%%%%%%%%%%%%%%%%
%% V.V. Shokurov
%% Prelimiting flips
%%%%%%%%%%%%%%%%%

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\title{Prelimiting flips
\thanks{First draft completed on August 16th, 1999, Moscow. \copyright V.V.
Shokurov, 2000. All rights reserved. Unauthorized copying, reproduction,
hiring, lending, and public performance prohibited.}
\thanks{Current draft, edited by Miles Reid. I haven't understood
everything. I've tried to make the table of contents useful, but it
could probably still be improved. I've started Appendix~\ref{app!b-d}
discussing {\bi}divisors, but it should be expanded. I have started on a
list of index keywords, and it will be quite easy to make an index from
this. Maybe a glossary to explain the new terms would also be useful.
Most of the footnotes are questions to myself or to Slava.}
 }

\date{August 3rd, 2000}
\author{V.V. Shokurov\thanks{Partially supported by the grant NSF-9800807.}}

 \begin{document}
 \maketitle
 \begin{abstract} The paper discusses an inductive approach to constructing
log flips. In addition to special termination and thresholds, we introduce
two new ingredients: saturation of linear systems, and families of divisors
with bounded singularities. We state conjectures concerning these notions
in any dimension, and prove these in general in dimension $\le2$. This
allows us to construct prelimiting flips (pl flips) and all log flips in
dimension~4, and to prove the stabilization of an asymptotically saturated
family of birationally free ({\bi}free) divisors under certain conditions in
dimension~3. In dimension~3, this stabilization generalizes special flips
and also gives for the first time a proof of the existence of log flips that
is algebraic in character, that is, via f.g.\ algebras, as opposed to
geometric flips. It also accounts for all the currently known flips and
flops.
 \end{abstract}

\clearpage
\tableofcontents
\clearpage

\section{Introduction} \label{int}

\subsection{Prelimiting contractions}\label{specont} By $f\colon X\to
X_\vee$ we denote a {\em prelimiting contraction} or {\em pl
contraction} with respect to a divisor $S$. By this we mean that $f$ is a
birational contraction such that
 \begin{enumerate}
 \item\label{svoS} $S=\sum S_i$ is a sum of $s\ge1$ prime Weil divisors
$S_1,\dots,S_s$ on $X$ that are $\Q$-Cartier and {\em proportional\/},
that is, $S_i\sim_\Q r_{i,j} S_j$ for some rational $r_{i,j}>0$;
 \item\label{logterm} there is a boundary $B$ such that $\rddown{B}=0$
and $K+S+B$ is divisorially log terminal, and purely log terminal if
$s=1$; and
 \item \label{negativ} $K+S+B$ is numerically\footnote{numerically
negative means simply that minus it is an ample $\Q$-Cartier divisor. No
clever distinction between numerically ample and ample is involved.}
negative$/X_\vee$.
 \end{enumerate}
 By Conditions~\ref{svoS}, \ref{logterm} and \cite[Corollary~3.8]{sh92}
(see also \cite[Chap.~16]{ko}) the divisors $S_i$ are normal varieties,
with normal varieties in their intersections and normal crossing at the
generic points of these intersections. In particular, $Y=\bigcap S_i$ is
a normal variety.

Conditions~\ref{logterm} and \ref{negativ} together mean that
$(X/X_\vee,S+B)$ is a weak log Fano contraction\footnote{{\em contraction}
is my addition, for grammar.} with only {\em log terminal\/}
singularities. Thus by Connectedness of LCS (Koll\'ar and others
\cite[Theorem~17.7]{ko}), $Y$ is irreducible of dimension $d=n-s'$ near
each fibre of $f$, where $s'$ is the number of $S_i$ that intersect that
fibre. Moreover, $f$ induces a contraction $Y\to f(Y)\subset X_\vee$,
and $(Y,B_Y)$ is Klt (Kawamata log terminal) for the adjoint log divisor
$K_Y+B_Y=(K+S+B)\rest Y$. These properties guarantee an induction on $Y$
in our construction of pl flips below (see Section~\ref{divalg} for
inductive families).

We always consider the {\em local\/} situation, where $f$ is a germ over a
neighborhood of a given point $P\in X_\vee$. In particular, all the $S_i$
intersect the {\em central fibre\/} $f\1P$ of $X/P$. Thus $\dim Y=d=n-s\le
n-1$ by Condition~\ref{svoS}. The {\em core dimension} $d$ measures
the difficulty of constructing pl flips, in a sense indicated in Induction
Theorem~\ref{ith}.

It is sometimes also reasonable to consider a contraction $f$ that is not
birational. Then $S$ is not numerically negative$/X_\vee$, in contrast to
the case of elementary pl flips. This situation occurs naturally in
Section~\ref{divalg} in certain applications of Main Lemma~\ref{mainl}
(see Example~\ref{ample}).

However, in Reduction Theorem~\ref{rth} below, we can assume that the pl
contractions $f$ of most concern to us are {\em elementary}, that is,
satisfy in addition:
 \begin{enumerate}
 \setcounter{enumi}{3}
 \item $S$ is numerically negative$/X_\vee$;
 \item $K+S+B$ is strictly log terminal$/X_\vee$, that is, $X$ is
$\Q$-factorial and projective$/X_\vee$;
 \item \label{pic1} (EL.\ref{pic1}) $\rho (X/X_\vee)=1$, where $\rho$
denotes the relative Picard number; and
 \item \label{small} (EL.\ref{small}) $f$ is small, or equivalently under
the current assumptions, $S$ is not $f$-exceptional.
 \end{enumerate}
 We can omit the final assumption (EL.\ref{small}), but then we meet an
additional well-known situation, when the contraction is divisorial, and
is its own $S$-flip (cf.\ Example~\ref{fbrcont}). Note also that by
(EL.\ref{pic1})
$f$ is nontrivial, that is, not an isomorphism.

A {\em pl flip\/} is the $S$-flip of $f$ (\cite[Section~5]{sh96b}, see
also Example~\ref{flipal} below). By a {\em log flip\/}, we mean the
$D$-flip of a log contraction $(X/T,B)$ over a\footnote{At present $T$ is
not defined. There is a small problem of consistency; you use $T,Z$
interchangeably in Chap~1. On the other hand, you also introduce the
slightly crazy notation $X_\vee$, which appears only in a few sections. Is
there some reason for using three different notations $X/X_\vee$, $/Z$,
$/T$?} base $T$ with $D=(K+B)$, assuming (cf.\ \cite[Theorem~p.~96]{sh92})
that
 \begin{itemize}
 \item $K+B$ is Klt, and
 \item $-(K+B)$ is nef$/T$.
 \end{itemize}
 Thus each elementary pl flip is also the log flip of
$(X/T,B)=(X/X_\vee,aS+B)$ for some real number $0\ll a<1$, since $-(K+aS)$
is Klt, and $\sim_\R r S$ for some real $r>0$. Note also that a $D$-flip
is uniquely determined up to isomorphism by the class of $D$ up to
$\sim_\R$ and multiplication by positive real numbers (see
Corollary~\ref{uniqf}).

Interest in pl flips rests on the following result:

 \begin{rtheorem}\label{rth} Log flips exist in dimension $n$ provided
that:
 \begin{q}
 \item[$\PLF_n^{\mathrm{el}}$] {\em elementary\/} pl flips exist in dimension
$n$; and
 \item[$\ST_n$] special termination holds in dimension $n$.
 \end{q}
 \end{rtheorem}

We discuss Termination (ST) presently in Section~\ref{tt}, where we sketch
a proof of Reduction Theorem~\ref{rth}. In Special Termination~\ref{ster},
we also prove that (ST)$_n$ follows from LMMP in dimension $n-1$. By
our reduction, this is sufficient for the existence of $3$-fold and
$4$-folds log flips.

 \begin{cor}\label{lffromsf} In dimension $n\le4$, $\PLF_n^{\mathrm{el}}$
implies the existence of all log flips.
 \end{cor}

See the proof at the end of Section~\ref{tt}, p.~\pageref{pf13}. The
following inductive statement is similar in nature, but its proof is more
sophisticated. However, we are obliged to drop\footnote{Slava: every time
you say ``drop'', you introduce an ambiguity: it could mean that you omit
the condition in the assumptions, or in the conclusions, or both. In many
cases, I have not been able to sort out which you mean to clear up the
ambiguity. ``drop'' $\mapsto$ ``omit'' is better English.} the elementary
property in our approach to pl flips, because it is not preserved on
passing to covers (cf.\ Lemma~\ref{covtric}).

 \begin{intheorem}\label{ith} The statement
 \addcontentsline{toc}{subsection}{\ref{ith} Induction Theorem}
 \begin{qq}
 \item[$\PLF_n\sm$] {\em small\/} pl flips exist in dimension $n$
 \end{qq}
 follows from
 \begin{qq}
 \item[$\FGA_m$] the finite generation (f.g.)\ of certain
sheaves\footnote{Is singular or plural intended? It might be better
to say f.g.\ of algebras of type (FGA)$_{n-1}$.} of algebras
over algebraic varieties of dimension $m\le n-1$.
 \end{qq}

More precisely, for any pl flip of core dimension $d=n-s$, we need the
local version $\FGA_d\bir$ in dimension $d$. Moreover, for the existence
of these pl flips, it is enough to assume the local restricted version
$\RFA_{n,d}\bir$ in dimension $d$ (we refer forwards to
Definition\ref{rfad} and Conjecture~\ref{rfac} for $\RFA$).
 \end{intheorem}

Sections~\ref{divalg}--\ref{fgalg} explain all the statements $\FGA$,
$\RFA$ together with variations on them, and relations between them.

 \begin{cor}\label{mainc1} LMMP and $\FGA_m$ in dimension $m\le n-1$
imply the existence of log flips in dimension $n$. More precisely, for
the existence of these flips, it is enough to assume $\FGA_m\bir$ or even
$\RFA_{n,m}\bir$ in dimension $m\le n-1$.
 \end{cor}

See the proof at the end of Section~\ref{tt}. Since LMMP is known in
dimension $\le3$ \cite[Theorem~5.2]{sh96b}, for $3$-fold and $4$-folds
log flips, we can focus on (FGA) and (RFA).

 \begin{cor}\label{mainc2} $\FGA_m$ in dimension $m\le3$ implies the
existence of log flips in dimension $n\le4$. More precisely, for the
existence of these flips, it is enough to assume $\FGA_m\bir$ or even
$\RFA_{n,m}\bir$ in dimension $m\le n-1$.
 \end{cor}

 \begin{proof} Immediate by Corollary~\ref{lffromsf} and Induction
Theorem~\ref{ith}.
 \qed\end{proof}

 \begin{mtheorem}\label{mthm} $\FGA$ and $\RFA$ hold in low dimension:
 \addcontentsline{toc}{subsection}{\ref{mthm} Main Theorem}
 \begin{qq}
 \item[$\FGA_m$] holds for $m\le2$; and
 \item[$\RFA_{n,m}\bir$] holds for all $n\le4$ and $m\le n-1$.
 \end{qq}
 \end{mtheorem}

These are among the main results of the paper, and they imply

 \begin{cor} Log flips exist in dimension $\le4$.
 \end{cor}

 \begin{proof} Immediate by Corollary~\ref{mainc2} and the Main Theorem.
 \qed\end{proof}

The (RFA) statement of Induction Theorem~\ref{ith} is proved in
Section~\ref{divalg}, and the (FGA) statement in Section~\ref{fgalg}.
These two sections explain the role of pl flips in LMMP as a link
between the geometry and algebra of flips. Section~\ref{divalg} introduces
graded divisorial sheaves of algebras, and the restricted algebras
appearing in (RFA). Section~\ref{fgalg} introduces more general functional
algebras, and we state a conjecture that we need about finitely generated
algebras (FGA) of a certain type.

The Main Theorem needs more preparations; we carry these out for (FGA) in
Sections~\ref{satdes}--\ref{canbound}, and Section~\ref{canbound} contains
the proof of (FGA). For (RFA), Sections~\ref{apprx}--\ref{dstb} contain
preparations, and Section~\ref{mr} the proof.

\subsection{History} Kulikov, Reid, Mori, Kawamata, Tsunoda, Koll\'ar,
Kachi, Takagi.

 \begin{rem} Minor modifications of our arguments prove the existence of
3-fold and 4-fold log flips in the analytic category over a neighborhood
of any compact subset of $Z$.
\end{rem}

\subsection{Conventions} Since the construction of flips is local, we fix
$P\in S_\vee\subset X_\vee$ and view $f\colon X\to X_\vee$ throughout as a
germ$/P$. All other varieties and their objects are considered locally$/P$.
For example, if $D$ is a divisor on a variety $Y/X_\vee$ then $\linsys{D}$
denotes its linear system in a neighborhood of a fibre of $Y/P$.

$\Mov{D}$ and $\Fix{D}$ denote the {\em mobile\/} and {\em fixed\/}
components of an $\R$-Weil divisor $D$, that is, divisors $M=\Mov{D}$ and
$F=\Fix{D}$ such that $\linsys{D}=\linsys{M}+F$ and $\linsys{M}$ does not
have fixed components. These are well defined if $\linsys{D}\ne\emptyset$.
Then $D=M+F$ with $F\ge0$ and $M$ integral (Cartier in codimension~$1$). If
$\linsys{D}=\emptyset$ we set $F=+\infty$; then $M=-\infty$ and
$\Oh_X(M)=0$.

To say that an $\R$-divisor $D$ is nef always means in particular that
$D$ is $\R$-Cartier (compare {\bi}nef in Lemma~\ref{divb}). This last
condition holds for every $\R$-divisor if $Y$ is $\Q$-factorial.

The base field $k$ is of characteristic $0$. For example, $k=\C$.

\subsection{Temporary Appendix: {\bi}divisors}\label{app!b-d}
Much of the paper uses the language of {\bi}divisors; see \cite{sh96b} and
Appendix~\ref{app!b-d} for a discussion of these ideas.
$\sA=\sA^X=\sA(X,B)$ denotes the {\em discrepancy\/} or {\em coboundary
{\bi}divisor\/} of the pair $(X,B)$ for some $\R$-divisor $B$ on $X$.
(This is $R$ in \cite[Definition~1.1.4]{sh96b}, with $C=0$ as a
{\bi}divisor, and $C_X=B$ in the formula.) $\sB=\sB^X=\sB(X,B)=-\sA$
denotes the {\em codiscrepancy\/} or {\em pseudo-boundary\/} {\bi}divisor
of $(X,B)$.

The ``{\bi}'' of {\bi}divisor stands for {\em birational}, that is, a
{\em {\bi}divisor} is a system of divisors on all the different models
of $X$ compatible with birational equivalence. If $Y\broken X$ is an
isomorphism in codimension~1, we can walk a divisor from $X$ to $Y$ by
birational transform. Otherwise we can do various things, sensible or
otherwise, with the exceptional prime divisors. The notion of
{\bi}divisor is introducted in \cite[Section~1]{sh96a} to do this
systematically.\footnote{This appendix will be rewritten formally. What
follows is based on an e-mail correspondence with FAQ by Iskovskikh,
Kachi, Reid, etc.}

The notion of {\bi}divisor hides an exceptional divisor on each concrete
model, which may be extremely infinite.

The definition in \cite[Section~1]{sh96a} has at least one defect: ``the
group of {\bi}divisors is the Abelian group generated by prime Weil
{\bi}divisors''. Here ``generated'' is in the same sense as the formal power
series ring is generated by monomials. The group of {\bi}divisors is in fact
between the sum $\sum Z D_i$ and the product $\times Z D_i$ where $D$ runs
through prime {\bi}divisors (geometric or divisorial valuations). So, each
{\bi}divisor $D=\sum d_iD_i$ and this sum can be infinite, but on each
model $Y$ of $X$ its visible (divisorial) component $D_Y=\sum d_iD_i$ is
a usual divisor on $Y$, where now $D_i$ only runs through prime Weil
divisors on $Y$. Thus, the latter sum $D_Y$ is finite in the algebraic
category (locally finite in the analytic one). Then this condition is
satisfied on each model $Y$.

Thus, in a formal sense, a {\bi}divisor is a projective limit of
ordinary divisors.

 \paragraph{Frequently asked questions} 
 \addcontentsline{toc}{subsection}{Frequently asked questions}
 \begin{description}
 \item[Q] Give some examples of {\bi}divisor.
 \item[A] The total transform of a Cartier divisor, the birational
transform of a mobile linear system, the canonical class of $Y$, or
the discrepancy divisor $K_Y-K_XX$ over $X$ are divisors defined on
every model $Y\to X$ that are compatible with pushdown.
 \item[Q] Is the canonical class a bidivisor?
 \item[A] The canonical {\bi}divisor $\sK$ is a typical example. Just as
the usual canonical divisor, $\sK$ is associated by definition with a
rational canonical differential form $\om$ as follows. If $D_i$ is a usual
prime divisor on a model $Y$ then $d_i$ is the multiplicity of $K_Y=(\om)$
in $D_i$. Thus, on any model $Y$, $\sK_Y=K_Y=(\omega)$ (note that $\om$ is
well defined on any model). If in addition $K_X$ is $\Q$-Cartier, then
Reid's discrepancy $R=\sK_X-\overline{K_X}$ is a {\bi}divisors that is
exceptional and measures the deviation of $\sK_Y$ from being a Cartier
{\bi}divisor $\overline{K_X}$ over $X$.
 \item[Q] Can a {\bi}boundary have infinitely many components?
 \item[A] Yes, it can. Normally a log boundary $B$ is suppose to have
multiplicity~1 in all exceptional prime Weil divisors, but multiplicity
0 at almost all nonexceptional divisors. The discrepancy {\bi}divisor
has bigger and bigger multiplicities at exceptional divisors on higher
and higher models $Y/X$.
 \item[Q] Is the following interpretation right? Consider the category
$\bir/X$ of birational morphisms $Y\to X$. Then a Weil {\bi}divisor is
any map $\sD\colon Y\mapsto D_Y$ that assigns a Weil divisor $D_Y$ to
each $Y$, in a way that is compatible with push down (that is,
$f_*(D_{Y'})=D_Y$ for every morphism $f\colon Y'\to Y$ of $\bir/X$. For
example, we could choose some model $Y$ and a Cartier divisor $D_Y$ on
$Y$ that pushes down to $D$ on $X$, and just take $D_{Y'}=f^*D_Y$ for
every $f\colon Y'\to Y$, and the obvious pushdown from a hut
$Y'\swarrow W\searrow Y''$ for models not dominating $Y$. (Or maybe
$\bir/X$ is only considered as a limiting system? Are {\bi}divisors
equal (or identified) if they coincide from some $Y/X$ upwards?)
 \item[A] \dots? {\bi}divisors turn up at many points of the paper,
notably Definition~\ref{bssd}, the discussion after Theorem~\ref{fgal},
Definition~\ref{pbdad}, Section~\ref{rstr},
etc., and the correct interpretation of these matters can be determined
from the treatment given there. There are accompanying notions of
{\bi}nef, {\bi}semiample, {\bi}free, {\bi}Cartier, {\bi}divisorial sheaf,
{\bi}divisorial algebra, {\bi}restriction, etc. In short, it's a whole
bundle of fun.
 \end{description}

\section{Special termination} \label{tt}

Let $(X/Z,B)$ be a log pair such that
 \begin{itemize}
 \item $f\colon X\to Z$ is a proper morphism; and
 \item $(X,B)$ is log canonical.
 \end{itemize}
 Let $g\colon X\to Y/Z$ be a {\em birational contraction\/} of
$(X/Z,B)$ that is {\em log canonical\/}, that is, $g$ satisfies:
 \begin{qq}
 \item[log canonical] $K+B$ is numerically negative$/Y$.
 \end{qq}
 Equivalently, this is a contraction $\cont_F=g$ of an extremal face
$F$ of $\NEbar(X/Z$ with $(K+B)\cdot F<0$. We say that the
$(K+B)$-flip of $g$ is a {\em log canonical\/} flip (see
\cite[Section~5]{sh96b}.)

 \begin{warn} These flips are log flips in the sense of Section~\ref{int}
only if $K+B$ is Klt.\footnote{Note that \cite{sh96b} develops LMMP in
the log canonical category, more general than that of \cite{ko}; in
particular, its contractions may involve contracting a locus of log
Kodaira dimension $\ge0$ along its Iitaka fibration.}
\end{warn}

{\em Special termination\/} in dimension $n$ is the statement
 \begin{q}
 \item[$\ST_n$] for arbitrary $(X/Z,B)$ with $\dim X=n$, any chain of
successive log canonical flips in extremal faces $F$ with $|F|\cap
\rddown{B}\ne\emptyset$ terminates.
 \end{q}
 Here the {\em support\/} $|F|$ of an extremal face $F$ is the exceptional
subvariety of $\cont_F{}$, that is, the union of curves
$C/Z$ with $\cont_F{C}=\pt$

 \begin{exa}\label{inclus} Suppose that we have a chain of successive log
canonical flips of $(X/Z,B)$ in extremal rays $R$ with $S\cdot R<0$, where
$S$ is reduced and $0<S\le\rddown{B}$. Then $|R|\subset
\rddown{B}$ and $|R|\cap \rddown{B}=|R|\ne\emptyset$. If (ST)$_n$ holds
then any chain of this form terminates. We meet this situation below in
the proof of Reduction Theorem~\ref{rth}.
 \end{exa}

 \begin{sth}\label{ster}
 \addcontentsline{toc}{subsection}{\ref{ster} Special Termination}
LMMP in dimension $\le n-1$ implies $\ST_n$.
 \end{sth}

 \begin{rem} \label{sterem} For the proof of Reduction Theorem~\ref{rth},
we need the situation where
 \begin{itemize}
 \item $f\colon X\to Z$ is projective;
 \item $(X,B)$ is log terminal;
 \item $X$ is $\Q$-factorial; and
 \item the contractions of $(X/Z,B)$ and their flips are {\em extremal\/},
that is, they are contractions $g$ satisfying
 \begin{qq}
 \item[extremal] $\rho(X/Y)=1$.
 \end{qq}
 \end{itemize}
 Flips preserve this situation. Thus, in Special Termination~\ref{ster},
it is enough to assume log termination in LMMP (cf.\
Remark~\ref{logter} below).
 \end{rem}

 \begin{proof} First, as in \cite[proof of Theorem~4.1]{sh92}, after a
finite number of flips, we can assume that
 \begin{itemize}
 \item for flips, the support $|F|$ does not contain log canonical centers,
 \end{itemize}
 because the set of log canonical centers is finite and each flip in
$F$ eliminates any log canonical center in $|F|$.

We prove termination by induction on the dimension $\dim S=d\le n-1$ of
log canonical centers $S\subset \rddown{B}$ of $(X,B)$ that intersect
$|F|$. We state that such flips terminate, that is, after a finite number
of steps, they do not intersect the log canonical centers $S\subset
\rddown{B}$ of dimension $d$. In particular, this gives (ST)$_n$ for
$d=n-1$.

Thus by the above we know this for $d=0$. Suppose now that
 \begin{itemize}
 \item for flips, the support $|F|$ does not intersect any log canonical
center
$S\subset \rddown{B}$ of dimension $<d$.
 \end{itemize}
 For our applications, the only difficult case is $d=1$ and $n=4$: that
is, a curve $S\subset \rddown{B}$ in a 4-fold $X$.

Note that, by our assumptions, $S$ is a minimal center, and is normal as a
subvariety near $|F|$. By local adjunction, $(K+B)\rest S=K_S+B_S$ is
Klt near each point $P\in|F|\cap S$, and all the possible multiplicities
of $D_S$ belong to a finite subset of the real interval $[0,1)$. This
subset only depends on the local structure of $(X,B)$ near $S$, and
includes the multiplicities of $B$ near $P$. Moreover, under the
restrictions of Remark~\ref{sterem}, these multiplicities are all of the
form
 \[
 \frac{m-1}m+\sum\frac{l_i}m b_i,
 \]
 where $b_i$ are the multiplicities of $B$ and $m$ and $l_i$ are natural
numbers. This follows from \cite[Corollary~3.10 and Lemma~4.2]{sh92}.


In particular, there is only\footnote{Maybe something like: we introduce
the notation $\mu=\lld(S,B_S)$ (least log discrepancy); then in the interval
$[0,1-\mu]$, the starting model $(X,B)$ has only a finite set of \dots?}
a finite set of such numbers that are $\le1$ minus the least log
discrepancy of $(S,B_S)$ for the starting model $(X,B)$, in the Klt part
of $(S,B_S)$. Another approach is to let $(W,B_W)\to S$
be the Iitaka fibration of $S$, with $\dim W=n-2$, replace $S$ by $W$ and
use divisorial adjunction on $W$ for a log terminal resolution of
$S\subset \rddown{B}$. Then finiteness means the finiteness of such
models with the above multiplicities $\le1-\lld(S,B_S)$. The only
interesting case is $d=1$ and $n=4$: then $W$ is a log elliptic Iitaka
fibrations over a curve, with a positive lower bound for the least log
discrepancy and given boundary multiplicities.

Each flip restricts to a birational transformation of $(S/Z,B_S)$ or of
the Iitaka fibration $(W/Z,B_W)$. Unfortunately, restricted to these, it
may not be a flip; this happens exactly if the flipped contraction on
$(S^+/Z,B^+_S)$ or $(W^+/Z,B^+_W)$ is not small. In this case, it blows up
prime divisors that again have multiplicities of $B^+_S$ or $B^+_W$ in our
finite set, and $<1-{}$ the log discrepancy in them for the previous model
$(S/Z,B_S)$ or $(W/Z,B_W)$. However there only exist a finite number of
the latter transformations (cf.\ \cite[proof of Theorem~4.1]{sh92}). The
final monotonicity follows from adjunction and the opposite monotonic
property of log discrepancies for flips, strict\footnote{What? Presumably
strictly monotonic under some assumption is intended.} in $|F|$
\cite[2.13.3]{sh85}. Only such transformations occur in our main case
$d=1$ and $n=4$.

Then we use termination of log flips on $(S/Z,B_S)$ or $(W/Z,B_W)$ by LMMP.
 \qed\end{proof}

 \begin{rem}\label{logter} The termination at the end of the proof of
special termination is that of any chain of successive log flips. They are
{\em not necessarily extremal\/}, that is, they may contract extremal faces
of dimension $\ge2$. Using LMMP, we can reduce this to a chain of elementary
flips. Thus we need LMMP in dimension $n-1$ in an essential
way.
 \end{rem}

 \begin{pfof}{Reduction Theorem~\ref{rth}}
 \addcontentsline{toc}{subsection}{Proof of Reduction Theorem~\ref{rth}}
We follow the part of the argument of
\cite[proof of Reductions~6.4--5]{sh92} that concerns limiting flips, with
improvements by Koll\'ar and others \cite[(18.12.1.3--4)]{ko}. In this
reduction, each small contraction $\cont_R$ is elementary pl, and then
we discard all the reduced irreducible components $S_i$ in $B$ with
$S_i\cdot R\ge0$. Indeed, by construction, $B$ includes a reduced
component $S_i$ with $S_i\cdot R<0$. Thus log flips exists by (PLF)$_n$
and are the same as before discarding some of the $S_i$. They terminate by
Example~\ref{inclus} and (ST)$_n$ (cf.\ \cite[Corollary~4.6]{sh92}).
 \qed\end{pfof}

 \begin{pfof}{Corollary~\ref{lffromsf}}\label{pf13} Immediate by Reduction
Theorem~\ref{rth}, Special Termination~\ref{ster} and LMMP in
dimension $\le3$ \cite[Theorem~5.2]{sh96b}.
 \qed\end{pfof}

 \begin{pfof}{Corollary~\ref{mainc1}} Immediate by Reduction
Theorem~\ref{rth} and Induction Theorem~\ref{ith}, and by Special
Termination~\ref{ster}.
 \qed\end{pfof}

\section{Divisorial $\Oh_X$-algebras and flips}\label{divalg}

In this section we explain the relation between the geometry and algebra of
$D$-flips, and prove Induction Theorem~\ref{ith} in the case of (RFA)
algebras. We start with geometry. The following clumsy definition aims to
unify some standard geometric constructions, in particular, flips and
contractions.

 \begin{defn}\label{bssd}
 \addcontentsline{toc}{subsection}{\ref{bssd} Definition of bss ample (BSS)}
We say that a Weil $\R$-divisor $D$ on $X$ is {\em {\bi}sup-semiample\/}$/Z$
or {\em bss ample\/} if one of the following two equivalent conditions hold
 \begin{q}
 \item[\BSS] there is a {\em rational $1$-contraction\/} $g\colon X\broken
Y/Z$ and a numerically ample $\R$-divisor $H$ on $Y/T$ such\footnote{must
be $Y/Z$. $T$ and $Z$ seem to be interchangeable} that $D\sim_\R g^*H+E/Z$,
where $E$ is effective and {\em essentially exceptional\/} on $Y$; or
 \item[(ZDC)] there is a hut
 \[
 \renewcommand{\arraycolsep}{0.2em}
 \begin{matrix}
 &&W \\
 &^g\swarrow &&\searrow^h \\
 X\kern-0.6em &&&& \kern-0.6emY& /Z
 \end{matrix}
 \]
 with contractions $g,h$ and\footnote{Here the letter $g$ is used for two
different maps $g\colon X\broken Y$ and $g\colon W\to X$. Change one of
them to $i$ or $\ga$ or something.} a numerically ample $\R$-divisor
$H$ on $Y/T$ such\footnote{must be $Y/Z$.} that
 $g$ is birational,
 each $g$-exceptional divisor is {\em contractible\/} for $h$, and
 $g\1D+F\sim_\R h^*H+E/Z$, where $F,E$ are Weil $\R$-divisors on $W$ that
are exceptional on $X$ and $Y$ respectively,
 $E$ is effective, and
 the complement to $\Supp E$ in the fibre over each divisor in $Y$ is not
exceptional on $X$, so that in particular, $g(E)$ is essentially
exceptional on $Y$.
 \end{q}
 \end{defn}

 \paragraph{Proof--Explanation} By a {\em rational contraction\/}
$X\broken Y/Z$ we mean a composite $g:=h\circ g\1$ of two contractions as
in (ZDC). This is a dominant rational map with connected fibres $g(h\1P)$.
For any $\R$-Cartier divisor $H$, we can define a Weil $\R$-divisor
$g^*H:=g(h^*H)$ on $X$ that is independent of the Hironaka hut. In terms
of {\bi}divisors, $g^*H=\sH_X$, where $\sH=\overline{h^*H}$
\cite[Example~1.1.1]{sh96b}.

By a {\em rational\/ $1$-contraction\/} we mean a rational contraction
$g\colon X\broken Y$ such that
 \begin{itemize}
 \item $g$ does {\em not blow up\/} divisors {\em downstairs\/}: every
exceptional prime {\bi}divisor $E$ of $X$ is {\em contractible\/} on $Y$,
that is,
 \[
 \dim \cent_Y{E}<\dim E=\dim X-1.
 \]
 \end{itemize}
 Thus each hut in (ZDC) defines a rational 1-contraction and vice versa.

With respect to $g\colon X\broken Y$, we say that a divisor $E$ {\em on\/}
$X$ is {\em exceptional on $Y$\/} if it is contractible and {\em
nonhorizontal\/}, that is, has $g(E)\ne Y$; we only need this when the
rational contraction $g$ is not birational or equivalently, $h$ is not
birational. For a rational contraction of fibre type, it contracts to a
divisor again. In this case, it is {\em essentially exceptional\/} if its
support does not contain entire fibres over divisors in $Y$. The divisor is
{\em really exceptional\/} if its image is supported in codimension~$\ge2$.
For birational contractions all flavors of contractibility are equivalent.

The divisor $E$ in (ZDC) gives $E:=g(E)$ in the statement \BSS, since
$F$ is exceptional on $X$ and $g(E)$ is still essentially exceptional on
$Y$. Conversely, the equivalence $D\sim_\R g^*H+E/Z$ gives an equivalence
$g\1D+F\sim_\R h^*H+E/Z$ in which $E:=g\1E\ge0$ and $F$ are still
essentially exceptional on $Y$ and $X$ respectively. \qed\par\medskip

Each bss ample divisor is effective modulo $\sim_\R$:
 \[
 D\sim_\R h^*H+E\sim_\R h^*H'+E\quad\text{and}\quad\ge0,
 \]
 where the $\R$-divisor $H'\sim_\R H/Z$ is effective. Thus not every
divisor
$D$ is bss ample. However, the following uniqueness result holds.

 \begin{prop}\label{uniqbss} Suppose that $D$ is bss ample. Then the
rational contraction $g$ (up to isomorphism) and the divisor $E$ are
uniquely determined by $(X/Z,D)$. The divisor $H$ is defined up to
$\sim_\R/Z$.
 \end{prop}

 \begin{proof} Since $E\sim_\R D-\sH_X/Z$ is effective and essentially
exceptional on $Y$, by \cite[Negativity~1.1]{sh92}, it is unique if
$\sH$ is unique up to ${\sim_\R}/Z$. Thus it is enough to establish the
uniqueness of $\sH$. If $\sH'$ corresponds to another $g', Y'$ and
$E'$, then $\sH+\sE\sim_\R\sH'+\sE'/Z$, where $\sE,\sE'$ are effective
and essentially exceptional on $Y,Y'$ respectively. Note that $\sH'$ is
free modulo ${\sim_\R}/Z$, $\sH$ is numerically trivial$/Y$,
$\sH-\sH'\equiv\sE'-\sE/Z$ is numerically seminegative over $Y$, and the
negative components of $\sE'-\sE$ are exceptional on $Y$. Thus if $h$ is
not birational, that is, $h$ is of fibre type, the whole of $\sE'-\sE$ is
exceptional, and $\sH'\equiv0$ over the generic point of $Y$. Moreover,
the latter holds over generic points of every divisor in $Y$, because we
can assume that $\sH'$ is effective without fixed components and
nonhorizontal. Hence $\sE'-\sE\equiv 0$ over generic points of divisors.
By (ZDC) $\sE$ does not support entire fibres over generic points of
divisors. Hence $\sE'-\sE\ge0$ over generic points of divisors, and the
negative part of $\sE'-\sE$ is really exceptional. This also holds if $h$
is birational.

Hence, again by \cite[Negativity~1.1]{sh92}, $\sE'-\sE\ge0$ or $\sE'\ge
\sE$ since, for every model $W/Y$ and $/Y'$ of $X$, we have that
$-(\sH_W-\sH'_W)/Y$ is nef and $\sE'-\sE$ is effective on prime divisors
that are really nonexceptional on $Y$, in particular on
$\sE'$. Similarly, $\sE\ge\sE'$. Hence $\sE'=\sE$ and $\sH\sim_\R\sH'$.
 \qed\end{proof}

 \begin{cor}\label{uniqbssc} If $D'\sim_\R rD$ for some positive
$r\in\R$, then $D'$ is also bss ample with the same rational
$1$-contraction $g$, $E'=r E$, and $H'\sim_\R rH$. Thus bss ample is a
property of the ray\/ $\R^{>0}\cdot D/{\sim_\R}$ in the space of
divisors.
 \end{cor}

 \begin{proof} Immediate by Proposition~\ref{uniqbss}, because $rE>0$ and
$H'=r H$ is ample.
 \qed\end{proof}

The most important applications have $E=0$; then we say that $D$ is {\em
{\bi}semiample\/}$/Z$, that is, semiample\footnote{Slava: You might
consider changing ``semiample'' $\mapsto$ ``eventually free'', which is
more logical terminology, and sounds better in English to me; at two
points later you say ``very semiample'' for ``free'', which seems quite
bizarre.} on some model $W$ of $X$ after adding certain divisors that are
exceptional on $X$. For example, if $Y=\pt$, then $H=0$, $E=0$,
$D\sim_\R\sH=0$.  

 \begin{cor} \label{uniqf} Suppose that $f\colon X\to Z=T$ is a birational
contraction. Then any $D$-flip is uniquely determined by $D$ given up to
$\sim_\R$ and multiplication by positive real numbers.
 \end{cor}

 \begin{lem}\label{exfl} The $D$-flip of a contraction $f\colon X\to T$
exists if and only if
 \begin{q}
 \item[\BSA] the divisor $f(D)$ on $T/T$ is bss ample$/T$,
 \end{q}
 and then the flipped contraction\/ $g\1\colon Y\to Z=T$ is small, and the
flipping modification is $g\circ f$; we have $E=0$ in \BSS.

The same holds for $D$ if $f$ is small.
 \end{lem}

In \BSA\ $E=0$, so $f(D)$ is {\bi}semiample$/T$.

 \begin{proof} If the $D$-flip exists, then it defines a small contraction
$Y=X^+/T$ with $H=D^+=g(f(D))$ numerically ample$/T$ and a flipping
modification $X\broken X^+=Y/T$. Then $f(D)$ is bss ample with the
rational $1$-contraction $g\colon T\broken X^+$, given $H$ and $E=0$.
Indeed, $f(D)=g^*H=g^*D^+$.

If $f(D)$ is bss ample, then $Y/T=Z$ is small, $E=0$ and
 \[
 D^+=g(f(D))\sim_\R g(g^*H)=H.
 \]

The same arguments apply to $D$ itself for any small $f$.
 \qed\end{proof}

 \begin{pfof}{Corollary~\ref{uniqf}} Immediate by Lemma~\ref{exfl},
Proposition~\ref{uniqbss} and Corollary~\ref{uniqbssc}.
 \qed\end{pfof}

 \begin{cor} \label{bssasa} Suppose that $D$ is nef$/Z$. Then $D$ is bss
ample if and only if it is semiample\/ {\em
\cite[Definition~2.5]{sh96b}\/}. In addition $E=0$. In particular, every
semiample divisor $D$ is bss ample.
 \end{cor}

 \begin{proof} We can replace $(X/Z,D)$ by $(W/Z,g^*D)$ in (ZDC). Then
$E=0$ by \BSS\ and \cite[Negativity~1.1]{sh92}. Thus $D\sim_\R h^*H$,
$\Dbar\sim_\R\sH$ and $h$ contracts all curves $C$ on $W/Z$ and
$X/Z$ with $D\cdot C=0$.
 \qed\end{proof}

Thus if $D$ is bss ample, Proposition~\ref{uniqbss} define a unique
decomposition
 \[
 D=D^m+D^e.
 \]
 as a sum of two Weil $\R$-divisors $D^e:=E\ge0$ and $D^m:=D-E$. If $X/\pt$
is a surface, this is a {\em Zariski decomposition\/}; then $g$ in \BSS\ is
a regular contraction. It is also useful to consider the unique
{\bi}divisors $\sD^m\sim_\R\sH$ with $\sD^m_X=D^m$, and $\sD^e=E$
(considered as a {\bi}divisor) with $\sD^e_X=D^e$. The divisor
 \begin{equation}
 D\sm=\sum(\mult_{D_i}{D})D_i, \quad\hbox{for $D_i$ not in
$\Supp{D^e}=\Supp E$}
 \label{eq!Dsmall}
 \end{equation} is also important. We define generalizations of these for
any $D$ presently in Section~\ref{fgalg}.

Again if $Y$ is a point, then $D^e=\sD^e=0$, $D\sm=D^m\sim_\R0$ and
$\sD^m\sim_\R0$.

 \subsection{The algebra of flips} Here we interpret a flip in terms of a
certain graded functional sheaf of algebras. Its {\em homogeneous}
elements (that is, elements of {\em pure degree}) are sections of {\em
divisorial\/} $\Oh_X$-sheaves
$\Oh_X(D)=\Oh_X(\rddown{D})$ with
 \[
 \Ga(U,\Oh_X(D))=\bigl\{a\in k(X) \bigm| (a)+D\ge0 \bigr\},
 \]
 where $D$ is a Weil $\R$-divisor, and $k(X)$ denotes the field of
rational functions of $X$; we write $(a)$ for the divisor of a nonzero
element
$a\in k(X)$, with the convention that $0$ has divisor $(0)=+\infty$. An
axiomatic treatment of divisorial sheaves is given by Reid
\cite[Appendix to \S1]{r80}. The sheaves $\Oh_X(D)$ are {\em functional
sheaves\/}, that is, subsheaves of the constant sheaf $k(X)$, and are
coherent. They are well defined if $D$ is {\em nonsingular\/}, that is,
every generic point of $\Supp D$ is nonsingular in $X$. (At a prime divisor
$D_i$ where $X$ has a nonnormal singularity, the above sheaf condition means
that $f$ is regular\footnote{This is of course wrong. It's not very subtle
either.} at $D_i$\/.) However, since $X$ is assumed to be normal, we can
disregard these subtleties. Any Weil $\R$-divisor on a normal variety $X$ is
nonsingular, because $S$ is nonsingular in codimension~$1$.

 \begin{defn}\label{divald} Let $g\colon X\to Z$ be a proper morphism. We
define the {\em $\N$-graded $\Oh_Z$-algebra\/} of a Weil $\R$-divisor $D$
on $X$ as
 \[
 \dival_g{D}=\dival_{X/Z}{D}
\eqdef \directsum_{i=0}^\infty g_*\Oh_X(iD).
 \]
 This is an $\N$-graded {\em functional\/} $\Oh_Z$-subalgebra of the
constant $\Oh_Z$-algebra
 \[
 k(X)_\bull=\directsum_{i=0}^\infty k(X).
 \]
 Thus the natural multiplication in $\dival_{X/Z}{D}$ is induced by
multiplication in $k(X)$: for homogeneous elements
$a\in\Oh_X(iD)\subset k(X)$ and $b\in\Oh_X(jD)\subset k(X)$, we have
$ab\in\Oh_X((i+j)D)\subset k(X)$. Indeed, $(ab)+(i+j)D=(a)+i D+(b)+j
D\ge0$.

An $\N$-graded $\Oh_Z$-algebra is {\em (relatively) divisorial} for
$X/Z$ if it is isomorphic to $\dival_{X/Z}{D}$ for some $D$. An
$\N$-graded
$\Oh_Z$-algebra is {\em (relatively) {\bi}divisorial} for $X/Z$ if it is
isomorphic to $\dival_{Y/Z}{D}$ for some model $Y/Z$ of $X/Z$ and for some
divisor $D$ on $Y$. Section~\ref{fgalg} generalizes these algebras and the
notation (see Definition~\ref{pbdad} and Example~\ref{rdivalg} below).
 \end{defn}

 \begin{rem} If $X=Z$, so that $X/Z=X/X$ is the identity, each
homo\-geneous component $g_*\Oh_X(iD)=\Oh_X(iD)$ is a divisorial sheaf.
The algebras $\dival_{X/X}{D}$ have a similar axiomatic description. If
$X/Z$ is a birational contraction, then $g_*\Oh_X(iD)$ is a {\em
{\bi}divisorial} sheaf. They also have an axiomatic description, as do the
{\bi}divisorial algebras.
 \end{rem}

 \begin{exa}
 If $D$ is a Cartier divisor on $X$, its algebra $\sR_{X/X}(D)$ is
canonically isomorphic to the {\em tensor algebra\/} of $D$
 \[
 \sT(\Oh_X(D))=\directsum_{i=0}^\infty\Oh_X(D)^{\tensor i}.
 \]
 This graded algebra is defined for any $\R$-divisor $D$. However the
above isomorphism does not hold for general $\R$- or $\Q$-Cartier divisors
$D$, even in the local case $X/X$, because the tensor algebra is f.g.,
and, moreover, generated by homogeneous elements of degree $1$. For any
$\R$-divisor $D$, the divisorial algebra $\sR_{X/X}(D)$ is canonically
isomorphic to the {\em reflexive tensor algebra\/} of $D$
 \[
 \sA(\Oh_X(D))=\sT(\Oh_X(D))^{\vee \vee}.
 \]
 But this algebra may be infinitely generated.

If $X/Z$ is not the identity then divisorial and {\bi}divisorial algebras
are more complicated. But they are sometimes f.g., as we see in
Section~\ref{fgalg}.
 \end{exa}

 \begin{defn}
 An $\Oh_Z$-algebra has {\em finite type\/} if it is {\em finitely
generated\/} or {\em f.g.\/} The latter means that it is locally$/Z$
generated as an $\Oh_Z$-algebra by a finite set of sections ({\em
generators\/}).
 \end{defn}

 \begin{exa} \label{flipal} Let $f\colon X\to T$ be a birational
contraction and $D$ an $\R$-Weil divisor. Then the {\em flipping} algebra
of
$(X/T,D)$ is the divisorial algebra
 \[
 \flpal_{X/T}{D}=\dival_{T/T}{f(D)}.
 \]
 For any $\Q$-divisor $D$, by Corollary~\ref{fgflpal}, the $D$-flip exists
if and only if $\flpal_{X/T}{D}$ is f.g.

Note that if $f$ is small then
 \[
 \flpal_{X/T}{D}\iso\dival_{T/T}{D}
 \]
 by Example~\ref{smbiriso}, (b). But this is not true in general. For
example, the isomorphism does not hold if $f$ is divisorial and $D=H$ is
a hyperplane section of $X/T$ (use Theorem~\ref{fgal} below).
 \end{exa}

 \begin{exa}[Stupid Example]\label{exa!st} Let $P$ be a point on a
complete curve $C$. Take $D=d P$ with nonnegative $d\in\R$. Then
$\dival_{C/\pt}{D}$ has finite type if and only if $d$ is rational (cf.\
Theorem~\ref{fgal} below). Nonetheless $D$ is semiample for any real
$d\ge0$ and bss ample by Corollary~\ref{bssasa}.

Note that $\dival_{C/\pt}{D}$ is also f.g.\ for any real number $d<0$
because then it is {\em trivial\/}, that is, it has no sections of {\em
positive degree\/}.
\end{exa}

Thus f.g.\ is somewhat different from ample, or from bss ample, although
the notions are intimately related. Here the important phenomena are
rationality of multiplicities and nonvanishing in positive degrees.
However, to explain f.g.\ in geometric terms of bss ampleness, we need
some preparations.

 \begin{thm} \label{fgal} The divisorial algebra $\sR=\dival_{X/Z}{D}$ is
f.g.\ and {\em globally almost generated\/} if and only if $D$ is bss
ample$/Z$ and $D\sm$ is a $\Q$-divisor (see (\ref{eq!Dsmall})); this last
{\em rationality condition\/} holds in particular if $D$ is a $\Q$-divisor.

Moreover, if\/ $\sR$ is f.g., then, in the notation of
Definition~\ref{bssd}, $Y/Z\iso\Proj_Z{\sR}$, $E$ is the {\em stable
base divisor\/} of $\sR$ and\/ $\Supp E$ is the {\em stable divisorial
base locus\/} of\/ $\sR$.
 \end{thm}

The proof is given on p.~\pageref{pf!fgal}.

A divisorial algebra $\sR=\dival_{X/Z}{D}$ is {\em globally almost
generated\/} or {\em g.a.g.\/} if there is a natural number $N$ and
sections of $f_*\Oh(ND)$ that generate the algebra $\dival_{X/X}{ND}$ in
codimension~$1$ except for a divisorial subset that is exceptional on
$Y=\Proj_Z{\sR}$. (In this definition, we need to assume that $\sR$ is
finitely generated. But one can generalize it to get rid of this
assumption.) In particular, this implies that $\sR$ is {\em nontrivial\/},
that is, it has a nonzero element of some positive degree
$d>0$. For a functional algebra, this implies that the algebra has a
nonzero element in each degree $i$ {\em divisible by\/} $d$. By the
theorem, we can define divisors $E,D^m$ and {\bi}divisors $\sD^m,\sD^e$,
etc., for $\sR$. These are well known: $E$ is the {\em stable base
divisor\/} of $D$, and it gives the {\em stable divisorial base locus\/}
$\Supp E$ as the above exceptional divisorial subset. Either of these can
be referred to $D$ or to the {\em divisorial algebra\/} $\sR$, but as an
algebra associated with $D$, not an abstract algebra (cf.\ Truncation
Principle~\ref{trprinc}).

Note that if we replace bss ample by semiample in the theorem, we should
replace g.a.g.\ by the usual global generation of the algebra
$\dival_{X/X}{ND}$. The next result shows under what circumstances we can
add an effective divisor to the mobile part of a multiple linear system.

 \begin{lem}\label{adjd} Let\/ $D$ be a semiample\/ $\Q$-divisor, and
suppose that\/ $E$ is an effective Weil\/ $\Q$-divisor that is not
contractible, or is exceptional but not essentially
exceptional\footnote{See the explanation after \ref{bssd}.} for the
contraction given by\/ $D$. Then for some natural numbers\/ $N\gg0$ and\/
$M$, the base locus\/ $\Bs{(ND+M E)}$ is a strict subset of\/
$\Supp E$, that is,\/
 \[
 \Supp{\Fix{(ND+ME)}}\subsetneq\Supp E.
 \]
 \end{lem}

 \begin{add} Moreover, if $E$ is integral, we can take $M=1$ except for
the following two cases:
 \begin{itemize}
 \item $E$ has no contractible or nonhorizontal components,
$\nu(D)=\dim X-1$ (the numerical dimension) and the generic fibre of the
contraction is a curve of genus $g\ge1$; then we can take $M=g+1$; or
 \item $E$ is contained in multiple fibres of the contraction; then we can
take $M$ to be the least common multiple of the multiplicities.
 \end{itemize}
 \end{add}

 \begin{proof} We can assume that all components of $\Supp E$ have a
single irreducible image for the contraction $h\colon X\to Y/Z$ given by
$D$. Then we have three cases:

\case{$E$ is not contractible and horizontal} Then $\nu(D)=\dim
\Supp E=\dim X-1$. The generic fibre of $h$ is a curve of genus $g$, and
$ND+ME$ cuts out a divisor of positive degree $\ge M$. Thus we have the
required result in the generic curve by RR for curves. This implies the
lemma in this case for $N\gg0$.

\case{$E$ is not contractible and not horizontal} Then $\nu(D)=\dim X$ and
$h$ is birational. In this case the lemma follows from the general RR
(compare Reid \cite[proof of Lemma~(1.6)]{r80}).

\case{$E$ is contractible but not essentially exceptional} This means that
$\nu(D)<\dim X$, that is, $h$ is fibred, and $\Supp E$ forms the fibre
over the prime divisor $E'=g(\Supp {E})$. Let $M$ be the lcm of the
multiplicities of $g^*E'$. Then there exists a natural number $m$ such
that $ME-mg^*E'$ is effective and essentially exceptional. Then we obtain
this case from the second one with $E=mE'$.
 \qed\end{proof}

On the other hand, in certain cases we can't add an effective divisor to
the mobile part of a multiple of the linear systems.

 \begin{prop}\label{isom} Let $g\colon X\broken Y/Z$ be a rational
$1$-contraction, $D$ an $\R$-Cartier divisor on $Y$, and $E$ an effective
divisor on $X$ that is essentially exceptional on $Y$. Then
$g$ induces an isomorphism
 \[
 g^*\colon i_*\Oh_Y(D)\to f_*\Oh_X(g^*D+E),
 \]
 where $i\colon Y\to Z$.
 \end{prop}

This statement has the following applications, that can also be proved
directly and generalized:
 \begin{exa}\label{smbiriso} \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item If $g\colon X\broken X^+/T$ is a $D$-flip$/Z$ and $X/T$ is small
then $g^*$ induces an isomorphism
 \[
 g^*\colon f^+_*\Oh_Y(D^+)\to f_*\Oh_X(D)
 \]
 where $D^+=g(D)$, $f\colon X\to Z$ and $f^+\colon X^+\to Z$. Indeed,
$E=0$ in this case. The proof is a direct verification; it is enough to do
it for $Z=T$.
 \item Moreover, we can replace the $D$-flip by any small birational
trans\-formation, that is, assume that it is a $1$-contraction in each
direction. Here we can omit the assumption that $D$ is $\R$-Cartier.
 \end{enumerate}
 Thus in either case $g^*$ induces an isomorphism of $\Oh_Z$-algebras
 \[
 g^*\colon \dival_{X^+/Z}{D^+}\to\dival_{X/Z}{D}
 \]
 because $g^*$ is compatible with multiplication.
 \end{exa}

 \begin{lem}\label{negat} Let
 \[
 \renewcommand{\arraycolsep}{0.2em}
 \begin{matrix}
 &&W \\
 &^g\swarrow &&\searrow^h \\
 X\kern-0.6em &&&& \kern-0.6em Y& /Z
 \end{matrix}
 \]
 be a hut with contractions $g,h$, where $g$ is birational, and $D$ an
$\R$-Cartier divisor on $W$ such that
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item each prime divisor $E_i$ with negative multiplicity in $D$ is
exceptional on $X$ and is contractible$/Y$; and
 \item $-D$ is nef at general curves of $E_i/h(E_i)$ for all $E_i$ in (i).
 \end{enumerate}
 Then $D$ is effective.
 \end{lem}

 \begin{proof} This follows from the negativity of a proper modification
\cite[Negativity~2.15]{sh95}. Indeed, each prime $E_i$ with negative
multiplicity has a sufficiently general curve$/Z$ on which $D$ is
numerically nonpositive, namely, a general curve in the generic fibre
of $E_i/h(E_i)$. By (i--ii), this exists because $E_i$ is contractible$/Y$.
 \qed\end{proof}

 \begin{pfof}{Proposition~\ref{isom}} First, $g^*$ induces an inclusion
because, for any nonzero rational function $s\in i_*\Oh_Y(D)\subset k(Y)$,
the function $g^*s\in k(X)$ belongs to $f_*\Oh_X(g^*D+E)$. Indeed,
$(s)+D\ge0$. Hence
 \[
 (g^*s)+g^*D+E\ge g^*(s)+g^*D=g^*((s)+D)\ge0.
 \]

Surjectivity uses Lemma~\ref{negat}. For this, we decompose $g$ into
$h\circ g\1$ as in the hut of Lemma~\ref{negat}. Then we need to show that
 \[
 (g\1)^*\circ h^*\colon i_*\Oh_Y(D)\to f_*\Oh_X(g_*(h^*(D))+E).
 \]
 is surjective. Thus take a nonzero section $s\in f_*\Oh_X(g_*(h^*(D))+E)$.
Then $(s)+g_*(h^*(D))+E\ge0$. Hence $(g^*s)+h^*(D)+g\1E\ge0$ in any prime
divisor $E_i$ nonexceptional on $X$. We claim that the horizontal part $H$
of the latter divisor is $0$. We can of course suppose that $W$ is
nonsingular, so that any $\R$-divisor is $\R$-Cartier. In addition, $H$
satisfies Lemma~\ref{negat}, (i), because $h$ contracts each $E_i$
exceptional on $X$. Finally, since $g\1E$ is exceptional on $Y$, then
$H\equiv (g^*s)\equiv 0$ over the generic point of $Y=h(H)$. Hence $H\ge0$
by Lemma~\ref{negat}, and $H=0$ because it is $\equiv 0$ over the generic
point of $Y$.

Therefore $g^*s$ does not have horizontal zeros or poles. Thus
$g^*s=h^*s'$ for some rational function $s'\in k(Y)$. Moreover,
$(g\1)^*\circ h^*s'=(g\1)^*g^*s=s$, and
 \begin{align*}
 h^*((s')+D)+g\1E&=(h^*s')+h^*(D)+g\1E \\
 &=(g^*s)+h^*(D)+g\1E\ge0
 \end{align*}
 in any prime divisor $E_i$ nonexceptional on $X$. In particular in any
prime $E_i$ over a prime divisor $h(E_i)$ but not in $g\1E$ and
nonexceptional on $X$. Since $g\1E$ is essentially exceptional, such
$E_i$ exists over each prime divisor in $Y$. Hence $(s')+D\ge0$ and
$s'\in i_*\Oh_Y(D)$.
 \qed\end{pfof}

 \begin{exa}\label{smbiriso2} Let $g\colon X\broken X^+/T$ be a
$D$-flip$/Z$, where $X/T$ is a $D$-contraction$/Z$, that is, $D$ is
numerically negative$/T$. Then $g^*$ induces a canonical isomorphism
 \[
 g^*\colon f^+_*\Oh_Y(D^+)\to f_*\Oh_X(D)
 \]
 where $D^+=g(D)$, $f\colon X\to Z$ and $f^+\colon X^+\to Z$ (see
Example~\ref{smbiriso}, (a)). Indeed, $D=g^*D^++E$, where $E$ is effective
by Lemma~\ref{negat}. We can decompose $g$ into birational contractions $g$
and $h$ as in Lemma~\ref{negat}. Then $g^*D=h^*D^++E_W$, where $E_W$ is
exceptional on $X^+$ and $-E_W=h^*D^+-g^*D$ is nef$/X^+$. Thus the lemma
applied to $X^+\gets W\to X^+$ give $E_W\ge0$. Hence $E=g(E_W)\ge0$, and
the required isomorphism follows by Proposition~\ref{isom}.

Thus $g^*$ induces a canonical isomorphism of $\Oh_Z$-algebras
 \[
 g^*\colon \dival_{X^+/Z}{D^+}\to
\dival_{X/Z}{D}.
 \]
 In fact we can replace the $D$-flip $X^+/T$ by any small contraction
(cf.\ Example~\ref{smbiriso}, (b)).
 \end{exa}

By the following result, Proposition~\ref{isom} can also be applied in
cases where $g^*D+E$ is replaced by a linearly equivalent divisor.

 \begin{lem}\label{sim} Let $D\sim D'$ be two divisors that are linearly
equivalent; that is, $D=D'+(s)$, where $s\in k(X)$ is a nonzero rational
function. Then the multiplication map $t\mapsto st$ gives an isomorphism
 \[
 \sim\colon f_*\Oh_X(D) \to f_*\Oh_X(D')
 \]
 and an isomorphism of $\Oh_Z$-algebras
 \[
 \sim \colon \dival_{X/Z}{D}\to \dival_{X/Z}{D'}.
 \]
 \end{lem}

These isomorphisms are not unique, and depend on the choice of $s$. But
they are compatible with multiplication provided we take $s^i$ for
$iD'\sim iD$.

 \begin{proof} Immediate from the definitions.
 \qed\end{proof}

 \begin{cor}\label{isoc} Under the assumptions of Proposition~\ref{isom},
suppose that $D'\sim g^*D+E$. Then $g$ and $\sim$ induce isomorphisms
 \[
 \sim g^*\colon i_*\Oh_Y(D)\to f_*\Oh_X(D')
 \quad\hbox{and}\quad
 \sim g^*\colon \dival_{Y/Z}{D}\to\dival_{X/Z}{D'}.
 \]
 \end{cor}

 \begin{proof} Immediate by Proposition~\ref{isom}, since $g^*$ is
compatible with multi\-plication.
 \qed\end{proof}

However, we can't replace $\sim$ by $\sim_\R$ in Lemma~\ref{sim} and
Corollary~\ref{isoc}, or even by $\sim_\Q$. This last equivalence only
gives a quasi-isomorphism of algebras (see Example~\ref{rdivalg}). Together
with an application of the Truncation Principle~\ref{trprinc} (see
Corollary~\ref{stupidc}), this is enough to prove Theorem~\ref{fgal}. For
this, we first clarify one definition.

 \begin{defn}[compare {\cite[Definition~2.5]{sh96b}}]\label{qlin} We say that
$D$ and $D'$ are $\Q$-{\em linearly equivalent\/}, and write
$D\sim_\Q D'$, if $D-D'$ is a $\Q$-{\em principal\/} divisor, that is, a
rational linear combination of principal divisors. Equivalently,
$iD\sim iD'$ for some nonzero integer $i$.

Note that $\sim_*$ does not need $/Z$ (even if we have one!).
 \end{defn}

Since every principal divisor is integral, the $\R$-vector space of
$\R$-principal divisors is defined over $\Q$, and all its
$\Q$-divisors are
$\Q$-principal. Thus any two $\Q$-divisors $D$ and $D'$ are
$\sim_\R$ if and only if they are $\sim_\Q$. Note also that if
$D\sim_\Q D'$ and $D$ is a $\Q$-divisor, then $D'$ is also a
$\Q$-divisor.

 \begin{prop}\label{lin0} Suppose that $D$ is bss ample$/Z$, and $D\sm$ is
a\/ $\Q$-divisor (see (\ref{eq!Dsmall})). Then there is a\/ $\Q$-divisor\/
$H$ on $Y$ and a natural number $I$ such that
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item $H$ is numerically ample$/Z$;
 \item $D\sim_\Q g^*H+E$; and moreover,
 \item $i D\sim i g^*H+i E$ if and only if $I\mid i$; equivalently,
 \item for all $i\gg0$, the $i$th component $\sR_i$ of\/
$\sR=\dival_{X/Z}{D}$ is nonzero.
 \end{enumerate} The natural number $I$ is unique, and $H$ is unique up to
$\sim$.
 \end{prop}

 \begin{add} The divisorial algebra $\sR=\dival_{X/Z}{D}$ is equal to its
truncation
 \[
 \sR^{[I]}=\sum_{I\mid i}\sR_i,
 \]
 and isomorphic to the truncation
 \[ (\dival_{Y/Z}{H})^{[I]},
 \]
 where $[I]$ means that we preserve degrees.
 \end{add}

Note that $(\dival_{Y/Z}{H})^{[I]}$ is usually not equal to
$\dival_{Y/Z}{H}$ for $I\ge2$.

 \begin{lem}\label{lin0l} Let $f\colon X\to T$ be a contraction and let
$D,E$ be $\R$-divisors on $T,X$ respectively such that
 \begin{itemize}
 \item $D$ is $\R$-Cartier;
 \item $E$ is vertical, and
 \item $E\sim_\R f^*D$.
 \end{itemize}
 Then there is a unique $\R$-divisor $F$ on $T$ such that
 \begin{itemize}
 \item $F\sim_\R D$,
 \item in particular, $F$ is also $\R$-Cartier; and
 \item $E=f^*F$.
 \end{itemize}
 \end{lem}

 \begin{proof} We can assume that $D=0$. Indeed, if we replace $D$ by
$E-f^*D$ and $D$ by $0$, then the required $F$ is $F-D\sim_\R0$.

Thus $E$ is $\R$-principal and vertical. In other words, $E=\sum d_i(f_i)$,
where all $d_i\in\R$ and all $f_i$ are nonzero rational functions on $X$.
We need to find a presentation in this form with functions $f_i=f^*g_i$,
where the $g_i$ are nonzero rational functions on $T$. Indeed, we can then
take $F=\sum d_i(g_i)$.

Since $E$ is vertical, a presentation of the required form is a
presentation with $\Q$-linear independent and nonzero $(f_i)$ over the
generic point of $T$, that is, in the horizontal components.

A presentation of this form exists by the following inductive contraction.
Suppose that $\sum r_i(f_i)=0$ over the generic point of $T$, where all
$r_i\in\Q$ and one of them is nonzero, say, $r_0\ne0$. Then the same holds
for some integral coefficients $r_i$. Hence $\Bigl(\prod f_i^{r_i}\Bigr)=0$
over the generic point of $T$ and the rational function $g=\prod f_i^{r_i}$
does not have horizontal zeros or poles over the generic point of $T$, that
is, $0$ over the generic point of $T$. Thus in our presentation, we can
replace $(f_0)$ by a rational linear combination of other divisors $(f_i)$
and $(g)$. But $(g)=0$ over the generic point of $T$.

Uniqueness follows because $f^*$ is injective, since $f$ is surjective.
 \qed\end{proof}

 \begin{pfof}{Proposition~\ref{lin0}} We first check that the extension
$D_W\sm=\sD_W\sm$ is also a $\Q$-divisor. Indeed, by definition
$D_W^m\sim_\R h^*H$. Thus $D_W^m\sim_\R0$ over the generic point of
$T$ and $\equiv 0/T$; again by definition, $D_W^m=D_W\sm$ outside
$\Supp E$. On the other hand, over the generic point on $Y$,
$E$ is $0$ and the prime divisors $E_i$ with nonrational multiplicities in
$D_W^m$ are exceptional on $X$ and contractible on
$Y$. They are thus uniquely determined by the linear equations with
rational coefficients $D_W^m\cdot C_i=0$ for general curves $C_i$ of
$E_i/h(E_i)$. They are unique and thus rational by Lemma~\ref{negat}.
Therefore $D_W^m$ is rational and $D_W^m\sim_\Q0$ over the generic
point of $T$. In particular, $D_W^m$ is vertical if $D^m$ is.

Thus over the generic point of any prime divisor in $Y$, the prime
divisors $E_i$ with nonrational multiplicities in $D_W^m$ and not in
$\Supp E$ are exceptional on $X$ and exceptional contractible on $Y$.
Again they are uniquely determined by the linear equations with rational
coefficients $D_W^m\cdot C_i=0$ for general curves $C_i$ of
$E_i/h(E_i)$. Hence they are rational and $D_W^m$ is rational over such
points because $E$ is exceptionally singular.

Since $D_W^m\sim_\Q0$ over the generic point of $T$ and $E$ is vertical,
we can suppose that the divisors $D_W^m$ and $D$ are themselves
vertical$/Y$ up to $\Q$-linear equivalence. By Lemma~\ref{lin0l}, we can
replace $H$ by an $\R$-linearly equivalent divisor such that $D_W^m=h^*H$
and it is also ample$/Z$. Then $H$ is also rational because $h^*$ is
$\Q$-linear. This is our choice of $H$.

Take $I$ as the minimal natural number such that $I D_W\sm$ is
vertical$/Y$ up to $\sim$. The latter vertical divisor is $I D_W^m=I
h^*H=h^*(IH)$. Then $H$ and $I$ satisfy the required conditions. Indeed,
for any natural number $i$ divisible by $I$, $i D\sim i D_W\sm+i E=I
h^*H+i E$ on $W$. This induces
 \[
 iD\sim iD\sm+iE=ig^*H+iE \quad\hbox{on $X$.}
 \]
 On the other hand if this holds for any integer $i$, it holds for its
nonnegative remainder $r$ by $I$, and $r=0$ by our choice of $I$ since
then $rD_W\sm=rh^*H$ is vertical.

This implies that $\sR_i=0$ whenever $I\nmid i$. Thus
Addendum~\ref{lin0}.1 follows by Corollary~\ref{isoc}. In addition,
$\sR^{[I]}$ has a nonzero element in each $i\gg0$ divisible by $I$,
because this holds for $\dival_{Y/Z}{H}$. This gives
Proposition~\ref{lin0}, (iv).

As a minimal natural number, $I$ is unique. This implies the uniqueness of
$H$ up to linear equivalence $\sim$.
 \qed\end{pfof}

In Section~\ref{fgalg} we derive the following result from Truncation
Principle~\ref{trprinc}:

 \begin{cor}[cf.\ Stupid example~\ref{exa!st}] \label{stupidc} Let $H$ be an
$\R$-divisor that is numerically ample$/Z$. Then $H$ is a $\Q$-divisor if
and only if its algebra $\dival_{X/Z}{H}$ is f.g. Moreover, it is g.a.g.\
with the empty stable base locus, and with $E=0$ whenever it is f.g.
 \end{cor}

 \begin{pfof}{Theorem~\ref{fgal}}\label{pf!fgal} Suppose that
$\sR=\dival_{X/Z}{D}$ is f.g.\ and g.a.g. Then we can take
$Y/Z\iso\Proj_Z{\sR}$. It is well known that the correspondence $g$ from
$X$ to $Y/Z$ is then a rational $1$-contraction by Lemma~\ref{adjd}, and
$D=D^m+D^e$, where, for some natural number
$N\gg0$,
 \begin{gather*}
 D^m=\Mov{(ND)}/N, \quad D^e=\Fix{(ND)}/N, \\
 \hbox{and}\quad \linsys{ND}=\linsys{\Mov{(ND)}}+\Fix{(ND)}
 \end{gather*}
 is a decomposition of the linear system$/Z$ into its {\em mobile\/} and
{\em fixed} components. More precisely, this holds for any $N>0$ such that
the $N$th component $\sR_N=g_*\Oh_X(ND)$ generates the subalgebra
 \[
 \sR^N=\directsum_{i=0}^\infty\sR_{iN}
 \]
 (cf.\ \ref{raral}). G.a.g.\ implies that $E=D^e$ is exceptional on $Y$,
and essentially exceptional again by Lemma~\ref{adjd}. Indeed, in
codimension~$1$, $\dival_{X/X}{ND}$ is not generated by global sections
exactly in $\Supp E$. Then we apply Lemma~\ref{adjd} to a divisor
$h^*g(D')$ for $D'\in \linsys{ND}$ on any model $W/Z$ of $X/Z$ that is
regular$/Y$ with a contraction $h\colon W\to Y/Z$. If $E=D^e$ is
exceptional but not essentially exceptional on $Y$, there is an effective
but not essentially exceptional $\Q$-divisor $E'\le E$. By Lemma~\ref{adjd},
this contradicts f.g. Thus $D$ is bss ample and $D\sm$ is $\Q$-divisor equal
to $D^m$ outside $\Supp{D^e}$.

Conversely, suppose that $D$ is bss ample$/Z$ and $D\sm$ is a $\Q$-divisor.
By Proposition~\ref{lin0} and the description of its isomorphism, the
divisorial algebra $\sR=\dival_{X/Z}{D}=\dival_{X/Z}{(D-E)}$ is isomorphic
to its truncation $(\dival_{Y/Z}{H})^{[I]}$. On the other hand, the latter
is f.g.\ by Corollary~\ref{stupidc}, because $H$ is a $\Q$-divisor and is
numerically ample$/Z$. Moreover, $H$ is g.a.g.\ with the empty stable base
locus on $Y$. Since $E$ is an effective divisor on $X$ and
$\dival_{X/Z}{D}=\dival_{X/Z}{(D-E)}$, g.a.g.\ holds for $D$ on $X$ with
exceptional stable base divisor $E$ on $X$.
 \qed\end{pfof}

 \begin{rem}\label{gzardec}
 \addcontentsline{toc}{subsection}{\ref{gzardec} Remark: generalized
Zariski decomposition}
 By the proof of the theorem, every nontrivial
f.g.\ algebra $\dival_{X/Z}{D}$ gives a decomposition $D=D^m+E$, where
$D^m$ is {\bi}semiample$/Z$, $E=D^e\ge0$ is a stable base divisor and
$\dival_{X/Z}{D}=\dival_{X/Z}{D^m}$ (cf.\ Example~\ref{moval}). $D^m$ is
the {\em minimal divisor\/} with these properties; it is of course a
$\Q$-divisor.

