diff 3ff9228..b34550f nef_adjoint_numerical_divisors_v2.txt
diff --git a/nef_adjoint_numerical_divisors_v2.txt b/nef_adjoint_numerical_divisors_v2.txt
index 07c72a3..3f17803 100644
--- a/nef_adjoint_numerical_divisors_v2.txt
+++ b/nef_adjoint_numerical_divisors_v2.txt
@@ -1,53 +1,53 @@
-\documentclass[11pt]{amsart}
-\usepackage[T1]{fontenc}
-\usepackage{lmodern,amsmath,amssymb,mathtools}
-\usepackage{tikz-cd,microtype}
-\usepackage[a4paper,margin=30mm]{geometry}
-\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
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-\numberwithin{equation}{section}
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{remark}[theorem]{Remark}
-\newtheorem{example}[theorem]{Example}
-\newcommand{\Q}{\mathbb Q}
-\newcommand{\R}{\mathbb R}
-\newcommand{\C}{\mathbb C}
-\newcommand{\Z}{\mathbb Z}
-\newcommand{\OO}{\mathcal O}
-\newcommand{\PP}{\mathbb P}
-\DeclareMathOperator{\Pic}{Pic}
-\DeclareMathOperator{\Supp}{Supp}
-\DeclareMathOperator{\rk}{rk}
-\DeclareMathOperator{\ord}{ord}
-\DeclareMathOperator{\mult}{mult}
-\DeclareMathOperator{\vol}{vol}
-\DeclareMathOperator{\Sym}{Sym}
-\DeclareMathOperator{\Bir}{Bir}
-\DeclareMathOperator{\codim}{codim}
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\title[Numerical semiampleness of nef adjoint divisors]{Numerical Semiampleness of Nef Adjoint Divisors}
\subjclass[2020]{Primary 14E30; Secondary 14C20, 14J32}
\keywords{generalised abundance, numerical semiampleness, nef divisor, minimal model, Frobenius morphism}
-\begin{document}
-\begin{abstract}
+
+Abstract:
We prove the Generalised Abundance Conjecture: if $(X,B)$ is a projective
klt $\Q$-pair over an algebraically closed field of characteristic zero,
$K_X+B$ is pseudo-effective, $M$ is a nef $\Q$-Cartier divisor on $X$,
and $K_X+B+M$ is nef, then $K_X+B+M$ is numerically equivalent to a
semiample $\Q$-Cartier divisor on $X$.
-\end{abstract}
+
\maketitle
-\section{Introduction}
-\label{sec:introduction}
+# Introduction
-A divisor $D$ on a projective variety is \emph{nef} if $D\cdot C\ge 0$
-for every integral curve $C$, and \emph{semiample} if some positive
+
+A divisor $D$ on a projective variety is nef if $D\cdot C\ge 0$
+for every integral curve $C$, and semiample if some positive
multiple $mD$ is globally generated, so that $mD$ defines a morphism to
projective space. Nefness is numerical and semiampleness is geometric;
passing from one to the other is the central concern of the minimal
@@ -60,78 +60,78 @@ input. Adding an independent nef summand $M$ changes what can be
expected: even if $D=K_X+B+M$ is nef, $D$ itself need not be semiample,
and one is forced to require only a numerical semiample representative.
-\begin{definition}\label{def:numsemiample}
+
A rational Cartier divisor $D$ on a projective variety $X$ is
-\emph{numerically semiample} if it is numerically equivalent to a
+numerically semiample if it is numerically equivalent to a
semiample rational Cartier divisor on the same $X$. Two rational
Cartier divisors are numerically equivalent, written $\equiv$, if their
difference meets every integral curve in zero.
-\end{definition}
-\begin{theorem}[Generalised Abundance; {\cite{LP}}]\label{thm:main}
+
+[Generalised Abundance; {[cite]}]
Let $(X,B)$ be a projective klt $\Q$-pair over an algebraically closed
field of characteristic zero. Suppose that $K_X+B$ is pseudo-effective,
and let $M$ be a nef $\Q$-Cartier divisor on $X$. If $D=K_X+B+M$ is
nef, then there exists a semiample $\Q$-Cartier divisor $L$ on $X$ with
$D\equiv L$.
-\end{theorem}
+
Both the boundary $B$ and the nef summand $M$ sit on $X$ itself. The
proof uses two named inputs: the log-abundance theorem of
\cite[Theorem~1.1]{LA}, and the terminating minimal-model programs of
\cite[Theorem~1.1 and Section~3.3]{MM}. Both are stated precisely in
-Section~\ref{sec:preliminaries}.
+Section~sec:preliminaries.
-\paragraph{Why numerical equivalence is necessary.} Let $E$ be a
+Why numerical equivalence is necessary.: Let $E$ be a
complex elliptic curve and let $L_\alpha$ be a nontorsion line bundle
of degree zero on $E$. Then $L_\alpha$ is nef and numerically
trivial, but no positive multiple has a nonzero section. Since
-$K_E\sim0$, taking $B=0$ and $M=L_\alpha$ in Theorem~\ref{thm:main}
+$K_E\sim0$, taking $B=0$ and $M=L_\alpha$ in Theorem~thm:main
gives a valid hypothesis in which $K_E+M$ is nef and numerically
trivial; the conclusion only asks for a semiample divisor numerically
equivalent to $K_E+M$, which the zero divisor provides. Linear
equivalence would force a section, and there is none.
-\paragraph{The ordinary adjoint case.} For $M=0$, the conclusion of
-Theorem~\ref{thm:main} is equivalent to ordinary klt abundance. One
+The ordinary adjoint case.: For $M=0$, the conclusion of
+Theorem~thm:main is equivalent to ordinary klt abundance. One
direction is clear. For the other, Fukuda \cite[Theorem~0.1]{Fukuda}
proves that if $K_X+B$ is numerically equivalent to a semiample
-divisor, then $K_X+B$ itself is semiample. Hence Theorem~\ref{thm:main}
+divisor, then $K_X+B$ itself is semiample. Hence Theorem~thm:main
with $M=0$ recovers ordinary klt abundance, which is an input to the
-present proof via \cite{LA}.
+present proof via [cite].
+
-\begin{corollary}\label{cor:cy}
Let $(X,B)$ be a projective klt $\Q$-pair over $\C$ with $K_X+B\equiv 0$.
Then every nef $\Q$-Cartier divisor on $X$ is numerically semiample.
-\end{corollary}
-\begin{proof}
+
+Proof.
For nef $M$, the divisor $K_X+B+M\equiv M$ is nef. Apply
-Theorem~\ref{thm:main}.
-\end{proof}
+Theorem~thm:main.
-\paragraph{The key new ingredient.} The heart of the proof reduces,
+
+The key new ingredient.: The heart of the proof reduces,
step by step, to the following positivity statement about Calabi--Yau
pairs.
-\begin{theorem}[Full nef dimension forces bigness]\label{thm:key-ingredient}
+[Full nef dimension forces bigness]
Let $(X,B)$ be a projective klt Calabi--Yau $\Q$-pair of dimension $n$
over $\C$: that is, $K_X+B\equiv0$. Let $M$ be a nef $\Q$-Cartier
divisor on $X$ with full nef dimension, meaning every curve through a
very general point of $X$ has positive $M$-degree. Then $M$ is big.
-\end{theorem}
-Recall that $M$ is \emph{big} if the top self-intersection $M^n$ is
+
+Recall that $M$ is big if the top self-intersection $M^n$ is
positive, equivalently if the linear systems $|kM|$ grow like $k^n$ for
large $k$. The gap between strict positivity along very general curves
and positive volume is exactly where the geometry of the paper lives.
-Sections~\ref{sec:jet-tools} through \ref{sec:completion} are devoted to
-the proof of Theorem~\ref{thm:key-ingredient} for every $n\ge 2$.
+Sections~sec:jet-tools through sec:completion are devoted to
+the proof of Theorem~thm:key-ingredient for every $n\ge 2$.
-\paragraph{Shape of the proof.} Two simultaneous inductions on the
+Shape of the proof.: Two simultaneous inductions on the
dimension $n=\dim X$ organise the argument.
-\begin{enumerate}
+
\item[(P$_n$)] Suppose $T=K_X+B$ is pseudo-effective, $M$ is a nef
$\Q$-Cartier divisor on a projective klt $\Q$-pair $(X,B)$ of dimension
$n$, and there is a rational $a>0$ such that $(T+aM)\cdot C>0$ for
@@ -143,7 +143,7 @@ $w\colon W\to X$ from a smooth projective $W$ such that the positive
part $P_\sigma(w^*(T+M))$ of the divisorial Zariski decomposition is a
rational divisor numerically equivalent to a semiample $\Q$-divisor on
$W$.
-\end{enumerate}
+
Statement (P$_n$) produces bigness and (Z$_n$) produces numerical
semiampleness; both are proved by induction on $n$. The case $n=0$ is
@@ -154,84 +154,84 @@ numerically semiample.
For the inductive step, three cases arise.
-\begin{itemize}
-\item \emph{Positive Iitaka dimension.} If $\kappa(T)>0$, the
+
+\item Positive Iitaka dimension. If $\kappa(T)>0$, the
semiample adjoint on a suitable birational model has a
positive-dimensional base and the induction hypothesis applies to the
-fibres of its contraction. This is Proposition~\ref{prop:full-positive}.
+fibres of its contraction. This is Proposition~prop:full-positive.
-\item \emph{Reduction to Calabi--Yau.} A compatible pair of
+\item Reduction to Calabi--Yau. A compatible pair of
minimal-model programs replaces $(X,B)$ by a model $(Y,B_Y)$ on which
the ordinary adjoint $U=K_Y+B_Y$ is semiample. If $\kappa(T)=0$ then
$U\sim_\Q 0$, so $(Y,B_Y)$ is a klt Calabi--Yau pair and $(P_n)$
-reduces to Theorem~\ref{thm:key-ingredient}. This is
-Proposition~\ref{prop:reduction-cy}.
+reduces to Theorem~thm:key-ingredient. This is
+Proposition~prop:reduction-cy.
-\item \emph{The full-nef-dimension step.} Theorem~\ref{thm:key-ingredient}
+\item The full-nef-dimension step. Theorem~thm:key-ingredient
is the technical core. Its proof uses two incompatible jet estimates,
-produced in Section~\ref{sec:geometry} and made to conflict in
-Section~\ref{sec:frobenius}.
-\end{itemize}
+produced in Section~sec:geometry and made to conflict in
+Section~sec:frobenius.
+
-\paragraph{Two incompatible jet estimates.} Assume, towards a
+Two incompatible jet estimates.: Assume, towards a
contradiction, that $X$ is projective $\Q$-factorial and terminal with
$K_X\sim0$, and that $L$ is a nef Cartier divisor of full nef dimension
that is not big. Since $L$ is nef and not big, $L^n=0$.
Fix an ample Cartier divisor $A$ and let $t\to 0$ through positive
-rationals. Let $\epsilon>0$ be small enough that $(14\epsilon)^n<1/4$
-and $80\epsilon<3/4$. For suitable integers $r=r(t)\to\infty$ the
-divisor $P=r(L+tA)$ is ample rational Cartier with $P^n\to\epsilon^n$;
+rationals. Let $ε>0$ be small enough that $(14ε)^n<1/4$
+and $80ε<3/4$. For suitable integers $r=r(t)\to∞$ the
+divisor $P=r(L+tA)$ is ample rational Cartier with $P^n\toε^n$;
for small $t$ the volume window
-$\epsilon^n/2\le P^n\le 2\epsilon^n$ holds. Then:
+$ε^n/2\le P^n\le 2ε^n$ holds. Then:
-\begin{itemize}
-\item \emph{Small scalar bound.} At a very general point $x\in X$,
+
+\item Small scalar bound. At a very general point $x\in X$,
every section of every $kP$ with $kP$ Cartier vanishes to order at most
$4k\rho$ at $x$, where $\rho=(P^n)^{1/n}$; this is
-Lemma~\ref{lemma:scalar-bound}.
+Lemma~lemma:scalar-bound.
-\item \emph{Large jet system.} Over $X\times X$, define the
-projective bundle $Z=\PP(\OO_X(L)_1\oplus \OO_X(L)_2)$ with
+\item Large jet system. Over $X\times X$, define the
+projective bundle $Z=\PP(\OO_X(L)_1 ⊕ \OO_X(L)_2)$ with
tautological class $\xi$, and the ample divisor $R=P_1+P_2+q\xi$ for
suitable integer $q>0$. There is a smooth point $z_*$ off the two axes
-with Seshadri constant $\varepsilon(R;z_*)>2n+2$; this is
-Proposition~\ref{prop:large-seshadri}. A Seshadri constant above
+with Seshadri constant $ε(R;z_*)>2n+2$; this is
+Proposition~prop:large-seshadri. A Seshadri constant above
$2n+2$ lets us generate jets at $z_*$ to that order from sections of a
suitable Cartier multiple of $R$, via Serre vanishing and the cohomology
of a point blowup.
-\end{itemize}
+
The proof of the large jet system is the longest part of the paper.
Starting from the scalar bound, together with the moving-kernel
-machinery of Section~\ref{sec:jet-tools}, one rules out every possible
+machinery of Section~sec:jet-tools, one rules out every possible
failure of the large Seshadri constant. Each failure produces a small
base component of a suitable jet kernel, and each such component — plus
its slices, its swept branch divisors, and its associated
correspondences between finite covers — gives a bounded-degree curve
-of the kind forbidden by Lemma~\ref{lemma:bounded-curves}.
+of the kind forbidden by Lemma~lemma:bounded-curves.
-\paragraph{The Frobenius comparison.} With the two estimates in hand,
-Section~\ref{sec:frobenius} derives a contradiction. A fixed finite
+The Frobenius comparison.: With the two estimates in hand,
+Section~sec:frobenius derives a contradiction. A fixed finite
amount of complex data is spread out to an integral finitely generated
$\Z$-algebra of characteristic zero. For all large residual
characteristics $p$, the two jet systems evaluate on the same
determinant on the diagonal of the Frobenius fiber product
$W\times_{W'} W$. The scalar bound makes that determinant vanish at
-the diagonal to order at most $80\epsilon+O(1/p)$ in the relevant
+the diagonal to order at most $80ε+O(1/p)$ in the relevant
normalization, while the diagonal-rank bound from the Frobenius
filtration makes the same order of vanishing at least $3/4$. Since
-$80\epsilon<3/4$ and $O(1/p)\to 0$, the two bounds are incompatible for
-all large $p$; this is Theorem~\ref{thm:finite-data}, whose proof
-occupies Section~\ref{sec:frobenius}.
+$80ε<3/4$ and $O(1/p)\to 0$, the two bounds are incompatible for
+all large $p$; this is Theorem~thm:finite-data, whose proof
+occupies Section~sec:frobenius.
-\paragraph{Extension to klt Calabi--Yau pairs.} Once
-Theorem~\ref{thm:key-ingredient} is proved for terminal $X$ with trivial
-boundary, Section~\ref{sec:completion} extends it in two steps. First,
+Extension to klt Calabi--Yau pairs.: Once
+Theorem~thm:key-ingredient is proved for terminal $X$ with trivial
+boundary, Section~sec:completion extends it in two steps. First,
a klt $X$ with $K_X\equiv 0$ admits a finite quasi-\'etale index-one
-cover $\pi\colon X^{\mathrm{ind}}\to X$ with $K_{X^{\mathrm{ind}}}\sim 0$
+cover $\pi\colon X^{ind}\to X$ with $K_{X^{ind}}\sim 0$
and canonical singularities; a crepant terminalization then gives a
-terminal $\Q$-factorial $\widetilde X\to X^{\mathrm{ind}}$. Full nef
+terminal $\Q$-factorial $\widetilde X\to X^{ind}$. Full nef
dimension and bigness both survive the finite quasi-\'etale cover, so
the terminal trivial-boundary case gives bigness of $M$ on $X$. Second,
for an arbitrary effective boundary $B$ with $K_X+B\equiv 0$, a small
@@ -240,8 +240,8 @@ reduces to the positive-Iitaka case or forces $B_Y=0$; in the latter
case $K_Y\sim_\Q0$ and $M$ remains nef of full nef dimension, and the
terminal trivial-boundary case makes $M$ big.
-\paragraph{Positive parts and numerical semiampleness.} Once both
-inductions hold, Section~\ref{sec:positive} proves (Z$_n$). First, a
+Positive parts and numerical semiampleness.: Once both
+inductions hold, Section~sec:positive proves (Z$_n$). First, a
large-nef-adjoint MMP trivial on a multiple of $M$ may be arranged so
that $T+mL$ becomes nef. On the resulting model either $T+mL$ has full
nef dimension, in which case (P$_n$) gives bigness of $T+M$ and
@@ -257,135 +257,135 @@ $\Pic^0$ matches along the pullback of any birational model. A
finite-type spread argument completes the passage from $\C$ to any
algebraically closed field of characteristic zero.
-\paragraph{Organization.} Section~\ref{sec:preliminaries} records
-conventions and the two input theorems. Section~\ref{sec:reduction}
+Organization.: Section~sec:preliminaries records
+conventions and the two input theorems. Section~sec:reduction
sets up the two inductions and reduces (P$_n$) to the Calabi--Yau
-case. Section~\ref{sec:jet-tools} proves the bounded-degree
-obstruction and records the moving-kernel tools from \cite{LA}.
-Sections~\ref{sec:geometry} and~\ref{sec:frobenius} prove the terminal
-Calabi--Yau case of Theorem~\ref{thm:key-ingredient}, and
-Section~\ref{sec:completion} extends it to all klt Calabi--Yau pairs.
-Section~\ref{sec:positive} completes the inductions and descends to
-$X$; Section~\ref{sec:field} handles general algebraically closed
+case. Section~sec:jet-tools proves the bounded-degree
+obstruction and records the moving-kernel tools from [cite].
+Sections~sec:geometry and~sec:frobenius prove the terminal
+Calabi--Yau case of Theorem~thm:key-ingredient, and
+Section~sec:completion extends it to all klt Calabi--Yau pairs.
+Section~sec:positive completes the inductions and descends to
+$X$; Section~sec:field handles general algebraically closed
fields.
-\paragraph{Relation to earlier work.} Lazi\'c--Peternell \cite{LP}
+Relation to earlier work.: Lazi\'c--Peternell [cite]
formulated generalised abundance as the numerical extension of
abundance and established it on a Calabi--Yau variety under a
Semiampleness Conjecture plus termination and abundance in lower
dimensions. Their Corollaries C and D settle surfaces and threefolds
with $\kappa(K_X+B)>0$ unconditionally. Subsequent work by
-Lazi\'c--Peternell \cite{LPII}, Chaudhuri, and Fontanari addressed
+Lazi\'c--Peternell [cite], Chaudhuri, and Fontanari addressed
numerical nonvanishing, small-dimensional nef reductions, and
semiampleness for surfaces. The strategy here follows
\cite[Section~7]{LP} for the reduction; the new geometric ingredient is
-Theorem~\ref{thm:key-ingredient}. The moving-jet and Frobenius tools
+Theorem~thm:key-ingredient. The moving-jet and Frobenius tools
build on Musta\c{t}\u{a}--Schwede, Sun, Kitadai--Sumihiro, and
Langer's Frobenius-slope theory; the local-positivity framework
descends from Ein--K\"uchle--Lazarsfeld. The polarized birationality
theorem of Birkar--Zhang turns accumulated numerical bounds into
bounded-degree curves for the moving-jet machinery.
-\section{Conventions and standing inputs}
-\label{sec:preliminaries}
+# Conventions and standing inputs
+
Throughout, unless stated otherwise, the base field is $\C$. Varieties
are integral and projective. Grading of local inputs is stated at
each appearance.
-\subsection{Log pairs and singularities}
+## Log pairs and singularities
-A \emph{$\Q$-pair} $(X,B)$ consists of a normal projective variety $X$,
+A $\Q$-pair $(X,B)$ consists of a normal projective variety $X$,
an effective rational boundary $B$, and a $\Q$-Cartier divisor $K_X+B$.
Log discrepancies are defined on a log resolution $h\colon Y\to X$,
via the equality $K_Y+B_Y=h^*(K_X+B)$: a prime divisor $E\subset Y$
with coefficient $b_E$ in $B_Y$ carries log discrepancy $1-b_E$.
-The pair is \emph{klt} if all log discrepancies are positive and
-\emph{log canonical} if they are nonnegative. Standard cone,
+The pair is klt if all log discrepancies are positive and
+log canonical if they are nonnegative. Standard cone,
contraction, negativity, and resolution theorems are used as in
-\cite{KM,BCHM}.
+[cite].
The symbols $\sim_\Q$ and $\equiv$ denote rational linear and numerical
-equivalence. A divisor $D$ is \emph{pseudo-effective} if its class in
+equivalence. A divisor $D$ is pseudo-effective if its class in
$N^1(X)_\R$ lies in the closure of the cone generated by effective
-divisors; \emph{big} if its class lies in the interior of that cone;
-\emph{nef} if $D\cdot C\ge0$ for every integral curve $C$. A divisor
-is \emph{semiample} if some positive multiple is globally generated.
-The \emph{Iitaka dimension} $\kappa(D)$ is the maximal dimension of
+divisors; big if its class lies in the interior of that cone;
+nef if $D\cdot C\ge0$ for every integral curve $C$. A divisor
+is semiample if some positive multiple is globally generated.
+The Iitaka dimension $\kappa(D)$ is the maximal dimension of
the image of a map defined by a linear system $|kD|$, for $k$ verifying
$h^0(kD)>0$.
-Over $\C$, a \emph{very general} point is a point outside a specified
+Over $\C$, a very general point is a point outside a specified
countable union of proper closed subsets. Most statements below
specify the countable union only implicitly; iterated applications
remain valid because a countable union of countable unions remains
countable.
-\subsection{Nef reduction}
+## Nef reduction
-For a nef divisor $N$ on a projective variety, the \emph{nef
-reduction} \cite{NefReduction} is an almost holomorphic dominant
+For a nef divisor $N$ on a projective variety, the nef
+reduction [cite] is an almost holomorphic dominant
rational map $\pi\colon X\dashrightarrow Z$ with connected compact
general fibres such that:
-\begin{enumerate}
+
\item $N|_F\equiv0$ on a very general compact fibre $F$;
\item if $C$ is a curve through a very general point of $X$ not
contracted by $\pi$, then $N\cdot C>0$.
