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%\begin{document}

%\title{Log smooth curves and their moduli}
%\author{Matthew Satriano}
%\date{}
%\maketitle
In this section we discuss F. Kato's paper \cite{FKato} in which he gives a log geometry theoretic construction of the 
Deligne-Mumford compactification $\mgnb$ of the moduli space $\mgn$ of curves.  He states in the introduction a motiving 
philosophy which relates log geometry to compactifications of moduli spaces:
\begin{philosophy}
 Since log smoothness already incorporates degenerate objects, one
should expect that a moduli space of log smooth objects   
is already compact, and so provides a compactification of the moduli
of objects where the log structure is trivial.  
\end{philosophy}
Along the lines of this philosophy, to compactify $\mgn$, we want to
introduce a notion of \emph{log curve} which extends  
the notion of smooth curve.  We do so after some preliminaries.

\subsection{Relative characteristic sheaves}
\begin{definition}
\label{def:char}
Given a log scheme $(X,\sM_X)$, the \emph{characteristic} $\bar{\sM}_X$ is defined as $\sM_X/\alpha(\sO_X^*)$.  Given a 
morphism $(f,h):(X,\sM_X)\rightarrow (Y,\sM_Y)$ of log schemes, the \emph{relative characteristic} $\bar{\sM}_{X/Y}$ is 
defined as $\sM_X/\im(f^*\sM_Y\rightarrow\sM_X)$.
\end{definition}
\begin{example}
\label{ex:char}
Let $X=\Spec k[\N^n]$ with its canonical log structure.  Given $t\in X$, let $d(t)$ be the number of coordinate hyperplanes 
on which $t$ lies.  Then
\[
\bar{\sM}_{X,\bar{t}}=\bar{\sM}_{X,t}=\N^{d(t)},
\]
where $\bar{t}$ is any geometric point mapping to $t$.  More generally, if $X$ is a smooth scheme and $D$ is a normal 
crossing divisor on $X$, then we obtain a log structrue $\sM_X$ defined by $D$.  Then
\[
\bar{\sM}_{X,\bar{t}}=\N^{d(\bar{t})},
\]
where $d(\bar{t})$ denotes the number of irreducible components of the inverse image of $D$ in $\Spec\sO_{X,\bar{t}}$.
%branches of $D$ on which $\bar{t}$ lies.
\end{example}
The relative characteristic, as one might imagine, tells how much log structure one log scheme has relative to another:
\begin{lemma}[{\cite[Lemma 1.6]{FKato}}]
If $(f,h):(X,\sM_X)\rightarrow (Y,\sM_Y)$ is an integral morphism of fine log schemes, then $\bar{\sM}_{X/Y,\bar{x}}=0$ 
if and only if $(f,h)$ is strict in an \'etale neighborhood of $x$.
\end{lemma}
As the following example illustrates, the integrality assumption on $(f,h)$ is necessary.
\begin{example}
\label{ex:intrel}
Let $P$ be the monoid on three generators $x$, $y$, and $z$ subject to the relation $x+y=2z$.  We have an injection
\[
i:P\longrightarrow \N^2
\]
sending $x$ to $(2,0)$, $y$ to $(1,1)$, and $z$ to $(0,2)$.  Let $X=\Spec k[\N^2]$ and $Y=\Spec k[P]$ with their canonical 
log structures.  Then it follows from \cite[Prop 3.4]{KKato} that the morphism $(f,h):(X,\sM_X)\rightarrow (Y,\sM_Y)$ 
induced by $i$ is log \'etale, but it is not flat, and hence, by \cite[Cor 4.5]{KKato}, not integral.  It is easy to 
check that
\[
\sM_{X/Y,0}=\N^2/P\simeq\Z/2,
\]
so $\bar{\sM}_{X/Y,0}=0$, but $(f,h)$ is not strict.
\end{example}
We are now ready to introduce and discuss log curves.