It is not difficult to show that this is equivalent to a geometric
decomposition $D=D^m+E$ for an $\R$-divisor $D$ with $E\ge0$ and with {\em
maximal\/} {\bi}semiample $D^m$, under the assumption that the maximal
divisor $D^m$ is a $\Q$-divisor. A decomposition of this form can be
defined even if the maximum $D^m$ is not a $\Q$-divisor; it is unique if it
exists, and can be considered as a {\em generalized Zariski
decomposition\/}. In the case $D$ bss ample$/Z$, $E$ is essentially
exceptional. However in general there is no numerical criterion to be
maximal, e.g., being exceptional or negative (cf.\
\cite[Examples~1.1--2]{sh95}). But this can be done in certain situations,
namely, if
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $D$ is big$/Z$ (cf.\ Corollary~\ref{big} below), or
 \item (cf.\ Theorem~\ref{existd} and Corollary~\ref{existc} below) there
is a boundary $B$ on $X$, such that $(X/Z,B)$ is a $0$-{\em log pair\/},
that is,
 \begin{itemize}
 \item $X/Z$ is projective,
 \item $(X,B)$ is Klt, and
 \item $K+B\equiv 0/Z$.
 \end{itemize}
 \end{enumerate}
 Moreover, we then expect the existence of the decompositions for all
divisors $D\sim_\R D'\ge0$ that are effective up to $\sim_\R$.

 For example, (1) holds if $X/Z$ is finite or birational; and, by the base
point free theorem, (2) holds if $X/Z$ has the structure of a weak log
Fano contraction $(X/Z,B)$, with only Klt singularities.
 \end{rem}

 \begin{cor}\label{big} Suppose that $D$ is big$/Z$. Then $D$ is bss
ample$/Z$ and has a $\Q$-divisor $D\sm$ if and only if
$\dival_{X/Z}{D}$ is f.g.

In particular, we can replace\footnote{Why replace? how replace? replace
as an assumption in a theorem?} rationality of $D\sm$ by rationality of
$D$.
 \end{cor}

 \begin{proof} Indeed, for each $D$ big$/Z$ with a f.g.\ divisorial
algebra $\sR=\dival_{X/Z}{D}$, the algebra $\sR$ is big$/Z$ (that is, its
field of fractions\footnote{Surely the opposite of what is intended:
$D$ is big, so $k(X)=k(\sR(D))$?} is finite$/k(Z)$) and g.a.g.\ with empty
stable divisorial base locus. This follows by Lemma~\ref{adjd} in the
noncontractible case. Thus we can omit the g.a.g.\ assumption in
Theorem~\ref{fgal}.
 \qed\end{proof}

 We can for example apply the corollary to a birational contraction $f$.

 \begin{cor} \label{fgflpal} Suppose that $f$ is birational and $D$ is a
$\Q$-divisor. Then the $D$-flip exists if and only if $\flpal_{X/T}{D}$
is f.g.

If $X/T$ is small, $\flpal_{X/T}{D}=\dival_{X/T}{D}$, so that the
$D$-flip exists if and only $\dival_{X/T}{D}$ is f.g.
 \end{cor}

 \begin{proof} Immediate by Lemma~\ref{exfl}, Example~\ref{flipal} and
Corollary~\ref{big}. Indeed, each divisor on $X$ is big$/T$, since
$f\colon X\to T$ is birational.
 \qed\end{proof}

 \begin{thm}\label{existd} LMMP in dimension $n$ implies the existence
of both decompositions of Remark~\ref{gzardec} under (2) in
$\dim X=n$, for any divisor $D$ that is effective up to $\sim_\R$.

In particular, the algebra $\dival_{X/Z}{D}$ is f.g.\ for any
$\Q$-divisor
$D$.
 \end{thm}

 This follows essentially because LMMP is compatible with the
decompositions.

 \begin{proof} By Corollary~\ref{uniqbssc}, we can assume that $D$ is
effective and $(X,B+D)$ is still Klt. Moreover, the new $D$ and $D\sm$
are $\Q$-divisors again whenever the old $D$ and $D\sm$ are; and the
old algebra $\dival_{X/Z}{D}$ is a truncation of the new one. Thus it
is enough to establish the theorem for effective $D$.

For this, we apply LMMP to the pair $(X/Z,B+D)$. This is the same as
the $D$-MMP \cite[Section~5]{sh96b} since $K+B+D\equiv D/Z$ by
Remark~\ref{gzardec}, (2). Thus if $D$ is not nef then we have an
(extremal) $D$-contraction $h\colon X\to Y/Z$, that is birational since
$D$ is effective. By LMMP we can make a $D$-flip $g\colon X\broken
X^+/Z$, where $D=g^*D^++E$ with $E\ge0$ by Lemma~\ref{negat} (cf.\
Example~\ref{smbiriso2}). Lemma~\ref{negat} also implies that the maximal
$D^m$, if it exists, is $\le g^*D^+$; and it exists and is equal to
$g^*D^+{}\sm$ if and only if it exists for $D^+$. By
Example~\ref{smbiriso2}, the same holds for f.g.\ of the algebra
$\dival_{X/Z}{D}=\dival_{X^+/Z}{D^+}$ and the rationality of $D\sm$,
because the divisorial $\Supp E$ is contractible on $X^+$.

By termination in LMMP, we can suppose that $D$ is nef. Then it is
semiample by semiampleness of log canonical divisors, and we have its bss
decomposition by Lemma~\ref{bssasa} with $D^m=D$ and $E=0$. Thus if
$D\sm=D$ is a $\Q$-divisor then $\dival_{X/Z}{D}$ is f.g.\ by
Theorem~\ref{fgal}. Semiample follows by Kawamata and
\cite[Remark~6.23.5]{sh96b}, since $D\ge0$.

Finally, note that if $D$ is not effective up to $\sim_\R$ then
$\dival_{X/Z}{D}=k(X)$ is trivial\footnote{Surely not? You must mean
$k$ or $k(Z)$ or $\Oh_Z$} and f.g.\ but not g.a.g.
 \qed\end{proof}

 \begin{cor}\label{existc} In dimension $n\le3$, we can omit LMMP as
an assumption in Theorem~\ref{existd}.
 \end{cor}

 \begin{proof} Immediate by Theorem~\ref{existd} and
\cite[Theorem~5.2]{sh96b}.
 \qed\end{proof}

Thus we expect in particular the following:

 \begin{conj}\label{plfconj} Let $f\colon X\to X_\vee$ be a pl contraction
with respect to $S$. Then\footnote{Who is $S$ in the display? I think
$S=S_1,S_2,\dots,S_s$ is part of the inductive apparatus, whereas there
must be a divisor $D$ and $\dival_{X/X_\vee}{D}$ is f.g.}
 \begin{q}
 \item[$\PLF_n$] the divisorial algebra $\dival_{X/X_\vee}{S}$ is f.g.
 \end{q}
 \end{conj}

On the other hand,

 \begin{cor}\label{plfcol} {$\PLF_n$} implies Induction Theorem~\ref{ith},
{$\PLF_n\sm$} and the existence of elementary pl flips in dimension $n$.
 \end{cor}

 \begin{proof} Immediate by Corollary~\ref{fgflpal}.
 \qed\end{proof}

In turn we have the following result.

 \begin{thm}\label{plfth} $\RFA_{n,d}\bir$ implies {$\PLF_n$} of
Conjecture~\ref{plfconj}.
 \end{thm}

This is the main result of this section. We explain and prove it below,
but we start with the following:

 \begin{pfof}{Induction Theorem~\ref{ith} for $\RFA$} Immediate by
Corollary \ref{plfcol} and Theorem~\ref{plfth}.
 \qed\end{pfof}

 \begin{exa}\label{ample} Let $X$ be a projective variety with an ample,
effective and reduced divisor $S$ and $H$ an $\R$-divisor such that
$S\sim r H$ for some natural number $r$ and the divisorial
$\Supp H\cap\Supp{S}=\emptyset$. Then it is known that $H$ is a
$\Q$-divisor and $\dival_XH$ is f.g.\ (cf.\ Corollary~\ref{stupidc}).
However, here the essential ingredient is not just Truncation
Principle~\ref{trprinc} but rather the following induction on $n=\dim X$.

Indeed, restricting rational functions without poles to $S$ gives a
canonical homomorphism of graded $k$-algebras
 \[
 \rest S \colon \dival_XH\to \dival_SH\rest S.
 \]
 Moreover, to prove that $\dival_XH$ is f.g., it is enough to prove
that the image subalgebra $\sR=\dival_XH\rest S\subset
\dival_SH\rest S$ is f.g.\ (use Main Lemma~\ref{mainl} below).

On the other hand, by Serre vanishing, the truncation $\sR^{[I]}$ of $\sR$
is equal to that of $\dival_SH\rest S$. Thus by Truncation
Principle~\ref{trprinc} it is enough to verify that
$\dival_SH\rest S$ is f.g. This gives an induction using the base
point free theorem for multiples of $H\rest S$.
 \end{exa}

We use the same idea in the diametrically opposite situation, when $-S$
can be ample. But as for pl contractions, this is only possible in the
local case if $X/Z$ is generically finite or birational. For a more
general situation than the above example, in Main Lemma~\ref{mainl} below,
the f.g.\ of $\dival_{X/Z}{D}$ reduces to that of its restriction
$\sR=\dival_{X/Z}{D}\rest S$. It is a subalgebra of the divisorial algebra
$\dival_{S/Z}{D\rest S}$. Truncation Principle~\ref{trprinc} applies
again, but now Serre vanishing does not work. Replacing it by
Kawamata--Viehweg vanishing is a natural guess; the idea is right, but,
rather than divisorial algebras up to truncation, it leads us to the (FGA)
algebras of Section~\ref{fgalg}.

 \begin{defn}\label{indfam} A {\em normal (irreducible) ladder\/}
$\{S^i\}$ in $X$ is a chain $S^0=X\supset S^1\supset S^2\supset \dots
\supset S^s$, where
 \begin{itemize}
 \item each $S^i$ is a normal (irreducible) proper subvariety of pure
codimension~$i$ in $X$, and
 \item the generic point of $S^i$ is a nonsingular point of $S^j$ for all
$j\le i$.
 \end{itemize}

Let $D$ be an $\R$-Cartier divisor of $X$. A sequence $S_1,\dots,S_s$ of
reduced Weil divisors is {\em inductive\/} with respect to $D$ if
 \begin{itemize}
 \item the subvarieties $S^i=S_1\cap \dots \cap S_i$ form a normal
(irreducible) ladder with normal crossing at the generic point of each
$S^i$;
 \item $\Supp D$ does not contain any $S^i$; and
 \item each $S_i\sim r_iD$ for some natural number $r_i\ge1$.
 \end{itemize}
 The final condition means that for each $i=1,\dots,s$, there exists a
rational function $0\ne t_i\in k(X)$ such that $(t_i)=S_i-r_iD$. In
particular, $r_iD$ must be integral divisors. Thus $D$ itself is a
$\Q$-Cartier divisor, possibly not integral, and each $S_i$ is also
$\Q$-Cartier.

Note also that $t_i\in\Oh_X(r_iD)$, and so $t_i\in \dival_XD$ has degree
$r_i$, because $(t_i)+r_iD=S_i> 0$. Moreover, for all $s\ge i>j\ge0$, the
rational function $0\ne t_i\rest{S^j}\in k(S^j)$ (otherwise $S^j\subset
S_i$, contradicting the assumption on a normal ladder), and it gives the
rational equivalence $S_i\rest{S^j}\sim r_iD\rest{S^j}$. We call $t_i$ a
{\em translation function\/} for $S_i$ of {\em degree\/} $r_i$.
 \end{defn}

 \begin{exa}\label{resralge} Let $(X/T,S+B)$ be a log pair such that
 \begin{itemize}
 \item $X/T$ is a (birational) contraction;
 \item $S=\sum S_i$ is a sum of $s\ge1$ prime Weil $\Q$-Cartier divisors
$S_1,\dots,S_s$;
 \item the $S_i$ are {\em linearly proportional\/} to an integral divisor
$D$, for $1\le t\le i\le s$, that is, each $S_i\sim r_iD$ for some natural
number $r_i\ge1$ if $t\le i\le s$ (compare\ \ref{specont}.\ref{svoS} for
pl contractions);\footnote{You don't need $t$ here. You don't need to repeat
$1\le t\le i\le s$ twice. On the other hand, you intend to restrict
everything down to a surface, so $t=n-2$.}
 \item $K+S+B$ is divisorially log terminal with $\rddown{B}=0$; and
 \item $K+S+B$ is numerically negative$/T$.
 \end{itemize}
 This is a slight twist on the definition of pl contraction, that we use
as follows: the restrictions $S_i\rest{S^{t-1}}$ for $t\le i\le s$ form an
inductive family with respect to a divisor $D'\rest{S^{t-1}}$, where
$D'\sim D$. Moreover, the normal ladder $S^j$ for $t\le i\le s$ is {\em
irreducible\/} and $X/T$ induces a contraction $S^j/f(S^j)$ for each $j$.

Indeed, the latter follows by induction on $s$. By
\cite[Corollary~3.8]{sh92}, $S^1=S_1$ is normal and irreducible$/T$, even
formally so (that is, single branched). The numerical negativity of
$K+S+B$ and the other assumptions imply the numerical negativity of
$K+S^1+B'$ for a boundary $S^1+B'\le S+B$ with $\rddown{B'}=0$. In other
words, $(X,S^1+B')$ is purely log terminal with the reduced divisor $S^1$
in the boundary. Then any linear system that defines the contraction $X/T$
restricts surjectively to $S^1$, and thus induces a contraction
$S^1/f(S^1)$; and $P\in f(S^1)$ since $S_1$ intersects the fibre $f\1P$.

Each $S_i$ with $i\ge2$ gives a prime Weil divisor $S_i\rest{S^1}=
S_i\cap S_1$ in $S^1$. Indeed, again by \cite[Corollary~3.8]{sh92}, this
last intersection is normal as a subvariety and transverse at generic
points. In other words, it gives a normal ladder $S^1\supset S_1\cap S_i$
and $S_i\rest{S^1}=\sum T_j$ is a sum of prime Weil divisors. In fact, we
have a single $T_i$. First, some $T_j$ exists by Connectedness of LCS
(Koll\'ar and others \cite[Theorem~17.7]{ko}) for $K+S+B-\ep S'$, where
$0\ll\ep<1$ and $S'=\sum_{j\ne1, i} S_j$. In other words, $S_1$ intersects
any other $S_i$ in $f\1P$, since each $S_i$ intersects $f\1P$. Second,
there is only one $T_i$ for the same reasons: namely, $\bigcup T_i$ is a
normal subvariety with disjoint components $T_j$ and by adjunction
\cite[3.2.3]{sh92} $(K+S+B-\ep S')\rest{S^1}=K_{S^1}+B''+\sum T_j$ is
divisorially log terminal with $\rddown{B''}=0$, where the LCS is
$\bigcup T_j$. Finally, each restrictions $T_i=S_i\rest{S^1}$ is
$\Q$-Cartier.

Thus by adjunction we preserve the situation for $T_i=S_i\rest{S^1}$ for
$s\ge2$ and for an appropriate choice of $D$. Since $D$ is integral and
the generic point of $S^s$ is nonsingular in $X$, by the Moving Lemma we
can assume up to $\sim$ that $\Supp D$ does not contain $S^s$ or any other
$S^i$ with $i\ge2$.

The new contraction $S^j/f(S^j)$ may not be birational even if $X/S$ is
(cf.\ Example~\ref{fbrcont}).

Note also that, by definition, pl contractions do not always give the
above situation, since they do not necessarily satisfy linear
proportionality. Indeed, the proportionality~\ref{specont}.\ref{svoS} only
implies that there exists an integral divisor $D$ such that each
$S_i\sim_\Q r_iD$ for natural number $r_i$. Fortunately, we can upgrade
this to $\sim$ using the covering trick (cf.\ Lemma~\ref{covtric} and the
proof of Theorem~\ref{plfth}).
 \end{exa}

To generalize Example~\ref{ample}, we need to consider restrictions of
divisorial algebras.

 \begin{defn}\label{restal} Let $Y\subset X$ be a subvariety and $D$ an
$\R$-Cartier divisor such that
 \begin{itemize}
 \item $\Supp D$ does not contain the generic points of $Y$.
 \end{itemize}
 Then restricting functions induces the {\em restriction\/} of divisorial
algebras
 \[
 \rest Y \colon \dival_{X/Z}{D}\to \dival_{Y/Z}{D\rest Y}
 \]
 (see Remark~\ref{restd} for the restrictions of divisors). This map is an
$\Oh_Z$-homo\-morphisms of $\N$-graded algebras. In particular, it induces
a restriction homomorphism $\rest Y\colon\sL\to
\dival_{Y/Z}{D\rest Y}$ of any $\N$-graded $\Oh_Z$-subalgebra
$\sL\subset \dival_{X/Z}{D}$ (functional in the terminology of
Definition~\ref{funcal}).
 \end{defn}

 \begin{rem}\label{restd} The above restriction is defined provided that
functions in $\dival_{X/Z}{D}$ have no poles at generic points of
$Y$, that is, the support of positive components of $D$ does not contain
any generic point of $Y$. The restriction can however be zero; and this
may happen even under our assumption on $D$. The important point in the
definition is that if $D\rest Y$ is defined, then each restricted nonzero
function $f\rest Y$ for $f\in\dival_{X/Z}{D}$ belongs to a divisorial
algebra in the same degree, so that its zeros and poles are controllable.

Restriction generalizes of course to $D$ that are not $\R$-Cartier,
assuming that we restrict $\sL$ either
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item as a subalgebra for a divisorial algebra of $\R$-Cartier $D'\ge D$;
or
 \item $Y$ is the final step $S^s$ of a normal ladder (or $\rest{Y^\nu}$
on the normalization $Y^\nu$ for a nonnormal\ ladder).
 \end{enumerate}

See \cite[Section~3]{sh92} for restricting Weil $\R$-divisors to a
normal\footnote{Probably the {\em normalization of a divisor} is
intended.} divisor. Note that a $\Q$-divisor restricts to a $\Q$-divisor,
but an integral divisor may not restrict to a $\Z$-divisor (cf.\ $D$ in
Definition~\ref{indfam}). However, an integral divisor restricts to an
integral divisor along any nonsingular locus of $X$ (the ambient space).
Inequalities between divisors such as $D_1\ge D_2$ are preserved, and in
particular, an effective divisor remains effective. (This follows from
\cite[Negativity~1.1]{sh92}.) This implies an inclusion
$\dival_{X/Z}{D_2\rest Y}\subset \dival_{X/Z}{D_1\rest Y}$.
 \end{rem}

 \begin{mainl}\label{mainl}
 \addcontentsline{toc}{subsection}{\ref{mainl} Main Lemma} \
 \begin{q}
 \item[\SDA] Let $\sL$ be an $\Oh_Z$-subalgebra of a divisorial algebra
$\dival_{X/Z}{D}$,
 \item[\IND] let $\bigl\{S_i\bigm| 1\le i\le s\bigr\}$ be an inductive
family with respect to $D$ with translations $t_i$ satisfying:
 \item[\TRL] $t_i\in\sL_{r_i}$ for each $i$ and
 \[
 t_i\1\colon\sL_N(-S_i)\to\sL_{N-r_i} \quad\hbox{for every $N\ge r_i$;}
 \]
 that is, $a t_i\1\in\sL_{N-r_i}$ for a function $a\in\sL_N(-S_i)$,
where
 \[
 \sL_j=\sL\cap f_*\Oh_X(ND),
 \]
 is the $j$th component of $\sL$, and
 \begin{align*}
 \sL_N(-S_i) &=\sL_N\cap f_*\Oh_X(ND-S_i) \\
 &=\bigl\{a\in\sL_N\bigm|(a)+ND-S_i\ge0\bigr\}.
 \end{align*}
 \end{q}
 Then, for $Y=S^s$, the algebra $\sL$ is f.g.\ near $f(Y)$ if and only if
its restriction $\sL\rest Y$ is.
 \end{mainl}

 \begin{add} The required translations $t_i$ exist if $\sL=\dival_{X/Z}{D}$
is divisorial.
 \end{add}

 \begin{proof} We only need to check that $\sL$ is of finite type if
$\sL\rest Y$ is, and by induction on $s$, we only only need to consider
the divisorial case: that is, $s=1$ and $Y=S=S_1=S^1$ with a single
translation $t=t_1$ of degree $r=r_1\ge1$. Indeed, for $s\ge2$, the
restrictions
 \[
 \sL\rest{S^1},\quad \bigl\{S_i\rest{S^1}\bigm|2\le1\le s\bigr\},\quad
D\rest{S^1} \quad\hbox{and}\quad t_i\rest{S^1} \hbox{ for } 2\le i\le s
 \]
 again satisfy the assumptions of the lemma (with the same $r_i$). Thus
$\sL\rest{S^1}$ is f.g.\ by induction.

The problem is local. Thus we fix a point $P\in f(S)\subset Z$ and check
that $\sL$ is f.g.\ near $P$. Since the algebra $\sL\rest S$ has finite
type near $P$, it has a system of homogeneous generators $s_1,\dots,s_m$.
Recall that a homogeneous element of a graded algebra is an element in a
homogeneous component; in our case, $(\sL\rest S)_j=\sL_j\rest S$. The
sections can be presented as restrictions $s_i=t_i\rest S$, where $t_i\in
\sL_j$. We claim that $t$, the sections $t_i$ and the generators of the
$\Oh_Z$-modules $\sL_j$ for
$j< r$ generate $\sL$ near $P$.

Indeed, for any homogeneous $a\in\sL$ of degree $N\ge r$, we can find a
polynomial $p(x_1,\dots,x_m)\in\Oh_{f(S),P}[x_1,\dots,x_m]$ such that
$a\rest S=p(s_1,\dots,s_m)$ and is homogeneous, that is, all its monomials
are homogeneous of weighted degree $N$, where $\deg x_i=\deg s_i$. The
coefficients of $p$ are in the local ring $\Oh_{f(S),P}$ of $P\in f(S)$.
Thus $p$ can also be obtained as the restriction $p=q\rest S$ of a similar
polynomial $q(x_1,\dots,x_m)\in\Oh_{Z,P}[x_1,\dots,x_m]$ with coefficients
in the local ring $\Oh_{Z,P}$. Then $a-q(t_1,\dots,t_m)$ vanishes on $S$,
and belongs to $\sL_N(-S)$. This last conclusion holds because $S\not\subset
\Supp D$. Hence, by \TRL, $b=t\1(a-q(t_1,\dots,t_m))\in\sL_{N-r}$, and
$a=q(t_1,\dots,t_m)+t b$, where $b$ is homogeneous of degree $(N-r)<N$.
Induction on $N$ completes the proof.

For Addendum~\ref{mainl}.1, we need to verify \TRL\ for the divisorial
algebra $\sL=\dival_{X/Z}{D}$ with any translations $t_i$. By
Definition~\ref{indfam}, $t_i\in \dival_{X/Z}{D}$ is homogeneous of degree
$r_i$. Moreover,
 \[
 (a)+ND-S_i\ge0 \quad\hbox{for any $a\in
\sL_N(-S_i)=g_*\Oh_X(ND-S_i)$.}
 \]
 Thus $a t_i\1$ belongs to $\sL_{N-r_i}=g_*\Oh_X((N-r_i)D)$:
 \begin{align*}
 (a t_i\1)+(N-r_i)D&=(a)-(t_i)+(N-r_i)D\\
 &=(a)-S_i+r_iD+(N-r_i)D\\
 &=(a)-S_i+ND\ge0. \qed
 \end{align*}
 \end{proof}

 \begin{rem} We have proved more: namely, the kernel of the restriction
is f.g.\ as an algebra (without $1$).\footnote{This is nonsense. You
probably mean f.g.\ as an ideal. Even a principal ideal such as
$(x)\subset k[x,y]$ is not f.g.\ as a ring: you need $xy^i$ for every
$i$.} Of course, if $A\onto B$ is a surjection of algebras and its kernel
is f.g.\ as an algebra then $A$ is f.g.\ if and only if $B$ is.
 \end{rem}

 \begin{exa}\label{fbrcont} Let $g\colon Y\to g(Y)$ be a contraction of
fibre type, and suppose that $D$ is numerically negative on general curves
of its generic fibres. Then $\sL\rest Y=\Oh_{f(Y)}$ is trivial and needs
$0$ generators. Thus $\sL$ is f.g.$/Z$. More precisely, it is generated by
the translations $t_i$ and generators of $\sL_j$ with $j\le\max\{r_i\}$.

If we replace Condition~(EL.\ref{small}) for elementary flips by its
opposite, that is, by the condition
 \begin{itemize}
 \item $f$ is divisorial,
 \end{itemize}
 or, more generally, if we consider a divisorial pl contraction with
$S$ numerically negative$/X_\vee$, then $S=S_1=S^1$ and
$\dival_{X/X_\vee}{S}=\dival_{X/X_\vee}{0}$ is f.g., that is,
(PLF)$_n$ holds in this situation. This is of course well known. But from
our point of view $S\sim D$ with $S\not\subset\Supp D$ and $D$ is
numerically negative on $S/X_\vee$. Hence $\dival_{X/X_\vee}{S}$ is
isomorphic to $\sL=\dival_{X/X_\vee}{D}$, which is generated by $t$ with
$(t)=S-D$ and $1\in\sL_0=\Oh_{X_\vee}$.

Of course, Main Lemma~\ref{mainl} only proves f.g.\ near $f(S)$, but it
holds outside $f(S)$ because $\dival_{X/X_\vee}{S}=\dival_{X/X_\vee}{0}$
there. In general, {\em the lemma is entirely sufficient for local
purposes\/}, since $P\in f(S)$, as we assume throughout what follows.
 \end{exa}

 \begin{cor} Under the assumptions of Example~\ref{resralge}, there exists
$D'\sim D$ such that the algebra $\sL=(\dival_{X/T}{D'})\rest{S^{t-1}}$ is
f.g.\ if and only if $\sL\rest{S^s}$ is.
 \end{cor}

 \begin{proof} Immediate by Main Lemma~\ref{mainl} and
Example~\ref{resralge}, if we take $D$ such that
$S^s\not\subset\Supp{D'}$. The restricted algebra $\sL$ is a
$\Oh_{f(S^{t-1})}$-subalgebra of
$\dival_{S^{t-1}/f(S^{t-1})}{D'\rest{S^{t-1}}}$. The condition \TRL\ on
$S^{t-1}$ is induced by the same on $X$ by Addendum~\ref{mainl}.1.
 \qed\end{proof}

 \begin{defn}\label{rfad}
 \addcontentsline{toc}{subsection}{\ref{rfad} Definition of restricted
algebras (RFA)}
Under the assumptions of Example~\ref{resralge}, suppose that $X/T$ is
birational, $t=1$ and $S^s\not\subset\Supp D$. Then we say that the {\em
restricted algebra\/} $\sL=(\dival_{X/T}{D})\rest Y$ is of type
$\RFA_{n,d}$, where $Y=S^s$, $n=\dim X$ and $d=n-s=\dim Y$. If $Y/f(Y)$ is
again birational, we say that $\sL$ is of type $\RFA_{n,d}\bir$.
 \end{defn}

 \begin{conj}\label{rfac} \
 \begin{qq}
 \item[$\RFA_{n,d}\bir$] every algebra of type $\RFA_{n,d}\bir$ is f.g.
 \end{qq}
 \end{conj}

 \begin{cor}\label{rfaco} $\RFA_{n,d}\bir$ of Conjecture~\ref{rfac} implies
 \begin{qq}
 \item[$\RFA_{n,d}$] every algebra of type $\RFA_{n,d}$ is f.g.
 \end{qq}
 \end{cor}

 \begin{proof} In the other words, $(\dival_{X/T}{D})\rest Y$ is still
f.g., even if $Y/f(Y)$ is not birational. Indeed, since $X/T$ is birational
but $Y=S^s/f(Y)=f(S^s)$ is not, then for some $1\le i<s$, $S^i/f(S^i)$ is
still birational but $E=S^{i+1}/f(S^{i+1})$ is not. Therefore $E$ is a Weil
divisor of $S^i$, and is exceptional on $f(S^{i+1})$. By our assumptions,
it is $\Q$-Cartier and $\sim r_{i+1} D$. Thus $D$ is numerically negative
on general curves of the generic fibre of $S^{i+1}/f(S^{i+1})$. Therefore
the algebras $(\dival_{X/T}{D})\rest{S^{i+1}}$ and
$(\dival_{X/T}{D})\rest Y$ are trivial by Example~\ref{fbrcont}.
 \qed\end{proof}

\subsection{The covering trick} Further applications needs the covering
trick.

 \begin{lem} \label{covtric} Let $\pi \colon \wX\to X$ be a finite cover
that is etale in codimension~$1$.
 \begin{enumerate}
 \renewcommand{\labelenumi}{$(\arabic{enumi})$}
 \item The pair $(\wX,\wD=\pi\1D)$ is divisorially log terminal if
$(X,D)$ is; and $(\wX,\wD=\pi\1D)$ is log canonical, respectively Klt or
purely log terminal if and only if $(X,D)$ is;
 \item $\pi\1D=f^*D$ is $\Q$-Cartier (or $\R$-Cartier) if and only if $D$
is;
 \item $\pi\1D\sim_\R \pi\1D'$ if and only if the same holds for
$D,D'$;
 \item $\pi\1D=\pi^*D$ is bss ample$/Z$ (or {\bi}semiample, semiample,
ample) if and only if $D/Z$ is; and
 \item a pl contraction $f\colon X\to X_\vee$ induces a pl contraction
$\wf\colon \wX\to \wX_\vee$ with $\wS=\pi\1S$ and finite
$\wX_\vee/X_\vee$; the same holds for $f\colon X\to T$ of
Example~\ref{resralge}; in addition, $f$ has a pl flip or even a $D$-flip
for any contraction $f$ if and only if the same holds for $\wf$.
 \end{enumerate}
 \end{lem}

 \begin{proof} (1) is immediate from the pullback formula \cite[1.1]{sh92}
(cf.\ the proof of \cite[Corollary~2.2]{sh92}). For divisorially log
terminal (which means that the log discrepancies are only $0$ over normal
crossing intersections of $\rddown{B}$), note that $\pi$ is etale over the
nonsingular points, and preserves the nonsingularity of the log canonical
centers and normal crossings of $\rddown{B}$ in their generic points. The
nonsingularity in codimension~1 of irreducible components of
$\rddown{\wB}$ follows from \cite[Lemma~3.6 and Corollary~2.2]{sh92}.

For (2--3), see \cite[proof of Corollary~2.2]{sh92}. (4) is immediate by
Definition~\ref{bssd} and the uniqueness of Proposition~\ref{uniqbss}.

(5) follows by the connectedness arguments of Example~\ref{resralge}, and
again by \cite[Corollary~2.2]{sh92}. They imply that $\wS_i=\pi\1S_i$ are
again prime. The rational numbers $r_{i,j}$ and natural number $r_i$ are
the same on $\wX$. The final statement on pl and $D$-flips follows from
(4) (cf.\ \cite[Lemma~2.5]{sh92}).
 \qed\end{proof}

 \begin{pfof}{Theorem~\ref{plfth}} Let $f\colon X\to X_\vee$ be a pl
contraction with respect to $S$. By Corollary~\ref{big},
$\dival_{X/X_\vee}{S}$ is f.g.\ if and only if $S$ is bss ample. Thus by
Lemma~\ref{covtric} we can replace $X$ by any finite cover that is etale
in codimension~$1$.

On the other hand, by Condition~\ref{specont}.\ref{svoS}, the divisors
$S_i$ generate an Abelian subgroup of rank $1$ in the group of integral
Weil divisors modulo $\sim_\Q$. This is torsion free. Hence there is a
generator of this subgroup that is the class modulo $\sim_\Q$ of an
integral Weil divisor $D$. In other words, each $S_i\sim_\Q r_iD$ for
some natural number $r_i$; and $S\sim_\Q(\sum r_i)D$. By
Corollary~\ref{uniqbssc}, $\dival_{X/X_\vee}{S}$ is f.g.\ if and only if
$D$ is bss ample, and hence if and only if $\dival_{X/X_\vee}{D}$ is f.g.

After a finite cover etale in codimension~1, we can assume that each
$S_i\sim r_iD$ for some natural number $r_i$; and $S\sim (\sum r_i)D$.
Indeed, if $S_i\sim_\Q r_iD$ then it defines a cyclic cover $\pi$ that is
etale in codimension~$1$ such that $\pi\1S_i\sim r_i\pi\1D$
\cite[Construction~2.3]{sh92}. Then we use induction on $i$. In addition,
we can choose $D$ up to $\sim$ such that $\Supp D$ does not contain all
$S^j$ for $j\ge1$. We are now in the situation of Definition~\ref{rfad}
and by Main Lemma~\ref{mainl}, $\sR=\dival_{X/X_\vee}{D}$ is f.g.\ if
$\sL=\sL\rest{S^s}$ is. This last algebra is of type $\RFA_{n,d}$. Hence
if $S^s/f(S^s)$ is birational, then $\sL$ is f.g.\ by our assumption.
Otherwise we use Corollary~\ref{rfaco}.
 \qed\end{pfof}

 \begin{cor}\label{plff} $\PLF_n$ holds for a pl contraction $X/X_\vee$
if $Y=S^i=E$ or $Y\subset f\1P$, where $E$ is the exceptional locus for
$X/X_\vee$ and $f\1P$ is the fibre of $X/X_\vee$ over $P$.
 \end{cor}

 \begin{proof} Immediate by the proof of Theorem~\ref{plfth}, because then
$Y=S^s$ is a point whenever it is birational$/f(Y)$. But if $Y$ is a
point, any subalgebra of $\dival_Y{D}=k_\bull$ is f.g.$/k$ (and is
quasi-isomorphic to a divisorial algebra whenever it is nontrivial).
 \qed\end{proof}

For example, the latter case in the proof applies to Example~\ref{ample}
even if $f$ is not a pl contraction and is not birational.

However even if $Y$ is a curve, a subalgebra $\sL\subset\dival_Y{D}$
need not be divisorial in general (cf.\
Examples~\ref{toosl}--\ref{tooqu}). Thus further investigations are needed
to specify restricted subalgebras. We do this in Section~\ref{fgalg}.

 \begin{exa}[cf.\ {\cite[flips of type (6.6.2)]{sh92}}]\label{plneg} Let
$X/X_\vee$ be a pl contraction with respect to $S$ such that:
 \begin{itemize}
 \item $S$ is numerically negative$/X_\vee$.
 \end{itemize}
 For example, this holds for an elementary pl contraction. Then
$X/P\subset E\subset Y=S^s$, where $X/P=f\1P$ is the fibre$/P$ and $E$
denotes the exceptional locus of $f$.

Indeed, $C\cdot S_i<0$ for any curve $C/P$ and for each $S_i$. Hence
$C\subset Y$, and $s\le n-1$, where $n=\dim X$. Moreover,
 \begin{q}
 \item[\DPT] $s\le n-\dim E\le n-\dim X/P$. Moreover, $s=n-\dim E$ holds
only if $E=Y$; and $\dim E=\dim X/P$ holds only if $E=X/P$.
 \end{q}

Thus if $E\ne\emptyset$ and $s=n-\dim E\le n-1$, then
$\dival_{X/X_\vee}{S}$ is f.g.\ by Corollary~\ref{plff}. In particular,
elementary flips exist under this assumption. For example, this holds if
$s=n-1$ and $f$ is not an isomorphism.
 \end{exa}

Another trivial case is when $f\colon X\to T$ is an {\em isomorphism\/}.
Then any divisorial algebra $\dival_{X/T}{D}$ is f.g.\ for $D$ a
$\Q$-Cartier divisor by Corollary~\ref{stupidc}. Our restriction arguments
apply again to this case because the restrictions are surjective.

 \begin{exa} The same arguments apply to any toric contraction
$(X/T,S)$ provided that
 \begin{itemize}
 \item $X$ is $\Q$-factorial, and
 \item $X/T$ is birational and extremal, that is, $\rho (X/T)=1$;
 \end{itemize}
 $S=\sum S_i$ denotes the invariant divisor, where the $S_i$ are the
invariant prime divisors, and $1\le i\le \dim X+1=n+1$ (cf.\
\cite[Theorem~6.4 and the conjecture after it]{sh95}). We order the $S_i$
so that those with $1\le i\le s$ are the only divisors that are
numerically negative$/T$. It is known that the exceptional locus $E$ of
$X/T$ is then also invariant, and thus of the form $E=\bigcap_{1\le i\le
s} S_i$ with $s\le n-1$, and one of the other divisors, say $S_n$, is
numerically positive$/T$. Unfortunately, the contraction may not be pl
with respect to $S^-=\sum_{1\le i\le s} S_i$ since the pair $(X,S^-)$ may
not be divisorially log terminal, having singularities along log centers.
However we can eliminate all these after a finite covering (even in the
toric category) using the positive $S_n$ and get a pl contraction. Then
Example~\ref{plneg} implies that $S^-$ is bss ample or equivalently, that
$\dival_{X/T}{S^-}$ is f.g. This gives a construction of toric flips, or
it would be better to say flops.
 \end{exa}

This is the best we can do using divisorial algebras.

\section{Finitely generated pbd-algebras} \label{fgalg}

In this section\footnote{I have corrected several errors in
Sections~4.1--4.10, but there are possibly several more left. The material
is basically very elementary.} we focus on $\N$-graded $\Oh_Z$-algebras
over a normal algebraic variety $Z$. We again consider the local case near
$P\in Z$.

 \begin{defn}\label{funcal}
 \addcontentsline{toc}{subsection}{\ref{funcal} Definition of functional
algebra}
We say that a $\N$-graded $\Oh_Z$-algebra
$\sL=\directsum_{i\ge0}\sL_i\subset k(X)_\bull$ is a (coherent) {\em
functional algebra\/} if each homogeneous component $\sL_i$ (of degree
$i$) is a coherent $\Oh_Z$-submodule $\sL_i\subset k(X)$, where
$X/Z$ is a proper morphism. We always assume that $\sL_0=\Oh_Z$.
 \end{defn}

In slightly more abstract form, we have:

 \begin{prop} Any functional algebra $\sL$
 \begin{itemize}
 \item is commutative;
 \item is {\em integral\/}, that is, has no zerodivisors;
 \item each $\sL_i$ is a coherent $\Oh_Z$-module, and $\sL_0=\Oh_Z$; and
 \item $\sL$ has field of fractions of finite type over $k(Z)$.
 \end{itemize}
 Conversely, each $\N$-graded $\Oh_Z$-algebra with these properties is
functional.
 \end{prop}

 \paragraph{Proof--Explanation} The homogeneous field of fractions of an
integral graded algebra $\sL$ is a field $L$ such that $\sL$ is a graded
subalgebra of $L_\bull$; each element (function) of $L$ is a fraction
$l/l'$, where $l,l'\in\sL_i$ have the same degree $i$ and
$l'\ne0$. For a functional algebra $\sL$, it is a subfield of $k(X)$ and
of finite type over $k(Z)$. Thus the properties of $\sL$ are immediate
from the definition.

Conversely, every integral graded algebra has a field of fractions $L$. It
has an inclusion $\sL\into L_\bull$ given by fractions:
 \[
 \sL_i\into L \quad\hbox{defined by}\quad
 l_i\mapsto\frac{l_i}{t^{i/d}}=\frac{l_il_m^{i/d}}{l_n^{i/n}},
 \]
 where for a nontrivial algebra $\sL$, $d\mid i$, $d=\gcd
\bigl\{i\bigm|\sL_i\ne0\bigr\}$, and $t=l_n/l_m$ with $n-m=d, 0\ne l_n\in
\sL_n, 0\ne l_m\in\sL_m$. For a trivial algebra we take $t=1$.

Since $L$ has {\em finite type} over $k(Z)$, then $L=k(X)$ for a normal
algebraic variety $X$, and we can assume that $X/Z$ is proper.

Note that any $\Oh_Z$-subalgebra of finite type of $k(X)$ is coherent,
since $\Oh_Z$ is Noetherian. We give a more explicit form soon (see
Proposition~\ref{texa}). \qed\par\medskip

 \begin{exa}\label{raral} Let $\sL=\directsum\sL_i$ be a $\N$-graded
algebra and $I$ a natural number. The {\em truncation\/} of $\sL$ is the
algebra $\sL^{[I]}=\directsum_{I\mid i}\sL_i$. Our convention is to give
elements the same degree (traditionally, one often gives them degree
$i/I$). Any truncation of a functional algebra is also functional.

Two algebras are {\em quasi-isomorphic\/} if they have isomorphic
truncations.
 \end{exa}

 \begin{exa} Let $\sL$ such graded $k$-algebra that
 \[
 \sL_i=\left\{
 \begin{array}{ll} k x^{i/2}&\text{for $i$ even,}\\ k
\ep_i&\text{\ otherwise}\\
 \end{array}
\right.
 \]
 with $x^i x^j=x^{i+j}, \ep_ix^j=\ep_i\ep_j=0$. Then $\sL^{[2]}$ is
isomorphic to $k[x]$, that is f.g.\ and generated by $x=x^1$. But
$\sL$ itself is not f.g.\ Note that $\sL$ is not integral, in particular,
not functional.
 \end{exa}

 \begin{thm}[Truncation Principle]\label{trprinc}
 \addcontentsline{toc}{subsection}{\ref{trprinc} Truncation Principle}
Quasi-isomorphism preserves f.g.\ of a functional algebra.
 \end{thm}

 \begin{proof} It is enough to prove this for a truncation
$\sL^{[I]}\subset\sL$.

If $\sL$ is f.g., it is well known that $\sL^{[I]}$ is also f.g. If
$s_1,\dots,s_m$ generate $\sL$, then $\sL^{[I]}$ is generated by
$s_1^I,\dots,s_m^I$ and the generators of $\sL_{Ii}$ with $i<\sum d_i$,
where $d_i=\deg s_i$.

Conversely, if $\sL^{[I]}$ is f.g., each coherent $\Oh_Z$-submodule of
$\sL^{[I]}$ is of finite type because $\Oh_Z$ is Noetherian. On the other
hand, $\sL$ f.g.\ means that $\sL$ is of finite type as
$\sL^{[I]}$-module. This follows from the Noetherian property since it is
a direct sum $\sL=\directsum_{0\le r\le I-1}\sL^r$ of modules
$\sL^r=\directsum_{i\equiv r\mod{I}}\sL_i$ each of which is of finite
type. Indeed, we can assume that $\sL^r\ne0$. Then there is some
$0\ne s\in\sL^r$ of degree $i$ such that multiplication $x\mapsto s^{I-1}x$
gives an inclusion of $\sL^r$ into $\sL^{[I]}$.
 \qed\end{proof}

 \begin{exa}[Blowup of an ideal] Let $\sI\subset k(Z)$ be a {\em
fractional ideal\/}, that is, up to a multiple, a coherent $\Oh_Z$-ideal
$\sI\subset\Oh_Z$. It defines a functional algebra $\sT=\sT(\sI)$,
the\footnote{This is nonsense. $\sI^{\tensor i}\ne\sI^i$ in general.}
tensor algebra of $\sI$, with $\sT_i=\sI^{\tensor i}=\sI^i
\subset\Oh_Z^{\tensor i}=\Oh_Z$. This algebra is always f.g.\ and its
projective spectrum $\sigma \colon \Proj_Z{\sT}\to Z$ is the {\em
blowup\/} of $Z$ in $\sI$ or in its subscheme. It is a projective
birational morphism, possibly not small, since any projective birational
morphism to $Z$ can be obtained in this form.

If $\sI=\Oh_T(f(D))$ for a birational contraction $f\colon X\to T=Z$ as in
Example~\ref{flipal}, the corresponding tensor algebra $\sT(\sI)$ is a
subalgebra of the flipping algebra $\flpal_{X/T}{D}=\dival_{T/T}{f(D)}$,
and is a proper subalgebra in many interesting cases. Indeed, for integral
$D$, $\flpal_{X/T}{D}=\sA(\sI)=\sT(\sI)^{\vee\vee}$ is the reflexive tensor
algebra. If this last algebra is f.g., it gives a {\em small\/}
contraction: namely, the $D$-flip $X^+/T$. However if $D$ is not Cartier
on $X^+$, the blowup in $\sI$ is bigger, and even divisorial if $X^+$ is
$\Q$-factorial.
 \end{exa}

Blowups in ideals have other drawbacks: they can lead to nonnormal
varieties, which are outside our category. It also relates to other
functional algebras and can be improved in terms of functional algebras.

 \begin{exa}[Integral closure] Let $\sL$ be a functional algebra. Then
$\sLbar=\directsum\sLbar_i$ with
 \[
 \sLbar_i=\left\{a\in k(X)\left| \begin{array}{l}
a^m+l_1a^{m-1}+\dots+l_{m-1}a+l_m=0 \\
\hbox{for some elements $l_j\in\sL_i^j$}
 \end{array} \right\}\right.,
 \]
 where $\sL_i^j$ denotes the product $\Oh_Z$-submodule $\sL_i\cdots\sL_i$
($j$ times) in $k(X)$ (or in any other field $F$), is also a functional
algebra in $k(X)_\bull$ (or in $F_\bullet$, that depends on $F$). We call
it the {\em integral closure\/} of $\sL$ in $k(X)$ (resp.\ in $F$). We have
$\overline{\sLbar}=\sLbar$. The proof is given in Proposition~\ref{texa}
below, together with another description of integral closure. We also prove
that $\sL$ is f.g.\ if and only if $\sLbar$ is; in this case, the
projective spectrum morphism $\Proj_Z{\sLbar}\to \Proj_Z{\sL}$ is a
normalization. It is birational (see Corollary~\ref{norm}). We say that an
algebra is {\em normal\/} if it is integrally closed in its field of
fractions. By Proposition~\ref{texa}, these are the pbd algebras (up to
a truncation).
 \end{exa}

 \begin{exa}[Twist]\label{final} Let $\sA$ be a coherent $\Oh_X$-algebra.
It gives a $\N$-graded $\Oh_X$-algebra $\sA_\bull$ with components
$\sA_i=\sA$ for $i\ge1$ and $\sA_0=\Oh_X$. We define its {\em twist\/}
$\sA[D]$ by a Cartier divisor $D$ on $X$ by setting
$\sA[D]_i=\sA_i(iD)=\sA_i\tensor_{\Oh_X}\Oh_X(iD)$. Since $\sA$ is
coherent, the twisted algebra is finite as a module over
$\dival_{X/X}{D}$. Moreover, $f_*\sA[D]$ is also finite as a module over
the algebra $\dival_{X/Z}{D}=f_*\dival_{X/X}{D}$ if $D$ is ample$/Z$.

The algebra $f_*\sA[D]$ is functional if $\sA$ does not have torsion: that
is, is a submodule of $k(Y)/k(X)$. For example, if $g\colon Y\to X$ is a
proper morphism, then $g_*\Oh_Y$ is a coherent $\Oh_X$-subalgebra of
$k(Y)$ with the natural multiplication. Its twist $g_*\Oh_Y[D]$ is the
divisorial algebra $\dival_{Y/X}{g^*D}$ by the Projection Formula.
 \end{exa}

However, the most important functional algebras for us are geometric, that
is, associated to divisors.

 \begin{defn}\label{pbdad}
 \addcontentsline{toc}{subsection}{\ref{pbdad} Definition of pbd algebra}
 Let $\sD_\bull=(\sD_i\mid i\in \N)$ be a system (sequence) of
$\R$-{\bi}divisors of $X/Z$ such that:
 \begin{itemize}
 \item $\sD_i+\sD_j\le
\sD_{i+j}$;
 \item $\sD_0=0$; and
 \item each $f_*O_X(\sD_n)$ is a coherent $\Oh_Z$-module,
 \end{itemize}
 where the sections of the $\Oh_X$-sheaf
$\Oh_X(\sD_i)=\Oh_X(\rddown{\sD_i})$ are
 \[
 \Ga(U,\Oh_X(\sD_i))=\bigl\{a\in k(X)\bigm|(a)+\sD_i\ge0 \hbox{ on $U$}
\bigr\},
 \]
 and the principal divisor $(a)$ is considered as a {\bi}divisor, namely,
the closure of the usual $(a)$ \cite[Example~1.1.1]{sh95}. Since $X/Y$ is
proper, it is enough\footnote{What on earth does this mean? The assumption
$X/Y$ proper is enough to ensure that $f_*\Oh_X(iD)$ is coherent? Or, the
assumption that $f_*\Oh_X(iD)$ is coherent is enough for subsequent
purposes?} that the latter sheaf is coherent. Every such sheaf is torsion
free of rank $1$ (a fractional ideal sheaf), integrally closed and known
as {\em {\bi}divisorial\/}. In the {\em noncoherent\/} case it is
reasonable to {\em set\/} $\sD_i=-\infty$, then again $\Oh_X(\sD_i)=0$ is
coherent (cf.\ the proof of Proposition~\ref{texa} below). Note also that
$f_*\Oh_X(\sD_0)=f_*\Oh_X=\Oh_Z$.

Then we can associate to the system a functional $\Oh_Z$-algebra
 \[
 \dival_{X/Z}{\sD_\bull}=\dival_f{\sD_\bullet}\eqdef
\directsum_{n=0}^\infty f_*O_X(\sD_n)
 \]
 that we call a {\em pseudo {\bi}divisorial\/} algebra ({\em pbd
algebra\/}).

Multiplication is well defined, since it is for $X/X$: for
$a\in\Oh_X(\sD_i)$ and $b\in\Oh_X(\sD_j)$, $ab\in\Oh_X(\sD_{i+j})$,
because $(ab)+\sD_{i+j}\ge (a)+(b)+\sD_i+\sD_j$ (cf.\
Definition~\ref{divald}).
 \end{defn}

This definition gives a functorial homomorphism $\dival_{X/Z}{(-)}$ of
systems of {\bi}divisors into functional algebras. It is compatible with
truncations, where the {\em truncation\/} of $\sD_\bull$ is the system
$\sD_i^{[I]}=\sD_{iI}$. To preserve degrees, we make the convention
$\sD_i^{[I]}=\sD_i$ for $I\mid i$ and $\sD_i^{[I]}=-\infty$
otherwise. We say that two systems having identical truncations are {\em
similar\/}; we can replace identity by linear equivalence of truncations.
Two systems $\sD_\bull$ and $\sD_\bullet'$ are {\em linearly
equivalent\/} if $\sD_i=\sD'_i+i\overline{(a)}$ for some $0\ne a\in k(X)$.
Linear equivalence induces an isomorphism of algebras (cf.\
Lemma~\ref{sim}). Moreover, this is the {\em identity\/} on its field of
fractions.\footnote{More precisely, it is multiplication by $a^i$ on
$\sR_i$ and so the identity on the field of homogeneous fractions.} We
always assume this condition for isomorphisms of functional algebras (cf.\
Proposition~\ref{texa}, (8)) that is relevant to general isomorphisms if
we add isomorphisms of systems.

 \begin{cor}\label{pbdrar} Quasi-isomorphism preserves f.g.\ of a pbd
algebra. In particular, a pbd algebra is f.g.\ if and only if any of its
truncation is; or, more generally, similar systems have the same f.g.\
properties.

Moreover, if
 \begin{itemize}
 \item $D_\bull$ is big$/Z$ {\em (cf.\ Remark~\ref{gzardec}, (1))\/}; and
 \item $\sR=\dival_{X/Z}{D_\bull}$ is f.g.,
 \end{itemize}
 then the algebra is g.a.g.\ {\em (cf.\ Corollary~\ref{big})\/}.
 \end{cor}

 \paragraph{Proof--Explanation} F.g.\ is immediate by Truncation
Principle~\ref{trprinc}, and g.a.g.\ follows by Lemma~\ref{adjd} and the
following definitions.

A pbd algebra $\sR=\dival_{X/Z}{\sD_\bull}$ is {\em globally almost
generated\/} or {\em g.a.g.\/} if there is a natural number $N$ and
sections of $\Oh_X(N\sD)$ that generate the algebra
$\dival_{X/X}{\sD_\bull^{[N]}}$ in codimension~$1$ except for a
divisorial subset that is exceptional on $Y=\Proj_Z{\sR}$.

A system $D_\bull$ is {\em big\/}$/Z$ if its algebra $\sR$ is big, that
is, $k(X)/F$ is finite where $F$ is the field of fractions for $\sR$; in
particular, $F=k(Y)$, where $Y=\Proj_Z{\sR}$ whenever $\sR$ is f.g.
\qed\par\medskip

 \begin{exa}\label{rdivalg} Let $\sD$ be an $\R$-{\bi}divisor such that:
 \begin{itemize}
 \item $O_X(i\sD)$ is a coherent $\Oh_X$-module for all $i\in\N$.
 \end{itemize}
 Then the system $(\sD_i=i\sD)$ satisfies the assumptions of
Definition~\ref{pbdad}, and gives a {\em {\bi}divisorial algebra\/}
$\dival_{X/Z}{\sD}=\dival_{X/Z}{(i\sD)}$. The divisorial algebras of
Definition~\ref{divald} for some {\bi}divisor $\sD$ with $\sD_Y=D$ are a
particular case. For example, if $D$ is $\R$-Cartier on model $Y/Z$ of
$X/Z$ we can take $\sD=\Dbar$ by Proposition~\ref{isom}. (For any $\sD$,
we can replace each $\Oh_X(i\sD)$ by its maximal coherent subsheaf
$\Oh_X(\sD)^{ch}$; cf.\ Remark~\ref{cbrm}, (4).)

In general, we need infinitely many models, as in the more typical case
Example~\ref{texam} below.

Thus the difference between (pbd) {\bi}divisorial and divisorial algebras
is the minor one that our divisor is not on $X$ but rather on a different
model $Y$ (models $X_i$). However divisorial is not so helpful from the
f.g.\ point of view (cf.\ Example~\ref{flipal}).

Finally, note that as for divisors, $\sD\sim_{\Q}\sD'$ if and only if the
corresponding systems $(i\sD)$ and $(i\sD')$ are similar. The equivalences
$\sim_{*}$ can be defined for {\bi}divisors as for divisors (see
Definition~\ref{qlin}): we replace $*$-principal divisors $(a)$ by their
{\bi}divisors $\overline{(a)}$.

Thus an equivalence $D\sim_\Q D'$ gives a quasi-isomorphism of the
algebras $\dival_{X/Z}{D}$ and $\dival_{X/Z}{D'}$ provided that both are
functional (cf.\ Corollary~\ref{uniqbssc}). For the converse, see
Proposition~\ref{texa}, (8).
 \end{exa}

 \begin{exa}\label{texam} Let $\bigl(X_i/Z,D_i\bigm|i\in\N\bigr)$ be a
sequence of models $X_i/Z$ of $X/Z$ and their $\R$-Cartier divisors such
that:
 \begin{itemize}
 \item $\sD_i+\sD_j\le \sD_{i+j}$, where $\sD_i=\Dbar_i$.
 \end{itemize}
 Then the system $(\sD_i)$ again satisfies the assumptions of
Definition~\ref{pbdad}.

In fact, by Proposition~\ref{texa} below, an integrally closed functional
algebra $\sL$ is always of this form, but is usually not {\bi}divisorial.
Moreover, we can assume that each $D_i$ is Cartier with
$\Bs{\linsys{D_i}}=\emptyset$ whenever $\sL_i\ne0$. A system with this
property for all $\sD_i$ or for some truncation is {\em birationally
free\/} ({\bi}free). In particular, any {\bi}free system or its truncation
is integral, that is, has integral $\sD_i$ or $D_i$.
 \end{exa}

We now construct an inverse of $\dival_{X/Z}(-)$: namely, the functorial
homo\-morphism sending a functional algebra $\sL$ to its {\em mobile
system\/} $\Mov{\sL}=\sM_\bull$. But first we make the convention either to
accept $-\infty$ as a divisor and {\bi}divisor, or to define $\sM_\bull$
only as a truncation, and only for nontrivial $\sL$.

 \begin{prop}\label{texa}
 \addcontentsline{toc}{subsection}{\ref{texa} Proposition: the mobile system
$\sM_\bull$ of a functional algebra}
For each functional algebra $\sL$ in $k(X)_\bull$ there is a unique system
$\sM_\bull$ of {\bi}divisors of\/ $X$ with the following properties:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item each $(\sM_i)$ is {\bi}free, and in particular, integral;
 \item $\sL\subset \dival_{X/Z}{\sM_\bull}$;
 \item for any system $\sD_\bull$, if $\sL\subset
\dival_{X/Z}\sD_\bull$, then $\sM_\bullet\le\sD_\bullet$ {\em (each
$\sM_i\le\sD_i$)\/}; that is, $\sD_\bull$ is the minimal system that
gives the inclusion of $\sL$;
 \item $\sLbar=\dival_{X/Z}{\sM_\bull}$ and
$\Mov{\sLbar}=\Mov{\sL}=\sM_\bull$; thus
$\sL=\dival_{X/Z}{\sM_\bull}$ if $\sL$ is integrally closed in
$k(X)$;
 \item if\/ $\sD_\bull$ is {\bi}free, then
$\Mov{\dival_{X/Z}{\sD_\bull}}=\sD_\bullet$;
 \item $\sL$ is f.g.\ if and only if\/ $\dival_{X/Z}{\sM_\bull}$ is f.g.;
 \item $\dival_{X/Z}{\sM_\bull}$ is g.a.g.\ with $E=0$ on each model
$Y/Z$ of $X/Z$; and finally,
 \item quasi-isomorphic algebras give similar systems.
 \end{enumerate}
 \end{prop}

 \paragraph{Proof--Construction} We first construct $\sM_i$ for each
$\sL_i\ne0$. (We set $\sM_i=-\infty$ if $\sL_i=0$.) We set
 \[
 \sM_i=\sup\bigl\{-\overline{(s)}\bigm|0\ne s\in\sL_i\bigr\}=-\inf
 \bigl\{\overline{(s)}\bigm|0\ne s\in\sL_i\bigr\},
 \]
 where the sup and inf are taken componentwise, as for {\bi}divisors (the
same applies to max and min in what follows); in particular, the result
is not necessarily a principal divisor. Since $\sL_i\subset k(X)$ is a
finite $\Oh_Z$-module, locally$/Z$, there is a finite set of generators
$0\ne s_i\in k(X)$ for $\sL_i$, and the sup and inf are in fact the max and
min respectively, for {\bi}divisors $\overline{(s_i)}$ considered as
functions over prime {\bi}divisors. In particular, it is a {\bi}divisor,
and its restriction to each model of $X/Z$ is an ordinary divisor.

Indeed, the analogous results for usual principal divisors imply
 \begin{itemize}
 \item if $s\in\Oh_Z\subset k(X)$, then $\overline{(s)}\ge0$; and
 \item for any $s,s'\in k(X)$,
$\overline{(s+s')}\ge\min\bigl\{\overline{(s)},
\overline{(s')}\bigr\}$.
 \end{itemize}
 Therefore, for any $0\ne s\in\sL_i$, since $s=\sum a_is_i$ with all
$a_i\in\Oh_Z$,
 \[
 \overline{(s)}=\overline{(\sum a_is_i)}\ge
 \min\bigl\{\overline{(a_is_i)}\bigr\}=
\min\bigl\{\overline{(a_i)}+\overline{(s_i)}\bigr\}\ge
 \min\{\overline{(s_i)}\}.
 \]

 For each $0\ne s\in\sL_i$ we have $\sM_i\ge-(s)$ or $(s)+\sM_i\ge0$,
which gives the inclusion $\sL_i\subset f_*\Oh_X(\sM_i)$ of (2). However,
for $\dival_{X/Z}{\sM_\bull}$ to be functional, we need it to be
coherent. For this, (1) is enough.

Next, $\sM_i$ is the least divisor for which the inclusion $\sL_i\subset
f_*\Oh_X(\sM_i)$ holds, which proves (3). Indeed, if the same holds for
$\sD_i$ then $\sD_i\ge-(s)$ or $\sD_i+(s)\ge0$ for any $0\ne s\in\sL_i$.
Hence $\sD_i\ge\sM_i$.

On each model $X_i/Z$ of $X/Z$, the restriction $M_i=(\sM_i)_Y$ is {\em
mobile\/}$/Z$, provided that $\sM_i\ne-\infty$ or $\sL_i\ne0$. (In a
certain sense this always holds.) This means that $\Mov{M_i}=M_i$ or,
equivalently, the linear system $\linsys{M_i}$ does not have fixed
components. This proves (7) with $E=0$ on each model of $X/Z$. Moreover,
$\sL_i$ gives its nonempty linear subsystem $L_i$ also without fixed
components (in that sense any functional algebra $\sL$ is g.a.g.\ with
$E=0$).

The linear system $L_i$ only depends on $\sL_i$: every element of
 \[
 L_i=\bigl\{(s)+M_i\bigm|0\ne s\in\sL_i\bigr\}\subset \linsys{M_i}
 \]
 is the restriction to $X_i$ of $\overline{(s)}+\sM_i$, and the general
element does not have fixed components. Indeed, by the definitions
$\Bs{L_i}=\min L_i=0$.

If we take a model $X_i/Z$ on which $L_i$ is free (which exists
by Hironaka), then $M_i$ is also free on $X_i$, and
$\sM_i=\overline{M_i}$. Indeed, for any other model $g\colon X_i'\to X_i$
and general $D_i=(\sD_i)_{X_i}$ in $L_i$, the divisor does not contain the
center of any exceptional divisor, and its birational transform
$D_i'=(\sD_i)_{X_i'}$ does not contain any exceptional divisor. Thus
 \[
 D_i'=g\1D_i=g^*D_i, \quad\sD_i=(s)+\sM_i=\Dbar_i\quad\hbox{and}\quad
\sM_i=\overline{M_i}.
 \]
 Hence the components $\sL_i\ne0$ of functional algebras are equivalent to
linear systems $L_i$ without fixed components.

Now $\sM_{i+j}\ge\sM_i+\sM_j$ for all $i,j$, which completes the proof of
(1). Indeed, since $\sL_{i+j}\supset\sL_i\sL_j$, then
 \begin{align*}
\sM_{i+j} &=\sup\Bigl\{-\overline{(s)}\Bigm|0\ne s\in\sL_{i+j}\Bigr\}\ge
\sup\Bigl\{-\overline{(s)}\Bigm|0\ne s\in\sL_i\sL_j\Bigr\} \\[4pt]
 &\ge\sup\Bigl\{-\overline{(s s')}=-\overline{(s)}-\overline{(s')}
\Bigm|0\ne s\in\sL_i\hbox{ and } 0\ne s'\in\sL_j\Bigr\} \\[4pt]
 &=\sup\bigl\{-\overline{(s)}
\bigm|0\ne s\in\sL_i\bigr\}+\sup\bigl\{-\overline{(s')}\bigm|0\ne s\in
\sL_j\bigr\}=\sM_i+\sM_j.
 \end{align*}
 Each $\sLbar_i=f_*\Oh_X(\sM_i)$, which proves the first statement in (4).
Indeed, $\sLbar_i\subset \overline{f_*\Oh_X(\sM_i)}=f_*\Oh_X(\sM_i)$ by
(2). This last equation corresponds to the completeness of the linear
system for $f_*\Oh_X(\sM_i)$: if $0\ne s\in k(X)$ such that
$s^m+l_1s^{m-1}+\dots+l_{m-1}s+l_m=0$ with all $l_j\in
f_*\Oh_X(j\sM_i)$, or equivalently, $\overline{(l_j)}+j\sM_i\ge0$, then
 \begin{align*}
 m(\overline{(s)}+\sM_i)&=\overline{(s^m)}+m\sM_i
 =\overline{(l_1s^{m-1}+\dots+l_{m-1}s+l_m)}+m\sM_i \\[4pt]
 &\ge \min\Bigl\{\overline{(l_1s^{m-1})},\dots,
\overline{(l_{m-1}s)},\overline{(l_m)}\Bigr\}+m\sM_i \\[4pt]
 &= \min\Bigl\{\overline{(l_1)}+\sM_i+(m-1)(\overline{(s)}+\sM_i),\dots,
 \\
 & \kern1.6cm \overline{(l_{m-1})}+(m-1)\sM_i+(\overline{(s)}+\sM_i),
\overline{(l_m)}+m\sM_i\Bigr\} \\[4pt]
 &\ge \min\Bigl\{(m-1)(\overline{(s)}+\sM_i),\dots,
\overline{(s)}+\sM_i\Bigr\},
 \end{align*}
 and $\overline{(s)}+\sM_i\ge0$. Hence $s\in f_*\Oh_X(\sM_i)$.

On the other hand, each element $s\in f_*\Oh_X(\sM_i)$ is integral for
$\sL_i$, which gives the opposite inclusion $f_*\Oh_X(\sM_i)\subset
\sLbar_i$. The linear systems $L_i$ and $\linsys{M_i}$ induce a morphism
$g\colon \Proj_Z{\dival_{X/Z}{\sM_i}}=Y'\to\Proj_Z{\directsum\sL_i^j}=Y$
that is finite, because it is quasi-finite and proper. Hence the
{\bi}divisorial algebra $\dival_{X/Z}{\sM_i}$, isomorphic to
$\dival_{Y/Z}{\sA[D]}$, is finite as a module over the product functional
algebra $\directsum\sL_i^j\into \dival_{Y/Z}{D}$ (under the above
isomorphism) by Example~\ref{final}, where $\sA=g_*\Oh_Y=h_*\Oh_{X_i}$,
$h\colon X_i\to Y$, and $\Oh_Y(D)=\Oh_Y(1)$ or $\sM_i\sim \overline{h^*D}$
with very ample $D$ on $Y/Z$. Indeed, $\directsum
\sL_i^j\iso\dival_{Y/Z}{D}$ for all degrees $i\gg1$ because $\sL_i$ is
isomorphic to $e_*\Oh_Y(1)$, $e\colon Y\to Z$. In particular, each $s\in
f_*\Oh_X(\sM_i)$ is integral, because the sequence
 \[
 (1)\subset (1,s)\subset \dots\subset (1,s,\dots,s^{m-1})=(1,s,\dots,
s^{m-1},s^m)
 \]
 of $\directsum\sL_i^j$-submodules terminates by the Noetherian property.

We have $\Mov{\sLbar}\le\sM_\bull$ by (3), since $\dival_{X/Z}=\sLbar$.
Conversely, $\sM_\bull\le\Mov{\sLbar}$ by (3) again, because
$\sL\subset\sLbar$ implies $\sM_\bull=\Mov{\sL}\le\Mov{\sLbar}$. This
completes the proof of (4).

 (5) follows by construction: a {\bi}free divisor is the minimal one
giving any linear system.

If $\sL$ is f.g., by Truncation Principle~\ref{trprinc}, we can assume that
it is generated by $\sL_1$. In addition, we can assume that
$\dival_{X/Z}{\sM_1}$ gives the normalization
$\Proj_Z{\dival_{X/Z}{\sM_1}}$ of $Y=\Proj_Z{\sL}$ in $k(X)$. Then
$\sLbar=\dival_{X/Z}{\sM_\bull}=\dival_{X/Z}{\sM_1}$ up to a finite
number of components (by normality for high multiples of a very ample
divisor). Hence $\dival_{X/Z}{\sM_\bull}$ is a finite module over
$\dival_{X/Z}{\sM_1}$. But in turn, this last algebra is a finite
$\sL$-module by Example~\ref{final} and the above arguments. Hence
$\dival_{X/Z}{\sM_\bull}$ is also a finite $\sL$-module and is f.g.\ by
the Hilbert basis theorem.

Conversely, suppose that $\dival_{X/Z}{\sM_\bull}$ and $f_*\Oh_X(M_1)$
generate the algebra; in particular, $\dival_{X/Z}{\sM_\bull}=
\dival_{X_1/Z}{M_1}$. Then each $\sL_i$ defines a finite morphism
$Y=\Proj_Z{\dival_{X_1/Z}{M_1}}\to Y_i=\Proj_Z{\directsum\sL_i^j}$.
Moreover, the inclusions $\sL_i^j\subset\sL_{i j}$ define finite
morphisms $Y_{ij} \to Y_i$. However, this is only possible under a
stabilization: $Y_{ij}\iso Y_i$ for some $i\gg1$ and all $j\ge1$. This
implies that $\sL^{[i]}$ is f.g.\ because $\sL_i\iso e_*\Oh_{Y_i}(1)$
generates $\sL_{ij}$ for all $j\gg1$, and completes the proof of (6).

Finally, suppose that algebras $\sL$ and $\sL'$ are quasi-isomorphic, that
is, isomorphic up to a truncation. Since we assume that the isomorphism
induces the identity on their fields of fractions, each nontrivial $\sL_i$
is isomorphic to $\sL_i'$ under a multiplication $t\mapsto s_it$ for a
unique $0\ne s_i\in k(X)$. Since this is a homomorphism of algebras,
$s_is_j=s_{ij}$. Then we can find $s_d\in k(X)$ (even if $\sL_d=0$) such
that each $s_{i d}=s_d^i$, where $d$ is the gcd of $j$ with $\sL_j\ne0$.
This implies the linear equivalence $\sM_\bull\sim
\sM_\bull'=\Mov{\sL'}$ and completes the proof of (8).
\qed\par\medskip

On the way we have proved the following two corollaries:

 \begin{cor}\label{sacr} $\sL$ is f.g.\ if and only if its mobile system
$\sM_\bull$ is {\bi}divisorial up to a truncation: there exists a
natural number $I$ such that $\sM_{iI}=i\sM_I$ for all $i\ge1$.
 \end{cor}

 \begin{cor}\label{norm} If $\sL$ or $\sLbar$ is f.g., the operation of
taking projective spectrum induces the normalization\/ $\Proj_Z{\sLbar}\to
\Proj_Z{\sL}$ of\/ $\Proj_Z{\sL}$ in $k(X)$. In particular, if it is
birational, it is the normalization of\/ $\Proj_Z{\sL}$.
 \end{cor}

By Proposition~\ref{texa}, (6), it is enough to consider pbd algebras to
establish a criterion for f.g.\ (cf.\ Theorem~\ref{limcr}). By
Corollary~\ref{sacr} most pbd algebras are not f.g.

 \begin{exa}\label{toosl} Let $C/C$ be a curve locally. If
$D_\bull=(-P,-P,\dots,-P,\dots)$, the pbd algebra $\dival_{C/C}{D_\bull}$
is not f.g.
 \end{exa}

 \begin{exa}\label{tooqu} Let $C/C$ be a curve locally. If
$D_\bull=(P,2^2 P,\dots,n^2 P,\dots)$, the pbd algebra
$\dival_{C/C}{D_\bull}$ is not f.g.
 \end{exa}

In this last case, the algebra is not f.g.\ since its mobile system grows
too fast.

 \begin{defn} We say that a functional algebra $\sL$ is {\em bounded\/} by
a {\bi}divisor $\sD$ of $X$ if $\sL$ is a subalgebra of $\dival_{X/Z}{\sD}$.
 \end{defn}

We will see in Proposition~\ref{bpbdal} that it is equivalent for $\sL$
to be bounded by a divisor.

 \begin{exa}[Restricted algebra]\label{bra} If we compare
Definitions~\ref{restal} and \ref{rfad}, we see that in the first, the
restriction to $Y$ of the divisorial algebra $\dival_{X/Z}{D}$ is bounded:
$\dival_{X/Z}{D}\rest Y\subset \dival_{Y/Z}{D\rest Y}$. Moreover, in this
inclusion, we can replace $D$ by its {\bi}divisor $\sD=\Dbar$, and $D\rest
Y$ by $\sD\rest Y=\Dbar\rest Y$ (see \ref{frest} for fixed restriction of
{\bi}divisors).

