\Big{|}\,\begin{subarray}{c}\hbox{${\boldsymbol{\tau}}$
is an isomorphism class of admissible decorated}\\\ \hbox{broken line types
with $x\in\operatorname{Int}(h(\tau_{\mathrm{out}}))$ and $p_{\tau}=p$
}\end{subarray}\right\\}$

given by $\beta\mapsto{\boldsymbol{\tau}}_{\beta}$,
${\boldsymbol{\tau}}\mapsto\beta_{{\boldsymbol{\tau}},x}$.

The key result, to be proved via gluing of punctured curves in the next
subsection, is then:

######  Theorem 4.14.

Let ${\boldsymbol{\tau}}$ be an admissible decorated broken line type, and let
$\beta_{{\boldsymbol{\tau}},x}$ be the corresponding broken line with endpoint
$x\in\operatorname{Int}(h(\tau_{\mathrm{out}}))$. Then we have

$a_{\beta_{{\boldsymbol{\tau}},x}}z^{m_{\beta_{{\boldsymbol{\tau}},x}}}=k_{\tau}N_{{\boldsymbol{\tau}}}t^{A}z^{-u_{\tau}},$

where $A$ is the total degree of ${\boldsymbol{\tau}}$.

###### Proof of Theorem 4.5 given Theorem 4.14..

This follows immediately from the definitions Definition 4.4 and 3.21 of the
two types of theta functions, Proposition 4.13, and Theorem 4.14. ∎

###  4.2. Proof of the correspondence theorem, Theorem 4.14

The proof is by induction on the length of $\beta_{{\boldsymbol{\tau}},x}$,
that is, the number $r\geq 1$ in Definition 4.2. The base case $r=1$ is the
case of a straight broken line ending at some point $x$ in the interior of a
maximal cell. The corresponding broken line type is then a trivial one
${\boldsymbol{\tau}}_{\mathrm{in}}$ without a vertex and trivial curve class
$A=0$. Thus $N_{\boldsymbol{\tau}}=1$, $k_{\boldsymbol{\tau}}=1$ by definition
and the claimed equality holds trivially.

For the inductive step, let ${\boldsymbol{\tau}}=(\tau,\bf A)$,
$\tau=(G,{\boldsymbol{\sigma}},\bf u)$ be an admissible decorated broken line
type as in the theorem, with associated broken line
$\beta_{{\boldsymbol{\tau}},x}$ of length $r$. Let $v\in V(G)$ be the vertex
adjacent to $L_{\mathrm{out}}$, $\tau_{v}\in\Gamma(G,\ell)$ the corresponding
cone and $\sigma_{\mathfrak{p}}={\boldsymbol{\sigma}}(v)$ the smallest cell
containing $h(\tau_{v})$. Here we denote by $\mathfrak{p}$ the set-theoretic
intersection of all walls or codimension one cells of $\mathscr{P}$ containing
$h(\tau_{v})$. Thus $\mathfrak{p}$ is a polyhedral subset of $B$ of dimension
$n-1$. Note that $\dim\sigma_{\mathfrak{p}}\in\\{n-1,n\\}$, and we refer to
the two cases as codimension one and codimension zero, respectively. We work
in an affine chart of $B_{0}$ containing
$\operatorname{Int}\sigma_{\mathfrak{p}}$, with $\Lambda$ denoting the lattice
of integral tangent vector fields by $\Lambda_{\mathfrak{p}}\subset\Lambda$
the corank one lattice of vectors tangent to $h(\tau_{v})$. Let
$E_{\mathrm{in}}\in E(G)$ be the unique edge adjacent to $v$ and belonging to
the spine of $\tau$, and $E_{1},\ldots,E_{l}$ the remaining edges adjacent to
$v$, all oriented toward $v$. The case $l=0$ means that no such edges are
present and then $E_{1},\ldots,E_{l}$ denotes the empty sequence. Denote
further $u_{\mathrm{in}}=\mathbf{u}(E_{\mathrm{in}})$,
$u_{\mathrm{out}}=\mathbf{u}(L_{\mathrm{out}})=u_{\tau}$,
$u_{i}=\mathbf{u}(E_{i})$, and
$\sigma_{\mathrm{in}}={\boldsymbol{\sigma}}(E_{\mathrm{in}})$ the maximal cell
containing $h_{s}(E_{\mathrm{in}})$ for all $s\in\tau$.

We now split $G$ at all the edges $E_{\mathrm{in}},E_{1},\ldots,E_{l}$
adjacent to $v$. See [ACGS2, §5.1] for a formal treatment of splitting.
Splitting an edge $E$ with vertices $v_{1},v_{2}$ leads to a pair of legs that
we denote $(E,v_{1})$, $(E,v_{2})$. By Construction 4.12 the types obtained
from ${\boldsymbol{\tau}}$ after splitting are as follows.

  1. (1)

A decorated broken line type
${\boldsymbol{\tau}}_{\mathrm{in}}=(G_{\mathrm{in}},{\boldsymbol{\sigma}}_{\mathrm{in}},\mathbf{u}_{\mathrm{in}},\mathbf{A}_{\mathrm{in}})$,
with the unique leg $(E_{\mathrm{in}},v_{\mathrm{in}})$ obtained from
splitting and
$\mathbf{u}_{\mathrm{in}}(E_{\mathrm{in}},v_{\mathrm{in}})=u_{\mathrm{in}}$.

  2. (2)

Decorated wall types
${\boldsymbol{\tau}}_{i}=(G_{i},{\boldsymbol{\sigma}}_{i},\mathbf{u}_{i},\mathbf{A}_{i})$,
$i=1,\ldots,l$, with legs $(E_{i},v_{i})\in L(G_{i})$ obtained from splitting,
and $\mathbf{u}_{i}(E_{i},v_{i})=u_{i}$.

  3. (3)

${\boldsymbol{\tau}}_{0}=(G_{0},{\boldsymbol{\sigma}}_{0},\mathbf{u}_{0},\mathbf{A}_{0})$,
the type of a decorated punctured map with only one vertex $v$ with
${\boldsymbol{\sigma}}_{0}(v)=\sigma_{\mathfrak{p}}={\boldsymbol{\sigma}}(v)$,
no edges, and legs $(E_{\mathrm{in}},v)$, $(E_{1},v),\ldots,(E_{l},v)$,
$(E_{\mathrm{out}},v)=L_{\mathrm{out}}$ with $\mathbf{A}_{0}(v)=\mathbf{A}(v)$
and

$\mathbf{u}_{0}(E_{\mathrm{in}},v)=-u_{\mathrm{in}},\quad\mathbf{u}_{0}(E_{\mathrm{out}},v)=u_{\mathrm{out}},\quad\mathbf{u}_{0}(E_{i},v)=-u_{i},\
i=1,\ldots,l.$

As usual the corresponding undecorated types are denoted $\tau_{\mathrm{in}}$
and $\tau_{i}$, $i=0,1,\ldots,l$. Grouping together identical
${\boldsymbol{\tau}}_{i}$ we may relabel and assume the
${\boldsymbol{\tau}}_{i}$ are pairwise non-isomorphic, but each
${\boldsymbol{\tau}}_{i}$ to occur $\mu_{i}$-times in the splitting for some
$\mu_{i}\in\mathbb{N}$.

The propagation rule (4.1) for monomials in the definition of decorated broken
lines (Definition 4.6) together with the induction hypothesis shows that the
claimed equality
$a_{\beta_{{\boldsymbol{\tau}}}}z^{m_{\beta_{{\boldsymbol{\tau}}}}}=k_{\tau}N_{{\boldsymbol{\tau}}}t^{A}z^{-u_{\mathrm{out}}}$
is equivalent to the following

(4.2)
$k_{\tau}N_{\boldsymbol{\tau}}t^{A}z^{-u_{\mathrm{out}}}=k_{\tau_{\mathrm{in}}}N_{{\boldsymbol{\tau}}_{\mathrm{in}}}t^{A_{\mathrm{in}}}t^{d[X_{\sigma_{\mathfrak{p}}}]}z^{-u_{\mathrm{in}}}\cdot\prod_{i=1}^{l}\frac{\big{(}dk_{\tau_{i}}W_{{\boldsymbol{\tau}}_{i}}t^{A_{i}}z^{-u_{\tau_{i}}}\big{)}^{\mu_{i}}}{\mu_{i}!}.$

Note that $[X_{\sigma_{\mathfrak{p}}}]=0$ in the codimension zero case and
$[X_{\sigma_{\mathfrak{p}}}]$ is the class of the curve corresponding to the
$(n-1)$-cell containing $\mathfrak{p}$ in the codimension one case.
Furthermore,

(4.3) $d=|\delta(u_{\mathrm{in}})|$

for $\delta:\Lambda\longrightarrow\mathbb{Z}$ the quotient by
$\Lambda_{\mathfrak{p}}$, and $k_{\tau_{\mathrm{in}}}$, $k_{\tau_{i}}$ are
defined in (3.20) and (3.10), respectively.

To prove this equality, we employ the numerical splitting formula for
punctured Gromov-Witten invariants proved by Yixian Wu [Wu] and recalled in
the appendix. Assumption A.1 of toric gluing strata is trivially fulfilled in
the codimension zero case, while in codimension one it follows from Lemma
1.10. In the following discussion, we freely use the notation introduced in
the appendix, but note that we do not split all edges of $\tau$, but just the
subset $E(G,v)\subseteq E(G)$ of $l+1$ edges adjacent to $v$. Thus for any
$E\in E(G,v)$ the vector space $(\Lambda_{E})_{\mathbb{R}}$ holding the
$E$-component of the displacement vector equals $\Lambda_{\mathbb{R}}$. For
the time being we also revert to the original labelling of the
${\boldsymbol{\tau}}_{i}$ with possibly several identical wall types. The
numerator $\mu_{i}!$ in (4.2) arises only at the very end when taking the
automorphism group of the glued decorated type ${\boldsymbol{\tau}}$ into
account.

To invoke Theorem A.4 we need to choose a general displacement vector
$\nu=(\nu_{E})_{E\in E(G,v)}$ and then determine the corresponding set
$\Delta(\nu)$ of decorated transverse types

$({\boldsymbol{\omega}}_{\mathrm{in}},{\boldsymbol{\omega}}_{0},{\boldsymbol{\omega}}_{1},\ldots,{\boldsymbol{\omega}}_{l})$

for $\nu$, with ${\boldsymbol{\omega}}_{\bullet}$ a decorated type marked by
$\tau_{\bullet}$. Lemma 4.15 below shows that for a certain choice of $\nu$
there is a unique decorated transverse type with
${\boldsymbol{\omega}}_{\mathrm{in}}={\boldsymbol{\tau}}_{\mathrm{in}}$,
${\boldsymbol{\omega}}_{i}={\boldsymbol{\tau}}_{i}$ for all $i$, and with

(4.4)
${\boldsymbol{\omega}}_{0}=(\hat{G}_{0},\hat{\boldsymbol{\sigma}}_{0},\hat{\mathbf{u}}_{0},\hat{\mathbf{A}}_{0})=(\omega_{0},\hat{\mathbf{A}}_{0})$

a certain maximal decorated global type marked by ${\boldsymbol{\tau}}_{0}$.

$v^{\prime}_{3}$$v^{\prime}_{l-1}$$v^{\prime}_{1}$$v^{\prime}_{2}$$v^{\prime}_{l}$$(E_{1},v^{\prime}_{1})$$(E_{\mathrm{out}},v^{\prime}_{l})$$(E_{\mathrm{in}},v^{\prime}_{1})$$E^{\prime}_{1}$$E^{\prime}_{2}$$(E_{3},v^{\prime}_{3})$$(E_{l-1},v^{\prime}_{l-1})$$E^{\prime}_{l-1}$$(E_{l},v^{\prime}_{l})$$(E_{2},v^{\prime}_{2})$

Figure 4.1. The tropical type $\omega_{0}$ (codimension zero case).

