==== TABLE OF CONTENTS ====
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//cartier.tex
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//introduction.tex
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//jets.tex
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//laurent.tex
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//paper.tex
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//preamble.tex
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//preparation.tex
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//slopes.tex
Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//references.bib
===========================

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//cartier.tex ====
\section{Cartier descent and the final degree contradiction}
\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 this section, a line bundle means a locally free
module of rank one for the indicated sheaf of rings, in this Zariski
topology.  We write \(\Omega^j_{\cR}\) for the actual Laurent
differential forms, and \(\cB^1\subset\cZ^1\subset\Omega^1_{\cR}\)
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}
 0\longrightarrow\cB^1\longrightarrow\cZ^1
   \xrightarrow{\ C\ }\Omega^1_{\cR}\longrightarrow0,
 \label{eq:laurent-cartier-sequence}
\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 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
\(C-\id:W\to W\) is surjective as an additive map.
Moreover, every additive \(\cZ^1\)-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 \(\cB^1\)
and \(\cZ^1\), proving the last assertion.  To lift a global form
through \(C\), choose local lifts in \(\cZ^1\) using
Equation~\eqref{eq:laurent-cartier-sequence}.  Their differences
form a \(\cB^1\)-cocycle.  Subtracting a coboundary makes the
lifts agree, so
\[
 C:H^0(|D|,\cZ^1)\longrightarrow W
\]
is surjective.  Its source is a vector subspace of \(W\).
Since the ground field is perfect, a semilinear map has the same
dimension properties as a linear map.  Finite-dimensionality
therefore implies \(H^0(\cZ^1)=W\) and bijectivity of \(C\).

For the last assertion about \(C\), set \(F=C^{-1}\), a bijective
\(p\)-semilinear map.  Choose coordinates on \(W\simeq k^n\).
Then \(F(x)=A x^{[p]}\) with \(A\in\operatorname{GL}_n(k)\),
where \(x^{[p]}\) means componentwise \(p\)-th powers.  The
polynomial map \(F-\id\) extends to
\begin{equation}
 \Psi:\PP^n\longrightarrow\PP^n,
 \qquad
 [x:t]\longmapsto
 [A x^{[p]}-t^{p-1}x:t^p].
 \label{eq:semilinear-projective-map}
\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, a contradiction.  Hence \(\Psi\) is finite.
Its image is closed and has 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.  The case \(n=0\) is immediate.
Applying \(F\) to \((C-\id)\beta=\eta\) gives the equivalent
equation
\[
 (F-\id)\beta=-F\eta,
\]
so \(C-\id\) 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.
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.  These observations also apply after finite refinements
for local Cartier lifts.  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 and 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 its 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 by their retained \(p\)-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}
 R=\bigoplus_{0\le I_1,I_2,I_3<p}Sx^I,
 \qquad
 \Omega^1_{R/k}=\bigoplus_{i=1}^3R\,dx_i.
 \label{eq:cartier-pbasis}
\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}
 D_i^p=0,
 \qquad [D_i,x_j]=\delta_{ij}.
 \label{eq:horizontal-relations}
\end{equation}
Here 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}
 \partial_i^{p-1}a_i
   =(p-1)!\,b_i^p=-b_i^p.
 \label{eq:cartier-coefficient}
\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}
                 (\partial+a)^p=a^p+\partial^{p-1}a.
 \label{eq:rank-one-pcurvature}
\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 \(\partial^p b=0\).
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
\(\partial^{j-1}a\).  This formula follows by adjoining the last
label either as a singleton or to one existing block.
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.  To check the exceptions, in an
invariant partition a block is either the whole set or has an orbit
of \(p\) members,
in which case all blocks are singletons.  The summands have the
same value throughout an orbit, so only \(a^p\) and
\(\partial^{p-1}a\) survive in characteristic \(p\).
This proves Equation~\eqref{eq:rank-one-pcurvature} and 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 Equation~\eqref{eq:horizontal-relations}
and commute.  Its horizontal module
\[
 H_M=\bigcap_i\ker(D_i:M\to M)
\]
is a finite projective rank-one \(S\)-module, and multiplication
gives an isomorphism
\begin{equation}
                 R\otimes_S H_M\xrightarrow{\ \sim\ }M.
 \label{eq:horizontal-descent}
\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}
 P_i=\sum_{n=0}^{p-1}\frac{(-1)^n}{n!}x_i^nD_i^n,
 \qquad P=P_1P_2P_3.
 \label{eq:horizontal-projector}
\end{equation}
The multiplication operator \(x_i^n\) is placed on the left.
All factorials here 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}
 \sum_{n=0}^{p-1}\frac{x_i^n}{n!}P_iD_i^n=\id_M.
 \label{eq:horizontal-taylor-one}
\end{equation}
Indeed, after substituting Equation~\eqref{eq:horizontal-projector},
the coefficient of \(x_i^rD_i^r\), for \(0\le r<p\), is
\[
 \sum_{n+j=r}\frac{(-1)^j}{n!j!}=\begin{cases}1&r=0,\\0&r>0.
 \end{cases}
\]
All terms with \(r\ge p\) vanish by \(D_i^p=0\).
Multiplying Equation~\eqref{eq:horizontal-taylor-one} over the
three coordinates gives the explicit expansion
\begin{equation}
 m=\sum_{0\le I_j<p}x^Ih_I,
 \qquad h_I=\frac1{I!}P(D^I m)\in H_M.
 \label{eq:horizontal-taylor}
\end{equation}
Here \(I!=I_1!I_2!I_3!\) and
\(D^I=D_1^{I_1}D_2^{I_2}D_3^{I_3}\).
The coefficients are unique: for \(h\in H_M\),
\[
 P D^J(x^Ih)=\begin{cases}I!h&I=J,\\0&I\ne J,
 \end{cases}
 \qquad 0\le I_i,J_i<p.
\]
This follows by differentiation and the same binomial sum, which
makes \(P(x^Kh)=0\) for every nonzero \(K\) in that range.
Together with Equation~\eqref{eq:cartier-pbasis}, this proves
Equation~\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;
Equation~\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 set \(S'=(R')^p\).  The shared monomial bases give
an isomorphism
\[
 S'\otimes_S R\xrightarrow{\ \sim\ }R'.
\]
The restricted module is consequently \(S'\otimes_S M\), 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
\(S'\otimes_S H_M\).  This argument does not require a separate
flatness assertion for the restriction homomorphism.  Since
horizontality means \(\nabla m=0\), the construction is also
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.  More generally, an element
invertible modulo \(f\) is invertible in \(B\)
\cite[Lemma~10.96.6]{stacks}.

Every maximal ideal \(\mathfrak n\subset B\) thus corresponds
to a maximal ideal \(\mathfrak m\subset A\) containing \(f\),
and \(\mathfrak n=\mathfrak mB\).  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.  Localization now 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.  Its line bundle restricts to the original one.
Doing this on all components gives the asserted 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 on this formal neighborhood, and
remains free 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}

\subsection{Orders of units and an integral degree}

Infinite Frobenius divisibility forces integral numerical invariants
to vanish; this is also 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.

\begin{lemma}
\label{lem:laurent-unit-orders}
Let \(\iota_\alpha:D_\alpha\hookrightarrow|D|\) be the
inclusion of a prime component.  There is an exact sequence of
abelian sheaves
\begin{equation}
 1\longrightarrow\widehat{\cO}^{\,\times}
 \longrightarrow\cR^\times
 \xrightarrow{\ \operatorname{ord}\ }
 \bigoplus_\alpha(\iota_\alpha)_*\Z
 \longrightarrow0.
 \label{eq:laurent-unit-orders}
\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}
 B/(f_\alpha)\simeq A/(f_\alpha).
 \label{eq:prime-completion-quotient}
\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}
 u=v\prod_\alpha f_\alpha^{n_\alpha},
 \qquad v\in B^\times,\quad n_\alpha\in\Z.
 \label{eq:laurent-unit-factorization}
\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
Equation~\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
Equation~\eqref{eq:laurent-unit-orders} consequently proves
surjectivity of the Picard restriction map.  The image of the
connecting map from the 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}

\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}
 \Phi(\cL)=
 \sum_\alpha m_\alpha
   \bigl(c_1(\cL|_{D_\alpha})\cdot H|_{D_\alpha}\bigr).
 \label{eq:formal-integral-degree}
\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}
            \overline\Phi(\cO_V(H)|_{\cR})=DH^2>0.
 \label{eq:positive-laurent-degree}
\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
Equation~\eqref{eq:positive-laurent-degree}.
\end{proof}

\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
The assumption of maximal nef dimension 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.

To 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 \(\kappa(X,K_X)=1\), and the established
positive-Kodaira-dimension abundance theorem
\cite[Theorem~1.6]{xu2024} also gives semiampleness over the
original field.
\end{proof}
==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//cartier.tex ====

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//introduction.tex ====
\section{Introduction}
\label{sec:introduction}

The abundance conjecture predicts that a nef canonical divisor on a
projective variety with mild singularities is semiample: some positive
Cartier multiple is generated by its global sections.  Such sections
turn numerical information about the canonical divisor into a morphism
that records the canonical geometry of the variety.  For a nef divisor
$L$ on a threefold, numerical dimension one means
$L\cdot H^2>0$ and $L^2\cdot H=0$ for an ample Cartier divisor $H$.
We prove this case of abundance for terminal threefolds in
characteristic greater than three.

\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
$\nu(K_X)=1$.  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}

No assumption is made on the first Betti number, irregularity,
or Albanese variety.  Thus Theorem~\ref{thm:main} resolves the
numerical-dimension-one case of the abundance conjecture under the
stated terminal and characteristic hypotheses.

Geometrically, the theorem gives a morphism $f:X\to C$ with connected
fibers to a normal projective curve and an ample Cartier divisor $A$
on $C$ such that $mK_X\sim f^*A$ for some $m>0$.  Indeed, the image
of a morphism defined by a globally generated pluricanonical multiple
is a curve, since $\nu(K_X)=1$.  Its Stein factorization gives $f$;
the finite pullback to $C$ of a hyperplane divisor on the image gives $A$.

\subsection{Context}

Over the complex numbers, Miyaoka proved the numerical-dimension-one
case of threefold abundance by studying effective pluricanonical
divisors, primitive connected cycles, and their infinitesimal
neighborhoods \cite{miyaoka1988}.  Kawamata's abundance theorem for
minimal threefolds includes another proof of this case using the log
minimal model program and formal thickenings
\cite[Section~4]{kawamata1992}.  The present argument develops this
canonical-cycle framework in positive characteristic.
More recently, Liu--Xu developed higher-dimensional
numerical-dimension-one abundance over $\mathbb C$ under nonvanishing
hypotheses \cite{liuxu2025}.  The ground field in the present theorem
requires the positive-characteristic inputs described next.

The positive-characteristic minimal model program supplies the
birational operations needed to study abundance for threefolds.
Over algebraically closed fields of characteristic greater than five,
Hacon--Xu established standard-coefficient flips and minimal models
for projective $\Q$-factorial terminal threefolds with pseudoeffective
canonical divisor \cite[Theorems~1.1 and~1.2]{haconxu2015}.
Birkar proved flips for $\Q$-factorial dlt threefold pairs and log
minimal models for projective klt threefold pairs with pseudoeffective
log canonical divisor \cite[Theorems~1.1 and~1.2]{birkar2016}.
The relative log minimal model program in characteristic greater than
five was established by Hashizume, Nakamura and Tanaka
\cite{hnt2020}; Hacon and Witaszek treated the characteristic-five
case \cite{hw2022}.

In characteristic greater than five,
Xu--Zhang proved nonvanishing for terminal minimal threefolds
\cite{xuzhang2019}; Witaszek obtained nonvanishing for klt threefold
pairs and abundance in nef dimension at most two through a canonical
bundle formula \cite{witaszek2021}.  Zhang proved abundance for
minimal threefolds with nontrivial Albanese map \cite{zhang2020}.
Xu proved nonvanishing for pseudoeffective log
canonical threefold pairs in characteristic greater than three and
semiampleness when the nef log canonical divisor has positive Kodaira
dimension \cite[Theorems~1.4 and~1.6]{xu2024}.  His theorem for nef
dimension at most two \cite[Theorem~1.8]{xu2024} and his subsequent
numerical-dimension-two abundance theorem \cite[Theorem~1.1]{xu2026}
leave a specific issue in the present setting: an effective canonical
divisor of numerical dimension one could a priori still have Kodaira
dimension zero and maximal nef dimension.  Theorem~\ref{thm:main}
excludes that possibility.  Here nef dimension is the dimension of the
nef reduction, and maximal nef dimension means that no curve of
canonical degree zero passes through a very general point.  It is a
different invariant from numerical dimension.  The new step is the
production of a second pluricanonical section; the existence of a first
section and the passage from positive Kodaira dimension to semiampleness
are Xu's results.