In particular, each algebra of type $\RFA_{n,d}$ is bounded. This is a
good sign, but not surprising, because it holds for any bounded algebra
(cf.\ Corollary~\ref{brest}).
 \end{exa}

 \begin{prop}\label{bpbdal} Let $\sL$ be a functional algebra and
$\sM_\bull=\Mov{\sL}$ its mobile system. Then $\sL$ is bounded if and
only if any one of the following holds:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $\sL$ is\/ {\em divisorially bounded\/}, that is, there exists a
Cartier divisor $D$ on $X$ such that\/
$\sL\subset\dival_{X/Z}{D}=\dival_{X/Z}{\Dbar}$;
 \item the {\em characteristic system\/} $(\sD_i)=(\sM_i/i)$ is bounded,
that is,\/ {\em there exists a {\bi}divisor $\sD$ such that all
$\sD_i\le\sD$\/}; and
 \item the algebra is\/ {\em convergent\/}, that is, {\em the limit
 \[
 \sD=\lim_{i\to\infty}\sD_i,
 \]
 exists}, where we consider only $\sD_i\ne-\infty$.
 \end{enumerate}
 \end{prop}

 \begin{add} In addition, the limit $\sD$ in (3) satisfies
 \begin{q}
 \item[\MXD] each $\sD_i=\sM_i/i\le\sD$; and
 \item[\BSD] all the $\sD_i$ are supported in a {\bi}divisor, that is,
$\Supp{\sD_i} \le S$ for a fixed reduced {\bi}divisor $S$. (Here we set\/
$\Supp(-\infty)=0$.)
 \end{q}
 \end{add}
Thus to say that an algebra is bounded restricts its characteristic divisors
$\sD_i$ in two ways: i.e., in multiplicities and in supports.

 \begin{lem}\label{divb} Let $\sM$ be a $\R$-{\bi}divisor that is {\em
{\bi}nef\/}$/X$, and $H$ an $\R$-Cartier divisor on $X$ such that $\sM\le
H$. Then $\sM\le\overline{H}$.

In particular, this holds if $\sM$ is {\bi}semiample.
 \end{lem}

 \paragraph{Proof--Explanation} In general, we say that an $\R$-{\bi}divisor
$M$ is {\em {\bi}nef\/}$/Z$ (or {\em {\bi}semipositive\/}) if it is nef
$\R$-Cartier, that is, there is a model $Y/Z$ of $X/Z$ such that
 \begin{itemize}
 \item $\sM_Y$ is a nef$/Z$ $\R$-Cartier divisor on $Y$, and
 \item $\sM=\overline{M_Y}$.
 \end{itemize}
 {\bi}semiample (see Section~\ref{divalg}) implies {\bi}nef. {\em Numerically
{\bi}seminegative\/} is defined in a similar way.

For the lemma, we need to prove that $\overline{H}-\sM$ is effective. This
is immediate by Negativity Lemma~\ref{negat}, because $\overline{H}-\sM$ is
numerically {\bi}seminegative$/X$, and $(\overline{H}-\sM)_X=H-\sM_X\ge0$.
\qed\par\medskip

 \begin{lem}\label{arithm} Let $\sM_\bull$ be a system of\/
$\R$-{\bi}divisors satisfying
 \[
 \sM_i+\sM_j\le\sM_{i+j}.
 \]
 (compare Example~\ref{texam}). Then the system $(\sD_i)=(\sM_i/i)$ has
the following properties:
 \begin{description}
 \item[convexity]
 $\sD_{i+j}\ge (i\sD_i+j\sD_j)/(i+j)$ for all $i,j$; thus
 \item[arithmetic monotonicity] $\sD_{ij}\ge\sD_i$;
 and
 \item[convergence] $\sup\{\sD_i\}=\lim_{i\to\infty}\sD_i$; in particular,
$\sD_\bull$ is convergent if and only if it is bounded from above. In this
case, all the\/ $\sD_i$ have common support as in \BSD\ of
Addendum~\ref{bpbdal}.1
 \end{description}
 \end{lem}

 \begin{warn}\label{wlsup} We should really write $\limsup$ rather than
$\lim$; so, to be more precise, we make the convention that we {\em omit
all\/} $\sD_i=-\infty$ in the sequence! The limits are taken componentwise
and possibly {\em not uniformly\/}; and the limit itself may not be a
{\bi}divisor (cf.\ \BSD\ of Addendum~\ref{bpbdal}.1).
 \end{warn}

If $\sM_\bull$ is a {\bi}free system, that is, $\sM_\bullet=\Mov{\sL}$
for a pbd algebra $\sL=\dival_{X/Z}{\sM_\bull}$, then by arithmetic
monotonicity $\sL$ has subalgebras
 \[
 \sL^i=\dival_{X/Z}{(j\sM_i/i)'},
 \]
where $(j\sM_i/i)'$ is the system $(j\sM_i/i)$ with $j\sM_i/i$ replaced
by $-\infty$ whenever $i\nmid j$. The algebra is quasi-isomorphic to a
divisorial algebra $\dival_{X/Z}{\sM_i}$, and is bounded by
$\dival_{X/Z}{\sM_i/i}$. The total algebra $\sL$ is an inductive limit
$\lim_{-|-}\sL^i$ for inclusions $\sL^i\subset\sL^j$ with $i\mid j$.
{From} this point of view, it is {\em pseudo {\bi}divisorial\/}, but not
{\bi}divisorial, since the limit of {\bi}divisors does not always exist,
and, if it exists under the convergence of the lemma, its algebra may
be different from $\sL$ (even up to quasi-isomorphism, cf.\
Example~\ref{toosl}), or worse, may not be defined. The only case the
inductive limit is well defined as a {\bi}divisorial algebra up to a
quasi-isomorphism is when it {\em stabilizes\/}, that is, $\sL$ is
quasi-isomorphic to $\sL^i$ for some $i\gg1$ (cf.\ Limiting Criterion
below). This means f.g.

 \begin{proof} Convexity means
 \[
 \sM_{i+j}=(i+j)\sD_{i+j}\ge i\sD_i+j\sD_j=\sM_i+\sM_j.
 \]
 Thus by induction on $j$
 \[
 \sD_{ij}\ge (i(j-1)\sD_{i(j-1)}+i\sD_i)/ij\ge
(i(j-1)\sD_i+i\sD_i)/ij=\sD_i.
 \]
 It is enough to prove convergence componentwise, that is, for a numerical
sequence $\sD_i$. Moreover, it follows from the existence of
$\sD=\lim_{i\to\infty}\sD_i$ and from its upper bound (cf.\
Proposition~\ref{bpbdal}, \MXD), that is, for all $i$,
 \[
 \sD_i\le\sD,
 \]
 where in the limit we consider only for $\sD_i\ne-\infty$, but the limit
can be $+\infty$. The case that all $\sD_i=-\infty$ is trivial.

Otherwise, by monotonicity, for the gcd $d$ of the indices $i$ with
$\sD_i\ne-\infty$, all $\sD_j\ne-\infty$ with $d\mid j\gg1$, and all
$\sD_j=-\infty$ with $d\nmid j$. After truncating by $d$, we can assume
that all $\sD_i\ne-\infty$ with $i\gg1$.

Again by monotonicity, we have an increasing subsequence $\sD_{i^j}$ for
$\sD_i\ne-\infty$. It converges to $\sD$ that is a real number if the
sequence is bounded from above, or to $+\infty$. Thus we have a convergent
subsequence $\sD_{i^j}\to\sD$.

This gives the required limit. For any $\ep>0$, there exists $j\ge1$ such
that $\sD_j>\sD-\ep /2$ (or $>\ep/2$ if $\sD=+\infty$), and all
$\sD_i\ne-\infty$ with $i\ge j$. Then, for all $i\gg1$, $i=j q+r$ with
$j\le r\le2 j$, $0<j q$, and
 \[
 \sD_i\ge \frac{j q} i\sD_{j q}+\frac r i\sD_r\ge
\frac{j q} i\sD_j+\frac r i\sD_r>
\frac{j q} i (\sD-\frac \ep 2)+\frac r i\sD_r>\sD-\ep
 \]
 (or $>\frac{j q} i\frac \ep 2+\frac r i\sD_r>\ep$). Thus
$\liminf_{i\to\infty}\sD_i\ge\sD$.

Since this holds for any convergent subsequence, $\lim_{i\to\infty}\sD_i$
exists, is equal to $\sD$, and by arithmetic monotonicity we have the
inequality $\sD_i\le\sD$.

Finally, to find the common support $\sS$ of the {\bi}divisors $\sD_i$, we
can again assume that $\sD_i\ne-\infty$ for all $i\gg1$, or say, for all
$i\ge j\ge1$. Then as above, for all $i\gg1$
 \[
 \sD_i\ge\frac{j q} i\sD_j+\frac r i\sD_r,
 \]
 where $j\le r\le2 j$. On the other hand, if the characteristic system is
bounded by $\sD$, then all $\sD_i\le\sD$. Thus for all $i\gg1$,
$\Supp{\sD_i}$ is in the union of $\Supp{\sD}$ and of all $\Supp{\sD_r}$.
(If $\sD\ge\sD_i\ge\sD'$, then $\Supp{\sD_i}$ is in the union of
$\Supp{\sD}$ and $\Supp{\sD'}$.) Taking the union of $\Supp{\sD_i}$ not
included in the $i\gg1$ gives $\sS$.
 \qed\end{proof}

 \begin{pfof}{Proposition~\ref{bpbdal}} Let $\sL$ be an algebra bounded by
a {\bi}divisor $\sD$. Then, by Proposition~\ref{texa}, (3) each $\sM_i\le
i\sD$, which implies (2). Conversely, (2) together with
Proposition~\ref{texa}, (2) imply the inclusion $\sL\subset
\dival_{X/Z}{\sD}$, provided that $\sD$ satisfies the coherence of
Example~\ref{rdivalg}. In fact, we can choose the required $\sD$ as the
closure $\Dbar$, where $D$ is a Cartier divisor on $X$ such that $D\ge D_X$.
Moreover such $D$ exists among hyperplane sections of $X/Z$.

Thus each $(\sM_i)_X\le(i\sD)_X=i(\sD_X)\le i D$. Hence each $\sM_i\le
\overline{iD}=i\Dbar$ by Lemma~\ref{divb}, which implies (1) and boundedness.

Finally, by Lemma~\ref{arithm}, (2) is equivalent to (3) and
Addendum~\ref{bpbdal}.1 with a {\bi}divisor $\sD$.
 \qed\end{pfof}

 \begin{cor} Quasi-isomorphism of algebras preserves their boundedness.
Moreover, the corresponding {\em characteristic limit\/} satisfies
$\sD'\sim_\Q r\sD$ for some positive $r\in\Q$ {\em (cf.\
Corollary~\ref{uniqbssc})\/}.
 \end{cor}

 \begin{proof} Immediate by Proposition~\ref{texa}, (8) and
Proposition~\ref{bpbdal}, (2--3).
 \qed\end{proof}

 \begin{cor}\label{brest} The restriction of a bounded algebra is also
bounded\/ {\em (provided it is well defined)\/}.
 \end{cor}

 \paragraph{Proof--Explanation} Immediate by Proposition~\ref{bpbdal}, (1),
because $\sL\rest Y\subset \dival_{Y/Z}{D\rest Y}$. The existence of a
Cartier divisor $D$ such that $D\rest Y$ is well defined follows from the
proof of Proposition~\ref{texa}, (1). Indeed, $\sL\rest Y$ is well defined
if $Y\subset X$ is not contained in the poles of any $0\ne s\in\sL$. We
then say that $Y$ is {\em in general position\/} with respect to $\sL$, or
vice versa, that $\sL$ is in general position with respect to $Y$ (cf.\
Definitions~\ref{restal} and \ref{rfad}). Equivalently, the positive
component of each {\bi}divisor $\sM_i$ in $\Mov{\sL}$ or of its
characteristic system does not contain $Y$. The same should hold for the
limit $\sD$. Then we can find $D\ge\sD_X$ such that its positive component
does not contain $Y$.

Note that the restriction $\sD\rest Y$ is usually not well defined, but
$\lim_{i\to\infty}\sD_i\frest Y$ exists if $\sL$ is bounded.\footnote{cross
ref to Fixed Restriction~\ref{frest} for the future notation $\frest Y$.}
\qed\par\medskip

 \begin{thm}[Limiting Criterion]\label{limcr}
 \addcontentsline{toc}{subsection}{\ref{limcr} Limiting criterion}
A functional algebra $\sL$ is f.g.\ if and only if the limit\/
$\sD=\lim_{i\to\infty}\sD_i$ {\em stabilizes\/}, that is, {\em
$\sD=\sD_i$ for some $i\gg1$\/}.

Under either condition, the limit\/ $\sD$ is a {\bi}semiample $\Q$-divisor;
the algebra and its characteristic system are bounded.
 \end{thm}

 \begin{proof} Immediate by Corollary~\ref{sacr} and the arithmetic
monotonicity of Lemma~\ref{arithm}. Note that stabilization is preserved
under similarity of systems.

In general, stabilization does not mean that $\sD=\sD_i$ for all $i\gg1$.
However, this holds up to a truncation, essentially by the above
monotonicity (cf.\ Corollary~\ref{assdesc}).
 \qed\end{proof}

By Example~\ref{toosl}, for an algebra to be f.g., it is not enough that
its limit $\sD$ is a {\bi}semiample $\Q$-divisor.

 \begin{exa}\label{pbsa} \label{moval} We now discuss the case of
{\bi}divisorial algebras. Let $\sD$ be an $\R$-{\bi}divisor as in
Example~\ref{rdivalg}. Its {\em characteristic limit\/} is the
characteristic limit of its algebra $\dival_{X/Z}{\sD}$, that is,
$\sD^m\eqdef\lim_{i\to\infty}\sD_i$. This exists because $\dival_{X/Z}{\sD}$
is bounded by itself, and $\sD^m\le\sD$ by Proposition~\ref{texa}, (3) (but
not always $=\sD$). Thus we have a decomposition
 \[
 \sD=\sD^m+\sD^e,
 \]
 where $\sD^e=\sD-\sD^m\ge0$, and $\sD^m$ is the {\em maximal pseudo
{\bi}semiample\/} part (or {\bi}divisor) of $\sD$ (or {\em maximal pbs
ample\/} part of $\sD$ for short), and $\sD^e$ is the {\em fixed part\/} of
$\sD$. Indeed, the {\em {\bi}semiample part} of $\sD'$ is a {\bi}semiample
$\R$-{\bi}divisor $\sD'\le\sD$. A {\em pseudo {\bi}semiample\/} divisor is
a limit of these. In particular, $\sD=\sD^m$ itself is pbs ample if and
only if $\sD^e=0$.

This generalizes the decompositions of Remark~\ref{gzardec}. Indeed, if
$D=D^m+E$ is as in the remark, then $D=\sD^m_X$ for some {\bi}semiample
$\R$-{\bi}divisor $\sD^m$. Thus if we take $\sD=\sD^m+E$, then $D=\sD_X$,
$D^m=\sD_X^m$, and $E=\sD^e$.

In general, $\sD^m$ is not always {\bi}semiample.

If $\dival_{X/Z}{\sD}$ is f.g., then $\sD^m$ is {\bi}semiample by Limiting
Criterion~\ref{limcr} (cf.\ Remark~\ref{gzardec}); and, moreover, $\sD^m$ is
then a $\Q$-{\bi}divisor. However g.a.g.\ can fails, even if $\sD$ is
big$/Z$, because Lemma~\ref{adjd} does not hold for any prime {\bi}divisor
$E$. The latter is related to saturation, as we discuss below.

We do not expect f.g.\ for an arbitrary {\bi}divisor $\sD$, even in the
situation of Remark~\ref{gzardec}, (2) (cf.\ Theorem~\ref{existd}). This
can again be attributed to saturation (cf.\ Conjecture~\ref{conj!fga} and
Example~\ref{satur} below).
 \end{exa}

But before we finish

 \begin{pfof}{Corollary~\ref{stupidc}} For $\Q$-divisors, immediate by
Corollary~\ref{pbdrar}; and g.a.g.\ holds with empty base divisorial locus
or $E=0$ (cf.\ Proposition~\ref{texa}, (7)).

For a non $\Q$-divisor $D$, ample$/Z$, $\dival_{X/Z}{D}$ is never f.g.\
(cf.\ Stupid Example~\ref{exa!st}). Indeed, since $\sD=\Dbar$ is
({\bi}semi-)ample, and $\dival_{X/Z}{\sD}=\dival_{X/Z}{D}$ is f.g., then
$\sD=\sD^m$ is a $\Q$-divisor by Example~\ref{pbsa}.
 \qed\end{pfof}

 \subsection{Saturation of linear systems}
 Although saturation can sometimes be explained in terms of sheaves and
their algebras, essentially in the situations we need, a better
explanation (arguably the only possible explanation) uses divisors and
linear systems. Let $D$ be an $\R$-Weil divisor on $X/Z$. We recall that
its linear system is the set
 \[
 \linsys{D}=\linsys{D}_{X/Z}=\bigl\{D'\bigm| D'\ge0 \text{ and }D'\sim
 D/Z\bigr\}.
 \]
 In its local version over a point $P\in Z$, $\linsys{D}$ is defined up to
the equivalence relation that identifies divisors $D'$ that are equal over
a neighborhood of $P$. The most important thing for us is its decomposition
into mobile and fixed parts.

Similarly, for any $\R$-{\bi}divisor, we can define $\R$-{\bi}divisors
$\Mov{\sD}$ and $\Fix{\sD}$, the {\em mobile\/} and {\em fixed parts\/}
of $\sD$ (or of its linear system $\linsys{\sD}$, or sheaf
$f_*\Oh_X(\sD)\subset k(X)$); for the definition, see the
proof--construction of Proposition~\ref{texa}). By definition,
$(\Mov{\sD})_Y\le\Mov{(\sD_Y)}$ on any model $Y/Z$ of $X/Z$. However, if the
sheaf $f_*\Oh_X(\sD)$ is coherent, then, by Proposition~\ref{texa}, (1) and
(4), we have the following stabilization on a {\em sufficiently high\/}
resolution $X\hr$. There is a model $g\colon X\hr\to Z$ of $X/Z$
such that $f_*\Oh_X(\sD)=g_*\Oh_{X\hr}(M)$, where
$M=(\Mov{\sD})_{X\hr}=\Mov{(\sD_{X\hr})}$, $\Mov{\sD}=\Mbar$,
$\Bs{\linsys{M}}=\emptyset$, and
$\linsys{M}+\Fix{\sD_{X\hr}}=\linsys{\sD_{X\hr}}=\linsys{\sD}_{X\hr}$.

 \begin{defn}\label{asatur} Let $D$ and $C$ be two $\R$-divisors on $X$. We
say that $D$ is {\em saturated} with respect to $C$ or $C$-{\em saturated\/}
if $M=\Mov{\rdup{D+C}}\le D$. By convention, we agree that the condition
holds if $\linsys{\rdup{D+C}}=\emptyset$, i.e., $\mult_{D_i}{M}=-\infty$;
we also don't need to verify the condition if $\rdup{D+C}\le D$.

We can attribute the same saturation to the linear system $\linsys{D}$, or
even to any (incomplete) linear system, as well as to a functional sheaf
$\sF\subset k(X)$. In this last case, we take $D=\Mov{\sF}$. Moreover, if
$\sF=f_*\Oh_X(D)$ then $C$-saturation of $D$ implies the same for $\sF$
(cf.\ Remark~\ref{asaturr}, (1) below).

For {\em $\R$-{\bi}divisors\/} $\sD$ and $\sC$, the {\em saturation\/} of
$\sD$ with respect to $\sC$ or $\sC$-saturation means the
$C=\sC_{X\hr}$-{\em saturation\/} of $D=\sD_{X\hr}$ on any {\em
sufficiently high models\/} $X\hr/Z$ of $X/Z$. This means that there is
a model $Y/Z$ of $X/Z$ such that saturation holds for any model $X\hr/Z$
that is also $/Y$.

For asymptotic saturation, we consider a system $\sD_\bull$ of
$\R$-{\bi}divisors. This system is {\em asymptotically saturated\/} with
respect to $\sC$ or {\em asymptotically $\sC$-saturated} if there exists
a natural number $I$, the {\em saturation index\/}, such that, for all
natural numbers $i,j$ with $I\mid i,j$, and on any sufficiently high model
$X\hr/Z$ the decomposition into mobile and fixed parts
satisfies:\footnote{Should it be $j\sD_j\mapsto j\sD_i$ on the right-hand
side?}
 \[
 \Mov{(\rdup{j\sD_i+\sC}}_{X\hr})\le j(\sD_j)_{X\hr}
 \]
 (cf.\ Remark~\ref{asaturr}, (2--3) below).

A functional algebra is {\em asymptotically $\sC$-saturated\/} if this
holds for its characteristic system $\sD_\bull$. Moreover, a pbd algebra
associated with a system $\sM_\bull$ is asymptotically $\sC$-saturated
if this holds for its normalized system $(\sD_i=\sM_i/i)$
(cf.\ Remark~\ref{asaturr}, (1)).

Finally, we say that an $\R$-{\bi}divisor $\sD$ is {\em asymptotically
$\sC$-saturated\/} if this holds for its system $(\sD_i=\sD)$ (cf.\
Example~\ref{rdivalg} above). Even for a Cartier {\bi}divisor $\sD=\Dbar$,
we need to run through all sufficiently high models $X\hr/Z$ (cf.\
Example~\ref{satur} below). The point is that we have no universal
$X\hr/Z$ unless the {\bi}divisorial algebra of $\sD$ is f.g.\ (cf.\
Remark~\ref{asaturr}, (2) below).
 \end{defn}

 \begin{rem}\label{asaturr}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item It is enough to verify the inequality
 \[
 M=\Mov{\rdup{D+C}}\le D
 \]
 over $\rdup{D+C}>D$, that is, to check that $\mult_{D_i}{M}\le
\mult_{D_i}{D}$ for any prime divisors $D_i$ with
$\rdup{\mult_{D_i}{D}+\mult_{D_i}{C}}>\mult_{D_i}{D}$. The same holds for
the other saturations.

This implies that $\Mov{\rdup{D+C}}\le\Mov{D}$. In most of our applications,
$D$ is mobile, or even free. Then $\Mov{D}=D$ and saturation
is equivalent to $\Mov{(D+\rdup{C})}=\Mov{\rdup{D+C}}\le \Mov{D}=D$. Thus
if in addition $\rdup{C}\ge0$, then $D\le \Mov{(D+\rdup{C})}\le D$, and
$\Mov{(D+\rdup{C})}=D$ or, in terms of linear systems,
 \[
 \linsys{D+\rdup{C}}=\linsys{D}+\rdup{C},
 \]
 where $\rdup{C}=\Fix{(D+\rdup{C})}$.

 \item In general, $(\Mov{\rdup{j (\sD_i)+\sC})}_{X\hr}\le\Mov{(\rdup{j
(\sD_i)+\sC}}_{X\hr})$, but equality does not necessarily hold. In
important cases, equality holds on any sufficiently high model $X\hr$;
this means that $\Mov{(\rdup{j (\sD_i)+\sC}_{X\hr})}$ stabilizes on {\em
all\/} sufficiently high models, but only for given $i,j$ (cf.\
Examples~\ref{satur} and \ref{coha} below). In applications, however we
need {\em some\/} sufficiently high model $X\hr$ on which, say, some
divisors have normal crossing, and some {\bi}divisors are Cartier (cf.\
the proof of Proposition~\ref{surbound}). But we prefer to consider all
sufficiently high models $X\hr$ because in most cases it is easy to
verify saturation for all such $X\hr$ by stabilization (cf.\
Proposition~\ref{coher} below).

 \item If $\sD_\bull$ is the characteristic system of a functional algebra
with mobile system $\sM_\bull$, and $i\mid j$, then as in (1) above,
asymptotic saturation means that
 \[
 M=\Mov{(j D_i+\rdup{C})}=\Mov{\rdup{j D_i+C}}\le\Mov{j D_j}=j D_j
 \]
 with $C=\sC_Y$, $D_i=(\sD_i)_Y$, $D_j=(\sD_j)_Y$ on any sufficiently high
model $X\hr=Y/X$, where $iD_i=\Mov{(\sM_i)}_Y$, $j D_j=\Mov{(\sM_j)}_Y$
and both linear systems $\linsys{iD_i}$ and $\linsys{jD_j}$ are free (cf.\
Proposition~\ref{texa}, (1)). Thus in terms of linear systems,
 \[
 \linsys{\Mov{(j D_i+\rdup{C})}}=\linsys{M}+F,
 \]
 where $M\le M_j=j D_j$ and $F=\Fix{(j D_i+\rdup{C})}$. This can be
interpreted in terms of sheaves: namely
 \begin{align*}
 f_*\Oh_Y(M_i)^{\tensor\frac{j}{i}} \tensor_{\Oh_Y}\Oh_Y(\rdup{C})
 &= f_*\Oh_Y(j D_i+\rdup{C}) \\
 &\subset f_*\Oh_Y(j D_j)=f_*\Oh_Y(M_j),
 \end{align*}
 where $M_i=i D_i$ and $\rdup{C}$ is assumed to be Cartier. However in
most applications we consider $i\gg j$, and then asymptotic saturation has
no sheaf theoretic interpretation (cf.\ Example~\ref{satur} below) because
the integral parts depend on divisors up to a linear equivalence, not up
to $\Q$-linear equivalence.

 \item Again, if $\sD_\bull$ is the characteristic system of a functional
algebra, then by the arithmetic monotonicity of Lemma~\ref{arithm}, we
have inequalities
 \[
 \Mov{(\rdup{j (\sD_i)+\sC}}_{X\hr})\le j(\sD_j)_{X\hr}\le j(\sD_l)_{X\hr}\le j(\sD)_{X\hr}
 \]
 with any $j\mid l$ and limiting divisor $\sD$. In fact, this last inequality
for $\sD$ is the most important (cf.\ Example~\ref{1dfga}).

 \item For $j=i$ the asymptotic saturation inequality implies
 \[
 \Mov{(\rdup{\sM_i+\sC}}_{X\hr})=
 \Mov{(\rdup{i (\sD_i)+\sC}}_{X\hr})\le
i(\sD_i)_{X\hr}=(\sM_i)_{X\hr}
 \]
 that means the $\sC$-saturation for $\sM_i=i\sD_i$. The same holds for
each component $\sL_i$ of a functional algebra provided that $\sL$ is
asymptotically saturated.

 \item In general, asymptotic $\sC$-saturation does not imply that
$\sD_\bull$ is bounded (cf.\ Example~\ref{exc}). But this does holds for
functional algebras and their characteristic systems. By Lemma~\ref{divb},
it is enough to establish this on any model $Y/Z$ of $X/Z$, for example,
on $X$ itself. Then $\sD_\bull$ is convergent and has other properties
(cf.\ $\FGA_n$ and Conjecture~\ref{conj!fga} below). However we always
consider bounded algebras, as they appear in our applications.

 \item Note also that {\em similarity\/} of characteristic systems is induced
by similarity of $(\sM_i=i\sD_i)$: e.g.,
 \begin{itemize}
 \item $\sD_\bull\sim\sD_\bullet'$, where
$\sD_\bull=\sD_\bullet'+\overline{(a)}$; and
 \item truncation means
$\sD_\bull^{[I]}=I(\sD_{iI})$, that is, {\em each\/}
$\sD_i^{[I]}=I\sD_{iI}$.
 \end{itemize}
 Similarity preserves asymptotic $\sC$-saturation for any system
$\sD_\bull$ (see Proposition~\ref{linvsat} below).
 \end{enumerate}
 \end{rem}

We discuss other properties of saturations below. In most applications,
the saturation is {\em essentially exceptional\/}: i.e., $\sC$ or
$\rdup{\sC}$ are exceptional on $X$.

 \begin{exa}[Exceptional saturation]\label{satur} We say that the
saturation of Definition~\ref{asatur} is {\em exceptional$/X$\/} if it
holds for any $\sC$ that is exceptional on $X$.

Exceptional saturation holds in particular for any {\em integral\/} $D$ on
$X/X$ (cf.\ Example~\ref{1dfga} and\footnote{I don't understand what this
has to do with Example~\ref{1dfga} and I can't find any related ``remark
below''.} the remarks below); this means that it is also exceptionally
saturated as a {\bi}divisor. Since $\sC$ is quite arbitrary, we only need
to worry about the inequality
$\Mov{\rdup{D+\sC_{X\hr}}}=\Mov{(D+\rdup{\sC_{X\hr}})}\le D$ for a
prime divisor nonexceptional on $X$. Thus on $X\hr$ we add to $D$ any
big divisor that is exceptional on $X$. This usually taken for granted in
classical algebraic geometry when we consider Cartier divisors (cf.\
Lemmas~\ref{adjd} and \ref{divb}). Indeed, if $D$ is Cartier, then its
closure $\sD=\Dbar$ is saturated and $\Oh_X(D)=\Oh_X(\sD)$ but this last
equality does not hold if $D$ is considered as a Weil {\bi}divisor.
Exceptional saturation makes it.\footnote{This does not make sense: makes
it \dots do what? Possibly: Assuming that $D$ is exceptionally
saturated/$X$ makes the inequality $\Oh_X(D)=\Oh_X(\sD)$ true?}

The sheaf $\Oh_X(D)$ is also saturated. For divisorial sheaves, this
occurs as
 \[
 \Oh_X(D)^{\vee\vee}=\Oh_X(D)
 \]
 (or \cite[Proposition~2, (iv)]{r80}). The divisorial algebra
$\dival_{X/X}{D}$ is also exceptionally asymptotically saturated with
$I=1$. For an $\R$-divisor $D$, this last condition holds with some $I\ge
1$ if and only if $D$ is a $\Q$-divisor (cf.\ Stupid Example~\ref{exa!st}
and Theorem~\ref{fgal}).

This holds also for $X/Z$, and we can explain exceptional saturation for
$D/Z$ and its algebra $\dival_{X/Z}{D}$ in the same style in terms of
double dual. However for divisorial functional sheaves, it has three
different versions, namely:
 \[
 (f_*\Oh_X(D))^{\vee\vee}\subset (f_*(\Oh_X(D)^{\vee}))^{\vee}\subset
f_*(\Oh_X(D)^{\vee\vee})=f_*\Oh_X(D).
 \]
 In general, for a functional subsheaf $\sL\subset k(X)$, we can take any
of these. We use the {\em minimal saturation\/} $\sL^{\vee\vee}=\sL$
(independent of $X$) or the {\em maximal saturation\/}
$f_*\sL^{\vee\vee}=\sL$, where we take twice $^\vee$ for $\Oh_X$. Thus the
maximal saturation depends on the choice of $X$. However the minimal
saturation corresponds to {\em essentially exceptional\/} saturation on
divisors, that is, we can add any divisors that are essentially exceptional
on $Z$. Usually, $\dival_{X/Z}{D}$ is not maximally or essentially
exceptionally saturated: for example, if $X/Z$ is birational, because
there is a difference between divisorial and {\bi}divisorial sheaves and
algebras. The maximal saturation agrees with the exceptional asymptotic
saturation$/X$, and it holds for any integral or $\Q$-divisor $D$ (only
integral in the case of sheaves).

This does not hold in general for {\bi}divisorial, pbd-divisorial or
functional algebras $\sL\subset k(X)$. But in the cases we are interested
in, maximal saturation and exceptionally asymptotic saturation hold {\em
over another birational model\/} $Y/Z$ of $X/Z$: for this, see predictions,
triples, Addendum~\ref{assdes}.2 and (BIR) of Conjecture~\ref{cbcon}.
 \begin{itemize}
 \item the flipping algebra of Example~\ref{flipal} with $Y=X$ and the
$\Q$-divisor $D$ (here $I=1$ if $D$ is integral);
 \item the (pre-restriction) algebra $\dival_{X/T}{D}$ with $Y=X$ and
$I=1$ of Example~\ref{resralge} (cf.\ proof of Induction Theorem~\ref{ith}
below);
 \item any {\bi}divisorial algebra as in Example~\ref{rdivalg} with a
Cartier $\Q$-{\bi}divisor $\sD$, on a model $Y$ over which $\sD=\Dbar$
for a $\Q$-Cartier divisor $D$ on $Y$; and
 \item any f.g.\ functional algebra $\sL$ on any model of $Y/\Proj_Z{\sL}$,
by Corollary~\ref{sacr}.
 \end{itemize}
 In this sense, asymptotic saturation is a necessary but not sufficient
condition for f.g. The latter also needs certain conditions on $X/Z$ (cf.\
Remark~\ref{gzardec}, (2), Theorem~\ref{existd}, Conjectures~\ref{plfconj},
\ref{conj!fga}, Proposition~\ref{exsatcan} and Remark~\ref{fga0lm}, (2)).
However, it helps to find a model $Y$ where we expect f.g. Indeed, if $\sL$
is exceptionally asymptotically saturated$/Y$ and f.g., then $\Proj$
defines a rational contraction of $X$ to $\Proj_Z{\sL}$ that is rational
$1$-contraction on $Y$; in particular, $Y$ and $\Proj_Z{\sL}$ are
isomorphic in codimension~$1$ when $\sL$ is big. In the case of a pbd
algebra it gives the g.a.g.\ of $\sD^m$ or even $\sD_\bull$ over $Y$ (cf.\
bss ampleness in Theorem~\ref{fgal}, where the $\Q$-divisor condition for
$D\sm$ can be replaced by exceptionally asymptotic saturation$/X$); this
generalizes Zariski decomposition (cf.\ Example~\ref{pbsa} above).
 \end{exa}

Unfortunately, saturation conditions are no good for induction on the
dimension. It is well known that the usual saturation is not preserved
under restrictions when they are not surjective (cf.\ reasons (2--3)
in our motivation for introducing log canonical saturations below). The
same goes for our saturations for an arbitrary $\sC$.

 \begin{defn}\label{lca} A $\sC$-saturation is {\em log canonical\/}$/X$
or, to be more precise, log canonical over $(X,B)$ if
$\sC=\sA=\sA^X=\sA(X,B)$ is the {\em discrepancy\/} {\bi}divisor of the
pair $(X,B)$ for some $\R$-divisor $B$ on $X$.

In the divisorial case we take $C=\sA_X=-B$.

We abbreviate log canonical asymptotic saturation to {\em lca\/} saturation.
 \end{defn}

 \begin{warn}\label{warn!lca} For another pair $(Y,B_Y)$ with a model $Y/Z$
of $X/Z$, in general $\sA^Y=\sA(Y,B_Y)\ne\sA$, and
$\sB^Y=\sB(Y,B_Y)\ne\sB$. Thus the log canonical saturation can be
different over $(Y,B_Y)$.
 \end{warn}

As it used\footnote{What? Does this mean ``as before'' or ``as usual''?}
to be, $\sA_Y$ denotes the restriction divisor of the {\bi}divisor $\sA$
on a model $Y$ of $X$. In general, $\sB_Y=B^Y\ne B_Y$. The latter is the
codiscrepancy of $(Y,B_Y)$ on $Y$, and $\sB_Y=B^Y=B_Y$ if and only if
$(Y,B_Y)$ is a {\em crepant\/} birational transform of $(X,B)$ (cf.\
Definition~\ref{tri}).

By Example~\ref{toosl}, for a nondivisorial algebra, saturation on its own
is not enough for f.g. But we hope that asymptotic saturation works
better, especially lca saturation (cf.\ Conjecture~\ref{conj!fga} and
Example~\ref{1dfga} below). There are four reasons for this according to
our applications:\footnote{I don't understand ``according to'': {\em in
view} of our applications, or {\em depending on} the application we have in
mind, or what?}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the restriction of a flip can be a nonflip;
 \item canonical saturation is compatible with restrictions by
Kawamata--Viehweg vanishing (cf.\ Proposition~\ref{ressat});
 \item $\sA$ is sufficiently effective: e.g., for a Klt pair $(X,B)$,
$\rdup{\sA}\ge0$ is exceptional, and at the same time grows at least
linearly with blowups (cf.\ Proposition~\ref{coher}, \CGR, and (LGD) for
predictions in Section~\ref{satdes} below); and
 \item $\sA$-saturation is well defined (cf.\ Example~\ref{coha} below).
 \end{enumerate}

 \begin{defn}\label{defn!fga}
 \addcontentsline{toc}{subsection}{\ref{defn!fga} Definition of algebra
of type $\FGA_n$}
 Suppose that a pair $(X/T,B)$ satisfies:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item $X/T$ is a contraction;
 \item $(X,B)$ is Klt;
 \item $-(K+B)$ is nef and big$/Z$; and, finally,
 \item $\dim X=n$.
 \end{enumerate}
 The two middle conditions mean that $(X/T,B)$ is a weak log Fano
contraction (cf.\ \WLF\ in Proposition~\ref{exsatcan}).

 Then we say that a functional $\Oh_T$-algebra $\sL\subset k(X)_\bull$ or
its system $\sD_\bull$ is {\em of type $\FGA_n$} if
 \begin{q}
 \item[$\FGA_n$]
 it is bounded and\/ {\em lca saturated\/} over $(X/T,B)$.
 \end{q}
It is of type $\FGA_n\bir$ if\/ $X/T$ is birational.\footnote{I have tried
to untangle the definition and the following Conjecture: ``We also use
these to refer to the conjecture (cf.\ Induction Theorem~\ref{ith}).''}

 \end{defn}

 \begin{conj}\label{conj!fga} An algebra of type $\FGA_n$ is f.g.$/T$.
 \end{conj}

By Limiting Criterion~\ref{limcr}, an equivalent condition is the
stabilization of $\lim_{i\to\infty}\sD_i$ for the characteristic system
$\sD_\bull$ of $\sL$. Thus the system stabilizes if
 \begin{q}
 \item[\rm(LIM)] there is a (finite) limit $\sD=\lim_{i\to\infty}\sD_i$; and
 \item[\LCA] it is lca saturated.
 \end{q}

However, we prefer to work with {\bi}divisors (cf.\ Remark~\ref{fga0lm},
(5) below). Note for this that the system $\sD_\bull$ also satisfies
 \begin{q}
 \item[\MXD] each $\sD_i=\sM_i/i\le\sD$ by Addendum~\ref{bpbdal}.1;
 \item[\rm(LBF)] each $\sM_i=i\sD_i$ is {\bi}free, by
Proposition~\ref{texa}, (1); and
 \item[\AMN] {\em arithmetic monotonicity\/}: for any $j\mid i$ we have
$\sD_i\ge\sD_j$ by Lemma~\ref{arithm}.
 \end{q}

Moreover, one can expect that LMMP in dimension $m\le n$ implies
$\FGA_n$ (cf.\ Theorem~\ref{existd}).
 % \end{conj}

By Corollary~\ref{mainc1}, the conjecture implies the existence of log
flips in dimension $n$ modulo LMMP in dimension $m\le n-1$ (cf.\
Remark~\ref{fga0lm}, (6) below). This would be an inductive construction
of log flips.

 \begin{rem}\label{fga0lm} Some of the assumptions in the conjecture can
be relaxed or modified:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item We can replace the contraction $X/T$ by any proper morphism $X/Z$
of normal algebraic varieties or normal algebraic spaces (certainly), and
probably also normal analytic spaces (use the Zariski decomposition of
$X\to Z$).

 \item We expect also that Conjecture~\ref{conj!fga} holds for the $0$-log
pairs $(X/Z,B)$ of Remark~\ref{gzardec}, (2) (cf.\ Theorem~\ref{existd}).

 \item Moreover, Conjecture~\ref{conj!fga} may hold for pairs $(X/Z,B)$
with $\sD_X=D=K+B+H$, where $H$ is a nef and big$/Z$ $\R$-divisor; or the
same on some crepant model $(Y/Z,B_Y)$ of $(X/Z,B)$.

 \item Possibly everything works for more general gradings, e.g.,
$\N^r$-gradings.

 \item It can also be generalized to worse singularities, that is, when
$(X,B)$ is not Klt, $B$ is not a boundary, and $X$ is nonnormal, or not
even seminormal. However we still need to assume that $B$ is effective,
and that the algebra $\sL$ behaves well (f.g.)\ or the limit
$\sD=\lim_{i\to\infty}\sD_i$ stabilizes near the locus of log canonical
singularities $\LCS{(K+B)}$ (see \cite[Definition~3.14]{sh92} and
Example~\ref{1dfga} below).

 \item The restriction of a special flip or a pl flip is not necessarily
a flip. However, we expect that $\FGA_{n-1}\bir=\RFA_{n,n-1}\bir$. For
algebras, this means that each normal algebra $\sL$ of type
$\FGA_{n-1}\bir$ up to a quasi-isomorphism is a restricted algebra of
type $\RFA_{n,n-1}\bir$ as in Definition~\ref{rfad} (cf.\ proof of
Induction Theorem~\ref{ith} below). Then the existence of $n$-fold
flips would be equivalent to $\FGA_{n-1}\bir$.

 \item Finally, if $\sD$ is big, the {\em stable\/} crepant model
$(X_{st}/T,B_{st})$ of $(X/Z,B)$ in the conjecture with
$X_{st}/T=\Proj_{T}{\sL}$ is isomorphic in codimension~$1$ to a crepant
weak log Fano contraction (cf.\ Addendum~\ref{assdes}.2). According to
conjectures of Alexeev and Borisov,\footnote{Reference?\label{ABfoot}
Possibly Valery Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J.
Math. {\bf5} (1994) 779--810?} $X_{st}$ should then have bounded modulus
(cf.\ canonical boundedness and triples in (BIG) of Conjecture~\ref{cbcon}
below).
 \end{enumerate}
 \end{rem}

 \begin{exa}[A Pythagorian dream]\label{1dfga}
 \addcontentsline{toc}{subsection}{\ref{1dfga} A Pythagorian dream: (FGA)
in the 1-dimensional case}
Suppose that $X=C$ is a normal algebraic curve and $B=\sum b_m P_m$ a
Klt $\R$-boundary; in other words, each $b_i\in [0,1)$. Then any lca
saturated$/(C,B)$ algebra $\sL$ is f.g., or the limit
$\sD=\lim_{i\to\infty}\sD_i$ stabilizes. Note that in this case $D=\sum
d_m P_m=\sD=\lim_{i\to\infty}\sD_i$ is a limit of divisors $D_i=\sD_i=\sum
d_{m,i} P_{m,i}$. Equivalently, each $d_m=\lim_{i\to\infty} d_{m,i}$ with
$d_{m,i}\in \Q$. In particular, stabilization means that each $d_m$ is
{\em also\/} $\in\Q$. This is the point.

All the divisors $D$ and $D_i$ are supported in a finite subset of $C$ by
Lemma~\ref{arithm}. Thus the limit takes place for the vector of
coefficients in $\R^n$.

On the other hand, the discrepancy $\sum a_m P_m=\sA=A=-B\le0$. Moreover,
by Klt, all $a_m=-b_m\in [0,-1)$. In the one dimensional case, any
$X\hr=C\hr=C$. After truncating $\sL$, we can assume that the saturation
has index $I=1$. Thus lca saturation gives the inequality
$\Mov{\rdup{jD_i+A}}\le j D_j$ for any natural numbers $i,j$. In our
situation, we can assume that $\Mov{\rdup{j D_i+A}}=\rdup{j D_i+A}$ for
all $i,j\gg1$.

By RR on curves, this holds unless $C$ is complete, $Z=\pt$, and $\deg
D\le0$. Moreover, then $\deg D=0$, because we consider the limit only for
nonempty linear systems $\linsys{M_i}=\linsys{iD_i}$. Thus $\deg D_i\ge0$.
Hence $D=0$, and we have the stabilization by \MXD (see
Addendum~\ref{bpbdal}.1).
The stable model in this case is $C_{st}=\Proj_Z{\sL}=\pt/Z=\pt$

Otherwise, we have the inequality $\rdup{j D_i+A}\le j D_j$ or,
componentwise, for each $P_m$ and all $i,j\gg1$,
 \[
 \rdup{j d_{m,i}+a_m}\le j d_{m,j}.
 \]
 Again by \MXD\ this gives, for each $P_m$ and all $j\gg1$,
 \[
 \rdup{j d_m+a_m}=\rdup{j d_{m,i}+a_m}\le j d_{m,j}\le j d_m,
 \]
 when we take $i\gg j$! By Diophantine geometry, for any real number
$a=a_m>-1$, any real number $d=d_m$ satisfying
 \[
 \rdup{j d+a}\le j d
 \]
 for all natural numbers $j\gg1$ is rational. (See Cassels
\cite[Lemma~1A]{cas57}; take the best upper approximations, for $\theta=d$;
cf.\ proof of Approximation Lemma~\ref{aproxl} below.) This explains also that
positive $a=a_m$ are also not good (cf.\ Remark~\ref{fga0lm}, (5) above).
However $A$ with $\rdup{A}=0$, if even the total $A=0$, does have an
effect!

Since $d_m$ is rational, then, for any $j\gg1$ with integral $j d_m$, we
get stabilization
 \[
 \rdup{j d_m+a_m}=j d_{m,j}=j d_m.
 \]
 By \AMN\ this implies stabilization: each $d_{m,i}=d_m$ for all $i$
divisible by some $j$.

Finally, since the divisors are bounded in their supports, we get
stabilization by a truncation of $D_\bull$.

In particular, this establishes $\FGA_1$.

By these arguments, we can generalize f.g.\ for any effective $B$ assuming
that the limit stabilizes near the $\LCS{(K+B)}=\bigl\{P_m\bigm|b_m\ge
1\bigr\}$; e.g., $d_m=\lim_{i\to\infty} d_{m,i}$ stabilizes in each
such point $P_m$.

For nonnormal $C$, stabilization near bad singularities means that, up to
a similarity, all $D_i$ and $D$ are nonsingular, that is, are supported in
nonsingular points of $C$.
 \end{exa}

To establish even $\FGA_2$ is not so simple; this is discussed below after
some preparations. Now we turn to the proof of Induction Theorem~\ref{ith}.
For this we need some results on saturations.

 \begin{prop}\label{exsatcan}
 \addcontentsline{toc}{subsection}{\ref{exsatcan} Proposition on log
canonical saturation}
For any effective $\R$-divisor $B$ such that $K+B$ is $\R$-Cartier, any
exceptional saturation$/X$ implies lca saturation$/(X,B)$.

Thus $\FGA_n$ implies f.g.\ of any algebra $\dival_{X/T}{D}$, provided
that
 \begin{q}
 \item[\WLF] $(X/T,B)$ is a {\em weak log Fano contraction\/}, that is,
$(X,B)$ is Klt, and $-(K+B)$ is nef and big$/T$
 \end{q}
(cf.\ Theorem~\ref{existd} and Remark~\ref{fga0lm}, (2) above).
 \end{prop}

 \begin{add} Any exceptional saturation$/X$ implies the corresponding
saturation for $\sA'=\sA(X,B)+\sE$, where $\sE$ is a reduced {\bi}divisor
such that:
 \begin{itemize}
 \item $\sE_X$ is supported over $\Supp{\sA_X}=\Supp{B}$, and
 \item $\sA$-saturation is {\em integral\/} over $\sE$.
 \end{itemize}
 \end{add}

 \begin{defn}\label{ints} The $\sC$-saturation is {\em integral\/} over
$\sE$ means that $j\sD_i+\sC$ is integral over $\sE$.

The same applies to other types of saturations for $\sD+\sC$.
 \end{defn}

The proposition applies in particular to a flipping algebra
$\flpal_{X/T}{D}$. Models of algebras, with exceptional asymptotic
saturations, as models for bss ample divisors are obtained by a rational
$1$-contraction of $X$. The former $\FGA_n$ algebras are more general and
their models can blow up some divisors exceptional on $X$. This is the main
difference between (FGA) (former) and Zariski (latter) approaches to f.g.\
algebras.

 \begin{lem} \label{monots} Let $C_1\ge C_2$ be divisors on $X/T$, and
$\sC_1\ge\sC_2$ {\bi}divisors on $X/T$. Then $C_1$-saturation implies
$C_2$-saturation, $\sC_1$-saturation implies $\sC_2$-saturation, and
asymptotic $\sC_1$-saturation implies asymptotic $\sC_2$-saturation.
 \end{lem}

 \begin{add} Asymptotic $\sC_1$-saturation implies asymptotic
$\sC'_2$-satur\-a\-tion with $\sC'_2=\sC_2+\sE$, where $\sE$ is a reduced
{\bi}divisor such that:
 \begin{itemize}
 \item $\sC_1>\sC_2$ over $\sE$; and
 \item the $\sC_2$-saturation is integral over $\sE$.
 \end{itemize}

 The same holds for the other types of saturation.
 \end{add}

 \begin{proof} We use two facts:
 \begin{itemize}
 \item $C_1\ge C_2$ implies that $\rdup{D+C_1}\ge\rdup{D+C_2}$; and
 \item $C_1\ge C_2$ implies the inclusion
$\linsys{\rdup{D+C_1}}\supset\linsys{\rdup{D+C_2}}+E$ with
$E=\rdup{D+C_1}-\rdup{D+C_2}
\ge0$.
 \end{itemize}
 This last fact implies that if $\linsys{\rdup{D+C_2}}\ne\emptyset$,
then $\linsys{\rdup{D+C_1}}\ne\emptyset$ and $D\ge\Mov{\rdup{D+C_1}}\ge
\Mov{\rdup{D+C_2}}$. This gives the first statement.

The {\bi}divisorial case and asymptotic saturation follows from the same
estimate. $\sC'_2$-saturation follows from two facts:
 \begin{itemize}
 \item we can replace $\sC_2$ by $\sC_2+\sum\ep_iE_i$ with
$0<\ep_i\ll 1$ by the lemma; and
 \item the effect of this is equivalent to replacing $\sC_2$ by $\sC'_2$.
 \end{itemize}
 This last fact follows from equations $\rdup{d+\ep_i}=d+1=\rdup{d+1}$ for
any integer $d$.
 \qed\end{proof}

 \paragraph{Proof--Explanation of Proposition~\ref{exsatcan}} By any
saturation in the statement, we mean any type of saturation in
Definition~\ref{asatur}: e.g., for a divisor, for a {\bi}divisor, etc.

Since $B$ is effective, the codiscrepancy $\sB=\sB(X,B)$ is {\em also\/}
effective up to components exceptional on $X$. There is thus an exceptional
{\bi}divisor $\sC$ such that $\sC\ge\sA=-\sB$; in the divisorial case, we
take $C=\sC_X=0$. Then the proposition follows immediately from
Lemma~\ref{monots}, and Addendum~\ref{exsatcan}.1 follows from
Addendum~\ref{monots}.1.
 \qed\par\medskip

We apply this presently in the following situation.

 \begin{exa} \label{monotsi} Let $(X/T,B)$ and $(X/T,B^+)$ be two log
pairs such that
 \begin{itemize}
 \item $(X,B)$ is Klt;
 \item $(X,B^+)$ is log canonical;
 \item $B^+\ge B$, or equivalently, $A=\sA_X\ge A_X^+=\sA_X^+$, where
$\sA^+=\sA(X,B^+)$; and
 \item $\sA^+$-saturation is integral over divisors in $\LCS{(X,B^+)}$.
 \end{itemize}
 Then lca saturation$/(X,B)$, e.g., of Proposition~\ref{exsatcan}, implies
asymptotic $\sA'$-saturation with $\sA'=\sA^++\sE$, where $\sE=\sum E_i$ is
a sum of prime {\bi}divisors with discrepancy $-1$ for $(X,B^+)$. The integral
saturation for $\sA^+$ means that each divisor $j D_i=j\sD_i\rest X$ is
integral over $\LCS{(X,B^+)}$. For example, this holds when each
$\sD_i$ is in general position with respect to each divisorial component
of $\LCS{(X,B^+)}$ (see \GNP\ in Proposition~\ref{ressat} below).

Indeed, by monotonicity \cite[1.3.3]{sh92} we have $\sB^+=\sB(X,B^+)\ge\sB$,
or equivalently, $\sA\ge\sA^+$; and $A>A^+$ everywhere over $\LCS{(X,B^+)}$.
Hence lca saturation implies $\sA^+$ and $\sA'$-saturations by
Lemma~\ref{monots} and Addendum~\ref{monots}.1 respectively.
 \end{exa}

In general, the {\bi}divisorial sheaf $\Oh_X(\sD)$ may not be a coherent
sheaf (see Definition~\ref{pbdad} and Examples~\ref{rdivalg}--\ref{texam}).
In other words, this means a stabilization, that is, $=$ for inclusions
$\Oh_Y(\sD_Y)\supseteq\Oh_X(\sD)$ on a sufficiently high model $Y/X$. For
example, it is true for any $\R$-Cartier divisor $\sD$ by
Proposition~\ref{isom} over any model $Y$, where $\sD=\Dbar$ for some
$\R$-Cartier divisor $D$ on $Y$.

 \begin{prop}\label{coher} Let $\sD$ be an $\R$-Cartier {\bi}divisor on
$X$, $\sE$ a {\em finite\/} reduced {\bi}divisor, and $\sC$ an
$\R$-{\bi}divisor on $X$ such that:
 \begin{q}
 \item[\INO] $\sD$ and $\sC$ (or just $\sD+\sC$) are {\em integral
over\/} $\Supp{\sE}$; and
 \item[\CGR] $\sC$ {\em grows canonically:\/} $\sC_{X\hr}\ge
g^*(\sC_Y)+\sum a_iE_i$ for any further resolution $g\colon {X\hr}\to Y$
of some (sufficiently high) resolution $Y$ of $X$, where $a_i=a(Y,0,E_i)$
are the standard discrepancies.
 \end{q}
 Then the sheaf $\Oh_X(\rdup{\sD+\sC+\sE})$ is coherent.
 \end{prop}

 \begin{proof} Take a sufficiently high resolution
$Y$ such that:
 \begin{itemize}
 \item $\sD=\Dbar$ for an $\R$-divisor $D$ on $Y$;
 \item \CGR\ is satisfied for $Y$;
 \item $\sC_Y$ has normal crossing support on $Y$ (a property of all
{\bi}divisors on some resolution);
 \item moreover, the supports of all divisors under consideration have
normal crossings; and
 \item $\sE=\sE_Y$ is divisor on $Y$ whose prime components are all
pairwise disjoint.
 \end{itemize}
 It is enough to verify that, for any other model
$g\colon X\hr\to Y$,
 \[
 g^*(\rdup{\sD+\sC+\sE}_Y)\le\rdup{\sD+\sC+\sE}_{X\hr},
 \]
 because then
 \[
 \Oh_Y(\rdup{\sD+\sC+\sE}_Y)\subset g_*\Oh_{X\hr}(\rdup{(\sD+\sC+\sE)_{X\hr}})
 \]
 by Proposition~\ref{isom}, which even gives $=$, the stabilization and the
coherence of $\Oh_X(\rdup{\sD+\sC+\sE})$.

Set
 \[
 \rdup{\sD+\sC+\sE}_Y=(\sD+\sC+\sE)_Y+F,
 \]
 where $F=f_iE_i$ is the {\em fractional\/} part, that is, all $f_i\in
[0,1)$. Note that $(Y,\sE+F)$ is purely log terminal by \INO, the fact
that the $\sE$ are disjoint, and the normal crossings.

Thus by \INO, because $\sD$ is Cartier, and by \CGR, we get the inequality
 \begin{align*}
 g^*(\rdup{\sD+\sC+\sE}_Y)&=g^*(\sD_Y+\sC_Y+\sE_Y)+g^*F\\
 &= g^*(\sD_Y)+g^*(\sC_Y)+g^*(\sE+F) \\
 &\le\sD_{X\hr}+\sC_{X\hr}+g^*(\sE+F)-\sum a_iE_i \\
 &= (\sD+\sC+\sE)_{X\hr}+g^*(\sE+F)-\sE-\sum a_iE_i.
 \end{align*}
 Since we need to verify the above inequality for integral parts over the
exceptional $E_i$, it is enough to verify that
 \[
 \mult_{E_i}{(g^*(\sE+F)-\sE-\sum a_iE_i)}=
 \mult_{E_i}{(g^*(\sE+F)-\sum a_iE_i)}<1
 \]
 for each exceptional $E_i$. (For any integer $i$ and real numbers $f<1$
and $r$, the inequality $i\le r+f$ implies $i\le\rdup{r}$.) But this means
that $(Y,\sE+F)$ is purely log terminal:
$\sA(Y,\sE+F)=\sA(Y,0)-g^*(\sE+F)$ has each exceptional multiplicity $>-1$.
 \qed\end{proof}

 \begin{exa}\label{coha} An discrepancy divisor $\sC=\sA=\sA(X,B)$ is the
typical example when \CGR\ in Proposition~\ref{coher} holds. Indeed,
 \begin{align*}
 \sA_{X\hr}&=K_{X\hr}-(\overline{K+B})_{X\hr}
 =K_{X\hr}-g^*(\overline{K+B}_Y) \\
 &=K_{X\hr}-g^*(K_Y-\sA_Y)
 =g^*\sA_Y+K_{X\hr}-\overline{K}_Y=g^*\sA_Y+\sA(Y,0)_{X\hr}.
 \end{align*}
 Thus for
$\sC=\sA$ in Proposition~\ref{coher} we get that
$\Oh_X(\rdup{\sD+\sA+\sE})$ is coherent.

In particular, for $\sD=0$ and $\sE=0$, the fractional ideal
$\Oh_X(\rdup{\sA})$ is always coherent. If $B$ is effective, it is the
well-known ideal sheaf for $\LCS{(X,B)}$.
 \end{exa}

 \begin{prop}\label{ressat} Let\/ $(X/Z,B)$ be a pair with $B$ an
$\R$-divisor, $Y$ a prime divisor of $X$, and $\sD_\bull$ a system of
$\R$-{\bi}divisors such that:
 \begin{q}
 \item[\GLF] $(X/Z,B)$ is a {\em general log Fano contraction:\/} $-(K+B)$
is nef and big$/Z$;
 \item[\LCC] $Y$ is a\/ {\em log canonical center:\/} $a(X,B,Y)=-1$;
 \item[\ASAprime] the system $\sD_\bull$ is asymptotically saturated with
respect to $\sA'=\sA(X,B)+Y$;
 \item[\BSA] each\/ $\sD_i$ is {\em {\bi}semiample:\/} $\sD_i=\Dbar_i$
for a semiample $\R$-divisor $D_i$ on a model\/ $W/Z$ of $X/Z$ {\em (cf.\
Lemma~\ref{exfl});} and
 \item[\GNP] $\sD_\bull$ is\/ {\em in general position\/} with respect to
$Y$: for each $\sD_i$, $Y\not\subset\Supp{\sD_i}$.
 \end{q}
 Then the restriction $\sD_\bull\frest{Y^\nu}$ is lca saturated over
$(Y^\nu,B_{Y^\nu})$, where $\nu\colon Y^\nu\to Y$ is the normalization
of $Y$, $\sD_\bull\frest{Y^\nu}$ is the\/ {\em fixed restriction} (see
the proof and Fixed Restriction~\ref{frest} below), and\/
$B_{Y^\nu}=(B-Y)_{Y^\nu}$ is the different of\/ $B-Y$ {\em (see
\cite[\S3]{sh92} and \cite[Prop.~16.5]{ko})\/}.
 \end{prop}

 \begin{add}
$\sL\subset k(X)_\bull$ be a functional algebra such that:
 \begin{q}
 \item[\ASAprime] $\sL$ is asymptotically saturated with respect to
$\sA'$; and
 \item[\GNP] $\sL$ is {\em in general position\/} with respect to
$Y$, as is the mobile system $\sM_\bull$.
 \end{q}

Then $\sL\rest{Y^\nu}$ is lca saturated$/(Y^\nu,B_{Y^\nu})$. If, in
addition, $(X,B)$ is purely log terminal, then $\sL\rest{Y^\nu}$ is
normal$/f(Y)$ if $\sL/Z$ is.
 \end{add}

For a log canonical center $Y$ of a higher codimension, we need an
adjunction formula.

 \begin{proof} Take a log resolution $W/Z$ of $(X/Z,B)$ that is sufficiently
high over $X$ and over $Y^\nu$, and take any natural number $i$ and $j$. Let
$Y\hr$ be the birational transform of $Y$ on $W$. By construction,
$Y\hr$ is normal and we have a decomposition $g\colon Y\hr\to Y^\nu
\to Y$. Then the restriction
 \[
 \linsys{\rdup{j(\sD_i)_W+\sA_W'}} \xrightarrow{\rest{Y\hr}}
 \linsys{\rdup{j(\sD_i)_W+\sA_W'}\rest{Y\hr}}=
 \linsys{\rdup{j(\sD_i)_W\rest{Y\hr}+(\sA^{Y^\nu})_{Y\hr}}}
 \]
 is surjective, where $\,\rest{X\hr}=g^*$, and
$\sA^{Y^\nu}=\sA(Y^\nu,B_{Y^\nu})$. Indeed, by adjunction,
$K_{Y^\nu}+B_{Y^\nu}=(K+B)\rest{Y^\nu}$ (\cite[3.1]{sh92} and
\cite[Prop.~16.5]{ko}). On the other hand, by definition
$\sA=\sA(X,B)=\sK-\overline{K+B}$ and
$\sA^{Y^\nu}=\sK^{Y^\nu}-\overline{K_{Y^\nu}+B_{Y^\nu}}$, where $\sK$ and
$\sK^{Y^\nu}$ are canonical {\bi}divisors of $X$ and $Y^\nu$ respectively
\cite[Example~1.1.3]{sh96b}, that is, $\sK_W=K_W$ and
$\sK^{Y^\nu}_{Y^\nu}=K_{Y^\nu}$. Hence
 \begin{multline*}
 \sA_W'\rest{Y\hr}=(\sA_W+Y\hr)\rest{Y\hr}=
(K_W+Y\hr-\overline{K+B})\rest{Y\hr}=\\
 (K_W+Y\hr)\rest{Y\hr}-(\overline{(K+B)\rest{Y_{\nu}}})_{Y\hr}=
K_{Y\hr}-(\overline{K_{Y_{\nu}}+B_{Y_{\nu}}})_{Y\hr}=
(\sA^{Y_{\nu}})_{Y\hr}.
 \end{multline*}
 (In terms of fixed restrictions, this amounts to
$\sA'\frest{Y^{\nu}}=\sA^{Y_{\nu}}$; see \ref{frest} below.)

By normal crossings of the supports in the following formula, the roundup
$\rdup{\ }$ commutes with the restriction:
 \[
 \rdup{j(\sD_i)_W+\sA_W'}\rest{Y\hr}=
 \rdup{j(\sD_i)_W\rest{Y\hr}+\sA_W'\rest{Y\hr}}=
 \rdup{j(\sD_i)_W\rest{Y\hr}+(\sA^{Y^\nu})_{Y\hr}}.
 \]
 For surjectivity, it is enough to establish the vanishing
 \[
 R^1h_*\Oh_W (\rdup{j(\sD_i)_W+\sA_W'}-Y\hr),
 \]
 where $h\colon W\to Z$. It is natural to use the Kawamata--Viehweg
vanishing for this:
 \begin{align*}
 \rdup{j(\sD_i)_W+\sA_W'}-Y\hr&=
 \rdup{j(\sD_i)_W+\sA_W+Y\hr}-Y\hr \\
 &=\rdup{j(\sD_i)_W+\sA_W}= \rdup{j(\sD_i)_W+K_W-r^*(K+B)} \\
 &=K_W+\rdup{j(\sD_i)_W- r^*(K+B)},
 \end{align*}
 because $Y\hr$ and $K_W$ are integral, and $\sA_W=K_W-r^*(K+B)$, where
$r\colon W\to X$ is the resolution. Now we get the required vanishing,
since $(\sD_i)_W$ is nef$/Z$ by \BSA, and $-(K+B)$ is nef and big $/Z$ by
\GLF. The former holds on $W$ over any model where $\sD_i=\Dbar_i$ for
semiample $D_i$.

Thus by the surjectivity of the above restriction,
 \[
 \linsys{\rdup{j(\sD_i)_W+\sA_W'}}\ne\emptyset
 \quad\hbox{provided that}\quad
 \linsys{\rdup{j(\sD_i)_W\rest{Y\hr}+(\sA^{Y^\nu})_{Y\hr}}}\ne
 \emptyset.
 \]
 Moreover, in this case, by Hironaka, and because
$\Oh_X(\rdup{j(\sD_i)+\sA'})$ is coherent by Proposition~\ref{coher} and
Example~\ref{coha}, we can assume that
 \[
 \Bs{\linsys{\Mov{\rdup{j(\sD_i)_W+\sA_W'}}}}=\emptyset
 \]
 on a sufficiently high resolution $W$ (cf.\ the proof of
Proposition~\ref{texa}, (1)) and both $\Mov$ are stabilized. Therefore the
surjectivity gives
 \[
 \Mov{\rdup{j(\sD_i)_W\rest{Y\hr}+(\sA^{Y^\nu})_{Y\hr}}}=
 \Bigl(\Mov{\rdup{j(\sD_i)_W+\sA_W'}}\Bigr)\rest{Y\hr},
 \]
 (in general, only $\le$ holds), and
 \[
 \Bigl(\Mov{\rdup{j(\sD_i)_W+\sA_W'}}\Bigr)\rest{Y\hr}
 \,\le\, j(\sD_j)_W\rest{Y\hr}=j\Bigl(\sD_j\frest{Y^\nu}\Bigr)_{Y\hr}
 \]
 provided that
 \[
 \Mov{\rdup{j(\sD_i)_W+\sA_W'}}\le j(\sD_j)_W.
 \]
 The previous equality holds by definition of fixed restrictions of
{\bi}divisors (see \ref{frest} below) if $Y\hr$ lies over a model such
that $\sD_i=\Dbar_i$ for semiample $D_i$. This implies the lca saturation
for $\sD_\bull\frest{Y^\nu}$ and on any model$/Y\hr$ by \ASAprime\ with
the same index $I$.

Addendum~\ref{ressat}.1 follows from the proposition applied to the
characteristic system $\sD_\bull$ of $\sL$. Indeed, the characteristic
system of $\sL\rest{Y^\nu}$ is the restriction $\sD_\bull\frest{Y^\nu}$
(cf.\ Fixed restriction~\ref{frest}), and \BSA\ holds by
Proposition~\ref{texa}, (1). The normality of restricted algebras follows
from the surjectivity of mobile parts when $j=i$.

Indeed, \GLF\ and the connectedness of $\LCS{(X,B)}/Z$ (Koll\'ar and
others \cite[Theorem~17.7]{ko}) implies that $\sA$ has (locally) only one
divisor $Y$ with $a(X,B,Y)=-1$ if $(X,B)$ is purely log terminal. Then
$\rdup{\sA'}\ge0$, but $i\sD_i=\sM_i$ is integral as a {\bi}free
{\bi}divisor in Proposition~\ref{texa}, (1). Hence
$\linsys{\rdup{i(\sD_i)_W+\sA_W'}}=\linsys{(\sM_i)_W}+\rdup{\sA_W'}$ by
saturation (cf.\ Remarks~\ref{asaturr}, (1) and (3) above).

On the other hand, by the adjunction of \cite[3.2.3]{sh92},
$(Y^\nu,B_{Y^\nu})$ is Klt. Thus $\sA^{Y^\nu}\ge0$ and
 \[
 i(\sD_i)_W\rest{Y\hr}=(\sM_i)_W\rest{Y\hr}=
(\Mov{\sL\rest{Y^\nu}})_{Y\hr}
 \]
 is integral and mobile. Thus again
 \[
 \linsys{\rdup{i(\sD_i)_W\rest{Y\hr}+(\sA^{Y^\nu})_{Y\hr}}}=
 \linsys{(\Mov{\sL\rest{Y^\nu}})_{Y\hr}}+
 \linsys{\rdup{(\sA^{Y^\nu})_{Y\hr}}}.
 \]
 Thus we get surjectivity of the restriction
 \[
 \linsys{(M_i)_W} \xrightarrow{\rest{Y\hr}}
 \linsys{(\Mov{\sL\rest{Y^\nu}})_{Y\hr}},
 \]
 and by Proposition~\ref{texa}, (4) this implies that $\sL\rest{Y^\nu}$ is
normal if $\sL$ is.
 \qed\end{proof}

 \begin{exa}\label{fgapl} Consider a pl contraction $X/X_\vee$, and assume
that
 \begin{itemize}
 \item $X/X_\vee$ satisfies the inductive assumptions of
Definition~\ref{indfam} with respect to a divisor $D$ on $X$; and
 \item let $\sL$ be a functional algebra bounded by $D$ and satisfying the
other assumptions of Main Lemma~\ref{mainl}.
 \end{itemize}
 We say that such an algebra $\sL$ is of type $\FGA_n^{pl}$ if, in
addition,
 \begin{itemize}
 \item $\sL$ is lca saturated over $(X,B+S-\ep S)$ for some $\ep>0$; and
 \item $\dim X=n$.
 \end{itemize}
 As in Conjecture~\ref{conj!fga}, we expect such algebras to be f.g. More
precisely,
 \[
 \FGA_d \implies\FGA_n^{pl},
 \]
 where $d=\dim Y$ is the dimension of the irreducible normal variety
$Y=S^s=\bigcap S_i$ (cf.\ Example~\ref{resralge} with $t=1$ and $s=n-d$).
The implication means that the first conjectural f.g.\ implies the second.

