# Remarks on gluing punctured logarithmic maps

Mark Gross

###### Abstract.

We consider some well-behaved cases of the gluing formalism for punctured
stable log maps of [ACGS1, ACGS2]. This gives a gluing formula for log Gromov-
Witten invariants in a diverse set of cases; in particular, the gluing
formulae of [LR01, Li02, KLR] become an easy special case. The last section
gives an application of this gluing formalism to canonical wall structures for
K3 surfaces as constructed in [GS22].

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Tropical maps and moduli of punctured curves
    2. 2.2 The gluing formalism
  3. 3 Four-point lemmas: fibre products of log points
  4. 4 Gluing one curve
  5. 5 Gluing moduli spaces
    1. 5.1 The general situation
    2. 5.2 Gluing from rigid tropical curves
  6. 6 Punctured versus relative
  7. 7 The classical degeneration formula
  8. 8 Applications to wall structures for type III degenerations of K3 surfaces
    1. 8.1 Review of wall types and balancing
    2. 8.2 An invariant of Looijenga pairs
    3. 8.3 Gluing via Parker and Ranganathan’s triple point method
    4. 8.4 Behaviour of $N_{\beta}$ under blow-ups.

##  1\. Introduction

One of the key motivations for developing logarithmic Gromov-Witten theory
[AC, Ch, GS13] was to generalize the gluing formulae of Li–Ruan [LR01] and Jun
Li [Li02] to more general degenerations. The original gluing formulae consider
flat families $\pi:X\rightarrow B$, where $B$ is a non-singular curve with a
special point $b_{0}\in B$, $\pi$ is a normal crossing morphism with
$\pi|_{\pi^{-1}(B\smallsetminus\\{b_{0}\\})}$ smooth and
$\pi^{-1}(b_{0})=:X_{0}$ a union of two irreducible divisors $Y_{1},Y_{2}$
meeting transerversally. The above-mentioned gluing formulae then relate the
Gromov-Witten invariants of the general fibre of $\pi$ to relative Gromov-
Witten invariants of the pairs $(Y_{1},D)$, $(Y_{2},D)$, where $D=Y_{1}\cap
Y_{2}$. In principal, one would like to allow the fibre $X_{0}$ to have many
irreducible components and deeper strata, where more than two irreducible
components meet.

Logarithmic Gromov-Witten theory defines the notion of a logarithmic stable
map to such targets $X\rightarrow B$; more generally, this morphism just needs
to be toroidal rather than normal crossings, i.e., log smooth. However,
completely satisfactory generalizations of the gluing formulae have remained
elusive. The work of Abramovich, Chen, Gross and Siebert [ACGS1, ACGS2] sets
up a framework for thinking about gluing formulae. This requires, in
particular, developing _punctured_ stable maps, further generalizing
logarithmic stable maps. The essential reason for this is, if given a log
smooth curve, the restriction of the log structure to an irreducible component
need not yield a log smooth curve. In particular, this requires allowing
somewhat more general domains. As a side benefit, this generalization
introduces the notion of negative contact order. In turn, this gives a richer
set of invariants which have proved invaluable for mirror symmetry
constructions, see e.g., [GS19, GS22].

The basic setup for a gluing problem for log or punctured maps with log smooth
target $X\rightarrow B$ is given by the data of a _decorated tropical type_ ,
reviewed in §2. This is data
${\boldsymbol{\tau}}:=(G,\boldsymbol{\sigma},\mathbf{u},\mathbf{A})$. Here,
$G$ is a dual intersection graph for a domain curve, with vertices $V(G)$
corresponding to (unions of) irreducible components, edges $E(G)$
corresponding to nodes and legs $L(G)$ corresponding to marked or punctured
points. The map $\boldsymbol{\sigma}:V(G)\cup E(G)\cup
L(G)\rightarrow\Sigma(X)$ records which stratum of $X$ (strata of $X$ being
indexed by cones in the tropicalization $\Sigma(X)$ of $X$) the corresponding
curve feature maps into. The data $\mathbf{u}$ records the contact orders
associated to edges and legs, and the data $\mathbf{A}$ associates a curve
class in $X$ to each vertex of $G$. Together, this data determines a moduli
space $\mathscr{M}(X/B,{\boldsymbol{\tau}})$ of punctured log maps marked by
${\boldsymbol{\tau}}$, as defined in [ACGS2]. This moduli space is a proper
Deligne-Mumford stack over $B$, assuming $X$ is projective over $B$, and
carries a virtual fundamental class
$[\mathscr{M}(X/B,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}$.

The question of gluing is then as follows. Suppose given a decorated tropical
type ${\boldsymbol{\tau}}$ as above, and a subset of edges
$\mathbf{E}\subseteq E(G)$. By splitting $G$ at the edges of $\mathbf{E}$, we
obtain connected graphs $G_{1},\ldots,G_{r}$. Here, each edge $E\in\mathbf{E}$
with endpoints $v_{1},v_{2}$ is replaced by two legs with endpoints
$v_{1},v_{2}$ respectively. By restricting $\boldsymbol{\sigma},\mathbf{u}$
and $\mathbf{A}$ to $G_{i}$, we obtain types
${\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{r}$. Further, there is a
canonical splitting map

$\mathscr{M}(X/B,{\boldsymbol{\tau}})\rightarrow\prod_{i=1}^{r}\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i}).$

This splitting map is defined by normalizing the domain curves at the nodes
corresponding to edges of $\mathbf{E}$. The key question is then: can we
describe $[\mathscr{M}(X/B,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}$ in
terms of
$\prod_{i=1}^{r}[\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}$?

There have been a number of approaches to this question. First, Kim, Lho and
Ruddat [KLR] proved the Li–Ruan and Jun Li degeneration formula in the context
of logarithmic Gromov-Witten theory. Yixian Wu, in [Wu], gave a very general
gluing formula, under the hypothesis that all gluing strata are toric. This
has already proven to be very useful in [GS22], but doesn’t give a proof of
the Li–Ruan/Jun Li formula (unless the divisor $D$ is in fact a point).

A very different approach has been pursued by Dhruv Ranganathan, using
expanded degenerations, in [Ra]. There, the theoretical difficulties of gluing
are removed by allowing target expansions in such a way that the gluing always
happens along codimension one strata. However, this can result in an explosion
in combinatorial complexity of the problem. It is not clear whether such an
explosion can be avoided in any general approach to the gluing problem.

The basic problem is that the description of the glued moduli space
$\mathscr{M}(X/B,{\boldsymbol{\tau}})$ in terms of the moduli spaces
$\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})$ involves a fibre product in the
category of fs log schemes; this is roughly encapsulated in one of the main
gluing theorems of [ACGS2], quoted here in Theorem 2.5. One of the basic
difficulties in log geometry is that the underlying scheme of an fs fibre
product can be quite far from the underlying scheme of the ordinary fibre
product. Indeed, the classical gluing situation investigated in the log
category in [KLR] works because in this case the relevant fs fibre product is
étale over the ordinary fibre product, while this is definitely not true in
complete generality.

Here, we consider a case of gluing in which, at the virtual level, the
ordinary fibre product and fs fibre product are still not wildly divergent.
Understanding fs fibre products even of log points is non-trivial. Results
about fibre products of points in the category of schemes are known as four-
point lemmas, and generalisations of such to the fs log category have been
explored to some extent in [Og]. In §3, we give some general results in a
slightly different direction about such fibre products helpful for our
situation. In particular, we give a sufficient criteria for non-emptiness of
an fs fibre product of log points, as well as a computation for the number of
connected components of this fs fibre product if it is non-empty.

Happily, these criteria for non-emptiness have a simple tropical
interpretation. In §4, we consider gluing a single curve. In other words, we
consider a type ${\boldsymbol{\tau}}$ as described above, a set of splitting
edges $\mathbf{E}$, and the types
${\boldsymbol{\tau}}_{1},\ldots,{\boldsymbol{\tau}}_{r}$ obtained from
splitting at the edges of $\mathbf{E}$. We consider log points $W_{i}$ and
punctured log maps $f_{i}:C_{i}^{\circ}/W_{i}\rightarrow X$ of type
${\boldsymbol{\tau}}_{i}$, $1\leq i\leq r$. If $E\in\mathbf{E}$ is an edge
with vertices $v_{1},v_{2}$, $v_{j}\in V(G_{i_{j}})$, $j=1,2$, let
$p_{v_{j},E}\in C_{i_{j}}^{\circ}$ be the punctured point corresponding to the
leg of $G_{i_{j}}$ indexed by the flag $v_{j}\in E$. Assume that we have for
each $E$ the equality $f_{i_{1}}(p_{v_{1},E})=f_{i_{2}}(p_{v_{2},E})$. Then
the maps $f_{i}$ may be glued schematically. The question is then: is there a
logarithmic gluing and if so, how many logarithmic gluings of the $f_{i}$’s
are there?

In Definition 4.2, we define a map of lattices, the _tropical gluing map_
$\Psi$, which depends only on the tropical data of
${\boldsymbol{\tau}},\mathbf{E}$, and define the _tropical multiplicity_
$\mu({\boldsymbol{\tau}},\mathbf{E})$ as the order of the torsion part of
$\operatorname{coker}\Psi$. This lattice map gives the obstruction to gluing
tropical maps of types $\tau_{1},\ldots,\tau_{q}$ to obtain a tropical map of
type $\tau$. In addition, let $f:C^{\circ}/W\rightarrow X$ be the universal
gluing of the punctured maps $f_{i}$. Using the four-point lemmas of §3, we
obtain the first result, Theorem 4.4, whcih is:

######  Theorem 1.1.

If $W$ is non-empty, then it has $\mu({\boldsymbol{\tau}},\mathbf{E})$
connected components.

We remark that this is not complete information about $W$, as it may have some
non-reduced structure. However, when gluing questions are set up properly,
this becomes unimportant, as is seen in [Wu] or §5.

We say the gluing situation is _tropically transverse_ if
$\operatorname{coker}\Psi$ is in fact finite. In this case, we have Theorem
4.9, again from the four-point lemmas of §3:

######  Theorem 1.2.

If the gluing situation is tropically transverse, then $W$ is non-empty.

These results can be viewed as a generalization of the more hands-on
constructions of stable log maps beginning with work of Nishinou–Siebert [NS],
Argüz [Ar], Cheung–Fantini–Park–Ulirsch [CFPU], Mandel–Ruddat [MR20], and
[ACGS1, §4.2]. In fact, versions of tropical gluing maps already appeared in
[NS].

In §5, we now apply these observations to gluing moduli spaces, with an aim to
describe $[\mathscr{M}(X/B,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}$ in
terms of
$\prod_{i=1}^{r}[\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}$.
We define an intermediate moduli space
$\mathscr{M}^{\mathrm{sch}}(X,{\boldsymbol{\tau}})$. A point in this moduli
space is represented by a point in
$\prod_{i=1}^{r}\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})$ corresponding to a
collection of punctured maps $f_{i}:C_{i}^{\circ}/W_{i}\rightarrow X$, such
that the $f_{i}$ glue schematically. This moduli stack can be defined via a
Cartesian diagram in the category of ordinary stacks, see Theorem 5.1 for
details. The results of §4 then apply to give us some information about the
natural map
$\phi^{\prime}:\mathscr{M}(X/B,{\boldsymbol{\tau}})\rightarrow\mathscr{M}^{\mathrm{sch}}(X/B,{\boldsymbol{\tau}})$.
Unfortunately, in general it is difficult to extract useful results from this.
However, tropical transversality of the gluing situation, along with a
flatness hypothesis which can also be tested tropically, implies virtual
surjectivity of this map, with degree given by the tropical multiplicity, so
that

$\phi^{\prime}_{*}[\mathscr{M}(X/B,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}=\mu({\boldsymbol{\tau}},\mathbf{E})[\mathscr{M}^{\mathrm{sch}}(X/B,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}.$

On the other hand, if the gluing strata are sufficiently nice (e.g., smooth)
and certain other conditions hold, then
$[\mathscr{M}^{\mathrm{sch}}(X/B,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}$
can be calculated as an ordinary Gysin pull-back of
$\prod_{i=1}^{r}[\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}$.
See Remark 5.2 for details.

In §5.2, we specialize to the degeneration situation considered in [ACGS1].
Here, $B$ is a curve or spectrum of a DVR over $\Bbbk$, with divisorial log
structure coming from a closed point $b_{0}\in B$. Let $X_{0}$ be the fibre
over $b_{0}$. In [ACGS1], we showed that virtual irreducible components of
$\mathscr{M}(X_{0}/b_{0})$ were indexed by rigid tropical curves. We recast
the earlier discussion in the gluing situation provided by a rigid tropical
curve.

While these results are not yet the dreamed-of general gluing formula, they in
fact appear to be strong enough to be useful in many circumstances. In
particular, in §7, we give a very short proof of the Li-Ruan/Jun Li
degeneration formula. This is not new even in the logarithmic setup: [KLR]
first obtained this result. However, it is pleasant to see that the more
general setup proves this special case without pain. The reason this works
easily is that the gluing situation is always tropically transverse in this
case.

Along the way, in §6, we first prove a more generally useful comparison result
between punctured and log invariants for irreducible components of
degenerations. Explicitly, given $X\rightarrow B$ as in the degeneration
situation, often one needs to look at moduli spaces of punctured maps into
strata of $X_{0}$. In general, this may involve additional information, but if
the stratum is an irreducible component $Y\subseteq X_{0}$, life becomes
simpler. In particular, $Y$ carries two possible log structures, one induced
from $X$, and one the divisorial log structure coming from the union of
substrata of $Y$. We write this latter log structure as $\overline{Y}$. We
then obtain in Theorem 6.1 an isomorphism of underlying stacks
$\underline{\mathscr{M}(Y/b_{0},{\boldsymbol{\tau}})}\cong\underline{\mathscr{M}(\overline{Y},\bar{\boldsymbol{\tau}})}$,
where the type $\bar{\boldsymbol{\tau}}$ is derived from the type
${\boldsymbol{\tau}}$. Happily, the latter type does not involve punctures,
and hence gives a more familiar moduli space. We note these stacks are not
isomorphic as log stacks, but they do carry the same virtual fundamental
class.

The final section is an extended application of the gluing techniques given in
this paper. We study the genus zero punctured Gromov-Witten theory of
maximally unipotent degenerations of K3 surfaces, giving an inductive
description of the so-called canonical wall structure of [GS22]. This, along
with a number of other results proved using our gluing technology, will be of
use in [GHKS], which explores mirror symmetry for K3 surfaces and uses the
mirror construction of [GS22] to build geometrically meaningful
compactifications of the moduli space of K3 surfaces.

One of the key reasons our approach to gluing is applicable in this case is
that the degenerate varieties being considered only have at worst triple
points. It was originally observed by Brett Parker in [Pa] that this was a
particularly amenable situation for gluing.

_Conventions_ : All logarithmic schemes and stacks are defined over an
algebraically closed field $\Bbbk$ of characteristic $0$. We follow the
convention that if $X$ is a log scheme or stack, then $\underline{X}$ is the
underlying scheme or stack. We almost always write $\mathcal{M}_{X}$ for the
sheaf of monoids on $X$ and
$\alpha_{X}:\mathcal{M}_{X}\rightarrow\mathcal{O}_{X}$ for the structure map.
If $P$ is a monoid, we write $P^{\vee}:={\operatorname{Hom}}(P,\mathbb{N})$
and $P^{*}={\operatorname{Hom}}(P,\mathbb{Z})$.

_Acknowledgements_ : Some of the material here, in an earlier form, was
originally written with the intention of appearing in [ACGS2]. So it has had a
lot of influence from my co-authors of that project, Dan Abramovich, Qile Chen
and Bernd Siebert. In addition, the last section was greatly influenced by
discussions with my coauthors on [GHKS], i.e., Paul Hacking, Sean Keel, and
Bernd Siebert. This paper has also benefited very much from discussions with
Evgeny Goncharov, Sam Johnston, Dhruv Ranganathan, and Yu Wang and comments
from Helge Ruddat. It was supported by the ERC Advance Grant MSAG.

##  2\. Preliminaries

###  2.1. Tropical maps and moduli of punctured curves

We will work with a relative target space $X\rightarrow B$, a projective log
smooth morphism. We further assume that the log structure on $X$ is Zariski,
and that $X$ satisfies assumptions required to guarantee finite type moduli
spaces of punctured curves. At the moment, [ACGS2] requires that
$\overline{\mathcal{M}}_{X}^{{\mathrm{gp}}}\otimes_{\mathbb{Z}}\mathbb{Q}$ be
generated by global sections, so what follows will be written with this
assumption. However, see [J22] for finiteness results without this condition.
Typically $B$ itself is taken to be an affine scheme and $B$ is either log
smooth over $\operatorname{Spec}\Bbbk$ or a log point
$\operatorname{Spec}(Q_{B}\rightarrow B)$.

We briefly review notation from [ACGS1, ACGS2] for tropical maps to
$\Sigma(X)$ and punctured log maps to $X$ as developed in [GS13], [ACGS1,
§2.5] and [ACGS2, §2.2].

In what follows, $\mathbf{Cones}$ denotes the category of rational polyhedral
cones with integral structure, i.e., objects are rational polyhedral cones
$\omega\subseteq N_{\omega}\otimes_{\mathbb{Z}}\mathbb{R}$ for $N_{\omega}$
the lattice of integral tangent vectors to $\omega$. Morphisms are maps of
cones induced by maps of the corresponding lattices. We write
$\omega_{\mathbb{Z}}=\omega\cap N_{\omega}$ for the set of integral points of
$\omega$.

A _generalized cone complex_ is a topological space with a presentation as the
colimit of an arbitrary diagram in the category $\mathbf{Cones}$ with all
morphisms being face morphisms. If $\Sigma$ is such a generalized cone
complex, we write $\sigma\in\Sigma$ if $\sigma$ is a cone in the presentation
and $|\Sigma|$ for the underlying topological space. A morphism of generalized
cone complexes is a continuous map $f:|\Sigma|\rightarrow|\Sigma^{\prime}|$
such that for each $\sigma\in\Sigma$, the induced map
$\sigma\rightarrow|\Sigma^{\prime}|$ factors through a cone map
$\sigma\rightarrow\sigma^{\prime}\in\Sigma^{\prime}$.

There is a functor from fine saturated log schemes to generalized cone
complexes, written as $X\mapsto\Sigma(X)$. There is a one-to-one
correspondence between elements in the presentation $\Sigma(X)$ and
logarithmic strata of $X$. If $\mathcal{M}_{X}$ denotes the log structure on
$X$ with ghost sheaf $\overline{\mathcal{M}}_{X}$, and $\bar{\eta}$ is a
geometric generic point of a log stratum, then the corresponding cone is
${\operatorname{Hom}}(\overline{\mathcal{M}}_{X,\bar{\eta}},\mathbb{R}_{\geq
0})$. If $\sigma\in\Sigma(X)$, we write $X_{\sigma}\subseteq X$ for the
corresponding (closed) stratum.

