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\title{Log abundance in numerical dimension one\\for threefolds in positive characteristic\\[2mm]
\large A reader-friendly edition}
\author{OpenAI}
\date{October 5, 2026}

\begin{document}
\maketitle

\begin{abstract}
Let $k$ be an algebraically closed field of characteristic $p>3$, and let
$(X,B)$ be a projective log canonical threefold pair over $k$ with
effective rational boundary $B$. If the log canonical divisor $K_X+B$ is
nef and has numerical dimension one, then it is semiample: some positive
Cartier multiple is generated by its global sections. Neither
terminality nor $\Q$-factoriality is assumed for the original $X$. This
edition states the theorem, explains the strategy in plain language,
and gives the complete proof. The technical engine is a structural
result of the companion paper: a certain kind of nef cycle cannot exist
on a smooth threefold. The contribution here is to build that cycle from
an arbitrary log canonical pair, controlling the boundary throughout.
\end{abstract}

\tableofcontents

\section*{How to read this paper}
\addcontentsline{toc}{section}{How to read this paper}

This paper proves a case of the \emph{abundance conjecture} in algebraic
geometry, in positive characteristic. The result is advanced, but the
shape of the argument can be described in ordinary language.

\paragraph{The goal.}
A \emph{threefold} is a space of dimension three; over the complex
numbers it looks locally like $\C^3$. To such a space and an effective
\emph{boundary} divisor $B$ one associates the \emph{log canonical
divisor} $K_X+B$, a bookkeeping object recording how differential forms
twist. A divisor is \emph{nef} if it never meets a curve negatively;
think of it as a direction in which the space does not bend downward.
We assume $K_X+B$ is nef and has \emph{numerical dimension} one. This
means $K_X+B$ is positive in exactly one direction and numerically zero
in the other two.

Abundance predicts that some positive multiple of $K_X+B$ has enough
global sections to define a map to a curve. We prove exactly that: some
positive Cartier multiple of $K_X+B$ is globally generated, so $K_X+B$
is \emph{semiample} and the associated map lands on a curve.

\paragraph{Why this is hard.}
The hypothesis only says the divisor is not negative and is positive in
one direction. It does not by itself produce a single section upstairs.
The proof must rule out a hidden bad case: \emph{maximal nef dimension}.
If that bad case occurred, the positive direction would be locally
invisible, so that no curve of degree zero would pass through a very
general point. The body of the paper derives a contradiction from this
assumption.

\paragraph{The contradiction in one sentence.}
In the bad case one constructs a connected nef cycle $D$ on a smooth
projective threefold $V$, living near a log canonical component, with a
torsion normal bundle and a local relation between a power of the
canonical bundle and $D$. The companion paper proves that no such cycle
can exist: its torsion normal line and canonical relation make the
Picard group of the formal neighborhood divisible by every power of
$p$, while intersection with an ample divisor supplies a nonzero
integer-valued degree, and a $p$-divisible group admits no nonzero map
to $\Z$. Contradiction.

\paragraph{A small dictionary.}
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Term & Plain meaning \\
\midrule
Threefold & A space of dimension $3$. \\
Nef & Never negative against curves. \\
Ample & Positive enough to embed the space in projective space. \\
Cartier divisor & A codimension-one object cut out locally by one equation. \\
$\Q$-Cartier & Some positive multiple of the divisor is Cartier. \\
$\Q$-factorial & Some positive multiple of every divisor is Cartier. \\
Log canonical (lc) & The mild singularities allowed by the log MMP. \\
Terminal & The mildest singularities for the minimal model program. \\
klt / dlt & Slightly stronger variants of log canonical. \\
Numerical dimension & The number of directions in which a nef divisor is positive. \\
Nef dimension & The dimension of the target of the nef reduction. \\
Semiample & Some positive multiple is generated by global sections. \\
Log canonical threshold & The largest coefficient keeping a pair lc. \\
Frobenius & The map $x\mapsto x^p$ in characteristic $p$. \\
Cartier descent & Recovering data from its Frobenius pullback. \\
Formal neighborhood & The completion along a divisor, remembering its thickening. \\
Picard group & The group of line bundles up to isomorphism. \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{How the sections fit together.}
Section~\ref{sec:introduction} states the theorem and the proof map.
Section~\ref{sec:foundations} states the companion's cycle obstruction,
records elementary numerical facts, and reduces the theorem to a
terminal $\Q$-factorial pair. Section~\ref{sec:boundary} separates the
boundary from the cycle using quantitative bend-and-break.
Section~\ref{sec:isolation} isolates a single log canonical component by
a minimal model program that is trivial for the restored divisor.
Section~\ref{sec:torsion} proves torsion on the full nonreduced Cartier
thickening of that component. Section~\ref{sec:cycle} assembles the
prepared cycle and completes the proof.

Readers who want the shortest path can read
\[
\text{Theorem~\ref{thm:main}}
\longrightarrow
\text{Proposition~\ref{prop:reduction}}
\longrightarrow
\text{Proposition~\ref{prop:separation}}
\longrightarrow
\text{Proposition~\ref{prop:isolation}}
\longrightarrow
\text{Proposition~\ref{prop:boundary-torsion}}
\longrightarrow
\text{Proposition~\ref{prop:prepared}}.
\]
The intervening lemmas explain why each arrow is valid.

\section{Introduction}
\label{sec:introduction}

\subsection{The theorem}

Throughout, a \emph{variety} is integral and separated of finite type
over a field. A \emph{pair} $(X,B)$ has $X$ normal and $B$ an effective
$\Q$-divisor with $K_X+B$ $\Q$-Cartier. We use the standard discrepancy
definitions of terminal, klt, dlt, and lc singularities; see
\cite{KM}. For a nef $\Q$-Cartier divisor $L$ on a threefold, we say its
\emph{numerical dimension} is one when, for an ample Cartier divisor
$H$,
\begin{equation}\label{eq:numdim}
L H^2>0
\qquad\text{and}\qquad
L^2 H=0.
\end{equation}

\begin{theorem}\label{thm:main}
Let $k$ be an algebraically closed field of characteristic $p>3$. Let
$(X,B)$ be a projective log canonical threefold pair over $k$, where
$X$ is normal, $B$ is an effective $\Q$-divisor, and $K_X+B$ is
$\Q$-Cartier. If $K_X+B$ is nef and has numerical dimension one, then
$K_X+B$ is semiample.
\end{theorem}

The boundary coefficients are arbitrary rationals allowed by log
canonicity. Neither terminality nor $\Q$-factoriality is assumed for the
original variety. The theorem proves the numerical-dimension-one case of
threefold log abundance in this characteristic range.

Geometrically, semiampleness gives a morphism $f:X\to C$ with connected
fibers to a normal projective curve together with an ample divisor $A$
on $C$ such that $m(K_X+B)\sim f^*A$ for some $m>0$. The target is a
curve because the numerical dimension is one; Stein factorization
produces $f$ from any globally generated pluricanonical multiple.

\subsection{Background and context}

Over the complex numbers, Miyaoka proved abundance for minimal
threefolds of numerical dimension one \cite{Miyaoka}, and Kawamata gave
another proof and completed abundance for minimal threefolds
\cite{Kawamata}. Keel, Matsuki, and McKernan established log abundance
for log canonical threefold pairs \cite{KMM,KMMCorrection}.

In positive characteristic, Keel's semiampleness criterion \cite{Keel}
and the basepoint-freeness methods of Cascini, Tanaka, and Xu \cite{CTX}
provided the foundations. The threefold minimal model program was
developed by Hacon--Xu \cite{HX}, Birkar \cite{Birkar}, and
Birkar--Waldron \cite{BW} in characteristic greater than five, with log
canonical and relative extensions by Waldron \cite{Waldron} and
Hashizume--Nakamura--Tanaka \cite{HNT}, and the characteristic-five
case by Hacon--Witaszek \cite{HW}.