Indeed, by Example~\ref{monotsi} with $B^+=B+S-\ep S'$ and
$S'=\sum_{i\ge2}S_i$, lca saturation of $\sL$ implies the asymptotic
saturation of $\sL$ for $\sA'=\sA^++S_1$. Thus by
Addendum~\ref{ressat}.1, this implies lca saturation$/(S_1,B_{S_1})$ with
the different $B_{S_1}=(B^+-S_1)_{S_1}=(B+S'-\ep S')_{S_1}$. Since
$K+B+S$ is numerically negative$/X_\vee$, \GLF\ holds for $B^+/X_\vee$ if
$0<\ep\ll 1$. Because $(X,B)$ is divisorially log terminal, $(X,B^+)$ is
purely log terminal for such $\ep$, and $S_1$ is the only log canonical
center, with $a(X,B^+,S_1)=-1$. The algebra $\sL$ is in general position
to $S_1$ by our choice of $D$ (cf.\ Definition~\ref{indfam} and Main
Lemma~\ref{mainl}).

Then, by Example~\ref{resralge}, we can use induction on $d$\/ because
 \[
 B_{S_1}=(B+S'-\ep S')_{S_1}=(B+S')_{S_1}-\ep (S'\rest{S_1})
 \]
 by semiadditivity \cite[3.2.1]{sh92}. The pair $(S_1/f(S_1),(B+S')_{S_1})$
satisfies the same assumption as $(X/X_\vee,B)$ by Example~\ref{resralge};
in particular,
 \[
 (S_1,(B+S')_{S_1}-\ep (S'\rest{S_1}))
 \]
 is Klt, and the restriction $\sL\rest{S_1}$ is of type $\FGA_{n-1}^{pl}$.

Hence by induction, $\sL\rest Y$ is lca saturated$/(Y,B_Y)$, where $B_Y$
corresponds to the successive adjunction. Moreover $(Y,B_Y)$ is Klt. Thus
by $\FGA_d$, the algebra $\sL\rest Y$ is f.g. Then $\sL$ is f.g.\ by
Main Lemma~\ref{mainl} (of course, locally near $P\in X_\vee$). In
addition, $\sL\rest Y$ is normal if $\sL$ is.

This is a generalization of Induction Theorem~\ref{ith} for (FGA) algebras
(cf.\ its proof below).

If, for $d=n$ and for big $\sL$, we knew the equivalence
 \[
 \text{(FGA)}_n\iff
\text{(FGA)}_n^{pl},
 \]
 this would imply the existence of log flips in dimension $n$ by induction
on $n$ (cf.\ the Reduction Theorem and Conjecture~\ref{conj!fga}).
 \end{exa}

In Section~\ref{canbound}, as an application of the example, we prove
$\FGA_3^{pl}$ (cf.\ Corollary~\ref{fgapl3} and Remark~\ref{fgaa} below).

Now we are ready to prove the main result of this section.

 \begin{cor}\label{fgarfa}
 $\FGA_d \implies\FGA_{n,d}$.
 \end{cor}

 \paragraph{Proof--Explanation} We verify that a restricted algebra
$\sL=(\dival_{X/T}{D})\rest Y$ is of type $\RFA_{n,d}$ and satisfies
$\FGA_d$. Indeed, the bounded algebra $\dival_{X/T}{D}$ is exceptionally
asymptotically saturated$/X$ by Example~\ref{satur}. Thus by
Proposition~\ref{exsatcan}, $\dival_{X/T}{D}$ is lca
saturated$/(X,B+S-\ep S)$ for any $\ep\in [0,1]$. In addition, by
Addendum~\ref{mainl}.1, the algebra $\dival_{X/T}{D}$ is of type
$\FGA_n^{pl}$. Thus by Example~\ref{fgapl}, the restriction $\sL$ is of
type $\FGA_d$.
 \qed\par\medskip

 \begin{pfof}{Induction Theorem~\ref{ith} for $\FGA$} Immediate by the
corollary and by case (RFA) in Section~\ref{divalg} or by
Theorem~\ref{plfth}. Moreover, we only need $\FGA_d\bir$ as for (RFA).
 \qed\end{pfof}

As a first application of Induction Theorem~\ref{ith} and its proof, by
Examples~\ref{1dfga}, \ref{fgapl} and by Corollary~\ref{fgarfa}, we get the
f.g.\ of algebras of type $\FGA_n^{pl}$ and (PLF)$_n$ with $d=1$. By the
Reduction Theorem this gives all log flips in dimension~2, which is well
known since they are divisorial contractions. However, the same methods
also allows us to construct log flips for seminormal surfaces and these
in fact may not be contractions (cf.\ modifications of surfaces $D$ of the
semistable models for $D$ \cite{sh93}).

In higher dimensions, pl flips with $d=1$ are also trivial since they are
negative on $Y$ (cf.\ Example~\ref{plneg}). Nonetheless $\FGA_1$ also gives
f.g.\ for $\sL$ in $\FGA_n$ over the generic points $P\in T$ when $\dim
f\1P=1$, and over the generic points of the prime divisors $D\subset T$
when $X/T$ is birational (cf.\ condition \BED\ in
Proposition~\ref{prop!fga} and in Theorem~\ref{stabin}). This follows by
induction on $n$ after taking general hyperplane sections.

 \begin{prop}\label{prop!fga} Over codimension~$d$ points $\FGA_n$ follows
from $\FGA_d$. Thus by induction on $n$, it is enough to prove $\FGA_n$
over closed points in $T$ assuming after a truncation that $\sL$ locally
has stabilized characteristic system:
 \begin{q}
 \item[\BED] for all $i$, $\sD=\sD_i$ outside $f\1P$ over $T$.
 \end{q}

The same holds for $\RFA$ in place of $\FGA$.
 \end{prop}

For example, $\sL=\dival_{X/T}{D}$ has \BED\ in codimension~$1$ when $D$ is
integral and $X/T$ is birational. After a truncation this holds even in
codimension~$2$, since normal $X$ is $\Q$ factorial in codimension~$2$.
(Any flipping algebras satisfies this over $X=T$.) By $\FGA_1$ and the
proposition this holds, but only in codimension~$1$, for any $\FGA_n$
algebra when $X/T$ is birational. In Corollary~\ref{stabc2} this is
established in codimension~$2$ on a similar basis.

 \begin{proof} Take a general hyperplane section $H$ of $T$ and a point
$P\in T$ and $Y=f\1H$. Then $\sL\rest Y$ is of type $\FGA_{n-1}$. The lca
saturation$/(Y/H,B\rest Y)$ follows from Addendum~\ref{ressat}.1 with
$X/Z=X/T$ and $B:=B+H$, where \ASAprime\ holds by Example~\ref{monotsi}.
Then local f.g.\ of $\sL\rest Y$ near $H\cap P$ implies local f.g.\ of
$\sL$ near $P$ by general properties of restrictions and the definition of
stabilization. Then we use induction on $n$.

For (RFA), note that the restriction $\rest Y$ also preserves (RFA).
 \qed\end{proof}

We remark finally that in the proof of $\FGA_1$ it was very important that
every prime divisor, and every effective divisor of rather high degree up to
linear equivalence, was nonsingular, that is, each is a reduced, nonsingular
subvariety (cf.\ Example~\ref{crv} below). In Section~\ref{satdes} we
explain to what extent we need this property (``predictions''), and in
Section~\ref{canbound} we state when we may expect this property to hold
(``triples'' and Conjecture~\ref{cbcon} below).

\section{Saturation and descent} \label{satdes}

\subsection{Descent of divisors} In this section we consider the following,
which is a key problem in our construction of pl flips. Let $f\colon Y\to
X$ be a morphism of algebraic varieties and $D$ an $\R$-Cartier divisor on
$Y$. When can we find an $\R$-Cartier divisor $D_X$ on $X$ such that
$D=f^*D_X$? If we can do this, we say that $D_X$ is the {\em descent\/} of
$D$; this is thus a descent problem. Note that we solve this problem in a
very special circumstances when our divisor is a limit or a rather high
element in a sequence of divisors on birational models of $X$. Thus in fact
we consider this problem for {\bi}divisors~\cite{sh96b}. Therefore we
replace $D$ by its completion $\sD=\Dbar$ or any other $\R$-{\bi}divisor,
and we can replace $Y$ by any birational model. The problem is to know
whether $\sD=\overline{\sD_X}$ then holds.

 \begin{rem} If we replace equality of divisors by linear equivalence, we
get a different problem, which is better considered in terms of sheaves:
$\Oh_Y(\sD)=f^*\Oh_X(\overline{\sD_X})$. If $Y/X$ is birational, this is
equivalent to the above problem. Otherwise the latter is quite different
and more flexible. However we can relax the problem with $=$ up to the
problem with $\sim$ by considering $\sD$ up to linear equivalence:
$\sD'\sim\sD$ and $\sD'$ has a descent (cf.\ Lemma~\ref{lin0l}). It is
even reasonable to consider $\sim_\Q$ or $\sim_\Q$ (cf.\ the bss and
{\bi}semiampleness above, as well as the triples of Lemma~\ref{split}
below).
 \end{rem}

The obstruction to the problem is easy to find.

\subsection{Descent data} The descent of $\sD$ on $X$ exists if only if
$\sE=\sE(\sD)=\overline{\sD_X}-\sD$ is $0$ as a {\bi}divisor. The
{\bi}divisor $\sE$ is the {\em descent data\/} of $\sD$ {\em over\/} $X$.

 \begin{prop}[Properties of descent data]\label{desd} Let $\sD$ be an
$\R$-{\bi}divisor of $X$. Then
 \begin{q}
 \item[\EXI] the descent data $\sE$ of\/ $\sD$ over $X$ {\em exists\/} if\/
$\sD_X$ is $\R$-Cartier, in particular, if\/ $X$ is $\Q$-factorial;
 \item[\EXC] $\sE/X$ is {\em exceptional\/} on $X$;
 \item[\ADD] {\em additivity:\/}
$\sE(\sD+\sD')=\sE(\sD)+\sE(\sD')/X$;
 \item[\HOM] {\em homogeneity:\/} for any real number $r$,
$\sE(r\sD)=r\sE(\sD)/X$;
 \item[\DEP] the data $\sE$ depends on $\sD$ only {\em up to $\R$-linear
equivalence\/}, and even only up to numerical equivalence$/X$; and
 \item[\EFF] $\sE$ is {\em effective\/} if $\sD$ is {\bi}semiample, or
even if $\sD$ is {\bi}nef$/X$.
 \end{q}
 \end{prop}

Thus for $\sD$ as in \EFF, if we know that $\sE\le0$ then $\sE=0$ and
the descent exists.

 \begin{proof} \EXI, \EXC, \ADD\ and \HOM\ follow directly from the
definitions; \EXC\ means that $\sE_X=0$.

 The assertion \DEP\ for $\sim_\R$ follows because the descent data for
a principal $\R$-{\bi}divisor is trivial. In turn by \ADD\ and \HOM, this
follows from the same fact for principal {\bi}divisors:
 \[
 \sE(\overline{(f)})=\overline{\overline{(f)}_X}-\overline{(f)}=
 \overline{(f)}-\overline{(f)}=0.
 \]
For numerical equivalence, by \ADD\ we can apply \EFF\ to the difference.

For \EFF, it is enough to check that $E_Y=f^*\sD_X-\sD_Y$ is effective for
any resolution $f\colon Y\to X$. For {\bi}semiample $\sD$, this follows
from \DEP\ and the effectiveness of $f^*D$ for any effective
$D$. For a {\bi}divisor $\sD$ that is {\bi}free$/X$, we get this by
Lemma~\ref{divb} or by the negativity~\cite[2.15]{sh95}.
 \qed\end{proof}

For a general divisor $D$, as well as for a general {\bi}divisor $\sD$,
descent may not exist even on a sufficiently high model. However, we are
interested in two related cases: $\sD=\lim_{i\to\infty}\sD_i$, where
each $\sD_i$ is the completion of some $\R$-Cartier divisor $D_i$ on a
model $Y_i/X$, that is, $\sD_i=\Dbar_i$ (cf.\ Example~\ref{texam}). Thus
each $\sD_i$ has a descent $D_i$ on $Y_i$. We expect also that under
certain conditions the {\bi}divisor $\sD$ may also have a descent; or
even that all $\sD_i$ together with $\sD$ have descents on one model.

\subsection{Asymptotic descent problem}\label{asdepr} Suppose that
$\sD=\lim_{i\to\infty}\sD_i$ is a limit of {\bi}divisors on $X/Z$. Find
a model $Y/Z$ of $X/Z$ such that
 \begin{itemize}
 \item $\sD=\overline{(\sD)_Y}$;
 \item the limit {\em stabilizes\/}, that is,
$\sD_i=\sD$ for some $i\gg0$; in particular,
 \item for every such $i$, also $\sD_i=\overline{(\sD_i)_Y}=\sD$.
 \end{itemize}
 Thus in this case, infinitely many $\sD_i$ have descents on a single
model $Y$ (cf.\ Remark~\ref{minar}, (4) below). Our approach gives a {\em
strict\/} asymptotic descent: e.g., stabilization holds for some $i\gg0$
after any truncation of $\sD_\bull$. But, in fact, in applications we get
a {\em complete\/} asymptotic descent: e.g., stabilization holds for all
$i$ after a truncation of $\sD_\bull$ (cf.\ Theorem~\ref{assdes} vs
Corollary~\ref{assdesc}).

A sheaf version of this conditions is related to f.g.\ and is given in our
standard Example~\ref{stexm} below.

The {\em minimal\/} assumptions on the system $\sD_\bull$ are the following:
 \begin{q}
 \item[\FDS] {\em finite divisorial support\/}: the divisors
$(\sD_i)_X$ are supported in one reduced divisor $F$; and
 \item[\MXD] of Addendum~\ref{bpbdal}.1: each $\sD_i\le\sD$.
 \end{q}
 However these assumptions with two standard ones (saturation and
boundedness) are only enough for the {\em rationality\/} of $\sD$ (cf.\
Examples~\ref{1dfga}, \ref{exc}, and Theorem~\ref{assdes}).

Descent needs more. The {\em additional\/} assumption on the system
$\sD_\bull$ is:
 \begin{q}
 \item[\BNF] each $\sD_i$ is {\em {\bi}nef\/} in the sense of
Lemma~\ref{divb}; in particular, this includes
 \item[(CAR)] each $\sD_i$ is $\R$-{\em Cartier\/}:
$\sD_i=\Dbar_i$ for the $\R$-Cartier divisor $D_i=(\sD_i)_{X_i}$ over some
model $X_i/Z$ of $X/Z$ (cf.\ Example~\ref{texam}).
 \end{q}
In the above assumptions {\em each\/} means {\em up to a truncation\/}.

 \begin{rem}\label{minar} \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item As in Remark~\ref{asaturr}, (7), similarity of characteristic
type preserves the minimal and additional assumptions.

 \item In our applications {\bi}nef follows from {\bi}semiample; and
usually each $\sD_i$ is $\Q$-Cartier (cf.\ Example~\ref{stexm} below).

 \item Since every modification blows up at most a finite number of
divisors, the assertion \FDS\ is birationally invariant: if it holds on one
model of $X$ then it also holds on any other. It is equivalent to \BSD\
in Addendum~\ref{bpbdal}.1. But it is weaker than
 \begin{q}
 \item[(FSP)] {\em finite support\/:} there exists a proper subvariety
$S\subset X$ such that all $\sD_i$ are supported over $S$.
 \end{q}
 However under \BNF, which holds in most of our applications, both
conditions are equivalent by Lemma~\ref{trivbd} below when each
$\sD_i\ge0$.

 \item A limit of $\R$-Cartier {\bi}divisors is not necessarily
$\R$-Cartier in general. For example, this used to be\footnote{probably
``This is usually the case for \dots''} for the limit of the characteristic
system of an infinitely generated functional algebra. However under \FDS,
if (CAR) holds on a single model $Y=X_i$ for infinitely many $i$, the
limit $\sD$ is also $\R$-Cartier$/Y$. By definition it is enough to
establish this (locally) for principal divisors. These are $\R$-Cartier$/Y$
because the $\R$-principal divisors, with support in a fixed reduced
divisor, form an $\R$-subspace (defined$/\Q$) in the space of all
$\R$-divisors within the given support.
\end{enumerate}
 \end{rem}

We can resolve the asymptotic descent problem for $\sD_\bull$ under
asymptotic saturation and the following boundedness.

 \begin{defn}[Boundedness for descent data]\label{bdd} Let $\sC$ and
$\sD$ be two $\R$-{\bi}divisors of $X$. We say that the descent data
$\sE$ of $\sD$ {\em over\/} $X$ is {\em bounded\/} by $\sC$ if
 \begin{itemize}
 \item the data $\sE=\overline{\sD_X}-\sD$ is {\em defined\/}, in
particular, $\sD_X$ is
$\R$-Cartier, and
 \item $\sE\le\sC$ {\em over}
$Y$, that is, $\mult_{E_i}{\sE}\le
\mult_{E_i}{\sC}$ in each prime {\bi}divisor $E_i$ that is {\em
exceptional\/} on $X$.
 \end{itemize}

Now let $\sD_\bull$ be a system of $\R$-{\bi}divisors of $X$: e.g., the
sequence of a limit $\sD=\lim_{i\to\infty}\sD_i$. The system $\sD_\bull$
is {\em asymptotically bounded\/} by $\sC$ if there exists a sequence of
positive real numbers $r_i$ for some $i\gg0$ such that:
 \begin{itemize}
 \item $\lim_{i\to\infty}r_i=+\infty$, and
 \item for each $r_i$, the descent data $\sE_i=\overline{(\sD_i)_X}-\sD_i$
over $X$ is bounded by $\sC/r_i$, or equivalently, $r_i\sE_i\le\sC$ over $Y$.
 \end{itemize}

We say that an asymptotic bound is {\em strict\/} if we can choose a
sub\-sequence of real numbers $r_i$ with the required property for any
truncation of $\sD_\bull$.
 \end{defn}

 \begin{rem}\label{asb}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item Under \FDS, $\lim_{i\to\infty}\sE_i=\sE=\overline{\sD_X}-\sD$ is
the descent data for $\sD$ on $X$. Indeed, $\sD_X$ is also $\R$-Cartier
by Remark~\ref{minar}, (4), and passing to the limit commutes with
restriction $(\ )_X$ and Cartier closure $\overline{(\ )}$.

 \item If, in addition, the sequence of the limit
$\sD=\lim_{i\to\infty}\sD_i$ satisfies \BNF, then by \EFF\ in
Proposition~\ref{desd} each $\sE_i\ge0$, and so $\sE\ge0$. On the other
hand, for chosen $i$, $\sE_i\le\sC/r_i$ over $X$, and
$\sE=\lim_{i\to\infty}\sE_i\le0$. (This means that the limits
$\sE=\lim_{i\to\infty}\sE_i$ and $\sD=\lim_{i\to\infty}\sD_i$ are uniform
with respect to $\sC$.) Hence $\sE=0$ and $\sD=\overline{\sD_X}$ has the
descent $\sD_X$ on $X$. Thus the limit $\sD=\lim_{i\to\infty}\sD_i$ has
asymptotic descent if and only if it stabilizes; and the latter descent is
also on $X$.

 \item Asymptotic boundedness is also necessary for the existence of
asymptotic descent on $X$, namely, for any $\sC\ge0$, and any sequence of
$r_i$ with $i$ such that $\sE_i=0$.

 \item Similarity of the characteristic type as in Remarks~\ref{asaturr},
(7) preserves strict asymptotic bounds. For the truncation $I\sD_{iI}$,
we can take the same $\sC$ and new real numbers $r_{i/I}:=r_i/I$, where
we can take the same (truncated) real numbers $r_{i/I}:=r_i$ for the usual
truncation $\sD_{iI}$.
 \end{enumerate}
 \end{rem}

Remark~\ref{asb}, (2) is very close to the stabilization that we require
for f.g. The only difference is in a {\em universal\/} descent: i.e.,
 \begin{itemize}
 \item infinitely many $\sD_i$ have descent on the single model $X$.
 \end{itemize}
 To establish this we need asymptotic $\sC$-saturation (cf.\
Theorem~\ref{assdes}). In addition, asymptotic $\sC$-saturation gives
 \begin{itemize}
 \item {\em rationality:\/}
$\sD$ is a $\Q$-divisor.
 \end{itemize}
 Both of the properties are essential for f.g.\ of functional algebras
(cf.\ Limiting Criterion~\ref{limcr} above and Example~\ref{stexm} below).

For {\bi}divisors, the choice of an appropriate model for descent is
crucial (cf.\ Example~\ref{exc}, triples and Conjecture~\ref{cbcon}). A
criterion for an appropriate model is asymptotic boundedness over it
(cf.\ Remarks~\ref{asb}, (2) and~(3)).

\subsection{Choice of $Y$ in the asymptotic descent}\label{schzy} In our
approach to the descent problem (cf.\ Theorems~\ref{assdes} and
\ref{fnonvan} below), it is easier to present a {\bi}divisor by a
semiample $\R$-Cartier divisor than by any other. Because by definition,
semiample gives some model and an $\R$-Cartier divisor on it! Thus we
need a model $Y/Z$ of $X/Z$ such that:
 \begin{q}
 \item[(SAM)] $\sD_Y$ is {\em semiample\/}$/Z$, in particular, nef$/Z$.
 \end{q}
 Semiampleness$/Z$ means that of $\R$-divisors \cite[Definition
2.5]{sh96b}. If $D=\sD_Y$ is a $\Q$-divisor, it is equivalent to {\em
stably free\/}: $\Bs{\linsys{ND}}=\emptyset$ for some natural
number $N$.

We also need an $\R$-divisor $\sC$ on $X$, a reduced $\Q$-Cartier divisor
$F$ on $Y$, and a positive real number $\ga$ such that:
 \begin{q}
 \item[(EEF)] {\em exceptional effectiveness:\/} $\sC\ge0/Y$, that is,
$\mult_{E_i}{\sC}\ge0$ in each exceptional$/Y$ prime {\bi}divisor $E_i$;
and
 \item[(LGD)] {\em linear growth for divisor:\/} in each prime {\bi}divisor
$P_i$
 \[
 \mult_{P_i}{\sC}>-1+\ga\mult_{P_i}{F},
 \]
 or, equivalently,
 \item[(AEF)] $\sC-\ga \Fbar$ is {\em almost effective\/}:
$\rdup{\sC-\ga\Fbar}\ge0$.
 \end{q}
 The latter implies that $\sC$ is also almost effective: $\rdup{\sC}\ge0$,
whereas (EEF) states more: exceptional effectiveness$/Y$.

 \begin{defn}\label{defn!prd}
 \addcontentsline{toc}{subsection}{\ref{defn!prd} Definition of prediction}
 We say that the asymptotic descent problem has a {\em prediction\/}
$(Y/Z,\sC,F,\ga)$ if $Y,\sC, F$ and $\ga$ exist, and if
 \begin{q}
 \item[(SAB)] the descent data for $\sD_\bull$ is {\em strictly
asymptotically bounded\/}$/Y$ by $\sC$; and
 \item[(UAD)] $F$ includes the supports of the irrational part of $\sD_Y$,
and the {\em divisorial\/} locus where the {\em strict asymptotic
descent\/} is {\em unknown (not given)\/} (cf.\ Remark~\ref{casd}, (3)
below); that is, for any prime divisor $P_i$ in $Y$, $\mult_{P_i}{\sD}$
is irrational, or after any truncation $\mult_{P_i}{\sD}$ is an
accumulation point of $\mult_{P_i}{\sD_i}$.
 \end{q}
 Under \FDS, by Definition~\ref{bdd} and Remark~\ref{minar}, (4), (SAB)
implies (SAM).
 \end{defn}

 \begin{rem} \label{casd}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item In applications under \BNF, boundedness implies that
$\be\sC\ge\sE_i\ge0/Y$ for some natural number $i$ and positive real
$\be$, (cf.\ Proposition~\ref{desd}, \EFF\ and Example~\ref{stexm} below).
Thus $\sC\ge0/Y$ as in (EEF).

 \item If $\sC\ge \ep \Fbar/Y$ for some $\ep>0$, and
$\rdup{\sC_Y}\ge0$ then (LGD) holds for some $0<\ga\ll \ep$.

 \item For simplicity, we can assume in (LGD) that for {\em any\/} $F$,
there exists a positive $\ga$ for which it holds. In fact, we need a
bounded family of $F$ (cf.\ triples), or just a single $F$ in most
applications. The minimal assumption on $F$ is (UAD), but it is enough
that $F$ contains the support of the fractional part for every divisor
$(\sD_i)_Y$ (cf.\ \FDS\ in the minimal assumptions above). We need this
assumption even though a posteriori $F=0$, as in Theorem~\ref{assdes}
below.

 \item As in Remarks~\ref{asaturr}, (7), similarity of characteristic
types preserves boundedness of descent data for predictions with new
$\sC:=\sC- \overline{(a)}$. For (UAD) we can use the invariance of
irrational part of $\sD$ under similarity (cf.\ Lemma~\ref{intpart}).
 \end{enumerate}
 \end{rem}

Now we are ready to state the main result of the section.

 \begin{thm} \label{assdes}
 \addcontentsline{toc}{subsection}{\ref{assdes} Main theorem on asymptotic
descent}
Let $\sD=\lim_{i\to\infty}\sD_i$ be an asymptotic descent problem such that
 \begin{itemize}
 \item the problem has a prediction $(Y/Z,\sC,F,\ga)$;
 \item $\sD_\bull$ satisfies the minimal assumptions of \ref{asdepr}:
{\em\FDS\ on $Y$ (maybe, for some reduced divisor $\ne F$)\/} and \MXD; and
 \item $\sD_\bull$ is asymptotically $\sC$-saturated.
 \end{itemize}
 Then $\sD_Y$ is a $\Q$-divisor, and\/ {\em strict stabilization\/} holds
on $Y$: after any truncation, $(\sD_i)_Y=\sD_Y$ for some $i\gg0$.
 \end{thm}

 \begin{add} If \BNF\ also holds, then the strict asymptotic descent
problem has a solution on $Y$.
 \end{add}

 \begin{add} Moreover, if \CGR\ of Proposition~\ref{coher} also holds,
then $\sD_Y$ (or $\sD$) contracts all prime divisors $P_i$ on $Y$ with
$\mult_{P_i}{\sC}> 0$. Thus on the {\em stable\/} model
 \[
 Xst/Z=\Proj_Z{\dival_Z{
\overline{\sD_Y}}}=\Proj_Z{\dival_Z{\sD}},
 \]
 $\sC_{Xst}\le0$ (cf.\ Conjecture~\ref{cbcon}).
 \end{add}

The main part of the theorem is a rationality theorem, while
\ref{assdes}.1--2 treat the descent and the structure of the stable model.

 \begin{cor}\label{assdesc} Under arithmetic monotonicity (assertion \AMN\
of Conjecture~\ref{conj!fga}), {\em complete stabilization\/} on $Y$ holds:
i.e., for all $i$ after a truncation of $\sD_\bull$. More precisely,
there exist a stabilization index $I$ such that
$(\sD_i)_Y=\sD_Y$ for all $I\mid i\gg0$.
 \end{cor}

 \begin{add} Again under arithmetic monotonicity \AMN, {\em complete
stabilization\/} holds for $\sD_\bull$. More precisely, there exist a
stabilization index $I$ such that $\sD_i=\sD=\overline{\sD_Y}$ for all
$I\mid i\gg0$.
 \end{add}

 \begin{proof} Immediate by the theorem, \MXD\ and \AMN.
 \qed\end{proof}

 \begin{exa}\label{exc} Let $Y\to X$ be a divisorial contraction with an
exceptional Cartier divisor $E$. For example, it might be a blowup of a
smooth locus in $X$. Take a system $\sD_i=d_i\Ebar$, where $d_i$
is a sequence of real numbers with $\lim_{i\to\infty}d_i=d$. If all $d_i\le
d$ and $d>0$ then $\sD_\bull$ satisfies all the conditions of
Theorem~\ref{assdes} on $X$, except for \BNF. The descent problem has a
solution on $Y$ but it does not stabilize and may not be rational.
Nonetheless, for a prediction $X$, $\sD_X=0$ is always rational as in the
main part of the theorem.

Note that the same holds even for $d=+\infty$.
 \end{exa}

 \begin{lem}[Approximation] \label{aproxl}
 \addcontentsline{toc}{subsection}{\ref{aproxl} Lemma on approximation}
 Let $\fQ\subset\R^N=\Z^N\tensor_\R\R$ be a rational
polyhedron,\footnote{Convex is probably is intended.} and
$D=(d_1,\dots,d_N)\in\fQ$ a point in it. Then there exist positive real
numbers $r$ and $\ep$, an infinite set of natural numbers $m$ and a
sequence of rational points $Q_m\in\fQ$ such that:
 \begin{itemize}
 \item each $m Q_m$ is {\em integral:\/}
$m Q_m\in \Z^N$,
 \item $\Vert D-Q_m\Vert< \ep/m^{1+r}$.
 \end{itemize}
 Moreover, if $D$ is irrational then there exists $D'\in\fQ$, and a
sequence $Q_m=(q_{m,1},\dots,q_{m,N})$ as above for $D'$ such that:
 \begin{itemize}
 \item $D'\le D$; but
 \item $q_{m,i}>d_i$ at least in one coordinate (the $i$th); and
 \item $q_{m,i}=d_i$ for the rational coordinates $d_i$.
 \end{itemize}
 \end{lem}

 \begin{nota} We write $\Vert (x_1,\dots,x_N) \Vert=\max\{|x_i|\}$ for the
{\em maximal absolute value\/} norm in $\R^N$; it induces the usual real
topology of $\R^N$. The inequality
 \[
 D'=(d_1',\dots,d_N') \le D=(d_1,\dots,d_N)
 \]
means that each $d_i'\le d_i$.
 \end{nota}

 \begin{proof} First of all, we can assume that $D$ is irrational; that
is, it has an irrational coordinate $d_i$. Otherwise we take $Q_m=D$
for $m$ such that $m Q_m$ are integral, and take any $r$ and $\ep>0$. In
particular, $N\ge1$ if $D$ is irrational.

Secondly, we can assume that each coordinate $d_i$ of $D$ is irrational,
and, moreover, that $D$ does not belong to any proper rational affine
subspace $H$. (If $d_i$ is rational then $D\in H=\{x_i=d_i\}$.) Indeed, if
$D\in H$ then there is a rational affine embedding $A\colon\R^m\into\R^N$
with
 \begin{itemize}
 \item $A(\R^m)=H$, and
 \item $M\le N-1$.
 \end{itemize}
 Note that $\fP=A\1\fQ\subset\R^m$ is also a rational polyhedron with
$C=A\1D\in\fP$. Thus if $P_m\in\fP$ is a sequence with the required
property for $C$, then $Q_{Im}=A(P_m)$ gives the required sequence for
$D=A(C)\in\fQ$, where $I$\/ is an {\em index\/} for $A$, that is, $IA$
is integral. The set of natural numbers $m$ is replaced by $Im$
(truncation). Hence $ImQ_m=(IA)(m P_m)\in\Z^N$; and
 \[
 \Vert D-Q_{Im}\Vert=\Vert A(C-P_m)\Vert<
\Vert A\Vert \ep/ m^{1+r}=\Vert A\Vert I^{1+r}\ep/ (I m)^{1+r};
 \]
 and we can take $\ep:=\Vert A\Vert I^{1+r}\ep$ for $Q_{Im}$, where
$\Vert A\Vert=\max \Vert A x\Vert/ \Vert x\Vert$, for $x\ne0$, is the usual
norm of the linear part of $A$. However we need to replace the conditions
on $D'$ by geometric ones: namely, for a polyhedral cone $\fD$ with the
single vertex $D$
 \begin{itemize}
 \item $D'\in\fD$; but
 \item each $Q_m\notin\fD$; and
 \item $D'$ is still in each rational face of $\fD$.
 \end{itemize}
 For example, the original cone $\fD$ is given by the inequalities
$x_i\le d_i$. But the cone $\fC=A\1\fD$ with the only vertex $C$ is more
general. Nonetheless, if $C'\in\fC$ has a required sequence $P_m$, then
$Q_{Im}$ satisfies the lemma with $D'=A(C')$. Note that each rational
face of $\fD$ gives a (nonempty) rational face of $\fC$ (because $C$
belongs to it).

Thirdly, we can assume that $\dim\fQ=N$ near $D$, and $D$ is in interior
of $\fQ$. Thus we can assume that $\fQ=\R^N$ locally near $D$. Otherwise
there is a rational face $H\subset\R^N$ of $\fQ$ such that $D\in H$. Then
we can reduce the lemma to $H$ as in Step~Two. Therefore by induction
on $N$ we get the interior.

The required approximations then exist by Cassels \cite[Chap.~I,
Theorem~VII]{cas57} with any $0<r<1/N$ and $\ep\ge N/(N+1)$, where
$N\ge1$ is the dimension of the approximation space $\R^N$.

Fourth, by the arguments of Step~Three, we can assume that the cone $\fD$
has no rational faces.

Finally, for the last statement of the lemma we need a more accurate
choice of $m$ and $Q_m$, namely, $Q_m\notin\fD$. Otherwise there is an
infinite subsequence $Q_m\in\fD$. Their affine span $H$ is rational and
contains $D$ since $D=\lim_{m\to\infty} Q_m$. Thus by Step~Two, $H=\R^N$,
and the cone $\fD$ is {\em big}: e.g., $\dim\fD=N$. Since $N\ge1$, there
is a rational line $L$ in $\R^N$ that the polyhedron
$\fD\cap L=[D',\infty)$ is a ray, and its vertex $D'$ is close to $D$ and
irrational on $L$. By Step~Two we can reduce the last statement of the
lemma to $L$ with $D'$. In the latter case we use continued fractions for
the best approximations in $L$ from outside $[D,\infty)$
\cite[Chap.~I, Theorem~II]{cas57}.
 \qed\end{proof}

We apply the lemma to a polyhedron in an $\R$-vector space
 \[
 \fD_F=\directsum_{i=1}^N\R P_i,
 \]
 of $\R$-divisors on $X$ supported in $F$, where $F=\sum_{i=1}^N P_i$ is a
reduced divisor on $X$. Multiplicities of divisors define a canonical
isomorphism$/\Q$ of the space with $\R^N$. It induces the well-known
topology in $\fD_F$ and norm, the {\em maximal absolute value\/} of the
multiplicities in prime divisors (or in prime {\bi}divisors in the case of
$\R$-{\bi}divisors). In particular, the limits of {\em divisors\/} (but
not {\bi}divisors) are limits in the norm (cf.\ Caution~\ref{wlsup}). The
isomorphism transforms the ordering of divisors into $\le$ in $\R^N$.

In the space we take a polyhedron $\fD$ of semiample$/Z$ divisors under
the following extra assumption.

 \begin{defn}\label{bnd} For a given set of divisors $D$, e.g., for a cone
of divisors in Example~\ref{sac} below,
 \begin{q}
 \item[(BND)] the semiampleness is {\em bounded (effective)\/} if there
exists a natural number $M$ such that $N\le MI$ for the stable base
freedom of $D$, where $I$ is the index $D$
 \end{q}
 (cf.\ triples and Lemma~\ref{split}, (d)). Of course, this concerns {\em
only\/} $\Q$-divisors in the set.
 \end{defn}

 \begin{exa}\label{sac} Any rational polyhedral cone $\fD$ generated by a
finite set of semiample$/Z$ divisors $D_i$ satisfies (BND) of
Definition~\ref{bnd}. Indeed, we can assume that
 \begin{itemize}
 \item each $D_i$ is a $\Q$-divisor;
 \item $F$ is the common support of all divisors $D_i$; and
 \item the cone is simplicial.
 \end{itemize}
 Thus each $D\in\fD$ has a unique presentation $D=\sum r_iD_i$ with {\em
coordinates\/} $r_i\in\R$, and each $r_i\ge0$ and $D$ is rational if
and only if each $r_i$ is rational. There is a natural number $J$ such
that each $J I D$ has integer coordinates $r_i$, where $I$ is the index
of $D$. Then we can take $M:=JM$ in (BND) where $M$ is the minimal natural
number for which every $\linsys{MD_i}$ is free.
 \end{exa}

 \begin{cor}[Approximation] \label{aprox}
 \addcontentsline{toc}{subsection}{\ref{aproxl} Corollary on free
linear systems}
 Let $D$ be a semiample$/Z$ $\R$-divisor on $X$. Then there exist
positive real numbers $r$ and $\ep$, an infinite set of natural numbers
$m$ and a sequence of\/ $\Q$-divisors $Q_m$ on $X$ such that:
 \begin{itemize}
 \item each $\linsys{mQ_m}$ is free, and {\em in particular, each
$Q_m$ is semiample;\/}
 \item each $\Supp{(D-Q_m)}\subset\Supp{D\irr}$, where $D\irr$ is the
irrational part of $D$; that is, the approximation concerns only the
irrational multiplicities of $D$;
 \item $\Vert D-Q_m\Vert< \ep/m^{1+r}$.
 \end{itemize}
 Moreover, if $D$ is not a $\Q$-divisor, then there exists an $\R$-divisor
$D'$, and a sequence of $\Q$-divisors $Q_m$ on $X$ as above for $D'$ such
that:
 \begin{itemize}
 \item $\Supp{(D-D')}\subset\Supp{D\irr}$;
 \item $D'\le D$; but
 \item $\mult_{P_i}{Q_m}>\mult_{P_i}{D}$ at least in one prime divisor
$P_i$ on $X$; and
 \item $\mult_{P_i}{Q_m}=\mult_{P_i}{D}$ for the rational multiplicities
$\mult_{P_i}{D}$.
 \end{itemize}
 \end{cor}

 \begin{proof} To apply Lemma~\ref{aproxl} we include $D$ into a rational
polyhedron $\fQ$.

Since $D$ is semiample$/Z$,
 \begin{q}
 \item[(NBH)] each divisor in a {\em neighborhood\/} of $D$ is (nef and)
semiample$/Z$;
 \end{q}
moreover, (BND) holds in it. Indeed, $D\sim_\R g^*H$, where
 \begin{itemize}
 \item $g\colon X\to T/Z$ is a contraction; and
 \item $H$ is a numerically ample$/Z$
$\R$-divisor on $T$.
 \end{itemize}
 In other words, $D=g^*H+\sum r_i(f_i)$, where $r_i\in\R$ and $0\ne f_i\in
k(X)$. Let $F$ be a reduced divisor on $X$ that contains the supports of
divisors $(f_i)$ and $g^*H$. Take any rational polyhedral cone $\fH$ of
numerically ample$/Z$ $\R$-divisors on $T$ with $H\in\fH$, and with
$\Supp{g^*\fH}\subset F$. Then the rational polyhedral cone
 \[
 \fD=g^*\fH\oplus\Bigl(\directsum\R(f_i)\Bigr)\subset\fD_F
 \]
 is generated by semiample divisors, because each numerically ample$/Z$
divisor is semiample$/Z$ by the Kleiman criterion, and each principal
divisor is also semiample$/Z$. By Example~\ref{sac} it satisfies (BND)
with some natural number $M$. (In addition, we can suppose that $\fH$ and
$\fD$ are respectively neighborhoods of $H$ and $D$ respectively in their
linear spans$/\Q$.) By construction $D\in\fD$.

To preserve the rational multiplicities (coordinates) under the
approximation, set $\fQ=\fD\cap Q$, where $Q$ is the subspace of $\fD_F$
given by the system of affine equations$/\Q$:
$\mult_{P_i}{X}=\mult_{P_i}{D}$, provided that the latter multiplicity is
rational. By construction
 \begin{itemize}
 \item $\fQ$ is a rational polyhedron; and
 \item $D\in\fQ$.
 \end{itemize}

Then Lemma~\ref{aproxl} gives real numbers $r$ and $\ep$, and a sequence of
$\Q$-divisors $Q_m\in\fQ$. They satisfy the corollary except for
$\linsys{mQ_m}$ free. However $mQ_m$ is {\em integral\/} divisor,
and $\linsys{MmQ_m}$ has no base points by Example~\ref{sac}. Thus as in
Step~2 in the proof of Lemma~\ref{aproxl}, we can replace $Q_m$ by
$Q_{Mm}:=Q_m$ and take $\ep:=M^{1+r}\ep$.
 \qed\end{proof}

However, to apply the corollary to {\bi}divisors we need the following
result.

 \begin{lem}\label{movest} Let $\sC$ and $\sD$ be two {\bi}divisors of
$X$, $F$ a reduced divisor, and $\al\le1,\tau$ positive real numbers
such that:
 \begin{itemize}
 \item $F$ is $\Q$-Cartier and
 \begin{equation}
 \mult_{P_i}{\al\sC}>-\al+\tau \mult_{P_i}{F}; \tag{$*$}
 \end{equation}
in each prime {\bi}divisor $P_i$,
 \item the descent data $\sE$ of $\sD$ over $X$ is bounded by $(1-\al)
\sC$, {\em that is,}
 \begin{equation}
 \sE\le(1-\al)\sC/X; \tag{$**$}
 \end{equation}
 and
 \item $\sD_X=D\bof+D\fr$, where $\linsys{D\bof}$ is free,
$\Supp{D\fr}\subset F$, and $\Vert D\fr\Vert<\tau$.
 \end{itemize}
 Then the following estimate for $\Mov{\rdup{\sD+\sC}}/Z$ holds: on any
model $W/X$
 \[
 \Mov{\rdup{\sD+\sC}_W}\ge (\Dbar\bof)_W;
 \]
 in particular, the nonvanishing
$\linsys{\rdup{\sD+\sC}_W}\ne\emptyset/Z$ holds.
 \end{lem}

Note that $\sD_X$ is $\R$-Cartier since its descent data is bounded (cf.\
Definition~\ref{bdd}).

 \begin{proof} By definition of descent data, $\sD=\overline{\sD_X}-\sE$.
Then by the decomposition of $\sD_X$,
 \[
 \sD=\overline{D\bof+D\fr}-\sE=\overline{D\bof}+\overline{D\fr}-\sE.
 \]
Hence $\sD+\sC=\overline{D\bof}+\overline{D\fr}+\sC-\sE$. Since
$\overline{D\bof}$ is integral, it is enough to verify that
$\rdup{\overline{D\fr}+\sC-\sE}\ge0$ (cf.\ the proof of
Proposition~\ref{coher}), or $\overline{D\fr}+\sC-\sE$ is almost
effective (cf.\ (AEF) in \ref{schzy}, Choice of $Y$ in the asymptotic
descent).

We make it into two steps. First, on $X$. Since $0<\al\le1$, $(*)$ implies
 \begin{equation}
 \rdup{\sC_X-\tau F}\ge0. \tag{$*{*}*$}
 \end{equation}
 Indeed, $(*)$ times $\al\1$ gives $(*{*}*)$ with $\tau/\al$ instead of
$\tau$. Since $F\ge0$ and $1/\al\ge1$ we can replace $\tau/\al$ by
$\tau$.

Since $\Supp{D\fr}\subset F$, $F$ is reduced, and $\Vert D\fr\Vert<\tau$,
then $F=\sum P_i$ and $D\fr=\sum d\fr{}_{,i}P_i$ with each
$|d\fr{}_{,i}|<\tau$. Thus each $d\fr{}_{,i}>-\tau=-\tau\mult_{D_i}{F}$
for $P_i$ in $F$. In other words, $D\fr\ge-\tau F$. Therefore
$(\overline{D\fr}+\sC-\sE)_X=D\fr+\sC_X\ge \sC_X-\tau F$, and by
$(*{*}*)$, $\rdup{\overline{D\fr}+\sC-\sE}_X\ge0$ on $X$.

Second, $/X$. Note that $D\fr\ge-\tau F$ implies
$\overline{D\fr}\ge-\tau\Fbar$. Therefore by $(**)$,
 \[
 \overline{D\fr}+\sC-\sE\ge
\sC-\tau F- (1-\al)\sC=\al C-\tau F/X.
 \]
 Hence by $(*)$
$\rdup{\overline{D\fr}+\sC-\sE}\ge0$ over $X$ because $-\al\ge-1$.

Finally, the nonvanishing follows since $\linsys{D\bof}$ is free.
 \qed\end{proof}

 \begin{pfof}{Theorem~\ref{assdes}}
 \addcontentsline{toc}{subsection}{Proof of Theorem~\ref{assdes}}
After a truncation of $\sD_\bull$ we can assume that the saturation index
is $I=1$ and all other assumptions hold for each $i$ literally. The
conditions of the theorem are preserved by Remarks~\ref{asb}, (4),
\ref{casd}, (4), \ref{minar}, (1), and \ref{asaturr}, (7). To prove
strict asymptotic descent, we can do it after any truncation.

First, we verify that $D:=\sD_Y$ is a $\Q$-divisor. Thus suppose that
$D$ is not a $\Q$-divisor. Then we find $i$ and $j$ such that: on any model
$W/Y/Z$ of $X/Z$
 \[
 \Mov{\rdup{j\sD_i+\sC}_W} \not\le j\sD_W,
 \]
 and the mobile part is finite, that is, a {\em nonvanishing\/}$/Z$:
$\linsys{\rdup{j\sD_i+\sC}_W}\ne\emptyset$, holds. This contradicts
asymptotic $\sC$-saturation. Indeed, by \MXD\ this implies the mobile
part is $\not\le j(\sD_j)_W$ for such $i$ and $j$ on any sufficiently
high model $W=X\hr$.

A estimate for the mobile part can be done by Lemma~\ref{movest} for
$\sD:=j\sD_i$ and $F$ in the prediction with some $j$ and $i\gg0$ on any
$W/Y$.

By Corollary~\ref{aprox} and (SAM), for $D:=\sD_Y$, there exist positive
real numbers $r$ and $\ep$, an $\R$-divisor $D'$ on $Y$, an infinite set of
natural numbers $m$ and a sequence of $\Q$-divisors $Q_m$ on $Y$ such that:
 \begin{itemize}
 \item each $\linsys{mQ_m}$ is free;
 \item each $\Supp{(mD'-mQ_m)}\subset\Supp{(m D\irr)\subset F}$ by
(UAD) for the prediction;
 \item $\Vert m D'-m Q_m\Vert< \ep/m^r$.
 \item $D'\le\sD_Y$; but
 \item $m Q_m\not\le m\sD_Y$; and
 \item $m Q_m=m D'=m\sD_Y$ in the rational multiplicities of $\sD_Y$.
 \end{itemize}

It is enough to verify that, for some $j=m$ and $i\gg0$, on any model
$W/Y/Z$ of $X/Z$
 \[
 \Mov{\rdup{j\sD_i'+\sC}_W}
\ge m (\overline{Q_m})_W,
 \]
 where $\sD_i'=\sD_i-\Ebar$, and $E=\sD_Y- D'\ge0$ is an effective
divisor on $Y$ by construction. Since $\Ebar$ is also effective,
then, for the same $j$ and $i\gg0$, on any model $W/Y/Z$ of $X/Z$
 \[
 \Mov{\rdup{j\sD_i+\sC}_W}=\Mov{\rdup{j\sD_i'+j\Ebar+\sC}_W}\ge
\Mov{\rdup{j\sD_i'+\sC}_W}
\ge m (\overline{Q_m})_W.
 \]
 On the other hand, $m (\overline{Q_m})_W\not\le m\sD_W=j\sD_W$ even on
$W=Y$.

Note that the {\bi}divisors $\sD_i'$ share most of their properties with
{\bi}divisors $\sD_i$: e.g.,
 \begin{itemize}
 \item \FDS\ and (UAD) hold for $\sD_\bull'$; moreover, each
$\Supp{(j\sD_i'-j\sD')_Y}=\Supp{(\sD_i-\sD)_Y}\subset F$, where
$\sD'=\sD-\Ebar$; usually, in \FDS\ we need a reduced divisor
$\ge F$;
 \item $\lim_{i\to\infty}\sD_i'=\sD'$, whereas $\sD_Y'=D'$; and
 \item (SAB) holds for $\sD_\bull'$ with the same $i$, because
$\sE(\Ebar)=0$ and $\sE(\sD_i')=\sE(\sD_i-\Ebar)=\sE(\sD_i)/Y$
by \EFF\ and \ADD\ in Proposition~\ref{desd}
 \end{itemize}

Now take any positive $\al<1$, and set $\tau=\al\ga$. Then (LGR) for
the prediction (multiplied by $\al$) implies $(*)$ in Lemma~\ref{movest}.
Take any $j=m\gg0$ such that $\ep/m^r\le
\tau$. Then $\sD_Y=m D'=D\bof+D\fr$ satisfies the assumptions for the
decomposition in the lemma where $\sD:=m\sD'$, $D\bof=m Q_m$ and
$D\fr=m D'-m Q_m$. Indeed, by construction, $\Vert D\fr\Vert<\ep/
m^r\le\tau$ and $\Supp{D\fr}\subset F$.

By \FDS\ the limit $\lim_{i\to\infty}(\sD_i')_Y=\sD_Y'=D'$ takes place on
$Y$ in the norm $\Vert\ \Vert$. Thus by (UAD), for such $j=m$ and {\em
any\/} $i\gg0$, the decomposition $\sD_Y=j (\sD_i')_Y=D\bof+D\fr$
satisfies the assumptions of Lemma~\ref{movest}, where $D\bof=m Q_m$ is
still the same, but $\sD:=j\sD_i'$, and
$D\fr:=j(\sD_i')_Y-m Q_m=j(\sD_i)_Y-j E-m Q_m$ with $\Vert
D\fr\Vert<\tau$ and $\Supp{D\fr}\subset F$.

The boundedness $(**)$ in the lemma holds for {\em some\/} $i\gg0$. More
precisely, take any $i\gg0$ with $m/r_i\le(1-\al)$; the latter is
$>0$. Then $(**)$ holds for the $i$. Indeed, by (EEF) in the choice of
$Y$, $m\sC/r_i\le(1-\al)\sC/Y$. Thus by \ADD\ in Proposition~\ref{desd} and
by the definition of the boundedness,
 \[
 \sE(\sD)=\sE(j\sD_i')=m\sE(\sD_i')=m\sE(\sD_i)=m\sE_i\le m\sC/r_i\le
(1-\al)
\sC/Y.
 \]
 Hence by Lemma~\ref{movest}
 \[
 \Mov{\rdup{j\sD_i'+\sC}_W}=\Mov{\rdup{\sD+\sC}_W}\ge
(\overline{D\bof})_W=m (\overline{Q_m})_W.
 \]
 This completes the proof that $\sD_Y$ is a $\Q$-divisor.

Second, we check that the limit $\lim_{i\to\infty}\sD_i=\sD$ stabilizes on
$Y$. Using Corollary~\ref{aprox} again, but now in the trivial case that
$D:=\sD_Y$ is a $\Q$-divisor, and by above arguments, this time we can't
derive a contradiction. But by the asymptotic saturation we get that, for
some $j=m$ (with integral $m\sD_Y$) and some $i\gg0$, on any $W/Y/X$
 \[
 j(\sD_j)_W\ge \Mov{\rdup{j\sD_i+\sC}_W}\ge m(\overline{Q_m})_W,
 \]
 whereas $m(\overline{Q_m})_Y=m Q_m=m D=m\sD_Y$ because the approximation
is trivial. Hence, for such $j=m\gg0$, $\overline{Q_m}\le\sD_j$, and, on
$Y$, $\sD_Y=Q_m\le(\sD_j)_Y$. Thus \MXD\ gives the stabilization of the
limit on $Y$, namely, $(\sD_j)_Y=\sD_Y$.

Third, we assume \BNF\ and prove the stabilization in
Addendum~\ref{assdes}.1. It is enough to verify that $\sE(\sD)=0/Y$.
Indeed, then, for some $j=m\gg0$, $\overline{Q_m}=\sD$ and
$\sD_j\ge\overline{Q_m}=\sD$. Hence \MXD\ gives stabilization, namely,
$\sD_j=\sD$.

By Remarks~\ref{asb}, (1--2), under \FDS\ and (SAB),
 \[
 \sE(\sD)=\sE(\lim_{i\to\infty} \sD_i)=\lim_{i\to\infty}\sE (\sD_i)\le
 \lim_{i\to\infty}\sC /r_i=\sC /\lim_{i\to\infty} r_i=0/Y.
 \]
 On the other hand, under \BNF, $\sE(\sD)\ge0/Y$ again by
Remarks~\ref{asb}, (1--2). Therefore $\sE(\sD)=0/Y$.

Finally, we assume also (CRG) and verify Addendum~\ref{assdes}.2, i.e.,
$D:=\sD_Y$ contracts every prime divisor $P_i$ on $Y$ with
$\mult_{P_i}{\sC}>0$.  (Equivalently, $D\rest{P_i}$ is semiample but is
not big on every such $P_i$. For {\bi}divisors $P_i$ exceptional on $Y$,
the same follows from the above descent.) Here we use the arguments of
Reid \cite[Proposition~1.2]{r80}. By our assumptions, $\de P_i\le\sC$
for some $\de>0$. By asymptotic descent Addendum~\ref{assdes}.1 and by
Reid (or by Addendum~\ref{adjd}.1), if $\mult_{P_i}{\sC}>0$, on any
model $W/Y/Z$ of $X/Z$, and for all $j=m\gg0$ such that $\linsys{mD}$ is
free, we have the following lower bound:
 \[
 \Mov{(j(\sD_j)_W+P_i)}=\Mov{(j\sD_W+P_i)}=j\sD_W+P_i=j(\sD_j)_W+P_i.
 \]
 This contradicts asymptotic $\sC$-saturation whenever the upper bound
 \[
 \Mov{\rdup{j\sD_j+\sC}_W}\le j(\sD_j)_W
 \]
 with $i:=j$ holds in asymptotic saturation over some $W$ (cf.\
Remarks~\ref{asaturr}, (2) and (5)) that does not depend on $j$. More
precisely, the estimate holds on any $X\hr/W$ and any $j=m$ as above.
Indeed, since $j\sD_j=j\sD$ is integral,
 \[ j(\sD_j)_W+P_i\le
\rdup{j(\sD_j)_W+\de P_i}\le
\rdup{j\sD_j+\sC}_W
 \]
 and
 \[ j(\sD_j)_W+P_i=\Mov{(j(\sD_j)_W+P_i)}\le
\Mov{\rdup{j\sD_j+\sC}_W},
 \]
 which contradicts asymptotic $\sC$-saturation.

The independence of the upper estimate on $X\hr/W$ follows from
Proposition~\ref{coher} for $\sD:=j\sD_j$ and $\sE:=0$. Note only that the
latter for some $j=m$ implies the same with the same $W$ for any other
natural number $j:=m'$ under our assumptions.\footnote{I give up. I have
absolutely no idea what ``the latter'' or ``the same'' refers to.} Indeed,
we can assume that $m'\ge m$ and $\linsys{(m'-m)D}$ is free. Then
 \[
 \rdup{m'\sD_{m'}+\sC}=\rdup{m'\sD+\sC}=(m'-m)\sD+\rdup{m\sD_m+\sC}.
 \]
 Thus by the projection formula if
 \[
 \Oh_W(
\rdup{m\sD_m+\sC}_W)=\Oh_W(
\rdup{m\sD_m+\sC})
 \]
 then the same holds for other natural numbers $m'$ on $W$, because
$(m-m')\sD=\overline{(m'-m)D}$ is a Cartier {\bi}divisor$/Y$ (cf.\
{\bi}free in Example~\ref{texam}).
 \qed\end{pfof}

Now we are ready to explain what one needs to prove (FGA) and how to apply
it to flips (cf.\ Corollary~\ref{mainc1}).

 \begin{exa} \label{stexm} Let $\sL$ be an algebra of type (FGA) over
$(X/T,B)$ (see Definition~\ref{defn!fga} above), and $\sD_\bull$ its
characteristic system. We expect that such $\sL$ is always f.g. By
Theorem~\ref{limcr}, this is equivalent to an affirmative solution of
the problem of asymptotic descent for the limit
$\sD=\lim_{i\to\infty}\sD_i$ on some model $Y/\Proj_T{\sL}/T$ of $X/T$,
where $\sD$ is {\bi}semiample$/T$ and is a semiample$/Y/T$
$\Q$-{\bi}divisor. This last condition means that $D=\sD_Y$ is a
$\Q$-Cartier divisor that is semiample$/T$, and $\sD=\Dbar$.

By Addendum~\ref{bpbdal}.1 and Proposition~\ref{texa}, (1) respectively,
the system $\sD_\bull$ satisfies the minimal and additional assumptions in
Theorem~\ref{assdes}. Indeed, \BSD\ in Addendum~\ref{bpbdal}.1 implies \FDS.
Moreover, by (FGA) of Definition~\ref{defn!fga}, asymptotic saturation
holds for $\sD_\bull$ with $\sC=\sA=\sA(X,B)$, that is, lca saturation
holds. By Example~\ref{coha}, \CGR\ holds for the chosen $\sC=\sA$.

Thus the main obstacle to apply Theorem~\ref{assdes} is to choose a
prediction $(Y/Z=T,\sC,F,\ga)$. Our choice of $\sC=\sA$ is canonical.
On the other hand, because $(X,B)$ is Klt, on any model $Y/T$ of $X/T$
and for any $\Q$-Cartier divisor $F$ on $Y$, (LGD) holds for some
$\ga>0$ (cf.\ Example~\ref{fnonvex} below). In particular, \FDS\ implies
(UAD) on any $Y/T$ for some $F$ (cf.\ Remarks~\ref{minar}, (3) and
\ref{casd}, (3) above).

However to satisfy the effectiveness (EEF) we need to take a model
$Y/T$ on which every prime {\bi}divisor $P_i$ with nonpositive discrepancy
$a_i=d_i=d(P_i,B,X)=\mult_{P_i}{\sA}$ \cite[Example~1.1.4]{sh96b} is
blown up, that is, $P_i$ is a divisor on $Y$. Again since $(X,B)$ is
Klt, such models exists by \cite[Lemma~1.6]{sh96b}. In addition, we can
assume that $Y$ is $\Q$-factorial.

Using LMMP we can make a more accurate choice of $Y$, where infinitely
many $D_i=(\sD_i)_Y$ are nef$/T$ and even semiample$/T$. This implies
(SAM) for $D=\sD_Y=\lim_{i\to\infty} D_i$ on such $Y/T$ (cf.\
Remark~\ref{minar}, (4)), and indicates where one can choose a prediction
(cf.\ triples in (MOD) of Conjecture~\ref{cbcon} and Corollary~\ref{mainb}
below).

Note that $D$ as the above limit $D$ is only nef$/T$ in particular
$\R$-Cartier by Remark~\ref{minar}, (4). But, in general, nef$/T$ does not
implies semiample$/T$. This is the difference between numerical and linear
geometry. However, by the log semiampleness conjecture
\cite[Conjecture~2.6]{sh96b} (proved in dimension $\le3$ \cite[Theorem
2.7]{sh96b}), nef is equivalent to the semiample when
 \begin{q}
 \item[(0LP)] $X/T$ is a $0$-log pair for some boundary $B$ as in
Remark~\ref{gzardec}, (2); in particular, if $(X/T,B)$ has an
$\R$-complement (cf.\ \cite[(EC) in Conjecture~1.3]{sh95})
 \end{q}
for example, by \cite[Proposition~5.5]{sh92}, this holds if
 \begin{q}
 \item[\WLF] $(X/T,B)$ is a weak log Fano contraction as in
Proposition~\ref{exsatcan}.
 \end{q}
 Moreover, under \WLF\ we do not need the log semiampleness conjecture,
and \WLF\ implies
 \begin{q}
 \item[(RPF)] the nef cone $\NEbar{X/T}$ is {\em rationally polyhedral\/}
and has {\em contractible\/} and {\em flippable birational\/} faces$/T$
(cf.\ \cite[(RPC) in Inductive Theorem~2.3]{sh95}).
 \end{q}
 In addition, the \WLF\ for some boundary and (RPF) are preserved under
modifications in any face of $\NEbar{X/T}$, that is, under the $D$-flips
of any birational contraction of $X/T$ with respect to which $D$ is
negative.

Indeed, let $X\broken X^+/T$ be such a modification, $H$ an ample$/T$
divisor on $X^+$, and $H^-$ its birational transform on $X$. Then there is
some $\ep>0$ and a boundary $B'\ge B$ such that $(X/T,B'+\ep H^-)$ is a
$0$-log pair. Then the modification is a flop of such pair,
$(X^+/T,(B')^++\ep H)$ is again a $0$-log pair, and $(X^+/T,(B')^+)$ is a
weak log Fano contraction. Therefore it again satisfies (RPF).

Since $(X/T,B)$ satisfies \WLF, under LMMP, we can apply this to modify
$X/T$ making a divisor semiample (cf.\ Theorem~\ref{existd}). For any
effective $\R$-Cartier divisor $D$ (even up to $\sim_\R$, cf.\ the
effectiveness of bss divisors explained before Proposition~\ref{uniqbss}),
there is a model $Y/T$ of $X/T$, where the birational transform
(composition of above modifications in faces) of $D$ is nef$/T$, and so
by (RPF) is semiample$/T$. Moreover, if
 \begin{q}
 \item[(SA1)] $D$ is semiample in {\em codimension\/} $1$, that is, for
any effective divisor $E$ on $X$, there is effective $D'\sim_\R D$ with
disjoint $\Supp E$ and $\Supp{D'}$; hence such $D$ is nef on general
curves$/T$ of any divisor in $X$,
 \end{q}
 then the model $Y/T$ can be taken isomorphic to $X/T$ in codimension~$1$.

Indeed, if $D$ is nef, we take $Y=X$ and we are done. Otherwise by (RPF)
there is a contraction $X\to Z/T$ on which $D$ is negative. It is
birational because $D\ge0$. Moreover, it is small under (SA1). Thus we can
modify $X/T$ by a $D$-flips, and the $D$-MMP gives a required model $Y/T$.
The modification is small under (SA1). The required termination is as in
the proof of Theorem~\ref{existd}, that is, by LMMP for
$(X/T,B'+\ep D)$, where $(X/T,B')$ is a $0$-log pair.

Note that the use of LMMP is justified by our approach (cf.\
Corollary~\ref{mainc1} and the remark at the end of
Conjecture~\ref{conj!fga}).

Since f.g.\ dependends on $\sL$ only up to quasi-isomorphism, or
equivalently, stabilization dependends on $\sD_\bull$ only up to
similarity, we can assume that all $\sD_i$ are effective. Moreover, by
(LBF) in Conjecture~\ref{conj!fga} we can assume (SA1) for
$D:=D_i:=(\sD_i)_X$. Thus if $D_i$ is $\Q$-factorial, we can construct
a model $Y_i/T$ such that:
 \begin{itemize}
 \item $Y_i$ is isomorphic to $X$ in codimension~$1$; and
 \item $D_i$ is nef, and even semiample$/T$ on $Y_i$ by (RPF).
 \end{itemize}

After taking a complement $(X/T,B')$, by the Borisov--Alexeev
conjecture\footnote{See footnote~\ref{ABfoot} on p.~\pageref{ABfoot}},
we expect that there only a finite number of models $Y_i/T$ satisfying
the above conditions. The same would follow from Kawamata's
conjecture\footnote{Reference?} on finiteness of log minimal models.
However, for our purposes, it is sufficient to have (LMMP by (RPF)
in the proof of) the Second Main Theorem \cite[Theorem~6.20]{sh96b}
with prime {\bi}divisors $D_i:=P_i$ of $\Supp{F}$ in \FDS\ and exceptional
on $X$ $P_i$ with $a_i=\mult_{P_i}{\sA(X,B)}\le0$. Then the equivalence
of weakly log canonical models $(Y_i/T,B'+\ep D_i)$ means that the
divisors $D_i$ are nef on both models. This gives a model $Y/T$ with
(SAM) and infinitely many semiample $D_i=(\sD_i)_Y$.

Finally, to secure the other properties of a prediction, e.g., (EEF), we
need to make a crepant terminal (in codimension~$2$) resolution of
$(Y/T,B')$, or of $(X/T,B')$, or even of $(X,B)$ before the modifications.
We can assume also that the resolution is $\Q$-factorial. The existence of
such a resolution follows from \cite[Theorem~3.1]{sh96b} again up to the
LMMP. (Apply it to $(X,B')$ with $B'\ge B$ and such that the new
discrepancies $a_i:=\mult_{P_i}{\sA(X,B')}$ are negative for all
exceptional $P_i$ with old $a_i:=\mult_{P_i}{\sA(X,B)}\le0$.)

This satisfies all the requirements for a prediction, except for boundedness
(SAB)! This may actually fail (cf.\ Addendum~\ref{surbound}.2 and
Example~\ref{esurbound}). However we expect that it holds for $Y/T$
provided that $\sD$ is big (cf.\ Conjecture~\ref{cbcon}, and
Proposition~\ref{surbound}). Moreover, this is the only possible choice
by Addendum~\ref{assdes}.2 and Example~\ref{fcanb}.
 \end{exa}

Thus the main difficulty in constructing pl flips is boundedness (SAB).
We start to attack it in the next section where we propose a conjecture
to resolve the difficulty and solve it in dimension 2. The latter is
enough for $3$-folds log flips.

\section{Canonical bounds and saturation} \label{canbound}

We start with another interpretation of (asymptotic) boundedness in the
{\em canonical\/} case, where $\sC=\sA=\sA(X,B)$ is the discrepancy
{\bi}divisor (cf.\ Example~\ref{stexm}, Definition~\ref{cbd} and
Proposition~\ref{aboundd} below).

 \begin{lem} \label{discr} Let $(X,B)$ be a log pair, $\sD$ an
$\R$-{\bi}divisor, and $\{\eta\}$ a set of (scheme theoretic) points of
$X$ such that:
 \begin{itemize}
 \item the divisors $K+B$ and\/ $\sD_X$ of $X$ are $\R$-Cartier;
 \item the codimension of each point $\eta$ is $\ge2$ in $X$; and
 \item $\sD\ge0$ over each $\eta$, {\em that is,
$\mult_{E_i}{\sD}\ge0$ in each prime {\bi}divisor $E_i$ with
$\cent_X{E_i}=\eta$}.
 \end{itemize}
 Then for any real number $c\ge0$ such that $(X,B+c\sD_X)$ is canonical in
each $\eta$, we have
 \[
 c\sE\le\sA
 \]
 over each $\eta$, where $\sE$ is the descent data of $\sD$ over $X$, and
$\sA=\sA(X,B)$ is the discrepancy {\bi}divisor of\/ $(X,B)$.
 \end{lem}

The inequalities {\em hold\/}$/X$ if $\{\eta\}$ is the set of all
codimension~$\ge2$ points in $X$ (cf.\ (EEF) in \ref{schzy} Choice of
prediction). For example, for curves, namely, when $\dim X=1$, the
inequalities always hold because $\{\eta\}=\emptyset$ -- there no
exceptional divisors.

 \begin{rem}\label{discrm} It is easy to generalize the lemma to the
situation when
 \begin{itemize}
 \item the codimension of $\eta$ is arbitrary; and
 \item canonical $\eta$ is replaced by $\ep$-log canonical or $\ep$-log
terminal (where $\ep$ may depend on $\eta$).
 \end{itemize}
 However the inequality $c\sE\le\sA$ should be replaced by
 \[
 \mult_{P_i}{\sA}\ge \ep-1+c\mult_{P_i}{\sE}
 \]
 for each prime {\bi}divisor $P_i$ with $\cent_X{P_i}=\eta$ in the
$\ep$-log canonical case (respectively $>$ in $\ep$-log terminal case,
cf.\ (LGD) in \ref{schzy}, Choice of prediction), or replace $\sA$ by
its $\ep$-log version if it is defined as a {\em {\bi}divisor\/}.

In the lemma $\ep=1$. The log canonical and log terminal cases have
$\ep=0$ (cf.\ Example~\ref{fnonvex}).
 \end{rem}

The converse of the lemma usually fails, essentially because $\sA$ and
$\sE$ are independent of $\sD$ up to $\sim_\R$ by \DEP\ in
Proposition~\ref{desd}, whereas canonical {\em depends\/} on $\sD$.

 \begin{exa}[cf.\ Addendum~\ref{mainb}.1 below] Let $D$ be an $\R$-Cartier
divisor on $X$ and $\sD=\Dbar$. Then $\sD_X=D$, $\sE=0$, and for any $c$,
$0=c\sE\le\sA/X$ if and only if $(X,B)$ is canonical in codimension~$2$.
Of course, this does not imply that $(X,B+c\sD_X)$ is always canonical in
codimension~$2$: e.g., when $c>1$, $D$ is effective Cartier
passing,\footnote{I don't understand ``passing'': maybe ``passing
through $\eta$.''} and $\dim X\ge2$ (cf.\ Example~\ref{tre}, (3)).

However, if $\sD=D_X$ then by the proof of Lemma~\ref{discr}, $c\sE\le
\sA$ over each $\eta$ if and only if $K+B+c\sD_X$ is canonical in each
$\eta$ of Lemma~\ref{discr}. Indeed, then $\sK=\sK+c\sD$ over each $\eta$.
 \end{exa}

 \begin{pfof}{Lemma~\ref{discr}} By definition
$\sK=\overline{K+B}+\sA$ and $\sD=\overline{\sD_X}-\sE$ (cf.\ the proof
of Proposition~\ref{ressat}). Thus for each $\eta$, since
$\sD\ge0/\eta$,
 \[
 \sK\le\sK+c\sD=\overline{K+B+c\sD_X}+\sA-c\sE/\eta
 \]
 and $\sK-\overline{K+B+c\sD_X}\le\sA-c\sE/\eta$. Therefore if $K+B+c
\sD_X$ is canonical over $\eta$, then by definition
$\sA(X,B+c\sD_X)=\sK-\overline{K+B+c\sD_X}\ge0/\eta$ as the discrepancy
{\bi}divisor. Hence $\sA-c\sE\ge0$ and $c\sE\le\sA/\eta$.
 \qed\end{pfof}

 \begin{defn}\label{cbd}
 \addcontentsline{toc}{subsection}{\ref{cbd} Definition of a canonically
bounded set of divisors}
 Let $\fD$ be a set of (effective) $\R$-{\bi}divisors and $(X,B)$ a log
pair such that:
 \begin{itemize}
 \item $B$ may not be a boundary, but
 \item $K+B$ and $\sD_X$ for every $\sD\in\fD$ are $\R$-Cartier.
 \end{itemize}
 We say that $\fD$ has {\em canonically bounded singularities\/} on
$(X,B)$, or simply that it is {\em canonically bounded\/} on $(X,B)$ if
there exists a real number $c>0$ such that each pair $(X,B+c\sD_X)$ is
canonical in codimension~$2$. More precisely, the family is {\em
bounded\/} by $c$ (from {\em below\/}).