-\end{enumerate}
-The base dimension $\dim Z$ is the \emph{nef dimension} $n(X,N)$. It
+
+The base dimension $\dim Z$ is the nef dimension $n(X,N)$. It
depends only on the numerical class of $N$. By construction,
$n(X,N)=\dim X$ if and only if every curve through a very general
point has positive $N$-degree. This is the condition called
-\emph{full nef dimension}. Crucially, full nef dimension is purely
+full nef dimension. Crucially, full nef dimension is purely
numerical: it does not assert or imply bigness. The gap between full
nef dimension and bigness is exactly the gap that
-Theorem~\ref{thm:key-ingredient} closes for Calabi--Yau pairs.
+Theorem~thm:key-ingredient closes for Calabi--Yau pairs.
-\subsection{Divisorial Zariski decomposition}
+## Divisorial Zariski decomposition
For a pseudo-effective real divisor $D$ on a smooth projective
variety, and a prime divisor $\Gamma$ on it, fix an ample divisor $A$
and put
\[
- \sigma_\Gamma(D)=\lim_{\delta\downarrow0}
+ \sigma_\Gamma(D)=\lim_{\delta↓0}
\inf\{\mult_\Gamma D':D'\ge0,\ D'\sim_\R D+\delta A\}.
\]
These asymptotic divisorial orders are finite, nonnegative, and
independent of $A$. Only finitely many of them are nonzero, and each
depends only on the numerical class of $D$. The
-\emph{divisorial Zariski decomposition} is
+divisorial Zariski decomposition is
\[
N_\sigma(D)=\sum_\Gamma\sigma_\Gamma(D)\Gamma,
\qquad P_\sigma(D)=D-N_\sigma(D).
\]
-We call $N_\sigma(D)$ the \emph{negative part} and $P_\sigma(D)$ the
-\emph{positive part}. The sum is a finite $\R$-linear combination of
+We call $N_\sigma(D)$ the negative part and $P_\sigma(D)$ the
+positive part. The sum is a finite $\R$-linear combination of
prime divisors; the difference $P_\sigma(D)$ is a pseudo-effective
$\R$-divisor whose negative part is trivial. A divisor $D$ is nef if
and only if $N_\sigma(D)=0$. Otherwise, the negative part captures
exactly the ``asymptotically fixed'' divisorial component of $D$.
-\subsection{The very general point lemma}
+## The very general point lemma
+
-\begin{lemma}\label{lemma:pseudo-vg}
Let $D$ be a pseudo-effective rational Cartier divisor on a projective
variety $X$ over $\C$. Then every integral curve through a
sufficiently very general point of $X$ has nonnegative $D$-degree.
-\end{lemma}
-\begin{proof}
+
+Proof.
Fix an ample Cartier divisor $A$ and rationals $\delta_i>0$ with
$\delta_i\to 0$. Each $D+\delta_iA$ is big, so there is an effective
rational divisor $E_i\sim_\Q D+\delta_iA$. A point outside the
-countable union $\bigcup_i\Supp E_i$ is very general, and every curve
+countable union $∪_i\Supp E_i$ is very general, and every curve
$C$ through such a point meets no $\Supp E_i$. Therefore
$(D+\delta_iA)\cdot C\ge0$ for every $i$, and $D\cdot C\ge0$ after
-letting $i\to\infty$.
-\end{proof}
+letting $i\to∞$.
+
-\subsection{Two input theorems}
+## Two input theorems
We record the two external inputs. Both are theorems in characteristic
zero; both are used as black boxes.
@@ -394,39 +394,39 @@ The first input is ordinary log abundance. Recall that a klt variety
has rational singularities, so pullbacks from an arbitrary birational
model do not affect $\Pic^0$; the distinction between a rational
divisor and the line bundle it defines is thus controlled on smooth
-models (see Lemma~\ref{lemma:numsemiample-descent}). The log
+models (see Lemma~lemma:numsemiample-descent). The log
abundance statement asserts that this subtlety disappears entirely
for nef log canonical adjoints.
-\begin{theorem}[Log abundance, {\cite[Theorem~1.1]{LA}}]
-\label{input:abundance}
+[Log abundance, {\cite[Theorem~1.1]{LA}}]
+
Let $(X,B)$ be a projective log canonical $\Q$-pair over an
algebraically closed field of characteristic zero. If $K_X+B$ is
nef, then it is semiample.
-\end{theorem}
+
The second input provides the minimal-model programs used in
-Section~\ref{sec:reduction}, both for ordinary adjoints and for
+Section~sec:reduction, both for ordinary adjoints and for
generalized pairs whose only polarized data is a nef
-$\Q$-Cartier $b$-divisor. The statement records a \emph{terminating
-choice} of program: at each step there are many possible
-$(K_X+B+\mathbf{M})$-negative extremal rays, but the theorem gives a
+$\Q$-Cartier $b$-divisor. The statement records a terminating
+choice of program: at each step there are many possible
+$(K_X+B+M)$-negative extremal rays, but the theorem gives a
fixed order in which they have been contracted that terminates after
finitely many steps.
-A \emph{nef rational Cartier b-divisor} on a normal projective variety
+A nef rational Cartier b-divisor on a normal projective variety
$X$ is specified by a nef rational Cartier divisor $N'$ on a projective
birational model $Y\to X$; throughout the model $Y$, and on every
higher birational model, $N'$ pulls back unchanged. Generalized
discrepancies are computed by pulling back the generalized adjoint
while keeping $N'$ fixed on all higher models; the generalized pair is
generalized klt when all generalized discrepancies are positive. The
-trace of $\mathbf{M}$ on a lower model is its pushforward, which need
+trace of $M$ on a lower model is its pushforward, which need
not be nef.
-\begin{theorem}[Minimal-model programs,
+[Minimal-model programs,
{\cite[Theorem~1.1 and Section~3.3]{MM}}]
-\label{input:models}
+
Let $(X,B+\mathbf M)$ be a projective generalized klt $\Q$-pair over
$\C$, with $X$ $\Q$-factorial and with fixed nef rational Cartier
b-divisor data $\mathbf M$. If $K_X+B+M_X$ is pseudo-effective, there
@@ -436,19 +436,19 @@ extracts no divisors and generalized log discrepancies do not decrease.
On a common resolution of its initial and final models, the pullback of
the initial adjoint is the pullback of the final adjoint plus an
effective divisor exceptional over the final model.
-\end{theorem}
+
Two remarks. First, the ordinary klt case of
-Theorem~\ref{input:models} arises by taking $\mathbf M=0$; then
-Theorem~\ref{input:abundance} makes the endpoint a good minimal model.
+Theorem~input:models arises by taking $\mathbf M=0$; then
+Theorem~input:abundance makes the endpoint a good minimal model.
Second, neither theorem above asserts abundance for a nonzero
generalized nef summand. Their combination still exposes the
full-nef-dimension obstruction, which is exactly the content of
-Theorem~\ref{thm:key-ingredient}.
+Theorem~thm:key-ingredient.
-\section{The two inductions and reduction to Calabi--Yau}
-\label{sec:reduction}
+# The two inductions and reduction to Calabi--Yau
+
We work over $\C$ in this section. The reductions follow the strategy
of Lazi\'c--Peternell \cite[Sections~4--5]{LP}, with the terms made
@@ -459,40 +459,40 @@ well-behaved full nef dimension lets bigness descend from a
Calabi--Yau model. The two adjoints live on the same birational model
and must remain tractable simultaneously.
-\subsection{The two induction statements}
+## The two induction statements
For a projective klt $\Q$-pair $(X,B)$ and a nef $\Q$-Cartier divisor
$M$ on $X$, write $T=K_X+B$. Assume $T$ is pseudo-effective, and
consider the following condition: for some $a\in\Q_{>0}$,
-\begin{equation}\label{eq:full-condition}
+
(T+aM)\cdot C>0
- \quad\text{for every integral curve $C$ through a very general point of $X$.}
-\end{equation}
+ \quadfor every integral curve $C$ through a very general point of $X$.
+
Note that $T+aM$ need not be nef; the condition is on the class it
defines, and only at the very general locus.
-Lemma~\ref{lemma:pseudo-vg} implies that
-\eqref{eq:full-condition} holds whenever $M$ has full nef dimension.
-If $T+aM$ is nef, then \eqref{eq:full-condition} is exactly the
+Lemma~lemma:pseudo-vg implies that
+(eq:full-condition) holds whenever $M$ has full nef dimension.
+If $T+aM$ is nef, then (eq:full-condition) is exactly the
assertion that $T+aM$ has full nef dimension.
We prove the following two statements simultaneously by induction on
$n=\dim X$.
-\begin{description}
+
\item[(P$_n$)] Let $(X,B)$ be a projective klt $\Q$-pair of dimension
$n$, let $M$ be a nef $\Q$-Cartier divisor on $X$, let $T=K_X+B$ be
-pseudo-effective, and suppose \eqref{eq:full-condition} holds for some
+pseudo-effective, and suppose (eq:full-condition) holds for some
rational $a>0$. Then $T+cM$ is big for every rational $c>0$.
\item[(Z$_n$)] Under the same hypotheses on $(X,B)$, $M$, $T$ but
-without \eqref{eq:full-condition}, there is a projective birational
+without (eq:full-condition), there is a projective birational
morphism $w\colon W\to X$ from a smooth projective variety such that
$P_\sigma(w^*(T+M))$ is rational and numerically equivalent to a
semiample $\Q$-divisor on $W$.
-\end{description}
+
Both statements hold in dimensions $0$ and $1$. For $n=0$ there is
nothing to prove. For $n=1$, pseudo-effectivity of $T$ and nefness of
$M$ both say their degrees are nonnegative; then
-\eqref{eq:full-condition} makes every positive combination have
+(eq:full-condition) makes every positive combination have
positive degree, hence ample, and (P$_1$) is immediate. Similarly a
nef divisor on a curve has zero negative part in the divisorial
Zariski decomposition, so it is itself semiample in its own numerical
@@ -503,14 +503,14 @@ verify (P$_n$) and (Z$_n$). Along the way we may freely replace $X$
by a small projective $\Q$-factorialization (which pulls back the
adjoint crepantly and the nef divisor $M$ to a nef divisor), and
replace a statement about $X$ by the corresponding statement about a
-smooth birational model. Indeed \eqref{eq:full-condition} persists
+smooth birational model. Indeed (eq:full-condition) persists
to the isomorphism locus, bigness descends under the birational
pullback, and a smooth model on which $P_\sigma$ is rational and
numerically semiample descends to $X$ by
-Lemma~\ref{lemma:numsemiample-descent}. So we are free to impose
+Lemma~lemma:numsemiample-descent. So we are free to impose
$\Q$-factoriality when convenient.
-\subsection{The elementary ray bound for generalized pairs}
+## The elementary ray bound for generalized pairs
We start with the only new cone-theoretic input: a uniformity statement
for negative extremal rays of generalized klt pairs. Recall that a
@@ -520,7 +520,7 @@ $\mathbf N$ determined on some projective birational model of $V$.
The notation $N_V$ denotes the trace of $\mathbf N$ on $V$; this trace
need not be nef, only the polarized data on the fixing model is.
-\begin{lemma}\label{lemma:generalized-ray-bound}
+
Let $(V,\Delta+\mathbf N)$ be a projective generalized klt $\Q$-pair
of dimension $n$, with effective boundary, $\Q$-factorial $V$, and
nef rational Cartier b-divisor data. Every
@@ -535,7 +535,7 @@ minimal-model outputs remain $\Q$-factorial. On a common resolution
of a birational step, the pullback of the input generalized adjoint
equals the pullback of its output transform plus an effective divisor
exceptional over the output.
-\end{lemma}
+
The constant $4n$ is twice the usual bound $2n$ for an ordinary klt
pair. The factor $2$ arises because the argument passes through a
@@ -543,7 +543,7 @@ small perturbation of $G=K_V+\Delta+N_V$ by an ample divisor to reach
a genuine klt adjoint; the perturbation inflates the negativity of
$G$ along the chosen ray by at most a factor of two.
-\begin{proof}
+Proof.
Set $G=K_V+\Delta+N_V$ and fix a $G$-negative extremal ray generated
by $R$. Choose an ample rational divisor $A$ on $V$ and a rational
$\eta>0$ small enough that $G+\eta A$ is still negative on $R$, and
@@ -560,21 +560,21 @@ $\mathbf N$ as a locally free sheaf, and write
\]
with coefficients of $\Delta'$ strictly less than one; this is
generalized klt. Choose $V'$ with an effective exceptional $F$ such
-that $-F$ is $h$-ample. For small $\varepsilon>0$ the divisor
-$N'+\eta h^*A-\varepsilon F$ is ample (being the sum of an ample and a
+that $-F$ is $h$-ample. For small $ε>0$ the divisor
+$N'+\eta h^*A-ε F$ is ample (being the sum of an ample and a
nef, adjusted by an $h$-ample), and a general effective rational
representative of this ample class can be chosen with coefficients
-small enough that, after adding $\varepsilon F$, it preserves the sub-klt
+small enough that, after adding $ε F$, it preserves the sub-klt
condition on $\Delta'$. Pushing that representative to $V$ gives an
-effective rational divisor $\Theta$ on $V$ with
+effective rational divisor $Θ$ on $V$ with
\[
- K_V+\Theta \sim_\Q G+\eta A.
+ K_V+Θ \sim_\Q G+\eta A.
\]
The crepant equality is checked upstairs: its difference is
exceptional and relatively numerically trivial, hence zero by the
negativity lemma.
-The ordinary cone theorem for the klt pair $(V,\Theta)$ gives a
+The ordinary cone theorem for the klt pair $(V,Θ)$ gives a
rational curve generator $C$ of the ray with
\[
0<-(G+\eta A)\cdot C\le2n.
@@ -583,10 +583,10 @@ The chain
$-(G+\eta A)\cdot C\ge-\tfrac12 G\cdot C$
gives $-G\cdot C\le 4n$.
The contraction $c\colon V\to S$ of the ray for $G$ is the contraction
-of the same ray for the ordinary klt pair $(V,\Theta)$, by
+of the same ray for the ordinary klt pair $(V,Θ)$, by
cone-contraction uniqueness, and the flip behaves identically: since
$\Pic(V/S)$ is one-dimensional, there is a rational $\lambda>0$ with
-$G-\lambda(K_V+\Theta)$ numerically trivial over $S$, and Cartier
+$G-\lambda(K_V+Θ)$ numerically trivial over $S$, and Cartier
descent for the ordinary pair gives a pullback divisor from $S$ in
the appropriate class; on the flipped model $V^+$, the same
observation shows $G^+$ is ample over $S$. The $\Q$-factoriality of
@@ -595,19 +595,19 @@ exceptional comparison of $G$ is the sum of its ordinary counterpart
with $\lambda$, since the $S$-pullback parts cancel.
This proves the lemma.
-\end{proof}
-\subsection{Keeping the ordinary adjoint semiample}
-The heart of Section~\ref{sec:reduction} is a pair of MMPs organised
+## Keeping the ordinary adjoint semiample
+
+The heart of Section~sec:reduction is a pair of MMPs organised
so that the ordinary adjoint becomes a pullback of an ample divisor
on a lower-dimensional base, while a small multiple of the nef summand
stays tractable on the same model.
-\begin{lemma}\label{lemma:two-nef-model}
+
Let $(X,B)$ be a projective $\Q$-factorial klt $\Q$-pair of dimension
$n$, let $T=K_X+B$ be pseudo-effective, and let $M$ be a nef
-$\Q$-Cartier divisor on $X$. Suppose \eqref{eq:full-condition} holds.
+$\Q$-Cartier divisor on $X$. Suppose (eq:full-condition) holds.
Then there are a birational contraction $X\dashrightarrow Y$, a number
$s_0\in\Q_{>0}$, and a common smooth projective resolution
\[
@@ -616,34 +616,34 @@ $s_0\in\Q_{>0}$, and a common smooth projective resolution
with the following properties. If $B_Y$, $N$ denote the transforms of
$B$ and $M$, and $U=K_Y+B_Y$, then:
-\begin{enumerate}
+
\item $(Y,B_Y)$ is klt and $\Q$-factorial, $U$ is semiample, and
$\kappa(Y,U)=\kappa(X,T)\ge0$.
\item $N$ is pseudo-effective, and $U+s_0N$ is nef of full nef
dimension.
\item There are effective $q$-exceptional rational divisors $E_0,E_1$
on $W$ with
-\begin{equation}\label{eq:two-comparisons}
+
p^*T=q^*U+E_0,
\qquad
p^*(T+s_0M)=q^*(U+s_0N)+E_1.
-\end{equation}
-\end{enumerate}
-\end{lemma}
-The two comparisons of \eqref{eq:two-comparisons} are the effective
+
+
+
+The two comparisons of (eq:two-comparisons) are the effective
exceptional adjoint comparisons of the two minimal-model programs; the
constant $s_0$ is chosen so that the second program's rays are already
negative for the first program's adjoint at coefficient $s$, and $s_0$
is small enough that a proportionality argument with a free Cartier
multiple $dU_0$ rules out any positivity of $U_0$ along such a ray.
-\begin{proof}
-Apply Theorem~\ref{input:models} to the ordinary klt pair $(X,B)$
+Proof.
+Apply Theorem~input:models to the ordinary klt pair $(X,B)$
(i.e.\ with $\mathbf M=0$) and pseudo-effective adjoint $T$. The
output is a terminating $T$-MMP ending on a $\Q$-factorial klt pair
$(Y_0,B_0)$ with nef ordinary adjoint $U_0=K_{Y_0}+B_0$.
-Theorem~\ref{input:abundance} makes $U_0$ semiample. The
+Theorem~input:abundance makes $U_0$ semiample. The
comparisons along the finitely many birational steps give
$\kappa(X,T)=\kappa(Y_0,U_0)\ge0$.
@@ -664,11 +664,11 @@ proves the claim for classes. Thus $N_0$ is pseudo-effective.
Choose a positive integer $d$ such that $dU_0$ is Cartier and
globally generated. Pick $s_0\in(0,s)$ rational with
-\begin{equation}\label{eq:small-nef-coefficient}
- \frac{1-s_0/s}{d}>4n\frac{s_0}{s}.
-\end{equation}
+
+ (1-s_0/s)/(d)>4n(s_0)/(s).
+
The generalized adjoint $U_0+s_0N_0$ is pseudo-effective.
-Theorem~\ref{input:models} gives a terminating program for this
+Theorem~input:models gives a terminating program for this
generalized adjoint. We claim that every step of this program is
$U_0$-trivial and is also negative for the generalized adjoint with
coefficient $s$, where throughout the claim the symbols denote their
@@ -682,13 +682,13 @@ $(U_i+s_0N_i)$-negative ray, the decomposition
\]
together with nefness of $U_i$ shows the ray is
$(U_i+sN_i)$-negative. Pick its rational curve generator $C$ from
-Lemma~\ref{lemma:generalized-ray-bound}. If $U_i\cdot C>0$,
+Lemma~lemma:generalized-ray-bound. If $U_i\cdot C>0$,
integrality of the free $dU_i$ gives $U_i\cdot C\ge1/d$, so
\[
(U_i+s_0N_i)\cdot C
- \ge\frac{1-s_0/s}{d}-4n\frac{s_0}{s}>0
+ \ge(1-s_0/s)/(d)-4n(s_0)/(s)>0
\]
-by \eqref{eq:small-nef-coefficient}; contradiction. Thus
+by (eq:small-nef-coefficient); contradiction. Thus
$U_i\cdot C=0$.
The morphism defined by the free $dU_i$ is constant on each fibre of
@@ -703,7 +703,7 @@ $U=U_0$ is pulled back unchanged, hence semiample with the same
Iitaka dimension $\kappa(X,T)$. The transform of the generalized
adjoint is $U+s_0N$, which is nef by the construction of the program.
Combining the pullback comparisons of the two programs gives
-\eqref{eq:two-comparisons}.
+(eq:two-comparisons).
It remains to prove that $U+s_0N$ has full nef dimension. Suppose
toward a contradiction that its nef reduction has a positive-dimensional
@@ -714,37 +714,37 @@ $\ge2$; a very general fibre $F$ therefore misses it in codimension
$\ge2$ at every component, by the fibre dimension theorem applied to
the components that dominate the base. We can pick a curve $C\subset
F$ passing through a point chosen with the very general properties of
-\eqref{eq:full-condition}: outside the exceptional image, and in the
+(eq:full-condition): outside the exceptional image, and in the
common isomorphism locus of the first program. If $\dim F>1$, take
$C$ to be a sufficiently general complete intersection in $F$
through that point; if $\dim F=1$, take $C=F$ itself.
Because $U+s_0N$ is nef with positive-dimensional nef reduction and
$U$ is semiample, both $U$ and $N$ have degree zero on $C$: this
-follows from Lemma~\ref{lemma:pseudo-vg} applied to each, since $U$ is
+follows from Lemma~lemma:pseudo-vg applied to each, since $U$ is
nef and $N$ pseudo-effective and their positive combination gives a
nonnegative, hence zero, degree along $F$. The strict lift
$\widetilde C\subset W$ avoids the effective exceptional divisors
-$E_0,E_1$. Both equalities in \eqref{eq:two-comparisons} then give
+$E_0,E_1$. Both equalities in (eq:two-comparisons) then give
\[
p^*T\cdot\widetilde C=p^*M\cdot\widetilde C=0.
\]
The image $p(\widetilde C)$ passes through the chosen very general
point of $X$. Hence $(T+aM)\cdot p(\widetilde C)>0$ by
-\eqref{eq:full-condition}, but the two equalities above give
+(eq:full-condition), but the two equalities above give
$(T+aM)\cdot p(\widetilde C)=0$.
-\end{proof}
-\subsection{Positive Iitaka dimension}
-\begin{proposition}\label{prop:full-positive}
+## Positive Iitaka dimension
+
+
Assume (P$_j$) for every $j<n$. Let $(X,B)$ be a projective klt
$\Q$-pair of dimension $n$, let $T=K_X+B$ be pseudo-effective, and
let $M$ be a nef $\Q$-Cartier divisor on $X$. Suppose
-\eqref{eq:full-condition} holds for some rational $a>0$, and that
+(eq:full-condition) holds for some rational $a>0$, and that
$\kappa(X,T)>0$. Then $T+cM$ is big for every rational $c>0$.
-\end{proposition}
+
The argument reduces to the fibres of the Iitaka fibration of the
ordinary semiample adjoint: on a general positive-dimensional fibre
@@ -752,9 +752,9 @@ the ordinary adjoint contracts, while the nef summand restricts to a
full-nef-dimension divisor of smaller dimension, and the induction
hypothesis applies.
-\begin{proof}
+Proof.