For abundance itself, Xu--Zhang proved nonvanishing for terminal minimal
threefolds in characteristic greater than five \cite{XZ}, and Witaszek
obtained klt nonvanishing and abundance in nef dimension at most two in
that range \cite{Witaszek}. Zheng Xu proved nonvanishing for
pseudoeffective log canonical divisors on projective lc threefold pairs
in characteristic greater than three, together with abundance in nef
dimension at most two \cite[Theorems~1.4 and~1.8]{XuNonvanishing}, and
later proved abundance in numerical dimension two \cite{XuNuTwo}.
The case left open, and treated here, has numerical dimension one and
maximal nef dimension three.

The companion paper \cite{Companion} treats terminal threefolds whose
nef canonical divisor has numerical dimension one. Its proof isolates a
stronger structural fact: a certain connected nef cycle cannot exist on
a smooth threefold. We state this \emph{cycle obstruction} in
Theorem~\ref{thm:obstruction}, with all its hypotheses. The present paper
constructs such a cycle from a log canonical pair with arbitrary rational
boundary; the obstruction is an essential input, not reproved here.

\subsection{The proof strategy}

After an uncountable field extension and a crepant birational change, we
may suppose the underlying variety is terminal and $\Q$-factorial.
Nonvanishing then gives a nonzero effective Cartier divisor
$N_0\sim m(K_X+B)$. We assume for contradiction that the nef dimension
of $N_0$ is three. The new difficulty, compared with the companion, is to
obtain a relation for a power of the canonical bundle near a component of
this \emph{log} canonical divisor, controlling both the boundary and the
exceptional divisors of a resolution.

Proposition~\ref{prop:separation} handles the boundary. It proves
$B N_0 H=0$ and shows that every boundary component is either contained
in $\Supp N_0$ or disjoint from it. If the intersection were positive,
complete-intersection curves would have arbitrarily small ratios of
$N_0$-degree to anticanonical degree, and quantitative bend-and-break
\cite[Theorem~1.1]{JLR} would then produce $N_0$-trivial rational curves
through very general points.

Next, adapting \cite[Section~2.3]{Companion}, we raise the boundary to
its log canonical threshold along $N_0$, obtaining a coefficient-one
component on a dlt model. A partial MMP, trivial for the resulting full
log canonical class, makes that component disjoint from the rest of the
divisor. Reduced-boundary adjunction and surface abundance
\cite{Tanaka,Posva} then yield torsion on a Cartier multiple of the
component, and the companion's extension argument passes from the reduced
surface to the full nonreduced Cartier thickening.

Finally, terminality makes the singular locus finite, so the exceptional
divisors of a common resolution obey the same support separation as the
boundary. A finite separable cover removes the prime-to-$p$ part of the
pluricanonical exponent. A connected component of the pulled-back Cartier
divisor, divided by the gcd of its multiplicities, has exactly the
numerical, torsion, and canonical properties forbidden by
Theorem~\ref{thm:obstruction}. This contradiction completes the proof.

Figure~\ref{fig:proof-map} records the structure.

\begin{figure}[ht]
\centering
\small
\begin{tikzcd}[column sep=small,row sep=large]
& \text{lc pair }(X,B),\ N_0\sim m(K_X+B),\ n(N_0)=3
  \arrow[dl,"\text{reduce to terminal}"']
  \arrow[dr,"\text{boundary separation}"'] & \\
\begin{gathered}\text{terminal }\Q\text{-factorial }(X,B)\\N_0\text{ nef, }\nu=1,\ n=3\end{gathered}
  \arrow[d,"\text{isolation}"']
&& \begin{gathered}B N_0 H=0\\ \Supp B\subset \Supp N_0\sqcup P_{\mathrm{out}}\end{gathered}
  \arrow[d,"\text{torsion}"] \\
\begin{gathered}\text{isolated }(Y,S_Y)\\S_Y\perp\text{other components}\end{gathered}
  \arrow[dr,"\text{dlt modification}"] &&
\begin{gathered}f^*S_Y\text{ torsion on Cartier thick }M\\ \cO_M(M)\text{ torsion}\end{gathered}
  \arrow[dl] \\
& \text{prepared cycle }(V,D)\ \Rightarrow\ \text{contradiction with Theorem~\ref{thm:obstruction}} &
\end{tikzcd}
\caption{The maximal-nef-dimension assumption produces the forbidden
cycle, contradicting the companion obstruction.}
\label{fig:proof-map}
\end{figure}

\section{The cycle obstruction and the initial reductions}
\label{sec:foundations}

We first state precisely the input taken from the companion paper
\cite{Companion}. We then reduce Theorem~\ref{thm:main} to the task of
constructing the forbidden cycle. Boundaries are effective
$\Q$-divisors, and canonical divisors on birational models are chosen
compatibly. Restrictions and intersections of $\Q$-Cartier divisors are
computed using Cartier multiples.

\subsection{The obstruction supplied by the companion}

For a nef divisor $L$ over an uncountable algebraically closed field,
its \emph{nef dimension} $n(L)$ is the dimension of the target of its
nef reduction. In dimension three we use only the following
characterization: $n(L)=3$ if and only if no curve of $L$-degree zero
passes through a very general point, where \emph{very general} means
outside a countable union of proper closed subsets. These properties and
the existence of the nef reduction are recalled in
\cite[Definition~2.1 and Theorem~2.2]{XuNuTwo}. Numerical dimension and
nef dimension need not agree; both appear below.

\begin{theorem}[Cycle obstruction of \cite{Companion}]\label{thm:obstruction}
Let $k$ be an uncountable algebraically closed field of characteristic
$p>3$. There do not exist a smooth projective threefold $V$, a very
ample Cartier divisor $H_V$, and a nonzero effective Cartier divisor
$D=\sum_\alpha m_\alpha D_\alpha$ satisfying all of the following:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the $D_\alpha$ are distinct prime divisors, $m_\alpha\in\Z_{>0}$,
the support of $D$ is connected with simple normal crossings, and
$\gcd_\alpha m_\alpha=1$;
\item $D$ is nef, its restriction to every $D_\alpha$ is numerically
trivial, and
\[
D H_V^2>0,\qquad D^2 H_V=0,\qquad n(D)=3;
\]
\item the line bundle $\cO_D(D)$ on the possibly nonreduced Cartier
scheme $D$ is torsion;
\item for some integer $b\ge0$ and integers $b_\alpha$, there is an
isomorphism on a neighborhood of $\Supp D$,
\begin{equation}\label{eq:canonical-cycle}
\omega_V^{\otimes p^b}\simeq
\cO_V\Bigl(\sum_\alpha b_\alpha D_\alpha\Bigr).
\end{equation}
\end{enumerate}
\end{theorem}

Theorem~\ref{thm:obstruction} is the structural statement proved in
\cite[Sections~3--6]{Companion} for the data of its Proposition~2.1. The
end of Section~2 of that paper retains only $V$, $D$, $H_V$, the torsion
order, and the local canonical relation. Under these hypotheses its
Sections~3--5 prove Corollary~5.4, and its Proposition~6.8 gives the
contradiction. Hence the obstruction applies to any cycle with the
displayed properties; in particular, the construction below may start
from a \emph{log} canonical divisor with boundary.

For orientation, the companion uses the torsion normal line and the
local canonical relation to control differential forms on the completion
along $D$, with poles allowed along $D$. Cartier descent then makes the
Picard group of the punctured formal neighborhood divisible by every
power of $p$. The numerical conditions give a nonzero integer-valued
degree homomorphism on that same group, and a $p$-divisible group admits
no nonzero homomorphism to $\Z$. We take this as a theorem and supply
all its input data for a log canonical pair.