If $l=0$ we take ${\boldsymbol{\omega}}_{0}={\boldsymbol{\tau}}_{0}$. To
define ${\boldsymbol{\omega}}_{0}$ for $l\geq 1$, assume first
$\sigma_{\mathfrak{p}}=\sigma_{\mathrm{in}}$, that is, we are in the situation
of codimension zero. Take for $\hat{G}_{0}$ the graph with $l$ trivalent
vertices $v^{\prime}_{1},\ldots,v^{\prime}_{l}$ connected in the given order
by $l-1$ edges $E^{\prime}_{1},\ldots,E^{\prime}_{l-1}$ to form a chain, and
with legs distributed as

$(E_{\mathrm{in}},v^{\prime}_{1}),\quad(E_{\mathrm{out}},v^{\prime}_{l}),\quad(E_{i},v^{\prime}_{i}),\,i=1,\ldots,l,$

see Figure 4.1. The notation $E_{\mathrm{in}}$, $E_{i}$, $E_{\mathrm{out}}$
indicates that we identify these legs with the legs of $\tau_{0}$ via the
contraction morphism $\omega_{0}\longrightarrow\tau_{0}$ contracting all edges
of $\omega_{0}$. All strata are given by $\sigma_{\mathrm{in}}$ and the
contact orders for each edge are determined inductively from the contact
orders $\mathbf{u}_{0}$ of $\tau_{0}$ by enforcing the balancing condition
Lemma 2.1. Necessarily $\hat{\mathbf{A}}_{0}=0$.

If $\sigma_{\mathfrak{p}}$ is of codimension one, we further split
$v^{\prime}_{l}$ into two vertices $v^{\prime}_{l}$ and
$v^{\prime}_{\mathrm{out}}$ connected by another edge $E^{\prime}_{l}$, with
$\hat{\boldsymbol{\sigma}}_{0}(v^{\prime}_{\mathrm{out}})=\sigma_{\mathfrak{p}}$,
and the legs at $v^{\prime}_{l}$ and $v^{\prime}_{\mathrm{out}}$ distributed
as $(E_{l},v^{\prime}_{l})$, $(E_{\mathrm{out}},v^{\prime}_{\mathrm{out}})$.
For $l=1$ the leg
$(E_{\mathrm{in}},v^{\prime}_{1})=(E_{\mathrm{in}},v^{\prime}_{l})$ is also
adjacent to $v^{\prime}_{l}$. The other strata are forced to be one of the two
maximal cells $\sigma_{\mathrm{in}}={\boldsymbol{\sigma}}(E_{\mathrm{in}},v)$
or ${\boldsymbol{\sigma}}(L_{\mathrm{out}})$ containing
$\sigma_{\mathfrak{p}}$, namely
$\hat{\boldsymbol{\sigma}}_{0}(E_{\mathrm{out}},v^{\prime}_{\mathrm{out}})={\boldsymbol{\sigma}}(L_{\mathrm{out}})$
and all other strata equal to $\sigma_{\mathrm{in}}$. The curve classes are
determined by the contraction morphism as

$\hat{\mathbf{A}}(v^{\prime}_{\mathrm{out}})=\mathbf{A}(v)=d\cdot[X_{\sigma_{\mathfrak{p}}}]$

and all other curves classes trivial. Note that in any case the curve classes
are determined by the contraction morphism. It is therefore enough to drop the
decoration in the following classification of transverse types according to
Definition A.2.

######  Lemma 4.15.

There exists a displacement vector $\nu=(\nu_{E})_{E}\in\prod_{E\in
E(G)}\Lambda_{\mathbb{R}}$ that is general for $\tau$ such that any
$\nu$-transverse type $\omega$ for $\tau$ (Definition A.2,2) with
$\mathscr{M}(X,\omega^{\prime}_{0})\neq\emptyset$, for $\omega^{\prime}_{0}$
the part of $\omega$ marked by $\tau_{0}$, is isomorphic to

$\omega=(\tau_{\mathrm{in}},\tau_{1},\ldots,\tau_{l},\omega_{0}),$

where $\omega_{0}$ is as described in (4.4).

###### Proof.

Take the quotient $\delta:\Lambda\longrightarrow\mathbb{Z}$ by
$\Lambda_{\mathfrak{p}}$ to evaluate non-negatively on vectors pointing from
$\mathfrak{p}$ to the incoming direction, and let $m\in\Lambda$ be a vector
with $\delta(m)=1$. Then in particular
$\Lambda=\Lambda_{\mathfrak{p}}\oplus\mathbb{Z}m$. Choose
$\nu_{1},\ldots,\nu_{l}\in\mathbb{R}$ with

$\nu_{1}>\ldots>\nu_{l}>0,$

and define

$\nu_{E_{\mathrm{in}}}=0,\quad\nu_{E_{i}}=\nu_{i}\cdot m,\quad i=1,\ldots,l.$

Thus $\nu_{E_{i}}=(0,\nu_{i})$ when identifying $\Lambda$ with
$\Lambda_{\mathfrak{p}}\oplus\mathbb{Z}$. Following the interpretation in
Remark A.3 of $\Delta(\nu)$, we have to show that any $\nu$-broken tropical
punctured map marked by $\tau$ fulfilling
$\mathscr{M}(X,\omega^{\prime}_{0})\neq\emptyset$ has components of types

$\omega_{\mathrm{in}}=\tau_{\mathrm{in}},\
\omega_{1}=\tau_{1},\ldots,\omega_{l}=\tau_{l},\ \omega_{0}.$

We first consider the component marked by $\tau_{0}$. Denote this component by
$\omega^{\prime}_{0}=(G^{\prime}_{0},{\boldsymbol{\sigma}}^{\prime}_{0},\mathbf{u}^{\prime}_{0})$
and by

$(E_{\mathrm{in}},v^{\prime}_{\mathrm{in}}),\
(E_{\mathrm{out}},v^{\prime}_{\mathrm{out}}),\
(E_{1},v^{\prime}_{1}),\ldots,(E_{l},v^{\prime}_{l})$

the legs, with some of the vertices
$v^{\prime}_{\mathrm{in}},v^{\prime}_{1},\ldots,v^{\prime}_{l},v^{\prime}_{\mathrm{out}}$
possibly coinciding. Denote further by
$E^{\prime}_{1},\ldots,E^{\prime}_{r}\in E(G^{\prime}_{0})$ the unique
sequence of edges connecting $v^{\prime}_{\mathrm{in}}$ to
$v^{\prime}_{\mathrm{out}}$, oriented away from $v^{\prime}_{\mathrm{in}}$.
Note also that $d=-\delta(u_{\mathrm{in}})$ for $d$ as introduced in (4.3) and
that $E_{\mathrm{in}}\in E(G)$ is oriented away from $v$. Now since
$\mathscr{M}(X,\omega^{\prime}_{0})\neq\emptyset$, Lemma 2.1 shows that
$\omega^{\prime}_{0}$ fulfills the balancing condition. Thus
$u_{1},\ldots,u_{l}\in\Lambda_{\mathfrak{p}}=\operatorname{ker}(\delta)$
implies that the images under $\delta$ of the contact orders of all edges are
determined as follows:

$-\delta(\mathbf{u}^{\prime}_{0}(E))=\begin{cases}d,&E=E^{\prime}_{1},\ldots,E^{\prime}_{r}\\\
0,&\text{otherwise}.\end{cases}$

Moreover, by the perturbed matching condition (A.9), the fact that the
outgoing vertex for each $\tau_{i}$ maps to $\mathfrak{p}$ and the choice of
$\nu_{1},\ldots,\nu_{l}$, it follows that $\delta$ is constant on the subtree
connecting the leg $(E_{i},v_{i})$ to the chain of edges
$E^{\prime}_{1},\ldots,E^{\prime}_{r}$, with value $\nu_{i}$. Since all
$\nu_{i}$ are disjoint, it follows that all of these subtrees are disjoint.
Note that there are $r+1$ vertices on the path connecting
$v^{\prime}_{\mathrm{in}}$ and $v^{\prime}_{\mathrm{out}}$ (including these
latter two vertices). Thus necessarily $r+1\geq l$. However, if
$\dim\sigma_{\mathfrak{p}}=n-1$,
${\boldsymbol{\sigma}}_{0}^{\prime}(E_{\mathrm{in}},v^{\prime}_{\mathrm{in}})$
and
${\boldsymbol{\sigma}}_{0}^{\prime}(E_{\mathrm{out}},v^{\prime}_{\mathrm{out}})$
are distinct maximal cones containing $\sigma_{\mathfrak{p}}$, and thus there
must be at least one vertex $v_{j}^{\prime}$ on this path contained in
$\sigma_{\mathfrak{p}}$. By the choice of displacement vectors, none of the
subtrees connecting an edge $(E_{i},v_{i})$ to the chain of edges
$E^{\prime}_{1},\ldots,E^{\prime}_{r}$ is connected to $v_{j}^{\prime}$. Thus
in this case, $r\geq l$.

Now by the dimension formula (A.8) we require

(4.5)
$\textstyle\dim\widetilde{\omega}_{\mathrm{in}}+\dim\widetilde{\omega}^{\prime}_{0}+\sum_{i=1}^{l}\dim\widetilde{\omega}_{i}=\dim\widetilde{\tau}+\sum_{E}\operatorname{rk}\Lambda=(n+l)+(l+1)\cdot
n=ln+2n+l.$

Note that if $\dim\sigma_{\mathfrak{p}}=n$, then $\dim\omega_{0}^{\prime}\geq
n+r$, as there is an $n$-dimensional choice of location for one vertex, and an
additional choice of length for the edges
$E_{1}^{\prime},\ldots,E_{r}^{\prime}$. Thus, in this first case,
$\dim\widetilde{\omega}_{0}^{\prime}\geq n+r+l+1$. On the other hand, if
$\dim\sigma_{\mathfrak{p}}=n-1$, then $\dim\omega_{0}^{\prime}\geq n-1+r$, as
there is an $(n-1)$-dimensional choice of location for the vertex
$v_{j}^{\prime}$. Thus, in this second case,
$\dim\widetilde{\omega}_{0}^{\prime}\geq n+r+l$. The left-hand side of (4.5)
now has dimension at least

$\begin{cases}n+(n+l+r+1)+l\cdot(n-1)=ln+2n+r+1,&\dim\sigma_{\mathfrak{p}}=n\\\
n+(n+l+r)+l\cdot(n-1)=ln+2n+r,&\dim\sigma_{\mathfrak{p}}=n-1.\end{cases}$

Note in both cases, this quantity is then $\geq ln+2n+l$, with equality with
the right-hand side of (4.5) precisely if $r+1=l$ or $r=l$ in the two cases,
$\omega_{\mathrm{in}}=\tau_{\mathrm{in}}$, $\omega_{i}=\tau_{i}$,
$i=1,\ldots,l$, and the mentioned subtrees in $\omega^{\prime}_{0}$ trivial.
Comparing with the right-hand side of (4.5) shows that we indeed have
equality. In particular, $G^{\prime}_{0}$ agrees with the graph $G_{0}$ in
(4.4). Then also
${\boldsymbol{\sigma}}^{\prime}_{0}=\hat{\boldsymbol{\sigma}}_{0}$ since each
vertex, except $v^{\prime}_{\mathrm{out}}$ in the codimension one case, maps
to the maximal cell ${\boldsymbol{\sigma}}(E_{\mathrm{in}})$. Furthermore, the
contact orders $\mathbf{u}^{\prime}_{0}$ agree with $\hat{\mathbf{u}}_{0}$ due
to the balancing condition.

