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\title{Abundance in numerical dimension one\\for terminal threefolds in positive characteristic\\[2mm]
\large A reader-friendly edition}


\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.
\end{abstract}



\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 $V^{(1)}$.

Remove the codimension-at-least-two loci where $A$ or $T_V/A$ fails to be locally free. Near any remaining point, choose functions $x_1,\ldots,x_d$ whose differentials restrict to a basis of $A^\vee$, and let $u_1,\ldots,u_d$ be the dual local derivations, so $u_i(x_j)=\delta_{ij}$. Bracket closure forces $[u_i,u_j]=0$: their bracket lies in $A$ but annihilates every $x_l$. Likewise $u_i^p\in A$ annihilates every $x_l$, so $u_i^p=0$.

These commuting nilpotent derivations give an explicit invariant-ring description. For one derivation $u$ with $u(x)=1$, the coefficient $(p-1)!^{-1}u^{p-1}(f)$ belongs to $\ker u$. Subtracting its product with $x^{p-1}$ and iterating writes every $f$ uniquely as
\[
f = \sum_{k=0}^{p-1} x^k\, g_k,
\qquad g_k \in \ker u.
\]
Applying this successively to the commuting $u_i$ gives
\begin{equation}\label{eq:foliation-local-presentation}
R = \bigoplus_{0\le I_1,\ldots,I_d<p} P x^I
\simeq P[X_1,\ldots,X_d]/(X_1^p-c_1,\ldots,X_d^p-c_d),
\qquad c_i = x_i^p \in P.
\end{equation}
Uniqueness follows by iterating the highest-derivative extraction. In particular $\pi$ is locally free of degree $p^d$ on this open set.

Remove further the singular locus of $W$ and its inverse image; normality and finiteness make these also codimension at least two. Then \eqref{eq:foliation-local-presentation} produces an exact sequence of vector bundles
\begin{equation}\label{eq:foliation-cotangent}
0 \longrightarrow F_V^*(A^\vee)
  \longrightarrow \pi^*\Omega_W^1
  \longrightarrow \Omega_V^1
  \longrightarrow A^\vee \longrightarrow 0,
\end{equation}
where the last map is evaluation on $A$. To verify local freeness of the first image, put $N=\langle dc_1,\ldots,dc_d\rangle$ inside $R\otimes_P\Omega_{P/k}^1$. Then
\[
\Omega_{R/k}^1 \simeq \bigl((R\otimes_P\Omega_{P/k}^1)/N\bigr)\oplus\bigoplus_{i=1}^d R\,dx_i .
\]
Smoothness of $R$ makes the first summand locally free of rank $3-d$; smoothness of $P$ then makes $N$ a locally free direct summand of rank $d$ in the rank-three bundle.

The transition maps match those of $F_V^*(A^\vee)$. Let $x'_j$ be another such set of functions and write $x'_j$ in the basis $x^I$ over $P$. Taking $p$-th powers and differentiating downstairs, the coefficient of $dc_l$ in $d((x'_j)^p)$ is $u_l(x'_j)^p$; the coefficient of the $l$-th basis vector of $A^\vee$ in $dx'_j|_A$ is $u_l(x'_j)$. Hence the first term of \eqref{eq:foliation-cotangent} is indeed $F_V^*(A^\vee)$.

Taking determinants in \eqref{eq:foliation-cotangent} gives
\[
\pi^*\omega_W \simeq (\det A)^{-p}\otimes\omega_V\otimes\det A,
\]
which is \eqref{eq:foliation-canonical}.
\end{proof}

\subsection{Strong semistability of the tangent bundle}

\begin{proposition}[Strong semistability]\label{prop:strong-semistability}
The tangent bundle $T_V$ is strongly semistable for the slope $c_1(-)HD/\rk(-)$, and
\begin{equation}\label{eq:c2-nonnegative}
c_2(V)D \ge 0.
\end{equation}
\end{proposition}

\begin{proof}
We first prove semistability by the foliation and rational-curve argument of \cite[Lemma~2.10]{xuzhang2019}, upgraded by the quantitative bend-and-break of \cite[Theorem~1.1]{jlr2026}. Suppose $A\subset T_V$ is the first destabilizing term, with $a=\mu(A)>0$ and $d=\rk A$. By Lemmas~\ref{lem:destabilizing-foliation} and \ref{lem:foliation-quotient}, it gives a finite purely inseparable quotient $\pi:V\to W$. On the big smooth open set where \eqref{eq:foliation-canonical} holds, set $\Lambda=-\pi^*K_W$. As a divisor class on $V$,
\[
\Lambda = (p-1)c_1(A)-K_V,
\qquad
\Lambda HD = (p-1)da>0.
\]
No global Cartier property of $K_W$ is required: its degree is only taken on curves inside this smooth open set.

Let $H^{(1)},D^{(1)}$ be the corresponding classes on $V^{(1)}$, and set $H'=\rho^*H^{(1)}$, $D'=\rho^*D^{(1)}$. Then $H'$ is ample Cartier, $D'$ is nef Cartier, and
\[
\pi^*H'=pH,
\qquad
\pi^*D'=pD.
\]
For a positive integer $t$, take sufficiently divisible very ample multiples of $H'$ and $H'+tD'$; their general complete intersection $C_t$ is a smooth projective integral curve.

Why $C_t$ can be chosen in $W_{\mathrm{sm}}$: the bad complement in $W$ has dimension at most one. A general first divisor meets it in finitely many points and contains none of its curve components; a general second divisor avoids those points. Bertini on the smooth open set, plus irreducibility for the successive very ample sections, gives the required curve.

Projection formula, computed over the locally free locus of $\pi$, cancels the degree of $\pi$ and the chosen multiples, giving
\begin{align}
\frac{D'C_t}{-K_WC_t}
&=\frac{pD\,H(H+tD)}{\Lambda H(H+tD)} \notag\\
&=\frac{pDH^2}{\Lambda H^2+t\Lambda HD}
\longrightarrow 0
\qquad(t\to\infty).
\label{eq:bend-break-ratio}
\end{align}
The denominator is positive for all large $t$.

Quantitative bend-and-break \cite[Theorem~1.1]{jlr2026}: if $Y$ is projective over an algebraically closed field, $C\subset Y_{\mathrm{sm}}$ is an integral curve with $K_YC<0$, and $N$ is a nef real Cartier divisor, then every closed point of $C$ lies on a rational curve $\Gamma\subset Y$ with
\[
N\Gamma \le (\dim Y+1)\frac{NC}{-K_YC}.
\]
The statement is valid in arbitrary characteristic. Choose $t$ so large that \eqref{eq:bend-break-ratio} is less than $1/4$. Applied on $W$ with $N=D'$, it gives at each closed point of $C_t$ a rational curve $\Gamma$ with $0\le D'\Gamma<1$. Since $D'$ is Cartier, this degree is an integer, so $D'\Gamma=0$.

A separate very-general-point argument upgrades this to a $D$-trivial curve through any prescribed very general $y\in V$. Fix the product of the two very ample linear systems and let $\mathcal T$ be the nonempty open locus of smooth integral complete intersections avoiding the bad locus. The parameter space $\mathcal T$ is irreducible, its universal curve $\mathcal C\to\mathcal T$ is smooth, and the evaluation $\mathcal C\to W$ is dominant: the ambient incidence of points and pairs of divisors through them is a product of projective bundles over $W$, hence irreducible.

Let $Z_1,Z_2,\ldots\subset V$ be proper closed subsets outside whose union no $D$-trivial curve passes; take $Z_j$ to be the countable family furnished by the maximal-nef-dimension hypothesis. For each $j$, the locus in $\mathcal T$ of curves not contained in $\pi(Z_j)$ is a nonempty open (the image under the open map $\mathcal C\to\mathcal T$ of the complement of $\operatorname{ev}^{-1}(\pi(Z_j))$). Over the uncountable field there is a common $k$-point, whose curve $C_t$ is integral and not contained in any $\pi(Z_j)$. Its points outside $\bigcup\pi(Z_j)$ are nonempty.

Choose a closed point $x\in C_t$ outside that union, and let $y\in V$ be the unique inverse-image point. The rational curve $\Gamma$ through $x$ has $D'\Gamma=0$. Its reduced inverse image contains a curve $\widetilde\Gamma\subset V$ through $y$, finite and surjective over $\Gamma$. Then
\[
pD\widetilde\Gamma = (\pi^*D')\widetilde\Gamma
= \deg(\widetilde\Gamma/\Gamma)\,D'\Gamma=0,
\]
contradicting the choice of $y$. Hence $T_V$ is semistable.

Strong semistability follows by Frobenius iteration. Suppose $e\ge1$ is the first integer for which $F_V^{e*}T_V$ is unstable, with first HN term $A'$ and quotient $Q'=F_V^{e*}T_V/A'$. Applying Lemma~\ref{lem:connection-destabilization} to the semistable sheaf $F_V^{(e-1)*}T_V$ gives a nonzero map
\[
A'\longrightarrow Q'\otimes\Omega_V^1,
\qquad\text{equivalently}\qquad
A'\otimes(Q')^\vee\longrightarrow\Omega_V^1
\]
on a big open set. One of the tensor factors has rank one. Semistability of $\Omega_V^1$ (equally, of $T_V$) and the rank-one twisting rule give
\[
\mu_{\min}\bigl(A'\otimes(Q')^\vee\bigr)
=\mu(A')-\mu_{\max}(Q')>0,
\]
contradicting \eqref{eq:slope-hom-vanishing}. All iterated Frobenius pullbacks of $T_V$ are therefore semistable.

Finally, apply Langer's Bogomolov inequality \cite[Theorem~3.2]{langer2004},\cite{langeradd2004}: for a strongly semistable torsion-free sheaf with a nef tuple whose product is nonzero, $\Delta(E)D\ge0$ where $\Delta(E)=2\rk(E)c_2(E)-(\rk(E)-1)c_1(E)^2$. Here the tuple is $(H,D)$, which has $H^2D>0$. For $T_V$ this gives
\[
\bigl(6c_2(V)-2K_V^2\bigr)D\ge0,
\]
and \eqref{eq:slope-numerical-input} gives $K_V^2D=0$, proving \eqref{eq:c2-nonnegative}.
\end{proof}

\subsection{Polar-part bounds}

\begin{lemma}[Zero-degree line bundles]\label{lem:degree-zero-line-sections}
If a line bundle $L$ has $\deg L\le0$, then $h^0(V,L)\le1$.
\end{lemma}

\begin{proof}
A nonzero section has an effective zero divisor, so its degree is nonnegative and hence zero. Suppose two independent sections exist. Remove the base locus of their pencil. Through every point $x$ of the remaining open set there is a pencil member; its divisor has degree zero. Each prime component $S$ through $x$ therefore satisfies $DHS=0$, because all component degrees are nonnegative.

The restriction $D|_S$ is nef and numerically trivial. To see this for possibly singular $S$, take a projective resolution $\sigma:\widetilde S\to S$. The class $L_S=\sigma^*(D|_S)$ is nef and orthogonal to the big-and-nef class $A_S=\sigma^*(H|_S)$ with $A_S^2>0$. The surface Hodge index theorem makes the form negative definite on $A_S^\perp$, while nefness gives $L_S^2\ge0$, so $L_S$ is numerically zero. The projection formula gives the assertion on $S$.

A projective integral curve in $S$ through $x$ (e.g. a component of a sufficiently ample section through $x$) then has $D$-degree zero. Choosing $x$ very general contradicts the prepared-cycle property.
\end{proof}

\begin{lemma}[Hom bound]\label{lem:semistable-hom-bound}
If $G_1,G_2$ are torsion-free semistable sheaves of slope zero, then
\[
\dim_k\Hom(G_1,G_2)\le\rk G_1\cdot\rk G_2.
\]
\end{lemma}

\begin{proof}
First suppose $G_1,G_2$ are stable of slope zero. For a nonzero map, its image has slope at least zero as a quotient of $G_1$ and at most zero as a subsheaf of $G_2$. A positive-rank kernel makes the first inequality strict; an image of smaller rank than $\rk G_2$ makes the second strict. Hence every nonzero map is full rank and $\rk G_1=\rk G_2=n$.

