%%%%%%%%%%%%%%%%% %% V.V. Shokurov %% Prelimiting flips %%%%%%%%%%%%%%%%% \documentclass[12pt]{article} \usepackage{amsmath,amsfonts,amssymb,theorem} % \usepackage{amsthm} % \usepackage{latexsym} % \usepackage{amscd} % \input cyracc.def % \font\tencyr=wncyr10 % \def\cyr{\tencyr\cyracc} % \newcommand{\dival}[2]{\mathop{\mathcal R_{#1}}#2} % \newcommand{\cent}[2]{\operatorname{center}_{#1}#2} % \newcommand{\cont}[2]{\operatorname{cont}_{#1}#2} % \newcommand{\flpal}[2]{\mathop{\mathcal F\mathcal R_{#1}}#2} % \newcommand{\mult}[2]{\operatorname{mult}_{#1}#2} % \newcommand{\Proj}[2]{\operatorname{Proj}_{#1}#2} % \newcommand{\BDiv}[2]{\mathop{\mathcal Div_{#1}}#2} % \DeclareMathOperator{\BDiv}{\mbox{$\mathcal{D}iv$}} % \newcommand{\dival}[2]{\mathop{\mathcal R_{#1}}#2} \newcommand{\bi}{b-} % b- for birational, replaces original bi- \newcommand{\ADD}{{\rm(ADD)}} \newcommand{\AMN}{{\rm(AMN)}} \newcommand{\BED}{{\rm(BED)}} \newcommand{\BNF}{{\rm(BNF)}} \newcommand{\BSA}{{\rm(BSA)}} \newcommand{\BSD}{{\rm(BSD)}} \newcommand{\BSS}{{\rm(BSS)}} \newcommand{\BWQ}{{\rm(BWQ)}} \newcommand{\CGR}{{\rm(CGR)}} \newcommand{\CMD}{{\rm(CMD)}} \newcommand{\DEP}{{\rm(DEP)}} \newcommand{\DPT}{{\rm(DPT)}} \newcommand{\EFF}{{\rm(EFF)}} \newcommand{\EXC}{{\rm(EXC)}} \newcommand{\EXI}{{\rm(EXI)}} \newcommand{\FDS}{{\rm(FDS)}} \newcommand{\GLF}{{\rm(GLF)}} \newcommand{\GNP}{{\rm(GNP)}} \newcommand{\HOM}{{\rm(HOM)}} \newcommand{\IND}{{\rm(IND)}} \newcommand{\INO}{{\rm(INO)}} \newcommand{\LCC}{{\rm(LCC)}} \newcommand{\LCA}{{\rm(LCA)}} \newcommand{\MXC}{{\rm(MXC)}} \newcommand{\MXD}{{\rm(MXD)}} \newcommand{\SAT}{{\rm(SAT)}} \newcommand{\SDA}{{\rm(SDA)}} \newcommand{\TRL}{{\rm(TRL)}} \newcommand{\WLF}{{\rm(WLF)}} \newcommand{\rfabir}{(\mathrm{rfa},\mathrm{bir})} \newcommand{\fgabir}{(\mathrm{fga},\mathrm{bir})} \newcommand{\ASAprime}{$(\mathrm{ASA}')$} \newcommand{\CBS}{(\mathrm{CBS})} \newcommand{\fga}{(\mathrm{fga})} \newcommand{\gl}{(\mathrm{gl})} \newcommand{\FGA}{(\mathrm{FGA})} \newcommand{\PLF}{(\mathrm{PLF})} \newcommand{\RFA}{(\mathrm{RFA})} \newcommand{\SSB}{(\mathrm{SSB})} \newcommand{\ST}{(\mathrm{ST})} \newcommand{\bir}{(\mathrm{bir})} % special quote environment for logical statements such as (RFA) \newenvironment{q}{\begin{list}% {}{\setlength{\labelwidth}{2.00cm}\setlength{\labelsep}{1.00em} \setlength{\leftmargin}{2.00cm}\setlength{\rightmargin}{.65cm}}} {\end{list}} \newenvironment{qq}{\begin{list}% {}{\setlength{\labelwidth}{3.5cm}\setlength{\labelsep}{1.5em} \setlength{\leftmargin}{3.5cm}\setlength{\rightmargin}{1.0cm}}} {\end{list}} \DeclareMathOperator{\Bs}{Bs} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\CDiv}{CDiv} \DeclareMathOperator{\CS}{CS} \DeclareMathOperator{\reg}{reg} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\LCS}{LCS} \DeclareMathOperator{\Mov}{Mov} \DeclareMathOperator{\NEbar}{\overline{\mathrm{NE}}} \DeclareMathOperator{\NonSing}{NonSing} \DeclareMathOperator{\cent}{center} \DeclareMathOperator{\cont}{cont} \DeclareMathOperator{\dival}{\mbox{$\mathcal R$}} \DeclareMathOperator{\flpal}{\mbox{$\mathcal{FR}$}} \DeclareMathOperator{\mult}{mult} \DeclareMathOperator{\Proj}{Proj} \newcommand{\sm}{^{\mathrm{small}}} % D^sm = nonexcept. part of D \newcommand{\rest}[1]{{}_{{\textstyle{|}}#1}} % restriction to subset \makeatletter \newcommand{\frestdots} {\vbox{\baselineskip3\p@\lineskiplimit\z@\kern6\p@\hbox{.}\hbox{.}\hbox{.}}} \makeatother \newcommand{\frest}[1]{{}_{\frestdots #1}} % free restriction \newcommand{\1}{^{-1}} \newcommand{\bull}{{\scriptscriptstyle{\bullet}}} % \newcommand{\bull}{\mathbf\cdot} \newcommand{\ndiv}{\mathrel{\not|}} \newcommand{\broken}{\dasharrow} \newcommand{\into}{\hookrightarrow} \newcommand{\onto}{\twoheadrightarrow} % \newcommand{\ssw}{{\scriptstyle\swarrow}} % \newcommand{\sse}{{\scriptstyle\searrow}} \newcommand{\directsum}{\bigoplus} \newcommand{\tensor}{\otimes} \newcommand{\iso}{\cong} \newcommand{\pt}{\mathrm{pt.}} \newcommand{\wave}{\widetilde} \newcommand{\eqdef}{\mathrel{\,\stackrel{\mathrm{def}}{=}\,}} \newcommand{\rddown}[1]{\left\lfloor{#1}\right\rfloor} % round-down \newcommand{\rdup}[1]{\left\lceil{#1}\right\rceil} % round-up \newcommand{\linsys}[1]{\left|{#1}\right|} % linear system |D| \newcommand{\ep}{\varepsilon} \newcommand{\Ga}{\Gamma} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\de}{\delta} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\PP}{\mathbb P} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\Z}{\mathbb Z} \newcommand{\fC}{\mathfrak C} \newcommand{\fD}{\mathfrak D} \newcommand{\fF}{\mathfrak F} \newcommand{\fH}{\mathfrak H} \newcommand{\fL}{\mathfrak L} \newcommand{\fM}{\mathfrak M} \newcommand{\fN}{\mathfrak N} \newcommand{\fP}{\mathfrak P} \newcommand{\fQ}{\mathfrak Q} \newcommand{\fS}{\mathfrak S} \newcommand{\fU}{\mathfrak U} \newcommand{\sA}{\mathcal A} \newcommand{\sB}{\mathcal B} \newcommand{\sC}{\mathcal C} \newcommand{\sD}{\mathcal D} \newcommand{\sE}{\mathcal E} \newcommand{\sF}{\mathcal F} \newcommand{\sH}{\mathcal H} \newcommand{\sI}{\mathcal I} \newcommand{\sK}{\mathcal K} \newcommand{\sL}{\mathcal L} \newcommand{\sM}{\mathcal M} \newcommand{\Oh}{\mathcal O} \newcommand{\sR}{\mathcal R} \newcommand{\sS}{\mathcal S} \newcommand{\sT}{\mathcal T} \newcommand{\Dbar}{\overline{D}} \newcommand{\sLbar}{\overline{\sL}} \newcommand{\wB}{\wave{B}} \newcommand{\wD}{\wave{D}} \newcommand{\wf}{\wave{f}} \newcommand{\wS}{\wave{S}} \newcommand{\wX}{\wave{X}} % \newcounter{subsec}[section] % all theorems are numbered by the subsection counter \numberwithin{equation}{subsection} \numberwithin{footnote}{section} \newtheorem{rtheorem}[subsection]{Reduction Theorem} \newtheorem{intheorem}[subsection]{Induction Theorem} \newtheorem{mtheorem}[subsection]{Main Theorem} \newtheorem{sth}[subsection]{Special Termination} \newtheorem{mainl}[subsection]{Main Lemma} \newtheorem{cor}[subsection]{Corollary} \newtheorem{lem}[subsection]{Lemma} \newtheorem{prop}[subsection]{Proposition} \newtheorem{thm}[subsection]{Theorem} \newtheorem{conj}[subsection]{Conjecture} \newtheorem{sta}{}[subsection] { \theorembodyfont{\rmfamily} \newtheorem{exa}[subsection]{Example} \newtheorem{exa-cr}[subsection]{Example--Construction} \newtheorem{rem}[subsection]{Remark} \newtheorem{warn}[subsection]{Caution} \newtheorem{quest}[subsection]{Question} \newtheorem{defn}[subsection]{Definition} } \newcommand{\qed}{\ifhmode\unskip\nobreak\fi\quad\ensuremath\square} \newenvironment{proof}{\paragraph{Proof}}{\qed\par\medskip} \newenvironment{pfof}[1]{\paragraph{Proof of #1}}{\qed\par\medskip} \newcommand{\case}[1]{\paragraph{{\sc Case:} \rm #1}} \newcommand{\step}[1]{\paragraph{{\sc Step #1:}}} \title{Prelimiting flips \thanks{First draft completed on August 16th, 1999, Moscow.\copyright V.V. Shokurov, 2000. All rights reserved. Unauthorised copying, reproduction, hiring, lending, and public performance prohibited.} \thanks{Current draft in progress, edited by Miles Reid. I've only read it line by line up to about p.~60, and I haven't really understood anything. Forthcoming steps include making the table of contents more useful, and making an index, glossary and road map. Most of the footnotes are really questions to myself.} } \date{August 3rd, 2000} \author{V.V. Shokurov\thanks{The author was 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: the saturation of linear systems, and bounds on singularities for families of divisors. We state conjectures corresponding to 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 divisors under certain conditions in dimension~3. In dimension~3, the latter 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 \cleardoublepage \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}, the\footnote{find ref to this in \cite{ko}} 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, but {\em fibre space} or something else may be intended.} with only {\em log terminal\/} singularities. Thus by Connectedness of LCS (Koll\'ar and others \cite[Theorem~14.7]{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 Kawamata log terminal for the adjoint log divisor $K_Y+B_Y=(K+S+B)\rest Y$. These properties secure 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 dimension or {\em height} $d$ 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 an\footnote{I added this clause, because $T$ is otherwise not defined.} arbitrary base $T$ with $D=(K+B)$, assuming (cf.\ \cite[Theorem~p.~96]{sh92}) that \begin{itemize} \item $K+B$ is Kawamata log terminal, 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 $0\ll a<1$, since $-(K+aS)$ is Kawamata log terminal, 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 reals (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[$\PLF_n^{el}$] {\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 the 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 $n\le 4$, {$\PLF_n^{el}$} 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 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 a certain sheaf\footnote{Is singular or plural intended?} of algebras over algebraic varieties of dimension $m\le n-1$. \end{qq} More precisely, for any pl flip of height $d=n-s$, 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} The 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 the LMMP is known in dimension $\le 3$ \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 $m\le 3$ implies the existence of log flips in dimension $n\le 4$. 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}. \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 $m\le 2$; and \item[$\RFA_{n,m}\bir$] holds for all $n\le 4$ 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 $\le 4$. \end{cor} \begin{proof} Immediate by Corollary~\ref{mainc2} and the Main Theorem. \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 the 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 movable\/} 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 $\Oh_X(M)=0$. $\sA=\sA^X=\sA(X,B)$ denotes the {\em discrepancy\/} or {\em coboundary {\bi}divisor\/} of\footnote{The ``{\bi}'' stands for {\em birational}, that is, a {\em {\bi}divisor} is divisor on some model of $X$ up to birational equivalence. See \cite[Section~1]{sh96a}} 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 $(X,B)$. 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 each $\R$-divisor if $Y$ is $\Q$-factorial. The base field $k$ is of characteristic $0$. For example, $k=\C$. \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 Kawamata log terminal.\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 $00}\cdot D/{\sim_{\R}}$ in the space of divisors. \end{cor} \begin{proof} Immediate by Proposition~\ref{uniqbss}, because $rE>0$ and $H'=r H$ is ample. \end{proof} The most important applications have $E=0$; then we say that $D$ is {\em {\bi}semiample\/}$/Z$, that is, semiample 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 reals. \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 $f$. \end{proof} \begin{pfof}{Corollary~\ref{uniqf}} Immediate by Lemma~\ref{exfl}, Proposition~\ref{uniqbss} and Corollary~\ref{uniqbssc}. \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 $D\cdot C=0$. \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 $D^e=\sD^e=0$, $D\sm=D^m\sim_{\R} 0$ and $\sD^m\sim_{\R} 0$. \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 $d<0$ 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 movable 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{sta} 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{sta} \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 $E=mE'$. \end{proof} On the other hand, in certain cases we can't add an effective divisor to the movable 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$/Y$. \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 $s'\in i_*\Oh_Y(D)$. \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. \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. \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{sta} 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{sta} 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 $F-D\sim_{\R} 0$. 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. \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 $D_W^m\sim_{\R} 0$ 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 $D_W^m\sim_{\Q} 0$ 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 $D_W^m\sim_{\Q}0$ 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 $I\ndiv i$. Thus \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 $\sim$. \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 movable\/} 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 \[ \sR^N=\directsum_{i=0}^\infty\sR_{i N} \] (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 $X$. \end{pfof} \begin{rem}\label{gzardec} 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 Kawamata log terminal, 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 Kawamata log terminal 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}. \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. \end{proof} \begin{thm}\label{existd} The 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 the 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 Kawamata log terminal. 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 the 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 the LMMP we can make a $D$-flip $g\colon X\broken X^+/Z$, whereas $D=g^*D^++E$ 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 the 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. \end{proof} \begin{cor}\label{existc} In dimension $n\le 3$, we can omit the LMMP as an assumption in Theorem~\ref{existd}. \end{cor} \begin{proof} Immediate by Theorem~\ref{existd} and \cite[Theorem~5.2]{sh96b}. \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}. \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}. \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 $\dival_{X}{H}$ 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 \[ \rest S \colon \dival_{X}{H}\to \dival_{S}{H\rest S}. \] Moreover, to prove that $\dival_{X}{H}$ is f.g., it is enough to prove that the image subalgebra $\sR=\dival_{X}{H}\rest S\subset \dival_{S}{H\rest S}$ 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 $\dival_{S}{H\rest S}$. Thus by Truncation Principle~\ref{trprinc} it is enough to verify that $\dival_{S}{H\rest S}$ 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 $t_i\in \dival_{X}{D}$ 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~14.7]{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 $\rddown{B''}=0$, whereas 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 normalisation 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{sta} The required translations $t_i$ exist if $\sL=\dival_{X/Z}{D}$ is divisorial. \end{sta} \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 \[ \sL\rest{S^1},\quad \bigl\{S_i\rest{S^1}\bigm|2\le 1\le s\bigr\},\quad D\rest{S^1} \quad\hbox{and}\quad t_i\rest{S^1} \hbox{ for } 2\le i\le s \] 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 $(N-r)0$, 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 \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 $\sS$. \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 $\sM_i\le \overline{i D}=i\Dbar$ by Lemma~\ref{divb}, which implies (1) and boundedness. Finally, by Lemma~\ref{arithm}, (2) is equivalent to (3) and \ref{bpbdal}.1 with a {\bi}divisor $\sD$. \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). \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 $\vdots$.} \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}). \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{fgalmmp} 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}. \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 movable and fixed parts. Similarly, for any $\R$-{\bi}divisor, we can define $\R$-{\bi}divisors $\Mov{\sD}$ and $\Fix{\sD}$, the {\em movable\/} 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 $M=(\Mov{\sD})_{X_{hr}}=\Mov{(\sD_{X_{hr}})}$, $\Mov{\sD}=\overline{M}$, $\Bs{\linsys{M}}=\emptyset$, 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 (noncomplete) 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 movable 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 over 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 movable, or even base point 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 movable 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 base point free (cf.\ (1) in Proposition~\ref{texa}). 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{fgalmmp} 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 $\sD_\bull^{[I]}=I(\sD_{i I})$, that is, {\em each\/} $\sD_i^{[I]}=I\sD_{i I}$. \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, \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{fgalmmp}, 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 (ii--iii) 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} 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{fgalmmp} 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}{(\roman{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 Kawamata log terminal 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{conj}\label{fgalmmp} A functional $\Oh_T$-algebra $\sL\subset k(X)_\bull$ is f.g.$/T$ if \begin{q} \item[$\FGA_n$] it is bounded and\/ {\em lca saturated\/} over a pair $(X/T,B)$ such that \begin{itemize} \item $X/T$ is a contraction; \item $(X,B)$ is Kawamata log terminal; \item $-(K+B)$ is nef and big$/Z$; and, finally, \item $\dim X=n$. \end{itemize} \end{q} The two middle conditions mean that $(X/T,B)$ is a weak log Fano contraction (cf.\ \WLF\ in Proposition~\ref{exsatcan}). 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} Under these assumptions, we say that $\sL$, or its system $\sD_\bull$ are of {\em type} $\FGA_n$; they are of type $\FGA_n\bir$ if\/ $X/T$ is birational. We also use these to refer to the conjecture (cf.\ Induction Theorem~\ref{ith}). 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~\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 the 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 the 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{fgalmmp} holds for the $0$-log pairs $(X/Z,B)$ of Remark~\ref{gzardec}, (2) (cf.\ Theorem~\ref{existd}). \item Moreover, Conjecture~\ref{fgalmmp} 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 Kawamata log terminal, $B$ 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.\ \ref{assdes}.2). According to conjectures of Alexeev and Borisov,\footnote{Reference?} $X_{st}$ 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 Kawamata log terminal $\R$-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 Kawamata log terminal, 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 $\Mov{\rdup{j D_i+A}}\le j D_j$ 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 \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 $a=a_m>-1$, any real $d=d_m$ 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} For any effective $\R$-divisor $B$ such that $K+B$ is $\R$-Cartier, any exceptional saturation$/X$ implies lc 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 Kawamata log terminal, and $-(K+B)$ is nef and big$/T$ \end{q} (cf.\ Theorem~\ref{existd} and Remark~\ref{fga0lm}, (2) above). \end{prop} \begin{sta} Moreover, 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{sta} \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{sta} Asymptotic $\sC_1$-saturation implies asymptotic $\sC'_2$-saturation 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{sta} \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 $\sC_2+\sum\ep_i E_i$ 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 $d$. \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 directly from Lemma~\ref{monots}, and \ref{exsatcan}.1 follows from \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 Kawamata log terminal; \item $(X,B^+)$ is log canonical; \item $B^+\ge B$, 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 \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 it 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:\/} $\sC_{X_{hr}}\ge g^*(\sC_Y)+\sum a_i E_i$ 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 $F=f_i E_i$ 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_i E_i \\ &= (\sD+\sC+\sE)_{X_{hr}}+g^*(\sE +F)-\sE-\sum a_i E_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 \[ \mult_{E_i}{(g^*(\sE +F)-\sE-\sum a_i E_i)}= \mult_{E_i}{(g^*(\sE +F)-\sum a_i E_i)}<1 \] for each exceptional $E_i$. (For any integer $i$ and reals $f<1$ 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 $>-1$. \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. When $B$ 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{sta} $\sL\subset k(X)_\bull$ 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 movable 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{sta} 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 $I$. Statement~\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 movable parts when $j=i$. Indeed, \GLF\ and the connectedness of $\LCS{(X,B)}/Z$ (Koll\'ar and others \cite[Theorem~14.7]{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 Kawamata log terminal. 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 movable. 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. \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{fgalmmp}, 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~\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 $B^+/X_\vee$ when $0<\ep\ll 1$. 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 Kawamata log terminal, and the restriction $\sL\rest{S_1}$ has 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 Kawamata log terminal. 