Taken together this shows $\omega^{\prime}_{0}=\omega_{0}$, and hence
$(\tau_{\mathrm{in}},\tau_{1},\ldots,\tau_{l},\omega_{0})$ is indeed the only
transverse type for $\nu$. ∎

It remains to compute the multiplicity $m({\boldsymbol{\omega}})=m(\omega)$
occurring in Theorem A.4.

######  Lemma 4.16.

The multiplicity according to Definition A.2,(3) for the single element
$\omega=(\tau_{\mathrm{in}},\tau_{1},\ldots,\tau_{l},\omega_{0})$ of
$\Delta(\nu)$ in Lemma 4.15 equals

$m(\omega)=\begin{cases}k_{\tau}^{-1}k_{\tau_{\mathrm{in}}}\prod_{i=1}^{l}(dk_{\tau_{i}}),&\dim\sigma_{\mathfrak{p}}=n\\\
dk_{\tau}^{-1}k_{\tau_{\mathrm{in}}}\prod_{i=1}^{l}(dk_{\tau_{i}}),&\dim\sigma_{\mathfrak{p}}=n-1.\end{cases}$

###### Proof.

The multiplicity in question is the index of
$\operatorname{im}(\Phi^{\operatorname{gp}})$, where

$\textstyle\Phi:(\widetilde{\tau}_{\mathrm{in}})_{\mathbb{Z}}\times(\widetilde{\omega}_{0})_{\mathbb{Z}}\times\prod_{i=1}^{l}(\widetilde{\tau}_{i})_{\mathbb{Z}}\longrightarrow\Lambda\times\Lambda^{l}$

is the map describing the matching at the gluing edges, denoted
$\varepsilon_{\omega_{0}}$ in (A.7). In particular,
$\Phi^{-1}(0)=\widetilde{\tau}$ is the enlarged basic cone for the glued type
$\tau$. For the case of a trivial incoming broken line type define
$\widetilde{\tau}_{\mathrm{in}}$ as the maximal cone containing the incoming
vector and $\Phi$ on this factor as the inclusion to the first copy of
$\Lambda$.

We first treat the case $\dim\sigma_{\mathfrak{p}}=n$. In this case, by the
description of the type $\omega_{0}$, we can write

(4.6)
$(\widetilde{\omega}_{0}^{\operatorname{gp}})_{\mathbb{Z}}=\Lambda\times\mathbb{Z}^{l}\times\mathbb{Z}\times\mathbb{Z}^{l-1}=\Lambda\times\mathbb{Z}^{l}\times\mathbb{Z}^{l},$

with the $\mathbb{Z}$-factors holding the lengths

$\ell_{1},\ldots,\ell_{l},\ell_{\mathrm{in}},\ell^{\prime}_{1},\ldots,\ell^{\prime}_{l-1}$

of the legs $(E_{1},v^{\prime}_{1}),\ldots,(E_{l},v^{\prime}_{l})$,
$(E_{\mathrm{in}},v^{\prime}_{1})$ and edges
$E^{\prime}_{1},\ldots,E^{\prime}_{l-1}$, in this order, while the
$\Lambda$-factor records the image of the point on the incoming leg. For
uniformity of notation write
$E^{\prime}_{0}=(E_{\mathrm{in}},v^{\prime}_{1})$,
$u^{\prime}_{0}=u_{\mathrm{in}}$, $\ell^{\prime}_{0}=\ell_{\mathrm{in}}$ and

$u^{\prime}_{i}=\hat{\mathbf{u}}_{0}(E^{\prime}_{i})=u_{\mathrm{in}}+u_{1}+\cdots+u_{i},\quad
i=1,\ldots,l-1.$

Thus the image of $v^{\prime}_{i}\in V(\hat{G}_{0})$ under the tropical
punctured map of type $\omega_{0}$ defined by

$(m,\ell_{1},\ldots,\ell_{l},\ell^{\prime}_{0},\ldots,\ell^{\prime}_{l-1})\in\Lambda\times\mathbb{Z}^{l}\times\mathbb{Z}^{l}$

equals

$m+\ell^{\prime}_{0}u^{\prime}_{0}+\ell^{\prime}_{1}u^{\prime}_{1}+\ldots+\ell^{\prime}_{i-1}u^{\prime}_{i-1}.$

Denote further by
${\operatorname{ev}}_{\mathrm{in}}:(\widetilde{\tau}_{\mathrm{in}}^{\operatorname{gp}})_{\mathbb{Z}}\to\Lambda$
and
${\operatorname{ev}}_{i}:(\widetilde{\tau}_{i}^{\operatorname{gp}})_{\mathbb{Z}}\to\Lambda_{\mathfrak{p}}\subset\Lambda$
the linear extensions of the evaluation maps at the point on the respective
unique leg.

We first compute the cokernel $Q$ of the restriction of
$\Phi^{\operatorname{gp}}$ to

$(\widetilde{\tau}_{\mathrm{in}})^{\operatorname{gp}}\times(\\{0\\}\times\mathbb{Z}^{l})\times\prod_{i=1}^{l}(\widetilde{\tau}_{i})^{\operatorname{gp}}.$

The further restriction to the subspace spanned by the last copy of
$\mathbb{Z}$ in (4.6) and $(\widetilde{\tau}_{l})^{\operatorname{gp}}$ leads
to the map

$\varepsilon_{l}:\mathbb{Z}\times(\widetilde{\tau}_{l}^{\operatorname{gp}})_{\mathbb{Z}}\longrightarrow\Lambda,\quad(\ell^{\prime}_{l-1},h_{l})\longmapsto{\operatorname{ev}}_{l}(h_{l})-\ell^{\prime}_{l-1}\cdot
u^{\prime}_{l-1}$

to the last copy of $\Lambda$ in the codomain of $\Phi^{\operatorname{gp}}$.
Now since $|\delta(u^{\prime}_{l-1})|=d$ we can fit $\varepsilon_{l}$ into the
diagram with exact rows and injective columns

(4.7)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{\tau}_{l}^{\operatorname{gp}})_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{ev}}_{l}}$$\textstyle{\mathbb{Z}\times(\widetilde{\tau}_{l}^{\operatorname{gp}})_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{l}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
d}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{\mathfrak{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The cokernel of ${\operatorname{ev}}_{l}$ having order $k_{\tau_{l}}$, the
snake lemma implies $|\operatorname{coker}(\varepsilon_{l})|=d\cdot
k_{\tau_{l}}$. An induction on $l$ thus shows that $|Q|$ equals
$\prod_{i=1}^{l}\big{(}dk_{\tau_{i}}\big{)}$ times the order of the cokernel
of the case $l=0$, which is
$|\operatorname{coker}({\operatorname{ev}}_{\mathrm{in}})|=k_{\tau_{\mathrm{in}}}$.
We have thus shown

(4.8)
$\textstyle\big{|}Q\big{|}=k_{\tau_{\mathrm{in}}}\prod_{i=1}^{l}(dk_{\tau_{i}}).$

To finish the computation of
$\big{|}\operatorname{coker}\Phi^{\operatorname{gp}}\big{|}$, consider the
following diagram with exact rows and columns and with the obvious maps.

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{\tau}^{\operatorname{gp}}_{\mathrm{in}})_{\mathbb{Z}}\times\\{0\\}\times\mathbb{Z}^{l}\times\prod_{i=1}^{l}(\widetilde{\tau}^{\operatorname{gp}}_{i})_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\times\Lambda^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\tau}^{\operatorname{gp}}_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{(\widetilde{\tau}^{\operatorname{gp}}_{\mathrm{in}})_{\mathbb{Z}}\times(\widetilde{\omega}^{\operatorname{gp}}_{0})_{\mathbb{Z}}\times\prod_{i=1}^{l}(\widetilde{\tau}^{\operatorname{gp}}_{i})_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi^{\operatorname{gp}}}$$\textstyle{\Lambda\times\Lambda^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}(\Phi^{\operatorname{gp}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\times\mathbb{Z}^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\times\mathbb{Z}^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

The snake lemma yields the exact sequence

(4.9)
$0\longrightarrow\widetilde{\tau}^{\operatorname{gp}}_{\mathbb{Z}}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\Lambda\times\mathbb{Z}^{l}\longrightarrow
Q\longrightarrow\operatorname{coker}(\Phi^{\operatorname{gp}})\longrightarrow
0.$

Note also that
$\widetilde{\tau}^{\operatorname{gp}}_{\mathbb{Z}}=\tau^{\operatorname{gp}}_{\mathbb{Z}}\times\mathbb{Z}\times\mathbb{Z}^{l}$
with the $\mathbb{Z}$-factors from the additional points at the $l+1$ gluing
edges, and the map $\alpha$ is a product of the injection of lattices

$\bar{\alpha}:\tau^{\operatorname{gp}}_{\mathbb{Z}}\times\mathbb{Z}\longrightarrow\Lambda,\quad(h,\lambda)\longmapsto
V(h)-\lambda\cdot u_{\mathrm{in}}$

and $\operatorname{id}_{\mathbb{Z}^{l}}$. Here
$V:\tau^{\operatorname{gp}}_{\mathbb{Z}}\longrightarrow\Lambda_{\mathfrak{p}}$
is the evaluation at the outgoing vertex. Replacing $u_{\mathrm{in}}$ in this
formula by $u_{\mathrm{out}}$ yields
${\operatorname{ev}}_{\mathrm{out}}:\tau^{\operatorname{gp}}_{\mathbb{Z}}\times\mathbb{Z}\longrightarrow\Lambda$
with index of the image defining $k_{\tau}$. But
$|\delta(u_{\mathrm{out}})|=|\delta(u_{\mathrm{in}})|=d$. By considering two
diagrams similar to (4.7), it follows that

$|\operatorname{coker}(\alpha)|=|\operatorname{coker}(\bar{\alpha})|=|\operatorname{coker}({\operatorname{ev}}_{\mathrm{out}})|=k_{\tau}.$

Together with (4.8) the exact sequence (4.9) thus yields

$\textstyle
m(\omega)=\big{|}\operatorname{coker}(\Phi^{\operatorname{gp}})\big{|}=k_{\tau}^{-1}\cdot|Q|=k_{\tau}^{-1}k_{\tau_{\mathrm{in}}}\prod_{i=1}^{l}(dk_{\tau_{i}}),$

as claimed.