Our tangent-sheaf argument follows the positive-characteristic
foliation method appearing in Xu--Zhang \cite[Section~2]{xuzhang2019}.
It uses the Bogomolov inequality for strongly semistable sheaves with
respect to a nef tuple, in Langer's formulation
\cite[Theorem~3.2]{langer2004}; the corrections in
\cite{langeradd2004} preserve that theorem.  Surface abundance enters
through abundance for projective semi-log-canonical surfaces, proved
by Tanaka over algebraically closed fields \cite[Theorem~0.1]{tanaka2016}
and by Posva over arbitrary fields \cite[Theorem~3.1]{posva2024}.

The descent mechanism is classical Cartier theory.  In particular,
Katz's proof of Cartier descent supplies the horizontal projector and
truncated Taylor expansion \cite[Theorem~5.1]{katz1970}.
We establish the completed-ring and bounded-pole statements that allow
this algebra to be used on the punctured formal neighborhood of a
nonreduced cycle.  The three-row jet calculation and its uniform pole
control are the bridge between the canonical-cycle geometry and this
descent argument.

\subsection{The argument}

After nonvanishing and the birational preparation in
Section~\ref{sec:preparation}, the hypothetical
maximal-nef-dimension case produces a smooth projective threefold $V$
and a connected effective nef Cartier cycle $D$ with primitive
multiplicities.  The cycle is numerically trivial on each of its
components, its normal line $\cO_D(D)$ is torsion, and, on a neighborhood
of $D$, a $p$-power of $\omega_V$ has a divisor supported on $D$.
The preparation proves
torsion on the full nonreduced cycle, which is needed in the subsequent
calculation.

Section~\ref{sec:slopes} then proves strong semistability of the tangent
and cotangent bundles for the nef degree $c_1(-)\cdot H\cdot D$.
The foliation quotient of a destabilizing sheaf would produce
$D$-trivial curves through very general points.  The resulting
semistability gives $c_2(V)\cdot D\geq0$ and uniform bounds on polar
parts, which are the inputs to the jet calculation.

The central calculation in Section~\ref{sec:jets} concerns finite jets
along $D$, that is, sections on successive Cartier thickenings.  It keeps all
three cohomological rows of the pole filtration and compares their
ranks over the periods $rp^e$, where $r$ is the order of
$\cO_D(D)$.  The scalar rows identify the completed canonical line with
the line of an integral divisor supported on $D$, and force both
$c_2(V)\cdot D=0$ and the
precise limiting ratio $p/(p+1)$ for the last trivialization widths.
These facts control the positions and lengths of differentials for
vector bundles as well.

Section~\ref{sec:laurent} turns the calculation into a
formal-neighborhood criterion,
Theorem~\ref{thm:laurent-vanishing}: a bundle satisfying an Euler
identity, bounded polar parts, and a concentration condition for
leading jets has vanishing first Laurent cohomology.  A bound on the
source cuts of incoming differentials ensures that the constructed
primitives actually converge with bounded poles.  We prove this
criterion directly for finite covers and completed lattices.
The criterion applies to one-forms and exact forms.

Section~\ref{sec:cartier} uses these vanishings to make every line
bundle on the punctured formal neighborhood a $p$th power, by Cartier
descent.  On the other hand, intersection with $H$ on the components
of $D$ defines a nonzero integer-valued homomorphism on its Picard
group.  Such a homomorphism cannot exist on a $p$-divisible group.
The contradiction proves Theorem~\ref{thm:main}.

Figure~\ref{fig:proof-map} records the two conclusions about the same
Picard group that give the contradiction.  In the figure,
$\cR=\widehat{\cO}_V(*D)$ denotes functions on the completion along
$D$ with a finite pole bound; its Picard group consists of locally free
rank-one $\cR$-modules on $|D|$.
\begin{figure}[ht]
\centering
\small
\begin{tikzcd}[column sep=small,row sep=large]
& \text{Prepared cycle }(V,D)
  \arrow[dl,"\text{Sections~\ref{sec:slopes}--\ref{sec:laurent}}"']
  \arrow[dr,"\text{Section~\ref{sec:cartier}}"] & \\
\begin{gathered}\text{Slope bounds and finite jets}\\
            \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 produces a cycle for
which Laurent cohomology forces Picard divisibility, while intersection
theory supplies a nonzero integral degree.}
\label{fig:proof-map}
\end{figure}

All arguments take place in characteristic $p$.  The proof uses no
lifting to characteristic zero and no orbifold Iitaka theorem.

\subsection{Conventions}

Varieties are integral and all divisors are rational Cartier divisors
when a pullback or intersection requires this, unless they are
explicitly declared integral Cartier divisors.  On a smooth variety
we identify Cartier and Weil divisors.  Linear and numerical
equivalence are denoted by $\sim$ and $\equiv$, with a subscript
$\Q$ for rational linear equivalence.  For a nef divisor $L$, its
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.  We use an uncountable algebraically
closed field when making this formulation, and descend the final
section-space statement to the original field.

If $E$ is a vector bundle on a smooth threefold $V$, write
$E^!=E^\vee\otimes\omega_V$.  All completions below are along an
effective Cartier divisor, and all Laurent sheaves are considered on
its Zariski support.  The term \emph{width} refers to the number of
successive Cartier layers in a jet.
==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//introduction.tex ====

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//jets.tex ====
\section{Jets and the three rows}\label{sec:jets}

We retain the prepared data of Proposition~\ref{prop:prepared-cycle}.
Thus $V$ is a smooth projective threefold, and
\[
 D=\sum_\alpha m_\alpha D_\alpha
\]
is an effective Cartier divisor with connected support,
$\gcd_\alpha m_\alpha=1$, and $D|_{D_\alpha}$ 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 shall use
$c_2(V)D\geq0$ from Proposition~\ref{prop:strong-semistability}
and the absence of scalar polar parts from
Corollary~\ref{cor:no-scalar-poles}.

The argument in this section concerns finite jets and their ranks.
Its two outputs are a trivialization of the canonical line on the
completion up to an integral divisor supported on $D$, and 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 $sD$.
Its only passage to a formal limit will be 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 $V$, refining it 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.

Here are the elementary completeness facts we need. On an affine
chart with local equation $f$ for $D$, adic completion is flat and
$f$ remains a non-zero-divisor. Thus the completed lattices embed
in their localization. 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 $C_{i-s}^\bullet(E)$, $s\geq1$. Exactness of
completion for finite modules gives
\begin{equation}\label{eq:jet-quotient}
 \begin{gathered}
 C_i^\bullet(E)/C_{i-s}^\bullet(E)
 \quad\text{computes}\quad H^\bullet\bigl(V,P_i^s(E)\bigr),\\
 P_i^s(E)=\frac{E(iD)}{E((i-s)D)}.
\end{gathered}
\end{equation}
Indeed its terms are exactly the sections of this algebraic
quotient on the affine intersections. Thus $P_i^s(E)$ retains
the $s$ layers at weights $i-s+1,\ldots,i$; we also call $i$
the upper cut. We abbreviate
$E^!=E^\vee\otimes\omega_V$.

For $s\geq1$, consider a leading cochain, that is, a class in
$C_i^a(E)/C_{i-1}^a(E)$, where $a$ is its cohomological degree.
The cycle condition below 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 the index $s$ 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
\[
 E_s^{i,a}(E)=Z_s^{i,a}/U_s^{i,a},\qquad
 g_s^a(i;E)=\dim_k E_s^{i,a}(E).
\]
The notation $E_s^{i,a}(E)$ denotes 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
$a=0,1,2$. In particular, with
$\delta_s^a(i)=g_s^a(i;E)-g_{s+1}^a(i;E)$ for $a=0,2$,
\begin{align}
 g_s^0(i;E)
 &=\dim\im\bigl(H^0(P_i^s(E))\longrightarrow 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
Equation~\ref{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 coming from
$C_{i-1}=C_{(i-s)+s-1}$. Adding an exact representative has no
effect either. This proves that $d_s$ 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~\ref{eq:jet-quotient}
identifies page one. Since the quotients are supported on the
projective surface scheme $D$, their cohomology is zero outside
degrees $0,1,2$; the same is true of every subsequent page.

The differentials through a middle-row term are arranged as
\[
 \begin{tikzcd}[column sep=large]
 E_s^{i+s,0}(E) \arrow[r,"d_s"] &
 E_s^{i,1}(E) \arrow[r,"d_s"] & E_s^{i-s,2}(E).
 \end{tikzcd}
\]
Moving right lowers weight by $s$ and raises cohomological degree
by one; the composite is zero. There is no incoming differential
in degree zero. Its representatives are exactly the sections of
$P_i^s(E)$, proving Equation~\ref{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. More explicitly,
the kernel calculation above and vanishing of the degree-three
page imply successively 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 therefore
$U_s^{i,2}/U_1^{i,2}$, and 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 $H^0(P_{s-j}^s(E^!))$
\cite[Chapter~III, Section~7]{hartshorne1977}.
Under this identification the dual of the displayed inclusion is
\[
 H^0(P_{1-i}^s(E^!))\longrightarrow H^0(P_{1-i}^1(E^!)).
\]
One can check the map, as well as its index, directly: the map of
the Cartier presentations is the identity on $E((i-1)D)$ and
the inclusion on their upper terms, so dualizing gives precisely
the indicated projection. This proves Equation~\ref{eq:top-endpoint}.
The ranks of the two differentials adjacent to the middle term
are $\delta_s^0(i+s)$ and $\delta_s^2(i-s)$, respectively.
Taking its cohomology proves Equation~\ref{eq:middle-recurrence}.

Triviality of $\cO(MD)$ on $sD$ identifies each quotient of width
$t\leq s$ with its translate by $M$, compatibly with its layer
maps. Equations~\ref{eq:bottom-endpoint} and
\ref{eq:top-endpoint} therefore give periodicity of both outer
rows at every such width. Page one is periodic in all degrees;
Equation~\ref{eq:middle-recurrence} then gives periodicity of
the middle row by induction. If
$e_t(i)=g_t^0(i)-g_t^1(i)+g_t^2(i)$, that equation 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 $M$.
Their sums cancel, proving Equation~\ref{eq:periodic-euler}
by induction from page one.

For the last assertion, set $A_s=H^0(P_i^s(E))$ and
\[
 B_s=\bigcap_{t\geq s}\im(A_t\longrightarrow A_s).
\]
Each $A_s$ is finite dimensional, so the intersection defining
$B_s$ is attained at a finite stage. Choosing one stage beyond
stabilization at both $s$ and $s+1$ shows that
$B_{s+1}\longrightarrow B_s$ is surjective. Every element of
$B_1$ therefore admits compatible lifts in all the $B_s$.
Their inverse limit is a global section of $\widehat{E(iD)}$,
since sections commute with inverse limits of sheaves. Conversely
a completed section restricts to the image of every $A_s$.
This proves the assertion directly.
\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 let
$\xi\in H^0(\cO(iD)/\cO(uD))$ be nonzero. There is an integer
$t$ with $u<t\leq i$ such that local lifts of $\xi$ have pole
order exactly $tm_\alpha$ at every $D_\alpha$. Those lifts
give a trivializing section of $\cO(tD)$ 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, variety $V$.
Thus some component detects the nonzero quotient section. The
maximum 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.
All lifts belong to $\cO(tD)$ in the sense of these rational
divisor bounds.

Choose a positive integer $n$ with $nt\in\Z$. Then
$f_\lambda^n\in\cO(ntD)$, and these sections agree on
$D_{\mathrm{red}}$. Indeed
$f_\lambda-f_\mu\in\cO(uD)$, 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 therefore 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. It is consequently nowhere
zero on that component. At an intersection with another
component its common value is nonzero, so its restriction to
that component is also nonzero and nowhere zero. Connectedness
of $|D|$ propagates this conclusion 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, or an integral linear
combination of the $m_\alpha$ equal to one, gives $t\in\Z$.
The original lifts are now sections of the honest line bundle
$\cO(tD)$, are units in it along $D$, and agree modulo
$\cO(uD)$. Since
\[
 \frac{\cO(tD)}{\cO(uD)}
 =\cO(tD)|_{(t-u)D},
\]
they give the asserted trivialization.

