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\title{Abundance in numerical dimension one\\for terminal threefolds in positive characteristic\\[2mm]
\large A reader-friendly edition}
\author{OpenAI}
\date{October 28, 2026}

\begin{document}
\maketitle

\begin{abstract}
Let $X$ be a projective $\Q$-factorial terminal threefold over an algebraically closed field of characteristic $p>3$. If the canonical divisor $K_X$ is nef and has numerical dimension one, then some positive multiple of $K_X$ has at least two independent global sections. Hence $K_X$ is semiample and has Kodaira dimension one. This edition explains the theorem, the strategy, and every main step in plain language, while keeping the mathematical details needed for a complete proof.
\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. The result is advanced, but the shape of the argument is simple enough to explain in ordinary language.

\paragraph{The goal.}
We have a geometric object $X$ called a \emph{threefold}: over the complex numbers it looks locally like $\C^3$. To each such object one attaches a line bundle $K_X$, the \emph{canonical bundle}, which records how volume and differential forms behave. A divisor is \emph{nef} if it never has negative intersection with any curve; think of it as a direction in which the space is not bending negatively. We assume $K_X$ is nef and that its \emph{numerical dimension} is $1$. This means $K_X$ is positive in one direction and zero in the other two.

Abundance predicts that some positive multiple $mK_X$ has enough global sections to define a map to a curve. We prove exactly that: there are at least two independent sections of some $mK_X$. Equivalently, the \emph{Kodaira dimension} is $1$ and $K_X$ is \emph{semiample}.

\paragraph{Why this is hard.}
The hypothesis only says $K_X$ is not negative and has one positive direction. It does not directly produce a section. The proof therefore has to rule out a hidden bad case: \emph{maximal nef dimension}. If that bad case occurred, the positive direction of $K_X$ would be locally invisible---no curve of $K_X$-degree zero would pass through a very general point. The main body of the paper shows that this bad case leads to a contradiction.

\paragraph{The contradiction in one sentence.}
Inside the bad case one builds a formal neighborhood of a cycle $D$ on a smooth model $V$. The geometry of jets and the semistability of the tangent bundle make the Picard group of that formal neighborhood divisible by every power of $p$. On the other hand, intersection with an ample divisor $H$ gives a nonzero integer-valued degree on that same Picard group. A group that is divisible by every $p^n$ cannot map nontrivially to $\Z$. Contradiction.

\paragraph{A small dictionary.}
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Term & Plain meaning \\
\midrule
Threefold & A space of dimension $3$; locally like $\A^3$ or $\C^3$. \\
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. \\
$\Q$-factorial & Some positive multiple of every divisor is Cartier. \\
Terminal & The mildest kind of singularity for the minimal model program. \\
Numerical dimension & The number of directions in which a nef divisor is positive. \\
Semiample & Some positive multiple is generated by global sections. \\
Kodaira dimension & The growth rate of spaces of pluricanonical sections. \\
Frobenius & The map $x\mapsto x^p$ in characteristic $p$. \\
Cartier descent & Recovering data from its Frobenius pullback. \\
Jet & A section remembered only up to a finite infinitesimal order. \\
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:conventions} fixes notation. Section~\ref{sec:preparation} performs the birational reduction to a smooth threefold $V$ and a connected cycle $D$. Section~\ref{sec:slopes} proves strong semistability of the tangent bundle and derives polar-part bounds. Section~\ref{sec:jets} computes finite jets along $D$, proves the limiting ratio $p/(p+1)$, and obtains $c_2(V)D=0$. Section~\ref{sec:laurent} proves the Laurent-cohomology vanishing criterion. Section~\ref{sec:cartier} uses Cartier descent to make the Picard group of the formal neighborhood $p$-divisible. Section~\ref{sec:degree} constructs the nonzero integer degree and derives the contradiction. Section~\ref{sec:descent} descends the result back to the original field.

Readers who want the shortest path can read:
\[
\text{Theorem~\ref{thm:main}}
\longrightarrow
\text{Proposition~\ref{prop:prepared-cycle}}
\longrightarrow
\text{Proposition~\ref{prop:strong-semistability}}
\longrightarrow
\text{Proposition~\ref{prop:scalar-widths}}
\longrightarrow
\text{Theorem~\ref{thm:laurent-vanishing}}
\longrightarrow
\text{Proposition~\ref{prop:picard-divisibility}}
\longrightarrow
\text{Proposition~\ref{prop:laurent-degree-contradiction}}.
\]
The intervening lemmas explain why each arrow is valid.

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

\subsection{The theorem}

\begin{theorem}\label{thm:main}
Let $X$ be a normal projective $\Q$-factorial threefold over an algebraically closed field $k$ of characteristic $p>3$. Suppose that $X$ has terminal singularities and that $K_X$ is nef with numerical dimension
\[
\nu(K_X)=1.
\]
Then some positive Cartier multiple of $K_X$ has at least two independent global sections. In particular,
\[
\kappa(X,K_X)=1
\]
and $K_X$ is semiample.
\end{theorem}

Recall that $\nu(L)=1$ for a nef divisor $L$ means
\[
L\cdot H^2>0,
\qquad
L^2\cdot H=0
\]
for an ample Cartier divisor $H$. The theorem makes no assumption on the irregularity, the Albanese map, or the first Betti number.

Geometrically, semiampleness gives a morphism
\[
f:X\longrightarrow C
\]
with connected fibers to a normal projective curve $C$, together with an ample divisor $A$ on $C$ such that
\[
mK_X\sim f^*A
\]
for some $m>0$. The image of the map defined by a globally generated pluricanonical multiple is a curve because $\nu(K_X)=1$; the Stein factorization produces $f$.