If $\dim\sigma_{\mathfrak{p}}=n-1$ we have an additional vertex
$v^{\prime}_{\mathrm{out}}$ mapping to $\sigma_{\mathfrak{p}}$ and an
additional edge $E^{\prime}_{l}$ connecting $v^{\prime}_{l}$ to
$v^{\prime}_{\mathrm{out}}$, with contact order $u^{\prime}_{l}$. Thus in this
case we have

$(\widetilde{\omega}_{0})^{\operatorname{gp}}_{\mathbb{Z}}=\Lambda_{\mathfrak{p}}\times\mathbb{Z}^{l}\times\mathbb{Z}\times\mathbb{Z}^{l},$

with $\Lambda_{\mathfrak{p}}$ recording the position of
$v^{\prime}_{\mathrm{out}}$ in $\mathfrak{p}$. We can now exhibit
$(\widetilde{\omega}_{0})^{\operatorname{gp}}_{\mathbb{Z}}$ as a sublattice of
$\Lambda\times\mathbb{Z}^{l}\times\mathbb{Z}^{l}$ from (4.6) by the map

$\displaystyle\Psi:\Lambda_{\mathfrak{p}}\times\mathbb{Z}^{l}\times\mathbb{Z}^{l+1}$
$\displaystyle\longrightarrow$
$\displaystyle\Lambda\times\mathbb{Z}^{l}\times\mathbb{Z}^{l},$
$\displaystyle(m,\ell_{1},\ldots,\ell_{l},\ell^{\prime}_{0},\ldots,\ell^{\prime}_{l})$
$\displaystyle\longmapsto$
$\displaystyle\textstyle\big{(}m-\sum_{i=0}^{l}\ell^{\prime}_{i}u^{\prime}_{i},\ell_{1},\ldots,\ell_{l},\ell^{\prime}_{0},..,\ell^{\prime}_{l-1}\big{)}.$

Here $u^{\prime}_{0}=u_{\mathrm{in}}$, $\ell^{\prime}_{0}=\ell_{\mathrm{in}}$
are defined analogously to the codimension zero case. Since the images of
$u^{\prime}_{i}$ in $\Lambda/\Lambda_{\mathfrak{p}}\simeq\mathbb{Z}$ generate
$d\mathbb{Z}$, we see that $\Psi$ is an inclusion of lattices of index $d$.
The stated formula in codimension one now follows by observing that
$\Phi^{\operatorname{gp}}$ is the composition of $\Phi^{\operatorname{gp}}$ in
the codimension zero case with
$\operatorname{id}_{(\tilde{\tau}_{\mathrm{in}})_{\mathbb{Z}}}\times\Psi\times\operatorname{id}_{\prod_{i}(\tilde{\tau}_{i})_{\mathbb{Z}}}$.
∎

Finally we need to compute one simple punctured invariant.

######  Lemma 4.17.

$\deg\big{[}\mathscr{M}(X,{\boldsymbol{\omega}}_{0})\big{|}^{\mathrm{virt}}=\begin{cases}1,&\operatorname{codim}\sigma_{\mathfrak{p}}=n\\\
1/d,&\operatorname{codim}\sigma_{\mathfrak{p}}=n-1.\end{cases}$

###### Proof.

The graph $\hat{G}_{0}$ given by ${\boldsymbol{\omega}}_{0}$ is a chain with
$l$ or $l+1$ trivalent vertices of genus $0$, depending if
$\dim\sigma_{\mathfrak{p}}=n$ or $\dim\sigma_{\mathfrak{p}}=n-1$. In any case,
$l$ vertices are marked by $\sigma_{\mathrm{in}}$, the maximal cell containing
the image of $E_{\mathrm{in}}$. Thus if $\dim\sigma_{\mathfrak{p}}=n$, the
underlying stable maps in $\mathscr{M}(X,{\boldsymbol{\omega}}_{0})$ are
contracted trees of $l$ copies of $\mathbb{P}^{1}$ with three special points
each, with $l-1$ pairs of such special points identified at the nodes. With
the labelling of the $l+2$ remaining points, this is a rigid curve with
trivial automorphism group. Moreover, the pull-back of $\Theta_{X}$ is
trivial. Hence $\mathscr{M}(X,{\boldsymbol{\omega}}_{0})$ is a reduced point,
giving the result in the case $\dim\Lambda_{\mathfrak{p}}=n$.

In the codimension one case, $\mathfrak{p}$ is contained in the $(n-1)$-cell
$\rho=\sigma_{\mathfrak{p}}$ with stratum $X_{\rho}\subset X$ isomorphic to
$\mathbb{P}^{1}$ and $\mathcal{M}_{X}|_{X_{\rho}}$ admitting a lift of the
torus action. In particular, it holds
$\Theta_{X}|_{X_{\rho}}\simeq\mathcal{O}_{\mathbb{P}^{1}}^{n}$. The incoming
and outgoing legs of $G_{0}$ are labelled by the maximal cells
$\sigma_{\mathrm{in}},{\boldsymbol{\sigma}}(L_{\mathrm{out}})\in\Sigma(X)$
containing $\rho$. By the description of ${\boldsymbol{\omega}}_{0}$, only the
outgoing vertex maps to $\rho$, all others map to $\sigma_{\mathrm{in}}$. Thus
the stable maps underlying objects in
$\mathscr{M}(X,{\boldsymbol{\omega}}_{0})$ are contracted trees of $l$ copies
of $\mathbb{P}^{1}$ with three special points each as before, joined at the
outgoing point with another copy of $\mathbb{P}^{1}$ mapping to
$X_{\rho}\simeq\mathbb{P}^{1}$. The tropical picture shows that the map
$\mathbb{P}^{1}\to\mathbb{P}^{1}$ is totally branched at the two zero-
dimensional strata
$X_{\sigma_{\mathrm{in}}},X_{{\boldsymbol{\sigma}}(L_{\mathrm{out}})}\subset
X_{\rho}$ with branching order $d=|\delta(u_{\mathrm{in}})|$ the constant from
(4.3). Hence the associated curve class is $d\cdot[X_{\rho}]$ (as follows also
from Lemma 2.3). Thus there is a unique underlying stable map over
$\underline{W}=\operatorname{Spec}\Bbbk$. Arguing similarly as in [GS7, Claim
3.22], there is then a unique enhancement, up to isomorphism, of this
underlying stable map to a basic punctured map $f:C^{\circ}/W\rightarrow X$ of
type ${\boldsymbol{\omega}}_{0}$ with
$\underline{W}=\operatorname{Spec}\Bbbk$. Moreover, by the triviality of
$\Theta_{X}|_{X_{\rho}}$ this is again a logarithmically unobstructed and
rigid situation, now with automorphism group $\mu_{d}$. Hence
$\mathscr{M}(X,{\boldsymbol{\omega}}_{0})$ is the quotient of a reduced point
by $\mu_{d}$, proving the stated result. ∎

###### Proof of (4.2).

For applying the gluing formula we now use the labelling with pairwise
different $\tau_{i}$ and multiplicities $\mu_{i}$. Let

$\delta:\mathscr{M}(X,{\boldsymbol{\tau}})\longrightarrow\mathscr{M}(X,{\boldsymbol{\tau}}_{\mathrm{in}})\times\mathscr{M}(X,{\boldsymbol{\tau}}_{0})\times\prod_{i=1}^{l}\mathscr{M}(X,{\boldsymbol{\tau}}_{i})^{\mu_{i}}$

be the splitting map from (A.1) for ${\boldsymbol{\tau}}$. Applying Theorem
A.4 with the general displacement vector $\nu$ from Lemma 4.15 and noting that
$\operatorname{Aut}({\boldsymbol{\omega}}/{\boldsymbol{\tau}})=\\{1\\}$ yields

(4.10)
$\delta_{*}\big{[}\mathscr{M}(X,{\boldsymbol{\tau}})\big{]}^{\mathrm{virt}}=m(\omega)\cdot\big{[}\mathscr{M}(X,{\boldsymbol{\tau}}_{\mathrm{in}})^{\mathrm{virt}}\big{]}\times(j_{{\boldsymbol{\omega}}_{0}})_{*}[\mathscr{M}(X,{\boldsymbol{\omega}}_{0})]^{\mathrm{virt}}\times\prod_{i=1}^{l}\big{[}\mathscr{M}(X,{\boldsymbol{\tau}}_{i})^{\mu_{i}}\big{]}^{\mathrm{virt}}.$

Next observe

$\operatorname{Aut}({\boldsymbol{\tau}})\simeq\operatorname{Aut}({\boldsymbol{\tau}}_{\mathrm{in}})\times\prod_{i=1}^{l}\big{(}\operatorname{Aut}({\boldsymbol{\tau}}_{i})\rtimes
S_{\mu_{i}}),$

with the symmetric group $S_{\mu_{i}}$ acting by permutation of the $\mu_{i}$
copies of ${\boldsymbol{\tau}}_{i}$ attached at $v$. Taking degrees in (4.10)
and applying Lemma 4.17, plugging in $m(\omega)$ from Lemma 4.16, multiplying
by $k_{\tau}$, and dividing by $|\operatorname{Aut}({\boldsymbol{\tau}})|$ now
leads to

$\displaystyle k_{\tau}N_{\boldsymbol{\tau}}$
$\displaystyle=k_{\tau_{\mathrm{in}}}\cdot{\textstyle\prod_{i=1}^{l}(dk_{\tau_{i}})^{\mu_{i}}}\cdot\frac{\deg[\mathscr{M}(X,{\boldsymbol{\tau}}_{\mathrm{in}})]^{\mathrm{virt}}}{|\operatorname{Aut}({\boldsymbol{\tau}}_{\mathrm{in}})|}\cdot\prod_{i=1}^{l}\frac{\deg[\mathscr{M}(X,{\boldsymbol{\tau}}_{i})^{\mu_{i}}]^{\mathrm{virt}}}{|\operatorname{Aut}({\boldsymbol{\tau}}_{i})\rtimes
S_{\mu_{i}}|}$
$\displaystyle=k_{\tau_{\mathrm{in}}}N_{\tau_{\mathrm{in}}}\cdot\prod_{i=1}^{l}\frac{(dk_{\tau_{i}}W_{{\boldsymbol{\tau}}_{i}})^{\mu_{i}}}{\mu_{i}!}.$

The total curve class and contact orders are computed from the decorations as

$A=A_{\mathrm{in}}+d[X_{\sigma_{\mathfrak{p}}}]+\sum_{i=1}^{l}\mu_{i}A_{i},$

and

$u_{\mathrm{out}}=u_{\mathrm{in}}+\sum_{i=1}^{l}\mu_{i}u_{i}.$

The last three equalities together imply the claimed scattering equation
(4.2). ∎

##  5\. Consistency of the canonical wall structure

We now prove the main result of the paper, namely that
$\mathscr{S}_{{\mathrm{can}}}$ is a consistent wall structure. The actual
definition of consistency, while summarized in [GHS, Def. 3.9], is a bit
complicated. However, the main point of the definition is to enable the
construction of so-called theta functions. In other words, consistency
guarantees that the local description $\vartheta_{p}(x)$ for theta functions
patch together to give global functions on a scheme constructed from the wall
structure.

Here, we proceed in the opposite direction. We will first show that the
$\vartheta_{p}(x)$ satisfy the necessary patching criteria, and then show this
patching implies consistency as defined in [GHS, Def. 3.9].

###  5.1. Patching of theta functions

As the discussion from now on largely follows the notions of [GHS], we now
adopt the conventions and notation of that paper with regards to wall
structures. In particular, we now work with a wall structure $\mathscr{S}$
satisfying precisely the properties of [GHS, Def. 2.11,2], by assuming that
the walls of $\mathscr{S}$ comprise the codimension one cells of a polyhedral
cone refinement $\mathscr{P}_{\mathscr{S}}$ of $\mathscr{P}$. As mentioned in
Remark 3.4, this may be achieved for $\mathscr{S}_{{\mathrm{can}}}$ by
subdividing walls, combining walls with the same support, and adding walls
with attached function $1$ if necessary. In particular, as in [GHS], we use
the notation $\mathfrak{u}$ for a maximal cell of $\mathscr{P}_{\mathscr{S}}$
and $\mathfrak{b}$ for a codimension one cell of $\mathscr{P}_{\mathscr{S}}$
contained in $\rho\in\mathscr{P}^{[n-1]}$. We call $\mathfrak{u}$ a _chamber_
and $\mathfrak{b}$ a _slab_ or _codimension one wall_ , while we call those
walls intersecting the interior of a maximal cell a _codimension zero wall_. A
_joint_ $\mathfrak{j}$ is a codimension two cell in
$\mathscr{P}_{\mathscr{S}}$, and the codimension of $\mathfrak{j}$ is the
codimension of the smallest cell of $\mathscr{P}$ containing $\mathfrak{j}$.

