We then have the corresponding basic monoid

$Q_{\bar{w}}:=\tau_{\bar{w}}^{\vee}\cap(N_{\bar{\tau}_{\bar{w}}}^{*}\oplus\mathbb{Z}).$

Note that

$Q_{\bar{w}}\subseteq N_{\bar{\tau}_{\bar{w}}}^{*}\oplus\mathbb{N}$

since $(0,1)\in\tau_{\bar{w}}$. The monoid $Q_{\bar{w}}$ will be the basic
monoid for the point $\bar{w}$ for the punctured log map to $X_{\sigma}$ we
are going to construct. Note the projection
$\tau_{\bar{w}}\rightarrow\bar{\tau}_{\bar{w}}$ dualizes to an inclusion
$\overline{Q}_{\bar{w}}\hookrightarrow Q_{\bar{w}}$ which identifies
$\overline{Q}_{\bar{w}}$ with the facet $\\{(q,0)\in Q_{\bar{w}}\\}$ of
$Q_{\bar{w}}$.

It is easy to check that if $\bar{w}^{\prime}$ is a generization of $\bar{w}$,
the above construction of $Q_{\bar{w}}$ is compatible with generization maps,
i.e., the generization map
$\overline{Q}_{\bar{w}}\rightarrow\overline{Q}_{\bar{w}^{\prime}}$ induces a
generization map $Q_{\bar{w}}\rightarrow Q_{\bar{w}^{\prime}}$. Indeed,
dually, we have an inclusion of faces
$\bar{\tau}_{w^{\prime}}\subseteq\bar{\tau}_{w}$, and
$\alpha:\bar{\tau}_{w}\rightarrow\mathbb{R}_{\geq 0}$ restricts to the
corresponding map for $\bar{\tau}_{w^{\prime}}$. Hence we obtain an inclusion
of faces $\tau_{\bar{w}^{\prime}}\subseteq\tau_{\bar{w}}$. From this, we see
that the monoids $Q_{\bar{w}}$ for various $\bar{w}$ define a fine subsheaf
$\overline{\mathcal{M}}_{W}$ of
$\overline{\mathcal{M}}_{\overline{W}}^{{\mathrm{gp}}}\oplus\mathbb{N}$
containing $\overline{\mathcal{M}}_{\overline{W}}\oplus\mathbb{N}$.

Let $W^{\dagger}=\overline{W}\times b_{0}$, so that
$\overline{\mathcal{M}}_{W^{\dagger}}=\overline{\mathcal{M}}_{\overline{W}}\oplus\mathbb{N}$,
and set

$\mathcal{M}_{W}:=\overline{\mathcal{M}}_{W}\times_{\overline{\mathcal{M}}_{W^{\dagger}}^{{\mathrm{gp}}}}\mathcal{M}_{W^{\dagger}}^{{\mathrm{gp}}}.$

We may then define a structure morphism
$\alpha_{W}:\mathcal{M}_{W}\rightarrow\mathcal{O}_{W}$ by taking
$\alpha_{W}=\alpha_{\overline{W}}$ on $\mathcal{M}_{\overline{W}}\oplus
0\subseteq\mathcal{M}_{W}$ and $\alpha_{W}$ taking the value $0$ on
$\mathcal{M}_{W}\smallsetminus(\mathcal{M}_{\overline{W}}\oplus 0)$. Thus we
obtain an fs log scheme $W$.

We have morphisms $W\rightarrow W^{\dagger}\rightarrow\overline{W}$ by
construction, and the log smooth curve $\overline{C}\rightarrow\overline{W}$
pulls back to give $C^{\dagger}\rightarrow W^{\dagger}$ and $C\rightarrow W$.
Note further $\bar{f}:\overline{C}\rightarrow\overline{X}_{\sigma}$ induces a
morphism $f^{\dagger}:C^{\dagger}\rightarrow\overline{X}_{\sigma}\times b_{0}$
defined over $b_{0}$. We wish to define a punctured structure $C^{\circ}$ on
$C$ yielding a commutative diagram

$\textstyle{C^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\dagger}}$$\textstyle{\overline{X}_{\sigma}\times
b_{0}}$

As
$\mathcal{M}_{W^{\dagger}}^{{\mathrm{gp}}}=\mathcal{M}_{W}^{{\mathrm{gp}}}$,
so that
$\mathcal{M}^{{\mathrm{gp}}}_{C^{\circ}}=\mathcal{M}^{{\mathrm{gp}}}_{C^{\dagger}}$,
and
$\mathcal{M}_{X_{\sigma}}^{{\mathrm{gp}}}=\mathcal{M}_{\overline{X}_{\sigma}\times
b_{0}}^{{\mathrm{gp}}}$, $\bar{f}^{\flat}$ and
$\overline{(f^{\dagger})}^{\flat}$ agree at the level of groups, and it is
sufficient to construct a pre-stable punctured log structure $C^{\circ}$ on
$C$ so that

(6.4)
$\overline{(f^{\dagger})}^{\flat}(\overline{\mathcal{M}}_{X_{\sigma}})\subseteq\overline{\mathcal{M}}_{C^{\circ}}.$

Note that the saturation of $\overline{\mathcal{M}}_{C^{\circ}}$ over
$\bar{w}\in W$, as well as the map
$\overline{(f^{\dagger})}^{\flat}:f^{-1}\overline{\mathcal{M}}_{X_{\sigma}}\rightarrow\overline{\mathcal{M}}_{C^{\circ}}^{\mathrm{sat}}$,
are completely determined locally near $\bar{w}$ by the data of the tropical
family of maps $h_{\bar{w}}$ of (6.3). Thus we only need to consider the
punctured points.

So let $p\in C_{\bar{w}}$ be a marked point corresponding to a leg $L$ of
$\bar{G}_{\bar{w}}$ (or the corresponding leg of $G_{\bar{w}}$). We have the
monoids $\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}$ (resp.
$P_{\boldsymbol{\sigma}(L)}$) which are the stalks of
$\overline{\mathcal{M}}_{\overline{X}_{\sigma}}$ (resp.
$\overline{\mathcal{M}}_{X_{\sigma}}$) at the generic point of the strata of
$\overline{X}_{\sigma}$ (resp. $X_{\sigma}$) corresponding to
$\bar{\boldsymbol{\sigma}}(L)$ (resp. $\boldsymbol{\sigma}(L)$). Then
$\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}$ is a codimension one face of
$P_{\boldsymbol{\sigma}(L)}$, and we have a commutative diagram with
horizontal arrows induced by $\bar{f}$, $f^{\dagger}$ and the family of
tropical maps parameterized by $\tau_{\bar{w}}$:

$\textstyle{\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{\overline{C},p}=\overline{Q}_{\bar{w}}\oplus\mathbb{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}\oplus\mathbb{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{C^{\dagger},p}=(\overline{Q}_{\bar{w}}\oplus\mathbb{N})\oplus\mathbb{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\boldsymbol{\sigma}(L)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{C^{\circ},p}\subseteq
Q_{\bar{w}}\oplus\mathbb{Z}}$

Here the inclusion
$\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}\oplus\mathbb{N}\rightarrow
P_{\boldsymbol{\sigma}(L)}$ identifies $P_{\boldsymbol{\sigma}(L)}$ with a
submonoid of
$\overline{P}^{{\mathrm{gp}}}_{\bar{\boldsymbol{\sigma}}(L)}\oplus\mathbb{N}$
with the facet $\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}$ of
$P_{\boldsymbol{\sigma}(L)}$ identified with
$\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}\oplus 0$. To obtain a pre-stable
puncturing, we take the stalk of the ghost sheaf of
$\overline{\mathcal{M}}_{C^{\circ}}$ at $p$ to be the submonoid of
$Q_{\bar{w}}\oplus\mathbb{Z}$ generated by $Q_{\bar{w}}\oplus\mathbb{N}$ and
the image of $P_{\boldsymbol{\sigma}(L)}$ under
$\overline{(f^{\dagger})}^{\flat}$. We just need to make sure that this
defines a puncturing by showing that if
$(\bar{m}_{1},\bar{m}_{2})\in\overline{\mathcal{M}}_{C^{\circ}}\smallsetminus\overline{\mathcal{M}}_{C}$
is in the image of $P_{\boldsymbol{\sigma}(L)}$, and $m_{1}$ is a lift of
$\bar{m}_{1}$ to a local section of $\mathcal{M}_{W}$, then
$\alpha_{W}(m_{1})=0$.

Let
$\varphi_{L}:\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}\oplus\mathbb{N}\rightarrow\overline{Q}_{\bar{w}}\oplus\mathbb{N}$
be the composition of $\overline{(f^{\dagger})}^{\flat}$ with the projection
to $Q_{\bar{w}}\oplus\mathbb{N}$. By construction, $\varphi_{L}$ preserves the
second component, i.e., $\varphi_{L}(p,r)=(q,r)$ for some $q$. Also by
construction, $\varphi_{L}^{{\mathrm{gp}}}$ coincides with the similarly
defined map on groups
$P^{{\mathrm{gp}}}_{\boldsymbol{\sigma}(L)}\rightarrow\overline{\mathcal{M}}_{C^{\circ},p}^{{\mathrm{gp}}}$
coming from $\overline{f}^{\flat}$, and
$\varphi_{L}^{{\mathrm{gp}}}(P_{\boldsymbol{\sigma}(L)}\smallsetminus\overline{P}_{\bar{\boldsymbol{\sigma}}(L)})\subseteq
Q_{\bar{w}}\smallsetminus\overline{Q}_{\bar{w}}$. Suppose
$(\bar{m}_{1},\bar{m}_{2})$ is in the image of $P_{\boldsymbol{\sigma}(L)}$
and $\bar{m}_{2}<0$. Then
$\bar{m}_{1}=\varphi_{L}^{{\mathrm{gp}}}(\bar{n}_{1})$ for some
$\bar{n}_{1}\in P_{\boldsymbol{\sigma}(L)}$. As $\mathbf{u}(L)$ is positive on
$\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}$, $p$ being a marked point on
$\overline{C}$, we see $\bar{n}_{1}\in
P_{\boldsymbol{\sigma}(L)}\smallsetminus\overline{P}_{\bar{\boldsymbol{\sigma}}(L)}$,
and hence $\varphi_{L}^{{\mathrm{gp}}}(\bar{n}_{1})\in
Q_{\bar{w}}\smallsetminus\overline{Q}_{\bar{w}}$. Hence, by definition of the
structure map $\alpha_{W}$, we have $\alpha_{W}(m_{1})=0$, as desired.

We have thus now constructed a basic punctured map $f:C^{\circ}/W\rightarrow
X_{\sigma}$. It is easy to check that this construction is functorial and
hence defines a morphism

$\underline{\mathscr{M}(\overline{X}_{\sigma},\bar{\boldsymbol{\tau}})}\rightarrow\underline{\mathscr{M}(X_{\sigma}/B,{\boldsymbol{\tau}})}.$

It is easy to check that this morphism is then inverse to the morphism
$\underline{\mathscr{M}(X_{\sigma}/B,{\boldsymbol{\tau}})}\rightarrow\underline{\mathscr{M}(\overline{X}_{\sigma},\bar{\boldsymbol{\tau}})}$
previously constructed. Hence these stacks are isomorphic. Note they carry
different log structures. Further, one easily checks the induced obstruction
theories are the same, as the construction of these obstruction theories does
not depend on the log structure. ∎

##  7\. The classical degeneration formula

The classical degeneration situation, originally considered by Li and Ruan in
[LR01] and developed in the algebro-geometric context by Jun Li in [Li02] is a
special case of the degeneration situation of the previous section. We have
already discussed this case in the context of the decomposition formula in
[ACGS1, §6.1]. We consider $X\rightarrow B$ a simple normal crossings
degeneration with $X_{0}=Y_{1}\cup Y_{2}$ a reduced union of two irreducible
components, with $Y_{1}\cap Y_{2}=D$ a smooth divisor in both $Y_{1}$ and
$Y_{2}$. In this case $\Sigma(X)=(\mathbb{R}_{\geq 0})^{2}$ with map
$\Sigma(X)\rightarrow\Sigma(B)$ given by $(x,y)\mapsto x+y$, so that
$\Delta(X)$ is a unit interval. It was shown in [ACGS1, Prop. 6.1.1] that if
$f:\Gamma\rightarrow\Delta(X)$ is a rigid tropical map, then all vertices of
$\Gamma$ map to endpoints of $\Delta(X)$ and all edges of $\Gamma$ surject
onto $\Delta(X)$. In this case, giving a rigid tropical map with target
$\Delta(X)$ is the same information as an admissible triple of [Li02].

Using the setup of §5, we immdiately obtain the logarithmic stable map version
of the main result of [Li02], as proved in [KLR]:

######  Theorem 7.1.

In the situation described above, let ${\boldsymbol{\tau}}$ be a decorated
type of rigid tropical map in $\Delta(X)$. There is a diagram of (non-
logarithmic) stacks

$\textstyle{\underline{\mathscr{M}(X_{0}/b_{0},{\boldsymbol{\tau}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{\underline{\mathscr{M}^{\mathrm{sch}}(X_{0}/b_{0},{\boldsymbol{\tau}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
V(G)}\underline{\mathscr{M}(\overline{X}_{\boldsymbol{\sigma}(v)},\bar{\boldsymbol{\tau}}_{v})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{E\in
E(G)}\underline{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\prod_{v\in
E\in E(G)}\underline{D}}$

with the square Cartesian and defining the space
$\underline{\mathscr{M}^{\mathrm{sch}}(X_{0}/b_{0},{\boldsymbol{\tau}})}$.
Further, $\phi^{\prime}$ is finite and

$\phi^{\prime}_{*}[\mathscr{M}(X_{0}/b_{0},{\boldsymbol{\tau}})]^{{\operatorname{virt}}}=m_{{\boldsymbol{\tau}}}^{-1}\left(\prod_{E\in
E(G)}w(E)\right)\Delta^{!}\left(\prod_{v\in
V(G)}[\mathscr{M}(\overline{X}_{\boldsymbol{\sigma}(v)},\bar{\boldsymbol{\tau}}_{v})]^{{\operatorname{virt}}}\right),$

where $w(E)$ is the index (degree of divisibility) of $\mathbf{u}(E)$.

###### Proof.

Using Theorem 6.1, the given diagram is a part of the diagram of Theorem 5.1.
The result will follow from Theorem 5.9. Thus we first verify the tropical
transversality condition, and calculate $\mu({\boldsymbol{\tau}})$.

Note that $\overline{N}_{\boldsymbol{\sigma}(v)}=0$, while each
$\overline{N}_{\boldsymbol{\sigma}(E)}$ can be identified with
$\mathbb{Z}\cong\mathbb{Z}(1,-1)\subseteq\mathbb{Z}^{2}$ for any vertex $v$,
edge $E$. Thus the morphism $\Psi$ of Definition 5.6 takes the form

$\Psi:\bigoplus_{E\in E(G)}\mathbb{Z}\rightarrow\bigoplus_{E\in
E(G)}\mathbb{Z}$

given by $\Psi\big{(}(\ell_{E})_{E\in
E(G)}\big{)}=\big{(}\ell_{E}\mathbf{u}(E)\big{)}$. In particular, the image
has finite index, and this index is $\prod_{E}w(E)$.

By Theorem 5.9, it is thus sufficient to show that $\underline{\mathrm{ev}}$
is flat. However, this follows immediately from Theorem 5.3, as
$\boldsymbol{\sigma}^{\prime}(L)=\boldsymbol{\sigma}(L)$ always in this
situation. ∎

##  8\. Applications to wall structures for type III degenerations of K3
surfaces

In this section, we will work with a specific kind of degeneration
$g:X\rightarrow S$. Here, we will use $S$ for the base log scheme rather than
$B$ as is done in the previous sections and in [ACGS2], as $B$ will
notationally play a different role. We make the following assumptions:

######  Assumptions 8.1.

  1. (1)

$\underline{S}$ is a one-dimensional non-singular scheme with closed point
$0\in\underline{S}$ inducing the divisorial log structure on $S$.

