Canonical models and stable reduction for plurifibered varieties
Dan Abramovich
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\begin{document}

\title[Plurifibered varieties]{Canonical models and stable reduction for
plurifibered varieties} 
\author[D. Abramovich]{Dan Abramovich}
\thanks{Research partially supported by NSF grant DMS-0070970, a
Forscheimer Fellowship and Landau Center Fellowship}  
\address{Department of Mathematics\\ Boston University\\ 111 Cummington
         Street\\ Boston, MA 02215\\ U.S.A.} 
\email{abrmovic@math.bu.edu}\maketitle


\section{Introduction}
All schemes considered here are over some noetherian base scheme $\bbS$ of pure   
characteristic 0.
\subsection{Prologue: semistable reduction over a base of higher dimension}

One result we prove here is 
the
following weak variant of the main theorem of \cite{A-K}:

\begin{theorem}\label{Th:semistable} Let $X \to B$ be a morphism of
projective varieties, with 
geometrically integral generic fiber. Then there is an alteration $B_1
\to B$, a modification of the main component 
$Y\to X\times_B B_1$, and a divisor $D\subset Y$  such that 
\begin{enumerate}
\item both $Y
\to B_1$ and $D\to B_1$ are flat, 
\item the singularities of $(Y, D) $ are log-canonical, and
\item  for every
$b\in B_1$, the fiber $(Y_b, D_b)$ has at most semi-log-canonical
singularities. 
\end{enumerate}
\end{theorem}

This result is weaker that \cite{A-K}, Theorem 1, since the
singularities of $(Y, D) $ are only log-canonical, and we do not claim
that the singularities of $Y$ itself are Gorenstein, or toroidal, or
canonical. However, as we will see, the procedure that leads to $(Y,D)$ is more
canonical than that of \cite{A-K}, since the only data required is a
rational plurifibration, and  once this is chosen everything is
canonically defined. The main idea is here inspired by
A.J.\,de Jong's work on alterations \cite{DeJong}.  


The contents of this note are centered around stable models and stable
reduction for plurifibrations. This  is an expansion of remarks which apeared
in 
\cite{stablemaps} and \cite{modfam}, on natural extensions of the
results of \cite{fibsurf}. Related ideas, but with quite a
different bent, were independently discovered by Mochizuki
\cite{Mochizuki}. 


\subsection{The minimal model program}
One could say that the basic goal of the minimal model program (MMP)
is to find, in the birational equivalence class of each variety $X$ of
general type, a canonical model 
$$X \das X^\can,$$ where $X^\can$ has canonical singularities and ample
canonical divisor.  A more ambitious goal proposed by V. Alexeev is that
of stable maps: given a triple
$$( X, D, f: X \to \PP),$$
where $X$ is, say, a smooth projective variety, $D$ a normal-crossings
divisors, and $f:  X \to \PP$ is a morphism to a fixed projective
space, with the ``pre-stability assumption'' $$L_{( X, D, f: X \to
\PP),n}:=\ K_X(D) \otimes
f^*\cO_\PP(n)\ \ \mbox{ is a big line bundle for }\ n >> 0, $$
one expects to find a suitable birational representative
$$( X^\st, D^\st, f^{\st}: X \to \PP)$$ which is a {\em stable map},
that is, $( X^\st, D^\st)$ log-canonical and $$K_{X^\st}(D^\st) \otimes
(f^\st)^*\cO_\PP(n)\ \ \mbox{ is an {\bf ample} line bundle for } \ n>>0. $$ 

Following Deligne-Mumford \cite{Deligne-Mumford},
Alexeev\cite{Alexeev:stable-maps}, 
Koll\'ar, Shepherd-Barron \cite{Kollar,Kollar-Shepherd-Barron} and
Viehweg\cite{Viehweg}, one also considers moduli. One defines certain
degenerate 
versions of stable $( X, D, f: X \to \PP)$  above, where $(X, D)$
becomes {\em semi-log-canonical}. Then one conjectures that, fixing
suitable numerical invariants, stable
maps admit a proper moduli stack having a projective coarse moduli
space. It should be pointed out that, following Koll\'ar \cite{Kollar},
Alexeev\cite{Alexeev:stable-maps} and 
Karu \cite{Karu}, the projectivity of moduli in a sense follows from a strong 
enough version of the MMP in one dimension higher.