We consider graphs $G$, with sets of vertices $V(G)$, edges $E(G)$ and legs
$L(G)$. In what follows, we will frequently confuse $G$ with its topological
realisation $|G|$. Legs will correspond to marked or punctured points of
punctured curves, and are rays in the marked case and compact line segments in
the punctured case. We view a compact leg as having only one vertex. An
abstract tropical curve over $\omega\in\mathbf{Cones}$ is data
$(G,{\mathbf{g}},\ell)$ where ${\mathbf{g}}:V(G)\rightarrow\mathbb{N}$ is a
genus function and
$\ell:E(G)\rightarrow{\operatorname{Hom}}(\omega_{\mathbb{Z}},\mathbb{N})\smallsetminus\\{0\\}$
determines edge lengths.

Associated to the data $(G,\ell)$ is a generalized cone complex
$\Gamma(G,\ell)$ along with a morphism of cone complexes
$\Gamma(G,\ell)\rightarrow\omega$ with fibre over
$s\in\operatorname{Int}(\omega)$ being a tropical curve, i.e., a metric graph,
with underlying graph $G$ and affine edge length of $E\in E(G)$ being
$\ell(E)(s)\in\mathbb{R}_{\geq 0}$. Associated to each vertex $v\in V(G)$ of
$G$ is a copy $\omega_{v}$ of $\omega$ in $\Gamma(G,\ell)$. Associated to each
edge or leg $E\in E(G)\cup L(G)$ is a cone $\omega_{E}\in\Gamma(G,\ell)$ with
$\omega_{E}\subseteq\omega\times\mathbb{R}_{\geq 0}$ and the map to $\omega$
given by projection onto the first coordinate. This projection fibres
$\omega_{E}$ in compact intervals or rays over $\omega$ (rays only in the case
of a leg representing a marked point).

A _family of tropical maps_ to $\Sigma(X)$ over $\omega\in\mathbf{Cones}$ is a
morphism of cone complexes

$h:\Gamma(G,\ell)\rightarrow\Sigma(X).$

If $s\in\operatorname{Int}(\omega)$, we may view $G$ as the fibre of
$\Gamma(G,\ell)\rightarrow\omega$ over $s$ as a metric graph, and write

$h_{s}:G\rightarrow\Sigma(X)$

for the corresponding tropical map with domain $G$. The _type_ of such a
family consists of the data
$\tau:=(G,\mathbf{g},\boldsymbol{\sigma},\mathbf{u})$ where

$\boldsymbol{\sigma}:V(G)\cup E(G)\cup L(G)\rightarrow\Sigma(X)$

associates to $x\in V(G)\cup E(G)\cup L(G)$ the minimal cone of $\Sigma(X)$
containing $h(\omega_{x})$. Further, $\mathbf{u}$ associates to each
(oriented) edge or leg $E\in E(G)\cup L(G)$ the corresponding _contact order_
$\mathbf{u}(E)\in N_{\boldsymbol{\sigma}(E)}$, the image of the tangent vector
$(0,1)\in N_{\omega_{E}}=N_{\omega}\oplus\mathbb{Z}$ under the map $h$. Legs
are always oriented away from their unique vertex.

As we shall only consider tropicalizations of pre-stable punctured curves (see
[ACGS2, Def. 2.5], following [ACGS2, Prop. 2.21] we may assume that for $L\in
L(G)$ with adjacent vertex $v\in V(G)$ giving
$\omega_{L},\omega_{v}\subseteq\Gamma(G,\ell)$, we have

(2.1) $h(\omega_{L})=(h(\omega_{v})+\mathbb{R}_{\geq 0}{\bf
u}(L))\cap\boldsymbol{\sigma}(L)\subseteq
N_{\boldsymbol{\sigma}(L),\mathbb{R}}.$

In other words, the images of legs extend as far as possible inside their
cones.

A _decorated type_ is data
${\boldsymbol{\tau}}=(G,\mathbf{g},\boldsymbol{\sigma},\mathbf{u},\mathbf{A})$
where $\mathbf{A}:V(G)\rightarrow H_{2}(X)$ associates a curve class to each
vertex of $G$. The _total curve class_ of ${\bf A}$ is $A=\sum_{v\in V(G)}{\bf
A}(v)$.111We recall that $H_{2}(X)$ represents some choice of group of curve
classes. It could be integral homology of $X$, or the group of curve classes
modulo algebraic or numerical equivalence, but other choices are also
possible. See e.g., [ACGS1, §2.3.8] for a discussion.

We also have a notion of a contraction morphism of types
$\phi:\tau\rightarrow\tau^{\prime}$, see [ACGS1, Def. 2.24]. This is a
contraction of edges on the underlying graphs, and the additional data
satisfies some relations as follows. If $x\in V(G)\cup E(G)\cup L(G)$, then
$\boldsymbol{\sigma}^{\prime}(\phi(x))\subseteq\boldsymbol{\sigma}(x)$ (if $x$
is an edge, it may be contracted to a vertex by $\phi$). Further, if $E\in
E(G)\cup L(G)$ then $\mathbf{u}(E)=\mathbf{u}^{\prime}(\phi(E))$ under the
inclusion $N_{\boldsymbol{\sigma}^{\prime}(\phi(E))}\subseteq
N_{\boldsymbol{\sigma}(E)}$, provided that $E$ is not an edge contracted by
$\phi$.

We say a type $\tau$ is _realizable_ if there exists a family of tropical maps
to $\Sigma(X)$ of type $\tau$. We also say ${\boldsymbol{\tau}}=(\tau,{\bf
A})$ is realizable if $\tau$ is realizable. In this paper, we will only deal
with realizable types. As a consequence, we will not need the more general
notion of global type discussed in [ACGS2, §3]. However, in the case that
there are no punctures, but only marked points, we will use the notion of a
_class of logarithmic map_ $\beta$, consisting of data of a genus $g$, a curve
class $A$, and contact orders $u_{i}\in\operatorname{Int}(\sigma_{i})$ with
$\sigma_{i}\in\Sigma(X)$ for $1\leq i\leq k$ for $k$ marked points. This may
be viewed as a decorated type
$(G,\mathbf{g},\boldsymbol{\sigma},\mathbf{u},\mathbf{A})$ where the
underlying graph $G$ has only one vertex $v$ and no edges, $\mathbf{g}(v)=g$,
$\boldsymbol{\sigma}(v)=\\{0\\}$, $\boldsymbol{\sigma}(L_{i})=\sigma_{i}$,
$\mathbf{u}(L_{i})=u_{i}$, and $\mathbf{A}(v)=A$.

If a type $\tau$ is realizable, then there is a universal family of tropical
maps of type $\tau$, parameterized by an object of $\mathbf{Cones}$. Hopefully
without confusion, we will generally write this cone as $\tau$. Hence we have
a cone complex $\Gamma(G,\ell)$ equipped with a map to $\tau$ and a map of
cone complexes $h=h_{\tau}:\Gamma(G,\ell)\rightarrow\Sigma(X)$. Generally we
write $h$ rather than $h_{\tau}$ when unambiguous. Note that for each $x\in
E(G)\cup L(G)\cup V(G)$, we thus obtain $\tau_{x}\in\Gamma(G,\ell)$ the
corresponding cone.

We write $\mathcal{A}_{X}$ for the Artin fan of $X$, see [ACMW], as well as
[ACGS1, §2.2] for a summary. With $X\rightarrow B$ log smooth with $X$
Zariski, we obtain a morphism of Artin fans
$\mathcal{A}_{X}\rightarrow\mathcal{A}_{B}$ and define

$\mathcal{X}:=\mathcal{A}_{X}\times_{\mathcal{A}_{B}}B.$

We refer to [ACGS2, Defs. 2.10, 2.13, 2.14] for the notion of a family
$\pi:C^{\circ}\rightarrow W$ of punctured curves and pre-stable or stable
punctured log maps $f:C^{\circ}/W\rightarrow X$ or
$f:C^{\circ}/W\rightarrow\mathcal{X}$ defined over $B$.

Given a punctured log map with domain $C^{\circ}\rightarrow W$ and
$W=\operatorname{Spec}(Q\rightarrow\kappa)$ for $\kappa$ an algebraically
closed field and target $X$ or $\mathcal{X}$, we obtain by functoriality of
tropicalizations a family of tropical maps

(2.2)
$\textstyle{\Sigma(C)=\Gamma(G,\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma(W)=\omega=Q^{\vee}_{\mathbb{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma(B)}$

parameterized by $W$. The _type_ of the punctured map is then the type
$\tau=(G,\mathbf{g},\boldsymbol{\sigma},\mathbf{u})$ of this family of
tropical maps. We recall that the punctured map $f:C^{\circ}/W\rightarrow X$
is _basic_ if (2.2) is the universal family of tropical maps of type $\tau$.

Given a type $\tau=(G,\mathbf{g},\boldsymbol{\sigma},\mathbf{u})$, [ACGS2,
Def. 3.7] defines the notion of a _marking_ or _weak marking_ of a punctured
map by $\tau$.222The cited reference applies to global types, but by [ACGS2,
Lem. 3.5], giving a realizable global type is the same as giving a realizable
type. Since all our types are realizable here, we ignore the notion of global
type. Roughly, a _weak marking_ of a punctured map $f:C^{\circ}/W\rightarrow
X$ involves the following information. (1) A marking of the underlying domain
curve $\underline{C}$ by $G$. In other words, we have a pre-stable curve
$\underline{C}_{v}$ for each $v\in V(G)$ of genus $\mathbf{g}(v)$, a marked
point $p_{L}\in\underline{C}_{v}$ for each leg $L\in L(G)$ adjacent to $v$,
and marked points $q_{E,1},q_{E,2}$ in
$\underline{C}_{v_{1}},\underline{C}_{v_{2}}$ for each edge $E$ connecting
$v_{1}$ to $v_{2}$. Further, $\underline{C}$ is obtained as a marked curve by
identifying pairs $q_{E,1},q_{E,2}$ for all $E\in E(G)$. (2) For each subcurve
or punctured or nodal section $Z$ of $\underline{C}$, indexed by an element
$x\in V(G)\cup L(G)\cup E(G)$, the morphism $\underline{f}|_{Z}$ factors
through the closed stratum $X_{\boldsymbol{\sigma}(x)}$ of $X$. (3) For any
geometric point $\bar{w}\rightarrow W$ giving a curve of type
$\tau_{\bar{w}}=(G_{\bar{w}},\mathbf{g}_{\bar{w}},\boldsymbol{\sigma}_{\bar{w}},\mathbf{u}_{\bar{w}})$,
the contraction morphism $G_{\bar{w}}\rightarrow G$ induced by the marking of
the domain yields a contraction morphism of types
$\tau_{\bar{w}}\rightarrow\tau$.

A _marking_ of a punctured map $f:C^{\circ}/W\rightarrow X$ is a weak marking
satisfying an additional requirement that a certain natural monoid ideal on
$\mathcal{M}_{W}$ defines an idealized log structure on $W$. See [ACGS2, Def.
3.7] for full details.

In either case, if further ${\boldsymbol{\tau}}=(\tau,\mathbf{A})$ is a
decoration of $\tau$, then $f:C^{\circ}/W\rightarrow X$ is (weakly)
${\boldsymbol{\tau}}$-marked if in addition to being (weakly) $\tau$-marked,
for each $v\in V(G)$, the curve class associated the the stable map
$\underline{f}$ restricted to the subcurve indexed by $v$ is $\mathbf{A}(v)$.

In particular, this gives rise to the following moduli spaces:

  1. (1)

$\mathscr{M}(X,{\boldsymbol{\tau}})$ (resp.
$\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}})$) the moduli space of (weakly)
${\boldsymbol{\tau}}$-marked stable punctured maps.

  2. (2)

$\mathfrak{M}(\mathcal{X},\tau)$ (resp.
$\mathfrak{M}^{\prime}(\mathcal{X},\tau)$) the moduli space of (weakly)
${\boldsymbol{\tau}}$-marked punctured maps to $\mathcal{X}$.

  3. (3)

$\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}})$ (resp.
$\mathfrak{M}^{\prime}(\mathcal{X},{\boldsymbol{\tau}})$) the moduli space of
(weakly) ${\boldsymbol{\tau}}$-marked punctured maps to $\mathcal{X}$. Note
here that while curve classes in $\mathcal{X}$ are meaningless, the decoration
$\mathbf{A}$ on $\tau$ affects the notion of isomorphism in the categories
$\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}})$ or
$\mathfrak{M}^{\prime}(\mathcal{X},{\boldsymbol{\tau}})$, and there is an
étale morphism
$\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}})\rightarrow\mathfrak{M}(\mathcal{X},\tau)$.

In general, we are always working over $B$, but when we need to be more
precise, we write $\mathscr{M}(X/B,{\boldsymbol{\tau}})$. We say type $\tau$
is _realizable over $B$_ if $\mathfrak{M}(\mathcal{X}/B,\tau)$ has a geometric
point corresponding to a map of type $\tau$, see [ACGS2, Def. 3.26]. By
[ACGS2, Prop. 3.27], this is equivalent to a combinatorial condition requiring
$\tau$ to be realisble by a family of tropical maps defined over $\Sigma(B)$
with an additional condition which is easily checked.

By [ACGS2, Thm. 3.10], all of the above moduli spaces are algebraic stacks and
$\mathscr{M}(X,{\boldsymbol{\tau}})$ and
$\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}})$ are Deligne-Mumford. Further,
there are natural morphisms

(2.3)
$\begin{split}\varepsilon:\mathscr{M}(X,{\boldsymbol{\tau}})&\rightarrow\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}})\\\
\varepsilon:\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}})&\rightarrow\mathfrak{M}^{\prime}(\mathcal{X},{\boldsymbol{\tau}})\end{split}$

given by composing a punctured log map $C^{\circ}\rightarrow X$ with the
canonical map $X\rightarrow\mathcal{X}$. [ACGS2, §4] then gives a perfect
relative obstruction theory for $\varepsilon$.

In general, it appears that the ${\boldsymbol{\tau}}$-marked moduli spaces are
more important than the weakly ${\boldsymbol{\tau}}$-marked moduli spaces. In
particular, [ACGS2, Prop. 3.31] shows that $\mathfrak{M}(\mathcal{X},\tau)$ is
a closed substack of $\mathfrak{M}^{\prime}(\mathcal{X},\tau)$ defined by a
nilpotent ideal. Further, in the cases of greatest interest for this paper
(${\boldsymbol{\tau}}$ realizable, $B$ log smooth over
$\operatorname{Spec}\Bbbk$ or $B=\operatorname{Spec}\Bbbk^{\dagger}$, the
standard log point), $\mathfrak{M}(\mathcal{X},\tau)$ is in fact reduced and
pure-dimensional, see [ACGS2, Prop. 3.28]. In fact, while
$\mathfrak{M}(\mathcal{X},\tau)$ may be quite poorly behaved globally, it has
a simple local structure coming from the fact that it is idealized log smooth
over $B$, see [ACGS2, Thm. 3.24, Rem. 3.25]. The main point for including the
weakly marked moduli spaces is that they naturally occur in the gluing
formalism.

If there is a contraction morphism between decorated global types
$\phi:{\boldsymbol{\tau}}\rightarrow{\boldsymbol{\tau}}^{\prime}$, we obtain a
forgetful map
$\mathscr{M}(X,{\boldsymbol{\tau}})\rightarrow\mathscr{M}(X,{\boldsymbol{\tau}}^{\prime})$.
This gives rise to a stratified description of these moduli spaces, see
[ACGS2, Rem. 3.29].

If ${\boldsymbol{\tau}}=(G,\boldsymbol{\sigma},{\bf u},\mathbf{A})$ denotes a
choice of decorated type, and $I\subseteq E(G)\cup L(G)$ is a collection of
edges and legs, then we write

$\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},{\boldsymbol{\tau}})=\mathfrak{M}^{\mathrm{ev}(I)}(\mathcal{X},{\boldsymbol{\tau}}):=\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}})\times_{\underline{\mathcal{X}}^{I}}\underline{X}^{I}.$

Here $\underline{\mathcal{X}}^{I}$ denotes the product of $|I|$ copies of
$\underline{\mathcal{X}}$ over $\underline{S}$, and similarly
$\underline{X}^{I}$; the morphism
$\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}})\rightarrow\underline{\mathcal{X}}^{I}$
is given by evaluation at the nodes and punctured points indexed by elements
of $I$, and $\underline{X}^{I}\rightarrow\underline{\mathcal{X}}^{I}$ is
induced by the canonical smooth map
$\underline{X}\rightarrow\underline{\mathcal{X}}$. The map $\varepsilon$ then
factors as

(2.4)
$\textstyle{\mathscr{M}(X,{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{\mathrm{ev}}}$$\textstyle{\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{M}(\mathcal{X},{\boldsymbol{\tau}}).}$

The second morphism is smooth, while $\varepsilon^{\mathrm{ev}}$ also
possesses a relative obstruction theory compatible with the morphism
$\varepsilon$ of (2.3), see [ACGS2, §4.2].

We end by reviewing a basic result which encodes a generalisation of the
tropical balancing condition.