Its determinant is a global section of $L=\det(G_2)\otimes\det(G_1)^{-1}$ with $\deg L=0$, so $h^0(L)\le1$ by Lemma~\ref{lem:degree-zero-line-sections}. If $f,g$ were linearly independent, write $\det(sf+tg)=P(s,t)\tau$ with $\tau$ a global generator and $P$ homogeneous of degree $n$. Since $f$ has full rank, $P\ne0$; over the algebraically closed field, $P$ vanishes at some $(s:t)\in\PP^1(k)$. Then $sf+tg$ is nonzero (by independence) but has zero determinant, a contradiction. Hence $\dim\Hom\le1$.

Every slope-zero semistable sheaf has a finite filtration by saturated subsheaves whose factors are torsion-free stable of slope zero: if it is not stable, a smaller-rank equal-slope subsheaf can be saturated, its degree not increasing strictly, and induction on rank gives the filtration. Left-exactness of $\Hom$ gives the bound by summing one-dimensional contributions over the pairs of stable factors.
\end{proof}

\begin{proposition}[Polar-part bounds]\label{prop:polar-bounds}
Let $G$ be a semistable vector bundle of slope zero. For every integer $i$,
\[
h^0(V,G(iD))\le\rk G,
\qquad
h^0(V,G^{\otimes2}(iD))\le(\rk G)^2.
\]
For each fixed $J=G$ or $J=G^{\otimes2}$, the increasing spaces $H^0(V,J(iD)/J)$ for $i=1,2,\ldots$ have bounded dimension, eventually stabilize, and have a uniform pole bound along every component of $D$.
\end{proposition}

\begin{proof}
Apply Lemma~\ref{lem:semistable-hom-bound} to $\cO_V, G(iD)$ and to $G^\vee, G(iD)$, using
\[
H^0(G(iD))=\Hom(\cO_V, G(iD)),
\qquad
H^0(G^{\otimes2}(iD))=\Hom(G^\vee, G(iD)).
\]

For $i>0$, the sequence $0\to J\to J(iD)\to J(iD)/J\to0$ gives
\[
h^0(V,J(iD)/J)
\le h^0(V,J(iD))+h^1(V,J),
\]
bounded independently of $i$. Effectivity of $D$ gives injections of the quotient sheaves as $i$ increases, hence injections of section spaces. A bounded increasing sequence of dimensions stabilizes.

More explicitly, choose $i_0$ after stabilization. Every global polar part, being the image of an element of $H^0(V,J(i_0D)/J)$, has pole order bounded by $i_0\operatorname{mult}_{D_\alpha}D$ at the generic point of $D_\alpha$, uniformly in the cut and the chosen section.
\end{proof}

\begin{corollary}[No scalar polar parts]\label{cor:no-scalar-poles}
For every $i>0$, $H^0(V,\cO_V(iD)/\cO_V)=0$.
\end{corollary}

\begin{proof}
Apply Proposition~\ref{prop:polar-bounds} to $G=\cO_V$. A nonzero polar part $\eta$ has genuine negative valuation at some component of $D$: if all component valuations were nonnegative, normality and the absence of poles away from $D$ would make the local representatives regular, causing the quotient section to vanish.

For every $e\ge0$, local representatives $f^{p^e}$ define a global polar part $\eta^{p^e}\in H^0(V,\cO_V(p^eiD)/\cO_V)$: well defined because two representatives differing by a regular function have $p^e$-th powers differing by a regular function. At the chosen component the negative valuation is multiplied by $p^e$, so these pole orders are unbounded. This contradicts the uniform bound of Proposition~\ref{prop:polar-bounds}.
\end{proof}

\section{Jets and the three rows}
\label{sec:jets}

We retain the prepared data of Proposition~\ref{prop:prepared-cycle}: $V$ is a smooth projective threefold, $D=\sum_\alpha m_\alpha D_\alpha$ is an effective Cartier divisor with connected support, $\gcd_\alpha m_\alpha=1$, and $D|_{D_\alpha}$ is numerically trivial for every $\alpha$. In particular $D^2A=0$ for every divisor $A$. The line bundle $\cO_D(D)$ is torsion. Near $D$ there is an isomorphism
\begin{equation}\label{eq:canonical-p-power}
\omega_V^{\otimes p^b}\simeq\cO_V(B_*),
\qquad
B_*=\sum_\alpha b_\alpha D_\alpha,
\quad
b_\alpha\in\Z,
\end{equation}
and $K_V^2D=0$. We use $c_2(V)D\ge0$ from Proposition~\ref{prop:strong-semistability} and the absence of scalar polar parts from Corollary~\ref{cor:no-scalar-poles}.

The argument computes finite jets and their ranks. Its two outputs are (a) a trivialization of the canonical line on the completion up to an integral divisor supported on $D$, and (b) the equality $c_2(V)D=0$ together with the limit $p/(p+1)$ for the normalized last trivialization widths. The width at period $rp^e$ is the largest integer $s$ for which $\cO(rp^eD)$ is trivial on $sD$. The only passage to a formal limit is an explicit inverse-limit argument for finite-dimensional spaces of sections.

\subsection{Finite pages and duality}

Write $\widehat\cO_V$ for the completion along $D$, regarded as a sheaf on $|D|$, and set
\[
\cR=\widehat\cO_V(*D).
\]
Choose a finite cover by the restrictions to $|D|$ of affine open subsets of $V$, refined so that $D$ is principal and the bundles under consideration are free on each member. All intersections of these affine opens are affine, since $V$ is separated. For a vector bundle $E$, let $C_i^\bullet(E)$ be the resulting \v{C}ech complex with coefficients in the completion of $E(iD)$. These complexes give an increasing filtration, indexed by $i\in\Z$, on the Laurent \v{C}ech complex.

Two basic completeness facts. First, on an affine chart with local equation $f$ for $D$, adic completion is flat and $f$ remains a non-zero-divisor, so the completed lattices embed in their localization. Second, a section of the localization on a quasi-compact open lies in one lattice: finitely many local pole bounds have a common upper bound and the resulting lattice sections glue. Consequently every Laurent cochain has a finite upper weight. Each $C_i^\bullet(E)$ is complete and separated for the filtration by $C_{i-s}^\bullet(E)$ with $s\ge1$. Exactness of completion for finite modules gives
\begin{equation}\label{eq:jet-quotient}
C_i^\bullet(E)/C_{i-s}^\bullet(E)
\quad\text{computes}\quad
H^\bullet\bigl(V,P_i^s(E)\bigr),
\qquad
P_i^s(E)=\frac{E(iD)}{E((i-s)D)}.
\end{equation}
Thus $P_i^s(E)$ retains the $s$ layers at weights $i-s+1,\ldots,i$; call $i$ the upper cut. We abbreviate $E^!=E^\vee\otimes\omega_V$.

For $s\ge1$, consider a leading cochain: a class in $C_i^a(E)/C_{i-1}^a(E)$, where $a$ is its cohomological degree. The cycle condition asks whether this class has a representative closed modulo $C_{i-s}^\bullet(E)$. The boundary condition asks whether it is the leading term of a coboundary with a primitive in $C_{i+s-1}^{a-1}(E)$. Thus $s$ records both the precision of closedness and the allowed upper cut of a primitive. Define
\begin{align*}
Z_s^{i,a}&=\{[x]:x\in C_i^a(E),\ \partial x\in C_{i-s}^{a+1}(E)\},\\
U_s^{i,a}&=\{[\partial y]:y\in C_{i+s-1}^{a-1}(E),\
                        \partial y\in C_i^a(E)\}.
\end{align*}
Every exact cochain is closed, so $U_s^{i,a}\subseteq Z_s^{i,a}$. Put
\[
E_s^{i,a}(E)=Z_s^{i,a}/U_s^{i,a},
\qquad
g_s^a(i;E)=\dim_k E_s^{i,a}(E).
\]
Here $E_s^{i,a}(E)$ is a page space, not a new bundle.

\begin{lemma}[Finite-page calculus]\label{lem:page-calculus}
The operation $x\mapsto\partial x$ induces maps
\[
d_s:E_s^{i,a}(E)\longrightarrow E_s^{i-s,a+1}(E)
\]
whose cohomology is the next page. At page one, $E_1^{i,a}(E)=H^a(V,P_i^1(E))$. Every page has only rows $a=0,1,2$. Writing $\delta_s^a(i)=g_s^a(i;E)-g_{s+1}^a(i;E)$ for $a=0,2$, we have
\begin{align}
g_s^0(i;E)
&=\dim\im\Bigl(H^0(P_i^s(E))\to H^0(P_i^1(E))\Bigr),
\label{eq:bottom-endpoint}\\
g_s^2(i;E)&=g_s^0(1-i;E^!),\label{eq:top-endpoint}\\
g_{s+1}^1(i;E)
&=g_s^1(i;E)-\delta_s^0(i+s)-\delta_s^2(i-s).
\label{eq:middle-recurrence}
\end{align}
If $M>0$ and $\cO(MD)|_{sD}$ is trivial, the page-$s$ ranks are periodic in $i$ modulo $M$, and
\begin{equation}\label{eq:periodic-euler}
\sum_{i\bmod M}\bigl(g_s^0(i;E)-g_s^1(i;E)+g_s^2(i;E)\bigr)
=\sum_{i\bmod M}\chi(P_i^1(E)).
\end{equation}
Finally, at each fixed weight $i$, the stable image in \eqref{eq:bottom-endpoint} is precisely the image of the global sections of the completed bundle $\widehat{E(iD)}$ in $H^0(P_i^1(E))$.
\end{lemma}

\begin{proof}
If two representatives of a leading cochain differ by $y\in C_{i-1}^a$ and both have coboundary in $C_{i-s}$, their coboundaries differ by a target boundary from $C_{i-1}=C_{(i-s)+s-1}$. Adding an exact representative has no effect either. So $d_s$ is well defined, and $d_s^2=0$ follows from $\partial^2=0$.

If $d_s[x]=0$, the leading term of $\partial x$ is represented by $\partial y$ with $y\in C_{i-1}^a$ and $\partial y\in C_{i-s}^{a+1}$. Replacing $x$ by $x-y$ makes its coboundary lie in $C_{i-s-1}^{a+1}$. Thus the kernel is represented by $Z_{s+1}^{i,a}$. The incoming image enlarges $U_s^{i,a}$ to $U_{s+1}^{i,a}$: its representatives are exactly the leading terms of $\partial z\in C_i^a$ with $z\in C_{i+s}^{a-1}$. This proves the next-page assertion. Equation~\eqref{eq:jet-quotient} identifies page one. Since these quotients are supported on the projective surface scheme $D$, their cohomology is zero outside degrees $0,1,2$, and so is every subsequent page.

The differentials through a middle-row term are
\[
\begin{tikzcd}[column sep=large]
E_s^{i+s,0}(E) \arrow[r,"d_s"] &
E_s^{i,1}(E) \arrow[r,"d_s"] & E_s^{i-s,2}(E).
\end{tikzcd}
\]
Moving right lowers weight by $s$ and raises cohomological degree by one; the composite is zero. There is no incoming differential in degree zero, and its representatives are exactly the sections of $P_i^s(E)$, giving \eqref{eq:bottom-endpoint}.

For the other endpoint, consider the inclusion of the bottom layer
\[
H^2(P_i^1(E))\longrightarrow H^2(P_{i+s-1}^s(E)).
\]
There is no outgoing differential in degree two. The kernel calculation above and the vanishing of the degree-three page show that $Z_s^{i,2}=Z_1^{i,2}$. A class represented by $x\in C_i^2$ with $\partial x\in C_{i-1}^3$ dies under this inclusion exactly when $x-\partial y\in C_{i-1}^2$ for some $y\in C_{i+s-1}^1$. Its kernel is $U_s^{i,2}/U_1^{i,2}$, so the image has dimension $g_s^2(i;E)$.