A jet of width $s$ at upper weight $i$ has nonzero image in
the leading layer exactly when the uniform ratio just proved
is $t=i$. This proves existence in
Equation~\ref{eq:scalar-endpoint}, in both directions.
To see that the image dimension is 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. Subtract the appropriate
multiple. If its leading image were still nonzero, the preceding
argument would again give a unit on the reduction with ratio
$i$, contradicting the cancellation. Thus the two leading
images are proportional.
\end{proof}

The organizing viewpoint of tracking torsion and triviality on infinitesimal
thickenings was informed by Totaro's treatment of these obstructions
\cite[Lemma~4.1 and Section~6]{totaro2007}. The elementary width properties
are proved below; the later three-row rank calculation and Cartier descent
are not inputs from that work.

Let $r$ be the order of $\cO_D(D)$. For $e\geq0$ put
\begin{equation}\label{eq:width-definition}
 M_e=rp^e,\qquad
 d_e=\max\{s\geq1:\cO(M_eD)|_{sD}\simeq\cO_{sD}\}.
\end{equation}
These maxima exist and satisfy
\begin{equation}\label{eq:width-bounds}
 1\leq d_e<M_e,\qquad d_{e+1}\geq p d_e,
 \qquad \frac{d_e}{M_e}\nearrow a_*\in(0,1].
\end{equation}
In fact 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 $d_{e+1}\geq pd_e$. Division by
$M_{e+1}=pM_e$ proves monotonicity and the positive limit.
If $r=1$, already $1\leq d_0<M_0=1$ is a contradiction;
we may therefore continue with $r>1$.

We will repeatedly use the following precise 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 $x\in J$.
Thus $L^{\otimes p^N}$ is trivial. Multiplication by an
integer prime to $p$ is injective on such classes, by
B\'ezout's identity, as well as 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\geq0.
\end{equation}
Indeed the terms involving $D^3$, $D^2K_V$, and $DK_V^2$
vanish, leaving the displayed linear term in the threefold
Riemann--Roch polynomial \cite[Section~15]{fulton1998}.

\subsection{Canonical jets and a formal canonical root}

For $E=\cO_V$, Equation~\ref{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. We first show that canonical jets
survive to arbitrarily large widths. Their $p^b$th powers then let the
scalar rule identify $\widehat{\omega}_V$ with the line of an integral
divisor supported on $D$.

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

\begin{proof}
Suppose instead that $g_s^0(l;\omega_V)=0$ for every $l$
and every $s\geq S$, for some $S$. By
Equation~\ref{eq:top-endpoint}, the top row for $E=\cO_V$
then vanishes at all these widths. Take $e$ sufficiently
large that $s_e=d_{e-1}+1\geq S$. At page $s_e$ the bottom
row is one precisely at multiples of $M_e$, by
Equation~\ref{eq:scalar-last-width} and
$s_e\leq d_e$. This page is periodic modulo $M_e$.
Equations~\ref{eq:periodic-euler} and
\ref{eq:scalar-euler} give
\[
 \sum_{i\bmod M_e}g_{s_e}^1(i;\cO_V)
 =1-M_e\gamma\leq1.
\]
At differential length $d_e$, the bottom entries at $jM_e$,
$p\nmid j$, each drop by one. By
Equation~\ref{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 that it must instead lie at
$-d_{e+1}\pmod{M_{e+1}}$. But
$0<d_e<d_{e+1}<M_{e+1}$, so these residues are different.
This contradiction proves the lemma.
\end{proof}

\begin{proposition}[The canonical formal root]\label{prop:canonical-formal-root}
Every coefficient of $B_*$ in
Equation~\ref{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 choose 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 Equation~\ref{eq:canonical-p-power}
gives a scalar jet with bounds
\[
 A=PlD+B_*,\qquad B=P(l-s)D+B_*.
\]
The lower 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
$c_*\geq\max_\alpha\beta_\alpha$. Enlarging both bounds
gives a section of
\[
 \frac{\cO((Pl+c_*)D)}{\cO((P(l-s)+c_*)D)}.
\]
This section is nonzero for all sufficiently large $s$,
uniformly in $l$. To check this, 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
\[
 -P\frac{\ord_{D_\alpha}(f)}{m_\alpha}
       +\beta_\alpha>P(l-1)+\min_\alpha\beta_\alpha.
\]
The new lower bound is $P(l-s)+c_*$, which is smaller for
large $s$.

Lemma~\ref{lem:scalar-jets} now shows that all these powered
coefficients have one integral pole ratio $t$. The preceding
lower estimate and the upper bound give a constant $C$, independent
of $s,l$, such that
\begin{equation}\label{eq:canonical-ratio-bound}
 \begin{gathered}
 |t-Pl|\leq C,\\
 \cO(tD)\text{ is trivial to a width at least }Ps-C.
 \end{gathered}
\end{equation}
For each fixed $J$, widths greater than $d_{J-1}$ can occur
only at multiples of $M_J$, by the scalar classification;
zero is allowed as a multiple. Since the widths in
Equation~\ref{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 $-b_\alpha$. Consequently
\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 $P$.
Equation~\ref{eq:canonical-order-identity} therefore shows
that $P\mid b_\alpha$ for every $\alpha$. This proves
$B_*=PB_0$ with $B_0$ integral.

For the same sequence put $h_0=t/P$. The order identity
says 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 by this line bundle. 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
$\omega_V((l-s)D)$. Hence they trivialize $L_{h_0}$ on
the effective thickness
\[
 T=(h_0-l+s)D-B_0.
\]
By Equation~\ref{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 $J\geq b$.
The line bundle $\cO(h_0D)$ is therefore trivial on
$d_{J-b}D$, and $d_{J-b}\to\infty$ by
Equation~\ref{eq:width-bounds}. Removing this twist shows
that $L=\omega_V(-B_0)$ is trivial on arbitrarily large
finite thickenings $nD$.

We finally choose these trivializations compatibly. Apply
the stabilized-image argument in the proof of
Lemma~\ref{lem:page-calculus} to the spaces
$H^0(nD,L|_{nD})$, also including 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 Equation~\ref{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 now use
$h$ for the integer offset
\begin{equation}\label{eq:canonical-offset}
 h=-\left\lfloor\min_\alpha
                   \frac{b_{0,\alpha}}{m_\alpha}\right\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 $D$.
Indeed the localized $\widehat{\cO}_V$-linear isomorphism
respects every nested divisor lattice. Each width-$s$
quotient is killed by $\cI_D^s$, where $\cI_D=\cO_V(-D)$;
since $\widehat{\cO}_V/\cI_D^s\widehat{\cO}_V=\cO_{sD}$,
the induced map is an algebraic $\cO_{sD}$-linear map,
compatible with the leading-layer projections.

We first give the exact canonical counterpart of the scalar
rule. 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 sufficiently 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
\[
 l-1+\beta_0<t\leq l+\beta_0.
\]
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. Indeed $tD\leq
lD+B_0$, whereas the inequalities above prevent inclusion
in the preceding upper lattice. 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 that reduction cancels, by the
scalar rule. 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
\[
 C_0=h+\left\lceil\max_\alpha
                    \frac{b_{0,\alpha}}{m_\alpha}\right\rceil
 \geq0.
\]
Then, coefficient by coefficient,
\begin{equation}\label{eq:thickness-comparison}
 (s-C_0)D\leq T_s\leq 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~\ref{eq:thickness-comparison}
and the definition of $d_e$ give
\begin{equation}\label{eq:canonical-width-comparison}
 d_e\leq d'_e\leq d_e+C_0.
\end{equation}
In particular this last width is finite and exists in the
range where Equation~\ref{eq:canonical-endpoint-rule}
applies. Once $T_s$ contains $D$, the nilpotent-kernel
argument used above shows that the last width is the same
for $t=jM_e$ whenever $p\nmid j$. Also such a trivialization
requires $r\mid t$; for $t\ne0$ the unique expression
$t=jM_e$, $p\nmid j$, classifies its last width. The value
$t=0$ persists forever. To check that the large-$e$ rules
exhaust the entries at large width, observe from
Equation~\ref{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 $M_J\mid t$. Thus every nonzero index occurring at
sufficiently large width belongs to the stated range of
last-width indices.

\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~\ref{eq:width-bounds} and
\ref{eq:canonical-width-comparison} show that these are
strictly increasing and
$s_e\leq\min(d_e,d'_e)$. 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 Equation~\ref{eq:scalar-last-width}.
For the top entry, Equations~\ref{eq:top-endpoint} and
\ref{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.

\begin{samepage}
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. The omitted lifts are illustrative. If they
coincide, their column is omitted twice and every other
column receives both drops. Multiplicities are always added.
\end{samepage}

In general, since $p>2$, some lift receives all $m(a)$ drops.
Nonnegativity of its final rank gives $n(a)\geq m(a)$.
Summing over $a$, and then using
Equations~\ref{eq:periodic-euler} and
\ref{eq:scalar-euler}, gives
\[
 2\leq\sum_{a\bmod M_e}n(a)
 =2-M_e\gamma\leq2.
\]
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 Equation~\ref{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
$d'_e\leq d_e+C_0<M_e+C_0$, 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~\ref{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
Equation~\ref{eq:limiting-width}.
\end{proof}
==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//jets.tex ====

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//laurent.tex ====
\section{Laurent cohomology of vector bundles}
\label{sec:laurent}

We retain the smooth threefold $V$, the cycle $D$, and the notation of
the preceding sections.  In particular,
\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)\cdot D=0.
\end{equation}
The line bundle $\cO_V(M_eD)$ is trivial through width $d_e$.
Choose, once and for all, real numbers
\begin{equation}\label{eq:phase-constants}
 1-\frac2p<c_0<c<2a_*-1.
\end{equation}
Such choices exist since
\[
 2a_*-1-\left(1-\frac2p\right)=\frac{2}{p(p+1)}>0.
\]
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 the use of 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 that 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 follows 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 of these sheaves is
the one used in the jet construction.

\begin{theorem}\label{thm:laurent-vanishing}
For the prepared data and periods fixed above, let $E$ be a vector
bundle on $V$ with the following properties:
\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 the positive integer $n$.
\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 separate the argument into the endpoint patterns, their drops,
the middle row, and convergence.

\medskip\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 $w$.  We 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.  These quotients give a finite filtration of
the space of global sections.  The same observation applies to $E^!$.

\medskip\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 must control their lengths, with
multiplicities, independently of $j$ 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 permits no 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 that its drops occur at lengths
\begin{equation}\label{eq:drop-range}
 d_e\le s\le M_e+C
\end{equation}
for a fixed $C$, and that their multiplicities at each length are
independent of $j$ among integers prime to $p$.

There is no drop before length $d_e$, since translation by a unit
of $\cO(M_eD)|_{d_eD}$ identifies the 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 $C$ for the moment and a width
\[
 d_e<k\le M_e+C.
\]
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 Equation~\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 it to a completed global section
and subtract.  A finite induction constructs the asserted $\beta$.

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, Equation~\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 $t_\lambda^{j'}$ therefore gives a
width-$k$ jet at $v+j'M_e$.  It has 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 consequently equal at every width under consideration.
Their ranks, and hence their drops at each length, 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.  Indeed 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.
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 corresponding top-row rule.

