$(\chi_{\mathfrak{b}_{i},\sigma},\chi_{\mathfrak{b}_{i},\sigma^{\prime}}):R_{\mathfrak{b}_{i}}\rightarrow
R_{\sigma}\times R_{\sigma^{\prime}}$ is injective. Thus we conclude that
$\vartheta_{p}(\mathfrak{b}_{i})-z^{p}$ is nilpotent in
$R_{\mathfrak{b}_{i}}$, and hence $\vartheta_{p}(\mathfrak{b}_{i})$ is
invertible, as $z^{p}$ is invertible. Further, if
$p\in\sigma_{\mathbb{Z}}\setminus\\{0\\}$, we may write $p=p^{\prime}+a\xi$
with $a\geq 0$ $p^{\prime}\in\Lambda_{\rho}$, and then
$\vartheta_{p}(\mathfrak{b}_{1})=z^{p^{\prime}}Z_{+}^{a}\mod\mathfrak{m}$, and
similarly if $p\in\sigma^{\prime}_{\mathbb{Z}}\setminus\\{0\\}$,
$p=p^{\prime}-a\xi$, we have
$\vartheta_{p}(\mathfrak{b}_{1})=z^{p^{\prime}}Z_{-}^{a}\mod\mathfrak{m}$.
From this we see that any element of $R_{\mathfrak{b}_{1}}$ may be written as
a $\Bbbk[Q]/I$-linear combination of Laurent monomials in theta functions of
the form
$(\prod\vartheta_{p_{i}}(\mathfrak{b}_{1})^{a_{i}})/\vartheta_{p}(\mathfrak{b}_{1})$
with $p\in\rho_{\mathbb{Z}}$,
$p_{i}\in\sigma_{\mathbb{Z}}\cup\sigma^{\prime}_{\mathbb{Z}}$. Note here we
only have control over the expressions $\vartheta_{p}(\mathfrak{b}_{1})$ for
$p\in\sigma_{\mathbb{Z}}\cup\sigma^{\prime}_{\mathbb{Z}}$, so we need such
Laurent monomials in order to get all elements of $R_{\mathfrak{b}_{1}}$.

Now it follows from Theorem 5.1,(2) that for $p\in
B(\mathbb{Z})\setminus\\{0\\}$

$\displaystyle(\theta\times\theta^{\prime})\big{(}(\chi_{\mathfrak{b}_{1},\sigma},\chi_{\mathfrak{b}_{1},\sigma^{\prime}})(\vartheta_{p}(\mathfrak{b}_{1}))\big{)}=(\theta\times\theta^{\prime})\big{(}\vartheta_{p}(x_{1}),\vartheta_{p}(x^{\prime}_{1})\big{)}$
$\displaystyle={}$
$\displaystyle\big{(}\vartheta_{p}(x_{2}),\vartheta_{p}(x_{2}^{\prime})\big{)}=(\chi_{\mathfrak{b}_{2},\sigma}\times\chi_{\mathfrak{b}_{2},\sigma^{\prime}})(\vartheta_{p}(\mathfrak{b}_{2})).$

Combined with the generation statement of the previous paragraph, we obtain
the desired equality.

This shows consistency at codimension one joints, noting there is no condition
for codimension one joints contained in $\partial B$. Thus we obtain
consistency in codimension one.

Consistency in codimension two. Consistency at codimension two joints is
defined in [GHS, §3.2]. If $\mathfrak{j}$ is a boundary joint, i.e., contained
in $\partial B$, then it is easy to check that $\mathscr{S}_{{\mathrm{can}}}$
is convex at $\mathfrak{j}$ in the sense of [GHS, Def. 3.12]. Indeed, if
$\partial B\not=\emptyset$, it follows from Propositions 1.15 and 3.7 that if
$\tau$ is a wall type, then $u_{\tau}$ is tangent to $\partial B$. Thus by
[GHS, Thm. 3.13], the joint is consistent.

In case $\mathfrak{j}$ is an interior joint, we refer the reader to [GHS,
§3.2] for the definition of a corresponding polyhedral affine manifold
$(B_{\mathfrak{j}},\mathscr{P}_{\mathfrak{j}})$, with
$\mathscr{P}_{\mathfrak{j}}$ consisting of tangent cones to elements of
$\mathscr{P}$ containing $\mathfrak{j}$ along $\mathfrak{j}$. Further,
$\mathscr{S}_{{\mathrm{can}}}$ induces a wall structure on $B_{\mathfrak{j}}$
we denote as $\mathscr{S}_{\mathfrak{j}}$. For $p\in
B_{\mathfrak{j}}(\mathbb{Z})\setminus\\{0\\}$, broken lines on
$B_{\mathfrak{j}}$ then define for general $x\in B_{\mathfrak{j}}$ theta
functions $\vartheta^{\mathfrak{j}}_{p}(x)$ as before. Consistency is then the
statements (1)–(3) of Theorem 5.1 for $B_{\mathfrak{j}}$ and
$\mathscr{S}_{\mathfrak{j}}$ rather than $B$ and
$\mathscr{S}_{{\mathrm{can}}}$.

We do not prove consistency directly from Theorem 5.1 as it is not clear what
the relationship between theta functions on $B$ and on $B_{\mathfrak{j}}$ is.
Instead, we proceed as follows. Write $\mathrm{Star}(\mathfrak{j})$ for the
open star of $\mathfrak{j}$ with respect to the polyhedral cone decomposition
$\mathscr{P}_{\mathscr{B}}$. Then there is a natural embedding of
$\mathrm{Star}(\mathfrak{j})$ in $B_{\mathfrak{j}}$, identifying a cone of
$\mathscr{P}_{\mathscr{B}}$ containing $\mathfrak{j}$ and contained in
$\sigma\in\mathscr{P}$ with a cone in the tangent wedge of $\sigma$ along
$\mathfrak{j}$, with vertex at the origin of the tangent wedge. Note the walls
of $\mathscr{S}_{\mathfrak{j}}$ are in one-to-one correspondence with the
walls of $\mathscr{S}_{{\mathrm{can}}}$ intersecting
$\mathrm{Star}(\mathfrak{j})$.

The tangent space $\Lambda_{\mathfrak{j},\mathbb{R}}$ to $\mathfrak{j}$ acts
by translation on $B_{\mathfrak{j}}$ and $\Lambda_{\mathfrak{j}}$ acts on
$B_{\mathfrak{j}}(\mathbb{Z})$. Further, all walls of
$\mathscr{S}_{\mathfrak{j}}$ are invariant under this translation, and
translation takes broken lines to broken lines. In particular, if $p\in
B_{\mathfrak{j}}(\mathbb{Z})$, $v\in\Lambda_{\mathfrak{j},\mathbb{R}}$, it is
immediate that

$\vartheta^{\mathfrak{j}}_{p}(x)=\vartheta^{\mathfrak{j}}_{p}(x+v)$

for general $x$. Further, since the monomials $z^{v}$ for
$v\in\Lambda_{\mathfrak{j}}$ are invariant under crossing all walls in
$\mathscr{S}_{\mathfrak{j}}$, we in fact have

$\vartheta_{p+v}^{\mathfrak{j}}(x)=z^{v}\vartheta_{p}^{\mathfrak{j}}(x).$

Note that the broken lines defining the two theta functions are not precisely
the same; rather, if $a_{i}z^{m_{i}}$ is a monomial attached to a segment of a
broken line contributing to $\vartheta_{p}^{\mathfrak{j}}(x)$, then the
corresponding broken line contributing to $\vartheta_{p+v}^{\mathfrak{j}}(x)$
has attached monomial $a_{i}z^{m_{i}+v}$, and thus the actual maps $\beta$ are
different as they have different derivatives. However, there is clearly a one-
to-one correspondence between such broken lines. In particular, if items
(1)–(3) of Theorem 5.1 hold for a given choice of $p$, for the functions
$\vartheta_{p}^{\mathfrak{j}}(x)$ rather than $\vartheta_{p}(x)$, they will
also hold for $p+v$ for any $v\in\Lambda_{\mathfrak{j}}$.

We will now show items (1)–(3) of Theorem 5.1 for the functions
$\vartheta^{\mathfrak{j}}_{p}(x)$ for arbitrary non-zero $p\in
B_{\mathfrak{j}}(\mathbb{Z})$, which is fixed throughout the following
discussion. Because we may replace $p$ with $p+v$ for any
$v\in\Lambda_{\mathfrak{j}}$, we may assume, as we do from now on in this
proof, that $p\in\mathrm{Star}(\mathfrak{j})$. Indeed, for any point $x\in
B_{\mathfrak{j}}(\mathbb{Z})$, there is some $v\in\mathfrak{j}\cap
B(\mathbb{Z})$ such that $x+v\in\mathrm{Star}(\mathfrak{j})$ under the
canonical embedding of $\mathrm{Star}(\mathfrak{j})$ in $B_{\mathfrak{j}}$.

More generally, it follows easily from this observation that if $Z\subseteq
B_{\mathfrak{j}}$ is any compact subset, then there is a $v\in\mathfrak{j}\cap
B(\mathbb{Z})$ such that $Z+v\subseteq\mathrm{Star}(\mathfrak{j})$.

Now suppose given an open subset $U\subseteq B_{\mathfrak{j}}$ whose closure
$Z$ is compact. For each $x\in Z$ not contained in
$|\mathscr{S}_{\mathfrak{j}}|$, we have a finite set of broken lines
contributing to $\vartheta^{\mathfrak{j}}_{p}(x)$, and these all have
asymptotic direction $p$ contained in $\mathrm{Star}(\mathfrak{j})$. Thus if
$\beta$ is a broken line contributing to this theta function,
$\beta([t_{1},0])$ is a compact subset of $B_{\mathfrak{j}}$ and we may
translate $x$ by some $v\in\mathfrak{j}$ so that
$\beta([t_{1},0])\subseteq\mathrm{Star}(\mathfrak{j})$. As the asymptotic
direction $p$ necessarily lies in a cone of $\mathscr{P}_{\mathscr{B}}$
containing $\beta(t_{1})$, we see also that
$\beta((-\infty,t_{1}])\subseteq\mathrm{Star}(\mathfrak{j})$. By summing over
the $v$’s for the finite number of broken lines contributing to
$\vartheta^{\mathfrak{j}}_{p}(x)$ we may thus translate $x$ by some
$v\in\mathfrak{j}$ to guarantee that all broken lines contributing to
$\vartheta^{\mathfrak{j}}_{p}(x)$ have image contained in
$\mathrm{Star}(\mathfrak{j})$. Noting that broken lines can be deformed
continuously inside some polyhedron without changing their type (see [GHS,
Prop. 3.5]), we may further in fact assume that all broken lines contributing
to $\vartheta^{\mathfrak{j}}_{p}(x)$ for $x\in
Z\setminus|\mathscr{S}_{\mathfrak{j}}|$ have image in
$\mathrm{Star}(\mathfrak{j})$.