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{fgalmmp}). \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 \ref{mainl}.1, the algebra $\dival_{X/T}{D}$ has type $\FGA_n^{pl}$. Thus by Example~\ref{fgapl}, the restriction $\sL$ has 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 need only $\FGA_d\bir$ as for (RFA). \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 $\FGA_1$ gives also 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{fgadi} and in Theorem~\ref{stabin}). This follows by induction on $n$ after taking generic hyperplane sections. \begin{prop}\label{fgadi} 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 generic hyperplane section $H$ of $T$ and a point $P\in T$ and $Y=f\1H$. Then $\sL\rest Y$ has type $\FGA_{n-1}$. The lca saturation$/(Y/H,B\rest Y)$ follows from~\ref{ressat}.1 with $X/Z=X/T$ and $B:=B+H$, whereas \ASAprime\ holds by Example~\ref{monotsi}. Then the local f.g.\ of $\sL\rest Y$ near $H\cap P$ implies the 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). \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{curves} 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 $=$ up to the $\sim$ problem if we consider $\sD$ 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 $r$, $\sE(r\sD)=r\sE(\sD)/X$; \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 its 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}. \end{proof} For a generic divisor $D$, as well as for a generic {\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} 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~\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, that is, if it holds on one model of $X$ then it also holds on any other. It is equivalent to \BSD\ in~\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 characteristic system of a infinitely generated functional algebra. However under \FDS, and when, for infinitely many $i$, (CAR) holds on a single model $Y=X_i$, 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 reals $r_i$ 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 $\sC/r_i$, equivalently, $r_i\sE_i\le \sC$ over $Y$. \end{itemize} We say that asymptotic boundedness is {\em strict\/} if we can choose a sub\-sequence of reals $r_i$ 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 boundedness. For the truncation $I\sD_{i I}$, we can take the same $\sC$ and new reals $r_{i/I}:=r_i/I$, whereas we can take the same (truncated) reals $r_{i/I}:=r_i$ for the usual truncation $\sD_{i I}$. \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} 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. (Semiample gives some model and an $\R$-Cartier divisor on it by definition!) Thus we need a model $Y/Z$ of $X/Z$ such that: \begin{description} \item{(SAM)} $\sD_Y$ is {\em semiample\/}$/Z$, in particular, nef$/Z$. \end{description} 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 base point free\/}: $\Bs{\linsys{ND}}=\emptyset$ for some natural number $N$. We also need an $\R$-divisor $\sC$ of $X$, a reduced $\Q$-Cartier divisor $F$ on $Y$, and a positive real $\gamma$ 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 $P_i$ \[ \mult_{P_i}{\sC}>-1+\gamma\mult_{P_i}{F}, \] or, equivalently, \item[(AEF)] $\sC-\gamma \overline{F}$ is {\em almost effective\/}: $\rdup{\sC-\gamma \overline{F}}\ge0$. \end{q} The latter implies that $\sC$ is also almost effective: $\rdup{\sC}\ge0$, whereas (EEF) states more: exceptional effectiveness$/Y$. If such $Y,\sC, F$ and $\gamma$ exist, we say that the asymptotic descent problem has a {\em prediction\/} $(Y/Z,\sC, F,\gamma)$ 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). \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 $\sC\ge \ep \overline{F}/Y$ for some $\ep>0$, and $\rdup{\sC_Y}\ge0$ then (LGD) holds some $0<\gamma\ll \ep$. \item For simplicity, we can assume in (LGD) that for {\em any\/} $F$, there exists a positive $\gamma$ 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 the similarity (cf.\ Lemma~\ref{intpart} below). \end{enumerate} \end{rem} Now we are ready to state the main result of the section. \begin{thm} \label{assdes} Let $\sD=\lim_{i\to\infty}\sD_i$ be an asymptotic descent problem such that \begin{itemize} \item the problem has a prediction $(Y/Z, \sC, F,\gamma)$; \item $\sD_\bull$ satisfies the minimal assumptions: {\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{sta} If, in addition, \BNF\ holds, then the strict asymptotic descent problem has a solution on $Y$. \end{sta} \begin{sta} If, moreover, Proposition~\ref{coher}, \CGR\ 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 \[ X_{st}/Z=\Proj_Z{\dival_Z{ \overline{\sD_Y}}}=\Proj_Z{\dival_Z{\sD}}, \] $\sC_{X_{st}}\le0$ (cf.\ Conjecture~\ref{cbcon}). \end{sta} 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{fgalmmp}, {\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{sta} 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{sta} \begin{proof} Immediate by the theorem by \MXD\ and \AMN. \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 $\sD_i=d_i \overline{E}$, where $d_i$ is a sequence of reals 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} Let $\fQ\subset\R^N=\Z^N\tensor_{\R}\R$ be a rational polyhedron, and $D=(d_1,\dots,d_N)\in \fQ$ a point in it. Then there exist positive reals $r$ 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} Notation: The norm $\Vert (x_1,\dots,x_N) \Vert$ in $\R^N$ denotes the {\em maximal absolute value\/} norm for the coordinates: \[ \Vert (x_1,\dots,x_N)\Vert=\max\{|x_i|\}. \] (The norm 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 $d_i'\le d_i$. \begin{proof} First, 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. Second, we can assume that each coordinate $d_i$ of $D$ is irrational; and, moreover, that $D$ does not belong to any rational proper (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 imbedding $A\colon\R^m\into\R^N$ 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 $P_m$ is a required sequence $P_m\in\fP$ for $C$, then $Q_{I m}=A(P_m)$ gives a required sequence for $D=A(C)\in\fQ$, where $I$ is an {\em index\/} of $A$, that is, $I A$ is integral. Respectively, the set of natural numbers $m$ is replaced by $I m$ (truncation). Hence $I m Q_m=(I A)(m P_m)\in \Z^N$; and \[ \Vert D-Q_{I m}\Vert=\Vert A(C-P_m)\Vert< \Vert A\Vert \ep/ m^{1+r}=\Vert A\Vert I^{1+r}\ep/ (I m)^{1+r}; \] and we can take $\ep:=\Vert A\Vert I^{1+r}\ep$ for $Q_{I m}$, 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 the conditions for $D'$ we need to replace by geometric ones: e.g., 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 $Q_{I m}$ 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 the latter). Third, 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 by Step two we can reduce the lemma to $H$. Therefore by induction on $N$ we got the interior. Then an existence of required approximations follows from Cassels \cite[Theorem VII of Chapter I]{cas57} with any $0\mult_{P_i}{D}$ 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{description} \item{(NBH)} each divisor in a {\em neighborhood\/} of $D$ is (nef and) semiample$/Z$; moreover, \item{(BND)} holds in it. \end{description} 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 \[ \fD=g^*\fH\oplus(\directsum\R(f_i))\subset \fD_F \] 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 the 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$/\Q$: $\mult_{P_i}{X}=\mult_{P_i}{D}$ whenever the latter multiplicity is rational. By the construction \begin{itemize} \item $\fQ$ is a rational polyhedron; and \item $D\in \fQ$. \end{itemize} Then Lemma~\ref{aproxl} gives reals $r$ and $\ep$, and a sequence of $\Q$-divisors $Q_m\in\fQ$. They satisfy the corollary except for the base freeness of $\linsys{mQ_m}$. 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 $Q_{M m}:=Q_m$ and take $\ep:=M^{1+r}\ep$. \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 $X$, let $F$ be a reduced divisor, and $\al\le 1,\tau$ be positive reals such that: \begin{itemize} \item $F$ is $\Q$-Cartier and \begin{itemize} \item[$(*)$] in each prime {\bi}divisor $P_i$, $\mult_{P_i}{\al \sC}>-\al+\tau \mult_{P_i}{F}$; \item the descent data $\sE$ of $\sD$ over $X$ is bounded by $(1-\al) \sC$, {\em that is, \item[$(**)$] $\sE\le(1-\al) \sC/X$;} \end{itemize} and \item $\sD_X=D_{bf}+D_{fr}$, where $\linsys{D_{bf}}$ is base point 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 $W/X$ \[ \Mov{\rdup{\sD+\sC}_W}\ge (\Dbar_{bf})_W; \] 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 $\sD_X$, $\sD=\overline{D_{bf}+D_{fr}}-\sE=\overline{D_{bf}}+\overline{D_{fr}}-\sE$. Hence $\sD+\sC=\overline{D_{bf}}+\overline{D_{fr}}+\sC-\sE$. Since $\overline{D_{bf}}$ 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 the Choice of $Y$ in the asymptotic descent above). We make it into two steps. First, on $X$. Since $0<\al\le 1$, (*) implies \begin{description} \item{(***)} $\rdup{\sC_X-\tau F}\ge0$. \end{description} 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 $\overline{D_{fr}}\ge-\tau\overline{F}$. 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 $\linsys{D_{bf}}$ is base point free. \end{proof} \begin{pfof}{Theorem~\thesection.\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 the strict asymptotic descent we can make 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 movable 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 movable part is $\not\le j(\sD_j)_W$ for such $i$ and $j$ on any sufficiently high model $W=X_{hr}$. A movable part estimate 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 reals $r$ 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 base point 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 $\sD_i'=\sD_i- \overline{E}$, and $E=\sD_Y- D'\ge0$ is an effective divisor on $Y$ by the construction. Since $\overline{E}$ is also effective, then, for the same $j$ and $i\gg0$, on any model $W/Y/Z$ of $X/Z$ \[ \Mov{\rdup{j\sD_i+\sC}_W}=\Mov{\rdup{j\sD_i'+j\overline{E}+\sC}_W}\ge \Mov{\rdup{j\sD_i'+\sC}_W} \ge m (\overline{Q_m})_W. \] 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 $\sD'=\sD-\overline{E}$; 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 $\sE(\overline{E})=0$ and $\sE(\sD_i')=\sE(\sD_i-\overline{E})=\sE(\sD_i)/Y$ by \EFF\ and \ADD\ in Proposition~\ref{desd} \end{itemize} Now take any positive $\al<1$, and set $\tau=\al\gamma$. 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 $\sD_Y=m D'=D_{bf}+D_{fr}$ satisfies the assumptions for the decomposition in the lemma where $\sD:=m\sD'$, $D_{bf}=m Q_m$ 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 $\sD_Y=j (\sD_i')_Y=D_{bf}+D_{fr}$ satisfies the assumptions of Lemma~\ref{movest}, where $D_{bf}=m Q_m$ 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} \[ \Mov{\rdup{j\sD_i'+\sC}_W}=\Mov{\rdup{\sD+\sC}_W}\ge (\overline{D_{bf}})_W=m (\overline{Q_m})_W. \] 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 again Corollary~\ref{aprox}, but now in the trivial case that $D:=\sD_Y$ is a $\Q$-divisor, and by above arguments, we can't derive this time 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, such $(\sD_j)_Y=\sD_Y$. Third, we assume \BNF\ and prove the stabilization in \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 the stabilization, namely, such $\sD_j=\sD$. 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 \ref{assdes}.2: e.g., $D:=\sD_Y$ contracts all prime divisors $P_i$ 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 Reid's arguments \cite[Proposition (1.2)]{r80}. By our assumptions, $\de P_i\le \sC$ for some $\de>0$. By asymptotic descent \ref{assdes}.1 and by Reid (or by \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 base point 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 \[ j(\sD_j)_W+P_i=\Mov{(j(\sD_j)_W+P_i)}\le \Mov{\rdup{j\sD_j+\sC}_W}, \] that 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. Indeed, we can assume that $m'\ge m$ and $\linsys{(m'-m)D}$ is base point 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}). \end{pfof} Now we are ready to explain what one needs to prove (FGA) and apply to flips (cf.\ Corollary~\ref{mainc1}). \begin{exa} \label{stexm} Let $\sL$ be an algebra of type (FGA) over $(X/T,B)$, and $\sD_\bull$ its characteristic system (cf.\ Conjecture~\ref{fgalmmp} above). We expect that such $\sL$ is always f.g. By Theorem~\ref{limcr} it is equivalent to an affirmative solution of the asymptotic descent problem for the limit $\sD=\lim_{i\to\infty}\sD_i$ on some model $Y/\Proj_{T}{\sL}/T$ of $X/T$, whereas $\sD$ is {\bi}semiample$/T$ and semiample$/Y/T$ $\Q$-{\bi}divisor. The latter means that $D=\sD_Y$ is a $\Q$-Cartier and semiample$/T$ divisor, and $\sD=\Dbar$. By \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~\ref{bpbdal}.1 implies \FDS. Moreover, by (FGA) in Conjecture~\ref{fgalmmp} the asymptotic saturation holds for $\sD_\bull$ with $\sC=\sA=\sA(X,B)$, that is, the lca saturation holds. By Example~\ref{coha}, \CGR\ holds for chosen $\sC=\sA$. Thus the main obstacle to apply Theorem~\ref{assdes} is to choose a prediction $(Y/Z=T,\sC,F,\gamma)$. Our choice of $\sC=\sA$ is canonical. On the other hand, because $(X,B)$ is Kawamata log terminal, on any model $Y/T$ of $X/T$ and for any $\Q$-Cartier divisor $F$ on $Y$, (LGD) holds for some $\gamma>0$ (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 fulfil 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 blownup, that is, $P_i$ is a divisor on $Y$. Again since $(X,B)$ is Kawamata log terminal, such models exists by \cite[Lemma~1.6]{sh96b}. In addition, we can assume that $Y$ is $\Q$-factorial. Using the 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 $\le 3$ \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 the 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 also $0$-log pair, and $(X^+/T,(B')^+)$ is a weak log Fano contraction. Therefore it again satisfies (RPF). Since $(X/T,B)$ satisfies \WLF, under the LMMP, we can apply this to modify $X/T$ making a divisor semiample (cf.\ Theorem~\ref{existd}). For any effective $\R$-Cartier (even up to $\sim_{\R}$, cf.\ the same effectiveness of bss divisors before Proposition~\ref{uniqbss}) divisor $D$, 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{description} \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 in generic curves$/T$ of any divisor in $X$, \end{description} 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 needed termination is as in the proof of Theorem~\ref{existd}, that is, by the LMMP for $(X/T,B'+\ep D)$, where $(X/T,B')$ is a $0$-log pair. Note that the use of the LMMP is justified by our approach (cf.\ Corollary~\ref{mainc1} and the remark at the end of Conjecture~\ref{fgalmmp}). Since f.g.\ is independent on the quasi-isomorphism of $\sL$, equivalently, the stabilization does so on the similarity for $\sD_\bull$, we can assume that each $\sD_i$ is effective. Moreover, by (LBF) in Conjecture~\ref{fgalmmp} 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 by (RPF) even is semiample$/T$ on $Y_i$. \end{itemize} After taking a complement $(X/T,B')$, by the Borisov--Alexeev conjecture we expect a finite number of such models $Y_i/T$. The same follows from the Kawamata conjecture on finiteness of the log minimal models. However it is enough (the 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 other properties of 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 resolution follows from \cite[Theorem~3.1]{sh96b} again up to the LMMP. (Apply to $(X,B')$ with $B'\ge B$ and such that new discrepancies $a_i:=\mult_{P_i}{\sA(X,B')}$ is negative for all exceptional $P_i$ with old $a_i:=\mult_{P_i}{\sA(X,B)}\le0$.) This fulfills all the conditions of a prediction except for the boundedness (SAB)! Actually, this may not always be so (cf.\ \ref{surbound}.2 and Example~\ref{esurbound}). However we expect that this hold for such a model $Y/T$ whenever $\sD$ is big (cf.\ Conjecture~\ref{cbcon}, and Proposition~\ref{surbound}). Moreover, this is the only possible choice by \ref{assdes}.2 and Example~\ref{fcanb}. \end{exa} Thus the main difficulty in construction of pl flips is the 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 boundedness and saturation} \label{canbound} We start with another interpretation for (asymptotic) boundedness in the {\em canonical\/} case where $\sC=\sA=\sA(X,B)$ is the discrepancy {\bi}divisor (cf.\ Example~\ref{stexm} above, and 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 $c\ge0$ 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 the Choice of prediction in Section~\ref{satdes}). 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 the Choice of prediction in Section~\ref{satdes}), or replace $\sA$ by its $\ep$-log version whenever 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.\ \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, 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 such $\eta$, 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 $c\sE\le \sA/\eta$. \end{pfof} \begin{defn}\label{cbd} 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 the set has {\em canonically bounded\/} singularities on $(X,B)$, or, simply, that it is {\em canonically bounded\/} on $(X,B)$ when there exists a real $c>0$ 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 $(X,B)$ is replaced by a family $(X_i,B_i)$ of log pairs for each member of which $X_i$ is a model of $X$, the boundedness of $\fD$ on the {\em 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 their family (cf.\ Remark~\ref{discrm}). Thus in case of $\ep=1$, and for the points $\eta$ of codimension~$\ge2$, we get again the canonical (terminal) boundedness. For $\ep=0$, we get a 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 for 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, by a {\em bounded\/} set of divisors we mean a set of divisors $\fD$ 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 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 a 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 generic element in a base point 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 rather generic {\bi}free, that is, as in (2) on some model $Y/X$ of $X$, and $(X,B)$ is Kawamata log terminal, 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 apply the canonical boundedness to asymptotic boundedness. \begin{prop} \label{aboundd} Under 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 the canonical boundedness of $\fD$ 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{sta} Let $\sD_i$ be a sequence of {\bi}divisors, and $p_i$ a sequence of positive reals 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 numerically equivalent), where \item all the $\sM_i$ are canonically bounded by $c$ on a {\em common\/} model $(Y=X_j,\linebreak[2] B_Y=B_j)$ in the family. \end{itemize} Then the descent data for sequence $\sD_i$ is asymptotically bounded by $\sA/Y$. \end{sta} \begin{proof} The boundedness $c\sE\le \sA/X_i$ follows from Lemma~\ref{discr} for $(X,B):=(X_i,B_i)$, for the set $\{\eta\}$ of points of codimension~$\ge2$, and for $c$ given in the canonical boundedness, where $(X_i,B_i)$ itself corresponds to $\sD$ under the latter boundedness. By the \HOM\ in Proposition~\ref{desd} $\sE(c\sD)=c \sE(\sD)/X:=X_i$. Since $c>0$, then each $r_i=c p_i>0$ and also $\lim_{i\to\infty}c p_i=+\infty$. On the other hand, by Proposition~\ref{desd}, \DEP\ and \HOM\, $r_i\sE_i=c p_i \sE_i=c\sE(p_i\sD_i)=c\sE(\sM_i)/Y$. Hence the above boundedness for $\fD$ implies $r_i\sE_i=c\sE(\sM_i)\le \sA/Y$, that means the asymptotic boundedness. \end{proof} We define wanted pairs $(X_i,B_i)$ of bounding families in terms of triples. \begin{defn}\label{tri} 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 wanted\/} 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. Its {\em wanted\/} triple is a triple $(Y/T,B_Y,\fF)$ such that \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 wanted triples, we need to weaken conditions: e.g., $X/T$ may be a noncomplete morphism of normal varieties, cutting log singularities, or $(X,B)$ may not be Kawamata log terminal. \item Maybe, the boundedness of $\fF$ is better to spread into a 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.\ the warning after Definition~\ref{lca}). \item Perhaps, we can drop (QFC) but then we can only estimate singularities of $\sD$ having an $\R$-Cartier restriction $\sD_Y$ (cf.\ Lemma~\ref{discr} vs Lemma~\ref{split}). In many applications this holds. For example, $\sM_Y$ is $\Q$-Cartier by definition on its wanted triple. \item In (TER) the divisor $B_Y$ may not be a boundary. However, since $(X,B)$ is Kawamata log terminal, $B_Y$ is a {\em Kawamata log terminal subboundary\/}: that is, each its multiplicity $b_i<1$. \item (RPC) is in the middle of the following conditions on $T/Z$ in order of increasing strength: \end{enumerate} \begin{q} \item[(LSA)] the limit of any semiample on $T/Z$ is semiample on $T/Z$, and assume that $M$ is {\em semiample\/} in the definition of wanted triples (cf.\ Remark~\ref{minar}, (4)); \item[(NSA)] each nef$/Z$ divisor $M$ on $T$ is semiample; \item[(0LP)] $T/Z$ is a $0$-log pair for some boundary $B_T$ as in Remark~\ref{gzardec}, (2); in particular, then (NSA) holds for each effective divisor, and we can assume that $\sM$ is effective, any limit of effective divisors is also effective under \MXD; \item[(RPC)] of the definition (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}), it is enough (LSA). But for approximations we need at least (RPC) (cf.\ Theorem~\ref{bnonvan}). In addition, we prefer numerical properties vs linear ones. \end{rem} The reader may wonder what is the purpose of $\fF$ in triples? In fact, we need very special sets $\fF$ that we call {\em standard\/}. First, in typical examples, that appear below in Corollary~\ref{mainb} and in Proposition~\ref{bounds} as fixed and of fractional parts, the standard families are bounded. Second, they generalize the canonical boundedness 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 pair $(X/Z,B)$ if it contains only standard divisors on the pair. 