If we replace $(X,B)$ by a family $(X_i,B_i)$ of log pairs, with each
$X_i$ a model of $X$, $\fD$ bounded {\em on the family\/} means that, for
each $\sD\in\fD$, there is a pair $(X_i,B_i)$ on which $\sD$ is bounded
by the {\em same\/} $c$. Thus $c$ is {\em universal\/}.
 \end{defn}

 \begin{rem}\label{cbr}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item Similarly, we can define the $\ep$-log canonical (terminal)
boundedness of $\fD$ on $(X,B)$ in a point $\eta$ or family of points
(cf.\ Remark~\ref{discrm}). Thus in the case $\ep=1$, and for $\eta$ of
codimension~$\ge2$, we get again the canonical (terminal) boundedness.
For $\ep=0$, we get log canonical (terminal) boundedness (cf.\
Example~\ref{fnonvex}).

 \item If $\fD$ is bounded, we can define its {\em threshold} as
$\sup\{c\}$. This is a more precise characteristic of the singularities
of $\fD$.
 \end{enumerate}
 \end{rem}

 \begin{defn} A set $\fS$ of reduced divisors $S=\sum P_i$ on $X$ is {\em
bounded\/} if its elements are bounded as reduced subvarieties of $X$
(cf.\ the proof of Theorem~\ref{bndbf} and Fixed restriction~\ref{frest}).
More generally, a {\em bounded set of divisors\/} is\footnote{As
terminology, this has the disadvantage that it already has two other
ordinary meanings (family having bounded moduli, finite), which are quite
different, and both of which you use later in the paper. I think here you
should say a {\em set of divisors with bounded singularities\/}.} a set of
divisors $\fD$ such that all $S=\Supp D$ of $D\in\fD$ are {\em bounded\/},
that is, belong to a bounded family of reduced divisors $\fS$. In this case
we say that $\fD$ is {\em bounded\/} by $\fS$, and $D$ is bounded by $S$
respectively.

We add\footnote{I can't make any sense of this sentence.} to the
boundedness with {\em multiplicities\/} on $Y$, whenever it includes the
boundedness of the multiplicities for each $\sD_Y$, where $Y$ is a model
of $X$.
 \end{defn}

 \begin{exa}[Trivial]\label{tre}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item If $\fD$ is a finite or bounded set of {\bi}divisors $\sD$,
including {\em multiplicities\/} on $X$, and $(X,B)$ is terminal in
codimension~$2$, then $\fD$ is bounded on $(X,B)$.

 \item If each {\bi}divisor $\sD$ of $\fD$ is the Cartier closure
$\sD=\Dbar$ of a sufficiently general element in a free linear
system, and $(X,B)$ is terminal in codimension~$2$, then $\fD$ is also
bounded on $(X,B)$.

 \item If each {\bi}divisor of $\fD$ is sufficiently general and {\bi}free,
that is, as in (2) on some model $Y/X$ of $X$, and $(X,B)$ is Klt, then
$\fD$ is bounded on the family $(X_i,B_i)$ which consists of the crepant
models $(X_i/X,B_i)$ terminal in codimension~2. Note that $c=1+\min\sA$
is the minimal {\em log\/} discrepancy of $(X,B)$ (cf.\
Example~\ref{tdb}). In particular, $c=1$ when $(X,B)$ is canonical.
 \end{enumerate}
 \end{exa}

Now we show how to apply canonical boundedness to obtain asymptotic
boundedness.

 \begin{prop}\label{aboundd} Under the assumptions of Definition~\ref{cbd},
suppose also that%
\begin{q}
 \item[\CMD] each model $(X_i/Z,B_i)$ is a {\em crepant model\/} of
$(X/Z,B)$ (cf.\ {\em Example~\ref{tre}, (3)\/}); and
 \item[\EFF] each $\sD\in\fD$ is effective.
 \end{q}
 Then $\fD$ canonically bounded by $c$ implies that, for each $\sD\in\fD$,
there exists a pair $(X_i,B_i)$ such that the descent data $c\sE$ of $c\sD$
over $X_i$ is bounded by $\sA/X_i$, where $\sA=\sA(X,B)=\sA(X_i,B_i)$ is
the discrepancy {\bi}divisor.
 \end{prop}

 \begin{add} Let $\sD_i$ be a sequence of {\bi}divisors, and $p_i$ a
sequence of positive real numbers respectively such that:
 \begin{itemize}
 \item $\lim_{i\to\infty}p_i=+\infty$; and
 \item each $p_i\sD_i\sim_\R\sM_i\in\fD$ (or even just numerically
equivalent), where
 \item all the $\sM_i$ are canonically bounded by $c$ {\em on a common
model\/} $(Y=X_j,\linebreak[2] B_Y=B_j)$ in the family.
 \end{itemize}
 Then the descent data for the sequence $\sD_i$ is asymptotically bounded
by $\sA/Y$.
 \end{add}

 \begin{proof} The bound $c\sE\le\sA/X_i$ follows from Lemma~\ref{discr}
for $(X,B):=(X_i,B_i)$, for the set $\{\eta\}$ of points of codimension
$\ge2$, and for $c$ the given canonical bound, where $(X_i,B_i)$ itself
corresponds to $\sD$ under the canonical bound. By \HOM\ of
Proposition~\ref{desd}, $\sE(c\sD)=c\sE(\sD)/X:=X_i$.

Since $c>0$, each $r_i=c p_i>0$, and also $\lim_{i\to\infty}cp_i=+\infty$.
On the other hand, by Proposition~\ref{desd}, \DEP\ and \HOM\,
$r_i\sE_i=cp_i\sE_i=c\sE(p_i\sD_i)=c\sE(\sM_i)/Y$. Hence the above
boundedness\footnote{Does ``above boundedness'' just mean the given
canonical bound for $\fD$?} for $\fD$ implies
$r_i\sE_i=c\sE(\sM_i)\le\sA/Y$, which means asymptotic boundedness.
 \qed\end{proof}

We define \w\ pairs $(X_i,B_i)$ of bounding families in terms of
triples. \footnote{Slava: You can't have ``wanted triples'' -- you can
sometimes use a past participle as an adjective, but not of an auxiliary
verb such as ``want''. ``desirable'' is unambiguous and probably
translates the intention of the Russian, although it has a slightly
humorous sound. It's put in by a macro $\tt{\backslash w}$. You could
have ``ideal triples'' or ``target triples'' if you don't like my
solution.}

 \begin{defn}\label{tri}
 \addcontentsline{toc}{subsection}{\ref{tri} Definition of \w\ triple}
 A {\em triple\/} $(X/T/Z,B,\fF)$ includes
 \begin{itemize}
 \item a log pair $(X/Z,B)$, for which $K+B$ is $\R$-Cartier, but $B$
may not be a boundary;
 \item a contraction $g\colon X\to T/Z$; and
 \item a bounded set $\fF$ of reduced divisors on $X$.
 \end{itemize}

Let $(X/Z,B)$ be a weak log Fano contraction (see \WLF\ in
Proposition~\ref{exsatcan}). A {\em \w\ triple\/} for $(X/Z,B)$ is a
triple $(Y/T/Z,B_Y,\fF)$ such that:
 \begin{q}
 \item[(CRP)]
$(Y/Z,B_Y)$ is a crepant model of
$(X/Z,B)$;
 \item[(QFC)]
$Y$ is $\Q$-factorial;
 \item[(TER)]
$(Y,B_Y)$ is terminal in codimension~$2$; and
 \item[(RPC)] the nef cone $\NEbar{T/Z}$ is {\em rationally polyhedral\/}
and has {\em contractible\/} faces$/T$ (cf.\
\cite[Inductive Theorem~2.3]{sh95}).
 \end{q}

Let $\sM$ be a {\bi}free {\bi}divisor. A triple $(Y/T/Z,B_Y,\fF)$ is a
{\em \w\ triple\/} for $\sM$ if in addition\footnote{I change the wording
of the definition, since you clearly want to include all of the above
conditions.}

 \begin{itemize}
 \item $\sM_Y=g^*M$, where
$M$ is nef and big on $T/Z$.
 \end{itemize}
 \end{defn}

 \begin{rem}\label{trir}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $T/Z$ is a contraction if $X/Z$ is.

 \item Sometimes, even for \w\ triples, we need to weaken the assumptions:
e.g., $X/T$ may be a incomplete morphism of normal varieties, cutting log
singularities, or $(X,B)$ may not be Klt.

 \item It might be better to spread the boundedness of $\fF$ into
boundedness of the family of {\em simple\/} triples $(X/T/Z,B,F)$ having
a $1$-element set $\{F\}\subset\fF$ for $F\in\fF$. The {\em
boundedness\/} of a family of triples means that their moduli are bounded.

 \item (CRP) in the definition means that $\sA(Y,B_Y)=\sA(X,B)=\sA$ (cf.\
Caution~\ref{warn!lca}).

 \item We could possibly drop (QFC), but then we can only bound
singularities of a divisor $\sD$ having $\R$-Cartier restriction $\sD_Y$
(cf.\ Lemma~\ref{discr} vs Lemma~\ref{split}). This holds in many
applications. For example, $\sM_Y$ is $\Q$-Cartier on a \w\ triple for
$\sM$ by definition.

 \item In (TER), the divisor $B_Y$ need not be a boundary. However, since
$(X,B)$ is Klt, $B_Y$ is a {\em Klt subboundary\/}: that is, each of its
multiplicity $b_i<1$.

 \item (RPC) is intermediate among the following conditions on $T/Z$, that
we list in order of increasing strength:
 \begin{q}
 \item[(LSA)] the limit of any semiample sequence of $\R$-divisors on
$T/Z$ is a semiample $\R$-divisor on $T/Z$, and we assume that $M$ is
{\em semiample\/} in the definition of \w\ triples (cf.\
Remark~\ref{minar}, (4));
 \item[(NSA)] each divisor $M$ on $T$ that is nef$/Z$ is semiample;
 \item[(0LP)] $T/Z$ is a $0$-log pair for some boundary $B_T$ as in
Remark~\ref{gzardec}, (2); in particular, (NSA) then holds for each
effective divisor, and we can assume that $\sM$ is effective, and any
limit of effective divisors is also effective under \MXD;
 \item[(RPC)] as in Definition~\ref{tri} (cf.\ Lemma~\ref{split}, (2)); and
 \item[\WLF] $(T,B_T)$ is a weak log Fano contraction for some boundary
$B_T$.
 \end{q}
 For the main application of (CBS) below (cf.\ Theorem~\ref{cbsfg}), (LSA)
is enough. But for approximations we need at least (RPC) (cf.\
Theorem~\ref{bnonvan}). In addition, we prefer\footnote{Maybe it would be
better to say ``we prefer to work with notions defined up to numerical
equivalence rather than up to linear equivalence''.} to work with numerical
properties rather than linear properties.
 \end{enumerate}
 \end{rem}

The reader may wonder what role the set of divisors $\fF$ plays in
triples. In fact, we need very special sets $\fF$ that we call {\em
standard\/}. First, the standard families are bounded in typical examples,
such as the fixed and fractional parts that appear in Corollary~\ref{mainb}
and in Proposition~\ref{bounds}. Second, they generalize the canonical
bound conjecture (cf.\ (GCB) in Conjecture~\ref{cbcon}). Finally, they are
crucial in our construction of $4$-fold flips (cf.\ Theorem~\ref{cbrfa}).

 \begin{defn}\label{ssd} Let $(X/Z,B)$ be a log pair. We say that a set
$\fF$ is {\em standard\/} on $(X/Z,B)$ if it contains only standard
divisors. A {\em standard divisor\/} $S$ on $(X/Z,B)$ is a reduced divisor
$S=\Supp{\sD_X}$, where $\sD$ is an {\em effective\/} $\R$-{\bi}divisor
such that:
 \begin{q}
 \item[(SEF)] $\sD$ is {\em strictly effective\/}$/X$; that is, it has
$\mult_{E_i}{\sD}\ge0$ in each exceptional$/X$ prime {\bi}divisor $E_i$ and
$>0$ {\em over\/} $\Supp{\sD_X}$, where the support is considered as a
{\em divisorial subvariety\/} (cf.\ (EEF) in \ref{schzy}, Choice of
prediction); and
 \item[(STD)] $0$ is saturated with respect to $\sA+\sD$ on any sufficiently
high model $X\hr/Z$ of $X/Z$.
 \end{q}

The {\em standard set\/} $\fS$ is the set of all standard divisors. In
general, it may be unbounded. However by $\SSB$ in Conjecture~\ref{cbcon},
we expect that this is true\footnote{You mean the opposite, namely ``We
expect that $\fS$ is bounded.''} when $(X/Z,B)$ is a weak log Fano
contraction. In particular, then $(X/X/Z,B,\fS)$ is a triple. For any
other triple $(Y/T/Z,B_Y,\fS_Y)$, the {\em induced standard set\/} $\fS_Y$
is defined as the log birational transform of the set $\fS$ of $X$.

A \w\ triple for $(X/Z,B)$ with the {\em induced set\/} $\fS_Y$ is a \w\
triple $(Y/T/Z,B_Y,\fS_Y)$ with the induced standard set.
 \end{defn}

 \begin{warn} It may happen that $\fS_Y$ itself is actually {\em
nonstandard\/} (cf.\ Remark~\ref{ssr}, (2)).
 \end{warn}

 \begin{rem}\label{ssr}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item If $D$ is an effective $\R$-Cartier divisor on $X$ then $\Dbar$
satisfies (SEF).

 \item The induced set $\fS_Y$ is well defined, namely, it is bounded when
$\fS$ is bounded. Indeed, the log birational transform of $S\in\fS$ is
$g\1S+\sum E_i$, where each $E_i$ is a prime divisor on $Y$ that is
exceptional on $X$. Note that $\sum E_i$ is finite and fixed for any such
transform.

 \item In general, $\fS_Y$ is not standard even up to the additive log
contribution $\sum E_i$. Let $X$ be a del Pezzo surface having two
$-1$-curves $E_1,E_2$ intersecting transversally in a single point $P$,
for example, two intersecting lines on a cubic surface in $\PP^3$. Then
$E_1$ and $E_2$ belongs to the standard set $\fS(X/Z=\pt,0)$, but
$E_1+E_2$ does not (cf.\ Example~\ref{esurbound}). For the blowup
$g\colon Y\to X$ in $P$, the blowup $E_3$ of $P$ (the log transform of 0)
is standard. The birational transform $g\1E_i$ of each $E_i$ is also
standard (in addition to their log transforms $g\1E_i+E_3$). But now
$g\1E_1+g\1E_2$ is also standard since
$g\1E_1+g\1E_2+a(E_3)E_3=g\1E_1+g\1E_2+E_3$ is not mobile on $Y$.
 \end{enumerate}
Nonetheless the difference between $\fS_Y$ and $\fS(Y/Z,B_Y)$ is not great
(cf.\ (GFC) in Conjecture~\ref{cbcon} below).
 \end{rem}

Other facts about standard sets and equivalent forms of the standard
property (SEF)$+$(STD) are given below in Proposition~\ref{ssp}. In
general, a bounding family $(X_i,B_i)$ may itself be unbounded (cf.\
Example~\ref{tre}, (3)), and an infinite subset in $\fD$ may not be bounded
on any bounded subfamily of $(X_i,B_i)$. However we need and expect the
following:

 \begin{conj}[Canonical bounds]\label{cbcon}
 \addcontentsline{toc}{subsection}{\ref{cbcon} Conjecture on canonical
bounds}
 Let $(X/Z,B)$ be a weak log Fano contraction with a contraction $X/Z$. We
expect boundedness of the following sets. First,
 \begin{q}
 \item[$\SSB$] {\em the standard set $\fS=\fS(X/Z,B)$ of $(X/Z,B)$ is
bounded\/}.
 \end{q}
 In particular, this includes boundedness of prime components of the
standard divisors. Moreover, we expect
 \begin{q}
 \item[\rm(PRM)] the {\em prime standard set\/} $\fP=\fP(X/Z,B)$ {\em is
bounded\/}, where $\fP$ is the set of\/ {\em prime standard divisors\/}
$P=\Supp{\sD_X}$ on $X$ with $\sD\ge0$ and $\sD$ satisfying saturation
(STD) of Definition~\ref{ssd}.
 \end{q}

The third set is the {\em sat set\/} $\fD=\fD(X/Z,B)$ of\/ {\em effective\/}
$\R$-{\bi}divisors $\sD$ under saturation
 \begin{q}
 \item[\SAT] $\sD$ is {\em log canonically saturated\/}$/(X,B)$ on any
sufficiently high model $X\hr/Z$ of $X/Z$ (cf.\ Definition~\ref{lca}); that
is, it is $\sA$-saturated, where $\sA=\sA(X,B)$ is the discrepancy
{\bi}divisor.
 \end{q}
 In most cases, it is unbounded even on $X$. However we expect that
 \begin{q}
 \item[\rm(PFC)] the set of\/ {\em fixed prime components\/}, that is, the
set of prime components of $(\Fix{\sD})_X$ for $\sD\in\fD$ on $X$
(cf.\ Remark~\ref{cbrm}, (4)) is {\em bounded}.
 \end{q}
 (PFC) is equivalent to (PRM). Actually, the set is equal to $\fP$ (cf.\
Theorem~\ref{cbsfg}, (1) and its proof). By definition, (PRM) and (PFC)
hold on any \w\ triple $(Y/T/Z,B_Y,\fF)$ if they hold on $(X/Z,B)$.
Indeed, $\fP(Y/Z,B_Y)$ is the birational transform of the\/ $\fP$ plus
some $E_i$ on $Y$ that are exceptional on $X$. Moreover, we expect that
 \begin{q}
 \item[\rm(GFC)] boundedness (PRM), (PFC) and $\SSB$ each holds on any
{\em general\/} log Fano contraction $(X/Z,B)$ (see \GLF\ in
Proposition~\ref{ressat}) with {\em Klt\/} subboundary $B$ for $\SSB$.
 \end{q}

Each $\sD\in\fD$ has the form $\sD=\sM+\sF$ with {\bi}free $\sM\ge0,
\sM\in\linsys{\Mov{\sD}}$, and fixed $\sF=\Fix{\sD}\ge0$. This splits $\fD$
as a {\em partial\/} sum (cf.\ Remark~\ref{cbrm}, (6))
 \[
 \fD(X/Z,B)\subset\fM(X/Z,B)\oplus\fF(X/Z,B)
 \]
 where $\fM=\fM(X/Z,B)$ and $\fF=\fF(X/Z,B)$ are the {\em subsets\/} of
{\bi}free and fixed (with $\sM=0$) {\bi}divisors in $\fD$. Indeed, under
Klt, \SAT\ is equivalent to
 \begin{q}
 \item[\rm(SAF)]
$\sM$ is {\em saturated\/} with respect to $\sA+\sF$ on any sufficiently high
model $X\hr/Z$ of $X/Z$ by Proposition~\ref{ssp}, (2);
 \end{q}
and in turn by Proposition~\ref{ssp}, (3), (SAF) is equivalent to
 \begin{q}
 \item[\rm(STD)] as in Definition~\ref{ssd} for {\em fixed\/} $\sD:=\sF$.
 \end{q}
 Hence $\sF$ also satisfies \SAT, and by Lemma~\ref{monots} with
$\sC_1=\sA+\sF$ and $\sC_2=\sA$, so does $\sM$. We use the splitting
$\sD=\sM+\sF$ to bound the singularities of the sum in terms of its two
summands. Moreover, we can consider any such sum (cf.\ Remark~\ref{cbrm},
(6)).

Indeed, $\sM$ and $\sF\ge0$, and, on each \w\ triple $(Y/T/Z,B_Y,\fF)$
of $(X/Z,B)$, $\sD_Y\sim\sM_Y+\fF_Y$ with a $\Q$-Cartier Weil divisor
$\sM_Y\ge0$ and an $\R$-Cartier $\sF_Y\ge0$. Thus because the canonical
property is convex (cf.\ \cite[1.3.1]{sh92}), it is enough to prove
the latter\footnote{What? possibly ``enough to prove that
$\sD_Y\sim\sM_Y+\fF_Y$ is canonical, or is canonically bounded.''} for
each summand. If $\sF$ is bounded by $c_f$ from below, and $\sM$ by $c_m$
then $\sD$ is bounded by any weighted combination of them: e.g., half the
sum $c=(c_m+c_f)/2$.

On any desirable triple $(Y/T/Z,B_Y,\fF)$, by (PRM) and Example~\ref{tre},
(1), each irreducible component $f_iP_i$ of $\sF_Y$ (with prime $P_i$) has
canonically bounded singularities if the multiplicities $f_i$ are bounded.
For the whole $\sF$, we need more, namely, boundedness of the number of
irreducible components, which is equivalent to $\SSB$ and holds for
$\sD:=\sF$ under condition (SEF)$/Y$ of Definition~\ref{ssd}, or for the
log transforms in $\fS_Y$ from $(X/Z,B)$.

The mobile part $\sM$ behaves worse (cf.\ Remark~\ref{cbrm}, (7)):
 \begin{itemize}
 \item canonical boundedness from below holds for {\em some\/} effective
$\sM$ (usually meaning {\em sufficiently general\/}); and
 \item on some triples.
 \end{itemize}

With a certain dose of {\em optimism\/} we expect
 \begin{q}
 \item[$\CBS$] there is a {\em algebraic\/} (throughout the paper; cf.\
Remark~\ref{bndf}) {\em bounded\/} family of \w\ triples
$(Y/W/T,B_Y,\fF)$ on which $\fM=\fM(X/Z,B)$ has {\em canonically bounded
singularities\/} up to $\sim$, where $\fM$ is the set of {\bi}free
{\bi}divisors satisfying log saturation \SAT. More precisely, for each
$\sM\in\fM$ there is $\sD\in\linsys{\sM}$ and a \w\ triple
$(Y/W/T,B_Y,\fF)$ for $\sM$ in the family such that $\sD$ is canonically
bounded on $(Y,B_Y)$.
 \end{q}

Thus jointly
 \begin{q}
 \item[\rm(GCB)] {\em canonical boundedness\/} for some $\sD+\sF\sim
\sM+\sF$ over the family
 \end{q}
 is equivalent to (CBS)+$\SSB$. But the latter needs extra assumptions on
$\sF$ as (SEF). Without them (GCB) means (CBS)+(PRM).

Finally, we expect (cf.\ Remark~\ref{cbrm}, (8)) that
 \begin{q}
 \item[\rm(BIG)] for {\em big\/} $\sM$, the subfamily of \w\ triples is
finite; in particular,
 \item[\rm(BIR)] for {\em birational\/} $X/Z$, the whole family of \w\
triples can be taken to be finite; and, more generally,
 \item[\rm(MOD)] among {\em birationally equivalent\/} $\sM$, that is,
$\sM$ that define birationally the same contractions $Y\to T/Z$ (see
Definition~\ref{defn!bireq}, the subfamily of \w\ triples $\sM$ is finite;
in addition, $\dim T/Z=\ka(X/Z,\sM)$ is the Iitaka dimension of $\sM/Z$.
 \end{q}

However {\em to be more realistic\/} we can restrict ourself to very
special subsets in $\fM$ and impose further conditions. We consider mobile
systems $\sM_\bull\subset\fM$ at least under {\em asymptotic saturation\/}
\LCA: e.g.,
 \begin{qq}
 \item[\rm(CBS)(fga)] (CBS) for $\sM_\bull=\Mov{\sL}$ for each
\footnote{Possibly you mean ``condition for each individual
functional algebra''.} (separate) functional algebra $\sL$ in
Conjecture~\ref{conj!fga}; and
 \item[\rm(CBS)(rfa)] (CBS) for each algebra in Definition~\ref{rfad}.
 \end{qq}
(CBS)(rfa,bir) is what we really need (see the notation below, and
Conjecture~\ref{rfac}).
 \end{conj}

 \begin{rem}\label{cbrm}
 \addcontentsline{toc}{subsection}{\ref{cbrm} Remark: Generalizations of
the conjecture}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item We can expect generalizations of the conjectures when $X/Z$ is only
a proper morphism of normal varieties but possible not a contraction, or
$X/Z$ has other extra structures: e.g., a group action, a morphism $X/W$,
etc. The latter relates in particular to the nonlocal case $X/W$ with a
proper morphism $W\to Z$ and {\em local\/}$/Z$, as we always assume.

 \item (PRM) drops (SEF) in Definition~\ref{ssd} but assumes that $S=P$
is prime.

 \item We can drop the effective condition of $\sD\in\fD$ in (PFC). Indeed,
if $|\sD|=\emptyset$, the prime component of the base locus is a single
prime variety $X$ that we can add to $\fP$. Otherwise, $D\ge0$ up to
$\sim$ that preserves the saturation \SAT\ and the fixed components.
Perhaps, we can add as well other prime maximal fixed centers.

Again, in (CBS) and (GBC), we need the effectiveness of $\sM$ and
$\sM+\sF$ respectively up to $\sim$. Otherwise $\sM=-\infty$ is canonically
bounded on any model. Thus we can also drop effectiveness.

 \item Any $\R$-{\bi}divisor $\sD$ has a decomposition $\sD=\sM+\sF$ with
{\bi}free {\em mobile part} $\sM=\Mov{\sD}$ and {\em fixed part}
$\sF=\Fix{\sD}\ge0$. We explained this in Proposition~\ref{texa}, (1) and
(3) when $\Oh_Z(\sD)$ is coherent. In general, there is the {\em maximal
coherent\/} subsheaf
 \[
 \Oh_Z(\sM)=\Oh_Z(\sD)^{ch}\subset\Oh_Z(\sD) \subset\Oh_Z(\sD_X)\subset k(X)
 \]
 (because the coherent sheaf $\Oh_Z(\sD_X)$ is Noetherian). Note that in
the {\em global\/} case with $Z=\pt$, each $\Oh_Z(\sD)$ is coherent.

An $\R$-{\bi}divisor $\sD\ge0$ up to $\sim$ if and only if $\sM\ne-\infty$,
or equivalently, $|\sM|\ne\emptyset$, or $\Oh_Z(\sM)=\Oh_Z(\sD)^{ch}\ne0$.
If $\sD\ge0$ then $\sD\ge\sF\ge0$. In the same way, $\sD\ge0$ is {\em
mobile\/} if and only if $|\sM|\ne|0|$, or equivalently,
$\Oh_Z(\sM)=\Oh_Z(\sD)^{ch}$ has $>1$ generator (is nonprincipal).

 \item For general log Fano contractions in \GLF\ of
Proposition~\ref{ressat}, we drop any properties of $B$. In general,
$\SSB$, (PRM) and (PFC) do not hold for contractions that are not general
log Fano contractions, even for rational varieties $X$. For example, let
$(X/\pt,0)$ be a sufficiently high resolution of $\PP^2$: e.g., such that
the set of exceptional (= contractible) curves on $Y$ is
infinite.\footnote{``unbounded'' contradicts your earlier usage ``bounded
set of divisors'', and I've replaced it by infinite, which means the same.}
Then $\fS$ and $\fP$ include at least the exceptional curves $C$ (take
$\sD=\overline{C}$ for $\SSB$). We expect that (PRM) and (PFC) hold on any
general log Fano contraction but for $\SSB$ we need Klt; in particular,
$B$ is a subboundary (cf.\ bad singularities in Remark~(9)
below).

 \item We say that $\fD$ splits only as a {\em partial sum\/} because for
rather powerful $\sM$, $\Mov{\rdup{\sM+\sF+\sA}}$ can certainly be $>\sM$,
which gives a {\em different\/} splitting.\footnote{``Powerful'' is not
defined. Presumably it means ``very big'' linear system. The remark is too
cryptic, and much too far away from the statement in Conjecture~\ref{cbcon}
to which it refers.}

 \item Write $l=\lld(X,B)$ for the least log discrepancy of $(X,B)$. If some
element $\sD\in|\sM|$ is canonically bounded by $c$ with $0\le c\le l$, then
so is the general $\sD'\in|\sM|$. Hence (CBS) for some $\sD$ implies (CBS)
for general $\sD$, in fact with the same $c$. Indeed, $\sD$ canonically
bounded on $(X,B)$ means that $(X,B+c\sD_X)$ has canonical singularities in
codimension~$2$. Equivalently, on {\em any\/} crepant model $(Y/X/Z,B)$, the
crepant subboundary
 \[
 B_Y:=\sB(X,B+c\sD_X)_Y=\sB(X,B)_Y+(\overline{c\sD_X})_Y
 \]
 has only nonpositive multiplicities in divisors that are exceptional on
$X$, and $(Y,B_Y)$ is canonical in codimension~$2$. Note that, for any
other $\sD'$, if $B'_Y:=\sB(X,B+c\sD'_X)_Y$ is a subboundary on $Y$ such
that $B'_Y=cL+F$, where $F\le B_Y$ in the divisors that are exceptional on
$X$, and $\Bs{|L|}=\emptyset$, then by monotonicity (cf.\
\cite[1.3.3]{sh92}), for sufficiently general $L$ (that is, $\sD$),
$(X,B+c\sD'_X)$ is also canonical in codimension~2. This holds on {\em
some\/} good model $Y/Z$, where $\linsys{\sD'}=\overline{\linsys{\sD'_Y}}$,
with $\Bs{\linsys{\sD'_Y}}=\emptyset$, and $L=\sD'_Y=0/X$. Indeed, for such
fixed model and for a general $\sD'$,
 \begin{gather*}
 B_Y'=\sB(X,B)_Y+(\overline{c\sD_X'})_Y \quad\hbox{and} \\
(\overline{c\sD_X'})_Y=c(\overline{\sD_X'})_Y\le
c(\overline{\sD_X})_Y=(\overline{c\sD_X})_Y/X.
 \end{gather*}
This can also be established by inversion of adjunction (cf.\ \cite[after
Proposition~5.13]{sh92}).

Klt implies that $\lld(X,B)>0$ (least log discrepancy). On the other
hand, $c\le\lld$ on $\fM$ by Example~\ref{tre}, (3).

 \item We could perhaps expect more for the triples $(Y/T/Z,B_Y,\fF)$ that
are required for the conjecture: e.g.,
 \begin{itemize}
 \item each model $Y/Z$ is a contraction over a crepant terminal resolution
of $(X/Z,B)$, and is thus a rational $1$-contraction over $X/Z$; and
 \item in (BIG), $Y/Z$ is isomorphic in codimension~$1$ to a crepant
terminal resolution of $(X/Z,B)$.
 \end{itemize}
 The discussion in Example~\ref{stexm} sheds light on the finiteness in
(BIG). In fact, each of the finiteness assertions in (BIG), (BIR), (MOD)
and (CBS)(fga), (CBS)(rfa) amounts to a {\em single\/} triple.

To prove (CBS) in general, it is enough to consider a {\em \w\/}
triple $(X/T/Z,B,\fF)$ with the subset $\fN=\fN(X/T/Z,B)$ of $\sM\in\fM$
satisfying the following conditions:
 \begin{q}
 \item[(NEF)] $\sM_X$ is nef$/Z$ and
 \item[(SFB)] $\sM_X$ is {\em supported in fibres\/} of $X/T$ of
{\em Iitaka dimension$/Z$\/} $\ka(X,\sM_X)=\dim T/Z$.
 \end{q}
 In particular, both hold when the triple is \w\ for $\sM$ (conversely,
$\sM$ is \w\ in codimension~$1$ on $T$, so if $T$ is $\Q$-factorial this
always holds). Then to prove that $\fN$ is {\em canonically bounded\/} on
$(X,B)$ modulo $\sim$, we can assume that moreover
 \begin{q}
 \item[(IRR)]
$\sM_X$ is {\em irreducible\/}
 \end{q}
 except for pencils\footnote{I don't understand this. Possibly ``Except
for the case of a pencil, when all the conditions of bounded singularities
are known.''} when any boundedness of singularities is known. Since $\sM_X$
is integral, and by the cone property (RPC), the divisors for which the
triple is \w, correspond modulo $\sim$ to integral points in a rational
cone dual to the Kleiman--Mori cone $\NEbar{T/Z}$. As in Truncation
Principle~\ref{trprinc}, it is enough to establish the canonical bound:
 \begin{itemize}
 \item for a {\em truncated\/} subcone: for $\sM$ with a rather high {\em
index of height\/} on $X$, namely, for some natural number $I$, every such
$\sM_X\sim\sum h_ig^*H_i$, where each $H_i$ is nef and big on $T/Z$, and
$I\mid\sum h_i$, and
 \item for {\em bounded\/} heights $\sum h_i$.
 \end{itemize}
 We hope that, for some $I\gg0$, the former $\sM$ are {\em free\/} over
$X$, that is, $\sM=\overline{\sM_X}$ and $\Bs{\linsys{\sM_X}}=\emptyset$.
Perhaps, in the latter case, bounded heights implies bounded singularities
by reduction to lower dimension of $T$. Finally, for general elements in
$\fN$ we can use the birational case after a localization over
non$\Q$-factorial points of $T$, or to assume that $T$ is $\Q$-factorial.

Example~\ref{stexm}, explains how to find such triples and to secure their
finiteness if $\sM$ is big and $T=X$. In particular, this reveals why we
expect Conjecture~\ref{cbcon} to hold in full. But for (FGA) we only need
the boundedness of triples (cf.\ of Theorem~\ref{cbsfg}, (3)).

Canonical boundedness really requires the terminal property (TER) of \w\
triples (cf.\ Remark (11) below). For example, if $\sM_i=i\overline{H}$
over a terminal resolution $Y/X$ of $(X/Z,B)$ with ample $H/Z$, then
$\sM_\bull\subset\sM$, but $\sM_\bullet$ has bounded singularities on $X$
only when $X=Y$.

Note also that we need fibred triples, with $X/T$ not birational (cf.\
Example~\ref{esurbound} below).

 \item Possible important generalizations concern bad singularities of
$(X,B)$: e.g., we could weaken the Klt condition, as for general log
Fano varieties \GLF, but still assume that $B\ge0$. But then we need
extra assumptions to ensure good behavior on the bad singularities,
namely:
 \begin{q}
 \item[(GEN)] On a universal crepant model $(Y/Z,B_Y)$ of $(X/Z,B))$,
each $\sD_Y$ is {\em free\/} or even {\em (very) ample\/}$/Z$
on $\LCS{(Y,B_Y)}$.
 \end{q}
 Therefore in this case, we do care about canonical singularities outside
$\LCS{(Y,B_Y)}$.

 \item Another generalization concerns the numerical conditions of Fano type
\GLF\ that can be replaced by
 \begin{q}
 \item[(ADJ)] $\sD_X\equiv (K+B+D)/T$, where $D$ is a sum $D=F+H$ of {\em
effective\/} $F$ plus {\em nef and big\/} $H/T$.
 \end{q}

 \item We expect similar results in the $\ep$-log category, for example,
$\ep$-log canonical boundedness. But this requires appropriate
restrictions: e.g., $\ep$-log saturation with respect to $\ep$-log
discrepancy. For triples, (TER) can be replaced by $\ep$-log terminal in
codimension~$2$.
 \end{enumerate}
 \end{rem}

 \begin{quest} Could we replace the condition that $\fM$ is {\bi}free by
the {\bi}nef assertion of Lemma~\ref{divb}?
 \end{quest}

 \begin{phil}
 Saturation \SAT\ improves properties of any $\sD$ up to $\sim$, even if
it is not {\bi}free. (By analogy with regularizing solutions of elliptic
differential equations.) On the other hand, by \DEP\ in
Proposition~\ref{desd} and by Proposition~\ref{aboundd}, we can bound the
descent data for $\sM$ in terms of the canonical bound of an improved
divisor. The same applies when $p\sD\sim_\R\sM$ with real number $p>0$
and $\sM$ has good canonical singularities (cf.\
Addendum~\ref{aboundd}.1), for example, by Conjecture~\ref{cbcon}.
 \end{phil}

 \begin{nota} We use the following specifications:
 \begin{itemize}
 \item (--)$_n$ means (--) with $\dim X=n$;
 \item (--)($X/Z,B$) means (--) for $(X/Z,B)$;
 \item (--)(big) means that (--) concerns only big$/Z$ {\bi}divisors;
 \item (--)(bir) means that $X/Z$ is birational in (--);
 \item (--)(fga) means that we consider (--) only on a subset
$\sM_\bull=\Mov{\sL}$ for an algebra $\sL$ of type (FGA) in
Definition~\ref{defn!fga};
 \item (--)(gl) means that $Z=\pt$, that is, $X$ is global in (--); and
 \item (--)(rfa) means that we consider (--) only on a subset
$\sM_\bull=\Mov{\sL}$ for an algebra $\sL$ of type (RFA) in
Definition~\ref{rfad}.
 \end{itemize}
 \end{nota}

 We apply these mainly to (CBS). For example,
 \begin{itemize}
 \item $\CBS_n$ means (CBS) for all $(X/T,B)$ with $\dim X=n$;
 \item $\CBS_n\fgabir$ means (CBS) for any system $\sM_\bull$ of type
(FGA) with birational $X/Z$ and with $\dim X=n$; and
 \item $\CBS_n\mathrm{(bir,gl)}=\emptyset$ for $n\ge1$.
 \end{itemize}

The main result of this section is

 \begin{thm}\label{cbsfg}
 \addcontentsline{toc}{subsection}{\ref{cbsfg} Main Theorem on canonical
bounds}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item {\rm(PFC)\/}${}={}${\rm(PRM)\/} (equality of sets)\/.

 \item {\rm(CBS)\/}$_n$ implies the same with any specifications
{\rm(big), (bir), (gl), (fga)\/} and {\rm(rfa)\/}.

 \item $\CBS(X/Z,B)$ implies $\FGA(X/Z,B)$, and $\CBS\fga(X/Z,B)$ is
equivalent to $\FGA(X/Z,B)$.

 \item $\CBS_n\fga$ is equivalent to $\FGA_n$.
 \end{enumerate}
 \end{thm}

We start by clarifying (MOD) of Conjecture~\ref{cbcon}.

 \begin{defn}\label{defn!bireq} Two contractions $X_1\to Y_1$ and $X_2\to
Y_2$ are {\em birationally equivalent\/}, if they are isomorphic over an
open nonempty subsets $U_i\subset Y_i$. In the definition we can assume
that $X_1=X_2$ after some resolution.
 \end{defn}

 \begin{exa}
Suppose that $X_1$ and $X_2$ are birationally equivalent. Then

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item any two big contractions are birationally equivalent;
 \item any two contractions to $\pt$ are birationally equivalent; and
 \item any two contractions over a curve are birationally equivalent if
they define the same pencil composed of their fibres.
 \end{enumerate}
 \end{exa}

 \begin{lem} \label{monnd} Let $\sD_1\ge\sD_2$ be two $\R$-{\bi}divisors
on $X/Z$. Then
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the Iitaka dimensions satisfy $\ka(X/Z,\sD_1)\ge\ka(X/Z,\sD_2)$, with
equality if and only if they are birationally equivalent;
 \item if both $\sD_1$ and $\sD_2$ are {\bi}nef, then
$\nu(\sD_1/Z)\ge\nu(\sD_2/Z)$ with equality if and only if
$\nu(\sD_2\rest{(\sD_1-\sD_2)}/Z)\le\nu(\sD_2)-1$; and
 \item if $\sD_1$ and $\sD_2$ are both {\bi}semiample, $\nu(\sD_1/Z)\ge
\nu(\sD_2/Z)$ with equality if and only if they are birationally equivalent.
 \end{enumerate}
 \end{lem}

 \begin{proof} (1) Immediate from the definitions.

(2) Taking an appropriate model of $X/Z$, we can assume that each
$\sD_i=\Dbar_i$ for a nef$/Z$ divisor $D_i=(\sD_i)_X$. By definition each
$\nu(\sD_i)=\nu(D_i)$. Since $D_1\ge D_2$, it is enough to verify that
$\nu(D_1)\ge \nu(D_2)$. Indeed, $D_1=D_2+F$ with $F\ge0$. Taking a general
point of $Z$, we can assume that $Z=\pt$ and each $\nu(D_i/Z)=\nu(D_i)$.

It is enough to consider the case $\nu(D_1)\le\nu=\nu(D_2)$. Thus for some
cycle $C_\nu$ of dimension $\nu$, $C_\nu D_2^\nu>0$. Equivalently, the
same holds for mobile cycles $H^{n-\nu}$, where $n=\dim X$ and $H$ is a
hyperplane section of $X$, because $H^{n-\nu}$ is rationally equivalent to
$C_\nu$ plus an effective cycle. Taking a hyperplane section gives an
induction on the dimension $n\ge\nu$ with divisors $D_i\rest H$. Hence we
can assume that $\nu=n$. Then
 \begin{align*}
 D_1^n&=D_2D_1^{n-1}+F D_1^{n-1} \\
 & \ge D_2 D_1^{n-1}\ge\cdots\ge D_2^{n-1}D_1 \\
 & =D_2^n+D_2^{n-1}F\ge D_2^n>0.
 \end{align*}
This means that also $\nu(D_1)=\nu$. In addition,
 \[
 \nu(D_2\rest{(D_1-D_2)})=\nu(D_2\rest F)\le\dim F\le n-1=\nu-1.
 \]

(3) is immediate by (1) because $\nu(\sD/Z)=\ka(\sD/Z)$ for any
{\bi}semiample divisor $\sD$.
 \qed\end{proof}

 \begin{pfof}{Theorem~\ref{cbsfg}} (1) Compare the proof of
Proposition~\ref{ssp}, (3). For $\sD\ge0$, Klt implies that
 \[
 \Mov{\rdup{\sA+\sD}}\ge\Mov{\sD}\ge0.
 \]
Hence (PRM) for $P=\sD_X$ implies (PFC) for the $\sD$ with prime
$P=(\Fix{\sD})_X$, because $\Fix{\sD}=\sD$. Conversely, for any
$\sD=\sM+\sF\ge0$ under \SAT\ and with $\sM=\Mov{\sD}$ and
$\sF=\Fix{\sD}$,
 \begin{align*}
 \sM+\Mov{\rdup{\sA+\sF}} & \le \Mov{(\sM+\rdup{\sA+\sF})} \\
 &=\Mov{\rdup{\sD+\sA}}\le\Mov{\sD}=\sM,
 \end{align*}
 since it is $\le\sD$. Hence $\Mov{\rdup{\sA+\sF}}\le0$, and actually $=0$,
by Klt again. This means (STD) of Definition~\ref{ssd} for $\sD:=\sF$. By
Lemma~\ref{monots}. The same holds for each prime component of
$P=\sD_X:=\sF_X$, that is, $P\in\fP$.

(2) Immediate from the definitions, except for (fga) and (rfa). In these
cases we only need to verify that $\sM_\bull=\Mov{\sL}\subset\sM$. More
precisely, this concerns only effective $\sM_i$ up to $\sim$. Indeed,
each $\sM_i$ satisfies \SAT\ by \LCA\ and Remark~\ref{asaturr}, (5).

(3) Let $\sM_\bull=\Mov{\sL}$ be a mobile system on $(X/Z,B)$ for an
algebra $\sL$ of type (FGA). By Limiting Criterion~\ref{limcr}, the f.g.\
of $\sL$ is equivalent to stabilization of its characteristic system, that
is, of the limit $\sD=\lim_{i\to\infty}\sD_i$ with $\sD_i=\sM_i/i$. In
turn stabilization is equivalent to the asymptotic descent problem. To
solve the problem we apply Theorem~\ref{assdes}.

First, we choose a prediction (Definition~\ref{defn!prd}). By
Lemma~\ref{monnd}, (3) and the arithmetic monotonicity in
Lemma~\ref{arithm}, the numerical dimension $\nu(\sD_i/Z)$ stabilizes.
After a truncation, we can assume that each $\nu(\sD_i/Z)=\nu$, the Iitaka
dimension of $\sL$. Moreover, up to a similarity, corresponding to a
quasi-isomorphism of algebras by Proposition~\ref{texa}, (8), we can
assume that $\nu\ge0$ and entire $\sM_\bull\subset\fM$. Otherwise
$\sL=k\oplus 0_\bull$ and is f.g. Besides the effectiveness of each
$\sM_i$ we use the saturation \LCA\ of (FGA) as in (2) above.

Since by Lemma~\ref{monnd}, (3), the {\bi}divisors $\sM_i/Z$ are
birationally equivalent, by (MOD) there is a finite family of \w\ triples
$(Y/T/Z,B_Y,\fF)$ {\em for\/} them. (Actually, it can be replaced by a
single triple; cf.\ the remark below.) Moreover, this can be in the {\em
strict\/} form: each of the triple has an {\em infinite\/} subset of
$\sM_i$ for itself, and the same hold after any truncation. Indeed, we can
discard all triples in our finite family that does not satisfy the former
infiniteness and replace $\sM_\bull$ by a truncation for which the
corresponding $\sM_i$ was discarded with triples. Then we take a
truncation such that some triple does not satisfy the same infiniteness,
etc. Finally, we get a family with the strict property.

Now any remaining triple $(Y/T/Z,B_Y,\fF)$ predicts $(Y/Z,\sC=\sA,F,\ga)$
for the asymptotic descent problem $\sD=\lim_{i\to\infty}\sD_i$. If we
consider only $i$ for which this is a \w\ triple then (SAM) for
$\sD_Y=\lim_{i\to\infty}(\sD_i)_Y$ follows from the cone property (RPC)
because, by our choice of {\bi}divisors $\sD_i$, each $(\sD_i)_Y$ is
semiample$/Z$ (cf.\ (LSA) in Remark~\ref{trir}, (7) and the discussion in
Example~\ref{stexm}). The effectiveness (EEF) for the {\bi}discrepancy
$\sC=\sA$ means (TER) for the triples. The boundedness of algebra $\sL$
and \BSD\ of Addendum~\ref{bpbdal}.1 imply that there is a reduced divisor
$F$ that contains the support of each $(\sD_i)_Y$. Thus (UAD) holds for $F$
(cf.\ Remark~\ref{casd}, (3)). On the other hand, the growth (LGD) for
$F$ and some $\ga>0$ follows from (QFC), (CRP) and $(X,B)$ Klt.

The harder property of predictions is boundedness (SAB); by
Addendum~\ref{aboundd}.1, this follows from $\CBS\fga(X/Z,B)$ with the
common model $(X_i/Z,B_i)=(Y/Z,B_Y)$, for $\sD_i$ with $i$ for which
this is a \w\ triple, and with $p_i=i$. Indeed, the descent data for
$\sD_\bull$ is asymptotically bounded by $\sA/Y$. This bound is strict
because each truncation leaves infinitely many {\bi}divisors $\sD_i$ for
the required triple in the descent problem depending on the choice of
the triple.

Secondly, $\sD_\bull$ satisfies the minimal assumptions \FDS\ by
\BSD\ and \MXD\ by Addendum~\ref{bpbdal}.1, and the additional assumption
\BNF\ because each $\sM_i$ is {\bi}free.

Thirdly, as we know, \LCA\ in (FGA) implies the required saturation. Thus
by Addendum~\ref{assdes}.1, the asymptotic descent problem has a solution
on $Y$.

Finally, \CGR\ of Proposition~\ref{coher} holds for $\sC=\sA$ by
Example~\ref{coha}. Hence by Addendum~\ref{assdes}.2 $\sA_{Xst}\le0$ on
the stable model $Xst/Z$, which proves Remark~\ref{fga0lm}, (7). Indeed,
by Klt, the crepant boundary $Bst$ is actually a boundary and
$(Xst,Bst)$ is Klt. On the other hand, by the improved (BIG) in
Remark~\ref{cbrm}, (8) it is dominated by a rational 1-contraction of a
crepant weak log Fano contraction for $(X/Z,B)$. Thus $-(K_{Xst}+Bst)/Z$
is big and {\bi}semiample. Hence by LMMP, $(Xst,Bst)$ is isomorphic in
codimension~$1$ to a weak log Fano contraction, with the same boundary,
on which $-(K_{Xst}+Bst)$ is semiample = nef and big (cf.\
Example~\ref{stexm}).

 \begin{rem} We can use the boundedness of triples instead of the
finiteness in (MOD) in the descent of Theorem~\ref{assdes}. Indeed, an
irreducible family of birational models is birationally a single model
over a function field, that can have infinitely many special points
corresponding to elements of the family and also for any truncation. The
canonical bound at any special point implies the same over the generic
point by inversion of adjunction. The saturation \SAT\ also holds over the
generic point, because it holds over each point as a birational property
over a {\em sufficiently high\/} model. The latter need just a good
resolution as in the proof of Proposition~\ref{coher}.
 \end{rem}

Conversely, the stabilization in Limiting Criterion~\ref{limcr} gives (CBS)
for a truncation of $\sM_\bull$. Then by Truncation Principle~\ref{trprinc},
the same holds for any system $\sM_\bull$ (cf.\ Remark~\ref{cbrm}, (7)).

(4) Immediate from (3).
 \qed\end{pfof}

The following examples illustrate Conjecture~\ref{cbcon}.

 \begin{exa}[Global curve]\label{crv}
 \addcontentsline{toc}{subsection}{\ref{crv}--\ref{tdb} Examples in
dimension~$\le2$ for Conjecture~\ref{cbcon}} 
 Let $(X/Z=\pt,B)$ be a complete log curve $X$ with arbitrary $B$, reduced
but possibly reducible. Then it is terminal in codimension~$2$. The
{\bi}divisors are divisors $\sD$ on the normalization of $X$, that is the
space of a triple of the required form. Conjecture~\ref{cbcon} holds in this
case. Note that $\sD_X$ is well defined for nonsingular divisors $D$ (with
$\Supp D\subset\NonSing X$), and $\sD=\sD_X=D$. Set $D=M+F\ge0$ with
$M=\Mov{D}$ and $F=\Fix{D}$. Then $M$ satisfies (CBS) with $c=1$, which
does not need any saturation. (Even \SAT\ with $\deg B\gg0$ can exclude
some $M$.) But the boundedness of $F$ follows from the saturation \SAT\
and RR. In addition, the multiplicities of $F$ are bounded. For example,
if $B=0$, then $\rdup{F}\le g-1$, where $g=g(X)$ is the arithmetic genus
of $X$.
 \end{exa}

 \begin{exa}[Two dimensional birational case]\label{tdb} Let $f\colon X\to
Z$ be a birational contraction of a normal surface $X$. As we will soon
see (cf.\ the proof of (FGA) in the Main Theorem at the end of this
section), this is the key point in our construction of 3-fold log flips.
Except for (CBS), Conjecture~\ref{cbcon} holds for {\em any\/} birational
contraction with finite $\fS,\fP$ and $\Supp{\fF_X}$ supported\footnote{I
guess that $\fS$ is the standard set of divisors of Definition~\ref{ssd}
and $\fP$ is the standard prime set of Conjecture~\ref{cbcon}. Who is
$\fF$? In the next sentence, who is $\fM$?} in divisors of $X$ that are
exceptional on $Z$, and in the fractional part of $B$.

Suppose first that $B=0$, $X$ is nonsingular, and $X/Z$ is
co\-homo\-logic\-ally rational, that is, $Z$ has rational singularities.
We claim that $(X/X/Z,0,\fF)$ is a \w\ triple for any
$\sM\in\fM=\fM(X/Z,B)$.

Indeed, $\sA=\sA(X,0)=\rdup{\sA}\ge\sE\ge0$, where $\sE$ is the reduced
{\bi}divisor of all the prime {\bi}divisors that are exceptional on $X$,
the {\em support of the exceptional locus}. By the rationality of $X/Z$,
on any model $Y/Z$ of $X/Z$, each Cartier divisor $M$ that is nef$/Z$ is
free$/X$. In particular, this holds for any integral $M\ge0$
for which $\Supp M$ has no exceptional components of $Y/Z$. In addition,
if $C$ is a $-1$-curve on $Y/X$ and $M\cdot C\ge1$ (intersects any $M$
modulo $\sim$), then $M+C$ is also free, and 
 \begin{align*}
 \Mov{\rdup{M+\sA_Y}}&=\Mov{(M+C+\sA_Y-C)} \\
 &\ge\Mov{(M+C)}+\Mov{(\sA_Y-C)} \\
 &\ge M+C>M,
 \end{align*}
 because $\sA_Y-C\ge0$. Hence no {\bi}free $\sM=\Mbar\in\fM$, since
it does not satisfy \SAT. Conversely, by \SAT\ every $\sM\in\fM$ descends
to $X$ as a free divisor$/Z$. Thus we can take $c=1$ in (CBS)
of Conjecture~\ref{cbcon}. This explains the role of saturation.

Indeed, if $Y/X$ is a minimal resolution of the base locus for $|\sM|_X$,
then a $-1$-curve $C$ as above exists over the locus. If $Y:=X\hr$ is also
a sufficiently high resolution for \SAT\ and $M=\sM_Y$ is sufficiently
general in $\linsys{\sM_Y}=|\sM|_Y$, then we replace $C$ by
$\overline{C}_Y\le\sA_Y$. This again contradicts \SAT. Thus $\sM$ is free
on $X/Z$.

Moreover, the same holds for $(X/Z,B)$ under (TER). But in this case we
may have $c<1$, namely, $c=\min \{1-b_i\}$.

If $(X/Z,B)$ is Klt, then (CBS) holds on a crepant terminal resolution
$(Y/X,B_Y)$. Since $(X/Z,B)$ is not assumed to be a weak log Fano
contraction, Remark~\ref{cbrm}, (10) holds, even without (ADJ).

Another approach that we discuss below reduces our problem to freedom in
dimension $1$ (cf.\ Example~\ref{crv} above). It is not so effective, and
moreover has some unpleasant new features (see Remark~\ref{rsurbound}.(2)),
but this finally leads to $4$-fold flips.
 \end{exa}

The main techniques to establish Conjecture~\ref{cbcon} in dimension~2 are
contained in the following proposition.

 \begin{prop}[General two dimensional case]\label{surbound}
 \addcontentsline{toc}{subsection}{\ref{surbound} Main Proposition for
Conjecture~\ref{cbcon} in dimension~2} 
 Let $(X/Z,B)$ be a log pair with a surface $X$, and $\sD\ge0$ an {\rm
integral\/} {\bi}divisor such that:
 \begin{q}
 \item[\GLF] $(X/Z,B)$ is a general log Fano contraction as in
Proposition~\ref{ressat} with arbitrary $B$, not even assumed to be a
boundary; and
 \item[\rm(RIR)] on any model $Y/Z$ of $X/Z$, the general $\sD_Y\in|\sD|_Y$
is {\em reduced and irreducible\/}; and
 \item[\SAT] $\sD\in\fD=\fD(X/Z,B)$; {\em that is, $\sD$ is saturated with
respect to $\sA=\sA(X,B)$\/}.
 \end{q}
 Then except for a bounded set of complete divisors $\sD_X$, $|\sD|_X$ is
free$/Z$ outside $\CS{(X/Z,B)}$ on $X$, where $\CS{(X/Z,B)}$ denotes the
{\em locus of canonical singularities.} In other words, $(X/Z,B)$ satisfies
(TER) outside $\CS{(X/Z,B)}$.
 \end{prop}

 \paragraph{Commentary:} We only need two essential cases:
 \begin{itemize}
 \item $\sD$ is a {\bi}free nonpencil (or an irreducible member of
pencil); or
 \item $|\sD|=\{ P\}$ is a fixed prime {\bi}divisor.
 \end{itemize}
 By (RIR), $|\sD|_X$ free$/Z$ somewhere on $X$ means that $\sD$ has a
descent $\sD_X$ over the locus. (In the proposition, outside $\CS$.)
{From} this point of view ``sufficiently general'' $\sD_Y$ means in
particular that $\sD_Y$ has minimal multiplicities in every divisor that is
exceptional on $X$: e.g., $0$ for {\bi}free $\sD$.

 \begin{add} If the assumptions {\rm(RIR)\/} and \SAT\ in the proposition
are replaced respectively by
 \begin{q}
 \item[\rm(RED)] on any model $Y/Z$ of $X/Z$, a general $\sD_Y\in|\sD|_Y$
is {\em reduced\/}; and
 \item[\rm(SA$'$T)] $\sD$ is saturated with respect to
$\sA'=\sA(X,B)+\sum E_i$, where {\bi}divisors $E_i$ are the exceptional on
$X$ prime over {\em integral components of $\sA$\/}
 \end{q}
 then any two prime components (or even two branches) $D_1\ne D_2$ of
general elements $\sD_X\in|\sD|_X$ intersect each other only in points
$Q\in X$ at which $K+B$ is not Cartier or in $\LCS{(X,B)}$, with a finite
set of exceptions for $D_1$ {\em or\/} $D_2$. More precisely, any set of
exceptions to this can only include $D_i$ with $(K+B)\cdot D_i=0$ (in
particular, $D_i$ is complete, so lies over $P$). Thus these exceptions
$D_i$ belong to a bounded family of reduced divisors $\fF$.

Moreover, we can replace {\rm(SA$'$T)\/} by a weaker condition {\rm(cf.\
Proposition~\ref{ssp}, (4))\/}.
 \begin{q}
 \item[\rm(Sa$'$T)] as in {\rm(SA$'$T)\/}, but assume that the primes
$E_i$ are over $\Bs{|\sD|_X}$.
 \end{q}
 \end{add}

 \begin{add} Suppose in addition that
 \begin{itemize}
 \item $(X,B)$ is Klt and
 \item $\sD$ is {\bi}free.
 \end{itemize}
 Then $|\sD|_X$ has at most one base point on $X$ outside $\CS{(X,B)}$;
in particular, it has at most one base point on a terminal resolution
of\/ $(X/Z,B)$. In addition, $|\sD|_X$ is free in $\CS$ after the point
is blown up. Moreover, any pencil $\sD$ is
elliptic, that is, its general member is a curve of geometric genus $0$.
 \end{add}

 \begin{cor} Locally$/Z$, under the assumptions of {\em
Proposition~\ref{surbound}\/}, the singularities of a general
$\sD_X\in|\sD|_X$ are bounded outside $\CS$; in particular, outside
$\LCS{(X,B)}$ on a terminal resolution.
 \end{cor}

 \begin{proof} The singularities of $\sD_X$ are bounded for any free
$|\sD|_X$. The same holds for any bounded set of divisors $\sD_X$
(Example~\ref{tre}, (1)).
 \qed\end{proof}

 \begin{rem} \label{rsurbound}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item (RIR) and (1--2) in the proof of Proposition~\ref{surbound} below
hold when $\sD$ is {\bi}free and big. For surfaces, the latter means that
$\sD\ne0$ and is not a pencil.

 \item Sometimes, $\Bs{|\sD|_X}\ne\emptyset$, and requires $c<1$ even if
$(X,B)$ is terminal in codimension~$2$ (see Example~\ref{esurbound} below).

 \item However, for {\bi}free $\sD$, we expect only a {\em finite\/} set
of linear systems $|\sD|$ having base points outside $\CS$.
 \end{enumerate}
 For example, one verifies that this finiteness holds when $(X/Z=\pt,B)$
is a del Pezzo surface with a standard boundary $B$ and of nonexceptional
type.
 \end{rem}

 \begin{lem}[Lifting $\R$-Cartier divisors]\label{restdiv} $f^*D$ is
defined up to $\sim$ under the following conditions:
 \begin{itemize}
 \item if $D$ is an $\R$-Cartier divisor defined up to $\sim$, or
 \item for any morphism $f\colon X\to Y$ such that $f(X)$ is not contained
in the fractional prime components of $D$.
 \end{itemize}
 \end{lem}

 \begin{add}[Functoriality] For $g\circ f\colon X\to Y\to Z$,
 \[
 (g\circ f)^*D=f^*(g^*D).
 \]
 \end{add}

 \begin{proof} Since $D$ is given up to $\sim$, we can assume that $f$ and
$D$ are in {\em general position\/}, i.e., $\Supp{f(X)}\cap\Supp D$ is
proper in each irreducible component of $f(X)$.
 \qed\end{proof}

 \begin{lem}[Invariance of integral part under $\sim$]\label{intpart} If
$D\sim D'$ then
 \[
 \rddown{D}\sim \rddown{D'}
 \quad\hbox{and}\quad
 \rdup{D}\sim \rdup{D'}.
 \]
 Thus the same holds for the fractional parts.
 \end{lem}

 \begin{proof}
$F=D-D'$ is principal, in particular integral, and
 \[
 \rddown{D}-\rddown{D'}=\rddown{F+D'}-\rddown{D'}=F=\rdup{D}-\rdup{D'}
 \]
 is also principal.
 \qed\end{proof}

 \begin{lem} \label{inequol} Let $D$ be a divisor on a complete
nonsingular curve $C$ with $\deg D\ge d$. Then $\deg \rdup{D}\ge \rdup{d}$.
 \end{lem}

 \begin{proof} We need to verify that if $\{d_i\}$ is a set of real numbers
such that $\sum d_i\ge d$, then $\sum \rdup{d_i}\ge\rdup{d}$. We can apply
this to the set of multiplicities of divisor $D=\sum d_iP_i$ on $C$.

Indeed, each $\rdup{d_i}\ge d_i$ (and $=$ holds only if $d_i$ is
integral). The sum of these gives
 \[
 \sum \rdup{d_i}\ge\sum d_i\ge d
 \]
 and the required inequality, because $\sum \rdup{d_i}$ is integral, and
any integer $\ge d$ is $\ge \rdup{d}$.
 \qed\end{proof}

 \begin{pfof}{Proposition~\ref{surbound}} We restrict to $D=\sD_X$ after a
resolution. Up to a bounded (even finite) set of divisors $D$ we can assume
that
 \begin{description}
 \item{(1)} $\Supp D\cap\Supp{B}$ is {\em small\/},
 \end{description}
 that is, there no divisors in the intersection.

We consider a sufficiently high resolution $g\colon Y\to X$, where we have
the saturation \SAT. Since $\sim$ preserves \SAT\ (cf.\ Remark~\ref{asaturr},
(7)) we can assume by (RIR) that $D:=\sD_Y$ is general, irreducible and
reduced. Let $g_D\colon D\to X$ be the induced morphism. After an additional
resolution we can assume also that $Y$ is a log resolution for $(X,B+D)$.
In particular, $D$ is nonsingular.

We have a surjective restriction$/Z$
 \[
 \linsys{D+\rdup{\sA_Y}}\broken \linsys{(D+\rdup{\sA_Y})\rest D}=
 \linsys{K_D+\rdup{-g_D^*(K+B)}}.
 \]
 Indeed, here we can use Lemma~\ref{restdiv}, because $D$ is reduced by
(RIR), and $K+B$ is integral on $D$ by (1). Note also that
$\sA_Y\sim K_Y-g^*(K+B)$, whereas $\rdup{\sA_Y}\sim K_Y+\rdup{-g^*(K+B)}$
by Lemma~\ref{intpart}. Thus $D+\rdup{\sA_Y}\sim D+K_Y+\rdup{-g^*(K+B)}$
and by adjunction, normal crossings on $Y$ and Addendum~\ref{restdiv}.1,
we have
 \[
 (D+\rdup{\sA_Y})\rest D\sim (D+K_Y)\rest D+\rdup{-g_D^*(K+B)}=
 K_D+\rdup{-g_D^*(K+B)}.
 \]
 Also we use Kawamata--Viehweg vanishing: $R^1h_*\Oh_Y(\rdup{\sA_Y})=0$
with $h=f\circ g\colon Y\to Z$, because $\rdup{\sA_Y}\sim
K_Y+\rdup{-g^*(K+B)}$ and \GLF.

If $|\sD|_X$ has a base point $Q\in X/Z$ outside $\CS$, then by \SAT\
 \[
 \Mov{(D+\rdup{\sA_Y})}=\Mov{\rdup{\sD+\sA}_Y} \le\sD_Y=D.
 \]
 Thus $\Fix{(D+\rdup{\sA_Y})}\ge E_i$ for some exceptional divisor $E_i$
with discrepancy $a_i=a(X,B,E_i)>0$ and such that $E_i$ intersects $D$ on
$Y$. Hence $\Fix{((D+\rdup{\sA_Y})\rest D)}\ge E_i\rest D>0$ and
$\Bs{\linsys{(D+\rdup{\sA_Y})\rest D}}\ne\emptyset$ -- the restriction is
not free by the above surjection. Hence $D$ is complete.
Otherwise $\linsys{(D+\rdup{\sA_Y})\rest D}$ is always free.

Thus $D$ is complete and $\linsys{(D+\sA_Y)\rest D}=
\linsys{K_D+\rdup{-g_D^*(K+B)}}$ must have base points or be $\emptyset$.
This is impossible if $\deg B'>1$, where $B'=-g_D^*(K+B)$ because then by
the Lemma~\ref{inequol} $\deg\rdup{B'}>1$ and so $\ge2$. But for such a
divisor $B'$, the linear system $\linsys{K_D+\rdup{B'}}$ is free. Hence
each $D$ having fixed points on $X$ outside $\CS$ should have bounded degree
on $X/Z$, namely $-(K+B)\cdot D\le1$. Such divisors are bounded. Indeed,
we need to consider only complete divisors $D/Z$. In particular, they
include the contracted divisors $D$ of $X$:
 \begin{description}
 \item{(2)} $-(K+B)$ is not big$/Z$ on $D$
 \end{description}
this adds a finite set of divisors with discrepancy $0$ to $\Supp{B}$;
cf.\ \GLF\ above.
 \qed\end{pfof}

 \begin{prop} \label{monotd} If $D_1\ge D_2$, then $D_1$ $C$-saturated
implies $D_2$ $C$-saturated over prime divisors at which $D_1=D_2$. The
same holds in the {\bi}divisorial versions.
 \end{prop}

 \paragraph{Proof--Explanation} Saturation over the prime divisors at which
$D_1=D_2$ (or over any other set of divisors), means that
$\Mov{\rdup{D_2+C}}\le D_2$ over this set of divisors, in other words, the
inequality concerns only the prime divisors $P_i$ in the set: e.g., in our
situation, $\mult_{P_i}{D_1}=\mult_{P_i}{D_2}$. Thus the proposition is
immediate from the definition. \qed\par\medskip

 \begin{pfof}{Addendum~\ref{surbound}.1} Suppose that $K+B$ is Cartier and
canonical at a point $Q\in D_1\cap D_2$. Then $K+B$ Cartier implies that
each $D_i$ lies over an integral component of $B$. Again by (1) in the
proof of Proposition~\ref{surbound}, we assume either that $(K+B)\cdot
D_i<0$ for $i=1,2$, or that $D_i$ is {\em not complete\/}. Then we verify
that $D$ is not $\sA'$-saturated on any sufficiently high $Y/X$. Since
$-(K+B)$ is nef and big$/Z$, there exists only a finite number of complete
$D_i$ with $(K+B)\cdot D_i=0$. Thus up to finitely many exceptions, the
contradiction with $\sA'$-saturation means that $Q$ is not Cartier or not
canonical.

By (RED) and Remark~\ref{asaturr}, (1) (cf.\ the proof of
Proposition~\ref{ssp}, (ii)) we can replace $D$ by its components that are
not exceptional on $X$, or in other words, omit the divisors that are
exceptional on $X$ (because they are fixed in $|\sD|_Y$). Then by (RED) and
Proposition~\ref{monotd} it is enough to verify that the {\em new\/}
$D:=D_1+D_2$ is not $\sA'$-saturated on any sufficiently high $Y/X$ {\em
over\/} $D_1\cup D_2$ and some exceptional prime {\bi}divisors $E_i/Q$. We
assume also that each $g(D_i)$ passes through $Q$ and that either
$(K+B)\cdot D_i<0$, or $D_i$ is not complete. We derive a contradiction
with $\sA'$-saturation on any log resolution $g\colon Y\to X/Z$ for $B+D$.

On $Y$ we put together a chain of curves (chain maybe after additional
blowups) $C=D_1\cup(\bigcup E_i)\cup D_2$ with only nodal singularities,
where $E_i$ are the exceptional curves over $Q$ and $D_i$ the end curves.
We verify that $\linsys{C+\rdup{\sA_Y}}=\linsys{\rdup{D+\sA_Y+\sum E_i}}$
has no base points in a neighborhood of $C$ on $Y/Z$. Since
$\sA_Y+\sum E_i\le\sA'_Y$ even under (Sa$'$T), then by Lemma~\ref{monots}
this contradicts $\sA'$-saturation for $D$ on $Y/X$: 
 \[
 \Mov{\rdup{D+\sA_Y+\sum E_i}}\ge D+\sum E_i>D
 \]
 over each $E_i$. Note that $\sA_Y\ge0$ and integral over $Q$, because $Q$
is canonical for $K+B$.

As in the proof of Proposition~\ref{surbound}, for the restriction to $D$,
we get a surjective restriction$/Z$ on $C$:
 \[
 \linsys{C+\rdup{\sA_Y}}\broken \linsys{(C+\rdup{\sA_Y})\rest C}=
 \linsys{K_C+\rdup{-g_C^*(K+B)}}
 \]
 with $g_C\colon C\to X/T$. Indeed, here we can use Lemma~\ref{restdiv},
because $C$ is reduced and $K+B_Y$ is integral on $C$ (since $K+B$ is Cartier
at $Q$). As above, by Lemma~\ref{intpart},
 \[
 C+\rdup{\sA_Y}\sim C+K_Y+\rdup{-g^*(K+B)}
 \]
Thus by Lemma~\ref{intpart}, adjunction, normal crossings on $Y$ together
with Addendum~\ref{restdiv}.1, we get
 \[
 (C+\rdup{\sA_Y})\rest C\sim (C+K_Y)\rest C+\rdup{-g_C^*(K+B)}=
 K_C+\rdup{-g_C^*(K+B)}.
 \]
 Now, as before, Kawamata--Viehweg vanishing gives
$R^1h_*\Oh_Y(\rdup{\sA_Y})=0$.