From the construction of $\mathscr{S}_{\mathfrak{j}}$, it is now clear that
provided $x$ has been translated sufficiently,
$\vartheta^{\mathfrak{j}}_{p}(x)$ is the sum of contributions of those broken
lines contributing to $\vartheta_{p}(x)$ with image wholly contained in
$\mathrm{Star}(\mathfrak{j})$.

One now checks that the proof of Theorem 5.1 applies equally well when
$p\in\mathrm{Star}(\mathfrak{j})$ if one restricts attention to endpoints
$x\in\mathrm{Star}(\mathfrak{j})$ translated “sufficiently far” by an element
$v\in\mathfrak{j}$. Indeed, there are two key points of the argument in the
proof of Theorem 5.1 where one needs to control the behaviour of the broken
lines being considered.

First, in Step 2, the proof of (5.3) and (5.4) works for any wall structure
(not necessarily a consistent one) and hence works for
$\mathscr{S}_{\mathfrak{j}}$.

Second, in Steps 3 and following, we focus on a single degenerate broken line
type $\tau^{\prime}$ and the result follows from the relation (5.8). However,
if we fix a point $x\in h_{\tau^{\prime}}(\tau^{\prime}_{\mathrm{out}})$ and a
neighbourhood $U$ of $x$ with compact closure $Z$, we then may translate $Z$
by some $v\in\mathfrak{j}$ so that all broken lines with endpoints in
$Z\setminus|\mathscr{S}_{\mathfrak{j}}|$ lie in $\mathrm{Star}(\mathfrak{j})$,
and thus in (5.8), we only need to consider those types $\tau$ with
contraction morphisms $\tau\rightarrow\tau^{\prime}$ whose corresponding
broken lines with endpoint in $Z$ lie entirely in
$\mathrm{Star}(\mathfrak{j})$. ∎

We end this section with a brief observation in the relative case
$g:X\rightarrow S$, following on from Proposition 1.16 and Remarks 1.17,1.18.
Assume that the hypotheses of Proposition 1.16 hold, so that
$(B^{\prime},\mathscr{P}^{\prime})$ is a polyhedral affine pseudomanifold. In
this case, the canonical wall structure $\mathscr{S}_{{\mathrm{can}}}$ induces
a wall structure $\mathscr{S}^{\prime}_{{\mathrm{can}}}$ on
$(B^{\prime},\mathscr{P}^{\prime})$. Indeed, first note that
$\Lambda_{B^{\prime}}$, the local system of integral tangent vectors on
$B^{\prime}$, is a subsheaf of $\Lambda_{B}|_{B^{\prime}}$ consisting of those
tangent vectors $v$ with $(g_{\mathrm{trop}})_{*}(v)=0$. Using $\varphi$ the
MPL function on $B$ given by Construction 1.14, we obtain also an MPL function
$\varphi|_{B^{\prime}}$ as in Remark 1.17. This induces a local system
$\mathcal{P}_{B^{\prime}}$, and
$\mathcal{P}_{B^{\prime}}\subseteq\mathcal{P}_{B}|_{B^{\prime}}$ is the
subsheaf consisting of those sections $m$ of $\mathcal{P}_{B}|_{B^{\prime}}$
such that $(g_{\mathrm{trop}})_{*}(\bar{m})=0$.

Next, given a wall
$(\mathfrak{p},f_{\mathfrak{p}})\in\mathscr{S}_{{\mathrm{can}}}$, we define
the _index_ of $\mathfrak{p}$ to be

$\operatorname{ind}(\mathfrak{p}):=|\operatorname{coker}(g_{\mathrm{trop},*}:\Lambda_{\mathfrak{p}}\rightarrow\mathbb{Z})|.$

We may now define

$\mathscr{S}_{{\mathrm{can}}}^{\prime}:=\\{(\mathfrak{p}\cap
B^{\prime},f_{\mathfrak{p}}^{\operatorname{ind}(\mathfrak{p})})\,|\,(\mathfrak{p},f_{\mathfrak{p}})\in\mathscr{S}_{{\mathrm{can}}}\\},$

noting that for any $x\in B^{\prime}\setminus\Delta$,
$f_{\mathfrak{p}}\in\Bbbk[\mathcal{P}^{+}_{B^{\prime},x}]\subseteq\Bbbk[\mathcal{P}^{+}_{B,x}]$
by Proposition 3.7 and the construction of the canonical wall structure.

######  Proposition 5.3.

Assuming $g:X\rightarrow S$ satisfies the hypotheses of Proposition 3.7, then
$\mathscr{S}_{{\mathrm{can}}}^{\prime}$ is a consistent wall structure.

###### Proof.

This follows from the definition of consistency and the consistency of
$\mathscr{S}_{{\mathrm{can}}}$. Indeed, checking consistency of
$\mathscr{S}_{{\mathrm{can}}}^{\prime}$ in codimensions zero, one and two
involves checking certain properties in rings which are subrings of the
corresponding rings for $B$. For example, for consistency in codimension zero,
to verify (5.9) in the ring $R_{\sigma\cap B^{\prime}}$ defined using the data
$(B^{\prime},\mathscr{P}^{\prime},\varphi|_{B^{\prime}})$, it is enough to
note that $R_{\sigma\cap B^{\prime}}$ is a subring of $R_{\sigma}$ and that
the ring automorphism $\vartheta_{\mathfrak{p}}$ of $R_{\sigma}$ restricts to
an automorphism of $R_{\sigma\cap B^{\prime}}$ given by the wall
$(\mathfrak{p}\cap
B^{\prime},f_{\mathfrak{p}}^{\operatorname{ind}(\mathfrak{p})})$. This is
straightforward: see the discussion of [GHS, §4.2] as to why the power of
$f_{\mathfrak{p}}$ is necessary. Indeed, [GHS, §4.2] goes the opposite way,
from a wall structure for $B^{\prime}$ to a wall structure for
$\mathbf{C}B^{\prime}=B$, see Remark 1.18. Thus there one must pass to a
$\operatorname{ind}(\mathfrak{p})^{th}$ root rather than power. We leave the
remaining details to the reader. ∎

##  6\. Comparison with intrinsic mirror symmetry

We continue with $g:X\rightarrow S$ in the absolute or relative cases
satisfying Assumptions 1.1 or 1.2. Having now constructed a consistent wall
structure $\mathscr{S}_{{\mathrm{can}}}$ on $B$, we observe that the data
$(B,\mathscr{P})$ and $\mathscr{S}_{{\mathrm{can}}}$ are _conical_ in the
sense of [GHS, Def. 3.20]. Thus this data first specifies a scheme999This
scheme is written as $\mathfrak{X}^{\circ}$ in [GHS]; we include the check
here to indicate it is the mirror to $(X,D)$. $\check{\mathfrak{X}}^{\circ}$
flat over $\Bbbk[Q]/I$, as constructed in [GHS, Prop. 2.4] by gluing together
copies of spectra of rings $R_{\mathfrak{u}}$ and $R_{\mathfrak{b}}$, as well
as some additional rings corresponding to $n-1$ dimensional cells of
$\mathscr{P}_{\mathscr{S}}$ contained in $\partial B$. We refer the reader to
the cited result for details. Most importantly for us, [GHS, Prop. 3.21] tells
us that
$R:=\Gamma(\check{\mathfrak{X}}^{\circ},\mathcal{O}_{\check{\mathfrak{X}}^{\circ}})$
is freely generated as a $\Bbbk[Q]/I$-module by a basis
$\\{\vartheta_{p}\,|\,p\in B(\mathbb{Z})\\}$. Further,
$\check{\mathfrak{X}}:=\operatorname{Spec}R$ contains
$\check{\mathfrak{X}}^{\circ}$ as a dense open subscheme and is also flat over
$\operatorname{Spec}\Bbbk[Q]/I$. The theta functions $\vartheta_{p}$ restrict
on standard pieces $\operatorname{Spec}R_{\mathfrak{u}}$ to the expressions
$\vartheta_{p}(x)$ for $x\in\operatorname{Int}(\mathfrak{u})$ defined in
Definition 4.4.

In the relative case, $R$ is naturally a graded ring via
$\deg(\vartheta_{p})=g_{\mathrm{trop}}(p)\in\mathbb{N}$. Thus to obtain the
mirror in the relative case, one considers instead the flat family
$\operatorname{Proj}R\rightarrow\operatorname{Spec}\Bbbk[Q]/I$; see [GHS, §4]
and [GS7, §1] for more details on this point of view.

[GHS, Thm. 3.24] gives a description of the structure constants for $R$.
Write, for $p_{1},p_{2}\in B(\mathbb{Z})\setminus\\{0\\}$,

$\vartheta_{p_{1}}\cdot\vartheta_{p_{2}}=\sum_{r\in
B(\mathbb{Z})}\alpha^{\mathrm{trop}}_{p_{1}p_{2}r}\cdot\vartheta_{r}$

with $\alpha^{\mathrm{trop}}_{p_{1}p_{2}r}\in\Bbbk[Q]/I$. These structure
constants, written as $\alpha_{r}(p_{1},p_{2})$ in [GHS], are defined
tropically as follows. Let $\mathfrak{u}$ be a chamber of
$\mathscr{S}_{{\mathrm{can}}}$ such that $r\in\mathfrak{u}$, and let
$x\in\mathfrak{u}$ be general. Then

(6.1)
$\alpha^{\mathrm{trop}}_{p_{1}p_{2}r}=\sum_{(\beta_{1},\beta_{2})}a_{\beta_{1}}a_{\beta_{2}}$

where the sum is over all pairs $(\beta_{1},\beta_{2})$ of broken lines with
asymptotic monomials $p_{1},p_{2}$ respectively, with endpoint $x$, and such
that $m_{\beta_{1}}+m_{\beta_{2}}=r$, viewed as an equation in
$\Lambda_{\mathfrak{u}}$. We note that the restriction that
$p_{1},p_{2}\not=0$ is not important as $\vartheta_{0}$ was defined as the
unit in the ring $R$.