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 the Choice of prediction in Section~\ref{satdes}); 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 standard divisors. In general, it may be unbounded. However by $\SSB$ in Conjecture~\ref{cbcon} we expect that this is true 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 $\fS$ from $X$. A wanted triple for $(X/Z,B)$ with {\em the induced\/} set $\fS_Y$ is a wanted triple $(Y/T/Z,B_Y,\fS_Y)$ with the induced standard set. \end{defn} \begin{warn} $\fS_Y$ itself can be actually {\em non\/}-standard (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 $\fS_Y$ 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 $Y$ which is exceptional on $X$. Note that $\sum E_i$ is finite and fixed for such transform. \end{enumerate} In general, $\fS_Y$ is not standard even up to the log additive $\sum E_i$. Let $X$ be a Del Pezzo surface having two $-1$-curves $E_1,E_2$ with the simple intersection in a single point $P$: e.g., 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 so (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 movable on $Y$. 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 $(X_i,B_i)$ itself (cf.\ Example~\ref{tre} (3)) may be unbounded, 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}[On canonical boundedness]\label{cbcon} 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, it includes the 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) in 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 prime fixed components\/} of $\sD\in\fD$ on $X$ is {\em bounded}, that is, of prime components of $(\Fix{\sD})_X$ (cf.\ Remark~\ref{cbrm}, (4)) \end{q} (PFC) is equivalent to (PRM). Actually, the set is equal to $\fP$ (cf.\ Theorem~\ref{cbsfg}, (1) and its proof). (PRM) and (PFC) hold on any wanted triple $(Y/T/Z,B_Y,\fF)$ if they hold on $(X/Z,B)$, by definition. Indeed, $\fP(Y/Z,B_Y)$ is the birational transform of the $\fP$ plus some $E_i$ on $Y$, exceptional on $X$. Moreover, we expect that \begin{q} \item[\rm(GFC)] the boundednesses (PRM), (PFC) and $\SSB$ hold on any {\em general\/} log Fano contraction $(X/Z,B)$ (see \GLF\ in Proposition~\ref{ressat}) with {\em Kawamata log terminal\/} 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 $\fD$ into the {\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 Kawamata log terminal, \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); and ((SAF) in its turn is equivalent to) \item[\rm(STD)] of Definition~\ref{ssd} for {\em fixed\/} $\sD:=\sF$ by Proposition~\ref{ssp}, (3). \end{q} Hence $\sF$ and, by Lemma~\ref{monots} with $\sC_1=\sA+\sF$ and $\sC_2=\sA$, $\sM$ also satisfy \SAT. We use the splitting $\sD=\sM+\sF$ to estimate singularities of the sum in terms of both components. Moreover, we can consider any such sum (cf.\ Remark~\ref{cbrm} (6)). Indeed, $\sM$ and $\sF\ge0$, and, on each wanted triple $(Y/T/Z,B_Y,\fF)$ of $(X/Z,B)$, $\sD_Y\sim \sM_Y+\fF_Y$ with $\Q$-Cartier Weil divisor $\sM_Y\ge0$ and with $\R$-Cartier $\sF_Y\ge0$. Thus by the convexity of canonical property (cf.\ \cite[1.3.1]{sh92}) it is enough to establish the latter for each summand. If $\sF$ is bounded by $c_f$ from below, and $\sM$ does so for $c_m$ then $\sD$ is bounded for any weighted combination of them: e.g., half sum $c=(c_m+c_f)/2$. On any triple $(Y/T/Z,B_Y,\fF)$ of the required type, by (PRM) and Example~\ref{tre}, (1), each irreducible component $f_i P_i$ of $\sF_Y$ (with prime $P_i$) has canonically bounded singularities when the multiplicities $f_i$ are bounded. For the whole $\sF$, we need more: e.g., a boundedness of the number of irreducible components that is equivalent to $\SSB$ and holds for $\sD:=\sF$ under (SEF)$/Y$ in Definition~\ref{ssd}, or for the log transforms in $\fS_Y$ from $(X/Z,B)$. The movable part $\sM$ behavior worse: (cf.\ Remark~\ref{cbrm}, (7)) \begin{itemize} \item the canonical boundedness from below holds for {\em some\/} effective $\sM$, usually means rather {\em 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\/} (always in the paper; cf.\ Remark~\ref{bndf}) {\em bounded\/} family of wanted 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 the log saturation \SAT. More precisely, for each $\sM\in \fM$ there is $\sD\in\linsys{\sM}$ and a wanted {\em for\/} $\sM$ triple $(Y/W/T,B_Y,\fF)$ in the family such that $\sD$ is canonically bounded on $(Y,B_Y)$. \end{q} Thus jointly \begin{description} \item{(GCB)} the {\em canonical boundedness\/} for some $\sD+\sF\sim \sM+\sF$ over the family \end{description} is equivalent to (CBS)+$\SSB$. But the latter needs extra assumptions on $\sF$ as (SEF). Without them (GCB) means (CBS)+(PRM). Finally, we expect that (cf.\ Remark~\ref{cbrm}, (8)) \begin{q} \item[\rm(BIG)] for {\em big\/} $\sM$, the subfamily of wanted triples is finite; in particular, \item[\rm(BIR)] for {\em birational\/} $X/Z$, the whole family of wanted triples can be taken finite; and, more generally, \item[\rm(MOD)] for {\em birationally equivalent\/} $\sM$, that is, for $\sM$ that define the same contractions $Y\to T/Z$ birationally, the subfamily of wanted triples $\sM$ is finite; in addition, $\dim T/Z=$ the Iitaka dimension of $\sM/Z$. \end{q} However to be more {\em realistic\/} we can restrict ourself with very special subsets in $\fM$ and load more conditions. We consider movable systems $\sM_\bull\subset \fM$ at least under the {\em asymptotic\/} saturation \LCA: e.g., \begin{itemize} \item (CBS)(fga) means the (CBS) for $\sM_\bull=\Mov{\sL}$ (separately) for each functional algebra $\sL$ in Conjecture~\ref{fgalmmp}; and \item (CBS)(rfa) means the (CBS) for each algebra in Definition~\ref{rfad}. \end{itemize} (CBS)(rfa,bir) is what we really need (see the notation below, and Conjecture~\ref{rfac})! \end{conj} \begin{rem}\label{cbrm} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item One can expect generalizations of the conjectures when $X/Z$ is only a proper morphism of normal varieties but possible not a contraction, or/and $X/Z$ has other extra structures: e.g., a group action, a morphism $X/W$, etc. In particular, the latter concerns to the nonlocal case $X/W$ with a proper morphism $W\to Z$ and {\em local\/}$/Z$ as we assume always. \item (PRM) drops (SEF) in Definition~\ref{ssd} but assumes that $S=P$ is prime. \item The effectiveness of $\sD\in\fD$ of (PFC) can be dropped. Indeed, if $|\sD|=\emptyset$, then 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 the effectiveness. \item For any $\R$-{\bi}divisor $\sD$, there is a decomposition $\sD=\sM+\sF$ with the {\bi}free {\em movable} component $\sM=\Mov{\sD}$ and {\em fixed} $\sF=\Fix{\sD}\ge0$. It was explained 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. A $\R$-{\bi}divisor $\sD\ge0$ up to $\sim$ if and only if $\sM\ne-\infty$, equivalently, $|\sM|\ne\emptyset$, or $\Oh_Z(\sM)=\Oh_Z(\sD)^{ch}\ne0$. If $\sD\ge0$ then $\sD\ge \sF\ge0$. Respectively, $\sD\ge0$ is {\em movable\/} if and only if $|\sM|\ne|0|$, equivalently, $\Oh_Z(\sM)=\Oh_Z(\sD)^{ch}$ has more than $1$ generator (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 nongeneral 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 unbounded. Then $\fS$ and $\fP$ include at least the exceptional curves $C$ (take $\sD=\overline{C}$ for $\SSB$). It is expected that (PRM) and (PFC) hold on any general log Fano contraction but for $\SSB$ we need Kawamata log terminal, in particular, $B$ is subboundary (cf.\ bad singularities in Remark~(9) below). \item The splitting of $\fD$ is partial because \begin{itemize} \item for rather powerful $\sM$, $\Mov{\rdup{\sM+\sF+\sA}}$ can be $>\sM$, that gives a {\em different\/} splitting. \end{itemize} \item For any $0\le c\le $ the m.l.d. of $(X,B)$, if some $\sD\in |\sM|$ is canonically bounded by $c$ then so is the generic $\sD'\in|\sM|$. Hence (CBS) for some $\sD$ implies the same for generic and actually with the same $c$. Indeed, the canonical boundedness of $\sD$ 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 the divisors 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 subboundary on $Y$ such that $B'_Y=c L+F$, where $F\le B_Y$ in the exceptional on $X$ divisors, and $\Bs{|L|}=\emptyset$, then by monotonicity (cf.\ \cite[1.3.3]{sh92}), for rather generic $L$, that is, $\sD$, $(X,B+c\sD'_X)$ is also canonical in codimension~2. This holds on {\em some\/} good model $Y/Z$, where $|\sD'|=\overline{|\sD'_Y|}$, with $\Bs{|\sD'_Y|}=\emptyset$, and $L=\sD'_Y=0/X$. Indeed, for such fixed model and for a generic $\sD'$, $B_Y'=\sB(X,B)_Y+(\overline{c\sD_X'})_Y$ and $(\overline{c\sD_X'})_Y=c(\overline{\sD_X'})_Y\le c(\overline{\sD_X})_Y=(\overline{c\sD_X})_Y/X$. This can also be established by inversion of adjunction (cf.\ \cite[after Proposition~5.13]{sh92}). Kawamata log terminal implies that the m.l.d. of $(X,B)$ is $>0$. On the other hand, $c\le$ the m.l.d. on $\fM$ by Example~\ref{tre}, (3). \item could perhaps expect more for 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} By the discussion in Example~\ref{stexm}, it sheds light on the finiteness in (BIG). In fact, each finiteness in (BIG), (BIR), (MOD) and (CBS)(fga), (CBS)(rfa) amounts to a {\em single\/} triple. \end{enumerate} To prove (CBS) in general, it is enough to consider a {\em wanted\/} triple $(X/T/Z,B,\fF)$ with the following subset $\fN=\fN(X/T/Z,B)$ of $\sM\in\fM$: \begin{q} \item[(NEF)] $\sM_X$ is nef$/Z$ and \item[(SFB)] {\em supported in fibres\/} of $X/T$ with the {\em Iitaka\/} dimension$/Z$ $=\dim T/Z$. \end{q} In particular, both hold when the triple is wanted for $\sM$ (conversely, $\sM$ is wanted 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 $\sim$. Moreover, we can assume that \begin{q} \item[(IRR)] $\sM_X$ is {\em irreducible\/} \end{q} except for the pencil 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 wanted, 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 boundedness \begin{itemize} \item for a {\em truncated\/} subcone: for $\sM$ with a rather high {\em index of xheight\/} on $X$, namely, for some natural number $I$, every such $\sM_X\sim \sum h_i g^*H_i$, where each $H_i$ is nef and big on $T/Z$, and $I\mid \sum h_i$, and \item for {\em bounded\/} xheights $\sum h_i$. \end{itemize} We hope that, for some $I\gg0$, the former $\sM$ are {\em free\/} over $X$, that is, $\sM=\overline{\sM_X}$ and $\Bs{|\sM_X|}=\emptyset$. Perhaps, in the latter case, the boundedness of xheights implies the boundedness of singularities by a 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 the finiteness of them when $\sM$ is big and $T=X$. In particular, this reveals why we expect Conjecture~\ref{cbcon} in full. But (FGA) needs just the boundedness of triples (cf.\ the proof of (3) in Theorem~\ref{cbsfg}). The canonical boundedness really requires the terminal property (TER) of wanted 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). (9) Possible important generalizations concern bad singularities of $(X,B)$: e.g., we could weaken the Kawamata log terminal 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 base point free\/} or even {\em (very) ample\/}$/Z$ on $\LCS{(Y,B_Y)}$. \end{q} Thus in this case, we care about canonical singularities outside $\LCS{(Y,B_Y)}$. (10) 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} (11) One expects similar results in the $\ep$-log category: e.g., $\ep$-log canonical boundedness. But this required appropriate restrictions: e.g., $\ep$-log saturation with respect to $\ep$-log discrepancy. Respectively, for triples, (TER) can be replaced by $\ep$-log terminal in codimension~$2$. \end{rem} \begin{quest} Could the condition that $\fM$ is {\bi}free be replaced by the {\bi}nef assertion of Lemma~\ref{divb}? \end{quest} Philosophy: Saturation \SAT\ improves properties of any $\sD$ modulo $\sim$ even it is not {\bi}free. (Compare the regularization for solutions of elliptic differential equations.) On the other hand, by \DEP\ in Proposition~\ref{desd} and by Proposition~\ref{aboundd} the descent data for $\sM$ can be bounded in terms of the canonical boundedness of an improved divisor. The same can be applied when $p\sD\sim_{\R}\sM$ with real $p>0$ and $\sM$ has good canonical singularities (cf.\ \ref{aboundd}.1), for example, by Conjecture~\ref{cbcon}. Notation: 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 Conjecture~\ref{fgalmmp}; \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} We apply them 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 $\CBS_n$(bir,gl)=$\emptyset$ for $n\ge1$. \end{itemize} The main result of this section is \begin{thm}\label{cbsfg} \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 with a clarification of (MOD). \begin{defn} 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} Examples: 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 define the same pencil combine from their fibres. \end{enumerate} \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 dimension of $\sD_1/Z\ge$ the Iitaka dimension of $\sD_2/Z$ with $=$ only when 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 $=$ only when $\nu(\sD_2 \rest{(\sD_1-\sD_2)}/Z)\le \nu(\sD_2)-1$; and \item if $\sD_1$ and $\sD_2$ are both {\bi}semiample, $\nu(\sD_1/Z)\ge \nu(\sD_2/Z)$ with $=$ only if they are birationally equivalent. \end{enumerate} \end{lem} \begin{proof} (1) Immediate from 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 the 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 generic point of $Z$ we can assume that $Z=\pt$ and each $\nu(D_i/Z)=\nu(D_i)$. Then it is enough to consider the case with $\nu(D_1)\le\nu=\nu(D_2)$. Thus for some cycle $C_{\nu}$ of dimension $\nu$, $C_{\nu}D_2^\nu>0$. Equivalently, the same holds for movable 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 we get an induction on the dimension $n\ge\nu$ with divisors $D_i\rest H$. Hence we can assume that $\nu=n$. Then $D_1^n=D_2D_1^{n-1}+F D_1^{n-1}\ge D_2 D_1^{n-1}\ge\dots\ge D_2^{n-1}D_1=D_2^n+D_2^{n-1}F\ge D_2^n>0$. 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) Immediate by (1) because $\nu(\sD/Z)={}$ the Iitaka dimension of $\sD/Z$ for any {\bi}semiample divisor $\sD$. \end{proof} \begin{pfof}{Theorem~\ref{cbsfg}} (1) Compare the proof of Proposition~\ref{ssp}, (3). For $\sD\ge0$, Kawamata log terminal 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 $\sF=\Fix{\sD}$, $\sM+\Mov{\rdup{\sA+\sF}}\le \Mov{(\sM+\rdup{\sA+\sF})}=\Mov{\rdup{\sD+\sA}}\le \Mov{\sD}=\sM$ since it is $\le \sD$. Hence $\Mov{\rdup{\sA+\sF}}\le0$, and actually $=0$ again by Kawamata log terminal. This means (STD) 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 need only 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 movable system on $(X/Z,B)$ for an algebra $\sL$ of type (FGA). By Limiting Criterion~\ref{limcr}, f.g.\ of $\sL$ is equivalent to the stabilization of its characteristic system, that is, of the limit $\sD=\lim_{i\to\infty} \sD_i$ with $\sD_i=\sM_i/i$. In its turn the stabilization is equivalent to the asymptotic descent problem. To solve the problem we apply Theorem~\ref{assdes}. First, we choose a prediction. 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, that corresponds to a quasi-isomorphism of algebras by (8) in Proposition~\ref{texa}, 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, then by (MOD) there is a finite family of wanted 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 the family with the strict property. Now any of the remaining triple $(Y/T/Z,B_Y,\fF)$ predicts $(Y/Z,\sC=\sA, F,\linebreak[2]\gamma)$ for the asymptotic descent problem $\sD=\lim_{i\to\infty} \sD_i$. If we consider only $i$ for which this is a wanted 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 \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 $\gamma>0$ follows from (QFC), (CRP) and $(X,B)$ Kawamata log terminal. The harder property of predictions is the boundedness (SAB); this follows from $\CBS\fga(X/Z,B$) by \ref{aboundd}.1 with the common model $(X_i/Z,B_i)=(Y/Z,B_Y)$, for $\sD_i$ with $i$ for which this is the wanted triple, and with $p_i=i$. Indeed, the descent data for $\sD_\bull$ is asymptotically bounded by $\sA/Y$. The boundedness 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 \ref{bpbdal}.1, and the additional one \BNF\ because each $\sM_i$ is {\bi}free. Thirdly, as we know \LCA\ in (FGA) implies the required saturation. Thus by \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 \ref{assdes}.2 $\sA_{X_{st}}\le0$ on the stable model $X_{st}/Z$ that proves Remark~\ref{fga0lm}, (7). Indeed, by Kawamata log terminal, the crepant boundary $B_{st}$ is actually a boundary and $(X_{st}, B_{st})$ is Kawamata log terminal. 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 $-(K_{X_{st}}+B_{st})/Z$ is big and {\bi}semiample. Hence by the LMMP $(X_{st},B_{st})$ is isomorphic in codimension~$1$ to a weak log Fano contraction, with the same boundary, on which $-(K_{X_{st}}+B_{st})$ 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 that correspond to elements of the family and also for any truncation. The canonical boundedness in 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). \end{pfof} The following examples illustrate Conjecture~\ref{cbcon}. \begin{exa}[global curve]\label{crv} \label{curves} 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 $c=1$ that 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 $\Supp{\fF_X}$ that are supported in exceptional on $Z$ divisors of $X$, and in the fractional part of $B$. Suppose, first, that $B=0$, $X$ is nonsingular, and $X/Z$ is cohomologically rational, that is, the singularities of $Z$ are rational. Then we contend that $(X/X/Z,0,\fF)$ is a wanted 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 exceptional on $X$ prime {\bi}divisors, the {\em support of exceptional locus}. By rationality of $X/Z$, on any model $Y/Z$ of $X/Z$, each Cartier nef$/Z$ divisor $M$ is base point free$/X$. In particular, this holds for any integral $M\ge0$ with $\Supp{M}$ having no exceptional components of $Y/Z$. In addition, if $C$ is an exceptional curve of the first kind on $Y/X$ and $M\cdot C\ge1$ (intersects any $M$ modulo $\sim$), then $M+C$ is also base point free, and $\Mov{\rdup{M+\sA_Y}}=\Mov{(M+C+\sA_Y-C)}\ge \Mov{(M+C)}+\Mov{(\sA_Y-C)}\ge M+C>M$, because $\sA_Y-C\ge0$. Hence {\bi}free $\sM=\overline{M} \notin\fM$ since does not satisfy \SAT. Conversely, by \SAT\ all $\sM\in\fM$ can descend to $X$ as base point free divisors$/Z$. Thus we can take $c=1$ in (CBS). This explains the role of saturation. Indeed, if $Y/X$ is a minimal resolution of the base locus for $|\sM\rest X$ then such $C$ exists over the locus. If $Y:=X_{hr}$ is also a sufficiently high resolution for \SAT\ and $M=\sM_Y$ is rather generic in $|\sM_Y|=|\sM\rest Y$, then we replace $C$ by $\overline{C}_Y\le\sA_Y$. This again contradicts \SAT. Thus $\sM$ is base point free on $X/Z$. Moreover, the same holds for $(X/Z,B)$ under (TER). But in this case $c$ may be $<1$, namely, $c=\min \{1-b_i\}$. If $(X/Z,B)$ is Kawamata log terminal, the above (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, and even without (ADJ). Another approach that we discuss below reduces our problem to base point freeness in dimension $1$ (cf.\ Example~\ref{curves} above). It is not so effective and moreover has some new unpleasant 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 given in \begin{prop}[General two dimensional case]\label{surbound} Let $(X/Z,B)$ be a log pair with a surface $X$, and $\sD\ge0$ an {\rm integral\/} {\bi}divisor such that: \begin{description} \item[\GLF] $(X/Z,B)$ is a general log Fano contraction as in Proposition~\ref{ressat} with arbitrary $B$ even not assuming a boundary; and \item{\rm(RIR)} on any model $Y/Z$ of $X/Z$, generic $\sD_Y\in |\sD\rest Y$ is {\rm reduced, irreducible\/}; and \item[\SAT] $\sD\in\fD=\fD(X/Z,B)$, {\rm that is, saturated with respect to $\sA=\sA(X,B)$\/} \end{description} Then except for a bounded set of complete divisors $\sD_X$, $|\sD|_X$ is base point free$/Z$ outside $\CS{(X/Z,B)}$ on $X$, where $\CS{(X/Z,B)}$ denotes the {\rm locus of canonical singularities. In other words, $(X/Z,B)$ satisfies (TER) outside $\CS{(X/Z,B)}$\/}. \end{prop} Commentary: Essentially we need only two cases: \begin{itemize} \item $\sD$ is a {\bi}free nonpencil (or an irreducible member of pencil); or \item $|\sD|=\{ P\}$ is a prime fixed {\bi}divisor. \end{itemize} By (RIR), $|\sD\rest X$ is base point 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 ``rather generic'' $\sD_Y$, in particular, means that $\sD_Y$ has the minimal multiplicities in the exceptional on $X$ divisors: e.g., $0$ for {\bi}free $\sD$. \begin{sta} If we change assumptions {\rm(RIR)\/} and \SAT\ in the proposition respectively on \begin{q} \item[\rm(RED)] on any model $Y/Z$ of $X/Z$, generic $\sD_Y\in |\sD\rest Y$ is {\rm 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 {\rm integral components of $\sA$\/} \end{q} then any two prime components (or even two branches) $D_1\ne D_2$ of generic $\sD_X\in |\sD\rest X$ intersect each other only in points $Q$ in $X$, where $K+B$ is not Cartier or in $\LCS{(X,B)}$, with a finite set of exceptions for $D_1$ {\em or\/} $D_2$. More precisely, any such set of exceptions includes only $D_i$ $(K+B)\cdot D_i=0$ (in particular, such $D_i$ is complete, that is, over $P$). Thus such 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{description} \item{\rm(Sa$'$T)} assuming in {\rm(SA$'$T)\/} that the primes $E_i$ are over $\Bs{|\sD\rest X}$. \end{description} \end{sta} \begin{sta} Suppose in addition that \begin{itemize} \item $(X,B)$ is Kawamata log terminal and \item $\sD$ is {\bi}free. \end{itemize} Then $|\sD\rest X$ has at most one base point on $X$ outside $\CS{(X,B)}$; in particular, at most one point on a terminal resolution of $(X/Z,B)$. In addition, $|\sD\rest X$ is base point free in the {\rm CS\/} after the blowup of the point. Moreover, any pencil $\sD$ is elliptic, that is, its generic member is a curve of geometric genus $0$. \end{sta} \begin{cor} Locally$/Z$, under the assumptions of {\rm Proposition~\ref{surbound}\/}, the singularities of generic $\sD_X\in |\sD\rest X$ are bounded outside the {\rm CS\/}; in particular, outside $\LCS{(X,B)}$ on a terminal resolution. \end{cor} \begin{proof} The singularities of $\sD_X$ are bounded for any base point free $|\sD\rest X$. The same holds for any bounded set of divisors $\sD_X$ (Example~\ref{tre}, (1)). \end{proof} \begin{rem} \label{rsurbound} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item (RIR) and (1--2) below in the proof of the proposition hold when $\sD$ is {\bi}free and big. For surfaces, the latter means that $\sD\ne0$ and is not a pencil. \item Sometimes, $\Bs{|\sD\rest X}\ne\emptyset$, and requires $c<1$ even when $(X,B)$ is terminal in codimension~$2$ (see Example~\ref{esurbound} below). However \item for {\bi}free $\sD$, we expect only a {\em finite\/} set of linear systems $|\sD|$ that have base points outside CS. \end{enumerate} For example, one verifies that the finiteness holds when $(X/Z=\pt,B)$ is a del Pezzo surface with a standard boundary $B$ and is of the nonexceptional type. \end{rem} \begin{lem}[Lifting of $\R$-Cartier divisors]\label{restdiv} $f^*D$ is defined up to $\sim$ if \begin{itemize} \item $D$ is an $\R$-Cartier divisor defined up to $\sim$, and \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{sta}[Functoriality] For $g\circ f\colon X\to Y\to Z$, \[ (g\circ f)^*D=f^*(g^*D). \] \end{sta} \begin{proof} Since $D$ is given up to $\sim$, we can assume that $f$ and $D$ are {\em in general\/} position, that is, $\Supp{f(X)}\cap \Supp{D}$ is proper in each irreducible component of $f(X)$. \end{proof} \begin{lem}[Invariance of integral part for $\sim$]\label{intpart} If $D\sim D'$ then \[ \rddown{D}\sim \rddown{D'} \] and \[ \rdup{D}\sim \rdup{D'}. \] 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. \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 reals 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 $D=\sum d_i P_i$ 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 $\ge \rdup{d}$. \end{proof} \begin{pfof}{Proposition~\ref{surbound}} A restriction on $D=\sD_X$ 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\/}, that is, there no divisors in the intersection. \end{description} 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 generic, irreducible and reduced. Let $g_D:D\to X$ 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 an epimorphic restriction$/Z$ \[ \left|D+\rdup{\sA_Y}\right|\broken \left|(D+\rdup{\sA_Y})\rest D\right|=\left|K_D+\rdup{-g_D^*(K+B)}\right|. \] Indeed, we can use here Lemma~\ref{restdiv} because $D$ is reduced and $K+B$ is integral on $D$ by (RIR) and (1) respectively. 