The canonical relation also implies an intersection identity used later
in the companion. A rational section realizing
\eqref{eq:canonical-cycle} has global divisor
$\sum b_\alpha D_\alpha+E$ with $\Supp E$ disjoint from $\Supp D$. Since
$D|_{D_\alpha}\equiv0$, for every Cartier divisor $A$ on $V$ we get
\begin{equation}\label{eq:canonical-intersection}
K_V D A=0.
\end{equation}

\subsection{Elementary numerical facts}

The next observation converts zero intersection numbers into restrictions
that are numerically trivial.

\begin{lemma}\label{lem:surface-nef}
Let $T$ be an integral projective surface, $H$ an ample Cartier divisor,
and $P$ a nef $\Q$-Cartier divisor on $T$. If $P H=0$, then
$P\equiv0$. Moreover, a nonzero effective Cartier divisor on $T$ has
positive intersection with $H$.
\end{lemma}
\begin{proof}
For any integral curve $C\subset T$, a sufficiently large multiple of
$H$ has an effective Cartier member $E$ containing $C$; the effective
one-cycle of $E$ has positive coefficient along $C$. Every component of
that cycle has nonnegative $P$-degree, while $PE=0$, so $PC=0$. This
proves the first assertion. The second is positivity of the ample degree
of each component of the effective one-cycle of a nonzero Cartier
divisor.
\end{proof}

In particular, if $N$ is a nonzero effective nef Cartier divisor on a
projective threefold and $N^2H=0$ for an ample Cartier divisor $H$, then
\begin{equation}\label{eq:restriction-zero}
N|_T\equiv0\qquad\text{for every prime component }T\text{ of }N.
\end{equation}
Indeed, writing $N=\sum a_T T$ with $a_T>0$, the nonnegative numbers
$NHT$ have positive-coefficient sum $N^2H=0$, so each $NHT=0$, and
Lemma~\ref{lem:surface-nef} gives $N|_T\equiv0$.

\subsection{Reduction to a terminal underlying variety}

We use three results of Xu for projective lc threefolds in characteristic
greater than three: nonvanishing for a pseudoeffective log canonical
divisor, termination of flips in the pseudoeffective case, and abundance
when the divisor is nef of nef dimension at most two
\cite[Theorems~1.4, 1.5, and~1.8]{XuNonvanishing}. All permit rational
boundaries. Relative MMPs and crepant $\Q$-factorial dlt modifications
are available in this range, including characteristic five
\cite[Theorem~1.2]{HW}, \cite[Theorems~2.20 and~2.27]{XuNuTwo}. We also
use projective resolution of threefolds preserving the regular locus,
and resolution of divisor supports preserving their complement on a
regular variety; the required forms are recalled from
\cite{CPFirst,CP,CJS} in \cite[Section~2.1]{Companion}.

\begin{proposition}\label{prop:reduction}
To prove Theorem~\ref{thm:main} it suffices to rule out the following
data: an uncountable algebraically closed field $k$ of characteristic
$p>3$, a terminal $\Q$-factorial projective threefold $X$, an lc pair
$(X,B)$, and a nonzero effective Cartier divisor
\begin{equation}\label{eq:N0}
N_0\sim m(K_X+B),\qquad m\in\Z_{>0},
\end{equation}
such that $mK_X$ and $mB$ are Cartier, and $N_0$ is nef with
$\nu(N_0)=1$ and $n(N_0)=3$.
\end{proposition}
\begin{proof}
First extend the field to an uncountable algebraically closed one.
Integrality and normality persist, and a log resolution with its
discrepancy computation shows the base-changed pair is still lc. The
canonical divisor is defined by a rational top form on the smooth locus,
so the log canonical class also base changes. Nefness persists by ample
approximation, and the prescribed intersection numbers are unchanged.

Semiampleness descends along this extension. After making a generated
multiple divisible by the original Cartier index, it is the base change
of a line bundle on the original variety. Global sections commute with
field extension, and faithful flatness detects surjectivity of the
evaluation map. We may thus work over the enlarged field.

Write $L=K_X+B$. If $n(L)\le2$, abundance in nef dimension at most two
applies. Suppose $n(L)=3$. Take a crepant projective $\Q$-factorial dlt
modification $(X_d,B_d)$ and resolve $X_d$ projectively. Run a relative
canonical MMP of this resolution over $X_d$; the MMP with scaling applies
by \cite[Theorem~1.2]{HW}. Its output is not of fiber type over the
three-dimensional base, and it gives a projective birational morphism
$g:X_t\to X_d$ with $X_t$ terminal and $\Q$-factorial and $K_{X_t}$ nef
over $X_d$. Terminality is preserved by a canonical MMP: discrepancies do
not decrease, and a divisor contracted in a negative divisorial step has
strictly positive ordinary discrepancy over the output. Relative
pseudoeffectivity causes no obstruction for this birational morphism,
since adding a large pullback of an ample divisor makes the canonical
class big.

The exceptional divisor $F=K_{X_t}-g^*K_{X_d}$ is $g$-nef, so the
negativity lemma gives $F\le0$. Since $X_d$ is $\Q$-factorial, the
divisor $B_t=g^*B_d-F$ is effective and satisfies
$K_{X_t}+B_t=g^*(K_{X_d}+B_d)$. Crepancy makes $(X_t,B_t)$ lc.

The new log canonical class is nef of numerical dimension one and nef
dimension three. For the square, note that $L^2T=0$ for every prime
surface $T$ on the original variety: place $T$ in an effective member of
a multiple of the ample divisor in the hypothesis and use nonnegativity
of $L^2$ on its components. The projection formula gives the vanishing
upstairs. Numerical nontriviality persists because a curve downstairs is
dominated by a curve upstairs, and the no-zero-degree-curve property
persists over the birational isomorphism locus.

Relabel this pair as $(X,B)$. By nonvanishing
\cite[Theorem~1.4]{XuNonvanishing}, a positive multiple of $K_X+B$ has an
effective representative $N_0$; it is nonzero because $K_X+B$ is not
numerically trivial. Increasing the multiple makes $N_0$, $mK_X$, and
$mB$ Cartier, as required.
\end{proof}

From now until the proof of Theorem~\ref{thm:main} we work with the data
of Proposition~\ref{prop:reduction}. Fix a very ample divisor $H$ on $X$.
Then $N_0H^2>0$, $N_0^2H=0$, and \eqref{eq:restriction-zero} holds. The
next task is to establish the same numerical triviality on the boundary
components.

\section{Separating the boundary from the log canonical cycle}
\label{sec:boundary}

The numerical relation $N_0\sim m(K_X+B)$ controls the \emph{log}
canonical divisor, whereas Theorem~\ref{thm:obstruction} needs a relation
for a power of the \emph{canonical} bundle. The gap is the boundary $B$.
We now show that the boundary contributes only components of the cycle or
divisors entirely disjoint from it. The argument uses maximal nef
dimension and quantitative bend-and-break, and does not involve the MMP.

\begin{proposition}[Boundary separation]\label{prop:separation}
Let $X$ be a normal $\Q$-factorial projective threefold over an
uncountable algebraically closed field, and let $B$ be an effective
$\Q$-divisor. Suppose $N$ is a nonzero effective nef Cartier divisor with
$\nu(N)=1$, $n(N)=3$, and $N\sim_{\Q}a(K_X+B)$ for some rational $a>0$.
Then for every ample Cartier divisor $H$ one has
\begin{equation}\label{eq:boundary-zero}
B N H=0.
\end{equation}
Consequently $N$ is numerically trivial on each prime component of $B$,
and every such component is either contained in $\Supp N$ or disjoint
from $\Supp N$.
\end{proposition}
\begin{proof}
Replace $H$ by a very ample multiple. The number $BNH$ is nonnegative.
If it were positive, then $B$ has a component on which $N$ is not
numerically trivial, and the identity
$(K_X+B)NH=N^2H/a=0$ would give $K_X N H<0$.

