% dss.tex Yuhao Huang

\bg{conv}
Throughout this section, $X$ will be a normal crossing variety over an algebraically closed field $k$. $x\in X$ means a closed point of $X$. By a \std\nbhd of $x\in X$, we mean an \'etale \nbhd of $x$ of the form $\Spec\dfrac{k[x_1,\cdots,x_n]}{(x_1\cdots x_r)}\ra X$ such that $x$ corresponds to the point with coordinate $(0,\cdots,0)$. $\sM_k$ denotes the log \str of the \std log \pt $\NN\ra k, n\mapsto 0^n$.
\ed{conv}

\subsection{Normal crossings and d-semistability}

To study the geometry of a normal crossing variety $X$, one would like to ask the following question:

\bg{enum}
\item\label{1}{Can we embed $X$ into another variety with codimension 1, i.e. an embedding $i:X\ra\scr{X}$as a normal crossing divisor?}
\item\label{2}{Can we find a semi-stable smoothing of $X$, i.e. embed $X\ra\Spec{k}$ in a flat family over a curve $\scr{X}\ra C$,
s.t. there exists a diagram 
\[
{
   \xymatrix{
  X \ar[d] \ar[r]          & \scr{X} \ar[d]  & \ar[l] \scr{X}^*=\scr{X}\backslash X\ar[d]_{f^*}\\
  \Spec{k}  \ar[r]^{0}         & C               & \ar[l] C^*=C\backslash{0}
  }
}
\]\newline
 where $\scr{X}$ is smooth, the squares are cartesian and $f^*$ is smooth?}
\end{enum}

\zz
The answer to these questions are not always yes, because the
existence of such maps would imply the existence of certain log
structures on $X$, which in turn would imply intrinsic condition on $X$,
 so their existence is not guaranteed.

\bg{eg}
[log structure of embedding type]\label{emb}
If we can find an embedding as in \ref{1} above, then $i^*(j_* \mathcal{O}^{\times}
_{\scr{X}^*})\ra\ox$ defines a log \str $\sM_X$ on $X$, which \etale locally has a chart $\mathbb{N}^r\ra\Spec\dfrac{k[x_1,\cdots,x_n]}{(x_1\cdots
x_r)}$ sending the \elt $e_i$ in the \std basis of $\NN^r$ to $x_i$, making $X$ a log smooth variety over $\Spec k$ with the trivial log structure. This is called a log \str of embedding type.
\end{eg}

\bg{rmk}This construction is functorial in $X$.
\end{rmk}

\bg{eg}[log \str of semi-stable type]\label{sm}If we can find a semi-stable smoothing as in \ref{2} above, then by the remark on functoriality, what we have in this case is not only a log \str of embedding type on $X$, but also a morphism (of sheaf of monoids) $f^\flat: f^*\sM_k\ra \sM_X$, which makes $X$ a log smooth variety over ($\Spec k, \sM_k$). \'Etale locally a chart for the log \str on X can be put in the form $\big(\Spec\dfrac{k[x_1,\cdots,x_n]}{(x_1\cdots
x_r)}, \NN^r, e_i\mapsto x_i, i=1,\cdots,r\big)$. Modulo the units in the monoid, the morphsim of quotient monoids induced by $f^\flat:f^*\sM_k\ra \sM_X$ is just the diagonal $\Delta:\NN\ra\NN^r$. Such a pair $(\sM_X,f^\flat:f^*\sM_k\ra \sM_X)$ is called a log \str of semi-stable type on $X$.
\end{eg}

\bg{rmk}
The log structure $\sM_k$ on $\Spec k$ can be defined by the pullback of the log \str $(\sM_0=j_* \mathcal{O}^{\times}_{C^*}\ra\oo_C)$ on $C$, i.e. the log \str defined by the divisor $0$ of $C$. We have an \isom $\overline{\sM_k}:=\sM_k/k^\times\iso (\sM_0/\oo_C^\times)_0\iso\NN$, where the second \isom assigns each function to its vanishing order at the point $0$ in $\NN$. This gives a geometric interpretation of the \std log point.
\end{rmk}