\medskip\noindent
\emph{The middle row.}
We use the nonnegativity and Euler comparison from
Proposition~\ref{prop:scalar-widths}, now with a multiset of drop
positions.
Let $T_e$ be the multiset of the numbers $v-s$ for the bottom
drops and $w+s$ for the top drops.  Each is 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
in 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, which do not drop.
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 in Lemma~\ref{lem:page-calculus} and property (i).
The inequality is therefore equality at every residue.
Equation~\eqref{eq:middle-distribution} now says that $H_{e+1}$
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}

\medskip\noindent
\emph{Convergence of primitives.}
Fixed-weight page vanishing alone does not bound the poles of
successive primitives.  Fix an integer upper bound $l$ for a
Laurent cocycle.  We will 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 $x\ge M_e-A-C$, which exceeds $l$ for
all sufficiently 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 $e\ge e_0$; the preceding exclusions
leave only $j\le-1$.  In that case $z\le-M_e+A$, and hence
$M_e\le-z+A$.  Thus $x=z-s\ge2z-A-C$, or
\[
 z\le\frac{x+A+C}{2}.
\]
The real-valued upper bound
\[
 B_l(x)=\max\left\{x+S_l,\frac{x+A+C}{2}\right\}
 \qquad(x\le l)
\]
therefore bounds every possible source and satisfies both assertions
of Equation~\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
\[
 U_{s+1}^{x,1}/U_s^{x,1}
 =\im\bigl(d_s:E_s^{x+s,0}\longrightarrow E_s^{x,1}\bigr),
\]
where $E_s^{i,a}=Z_s^{i,a}/U_s^{i,a}$.
Thus $U_s^{x,1}$ stops increasing once $x+s>B_l(x)$.
By its definition, each increase from a source cut $z=x+s$
has a representative primitive in $C_z^0(E)$; the initial space
$U_1^{x,1}$ 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.  The differential is continuous,
and the filtration is separated, so its coboundary is $\gamma$.
All arguments use finite covers by affine charts and their affine
intersections, and remain valid after refinement.  This proves the
theorem.
\end{proof}

\subsection{Cartier algebra on the Laurent charts}

We specify the module structures before applying the criterion to
exact forms.  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 \'{e}tale coordinates $x_1,x_2,x_3$
and a local equation $f$ for $D$, put
$R=\widehat A[1/f]$.  The monomials
$x^I=x_1^{I_1}x_2^{I_2}x_3^{I_3}$, $0\le I_j<p$, 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 first check that its finite basis
survives the completion and localization used here.
For an \'{e}tale algebra the relative Frobenius square is a base
change square.  One can check this because the relative Frobenius
is \'{e}tale 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 then 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 $dx_j$.
There is no additional continuity assumption on differentials.

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, while 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).
Indeed 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 is 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\longhookrightarrowactual F_*\Omega,
 \qquad
 (B^1)^!\longhookrightarrowactual 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 $F_*(G(pnD)/G)$.  The polar bounds for $G=\Omega$
and $G=\omega_V$ therefore 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 that is useful to make explicit.  A width-$u_e$
jet of $E\subset F_*G$ with a nonzero leading term maps to a
width-$pu_e$ jet of $G$ at upper weight $pi$.  Its leading term
is nonzero 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 all sufficiently large $e$ we have
\[
 pu_e-p\ge\lfloor c_0M_{e+1}\rfloor.
\]
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}; refining the finite cover
if necessary preserves every preceding argument.  This proves
the assertion for $Z^1$.
\end{proof}

The next section uses both conclusions: cocycle vanishing supplies
compatible closed connection forms, and finite-dimensionality permits
a global adjustment that makes those forms Cartier-fixed.
==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//laurent.tex ====

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//paper.tex ====
\documentclass[11pt]{article}
\pdftrailerid{}
\input{preamble}
\hypersetup{pdftitle={Abundance in numerical dimension one for terminal threefolds in positive characteristic},pdfsubject={Pluricanonical sections, formal neighborhoods and Cartier descent},pdfauthor={OpenAI}}
\title{Abundance in numerical dimension one\\for terminal threefolds in positive characteristic}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $X$ be a projective $\Q$-factorial terminal threefold over an
algebraically closed field of characteristic $p>3$. We prove the
numerical-dimension-one case of abundance: if $K_X$ is nef with
$\nu(K_X)=1$, then $K_X$ is semiample and $\kappa(X,K_X)=1$.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\input{sections/introduction}
\input{sections/preparation}
\input{sections/slopes}
\input{sections/jets}
\input{sections/laurent}
\input{sections/cartier}
\begingroup
\raggedright
\bibliographystyle{plain}
\bibliography{references}
\endgroup
\end{document}
==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//paper.tex ====

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//preamble.tex ====
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==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//preamble.tex ====

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//preparation.tex ====
\section{Birational preparation}
\label{sec:preparation}

The purpose of this section is to replace the hypothetical case of maximal
nef dimension by a smooth threefold carrying a single connected nef cycle.
The torsion assertion below concerns the entire Cartier cycle, with its
nilpotents.

\begin{proposition}[The prepared cycle]
\label{prop:prepared-cycle}
Suppose that \(k\) is uncountable and algebraically closed of characteristic
\(p>3\), and that \(X\) is a projective terminal threefold with \(K_X\) nef,
\(\nu(K_X)=1\), and \(n(X,K_X)=3\).
There exist a smooth projective threefold \(V\), a generically finite
separable morphism \(\rho:V\to X\), a very ample divisor \(H\) on \(V\), and
a nonzero effective Cartier divisor
\[
 D=\sum_{\alpha}m_\alpha D_\alpha,\qquad
 m_\alpha\in\Z_{>0},\qquad \gcd_\alpha m_\alpha=1,
\]
with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The support of \(D\) is connected and has simple normal crossings.
The divisor \(D\) is nef, and \(D|_{D_\alpha}\) is numerically trivial
for every \(\alpha\).
\item There is no \(D\)-trivial curve through a very general point of \(V\).
Moreover \(D\cdot H^2>0\) and \(D^2\cdot H=0\).
\item The invertible sheaf \(\cO_D(D)\) is torsion.  Its order will be denoted
by \(r\).
\item For an integer \(b\geq0\), an open neighborhood \(U\) of \(|D|\), and
integers \(b_\alpha\), there is an isomorphism
\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 case $n(X,K_X)\leq2$ is already covered by
\cite[Theorem~1.8]{xu2024}, so this is the remaining nef-dimension case.
The rest of this section proves Proposition~\ref{prop:prepared-cycle}.
Starting with an effective pluricanonical divisor, we isolate a reduced
boundary component by the relative minimal model program.  Surface
adjunction supplies torsion on its reduction; local cohomology and
$p$-power gluing then give torsion on the full Cartier cycle.
A tame cover and primitive division produce $(V,D)$, and numerical
proportionality transfers the maximal-nef-dimension condition to $D$.
The intervening local surface arguments establish regularity in
codimension two and justify reduced-boundary adjunction.
The use of a connected primitive cycle follows the canonical-divisor
viewpoint of Miyaoka \cite[Definition~2.4 and Lemma~2.5]{miyaoka1988};
the torsion and covering arguments here take place in characteristic $p$.

\subsection{Resolution and field extension}

We use projective resolution in dimension three
\cite{cossartpiltant2008,cp2009,cp2019}.
The projective form that is unchanged over the regular locus is stated
explicitly in \cite[p.~1, introductory theorem]{cossart2008notes}.
It applies to our quasiprojective threefolds over the perfect field $k$.
For divisor supports on a regular threefold we use embedded resolution
\cite[Corollary~0.4]{cjs2009}: a reduced closed subscheme of dimension at
most two in a regular excellent scheme admits a projective modification,
unchanged off that subscheme, whose reduced inverse image is snc.
The resolutions may be chosen to be isomorphisms over the specified regular
open set.  In particular, pairs admit the log resolutions used below.

We also use the discrepancy criterion for a simple normal crossing
subpair on a smooth variety, allowing negative coefficients: coefficients
at most one give log canonicity, and coefficients strictly less than one
give the klt property.  One can see the relevant inequalities directly.
At the generic point of a center, use logarithmic differentials for the
components through it and regular differentials for the other coordinates.
At the generic point of a divisor on a smooth model, with uniformizer \(t\),
each pulled-back logarithmic differential is a regular differential plus
a multiple of \(dt/t\).  Their exterior product has at most a simple pole:
two occurrences of \(dt/t\) have zero exterior product.  The relative
Jacobian formula gives the discrepancy inequality.  Lowering a coefficient
adds its positive difference times the order of that component.
For an exceptional divisor whose center misses the boundary, the relative
Jacobian has positive order.  Indeed a birational morphism between smooth
models cannot be generically unramified along an exceptional divisor,
since it would then be quasi-finite there.  These observations also explain
why it suffices to compute thresholds on a log resolution.

We may replace \(k\) by an uncountable algebraically closed extension.
Here are the descent and preservation points needed for this reduction.
A normal variety over an algebraically closed field is geometrically
normal and geometrically integral.  In the present setting, geometric
normality also follows from Serre's criterion: the field is perfect,
residue fields of finite type are separably generated, and the relevant
flat base-change maps have regular fibers.  A smooth projective resolution
remains smooth and integral after extension.  The canonical rational line
bundle base-changes, since its restriction to the smooth open set does,
and reflexive extension determines it elsewhere.

On a resolution \(g:T\to X\), terminality gives
\[
 K_T=g^*K_X+\sum_E a_EE,\qquad a_E>0.
\]
The base-changed resolution has the same positive exceptional
discrepancies.  A further exceptional divisor has a positive relative
discrepancy over the smooth model and nonnegative contributions from
the effective divisor \(\sum_Ea_EE\); hence terminality is preserved.
Nefness is preserved, for example by writing a nef class as a limit of
ample rational classes, and intersection numbers preserve the numerical
dimension.  Finally, for every Cartier multiple \(qK_X\),
\[
 H^0(X_{k'},\cO_{X_{k'}}(qK_{X_{k'}}))
   =H^0(X,\cO_X(qK_X))\otimes_k k'.
\]
Thus pluricanonical sections descend in the required sense of equality
of dimensions.  If necessary, take a further multiple divisible by the
original Cartier index; the powers of two linearly independent sections
remain linearly independent, since their nonconstant ratio cannot acquire
a constant positive power in a function field with algebraically closed
constant field.  We do not assume that \(\Q\)-factoriality is preserved by
this extension: the modifications below supply it when needed.

\subsection{The maximal nef-dimension branch}

Nonvanishing \cite[Theorem~1.4]{xu2024} supplies an effective Cartier divisor
\[
 0\neq N_0\sim mK_X,\qquad m>0.
\]
It is nonzero because \(K_X\cdot H_X^2>0\) for an ample divisor \(H_X\).
It is nef, and \(N_0^2\cdot H_X=0\).
Every prime component \(T\) of \(N_0\) satisfies
\[
 N_0\cdot H_X\cdot T=0,
\]
because these nonnegative numbers have a positive weighted sum equal to
zero.  In fact \(N_0|_T\equiv0\).  To verify this for an integral curve
\(C\subset T\), choose a sufficiently large multiple of \(H_X|_T\) with
an effective Cartier member containing \(C\).  Its intersection with the
nef restriction is zero; all components contribute nonnegatively, so the
contribution of \(C\) is zero.

The nef reduction exists over our uncountable algebraically closed field
\cite[Definition~2.1 and Theorem~2.2]{xu2026}.
If \(n(X,K_X)\leq2\), the low-nef-dimension abundance theorem
\cite[Theorem~1.8]{xu2024} makes \(K_X\) semiample.  Since its numerical
dimension is one, some Cartier multiple has at least two sections.
We may therefore assume \(n(X,K_X)=3\).
The defining curve property of the nef reduction says in this case that
there is no \(N_0\)-trivial curve through a very general point of \(X\).
Here \emph{very general} means outside a countable union of proper closed
subsets.

\subsection{Isolating a boundary component}

We use the relative threefold MMP in characteristic greater than three,
including preservation of \(\Q\)-factorial dlt pairs for the boundary
driving the MMP, crepant dlt modifications, and the partial MMP with
nef restored boundary
\cite[Theorem~2.20, Proposition~2.21, Definition~2.23,
Lemma~2.24, and Theorem~2.27]{xu2026}.
The characteristic-five case is included through \cite{hw2022};
the corresponding results in characteristic greater than five include
\cite{hnt2020}.  Termination for the pseudoeffective lc divisor we use
is \cite[Theorem~1.5]{xu2024}.
The comparisons of pullbacks use the negativity lemma
\cite[Lemma~3.39]{kollarMori1998}.

We recall why the dlt preservation used here concerns negative steps.
On a common resolution of a negative divisorial contraction or flip,
the old log pullback minus the new one is effective by negativity.
On curves over the contraction target its degree is nonpositive, and
is strictly negative if the curve has a nonconstant image on a modified
side.  A component of a projective fiber not contained in this effective
divisor cannot meet it: its effective restriction is anti-nef, hence zero.
Connectedness of fibers then makes any fiber meeting the support lie
entirely in it.  Fibers over the modification locus do meet the support,
by the strict degree inequality.  Thus discrepancies strictly increase
for valuations centered in that locus.  A zero log discrepancy on the
output of a dlt step must consequently come from the unchanged open set,
where the required simple normal crossing behavior persists.
We will apply this to the lowered boundary.  For a larger boundary whose
log divisor is trivial along the steps we retain log canonicity by
crepancy, and later take a new dlt modification.