For each sufficiently large integer $l$, choose positive integers
$u_l,v_l$ such that $u_lH$ and $v_l(H+lN)$ are very ample. The general
complete intersection of members of these two linear systems is an
integral curve $C_l$ contained in the smooth locus of $X$. Indeed,
normality makes the singular locus of $X$ at most one-dimensional; a
general member of the first system meets that locus in finitely many
points, and a general member of the second avoids them. Bertini on the
smooth open then gives the integral curve. Its intersection numbers are
\[
NC_l=u_l v_l\,NH^2,\qquad
-K_XC_l=u_l v_l\bigl(-K_XH^2-lK_XNH\bigr),
\]
so
\begin{equation}\label{eq:bend-ratio}
\frac{NC_l}{-K_XC_l}
=\frac{NH^2}{-K_XH^2-lK_XNH}\longrightarrow0,
\end{equation}
and $-K_XC_l>0$ for large $l$.

The quantitative bend-and-break theorem of Jovinelly, Lehmann, and Riedl
\cite[Theorem~1.1]{JLR} applies to an integral curve in the smooth locus
of a projective variety over an algebraically closed field, with negative
canonical degree, and any nef Cartier test divisor. In dimension three it
supplies, through every closed point of $C_l$, a rational curve $\Gamma$
with
\[
0\le N\Gamma\le4\,\frac{NC_l}{-K_XC_l}.
\]
Fix $l$ so large that the right-hand side is less than one. Since $N$ is
Cartier, it follows that $N\Gamma=0$.

These complete intersections pass through points to which the
very-general-point condition applies. For the two fixed very ample
systems, the incidence of a point of $X$ and a pair of members through it
is a product of projective bundles over $X$, hence irreducible and
dominating $X$. Restricting to the nonempty open set of pairs giving the
curves just considered leaves a dense open incidence and a dominant
evaluation map. Its image is constructible and contains a nonempty open
subset of $X$. Over the uncountable field this open contains a closed
point outside the countable union excluded by $n(N)=3$. The curve
$\Gamma$ through that point is a curve of $N$-degree zero through a very
general point, contradicting $n(N)=3$. This proves
\eqref{eq:boundary-zero}.

Write $B=\sum b_T T$ with $b_T>0$. All numbers $NHT$ are nonnegative, so
\eqref{eq:boundary-zero} gives $NHT=0$ for each $T$, and
Lemma~\ref{lem:surface-nef} gives $N|_T\equiv0$. If $T$ is not a
component of $N$, the canonical section of $\cO_X(N)$ restricts to a
nonzero section on the integral surface $T$. Were $T\cap\Supp N$
nonempty, its zero scheme would be a nonzero effective Cartier divisor on
$T$, with positive ample degree, contradicting $N|_T\equiv0$. Hence the
two supports are disjoint.
\end{proof}

Apply Proposition~\ref{prop:separation} to $N_0$ and set
\begin{equation}\label{eq:Pout}
P_{\mathrm{out}}=
\bigcup_{\substack{T\text{ a prime component of }B\\T\not\subset\Supp N_0}}T.
\end{equation}
This is a closed subset disjoint from $\Supp N_0$. Thus
$\Supp B\subset\Supp N_0\cup P_{\mathrm{out}}$ is a union of two disjoint
closed sets. We keep this separation while isolating one component of
$N_0$ in the next section.

\section{Isolating a log canonical component}
\label{sec:isolation}

The effective divisor $N_0$ may have several intersecting components. We
now make one of them disjoint from the others while keeping the pullback
of the nef divisor. The construction adapts \cite[Section~2.3]{Companion}:
raise the boundary to a log canonical threshold, lower the coefficient of
one component, and run an MMP that is trivial for the restored log
canonical divisor. The resulting nefness forces the desired disjointness.

\begin{proposition}\label{prop:isolation}
Let $k$ be an algebraically closed field of characteristic $p>3$. Let $X$
be a terminal $\Q$-factorial projective threefold, let $(X,B)$ be log
canonical with rational boundary, and let $0\ne N_0\sim m(K_X+B)$ be an
effective nef Cartier divisor, with $m$ a positive integer. Assume $N_0$
is numerically trivial on each component of its support, and every
component of $B$ is contained in or disjoint from $\Supp N_0$. Let
$P_{\mathrm{out}}$ be the union of the latter components and put
\[
U_X=X_{\mathrm{reg}}\setminus\bigl(\Supp N_0\cup P_{\mathrm{out}}\bigr).
\]
Then there exist a projective $\Q$-factorial klt threefold $Y$ birational
to $X$, a rational boundary $G_Y$, an effective nef $\Q$-Cartier divisor
$N_Y$, and a prime divisor $S_Y$ with the following properties.
\begin{enumerate}
\item The pair $(Y,G_Y)$ is log canonical, $S_Y$ has coefficient one in
$G_Y$, and $s:=\operatorname{mult}_{S_Y}N_Y$ is positive. For every
rational $\delta>0$,
\begin{equation}\label{eq:isolation-nef}
K_Y+G_Y\sim_{\Q}\tau N_Y
\qquad\text{and}\qquad
K_Y+G_Y-\delta S_Y\ \text{is nef},
\end{equation}
for $\tau>0$ specified at the end.
\item The divisor $N_Y$ is numerically trivial on every component of its
support, and every component of $G_Y$ is contained in or disjoint from
$\Supp N_Y$.
\item The divisor $S_Y$ is disjoint from every other component of $N_Y$
and from $\Supp(G_Y-S_Y)$. In particular,
\begin{equation}\label{eq:isolation-local}
N_Y=sS_Y,\quad G_Y=S_Y,\quad
K_Y+S_Y\sim_{\Q}\tau s\,S_Y
\end{equation}
on a neighborhood of $S_Y$, and $S_Y|_{S_Y}\equiv0$.
\item The birational correspondence identifies $U_X$ with an open subset
of $Y$. On a smooth projective common resolution $q:V_0\to X$,
$r:V_0\to Y$ that is an isomorphism over this open,
\begin{equation}\label{eq:isolation-pullbacks}
q^*N_0=r^*N_Y,
\end{equation}
and the same equality holds after passing to a further common resolution.
\end{enumerate}
One may take $\tau=m^{-1}+\lambda$, where
$\lambda=\lct_{(X,B)}(N_0)\ge0$.
\end{proposition}

\begin{proof}
\emph{A coefficient-one component over $\Supp N_0$.}
On a log resolution of $(X,B+N_0)$ the log canonical threshold is the
minimum of the finitely many ratios
\[
\frac{a(E,X,B)}{\operatorname{ord}_E(N_0)}
\qquad\bigl(\operatorname{ord}_E(N_0)>0\bigr),
\]
where $a(E,X,B)$ is the log discrepancy and the divisors considered
include the strict transforms of the components of $N_0$. Thus $\lambda$
is finite and rational, possibly zero, and $(X,B+\lambda N_0)$ is log
canonical. There is a log canonical place $E$ of this pair with
$\operatorname{ord}_E(N_0)>0$.

Take a projective crepant $\Q$-factorial dlt modification
$\mu:(W,G)\to(X,B+\lambda N_0)$ with every extracted divisor of
coefficient one \cite[Theorem~2.27]{XuNuTwo}. Set
\[
N=\mu^*N_0,\qquad \tau=m^{-1}+\lambda.
\]
The relation
\begin{equation}\label{eq:isolation-restored}
K_W+G\sim_{\Q}\tau N
\end{equation}
holds since $\mu$ is crepant. The divisor $N$ is nef and numerically
trivial on every component of its support: a curve in that support maps
to a point or to a curve in $\Supp N_0$, and the projection formula
finishes.