After a small $\Q$-factorialization, apply
-Lemma~\ref{lemma:two-nef-model} to obtain the model $(Y,B_Y)$, its
+Lemma~lemma:two-nef-model to obtain the model $(Y,B_Y)$, its
transforms $N$, $U$, and the rational $s_0>0$. Let
$h\colon Y\to S$ be the contraction defined by the semiample divisor
$U$. Since $U$ is semiample, there is an ample rational divisor
@@ -775,7 +775,7 @@ on $Y$, and note that such a point exists for every nonempty fibre.
Since $U|_F\equiv0$, we have $Q|_F\equiv s_0N|_F$, so $N|_F$ is nef
of full nef dimension. Apply $(P_{\dim F})$ to the restriction pair
$(F,B_Y|_F)$ and the nef divisor $s_0N|_F$; the full condition holds
-for $(N|_F)^{}$ by Lemma~\ref{lemma:pseudo-vg}. This yields bigness
+for $(N|_F)^{}$ by Lemma~lemma:pseudo-vg. This yields bigness
of $K_F+B_Y|_F+c(s_0N|_F)$ for every rational $c>0$, and in
particular of $Q|_F$.
@@ -801,68 +801,68 @@ pullback of $A_S$,
\]
is big for some rational $b>0$.
-By \eqref{eq:two-comparisons}(ii), this makes $(1+b)T+s_0M$ big on
+By (eq:two-comparisons)(ii), this makes $(1+b)T+s_0M$ big on
$X$. Finally, fix a rational $c>0$ and choose rational
$\lambda>0$ with $\lambda(1+b)<1$ and $\lambda s_0<c$. Then
\[
T+cM=
- \lambda\bigl((1+b)T+s_0M\bigr)
- +\bigl(1-\lambda(1+b)\bigr)T
+ \lambda((1+b)T+s_0M)
+ +(1-\lambda(1+b))T
+(c-\lambda s_0)M
\]
is the sum of a big divisor and two pseudo-effective divisors, hence
big.
-\end{proof}
-\subsection{The Calabi--Yau reduction}
-\begin{proposition}\label{prop:reduction-cy}
+## The Calabi--Yau reduction
+
+
Assume (P$_j$) for every $j<n$. Suppose that on every projective
klt $\Q$-pair $(V,\Delta)$ of dimension $n$ with $K_V+\Delta\equiv0$,
every nef $\Q$-Cartier divisor of full nef dimension is big. Then
(P$_n$) holds.
-\end{proposition}
+
The proposition makes the reduction to
-Theorem~\ref{thm:key-ingredient} explicit: away from the case
+Theorem~thm:key-ingredient explicit: away from the case
$\kappa(T)=0$, the positive-dimensional Iitaka argument of
-Proposition~\ref{prop:full-positive} already gives $(P_n)$; the only
+Proposition~prop:full-positive already gives $(P_n)$; the only
remaining case is a klt Calabi--Yau pair with a full-nef-dimension
nef summand.
-\begin{proof}
+Proof.
Use a small $\Q$-factorialization and
-Lemma~\ref{lemma:two-nef-model}. The Iitaka dimension
+Lemma~lemma:two-nef-model. The Iitaka dimension
$\kappa(X,T)\ge0$ is either positive, in which case
-Proposition~\ref{prop:full-positive} applies directly, or zero.
+Proposition~prop:full-positive applies directly, or zero.
Suppose $\kappa(X,T)=0$. Then $U$ is semiample with Iitaka
dimension zero, so $U\sim_\Q0$. The pair $(Y,B_Y)$ is a klt
Calabi--Yau pair of dimension $n$. The nef divisor $U+s_0N$ has
-full nef dimension by Lemma~\ref{lemma:two-nef-model}(2), and since
+full nef dimension by Lemma~lemma:two-nef-model(2), and since
$U\sim_\Q0$ we have $U+s_0N\sim_\Q s_0N$; thus $N$ is nef of full
nef dimension. By hypothesis $N$ is big, and so is $s_0N$.
-By \eqref{eq:two-comparisons}(ii), the pullback $p^*(T+s_0M)$ is
+By (eq:two-comparisons)(ii), the pullback $p^*(T+s_0M)$ is
the sum of a $q$-exceptional effective divisor and the pullback of
$U+s_0N$; hence $T+s_0M$ is big on $X$. The positive-combination
-argument at the end of the proof of Proposition~\ref{prop:full-positive}
+argument at the end of the proof of Proposition~prop:full-positive
(which requires only $T$ pseudo-effective and $M$ nef) then gives
that $T+cM$ is big for every rational $c>0$.
-\end{proof}
+
What remains for (P$_n$) is therefore the hypothesis of
-Proposition~\ref{prop:reduction-cy}: on a projective klt Calabi--Yau
+Proposition~prop:reduction-cy: on a projective klt Calabi--Yau
$\Q$-pair, a nef $\Q$-Cartier divisor of full nef dimension is big.
-This is Theorem~\ref{thm:key-ingredient}. We prove it first for
+This is Theorem~thm:key-ingredient. We prove it first for
terminal $X$ with $K_X\sim0$ in
-Sections~\ref{sec:jet-tools}--\ref{sec:frobenius}, and then extend
-to all klt Calabi--Yau pairs in Section~\ref{sec:completion}.
+Sections~sec:jet-tools--sec:frobenius, and then extend
+to all klt Calabi--Yau pairs in Section~sec:completion.
+
+# Bounded-degree obstruction and moving kernels
-\section{Bounded-degree obstruction and moving kernels}
-\label{sec:jet-tools}
This section assembles the technology of moving jets that
-Section~\ref{sec:geometry} will use. Three things are needed.
+Section~sec:geometry will use. Three things are needed.
First, a way for numerical bounds on a subvariety (bounded volume,
bounded canonical ratio) to forbid a growing lower bound on curve
degrees. Second, a way for non-uniruledness and quasi-finiteness to
@@ -872,14 +872,14 @@ those bounds from a jet kernel with positive-dimensional base locus.
All varieties in this section are over $\C$.
-\subsection{The bounded-degree obstruction}
+## The bounded-degree obstruction
The following lemma is the numerical engine: it says a curve class
that meets a targeted point with positive integer degree cannot be
arbitrarily thin in degree, when its ambient variety satisfies a
bounded volume and bounded canonical ratio.
-\begin{lemma}[Bounded-degree curves]\label{lemma:bounded-curves}
+[Bounded-degree curves]
For every integer $f\ge1$ there is an integer $m_f>0$ with the
following property. Let $Y$ be a smooth integral projective variety
of dimension $f$, let $D_0$ be a reduced simple normal crossing
@@ -888,27 +888,27 @@ divisor of full nef dimension on $Y$. Suppose that $P_0$ is a nef
rational Cartier divisor, and that $r,C_0,C_1$ are positive real
numbers with
\[
- 0<P_0^f\le C_0,\qquad P_0-rL_0\text{ is nef}.
+ 0<P_0^f\le C_0,\qquad P_0-rL_0 is nef.
\]
Suppose also $A_0=K_Y+D_0+L_0$ is big and
-\begin{equation}\label{j:adjoint-ratio}
+
A_0\cdot P_0^{f-1}\le C_1 P_0^f.
-\end{equation}
+
Then
-\begin{equation}\label{j:bounded-curve-obstruction}
+
r\le(m_f C_1)^{f-1} C_0.
-\end{equation}
+
The integer $m_f$ depends only on $f$ and not on the variety or the
divisors. For all $1\le f\le n$, one may take a single integer $m$
depending only on $n$.
-\end{lemma}
+
The quantity $m_f$ is not explicit: it comes from the Birkar--Zhang
effective birationality theorem for polarized pairs
\cite[Theorem~1.3]{BZ}. What matters is that it depends only on the
dimension, not on the variety or the divisors.
-\begin{proof}
+Proof.
The pair $(Y,D_0)$ is log canonical with boundary coefficients in
$\{0,1\}$, and the polarizing divisor $L_0$ is a nef integral Cartier
divisor. By \cite[Theorem~1.3]{BZ}, an integer $m_f$ exists such
@@ -924,9 +924,9 @@ for the basepoint-free moving part $H$ and an effective $F$. The
map defined by $H$ is birational onto its image; $H$ is nef and big.
In the rest of this proof, $P_0$ and $L_0$ denote pullbacks to
$\widetilde Y$. Effectivity of $F$ and nefness of $P_0$ give
-\begin{equation}\label{j:moving-adjoint-degree}
+
H\cdot P_0^{f-1}\le m_f C_1 P_0^f.
-\end{equation}
+
For $f\ge2$, put $s_i=P_0^{f-i}\cdot H^i$ for $0\le i\le f$.
Both divisors are nef and big, so each $s_i$ is positive. The
@@ -934,11 +934,11 @@ Khovanskii--Teissier inequalities
\cite[Corollary~1.6.3(i) and Example~1.6.4]{LazarsfeldPAGI}, namely
$s_i^2\ge s_{i-1}s_{i+1}$, show that the ratios $s_i/s_{i-1}$ are
decreasing; hence
-\begin{equation}\label{j:mixed-degree}
+
P_0\cdot H^{f-1}
- \le P_0^f\left(\frac{H\cdot P_0^{f-1}}{P_0^f}\right)^{f-1}
+ \le P_0^f(\frac{H\cdot P_0^{f-1}}{P_0^f})^{f-1}
\le (m_f C_1)^{f-1} C_0.
-\end{equation}
+
Choose a very general point $y\in\widetilde Y$ in the isomorphism
open of $\mu$, and such that $\mu(y)$ is very general with respect
to the full-nef-dimension condition on $L_0$. Pull back $f-1$
@@ -954,27 +954,27 @@ The curve $C$ is not contracted by $\mu$, and its image passes
through the very general point $\mu(y)$. Full nef dimension and
integrality therefore give $L_0\cdot C\ge1$. Since $P_0-rL_0$ is
nef, we obtain $P_0\cdot C\ge r$, which together with
-\eqref{j:mixed-degree} proves \eqref{j:bounded-curve-obstruction}.
+(j:mixed-degree) proves (j:bounded-curve-obstruction).
If $f=1$, take $C=Y$ directly; then
$P_0\cdot Y\ge rL_0\cdot Y\ge r$, and $r\le C_0$ is the claim.
Finally, \cite[Theorem~1.3]{BZ} applies to every multiple of its
prescribed integer; a common multiple of the finitely many integers
-$m_1,\dots,m_n$ gives the last assertion.
-\end{proof}
+$m_1,...,m_n$ gives the last assertion.
+
The lemma is uniform in the sense that varieties, divisors, and $r$
may vary, provided $C_0$ and $C_1$ stay bounded. In the geometric
-section, $r\to\infty$ along a family, so a sequence satisfying the
+section, $r\to∞$ along a family, so a sequence satisfying the
hypotheses of the lemma will automatically be absent for large $r$.
-\subsection{Very general images}
+## Very general images
The second ingredient is that full nef dimension, non-uniruledness,
and pseudo-effectivity of the canonical divisor are compatible with
passing to sufficiently general subvarieties.
-\begin{lemma}[Very general images]\label{j:very-general-images}
+[Very general images]
Let $X$ be a normal integral projective variety that is not
uniruled, and let $L$ be a nef Cartier divisor of full nef dimension
on $X$. Then there is a complement $X^\circ$ of a countable union of
@@ -983,12 +983,12 @@ a smooth integral projective variety and $g\colon Y\to X$ is
generically finite onto a positive-dimensional image that meets
$X^\circ$, then $g^*L$ has full nef dimension, $Y$ is not uniruled,
and $K_Y$ is pseudo-effective.
-\end{lemma}
+
The point is that very general conditions simultaneously test full
nef dimension of $L$ and avoid all rational curves.
-\begin{proof}
+Proof.
Choose $X^\circ$ so that every curve through any of its points has
positive $L$-degree and no rational curve meets $X^\circ$. The first
condition characterizes full nef dimension at a point. For the
@@ -1004,17 +1004,17 @@ $g^*L$-degree is positive. This gives full nef dimension of $g^*L$.
A covering family of rational curves on $Y$ would similarly give a
nonconstant rational curve meeting $X^\circ$, a contradiction; so
$Y$ is not uniruled. Pseudo-effectivity of $K_Y$ now follows from
-\cite{BDPP}.
-\end{proof}
+[cite].
+
-\subsection{Moving kernels and the LA inputs}
-\label{sec:moving-kernel-inputs}
+## Moving kernels and the LA inputs
+
The third ingredient, and the most technical, is a set of five results
-from the section \emph{Tools for moving base components} of
-\cite{LA}. We state them in full, because they are used, not just
-cited, in Section~\ref{sec:geometry}. Their hypotheses separate the
+from the section Tools for moving base components of
+[cite]. We state them in full, because they are used, not just
+cited, in Section~sec:geometry. Their hypotheses separate the
generic information along a moving base component from the global
information about the ambient variety. Intersections with a
subvariety are computed after pullback to its normalization and a
@@ -1022,9 +1022,9 @@ smooth projective resolution, because the subvariety itself may be
singular.
A linear system $\mathcal H\subset H^0(Z,\OO_Z(kR))$ has a
-\emph{base ideal} $\mathfrak b$ defined as the ideal generated by its
+base ideal $\mathfrak b$ defined as the ideal generated by its
sections in $\OO_Z$; its zero locus is the base locus. A base
-component $V$ is \emph{isolated at the generic point} of $V$ when the
+component $V$ is isolated at the generic point of $V$ when the
localized ideal $\mathfrak b\OO_{Z,\eta_V}$ is primary for the
maximal ideal $\mathfrak m$ of $\OO_{Z,\eta_V}$. In this case there
is a well-defined multiplicity $h$ with
@@ -1034,23 +1034,23 @@ strongly $\mathcal H$ vanishes along $V$ at its generic point.
The first input is a numerical version of the
Kawakita--Ambro adjunction for a lc-trivial fibration:
-\begin{lemma}[Numerical generic subadjunction, {\cite[Lemma~7.1]{LA}}]
-\label{j:subadjunction}
-Let $Z$ be a projective $\Q$-factorial klt variety, let $\Theta\ge0$
+[Numerical generic subadjunction, {\cite[Lemma~7.1]{LA}}]
+
+Let $Z$ be a projective $\Q$-factorial klt variety, let $Θ\ge0$
be a rational divisor, and let $V\subset Z$ be integral of dimension
-$f>0$. Suppose $(Z,\Theta)$ is log canonical at the generic point of
+$f>0$. Suppose $(Z,Θ)$ is log canonical at the generic point of
$V$, and admits a divisor of log discrepancy zero centered on $V$. If
$P$ is a nef rational Cartier divisor on $V$ and $\rho\colon V^*\to V$
is a smooth projective resolution factoring through its
normalization, then
\[
K_{V^*}\cdot(\rho^*P)^{f-1}
- \le (K_Z+\Theta)|_V\cdot P^{f-1}.
+ \le (K_Z+Θ)|_V\cdot P^{f-1}.
\]
-\end{lemma}
+
Only generic log canonicity is required: the inequality lives at the
-generic point of $V$. The proof in \cite{LA} passes through the dlt
+generic point of $V$. The proof in [cite] passes through the dlt
modification of \cite[Theorem~2.10]{FH}, whose truncated boundary
leaves an effective surplus supported over the non-lc locus. On a
minimal stratum over $V$, adjunction gives a pair that is klt over
@@ -1064,59 +1064,59 @@ $\Q$-Gorenstein assumption on the normalization of $V$.
The second input transfers a base component to a genuine pair:
-\begin{lemma}[High-multiplicity base component, {\cite[Lemma~7.2]{LA}}]
-\label{j:base-component}
+[High-multiplicity base component, {\cite[Lemma~7.2]{LA}}]
+
Let $Z$ be a projective $\Q$-factorial klt variety of dimension $d$,
let $R$ be a nef rational Cartier divisor, and let $k>0$ be an integer
such that $kR$ is Cartier. Let
$0\ne\mathcal H\subset H^0(Z,\OO_Z(kR))$ be a linear subspace with
proper base locus and base ideal $\mathfrak b$. Suppose that
$V$ is an integral base-locus component of dimension $f<d$, isolated
-at its generic point $\eta_V$, and that $\eta_V\in Z_{\mathrm{sm}}$.
+at its generic point $\eta_V$, and that $\eta_V\in Z_{sm}$.
Let $\mathfrak m$ be the maximal ideal of $\OO_{Z,\eta_V}$. If
$\mathfrak b\OO_{Z,\eta_V}\subset\mathfrak m^h$ for an integer $h>0$,
there are a rational $0<c\le(d-f)k/h$ and an effective rational
-divisor $\Theta\sim_\Q cR$ such that $(Z,\Theta)$ is log canonical
+divisor $Θ\sim_\Q cR$ such that $(Z,Θ)$ is log canonical
at $\eta_V$ and has a log canonical place centered on $V$. The
degree and canonical bounds are
-\begin{equation}\label{j:base-bounds}
- \begin{split}
+
+
R^f\cdot V&\le(k/h)^{d-f}R^d,\\
K_{V^*}\cdot R^{f-1}
&\le(K_Z+cR)|_V\cdot R^{f-1}\qquad(f>0),
- \end{split}
-\end{equation}
+
+
where $V^*$ is a smooth projective resolution. The degree bound
also holds at $f=0$.
-\end{lemma}
+
The degree estimate is the local multiplicity bound $h^{d-f}$
combined with successive general cuts by the given linear series.
The boundary is obtained by averaging general members with total
coefficient equal to the generic log canonical threshold of
$\mathfrak b$; thus it is an actual divisor to which
-Lemma~\ref{j:subadjunction} applies.
+Lemma~j:subadjunction applies.
The third input produces the moving base components in the first
place:
-\begin{lemma}[Moving multiplicity, {\cite[Lemma~7.3]{LA}}]
-\label{j:moving-kernel}
+[Moving multiplicity, {\cite[Lemma~7.3]{LA}}]
+
Let $Z$ be an integral projective variety of dimension $d$, let $R$
be a nef, big, semiample rational Cartier divisor, and fix an integer
$k>0$ such that $kR$ is Cartier. Choose positive real numbers
-$\tau_0<\dots<\tau_d$ with common gap $\delta>0$. For $x\in
-Z_{\mathrm{sm}}$, put
+$\tau_0<...<\tau_d$ with common gap $\delta>0$. For $x\in
+Z_{sm}$, put
\[
- \mathcal H_i(x)=H^0\bigl(Z,\OO_Z(kR)
- \otimes\mathfrak m_x^{\lceil k\tau_i\rceil}\bigr).
+ \mathcal H_i(x)=H^0(Z,\OO_Z(kR)
+ ⊗ \mathfrak m_x^{\lceil k\tau_i\rceil}).
\]
Suppose that for very general $x$ all these spaces are nonzero, and
that the base locus of $\mathcal H_0(x)$ has a positive-dimensional
component through $x$. If $k\delta\ge4$, then, after an algebraic
parameter extension and restriction to nonempty open subsets, there
are: an integral parameter variety $T$ with a dominant marked-point
-map $x\colon T\to Z_{\mathrm{sm}}$; an index $i\in\{0,\dots,d-1\}$
+map $x\colon T\to Z_{sm}$; an index $i\in\{0,...,d-1\}$
fixed throughout; and a family of integral subvarieties $V_t$
containing $x(t)$ and common to the base loci of $\mathcal
H_i(x(t))$ and $\mathcal H_{i+1}(x(t))$. The incidence
@@ -1136,17 +1136,17 @@ The incidence base ideal is likewise contained in
$\mathcal I_\Gamma^h$ on a dense open of $\Gamma$. These
containments are taken in the smooth ambient open. If $Z$ is also
$\Q$-factorial and klt, then
-\begin{equation}\label{j:moving-bounds}
- \begin{split}
+
+
R^f\cdot V_t&\le(2/\delta)^{d-f}R^d,\\
K_{V_t^*}\cdot R^{f-1}
&\le(K_Z+cR)|_{V_t}\cdot R^{f-1},\qquad
0<c\le2(d-f)/\delta.
- \end{split}
-\end{equation}
+
+
The top intersection on $V_t$ is positive when $V_t$ meets the
open on which the big semiample contraction is an isomorphism.
-\end{lemma}
+
The use of full kernels matters. Differentiation in the parameter
lowers vanishing at the moving mark and still gives a global section
@@ -1158,10 +1158,10 @@ containment.
The fourth input restricts an isolated base ideal to a fibre of a
morphism:
-\begin{lemma}[Restriction of an isolated base ideal,
+[Restriction of an isolated base ideal,
{\cite[Lemma~7.4]{LA}}]
-\label{j:fiber-restriction}
-In the setting of Lemma~\ref{j:moving-kernel}, let $g\colon Z\to B$
+
+In the setting of Lemma~j:moving-kernel, let $g\colon Z\to B$
be a morphism. There is a dense open of the incidence with the
following property. Choose $(t,z)$ in this open, put $b=g(z)$, and
suppose $V_t$ is smooth over a smooth open of its actual image at
@@ -1173,10 +1173,10 @@ in the ordinary power $\mathcal I_F^h$.
If $F$ is proper in an integral ambient fibre component containing
it, the restricted series is nonzero there and $F$ is an isolated
base component at its generic point. Applying
-Lemma~\ref{j:base-component} on that ambient component additionally
+Lemma~j:base-component on that ambient component additionally
requires its projectivity, $\Q$-factoriality, klt singularities, and
smoothness at the generic point of $F$.
-\end{lemma}
+
The fibre is chosen through a general incidence point; the statement
does not assert the same conclusion for an arbitrary special fibre.
@@ -1187,9 +1187,9 @@ powers.
The fifth input turns a bounded-degree family of finite covers into
a finite list once the branch divisors do not sweep:
-\begin{lemma}[Finite covers after excluding sweeping branch divisors,
+[Finite covers after excluding sweeping branch divisors,
{\cite[Lemma~7.5]{LA}}]
-\label{j:finite-covers}
+
Let $Y$ be a normal integral projective variety, let
$\mathcal V\to T$ be a smooth projective family with integral fibres
over an integral parameter variety, and let $g\colon\mathcal V\to Y$
@@ -1199,13 +1199,13 @@ at most a fixed integer $e$.
After a finite parameter extension and shrinking, spread the
geometric generic components of the zero divisor of the relative
-Jacobian over $Y_{\mathrm{sm}}$ to divisors
-$\mathcal R_1,\dots,\mathcal R_N$ in $\mathcal V$. Retain this
+Jacobian over $Y_{sm}$ to divisors
+$\mathcal R_1,...,\mathcal R_N$ in $\mathcal V$. Retain this
notation after the base change. Let $I$ consist of those $j$ for
-which $\overline{g_t((\mathcal R_j)_t)}$ is a divisor in $Y$ for
+which $g_t((\mathcal R_j)_t)bar$ is a divisor in $Y$ for
general $t$. Assume
\[
- \overline{g(\mathcal R_j)}\ne Y\qquad(j\in I).