  2. (2)

$\underline{X}\rightarrow\underline{S}$ is a simple normal crossings
degeneration of K3 surfaces.

  3. (3)

The fibre over $0$, $X_{0}$, is the only singular fibre, necessarily a simple
normal crossings divisor, and the log structure on $X$ is the divisorial log
structure induced by $X_{0}$.

  4. (4)

With $D:=(X_{0})_{{\mathrm{red}}}$ the reduction, we have $K_{X/S}+D=0$.

  5. (5)

Any intersection of irreducible components of $D$ is connected.

  6. (6)

$D$ has a zero-dimensional stratum.

Here we will give a useful inductive description for the so-called _canonical
wall structure_ defined in [GS22]. We make use of an observation of
Ranganathan in [Ra, §6.5.2], following Parker [Pa], that gluing remains fairly
easy in the case the normal crossings divisor $D$ has at worst triple points
and the domain curve is genus $0$. Rather than give the general description of
this approach in our language, we just carry out the procedure in our
particular application.

###  8.1. Review of wall types and balancing

We begin by recalling certain concepts from [GS22]. By the assumptions on
$X\rightarrow S$ made above, all irreducible components of $D$ are good in the
sense of [GS22, §1.1]. Further, Assumptions 1.1 and 1.2 of [GS22] hold. Thus,
in the notation of that paper, we may take $B=|\Sigma(X)|$ and $\mathscr{P}$
the set of cones of $\Sigma(X)$, so that $(B,\mathscr{P})$ is a pseudo-
manifold as explained in [GS22, Prop. 1.3]. We set

$\Delta:=\bigcup_{\sigma\in\Sigma(X)\atop\operatorname{codim}\sigma\geq
2}\sigma,$

and $B_{0}:=B\smallsetminus\Delta$. Then [GS22, §1.3] gives the structure of
integral affine manifold to $B_{0}$. Further, by [GS22, Prop. 1.15], the
tropicalization of $g$ induces a map
$g_{{\mathrm{trop}}}:B\rightarrow\Sigma(S)=\mathbb{R}_{\geq 0}$ which is an
affine submersion. We set $B^{\prime}=g_{{\mathrm{trop}}}^{-1}(1)$, and

$\mathscr{P}^{\prime}=\\{\sigma\cap
g_{{\mathrm{trop}}}^{-1}(1)\,|\,\sigma\in\mathscr{P}\\},$

a polyhedral decomposition of $B^{\prime}$. In the notation of the previous
sections, $\mathscr{P}^{\prime}$ is the set of polyhedra in $\Delta(X)$, but
we wish to avoid this notation now to avoid conflict with $\Delta$ as the
discriminant locus. Here we also set

$\Delta^{\prime}:=B^{\prime}\cap\Delta,\quad\quad
B_{0}^{\prime}:=B^{\prime}\smallsetminus\Delta.$

We use the convention that for $\sigma\in\mathscr{P}$ and
$\sigma^{\prime}=\sigma\cap B^{\prime}\in\mathscr{P}^{\prime}$, we write
either $X_{\sigma}$ or $X_{\sigma^{\prime}}$ for the stratum of $X$
corresponding to $\sigma$. Under this convention, irreducible components of
$D$ correspond to vertices of $\mathscr{P}^{\prime}$, and a vertex $v$ has
integral coordinates if and only if the corresponding irreducible component
$X_{v}$ of $X_{0}$ appears with multiplicity $1$ in $X_{0}$. Thus the cells of
$\mathscr{P}^{\prime}$ are not in general lattice polytopes. In case $X_{0}$
is reduced, then [GS22, Prop. 1.16] applies, but we do not wish to make this
restriction. In any event, $B^{\prime}$ is still an affine manifold with
singularities.

We recall from [GS22, Lem. 2.1]:

######  Lemma 8.2.

Let $f:C^{\circ}/W\rightarrow X$ be a stable punctured map to $X$, with
$W=\operatorname{Spec}(Q\rightarrow\kappa)$ a geometric log point. For
$s\in\operatorname{Int}(Q^{\vee}_{\mathbb{R}})$, let $h_{s}:G\rightarrow B$ be
the corresponding tropical map. If $v\in V(G)$ satisfies $h_{s}(v)\in B_{0}$,
then $h_{s}$ satisfies the balancing condition at $v$. More precisely, if
$E_{1},\ldots,E_{n}$ are the legs and edges adjacent to $v$, oriented away
from $v$, then the contact orders $\mathbf{u}(E_{i})$ may be interpreted as
elements of $\Lambda_{h_{s}(v)}$, the stalk of the local system $\Lambda$ of
integral tangent vectors $h_{s}(v)$. In this group, the balancing condition

(8.1) $\sum_{i=1}^{m}\mathbf{u}(E_{i})=0$

is satisfied.

As a consequence, we say a realizable tropical type $\tau$ of tropical map to
$\Sigma(X)/\Sigma(S)$ is _balanced_ if, for any
$s\in\operatorname{Int}(\tau)$, the corresponding tropical map
$h_{s}:G\rightarrow B$ satisfies the balancing condition of this lemma.

We then define:

######  Definition 8.3.

A _wall type_ is a type $\tau=(G,\boldsymbol{\sigma},{\mathbf{u}})$ of
tropical map to $\Sigma(X)$ defined over $\Sigma(S)$ such that:

  1. (1)

$G$ is a genus zero graph with $L(G)=\\{L_{\mathrm{out}}\\}$ and
$u_{\tau}:={\bf u}(L_{\mathrm{out}})\not=0$.

  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 $\tau_{\mathrm{out}}\in\Gamma(G,\ell)$ the cone
corresponding to $L_{\mathrm{out}}$. Then $\dim\tau=1$ and $\dim
h(\tau_{\mathrm{out}})=2$.

A _decorated wall type_ is a decorated type ${\boldsymbol{\tau}}=(\tau,{\bf
A})$ with $\tau$ a wall type and

${\bf A}:V(G)\rightarrow\coprod_{\sigma\in\Sigma(X)}H_{2}(X_{\sigma})$

is a _refined decoration_ , i.e., ${\bf A}(v)\in
H_{2}(X_{\boldsymbol{\sigma}(v)})$ for $v\in V(G)$.

######  Remarks 8.4.

(1) The notion of refined decoration, in which curve classes lie in the group
of curve classes of the relevant stratum rather than of $X$, gives more
control over moduli spaces. In particular, the moduli space
$\mathscr{M}(X/S,{\boldsymbol{\tau}})$ for ${\boldsymbol{\tau}}$ carrying a
refined decoration is simply a union of connected components of the moduli
space where the decoration is obtained by pushing forward all curve classes
$\mathbf{A}(v)$ to $H_{2}(X)$.

(2) We note that, other than the issue of refined decorations, this definition
is slightly simpler than the one given in [GS22, Def. 3.6] in that (1) we have
specialized to the case that $\dim X=3$ and all irreducible components of $D$
are good; (2) $B$ does not have a boundary in our case; and (3) here we insist
that the type be defined over $\Sigma(S)$, but by [GS22, Prop. 3.7], this is
implied by [GS22, Def. 3.6]. As a consequence, the two notions of wall type
coincide in this particular case.

(3) We note that because of our restriction to $\dim X=3$, given a wall type
$\tau$, there is a unique tropical map $h^{\prime}_{\tau}:G\rightarrow B$ of
type $\tau$ factoring through $B^{\prime}$. We will write this as
$h^{\prime}_{\tau}:G\rightarrow B^{\prime}$, or $h^{\prime}$ when clear from
context. Such a tropical map is rigid.

It is shown in [GS22, Const. 3.13] that if ${\boldsymbol{\tau}}$ is a
decorated wall type, then
$\mathscr{M}(X/\Bbbk,{\boldsymbol{\tau}})\cong\mathscr{M}(X/S,{\boldsymbol{\tau}})$,
and this stack is proper over $\operatorname{Spec}\Bbbk$ and carries a zero-
dimensional virtual fundamental class. As a consequence, from now on we will
work over $\operatorname{Spec}\Bbbk$ rather than $S$, writing
$\mathscr{M}(X,{\boldsymbol{\tau}})$, $\mathfrak{M}(\mathcal{X},\tau)$ for the
relevant moduli spaces.

We then define

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

We also define the number $k_{\tau}$ as follows. The map
$h:\Gamma(G,\ell)\rightarrow\Sigma(X)$ induces a homomorphism

$h_{*}:N_{\tau_{\mathrm{out}}}\rightarrow
N_{\boldsymbol{\sigma}(L_{\mathrm{out}})}.$

We then define

$k_{{\boldsymbol{\tau}}}:=|\operatorname{coker}(h_{*})_{{\mathrm{tors}}}|.$

######  Example 8.5.

Let $v\in\mathscr{P}^{\prime}$ be a vertex with corresponding irreducible
component $X_{v}$. Let $D_{v}\subseteq X_{v}$ be the union of lower-
dimensional strata of $D$ contained in $X_{v}$. Suppose $X_{v}$ contains a
rational curve $E$ with self-intersection $-1$. Suppose further that $E$ meets
$D_{v}$ transversally at one point. This point is contained in a one-
dimensional stratum $X_{\rho}$ for $\rho\in\mathscr{P}^{\prime}$ an edge.
Necessarily, $v$ is one endpoint of $\rho$. Let $v^{\prime}$ be the other
endpoint of $\rho$, and $\mu,\mu^{\prime}$ be the multiplicities of the
irreducible components $X_{v},X_{v^{\prime}}$ in the fibre $X_{0}$. Let $k$ be
such that $\mu|k\mu^{\prime}$.

Consider a type $\tau$ where $G$ has one vertex $w$, one leg $L$, and no
edges. We take $\boldsymbol{\sigma}(w)$ to be the ray of $\mathscr{P}$
corresponding to $v$ and $\boldsymbol{\sigma}(L)$ to be the cone of
$\mathscr{P}$ corresponding to $\rho$. Before specifying $\mathbf{u}(L)$, we
decorate $\tau$ by taking $\mathbf{A}(w)=k[E]\in H_{2}(X_{v})$. Then
$\mathbf{A}(w)\cdot X_{v^{\prime}}=k$. Since $E\subseteq X_{v}\cup
X_{v^{\prime}}$ and $E\cdot X_{0}=0$, necessarily $\mathbf{A}(w)\cdot(\mu
X_{v}+\mu^{\prime}X_{v^{\prime}})=0$. From this we conclude that
$\mathbf{A}(w)\cdot X_{v}=-k\mu^{\prime}/\mu$, which is an integer by
assumption. Thus there is a unique choice of $\mathbf{u}(L)$ compatible with
these intersections, by Corollary 2.2.

In this case, we may apply Theorem 6.1 to see that
$\mathscr{M}(X/S,{\boldsymbol{\tau}})\cong\mathscr{M}(\overline{X}_{v},\bar{\boldsymbol{\tau}})$
in the notation of §6. Here $\overline{X}_{v}$ will be the log scheme
structure on $\underline{X}_{v}$ induced by the divisor
$D_{v}\subseteq\underline{X}_{v}$. Note that any stable log map of curve class
$kE$ is a multiple cover of $E$. Via the comparison of logarithmic and
relative invariants of [AMW], we may use the calculation of [GPS, Prop. 5.2]
to obtain that
$\deg[\mathscr{M}(X/S,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}=(-1)^{k+1}/k^{2}$.

On the other hand, a simple calculation shows that $k_{\tau}=k$.

Our goal here is to give an inductive method for computing $k_{\tau}W_{\tau}$,
the quantity which plays a key role in the construction of the canonical wall
structure of [GS22]. The methods here are particular to relative dimension
two, and the structure of these invariants in higher dimensions is more
subtle.

###  8.2. An invariant of Looijenga pairs

Recall a _Looijenga pair_ is a pair $(X,D)$ where $X$ is a non-singular
projective rational surface and $D$ is a reduced nodal anti-canonical divisor
with at least one node. In what follows, assume that $D$ has at least three
irreducible components. In this case, $D$ will be a cycle of rational curves,
and we write $D=D_{1}+\cdots+D_{n}$. We are again in the situation of [GS22],
and may again take $B=\Sigma(X)$ and $\mathscr{P}$ the set of cones of
$\Sigma(X)$. Then $B_{0}:=B\smallsetminus\\{0\\}$ carries the structure of an
integral affine manifold, see [GS22, §1.3] again, but the original
construction was given in [GHK, §1.2]. Then Lemma 8.2 still holds in this
case.

######  Example 8.6.

Suppose we are in the situation of Assumptions 8.1, with
$v\in\mathscr{P}^{\prime}$ a vertex and $X_{v}$ the corresponding irreducible
component of $D$. With $D_{v}$ the union of lower-dimensional strata of $D$
contained in $X_{v}$, it follows from the assumption that $K_{X/S}+D=0$ and
adjunction that $K_{X_{v}}+D_{v}=0$. Note that $D_{v}$ is nodal, and then it
is easy to see, e.g., by the classification of surfaces, that $(X_{v},D_{v})$
is a Looijenga pair.

Fix a class $\beta$ of logarithmic map of genus $0$ to a Looijenga pair
$(X,D)$, with $q+1$ marked points, and contact orders
$u_{1},\ldots,u_{q},u_{\mathrm{out}}$. Here, we assume that
$u_{k}=w_{k}\nu_{i_{k}}$,
$u_{\mathrm{out}}=w_{\mathrm{out}}\nu_{\mathrm{out}}$, where $\nu_{i_{k}}$ is
a primitive generator of a ray $\rho_{i_{k}}$ of $\mathscr{P}$ corresponding
to the irreducible component $D_{i_{k}}$ of $D$, and $w_{k}$ is a positive
integer, and similarly for $w_{\mathrm{out}}$ and $\nu_{\mathrm{out}}$. We do
not assume the irreducible components $D_{i_{k}}$ are distinct.

We then have a schematic evaluation map

(8.2)
$\underline{\mathrm{ev}}:\mathscr{M}(X,\beta)\rightarrow\prod_{k=1}^{q}D_{i_{k}}$

given by evaluating at the marked points with contact orders
$u_{1},\ldots,u_{q}$. It follows from a standard virtual dimension calculation
via Riemann-Roch that the virtual dimension of $\mathscr{M}(X,\beta)$ is $q$.
We define $N_{\beta}$ by the identity

(8.3)
$\underline{\mathrm{ev}}_{*}[\mathscr{M}(X,\beta)]^{{\operatorname{virt}}}=N_{\beta}\prod_{k=1}^{q}[D_{i_{k}}].$

We recall that subdivisions of $(B,\mathscr{P})$ correspond to log étale
birational morphisms $(\widetilde{X},\widetilde{D})\rightarrow(X,D)$, see
e.g., [GHK, Lem. 1.6] for this case. As the contact orders in $\beta$ for $X$
are integral points of $B$, they also determine a set of contact orders for
$\widetilde{X}$. This set of contact orders along with a curve class
$\widetilde{A}\in H_{2}(\widetilde{X})$ determines a class $\tilde{\beta}$ of
logarithmic map to $\widetilde{X}$. We have:

######  Lemma 8.7.

Let $\pi:\widetilde{X}\rightarrow X$ be a log étale birational morphism with
$\widetilde{X}$ non-singular. If $\mathscr{M}(X,\beta)$ is non-empty, then
there exists a unique curve class $\widetilde{A}\in H_{2}(\widetilde{X})$ with
$\pi_{*}\widetilde{A}=A$ such that $\mathscr{M}(\widetilde{X},\tilde{\beta})$
is non-empty. Further, with this choice of curve class,
$N_{\beta}=N_{\tilde{\beta}}$.