The two-term locally free presentation of $P_j^s(E)$ gives
\[
\mathcal E xt^1(P_j^s(E),\omega_V)=\frac{E^!((s-j)D)}{E^!(-jD)}=P_{s-j}^s(E^!),
\]
and all other sheaf Ext groups vanish. Serre duality on the smooth projective threefold consequently identifies $H^2(P_j^s(E))^\vee$ with $H^0(P_{s-j}^s(E^!))$. Under this identification, the dual of the displayed inclusion is
\[
H^0(P_{1-i}^s(E^!))\longrightarrow H^0(P_{1-i}^1(E^!)),
\]
and the map is the projection. This proves \eqref{eq:top-endpoint}. The adjacent differentials to the middle term have ranks $\delta_s^0(i+s)$ and $\delta_s^2(i-s)$, giving \eqref{eq:middle-recurrence}.

Triviality of $\cO(MD)$ on $sD$ identifies each quotient of width $t\le s$ with its translate by $M$, compatibly with layer maps. Equations \eqref{eq:bottom-endpoint} and \eqref{eq:top-endpoint} therefore give periodicity of both outer rows at every such width; page one is periodic in all degrees; \eqref{eq:middle-recurrence} gives periodicity of the middle row by induction.

For \eqref{eq:periodic-euler}, let $e_t(i)=g_t^0(i)-g_t^1(i)+g_t^2(i)$. The recurrence yields
\[
e_{t+1}(i)-e_t(i)=-\delta_t^0(i)-\delta_t^2(i)+\delta_t^0(i+t)+\delta_t^2(i-t).
\]
For $t<s$ all four functions are periodic modulo $M$, and their sums cancel, proving the Euler identity by induction from page one.

For the last assertion, set $A_s=H^0(P_i^s(E))$ and $B_s=\bigcap_{t\ge s}\im(A_t\to A_s)$. Each $A_s$ is finite dimensional, so the intersection is attained at a finite stage. Choosing one stage beyond stabilization at both $s$ and $s+1$ shows that $B_{s+1}\to B_s$ is surjective. Every element of $B_1$ therefore admits compatible lifts in all $B_s$, and their inverse limit is a global section of $\widehat{E(iD)}$ (sections commute with inverse limits of sheaves). Conversely a completed section restricts to the image of every $A_s$.
\end{proof}

\subsection{Scalar jets and their last widths}

\begin{lemma}[The scalar jet rule]\label{lem:scalar-jets}
Let $u<i$ be integers and $\xi\in H^0(\cO(iD)/\cO(uD))$ nonzero. There is an integer $t$ with $u<t\le i$ such that local lifts of $\xi$ have pole order exactly $tm_\alpha$ at every $D_\alpha$. Those lifts trivialize $\cO(tD)$ on $(t-u)D$. In particular,
\begin{equation}\label{eq:scalar-endpoint}
g_s^0(i;\cO_V)=
\begin{cases}
1,&\cO(iD)|_{sD}\simeq\cO_{sD},\\
0,&\text{otherwise}.
\end{cases}
\end{equation}
\end{lemma}

\begin{proof}
Choose local algebraic lifts $f_\lambda$ of $\xi$ on a finite affine cover. Membership in $\cO(uD)$ is checked by valuations at the prime divisors of the smooth, hence normal, $V$. Some component therefore detects the nonzero quotient: the maximum over components of the ratios $-\ord_{D_\alpha}(f_\lambda)/m_\alpha$ that exceed $u$ is a well-defined rational number $t\in(u,i]$. Changing a lift adds a section of $\cO(uD)$ and cannot alter such a valuation.

Choose a positive integer $n$ with $nt\in\Z$. Then $f_\lambda^n\in\cO(ntD)$, and these sections agree on $D_{\mathrm{red}}$: the difference $f_\lambda-f_\mu$ lies in $\cO(uD)$, and in the factorization of $f_\lambda^n-f_\mu^n$ each term has one factor of pole ratio at most $u$ and $n-1$ factors of pole ratio at most $t$. After normalization in $\cO(ntD)$ its valuation at every $D_\alpha$ is at least $(t-u)m_\alpha>0$. They define a section of $\cO(ntD)|_{D_{\mathrm{red}}}$, nonzero on a component where the maximum is attained.

The restriction of this line bundle to every integral component is numerically trivial. A nonzero section of a numerically trivial line bundle on a projective integral surface has no zeros: a nonempty effective Cartier zero divisor has positive intersection with an ample divisor. Hence the section is nowhere zero on that component. At an intersection with another component its common value is nonzero; connectedness of $|D|$ propagates this to all components.

It follows that $\ord_{D_\alpha}(f_\lambda)=-tm_\alpha$ for every $\alpha$. In particular every $tm_\alpha$ is an integer. Primitivity of the cycle ($\gcd_\alpha m_\alpha=1$) gives $t\in\Z$.

The original lifts are sections of the honest line bundle $\cO(tD)$, units in it along $D$, agreeing modulo $\cO(uD)$. Since $\cO(tD)/\cO(uD)=\cO(tD)|_{(t-u)D}$, they give the asserted trivialization.

A jet of width $s$ at upper weight $i$ has nonzero leading image exactly when the uniform ratio is $t=i$, giving existence in \eqref{eq:scalar-endpoint} in both directions. For dimension one, take two such jets; their normalized unit sections on $D_{\mathrm{red}}$ are proportional (a connected reduced projective scheme over $k$ has only constant global functions). Subtracting the appropriate multiple leaves a leading image that would give a unit on $D_{\mathrm{red}}$ with ratio $i$, contradicting the cancellation.
\end{proof}

Let $r$ be the order of $\cO_D(D)$. For $e\ge0$ put
\begin{equation}\label{eq:width-definition}
M_e=rp^e,
\qquad
d_e=\max\{s\ge1:\cO(M_eD)|_{sD}\simeq\cO_{sD}\}.
\end{equation}
These maxima exist and satisfy
\begin{equation}\label{eq:width-bounds}
1\le d_e<M_e,
\qquad
d_{e+1}\ge p d_e,
\qquad
\frac{d_e}{M_e}\nearrow a_*\in(0,1].
\end{equation}
A trivialization on $M_eD$ would give, by Lemma~\ref{lem:scalar-jets}, a nonzero section of $\cO(M_eD)/\cO$, contrary to Corollary~\ref{cor:no-scalar-poles}. Triviality at width one follows from the definition of $r$. Taking $p$th powers of local trivializing sections multiplies the gluing precision by $p$ and gives $d_{e+1}\ge pd_e$. Division by $M_{e+1}=pM_e$ proves monotonicity and the positive limit. If $r=1$, the inequality $1\le d_0<M_0=1$ is a contradiction; hence $r>1$.

We will repeatedly use the following consequence of nilpotence. If a line bundle $L$ on a finite thickening $T$ of $D$ is trivial on $D$, then its class has $p$-power order. Indeed its transition functions can be chosen in $1+J$, where $J$ is the nilpotent ideal of $D$ in $T$. For $p^N$ at least a nilpotence exponent, $(1+x)^{p^N}=1$ for $x\in J$, so $L^{\otimes p^N}$ is trivial. Multiplication by an integer prime to $p$ is injective on such classes by B\'ezout and surjective on their cyclic subgroups. It follows that
\begin{equation}\label{eq:scalar-last-width}
\text{the last width at }i=jM_e\text{ is }d_e
\quad\text{if }p\nmid j.
\end{equation}
This includes negative $j$. If $r\nmid i$, no scalar leading jet occurs even at width one. Every nonzero multiple of $r$ has a unique expression $jM_e$ with $p\nmid j$; at $i=0$ the scalar entry persists at every width. These observations classify all scalar outer-row entries and drops.

Finally, Hirzebruch--Riemann--Roch on the smooth threefold gives
\begin{equation}\label{eq:scalar-euler}
\chi(P_i^1(\cO_V))
=\chi(\cO(iD))-\chi(\cO((i-1)D))
=\frac{Dc_2(V)}{12}=:\gamma\ge0.
\end{equation}
The terms involving $D^3$, $D^2K_V$, and $DK_V^2$ vanish, leaving the displayed linear term in the threefold Riemann--Roch polynomial.

\subsection{Canonical jets and a formal canonical root}

For $E=\cO_V$, \eqref{eq:top-endpoint} computes the top-row rank at weight $i$ from leading jets of $\omega_V$ at weight $1-i$. We need to describe that row as explicitly as the scalar bottom row. First, canonical jets must survive to arbitrarily large widths.

\begin{lemma}\label{lem:forced-canonical-jets}
There are nonzero leading jets of $\omega_V$ of arbitrarily large width, with upper weight allowed to vary.
\end{lemma}

\begin{proof}
Suppose instead that $g_s^0(l;\omega_V)=0$ for every $l$ and every $s\ge S$, for some $S$. By \eqref{eq:top-endpoint}, the top row for $E=\cO_V$ then vanishes at all these widths. Take $e$ large enough that $s_e=d_{e-1}+1\ge S$. At page $s_e$ the bottom row is nonzero precisely at multiples of $M_e$ (by \eqref{eq:scalar-last-width} and $s_e\le d_e$). This page is periodic modulo $M_e$. Equations \eqref{eq:periodic-euler} and \eqref{eq:scalar-euler} give
\[
\sum_{i\bmod M_e}g_{s_e}^1(i;\cO_V)
=1-M_e\gamma\le1.
\]
At differential length $d_e$, the bottom entries at $jM_e$ with $p\nmid j$ each drop by one. By \eqref{eq:middle-recurrence} these drops consume one middle-row dimension at $-d_e+jM_e$. Middle-row dimensions can only decrease as the page increases. Periodicity and the preceding upper bound therefore force the initial middle row to consist exactly of one entry at $-d_e\pmod{M_e}$.

Between pages $s_e$ and $s_{e+1}=d_e+1$ these are the only drops: there is no top row, and all scalar entries with a different last-width index either disappeared earlier or have last width at least $d_{e+1}$. Modulo $M_{e+1}=pM_e$, the drops remove all the $p$ lifts of the middle entry except its unshifted lift. Thus at page $s_{e+1}$ the middle row consists of one entry at $-d_e\pmod{M_{e+1}}$. Applying the same argument with $e+1$ says it must lie at $-d_{e+1}\pmod{M_{e+1}}$. But $0<d_e<d_{e+1}<M_{e+1}$, so these residues are different.
\end{proof}

\begin{proposition}[The canonical formal root]\label{prop:canonical-formal-root}
Every coefficient of $B_*$ in \eqref{eq:canonical-p-power} is divisible by $p^b$. Writing $B_*=p^bB_0$ with $B_0$ integral, there is an isomorphism on the completion along $D$,
\begin{equation}\label{eq:canonical-formal-root}
\widehat\omega_V\simeq\widehat\cO_V(B_0).
\end{equation}
\end{proposition}

\begin{proof}
Put $P=p^b$, and take a sequence of nonzero leading canonical jets of widths $s\to\infty$, with upper weights $l$ that may vary. Raising local representatives to the $P$th power and using \eqref{eq:canonical-p-power} gives a scalar jet with bounds
\[
A=PlD+B_*,
\qquad
B=P(l-s)D+B_*.
\]
The lower pole bound is multiplied by $P$ exactly because $P$ is a power of the characteristic. Set $\beta_\alpha=b_\alpha/m_\alpha$ and choose an integer $c_*\ge\max_\alpha\beta_\alpha$. Enlarging both bounds gives a section of
\[
\frac{\cO((Pl+c_*)D)}{\cO((P(l-s)+c_*)D)},
\]
nonzero for all sufficiently large $s$, uniformly in $l$. Indeed, at a component detecting the original leading layer, a local coefficient $f$ in an ordinary frame of $\omega_V$ has $-\ord_{D_\alpha}(f)/m_\alpha>l-1$. Its powered scalar coefficient has pole ratio
\[
-P\frac{\ord_{D_\alpha}(f)}{m_\alpha}+\beta_\alpha
>P(l-1)+\min_\alpha\beta_\alpha,
\]
while the new lower bound is $P(l-s)+c_*$, smaller for large $s$.

Lemma~\ref{lem:scalar-jets} shows that all these powered coefficients have one integral pole ratio $t$. The preceding inequality and the upper bound give a constant $C$, independent of $s,l$, such that
\begin{equation}\label{eq:canonical-ratio-bound}
|t-Pl|\le C,
\qquad
\cO(tD)\text{ is trivial to a width at least }Ps-C.
\end{equation}
For each fixed $J$, widths greater than $d_{J-1}$ can occur only at multiples of $M_J$ (zero is allowed as a multiple). Since the widths in \eqref{eq:canonical-ratio-bound} tend to infinity, we can assign integers $J\to\infty$ along this sequence such that $M_J\mid t$.