+ g(\mathcal R_j)bar\ne Y\qquad(j\in I).
\]
After shrinking again, there is a smooth nonempty open
$Y^\circ\subset Y$ over which every general finite normal Stein
@@ -1213,10 +1213,10 @@ cover of $g_t$ is finite \'etale. Only finitely many such normal
covers of $Y$ occur, up to isomorphism over $Y$. The conclusion
holds simultaneously for finitely many projections; the open and the
finite lists may depend on all fixed parameters of the family.
-\end{lemma}
-The hypothesis removes $Y_{\mathrm{sing}}$ and the finitely many
-proper image closures $\overline{g(\mathcal R_j)}$ for $j\in I$.
+
+The hypothesis removes $Y_{sing}$ and the finitely many
+proper image closures $g(\mathcal R_j)bar$ for $j\in I$.
Every branch prime of a general Stein cover is accounted for by one
of these relative Jacobian components, so purity makes the
restricted covers \'etale. The fundamental group of the resulting
@@ -1225,7 +1225,7 @@ finitely many covers of degree at most $e$. Finally, a finite
normal cover of $Y$ is determined by its function field, hence by
its restriction to this fixed open.
-\subsection{$\Q$-factorialization of a klt ambient variety}
+## $\Q$-factorialization of a klt ambient variety
The base-component estimates require a $\Q$-factorial ambient
variety. When the ambient is only klt, we may pass to a small
@@ -1242,57 +1242,57 @@ because their additional exceptional divisors have zero
intersection with the pulled-back testing class to the required
power.
-\section{The terminal Calabi--Yau case: two jet estimates}
-\label{sec:geometry}
+# The terminal Calabi--Yau case: two jet estimates
+
-We now prove Theorem~\ref{thm:key-ingredient} in the terminal case
+We now prove Theorem~thm:key-ingredient in the terminal case
with trivial canonical divisor. This is where the geometry of the
paper is concentrated.
-\begin{theorem}[Terminal Calabi--Yau case]\label{thm:terminal-big}
+[Terminal Calabi--Yau case]
Let $n\ge2$ and assume $(P_j)$ for every $j<n$. Let $X$ be a
projective $\Q$-factorial terminal complex variety of dimension $n$
with $K_X\sim0$. If $L$ is a nef Cartier divisor on $X$ of full
nef dimension, then $L$ is big.
-\end{theorem}
-The proof of Theorem~\ref{thm:terminal-big} occupies this section and
-Section~\ref{sec:frobenius}. Assume $L$ is not big. We produce two
+
+The proof of Theorem~thm:terminal-big occupies this section and
+Section~sec:frobenius. Assume $L$ is not big. We produce two
incompatible jet estimates: a vanishing-order bound for sections of
-big multiples of a rescaled ample approximation (Lemma~\ref{lemma:scalar-bound}),
+big multiples of a rescaled ample approximation (Lemma~lemma:scalar-bound),
and a large jet system on a projective bundle over $X\times X$
-(Proposition~\ref{prop:large-seshadri}). The incompatibility is made
-precise by the Frobenius comparison of Section~\ref{sec:frobenius}.
+(Proposition~prop:large-seshadri). The incompatibility is made
+precise by the Frobenius comparison of Section~sec:frobenius.
-\subsection{Normalization and the curve obstruction}
+## Normalization and the curve obstruction
Throughout, fix $X,L$ forming a counterexample to
-Theorem~\ref{thm:terminal-big}. Thus: $X$ is projective,
+Theorem~thm:terminal-big. Thus: $X$ is projective,
$\Q$-factorial, terminal, of dimension $n\ge2$, with $K_X\sim0$; and
$L$ is a nef Cartier divisor of full nef dimension but not big. In
particular, $L^n=0$, and every curve through a very general point of
$X$ has positive integral $L$-degree.
Fix an ample Cartier divisor $A$ on $X$ and choose a rational
-$\epsilon>0$ with
-\begin{equation}\label{j:epsilon}
- (14\epsilon)^n<\tfrac14, \qquad 80\epsilon<\tfrac34.
-\end{equation}
+$ε>0$ with
+
+ (14ε)^n<\tfrac14, \qquad 80ε<\tfrac34.
+
Let $t>0$ tend to zero through positive rationals. Since $L$ is nef
and not big, $L^n=0$; we choose positive integers $r=r(t)$ by
-\begin{equation}\label{eq:scaling}
+
P=P_t=r(L+tA),\qquad
- r\longrightarrow\infty,\qquad P^n\longrightarrow\epsilon^n.
-\end{equation}
+ r→∞,\qquad P^n→ε^n.
+
For instance, $r$ may be the nearest integer to
-$\epsilon/((L+tA)^n)^{1/n}$; if the expansion of $(L+tA)^n$ is
+$ε/((L+tA)^n)^{1/n}$; if the expansion of $(L+tA)^n$ is
nonzero at every order up to $t^{n-1}$ (and it is, since
-$L^n=0$ and $L^{n-1}A>0$), this makes $r\to\infty$ as $t\to0$.
+$L^n=0$ and $L^{n-1}A>0$), this makes $r\to∞$ as $t\to0$.
The divisor $P$ is ample and rational Cartier, and $P-rL=rtA$ is nef
for all $t$. After discarding finitely many $t$ we may assume
-\begin{equation}\label{j:volume-window}
- \epsilon^n/2\le P^n\le 2\epsilon^n.
-\end{equation}
+
+ ε^n/2\le P^n\le 2ε^n.
+
Every curve through a very general point of $X$ has positive
integral $L$-degree, hence $P$-degree at least $r$ (on such a curve,
$P\cdot C = r(L+tA)\cdot C \ge r$ because $L\cdot C\ge1$ and
@@ -1306,53 +1306,53 @@ A smooth projective resolution $W\to X$ admits an effective
canonical divisor: pull back a nowhere-zero canonical form from the
smooth locus of $X$, and use terminality of $X$ to represent
$K_W$ by an effective $L$-exceptional divisor. In particular $W$,
-and therefore $X$, is not uniruled \cite{BDPP}.
+and therefore $X$, is not uniruled [cite].
-By Lemma~\ref{j:very-general-images}, if $Y$ is a smooth projective
+By Lemma~j:very-general-images, if $Y$ is a smooth projective
variety with a generically finite map $g\colon Y\to X$ onto a
positive-dimensional image that meets a sufficiently very general
point of $X$, then $Y$ is not uniruled, $K_Y$ is pseudo-effective,
and the pullback $L_Y=g^*L$ has full nef dimension. If
$f=\dim Y<n$, the induction hypothesis $(P_f)$ applied to $(Y,0)$
gives
-\begin{equation}\label{j:lower-big}
- K_Y+L_Y\text{ is big}.
-\end{equation}
+
+ K_Y+L_Y is big.
+
All uses of ``very general'' below include the fixed conditions of
-Lemma~\ref{j:very-general-images}.
+Lemma~j:very-general-images.
Consequently, for $f<n$ it suffices to find a nef testing divisor
$P_Y$ on such $Y$ with
\[
- 0<P_Y^f\le C_0,\qquad P_Y-rL_Y\text{ nef},\qquad\text{and }
+ 0<P_Y^f\le C_0,\qquad P_Y-rL_Y nef,\qquadand
K_YP_Y^{f-1}\le C\, P_Y^f,
\]
where $C_0$ and $C$ are independent of $t$. Indeed
-$L_YP_Y^{f-1}\le P_Y^f/r$, so Lemma~\ref{lemma:bounded-curves}
+$L_YP_Y^{f-1}\le P_Y^f/r$, so Lemma~lemma:bounded-curves
applies. We use that lemma in dimension $n$ as well, after proving
the needed bigness separately.
-Call a bound \emph{uniform} if it does not depend on $t$; it may
+Call a bound uniform if it does not depend on $t$; it may
depend on parameters fixed before $t$ tends to zero. Since
-dimensions lie in the finite range $[1,n]$, Lemma~\ref{lemma:bounded-curves}
+dimensions lie in the finite range $[1,n]$, Lemma~lemma:bounded-curves
excludes all such configurations once $r$ is large enough.
-\subsection{The scalar jet estimate}
+## The scalar jet estimate
Put $\rho=(P^n)^{1/n}$. The first incompatible estimate is the
following vanishing-order bound. It is stronger than a bound on a
single linear system.
-\begin{lemma}\label{lemma:scalar-bound}
-In the setting of \eqref{eq:scaling}, for all sufficiently small
+
+In the setting of (eq:scaling), for all sufficiently small
$t$, at a very general point $x\in X$, every nonzero section $s$ of
every $kP$ with $k$ a positive integer and $kP$ Cartier satisfies
-\begin{equation}\label{j:scalar-order}
+
\ord_x(s)\le 4k\rho.
-\end{equation}
-\end{lemma}
-\begin{proof}
+
+
+Proof.
Suppose violations occur for arbitrarily small $t$. Fix such a $t$.
At a very general point, a violation for some $k$ means the
corresponding complete jet kernel is nonzero at the general point.
@@ -1364,7 +1364,7 @@ powers makes $k$ arbitrarily large and divisible.
Choose $n+1$ levels
\[
\tau_i=\frac32\rho+i\delta\quad(0\le i\le n),
- \qquad\delta=\frac{2\rho}{n}.
+ \qquad\delta=(2\rho)/(n).
\]
These lie strictly between $\rho$ and $4\rho$. For a sufficiently
large divisible $k$, the corresponding full jet kernels are nonzero
@@ -1374,7 +1374,7 @@ Consider the lowest kernel. If its base locus at the marked point
were zero-dimensional, the isolated base component at that point
would have multiplicity at least
$h_0=\lceil k\tau_0\rceil$. The zero-dimensional case of
-Lemma~\ref{j:base-component} (its degree bound, valid at $f=0$) would
+Lemma~j:base-component (its degree bound, valid at $f=0$) would
then force
\[
1\le(k/h_0)^nP^n < 1,
@@ -1385,7 +1385,7 @@ $(k/h_0)^nP^n\le(2/3)^n(P^n/\rho^n)=(2/3)^n<1$, since $n\ge2$.
Hence the lowest base locus has a positive-dimensional component
through its marked point.
-Lemma~\ref{j:moving-kernel} now supplies a moving family of proper
+Lemma~j:moving-kernel now supplies a moving family of proper
base components $V$ with dominant incidence and dimension
$1\le f<n$. On a resolution $V^*$, let $P_V,L_V$ be the
restrictions followed by pullback. The lemma gives
@@ -1394,20 +1394,20 @@ restrictions followed by pullback. The lemma gives
\qquad
K_{V^*}P_V^{f-1}\le cP_V^f,
\qquad
- 0<c\le\frac{2(n-f)}{\delta}.
+ 0<c\le(2(n-f))/(\delta).
\]
Here the canonical term of the ambient vanishes because
-$K_X\sim0$. The volume window \eqref{j:volume-window} bounds
+$K_X\sim0$. The volume window (j:volume-window) bounds
$\delta$ above and away from zero, so these are uniform estimates.
The incidence point may be chosen over a very general point of $X$;
-then Lemma~\ref{j:very-general-images} makes $V^*$ non-uniruled
+then Lemma~j:very-general-images makes $V^*$ non-uniruled
and $K_{V^*}$ pseudo-effective, with $L_V$ of full nef dimension.
-Equation~\eqref{j:lower-big} gives bigness of $K_{V^*}+L_V$, and
-$P_V-rL_V$ is nef; Lemma~\ref{lemma:bounded-curves} contradicts
-$r\to\infty$.
-\end{proof}
+Equation~(j:lower-big) gives bigness of $K_{V^*}+L_V$, and
+$P_V-rL_V$ is nef; Lemma~lemma:bounded-curves contradicts
+$r\to∞$.
+
-\subsection{The projective bundle and its volume}
+## The projective bundle and its volume
We now seek a jet system whose order is much larger than the
scalar bound $4\rho$. The extra positivity comes from the fibres of
@@ -1416,51 +1416,51 @@ $X$ in its two summands. For a divisor $D$ on $X$, we write
$D_1,D_2$ for its pullback from the two factors of $X\times X$.
Set
-\begin{equation}\label{eq:bundle}
- Z=\PP\bigl(\OO_X(L)_1\oplus\OO_X(L)_2\bigr)
+
+ Z=\PP(\OO_X(L)_1 ⊕ \OO_X(L)_2)
\xrightarrow{\pi}X\times X,
\qquad\xi=c_1(\OO_Z(1)),
\qquad R=P_1+P_2+q\xi,
-\end{equation}
+
where $q$ is a positive integer to be fixed shortly. We use the
convention
\[
- \pi_*\OO_Z(a)=\Sym^a\bigl(\OO_X(L)_1\oplus\OO_X(L)_2\bigr)
+ \pi_*\OO_Z(a)=\Sym^a(\OO_X(L)_1 ⊕ \OO_X(L)_2)
\quad(a\ge0).
\]
The tautological class $\xi$ is nef, and $R$ is ample. As a
projective bundle over $X\times X$ with klt base, and with an
unramified flag or torus-part description on the complement of its
natural sections, $Z$ is klt. The two natural sections of $\pi$,
-called the \emph{axes}, have divisor classes $\xi-L_1$ and
+called the axes, have divisor classes $\xi-L_1$ and
$\xi-L_2$. The canonical-bundle formula for a projective bundle
gives
-\begin{equation}\label{j:axes-canonical}
- K_Z\sim-\bigl(\text{(sum of the two axes)}\bigr).
-\end{equation}
-The complement of the two axes is called the \emph{torus open}; a
+
+ K_Z\sim-((sum of the two axes)).
+
+The complement of the two axes is called the torus open; a
general point of $Z$ lies there.
The fibre of either projection $Z\to X$ over a smooth point is, up
to a constant one-dimensional factor,
\[
- Z_{\mathrm{sl}}=\PP(\OO_X\oplus\OO_X(L)),
- \qquad R_{\mathrm{sl}}=P+q\xi_{\mathrm{sl}}.
+ Z_{sl}=\PP(\OO_X ⊕ \OO_X(L)),
+ \qquad R_{sl}=P+q\xi_{sl}.
\]
-Its axes have classes $\xi_{\mathrm{sl}}$ and
-$\xi_{\mathrm{sl}}-L$, and its canonical divisor is the negative of
+Its axes have classes $\xi_{sl}$ and
+$\xi_{sl}-L$, and its canonical divisor is the negative of
their sum. The slice is projective and klt, so it satisfies the
singularity hypotheses needed to apply the base-component estimates
-of Section~\ref{sec:jet-tools}, using a small $\Q$-factorialization
+of Section~sec:jet-tools, using a small $\Q$-factorialization
if necessary.
Write $d=2n+1$. There are positive constants depending only on $n$
such that, for each fixed $q$ and all sufficiently small $t$,
-\begin{equation}\label{j:bundle-volumes}
- c_nq\epsilon^{2n}\le R^d\le C_nq\epsilon^{2n},
+
+ c_nqε^{2n}\le R^d\le C_nqε^{2n},
\qquad
- R_{\mathrm{sl}}^{n+1}\le C_{\mathrm{sl},n}q\epsilon^n.
-\end{equation}
+ R_{sl}^{n+1}\le C_{sl,n}qε^n.
+
Here is the computation. Expanding $R^d=(P_1+P_2+q\xi)^d$ in nef
classes and using
@@ -1469,9 +1469,9 @@ classes and using
\]
the term containing exactly one $\xi$ is
$(2n+1)\binom{2n}{n}q(P^n)^2$, which gives the lower bound in
-\eqref{j:bundle-volumes}. For the upper bound, the mixed
+(j:bundle-volumes). For the upper bound, the mixed
intersections of $L_i$ can be bounded by $P_i/r$ (since
-$\frac{1}{r}P_i=L_i+tA_i$, and $A_i$ can be absorbed by nonnegativity
+$(1)/(r)P_i=L_i+tA_i$, and $A_i$ can be absorbed by nonnegativity
of the mixed-degree coefficients); after factoring out $q$, the
remaining powers of $q/r$ are bounded by one once $r\ge q$, and
there are only finitely many monomials depending on $n$. The slice
@@ -1479,46 +1479,46 @@ calculation is identical, using
$\pi_*(\xi^j)=L^{j-1}$.
For a nef rational Cartier divisor $R$ and a smooth point $z$, its
-\emph{Seshadri constant} is
+Seshadri constant is
\[
- \varepsilon(R;z)=\inf_{C\ni z}\frac{R\cdot C}{\mult_z C},
+ ε(R;z)=\inf_{C\ni z}(R\cdot C)/(\mult_z C),
\]
where $C$ ranges over integral curves through $z$. A lower bound
for this constant controls the degree of curves through $z$ relative
to their multiplicity at $z$, which is what asymptotic Riemann--Roch
needs to make jets surjective.
-\begin{proposition}\label{prop:large-seshadri}
+
For a suitable fixed positive integer $q$, and all sufficiently
-small $t$ in \eqref{eq:scaling}, there is a smooth point $z_*\in Z$
+small $t$ in (eq:scaling), there is a smooth point $z_*\in Z$
in the torus open, lying over smooth points of both factors of
$X\times X$, such that
-\begin{equation}\label{j:large-seshadri}
- \varepsilon(R;z_*)>2n+2.
-\end{equation}
+
+ ε(R;z_*)>2n+2.
+
More generally, for any prescribed dense open subsets $U_1,U_2$ of
the smooth locus of $X$, the point may be chosen over $U_1\times U_2$.
In particular, these opens may be the isomorphism loci of fixed
resolutions of $X$.
-\end{proposition}
-\subsection{Proof of the large Seshadri constant}
+
+## Proof of the large Seshadri constant
We prove the proposition in the remainder of this section.
-\paragraph{Choices before the limit.} Choose a positive rational gap
+Choices before the limit.: Choose a positive rational gap
$\delta$ such that
-\begin{equation}\label{j:slice-gap}
- (2/\delta)^nC_{\mathrm{sl},n}\epsilon^n<1.
-\end{equation}
+
+ (2/\delta)^nC_{sl,n}ε^n<1.
+
Next choose $q$ so large that $d+1$ increasing levels of gap
$\delta$, with the lowest level strictly greater than $2n+2$, all lie
-strictly below $(c_nq\epsilon^{2n})^{1/d}$. Both $q,\delta$ are
+strictly below $(c_nqε^{2n})^{1/d}$. Both $q,\delta$ are
fixed before $t$ tends to zero. Fix the prescribed opens
$U_1,U_2$ if they are given.
-\paragraph{The reduction to a moving kernel.} Suppose
-\eqref{j:large-seshadri} fails throughout the smooth torus preimage
+The reduction to a moving kernel.: Suppose
+(j:large-seshadri) fails throughout the smooth torus preimage
of $U_1\times U_2$ for arbitrarily small $t$. Fix such a $t$.
Asymptotic Riemann--Roch and the elementary jet count at a smooth
@@ -1529,9 +1529,9 @@ length of the quotient by jets of order $<k\tau$ is
$\binom{k\tau+d}{d}\sim(k\tau)^d/d!$, and the number of sections is
$\chi(\OO(kR))\sim(kR)^d/d!=(k\tau)^d/d!\cdot (R/\tau)^d$, which
dominates as long as $\tau^d<R^d$. For our chosen levels
-$\tau_i<(c_nq\epsilon^{2n})^{1/d}\le (R^d)^{1/d}$, this holds.
+$\tau_i<(c_nqε^{2n})^{1/d}\le (R^d)^{1/d}$, this holds.
-Failure of \eqref{j:large-seshadri} supplies, through each very
+Failure of (j:large-seshadri) supplies, through each very
general point, a curve $C$ with $R\cdot C/\mult_zC$ smaller than the
lowest level $\tau_0$. Every section in the lowest kernel vanishes
identically on $C$: otherwise its intersection with $C$ would be at
@@ -1541,164 +1541,164 @@ Thus the lowest base locus has a positive-dimensional component
through the marked point.
Take $k$ divisible enough that $k\delta\ge4$ and apply
-Lemma~\ref{j:moving-kernel}. We obtain a family of proper integral
+Lemma~j:moving-kernel. We obtain a family of proper integral
components $V\subset Z$ whose incidence dominates $Z$. For general
members of dimension $f$, the estimates are
-\begin{equation}\label{j:total-component-bounds}
- \begin{split}
+
+
0<R^f\cdot V&\le(2/\delta)^{d-f}R^d,\\
K_{V^*}R^{f-1}&\le(K_Z+cR)|_V\cdot R^{f-1},
\qquad 0<c\le 2(d-f)/\delta.
- \end{split}
-\end{equation}
+
+
Here and below, classes on a resolution are understood by pullback.
The positivity $R^f\cdot V>0$ follows from ampleness of $R$. General
-members meet the torus open, so \eqref{j:axes-canonical} makes the
+members meet the torus open, so (j:axes-canonical) makes the
ambient canonical contribution nonpositive. All constants in these
-bounds are uniform once $q,\epsilon,\delta$ have been fixed.
+bounds are uniform once $q,ε,\delta$ have been fixed.
The remainder of the proof rules out all possible $(V,f)$. We first
constrain the two projections of $V$ to $X$.
-\subsection{Excluding positive-dimensional slice fibres}
+## Excluding positive-dimensional slice fibres
Consider either projection $V\to X$. Let $F$ be the component of a
-general fibre of $V\to\operatorname{im}(V\to X)$ through a general
+general fibre of $V\toim(V\to X)$ through a general
incidence point of $V$. Choose the incidence point in the opens
-given by Lemma~\ref{j:fiber-restriction}, over very general points of
+given by Lemma~j:fiber-restriction, over very general points of
both factors of $X$, and in the smooth torus open. We claim
-\begin{equation}\label{j:no-intermediate-fibre}
- \dim F\notin\{1,\ldots,n\}.
-\end{equation}
-Suppose, toward a contradiction, that $1\le f_{\mathrm{sl}}=\dim F
+ \dim F\notin\{1,...,n\}.
+
+
+Suppose, toward a contradiction, that $1\le f_{sl}=\dim F
\le n$. The ambient fibre of either projection $Z\to X$ over a
-smooth point is the slice $Z_{\mathrm{sl}}$ of dimension $n+1$.
-Lemma~\ref{j:fiber-restriction} gives a nonzero restricted subseries
+smooth point is the slice $Z_{sl}$ of dimension $n+1$.