\emph{Support separation on $W$.}
Every component of $G$ is contained in or disjoint from $\Supp N$. For
strict transforms of components of $B$ this is the hypothesis, while
components coming from $\lambda N_0$ are contained in the support. An
extracted divisor has center in $\Supp N_0\cup P_{\mathrm{out}}$, since
$X$ is klt and the boundary vanishes outside this union; its irreducible
center lies in one of these two disjoint closed sets. In the first case
its order on $N_0$ is positive, so it is contained in $\Supp\mu^*N_0$; in
the second it is disjoint from that support.

The center on $W$ of the place $E$ lies in $\Supp N$ and, by crepancy and
the dlt property, in $\lfloor G\rfloor$. Choose a component $S$ of
$\lfloor G\rfloor$ containing this center. The preceding separation gives
$S\subset\Supp N$. This covers $\lambda=0$ as well; it does not require
all components of $N$ to occur in $G$.

\emph{The modification is an isomorphism off the support.}
The map $\mu$ has no exceptional divisors outside
$\Supp N_0\cup P_{\mathrm{out}}$, and a projective small birational
morphism to a $\Q$-factorial variety is an isomorphism. To see the latter,
push down a relatively ample Cartier divisor. A Cartier multiple of its
pushdown pulls back to the same multiple upstairs, since there are no
exceptional divisors; it has degree zero on every contracted curve,
contradicting relative ampleness if such a curve existed.

\emph{The partial MMP.}
Choose a rational $\epsilon>0$ small enough that
\begin{equation}\label{eq:isolation-effective}
G-\epsilon S\ge0,\qquad \tau N-\epsilon S\ge0.
\end{equation}
Apply the partial MMP with lowered boundary $G-\epsilon S$ and restoration
divisor $\epsilon S$, in the form of
\cite[Proposition~2.21, Definition~2.23, and Lemma~2.24]{XuNuTwo}. Its
steps are $(K_W+G-\epsilon S)$-negative and $(K_W+G)$-trivial. The lowered
pair is dlt and its log canonical divisor is pseudoeffective by
\eqref{eq:isolation-restored} and \eqref{eq:isolation-effective}, so the
required termination is supplied by \cite[Theorem~1.5]{XuNonvanishing}.
The threefold MMP package used here, including in characteristic five, is
recalled in \cite[Section~2.3]{Companion}.

The partial-MMP conclusion is either a Mori fiber contraction for the
lowered divisor, trivial for the restored divisor, or nefness of the
restored divisor minus $\delta$ times the transform of $S$ for every small
rational $\delta>0$. We track $S$, the pullbacks of $N$, and support
separation through the steps, and then exclude the fiber alternative.

\emph{Tracking.}
Write $W_i\dashrightarrow W_{i+1}$ for a step, with transforms
$G_i,N_i,S_i$. The equality $K_{W_i}+G_i\sim_{\Q}\tau N_i$ persists by
pushforward, so the contracted ray $R_i$ satisfies
\begin{equation}\label{eq:isolation-ray}
N_i\cdot R_i=0,\qquad S_i\cdot R_i>0.
\end{equation}
Thus $S_i$ cannot be the exceptional divisor of a divisorial step: the
negativity lemma would then give a contracted curve of negative
$S_i$-degree. Hence $S$ survives throughout the MMP.

Let $a:T\to W_i$ and $b:T\to W_{i+1}$ be a common resolution of a step.
The divisor $F=a^*N_i-b^*N_{i+1}$ is $b$-exceptional. If a curve on $T$ is
contracted by $b$, its image on $W_i$ is a point or is a curve contracted
to the common target, so $F$ has degree zero on every such curve by
\eqref{eq:isolation-ray}. Applying the negativity lemma to $F$ and $-F$
gives
\begin{equation}\label{eq:isolation-step-pullback}
a^*N_i=b^*N_{i+1}.
\end{equation}
This covers divisorial contractions and flips. The same argument for the
full log divisors gives
$a^*(K_{W_i}+G_i)=b^*(K_{W_{i+1}}+G_{i+1})$, so the restored pair stays
log canonical.

Equation~\eqref{eq:isolation-step-pullback} preserves nefness and
numerical triviality on the support. For the latter, lift a curve in
$\Supp N_{i+1}$ to a curve on $T$ dominating it. Its image on $W_i$ lies
in $\Supp N_i$ or is a point, because the inverse image of the support of
an effective $\Q$-Cartier divisor is exactly the support of its pullback.
The projection formula gives degree zero. A boundary component contained
in $\Supp N_i$ stays contained in the transformed support. A component
disjoint from $\Supp N_i$ has strict transform on $T$ disjoint from
$a^*N_i$, hence from $b^*N_{i+1}$, and since it maps onto its image on
$W_{i+1}$, the transformed component is disjoint from $\Supp N_{i+1}$.

\emph{The open complement is preserved.}
Every contracted curve has negative degree against the effective divisor
$\tau N_i-\epsilon S_i$, so lies in $\Supp N_i$. A connected
positive-dimensional projective fiber contains a curve through each of its
closed points, so a fiber meeting the complement of $\Supp N_i$ consists
of a single point and does not meet the support. Properness and normality
show the contraction is an isomorphism from this complement onto an open
subset of its target. For a flip, this open in the target is
$\Q$-factorial, being isomorphic to the corresponding open of $W_i$, so
the new small morphism is also an isomorphism over it.

\emph{No Mori fiber output.}
The effective representative $\tau N_i-\epsilon S_i$ persists on each
model, as does $\tau N_i-tS_i$ for every rational $0<t\le\epsilon$. A
curve in a fiber through a general point outside this effective support has
nonnegative degree, whereas the driving divisor of a Mori fiber
contraction is negative on every contracted curve. This rules out the
fiber alternative for the lowered class and for every smaller
perturbation of the restored class.

\emph{The output model.}
Let $Y$ be the resulting model. The partial-MMP conclusion gives
\eqref{eq:isolation-nef} for some rational $0<\delta<\epsilon$. Negative
MMP steps preserve the $\Q$-factorial dlt pair with lowered boundary, so
$(Y,G_Y-\epsilon S_Y)$ is dlt. Its $\Q$-factorial underlying variety $Y$
is then klt: every place of zero log discrepancy for a dlt pair has center
in the reduced boundary, and removing that effective $\Q$-Cartier
boundary strictly increases the discrepancy. Dlt preservation here is
used for the lowered boundary; for the restored boundary we use the
crepant comparison just proved.

\emph{Nefness isolates $S_Y$.}
Let $T\ne S_Y$ be a component of $N_Y$. Numerical triviality of $N_Y|_T$
and \eqref{eq:isolation-nef} make $-S_Y|_T$ nef. On the other hand
$S_Y|_T$ is an effective $\Q$-Cartier divisor; if it were nonzero, its
intersection with an ample Cartier divisor on the integral projective
surface $T$ would be positive, contradicting anti-nefness. Hence
$S_Y\cap T=\varnothing$. Each other component of $G_Y$ is either such a
component of $N_Y$ or is disjoint from its support. This proves the stated
separation and the local identities \eqref{eq:isolation-local}. The
equality $N_Y|_{S_Y}\equiv0$ gives $S_Y|_{S_Y}\equiv0$.