\subsection{Tweaking with the data}
The stable models above are supposed to arise by a prescribed
procedure: $X^\st$ is the projective spectrum of the algebra of
sections  $$\otimes_{i\geq 0}L_{( X, D, f: X \to
\PP),n}^{\otimes i},$$ and the divisor $D^\st$ and morphism $f^\st$
are induced. Without this requirement, one can easily cheat, i.e by
throwing in a sufficiently  ample divisor $D^\st$, or by replacing
$f$ by an embedding. Moreover, such ``cheat solutions'' are far from
canonical, are not birational in nature, and certainly are not
sufficient in 
finding a good compactification of moduli. Nevertheless, the idea of
temporarily ``tweaking'' with the given data has been quite fruitful:
methods of changing the divisor $D$ a little bit are fundamental in the
work of an illustrious list of authors, specifically Shokurov; and the
addition of a map $f:X\to Z$ 
(specifically a small contraction of an extremal ray)
puts the study of flips right into this framework.

In this note, we propose a situation where the MMP has a complete
solution - stable models and complete moduli included - which is cheap
given existing theory (though the existing theory is far from
cheap). It is still a ``cheat'' solution, but in a milder sense: the
additional data we add to the given variety is {\em birational} in nature,
and the final maps and divisors arise {\em canonically} from the
data, no further choices needed. As with other ``cheat'' solutions,
one hopes that this can serve as a stepping stone for further
undferstanding of the MMP. A situation where this can be
done is studied in the paper \cite{Lanave}.

\subsection{The setup}
We are given a function field $K$ of transcendence
degree $d$ over a field $k$ of characteristic 0. A {\em rational
plurifibration} $(K_i)$ is a sequence 
$$ K = K_0 \supset K_1\supset \cdots \supset K_d = k$$
where each $K_i \supset K_{i+1}$ is an extension
of transcendence degree 1 such that $K_{i+1}$ is algebraically closed
in $K_i$. We denote by $C_i \to \Spec K_{i+1}$ the smooth projective
curve associated to the function field extension $K_i \supset K_{i+1}$. 
%A {\em marked rational plurifibration} $(K_i, D_i)$ is a rational
%plurifibration $(K_i)$ along with a reduced effective divisor $$D_i
%\subset C_i, \ i= 1,\ldots,d-1,$$
%where $C_i/K_{i+1}$ is the smooth projective curve associated to
%$K_i \supset K_{i+1}$. 


Suppose in addition we are given a 
 projective variety $\bM_0$. A {\em rational
plurifibered map} with target $\bM_0$ is a  rational
 plurifibration $(K_i)$ and a morphism 
$\Spec K_0\to \bM_0$. 

A slightly more general notion is the following: suppose $\cM_0$ is a
proper Deligne--Mumford stack having $\bM_0$ as its coarse moduli
space. Then   a {\em rational
plurifibered map} with target $\cM_0$  is a marked rational
plurifibration $(K_i)$ 
and an object
$\cM_0(\Spec K_0)$ (equivalently, a morphism $\Spec K_0\to \cM_0$).

We need to impose a pre-stability condition. The most general
situation allowed is a bit subtle to describe - essentially it is
tautologically  the situation where the main result works. In order to
be more specific we will use  a sufficient condition for
pre-stability: we say  that $(K_i,\Spec K_0\to \cM_0 )$ is {\em
sufficiently good}
if one of the following conditions holds:
\begin{enumerate}
\item[\bf G1.] the morphism $\Spec K_0\to \cM_0$ is finite to its image, or
\item[\bf G2.] the genus $g(C_i)>1$, for $i=0,\ldots,d-2$, and
either $g(C_{d-1})>1$ or $C_{d-2} \to \Spec K_{d-1}$ is nonisotrivial, or
\item[\bf G3.] the function field $K_0$ is of general type over $K_d$.
\end{enumerate}



There are some natural extensions, where
$C_i$ are allowed to have a-priori markings and where more maps $\Spec
K_i \to \cM_i$ are a-priori given. We delay facing these to a later version.