Let \(\lambda>0\) be the log canonical threshold of \(N_0\) on \(X\).
It is finite and rational, and is attained: on a log resolution it is
the minimum of finitely many positive rational discrepancy-to-order
ratios.  Thus \((X,\lambda N_0)\) is lc and is 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.
One construction starts on a log resolution with the strict boundary
and all exceptional divisors reduced.  Its log divisor is the desired
pullback plus an effective exceptional discrepancy divisor.  The relative
dlt MMP removes that difference by negativity, leaving precisely the
crepant divisors.  This is the modification theorem cited above.

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}
Indeed an extracted log-canonical place must have center in \(\Supp N_0\),
since \(X\) is klt away from that divisor.  Its order on \(N_0\) is positive.
The modification is an isomorphism over the smooth open set
\(X_{\mathrm{reg}}\setminus\Supp N_0\): there it has no exceptional divisors,
and a projective small birational morphism to a locally factorial variety
is an isomorphism.  For the latter assertion, push down a relatively ample
Cartier divisor.  A multiple of its pushdown is Cartier, and its pullback
agrees with that multiple upstairs because there are no exceptional
divisors.  It therefore has degree zero on any contracted curve,
contradicting relative ampleness unless there are no such curves.

Since \((W,B)\) is dlt and not klt, choose a coefficient-one component \(S\).
Choose a rational \(\epsilon>0\) so small that both
\[
 B-\epsilon S\geq0,\qquad \tau N-\epsilon S\geq0.
\]
Run a \((K_W+B-\epsilon S)\)-MMP which is
\((K_W+B)\)-trivial.  In the partial-MMP notation the boundary is
\(B-\epsilon S\), and the restoration divisor is \(\epsilon S\).
The lowered pair is dlt and its log divisor is pseudoeffective by the
displayed effective representative.  Termination therefore applies.
There is no Mori fiber output, since a pseudoeffective divisor cannot be
anti-ample on the covering family of curves in a Mori fiber space.

We spell out what happens to \(N\).  On a common resolution of each step,
its two pullbacks agree.  Their difference is exceptional over the new
model.  A curve contracted over that model maps on the old model to a
curve over the step's contraction target, and has \(N\)-degree zero,
because the extremal ray is \(N\)-trivial.  The difference is therefore
numerically trivial over the new model, and is zero by negativity applied
to it and to its negative.  This proves the pullback assertion for flips
as well as divisorial contractions.

It follows that the transform of \(N\) remains nef and numerically trivial
on every component of its support.  For the latter assertion, lift a
curve on such a component to the common resolution.  Its image on the
old side is a point or a curve in the old support, where \(N\) is
numerically trivial.  The same pullback equality shows that the full
pair in \eqref{eq:initial-dlt-pair} remains crepant and lc.
The component \(S\) is not contracted: every contracted ray \(R\) satisfies
\[
 S\cdot R>0,
\]
whereas an exceptional divisor of a divisorial contraction has negative
degree on some contracted curve, again by negativity.
Moreover a negative curve for the effective divisor
\(\tau N-\epsilon S\) lies in its support.  Thus all steps are
isomorphisms off the support under consideration.

Write \(Y,B_Y,S_Y,N_Y\) for the output and its transforms.
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 \(\delta>0\).
The variety \(Y\) is \(\Q\)-factorial and \((Y,B_Y)\) is lc.
If \(T\neq S_Y\) is a component of \(N_Y\), then
\(-S_Y|_T\) is nef by \eqref{eq:isolated-nef-pair} and \(N_Y|_T\equiv0\).
But \(S_Y|_T\) is effective and \(\Q\)-Cartier.  If nonzero, it has
positive degree against an ample divisor on the integral projective
surface \(T\), contradicting anti-nefness.  Consequently
\[
 S_Y\cap T=\varnothing\qquad(T\neq S_Y).
\]
The pair \((Y,S_Y)\) is lc, and, with \(s=\operatorname{mult}_{S_Y}N_Y>0\),
\begin{equation}
\label{eq:isolated-boundary}
 S_Y|_{S_Y}\equiv0,\qquad
 K_Y+S_Y\sim_{\Q}\tau s S_Y
 \quad\text{on a neighborhood of }S_Y .
\end{equation}

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 the
exceptional divisors, all reduced.  We claim
\begin{equation}
\label{eq:second-dlt-support}
 \Supp f^*S_Y=\Supp\Sigma .
\end{equation}
For an extracted divisor \(E\), let \(a_E\) denote log discrepancy.
Log canonicity gives
\[
 0\leq a_E(Y,B_Y)
 =a_E(Y,S_Y)-\operatorname{ord}_E(B_Y-S_Y)
 =-\operatorname{ord}_E(B_Y-S_Y).
\]
Both terms on the right must vanish.
Crepancy with \((X,\lambda N_0)\) then gives
\(a_E(X,\lambda N_0)=0\); since \(X\) is klt,
\(\operatorname{ord}_E N_0>0\).
The pullback equalities just proved give
\(\operatorname{ord}_E N_Y>0\).
The other components of \(N_Y\) have the same support as \(B_Y-S_Y\)
and order zero at \(E\), so \(\operatorname{ord}_E S_Y>0\).
This proves \eqref{eq:second-dlt-support}.
The morphism \(f\) is an isomorphism off \(S_Y\), by the small-morphism
argument and \(\Q\)-factoriality of \(Y\).
Near \(\Sigma\) we therefore have
\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}

\subsection{Local surface intersections and terminal regularity}

The adjunction calculation for \(\Sigma\) uses an intersection inequality
on excellent local surfaces, whose residue fields need not be perfect.
We prove that inequality first and also derive regularity of \(X\) in
codimension two.  For a resolution unchanged over
\(X_{\mathrm{reg}}\setminus\Supp N_0\), this regularity ensures that
exceptional divisors with centers outside \(\Supp N_0\) are disjoint
from its total pullback.

\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
which 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, by proper birationality and normality
of \(P\).  Choose a nonzero nonunit \(t\in\cO_P\) and write
\[
 \operatorname{div}_T(t)=A+\sum_i c_iG_i,\qquad c_i>0,
\]
where \(A\) is the strict-transform part.  It meets the exceptional fiber:
the strict transform of a component of the divisor of \(t\) has a point
over the closed point by properness.  For \(C=\sum_i c_iG_i\) we have
\[
 C\cdot G_i=-A\cdot G_i\leq0.
\]
Distinct curves have nonnegative intersections.  Connectedness therefore
shows \(G_i^2<0\): 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 this maximum is attained.  Choose \(j\in I\)
which either meets \(A\) or has a neighbor outside \(I\).
Such a vertex exists by connectedness and the fact that \(A\) meets the
fiber.  Then
\[
 Z\cdot G_j
 =aC\cdot G_j+\sum_i(z_i-ac_i)G_i\cdot G_j<0.
\]
The term with \(i=j\) is zero; all other terms are nonpositive, and the
chosen vertex makes at least one term strictly negative.
\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
\(g:T\to P\) obtained by localizing a threefold resolution.
The residue field \(l\) need not be perfect.  We keep all intersection
degrees over \(l\).

For an integral exceptional curve \(G\), divisor adjunction and curve
duality give
\begin{equation}
\label{eq:local-surface-adjunction}
 (K_T+G)\cdot G
 =2\dim_lH^1(G,\cO_G)-2\dim_lH^0(G,\cO_G).
\end{equation}
To specify the duality in this formula, use the dualizing complex on \(P\)
localized from the threefold and its exceptional inverse image on \(T\).
Transitivity for \(G\to T\to P\) and
\(G\to\Spec l\to P\) identifies the adjunction sheaf on \(G\) with
its dualizing sheaf over \(l\), up to tensoring with a one-dimensional
\(l\)-vector space.  This does not change degree.
Proper duality and the formula
\(\chi(\cL)-\chi(\cO_G)=\deg_l\cL\) prove
\eqref{eq:local-surface-adjunction}
\cite{hartshorne1966,hartshorne1977}.

Suppose \(K_T\cdot G<0\), and set
\(l'=H^0(G,\cO_G)\) and \(d=[l':l]\).
Both \(G^2\) and \(K_T\cdot G\) are negative multiples of \(d\).
Equation \eqref{eq:local-surface-adjunction} and
Lemma~\ref{lem:exceptional-intersections} force
\[
 H^1(G,\cO_G)=0,\qquad G^2=K_T\cdot G=-d.
\]
The line bundle \(\cO_G(-G)\) consequently has degree one over \(l'\).
Riemann--Roch gives at least two independent sections over \(l'\).
Every nonzero section has a zero scheme of length one.  Two independent
sections have no common zero: otherwise their effective Cartier divisors
coincide, so their ratio is a global unit, an element of \(l'^\times\).
They define a nonconstant map \(G\to\PP^1_{l'}\) of degree one.
It is finite, and a finite birational morphism to a normal curve is an
isomorphism.  Hence
\[
 G\simeq\PP^1_{l'},\qquad \cO_G(G)\simeq\cO_{\PP^1_{l'}}(-1).
\]

For completeness this curve can be contracted while retaining a
projective regular resolution over \(P\).  Choose a relatively very ample
Cartier divisor \(A\), put \(a=\deg_{l'}(A|_G)>0\), and set \(Q=A+aG\).
The divisor \(Q-G\) has positive degree on every component of the closed
fiber, and is relatively ample by the fiber criterion for ampleness
\cite{egaIII}.  The restriction \(Q|_G\) is trivial.
For every \(n\geq1\), a trivializing section of \(\cO_G(nQ)\) extends
successively to \(nG\), because the obstruction groups are
\[
 H^1\bigl(G,\cO_G(nQ-jG)\bigr)
   =H^1(\PP^1_{l'},\cO(j))=0\qquad(j\geq1).
\]
For large \(n\), relative Serre vanishing applied to \(n(Q-G)\) lifts
these sections from \(nG\) to \(T\).  They generate along \(G\).
Away from \(G\), the sections of \(n(Q-G)\), multiplied by the canonical
section of \(nG\), generate and separate points and tangent directions.
Thus \(nQ\) defines a projective contraction whose only nontrivial fiber
is \(G\).  Its Stein factorization has normal target and is an isomorphism
off \(G\).

Let \(q\) be the image of \(G\).  Formal functions identifies the completed
local ring at \(q\) with
\[
 \varprojlim_n H^0(nG,\cO_{nG}).
\]
The \(G\)-adic and inverse-image \(q\)-adic neighborhoods are cofinal,
since they have the same support.  The successive kernels have graded
pieces
\[
 H^0(G,\cO_G(-nG))=H^0(\PP^1_{l'},\cO(n)).
\]
There is no obstruction in \(H^1\), so the associated graded algebra is
the polynomial ring \(l'[x,y]\).
More explicitly, let \(F_n\) be the kernel of the map from the completed
local ring to \(H^0(nG,\cO_{nG})\).  Then \(F_1\) is its maximal ideal,
and the \(F_n\)-filtration is cofinal with the maximal-ideal-adic filtration.
Choose lifts of the two degree-one generators.
Generation of the graded algebra gives
\(F_1\subset(x,y)+F_n\) for every \(n\), by successive approximation,
using arbitrary lifts of coefficients from \(l'\).
Finitely generated ideals in the complete Noetherian local ring are
closed, so \(F_1=(x,y)\).  No embedding of the residue field \(l'\) into
the completed local ring is required.
Its dimension is two; hence
it, and then the local ring at \(q\), is regular.  This proves the claimed
contraction using formal functions and Serre vanishing
\cite[III, Sections~5 and~11]{hartshorne1977}.

There are only finitely many exceptional curves.  Contracting a
\(K_T\)-negative one whenever possible therefore yields a regular
resolution with \(K_T\) relatively nef.  If any exceptional curves remain,
terminality gives
\[
 E=K_T-g^*K_P=\sum_i a_iG_i,\qquad a_i>0.
\]
The discrepancies of surviving curves have not changed, since they are
read at their generic points.  Lemma~\ref{lem:exceptional-intersections}
gives \(E\cdot G_j<0\) for some \(j\), contradicting relative nefness.
There are no exceptional curves left, and the proper birational map is
an isomorphism.  Thus the codimension-two point was regular.
Normality already gives regularity in codimension one, so the remaining
singular locus of the projective threefold is finite.
\end{proof}

\subsection{Adjunction on the reduced boundary}

We return to the dlt pair \((Y_d,\Sigma)\) satisfying
\eqref{eq:dlt-adjunction-input}.  To apply surface abundance, we must
identify its restricted log canonical class with that of a projective
slc surface.