The construction gives a Cartier divisor $L=(C+\rdup{\sA_Y})\rest C
\sim K_C+M/Z$ such that
 \begin{itemize}
 \item $M$ is nef on $C/Z$;
 \item each $M\rest{E_i}\sim 0$; and
 \item for each $D_i$, either $\deg M\rest{D_i}\ge1$ or $D_i$ is not complete.
 \end{itemize}
 Indeed, $M=\rdup{g^*(-K-B)}\rest C=\rdup{-g_C^*(K+B)}$ by the normal
crossings and by Addendum~\ref{restdiv}.1, and we have:
 \begin{itemize}
 \item $-(K+B)$ is nef on $X/Z$;
 \item each $K+B$ is Cartier in $Q$; and
 \item for each $D_i$, either $(K+B)\cdot D_i<0$ or $D_i$ is not complete.
 \end{itemize}

Then $\Bs{|L|}=\emptyset$ and this implies that $\linsys{C+\rdup{\sA_Y}}$ is
free near $C$ on $Y/T$. Indeed, by (2) we can consider a Cartier
divisor $L'$ on a nodal curve $C'$ with a single node, that is a result of a
contraction $h\colon C\to C'$ of all $E_i$ and $L=h^*L'$. Note that $C'$ has
$\ge2$ components $D_i$, and $\deg(K_{C'}+L')\rest{D_i}\ge1$ on each
complete $D_i$. Thus $\Bs{|L'|}=\emptyset$ by\footnote{Something missing.}

(In the case of two branches, this gives a nonvanishing and a contradiction
to the $\sA'$-saturation near $\bigcup E_i$.)
 \qed\end{pfof}

 \begin{quest} State and prove a semi log canonical version of the last
freedom result in higher dimensions.
 \end{quest}

 \begin{pfof}{Addendum~\ref{surbound}.2} Since $K+B$ is Klt,
$\rdup{\sA_Y}\ge0$. Thus because $D$ is {\bi}free, in the proof of
Proposition~\ref{surbound} we have an effective Cartier divisor
$L\in\linsys{K_D+\rdup{B'}}$, this time with $\deg B'>0$. Thus $L$ has
at most a single base point $Q$ of multiplicity $1$. Moreover, $L$ has
trivial mobile part ($0$) and has $Q$ as a base point only if $K_D\equiv
0$ and $\deg L=1$, that is, when $D$ belongs to an elliptic pencil.
Since $\deg L=1$ we only need one blowup outside $\CS$ to resolve the
base point, namely, the blowup in $Q$.
 \qed\end{pfof}

 \begin{prop}\label{ssp} Let $(X/Z,B)$ be a Klt pair. Then the following
holds.

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item For any crepant birational contraction $g\colon X\to Y/Z$ is a
birational contraction, $\fS_Y=g(\fS)=\{g(S),
S\in\fS\}\supset\fS(Y/Z,g(B)=B_Y)$.
 \item \SAT${}={}${\rm(SAF)\/} in Conjecture~\ref{cbcon} with the fixed
$\sF$.
 \item {\rm(SAF)\/}$=${\rm(STD)\/} in Definition~\ref{ssd} for any
$\sF\ge0$.
 \item The standard set $\fS=\fS(X/T,B)$ has the following equivalent
description, up to a component over the fractional part of $B$ (see
below),
 \begin{q}
 \item[\rm(Sa$'$F)] $S=\Supp{(\Fix{\sD})}_X$ for some $\sD\ge0$
satisfying {\rm(Sa$'$T)\/} of Addendum~\ref{surbound}.1; and includes
 \item[\rm(SA$'$F)] a reduced divisor $F$ is contained in the\/ {\em
substandard\/} set\/ $\fS'=\fS'(X/Z,B)$ if $\Supp{F}$ only lies over
integral components of $B$, and some {\bi}free {\bi}divisor $\sM$ is
saturated for $\sA'+F$ on {\em some\/} sufficiently high model $X\hr/X$.

$\sA'=\sA+\sum E_i$ with the exceptional divisors $E_i$ on $X$ over
integral components of $\sA$.
 \end{q}
 \end{enumerate}
 \end{prop}

 \begin{rem}\label{sspr}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item {\em Up to a component over the fractional part of $B$} means that
each $S\in\fS$ is $S'+F$, whereas $S'$ under (Sa$'$F) and $F$ is over the
fractional part. Thus bounds on the two sets are equal.
 \item $F$ is considered as a {\bi}divisor.
 \end{enumerate}
 \end{rem}

 \begin{lem}\label{as} Let $\sC$ and $\sM$ be {\bi}divisors such that:
 \begin{itemize}
 \item $\rdup{\sC}\ge0$, and
 \item $\sM$ is {\bi}free.
 \end{itemize}
 Then $\sC$-saturation of $\sM$ on {\em some\/} sufficiently high model
of $X/Z$ is equivalent to $\sC$-saturation on {\em every\/} sufficiently
high model.
 \end{lem}

 \begin{proof} Some sufficiently high model $Y/Z$ means any model of
$X/Z$ on which $\sM=\overline{\sM_Y}$ is free$/Z$, that is,
$\Bs{\linsys{\sM_Y}}=\emptyset$. Then $\sC$-saturation of $\sM$ on any
model $X\hr/Y/Z$ follows from that on $Y$. Indeed, since
$\rdup{\sC}\ge0$, and $\sM$ is free$/Y/Z$, $\sC$-saturation on $Y$
implies that
 \[
 \Oh_Z(\rdup{\sM+\sC})=\Oh_Z^{ch}(\rdup{\sM+\sC})=\Oh_Z(\sM)=
f_*\Oh_Y(\sM_Y)=f_*\Oh_{X\hr}(\sM_{X\hr}).
 \]
 This is the $\sC$-saturation on $X\hr$ (cf.\ Remarks~\ref{asaturr},
(1--2)).
 \qed\end{proof}

 \begin{pfof}{Proposition~\ref{ssp}}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \begin{enumerate}
 \item For each $s\in\fS(Y/Z,B_Y)$, the log transform $S$ of $s$ is an
element of $\fS(X/Z,B)$. Thus $s=g(S)$.

 \item By definition $\Mov{\rdup{\sM+\sF+\sA}}\le\sM$. This means
(SAF). Conversely, assuming Klt,
 \[
 \rdup{\sA+\sF}\ge0, \quad \Mov{\rdup{\sM+\sF+\sA}}\ge\sM,
 \]
 and $=\sM$ by (SAF). On the other hand, since $\sF\ge0$, $\sM\le\sM+\sF$,
and this gives \SAT\ for $\sM+\sF$.

 \item $\sL=\Mov{\rdup{\sA+\sF}}$. Again $\rdup{\sA+\sF}\ge0$, and
$\sL\ge0$. Since $\sM$ is {\bi}free, by (SAF),
 \[
 \sM\ge\Mov{\rdup{\sM+\sF+\sA}}\ge \Mov{\sM}+\Mov{\rdup{\sF+\sA}}=\sM+\sL,
 \]
and $\sL\le0$. Thus $\sL=0$, which means (STD) for $\sD=\sF$. Conversely,
(STD) for $\sD$ means (SAF) for $\sM=0$ and $\sF=\sD$.

 \item If $S=\Supp{\sD_X}\in\fS$, then by Lemma~\ref{monots}, (STD) implies
that $0$ is saturated with respect to $\rdup{S'+\sA'}\le\rdup{\sA+\sD}$,
where $S$ is fixed and $S'$ is as in Remark~\ref{sspr}, (1). The inequality
follows from (SEF), since $\Bs{|S'|_X}=S'\subset S$. Thus (Sa$'$F) holds
for $\sD=S'$.
 \end{enumerate}

Conversely, if $\sD$ satisfies (Sa$'$F) then, as in (ii--iii) above,
$\sF+\sum E_i+E'$ satisfies (STD) and (SEF), where $\sF=\Fix{\sD}$ and
where $E'=\sum \ep_iE_i$ with $0<\ep_i\ll 1$ for the exceptional divisors
over the {\em fractional\/} components of $\sA$. Note that $E'$ does not
affect (Sa$'$F) (cf.\ Addendum~\ref{monots}.1):
 \[
 \rdup{\sF+\sA+\sum E_i+E'}=\rdup{\sF+\sA'},
 \]
 but (SEF) holds because $\sF_X\subset\Bs{|\sD|_X}$. Thus
$\Supp{\sF_X}\in\fS$.

By Lemma~\ref{monots}, (SA$'$F) implies (Sa$'$F) for $\sD=\sM+F$, where
$\Fix{\sD}=F$ and $\Supp{F}=F\in\fS$. In (SA$'$F), we can replace ``for
some model'' by ``for every model'' by Lemma~\ref{as} with $\sC=\sA'$.
 \qed\end{pfof}

 \begin{cor} \label{boundsf} Let $(X/T,B)$ be as in
Proposition~\ref{surbound}, and suppose that it is Klt (but $B$ may only
be a subboundary). Then the standard set $\fS=\fS(X/Z,B)$ is bounded.

In particular, $\fS'$ is also bounded in {\rm(SA$'$F)}.

 \end{cor}

 \begin{proof} By Proposition~\ref{ssp}, (3--4) and Remark~\ref{sspr}, (1),
it is enough to establish the boundedness for a {\em fixed reduced\/}
$D=\sum D_i=\sD_X=\sD$ that satisfies (Sa$'$F), and in particular, (SA$'$F)
with $\sL=0$ and $F=D$; we can also assume that each component
$D_i\not\subset\Supp{B}$.

Since $F$ is fixed and satisfies (Sa$'$F), by Lemma~\ref{monots} each
component $D_i$ is $\sA'$-saturated, and even $\sA$-saturated (cf.\
Proposition~\ref{ssp}, (3)). The same holds for any reduced $0\le D'\le
D$. In particular, these components are complete by
Proposition~\ref{surbound} applied to a terminal resolution of $(X/T,B)$.
Moreover, the prime components $D_i$ belong to a bounded set. Thus we
need to verify that the number of components in $D$ is bounded. It is
enough to consider only the global case with $Z=\pt$

The boundedness of $D_i$ implies that they have bounded intersections.
Thus any two components can be disconnected by a bounded number of
resolutions. Suppose that the set of $D$ is unbounded. We derive from this
a contradiction with the boundedness of resolutions to disconnect two
components $D_i$ and $D_j$. (Equivalently, the intersection numbers
$D_i\cdot D_j$ are bounded.)

Since the number of components is unbounded, there exist a set of divisors
$D$ containing unboundedly many algebraically equivalent irreducible
components $D_i$; clearly, $D_i^2\ge0$, for otherwise there can only be one
such $D_i$. If $D_i^2=0$, the divisors $D_i$ are all fibres of a fibration
$X\to C$ over a curve $C$. Then for sufficiently many such components,
there is an unbounded number of nonsingular fibres $0<D'=\sum D_i\le D$
and $\sum D_i$ is free, which contradicts $\sA$-saturation. Moreover,
$D:=D'$ is never sufficiently general on any model $Y/Z$ of $X/Z$, that
is, general in the free linear system $\linsys{D}$ on $Y/Z$. Note that
two {\bi}free divisors have the same $0$ multiplicities in any fixed
finite set of prime {\bi}divisors. Indeed, then $K+B_Y+D$ is again
Klt. Hence $\Dbar\le\rdup{D+\sA}$; this holds at each prime $P_i$ on $Y$,
because $D$ is integral and $\rdup{\sA_Y}\ge0$ by Klt. If $P_i$ is
exceptional on $Y$, each $\mult_{P_i}{(\sA-\Dbar)}>-1$ and so each
$\mult_{P_i}{\sA}> \mult_{P_i}{\Dbar}-1$. Taking $\rdup{\ }$ we get
$\mult_{P_i}{\rdup{\sA}}\ge \mult_{P_i}{\Dbar}$, because $D$ is Cartier
on $Y$ with only integral multiplicities. Thus on any fixed sufficiently
high model $X\hr/Y$, by the $\sA$-saturation for $D:=D'$ as {\bi}divisor,
$D=D_{X\hr}\ge \Mov{\rdup{D+\sA}_{X\hr}}\ge \Dbar_{X\hr}$, which
contradicts $D$ fixed. (Compare the proof of Proposition~\ref{surbound}.)

Hence $D_i^2>0$ and the divisors $D_i$ intersect each other. By
Addendum~\ref{surbound}.1, any of these $D_i$ intersects any other
$D_j\iso D_i$ in a finite number of points, namely, in
$D_i\cap(\Supp{B}\cup\CS)$, where $\CS$ denotes non-Gorenstein or
noncanonical points. Therefore, an unbounded number of prime components
$D_i$ pass through the same point $Q$. This point depends on $D$. We can
make a bounded resolution in such a points $Q$, e.g., the minimal
resolution if $Q$ is singular, or the usual blowup if $Q$ is nonsingular
on $X$. Then we replace $(X,B)$ by its crepant transform.

The $\sA$-saturation is birational, and the intersection points of $D_i$
lie only over the log transform of $D_i\cap(\Supp{B}\cup\CS)$. We
consider the proper birational transform of $D$ and of the prime components
$D_i$. We again have an unbounded family of algebraically equivalent $D_i$,
and again $D_i^2>0$. Again we have some (new) point $Q$ through which an
unbounded number of components $D_i$ are passing. Again we make a bounded
resolution, etc.

But this process must terminate, since any two components are disconnected
by a bounded resolution. Or we can see that the new self-intersection
$D_i^2\le$ the old self-intersection $D_i^2-\hbox{const.}$, where
$\hbox{const.}=1$ for the usual blowup of nonsingular points, and some
positive numbers at other points.
 \qed\end{proof}

 \begin{exa} \label{esurbound} Let $(X/\pt,B=0)$ be a (terminal)
nonsingular del Pezzo surface. Then in Proposition~\ref{surbound}, every
$D=\sD_X$ is either free or is a $-1$-curve, except for del Pezzo surfaces
of degree 1 with $D\in|{-}K|$. The $-1$-curves $D$ give fixed $D$ of
degree $1$: $-K\cdot D=1$. The standard sets are the disjoint sums of
these $-1$-curves.

Then (FGA)($X/\pt,0$) gives the models $Xst=X$, except for $=\PP^1$
in the case of a pencil, and $=\pt$ (cf.\ Example~\ref{prs}).

\paragraph{Pencils and irreducibility in (RIR) of
Proposition~\ref{surbound}} There is no pencil of elliptic curves every
element of which is tangent to the $1$-complement boundary $C$, because
the restriction of the pencil on $C$ is an isomorphism for a general
divisor. Thus we have only a single element in the pencil with a single
intersection (and tangent) point. However for higher genus these are
possible.
 \end{exa}

 \begin{exa}\label{fcanb}
 \addcontentsline{toc}{subsection}{\ref{fcanb} Example: log canonical
bounds and the nef condition}
 For (CBS) on $X/Z$ or on a triple $Y/Z$ in the 3-fold or higher
dimensional case (even the local case), we need the numerical condition
that $\sM_Y/Z$ is {\em nef} (cf.\ (NEF) in Remark~\ref{cbrm}, (8)). In
the one dimensional case, it is implied because $\sD(X/Z,B)$ is effective
in Conjecture~\ref{cbcon}; in the two dimensional case, because $\sM$ is
{\bi}free or $|\sM|_Y$ is free in codimension~1. But nef is important in
dimension $\ge3$. Indeed, let $f\colon X\to Z$ be a small extremal
contraction that is negative with respect to $D$. Then any infinite set
of {\bi}divisors $\sD$ of the form $\sD_X=nD$ for a natural number $n$
is not canonically bounded, or equivalently, not log canonically
bounded. Otherwise, we have a very negative curve $C$ contracted by $f$
for $K+c D_X$ with $C$ as a LCS (center of log canonical singularities),
and log threshold $c$ along $C$ for all $D_X$. This is impossible by
anticanonical boundedness~\cite[Theorem]{sh96a}. However we can take
another model $Y/Z$ (for example, the $D$-flip) of $X/Z$, where
$\{\sD_Y\}$ is bounded (if the flip conjecture holds).
 \end{exa}

 \begin{cor} \label{mainb}
 \addcontentsline{toc}{subsection}{\ref{mainb} Corollary:
Conjecture~\ref{cbcon} in dimension $2$}
 Conjecture~\ref{cbcon} holds in dimension $2$: for any weak log Fano
contraction $(X/Z,B)$ with $\dim X=2$, there exists a bounded family of
\w\ triples $(Y/T/Z,B_Y,\fF)$ with induced standard $\fF=\fS_Y$ such
that, for each {\bi}free $\sM\in\fM(X,B)$, the singularities of $\sM_Y$
are bounded {\em for some \w\/} $Y/T$. In particular, they are
canonically bounded with respect to $B_Y$.
 \end{cor}

 \begin{add} In general, the only estimate is $0<c\le\lld(X,B)$.
 \end{add}

 \begin{rem}\label{rem!exin} In applications, we can extend the induced
standard set $\fS_Y$ to some $\fF$ by adding some fixed reduced
divisors, e.g., $\Supp{B_Y}$, or even bounded sets (cf.\ proof of
Proposition~\ref{bounds}). However, we then lose (SA$'$F), for example, on
triples other than $(X/X,B,\fS)$. Otherwise to preserve (SA$'$F), we need
to take $\fS_Y$ as an integral log transform: with $\sum E_i$, where $E_i$
lie over integral components of $\sA_Y$. Thus we need (SA$'$F) only as a
condition to establish that $\fF$ is bounded.

The integral condition on $F$ in (SA$'$F) of Proposition~\ref{ssp} can be
replaced by $\Supp{F}\cap\Supp{B_Y}=\emptyset$ when $B_Y$ is a boundary
and $(X,B)$ is Klt.
 \end{rem}

 \begin{proof} We assume that the surface $X$ is complete. This is
important for the applications below, and nontrivial.

We associate \w\ triples with respect to the bigness of $\sM$, which
proves (MOD). As in (CRP), each $(Y/T,B_Y)$ is a crepant model over
$(X,B)$ and satisfies (QFC). The corresponding set $\fF$ on $Y$ is the
log birational transform of the standard set $\fS=\fS(X,B)$. This is
bounded by Corollary~\ref{boundsf}. We need to satisfy (TER) and on $T$
(RPC), choose a triple {\em for each\/} $\sM\in\fM(X,B)$.

Thus for big $\sM$, we take a terminal resolution $(Y/T,B_Y)$ as a \w\
triple with the identity $Y/Y=T$. In particular, it is a weak log Fano
contraction satisfying (TER) and (RPC). In addition, $\sM_Y=id^*M$, where
$M=\sM_Y$ is big. Since $\dim X=2$ and $\sM$ is {\bi}free, $M$ is nef. On
any model, all such sufficiently general $\sM_Y$ are reduced and
irreducible. Hence by Proposition~\ref{surbound}, all such sufficiently
general divisors $\sM_Y$ have bounded singularities everywhere, because
$(Y,B_Y)$ is terminal in points. We can take just one such triple.

The next case is when $\sM$ gives a $1$-dimension image, that is, there
exists a rational contraction $X\broken C$ onto a curve $C$ with $\sM_X$
in the fibres. In classical terminology, $|\sM|_X$ is {\em composed of
a pencil\/}. In this case, we define $Y/T$ as the regular contraction
defined by this pencil. In particular, $T=C$, and $\sM_Y=g^*M$ for some
divisor $M>0$. Hence $M$ is nef and big, and by construction, all such
sufficiently general $\sM_Y$ have bounded singularities everywhere. This
family of triples is bounded, because the intersection of the elements of
the pencil are bounded by Proposition~\ref{surbound}. More precisely, by
Addendum~\ref{surbound}.2, every such pencil is elliptic and has a
regularization in one blowup of a terminal resolution. In this case
$C=T=\PP^1$ by the rational connectedness of $X$, satisfies (RPC), and
$(Y,B_Y)$ satisfies (TER).

Finally, let the dimension of $|\sM|_X$ be $0$, that is, the effective
divisor $\sM=0$. Take $Y/T$ to be $X\to\pt$; then $M=0$. This triple is
unique.
 \qed\end{proof}

 \begin{pfof}{$\FGA$ in the Main Theorem} Immediate by Theorem~\ref{cbsfg},
(3) and Corollary~\ref{mainb}. The birational case was done in
Example~\ref{tdb}.
 \qed\end{pfof}

 \begin{cor}\label{stabc2} Any $\FGA_n\bir$ algebra $\sL=\dival_{X/T}{D}$,
is f.g., up to codimension~$2$ over $T$, {\em that is, over points of
codimension $\le2$\/}.
 \end{cor}

 \begin{proof} Immediate by $\FGA_n$ with $n\le2$ in the Main Theorem.
 \qed\end{proof}

 \begin{cor} \label{fgapl3} $\FGA_3^{pl}$ of {\em Example~\ref{fgapl}\/}
holds.

Moreover, we can drop the condition that $\sL$ is bounded by $D$ and
satisfies Main Lemma~\ref{mainl} (cf.\ Remark~\ref{fgaa}).
 \end{cor}

 \begin{proof} Immediate by $\FGA_d\bir$ with $d\le2$, which was
essentially done in Example~\ref{tdb}, and Example~\ref{fgapl}.
 \qed\end{proof}

Of course, by Corollary~\ref{fgarfa} (CBS) also implies (RFA), but we can
prove more.

 \begin{thm}\label{cbrfa}
 \addcontentsline{toc}{subsection}{\ref{cbrfa} Theorem: LMMP$_d$ implies
$\CBS_{d-1}$ and $\SSB_{d-1}$}
 LMMP in dimensions $d\le n-1$ implies $\CBS_{d-1}^*$ and $\SSB_{d-1}\gl$,
where $^*$ means a modification in dimensions $\le d-1$ including log
singularities.

Moreover, we can drop $\CBS_{d-1}^*$ and $\SSB_{d-1}\gl$ with $d\le3$,
i.e, for $n\le4$. $\RFA_{n,m}\bir$ holds with $m\le n-1$.
 \end{thm}

Essentially, we prove the theorem for $n=4$, that is, $\RFA_{4,m}\bir$ in
Section~\ref{mr}; the $^*$ modification of (CBS) is explained in the proof
of Corollary~\ref{stabg}. The proof of the theorem is based on the
stabilization of Theorem~\ref{stabin} in Section~\ref{stbd} and
destabilization results in Section~\ref{dstb}. However before we need to
clarify restrictions of {\bi}divisors in Section~\ref{rstr}, and develop
approximations in Section~\ref{apprx}.

\section{Restrictions of {\bi}divisors} \label{rstr}

The\footnote{You could explain that this notion corresponds to the
restriction of mobile linear systems.} new tools we introduce here are
different flavors of {\bi}restriction from $X$ to a {\bi}divisor $E$.
The proof of Theorem~\ref{stabin} uses systematically restrictions of
{\bi}divisors (cf.\ also Proposition~\ref{ressat}). There are essentially
the two following types, or some mixture of them.

\subsection{Mobile restriction}\label{sec!mrest} Let $E$ be a prime
{\bi}divisor on $X$ and $\sD$ a {\bi}free divisor$/Z$. The {\em mobile
restriction} of $\sD$ to $E$ is\footnote{The broken bar for
{\bi}restriction is given by the macro $\tt\backslash$frestbar, and I may
redesign it to give better line spacing.} the {\bi}free divisor
$\sM=\sD\frest E$ on $E$ defined by the formula
 \[
 \sM_Y=\sD_Y\rest E
 \]
 where $Y/X$ is a model such that $E$ is a normal divisorial subvariety of
$Y$ and $\sD$ is free on $Y/Z$. Thus $\sM_Y$ is also free on $E/Z$, and
$\sM=\overline{\sM_Y}$. To cut out the restriction as a Weil divisor it is
enough to take a normal model $Y/X$, where $E$ is a divisorial
variety,\footnote{I think this means: It is enough to take a model $Y/X$
on which $E$ appears as a prime divisor, and such that $Y$ is nonsingular
at every prime divisor of $E$.} nonsingular in $X$ in divisors of $E$.
Similarly, we can define a Cartier {\bi}divisor $\sD\frest E$ for any
Cartier {\bi}divisor $\sD$ on $X$ (cf.\ (CAR) in the additional
assumptions of \ref{asdepr}). Thus mobile restriction is well defined,
because the divisor has only $0$ multiplicities up to linear equivalence.
mobile restriction is {\em defined up to linear equivalence\/}.

\subsection{Fixed restriction}\label{frest} When we define a fixed
restriction $D\rest E$, we usually require $D$ to be Cartier and
$E\not\subset\Supp D$. Moreover, by Lemma~\ref{restdiv}, we can assume
that $D$ is $K$-Cartier (where $K=\Z,\Q$ or $\R$); then $D\rest E$ is
also $K$-Cartier and restriction preserves linear equivalence, since it
extends it. (Locally, each $K$-Cartier divisor is $K$-principal; cf.\
Definition~\ref{qlin}. The restriction of any $K$-principal divisor is
also $K$-principal.) In the same way we can define the {\em fixed
restriction} $\sD\frest E$ of an $K$-Cartier {\bi}divisor $\sD$
whenever $\mult_E{\sD}=0$. Again we take
 \[
 \sM_Y=\sD_Y\rest E,
 \]
 where $Y/X$ is a model such that $E$ is normal divisorial subvariety of $Y$
and $\sD$ is $K$-Cartier$/Y$, that is, $\sD=\overline{\sD_Y}$. The
{\bi}divisor $\sM=\overline{\sM_Y}$ is well defined a $K$-Cartier
divisor$/E$. In this case some multiplicities grow on subsequent blowups.
This restriction also preserves {\em linear equivalence\/}.

\subsection{Mixed restriction} If $\sD$ is a $K$-Cartier {\bi}divisor having
$\mult_E{\sD}\in K'\subset K$, we can define the {\em mixed restriction}
$\sM=\sD\frest E$ by setting
 \[
 \sM_Y=\bigl(\sD_Y-(\mult_E{\sD})E\bigr)\rest E+(\mult_E{\sD})E'\rest E,
 \]
 where $Y/X$ is a model such that $E$ is a normal divisorial subvariety of
$Y$, $\sD$ is $K$-Cartier$/Y$, and $E'\sim E$ on $Y$, but
$E\not\subset\Supp E'$. This is again a $K$-Cartier {\bi}divisor.

 In other words, if $E$ is $\Q$-Cartier on $Y$, $K$ has characteristic $0$,
we obtain
 \[
 \sM=\bigl(\sD-(\mult_E{\sD}\bigr)\Ebar)\frest E
 +(\mult_E{\sD})\overline{E'}\frest E.
 \]
 Of course, this divisor depends on the choice of model $Y/X$ and on the
equivalence $E'\sim E$. Nonetheless this restriction is compatible with
{\em $K'$-linear equivalence\/}, in particular, with $\sim$ for $K'=\Z$,
by Lemma~\ref{restdiv} and because $E'\sim E$ implies
$\overline{E'}\sim\Ebar$.

 \begin{warn} In general $E'$ is not effective or prime. Note also, that
$E$ is automatically $\Q$-Cartier, provided that $Y$ is $\Q$-factorial.
 \end{warn}

{\em Where is the support of\/} $\sD\frest E$? For saturations, we need to
take integral parts of fractional divisors, and to estimate their
fractional parts, in particular, for restrictions. For this purpose we
now describe the support of restrictions on a case-by-case basis. In
addition, for a birational contraction $f\colon X\to T$, we consider the
case (df) that recalls discrepancy.\footnote{I omit ``such that
$f_*(\sD_1)=f_*(\sD_2)$,'' because it doesn't seem to make sense without
any $\sD_1$ and $\sD_2$ defined.}

 \begin{defn}
 A {\bi}divisor $\sD$ on $X$ is $K$-Cartier at $p\in X$ if
$\sD=\overline{D'}$ locally near $p$, for some $K$-Cartier divisor $D'$.
Note that in this case $D'=\sD_X$ near $p$.
 \end{defn}

 \begin{exa} \label{bbasep} Suppose that $\sD$ is an {\em effective}
{\bi}free divisor$/Z$, or a $K$-linear combination $\sum k_i\sD_i$ of
such (for example, a difference). We assume that the $\sD_i$ are in
general position on $X$, and in particular,
$\Supp\bigl(\sum k_i\sD_i\bigr)_X=\bigcup\Supp(\sD_i)_X$. This condition is
equivalent to requiring that $\sD$ is a $K$-Cartier {\bi}divisor up to
$\sim$. Then $\sD$ is {\em Cartier\/} outside $\Supp(\sD_X)$ (a divisorial
sub\-variety on $X$). Indeed, $\sD\sim0$ outside the base locus of the
linear system of $|\sD|_X$. The same holds for $K$-Cartier divisors $\sD$
on $X$. They are $K$-Cartier and even {\em Cartier\/} outside
$\Supp(\sD_Y)$ on any model $Y/Z$ of $X/Z$ whenever $D_Y$ has above
presentation with $\sD_i$ in general position on $Y$.
Moreover, we can replace $\Supp{\sD_Y}$ by the intersection
 \[
 \bigcap\Supp(\sD'_Y),
 \]
 where $\sD'$ runs through the {\bi}divisors $\sD'\sim_K\sD$ at least
locally$/Y$, and $D'$ is a $K$-linear combination as above, also general
on $Y$, although not necessary effective.

For example, let $X\to Y=\PP^2$ be the blowup of $P\in\PP^2$, with
$-1$-curve $E$ over $P$, and let $L,L_P$ be distinct lines through $P$.
Then $\Ebar=\sL-\sL_P$, where $\sL=\Lbar$ and
$\linsys{\sL_P}=\linsys{\sL-P}$ are {\bi}free. Thus $\Ebar$ is Cartier
outside $\Supp\bigl((\sL-\sL_P)_Y\bigr)=L\cup L_P$. Note also that
$\sL_P\sim\sL-\Ebar\not\ge0$ if\/ $P\notin L$. However, $\sL_P$ is not
Cartier at $P$ outside $\Supp\bigl((\sL-\Ebar)_Y\bigr)=L$.
 \end{exa}

 \begin{prop} \label{rest} Let $X$ be a model where $E$ is a (normal)
divisorial sub\-variety, and let $\sD,\sD_1,\sD_2$ be $K$-Cartier
{\bi}divisors that are $K$-Cartier outside $\Supp\bigl((\sD_*)_X\bigr)$.
Then there are just two possible cases\footnote{I can't figure out if this
means $\Supp\Bigl((\sD\frest E)_E\Bigr)$ or
$\Bigl(\Supp(\sD\frest E)\Bigr)_E$. All of this would be easier to read
if we put appropriate sized brackets $\Bigl(\bigl(()\bigr)\Bigr)$ around
the arguments, although that is time-consuming.} for
$\Supp{(\sD\frest E)_E}$:
 \begin{q}
 \item[\rm(mv)] $\Supp{(\sD\frest E)_E}\sim\emptyset$ if\/ $\sD$ is either
Cartier or {\bi}free over $X$; in other words, a Cartier {\bi}divisor on
$X$ restricts to a Cartier {\bi}divisor on $E$;
 \item[\rm(fx)] $\Supp{(\sD\frest E)_E}=\bigcap\Supp{\sD_X}$ if\/
$\mult_E{\sD}=0$.
 \end{q}

Suppose in addition that $E$ is $\Q$-Cartier. Write $\sD_*$ for any of
$\sD$ or $\sD_i$, and set $\sD'_*\sim\sD_*-(\mult_E{\sD_*})\Ebar$, so
that $\mult_E{\sD'_*}=0$. Suppose that each
$\sD_*-(\mult_E{\sD_*})\Ebar$ is $K$-Cartier outside $\Supp{(\sD'_*)_X}$.
Then, modulo $\sim$, there are just two possible cases for
$\Supp(\sD\frest E)_E$:
 \begin{q}
 \item[\rm(mx)] $\Supp{(\sD\frest E)_E}$ equals the union of
 \[
 E\cap\Supp{(\sD_X-(\mult_E{\sD})E)}= E\cap\Supp{\sD'_X}
 \]
and $E\cap\Supp{E'}$; or
 \item[\rm(df)] $\Supp{(\sD\frest E)_E}$ is contained in the union of
 \[
 E\cap\Supp((\sD_1)_X-(\mult_E{\sD_1})E)=E\cap\Supp{(\sD'_1)_X}
 \]
 and
 \[
 E\cap\Supp((\sD_2)_X-(\mult_E{\sD_2})E)=E\cap\Supp{(\sD'_2)_X}
 \]
 for $\sD:=\sD_1-\sD_2-\mult_E{(\sD_1-\sD_2)}\Ebar$.
 \end{q}
 \end{prop}

 \begin{rem} If $E$ is $\Q$-Cartier, by Example~\ref{bbasep}, the condition
that a {\bi}divisor $\sD$ is $K$-Cartier away from $\Supp{\sD}$ holds if
$\sD$ is {\bi}free or a $K$-linear combination of such {\bi}divisors in
general position on $X$.
 \end{rem}

 \begin{proof} (mv) and (fx) hold by definition (cf.\
Example~\ref{bbasep} above). (fx) implies (mx) and (df). Indeed, $\sD-
(\mult_E{\sD})\Ebar$ is nontrivial and not $K$-Cartier$/E$ only
over
 \[
 E\cap\Supp\bigl(\sD-(\mult_E{\sD})\Ebar\bigr)_X=
 E\cap\Supp\bigl(\sD_X-(\mult_E{\sD})E\bigr),
 \]
 that is, modulo $\sim$, only over $E\cap\Supp{\sD'_X}$. In (df) we take
$\sD'=\sD'_1-\sD'_2$. Note that $E'$ is $\Q$-Cartier everywhere on $X$
together with $E$.
 \qed\end{proof}

 \begin{cor} \label{resdf} Suppose that
 \begin{itemize}
 \item $K$ is a field;
 \item $a_2\sD_2$ is {\bi}free$/Z$, for some $a_2\in K$;
 \item $X/T$ is a birational contraction$/Z$, and $(\sD_1)_T=(\sD_2)_T$
on $T$.
 \end{itemize}
 Then modulo $\sim$, we can replace the union in {\rm(df)\/} by the union
 \[
 \Bigl(E\cap\Supp(\sD_1)_X\Bigr)\cup
 \Bigl(E\cap\Bs{\linsys{a_2\sD_2}_X}\Bigr)\cup
 \biggl(E\cap\left(
 \begin{smallmatrix}\hbox{other exceptional} \\
 \hbox{divisors of $X/T$}\end{smallmatrix}
 \right)\biggr).
 \]
In particular, for $a_1=1$, the\/ {\em non-Cartier points\/} on $E$ and
the\/ {\em fractional part\/} of $(\sD\frest E)_E$ are all contained in
$E\cap \Bs\linsys{\sD_1}_X$, $E\cap \Bs\linsys{a_2\sD_2}_X$, and $E\cap{}$
other exceptional divisors of $X/T$.

 \begin{add} In the local case over $P\in Z$ (which is more general),
modify the above assumptions to
 \begin{itemize}
 \item $K$ is a field;
 \item $a_1\sD_1$ is sufficiently general effective {\bi}free$/Z$
for some $a_1\in K$;
 \item $a_2\sD_2$ is {\bi}free$/Z$, for some $a_2\in K$;
 \item $X/T$ is a birational contraction$/Z$, and $(\sD_1)_T=(\sD_2)_T$
outside $f\1P$.
 \end{itemize}
 Then modulo $\sim$, the union in {\rm(df)\/} is contained in
 \[
 \Bigl(E\cap\Supp(\sD_1)_X\Bigr)\cup
 \Bigl(E\cap\Bs{\linsys{a_2\sD_2}_X}\Bigr)\cup
 \biggl(E\cap(\hbox{divisors of $X/P$})\biggr)
 \]
 \end{add}

In particular, for $a_1=1$, the\/ {\em non-Cartier points\/} on $E$ and
the\/ {\em fractional part\/} of $(\sD\frest E)_E$ are all contained in
$E\cap \Bs\linsys{\sD_1}_X$, $E\cap \Bs\linsys{a_2\sD_2}_X$, and $E\cap{}$
divisors of $X/P$).
 \end{cor}

 \begin{proof} $\sD_1-\sD_2$ is exceptional$/T$ (and respectively, in
divisors$/P$). The same holds over $X$ up to exceptional divisors of
$X/T$. More precisely, by (df) (with $\sD'_1=\sD_1$, when $a_1\sD_1$ is
sufficiently general effective {\bi}free) and because
$(\sD_2)_X=(\sD_1)_X$ up to exceptional divisors (and respectively,
divisors$/P$), $\sD\frest E$ is nontrivial modulo $\sim$ only over
$E\cap{}$ the exceptional divisors ($\ne E$) of $X/T$ (or respectively
$E\cap{}$ divisors$/P$), $E\cap\Supp{(\sD_1)_X}$ or
$E\cap\Supp{(\sD^g_2)_X}$, where $\sD^g_2\sim a_2\sD_2$ is sufficiently
general effective. The case with $\mult_E{\sD_1}\ne0$ is trivial:
$E\subset\Supp{(\sD_1)_X}$.

Indeed, $\sD_X=0$ near any point $Q\in E$ outside $E\cap{}$ exceptional
divisors ($\ne E$) of $X/T$ (and respectively in $E\cap{}$ divisors of
$X/P$) and $E\cap\Supp{(\sD_1)_X}$. Hence $(\sD_2)_X=e E$ near $Q$ with
$e=\mult_E{\sD_2}$. Since $\sD_2$ is {\bi}free$/X$ (near $Q$, in
particular), then $\sD\frest E$ is nontrivial over $Q$, only when $\sD\ne
e\Ebar$ near $Q$ and therefore, by Example~\ref{bbasep} above, only
when all $(D^g_2)_X$ pass through $Q$.

Taking the intersection of $E\cap\Supp{(\sD^g_2)_X}$ for all effective
$\sD^g_2\sim a_2\sD_2$ we obtain that $Q\in E\cap \Bs{(a_2\sD_2)_X}$,
whenever $\sD\frest E$ is nontrivial$/Q$.
 \qed\end{proof}

\section{Approximations} \label{apprx}
 \begin{defn}\label{bnds} Let $F\subset X$ be a reduced
divisor.\footnote{This material is very similar to that of Section~5,
especially Lemma~5.15 and Corollary~5.19. Maybe it would be better
to organize them together.} A {\em set of (effectively) bounded
semiample$/Z$ divisors\/} (or an {\em bnd set\/}$/Z$) is defined to be
a set $\fN\bnd(F)=\fN\bnd(X/Z,F)$ of $\R$-divisors of $X$ that are
semiample$/Z$ and satisfy the following conditions:
 \begin{q}
 \item[(BFP)] the {\em fractional part\/} of $D$ is {\em bounded\/} by $F$:
$D=D\inte+D\fr$, where $D\inte$ is integral, and
$\Supp{D\fr}\subset F$;
 \item[(CFG)] $\fN\bnd(F)$ has a set of generators that is {\em compact
plus finite\/}: there is a compact rational polyhedron $\fN_c$ and a
(finite) set of integral divisors $D_i$ in $\fN\bnd(F)$ such that each
$D\in\fN\bnd(F)$ has a decomposition $D=D_c+\sum n_iD_i$, or even {\em
the same up to\/} $\sim$, where $D_c\in\fN_c$ and every $n_i\in\N$; and
 \item[(BND)] the same as condition (BND) in Definition~\ref{bnd} for
$\fN\bnd(F)$, or equivalently, the same holds for $\fN_c$.
 \end{q}
 Taking a bigger $F$ we can assume that the polyhedron of (CFG) is
contained in the $\R$-space $\fD_F$ (for the notation, see
Section~\ref{satdes}, in proof of Corollary~\ref{assdesc}). Note that
(CFG) implies (BND) when $\{D_i\}$ is finite.

The {\em uniform neighbourhood\/} of $\fN\bnd(F)$ with {\em diameter\/}
$\de\in\R$ is the set of Weil $\R$-divisors
 \begin{align*}
 \fU\bndde(X/Z,F)&=\fU\bndde(F) \\
 &=\bigl\{D=\sum d_iP_i\bigm| d_i\in\R \hbox{ and } \Vert D-\fN\bnd(F)
\Vert <\de\bigr\},
 \end{align*}
 assuming that $\Vert\ \Vert$ is taken over $\Supp{(D-\fN\bnd(F))}\subset
F$. In other words, the following two equivalent conditions holds:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item $D=D\inte+D\fr$ such that $D\fr$ is supported in $F$ and there
exists $D'=D\inte+D'\fr\in\fN\bnd(F)$ with $\Vert D-D'\Vert=\Vert
D\fr-D'\fr\Vert <\de$;
 \item there exists $D'\in\fN\bnd(F)$ such that $D-D'$ is supported in
$F$ and $\Vert D-D'\Vert<\de$.
 \end{enumerate}
 In particular, in either case $\Vert D-\fN\bnd(F)\Vert \le\Vert
D-D'\Vert<\de$. (i) implies (ii), because $D-D'=D\fr-D'\fr$ is supported
in $F$. (ii) implies (i) for $D\fr=D-D'+D'\fr$, because then
$D=D\fr+D'-D'\fr=D\fr+D\inte$ and $D-D'=D\fr-D'\fr$.
 \end{defn}

 \begin{exa}\label{bndc}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item Any rational polyhedral cone generated by a finite set of
semiample$/Z$ divisors $D_i$ is a {\em bnd cone\/}$/Z$. We can assume
that each $D_i$ is integral. Then we can take
 \begin{itemize}
 \item $F$ as the common support of all divisors $D_i$; and
 \item $\fN_c=\bigl\{\sum r_iD_i\bigm| r_i\in[0,1]\bigr\}$.
 \end{itemize}

 \item Let $X\to T/Z$ be a contraction. The most important example of a
bnd set is the Abelian semigroup $\fN(F)=\fN(X/T/Z,F)$ of divisors $D$
that are nef$/Z$ and $\sim_\R0/T$, and have fractional part bounded by
$F$. (In particular, $D$ is assumed to be an $\R$-Cartier divisor.) In
this case, a {\em uniform\/} neighbourhood $\fU_\de(F)=\fU_\de(X/T/Z,F)$
of $\fN(F)$ with {\em diameter\/} $\de$ is also defined as in
Definition~\ref{bnds}.

In fact, $\fN(F)$ is a bnd set, and is even semiample in several cases
which, although very special, are crucial for our purposes:
 \begin{q}
 \item[\WLF] $(X/Z,B)$ is a weak log Fano contraction for some boundary
$B$ and $T=X$;
 \item[(0LP)] $(X/Z,B)$ is a $0$-log pair for some boundary $B$ (cf.\
Remark~\ref{gzardec}, (2)), and $T=Z$; and
 \item[(TRP)] triples $(X/T/Z,B,\fF)$ with $F\in\fF$.
 \end{q}
 In (0LP), the bound (BND) does not hold in general: torsion points of
Abelian varieties contradict (BND). To satisfy (CGF) and (BND), we need
to impose irregularity $=0$, so that the Picard group is finitely
generated. Thus integral points of $\fF(X/Z/Z,F)$ are finitely generated
modulo $\sim$, and $\fN_c$ is a cube with bounded multiplicities as in
(1) above. In particular, this is true if $X/Z$ is a weak log Fano
contraction. Moreover, the bnd property for \WLF\ follows from (TRP) in
the case of \w\ triples (see Lemma~\ref{split}, (5) below).
 \end{enumerate}
 \end{exa}

 \begin{defn}[Freedom with tolerance $\tau>0$] \label{abf}
 \addcontentsline{toc}{subsection}{\ref{abf} Definition of freedom with
tolerance $\tau$}
 We fix a reduced divisor $F$; then $D\sim D\bof+D\fr$, where
$\Bs{\linsys{D\bof}}=\emptyset$. In particular, $D\bof$ is Cartier, and
$D\fr$ is supported in $F$ with $\Vert D\fr\Vert<\tau$ (cf.\
Lemma~\ref{movest}). We denote this by $\Bs{\linsys{D}}=\emptyset
\mod{\tau}$, and pronounce the linear system to be free {\em with
tolerance\/} $\tau$.

{\em Nonvanishing with tolerance\/} $\tau>0$ is similar: $D\sim
D\inte+D\fr$, where $D\inte$ is effective integral and $D\fr$ is supported
in $F$ with $\Vert D\fr\Vert<\tau$. We denote this by
$\linsys{D}\ne\emptyset \mod{\tau}$ and pronounce the linear system
to be {\em nonempty with tolerance $\tau$\/}.
 \end{defn}

The main result of the section is

 \begin{thm}\label{bndbf}
 \addcontentsline{toc}{subsection}{\ref{abf} Main theorem on freedom with
tolerance}
Suppose given:
 \begin{itemize}
 \item a bounded family $(X_u/Z_u,F_u)$ for $u\in U$ of pairs with reduced
divisor $F_u$ on $X_u$; 
 \item a {\em family\/} $\fN_{nbf}(X_u/Z_u,F_u)$ of bnd sets$/U$ of
divisors {\em (see (7) and (7$'$) in the proof below);\/} and
 \item a tolerance $\tau>0$.
 \end{itemize}
 Then there exists a natural number $M$ (depending on the family
$(X_u/Z_u, F_u)$ and $\tau$) such that, for any $u\in U$ and any $D\in
\fN\bnd(X_u/Z_u,F_u)$, $\Bs{|mD|}=\emptyset\mod{\tau}$, in particular,
$|mD|\ne\emptyset\mod{\tau}$, for some $1\le m\le M$ (depending on
$D$).
 \end{thm}

 \begin{add} Moreover, there is a real number $\de>0$ (depending on the
family $(X_u/Z_u,F)$ and $\tau$), such that the same freedom
and nonvanishing hold for any $u\in U$ and any
$D\in\fU\bndde(X_u/Z_u,F_u)$.
 \end{add}

 \begin{rem} Both choices of $M$ and $\de$ are very ineffective, as in
Kronecker's theorem.
 \end{rem}

We start with properties of freedom with tolerance.

 \begin{prop} \label{pnonvan}
 \addcontentsline{toc}{subsection}{\ref{pnonvan} Proposition: Properties
of freedom with tolerance}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item Freedom with tolerance $\tau$ is invariant under linear
equivalence: if $D\sim D'$, then $\Bs{\linsys{D}}=\emptyset\mod{\tau}$
implies $\Bs{|D'|}=\emptyset\mod{\tau}$.

 \item Freedom with tolerance is an {\em open property\/}: if
$\Bs{\linsys{D}}=\emptyset\mod{\tau}$, then there is an $\ep>0$ such
that, for any $D'$ with $D-D'$ supported in $F$ and $\Vert D-D'\Vert<\ep$,
also $\Bs{|D'|}=\emptyset\mod{\tau}$.

 \item If $D=D\bof+D'$, where $\Bs{\linsys{D\bof}}=\emptyset$ and
$\Bs{|D'|}=\emptyset\mod{\tau}$, then $\Bs{\linsys{D}}=\emptyset\mod{\tau}$.
 \item[] The same holds for nonvanishing with tolerance $\tau$.
 \end{enumerate}
 \end{prop}

 \begin{proof} (1) holds by definition. In (2), we can take any
$0<\ep\le\tau-\Vert D\fr\Vert$, where $D\sim D\bof+D\fr$ as in
Definition~\ref{abf} and $D'\sim D\bof+D\fr'$. (3) holds because the sum
of two free linear systems is also free.
 \qed\end{proof}

 \begin{pfof}{Theorem~\ref{bndbf}} Let $(X/Z,F)$ be a point of the family,
possibly nonclosed (= prime subfamily). Let $F=\sum P_i$ and
$\fD_F=\bigl\{\sum d_iP_i\bigm| d_i\in\R\bigr\}$ be the set of divisors
generated over $F$.

For $\fN\bnd(F)$, suppose that we have a compact subspace
$\fN_c\subset\fN\nef(F)$, a (finite) set of Cartier divisors
$T_i\in\fN\bnd(F)$ and a finite set of $\Q$-Cartier divisors $N_i\in\fN_c$
with open neighbourhoods $U_i$ of $N_i\in\fD_F$, such that
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item each $T_i$ is free$/Z$;
 \item each $N_i$ has a {\em freedom index\/} $n_i>0$ for which
$\Bs{\linsys{n_iN_i}}=\emptyset$;
 \item in addition, for each $D\in U_i$, $\Bs{\linsys{n_iD}}=
\emptyset\mod{\tau}$; more precisely, $n_iD=n_iN_i+D\fr$ with $D\fr$
supported in $F$ with $\Vert D\fr\Vert <\tau$;
 \item $\fN_c$ is a compact rational polyhedron with generators $N_i$;
 \item each $D\in\fN\bnd(F)$ decomposes as $D\sim D_c+\sum t_iT_i$ with
natural numbers $t_i$ and $D_c\in\fN_c$; and in turn
 \item $\fN_c\subset\bigcup U_i$; and
 \item each $T_i$ is defined and is free$/Z_u$ for each specialization
of $(X/Z,F)$ and for any general point of $(X/Z,F)$. That is, for every
$u$ in some nonempty Zariski open subset $U$, $\fN\bnd(F)$ specializes
isomorphically to $\fN\bnd(F_u)$ preserving the structures $T_i,N_i$
and $\fN_c$ in (1--4) above.
 \end{enumerate}
 Then Theorem~\ref{bndbf} holds for $\fN\bnd(F)$, and there exists $\de>0$
such that Addendum~\ref{bndbf}.1 holds for $\fU\bndde(F)$. Moreover, the
theorem and addendum both hold for every $u$ in a nonempty $V\subset U$.
In particular, in the proof of the theorem and Addendum~\ref{bndbf}.1, we
can use Noetherian induction; that is, it is enough to verify both on a
single element $(X/Z,F)$ of the family. Thus if there exist a natural number
$M_V=M_F$ and real number $\de_V=\de_F$ giving the theorem and addendum
at $(X/Z,F)$, and numbers $M_{(X/Z,F)\setminus V}$ and
$\de_{(X/Z,F)\setminus V}$ that give the same over the closed proper
subfamily $(X/Z,F)\setminus V$, then both hold over all points
(= specializations) of $(X/Z,F)$ with
$M=\max\bigl\{M_V,M_{(X/Z,F)\setminus V}\bigr\}$ and
$\de=\min\bigl\{\de_V,\de_{(X/Z,F)\setminus V}\bigr\}$.

Now (1--6) holds over some $V$. (1) holds by our assumption in (7). (2)
and (5) define a nontrivial affine open subset $V$ over which all $n_iN_i$
are free, and restriction (= specialization) of $\sim$ on $X_u/Z_u$ is
correct (= in general position; cf.\ (GPN) of Proposition~\ref{ressat}).
To have good specialization of $\sim$, it is enough to consider $X_u/Z_u$
that are smooth at the generic point of $X_u$ (= multiplicity $1$), or
just on the moduli space by Lemma~\ref{restdiv}. Thus (2) and (5) hold in
each $u\in V$. The inequality in (3), the generators (4), and the
inclusion in (6) concern multiplicities of divisors $P_i$ and are uniform
over the connected component of $F$ in $U$, because we assume that $F_U$
is reduced everywhere over $U$ by (7$'$) below. This establishes the
Noetherian induction by (7).

Now derive the theorem and the addendum for a fixed $(X/Z,F)$ from (1--6),
and then verify these together with (7). We take $M_F=\max\{n_i\}$ for the
indexes in (2); $\de_F$ is chosen later.

Indeed, Proposition~\ref{pnonvan}, (1) and (3) and the above (1) and (5)
imply that it is enough to verify the theorem and the addendum
respectively for $\fN_c$ and for
 \[
 \fU_{c,\de}=\bigl\{ D\bigm| D-D'\in\fD_F,\ D'\in\fN_c \hbox{ and }
 \Vert D-\fN_c\Vert\le\Vert D-D'\Vert<\de\bigr\}.
 \]
 In turn both the theorem and the addendum follow from (2--3) and (6). For
the addendum we need to choose such $\de=\de_F$ that $\fU_{c,\de_F}\subset
\bigcup U_i$. Such $\de_F$ exists since $\bigcup U_i$ is open.

Finally, we verify (1--7). Note for this that the boundedness of the
family means
 \begin{description}
 \item{(7$'$)} on the total space of the family, there are a reduced
divisor $F$, integral semiample divisors $D_i$, and semiample
$\Q$-divisors $V_i$ such that, for each $u$,
 \begin{itemize}
 \item $F_u=F\rest{X_u}$ is reduced;
 \item $V_i\rest{X_u}$ are vertices of
$\fN_c(F_u)$ (= a simplicial structure with the vertexes);
 \item $D_i\rest{X_u}$ and $\fN_c(F_u)$ generate $\fN\bnd(F_u)$ as in
(CFG); and, finally,
 \item (BND) holds {\em uniformly\/} for the family.
 \end{itemize}
 \end{description}
 (BND) means that there is natural number $I>0$ such that, for any
semiample integral divisor $D$, $\Bs|ID|=\emptyset$. To satisfy (1) and
(7), we take $T_i=I D_i$. We set $\fN_c=I\fN_c(F)$, and $N_i=I V_i\rest
X$. This implies (2) with some $n_i$ for each $N_i$, also by (BND).
More precisely, $n_i\le Id_i$, where $d_i$ is the minimal natural number
such that $d_iN_i$ is integral. Thus (BFP) and
Proposition~\ref{pnonvan}, (2--3) implies (3) for some $U_i$. By definition
we can take $U_i$ as an open disc with the center $N_i$ in a rational
translate of $\fD_F$ and of the radius $r_i=\tau/n_i\ge \tau/I d_i$.

(4) and (7), with $U$ as a normal points in the subfamily given by point
$(X/Z,F)$, holds by (7$'$).

(5) holds by (CFG) for $(X/Z,F)$ in (7$'$).

To satisfy (6) we need to add $\Q$-divisors $N_i$ from $\fN_c$ that does
not affect (4). Since each such $N_i$ is semiample, as above we can find
$n_i$ and $U_i$ for (2--3). If it is an open covering then by the
compactness of $\fN_c$ we can take a finite subset of $N_i$ that satisfies
(6) and still (2--4). Actually, the discs form a covering. To verify this
we can assume that $\fN_c$ is a simplex, and all its faces are covered by
an induction.

Indeed, the $\Q$-points of the rational simplex $\fN_c$ are $\Q$-Cartier
$\Q$-divisors, so these are covered.

On the other hand, by simultaneous approximation, Cassels \cite[Chap.~1,
Theorem~VII]{cas57}, for any $\ep>0$, each point of the affine span $\fL$
of $\fN_c$, in particular, each point $D\in\fN_c$, has a rational
approximation $N_i$ in $\fL$ such that $\Vert D-N_i\Vert< \ep/d_i$. Hence
$N_i\in\fN_c$ and $D\in U_i$ respectively whenever $\ep/d_i\le\ep<$ the
distance from $D\in\fN_c$ to the complement $\fL\setminus\fN_c$ and
$\ep<\tau/I$. Thus the internal points of $\fN_c$ are covered by $U_i$.
This completes the induction and the proof of the theorem.
 \qed\end{pfof}

Freedom with tolerance sometimes implies the usual nonvanishing, but
typically this is still far from freedom.

 \begin{cor}\label{bndnv} Let $C_u$ be a divisor on $X_u$ with
$\mult_{P_i}{C_u}>-1+\tau$ for all prime divisors $P_i$ of $X$. Then,
for any $D\in\fU\bndde(F_u)$, there exists some natural number
$1\le m\le M$ such that $\linsys{\rdup{mD+C_u}}\ne\emptyset$.
 \end{cor}

 \begin{add} If $\mult_{P_i}{F_u}=0$ we can assume just
$\mult_{P_i}{C_u}>-1$.
 \end{add}

 \begin{proof} In the proof below ``all $\dots$'' means for all prime
divisors $P_i$ of $X$. We drop the subscripts $_u$.

Take $m$ in Addendum~\ref{bndbf}.1 such that $|mD|\ne\emptyset\mod{\tau}$,
that is, $mD\sim D\inte+D\fr$, where $D\fr=\sum d\fr{}_{,i}P_i$ with all
$\linsys{d\fr{}_{,i}}<\tau$, in particular $d\fr{}_{,i}>-\tau$. Hence
 \[
 \mult_{P_i}{(D\fr+C)}=d\fr{}_{,i}+\mult_{P_i}{C}>\mult_{P_i}{C}-\tau>-1
 \]
 and all $\mult_{P_i}{\rdup{D\fr+C}}=\rdup{\mult_{P_i}{(D\fr+C)}}\ge0$.
Thus $\rdup{D\fr+C}\ge0$ and $\rdup{D\inte+D\fr+C}=
D\inte+\rdup{D\fr+C}\ge0$. This proves the required nonvanishing:
$\rdup{m D+C}\sim \rdup{D\inte+D\fr+C}\ge0$ by Lemma~\ref{intpart}, since
$mD+C\sim D\inte+D\fr+C$.

Since $d\fr{}_{,i}=0$ when $\mult_{P_i}{F}=0$, we get
Addendum~\ref{bndnv}.1, because then again
$\mult_{P_i}{(D\fr+C)}=d\fr{}_{,i}+\mult_{D_i}{C}>0-1>-1$.
 \qed\end{proof}

We can use Corollary~\ref{bndnv} as a restriction on $\tau$ for which the
former\footnote{I don't know what ``the former'' refers to.} holds.

 \begin{exa}
 \begin{enumerate}
 \item Any $\tau<1$, when $C\ge0$.
 \item Let $C=\sA_E$, the discrepancy divisor. Then all
 \[
 \tau<\min\{1-b_i\}\le\lld(X,B).
 \]
 In particular, there exists $\tau>0$ provided that all $b_i<1$, that
is, $(X,B)$ is Klt in divisors.
 \end{enumerate}
 \end{exa}

 \begin{cor}\label{nonvc} Suppose that each $F_u$ is $\Q$-Cartier on
$X_u$. Let $\al<1$ and $\tau$ be positive real numbers, and $\sC_u$ an
$\R$-{\bi}divisor of $X_u$ such that $\sC_u\ge0/X_u$ and
 \begin{equation}
 \mult_{P_i}{\al\sC_u}>-\al+\tau \mult_{P_i}{F_u}
 \quad\hbox{for all $P_i$.} \tag{$*$}
 \end{equation}
 Then there exists a natural number $M$ and positive real numbers
$\be,\de$ giving the following nonvanishing$/Z_u$ on any $Y_u/X_u$
(uniformly in $u$). For any $u$, and for all {\bi}divisors $\sD$ such that
the descent data $\sE$ of $\sD$ over $X_u$ is bounded by $\be\sC_u$, that
is, $\sE\le\be\sC_u/X_u$, and $\sD_X\in\fU\bndde(F_u)$, we have
 \[
 \linsys{\rdup{m\sD_Y+(\sC_u)_Y}}\ne\emptyset \quad \hbox{for some
 $1\le m\le M$.}
 \]
 \end{cor}

 \begin{proof} Again we drop the subscripts $_u$. By
Addendum~\ref{bndbf}.1, we can choose a natural number $M$ and $\de>0$,
depending on the family $(X/Z,F)$ and $\tau$. Now take any
 \[
 \be\in\R \quad \hbox{with} \quad
 0\le\be\le\frac{1-\al}M\,.
 \]
 Take now any {\bi}divisor $\sD$ satisfying the nonvanishing conditions.
Let $m$ be a natural number in $[1,M]$ for which
$\Bs{\linsys{m\sD_X}}=\emptyset\mod{\tau}$. We verify that
$\linsys{\rdup{m\sD_Y+\sC_Y}}\ne\emptyset$ for arbitrary $Y/X$; this
follows from Lemma~\ref{movest} with $\sD:=m\sD$. Assumption $(**)$ of
the lemma follows from the boundedness of the descent data of the
original $\sD$. Since $\sC\ge0/X$ and by \HOM\ of
Proposition~\ref{desd}, for any natural number $m\le M$ (or even
for any positive real number), the descent data $\sE$ of $\sD:=m\sD$ is
bounded by $m\be\sC\le M\be\sC\le(1-\al)\sC/X$, that is,
 \begin{equation}
 \sE\le(1-\al)\sC/X. \qed \tag{$**$}
 \end{equation}
 \end{proof}

 \begin{rem} \label{constnv} For a given family $(X_u/Z_u,F_u)$ with
$\sC_u$, we find constants $\al,\be,\de,\tau$ and $M$ in the following
order. First, we take any positive $\al<1$. Next, we choose an
appropriate positive $\tau$ (cf.\ Example~\ref{fnonvex} below). Next, by
Theorem~\ref{bndbf} and Addendum~\ref{bndbf}.1 we can find $\de$ and $M$.
Finally, we take $\be$ as in the preceding proof. Note that to find
$\tau$ we need to assume that each $F_u$ is $\Q$-Cartier. For this, we
usually assume that $X$ is $\Q$-factorial. In applications we only need
$\be,\de>0$ and a natural number $M$, and so can drop $\al,\tau$ (cf.\
Corollary~\ref{fnonvcon} below).
 \end{rem}

 \begin{rem} \label{edesd} If $\sD_X$ is $\R$-Cartier, the descent
data $\sE$ exists by \EXI\ in Proposition~\ref{desd}; hence it exists
for all $\sD$ if $X$ is $\Q$-factorial.
 \end{rem}

Now we apply the results obtained above to triples $(X/T/Z,B,\fF)$ (see
Definition~\ref{tri}, and cf.\ Corollary~\ref{mainb}). A {\em bounded\/}
family of triples is a family with bounded moduli; for the details, see
the proof of Theorem~\ref{bnonvan} below.

The next preliminary result explains also the role of the conditions
(CRP) and (RPC) in Definition~\ref{tri} (cf.\ Example~\ref{noncrm} below).
We first fix notation: Let
 \[
 \CDiv^0_\R(X/T/Z)=
 \bigl\{D\in \CDiv_\R(X/Z)\bigm| D\sim_\R0/T\bigr\}.
 \]
 Equivalently, every such divisor $D\sim_\R g^*M/Z$ for some $\R$-Cartier
divisor $M\in \CDiv_\R(T/Z)$. Moreover, $M$ is defined up to $\sim_\R/Z$
(cf.\ Lemma~\ref{lin0l}), and $\R$-linear equivalence $\sim_\R/Z$ and
numerical equivalence $\equiv/Z$ are the same on $T/Z$ (cf.\
Lemma~\ref{split}, (1) below). Thus we have an $\R$-linear projection
 \[
 g_*\colon \CDiv^0_\R(X/T/Z) \to\CDiv_\R(T/Z)/\equiv.
 \]
 This map is defined over $\Q$, but not over $\Z$ in general, since there
may be multiple fibres. Thus on each $\R$-linear subspace
$L\subset
\CDiv^0_\R(X/T/Z)$ we have an {\em induced\/} splitting
 \[
 L=L^0\oplus L^1,
 \]
 where $L^0=\ker g_*\rest L$ and $L^1$ is isomorphic to $g_*L$. Note that
$L^0$ is a subspace of $\ker g_*=\CDiv^0_\R(X/Z/Z)$. Note also that $L^1$
is not defined uniquely. Nonetheless, we use it to identify the second
summand, and denote $L^1$ by $g_*L$. If $L$ is defined$/\Q$, the splitting
can also be defined over $\Q$. For example, if
 \[
 L=\fC=\fC(X/T/Z,F)=\bigl\{D=\sum d_iP_i\bigm| d_i\in\R
\hbox{ and }D\sim_\R0/T\bigr\}\subset\fD_F
 \]
with $F=\sum P_i$, we obtain a splitting
 \begin{gather*}
 \fC=\fC^0\oplus g_*\fC, \quad\hbox{where} \\
 \fC^0=\fC^0(X/T/Z,F)=\bigl\{D=\sum d_iP_i\bigm| d_i\in\R
 \hbox{ and } D\sim_\R0/Z\bigr\}.
 \end{gather*}
 \begin{lem} \label{split} Let $(X/T/Z,B,\fF)$ be a \w\ triple for a
weak log Fano contraction and $F=\sum P_i\in\fF$ a reduced divisor. Then
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $g_*$ induces a splitting$/\Q$
 \begin{gather*}
 \fC=\fC^0\oplus g_*\fC, \quad\hbox{where now} \\
 \fC^0=\fC^0(X/T/Z,F)=\bigl\{D=\sum d_iP_i\bigm| d_i\in\R
 \hbox{ and } D\equiv 0/Z\bigr\}.
 \end{gather*}
 \item Cone theorem: the nef or even {\em semiample cone}
 \begin{align*}
 \fN\nef &=\fN\nef(X/T/Z,F) \\
 &=\bigl\{D=\sum d_iP_i\bigm| d_i\in\R, D\hbox{ is nef and }
\sim_\R0/T
 \bigr\}
 \end{align*}
 is a rational polyhedral cone in $\fC$.
 \item $g_*\fC$ is isomorphic$/\Q$ to $\CDiv_\R(T/Z)/\equiv$, whenever
the components $P_i$ generate the group of Weil divisors of $X$ $/\sim$.
 \item There exists a uniform constant, the {\em freedom index\/} $I$,
which is a natural number such that for each nef integral $\Q$-Cartier
$D\in\CDiv^0_\R(X/T/Z)$, the multiple $ID$ is free on $X/Z$.
 \item $\fN(X/T/Z,F)$ is a bnd set {\em (cf.\ (TRP) in
Example~\ref{bndc})\/}.
 \end{enumerate}
 \end{lem}

 \begin{proof} (1) Since the triple is a \w\ triple for a weak log
Fano contraction $(X'/Z,B')$, we have a diagram of contractions
 \[
 \renewcommand{\arraycolsep}{0.2em}
 \begin{matrix}
 &&W \\
 &^i\swarrow &&\searrow^h \\
 X'\kern-0.6em &&&& \kern-0.6emX&\xrightarrow{\,g\,} T/Z
 \end{matrix}
 \]
 where $i\circ h\1\colon X\to X'$ is a birational map to a weak log
Fano contraction $X'$. We need to prove that each $\R$-Cartier divisor
$D\equiv 0$ on $X/Z$ is $\sim_\R/Z$. It is enough to verify this for
$h^*D$ on $W/Z$. Since $X'$ has rational singularities, the problem
descends to $X'/Z$ up to $\sim_\R$. However each $\R$-Cartier divisor
$\equiv 0/Z$ is $\sim_\R0/Z$ on any weak log Fano contraction$/Z$.

In addition, if $D$ is a Cartier divisor and $D\equiv 0$, then $D\sim 0$
essentially by the Contraction Theorem.

 \begin{rem}\label{rem!rats} Apart from the final statement, the above
holds if we assume only that $X'$ has rational singularities and
irregularity $0/Z$.
 \end{rem}

(2) follows from the numerical description of the splitting in (1):
 \[
 \fN\nef=\fC^0\oplus g_*\fN\nef.
 \]
 By (1) and (RPC), this means that each nef$/Z$ divisor in
$\CDiv^0_\R(X/T/Z)$ is also semiample$/Z$, that is, the cone is also
semiample$/\sim_\R$. Since
 \[
 g_*\fN\nef=g_*\fC\cap{} \hbox{the nef cone in $\CDiv_\R(T/Z)/\equiv$,}
 \]
 the nef cone $g_*\fN\nef$ is rational polyhedral by (RPC) again. Hence
$\fN\nef$ is also rational polyhedral.

(3) follows from the surjectivity of the projection $g_*$, since the
integral Cartier divisors are also generated by the components $D_i$.

To prove (4), as in (3), we can assume that an integral Weil divisor
$D/Z$ that is nef and $\Q$-Cartier is contained in $\fC$ up to $\sim$.
By (2) we can decompose $D$ into $D_0+D_1$, where $D_0\in\fC^0$ and
$D_1\in g_*\fN\nef$. This is a decomposition$/\Q$ but with bounded
denominators, that we include in $I$; thus we can assume that both $D_0$
and $D_1$ are integral. Moreover, (4) for $D_1$ follows from a similar
decomposition by the cone property (RPC). Now $D_0$ is free provided it
is Cartier, as we remarked in the proof of (1). Hence we can take $I$ as
Cartier index for $D_0$: $ID_0$ is Cartier. This exists because $\fC^0$
is finite dimensional.

(5) Each divisor in $\fN(X/T/Z,F)$ is semiample by (2). (BND) follows from
(4). If $F$ is big as in (3), the (CFG) follows from (2) and (3) as in
Example~\ref{bndc}, (1). For any subset of $F$ we need to cut $\fN_c$ by
equations $\mult_{P_i}{D}=i$ for $i\in \Z$, to drop $P_i$,
and\footnote{Is this right? Maybe $\mult_{P_i}{D}=d_i$ for $i\in\Z$ for
all $i$, then to omit some of the $P_i$ implicit in the argument?} keep
integral generators $D_i$. In each compact rational polyhedron $\fN_c$,
this also cuts out a rational compact polyhedron.
 \qed\end{proof}

 \begin{nota} For a triple $(X/Z,B,\fF)$ with a bounded family $\fF$,
we set
 \[
 \fN=\fN(X/T/Z,\fF)=\bigcup_{F\in\fF}\fN(F)
 \]
 (cf.\ Example~\ref{bndc}, (2)), and $\fN=\fN(X/Z,\fF)=\fN(X/X/Z,\fF)$
if $T=X$ (cf.\ Theorem~\ref{fnonvan}). In other words, each $D\in\fN$ is
nef$/Z$, $\sim_\R 0/T$, and $D=D\inte+D\fr$, where $D\inte$ is integral,
and $\Supp{D\fr}$ is in $\fF$. We set
 \[
 \fU_\de=\fU_\de(X/T/Z,\fF)=\bigcup_{F\in\fF}\fU_\de(F),
 \]
 and $\fU_\de=\fU_\de(X/Z,\fF)=\fU_\de(X/X/Z,\fF)$.
 \end{nota}

 \begin{thm} \label{bnonvan}
 \addcontentsline{toc}{subsection}{\ref{bnonvan} Theorem on freedom with
tolerance in families}
 Let $(X_u/T_u/Z_u,B_u,\fF_u)$ be a bounded family of triples that are
\w\ for some weak log Fano contractions {\em (see the proof);\/} and
let $\tau>0$ be a tolerance.

 Then there exists a natural number $M$ (depending on the family and
$\tau$) such that, for any $(X_u/T_u,B_u,\fF_u)$ in the family and for
each divisor $D\in\fN=\fN(X_u/T_u/Z_u,\fF_u)$, we have
$\Bs{|mD|}=\emptyset\mod{\tau}$ for some $1\le m\le M$ (depending on
$D$), and in particular $|mD|\ne\emptyset\mod{\tau}$.

Moreover, there exists some $\de>0$ such that the same nonvanishing and
freedom results hold uniformly in the neighbourhood
$\fU_\de(X_u/T_u/Z_u,\fF_u)$.
 \end{thm}

 \begin{proof} We reduce to the case of bnd sets in Theorem~\ref{bndbf} and
Addendum~\ref{bndbf}.1 using Lemma~\ref{split}, (5). To apply this to a
bounded family of triples we need to verify (7$'$) in the proof of
Theorem~\ref{bndbf}.