+Lemma~j:fiber-restriction gives a nonzero restricted subseries
with $F$ an isolated base component, whose base ideal is contained in
$\mathcal I_F^h$ with $h=\lceil k\delta/2\rceil$. Generic smoothness
is imposed over the actual image of $V$, which is why a general fibre
through the incidence point was chosen; the chosen point is smooth
in the ambient slice by the very general choice. Applying
-Lemma~\ref{j:base-component} on that slice gives
-\begin{equation}\label{j:slice-bounds}
- \begin{split}
- 0<R_{\mathrm{sl}}^{f_{\mathrm{sl}}}\cdot F
- &\le(2/\delta)^{n+1-f_{\mathrm{sl}}}
- R_{\mathrm{sl}}^{n+1},\\
- K_{F^*} R_{\mathrm{sl}}^{f_{\mathrm{sl}}-1}
- &\le(K_{Z_{\mathrm{sl}}}+cR_{\mathrm{sl}})|_F
- R_{\mathrm{sl}}^{f_{\mathrm{sl}}-1},
- \end{split}
-\end{equation}
-with $c\le 2(n+1-f_{\mathrm{sl}})/\delta$. Cartier multiples are
+Lemma~j:base-component on that slice gives
+
+
+ 0<R_{sl}^{f_{sl}}\cdot F
+ &\le(2/\delta)^{n+1-f_{sl}}
+ R_{sl}^{n+1},\\
+ K_{F^*} R_{sl}^{f_{sl}-1}
+ &\le(K_{Z_{sl}}+cR_{sl})|_F
+ R_{sl}^{f_{sl}-1},
+
+
+with $c\le 2(n+1-f_{sl})/\delta$. Cartier multiples are
chosen before restriction, so the constant line from the fixed
factor causes no change in this calculation.
Let $W_0\subset X$ be the image of $F$ in the other factor. Three
cases arise.
-\paragraph{Case 1: $F$ is generically finite over a proper
-subvariety.} Suppose $F\to W_0$ is generically finite with
-$\dim W_0<n$. Let $P_Y=R_{\mathrm{sl}}|_{F^*}$, and let $L_Y$ be the
+Case 1: $F$ is generically finite over a proper
+subvariety.: Suppose $F\to W_0$ is generically finite with
+$\dim W_0<n$. Let $P_Y=R_{sl}|_{F^*}$, and let $L_Y$ be the
pullback of $L$ from $X$. The first inequality in
-\eqref{j:slice-bounds} gives a uniform positive volume bound; since
-$F$ meets neither axis generically, $K_{Z_{\mathrm{sl}}}|_F\le0$, so
+(j:slice-bounds) gives a uniform positive volume bound; since
+$F$ meets neither axis generically, $K_{Z_{sl}}|_F\le0$, so
the second inequality gives a uniform canonical ratio
-$K_{F^*}P_Y^{f_{\mathrm{sl}}-1}\le cP_Y^{f_{\mathrm{sl}}}$. Moreover
+$K_{F^*}P_Y^{f_{sl}-1}\le cP_Y^{f_{sl}}$. Moreover
$P_Y-rL_Y$ is nef, since
-$P_Y-rL_Y=(P|_{W_0}-rL|_{W_0})+q\xi_{\mathrm{sl}}|_{F^*}$ is the sum
+$P_Y-rL_Y=(P|_{W_0}-rL|_{W_0})+q\xi_{sl}|_{F^*}$ is the sum
of two nef classes. The image $W_0$ contains the very general
-projection of the incidence point, so \eqref{j:lower-big} applies and
-gives bigness of $K_{F^*}+L_Y$. Lemma~\ref{lemma:bounded-curves}
+projection of the incidence point, so (j:lower-big) applies and
+gives bigness of $K_{F^*}+L_Y$. Lemma~lemma:bounded-curves
excludes this case for large $r$.
-\paragraph{Case 2: $F$ is a projective-line bundle over a proper
-subvariety.} Suppose $F\to W_0$ is not generically finite. Its
+Case 2: $F$ is a projective-line bundle over a proper
+subvariety.: Suppose $F\to W_0$ is not generically finite. Its
general fibre has dimension one, and since the ambient projection
-$Z_{\mathrm{sl}}\to X$ is a $\PP^1$-bundle, the integral $F$ must
+$Z_{sl}\to X$ is a $\PP^1$-bundle, the integral $F$ must
coincide with the full $\PP^1$-bundle over $W_0$. Write
-$w=\dim W_0=f_{\mathrm{sl}}-1$.
+$w=\dim W_0=f_{sl}-1$.
If $w=0$, then $F$ is a projective-line fibre and
-$R_{\mathrm{sl}}\cdot F=q$. The first inequality in
-\eqref{j:slice-bounds}, together with \eqref{j:bundle-volumes} and
-\eqref{j:slice-gap}, gives
+$R_{sl}\cdot F=q$. The first inequality in
+(j:slice-bounds), together with (j:bundle-volumes) and
+(j:slice-gap), gives
\[
- q\le(2/\delta)^nR_{\mathrm{sl}}^{n+1}
- \le(2/\delta)^n C_{\mathrm{sl},n}q\epsilon^n<q,
+ q\le(2/\delta)^nR_{sl}^{n+1}
+ \le(2/\delta)^n C_{sl,n}qε^n<q,
\]
impossible.
Suppose $w>0$. Resolve $W_0$ to $W_0^*$, and take as a resolution
of $F$ the induced projective bundle $p_0\colon F^*\to W_0^*$. Its
relative canonical divisor is $-2\xi+p_0^*L$, exactly the
-restriction of $K_{Z_{\mathrm{sl}}}$. Cancelling this term in
-\eqref{j:slice-bounds} gives
-\begin{equation}\label{j:saturated-canonical}
- p_0^*K_{W_0^*}R_{\mathrm{sl}}^w
- \le cR_{\mathrm{sl}}^{w+1}\cdot F.
-\end{equation}
+restriction of $K_{Z_{sl}}$. Cancelling this term in
+(j:slice-bounds) gives
+
+ p_0^*K_{W_0^*}R_{sl}^w
+ \le cR_{sl}^{w+1}\cdot F.
+
The divisor $K_{W_0^*}$ is pseudo-effective by the very general point
-condition. On expanding the left side of \eqref{j:saturated-canonical}
+condition. On expanding the left side of (j:saturated-canonical)
and pushing down, its term with exactly one $\xi$ is
$wqK_{W_0^*}P^{w-1}$; every other nonzero term is nonnegative, since
it pairs the pseudo-effective $K_{W_0^*}$ with nef classes. On the
-right, the expansion used for \eqref{j:bundle-volumes} gives
-\begin{equation}\label{j:saturated-volume}
+right, the expansion used for (j:bundle-volumes) gives
+
(w+1)qP^w\cdot W_0
- \le R_{\mathrm{sl}}^{w+1}\cdot F
+ \le R_{sl}^{w+1}\cdot F
\le C'_n qP^w\cdot W_0,
-\end{equation}
+
using $r\ge q$. The leading coefficient in the lower bound is the
term with exactly one $\xi$, as before. The first inequality in
-\eqref{j:slice-bounds} then bounds $0<P^w\cdot W_0$ uniformly from
-above, and \eqref{j:saturated-volume}'s leading coefficient bounds it
-from below. Equations \eqref{j:saturated-canonical}--\eqref{j:saturated-volume}
+(j:slice-bounds) then bounds $0<P^w\cdot W_0$ uniformly from
+above, and (j:saturated-volume)'s leading coefficient bounds it
+from below. Equations (j:saturated-canonical)--(j:saturated-volume)
together give
\[
- K_{W_0^*}P^{w-1}\le\frac{cC'_n}{w}P^w\cdot W_0.
+ K_{W_0^*}P^{w-1}\le(cC'_n)/(w)P^w\cdot W_0.
\]
-Now apply \eqref{j:lower-big} on $W_0^*$, with the pullback of $P$ as
+Now apply (j:lower-big) on $W_0^*$, with the pullback of $P$ as
the testing class and $L$ as the full-nef-dimension divisor.
$W_0$ contains a very general point of $X$, so the required bigness
-and full nef dimension hold by Lemma~\ref{j:very-general-images}.
-Lemma~\ref{lemma:bounded-curves} excludes this case for large $r$.
+and full nef dimension hold by Lemma~j:very-general-images.
+Lemma~lemma:bounded-curves excludes this case for large $r$.
-\paragraph{Case 3: $F$ is a multisection over $X$.} The only
-remaining possibility is that $f_{\mathrm{sl}}=n$ and $F\to X$ is
+Case 3: $F$ is a multisection over $X$.: The only
+remaining possibility is that $f_{sl}=n$ and $F\to X$ is
generically finite of degree $e>0$. Intersect $F$ with each of the
two Cartier axes and push the resulting cycles down to $X$. This
gives effective integral Weil divisors $D_1,D_2$. The axes do not
contain $F$, and their classes differ by $L$; taking the norm of the
associated rational section, or equivalently its projection, gives
-\begin{equation}\label{j:signed-L}
+
D_1-D_2\sim\pm eL.
-\end{equation}
-For either axis $S$, the classes $R_{\mathrm{sl}}-qS$ and
-$R_{\mathrm{sl}}-P$ are nef. Intersecting with the effective cycle
+
+For either axis $S$, the classes $R_{sl}-qS$ and
+$R_{sl}-P$ are nef. Intersecting with the effective cycle
$S\cdot F$ and then using nefness on $F$ yields
-\begin{equation}\label{j:axis-degrees}
- qD_iP^{n-1}\le qS\cdot F\cdot R_{\mathrm{sl}}^{n-1}
- \le R_{\mathrm{sl}}^n\cdot F.
-\end{equation}
+
+ qD_iP^{n-1}\le qS\cdot F\cdot R_{sl}^{n-1}
+ \le R_{sl}^n\cdot F.
+
Thus both axis-image degrees are uniformly bounded.
Set $G=D_1+D_2$. It is $\Q$-Cartier because $X$ is $\Q$-factorial.
Choose $b>0$ rational sufficiently small that $(X,bG)$ is klt. We
claim
-\begin{equation}\label{j:positive-boundary-kappa}
+
\kappa(K_X+bG)>0.
-\end{equation}
+
The Iitaka dimension is at least zero, since $K_X+bG\sim_\Q bG$.
-If it were zero, Theorems~\ref{input:models} and~\ref{input:abundance}
+If it were zero, Theorems~input:models and~input:abundance
would give an ordinary MMP ending at a good model $Y$ whose
transformed adjoint is rationally linearly trivial. The transform of
$K_X$ is then linearly trivial as well, so the effective transform
of $G$ must vanish: every component of $G$ has been contracted.
On a common resolution $a\colon U\to X$ and $b_Y\colon U\to Y$,
-relation \eqref{j:signed-L} then expresses $a^*L$ up to rational
+relation (j:signed-L) then expresses $a^*L$ up to rational
linear equivalence as a signed $b_Y$-exceptional divisor $E$.
Divisors exceptional over $X$ remain exceptional over $Y$, because
the MMP extracts no divisors. Nefness of $a^*L$ and the negativity
@@ -1706,12 +1706,12 @@ lemma give $E\le0$. Its class is also pseudo-effective.
A nonzero anti-effective divisor has negative intersection with a
sufficiently high power of an ample divisor, so $E=0$. This
contradicts full nef dimension of $L$, and thus proves
-\eqref{j:positive-boundary-kappa}.
+(j:positive-boundary-kappa).
The already proved positive-Iitaka-dimension case
-(Proposition~\ref{prop:full-positive}) now gives
+(Proposition~prop:full-positive) now gives
\[
- K_X+bG+L\ \text{big}.
+ K_X+bG+L\ big.
\]
Its full-positivity hypothesis holds because $L$ has full nef
dimension and $K_X+bG$ is pseudo-effective.
@@ -1719,7 +1719,7 @@ dimension and $K_X+bG$ is pseudo-effective.
Take a log resolution $a\colon Y\to X$ of $G$, and let $D_0$ be the
reduced simple normal crossing divisor consisting of the strict
support of $G$ together with every exceptional divisor. Decrease $b$
-if necessary; Proposition~\ref{prop:full-positive} still applies.
+if necessary; Proposition~prop:full-positive still applies.
Terminality of $X$ and the support of $a^*G$ then give
\[
K_Y+D_0+a^*L\ \ge\ a^*(K_X+bG+L)
@@ -1733,41 +1733,41 @@ is big.
Test on $Y$ with $a^*P$. Exceptional divisors contribute zero to
intersection with $(a^*P)^{n-1}$, and $K_X\sim0$. The degree of the
reduced strict support is no larger than $GP^{n-1}$, which is bounded
-by \eqref{j:axis-degrees}; also $LP^{n-1}\le P^n/r$. Since
-\eqref{j:volume-window} bounds $P^n$ away from zero, we obtain a
+by (j:axis-degrees); also $LP^{n-1}\le P^n/r$. Since
+(j:volume-window) bounds $P^n$ away from zero, we obtain a
uniform constant $C$ such that
\[
(K_Y+D_0+a^*L)(a^*P)^{n-1}\le C(a^*P)^n.
\]
-Lemma~\ref{lemma:bounded-curves}, now with its reduced boundary
+Lemma~lemma:bounded-curves, now with its reduced boundary
$D_0$, gives the final contradiction. This proves
-\eqref{j:no-intermediate-fibre}.
+(j:no-intermediate-fibre).
-\subsection{Dimension reduction for moving components}
+## Dimension reduction for moving components
Thus a general moving component $V$ has fibre dimension either zero
or $n+1$ over its image in either factor. Fibre dimension $n+1$
would make $V$ the full inverse image of its image in that factor.
Since $V$ is proper, the image has dimension at most $n-1$, so the
other projection would have fibre dimension between $1$ and $n$,
-already excluded by \eqref{j:no-intermediate-fibre}. Both
+already excluded by (j:no-intermediate-fibre). Both
projections of $V$ are therefore generically finite onto their
images, and $\dim V\le n$. If $\dim V<n$, the bounds
-\eqref{j:total-component-bounds}, the nef class $R-rL_i$, and
-\eqref{j:lower-big} again contradict Lemma~\ref{lemma:bounded-curves}.
+(j:total-component-bounds), the nef class $R-rL_i$, and
+(j:lower-big) again contradict Lemma~lemma:bounded-curves.
Only $n$-dimensional correspondences, dominant over both factors of
$X\times X$, remain for further analysis.
-\subsection{Correspondences and their branch divisors}
+## Correspondences and their branch divisors
For the remaining $V$: dimension $n$, dominant over both factors of
-$X\times X$. Equation~\eqref{j:total-component-bounds} gives uniform
+$X\times X$. Equation~(j:total-component-bounds) gives uniform
bounds
-\begin{equation}\label{j:correspondence-bounds}
+
R^n\cdot V\le C,
\qquad
K_{V^*}R^{n-1}\le cR^n\cdot V\le C,
-\end{equation}
+
after increasing $C$. Each projection has bounded degree:
$R-P_i$ is nef, and $P^n$ is bounded below.
@@ -1776,8 +1776,8 @@ family and shrink the parameter space so that the resolutions form a
smooth projective family with integral fibres and morphisms to $X$ in
both slots. The dominant incidence and all numerical bounds
persist. Finite extensions of the parameter space are allowed.
-Over $X_{\mathrm{sm}}$, name the components of the relative Jacobian
-divisors as in Lemma~\ref{j:finite-covers}. Consider a component
+Over $X_{sm}$, name the components of the relative Jacobian
+divisors as in Lemma~j:finite-covers. Consider a component
whose image on a general member is a divisor $J\subset X$.
Such divisors $J$ cannot sweep $X$. If they did, choose a general
@@ -1786,22 +1786,22 @@ divisor over it on the corresponding $V^*$. Pullback of a
canonical form from $X$ gives an effective canonical divisor on
$V^*$. Regularity follows by factoring through a resolution of $X$
and using its canonical singularities; the coefficient of $E_J$ is
-positive because its image meets $X_{\mathrm{sm}}$ generically. The
-difference $R-P_i$ is nef, so \eqref{j:correspondence-bounds} and the
+positive because its image meets $X_{sm}$ generically. The
+difference $R-P_i$ is nef, so (j:correspondence-bounds) and the
projection formula bound
\[
0<d_J:=P^{n-1}\cdot J\le C.
\]
-On a resolution $J^*$, Lemma~\ref{j:subadjunction}, applied to
+On a resolution $J^*$, Lemma~j:subadjunction, applied to
$(X,J)$ at the generic point of $J$, gives
-\begin{equation}\label{j:branch-canonical}
- \begin{split}
+
+
K_{J^*}P^{n-2}
&\le(K_X+J)\cdot J\cdot P^{n-2}\\
&=J^2P^{n-2}
\le\frac{(JP^{n-1})^2}{P^n}\le C' d_J.
- \end{split}
-\end{equation}
+
+
The pair is lc at that generic point because $X$ is normal and hence
smooth there. The middle inequality is the Hodge index inequality
on a resolution, obtained by ample approximation of the pulled-back
@@ -1809,13 +1809,13 @@ testing class; it does not require $J$ to be nef. The final constant
is uniform by the volume lower bound and the bound for $d_J$.
Since $J$ sweeps $X$, its resolution satisfies the very general point
-conditions used in \eqref{j:lower-big}. With testing class
+conditions used in (j:lower-big). With testing class
$P|_{J^*}$, whose top self-intersection is $d_J$,
-Equation~\eqref{j:branch-canonical} gives precisely the hypotheses
-of Lemma~\ref{lemma:bounded-curves} in dimension $n-1$. This
+Equation~(j:branch-canonical) gives precisely the hypotheses
+of Lemma~lemma:bounded-curves in dimension $n-1$. This
contradiction excludes sweeping branch divisors.
-Lemma~\ref{j:finite-covers} now gives, for each slot and this fixed
+Lemma~j:finite-covers now gives, for each slot and this fixed
$t$, only finitely many possible finite normal Stein covers of $X$,
up to isomorphism over $X$. The point of the preceding step is that
degree bounds alone would not give this conclusion: non-sweeping
@@ -1823,7 +1823,7 @@ branch images provide a fixed smooth branch complement for the
family. Neither that complement nor the resulting finite lists need
be uniform in $t$.
-\subsection{Fixed covers and an abelian model}
+## Fixed covers and an abelian model
Each $V^*$ maps birationally to both of its normal Stein covers, and
hence determines a graph of a birational map between a pair of
@@ -1831,7 +1831,7 @@ covers drawn from the two finite lists. We explain why some fixed
pair has a family of graphs dominating its product.
Restrict the finite set of pairs to those realized by at least one
-original member $V^*$ satisfying \eqref{j:correspondence-bounds}. The
+original member $V^*$ satisfying (j:correspondence-bounds). The
original incidence dominates $Z$, so the projected incidences for
these pairs cover a dense open of $X\times X$. For a fixed pair,
graphs of birational maps lie in countably many Hilbert schemes.
@@ -1869,18 +1869,18 @@ used.
This structural conclusion depends only on the two covers. We may
therefore return to an original member $V^*$ of this pair, which
-retains \eqref{j:correspondence-bounds}, and resolve further to
+retains (j:correspondence-bounds), and resolve further to
obtain a birational morphism
\[
- p_0\colon V^*\longrightarrow A_0
+ p_0\colon V^*→ A_0
\]
to an abelian variety. Higher resolutions do not change the
canonical intersection with the pulled-back $(n-1)$-st power of $R$.
Let $L_0$ be the pullback of $L$ from either slot. It is nef,
Cartier, and of full nef dimension. We claim
-\begin{equation}\label{j:abelian-big}
- K_{V^*}+L_0\text{ is big}.
-\end{equation}
+
+ K_{V^*}+L_0 is big.
+
The pushforward $N_0=(p_0)_*L_0$ is pseudo-effective on $A_0$, and
is therefore nef; on an abelian variety, translations make the
@@ -1910,44 +1910,44 @@ $E_K-uE_0\ge0$. Then
K_{V^*}+uL_0\sim_\Q u p_0^*N_0+(E_K-uE_0)
\]
is big. Adding the nef divisor $(1-u)L_0$ proves
-\eqref{j:abelian-big}.
+(j:abelian-big).
-Now apply Lemma~\ref{lemma:bounded-curves} in dimension $n$ on $V^*$,
+Now apply Lemma~lemma:bounded-curves in dimension $n$ on $V^*$,
with testing class $R$ and empty boundary. The bounds
-\eqref{j:correspondence-bounds} give both the top intersection bound
+(j:correspondence-bounds) give both the top intersection bound
and the canonical ratio, while $R-rL_0$ is nef and
-\eqref{j:abelian-big} supplies bigness. This final contradiction
+(j:abelian-big) supplies bigness. This final contradiction
excludes the correspondence case.
-\subsection{Conclusion of the proof of Proposition~\ref{prop:large-seshadri}}
+\subsection{Conclusion of the proof of Proposition~prop:large-seshadri}
The alternatives for every moving base component have been
exhausted: first its slice fibres, then components of dimension less
than $n$, then sweeping branch divisors, and finally the fixed
covers. In each case the contradiction comes from
-Lemma~\ref{lemma:bounded-curves} with constants uniform in $t$. The
+Lemma~lemma:bounded-curves with constants uniform in $t$. The
finite-cover lists were used only for one fixed $t$, and their sizes
do not enter the degree bounds. Thus failures of
-\eqref{j:large-seshadri} cannot occur along a sequence $t\to0$. This
+(j:large-seshadri) cannot occur along a sequence $t\to0$. This
proves the proposition, including the additional open conditions on
the chosen point.
We have now constructed, under the hypothetical failure of
-Theorem~\ref{thm:terminal-big}, the scalar estimate of
-Lemma~\ref{lemma:scalar-bound} and the projective-bundle estimate of
-Proposition~\ref{prop:large-seshadri}. Section~\ref{sec:frobenius}
+Theorem~thm:terminal-big, the scalar estimate of
+Lemma~lemma:scalar-bound and the projective-bundle estimate of
+Proposition~prop:large-seshadri. Section~sec:frobenius
fixes one sufficiently small value of $t$ and places these data in
the finite-data Frobenius comparison.
-\section{Frobenius incompatibility}
-\label{sec:frobenius}
+# Frobenius incompatibility
+
-The scalar estimate of Lemma~\ref{lemma:scalar-bound} and the
-projective-bundle estimate of Proposition~\ref{prop:large-seshadri}
+The scalar estimate of Lemma~lemma:scalar-bound and the
+projective-bundle estimate of Proposition~prop:large-seshadri
are incompatible. Their incompatibility is expressed by the following
theorem about fixed complex data, whose proof occupies this section.
The reduction to positive characteristic is managed so that every
-constant is uniform in the residual characteristic $p\to\infty$.