######  Proposition 2.1.

Suppose given a punctured map $f:C^{\circ}/W\rightarrow X$ with $W$ a log
point. Let $\tau=(G,\boldsymbol{\sigma},\mathbf{u})$ be the corresponding
tropical type. If $v\in V(G)$, let $\underline{C}_{v}\subseteq\underline{C}$
be the corresponding irreducible component, and let $E_{1},\ldots,E_{n}$ be
the edges and legs adjacent to $v$, oriented away from $v$. Let
$s\in\Gamma(X,\overline{\mathcal{M}}_{X}^{{\mathrm{gp}}})$, and let
$\mathcal{L}_{s}$ be the corresponding line bundle, i.e., the line bundle
associated to the torsor given by the inverse image of $s$ under the quotient
map $\mathcal{M}_{X}\rightarrow\overline{\mathcal{M}}_{X}$. For an integral
tangent vector $v$ of
$\sigma={\operatorname{Hom}}(\overline{\mathcal{M}}_{X,x},\mathbb{R}_{\geq
0})\in\Sigma(X)$, we write $\langle v,s\rangle$ for the evaluation of $v$, as
an element of ${\operatorname{Hom}}(\overline{\mathcal{M}}_{X,x},\mathbb{Z})$,
on the germ of $s$ at $x$. Then

$\deg(\underline{f}^{*}\mathcal{L}_{s})|_{\underline{C}_{v}}=-\sum_{i=1}^{n}\langle\mathbf{u}(E_{i}),s\rangle.$

###### Proof.

This is [ACGS2, Prop. 2.27], applied after splitting the punctured map at all
nodes contained in $\underline{C}_{v}$ via [ACGS2, Prop. 5.2] and restricting
the punctured map to the component indexed by $v$. ∎

In the case that $\underline{X}$ is non-singular and the log structure on $X$
comes from a simple normal crossings divisor $D=D_{1}+\cdots+D_{s}$ such that
all intersections of irreducible components of $D$ are connected, then this
has a particularly nice interpretation as saying that the contact orders
determine intersection numbers of the curve class with the divisors $D_{i}$.
More precisely, in this case we may view $\Sigma(X)$ as in [GS19, Ex. 1.4] as
follows. Let $\mathrm{Div}_{D}(X)=\bigoplus_{i}\mathbb{Z}D_{i}$ be the group
of divisors supported on $D$, and write
$\mathrm{Div}_{D}(X)^{*}={\operatorname{Hom}}(\mathrm{Div}_{D}(X),\mathbb{Z})=\bigoplus_{i}\mathbb{Z}D_{i}^{*}$
for the dual lattice and
$\mathrm{Div}_{D}(X)^{*}_{\mathbb{R}}=\mathrm{Div}_{D}(X)^{*}\otimes_{\mathbb{Z}}\mathbb{R}$.
Then

$\Sigma(X)=\left\\{\sum_{i\in I}\mathbb{R}_{\geq
0}D_{i}^{*}\Big{|}\hbox{$I\subseteq\\{1,\ldots,s\\}$ an index set with
$\bigcap_{i\in I}D_{i}\not=\emptyset$}\right\\}.$

Thus a contact order $u$ may be written as $\sum_{i}a_{i}D_{i}^{*}$, with
$a_{i}\in\mathbb{Z}$, with $a_{i}$ denoting the order of tangency with
$D_{i}$. We then have the immediate corollary

######  Corollary 2.2.

Let $X,D$ be as above, and $f,\tau,E_{1}\ldots,E_{n}$ as in Proposition 2.1.
Write $\mathbf{u}(E_{i})=\sum_{j}a_{ij}D_{j}^{*}$. Then

$\deg(\underline{f}^{*}\mathcal{O}_{X}(D_{j}))|_{\underline{C}_{v}}=\sum_{i=1}^{n}a_{ij}.$

###  2.2. The gluing formalism

We may now describe the key gluing formalism of [ACGS2]. We begin with the
_standard gluing situation_.

######  Notation 2.3 (The standard gluing situation).

We fix a target $X\rightarrow B$, a proper log smooth morphism. We further
assume that the log structure on $X$ is Zariski, and that $X$ satisfies
assumptions required to guarantee finite type moduli spaces of punctured
curves. Further we assume that $B$ is either log smooth over
$\operatorname{Spec}\Bbbk$ or $B=\operatorname{Spec}\Bbbk^{\dagger}$, the
standard log point. Fix a realizable type
$\tau=(G,\mathbf{g},\boldsymbol{\sigma},\mathbf{u})$ of tropical map to
$\Sigma(X)/\Sigma(B)$. We select a set of splitting edges $\mathbf{E}\subseteq
E(G)$, and let $G_{1},\ldots,G_{r}$ be the connected components of the graph
obtained by splitting $G$ at the edges of $\mathbf{E}$, i.e., replacing each
edge $E\in\mathbf{E}$ with endpoints $v_{1},v_{2}$ with two legs with
endpoints $v_{1},v_{2}$ respectively. We write these two legs as flags
$(E,v_{1})$ and $(E,v_{2})$. We then let $\tau_{1},\ldots,\tau_{r}$ be the
induced set of decorated types with underlying graphs $G_{1},\ldots,G_{r}$.
Let $\mathbf{L}\subseteq\bigcup_{i=1}^{r}L(G_{i})$ be the subset of all legs
obtained from splitting edges, and $\mathbf{L}_{i}=\mathbf{L}\cap L(G_{i})$.
For $v\in V(G)$, let $i(v)\in\\{1,\ldots,r\\}$ denote the connected component
$G_{i}$ containing $v$.

For each $E\in\mathbf{E}$ denote by
$\mathfrak{M}^{\prime}_{E}(\mathcal{X},\tau)$ the image of the nodal section
$s_{E}:\underline{\mathfrak{M}^{\prime}}(\mathcal{X},\tau)\longrightarrow\underline{\mathfrak{C}^{\prime}}^{\circ}(\mathcal{X},\tau)$
with the restriction of the log structure on the universal domain
$\mathfrak{C}^{\prime\circ}(\mathcal{X},\tau)$. Denote further by
$\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau)$ the fs fiber product

(2.5)
$\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau)=\mathfrak{M}^{\prime}_{E_{1}}(\mathcal{X},\tau)\times^{\mathrm{fs}}_{\mathfrak{M}^{\prime}(\mathcal{X},\tau)}\cdots\times^{\mathrm{fs}}_{\mathfrak{M}^{\prime}(\mathcal{X},\tau)}\mathfrak{M}^{\prime}_{E_{r}}(\mathcal{X},\tau),$

where $E_{1},\ldots,E_{r}\in E(G)$ are the edges in $\mathbf{E}$. With this
enlarged log structure, the pull-back
$\widetilde{\mathfrak{C}}^{\prime\circ}(\mathcal{X},\tau)\longrightarrow\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau)$
of the universal domain has sections $\tilde{s}_{E}$, $E\in\mathbf{E}$, in the
category of log stacks. Moreover, $\underline{\mathrm{ev}}_{\mathbf{E}}$ lifts
to a logarithmic evaluation morphism

(2.6)
$\textstyle\mathrm{ev}_{\mathbf{E}}:\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau)\longrightarrow\prod_{E\in\mathbf{E}}\mathcal{X},$

with $E$-component equal to $\tilde{f}\circ\tilde{s}_{E}$ for
$\tilde{f}:\widetilde{\mathfrak{C}}^{\prime\circ}(\mathcal{X},\tau)\longrightarrow\mathcal{X}$
the universal punctured morphism.

Similarly, for each of the types
$\tau_{i}=(G_{i},\mathbf{g}_{i},\boldsymbol{\sigma}_{i},\mathbf{u}_{i})$
obtained by splitting and $L\in L(G_{i})$, denote by
$\mathfrak{M}^{\prime}_{L}(\mathcal{X},\tau_{i})$ the image of the punctured
section
$s_{L}:\underline{\mathfrak{M}}^{\prime}(\mathcal{X},\tau_{i})\longrightarrow\underline{\mathfrak{C}^{\prime}}^{\circ}(\mathcal{X},\tau_{0})$
defined by $L$, again endowed with the pull-back of the log structure on
${\mathfrak{C}^{\prime}}^{\circ}(\mathcal{X},\tau_{0})$. With
$L_{1},\ldots,L_{s}$ the legs of $\mathbf{L}_{i}$, define the stack

$\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau_{i})=\big{(}\mathfrak{M}^{\prime}_{L_{1}}(\mathcal{X},\tau_{i})\times^{\mathrm{f}}_{\mathfrak{M}^{\prime}(\mathcal{X},\tau_{i})}\cdots\times^{\mathrm{f}}_{\mathfrak{M}^{\prime}(\mathcal{X},\tau_{i})}\mathfrak{M}^{\prime}_{L_{s}}(\mathcal{X},\tau_{i})\big{)}^{\mathrm{sat}},$

where $\mathrm{sat}$ denotes saturation and $\times^{\mathrm{f}}$ denotes
fibre product in the category of fine log stacks. This stack differs from
$\mathfrak{M}^{\prime}(\mathcal{X},\tau_{i})$ by adding the pull-back of the
log structure of each puncture, so that the pull-back
$\widetilde{\mathfrak{C}}^{\prime\circ}(\mathcal{X},\tau_{i})\longrightarrow\widetilde{\mathfrak{M}}^{\prime}\mathcal{X},\tau_{i})$
of the universal curve now has punctured sections in the category of log
stacks. We define the evaluation morphism

(2.7)
$\textstyle\mathrm{ev}_{\mathbf{L}}:\prod_{i=1}^{r}\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau_{i})\longrightarrow\prod_{E\in\mathbf{E}}\mathcal{X}\times\mathcal{X},$

by taking as $E$-component the evaluation at the corresponding two sections
$s_{E,v_{1}},s_{E,v_{2}}$, where $v_{1},v_{2}$ are the endpoints of $E$. Note
that this involves a choice of ordering of the endpoints of $E$, i.e., a
choice of orientation of $E$.

It is worth noting the following (see [ACGS2, Prop. 5.5]).

######  Proposition 2.4.

The canonical map
$\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau)\rightarrow\mathfrak{M}^{\prime}(\mathcal{X},\tau)$
induces an isomorphism of underlying stacks, while the canonical maps
$\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau_{i})\rightarrow\mathfrak{M}^{\prime}(\mathcal{X},\tau_{i})$
induces an isomorphism on reductions.

There are evaluation space versions of this. We set

$\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}}(\mathcal{X},\tau_{i})=\mathfrak{M}^{\prime\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X},\tau_{i})\times_{\mathfrak{M}^{\prime}(\mathcal{X},\tau_{i})}\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau_{i})$

and

$\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}}(\mathcal{X},\tau)=\mathfrak{M}^{\prime\mathrm{ev}(\mathbf{E})}(\mathcal{X},\tau_{i})\times_{\mathfrak{M}^{\prime}(\mathcal{X},\tau_{i})}\widetilde{\mathfrak{M}}^{\prime}(\mathcal{X},\tau_{i}).$

There is a natural splitting map

(2.8)
$\delta_{\mathfrak{M}}:\mathfrak{M}^{\prime}(\mathcal{X},\tau)\rightarrow\prod_{i=1}^{r}\mathfrak{M}^{\prime}(\mathcal{X},\tau_{i})$

as defined in [ACGS2, Prop. 5.4], taking a $\tau$-marked map and splitting it
at the nodes marked by the edges in $\mathbf{E}$. This lifts to

(2.9)
$\delta_{\mathfrak{M}}:\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}(\mathbf{E})}(\mathcal{X},\tau)\rightarrow\prod_{i=1}^{r}\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X},\tau_{i}),$

and is shown to be a finite morphism in [ACGS2, Cor. 5.13]. Passing to the
$\mathrm{ev}$-spaces is key here: the morphism in (2.8) is rarely finite or
even proper. This also gives an upgrading of the evaluation morphisms:

(2.10)
$\displaystyle\begin{split}\mathrm{ev}_{\mathbf{E}}:&\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}(\mathbf{E})}(\mathcal{X},\tau)\longrightarrow\prod_{E\in\mathbf{E}}X,\\\
\mathrm{ev}_{\mathbf{L}}:&\prod_{i=1}^{r}\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X},\tau_{i})\longrightarrow\prod_{E\in\mathbf{E}}X\times
X.\end{split}$

We review the key gluing results of [ACGS2]. We first describe the glued
moduli space as an fs fibre product:

######  Theorem 2.5.

Suppose given a gluing situation as in Notation 2.3. Then the commutative
diagram

$\textstyle{\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}(\mathbf{L})}(\mathcal{X},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\mathfrak{M}}}$$\scriptstyle{\mathrm{ev}_{\mathbf{E}}}$$\textstyle{\prod_{i=1}^{r}\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X},\tau_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\mathbf{L}}}$$\textstyle{\prod_{E\in\mathbf{E}}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\prod_{E\in\mathbf{E}}X\times
X}$

with $\Delta$ the product of diagonal embeddings and the other arrows defined
in (2.9) and (2.10), is cartesian in the category of fs log stacks. We remind
the reader that all products in this square are taken over $B$.

An analogous statement holds for $\tau$ replaced by a decorated global type
${\boldsymbol{\tau}}=(\tau,\mathbf{A})$, or replacing
$\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}}(\mathcal{X},\tau)$,
$\widetilde{\mathfrak{M}}^{\prime\mathrm{ev}}(\mathcal{X},\tau_{i})$ with the
analogous moduli spaces of stable maps to $X/B$,
$\widetilde{\mathscr{M}}^{\prime}(X,\tau)$,
$\widetilde{\mathscr{M}}^{\prime}(X,\tau_{i})$.

We remark that in this theorem, the fact that all fibre products are over $B$
can make the actual calculation of fibre products more difficult than they
need to be. But it is often enough to work over $\operatorname{Spec}\Bbbk$, as
the following proposition (see [ACGS2, Prop. 5.11]) shows:

######  Proposition 2.6.

Let $B$ be an affine log scheme equipped with a global chart
$P\rightarrow\mathcal{M}_{B}$ inducing an isomorphism
$P\cong\Gamma(B,\overline{\mathcal{M}}_{B})$. Let $\tau$ be a type of tropical
map for $X/B$, with underlying graph connected. Then there are isomorphisms
$\mathfrak{M}(\mathcal{X},\tau)\cong\mathfrak{M}(\mathcal{X}/\operatorname{Spec}\Bbbk,\tau)$
and
$\mathscr{M}(X,\tau)\cong\mathscr{M}(X/\operatorname{Spec}\Bbbk,\tau)$.333We
recall our convention that $\mathscr{M}(X,\tau)=\mathscr{M}(X/B,\tau)$.

Finally, to make contact with stable punctured maps to $X$, we have [ACGS2,
Prop. 5.15]:

######  Theorem 2.7.

In the situation of Theorem 2.5 there is a cartesian diagram

(2.11)
$\textstyle{\mathscr{M}(X,\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\varepsilon^{\mathrm{ev}}}$$\textstyle{\prod_{i=1}^{r}\mathscr{M}(X,\tau_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\varepsilon}=\prod_{i}\varepsilon^{\mathrm{ev}}_{i}}$$\textstyle{\mathfrak{M}^{\mathrm{ev}(\mathbf{E})}(\mathcal{X},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\mathrm{ev}}}$$\textstyle{\prod_{i=1}^{r}\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X},\tau_{i})}$

with horizontal arrows the canonical splitting maps (see [ACGS2, Prop. 5.4],
and the vertical arrows the canonical strict morphisms of (2.4).

Analogous statements hold for decorated and for weakly marked versions of the
moduli stacks, see [ACGS2, Def. 3.8].

The evaluation spaces $\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\tau)$ etc. play
a crucial role here. First, if instead one used the spaces
$\mathfrak{M}(\mathcal{X},\tau)$, there would be no way to obtain a Cartesian
diagram, as the splitting map on the level of punctured maps to Artin fans has
no way to impose a matching condition at the schematic level. Using the
evaluation spaces allows $\delta^{\mathrm{ev}}$ to impose both schematic and
logarithmic matching conditions.

Second, the splitting map at the level of the spaces of punctured maps to
Artin fans is very poorly behaved, being neither representable nor proper.
However, $\delta^{\mathrm{ev}}$, as this theorem states, is in fact finite and
representable. Hence we may use it to push-forward Chow classes, and in
particular, as [ACGS2, Thm. 5.17] points out, the compatibility of obstruction
theories then allows a calculation

$\delta_{*}([\mathscr{M}(X,\tau)]^{{\operatorname{virt}}})=\hat{\varepsilon}^{!}\delta^{\mathrm{ev}}_{*}[\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\tau)]$

of the virtual fundamental class of $\mathscr{M}(X,\tau)$.

Thus the main task is finding a useful expression for
$\delta^{\mathrm{ev}}_{*}[\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},{\boldsymbol{\tau}})]$.
While the Artin stacks of the type
$\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\tau)$ may seem very forbidding, in a
certain sense they are very well-behaved: they are idealized log smooth over
$B$, see [ACGS2, Thm. 3.24]. This means that there are local descriptions of
these stacks as unions of strata of toric varieties. Further, these local
descriptions can be determined very explicitly from the tropical description
of the types $\tau$ and $\tau_{i}$.

This effective description of these moduli spaces has led Yixian Wu [Wu], in
the case that all gluing strata are toric, to give an effective formula for
the Chow class $\delta_{*}[\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\tau)]$ as a
weighted sum of strata of
$\prod_{i=1}^{r}\mathfrak{M}^{\mathrm{ev}}(\mathcal{X},\tau_{i})$. This
formula has already proved to be very useful, see e.g., [GS22].

Here we wish to develop a gluing formalism in a complementary direction,
which, although very far from general, is also useful.

##  3\. Four-point lemmas: fibre products of log points

The key point is to understand the gluing fibre diagram of Theorem 2.5 by
studying fibre products of log points. We carry this study out in this
section; unfortunately, this is rather dry. An fs fibre product of log points
can be quite subtle. Even determining whether such a fibre product is non-
empty is difficult, as a “four-point lemma” does not hold widely in log
geometry, see [Og, III Prop. 2.2.3] for some results. Here we will generally
be interested in the number of connected components of a fibre product for
application to specific gluing situations.

We start with a small lemma:

######  Lemma 3.1.

Let $W:=\operatorname{Spec}(Q\rightarrow\kappa)$ be a log point with $Q$ a
sharp fine monoid. Then $W^{\mathrm{sat}}$ is a disjoint union of possibly
non-reduced points, and the number of connected components of
$W^{\mathrm{sat}}$ is $|(Q^{{\mathrm{gp}}})_{{\mathrm{tors}}}|$.

###### Proof.

By construction of the saturation, [Og, III Prop. 2.1.5],

$W^{\mathrm{sat}}=W\times_{\operatorname{Spec}\kappa[Q]}\operatorname{Spec}\kappa[Q^{\mathrm{sat}}],$

where $Q^{\mathrm{sat}}$ is the saturation of $Q$ inside $Q^{{\mathrm{gp}}}$.
Note that the morphism $W\rightarrow\operatorname{Spec}\kappa[Q]$ identifies
$W$ with the closed point of $\operatorname{Spec}\kappa[Q]$ corresponding to
the maximal monomial ideal $\mathfrak{m}=\langle z^{q}\,|\,q\in
Q\smallsetminus\\{0\\}\rangle$.

The monomial ideal $I\subseteq\kappa[Q^{\mathrm{sat}}]$ generated by the image
of $\mathfrak{m}$ then satisfies $\sqrt{I}=\mathfrak{m}^{\prime}=\langle
z^{q}\,|\,q\in
Q^{\mathrm{sat}}\smallsetminus(Q^{\mathrm{sat}})^{\times}\rangle$. Indeed, for
any element $q\in Q^{\mathrm{sat}}\smallsetminus(Q^{\mathrm{sat}})^{\times}$,
there exists a positive integer $n$ such that $nq\in Q$. Further, $nq\not=0$
since otherwise $q$ is torsion in $Q^{\mathrm{sat}}$ and hence invertible.
Thus $z^{nq}$ is a generator of $\mathfrak{m}$, so $z^{nq}\in I$. This shows
that $\mathfrak{m}^{\prime}\subseteq\sqrt{I}$. The converse holds as
$I\subseteq\mathfrak{m}^{\prime}$ and $\mathfrak{m}^{\prime}$ is a radical
ideal, as
$\kappa[Q^{\mathrm{sat}}]/\mathfrak{m}^{\prime}=\kappa[(Q^{{\mathrm{gp}}})_{{\mathrm{tors}}}]$
is reduced.