\subsection{Twisted stable maps} Our main tool is the moduli stack of
{\em twisted stable maps}. Recall that a {\em twisted curve} is a
marked Deligne--Mumford stack of dimension 1 with simple local structure
(\cite{stablemaps}, Definition 4.1.2).  A {\em twisted stable map} $f: \cC
\to \cM$ is a representable morphism from a twisted curve $\cC$ to a
Deligne--Mumford stack $\cM$ such that the resulting map $C \to \bM$
of course moduli spaces is stable (\cite{stablemaps}, Definition
4.3.1). It is proven in \cite{stablemaps}, Theorem 1.4.1 that
families of twisted stable with fixed genus $g$, number of markings $n$, and image
class $\beta$  form a Deligne--Mumford stack $\TSM$ having a
projective coarse moduli space $\tsm$. In our discussions below we
will restrict attension to the stack of {\em balanced} twisted stable
maps $\TSMB$. From now on we will assume this and suppress the
superscript ``$\bal$''. In addition, we will ignore the ordering of the markings,
thus consider the stack $$\TSMM := [\TSM/\cS_n],$$ where the symmetric group
$\cS_n$ acts by permuting the numbers of the markings. Thus the stack $\TSMM$
parametrizes maps where the twisted curves $\cC$ are marked by
$\Sigma^\cC$, which is a reduced Cartier divisor which is $n$-sheeted finite
\'etale over the base. In most of what follows the
marking $\Sigma^\cC$ will all be twisted, i.e. the inertia group at
any point of $\Sigma^\cC$ is nontrivial.


\subsection{The outcome: twisted version}
We now describe the type of objects that arise as the stable models in
a certain sense of a rational plurifibered map.

By a {\em twisted stable plurifibered map} over a scheme $S$ we mean  a gadget
$$ (\cX_i \to \cX_{i+1},\Sigma_i, \cX_i\to \cM_i)$$
where 
\begin{enumerate}
\item $\cX_i$ are Deligne--Mumford stacks, $i=0,\ldots,d-1$, and $\cX_d
= S$
\item $ (\pi_1:\cX_i \to \cX_{i+1},\Sigma_i\subset\cX_i, f_i:\cX_i\to
\cM_i)$ is a family 
of twisted
stable maps of certain genus $g_i$, unordered twisted marking of
degree $n_i$, and 
image class $\beta_i$ for each $i=0,\ldots,d-1$, and
\item $\cM_{i+1} = \cK_{g_i, \{n_i\}, \beta_i}$ for $i=0,\ldots,d-2$.
\end{enumerate}

We remark that, almost by definition of twisted stable maps, each $\cX_i$ has a
representable dense open substack, and the coarse moduli space $X_i$
is projective over $S$. 

We use the notation $q_i: \cX_i \to X_i$
for the canonical map.
Also, for each $0\leq i\leq j\leq d$ we denote by $\pi_{i,j}: \cX_i \to
\cX_j$ the composite $\pi_{j-1}\circ\cdots \circ \pi_i$.