\emph{The common open.}
All models share the open $U_X$. Resolve the graph of the correspondences
among them, keeping this smooth open unchanged, to obtain $V_0$.
Composing \eqref{eq:isolation-step-pullback} with $N=\mu^*N_0$ proves
\eqref{eq:isolation-pullbacks}, and pullback preserves the equality on any
further common resolution.
\end{proof}

For adjunction we replace the isolated log canonical pair by a dlt pair.
The klt property of $Y$ ensures that every divisor extracted below belongs
to the total transform of $S_Y$.

\begin{corollary}\label{lem:dlt-boundary}
With the notation of Proposition~\ref{prop:isolation}, there is a
projective crepant $\Q$-factorial dlt modification
$f:(Y_d,\Sigma)\to(Y,S_Y)$ such that $\Sigma$ is reduced,
\begin{equation}\label{eq:dlt-support}
\Supp\Sigma=\Supp f^*S_Y,
\end{equation}
and $f$ is an isomorphism off $S_Y$. On a neighborhood of $\Sigma$,
\begin{equation}\label{eq:dlt-local}
K_{Y_d}+\Sigma\sim_{\Q}\tau s\,f^*S_Y.
\end{equation}
Moreover $f^*S_Y$ is numerically trivial on each component of $\Sigma$,
and $Y_d$ shares the open $U_X$ with $X$ and $Y$.
\end{corollary}
\begin{proof}
The pair $(Y,S_Y)$ is log canonical, since $G_Y\ge S_Y$ and $(Y,G_Y)$ is
log canonical. Take a crepant dlt modification extracting only log
canonical places. Its boundary $\Sigma$ consists of the strict transform
of $S_Y$ and its exceptional divisors, all of coefficient one. Every
extracted divisor has center in $S_Y$: outside $S_Y$ the pair is the klt
variety $Y$. Such a divisor has positive order on $S_Y$, which proves
\eqref{eq:dlt-support}. There are no exceptional divisors off $S_Y$, and
the small-morphism argument of the preceding proof shows $f$ is an
isomorphism there.

Crepancy and \eqref{eq:isolation-local} give \eqref{eq:dlt-local}. Every
curve in $\Sigma$ maps to a point or to a curve in $S_Y$, where $S_Y$ is
numerically trivial, so the projection formula gives the numerical
assertion. Under the identification of Proposition~\ref{prop:isolation},
$U_X$ is disjoint from $S_Y$ and hence is unchanged by $f$.
\end{proof}

\section{Torsion on the full boundary cycle}
\label{sec:torsion}

The reduced dlt boundary $\Sigma$ of Corollary~\ref{lem:dlt-boundary} is a
surface pair to which adjunction and surface abundance apply. The
prepared-cycle obstruction, however, needs torsion on the \emph{effective
Cartier divisor} including its multiplicities. We first give the
extension argument that passes from the reduced surface to the possibly
nonreduced Cartier thickening.

\begin{lemma}[{\cite[Lemma~2.4]{Companion}}]\label{lem:torsion}
Let $T$ be a normal projective threefold over an algebraically closed
field of characteristic $p>0$, and let $E$ be a nonzero effective Cartier
divisor on $T$. Suppose $E$ is numerically trivial on every integral
component of its support. If
\[
\cO_T(E)|_{E_{\mathrm{red}}\setminus Z}
\simeq\cO_{E_{\mathrm{red}}\setminus Z}
\]
for a finite set $Z$ of closed points, then $\cO_E(E)$ is torsion.
\end{lemma}
\begin{proof}
We reproduce the proof because both the finite exceptional set and the
nonreduced divisor occur in our application. Replace $Z$ by
$Z\cap\Supp E$. At $z\in Z$ write $R=\cO_{T,z}$, with maximal ideal
$\mathfrak m$, and let $t_z$ be a local equation of $E$. Normality gives
\[
H^1_{\mathfrak m}(R)=0,
\qquad
\ell_R\bigl(H^2_{\mathfrak m}(R)\bigr)<\infty.
\]
The second assertion uses the dimension-three hypothesis. Present $R=A/I$
with $A$ regular local of dimension $n$, essentially of finite type over
the field. Each nonmaximal localization of $R$ is normal of dimension at
most two, hence Cohen--Macaulay. Catenarity gives
$\dim A_{\mathfrak q}-\dim R_{\mathfrak p}=n-3$ for a prime
$\mathfrak q$ above $\mathfrak p$, so the Auslander--Buchsbaum formula
gives projective dimension $n-3$ over $A_{\mathfrak q}$. Thus
$\operatorname{Ext}^{n-2}_A(R,A)$ is supported only at the closed point.
It is a finite module, hence of finite length, and local duality gives the
assertion for $H^2_{\mathfrak m}(R)$.

Choose $j\ge1$ such that $t_z^{j-1}$ annihilates $H^2_{\mathfrak m}(R)$
at every $z\in Z$; a single $j$ works because $Z$ is finite. For every
integer $u$, restriction sequences give a commutative diagram
\[
\begin{tikzcd}[column sep=small]
0\arrow[r]&\cO_T((u-j)E)\arrow[r]\arrow[d]
 &\cO_T(uE)\arrow[r]\arrow[d,equal]
 &\cO_{jE}(uE)\arrow[r]\arrow[d]&0\\
0\arrow[r]&\cO_T((u-1)E)\arrow[r]
 &\cO_T(uE)\arrow[r]
 &\cO_E(uE)\arrow[r]&0.
\end{tikzcd}
\]
In local trivializations the left vertical map is multiplication by
$t_z^{j-1}$. Local cohomology with support in the finite set $Z$ is the
direct sum of the corresponding local groups. The vanishing of the ambient
$H^1_Z$ makes the connecting maps from the $H^1_Z$ of the rightmost terms
to the ambient $H^2_Z$ injective. Naturality and the choice of $j$ give
\begin{equation}\label{eq:torsion-obstruction-zero}
H^1_Z\bigl(T,\cO_{jE}(uE)\bigr)
\longrightarrow H^1_Z\bigl(T,\cO_E(uE)\bigr)
\quad\text{is zero for every }u\in\Z.
\end{equation}
The same $j$ works for every twist: locally the invertible sheaves in the
diagram are free of rank one, and the annihilating map is still
multiplication by $t_z^{j-1}$.

Fix this thickening $jE$. The ideal of $E_{\mathrm{red}}\setminus Z$ in
$jE\setminus Z$ is nilpotent. Lift a chosen generator of
$\cO_T(E)|_{E_{\mathrm{red}}\setminus Z}$ to local generating sections on
a finite open cover of $jE\setminus Z$. On overlaps the lifts differ by
sections in that nilpotent ideal times the line bundle. For $u=p^e$
sufficiently large, their $u$th tensor powers agree: in local frames this
is the identity $(a+b)^{p^e}=a^{p^e}+b^{p^e}$ with $b^{p^e}=0$. They
therefore glue to a generator
\[
s_j\in H^0\bigl(jE\setminus Z,\cO_{jE}(uE)\bigr).
\]
Let $s$ be its restriction to $E\setminus Z$. The obstruction to extending
$s$ over $E$ is the image of the obstruction to extending $s_j$ over
$jE$. Equation~\eqref{eq:torsion-obstruction-zero}, applied to the exact
sequences for sections with support in $Z$, kills that image, so $s$
extends to a section of $\cO_E(uE)$.

