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\title{Main components of moduli spaces}
\author{Matthew Satriano}
\date{}
\maketitle
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In the section ``Log smooth curves and their moduli'', we gave an overview of F. Kato's work \cite{Fkato} in which 
he uses log geometry to compactify the moduli space $\mgn$ of curves.  Specifically, he shows that the moduli 
space of log smooth curves agrees with the Deligne-Mumford compactification $\mgnb$.  The key philosophic idea in 
that section was that since moduli spaces of log smooth objects already includes degenerate objects, it is reasonable 
to expect that such a moduli space is a compactification of the moduli of objects with trivial log structure.\\
\\
While F. Kato's moduli space of 
log smooth curves turns out to be irreducible, it is an unfortunate fact of life that if $\mf{X}$ is a 
moduli space of higher dimensional objects, then moduli-theoretic compactifications $\bar{\mf{X}}$ of $\mf{X}$ tend 
to have many irreducible components.  If $\mf{X}$ is irreducible, then it sits entirely within one of these many 
components of $\bar{\mf{X}}$ and so it is natural then to ask if this ``main component'' can itself be given 
a moduli interpretation.\\
\\
%The point of this section is to elaborate upon the second of the two philosophic principles mentioned in the section 
%``Log smooth curves and their moduli''.  This second philosophy states that log geometry controls degenerations; that 
At the end of ``Log smooth curves and their moduli'', we stated a second philosophic principle: log geometry controls 
degenerations; that 
is, moduli of log smooth objects does not incorporate ``too many'' degenerate objects.  This provides a type of 
converse to the afformentioned philosophy that moduli of log smooth objects should be compact.  Combining the two 
principles, one would expect that moduli of log smooth objects will not only be compact, it will also be 
\emph{irreducible}.  In particular, one would imagine that if $\mf{X}$ is an irreducible moduli space and $\bar{\mf{X}}$ 
a moduli-theoretic compactification, then by appropriately incorporating log structures into the objects parameterized 
by $\bar{\mf{X}}$, one will isolate the main component.\\
\\
This technique of using log geometry to isolate the main component of a moduli space has been carried out by 
M. Olsson in several different settings.  In \cite{toricHilb}, Olsson gives a moduli interpretation to the 
normalization of the main component of the toric Hilbert scheme.  In \cite{cancmpt}, he isolates the 
normalization of the main component of V. Alexeev's compactification of the moduli space of principally polarized 
abelian varieties given in \cite{alexeev}; he further constructs a moduli-theoretic irreducible compactification of 
the moduli space of abelian varieties with higher degree polarization.  In \cite{K3}, he gives an irreducible modular 
compactification of the moduli space of polarized K3 surfaces.\\
\\
Our goal in this section is to explain the technique of isolating the main component of a moduli space by following 
Olsson's work \cite{toricHilb}.  We begin with the definition of the toric Hilbert scheme.  Let $k$ be a field 
and let $P$ and $Q$ be finitely-generated integral monoids with $Q$ sharp and $P^{gp}$ and $Q^{gp}$ torsion-free.  
Fix a surjective morphism $\pi:P\rightarrow Q$.  This yields a closed immersion from $A_Q:=\Spec k[Q]$ to 
$A_P:=\Spec k[Q]$, which is $T_Q$-equivariant, where $T_Q$ (resp. $T_P$) denotes the torus associated to $Q^{gp}$ 
(resp. $P^{gp}$).  Consider the functor $\sH$ whose $S$-valued points are diagrams 
\[
\xymatrix{
Z\ar[r]^{i}\ar[dr]_{g} & A_{P,S}\ar[d]\\
 & S
}
\]
where $i$ is a $T_Q$-invariant closed immersion and for every $q\in Q^{gp}$, the $q$-eigenspace of $g_*\sO_Z$ is a 
finitely-presented projective $\sO_S$-module of rank $1$ if $q\in Q$ and rank $0$ otherwise.  By 