Recalling further notation from [GHS], if $\mathfrak{u}$ is a chamber
contained in $\sigma\subseteq\mathscr{P}_{{\operatorname{max}}}$, we set

$R_{\mathfrak{u}}=R_{\sigma}:=\Bbbk[\mathcal{P}_{x}^{+}]/I_{x}=(\Bbbk[Q]/I)[\Lambda_{\sigma}]$

for any point $x\in\operatorname{Int}(\mathfrak{u})$.

Given a codimension zero wall $\mathfrak{p}$ separating chambers
$\mathfrak{u},\mathfrak{u}^{\prime}$, we obtain a $\Bbbk[Q]/I$-algebra
homomorphism

(5.1) $\theta_{\mathfrak{p}}:R_{\mathfrak{u}}\rightarrow
R_{\mathfrak{u}^{\prime}},\quad z^{m}\mapsto f_{\mathfrak{p}}^{\langle
n_{\mathfrak{p}},m\rangle}z^{m},$

where $n_{\mathfrak{p}}\in\operatorname{Hom}(\Lambda_{\sigma},\mathbb{Z})$ is
the primitive normal vector to $\mathfrak{p}$ positive on $\mathfrak{u}$.
Since $f_{\mathfrak{p}}\equiv 1\bmod\mathfrak{m}$ and $\sqrt{I}=\mathfrak{m}$,
$f_{\mathfrak{p}}$ is in fact invertible so that $\theta_{\mathfrak{p}}$ is an
automorphism.

For a slab $\mathfrak{b}$, after choosing $\xi$ as chosen for the isomorphism
(3.3), we define

$R_{\mathfrak{b}}:=(\Bbbk[Q]/I)[\Lambda_{\rho}][Z_{+},Z_{-}]/(Z_{+}Z_{-}-f_{\mathfrak{b}}t^{\kappa_{\rho}}),$

where $f_{\mathfrak{b}}$ is the function attached to $\mathfrak{b}$. If
$\mathfrak{u},\mathfrak{u}^{\prime}$ are the adjacent chambers to
$\mathfrak{b}$, there are natural localization maps at $Z_{+}$ and $Z_{-}$
respectively

(5.2) $\chi_{\mathfrak{b},\mathfrak{u}}:R_{\mathfrak{b}}\rightarrow
R_{\mathfrak{u}},\quad\chi_{\mathfrak{b},\mathfrak{u}^{\prime}}:R_{\mathfrak{b}}\rightarrow
R_{\mathfrak{u}^{\prime}},$

given by

$\chi_{\mathfrak{b},\mathfrak{u}}(t^{A}z^{m}Z_{+}^{a}Z_{-}^{b})=t^{A+b\kappa_{\rho}}z^{m+(a-b)\xi}f_{\mathfrak{b}}^{b},\quad\chi_{\mathfrak{b},\mathfrak{u}^{\prime}}(t^{A}z^{m}Z_{+}^{a}Z_{-}^{b})=t^{A+a\kappa_{\rho}}z^{m+(a-b)\xi}f_{\mathfrak{b}}^{a},$

where $A\in Q$ and $m\in\Lambda_{\rho}$. Note that these maps differ from
those given in (3.4), (3.5) only in the powers of $f_{\mathfrak{b}}$ appearing
here.

We may now make precise the notion that the $\vartheta_{p}(x)$ patch.

######  Theorem 5.1.

For the canonical wall structure $\mathscr{S}_{{\mathrm{can}}}$ and any $p\in
B(\mathbb{Z})\setminus\\{0\\}$, we have

  1. (1)

If $x,x^{\prime}\in\operatorname{Int}(\mathfrak{u})$ for a chamber
$\mathfrak{u}$, then

$\vartheta_{p}(x)=\vartheta_{p}(x^{\prime}).$

  2. (2)

If $\mathfrak{p}$ is a codimension zero wall separating chambers
$\mathfrak{u},\mathfrak{u}^{\prime}$, and
$x\in\operatorname{Int}(\mathfrak{u})$,
$x^{\prime}\in\operatorname{Int}(\mathfrak{u}^{\prime})$, then

$\vartheta_{p}(x^{\prime})=\theta_{\mathfrak{p}}\big{(}\vartheta_{p}(x)\big{)}.$

  3. (3)

If $\mathfrak{b}$ is a slab with adjacent chambers
$\mathfrak{u},\mathfrak{u}^{\prime}$, $x\in\operatorname{Int}(\mathfrak{u})$,
$x^{\prime}\in\operatorname{Int}(\mathfrak{u}^{\prime})$, then there exists a
unique $\vartheta_{p}(\mathfrak{b})\in R_{\mathfrak{b}}$ such that

$\chi_{\mathfrak{b},\mathfrak{u}}(\vartheta_{p}(\mathfrak{b}))=\vartheta_{p}(x),\quad\chi_{\mathfrak{b},\mathfrak{u}^{\prime}}(\vartheta_{p}(\mathfrak{b}))=\vartheta_{p}(x^{\prime}).$

###### Proof.

Step 1. _Establishing the setup._ We fix $p\in B(\mathbb{Z})\setminus\\{0\\}$
once and for all. We work throughout with $\vartheta^{\log}_{p}(x)$ using
Theorem 4.5. Let $\mathscr{B}$ be the set of isomorphism classes of decorated
broken line types given as

$\mathscr{B}:=\\{{\boldsymbol{\tau}}=(\tau,\mathbf{A})\,|\,\hbox{$p_{\tau}=p$,
the total curve class of $\mathbf{A}$ lies in $Q\setminus I$ and
$N_{{\boldsymbol{\tau}}}\not=0$}\\}.$

$\mathscr{B}$ is a finite set by Lemma 3.22. Thus we may choose a polyhedral
cone complex $\mathscr{P}_{\mathscr{B}}$ refining $\mathscr{P}_{\mathscr{S}}$
with the property that for each ${\boldsymbol{\tau}}\in\mathscr{B}$,
$h({\boldsymbol{\tau}}_{\mathrm{out}})$ is a union of maximal cells
$\mathfrak{u}\in\mathscr{P}_{\mathscr{B}}$. At the same time, we refine the
wall structure $\mathscr{S}$ so that we may take
$\mathscr{P}_{\mathscr{S}}=\mathscr{P}_{\mathscr{B}}$.

It is immediate from the definition of the log theta functions that if
$\mathfrak{u}\in\mathscr{P}_{\mathscr{B}}$ is a maximal cone then
$\vartheta^{\log}_{p}(x)=\vartheta^{\log}_{p}(x^{\prime})$ for any
$x,x^{\prime}\in\operatorname{Int}(\mathfrak{u})$. Thus, it is now sufficient
to verify that if
$\mathfrak{u},\mathfrak{u}^{\prime}\in\mathscr{P}_{\mathscr{B}}$ are two
chambers separated by a codimension one cell $\mathfrak{p}$, then statements
(2) or (3) of the theorem hold, depending on whether $\mathfrak{p}$ is a
codimension zero wall or a slab.

Step 2. _Broken lines transversal to $\mathfrak{p}$._ For
$y\in\operatorname{Int}(\mathfrak{p})$, let
$n_{\mathfrak{p}}\in\check{\Lambda}_{y}$ be a primitive normal vector to
$\mathfrak{p}$, positive on $\mathfrak{u}$. We may then decompose, for
$x\in\mathfrak{u}$, $x^{\prime}\in\mathfrak{u}^{\prime}$,

$\displaystyle\vartheta^{\log}_{p}(x)={}$
$\displaystyle\vartheta_{+}+\vartheta_{-}+\vartheta_{0}$
$\displaystyle\vartheta^{\log}_{p}(x^{\prime})={}$
$\displaystyle\vartheta^{\prime}_{+}+\vartheta^{\prime}_{-}+\vartheta^{\prime}_{0}$

where $\vartheta_{+}$ (resp. $\vartheta_{-}$, $\vartheta_{0}$) consists of a
sum of those monomials $at^{A}z^{m}$ appearing in $\vartheta^{\log}_{p}(x)$
with $\langle n_{\mathfrak{p}},m\rangle>0$ (resp. $\langle
n_{\mathfrak{p}},m\rangle<0$ and $\langle n_{\mathfrak{p}},m\rangle=0$). The
terms $\vartheta^{\prime}_{\pm}$, $\vartheta^{\prime}_{0}$ are defined
analogously.

Using the broken line description of these theta functions, the now standard
argument of the proof of [Gr2, Thm. 4.12] or [CPS, Lem. 4.9], shows
immediately that if $\mathfrak{p}$ is a codimension zero wall, then

(5.3)
$\theta_{\mathfrak{p}}(\vartheta_{+})=\vartheta_{+}^{\prime},\quad\theta_{\mathfrak{p}}(\vartheta_{-})=\vartheta_{-}^{\prime}.$

We briefly recall the argument. Given a decorated broken line contributing to
$\vartheta_{+}$ with endpoint in $\mathfrak{u}$ very close to $\mathfrak{p}$,
we may perturb the broken line by moving its endpoint to a nearby point in
$\mathfrak{p}$. We may then add an additional line segment (or simply extend
the final line segment through $\mathfrak{p}$) following the definition of a
decorated broken line to obtain a decorated broken line with endpoint nearby
in $\mathfrak{u}^{\prime}$. If this is done in all possible ways, it then
follows from the propagation rule for monomials (4.1) that
$\theta_{\mathfrak{p}}(\vartheta_{+})=\vartheta_{+}^{\prime}$. Finally, any
monomial in $\vartheta_{0}$ or $\vartheta_{0}^{\prime}$ has exponent tangent
to $\mathfrak{p}$, and hence is left invariant under transport from
$\mathfrak{u}$ to $\mathfrak{u}^{\prime}$. Thus in this case, it is sufficient
to show that $\vartheta_{0}=\vartheta_{0}^{\prime}$ under transport
$\mathfrak{t}_{\sigma,\sigma^{\prime}}$.

If $\mathfrak{p}=\mathfrak{b}$ is a slab contained in $\rho$,
$\mathfrak{u}\subseteq\sigma$,
$\mathfrak{u}^{\prime}\subseteq\sigma^{\prime}$, then we may write

$\vartheta_{+}=\sum_{i}a_{i}z^{m_{i}+\alpha_{i}\xi},\quad\vartheta_{-}^{\prime}=\sum_{i}a_{i}^{\prime}z^{m_{i}^{\prime}-\alpha_{i}^{\prime}\xi}$

with $a_{i},a_{i}^{\prime}\in\Bbbk[Q]/I$,
$m_{i},m_{i}^{\prime}\in\Lambda_{\rho}$ and
$\alpha_{i},\alpha_{i}^{\prime}>0$. We may then define

$\vartheta_{+}(\mathfrak{b}):=\sum_{i}a_{i}z^{m_{i}}Z_{+}^{\alpha_{i}},\quad\quad\vartheta_{-}(\mathfrak{b}):=\sum_{i}a^{\prime}_{i}z^{m^{\prime}_{i}}Z_{-}^{\alpha^{\prime}_{i}}.$

By construction,
$\chi_{\mathfrak{b},\mathfrak{u}}(\vartheta_{+}(\mathfrak{b}))=\vartheta_{+}$
and
$\chi_{\mathfrak{b},\mathfrak{u}^{\prime}}(\vartheta_{-}(\mathfrak{b}))=\vartheta_{-}^{\prime}$.
On the other hand, it follows from exactly the same argument as outlined above
and the formulas for $\chi_{\mathfrak{b},\mathfrak{u}^{\prime}}(Z_{+})$ and
$\chi_{\mathfrak{b},\mathfrak{u}}(Z_{-})$ of (5.2) that

(5.4)
$\chi_{\mathfrak{b},\mathfrak{u}^{\prime}}(\vartheta_{+}(\mathfrak{b}))=\vartheta^{\prime}_{+},\quad\quad\chi_{\mathfrak{b},\mathfrak{u}}(\vartheta_{-}(\mathfrak{b}))=\vartheta_{-}.$

As before, any monomial in $\vartheta_{0}$ or $\vartheta_{0}^{\prime}$ is left
invariant under transport from $\mathfrak{u}$ to $\mathfrak{u}^{\prime}$. Thus
again it is sufficient to show that $\vartheta_{0}=\vartheta_{0}^{\prime}$
under transport $\mathfrak{t}_{\sigma,\sigma^{\prime}}$, as the expression

$\vartheta_{p}(\mathfrak{b}):=\vartheta_{+}(\mathfrak{b})+\vartheta_{-}(\mathfrak{b})+\vartheta_{0}$

then satisfies the desired properties.