\subsection{Background and context}

Over $\C$, Miyaoka proved this case of threefold abundance by studying effective pluricanonical divisors and their primitive connected cycles \cite{miyaoka1988}. Kawamata gave another proof using the log minimal model program and formal thickenings \cite{kawamata1992}. Over fields of positive characteristic, the minimal model program for threefolds is now available in the necessary generality: Hacon--Xu \cite{haconxu2015} and Birkar \cite{birkar2016} established flips and minimal models in large characteristic, and the relative log MMP was proved by Hashizume--Nakamura--Tanaka \cite{hnt2020} and Hacon--Witaszek \cite{hw2022}.

Nonvanishing and abundance in neighboring cases are due to Xu--Zhang \cite{xuzhang2019}, Witaszek \cite{witaszek2021}, Zhang \cite{zhang2020}, Xu \cite{xu2024,xu2026}, and Posva \cite{posva2024}. The remaining gap addressed here is a possible effective canonical divisor of numerical dimension one whose nef dimension is maximal. Theorem~\ref{thm:main} rules out that possibility.

The proof is in the tradition of the positive-characteristic foliation method of Xu--Zhang \cite{xuzhang2019}. It uses Langer's Bogomolov inequality \cite{langer2004,langeradd2004}, surface abundance \cite{posva2024,tanaka2016}, and Katz's Cartier descent \cite{katz1970}.

\subsection{The proof map}

The argument has two halves that meet in a contradiction.

\begin{enumerate}[label=\textbf{Step \arabic*.}]
\item \textbf{Birational preparation.} Assume the theorem is false in the worst way: the nef dimension $n(X,K_X)$ is $3$. After a birational modification and a finite tame cover, produce a smooth projective threefold $V$, a very ample divisor $H$, and a nonzero effective Cartier divisor
\[
D=\sum_\alpha m_\alpha D_\alpha
\]
with connected simple-normal-crossing support, primitive multiplicities, $D$ nef and numerically trivial on each component, $\cO_D(D)$ torsion, and a local relation $\omega_V^{\otimes p^b}\simeq\cO_V(B_*)$ supported on $D$. See Proposition~\ref{prop:prepared-cycle}.

\item \textbf{Slopes.} Prove that the tangent bundle $T_V$ is strongly semistable for the slope $c_1(-)HD/\rk(-)$. Deduce $c_2(V)D\ge0$ and uniform bounds on polar parts of sections of semistable bundles. See Proposition~\ref{prop:strong-semistability}, Proposition~\ref{prop:polar-bounds}, and Corollary~\ref{cor:no-scalar-poles}.

\item \textbf{Jets.} Study finite jets of line bundles along $D$. A three-row spectral sequence relates scalar, middle, and canonical jets. Prove the scalar last width is asymptotically $p/(p+1)$ and consequently $c_2(V)D=0$. Prove a canonical formal root $\widehat\omega_V\simeq\widehat\cO_V(B_0)$. See Proposition~\ref{prop:scalar-widths} and Proposition~\ref{prop:canonical-formal-root}.

\item \textbf{Laurent cohomology.} Turn the jet estimates into a criterion: for a bundle satisfying an Euler-characteristic identity, bounded polar parts, and concentration of leading jets, every Laurent \v Cech one-cocycle has a primitive. Apply this to one-forms, exact forms, and closed forms. See Theorem~\ref{thm:laurent-vanishing} and Corollary~\ref{cor:form-vanishing}.

\item \textbf{Cartier descent.} Use the vanishing of Laurent cocycles to give every line bundle on the formal punctured neighborhood an integrable connection whose forms are Cartier-fixed. Cartier descent then shows that every line bundle is a $p$th power. Hence $\Pic(|D|,\cR)$ is $p$-divisible. See Proposition~\ref{prop:picard-divisibility}.

\item \textbf{Degree contradiction.} Orders of Laurent units and intersection with $H$ give a nonzero homomorphism
\[
\overline\Phi:\Pic(|D|,\cR)\longrightarrow\Z.
\]
A $p$-divisible group admits no nonzero homomorphism to $\Z$. Contradiction. See Proposition~\ref{prop:laurent-degree-contradiction}.

\item \textbf{Descent.} The contradiction proves abundance over an uncountable algebraic closure. Standard base-change and Cartier-index arguments descend the conclusion to the original field. See Section~\ref{sec:descent}.
\end{enumerate}

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

\begin{figure}[ht]
\centering
\small
\begin{tikzcd}[column sep=small,row sep=large]
& \text{Prepared cycle }(V,D)
  \arrow[dl,"\text{slopes and jets}"']
  \arrow[dr,"\text{Cartier descent}"] & \\
\begin{gathered}\text{Strong semistability}\\
\text{Laurent cocycles have primitives}\end{gathered}
  \arrow[d,"\text{Cartier descent}"']
&& \begin{gathered}\text{Orders of Laurent units}\\
             \text{Intersection with }H\end{gathered}
  \arrow[d] \\
\begin{gathered}\Pic(|D|,\cR)\\\text{is }p\text{-divisible}\end{gathered}
  \arrow[dr]
&& \begin{gathered}\overline\Phi:\Pic(|D|,\cR)\to\Z\\
             \overline\Phi(\cO(H))=DH^2>0\end{gathered}
  \arrow[dl] \\
& \text{Contradiction} &
\end{tikzcd}
\caption{The maximal-nef-dimension assumption forces the Picard group of the formal neighborhood to be $p$-divisible, while intersection theory supplies a nonzero integer degree.}
\label{fig:proof-map}
\end{figure}

\section{Conventions and background}
\label{sec:conventions}

\subsection{Varieties and divisors}
A \emph{variety} is an integral separated scheme of finite type over a field. A \emph{threefold} has dimension three. A \emph{divisor} is a formal integer combination of prime divisors; it is \emph{Cartier} if it is locally defined by one equation. We write $\sim$ for linear equivalence and $\equiv$ for numerical equivalence. A divisor $L$ is \emph{nef} if $L\cdot C\ge0$ for every integral curve $C$. It is \emph{ample} if some positive multiple is very ample. It is \emph{semiample} if some positive multiple is globally generated.