On the other hand, [GS7] defines structure constants on the free
$\Bbbk[Q]/I$-module with basis $\\{\vartheta_{p}\,|\,p\in B(\mathbb{Z})\\}$
directly in terms of punctured Gromov-Witten invariants. Following the
notation of [GS7], we have a potentially different product rule

$\vartheta_{p_{1}}\cdot\vartheta_{p_{2}}=\sum_{r\in
B(\mathbb{Z})}\alpha^{\log}_{p_{1}p_{2}r}\cdot\vartheta_{r}.$

These structure constants are written simply as $\alpha_{p_{1}p_{2}r}$ in
[GS7]. They are given by a formula

(6.2)
$\alpha^{\log}_{p_{1}p_{2}r}=\sum_{A}N^{A}_{p_{1}p_{2}r}t^{A}\in\Bbbk[Q]/I,$

where the numbers $N^{A}_{p_{1}p_{2}r}$ are defined in [GS7, Def. 3.21] as
certain punctured invariants with contact orders $p_{1},p_{2}$ and $-r$.

In this section, we show that in fact we obtain the same algebra, i.e.,

######  Theorem 6.1.

For all $p_{1},p_{2}\in B(\mathbb{Z})\setminus\\{0\\}$, $r\in B(\mathbb{Z})$,

$\alpha^{\mathrm{trop}}_{p_{1}p_{2}r}=\alpha^{\log}_{p_{1}p_{2}r}.$

Before proving this theorem, let us introduce some additional terminology.

######  Definition 6.2.

A _product type_ with inputs $p_{1},p_{2}\in B(\mathbb{Z})$ and output $r\in
B(\mathbb{Z})$ 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 $L(G)=\\{L_{1},L_{2},L_{\mathrm{out}}\\}$ with
$p_{i}\in{\boldsymbol{\sigma}}(L_{i})$, $\mathbf{u}(L_{i})=p_{i}$,
${\boldsymbol{\sigma}}(L_{\mathrm{out}})\in\mathscr{P}$,
$r\in{\boldsymbol{\sigma}}(L_{\mathrm{out}})$ and ${\bf
u}(L_{\mathrm{out}})=-r$.

  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 $\tau_{v_{\mathrm{out}}}\in\Gamma(G,\ell)$ be
the cone corresponding to the vertex $v_{\mathrm{out}}$ of $G$ adjacent to
$L_{\mathrm{out}}$. Then $\dim\tau=\dim h(\tau_{v_{\mathrm{out}}})=n$.

A _decorated product type_ is a decorated type
${\boldsymbol{\tau}}=(\tau,\mathbf{A})$ with $\tau$ a product type.

Identically to Lemmas 3.9 and 3.20, we have

######  Lemma 6.3.

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

We now define

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

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

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

We split the proof of Theorem 6.1 into several steps.

######  Theorem 6.4.

Fix $p_{1},p_{2},r\in B(\mathbb{Z})$, and let $\mathfrak{u}$ be a chamber of
$\mathscr{P}_{\mathscr{B}}$ containing $r$. Then there exists a top-
dimensional subcone $\mathfrak{u}^{\prime}\subseteq\mathfrak{u}$ containing
$r$ such that

$\alpha^{\log}_{p_{1}p_{2}r}=\sum_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}k_{\tau}N_{{\boldsymbol{\tau}}}t^{A}\in\Bbbk[Q]/I,$

where the sum is over all isomorphism classes of decorated product types with
inputs $p_{1},p_{2}$, output $r$, and $\mathfrak{u}^{\prime}\subseteq
h(\tau_{v_{\mathrm{out}}})$. Here $A$ is the total class of ${\bf A}$.

###### Proof.

Throughout the proof, we assume familiarity with the notation of [GS7], and do
not review it, giving only references where appropriate.

Step 1. _Alternative description of $N^{A}_{p_{1}p_{2}r}$._ Fix a curve class
$A\in Q\setminus I$, and let $\beta$ be the type of punctured map of curve
class $A$, with three punctured points $x_{1},x_{2},x_{\mathrm{out}}$ with
contact orders given by $p_{1},p_{2}$ and $-r$ respectively. Thus we obtain
moduli spaces of punctured maps $\mathscr{M}(X,\beta)$ and
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)$, where evaluation is
at $x_{\mathrm{out}}$ only.

Let $\sigma_{\mathfrak{u}}\in\mathscr{P}$ be the maximal cell containing
$\mathfrak{u}$. Let $P_{\mathfrak{u}}$ be the stalk of
$\overline{\mathcal{M}}_{X}$ at the zero-dimensional stratum
$X_{\sigma_{\mathfrak{u}}}$. Let $R:=\mathbb{N}^{n}$ be generated by
$e_{1},\ldots,e_{n}$, with dual monoid generated by
$e_{1}^{*},\ldots,e_{n}^{*}$. Choose a monoid homomorphism
$\psi:P_{\mathfrak{u}}\rightarrow R$ in such a way that the transpose
$\psi^{t}:R^{\vee}\rightarrow
P_{\mathfrak{u}}^{\vee}=\sigma_{\mathfrak{u}}\cap\Lambda_{\mathfrak{u}}$ is
(1) injective; (2) has image contained in $\mathfrak{u}$. Further, if
$r\not=0$, we also require (3) $\psi^{t}(e_{1}^{*})=r$.

If $r\not=0$, define $W$ as the log stack quotient of
$\operatorname{Spec}(R\rightarrow\Bbbk)$ by the $\mathbb{G}_{m}$ action which
acts on the torsor associated to $m\in R$ with weight $\langle
e_{1}^{*},m\rangle$. Put another way, $\underline{W}=B\mathbb{G}_{m}$,
$\overline{\mathcal{M}}_{W}=R$, and the torsor associated to $m\in R$ is
$\mathcal{U}^{\otimes\langle e_{1}^{*},m\rangle}$ where $\mathcal{U}$ is the
universal torsor on $B\mathbb{G}_{m}$. If, on the other hand, $r=0$, we set

$W:=\operatorname{Spec}(R\rightarrow\Bbbk)\times B\mathbb{G}_{m}^{\dagger},$

where $B\mathbb{G}_{m}^{\dagger}$ denotes the log structure on
$B\mathbb{G}_{m}$ induced by the inclusion as a divisor
$B\mathbb{G}_{m}\subseteq[\mathbb{A}^{1}/\mathbb{G}_{m}]$.

We may then define a morphism $g:W\rightarrow\mathscr{P}(X,r)$, where the
latter stack is as defined in [GS7, §3.1]. Using the notation of [GS7], write
$Z_{r}$ for the stratum $X_{\sigma}$ where $\sigma\in\mathscr{P}$ is the
minimal cone containing $r$. Then $\mathscr{P}(X,r)$ may be described as
$[Z_{r}/\mathbb{G}_{m}]$, where $\mathbb{G}_{m}$ acts trivially on
$\underline{Z}_{r}$ and acts on the log structure on $Z_{r}$ as described in
[GS7, Rem. 3.1]. Note that since $r\in\sigma_{\mathfrak{u}}$,
$X_{\sigma_{\mathfrak{u}}}\subseteq Z_{r}$. There then exists a morphism
$g^{\prime}:\operatorname{Spec}(R\rightarrow\Bbbk)\rightarrow
X_{\sigma_{\mathfrak{u}}}$ which is an isomorphism on underlying schemes and
such that $\overline{(g^{\prime})}^{\flat}=\psi$. We also denote by
$g^{\prime}$ the composition with the inclusion into $Z_{r}$. If $r\not=0$,
then the action of $\mathbb{G}_{m}$ on
$\operatorname{Spec}(R\rightarrow\Bbbk)$ used to define $W$ is compatible with
the action of $\mathbb{G}_{m}$ on $Z_{r}$ using the fact that
$\psi^{t}(e_{1}^{*})=r$ and the description of [GS7, Rem. 3.1], and hence
$g^{\prime}$ descends to the desired map $g:W\rightarrow\mathscr{P}(X,r)$. If,
on the other hand, $r=0$, $\mathscr{P}(X,r)=X\times B\mathbb{G}_{m}$, and
hence we obtain the desired morphism $g:W\rightarrow\mathscr{P}(X,r)$ as the
product of $g^{\prime}$ with the morphism
$B\mathbb{G}_{m}^{\dagger}\rightarrow B\mathbb{G}_{m}$ which is an isomorphism
on underlying stacks.

We have canonical morphisms
${\operatorname{ev}}_{\mathcal{X}}:\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)\rightarrow\mathscr{P}(X,r)$
and ${\operatorname{ev}}_{X}:\mathscr{M}(X,\beta)\rightarrow\mathscr{P}(X,r)$,
see [GS7, Def. 3.5]. We may then define

$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}=W\times_{\mathscr{P}(X,r)}^{\mathrm{fs}}\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta),\quad\mathscr{M}(X,\beta)_{W}=W\times_{\mathscr{P}(X,r)}^{\mathrm{fs}}\mathscr{M}(X,\beta).$

By choosing the image of $\psi^{t}$ to be sufficiently small, we may assume
that $g$ is transverse to ${\operatorname{ev}}_{\mathcal{X}}$ in the sense of
[GS7, Def. 2.6], see [GS7, Thm. 2.9] for the necessary tropical criterion. By
[GS7, Def. 7.9], we then obtain a number

$N^{A,W}_{p_{1}p_{2}r}=\deg[\mathscr{M}(X,\beta)_{W}]^{\mathrm{virt}}$

with the virtual fundamental class defined via an obstruction theory for

$\varepsilon_{W}:\mathscr{M}(X,\beta)_{W}\rightarrow\mathfrak{M}^{{\operatorname{ev}}}(X,\beta)_{W}$

pulled back from that for
$\mathscr{M}(X,\beta)\rightarrow\mathfrak{M}^{{\operatorname{ev}}}(X,\beta)$.
By [GS7, Lem. 7.8], $\mathfrak{M}^{{\operatorname{ev}}}(X,\beta)_{W}$ is pure-
dimensional of dimension zero, so that the virtual pull-back of its
fundamental class gives the virtual fundamental class of
$\mathscr{M}(X,\beta)_{W}$.

The hypotheses of [GS7, Thm. 7.10] now hold, and we obtain
$N^{A,W}_{p_{1}p_{2}r}=N^{A}_{p_{1}p_{2}r}$. Thus by (6.2), to prove the
theorem it is sufficient to take
$\mathfrak{u}^{\prime}:=\psi^{t}(R^{\vee}_{\mathbb{R}})$ and show that

$N^{A,W}_{p_{1}p_{2}r}=\sum_{{\boldsymbol{\tau}}}k_{\tau}N_{{\boldsymbol{\tau}}}$

where the sum is over all isomorphism classes of decorated product types with
total class $A$ as in the statement of the theorem.