###### Proof.

We recall from [AW, Eq. (1)] that there is a Cartesian diagram (in all
categories) with strict vertical arrows

$\textstyle{\mathscr{M}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{M}(\pi)}$$\textstyle{\mathscr{M}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{M}(\widetilde{\mathcal{X}}\rightarrow\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{M}(\pi)}$$\textstyle{\mathfrak{M}(\mathcal{X})}$

(where the moduli spaces have no restriction on type). Further, [AW, Prop.
5.2.1] shows that $\mathfrak{M}(\pi)$ is of pure degree $1$, while [HW, Lem.
4.2] shows that $\mathfrak{M}(\pi)$ is proper. Taken together, this implies
$\mathfrak{M}(\pi)$ is surjective, and hence the union of connected components
of $\mathscr{M}(\widetilde{X})$ lying over $\mathscr{M}(X,\beta)$ is non-
empty. Given a class $\tilde{\beta}$ of such a stable log map lying over
$\mathscr{M}(X,\beta)$, the contact orders of $\tilde{\beta}$ are determined
by the contact orders of $\beta$, and hence the only unknown is the curve
class $\widetilde{A}\in H_{2}(\widetilde{X})$, which necessarily satisfies
$\pi_{*}\widetilde{A}=A$ by construction of the map $\mathscr{M}(\pi)$. On the
other hand, for any irreducible component $\widetilde{D}_{\rho}$ of
$\widetilde{D}$, the intersection number
$\widetilde{D}_{\rho}\cdot\widetilde{A}$ is completely determined by the
contact orders of $\tilde{\beta}$, by Corollary 2.2. Since the intersection
matrix of the set of exceptional curves of $\pi$ is negative definite, and any
two choices of lifts of $A$ differ by a linear combination of exceptional
curves, it follows that $\widetilde{A}$ is uniquely determined.

Identifying $\widetilde{D}_{i}$ with $D_{i}$ using $\pi$, we now have a
commutative diagram

$\textstyle{\mathscr{M}(\widetilde{X},\tilde{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\underline{\mathrm{ev}}^{\prime}}}$$\scriptstyle{\mathscr{M}(\pi)}$$\textstyle{\mathscr{M}(X,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mathrm{ev}}}$$\textstyle{\prod_{k=1}^{q}D_{i_{k}}}$

By [AW, Thm. 1.1.1],
$\mathscr{M}(\pi)_{*}[\mathscr{M}(\widetilde{X},\tilde{\beta})]^{{\operatorname{virt}}}=[\mathscr{M}(X,\beta)]^{{\operatorname{virt}}}$.
The result follows. ∎

Alternatively, we may define $N_{\beta}$ as follows. Choose points $x_{k}\in
D_{i_{k}}$, $1\leq k\leq q$, and write $\mathbf{x}:=(x_{1},\ldots,x_{q})$.
Define

(8.4)
$\mathscr{M}(X,\beta,\mathbf{x}):=\mathscr{M}(X,\beta)\times_{\prod_{k}D_{i_{k}}}\mathbf{x}.$

######  Proposition 8.8.

$\mathscr{M}(X,\beta,\mathbf{x})$ carries a virtual fundamental class of
virtual dimension zero, and
$\deg[\mathscr{M}(X,\beta,\mathbf{x})]^{{\operatorname{virt}}}=N_{\beta}$.

###### Proof.

Let $D_{i}^{\circ}\subseteq D_{i}$ be the open subset obtained by deleting the
two double points of $D$ contained in $D_{i}$. We then have a diagram:

$\textstyle{\mathscr{M}(X,\beta,\mathbf{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{\mathbf{x}}}$$\textstyle{\mathscr{M}(X,\beta)^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime\prime}}$$\scriptstyle{\varepsilon^{\circ}}$$\textstyle{\mathscr{M}(X,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\beta,\mathbf{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mathrm{ev}}_{\mathbf{x}}}$$\scriptstyle{\iota^{\prime}}$$\textstyle{\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\beta)^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mathrm{ev}}^{\circ}}$$\scriptstyle{j^{\prime}}$$\textstyle{\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mathrm{ev}}}$$\textstyle{\mathbf{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\prod_{k=1}^{q}D_{i_{k}}^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\prod_{k=1}^{q}D_{i_{k}}}$

Here the evaluation space $\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\beta)$ is
given by evaluation at the marked points with contact orders
$u_{1},\ldots,u_{q}$, and all other stacks are defined by the requirement that
all squares are cartesian in the category of ordinary stacks. The relative
obstruction theory for $\varepsilon$ then pulls back to give relative
obstruction theories for $\varepsilon^{\circ}$ and $\varepsilon_{\mathbf{x}}$.

By compatibility of flat pullback with proper pushfoward, and compatibility of
virtual pullback with flat pullback, we have

$\displaystyle
N_{\beta}\prod_{k}[D_{i_{k}}^{\circ}]=j^{*}\left(N_{\beta}\prod_{k}[D_{i_{k}}]\right)={}$
$\displaystyle
j^{*}(\underline{\mathrm{ev}}\circ\varepsilon)_{*}[\mathscr{M}(X,\beta)]^{{\operatorname{virt}}}$
$\displaystyle={}$
$\displaystyle(\underline{\mathrm{ev}}^{\circ}\circ\varepsilon^{\circ})_{*}(j^{\prime\prime})^{*}[\mathscr{M}(X,\beta)]^{{\operatorname{virt}}}$
$\displaystyle={}$
$\displaystyle(\underline{\mathrm{ev}}^{\circ}\circ\varepsilon^{\circ})_{*}[\mathscr{M}(X,\beta)^{\circ}]^{{\operatorname{virt}}}.$

Further, $\underline{\mathrm{ev}}^{\circ}$ is flat, being a base-change of an
evaluation map
$\mathfrak{M}(\mathcal{X},\beta)^{\circ}\rightarrow\prod_{k}[D_{i_{k}}^{\circ}/\mathbb{G}_{m}^{2}]$.
Hence, the Gysin maps $\iota^{!}$ and $(\iota^{\prime})^{!}$ agree, and by
compatibility of Gysin maps with virtual pullback, we have

$\displaystyle\deg[\mathscr{M}(X,\beta,\mathbf{x})]^{{\operatorname{virt}}}=\deg(\underline{\mathrm{ev}}_{\mathbf{x}}\circ\varepsilon_{\mathbf{x}})_{*}[\mathscr{M}(X,\beta,\mathbf{x})]^{{\operatorname{virt}}}={}$
$\displaystyle\deg(\underline{\mathrm{ev}}_{\mathbf{x}}\circ\varepsilon_{\mathbf{x}})_{*}\iota^{!}[\mathscr{M}(X,\beta)^{\circ}]^{{\operatorname{virt}}}$
$\displaystyle={}$
$\displaystyle\deg\iota^{!}(\underline{\mathrm{ev}}^{\circ}\circ\varepsilon^{\circ})_{*}[\mathscr{M}(X,\beta)^{\circ}]^{{\operatorname{virt}}}$
$\displaystyle={}$ $\displaystyle N_{\beta},$

as desired. ∎

While we will only need the following result in [GHKS], its method of proof
will be used in §8.4.

######  Lemma 8.9.

Let $(X,D)$, $\beta$ be as above. Suppose given a stable log map
$f:C\rightarrow X$ in $\mathscr{M}(X,\beta)$ lying in
$\underline{\mathrm{ev}}^{-1}(x_{1},\ldots,x_{q})$ where none of the $x_{i}$
are double points of $D$. Then $f(C)\cap D$ is a finite set.

###### Proof.

Denote by $p_{1},\ldots,p_{q},p_{\mathrm{out}}\in C$ the marked points with
contact orders $u_{1},\ldots,u_{q},u_{\mathrm{out}}$ respectively, so that
$f(p_{k})=x_{k}$. However, we do not know the value of $f(p_{\mathrm{out}})$.

Let $f$ be of tropical type $\tau=(G,\boldsymbol{\sigma},\mathbf{u})$, and fix
$s\in\operatorname{Int}(\tau)$, yielding a tropical map $h_{s}:G\rightarrow
B$. We write the legs of $G$ corresponding to the marked points as
$L_{1},\ldots,L_{q},L_{\mathrm{out}}$. Of course $G$ may have many vertices,
but $h_{s}$ is balanced at those vertices which don’t map to the origin of
$B$. Also, as by assumption the marked points $p_{1},\ldots,p_{q}$ of $C$ do
not map to a double point of $D$, the image under $h_{s}$ of each leg $L_{k}$
of $G$ lies in a ray of $\mathscr{P}$. Now note that if $f(C)\cap D$ is not
finite, then $f(C)$ necessarily contains a double point of $D$, and then the
image of $h_{s}$ must intersect the interior of a two-dimensional cone of
$\mathscr{P}$. Thus it is now enough to show that the image of $h_{s}$ is
contained in the one-skeleton of $\mathscr{P}$.

Suppose this is not the case. First suppose there is a $v\in V(G)$ such that
$h_{s}(v)\in\operatorname{Int}(\sigma)$ for a two-dimensional cone
$\sigma\in\mathscr{P}$. As such a vertex corresponds to a contracted
irreducible component of $C$, by stability, there must be at least three edges
or legs adjacent to $v$. Further, the only possible leg adjacent to $v$ is
$L_{\mathrm{out}}$. However, necessarily $h_{s}(L_{\mathrm{out}})$ is parallel
to the ray $\boldsymbol{\sigma}(L_{\mathrm{out}})$ of $\mathscr{P}$, and in
particular, is not contained in the ray $\mathbb{R}_{\geq 0}h_{s}(v)$. Then it
follows easily from the balancing condition that one of the two possibilities
occur: (1) There are at least two edges or legs $E_{1},E_{2}$ adjcaent to $v$
with $h_{s}(E_{i})$ not contained in the ray $\mathbb{R}_{\geq 0}h_{s}(v)$,
with $h_{s}(E_{1})$ mapping into one side of this ray and $h_{s}(E_{2})$
mapping into the other. (2) No leg is adjacent to $v$ and all edges $E_{i}$
adjacent to $v$ satisfy $h_{s}(E_{i})\subseteq\mathbb{R}_{\geq 0}h_{s}(v)$. In
the latter case, again by balancing, there must be another vertex $v^{\prime}$
with $h_{s}(v^{\prime})\in\mathbb{R}_{>0}h_{s}(v)$ with $h_{s}(v)$ lying
between $0$ and $h_{s}(v^{\prime})$. Repeating this argument, eventually we
will come to a vertex where case (1) holds.

Thus we may assume case (1) holds at the vertex $v$. Choose one of these two
edges which is not a leg, say $E_{1}$. It has another vertex $v^{\prime}$.
Note that $h_{s}(v^{\prime})$ is not the origin in $B$, so the balancing
condition holds for $h$ at $v^{\prime}$. Thus, again there must be another
edge or leg adjacent to $v^{\prime}$ which maps to the other side of the ray
$\mathbb{R}_{\geq 0}h_{s}(v^{\prime})$. Continuing in this fashion, there are
three possibilities: (1) We arrive at $L_{\mathrm{out}}$, in which case we
made the wrong initial choice of edge $E_{i}$. Choose the other edge instead.
(2) We get an infinite sequence of edges, of course impossible. (3) We obtain
a loop, in which case the genus of $C$ is positive. But $C$ is assumed to be
genus zero. Thus we arrive at a contradiction.

If there are no vertices of $G$ mapping into the interior of a two-dimensional
cone, but there is an edge $E$ of $G$ with $h_{s}(E)$ intersecting the
interior of a two-dimensional cone, then the endpoints of $E$ must map to
different rays of $\mathscr{P}$. We can then repeat the same argument as
above, starting at either endpoint of $E$.

So in either case we obtain a contradiction, showing the result. ∎

We observe that in the case $(X,D)$ is a toric pair, the invariant $N_{\beta}$
has already been encountered in [GPS]. In [GPS, §3], a number
$N^{\mathrm{hol}}_{\mathbf{m}}(\mathbf{w})$ is defined. Here, we start with a
toric surface $X$ with fan $\Sigma$ in a two-dimensional vector space
$N_{\mathbb{R}}$; of course, the tropicalization of the toric pair $(X,D)$ is
$(N_{\mathbb{R}},\Sigma)$. Let $m_{1},\ldots,m_{n}$ be primitive generators of
distinct rays $\rho_{1},\ldots,\rho_{n}$ in $\Sigma$. Write
$\mathbf{m}=(m_{1},\ldots,m_{n})$. Further, write
$\mathbf{w}=(\mathbf{w}_{1},\ldots,\mathbf{w}_{n})$ with
$\mathbf{w}_{i}=(w_{i1},\ldots,w_{in_{i}})$. Suppose we may write

$w_{\mathrm{out}}m_{\mathrm{out}}=-\sum_{i,j}w_{ij}m_{i}$

with $m_{\mathrm{out}}$ a primitive generator of a ray $\rho_{\mathrm{out}}$
of $\Sigma$. Let $D_{1},\ldots,D_{n},D_{\mathrm{out}}$ be the toric divisors
of $X$ corresponding to $\rho_{1},\ldots,\rho_{n},\rho_{\mathrm{out}}$. Then
it is standard (see e.g., [GHK, Lem. 1.13]) that this data determines a unique
curve class $A$ such that for each prime toric divisor $D$, we have

(8.5) $A\cdot D=\begin{cases}\sum_{j}w_{ij}&\hbox{$D=D_{i}$ and
$D\not=D_{\mathrm{out}}$;}\\\
w_{\mathrm{out}}+\sum_{j}w_{ij}&D=D_{i}=D_{\mathrm{out}};\\\
w_{\mathrm{out}}&\hbox{$D=D_{\mathrm{out}}$ and $D\not=D_{i}$ for any $i$;}\\\
0&\hbox{otherwise}.\end{cases}$

Thus the above data determines a class of log curve $\beta$ with underlying
curve class $A$ and marked points with set of contact orders
$\\{w_{ij}m_{ij}\,|\,1\leq i\leq n,1\leq j\leq
n_{i}\\}\cup\\{w_{\mathrm{out}}m_{\mathrm{out}}\\}$.

######  Lemma 8.10.

In the above situation, $N_{\beta}=N^{\mathrm{hol}}_{\mathbf{m}}(\mathbf{w})$.

###### Proof.

This follows from the description of $N_{\beta}$ from Proposition 8.8, the
definition of $N_{\mathbf{m}}^{\mathrm{rel}}(\mathbf{w})$ of [GPS, (4.7)] and
[GPS, Thm. 4.4], as well as the comparison between logarithmic and relative
moduli spaces of [AMW]. In more modern language, one proceeds as follows. By
adapting the argument of [GPS, Prop. 4.3] to stable log maps, one shows that
in fact for general choice of $\mathbf{x}=(x_{ij})_{1\leq i\leq n,1\leq j\leq
n_{i}}$, the moduli space $\mathscr{M}(X,\beta,\mathbf{x})$ is particularly
simple. It consists of a finite number of reduced points, each corresponding
to a log map from an irreducible domain curve $C$ with $f^{-1}(D)$ consisting
only of the marked points on $C$. The number
$N^{\mathrm{hol}}_{\mathbf{m}}(\mathbf{w})$ is defined precisely to be the
count of such maps. ∎

We give one further result required for [GHKS]. Suppose given an inclusion
$\widetilde{N}\subseteq N$ of rank two lattices and a fan $\Sigma$ in $N$
determining a toric surface $X$. This also determines a fan
$\widetilde{\Sigma}$ for $\widetilde{N}$, with corresponding toric surface
$\widetilde{X}$. The inclusion $\widetilde{N}\subseteq N$ determines a
covering $p:\widetilde{X}\rightarrow X$ of degree $\mu$ the lattice index
$[\widetilde{N}:N]$. Suppose given a class $\tilde{\beta}$ of log curve with
underlying curve class $\widetilde{A}$ for the toric pair
$(\widetilde{X},\widetilde{D})$ with contact orders
$\tilde{u}_{1},\ldots,\tilde{u}_{n},\tilde{u}_{\mathrm{out}}\in\widetilde{N}$
contained in rays of $\widetilde{\Sigma}$, with
$\tilde{u}_{i}=\tilde{w}_{i}\tilde{\nu}_{i}$ for $\tilde{\nu}_{i}$ primitive.
Suppose $\tilde{u}_{\mathrm{out}}+\sum_{i}\tilde{u}_{i}=0$, and
$\widetilde{A}$ is the unique curve class determined by this data as in (8.5).
Then via the inclusion $\widetilde{N}\subseteq N$, the contact orders
$\tilde{u}_{i}$, $\tilde{u}_{\mathrm{out}}$ may be viewed as contact orders
$u_{i}$ for the toric pair $(X,D)$. (Note that under the inclusion
$\widetilde{N}\subseteq N$, we have $\tilde{u}_{i}=u_{i}$.) This determines
similarly a class $\beta$ of log curve for the pair $(X,D)$. Write
$u_{i}=w_{i}\nu_{i}$ as bofore, with $\nu_{i}$ primitive in $N$. Of course,
$\tilde{w}_{i}|w_{i}$.