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 the adjunction, the normal crossings on $Y$ and \ref{restdiv}.1: \[ (D+\rdup{\sA_Y})\rest D\sim (D+K_Y)\rest D+\rdup{-g_D^*(K+B)}= K_D+\rdup{-g_D^*(K+B)}. \] 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 $|\sD\rest X$ 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 $\Bs{|(D+\rdup{\sA_Y})_D|}\ne\emptyset$ -- the restriction is not base point free by the above epimorphism. Hence $D$ is complete. Otherwise $|D+\rdup{\sA_Y}\rest D$ is always base point free. Thus $D$ is complete and $|(D+\sA_Y)\rest D|=|K_D+\rdup{-g_D^*(K+B)}|$ should have base points or even to be $\emptyset$. This is impossible when $\deg B'>1$, where $B'=-g_D^*(K+B)$ because then by the Lemma \ref{inequol} $\deg \rdup{B'}>1$ and so $\ge2$. But for such divisors $B'$ the linear system $|K_D+\rdup{B'}|$ is base point free. Hence each $D$, that has fixed points on $X$ outside the CS, should have a bounded degree on $X/Z$: e.g., $-(K+B)\cdot D\le 1$. 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 $D$ (this adds to $\Supp{B}$ a finite set of divisors with discrepancy $0$; cf.\ \GLF\ above). \end{description} \end{pfof} \begin{prop} \label{monotd} If $D_1\ge D_2$, then the $C$-saturation of $D_1$ implies the $C$-saturation of $D_2$ over $D_1=D_2$. The same holds for {\bi}divisorial versions. \end{prop} \paragraph{Proof--Explanation} The saturation over $D_1=D_2$, or over any other set of divisors, means that $\Mov{\rdup{D_2+C}}\le D_2$ over the set, that is, $\le$ concerns only prime divisors $P_i$ in the set: e.g., $\mult_{P_i}{D_1}=\mult_{P_i}{D_2}$ in our situation. Then the proposition is immediate from the definition. \qed\par\medskip \begin{pfof}{\ref{surbound}.1} Suppose that a point $Q\in D_1\cap D_2$ is Cartier and canonical for $K+B$. The former implies that each $D_i$ is over an integral component of $B$. Again by (1) in the proof of Proposition~\ref{surbound} we assume that each $(K+B)\cdot D_i<0$ or $D_i$ is {\em non\/}-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 the finite number of the 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 (2) in Proposition~\ref{ssp}) we can replace $D$ by its nonexceptional on $X$ component, that is, drop the exceptional on $X$ divisors (because they are fixed in $|\sD\rest Y$). Then by (RED) and Proposition~\ref{monotd} it is enough to verify that {\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 $(K+B)\cdot D_i<0$ or $D_i$ is noncomplete. We derive the contradiction with $\sA'$-saturation on any log resolution $g\colon Y\to X/Z$ for $B+D$. On $Y$ we combine a chain-curve (chain maybe after additional blowups) $C=D_1\cup (\bigcup E_i)\cup D_2$ with the edge curves $D_i$, and exceptional curves $E_i$ over $Q$. $C$ has only nodal singularities. We verify that $|C+\rdup{\sA_Y}|=|\rdup{D+\sA_Y+\sum E_i}|$ 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 $Y/X$: e.g., $\Mov{\rdup{D+\sA_Y+\sum E_i}}\ge D+\sum E_i>D$ 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 on $D$, we get an epimorphic restriction$/Z$ on $C$: \[ |C+\rdup{\sA_Y}|\broken |(C+\rdup{\sA_Y})\rest C|=|K_C+\rdup{-g_C^*(K+B)}| \] with $g_C:C\to X/T$. Indeed, we can use here Lemma~\ref{restdiv} because $C$ is reduced and $K+B_Y$ is integral on $C$ since $K+B$ is Cartier in $Q$. As above by Lemma~\ref{intpart}, $C+\rdup{\sA_Y}\sim C+K_Y+\rdup{-g^*(K+B)}$. Thus by Lemma~\ref{intpart}, by the adjunction, the normal crossings on $Y$ and \ref{restdiv}.1: $(C+\rdup{\sA_Y})\rest C\sim (C+K_Y)\rest C+\rdup{-g_C^*(K+B)}=K_C+\rdup{-g_C^*(K+B)}$. We use here the same Kawamata--Viehweg vanishing: $R^1h_*\Oh_Y(\rdup{\sA_Y})=0$. 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 each $\deg M\rest{D_i}\ge1$ or $D_i$ is noncomplete. \end{itemize} Indeed, $M=\rdup{g^*(-K-B)}\rest C=\rdup{-g_C^*(K+B)}$ by the normal crossings and by \ref{restdiv}.1, whereas respectively \begin{itemize} \item $-(K+B)$ is nef on $X/Z$; \item each $K+B$ is Cartier in $Q$; and \item each $(K+B)\cdot D_i<0$ or $D_i$ is noncomplete. \end{itemize} Then $\Bs{|L|}=\emptyset$ and this implies the base point freeness of $|C+\rdup{\sA_Y} |$ 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 $h:C\to C'$ of all $E_i$ and $L=h^*L'$. Note that $C'$ prime has at least two components $D_i$, and $\deg (K_{C'}+L')\rest{D_i}\ge1$ on each complete $D_i$. Thus $\Bs{|L'|}=\emptyset$ by (In the case of two branches this gives a nonvanishing and a contradiction to the $\sA'$-saturation near $\bigcup E_i$.) \end{pfof} Quiz: State and prove a similar higher dimensional semi log canonical version of the last base point freeness. \begin{pfof}{\ref{surbound}.2} Since $K+B$ is Kawamata log terminal, $\rdup{\sA_Y}\ge0$. Thus because $D$ is {\bi}free, in the proof of Proposition~\ref{surbound} we have an effective Cartier divisor $L$ in $|K_D+\rdup{B'} |$ with $\deg B'>0$ this time. Thus $L$ has at most a single base point $Q$ of multiplicity $1$. Moreover, $L$ has a trivial movable part ($0$) and the base point $Q$ only when $K_D\equiv 0$ and $\deg L=1$, that is, when $D$ belongs to an elliptic pencil. Since $\deg L=1$ we need only one blowup outside $CS$ to resolve the base point, namely, the blowup in $Q$. \end{pfof} \begin{prop}\label{ssp} Let $(X/Z,B)$ be a Kawamata log terminal 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 The {\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, {\rm up to a component over the fractional part of $B$,\/} \end{enumerate} \begin{q} \item[\rm(Sa$'$F)] $S=\Supp{(\Fix{ \sD})}_X$ for some $\sD\ge0$ satisfying {\rm(Sa$'$T)\/} of \ref{surbound}.1; and includes \item[\rm(SA$'$F)] a reduced divisor $F$ is contained in the\/ {\em substandard\/} set\/ $\fS'=\fS'(X/Z,B)$ whenever $\Supp{F}$ is only over integral components of $B$, and some {\bi}free {\bi}divisor $\sM$ is saturated for $\sA'+F$ on {\rm 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{prop} \begin{rem}\label{sspr} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item 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 boundednesses of both sets are equal. \item $F$ is considered as a {\bi}divisor. \end{enumerate} \end{rem} \begin{lem}\label{as} Let $\sC$ and $\sM$ be such {\bi}divisors that: \begin{itemize} \item $\rdup{\sC}\ge0$, and \item $\sM$ is {\bi}free. \end{itemize} Then $\sC$-saturation on {\rm any\/} sufficiently high model is equivalent to that of on {\rm some\/} sufficiently high model. \end{lem} \begin{proof} Some sufficiently high model is any model $Y/Z$ of $X/Z$, where $\sM=\overline{\sM_Y}$ is free$/Z$, that is, $\Bs{|\sM_Y|}=\emptyset$. Then on any model $X_{hr}/Y/Z$, the $\sC$-saturation for $\sM$ follows from that of on $Y$. Indeed, since $\rdup{\sC}\ge0$, and $\sM$ is free$/Y/Z$, the $\sC$-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)). \end{proof} \begin{pfof}{Proposition~\ref{ssp}} (1) For each $s\in\fS(Y/Z,B_Y)$, the log transform $S$ of $s$ belongs to $\fS(X/Z,B)$. Thus $s=g(S)$. (2) By definition $\Mov{\rdup{\sM+\sF+\sA}}\le\sM$. This means (SAF). Conversely, assuming Kawamata log terminal, \[ \rdup{\sA+\sF}\ge0, \quad \Mov{\rdup{\sM+\sF+\sA}}\ge \sM \quad \hbox{and} =\sM \] by (SAF). On the other hand, since $\sF\ge0$, $\sM\le \sM+\sF$, and this gives \SAT\ for $\sM+\sF$. (3) Set $\sL=\Mov{\rdup{\sA+\sF}}$. Again $\rdup{\sA+\sF} \ge0$, and $\sL\ge0$. Since $\sM$ is {\bi}free, then, by (SAF), $\sM\ge\Mov{\rdup{\sM+\sF+\sA}}\ge \Mov{\sM}+\Mov{\rdup{\sF+\sA}}=\sM+\sL$, and $\sL\le0$. Thus $\sL=0$, that means (STD) for $\sD=\sF$. Conversely, (STD) for $\sD$ means (SAF) for $\sM=0$ and $\sF=\sD$. (4) If $S=\Supp{\sD_X}\in \fS$, then by Lemma~\ref{monots}, (STD) implies that $0$ is saturated with respect to $\rdup{S'+\sA'}\le\rdup{\sA+\sD}$, whereas $S$ is fixed and $S'$ as in Remark~\ref{sspr}, (1). The inequality follows from (SEF) and because $\Bs{|S'\rest X}=S'\subset S$. Thus (Sa$'$F) holds for $\sD=S'$. Conversely, if $\sD$ satisfies (Sa$'$F) then, as in (2--3) above, $\sF+\sum E_i+E'$ satisfies (STD) and (SEF), where $\sF=\Fix{\sD}$ and $E'=\sum \ep_i E_i$ with $0<\ep_i\ll 1$ for the exceptional divisors over the {\em fractional\/} components of $\sA$. Note that $E'$ does not effect (Sa$'$F) (cf.\ \ref{monots}.1): $\rdup{\sF+\sA+\sum E_i+E'}=\rdup{\sF+\sA'}$, but (SEF) holds because $\sF_X\subset\Bs{|\sD\rest X}$. Thus $\Supp{\sF_X}\in\fS$. By Lemma~\ref{monots}, (SA$'$F) implies (Sa$'$F) for $\sD=\sM+F$, whereas $\Fix{\sD}=F$ and $\Supp{F}=F\in\fS$. ``Any'' for models can be replaced by ``some'' in (SA$'$F) by Lemma~\ref{as} with $\sC=\sA'$. \end{pfof} \begin{cor} \label{boundsf} Let $(X/T,B)$ be as in Proposition~\ref{surbound} and suppose that is Kawamata log terminal (but $B$ may only be a subboundary). Then the standard set $\fS=\fS(X/Z,B)$ is bounded. In particular, $\fS'$ as in {\rm(SA$'$F)} is also bounded. \end{cor} \begin{proof} By Proposition~\ref{ssp}, (3--4) and Remark~\ref{sspr}, (1) it is enough to establish the boundedness of {\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 assume also that each component $D_i\not\subset\Supp{B}$. Since $F$ is fixed and satisfies (Sa$'$F), then by Lemma~\ref{monots} each component $D_i$ is $\sA'-$ 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, primes 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 disjoint 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 disjoint 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 such set of divisors $D$ that they have unbounded number of algebraically equivalent irreducible components $D_i$. Moreover, then $D_i^2\ge0$. (Otherwise only one such $D_i$.) If $D_i^2=0$ then there exists a fibering $X\to C$ over a curve $C$ such that divisors $D_i$ are in fibres. And then for sufficiently many such components there is an unbounded number of nonsingular fibres $0-1$ 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 the only integral multiplicities. Thus on any fixed sufficiently high model $X_{hr}/Y$, by the $\sA$-saturation for $D:=D'$ as {\bi}divisor, $D=D_{X_{hr}}\ge \Mov{\rdup{D+\sA}_{X_{hr}}}\ge \Dbar_{X_{hr}}$, that contradicts $D$ fixed. (Compare the proof of Proposition~\ref{surbound}.) Hence $D_i^2>0$ and divisors $D_i$ intersect each other. By \ref{surbound}.1 any of these $D_i$ intersects any other $D_j\iso D_i$ in a finite number of points, namely, in $D_i\cap(\Supp{B}\cup {\rm CS})$, where the CS denotes nonGorenstein 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 points $Q$, e.g., minimal when $Q$ is singular, or the usual blowup when $Q$ is nonsingular on $X$. Then we replace $(X,B)$ by its crepant transform. The $\sA$-saturation is birational, and intersections of $D_i$ only over the log transform of $D_i\cap(\Supp{B}\cup {\rm CS})$. We consider the proper birational transform of $D$ and of prime components $D_i$. Again we have an unbounded family of algebraically equivalent $D_i$. 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 should terminates, since any two components have a bounded resolution to disjoint them. Or we can see that new $D_i^2\le$ old $D_i^2-const$, where $const=1$ for usual blowups of nonsingular points, and some positive numbers in other points. \end{proof} \begin{exa} \label{esurbound} Let $(X/\pt,B=0)$ be a (terminal) nonsingular del Pezzo surface. Then in Proposition~\ref{surbound} each $D=\sD_X$ is base point free or ($-1$)-curve except for Del Pezzos 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 the latter curves. The (FGA)($X/\pt,0$) gives the models $X_{st}=X$, except for $=\PP^1$ in the case of the pencil, and $=\pt$ (cf.\ Example~\ref{prs}). Pencils and irreducibility in (RIR) of Proposition~\ref{surbound}: There are no pencils of elliptic curves that always tangent the $1$-complement boundary $C$, because the restriction of the pencil on $C$ is isomorphism for a generic divisor. Thus we have only single element in the pencil with a single intersection (and tangent) point. However for higher genera they are possible. \end{exa} \begin{exa}\label{fcanb} For (CBS) on $X/Z$ or on a triple $Y/Z$ in the 3-fold or higher dimensional (even local) case we need a numerical condition: {\em nef} of $\sM_Y/Z$ (cf.\ (NEF) in Remark~\ref{cbrm}, (8)). In the one dimensional case: it is implied by the effectiveness of $\sD(X/Z,B)$ in Conjecture~\ref{cbcon}; in the two dimensional case by {\bi}freeness of $\sM$ or by the freeness of $|\sM\rest Y$ in codimension~1. But nef is important in dimension $\ge3$. Indeed, let $f\colon X\to Z$ be an extremal and small contraction negative with respect to $D$. Then any infinite set of {\bi}divisors $\sD$ such that $\sD_X=nD$ for a natural number $n$ is not canonically bounded. 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 center of a LCS and $c$ is the log threshold along $C$ for $D_X$. This is impossible by the 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} 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 wanted 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 {\rm for\/} some wanted $Y/T$. In particular, they are canonically bounded with respect to $B_Y$. \end{cor} \ref{mainb}.1: In general, the only estimation is $00$. Hence $M$ is nef and big, and all such rather general $\sM_Y$ have bounded singularities everywhere by the construction. This family of triples is bounded, because intersections of the elements of the pencil are bounded by Proposition~\ref{surbound}. More precisely, by \ref{surbound}.2 each 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 $|\sM\rest X$ be $0$, that is, such effective $\sM=0$. Take $Y/T$ to be $X\to\pt$ Then $M=0$. Such triple is unique. \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}. \end{pfof} \begin{cor}\label{stabc2} Any $\FGA_n\bir$ algebra $\sL=\dival_{X/T}{D}$, is f.g., up to codimension~$2$ over $T${ \rm, that is, over the point of codimension~$\le 2$\/}. \end{cor} \begin{proof} Immediate by $\FGA_n$ with $n\le 2$ in the Main Theorem. \end{proof} \begin{cor} \label{fgapl3} $\FGA_3^{pl}$ of {\rm Example~\ref{fgapl}\/} holds. \end{cor} Moreover, we can drop the condition that $\sL$ is bounded by $D$ and satisfies Main Lemma~\ref{mainl} (cf.\ Remark~\ref{fgaa}). \begin{proof} Immediate by $\FGA_d\bir$ with $d\le 2$, that was essentially done in Example~\ref{tdb}, and Example~\ref{fgapl}. \end{proof} Of course, by Corollary~\ref{fgarfa} (CBS) implies also (RFA) but we can prove more. \begin{thm}\label{cbrfa} Under the {\rm LMMP\/} in dimensions $d \le n-1$, $\CBS_{d-1}^*$ 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 $d\le 3$, i.e, for $n\le 4$. The $\RFA_{n,m}\bir$ 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 new tools introduced here are birational restrictions with different flavors. For example, in the proof of Theorem~\ref{stabin} (cf.\ also Proposition~\ref{ressat}), we use systematically restrictions of {\bi}divisors. These are essentially of one of the following two types, or some mixture. \subsection{Movable restriction} Let $E$ be a prime {\bi}divisor on $X$ and $\sD$ a {\bi}free divisor$/Z$. Then we can define a {\bi}free divisor $\sM=\sD\frest E$ of $E$ by the formula \[ \sM_Y=\sD_Y\rest E \] where $Y/X$ is a model such that $E$ is normal divisorial subvariety of $Y$ and $\sD$ is base point free$/Y/Z$. Thus $\sM_Y$ is also base point free$/E/Z$, and $\sM=\overline{\sM_Y}$. To cut the restriction as a Weil divisor it is enough to take a normal model $Y/X$, where $E$ is divisorial variety, nonsingular in $X$ in divisors of $E$. Similarly, we can defined a Cartier {\bi}divisor (cf.\ (CAR) in the additional assumptions of Section~\ref{satdes}) $\sD\frest E$ for any Cartier {\bi}divisor $\sD$ on $X$. Thus the restriction is well defined because the divisor has only $0$ multiplicities up to a linear equivalence. Such restrictions are defined up to the {\em linear\/} equivalence. \subsection{Fixed restriction}\label{frest} When defining a fixed restriction $D\rest E$, it is usual to require that $D$ is Cartier and not supported in $E$, that is, $E\not\subset \Supp{D}$. Moreover, by Lemma~\ref{restdiv}, we can assume that $D$ is $K$-Cartier; 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 a 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 $K$-Cartier$/E$. In this case some multiplicities are growing for subsequent blowups. This restriction also preserves {\em linear\/} equivalence. Finally, \subsubsection{Mixed} We can defined a restriction $\sM=\sD\frest E$, when $\sD$ is $K$-Cartier {\bi}divisor having $\mult_{E}{\sD}\in K'\subset K$ such that \[ \sM_Y=(\sD_Y- (\mult_{E}{\sD})E)\rest E+(\mult_{E}{\sD})E'\rest E \] is again $K$-Cartier {\bi}divisor, where $Y/X$ is a model such that $E$ is normal divisorial subvariety of $Y$, $\sD$ is $K$-Cartier$/Y$, and $E'\sim E$ on $Y$, but $E$ is not in $\Supp{E'}$. (Warning: $E'$ may be neither prime nor effective. Note also, that $E$ is $\Q$-Cartier automatically, whenever $Y$ is $\Q$-factorial.) In other words, if $E$ is $\Q$-Cartier on $Y$, $K$ has characteristic $0$, we obtain $\sM=(\sD-(\mult_{E}{\sD})\overline{E})\frest E +(\mult_{E}{\sD})\overline{E'}\frest E$. 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 the {\em $K'$-linear equivalence\/}, in particular, with $\sim$ for $K'=\Z$, by Lemma~\ref{restdiv} and because $E'\sim E$ implies $\overline{E'}\sim\overline{E}$. {\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, we consider case (df) that reminds the discrepancy, when $f\colon X\to T$ is a birational contraction and $f_*(\sD_1)=f_*(\sD_2)$. Definition: a {\bi}divisor $\sD$ on $X$ is $K$-Cartier {\em in\/} a point $p\in X$ or {\em near\/} $p$, when locally near $p$, for some $K$-Cartier divisor $D'$, $\sD=\Dbar'$. Note that in this case $D'=\sD_X$ near $p$. \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, assuming they are in general position on $X$, in particular, $\Supp({\sum k_i\sD_i)_X}=\bigcup\Supp{(\sD_i})_X$. This is {\em equivalent\/} to be a $K$-Cartier {\bi}divisor up to $\sim$. Then $\sD$ is {\em Cartier\/} outside $\Supp{\sD}_X$ (the divisorial subvariety on $X$). Indeed, it is $0$ modulo $\sim$ outside the base locus of the linear system of $|\sD\rest X$. 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 \[ \cap\Supp{\sD'_Y}, \] where $\sD'$ runs the {\bi}divisors $\sD'\sim_{K}\sD$ at least locally$/Y$, and $D'$ is not necessary effective but it is a $K$-linear combination as above, also generic on $Y$. For example, let $X\to Y=\PP^2$ be a blowup of a point $P$, $E$ the blowup of $P$, and $L, L_P$ lines on $Y$ through $P$. Then $\overline{E}=\sL-\sL_P$, where $\sL=\overline{L}$ and $\sL_P$ (with $|\sL_P|=|\sL-P|$) are {\bi}free. Thus $\overline{E}$ is Cartier outside $\Supp{(\sL-\sL_P)_Y}=L\cup L_P$, whereas lines $L$ and $L_P$ through $P$ should be different (=generic). Note also that $\sL_P\sim \sL-\overline{E}\not\ge0$ when $L$ is not through $P$. Nonetheless $\sL_P$ is not Cartier in $P$ out of $\Supp{(\sL-\overline{E})_Y}=L$. \end{exa} \begin{prop} \label{rest} Let $X$ be a model where $E$ is a (normal) divisorial subvariety, and let $\sD,\sD_1,\sD_2$ be $K$-Cartier {\bi}divisors that are $K$-Cartier out of $\Supp{(\sD_*)_X}$. Then \[ \Supp{(\sD\frest E)_E} \] is in \begin{description} \item{\rm(mv)} $\emptyset$ modulo $\sim$, whenever $\sD$ is Cartier or {\bi}free over $X$, that is, the restriction of Cartier is Cartier; and \item{\rm(fx)} $E\cap\Supp{\sD_X}$, whenever $\mult_{E}{\sD}=0$. \end{description} If, in addition, $E$ is $\Q$-Cartier, and each $\sD_*- (\mult_{E}{\sD_*})\overline{E}$ is $K$-Cartier out of $\Supp{(\sD'_*)_X}$, where $\sD'_*\sim\sD_*- (\mult_{E}{\sD_*})\overline{E}$ with $\mult_{E}{\sD'_*}=0$, then \[ \Supp{(\sD\frest E)_E} \] is in, modulo $\sim$, \begin{description} \item{\rm(mx)} union of $E\cap\Supp{(\sD_X- (\mult_{E}{\sD})E)}=E\cap\Supp{\sD'_X}$ and $E\cap \Supp{E'}$; and \item{\rm(df)} in 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 $\sD:=\sD_1-\sD_2-\mult_{E}{(\sD_1-\sD_2)} \overline{E}$. \end{description} \end{prop} Note: if $E$ is $\Q$-Cartier, by Example~\ref{bbasep} the condition to be $K$-Cartier away from $\Supp{\sD}$ is satisfied whenever the {\bi}divisor $\sD$ is {\bi}free or a $K$-linear combination of such {\bi}divisors in general position on $X$. \begin{proof} (mv) and (fx) hold by definition (cf.