\subsection{Log curves}
\begin{definition}
\label{def:logcurve}
A \emph{log curve} is a log smooth integral morphism $(f,h):(X,\sM_X)\rightarrow (S,\sM_S)$ of fs log schemes such that the 
geometric fibers of $f$ 
%the underlying morphism of schemes 
are reduced connected curves.
\end{definition}
We require $(f,h)$ to be integral so that, by \cite[Cor 4.5]{KKato}, $f$ is flat.  The reason for the fs assumption is to 
avoid cusps, as the following example shows.
\begin{example}
\label{ex:cusp}
If $X=\Spec k[\N-\{1\}]$ is given its canonical log structure and $S=\Spec k$ is given the trivial log structure, then 
$(X,\sM_X)\rightarrow (S,\sM_S)$ is log smooth and integral; however, $X=\Spec k[x,y]/(y^2-x^3)$ has a cusp.
\end{example}
It is a remarkable fact that by endowing our curves with log structures as in Definition \ref{def:logcurve}, this is 
enough to control the singularities of the curve.
\begin{theorem}[{\cite[Thm 1.3]{FKato}}]
\label{thm:node}
If $k$ is seperably closed and $(f,h):(X,\sM_X)\rightarrow(S,\sM_S)$ is a log curve with $S=\Spec k$, then $X$ has at 
worst nodal singularities.  Moreover, if $r_1,\dots,r_\ell$ are the nodes of $X$, then there exist smooth points 
$s_1,\dots,s_n$ of $X$ such that
\[
\bar{\sM}_{X/S}=\Z_{r_1}\oplus\dots\oplus\Z_{r_\ell}\oplus\N_{s_1}\oplus\dots\N_{s_n};
\]
here $M_x$ denotes the skyscraper sheaf for a monoid $M$ supported at a point $x\in X$.
\end{theorem}
%SHOULD MENTION: In particular, note that if $\sM_S$ is the trivial log structure, then $X$ has no nodes.
The reader should think of the $s_i$ in the above theorem as marked points.  So we can already see how $n$-pointed curves 
emerge naturally from the log geometry perspective.
\begin{example}
\label{ex:logcurve}
Consider the closed subscheme $X$ of $\mathbb{P}^2_k\times_{k}\mathbb{A}^1_k$ defined by $xz=ty$, where $t$ is the 
coordinate of $\mathbb{A}^1_k$ and $x$, $y$, and $z$ are the coordinates of $\mathbb{P}^2_k$.  Then $X$ has a natural 
log structure $\sM_X$.  For example, on the locus where $z$ is invertible, $X$ is given by $\Spec k[P_z]$ with $P_z$ 
a monoid on five generators $a$, $b$, $c$, $c'$, $u$ subject to the relations $c+c'=0$ and 
$a+c=b+u$; here $\sM_X$ is given by the canonical log structure associated to $P_z$.  Then the projection
\[
(X,\sM_X)\longrightarrow \mathbb{A}^1_k
%=\Spec k[\N]
\]
is a log curve, where $\mathbb{A}^1_k$ 
%$\Spec k[\N]$ is given its canonical log structure
is given the log structure defined by the divisor $t=0$.  We see that every fiber above $t\neq0$ is 
isomorphic to $\mathbb{P}^1_k$ with log structure given by the divisor at 0 and $\infty$; the fiber above $t=0$ is nodal.  
The $n$ in Theorem \ref{thm:node} is equal to 2 for all geometric fibers.
\end{example}
Since our goal is to give a log geometric 
description of $\mgnb$, we would like to express the stability condition purely in terms of log geometry.  The following 
proposition provides the key.
\begin{proposition}[{\cite[Prop 1.13]{FKato}}]
\label{prop:dualizing}
With notation as in Theorem \ref{thm:node}, there is a natural isomorphism
\[
\Omega^1_{(X,\sM_X)/(S,\sM_S)}\longrightarrow\omega_{X}(s_1+\dots+s_n),
\]
where $\omega_X$ is the dualizing sheaf of $X$.
\end{proposition}
We therefore make the following definition.
\begin{definition}
\label{def:stable}
Let $(f,h):(X,\sM_X)\rightarrow(S,\sM_S)$ be a log curve and for all geometric points $\bar{t}$ of $S$, let $\ell(\bar{t})$ 
and $n(\bar{t})$ be such that
\[
\bar{\sM}_{X_{\bar{t}}/\bar{t}}=\Z_{r_1}\oplus\dots\oplus\Z_{r_{\ell(\bar{t})}}\oplus\N_{s_1}\oplus\dots\N_{s_{n(\bar{t})}}.
\]
We say $(f,h)$ is \emph{of type $(g,n)$} if $f$ is proper, $X$ has genus $g$, and $n(\bar{t})=n$ for all $\bar{t}$.  We say 
$(f,h)$ is \emph{stable of type of $(g,n)$} if it of type $(g,n)$ and 
\[
H^0(X_{\bar{t}},T_{(X_{\bar{t}},\sM_{X_{\bar{t}}})/(\bar{t},\sM_{\bar{t}})})=0
\]
for all geometric points $\bar{t}$ of $S$.
\end{definition}