######  Lemma 8.11.

In the above situation,

$\mu N_{\beta}=N_{\tilde{\beta}}\prod_{i=1}^{n}{\mu\tilde{w}_{i}\over w_{i}}.$

###### Proof.

One first notes via elementary toric geometry that the degree of
$p_{i}=p|_{\widetilde{D}_{i}}:\widetilde{D}_{i}\rightarrow D_{i}$ is the
lattice index $[\widetilde{N}/\tilde{\nu}_{i}\mathbb{Z}:N/\nu_{i}\mathbb{Z}]$,
which is $\mu w_{i}/w_{i}$. Further, in a neighbourhood of a point of
$\widetilde{D}_{i}$, the map $p$ has ramification of degree
$w_{i}/\tilde{w}_{i}$ along $\widetilde{D}_{i}$.

Composing stable log maps with $p$ gives a map
$q:\mathscr{M}(\widetilde{X},\tilde{\beta})\rightarrow\mathscr{M}(X,\beta)$.
Note that since the targets are toric pairs, and the log tangent bundle of a
toric variety is trivial, these moduli spaces are unobstructed. Hence their
virtual fundamental class and fundamental class coincide. Further, the degree
of the map $q$ is $\mu$. Indeed, choose a general closed point in
$\mathscr{M}(X,\beta)$, corresponding to a stable log map $f:C\rightarrow X$.
Because it is general, then as in the proof of Lemma 8.10, $C$ is an
irreducible curve with $f^{-1}(D)$ consisting only of the marked points of
$C$. Then $C^{\prime}:=C\times_{X}\widetilde{X}$ is a (in general) non-normal
curve. In a local coordinate $t$ near the marked point $p_{i}$ with contact
order $u_{i}$ and local coordinates near $f(p_{i})$, $\widetilde{X}\rightarrow
X$ takes the form $(x,y)\mapsto(x^{w_{i}/\tilde{w}_{i}},y)$ (with
$\widetilde{D}_{i}$ given by $x=0$) and $f$ is given by
$t\mapsto(t^{w_{i}}\varphi(t),\psi(t))$ , where $\varphi(t)$ is invertible and
$\psi(t)$ is arbitrary. Then the local equation for $C^{\prime}$ in
$\mathbb{A}^{2}$ with coordinates $x,t$ is

$x^{w_{i}/\tilde{w}_{i}}-t^{w_{i}}\varphi(t)=0.$

Note that étale locally we may find a function $\bar{\varphi}(t)$ with with
$\bar{\varphi}^{w_{i}/\tilde{w}_{i}}=\varphi$, so that the left-hand side of
the above equation factors as
$\prod_{i=1}^{w_{i}/\tilde{w}_{i}}(x-\zeta^{i}t^{\tilde{w}_{i}}\bar{\varphi}(t))$,
with $\zeta$ a primitive root of unity. Thus, after normalizing $C^{\prime}$
to obtain a curve $\widetilde{C}$, we see that $\widetilde{C}\rightarrow C$ is
an étale map, necessarily of degree $\mu$. Since $C\cong\mathbb{P}^{1}$,
$\widetilde{C}$ must split into $\mu$ connected components, each isomorphic to
$\mathbb{P}^{1}$. In addition, the composition $\widetilde{C}\rightarrow
C^{\prime}\rightarrow\widetilde{X}$ of the normalization with the projection
to $\widetilde{X}$ then induces a stable map on each irreducible component.
From the equations above, one sees that each such stable map has contact order
$\tilde{w}_{i}$ with $\widetilde{D}_{i}$ and hence gives an element of the
fibre of $q$. Conversely, by the universal property of the fibre product, any
map in the fibre of $q$ must arise in this way. Thus the degree of $q$ is
$\mu$.

We now consider the commutative diagram

$\textstyle{\mathscr{M}(\widetilde{X},\tilde{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\widetilde{\underline{\mathrm{ev}}}}$$\textstyle{\mathscr{M}(X,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mathrm{ev}}}$$\textstyle{\prod_{i=1}^{n}\widetilde{D}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
p_{i}}$$\textstyle{\prod_{i=1}^{n}D_{i}}$

We then have

$\left(\prod
p_{i}\right)_{*}\widetilde{\underline{\mathrm{ev}}}_{*}[\mathscr{M}(\widetilde{X},\tilde{\beta})]=N_{\tilde{\beta}}\deg\left(\prod
p_{i}\right)\prod[D_{i}]$

while

$\underline{\mathrm{ev}}_{*}q_{*}[\mathscr{M}(\widetilde{X},\tilde{\beta})]=\mu
N_{\beta}\prod[D_{i}].$

Since $\deg\left(\prod p_{i}\right)=\prod(\mu\tilde{w}_{i}/w_{i})$, the result
follows. ∎

###  8.3. Gluing via Parker and Ranganathan’s triple point method

We now return to the situation of Assumptions 8.1. To state the main result of
this section, fix a decorated wall type
${\boldsymbol{\tau}}=(G,\boldsymbol{\sigma},\mathbf{u},\mathbf{A})$, and let
$v_{\mathrm{out}}\in V(G)$ be the vertex adjacent to $L_{\mathrm{out}}$. Let
$h:\Gamma(G,\ell)\rightarrow\Sigma(X)$ be the universal family of tropical
maps of type ${\boldsymbol{\tau}}$. Let $E_{1},\ldots,E_{q}\in E(G)$ be the
edges adjacent to $v_{\mathrm{out}}$ (with $q\geq 0$). If we split $G$ at the
edges $E_{1},\ldots,E_{q}$, we obtain connected components
$G_{1},\ldots,G_{q}$ and $G_{\mathrm{out}}$, where $v_{\mathrm{out}}$ is the
unique vertex of $G_{\mathrm{out}}$ and $E_{i}$ becomes the unique leg of
$G_{i}$, which we will denote as $L_{i,\mathrm{out}}$. This in turn gives rise
to decorated types
${\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{q},{\boldsymbol{\tau}}_{v_{\mathrm{out}}}$.

######  Lemma 8.12.

${\boldsymbol{\tau}}_{i}$ is a decorated wall type.

###### Proof.

Recall we write $h^{\prime}:G\rightarrow B^{\prime}$ for the unique tropical
map to $B^{\prime}$ of type $\tau$. If
$u_{i,\mathrm{out}}:=\mathbf{u}(L_{i,\mathrm{out}})=0$, then as this coincides
with $\mathbf{u}(E_{i})$, $h^{\prime}$ contracts the edge $E_{i}$. However,
the length of the edge $E_{i}$ is a parameter in the moduli space of tropical
maps of type $\tau$, and hence this edge length may vary freely. This
contradicts the rigidity of $h^{\prime}$. Thus Definition 8.3,(1) holds. Item
(2) of that definition holds because it holds for ${\boldsymbol{\tau}}$.

For item (3) of Definition 8.3, to show $\dim\tau_{i}=1$, it is sufficient to
show that $h^{\prime}_{i}:=h^{\prime}|_{G_{i}}$ is rigid. If it deformed as a
tropical map to $B^{\prime}$ in a way so that the image of
$L_{i,\mathrm{out}}$ also deforms to a line segment not passing through
$h^{\prime}(v_{\mathrm{out}})$, then this violates [GS22, Lem. 2.5,(1)].
Otherwise, if $h^{\prime}_{i}$ deforms so that the image of the leg
$L_{i,\mathrm{out}}$ continues to pass through $h^{\prime}(v_{\mathrm{out}})$,
then this deformation can be glued to produce a deformation of $h^{\prime}$,
again a contradiction. Thus $\dim\tau_{i}=1$ (remembering that $\tau_{i}$
parameterizes maps to $B$ rather than $B^{\prime}$) and, as
$u_{i,\mathrm{out}}\not=0$, one sees that $\dim
h_{i}(\tau_{i,\mathrm{out}})=2$.

Thus ${\boldsymbol{\tau}}_{i}$ is a decorated wall type. ∎

######  Construction 8.13.

Continuing with a fixed decorated wall type ${\boldsymbol{\tau}}$, our next
goal is to associate an enumerative invariant of the form defined in the
previous subsection to the vertex $v_{\mathrm{out}}$ of $G$. Let
$x:=h^{\prime}(v_{\mathrm{out}})\in B^{\prime}$. We will first construct: (1)
a pair $(\overline{B}_{x},\widetilde{\Sigma}_{x})$ of integral affine surface
with singularity $\overline{B}_{x}$ and a polyhedral cone decomposition
$\widetilde{\Sigma}_{x}$; (2) a Looijenga pair
$(\widetilde{X}_{x},\widetilde{D}_{x})$ which tropicalizes to
$(\overline{B}_{x},\widetilde{\Sigma}_{x})$; (3) a class
${\boldsymbol{\tau}}_{\mathrm{out}}$ of logarithmic map to
$(\widetilde{X}_{x},\widetilde{D}_{x})$.

In what follows, let $\sigma^{\prime}_{x}\in\mathscr{P}^{\prime}$ be the
minimal cell containing $x$, $\sigma_{x}\in\mathscr{P}$ the minimal cone of
$\mathscr{P}$ containing $x$. We write $X_{x}$ for the stratum
$X_{\sigma_{x}}$ of $X$.

Construction of a pair $(\overline{B}_{x},\Sigma_{x})$. First, we define
$\overline{B}_{x}$ an affine manifold with singularities along with a
decomposition $\Sigma_{x}$ into not-necessarily strictly convex polyhedral
cones. If $x\in B^{\prime}_{0}$, let $v_{x}\in\Lambda_{B,x}$ be a primitive
integral tangent vector to the ray $\mathbb{R}_{\geq 0}x\subseteq B_{0}$. Then
$\overline{B}_{x}$ is identified with
$\Lambda_{B,\mathbb{R},x}/\mathbb{R}v_{x}$, with lattice structure coming from
$\Lambda_{B,x}/\mathbb{Z}v_{x}$. In this case, we define

(8.6)
$\Sigma_{x}:=\\{(\sigma+\mathbb{R}v_{x})/\mathbb{R}v_{x}\,|\,\sigma_{x}\subseteq\sigma\in\mathscr{P}\\}.$

Specifically, if $\dim\sigma^{\prime}_{x}=2$, then $\Sigma_{x}$ just consists
of one cone, namely all of $\overline{B}_{x}$. If $\dim\sigma^{\prime}_{x}=1$,
then $\Sigma_{x}$ consists of two half-spaces and their common face, the image
of the tangent space to $\sigma^{\prime}_{x}$.

If $x\in\Delta^{\prime}$, i.e., $x$ is a vertex of $\mathscr{P}^{\prime}$, we
obtain $(\overline{B}_{x},\Sigma_{x})$ as the tropicalization of the
corresponding irreducible component $X_{x}$ of $D$. Here $X_{x}$ carries the
divisorial log structure coming from the divisor $D_{x}:=\partial X_{x}$, as
in §6. Note that by the discussion in §6, in this case, (8.6) is still the
correct description of $\Sigma_{x}$. However, further, as in §8.2,
$\overline{B}_{x}$ also carries an integral affine structure with a
singularity at the origin.

Refinement $\widetilde{\Sigma}_{x}$ of $\Sigma_{x}$. Note that by construction
and the assumption that ${\boldsymbol{\tau}}$ is a wall-type, for any edge or
leg $E$ adjacent to $v_{\mathrm{out}}\in V(G)$, $h(\tau_{E})$ is two-
dimensional and $\rho_{E}:=(h(\tau_{E})+\mathbb{R}v_{x})/\mathbb{R}v_{x}$ is a
ray in $\overline{B}_{x}$.

We now let $\widetilde{\Sigma}_{x}$ be a refinement of $\Sigma_{x}$ chosen so
that (1) every cone in $\widetilde{\Sigma}_{x}$ is a strictly convex rational
polyhedral cone integral affine isomorphic to the standard cone
$\mathbb{R}_{\geq 0}^{2}$, and (2) the rays $\rho_{E_{i}}$, $1\leq i\leq q$
and $\rho_{L_{\mathrm{out}}}$ are one-dimensional cones of
$\widetilde{\Sigma}_{x}$. In what follows, the precise choice of
$\widetilde{\Sigma}_{x}$ will be unimportant, see Lemma 8.7.

The type $\tau_{\mathrm{out}}$. Note that the adjacent edges to
$v_{\mathrm{out}}\in V(G)$ determine a type
$\tau_{\mathrm{out}}=(G_{\mathrm{out}},\boldsymbol{\sigma}_{\mathrm{out}},\mathbf{u}_{\mathrm{out}})$
of tropical map to $(\overline{B}_{x},\widetilde{\Sigma}_{x})$, as follows.
First, $G_{\mathrm{out}}$ is the graph underlying
${\boldsymbol{\tau}}_{v_{\mathrm{out}}}$ as previously described, with only
one vertex $v_{\mathrm{out}}$ and legs $E_{1},\ldots,E_{q},L_{\mathrm{out}}$.
We set
$\boldsymbol{\sigma}_{\mathrm{out}}(v_{\mathrm{out}})=\\{0\\}\in\widetilde{\Sigma}_{x}$,
$\boldsymbol{\sigma}_{\mathrm{out}}(E_{i})=\rho_{E_{i}}$ and
$\boldsymbol{\sigma}_{\mathrm{out}}(L_{\mathrm{out}})=\rho_{L_{\mathrm{out}}}$.
Finally, the images of $\mathbf{u}(E_{i})$ (with the edges $E_{i}$ oriented
away from $v_{\mathrm{out}}$) and $\mathbf{u}(L_{\mathrm{out}})$ under the
quotient map given by dividing out by $\mathbb{R}v_{x}$ yield
$\mathbf{u}_{\mathrm{out}}(E_{i}),\mathbf{u}_{\mathrm{out}}(L_{\mathrm{out}})$.

The pair $(\widetilde{X}_{x},\widetilde{D}_{x})$. If
$\dim\sigma^{\prime}_{x}=2$ or $1$, then we may interpret the pair
$(\overline{B}_{x},\widetilde{\Sigma}_{x})$ as a fan, hence defining a toric
variety $\widetilde{X}_{x}$ with toric boundary $\widetilde{D}_{x}$. In
particular, $(\overline{B}_{x},\widetilde{\Sigma}_{x})$ is the tropicalization
of the pair $(\widetilde{X}_{x},\widetilde{D}_{x})$. In both these cases,
there is a morphism $\pi:\widetilde{X}_{x}\rightarrow X_{x}$. Indeed, if
$\dim\sigma^{\prime}_{x}=2$, this is just a constant map to a point. If
$\dim\sigma^{\prime}_{x}=1$, consider the quotient map
$\overline{B}_{x}\rightarrow\mathbb{R}$ given by dividing out by the tangent
space to $\sigma^{\prime}_{x}$. In this case $\mathbb{R}$ carries the fan
$\Sigma_{\mathbb{P}^{1}}$ defining $\mathbb{P}^{1}$, and the quotient map
induces a map of fans
$\widetilde{\Sigma}_{x}\rightarrow\Sigma_{\mathbb{P}^{1}}$, hence defining a
morphism $\pi:\widetilde{X}_{x}\rightarrow\mathbb{P}^{1}$. This
$\mathbb{P}^{1}$ can be identified with $X_{x}$.

If, on the other hand, $\dim\sigma^{\prime}_{x}=0$, then the refinement
$\widetilde{\Sigma}_{x}$ of $\Sigma_{x}$ defines a toric blow-up
$\pi:\widetilde{X}_{x}\rightarrow X_{x}$. We take $\widetilde{D}_{x}$ to be
the strict transform of $D_{x}$.