\ Example~\ref{bbasep} above). (fx) implies (mx) and (df). Indeed, $\sD- (\mult_{E}{\sD})\overline{E}$ is nontrivial and not $K$-Cartier$/E$ only over \[ E\cap\Supp{(\sD-(\mult_{E}{\sD})\overline{E})_X}= E\cap\Supp{(\sD_X-(\mult_{E}{\sD})E)}, \] that is, modulo $\sim$, only over $E\cap\Supp{\sD'_X}$. In (df) we take $\sD'=\sD'_1-\sD'_2$. Note that $E'$ as $E$ is $\Q$-Cartier everywhere on $X$. \end{proof} \begin{cor} \label{resdf} Suppose that \begin{itemize} \item $K$ is a field; \item ($a_1\sD_1$ is rather generic 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 $(\sD_1)_T=(\sD_2)_T$ on $T$, (or outside $f\1P,P\in Z,$ in the more general local case). \end{itemize} Then modulo $\sim$, we can replace the union in {\rm(df)\/} by the union of\/ \[ E\cap\Supp{(\sD_1)_X}, \quad E\cap\Bs{|a_2\sD_2\rest X}, \] and $E\cap{}$ exceptional divisors ($\ne E$) of $X/T$ (and respectively in $E\cap{}$ divisors of $X/P$ in the local case). In particular, for $a_1=1$, the {\rm non-Cartier points\/} on $E$ and {\rm fractional\/} parts of $(\sD\frest E)_E$ are in $E\cap \Bs|\sD_1|_X$, $E\cap \Bs|a_2\sD_2|_X$, and $E\cap$ exceptional divisors ($\ne E$) of $X/T$ (and respectively in $E\cap$ 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 rather generic 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 rather generic 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 $\sD\ne e\overline{E}$ 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$/Q$. \end{proof} \section{Approximations} \label{apprx} \begin{defn}\label{bnds} Let $F$ be a reduced divisor of $X$. A {\em set\/} of semiample$/Z$ $\R$-divisors of $X$, $\fN_{bnd}(F)=\fN_{bnd}(X/Z,F)$, is a set of (effectively) {\em bounded\/} semiample$/Z$ divisors or {\em bnd\/}-set$/Z$ when \begin{q} \item[(BFP)] a {\em fractional part\/} of $D$ is {\em bounded\/} by $F$: $D=D_{int}+D_{fr}$, where $D_{int}$ is integral, and $\Supp{D_{fr}}\subset F$; \item[(CFG)] the set has a {\em compact plus finite\/} set of {\em generators:\/} there are 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 $D=D_c+\sum n_i D_i$, or even {\em up to\/} $\sim$, where $D_c\in \fN_c$ and every $n_i\in\N$; and \item[(BND)] of Definition~\ref{bnd} in $\fN_{bnd}(F)$, or, equivalently, the same holds for $\fN_c$. \end{q} Taking bigger $F$ we can assume that the polyhedron of (CFG) belongs to the $\R$-space $\fD_F$ (see notations in Section~\ref{satdes}). Note that (CFG) implies (BND) when $\{D_i\}$ is finite. A {\em uniform\/} neighbourhood of $\fN_{bnd}(F)$ with {\em diameter\/} $\de\in\R$ is the set of Weil $\R$-divisors \begin{multline*} \fU_{bnd,\de}(X/Z,F)=\fU_{bnd,\de}(F)= \\ \bigl\{D=\sum d_i P_i\bigm| d_i\in\R \hbox{ and } \Vert D-\fN_{bnd}(F) \Vert <\de\bigr\}, \end{multline*} assuming that $\Vert-\Vert$ is taken under $\Supp{(D-\fN_{bnd}(F))}\subset F$. In other words, one of the following two equivalent conditions is satisfied: \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $D=D_{int}+D_{fr}$ such that $D_{fr}$ is supported in $F$ and there exists $D'=D_{int}+D'_{fr}\in \fN_{bnd}(F)$ with $\Vert D-D'\Vert=\Vert D_{fr}-D'_{fr}\Vert <\de$; or \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 $\Vert D-\fN_{bnd}(F)\Vert \le \Vert D-D'\Vert<\de$. (1) implies (2), because $D-D'=D_{fr}-D'_{fr}$ is supported on $F$. (2) implies (1) for $D_{fr}=D-D'+D'_{fr}$, because then $D=D_{fr}+D'-D'_{fr}=D_{fr}+D_{int}$ 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 bnd-{\em cone\/}$/Z$. We can assume that each $D_i$ is integral. Then we can take \begin{itemize} \item $F$ as the joint 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 for us of a bnd set is an Abelian semigroup $\fN(F)=\fN(X/T/Z,F)$ of the nef$/Z$ and $\sim_{\R} 0/T$ divisors $D$ having fractional parts bounded by $F$. (In particular, $D$ is assumed to be an $\R$-Cartier divisor.) In such situation, 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}. Actually, $\fN(F)$ is a bnd-set and even semiample in very special but crucial cases for us: \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 $0$-log pairs 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 boundedness (BND) does not hold in general: torsions of Abelian varieties contradict to (BND). To fulfil (CGF) and (BND) we need to ask the irregularity to be zero. Thus the Picard group is finitely generated. Hence integral points in $\fF(X/Z/Z,F)$ are finitely generated modulo $\sim$, and $\fN_c$ is a cube with bounded multiplicities as in (1) above. In particular, this is true for $(X/Z)$ having the structure of a weak log Fano contraction. Moreover, the bnd-property for \WLF\ follows from (TRP) in the case of wanted triples (see Lemma~\ref{split}, (5) below). \end{enumerate} \end{exa} \begin{defn}\label{abf} Base point freeness with a {\em tolerance\/} $\tau>0$: fix a reduced divisor $F$, then $D\sim D_{bf}+D_{fr}$, where $\Bs{|D_{bf}|}=\emptyset$, in particular, $D_{bf}$ 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 that the linear system is base point free {\em within the tolerance\/} $\tau$. Similarly, the nonvanishing with a {\em tolerance\/} $\tau>0$: $D\sim D_{int}+D_{fr}$, where $D_{int}$ 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 that the liner system is $\ne\emptyset$ {\em within the tolerance\/} $\tau$. \end{defn} The main result of the section is \begin{thm}\label{bndbf} Let \begin{itemize} \item $(X_u/Z_u,F_u), u\in U$ be a bounded family of pairs with reduced divisor $F_u$ on $X_u$; \item $\fN_{nbf}(X_u/Z_u,F_u)$ be a {\rm family\/} of bnd-sets$/U$ of divisors {\rm(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 $D\in \fN_{bnd}(X_u/Z_u,F_u)$, $\Bs{|m D|}=\emptyset\mod{\tau}$, in particular, $|m D|\ne\emptyset\mod{\tau}$, for some $1\le m\le M$ (depending on $D$). \end{thm} \begin{sta} Moreover, there is a real $\de>0$ (depending on the family $(X_u/Z_u,F)$ and $\tau$), such that the same base point freeness and nonvanishing hold for any $u\in U$ and any $D\in \fU_{bnd,\de}(X_u/Z_u,F_u)$. \end{sta} Note: both choices of $M$ and $\de$ are very ineffective as the Kronecker theorem. We start with properties of the approximate base point freeness. \begin{prop} \label{pnonvan} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item The base point freeness within a tolerance $\tau$ is invariant for the linear equivalence: if $D\sim D'$, then $\Bs{\linsys{D}}=\emptyset\mod{\tau}$ implies $\Bs{|D'|}=\emptyset\mod{\tau}$. \item The base point freeness is an {\rm open\/} property: if $\Bs{\linsys{D}}=\emptyset\mod{\tau}$, then there is $\ep>0$ such that, for any $D'$ such that $D-D'$ is supported in $F$ and $\Vert D-D'\Vert<\ep$, also $\Bs{|D'|}=\emptyset\mod{\tau}$. \item If $D=D_{bf}+D'$, where $\Bs{|D_{bf}|}=\emptyset$ and $\Bs{|D'|}=\emptyset\mod{\tau}$, then $\Bs{\linsys{D}}=\emptyset\mod{\tau}$. \end{enumerate} The same holds for the nonvanishing within a tolerance $\tau$. \end{prop} \begin{proof} (1) By definition. In (2) we can take any $0<\ep\le \tau-\Vert D_{fr}\Vert$, where as $D\sim D_{bf}+D_{fr}$ as in Definition~\ref{abf} and $D'\sim D_{bf}+D_{fr}'$. (3) because the sum of two base point free linear systems is also base point free. \end{proof} \begin{pfof}{Theorem~\ref{bndbf}} Let $(X/Z,F)$ be a point of the family, that can be nonclosed (=prime subfamily). Let $F=\sum P_i$ and $\fD_F=\bigl\{\sum d_i P_i\bigm| d_i\in\R\bigr\}$, divisors generated over $F$. Suppose that, for $\fN_{bnd}(F)$, there are 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}{(\roman{enumi})} \item each $T_i$ is base point free$/Z$; \item each $N_i$ has a natural {\em index\/} (of base point freeness) $n_i>0$ for which \[ \Bs{|n_i N_i|}=\emptyset; \] \item in addition, for each $D\in U_i$, $\Bs{|n_i D|}= \emptyset\mod{\tau}$, more precisely, $n_i D=n_i N_i+D_{fr}$, $D_{fr}$ is supported in $F$ with $\Vert D_{fr}\Vert<\tau$; \item $\fN_c$ is a compact rational polyhedron with generators $N_i$; \item each $D\in \fN_{bnd}(F)$ can be decomposed as follows $D\sim D_c+\sum t_i T_i$ 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 base point free$/Z_u$ for each specialization of $(X/Z,F)$, and, for any generic point of $(X/Z,F)$, that is, over some Zariski open subset $U\ne\emptyset$, or for each $u\in U$, $\fN_{bnd}(F)$ specializes isomorphically to $\fN_{bnd}(F_u)$ preserving the above structures $T_i,N_i$ and $\fN_c$ in (4). \end{enumerate} Then the theorem holds for $\fN_{bnd}(F)$ and there exists $\de>0$ that addition~\ref{bndbf}.1 holds for $\fU_{bnd,\de}(F)$. Moreover, both hold for each $u$ in nonempty $V\subset U$. In particular, in such proof of the theorem and of \ref{bndbf}.1 we can use the Noetherian induction, that is, enough to verify both on a one element family $(X/Z,F)$. Thus if there exist a natural number $M_V=M_F$ and real $\de_V=\de_F$ and respectively $M_{(X/Z,F)\setminus V}$ and $\de_{(X/Z,F)\setminus V}$ over the closed proper subfamily $(X/Z,F)\setminus V$ in the theorem and in the addition, then the both hold over all points (=specializations) of $(X/Z,F)$ with $M=\max\bigl\{M_V, M_{(X/Z,F)\setminus V}\bigr\}$ and $\de=\min\bigl\{\de_V,\de_{(X/Z,F) \setminus V}\bigr\}$. Indeed, (1--6) holds over some $V$. (1) by our assumption in (7). (2) and (5) define a nontrivial affine open subset $V$ over which all $n_i N_i$ are base point free, and a restriction (=specialization) of $\sim$ on $X_u/Z_u$ is correct (= in general position; cf.\ (GPN) of Proposition~\ref{ressat}). To have the good specialization of $\sim$ it is enough to consider $X_u/Z_u$ that are smooth in 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 generation (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 we fix $(X/Z,F)$ and derive for it the theorem and addition from (1--6) and then verify them and (7). We take $M_F=\max\{n_i\}$ for the indexes in (2); $\de_F$ is chosen later. Indeed, Proposition~\ref{pnonvan}, (1), (3) and the above (1) (5) imply that the theorem and the addition is enough to verify respectively for $\fN_c$ and for \[ \fU_{c,\de}=\bigl\{ D\bigm| D-D'\in\fD_F, D'\in \fN_c, \quad {\rm and} \quad \Vert D-\fN_c \Vert\le \Vert D-D' \Vert<\de\bigr\}. \] In its turn both, the theorem and the addition, follow from (2--3) and (6). For the addition 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 family space, 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} The (BND) means that there is natural number $I>0$ such that, for any semiample integral divisor $D$, $\Bs|I D|=\emptyset$. To fulfil (1) and (7), we take $T_i=I D_i$. Respectively, 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 $N_i$ by (BND), too. Moreover precisely, $n_i\le I d_i$, where $d_i$ is the minimal natural number such that $d_i N_i$ 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) by (CFG) for $(X/Z,F)$ in (7$'$). To fulfil (6) we need to add $\Q$-divisors $N_i$ from $\fN_c$ that does not effect (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 that we can assume that $\fN_c$ is a simplex, and all its faces are covered by an induction. Indeed, $\Q$-points of the rational simplex $\fN_c$ are $\Q$-Cartier $\Q$-divisors. They are covered. On the other hand, by simultaneous approximation, Cassels \cite[Th. VII of Ch. I]{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. \end{pfof} Base point freeness within a tolerance sometimes implies the usual nonvanishing but the latter is typically far away from the base point freeness. \begin{cor}\label{bndnv} Let $C_u$ be a divisor on $X_u$ such that all $\mult_{P_i}{C_u}>-1+\tau$. Then, for any $D\in \fU_{bnd,\de}(F_u)$, there exists some natural number $1\le m\le M$, that $|\rdup{m D+C_u}|\ne\emptyset$. \end{cor} \begin{sta} If $\mult_{P_i}{F_u}=0$ we can assume just $\mult_{P_i}{C_u}>-1$. \end{sta} \begin{proof} In the proof below ``all $\dots$'' means for all prime divisors $P_i$ of $X$. We drop the low scripts $_u$ below. Take such $m$ in \ref{bndbf}.1, that $|mD|\ne\emptyset\mod{\tau}$, that is, $mD\sim D_{int}+D_{fr}$, where $D_{fr}=\sum d_{fr,i}P_i$ with all $|d_{fr,i}|<\tau$, in particular, all $d_{fr,i}>-\tau$. Hence all $\mult_{P_i}{(D_{fr}+C)}=d_{fr,i}+\mult_{P_i}{C}> \mult_{P_i}{C}-\tau>-1$ and all $\mult_{P_i}{\rdup{D_{fr}+C}}=\rdup{\mult_{P_i}{(D_{fr}+C)}}\ge0$. Thus $\rdup{D_{fr}+C}\ge0$ and $\rdup{D_{int}+D_{fr}+C}=D_{int}+\rdup{D_{fr}+C}\ge0$. That proves the required nonvanishing: $\rdup{m D+C}\sim \rdup{D_{int}+D_{fr}+C}\ge0$ by Lemma~\ref{intpart}, since $m D+C\sim D_{int}+D_{fr}+C$. Since $d_{fr,i}=0$ when $\mult_{P_i}{F}=0$, we get \ref{bndnv}.1, because then again $\mult_{P_i}{(D_{fr}+C)}=d_{fr,i}+\mult_{D_i}{C}>0-1>-1$. \end{proof} We can use Corollary~\ref{bndnv} as a restriction on $\tau$ for which the former holds. Examples: 1. Any $\tau<1$, when $C\ge0$. 2. Let $C=\sA_E$, the discrepancy divisor, then any $\tau<\min\{1-b_i\}\le$ the m.l.d. of $(X,B)$. In particular, there exists $\tau>0$ whenever all $b_i<1$, that is, $(X,B)$ is Kawamata log terminal in divisors. \begin{cor}\label{nonvc} Suppose that each $F_u$ is $\Q$-Cartier on $X_u$. Let $\al<1,\tau$ be positive reals, and $\sC_u$ an $\R$-{\bi}divisor of $X_u$ such that $\sC_u\ge0/X_u$ and \begin{description} \item{\rm(*)} all $\mult_{P_i}{\al \sC_u}>-\al+\tau \mult_{P_i}{F_u}$. \end{description} Then there exist a natural number $M$, and positive reals $\be,\de$, which give 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 $\sD_X\in \fU_{bnd,\de}(F_u)$, $|\rdup{m \sD_Y+(\sC_u)_Y}|\ne\emptyset$ for some $1\le m\le M$. \end{cor} \begin{proof} Again we drop the low scripts $_u$ below. By \ref{bndbf}.1, we can choose natural number $M$ and positive $\de$. They depend on the family $(X/Z,F)$ and $\tau$. Then we can take any positive \[ \be\le \frac{1-\al}M. \] Take now any {\bi}divisor $\sD$ satisfying the nonvanishing conditions. Let $1\le m\le M$ be a natural number such that $\Bs{|m\sD_X|}=\emptyset\mod{\tau}$. We verify that $|\rdup{m \sD_Y+\sC_Y}|\ne\emptyset$ for arbitrary $Y/X$. This follows from Lemma~\ref{movest} with $\sD:=m\sD$. The (**) in the lemma follows from the boundedness of the descent data of the original $\sD$. Since $\sC\ge0/X$ and by \HOM\ in Proposition~\ref{desd}, for any natural number (even positive real number) $m\le M$, 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{description} \item{(**)} $\sE\le (1-\al)\sC/X$. \end{description} \end{proof} \begin{rem} \label{constnv} Whenever the family $(X_u/Z_u,F_u)$ with $\sC_u$ is given, we use the following order to find constants $\al,\be,\de,\tau$ and $M$. First, we take any positive $\al<1$. Second, we choose an appropriate positive $\tau$ (cf.\ Example~\ref{fnonvex} below). Third, by Theorem~\ref{bndbf} and~\ref{bndbf}.1 we can find $\de$ and $M$. Finally, we take $\be$ as in the last 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 need only $\be,\de>0$ and a natural number $M$, and so can drop $\al,\tau$ (cf.\ Corollary~\ref{fnonvcon} below). \end{rem} \begin{rem} \label{edesd} The descent data $\sE$ exists when $\sD_X$ is $\R$-Cartier by \EXI\ in Proposition~\ref{desd}. Thus for all $\sD$, when $X$ is $\Q$-factorial. \end{rem} Now we apply above results to triples $(X/T/Z,B,\fF)$ (see Definition~\ref{tri}, and cf.\ Corollary~\ref{mainb}). A {\em boundedness\/} of triples means that moduli of them are bounded; details see in the proof of Theorem~\ref{bnonvan} below. The next preliminary result explains also a role of conditions (CRP) and (RPC) in Definition~\ref{tri} (cf.\ Example~\ref{noncrm} below). Before we fix notation: Let \[ \CDiv^0_{\R}(X/T/Z)=\bigl\{D\in \CDiv_{\R}(X/Z)\bigm| D\sim_{\R}0/T\bigr\}. \] Equivalently, each such divisor $D\sim_{\R}g^*M/Z$ for some $\R$-Cartier divisor $M\in \CDiv_{\R}(T/Z)$. In addition, $M$ is defined up to $\sim_{\R}/Z$ (cf.\ Lemma~\ref{lin0l}). Moreover, $\sim_{\R}/Z$ on $T/Z$ is the same as numerical equivalence $\equiv/Z$ (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 used to identify and denote $L^1$ by $g_*L$. If $L$ is defined$/\Q$, the splitting can be defined also over $\Q$. For example, if $L=\fC=\fC(X/T/Z,F)=\bigl\{D=\sum d_i P_i\bigm| d_i\in\R, \text{ and } D\sim_{\R}0/T\bigr\}\subset \fD_F$ with $F=\sum P_i$, we have a splitting \[ \fC=\fC^0\oplus g_*\fC \] where \[ \fC^0=\fC^0(X/T/Z,F)=\bigl\{D=\sum d_i P_i\bigm| d_i\in\R \text{ and } D\sim_{\R}0/Z\bigr\}. \] \begin{lem} \label{split} Let $(X/T/Z,B,\fF)$ be a wanted triple for a weak log Fano contraction, and $F=\sum P_i\in \fF$ be a reduced divisor. Then \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item the $g_*$ induces a splitting$/\Q$ \[ \fC=\fC^0\oplus g_*\fC \] where {\rm now\/} \[ \fC^0=\fC^0(X/T/Z,F)=\bigl\{D=\sum d_i P_i\bigm| d_i\in\R \text{ and } D\equiv 0/Z\bigr\}; \] \item Cone theorem: the nef or even {\rm semiample} cone \begin{multline*} \fN_{nef}=\fN_{nef}(X/T/Z,F)= \\ \bigl\{D=\sum d_i P_i\bigm| d_i\in\R, D\hbox{ is nef and } \sim_{\R}0/T \bigr\} \end{multline*} is a rational polyhedral cone in $\fC$; \item $g_*\fC$ is isomorphic$/\Q$ to $\CDiv_{\R}(T/Z)/\equiv$, whenever components $P_i$ generate the group of Weil divisors of $X$ $/\sim$; and \item there exists a uniform natural constant, the {\rm base point freeness\/} index $I$, such that for each nef integral $\Q$-Cartier $D\in \CDiv^0_{\R}(X/T/Z)$, multiple $I D$ is base point free on $X/Z$. \item $\fN(X/T/Z,F)$ is a bnd-set {\rm(cf.\ (TRP) in Example~\ref{bndc})\/}. \end{enumerate} \end{lem} \begin{proof} (1) Since the triple is a wanted triple for a weak log Fano contraction $(X'/Z,B')$, we have a diagram of contractions \[ X'\xleftarrow{\,i\,} W\xrightarrow{\,h\,} X\xrightarrow{\,g\,} T/Z \] where $i\circ h\1:X\to X'$ is a birational isomorphism onto a weak log Fano contraction $X'$. We need to verify that each $\R$-Cartier divisor $D \equiv 0$ on $X/Z$ is $\sim_{\R} /Z$. It is enough to verify the same for $h^*D$ on $W/Z$. By the rationality of the singularities of $X'$ can be descend up to $\sim_{\R}$ the problem onto $X'/Z$. However each $\R$-Cartier divisor, that $D\equiv 0/Z$, is $\sim_{\R}0/Z$ on any weak log Fano contraction$/Z$. In addition, if $D$ is Cartier, that $D\equiv 0$, then $D\sim 0$ essentially by the Contraction Theorem. \begin{rem} Except for the last statement, the above holds if we just assume that $X'$ has only rational singularities and the 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 also that each nef$/Z$ divisor in $\CDiv^0_{\R}(X/T/Z)$ is semiample$/Z$, that is, the cone is also semiample$/\sim_{\R}$. Since $g_*\fN_{nef}=g_*\fC\cap$ 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 each integral Cartier divisor is also generated by components $D_i$. To prove (4) we can assume as in (3) that up to $\sim$ integral $\Q$-Cartier and nef $D/Z\in\fC$. 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 can be included into $I$. 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). For $D_0$, we have base point freeness whenever $D_0$ is Cartier as it was remarked in the proof of (1). Hence for $D_0$ we can take $I$ as a Cartier index: $I D_0$ is Cartier. The latter 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 $\mult_{P_i}{D}=i,i\in \Z,$ for dropping $P_i$, and keep integral generators $D_i$. In each compact rational polyhedron $\fN_c$, this also cuts out a rational compact polyhedron. \end{proof} Notation: For a triple $(X/Z,B,\fF)$ with bounded family $\fF$, set (cf.\ Example~\ref{bndc}, (2)) \[ \fN=\fN(X/T/Z,\fF)=\bigcup_{F\in \fF}\fN(F), \] and $ \fN=\fN(X/Z,\fF)=\fN(X/X/Z,\fF)$ when $T=X$ (cf.\ Theorem~\ref{fnonvan}). In other words, each $D\in \fN$ is nef$/Z$, $\sim_{\R} 0/T$, and $D=D_{int}+D_{fr}$, where $D_{int}$ is integral, and $\Supp{D_{fr}}$ is in $\fF$. Respectively, we set \[ \fU_{\de}=\fU_{\de}(X/T/Z,\fF)=\bigcup_{F\in \fF} \fU_{\de}(F), \] and $\fU_{\de}=\fU_{\de}(X/Z,\fF)=\fU_{\de}(X/X/Z,\fF)$. \begin{thm} \label{bnonvan} Let \begin{itemize} \item $(X_u/T_u/Z_u,B_u,\fF_u)$ be a bounded family of triples, wanted for some weak log Fano contractions {\rm(see the proof);\/} and \item a tolerance $\tau>0$. \end{itemize} 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 $D\in \fN=\fN(X_u/T_u/Z_u,\fF_u)$, $\Bs{|mD|}=\emptyset\mod{\tau}$, in particular, $|mD|\ne\emptyset\mod{\tau}$, for some $1\le m\le M$ (depending on $D$). Moreover, there exists some $\de>0$ that the same base point freeness and nonvanishing hold in any uniform 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 \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 $(X_u/T_u/Z_u,B_u,F_u),u\in U$, 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, pairs $(X_u/Z_u,F_u)$ belong to a bounded family $(X/Z,F)$. To fulfil (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, finally, this gives a family satisfying (7$'$). By definition each $F_u=F\rest{X_u}$ is reduced. After a base change we can assume that each prime component $P_i$ of $F$ specializes into a prime $P_{u,i}$. Thus the finite dimensional $\R$-linear space $\fD_F$ (generated by $F$) is locally constant (with respect to the basis $P_i$) over $U$, that is, specializes isomorphically on $\fD_{F,u}$ from its generic point. By Lemma~\ref{split}, (1) with $T=Z$, $\sim_{\R}0/T$ is defined geometrically: e.g., 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 constant$/V$ subspace of $\fD_F$ that specializes isomorphically onto $\fC_u=\fC(X_u/T_u/Z_u,F_u)$ in 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 onto the rational polyhedral cone $\fN_{nef,u}=\fN_{nef}( X_u/T_u/Z_u,F_u)$ in 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 satisfies with bounded (by degree) generating curves. Otherwise the complement to the union stratified by the cones, that are generated by the bounded (degree) curves, 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 bounded curves, is {\em constructive\/}; see. Remark~\ref{bndf} below.) To find the decomposition, in (CFG) we can add some prime components $P_i$ to $F$ that they together generate the group of Weil divisors of $X/\sim$ and also for the 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 onto $\fN_{c,u}$ for $(X_u/Z_u,F_u)$ and $D_i$ specializes into $D_{u,i}=D_i\rest{X_u}$ respectively. Since we have a finite set of generators $D_i$, (CFG) implies (BND) and 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. \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 the anticanonical boundedness \cite{sh96a}. It 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 generic $u\in U$. 