Fix a component $D_\alpha$. If $f$ is the original local coefficient in a frame of $\omega_V$, the local generator of $\cO(B_*)$ has order $-b_\alpha$, so
\begin{equation}\label{eq:canonical-order-identity}
b_\alpha-tm_\alpha=P\ord_{D_\alpha}(f).
\end{equation}
For large $J$, the integer $t$ is divisible by $P$, so \eqref{eq:canonical-order-identity} shows $P\mid b_\alpha$ for every $\alpha$. This proves $B_*=PB_0$ with $B_0$ integral.

For the same sequence put $h_0=t/P$. The order identity shows that the original local representatives are sections of
\[
L_{h_0}=\omega_V(-B_0+h_0D)
\]
near $D$: their coefficient orders are exactly those required. They have no additional poles near $D$, since the original representatives were sections of $\omega_V(lD)$ and membership in these divisor lattices is checked on the normal variety in codimension one. Moreover $L_{h_0}^{\otimes P}\simeq\cO(tD)$, and the $P$th powers of these sections are units along all of $D$ by Lemma~\ref{lem:scalar-jets}. The original sections are therefore themselves units along $D$. They agree modulo the original lower lattice $\omega_V((l-s)D)$, and so trivialize $L_{h_0}$ on the effective thickness
\[
T=(h_0-l+s)D-B_0.
\]
By \eqref{eq:canonical-ratio-bound}, $|h_0-l|$ is bounded independently of the sequence. Thus $T$ contains $nD$ for integers $n\to\infty$. Also $M_J\mid t$ implies $M_{J-b}\mid h_0$ once $J\ge b$. The line bundle $\cO(h_0D)$ is therefore trivial on $d_{J-b}D$, and $d_{J-b}\to\infty$ by \eqref{eq:width-bounds}. Removing this twist shows that $L=\omega_V(-B_0)$ is trivial on arbitrarily large finite thickenings $nD$.

Finally, choose the trivializations compatibly. Apply the stabilized-image argument in the proof of Lemma~\ref{lem:page-calculus} to the spaces $H^0(nD,L|_{nD})$, additionally imposing restriction to $D_{\mathrm{red}}$. The latter line bundle is trivial and its section space is one dimensional because $D_{\mathrm{red}}$ is connected, reduced, and projective. The image from every sufficiently large thickening contains a unit, hence is that full one-dimensional space. The surjective maps between stabilized images therefore lift a chosen nonzero section on $D_{\mathrm{red}}$ compatibly to all thickenings. Each lift is a unit (as may be checked after reduction), and their inverse limit trivializes $\widehat L$. This gives \eqref{eq:canonical-formal-root}.
\end{proof}

\subsection{The two endpoint residues and the limiting width}

Write $B_0=\sum_\alpha b_{0,\alpha}D_\alpha$, and define
\begin{equation}\label{eq:canonical-offset}
h=-\Bigl\lfloor\min_\alpha\frac{b_{0,\alpha}}{m_\alpha}\Bigr\rfloor,
\qquad
q=1-h.
\end{equation}
The formal isomorphism of Proposition~\ref{prop:canonical-formal-root} identifies the finite canonical jet lattices with scalar lattices having upper and lower bounds $lD+B_0$ and $(l-s)D+B_0$. All these finite quotients are algebraic sheaves supported on finite thickenings of $D$: the localized $\widehat\cO_V$-linear isomorphism respects every nested divisor lattice, and each width-$s$ quotient is killed by $\cI_D^s$. Since $\widehat\cO_V/\cI_D^s\widehat\cO_V=\cO_{sD}$, the induced map is $\cO_{sD}$-linear and compatible with the leading-layer projections.

The exact canonical counterpart of the scalar rule is: for all sufficiently large $s$, a canonical jet at weight $l$ with nonzero leading image has uniform integral scalar pole ratio
\begin{equation}\label{eq:canonical-scalar-ratio}
t=l-h.
\end{equation}
To prove this, enlarge the two bounds to uniform multiples of $D$ as in the preceding proof. A component detecting the leading image has ratio within a fixed distance of $l$, so for large $s$ the enlarged quotient still detects it. Lemma~\ref{lem:scalar-jets} gives a uniform integral ratio $t$. Set $\beta_0=\min_\alpha b_{0,\alpha}/m_\alpha$. Fitting under every upper bound but not under the preceding layer is exactly the pair of inequalities
\[
l-1+\beta_0<t\le l+\beta_0.
\]
Its unique integral solution is $t=l+\lfloor\beta_0\rfloor=l-h$.

The normalized unit sections of $\cO(tD)$ glue on the irregular thickness
\begin{equation}\label{eq:irregular-thickness}
T_s=(s-h)D-B_0,
\end{equation}
because this is the difference between $tD$ and the lower bound $(l-s)D+B_0$. Conversely a trivialization of $\cO(tD)$ on $T_s$, with $t=l-h$, includes in the canonical quotient and has nonzero leading image. The leading image has dimension one: normalized unit sections are proportional on $D_{\mathrm{red}}$, and their difference cannot retain the same leading ratio after reduction cancels. We have proved
\begin{equation}\label{eq:canonical-endpoint-rule}
g_s^0(l;\omega_V)=
\begin{cases}
1,&\cO((l-h)D)|_{T_s}\simeq\cO_{T_s},\\
0,&\text{otherwise},
\end{cases}
\qquad(s\text{ sufficiently large}).
\end{equation}

The comparison with ordinary widths is uniform and quite explicit. Put
\[
C_0=h+\Bigl\lceil\max_\alpha\frac{b_{0,\alpha}}{m_\alpha}\Bigr\rceil\ge0.
\]
Then, coefficient by coefficient,
\begin{equation}\label{eq:thickness-comparison}
(s-C_0)D\le T_s\le sD.
\end{equation}
For $e$ sufficiently large, let $d'_e$ be the last width $s$ for which $\cO(M_eD)|_{T_s}$ is trivial. The thicknesses are nested as $s$ increases. Equation~\eqref{eq:thickness-comparison} and the definition of $d_e$ give
\begin{equation}\label{eq:canonical-width-comparison}
d_e\le d'_e\le d_e+C_0.
\end{equation}
In particular the last width is finite and exists in the range of \eqref{eq:canonical-endpoint-rule}. Once $T_s$ contains $D$, the nilpotent-kernel argument used above shows that the last width is the same for $t=jM_e$ with $p\nmid j$. Also such a trivialization requires $r\mid t$; for $t\ne0$ the unique expression $t=jM_e$ with $p\nmid j$ classifies its last width. The value $t=0$ persists forever. To see that the large-$e$ rules exhaust the entries at large width, observe from \eqref{eq:thickness-comparison} that an entry at width $s$ gives an ordinary scalar trivialization to width $s-C_0$. For each fixed $J$, taking $s-C_0>d_{J-1}$ forces $M_J\mid t$.

\begin{proposition}[The limiting scalar widths]\label{prop:scalar-widths}
The scalar widths satisfy
\begin{equation}\label{eq:limiting-width}
a_*=\lim_{e\to\infty}\frac{d_e}{M_e}
=\frac{p}{p+1},
\qquad
Dc_2(V)=0.
\end{equation}
\end{proposition}

\begin{proof}
For all sufficiently large $e$ set
\[
s_e=\max(d_{e-1},d'_{e-1})+1.
\]
Equations \eqref{eq:width-bounds} and \eqref{eq:canonical-width-comparison} make these strictly increasing, with
$s_e\le\min(d_e,d'_e)$. Work with the three rows for $E=\cO_V$. At page $s_e$ all three rows are periodic modulo $M_e$, and each outer row has a single entry per period. The bottom entry is at $0$, by \eqref{eq:scalar-last-width}. For the top entry, \eqref{eq:top-endpoint} and \eqref{eq:canonical-scalar-ratio} associate to a top weight $i$ the scalar ratio $t=1-i-h=q-i$. Its entry is therefore at $q\pmod{M_e}$.

As the page runs from $s_e$ through $s_{e+1}$, the only outer drops are those with last-width index $e$. The bottom drops, of length $d_e$, consume middle-row dimensions at
\[
-d_e+jM_e,\qquad p\nmid j.
\]
The top drops, of length $d'_e$, consume middle-row dimensions at
\[
q+d'_e+jM_e,\qquad p\nmid j;
\]
here replacing $j$ by $-j$ accounts for the sign in the top ratio. Put
\[
A_e=-d_e,
\qquad
B_e=q+d'_e.
\]
The two lists, including their multiplicities when they meet, are thus $A_e+jM_e$ and $B_e+jM_e$ with $p\nmid j$.

We spell out the rank count also when the two centers coincide modulo $M_e$. Fix a residue $a\pmod{M_e}$ and let $m(a)\in\{0,1,2\}$ be its multiplicity in the multiset $\{A_e,B_e\}$. Its initial middle-row rank $n(a)$ is constant on all $p$ lifts modulo $pM_e$. Each of the $m(a)$ drop lists omits exactly one lift, namely its unshifted center modulo $pM_e$; at most two lifts are omitted. For example, when $p=5$ and $m(a)=2$, two distinct omitted lifts can be pictured as follows:
\[
\begin{array}{c|ccccc}
 & a&a+M_e&a+2M_e&a+3M_e&a+4M_e\\
\text{first list}&\circ&\bullet&\bullet&\bullet&\bullet\\
\text{second list}&\bullet&\bullet&\circ&\bullet&\bullet
\end{array}
\]
A filled dot consumes one rank; a circle marks that list's unshifted lift. If the omitted lifts coincide, the column is omitted twice and every other column receives both drops. Multiplicities are always added.

Since $p>2$, some lift receives all $m(a)$ drops. Nonnegativity of its final rank gives $n(a)\ge m(a)$. Summing over $a$ and using \eqref{eq:periodic-euler} and \eqref{eq:scalar-euler} gives
\[
2\le\sum_{a\bmod M_e}n(a)
=2-M_e\gamma\le2.
\]
Consequently $\gamma=0$, so $Dc_2(V)=0$, and $n(a)=m(a)$ for every residue.

After subtracting the drop lists, the final rank at each lift is exactly the number of lists omitting that lift. Thus at page $s_{e+1}$ the middle row is described by the same two-element multiset $\{A_e,B_e\}$, now modulo $M_{e+1}=pM_e$. Applying the just-proved description at index $e+1$ gives the multiset equality
\begin{equation}\label{eq:residue-swap}
\{-d_e,q+d'_e\}
=\{-d_{e+1},q+d'_{e+1}\}\pmod{M_{e+1}}.
\end{equation}
The entries $-d_e$ and $-d_{e+1}$ are distinct modulo $M_{e+1}$ because $0<d_e<d_{e+1}<M_{e+1}$. Therefore the matching in \eqref{eq:residue-swap} must be the swapped matching, even if a multiset initially has repeated entries. In particular
\[
d_{e+1}+q+d'_e\equiv0\pmod{M_{e+1}}.
\]
For large $e$ its left side is positive and, by $d_{e+1}<M_{e+1}$ and $d'_e\le d_e+C_0<M_e+C_0$, is less than $2M_{e+1}$. Hence
\begin{equation}\label{eq:width-swap-equality}
d_{e+1}+q+d'_e=M_{e+1}.
\end{equation}
Divide by $M_{e+1}=pM_e$ and let $e\to\infty$. Equation~\eqref{eq:canonical-width-comparison} gives $d'_e/M_{e+1}\to a_*/p$, while $d_{e+1}/M_{e+1}\to a_*$. Thus $a_*+a_*/p=1$, which proves \eqref{eq:limiting-width}.
\end{proof}

\section{Laurent cohomology of vector bundles}
\label{sec:laurent}

We work with the smooth threefold $V$, the cycle $D$, and the notation of the previous sections. The jet analysis produced
\begin{equation}\label{eq:laurent-width-input}
M_e=rp^e,
\qquad
\frac{d_e}{M_e}\longrightarrow a_*:=\frac{p}{p+1},
\qquad
c_2(V)D=0,
\end{equation}
with $\cO_V(M_eD)$ trivial through width $d_e$. Choose real numbers
\begin{equation}\label{eq:phase-constants}
1-\frac2p<c_0<c<2a_*-1.
\end{equation}
Such choices exist since
\[
2a_*-1-\Bigl(1-\frac2p\Bigr)=\frac{2}{p(p+1)}>0.
\]
As $p\ge5$, we also have $1/2<c_0<c<a_*$. Put $u_e=\lfloor cM_e\rfloor$. The lower bound on $c_0$ permits using only first and second tensor powers in Lemma~\ref{lem:phase-bound}; the upper bound on $c$ leaves the precision required for matching jets in Theorem~\ref{thm:laurent-vanishing}.