For a nef divisor $L$, the \emph{nef dimension} $n(L)$ is the dimension of its nef reduction. In particular, $n(L)=\dim X$ means that no $L$-trivial curve passes through a very general point. This is a different invariant from the numerical dimension $\nu(L)$.

\subsection{Completion and Laurent sheaves}
Let $V$ be a smooth variety and $D$ an effective Cartier divisor. We write $\widehat\cO_V$ for the completion along $D$, regarded as a sheaf on the Zariski space $|D|$. Set
\[
\cR=\widehat\cO_V(*D).
\]
Thus a section of $\cR$ is a formal function on a formal neighborhood of $D$ that is allowed a finite pole along $D$. A \emph{line bundle} on $(|D|,\cR)$ is a locally free rank-one $\cR$-module in the Zariski topology. Its Picard group is denoted $\Pic(|D|,\cR)$.

For a vector bundle $E$, we write $\widehat E$ for its completion and
\[
E_{\cR}=\widehat E\otimes_{\widehat\cO_V}\cR.
\]
A \emph{Laurent \v Cech cocycle} is a \v Cech cocycle with values in $E_{\cR}$ on a finite affine cover adapted to $D$ and $E$. A \emph{primitive} is a cochain whose coboundary is the given cocycle.

\subsection{Frobenius and Cartier}
Let $k$ be perfect of characteristic $p>0$. Absolute Frobenius is the map
\[
F:V\longrightarrow V,
\qquad
F^\sharp(a)=a^p.
\]
The inverse Cartier operator is the map
\[
C^{-1}:\Omega^1_V\longrightarrow\Omega^1_V/(\text{exact forms}),
\qquad
C^{-1}(a^p db)=a\,db.
\]
It is inverse-Frobenius-semilinear: $C^{-1}(a^p\alpha)=aC^{-1}(\alpha)$. On a smooth chart with \'etale coordinates $x_1,x_2,x_3$, the monomials $x^I$ with $0\le I_j<p$ form a basis over $p$th powers. The de Rham complex decomposes accordingly, and the Cartier operator can be computed monomial by monomial.

\subsection{Jets and widths}
Let $D$ be an effective Cartier divisor and $E$ a vector bundle. For integers $u<i$, the quotient
\[
P_i^s(E)=\frac{E(iD)}{E((i-s)D)}
\]
is the bundle of \emph{width-$s$ jets} with upper weight $i$. Its sections remember the first $s$ infinitesimal layers along $D$ above the weight $i$. The \emph{last width} of a line bundle $L$ at weight $i$ is the largest $s$ for which $L(iD)$ is trivial on the $s$-th thickening $sD$. We write $d_e$ for the last width of $\cO_V(M_eD)$ when $M_e=rp^e$ and $r$ is the order of $\cO_D(D)$.

\section{Birational preparation}
\label{sec:preparation}

The goal of this section is to reduce the maximal-nef-dimension case to a clean geometric situation. The output is the following proposition.

\begin{proposition}[The prepared cycle]\label{prop:prepared-cycle}
Suppose $k$ is uncountable and algebraically closed of characteristic $p>3$, and $X$ is a projective terminal threefold with $K_X$ nef, $\nu(K_X)=1$, and $n(X,K_X)=3$. Then there exist:
\begin{itemize}
\item a smooth projective threefold $V$;
\item a generically finite separable morphism $\rho:V\to X$;
\item a very ample divisor $H$ on $V$;
\item a nonzero effective Cartier divisor
\[
D=\sum_\alpha m_\alpha D_\alpha,
\qquad
m_\alpha\in\Z_{>0},
\qquad
\gcd_\alpha m_\alpha=1,
\]
\end{itemize}
such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\Supp D$ is connected and has simple normal crossings. The divisor $D$ is nef, and $D|_{D_\alpha}$ is numerically trivial for every $\alpha$.
\item No $D$-trivial curve passes through a very general point of $V$. Moreover
\[
D\cdot H^2>0,
\qquad
D^2\cdot H=0.
\]
\item $\cO_D(D)$ is torsion. Let $r$ be its order.
\item There are an integer $b\ge0$, an open neighborhood $U$ of $|D|$, and integers $b_\alpha$ such that
\begin{equation}\label{eq:prepared-canonical}
\omega_V^{\otimes p^b}|_U
\simeq \cO_U(B_*),
\qquad
B_*=\sum_\alpha b_\alpha D_\alpha.
\end{equation}
Consequently $K_V\cdot D\cdot A=0$ for every Cartier divisor $A$ on $V$.
\end{enumerate}
\end{proposition}

The proof has several steps. We give them in a form that highlights the geometry and records the technical inputs.

\subsection{Nonvanishing and the maximal nef-dimension branch}
By nonvanishing \cite[Theorem~1.4]{xu2024}, there is a nonzero effective Cartier divisor
\[
0\ne N_0\sim mK_X,
\qquad
m>0.
\]
It is nef and $N_0^2\cdot H_X=0$ for an ample divisor $H_X$. Every prime component $T$ of $N_0$ satisfies $N_0\cdot H_X\cdot T=0$, because these numbers are nonnegative and have positive weighted sum zero. In fact $N_0|_T\equiv0$: for an integral curve $C\subset T$, take a large multiple of $H_X|_T$ with an effective member containing $C$; the nef restriction has intersection zero with it, so every component contributes zero.