Concerning the existence of such log \str on the normal crossing variety $X$, we have:

\begin{thm}\label{embed}
Let X be a normal crossing variety over the spectrum of
an algebraically closed field, then X can be equipped with
a log structure of embedding type iff there exists a line bundle $\mathcal{L}$ on $X$ such that
\begin{equation*}
\mathcal{E}xt_X^1(\Omega^1_X,\mathcal{O}_X)|_D\iso\mathcal{L}|_D
\end{equation*}
where D is the non-smooth locus of X.
\end{thm}

\begin{thm}[d-semistability]\label{d-semi}
Let $X$ be a normal crossing variety over the spectrum of
an algebraically closed field, then $X$ can be equipped with
a log structure over the standard log point, such that the structure
morphism is log smooth iff it is \underline{d-semistable}, i.e.,
\begin{equation*}
\mathcal{E}xt_X^1(\Omega^1_X,\mathcal{O}_X)|_D\iso\oo_D
\end{equation*}
where D is the non-smooth locus of X.
\end{thm}

\bg{rmk}
It is not hard to see $\mathcal{E}xt_X^1(\Omega^1_X,\mathcal{O}_X)|_D$ is a line bundle on $D$.
\ed{rmk}

Generalization of these theorems can be found in
\cite{Olsson2003}, section 3.

\bg{cor}
If $X$ has a smoothing, then $X$ is d-semistable.
\ed{cor}

\begin{rmk}
Being d-semistable is not equivalent to being semi-stably smoothable. See \cite{MR705035} for counter examples.
\end{rmk}

\begin{eg}[a normal crossing variety that is not d-semistable\cite{Friedman}]
Let $X$ be the subvariety of $\mathbf{P}^3$ defined by the product
of 4 linear equations $f=L_1L_2L_3L_4=0$, it is of normal crossing,
provided the four planes has no points in common. Then $D$ is
defined by the homogeneous ideal $(L_iL_j|1\leq i<j\leq 4)$, and it
is not hard to calculate that
$\mathcal{E}xt_X^1(\Omega^1_X,\mathcal{O}_X)|_D\cong
\mathcal{O}_D(4)$, which is not trivial. So $X$ is not d-semistable.
\end{eg}

\bg{cor}
The $X$ above has no semi-stable smoothings.
\ed{cor}

\begin{rmk}
If we put $X$ in the $1-$dimensional familly $\scr{X}$ defined by $f+tg=0$ with parameter $t$, where $g$ is a smooth quartic, then $X$ is the fiber over $t=0$ and for generic $g$, $\scr{X}$ over $(t\neq 0)$ is smooth. But the whole space of this family is not smooth: in fact, for a generic $g$, this family is singular at the 24 points of $D\cap\{g=0\}$. However, a single blowing up at such a point gives a $\mathbf{P}^1\times\mathbf{P}^1$. Contract along either ruling gives back a family with parameter $t$ which has one less singularity. If we do this process to all 24 points, we will get a family which is a semi-stable smoothing of $\tilde{X}$, the blowing-up of $X$ at those $24$ points. $\tilde{X}$ is d-semistable. For details, see \cite{Friedman}.
\end{rmk}

\subsection{Strategy of the Proof}
Let's analyze the situation. we want to:

\begin{enumerate}
\item Put a log structure of embedding type on $X$.
\item See if it is possible to make that log structure semi-stable.
\end{enumerate}

We will see that two related obstructions arise naturally, where the
vanishing of the first corresponds to the first step, and the
vanishing of the second, which means precisely being d-semistable,
allows us to do the second step.
\zz

Since \'etale locally a log \str of embedding type always exists, let's consider the stack $\mcal{G}$, which to each $U\in \xet$ associates the \gpd of log \str of embedding type on $U$. Then it is not hard to show (using Artin's approximation theorem) that any two \elt of $U$ is locally isomorphic, which means $\mcal{G}$ is a gerbe. Since $\Aut{\mcal{G}}\iso\mcal{K}$, where $\mcal{K}$ is the kernel of the restriction map $\ox^\times\ra\oo_D^\times$, we have:

\bg{prop}
There is an obstruction $\eta$ in $H^2(\xet,\mcal{K})$ 
whose vanishing is equivalent to the existence of a log \str of embedding type on $X$.
\ed{prop}

For the calculation of this obstruction, we state the following result (See \cite{FKato-def}
\footnote{An early version of this paper, \itshape{Logarithmic embeddings and logarithmic semistable reductions}\normalfont, 
 \url{http://arxiv.org/PS_cache/alg-geom/pdf/9411/9411006v2.pdf}, is probably more relevant to our topic here.
 }
\cite{Olsson2003}):

\bg{prop}\label{gerbe}
In the \les of cohomology associated to the \ses $1\ra\mcal{K}\ra\ox^\times\ra\oo_D^\times\ra1$, the line bundle $\mathcal{E}xt_X^1(\Omega^1_X,\mathcal{O}_X)|_D$ maps to $-\eta\in H^2(\xet,\mcal{K})$ 
\end{prop}

Combining these results with the exactness of the \les, we have proved \Thm\ref{embed}.