Let \(\sigma:\Sigma'\to\Sigma\) be the finite \(S_2\)-ification.
It is an isomorphism away from finitely many closed points
\cite[Proposition~2.14]{xu2026}.
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}.
Restrictions here and below mean rational line bundles, or sufficiently
divisible Cartier multiples.  We give the local verification needed
in this dlt situation.

Choose a log resolution and write
\begin{equation}
\label{eq:boundary-log-pullback}
 K_{\widetilde Y}+\widetilde\Sigma+\sum_i\beta_iE_i
   =h^*(K_{Y_d}+\Sigma),\qquad \beta_i\leq1.
\end{equation}
At a codimension-one point of \(\Sigma\), localize the ambient threefold
to a normal local surface \(P\), and write \(h:T\to P\) for the
localized regular resolution.  If the pair is not snc at this point,
dlt gives \(\beta_i<1\) on the exceptional curves over it.
Set
\[
 J=-\sum_i\lfloor\beta_i\rfloor E_i,\qquad
 M=J-\widetilde\Sigma,\qquad
 A_{\mathrm{fr}}=\sum_i\{\beta_i\}E_i.
\]
Here \(J\) is effective, and
\begin{equation}
\label{eq:surface-relative-vanishing-input}
 M-K_T-A_{\mathrm{fr}}\equiv_P0 .
\end{equation}

We claim \(R^1h_*\cO_T(M)=0\).
For every nonzero effective integral exceptional cycle \(Z\), the divisor
\(Z-A_{\mathrm{fr}}\) has a positive coefficient.
Lemma~\ref{lem:exceptional-intersections} gives a component \(E_j\) with
positive coefficient in \(Z\) such that
\((Z-A_{\mathrm{fr}})\cdot E_j<0\).
Using \eqref{eq:surface-relative-vanishing-input},
\[
 (M-Z+E_j)\cdot E_j>(K_T+E_j)\cdot E_j
   =\deg_l\omega_{E_j/l}.
\]
Curve duality gives
\(H^1(E_j,\cO_{E_j}(M-Z+E_j))=0\):
the dual space consists of sections of a line bundle of negative degree.
The exact sequence
\[
 0\longrightarrow\cO_{E_j}(M-Z+E_j)
 \longrightarrow\cO_Z(M)
 \longrightarrow\cO_{Z-E_j}(M)\longrightarrow0
\]
then proves \(H^1(Z,\cO_Z(M))=0\) by induction on the sum of
the coefficients of \(Z\).
Exceptional cycles give a cofinal system of neighborhoods of the
exceptional fiber.  Formal functions, followed by faithful completion
for the coherent higher direct image, proves the claim.

Normality gives \(h_*\cO_T(J)=\cO_P\): a rational function with possible
poles only along exceptional divisors is regular at every codimension-one
point of \(P\), hence regular on \(P\).
The preceding vanishing and
\[
 0\longrightarrow\cO_T(M)\longrightarrow\cO_T(J)
 \longrightarrow\cO_{\widetilde\Sigma}(J)\longrightarrow0
\]
give a surjection
\(\cO_P\to h_*\cO_{\widetilde\Sigma}(J)\).
It factors through \(h_*\cO_{\widetilde\Sigma}\), which is a subsheaf of
the target.  The strict boundary has no exceptional component, so its
proper map to the local boundary is quasi-finite and hence finite.
The kernel of
\(\cO_P\to h_*\cO_{\widetilde\Sigma}\) is precisely the ideal of the
reduced boundary.  The surjection thus identifies
\(\widetilde\Sigma\) with that boundary and also gives
\(\cO_{\widetilde\Sigma}=\cO_{\widetilde\Sigma}(J)\)
under their natural inclusion.
In particular \(J\) misses the strict boundary.

Two strict branches cannot cross above this non-snc point.
In a nontrivial regular surface resolution any such intersection over the
closed point lies on the exceptional fiber; the total divisor would then
have three components through a point on a surface, contrary to snc.
The trivial-resolution case would already be snc downstairs.
Thus the boundary is regular at a non-snc point of codimension one,
and every exceptional coefficient meeting its strict transform is
nonnegative.  At the snc points, the reduced boundary is nodal.
This proves that \(\Sigma'\) is nodal in codimension one.

Here is the log-line identification in \eqref{eq:slc-adjunction}.
Use rational residue forms to identify the plurilog-canonical line with
pluricanonical forms on each boundary component.
At snc points, Cartier divisor adjunction gives this identification,
including along the nodes.
At the other codimension-one points, restrict
\eqref{eq:boundary-log-pullback} to the smooth strict component.
The preceding local calculation identifies it with the boundary there
and shows that the resulting different is effective, since the exceptional
coefficients meeting it are nonnegative.
It has coefficient zero along the nodes.
The identification extends by the \(S_2\) property, using reflexive powers
with the indicated divisorial twist.  Its right side is therefore
\(\Q\)-Cartier with exactly the rational line bundle on the left side
of \eqref{eq:slc-adjunction}.

On the normalization of \(\Sigma'\), the boundary is the transform of
\(\Delta'\) together with the conductor, the latter with coefficient one.
This is the residue formula at a node, extended from codimension one.
Its log pullback to the smooth strict component
\(\widetilde\Sigma_i\) is
\[
 K_{\widetilde\Sigma_i}
 +\left((\widetilde\Sigma-\widetilde\Sigma_i)
                   +\sum_j\beta_jE_j\right)|_{\widetilde\Sigma_i}.
\]
This follows from smooth adjunction applied to
\eqref{eq:boundary-log-pullback}, with compatible rational residue forms,
so it is an equality of log pullbacks, not just numerical classes.
The displayed subpair is snc with every coefficient at most one.
The discrepancy criterion at the start of the section makes it lc.
Thus \(\Sigma'\) is \(S_2\), nodal in codimension one, and lc after
normalization with conductor; this proves the asserted slc property.

By \eqref{eq:dlt-adjunction-input}, the log canonical class in
\eqref{eq:slc-adjunction} is numerically trivial.
The surface \(\Sigma'\) is projective, since it is finite over \(\Sigma\).
Abundance for projective slc surfaces in positive characteristic
\cite[Theorem~3.1]{posva2024} makes this class semiample, hence
\(\Q\)-linearly trivial.  Indeed a globally generated numerically trivial
line bundle gives a constant morphism on each connected component and
is trivial there.
It follows from \eqref{eq:dlt-adjunction-input}, whose coefficient
\(\tau s\) is positive rational, that 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.
Its support is \(\Sigma\), and \(L|_{\Sigma_i}\equiv0\) for every component.

\subsection{Torsion on the full Cartier cycle}

We now pass from \eqref{eq:reduced-boundary-triviality} to the actual
nonreduced divisor \(L\).

\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 that \(L\) is numerically trivial on every integral
component of its support and that \(\cO_Y(L)|_{L_{\mathrm{red}}\setminus Z}\)
is trivial for a finite set of closed points \(Z\).
Then \(\cO_L(L)\) is torsion.
\end{lemma}

\begin{proof}
Replacing \(Z\) by \(Z\cap|L|\), we may assume \(Z\subset|L|\).
For \(z\in Z\), put \(R=\cO_{Y,z}\) and let \(\mathfrak m\) be its maximal
ideal.  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, and is consequently Cohen--Macaulay.
This implies that \(H^2_{\mathfrak m}(R)\) has finite length.
Explicitly, present \(R=A/I\), where \(A\) is a regular local ring of
dimension \(n\), essentially of finite type over the field.
The height of \(I\) is \(n-3\).
At a nonmaximal prime in \(\Spec R\), catenarity and
Auslander--Buchsbaum give projective dimension \(n-3\) over the
corresponding localization of \(A\).  Hence
\(\Ext_A^{n-2}(R,A)\) is supported only at the closed point.
It is a finite module, thus has finite length.
Local duality identifies \(H^2_{\mathfrak m}(R)\) with its Matlis dual,
up to completion, and proves the assertion
\cite{hartshorne1967}.

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 \(z\in Z\).
There are only finitely many points, so the same \(j\) works at all of
them.  The restriction sequences fit into a commutative diagram
\[
\begin{tikzcd}[column sep=small]
0 \arrow[r] &
\cO_Y((u-j)L) \arrow[r] \arrow[d] &
\cO_Y(uL) \arrow[r] \arrow[d,equal] &
\cO_{jL}(uL) \arrow[r] \arrow[d] & 0\\
0 \arrow[r] &
\cO_Y((u-1)L) \arrow[r] &
\cO_Y(uL) \arrow[r] &
\cO_L(uL) \arrow[r] & 0 .
\end{tikzcd}
\]
In local trivializations the left vertical map is multiplication by
\(t_z^{j-1}\).
The vanishing of the ambient \(H^1_Z\) makes the connecting maps from
the \(H^1_Z\) of the rightmost terms to the ambient \(H^2_Z\) injective.
The choice of \(j\) therefore gives
\begin{equation}
\label{eq:torsion-zero-comparison}
 H^1_Z(Y,\cO_{jL}(uL))
       \longrightarrow H^1_Z(Y,\cO_L(uL))
 \quad\text{equal to zero for every integer }u .
\end{equation}
The assertion is uniform in \(u\): twisting by \(uL\) only changes local
trivializations of these finite-length modules.

Fix this \(j\).
The kernel of
\(\cO_{jL\setminus Z}\to\cO_{L_{\mathrm{red}}\setminus Z}\)
is nilpotent.
Lift a chosen 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 are 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.  The powers therefore give a trivializing section
\[
 s_j\in H^0(jL\setminus Z,\cO_{jL}(uL)).
\]
Project it to a section \(s\) on \(L\setminus Z\).
In the exact sequence of sections with support in \(Z\), its obstruction
to extending over \(L\) is the image of the obstruction for \(s_j\).
Equation \eqref{eq:torsion-zero-comparison} kills that image.
Thus \(s\) 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: a nonzero section of an invertible sheaf
on an integral scheme is a non-zero-divisor in local trivializations.
Such a divisor has positive degree against an ample class, whereas
\(\cO_T(uL)\) is numerically trivial.  This is impossible.
No normality of \(T\) is needed for this intersection argument.
Nonvanishing after reduction implies invertibility on \(L\), by the
local criterion for a unit modulo a nilpotent ideal.
We conclude \(\cO_L(uL)\simeq\cO_L\), as required.
\end{proof}

Applying Lemma~\ref{lem:full-cycle-torsion} to \(Y_d\) and \(L\) proves
that \(\cO_L(L)\) is torsion.

\subsection{A tame cover and the primitive cycle}

Choose a smooth projective common resolution \(V_0\) of \(Y_d\) and \(X\).
It may be chosen to be an isomorphism over the common open set
\[
 X_{\mathrm{reg}}\setminus\Supp N_0
\]
and to resolve the relevant divisor supports.
Let \(\rho_0:V_0\to X\) be its map to \(X\).
The pullbacks of \(N_Y\) and \(N_0\) agree on \(V_0\), by the pullback
equalities through the MMP.

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

Write
\[
 m=p^bm',\qquad b\geq0,\qquad (m',p)=1.
\]
Choose a nonzero rational generator \(\eta\) of
\(\omega_{V_0}^{\otimes p^b}\), so that
\(\theta=a\eta^{m'}\) for \(a\in k(V_0)^\times\).
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 \(m'\).  Its normalization is projective.
Away from \(\Supp\operatorname{div}(\theta)\), this construction is an
integral component of the torsor of \(m'\)-th roots of a nowhere-vanishing
section of a line bundle.  It is finite étale there, since \(m'\) is
invertible in \(k\).

Resolve this normalization, and then the total divisor support, keeping
that regular étale open set unchanged.  Denote the resulting smooth
projective variety by \(V\), and its generically finite separable map to
\(V_0\) by \(\pi\).
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 nonzero rational
section of \(\omega_V\otimes\pi^*\omega_{V_0}^{-1}\);
it is nonzero because \(\pi\) is separable.
Multiplying \(v\eta\) by the \(p^b\)-th power of this Jacobian section produces a rational section
of \(\omega_V^{\otimes p^b}\).
Its divisor is supported over
\(\Supp\operatorname{div}(\theta)\), since \(\pi\) is étale off that
support.  In particular it is supported on the total pullback of \(N_0\),
together with divisors disjoint from it.

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 remainder
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, since \(V\)
is smooth.  Its support is connected and snc by the final resolution.