If $n(X,K_X)\le2$, the low-nef-dimension abundance theorem \cite[Theorem~1.8]{xu2024} makes $K_X$ semiample. Since $\nu(K_X)=1$, some multiple has at least two sections, and we are done. Hence we may assume
\[
n(X,K_X)=3.
\]
The nef reduction exists over our uncountable algebraically closed field \cite[Theorem~2.2]{xu2026}. The assumption $n(X,K_X)=3$ says that no $N_0$-trivial curve passes through a very general point.

\subsection{Isolating a boundary component}
Let $\lambda>0$ be the log canonical threshold of $N_0$ on $X$. It is finite, rational, and attained on a log resolution. Thus $(X,\lambda N_0)$ is lc and not klt. Take a crepant projective $\Q$-factorial dlt modification
\[
\mu:(W,B)\longrightarrow(X,\lambda N_0).
\]
All $\mu$-exceptional divisors occur in $B$ with coefficient one. Put $N=\mu^*N_0$ and $\tau=m^{-1}+\lambda>0$. Then
\begin{equation}\label{eq:initial-dlt-pair}
K_W+B\sim_{\Q}\tau N,
\qquad
\Supp B=\Supp N.
\end{equation}

Choose a coefficient-one component $S$ of $B$. For small rational $\epsilon>0$,
\[
B-\epsilon S\ge0,
\qquad
\tau N-\epsilon S\ge0.
\]
Run a $(K_W+B-\epsilon S)$-MMP that is $(K_W+B)$-trivial. Termination is available because the lowered pair is dlt and its log divisor is pseudoeffective. There is no Mori fiber output, since a pseudoeffective divisor cannot be anti-ample on a covering family.

On a common resolution of each step, the pullbacks of $N$ agree. The difference is exceptional over the new model. A curve contracted over the new model maps to a curve of $N$-degree zero on the old model, because the extremal ray is $N$-trivial. Negativity therefore makes the difference numerically trivial, hence zero. Thus $N$ remains nef and numerically trivial on every component of its support.

Let $Y,B_Y,S_Y,N_Y$ be the output. The partial-MMP conclusion gives
\begin{equation}\label{eq:isolated-nef-pair}
K_Y+B_Y\sim_{\Q}\tau N_Y,
\qquad
K_Y+B_Y-\delta S_Y\ \text{nef}
\end{equation}
for some rational $\delta>0$. If $T\ne S_Y$ is a component of $N_Y$, then $-S_Y|_T$ is nef by \eqref{eq:isolated-nef-pair}, while $N_Y|_T\equiv0$. But $S_Y|_T$ is effective and $\Q$-Cartier. If nonzero, it has positive degree against an ample divisor on the integral projective surface $T$, contradiction. Hence
\[
S_Y\cap T=\varnothing
\qquad
(T\ne S_Y).
\]
Writing $s=\operatorname{mult}_{S_Y}N_Y>0$, we get
\begin{equation}\label{eq:isolated-boundary}
S_Y|_{S_Y}\equiv0,
\qquad
K_Y+S_Y\sim_{\Q}\tau s S_Y
\end{equation}
on a neighborhood of $S_Y$.

\subsection{Adjunction on the reduced boundary}
Take a new crepant projective $\Q$-factorial dlt modification
\[
f:(Y_d,\Sigma)\longrightarrow(Y,S_Y),
\]
where $\Sigma$ is the strict transform of $S_Y$ together with all exceptional divisors, all reduced. One checks
\[
\Supp f^*S_Y=\Supp\Sigma.
\]
For an extracted divisor $E$, log canonicity gives
\[
0\le a_E(Y,B_Y)=a_E(Y,S_Y)-\operatorname{ord}_E(B_Y-S_Y)
=-\operatorname{ord}_E(B_Y-S_Y).
\]
Both terms vanish. Crepancy with $(X,\lambda N_0)$ gives $a_E(X,\lambda N_0)=0$. Since $X$ is klt, $\operatorname{ord}_E N_0>0$, hence $\operatorname{ord}_E N_Y>0$ and $\operatorname{ord}_E S_Y>0$. This proves the support assertion.

Near $\Sigma$,
\begin{equation}\label{eq:dlt-adjunction-input}
K_{Y_d}+\Sigma\sim_{\Q}\tau s f^*S_Y,
\qquad
f^*S_Y|_\Sigma\equiv0.
\end{equation}

Let $\sigma:\Sigma'\to\Sigma$ be the finite $S_2$-ification, an isomorphism away from finitely many closed points. Reduced-boundary adjunction gives
\begin{equation}\label{eq:slc-adjunction}
(K_{Y_d}+\Sigma)|_{\Sigma'}=K_{\Sigma'}+\Delta',
\end{equation}
where $(\Sigma',\Delta')$ is an slc surface pair \cite[Lemma~2.19]{xu2026}. The local verification uses the following lemma.

\begin{lemma}[Exceptional surface intersections]\label{lem:exceptional-intersections}
Let $P$ be an excellent normal local surface with closed residue field $l$, and let $g:T\to P$ be a nontrivial projective regular resolution that is an isomorphism away from the closed point. Write $G_i$ for its exceptional integral curves, and take intersection degrees over $l$. Then $G_i^2<0$ for every $i$. More generally, if an exceptional real divisor $Z=\sum_i z_iG_i$ has a positive coefficient, there is an index $j$ with $z_j>0$ and $Z\cdot G_j<0$.
\end{lemma}

\begin{proof}
The exceptional fiber is connected. Choose a nonzero nonunit $t\in\cO_P$ and write
\[
\operatorname{div}_T(t)=A+\sum_i c_iG_i,
\qquad c_i>0.
\]
The strict transform $A$ meets the exceptional fiber. For $C=\sum_i c_iG_i$,
\[
C\cdot G_i=-A\cdot G_i\le0.
\]
Distinct curves have nonnegative intersections. Connectedness implies $G_i^2<0$: every vertex has either a neighbor or, if it is the only vertex, a positive intersection with $A$.