We now make a slight change in the definition of $W$,
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$ and
$\mathscr{M}(X,\beta)_{W}$ in the case that $r=0$, to make the remainder of
the proof more uniform. Note that in this case,

$\displaystyle\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}={}$
$\displaystyle(\operatorname{Spec}(R\rightarrow\Bbbk)\times
B\mathbb{G}_{m}^{\dagger})\times^{\mathrm{fs}}_{X\times
B\mathbb{G}_{m}}\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)$
$\displaystyle={}$ $\displaystyle
B\mathbb{G}_{m}^{\dagger}\times_{B\mathbb{G}_{m}}((\operatorname{Spec}(R\rightarrow\Bbbk)\times
B\mathbb{G}_{m})\times^{\mathrm{fs}}_{\mathscr{P}(X,r)}\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)).$

The stack $\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$ then has
the same underlying stack as

$(\operatorname{Spec}(R\rightarrow\Bbbk)\times
B\mathbb{G}_{m})\times_{\mathscr{P}(X,r)}^{\mathrm{fs}}\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta),$

but with the ghost sheaf on the former being a sum of the ghost sheaf of the
latter and the constant sheaf $\underline{\mathbb{N}}$. As we are only
concerned about virtual fundamental classes, which are defined using the
underlying morphism of stacks
$\underline{\mathscr{M}(X,\beta)}_{W}\rightarrow\underline{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)}_{W}$,
we may thus replace $W$ with $\operatorname{Spec}(R\rightarrow\Bbbk)\times
B\mathbb{G}_{m}$, which also changes
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$ and
$\mathscr{M}(X,\beta)_{W}$. Note that now the definition of $W$ is uniform
across the two cases, and whether or not $r=0$, the ghost sheaf of $W$ is $R$.

Step 2. _The structure of $\mathfrak{M}^{{\operatorname{ev}}}(X,\beta)_{W}$._
We recall from [GS7, Lem. 7.8] and its proof that the projection
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}\rightarrow W$ is
integral, log smooth and of fibre dimension and log fibre dimension one. For
the latter notion, see the review of [GS7, Def. A.5].

Let $\bar{\eta}$ be a generic point of an irreducible component
$\mathfrak{M}_{\bar{\eta}}$ of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$. We view the
former as an integral closed substack of the latter, so that

$[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}]=\sum_{\bar{\eta}}\mu_{\bar{\eta}}[\mathfrak{M}_{\bar{\eta}}]$

in the Chow group of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$. Here the sum is
over all generic points $\bar{\eta}$, and $\mu_{\bar{\eta}}$ is the
multiplicity of $\mathfrak{M}_{\bar{\eta}}$ in
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$.

Let $Q_{\bar{\eta}}$ be the stalk of the ghost sheaf of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$ at $\bar{\eta}$.
We then have

(6.3)
$\operatorname{rk}Q^{{\operatorname{gp}}}_{\bar{\eta}}=\operatorname{rk}\overline{\mathcal{M}}_{W}^{{\operatorname{gp}}}=n;$

the first equality comes from [GS7, (A.1)] and the above-mentioned equality of
log fibre dimension and fibre dimension, while the second equality is from the
definition of $W$, as $R=\mathbb{N}^{n}$. Let
$\tau=(G,{\boldsymbol{\sigma}},{\bf u})$ be the type of the punctured map
corresponding to $\bar{\eta}$.

Note that $\varepsilon_{W}^{!}[\mathfrak{M}_{\bar{\eta}}]$ is a zero-
dimensional cycle in the Chow group of $\mathscr{M}(X,\beta)_{W}$. Assume that
$\deg\varepsilon_{W}^{!}[\mathfrak{M}_{\bar{\eta}}]\not=0$. We claim that in
this case, $\tau$ is a product type. Definition 6.2,(1) is immediate by the
definition of $\beta$. For Condition (2) of the definition, note that $\tau$
is necessarily realizable as it is the type of a punctured map to
$\mathcal{X}$. On the other hand, the assumption of non-zero virtual degree
implies that for some decoration ${\boldsymbol{\tau}}$ of $\tau$ with total
curve class $A$, $\mathscr{M}(X,{\boldsymbol{\tau}})$ is non-empty. In
particular, one may find a punctured map $C^{\circ}\rightarrow X$ defined over
a geometric point whose type is $\tau^{\prime}$ with a contraction
$\tau^{\prime}\rightarrow\tau$. Hence by Lemma 2.1, $\tau^{\prime}$ is a
balanced type, and hence $\tau$ is also balanced.

For Condition (3) of the definition of product type, let $Q_{\tau}$ be the
basic monoid associated to the type $\tau$, so that
$Q^{\vee}_{\tau,\mathbb{R}}$ coincides with the cone $\tau$. Note that in
general $Q_{\bar{\eta}}\not=Q_{\tau}$, as the fibre product construction of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$ changes the log
structure. We wish to show that $\dim\tau=n$.

From [ACGS1, Prop. 5.2], we have
$Q_{\bar{\eta},\mathbb{R}}^{\vee}=\tau\times_{\sigma_{\mathfrak{u}}}R^{\vee}_{\mathbb{R}}$
where the map $R^{\vee}_{\mathbb{R}}\rightarrow\sigma_{\mathfrak{u}}$ is just
the inclusion $\psi^{t}$ and the map $\tau\rightarrow\sigma_{\mathfrak{u}}$,
induced by the morphism ${\operatorname{ev}}_{\mathcal{X}}$, is given by
evaluation at the vertex $v_{\mathrm{out}}$ adjacent to $L_{\mathrm{out}}$,
i.e., is $h|_{\tau_{v_{\mathrm{out}}}}$. See [GS7, Lem. 3.23] for this latter
statement.

Note the images of the two projections
$Q_{\bar{\eta},\mathbb{R}}^{\vee}\rightarrow\tau,R^{\vee}_{\mathbb{R}}$
intersect the interiors of $\tau$ and $R^{\vee}_{\mathbb{R}}$ respectively.
This is because these morphisms of cones are dual to local morphisms of
monoids, being induced by the two projections of log stacks
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}\rightarrow\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta),W$.
From this we may calculate the dimension of $Q^{\vee}_{\bar{\eta},\mathbb{R}}$
by computing its tangent space at a point in the interior of
$Q^{\vee}_{\bar{\eta},\mathbb{R}}$ mapping to the interiors of $\tau$ and
$R^{\vee}_{\mathbb{R}}$ via the two projections. This tangent space is
$Q^{*}_{\bar{\eta},\mathbb{R}}=\tau^{{\operatorname{gp}}}\times_{\sigma_{\mathfrak{u}}^{{\operatorname{gp}}}}R_{\mathbb{R}}^{*}=\tau^{{\operatorname{gp}}}$
as $R^{*}_{\mathbb{R}}\rightarrow\sigma^{{\operatorname{gp}}}_{\mathfrak{u}}$
is an isomorphism of real vector spaces. Thus $\dim\tau=\dim
Q^{\vee}_{\bar{\eta},\mathbb{R}}=n$, the latter equality by (6.3). Hence
$\dim\tau=\dim h(\tau_{v_{\mathrm{out}}})=n$, as desired.

Step 3. _The multiplicity of $\mathfrak{M}_{\bar{\eta}}$ in
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$._ We use the log
smoothness of the projection
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\eta)_{W}\rightarrow W$
already mentioned in Step 2. Note that $W$ carries a natural idealized log
structure given by the ideal $R\setminus\\{0\\}\subseteq R$. This ideal pulls
back to give an idealized structure on
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$, and an idealized
strict morphism101010We recall a morphism of idealized log schemes
$f:(X,\mathcal{M}_{X},\mathcal{K}_{X})\rightarrow(Y,\mathcal{M}_{Y},\mathcal{K}_{Y})$
is idealized strict if $\mathcal{K}_{X}$ agrees with the monoid ideal
generated by $f^{\flat}(\mathcal{K}_{Y})$. is idealized log smooth if and only
if it is log smooth, see [Og, IV,Variant 3.1.22]. Thus
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}\rightarrow W$ is
idealized log smooth, and hence, smooth locally at $\bar{\eta}$, is given as
$\mathcal{A}_{Q_{\bar{\eta}},K}\rightarrow\mathcal{A}_{R,R\setminus\\{0\\}}$
by [ACGS2, Prop. B.4], where $K\subseteq Q_{\bar{\eta}}$ is the monoid ideal
generated by the image of $R\setminus\\{0\\}$ in $Q_{\bar{\eta}}$ under the
map $R\rightarrow Q_{\bar{\eta}}$.

The multiplicity $\mu_{\bar{\eta}}$ of the component
$\mathfrak{M}_{\bar{\eta}}$ may now be calculated as the multiplicity of
$\mathcal{A}_{Q_{\bar{\eta}},K}$, or equivalently,
$\dim_{\Bbbk}\Bbbk[Q_{\bar{\eta}}]/K$. Note that
$Q^{\vee}_{\bar{\eta},\mathbb{R}}\rightarrow R^{\vee}_{\mathbb{R}}$ is
surjective, as follows from the integrality of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}\rightarrow W$ and
[GS7, Prop. 2.4]. Further, $Q^{\vee}_{\bar{\eta},\mathbb{R}}$ and
$R^{\vee}_{\mathbb{R}}$ are of the same dimension. It thus follows that
$R_{\mathbb{R}}\rightarrow Q_{\bar{\eta},\mathbb{R}}$ is an isomorphism of
real cones. Identifying $Q^{{\operatorname{gp}}}_{\bar{\eta}}$ with a super-
lattice of $R^{{\operatorname{gp}}}=\mathbb{Z}^{n}$, the multiplicity is then
the number of points of $Q^{{\operatorname{gp}}}_{\bar{\eta}}$ contained in
$[0,1)^{n}\subseteq R_{\mathbb{R}}$. This may be expressed as

(6.4)
$\mu_{\tau}:=\mu_{\bar{\eta}}=|\operatorname{coker}(R^{{\operatorname{gp}}}\rightarrow
Q^{{\operatorname{gp}}}_{\bar{\eta}})|,$

noting this expression only depends on the type $\tau$.