\bg{rmk}
General theory tells us if $\eta=0$, then the set of all log \str of embedding type on $X$ is naturally a torsor under $H^1(\xet,\mcal K)$. In general the set of all log \str of embedding type is only a pseudo torsor under this group.
\ed{rmk}
\zz

Suppose $\eta=0$, then we can put a log structure of embedding type on $X$, which maps to $(\Spec k,k^\times)$, to make it a log \str of semistable type, we need a morphism of monoids $f^*\sM_k\ra \sM_X$, such that in an \std \nbhd of $x\in X$, on the level of charateristic\footnote{Here we used the terminology of F.Kato, the characteristic of a log \str $\alpha:\sM_X\ra\ox$ is $ \overline{\sM_X}:=\sM_X/\alpha(\ox^\times)$}, is just the diagonal $\Delta:\NN\ra\NN^r$, where $r$ is the number of irrducible components passing through $x$.
 
Since $\sM_k\iso\NN\oplus k^\times$ (non-canonically), and the image of $k^\times\subset \sM_k$ is determined by the underlying morphism of schemes. To give the morphism wanted from $\sM_k$ to $\sM_X$, we only have to specify the image of an \elt in $\sM_k$ having vanishing order 1.

Now the question becomes a lifting problem for morphism of sheaves in monoids:
\[
   \xymatrix{
                               & \sM_X \ar[d]^{\beta}  \\
  %\undeline
  {\NN}\iso f^{-1}\overline{\sM_k} \ar@{..>}[ur] \ar[r]^(.65){ \overline{f^\flat}}         & \overline{\sM_X}   
  }
\]
where \'etale locally $\overline{f^\flat}$ is the diagonal $\Delta$. To lift it is equivalent to lift $\Delta(1)$.

Consider the sheaf of all the possible local liftings of $\Delta(1)$, $T=\beta^{-1}(\Delta(1))$, then $T$ is a torsor under $\ox^\times$. To find a lifting of $\Delta(1)$ is equivalent to find a global section of $T$, i.e. a trivialization of $T$. 

\zz
It seems like we've got an obstruction of finding a log \str on $X$ which is semi-stable in $H^1(\xet,\ox)=\Pic(X)$. This is, however, not quite true. In fact, what we got is for each \underline{log scheme} $X$ with a log \str of embedding type, the obstruction of making it semi-stable. And our original question (on the existence of log \str of semi-stable type) allows some ambiguity of choosing the log \str of embedding type $\sM_X$ on $X$. As we remarked after Prop.\ref{gerbe}, in this case the set of all log \str of embedding type on $X$ is an $H^1(\xet,\mcal K)$-torsor, which implies\footnote{Some compatibility should be checked here, which we left to the readers.}:

\bg{prop}
If $\eta=0$, i.e. there exists a log \str of embedding type on $X$. Then there is an obstruction of finding a log \str of semi-stable type on $X$, $\eta'\in H^1(\xet,\ox^\times)/H^1(\xet,\mcal K)$. There is a log \str of semi-stable type iff $\eta'=0$.  
\end{prop}

By the \les of cohomology, $H^1(\xet,\ox^\times)/H^1(\xet,\mcal K)$ embeds into $H^1(D_{\acute et},\oo_D^\times)\iso\Pic(D)$. For the calculation of $\eta'$ as an \elt of $\Pic(D)$, we state the following proposition (See \cite{Olsson2003}).

\bg{prop}
We have $-\eta'=[\mathcal{E}xt_X^1(\Omega^1_X,\mathcal{O}_X)|_D]\in \Pic(D)$.
\ed{prop}

Combining these two proposition, we have proved \Thm \ref{d-semi}. 
>>>>>>> .r54
