%%%%%%%%%Introduce Kim's Log structure on expanded degeneration

\subsection{Logarithmic structure on expanded degeneration}
As pointed in \cite[Lemma 2.7]{Li}, the predeformability is a closed condition, which makes the deformation theory extremely hard. The logarithmic structure introduced in \cite{Kim} are aimed at amend this situation. It was shown in \cite{Kim} that with suitable log structure put on both source and target, the predeformability will be replaced by several conditions on the log structure, and the deformation theory can be replaced by logarithmic deformations. Next, following \cite{Kim}, we introduce log structures for expanded degeneration.

\subsubsection{Minimal log prestable curves}
We first introduce the log structure on source curves. Consider a proper flat family of $n$-marked prestable curves $(\pi:C\to S,\{\Sigma_{i}\}_{i=1}^{n})$, where $\{\Sigma_{i}\}_{i=1}^{n}$ are the $n$ markings. Since Kim's log structure are mainly focus on the nodes in order to amend the predeformable condition, he put no interesting log structures on the markings. By dropping the $n$ markings, we still have a family of prestable curves $C\to S$ with no marked points. We have already seen in section \ref{Satriano} that such family of prestable curves has a unique (up to a unique isomorphism) pair of canonical log structure $\sM_{C}^{C/S}$ and $\sM_{S}^{C/S}$ on fiber and base respectively. Note that there is no interesting log structures on the marked points, therefore, for any other log structure $\sM_{S}$ and $\sM_{S}$ such that the log map $(C,\sM_{C})\to (S,\sM_{S})$ is log smooth and integral, the universal property of the canonical structure implies that there is a unique map of log structures $\sM_{S}^{C/S}\to\sM_{S}$ such that the following commutative diagram is cartesian:
\[
\xymatrix{
(C,\sM_{C}) \ar[r] \ar[d] & (C,\sM_{C}^{C/S}) \ar[d] \\
(S,\sM_{S}) \ar[r]  & (S,\sM_{S}^{C/S}).
}
\]

\begin{defn}(\cite[3.5]{Kim})
A pair $(\pi:C\to S, \sM_{S}^{C/S}\to \sM_{S})$ is called a log prestable curve over $(S,\sM_{S})$ if $C\to S$ is a usual prestable curve over $S$, and $\sM_{S}^{C/S}\to \sM_{S}$ is a morphism from the canonical log structure $\sM_{S}^{C/S}$ to a fine log structure $\sM_{S}$ on $S$. In this case, the log structure on the curve are given by 
\begin{equation}
\sM_{C} = \sM_{C}^{C/S}\oplus_{\pi^{-1}\sM_{S}^{C/S}}\pi^{-1}\sM_{S}.
\end{equation}
\end{defn}

In general, the category of fine log structures is too big. The following minimal condition restrict us to a much smaller class which is a slight modification of the canonical log structure. Moreover, the minimal condition is an open condition.

\begin{defn}(\cite[3.7]{Kim})
We call a log prestable curve $(\pi:C\to S, \sM_{S}^{C/S}\to \sM_{S})$ minimal if 
\begin{enumerate}
 \item the log structure $\sM_{S}$ is locally free, and there is no proper locally free submonoid of $\sM_{S}$ containing the image of $\sM_{S}^{C/S}$;
 \item for all $s\in S$ and any irreducible $b\in \overline{\sM}_{S,\overline{s}}$, there is an irreducible element $a \in \overline{\sM}_{S,\overline{s}}^{C/S}$ such that $a=lb$ for some positive integer $l$.
\end{enumerate}
\end{defn}

\subsubsection{Logarithmic structure on the expanded degeneration}
We now introduce our targets and their log structures. Consider $\pi: W\to B$, a flat and projective family of schemes over a smooth irreducible curve $B$. Let $0\in B$ be a closed point such that $\pi$ is smooth away from $W_{0}=W\times_{B}0$, and the central fiber $W_{0}$ is reducible with two smooth components $Y_{1}$ and $Y_{2}$ intersect transversally along the smooth divisor $\DD\subset W_{0}$. We view $\DD$ as a smooth divisor in $Y_{i}$, and denote $\DD_{i}\subset Y_{i}$.