\subsection{Concentration of leading jets}

The first step restricts the positions at which a long jet of a semistable bundle can have a nonzero leading term.

\begin{lemma}\label{lem:phase-bound}
Let $G$ be a vector bundle that is semistable of slope zero for the degree $c_1(-)\cdot H\cdot D$. There is a constant $A_G$ such that, for every sufficiently large $e$,
\[
g^0_{\lfloor c_0M_e\rfloor}(i;G)>0
\quad\Longrightarrow\quad
\dist(i,M_e\Z)\le A_G.
\]
\end{lemma}

\begin{proof}
Write $s=\lfloor c_0M_e\rfloor$. For large $e$ we have $s\le d_e$. A unit of $\cO(M_eD)|_{sD}$ translates the upper weight of a jet by $M_e$. We can therefore replace $i$ by a centered representative $v$, with $|v|\le M_e/2$, without changing the existence of its leading class.

Suppose no uniform bound on $|v|$ exists. Passing to a sequence, we may assume that $v\to+\infty$ or $v\to-\infty$.

In the first case $s>v$ for large $e$. The jet maps to a global section of $G(vD)/G$ and retains its nonzero leading term. Its pole order tends to infinity, contrary to Proposition~\ref{prop:polar-bounds}.

In the second case choose a real number $\lambda$ such that
\begin{equation}\label{eq:phase-lambda}
\frac{1-c_0}{2}<\lambda<\frac1p,
\qquad
p\lambda>\frac12.
\end{equation}
The interval is nonempty by \eqref{eq:phase-constants}. Since the periods increase by the factor $p$, we can choose $j\le e$ with
\[
\lambda\le\frac{|v|}{M_j}\le p\lambda.
\]
Here $j\to\infty$ as $|v|\to\infty$. After a subsequence the ratios tend to a number $y\in[\lambda,p\lambda]$. There is an integer $m\in\{1,2\}$ for which
\begin{equation}\label{eq:phase-tensor-choice}
1-c_0<my<1.
\end{equation}
Indeed, if $y>1-c_0$, take $m=1$; then $y\le p\lambda<1$. Otherwise take $m=2$; its lower bound follows from $2y\ge2\lambda>1-c_0$, and its upper bound from $2y\le2(1-c_0)<4/p<1$.

Restrict the original jet to width $\lfloor c_0M_j\rfloor$, take its $m$th tensor power, and multiply by a unit section of $\cO(M_jD)$ to that width. Tensoring keeps at least the same width: in a difference of two tensor products, every summand contains a factor from the lower lattice. The resulting jet is a jet of $G^{\otimes m}$ at upper weight
\[
w=M_j+mv.
\]
By \eqref{eq:phase-tensor-choice}, $w\to+\infty$ and $w<\lfloor c_0M_j\rfloor$ with a margin tending to infinity. It therefore gives a global polar part of $G^{\otimes m}$.

This polar part has unbounded actual order. At some component of $D$ an original coefficient has pole ratio greater than $v-1$. The corresponding pure tensor coefficient, after multiplication by the unit, has pole ratio greater than $M_j+m(v-1)$. This tends to infinity. Proposition~\ref{prop:polar-bounds} applies to both $G$ and $G\otimes G$, giving the required contradiction.
\end{proof}

\subsection{A vanishing criterion}

We use $\widehat E$ for the completion of a bundle $E$ along $D$, and $E_{\cR}=\widehat E\otimes_{\widehat\cO}\cR$. The finite-cover, bounded-pole interpretation is the one used above.

\begin{theorem}\label{thm:laurent-vanishing}
For the prepared data and periods fixed above, let $E$ be a vector bundle on $V$ satisfying:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\chi(P_i^1(E))=0$ for every integer $i$.
\item For $J=E,E^!$ the spaces $H^0(V,J(nD)/J)$ have dimensions bounded independently of $n\ge1$.
\item For $J=E,E^!$ there is a constant $A_J$ such that
\[
g^0_{u_e}(i;J)>0\quad\Longrightarrow\quad
\dist(i,M_e\Z)\le A_J
\]
for all sufficiently large $e$.
\end{enumerate}
Then global sections of $E_{\cR}$ form a finite-dimensional $k$-vector space. Every Laurent \v Cech $1$-cocycle on a finite affine cover adapted to the bundles and the divisor has an actual Laurent primitive. The same assertion holds after refining such a cover.
\end{theorem}

\begin{proof}
We split the argument into endpoint patterns, drops, the middle row, and convergence.

\smallskip\noindent\emph{Endpoint patterns.}
For $e$ large, $u_e\le d_e$, so the page at width $u_e$ is periodic modulo $M_e$. By (iii), the bottom row can be supported only at a fixed bounded set of centered positions. At each such position the descending sequence of leading images stabilizes as the width tends to infinity. The stable-image statement of Lemma~\ref{lem:page-calculus} realizes every vector in the stabilized image by a global completed section at that cut. Consequently, for all sufficiently large $e$, there is one fixed bottom pattern, at positions $v$ with fixed multiplicities, repeated by translations by $M_e$. Duality gives a fixed top pattern at positions $w$. Enlarge a constant $A$ so that all these $v,w$ lie in $[-A,A]$.

This already proves finite-dimensionality of global Laurent sections. A nonzero section has a least upper cut $i$: its poles are bounded, and the filtration is separated toward negative infinity. Its nonzero leading class belongs to $g^0_{u_e}(i;E)$ for every large $e$. With $i$ fixed and $M_e\to\infty$, condition (iii) forces $i$ to be an unshifted position of the fixed pattern. There are only finitely many such cuts, and each leading quotient is finite-dimensional, giving a finite filtration of the space of global sections. The same observation applies to $E^!$.

\smallskip\noindent\emph{Endpoint drops.}
By the middle-row recurrence, a bottom drop of length $s$ at $v+jM_e$ removes middle-row rank at $v-s+jM_e$; a top drop at $w+jM_e$ removes rank at $w+s+jM_e$. To determine these losses over the next period we control their lengths, with multiplicities, uniformly in $j$ whenever $p\nmid j$.

Consider a bottom position $i=v+jM_e$. If $p\mid j$, the next period $M_{e+1}=pM_e$ retains exactly the same multiplicity there; monotonicity therefore forbids any drop between pages $u_e$ and $u_{e+1}$. If $p\nmid j$, all of this multiplicity disappears by page $u_{e+1}$. We claim the drops occur at lengths
\begin{equation}\label{eq:drop-range}
d_e\le s\le M_e+C
\end{equation}
for a fixed $C$, with multiplicities at each length independent of $j$ among integers prime to $p$.

No drop occurs before length $d_e$: translation by a unit of $\cO(M_eD)|_{d_eD}$ identifies leading images with those at the unshifted stable position $v$. Choose local formal unit lifts $t_\lambda$ of this trivializing section of $\cO(M_eD)|_{d_eD}$, and write on overlaps
\[
\frac{t_\lambda}{t_\mu}=1+b_{\lambda\mu},
\qquad b_{\lambda\mu}\in\cI_D^{d_e}\widehat\cO_V,
\qquad \cI_D=\cO_V(-D).
\]
Fix a constant $C$, and pick $d_e<k\le M_e+C$. For large $e$, $k<2d_e$ by \eqref{eq:laurent-width-input}. Normalize local representatives $\sigma_\lambda$ of a width-$k$ jet at $v+jM_e$ by putting $\alpha_\lambda=\sigma_\lambda/t_\lambda^j$. These are local sections of $\widehat{E(vD)}$ and satisfy
\begin{equation}\label{eq:normalized-jet-overlap}
\alpha_\mu=(1+jb_{\lambda\mu})\alpha_\lambda
\pmod{\cI_D^k\widehat{E(vD)}}.
\end{equation}
Indeed $(1+b_{\lambda\mu})^j=1+jb_{\lambda\mu}$ to this precision, for positive or negative $j$ alike, because $k<2d_e$. In particular, the cochain $\alpha$ glues to width $d_e$.

We next match $\alpha$ to a single global completed section $\beta$ in that untwisted lattice through width $k-d_e$. Every completed section and every formal lattice map restricts to the actual algebraic finite jets of \eqref{eq:jet-quotient}: on each affine intersection this is the identity $\widehat A/(f^s)=A/(f^s)$. These restrictions commute with the \v Cech maps and with projection to leading layers. Start with $\beta_{<0}=0$. After matching the first $n$ depths, the sum $\beta_{<n}$ of the completed global sections already chosen satisfies
\[
\alpha-\beta_{<n}\in C_{v-n}^0(E),
\qquad
\partial(\alpha-\beta_{<n})\in C_{v-d_e}^1(E).
\]
The second inclusion persists because each subtracted section is global. Thus, for $n<k-d_e$, the residual is a finite jet at upper weight $v-n$ of width at least $d_e-n$. The inequalities
\begin{equation}\label{eq:matching-margin}
d_e-n\ge 2d_e-M_e-O(1)>u_e,
\qquad k-d_e<M_e/2-O(1)
\end{equation}
hold with margins growing linearly in $M_e$, by $c<2a_*-1$ and $a_*>1/2$. Thus $v-n$ is centered, and condition (iii) says that its leading image is either zero or one of the unshifted stabilized images. In the latter case lift and subtract a completed global section. A finite induction constructs $\beta$.

If $p\nmid j,j'$, put $q=j'/j$ in the prime field and set $\alpha'=q\alpha+(1-q)\beta$. Since $\beta$ is global, \eqref{eq:normalized-jet-overlap} gives
\[
\alpha'_\mu-(1+j'b_{\lambda\mu})\alpha'_\lambda
=j'(1-q)b_{\lambda\mu}(\alpha_\lambda-\beta)
\pmod{\cI_D^k\widehat{E(vD)}}.
\]
The right side vanishes: its two factors have depths $d_e$ and $k-d_e$. Retwisting by $t_\lambda^{j'}$ gives a width-$k$ jet at $v+j'M_e$ with the same normalized leading term (since $k-d_e\ge1$). Also $\alpha'-\beta=q(\alpha-\beta)$, so the same $\beta$ reverses the construction with $q^{-1}$. The normalized leading-image subspaces are therefore equal at every width under consideration, and their ranks and drops are independent of $j$ prime to $p$.

For $j=1$, property (ii) kills the entire leading image by width $M_e+C$ once $C>A$ is sufficiently large: this width is at least the upper weight $M_e+v$, so a surviving leading jet would give a global polar part of order tending to infinity, and the increasing spaces in (ii) stabilize and forbid such polar parts. Since $u_{e+1}>M_e+C$ for large $e$, this proves \eqref{eq:drop-range} and the complete drop rule. The identical argument for $E^!$, followed by the duality of Lemma~\ref{lem:page-calculus}, gives the top-row rule.