Decorating $\tau_{\mathrm{out}}$. Finally, we define a curve class
$\widetilde{A}_{\mathrm{out}}$ to turn $\tau_{\mathrm{out}}$ into a decorated
type ${\boldsymbol{\tau}}_{\mathrm{out}}$ for log maps to the pair
$(\widetilde{X}_{x},\widetilde{D}_{x})$. Note we already have a curve class
$A_{\mathrm{out}}=\mathbf{A}(v_{\mathrm{out}})\in H_{2}(X_{x})$. Of course, if
$\dim\sigma^{\prime}_{x}=2$, this curve class is $0$, and if
$\dim\sigma^{\prime}_{x}=1$, this curve class is some multiple of the class of
$X_{x}$.

We first introduce some additional notation. For each ray
$\rho\in\widetilde{\Sigma}_{x}$, denote by $\widetilde{D}_{\rho}$ the
corresponding irreducible component of $\widetilde{D}_{x}$. For each leg $L\in
L(G_{\mathrm{out}})$, write $w_{L}$ for the index of
$\mathbf{u}_{\mathrm{out}}(L)$, i.e., $\mathbf{u}_{\mathrm{out}}(L)$ is
$w_{L}$ times a primitive tangent vector. This represents the order of
tangency imposed by the contact order $\mathbf{u}_{\mathrm{out}}(L)$ with the
divisor $\widetilde{D}_{\boldsymbol{\sigma}_{\mathrm{out}}(L)}$.

######  Lemma 8.14.

There is at most one curve class $\widetilde{A}_{\mathrm{out}}\in
H_{2}(\widetilde{X}_{x})$ with
$\pi_{*}\widetilde{A}_{\mathrm{out}}=A_{\mathrm{out}}$ and for $\rho$ any ray
in $\widetilde{\Sigma}_{x}$,

(8.7) $\widetilde{A}_{\mathrm{out}}\cdot\widetilde{D}_{\rho}=\sum_{L\in
L(G_{\mathrm{out}})\atop\boldsymbol{\sigma}_{\mathrm{out}}(L)=\rho}w_{L}.$

If $W_{{\boldsymbol{\tau}}}\not=0$, then such an
$\widetilde{A}_{\mathrm{out}}$ exists.

###### Proof.

If $\dim\sigma^{\prime}_{x}=2$ or $1$, the unique existence of a class
$\widetilde{A}_{\mathrm{out}}$ satisfying (8.7) is easy. Indeed, there is a
unique tropical map of type $\tau_{\mathrm{out}}$, and by the balancing
condition at $v_{\mathrm{out}}$, this defines a balanced tropical map to
$\overline{B}_{x}$ with all legs mapping to rays of $\widetilde{\Sigma}_{x}$.
As is standard, this defines a curve class $\widetilde{A}_{\mathrm{out}}\in
H_{2}(\widetilde{X}_{x})$, see for example [GHK, Lem. 1.13]. Furthermore, it
is characterized precisely by the intersection numbers with the boundary
divisors as in (8.7).

If $\dim\sigma_{x}^{\prime}=2$, then
$\pi_{*}\widetilde{A}_{\mathrm{out}}=A_{\mathrm{out}}$ is trivial as
$A_{\mathrm{out}}=0$, and thus the last statement on the existence of
$\widetilde{A}_{\mathrm{out}}$ is vacuous.

If $\dim\sigma_{x}^{\prime}=1$, then we need to verify that if
$W_{{\boldsymbol{\tau}}}\not=0$, then
$\pi_{*}\widetilde{A}_{\mathrm{out}}=A_{\mathrm{out}}$. Let
${\boldsymbol{\tau}}_{v_{\mathrm{out}}}$ be the type of punctured map to $X/S$
corresponding to the vertex $v_{\mathrm{out}}$ after splitting
${\boldsymbol{\tau}}$ at the edges $E_{1},\ldots,E_{q}$. Necessarily, if
$W_{{\boldsymbol{\tau}}}\not=0$, then the moduli space
$\mathscr{M}(X/S,{\boldsymbol{\tau}}_{v_{\mathrm{out}}})$ is non-empty. The
requirement that this moduli space be non-empty then allows us to determine
$A_{\mathrm{out}}$, using Corollary 2.2. In particular,
$A_{\mathrm{out}}=d[X_{x}]$ for some $d\geq 0$, and $d$ can be determined by
intersecting $A_{\mathrm{out}}$ with an irreducible component of $D$
transverse to $X_{v}$. This is calculated as follows. Let
$\sigma_{1}^{\prime},\sigma_{2}^{\prime}\in\mathscr{P}^{\prime}$ be the two
two-cells containing $\sigma^{\prime}_{x}$, with additional vertices
$v_{1},v_{2}$ respectively not contained in $\sigma^{\prime}_{x}$. Also write
$\sigma_{1},\sigma_{2}\in\mathscr{P}$ for the corresponding cones in
$\mathscr{P}$. Then the corresponding irreducible components $X_{v_{i}}$ of
$D$ each meet $X_{x}$ transversally in one point.

By Corollary 2.2, we may now calculate
$d=X_{v_{1}}\cdot\mathbf{A}(v_{\mathrm{out}})$ as follows. Let
$\delta:\Lambda_{B,x}\rightarrow\mathbb{Z}$ be the quotient map by the tangent
space to $\sigma_{x}\in\mathscr{P}$, with sign chosen so that elements of
$\Lambda_{B,x}$ pointing into $\sigma_{1}$ map to positive integers. Then

$d=\sum_{E}\delta(\mathbf{u}(E)),$

where $E$ runs over edges and legs adjacent to $v_{\mathrm{out}}$ with
$\boldsymbol{\sigma}(E)=\sigma_{1}$. On the other hand, a simple toric
argument shows that given our definition of $\widetilde{A}_{\mathrm{out}}$,
$\pi_{*}\widetilde{A}_{\mathrm{out}}=d[X_{x}]$ for the same choice of $d$.
This completes the argument in the $\dim\sigma^{\prime}_{x}=1$ case.

Finally, consider the case $\dim\sigma^{\prime}_{x}=0$. For the uniqueness
statement, we argue as in the proof of Lemma 8.7: the intersection matrix of
the exceptional locus is negative definite. In particular, any two lifts
$\widetilde{A}_{\mathrm{out}}$ of $A_{\mathrm{out}}$ differ by a linear
combination of exceptional divisors. The uniqueness of the lift with the given
intersection numbers with the boundary divisors then follows.

For the second statement, suppose $W_{{\boldsymbol{\tau}}}\not=0$. Let
${\boldsymbol{\tau}}_{v_{\mathrm{out}}}$ be as in the
$\dim\sigma^{\prime}_{x}=1$ case, so that if $W_{{\boldsymbol{\tau}}}\not=0$,
then the moduli space
$\mathscr{M}(X/S,{\boldsymbol{\tau}}_{v_{\mathrm{out}}})$ is non-empty. By
Theorem 6.1, this moduli space is isomorphic to
$\mathscr{M}(X_{x},\bar{\boldsymbol{\tau}}_{v_{\mathrm{out}}})$, where
$\bar{\boldsymbol{\tau}}_{v_{\mathrm{out}}}$ is the type of tropical map to
$(\overline{B}_{x},\Sigma_{x})$ constructed from
${\boldsymbol{\tau}}_{v_{\mathrm{out}}}$ as in §6. On the other hand, the type
$\tau_{\mathrm{out}}$ is a type of tropical map to
$(\overline{B}_{x},\widetilde{\Sigma}_{x})$ with the same underlying graph and
contact orders as $\bar{\boldsymbol{\tau}}_{v_{\mathrm{out}}}$. It then
follows as in the proof of Lemma 8.7 that there must be at a curve class
$\widetilde{A}_{\mathrm{out}}$ with
$\pi_{*}\widetilde{A}_{\mathrm{out}}=A_{\mathrm{out}}$, determining a type
${\boldsymbol{\tau}}_{\mathrm{out}}$ with
$\mathscr{M}(\widetilde{X}_{x},{\boldsymbol{\tau}}_{\mathrm{out}})$ non-empty.
However, necessarily $\widetilde{A}_{\mathrm{out}}$ satisfies (8.7) by
Corollary 2.2. ∎

The invariant $N_{{\boldsymbol{\tau}}_{\mathrm{out}}}$. We have now
constructed a pair $(\widetilde{X}_{x},\widetilde{D}_{x})$, a curve class
$\widetilde{A}_{\mathrm{out}}\in H_{2}(\widetilde{X}_{x})$, and hence a class
${\boldsymbol{\tau}}_{\mathrm{out}}$ of log map to $\widetilde{X}_{x}$. Thus
we obtain an invariant $N_{{\boldsymbol{\tau}}_{\mathrm{out}}}$ as defined in
(8.3), independent of the choice of $\widetilde{\Sigma}_{x}$ by Lemma 8.7.

The application of our gluing formalism is then:

######  Theorem 8.15.

We have

(8.8)
$k_{\tau}W_{{\boldsymbol{\tau}}}={w_{L_{\mathrm{out}}}N_{{\boldsymbol{\tau}}_{\mathrm{out}}}\prod_{i=1}^{q}k_{\tau_{i}}W_{{\boldsymbol{\tau}}_{i}}\over|\operatorname{Aut}({\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{q})|},$

where the automorphism group of the denominator is the set of permutations
$\sigma$ of $\\{1,\ldots,q\\}$ such that ${\boldsymbol{\tau}}_{i}$ is
isomorphic to ${\boldsymbol{\tau}}_{\sigma(i)}$ as decorated types.

###### Proof.

We have a standard gluing situation obtained by splitting
${\boldsymbol{\tau}}$ at the edges $E_{1},\ldots,E_{q}$. In general this
gluing situation will not be tropically transverse. However, in this case a
relatively mild birational étale modification of $X$ takes care of this.
Flatness of the map $\underline{\mathrm{ev}}$ in Theorem 5.1 also is an issue,
but this is dealt with via Parker [Pa] and Ranganathan’s approach [Ra] to this
situation.

Step I. _Refining $\Sigma(X)$._ Note that a refinement of the polyhedral
decomposition $\mathscr{P}^{\prime}$ of $B^{\prime}$ into rational convex
polyhedra gives a refinement of $(B,\mathscr{P})$, i.e., of $\Sigma(X)$, and
hence a log étale modification $\pi:\widetilde{X}\rightarrow X$. In
particular, we choose a refinement $\widetilde{\mathscr{P}}^{\prime}$ of
$\mathscr{P}^{\prime}$ with the following properties: (1) $x$ is a vertex of
$\widetilde{\mathscr{P}}^{\prime}$; (2) the integral affine manifold with
singularity $(\overline{B}_{x},\Sigma_{x})$ coincides with
$(\overline{B}_{x},\widetilde{\Sigma}_{x})$ in the notation of Construction
8.13. It is not difficult to see that this can be done as a series of toric
blow-ups, and hence $\widetilde{X}\rightarrow X$ is projective. We omit the
details.

Step II. _Lifting the type ${\boldsymbol{\tau}}$._ Having chosen the log étale
modification $\widetilde{X}\rightarrow X$, we use [J22] to choose a _lift_ of
$\tau$ to $\Sigma(\widetilde{X})$. In general, there may be many choices of
lift of a type, but in the rigid case, the description of lifts given in [J22,
§4] reduces to a quite simple procedure. Given the map
$h^{\prime}:G\rightarrow B^{\prime}$, we first take the minimal refinement
$\widetilde{G}$ of the graph $G$ (i.e., subdivide edges or legs via the
addition of vertices) with the property that for any edge or leg $E$ of
$\widetilde{G}$, $h^{\prime}(E)$ is contained in an element of
$\widetilde{\mathscr{P}}^{\prime}$. There remains, however, some ambiguity as
to the treatment of $L_{\mathrm{out}}$, as it might have been subdivided into
a number of edges and one leg. We choose to discard all but the edge (or leg)
adjacent to $v_{\mathrm{out}}$, so that it is now a leg, which we take to be
the unique leg of $\widetilde{G}$. This provides a type
$\tilde{\tau}=(\widetilde{G},\tilde{\boldsymbol{\sigma}},\tilde{\mathbf{u}})$,
a lifting of $\tau$ in the terminology of [J22, §4].

Now it is shown in the proof of [J22, Cor. 9.4] that

(8.9)
$k_{\tau}W_{{\boldsymbol{\tau}}}=k_{\tilde{\tau}}\sum_{\tilde{\boldsymbol{\tau}}}W_{\tilde{\boldsymbol{\tau}}},$

where the sum is over all decorations $\tilde{\boldsymbol{\tau}}$ of
$\tilde{\tau}$ with $\pi_{*}\widetilde{\mathbf{A}}=\mathbf{A}$. What this
means is that if $v\in V(\widetilde{G})$ is contained in the interior of an
edge or leg $E$ of $G$, then
$\pi_{*}:H_{2}(\widetilde{X}_{\tilde{\boldsymbol{\sigma}}(v)})\rightarrow
H_{2}(X_{\boldsymbol{\sigma}(E)})$ satisfies
$\pi_{*}\widetilde{\mathbf{A}}(v)=0$. If on the other hand $v\in
V(\widetilde{G})$ is a vertex which is also a vertex of $G$, then
$\pi_{*}:H_{2}(\widetilde{X}_{\tilde{\boldsymbol{\sigma}}(v)})\rightarrow
H_{2}(X_{\boldsymbol{\sigma}(v)})$ satisfies
$\pi_{*}\widetilde{\mathbf{A}}(v)=\mathbf{A}(v)$.

We now note that for a given type $\tilde{\boldsymbol{\tau}}$ appearing in
(8.9), $W_{\tilde{\boldsymbol{\tau}}}=0$ unless
$\widetilde{\mathbf{A}}(v_{\mathrm{out}})$ coincides with the curve class
$\widetilde{A}_{\mathrm{out}}$ constructed from ${\boldsymbol{\tau}}$ in Lemma
8.14. Thus we may assume this equality and the condition of Lemma 8.14 in the
sequel.

The type $\tilde{\boldsymbol{\tau}}$ now gives rise to the invariant
$N_{\tilde{\boldsymbol{\tau}}_{\mathrm{out}}}$, and
$N_{{\boldsymbol{\tau}}_{\mathrm{out}}}=N_{\tilde{\boldsymbol{\tau}}_{\mathrm{out}}}$
by Lemma 8.7. We also have formula (8.9) for $W_{{\boldsymbol{\tau}}_{i}}$
using the lift $\tilde{\tau}_{i}$ of $\tau_{i}$ induced by restricting the
lift $\tilde{\tau}$ to $G_{i}$. Here we use what [J22] calls a maximally
extended lift, in that we don’t remove any segments from the subdivided leg
$L_{i,\mathrm{out}}$. We now observe it is sufficient to prove (8.8) after
replacing ${\boldsymbol{\tau}}$ with $\tilde{\boldsymbol{\tau}}$ and
${\boldsymbol{\tau}}_{i}$ with $\tilde{\boldsymbol{\tau}}_{i}$. Indeed, if
(8.8) has been proved in this case, then the left-hand-side of (8.8) can be
expanded using (8.9), giving

$k_{\tau}W_{{\boldsymbol{\tau}}}=\sum_{\tilde{\boldsymbol{\tau}}}w_{L_{\mathrm{out}}}N_{{\boldsymbol{\tau}}_{\mathrm{out}}}{\prod_{i=1}^{q}k_{\tilde{\tau}_{i}}W_{\tilde{\boldsymbol{\tau}}_{i}}\over|\operatorname{Aut}(\tilde{\boldsymbol{\tau}}_{1},\ldots,\tilde{\boldsymbol{\tau}}_{q})|}.$

Here $\tilde{\boldsymbol{\tau}}_{1},\ldots,\tilde{\boldsymbol{\tau}}_{q}$ are
the decorated types induces by $\tilde{\boldsymbol{\tau}}$. If instead, one
sums over all choices of decorations of the lifts
$\tilde{\tau}_{1},\ldots,\tilde{\tau}_{q}$, the same choice of
$\tilde{\boldsymbol{\tau}}$ will occur

$|\operatorname{Aut}({\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{q})|/|\operatorname{Aut}(\tilde{\boldsymbol{\tau}}_{1},\ldots,\tilde{\boldsymbol{\tau}}_{q})|$

times. Thus we obtain

$\displaystyle k_{\tau}W_{{\boldsymbol{\tau}}}={}$
$\displaystyle\sum_{\tilde{\boldsymbol{\tau}}_{1},\ldots,\tilde{\boldsymbol{\tau}}_{q}}w_{L_{\mathrm{out}}}N_{{\boldsymbol{\tau}}_{\mathrm{out}}}\prod_{i=1}^{q}{k_{\tilde{\tau}_{i}}W_{\tilde{\boldsymbol{\tau}}_{i}}\over|\operatorname{Aut}({\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{q})|}$
$\displaystyle={}$
$\displaystyle{w_{L_{\mathrm{out}}}N_{{\boldsymbol{\tau}}_{\mathrm{out}}}\over|\operatorname{Aut}({\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{q})|}\prod_{i=1}^{q}\sum_{\tilde{\boldsymbol{\tau}}_{i}}k_{\tilde{\tau}_{i}}W_{\tilde{\boldsymbol{\tau}}_{i}},$

which gives (8.8) using (8.9) again.