+constant is uniform in the residual characteristic $p\to∞$.
The shape of the argument is as follows. Two orders of vanishing are
compared for one determinant. Ordinary jets make the determinant
@@ -1957,41 +1957,41 @@ at each diagonal point. A fixed movable curve on a point blowup
bounds the vanishing of every section of each determinant factor, and
gives the opposite inequality.
-\subsection{The finite-data statement}
+## The finite-data statement
For a vector bundle on a smooth projective curve, write
$\mu_{\min}$ and $\mu_{\max}$ for its minimum and maximum
Harder--Narasimhan slopes, i.e.\ degrees divided by ranks of the
extremal sub- or quotient bundles. For a line bundle on a smooth
-variety, \emph{generate jets through order $a$ at a point} means the
+variety, generate jets through order $a$ at a point means the
evaluation map from sections to the quotient of the local ring by the
$(a+1)$-st power of the maximal ideal is surjective.
-\begin{theorem}[Finite-data Frobenius incompatibility]
-\label{thm:finite-data}
+[Finite-data Frobenius incompatibility]
+
Let $W$ be a smooth connected projective complex variety of dimension
$n\ge2$. Let $D,H$ be rational Cartier divisors with $H$ ample, let
$S$ be an integral Cartier divisor, and fix an integer $q>0$ and real
-numbers $r>1$, $\epsilon>0$. Suppose
-\begin{equation}\label{fr:intersection-bounds}
- \begin{gathered}
- H^n\le(2\epsilon)^n,\qquad
+numbers $r>1$, $ε>0$. Suppose
+
+
+ H^n\le(2ε)^n,\qquad
K_WH^{n-1}\le 2 H^n/r,\qquad
(2D+qS)H^{n-1}\le 4H^n,\\
- (14\epsilon)^n<\tfrac14,\qquad 80\epsilon<\tfrac34.
- \end{gathered}
-\end{equation}
+ (14ε)^n<\tfrac14,\qquad 80ε<\tfrac34.
+
+
Assume that the following fixed data exist.
-\begin{enumerate}
-\renewcommand{\labelenumi}{\textup{(\roman{enumi})}}
+
+\renewcommand{\labelenumi}{(\roman{enumi)}}
\item A smooth integral complete-intersection flag from $W$ to
a curve $C$, whose successive divisor classes are $h_i=l_iH$ for
fixed positive integers $l_i$ for which $l_iH$ is Cartier, with
-$\mu_{\min}(\Omega_W^1|_C)\ge0$.
+$\mu_{\min}(Ω_W^1|_C)\ge0$.
\item On the projective bundle
\[
- Z=\PP\bigl(\mathcal O_W(S)_1\oplus\mathcal O_W(S)_2\bigr)
- \longrightarrow W\times W,
+ Z=\PP(\mathcal O_W(S)_1 ⊕ \mathcal O_W(S)_2)
+ → W\times W,
\qquad \xi=c_1(\mathcal O_Z(1)),
\qquad M=D_1+D_2+q\xi,
\]
@@ -2006,19 +2006,19 @@ through order $(2n+2)k_0$ at $z_*$.
$\pi_x\colon\widehat W\to W$ with exceptional divisor $J$, and a
curve class
\[
- \gamma=f_*(A_1\cdots A_{n-1}),
+ \gamma=f_*(A_1... A_{n-1}),
\]
where $f\colon V\to\widehat W$ is one fixed birational morphism,
$V$ is smooth integral projective, and the $A_i$ are ample integral
Cartier divisors, such that $J\cdot\gamma>0$ and
-\begin{equation}\label{fr:movable-input}
+
\pi_x^*(D+K_W/2+\lambda S)\cdot\gamma
- \le 40\epsilon\, J\cdot\gamma\quad(0\le\lambda\le q).
-\end{equation}
+ \le 40ε\, J\cdot\gamma\quad(0\le\lambda\le q).
+
It suffices to check the two endpoints in this affine inequality.
-\end{enumerate}
+
These data cannot coexist.
-\end{theorem}
+
Note that $x$ is independent of the two base coordinates of $z_*$.
No nefness is required of $D$, $S$, or $K_W$, and the theorem does
@@ -2026,20 +2026,20 @@ not assume $K_W$ pseudo-effective. Its hypotheses specify the flag
and the movable curve class themselves. All these data will be
fixed before the residual characteristic tends to infinity.
-\subsection{Reduction to positive characteristic}
-\label{fr:reduction}
+## Reduction to positive characteristic
-We begin the proof of Theorem~\ref{thm:finite-data} by spreading its
+
+We begin the proof of Theorem~thm:finite-data by spreading its
hypotheses to finitely generated models and converting the fixed
ordinary jets into jets of order proportional to the residual
characteristic.
-Put $\Lambda=\prod_{i=1}^{n-1}l_i$. The complete-intersection flag
+Put $Λ=\prod_{i=1}^{n-1}l_i$. The complete-intersection flag
satisfies
-\begin{equation}\label{fr:flag-degrees}
- [C]=\Lambda H^{n-1},
- \qquad h_i\cdot C=l_i\Lambda H^n.
-\end{equation}
+
+ [C]=Λ H^{n-1},
+ \qquad h_i\cdot C=l_iΛ H^n.
+
Choose one sufficiently ample integral Cartier divisor $T_0$ so that
every $T_0+aS$, for integers $0\le a\le(k_0-1)q$, has a section
nonvanishing at each of the two base coordinates of $z_*$. Fix these
@@ -2056,7 +2056,7 @@ conditions persist. Birationality is preserved by spreading an
isomorphism between dense opens. All displayed intersection numbers
are constant on the spread.
-We also preserve $\mu_{\min}(\Omega_W^1|_C)\ge 0$. One route is to
+We also preserve $\mu_{\min}(Ω_W^1|_C)\ge 0$. One route is to
spread the characteristic-zero Harder--Narasimhan filtration, make its
factors locally free, and use openness of their semistability on
curves; the degrees and ranks are fixed. Equivalently, a single
@@ -2078,9 +2078,9 @@ birational morphism and intersect with the fixed ample divisors.
This tests effective divisors on the reduction itself. No
specialization of the characteristic-zero effective cone is used.
-\subsection{Positive-characteristic jets}
+## Positive-characteristic jets
+
-\begin{lemma}\label{fr:positive-characteristic-jets}
For a sufficiently large residual characteristic $p$, set
\[
N_p=\lfloor p/k_0\rfloor,
@@ -2090,9 +2090,9 @@ Then the sections of $pq\xi+B_{p,1}+B_{p,2}$ generate jets through
order $(2n+2)k_0N_p$ at $z_*$. In particular, they surject onto the
quotient of the local ring by the $p$-th powers of all regular
parameters.
-\end{lemma}
-\begin{proof}
+
+Proof.
Multiply $N_p$ copies of the fixed jet system. Every monomial of
total degree at most $(2n+2)k_0N_p$ is a product of $N_p$ monomials
of degree at most $(2n+2)k_0$. Taking sections with those leading
@@ -2124,10 +2124,10 @@ parameters has top total degree $(2n+1)(p-1)$. For large $p$, this
is at most $(2n+2)k_0N_p$. The underlying ordinary/Frobenius ideal
comparison is also recorded in
\cite[proof of Proposition~2.12, Equation~(2.8)]{MustataSchwede}.
-\end{proof}
-\subsection{Full rank from the two-slot jets}
-\label{fr:full-rank}
+
+## Full rank from the two-slot jets
+
Let $F\colon W\to W'$ be the relative Frobenius to the base-field
twist. It is finite flat because $W$ is smooth. Write $S'$ for the
@@ -2140,65 +2140,65 @@ twist of $S$, so $F^*S'=pS$, and define
For integers $0\le a\le q$, put
$U_a=H^0(W,\OO_W(B_p+paS))$. The projection formula gives an
evaluation map on $W'\times W'$:
-\begin{equation}\label{fr:two-slot-evaluation}
- \bigoplus_{\substack{a+b=q\\a,b\ge0}}
- U_a\otimes U_b\otimes
+
+ ⊕_{\substack{a+b=q\\a,b\ge0}}
+ U_a ⊗ U_b ⊗
\OO(-aS'_1-bS'_2)
- \longrightarrow \mathcal E\boxtimes\mathcal E.
-\end{equation}
+ → \mathcal E\boxtimes\mathcal E.
-\begin{lemma}\label{fr:generic-rank}
-The map in \eqref{fr:two-slot-evaluation} has generic rank $R$.
-\end{lemma}
-\begin{proof}
+
+The map in (fr:two-slot-evaluation) has generic rank $R$.
+
+
+Proof.
Trivialize the two summands defining $Z$ near the selected base
points. Let $z$ be the torus coordinate, whose value at $z_*$ is
$z_0\ne0$. By the projective-bundle formula, the sections in
-Lemma~\ref{fr:positive-characteristic-jets} are weight polynomials
+Lemma~fr:positive-characteristic-jets are weight polynomials
in $z$ with weight-$j$ coefficients in
\[
- H^0(W,B_p+jS)\otimes H^0(W,B_p+(pq-j)S),
+ H^0(W,B_p+jS) ⊗ H^0(W,B_p+(pq-j)S),
\qquad 0\le j\le pq.
\]
Let $A_*$ be the tensor product of the local algebras of the two base
Frobenius fibres. The local quotient in
-Lemma~\ref{fr:positive-characteristic-jets} is
+Lemma~fr:positive-characteristic-jets is
\[
A_*[z]/(z^p-z_0^p),
\]
-free over $A_*$ with basis $1,z,\ldots,z^{p-1}$. Project to the
+free over $A_*$ with basis $1,z,...,z^{p-1}$. Project to the
coefficient of $1$ in this basis. Exactly the weights $j=pa$
survive, with multipliers $z_0^{pa}$. Their coefficients are the
values of the corresponding summands of
-\eqref{fr:two-slot-evaluation}, in the chosen frames and the
+(fr:two-slot-evaluation), in the chosen frames and the
induced twisted frames. As the full jet map is surjective, these
coefficients span $A_*$, of dimension $p^{2n}=R$. Thus the
evaluation has full rank at this pair of base points, and hence
generically. The calculation uses a basis over the possibly
nonreduced algebra $A_*$, so reducedness of the Frobenius fibres is
unnecessary.
-\end{proof}
+
The jet calculation at $z_*$ proves generic full rank; the diagonal
estimate below will be tested at $(x',x')$, from the separately
fixed point $x$.
-\subsection{Sections on the Frobenius diagonal}
-\label{fr:filtration}
+## Sections on the Frobenius diagonal
+
The Frobenius fiber product has the cartesian square
-\begin{equation}\label{fr:frobenius-square}
-\begin{tikzcd}
- \Delta_F=W\times_{W'}W \arrow[r,"\mathrm{pr}_2"] \arrow[d,"\mathrm{pr}_1"'] & W \arrow[d,"F"]\\
+
+
+ \Delta_F=W\times_{W'}W \arrow[r,"pr_2"] \arrow[d,"pr_1"'] & W \arrow[d,"F"]\\
W \arrow[r,"F"'] & W'.
-\end{tikzcd}
-\end{equation}
+
+
Its reduced subscheme is the diagonal $W$. Set
\[
\mathcal L_p=\OO_{\Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
\]
-Write $h=F\circ\mathrm{pr}_1=F\circ\mathrm{pr}_2\colon
+Write $h=F\circpr_1=F\circpr_2\colon
\Delta_F\to W'$. On this fiber product,
\[
\OO(pS_1)\simeq h^*\OO_{W'}(S')\simeq\OO(pS_2).
@@ -2209,10 +2209,10 @@ Consequently every pair of weights $a+b=q$ gives the same line bundle:
\simeq\mathcal L_p.
\]
On the diagonal of $W'\times W'$, every source twist of
-\eqref{fr:two-slot-evaluation} becomes $-qS'$. Finite base change
+(fr:two-slot-evaluation) becomes $-qS'$. Finite base change
and the projection formula identify the twisted target as
\[
- (\mathcal E\otimes\mathcal E)(qS')\simeq h_*\mathcal L_p.
+ (\mathcal E ⊗ \mathcal E)(qS')\simeq h_*\mathcal L_p.
\]
The product sections defining the columns restrict to global
sections of this one bundle $\mathcal L_p$. Their values at a
@@ -2220,201 +2220,201 @@ diagonal point therefore lie in the image of
$H^0(\Delta_F,\mathcal L_p)$ in the corresponding fibre of
$h_*\mathcal L_p$. This proves that the rank at every diagonal point
is at most
-\begin{equation}\label{fr:diagonal-rank-upper}
+
h^0(\Delta_F,\mathcal L_p).
-\end{equation}
+
For a rank-$n$ bundle $V$ in characteristic $p$, denote by $A_j(V)$
the degree-$j$ part of its symmetric algebra modulo the $p$-th powers
of local generators. This construction is independent of frame: the
$p$-th power of a linear combination is the sum of the $p$-th powers
in characteristic $p$. Each $A_j(V)$ is locally free and a quotient
-of $V^{\otimes j}$. Set
+of $V^{ ⊗ j}$. Set
\[
u=n(p-1),\qquad e_j=\rk A_j(V),\qquad
\sum_{j=0}^u e_j=p^n.
\]
Filtering $\mathcal L_p$ by powers of the ideal of the reduced
diagonal in $\Delta_F$ gives the graded bundles
-\begin{equation}\label{fr:diagonal-graded}
- \mathcal G_j=\OO_W(2B_p+pqS)\otimes A_j(\Omega_W^1),
+
+ \mathcal G_j=\OO_W(2B_p+pqS) ⊗ A_j(Ω_W^1),
\qquad 0\le j\le u.
-\end{equation}
+
Indeed, in smooth local coordinates, the algebra is generated by
parameter differences with their $p$-th powers zero, and its
-degree-one conormal is $\Omega_W^1$. This coordinate calculation can
+degree-one conormal is $Ω_W^1$. This coordinate calculation can
be performed \'etale locally or in completions and identifies the
-global associated graded with \eqref{fr:diagonal-graded}. This is
+global associated graded with (fr:diagonal-graded). This is
the diagonal ideal form of the canonical Frobenius filtration
\cite[Corollary~3.5]{KS}. The same calculation, using only the
bundle from the second factor, filters $F^*\mathcal E$ with pieces
\cite[Theorem~3.7]{Sun}
-\begin{equation}\label{fr:Frob-pullback-graded}
- \OO_W(B_p)\otimes A_j(\Omega_W^1).
-\end{equation}
+
+ \OO_W(B_p) ⊗ A_j(Ω_W^1).
+
Complementary multiplication is a perfect pairing:
-\begin{equation}\label{fr:socle-pairing}
- A_{u-j}(V)\simeq A_j(V)^\vee\otimes(\det V)^{p-1}.
-\end{equation}
+
+ A_{u-j}(V)\simeq A_j(V)^\vee ⊗ (\det V)^{p-1}.
+
On monomials, each exponent vector is paired with its complement to
-$(p-1,\ldots,p-1)$. The top line transforms by the character
+$(p-1,...,p-1)$. The top line transforms by the character
$(\det V)^{p-1}$: this is immediate on diagonal matrices and hence
-identifies the character of $\mathrm{GL}_n$ on that line.
+identifies the character of $GL_n$ on that line.
+
+## A uniform bound for the diagonal rank
-\subsection{A uniform bound for the diagonal rank}
-\label{fr:rank-bound}
We bound the sections of every graded bundle in
-\eqref{fr:diagonal-graded} by its rank times the same scalar
+(fr:diagonal-graded) by its rank times the same scalar
polynomial in $p$. This uniformity lets us sum the ranks of the
graded pieces without an additional factor for their number. We
begin with a slope bound on the fixed curve; here slope means
ordinary degree divided by rank.
-\begin{lemma}\label{fr:truncated-slope}
+
There is a constant $c\ge0$, independent of the sufficiently large
residual characteristic $p$ and of $j$, such that for
-$V=\Omega_W^1|_C$ one has
+$V=Ω_W^1|_C$ one has
\[
\mu_{\max}(A_j(V))\le(p-1)K_W\cdot C+nc.
\]
-\end{lemma}
-\begin{proof}
+
+Proof.
Write $F_C$ for the absolute Frobenius of $C$. Langer's instability
estimate \cite[Corollary~2.5]{Langer} gives
-\begin{equation}\label{fr:Langer-min}
- L_{\min}(V):=\lim_{e\to\infty}p^{-e}\mu_{\min}(F_C^{e*}V)
- \ge-\frac{c}{p-1}.
-\end{equation}
+
+ L_{\min}(V):=\lim_{e\to∞}p^{-e}\mu_{\min}(F_C^{e*}V)
+ \ge-(c)/(p-1).
+
To make the uniformity explicit, let $g$ be the fixed genus of $C$.
One can choose a nef line bundle $A_C$ of degree at most
-$d_g=\max\{4g-2,0\}$ such that $T_C\otimes A_C$ is globally
+$d_g=\max\{4g-2,0\}$ such that $T_C ⊗ A_C$ is globally
generated; for $g\le1$, take $A_C=\OO_C$. Langer's bound is
\[
\mu_{\min}(V)-L_{\min}(V)
- \le\frac{(\rk V-1)\deg A_C}{p-1}.
+ \le((\rk V-1)\deg A_C)/(p-1).
\]
Since $\rk V=n$ and $\mu_{\min}(V)\ge0$, we may take
$c=(n-1)d_g$, independently of the reduction.
We also need, for every integer $i\ge0$,
-\begin{equation}\label{fr:tensor-min}
- \mu_{\min}(V^{\otimes i})\ge iL_{\min}(V).
-\end{equation}
+
+ \mu_{\min}(V^{ ⊗ i})\ge iL_{\min}(V).
+
Fix $p$ and $i$. Choose a line bundle $A_e$ of degree
$-\mu_{\min}(F_C^{e*}V)+O(1)$, rounded up with a fixed
genus-dependent additive constant, so that
-$F_C^{e*}V\otimes A_e$ is globally generated. This follows from
+$F_C^{e*}V ⊗ A_e$ is globally generated. This follows from
Serre duality: after subtracting any point, its minimum slope can
be made greater than $2g(C)-2$, which annihilates $H^1$ and makes
evaluation at that point surjective. For every vector-bundle
-quotient $Q$ of $V^{\otimes i}$, the bundle
-$F_C^{e*}Q\otimes A_e^{\otimes i}$ is then globally generated.
+quotient $Q$ of $V^{ ⊗ i}$, the bundle
+$F_C^{e*}Q ⊗ A_e^{ ⊗ i}$ is then globally generated.
Consequently
\[
p^e\mu(Q)+i\deg A_e\ge0.
\]
-Divide by $p^e$ and let $e\to\infty$ to prove
-\eqref{fr:tensor-min}. This limit is taken for each fixed $p,i$;
+Divide by $p^e$ and let $e\to∞$ to prove
+(fr:tensor-min). This limit is taken for each fixed $p,i$;
there is no interchange of Frobenius-iteration and characteristic
limits.
-Since $A_{u-j}(V)$ is a quotient of $V^{\otimes(u-j)}$, the perfect
-pairing \eqref{fr:socle-pairing} gives
-\begin{align*}
+Since $A_{u-j}(V)$ is a quotient of $V^{ ⊗ (u-j)}$, the perfect
+pairing (fr:socle-pairing) gives
+
\mu_{\max}(A_j(V))
&= (p-1)\deg\det V-\mu_{\min}(A_{u-j}(V))\\
- &\le(p-1)K_W\cdot C+\frac{(u-j)c}{p-1}\\
+ &\le(p-1)K_W\cdot C+((u-j)c)/(p-1)\\
&\le(p-1)K_W\cdot C+nc.
-\end{align*}
-\end{proof}
+
+
The divisors $B_p-pD=T_0-(p-k_0N_p)D$ range over a fixed finite
-list. Combining \eqref{fr:intersection-bounds},
-\eqref{fr:flag-degrees}, and Lemma~\ref{fr:truncated-slope},
+list. Combining (fr:intersection-bounds),
+(fr:flag-degrees), and Lemma~fr:truncated-slope,
therefore gives, uniformly for $0\le j\le u$,
-\begin{align}
+
\mu_{\max}(\mathcal G_j|_C)
- &\le p(4+2/r)\Lambda H^n+O(1)\notag\\
- &\le M_p:=7p\Lambda H^n+c_0,\label{fr:common-slope-bound}
-\end{align}
+ &\le p(4+2/r)Λ H^n+O(1)\notag\\
+ &\le M_p:=7pΛ H^n+c_0,
+
where $c_0\ge0$ is a fixed constant. The coefficient $7$ provides
harmless room in this common bound.
-\begin{lemma}\label{fr:diagonal-sections}
+
For all sufficiently large $p$,
\[
h^0(\Delta_F,\mathcal L_p)
- \le R\bigl(7^nH^n+O(1/p)\bigr)<R/4.
+ \le R(7^nH^n+O(1/p))<R/4.
\]
The error constant is independent of the filtration index $j$.
-\end{lemma}
-\begin{proof}
+
+Proof.
Write the fixed flag as
\[
- W=W_n\supset W_{n-1}\supset\cdots\supset W_1=C,
+ W=W_n\supset W_{n-1}\supset...\supset W_1=C,
\qquad W_{n-k}\in|h_k|_{W_{n-k+1}}
\quad(1\le k\le n-1).
\]
At step $k$, fix the preceding nonnegative integers
-$b_1,\ldots,b_{k-1}$ and put
+$b_1,...,b_{k-1}$ and put
\[
- \mathcal F_k=\mathcal G_j|_{W_{n-k+1}}\Bigl(-\sum_{i<k}b_ih_i\Bigr).
+ \mathcal F_k=\mathcal G_j|_{W_{n-k+1}}(-\sum_{i<k}b_ih_i).
\]
For each $b_k\ge0$, the divisor restriction sequence is
\[
- 0\longrightarrow\mathcal F_k(-(b_k+1)h_k)
- \longrightarrow\mathcal F_k(-b_kh_k)
- \longrightarrow\mathcal F_k(-b_kh_k)|_{W_{n-k}}
- \longrightarrow0.
+ 0→\mathcal F_k(-(b_k+1)h_k)
+ →\mathcal F_k(-b_kh_k)
+ →\mathcal F_k(-b_kh_k)|_{W_{n-k}}
+ →0.
\]
-Iterating it for $b_k=0,1,\ldots$ and then proceeding to the next
+Iterating it for $b_k=0,1,...$ and then proceeding to the next
member of the flag bounds the section space by
-\begin{equation}\label{fr:flag-lattice-sum}
+
h^0(W,\mathcal G_j)\le
- \sum_{b_1,\ldots,b_{n-1}\ge0}
- h^0\Bigl(C,\mathcal G_j|_C\bigl(-\sum_i b_ih_i|_C\bigr)\Bigr).