Thus we see that
$(W^{\mathrm{sat}})_{{\mathrm{red}}}=\operatorname{Spec}\kappa[Q^{\mathrm{sat}}]/\mathfrak{m}^{\prime}=\operatorname{Spec}\kappa[(Q^{{\mathrm{gp}}})_{{\mathrm{tors}}}]$.
However, the latter consists of $|(Q^{{\mathrm{gp}}})_{{\mathrm{tors}}}|$
points. ∎

######  Lemma 3.2.

Let $W_{1},W_{2},X$ be fs log schemes with morphisms $W_{1},W_{2}\rightarrow
X$. Then there is a canonical isomorphism

$(W_{1,{\mathrm{red}}}\times^{\mathrm{fs}}_{X_{{\mathrm{red}}}}W_{2,{\mathrm{red}}})_{{\mathrm{red}}}\cong(W_{1}\times_{X}^{\mathrm{fs}}W_{2})_{{\mathrm{red}}}.$

###### Proof.

Considering the strict closed immersion $(W_{1})_{{\mathrm{red}}}\rightarrow
W_{1}$, we obtain via base change a strict closed immersion
$(W_{1})_{{\mathrm{red}}}\times^{\mathrm{fs}}_{X}W_{2}\rightarrow
W_{1}\times^{\mathrm{fs}}_{X}W_{2}$. Repeating with $W_{2}$, we obtain a
strict closed immersion
$(W_{1})_{{\mathrm{red}}}\times^{\mathrm{fs}}_{X}(W_{2})_{{\mathrm{red}}}\rightarrow
W_{1}\times^{\mathrm{fs}}_{X}W_{2}$. Since the morphisms
$(W_{i})_{{\mathrm{red}}}\rightarrow X$ factor through $X_{{\mathrm{red}}}$,
the former log scheme is isomorphic to
$(W_{1})_{{\mathrm{red}}}\times^{\mathrm{fs}}_{X_{{\mathrm{red}}}}(W_{2})_{{\mathrm{red}}}$.

Thus we obtain a strict closed immersion
$(W_{1,{\mathrm{red}}}\times^{\mathrm{fs}}_{X_{{\mathrm{red}}}}W_{2,{\mathrm{red}}})_{{\mathrm{red}}}\rightarrow(W_{1}\times_{X}^{\mathrm{fs}}W_{2})_{{\mathrm{red}}}$,
which we must now prove is an isomorphism, which we do by showing this map
induces a bijection on geometric points. Indeed, the set of geometric points
of $W_{1}\times^{\mathrm{fs}}_{X}W_{2}$ and
$(W_{1}\times^{\mathrm{fs}}_{X}W_{2})_{{\mathrm{red}}}$ are the same, and a
strict geometric point point
$\operatorname{Spec}(Q\rightarrow\kappa)\rightarrow
W_{1}\times^{\mathrm{fs}}_{X}W_{2}$ clearly induces a strict closed point
$\operatorname{Spec}(Q\rightarrow\kappa)\rightarrow(W_{1})_{{\mathrm{red}}}\times^{\mathrm{fs}}_{X_{{\mathrm{red}}}}(W_{2})_{{\mathrm{red}}}$.
This shows the claim. ∎

######  Lemma 3.3.

Let $W_{1},W_{2},X$ be finite length connected fs log schemes over
$\operatorname{Spec}\kappa$ with $\kappa$ an algebraically closed field. Write
the ghost sheaf monoids as $Q_{1},Q_{2}$ and $P$ respectively. Suppose given
morphisms $f_{i}:W_{i}\rightarrow X$ inducing
$\theta_{i}=\bar{f}_{i}^{\flat}:P\rightarrow Q_{i}$. Set

$\theta:=(\theta^{{\mathrm{gp}}}_{1},-\theta^{{\mathrm{gp}}}_{2}):P^{{\mathrm{gp}}}\rightarrow
Q_{1}^{{\mathrm{gp}}}\oplus Q_{2}^{{\mathrm{gp}}}.$

Then

  1. (1)

If the fs fibre product $W_{1}\times^{\mathrm{fs}}_{X}W_{2}$ is non-empty, it
has $|\operatorname{coker}(\theta)_{\mathrm{tors}}|$ connected components.

  2. (2)

$|\operatorname{coker}(\theta)_{\mathrm{tors}}|=|\operatorname{coker}(\theta^{t})_{\mathrm{tors}}|,$
where $\theta^{t}:Q_{1}^{*}\oplus Q_{2}^{*}\rightarrow P^{*}$ denotes the
transpose map to $\theta$.

###### Proof.

(1) By Lemma 3.2, we may assume $W_{i}$ and $X$ are (reduced) log points.
According to the construction of the fs fibre product in [Og, III, §2.1], we
proceed in a couple of steps. Let $W,W^{\mathrm{int}}$ and $W^{\mathrm{fs}}$
denote the fibre product $W_{1}\times_{X}W_{2}$ in the category of log
schemes, fine log schemes, and fs log schemes. Then $W^{\mathrm{int}}$ is the
integralization of $W$ and $W^{\mathrm{fs}}$ is the saturation of
$W^{\mathrm{int}}$.

First, $\underline{W}$ agrees with the fibre product
$\underline{W}_{1}\times_{\underline{X}}\underline{W}_{2}=\operatorname{Spec}\kappa$.
Integralization involves passing to a closed subscheme of
$\operatorname{Spec}\kappa$. Thus either
$\underline{W}^{\mathrm{int}}=\operatorname{Spec}\kappa$ or is the empty
scheme. We rule out the latter as we have assumed that the fs fibre product is
non-empty. In this case,
$W^{\mathrm{int}}=\operatorname{Spec}(Q\rightarrow\kappa)$, where

$Q=Q_{1}\oplus^{\mathrm{fine}}_{P}Q_{2}$

and $\oplus^{\mathrm{fine}}$ denotes push-out in the category of fine monoids.
This is constructed (see [Og, I, Prop. 1.3.4] and its proof) as the fine
submonoid of

$\operatorname{coker}\theta=Q_{1}^{{\mathrm{gp}}}\oplus_{P^{\mathrm{gp}}}Q_{2}^{{\mathrm{gp}}}=Q^{{\mathrm{gp}}}$

generated by the images of $Q_{1}$ and $Q_{2}$.

As $W^{\mathrm{fs}}$ is the saturation of $W^{\mathrm{int}}$, (1) now follows
from Lemma 3.1.

(2) is easy homological algebra: $0\rightarrow\ker\theta\rightarrow
P^{{\mathrm{gp}}}\rightarrow Q_{1}^{{\mathrm{gp}}}\oplus
Q_{2}^{{\mathrm{gp}}}\rightarrow\operatorname{coker}(\theta)\rightarrow 0$ is
a free resolution of $\operatorname{coker}(\theta)$, and
$\operatorname{Ext}^{1}(\operatorname{coker}(\theta),\mathbb{Z})$ is
isomorphic to the torsion part of $\operatorname{coker}(\theta)$. However,
this $\operatorname{Ext}$ group is calculated as the middle cohomology of the
complex $Q_{1}^{*}\oplus Q_{2}^{*}\rightarrow
P^{*}\rightarrow\ker(\theta)^{*}$, and as the kernel of
$P^{*}\rightarrow\ker(\theta)^{*}$ is a saturated sublattice of $P^{*}$, the
torsion part of $\operatorname{coker}(\theta^{t})$ agrees with
$\operatorname{Ext}^{1}(\operatorname{coker}(\theta),\mathbb{Z})$. ∎

The following is a somewhat technically complicated criterion for non-
emptiness for $W_{1}\times^{\mathrm{fs}}_{X}W_{2}$, which is tailored for our
gluing needs.

######  Lemma 3.4.

Let $W_{i},X$ be as in Lemma 3.3, and suppose also given log points
$W_{i}^{\prime},X^{\prime}$ over $\operatorname{Spec}\kappa$ similarly with
maps $f_{i}^{\prime}:W_{i}^{\prime}\rightarrow X^{\prime}$. Write
$Q_{i}^{\prime},P^{\prime}$ for the corresponding monoids,
$\theta_{i}^{\prime}:P^{\prime}\rightarrow Q_{i}^{\prime}$ for the induced
maps. Suppose further given a commutative diagram

(3.1)
$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{f_{1}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\scriptstyle{g_{2}}$$\textstyle{W_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}^{\prime}}$$\textstyle{X^{\prime}}$$\textstyle{W_{2}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}^{\prime}}$

Suppose that:

  1. (1)

$W_{1}^{\prime}\times^{\mathrm{fs}}_{X^{\prime}}W_{2}^{\prime}$ is non-empty.

  2. (2)

The projections $Q_{1}^{\vee}\times_{P^{\vee}}Q_{2}^{\vee}\rightarrow
Q_{i}^{\vee}$ have image intersecting the interior of $Q_{i}^{\vee}$.

  3. (3)

The maps $\bar{g}^{\flat}:P^{\prime}\rightarrow P$ and
$\bar{g}^{\flat}_{i}:Q_{i}^{\prime}\rightarrow Q_{i}$ are all injective, and
the map induced by $\theta$,

(3.2)
$P^{{\mathrm{gp}}}/\bar{g}^{\flat}((P^{\prime})^{{\mathrm{gp}}})\rightarrow
Q_{1}^{{\mathrm{gp}}}/\bar{g}_{1}^{\flat}((Q_{1}^{\prime})^{{\mathrm{gp}}})\oplus
Q_{2}^{{\mathrm{gp}}}/\bar{g}_{2}^{\flat}((Q_{2}^{\prime})^{{\mathrm{gp}}})$

is injective.

Then $W_{1}\times^{\mathrm{fs}}_{X}W_{2}$ is non-empty.

###### Proof.

Given the hypotheses, we will construct a morphism
$\operatorname{Spec}\kappa^{\dagger}\rightarrow
W_{1}\times^{\mathrm{fs}}_{X}W_{2}$ from the standard log point. Recall that
giving a morphism of log points
$g:\operatorname{Spec}(Q\rightarrow\kappa)\rightarrow\operatorname{Spec}(P\rightarrow\kappa)$
is equivalent to giving $g^{\flat}:P\times\kappa^{\times}\rightarrow
Q\times\kappa^{\times}$ written as

$g^{\flat}(p,t)=(\bar{g}^{\flat}(p),\chi_{g}(p)\cdot t)$

for $\bar{g}^{\flat}:P\rightarrow Q$ a local homomorphism and
$\chi_{g}:P\rightarrow\kappa^{\times}$ an arbitrary homomorphism. Note also
that $\theta:P\rightarrow Q$ being a local homomorphism can be characterized
dually by the statement that $\theta^{t}(Q^{\vee})$ intersects the interior of
$P^{\vee}$.

First, let $Q=(Q_{1}\oplus^{\mathrm{fs}}_{P}Q_{2})/\mathrm{tors}$. Note that
$Q$ would be the stalk of the ghost sheaf of any point of
$W_{1}\times^{\mathrm{fs}}_{X}W_{2}$ if this log scheme were non-empty. Then
$Q^{\vee}=Q_{1}^{\vee}\times_{P^{\vee}}Q_{2}^{\vee}$ ([ACGS1, Proposition
6.3.5]), with a similar expression for $(Q^{\prime})^{\vee}$.

Choose an element $q\in\operatorname{Int}(Q^{\vee})$. Then necessarily the
image $q_{i}$ of $q$ in $Q_{i}^{\vee}$ lies in
$\operatorname{Int}(Q_{i}^{\vee})$ by condition (2). Further, $q_{1}$ and
$q_{2}$ have the same image in $P^{\vee}$. Let
$q_{i}^{\prime}=(\bar{g}_{i}^{\flat})^{t}(q_{i})$. Necessarily
$q_{i}^{\prime}\in\operatorname{Int}((Q_{i}^{\prime})^{\vee})$ as
$\bar{g}_{i}^{\flat}:Q_{i}^{\prime}\rightarrow Q_{i}$ is a local homomorphism.
Further, $q_{1}^{\prime}$ and $q_{2}^{\prime}$ have the same image in
$(P^{\prime})^{\vee}$ because of commutativity of (3.1). This gives an element
$q^{\prime}:=(q_{1}^{\prime},q_{2}^{\prime})$ of
$(Q^{\prime})^{\vee}=(Q_{1}^{\prime})^{\vee}\times_{(P^{\prime})^{\vee}}(Q_{2}^{\prime})^{\vee}$.
In particular $q^{\prime}$ lies in the interior of $(Q^{\prime})^{\vee}$.

Hence we obtain a commutative diagram

$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathbb{N}}$$\textstyle{Q^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime}}$

with $q,q^{\prime}$ local homomorphisms. Note that $Q^{\prime}$ is the stalk
of the ghost sheaf at any point of
$W_{1}^{\prime}\times^{\mathrm{fs}}_{X^{\prime}}W_{2}^{\prime}$. By condition
(1) this latter log scheme is non-empty, and so there is a morphism
$\operatorname{Spec}\kappa^{\dagger}\rightarrow
W_{1}^{\prime}\times^{\mathrm{fs}}_{X^{\prime}}W_{2}^{\prime}$ which induces
the map $q^{\prime}:Q^{\prime}\rightarrow\mathbb{N}$ on stalks of ghost
sheaves. Indeed, it is sufficient to construct a morphism
$\operatorname{Spec}\kappa^{\dagger}\rightarrow\operatorname{Spec}(Q^{\prime}\rightarrow\kappa)$,
which is equivalent to giving a local homomorphism
$Q^{\prime}\rightarrow\mathbb{N}$ and a homomorphism
$Q^{\prime}\rightarrow\kappa^{\times}$. We take the local homomorphism
$Q^{\prime}\rightarrow\mathbb{N}$ to be given by $q^{\prime}$.

The chosen morphism can also be viewed as arising in a commutative diagram

(3.3)
$\textstyle{\operatorname{Spec}\kappa^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W^{\prime}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}}$

At the level of ghost sheaves, this diagram is given by

$\textstyle{\mathbb{N}}$$\textstyle{Q_{2}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}^{\prime}}$$\textstyle{Q_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{1}^{\prime}}$$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{2}^{\prime}}$$\scriptstyle{\theta_{1}^{\prime}}$

and the morphisms $\operatorname{Spec}\kappa^{\dagger}\rightarrow
W^{\prime}_{i}$ are then determined by additional data of maps
$\psi_{i}^{\prime}:Q_{i}^{\prime}\rightarrow\kappa^{\times}$. Commutativity of
(3.3) then comes down to the equality

(3.4)
$(\psi_{1}^{\prime}\circ\theta_{1}^{\prime})\cdot\chi_{f_{1}^{\prime}}=(\psi_{2}^{\prime}\circ\theta_{2}^{\prime})\cdot\chi_{f_{2}^{\prime}}$

in ${\operatorname{Hom}}(P^{\prime},\kappa^{\times})$.

We now wish to construct an analogous commutative diagram

(3.5)
$\textstyle{\operatorname{Spec}\kappa^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$

given at the level of ghost sheaves by the commutative diagram

$\textstyle{\mathbb{N}}$$\textstyle{Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\textstyle{Q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{1}}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{2}}$$\scriptstyle{\theta_{1}}$

As all homomorphisms in this diagram are local, it is enough to construct
analogously $\psi_{i}:Q_{i}\rightarrow\kappa^{\times}$ such that

(3.6)
$(\psi_{1}\circ\theta_{1})\cdot\chi_{f_{1}}=(\psi_{2}\circ\theta_{2})\cdot\chi_{f_{2}}.$

By commutativity of (3.1), we have

(3.7)
$\chi_{g}\cdot(\chi_{f_{i}}\circ\bar{g}^{\flat})=\chi_{f_{i}^{\prime}}\cdot(\chi_{g_{i}}\circ\theta_{i}^{\prime})$

in ${\operatorname{Hom}}(P^{\prime},\kappa^{\times})$.

Using the assumed injectivity of $\bar{g}_{i}^{\flat}$ of condition (3), we
choose a lift $\psi_{i}:Q_{i}^{{\mathrm{gp}}}\rightarrow\kappa^{\times}$ of
$\chi_{g_{i}}^{-1}\cdot\psi_{i}^{\prime}:\bar{g}_{i}^{\flat}((Q_{i}^{\prime})^{{\mathrm{gp}}})\rightarrow\kappa^{\times}$.
As $\kappa$ is algebraically closed, this can always be done even if
$\bar{g}_{i}^{\flat}((Q_{i}^{\prime})^{{\mathrm{gp}}})$ is not saturated in
$Q_{i}^{{\mathrm{gp}}}$. Then for $p\in P^{\prime}$, we have, with the third
line by the definition of $\psi_{i}$ and (3.7),

$\displaystyle\big{(}(\psi_{i}\circ\theta_{i})\cdot\chi_{f_{i}}\big{)}(\bar{g}^{\flat}(p))={}$
$\displaystyle\psi_{i}(\theta_{i}(\bar{g}^{\flat}(p)))\cdot\chi_{f_{i}}(\bar{g}^{\flat}(p))$
$\displaystyle={}$
$\displaystyle\psi_{i}(\bar{g}_{i}^{\flat}(\theta_{i}^{\prime}(p)))\cdot\chi_{f_{i}}(\bar{g}^{\flat}(p))$
$\displaystyle={}$
$\displaystyle\left[\psi_{i}^{\prime}(\theta_{i}^{\prime}(p))\cdot\chi_{g_{i}}(\theta_{i}^{\prime}(p))^{-1}\right]\cdot\left[\chi_{g}(p)^{-1}\cdot\chi_{f_{i}^{\prime}}(p)\cdot\chi_{g_{i}}(\theta_{i}^{\prime}(p))\right]$
$\displaystyle={}$
$\displaystyle\left[\psi_{i}^{\prime}(\theta^{\prime}_{i}(p))\cdot\chi_{f_{i}^{\prime}}(p)\right]\cdot\chi_{g}(p)^{-1}.$

By (3.4), this is independent of $i$.

Now consider

$\left[(\psi_{1}\circ\theta_{1})\cdot\chi_{f_{1}}\right]\cdot\left[(\psi_{2}\circ\theta_{2})\cdot\chi_{f_{2}}\right]^{-1}\in{\operatorname{Hom}}(P^{{\mathrm{gp}}},\kappa^{\times}),$

and note this homomorphism is the identity on
$\bar{g}^{\flat}((P^{\prime})^{{\mathrm{gp}}})$ by the previous paragraph.
Thus it induces an element of
${\operatorname{Hom}}(P^{{\mathrm{gp}}}/\bar{g}^{\flat}((P^{\prime})^{{\mathrm{gp}}}),\kappa^{\times})$.
As $\kappa$ is algebraically closed, $\kappa^{\times}$ is a divisible group
and hence by the injectivity of (3.2), we obtain a surjective map

${\operatorname{Hom}}(Q_{1}^{{\mathrm{gp}}}/\bar{g}_{1}^{\flat}((Q_{1}^{\prime})^{{\mathrm{gp}}}),\kappa^{\times})\times{\operatorname{Hom}}(Q_{2}^{{\mathrm{gp}}}/\bar{g}_{2}^{\flat}((Q_{2}^{\prime})^{{\mathrm{gp}}}),\kappa^{\times})\rightarrow{\operatorname{Hom}}(P^{{\mathrm{gp}}}/\bar{g}^{\flat}((P^{\prime})^{{\mathrm{gp}}}),\kappa^{\times}).$

Thus we may find
$\varphi_{i}\in{\operatorname{Hom}}(Q_{i}^{{\mathrm{gp}}}/\bar{g}_{i}^{\flat}((Q_{i}^{\prime})^{{\mathrm{gp}}}),\kappa^{\times})$
such that if we replace $\psi_{i}$ with $\varphi_{i}\cdot\psi_{i}$, (3.6)
holds. ∎

##  4\. Gluing one curve

We give a first application of the material of the previous subsection. We
consider our standard gluing situation as in Notation 2.3.