Note that a bounded family of triples $(X_u/T_u/Z_u,B_u,\fF_u)$ is
equivalent to a bounded family of simple triples $(X_u/T_u/Z_u,B_u,F_u)$
for $u\in U$, where the reduced divisor $F_u=\sum P_{u,i}$ is considered
as a $1$-element set $\fF_u=\{F_u\}$ (cf.\ spreading in
Remark~\ref{trir}, (3)). This means that we have a bounded family
$X/T/Z/U$, of projective morphisms$/U$ and horizontal divisors
$F=\sum P_i$ and $B$ on $X$, such that each triple $(X_u/T_u,B_u,F_u)$ of
our family can be given as a specialization $(X/T,B,F)$ for some
$u\in U$.

In particular, the pairs $(X_u/Z_u,F_u)$ belong to a bounded family
$(X/Z,F)$. To satisfy (7$'$) we need to transform the family: e.g., change
the base $U$ and split it into an open subset and a closed complement,
etc. By Noetherian induction, this finally gives a family satisfying (7$'$).

Each $F_u=F\rest{X_u}$ is reduced by definition. After a base change, we
can assume that each prime component $P_i$ of $F$ specializes to a prime
$P_{u,i}$. Thus the finite dimensional $\R$-vector space $\fD_F$ (generated
by $F$) is locally constant (with respect to the basis $P_i$) over $U$, that
is, specializes isomorphically to $\fD_{F,u}$ from its generic point. By
Lemma~\ref{split}, (1) with $T=Z$, the equivalence $\sim_\R0/T$ is defined
geometrically, namely, by the contraction $g\colon X\to T/Z$ or by
$\equiv 0/T$. Thus for some nonempty open subset $V$ of $U$,
$\fC=\fC(X/T/Z,F)$ is a constant$/V$ subspace of $\fD_F$ that specializes
isomorphically to $\fC_u=\fC(X_u/T_u/Z_u,F_u)$ for each $u\in V$.

Similarly by Lemma~\ref{split}, (2) we can assume that over $V$ the
rational polyhedral cone $\fN\nef=\fN\nef( X/T/Z,F)$ specializes
isomorphically to the rational polyhedral cone
$\fN\nef{}_{,u}=\fN\nef(X_u/T_u/Z_u,F_u)$ for each $u\in V$. If the former
is given by inequalities $D\cdot C_i\ge0$ for a finite set of (bounded)
curves $C_i/Z$, then the latter is given by inequalities $D\cdot
C_{u,i}\ge0$ for their specializations $C_{u,i}$. (Here it is better to
assume that $k=\C$. Since each $T_u/Z_u$ satisfies (RPC), it also satisfies
(RPC) with generating curves of bounded degree. Otherwise, the complement
to the union stratified by the cones that are generated by curves of
bounded degree gives a very generic point where (RPC) does not hold.
Indeed, each subfamily (a priori nonalgebraic) that corresponds to cones
$\fN\nef{}_{,u}$ generated by curves of bounded degree, is {\em
constructive\/} (see Remark~\ref{bndf} below.)

To find the decomposition, in (CFG) we can add some prime components
$P_i$ to $F$ such that together they generate the group of Weil divisors
of $X/\sim$ and the same after specialization. In a {\em big\/} case such
as this with extended $F$ we get (CFG) and then for original $F$ as in the
proof of Lemma~\ref{split}, (5) (cf.\ Example~\ref{bndc}, (1)).

Thus the compact rational polyhedron $\fN_c$ specializes isomorphically
to $\fN_{c,u}$ for $(X_u/Z_u,F_u)$ and $D_i$ specializes to
$D_{u,i}=D_i\rest{X_u}$. Since we have a finite set of generators $D_i$,
(CFG) implies (BND), uniformly for the family. Again over some open $V$
of $U$. Then we repeat the same for a generic point of the closed
complement $U\setminus V$, etc.
 \qed\end{proof}

 \begin{rem}\label{bndf} As explained in the proof, the boundedness of
generators for $\NEbar{(T_u/Z_u)}$ is crucial. For weak log Fano
contractions $(X_u/Z_u,B_u)$ this follows from anticanonical boundedness
\cite{sh96a}. This is not enough for nonalgebraic families: nonalgebraic
subsets in algebraic families (cf.\ Example~\ref{noncrm}). However, for
triples any subfamily that corresponds to the bounded generators $C_i$
is constructive. Indeed, we can assume that $U$ is irreducible, and the
subfamily is {\em maximal\/}, that is, is not in a proper Zariski
subset. Then we need to verify that $\fN\nef$ is given by inequalities
$D\cdot C_i\ge0$. By maximality each $D\in\fN\nef$ is nef$/Z_u$ and
semiample$/Z_u$ for very general $u\in U$. Exclude $u$ with
$D\rest{X_u}\cdot C'<0$ for some curve $C'/Z$. To prove that $\fN\nef$ is
given by the above inequalities over the generic point it is enough to
verify that each $D\in\fN\nef$ is semiample over the generic point. We can
assume that $D$ is $\Q$-Cartier and even Cartier. Then it is very
semiample (free) over the very general point $u\in U$ by the standard
properties of $f_*\Oh_X(D)$. This implies that it is also very semiample
over the generic point of $U$.
 \end{rem}

As above we derive results similar to Corollaries~\ref{bndnv},
\ref{nonvc} and a new one
\ref{bnonvcon}.

 \begin{cor} \label{bnonvd} For some triple $(X_u/T_u/Z_u,B_u,\fF_u)$, let
$C_u$ be a divisor on $X_u$ such that all $\mult_{P_i}{C_u}>-1+\tau$.
Then, for any $D\in\fU_\de(X_u/T_u/Z_u,\fF_u)$, there exists some
natural number $1\le m\le M$ such that $\linsys{\rdup{m D+C_u}}\ne\emptyset$.
 \end{cor}

 \begin{add} For $D\in
\fU_\de(X_u/T_u/Z_u,F_u)$, if $\mult_{P_i}{F_u}=0$ we can assume just
$\mult_{P_i}{C_u}>-1$.
 \end{add}

 \begin{proof} The proof of Corollary~\ref{bndnv} with
Theorem~\ref{bnonvan} instead of Addendum~\ref{bndbf}.1.
 \qed\end{proof}

 \begin{exa} \label{noncrm} Let $(X/T=\pt/Z=\pt,B=0,\fF)$ be a triple such
that $X$ is a complete elliptic curve, and $\fF=\{p+q\mid p\ne q\in E\}$.
Then Theorem~\ref{bnonvan} does not hold for this triple. (Quiz time: why
not applied?)

Otherwise, by Corollary~\ref{bnonvd} with $C=0$ there exists a natural
number $M$ such that, for any $D\in\fN=\fN(X/\pt/\pt,\fF)$,
$\rdup{m D}\ne\emptyset$ for some $1\le m\le M$. Thus if $D=p-q$ is a
torsion of the elliptic curve, then $D\sim_\R0$ and $D\in\fN$. But then
$m(p-q)=m D=\rdup{m D}\sim 0$, or all torsions are bounded. This gives a
contradiction.
 \end{exa}

 \begin{cor} \label{bnonvbd} Let $\al<1,\tau$ be positive real numbers
and $\sC_u$ an $\R$-{\bi}divisor of $X_u$ such that $\sC_u\ge0/X_u$ and
 \begin{equation}
 \hbox{all $\mult_{P_i}{\al\sC_u}>-\al+\tau \mult_{P_i}{F_u}$
 for any $F_u\in\fF_u$.}
 \tag{$*$}
 \end{equation}
 Then there exists a natural number $M$, and positive real numbers $\be,\de$
which give the following nonvanishing$/Z_u$ on any $Y_u/X_u$ (uniformly in
$u$). For all {\bi}divisors $\sD$ such that the descent data $\sE$ of
$\sD$ over $X_u$ is bounded by $\be\sC_u$, that is, $\sE\le\be\sC_u/X_u$,
and $\sD_X\in\fU_\de( X_u/T_u/Z_u,\fF_u)$, we have
$\linsys{\rdup{m\sD_Y+(\sC_u)_Y}}\ne\emptyset$ for some $1\le m\le M$.
 \end{cor}

 \begin{proof} The proof of Corollary~\ref{nonvc} with
Theorem~\ref{bnonvan} instead of Addendum~\ref{bndbf}.1. Note that each
$F_u\in\fF_u$ is $\Q$-Cartier on $X_u$ by (QFC) of \w\ triples.
 \qed\end{proof}

 \begin{cor} \label{bnonvcon} There are real numbers $\be,\de>0$ and a
natural number $M$ giving the nonvanishing as in {\em
Corollary~\ref{bnonvbd}\/} for the discrepancy {\bi}divisors
$\sC_u=\sA_u=\sA(X_u,B_u)$.
 \end{cor}

 \begin{exa} \label{fnonvex} Let $(X,B)$ be a Klt pair with a subboundary
$B$, $\Q$-factorial $X$, and $\fF$ a {\em bounded\/} family of reduced
divisors on $X$. Then there exists a real number $\ga>0$ such that
$(X,B+\ga F)$ is Klt for any $F\in\fF$. Therefore all
 \[
 \mult_{P_i}{\sA}-\ga\mult_{P_i}{F}=
 \mult_{P_i}{(\sA-\ga\Fbar)}=a(X,B+\ga F,P_i)>-1
 \]
 (cf.\ the proof of Lemma~\ref{discr}, and Example~\ref{stexm}). Thus for
any real $\al>0$, we have $\al\mult_{P_i}{\sA}>-\al+\tau\mult_{P_i}{F}$
for $\tau=\al\ga$. This gives $(*)$ and the required $\tau$ in
Corollary~\ref{bnonvbd} for $\sC_u=\sA_u$.

If in addition $(X,B)$ is canonical in codimension~$2$, and $\al<1$ is a
positive real number (for example, $\al=1/2$), then there exists $\tau>0$
that satisfies the inequalities, including $(*)$, in
Corollary~\ref{bnonvbd} for $\sC_u=\sA_u$. Note that $\sA_u\ge0/X$,
because $(X,B)$ is canonical in codimension~$2$.
 \end{exa}

 \begin{lem} \label{existt} Let $\sC_u=\sA_u$ be discrepancy {\bi}divisors
on a bounded family of \w\ triples for weak log Fano contractions {\em
(for example, as in Corollary~\ref{bnonvcon}\/}), and let $\al\in (0,1)$.
Then there exists $\tau$ such that all $\sC_u$ satisfy the conditions of
{\em Corollary~\ref{bnonvbd}\/}, in particular, $(*)$.
 \end{lem}

 \begin{proof} Example~\ref{fnonvex} proves the lemma for each triple.
Then we can use a Noetherian induction over $U$ as in the proof of
Theorem~\ref{bnonvan}. Note that Klt is open in families. For any \w\
triple $(X_u/T_u/Z_u,B_u,\fF_u)$, each $X_u$ is $\Q$-factorial by (QFC)
of Definition~\ref{tri}, and each $(X_u,B_u)$ is terminal in
codimension~$2$ by (CRP) and (TER).
 \qed\end{proof}

 \begin{pfof}{Corollary~\ref{bnonvcon}} We can find $\be,\de>0$ and $M$ by
Lemma~\ref{existt} and then use Corollary~\ref{bnonvbd} (cf.\
Remarks~\ref{constnv} and \ref{edesd} above).
 \qed\end{pfof}

An explanation on bounded nonvanishing: in fact, in applications, we can
weaken the nef and big condition on $M$ in the definition of \w\ triples
to just nef. Thus we are interested in two types of \w\ triples
corresponding to cases when $\sM$ is big or a pencil in the proof of
Corollary~\ref{mainb}: (1) (weak) log del Pezzos $(X_t/\pt,B_t)$ with
only terminal closed points (= terminal resolution) corresponding to
triples $(X_t/X_t/\pt,B_t,\fF)$ with crepant $B_t$, and (2) elliptic
fibrations$/T_u=\PP^1$, corresponding to triples $(X_u/T_u/\pt,B_u,\fF)$
with an elliptic fibration $X_u\to T_u=\PP^1$, and crepant $B_u$. In
either case, $\fF$ is obtained from the standard family on $(X/\pt,B)$ by
log birational transform.

As usual, the simplest form of freedom is on log Fano contractions.

 \begin{thm} \label{fnonvan}
 \addcontentsline{toc}{subsection}{\ref{bnonvan} Theorem: the case of
log Fano contractions}
 \begin{itemize}
 \item Let $(X/Z,B)$ be a weak log Fano contraction;
 \item let $\fF$ be a bounded family of reduced divisors; and
 \item let $\tau>0$ be a tolerance.
 \end{itemize}
 Then there exists a natural number $M$ (depending on $(X/Z,B)$, $\fF$ and
$\tau$) such that, for each divisor $D\in\fN=\fN(X/Z,\fF)$, we have
$\Bs{|mD|}=\emptyset\mod{\tau}$, in particular, $|mD|\ne\emptyset
\mod{\tau}$ for some $1\le m\le M$ (depending on $D$).
 \end{thm}

 \begin{add} Moreover, there exists a real number $\de>0$ (depending on
$(X/Z,B)$, $\fF$ and $\tau$) such that the same nonvanishing holds for
all $D\in\fU_\de=\fU_\de(X/Z,\fF)$.
 \end{add}

 \begin{proof} Immediate by Theorem~\ref{bnonvan}. We consider the weak
log Fano contraction $(X/Z,B)$ with bounded $\fF$ as a triple
$(X/X/Z,B,\fF)$.
 \qed\end{proof}

 \begin{cor} \label{fnonvd} Let $C$ be a divisor such that all\/
$\mult_{P_i}{C}>-1+\tau$. Then, for any $D\in\fU_\de$, there exists
some natural number $1\le m\le M$, such that $\linsys{\rdup{m D+C}}\ne
\emptyset$.
 \end{cor}

 \begin{add} If $\mult_{P_i}{F}=0$ we can assume just $\mult_{P_i}{C}>-1$.
 \end{add}

 \begin{proof} Immediate by Corollary~\ref{bnonvd}.
 \qed\end{proof}

 \begin{cor} \label{fnonvbd} Suppose that each $F\in\fF$ is $\Q$-Cartier.
Let $\al<1,\tau$ be positive real numbers and $\sC$ a {\bi}divisor such
that $\sC\ge0/X$ and all
 \begin{equation}
 \mult_{D_i}{\al\sC}>-\al+\tau \mult_{D_i}{F}
 \quad\hbox{for any $F\in\fF$.}
 \tag{$*$}
 \end{equation}
 Then there exists a natural number $M$, and positive real numbers
$\be,\de$ giving the following nonvanishing$/Z$ on any $Y/X$
(uniformly in $F$). For all {\bi}divisors $\sD$ such that the descent
data $\sE$ of $\sD$ over $X$ is bounded by $\be\sC$, that is,
$\sE\le\be\sC/X$, and $\sD_X\in \fU_\de$,
$\linsys{\rdup{m\sD_Y+\sC_Y}}\ne\emptyset$ for some $1\le m\le M$.
 \end{cor}

 \begin{proof} Immediate by Corollary~\ref{bnonvbd}.
 \qed\end{proof}

 \begin{cor} \label{fnonvcon} Let $(X/Z,B)$ be a weak log Fano contraction,
with $X$ $\Q$-factorial and with $(X,B)$ canonical in codimension~$2$. Then,
for any given bounded family $\fF$ of reduced divisors, there are real numbers
$\be,\de>0$ and a natural number $M$ giving the nonvanishing$/Z$ as
in {\em Corollary~\ref{fnonvbd}\/} for $\sC=\sA$.
 \end{cor}

 \begin{proof} Immediate by Corollary~\ref{bnonvcon}.
 \qed\end{proof}

\section{Stabilization in divisor} \label{stbd}

We are now ready to establish the stabilization of a central
multiplicity. We first describe what we mean by an inductive model and
its central divisor $E_c$. Then we state under what condition we expect
a limit of {\bi}divisors to stabilize near $E_c$. Finally, we prove
this for 3-folds. This means a nonvanishing result on $E_c$ for the
limiting divisor. Afterwards, we extend this stabilization to the
freedom result of Corollary~\ref{stbklt} under Klt and in the realistic
situation (CBS)(rfa) (cf.\ (CBS)(fga) in Conjecture~\ref{cbcon}). This
is a freedom result on a Klt model that is terminal in codimension~2,
and this leads to the proof of Theorem~\ref{cbrfa} in Section~\ref{mr}.

\subsection{Conventions} In this section we assume LMMP in $n=\dim X$
whenever we state ``under LMMP''. We disregard this for $n\le3$. By
Corollary~\ref{mainb} we can drop (CBS), (MOD) and $\SSB$ respectively
for $n\le2$.

 \begin{defn} We say that a limit $\sD=\lim_{i\to\infty}\sD_i$ of
$\R$-{\bi}divisors {\em stabilizes near a set of prime {\bi}divisors\/}
$E_i$ if there exists a model $Y/X$ on which the limit stabilizes near
the centers of these {\bi}divisors (viewed as closed subvarieties). That
is, $\sD_i=\sD$ for some $i\gg0$ over a neighborhood\footnote{I think
this means as divisors on $Y$, not as {\bi}divisors.} of these centers
{\em in\/} $Y$.
 \end{defn}

 \begin{defn}\label{defn!ind} [cf.\ Prokhorov and Shokurov \cite{PSh}]
 An {\em inductive model\/} of a log pair $(X/Z,B)$ (respectively, {\em
strict inductive model\/}) is a log model $(Y/Z,B_Y)$ of $(X/Z,B)$
for which
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $(X,B_X)/(Y,B_Y)$ is {\em log proper\/}, that is, equivalently
 \begin{gather*}
 \sB^Y=\sB(Y,B_Y)\ge\sB^X=\sB=\sB(X,B), \quad\hbox{or} \\
 \sA^Y=\sA(Y,B_Y)\le \sA^X=\sA=\sA(X,B).
 \end{gather*}
 \item $(Y,B_Y)$ is {\em exceptional\/} in the sense that there is a
single divisor $E_c$ of $Y$ with $\mult_{E}{B_Y}=1$, and $(Y,B_Y)$ is
{\em purely\/} log terminal, respectively {\em strictly\/} log terminal;
here ``strictly'' means that $Y$ is $\Q$-factorial (\cite[p.~99]{sh92},
see also \cite{ko}, Definition~2.13).
 \item $-(K_Y+B_Y)$ is nef and big$/Z$ (big is automatic if $X/Z$ is
birational, and $-(K_Y+B_Y)$ is nef$/Z$) (respectively, ample$/Z$). And
 \item $-(K_Y+B_Y)$ is nef and big on $E_c/Z$ (respectively, ample on
$E_c/Z$).
 \end{enumerate}
 We say that $E_c$ is the {\em central {\bi}divisor\/} of the inductive
model. For a {\bi}divisor $\sD$, the multiplicity $d_c=\mult_{E_c}{\sD}$
is referred to as {\em central\/}. Note that (2--3) mean \WLF\ except with
Klt replaced by purely log terminal (cf.\ (PFN) in Lemma~\ref{incrb}
below); but it is still not a general log Fano contraction.
 \end{defn}

 \begin{warn}
 In this definition, we do not assume that $Y$ is$/X$.
 \end{warn}

 \begin{defn} A {\em local weak log Fano contraction\/} $(X/T,B)$ means
\WLF\ of Proposition~\ref{exsatcan}) in the local situation, that is, when
 \begin{itemize}
 \item $X/T$ is {\em local\/} contraction, that is, $\dim T\ge1$ and
$f\colon X\to T$ is onto $T$.
 \end{itemize}
 \end{defn}

 \begin{lem}[cf.\ \cite{PSh}] \label{indmod} {\em Under LMMP.\/} Let
$(X/T,B)$ be a local weak log Fano contractin. Then locally$/T$ there
exists an inductive model. Moreover, we can assume for this model that
 \begin{itemize}
 \item $B_Y$ is a $\Q$-divisor;
 \item $E_c$ is complete {\em (that is, $E_c/P$)\/}.
 \end{itemize}
 \end{lem}

 \begin{add} Thus we have a weak log Fano contraction $(E_c,B_c)$ such
that $K_{E_c}+B_c=(K_Y+B_Y)\rest{E_c}$. We call it the\/ {\em central
model\/}. We can assume that $B_c$ is $\Q$-Cartier.
 \end{add}

 \begin{add} We can upgrade any inductive model to a strict inductive
model.
 \end{add}

 \begin{proof} We consider here only the case that $X/T$ is birational,
which is all that we need. The general case can be found in Prokhorov
and Shokurov \cite[3.2]{PSh}. After taking a complement, we assume that
$K+B\equiv 0/T$. We can suppose in addition that $B$ is a $\Q$-divisor.
This increases $\sB$, which is important for (1) in the definition.
Hence after a making a contraction we can assume that $X=T$.

First, we can build a log canonical singularity by adding $\ep H$ for an
ample divisor through $P\in X$. Moreover, after a perturbation of $H$,
we can assume that $P$ is exceptionally log canonical for $B+H$ with
some effective $\Q$-Cartier divisor $H$. Again $\sB$ was increased, and
strictly over $P$. By the exceptional property we have the required
central divisor $E_c/P$ as a unique {\bi}divisor with log discrepancy $0$.

Now take a strict log terminal resolution $(Y/X,B_Y)$ resolving $E_c$.
The boundary $B_Y$ is given by the crepant modification of $(X,B+H)$.
In addition, we can assume that $g\1H$ is nef$/X$; otherwise, we can apply
LMMP to $B_Y+\ep g\1H$.

Finally, this give the required model if we replace $B_Y$ by $B_Y-\ep
g\1H$. Properties (1--3) follow from the construction. For (4), we need
to check that $g\1H$ is big on $E_c$. Otherwise, because $g\1H$ is
nef$/X$ and strictly effective near $E_c$ it defines a fibre contraction
$Y\to Z/X$, where $Z/X$ is small nontrivial and $H$ is positive on $Z/X$.
Since $H$ is a $\Q$-Cartier divisor this is impossible (by the projection
formula).

Addendum~\ref{indmod}.1 follows from the adjunction formula \cite[3.1
and 3.2.3]{sh92}, and the fact that $E_c$ is normal \cite[Lemma~3.6]{sh92}.
Addendum~\ref{indmod}.2 follows directly from the following result.
 \qed\end{proof}

 \begin{lem} \label{incrb} {\em (Under LMMP; cf.\ remark at the end
of the proof.)\/}
 \begin{q}
 \item[\rm(PFN)] Let $(X/T,B)$ be a {\em weak purely log terminal\/} Fano
contraction, {\em that is, \GLF\ of Proposition~\ref{ressat} with purely
log terminal $(X,B)$ and with a boundary $B$\/}
 \end{q}
 such that $X$ is $\Q$-factorial, and suppose that $-(K+B)$ is nef and
big on each reduced component of $B$ (there are {\em at most two\/}).
Then we can find another boundary $B^+$ such that:
 \begin{itemize}
 \item $(X/T,B^+)$ still satisfies {\rm(PFN)\/} with ample
$-(K+B^+)/T$;
 \item $B^+\ge B$; and
 \item $B$ and $B^+$ have the same reduced components.
 \end{itemize}
 \end{lem}

 \begin{proof} We can contract all $0$-curves $C$ with respect to $K+B$,
that is, $C\cdot(K+B)=0$, by a contraction $X\to Y/T$. The contraction is
given by $-(K+B)$. (PFN) is preserved because the reduced divisors of $B$
is not contracted. There exists an effective mobile divisor $D$ on $X$
which is negative$/Y$. In particular, for a general such $D$, the reduced
divisors of $B$ are not in $\Supp D$. Thus we can replace $B$ by the
boundary $B^+=B+\ep D$ for some $0<\ep\ll 1$. Then $-(K+B^+)/T$ is ample,
and $(X/T,B^+)$ satisfies all the required properties. For ampleness, we
use that the Kleiman--Mori cone of $(X/T,B)$ is polyhedral.
 \qed\end{proof}

 \begin{rem} We don't need the full strength of LMMP for
Lemma~\ref{incrb}; it is enough to know the cone and the contraction
theorems, which work under our conditions.
 \end{rem}

Now we are ready to state the stabilization of the limit
$\sD=\lim_{i\to\infty}\sD_i$ {\em at\/} some central divisor $E_c$:
 \[
 d_c=d_{c,j}
 \]
 for some $j\gg0$, where $d_{c,j}=\mult_{E_c}{\sD_j}$. This holds in
the following situation.

 \begin{thm} \label{stabin}
 \addcontentsline{toc}{subsection}{\ref{stabin} Theorem: existence of
inductive model}
{\em We assume LMMP and $\CBS_{n-1}\gl$,
$\SSB_{n-1}\gl$.\/} Suppose that $(X/T,B)$ is a log pair, and let
$\sD=\lim_{i\to\infty}\sD_i$ be a limit of {\bi}divisors such that:
 \begin{q}
 \item[\rm(LWF)] $(X/T,B)$ is a local weak log Fano contraction;
 \item[\rm(LBF)] {\em linear {\bi}freedom}: $i\sD_i\sim\sM_i/T$ for
all $i$, where $\sM_i$ is {\bi}free;
 \item[\LCA] $\sD_\bull$ is lca saturated;
 \item[\MXD] {\em maximality of the limit:\/} each $\sD_i\le\sD$; and
 \item[\BED] each $\sD_i=\sD$ outside $f\1P$ over $T$ {\em (cf.\
Proposition~\ref{prop!fga})\/}.
 \end{q}
 Then there exists an inductive model with central divisor $E_c$ such
that the limit $d_c=\lim_{i\to\infty} d_{c,i}$ is a rational number,
and the limit stabilizes; that is, $d_c=d_{c,j}$ for infinitely many
$j$.
 \end{thm}

Corollary~\ref{stabind} below improves this result slightly.

 \begin{rem} Under (LWF), {\em linear {\bi}freedom\/} (LBF) is
equivalent to {\em numerical freedom\/} $i\sD_i\equiv\sM_i/T$, provided
that $i\sD_i$ is Cartier, or just integral. It is enough to establish
this for $\overline{i(\sD_i)_Y}=i\sD_i\equiv 0/T$. Then the required
linear equivalence follows from descent of Cartier divisors for their
contractions on rational singularities, and from stable freedom on $Y=X$
as in (LWF) (cf.\ Remark~\ref{rem!rats} in the proof of
Lemma~\ref{split}, (1)).
 \end{rem}

The main steps in the proof of Theorem~\ref{stabin} are as follows.
First, we establish the slightly more general rationality result,
$d_c\in\Q$, and a stabilization result in Proposition~\ref{ration}. Then
using Proposition~\ref{bounds} on bounded presentation of the fixed
part of linear systems on the central {\bi}divisor, we reduce
Theorem~\ref{stabin} to Proposition~\ref{ration}.
Proposition~\ref{ration} itself and its proof are similar to the
proof of the rationality theorem in the Kleiman--Mori cone and the
nonvanishing theorem. However this time, we need a birational version of
nonvanishing: e.g., Corollary~\ref{bnonvcon} and Theorem~\ref{fnonvan}.
We interpret this nonvanishing as the stabilization of
$\sD=\lim_{i\to\infty}\sD_i$ in $E_c$. Some preparatory results in
Lemmas~\ref{indmod}--\ref{incrb} above and in Lemma~\ref{incra} and
Corollary~\ref{incrad} are needed to construct an appropriate inductive
model.

The following example illustrating the use of an inductive model is a
paradigm for Theorem~\ref{stabin} and for other results in this section
and in the paper.

 \begin{exa}[Projective space]\label{prs}
 \addcontentsline{toc}{subsection}{\ref{prs} Example: projective space
$\PP^n$}
Let $(X/Z,B)=(\PP^n/\pt,0)$ be projective space. Then (FGA), together
with $\SSB$, (PRM), (PFC), (CBS) and (MOD) of Conjecture~\ref{cbcon} hold
on it. Moreover, $\fS=\fP=\emptyset$, and in (CBS)
$\Bs{|\sM|_{\PP^n}}=\emptyset$ for each $\sM\in\fM$; so, $c=1$! Moreover,
for any $\R$-{\bi}divisor {\em under saturation\/} \SAT,
$\Bs{|\sD|_{\PP^n}}=\emptyset$. (FGA) gives as the stable models only
$\PP^n$ itself or $\pt$

It is possible to prove this using restriction to $D=\sD_Y$, as in the
proof of Proposition~\ref{surbound}. But this needs the Fujita estimate on
freedom, that is still not established. Another {\em realistic\/} approach
is to use inductive models. The first one is $(Y/\pt,E=P^{n-1})$, where
$Y\to\PP^n$ is the blowup in a closed point $p\in\sD_{\PP^n}\subset\PP^n$
and $E$ is the exceptional locus over $p$. If $\sD=\sD_Y$ is fixed
(prime) we use saturation (STD) for $\sD\rest E$ for generic $P\in\sD$;
the saturation is preserved (cf.\ Lemma~\ref{boundl} below). Thus by
induction on $n$, $\sD$ is mobile.

If $\sD$ is mobile, for example, $\sD=\sM\in\fM$, we can also use the
models $(\PP^n/\pt,H=\PP^{n-1})$ with (general) hyperplane sections $H$.
Again by induction $|\sD|_{\PP^n}$ only has {\em closed\/} base points
$p$ (cf.\ Proposition~\ref{prop!fga}). For such $p\in\Bs{|\sD|_{\PP^n}}$,
we use induction. Namely, $|\sM|_ Y\rest E$ is free where
$\sM=\Mov{\sD}$ is {\em general\/} in the linear system on $Y$ with
$\sM_{\PP^n}=\sD_{\PP^n}$ (cf.\ Proposition~\ref{bounds}). Hence
 \[
 \Mov{\rdup{\sD+\sA}}\ge\Mov{(\sM+\Ebar)}=\overline{(\sM)_Y}+\Ebar,
 \]
which contradicts saturation of $\sD$ with respect to $\sA=\sA(\PP^n,0)$.

The same works for any other nonsingular Fano variety such that:
 \begin{itemize}
 \item each blowup in a closed point,\footnote{Something missing? e.g.,
each blowup in a closed points is again a Fano variety?} and
 \item there are ladders of smooth log Fano varieties (down to
surfaces),
 \end{itemize}
 both of which give inductive models. For example, this is true for
nonsingular quadrics. All of this is aesthetically appealing, but
unfortunately not deep.
 \end{exa}

The following result generalizes the Rationality Theorem for Cones.

 \begin{nota}
 In the following proposition, $\sA'=\sA'(X,B)=\sA+E_c$ is an {\em
adjusted (truncated)\/} discrepancy where $(X,B)$ is {\em
exceptionally\/} log terminal with a single $E_c$ having
$\mult_{E_c}\sA=-1$.
 \end{nota}

 \begin{prop} \label{ration}
 \addcontentsline{toc}{subsection}{\ref{ration} Proposition: rationality and
stabilization}
 Let $(X/T,B)$ be a strict inductive model with central divisor $E_c$,
$E'\sim E_c$ a linear equivalence with $E_c\not\subset\Supp{E'}$, and
$\sD_\bull$ a system of {\bi}divisors $\sD_i$ such that
$d_c=\lim_{i\to\infty} d_{c,i}$. Then $d_c\in\Q$ under the following
conditions:
 \begin{q}
 \item[\BNF] each $\sD_i$ is {\em {\bi}nef\/}$/T$ in the sense of
Lemma~\ref{divb};
 \item[\rm($\ep$A$'$S)] {\em integral asymptotic saturation over $E_c$
(cf.\ Definition~\ref{ints})\/} holds for $\sD_\bull$ with respect to
$\sC=\sA'+\ep \overline{E_c}$, for some $\ep>0$;
 \item[\MXC] {\em maximality of the limit:\/} each $d_{c,i}\le d_c$;
 \item[\BWQ] there exists a bounded family of triples
$(E_u/T_u/\pt,B_u,\fF_u)$ as required\footnote{I don't know what this
means. Possibly \w\ triples?} for $(E_c/\pt,B_c)$, {\em (cf.\
Theorem~\ref{bnonvan})\/} such that: each $E_u/E_c$, each
$(\sD_i\frest{E_c})_{E_i}\in\fN(E_i/T_i/\pt,\fF_i)$ {\em (the mixed
restriction for $K=\R$, cf.\ (mx) in Proposition~\ref{rest})\/}, and the
descent data of the restrictions $\sD_i\frest{E_c}$ are asymptotically
bounded with respect to $\sA_c=\sA(E_c,B_c)$ over the {\em
corresponding models\/} $E_i$;
 \item[\rm(BRE$'$)] $\Supp{(\overline{E'\rest{E_c}})_{E_u}}\le F\in\fF_u$
for each $F\in\fF_u$ on each model $E_u$ of triples in \BWQ.
 \end{q}
 Moreover, the limit stabilizes: $d_c=d_{c,j}$ for infinitely many
$j$.
 \end{prop}

 \begin{rem}\label{rem!indptY}
 $E'\sim E_c$ {\em induces\/} $E''\sim E=g\1E_c$ on any other model
$g\colon Y\to X$ with $E\not\subset\Supp{E''}$. Take
$E''=g^*E'+E-g^*E_c$. Then the restriction (mx) defined by $E''$ on
$Y$ is independent of $Y/X$: in fact, since
$\mult_{E_c}\sD=\mult_{E}\sD=d$, we have
 \begin{align*}
\sD\frest{E_c}
&=(\sD-d\overline{E_c})\frest{E_c}+d\overline{E'}\frest{E_c}
\,=\,(\sD-d\overline{E_c})\frest{E_c}+d\overline{E'\rest{E_c}} \\
&=(\sD-d\overline{E_c})\frest E+d\overline{g^*E'\rest E}
\,=\,(\sD-d\overline{E_c})\frest E+d\overline{(E''+g^*E_c-E)}\frest E \\
&=(\sD-d(\overline{E_c}-\overline{g^*E_c}+\Ebar))\frest E-
d\overline{E''}\frest E
\,=\, (\sD-d\Ebar)\frest E-d\overline{E''}\frest E,
 \end{align*}
because $g^*E'=E''+g^*E_c-E$ and
$\overline{E_c}=\overline{g^*E_c}$.
 \end{rem}

 \begin{proof} By the monotonicity in Lemma~\ref{monots} and because
$(K+B)/T$ is negative, we can replace $\ep$ by a smaller positive value
so that the saturation ($\ep$A$'$S) still holds, and at the same time
$-(K+B)+\ep E_c$ is nef and big$/T$. Moreover, we can assume that the
same holds for any smaller $\ep\ge0$.

In particular, for any $0\le d\le\ep$, for any real number $j\ge0$, and
for any natural number $i$, on a sufficiently high log resolution
$g\colon Y\to X/T$ of $(X/T,B)$, we have the following vanishing
 \[
 R^1 g_*\Oh(\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y})=0.
 \]
 Note that the resolution $Y$ depends only on $i$. Indeed, we take a
log resolution {\em over\/} which $(\sD_i)_Y$ is nef$/T$ and $\R$-Cartier.
Then
 \begin{align*}
 \rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}&=K_Y+\rdup{-g^*(K+B)+d
g^*E_c+j(\sD_i)_Y} \\
 &=K_Y+\rdup{g^*(-(K+B)+d E_c)+j(\sD_i)_Y},
 \end{align*}
 where $-(K+B)+d E_c$ is nef and big$/T$ and $(\sD_i)_Y$ is nef$/T$. It
follows that $g^*(-(K+B)+d E_c)+j(\sD_i)_Y)$ is nef and big$/T$, and
Kawamata--Viehweg gives the required vanishing.

However to apply this vanishing to a restriction to the birational image
$E$ of $E_c$ in $Y$, we need to assume that $d\overline{E_c}+j\sD_i$ has
integral multiplicity in $E_c$, by integral saturation in ($\ep$A$'$S),
or equivalently, that the $d+jd_{c,i}$ are integers. More precisely,
then we have a surjective map of linear systems
 \[
 \linsys{E+\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}}\broken
 \linsys{(E+\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y})\rest E}.
 \]
 In addition, because the $d+j d_{c,i}$ are integers, normal crossing,
by Lemmas~\ref{restdiv}-.\ref{intpart} and adjunction, the linear
system
 \begin{align*}
 &\linsys{(E+\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y})\rest E}= \\[6pt]
 &\kern2.5cm=\linsys{\rdup{(E+\sA_Y)\rest E+(d(\overline{E_c})_Y+
 j(\sD_i)_Y)\rest E}}\\[6pt]
 &\kern2.5cm=\linsys{\rdup{(\sA_c)_E+((d\overline{E_c}+j\sD_i)
\frest{E_c})_E}}\\[6pt]
 &\kern2.5cm=\linsys{\rdup{(\sA_c)_E+(((d+j
u_{c,i})\overline{E'}+j\sD'_i)\frest{E_c})_E}} \\[6pt]
 &\kern2.5cm=\linsys{\rdup{(\sA_c)_E+(((d+j
d_{c,i})\overline{E''}+d(\overline{E_c}-\Ebar)+
j\sD''_i)\frest{E})_E}},
 \end{align*}
 for a sufficiently high resolution $Y$ (over which $\sD_i$ is nef
$\R$-Cartier$/X/T$), is defined on $E$ by the birational restriction
$\bigl(d\overline{E_c}+j\sD_i\bigr)\frest{E_c}$, that is, by
 \[
 \Bigl((d+j d_{c,i})\overline{E'}+j\sD'_i\Bigr)\frest{E_c}=
 \Bigl((d+j d_{c,i})\overline{E''}+d(\overline{E_c}-\Ebar)
 +j\sD''_i\Bigr)\frest{E_c}
 \]
(as in Remark~\ref{rem!indptY}), where $E$ is a sufficiently high
resolution of $E_c$ and $E'\sim E_c$ on $X/T$ with
$E_c\not\subset\Supp{E'}$ and where $E''\sim E$ is induced on $Y/T$
with $E\not\subset\Supp{E''}$, $\sD'_i=\sD_i-d_{c,i}\overline{E_c}$
and $\sD''_i=\sD_i-d_{c,i}\Ebar$. This last resolution only depends on
$i$. Hence if we have nonvanishing for the linear system restricted to
$E$, the corresponding linear system
 \[
 \linsys{E+\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}}=
 \linsys{\rdup{\sA'_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}}
 \]
 is free on $E$.

In the present situation this is impossible, because it contradicts
\MXC, the maximality of $d_c$. Namely,
 \begin{q}
 \item[\rm(NBF)] $\linsys{\rdup{\sA'_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}}$
cannot be free on $E$ for $j$ a positive integer and integers
$d+jd_{c,i}>jd_c$.
 \end{q}
 Indeed, then the multiplicity of the mobile part for the linear system is
 \[
 \mult_{E}{\rdup{\sA'_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}}=
 \mult_{E_c}{(d\overline{E_c}+j\sD_i)}=d+j d_{c,i}.
 \]
On the other hand, for any natural number $i$, by asymptotic saturation
(see Definition~\ref{asatur}), the multiplicity is
$\le\mult_{E_c}{j\sD_j}=j d_{c,j}$, that is, $d+j d_{c,i}\le j d_{c,j}$.
Hence $d_{c,j}\ge d_{c,i}+d/j>d_c$ by the inequality in (NBF). This
contradicts \MXC.

By the above surjectivity, (NBF) is equivalent to the following
vanishing:
 \begin{q}
 \item[\rm(VN)] on a sufficiently high resolution $E/E_c$,
 \[
 \linsys{\rdup{(\sA_c)_E+(d+j d_{c,i})\overline{E'\rest{E_c}}+
j\sD'_i\frest{E_c})_E}}=\emptyset,
 \]
 whenever $j$ is a positive integer and integer $d+j d_{c,i}>jd_c$.
 \end{q}
 However if $d_c\notin\Q$, using a bounded nonvanishing we can disprove
this for some $i\gg0$.

For triples $(E_u/T_u/\pt,B_u,\fF_u)$ in \BWQ, by Corollary~\ref{bnonvcon}
there exist real numbers $\be,\de>0$ and a natural number $M$ such that for
any {\bi}divisors $\sD$ with the descent data bounded by $\be\sA_c$ over
$E_u$ and $\sD_{E_u}\in\fU_\de$, we have bounded nonvanishing:
$\linsys{\rdup{m\sD_{E\hr}+(\sA_c)_{E\hr}}}\ne\emptyset$ on any
$E\hr/E_u$, in particular, for some rather high $E\hr/E_c$ and for some
$1\le m\le M$.

We apply this to the {\bi}divisor
 \[
 \sD=q(d/j+d_{c,i})\overline{E'\rest{E_c}}+q\sD'_i\frest{E_c}
 \]
 for some natural number $q>0$ and satisfying the nonvanishing
conditions. Equivalently,
$\sD=p\overline{E'\rest{E_c}}+q\sD'_i\frest{E_c}$, where now $j=mq$
and $p=d/m+q d_{c,i}$ under the following conditions:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $m>0,p$ are integral, and
 \item $p/q>d_c$.
 \end{enumerate}
 These imply that $mq=j>0$ and $mp=md/m+mqd_{c,i}=d+jd_{c,i}$ are
integral and $mp=d+jd_{c,i}>jd_c$ as in (VN).

Now we rewrite the nonvanishing conditions for $\sD$:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \setcounter{enumi}{2}
 \item $\sD\in\fU_\de(E_i/T_i/\pt,\fF_i)$, and
 \item the descent data of $\sD$ is bounded by $\be\sA_c/E_i$.
 \end{enumerate}

Since
 \[
 \sD=\bigl(p-q d_{c,i}\bigr)\overline{E'\rest{E_c}}
 +q\Bigl(d_{c,i}\overline{E'\rest{E_c}}+\sD'_i\frest{E_c}\Bigr),
 \]
 we have $q(d_{c,i}\overline{E'\rest{E_c}}+\sD'_i\frest{E_c})=
q\sD_i\frest{E_c}$ (this (mx) restriction is defined up to $\sim_\R$),
$(\sD_i\frest{E_c})_{E_i}\in\fN(E_i/T_i/\pt,F_i)$ for some
$F_i\in\fF_i$, and $\Supp (E'|E_c)_{E_i}\le F_i$ by \BWQ\ and (BRE$'$),
then, for a {\em natural number\/} $q$, (3) holds for
$\sD=p\overline{E'\rest{E_c}}+q\sD'_i\frest{E_c}$, as long as we ensure
that $\Vert (p-q d_{c,i}) (\overline{E'\rest{E_c}})_{E_i}\Vert <\de$.
(For a natural number $q$, note that $\sD\in\fN(E_i/T_i/\pt,F_i)$ implies
that $q\sD\in\fN(E_i/T_i/\pt,F_i)$ because $\fN(E_i/T_i/\pt,F_i)$ is an
Abelian semigroup, although it is not a convex body.) This last
inequality holds provided that
 \begin{enumerate}
 \item[(3$'$)] $0\le p-q d_{c,i}<\de/N$, where $N$ is $\ge$ the {\em
maximal\/} absolute multiplicity of $(\overline{E'\rest{E_c}})_{E_u}$ over
all $F\in\fF_u$,
 \end{enumerate}
 because $\Supp (E'|E_c)_{E_i}\le F_i\in\fF_i$ by (BRE$'$). $N$ is bounded
under \BWQ. The left inequality in (3$'$) has the stricter form
$0<p-q d_{c,i}$, which follows from (2) and \MXC.

Since $E_i/E_c$, the descent data for $(\overline{E'\rest{E_c}})$ is
$0/E_i$, and the descent data for $\sD$ is the same as for $q\sD_i
\frest{E_c}$ by \ADD\ in Proposition~\ref{desd}. Hence for fixed $q$,
the descent data for $\sD$ is asymptotically bounded with respect to
$\sA_c/E_i$ (cf.\ Remark~\ref{asb}, (4); the only difference with
Definition~\ref{bdd} is that we consider a sequence of models $E_i$).
In particular, for some $i\gg0$ the descent data of $\sD$ is bounded by
$\be\sA_c$, because by definition $\sA_c\ge0/E_i$. This gives (4) for
some $i\gg0$.

Thus we need to find integers $q>0$ and $p$ satisfying (2) and (3$'$).
This gives
 \[
 m\sD=m p\overline{E'\rest{E_c}}+m q\sD'_i\frest{E_c}=
 (d+jd_{c,i})\overline{E'\rest{E_c}}+j\sD'_i\frest{E_c},
 \]
which contradicts (VN) for $j=m q$ and $d=m(p-q d_{c,i})$ with some
natural number $1\le m\le M$ by Corollary~\ref{bnonvcon}.

Suppose that $d_c\notin\Q$ then, for any real numbers $N$ and $\ep>0$,
there exists a rational (chain) approximation $r=p/q$ with integers
$q>0$ and $p$ such that
 \begin{itemize}
 \item $r>d_c$, and
 \item $r-d_c<\ep/Nq$.
 \end{itemize}

We apply this result for certain real numbers: $0<\ep\le\de$, where
$\ep$ is $\le$ that at the start of the proof, and $N\ge M$ as in
(3$'$). Thus we get required integers $q>0$ and $p$ satisfying (2) and
(3$'$) for all $i\gg0$. Indeed, $p/q=r>d_c$ implies (2) and the left
inequality in (3$'$) by \MXC. On the other hand, since
$d_c=\lim_{i\to\infty} d_{c,i}$, then for all $i\gg0$, we have
 \[
 p/q-d_{c,i}=r-d_{c,i}<\ep/N q\le\de/N q
 \quad\hbox{or} \quad p-q d_{c,i}<\de/N.
 \]
We also need to verify that $0\le d\le\ep$: since $d=m(p-q d_{c,i})$
and we get above more than (3$'$) $0<p-q d_{c,i}<\ep/N$, we get
$0<d<m\ep/N\le M\ep/N\le\ep$, because $M\le N$. This leads to the
promised contradiction. Therefore $d_c\in \Q$.

Finally, we verify the stabilization of the limit
$d_c=\lim_{i\to\infty} d_{c,i}$. We use the same arguments, but now the
contradiction turns into an honest nonvanishing and stabilization. In
particular, we replace (2) by the equation
 \begin{enumerate}
 \item[(2$'$)] $p/q=d_c$.
 \end{enumerate}
 Nonetheless, we get (3$'$) and the nonvanishing of the linear system in
(VN) for some $i\gg0$ by (3) and (4), in particular, for some $i\ge n$.
As explained above this implies that
$\linsys{\rdup{\sA'_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}}$ is free on $E$.
This time, by asymptotic saturation, we get inequalities
$d_{c,j}\ge d_{c,i}+d/j=d_c$. Thus by \MXC\ $d_c=d_{c,j}$ for $j=m q$.
We can find infinitely many $j$ if we replace $p$ and $q$ by $lp$ and
$lq$ respectively for any natural number $l>0$. (They may not be
proportional to the first $j$ since $m$ depend on $\sD$.)
 \qed\end{proof}

To apply Proposition~\ref{ration} in the proof of Theorem~\ref{stabin},
we need to obtain bounded presentations on the central divisor $E_c$ of
the restrictions $(\sD_i\frest{E_c})_{E_i}$ in \BWQ\ for an appropriate
inductive model of $(X/T,B)$. For this we use Corollary~\ref{resdf} in
conjunction with the following result, for a certain set of
{\bi}divisors $\sM$, on the boundedness of their fixed components on
$E_c$, that is, the boundedness of the divisorial components of
$\Bs{|\sM|_X}\cap E_c$.

 \begin{prop} \label{bounds}
 \addcontentsline{toc}{subsection}{\ref{bounds} Proposition: restriction to
central divisor and bounded singularities}
 {\em Under $\CBS_{n-1}\gl$ and
$\SSB_{n-1}\gl$.} Let $(X/T,B)$ be a strict inductive model with
central divisor $E$, $\Bs{}\subset X$ a proper reduced subvariety, and
$\{\sM\}$ a set of (sufficiently general) {\bi}free {\bi}divisors $\sM$
such that:
 \begin{q}
 \item[\rm(BSP)]
$\Bs{|\sM|_X}\subset \Bs{}$; and
 \item[\rm(SA$'$)] $\sM$ is $\sA'$-saturated on sufficiently high models
$X\hr/X$ {\em with $'$ over $P$ and $\Bs{}$, that is, $\sA'=\sA+\sum E_i$
with $\sum E_i$ {\em exactly} over the integral components of $\sA$ over
$P$ and $\Bs{}$, that is, with $f(\cent_T{E_i})=P$ or with
$\cent_X{E_i}\subset \Bs{}$;\/}
 \end{q}
 then the set of {\bi}divisors $\sM$ satisfies
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item {\em boundedness of the fixed component\/} of each $|\sM|_X$ on
$E$, that is, the {\em whole\/} divisorial component of the
intersection $\Bs{|\sM|_X}\cap E$ belongs to a bounded set $\fF$ of
reduced divisors on $E$;
 \item in each linear system $\linsys{\sM\frest E}$, a {\em general
element} has bounded canonical singularities on a triple
$(E_u/T_u/\pt,B_u,\fF_u)$ with $E_u/E$, \w\ for $(E/\pt,B)$ {\em (and
also just bounded if $E$ is a surface);\/} and
 \item the \w\ triples in (2) belong to a bounded algebraic family.
 \end{enumerate}
 \end{prop}

Note that $\sM$ is free on $X$ outside $f\1P$ if $\Bs{}=\emptyset$
{\em there\/}.

 \begin{lem}[cf.\ Proposition~\ref{ressat} and its proof] \label{boundl}
Under the assumptions of {\em Proposition~\ref{bounds}, (SA$'$)} for $\sM$
implies {\rm(SA$'$F)} of {\em Proposition~\ref{ssp}, (4)\/} with $\sM:=\sM
\frest E$ and $F=\Supp{\Fix{(|\sM|_X\rest E)}}$, {\em except over
\/} $\Supp{B_E}$.
 \end{lem}

 \begin{proof} Suppose that $\sM_X$ is sufficiently general in its linear
system $|\sM|_X$, and set $M=\sM_X$.

Then, for any $0<\ep\ll 1$, on a sufficiently high log resolution
$g\colon Y\to X/T$ of $(X/T,B)$, we have the following vanishing
 \[ R^1 g_*\Oh(\rdup{\sA_Y+
\ep\Mbar_Y})=0;
 \]
 $\Mbar$ is well defined because $X$ is $\Q$-factorial. This resolution
$Y$ depends only on $\sM$. We take a log resolution such that
$\Bs{|\sM|_Y}=\emptyset$. For general $M$, $g\1M=\sM_Y$ is also general
in $|\sM|_Y$. Then the resolution is also a log resolution for
$B+\ep M$, and
 \begin{align*}
 \rdup{\sA_Y+\ep\Mbar_Y}&=K_Y+\rdup{-g^*(K+B)+\ep g^*M} \\
 &=K_Y+\rdup{g^*(-(K+B)+\ep M)},
 \end{align*}
 where $-(K+B)+\ep M$ is ample$/T$ for all $0<\ep \ll 1$, since
$-(K+B)/T$ is ample$/T$. Thus $g^*(-(K+B)+\ep M)$ is nef and big$/T$, and
Kawamata--Viehweg gives the required vanishing.

However, to apply this vanishing downstairs to the restriction to the
birational image $E_Y$ of $E$ in $Y$, we need to assume that $\ep M$
has rather small multiplicities $e_i$ in the exceptional divisors $E_i$
of $Y/X$. More precisely, each $e_i<1-\{a_i\}$ for each discrepancy
$a_i=\mult_{E_i}{\sA}$, where $\{a_i\}$ is the usual fractional part; in
particular, $e_i<1$ for integral $a_i$. Equivalently,
$\rdup{a_i+e_i}=\rdup{a^*_i}$, where $a^*_i=\mult_{E_i}{\sA^*}=a_i+1$
for integral $a_i$, when $E_i$ is over $M$ and even exactly over
$\Bs{|\sM|_X}$ for the {\em general\/} $M$, and $a^*_i=a_i$ otherwise.
Thus as compared to $'$, the operation $^*$ increases {\em exactly\/}
by $1$ those integral values $a_i$ of components that are exceptional
on $X$ and lie over $\Bs{|\sM|_X}$; in particular, $^*$ does not hold
for $E$ itself. (When $\sM_X$ is free outside $f\1P$ we can assume
that $^*$ increases {\em only\/} values over $P$.)

Hence
 \[
 \rdup{\sA_Y+ \ep\Mbar_Y}=\rdup{\sA_Y+\ep\sM_Y+\sum e_iE_i}=
\rdup{\sA^*_Y}+\sM_Y
 \]
 and we get the vanishing $R^1 g_*\Oh(\rdup{\sA^*_Y}+\sM_Y)=0$. This
gives the surjective map of linear systems
 \begin{multline*}
 \linsys{E_Y+\rdup{\sA^*_Y}+\sM_Y} \broken \\
 \linsys{\bigl(\rdup{\sA^*_Y+E_Y}+\sM_Y\bigr)\rest{E_Y}}=
 \linsys{\rdup{\bigl(\sA^*_E\bigr)_{E_Y}}+
 \bigl(\sM\frest E\bigr)_{E_Y}}.
 \end{multline*}
 Indeed, by definition and our choice of $Y$,
$\sM_Y\rest{E_Y}=(\sM\frest E)_{E_Y}$. The same holds for {\em any}
sufficiently high resolution $X\hr/X$ and {\em some\/} sufficiently
high resolution $E\hr/E$. In addition, by adjunction
$(\sA_Y+E_Y)\rest{E_Y}=(\sA_E)_{E_Y}$, where $\sA_E=\sA(E,B_E)$. Hence
$(\sA^*_Y+E_Y)\rest{E_Y}=(\sA^*_E)_{E_Y}$, where
$(\sA^*_E)_{E_Y}=(\sA_E)_{E_Y}+\sF_{E_Y}$ for $\sF_{E_Y}=\sum F_i$ with
$F_i=E_i\cap E_Y$ and $E_i$ under $^*$. This assumes that we can
extend $\sF_{E_Y}$ birationally to $\sF$, or on (any) sufficiently high
$E\hr/E$. Since $\Mbar\frest{E}$ has the same support on $E$ as
$M\rest E$, then $F=\sF_X$ is exactly the fixed divisorial component of
$E\cap \Bs{|\sM|_X}=\Supp{\Fix{(|\sM|_X\rest E)}}$ (in general,
$\ne\Fix{\linsys{\sM\frest E}_ E}=(\Fix{\sM\frest E})_E=0$ for {\bi}free
$\sM$!) over the locuse where $B_E$ is integral, that is, {\em except
over\/} $\Supp{B_E}$, because $(E,B_E)$ is Klt. Since the resolution is
divisorial, the whole $\sF$ is exactly over the integral components of
$\sA_E$ and over $F$ (cf.\ (Sa$'$F) in Proposition~\ref{ssp}, (4)).
Finally, by normal crossings,
$\rdup{\sA^*_Y+E_Y}\rest{E_Y}=\rdup{(\sA^*_E)_{E_Y}}$.

Now by Lemma~\ref{monots}, (SA$'$), and Remark~\ref{asaturr}, (1) (again
because $(E,B_E)$ is Klt), we get saturation on $Y/T$:
 \[
 \linsys{E_Y+\rdup{\sA^*_Y}+\sM_Y}=\linsys{\sM_Y}+E_Y+\rdup{\sA^*_Y},
 \]
 where $\linsys{\sM_Y}$ is free on $Y$. Indeed, $E_Y+\rdup{\sA^*_Y}\ge0$
and $\sM_Y$ is integral; $\mult_{E_Y}{(E_Y+\rdup{\sA^*_Y})}=0$. Hence,
by the above surjectivity, on $E_Y$
 \begin{align*}
 \linsys{\rdup{(\sA^*_E)_{E_Y}+(\sM\frest E)_{E_Y}}}&=
 \linsys{\rdup{(\sA^*_E)_{E_Y}}+(\sM\frest E)_{E_Y}} \\
 &=\linsys{(\sM\frest E)_{E_Y}}+\rdup{(\sA^*_E)_{E_Y}},
 \end{align*}
 where $\linsys{(\sM\frest E)_{E_Y}}$ is free on $E_Y$.
Moreover, we can add to $\sF$ exceptional$/X$ $F_i$ over the integral
components of $\sA_E$, but not over $F$, preserving the mobile part
$(\sM\frest E)_{E_Y}$. This gives $\sA^*_E=\sA_E+\sF=\sA'+F$ with
$F,\sA'$ and with $\sM:=(\sM\frest E)$ under the saturation (SA$'$F),
where $F$ is considered as a {\bi}divisor. Then
 \begin{align*}
 \linsys{\rdup{(\sM\frest E)_{E_Y}+F+(\sA'_E)_{E_Y}}} &=
 \linsys{\rdup{(\sA^*_E)_{E_Y}+(\sM\frest E)_{E_Y}}} \\
 &=\linsys{(\sM\frest E)_{E_Y}}+F+\rdup{(\sA'_E)_{E_Y}},
 \end{align*}
 which means the saturation (SA$'$F).
 \qed\end{proof}

The proof of the lemma used the invariance of saturations (SA$'$) and
(SA$'$F) under $\sim$.

 \begin{prop}[Invariance of saturations]\label{linvsat} If\/
$\sD\sim\sD'/Z$ then the\/ $\sC$-saturation of\/ $\sD$ is linearly
equivalent to that of\/ $\sD'$. The same holds for\/ $\R$-divisors.

For asymptotic saturation, we can replace each $\sD_i$ by
$\sD_i\sim\sD_i'/Z$ {\em uniformly\/} (that is, there exists a rational
function $a\ne0$ on $X/T$ such that each $\sD_i=\sD_i'+\overline{(a)}$).
The same holds for the similarity of {\em characteristic type\/} of
$\sD_\bull$ {\em (see Remark~\ref{asaturr}, (7));\/} after a truncation
by $I'$, the index $I$ of asymptotic saturation equals $I/\gcd(I,I')$.
 \end{prop}

 \begin{proof} It is enough to verify the divisorial version, when
$D=\sD_X\sim D'=\sD'_X/Z$ and $C=\sC_X$. Thus $D'=D+(a)$.

Indeed,
 \begin{align*}
 \Mov{\rdup{D'+C}a}&=\Mov{(\rdup{D+C}+(a))} \\
 &=\Mov{\rdup{D+C}}+(a)\le D+(a)=D',
 \end{align*}
because the fixed part is independent of $\sim$, and the mobile part
changes by $(a)$ under $\sim$.

For asymptotic saturation, on a sufficiently high model $Y/X$, we
replace $D'$ by $j(\sD'_i)_Y=j(\sD_i)_Y+j\overline{(a)}_Y$.

By the above, to prove invariance under similarity it is enough to
consider a truncation $\sD_\bull^{[I']}=I'\sD_{iI'}$. Then, by
asymptotic $\sC$-saturation of $\sD_\bull$ with index $I$, for any $i$
and $j$, divisible by $I$,
 \[
 \Mov{\rdup{j\sD_i^{[I']}+\sC}}=
 \Mov{\rdup{j I'\sD_{iI'}+\sC}}\le jI'\sD_{jI'}=j\sD_j^{[I']},
 \]
 that means that $\sD_\bull^{[I']}$ is asymptotically $\sC$-saturated
with the same index $I$. In fact, we can replace $I$ by $I/\gcd(I,I')$.
 \qed\end{proof}

 \begin{pfof}{Proposition~\ref{bounds}} By Lemma~\ref{boundl}, we obtain
(SA$'$F), and this then applies in particular to the fixed component of
$|\sM|_X$ on $E$, because $\Supp{B_E}$ is fixed (cf.\
Remark~\ref{rem!exin}). This gives (1) by $\SSB_{n-1}\gl$ and
Proposition~\ref{ssp}, (4). This gives also (2--3) by $\CBS_{n-1}\gl$
(cf.\ Corollary~\ref{mainb} for $n=3$) because (SA$'$F) implies \SAT\
for $\sD=\sM$ by Lemmas~\ref{monots} and~\ref{as}.

Finally, we can assume that each $E_u$ is$/E$ after bounded blowups
(e.g., resolutions) that preserves all the properties of \w\ triples.
This holds for (CBS) because any blowup decreases the descent data of
{\bi}free $\sM$. Indeed, let $g\colon Y\to X/Z$ be birational, and
$\sM$ sufficiently general, namely, $\sM_Y=\sM_X$ as {\bi}divisors.
Then
 \begin{align*}
 \sE(\sM)/X&=\overline{\sM_X}-\sM_X=\overline{g^*\sM_X}-\sM_Y=
 \overline{\sM_Y}-\sM_Y+\Ebar \\
 &\ge \overline{\sM_Y}-\sM_Y=\sE(\sM)/Y,
 \end{align*}
 where $E=g^*\sM_X-\sM_X\ge0$ by Lemma~\ref{divb}.
 \qed\end{pfof}

To secure the saturation ($\ep$A$'$S) we need to strength the inductive
model.

 \begin{lem} \label{incra} Let $(X/T,B)$ be a weak log Fano contraction,
and $D$ any $\R$-Cartier divisor on $X$. Then there exists a boundary
$B^+\ge B$ such that
 \begin{itemize}
 \item $(X/T,B^+)$ is again a weak log Fano contraction; and
 \item $\sA(X,B)\ge\sA(X,B^+)+\ep\Dbar$ for any real number $0<\ep\ll 1$.
 \end{itemize}
 \end{lem}

 \begin{proof} Since $(X/T,B)$ is a weak log Fano contraction, we can
find an effective $\R$-Cartier divisor $D'\sim_\R-(K+B)$. Since this is
big we can assume also that there is an {\em effective\/} $D^+=ND'\ge
D$ for some real number $N>0$. Then, for any $0<\de\ll 1$,
$(X,B^+=B+\de D^+)$ is again a weak log Fano contraction with the
required properties. Indeed, $\sA^+=\sA(X,B^+)=\sA-\de\Dbar^+$ by
definition. Thus for any $0<\ep\le\de$, we have
 \[
 \sA=\sA^++\de\Dbar^+\ge\sA^++\ep\overline{ND'}\ge\sA^++\ep\Dbar.
 \qed
 \]
 \end{proof}

 \begin{cor} \label{incrad} {\em Under the assumptions of
Lemma~\ref{incra}\/}, saturation \LCA\ implies
 \begin{q}
 \item[\rm($\ep$AS)] asymptotic saturation for $\{\sD_i\}$ with respect
to $\sC=\sA^++\ep \Dbar$.
 \end{q}
 \end{cor}

 \begin{proof} By Lemma~\ref{monots} we need the inequality
$\sA\ge\sA^++\ep\Dbar$, that we know by Lemma~\ref{incra}.
 \qed\end{proof}

 \begin{pfof}{Theorem~\ref{stabin}}
 \addcontentsline{toc}{subsection}{Proof of Theorem~\ref{stabin}}
 To prove stabilization, we use Proposition~\ref{ration}. But before
this application, we need to construct an appropriate inductive model
of $(X/T,B)$. Note that the assumptions (LBF), \MXD, \BED\ and \LCA\ are
birational$/T$. Thus they hold on any birational model of $X/T$. However
\LCA\ is sensitive to changes of the boundary $B$. This allows us to
improve \LCA\ on an inductive model of $(X/T,B)$ as follows.

Let $D$ be an effective Cartier divisor that contains the special fibre
$f\1P$ of $X/T$. We can find such one in our local case. By
Corollary~\ref{incrad} we can increase $B$ so that $(X/T,B)$ is still a
weak log Fano contraction (in particular, is still Klt), and ($\ep$AS)
holds:
 \begin{q}
 \item[\rm($\ep$AS)] asymptotic saturation holds for $D_\bull$ with
respect to $\sC=\sA+\ep \Dbar$ for some $\ep>0$.
 \end{q}

Now we modify our weak log Fano contraction $(X/T,B)$ to an inductive
model $(X/T,B)$. Such a thing exists by Lemma~\ref{indmod}. All our
assumptions are preserved, including ($\ep$AS); this follows from
monotonicity (1) in Definition~\ref{defn!ind} of inductive models and
Lemma~\ref{monots}. After a $\Q$-factorialization we can assume that $X$
is $\Q$-factorial. This time ($\ep$AS) is preserved since this
modification is crepant.

Note that this final change affects not only the boundary $B$, but also
the contraction $X/T$ itself. Thus $\Dbar$ in ($\ep$AS) is replaced by an
effective Cartier {\bi}divisor $\sD$ that may not be Cartier on the new
$X$ itself. However it contains the special fibre $f\1P$, that is,
$\sD\ge\overline{f^*H}$, where $H$ is a hypersurface through $P$. (The
latter is a birational invariant of modifications of $X/T$.) In
particular, for any prime $\Q$-Cartier divisor $F$ on $X$ over $P$,
$\sD\ge\Fbar$. By Lemma~\ref{monots} we can replace $\sD$ by $\Fbar$ in
($\ep$AS). Taking $F=E_c$ we get an inductive model that satisfies
($\ep$AS) with $D=E_c$.

Moreover, by Addendum~\ref{monots}.1 we have ($\ep$A$'$S) for the
integral {\em over\/} $E_c$ asymptotic saturations with respect to
$\sA'+\ep\overline{E_c}$ (cf.\ Example~\ref{monotsi}). Take $\sC_1=\sA+
\ep\overline{E_c}>\sC_2=\sA+d\overline{E_c}$ with $d<\ep$ and then set
$\ep:=d$. Since $\mult_{E_c}{\sA}=-1$ is integral for the inductive model
$(X/Z,B)$, the integral property means that $j\sD_i+\ep\overline{E_c}$
also has integral multiplicity in $E_c$.

Now by Lemma~\ref{incrb}, after an increasing of boundary $B$ we can
assume that $(X/T,B)$ is a purely log terminal Fano contraction. This
gives the required strict inductive model of Proposition~\ref{ration}
that satisfies \BNF, ($\ep$A$'$S) and \MXC.

($\ep$A$'$S) was proved above. And also ($\ep$AS), without the integral
condition. \BNF\ follows from (LBF). \MXC\ follows from \MXD.

Fix a linear equivalence $E'\sim E_c$ with $E_c\not\subset\Supp{E'}$.
Since the sets $\fF$ in \BWQ\ are defined up to bounded addition of a
fixed divisor (cf.\ Remark~\ref{sspr}, (1)), and $E'$ is fixed, we get
(BRE$'$) whenever \BWQ\ is established. Indeed, for a bounded family of
models $E$ of $E_c$, the divisors $(\overline{E'\rest{E_c}})_E$ have
bounded support, that is, {\em add\/} a bounded set (really a single
element!) to each element of $\fF$. The sum is included in the log
transform of that from $E_c$ whenever $E/E_c$.

Hence to use Proposition~\ref{ration}, we want to verify \BWQ, after
increasing $B$. This property is birational and it follows from
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item boundedness of the fixed component of each $|\sM_i|_X$ on $E_c$,
that is, the divisorial component of $\Bs{\linsys{\sM_i}}_X\cap E_c$
form a bounded family $\fF$;
 \item in each $\linsys{\sM_i\frest{E_c}}$, a general element has bounded
canonical singularities on a \w\ triple of $(E_c/\pt,B_c)$ for the element
(also just bounded when $E_c$ is a surface); and
 \item \w\ triples in (2) form a bounded algebraic family.
 \end{enumerate}
 Thus by (3) there exists a bounded family of triples
$(E_u/T_u/\pt,B_u,\fF_u)$ of the required form for $(E_c/\pt,B_c)$ (cf.\
Corollary~\ref{mainb}). On the other hand, by Proposition~\ref{bounds}, we
obtain (1--3), because $(X/T,B)$ is now a strict inductive model with
central divisor $E_c$, and ($\ep$AS) for {\bi}divisors $i\sD_i$ implies
(SA$'$) for the {\bi}divisors $\sM_i$. Indeed, by definition the asymptotic
saturation ($\ep$AS) for $\sD_i$ and $j=i$ means the saturation for
$i\sD_i$ with respect to $\sA+\ep\overline{E_c}$ (cf.\
Remark~\ref{asaturr}, (5)). Even increasing $B$, we can extend $E_c$ to
$E_c+M$ for any $0<\ep\ll 1$, where $M\ge M_1=(\sM_1)_X$ for a sufficiently
general $\sM_1$, and where $E_c+M$ contains the fibre $f\1P$. Then, by the
invariance of saturation in Proposition~\ref{linvsat} and by (LBF), the
saturation for $i\sD_i$ with respect to $\sA+\ep\overline{E_c+M}$ implies
the same saturation for $\sM_i$. The latter implies (SA$'$) by
Addendum~\ref{monots}.1 because each $\sM_i$ is integral with
$\Bs{}=\Supp{(E_c+M)}$ in (BSP) (cf.\ Example~\ref{monotsi}). Indeed,
then, by \BED, for all $\sM_i$, $\Bs{(\sM_i)_X}\subset\Supp M_1\cup
f\1P\subset \Bs{}$.

In fact, we proved (1--3) in a slightly more general settings: we can
consider an arbitrary family of {\bi}free {\bi}divisors $\sM=\sM_i$ that
satisfy (BSP) and the saturation (SA$'$).

Before applying Corollary~\ref{resdf}, note that we can assume that
$\sD_1\sim\sM_1$ is sufficiently general. Otherwise we can replace $\sD_1$
by general $\sD^g_1\sim\sD_1$, that is, by $\sD^g_1=\sD_1+\overline{(a)}$,
where $a\ne0$ is a rational functions on $X/T$. Then we replace each
$\sD_i$ uniformly by $\sD^g_i=\sD_i+\overline{(a)}\sim\sD_i$ (similarity!).
Thus
 \[
 \sD^g=\lim_{i\to\infty}\sD^g_i=\lim_{i\to\infty}\sD_i+\overline{(a)}=
 \sD+\overline{(a)}.
 \]
These changes preserve the conditions in the theorem. (LWF) is not
affected. (LBF) continues to hold because
$i\sD^g_i=i\sD_i+i\overline{(a)}\sim i\sD_i$; \LCA\ by
Proposition~\ref{linvsat}; \MXD\ because
$\sD^g=\sD+\overline{(a)}\ge\sD_i+\overline{(a)}=\sD^g_i$; \BED\ 
because, outside $f\1P$, each
$\sD^g_i=\sD_i+\overline{(a)}=\sD+\overline{(a)}=\sD^g$.