For the second assertion, put $a=\max_i(z_i/c_i)>0$, and let $I$ be the nonempty set where the maximum is attained. Choose $j\in I$ that either meets $A$ or has a neighbor outside $I$. Such a vertex exists by connectedness. Then
\[
Z\cdot G_j
=aC\cdot G_j+\sum_i(z_i-ac_i)G_i\cdot G_j<0.
\]
\end{proof}

\begin{lemma}[Terminal threefolds are regular in codimension two]\label{lem:terminal-codimension-two}
A terminal threefold in the present setting has only isolated singularities.
\end{lemma}

\begin{proof}
Localize at a codimension-two point. This gives an excellent normal local surface $P$ with residue field $l$ and a projective regular resolution $g:T\to P$. For an integral exceptional curve $G$, divisor adjunction and curve duality give
\[
(K_T+G)\cdot G
=2\dim_lH^1(G,\cO_G)-2\dim_lH^0(G,\cO_G).
\]
If $K_T\cdot G<0$, set $l'=H^0(G,\cO_G)$ and $d=[l':l]$. Both $G^2$ and $K_T\cdot G$ are negative multiples of $d$. The displayed formula and Lemma~\ref{lem:exceptional-intersections} force
\[
H^1(G,\cO_G)=0,
\qquad
G^2=K_T\cdot G=-d.
\]
The line bundle $\cO_G(-G)$ has degree one over $l'$, and Riemann--Roch gives at least two independent sections over $l'$. Two independent sections have no common zero, so they define a degree-one finite birational map $G\to\PP^1_{l'}$, hence an isomorphism. In particular $G\simeq\PP^1_{l'}$ and $\cO_G(G)\simeq\cO_{\PP^1_{l'}}(-1)$.

One can contract such a curve while retaining a projective regular resolution. Choose a relatively very ample Cartier divisor $A$, put $a=\deg_{l'}(A|_G)>0$, and set $Q=A+aG$. The divisor $Q-G$ is relatively ample. The restriction $Q|_G$ is trivial, and a trivializing section extends successively to $nG$ because
\[
H^1\bigl(G,\cO_G(nQ-jG)\bigr)
=H^1(\PP^1_{l'},\cO(j))=0
\qquad(j\ge1).
\]
For large $n$, relative Serre vanishing lifts the sections to $T$. They generate along $G$ and, away from $G$, the sections of $n(Q-G)$ generate and separate points and tangent directions. Thus $nQ$ defines a projective contraction whose only nontrivial fiber is $G$.

Let $q$ be the image of $G$. Formal functions identify the completed local ring at $q$ with
\[
\varprojlim_n H^0(nG,\cO_{nG}).
\]
The graded pieces are $H^0(G,\cO_G(-nG))=H^0(\PP^1_{l'},\cO(n))$, and there is no obstruction. The associated graded algebra is $l'[x,y]$, so the completed local ring, and then the local ring at $q$, is regular.

Contracting all $K_T$-negative exceptional curves yields a regular resolution with $K_T$ relatively nef. If any exceptional curves remain, terminality gives
\[
E=K_T-g^*K_P=\sum_i a_iG_i,
\qquad a_i>0.
\]
Lemma~\ref{lem:exceptional-intersections} gives $E\cdot G_j<0$ for some $j$, contradicting relative nefness. Thus the codimension-two point was regular. Normality already gives regularity in codimension one, so the singular locus is finite.
\end{proof}

Using these lemmas one proves that $\Sigma'$ is nodal in codimension one and that \eqref{eq:slc-adjunction} is an equality of log canonical classes. By \eqref{eq:dlt-adjunction-input}, the right side is numerically trivial. Surface abundance for projective slc surfaces \cite[Theorem~3.1]{posva2024} makes it semiample, hence $\Q$-linearly trivial. Therefore some positive Cartier multiple $L$ of $f^*S_Y$ satisfies
\begin{equation}\label{eq:reduced-boundary-triviality}
\cO_{Y_d}(L)|_{\Sigma\setminus Z}\simeq\cO_{\Sigma\setminus Z}
\end{equation}
for a finite set $Z$ of closed points, and $L|_{\Sigma_i}\equiv0$ for every component.

\subsection{Torsion on the full Cartier cycle}
The reduced boundary is now under control. We need torsion on the full nonreduced cycle.

\begin{lemma}[Extending torsion across finitely many points]\label{lem:full-cycle-torsion}
Let $Y$ be a normal projective threefold over an algebraically closed field of characteristic $p>0$, and let $L$ be an effective Cartier divisor. Suppose $L$ is numerically trivial on every integral component of its support and $\cO_Y(L)|_{L_{\mathrm{red}}\setminus Z}$ is trivial for a finite set of closed points $Z$. Then $\cO_L(L)$ is torsion.
\end{lemma}

\begin{proof}
Replace $Z$ by $Z\cap|L|$. For $z\in Z$, put $R=\cO_{Y,z}$. Normality gives depth at least two, so $H^1_{\mathfrak m}(R)=0$. Every nonmaximal localization of $R$ is a normal local ring of dimension at most two, hence Cohen--Macaulay. This implies $H^2_{\mathfrak m}(R)$ has finite length. Present $R=A/I$ with $A$ regular local of dimension $n$. The height of $I$ is $n-3$, and Auslander--Buchsbaum gives projective dimension $n-3$ at nonmaximal primes. Hence $\Ext_A^{n-2}(R,A)$ is supported only at the closed point and has finite length. Local duality identifies $H^2_{\mathfrak m}(R)$ with its Matlis dual, up to completion.