-\end{equation}
+ \sum_{b_1,...,b_{n-1}\ge0}
+ h^0(C,\mathcal G_j|_C(-\sum_i b_ih_i|_C)).
+
At each stage, the remainder is zero after sufficiently negative
ample twisting. Its stopping index need not be uniform in $j,p$ or
in the preceding twists: enlarging the nonnegative finite sums to
the displayed lattice sum preserves the inequality.
-A summand in \eqref{fr:flag-lattice-sum} vanishes when
+A summand in (fr:flag-lattice-sum) vanishes when
$\sum_i b_i(h_i\cdot C)>M_p$, because then its maximum slope is
negative. Each nonzero summand has at most $e_j(M_p+1)$ sections.
Indeed, evaluation at $\lfloor M_p\rfloor+1$ distinct points is
injective; the kernel has negative maximum slope after subtracting
-those points. By \eqref{fr:flag-degrees} it follows that
-\begin{align}
+those points. By (fr:flag-degrees) it follows that
+
h^0(W,\mathcal G_j)
&\le e_j(M_p+1)
- \prod_{i=1}^{n-1}\Bigl(1+\frac{M_p}{l_i\Lambda H^n}\Bigr)\notag\\
- &=e_jp^n\bigl(7^nH^n+O(1/p)\bigr).\label{fr:common-flag-polynomial}
-\end{align}
-The leading coefficient follows from $\Lambda=\prod_il_i$. More
+ \prod_{i=1}^{n-1}(1+(M_p)/(l_iΛ H^n))\notag\\
+ &=e_jp^n(7^nH^n+O(1/p)).
+
+The leading coefficient follows from $Λ=\prod_il_i$. More
importantly, the whole scalar polynomial on the first line is the
same for every $j$.
-Sum over the filtration in \eqref{fr:diagonal-graded}. The ranks
-satisfy $\sum_j e_j=p^n$, so \eqref{fr:common-flag-polynomial} gives
+Sum over the filtration in (fr:diagonal-graded). The ranks
+satisfy $\sum_j e_j=p^n$, so (fr:common-flag-polynomial) gives
the asserted bound with $R=p^{2n}$. There is no extra factor for
the number of graded pieces. Finally,
\[
- 7^nH^n\le(14\epsilon)^n<1/4
+ 7^nH^n\le(14ε)^n<1/4
\]
-by \eqref{fr:intersection-bounds}, proving the strict inequality for
+by (fr:intersection-bounds), proving the strict inequality for
all large $p$.
-\end{proof}
-\subsection{The two orders of the determinant}
-\label{fr:determinant}
+
+## The two orders of the determinant
+
Choose $R$ generically independent columns of
-\eqref{fr:two-slot-evaluation} from bases of its source vector spaces.
+(fr:two-slot-evaluation) from bases of its source vector spaces.
Their determinant is a nonzero section
\[
0\ne\sigma\in H^0(W'\times W',\mathcal Q_1\boxtimes\mathcal Q_2)
@@ -2425,46 +2425,46 @@ $i$. The target determinant is
$(\det\mathcal E)^s\boxtimes(\det\mathcal E)^s$, and the chosen source
determinant is $\OO(-m_1S')\boxtimes\OO(-m_2S')$. Thus
\[
- \mathcal Q_i=(\det\mathcal E)^{\otimes s}
- \otimes\OO_{W'}(m_iS'),
+ \mathcal Q_i=(\det\mathcal E)^{ ⊗ s}
+ ⊗ \OO_{W'}(m_iS'),
\qquad \lambda_i=m_i/R.
\]
Every column weight lies between $0$ and $q$, so $0\le\lambda_i\le q$,
independently of the columns chosen.
Under the numerical identification with the base-field twist,
-\begin{equation}\label{fr:slot-determinant-class}
+
\frac{c_1(\mathcal Q_i)}R
=\frac{c_1(\mathcal E)}s+\lambda_iS
- =\frac{B_p}p+\frac{p-1}{2p}K_W+\lambda_iS,
+ =\frac{B_p}p+(p-1)/(2p)K_W+\lambda_iS,
\qquad 0\le\lambda_i\le q.
-\end{equation}
+
The Frobenius first-Chern-class identity is also
\cite[Lemma~4.2]{Sun}; it is an identity of rational divisor classes.
-For the second equality, use \eqref{fr:Frob-pullback-graded}.
-Complementary degrees in \eqref{fr:socle-pairing} have total first
+For the second equality, use (fr:Frob-pullback-graded).
+Complementary degrees in (fr:socle-pairing) have total first
Chern class $e_j(p-1)K_W$. Summing gives
\[
- \sum_jc_1(A_j(\Omega_W^1))=\frac{s(p-1)}2K_W.
+ \sum_jc_1(A_j(Ω_W^1))=\frac{s(p-1)}2K_W.
\]
Thus $c_1(F^*\mathcal E)=sB_p+s(p-1)K_W/2$, and Frobenius pullback of
twisted divisor classes multiplies by $p$, as required.
-For $0\le\lambda\le q$, the fixed movable inequality \eqref{fr:movable-input}
+For $0\le\lambda\le q$, the fixed movable inequality (fr:movable-input)
gives
\[
- (D+K_W/2+\lambda S)\cdot\gamma\le40\epsilon J\cdot\gamma,
+ (D+K_W/2+\lambda S)\cdot\gamma\le40ε J\cdot\gamma,
\]
where pullback by $\pi_x$ is understood. This concerns only
intersections of fixed divisors and the fixed complete-intersection
witness, so it remains true on the reduction. The class in
-\eqref{fr:slot-determinant-class} differs from its left-hand divisor
+(fr:slot-determinant-class) differs from its left-hand divisor
at $\lambda=\lambda_i$ by
\[
- \frac1p\bigl(B_p-pD-K_W/2\bigr).
+ \frac1p(B_p-pD-K_W/2).
\]
Its numerator belongs to the fixed finite list
-$T_0-aD-K_W/2$, $a\in\{0,\ldots,k_0-1\}$. The resulting error after
+$T_0-aD-K_W/2$, $a\in\{0,...,k_0-1\}$. The resulting error after
intersection with $\gamma$ is therefore $O(p^{-1})$, uniformly over
the selected determinant columns.
@@ -2472,70 +2472,70 @@ Let $x'$ be the twisted point, and let $0\ne\tau_i\in H^0(W',\mathcal Q_i)$.
The strict transform of the effective divisor of $\tau_i$ on the
blowup at $x'$ pairs nonnegatively with the twist of $\gamma$. Since
$J\cdot\gamma>0$, it follows that
-\begin{equation}\label{fr:slot-order}
- \frac{\ord_{x'}(\tau_i)}R\le40\epsilon+O(1/p).
-\end{equation}
+
+ \frac{\ord_{x'}(\tau_i)}R\le40ε+O(1/p).
+
The constant is uniform in the columns and in $\tau_i$. This
argument tests sections on the reduction itself; it does not require
them to lift to characteristic zero.
K\"unneth identifies the space containing $\sigma$ with
-$H^0(W',\mathcal Q_1)\otimes H^0(W',\mathcal Q_2)$. Choose bases
+$H^0(W',\mathcal Q_1) ⊗ H^0(W',\mathcal Q_2)$. Choose bases
adapted to the orders of vanishing at $x'$ in each factor. In each
degree, leading terms are linearly independent. Tensor products of
these leading terms remain independent in each bidegree, since the
two sets of parameters are disjoint; terms of different bidegrees
cannot cancel. Every nonzero tensor consequently has order at most
the sum of the two maximum basis orders. Applying
-\eqref{fr:slot-order}, we obtain
-\begin{equation}\label{fr:determinant-order-upper}
- \ord_{(x',x')}\sigma\le R\bigl(80\epsilon+O(1/p)\bigr).
-\end{equation}
+(fr:slot-order), we obtain
+
+ \ord_{(x',x')}\sigma\le R(80ε+O(1/p)).
-On the other hand, \eqref{fr:diagonal-rank-upper} and
-Lemma~\ref{fr:diagonal-sections} bound the matrix rank at $(x',x')$
+
+On the other hand, (fr:diagonal-rank-upper) and
+Lemma~fr:diagonal-sections bound the matrix rank at $(x',x')$
by a number strictly smaller than $R/4$. Trivialize its line bundles
near that point. Invertible constant row operations make more than
$3R/4$ rows vanish at the point. Each entry of those rows belongs to
the maximal ideal, so every term of the determinant has order at
least $3R/4$. Therefore
-\begin{equation}\label{fr:determinant-order-lower}
+
\ord_{(x',x')}\sigma\ge3R/4.
-\end{equation}
-Equations \eqref{fr:determinant-order-upper} and
-\eqref{fr:determinant-order-lower} contradict $80\epsilon<3/4$ for
+
+Equations (fr:determinant-order-upper) and
+(fr:determinant-order-lower) contradict $80ε<3/4$ for
sufficiently large $p$.
-This contradiction proves Theorem~\ref{thm:finite-data}. The
+This contradiction proves Theorem~thm:finite-data. The
full-rank calculation used $z_*$, while the diagonal estimate holds
at every point and was tested at the separately chosen $x$. The
errors were controlled by one fixed finite list of divisor classes
and one fixed flag polynomial. Thus their constants do not depend
on $p$, the filtration index, or the chosen columns.
-\subsection{Application to the terminal case}
-\label{fr:application}
+## Application to the terminal case
+
-We return to Theorem~\ref{thm:terminal-big}. Thus $X$ is a projective
+We return to Theorem~thm:terminal-big. Thus $X$ is a projective
terminal $\Q$-factorial variety of dimension $n\ge2$ with $K_X\sim0$,
and $L$ is a nef Cartier divisor of full nef dimension. Assume $L$
is not big. Recall the ample rational divisors $P=r(L+tA)$, where
$A$ is a fixed ample Cartier divisor and $t>0$ tends to zero, with
\[
- r\longrightarrow\infty,
- \qquad P^n\longrightarrow\epsilon^n,
+ r→∞,
+ \qquad P^n→ε^n,
\qquad \rho=(P^n)^{1/n}.
\]
-The positive rational number $\epsilon$ was chosen so that
-$(14\epsilon)^n<1/4$ and $80\epsilon<3/4$. Choose the integer $q$
-supplied by Proposition~\ref{prop:large-seshadri}, and then choose
+The positive rational number $ε$ was chosen so that
+$(14ε)^n<1/4$ and $80ε<3/4$. Choose the integer $q$
+supplied by Proposition~prop:large-seshadri, and then choose
$t$ sufficiently small that this proposition and
-Lemma~\ref{lemma:scalar-bound} apply, that $r>q+1$, and that
-$P^n<(2\epsilon)^n$. We now fix $t$, and therefore also $P,r,\rho$
-and the projective bundle in \eqref{eq:bundle}. Every remaining
+Lemma~lemma:scalar-bound apply, that $r>q+1$, and that
+$P^n<(2ε)^n$. We now fix $t$, and therefore also $P,r,\rho$
+and the projective bundle in (eq:bundle). Every remaining
choice is made over $\C$ before reducing to positive characteristic.
-\paragraph{The divisors and the flag.} Choose a smooth projective
+The divisors and the flag.: Choose a smooth projective
resolution $w\colon W\to X$ and set
\[
D=w^*P,\qquad S=w^*L,\qquad H_0=w^*P.
@@ -2555,43 +2555,43 @@ Thus, by continuity of intersection products, a sufficiently small
rational ample perturbation $H=H_0+\eta A_W$, with $A_W$ an ample
Cartier divisor on $W$ and $\eta>0$, satisfies
\[
- H^n\le(2\epsilon)^n,
+ H^n\le(2ε)^n,
\qquad
K_WH^{n-1}\le 2H^n/r,
\qquad
(2D+qS)H^{n-1}\le4H^n.
\]
-This gives \eqref{fr:intersection-bounds}.
+This gives (fr:intersection-bounds).
The canonical divisor of $W$ is pseudo-effective. Generic
-semipositivity of the cotangent bundle, originating in \cite{Miyaoka}
+semipositivity of the cotangent bundle, originating in [cite]
and used here in the form \cite[Theorem~2.1]{CP}, together with
restriction to sufficiently general complete-intersection curves
\cite[Theorem~6.1]{MR}, gives a smooth integral flag with successive
divisor classes $l_iH$, for sufficiently large divisible integers
$l_i$, such that
\[
- \mu_{\min}(\Omega_W^1|_C)\ge0.
+ \mu_{\min}(Ω_W^1|_C)\ge0.
\]
-This is the flag in Hypothesis~\textup{(i)} of
-Theorem~\ref{thm:finite-data}.
+This is the flag in Hypothesis~(i) of
+Theorem~thm:finite-data.
-\paragraph{The finite jet system.} On the bundle in \eqref{eq:bundle},
+The finite jet system.: On the bundle in (eq:bundle),
write $M=D_1+D_2+q\xi$. The divisor $M$ is ample: $\xi$ is nef and
relatively ample, and $D_1+D_2$ is the pullback of an ample divisor on
-$X\times X$. Proposition~\ref{prop:large-seshadri} supplies a smooth
+$X\times X$. Proposition~prop:large-seshadri supplies a smooth
point $z_*$ in the complement of the two axes with
\[
- \varepsilon(M;z_*)>2n+2.
+ ε(M;z_*)>2n+2.
\]
The point can be chosen over the open set where $w$ is an
isomorphism. Choose a rational number $a$ strictly between
-$2n+2$ and $\varepsilon(M;z_*)$. On the blowup
+$2n+2$ and $ε(M;z_*)$. On the blowup
$\widetilde Z\to Z$ at $z_*$, with exceptional divisor $G$, the
rational divisor $b^*M-aG$ is ample. For sufficiently large
divisible $k_0$, Serre vanishing gives
\[
- H^1\bigl(\widetilde Z,\OO_{\widetilde Z}(k_0b^*M-k_0aG)\bigr)=0.
+ H^1(\widetilde Z,\OO_{\widetilde Z}(k_0b^*M-k_0aG))=0.
\]
Since $z_*$ is smooth, the sections on the thickened exceptional
divisor $k_0aG$ identify with the jet quotient at $z_*$ modulo
@@ -2601,17 +2601,17 @@ generate jets through order $k_0a-1$, and hence through order
$(2n+2)k_0$. Increase the divisibility of $k_0$ to make $k_0P$
Cartier. Pulling these sections to
\[
- Z_W=\PP\bigl(\OO_W(S)_1\oplus\OO_W(S)_2\bigr)
- \longrightarrow W\times W
+ Z_W=\PP(\OO_W(S)_1 ⊕ \OO_W(S)_2)
+ → W\times W
\]
preserves the jets at the corresponding point, because the induced
map $Z_W\to Z$ is an isomorphism near that point. The pulled-back
divisor is $D_1+D_2+q\xi$. We have obtained the finite system
-required in Hypothesis~\textup{(ii)}.
+required in Hypothesis~(ii).
-\paragraph{The movable curve class.} Choose a very general point
+The movable curve class.: Choose a very general point
$x\in W$ outside the exceptional locus, with image in the locus where
-Lemma~\ref{lemma:scalar-bound} holds. Let
+Lemma~lemma:scalar-bound holds. Let
$\pi_x\colon\widehat W\to W$ be the blowup at $x$ and $J$ its
exceptional divisor. The rational divisor
\[
@@ -2621,29 +2621,29 @@ is big. For every sufficiently divisible positive integer $m$,
normality of $X$ and exceptionality of $E$ give
$w_*\OO_W(mE)=\OO_X$. Thus every section of $mD^+$ comes from a
section of $2mP$, multiplied by the section of the fixed divisor
-$mE$. Since $x\notin\Supp E$, Lemma~\ref{lemma:scalar-bound}
+$mE$. Since $x\notin\Supp E$, Lemma~lemma:scalar-bound
implies
-\begin{equation}\label{fr:application-order}
- \ord_x(s)/m\le8\rho<16\epsilon
- \qquad\bigl(0\ne s\in H^0(W,\OO_W(mD^+))\bigr).
-\end{equation}
+
+ \ord_x(s)/m\le8\rho<16ε
+ \qquad(0\ne s\in H^0(W,\OO_W(mD^+))).
+
It follows that
-\begin{equation}\label{fr:application-not-pe}
- \pi_x^*D^+-40\epsilon J\text{ is not pseudo-effective}.
-\end{equation}
+
+ \pi_x^*D^+-40ε J is not pseudo-effective.
+
Indeed, if it were pseudo-effective, a convex combination with the
big divisor $\pi_x^*D^+$ would make $\pi_x^*D^+-cJ$ big for any
-rational $c$ with $16\epsilon<c<40\epsilon$. A nonzero section of a
+rational $c$ with $16ε<c<40ε$. A nonzero section of a
sufficiently divisible multiple would then give a section violating
-\eqref{fr:application-order}.
+(fr:application-order).
By the duality between pseudo-effective divisors and strongly
-movable curves \cite{BDPP}, the strict separation
-in~\eqref{fr:application-not-pe} is detected by a class
+movable curves [cite], the strict separation
+in~(fr:application-not-pe) is detected by a class
\[
- \gamma=f_*(A_1\cdots A_{n-1}),
- \qquad f\colon V\longrightarrow\widehat W,
+ \gamma=f_*(A_1... A_{n-1}),
+ \qquad f\colon V→\widehat W,
\]
where $V$ is smooth projective, $f$ is birational, and the $A_i$ are
ample integral Cartier divisors. To obtain this form, first choose a
@@ -2651,7 +2651,7 @@ generating strongly movable class with negative pairing, then
approximate its ample real classes by rational ones and clear
denominators. We have
\[
- (\pi_x^*D^+)\cdot\gamma<40\epsilon J\cdot\gamma.
+ (\pi_x^*D^+)\cdot\gamma<40ε J\cdot\gamma.
\]
The left side is nonnegative, because $\pi_x^*D^+$ is
pseudo-effective, so $J\cdot\gamma>0$.
@@ -2666,48 +2666,48 @@ $\gamma$ therefore gives
\[
\pi_x^*(D+K_W/2+\lambda S)\cdot\gamma
\le(\pi_x^*D^+)\cdot\gamma
- <40\epsilon J\cdot\gamma.
+ <40ε J\cdot\gamma.
\]
-This proves Hypothesis~\textup{(iii)}. All hypotheses of
-Theorem~\ref{thm:finite-data} now hold for one fixed collection of
+This proves Hypothesis~(iii). All hypotheses of
+Theorem~thm:finite-data now hold for one fixed collection of
complex data, a contradiction. Hence $L$ is big, completing the
-proof of Theorem~\ref{thm:terminal-big}.
+proof of Theorem~thm:terminal-big.
-\section{Completion of the full-dimension step}
-\label{sec:completion}
+# Completion of the full-dimension step
-By Proposition~\ref{prop:reduction-cy}, Theorem~\ref{thm:key-ingredient}
+
+By Proposition~prop:reduction-cy, Theorem~thm:key-ingredient
reduces to the following claim in dimension $n$:
-\begin{equation}\label{eq:completion-cy-claim}
- \begin{gathered}
- (X,B)\text{ projective klt},\qquad K_X+B\equiv0,\\
- M\text{ nef and $\Q$-Cartier of full nef dimension}
- \quad\Longrightarrow\quad M\text{ big}.
- \end{gathered}
-\end{equation}
-Theorem~\ref{thm:terminal-big} proves this claim for terminal $X$
+
+
+ (X,B) projective klt,\qquad K_X+B\equiv0,\\
+ M nef and $\Q$-Cartier of full nef dimension
+ \quad⇒\quad M big.
+
+
+Theorem~thm:terminal-big proves this claim for terminal $X$
with $B=0$. We now extend it in two steps: first to arbitrary
klt $X$ with $B=0$, using an index-one cover and a crepant
terminalization; then to arbitrary effective $B$, using a small
-perturbation of the boundary and Lemma~\ref{lemma:two-nef-model}.
+perturbation of the boundary and Lemma~lemma:two-nef-model.
-\subsection{The case of zero boundary}
+## The case of zero boundary
Suppose $X$ is klt with $K_X\equiv 0$. By
-Theorem~\ref{input:abundance}, $K_X$ is semiample, hence
+Theorem~input:abundance, $K_X$ is semiample, hence
$K_X\sim_\Q0$. Indeed, a semiample numerically trivial divisor
defines a morphism with zero-dimensional image, so a sufficiently
divisible multiple is linearly trivial.
Take the global index-one cover
\[
- \pi\colon X^{\mathrm{ind}}\longrightarrow X
+ \pi\colon X^{ind}→ X
\]
associated with a trivialization of the smallest positive multiple
of $K_X$ that is linearly trivial. This is a finite quasi-\'etale
cover with
\[
- K_{X^{\mathrm{ind}}}=\pi^*K_X\sim0.
+ K_{X^{ind}}=\pi^*K_X\sim0.
\]
The cover is klt; since its canonical divisor is Cartier, its log
discrepancies are positive integers, hence at least one, and it is
@@ -2715,10 +2715,10 @@ canonical. By the crepant terminalization theorem
\cite[Corollary~1.4.3]{BCHM}, there is a projective birational
morphism
\[
- h\colon \widetilde X\longrightarrow X^{\mathrm{ind}}
+ h\colon \widetilde X→ X^{ind}
\]
with $\widetilde X$ terminal and $\Q$-factorial and
-$K_{\widetilde X}=h^*K_{X^{\mathrm{ind}}}\sim 0$.
+$K_{\widetilde X}=h^*K_{X^{ind}}\sim 0$.
Set $\widetilde M=(\pi\circ h)^*M$. This divisor is nef. To verify
that it has full nef dimension, choose a very general point upstairs
@@ -2727,16 +2727,16 @@ locus used for $M$. Every curve through this point maps to a curve
through a point where $M$ has positive degree, so $\widetilde M$ has
positive degree there as well by the projection formula. A positive
Cartier multiple of $\widetilde M$ is therefore big by
-Theorem~\ref{thm:terminal-big}. Bigness is invariant under a
+Theorem~thm:terminal-big. Bigness is invariant under a
dominant generically finite pullback, so $M$ is big on $X$.
-\subsection{Introducing the boundary}
+## Introducing the boundary
+
+Now let $(X,B)$ satisfy (eq:completion-cy-claim) with $B\ne0$.
+By Theorem~input:abundance,
-Now let $(X,B)$ satisfy \eqref{eq:completion-cy-claim} with $B\ne0$.
-By Theorem~\ref{input:abundance},
-\begin{equation}\label{eq:cy-cy}
K_X+B\sim_\Q0.