Step 4. _Comparison with the $N_{{\boldsymbol{\tau}}}$._ Consider the
commutative diagram

$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}_{\tau}}$$\scriptstyle{k}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\beta)_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\tau}^{\prime}}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\tau}}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\tau}}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)}$

where

$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W}=W\times^{\mathrm{fs}}_{\mathscr{P}(X,r)}\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau),\quad\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\beta)_{W}=W\times^{\mathrm{fs}}_{\mathscr{P}(X,r)}\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\beta).$

In this diagram, all squares are Cartesian in both the log and fs log
categories, as all horizontal morphisms are strict. The morphism
$\iota_{\tau}$, hence also $\iota^{\prime}_{\tau}$, is a closed embedding,
while $j_{\tau}$, hence also $j^{\prime}_{\tau}$, is a finite morphism of
degree $|\operatorname{Aut}(\tau)|$. Further, as
$W\rightarrow\mathscr{P}(X,r)$ is a finite morphism, so is $k$.

For a stack $M$, denote by $M_{{\operatorname{red}}}$ the reduction. Then

(6.5)
$\iota^{\prime}_{\tau,*}[\mathfrak{M}^{{\operatorname{ev}}}_{\tau}(\mathcal{X},\beta)_{W,{\operatorname{red}}}]=\sum_{\bar{\eta}}[\mathfrak{M}_{\bar{\eta}}]$

where the sum is over all generic points $\bar{\eta}$ of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$ whose type is
$\tau$. Indeed, any such generic point $\bar{\eta}$ corresponds to a punctured
map $C^{\circ}/\bar{\eta}\rightarrow\mathcal{X}$ in
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)$ of type $\tau$, hence
lies in $\mathfrak{M}_{\tau}^{{\operatorname{ev}}}(\mathcal{X},\beta)$.
Combining (6.5) with $\deg j^{\prime}_{\tau}=|\operatorname{Aut}(\tau)|$ gives

(6.6) $(\iota_{\tau}^{\prime}\circ
j^{\prime}_{\tau})_{*}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W,{\operatorname{red}}}]=|\operatorname{Aut}(\tau)|\sum_{\bar{\eta}}[\mathfrak{M}_{\bar{\eta}}].$

Let
$k_{{\operatorname{red}}}:\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W,{\operatorname{red}}}\rightarrow\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)$
be the morphism induced by $k$. Bearing in mind that
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)$ is reduced, we may now
calculate the degree of the finite morphism $k_{{\operatorname{red}}}$ as
follows. We take a (strict) geometric generic point
$\operatorname{Spec}\kappa\rightarrow\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)$.
The desired degree is the length of the reduction of the fibre

(6.7)
$\displaystyle\begin{split}\operatorname{Spec}\kappa\times_{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)}\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W}={}&\operatorname{Spec}\kappa\times_{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)}(\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)\times^{\mathrm{fs}}_{\mathscr{P}(X,r)}W)\\\
={}&\operatorname{Spec}(Q_{\tau}\rightarrow\kappa)\times^{\mathrm{fs}}_{\mathscr{P}(X,r)}W.\end{split}$

We may replace $\mathscr{P}(X,r)$ with the strict closed substack $Z$ with
underlying closed substack $\underline{X}_{\sigma_{\mathfrak{u}}}\times
B\mathbb{G}_{m}$ as the morphism
$\operatorname{Spec}\kappa\rightarrow\mathscr{P}(X,r)$ factors through this
closed substack. Further, the morphism $\operatorname{Spec}\kappa\rightarrow
Z$ factors through $X_{\sigma_{\mathfrak{u}}}$ as all line bundles on
$\operatorname{Spec}\kappa$ are trivial. Thus we may rewrite (6.7) as

(6.8)
$\operatorname{Spec}(Q_{\tau}\rightarrow\kappa)\times^{\mathrm{fs}}_{X_{\sigma_{\mathfrak{u}}}}\operatorname{Spec}(R\rightarrow\Bbbk)=(A_{Q_{\tau},Q_{\tau}\setminus\\{0\\}}\times^{\mathrm{fs}}_{A_{P_{\mathfrak{u}},P_{\mathfrak{u}}\setminus\\{0\\}}}A_{R,R\setminus\\{0\\}})\times_{\operatorname{Spec}\Bbbk}\operatorname{Spec}\kappa,$

again following the notation of [ACGS2, §B]. By [GS7, Prop. A.4,(4)], (6.8)
may then be expressed as

$A_{Q_{\tau}\oplus^{\mathrm{fs}}_{P_{\mathfrak{u}}}R,J}\times_{\operatorname{Spec}\Bbbk}\operatorname{Spec}\kappa$

where $J$ is the ideal generated by the image of $Q_{\tau}\setminus\\{0\\}$
and $R\setminus\\{0\\}$. We pass to the reduction by replacing $J$ with its
radical, which, by the explicit description of the fs push-out of [GS7, Prop.
A.4,(3)], is the complement of the group of invertible elements of
$Q_{\tau}\oplus_{P_{\mathfrak{u}}}^{\mathrm{fs}}R$. Further, this group of
invertible elements is the torsion part of the cokernel of the map
$(\varphi_{1},-\varphi_{2}):P_{\mathfrak{u}}^{{\operatorname{gp}}}\rightarrow
Q_{\tau}^{{\operatorname{gp}}}\oplus R^{{\operatorname{gp}}}$, where
$\varphi_{1}:P_{\mathfrak{u}}^{{\operatorname{gp}}}\rightarrow
Q_{\tau}^{{\operatorname{gp}}}$ and
$\varphi_{2}:P_{\mathfrak{u}}^{{\operatorname{gp}}}\rightarrow
R^{{\operatorname{gp}}}$ are induced by
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)\rightarrow\mathscr{P}(X,r)$
and $W\rightarrow\mathscr{P}(X,r)$ respectively. Thus we see that the reduced
fibre is

$\operatorname{Spec}\kappa[\operatorname{coker}(P_{\mathfrak{u}}^{{\operatorname{gp}}}\rightarrow
Q_{\tau}^{{\operatorname{gp}}}\oplus
R^{{\operatorname{gp}}})_{\mathrm{tors}}],$

whose length of course is given by the order of this group of invertible
elements.

In conclusion, we see that

(6.9)
$k_{{\operatorname{red}},*}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W,{\operatorname{red}}}]=|\operatorname{coker}(P_{\mathfrak{u}}^{{\operatorname{gp}}}\rightarrow
Q_{\tau}^{{\operatorname{gp}}}\oplus
R^{{\operatorname{gp}}})_{\mathrm{tors}}|[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)].$

Bearing in mind that
$Q_{\bar{\eta}}=(Q_{\tau}\oplus^{\mathrm{fs}}_{P_{\mathfrak{u}}}R)/\mathrm{tors}$,
it again follows from the explict description of the fs pushout in [GS7, Prop.
A.4,(3)] that

$Q_{\bar{\eta}}^{{\operatorname{gp}}}=\operatorname{coker}(P_{\mathfrak{u}}^{{\operatorname{gp}}}\rightarrow
Q_{\tau}^{{\operatorname{gp}}}\oplus R^{{\operatorname{gp}}})/\mathrm{tors},$

and then a simple diagram chase gives

(6.10)
$\displaystyle\begin{split}|\operatorname{coker}(P_{\mathfrak{u}}^{{\operatorname{gp}}}\rightarrow
Q_{\tau}^{{\operatorname{gp}}}\oplus
R^{{\operatorname{gp}}})_{\mathrm{tors}}||\operatorname{coker}(R^{{\operatorname{gp}}}\rightarrow
Q_{\bar{\eta}}^{{\operatorname{gp}}})|={}&|\operatorname{coker}(P_{\mathfrak{u}}^{{\operatorname{gp}}}\rightarrow
Q_{\tau}^{{\operatorname{gp}}})|\\\
={}&|\operatorname{coker}(Q_{\tau}^{*}\rightarrow
P_{\mathfrak{u}}^{*})|=k_{\tau}.\end{split}$

We next consider the diagram

$\textstyle{\coprod_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}\mathscr{M}(X,{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{\tau}}$$\textstyle{\coprod_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}\mathscr{M}(X,{\boldsymbol{\tau}})_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{\tau,W}}$$\scriptstyle{k^{\prime}}$$\textstyle{\mathscr{M}(X,\beta)_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{W}}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\scriptstyle{\iota^{\prime}_{\tau}\circ
j^{\prime}_{\tau}}$$\textstyle{\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}}$

where the disjoint unions are over all decorations of $\tau$ with total degree
$A$. Now observe that

$\displaystyle
N^{A,W}_{p_{1}p_{2}r}=\deg[\mathscr{M}(X,\beta)_{W}]^{\mathrm{virt}}={}$
$\displaystyle\deg\varepsilon_{W}^{!}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}]$
$\displaystyle={}$
$\displaystyle\deg\sum_{\bar{\eta}}\mu_{\bar{\eta}}\varepsilon_{W}^{!}[\mathfrak{M}_{\bar{\eta}}].$

Here we are summing over all generic points $\bar{\eta}$ of
$\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\beta)_{W}$, but of course we
may restrict the sum to those $\bar{\eta}$ such that
$\deg\varepsilon^{!}_{W}[\mathfrak{M}_{\bar{\eta}}]\not=0$. Thus by (6.4),
(6.6) and Step 2, we may rewrite this equation as

$N^{A,W}_{p_{1}p_{2}r}=\deg\sum_{\tau}{\mu_{\tau}\over|\operatorname{Aut}(\tau)|}\varepsilon_{W}^{!}(\iota^{\prime}_{\tau}\circ
j^{\prime}_{\tau})_{*}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W,{\operatorname{red}}}]$

where, by Step 2, the sum is now over all product types $\tau$ with inputs
$p_{1},p_{2}$, output $r$, and $\mathfrak{u}^{\prime}\subseteq
h_{\tau}(\tau_{v_{\mathrm{out}}})$. Now using the push-pull formula for
virtual pull-back of [Mn, Thm. 4.1] for the first and third equality, as well
as (6.4), (6.9) and (6.10) for the third, we obtain

$\displaystyle N^{A,W}_{p_{1}p_{2}r}={}$
$\displaystyle\deg\sum_{\tau}{\mu_{\tau}\over|\operatorname{Aut}(\tau)|}\varepsilon_{\tau,W}^{!}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W,{\operatorname{red}}}]$
$\displaystyle={}$
$\displaystyle\deg\sum_{\tau}{\mu_{\tau}\over|\operatorname{Aut}(\tau)|}k^{\prime}_{*}\varepsilon_{\tau,W}^{!}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)_{W,{\operatorname{red}}}]$
$\displaystyle={}$
$\displaystyle\deg\sum_{\tau}{k_{\tau}\over|\operatorname{Aut}(\tau)|}\varepsilon_{\tau}^{!}[\mathfrak{M}^{{\operatorname{ev}}}(\mathcal{X},\tau)]$
$\displaystyle={}$ $\displaystyle\sum_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}k_{\tau}N_{{\boldsymbol{\tau}}},$

as desired. ∎

######  Lemma 6.5.

Let $\tau=(G,{\boldsymbol{\sigma}},{\mathbf{u}})$ be a product type with
$N_{{\boldsymbol{\tau}}}\not=0$ for some decoration ${\boldsymbol{\tau}}$ of
$\tau$, and let $G^{\prime}$ be the spine of $G$ in the sense of Definition
2.4. Then $v_{\mathrm{out}}$ is the unique trivalent vertex of $G^{\prime}$.
Further, $v_{\mathrm{out}}$ is also a trivalent vertex of $G$, and if
$v_{\mathrm{out}}$ and $L_{\mathrm{out}}$ are removed from $G$, then $\tau$
splits into two broken line types $\tau_{1},\tau_{2}$.