Step 3. _Reduction to a non-virtual linear equivalence relation on moduli
spaces._ Let $\mathscr{B}_{\mathfrak{u}}$ (resp.
$\mathscr{B}_{\mathfrak{u}^{\prime}}$) denote the collection of isomorphism
classes of decorated broken line types ${\boldsymbol{\tau}}$ contributing to
$\vartheta_{0}$ (resp. $\vartheta_{0}^{\prime}$) such that
$\mathfrak{u}\subseteq
h(\tau_{\mathrm{out}}),\mathfrak{u}^{\prime}\not\subseteq
h(\tau_{\mathrm{out}})$ (resp. $\mathfrak{u}^{\prime}\subseteq
h(\tau_{\mathrm{out}}),\mathfrak{u}\not\subseteq h(\tau_{\mathrm{out}})$). Any
other broken line type ${\boldsymbol{\tau}}$ contributing to either
$\vartheta_{0}$ or $\vartheta_{0}^{\prime}$ must then have
$\mathfrak{u}\cup\mathfrak{u}^{\prime}\subseteq h(\tau_{\mathrm{out}})$, and
hence contributes equally to both $\vartheta_{0}$ and
$\vartheta_{0}^{\prime}$. Hence it is sufficient to show that

$\sum_{{\boldsymbol{\tau}}\in\mathscr{B}_{\mathfrak{u}}}k_{\tau}N_{{\boldsymbol{\tau}}}t^{A}z^{-u_{\tau}}=\sum_{{\boldsymbol{\tau}}\in\mathscr{B}_{\mathfrak{u}^{\prime}}}k_{\tau}N_{{\boldsymbol{\tau}}}t^{A}z^{-u_{\tau}}.$

Let $\mathscr{D}$ be the set of isomorphism classes of degenerate decorated
broken line types (see Definition 3.19) ${\boldsymbol{\tau}}^{\prime}$ for
which there exists a ${\boldsymbol{\tau}}\in\mathscr{B}$ and a contraction
morphism ${\boldsymbol{\tau}}\rightarrow{\boldsymbol{\tau}}^{\prime}$. Recall
that for a degenerate broken line type $\tau^{\prime}$,
$\dim\tau^{\prime}=n-2$ and $\dim
h_{\tau^{\prime}}(\tau^{\prime}_{\mathrm{out}})=n-1$. Let

$\mathscr{D}_{\mathfrak{p}}:=\\{{\boldsymbol{\tau}}^{\prime}\in\mathscr{D}\,|\,\mathfrak{p}\subseteq
h_{\tau^{\prime}}(\tau^{\prime}_{\mathrm{out}})\\}.$

Note by the defining assumption on $\mathscr{P}_{\mathscr{B}}$, if
${\boldsymbol{\tau}}\in\mathscr{B}\cup\mathscr{D}$ is a broken line type or a
degenerate broken line type and $\dim\mathfrak{p}\cap
h(\tau_{\mathrm{out}})=n-1$, then $\mathfrak{p}\subseteq
h(\tau_{\mathrm{out}})$.

For any
${\boldsymbol{\tau}}\in\mathscr{B}_{\mathfrak{u}}\cup\mathscr{B}_{\mathfrak{u}^{\prime}}$,
since $u_{\tau}$ is tangent to $\mathfrak{p}$ and $\mathfrak{p}$ is contained
in a codimension one face of $h(\tau_{\mathrm{out}})$, necessarily there is a
unique choice of contraction morphism of decorated types
$\phi:{\boldsymbol{\tau}}\rightarrow{\boldsymbol{\tau}}^{\prime}$ with
${\boldsymbol{\tau}}^{\prime}\in\mathscr{D}_{\mathfrak{p}}$. Note further the
isomorphism class of ${\boldsymbol{\tau}}^{\prime}$ only depends on the
isomorphism class of ${\boldsymbol{\tau}}$. This gives maps
$\Psi:\mathscr{B}_{\mathfrak{u}}\rightarrow\mathscr{D}_{\mathfrak{p}}$,
$\Psi^{\prime}:\mathscr{B}_{\mathfrak{u}^{\prime}}\rightarrow\mathscr{D}_{\mathfrak{p}}$
taking ${\boldsymbol{\tau}}$ to the corresponding degenerate decorated broken
line type ${\boldsymbol{\tau}}^{\prime}$. It is now sufficient to show that
for any ${\boldsymbol{\tau}}^{\prime}\in\mathscr{D}_{\mathfrak{p}}$, we have

(5.5)
$\sum_{{\boldsymbol{\tau}}\in\Psi^{-1}({\boldsymbol{\tau}}^{\prime})}k_{\tau}N_{{\boldsymbol{\tau}}}=\sum_{{\boldsymbol{\tau}}\in(\Psi^{\prime})^{-1}({\boldsymbol{\tau}}^{\prime})}k_{\tau}N_{{\boldsymbol{\tau}}}.$

To prove this, we now fix
${\boldsymbol{\tau}}^{\prime}\in\mathscr{D}_{\mathfrak{p}}$ with underlying
type $\tau^{\prime}$. We consider the moduli space

$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime}):=\mathfrak{M}(\mathcal{X},\tau^{\prime})\times_{\underline{\mathcal{X}}}\underline{X},$

where the morphism
$\mathfrak{M}(\mathcal{X},\tau^{\prime})\rightarrow\underline{\mathcal{X}}$ is
given by schematic evaluation at the section of the universal curve
corresponding to $L_{\mathrm{out}}$. For any contraction morphism
$\tau\rightarrow\tau^{\prime}$, we obtain by [ACGS2, Prop. 5.19] a Cartesian
diagram

(5.6) $\textstyle{\coprod_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}\mathscr{M}(X,{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{\tau}}$$\scriptstyle{\iota_{\tau}^{\prime}}$$\textstyle{\mathscr{M}(X,{\boldsymbol{\tau}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{\tau^{\prime}}}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\tau}}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})}$

Here the disjoint union is over all decorations
${\boldsymbol{\tau}}=(\tau,{\bf A)}$ of $\tau$ such that the contraction
morphism $\tau\rightarrow\tau^{\prime}$ induces a contraction morphism
${\boldsymbol{\tau}}\rightarrow{\boldsymbol{\tau}}^{\prime}$. The maps
$\iota_{\tau},\iota^{\prime}_{\tau}$ are induced by the contraction morphisms
$\tau\rightarrow\tau^{\prime}$ and
${\boldsymbol{\tau}}\rightarrow{\boldsymbol{\tau}}^{\prime}$. Note that

(5.7)
$\displaystyle\begin{split}\sum_{{\boldsymbol{\tau}}\in\Psi^{-1}({\boldsymbol{\tau}}^{\prime})}k_{\tau}N_{{\boldsymbol{\tau}}}={}&\sum_{{\boldsymbol{\tau}}\in\Psi^{-1}({\boldsymbol{\tau}}^{\prime})}k_{\tau}{\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}({\boldsymbol{\tau}})|}\\\
={}&\sum_{\tau\rightarrow\tau^{\prime}\atop{\boldsymbol{\tau}}=(\tau,{\bf
A})}k_{\tau}{\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}({\boldsymbol{\tau}})||\operatorname{Aut}(\tau/\tau^{\prime})||\operatorname{Aut}({\boldsymbol{\tau}}/{\boldsymbol{\tau}}^{\prime})|^{-1}}\\\
={}&\sum_{\tau\rightarrow\tau^{\prime}\atop{\boldsymbol{\tau}}=(\tau,{\bf
A})}k_{\tau}{\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}\over|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime})||\operatorname{Aut}(\tau/\tau^{\prime})|}.\end{split}$

The last two sums are over (1) isomorphism classes of types $\tau$ which are
the underlying type of some element of
$\Psi^{-1}({\boldsymbol{\tau}}^{\prime})$ and (2) decorations
${\boldsymbol{\tau}}=(\tau,{\bf A})$ of $\tau$ yielding an induced contraction
map ${\boldsymbol{\tau}}\rightarrow{\boldsymbol{\tau}}^{\prime}$. The factor
$|\operatorname{Aut}(\tau/\tau^{\prime})||\operatorname{Aut}({\boldsymbol{\tau}}/{\boldsymbol{\tau}}^{\prime})|^{-1}$
arises in the third summation because we are now summing over multiple
representatives for an isomorphism class in
$\Psi^{-1}({\boldsymbol{\tau}}^{\prime})$. Indeed, giving a fixed
representative $\tau$ equipped with its unique contraction morphism
$\tau\rightarrow\tau^{\prime}$, $\operatorname{Aut}(\tau/\tau^{\prime})$ now
acts on the set of decorations ${\boldsymbol{\tau}}=(\tau,\mathbf{A})$ of
$\tau$ such that $\tau\rightarrow\tau^{\prime}$ induces a contraction morphism
${\boldsymbol{\tau}}\rightarrow{\boldsymbol{\tau}}^{\prime}$. The stabilizer
of this action is the subgroup
$\operatorname{Aut}({\boldsymbol{\tau}}/{\boldsymbol{\tau}}^{\prime})\subseteq\operatorname{Aut}(\tau/\tau^{\prime})$,
and hence each ${\boldsymbol{\tau}}\in\Psi^{-1}(\tau^{\prime})$ appears
$|\operatorname{Aut}(\tau/\tau^{\prime})|/|\operatorname{Aut}({\boldsymbol{\tau}}/{\boldsymbol{\tau}}^{\prime})|$
times in the last two sums. For the last equality, we use
$|\operatorname{Aut}({\boldsymbol{\tau}})|=|\operatorname{Aut}({\boldsymbol{\tau}}/{\boldsymbol{\tau}}^{\prime})||\operatorname{Aut}({\boldsymbol{\tau}}^{\prime})|$.