\cite[Thm 1.1]{hs}, this functor is representable by a quasi-projective scheme, which we call the 
\emph{toric Hilbert scheme}.\\
\\
Given a closed subscheme $Z$ of $A_{P,S}$ as above, we can move $Z$ by the action of $T_P$ on $A_{P,S}$.  This yields 
an action of $T_P$ on $\sH$.  Since $Z$ is $T_Q$-invariant, this action factors through $T_K=T_P/T_Q$, where 
$K$ denotes the kernel of $\pi^{gp}$.  We therefore we obtain a map
\[
T_K\longrightarrow \sH
\]
by letting $u\in T_K$ act on the distinguished point of $\sH(k)$ given by the closed immersion $A_Q\rightarrow A_P$.  
By \cite[3.6(2)]{chuv}, this map is an open immersion.  Therefore the normalization $\sS$ of the scheme-theoretic 
closure of its image is a normal toric variety, and hence carries a natural fs log structure $\sM_{\sS}$ which 
makes it log smooth over $\Spec k$ (endowed with the trivial log structure).  The goal of \cite{toricHilb} is to 
give a moduli-theoretic interpretation of $(\sS,\sM_{\sS})$.\\
\\
Consider the functor $\sH^{log}$ on the category of fs log schemes over $k$ whose $(S,\sM_S)$-valued points are 
given by diagrams 
\[
\xymatrix{
(Z,\sM_Z)\ar[r]^-{i}\ar[dr]_{g} & (A_P,\sM_{A_P})\times (S,\sM_S)\ar[d]\\
 & (S,\sM_S)
}
\]
where the underlying maps on schemes defines a point of $\sH(S)$, where $g$ is log smooth and integral, and 
where the map
\[
P\longrightarrow \sM_{(Z,\sM_Z)/(S,\sM_S)}:=\coker(g^*\sM_S\rightarrow\sM_Z)
\]
induced by $i$ factors through $Q$.  The main theorem of \cite{toricHilb} is then 
\begin{theorem}[{\cite[Thm 1.6]{toricHilb}}]
\label{thm:mainthm}
The functor $\sH^{log}$ is representable by $(\sS,\sM_{\sS})$.
\end{theorem}
The first step in proving this theorem is to obtain a map from the functor points $\sH_{\sS}$ of 
$(\sS,\sM_{\sS})$ to $\sH^{log}$.  This is done by showing that the pullback to $\sS$ of the universal family 
over $\sH$ yields a point of $\sH^{log}(\sS,\sM_{\sS})$.  Explicitly, if $i:\sZ\rightarrow A_P\times\sS$ is 
the pullback of the universal family, Olsson constructs a log structure on $\sZ$ as follows.  Since $\sS$ is a 
toric variety with torus $T_K$, it can be covered by open affines of the form $\Spec k[L]$ with $L$ a submonoid of $K$ 
whose associated group is $K$.  Olsson proves (\cite[2.4]{toricHilb}) that over such an open affine, the closed 
immersion $i$ is given by
\[
\sZ\times_{\sS}\Spec k[L]=\Spec k[E_L]\longrightarrow \Spec k[P\oplus L]=A_P\times\Spec k[L],
\]
where $E_L$ is the image of $P\oplus L$ in $P^{gp}$ under the map $(p,\ell)\mapsto p+\ell$.
\begin{example}
\emph{Let $\pi:\N^2\rightarrow\N$ send $e_1$ to $2$ and $e_2$ to $1$.  Then the kernel $K$ of $\pi^{gp}$ is all 
integer multiples of $2e_2-e_1$.  Let $L$ be the submonoid of $K$ consisting of the non-negative multiples 
of $e_1-2e_2$ and let $M$ be the submonoid consiting of the non-positive multiples.  Then $E_L$ is generated 
by $e_1$, $e_2$, and $2e_2-e_1$; hence,}
\[
\Spec k[E_L]\simeq k[x,y,z]/(xy-z^2).
\]
\emph{We see that $E_M$ is freely generated by $e_2$ and $e_1-2e_2$, so 
$\Spec k[E_M]\simeq\mathbb{A}^2_k$.}
\end{example}
We see then that 
$\sZ\times_{\sS}\Spec k[L]$ carries a natural log structure.  These log structures glue to give a log structure on 
$\sZ$ (by \cite[Lemma 2.8]{toricHilb}) which makes $i$ a closed immersion of log schemes.  We therefore obtain a 
morphism of functors
\[
F:\sH_{\sS}\longrightarrow \sH^{log}.
\]
The next step in proving Theorem \ref{thm:mainthm} above is to work out the deformation theory of $\sH^{log}$.  
Olsson shows (\cite[\S4]{toricHilb}) that if $(\Spec B_0,\sM_{B_0})\rightarrow (\Spec B,\sM_{B})$ is a strict closed 
immersion defined by a square zero ideal $\sI$, then every object of $\sH^{log}(\Spec B_0,\sM_{B_0})$ lifts to 
an object of $\sH^{log}(\Spec B,\sM_{B})$; moreover, the set of lifts forms a torsor under $\Hom(K,\sI)$.  Since 
$\sS$ is a toric variety with torus $T_K$, we see
\[
\Omega^1_{(\sS,\sM_{\sS})/k}=\sO_{\sS}\otimes_{\Z}K.