\smallskip\noindent\emph{The middle row.}
Let $T_e$ be the multiset of the numbers $v-s$ for bottom drops and $w+s$ for top drops, each counted with its drop multiplicity for one translating integer prime to $p$. Its total size is the sum of the ranks of the two fixed endpoint patterns. Put $H_e(i)=g^1_{u_e}(i;E)$. The middle-row recurrence of Lemma~\ref{lem:page-calculus} gives
\begin{equation}\label{eq:middle-distribution}
\begin{split}
H_{e+1}(i)=H_e(i)
&-\#\{t\in T_e:t\equiv i\pmod{M_e}\}\\
&+\#\{t\in T_e:t\equiv i\pmod{pM_e}\}.
\end{split}
\end{equation}
The last term restores the unshifted copies. All members of $T_e$ lie in the two intervals
\begin{equation}\label{eq:drop-bands}
[-M_e-O(1),-d_e+O(1)]
\quad\hbox{and}\quad
[d_e-O(1),M_e+O(1)].
\end{equation}
Each interval has length less than $M_e$ for large $e$. For any fixed residue modulo $M_e$, these intervals can occupy at most two of its lifts modulo $pM_e$. Since $p\ge5$, some lift is missed. Nonnegativity of $H_{e+1}$ at that lift and periodicity of $H_e$ show that
\[
H_e(i)\ge\#\{t\in T_e:t\equiv i\pmod{M_e}\}.
\]
Summing over a period gives equality of the two sides by the Euler identity of Lemma~\ref{lem:page-calculus} and property (i). The inequality is therefore equality at every residue. Equation~\eqref{eq:middle-distribution} now says $H_{e+1}$ is exactly the distribution of $T_e$ modulo $pM_e$. For each fixed integer $i$, the bands \eqref{eq:drop-bands} and $d_e\to\infty$ imply
\begin{equation}\label{eq:fixed-weight-vanishing}
g^1_{u_e}(i;E)=0\qquad\text{for all sufficiently large }e.
\end{equation}

\smallskip\noindent\emph{Convergence of primitives.}
Fix an integer upper bound $l$ for a Laurent cocycle. We bound the source $z=x+s$ of every nonzero incoming differential of length $s$ to a middle-row weight $x\le l$ by a function $B_l(x)$ satisfying
\begin{equation}\label{eq:primitive-cut-bound}
\sup_{x\le l}B_l(x)<\infty,
\qquad B_l(x)\longrightarrow-\infty\quad(x\to-\infty).
\end{equation}

At a sufficiently late period, the drop rule gives $z=v+jM_e$, $p\nmid j$, and $d_e\le s\le M_e+C$. If $j\ge2$, then $x\ge M_e-A-C$, exceeding $l$ for large $e$. If $j=1$, the target must lie in the fixed finite set of integers in $[-A-C,l]$. Choose one index $e_0$, depending on $l$, beyond the thresholds for the endpoint and drop rules, so that $M_e-A-C>l$ for $e\ge e_0$ and every middle rank in this finite set is zero at page $u_{e_0}$. Equation~\eqref{eq:fixed-weight-vanishing} permits this simultaneous choice; middle ranks only decrease, so no nonzero incoming image can have $j=1$ at any later page.

Put $S_l=u_{e_0}\ge1$. Earlier lengths $s<S_l$ give $z\le x+S_l$. Every later length belongs to a period $u_e\le s<u_{e+1}$ with $e\ge e_0$, and the preceding exclusions leave only $j\le-1$. In that case $z\le-M_e+A$, so $M_e\le-z+A$, giving $x=z-s\ge2z-A-C$, i.e.
\[
z\le\frac{x+A+C}{2}.
\]
The real-valued upper bound
\[
B_l(x)=\max\Bigl\{x+S_l,\frac{x+A+C}{2}\Bigr\}
\qquad(x\le l)
\]
therefore bounds every possible source and satisfies both assertions of \eqref{eq:primitive-cut-bound}.

Let $\gamma$ be an actual Laurent $1$-cocycle of upper cut $x\le l$. Its leading cochain belongs to every cycle space $Z_s^{x,1}$. By \eqref{eq:fixed-weight-vanishing}, it eventually belongs to the boundary space $U_s^{x,1}$. More precisely, the page construction identifies
\[
U_{s+1}^{x,1}/U_s^{x,1}
=\im\bigl(d_s:E_s^{x+s,0}\longrightarrow E_s^{x,1}\bigr),
\]
so $U_s^{x,1}$ stops increasing once $x+s>B_l(x)$. Each increase from a source cut $z=x+s$ has a representative primitive in $C_z^0(E)$, and the initial $U_1^{x,1}$ uses $C_x^0(E)$. Consequently the leading cochain of $\gamma$ has a primitive of cut at most $\max\{x,B_l(x)\}$. Subtract that primitive's coboundary. The new cocycle has strictly smaller upper cut. Repeating at successive descending cuts gives primitives whose upper cuts tend to negative infinity by \eqref{eq:primitive-cut-bound}. Their sum converges in a single bounded-pole completed lattice, and the differential is continuous, so the resulting cochain has coboundary $\gamma$. All arguments use finite covers by affine charts and their affine intersections, and remain valid after refinement.
\end{proof}

\subsection{Cartier algebra on the Laurent charts}

Write $F$ for absolute Frobenius and $\Omega=\Omega_V^1$. Let $B^1,Z^1\subset F_*\Omega$ be the sheaves of exact and closed algebraic one-forms, with the Frobenius module structure.

\begin{lemma}\label{lem:laurent-cartier-algebra}
On a smooth affine chart with \'etale coordinates $x_1,x_2,x_3$ and a local equation $f$ for $D$, put $R=\widehat A[1/f]$. The monomials $x^I=x_1^{I_1}x_2^{I_2}x_3^{I_3}$ with $0\le I_j<p$ form a basis of $R$ over $R^p$. Ordinary differentials and the Cartier isomorphism extend to this ring. The algebraic bundles $B^1,Z^1$ and their defining exact sequences extend to the corresponding Laurent exact and closed forms, with their Frobenius module structure. In particular,
\begin{equation}\label{eq:cartier-sequences}
0\longrightarrow\cO_V\longrightarrow F_*\cO_V
\longrightarrow B^1\longrightarrow0,
\qquad
0\longrightarrow B^1\longrightarrow Z^1
\xrightarrow{C}\Omega\longrightarrow0,
\end{equation}
are sequences of vector bundles and remain exact on Laurent charts. The images and kernels in the Frobenius de Rham complex are locally free, as are the relevant quotients. Moreover,
\begin{equation}\label{eq:frobenius-duality}
(F_*\cO_V)^\vee\otimes\omega_V\simeq F_*\omega_V.
\end{equation}
\end{lemma}

\begin{proof}
The algebraic calculation is the $p$-basis form of Cartier's theorem \cite[Theorem~1]{cartier1957}; see also \cite[Theorem~7.2]{katz1970}. We check that its finite basis survives the completion and localization.

For an \'etale algebra, the relative Frobenius square is a base change square: one checks this because the relative Frobenius is \'etale and induces isomorphisms on geometric fibers. The polynomial-coordinate monomial basis over $p$th powers therefore gives the asserted basis for $A$ over $A^p$. The field $k$ is perfect.

Complete the $p$-power subring along $f^p$ and use this finite basis. The $f$-adic and $f^p$-adic topologies on $A$ coincide, and Frobenius identifies the completion of $A^p$ with $(\widehat A)^p$. The basis thus persists after completion and after inverting $f$; in the latter step, $1/f=f^{p-1}/f^p$.

Every $k$-derivation kills $p$th powers. Since each element has a unique finite expansion in the displayed basis with coefficients in $R^p$, its differential is determined by ordinary differentiation of those monomials. Conversely the coordinate derivations are defined by that rule. Hence $\Omega^1_{R/k}$ is free on $dx_j$.

In one variable, the de Rham complex decomposes into the monomials of degrees less than $p$; its cohomology has representatives $1$ in degree zero and $x^{p-1}dx$ in degree one. Tensoring the three one-variable complexes proves local freeness of images, kernels, and quotients, and proves the inverse Cartier rule
\[
f\longmapsto f^p,
\qquad
df\longmapsto[f^{p-1}df].
\]
The rule is intrinsic. Leibniz holds directly; additivity for differentials follows modulo exact forms by differentiating the mixed terms of the integral polynomial $((X+Y)^p-X^p-Y^p)/p$ and reducing modulo $p$. Thus the local isomorphisms glue.

More explicitly, the scalar extension of the algebraic Frobenius complex to a Laurent chart is the actual Laurent de Rham complex with Frobenius action, through
\[
R\otimes_A F_*\Omega_A^\bullet\longrightarrow F_*\Omega_R^\bullet,
\qquad a\otimes\eta\longmapsto a^p\eta.
\]
The monomial and coordinate-wedge bases make this an isomorphism. This proves the assertions about \eqref{eq:cartier-sequences}. Finally, the pairing obtained by Cartier on top forms is perfect: in coordinates it extracts the coefficient of $x_1^{p-1}x_2^{p-1}x_3^{p-1}dx_1\wedge dx_2\wedge dx_3$. It gives \eqref{eq:frobenius-duality} and is intrinsic by the same Cartier rule.
\end{proof}

\subsection{Application to one-forms and exact forms}

\begin{corollary}\label{cor:form-vanishing}
Every Laurent \v Cech $1$-cocycle with values in $\Omega$, $B^1$, or $Z^1$ has a Laurent primitive. Global Laurent one-forms form a finite-dimensional $k$-vector space.
\end{corollary}

\begin{proof}
For $E=\Omega$, both $E$ and $E^!=T_V\otimes\omega_V$ are semistable of slope zero by Proposition~\ref{prop:strong-semistability} and the canonical intersection identities. Proposition~\ref{prop:polar-bounds} and Lemma~\ref{lem:phase-bound} give conditions (ii) and (iii) of Theorem~\ref{thm:laurent-vanishing}; passing from width $\lfloor c_0M_e\rfloor$ to $u_e$ can only decrease the leading images. Hirzebruch--Riemann--Roch gives condition (i): the degree-two terms in $\ch(\Omega)\td(T_V)$ are linear combinations of $K_V^2$ and $c_2(V)$, both killed by intersection with $D$, and every term involving $D^2$ is also zero.

For $E=B^1$, the first sequence in \eqref{eq:cartier-sequences} gives condition (i). By projection formula, a single cell for $F_*\cO_V$ has the Euler characteristic of $p$ successive scalar cells, each zero by Proposition~\ref{prop:scalar-widths}. Frobenius changes the $k$-module action but not these finite dimensions.

There are subbundle inclusions with locally free quotients
\begin{equation}\label{eq:exact-form-inclusions}
B^1\longhookrightarrow F_*\Omega,
\qquad
(B^1)^!\longhookrightarrow F_*\omega_V.
\end{equation}
The first follows from Lemma~\ref{lem:laurent-cartier-algebra}; the second follows by dualizing the first sequence in \eqref{eq:cartier-sequences} and using \eqref{eq:frobenius-duality}. For a subbundle $E\subset F_*G$, the injection remains injective on every quotient $E(nD)/E$. Projection formula embeds that quotient in $F_*(G(pnD)/G)$, so the polar bounds for $G=\Omega$ and $G=\omega_V$ give condition (ii) for both bundles in \eqref{eq:exact-form-inclusions}. All these bounds concern fixed bundles and their first Frobenius pushforwards; the growing index $e$ below changes the jet period, not the bundle in the polar-part estimate.

The same subbundle property gives condition (iii), with a small precision margin. A width-$u_e$ jet of $E\subset F_*G$ with nonzero leading term maps to a width-$pu_e$ jet of $G$ at upper weight $pi$, with nonzero leading term in the top $p$ ordinary layers. Choose its first nonzero ordinary layer, at a weight $a$ with $|a-pi|<p$; it has width at least $pu_e-p$. Since $c>c_0$ and $M_{e+1}=pM_e$, for large $e$
\[
pu_e-p\ge\lfloor c_0M_{e+1}\rfloor.
\]
Lemma~\ref{lem:phase-bound} at period $M_{e+1}$ implies that $a$, and hence $pi$, is at bounded distance from $M_{e+1}\Z$. Dividing by $p$ gives the required bounded distance from $M_e\Z$.