Let $t_z$ be a local equation of $L$. Choose $j$ so large that $t_z^{j-1}$ kills $H^2_{\mathfrak m}(R)$ for every $z\in Z$. Then the restriction diagram
\[
\begin{tikzcd}[column sep=small]
0 \arrow[r] &
\cO_Y((u-j)L) \arrow[r] \arrow[d] &
\cO_Y(uL) \arrow[r] \arrow[d,equal] &
\cO_{jL}(uL) \arrow[r] \arrow[d] & 0\\
0 \arrow[r] &
\cO_Y((u-1)L) \arrow[r] &
\cO_Y(uL) \arrow[r] &
\cO_L(uL) \arrow[r] & 0
\end{tikzcd}
\]
shows that the connecting map
\[
H^1_Z(Y,\cO_{jL}(uL))\longrightarrow H^1_Z(Y,\cO_L(uL))
\]
is zero for every integer $u$.

The kernel of $\cO_{jL\setminus Z}\to\cO_{L_{\mathrm{red}}\setminus Z}$ is nilpotent. Lift a generator of the trivial line $\cO_Y(L)|_{L_{\mathrm{red}}\setminus Z}$ to local generating sections on a finite open cover of $jL\setminus Z$. On overlaps their differences lie in the nilpotent ideal. For $u=p^e$ sufficiently large, their $u$-th powers agree, since the $p^e$-power map is additive in characteristic $p$ and kills that ideal. Thus we get a trivializing section
\[
s_j\in H^0(jL\setminus Z,\cO_{jL}(uL)).
\]
Project it to $s$ on $L\setminus Z$. Its obstruction to extending over $L$ is killed by the vanishing above, so $s$ extends to $H^0(L,\cO_L(uL))$.

This extension is a generator everywhere. On each integral projective surface component $T$ of $L_{\mathrm{red}}$ it is nonzero, since it is a generator off $Z$. If it vanished somewhere on $T$, its zero scheme would be a nonempty effective Cartier divisor, which has positive degree against an ample class, whereas $\cO_T(uL)$ is numerically trivial. This is impossible. Nonvanishing after reduction implies invertibility on $L$. Hence $\cO_L(uL)\simeq\cO_L$.
\end{proof}

Applying the lemma to $Y_d$ and $L$ proves that $\cO_L(L)$ is torsion.

\subsection{The tame cover and the primitive cycle}
Choose a smooth projective common resolution $V_0$ of $Y_d$ and $X$, isomorphic over the common open set
\[
X_{\mathrm{reg}}\setminus\Supp N_0.
\]
Let $\rho_0:V_0\to X$. The pullbacks of $N_Y$ and $N_0$ agree on $V_0$.

The section cutting out $N_0\sim mK_X$ gives a rational section $\theta$ of $\omega_{V_0}^{\otimes m}$ with
\[
\operatorname{div}(\theta)
=\rho_0^*N_0+m(K_{V_0}-\rho_0^*K_X).
\]
Every exceptional center outside $\Supp N_0$ is an isolated singular point of $X$, by Lemma~\ref{lem:terminal-codimension-two}. The divisors over those centers are disjoint from $\rho_0^*N_0$. Thus the support consists of the total pullback support and possibly divisors disjoint from it.

Write $m=p^bm'$ with $(m',p)=1$. Choose a nonzero rational generator $\eta$ of $\omega_{V_0}^{\otimes p^b}$, so $\theta=a\eta^{m'}$ for some $a\in k(V_0)^\times$. Adjoin one root $v$ of $v^{m'}=a$ and normalize $V_0$ in the resulting field. The extension is finite separable, possibly of degree smaller than $m'$. Away from $\Supp\operatorname{div}(\theta)$ it is finite \'etale, since $m'$ is invertible in $k$.

Resolve the normalization and then the total divisor support, keeping the regular \'etale open set unchanged. Denote the result by $V$, with map $\pi:V\to V_0$. The root $v\eta$ is a rational section of $\pi^*\omega_{V_0}^{\otimes p^b}$. The determinant of the differential of $\pi$ is a rational section of $\omega_V\otimes\pi^*\omega_{V_0}^{-1}$, nonzero because $\pi$ is separable. Multiplying $v\eta$ by the $p^b$-th power of this Jacobian gives a rational section of $\omega_V^{\otimes p^b}$ whose divisor is supported over $\Supp\operatorname{div}(\theta)$.

Let $L_V$ be the pullback of $L$ from $Y_d$, and choose one connected component $C$ of $\Supp L_V$. The isolation of $S_Y$ shows that $C$ is disjoint from the rest of the total pullback support of $N_0$. Write the part of $L_V$ supported on $C$ as
\[
aD,
\qquad
D=\sum_\alpha m_\alpha D_\alpha,
\qquad
a=\gcd_\alpha\bigl(\operatorname{mult}_{D_\alpha}L_V\bigr).
\]
Then $D$ is a nonzero primitive effective Cartier divisor. Its support is connected and snc by the final resolution.