###### Proof.

Since $G^{\prime}$ has three legs and no univalent vertices, $G^{\prime}$ has
a unique trivalent vertex. Suppose that $v_{\mathrm{out}}$ is not this
trivalent vertex. Necessarily ${\boldsymbol{\sigma}}(v_{\mathrm{out}})$ is a
maximal cell $\sigma\in\mathscr{P}^{{\operatorname{max}}}$ since $\dim
h(\tau_{\mathrm{out}})=n$, and thus for any punctured map
$f:C^{\circ}\rightarrow X$ in the non-empty moduli space
$\mathscr{M}(X,{\boldsymbol{\tau}})$, the union of irreducible components of
$C^{\circ}$ corresponding to $v_{\mathrm{out}}$ are mapped to the zero-
dimensional stratum $X_{\sigma}$. Hence by stability, $v_{\mathrm{out}}$ must
be at least trivalent in $G$, and thus there is an edge $E$ adjacent to
$v_{\mathrm{out}}$ which is not contained in $G^{\prime}$. Let
$\bar{\tau}=(\bar{G},\bar{{\boldsymbol{\sigma}}},\bar{\bf u})$ be the type
obtained by cutting $G$ at the edge $E$ and taking the connected component of
the resulting graph not containing $L_{1},L_{2}$ and $L_{\mathrm{out}}$. Thus
$E$ becomes the unique leg of $\bar{G}$. Let $h_{\bar{\tau}}$ be the universal
tropical map of type $\bar{\tau}$. If
$\bar{\tau}_{E}\in\Gamma(\bar{G},\bar{\ell})$ is the cone corresponding to
$E$, then by Lemma 2.5,(1), $\dim h_{\bar{\tau}}(\bar{\tau}_{E})\leq n-1$. But
since $\dim h(\tau_{v_{\mathrm{out}}})=n$, we obtain a contradiction. Thus
$v_{\mathrm{out}}$ is a trivalent vertex of $G^{\prime}$.

A similar argument applies to show that $v_{\mathrm{out}}$ is also trivalent
in $G$: if the valency is higher than three, with an adjacent edge $E$ not an
edge of $G^{\prime}$, we may again cut at $E$ to obtain a contradiction as
above.

Thus by removing $v_{\mathrm{out}},L_{\mathrm{out}}$ from $G$, we obtain, for
$i=1,2$, types $\tau_{i}$ with legs $L_{i},L_{i,\mathrm{out}}$, where
$L_{1,\mathrm{out}},L_{2,\mathrm{out}}$ arise from the edges (or legs)
$E_{1},E_{2}$ adjacent to $v_{\mathrm{out}}$ other than $L_{\mathrm{out}}$. We
only need to show that $\tau_{i}$ is a broken line type.

First consider Condition (1) of Definition 3.19. The only thing to check here
is that ${\boldsymbol{\sigma}}(L_{i,\mathrm{out}})\in\mathscr{P}$ and that
${\bf u}(L_{i,\mathrm{out}})\not=0$. For the first statement, note that as
${\boldsymbol{\sigma}}(v_{\mathrm{out}})=\sigma$ and
${\boldsymbol{\sigma}}(v_{\mathrm{out}})\subseteq{\boldsymbol{\sigma}}(E_{i})={\boldsymbol{\sigma}}(L_{i,\mathrm{out}})$,
necessarily ${\boldsymbol{\sigma}}(L_{i,\mathrm{out}})=\sigma$ also. For the
second statement, note that if ${\bf u}(L_{i,\mathrm{out}})=0$, then we may
vary $s\in\tau$ by changing only the affine length of the edge $E_{i}$ of $G$,
so that $h_{s}(v_{\mathrm{out}})$ remains unchanged. Since $\dim
h(\tau_{v_{\mathrm{out}}})=n$, this would imply that $\dim\tau>n$, a
contradiction.

Condition (2) of Definition 3.19 is immediate. Finally, for Condition (3), it
follows from $\dim h(\tau_{v_{\mathrm{out}}})=n$ that $\dim
h_{\tau_{i}}(\tau_{i,\mathrm{out}})=n$, and in particular $\dim\tau_{i}\geq
n-1$. However, if $\dim\tau_{i}>n-1$ for $i=1$ or $2$, it then immediately
follows by gluing together tropical maps in the families $\tau_{1},\tau_{2}$
that $\dim\tau>n$. Thus we get $\dim\tau_{i}=n-1$, as desired. ∎

######  Lemma 6.6.

Let ${\boldsymbol{\tau}}$ be a decorated product type with
$N_{{\boldsymbol{\tau}}}\not=0$, and let
${\boldsymbol{\tau}}_{1},{\boldsymbol{\tau}}_{2}$ be the decorated broken line
types obtained from splitting ${\boldsymbol{\tau}}$ at $v_{\mathrm{out}}$.
Then

$k_{\tau}N_{{\boldsymbol{\tau}}}=k_{\tau_{1}}k_{\tau_{2}}N_{{\boldsymbol{\tau}}_{1}}N_{{\boldsymbol{\tau}}_{2}}.$

###### Proof.

The proof is by a straightforward application of Yixian Wu’s gluing formula
Theorem A.4. We first assume that neither $\tau_{1}$ nor $\tau_{2}$ is a
trivial broken line, that is, has no vertex. Then $v_{\mathrm{out}}$ has two
adjacent edges $E_{1}$,$E_{2}$. Splitting ${\boldsymbol{\tau}}$ at these two
edges leads to, by Lemma 6.5, the decorated broken line types
${\boldsymbol{\tau}}_{1}$, ${\boldsymbol{\tau}}_{2}$, and a third decorated
type ${\boldsymbol{\tau}}_{0}=(\tau_{0},\mathbf{A}_{0})$,
$\tau_{0}=(G_{0},{\boldsymbol{\sigma}}_{0},\mathbf{u}_{0})$ with only one
vertex $v_{\mathrm{out}}$ and three adjacent legs $(E_{1},v_{\mathrm{out}})$,
$(E_{2},v_{\mathrm{out}})$ and $L_{\mathrm{out}}$, the outgoing leg of $\tau$.
Moreover, $\tau_{0}$ is a type entirely contained in the maximal cell $\sigma$
containing $h(\tau_{v_{\mathrm{out}}})$, that is,
${\boldsymbol{\sigma}}_{0}(v_{\mathrm{out}})=\sigma$. In particular, the curve
class $\mathbf{A}_{0}(v_{\mathrm{out}})$ is trivial. Thus any punctured map of
type ${\boldsymbol{\tau}}_{0}$ is constant on underlying schemes and has
domain a $\mathbb{P}^{1}$ with three punctured points. It then follows as in
[GS7, Claim 3.22] that there is a unique basic punctured map
$f:C^{\circ}/W\rightarrow X$ of type ${\boldsymbol{\tau}}_{0}$ with
$\underline{W}=\operatorname{Spec}\Bbbk$. Since further
$\mathscr{M}(X,{\boldsymbol{\tau}}_{0})$ is unobstructed over
$\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}}_{0})$,
$\mathscr{M}(X,{\boldsymbol{\tau}}_{0})$ is a reduced point.

It is now clear by stability of the tropical situation under small
perturbations, and can be checked also explicitly, that the tropical gluing
map $\varepsilon_{\omega}^{\operatorname{gp}}$ in (A.7) is surjective already
for $\omega=\tau$. Thus the trivial displacement vector $\nu=0$ in Definition
A.2 is general for $\tau$. Moreover, the dimension formula (A.8) for
$\omega=\tau$ holds:

$\textstyle\dim\widetilde{\tau}_{0}+\dim\widetilde{\tau}_{1}+\dim\widetilde{\tau}_{2}=(n+2)+n+n=(n+2)+2n=\dim\widetilde{\tau}+\sum_{E}\operatorname{rk}\Lambda_{E}.$

Thus the set of transverse types $\Delta(\nu)$ consist only of the one element
$\tau$.

To compute the splitting multiplicity $m({\boldsymbol{\tau}})$ (Definition
A.2,3) consider the following commutative diagram with exact rows and columns.

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\tau}^{\operatorname{gp}}_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{\tau}_{0}^{\operatorname{gp}})_{\mathbb{Z}}\times(\widetilde{\tau}_{1}^{\operatorname{gp}})_{\mathbb{Z}}\times(\widetilde{\tau}_{2}^{\operatorname{gp}})_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\varepsilon_{\tau}^{\operatorname{gp}})_{\mathbb{Z}}}$$\textstyle{\operatorname{im}(\varepsilon_{\tau}^{\operatorname{gp}})_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\times\mathbb{Z}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Lambda\times\mathbb{Z}^{2})\times\Lambda\times\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\times\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/k_{\tau}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/k_{\tau_{1}}\mathbb{Z}\times\mathbb{Z}/k_{\tau_{2}}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}(\varepsilon_{\tau}^{\operatorname{gp}})_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

Here $\Lambda=\sigma^{\operatorname{gp}}_{\mathbb{Z}}$, the first column is
the exact sequence defining $k_{\tau}$, along with a trivial $\mathbb{Z}^{2}$
factor in the first two entries, the middle column the product of the exact
sequences defining $k_{\tau_{1}}$ and $k_{\tau_{2}}$, and the isomorphism
$(\widetilde{\tau}_{0}^{\operatorname{gp}})_{\mathbb{Z}}\to\Lambda\times\mathbb{Z}^{2}$.
The third row now shows

$m({\boldsymbol{\tau}})=|\operatorname{coker}((\varepsilon_{\tau}^{\operatorname{gp}})_{\mathbb{Z}})|=\frac{k_{\tau_{1}}k_{\tau_{2}}}{k_{\tau}}.$

Applying Theorem A.4 we conclude

$\deg\big{[}\mathscr{M}(X,{\boldsymbol{\tau}})\big{]}^{\mathrm{virt}}=\frac{k_{\tau_{1}}k_{\tau_{2}}}{k_{\tau}}\cdot\deg\big{[}\mathscr{M}(X,{\boldsymbol{\tau}}_{1})\big{]}^{\mathrm{virt}}\cdot\deg\big{[}\mathscr{M}(X,{\boldsymbol{\tau}}_{2})\big{]}^{\mathrm{virt}}.$

The claimed equality
$k_{\tau}N_{{\boldsymbol{\tau}}}=k_{\tau_{1}}k_{\tau_{2}}N_{{\boldsymbol{\tau}}_{1}}N_{{\boldsymbol{\tau}}_{2}}$
now follows by the definition of $N_{{\boldsymbol{\tau}}}$,
$N_{{\boldsymbol{\tau}}_{1}}$, $N_{{\boldsymbol{\tau}}_{2}}$ and noting that
$\operatorname{Aut}({\boldsymbol{\tau}})=\operatorname{Aut}({\boldsymbol{\tau}}_{1})\times\operatorname{Aut}({\boldsymbol{\tau}}_{2})$.