\]
As a result, every object of $\sH_{\sS}(\Spec B_0,\sM_{B_0})$ also lifts to an object of 
$\sH_{\sS}(\Spec B,\sM_{B})$ and the set of lifts forms a torsor under $\Hom(K,\sI)$.  This shows 
\begin{theorem}[{\cite[4.4]{toricHilb}}]
\label{thm:loget}
The functor $F$ is formally log \'etale.
\end{theorem}
Olsson then makes use of Theorem \ref{thm:loget} to show that $F$ is injective and surjective, thereby proving 
Theorem \ref{thm:mainthm}.  We explain his proof of 
injectivity as it gives a concrete instance of the second philosophy and diagram drawn at the end of ``Log smooth 
curves and their moduli''.  Suppose $f$ and $g$ are two morphisms from an fs log scheme $(T,\sM_T)$ to 
$(\sS,\sM_{\sS})$ such that $Ff=Fg$.  We show that $f=g$.  Since $\sS$ is locally of finite presentation 
over $k$, a limit argument allows us to assume $T$ is of finite type over $k$.  Since $\sH^{log}$ is 
limit preserving (by \cite[\S6]{toricHilb}), another limit argument shows that it is enough to check $f=g$ at the 
completion of $T$ at every point.  Since $T$ is assumed to be of finite type over $k$ (hence Noetherian), we have 
reduced to the case $T=\Spec R$ for $R$ a complete local Noetherian ring.  To check $f=g$, it now suffices to show 
that this is true 
after precomposing with all maps $\Spec R/\mf{m}^n\rightarrow\Spec R$, where $\mf{m}$ is the maximal ideal of $R$.  
Hence, we may assume $T=\Spec B$, where $B$ is a local Artin ring.  If $i:(\Spec B_0,\sM_{B_0})\rightarrow 
(\Spec B, \sM_{B})$ is a strict closed immersion with square zero ideal, and if $fi=gi$, then the diagram
\[
\xymatrix{
(\Spec B_0,\sM_{B_0})\ar[r]^-{fi=gi}\ar[d]_i & (\sS,\sM_{\sS})\ar[d]^F\\
(\Spec B,\sM_{B})\ar[r]\ar@<+.5ex>[ur]^-{f} \ar@<-.5ex>[ur]_-{g} & \sH^{log}
}
\]
commutes.  By Theorem \ref{thm:loget}, we see then that $f=g$.  We can therefore assume that $T=\Spec k'$ with $k'$ 
a field.  It further suffices to assume that $k'$ is separably closed, in which case there is an fs sharp monoid $L$ 
and a morphism $L\rightarrow\sM_T$ inducing an isomorphism of $L$ with $\bar{\sM}_T$.\\
\\
Consider the complete local ring $R'=k'[[L]]$ defined as the completion of $k'[L]$ at the maximal ideal $\mf{m'}$ 
which is the kernel of the morphism $k'[L]\rightarrow k'$ sending all non-zero elements of $L$ to 0.  The scheme 
$\Spec R'$ carries a natural log structure $\sM_{R'}$ defined as the pullback of the canonical log structure on 
$\Spec \Z[L]$.  By Theorem \ref{thm:loget} and log smoothness of $(\sS,\sM_{\sS})$, we obtain morphisms 
$\til{f}$ and $\til{g}$ from $(\Spec R',\sM_{R'})$ to $(\sS,\sM_{\sS})$ such that $F\til{f}_n=F\til{g}_n$ for 
every $n$, where $\til{f}_n$ (resp. $\til{g}_n$) denotes the reduction of $\til{f}$ (resp. $\til{g}$) 
modulo ${\mf{m}'}^n$.\\
\\
In other words, we have replaced our original $T$, $f$, and $g$ by $\Spec R'$, $\til{f}$, and $\til{g}$.  So, we 
are once again in the case where $T$ is the spectrum of a complete local Noetherian ring.  What have we gained, then, 
over the previous case when $T=\Spec R$ above?  We have succeeded in replacing an arbitrary complete local Noetherian 
ring $R$ by a very special one, namely $k'[[L]]$.  We saw these special rings come up in the diagram at the 
end of ``Log smooth curves and their moduli''.\\
\\
Let $h:\sS\rightarrow\sH$ be the natural morphism (recall that $\sS$ is the normalization of the main component of 
$\sH$).  Since $F\til{f}_n=F\til{g}_n$ for all $n$, the very definition of $\sH^{log}$ shows that 
$f\til{f}_n=h\til{g}_n$ for all $n$.  Therefore, $h\til{f}=h\til{g}$.  Since $k'[[L]]$ is normal and $\sS$ is 
separated, to show that $\til{f}=\til{g}$, it suffices to show 
they are equal on a dense open subset.  Since the generic point of $k'[[L]]$ has trivial log structure, it factors 
through the torus $T_K$ of $\sS$.  The map $h$, however, is an open immersion over $T_K$, and 
so $\til{f}$ and $\til{g}$ agree on the generic point of $k'[[L]]$.