As discussed in section \ref{D-SS}, the central fiber $W_{0}$ is d'semistable over the closed point. Denote $N_{\DD_{i}/Y_{i}}$ be the normal bundle of $\DD_{i}$ in $Y_{i}$. Then we have $N_{\DD_{1}/Y_{1}}\cong N_{\DD_{2}/Y_{2}}^{\vee}$. Let $\PP=\PP roj_{\DD}(1_{\DD}\oplus N_{\DD_{1}/Y_{1}})$. Since we have $\PP roj_{\DD}(1_{\DD}\oplus N_{\DD_{1}/Y_{1}})\cong \PP roj_{\DD}(N_{\DD_{2}/Y_{2}}\oplus 1_{\DD})$, we denote $D^{+}$ and $D^{-}$ be the smooth divisor in $\PP$ which correspond to the normal bundle $N_{\DD_{1}/Y_{1}}$ and $N_{\DD_{2}/Y_{2}}$ respectively. Note that the divisor $D^{+}$ and $D^{-}$ are canonically isomorphic to $\DD$. Let $\PP_{i}$ for $i\in \N$ be copies of $\PP$. We can glue $Y_{1}$ and $\PP_{1}$ along $\DD_{1}$ and $D^{-}$ respectively, $\PP_{i}$ and $\PP_{i+1}$ along $D^{+}$ and $D^{-}$ respectively, and $\PP_{n}$ and $Y_{2}$ along $D^{+}$ and $\DD_{2}$ respectively. We denote the resulting gluing to be 
\begin{equation}
W_{0}[n]= Y_{1} \bigcup_{D_{1}} \PP_{1} \bigcup_{D_{2}} \cdots \bigcup_{D_{n}} \PP_{n} \bigcup_{D_{n+1}}Y_{2}, 
\end{equation}
where $D_{1},\cdots, D_{n+1}$ are the disjoint singular loci of $W_{0}[n]$. By our construction, the singular scheme $W_{0}[n]$ is d'semistable along its singular loci, hence has a canonical log structure described in section \ref{D-SS}. Note that we have a map $W_{0}[n]\to W_{0}$ by contracting all $\PP_{i}'$s to the divisor $D$. The singular scheme $W_{0}[n]$ with the contracting map $W_{0}[n]\to W_{0}$ is called an $n$-th accordion over $W_{0}$. 

\begin{defn}
An expanded degeneration of $W\to B$ over a reduced $B-$scheme $p:T\to B$ is a flat proper family $\cW\to T$ with a morphism $\cW\to T\times_{B}W$ such that for any point $t\in T$, $\cW_{t}\to W_{p(t)}$ is an isomorphism if $p(t)\neq 0$, and an accordion over $W_{0}$ if $p(t)=0$.
\end{defn}

Denote $\cB$ to be the fibered category over $(Sch/B)$, which associate to each reduced $B-$scheme the groupoid of expanded degeneration of $W$. And denote $\cU$ to be the universal family over $\cB$. In fact, $\cB$ is a smooth algebraic stack, whose boundary loci that corresponds to accordions over $W_{0}$ are divisor of simple normal crossing. We denote $\sM_{\cB}$ be the log structure accociated to the boundary on $\cB$. On the other hand, since the fiber of $\cU\to\cB$ is either smooth or an accordion hence d'semistable, there is a canonical log structure $\sM^{\cU/\cB}_{\cU}$ and $\sM^{\cU/\cB}_{\cB}$ on the fiber and base respectively. We remark that the two log structure $\sM^{\cU/\cB}_{\cB}$ and $\sM_{\cB}$ coincide.

Now given an expanded degeneration $\cW\to T$, we have the canonical log structure $\sM^{\cW/T}_{\cW}$ and $\sM^{\cW/T}_{T}$ on fiber and base by pulling back that from $\cU$ and $\cB$ respectively. The log structure on the base is clearly locally free. Same as the curves case, we give the following definition:

\begin{defn}(\cite[4.3]{Kim})\label{LogTarget}
An extended log twisted expanded degeneration is a pair $(\pi:\cW\to T,\sM_{T}^{\cW/T}\to \sM_{T})$ where $\cW\to T$ is an expanded degeneration, and $\sM_{T}^{\cW/T}\to \sM_{T}$ is a morphism from the canonical log structure $\sM_{T}^{\cW/T}$ to a finitely generated locally free log structure $\sM_{T}$ such that \'etale locally there is a commutative diagram of charts:
\[
\xymatrix{
\sM^{\cW/T}_{T} \ar[rr] && \sM_{T} \\
\N^{m} \ar[r]^{(d_{1},\cdots,d_{m})} \ar[u] & \N^{m} \ar[r]^{(id,0)\ \ \ } & \N^{m}\oplus\N^{m'} \ar[u],
}
\]
where the first map at the bottom is the diagonal map given by the sequence of positive integers $(d_{1},\cdots,d_{m})$. And the log structure on $\cW$ is given by 
\[\sM_{\cW}=\sM^{\cW/T}_{\cW}\oplus_{\pi^{-1}\sM^{\cW/T}_{T}}\pi^{-1}\sM_{T}.\]
\end{defn}

We have seen in section \ref{D-SS} that with the log structure in the definition, the log map $(\cW,\sM_{\cW})\to (T,\sM_{T})$ is log smooth and integral. We will explain later the meaning of this log structure after we introduce the log stable map.