The pullback \(L_V\) is numerically trivial on its support:
any curve there maps to a point or to a curve in \(\Sigma\), where
\(L\) is numerically trivial.  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.  We have
\[
 D^2\cdot H=0,\qquad D\cdot H^2>0
\]
for every very ample divisor \(H\).

If \(\cO_L(tL)\simeq\cO_L\), its pullback is trivial on the Cartier
scheme \(L_V\).  Near \(C\), the inclusion \(D\subset aD=L_V\) is an
inclusion of closed subschemes.  Restricting the pulled-back
trivialization gives
\[
 \cO_D(taD)\simeq\cO_D.
\]
Thus \(\cO_D(D)\) is torsion.  This argument does not claim that its
order is unchanged under the cover or primitive division.

The rational canonical section constructed above has, on a neighborhood
of \(C\), divisor supported on \(D\); all other components of its divisor
are disjoint from \(C\).  It therefore gives
\eqref{eq:prepared-canonical}.  For any Cartier divisor \(A\) on \(V\),
\[
 p^bK_V\cdot D\cdot A
   =B_*\cdot D\cdot A
   =\sum_\alpha b_\alpha\,
       c_1(\cO_{D_\alpha}(D))\cdot A|_{D_\alpha}=0.
\]
This includes \(K_V^2\cdot D=0\).

\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:
every curve there maps to a point or to a curve in \(\Supp N_0\).
We have
\[
 D^2\cdot H=N_{\mathrm{tot}}^2\cdot H
           =D\cdot N_{\mathrm{tot}}\cdot H=0,
 \qquad D\cdot H^2>0,\quad N_{\mathrm{tot}}\cdot H^2>0.
\]
The surface Hodge index theorem implies that
\begin{equation}
\label{eq:prepared-proportionality}
 D\equiv cN_{\mathrm{tot}},\qquad
 c=\frac{D\cdot H^2}{N_{\mathrm{tot}}\cdot H^2}>0.
\end{equation}
Here is a way to check the numerical equivalence on every curve rather
than just on one general surface.
Given an integral curve \(C_0\subset V\), take \(a\gg0\) and an integral
surface \(T\in|aH|\) containing \(C_0\).
Such a surface exists by Bertini irreducibility with this prescribed
base locus \cite{jouanolou1983}: for sufficiently large \(a\), generators
of the twisted ideal of \(C_0\), multiplied by very ample sections,
give a linear system with no fixed divisor and with a birational map
to its image off \(C_0\).
The image has dimension three, so a general member is integral.

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
\((1,\dim_{\R}N^1(\widetilde T)_{\R}-1)\)
\cite[V, Section~1]{hartshorne1977}; its restriction to the orthogonal
complement of any positive-square class is negative definite.
Two nonzero orthogonal isotropic classes with positive intersection
against that class are therefore positively proportional.
Their ratio is the number \(c\) in \eqref{eq:prepared-proportionality},
since intersection with the pullback of \(H|_T\) gives
\(aD\cdot H^2\) and \(aN_{\mathrm{tot}}\cdot H^2\).
Choose a curve on the resolution dominating \(C_0\); projection of
degrees proves \((D-cN_{\mathrm{tot}})\cdot C_0=0\).
This proves \eqref{eq:prepared-proportionality}.

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, a contradiction.
This establishes the remaining assertion of
Proposition~\ref{prop:prepared-cycle}.
The proof of that proposition is complete.
In the subsequent sections only $V,D,H$, the torsion order $r$, and the
local canonical relation \eqref{eq:prepared-canonical} are retained;
the auxiliary birational models are no longer needed.
==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//preparation.tex ====

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//slopes.tex ====
\section{Slopes, foliations, and polar parts}
\label{sec:slopes}

We retain the smooth projective threefold $V$ and the cycle $D$ of
Proposition~\ref{prop:prepared-cycle}.  Fix a very ample divisor $H$ on $V$.
The numerical properties used in this section are
\begin{equation}
  H^2D>0,\qquad HD^2=0,\qquad K_VDA=0
  \quad\text{for every divisor class }A.
  \label{eq:slope-numerical-input}
\end{equation}
We also use that $D$ is nef and that no curve of $D$-degree zero passes
through a very general point of $V$.  The ground field is uncountable,
algebraically closed, and of characteristic $p>3$.
We will obtain strong semistability and $c_2(V)D\geq0$, followed by
uniform bounds on vector-valued polar parts and the vanishing of scalar
polar parts.  The scalar vanishing starts the jet calculation in
Section~\ref{sec:jets}; the vector bounds control Laurent primitives in
Section~\ref{sec:laurent}.

\subsection{Slope calculus for the nef product}

For a torsion-free coherent sheaf $E$ of positive rank, set
\[
  \deg(E)=c_1(E)HD,\qquad
  \mu(E)=\frac{\deg(E)}{\rk(E)}.
\]
Stability and semistability will always refer to this slope.  A sheaf is
strongly semistable if all its iterated absolute Frobenius pullbacks are
semistable.  Although $D$ need not be ample, the elementary slope calculus
needed below still applies.  We give the details that are relevant here.

An effective divisor has nonnegative degree, since $H$ and $D$ are nef.
Saturating a subsheaf therefore does not decrease its degree.  Every
torsion-free sheaf $E$ 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 then 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 $\rk E$, the possible slopes form a discrete subset of
$\Q$.  Thus a maximal slope is attained.  Among saturated subsheaves of
maximal slope, take one of maximal rank.  The usual sum-and-intersection
argument shows that this subsheaf contains every other subsheaf of maximal
slope.  It is semistable, and the maximal slope of its torsion-free quotient
is strictly smaller.  Repeating on the quotient produces 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 the familiar implication
\begin{equation}
  \mu_{\min}(E)>\mu_{\max}(F)
  \quad\Longrightarrow\quad \Hom(E,F)=0.
  \label{eq:slope-hom-vanishing}
\end{equation}
Indeed a nonzero image is both a torsion-free quotient of the source and
a subsheaf of the target, which would force its slope to be at least the
left-hand extreme and at most the right-hand extreme.  Dualizing the
filtration on the open set where its terms and factors are locally free
gives
\[
  \mu_{\max}(E^\vee)=-\mu_{\min}(E),\qquad
  \mu_{\min}(E^\vee)=-\mu_{\max}(E).
\]
The omitted set has codimension at least two and does not change any
determinant or degree.  The same observation permits all ensuing bundle
calculations on such open sets, with saturation or reflexive extension
when returning to $V$.

Tensoring by a line bundle adds its degree to every slope.  More generally,
the same rule holds for a rank-one torsion-free factor after restriction
to the open set where it is invertible.  In particular,
Equation~\eqref{eq:slope-numerical-input} shows that twisting by $iD$ leaves
all slopes unchanged, for every $i\in\Z$.  We will use this rank-one rule;
no assertion about semistability of arbitrary tensor products is required.

\subsection{Connections and distributions of rank at most two}

Let $F_V\colon V\to V$ denote absolute Frobenius.  A Frobenius pullback
$F_V^*E$ has its canonical connection, which in a pulled-back local frame
differentiates the coefficients.  We first record the elementary descent
observation behind its use.

\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 of the ambient generic vector space.  In a
Grassmannian chart, represent this subspace by a matrix $(I\mid M)$.
Horizontality says that every entry of $M$ has zero differential.  Since
$K$ is separably generated over the perfect field $k$, the kernel of
$d\colon K\to\Omega_{K/k}^1$ is $K^p$.  Taking $p$-th roots of the entries
of $M$ therefore defines a subspace before Frobenius pullback.

Saturate this subspace to a subsheaf $B_0\subset E$.  At every
codimension-one point the relevant torsion-free quotients are free over
the discrete valuation ring.  Frobenius is flat on the smooth variety,
so $F_V^*B_0$ is saturated there and agrees with $B$ there: they have the
same generic subspace.  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}

The general connection criterion is also stated in
\cite[Corollary~2.4]{langer2004}.  The argument above makes explicit why
it applies to the present nef slope.

\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}

This rank-one and rank-two argument follows the foliation criterion in
\cite[Lemma~2.9]{xuzhang2019}; we give the slope calculation for the
nef product used here.

\begin{proof}
Put $Q=T_V/A$ and $a=\mu(A)$.  By
Equation~\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
$\bigwedge^2A\to Q$.  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} again makes the map zero.

Once bracket closure holds, the $p$-th power of derivations modulo $A$
defines an $\cO_V$-linear map
\begin{equation}
  F_V^*A\longrightarrow Q.
  \label{eq:p-curvature-obstruction}
\end{equation}
For completeness, the two restricted-Lie identities needed for this
assertion can be seen directly.  For local derivations $u,v\in A$, set
$w=tu+v$ with $t$ a formal scalar.  In characteristic $p$,
\[
  \frac{d}{dt}w^p
  =\sum_{j=0}^{p-1}w^juw^{p-1-j}
  =\operatorname{ad}(w)^{p-1}(u).
\]
The last equality follows from
$\binom{p-1}{j}\equiv(-1)^j\pmod p$.  Every coefficient of the right-hand
side lies in the bracket-closed module $A$.  The mixed coefficients of
$w^p-t^pu^p-v^p$, whose degrees in $t$ lie between $1$ and $p-1$, thus
belong to $A$.  This proves additivity modulo $A$.

For a local function $f$, expanding $(fu)^p$ as a differential operator
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 index $j\ge2$ 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 can remain.  The scalar discrepancy is therefore a
multiple of $u$, proving Frobenius linearity modulo $A$ and hence
Equation~\eqref{eq:p-curvature-obstruction}.

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

\subsection{The inseparable quotient and its canonical divisor}

We spell out the quotient construction to keep track of the sign in the
canonical formula.  These constructions and their use for tangent
semistability also appear in \cite[Section~2]{xuzhang2019}.

\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 $\pi\colon V\to W$
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, the canonical bundles satisfy
\begin{equation}
  \omega_V\simeq\pi^*\omega_W\otimes(\det A)^{\otimes(p-1)}.
  \label{eq:foliation-canonical}
\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 $R^p$,
$P$ is finite over $R^p$ and $R$ is finite over $P$.  These invariant
rings glue to the asserted factorization of Frobenius; in particular both
maps are finite and their underlying maps of topological spaces are
homeomorphisms.  The ring $P$ is normal.  Indeed, an element of
$\Frac(P)$ integral over $P$ is integral over $R$, hence belongs to the
normal ring $R$, and all the derivations annihilate it.  Thus it belongs
to $P$.  The variety $W$ is projective because it is finite over the
projective variety $V^{(1)}$.

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 choose the dual local derivations $u_1,\ldots,u_d$.  Thus
$u_i(x_j)=\delta_{ij}$.  Bracket closure implies $[u_i,u_j]\in A$; this
bracket annihilates every $x_l$ and is therefore zero.  Likewise
$u_i^p\in A$ annihilates all the $x_l$, so $u_i^p=0$.

These commuting nilpotent derivations give an explicit invariant-ring
presentation.  For one derivation $u$ with $u(x)=1$, the coefficient
$(p-1)!^{-1}u^{p-1}(f)$ lies in its kernel.  Subtract its product with
$x^{p-1}$ and repeat with successively smaller powers.  This writes every
$f$ uniquely as a polynomial of degree less than $p$ in $x$, with
coefficients in the kernel of $u$.  Applying this procedure successively
to the commuting $u_i$ gives
\begin{equation}
  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.
  \label{eq:foliation-local-presentation}
\end{equation}
Uniqueness follows by applying the same successive highest derivatives
to a putative relation.  In particular, $\pi$ is locally free of degree
$p^d$ on this open set.

We may further remove the singular locus of $W$ and its inverse image;
normality and finiteness ensure that these removed sets also have
codimension at least two.  The presentation in
Equation~\eqref{eq:foliation-local-presentation} now yields the exact
sequence of vector bundles
\begin{equation}
  0\longrightarrow F_V^*(A^\vee)
  \longrightarrow\pi^*\Omega_W^1
  \longrightarrow\Omega_V^1
  \longrightarrow A^\vee\longrightarrow0.
  \label{eq:foliation-cotangent}
\end{equation}
Here 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
$R\otimes_P\Omega_{P/k}^1$.  The presentation gives
\[
  \Omega_{R/k}^1\simeq
  \bigl((R\otimes_P\Omega_{P/k}^1)/N\bigr)
  \oplus\bigoplus_{i=1}^dR\,dx_i,
\]
because the relations have no differential in the $X_i$ directions.
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 $R\otimes_P\Omega_{P/k}^1$.  The displayed
$d$ generators of $N$ are a basis: locally their surjection from $R^d$
onto a free module of rank $d$ is an isomorphism.  This proves the
required exactness at every point of the chosen open set.