\begin{comment}
give def of toric Hilbert scheme, define main component, give Martin/Olga's explicit description of the 
universal family, so we get map from mc to the stack we hope paramaterizes mc.  Then explain how to show this 
morphism is ``log \'etale'' (and explain that log \'etale is in quotes b/c we don't yet know that the stack is 
algebraic), then say that Martin shows it's injective and surjective.  We explain injectivity as in gives a 
concrete example of the converse philosophy and the diagram we drew at the end of the ``log smooth curves'' section.
\end{comment}




\begin{comment}
\begin{thebibliography}{10}
\bibitem[Al]{alexeev} V. Alexeev, \emph{Complete moduli in the presence of semiabelian group action}, Ann. of 
Math. (2) 155 (2002), no. 3, 611--708.

\bibitem[Ch]{chuv} O. Chuvashova, \emph{The main component of the toric Hilbert scheme}, 
Tohoku Math. J. (2) 60 (2008), no. 3, 365--382. 

\bibitem[HS]{hs} M. Haiman and B. Sturmfels, \emph{Multigraded Hilbert schemes}, J. Algebraic Geom. 13 (2004), 725–769.

\bibitem[Ol1]{cancmpt} M. Olsson, \emph{Compactifying Moduli Spaces for Abelian Varieties}, 
Lecture Notes in Mathematics, Vol. 1958, (2008).

\bibitem[Ol2]{toricHilb} M. Olsson, \emph{Logarithmic interpretation of the main component in toric Hilbert schemes}, 
in Proceedings of the International Conference on Curves and Abelian Varieties,  231--252, Contemp. Math., 465, 
Amer. Math. Soc., Providence, RI, 2008.

\bibitem[Ol3]{K3} M. Olsson, \emph{Semi-stable degenerations and period spaces for polarized K3 surfaces}, 
Duke Math. J. 125 (2004), 121-203

%\bibitem[FKa]{fkato} F. Kato, \emph{Log smooth deformation and moduli of log smooth curves}, 
%Internat. J. Math.  11  (2000), no. 2, 215--232. 

\end{thebibliography}
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