This extended section has no zeros. On each integral projective surface
component $F$ of $E_{\mathrm{red}}$ it is nonzero because it is a generator
off $Z$. If its restriction to $F$ vanished anywhere, its zero scheme would
be a nonempty effective Cartier divisor on $F$, with positive intersection
with an ample divisor, contradicting the numerical triviality of
$\cO_F(uE)$. The argument uses only integrality of $F$, not normality.
The section is thus a generator on the reduction, and then on $E$, since
an element invertible modulo a nilpotent ideal is invertible. Hence
$\cO_E(uE)\simeq\cO_E$.
\end{proof}

We apply this lemma to the reduced dlt model obtained by isolation. Recall
that $s$ is the multiplicity of $S_Y$ in $N_Y$ and $\tau>0$ satisfies
$K_Y+G_Y\sim_{\Q}\tau N_Y$.

\begin{proposition}\label{prop:boundary-torsion}
For the projective crepant dlt modification
$f:(Y_d,\Sigma)\to(Y,S_Y)$ of Corollary~\ref{lem:dlt-boundary}, there is a
positive Cartier multiple $M$ of $f^*S_Y$ such that
\[
\Supp M=\Sigma,
\qquad M|_{\Sigma_i}\equiv0\ \text{for every component }\Sigma_i,
\qquad \cO_M(M)\text{ is torsion}.
\]
\end{proposition}
\begin{proof}
The relations furnished by isolation are
\begin{equation}\label{eq:adjunction-input}
K_{Y_d}+\Sigma\sim_{\Q}\tau s\,f^*S_Y
\quad\text{on a neighborhood of }\Sigma,
\qquad f^*S_Y|_{\Sigma_i}\equiv0.
\end{equation}
Let $\sigma:\Sigma'\to\Sigma$ be the finite $S_2$-ification; it is
unchanged away from a finite set $Z\subset\Sigma$
\cite[Proposition~2.14]{XuNuTwo}. Reduced-boundary adjunction gives an
effective rational divisor $\Delta'$ such that $(\Sigma',\Delta')$ is a
projective semi-log-canonical surface pair and
\begin{equation}\label{eq:slc-adjunction}
K_{\Sigma'}+\Delta'
\sim_{\Q}\sigma^*((K_{Y_d}+\Sigma)|_\Sigma)
\sim_{\Q}\tau s\,\sigma^*(f^*S_Y|_\Sigma).
\end{equation}
We use adjunction in the form of \cite[Lemma~2.19]{XuNuTwo}; see also
\cite[Section~2.5]{Companion}. The $S_2$-ification is needed because the
reduced boundary need not satisfy $S_2$. Its codimension-one
singularities are nodes, and on its normalization the adjunction boundary
includes the conductor with coefficient one, so the normalized pair is
log canonical. These are the semi-log-canonical conditions needed for
surface abundance.

The surface $\Sigma'$ is projective because it is finite over $\Sigma$.
The last class in \eqref{eq:slc-adjunction} is numerically trivial, by
\eqref{eq:adjunction-input} and the projection formula for curves.
Posva's abundance theorem for projective semi-log-canonical surfaces in
positive characteristic makes $K_{\Sigma'}+\Delta'$ semiample
\cite[Theorem~3.1]{Posva}. A semiample numerically trivial rational line
bundle is $\Q$-linearly trivial: a globally generated Cartier multiple
defines a morphism contracting every curve, hence constant on each
connected component, whose line bundle is trivial there.

Since $\tau s$ is positive and rational, clearing denominators in
\eqref{eq:slc-adjunction} yields a positive Cartier multiple $M$ of
$f^*S_Y$ with $\sigma^*\cO_\Sigma(M)$ trivial. In particular
\[
\cO_{Y_d}(M)|_{\Sigma\setminus Z}\simeq\cO_{\Sigma\setminus Z}.
\]
The divisor $M$ is effective, its reduction is $\Sigma$, and it is
numerically trivial on every component of its support. Applying
Lemma~\ref{lem:torsion} on the normal projective threefold $Y_d$ proves
torsion on the full Cartier scheme $M$.
\end{proof}

\section{Construction of the prepared cycle}
\label{sec:cycle}

We combine the isolated boundary with the original log canonical section.
The boundary supplies a divisor with torsion normal line; the section
supplies a canonical relation near its support. A finite cover removes the
prime-to-$p$ part of the canonical exponent. This is the preparation of
\cite[Sections~2.7--2.8]{Companion}, keeping the support of the original
boundary throughout.

\begin{proposition}\label{prop:prepared}
Let $(X,B,N_0)$ be the data of Proposition~\ref{prop:reduction} in the
maximal-nef-dimension case. Then there exist a smooth projective threefold
$V$, a very ample divisor $H_V$, and a nonzero effective Cartier divisor
\[
D=\sum_\alpha m_\alpha D_\alpha
\]
having all the properties of Theorem~\ref{thm:obstruction}. In particular
its support is connected with simple normal crossings,
$\gcd_\alpha(m_\alpha)=1$, and $\cO_D(D)$ is torsion.
\end{proposition}
\begin{proof}
Equation~\eqref{eq:restriction-zero} and Proposition~\ref{prop:separation}
give the hypotheses of Proposition~\ref{prop:isolation}. Choose its
isolated model $(Y,G_Y,N_Y,S_Y)$ and the reduced dlt modification
$f:(Y_d,\Sigma)\to(Y,S_Y)$ of Corollary~\ref{lem:dlt-boundary}. Let $M$ be
the positive Cartier multiple of $f^*S_Y$ supplied by
Proposition~\ref{prop:boundary-torsion}.

\emph{The canonical section and its support.}
Let $V_0$ be a smooth projective common resolution of $X$ and $Y_d$,
dominating the intervening models. Write
\[
q:V_0\to X,\qquad h_0:V_0\to Y_d,\qquad r=f\circ h_0:V_0\to Y.
\]
These models share the open
\[
U_X=X_{\mathrm{reg}}\setminus(\Supp N_0\cup P_{\mathrm{out}}),
\]
with $P_{\mathrm{out}}$ the union of boundary components disjoint from
$\Supp N_0$. Choose the common resolution to be an isomorphism over this
open; projective resolution and resolution of divisor supports in
dimension three permit this \cite{CP,CJS}. The pullback equalities
through the partial MMP give
\begin{equation}\label{eq:cycle-common-pullbacks}
q^*N_0=r^*N_Y.
\end{equation}

The integer $m$ satisfies $N_0\sim m(K_X+B)$, with $mK_X$ and $mB$
Cartier. It gives a rational section $\theta$ of
$\omega_{V_0}^{\otimes m}$ with
\begin{equation}\label{eq:log-canonical-section}
\operatorname{div}(\theta)
=q^*N_0+m(K_{V_0}-q^*K_X)-m\,q^*B.
\end{equation}
The right side is an integral divisor linearly equivalent to $mK_{V_0}$.
Put $E_0=\Supp q^*N_0$. Every prime component in
\eqref{eq:log-canonical-section} is contained in $E_0$ or disjoint from
it. For $q^*B$ this is boundary separation on $X$. For a $q$-exceptional
divisor, its irreducible center lies in
\[
X\setminus U_X
=\Supp N_0\cup P_{\mathrm{out}}\cup\operatorname{Sing}X.
\]
A terminal threefold is regular in codimension two, so
$\operatorname{Sing}X$ is finite \cite[Lemma~2.3]{Companion}. The center
is thus contained in $\Supp N_0$ or disjoint from it, and taking inverse
images gives the assertion.

Let $F_0$ be the union of the components of
$\operatorname{div}(\theta)$ outside $E_0$. Then
\begin{equation}\label{eq:cover-closed-set}
Z_0:=E_0\cup\Supp\operatorname{div}(\theta)=E_0\sqcup F_0.
\end{equation}
The disjointness is the point: components of $q^*N_0$ can cancel in
\eqref{eq:log-canonical-section}, so we retain all of $E_0$ when choosing
the open set preserved by the resolution below.

\emph{Removing the prime-to-$p$ exponent.}
Write $m=p^b m'$ with $b\ge0$ and $(m',p)=1$. Choose a nonzero rational
generator $\eta$ of $\omega_{V_0}^{\otimes p^b}$ and write
$\theta=u\eta^{m'}$ with $u\in k(V_0)^\times$. Adjoin one root $v$ of
$v^{m'}=u$ and normalize $V_0$ in the resulting field. This is a finite
separable cover, possibly of degree less than $m'$. Over
$V_0\setminus\Supp\operatorname{div}(\theta)$ it is an integral component
of the torsor of $m'$th roots of the invertible section $\theta$, a
description independent of $\eta$. Since $m'$ is invertible in $k$, that
torsor is finite \'etale.