Theorem~\ref{thm:laurent-vanishing} now applies to $\Omega$ and $B^1$, including finite-dimensionality for the former. For a $Z^1$ cocycle, first solve its image in $\Omega$ and lift the resulting local primitives through Cartier. After subtraction the remaining cocycle lies in $B^1$ and can also be solved. These lifts exist on affine Laurent charts by Lemma~\ref{lem:laurent-cartier-algebra}, and refining the finite cover if necessary preserves every preceding argument.
\end{proof}

\section{Cartier descent and Picard divisibility}
\label{sec:cartier}

We work on the Zariski topological space $|D|$, with
\[
\widehat\cO=\widehat\cO_V,
\qquad
\cR=\widehat\cO_V(*D).
\]
Here and throughout, a \emph{line bundle} means a locally free module of rank one for the indicated sheaf of rings. Write $\Omega_{\cR}^j$ for the actual Laurent differential forms, and $B^1\subset Z^1\subset\Omega_{\cR}^1$ for the additive sheaves of exact and closed one-forms. Lemma~\ref{lem:laurent-cartier-algebra} identifies these with the Laurent extensions of the algebraic Frobenius modules. In particular there is an exact sequence of additive sheaves
\begin{equation}\label{eq:laurent-cartier-sequence}
0\longrightarrow B^1\longrightarrow Z^1
\xrightarrow{\ C\ }\Omega_{\cR}^1\longrightarrow0,
\end{equation}
where $C$ is inverse-Frobenius-semilinear for the ordinary scalar structures:
\[
C(a^p\alpha)=aC(\alpha).
\]
The distinction between these scalar structures and the Frobenius-module structures is essential in using the preceding cohomology calculation.

Corollary~\ref{cor:form-vanishing} supplies the Laurent cocycle vanishings and the finite-dimensionality of global one-forms that we use to make $\Pic(|D|,\cR)$ $p$-divisible. Orders along the components of $D$ then let intersection with $H$ descend to a nonzero integer degree on that same group. These two properties are incompatible.

\subsection{Global Cartier adjustment}

\begin{lemma}\label{lem:cartier-adjustment}
Every global Laurent one-form is closed. On the finite-dimensional $k$-vector space
\[
W=H^0(|D|,\Omega^1_{\cR}),
\]
Cartier is a bijective inverse-Frobenius-semilinear map, and $C-\id\colon W\to W$ is surjective as an additive map. Moreover, every additive $Z^1$-valued \v Cech one-cocycle is a coboundary after refining its open cover.
\end{lemma}

\begin{proof}
Corollary~\ref{cor:form-vanishing} gives finite-dimensionality of $W$ and actual Laurent one-cocycle vanishing for both $B^1$ and $Z^1$, proving the last assertion.

To lift a global form through $C$, choose local lifts in $Z^1$ using \eqref{eq:laurent-cartier-sequence}. Their differences form a $B^1$-cocycle. Subtracting a coboundary makes the lifts agree, so
\[
C\colon H^0(|D|,Z^1)\longrightarrow W
\]
is surjective. Its source is a vector subspace of $W$. Since the ground field is perfect, a semilinear map has the same dimension behaviour as a linear map. Finite-dimensionality therefore implies $H^0(Z^1)=W$ and bijectivity of $C$.

For the surjectivity of $C-\id$, set $F=C^{-1}$, a bijective $p$-semilinear map. Choose coordinates on $W\simeq k^n$. Then $F(x)=Ax^{[p]}$ with $A\in\operatorname{GL}_n(k)$, where $x^{[p]}$ means componentwise $p$-th powers. The polynomial map $F-\id$ extends to
\begin{equation}\label{eq:semilinear-projective-map}
\Psi:\PP^n\longrightarrow\PP^n,
\qquad
[x:t]\longmapsto
[Ax^{[p]}-t^{p-1}x:t^p].
\end{equation}
There is no base point: if $t=0$, the first coordinates vanish only when $x=0$. Also $\Psi^*\cO_{\PP^n}(1)=\cO_{\PP^n}(p)$. A positive-dimensional fiber would contain a projective curve on which this line has both degree zero and positive degree, contradiction. Hence $\Psi$ is finite. Its image is closed of dimension $n$, so it is all of $\PP^n$. The inverse image of the standard affine chart is exactly that chart, proving surjectivity of $F-\id$ over the algebraically closed field. Finally, $C-\id$ is obtained from $F-\id$ by the substitution $(C-\id)\beta=\eta\iff(F-\id)\beta=-F\eta$, so it is surjective as well.
\end{proof}

\begin{lemma}\label{lem:cartier-fixed-connection}
Every $\cR$-line bundle admits an integrable connection whose local connection one-forms are fixed by Cartier.
\end{lemma}

\begin{proof}
Let $\cL$ be such a line bundle. Its frame cover can be refined by opens $U_i\cap|D|$ where $U_i\subset V$ is affine, has \'etale coordinates, makes $D$ principal, and trivializes the finitely many algebraic bundles in use. Quasi-compactness of $|D|$ permits a finite such cover, and its finite intersections are restrictions of affine opens since $V$ is separated. On each intersection, a Laurent coefficient has a finite pole bound: take a finite subcover of local bounds and glue in their common completed lattice, as in Section~\ref{sec:jets}. Taking the maximum over the finitely many coefficients gives a common upper cut for a cochain. Thus the cocycles below lie in the actual Laurent complexes to which the vanishing results apply.

Choose frames $e_i$ on this cover, and write $e_j=g_{ij}e_i$ on overlaps. Then $g_{ij}g_{jk}=g_{ik}$, so
\[
d\log g_{ij}=g_{ij}^{-1}dg_{ij}
\]
is a closed additive one-cocycle. It is Cartier-fixed. Indeed, the intrinsic inverse Cartier formula gives
\[
C^{-1}(g^{-1}dg)
=\bigl[g^{-p}g^{p-1}dg\bigr]=[g^{-1}dg]
\]
for every Laurent unit $g$.

By Lemma~\ref{lem:cartier-adjustment}, after refining the cover there are closed one-forms $\alpha_i$ such that
\[
\alpha_j-\alpha_i=d\log g_{ij}.
\]
These are the connection forms of a connection defined by $\nabla e_i=\alpha_i\otimes e_i$. Their closedness is precisely its integrability, since the bundle has rank one.

The forms $C\alpha_i-\alpha_i$ agree on overlaps, hence define a global form $\eta\in W$. Choose $\beta\in W$ with $(C-\id)\beta=\eta$ using Lemma~\ref{lem:cartier-adjustment}. Replace every $\alpha_i$ by $\alpha_i-\beta$. This preserves their differences and their closedness, and now $C\alpha_i=\alpha_i$ for all $i$.
\end{proof}

\subsection{Horizontal sections over a Laurent chart}

We give the local descent calculation, including compatibility with restriction. This is the algebra underlying Cartier descent \cite[Theorem~5.1 and formulas (5.1.2)--(5.1.7)]{katz1970}; the formulas below apply directly to our Laurent rings because they retain a \(p\)-basis, without a finite-type hypothesis on those rings.

Take a smooth affine chart $U=\Spec A\subset V$ with \'etale coordinates $x_1,x_2,x_3$, and let $f$ be an equation of $D$ on it. Put
\[
B=\varprojlim_n A/(f^n),
\qquad
R=B[1/f],
\qquad
S=R^p.
\]
Lemma~\ref{lem:laurent-cartier-algebra} gives
\begin{equation}\label{eq:cartier-pbasis}
R=\bigoplus_{0\le I_1,I_2,I_3<p}Sx^I,
\qquad
\Omega^1_{R/k}=\bigoplus_{i=1}^3 R\,dx_i.
\end{equation}
Write $\partial_i$ for the coordinate derivations. They commute, kill $S$, and satisfy $\partial_i^p=0$.

\begin{lemma}[The rank-one curvature calculation]\label{lem:cartier-curvature}
If $\alpha=\sum_i a_i\,dx_i$ is closed and $C\alpha=\alpha$, then the connection operators
\[
D_i=\partial_i+a_i:R\longrightarrow R
\]
commute and satisfy
\begin{equation}\label{eq:horizontal-relations}
D_i^p=0,
\qquad
[D_i,x_j]=\delta_{ij},
\end{equation}
where a ring element in an operator formula denotes multiplication by that element.
\end{lemma}

\begin{proof}
If $C\alpha=\sum_i b_i\,dx_i$, the $p$-basis Cartier calculation represents $\alpha$ in the form
\[
\alpha=dc+\sum_i b_i^p x_i^{p-1}\,dx_i
\qquad(c\in R).
\]
Consequently
\begin{equation}\label{eq:cartier-coefficient}
\partial_i^{p-1}a_i=(p-1)!\,b_i^p=-b_i^p.
\end{equation}
Exact terms disappear because $\partial_i^p c=0$, and $b_i^p$ is constant for every coordinate derivation. The Cartier-fixed condition therefore says
\[
a_i^p+\partial_i^{p-1}a_i=0.
\]

We recall the operator identity giving its significance. For a derivation $\partial$ with $\partial^p=0$ and an element $a\in R$,
\begin{equation}\label{eq:rank-one-pcurvature}
(\partial+a)^p=a^p+\partial^{p-1}a.
\end{equation}
First, the left side commutes with multiplication by any $b\in R$: in characteristic $p$, its commutator is the $p$-fold iterated commutator, namely multiplication by $\partial^pb=0$. It is therefore multiplication by its value on $1$. Over the integers, successive use of the product rule expands $(\partial+a)^p(1)$ as a sum indexed by set partitions of $\{1,\ldots,p\}$, a block of size $j$ contributing $\partial^{j-1}a$. A cyclic permutation of the $p$ labels has orbits of size $p$ on these partitions, except for the discrete partition and the partition with one block; the summands have the same value throughout an orbit, so only $a^p$ and $\partial^{p-1}a$ survive in characteristic $p$. This proves \eqref{eq:rank-one-pcurvature}, hence $D_i^p=0$. Finally,
\[
[D_i,D_j]=\partial_i a_j-\partial_j a_i=0
\]
by closedness, and the commutator with $x_j$ follows directly from the Leibniz rule.
\end{proof}

\begin{lemma}[Horizontal expansion]\label{lem:horizontal-expansion}
Let $M$ be a free rank-one $R$-module with a connection whose operators satisfy \eqref{eq:horizontal-relations} and commute. Its horizontal module
\[
H_M=\bigcap_i\ker(D_i\colon M\to M)
\]
is a finite projective rank-one $S$-module, and multiplication gives an isomorphism
\begin{equation}\label{eq:horizontal-descent}
R\otimes_S H_M\xrightarrow{\ \sim\ }M.
\end{equation}
Formation of $H_M$ commutes with restriction to a smaller Laurent chart retaining the coordinates.
\end{lemma}

\begin{proof}
Define $S$-linear operators
\begin{equation}\label{eq:horizontal-projector}
P_i=\sum_{n=0}^{p-1}\frac{(-1)^n}{n!}x_i^n D_i^n,
\qquad
P=P_1P_2P_3,
\end{equation}
with the multiplication operator $x_i^n$ on the left. All factorials are invertible in $k$. The commutator relation gives a telescoping sum
\[
D_iP_i=\frac{(-1)^{p-1}}{(p-1)!}x_i^{p-1}D_i^p=0.
\]
Also $P_i$ is the identity on $\ker D_i$, and distinct $P_i$'s commute with each other and with $D_j$ for $j\ne i$. Hence $P$ is an idempotent with image $H_M$.

For one coordinate, the binomial identity gives
\begin{equation}\label{eq:horizontal-taylor-one}
\sum_{n=0}^{p-1}\frac{x_i^n}{n!}P_iD_i^n=\id_M.
\end{equation}
Indeed, after substituting \eqref{eq:horizontal-projector}, the coefficient of $x_i^rD_i^r$ for $0\le r<p$ is
\[
\sum_{n+j=r}\frac{(-1)^j}{n!j!}
=\begin{cases}1&r=0,\\0&r>0,\end{cases}
\]
and all terms with $r\ge p$ vanish by $D_i^p=0$. Multiplying \eqref{eq:horizontal-taylor-one} over the three coordinates gives the explicit expansion
\begin{equation}\label{eq:horizontal-taylor}
m=\sum_{0\le I_j<p}x^Ih_I,
\qquad
h_I=\frac1{I!}P(D^I m)\in H_M,
\end{equation}
with $I!=I_1!I_2!I_3!$ and $D^I=D_1^{I_1}D_2^{I_2}D_3^{I_3}$. The coefficients are unique: for $h\in H_M$,
\[
PD^J(x^Ih)=\begin{cases}I!h&I=J,\\0&I\ne J,\end{cases}
\qquad 0\le I_i,J_i<p.
\]
Together with \eqref{eq:cartier-pbasis}, this proves \eqref{eq:horizontal-descent}.

As an $S$-module, $M$ is free of rank $p^3$. The image of the idempotent $P$ is therefore finite projective. The extension $S\subset R$ is finite free and faithfully flat; \eqref{eq:horizontal-descent} shows that this projective module has rank one.