The pullback $L_V$ is numerically trivial on its support. On a neighborhood of $C$, $L_V=aD$. Consequently $D|_{D_\alpha}\equiv0$. For a curve not contained in $\Supp D$, effectivity gives nonnegative intersection with $D$; for one contained in a component the degree is zero. Hence $D$ is nef. For every very ample $H$,
\[
D^2\cdot H=0,
\qquad
D\cdot H^2>0.
\]

If $\cO_L(tL)\simeq\cO_L$, its pullback is trivial on the Cartier scheme $L_V$. Near $C$, the inclusion $D\subset aD=L_V$ gives
\[
\cO_D(taD)\simeq\cO_D.
\]
Thus $\cO_D(D)$ is torsion. Finally, the rational canonical section above has, on a neighborhood of $C$, divisor supported on $D$. It gives \eqref{eq:prepared-canonical}. For any Cartier divisor $A$,
\[
p^bK_V\cdot D\cdot A
=B_*\cdot D\cdot A
=\sum_\alpha b_\alpha\,c_1(\cO_{D_\alpha}(D))\cdot A|_{D_\alpha}=0.
\]
In particular $K_V^2\cdot D=0$.

\subsection{Numerical proportionality and the final interface}
Put $\rho=\rho_0\circ\pi$ and $N_{\mathrm{tot}}=\rho^*N_0$. Both $D$ and $N_{\mathrm{tot}}$ are nef and nonzero. The latter is numerically trivial on every component of its support. We have
\[
D^2\cdot H=N_{\mathrm{tot}}^2\cdot H
=D\cdot N_{\mathrm{tot}}\cdot H=0,
\qquad
D\cdot H^2>0,
\qquad
N_{\mathrm{tot}}\cdot H^2>0.
\]
The surface Hodge index theorem gives a positive rational $c$ such that
\begin{equation}\label{eq:prepared-proportionality}
D\equiv cN_{\mathrm{tot}}.
\end{equation}
To check this on every curve, take an integral curve $C_0\subset V$ and a large $a$ with an integral surface $T\in|aH|$ containing $C_0$. On a smooth projective resolution $\widetilde T$ of $T$, the pullbacks of $D$ and $N_{\mathrm{tot}}$ are nef, isotropic, mutually orthogonal classes. The pullback of $H|_T$ has positive square. The Hodge index form on $N^1(\widetilde T)_\R$ has signature $(1,\dim-1)$, so two nonzero orthogonal isotropic classes with positive intersection against that class are positively proportional. The ratio is
\[
c=\frac{D\cdot H^2}{N_{\mathrm{tot}}\cdot H^2}.
\]
Projection of degrees then gives $(D-cN_{\mathrm{tot}})\cdot C_0=0$ for every curve.

Finally a very general point of $V$ lies in the generically finite locus of $\rho$ and maps outside the countable union excluded by the nef reduction on $X$. A curve through such a point has a curve as its image. If it were $D$-trivial, \eqref{eq:prepared-proportionality} and the projection formula would make its image $N_0$-trivial, contradiction. This finishes Proposition~\ref{prop:prepared-cycle}.

\section{Slopes, foliations, and polar parts}
\label{sec:slopes}

We now work on the smooth projective threefold $V$ with the cycle $D$ from Proposition~\ref{prop:prepared-cycle}. Fix a very ample divisor $H$. We use
\begin{equation}\label{eq:slope-numerical-input}
H^2D>0,
\qquad
HD^2=0,
\qquad
K_VDA=0
\quad\text{for every divisor }A.
\end{equation}
The divisor $D$ is nef, and no $D$-trivial curve passes through a very general point. The field is uncountable, algebraically closed, and has characteristic $p>3$.

\subsection{Slope calculus}
For a torsion-free coherent sheaf $E$ of positive rank, set
\[
\deg(E)=c_1(E)\cdot H\cdot D,
\qquad
\mu(E)=\frac{\deg(E)}{\rk(E)}.
\]
Stability and semistability refer to this slope. A sheaf is \emph{strongly semistable} if all its iterated Frobenius pullbacks are semistable.

The usual Harder--Narasimhan theory works for this nef slope. An effective divisor has nonnegative degree, because $H$ and $D$ are nef. Saturating a subsheaf does not decrease its degree. Every torsion-free sheaf embeds in a finite direct sum of copies of $\cO_V(aH)$ for some $a$: dualize a sufficiently positive global-generation map for $E^\vee$ and use $E\hookrightarrow E^{\vee\vee}$. If $B\subset E$ has rank $b$, taking exterior powers and determinants gives a nonzero map from $\det B$ to some copy of $\cO_V(baH)$, hence
\[
\mu(B)\le aH^2D.
\]
For ranks bounded by $\rk E$, slopes form a discrete subset of $\Q$, so a maximal slope is attained. The standard sum-and-intersection argument gives the Harder--Narasimhan filtration.

Write $\mu_{\max}(E)$ and $\mu_{\min}(E)$ for the largest and smallest slopes of its factors. Additivity of degree and the image of a morphism give
\begin{equation}\label{eq:slope-hom-vanishing}
\mu_{\min}(E)>\mu_{\max}(F)
\quad\Longrightarrow\quad
\Hom(E,F)=0.
\end{equation}
Dualizing on the locally free open set gives
\[
\mu_{\max}(E^\vee)=-\mu_{\min}(E),
\qquad
\mu_{\min}(E^\vee)=-\mu_{\max}(E).
\]
Tensoring by a line bundle adds its degree to every slope. In particular, twisting by $iD$ leaves all slopes unchanged.