\subsection{Main results: twisted version}

The existence of stable models is given by the following theorem:

\begin{theorem}\label{Th:twisted} Let $(K_i, f: \Spec K_0 \to \cM_0)$
be a sufficiently good rational 
plurifibered map. Then
there exists a canonical  twisted stable 
plurifibered map $ (\pi_1:\cX_i \to \cX_{i+1},\Sigma_i\subset\cX_i,
f_i:\cX_i\to 
\cM_i)$ over $S=\Spec K_d$,  where
%\begin{enumerate}
%\item 
$\cX_i$ is irreducible with function field $K_i$, with $\pi_i$
corresponding to $K_i\supset K_{i+1}$.
%\item the restriction of $\Sigma_i$ over $ \Spec K_{i+1}\subset \cX_{i+1}$
%is the union of (a) twisted markings of $\cX_i|_{ \Spec K_{i+1}}$ and
%(b) the closure in $\cX_i$ of the reduction of $D_i \subset C_i = X_i|_{ \Spec
%K_{i+1}}$ pulled 
%back to $\cX_i|_{ \Spec
%K_{i+1}}$.
%\end{enumerate}
\end{theorem}

Completeness of moduli is immediately given:
\begin{theorem} The category of families of twisted stable
plurifibered maps with fixed data $\cM_0, g_i, n_i, \beta_i$ is a
proper Deligne--Mumford stack having a projective coarse moduli space.
\end{theorem}
{\em Proof.} This category is equivalent to
$\cK_{g_{d-1},\{n_{d-1}\}}(\cM_{d-1},\beta_{d-1})$, for which the
claim follows from \cite{stablemaps}, Theorem 1.4.1. \qed



\subsection{Main results: coarse version} Let $ (\pi_1:\cX_i \to
\cX_{i+1},\Sigma_i\subset\cX_i, f_i:\cX_i\to 
\cM_i)$ be a twisted stable plurifibered variety.
% For each $i$
%write $\Sigma_i = \Sigma_i^t \cup \Sigma_i^u$, with $\Sigma_i^t$
%signifying the twisted markings and $\Sigma_i^u$ the untwisted
%ones. Let $\cD_i\subset \Sigma_i$ be a markings of
%degree $n_i'$ containing  $\Sigma_i^u$. Denote by $D_i\subset X_i$ the
%image in the coarse moduli space. This is a $\QQ$-divisor.

We define a divisor on $X_0 = X$: for each $i$ we denote by
$\Sigma^{X_i}$ the marking on $X_i$ thought of as the coarse curve of
$\cX_i \to \cX_{i+1}$.  
Let $D\subset X$ be given by
$$D = \sum_{i=0}^{d-1} (\pi_{0,i}^*\Sigma^{X_i})_\red.$$



We define a projective variety $\bM$ and a morphism $f:X \to \bM$: 
let $\bM_i$ be the  coarse moduli space of $\cM_i$ and let 
$$\bM = \prod_{i=0}^{d-1} \bM_i.$$
We define $X \to \bM$ to be the product of the morphisms obtained by
composing $X \to X_i$ with $X_i \to \bM_i$. 
 

\begin{proposition}\label{Prop:alexeev-sm} 
The pair $(X,D)$ is semi-log-canonical, and the triple $(X,D,f:X \to
\bM)$ is a stable map in the sense of Alexeev. 
\end{proposition} 

  We call $(X,D,f:X \to \bM)$ the Alexeev stable map associated to $
  (\pi_1:\cX_i \to  \cX_{i+1},\Sigma_i\subset\cX_i, f_i:\cX_i\to  
\cM_i)$.
  

If $(K_i, D_i, f: \Spec K_0 \to \cM_0)$ is a sufficiently good
rational plurifibered 
map, and $(\pi_1:\cX_i \to
\cX_{i+1},\Sigma_i\subset\cX_i, f_i:\cX_i\to 
\cM_i)$ the associated twisted stable plurifibered map, then we obtain:

\begin{corollary} Given a sufficiently good  rational plurifibered
map, with $K_0$ of 
general type over $K_d$,  there is a
canonical Alexeev stable map $(X,D,f:X \to
\bM)$ associated to the corresponding twisted stable plurifibered map.
\end{corollary}


\section{Proofs}
\subsection{Proof of Theorem \ref{Th:twisted}} The proof proceeds by
induction on $d$. For $d=0$ there is nothing to prove. Assume the
claim is proven for $d-1$, and denote the stable twisted plurifibered
map over $K_{d-1}$ associated to $$K_0\supset\cdots \supset K_{d-1}$$
by $$(\pi_i^{d-1}:\cX_i^{d-1} \to\cX_{i+1}^{d-1}, \Sigma_i^{d-1},
f_i^{d-1}: \cX_i^{d-1} \to \cM_i).$$

{\sc Construction of target.} Denoting $\cM_{d-1} =
\cK_{g_{d-2},\{n_{d-2}\}}(\cM_{d-2}, \beta_{d-2})$, 
we have by definition a canonical morphism $\Spec K_{d-1} \to
\cM_{d-1}$.