If one of $\tau_{1}$ or $\tau_{2}$ is a trivial broken line, the same analysis
omitting the trivial broken line factor applies to prove the result in this
case. If both $\tau_{1},\tau_{2}$ are trivial broken lines the statement is
trivially true. ∎

###### Proof of Theorem 6.1..

Take $x\in\operatorname{Int}(\mathfrak{u})$ in the definition of
$\alpha^{\mathrm{trop}}_{p_{1}p_{2}r}$; by consistency of
$\mathscr{S}_{{\mathrm{can}}}$, (6.1) is independent of this choice of $x$.
Further, in (6.1), we may sum over decorated broken lines, rather than broken
lines, and get the same value. Thus by Theorem 4.14, we may instead write

$\alpha^{\mathrm{trop}}_{p_{1}p_{2}r}=\sum_{{\boldsymbol{\tau}}_{1}=(\tau_{1},{\bf
A}_{1}),{\boldsymbol{\tau}}_{2}=(\tau_{2},{\bf
A}_{2})}k_{\tau_{1}}k_{\tau_{2}}N_{{\boldsymbol{\tau}}_{1}}N_{{\boldsymbol{\tau}}_{2}}t^{A_{1}+A_{2}}$

where now the sum is over admissible decorated broken line types
${\boldsymbol{\tau}}_{1},{\boldsymbol{\tau}}_{2}$ with $\mathfrak{u}\subseteq
h_{{\boldsymbol{\tau}}_{i}}(\tau_{i,\mathrm{out}})$, ${\bf
u}_{{\boldsymbol{\tau}}_{i}}(L_{i,\mathrm{in}})=p_{i}$, and
$u_{\tau_{1}}+u_{\tau_{2}}=r$. (Note that by definition of
$\mathscr{S}_{\mathscr{B}}$, if $x\in
h_{{\boldsymbol{\tau}}_{i}}(\tau_{i,\mathrm{out}})$ then
$\mathfrak{u}\subseteq h_{{\boldsymbol{\tau}}_{i}}(\tau_{i,\mathrm{out}})$.)

For such a pair ${\boldsymbol{\tau}}_{1},{\boldsymbol{\tau}}_{2}$, we obtain a
decorated product type by taking a vertex $v_{\mathrm{out}}$ and gluing
$G_{1}$ and $G_{2}$ to $v_{\mathrm{out}}$ via the legs $L_{1,\mathrm{out}}$,
$L_{2,\mathrm{out}}$, and add an additional leg $L_{\mathrm{out}}$ adjacent to
$v_{\mathrm{out}}$ with ${\bf u}(L_{\mathrm{out}})=-r$. The data
${\boldsymbol{\sigma}},{\bf u},{\bf A}$ are then determined by
${\boldsymbol{\tau}}_{1}$, ${\boldsymbol{\tau}}_{2}$, with ${\bf
A}(v_{\mathrm{out}})=0$.

We see ${\boldsymbol{\tau}}$ is a decorated product type. Indeed, we only need
to check conditions (2) and (3) of Definition 6.2. Balancing just needs to be
checked at $v_{\mathrm{out}}$, which follows from $r={\bf
u}_{1}(L_{1,\mathrm{out}})+{\bf
u}_{2}(L_{2,\mathrm{out}})=u_{\tau_{1}}+u_{\tau_{2}}$. For realizability and
the dimension conditions (3), note that by the corresponding dimension
conditions for broken line types, for each point
$y\in\operatorname{Int}(\mathfrak{u})$, there exists a unique
$s_{i}\in\tau_{i}$ such that $y\in h_{\tau_{i},s_{i}}(L_{i,\mathrm{out}})$.
Thus by gluing together $h_{\tau_{1},s_{1}}$ and $h_{\tau_{2},s_{2}}$, we
obtain a tropical map realizing $\tau$ which takes $v_{\mathrm{out}}$ to $y$.
Furthermore, this map is necessarily the unique such tropical map of type
$\tau$, giving both realizability and the dimension statements of (3).

Conversely, by Lemma 6.5, all product types ${\boldsymbol{\tau}}$ arise in
this way. Thus we obtain using Lemma 6.6 that

$\alpha^{\mathrm{trop}}_{p_{1}p_{2}r}=\sum_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}k_{\tau}N_{{\boldsymbol{\tau}}}t^{A},$

which coincides with $\alpha^{\log}_{p_{1}p_{2}r}$ by Theorem 6.4. ∎

##  Appendix A The gluing formula with toric gluing strata

Let $B=\operatorname{Spec}(Q_{B}\longrightarrow\Bbbk)$ be a logarithmic point,
$X$ a Zariski log scheme and $X\longrightarrow B$ a logarithmically smooth,
integral, projective morphism. We assume $\Sigma(X)$ monodromy-free for
simplicity. As before we write $\mathcal{X}$ for the Artin fan of $X$. Let
${\boldsymbol{\tau}}=(\tau,\mathbf{A})$ with
$\tau=(G,{\boldsymbol{\sigma}},\bar{\mathbf{u}})$ be a realizable decorated
global type of a punctured map of genus $0$.111111The restriction to genus $0$
is for convenience, since this is the case relevant to us and the general
statement is slightly more complicated. Recall from [ACGS2, Def. 3.8] the
stacks $\mathscr{M}(X,{\boldsymbol{\tau}})$ and
$\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}})$ of punctured stable maps to
$X$ and to $\mathcal{X}$ marked by ${\boldsymbol{\tau}}$.

For each vertex $v\in V(G)$ denote by
${\boldsymbol{\tau}}_{v}=(\tau_{v},\mathbf{A}_{v})$,
$\tau_{v}=(G_{v},{\boldsymbol{\sigma}}_{v},\bar{\mathbf{u}}_{v})$ the
decorated global type with $V(G_{v})=\\{v\\}$ and $E(G_{v})=\emptyset$
obtained by splitting $G$ at all edges.121212Splitting at only a subset of
edges works the same way, with a little more bookkeeping notation necessary.
For simplicity of presentation we only discuss splitting at all edges. Note
that each $\tau_{v}$ is realizable as well. Then
$\mathfrak{M}(\mathcal{X},\tau)$ is locally pure-dimensional, and hence
defines a virtual fundamental class for $\mathscr{M}(X,{\boldsymbol{\tau}})$
by means of the obstruction theory for $\mathscr{M}(X,{\boldsymbol{\tau}})$
over $\mathfrak{M}(\mathcal{X},\tau)$.

According to [ACGS2, Cor. 5.13, Prop. 5.15], the splitting morphism

(A.1) $\delta:\underline{\mathscr{M}}(X,{\boldsymbol{\tau}})\ \longrightarrow\
\textstyle\prod_{v\in V(G)}\underline{\mathscr{M}}(X,{\boldsymbol{\tau}}_{v})$

is finite and representable.131313We suppress here the log structure which is
irrelevant for the formulation of the statement, although it is central for
its proof. The gluing formula expresses
$\delta_{*}[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}$ in terms of
the virtual fundamental classes of strata of
$\mathscr{M}(X,{\boldsymbol{\tau}}_{v})$. A stratum is given by a
${\boldsymbol{\tau}}_{v}$-marked decorated global type
${\boldsymbol{\omega}}_{v}$ as the image of the finite map

$j_{{\boldsymbol{\omega}}_{v}}:\mathscr{M}(X,{\boldsymbol{\omega}}_{v})\longrightarrow\mathscr{M}(X,{\boldsymbol{\tau}}_{v})$

defined by changing the marking [ACGS2, Rem. 3.28].

We now explain some more notation needed to state the gluing formula. For each
edge $E\in E(G)$ we obtain a gluing stratum
$\sigma_{E}={\boldsymbol{\sigma}}(E)\in\Sigma(X)$ defined by $\tau$. Note that
if $v,v^{\prime}\in V(G)$ are the vertices adjacent to $E$ and $L\in
L(G_{v})$, $L^{\prime}\in L(G_{v^{\prime}})$ are the legs in
$G_{v},G_{v^{\prime}}$ obtained by splitting $E$, then by the definition of
$\tau_{v}$,

${\boldsymbol{\sigma}}_{v}(L)={\boldsymbol{\sigma}}_{v^{\prime}}(L^{\prime})=\sigma_{E}.$

We write $L=(E,v)$, $L^{\prime}=(E,v^{\prime})$ in the following, where the
notation indicates that $v\in V(G_{v})\subseteq V(G)$ and $v^{\prime}\in
V(G_{v^{\prime}})\subseteq V(G)$ are the vertices adjacent to $E$.

Now the restriction of a $\tau_{v}$-marked basic tropical punctured map,
defined say over $\omega_{v}\in\mathbf{Cones}$, to the leg $(E,v)\in L(G_{v})$
defines a map of cones

(A.2) $\omega_{E,v}=\big{\\{}(h,\lambda)\in\omega_{v}\times\mathbb{R}_{\geq
0}\,\big{|}\,\lambda\leq\ell(E,v)(h)\big{\\}}\longrightarrow\sigma,$

for some $\sigma\in\Sigma(X)$ containing
$\sigma_{E}={\boldsymbol{\sigma}}_{v}(E,v)$ as a face. Recall also that
$\omega_{v}$ is the basic cone, or real cone with integral points the dual of
the basic monoid [ACGS2, Def. 2.36], associated to $\omega_{v}$. We conflate
notations for types and associated basic cones and hence write $\omega_{v}$
also for this type unless there is a danger of confusion. Similarly,
$\tau_{v}$ denotes both a global type and the basic cone
$(Q_{\tau_{v}})_{\mathbb{R}}^{\vee}$ associated to the type. Note also that
the marking by $\tau_{v}$ induces a face embedding

(A.3) $\tau_{v}=(Q_{\tau_{v}})_{\mathbb{R}}^{\vee}\longrightarrow\omega_{v}.$

of basic cones.