\subsection{Moduli}
Let $\lmgnb$ be the stack of stable log curves of type $(g,n)$.  This requires some explanation.  Namely, there 
are two equivalent ways we can view $\lmgnb$: as a stack over 
the category of fs log schemes (with the strict \'etale topology), or as 
a stack over the category of schemes equipped with a log structure.  In the former, $\lmgnb((S,\sM_S))$ is the category of 
stable log curves of type $(g,n)$ over $(S,\sM_S)$.  In the latter, $\lmgnb(S)$ is the category of fs log structures 
$\sM_S$ on $S$ together with a stable log curve of type $(g,n)$ over $(S,\sM_S)$.  In this second case, we obtain an fs 
log structure $\sM$ on $\lmgnb$ defined by the following property.  If $\varphi:S\rightarrow\lmgnb$ is a morphism which 
corresponds to the choice $\sM_S$ of fs log structure on $S$ along with a stable log curve of type $(g,n)$ over $(S,\sM_S)$, 
then $\varphi^*\sM=\sM_S$.

It is, in fact, true (\cite[Prop 1.7]{FKato}) that if $(f,h):(X,\sM_X)\rightarrow(S,\sM_S)$ is a log curve of type $(g,n)$, 
then the $s_i$ in each geometric fiber fit together to yield $n$
sections $\sigma_i$ of $f$.  It follows then that  
every stable log curve of type $(g,n)$ is an $n$-pointed stable curve
of genus $g$ in the classical sense.  We therefore  
obtain a morphism
\[
F:\lmgnb\longrightarrow\mgnb,
\]
which is \emph{a priori} only a morphism of stacks over the category
of schemes.  However, $\mgnb$ carries a log structure  
as well given by the divisor at infinity.  The main theorem
of \cite{FKato} is then 
\begin{theorem}[{\cite[Thm 4.5]{FKato}}]
The morphism $F$ is an equivalence of stacks with log structures.
\end{theorem}
%These observations, in fact, hold more generally:
%\begin{lemma}
%\label{l:logstack}
%If \emph{\mathbf{LSch^{fs}}} denotes the category of fs log schemes with the strict \'etale topology, then 
%\end{lemma}
We elaborate upon the proof that $F$ is essentially surjective.  It is
shown that there is a canonical way to add log  
structure to any $n$-pointed stable curve $f:X\rightarrow S$ of genus
$g$ to realize it as a stable log curve of type  
$(g,n)$.  Lemmas 2.1 and 2.2 of \cite{FKato} show that it is enough to
consider the case when $S=\Spec A$ and $A$ is  
strict Henselian.  For every node $r_i$ of the closed fiber of $f$, we can find an \'etale neighborhood $U_i$ of the points 
specializing to $r_i$ and a diagram
\[
\xymatrix{
U_i\ar[d]\ar[r]^-{\psi_i} & \Spec A[x,y,t]/(xy-t)\ar[d]^{\pi}\\
S\ar[r]^-{\varphi_i} & \Spec A[t]
}
\]
which is cartesian.  Let $t_i\in A$ be the image of $t$ under the morphism induced by $\varphi_i$.  
%If we 
Endowing $\Spec A[t]$ with the log structure associated to the morphism $\N\rightarrow A[t]$ sending 1 to $t$, and 
$\Spec A[x,y,t]/(xy-t)$ with the log structure associated to $\N^2\rightarrow A[x,y,t]/(xy-t)$ sending $e_1$ (resp. $e_2$) 
to $x$ (resp. $y$), we see that $(\pi,\Delta)$ 
is a morphism of log schemes, where $\Delta:\N\rightarrow\N^2$ is the diagonal map.  Pulling back these log structures 
under $\varphi_i$ (resp. $\psi_i$), we obtain log structures $\mathcal{L}_i$ (resp. $\sM'_i$) on $S$ (resp. $U_i$).  
Away from the points specializing to $r_i$, we define a log structure $\sM''_i$ as the pullback of $\mathcal{L}_i$.  
The log structures $\sM'_i$ and $\sM''_i$ glue to yield a log structure $\sM_i$ on $X$.  Let $\mathcal{N}$ be the 
log structure on $X$ associated to the divisor defined by the marked points.  We let
\[
\sM_X=\sM_1\oplus_{\sO_X^*}\dots\oplus_{\sO_X^*}\sM_{\ell}\oplus_{\sO_X^*}\mathcal{N}
\]
and 
\[
\sM_S=\mathcal{L}_1\oplus_{\sO_S^*}\dots\oplus_{\sO_S^*}\mathcal{L}_{\ell}.
\]
It is not difficult to see that with these definitions, we have
endowed $f$ with the structure of a log curve.