\subsubsection{Log structures on predeformable maps}
Now we can gathering various results above, and discuss the log structure on predeformable maps for expanded degeneration. 

\begin{defn}
A morphism $f:\sM\to\sN$ of log structures on a scheme $T$ is called simple at a point $t\in T$, if $\overline{f}:\overline{\sM}_{\overline{t}}\to\overline{\sN}_{\overline{t}}$ is injective, and for any irreducible element $b\in \overline{\sN}_{\overline{t}}$, there is an irreducible element $a$ in $\overline{\sM}_{\overline{t}}$ such that $\overline{f}(a)=nb$ for some positive integer $n$.
\end{defn}

\begin{defn}\label{LogPrestableMap}
(\cite[5.2.2]{Kim})
A log morphism $(f:(C,\sM_{C})\to(\cW,\sM_{\cW}))/(T,\sM_{T})$ is called a $(g,n)$ log prestable map over $(T,\sM_{T})$ if 
 \begin{enumerate}
  \item the source $(C,\sM_{C})\to(T,\sM_{T})$ is a genus $g$, $n$-pointed minimal log prestable curve;
  \item the target $(\cW\to,\sM_{\cW})\to(T,\sM_{T})$ is an extended log twisted expanded degeneration;
  \item (Corank condition) the rank of Coker$(\sM^{\cW/T}_{T,\overline{t}}\to\sM_{T,\overline{t}})$ coincides with the number of non-distinguished nodes on $C_{\overline{t}}$, for any point $t\in T$,i.e. the nodes mapping to the smooth loci of $\cW$;
  \item the morphism $f:(C,\sM_{C})\to(\cW,\sM_{\cW})$ is a log map;
  \item (Log admissibility) the map of log structures $f^{\flat}:f^{*}\sM_{\cW}\to\sM_{C}$ is simple at every distinguished node, i.e. the nodes mapping to the singular loci of the fiber.
 \end{enumerate}
\end{defn}

A detailed discussion of this definition can be find in \cite{Kim}, but we would like to highlight the following facts.

\textcircled{1}The corank condition implies that the extended $m'$ copies of $\N$ in the definition of \ref{LogTarget} are one-to-one corresponding to the non-distinguished nodes. On the curve side, the condition above implies that the log structure along non-distinguished nodes are the canonical log structure, and we only modify the log structure along distinguished nodes. Moreover, if a curve maps to a smooth fiber of $\cW\to T$, then the log structure on the curve are the canonical one, and there is a unique log prestable map realizing this underlying prestable map.

\textcircled{2}With all other condition imposed, the log admissibility condition is equivalent to the following:
 \begin{enumerate}
  \item only nodes map to the singular loci of $\cW$, and moreover,
  \item the underlying map $\underline{f}$ satisfies the predeformable condition \cite[2.5]{Li} at each distinguished node.
 \end{enumerate}
In fact, let $p\in C$ be a distinguished node. By the discussion in section \ref{Satriano} and \ref{D-SS}, locally at $\overline{p}$ and $f(\overline{p})$, we have coordinates function $u$, $v$ and $x$, $y$ such that $\log u, \log v \in \sM_{C,\overline{p}}$, and $\log x, \log y \in \sM_{\cW,f(\overline{p})}$ are irreducible elements in the log structure respectively. And our definition forces $f^{\flat}(\log x)=l\log u$ and $f^{\flat}(\log y)=l\log v$, for some positive integer $l$. This implies that $\underline{f}$ is predeformable along $p$.

\textcircled{3}As we mensioned above that both source and target of $f$ are log smooth and integral over the base. Morever, all conditions list in the definition above are open conditions. Namely for an arbitrary log map $f$ as above, the set of points in $T$ whose fiber are log prestable maps forms an open subset of $T$. This implies that the deformation theory of log stable maps would comming from some logarithmic deformation theory.

\begin{defn}(\cite[5.2.4]{Kim})
A log prestable map in definition \ref{LogPrestableMap} is called stable, if for any $t\in T$, the group of automorphism $(\sigma,\tau)$ is finite, where 
 \begin{enumerate}
  \item $\sigma$ is an automorphism of $((C,\sM_{C}),(T,\sM_{T}))_{\overline{t}}$ preserving $n-$marked points;
  \item $\tau$ is an automorphism of $((\cW,\sM_{\cW}),(T,\sM_{T}))_{\overline{t}}$ preserving the contraction $\cW_{\overline{t}}\to W_{0}$;
  \item $f_{\overline{t}}\circ \sigma = \tau\circ f_{\overline{t}}$.
 \end{enumerate}
\end{defn}

It was shown in \cite[6.3.1(2)]{Kim} that given an underlying predeformable map $\underline{f}$, the number of isomorphism classes of log pre-stable maps realizing $\underline{f}$ is finite. Thus, given a log prestable map, it is log stable is equivalent to its underlying map is stable in the sense of predeformable maps \cite[3.2]{Li}. Now with the curve class imposed in the fiber of $\cU\to\cB$, we have:
\begin{theorem}
The stack $\bM_{g,n}^{log}(\cU/\cB,\beta)$ parametrizing $(g,n)$ log stable maps to expanded degeneration of $W\to B$ with curves class $\beta$ in the fiber is a proper DM stack.
\end{theorem}
\begin{proof}
This is proved in \cite[6.3.1,6.3.2]{Kim}.
\end{proof}



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