Finally, we derive \BWQ\ from (1--3). By Corollary~\ref{resdf} with $K=\Q$,
$a_1=1$, $a_2=i$ and $\sD_2=\sD_i$ by \BED, which gives the required
condition in Corollary~\ref{resdf}, and by (1), we obtain that each
restriction $(\sD_1-\sD_i)\frest{E_c}$ has support bounded by some
$F\in\fF$, where $\fF$ is a bounded set of reduced divisors. Since
$\sD_1\frest{E_c}$ is fixed, this last bound holds for each
$\sD_i\frest{E_c}$, and even on each $E_u/T_u$ of our bounded family of
triples. To be complete we include also $\overline{E'\rest{E_c}}$ into
consideration by definition of (mx) restrictions. Hence each
$(\sD_i\frest{E_c})_{E_i}\in\fN(E_i/T_i/\pt,\fF_i)$ for a \w\ triple
$(X_i/T_i/\pt,B_i,\fF_i)$, because by definition of such triple
$(\sM_i\frest{E_c})_{E_i}=g_i^*M_i$, where $g_i\colon E_i\to T_i/\pt$ and
$M_i$ is nef and big on $T_i/\pt$ In particular,
 \[
 (\sD_i\frest{E_c})_{E_i}\sim_\Q((\sM_i/i)\frest{E_c})_{E_i}=
 g_i^*M_i/i\sim_{\Q}0/T_i/\pt,
 \]
 and is nef on $E_i/\pt$

By (LBF), $i\sD_i\frest{E_c}\sim\sM_i\frest{E_c}$. So by (2) and
Addendum~\ref{aboundd}.1, the descent data of restrictions
$\sD_i\frest{E_c}$ are asymptotically bounded with respect to $\sA_c$
over \w\ models $E_i/T_i$ for $\sM_i$, because $B_i$ is the crepant
boundary for $B_c$ as in the definition of the \w\ triple, that is,
$\sA(E_i,B_i)=\sA_c$, and $\lim_{i\to\infty} i=\infty$. Note that in
Addendum~\ref{aboundd}.1, we replace a common model $Y$ by the sequence
$X_i/\pt$ as in Proposition~\ref{aboundd} itself.
 \qed\end{pfof}

Now we can prove stabilization of $\lim_{i\to\infty}\sD_i\frest{E_c}$
on $E_c$. It is enough to prove this for a similar system $\sD_\bull$
(such as the characteristic system, cf.\ Remark~\ref{asaturr}, (7)).

 \begin{cor} \label{trunc} {\em Under the assumptions of
Theorem~\ref{stabin},\/} the system $\sD_\bull$ up to similarity of
{\em characteristic type\/} satisfies the theorem with $d_{c,1}=d_c=0$.

In addition, we can assume that the {\em new\/} $\sD_1=\sM_1$ is sufficiently
general; in particular, $\sD_1\ge0$.
 \end{cor}

 \begin{proof} By the theorem there exists a natural number $j>0$ such that
$d_{c,j}=d_c$. Take sufficiently general $\sD_1^{[j]}=\sM_j\sim jD_j$. This
data defines the required similarity.

Indeed, as in the proof of Theorem~\ref{stabin}, we can verify that any
similarity with $j=1$ preserves the conditions of the theorem, and also
its conclusions. The same holds for any truncation: e.g., for new
$\sD_i:=\sD_i^{[j]}=j\sD_{ij}$. (LWF) is not affected by the change.
(LBF) continues to hold because
$i\sD_i^{[j]}=i j\sD_{ij}\sim \sM_i^{[j]}=\sM_{ij}$; \LCA\
by Proposition~\ref{linvsat}; \MXD\ because
 \[
 \sD^{[j]}=\lim_{i\to\infty}\sD^{[j]}_i= \lim_{i\to\infty}j\sD_{ij}=
j\sD\ge j\sD_{i j}=\sD_i^{[j]}
 \]
 for any natural number $i$; \BED\ because, outside $f\1P$, each
$\sD_i^{[j]}=j\sD_{ij}=j\sD=\sD^{[j]}$.

Finally, for general $\sM_j$, $\mult_{E_c}{\sM_j}=0$ by (LBF) in
Theorem~\ref{stabin}. Hence by our choice of $j$,
$d_c=d_{c,1}=\mult_{E_c}{\sM_j}=0$. Note that stabilization takes place
for exactly the same divisors (but their indexes may be different and the
values of multiplicities in $E_c$ are $j$ times the old ones).

The last sentence of Corollary~\ref{trunc} holds by our choice of {\em
new\/} $\sD_1$.
 \qed\end{proof}

 \begin{cor} \label{stabind} {\em Under the conditions of
Theorem~\ref{stabin},\/} assume in addition
 \begin{q}
 \item[\AMN] {\em arithmetic monotonicity\/}: $\sD_i\ge\sD_j$ for any
$j\mid i$; and
 \item[\rm(RRF)] $d_{c,1}=d_c=0$ as in {\em Corollary~\ref{trunc}.\/}
 \end{q}
 Then $\lim_{i\to\infty}\sD_i\frest{E_c}$ stabilizes {\em (that is,
$\sD_j\frest{E_c}=\sD\frest{E_c}$ for some $j\gg0$).\/}

More precisely, there exist a natural number $N>0$ and a {\bi}free divisor
$\sM$ of $E_c$ such that $\sM$ satisfies (birational) saturation with
respect to $\sA_c=\sA(E_c,B_c)$ {\em (cf.\ \SAT\ and
Addendum~\ref{surbound}.2)\/}, and if $N\mid j$, then
 \begin{itemize}
 \item $d_{c,j}=d_c=0$; and
 \item $j\sD_j\frest{E_c}(=j\sD\frest{E_c})=(j/N)\sM$.
 \end{itemize}
Thus each $\sM_j\frest{E_c}\sim j\sD_j\frest{E_c}=(j/N)\sM$.
 \end{cor}

 \begin{rem}\label{stbdinr}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item \AMN\ is preserved for any similarity in the same way as \MXD\
of Theorem~\ref{stabin}.

 \item \AMN\ implies \MXD. For this we need that $\sD_\bull$ is bounded
by a Cartier divisor that is nonzero on $E_c$, which follows from \BNF\
and Lemma~\ref{divb} (cf.\ Proposition~\ref{bpbdal}).

 \item We can take a truncation of $\sD_\bull$ such that the corollary
holds with $N=1$.

 \item We {\em only} expect $\sM_j\frest{E_c}=j\sD\frest{E_c}$ if
stabilization holds in a neighborhood of some model (cf.\
Theorem~\ref{stabnbh} below).
 \end{enumerate}
 \end{rem}

 \begin{rem} If $d_{c,n}=0$, $\sD_j\frest{E_c}$ is the canonical {\em fixed\/}
restriction as in (fx) of Proposition~\ref{rest}.
 \end{rem}

 \begin{lem} \label{restas} Let $(X/T,B)$ be an inductive model with
central divisor $E_c$ and $\sD_\bull$ a system of $\R$-{\bi}divisors such
that:
 \begin{q}
 \item[\BSA] each $\sD_i$ is {\em {\bi}semiample\/} as in
Proposition~\ref{ressat}, or just satisfies \BNF;
 \item[\ASAprime] the system $\sD_\bull$ is asymptotically saturated
with respect to $\sC=\sA'=\sA(X,B)+E_c$; and
 \item[\rm(FXR)] each $d_{c,i}=0$; {\em in particular, each fixed restriction
$\sD_i\frest{E_c}$ is well defined\/}.
 \end{q}
 Then the restricted system $\sD_\bull\frest{E_c}$ satisfies asymptotic
saturation \LCA\ with respect to $\sA_c=\sA(E_c,B_c)$.
 \end{lem}

 \begin{proof} Immediate by Proposition~\ref{ressat} with $Y=E_c$.
\GLF\ and \LCC\ follows from properties of inductive models. (FXR) ensures
general position \GNP. Moreover, \BSA\ can be replaced by \BNF\
because {\bi}nef is enough for Kawamata--Viehweg vanishing and saturation
in the proof of the proposition.
 \qed\end{proof}

 \begin{lem}\label{bslim} Let $\sD_\bull$ be a system on a log pair
$(X/Z,B)$ such that:
 \begin{itemize}
 \item $(X,B)$ is Klt;
 \item $\sD_\bull$ is bounded by a {\bi}divisor, satisfies {\rm\AMN,
(LBF)\/} and saturation \LCA.
 \end{itemize}
 Then up to a truncation it converges to a {\bi}divisor.
 \end{lem}

Thus \AMN\ works as the convexity of Lemma~\ref{arithm}.

 \begin{proof} (Compare the proof of Lemma~\ref{arithm}.) Up to a
similarity preserving the conditions, we can assume that $\sD_1\ge0$ by
(LBF), and $\sD_\bull$ satisfies \LCA\ with index $I=1$. Hence each
$\sD_i\ge0$ by \AMN. Since the system is bounded and under \AMN, it has
a convergent subsequence. In addition, for $i=jq+r$ with natural numbers
$i,j,q$ and $r$, \LCA\ and (LBF) imply the estimate
 \[
 \frac{j q}{j q+r}\sD_j\le\sD_i,
 \]
 that implies that the sequence has a limit in view of that for the
subsequence $\sD_j$ and the bound.

Indeed, by \LCA, (LBF) for $\sD_j$, and since $\rdup{r\sD_j+\sA}\ge0$ by
Klt, we get the required inequality
 \begin{align*}
 j q\sD_j&\le qj\sD_j+\Mov{\rdup{r\sD_j+\sA}} \\
 &\le i\Mov{\rdup{(jq+r)\sD_j+\sA}}\le(jq+r)\sD_i. \qed
 \end{align*}
 \end{proof}

 \begin{pfof}{Corollary~\ref{stabind}} First, by \AMN, \MXD\ and (RRF), 
we have $d_{c,i}=d_c=0$ for each natural number $i$. Thus the fixed
restriction $\sD_i\frest{E_c}$ is well defined for each $i$.

Secondly, after restriction we preserve properties (LBF), \LCA, \MXD\ and
\AMN\ of Theorem~\ref{stabin} and of the corollary, now for the new
objects
 \begin{gather*}
 (X/T,B):=(E_c/\pt,B_c), \quad T=\pt, \quad \sD:=\sD\frest{E_c}, \\
 \sD_i:=\sD_i\frest{E_c}, \quad \sM_i:=\sM_i\frest{E_c} \quad\hbox{and}\quad
 \sA:=\sA_c=\sA(E_c,B_c).
 \end{gather*}
 Indeed, we get (LBF), \MXD\ and \AMN\ by definition of restrictions;
\AMN\ implies \MXD\ by Remark~\ref{stbdinr}, (2) above. For \AMN, note
that a birational fixed restriction of an effective divisor is effective.

By Lemma,~\ref{restas} \LCA\ for $\sD_\bull\frest{E_c}$ on
$(E_c/\pt,B_c)$ follows from \ASAprime\ of Lemma~\ref{restas} for the
inductive model in the proof of Theorem~\ref{stabin}. In turn,
\ASAprime\ follows from ($\ep$SA$'$) of Proposition~\ref{ration} by
Addendum~\ref{monots}.1, and ($\ep$SA$'$) itself follows from \LCA\ of
Theorem~\ref{stabin} by construction in its proof (cf.\
Example~\ref{fgapl}).

By Lemma~\ref{bslim} and the bound on $\sD_\bull \frest{E_c}$ (cf.\
Remark~\ref{stbdinr}, (2)), the limit $\lim_{i\to\infty}\sD_i\frest{E_c}$
exists up to a truncation, and it is a {\em candidate\/} for
$\sD\frest{E_c}$ (cf.\ Remark~\ref{stbdinr}, (4)).

Now the corollary follows from the solution of the asymptotic descent
problem in Addenda~\ref{assdes}.1 and~\ref{assdesc}.1 for the limit
$\sD:=\lim_{i\to\infty}\sD_i$ with $\sD_i:=\sD_i\frest{E_c}$. \FDS\
follows from \MXD\ for effective $\sD_\bull$. Asymptotic saturation holds
for $\sC=\sA (E_c,B_c)$ by \LCA. (LBF) implies \BNF. A prediction exists
by (CBS)($E_c/\pt, B_c)$, or $\CBS_{n-1}\gl$. This can be done exactly
as in the proof of Theorem~\ref{cbsfg}, (3). The only difference between
our current situation and (FGA) is that we do not have the convexity of
Lemma~\ref{arithm}. (However in our applications, this is satisfied.)
But this is not at all required for Theorem~\ref{cbsfg}, (3) and
Addendum~\ref{assdes}.1. We only need \AMN\ under \LCA!
 \qed\end{pfof}

Stabilization {\em near\/} $E_c$ needs an opposite technique that we
discuss in the next section.

\section{Destabilization} \label{dstb}

We find a neighborhood of $E_c$ on a $0$-log pair, with a {\em certain\/}
boundary, of Remark~\ref{gzardec}, (2) over which the system
$\sD_\bull$ stabilizes. We also work with generalizations of such
pairs (cf.\ \WLF\ vs \GLF\ and (PFN)).

 \begin{defn}\label{eltp} Let $(X/T,B)$ be a log pair such that
 \begin{itemize}
 \item $X/T$ is a projective contraction;
 \item $K+B\equiv 0/T$;
 \item $(X/T,B)$ is {\em exceptionally\/} log terminal; and, moreover,
 \item there exists {\em at most one\/} prime {\bi}divisor $E_c$ with
log discrepancy $0$ for $K+B$, and $E_c/P$.
 \end{itemize}
 If there exists such an $E_c$ we say that $(X/T,B)$ is an {\em
$0$-exceptional log terminal pair\/} or 0-elt pair, and that the
{\bi}divisor $E_c$ is {\em central\/}. Otherwise $(X/T,B)$ is Klt and is
just a $0$-{\em log pair\/}.
 \end{defn}

 \begin{exa}\label{cmd} Usually a $0$-elt pair arises as a {\em
complement} as follows. Suppose that $B=B_\eta+\eta C$ such that
$(X/T,B_\eta)$ is a $0$-log pair and $C\ge0$ with
$\eta=\mult_{E_c}{C}>0$. Note that in this situation $C\sim_\R0/T$, and
it defines canonically a {\bi}divisor $\sC$ such that $C=\sC_X$ and
$\sC$ is {\bi}free over $T$ whenever $B_\eta$ and $\eta$ are rational.

A typical example is building a log singularity. Let $(X/T,B)$ be a {\em
birational\/} weak log Fano contraction, that is, $X/T$ is birational.
Then after taking a complement, we can assume that $K+B_\eta\equiv0/T$.
Thus after a contraction, $X/T$ is the identity. Now we can find the
required $C\ge0$ passing through $P$ such that, for some threshold $\eta>0$,
$K+B_\eta+\eta C$ has only one log discrepancy $0$ in $E_c$ and $E_c/P$.
This is how we constructed our inductive model in Lemma~\ref{indmod}.
 \end{exa}

To be more precise, we introduce neighbourhoods of $E_c$ on some special
and quite\footnote{Something missing? Quite nice, quite bad? Or maybe you
mean to say ``completely canonical'' or log canonical?} (log) canonical
models. They give an induction on the {\em rank} of non$\Q$-factoriality
and the number of seminegative discrepancies (a kind of {\em difficulty};
cf.\ Lemma~\ref{nind} below).

 \begin{key}
 For given {\bi}free $\sL$, we need to construct a birational model $Y/T$
(later, a neighbourhood in such a model) on which there exists an effective
$\R$-divisor $D=\sum d_iE_i$ such that:
 \begin{itemize}
 \item $d_i>0$ precisely on the prime divisors $E_i$ that are exceptional
on $T$;
 \item $L+\sum d_iE_i$ is nef and even semiample$/T$, where $L=\sL_Y$; and
 \item $\sL$ is free over $Y/T$.
 \end{itemize}
 We call such a model a {\em destabilizing model\/} for $\sL$ (destab
model) because we can add to it (or to its multiple) a divisor $D$ on $Y$
that violates (destabilizes) $D$-saturation (for example, the exceptional
saturation of Example~\ref{satur}). The divisor $D$ is {\em
destabilizing\/}, and it defines the {\bi}divisor $\sD=\Dbar$.
 \end{key}

 \begin{warn} The contraction $Y/T$ is not necessarily divisorial.
 \end{warn}

 \begin{exa}\label{edst}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item If $L=\sL_T=0$, we can take $Y=T$. More generally, if $L=\sL_T$
is $\Q$-factorial on $T$ we can take $Y=T$ and also $D=0$.
 \item If $\dim X=2$ and $L$ is a curve passing through $P$ on $T$ then
for any $Y/T$ we can find a divisor $D>0$ that converts $Y/X$ to a destab
model. Indeed, we can find a numerically ample divisor $(D+L)/T$ (in
the sense of Mumford) with $D>0$ and $\Supp D=f\1P$ on a $\Q$-factorial
model (cf.\ Lemma~\ref{2ds}).
 \item In some cases in dimension $\ge 3$ we can't construct a {\em
divisorial\/} contraction $Y/T$: e.g., if $X/T$ is a flopping $0$-elt
pair and $L=\sL_X$ its hyperplane section. It means that in this case we
can't destab $\sL$ on any divisorial $Y/T$.
 \end{enumerate}
Probably a destab model does not exists for every $\sL$ in dimensions
$\ge3$. However in certain important situations it exists by LMMP.
 \end{exa}

 \begin{prop} \label{distabm} Let $(X/T,B)$ be a local weak log Fano
contraction with birational contraction $X/T$, and $\sL$ a {\bi}free
divisor. Then LMMP in dimension $n=\dim X$ implies the
existence of a destab model for $\sL$.

The same holds for a purely log terminal model whose reduced part is
nonexceptional$/T$; moreover, if $(X,B)$ has only log terminal
singularities with reduced centers not over $P$ and $\sL$ is free$/T$
over $X\setminus f\1P$. Or we can assume log terminal singularities and
that $\sL$ is in {\em rather general position}, that is, not passing through
the log canonical centers of $(X,B)$ and under {\em LSEPD} (see
\cite[10.5]{sh92} or \cite[2.30]{ko}).

We only need flips as a {\em noninductive new objects in LMMP\/} under
the {\em LSEPD trick} \cite{sh92}.
 \end{prop}

However sometimes {\em very\/} special flips are enough (cf.\
Example~\ref{lcresol} and Theorem~\ref{stabnbh} below).

 \begin{add} If we can construct such a model over a neighborhood $U$ on
$Z/T$, where $Z$ is the model defined by $\sL/T$, then destab holds over
the neighbourhood. The construction in the proof is log canonical; thus
it can be done locally$/Z$.
 \end{add}

 \subsection{Construction of a destab (destabilizing) model}
 The proof of Proposition~\ref{distabm} is completely effective.
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item Increasing $B$, we can convert our model into a $0$-elt pair: e.g.,
as in Example~\ref{cmd}. Then we can identify $(X=T/T,B=B_T)$.
 \item We take a log resolution $(Y/T,B_Y)$ of $(T/T,B_T)$ such that:
 \begin{itemize}
 \item $B_Y=f\1B_X+\sum E_i$ (a noncrepant boundary!), where
$f\colon Y\to X=T$, and the divisors $E_i$ are exceptional on $T$; and
 \item $L=\sL_Y$ is free over $Y$, in particular,
$\sL=\Lbar$, where
$L=\sL_Y$.
 \end{itemize}
 \item[] For plt and log terminal cases not resolving the LCS centers we
proceed as follows:
 \item[(2$'$)] We suppose that $L$ is rather powerful: $L\equiv N H/T$,
where $H$ is Cartier (semiample$/T$): e.g., $N\ge2 \dim X+2$ is enough.
Otherwise we replace $L$ by its multiple modulo $\equiv/T$. Finally, we
get the same multiple for the destab divisor $D$.
 \item We apply LMMP to $(Y/T,B_Y+L)$. At each step, we contract an
extremal ray $R$ with $R\cdot(K_Y+B_Y+L)<0$ by a birational contraction
$g\colon Y\to Z/T$. Note: by (2$'$) and boundedness of negative
contractions, $R\cdot L=0$. Thus freedom of $L$ and (2$'$) is preserved
during the modifications. Finally, we obtain a model where $K_Y+B_Y+L$
is nef; it is always big because $Y/T$ is birational, or by (2$'$) again.
 \item Then we make a semiample contraction for $K_Y+B_Y+L$. We obtain
this because the divisor is always big, because $Y/T$ is birational. We
also need the LSEPD trick \cite{sh92} and can do it if we secure it; in
addition Addendum~\ref{distabm}.1 holds over $U$. It also preserves the
freedom of $L$, because again by (2$'$) if, for a curve $C$,
$C\cdot(K_Y+B_Y+L)=0$ then $C\cdot L=0$. This is a destabilizing model.
 \item Finally, we take a destab divisor $D=l\sA_Y$ (so, $\sD=l\sA(T,B_T)$)
where $l$ means that we take log discrepancies. By definition of log
discrepancies
 \[
 D=\sum d_iD_i=K_Y+B_Y-g^*(K+B) \equiv K_Y+B_Y,
 \]
 for $ g\colon Y\to T$, since $K+B\equiv 0/T$ for elt $0$-models. Since
it is Klt all $d_i>0$. The same holds for purely log terminal models if
their reduced part is not exceptional. For terminal models under (2).

$D+L\equiv K_Y+B_Y+L$ is ample$/T$ by construction. In particular, it is
nef and semiample$/T$.

Finally, from LMMP we only need flips. Indeed, the termination is special
because we have pl flips (cf.\ Example~\ref{lcresol}). Thus we get it by
induction by Special Termination. However, the main problem is with
flips. Really the model $Y$ is over $Z/T$, so if we have flips for $Y/Z$,
over some neighborhood $U\subset Z$, then we can construct a model $Y/U$
and get the destab divisor over $U/T$, where $\equiv/U$ and $K_Y+B_Y$ is
ample$/U$ now. Since the model is log canonical$/U$, it is {\em
unique\/}$/U$ for given {\em choice\/} of $B_Y$, and can be constructed
locally$/U$. \qed
 \end{enumerate}

 \begin{lem} \label{trivbd} Let $\sD$ be a {\bi}divisor$/X$ such that
 \begin{itemize}
 \item $\sD_X=0$;
 \item {\em effective:\/} $\sD\ge0$; and
 \item nef on general curves$/X$: on (some) sufficiently high models
$Y/X$, the restriction $\sD_Y$ is nef$/X$ on curves covering the
exceptional divisors of $Y/X$; in particular, $\sD_Y$ is nef$/X$ on
such models.
 \end{itemize}
 Then $\sD=0$.
 \end{lem}

 \begin{proof} Immediate from the negativity of Lemma~\ref{negat}.
 \qed\end{proof}

 \begin{exa-cr}\label{lcresol} Working in dimension $n=\dim X$, when we
do not yet know the existence of all flips, it would be very helpful to
know in advance what types of flips are needed. Suppose that we have a
$0$-Klt pair $(X/T,B)$ (that is, replace elt by log terminal in
Definition~\ref{eltp}). For example, $(X/T,B)$ may have come from a purely
log terminal Fano contraction by taking a complement. Suppose that
$\sL\sim_\R \sS=S+\sum e_iE_i/T$ is {\bi}free and that $Z/T$ is a model
for $\sL$ over $T$ such that:
 \begin{itemize}
 \item LSEPD is secured for $B_Z$ (the log transform of $B_T$ or $B$) on
a model $Z/T$ such that $Z$ is nonsingular outside $B_Z$;
 \item $S=\sum s_iS_i$ is supported in the reduced part of $B$, considered
as a {\bi}divisor; and
 \item $\sS\ge0$ with the $E_i$ exceptional on $T$.
 \end{itemize}
 Then over some neighborhood $U$ of $\Supp{\sS_Z}$, strictly log terminal
and extremal flips of {\em type} (S+$-$) are sufficient: with
$B_Y=S^-+S^++B'$, where $(Y/U,B_Y)$ is log terminal, $B'$ has no reduced
components, and $R\cdot S^-<0$, $R\cdot S^+\ge0$.

Indeed, in the construction,
 \[
 B_Y=g\1B_T+\sum E_i \quad \hbox{and}
 \quad R\cdot(K_Y+B_Y+L)<0 \quad \hbox{with $L=\sL_Y$.}
 \]
 Since $R/Z$, then $R\cdot L=0$ and $R\cdot(K_Y+B_Y)<0$. Thus the log
discrepancy is supported in the exceptional$/T$ locus:
 \[
 D=l\sA(X,B)_Y=K_Y+B_Y-g^*(K+B)\equiv K_Y+B_Y,
 \]
 and there exists $E_i$ with $R\cdot E_i\ne0$. Moreover, $R\cdot E_i<0$
since $D\ge0$.

On the other hand, for $\sS_Y=S+\sum e_iE_i$ on $Y$, $R\cdot(S+\sum
e_iE_i)=R\cdot L=0$. Hence if $R\cdot E_i\ne0$ with $e_i\ne0$, then there
exists another divisor $E_j$ or a component in $S$ having opposite
intersection with $R$. (In general, some $e_i$ may be $0$). $R$ has a
curve$/\Supp{\sS_Z}$ by Lemma~\ref{trivbd} (cf.\ Example~\ref{bbasep}).
Thus there is $E_j$ that intersects the curve with $e_j\ne0$, or a
component of $S$ does so. If this component has nonzero numerical
intersection with $R$, we get a (S+$-$) flip by the above. Otherwise the
intersection is $=0$, and we again get a (S+$-$) flip with $S^-\ge E_i$.
 \end{exa-cr}

 \begin{defn} Let $(X/T,B)$ be a $0$-log pair and $E$ a prime
{\bi}divisor. A (birational) {\em Klt neighbourhood\/} $(Y/T,B_Y)$ of
$\cent_Y{E}$ is a model $(Y/T,B)$, that is also a $0$-log pair, and is
a Zariski neighbourhood of $\cent_Y{E}$ in it.
 \end{defn}

 \begin{lem} \label{nind} Under LMMP in dimension $n=\dim X$ and for
birational $X/T$, Klt neighbourhoods terminate for inclusions.
 \end{lem}

 \begin{proof} On each model this is a Noetherian property. But there
only exist a finite number of $0$-log pair models (the finiteness of
minimal models in the big case) \cite{sh96b}.
 \qed\end{proof}

We are now ready to establish stabilization in a birational neighbourhood
of $E=E_c$.

 \begin{thm} \label{stabnbh} {\em Under LMMP in dimensions $\le n-1$.}
Let $(X/T,B)$ be a log pair, $E$ a prime {\bi}divisor over a neighbourhood
of $P$, and $\sL_\bull$ a system of {\bi}divisors such that:
 \begin{q}
 \item[\rm(0LP)]
$(X/T,B)$ is a birational\/ $0$-log pair with\/ $\dim X=d\le n-1$;
 \item[\rm(LBF)] each $\sL_i$ is {\bi}free; moreover,
 \item[\rm(STB)] the $\sL_i\frest E$ stabilize over $E$, and each
$\sL_i\frest E=i\sM$, where $\sM=\sL_1\frest E$ is {\bi}free; and
 \item[$\RFA$] the situation is a restriction of a pl contraction in
dimension $n$, {\em that is, $\sL_i$ is a restriction of the mobile
system for some $\RFA_{n,d}$ algebra (cf.\ Definition~\ref{rfad} and the
proof below)\/}.
 \end{q}
 Suppose also that flips of type {\rm(S+$-$)} as in {\em
Example~\ref{lcresol}\/} exist in dimension $n$. Then there exists a
Klt neighborhood $(Y/T,B_Y)$ of $\cent_Y{E}$ on which all $\sL_i$ with
$i\gg0$ are free, ample and $\sL_i/i$ stabilizes {\em (over this
neighbourhood all $\sL_i/i=\sD$, the limit).\/} Thus after a
truncation, this holds for each $i>0$, and each $\sL_i$ gives this
quasi-projective neighborhood on a contraction given by $\sL_i$.
 \end{thm}

 \begin{proof} First, we recall and explain (RFA). There exists an
extension
 \[
 (X/T,B)\subset (X'/T',B'),
 \]
 where $(X'/T',B'')$ is a pl contraction of dimension $n$, whereas $X$ is
the intersection of the reduced divisor in $B''$, and $B'\ge B''$. Such
a complement always exists preserving $(X',B'')$ log terminal, and we fix
it. Then we define $B$ by adjunction:
 \[
 (K_{X'}+B')\rest X=K+B.
 \]
 We assume also that the system $\sL_\bull$ is the mobile restriction of
a {\bi}free system $\sL'_\bull$ (see \ref{sec!mrest}), that is:
$\sL_i\sim\sL'_i\frest{X}$, where $\sL'_\bull$ is the mobile system for
the $\RFA_{n,d}\bir$ algebra given by the pl contraction. Equivalently,
the mobile system of the restriction for the algebra, where in either
case we assume general position.

Fix some $i\gg0$, and consider the model $Z'/T'$ defined by $\sL_i'$. By
the construction of the (RFA) algebra, we can assume that
 \[
 \sL_i'\sim\sS'=S'+\sum e_iE_i\ge0,
 \]
as in Example~\ref{lcresol}. Note also that since $X/T$ is birational,
the model $Z/T$ of $X/T$ for $\sL_i$ is in $\Supp{\sS'_{Z'}}$ (and $Z$
is normal by Addendum~\ref{ressat}.1 and Example~\ref{fgapl}). It is
completely contained in any neighborhood $U'\subset Z'$ of
$\Supp{\sS'_{Z'}}$. Thus the destab model$/T$ over $Z/T$ is complete$/T$.

Now by Addendum~\ref{distabm}.1 and Example~\ref{lcresol}, for each
$\sL'_i$, there exists a destab$/T'$ model $(Y'/U',B'_{Y'})$ over some
neighborhood of $\Supp{\sS'_{Z'}}$ in $Z'$ (possibly not complete$/T'$).
Then by exceptional saturation we can destabilize multiples of
$\sL_i'/U'$ over the destab divisor, whose support includes the
exceptional$/T$ divisors. By (STB), $\cent_{Y'}{E}$ does not intersect
the destab divisor, that is, the exceptional divisorial locus of $Y'/T'$
for the log canonical model for $(Y'/U',B'+N L')$ with $L'=(\sL'_i)_{Y'}$.
Note: $\cent_{Y'}{E}$ is well defined for a sufficiently high
resolution ($g'':Y''\to X'$ that is also regularly dominated over $Y'$
by a Hironaka hut and $E$ is a divisor on $(g'')\1X$) for a blowup of
$E$. In particular, $\cent_{Y'}{E}\subset X_0$, where $X_0$ is the
birational transform of $X$ on $Y'$; and $\cent_{Y'}{E}=\cent_{X_0}{E}$

This gives the required neighbourhood. Indeed, if we apply adjunction of
the subboundary $(B')^{Y'}=\sB(X',B')_{Y'}$ on the normalization of $X_0$
(in fact, $X_0$ is normal) that is denoted also by $X_0$ we get an {\em
effective\/} different divisor outside exceptional divisors of $Y'/T'$.
In particular, the latter holds in a neighborhood of
$\cent_{Y'}{E}=\cent_{X_0}{E}$. On the other hand, if we identify $X'$
with $T'$ as we did for $0$-log pair, we get the adjunction
$(K_{X'}+B')\rest X=K+B$ with effective $B$. In addition, by the
commutative diagram the above adjunction gives the subboundary
$B^{X_0}=\sB(X,B)_{X_0}$ as the adjunction for the subboundary that is
effective on a neighbourhood $U\subset U'\cap Z$ of $\cent_{X_0}{E}$.
$K_{X_0}+B^{X_0}$ is numerically trivial$/P$. In particular, $X_0/U$ is
the identity. The discrepancies$/U$ for $(X_0,B^{X_0})$ are the same as
for $(X,B)$. Thus if we replace the subboundary $B^{X_0}$ by a boundary
$B_0\ge B^{X_0}$ that is the same where the subboundary is effective and
$0$ (or even $1$) where it is negative we obtain $K_{X_0}+B_0$ that is
nef on $U$, that is, on a closure of each curve in $U$. If $K_{X_0}+B_0$
is not $\R$-Cartier outside $U$ we can make it after some modification
outside $U/Z$ by the uniqueness as in Addendum~\ref{distabm}.1. The
discrepancies over $U$ for $(X_0,B_0)$ are the same as for $(X,B)$. Now
we can apply LMMP to $(X_0/T,B_0)$ the modification does not affect $U$
because we increase a discrepancy that is the same as for $(X/T,B)$.
However, at the end we obtain a $0$-log pair of $(X/T,B)$ with the same
discrepancies.

Finally, we can give $\sL_i\sim\sL'_i\frest X$ that is free and ample on
$U$ for $i\gg0$. Indeed, by Proposition~\ref{distabm} it gives a multiple
of $N\sL'_i+\sD$ since in the proposition the destab
$D$ is ample$/P$ and $N\sL'_i$ is {\bi}free$/P$. Since the neighborhood on
$Y'$ is not divisorial$/P$ it stabilizes (we can't add exceptional divisors
by Lemma~\ref{trivbd}). (See also Addendum~\ref{ressat}.1 and
Example~\ref{fgapl}.)
 \qed\end{proof}

Now we can apply Theorem~\ref{stabnbh} to the inductive model in the proof
of Theorem~\ref{stabin} to get a complete stabilization. This is the main
result of this section.

 \begin{cor} \label{stabg} If we assume LMMP in dimensions $d'\le
d=\dim X$, we obtain $\CBS_{d'-1}^*$ and $\SSB_{d'-1}\gl$. (Moreover, we
can omit the assumptions $\CBS_{d'-1}^*$ and $\SSB_{d'-1}\gl$ for
$d'\le3$, since they are known.)

Assume also that flips of type {\rm(S+$-$)} exist in dimension $n$. Let
$\sL_\bull$ be a mobile system of type $\RFA_{n,d}\bir$ on $X/T$. Then
there exists a $0$-log pair $(Y/T,B_Y)$ over which each $\sL_i$ with
$i\gg0$ is free and ample, up to a truncation.

This gives a solution for $\CBS_n\rfabir$ under the assumptions.
 \end{cor}

 \begin{lem} \label{lcsprop} {\em Under LMMP in dimension $d=\dim X$.}
 \begin{q}
 \item[\rm(0LP)] Let $(X/T,B)$ be a birational $0$-log pair;
 \item[\rm(NBU)] let $U$ be a nonempty open subset in $X$, that is
incomplete$/P$; and
 \item[\rm(CMP)] let $(X/T,B+C)$ be a complement {\em nonlog canonical\/}
$0$-log pair with $C=\sC_X$ such that $\LCS{(X,B+C)}$ is completely contained
inside $U$ {\em (cf.\ Example~\ref{cmd}\/)}.
 \end{q}
 Then there exists a different complement with a {\em new\/} $\sC$ such that
the {\em new\/}\linebreak[3] $\LCS{(X,B+C)}$ is not completely contained
in $U$, but all its $\LCS$ centers are in $U$, and $(X,B+C)$ is log
canonical outside $U$. Thus the nonlog canonical part $X_s$ of the
$\LCS$ is inside $U${\em, that is, does not intersect the complement of
$U$\/}. In addition, there is a {\em transient LCS center\/} $E$ such that
$(X,B+C)$ is log canonical in its generic point, and $E$ is not completely
in $U$. After a perturbation of $C$, such $E$ is unique, and is purely log
terminal in its generic point and outside $U$.

Moreover, we can find $C^+$ such that $(X/Z,B+C^+)$ is a general log Fano
contraction with the same support of $\LCS$ and the same purely log
terminal center $E$. Thus adjunction on $E$ gives a {\em general
nonnormal\/} log Fano contraction $(E/T_E,B_E,J_E)$, where
 \begin{itemize}
 \item $E/T_E=f(E)$ is proper;
 \item $(E^\nu/T_E,B_E)$ is a normal general log Fano contraction with
$-(K_{E^\nu}+B_E)\equiv-(K+B+C^+)\rest E$ ample$/T_E$ with normal
$E^\nu/E$; and
 \item $J_E$ is the ideal sheaf of the $\LCS{(E,B_E,J_E)}$ on $E$ {\em:
the direct image of that on the normalization)\/}; it is supported in
the intersection of $E$ with the closure of $\LCS{(X,B+C^+)}\setminus E$.
In particular, $(E,B_E)$ is normal and Klt outside $U_E=U\cap E$, and
$E/T_E$ is a contraction where $U_E$ is complete$/T_E$.
 \end{itemize}
 \end{lem}

 \begin{proof} We can find a maximal $c>1$ such that $K+B+c C$ is log
canonical outside $U$; this is the required $c$. It exists by (NBU)
because $\sC_X=(\overline{\sC_T})_X=f^*\sC_T$ and $P\in\Supp{\sC_T}$.
Set $X_0=X\setminus X_s$, that is, this is an open subset of $X$, where
$(X,B)$ is log canonical. Note that $X_s$ is completely in $U$.

Indeed, we verify that the generic points of the log canonical centers are
in $U$. Suppose that the exists such a center $E_1$ outside $U$, in
particular, $E_1$ is completely in $X_0$. On the other hand, by (CMP),
$c>1$, there is a center $E_n$ in $X_s$. By LMMP the LCS is connected.
Hence we have a chain of LCS centers $E_1,E_2,\dots,E_n$ with $E_1$
completely in $X_0$, $E_2$ in general in $X_0$, and $E_n$ completely
outside, that is, in $X_s$. Note that $K+B+cC\equiv 0/P$ on this chain.
This is impossible. Now we consider only $d=3$, that is, $X$ is a $3$-fold.
Then $E_i$ are at most of dimension $2$. We check that each $E_i\subset
X_0$, which gives a contradiction. If $E_2$ is completely in $X_0$ we can
use induction on $n$. Thus suppose that $E_2$ intersects $X_s$. If $E_2$
is a curve then $E_2/f(E_2)=P$ is complete$/P$. Otherwise we drop $E_2$
from our chain. On the other hand, by adjunction \cite{ka}, on
the normalization $E_2^\nu$, there is a boundary $B_2\ge0$, such that
$K_{E_2^\nu}+B_2\equiv 0$. In addition, $E_2$ has two points
$\nu(P_1)\in E_1\cap E_2$ and $\nu(P_2)\in E_2\cap X_s$ with
$\mult_{P_1}{B_2}\ge1$ and $\mult_{P_2}{B_2}>1$. That is impossible,
because $\deg K_{E_2}\ge-2$. Thus $E_2$ is a surface that is in general in
$X_0$. Now $E/T_E$ is a complete surface, or a fibering of curves. Again
adjunction on a normalization gives a boundary $B_2\ge0$ that has a
reduced divisor $P_1$ in $E_2^0=E_2\cap X_0\subset E_2^\nu$, and the
nonlog canonical center $P_m$ outside. On the other hand
$K_{E_2}+B_2\equiv 0/T_E$, so we have a chain of LCS centers for
$(E_2^\nu/T_E,B_2)$ as above (if it is not a contraction$/T_E$ make a
base change, or argue in the connected component of the fibre $E/T_E$).
That gives a contradiction by an induction on the dimension of $E$.

For $d\ge 4$ we can do similarly, by LMMP.

Thus new $\sC:=c\sC$. After a perturbation of $C_T=\sC_T$ we can get a
single transient center $E$. Perturbation: there is $H\ge0$ a $\Q$-Cartier
divisor on $T$ with $P\in\Supp H$ such that on a (sufficiently high) log
resolution $g\colon Y\to T$ of $(T,B_T+C_T+H)$, $g^*H\ge H'$, where $H'$
is ample$/T$. Then we can perturb $C_T$ and so $\sC=\overline{C_T}$:
replace by $(1-\ep)C_T+(1-\de)H''$, where $g^*H''\ge H-H'+\sum e_iE_i$
with $0<\ep,\de, e_i\ll 1$ and each $E_i$ is exceptional$/T$ or is in the
support of $B_T$ and $C_T$.

Finally, before the complement we can improve $(X/T,B+C)$ to a Fano
contraction$/T$ as in Lemma~\ref{incra} (non$\Q$-factorial case). Again
as in the perturbation $B^+-B\ge0$ should be $\ll$ than $C$.

The properties of $(C/T_C,B_C,J_C)$ are standard.
 \qed\end{proof}

 \begin{pfof}{Corollary~\ref{stabg}} By definition, the mobile system
$\sL_\bull$ is defined on a log Fano contraction $(X/T,B)$. Its
characteristic system is $\sD_i=\sL_i/i$. Using complements, we can
convert the pair into a $0$-log pair $(X/T,B)$ with a new fixed $B$. It is
also a weak log Fano contraction because $X/T$ is birational. Now
Corollaries~\ref{trunc} and~\ref{stabind} applied to $\sD_\bull$ give the
stabilization of $\sD_\bullet$ on some prime {\bi}divisor $E=E_c/P$, that
is, (STB) of Theorem~\ref{stabnbh} for $\sL_\bull$. Indeed, (LWF) holds
by construction. (LBF), \LCA\ and \MXD\ hold by Corollary~\ref{fgarfa}
as for a (FGA) system. In addition, \AMN\ also holds. Note that any
complement preserves \LCA\ by Lemma~\ref{monots}. \BED\ for $\sD_\bull$
after a truncation follows from Proposition~\ref{prop!fga} and
Theorem~\ref{cbsfg}, (3--4) under the assumption $\CBS_{d-1}$.

Therefore by Theorem~\ref{stabnbh} there is a Klt neighborhood $(Y/T,B_Y)$
of $E$ on which after a truncation each $\sL_i$ is ample and free, and
$\sL_i/i$ stabilizes. If $U$ is complete, we are done (compare the next
corollary). If not, we extend the neighborhood. This is eventually
complete by Lemma~\ref{nind} on Noetherian induction. (The final destab
$D$ does not intersect $Y$ at all.)

Suppose that $U$ is still not complete$/T$. Thus it satisfies (NBU). By
Noetherian induction, $U\ne\emptyset$ (because it includes old centers),
and satisfies (CMP). Hence Lemma~\ref{lcsprop} applied to
$(X/T,B):=(Y/T,B_Y)$ with previous $\sC$ gives a new transient center
$E$ and a new $C$ and $C^+$. Therefore if $\sD_\bull\frest E$
stabilizes$/T_E$ or (STB) holds for $\sL_\bull\frest E$, then again by
Theorem~\ref{stabnbh} up to a truncation we get that each $\sL_i$ is free,
ample$/T$, and $\sD_\bull$ stabilizes in a Klt neighborhood that extends
the old one by complete $\cent_X{E}$! (Compare the proof of the theorem.)

Thus we need to verify (STB) on $E$. We do this in dimension $d=3$. Thus
$\dim E\le2$. (As we see later, this is enough for $4$-folds flips.) For
higher dimensions we only explain $\CBS^*$.

Unfortunately, we can't apply Corollary~\ref{stabind} directly in the
current situation because the general Fano contraction $(E/E_T,B_E,J_E)$
has singularities even worse than log canonical. However,
Addendum~\ref{assdes}.1 on descent and Corollary~\ref{assdesc} in its
proof can be modified as follows, which secures the proof.

Suppose that $E$ is a surface; then by Proposition~\ref{ressat}, together
with the original $\sD_\bull$, the new $\sD_\bull:=\sD_\bullet\frest E$
satisfies lca saturation \LCA. The only novelty is that we now have
essentially negative components of $\sA:=\sA(E^\nu,B_E)$ (that is,
components with discrepancies $\le-1$).  The negative components defined
the ideal $J_E$. Since each $j\sD_i$ is Cartier on the LCS, saturation
is equivalent to the inclusion
 \begin{q}
 \item[(JLC)] On any sufficiently high model $E\hr/T:=T_E$ of $E/T$ that
is the {\em identity\/} or {\em contractible\/} in the neighborhood
$U:=U_E$ of the LCS, we have
 \[
 J_E((j\sD_i+\rdup{\sA})_{E\hr})\subset\Oh_T((j\sD_j)_{E\hr}).
 \]
 \end{q}

We construct a prediction $(E/T,\sC=\sA,F,\ga)$ for the descent problem
$\sD=\lim_{i\to\infty}\sD_i$ on a modification of $(E/T,B:=B,J:=J_E)$
into a triple (we can even normalize it, so assume $E$ is normal). The
modification is {\em contractible\/} if it does not blow up exceptional
divisors over $U$. However it can contract divisors: extend $U$ and
contract complete subvariety in the extension (see a case below). This
gives (EEF) on any log terminal resolution of $E$.

The system $\sD_\bull$ is bounded, by Example~\ref{bra}. By construction
each $\sD_i$ is effective and $(\sD_i)_U=\sD_U$ is stable on $U$. The
unstable part is over an effective divisor $F$ supported in the complement
$E\setminus U$. This gives (UAD) and (LGD) for some real number $\ga>0$
(cf.\ Example~\ref{fnonvex}).

In the two dimensional case, $\sD_E$ is nef = pseudo-nef$/T$ (that is, nef
over general complete curves$/T$ in any prime divisor), because this holds
for each $(\sD_i)_E/T$. The latter follows from (LBF): each
$\sM_i:=i\sD_i=\sL_i\frest{E}$ is {\bi}free$/T$. Then (SAM) follows from
generalized semiampleness (Florin Ambro): if $D$ is an $\R$-Cartier divisor
on $E/T$ such that it is Cartier and ample on $U/T$ and nef$/T$, then it
is semiample$/T$. By the rationality of the nef cone for such divisors
(variations only over $F$; LMMP$^*$, contract all exceptional curves in
$F$) it is enough to verify for $\Q$-divisors. This is essentially a
consequence of the general Fano property. For more details,\footnote{I'm
guessing the missing reference -- it's possibly in his thesis? The
adjunction conjecture and its applications, Ph.D. dissertation, Johns
Hopkins University 1999.} see \cite{Ambro}.  

To verify (SAB) we can use Proposition~\ref{surbound}. However to apply
this to approximations we need more from the triple and the prediction:
e.g., they should be strictly \w\ (WAM) for $\sD$, as explained below.

If the Kodaira dimension of $\sD$ is $0$, then each $\sD_i=0$ and we get
(STB).

If the Kodaira dimension is $1$, then, after a truncation, $|\sM_i|_E$ is
a pencil for each $i$. Its base point are only at good points, outside
$U$. Resolving these gives a prediction in a form of a triple
$(E/W/T,B,F)$, with crepant $B$, single $F$, and $g\colon E\to W/T$ given
by the pencil. Moreover, each $(\sD_i)_E=g^*M$ for some ample $M$ on
$W/T$. In general, this allows us to improve \w\ triples for $\sM=\sD$
(not {\bi}free) and $=\sM_i$
 \begin{q}
 \item[(WAM)] a \w\ triple for $\sM$ is {\em strict\/} if $M$ is ample
on $T/Z$ (see Definition~\ref{tri}); or at least $C\cdot M>0$ on each
complete curve$/T$ that intersects $g(\Supp{J})$.
 \end{q}
 If $U$ dominates $W/P$ we get (STB) from $W$, that is induced in turn
from $U$. Otherwise $U$ and $\Supp{J}$ do not dominate $W$, so in the
approximation methods in the proof of \ref{assdes} we can disregard $J$ in
(JLC). This means that
 \begin{q}
 \item[(BFF)] for any ample divisor $H$, or $H$ in a certain cone of
semiample divisors under the weak assumption in (WAM), and any proper
subscheme given by ideal $J$, $|J(NH)|$ is free outside
$\Supp{J}$ for $N\gg0$.
 \end{q}
 We can also use this in general. Note that in both previous cases we get
(SAB) without (CBS) (cf.\ Example~\ref{crv}). But we need
Example~\ref{1dfga}.

Finally, suppose that $\sD$ is big, then up to a truncation the same holds
for each $\sD_i$. By the saturation, for example, (JLB) with $i=j$, and
Proposition~\ref{surbound}, each $\linsys{\sM_i}_E$ is free on a terminal
resolution (up to a truncation again). That gives (CBS) and (SAB). The
triple is the resolution $(E/E/T,B,F)$.

However to apply approximations of the proof of Theorem~\ref{assdes} we
need (WAM) and (BFF) for $\sD$: e.g., $D=\sD_E$ should be ample$/T$. If
this is not true, there is a complete curve $C/T$ such that $C\cdot D=0$.
If $C$ does not intersects $U$ we can contract it because $C$ is
exceptional (as in LMMP$^*$ with good divisors on bad singularities).
This gives birational $E\to W/T$. The triple $(E/W/T,B_T,F_W=g(F))$ is a
general log Fano contraction with the same properties as $(E/E/T,B,F)$,
except for possible {\em nonterminal\/} points near $F$ but still Klt.
After such contractions we get (WAM) in the weak form, that is, each
$C$ with $C\cdot D=0, D:=\sD_W,$ intersects $U$. Since $D\ge0$, $C$ does
not intersect $\Supp D$ whenever $\sD$ is taken general on $U$. Then
(STB) holds on $C$, we can extend $U$ by a neighborhood of $C$, and then
contract $C$ (together with all the curves in the same numerical class,
to preserve the algebraic category). Note that now even in the {\em
big\/} case we need to modify (BIG), (BIR), and (MOD), and consider
$E/W$ that are not the identity. After that the methods of the proof of
\ref{assdes} are applied to the prediction $(E/T,\sA,F,\ga)$ with
approximations on $W/T$ (cf.\ Example~\ref{sac}).

If $E$ is a curve, the same arguments leads to a (rational) pencil of
rational curves that do not intersect the LCS on a surface with log
discrepancy $0$ over the curve. Such a pencil is free, and we can
treat it as the cases of Kodaira dimensions $0$ and $1$ above. Or we can
use adjunction on $E$ cite{ka}.

In higher dimensions, it is better to use adjunction on $E$ and over
non-Klt strictly \w\ triples of general log Fano contractions. Then
$\CBS_{d'-1}^*$ means the existence of such triple for each {\bi}free
$M$, and boundedness of the required triples. Above we verified that
for $d'\le3$, so we can drop it as the assumption. The next unknown
case $d'=4$: e.g., $\CBS_3^*$ for a $3$-fold $E$ in a $4$-fold $X$,
which is unknown even without $^*$.
 \qed\end{pfof}

 \begin{cor}\label{stbklt} {\em Under the assumptions of
Corollary~\ref{stabg}\/}. The limit $\sD=\lim_{i\to\infty}\sD_i$ of the
characteristic system of any $\RFA_{n,d}\bir$ algebra stabilizes. Thus
f.g.\ for $\RFA_{n,d}\bir$ holds under the assumptions.
 \end{cor}

 \begin{proof} Immediate from the proof of Corollary~\ref{stabg} or by
Corollary~\ref{stabg} itself, Theorem~\ref{cbsfg}, (3--4), and Limiting
Criterion~\ref{limcr}.
 \qed\end{proof}

 \begin{cor}\label{eplf} {\em Under the assumptions of
Corollary~\ref{stabg}\/}. In dimension $n$ there exist the pl flips.
 \end{cor}

 \begin{proof} Immediate by Main Lemma~\ref{mainl}, Corollary~\ref{stbklt}
and Corollary~\ref{fgflpal}. \par
 \qed\end{proof}

In the next section, we get rid of the assumption on the existence of
flips of type {\rm(S+$-$)} under $\CBS^*$.

\section{The main result} \label{mr}

 \begin{thm}\label{s+-} {\em Under LMMP in dimension $\le n-1$ and
$\CBS_{n-2}^*\bir$}. Flips of type {\rm(S+$-$)} exist in dimension $n$.
We can moreover omit the assumption that the flips are extremal {\em
($\rho(X/T)=1$; thus, $X/T$ is just small)\/}, and $\CBS_{n-2}^*$ with
$n\le4$.
 \end{thm}

 \begin{lem}\label{2ds} Let $\sL=\Lbar$ be an ample {\bi}divisor on
$X/T$, and $E$ and $E'$ two prime {\bi}divisors such that $E$ intersects
$E'$ divisorially. Then there is an effective Cartier {\bi}divisor $\sE'$
supported over $E'$ that {\em destabilizes} $\sL\frest E$, {\em that is,\/}
$(\Mov(\sE'+N
\sL))\frest E>N\sL\frest E$.
 \end{lem}

 \begin{proof} Taking hyperplane sections of $X/T$ by $L$ reduces the
lemma to the two dimensional case, with $X$ a surface. Then we can use
Example~\ref{edst}, (2). The main problem is that $E'$ may not be
$\Q$-Cartier. This can be resolved by a $\Q$-factorialization and
perturbation in exceptional components to preserve numerical ampleness$/T$.

In higher dimensions, we close\footnote{What? We take the closure of, or
we choose? I guess that you mean you fibre your higher dimensional variety
in surfaces, construct the flip of the generic fibre, then take the
closure.} a family of $\sE'$, constructed in dimension $2$.
 \qed\end{proof}

 \begin{lem}\label{wqf} {\em Under LMMP in dimension $n$.\/} Let
$B$ be a subboundary on $X$ with $\dim X=n\ge 3$, and $S$ a prime divisor
in $X$ such that:
 \begin{itemize}
 \item $K+S+B$ is $\R$-Cartier,
 \item $B$ has no reduced components,
 \item $(K+S+B)\rest S$ is Klt, and
 \item the negative components of $B$ intersect $S$ only in points.
 \end{itemize}
 Then there exists a {\em weak $\Q$-factorialization}, that is, a model
on which all the divisors that are exceptional$/X$ and the divisors that
intersect $S$ in a point, including the negative components of $B$, do not
intersect $S$.
 \end{lem}

 \begin{rem} \label{rem!tiny}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $S$ is normal outside its intersection points with negative
components of $B$ \cite[Lemma~3.6]{sh92} because $(X,B)$ is purely
log terminal there.

 \item We expect a more perfect form of the lemma in any dimension $n$, with
the intersections in codimension~$\ge 3$ rather than in points (cf.\ the
remark at the end of Step~1 in the proof of Theorem~\ref{s+-}).
 \end{enumerate}
 \end{rem}

 \begin{proof} Take a strictly log terminal resolution of $(X,S+B)$ that
we do with {\em reduced\/} exceptional divisors, intersecting $S$ in a
point, and negative components, that is, we set or change their
multiplicities to $1$. The reduced components of the resolution do not
intersect new $S$ because the construction give a log terminal resolution
of $(S,B_S)$, under the adjunction of original $(X,B)$, and the resolution
of $(S,B_S)$ does not have reduced components.
 \qed\end{proof}

 \begin{pfof}{Theorem~\ref{s+-}} Since the flip is strictly log terminal,
after slightly decreasing the other reduced components, we can assume
that both $S^+$ and $S^-$ are irreducible (cf.\ Example~\ref{lcresol}).
Thus the boundary is $B+S^++S^-$ with $\rdup{B}=0$.

This is a flip in the exceptional locus of dimension $\le n-2$. However,
the contraction is birational on $S^+,S^-$ and on their intersection
$E=S^+\cap S^-$. Indeed, if $E$ is contractible, it contradicts the normal
crossing of $S^+$ and $S^-$ because then $S^+\rest{S^-}=E$ is nef$/T$ and
exceptional on $S^-$. This last conclusion is impossible by
\cite[Negativity~1.1]{sh92}.

Let $\sL_\bull$ be the mobile system of a divisorial algebra
$\dival_{X/T}{D}$ with $D\sim S^-$, numerically negative$/T$. Then the
algebra defines a flip if it is f.g. We have a translation $t$ by
Addendum~\ref{mainl}.1, that corresponds to the inductive family with
$s=t=2$ and single $S_2=S^-$ (cf.\ Example~\ref{resralge} with
$S_1=S^+$).

By Proposition~\ref{exsatcan} the system $\sL_\bull$ is of type
$\FGA_n\bir$ over the pair $(X/T,B+(1-\ep)(S^++S^-))$ for $0<\ep\ll 1$,
because
 \begin{q}
 \item[(EXS)] $\sL_\bull$ is exceptionally asymptotically saturated$/T$.
 \end{q}
 Unfortunately, we cannot preserve condition (EXS) when restricting, but
the (FGA) is preserved for the birational restrictions $\frest{}$ of
$\sL_\bull$ on $S^+,S^-$ and $E$ by Example~\ref{monotsi} and
Proposition~\ref{ressat} (cf.\ Example~\ref{fgapl}).

In particular, $\sL_i\frest E$ is lca saturated. Thus by our assumption,
f.g.\ in $\FGA_{n-2}\bir$ follows from $\CBS_{n-2}\bir$ and
Theorem~\ref{cbsfg}, (3--4). Thus the characteristic system of
$\sL_i\frest E$ stabilizes. Since the translation $t$ is preserved
under the restriction, we get f.g.\ in (FGA) for $\sL_i\frest{S^+}$ on
$S^+$ by Main Lemma~\ref{mainl} with a single element inductive family
$S^+\cap S^-$ (cf.\ Corollary~\ref{fgapl3}).

For the same reasons, the characteristic system of $\sL_\bull$ stabilizes
over points (in general) of dimension $1$. Thus Proposition~\ref{prop!fga}
applied just to $\sL_\bull$ gives a stabilization for the characteristic
system of $\sL_\bull$ over $T\setminus P$ with closed $P$ with \BED\ after
a truncation.

\subsection{Preamble} We also establish below f.g.\ in (FGA) for
$\sL_i\frest{S^-}$. Thus by Main Lemma~\ref{mainl} again, with a single
element inductive family $S^-$ on $X$ we get f.g.\ in (FGA) for $\sL_i$
itself and get the flip on $X$.

To prove f.g.\ on $S^-$ we construct a $0$-log pair
$(S_m^-/T^-=f(S^-),B_{S_m^-}^-)$ over which each $\sL_i\frest{S^-}$ with
$i\gg0$ is free and ample, up to a truncation (cf.\
Corollary~\ref{stabg}). This gives (CBS) and f.g.\ by Theorem~\ref{cbsfg},
(3--4) because we can secure lca saturation for the system $\sL_\bull
\frest{S^-}$ as above.

First, in Step~1 below, we construct a Klt neighborhood $U_m$ of $E$ (as
$E_c$ but now no longer$/P$) on which all $\sL_i\frest{S^-}$ with $i\gg0$
are free, ample and $\sL_i/i\frest{S^-}$ stabilize {\em (over this
neighbourhood all $\sL_i/i=\sD$, the limit).\/} Thus after a truncation
this holds for each $i>0$, and each $\sL_i$ gives this quasi-projective
neighborhood on a contraction given by $\sL_i$. To compare with
Theorem~\ref{stabnbh}, now we can't us the (S+$-$) flips.

By Lemma~\ref{nind} and LMMP assumption neighborhoods $U_m$ of this form
considerd under inclusions terminate. The maximal $U_m$ is complete$/T^-$
and is the required $0$-log pair.

Finally, in Step~2 we extend each incomplete $U_m$.

\step1 Take $\sL=\sL_m$ for $m\gg0$ for which the restricted linear
systems $\sL_m\frest E$ and $\sL_m\frest{S_m^+}$ are very semiample
and have compatible restrictions
 \[
 \sL_m/m\frest E=\sD\frest E \quad\hbox{and}\quad
 \sL_m/m\frest{S_m^+}=\sD.
 \]
Then they are stabilized. This follows from the normality of restricted
algebras in Addendum~\ref{ressat}.1. Then the linear system of $\sL_m$
defines a flag of normal varieties $E_m\subset S_m^+\subset X_m$,
whereas $\sL=\Lbar$ for {\em just\/} ample but free $L=(\sL)_{X_m}/T$.
This can be done by a normalization of the model given by
$\dival_{X/T}{L}$.

However, the birational transform of $S^-$ in $X_m$ may still be
non\-normal. We denote its normalization by $\nu\colon S_m^-\to X_m$. We
claim that $U_m$ is a neighbor\-hood of $E_m$ in $S_m^-$. Note that $E_m$
is also embedded in $S_m^-$ by the above normality Addendum~\ref{ressat}.1.

By \BED\ and \EXC, or since $\sL$ gives the flip over $T\setminus P$, the
only exceptional divisors $E_i$ of $X_m/T$ are $/P$ (cf.\
Lemma~\ref{adjd}).

In addition, no exceptional $E_i/T$ intersects $S_m^+$ {\em
divisorially\/}. For otherwise, an effective exceptional$/T$ {\bi}divisor
$\sE'/E'=E_i$ destabilizes $\sL\frest{E}$ by Lemma~\ref{2ds}. This is
impossible by (EXS) on $X$ and \MXD\ on $E$ since
$\sL/m\frest{S_m^+}=\sL_m/m\frest{S_m^+}=\sD$ stabilizes.

In particular, this implies that adjunction for the crepant subboundary
$B_{X_m}$ on $S_m^+$ has an {\em effective different\/} (as we expect due
to the stabilization; cf.\ Addendum~\ref{assdes}.2). More precisely, by
a perturbation, we convert $(X/T,B)$ into a $0$-elt pair $(X/T,B^+)$
(actually, it is purely log terminal) with only reduced $S^+$. Thus
$S_m^+$ is a $0$-log pair for the adjunction $(K_T+B_T^+)\rest{f(S_m^+)}$
on $T_m^+=f(S_m^+)$ (which is normal). Hence by adjunction and the
commutative diagram up to codimension~$2$, that is, for general surface
sections, and near $S_m^+$, $K_{X_m}+B^+_{X_m}$ is log canonical with {\em
birationally\/} transformed $B^+$. The adjunction can be extended on the
whole of $S_m^+$ to a $0$-log pair $(S_m^+/T_m^+,B_{S_m^+}^+)$ with
 \[
 K_{S_m^+}+B_{S_m^+}^+=(K_{X_m}+B^+_{X_m})\rest{S_m^+};
 \]
 in particular, the pair is Klt everywhere on $S_m^+$ with
$B_{S_m^+}^+\ge0$ (crepant, for $(S^+,B^+_{S^+})$) by \cite[3.2.2]{sh92}.

Moreover, every exceptional $E_i$ intersects $S_m^+$ in at most one point.
Indeed, we can assume that the subvarieties in $S_m^+$ of dimension $\ge1$
in general are {\em weakly\/} $\Q$-factorial {\em for the exceptional
divisors\/} $E_i/T$; that is, if the dimension of intersection is $\ge1$,
these divisors intersect $S_m^+$ divisorially (in codimension~$2$ in
$X_m$). Indeed, taking a hyperplane section up to the intersection with
$E_i$ by a point, by Lemma~\ref{wqf}, we get weakly $\Q$-factorial for
divisors $E_i$ on a weak $\Q$-factorialization. Note that LMMP in the
lemma holds under our assumptions of the theorem. On the other hand, by
the numerical and geometric properties of $S^+$ and $S^-$,
$af(S^+)+b(\sL_m)_T$ is an $\R$-Cartier divisor on $T$ for some $a>0$ and
$b\ge0$; $b>0$ when $R\cdot S^+>0$. Then
 \begin{q}
 \item[(PRS)] on any model over $T$, $a S_m^++b L+\sum e_iE_i\equiv 0/T$
and, in particular, {\em locally\/} on $X_m$, $\R$-Cartier with
multiplicities
$e_i\ge0$ in exceptional $E_i$; and $e_i>0$ on $E_i/P$.
 \end{q}
 The latter is preserved on restricting to any general hyperplane
section. This is impossible for some curve$/X_m$ (over a point in $X_m$)
on a weak $\Q$-factorialization and, if the latter is nontrivial, its
exceptional curves are on the modification of $S_m^+$, by the
connectedness of the fibre $/X_m$. Thus as in the $\Q$-factorial case
the divisorial part always intersects in a divisor if it intersect in
a curve. (In the same way, we could eliminate intersections with $S_m^+$
in points if we knew the existence and termination of $n$-fold flips!
Thus we expect this near $S_m^+$, and that $S_m^-$ is normal by the
argument below. But for now we just disregard such intersections in
points.)

Therefore the {\em exceptional divisors $E_i$ intersect $S_m^+$ at most in
points\/}. In particular, the negative components in $B_{X_m}^+$ intersect
$S_m^+$ at most in points. The same holds for $E_m\subset S_m^+$ with the
adjoint subboundary $B_E^+$ given by the adjunction of a log terminal
$0$-log pair $(T/T,B_T^{+-})$. The pair has two reduced divisors $S^+$ and
$S^-$, a {\em substantial\/} nonreduced part, that is, for some $C>0$, and
$(T/T,B_T^{+-}-C_T)$ is a $0$-log pair. The second of these induces the
log pair $(T^-/T^-,B_{T^-}^-)$ which corresponds to a Klt neighborhood
$U_m$ under construction.

Now if we consider $E_m\subset S_m^-$, by the above, the negative
components of crepant subboundary $B_{X_m}^{+-}$ intersects $E_m$ in
points. Thus we have an adjunction of $(T,B_T^{+-})$ on
$(S_m^-,B_{S_m^-}^{+-})$ with the negative components in
$B_{S_m^-}^{+-})$ intersecting $E_m$ at most in points, and in turn an
effective adjunction of $(S_m^-,B_{S_m^-}^{+-})$ on $(E_m,B_{E_m}^{+-})$
with $B_m\ge0$. Hence again by Lemma~\ref{wqf} and the presentation
(PRS) the intersections even in points on $S_m^-$ is impossible, and
$B_{S_m^-}^{+-}$ is a {\em boundary\/} near $E_m$. The same holds for
$B_{S_m^-}^-$ on $S_m^-$. Moreover, the support of each restriction
$E_i\rest{S_m^-}$ in the sense of Mumford (restricted to the
normalization $S_m^-$ and defined only divisorially) {\em does not
intersect $E_m$ as closed subvarieties\/}. This is a crucial point that
{\em gives\/} a neighborhood $U_m$ as the complement to the supports.
Indeed, in the neighborhood $B_{S_m^-}^-$ is a boundary and
$\sL_{jm}/j\frest{U_m}$ stabilizes for all $j\ge1$! The
boundary\footnote{Something missing.} by \cite[3.2.2]{sh92}. Thus as at
the end in the proof of Theorem~\ref{stabnbh} (based on the uniqueness
in Addendum~\ref{distabm}.1) we can convert (= embed) $U_m$ into a Klt
neighborhood (into a crepant log pair for $(T^-/T^-,B_{T^-}^-)$).
Stabilization holds by the following:

 \begin{trick} By \AMN\ and \BED, and since the exceptional $E_i$ in
$X_m$ are$/P$, $\sL_{jm}=j\sL_m+\sE_{m,j}$, where each $\sE_{m,j}\ge0$
is {\bi}free (in particular, {\bi}Cartier) {\em over\/} $X_m$ with
$\Supp{(\sE_{m,j})_{X_m}}$ in a union of exceptional $E_i$. Hence by
Proposition~\ref{rest} (fx) {\em divisorially\/}
$\Supp{(\sE_{m,j}\frest{U_m})_{U_m}}=0$ (over $X_m$), or
$(\sE_{m,j}\frest{U_m})_{U_m}=0$, and by Lemma~\ref{trivbd}
$\sE_{m,j}\frest{U_m}=0$. Then $\sL_{jm}/j\frest{U_m}=\sL_m\frest{U_m}$.
This implies the required stabilization on $U_m$.
 \end{trick}

 \begin{rem} The proof is essentially higher dimensional: i.e., it is better to
assume $n\ge 4$. For $n\le2$, with only identical flips, no $E_i$, and
complete $U_m$ in Step~1. For $n=3$, under LMMP in dimension $3$,
Lemma~\ref{wqf} and (PRS) imply that each $E_i$ does not intersect $E_m$
at all (cf.\ Remark~\ref{rem!tiny}, (2)) and we have the same stabilization
as in Step~1. However the existence of $3$-fold log flips follows from
$\FGA_2\bir$ (see the proof of (FGA) in the Main Theorem at the end of
Section~\ref{canbound}), or from \cite{sh92}.
 \end{rem}

\step2 Using Lemma~\ref{lcsprop} we can extend the
neighborhood $U_m$, whenever it is still incomplete, by a transient center
$E_t$ in a crepant model of $(T^-/T^-,B_{T^-}^-)$, and establish a
stabilization of the restricted system $\sL_{jm}/j\frest{E_t}$ as in the
proof of Corollary~\ref{stabg} by $\CBS_d^*$ with $d=\dim E_t\le n-2$ and
the $\FGA^*$ property (with singularities) of the restricted system.

Finally, we extend $U_m$ to $U_{jm}$ for $j\gg0$ such that the restricted
$\sL_{jm}/j\frest{E_t}$ reaches stabilization. Indeed, if
$E_t\not\subset U_{jm}$ on $S_{jm}^-$, then we can make a destabilization
as in the proof of Corollary~\ref{stabg}. But now we replace
Proposition~\ref{distabm} by Example~\ref{edst}, (1), because $\sum
e_iE_i$ is $\R$-Cartier and the $e_i$ are positive for $E_i/P$. This
follows from ampleness of $L/T$ (which is in particular Cartier), and
because the $\sum e_iE_i\rest{S_{jm}^-}$ are disjoint from
$E_{jm}=S_{j m}^+\rest{S_{jm}^-}$ (by the log terminal property of
$(X_{jm},B_{X_{jm}^{+-}})$, $B_{X_{jm}^{+-}}$ is a boundary in the generic
point of $E_{jm}=S_{jm}^+\cap S_{jm}^-$, and \cite[Corollary~3.11]{sh92},
because in general $E=S^+\cap S^-$ goes birationally to $E_{m j}$ and the
exceptional$/T$ prime {\bi}divisors with log discrepancy $0$ are
only$/$ the generic point of $E_{jm}$). Indeed, by (PRS)
$a S_{jm}^++b L+\sum e_iE_i$ is $\R$-Cartier with $e_i>0$, and
$\equiv 0/T$.

Thus if we set new $m:=jm$ we can return to Step~2 again. The
above-mentioned termination completes the proof.
 \qed\end{pfof}

 \begin{pfof}{Theorem~\ref{cbrfa}} Immediate by Corollary~\ref{eplf} and
Theorem~\ref{s+-}. \par
 \qed\end{pfof}

 \begin{rem}\label{fgaa} During the proof of Theorem~\ref{cbrfa} we
essentially established that each $\FGA_3\bir$ algebra restricted from any
{\em exceptionally\/} saturated algebra on a contraction of
Example~\ref{resralge} is f.g. In general, this does not imply f.g.\ of
the algebra under restriction. Perhaps, this affects all $\FGA_3\bir$
algebras (cf.\ Remark~\ref{fga0lm}, (6)).
 \end{rem}

 \begin{pfof}{$\RFA$ in the Main Theorem} Immediate by Theorem~\ref{cbrfa},
because for $n\le4$ we can drop LMMP, $\CBS^*$, and $\SSB$(gl).
 \qed\end{pfof}

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\bigskip
\noindent Department of Mathematics,\\ Johns Hopkins University,\\
Baltimore, MD--21218, USA\\ e-mail: shokurov@math.jhu.edu

 \end{document}



Valery Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J.
Math. {\bf5} (1994) 779--810