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 (base point free) over the very generic point $u\in U$ by the standard properties of $f_*\Oh_X(D)$. This implies 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 $1\le m\le M$, that $|\rdup{m D+C_u}|\ne\emptyset$. \end{cor} \begin{sta} 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{sta} \begin{proof} The proof of Corollary~\ref{bndnv} with Theorem~\ref{bnonvan} instead of \ref{bndbf}.1. \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 $\fF=\{p+q\mid p\neq\in E\}$. Then Theorem~\ref{bnonvan} does not hold for this triple. (Quiz: 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 $D\sim_{\R}0$ 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 reals and $\sC_u$ an $\R$-{\bi}divisor of $X_u$ such that: $\sC_u\ge0/X_u$ and \begin{description} \item{\rm(*)} all $\mult_{P_i}{\al \sC_u}>-\al+\tau \mult_{P_i}{F_u}$ for any $F_u\in \fF_u$. \end{description} Then there exist a natural number $M$, and positive reals $\be,\de$ 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 $\sD_X\in \fU_{\de}( X_u/T_u/Z_u,\fF_u)$, $|\rdup{m \sD_Y+(\sC_u)_Y}|\ne\emptyset$ for some $1\le m\le M$. \end{cor} \begin{proof} The proof of Corollary~\ref{nonvc} with Theorem~\ref{bnonvan} instead of \ref{bndbf}.1. Note that each $F_u\in\fF_u$ is $\Q$-Cartier on $X_u$ by (QFC) of wanted triples. \end{proof} \begin{cor} \label{bnonvcon} There are reals $\be,\de>0$ and a natural number $M$, which give the nonvanishing as in {\rm 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 Kawamata log terminal pair with a subboundary $B$, $\Q$-factorial $X$, and $\fF$ a {\em bounded\/} family of reduced divisors on $X$. Then there exists real $\gamma>0$ such that $(X,B+\gamma F)$ is Kawamata log terminal for any $F\in \fF$. Therefore any $\mult_{P_i}{\sA}-\gamma \mult_{P_i}{F}=\mult_{P_i}{(\sA-\gamma\overline{F})}=a(X,B+\gamma F,P_i) >-1$ (cf.\ the proof of Lemma~\ref{discr}, and Example~\ref{stexm}). Thus for any real $\al>0$, $\al \mult_{P_i}{\sA}>-\al+\tau\mult_{P_i}{F}$ for $\tau=\al\gamma$. That give (*) 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 (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 wanted triples for weak log Fano contractions {\em (for example, as in Corollary~\ref{bnonvcon} \/}, and let $\al\in (0,1)$. Then on there exists $\tau$ such that all $\sC_u$ satisfy the conditions of {\rm Corollary~\ref{bnonvbd} \/}, in particular, {\rm(*)\/}. \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 Kawamata log terminal is open in families. For any wanted triple $(X_u/T_u/Z_u,B_u,\fF_u)$, each $X_u$ is $\Q$-factorial, each $(X_u,B_u)$ is terminal in codimension~$2$ respectively by (QFC), (CRP) and (TER) in Definition~\ref{tri}. \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). \end{pfof} An explanation to the bounded nonvanishing: Really, in applications, we can weaken just to nef the nef and big condition for $M$ in the definition of wanted triples. Thus we are interested in two types of wanted triples that correspond to the cases, when $\sM$ is big or 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) that corresponds to triples $(X_t/X_t/\pt,B_t,\fF)$ with crepant $B_t$, and (2) elliptic fiberings $/T_u=\PP^1$, that corresponds to triples $(X_u/T_u/\pt,B_u,\fF)$ with elliptic fibering $X_u\to T_u=\PP^1$, and crepant $B_u$. In both cases, $\fF$ is obtained from the standard family on $(X/\pt,B)$ by the log birational transform. As usual, the simplest form of base point freeness is on log Fano contractions. \begin{thm} \label{fnonvan} \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 $D\in \fN=\fN(X/Z,\fF)$, $\Bs{|mD|}=\emptyset\mod{\tau}$, in particular, $|m D|\ne\emptyset\mod{\tau}$ for some $1\le m\le M$ (depending on $D$). \end{thm} \begin{sta} Moreover, there exists real $\de>0$ (depending on $(X/Z,B)$, $\fF$ and $\tau$), that the same nonvanishing holds for all $D\in \fU_{\de}=\fU_{\de}(X/Z,\fF)$. \end{sta} \begin{proof} Immediate by Theorem~\ref{bnonvan}. We consider the weak log Fano contraction $(X/Z,B)$ with bounded $\fF$ as a triple $(X/X/Z,B,\fF)$. \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 $|\rdup{m D+C}|\ne\emptyset$. \end{cor} \begin{sta} If $\mult_{P_i}{F}=0$ we can assume just $\mult_{P_i}{C}>-1$. \end{sta} \begin{proof} Immediate by Corollary~\ref{bnonvd}. \end{proof} \begin{cor} \label{fnonvbd} Suppose that each $F\in\fF$ is $\Q$-Cartier. Let $\al<1,\tau$ be positive reals and $\sC$ a {\bi}divisor such that $\sC\ge0/X$ and \begin{description} \item{\rm(*)} all $\mult_{D_i}{\al \sC}>-\al+\tau \mult_{D_i}{F}$ for any $F\in \fF$. \end{description} Then there exist a natural number $M$, and positive reals $\be,\de$ which give 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 $\sD_X\in \fU_{\de}$, $|\rdup{m \sD_Y+\sC_Y}|\ne\emptyset$ for some $1\le m\le M$. \end{cor} \begin{proof} Immediate by Corollary~\ref{bnonvbd}. \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 reals $\be,\de>0$ and a natural number $M$, which give the nonvanishing$/Z$ as in {\rm Corollary~\ref{fnonvbd}\/} for $\sC=\sA$. \end{cor} \begin{proof} Immediate by Corollary~\ref{bnonvcon}. \end{proof} \section{Stabilization in divisor} \label{stbd} Now we are ready to establish a stabilization of a central multiplicity. First, we describe what we mean by an inductive model and its central divisor $E_c$. Then we state under what condition we expect that a limit of {\bi}divisors stabilizes near $E_c$. Finally, we prove this for 3-folds. This means a nonvanishing on $E_c$ for the limit. Afterwards, we extend this stabilization to a base point freeness of Corollary~\ref{stbklt} under Kawamata log terminal and in the realistic situation (CBS)(rfa) (cf.\ (CBS)(fga) in Conjecture~\ref{cbcon}). This is a base point freeness on a Kawamata log terminal model, terminal in codimension~2, that leads to the proof Theorem~\ref{cbrfa} in Section~\ref{mr}. Agreement: in this section we assume the LMMP in $n=\dim X$ whenever state ``Under the LMMP''. We disregard this for $n\le 3$. By Corollary~\ref{mainb} we can drop (CBS), (MOD) and $\SSB$ respectively for $n\le 2$. \begin{defn} A limit $\sD=\lim_{i\to\infty} \sD_i$ of $\R$-{\bi}divisors stabilizes {\em near a set of prime {\bi}divisors\/} $E_i$ if it stabilizes near centers (as closed subvarieties) of these {\bi}divisors on some model $Y/X$, that is, $\sD_i=\sD$ for some $i\gg0$ over a neighborhood of these centers {\em in\/} $Y$. \end{defn} \begin{defn}[cf.\ Prokorov and Shokurov \cite{PSh}] A (strictly) {\em inductive model\/} of a log pair $(X/Z,B)$ is its log model $(Y/Z,B_Y)$ such that \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $(X,B_X)/(Y,B_Y)$ is {\em log\/} proper, that is, $\sB^Y=\sB(Y,B_Y)\ge \sB^X=\sB=\sB(X,B)$ or, equivalently, $\sA^Y=\sA(Y,B_Y)\le \sA^X=\sA=\sA(X,B)$; \item $(Y,B_Y)$ is {\em exceptional\/} in the following sense: there is a single divisor $E_c$ of $Y$ with $\mult_{E}{B_Y}=1$, and $(Y,B_Y)$ is (respectively {\em strictly\/}) purely log terminal; \item $-(K_Y+B_Y)$ is nef and big$/Z$ (the latter holds automatically when $X/Z$ is birational, and $-(K_Y+B_Y)$ is nef$/Z$) (respectively is ample$/Z$); and \item $-(K_Y+B_Y)$ is nef and big on $E_c/Z$ (respectively is ample$E_c/Z$). \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 Kawamata log terminal replace by purely log terminal (cf.\ (PFN) in Lemma~\ref{incrb} below); but it is still not a general log Fano contraction. Reminder: ``strictly'' in (2) means that $Y$ is $\Q$-factorial \cite[p.~99]{sh92}. \begin{warn} $Y$ may not be$/X$. \end{warn} \end{defn} \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} {\rm Under the 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 {\rm(that is, $E_c/P$)\/}. \end{itemize} \end{lem} \begin{sta} 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{sta} \begin{sta} We can upgrade any inductive model up to a strict one. \end{sta} \begin{proof} We consider here only the case we need when $X/T$ is birational. The general case can be found in Prokhorov and Shokurov \cite[3.2]{PSh}. After a complement we assume that $K+B\equiv 0/T$. In addition we can suppose that $B$ is $\Q$-divisor. This increases $\sB$, which is important for (1) in the definition. Hence after a contraction we can assume that $X=T$. First, we can build a log canonical singularity adding $\ep H$ for an ample divisor through the point $P\in X$. 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 the 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$/X$. If it is not so we can apply the 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 $E_c$. Indeed, 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). \ref{indmod}.1 follows from the adjunction formula \cite[3.1 and 3.2.3]{sh92}, and $E_c$ normal \cite[Lemma~3.6]{sh92}. \ref{indmod}.2 follows directly from the following result. \end{proof} \begin{lem} \label{incrb} {\rm(Under the LMMP; cf.\ a remark at the end of the proof.)\/} Let \begin{q} \item[\rm(PFN)] $(X/T,B)$ be a weak {\rm purely\/} log terminal Fano contraction, {\rm 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 $-(K+B)$ is nef and big on each reduced component {\rm(at most two)\/} in $B$. Then we can find another boundary $B^+$ $(X/T,B^+)$ such that: \begin{itemize} \item $(X/T,B^+)$ still satisfies {\rm(PFN)\/} with ample $-(K+B^+)/T$; \item $B^+\ge B$; and \item the reduced components of $B$ and $B^+$ are the same. \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 movable divisor $D$ on $X$ which is negative$/Y$. In particular, for such generic $D$, the reduced divisors of $B$ are not in $\Supp{D}$. Thus we can replace $B$ by 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 the ampleness, we use here that the Kleiman--Mori cone of $(X/T,B)$ is polyhedral. \end{proof} \begin{rem} We don't need here the LMMP on the whole: e.g., it is enough the cone and the contraction theorems which work under our conditions. \end{rem} Now we are ready to state a stabilization of the limit $\sD=\lim_{i\to\infty} \sD_i$ {\em in\/} 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} {\rm We assume the LMMP and $\CBS_{n-1}\gl$, $\SSB_{n-1}\gl$.\/} Let $(X/T,B)$ be a log pair, and $\sD=\lim_{i\to\infty} \sD_i$ 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)] for all $i$, $i\sD_i\sim\sM_i/T$, where $\sM_i$ is {\bi}free; \item[\LCA] $\sD_\bull$ is lca saturated; \item[\MXD] {\rm maximality of the limit:\/} each $\sD_i\le \sD$; and \item[\BED] each $\sD_i=\sD$ outside $f\1P$ over $T$ {\rm(cf.\ Proposition~\ref{fgadi})\/}. \end{q} Then there exists an inductive model with a central divisor $E_c$ such that the limit $d_c=\lim_{i\to\infty} d_{c,i}$ is rational number and the limit stabilizes: $d_c=d_{c,j}$ for infinitely many $j$. \end{thm} Corollary~\ref{stabind} below slightly improves this result. \begin{rem} Under (LWF) the {\em linear\/} birational freeness (LBF) is equivalent to the {\em numerical\/} one $i\sD_i\equiv\sM_i/T$, whenever $i\sD_i$ is Cartier or just integral. It is enough to establish it for $\overline{i(\sD_i)_Y}=i\sD_i\equiv 0/T$. Then the required $\sim$ follows from a descent of Cartier divisors for their contractions on rational singularities, and from the stable base point freeness on $Y=X$ as in (LWF) (cf.\ an addition 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 a little bit more general rationality, $d_c\in \Q$, and a stabilization in Proposition~\ref{ration}. Then using Proposition~\ref{bounds} about bounded presentations of fixed parts 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 rationality theorem for the Kleiman-Mori cone and to the nonvanishing theorem. However this time we need a birational version of nonvanishing: e.g., Corollary~\ref{bnonvcon} and Theorem~\ref{fnonvan}. Then we interpret the 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} with Corollary~\ref{incrad} below are needed to construct an appropriate inductive model. An ideal pattern for Theorem~\ref{stabin} and for other results in this section and in the paper (how we can use an inductive model) gives \begin{exa}[Projective space]\label{prs} Let $(X/Z,B)=(\PP^n/\pt,0)$ be a projective space. Then (FGA), and $\SSB$, (PRM), (PFC), (CBS) with (MOD) of Conjecture~\ref{cbcon} hold on it. Moreover, $\fS=\fP=\emptyset$, and in (CBS) $\Bs{|\sM\rest{\PP^n}}=\emptyset$ for each $\sM\in \fM$; so, $c=1$! Moreover, for any $\R$-{\bi}divisor {\em under\/} the saturation \SAT, $\Bs{|\sD\rest{\PP^n}}=\emptyset$. Respectively, (FGA) gives as the stable models only $\PP^n$ itself or $\pt$ It is possible to prove this using the restriction on $D=\sD_Y$ as in the proof of Proposition~\ref{surbound}. But this needs the Fujita estimate on the base point freeness that is still not established. Another {\em real\/} 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 blowup of the point. If $\sD=\sD_Y$ is fixed (prime) we use the 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 movable. If $\sD$ is movable, for example, $\sD=\sM\in \fM$, then we can use also models $(\PP^n/\pt,H=\PP^{n-1})$ with (generic) hyperplane sections $H$. Again by induction $|\sD\rest{\PP^n}$ has only {\em closed\/} base points $p$ (cf.\ Proposition~\ref{fgadi}). For such $p\in\Bs{|\sD\rest{\PP^n}}$, we use induction, namely, $|\sM\rest Y\rest E$ is base point free where $\sM=\Mov{\sD}$ is {\em generic\/} in the linear system on $Y$ with $\sM_{\PP^n}=\sD_{\PP^n}$ (cf.\ Proposition~\ref{bounds}). Hence $\Mov{\rdup{\sD+\sA}}\ge \Mov{(\sM+\overline{E})}=\overline{(\sM)_Y}+\overline{E}$, which contradicts the 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 their blowup in a closed point, and \item there are ladders of smooth log Fano varieties (up to surfaces) \end{itemize} both of which give inductive models. For example, this is true for nonsingular quadrics. All these aesthetically appealing but unfortunately is not deep. \end{exa} The following result generalizes the Rationality Theorem for Cones. Notation: In the proposition below $\sA'=\sA'(X,B)=\sA+E_c$ is an {\em adjusted (truncated)\/} discrepancy whereas $(X,B)$ is {\em exceptionally\/} log terminal with single $E_c$ having $\mult_{E_c}{\sA}=-1$. \begin{prop} \label{ration} Let $(X/T,B)$ be a strict inductive model with a central divisor $E_c$, $E'\sim E_c$ a linear equivalence with $E_c$ not in $\Supp{E'}$, 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)} the {\rm integral over $E_c$\/} asymptotic saturation holds for $\sD_\bull$ with respect to $\sC=\sA'+\ep \overline{E_c}$ for some $\ep>0$ {\rm(cf.\ Definition~\ref{ints})\/}; \item[\MXC] {\rm 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, for $(E_c/\pt,B_c)$, {\rm(cf.\ Theorem~\ref{bnonvan})\/} such that: each $E_u/E_c$, each $(\sD_i\frest{E_c})_{E_i}\in \fN(E_i/T_i/\pt,\fF_i)$ {\rm(the mixed restriction for $K=\R$, cf.\ (mx) in Proposition~\ref{rest})\/}, and the descent data of the restrictions $\sD_i\vdots _{E_c}$ are asymptotically bounded with respect to $\sA_c=\sA(E_c,B_c)$ over the {\rm 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} Note: $E'\sim E_c$ {\em induces\/} $E''\sim E=g\1E_c$ on any other model $g\colon Y\to X$ with $E$ not in $\Supp{E''}$. Take $E''=g^*E'+E-g^*E_c$. The restriction (mx) defined then by $E''$ on $Y$ is independent on $Y/X$: since $\mult_{E_c}{\sD}=\mult_{E}{\sD}=d$, then $\sD\frest{E_c}= (\sD-d\overline{E_c})\frest{E_c}+d\overline{E'}\frest{E_c}= (\sD-d\overline{E_c})\frest{E_c}+d\overline{E'\rest{E_c}}= (\sD-d\overline{E_c})\frest E+d\overline{g^*E'\rest E}= (\sD-d\overline{E_c})\frest E+d\overline{(E''+g^*E_c-E)}\vdots_E= (\sD-d(\overline{E_c}-\overline{g^*E_c}+\overline{E}))\frest E- d\overline{E''}\frest E=(\sD-d\overline{E})\vdots_E- d\overline{E''}\frest E$, because $g^*E'=E''+g^*E_c-E$ and $\overline{E_c}=\overline{g^*E_c}$. \begin{proof} By the monotonicity in Lemma~\ref{monots} and the negativity of $(K+B)/T$, we can replace $\ep$ by a smaller positive value so that the saturation ($\ep$A$'$S) is still satisfied, 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 $j\ge0$, 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{multline*} \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{multline*} where $-(K+B)+d E_c$ is nef and big$/T$, $(\sD_i)_Y$ is nef$/T$. Thus $g^*(-(K+B)+d E_c)+j(\sD_i)_Y)$ is nef and big$/T$, and we have the required vanishing due to Kawamata--Viehweg. However to apply this vanishing to a restriction on a 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), equivalently, $d+j d_{c,i}$ is integral. More precisely, then we have a surjectivity of linear systems \[ |E+\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}|- \to |(E+\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y})\rest E|. \] In addition, by the integral property of $d+j d_{c,i}$, the normal crossing, by Lemmas~\ref{restdiv}-.\ref{intpart} and the adjunction, the linear system \begin{align*} & \left |(E+\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y})\rest E\right|= \\[6pt] &\qquad= \left|\rdup{(E+\sA_Y)\rest E+(d(\overline{E_c})_Y+ j(\sD_i)_Y)\rest E}\right|=\\[6pt] &\qquad=\left|\rdup{(\sA_c)_E+((d\overline{E_c}+j\sD_i) \frest{E_c})_E}\right|=\\[6pt] &\qquad=\left|\rdup{(\sA_c)_E+ (((d+j d_{c,i})\overline{E'}+j\sD'_i)\frest{E_c})_E}\right|= \\[6pt] &\qquad=\left|\rdup{(\sA_c)_E+ (((d+j d_{c,i})\overline{E''}+d(\overline{E_c}-\overline{E})+ j\sD''_i)\frest{E})_E}\right|, \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 $(d\overline{E_c}+j\sD_i)\frest{E_c}$ or by $((d+j d_{c,i})\overline{E'}+ j\sD'_i)\frest{E_c}=((d+j d_{c,i})\overline{E''}+ d(\overline{E_c}-\overline{E})+ j\sD''_i)\frest{E_c}$ (as in the note before the proof), whereas $E$ is a sufficiently high resolution of $E_c$ and $E'\sim E_c$ on $X/T$ with $E_c$ not in $\Supp{E'}$, induced $E''\sim E$ on $Y/T$ (with $E$ not in $\Supp{E''}$), $\sD'_i=\sD_i-d_{c,i}\overline{E_c}$ and $\sD''_i=\sD_i-d_{c,i}\overline{E}$. The latter resolution depends only on $i$. Hence if we have a nonvanishing for the linear system restricted on $E$, then the corresponding linear system \[ |E+\rdup{\sA_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}= |\rdup{\sA'_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}| \] is (base) free in $E$. In the following situation this is impossible, because gives a contradiction with \MXC. Namely, \begin{q} \item[\rm(NBF)] the base point freeness of $|\rdup{\sA'_Y+d(\overline{E_c})_Y+j(\sD_i)_Y}|$ in $E$ is impossible, whenever $j$ is a positive integer and integer $d+j d_{c,i}>j d_c$. \end{q} Indeed, then the multiplicity of the movable 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 $d_{c,j}\ge d_{c,i}+d/j>d_c$ due to the inequality in (NBF). This contradicts \MXC. By the above surjectivity, (NBF) is equivalent to the following vanishing: \begin{description} \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 $d+j d_{c,i}>j d_c$. \end{description} 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 reals $\be,\de>0$ 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 the bounded nonvanishing: $|\rdup{m\sD_{E_{hr}}+ (\sA_c)_{E_{hr}}}|\ne\emptyset$ 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 {\bi}divisor $\sD=q (d/j+d_{c,i})\overline{E'\rest{E_c}}+ q\sD'_i\frest{E_c}$ with 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}{(\roman{enumi})} \item $m>0,p$ are integral, and \item $p/q>d_c$. \end{enumerate} They imply that $m q=j>0$ and $m p=m d/m+m q d_{c,i}=d+j d_{c,i}$ are integral and $mp=d+j d_{c,i}>j d_c$ as in (VN). Now we rewrite the nonvanishing conditions for $\sD$: \begin{enumerate} \renewcommand{\labelenumi}{(\roman{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 $\sD=(p-q d_{c,i})\overline{E'\rest{E_c}}+ q(d_{c,i}\overline{E'\rest{E_c}}+ \sD'_i\frest{E_c})$, $q(d_{c,i}\overline{E'\rest{E_c}}+ \sD'_i\frest{E_c})=q\sD_i\frest{E_c}$ (this (mx) restriction 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 {\em natural\/} $q$, (3) holds for $\sD=p\overline{E'\rest{E_c}}+ q\sD'_i\frest{E_c}$, if we secure that $\Vert (p-q d_{c,i}) (\overline{E'\rest{E_c}})_{E_i}\Vert <\de$. (Note that for natural number $q$, $q \sD\in \fN(E_i/T_i/\pt,F_i)$ whenever $\sD\in \fN(E_i/T_i/\pt,F_i)$, because $\fN(E_i/T_i/\pt,F_i)$ is an Abelian semigroup, but it is not a convex body.) The latter holds whenever \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 a stricter form $00$ and $p$ that satisfy (2) and (3$'$). This gives $m\sD=m p\overline{E'\rest{E_c}}+ m q\sD'_i\frest{E_c}=(d+j d_{c,i})\overline{E'\rest{E_c}}+ j\sD'_i\frest{E_c}$ that 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 reals $N$ and $\ep>0$, there exists such 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 reals: $0<\ep\le\de$, whereas $\ep$ is the same or smaller than that as in the beginning of the proof, and $N\ge M$ as in (3$'$). Thus we get required integers $q>0$ and $p$, which satisfy (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 $i\gg0$, $p/q-d_{c,i}=r-d_{c,i}<\ep/N q \le\de/N q$ or $p-q d_{c,i}<\de/N$. We need also to verify that $0\le d\le\ep$: since $d=m(p-q d_{c,i})$ and we get above more than (3$'$) $00$. (They may not be proportional to the first $j$ since $m$ depend on $\sD$.) \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 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 $\sM$, about a boundedness of their fixed components on $E_c$, that is, that of the divisorial components of $\Bs{|\sM\rest X}\cap E_c$. \begin{prop} \label{bounds} {\rm Under $\CBS_{n-1}\gl$ and $\SSB_{n-1}\gl$.