Using (5.6), if we fix $\tau\rightarrow\tau^{\prime}$, we may write

$\sum_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}=\deg(\varepsilon_{\tau})^{!}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)].$

Thus, with all sums below being over isomorphism classes of broken line types
$\tau$ which are the underlying type of elements of
$\Psi^{-1}({\boldsymbol{\tau}}^{\prime})$, the quantity of (5.7) may now be
written as

$\displaystyle\sum_{\tau}k_{\tau}{\deg\iota^{\prime}_{\tau,*}(\varepsilon_{\tau})^{!}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)]\over|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime})||\operatorname{Aut}(\tau/\tau^{\prime})|}={}$
$\displaystyle\sum_{\tau}k_{\tau}{\deg\varepsilon_{\tau^{\prime}}^{!}\iota_{\tau,*}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)]\over|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime})||\operatorname{Aut}(\tau/\tau^{\prime})|}$
$\displaystyle={}$
$\displaystyle\sum_{\tau}k_{\tau}{|\operatorname{Aut}(\tau/\tau^{\prime})|\deg\varepsilon_{\tau^{\prime}}^{!}[\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})]\over|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime})||\operatorname{Aut}(\tau/\tau^{\prime})|}$
$\displaystyle={}$
$\displaystyle\sum_{\tau}k_{\tau}{\deg\varepsilon_{\tau^{\prime}}^{!}[\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})]\over|\operatorname{Aut}({\boldsymbol{\tau}}^{\prime})|}.$

Here the first equality follows from [Mn, Thm. 4.1], while the second equality
follows from the morphism
$\iota_{\tau}:\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)\rightarrow\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})$
being of degree $|\operatorname{Aut}(\tau/\tau^{\prime})|$. Thus it will be
sufficient to show the following relation in the codimension one Chow group of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$

(5.8) $\sum_{\tau\rightarrow\tau^{\prime}\atop\mathfrak{u}\subseteq
h_{\tau}(\tau_{\mathrm{out}})}k_{\tau}[\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})]=\sum_{\tau\rightarrow\tau^{\prime}\atop\mathfrak{u}^{\prime}\subseteq
h_{\tau}(\tau_{\mathrm{out}})}k_{\tau}[\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})],$

where the sums are over isomorphism classes of broken line types $\tau$ with a
contraction morphism $\tau\rightarrow\tau^{\prime}$ and with the stated
inclusion.

Step 4. _Evaluation maps and local structure of moduli spaces._ We continue
with ${\boldsymbol{\tau}}^{\prime}\in\mathscr{D}_{\mathfrak{p}}$ fixed as in
the previous step. If
$\mathfrak{C}^{\circ}\rightarrow\mathfrak{M}(\mathcal{X},\tau^{\prime})$ is
the universal punctured curve over the moduli space
$\mathfrak{M}(\mathcal{X},\tau^{\prime})$, with section $x_{\mathrm{out}}$
corresponding to the leg $L_{\mathrm{out}}$, we write

$\widetilde{\mathfrak{M}}(\mathcal{X},\tau^{\prime}):=(\underline{\mathfrak{M}(\mathcal{X},\tau^{\prime})},x_{\mathrm{out}}^{*}\mathcal{M}_{\mathfrak{C}^{\circ}})^{{\operatorname{sat}}},$

the saturation of the log structure of $\mathfrak{C}^{\circ}$ pulled back via
$x_{\mathrm{out}}$: see [ACGS2, §5.2]. Denote the reduction by

$\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime}):=\widetilde{\mathfrak{M}}(\mathcal{X},\tau^{\prime})_{{\operatorname{red}}}.$

By [ACGS2, Prop. 5.5], the composition
$\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})\rightarrow\widetilde{\mathfrak{M}}(\mathcal{X},\tau^{\prime})\rightarrow\mathfrak{M}(\mathcal{X},\tau^{\prime})$
induces an isomorphism on underlying stacks, as
$\mathfrak{M}(\mathcal{X},\tau^{\prime})$ is already reduced by [ACGS2, Prop.
3.28].

By composing the universal morphism
$f:\mathfrak{C}^{\circ}\rightarrow\mathcal{X}$ with the section
$x_{\mathrm{out}}$, we obtain an evaluation map
${\operatorname{ev}}:\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})\rightarrow\mathcal{X}$.
As the former space is reduced, this evaluation map factors through the
stratum $\mathcal{X}_{\sigma}$ of $\mathcal{X}$ where $\sigma\in\mathscr{P}$
is the minimal cell containing $\mathfrak{p}$.

We recall from [ACGS2, Def. 3.22] that
$\mathfrak{M}(\mathcal{X},\tau^{\prime})$ carries an idealized structure given
by a coherent sheaf of monoid ideals
$\mathcal{I}\subseteq\mathcal{M}_{\mathfrak{M}(\mathcal{X},\tau^{\prime})}$.
This sheaf may be described by giving the stalks of the image sheaf
$\overline{\mathcal{I}}$ of $\mathcal{I}$ in the ghost sheaf
$\overline{\mathcal{M}}_{\mathfrak{M}(\mathcal{X},\tau^{\prime})}$. By [ACGS2,
Prop. 3.23], because $\tau^{\prime}$ is realizable, for a geometric point
$\bar{w}$ of $\mathfrak{M}(\mathcal{X},\tau^{\prime})$, the monoid ideal
$\overline{\mathcal{I}}_{\bar{w}}$ has the following description. Let
$Q_{\bar{w}}$ be the stalk of the ghost sheaf at $\bar{w}$, and
$Q_{\tau^{\prime}}$ the stalk of the ghost sheaf at a generic point of
$\mathfrak{M}(\mathcal{X},\tau^{\prime})$. Then there is a well-defined
generization map $\chi_{\bar{w}}:Q_{\bar{w}}\rightarrow Q_{\tau^{\prime}}$,
and
$\overline{\mathcal{I}}_{\bar{w}}=\chi_{\bar{w}}^{-1}(Q_{\tau^{\prime}}\setminus\\{0\\})$.

Similarly, we may put an idealized structure on
$\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})$, with ideal sheaf
$\mathcal{J}$. For a geometric point $\bar{w}$, let
$Q^{\circ,{\operatorname{sat}}}_{\bar{w}}$ be the stalk of the ghost sheaf of
this latter log structure; necessarily, this monoid is contained in
$Q_{\bar{w}}\oplus\mathbb{Z}$ and contains $Q_{\bar{w}}\oplus\mathbb{N}$. Then
we take the stalk of the corresponding monoid ideal
$\overline{\mathcal{J}}_{\bar{w}}$ to be the monoid ideal generated by
$\overline{\mathcal{I}}_{\bar{w}}\oplus 0$,
$Q^{\circ,{\operatorname{sat}}}\setminus(Q\oplus\mathbb{N})$ and $(0,1)$. It
is straightforward to check that $\mathcal{J}$ yields an idealized structure
on $\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})$. Further, the natural
morphism
$\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})\rightarrow\mathfrak{M}(\mathcal{X},\tau^{\prime})$
is idealized log étale.

[Wu, Cor. 2.9] now shows that the map
${\operatorname{ev}}:\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})\rightarrow\mathcal{X}$
is idealized log smooth. As this map factors through the idealized log étale
strict closed embedding $\mathcal{X}_{\sigma}\rightarrow\mathcal{X}$, we write
${\operatorname{ev}}:\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})\rightarrow\mathcal{X}_{\sigma}$,
also idealized log smooth.

After a base-change, we also obtain

$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime}):=\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})\times_{\mathcal{X}_{\sigma}}X_{\sigma}\rightarrow
X_{\sigma}$

idealized log smooth, with the underlying stack of
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$
isomorphic to the underlying stack of

$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})=\mathfrak{M}(\mathcal{X},\tau^{\prime})\times_{\underline{\mathcal{X}}_{\sigma}}\underline{X}_{\sigma}.$

Step 5. _Proof of the linear equivalence relation ( 5.8)_. Continuing with
$\sigma\in\mathscr{P}$ the minimal cone containing $\mathfrak{p}$, denote by
$\sigma_{\mathfrak{u}},\sigma_{\mathfrak{u}^{\prime}}$ the cones containing
$\mathfrak{u}$ and $\mathfrak{u}^{\prime}$ respectively. We write
$X_{\mathfrak{u}},X_{\mathfrak{u}^{\prime}}$ for the corresponding zero-
dimensional strata of $X$, and write
$P_{\sigma},P_{\mathfrak{u}},P_{\mathfrak{u}^{\prime}}$ for the stalks of the
ghost sheaf of $X_{\sigma}$, $X_{\mathfrak{u}}$ and
$X_{\mathfrak{u}^{\prime}}$ at their generic points.

Our goal is to construct a rational function $\psi$ on
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$ and
determine its divisor of zeros and poles. We note that as
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$ is
idealized log smooth over $\operatorname{Spec}\Bbbk$, it is isomorphic,
locally in the smooth topology, to a subscheme of a toric variety defined by a
monomial ideal, see [ACGS2, Prop. B.2]. Further, by the cited proposition and
the explicit description of the idealized structure of
$\overline{\mathfrak{M}}(\mathcal{X},\tau^{\prime})$, hence of
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$, in
fact
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$ is
smooth locally isomorphic to a toric stratum of a toric variety, and hence is
normal.

Let
$\bar{s}\in\Gamma(X_{\sigma},\overline{\mathcal{M}}^{{\operatorname{gp}}}_{X_{\sigma}})$
be defined as follows. Note that giving such a section is equivalent to giving
an integral piecewise linear function on $\mathrm{Star}(\sigma)\subseteq B$.
Using the affine structure on $\mathrm{Star}(\sigma)$ induced by that on $B$,
we take this function in fact to be linear in this affine structure,
primitive, vanishing on $\mathfrak{p}$ and positive on $\mathfrak{u}$. We may
then choose a lift of $\bar{s}$ to
$s\in\Gamma(X_{\sigma},\mathcal{M}^{{\operatorname{gp}}}_{X_{\sigma}})$. The
existence of a lift is obvious when $\dim X_{\sigma}=0$. When $\dim
X_{\sigma}=1$, one needs to check that the corresponding torsor is trivial.
However, using the isomorphism of Lemma 1.10, we may identify the line bundle
associated to the corresponding torsor with the restriction of a line bundle
on $X_{\Sigma_{\sigma}}$. The line bundle on $X_{\Sigma_{\sigma}}$ is defined
by the linear function $\bar{s}$ on $\Sigma_{\sigma}$. However, it is a
standard fact of toric geometry that the line bundle corresponding to a linear
(as opposed to piecewise linear) function is in fact trivial.

Let
$U\subseteq\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$
be the dense open stratum, i.e., the open substack whose geometric points are
precisely those whose corresponding type is $\tau^{\prime}$. Note that the
image of ${\operatorname{ev}}^{\flat}(s)$ in
$\overline{\mathcal{M}}_{\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})}^{{\operatorname{gp}}}$
vanishes on $U$. Indeed, this follows from the fact that
$h_{\tau^{\prime}}(\tau^{\prime}_{\mathrm{out}})\subseteq\mathfrak{p}$ and
$h_{\tau^{\prime}}|_{\tau^{\prime}_{\mathrm{out}}}$ is dual to
$\bar{\operatorname{ev}}^{\flat}:P_{\sigma}\rightarrow
Q^{\circ,{\operatorname{sat}}}_{\tau^{\prime}}$. Thus, viewing
$\mathcal{O}_{\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})}^{\times}$
as a subsheaf of
$\mathcal{M}_{\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})}^{{\operatorname{gp}}}$
via the inverse of the structure map $\alpha$, we see that
${\operatorname{ev}}^{\flat}(s)$ restricts to an invertible function on $U$,
and hence defines a rational function $\psi$ on
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$. We
complete the proof by determining the divisor of zeros and poles of $\psi$.

We now use the identification of underlying stacks of
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$ and
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$. Since the
complement of $U$ is a union of strata of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$, it is enough
to check the order of vanishing of $\psi$ along each codimension one stratum
of $\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$. These
strata are given by
$\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})$ with
$\phi:\tau\rightarrow\tau^{\prime}$ a contraction morphism with
$\dim\tau=n-1$, see [ACGS2, Rem. 3.28]. We write
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})$
for the corresponding stratum of
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$.