\subsection{Back to the big picture}\label{Curves-big-picture}
We end by mentioning a type of converse to the philosophic principal mentioned at the beginning of this 
section.  We have seen that since log smoothness includes degenerate objects, log geometry can naturally lead to 
compactifications; however, it is also generally true that we do not end up with ``too many'' degenerate objects.
\begin{philosophy}
\emph{
Log geometry controls degenerations.
}
\end{philosophy}
%It is fact of life that compactifications in 
In higher dimensions, compactifications tend to have unwanted extra components.  Log geometry helps 
to cut down on these components.  Let us give some inkling of an idea as to why this should be true.  Suppose $\mf{X}$ is 
an algebraic stack which is irreducible.  Suppose we can find a proper algebraic stack $\bar{\mf{X}}$ with a fine log structure 
and an open immersion $i:\mf{X}\rightarrow\bar{\mf{X}}$ such that $\mf{X}$ is the trivial locus of $\bar{\mf{X}}$.

As we now explain, log geometry provides us with a good method of trying to show that $\bar{\mf{X}}$ is irreducible 
as well.  If $k$ is separably closed and $x:\Spec k\rightarrow \bar{\mf{X}}$ is a morphism, then pulling back the log 
structure on $\bar{\mf{X}}$ endows $\Spec k$ with a fine log structure.  By \cite[Lemma 2.10]{KKato}, it follows that this 
is the log structure associated to a morphism of monoids $P\rightarrow k$, where $P$ is fine.  Hence, 
we have a strict closed immersion of log schemes $j:\Spec k\rightarrow \Spec k[[P]]$, where $\Spec k[[P]]$ is given its 
canonical log structure.  Note that the generic point $\Spec K$ of $\Spec k[[P]]$ carries the trivial log structure.  
Therefore, if $x$ factors as a morphism of log stacks through $j$, then we automatically obtain a commutative 
diagram
\[
\xymatrix{
\mf{X}\ar[rr]^i & & \bar{\mf{X}}\\
\Spec K\ar[r]\ar[u]^y & \Spec k[[P]]\ar[ur] & \Spec k\ar[u]_x\ar[l]^-{j}
}
\]
and hence $x$ is the specialization of the point $y$ of $\mf{X}$.  We see then that the log structure on $\Spec k$ obtained 
from $\bar{\mf{X}}$ somehow serves as a compass telling us which way to look in order to find a family degenerating to 
our given point $x$ of $\bar{\mf{X}}$.
%We elaborate upon this with an ``example''.




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