%%%%%%%%Moduli of fine log
\subsection{A motivating example}
Before introducing the stack $LOG_S$ classifying fine log 
structures on schemes over a fine log scheme $S,$ constructed 
by Olsson in \cite{LogStack}, let's look at an example, which will give 
the local covers of $LOG_S.$

For a fine log scheme $X$ with log structure $(\sM_X,\alpha: 
\sM_X\to\sO_X),$ we identify $\sO_X^*$ with its 
inverse image under $\alpha,$ regarded as a subsheaf of 
monoids in $\sM_X.$ Let $\overline{\sM}_X$ be the quotient sheaf 
$\sM_X/\sO_X^*$ for the \'etale topology, called the 
characteristic of $X.$ We write $\underline{X}$ for the 
underlying scheme of the log scheme $X.$ See \cite{KKato}.

Define the \textit{Deligne-Faltings log structure} of rank 
$r$ on $\underline{X}$ to be a family $(L_1,\cdots,L_r)$ 
of line bundles on $\underline{X}$ together with maps 
$s_i:L_i\rightarrow\sO_X$ of line bundles, for each $i.$ 

Consider the following three functors from schemes to 
groupoids:

\begin{enumerate}
\item
$\quad X\mapsto\{\text{Deligne-Faltings log 
structures of rank 1 on }X\};$

\item $\quad X\mapsto\{\text{fine log structures 
}\sM_X\overset{\alpha}{\rightarrow}\sO_X\text{ with a 
morphism of sheaves of monoids }\mathbb N\overset{\beta} 
{\to}\overline{\sM}_X\text{ that \'etale locally lifts to a 
chart: }\mathbb N\overset{\widetilde{\beta}}{\to}\sM_X\};$

\item $\quad X\mapsto[\mathbb A^1/\mathbb G_m](X),$ 
where the quotient stack is formed with respect to the 
multiplication action of $\mathbb G_m$ on $\mathbb A^1.$
\end{enumerate}

\begin{lemma}
These three functors are equivalent.
\end{lemma}

Let's sketch the proof. Given a DF log structure 
$(L,s:L\to\sO_X)$ of rank 1 on $X,$ define a sheaf of 
monoids $\sM'$ on $X$ to be 
$$
\coprod_{n\ge0}\underline{\text{Isom}}(\sO_X,L^{\otimes 
n}),
$$
the sheafification of the presheaf that takes $U$ to 
$\coprod_{n\ge0}\text{Isom}(\sO_U,(L|_U)^{\otimes n}).$ 
It comes with a natural morphism of sheaves of monoids 
$\sM'\to\mathbb N.$ The monoid structure on $\sM'$ is the obvious 
one: 
$$
(n,a:\sO\to L^{\otimes n})\cdot(m,b:\sO\to 
L^{\otimes m})=(n+m,a\otimes b).
$$
The map $s:L\to\sO_X$ induces a morphism 
$$
\underline{\text{Isom}}(\sO_X,L^{\otimes 
n})\overset{\otimes s}{\longrightarrow}\underline{\text{Hom}} 
(\sO_X,\sO_X)=\sO_X
$$
of sheaves, hence giving a pre-log structure on $\sM':$
$$
\sM'\to\sO_X.
$$
We take $\sM_X$ to be the log structure associated to this 
pre-log structure $\sM'.$ Note that $\sM'/\sO_X^*\cong\mathbb N,$ and we define $\beta:\mathbb N\to\overline{\sM}_X$ to be the composite
$$
\beta:\mathbb N\cong \sM'/\sO_X^*\to \sM/\sO_X^*.
$$
Locally the line bundle $L$ is trivial, and one can choose a 
trivialization of $L,$ which gives trivializations of all 
$L^{\otimes n}.$ Sending $n\in\mathbb N$ to this 
trivialization defines a section $\mathbb N\to \sM',$ and hence 
a section $\widetilde{\beta}:\mathbb N\to \sM'\to \sM_X.$ One can 
check that this is a chart. 

Conversely, given a fine log structure $(\sM_X,\sM_X\overset{ 
\alpha}{\to}\sO_X)$ on $X$ with a morphism  
$\beta:\mathbb N\to\overline{\sM}_X$ that \'etale locally lifts 
to a chart $\widetilde{\beta}:\mathbb N\to \sM_X,$ we have a 
section $\beta(1)$ of $\overline{\sM}_X,$ and its inverse image 
under $\pi:\sM_X\to\overline{\sM}_X$ is an $\sO_X^*$-torsor, which corresponds to a line bundle $L.$ The 
composition 
$$
\pi^{-1}(\beta(1))\subset \sM_X\overset{\alpha}{\longrightarrow} 
\sO_X
$$
gives a morphism of line bundles $s:L\to\sO_X.$ 

Giving a morphism $X\to[\mathbb A^1/\mathbb G_m]$ is 
equivalent to giving a $\mathbb G_m$-torsor (namely a line 
bundle $L$) with a $\mathbb G_m$-equivariant morphism to 
$\mathbb A^1:$
$$
\xymatrix@C=.7cm{
Y \ar[r] \ar[d] & \mathbb A^1 \ar[d] \\ 
X \ar[r] & [\mathbb A^1/\mathbb G_m].}
$$
This diagram is equivalent to the following one
$$
\xymatrix@C=.7cm{
Y \ar[r] \ar[d] & \mathbb A^1_X \ar[d] \\
X \ar[r] & [\mathbb A^1/\mathbb G_m]_X,} 
$$
and the top arrow induces a $\mathbb G_m$-equivariant morphism 
$Y\to\mathbb A^1_X,$ namely a morphism of line bundles 
$s:L\to\sO_X.$ This finishes the proof.