{\sc Construction of twisted curve and map.} Essentially by
\cite{stablemaps}, Lemma 
7.2.6 there exists a 
unique smooth twisted curve $\cX_{d-1}\to \Spec K_d$ having function
field $K_{d-1}$, with a representable morphism $f_{d-1}:\cX_{d-1} \to
\cM_{d-1}$ and all markings twisted. Since that lemma had extra
assumptions, we recall the 
construction: the morphism $\Spec K_{d-1} \to \bM_{d-1} $ extends
uniquely to $C_{d-1} = X_{d-1} \to \bM_{d-1}$, since the latter  is
proper. We have a morphism $$\psi:\Spec K_{d-1} \to
C_{d-1}\times_{\bM_{d-1}}\cM_{d-1}.$$ 
Let $$\cX_{d-1} = (\overline{ \psi(\Spec K_{d-1} )} )\norm,$$
i.e. the normalization of the closure of the image of $\psi$. This is
the required twisted curve. 

{\sc Numerical invariants.} We set $g_{d-1} = g(C_{d-1}); n_{d-1} = $
the degree of the marking on $\cX_{d-1}$, and $\beta_{d-1} = $ the
class of $f_{d-1\,*}[C_{d-1}]$.

{\sc Construction of $\cX_i\to \cM_i$.} We construct $\cX_i$ by descending
recursion, $\cX_{d-1}$ being already given. Assume $\cX_{i+1}\to
\cM_{i+1}$ given, we take $(\cX_i\to \cX_{i+1}, \Sigma_i,\cX_{i}\to
\cM_{i})$ to be the universal object corresponding to $\cM_i =
\cK_{g_i, \{n_i\}}(\cM_i, \beta_i)$. We are also given canonical isomorphisms 
$$(\cX_i\to \cX_{i+1}, \Sigma_i,\cX_{i}\to
\cM_{i})_{\Spec K_{d-1}} \ \ \cong \ \  (\cX_i^{d-1}\to
\cX_{i+1}^{d-1}, \Sigma_i^{d-1},\cX_{i}^{d-1}\to 
\cM_{i}), $$ as required.  

{\sc Stability of map.} We claim that $\cX_{d-1} \to \cM_{d-1}$ is
stable. We need to show: 
\begin{quote}  if $X_{d-1} \to \bM_{d-1}$ is constant,
then $2g_{d-1} - 2 + n_{d-1} > 0$, namely:
\begin{enumerate}
\item[claim 1:] if $g_{d-1} = 1$ then $ n_{d-1} > 0$, and
\item[claim 2:] if $g_{d-1} = 0$ then $ n_{d-1} > 2$.
\end{enumerate}
\end{quote} The claims are obvious under the ``sufficiently good''
assumptions (G1) and (G2). We need 
to prove claims (1),(2) under assumtion (G3) that $K_0$ is of general type
over $K_d$. The claims, as well as the general type assumption, are
geometric, so we may replace $K_d$ by its algebraic closure. We prove
the claim by contradiction. 
We now construct trivializations of $\cX_{d-1} \to \cM_{d-1}$ after
finite \'etale covers.

 Assuming we have  $g_{d-1} = 1$ and $
n_{d-1} = 0$, then $\cX_{d-1} = X_{d-1}$ is a smooth elliptic
curve. Since $X_{d-1} \to \bM_{d-1}$ is constant, the object
corresponding to $X_{d-1} \to \cM_{d-1}$ is isotrivial, and becomes
trivial after a finite \'etale cover $E \to X_{d-1}$, and therefore
$E$ is an elliptic curve. 