Thus, replacing $X$ with $\widetilde{X}$, we may assume that $x$ is a vertex
of $\mathscr{P}^{\prime}$ and that every edge or leg of $G$ adjacent to
$v_{\mathrm{out}}$ maps to an edge of $\mathscr{P}^{\prime}$ under
$h^{\prime}$.

Step III. _The gluing situation_. We now are assuming ${\boldsymbol{\tau}}$
satisfies $x=h^{\prime}(v_{\mathrm{out}})$ is a vertex of
$\mathscr{P}^{\prime}$ and every edge or leg adjacent to $v_{\mathrm{out}}$ is
mapped to an edge of $\mathscr{P}^{\prime}$ under $h^{\prime}$. Let
${\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{q},{\boldsymbol{\tau}}_{v_{\mathrm{out}}}$
be as usual, and denote by $\tilde{\boldsymbol{\tau}}_{i}$ the decorated type
obtained by gluing ${\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{i}$
and ${\boldsymbol{\tau}}_{v_{\mathrm{out}}}$. Alternatively, after splitting
${\boldsymbol{\tau}}$ at the edges $E_{i+1},\ldots,E_{q}$,
$\tilde{\boldsymbol{\tau}}_{i}$ is the decorated type corresponding to the
connected component containing $v_{\mathrm{out}}$. In particular,
$\tilde{\boldsymbol{\tau}}_{0}={\boldsymbol{\tau}}_{v_{\mathrm{out}}}$ and
$\tilde{\boldsymbol{\tau}}_{q}={\boldsymbol{\tau}}$. For each edge $E_{j}$,
write $D_{j}:=\underline{X}_{\boldsymbol{\sigma}(E_{j})}$.

Let
$\underline{\mathrm{ev}}_{i}:\mathscr{M}(X,\tilde{\boldsymbol{\tau}}_{i})\rightarrow\prod_{j=i+1}^{q}D_{j}$
be the schematic evaluation map at the punctured points corresponding to the
legs $E_{i+1},\ldots,E_{q}$ of $\tilde{\boldsymbol{\tau}}_{i}$. Define $N_{i}$
to be the rational number such that

$\underline{\mathrm{ev}}_{i,*}[\mathscr{M}(X,\tilde{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}=N_{i}\prod_{j=i+1}^{q}[D_{j}].$

Thus (using Theorem 6.1) $N_{0}=N_{{\boldsymbol{\tau}}_{\mathrm{out}}}$ and
$N_{q}=\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}=W_{{\boldsymbol{\tau}}}|\operatorname{Aut}({\boldsymbol{\tau}})|$.
We note a simple calculation shows that the virtual dimension of
$\mathscr{M}(X,\tilde{\boldsymbol{\tau}}_{i})$ is $q-i$, but this also follows
from the computations below.

We will now inductively determine $N_{i}$ from $N_{i-1}$. We consider the
gluing situation given by $\tilde{\boldsymbol{\tau}}_{i}$ with set of
splitting edges $\mathbf{E}=\\{E_{i}\\}$. After splitting, we obtain the types
$\tilde{\boldsymbol{\tau}}_{i-1}$ and ${\boldsymbol{\tau}}_{i}$.

Step IV. _Calculating the tropical multiplicity._ We calculate
$\mu(\tilde{\tau}_{i},\mathbf{E})$. Let $Q_{i}$ be the basic monoid for the
type $\tau_{i}$ and $\widetilde{Q}_{i}$ the basic monoid for the type
$\tilde{\tau}_{i}$. Note that by rigidity of the types $\tau_{i}$ we have
$Q_{i}^{*}=\mathbb{Z}$ for all $i$. Similarly, the types $\tilde{\tau}_{i}$
are rigid, so $\widetilde{Q}_{i}^{*}=\mathbb{Z}$. Further,
$N_{\tau_{i,\mathrm{out}}}$ can be identified with $Q_{i}^{*}\oplus\mathbb{Z}$
in such a way so that the map $Q_{i}^{*}\oplus\mathbb{Z}\rightarrow
P_{E_{i}}^{*}$ given by
$(q_{i},\ell_{i})\mapsto\mathrm{ev}_{v_{i}}(q_{i})+\ell_{i}\mathbf{u}(L_{i,\mathrm{out}})$
coincides with $h_{\tau_{i},*}:N_{\tau_{i,\mathrm{out}}}\rightarrow
N_{\boldsymbol{\sigma}_{i}(L_{i,\mathrm{out}})}$. We then obtain a commutative
diagram

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{Q}_{i}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{Q}_{i-1}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{\tau_{i,\mathrm{out}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\tau_{i},*}}$$\textstyle{\widetilde{Q}_{i-1}^{*}\times
N_{\tau_{i,\mathrm{out}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{i}}$$\textstyle{\widetilde{Q}_{i-1}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{E_{i}}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{P_{E_{i}}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\operatorname{coker}h_{\tau_{i},*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}\Psi_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

giving a long exact sequence

$0\longrightarrow\widetilde{Q}_{i}^{*}\longrightarrow\widetilde{Q}_{i-1}^{*}\longrightarrow\operatorname{coker}h_{\tau_{i},*}\longrightarrow\operatorname{coker}\Psi_{i}\longrightarrow
0.$

Thus we see that $\operatorname{coker}\Psi_{i}$ is finite, hence we are in a
tropically transverse gluing situation, and

(8.10)
$\mu(\tilde{\tau}_{i},\mathbf{E})=|\operatorname{coker}\Psi_{i}|=k_{\tau_{i}}|\widetilde{Q}_{i-1}^{*}/\widetilde{Q}_{i}^{*}|^{-1}.$

Step V. _Gluing._ We have two evaluation maps

$\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\tau_{i})\rightarrow\underline{X}_{\boldsymbol{\sigma}(E_{i})}$
and
$\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\tilde{\tau}_{i-1})\rightarrow\underline{X}_{\boldsymbol{\sigma}(E_{i})}$

whose product gives the morphism $\underline{\mathrm{ev}}$ in Theorem 5.1.
Since we are only evaluating at one leg, the criterion for flatness of Theorem
5.3 holds automatically. For example, for the leg $L_{i,\mathrm{out}}$ of
${\boldsymbol{\tau}}_{i}$ and contraction
$\phi:\tau^{\prime}_{i}\rightarrow\tau_{i}$, we necessarily have
$\dim\boldsymbol{\sigma}^{\prime}(L)-\dim\boldsymbol{\sigma}(L)\leq 1$.
Indeed, $\dim\boldsymbol{\sigma}(L)=2$ and
$\dim\boldsymbol{\sigma}^{\prime}(L)\leq 3$ as $B$ is three-
dimensional.666From our point of view, this is the fundamental point of
Parker’s and Ranganathan’s observation about gluing in the genus zero, triple
point case: the fact that the boundary $D$ only has triple points means this
numerical criterion for flatness always holds when gluing along one edge.

Thus we may apply Theorems 5.1 and 5.5 to see that

(8.11)
$\displaystyle\begin{split}\phi^{\prime}_{*}[\mathscr{M}(X,\tilde{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}={}&\mu(\tilde{\tau}_{i},\mathbf{E})[\mathscr{M}^{\mathrm{sch}}(X,\tilde{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}\\\
={}&\mu(\tilde{\tau}_{i},\mathbf{E})\Delta^{!}([\mathscr{M}(X,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}\times[\mathscr{M}(X,\tilde{\boldsymbol{\tau}}_{i-1})]^{{\operatorname{virt}}}).\end{split}$

On the other hand, we have a diagram

$\textstyle{\mathscr{M}^{\mathrm{sch}}(X,\tilde{\boldsymbol{\tau}}_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mathrm{ev}}^{\prime\prime}}$$\textstyle{\mathscr{M}(X,{\boldsymbol{\tau}}_{i})\times\mathscr{M}(X,\tilde{\boldsymbol{\tau}}_{i-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mathrm{ev}}^{\prime}}$$\textstyle{D_{i}\times\prod_{j=i+1}^{q}D_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{i}\times\prod_{j=i}^{q}D_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{D_{i}\times
D_{i}}$

where $\underline{\mathrm{ev}}^{\prime}$ is the product of the evaluation map
$\mathscr{M}(X,{\boldsymbol{\tau}}_{i})\rightarrow D_{i}$ at
$L_{i,\mathrm{out}}$ and the evaluation map $\underline{\mathrm{ev}}_{i-1}$.
The bottom two vertical arrows are the obvious projections, and the
composition of the right-hand vertical arrows agrees with the morphism
$\underline{\mathrm{ev}}\circ\hat{\varepsilon}$ of Theorem 5.1. Next, by
definition

$\underline{\mathrm{ev}}^{\prime}_{*}([\mathscr{M}(X,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}\times[\mathscr{M}(X,\tilde{\boldsymbol{\tau}}_{i-1})]^{{\operatorname{virt}}})=(\deg[\mathscr{M}(X,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}})N_{i-1}\left([p]\times\prod_{j=i}^{q}[D_{j}]\right),$

where $p\in D_{i}$ is any closed point. By compatibility of push-forward and
Gysin pull-back, we see that

$\displaystyle\underline{\mathrm{ev}}^{\prime\prime}_{*}[\mathscr{M}^{\mathrm{sch}}(X,\tilde{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}={}$
$\displaystyle\Delta^{!}\underline{\mathrm{ev}}^{\prime}_{*}([\mathscr{M}(X,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}\times[\mathscr{M}(X,\tilde{\boldsymbol{\tau}}_{i-1})]^{{\operatorname{virt}}})$
$\displaystyle={}$
$\displaystyle(\deg[\mathscr{M}(X,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}})N_{i-1}\left([p]\times\prod_{j=i+1}^{q}[D_{j}]\right).$

Putting this together with (8.10) and (8.11), we conclude that

$N_{i}=k_{\tau_{i}}|\widetilde{Q}_{i-1}^{*}/\widetilde{Q}_{i}^{*}|^{-1}\deg[\mathscr{M}(X,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}N_{i-1}.$

Hence, inductively we obtain

(8.12)
$\deg[\mathscr{M}(X,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}=N_{q}=N_{{\boldsymbol{\tau}}_{\mathrm{out}}}|Q_{0}^{*}/Q_{q}^{*}|^{-1}\prod_{i=1}^{q}k_{\tau_{i}}\deg[\mathscr{M}(X,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}.$

Now note that $Q_{0}^{*}$ can be identified with
$N_{\boldsymbol{\sigma}(v_{\mathrm{out}})}$, as the vertex $v_{\mathrm{out}}$
of $G_{\tau_{\mathrm{out}}}$ can be placed at any integral point of
$\boldsymbol{\sigma}(v_{\mathrm{out}})$. Further,
$N_{\boldsymbol{\sigma}(L_{\mathrm{out}})}/N_{\boldsymbol{\sigma}(v_{\mathrm{out}})}$
can be identified with the integral tangent vectors of the ray
$\boldsymbol{\sigma}_{\mathrm{out}}(L_{\mathrm{out}})$ in
$\widetilde{\Sigma}_{x}$, while $w_{L_{\mathrm{out}}}$ is the index of the
image $\mathbf{u}_{\mathrm{out}}(L_{\mathrm{out}})$ of
$\mathbf{u}(L_{\mathrm{out}})$ in
$N_{\boldsymbol{\sigma}(L_{\mathrm{out}})}/N_{\boldsymbol{\sigma}(v_{\mathrm{out}})}$.
Thus $w_{L_{\mathrm{out}}}|Q_{0}^{*}/Q_{q}^{*}|$ coincides with $k_{\tau}$.

Next note that
$\operatorname{Aut}({\boldsymbol{\tau}})=\operatorname{Aut}({\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{q})\times\prod_{i=1}^{q}\operatorname{Aut}({\boldsymbol{\tau}}_{i})$.
Thus multiplying both sides of (8.12) by $k_{\tau}$ and dividing by
$|\operatorname{Aut}({\boldsymbol{\tau}})|$ gives the desired result. ∎

###  8.4. Behaviour of $N_{\beta}$ under blow-ups.

Here we will give another sample application of our gluing formalism. This
material does not represent anything radically new; rather, it is a modern
version of an argument appearing in [GPS, §5], albeit carried out there using
older technology in a somewhat restrictive circumstance. There are also
related arguments in [M21]. The formula given here will be essential in
[GHKS].

As in §8.2, we fix a Looijenga pair $(X,D)$. For convenience, we will assume
$D$ has at least three nodes. Write $D=D_{1}+\cdots+D_{n}$ the irreducible
decomposition, in cyclic order, so that $D_{i}\cdot D_{i+1}=1$, with indices
taken modulo $n$. This gives rise as before to an integral affine manifold
with singularities $B$ along with its cone decomposition $\mathscr{P}$.

Consider a collection of distinct boundary divisors
$D_{j_{1}},\ldots,D_{j_{s}}$, which for convenience we assume are pairwise
disjoint. Choose, for each $k$, distinct points $p_{k1},\ldots,p_{kn_{k}}\in
D_{j_{k}}$, also distinct from the nodes of $D$. Denote by
$\pi:\widetilde{X}\rightarrow X$ the blow-up of $X$ at all of these points.
Denote by $E_{k\ell}$ the exceptional curve over $p_{k\ell}$. Let
$\widetilde{D}$ be the strict transform of $D$, so that
$(\widetilde{X},\widetilde{D})$ is also a Looijenga pair. This gives rise to
an integral affine manifold with singularities $\widetilde{B}$ along with its
cone decomposition $\widetilde{\mathscr{P}}$. Note that $\pi$ induces a
bijection between the components of $\widetilde{D}$ and $D$, which in turn
induces a natural piecewise linear identification between
$(\widetilde{B},\widetilde{\mathscr{P}})$ and $(B,\mathscr{P})$.

We fix a class $\tilde{\beta}$ of logarithmic map to $\widetilde{X}$ of genus
zero and $q+1$ marked points with non-zero contact orders
$u_{1},\ldots,u_{q},u_{\mathrm{out}}\in\widetilde{B}(\mathbb{Z})$, all non-
zero and contained in rays of $\widetilde{\mathscr{P}}$. As in §8.2, we assume
$u_{k}$ is contained in a ray $\rho_{i_{k}}$ corresponding to the divisor
$\widetilde{D}_{i_{k}}$. The attached curve class is $\widetilde{A}\in
H_{2}(\widetilde{X})$. We thus obtain the invariant $N_{\tilde{\beta}}$ of
(8.3).

In this situation, set $w_{k\ell}:=\widetilde{A}\cdot E_{k\ell}$. Assume from
now on that:

(8.13) $w_{k\ell}\geq 0$ for all $k,\ell$.