\subsection{Connections and foliations}
Let $F_V:V\to V$ be absolute Frobenius. A Frobenius pullback $F_V^*E$ has a canonical connection that differentiates the coefficients in a pulled-back local frame.

\begin{lemma}[The connection detects destabilization]\label{lem:connection-destabilization}
Let $E$ be a semistable torsion-free sheaf. If $F_V^*E$ is unstable and $B$ is its first Harder--Narasimhan term, then the connection induces a nonzero morphism on a big open set
\[
B\longrightarrow (F_V^*E/B)\otimes\Omega_V^1.
\]
\end{lemma}

\begin{proof}
The induced map is $\cO_V$-linear by the Leibniz rule. Suppose it vanishes. Then the generic subspace defined by $B$ is horizontal. Let $K=k(V)$ and choose a pulled-back basis. In a Grassmannian chart, represent the subspace by a matrix $(I\mid M)$. Horizontality says every entry of $M$ has zero differential. Since $K$ is separably generated over the perfect field $k$, the kernel of $d:K\to\Omega_{K/k}^1$ is $K^p$. Taking $p$-th roots of the entries of $M$ defines a subspace before Frobenius pullback.

Saturate this subspace to a subsheaf $B_0\subset E$. At every codimension-one point the relevant quotients are free over a discrete valuation ring. Frobenius is flat on the smooth variety, so $F_V^*B_0$ is saturated and agrees with $B$ there. Consequently
\[
\deg(B)=p\deg(B_0),
\qquad
\mu(B_0)=\frac{\mu(B)}p>
\frac{\mu(F_V^*E)}p=\mu(E),
\]
contradicting semistability of $E$.
\end{proof}

\begin{lemma}[The first destabilizing term is a foliation]\label{lem:destabilizing-foliation}
If $T_V$ is unstable, its first Harder--Narasimhan term $A$ is closed under Lie brackets and under $p$-th powers of derivations.
\end{lemma}

\begin{proof}
Put $Q=T_V/A$ and $a=\mu(A)$. By \eqref{eq:slope-numerical-input}, $\mu(T_V)=0$, so
\[
a>0,
\qquad
\mu_{\max}(Q)<a.
\]
The rank of $A$ is one or two. Work where $A$ and $Q$ are bundles. The bracket modulo $A$ is an $\cO_V$-linear map
\[
\bigwedge^2A\longrightarrow Q.
\]
It vanishes automatically in rank one. In rank two, $Q$ has rank one and slope $-2a$, whereas $\bigwedge^2A$ has slope $2a$. Equation~\eqref{eq:slope-hom-vanishing} makes the map zero.

Once bracket closure holds, the $p$-th power of derivations modulo $A$ defines an $\cO_V$-linear map
\begin{equation}\label{eq:p-curvature-obstruction}
F_V^*A\longrightarrow Q.
\end{equation}
For local derivations $u,v\in A$, set $w=tu+v$ with $t$ a formal scalar. In characteristic $p$,
\[
\frac{d}{dt}w^p
=\sum_{j=0}^{p-1}w^juw^{p-1-j}
=\operatorname{ad}(w)^{p-1}(u).
\]
Every coefficient lies in the bracket-closed module $A$. The mixed coefficients of $w^p-t^pu^p-v^p$ thus belong to $A$. This proves additivity modulo $A$.

For a local function $f$, expanding $(fu)^p$ gives
\[
(fu)^p-f^pu^p=\sum_{j=1}^{p-1}c_ju^j.
\]
The left side is a derivation. At the generic point, if $u\ne0$, choose $x$ with $u(x)\ne0$. For a largest $j\ge2$ with $c_j\ne0$, the $j$-fold commutator with multiplication by $x$ is $j!c_j(u(x))^j\ne0$, while it must vanish for a derivation. Hence only the $j=1$ term remains, so the scalar discrepancy is a multiple of $u$. This proves Frobenius linearity modulo $A$ and hence \eqref{eq:p-curvature-obstruction}.

If $\rk A=1$, its source has slope $pa>a>\mu_{\max}(Q)$, so the map is zero. Suppose $\rk A=2$. Then
\[
\mu(Q)=-2a,
\qquad
\mu_{\max}(\Omega_V^1)=2a.
\]
If $F_V^*A$ is semistable, its minimum slope is $pa>-2a$. Otherwise let $L$ be its first Harder--Narasimhan term and $R=F_V^*A/L$. Both have rank one. Lemma~\ref{lem:connection-destabilization} gives a nonzero map
\[
L\longrightarrow R\otimes\Omega_V^1.
\]
Since $R$ has rank one, the slope rules give
\[
\mu(L)-\mu(R)\le2a,
\qquad
\mu(L)+\mu(R)=2pa.
\]
Hence
\[
\mu_{\min}(F_V^*A)=\mu(R)\ge pa-a>-2a.
\]
Equation~\eqref{eq:slope-hom-vanishing} makes \eqref{eq:p-curvature-obstruction} zero in either case.
\end{proof}

\subsection{The inseparable quotient}
\begin{lemma}[Quotient by the distribution]\label{lem:foliation-quotient}
Let $A\subset T_V$ be a saturated bracket-closed and $p$-closed subsheaf of rank $d$. There is a finite purely inseparable map $\pi:V\to W$ to a normal projective variety, with a factorization
\[
V\xrightarrow{\pi}W\xrightarrow{\rho}V^{(1)}
\]
of relative Frobenius. On open sets whose complements have codimension at least two,
\begin{equation}\label{eq:foliation-canonical}
\omega_V\simeq\pi^*\omega_W\otimes(\det A)^{\otimes(p-1)}.
\end{equation}
\end{lemma}

\begin{proof}
On an affine open with ring $R$, let
\[
P=\{f\in R:u(f)=0\text{ for every local section }u\in A\}.
\]
The ring $P$ lies between $R^p$ and $R$. Since $R$ is finite over $R^p$, it is finite over $P$ and $P$ is finite over $R^p$. These invariant rings glue. The ring $P$ is normal: an element of $\Frac(P)$ integral over $P$ is integral over $R$, hence belongs to $R$, and all derivations annihilate it, so it belongs to $P$. The variety $W$ is projective because it is finite over $V^{(1)}$.

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