-\end{equation}
+
Choose a rational $\alpha>0$ sufficiently small that
$(X,(1+\alpha)B)$ is klt, and set
\[
@@ -2748,9 +2748,9 @@ outside $\Supp B$, $T_*\cdot C=\alpha B\cdot C\ge0$, while
$M\cdot C>0$ if that point is very general.
We do not use (P$_n$) here; the argument uses only
-Lemma~\ref{lemma:two-nef-model} and the $B=0$ case just proved.
+Lemma~lemma:two-nef-model and the $B=0$ case just proved.
-If $\kappa(K_X+B)>0$, Proposition~\ref{prop:full-positive} applied
+If $\kappa(K_X+B)>0$, Proposition~prop:full-positive applied
directly to the pair $(X,B)$ with $M$ gives $K_X+B+cM$ big for every
$c>0$; since $K_X+B\equiv0$, this makes $M$ big. We may therefore
assume $\kappa(K_X+B)=0$, so that $K_X+B\sim_\Q0$. Then $\kappa(T_*)$
@@ -2758,12 +2758,12 @@ may still be positive or zero; we treat the two cases with the same
machinery below.
Apply
-Lemma~\ref{lemma:two-nef-model} to $(X,(1+\alpha)B)$ and $M$. Write
+Lemma~lemma:two-nef-model to $(X,(1+\alpha)B)$ and $M$. Write
$Y$ for its final model, and let $B_Y$, $N$ denote the transforms of
$B$ and $M$. Set $U=K_Y+(1+\alpha)B_Y$. The lemma gives
$U\sim_\Q0$ and, for some rational $s>0$, that $U+sN$ is nef of full
nef dimension. The maps in the construction extract no divisors, so
-pushing forward the rational linear equivalence \eqref{eq:cy-cy}
+pushing forward the rational linear equivalence (eq:cy-cy)
also gives
\[
K_Y+B_Y\sim_\Q0.
@@ -2777,7 +2777,7 @@ $U+sN\sim_\Q sN$). The zero-boundary case just proved makes $N$
big.
Consider now the fixed birational model $W$ with maps $p$ to $X$
-and $q$ to $Y$ from Lemma~\ref{lemma:two-nef-model}. The effective
+and $q$ to $Y$ from Lemma~lemma:two-nef-model. The effective
exceptional comparisons are
\[
p^*T_*=q^*U+E_0,
@@ -2804,18 +2804,18 @@ Hence
as actual divisors. If $N$ is big, so is $q^*N$, and the
inequality makes $p^*(sM)$ big, hence $sM$ (and $M$) big.
-In the case $\kappa(T_*)>0$, Proposition~\ref{prop:full-positive}
+In the case $\kappa(T_*)>0$, Proposition~prop:full-positive
applied to the pair $(X,(1+\alpha)B)$ with $M$ gives $T_*+cM$ big
for every $c>0$. Combined with the effective exceptional comparison
-of $p$ to $q$ from Lemma~\ref{lemma:two-nef-model}, this yields
+of $p$ to $q$ from Lemma~lemma:two-nef-model, this yields
bigness of $M$ on $X$.
In either case, then, $M$ is big. This proves
-\eqref{eq:completion-cy-claim}, and hence Theorem~\ref{thm:key-ingredient}
+(eq:completion-cy-claim), and hence Theorem~thm:key-ingredient
in all dimensions.
-\section{Positive parts and numerical semiampleness}
-\label{sec:positive}
+# Positive parts and numerical semiampleness
+
We now prove (Z$_n$) by induction on $n$, using (P$_j$) and (Z$_j$)
for $j<n$ as inductive hypotheses. The argument runs in four
@@ -2829,11 +2829,11 @@ to a lower-dimensional base, on which the induction applies).
Fourth, we descend the semiampleness from a smooth birational model
to $X$ itself, using that $X$ has rational singularities.
-\subsection{Positive-part operations}
+## Positive-part operations
+
-\begin{lemma}\label{lemma:positive-properties}
The positive part $P_\sigma$ satisfies the following properties.
-\begin{enumerate}
+
\item If $p\colon V\to X$ is a projective birational morphism from a
smooth projective $V$ to a normal projective $X$, and $D$ is a
pseudo-effective rational divisor on $X$, and $E\ge0$ is
@@ -2850,10 +2850,10 @@ $P_\sigma(D)$ is nef, then
\item On a fixed smooth projective variety, rationality and
numerical semiampleness of $P_\sigma(D)$ depend only on the
numerical class of $D$.
-\end{enumerate}
-\end{lemma}
-\begin{proof}
+
+
+Proof.
Parts (i) and (ii) are \cite[Lemmas 2.4 and 2.5]{LP}, with the
target allowed to be singular in (i). For (iii): each asymptotic
order $\sigma_\Gamma(D)$ depends only on the numerical class, so
@@ -2864,37 +2864,37 @@ effective real divisor. For rational $D\equiv D'$,
\]
Thus one positive part is rational if and only if the other is, and
the two are numerically equivalent.
-\end{proof}
-\subsection{Descent along a birational model}
+
+## Descent along a birational model
The following production lemma descends numerical semiampleness
from a smooth birational model to a normal projective variety with
rational singularities. Recall that klt varieties have rational
singularities.
-\begin{lemma}\label{lemma:numsemiample-descent}
+
Let $p\colon W\to X$ be a projective birational morphism from a
smooth projective $W$ to a normal projective $X$ with rational
singularities. Let $D$ be a $\Q$-Cartier divisor on $X$. If
$p^*D$ is numerically equivalent to a semiample $\Q$-divisor on
$W$, then $D$ is numerically equivalent to a semiample $\Q$-divisor
on $X$.
-\end{lemma}
-\begin{proof}
+
+Proof.
Let $A_W$ be a semiample rational divisor on $W$ with
$A_W\equiv p^*D$. After clearing denominators, the difference of
the corresponding line bundles is numerically trivial. Numerically
trivial line bundles modulo $\Pic^0(W)$ are torsion. Thus, after
raising a positive integer $r$, there is
\[
- \OO_W(rA_W)\simeq p^*\OO_X(rD)\otimes P_W,
+ \OO_W(rA_W)\simeq p^*\OO_X(rD) ⊗ P_W,
\qquad P_W\in\Pic^0(W).
\]
Rational singularities identify $\Pic^0(X)$ with $\Pic^0(W)$ by
pullback. Write $P_W=p^*P_X$ for $P_X\in\Pic^0(X)$, and set
-$\mathcal A_X=\OO_X(rD)\otimes P_X$. Then
+$\mathcal A_X=\OO_X(rD) ⊗ P_X$. Then
\[
p^*\mathcal A_X\simeq\OO_W(rA_W).
\]
@@ -2906,46 +2906,46 @@ A_X$. These sections generate downstairs. Hence $\mathcal A_X$ is
globally generated after a sufficiently high multiple, so the
divisor representing $\mathcal A_X$, divided by $r$, is a semiample
$\Q$-divisor on $X$ numerically equivalent to $D$.
-\end{proof}
-\subsection{The large-nef-adjoint model}
+
+## The large-nef-adjoint model
We may therefore replace $X$ by a small projective
$\Q$-factorialization and, at the end, descend the resulting
-semiample divisor via Lemma~\ref{lemma:numsemiample-descent}. For
+semiample divisor via Lemma~lemma:numsemiample-descent. For
the rest of this section $X$ is klt and $\Q$-factorial.
Let $L$ be a positive multiple of $M$ that is Cartier. We write
$L=m_0M$ and eventually absorb the factor $m_0$ into the nef
summand.
-\begin{lemma}\label{lemma:large-nef-adjoint}
+
There are an integer $m_0>0$, a birational contraction
$\varphi\colon X\dashrightarrow Y$ between klt $\Q$-factorial
projective varieties, and a common smooth model $W$ with
$p\colon W\to X$ and $q\colon W\to Y$, satisfying the following
properties.
-\begin{enumerate}
+
\item The transform $T'$ of $T=K_X+B$ is pseudo-effective, $N$ (the
transform of $M$) is nef, and $T'+m_0 N$ is nef on $Y$.
\item The contraction $\varphi$ is $M$-trivial: on $W$,
$p^*M=q^*N$.
\item There is an effective $q$-exceptional rational divisor $E\ge0$
on $W$ with $p^*T=q^*T'+E$.
-\end{enumerate}
+
Consequently,
$p^*(T+M)=q^*(T'+N)+E$ and, by
-Lemma~\ref{lemma:positive-properties}(i),
+Lemma~lemma:positive-properties(i),
\[
- P_\sigma\bigl(p^*(T+M)\bigr)=P_\sigma\bigl(q^*(T'+N)\bigr)
- \quad\text{as divisors on }W.
+ P_\sigma(p^*(T+M))=P_\sigma(q^*(T'+N))
+ \quadas divisors on W.
\]
So it suffices to prove (Z$_n$) for the pair $(Y,B_Y)$ with nef
divisor $N$, and the adjoint $T'+m_0N$ is nef.
-\end{lemma}
-\begin{proof}
-Apply Theorem~\ref{input:models} to the generalized klt $\Q$-pair
+
+Proof.
+Apply Theorem~input:models to the generalized klt $\Q$-pair
$(X,B+m_0M)$ with nef data $m_0M$. Its generalized adjoint is
$T+m_0M$, which is pseudo-effective since $T$ is pseudo-effective
and $m_0M$ is nef. The theorem gives a terminating generalized
@@ -2953,7 +2953,7 @@ program; we verify $M$-triviality.
Consider a $(T+m_0M)$-negative extremal ray, and let $C$ be a
rational curve generating it. By the length bound
-(Lemma~\ref{lemma:generalized-ray-bound}), we may choose $C$ with
+(Lemma~lemma:generalized-ray-bound), we may choose $C$ with
$0<-T\cdot C\le2n$. If $M\cdot C>0$, then $m_0M\cdot C\ge1$ by
integrality (the divisor $m_0M$ is Cartier). Hence
\[
@@ -2976,16 +2976,16 @@ $p^*(T+m_0M)=q^*(T'+m_0N)+E_{m_0}$ with $E_{m_0}$ effective
$q$-exceptional; subtracting the $M$-pullback comparison
$p^*(m_0M)=q^*(m_0N)$ yields $p^*T=q^*T'+E_{m_0}$. Setting
$E=E_{m_0}$ gives the effective $q$-exceptional comparison.
-\end{proof}
-After Lemma~\ref{lemma:large-nef-adjoint}, we may replace $(X,B,M)$
+
+After Lemma~lemma:large-nef-adjoint, we may replace $(X,B,M)$
by $(Y,B_Y,N)$ and assume in addition that $T+mL$ is nef for some
integer $m>0$.
-\subsection{The case of full nef dimension}
+## The case of full nef dimension
Suppose $T+mL$ has full nef dimension. Then
-\eqref{eq:full-condition} holds with $a=m$, and (P$_n$) shows that
+(eq:full-condition) holds with $a=m$, and (P$_n$) shows that
$T+M$ is big. Write, by Kodaira's lemma,
\[
T+M\sim_\Q A_1+G_1,
@@ -2997,46 +2997,46 @@ klt. The divisor $M+vA_1$ is ample (sum of a nef and an ample
divisor). A general effective rational representative $A'$ of its
class may be chosen with all coefficients small enough that
$(X,B+vG_1+A')$ remains klt. Its adjoint is
-\begin{equation}\label{eq:big-ordinary-boundary}
+
K_X+B+vG_1+A'\sim_\Q(1+v)(T+M),
-\end{equation}
+
because $T+M=K_X+B+M\sim_\Q A_1+G_1$. This adjoint is big, since
$T+M$ is big.
The pair $(X,B+vG_1+A')$ is klt with big adjoint, so it has a good
-minimal model by the big case of \cite{BCHM}; equivalently, apply
-Theorems~\ref{input:models} and~\ref{input:abundance}. On a common
+minimal model by the big case of [cite]; equivalently, apply
+Theorems~input:models and~input:abundance. On a common
smooth resolution $w$, the left side of
-\eqref{eq:big-ordinary-boundary} is the pullback of a semiample
+(eq:big-ordinary-boundary) is the pullback of a semiample
divisor plus an effective exceptional divisor, by the effective
exceptional comparison of the program. Its positive part is
therefore the semiample pullback.
Since $P_\sigma$ is homogeneous,
\[
- P_\sigma\bigl(w^*(T+M)\bigr)
- =\frac1{1+v}P_\sigma\bigl(w^*\bigl(K_X+B+vG_1+A'\bigr)\bigr)
+ P_\sigma(w^*(T+M))
+ =\frac1{1+v}P_\sigma(w^*(K_X+B+vG_1+A'))
\]
is a rational divisor numerically equivalent to a semiample
$\Q$-divisor on $w$. This is (Z$_n$) in the full-nef dimension
case.
-\subsection{Descent along a nef reduction}
+## Descent along a nef reduction
Now suppose $T+mL$ does not have full nef dimension. Let
$h\colon X\dashrightarrow Z_1$ be the nef reduction of $T+mL$
-(Definition at the start of \S\ref{sec:preliminaries}), so
+(Definition at the start of \Ssec:preliminaries), so
$\dim Z_1=r<n$. Fix a general compact fibre $F$.
After shrinking $Z_1$, $F$ is klt and the pair $(F,B|_F)$ is klt
with adjoint $T|_F=K_F+B|_F$. Since $(T+mL)|_{\widetilde F}\equiv0$
on any component $\widetilde F$ of the fibre, $T|_{\widetilde F}$
is numerically trivial and therefore nef. By
-Theorem~\ref{input:abundance}, $T|_F$ is semiample; a semiample
+Theorem~input:abundance, $T|_F$ is semiample; a semiample
numerically trivial divisor has zero-dimensional image, so
-\begin{equation}\label{eq:fibre-adjoint-trivial}
+
K_F+B|_F\sim_\Q0.
-\end{equation}
+
We use the birational descent of nef divisors
\cite[Lemma~3.1]{LP}. After resolving $h$ and modifying its base
@@ -3047,9 +3047,9 @@ birationally, it supplies a diagram
in which $X_1,Z_1$ are smooth and projective, $p$ is birational,
$f$ is a contraction, and there is a rational divisor $M_1$ on
$Z_1$ with
-\begin{equation}\label{eq:nef-descent}
+
p^*M\equiv f^*M_1.
-\end{equation}
+
The divisor $M_1$ is nef: its pullback under a surjective morphism
is numerically equivalent to the nef divisor $p^*M$, and nefness
descends under a surjective morphism.
@@ -3061,27 +3061,27 @@ Since $B$ is effective, $E$ is $p$-exceptional. Klt singularities
make all coefficients of $B_1$ less than one; after further
shrinking, its support is simple normal crossing. Thus
$(X_1,B_1)$ is klt and
-\begin{equation}\label{eq:resolved-boundary}
+
K_{X_1}+B_1=p^*T+E,\qquad E\ge0.
-\end{equation}
+
In particular the adjoint of $(X_1,B_1)$ is pseudo-effective.
A general fibre $F_1$ of $f$ maps birationally onto the
corresponding compact fibre $F$ of the nef reduction. Every
component of $E$ that meets $F_1$ restricts to an exceptional
-divisor over $F$. Equations~\eqref{eq:fibre-adjoint-trivial}
-and~\eqref{eq:resolved-boundary} therefore give
-\begin{equation}\label{eq:fibre-kappa-zero}
- \kappa\bigl(F_1,(K_{X_1}+B_1)|_{F_1}\bigr)=0.
-\end{equation}
+divisor over $F$. Equations~(eq:fibre-adjoint-trivial)
+and~(eq:resolved-boundary) therefore give
+
+ \kappa(F_1,(K_{X_1}+B_1)|_{F_1})=0.
+
The restriction is an effective exceptional divisor up to rational
linear equivalence, and its section spaces are those of the trivial
divisor on the normal fibre $F$. In particular the general fibre
pair has pseudo-effective adjoint and, by
-Theorems~\ref{input:models} and~\ref{input:abundance}, a good
+Theorems~input:models and~input:abundance, a good
minimal model.
-By \eqref{eq:nef-descent} we may replace the nef term by
+By (eq:nef-descent) we may replace the nef term by
$f^*M_1$ before making positive-part comparisons; this pullback is
trivial on every step of the relative program.
@@ -3092,33 +3092,33 @@ Let
be the relative semiample fibration of $K_{X_2}+B_2$, with $g$ a
contraction. Its construction gives a $j$-ample $\Q$-Cartier
divisor $A_2$ on $Z_2$ such that
-\begin{equation}\label{eq:relative-iitaka}
+
K_{X_2}+B_2\sim_\Q g^*A_2.
-\end{equation}
+
The relative program preserves Iitaka dimension on the general
-fibres, so by \eqref{eq:fibre-kappa-zero} the relative Iitaka
+fibres, so by (eq:fibre-kappa-zero) the relative Iitaka
dimension is zero, and hence
\[
\dim Z_2=\dim Z_1<n.
\]
Apply Ambro's canonical bundle formula consequence
-\cite[Theorem~0.2]{Ambro} to \eqref{eq:relative-iitaka}. Its
+\cite[Theorem~0.2]{Ambro} to (eq:relative-iitaka). Its
hypotheses hold: $(X_2,B_2)$ is a projective klt pair with effective
rational boundary, $g$ is a contraction to a normal projective
variety, and the ordinary adjoint is rationally linearly pulled back
from a $\Q$-Cartier divisor. It gives an effective rational
boundary $B_{Z_2}$ such that $(Z_2,B_{Z_2})$ is klt and
-\begin{equation}\label{eq:base-adjoint}
+
K_{X_2}+B_2\sim_\Q g^*(K_{Z_2}+B_{Z_2}).
-\end{equation}
+
The base adjoint is pseudo-effective. One direct verification,
using only the inputs already in force, is the following. The
pseudo-effective ordinary adjoint on $X_2$ has a good minimal model
-by Theorems~\ref{input:models} and~\ref{input:abundance}, so it has
+by Theorems~input:models and~input:abundance, so it has
a nonzero section in some positive multiple. Since
$g_*\OO_{X_2}=\OO_{Z_2}$, the projection formula and
-\eqref{eq:base-adjoint} identify this section space with the
+(eq:base-adjoint) identify this section space with the
corresponding section space on $Z_2$. Thus $K_{Z_2}+B_{Z_2}$ even
has an effective rationally linearly equivalent representative.
@@ -3128,7 +3128,7 @@ $M_{Z_2}=j^*M_1$ be the pullback of $M_1$ to $Z_2$: it is nef
induction hypothesis (Z$_{\dim Z_2}$) applies and gives a smooth
model $n\colon W_{Z_2}\to Z_2$ with
\[
- P_\sigma\bigl(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2})\bigr)
+ P_\sigma(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2}))
\]
rational and numerically equivalent to a semiample $\Q$-divisor on
$W_{Z_2}$.
@@ -3136,26 +3136,26 @@ $W_{Z_2}$.
Pull this back. Let $g'\colon W_2\to X_2$ be a smooth model onto
which $W_{Z_2}$ pulls back via $g$, so that
\[
- g'^*\bigl(K_{X_2}+B_2+M_2\bigr)
- =g'^*g^*\bigl(K_{Z_2}+B_{Z_2}+M_{Z_2}\bigr),
+ g'^*(K_{X_2}+B_2+M_2)
+ =g'^*g^*(K_{Z_2}+B_{Z_2}+M_{Z_2}),
\]
where $M_2=g^*M_{Z_2}=g^*j^*M_1$. By
-Lemma~\ref{lemma:positive-properties}(ii),
+Lemma~lemma:positive-properties(ii),
\[
- P_\sigma\bigl(g'^*(K_{X_2}+B_2+M_2)\bigr)
- =g'^*P_\sigma\bigl(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2})\bigr)
+ P_\sigma(g'^*(K_{X_2}+B_2+M_2))
+ =g'^*P_\sigma(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2}))
\]
is a rational divisor numerically equivalent to a semiample
$\Q$-divisor on $W_2$.
Transfer to $X$: on $W_2$, the pullback
-$p_{\mathrm{new}}^*(T+M)$ of the original adjoint sum, under the
-composition $p_{\mathrm{new}}=p\circ(X_2\to X_1)\circ(X_2\to X_1)$
+$p_{new}^*(T+M)$ of the original adjoint sum, under the
+composition $p_{new}=p\circ(X_2\to X_1)\circ(X_2\to X_1)$
of birational morphisms, differs from $K_{X_2}+B_2+M_2$ by an
-effective $p_{\mathrm{new}}$-exceptional divisor $E_2$. By
-Lemma~\ref{lemma:positive-properties}(i),
+effective $p_{new}$-exceptional divisor $E_2$. By
+Lemma~lemma:positive-properties(i),
\[
- P_\sigma\bigl(p_{\mathrm{new}}^*(T+M)\bigr)
+ P_\sigma(p_{new}^*(T+M))
=P_\sigma(K_{X_2}+B_2+M_2+E_2)
=P_\sigma(K_{X_2}+B_2+M_2),
\]
@@ -3165,13 +3165,13 @@ holds in the nef-reduction case.
Combining the full-nef and nef-reduction cases, (Z$_n$) holds for
every $(X,B,M)$ satisfying the hypotheses of (Z$_n$). This closes
both simultaneous inductions and completes the proof of
-Theorem~\ref{thm:key-ingredient} for every $n$ via
-Section~\ref{sec:completion}.
+Theorem~thm:key-ingredient for every $n$ via
+Section~sec:completion.
-\subsection{Proof of the main theorem over $\C$}
+## Proof of the main theorem over $\C$
With both inductions established, we can now prove
-Theorem~\ref{thm:main} over $\C$. Let $(X,B)$ be a projective klt
+Theorem~thm:main over $\C$. Let $(X,B)$ be a projective klt
$\Q$-pair of dimension $n$, $M$ a nef $\Q$-Cartier divisor, and
$K_X+B$ pseudo-effective. The sum $D=K_X+B+M$ is nef by
hypothesis. On any smooth birational model $w\colon W\to X$,
@@ -3182,6 +3182,6 @@ because a nef divisor has zero negative part. Statement (Z$_n$)
applies and gives that this pullback is rational and numerically
equivalent to a semiample $\Q$-divisor on $W$. Since a klt variety
has rational singularities,
-Lemma~\ref{lemma:numsemiample-descent} descends the result to a
+Lemma~lemma:numsemiample-descent descends the result to a
semiample $\Q$-Cartier divisor on $X$ that is numerically equivalent
-to $D$. This is Theorem~\ref{thm:main} over $\C$.
+to $D$. This is Theorem~thm:main over $\C$.