We denote by $\mathbf{P}_{k\ell}=P_{k\ell 1}+\cdots+P_{k\ell\mu}$ an unordered
partition of $w_{k\ell}$ into $\mu$ positive integers, for some $\mu\geq 0$
(with $\mu=0$ only if $w_{k\ell}=0$). Write $\mathbf{P}=(\mathbf{P}_{k\ell})$
a collection of partitions of all $w_{k\ell}$. We write
$\operatorname{Aut}(\mathbf{P}_{k\ell})$ for the subgroup of permutations
$\sigma$ of $\\{1,\ldots,\mu\\}$ with $P_{k\ell\sigma(m)}=P_{k\ell m}$ for
$1\leq m\leq\mu$, and write
$\operatorname{Aut}(\mathbf{P})=\prod_{k\ell}\operatorname{Aut}(\mathbf{P}_{k\ell})$.

For a given $\mathbf{P}$, we write $\beta(\mathbf{P})$ for the class of
logarithmic map to $X$ defined as follows. The curve class is
$A(\mathbf{P})=\pi_{*}\widetilde{A}$. The class has $q+1$ marked points, still
with contact orders $u_{1},\ldots,u_{q},u_{\mathrm{out}}$ using the piecewise
linear identification of $(\widetilde{B},\widetilde{\mathscr{P}})$ and
$(B,\mathscr{P})$. It has an additional marked point for every $P_{k\ell m}$,
with contact order $P_{k\ell m}\nu_{j_{k}}$, where $\nu_{j_{k}}$ is the
primitive generator of the ray of $\mathscr{P}$ corresponding to $D_{j_{k}}$.

We then have:

######  Theorem 8.16.

(8.14)
$N_{\tilde{\beta}}=\sum_{\mathbf{P}}{N_{\beta(\mathbf{P})}\over|\operatorname{Aut}(\mathbf{P})|}\prod_{k,\ell,m}{(-1)^{P_{k\ell
m}}\over P_{k\ell m}},$

where the sum is over all collections of partitions $\mathbf{P}$.

###### Proof.

Step I. _Building the degeneration_. We build a log smooth degeneration
$\widetilde{\mathcal{X}}\rightarrow\mathbb{A}^{1}$ of $\widetilde{X}$,
following [AG, §3.1], which in turn is inspired by [GPS], as follows.777During
this proof, we waive the typographic convention that $\mathcal{X}$ is the
Artin fan of $X$. We first construct a blow-up $\mathcal{X}\rightarrow
X\times\mathbb{A}^{1}$, blowing up the closed subscheme
$\bigcup_{k=1}^{s}D_{j_{k}}\times\\{0\\}$. Next, we form the blow-up
$\widetilde{\mathcal{X}}\rightarrow\mathcal{X}$ with center the strict
transform of $\bigcup_{k,\ell}\\{p_{k\ell}\\}\times\mathbb{A}^{1}$. Thus
$\widetilde{\mathcal{X}}_{t}\cong\widetilde{X}$ for $t\not=0$. Let
$\widetilde{\mathcal{D}}\subseteq\widetilde{\mathcal{X}}$ be the union of
$\widetilde{\mathcal{X}}_{0}$ and the strict transform of
$D\times\mathbb{A}^{1}$. We also write $\widetilde{\mathcal{D}}_{i}$ for the
strict transform of $D_{i}\times\mathbb{A}^{1}$. Then
$(\widetilde{\mathcal{X}},\widetilde{\mathcal{D}})$ is a log Calabi-Yau pair,
and the composition of

$\tilde{\pi}:\widetilde{\mathcal{X}}\rightarrow\mathcal{X}\rightarrow
X\times\mathbb{A}^{1}$

with the projection to $\mathbb{A}^{1}$ gives a log smooth morphism
$g:\widetilde{\mathcal{X}}\rightarrow\mathbb{A}^{1}$, with $\mathbb{A}^{1}$
carrying the divisorial log structure given by $0\in\mathbb{A}^{1}$.

We have the map
$\Sigma(g):\Sigma(\widetilde{\mathcal{X}})\rightarrow\Sigma(\mathbb{A}^{1})=\mathbb{R}_{\geq
0}$, with $\Delta(\widetilde{\mathcal{X}})$ the inverse image of $1$, a
polyhedral complex. On the other hand, in the notation of [GS22],
$\widetilde{B}:=|\Sigma(\widetilde{\mathcal{X}})|$ carries an integral affine
structure with singularities, and we may write
$\mathscr{P}_{\widetilde{\mathcal{X}}}$ for the collection of cones of
$\Sigma(\widetilde{\mathcal{X}})$. Similarly, we write
$\widetilde{B}^{\prime}:=|\Delta(\widetilde{\mathcal{X}})|$ for the fibre of
$g_{{\mathrm{trop}}}=\Sigma(g):\widetilde{B}\rightarrow\mathbb{R}_{\geq 0}$
over $1$, carrying an integral affine structure with singularities and the
polyhedral decomposition $\mathscr{P}^{\prime}_{\widetilde{\mathcal{X}}}$
consisting of the cells of $\Delta(\widetilde{\mathcal{X}})$. See [GS22, Prop.
1.16].

$X$$\widetilde{\mathcal{D}}_{1}$$\widetilde{\mathcal{D}}_{2}$$\widetilde{\mathcal{D}}_{3}$$\widetilde{\mathcal{D}}_{4}$$\widetilde{\mathcal{D}}_{5}$$\widetilde{\mathcal{D}}_{6}$$\scriptstyle\widetilde{\mathbb{P}}_{1}$$\scriptstyle\widetilde{\mathbb{P}}_{2}$$\scriptstyle\widetilde{\mathbb{P}}_{3}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$\scriptstyle\tilde{\rho}_{1}$$\scriptstyle\tilde{\rho}_{1}^{\prime}$$\scriptstyle\tilde{\rho}_{2}$
Figure 1. The boundary divisor $\widetilde{\mathcal{D}}$ of
$\widetilde{\mathcal{X}}$ and the corresponding affine manifold
$\widetilde{B}^{\prime}$. The vertices are all singular points of the affine
structure.

See Figure 1 for what $\widetilde{\mathcal{D}}$ and $\widetilde{B}^{\prime}$
will look like. In particular, the central fibre is a union

$\widetilde{\mathcal{X}}_{0}=X\cup\bigcup_{k=1}^{s}\widetilde{\mathbb{P}}_{k},$

where $\mathbb{P}_{k}$, $1\leq k\leq s$, is the exceptional divisor of the
first blow-up lying over $D_{j_{k}}\times\\{0\\}$, and
$\widetilde{\mathbb{P}}_{k}$ is the strict transform of $\mathbb{P}_{k}$ under
the second blow-up. Note that $\mathbb{P}_{k}$ is a Hirzebruch surface and
$\widetilde{\mathbb{P}}_{k}\rightarrow\mathbb{P}_{k}$ is a blow-up at $n_{k}$
points. The exceptional curves of this blowup, which we write as
$e_{k1},\ldots,e_{kn_{k}}$, are by construction disjoint from $X$. We also
write $f_{k}\in H_{2}(\widetilde{\mathbb{P}}_{k})$ for the the class of a
fibre of the ruling $\widetilde{\mathbb{P}}_{k}\rightarrow D_{j_{k}}$.

We write the vertices of $\mathscr{P}^{\prime}_{\widetilde{\mathcal{X}}}$ as
$v_{0},\ldots,v_{s}$, corresponding to the irreducible components
$X,\widetilde{\mathbb{P}}_{1},\ldots,\widetilde{\mathbb{P}}_{s}$ respectively.
We also have rays $\tilde{\rho}_{i}$, $1\leq i\leq n$ corresponding to the
one-dimensional stratum $X\cap\widetilde{\mathcal{D}}_{i}$ if
$i\not\in\\{j_{1},\ldots,j_{s}\\}$, and corresponding to the one-dimensional
stratum $\widetilde{\mathbb{P}}_{k}\cap\widetilde{\mathcal{D}}_{i}$ if
$i=j_{k}$. Further, we have a segment $\tilde{\rho}_{j_{k}}^{\prime}$
connecting $v_{0}$ and $v_{k}$, corresponding to the stratum
$\widetilde{\mathbb{P}}_{k}\cap X$.

The structure of $\widetilde{B}^{\prime}$ near the vertices
$v_{1},\ldots,v_{s}$ has been analyzed in detail in [AG, §3.3.1]. We note that
in that reference, $X$ was a toric variety, but the analysis away from the
vertex $v_{0}$ remains the same. In particular, consider the two one-cells
$\tilde{\rho}_{j_{k}}$, $\tilde{\rho}_{j_{k}}^{\prime}$. Then by [AG, Cor.
3.7], the tangent spaces to these two one-cells are left invariant under
affine monodromy about $v_{k}$. More precisely, if $x$ is a point near
$v_{k}$, then we may view the group $\Lambda_{\tilde{\rho}_{j_{k}}}$ (resp.
$\Lambda_{\tilde{\rho}_{j_{k}}^{\prime}}$) of integral tangent vectors to
$\tilde{\rho}_{j_{k}}$ (resp. $\tilde{\rho}_{j_{k}}^{\prime}$) as a well-
defined sublattice of $\Lambda_{x}$ via parallel transport. Furthermore, under
these identifications, $\Lambda_{\tilde{\rho}_{j_{k}}}$ and
$\Lambda_{\tilde{\rho}_{j_{k}}^{\prime}}$ agree. Thus
$\tilde{\rho}_{j_{k}}\cup\tilde{\rho}_{j_{k}}^{\prime}$ may be viewed as a
straight line through $v_{k}$, despite the singularity in the affine structure
at $v_{k}$.

Given a punctured map $f:C^{\circ}/W\rightarrow\widetilde{\mathcal{X}}$
defined over $\mathbb{A}^{1}$ with $W$ a log point and type $\tau$, let
$t\in\tau$ be such that the tropicalization $h_{t}:G\rightarrow\widetilde{B}$
factors through $\widetilde{B}^{\prime}$. Let $v\in V(G)$ be a vertex with
$h_{t}(v)=v_{k}$ for some $k>0$. If $E_{1},\ldots,E_{\ell}$ are the edges or
legs adjacent to $v$, oriented away from $v$, let $u_{i}$ be the image of
$\mathbf{u}(E_{i})$ in $\Lambda_{x}/\Lambda_{\tilde{\rho}_{j_{k}}}$, well-
defined under parallel transport. Then $h_{t}$ satisfies the weaker balancing
condition

(8.15) $\sum_{i=1}^{\ell}u_{i}=0$ in
$\Lambda_{x}/\Lambda_{\tilde{\rho}_{j_{k}}}$

by [AG, Prop. 3.10].

As $\widetilde{\mathcal{X}}$ is non-compact, we should be more precise about
what group our curve classes live in. Here we will take, for any stratum
$Y\subseteq\widetilde{\mathcal{X}}$, $H_{2}(Y):=\mathrm{Pic}(Y)^{*}$. Note
that as $X$ is a rational surface, when $Y$ is a compact stratum of
$\widetilde{\mathcal{X}}$, $H_{2}(Y)$ coincides with the usual integral
singular homology group $H_{2}(Y,\mathbb{Z})$.

Step II. _Degenerating the enumerative problem._ We begin by choosing points
$x_{k}\in\widetilde{D}_{i_{k}}$, $1\leq k\leq q$, not coinciding with any
double point of $\widetilde{D}$ and not coinciding with any of the points
$p_{k\ell}$. With $\mathbf{x}=(x_{1},\ldots,x_{q})$, consider the moduli space
$\mathscr{M}(\widetilde{X},\tilde{\beta},\mathbf{x})$ of (8.4). Then
$N_{\tilde{\beta}}=\deg[\mathscr{M}(\widetilde{X},\tilde{\beta},\mathbf{x})]^{{\operatorname{virt}}}$
by Proposition 8.8. Similarly, we may consider sections
$\tilde{x}_{k}\subseteq\widetilde{\mathcal{X}}$ which are the strict
transforms of $x_{k}\times\mathbb{A}^{1}$, and obtain

$\mathscr{M}(\widetilde{\mathcal{X}}/\mathbb{A}^{1},\tilde{\beta},\tilde{\mathbf{x}}):=\mathscr{M}(\widetilde{\mathcal{X}}/\mathbb{A}^{1},\tilde{\beta})\times_{\prod_{k}\widetilde{\mathcal{X}}}\prod_{k=1}^{q}\tilde{x}_{k}.$

Of course, for $0\not=t\in\mathbb{A}^{1}$, we have

$\mathscr{M}(\widetilde{X},\tilde{\beta},\mathbf{x})=\mathscr{M}(\widetilde{\mathcal{X}}/\mathbb{A}^{1},\tilde{\beta},\tilde{\mathbf{x}})\times_{\mathbb{A}^{1}}t,$

so as in [ACGS1, Thm. 1.1], we have

$\deg[\mathscr{M}(\widetilde{X},\tilde{\beta},\mathbf{x})]^{{\operatorname{virt}}}=\deg[\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,\tilde{\beta},\tilde{\mathbf{x}}_{0})]^{{\operatorname{virt}}}.$

Here $\tilde{\mathbf{x}}_{0}=((\tilde{x}_{k})_{0})$ denotes the tuple of
points with $(\tilde{x}_{k})_{0}=\tilde{x}_{k}\cap g^{-1}(0)$. As in [ACGS1,
Thm. 5.4], we then have a decomposition

(8.16)
$N_{\tilde{\beta}}=\deg[\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,\tilde{\beta},\tilde{\mathbf{x}}_{0})]^{{\operatorname{virt}}}=\sum_{{\boldsymbol{\tau}}}{m_{\tau}\over|\operatorname{Aut}({\boldsymbol{\tau}})|}j_{{\boldsymbol{\tau}}*}\deg[\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}},\tilde{\mathbf{x}}_{0})]^{{\operatorname{virt}}},$

where the sum is over all rigid decorated tropical maps to
$\widetilde{B}^{\prime}$ which can arise as a degeneration of the class
$\tilde{\beta}$. Here,
$j_{{\boldsymbol{\tau}}}:\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}},\widetilde{\mathbf{x}}_{0})\rightarrow\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,\tilde{\beta},\tilde{\mathbf{x}}_{0})$
is the canonical map forgetting the marking by ${\boldsymbol{\tau}}$. As in
Definition 8.3, we assume the decoration functions $\mathbf{A}$ are refined,
i.e., have $\mathbf{A}(v)\in
H_{2}(\widetilde{\mathcal{X}}_{\boldsymbol{\sigma}(v)})$ rather than in
$H_{2}(\widetilde{\mathcal{X}})$.

Step III. _Classifying rigid tropical maps: first steps_. Thus let
$h:G\rightarrow\widetilde{B}^{\prime}$ be a rigid tropical map of some type
$\tau$ and decoration $\mathbf{A}$ contributing to the decomposition (8.16).
Let $L_{1},\ldots,L_{q}\in L(G)$ be the legs corresponding to the marked
points mapping to the tuple of points of $\tilde{\mathbf{x}}_{0}$. Note $G$
has one additional leg, denoted $L_{\mathrm{out}}$. Because of the assumption
that the $x_{k}$ are not double points of $\widetilde{D}$, necessarily
$h(L_{k})$ is contained in the ray $\tilde{\rho}_{i_{k}}$. We may now apply
the proof of Lemma 8.9 in this setting. The target affine manifold with
singularities $\widetilde{B}^{\prime}$ is slightly more complicated than the
$B$ of the proof of the lemma, but the same argument works as the weaker
balancing condition (8.15) at the vertices $v_{1},\ldots,v_{s}$ is still
sufficient. This allows us to conclude that in any event, whether or not $h$
is rigid, its image lies in
$\bigcup_{i=1}^{n}\tilde{\rho}_{i}\cup\bigcup_{k=1}^{s}\tilde{\rho}_{j_{k}}^{\prime}$.
However, rigidity also then implies that vertices of $G$ must map to vertices
of $\widetilde{\mathscr{P}}^{\prime}$, as otherwise the location of these
vertices may be freely moved along the edge containing their image.