In fact, in the three functors, one can replace $\mathbb N$ by 
$\mathbb N^r,$ and rank 1 DF-log structure by rank $r$ DF-log 
structure, and replace $[\mathbb A^1/\mathbb G_m]$ by 
$[\mathbb A^r/\mathbb G_m^r],$ and they are still equivalent. 

More generally, let $P$ be a fine monoid and $S$ a scheme, 
and let $S[P]$ be the product $S\times_{\text{Spec }\mathbb 
Z}\text{Spec }\mathbb Z[P],$ which has a fine log structure 
coming from the chart $P\to\mathbb Z[P].$ Let $P^{\text{gp}}$ 
be the associated group, then we have the following.

\begin{lemma}
The following two functors from $S$-schemes to groupoids are 
equivalent:

\begin{enumerate}
\item $\quad X\mapsto\{\text{fine log structures }
\sM_X\overset{\alpha}{\to}\sO_X\text{ with a morphism 
of sheaves of monoids }P\overset{\beta}{\to}\overline{\sM}_X 
\text{ that fppf locally lifts to a chart: }P\overset{ 
\widetilde{\beta}}{\to}\sM_X\};$

\item $\quad X\mapsto[S[P]/S[P^{\text{gp}}](X).$
\end{enumerate}

If in addition $P$ is fs, then one can replace ``fppf" by 
``\'etale".
\end{lemma}

The $S$-group scheme $S[P^{\text{gp}}]$ acts on $S[P]$ by 
translation. Note that for an affine $S$-scheme $\text{Spec 
}R,$ the set of $R$-points $S[P](R)$ is the set of monoid 
homomorphisms $\text{Hom}_{\text{mon}}(P,R),$ where $R$ is 
regarded as a multiplicative monoid. When $S=\text{Spec }k$ 
for a field $k$ and $P$ is saturated and torsion-free, the 
variety $S[P]$ with this action of the torus 
$S[P^{\text{gp}}]$ is a toric variety, and the stack quotient 
$\cS_P=[S[P]/S[P^{\text{gp}}]]$ is a toric stack.


\subsection{The stack of log structures}
Now we can discuss the stack $LOG_S$ parameterizing fine log 
structures.

Let $S$ be a fine log scheme. Define $LOG_S$ to be the 
category with 

$\bullet$ objects: morphisms $(X,\sM_X)\to(S,\sM_S)$ of fine log 
schemes, and 

$\bullet$ morphisms: strict morphisms $(X,\sM_X)\to(Y,\sM_Y)$ over 
$S.$

With the functor $(X\to S)\mapsto(\underline{X}\to\underline{ 
S})$ from $LOG_S\to\text{Sch}_{\underline{S}},$ this defines a 
fibered category over $\underline{S}.$ One of the main results 
in \cite{LogStack} is the following.

\begin{theorem}
$LOG_S$ is an algebraic stack locally of finite presentation 
over $\underline{S}.$
\end{theorem}

Here for an algebraic stack we use a slightly different 
definition from \cite[4.1]{LMB}. Namely the first axiom there 
that the diagonal is representable, separated and 
quasi-compact, is replaced by that the diagonal is 
representable and of finite presentation. In fact the stack 
$LOG_S$ is not quasi-separated \cite[3.17]{LogStack}. 

Here are two basic properties of $LOG_S.$

\begin{proposition}
(1). The natural map $i_S:\underline{S}\to LOG_S$ 
corresponding to the given fine log structure $\sM_S$ on $S$ is 
an open immersion;

(2). The 2-functor 
$$
S\mapsto LOG_S:\{\text{fine log 
schemes}\}\to\{\text{algebraic stacks}\}
$$
preserves fiber product. More precisely, if 
$$
\xymatrix{
X' \ar[r] \ar[d] & X \ar[d] \\
S' \ar[r] & S}
$$
is a Cartesian square of fine log schemes, then the induced 
diagram
$$
\xymatrix{
LOG_{X'} \ar[r] \ar[d] & LOG_X \ar[d] \\
LOG_{S'} \ar[r] & LOG_S}
$$
is a 2-Cartesian square of algebraic stacks.
\end{proposition}

We don't give their proof here, which can be found in \cite{LogStack}.