} Let $(X/T,B)$ be a strict inductive model with a central divisor $E$, $\Bs{}\subset X$ a proper reduced subvariety, and $\{\sM\}$ a set of (rather general) {\bi}free {\bi}divisors $\sM$ such that: \begin{q} \item[\rm(BSP)] $\Bs{|\sM\rest X}\subset \Bs{}$; and \item[\rm(SA$'$)] $\sM$ is $\sA'$-saturated on sufficiently high models $X_{hr}/X$ {\rm 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}{(\roman{enumi})} \item the {\rm boundedness\/} of the {\rm fixed component\/} of each $|\sM\rest X$ on $E$, {\rm that is, the {\em whole\/} divisorial component of the intersection $\Bs{|\sM\rest X}\cap E$ belongs to a bounded set $\fF$ of reduced divisors on $E$;\/} \item in each linear system $|\sM\frest E|$, {\rm a} generic element has bounded canonical singularities on a triple $(E_u/T_u/\pt,B_u,\fF_u)$ with $E_u/E$, wanted for $(E/\pt,B)$ {\rm(and also just bounded when $E$ is a surface);\/} and \item the wanted triples in (2) belong to a algebraic bounded family. \end{enumerate} \end{prop} Note that $\sM$ is base point free on $X$ outside $f\1P$ when $\Bs{}=\emptyset$ {\em there\/}. \begin{lem} \label{boundl}[cf.\ Proposition~\ref{ressat} and its proof] Under the assumptions of {\rm Proposition~\ref{bounds}, (SA$'$)} for $\sM$ implies {\rm(SA$'$F)} of {\rm Proposition~\ref{ssp}, (4)\/} with $\sM:=\sM \frest E$ and $F=\Supp{\Fix{(| \sM\rest X\rest E)}}$, {\rm except over \/} $\Supp{B_E}$. \end{lem} \begin{proof} Suppose that $\sM_X$ is rather generic in its linear system $|\sM\rest X$, 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 \[ R^1 g_*\Oh(\rdup{\sA_Y+ \ep\overline{M}_Y})=0; \] $\overline{M}$ is well defined because $X$ is $\Q$-factorial. That resolution $Y$ depends only on $\sM$. We take a log resolution such that $\Bs{|\sM\rest Y}=\emptyset$. Respectively, for the generic $M$, $g\1M=\sM_Y$ is also generic in $|\sM\rest Y$. Then the resolution is also a log resolution for $B+\ep M$, and $\rdup{\sA_Y+ \ep\overline{M}_Y}=K_Y+\rdup{-g^*(K+B)+\ep g^*M}=K_Y+\rdup{g^*(-(K+B)+\ep M)}$, 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 we have the required vanishing due to Kawamata--Viehweg. However to apply this vanishing to a restriction below on a 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 $a_i=\mult_{E_i}{\sA}$, whereas $\{a_i\}$ is its 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 $\Bs{|\sM\rest X}$ for the {\em generic\/} $M$, and $a^*_i=a_i$ otherwise. Thus $^*$, to compare with $'$, increases by $1$ {\em exactly\/} integral $a_i$ that are exceptional on $X$ and are over $\Bs{|\sM\rest X}$, in particular, $^*$ does not hold for $E$ itself. (When $\sM_X$ is base point free outside $f\1P$ we can assume that $^*$ increases {\em only\/} over $P$.) Hence $\rdup{\sA_Y+ \ep\overline{M}_Y}=\rdup{\sA_Y+ \ep\sM_Y+ \sum e_iE_i}=\rdup{\sA^*_Y}+\sM_Y$ and we got the vanishing \[ R^1 g_*\Oh(\rdup{\sA^*_Y}+ \sM_Y)=0. \] This gives a surjectivity of linear systems \begin{multline*} \left|E_Y+\rdup{\sA^*_Y}+ \sM_Y\right| \broken \\ \left|(\rdup{\sA^*_Y+E_Y}+\sM_Y)\rest{E_Y}\right|= \left|\rdup{(\sA^*_{E})_{E_Y}}+ (\sM\frest E)_{E_Y}\right|. \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 resolutions $X_{hr}/X$ and {\em some\/} one $E_{hr}/E$ respectively. In addition, by the 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 birationally extend $\sF_{E_Y}$ to $\sF$, or on (any) sufficiently high $E_{hr}/E$. Since $\overline{M}\frest{E}$ has the same support on $E$ as $M\rest E$, then $F=\sF_X$ is exactly the fixed divisorial component of $E\cap \Bs{|\sM\rest X}=\Supp{\Fix{(|\sM\rest X\rest E)}}$ (in general, $\ne \Fix{|\sM\frest E\rest E}=(\Fix{\sM\vdots_E})_E=0$ for {\bi}free $\sM$!) over where $B_E$ is integral, that is, {\em except over\/} $\Supp{B_E}$, because $(E,B_E)$ is Kawamata log terminal. 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 the 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 Kawamata log terminal, we get the saturation on $Y/T$: \[ |E_Y+\rdup{\sA^*_Y}+ \sM_Y|=|\sM_Y|+ E_Y+\rdup{\sA^*_Y}, \] whereas $|\sM_Y|$ is base point 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{multline*} \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{multline*} whereas $|(\sM\frest E)_{E_Y}|$ is base point 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 movable 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), whereas $F$ is considered as a {\bi}divisor. Then \begin{multline*} \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{multline*} that means the saturation (SA$'$F). \end{proof} In the proof of the lemma the $\sim$ invariance of the saturations (SA$'$) and (SA$'$F) was used. \begin{prop}[Invariance of saturations]\label{linvsat} If\/ $\sD\sim\sD'/Z$ then the\/ $\sC$-saturation of\/ $\sD$ is equivalent to the\/ $\sC$-saturation of\/ $\sD'$. The same holds for\/ $\R$-divisors. For the asymptotic saturation, we can replace each $\sD_i$ by $\sD_i\sim \sD_i'/Z$ {\rm 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 {\rm characteristic type\/} of $\sD_\bull$ {\rm(see Remark~\ref{asaturr}, (7));\/} the index $I$ of asymptotic saturation after a truncation by $I'$, is $I/$the gcd of $I$ and $I'$. \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, $\Mov{\rdup{D'+C}}=\Mov{(\rdup{D+C}+(a))}= \Mov{\rdup{D+C}}+(a)\le D+(a)=D'$, because the fixed part is independent on $\sim$, and the movable one changes by $(a)$ under $\sim$. For the asymptotic saturation, on 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 the invariance under the similarity it is enough to consider a truncation $\sD_\bull ^{[I']}=I'\sD_{i I'}$. Then, by asymptotic $\sC$-saturation of $\sD_\bull$ with index $I$, for any $i$ and $j$, divisible by $I$, \[ \Mov{\rdup{j\sD_i^{[I']}+ \sC}}=\Mov{\rdup{j I'\sD_{i I'}+ \sC}}\le j I'\sD_{j I'}=j\sD_j^{[I']}, \] that means the asymptotic $\sC$-saturation of $\sD_\bull^{[I']}$ with the same index $I$. Really, we can replace $I$ by $I/$the gcd of $I$ and $I'$. \end{proof} \begin{pfof}{Proposition~\ref{bounds}} By Lemma~\ref{boundl} we obtain (SA$'$F) and in particular it applies to the fixed component of $|\sM\rest X$ on $E$ because $\Supp{B_E}$ is fixed (cf.\ Remark after Corollary~\ref{mainb}). 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 $E_u/E$ after bounded blowups (e.g., resolutions) that preserves all the properties of wanted triples. The latter is true for (CBS) because any blowup decreases the descent date of {\bi}free $\sM$. Indeed, let $g\colon Y\to X/Z$ be birational, and $\sM$ rather generic, namely, $\sM_Y=\sM_X$ as {\bi}divisors. Then $\sE(\sM)/X=\overline{\sM_X}-\sM_X=\overline{g^*\sM_X}-\sM_Y= \overline{\sM_Y}-\sM_Y+ \overline{E}\ge \overline{\sM_Y}-\sM_Y=\sE(\sM)/Y$, where $E=g^*\sM_X-\sM_X \ge0$ by Lemma~\ref{divb}. \end{pfof} To secure the saturation ($\ep$A$'$S) we need to strength an 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 exist \begin{itemize} \item boundary $B^+\ge B$ \end{itemize} 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 $0<\ep\ll 1$. \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 it is big we can assume also that {\em effective\/} $D^+=ND'\ge D$ for some real $N>0$. 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 $0<\ep\le\de$, $\sA=\sA^++ \de\Dbar^+\ge \sA^++ \ep\overline{ND'}\ge \sA^++ \ep\Dbar$. \end{proof} \begin{cor} \label{incrad} Under the assumptions of {\rm 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 inequality $\sA\ge \sA^++ \ep\Dbar$ that we have by Lemma~\ref{incra}. \end{proof} \begin{pfof}{Theorem~\ref{stabin}} To prove the 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 Kawamata log terminal), and ($\ep$AS) holds: \begin{description} \item{\rm($\ep$AS)} the asymptotic saturation holds for $D_\bull$ with respect to $\sC=\sA+\ep \Dbar$ for some $\ep>0$. \end{description} Now we modify our weak log Fano contraction $(X/T,B)$ into an inductive model $(X/T,B)$. Such exists by Lemma~\ref{indmod}. Our assumptions is preserved including ($\ep$AS). The latter follows from the monotonicity (1) in the definition 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 the last change touches not only the boundary $B$, but also the contraction itself $X/T$. Thus $\Dbar$ in the ($\ep$AS) is replaced by a 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 birational invariant of modifications of $X/T$.) In particular, for any prime $\Q$-Cartier divisor $F$ on $X$ over $P$, $\sD\ge \overline{F}$. By Lemma~\ref{monots} we can replace $\sD$ by $\overline{F}$ in ($\ep$AS). Taking $F=E_c$ we get an inductive model that satisfies ($\ep$AS) with $D=E_c$. Moreover, by~\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 in $\Supp{E'}$, Since sets $\fF$ in \BWQ\ is defined up to a 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 a bounded support, that is, {\em add\/} a bounded (really one element!) set to each element of $\fF$. (The sum is included into 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}{(\roman{enumi})} \item boundedness of the fixed component of each $|\sM_i\rest X$ on $E_c$, that is, the divisorial component of $\Bs{|\sM_i|}_X\cap E_c$ belongs to bounded $\fF$; \item in each $|\sM_i\frest{E_c}|$, a generic element has bounded canonical singularities on a wanted triple of $(E_c/\pt,B_c)$ for the element (also just bounded when $E_c$ is a surface); and \item wanted triples in (2) belong a algebraic bounded 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 the 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 rather generic $\sM_1$, 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 \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 $\sM_i$, $\Bs{(\sM_i)_X}\subset \Supp{M}_1\cup f\1P\subset \Bs{}$. (Really, we proved (1--3) in a little bit more general settings: we can consider 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 rather 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 uniformly we replace each $\sD_i$ 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 do not effect the conditions in the theorem. (LWF) does not concern the change. (LBF) because $i\sD^g_i=i\sD_i+i \overline{(a)}\sim i\sD_i$. \LCA\ holds by Proposition~\ref{linvsat}. \MXD\ holds 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\ (that gives a required condition in Corollary~\ref{resdf}), and by (1), we obtain that each restriction $(\sD_1-\sD_i)\frest{E_c}$ has the support bounded by some $F\in \fF$, where $\fF$ is a bounded set of reduced divisors. Since $\sD_1\frest{E_c}$ is fixed, the latter boundedness 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 considerations 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 wanted 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 $g_i:E_i\to T_i/\pt$ 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 nef on $E_i/\pt$ By (LBF), $i\sD_i\frest{E_c}\sim \sM_i\frest{E_c}$. So by (2) and \ref{aboundd}.1, the descent data of restrictions $\sD_i\vdots _{E_c}$ are asymptotically bounded with respect to $\sA_c$ over wanted models $E_i/T_i$ for $\sM_i$, because $B_i$ is the crepant boundary for $B_c$ as in the definition of the wanted triple, that is, $\sA(E_i,B_i)=\sA_c$, and $\lim_{i\to\infty} i=\infty$. Note that in \ref{aboundd}.1, we replace a common model $Y$ by the sequence $X_i/\pt$ as in Proposition~\ref{aboundd} itself. \end{pfof} Now we can prove a stabilization of $\lim_{i\to\infty} \sD_i\frest{E_c}$ on $E_c$. It is enough to prove for a similar (as characteristic) system $\sD_\bull$ (cf.\ Remark~\ref{asaturr}, (7)). \begin{cor} \label{trunc} Under the assumptions of {\rm Theorem~\ref{stabin},\/} the system $\sD_\bull$ up to a similarity of {\rm characteristic type\/} satisfies the theorem with $d_{c,1}=d_c=0$. In addition, we can assume that {\rm new\/} $\sD_1=\sM_1$ is rather generic; 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 rather generic $\sD_1^{[j]}=\sM_j\sim j D_j$. These data defined a 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 so does for the results. The same holds for any truncation: e.g., for new $\sD_i:=\sD_i^{[j]}=j\sD_{i j}$. (LWF) does not concern the change. (LBF) because $i\sD_i^{[j]}=i j\sD_{i j}\sim \sM_i^{[j]}=\sM_{i j}$. \LCA\ holds by Proposition~\ref{linvsat}. \MXD\ holds because $\sD^{[j]}=\lim_{i\to\infty}\sD^{[j]}_i= \lim_{i\to\infty}j\sD_{i j}=j\sD\ge j\sD_{i j}=\sD_i^{[j]}$ for any natural number $i$. \BED\ because, outside $f\1P$, each $\sD_i^{[j]}=j\sD_{i j}=j\sD=\sD^{[j]}$. Finally, for generic $\sM_j$, $\mult_{E_c}{\sM_j}=0$ 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 the stabilization take place exactly for the same divisors (but their indexes may be different and the values of multiplicities in $E_c$ are $j$-multiples of old ones). The addition holds by our choice of {\em new\/} $\sD_1$. \end{proof} \begin{cor} \label{stabind} Under the assumptions of {\rm Theorem~\ref{stabin},\/} suppose in addition \begin{q} \item[\AMN] {\rm arithmetic monotonicity\/}: $\sD_i\ge \sD_j$ for any $j\mid i$; and \item[\rm(RRF)] $d_{c,1}=d_c=0$ as in {\rm Corollary~\ref{trunc}.\/} \end{q} Then $\lim_{i\to\infty} \sD_i\frest{E_c}$ stabilizes {\rm(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 the (birational) saturation with respect to $\sA_c=\sA(E_c,B_c)$ {\rm(cf.\ \SAT\ and \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} (1) \AMN\ is preserved for any similarity as \MXD. (2) \AMN\ implies \MXD. We need for this the boundedness of $\sD_\bull$ by a Cartier divisor, non$0$ on $E_c$, that follows from \BNF\ and Lemma~\ref{divb} (cf.\ Proposition~\ref{bpbdal}). (3) We can take a truncation of $\sD_\bull$ such that the corollary holds with $N=1$. (4) $=(j) \sD\frest{E_c}$ is {\em only\/} expected when the stabilization holds in a neighborhood of some model (cf.\ Theorem~\ref{stabnbh} below). \end{rem} Note: when $d_{c,n}=0$, $\sD_j\frest{E_c}$ is the canonical {\em fixed\/} restriction as in (fx) of Proposition~\ref{rest}. \begin{lem} \label{restas} Let $(X/T,B)$ be an inductive model with a central divisor $E_c$ and $\sD_\bull$ a system of $\R$-{\bi}divisors such that: \begin{q} \item[\BSA] each $\sD_i$ is {\rm {\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[(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 \LCA, the asymptotic saturation 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) secures the general position \GNP. Moreover, \BSA\ can be replaced by \BNF\ because {\bi}nef is enough for Kawamata--Viehweg vanishing and the saturation in the proof of the proposition. \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 Kawamata log terminal; \item $\sD_\bull$ is bounded by a {\bi}divisor, satisfies {\rm\AMN, (LBF)\/} and the saturation \LCA. \end{itemize} Then up to a truncation it is convergent 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 that preserves the conditions, we can assume that $\sD_1\ge0$ by (LBF), and 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 $i=j q+ r$ with natural numbers $i,j,q$ and $r$, the \LCA\ and (LBF) imply an estimate \[ \frac{j q}{j q+r}\sD_j\le \sD_i, \] that gives a limit of the sequence due to that of for subsequence $\sD_j$ and the boundedness. Indeed, by \LCA, (LBF) for $\sD_j$, and since $\rdup{r\sD_j+\sA}\ge0$ by Kawamata log terminal, we got the required inequality \[ j q\sD_j\le q j\sD_j+ \Mov{\rdup{r\sD_j+\sA}}\le \Mov{\rdup{(j q+r)\sD_j+ \sA}}\le (j q+r)\sD_i. \] \end{proof} \begin{pfof}{Corollary~\ref{stabind}} First, by \AMN, \MXD\ and (RRF), for each natural number $i$, $d_{c,i}=d_c=0$. 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 new $(X/T,B):=(E_c/\pt,B_c)$, $T=\pt$, $\sD:=\sD\frest{E_c}$, $\sD_i:=\sD_i\frest{E_c}$, $\sM_i:=\sM_i\frest{E_c}$ and $\sA:=\sA_c=\sA(E_c,B_c)$. 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 the Lemma for the inductive model in the proof of Theorem~\ref{stabin}. In its turn the \ASAprime\ follows from ($\ep$SA$'$) of Proposition~\ref{ration} by \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 boundedness of $\sD_\bull \frest{E_c}$ (cf.\ Remark~\ref{stbdinr}, (2)) the $\lim_{i\to\infty}\sD_i \frest{E_c}$ 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 a solution of the asymptotic descent problem in \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 $\sD_\bull$. The 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 in our current situation from (FGA) is that we have no convexity of Lemma~\ref{arithm}. (However in our applications this is satisfied.) But it is not needed on the whole for Theorem~\ref{cbsfg}, (3) and \ref{assdes}.1. We need just \AMN\ under \LCA! \end{pfof} A 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 at {\em most one\/} prime {\bi}divisor $E_c$ with the log discrepancy $0$ for $K+B$, and $E_c/P$. \end{itemize} If there exists such $E_c$ we say that $(X/T,B)$ is an {\em $0$-exceptionally log terminal $=$ $0$-elt\/} pair. We say that the {\bi}divisor $E_c$ is {\em central\/}. Otherwise $(X/T,B)$ is Kawamata log terminal and is just a $0$-{\em log pair\/}. \end{defn} \begin{exa}\label{cmd} Usually a $0$-elt pair arises as a {\em complement} one 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 such situation $C\sim_{\R}0/T$ 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 gives a building of a log singularity. Let $(X/T,B)$ be a {\em birational\/} weak log Fano contraction, that is, $X/T$ is birational. Then after a complement we can assume that $K+B_{\eta}\equiv 0/T$. Thus after a contraction, $X/T$ is identical. Now we can find the required $C\ge0$ passing through $P$ such that, for some $\eta>0$, $K+B_{\eta}+\eta C$ has only one log discrepancy $0$ in $E_c$ and $E_c/P$. In such way our inductive model was constructed in Lemma~\ref{indmod}. \end{exa} To be more precise, we introduce neighbourhoods of $E_c$ on some special and quite (log) canonical models. They give an induction on the {\em rank} of non$\Q$-factoriality and the number of seminegative discrepancies (a kind of difficulty; cf.\ Lemma~\ref{nind} below). Key: For given {\bi}free $\sL$, we need to construct (later, a neighbourhood in) a birational model $Y/T$ on which there exists an effective $\R$-divisor $D=\sum d_i E_i$ such that: \begin{itemize} \item $d_i>0$ on the exceptional (on $T$) prime divisors $E_i$ and only on them; \item $L+\sum d_i E_i$ is nef and even semiample$/T$, where $L=\sL_Y$; and \item $\sL$ is base point free over $Y/T$. \end{itemize} We call such a model a {\em destabilizing model\/} (destab model) for $\sL$ because we can add to it (or to its multiple) $D$ on $Y$ that violates (destabilises) $D$-saturation (for example, the exceptional one in Example~\ref{satur}). The divisor $D$ is {\em destabilizing\/} destab, and it defines the {\bi}divisor $\sD=\Dbar$. \begin{warn} The contraction $Y/T$ is not necessarily divisorial! \end{warn} \begin{exa}\label{edst} (1) $L=\sL_T=0$, then we can take $Y=T$. More generally, if $L=\sL_T$ is $\Q$-factorial on $T$ we can take $Y=T$ and also $D=0$. (2) $\dim X=2$ and $L$ is a curve passing through $P$ on $T$ then for any $Y/T$ we can find a divisor $D>0$ that converts $Y/X$ into a destab model. Indeed, we can find a numerically (in Mumford sense) ample $D+L/T$ with $D>0$ and $\Supp{D}=f\1P$ on a $\Q$-factorial model (cf.\ Lemma~\ref{2ds}). (3) In some cases in dimension $\ge 3$ we can't construct {\em divisorial\/} $Y/T$: e.g., if $X/T$ is a flopping $0$-elt pair and $L=\sL_X$ its hyperplane section. It means that we can't destab $\sL$ on any divisorial $Y/T$ in this case. Probably in dimensions $\ge 3$ a destab model does not exists for every $\sL$. However in certain important situations it exists by the 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 {\rm the LMMP\/} in dimension $n=\dim X$ implies the existence of a destab model for $\sL$. The same holds for the purely log terminal model whose reduced part is nonexceptional$/T$; moreover, if $(X,B)$ has only log terminal singularities with the reduced centers not over $P$ and $\sL$ is base point free$/T$ over $X\setminus f\1P$. Or we can assume log terminal singularities and that $\sL$ is in a rather {\rm general position, that is, not passing through the log canonical centers of $(X,B)$\/} and under the {\em LSEPD} \cite[10.5]{sh92}. We need only flips as a {\rm noninductive new objects in the LMMP\/} under the {\em LSEPD trick} \cite{sh92}. \end{prop} However sometimes it is enough {\em very\/} special flips (cf.