We now note that it is immediate that each such rigid type $\tau$ is
tropically transverse in the sense of Definition 5.8. As in Step V of the
proof of Theorem 8.15, since again $\widetilde{\mathcal{D}}$ only has triple
points, we do not need to worry about the flatness hypothesis of Theorem 5.9,
and will use this gluing result in what follows without reference to these
hypotheses.

Step IV. _Balancing at each vertex mapping to $v_{k}$_. Continue with the
notation of the previous step, and let $v\in V(G)$ be a vertex such that
$h(v)=v_{k}$ for some $k>0$. Let $E_{1},\ldots,E_{m}$ be the edges or legs
adjacent to $v$ mapping to $\tilde{\rho}_{j_{k}}$. Note these in fact must
necessarily be legs, as if $E_{i}$ were an edge, the other vertex of $E_{i}$
would map to the interior of $\tilde{\rho}_{j_{k}}$. Similarly, let
$E_{1}^{\prime},\ldots,E_{m^{\prime}}^{\prime}$ be the edges or legs adjacent
to $v$ mapping to $\tilde{\rho}_{j_{k}}^{\prime}$. Note these are necessarily
edges with opposite endpoint mapping to $v_{0}$. Let $w_{i}$ (resp.
$w_{i}^{\prime}$) be the index of $\mathbf{u}(E_{i})$ (resp.
$\mathbf{u}(E_{i}^{\prime})$), and let $w=\sum w_{i}$, $w^{\prime}=\sum
w_{i}^{\prime}$. We now make a sequence of observations.

_Observation A_. No edge or leg of $G$ maps to the one-dimensional cells of
$\mathscr{P}^{\prime}_{\widetilde{\mathcal{X}}}$ adjacent to $v_{k}$ which are
not $\tilde{\rho}_{j_{k}}$ or $\tilde{\rho}_{j_{k}}^{\prime}$. Indeed, as
these cells are unbounded, only legs may map to such cells. As the images of
the legs $L_{i}$ all map to cells of the form $\tilde{\rho}_{j}$, the only
possibility is that $L_{\mathrm{out}}$ maps to such a cell. However, that
would violate the balancing condition (8.15).

_Observation B._ Suppose $f:C\rightarrow\widetilde{\mathcal{X}}_{0}$ is a
punctured map defined over a log point in the moduli space
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}},\tilde{\mathbf{x}}_{0})$.
Let $\underline{C}_{v}\subseteq\underline{C}$ be the union of irreducible
components of $\underline{C}$ correponding to $v$ in the marking by
${\boldsymbol{\tau}}$. Then $f(\underline{C}_{v})$ is of curve class
$A:=\mathbf{A}(v)\in H_{2}(\widetilde{\mathbb{P}}_{k})$. By Corollary 2.2,
necessarily the intersection number of $A$ with
$\widetilde{\mathcal{D}}_{j_{k}}\cap\widetilde{\mathbb{P}}_{k}$ is $w$ and the
intersection number of $A$ with $\widetilde{\mathbb{P}}_{k}\cap X$ is
$w^{\prime}$. Meanwhile, by Observation A, the intersection number of $A$ with
the other two strata of $\widetilde{\mathbb{P}}_{k}$ must be zero. As a
consequence, $A$ is a linear combination of $f_{k}$ and the $e_{k\ell}$. Since
$\underline{C}_{v}$ is connected, there is no choice but for
$A=w(f_{k}-e_{k\ell})+w^{\prime}e_{k\ell}$ for some $\ell$. In particular, the
image $f(\underline{C}_{v})$ is contained in a fibre of
$\pi|_{\widetilde{\mathbb{P}}_{k}}:\widetilde{\mathbb{P}}_{k}\rightarrow
D_{j_{k}}$, reducible if $w\not=w^{\prime}$.

_Observation C_. We will now show that the contribution to $N_{\tilde{\beta}}$
is zero unless either one of the following hold:

  * (IV.1)

$m=0$, i.e., there are no legs adjacent to $v$;

  * (IV.2)

$m=1$ and $E_{1}=L_{\mathrm{out}}$; or

  * (IV.3)

$w=w^{\prime}$.

Case (IV.3) is really balancing at $v$, which can now be viewed as a well-
defined equality in $\Lambda_{\tilde{\rho}_{j}}$ by Step I.

Assume we are not in case (IV.1) or (IV.2), so that there is at least one leg
$L_{i}$ adjacent to $v$. Assume that
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}},\tilde{\mathbf{x}}_{0})$
is non-empty so that there exists a punctured map as in Observation B. By
Observation B, we have $f(\underline{C}_{v})$ is a fibre of
$\widetilde{\mathbb{P}}_{k}\rightarrow D_{j_{k}}$, reducible if
$w\not=w^{\prime}$. On the other hand, by the assumption that $L_{i}$ is
adjacent to $v$, there is a marked point of $C$ on $C_{v}$ map to
$(\tilde{x}_{i})_{0}$ under $f$. However, since the points $x_{i}$ were chosen
to be distinct from any $p_{k\ell}$, $f(\underline{C}_{v})$ cannot be
reducible. Hence $w=w^{\prime}$.

_Observation D_. We have $m^{\prime}>0$. Indeed, if $m^{\prime}=0$, then since
$G$ is connected, the only possibility is $G$ has one vertex $v$, which maps
to $v_{k}$, and a number of legs, mapping to $\tilde{\rho}_{j_{k}}$. By
Observation B, we then have $\mathbf{A}(v)=we_{k\ell}$ for some $\ell$, and
then necessarily the total curve class of the type is also $ce_{k\ell}$. But
this curve class deforms to $wE_{k\ell}$ on a general fibre of
$\widetilde{\mathcal{X}}\rightarrow\mathbb{A}^{1}$, and then $w_{k\ell}=-w<0$,
contradicting (8.13).

Step V. _Univalency or bivalency of each vertex mapping to $v_{j}$._

Continuing with the notation of Step IV, assume that
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}},\tilde{\mathbf{x}}_{0})$
is non-empty. We first observe that we can’t have two distinct legs
$L_{i},L_{i^{\prime}}$ adjacent to $v$. Indeed, by Observation B above,
$f(\underline{C}_{v})$ lies in a fibre of
$\widetilde{\mathbb{P}}_{j}\rightarrow D_{j_{k}}$. Thus the corresponding
marked points of $C$ contained in $\underline{C}_{v}$ must map to the same
point of $\widetilde{\mathcal{D}}_{j_{k}}\cap\widetilde{\mathbb{P}}_{k}$.
However, since the $x_{i}$ are chosen to be distinct, this is not possible.

Now suppose that one of the following two cases hold.

  * (V.1)

Either (a) $L_{\mathrm{out}}$ is adjacent to $v$ and another leg $L_{i}$ is
adjacent to $v$ or (b) $L_{\mathrm{out}}$ is adjacent to $v$ and balancing
fails at $v$, i.e., $w\not=w^{\prime}$ or (c) $m^{\prime}>1$ and balancing
fails at $v$.

  * (V.2)

$m^{\prime}>1$.

We will show the contribution to (8.16) from the type ${\boldsymbol{\tau}}$ is
zero in these cases.

We may split ${\boldsymbol{\tau}}$ along the edges
$E_{1}^{\prime},\ldots,E^{\prime}_{m^{\prime}}$, giving decorated types
${\boldsymbol{\tau}}_{v}$ and
${\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{m^{\prime}}$ with
underlying graphs $G_{v},G_{1},\ldots,G_{m^{\prime}}$. We note that here we
use the fact the genus of the underlying graph is zero, so that splitting at
any edge produces an additional connected component. In either Case (V.1) or
(V.2), since $m^{\prime}>0$ in any event by Observation D, there is a $j$ such
that $L_{\mathrm{out}}\not\in L(G_{j})$. Thus all legs in $G_{j}$ other than
the leg corresponding to $E_{j}^{\prime}$ correspond to marked points which
are constrained to map to some subset of the points
$\\{(\tilde{x}_{i})_{0}\\}$. From [ACGS2, Prop. 3.28 and (4.17)], taking into
account the just-mentioned point-constraints, the virtual dimension of
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{j},\tilde{\mathbf{x}}_{0})$
is zero.

If we are in Case (V.1), $\underline{C}_{v}$ maps to a fixed fibre of
$\widetilde{\mathbb{P}}_{k}\rightarrow D_{j_{k}}$ (a reducible fibre in the
unbalanced case). Thus the image of the evaluation map
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{v},\tilde{\mathbf{x}}_{0})\rightarrow\widetilde{\mathbb{P}}_{k}\cap
X=D_{j_{k}}$ at the punctured point corresponding to an edge $E_{i}^{\prime}$
is a point, being either the image of $(\tilde{x}_{i})_{0}$ under the
projection map $\widetilde{\mathbb{P}}_{k}\rightarrow D_{j_{k}}$ in the case
(a) or a point $p_{k\ell}$ in the case (b) and (c). It then follows
immediately in this case that

$\Delta^{!}\big{(}[\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{v},\tilde{\mathbf{x}}_{0})]^{{\operatorname{virt}}}\times[\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{j},\tilde{\mathbf{x}}_{0})]^{{\operatorname{virt}}}\big{)}=0,$

with $\Delta$ as in Theorem 5.1 for the gluing of ${\boldsymbol{\tau}}_{v}$
and ${\boldsymbol{\tau}}_{j}$. Hence by Theorem 5.9, the contribution from the
type ${\boldsymbol{\tau}}$ is zero.

Next suppose we are in Case (V.2), but not Case (V.1). The only remaining
possibility is that $m^{\prime}>1$, there is only one leg adjacent to $v$, and
the type ${\boldsymbol{\tau}}$ is balanced at $v$. Suppose $L_{\mathrm{out}}$
is the only leg adjacent to $v$ In this case, the evaluation map
$\underline{\mathrm{ev}}_{v}:\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{v},\tilde{\mathbf{x}}_{0})\rightarrow
D_{j_{k}}$ will be surjective. On the other hand, in this case
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{i},\tilde{\mathbf{x}}_{0})$
is virtual dimension $0$ for all $i$, and another application of Theorem 5.9
shows the contribution from the type ${\boldsymbol{\tau}}$ vanishes. A similar
argument applies if $L_{i}$ is the leg adjacent to $v$, this time because the
evaluation map $\underline{\mathrm{ev}}_{v}$ has image a point and the virtual
dimension of
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{i},\tilde{\mathbf{x}}_{0})$
is zero for at least one $i$.

In summary, if ${\boldsymbol{\tau}}$ does contribute non-trivially to (8.16),
either $v$ is univalent, necessarily with $\mathbf{A}(v)$ a positive multiple
of $f_{k}-e_{k\ell}$ for some $k$, or $v$ is bivalent and balanced, with
$\mathbf{A}(v)$ a positive multiple of $f_{k}$.

Step VI. _Calculating the non-trivial contributions from rigid tropical maps_.
The only remaining possibility for a rigid tropical map contributing non-
trivially to the right-hand side of (8.16) is now as follows. There is one
vertex $w$ with $h(w)=v_{0}$. Attached to $w$ are those legs $L$ amongst
$L_{1},\ldots,L_{q},L_{\mathrm{out}}$ for which
$\boldsymbol{\sigma}(L)=\tilde{\rho}_{k}$ with
$k\not\in\\{j_{1},\ldots,j_{s}\\}$. For those legs $L$ with
$\boldsymbol{\sigma}(L)=\tilde{\rho}_{j_{k}}$, we instead have an edge
adjacent to $w$ mapping surjectively to $\tilde{\rho}_{j_{k}}^{\prime}$, with
opposite vertex $v_{L}$ bivalent and adjacent to $L$, which maps surjectively
to $\tilde{\rho}_{j_{k}}$. Finally, there may be an additional set of edges
adjacent to $w$ mapping to the various $\tilde{\rho}_{j_{k}}^{\prime}$, with
opposite vertex being univalent. For each $k,\ell$, we will have some number
$m_{k\ell}$ of such univalent vertices mapping to $v_{j}$, with attached curve
classes $P_{k\ell 1}(f_{k}-e_{k\ell}),\ldots,P_{k\ell
m_{k\ell}}(f_{k}-e_{k\ell})$, for some positive integers $P_{k\ell
1},\ldots,P_{k\ell m_{k\ell}}$. Note these curve classes also determine the
contact order for the edge adjacent to such a univalent vertex, again by
Corollary 2.2.

Recall from Step I that curve classes take values in the duals of the Picard
groups of strata. In particular, $H_{2}(X\times\mathbb{A}^{1})\cong H_{2}(X)$.
If ${\boldsymbol{\tau}}$ is a decorated type contributing to (8.16) and
$A({\boldsymbol{\tau}})$ is the total curve class of the decorated type
${\boldsymbol{\tau}}$, then under the blow-down
$\tilde{\pi}:\widetilde{\mathcal{X}}\rightarrow X\times\mathbb{A}^{1}$,
$\pi_{*}A({\boldsymbol{\tau}})$ must agree with $A:=\pi_{*}\widetilde{A}$.
Since the curve class $\mathbf{A}(v)$ is contracted by $\tilde{\pi}$ for any
$v\not=w$, we see that $\mathbf{A}(w)=A$. Further, since $\widetilde{A}\cdot
E_{k\ell}=w_{k\ell}$, it follows by intersecting $A({\boldsymbol{\tau}})$ with
the exceptional divisor over the strict transform of
$p_{k\ell}\times\mathbb{A}^{1}$ that $\mathbf{P}_{k\ell}=P_{k\ell
1}+\cdots+P_{k\ell m_{k\ell}}$ is a partition of $w_{k\ell}$. In conclusion,
the type ${\boldsymbol{\tau}}_{w}$ associated with the vertex $w$ may now be
viewed as a class of map to $X$, and is precisely the type
$\beta(\mathbf{P})$, $\mathbf{P}=(\mathbf{P}_{k\ell})$.

We are now ready to compute the contribution of this type to the right-hand-
side of (8.16). First, we note that for each univalent vertex $v$ with
associated curve class $P_{k\ell m}(f_{k}-e_{k\ell})$, we may apply Theorem
6.1 and [GPS, Prop. 5.2] to see that
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{w},\tilde{\mathbf{x}}_{0})$
is virtual dimension zero with virtual fundamental class of degree
$(-1)^{P_{k\ell m}}/P_{k\ell m}^{2}$. On the other hand, if $v$ is a bivalent
vertex mapping to $v_{k}$ with associated curve class $df_{k}$, then
$\mathscr{M}(\widetilde{\mathcal{X}}_{0}/0,{\boldsymbol{\tau}}_{w},\tilde{\mathbf{x}}_{0})$
is easily seen to consist of one curve mapping $d:1$ to a non-singular fibre
of $\widetilde{\mathbb{P}}_{k}\rightarrow D_{j_{k}}$, totally branched over
$\widetilde{\mathbb{P}}_{k}\cap\widetilde{\mathcal{D}}_{j_{k}}$ and
$\widetilde{\mathbb{P}}_{k}\cap X$. This map is unobstructed, and the
fundamental class of the moduli space is degree $1/d$.

A simple calculation now shows that the multiplicity $\mu(\tau)$ is
$\prod_{E\in E(G)}w_{E}$, where $w_{E}$ is the index of the contact order
$\mathbf{u}(E)$. If $E$ is an edge adjacent to a bivalent vertex with
associated curve class $df_{k}$, then $w_{E}=d$, while if $E$ is an edge
adjacent to a univalent vertex with associated curve class $P_{k\ell
m}(f_{k}-e_{k\ell})$, we have $w_{E}=P_{k\ell m}$. The desired contribution to
$N_{\tilde{\beta}}$ from ${\boldsymbol{\tau}}$ in (8.16) now agrees with the
summand in (8.14) corresponding to the collection of partitions $\mathbf{P}$.
Indeed, this follows from the description of the virtual degrees of the moduli
spaces above, the calculation of the multiplicity $\mu(\tau)$, the definition
of $N_{\beta(\mathbf{P})}$, and Theorem 5.9. ∎

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