Now assume given basic punctured maps $f_{i}:C_{i}^{\circ}/W_{i}\rightarrow X$
of type ${\boldsymbol{\tau}}_{i}$ for $1\leq i\leq r$, with the
$W_{i}=\operatorname{Spec}(Q_{i}\rightarrow\kappa)$ being logarithmic points,
$Q_{i}$ the basic monoid associated to $f_{i}$, and $\kappa$ algebraically
closed. Suppose further that whenever $E\in\mathbf{E}$ with vertices
$v_{1},v_{2}$, with corresponding punctured points $p_{E,v_{i}}\in
C_{i(v_{i})}$, we have

(4.1) $f_{i(v_{1})}(p_{E,v_{1}})=f_{i(v_{2})}(p_{E,v_{2}}),$

i.e., the maps $f_{i}$ will glue schematically. We may then ask how many
gluings exist at the logarithmic level. More precisely, we would like to
understand the scheme $W$ defined as follows:

######  Definition 4.1.

The _gluing_ $f:C^{\circ}/W\rightarrow X$ of the punctured maps $f_{i}$ is
defined by

$W:=\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}})\times_{\prod_{i=1}^{r}\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}}_{i})}\prod_{i=1}^{r}W_{i},$

and $f:C^{\circ}/W\rightarrow X$ the pull-back of the universal map over
$\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}})$ to $W$.

In this situation, we introduce the following notation. For any punctured
point $p_{E,v}$ of $C^{\circ}_{i}$ indexed by a flag $v\in E\in\mathbf{E}$,
let $P_{E,v}$ be the stalk of $\overline{\mathcal{M}}_{X}$ at
$f_{i(v)}(p_{E,v})$. By (4.1), we have $P_{E,v_{1}}=P_{E,v_{2}}$ if
$v_{1},v_{2}$ are the vertices of $E$, and write both as $P_{E}$. For any
irreducible component of $C_{i}$ with generic point $\eta$ corresponding to a
vertex $v$ of $G_{i}$, write $P_{v}$ for the stalk of the ghost sheaf at
$f_{i}(\eta)$.

For each $i$, we have a family of tropical maps
$h_{i}:\Gamma(G_{i},\ell_{i})\rightarrow\Sigma(X)$ defined over $\tau_{i}$. If
$\omega_{v}\in\Gamma(G_{i},\ell_{i})$ is the cone corresponding to a vertex
$v\in V(G_{i})$, then we obtain by restriction a map

(4.2) $\mathrm{ev}_{v}:\omega_{v}\rightarrow\Sigma(X)$

mapping into the cone $P^{\vee}_{v,\mathbb{R}}\in\Sigma(X)$. Explicitly this
is defined as the transpose of

$\textstyle{P_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{f}_{i}^{\flat}}$$\textstyle{Q_{i},}$

where here $Q_{i}$ is identified with
$\overline{\mathcal{M}}_{C_{i},\bar{\eta}}$. Hence at the level of groups we
also obtain a map

$\mathrm{ev}_{v}:Q_{i}^{*}\rightarrow P_{v}^{*}.$

######  Definition 4.2.

With the notation as above, choose an orientation on each edge $E\in E(G)$ so
that $E$ has vertices $v_{E},v_{E}^{\prime}$ and is oriented from $v_{E}$ to
$v^{\prime}_{E}$. We define the _tropical gluing map_

$\Psi:\prod_{i=1}^{r}Q_{i}^{*}\times\prod_{E\in\mathbf{E}}\mathbb{Z}\rightarrow\prod_{E\in\mathbf{E}}P_{E}^{*}$

by

$\Psi\big{(}(q_{1},\ldots,q_{r}),(\ell_{E})_{E\in\mathbf{E}}\big{)}=\big{(}\mathrm{ev}_{v_{E}}(q_{i(v_{E})})+\ell_{E}\mathbf{u}(E)-\mathrm{ev}_{v^{\prime}_{E}}(q_{i(v_{E}^{\prime})})\big{)}_{E\in\mathbf{E}}.$

We define the _tropical multiplicity_ of the gluing situation to be

$\mu=\mu(\tau,\mathbf{E}):=|(\operatorname{coker}\Psi)_{{\mathrm{tors}}}|.$

######  Remark 4.3.

The map $\Psi$ is called the tropical gluing map for the following reason.
Suppose given $s=\big{(}(q_{i}),(\ell_{E})_{E\in E(G)}\big{)}\in\ker\Psi$ such
that $q_{i}\in Q_{i}^{\vee}$ for each $v$ and $\ell_{E}>0$ for each $E$. Then
we may construct a tropical map $h_{s}:G\rightarrow\Sigma(X)$ as follows.
First, for each $i$, let $G^{\prime}_{i}$ be the subgraph of $G_{i}$ obtained
by removing legs of the form $(E,v)$ for $E\in\mathbf{E}$. Then
$G_{i}^{\prime}$ is naturally identified with a subgraph of $G$, and we may
define $h_{s}|_{G_{i}^{\prime}}$ to agree with
$(h_{i})_{q_{i}}|_{G_{i}^{\prime}}$. On the other hand, if we give each
$E\in\mathbf{E}$ the length $\ell_{E}$, then $s\in\ker\Psi$ guarantees we can
extend $h_{s}$ across all edges $E\in\mathbf{E}$.

Thus it is reasonable to think of $\ker\Psi$ as the integral tangent space to
the family of glued tropical curves.

######  Theorem 4.4.

Suppose we are in the above situation. If the gluing $W$ is non-empty, then
$W$ has $\mu(\tau,\mathbf{E})$ connected components.

###### Proof.

By Theorem 2.5, we have an fs Cartesian diagram

(4.3)
$\textstyle{\widetilde{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{i=1}^{r}\widetilde{W}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\mathbf{L}}}$$\textstyle{\prod_{E\in\mathbf{E}}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\prod_{E\in\mathbf{E}}X\times
X}$

Here

$\widetilde{W}_{i}:=W_{i}\times_{\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}}_{i})}\widetilde{\mathscr{M}}^{\prime}(X,{\boldsymbol{\tau}}_{i})$

and

$\widetilde{W}:=W\times_{\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}})}\widetilde{\mathscr{M}}^{\prime}(X,{\boldsymbol{\tau}}).$

A priori all fibre products are over $B$, but by the assumption (4.1), the
composition of $f_{i}$ with the structure map $X\rightarrow B$ are all
constant with the same image, so we may replace $B$ by a suitable affine
neighbourhood of this image and apply Proposition 2.6. Thus we may replace $B$
with $\operatorname{Spec}\Bbbk$ in the above discussion and thus assume that
all products in (4.3) are defined over $\operatorname{Spec}\Bbbk$.

By Proposition 2.4, the underlying schemes of $\widetilde{W}$ and $W$ agree.
Thus we need to calculate the number of connected components of
$\widetilde{W}$. Further, by (4.1), for any edge $E\in\mathbf{E}$ with
endpoints $v_{1},v_{2}$, the evaluation maps $\widetilde{W}_{i(v)}\rightarrow
X$, $\widetilde{W}_{i(v^{\prime})}\rightarrow X$ both factor through the
strict closed point $f_{i(v_{1})}(p_{E,v_{1}})=f_{i(v_{2})}(p_{E,v_{2}})$.
Thus we may replace, for each edge $E\in\mathbf{E}$, the target $X$ with the
corresponding log point, and hence obtain an fs Cartesian diagram

(4.4)
$\textstyle{\widetilde{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{i=1}^{r}\widetilde{W}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\mathbf{L}}}$$\textstyle{\prod_{E\in\mathbf{E}}\operatorname{Spec}(P_{E}\rightarrow\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\prod_{E\in\mathbf{E}}\operatorname{Spec}(P_{E}\rightarrow\Bbbk)^{2}}$

Further, by Lemma 3.2, we may replace $\widetilde{W}_{i}$ by its reduction
without changing the number of connected components of $\widetilde{W}$. Again
by Proposition 2.4, the reduction of the underlying scheme of
$\widetilde{W}_{i}$ agrees with the underlying scheme of $W_{i}$ (being a
point). Thus now $\widetilde{W}$ is a fibre product of log points.

Note that for $L_{j}=(E,v)\in\mathbf{L}_{i}$,
$W_{i}\times_{\mathfrak{M}^{\prime}(X,{\boldsymbol{\tau}}_{i})}\mathfrak{M}^{\prime}_{L_{j}}(X,{\boldsymbol{\tau}}_{i})$
has ghost sheaf $Q_{i}^{L_{j}}\subseteq Q_{i}\oplus\mathbb{Z}$, with an
equality on the level of groups, and with the induced evaluation map
$f_{i}\circ p_{E,v}:W_{i}\rightarrow X$ yielding a map at the tropical level
of

$((f_{i}\circ p_{E,v})^{\flat})^{t}:(Q_{i}^{L_{j}})^{\vee}\rightarrow
P_{E}^{\vee}$

taking the value on $(s,\ell)\in(Q_{i}^{L_{j}})^{\vee}\subseteq
Q_{i}^{\vee}\oplus\mathbb{Z}$ given by

$\mathrm{ev}_{v}(s)+\ell\mathbf{u}(L_{i}).$

Further, from the fibre product description (2.5) we see that

$\overline{\mathcal{M}}_{\widetilde{W}_{i}}\subseteq
Q_{i}\oplus\bigoplus_{(E,v)\in L(G_{i})}\mathbb{Z},$

again with an equality on groups.

Of course $\Delta$ induces a map $P_{E}^{*}\rightarrow P_{E}^{*}\times
P_{E}^{*}$ given by the diagonal. Putting this together, the homomorphism
$\theta^{t}$ of Lemma 3.3 then takes the form

(4.5)
$\theta^{t}:\prod_{E}P_{E}^{*}\times\prod_{i=1}^{r}Q_{i}^{*}\times\prod_{v\in
E\in\mathbf{E}}\mathbb{Z}\rightarrow\prod_{v\in E\in\mathbf{E}}P_{E}^{*}$

given by

$\theta^{t}\big{(}(n_{E})_{E\in E(G)},(s_{i})_{1\leq i\leq
r},(\ell_{E,v})_{v\in
E\in\mathbf{E}}\big{)}=(\mathrm{ev}_{v}(s_{i(v)})+\ell_{E,v}\mathbf{u}(E,v)-n_{E})_{v\in
E\in\mathbf{E}}.$

If $\widetilde{W}$ is non-empty, then by Lemma 3.3 the number of its connected
components is the order of the torsion part of
$\operatorname{coker}\theta^{t}$. We next compare this with the order of the
torsion part of $\operatorname{coker}\Psi$.

We have a diagram

(4.6)
$\textstyle{\prod_{E\in\mathbf{E}}\mathbb{Z}\times\prod_{E\in\mathbf{E}}P_{E}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\pi}$$\textstyle{\prod_{E\in\mathbf{E}}P_{E}^{*}\times\prod_{i}Q_{i}^{*}\times\prod_{v\in
E\in\mathbf{E}}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{t}}$$\scriptstyle{\gamma}$$\textstyle{\prod_{i}Q_{i}^{*}\times\prod_{E\in\mathbf{E}}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\prod_{E\in\mathbf{E}}P_{E}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\prod_{v\in
E\in\mathbf{E}}P_{E}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\prod_{E\in\mathbf{E}}P_{E}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}\theta^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{coker}\Psi}$

Here $\pi$ is the surjective map given by

$\pi\big{(}(\ell_{E})_{E\in\mathbf{E}},(n_{E})_{E\in\mathbf{E}}\big{)}=\big{(}\ell_{E}\mathbf{u}(E,v_{E})-n_{E})_{E\in\mathbf{E}},$

where we use the chosen vertices $v_{E},v^{\prime}_{E}$ of Definition 4.2.
Further, $\alpha,\beta$ are injections defined by

$\displaystyle\alpha\big{(}(\ell_{E})_{E\in\mathbf{E}},(n_{E})_{E\in\mathbf{E}}\big{)}={}$
$\displaystyle\big{(}(n_{E})_{E\in\mathbf{E}},(0)_{1\leq i\leq
r},((-1)^{\delta_{v,v^{\prime}_{E}}}\ell_{E})_{v\in E\in\mathbf{E}}\big{)},$
$\displaystyle\beta\big{(}(n_{E})_{E\in\mathbf{E}}\big{)}={}$
$\displaystyle\big{(}(n_{E})_{v\in E\in\mathbf{E}}\big{)}.$

while $\gamma$ and $\delta$ are surjections defined by

$\displaystyle\gamma\big{(}(n_{E})_{E\in\mathbf{E}},(s_{i})_{1\leq i\leq
r},(\ell_{E,v})_{v\in E\in\mathbf{E}}\big{)}={}$
$\displaystyle\big{(}(s_{i})_{1\leq i\leq
r},(\ell_{E,v_{E}}+\ell_{E,v^{\prime}_{E}})_{E\in\mathbf{E}}\big{)},$
$\displaystyle\delta\big{(}(n_{E,v})_{v\in E\in\mathbf{E}}\big{)}={}$
$\displaystyle\big{(}n_{E,v_{E}}-n_{E,v^{\prime}_{E}}\big{)}_{E\in\mathbf{E}}.$

Finally, $\Psi$ is the tropical gluing map of Definition 4.2. One checks that
the diagram is commutative with the top two rows exact, and hence the snake
lemma implies that
$\operatorname{coker}\theta^{t}\cong\operatorname{coker}\Psi$, giving the
result. ∎

######  Example 4.5.

It is very important to note that the gluing parameter space $W$ need not be
reduced, something which is quite different from gluing ordinary stable maps.
This arises via saturation in the fs fibre product. For example, consider a
target space $X$ with $\Sigma(X)=\mathbb{R}_{\geq 0}^{2}$, and a gluing
situation where $\tau$ is a graph with two vertices, $v_{1}$ and $v_{2}$, and
two edges, $E_{1}$ and $E_{2}$, each connecting $v_{1}$ with $v_{2}$. We split
at the two edges, getting types $\tau_{1},\tau_{2}$. We assume
$\mathbf{u}(E_{1},v_{1})=-\mathbf{u}(E_{1},v_{2})=(-w_{1},w_{1})$ and
$\mathbf{u}(E_{2},v_{1})=-\mathbf{u}(E_{2},v_{2})=(-w_{2},w_{2})$ with
$\gcd(w_{1},w_{2})=1$ and $w_{1}<w_{2}$. Finally, we assume
$\boldsymbol{\sigma}(v_{1})=\mathbb{R}_{\geq 0}\times 0$ and
$\boldsymbol{\sigma}(v_{2})=0\times\mathbb{R}_{\geq 0}$.

A slightly tedious calculation shows that
$\underline{W}=\operatorname{Spec}\Bbbk[t]/(t^{w_{1}})$ in this case. Morally,
one can think of this as follows. If the glued curve smooths, there are
smoothing parameters $u_{1},u_{2}$ for the two nodes, in the sense that the
local structure at each node is of the form $xy=u_{1}$ or $xy=u_{2}$. The
relation $u_{1}^{w_{2}}=u_{2}^{w_{1}}$ is then forced by the logarithmic
geometry of the situation. Saturation normalizes this curve, and the inverse
image of the point with ideal $(u_{1},u_{2})$ under this normalization is the
non-reduced gluing.

######  Example 4.6.

As soon as the target space $X$ has strata of codimension at least three, it
is easy to find examples where the gluing does not exist. Here is a standard
local example. Let $X$ be the three-fold ordinary double point
$\operatorname{Spec}\Bbbk[x,y,z,w]/(xy-zw)$, with toric log structure. Let
$\underline{C}_{1}=V(x,z,y-\lambda_{1}w)\subseteq X$,
$\underline{C}_{2}=V(y,w,\lambda_{2}x-z)$, for
$\lambda_{1},\lambda_{2}\in\Bbbk^{\times}$. In this case, $\Sigma(X)$ is the
cone in $\mathbb{R}^{3}$ generated by $v_{x,z}=(0,0,1)$, $v_{x,w}=(1,0,1)$,
$v_{y,z}=(0,1,1)$ and $v_{y,w}=(1,1,1)$. Here each generator generates a ray
corresponding to a toric divisor of $X$, and the subscripts indicate which
coordinates are zero on that divisor. It is possible to give the domain curves
$\underline{C}_{i}$ the structure of punctured log curves (ignoring the fact
these curves are not proper) so that the inclusions
$\underline{C}_{i}\rightarrow\underline{X}$ become punctured log maps, with
punctures $p_{i}\in C_{i}$ mapping to $0\in X$ and contact orders
$u_{1}=(1,1,0)$ and $u_{2}=(-1,-1,0)$. The basic monoids $Q_{1}$, $Q_{2}$ are
both $\mathbb{N}$. One checks easily that this gluing situation is not
tropically transverse.

In this case, a careful analysis of the fibre product shows the fibre product
is only non-empty if $\lambda_{1}=\lambda_{2}$, i.e., $\underline{C}_{1}$ and
$\underline{C}_{2}$ have the same slope. Morally, this can be seen by blowing
up the divisor $x=z=0$ in $X$ to obtain a small resolution
$\widetilde{X}\rightarrow X$. The strict transforms of $\underline{C}_{1}$ and
$\underline{C}_{2}$ are then disjoint unless $\lambda_{1}=\lambda_{2}$. Before
the blow-up, the slopes are encoded in the evalaution maps
$\widetilde{W}_{i}\rightarrow X$; after the blow-up, the slopes are already
visible in the schematic evaluation maps.

######  Remark 4.7.

Suppose instead we are given a gluing situation of maps to $\mathcal{X}$
equipped with evaluation maps at the punctured points, i.e., the maps are
determined by morphisms
$W_{i}\rightarrow\mathfrak{M}^{\prime\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X},{\boldsymbol{\tau}}_{i})$
rather than $W_{i}\rightarrow\mathscr{M}^{\prime}(X,{\boldsymbol{\tau}}_{i})$,
still with $\underline{W}_{i}=\operatorname{Spec}\kappa$. In other words, we
are given pre-stable punctured maps
$f_{v}:C_{v}^{\circ}/W_{v}\rightarrow\mathcal{X}$ along with compatible
morphisms $\underline{W}_{i}\rightarrow\underline{X}$ for each punctured point
$p_{E,v}$. We assume further, as before, that the images of $p_{E,v_{1}}$,
$p_{E,v_{2}}$ under these maps to $\underline{X}$ agree for each edge $E$.
Then similarly, the number of connected components of the glued space $W$ is
$\mu(\tau,\mathbf{E})$. Indeed, the gluing is controlled by precisely the same
Cartesian diagram as in the situation of Theorem 4.4.