Resolve the normalization and the total inverse image of $Z_0$,
preserving the regular \'etale open over $V_0\setminus Z_0$. We obtain a
smooth projective threefold and a generically finite separable morphism
$\pi:V\to V_0$ for which the total pullback support of $N_0$ has simple
normal crossings. The morphisms comparing divisors are
\[
\begin{tikzcd}[column sep=large,row sep=large]
V\arrow[r,"\pi"] &V_0\arrow[r,"q"]\arrow[d,"h_0"']&X\\
&Y_d\arrow[r,"f"']&Y.
\end{tikzcd}
\]
Here $q,h_0,f$ are birational and $\pi$ is generically finite and
separable. Set $\rho=q\circ\pi$ and $h=h_0\circ\pi$. By
\eqref{eq:cycle-common-pullbacks},
$\rho^*N_0=(f\circ h)^*N_Y$, while the Cartier divisor of
Proposition~\ref{prop:boundary-torsion} pulls back as $h^*M$.

The rational section $v\eta$ is a section of
$\pi^*\omega_{V_0}^{\otimes p^b}$. The determinant of the differential is
a nonzero section of the rational line
$\omega_V\otimes\pi^*\omega_{V_0}^{-1}$, denoted $J_\pi$; it is nonzero at
the generic point by separability. Consequently
\begin{equation}\label{eq:canonical-root-section}
\psi=(v\eta)\,J_\pi^{p^b}
\quad\text{is a rational section of }
\omega_V^{\otimes p^b}.
\end{equation}
Both factors are invertible on $\pi^{-1}(V_0\setminus Z_0)$: the first is
a root of an invertible section, the second the Jacobian of an \'etale
morphism. Hence $\operatorname{div}(\psi)$ is supported on
$\pi^{-1}(Z_0)$. By \eqref{eq:cover-closed-set} each of its prime
components lies in $\Supp\rho^*N_0$ or is disjoint from that support.

\emph{A connected primitive Cartier divisor.}
Put $M_V=h^*M$ and choose a connected component $C$ of $\Supp M_V$. Since
$M$ is a positive multiple of $f^*S_Y$ and $S_Y$ is disjoint from the
other components of $N_Y$, the set $C$ is also a connected component of
$\Supp\rho^*N_0$. Write the part of $M_V$ supported on $C$ as
\begin{equation}\label{eq:primitive-cycle}
aD,\qquad D=\sum_\alpha m_\alpha D_\alpha,
\qquad a=\gcd_{D_\alpha\subset C}\operatorname{mult}_{D_\alpha}(M_V).
\end{equation}
The divisor $D$ is Cartier because $V$ is smooth, its multiplicities are
positive with gcd one, and its support is connected with simple normal
crossings.

Every curve in $\Supp M_V$ maps under $h$ to a point or to a curve in
$\Sigma$. The projection formula and Proposition~\ref{prop:boundary-torsion}
therefore make $M_V$ numerically trivial on every component of its
support. Near $C$ we have $M_V=aD$, so $D|_{D_\alpha}\equiv0$ for every
$\alpha$. A curve not contained in $\Supp D$ has nonnegative intersection
with $D$ by effectivity, while a curve in the support has intersection
zero. Hence $D$ is nef, and every very ample $H_V$ satisfies
\begin{equation}\label{eq:primitive-intersections}
D H_V^2>0,\qquad D^2 H_V=0.
\end{equation}

Torsion survives on the entire Cartier scheme. Choose $t>0$ with
$\cO_M(tM)\simeq\cO_M$. The pullback divisor $M_V$ is scheme-theoretically
$V\times_{Y_d}M$: its local equation is the pullback of a local equation
of $M$, which is nonzero since $V$ is integral and $h$ is dominant.
Pulling back the trivialization gives
$\cO_{M_V}(tM_V)\simeq\cO_{M_V}$, with no flatness assumption on $h$.
Restricting along $D\subset M_V$ and using $M_V=aD$ near $C$ gives
\begin{equation}\label{eq:primitive-torsion}
\cO_D(t\,a\,D)\simeq\cO_D.
\end{equation}
Hence $\cO_D(D)$ is torsion.

The section $\psi$ of \eqref{eq:canonical-root-section} has, on a
neighborhood $W$ of $C$, divisor supported on $D$: its other components are
disjoint from $C$ and can be removed from that neighborhood. Thus there are
integers $b_\alpha$ with
\begin{equation}\label{eq:prepared-canonical-relation}
\omega_V^{\otimes p^b}|_W
\simeq\cO_W\Bigl(\sum_\alpha b_\alpha D_\alpha\Bigr).
\end{equation}

\emph{Numerical proportionality and very general points.}
It remains to show no $D$-trivial curve passes through a very general
point of $V$. Put $N_V=\rho^*N_0$. It is a nonzero nef divisor,
numerically trivial on every component of its support by the projection
formula. Since $\Supp D$ is contained in that support,
\[
N_V H_V^2>0,\qquad N_V^2H_V=N_V D H_V=0.
\]
Together with \eqref{eq:primitive-intersections}, the Hodge index theorem
gives
\begin{equation}\label{eq:cycle-proportionality}
D\equiv cN_V,
\qquad c=\frac{D H_V^2}{N_V H_V^2}>0.
\end{equation}
We verify this is numerical equivalence on every curve, not only on a
fixed hyperplane section, following \cite[Section~2.8]{Companion}.

Let $\Gamma\subset V$ be any integral curve. For $l\gg0$ there is an
integral surface $T\in|lH_V|$ containing $\Gamma$. To see this, twist the
ideal of $\Gamma$ sufficiently far and multiply its generators by very
ample sections; the resulting linear system has no fixed divisorial
component and gives a birational map onto its image off $\Gamma$. Bertini
irreducibility gives an irreducible general member, generically reduced on
that birational open. As a Cartier divisor on smooth $V$ it has no
embedded components, hence is integral. On a smooth projective resolution
$\widetilde T\to T$, let $d,n,e$ be the pullbacks of $D|_T,N_V|_T,H_V|_T$.
Their intersections are
\[
d^2=n^2=dn=0,\qquad e^2>0,\qquad
de=l\,D H_V^2,\quad ne=l\,N_V H_V^2.
\]
Hence $d-cn$ has square zero and is orthogonal to $e$. The Hodge index
form is negative definite on the orthogonal complement of the
positive-square class $e$, so $d-cn$ is numerically zero. The projective
surjection $\widetilde T\to T$ supplies an integral curve dominating
$\Gamma$: take the closure of a closed point of the fiber over the generic
point of $\Gamma$. The projection formula then gives
$(D-cN_V)\cdot\Gamma=0$. This proves \eqref{eq:cycle-proportionality}.

Finally let $Z_i\subset X$ be the countably many proper closed subsets
excluded by the nef reduction of $N_0$. Remove from $V$ the
non-quasi-finite locus of $\rho$ and all sets $\rho^{-1}(Z_i)$. These are
proper closed subsets, since $\rho$ is dominant and generically finite,
and their complement contains very general points because $k$ is
uncountable. Every curve through a point of this complement maps to a
curve on $X$. If it were $D$-trivial, then \eqref{eq:cycle-proportionality}
and the projection formula would make its image $N_0$-trivial, contradicting
maximal nef dimension. All properties of Theorem~\ref{thm:obstruction} are
now established.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Theorem~\ref{thm:obstruction} contradicts Proposition~\ref{prop:prepared}.
The maximal-nef-dimension case is therefore impossible. By
Proposition~\ref{prop:reduction}, it remains to apply abundance in nef
dimension at most two \cite[Theorem~1.8]{XuNonvanishing}. This gives
semiampleness, and faithful descent along the algebraic closure extension
gives it over the original field.
\end{proof}

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\end{document}