\subsection{What is $LOG_S$ good for?}
One can use this stack $LOG_S$ to reinterprete many concepts 
in log geometry. Note that for a morphism $f:X\to S$ of fine 
log schemes, the induced morphism $LOG(f):LOG_X\to LOG_S$ is 
faithful, hence representable.

\begin{definition}
Let $P$ be a property of representable morphisms of 
algebraic stacks. Then we say that $f:X\to S$ has property 
$LOG(P)$ if $LOG(f):LOG_X\to LOG_S$ has property $P.$ We say that $f$ has property weak $LOG(P)$ if the map 
$\underline{X}\to LOG_S$ corresponding to the given log 
structure $\sM_X$ has property $P.$
\end{definition}

\textbf{Caution:} The diagram 
$$
\xymatrix{
\underline{X} \ar[r]^-{i_X} \ar[d]_-{\underline{f}} & LOG_X 
\ar[d]^-{LOG(f)} \\
\underline{S} \ar[r]_-{i_S} & LOG_S}
$$
does not necessarily commute. It commutes if and only if $f$ 
is strict.

Recall that a morphism of fine log schemes $f:X\to S$ is said 
to be \textit{formally log smooth} (resp. \textit{formally 
log \'etale}) if in the following commutative diagram of fine 
log schemes, \'etale locally on $T$ there exists (resp. 
exists a unique) a lifting 
$$
\xymatrix{
T_0 \ar[r] \ar[d]_-i & X \ar[d]^-f \\
T \ar@{..>}[ur] \ar[r] & S}
$$
for any strict square zero thickening $i:T_0\to T.$ We say 
that $f$ is \textit{log smooth} (resp. \textit{log \'etale}) 
if it is formally log smooth (resp. formally log \'etale) and 
$\underline{f}$ is locally of finite presentation. One defines 
(formal) smoothness and \'etaleness for representable 
morphisms of algebraic stacks in the similar way.

\begin{theorem}
For a morphism $f:X\to S$ of fine log schemes, $f$ is LOG 
smooth (resp. LOG \'etale) if and only if $f$ is log smooth 
(resp. log \'etale), if and only if $f$ is weakly LOG smooth 
(resp. weakly LOG \'etale).
\end{theorem} 

This is part of \cite[4.6]{LogStack}.

Another application is the following. Let $f_0:X_0\to T_0$ be 
a log smooth integral morphism of fine log schemes, and 
$T_0\overset{i}{\to}T$ a strict square zero thickening defined 
by the ideal $I\subset\sO_T.$ Then we can consider the 
deformation problem of classifying Cartesian squares 
$$
\xymatrix{
X_0 \ar@{..>}[r] \ar[d]_-{f_0} & X \ar@{..>}[d]^-f \\
T_0 \ar[r]_-i & T,}
$$
where $f:X\to T$ is also log smooth (integral). Then 
the solution to this deformation problem is the cohomology of 
the \textit{log tangent bundle} $T^{\text{log}}_{X_0/T_0},$ 
namely 

$\bullet$ there is an obstruction class $o\in 
H^2(\underline{X_0},T^{\text{log}}_{X_0/T_0}\otimes f_0^*I);$

$\bullet$ when the obstruction class vanishes so that 
deformations exist, the set of all isomorphism classes of 
deformations is a torsor under 
$H^1(\underline{X_0},T^{\text{log}}_{X_0/T_0}\otimes f_0^*I);$

$\bullet$ the automorphism group of each deformation is 
$H^0(\underline{X_0},T^{\text{log}}_{X_0/T_0}\otimes f_0^*I).$

The morphism $f_0:X_0\to T_0$ of fine log schemes gives a 
smooth morphism $\underline{X_0}\overset{f_0}{\to}LOG_{T_0},$ 
also denoted $f_0,$ and the above 
deformation problem is equivalent to the following 
$$
\xymatrix{
\underline{X_0} \ar@{..>}[r] \ar[d]_-{f_0} & \underline{X} 
\ar@{..>}[d]^-f \\ LOG_{T_0} \ar[r] & LOG_T.}
$$
The solution to this deformation problem is the cohomology of 
the \textit{ordinary tangent bundle} 
$T_{\underline{X_0}/LOG_{T_0}}.$ In fact, we have 
$\Omega^{\text{log}}_{X_0/T_0}\cong\Omega_{\underline{ 
X_0}/LOG_{T_0}},$ because they represent the same functor.

Finally, the stack $LOG_S$ can be covered by toric stacks 
discussed above. Let $S$ be a fine log scheme and $U\to S$ 
an \'etale map, and $U$ is given the inverse image log 
structure. Let $Q\overset{\beta}{\to}\sM_U$ be a chart, and 
$h:Q\to P$ a morphism of fine monoids, then we have a natural 
morphism $\cS_P\times_{\cS_Q}\underline{U}\to 
LOG_U.$

\begin{proposition}
For any fine log scheme $S,$ the natural morphism 
$$
\coprod_{(U,\beta,h)}\cS_P\times_{\cS_Q}\underline{U}\to LOG_S
$$
is a representable \'etale surjection, where the disjoint 
union is taken over all triples $(U,\beta,h)$ consisting of an 
\'etale morphism $U\to S,$ a chart $Q\overset{\beta}{\to}\sM_U,$ 
and a morphism $h:Q\to P$ of fine monoids, for some fine 
monoids $P$ and $Q.$
\end{proposition}