\ Example~\ref{lcresol} and Theorem~\ref{stabnbh} below). \begin{sta} Moreover, if we can construct such a model over a neighborhood $U$ on $Z/T$, where $Z$ is the model defined by $\sL/T$, then the destab holds over the neighbourhood. The construction in the proof is log canonical; so, can be done locally$/Z$. \end{sta} \begin{proof} It is quite effective. Construction of a destab (destabilizing) model: (1) Increasing $B$ we can convert our model into a $0$-elt pair: e.g., as in Example~\ref{cmd}. Then we can identify $(X=T/T,B=B_T)$. (2) 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 base point free over $Y$, in particular, $\sL=\overline{L}$, where $L=\sL_Y$. \end{itemize} For the plt and the log terminal cases not resolving the LCS centers. (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 $D$. (3) We apply the LMMP to $(Y/T,B_Y+L)$. On each step we have a birational contraction $g\colon Y\to Z/T$ given by an extremal ray $R$ with $R\cdot(K_Y+B_Y+L)<0$. Note: by (2$'$) and boundedness of negative contractions, $R\cdot L=0$. Thus the base point freeness of $L$ and (2$'$) is preserved during the modifications. Finally, we obtain a model where $K_Y+B_Y+L$ is nef, because it is always big due to $Y/T$ be birational, or by (2$'$) again. (4) Then we make a semiample contraction for $K_Y+B_Y+L$. We obtain it, because the divisor is always big due to be birational $Y/T$. We need also LSEPD trick \cite{sh92} and can do it if we secure it; in the addition~\ref{distabm}.1 over $U$. It also preserves the base point freeness 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. (5) Finally, we take a destab divisor $D=l\sA_Y$ (so, $\sD=l\sA(T,B_T)$) where $l$ means that we take the log discrepancies. By definition of log discrepancies $D=\sum d_i D_i=K_Y+B_Y-g^*(K+B) \equiv K_Y+B_Y, g\colon Y\to T$, since $K+B\equiv 0/T$ for elt $0$-models. Since it is Kawamata log terminal all $d_i>0$. The same holds for pure log terminal models if their reduced part is not exceptional. For terminal models under (2). The ampleness of $D+L\equiv K_Y+B_Y+L$ because $K_Y+B_Y+L$ is ample$/T$ by construction. In particular, nef and semiample$/T$. Finally, from the LMMP we need only flips. Indeed, the termination is special because we have the 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 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$/U$. \end{proof} \begin{lem} \label{trivbd} Let $\sD$ be a {\bi}divisor$/X$ such that \begin{itemize} \item $\sD_X=0$; \item {\rm effective:\/} $\sD\ge0$; and \item nef on generic curves$/X$: on (some) sufficiently high models $Y/X$, 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}. \end{proof} \begin{exa-cr}\label{lcresol} When we work in dimension $n=\dim X$ and do not know the existence of all flips, it would be helpful to know what types we need. Suppose that we have a $0$-klt pair (replace elt by log terminal in Definition~\ref{eltp}) $(X/T,B)$ (for example, obtained by a complement from a pure log terminal Fano contraction), {\bi}free $\sL\sim_{\R} \sS=S+\sum e_iE_i/T$, and $Z/T$ is the model for $\sL$ over $T$ such that: \begin{itemize} \item the LSEPD is secured for $B_Z$ (the log transform of $B_T$ or $B$) on the 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 enough: with $B_Y=S^-+S^++B'$, where $(Y/U,B_Y)$ is log terminal, $R\cdot S^-<0$, $R\cdot S^+\ge0$ (and $B'$ has no reduced components). Indeed, in the construction, $B_Y=g\1B_T+\sum E_i$, $R\cdot(K_Y+B_Y+L)<0$ with $L=\sL_Y$. Since $R/Z$, then $R\cdot L=0$ and $R\cdot(K_Y+B_Y)<0$. Thus for the log discrepancy $D=l\sA(X,B)_Y=K_Y+B_Y-g^*(K+B)\equiv K_Y+B_Y$ is supported in the exceptional$/T$ locus, 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 the numerical intersection of it $\ne0$ 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 $E$ be a prime {\bi}divisor. A (birational) klt {\em neighbourhood\/} $(Y/T,B_Y)$ of $\cent_Y{E}$ is a model $(Y/T,B)$, that is also a $0$-log pair, and a Zariski neighbourhood of $\cent_Y{E}$ on it. \end{defn} \begin{lem} \label{nind} Under the {\rm LMMP} in dimension $n=\dim X$ and for birational $X/T$, the klt neighbourhoods terminate for inclusions. \end{lem} \begin{proof} On each model this is a Noetherian property. But there only exists a finite number of $0$-log pair models (the finiteness of minimal models in the big case) \cite{sh96b}. \end{proof} We are now ready to establish stabilization in a birational neighbourhood of $E=E_c$. \begin{thm} \label{stabnbh} Under the {\rm 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)] $\sL_i\frest E$ stabilize over $E$, and each $\sL_i\frest E=i\sM$, where $\sM=\sL_1\vdots_E$ is {\bi}free; and \item[$\RFA$] the situation is a restriction of a pl contraction in dimension $n$, {\rm that is, $\sL_i$ is a restriction of the movable system for some $\RFA_{n,d}$ algebra (cf.\ Definition~\ref{rfad} and the proof below)\/}. \end{q} Suppose also that the flips of type {\rm(S+$-$)} as in {\rm 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 base point free, ample and $\sL_i/i$ stabilizes {\rm(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 complement always exist preserving $(X',B'')$ log terminal, and we fix it. Then by adjunction, we define $B$: \[ (K_{X'}+B')\rest X=K+B. \] We assume also that the system $\sL_\bull$ is a movable restriction of a {\bi}free system $\sL'_\bull$: $\sL_i\sim\sL'_i \frest{X}$, where $\sL'_\bull$ is the movable system for the $\RFA_{n,d}\bir$ algebra given by the pl contraction. Equivalently, the movable system of the restriction for the algebra, where in either case we assuming generic 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 $\Supp{\sS'_{Z'}}$. It is completely contained in any neighborhood $U'\subset Z'$ of $\Supp{\sS'_{Z'}}$. Thus the destab model$/T$ over $Z/T$ is complete$/T$. ($Z$ is normal by \ref{ressat}.1 and Example~\ref{fgapl}.) Now by \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 the exceptional saturation we can destabilize multiples of $\sL_i'/U'$ over the destab divisor the support of that 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: the $\cent_{Y'}{E}$ 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 blow up of $E$. In particular, such $\cent_{Y'}{E}\subset X_0$, where $X_{0}$ is the birational transform of $X$ on $Y'$; and such $\cent_{Y'}{E}=\cent_{X_0}{E}$ This gives the required neighbourhood. Indeed, if we apply the adjunction of the subboundary $(B')^{Y'}=\sB(X',B')_{Y'}$ on the normalization of $X_0$ (really, $X_0$ 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 used to do 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 identical. 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 \ref{distabm}.1. The discrepancies over $U$ for $(X_0,B_0)$ are the same as for $(X,B)$. Now we can apply the LMMP to $(X_0/T,B_0)$ the modification does not touch $U$ because then 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 \ref{ressat}.1 and Example~\ref{fgapl}.) \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 the {\rm LMMP\/} in dimensions $d'\le d=\dim X$, we obtain $\CBS_{d'-1}^*$ and $\SSB_{d'-1}\gl$. Moreover, we can drop $\CBS_{d'-1}^*$ and $\SSB_{d'-1}\gl$ with $d'\le 3$. Assume also that flips of type {\rm(S+$-$)} exist in dimension $n$. Let $\sL_\bull$ be a movable 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 base point free and ample, up to a truncation. This gives a solution for $\CBS_n\rfabir$ under the assumptions. \end{cor} \begin{lem} \label{lcsprop} Under the {\rm 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 noncomplete$/P$; and \item[\rm(CMP)] let $(X/T,B+C)$ be a complement {\rm nonlog canonical\/} $0$-log pair with $C=\sC_X$ such that $\LCS{(X,B+C)}$ is completely inside $U$ {\rm(cf.\ Example~\ref{cmd}\/)}. \end{q} Then there exists a different complement with a {\rm new\/} $\sC$ such that the {\rm new\/}\linebreak[3] $\LCS{(X,B+C)}$ is not completely contained in $U$, but all its {\rm LCS\/} centers are in $U$, and $(X,B+C)$ is log canonical outside $U$. Thus the nonlog canonical part $X_s$ of the {\rm LCS\/} is inside $U${\rm, that is, does not intersect the complement of $U$\/}. In addition, there is a {\rm 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 {\rm LCS} and the same purely log terminal center $E$. Thus the adjunction on $E$ gives a {\rm 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 $E$ {\rm: the direct image of that of on the normalization)\/}; it is supported in intersection of $E$ with closure of $\LCS{(X,B+C^+)}\setminus E$. In particular, $(E,B_E)$ is normal and Kawamata log terminal 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 $U$. Then it is the required $c$. By (NBU) such $c$ exists 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 the 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 the 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 the 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. \end{proof} \begin{pfof}{Corollary~\ref{stabg}} By definition, the movable 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 new fixed $B$. It is also a weak log Fano contraction because $X/T$ is birational. Now by Corollaries~\ref{trunc} and~\ref{stabind} for $\sD_\bull$ 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 the 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{fgadi} 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 base point free, and $\sL_i/i$ stabilizes. It $U$ is complete, we are done. (Compare the next corollary.) If not we extend the neighborhood. This is eventually complete by Lemma~\ref{nind} on the 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, $U\ne\emptyset$ (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} we obtain up to a truncation that each $\sL_i$ is base point 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 the (STB) on $E$. This is done in dimension $d=3$. Thus $\dim E\le 2$. (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. But the descent \ref{assdes}.1 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}, $\sD_\bull:=\sD_\bullet\frest E$ satisfies the lca saturation \LCA\ as original $\sD_\bull$. The novelty is only in essentially negative (with discrepancies $\le-1$) components of $\sA:=\sA(E^\nu,B_E)$ that we have now. The negative components defined the ideal $J_E$. Since each $j\sD_i$ is Cartier on the LCS, the 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 {\em identical\/} 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 $(E/T,\sC=\sA,F, \gamma)$ 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 (even we can normalize it and assume normal). The modification is {\em contractible\/} it does not blow up exceptional divisors over $U$. However it can contract: 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 the 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) with some real $\gamma>0$ (cf.\ Example~\ref{fnonvex}). In two dimensional case, $\sD_E$ is nef = pseudo-nef$/T$ (nef over generic complete curves$/T$ on divisors) 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 $\R$-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; the LMMP$^*$, contract all exceptional curves in $F$) it is enough to verify for $\Q$-divisors. Essentially, due to the general Fano property. See more details in [Florin]. To verify (SAB) we can use Proposition~\ref{surbound}. However to apply this to approximations we need from the triple and the prediction more: e.g., they should be strictly wanted (WAM) for $\sD$, that we 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, for each $i$ after a truncation, $|\sM_i\rest E$ is a pencil. Its base point are only in good ones, outside $U$. Their resolution 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 to improve wanted triples for $\sM=\sD$ (not {\bi}free) and $=\sM_i$ \begin{q} \item[(WAM)] a wanted triple for $\sM$ is {\em strict\/} when $M$ 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 its 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 $J$, $|J(N H)|$ is base point 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 $|\sM_i\rest E$ is base point 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 the 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 Kawamata log terminal. 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 $D\ge0$, $C$ do not intersects $\Supp{D}$ whenever $\sD$ is taken quite generic on $U$. Then (STB) holds on $C$, we can extend $U$ by a neighborhood of $C$, and then contract $C$ (with the other such curves to preserve the algebraic category). Note that now even in the {\em big\/} case we need to modify (BIG), (BIR), and (MOD), and consider nonidentical $E/W$! After that the methods of the proof of \ref{assdes} are applied to the prediction $(E/T,\sA,F,\gamma)$ 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 pencil is base point free, and can be done as the Kodaira dimensions $0$ and $1$ above. Or we can use an adjunction on $E$ [Kawamata]. In higher dimensions, it is better to use the adjunction on $E$ and above nonKawamata log terminal strict wanted triples of general log Fano contractions. Then $\CBS_{d'-1}^*$ means existence of such triple for each {\bi}free $M$, and boundedness of needed triples. Above we verified that for $d'\le 3$, 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 $X$ that is unknown even without $^*$. \end{pfof} \begin{cor}\label{stbklt} Under the assumptions of {\rm 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}. \end{proof} \begin{cor}\label{eplf} Under the assumptions of {\rm 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 \end{proof} In the next section, we eliminate the exist of flips {\rm(S+$-$)} under $\CBS^*$. \section{The main result} \label{mr} \begin{thm}\label{s+-} Under the {\rm LMMP\/} in dimension $\le n-1$ and $\CBS_{n-2}^*\bir$. The flips of type {\rm(S+$-$)} exist in dimension $n$. Moreover, we can drop the extremality of the flips {\rm($\rho(X/T)=1$; so, $X/T$ is just small)\/}, and $\CBS_{n-2}^*$ with $n\le 4$. \end{thm} \begin{lem}\label{2ds} Let $\sL=\overline{L}$ 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 {\rm destabilizes} $\sL\frest E$, {\rm that is,\/} $(\Mov(\sE'+N \sL))\frest E>N\sL\vdots_E$. \end{lem} \begin{proof} Taking hyperplane sections of $X/T$ by $L$ we reduce 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. It can be resolved by a $\Q$-factorialization and perturbation in exceptional components to preserve numerical ampleness$/T$. For higher dimensions we close a family of $\sE'$, constructed in dimension $2$. \end{proof} \begin{lem}\label{wqf} Under the {\rm LMMP\/} in dimension $n$. Let $B$ be a subboundary $B$ on $X$ with $\dim X=n\ge 3$, and let $S$ be 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 Kawamata log terminal, and \item the negative components of $B$ intersect $S$ only in points. \end{itemize} Then there exists a weak $\Q$-{\rm factorialization}, that is, a model where 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} Note: (1) $S$ is normal outside the intersection points with negative components of $B$ \cite[Lemma~3.6]{sh92} because $(X,B)$ is purely log terminal there. (2) We expect a more perfect form of the lemma in any dimension $n$ with the intersections in codimension~$\ge 3$ instead of in points (cf.\ the remark at the end of Step~1 in the proof of Theorem~\ref{s+-}). \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. \end{proof} \begin{pfof}{Theorem~\ref{s+-}} Since the flip is strictly log terminal, we can assume that both $S^+$ and $S^-$ are irreducible after small decreasing of other reduced components (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 movable system of a divisorial algebra $\dival_{X/T}{D}$ with $D\sim S^-$, numerically negative$/T$. Then the algebra defines the flip when the former is f.g. We have a translation $t$ by \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 $\sL_\bull$ has type $\FGA_n\bir$ over $(X/T,B+(1-\ep)(S^++S^-))$ with $0<\ep\ll 1$ because \begin{q} \item[(EXS)] $\sL_\bull$ is exceptionally asymptotically saturated$/T$. \end{q} Unfortunately, we can't preserve for restrictions (EXS), but the (FGA) is preserved for birational restrictions $\vdots$ 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 generic) of dimension $1$. Thus Proposition~\ref{fgadi} 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 base point 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 the 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 not $/P$) on which all $\sL_i\frest{S^-}$ with $i\gg0$ are base point free, ample and $\sL_i/i\frest{S^-}$ stabilize {\rm(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 the LMMP assumption neighborhoods $U_m$ of this form considerd under inclusions terminate. The maximal $U_m$ is complete$/T^-$ is the required $0$-log pair. Finally, in Step 2 we extend each noncomplete $U_m$. \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 semi very ample and have respectively the same $\sL_m/m\frest E=\sD\vdots_E$ and $\sL_m/m\vdots _{S_m^+}=\sD$, that is, stabilized. This follows from the normality of restricted algebras in \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 $\sL=\overline{L}$ for {\em just\/} ample but base point 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 nonnormal. We denote by $\nu\colon S_m^- \to X_m$ its normalization. We claim that $U_m$ is a neighborhood of $E_m$ in $S_m^-$. Note that $E_m$ is also imbedded into $S_m^-$ by the above normality \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, any exceptional $E_i/T$ does not intersect $S_m^+$ {\em divisorially\/}. 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.\ \ref{assdes}.2). More precisely, by a perturbation, we convert $(X/T,B)$ into a $0$-elt pair $(X/T,B^+)$ (actually, 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 $T_m^+=f(S_m^+)$ (that is normal). Hence by the adjunction and the commutative diagram up to codimension~$2$, that is, for generic surface sections, and near $S_m^+$, $K_{X_m}+B^+_{X_m}$ is log canonical with {\em birationally\/} transformed $B^+$. And the adjunction can be extended on the whole $S_m^+$ 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 Kawamata log terminal everywhere on $S_m^+$ with $B_{S_m^+}^+\ge0$ (crepant, for $(S^+,B^+_{S^+})$) by \cite[3.2.2]{sh92}. Moreover, each exceptional $E_i$ intersects $S_m^+$ at most in a point. Indeed, we can assume that the subvarieties in $S_m^+$ of dimension $\ge 1$ in general are {\em weakly\/} $\Q$-factorial {\em for the exceptional divisors\/} $E_i/T$, that is, these divisors intersect $S_m^+$ divisorially (in the codimension~$2$ in $X_m$) if by the dimension $\ge 1$. Indeed, taking a hyperplane sections 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 the LMMP in the lemma holds under our assumptions of the theorem. On the other hand, by the numerical and geometric properties of $S^+$ and $S^-$, $a f(S^+)+b (\sL_m)_T$ form 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 for a restriction on any general hyperplane section. That 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 the divisorial part always intersect as in the $\Q$-factorial case by a divisor if it intersect by a curve. (Similarly, we could eliminate the intersections in points with $S_m^+$ if we knew the existence of $n$-fold flips and their termination! Thus we expect this near $S_m^+$, and that $S_m^-$ is normal by the argument below. But now we just disregard such intersections by 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 $B_E^+$ that is 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 latter induced 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 its 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 Mumford sense (restricted on 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 $\sL_{j m}/j\frest{U_m}$ stabilizes for all $j\ge1$! The boundary by \cite[3.2.2]{sh92}. Thus as at the end in the proof of Theorem~\ref{stabnbh} (based on the uniqueness in \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 Trick: By \AMN\ and \BED, and since the exceptional $E_i$ in $X_m$ are$/P$, $\sL_{j m}=j\sL_m+ \sE_{m,j}$, whereas 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\/} $\Supp{(\sE_{m,j}\vdots _{U_m})_{U_m}}=0$ (over $X_m$), or $(\sE_{m,j}\vdots _{U_m})_{U_m}=0$, and by Lemma~\ref{trivbd} $\sE_{m,j}\frest{U_m}=0$. Then $\sL_{j m}/j\frest{U_m}=\sL_m\frest{U_m}$. That means the required stabilization on $U_m$. \begin{rem} The proof is essentially higher dimensional: i.e., it is better to assume $n\ge 4$. For $n\le 2$, with only identical flips, no $E_i$, and complete $U_m$ in Step 1. For $n=3$, under the LMMP in dimension $3$, Lemma~\ref{wqf} and (PRS) imply that each $E_i$ does not intersect $E_m$ at all (cf.\ Note (2) to the lemma) and we have the same stabilization of 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 $\sL_{j m}/j\frest{E_t}$ 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 $U_{j m}$ for $j\gg0$ such that the restricted $\sL_{j m}/j\frest{E_t}$ 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 $\sum e_iE_i\rest{S_{j m}^-}$ are disjoint from $E_{j m}=S_{j m}^+\rest{S_{j m}^-}$ (by the log terminal property of $(X_{j m},B_{X_{j m}^{+-}})$, $B_{X_{j m}^{+-}}$ is a boundary in the generic point of $E_{j m}=S_{j m}^+\cap S_{j m}^-$, and \cite[Corollary~3.11]{sh92}, because in generic $E=S^+\cap S^-$ goes birationally to $E_{m j}$ and the exceptional$/T$ prime {\bi}divisors with the log discrepancy $0$ are only$/$ the generic point of $E_{j m}$). Indeed, by (PRS) $a S_{j m}^++b L+\sum e_iE_i$ 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. \end{pfof} \begin{pfof}{Theorem~\ref{cbrfa}} Immediate by Corollary~\ref{eplf} and Theorem~\ref{s+-}. \par \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 $n\le 4$ we can drop the LMMP, $\CBS^*$, and $\SSB$(gl). \end{pfof} \begin{thebibliography}{50} \bibitem{cas57} Cassels J.~W.~S., {\em Introduction to Diophantine approximation\/}, Cambridge University Press, 1957. \bibitem{ka} Kawamata, Y., Subadjunction of log canonical divisors. I, in Birational algebraic geometry (Baltimore, MD, 1996), 79--88, Contemp. Math., {\bf207}, Amer. Math. Soc., Providence, RI, 1997. II, Amer. J. Math. {\bf120} (1998) 893--899 \bibitem{ko} Koll\'ar~J., et al., {\em Flips and abundance for algebraic threefolds\/}, A Summer Seminar at the University of Utah, Salt Lake City, 1991, Asterisque v. {\bf 211} (1992) \bibitem{PSh} Prokhorov~Yu.G., Shokurov~V.V., {\em The first main theorem on complements: from global to local\/}, preprint math.AG/9912200, 27 pp. \bibitem{r80} Reid~M., {\em Canonical 3-folds\/}, in Journe\'es de g\'eom\'etrie alg\'ebrique d'An\-gers, ed.\ A. Beauville, Sijthoff and Noordhoff, Alphen 1980, 273--310 \bibitem{shaf65} Shafarevich~I. et al., {\em Algebraic surfaces\/}, Proc. of Steklov Institute, {\bf 75}, 1965 \bibitem{sh85} Shokurov V.V., {\em The nonvanishing theorem}. Math.\ USSR Izvestija, {\bf 26} (1986) 591--604 \bibitem{sh92} Shokurov V.V., {\em 3-fold log flips\/}, Russian Acad. Sci. Izv. Math., {\bf40} (1993) 93--202 \bibitem{sh93} Shokurov V.V., {\em Semi-stable 3-fold flips\/}, Russian Acad. Sci. Izv. Math., {\bf57} (1993) 164--222 \bibitem{sh96a} Shokurov V.V., {\em Anticanonical boundedness for curves\/}. Appendix to V.V. Nikulin ``Hyperbolic reflection group methods and algebraic varieties'' in Higher dimensional complex varieties (Trento, June 1994) /ed. M. Andreatta; T. Peternell. -- Berlin: New York: de Gruyter, 1996, pp. 321--328 \bibitem{sh96b} Shokurov V.V., {\em 3-fold log models\/}, Journal of Mathematical Sciences {\bf81}, No. 3, September, 1996, Plenum Publishing Corporation, pp. 2667--2699. / Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i ee prilozheniya, Tematicheskie Obzory, vol. 33, Algebraic Geometry 4, Editor-in-Chief R.V. Gamkrelidze, Moscow, 1996 \bibitem{sh95} Shokurov V.V., {\em Complements on surfaces\/}, preprint alg-geom/9711024, 105~pp. \end{thebibliography} \bigskip \noindent Department of Mathematics,\\ Johns Hopkins University,\\ Baltimore, MD--21218, USA\\ e-mail: shokurov@math.jhu.edu \end{document} scrap Alexeev, Valery(1-GA) Boundedness and $K\sp 2$ for log surfaces. Internat. J. Math. 5 (1994), no. 6, 779--810.