 Assume, on the other hand, we have 
  $g_{d-1} = 0$ and $
n_{d-1} <2$. Then $X_{d-1}$ is rational and has two marked
  points. Since we have made the base field algebraically closed, we
may choose coordinates so that the points are $0$ and $\infty$. The
  map $X_{d-1} \setminus \{0, \infty\} \to \cM_{d-1}$ is isotrivial,
  which means that there is a map $E = \PP^1 \to X_{d-1}$ (branched only
  at $0$ and $\infty$) with a lifting $\PP^1 \to \cM_{d-1}$. It
  actually follows that this map can be chosen \'etale over
  $\cX_{d-1}$, but this will not be essential.

In both cases we got a trivialization over a curve $E$ with genus
$\leq 1$. By decending induction and the construction
of $E$ and $X_{i}$ we obtain that $E \times_{X_{d-1}} X_0 \to E$ is  a trivial 
family $E \times Y \to E$ . We have a finite morphism  $E
\times_{X_{d-1}} X_0 \to X_0$, and since the source is covered by
curves $E$ of genus $\leq 1$ we get a contradiction to the assumption that
$X_0$ is of general type. This completes claims (1),(2) and proves that
$\cX_{d-1} \to \cM_{d-1}$ is stable.

Thus the theorem is proven. \qed

\subsection{Proof of Proposition \ref{Prop:alexeev-sm}}

We need to show that $(X,D)$ has semi-log-canonical singularities and
that a certain line bundle is ample. It is convenient to study first
the situation on $\cX=\cX_0$, which has an analogous divisor 
$$\cD = \sum_{i=0}^{d-1} \pi_{0,i}^*\Sigma_i.$$ This divisor is
automatically reduced. 
\begin{lemma} The pair $(\cX, \cD)$ has semi-log-canonical
singularities. Moreover, when $\cX$ is normal, the pair has
log-canonical singularities.
\end{lemma}
{\em Proof.}  We use induction on $d$. Since $\cX$ is a composite of
a fibration by nodal curves it is Gorenstein, in particular Cohen-Macaulay.
Thus to prove the claim it suffices to look at the normalization, as follows:

Let $\cX_0\norm \to \cX_0$ be the
normalization, and let $\cE\subset \cX_0\norm$ be the conductor
divisor. Consider an irreducible component $\cX'
\subset\cX_0\norm$. Denote $\cD' =\cD|_{\cX'}+\cE$.   It suffices to
show that the pair $(\cX',\cD')$ has 
log-canonical singularities.  

By definition $\cX'$ is a composite of
fibrations by twisted curves $(\cX_i' \to \cX_{i+1}', \Sigma_i')$,
each with smooth generic fiber  (In particular $\cX_{d-1}'$ is a smooth
curve). 
%Let $p_0$ be a geometric point on $\cX_0'$ and $p_i$ its image in
%$\cX_i'$.  Now $p_{d-1}$ has an \'etale neighborhood which is a smooth marked
%curve, and similarly by descending induction each $p_i$ has an \'etale
%neighborhood which is a composition of fibrations by nodal curves over a
%neighborhood of any $p_j$ with $j>i$. In particular $p_0$ has an \'etale
%neighborhood which is a composite of 
%fibrations by marked nodal curves over $K_d$ with normal generic fiber. 
By
\cite{fiberproduct}, Corollary 4.5,  $\cX'$ has rational Gorenstein
singularities, in particular canonical.
 We are going to 
show that $(\cX', \frac{m-1}{m} \cD')$ is log-canonical for every positive
integer $m$. Taking the limit as $m \to \infty$ we get that the pair $(\cX',
\cD')$ is log-canonical. 