Letting $(E,v)$ in (A.2) run over all the gluing legs, we arrive at the
following enlargement of the cone $\omega_{v}$ that records a point on each
puncturing leg:

(A.4)
$\widetilde{\omega}_{v}=\big{\\{}(h,\lambda_{E,v})\in\omega_{v}\times\mathbb{R}_{\geq
0}^{L(G_{v})}\,\big{|}\,\lambda_{E,v}\leq\ell(E,v)(h)\big{\\}}.$

For each leg $(E,v)\in L(G_{v})$, the projection forgetting all
$\lambda$-components but $\lambda_{E,v}$ defines a map of cones

(A.5) $\widetilde{\omega}_{v}\longrightarrow\omega_{E,v}.$

Similarly, for a $\tau$-marked tropical punctured map defined over
$\omega\in\mathbf{Cones}$, recording a point on each edge $E\in E(G)$ leads to

(A.6)
$\widetilde{\omega}=\big{\\{}(h,\lambda_{E})\in\omega\times\mathbb{R}_{\geq
0}^{E(G)}\,\big{|}\,\lambda_{E}\leq\ell(E)(h)\big{\\}}.$

We now add the following assumption:

######  Assumption A.1.

Assume all gluing strata $\underline{Z}_{E}$, $E\in E(G)$, are complete toric
varieties, with $\overline{\mathcal{M}}_{X}|_{\underline{Z}_{E}}$ invariant
under the torus action.

Under this assumption there is an embedding of the star of
$\sigma_{E}\in\Sigma(X)$ as a face-fitting complex of cones in the real vector
space associated to a lattice that we denote $\Lambda_{E}$. Moreover, this
embedding identifies the integral affine structures on each cone, and the
composition with the quotient by the image of the real subspace with lattice
$(\sigma_{E})_{\mathbb{Z}}^{\operatorname{gp}}\cap\Lambda_{E}$ maps the face-
fitting complex of cones to a complete fan. If $Z_{E}$ is strictly embedded as
a toric stratum of a toric variety, $\Lambda_{E}$ is the lattice underlying
the describing fan. We now view each of the cones $\sigma\in\Sigma(X)$
containing $\sigma_{E}$ as embedded in $(\Lambda_{E})_{\mathbb{R}}$.

Let $\omega=(G_{\omega},{\boldsymbol{\sigma}}_{\omega},\mathbf{u}_{\omega})$
be a not necessarily realizable global type of punctured map marked by $\tau$.
Splitting $\omega$ at the edges not contracted by the marking leads to a
collection $(\omega_{v})_{v\in V(G)}$ of types of tropical punctured maps,
with $\omega_{v}$ marked by $\tau_{v}$ and with a leg $(E,v)\in
L(G_{\omega_{v}})$ for each edge $E\in E(G)$ adjacent to $v$. For each $E\in
E(G)$ with $v,v^{\prime}$ the adjacent vertices, the composition of the
embeddings
${\boldsymbol{\sigma}}_{\omega}(E,v),{\boldsymbol{\sigma}}_{\omega}(E,v^{\prime})\to(\Lambda_{E})_{\mathbb{R}}$
with the quotient by the diagonal embedding
$(\Lambda_{E})_{\mathbb{R}}\longrightarrow(\Lambda_{E})_{\mathbb{R}}\times(\Lambda_{E})_{\mathbb{R}}$
defines a map

$\varepsilon_{E}:{\boldsymbol{\sigma}}_{\omega}(E,v)\times{\boldsymbol{\sigma}}_{\omega}(E,v^{\prime})\longrightarrow(\Lambda_{E})_{\mathbb{R}}\times(\Lambda_{E})_{\mathbb{R}}\longrightarrow(\Lambda_{E})_{\mathbb{R}}.$

Explicitly, the arrow on the right can be taken as the difference map
$(a,b)\mapsto a-b$.

Taking the product over all $E\in E(G)$ of the composition of
$\varepsilon_{E}$ with the maps (A.5) and (A.2) yields the map

(A.7) $\textstyle\varepsilon_{\omega}:\prod_{v\in
V(G)}\widetilde{\omega}_{v}\longrightarrow\prod_{E,v}\omega_{E,v}\longrightarrow\prod_{E}(\Lambda_{E})_{\mathbb{R}}\times(\Lambda_{E})_{\mathbb{R}}\longrightarrow\prod_{E}(\Lambda_{E})_{\mathbb{R}}.$

Thus $\varepsilon_{\omega}$ measures the failure of a collection of tropical
punctured maps of types $\omega_{v}$, together with points on the legs arising
from splitting, to patch to a tropical punctured map of type $\omega$ together
with a choice of point on each of the splitted edges. We refer to an element
of $\prod_{E}(\Lambda_{E})_{\mathbb{R}}$ as a _displacement vector_.

######  Definition A.2.

1) We call a displacement vector $\nu=(\nu_{E})_{E\in
E(G)}\in\prod_{E}(\Lambda_{E})_{\mathbb{R}}$ _general_ for $\tau$ if for each
global type $\omega$ marked by $\tau$, either
$\varepsilon_{\omega}^{\operatorname{gp}}$ from (A.7) is surjective or
$\nu\not\in\operatorname{im}(\varepsilon_{\omega}^{\operatorname{gp}})$.
2) Let $\nu=(\nu_{E})_{E}\in\prod_{E}\Lambda_{E}$ be general for $\tau$.
Define the _set $\Delta(\nu)$ of transverse types for $\nu$_ as the set of
isomorphism classes of global types $\omega$ marked by $\tau$ with
$\nu\in\operatorname{im}(\varepsilon_{\omega})$, and such that

(A.8)
$\textstyle\sum_{v}\dim\widetilde{\omega}_{v}=\dim\widetilde{\tau}+\sum_{E}\operatorname{rk}\Lambda_{E}.$

The set of decorated global types ${\boldsymbol{\omega}}=(\omega,\mathbf{A})$
marked by ${\boldsymbol{\tau}}$ and with $\omega\in\Delta(\nu)$ is denoted by
$\hat{\Delta}(\nu)$. We confuse a transverse type $\omega$ with its splitting
$(\omega_{v})_{v\in V(G)}$, and similarly in the decorated case.
3) For a general displacement vector $\nu$ and
${\boldsymbol{\omega}}=(\omega,\mathbf{A})\in\hat{\Delta}(\nu)$ define the
_splitting multiplicity_ by

$m({\boldsymbol{\omega}})=m(\omega):=\big{[}\textstyle\prod_{E}\Lambda_{E}:\operatorname{im}(\varepsilon_{\omega}^{\operatorname{gp}})_{\mathbb{Z}}\big{]}.$

######  Remark A.3.

The set $\Delta(\nu)$ has an interpretation in terms of types of “broken”
tropical punctured maps, in the sense that the matching condition along an
edge $E$ is replaced by matching translated by $\nu_{E}$. Specifically, let
$\omega$ be a global type marked by $\tau$ and $(\omega_{v})$ the collection
of global types marked by $\tau_{v}$ obtained by splitting. Then
$\omega\in\Delta(\nu)$ iff $\omega$ satisfies (A.8) and there exist
$h_{v}\in\omega_{v}$ and $\lambda_{v},\lambda_{v^{\prime}}\in\mathbb{R}_{>0}$
with

(A.9) $V(h_{v})+\lambda_{v}\cdot
u_{E}=V^{\prime}(h_{v^{\prime}})-\lambda_{v^{\prime}}\cdot u_{E}+\nu_{E},$

as an equation in $(\Lambda_{E})_{\mathbb{R}}$. For the signs we assume $E$
oriented from $v$ to $v^{\prime}$. Here
$V:\omega_{v}\longrightarrow(\Lambda_{E})_{\mathbb{R}}$ is the restriction of
the map in (A.2) to the face $\lambda=0$, that is, the image of the vertex $v$
under the tropical punctured map $h_{v}$, and analogously for $V^{\prime}$ and
$v^{\prime}$.

From this description it is also obvious that $\Delta(\nu)$ does not change by
rescaling $\nu$ by a positive constant. Moreover, if $\widetilde{\tau}$ is
defined analogously to $\widetilde{\omega}$, with $\lambda_{E}$ recording the
position of a point on the edge $E$, then $\Delta(\nu)$ is also unchanged by
adding to $\nu$ an element of the image of the map
$\widetilde{\tau}\longrightarrow\prod_{E}\Lambda_{E}$ evaluating at all points
on the splitting edges.

The dimension formula (A.8) is equivalent to requiring
$\omega=(\omega_{v})_{v}$ to be a minimal type of punctured map with
$\varepsilon_{\omega}^{\operatorname{gp}}$ surjective.

We are now in position to state the gluing formula.

######  Theorem A.4.

[Wu] Let ${\boldsymbol{\tau}}=(\tau,\mathbf{A})$,
$\tau=(G,{\boldsymbol{\sigma}},\bar{\mathbf{u}})$ be a realizable decorated
global type of punctured maps of genus $0$ to $X$, and
${\boldsymbol{\tau}}_{v}$ the associated decorated global type defined at
$v\in V(G)$ obtained from splitting all edges. Assume that all gluing strata
are toric in the sense of Assumption A.1. Let $\nu=(\nu_{E})_{E\in E(G)}$ be a
general displacement vector for $\tau$, and $\hat{\Delta}(\nu)$ the set of
decorated transverse types for $\nu$ (Definition A.2). Denote by $\delta$ the
splitting morphism from (A.1).

Then the following equality in the Chow group of $\prod_{v\in
V(G)}\underline{\mathscr{M}}(X,{\boldsymbol{\tau}}_{v})$ holds:

$\delta_{*}[\mathscr{M}(X,{\boldsymbol{\tau}})]^{\mathrm{virt}}=\sum_{{\boldsymbol{\omega}}=({\boldsymbol{\omega}}_{v})_{v}\in{\hat{\Delta}}(\nu)}\frac{m({\boldsymbol{\omega}})}{|\operatorname{Aut}({\boldsymbol{\omega}}/{\boldsymbol{\tau}})|}\cdot(j_{{\boldsymbol{\omega}}_{v}})_{*}\big{[}\textstyle\prod_{v}\mathscr{M}(X,{\boldsymbol{\omega}}_{v})\big{]}^{\mathrm{virt}}.$

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