Note that Theorem 4.4 says nothing about whether $W$ is non-empty. Here is one
often useful criterion:

######  Definition 4.8.

We say a gluing situation is _tropically transverse_ if the map $\Psi$ of
Definition 4.2 has finite cokernel.

######  Theorem 4.9.

In the situation of Theorem 4.4, $W$ is non-empty if the gluing situation is
tropically transverse.

###### Proof.

We apply the non-emptiness criterion of Lemma 3.4 to the product of (4.4). In
that lemma, we take $W_{i}^{\prime}=X^{\prime}=\operatorname{Spec}\Bbbk$ so
that the first condition of the lemma is trivially satisfied, and the third
condition is equivalent to the injectivity of the map $\theta$ whose transpose
$\theta^{t}$ is given in (4.5). However, $\theta^{t}$ having a finite cokernel
implies $\theta$ is injective, and (4.6) implies
$\operatorname{coker}\theta^{t}\cong\operatorname{coker}\Psi$, and hence
tropical transversality implies the third condition. Finally, $\tau$
realizable implies the second condition. ∎

##  5\. Gluing moduli spaces

###  5.1. The general situation

We continue with a standard gluing situation for $X/B$, as in Notation 2.3.

######  Theorem 5.1.

There is a diagram

$\textstyle{\mathscr{M}(X/B,{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{\mathrm{ev}}}$$\scriptstyle{\phi^{\prime}}$$\textstyle{\mathscr{M}^{\mathrm{sch}}(X/B,{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{\mathrm{sch}}}$$\textstyle{\prod_{i=1}^{r}\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\varepsilon}}$$\textstyle{\mathfrak{M}^{\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathfrak{M}^{\mathrm{sch},\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta^{\prime}}$$\scriptstyle{\underline{\mathrm{ev}}^{\prime}}$$\textstyle{\prod_{i=1}^{r}\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mathrm{ev}}}$$\textstyle{\prod_{E\in\mathbf{E}}\underline{X}_{\boldsymbol{\sigma}(E)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\prod_{v\in
E\in\mathbf{E}}\underline{X}_{\boldsymbol{\sigma}(E)}}$

with all squares Cartesian in all categories, defining the moduli spaces
$\mathscr{M}^{\mathrm{sch}}(X/B,{\boldsymbol{\tau}})$ and
$\mathfrak{M}^{\mathrm{sch},\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)$.
Further, $\phi$ is a finite representable morphism.

###### Proof.

We define the morphism
$\prod_{i=1}^{r}\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})\rightarrow\prod_{v\in
E\in\mathbf{E}}\underline{X}_{\boldsymbol{\sigma}(E)}$ as follows. For flag
$v\in E\in\mathbf{E}$, consider the composition

$\prod_{i=1}^{r}\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})\rightarrow\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i(v)})}(\mathcal{X}/B,\tau_{i(v)})=\mathfrak{M}(\mathcal{X}/B,\tau_{i})\times_{\prod\underline{\mathcal{X}}}\prod\underline{X}\rightarrow\underline{X}$

where the first arrow is projection and the second is further projection onto
the factor $\underline{X}$ indexed by the leg $(E,v)\in\mathbf{L}_{i}$.
Necessarily, this morphism factors through
$\underline{X}_{\boldsymbol{\sigma}(E)}$ by the definition of a
$\tau_{i}$-marked curve, see [ACGS2, Def. 3.7,(1)]. This gives a morphism

$\underline{\mathrm{ev}}_{(E,v)}:\prod_{i=1}^{r}\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})\rightarrow\underline{X}_{\boldsymbol{\sigma}(E)},$

and we define

$\underline{\mathrm{ev}}:=\prod_{v\in
E\in\mathbf{E}}\underline{\mathrm{ev}}_{(E,v)}.$

The morphism $\Delta$ is the product of diagonals. It is then clear that the
Cartesian diagram of Theorem 2.7 factors as stated, as
$\mathfrak{M}^{\mathrm{sch},\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)$
captures those curves which glue schematically. Further, $\phi$ is a finite
and representable morphism as $\psi=\Delta^{\prime}\circ\phi$ is finite and
representable by Theorem 2.7. ∎

######  Remark 5.2.

As stated, this is not a significant improvement over Theorem 2.7: it merely
separates the gluing into two steps, the first step being schematic gluing and
the second step taking into account only the gluing at the logarithmic level.

For the first step, we are in luck if (1) $\Delta$ is lci, so that the Gysin
pull-back $\Delta^{!}$ exists, and (2) $\underline{\mathrm{ev}}$ is flat, so
that $\Delta^{!}$ and $(\Delta^{\prime})^{!}$ induce the same map

$\Delta^{!}=(\Delta^{\prime})^{!}:A_{*}\left(\prod_{i}\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})\right)\rightarrow
A_{*}\left(\mathscr{M}^{\mathrm{sch}}(X/B,{\boldsymbol{\tau}})\right).$

In any event, each
$\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})$ is pure-
dimensional by [ACGS2, Prop. 3.28], so each
$\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})$ carries a virtual fundamental
class. Further, the pull-back of the relative obstruction theory for
$\hat{\varepsilon}$ (the product over $i$ of the relative obstruction theories
for
$\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})\rightarrow\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})$)
yields a relative obstruction theory for $\varepsilon_{\mathrm{sch}}$. If
$\underline{\mathrm{ev}}$ is flat, then
$\mathfrak{M}^{\mathrm{sch},\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)$ is
also pure-dimensional, yielding a virtual fundamental class
$[\mathscr{M}^{\mathrm{sch}}(X/B,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}$,
and if $\Delta$ is lci, we have

(5.1)
$[\mathscr{M}^{\mathrm{sch}}(X/B,{\boldsymbol{\tau}})]^{{\operatorname{virt}}}=\Delta^{!}\left(\prod_{i}[\mathscr{M}(X/B,{\boldsymbol{\tau}}_{i})]^{{\operatorname{virt}}}\right).$

Note that $\Delta$ is lci if and only if the strata
$\underline{X}_{\boldsymbol{\sigma}(E)}$ are non-singular. However, a deepest
stratum of $X$ is always non-singular, as is a stratum of dimension one more
than a deepest stratum of $X$. If instead the log structure on $X$ arises from
an snc divisor, all strata are non-singular.

On the other hand, flatness of $\underline{\mathrm{ev}}$ is only automatic
when the strata $X_{\boldsymbol{\sigma}(E)}$ are always deepest strata.
Otherwise, more care needs to be taken. Again, when the gluing strata are non-
singular, there is a tropical characterization of flatness.

######  Theorem 5.3.

Let $\tau=(G,\mathbf{g},\boldsymbol{\sigma},\mathbf{u})$ be a realizable type
of punctured map over $B$, $\mathbf{L}\subseteq L(G)$ a subset of legs, and
let

$\underline{\mathrm{ev}}:\mathfrak{M}(\mathcal{X}/B,\tau)\rightarrow\prod_{L\in\mathbf{L}}\underline{\mathcal{X}}_{\boldsymbol{\sigma}(L)}$

be the schematic evaulation map at the punctured points indexed by
$\mathbf{L}$. Suppose the strata
$\underline{\mathcal{X}}_{\boldsymbol{\sigma}(L)}$ for $L\in\mathbf{L}$ are
non-singular. Then $\underline{\mathrm{ev}}$ is flat if and only if, for every
realizable type
$\tau^{\prime}=(G^{\prime},\mathbf{g}^{\prime},\boldsymbol{\sigma}^{\prime},\mathbf{u}^{\prime})$
with a contraction morphism $\phi:\tau^{\prime}\rightarrow\tau$, the
inequality

$\dim\tau^{\prime}\geq\dim\tau+\sum_{L\in\mathbf{L}}(\dim\boldsymbol{\sigma}^{\prime}(L)-\dim\boldsymbol{\sigma}(L))$

holds.

###### Proof.

Since $\mathfrak{M}^{\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)$ is equi-
dimensional (see [ACGS2, Prop. 3.28]), if $\underline{\mathrm{ev}}$ is flat
then all stack-theoretic fibres $\underline{\mathrm{ev}}^{-1}(\bar{x})$ of
$\underline{\mathrm{ev}}$ are equi-dimensional with dimension independent of
the choice of a geometric point
$\bar{x}\rightarrow\prod_{L\in\mathbf{L}}\underline{\mathcal{X}}_{\boldsymbol{\sigma}(L)}$.
Conversely, it follows from [ACGS2, Thm. 3.24] that
$\mathfrak{M}(\mathcal{X}/B,\tau)$ is idealized log smooth over
$\operatorname{Spec}\Bbbk$ (as $\mathbf{M}(G,\mathbf{g})\times B$ appearing in
that theorem is idealized log smooth over $\operatorname{Spec}\Bbbk$).
However, from the specific description of the idealized structure on
$\mathfrak{M}(\mathcal{X}/B,\tau)$ in [ACGS2, Prop. 3.23], it follows from
[ACGS2, Prop. B.4] that $\mathfrak{M}(\mathcal{X}/B,\tau)$ smooth locally is a
stratum of a toric variety, and hence in particular smooth locally is toric,
hence Cohen-Macaulay. Thus, by [Ma89, Thm. 23.1], if the fibres of
$\underline{\mathrm{ev}}$ are all equi-dimensional of dimension
$\dim\mathfrak{M}(\mathcal{X}/B,\tau)-\dim\prod_{L\in\mathbf{L}}\underline{\mathcal{X}}_{\boldsymbol{\sigma}(L)}$,
then $\underline{\mathrm{ev}}$ is flat. Thus the question reduces to
calculating the fibre dimension.

Note that because we assumed $\tau$ was realizable over $B$,
$\mathfrak{M}(\mathcal{X}/B,\tau)$ contains a dense open stratum consisting of
punctured maps to $\mathcal{X}$ of type $\tau$. Further, we have by [ACGS2,
Prop. 3.28] that

$\dim\mathfrak{M}(\mathcal{X}/B,\tau)=3|\mathbf{g}|-3+|L(G)|-\dim\tau+\delta_{B}$

where

$\delta_{B}:=\begin{cases}0&B=\operatorname{Spec}\Bbbk\\\
1&B=\operatorname{Spec}\Bbbk^{\dagger}\end{cases}$

and $|\mathbf{g}|=b_{1}(G)+\sum_{v\in V(G)}\mathbf{g}(v)$.

Let $\bar{x}$ be a geometric point of the open stratum of
$\prod_{L\in\mathbf{L}}\underline{\mathcal{X}}_{\boldsymbol{\sigma}(L)}$. Note
this open stratum is isomorphic, as an algebraic stack, to
$B\mathbb{G}_{m}^{N}$ where

$N=-\dim\prod_{L\in\mathbf{L}}\underline{\mathcal{X}}_{\boldsymbol{\sigma}(L)}=\sum_{L\in\mathbf{L}}(\dim\boldsymbol{\sigma}(L)-\delta_{B}),$

and the dimension of the fibre of $\underline{\mathrm{ev}}$ over $\bar{x}$ is
then

(5.2)
$\dim\mathfrak{M}(\mathcal{X}/B,\tau)+N=3|\mathbf{g}|-3+|L(G)|-\dim\tau+\delta_{B}+\sum_{L\in\boldsymbol{\sigma}(L)}(\dim\boldsymbol{\sigma}(L)-\delta_{B}).$

Thus we need to show all fibres, when non-empty, have this dimension. So now
choose some other geometric point
$\bar{x}=(\bar{x}_{L})\rightarrow\prod_{L\in\mathbf{L}}\underline{\mathcal{X}}_{\boldsymbol{\sigma}(L)}$,
and suppose $\bar{x}_{L}$ lies in the stratum of $\mathcal{X}$ indexed by a
cone $\sigma_{L}$. Let $\tau^{\prime}$ be a type of punctured map which
appears in $\underline{\mathrm{ev}}^{-1}(\bar{x})$, so that there is a
contraction map $\phi:\tau^{\prime}\rightarrow\tau$. Note that
$\sigma_{L}=\boldsymbol{\sigma}^{\prime}(L)$ for $L\in\mathbf{L}$. Then the
dimension of the stratum of the fibre over $x$ with of punctured maps with
this type is similarly calculated as

(5.3)
$\displaystyle\begin{split}&3|\mathbf{g}^{\prime}|-3+|L(G^{\prime})|-\dim\tau^{\prime}+\delta_{B}+\sum_{L\in\boldsymbol{\sigma}(L)}(\dim\sigma_{L}-\delta_{B})\\\
={}&3|\mathbf{g}|-3+|L(G)|-\dim\tau^{\prime}+\delta_{B}+\sum_{L\in\boldsymbol{\sigma}(L)}(\dim\boldsymbol{\sigma}^{\prime}(L)-\delta_{B}).\end{split}$

Now since fibre dimension is upper semi-continuous, it is sufficient to show
that the quantity of (5.2) is greater than or equal to the quantity of (5.3).
But this is the inequality of the theorem. ∎

######  Remark 5.4.

The criterion of Theorem 5.3 may seem imposing to check as it in theory
involves an arbitrary number of type $\tau^{\prime}$. But we may always
replace $\mathfrak{M}(\mathcal{X}/B,\tau)$ with an open subset obtained by
deleting those strata with type $\tau^{\prime}$ (equipped with a contraction
$\tau^{\prime}\rightarrow\tau$) such that there does not exist a punctured map
in $\mathscr{M}(X/B,{\boldsymbol{\tau}})$ of type $\tau^{\prime\prime}$ such
that the induced contraction map $\tau^{\prime\prime}\rightarrow\tau$ factors
through $\tau^{\prime}\rightarrow\tau$. Thus for any application, it is
sufficient to restrict attention to tropical types which are contractions of
types which actually occur.

The next step is to understand $\phi$. In general, we don’t yet know how to
say much about it, save for the next theorem, which gives us the degree of
$\phi$ onto its image. When the image is a proper closed substack, this tells
us there are logarithmic obstructions to gluing, and it is still not
understood how to deal with these obstructions in general. However, in the
tropically transverse case, $\phi$ is surjective.

######  Theorem 5.5.

The degree of $\phi$ onto its image is the tropical multiplicity
$\mu(\tau,\mathbf{E})$ defined in Definition 4.2. If the gluing situation is
tropically transverse and $\underline{\mathrm{ev}}$ is flat, then $\phi$ is
dominant.

###### Proof.

We note that as $\tau$ is realizable over $B$,
$\mathfrak{M}^{\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)$ has a non-empty
dense open stratum $U$, whose geometric points are precisely those geometric
points corresponding to a punctured map whose tropicalization is a family of
tropical maps of type $\tau$. Similarly,
$\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})$ has a
non-empty dense stratum $U_{i}$, with each geometric point in this stratum
corresponding to a punctured map whose tropicalization is a family of curves
of type $\tau_{i}$. Certainly then
$\Delta^{\prime}\circ\phi(U)\subseteq\prod_{i}U_{i}$. Further,
$(\phi\circ\Delta)^{-1}(\prod_{i}U_{i})=U$ as the type of punctured map
obtained by gluing together punctured maps of type $\tau_{i}$ is necessarily
$\tau$. It is thus sufficient to determine the degree of $\phi|_{U}$ onto its
image in $(\Delta^{\prime})^{-1}(\prod_{i}U_{i})$. To this end, pick a strict
geometric point
$W^{\prime}\rightarrow\mathfrak{M}^{\mathrm{sch},\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau)$
in the image of $\phi|_{U}$. Because $\Delta^{\prime}$ is strict, we can
decompose $W^{\prime}=\prod_{i}W_{i}$ as a product over $B$, with
$W_{i}\rightarrow\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})$
strict geometric points. Thus we may view this as a situation of §4. In
particular, the corresponding gluing is
$W^{\prime}\times_{\prod_{i}\mathfrak{M}^{\prime\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X}/B,\tau_{i})}\mathfrak{M}^{\prime\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)$.
By Lemma 3.2 and [ACGS2, Prop. 3.31], the reduction of this gluing agrees with
the reduction of

$W^{\prime}\times_{\prod_{i}\mathfrak{M}^{\mathrm{ev}(\mathbf{L}_{i})}(\mathcal{X},\tau_{i})}\mathfrak{M}^{\mathrm{ev}(\mathbf{L})}(\mathcal{X},\tau)\cong
W^{\prime}\times_{\mathfrak{M}^{\mathrm{sch},\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)}\mathfrak{M}^{\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau).$

By Remark 4.7, this fibre product has $\mu(\tau,\mathbf{E})$ connected
components. Since this number is independent of the choice of geometric point
in the image of $\phi|_{U}$ and
$\mathfrak{M}^{\mathrm{ev}}(\mathcal{X}/B,\tau)$ is reduced, it follows this
fibre must be always reduced. This shows the degree.

If $\underline{\mathrm{ev}}$ is flat, then so is
$\underline{\mathrm{ev}}^{\prime}$, and hence every irreducible component of
$\mathfrak{M}^{\mathrm{sch},\mathrm{ev}(\mathbf{L})}(\mathcal{X}/B,\tau)$
dominates $\prod_{E\in\mathbf{E}}\underline{X}_{\boldsymbol{\sigma}(E)}$. The
domination of $\phi$ then similarly follows from Theorem 4.9. ∎

###  5.2. Gluing from rigid tropical curves

One of the standard situations for gluing is the degeneration situation
studied in [ACGS1]. Here, one considers a base $B$ a smooth curve or spectrum
of a DVR, with log structure the divisorial log structure induced by a closed
point $b_{0}\in B$. As usual, we assume $X\rightarrow B$ is projective and log
smooth. We denote by $X_{0}$ the fibre over $b_{0}$, and we view $b_{0}$ with
its induced log structure, i.e., $b_{0}$ is a standard log point.