   So fix an integer $m$. Let $p_0$ be a 
geometric point on  
$\cX_0'$ and $p_i$ its image in   
$\cX_i'$.  Passing to an \'etale neighborhood of $p_0$  we may replace $\cX'$
by a 
scheme, which we still denote by $\cX'$. For each $i$ such that $p_i$ lies on
$\Sigma_i'$ we have that $\cX_i 
\to \cX_{i+1}$ is smooth and $\Sigma_i'$ is a section. Passing to a
neighborhood again, we may assume there exists a cyclic cover $Y_i \to
\cX_i'$ of degree $m$ totally branched along $\Sigma_i'$ and \'etale
ensewhere. Pulling back and taking the compositum, we get a branched cover
$\psi_i:Z_i 
\to \cX_i'$, totally branched along all the components of $\cD_i$ with index
$m$, and each $Z_i \to Z_{i+1}$ is a family of nodal curves. Again by
\cite{fiberproduct}, Corollary 4.5, $Z_0$ has canonical singularities.

Note that $K_{Z_0} = \psi_0^* (K_{\cX'} + \frac{m-1}{m}\cD')$. Since $Z_0$ has
canonical singularities, a result of
Koll\'ar (see \cite{flab}, 20.3 (2)) says that the pair $(\cX', \cD')$ is
log-canonical. 





% Write $\cD' = \cD^{d-1'}+ \cD_{d-1}'$,
%where $\cD_{d-1}'$ is the part coming from the markings of
%$\cX_{d-1}$. By the induction hypothesis $(\cX'\to \cX_{d-1}',\cD^{d-1'})$
%is 
%a family of semi-log-canonical pairs with log-canonical generic
%fiber. A lemma of Alexeev 
%implies that $(\cX',\cD')$ is 
%log-canonical.  
\qed

We continue with the proof of the proposition. The next step is
\begin{lemma} The pair $(X,D)$ has semi-log-canonical
singularities. Moreover, for a positive integer $m$ such that
$O_X(m(K_X+D))$ is Cartier, we have
$p^*(O_X(m(K_X+D))) = (\omega_{\cX}(\cD))^m$.   
\end{lemma} 
{\em Proof.}
Recall that when $\cX$
is a Deligne--Mumford stack with coarse moduli space  $X$, there is an
\'etale covering $\cup X_j \to X$, schemes $V_j$, and finite groups
$\Gamma_j$ such that $\cX\times_X X_j \cong [V_j / \Gamma_j]$. In our
case, we already have $(\cX,\cD)$ semi-log-canonical, so $(V_j,
\cD_{V_j})$ are semi-log-canonical. We also have that $\cD_{V_j}$
contains the divisorial part of the fixed-point-logus of
$\Gamma_j$. Denote by $D_j$ the image of $\cD_{V_j}$ in $V_j/\Gamma_j$. The
aforementioned result of Koll\'ar (see \cite{flab}, 20.3 (2))  implies that
$(V_j/\Gamma_j,D_j)$ has 
semi-log-canonical singularities, and the lemma follows. \qed 

Let $\cO_\bM(1)$ be an ample line bundle.
The proposition follows once we prove the following lemma:
\begin{lemma} For any large enough integer $n$
the sheaf $\omega_{\cX}(\cD) \otimes f^*\cO_\bM(n)$ is ample.
\end{lemma} 

{\em Proof.} There is an integer $n$ such that
$L_i : = \omega_{\cX_i/\cX_{i+1}}(\Sigma_i) \otimes f_i^*\cO_{\bM_i}(n)$ is
$\pi_i$-ample. It follows from a result of Koll\'ar (\cite{Kollar}, Proposition
4.7)  
that  $L_i$ is nef on $\cX_i/S$: indeed, $L_i^m$ is $\pi_i$-ample and
$\pi_i$-acyclic for large
$m$; Koll\'ar shows that the resulting vector bundle $\pi_{i\,*}L_i^m$ is
 semipositive, therefore the tautological line-bundle on its projectivization
is 
nef, and therefore the restriction of this tautological bundle to 
$\cX_i$, which is  $L_i^m$, is nef.