For restriction, let $R\to R'$ be the homomorphism of Laurent chart rings and $S'=(R')^p$. The shared monomial bases give $S'\otimes_S R\simeq R'$. The restricted module is $S'\otimes_S M$, its operators are $1\otimes D_i$, and its projector is $1\otimes P$. Taking the image of a split idempotent commutes with every scalar extension, so its horizontal module is $S'\otimes_S H_M$. Since horizontality means $\nabla m=0$, the construction is independent of the coordinate system and the chosen line frame.
\end{proof}

\subsection{The formal Zariski topology and Picard divisibility}

The projective module in Lemma~\ref{lem:horizontal-expansion} must be locally trivial on the topology of $|D|$. The next elementary completion fact supplies this step.

\begin{lemma}\label{lem:formal-line-extension}
For the smooth affine chart and rings $A,B,R$ above, $B$ is a regular Noetherian ring. Every finite projective rank-one $R$-module extends to a finite projective rank-one $B$-module. Its associated line is locally trivial after restriction to formal Zariski neighborhoods of every point of $U\cap|D|$. The same assertions hold for $B^p\subset R^p$.
\end{lemma}

\begin{proof}
The Noetherian completion theorem gives that $B$ is Noetherian and flat over $A$, and $B/(f^n)=A/(f^n)$ \cite[Lemmas~10.97.1--10.97.5]{stacks}. The element $f$ belongs to the Jacobson radical of $B$: an element congruent to $1$ modulo $f$ has an inverse by the convergent geometric series, and more generally an element invertible modulo $f$ is invertible in $B$ \cite[Lemma~10.96.6]{stacks}.

Every maximal ideal $\mathfrak n\subset B$ corresponds to a maximal ideal $\mathfrak m\subset A$ containing $f$, with $\mathfrak n=\mathfrak mB$. For each $s$, completion modulo $\mathfrak m^s$ does nothing, since $f^s$ already vanishes there. Hence
\[
B/\mathfrak n^s\simeq A/\mathfrak m^s,
\qquad
\widehat{B_{\mathfrak n}}\simeq\widehat{A_{\mathfrak m}}.
\]
The latter ring is regular, as $A_{\mathfrak m}$ is a smooth local ring. A Noetherian local ring is regular if and only if its maximal-ideal completion is regular \cite[Lemma~15.44.4]{stacks}, so every $B_{\mathfrak n}$ is regular, and localization gives regularity at every prime of $B$.

A regular Noetherian ring is locally factorial \cite[Lemma~15.123.2]{stacks}. Its finitely many irreducible components are disjoint, since each local ring is a domain. On each component choose a rational section of the given line on $\Spec R$ and take its Weil divisor. The closure of this finite Weil divisor on $\Spec B$ is Cartier by local factoriality. Doing this on all components gives the required extension. This is the Cartier--Weil correspondence on a regular scheme \cite[II, \S6]{hartshorne1977}; global unique factorization of $B$ is not required.

Let $z\in U\cap|D|$. The extended line is free on a basic open $\Spec B_g$ containing the prime corresponding to $z$. Choose $g_0\in A$ whose image modulo $f$ equals that of $g$. Then $g_0(z)\ne0$. On the formal basic neighborhood coming from $\Spec A_{g_0}$, its ring is $B'=\widehat{A_{g_0}}$, and the image of $g$ is invertible: modulo $f$ it equals the invertible element $g_0$. The line is therefore free there and after inverting $f$. Finally, Frobenius is an abstract ring isomorphism from $B$ onto $B^p$ and from $R$ onto $R^p$, because these rings are reduced. It identifies the corresponding formal basic neighborhoods. Thus the same proof applies to the power subrings.
\end{proof}

\begin{proposition}[Divisibility of the Laurent Picard group]\label{prop:picard-divisibility}
Multiplication by $p$ on $\Pic(|D|,\cR)$ is surjective.
\end{proposition}

\begin{proof}
Give an arbitrary $\cR$-line bundle $\cL$ the connection from Lemma~\ref{lem:cartier-fixed-connection}. On a chart with a line frame, Lemma~\ref{lem:cartier-curvature} gives commuting nilpotent connection operators, and Lemma~\ref{lem:horizontal-expansion} makes the horizontal module finite projective of rank one over $R^p$. Its restriction compatibility and intrinsic definition glue these modules into a sheaf $\cH=\ker\nabla$ over $\cR^p$. Lemma~\ref{lem:formal-line-extension} shows that $\cH$ is locally free of rank one in the required formal Zariski topology. The local multiplication isomorphisms glue to
\[
\cR\otimes_{\cR^p}\cH\simeq\cL.
\]

Transport $\cH$ through the sheaf isomorphism $\cR\to\cR^p$, $a\mapsto a^p$, to obtain a line bundle $\cN$ over $\cR$. Extension from $\cR^p$ to $\cR$ is its absolute Frobenius pullback. In local frames, that pullback raises the transition units of $\cN$ to their $p$-th powers, which are also the transition units of $\cN^{\otimes p}$. Thus
\[
\cL\simeq\cN^{\otimes p}.
\]
This is a global root on the stated ringed space, proving the surjectivity assertion.
\end{proof}

\section{An integral degree and the contradiction}
\label{sec:degree}

Infinite Frobenius divisibility forces integral numerical invariants to vanish; this is the numerical mechanism for stratified bundles in \cite[Lemma~2.1 and Corollary~2.2]{esnaultmehta2010}. Here the required degree is defined first on the completion. We must prove that it descends to the Laurent Picard group before applying that elementary divisibility obstruction.

\subsection{Orders of Laurent units}

\begin{lemma}\label{lem:laurent-unit-orders}
Let $\iota_\alpha\colon D_\alpha\hookrightarrow|D|$ be the inclusion of a prime component. There is an exact sequence of abelian sheaves
\begin{equation}\label{eq:laurent-unit-orders}
1\longrightarrow\widehat{\cO}^{\,\times}
\longrightarrow\cR^\times
\xrightarrow{\ \operatorname{ord}\ }
\bigoplus_\alpha(\iota_\alpha)_*\Z
\longrightarrow0.
\end{equation}
Consequently restriction gives a surjection
\[
\Pic(|D|,\widehat{\cO})\longrightarrow\Pic(|D|,\cR),
\]
whose kernel is generated by the completed restrictions of $\cO_V(D_\alpha)$.
\end{lemma}

\begin{proof}
Work near a point on an affine chart as above, small enough that each component through it has a prime equation $f_\alpha$, and write $f$ as a unit times $\prod_\alpha f_\alpha^{m_\alpha}$. Every $f_\alpha$ remains a non-zero-divisor in $B$ by flatness of completion. It also remains prime, because
\begin{equation}\label{eq:prime-completion-quotient}
B/(f_\alpha)\simeq A/(f_\alpha).
\end{equation}
Indeed, the induced $f$-adic topology on $A/(f_\alpha)$ is the zero-ideal topology, so completion leaves this integral ring unchanged.

If $u\in B[1/f]^\times$, write $u=a/f^r$ and $u^{-1}=b/f^s$, with $a,b\in B$. Since $f$ is a non-zero-divisor, $ab=f^{r+s}$ in $B$. Successively use primality to cancel each of the finitely many prime factors of the right side from either $a$ or $b$. At the end the remaining factors multiply to a unit. This expresses $u$ uniquely as
\begin{equation}\label{eq:laurent-unit-factorization}
u=v\prod_\alpha f_\alpha^{n_\alpha},
\qquad v\in B^\times,\quad n_\alpha\in\Z.
\end{equation}
Uniqueness follows from the same prime cancellation. The exponents agree on smaller charts and are locally constant along the respective $D_\alpha$. They define the order map. Its kernel consists exactly of formal units, and the local equations give its local surjectivity, proving \eqref{eq:laurent-unit-orders}.

On an irreducible space with its Zariski topology, the constant sheaf $\Z$ is flasque: every nonempty open is irreducible, its locally constant integer-valued functions are constant, and all restriction maps are surjective. Each pushed-forward sheaf $(\iota_\alpha)_*\Z$, and their finite direct sum, is therefore flasque. Its first cohomology vanishes. Line bundles on a ringed space are classified by the first cohomology of its unit sheaf. The cohomology sequence of \eqref{eq:laurent-unit-orders} consequently proves surjectivity of the Picard restriction map. The image of the connecting map from global integer sections is the subgroup generated by $\cO_V(D_\alpha)$, as is seen from their local equations. Exactness proves the kernel assertion.
\end{proof}

\subsection{The degree contradiction}

\begin{proposition}\label{prop:laurent-degree-contradiction}
The prepared cycle of Proposition~\ref{prop:prepared-cycle} cannot satisfy the conclusions of Corollary~\ref{cor:form-vanishing}.
\end{proposition}

\begin{proof}
For a line bundle $\cL$ on the completion, its restriction to each integral projective surface $D_\alpha$ is an ordinary line bundle. Define the integer
\begin{equation}\label{eq:formal-integral-degree}
\Phi(\cL)=\sum_\alpha m_\alpha
\bigl(c_1(\cL|_{D_\alpha})\cdot H|_{D_\alpha}\bigr).
\end{equation}
Intersection with the Cartier class $H$ gives an integral degree on each surface, so $\Phi$ is an additive homomorphism to $\Z$. The numerical properties of the prepared cycle give
\[
\Phi(\cO_V(D_\beta))
=\sum_\alpha m_\alpha D_\alpha D_\beta H
=DD_\beta H=0.
\]
Lemma~\ref{lem:laurent-unit-orders} therefore makes $\Phi$ descend to a homomorphism
\[
\overline\Phi:\Pic(|D|,\cR)\longrightarrow\Z.
\]
It is nonzero, since
\begin{equation}\label{eq:positive-laurent-degree}
\overline\Phi(\cO_V(H)|_{\cR})=DH^2>0.
\end{equation}

On the other hand, Proposition~\ref{prop:picard-divisibility} implies that every class in this Picard group is divisible by $p^n$ for every positive integer $n$. Its image under $\overline\Phi$ must therefore be an integer divisible by every $p^n$, hence zero. This contradicts \eqref{eq:positive-laurent-degree}.
\end{proof}

\subsection{Conclusion in the maximal-nef-dimension case}

The assumption of maximal nef dimension in Proposition~\ref{prop:prepared-cycle} supplied the prepared cycle and all the conclusions of Corollary~\ref{cor:form-vanishing}. Proposition~\ref{prop:laurent-degree-contradiction} rules out that assumption. Thus $n(X,K_X)\le2$ over the uncountable algebraically closed extension used in the preparation. The nef-dimension-at-most-two abundance theorem \cite[Theorem~1.8]{xu2024} applies to the projective log canonical threefold pair $(X,0)$ in characteristic $p>3$ with nef canonical divisor. It makes $K_X$ semiample there.

\section{Descent to the original field}
\label{sec:descent}

We now finish over the original field. Denote it by $k_0$ and the extension by $k$. Restore $X$ for the original threefold and write $X_k=X\times_{k_0}k$. Choose a globally generated Cartier multiple of $K_{X_k}$, then take a further positive multiple divisible by the Cartier index of $K_X$ over $k_0$. Call the resulting integer $m$. The line bundle $\cO_{X_k}(mK_{X_k})$ is still globally generated and is not numerically trivial, since $\nu(K_{X_k})=1$. It has at least two independent sections: if a globally generated line bundle has only one, that section trivializes it, contradicting its nonzero numerical class.

Compatibility of the canonical Cartier multiple with field extension and proper section-space base change give
\[
H^0\bigl(X,\cO_X(mK_X)\bigr)\otimes_{k_0}k
\simeq
H^0\bigl(X_k,\cO_{X_k}(mK_{X_k})\bigr).
\]
Their dimensions are equal, proving the required inequality over $k_0$. In particular $\kappa(X,K_X)\ge1$.

Nefness and $\nu(K_X)=1$ give $\kappa(X,K_X)=1$. The positive-Kodaira-dimension abundance theorem \cite[Theorem~1.6]{xu2024} also gives semiampleness over the original field. This completes the proof of Theorem~\ref{thm:main}.

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\end{document}