In this case, the morphism $X\rightarrow B$ tropicalizes to a morphism
$\delta:\Sigma(X)\rightarrow\Sigma(B)=\mathbb{R}_{\geq 0}$. This gives rise to
a polyhedral cell complex $\Delta(X)=\delta^{-1}(1)$. Given a class $\beta$ of
log curve for $X/B$, we have the logarithmic decomposition formula, see
[ACGS1, Thm. 1.2]:

$[\mathscr{M}(X_{0}/b_{0},\beta)]^{{\operatorname{virt}}}=\sum_{{\boldsymbol{\tau}}=(\tau,{\bf
A})}{m_{\tau}\over|\operatorname{Aut}({\boldsymbol{\tau}})|}j_{\tau*}[\mathscr{M}(X_{0}/b_{0},{\boldsymbol{\tau}})]^{{\operatorname{virt}}}.$

Here ${\boldsymbol{\tau}}$ runs over isomorphism classes of decorated _rigid
tropical types_. These are realizable tropical types ${\boldsymbol{\tau}}$
such that the moduli space of tropical maps of type ${\boldsymbol{\tau}}$ is
one-dimensional, with a unique member of this one-dimensional family factoring
through $\Delta(X)$, hence the term rigid. The quantity
$m_{\tau}\in\mathbb{N}$ is the smallest integer such that scaling $\Delta(X)$
by $m_{\tau}$ leads to a tropical map with integral vertices and edge lengths;
put another way,

$m_{\tau}=|\operatorname{coker}(N_{\tau}\rightarrow
N_{\Sigma(B)}=\mathbb{Z})|.$

We now explore how Theorem 5.1 may be applied in this situation.
Unfortunately, the morphisms $\phi,\phi^{\prime}$ of Theorem 5.1 need not be
surjective, but here we determine the length of the inverse image of a point
in the image of $\phi$.

We first introduce the following notation. For each $\sigma\in\Sigma(X)$, the
morphism $\delta:\Sigma(X)\rightarrow\Sigma(B)$ induces homomorphisms
$\delta_{*}:N_{\sigma}\rightarrow\mathbb{Z}$, and we define
$\overline{N}_{\sigma}$ to be the kernel of this homomorphism. Provided
$\delta|_{\sigma}$ surjects onto $\Sigma(B)=\mathbb{R}_{\geq 0}$, this kernel
can be identified with the space of integral tangent vectors of the
corresponding polyhedron in $\Delta(X)$.

######  Definition 5.6.

Let $\tau$ be a type of rigid tropical curve in $\Delta(X)$. Define

$\overline{\Psi}:\bigoplus_{v\in
V(G)}\overline{N}_{\boldsymbol{\sigma}(v)}\oplus\bigoplus_{E\in
E(G)}\mathbb{Z}\rightarrow\bigoplus_{E\in
E(G)}\overline{N}_{\boldsymbol{\sigma}(E)}$

to be the homomorphism

$\big{(}(n_{v})_{v\in V(G)},(\ell_{E})_{E\in
E(G)}\big{)}\mapsto\big{(}n_{v_{E}}+\ell_{E}\mathbf{u}(E)-n_{v^{\prime}_{E}}\big{)}_{E\in
E(G)}$

where for each $E\in E(G)$, we orient $E$ from a vertex $v_{E}$ to a vertex
$v_{E}^{\prime}$ to determine the sign of $\mathbf{u}(E)$, which necessarily
lies in $\overline{N}_{\boldsymbol{\sigma}(E)}$. We define the _tropical
multiplicity_ of $\tau$ to be

$\mu(\tau):=|(\operatorname{coker}\overline{\Psi})_{{\mathrm{tors}}}|.$

######  Theorem 5.7.

Let $X\rightarrow B$ be as above. Fix a rigid tropical type
${\boldsymbol{\tau}}=(G,\boldsymbol{\sigma},{\bf u})$, and let
$\\{{\boldsymbol{\tau}}_{v}\,|\,v\in V(G)\\}$ be the decorated tropical types
obtained by splitting ${\boldsymbol{\tau}}$ at all edges. This gives a
standard gluing situation, and hence a diagram as in Theorem 5.1. With
notation as in that theorem, $\phi$ is degree $\mu(\tau)/m_{\tau}$ onto its
image.

###### Proof.

By Theorem 5.5, it is sufficient to compare the multiplicity
$|\operatorname{coker}(\Psi)_{{\mathrm{tors}}}|$ as defined in Definition 4.2
and the multiplicity
$|\operatorname{coker}(\overline{\Psi})_{{\mathrm{tors}}}|$. Note that as each
split type $\tau_{v}$ consists of a single vertex with a number of adjacent
edges, the only moduli of tropical maps of type $\tau_{v}$ is given by the
location of $v$ in $\boldsymbol{\sigma}(v)$. Thus one sees that the basic
monoid associated to the type $\tau_{v}$ is

(5.4) $Q_{v}=P_{\boldsymbol{\sigma}(v)}.$

Thus $Q_{v}^{*}=P_{\boldsymbol{\sigma}(v)}^{*}=N_{\boldsymbol{\sigma}(v)}$ and
$P_{E}^{*}=N_{\boldsymbol{\sigma}(E)}$ in Definition 4.2. The map $\Psi$ of
that definition now becomes

$\Psi:\bigoplus_{v\in V(G)}N_{\boldsymbol{\sigma}(v)}\oplus\bigoplus_{E\in
E(G)}\mathbb{Z}\rightarrow\bigoplus_{E\in E(G)}N_{\boldsymbol{\sigma}(E)},$

and the degree of $\phi$ onto its image is
$|(\operatorname{coker}\Psi)_{{\mathrm{tors}}}|$.

We now have a commutative diagram of exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v}\overline{N}_{\boldsymbol{\sigma}(v)}\times\prod_{E}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\Psi}}$$\textstyle{\prod_{v}N_{\boldsymbol{\sigma}(v)}\times\prod_{E}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\delta_{*}}$$\textstyle{\prod_{v}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{E}\overline{N}_{\boldsymbol{\sigma}(E)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{E}N_{\boldsymbol{\sigma}(E)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{*}}$$\textstyle{\prod_{E}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Here the two maps labelled $\delta_{*}$ are induced by
$\delta_{*}:N_{\boldsymbol{\sigma}(v)}\rightarrow\mathbb{Z}$ and
$\delta_{*}:N_{\boldsymbol{\sigma}(E)}\rightarrow\mathbb{Z}$ for each vertex
$v$ and edge $E$. The map $\overline{\Psi}$ is as defined in Definition 5.6.
The map $\partial$ is defined by

$\partial\big{(}(n_{v})_{v\in
V(G)}\big{)}=\big{(}n_{v_{E}}-n_{v^{\prime}_{E}}\big{)}_{E\in E(G)}.$

In particular, $\partial:\prod_{v}\mathbb{Z}\rightarrow\prod_{E}\mathbb{Z}$ is
the complex calculating the simplicial cohomology of $G$, and thus
$\ker\partial=H^{0}(G,\mathbb{Z})=\mathbb{Z}$, as $G$ is assumed connected.
Note $\ker\partial$ is generated by $(1)_{v\in V(G)}$. Also
$\operatorname{coker}\partial=H^{1}(G,\mathbb{Z})=\mathbb{Z}^{b_{1}(G)}$. Thus
the snake lemma gives a long exact sequence

$0\rightarrow\ker\overline{\Psi}\rightarrow\ker\Psi\rightarrow
H^{0}(G,\mathbb{Z})\rightarrow\operatorname{coker}\overline{\Psi}\rightarrow\operatorname{coker}\Psi\rightarrow
H^{1}(G,\mathbb{Z})\rightarrow 0.$

Note that $\ker\overline{\Psi}$ and $\ker\Psi$ can be interpreted as the space
of integral tangent vectors to the moduli space of maps of type $\tau$ in
$\Delta(X)$ and $\Sigma(X)$ respectively. By the assumption of rigidity of
$\tau$, we thus have $\ker\overline{\Psi}=0$ and $\ker\Psi=\mathbb{Z}$.
Further, the map $\mathbb{Z}\cong\ker\Psi\rightarrow
H^{0}(G,\mathbb{Z})\cong\mathbb{Z}$ is multiplication by $m_{\tau}$ by
definition of the latter number. This gives an exact sequence

$0\rightarrow\mathbb{Z}/m_{\tau}\mathbb{Z}\rightarrow\operatorname{coker}\overline{\Psi}\rightarrow\operatorname{coker}\Psi\rightarrow
H^{1}(G,\mathbb{Z})\rightarrow 0.$

Since $H^{1}(G,\mathbb{Z})$ is torsion-free and
$\mathbb{Z}/m_{\tau}\mathbb{Z}$ is torsion, we easily obtain a short exact
sequence

$0\rightarrow\mathbb{Z}/m_{\tau}\mathbb{Z}\rightarrow(\operatorname{coker}\overline{\Psi})_{{\mathrm{tors}}}\rightarrow(\operatorname{coker}\Psi)_{{\mathrm{tors}}}\rightarrow
0.$

This shows that the multiplicity $\mu(\tau,\mathbf{E})$ defined in Definition
4.2 agrees with $\mu(\tau)/m_{\tau}$, and the result follows. ∎

Again, we get better behaviour in the tropically transverse case, where now we
have a slightly weaker definition for tropical transversality.

######  Definition 5.8.

Let $\tau$ be the combinatorial type of a rigid tropical curve in $\Delta(X)$.
We say that $\tau$ is _tropically transverse_ if the image of the map
$\overline{\Psi}$ of Definition 5.6 has finite index.

######  Theorem 5.9.

In the situation of Theorem 5.7, suppose that the rigid decorated type
${\boldsymbol{\tau}}$ is tropically transverse, $\underline{\mathrm{ev}}$ is
flat, and the central fibre $\underline{X}_{0}$ is reduced. Then $\phi$ is a
finite dominant morphism of degree $\mu(\tau)/m_{\tau}$. Further, suppose that
each $\underline{X}_{\boldsymbol{\sigma}(E)}$ is non-singular (so that
$\Delta$ is an lci morphism and the Gysin map $\Delta^{!}$ exists). Then

$\phi^{\prime}_{*}[\mathscr{M}(X_{0}/b_{0},{\boldsymbol{\tau}})]^{{\operatorname{virt}}}={\mu({\boldsymbol{\tau}})\over
m_{{\boldsymbol{\tau}}}}\Delta^{!}\left(\prod_{v\in
V(G)}[\mathscr{M}(X_{0}/b_{0},{\boldsymbol{\tau}}_{v})]^{{\operatorname{virt}}}\right).$

###### Proof.

The statement concerning the degree of $\phi$ follows from Theorem 5.7
provided that we know $\phi$ is dominant. To show this, we follow the argument
of Theorem 5.5 and modify the argument of Theorem 4.4. In particular, we may
assume given a strict geometric point
$W^{\prime}\rightarrow\mathfrak{M}^{\mathrm{sch},\mathrm{ev}}(\mathcal{X}_{0}/b_{0},\tau)$,
with an induced decomposition with $W^{\prime}=\prod_{v}W_{v}$ and
$W_{v}\rightarrow\mathfrak{M}^{\mathrm{ev}}(\mathcal{X}_{0}/b_{0},\tau_{v})$
strict geometric points with image in the dense open strata of the latter
moduli spaces. We need to show that the gluing
$W^{\prime}\times_{\mathfrak{M}^{\mathrm{sch},\mathrm{ev}}(\mathcal{X}_{0}/b_{0},\tau)}\mathfrak{M}^{\mathrm{ev}}(\mathcal{X}_{0}/b_{0},\tau)$
is non-empty for general choice of strict morphism
$W^{\prime}\rightarrow\mathfrak{M}^{\mathrm{sch},\mathrm{ev}}(\mathcal{X}_{0}/b_{0},\tau)$
from a log point. To show this non-emptiness, we return to the setup and
notation of the proof of Theorems 4.4 and use the non-emptiness criterion of
Lemma 3.4, applied to the commutative diagram obtained from the fact that all
spaces involved are defined over $b_{0}$:444We note here we work with fibre
products over $\operatorname{Spec}\Bbbk$ rather than $B=b_{0}$, as we did in
Theorem 4.4. Here we apply Proposition 2.6 to see that we may just as well
work with the moduli spaces over $\operatorname{Spec}\Bbbk$.

(5.5) $\textstyle{\prod_{E\in
E(G_{)}}X_{\boldsymbol{\sigma}(E)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
E\in
E(G)}X_{\boldsymbol{\sigma}(E)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
V(G)}\widetilde{W}_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{E\in
E(G)}b_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
E\in E(G)}b_{0}}$$\textstyle{\prod_{v\in
V(G)}b_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Now $\prod_{E}b_{0}\times_{\prod_{v\in E}b_{0}}\prod_{v}b_{0}$ is easily seen
to be non-empty: indeed, there is a morphism from $b_{0}$ to this fibre
product induced by the diagonal morphisms $b_{0}\rightarrow\prod_{E}b_{0}$ and
$b_{0}\rightarrow\prod_{v}b_{0}$. (In fact, the induced morphism is an
isomorphism, so the fibre product is $b_{0}$, but we don’t need this.) This
gives condition (1) of the hypotheses of Lemma 3.4.

Condition (2) follows from realizability the tropical type $\tau$. Indeed, at
the level of dual of monoids, the fibre product involving the top row of (5.5)
gives a Cartesian diagram of monoids

$\textstyle{\widetilde{Q}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
V(G)}\widetilde{Q}_{v}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma(\mathrm{ev})}$$\textstyle{\prod_{E\in
E(G)}P_{\boldsymbol{\sigma}(E)}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma(\Delta)}$$\textstyle{\prod_{v\in
E\in E(G)}P_{\boldsymbol{\sigma}(E)}^{\vee}}$

Here $\widetilde{Q}_{v}$ is the stalk of the ghost sheaf of
$\widetilde{W}_{v}$, with $\widetilde{Q}_{v}\subseteq
Q_{v}\oplus\bigoplus_{v\in E\in E(G)}\mathbb{Z}$. Recall (5.4) that
$Q_{v}=P_{\boldsymbol{\sigma}(v)}$. We need to show that the image of
$\widetilde{Q}^{\vee}$ in $\prod_{E}P_{\boldsymbol{\sigma}(E)}^{\vee}$ or
$\prod_{v}\widetilde{Q}_{v}^{\vee}$ intersects the interior of that monoid.

First we describe $\Sigma(\Delta)$ and $\Sigma(\mathrm{ev})$. Indeed,
$\Sigma(\Delta)$ is just the diagonal, while

$\Sigma(\mathrm{ev})\left((n_{v})_{v\in V(G)},(\ell_{v,E})_{v\in E\in
E(G)}\right)=(n_{v}+\ell_{v,E}\mathbf{u}_{v}(E))_{v\in E\in E(G)}$

where $\mathbf{u}_{v}(E)$ is the contact order of the leg $(v,E)$ of $G_{v}$.
Thus the above fibre product diagram allows us to view $\widetilde{Q}^{\vee}$
as a submonoid of $\prod_{v\in v(G)}\widetilde{Q}_{v}^{\vee}$ consisting of
those tuples $\left((n_{v})_{v\in V(G)},(\ell_{v,E})_{v\in E\in E(G)}\right)$
such that, for every edge $E\in E(G)$ with vertices $v_{1},v_{2}$, we have

(5.6)
$n_{v_{1}}+\ell_{v_{1},E}\mathbf{u}_{v_{1}}(E)=n_{v_{2}}+\ell_{v_{2},E}\mathbf{u}_{v_{2}}(E).$

We may then interpret a point $s\in\widetilde{Q}^{\vee}$ as giving the data
of:

  1. (1)

The choice of a tropical map $h_{s}:G\rightarrow\Sigma(X)$ of type $\tau$.
This tropical map is determined by $h_{s}(v)=n_{v}\in
P^{\vee}_{\boldsymbol{\sigma}(v)}$, and the condition (5.6) then guarantees
that the edge $E$ is still mapped to an edge in $\boldsymbol{\sigma}(E)$ with
tangent vector $\mathbf{u}(E)$.

  2. (2)

For each edge $E\in E(G)$, a choice of decomposition of the length of $E$ as a
sum $\ell_{v_{1},E}+\ell_{v_{2},E}$. This is given by (5.6): since
$\mathbf{u}_{v_{2}}(E)=-\mathbf{u}_{v_{1}}(E)$, (5.6) can be rewritten as
$n_{v_{2}}-n_{v_{1}}=(\ell_{v_{1},E}+\ell_{v_{2},E})\mathbf{u}_{v_{1}}(E)$,
and thus $\ell_{E}=\ell_{v_{1},E}+\ell_{v_{2},E}$.

However, since $\tau$ is a type of rigid curve, there is a unique tropical
curve $h_{s}:\Gamma\rightarrow\Sigma(X)$ of type $\tau$, up to scaling. By
scaling this curve sufficiently so it is integrally defined and all edge
lengths are at least $2$, we may also choose non-trivial integral
decompositions of the edge lengths. This yields an interior point of
$\widetilde{Q}^{\vee}$, which necessarily maps to the interior of
$\prod_{v}\widetilde{Q}_{v}^{\vee}$ and to an interior point of
$\prod_{E}P^{\vee}_{\boldsymbol{\sigma}(E)}$. This shows condition (2) of
Lemma 3.4.

Abusing notation, we denote by $\delta$ the generator of the ghost sheaf
$\mathbb{N}$ of $b_{0}$, and also denote by $\delta$ its image in any of the
monoids $P_{\boldsymbol{\sigma}(E)}$ under the structure map $X_{0}\rightarrow
b_{0}$. Note this is largely compatible with our previous use of the notation
$\delta:\Sigma(X)\rightarrow\Sigma(B)$. To verify condition (3) of the lemma,
using the above description of the monoids involved in the fibre product, we
need to show injectivity of

$\bigoplus_{v\in E\in
E(G)}P_{\boldsymbol{\sigma}(E)}^{{\mathrm{gp}}}/\mathbb{Z}\delta\rightarrow\left[\bigoplus_{E\in
E(G)}P_{\boldsymbol{\sigma}(E)}^{{\mathrm{gp}}}/\mathbb{Z}\delta\right]\times\left[\bigoplus_{v\in
V(G)}P_{\boldsymbol{\sigma}(v)}^{{\mathrm{gp}}}/\mathbb{Z}\delta\times\bigoplus_{v\in
E\in E(G)}\mathbb{Z}\right]$

Because of the hypothesis that the central fibre $X_{0}$ is reduced, all
groups $P^{{\mathrm{gp}}}_{\boldsymbol{\sigma}(E)}/\mathbb{Z}\delta$ are
torsion-free, and hence injectivity is equivalent to the transpose map having
finite cokernel. However, we have a variant of diagram (4.6) given by
replacing each $N_{\boldsymbol{\sigma}(E)},N_{\boldsymbol{\sigma}(v)}$ with
$\overline{N}_{\boldsymbol{\sigma}(E)},\overline{N}_{\boldsymbol{\sigma}(v)}$,
which shows that the cokernel of the transpose of the above map is finite if
and only if the cokernel of $\overline{\Psi}$ is finite, which is the
tropically transverse condition.