Given such a $\phi:\tau\rightarrow\tau^{\prime}$, we have several cases:

Case 1. $\dim h_{\tau}(\tau_{\mathrm{out}})=n-1$. In this case, necessarily
$\bar{s}$ still vanishes on $h_{\tau}(\tau_{\mathrm{out}})$, so as above,
$\psi$ extends as an invertible function across
$\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})$.

Case 2. _$h_{\tau}(\tau_{\mathrm{out}})$ intersects the interior of
$\mathfrak{u}$._ We adopt the notation from [ACGS2, App. B] that if $P$ is a
monoid, $K\subseteq P$ a monoid ideal, then
$A_{P,K}:=\operatorname{Spec}\Bbbk[P]/K$ and
$\mathcal{A}_{P,K}:=[A_{P,K}/\operatorname{Spec}\Bbbk[P^{{\operatorname{gp}}}]]$.

Let $Q_{\tau}^{\circ,{\operatorname{sat}}}$ be the stalk of the ghost sheaf of
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$ at
a geometric generic point $\bar{w}$ of
$\overline{\mathfrak{M}}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})$,
and let $J\subseteq Q_{\tau}^{\circ,{\operatorname{sat}}}$ be the monoid ideal
arising from the idealized structure. As ${\operatorname{ev}}$ is idealized
log smooth, smooth locally in a neighbourhood of $\bar{w}$,
${\operatorname{ev}}$ concides with the morphism
$\mathcal{A}_{Q^{\circ,{\operatorname{sat}}},J}\rightarrow\mathcal{A}_{P_{\mathfrak{u}},K}$,
by [ACGS2, Prop. B.4]. Here
$K=P_{\mathfrak{u}}\setminus\chi_{\sigma\mathfrak{u}}^{-1}(0)$ where
$\chi_{\sigma\mathfrak{u}}:P_{\mathfrak{u}}\rightarrow P_{\sigma}$ is the
generization map. In particular, the stratum $X_{\sigma}\subseteq X$, smooth
locally in a neighbourhood of the point $X_{\mathfrak{u}}$, is isomorphic to
$A_{P_{\mathfrak{u}},K}$.

Let $\chi_{\tau^{\prime}\tau}:Q^{\circ,{\operatorname{sat}}}_{\tau}\rightarrow
Q^{\circ,{\operatorname{sat}}}_{\tau^{\prime}}$ be the generization map,
necessarily a localization along a face $F$ of
$Q^{\circ,{\operatorname{sat}}}_{\tau}$. As $\dim\tau=\dim\tau^{\prime}+1$ we
see that $F\cong\mathbb{N}$. Note that $\bar{s}$ is non-negative on
$\mathfrak{u}$. Since $h_{\tau}(\tau_{\mathrm{out}})$ contains $\mathfrak{u}$,
on which $\bar{s}$ is non-negative, and
$h_{\tau^{\prime}}(\tau^{\prime}_{\mathrm{out}})$ is a face of
$h_{\tau}(\tau_{\mathrm{out}})$, on which $\bar{s}$ vanishes, it follows that
$\bar{s}$ is non-negative on $h_{\tau}(\tau_{\mathrm{out}})$. Dually, it
follows that the stalk of $\bar{\operatorname{ev}}^{\flat}(\bar{s})$ at
$\bar{w}$ lies in $Q^{\circ,{\operatorname{sat}}}_{\tau}\subseteq
Q^{\circ,{\operatorname{gp}}}_{\tau}$. Further, the stalk necessarily lies in
$\chi_{\tau^{\prime}\tau}^{-1}(0)=F$. The order of vanishing of $\psi$ along
the stratum determined by $\tau$ is then, by standard toric geometry, equal to
the germ of $\bar{\operatorname{ev}}^{\flat}(\bar{s})$ under the
identification of $F$ with $\mathbb{N}$. Dually, since by construction
$\bar{s}$ generates
$P_{\mathfrak{u}}^{{\operatorname{gp}}}\cap\Lambda_{\mathfrak{p}}^{\perp}$,
this number is the same as

$d_{\tau}:=|\operatorname{coker}(\Lambda_{\tau_{\mathrm{out}}}/\Lambda_{\tau^{\prime}_{\mathrm{out}}}\rightarrow\Lambda_{\mathfrak{u}}/\Lambda_{\mathfrak{p}})|.$

This is the order of vanishing of $\psi$ along the stratum
$\mathfrak{M}_{\tau}(\mathcal{X},\tau^{\prime})$.

Note an elementary diagram chase shows that

$k_{\tau}=d_{\tau}|\operatorname{coker}(\Lambda_{\tau^{\prime}_{\mathrm{out}}}\rightarrow\Lambda_{\mathfrak{p}})_{\mathrm{tors}}|.$

Case 3. _$h(\tau_{\mathrm{out}})$ intersects the interior of
$\mathfrak{u}^{\prime}$._ The same analysis as in Case 2 applies once
$\bar{s}$ is replaced by $-\bar{s}$. Hence $\psi$ has a pole of order
$d_{\tau}$ along $\mathfrak{M}_{\tau}(\mathcal{X},\tau^{\prime})$.

Putting these three cases together, we see that the divisor of zeros and poles
of $\psi$ gives the relation in the Chow group of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau^{\prime})$:

$\sum_{\tau\rightarrow\tau^{\prime}\atop\mathfrak{u}\subseteq
h_{\tau}(\tau_{\mathrm{out}})}d_{\tau}[\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})]=\sum_{\tau\rightarrow\tau^{\prime}\atop\mathfrak{u}^{\prime}\subseteq
h_{\tau}(\tau_{\mathrm{out}})}d_{\tau}[\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\tau^{\prime})].$

If we then multiply both sides by
$|\operatorname{coker}(\Lambda_{\tau^{\prime}_{\mathrm{out}}}\rightarrow\Lambda_{\mathfrak{p}})_{\mathrm{tors}}|$,
we get the desired (5.8). ∎

###  5.2. Consistency from theta functions

######  Theorem 5.2.

$\mathscr{S}_{{\mathrm{can}}}$ is a consistent wall structure.

###### Proof.

Consistency as defined in [GHS, Def. 3.9] involves checking consistency in
codimensions zero, one and two. We check each case in turn.

Consistency in codimension zero. Let $\mathfrak{j}$ be a codimension zero
joint, $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}$ the walls containing
$\mathfrak{j}$, taken in cyclic order, with $\mathfrak{p}_{i}$ contained in
chambers $\mathfrak{u}_{i}$ and $\mathfrak{u}_{i+1}$, with indices taken
modulo $r$. With $\theta_{\mathfrak{p}_{i}}:R_{\mathfrak{u}_{i}}\rightarrow
R_{\mathfrak{u}_{i+1}}$ as defined in (5.1), consistency at the joint means
[GHS, Def. 2.13] that

(5.9)
$\theta:=\theta_{\mathfrak{p}_{r}}\circ\cdots\circ\theta_{\mathfrak{p}_{1}}=\operatorname{id}$

as an automorphism of $R_{\sigma}$, where $\sigma\in\mathscr{P}$ is the cell
containing $\mathfrak{j}$. Note that for
$p\in\sigma_{\mathbb{Z}}\setminus\\{0\\}$ and
$x\in\operatorname{Int}(\sigma)\setminus|\mathscr{S}_{{\mathrm{can}}}|$,
$\vartheta_{p}(x)=z^{p}\mod\mathfrak{m}_{x}$, where $\mathfrak{m}_{x}$ is as
in (3.6). Indeed, the trivial broken line with no bends contributes the term
$z^{p}$, and any other broken line $\beta$ must have
$a_{\beta}\in\mathfrak{m}_{x}$. Since the ideal $\mathfrak{m}_{x}$ is
nilpotent in $R_{\sigma}$ and $z^{p}$ is invertible in $R_{\sigma}$, it
follows that $\vartheta_{p}(x)$ is invertible in $R_{\sigma}$. Working
inductively modulo powers of $\mathfrak{m}_{x}$, one then sees that any
element of $R_{\sigma}$ may be written as a $\Bbbk[Q]/I$-linear combination of
ratios $\vartheta_{p}(x)/\vartheta_{p^{\prime}}(x)$ for
$p,p^{\prime}\in\sigma_{\mathbb{Z}}$. By Theorem 5.1,(2), applied successively
for the walls $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}$, we obtain
$\theta\big{(}\vartheta_{p}(x)\big{)}=\vartheta_{p}(x)$, and hence $\theta$ is
the identity on $R_{\sigma}$.

Consistency in codimension zero follows, as this just means each codimension
zero joint is consistent.

Consistency in codimension one. Let $\mathfrak{j}$ be a codimension one joint,
contained in slabs
$\mathfrak{b}_{1},\mathfrak{b}_{2}\subseteq\rho\in\mathscr{P}^{[n-1]}_{\mathrm{int}}$,
maximal cones $\sigma,\sigma^{\prime}$, and codimension zero walls
$\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}\subseteq\sigma$ and
$\mathfrak{p}^{\prime}_{1},\ldots,\mathfrak{p}^{\prime}_{s}\subseteq\sigma^{\prime}$.
We order these so that

$\mathfrak{b}_{1},\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r},\mathfrak{b}_{2},\mathfrak{p}^{\prime}_{1},\ldots,\mathfrak{p}^{\prime}_{s}$

are cyclically ordered about $\mathfrak{j}$. Then consistency at the joint
$\mathfrak{j}$ is expressed in [GHS, Def. 2.14] as follows. We set

$\displaystyle\theta:=\theta_{\mathfrak{p}_{r}}\circ\theta_{\mathfrak{p}_{r-1}}\circ\ldots\circ\theta_{\mathfrak{p}_{1}}:$
$\displaystyle R_{\sigma}\to R_{\sigma}$
$\displaystyle\theta^{\prime}:=\theta_{\mathfrak{p}_{1}^{\prime}}\circ\theta_{\mathfrak{p}_{2}^{\prime}}\circ\ldots\circ\theta_{\mathfrak{p}_{s}^{\prime}}:$
$\displaystyle R_{\sigma^{\prime}}\to R_{\sigma^{\prime}}.$

We use the notation
$\chi_{\mathfrak{b}_{i},\sigma},\chi_{\mathfrak{b}_{i},\sigma^{\prime}}$ for
the localization maps of (5.2). Then consistency is the statement that

$(\theta\times\theta^{\prime})\big{(}(\chi_{\mathfrak{b}_{1},\sigma},\chi_{\mathfrak{b}_{1},\sigma^{\prime}})(R_{\mathfrak{b}_{1}})\big{)}=(\chi_{\mathfrak{b}_{2},\sigma},\chi_{\mathfrak{b}_{2},\sigma^{\prime}})(R_{\mathfrak{b}_{2}}).$

To show this, let $x_{i}\in\sigma$ be a point in the chamber adjacent to
$\mathfrak{b}_{i}$, and similarly $x_{i}^{\prime}\in\sigma^{\prime}$. If
$p\in\rho_{\mathbb{Z}}\setminus\\{0\\}$, then
$\vartheta_{p}(x_{i})=z^{p}\mod\mathfrak{m}_{x_{i}}$, and similarly for
$x_{i}^{\prime}$, as in the codimension zero case. Thus
$\vartheta_{p}(\mathfrak{b}_{i})$, which exists by Theorem 5.1, (3), satisfies
$\chi_{\mathfrak{b}_{i},\sigma}(\vartheta_{p}(\mathfrak{b}_{i})-z^{p})$
nilpotent, and the same for $\chi_{\mathfrak{b}_{i},\sigma^{\prime}}$. By the
discussion preceding [GHS, Def. 2.14], the map