Let $F \subset \cX$ be closed and irreducible of dimension $l$. We
need to show that $$\deg_F c_1(L)^l >0.$$

We use induction on $d$ (the case $d=0$ being trivial).


If $\dim( \pi_0(F)\subset \cX_1) = l$ then we can use the inductive
assumpltion on $d$ applied to $\cX_1 \to S$ and the fact that $L_0$ is
nef to conclude that $\deg_F c_1(L)^l >0.$ 

Otherwise we have  $\dim( \pi_0(F)) = l-1$. Denote $\pi_0(F) = F_1$.
Write $L = L_0 \otimes \pi_0^* L'$, where $L'$ is the analogous line
bundle for $\cX_1/S$. By assumption $\deg_F \pi_0^*c_1(L')^{l}=0$, but
the inductive assumption implies $\deg_{F_1}c_1(L')^{l-1}>0$.
Now using the projection formula
$$\deg_F c_1(L)^l > \deg_F (c_1(L_0)\cdot \pi_0^*c_1(L')^{l-1}) = \deg_{F_1}(
\pi_{F/F_1\,*}c_1(L_0)\ \cdot\ c_1(L')^{l-1})>0.$$
By the Kleiman-Nakai-Moishezon criterion for ampleness this completes
the Lemma.\qed 
\subsection{Proof of Theorem \ref{Th:semistable}}
Let $X\to B$ be as in the theorem. We set $\bbS = B$ as base scheme,
and $\bM_0 = X$ as target scheme. We set $K_0 = K(X), K_d = K(B)$, and
write $\eta = \Spec K_d$. Now
define $K_i,\ i=1,\ldots,d-1$ by descending recursion: given $K_i$ choose a
general pencil of hypersurfaces $H_t \to\PP^1_{K_i}$ of $X_{K_i}$. We
have $K(H) = K_0$, and we set $K_{i-1} = K( \PP^1_{K_i})$. (The
connectedness theorem implies that $X_{K_i}$ is geometrically integral
for all $i$.)

Since the morphism $$\Spec K_0 \to \bM = X$$ is an embedding, the rational
plurifibered map $(K_i, \Spec K_0 \to \bM_0)$ is sufficiently good. We
therefore have a twisted stable plurifibered map over $\eta$
$$(\cX_i^\eta, \Sigma_i^\eta, f_i^\eta: \cX_i^\eta \to \cM_i).$$

This gives a morphism $$\eta \to \cK_{g_{d-1}, \{n_{d-1}\}}(\cM_{d-1},
\beta_{d-1}).$$ Since the latter is proper, there exists an alteration
$B_1 \to B$ and a lifting 
$$B_1\to \cK_{g_{d-1}, \{n_{d-1}\}}(\cM_{d-1},
\beta_{d-1}),$$ in other words, a family of  stable plurifibered maps
over $B_1$
$$(\cX_i^1, \Sigma_i^1, f_i^1: \cX_i^1 \to \cM_i)$$ 
(extending the pullback of the family over $\eta$ to the generic point
of $B_1$). In particular we have a rational map $\cX_0 \to X$. 

We now have
 an associated Alexeev stable map
$$(Y, D, Y \to \bM),$$ 
where by construction $\bM = \bM_0 \times \bM'$ for some projective scheme
$\bM'$. A-priori we have a rational map $Y \to X$, but the
composition $Y \to \bM \to \bM_0= X$ shows that this is regular.
\qed
\subsection{Acknowledgements} I wish to thank the Hebrew University and its
Lady Davis Foundation and Landau Center for Mathematical Research for support
of a visit during which this note was written. I also thank the organizers of
the HDG program at the Newton Institute for an invitation for a visit during
which this note was completed, and for a number of participants, including
Koll\'ar, Mosta\c{t}a and Reid, who clarified a few murky points about
inversion of adjunction. Finally I thank 
A. Vistoli for many discussions over the years which set the stage for the
main idea here.


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\end{document}

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