For the last statement, the extra condition implies $\Delta$ is lci, hence
$\Delta^{!}$ makes sense. By flatness of $\underline{\mathrm{ev}}$, the Gysin
pull-backs $\Delta^{!}$ and $(\Delta^{\prime})^{!}$ agree. The result then
follows from Theorem 5.7 and properties of virtual pull-backs [Ma12, Thm.
4.1]. ∎

##  6\. Punctured versus relative

In a gluing situation arising from a degeneration situation $X\rightarrow B$
as reviewed in §5.2, one has the moduli spaces
$\mathscr{M}(X,{\boldsymbol{\tau}}_{v})$, classifying punctured maps marked by
${\boldsymbol{\tau}}_{v}$. In particular, such maps will factor through a
stratum $X_{\boldsymbol{\sigma}(v)}$ with its induced log structure. On the
other hand, $X_{\boldsymbol{\sigma}(v)}$ comes with a divisorial log structure
induced by $\partial X_{\boldsymbol{\sigma}(v)}\subseteq
X_{\boldsymbol{\sigma}(v)}$, the union of lower dimensional strata of $X$
contained in $X_{\boldsymbol{\sigma}(v)}$. We write this different log scheme
as $\overline{X}_{\boldsymbol{\sigma}(v)}$. It is then natural to compare
$\mathscr{M}(X,{\boldsymbol{\tau}}_{v})$ with a moduli space of stable log
maps to $\overline{X}_{\boldsymbol{\sigma}(v)}$. In general, this still a non-
trivial question, and is likely related to double ramification cycles. Here,
however, we deal with a special case, namely where
$X_{\boldsymbol{\sigma}(v)}$ is an irreducible component of $X_{b_{0}}$.

To set this up, let $\sigma\in\Sigma(X)$ be a non-zero cone, corresponding to
a stratum $X_{\sigma}$, and assume $X_{\sigma}\subseteq X_{b_{0}}$. If
$\dim\sigma=1$, then $X_{\sigma}$ is an irreducible component of $X_{b_{0}}$.
As above, we have the log scheme $\overline{X}_{\sigma}$, and there is a
canonical morphism of log schemes

$\psi:X_{\sigma}\rightarrow\overline{X}_{\sigma}$

induced by the natural inclusion
$\mathcal{M}_{\overline{X}_{\sigma}}\subseteq\mathcal{M}_{X_{\sigma}}=\mathcal{M}_{X}|_{X_{\sigma}}$.
Indeed, smooth locally at a geometric point $\bar{x}\in X_{\sigma}$, $X$ is
given as $\operatorname{Spec}\Bbbk[P]$ for
$P=\overline{\mathcal{M}}_{X,\bar{x}}$, and the monoid $P$ has a codimension
$\dim\sigma$ face $F\subseteq P$ such that $X_{\sigma}$ is smooth locally
$\operatorname{Spec}\Bbbk[F]$ and
$F=\overline{\mathcal{M}}_{\overline{X}_{\sigma},\bar{x}}$. The log structure
on $X_{\sigma}$ at $\bar{x}$ is given by a neat chart
$P\rightarrow\mathcal{O}_{X_{\sigma}}$, and the log structure on
$\overline{X}_{\sigma}$ is given by restricting this chart to $F$. Hence we
obtain the inclusion of log structures.

Note the inclusion of faces $F\subseteq P$ dualizes to a generization

$P^{\vee}_{\mathbb{R}}\rightarrow
F^{\vee}_{\mathbb{R}}=(P^{\vee}_{\mathbb{R}}+\mathbb{R}\sigma)/\mathbb{R}\sigma.$

This yields a description of the induced map of cone complexes

$\Sigma(\psi):\Sigma(X_{\sigma})\rightarrow\Sigma(\overline{X}_{\sigma})$

at the level of cones of $\Sigma(X_{\sigma})$. In particular, a type
$\tau=(G,\mathbf{g},\boldsymbol{\sigma},\mathbf{u})$ for tropical map to
$\Sigma(X_{\sigma})$ induces a type
$\bar{\tau}=(\bar{G},\bar{\mathbf{g}},\bar{\boldsymbol{\sigma}},\bar{\mathbf{u}})$
for tropical map to $\Sigma(\overline{X}_{\sigma})$. Indeed, $\bar{G}$ will
coincide with $G$, except some bounded legs of $G$ may become unbounded. Of
course $\bar{\mathbf{g}}=\mathbf{g}$. For $v\in V(G)$, we may take
$\bar{\boldsymbol{\sigma}}(v)$ to be the minimal cone of
$\Sigma(\overline{X}_{\sigma})$ containing
$\Sigma(\psi)(\boldsymbol{\sigma}(v))$. For an edge $E$ of $G$, we define
$\bar{\mathbf{u}}(E)=\Sigma(\Psi)_{*}(\mathbf{u}(E))$.

We remark that the strict inclusion $X_{\sigma}\hookrightarrow X$ induces a
map $\Sigma(X_{\sigma})\rightarrow\Sigma(X)$. Even though $X$ is assumed to be
Zariski, this need not be an inclusion of cone complexes. For example, if
$X_{0}$ is a union of two $\mathbb{P}^{1}$’s meeting at two points and
$X_{\sigma}$ is one of these $\mathbb{P}^{1}$’s, we have $\Sigma(X)$ a union
of two quadrants glued along their boundary, but $\Sigma(X_{\sigma})$ consists
of two quadrants glued together along one boundary ray. However, the map
$|\Sigma(X_{\sigma})|\rightarrow|\Sigma(X)|$ will always be injective in a
neighbourhood of $\sigma$.

Now suppose given a type $\tau$ of punctured maps to $X/B$, with underlying
graph $G$ having precisely one vertex $v$ and adjacent legs
$L_{1},\ldots,L_{n}$. Suppose further that $\boldsymbol{\sigma}(v)=\sigma$.
Then $\tau$ may also be viewed as a type of punctured map to $X_{\sigma}/B$,
and thus we also obtain a type $\bar{\tau}$ of punctured map to
$\overline{X}_{\sigma}$. However, in fact all contact orders are positive.
Indeed, because $\boldsymbol{\sigma}(v)=\sigma$, any $L_{i}$ must have contact
order $\mathbf{u}(L_{i})$ lying in the tangent wedge of
$\boldsymbol{\sigma}(L_{i})$ along the face $\sigma$ of
$\boldsymbol{\sigma}(L_{i})$. Hence the image of $\mathbf{u}(L_{i})$ in the
tangent space to
$(\boldsymbol{\sigma}(L_{i})+\mathbb{R}\sigma)/\mathbb{R}\sigma$ in fact lies
in this cone. Therefore we may view the type $\bar{\tau}$ as a type of
logarithmic map to $\overline{X}_{\sigma}$.

In addition, in this situation, we have a commutative diagram

(6.1)
$\textstyle{\mathscr{M}(X/B,{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{M}(\overline{X}_{\sigma},\bar{\boldsymbol{\tau}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{M}(\mathcal{X}/B,\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{M}(\overline{\mathcal{X}}_{\sigma},\bar{\tau})}$

where $\overline{\mathcal{X}}_{\sigma}$ is defined in the same way as
$\overline{X}_{\sigma}$. While the vertical arrows are the standard ones, the
horizontal ones are as follows. By the definition of a
${\boldsymbol{\tau}}$-marked curve, we have
$\mathscr{M}(X/B,{\boldsymbol{\tau}})\cong\mathscr{M}(X_{\sigma}/B,{\boldsymbol{\tau}})$,
i.e., any ${\boldsymbol{\tau}}$-marked curve factors through $X_{\sigma}$, by
[ACGS2, Def. 3.7,(1)]. Given a stable punctured map $f:C^{\circ}/W\rightarrow
X_{\sigma}$, we have $\psi\circ
f:C^{\circ}/W\rightarrow\overline{X}_{\sigma}$. In general, this is neither a
pre-stable nor basic log map, but we may pre-stabilize by [ACGS2, Prop. 2.4]
and replace with a basic log map by [ACGS2, Prop. 2.40].555Note that once we
pre-stabilize, all punctured points become marked as all contact orders in
$\bar{\mathbf{u}}$ are positive. Hence we could just as well apply the earlier
[GS13, Prop. 1.24]. This gives the upper horizontal arrow, and a similar
discussion gives the lower horizontal arrow.

######  Theorem 6.1.

Let $X\rightarrow B$, ${\boldsymbol{\tau}}$, $\sigma$ be as in the above
discussion, and suppose further that $\dim\sigma=1$. Then the horizontal
arrows of the diagram (6.1) are isomorphisms at the level of underlying stacks
(but not at the level of logarithmic stacks). Furthermore these isomorphisms
induce an isomorphism of obstruction theories.

###### Proof.

We work with the top horizontal arrow of (6.1), as the Artin fan case is
identical. We need to construct the inverse map. Explicitly, given a basic
$\bar{\boldsymbol{\tau}}$-marked stable log map
$\bar{f}:\overline{C}/\overline{W}\rightarrow\overline{X}_{\sigma}$, we need
to construct a stable punctured ${\boldsymbol{\tau}}$-marked map
$f:C/W\rightarrow X_{\sigma}$. The maps $f,\bar{f}$ will be the same on
schemes, but we need to modify the log structures on $\overline{W}$,
$\overline{C}$.

To this end, let $\overline{Q}_{\bar{w}}$ be the stalk of the ghost sheaf at a
geometric point $\bar{w}\in\overline{W}$, so that
$\bar{\tau}_{\bar{w}}=\overline{Q}^{\vee}_{\bar{w},\mathbb{R}}$ parametrizes a
universal family of tropical maps
$\bar{h}:\Gamma(\bar{G}_{\bar{w}},\bar{\ell})\rightarrow\Sigma(\overline{X}_{\sigma})$,
inducing for each $s\in\bar{\tau}$ a tropical map
$\bar{h}_{s}:\bar{G}_{\bar{w}}\rightarrow\Sigma(\overline{X}_{\sigma})$.

For $r\in\mathbb{R}_{\geq 0}$, write $\Delta_{r}:=\delta^{-1}(r)$, where
$\delta:\Sigma(X_{\sigma})\rightarrow\Sigma(B)$ is the tropicalization of
$X_{\sigma}\rightarrow B$. Of course $\Delta_{1}$ agrees with the underlying
topological space of $\Delta(X_{\sigma})$. Now note that we may restrict the
map $\Sigma(\psi)$ to $\Delta_{r}$ to obtain an inclusion of topological
spaces $\Delta_{r}\hookrightarrow|\Sigma(\overline{X}_{\sigma})|$. Indeed, it
is sufficient to check that for each $\omega\in\Sigma(X_{\sigma})$,
$\Sigma(\psi)$ induces an inclusion on $\omega_{r}:=\omega\cap\delta^{-1}(r)$.
But this follows immediately from the fact that $\Sigma(\psi)|_{\omega}$ is
given by localizing along the ray $\sigma\subseteq\omega$ and
$\delta|_{\sigma}$ surjects onto $\Sigma(B)$. We thus view $\Delta_{r}$ in
this way as a subspace of $\Sigma(\overline{X}_{\sigma})$.

One also easily sees that the spaces $\Delta_{r}$ exhaust
$|\Sigma(\overline{X}_{\sigma})|$, i.e., for
$y\in|\Sigma(\overline{X}_{\sigma})|$, there exists an $r_{0}\geq 0$ such that
$y\in\Delta_{r}$ for all $r\geq r_{0}$.

Let $\bar{G}^{\prime}_{\bar{w}}$ be the subgraph of $\bar{G}_{\bar{w}}$
obtained by deleting all legs of $\bar{G}_{\bar{w}}$. We define a function
$\alpha:\bar{\tau}_{\bar{w}}\rightarrow\mathbb{R}_{\geq 0}$ by

$\alpha(s):=\inf\\{r\in\mathbb{R}_{\geq
0}\,|\,\bar{h}_{s}(\bar{G}^{\prime}_{\bar{w}})\subseteq\Delta_{r}\\}.$

The set on the right-hand side is non-empty, and hence this makes sense.

######  Claim.

$\alpha$ is an upper convex piecewise linear function on
$\bar{\tau}_{\bar{w}}$ with rational slopes.

###### Proof.

Suppose $\omega\in\Sigma(X_{\sigma})$, necessarily containing $\sigma$. Let
$F^{1},\ldots,F^{p}\subseteq\omega$ be the codimension one faces of $\omega$
not containing $\sigma$ with $\delta|_{F^{i}}$ surjective. Let
$G=\delta^{-1}(0)\cap\omega$. Necessarily $G$ does not contain $\sigma$. We
then have the Minkowski decomposition

$\omega_{r}=\mathrm{Conv}\\{F^{1}_{r},\ldots,F^{p}_{r},\sigma_{r}\\}+G$

where $\mathrm{Conv}$ denotes convex hull.

Let $n_{F^{i}}\in\omega^{\vee}$ be a rational generator of the dual one-
dimensional face to $F^{i}$. Since $\sigma\not\subseteq F^{i}$, $n_{F^{i}}$ is
positive on $\sigma\smallsetminus\\{0\\}$. So we can normalize $n_{F^{i}}$ so
that $n_{F^{i}}|_{\sigma}$ agrees with $\delta|_{\sigma}$ by rescaling by a
rational number. Thus $\delta-n_{F^{i}}$ vanishes on $\sigma$ and hence
descends to a linear function on
$\bar{\omega}=(\omega+\mathbb{R}\sigma)/\mathbb{R}\sigma$. Further, $\delta-
n_{F^{i}}$ takes the value $r$ on $F^{i}_{r}$.

We now show that

(6.2) $\Sigma(\psi)(\omega_{r})=\\{m\in\bar{\omega}\,|\,\langle\delta-
n_{F^{i}},m\rangle\leq r,1\leq i\leq p\\}.$

To see the forward inclusion, we may write an element $w$ of $\omega_{r}$ as

$\sum_{i=1}^{p}a_{i}f_{i}+a_{p+1}s+g$

with $f_{i}\in F^{i}_{r}$, $s\in\sigma_{r}$, $\sum_{i=1}^{p+1}a_{i}=1$,
$a_{i}\geq 0$, and $g\in G$. Then

$(\delta-n_{F^{i}})(w)=\sum_{i=1}^{p}a_{i}r-\langle n_{F^{i}},g\rangle\leq r.$

Conversely, if $m$ lies in the right-hand side of (6.2), choose a lift
$m^{\prime}$ of $m$ to $\omega$. Write $\sigma_{1}=\\{s\\}$, and set
$m^{\prime\prime}=m^{\prime}+s(r-\delta(m^{\prime}))$. Then $m^{\prime\prime}$
is another lift of $m$ to $\omega^{{\mathrm{gp}}}$, but may not lie in
$\omega$. However, $\delta(m^{\prime\prime})=r$, and we would like to show
$m^{\prime\prime}\in\omega$. First, $\langle\delta-
n_{F^{i}},m^{\prime\prime}\rangle=\langle\delta-n_{F^{i}},m\rangle\leq r$, so
$\langle n_{F^{i}},m^{\prime\prime}\rangle\geq\delta(m^{\prime\prime})-r=0$.
If $H\subseteq\omega$ is a codimension one face of $\omega$ containing
$\sigma$, and $n_{H}$ is a generator of the dual ray, then $\langle
n_{H},m^{\prime\prime}\rangle=\langle n_{H},m^{\prime}\rangle\geq 0$. Finally,
if $G$ is a codimension one face of $\omega$, then $\delta$ is a generator of
the dual ray and $\delta(m^{\prime\prime})=r>0$. Thus we see that
$m^{\prime\prime}$ is positive on all generators of rays of $\omega^{\vee}$,
and hence lies in $\omega$, hence in $\omega_{r}$. So $m$ lies in
$\Sigma(\psi)(\omega_{r})$. We have now shown (6.2).

Now for each vertex $v\in V(\bar{G}_{\bar{w}})$, we have the evaluation map
$\mathrm{ev}_{v}:\bar{\tau}_{\bar{w}}\rightarrow\bar{\boldsymbol{\sigma}}(v)$
at $v$ given by (4.2). Note that
$\bar{h}_{s}(\bar{G}^{\prime}_{\bar{w}})\subseteq\Delta_{r}$ if and only if
$\bar{h}_{s}(v)\in\Sigma(\psi)(\boldsymbol{\sigma}(v)_{r})$ for all $v\in
V(\bar{G}_{\bar{w}})$, where $\boldsymbol{\sigma}(v)$ is the cone of
$\Sigma(X_{\sigma})$ mapping to $\bar{\boldsymbol{\sigma}}(v)$ under
$\Sigma(\psi)$. In particular, for a given $v$,
$\bar{h}_{s}(v)\in\Sigma(\psi)(\boldsymbol{\sigma}(v)_{r})$ if and only if
$\langle\delta-n_{v,F_{i}},\mathrm{ev}_{v}(s)\rangle\leq r$ for $n_{v,F_{i}}$
running over the set of normal vectors to codimension one faces of
$\boldsymbol{\sigma}(v)$ not containing $\sigma$ or contained in
$\delta^{-1}(0)$ as above. Thus $\alpha$ is given as

$\alpha(s)=\sup\\{\langle\delta-n_{v,F_{i}},\mathrm{ev}_{v}(s)\rangle\\},$

with the supremum running over all $v\in V(\bar{G}_{\bar{w}})$ and normal
vectors $n_{v,F_{i}}$, i.e., $\alpha$ agrees with the supremum of the linear
functionals $\mathrm{ev}_{v}^{*}(\delta-n_{v,F_{i}})$. This makes $\alpha$
piecewise linear and (upper) convex. ∎

We may now define

$\tau_{\bar{w}}:=\\{(s,r)\in\bar{\tau}_{\bar{w}}\times\mathbb{R}_{\geq
0}\,|\,r\geq\alpha(s)\\}.$

This is a rational polyhedral cone by the claim. From construction,
$\tau_{\bar{w}}$ parameterizes lifts of the tropical maps $\bar{h}_{s}$ to
$\Sigma(X_{\sigma})$. Indeed, if $(s,r)\in\tau_{\bar{w}}$, $\bar{h}_{s}$ maps
$\bar{G}^{\prime}_{\bar{w}}$ into $\Delta_{r}$, which we now view as a subset
of $|\Sigma(X_{\sigma})|$. We may then extend this map as far as possible
along each leg $L_{i}$ to define a domain graph $G_{\bar{w}}$, which coincides
with $\bar{G}_{\bar{w}}$ except that the legs may be replaced with line
segments. We then get $h_{s,r}:G_{\bar{w}}\rightarrow\Sigma(X_{\sigma})$. In
particular, a leg $L_{i}$ turns into a punctured leg, i.e., a line segment, if
only a portion of $h_{s}(L_{i})$ is contained in $\Delta_{r}$.

Note here we may now view $\tau_{\bar{w}}$ as type of tropical map to
$\Sigma(X_{\sigma})$, with $\boldsymbol{\sigma}$ defined in the obvious way
from $\bar{\boldsymbol{\sigma}}$ and $\mathbf{u}(E)$ a tangent vector of
$\boldsymbol{\sigma}(E)_{1}$ mapping under $\Sigma(\psi)_{*}$ to
$\bar{\mathbf{u}}(E)$. In particular, we obtain a universal family of tropical
maps of type $\tau_{\bar{w}}$,

(6.3) $h_{\bar{w}}:\Gamma(G,\ell)\rightarrow\Sigma(X_{\sigma}).$