To identify its transition maps, let $x'_j$ be another such set of
functions and write it in the basis $x^I$ over $P$.  Taking $p$-th powers
and differentiating downstairs shows that, after pullback, the
coefficient of $dc_l$ in $d((x'_j)^p)$ is $u_l(x'_j)^p$.  On the other
hand, the coefficient of the $l$-th basis vector of $A^\vee$ in
$dx'_j|_A$ is $u_l(x'_j)$.  The first term of
Equation~\eqref{eq:foliation-cotangent} is therefore precisely
$F_V^*(A^\vee)$.  Taking determinants in that sequence gives
\[
  \pi^*\omega_W
  \simeq(\det A)^{-p}\otimes\omega_V\otimes\det A,
\]
which is Equation~\eqref{eq:foliation-canonical}.
\end{proof}

\subsection{Strong semistability of the tangent bundle}

\begin{proposition}
\label{prop:strong-semistability}
The tangent bundle $T_V$ is strongly semistable for the slope
$c_1(-)HD/\rk(-)$.  Moreover,
\begin{equation}
  c_2(V)D\ge0.
  \label{eq:c2-nonnegative}
\end{equation}
\end{proposition}

\begin{proof}
We first prove semistability by the foliation and rational-curve
argument of \cite[Lemma~2.10]{xuzhang2019}, with the quantitative
bend-and-break statement specified below.  Suppose that $A$ 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 has a finite purely inseparable quotient
$\pi\colon V\to W$.  On the big smooth open set where
Equation~\eqref{eq:foliation-canonical} holds, put
$\Lambda=-\pi^*K_W$.  Its extension as a divisor class on $V$ is
\[
  \Lambda=(p-1)c_1(A)-K_V,
  \qquad
  \Lambda HD=(p-1)da>0.
\]
No global Cartier property of $K_W$ is needed: its degree will be taken
only on curves contained in this smooth open set.

Let $H^{(1)}$ and $D^{(1)}$ denote the corresponding divisor classes on
$V^{(1)}$, and set
\[
  H'=\rho^*H^{(1)},\qquad D'=\rho^*D^{(1)}.
\]
Then $H'$ is ample and Cartier, $D'$ is nef and Cartier, and
\[
  \pi^*H'=pH,\qquad \pi^*D'=pD.
\]
For a positive integer $t$, take sufficiently divisible positive
multiples of $H'$ and $H'+tD'$ which are very ample.  Their general
complete intersection is a smooth projective integral curve $C_t$ wholly
contained in the open set under consideration.  Here is the avoidance
point in this assertion.  The complement 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 finitely
many points.  Bertini on the smooth open set, and irreducibility for the
successive very ample sections, give the required curve.

Projection formula, computed where $\pi$ is locally free, cancels both
the degree of $\pi$ and the chosen complete-intersection multiples from
the degree ratio.  It gives
\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}
    \longrightarrow0
    \qquad(t\longrightarrow\infty).
  \label{eq:bend-break-ratio}
\end{align}
The denominator is positive for all sufficiently large $t$.

We use the following quantitative form of 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 $\Gamma\subset Y$ satisfying
\[
  N\Gamma\le(\dim Y+1)\frac{NC}{-K_YC}.
\]
This statement is valid in arbitrary characteristic.  Choose $t$ so large
that the ratio in Equation~\eqref{eq:bend-break-ratio} is less than $1/4$.
Applied on $W$ with $N=D'$, the theorem gives, through every closed point
$x\in C_t$, a rational curve $\Gamma$ with
\[
  0\le D'\Gamma<1.
\]
As $D'$ is Cartier, this degree is an integer and hence is zero.

We choose $C_t$ to witness the very-general-point contradiction as well.
Fix the product of the two chosen very ample linear systems, and let
$\mathcal T$ be its nonempty open locus of smooth integral complete
intersections avoiding the bad locus above.  This parameter space is
irreducible.  Its universal curve $\mathcal C\to\mathcal T$ is smooth:
each fiber is a transverse complete intersection inside $W_{\mathrm{sm}}$,
so the relative Jacobian criterion applies.  Its evaluation map
$\mathcal C\to W$ is dominant.  Indeed the ambient incidence of points
and pairs of divisors through them is a product of projective bundles
over $W$, hence is irreducible and surjects onto $W$; restricting it to
the nonempty open parameter locus $\mathcal T$ leaves a dense open
subset, whose evaluation still has dense image.

Let $Z_1,Z_2,\ldots$ be proper closed subsets of $V$ outside whose union
no $D$-trivial curve passes.  The finite map $\pi$ has proper closed
images $\pi(Z_j)$.  For each $j$, the locus in $\mathcal T$ of curves not
contained in $\pi(Z_j)$ is the image of
$\mathcal C\setminus\operatorname{ev}^{-1}(\pi(Z_j))$ under the smooth
map $\mathcal C\to\mathcal T$.  It is open because smooth maps are open,
and nonempty because evaluation is dominant.  Over the uncountable field,
the countably many resulting nonempty opens in $\mathcal T$ have a common
$k$-point.  On its integral curve $C_t$, each $C_t\cap\pi(Z_j)$ is finite.
Choose a closed point $x$ outside their union.  Its unique inverse-image
point $y\in V$ lies outside every $Z_j$.

The reduced inverse image of the rational curve $\Gamma$ through $x$
contains a curve $\widetilde\Gamma$ through $y$, finite and surjective
over $\Gamma$.  Projection formula gives
\[
  pD\widetilde\Gamma
  =(\pi^*D')\widetilde\Gamma
  =\deg(\widetilde\Gamma/\Gamma)D'\Gamma=0.
\]
This contradicts the choice of $y$.  Thus $T_V$ is semistable.

Suppose next that $e\ge1$ is the first integer for which
$F_V^{e*}T_V$ is unstable.  Let $A'$ be its first Harder--Narasimhan term
and $Q'=F_V^{e*}T_V/A'$.  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.  The cotangent bundle is semistable of slope zero,
because $T_V$ is.  Since $\rk A'+\rk Q'=3$, at least one tensor factor
in the second source has rank one.  If $\rk Q'=1$, semistability of
$A'$ and the rank-one twisting rule give the minimum slope of the source
as $\mu(A')-\mu(Q')$.  If $\rk A'=1$, the quotient $Q'$ need not be
semistable; use instead
\[
  \mu_{\min}((Q')^\vee)=-\mu_{\max}(Q')
\]
and twist by the rank-one factor $A'$.  In both cases,
\[
  \mu_{\min}\bigl(A'\otimes(Q')^\vee\bigr)
  =\mu(A')-\mu_{\max}(Q')>0.
\]
This contradicts Equation~\eqref{eq:slope-hom-vanishing}.  All Frobenius
pullbacks of $T_V$ are therefore semistable.

Finally apply Langer's Bogomolov inequality
\cite[Theorem~3.2]{langer2004}; see also the addendum
\cite{langeradd2004}.  This theorem is formulated for a nef tuple
$D_1,\ldots,D_{n-1}$ with numerically nonzero product, and for a strongly
semistable torsion-free sheaf gives
$\Delta(E)D_2\cdots D_{n-1}\ge0$, where
$\Delta(E)=2\rk(E)c_2(E)-(\rk(E)-1)c_1(E)^2$.
Here the tuple is $(H,D)$, whose product is nonzero since $H^2D>0$.
For the rank-three tangent bundle this yields
\[
  \bigl(6c_2(V)-2K_V^2\bigr)D\ge0.
\]
Equation~\eqref{eq:slope-numerical-input} gives $K_V^2D=0$ and proves
Equation~\eqref{eq:c2-nonnegative}.
\end{proof}

\subsection{Uniform section bounds}

\begin{lemma}
\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 must be zero.  Suppose that two independent
sections $s_0,s_1$ exist.  Remove the base locus of the resulting pencil.
Through every point $x$ of the remaining open set there is a member of
the pencil: take a nonzero linear combination vanishing at $x$.  Its
divisor has degree zero.  Each prime component $S$ through $x$ therefore
satisfies $DHS=0$, because all component degrees are nonnegative.

The restriction of $D$ to $S$ is nef and numerically trivial.  One way
to see the latter assertion, allowing $S$ to be singular, is to take a
projective resolution $\sigma\colon\widetilde S\to S$.  The class
$L_S=\sigma^*(D|_S)$ is nef and is orthogonal to the big and nef class
$A_S=\sigma^*(H|_S)$, with $A_S^2>0$.  The surface Hodge index theorem
\cite[Chapter~V, Section~1]{hartshorne1977}
makes the intersection form negative definite on $A_S^\perp$, whereas
nefness gives $L_S^2\ge0$.  Hence $L_S$ is numerically zero, and the
projection formula gives the assertion on $S$.  A projective integral
curve in $S$ through $x$, obtained for example as a component of a
sufficiently ample section through $x$, consequently has $D$-degree zero.
Choosing $x$ very general contradicts the prepared-cycle property.
\end{proof}

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

\begin{proof}
First suppose both sheaves are stable.  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 would make the first inequality
strict by stability of $G_1$.  An image of rank smaller than $\rk G_2$
would make the second inequality strict by stability of $G_2$.  Thus
every nonzero map has full rank, and the two sheaves have the same rank,
say $n$.

The determinant of each map is a global section of
\[
  L=\det(G_2)\otimes\det(G_1)^{-1},\qquad\deg L=0.
\]
It is initially defined on the locally free open set and extends across
its codimension-at-least-two complement.  By
Lemma~\ref{lem:degree-zero-line-sections}, $H^0(V,L)$ has dimension at
most one.  If $f,g$ were linearly independent maps, fix a nonzero section
$\tau$ spanning this space and write
\[
  \det(sf+tg)=P(s,t)\tau
\]
for a homogeneous polynomial $P$ of degree $n$.  Since $f$ has full rank,
$P$ is nonzero; since $n>0$ and $k$ is algebraically closed, $P$ vanishes
at some point of $\PP^1(k)$.  The corresponding linear combination is a
nonzero map by independence, but has zero determinant, a contradiction.
Thus the Hom space between stable factors has dimension at most one.

Every slope-zero semistable sheaf admits a finite filtration by saturated
subsheaves with torsion-free stable factors of slope zero.  Indeed if it
is not stable, a smaller-rank equal-slope subsheaf can be saturated; its
degree cannot increase strictly because of semistability.  Both it and
the quotient are semistable of slope zero, and induction on rank finishes
the construction.  Apply the left-exact Hom sequences successively to
such filtrations of $G_1$ and $G_2$.  The dimension of the Hom space is at
most the sum of the dimensions between all pairs of stable factors.
Each summand is at most one, and the numbers of factors are at most the
respective ranks.  This proves Equation~\eqref{eq:hom-rank-bound}.
\end{proof}

\begin{proposition}[Polar-part bounds]
\label{prop:polar-bounds}
Let $G$ be a semistable vector bundle of slope zero.  For every integer
$i$,
\begin{equation}
  h^0\bigl(V,G(iD)\bigr)\le\rk G,
  \qquad
  h^0\bigl(V,G^{\otimes2}(iD)\bigr)\le(\rk G)^2.
  \label{eq:uniform-bundle-sections}
\end{equation}
For each fixed bundle $J=G$ or $J=G^{\otimes2}$, the increasing spaces
\begin{equation}
  H^0\bigl(V,J(iD)/J\bigr),\qquad i=1,2,\ldots,
  \label{eq:polar-part-system}
\end{equation}
have bounded dimension and eventually stabilize under their natural
injections.  In particular their elements have a uniform bound on their
actual pole orders along every prime component of $D$.  This pole bound
may depend on $J$; it is uniform in the cut $i$ and in the chosen section.
\end{proposition}

\begin{proof}
All the twists $G(iD)$ are semistable of slope zero.  Apply
Lemma~\ref{lem:semistable-hom-bound} first to $\cO_V,G(iD)$ and then to
$G^\vee,G(iD)$.  The identities
\[
  H^0\bigl(V,G(iD)\bigr)=\Hom\bigl(\cO_V,G(iD)\bigr),
  \qquad
  H^0\bigl(V,G^{\otimes2}(iD)\bigr)
    =\Hom\bigl(G^\vee,G(iD)\bigr)
\]
give Equation~\eqref{eq:uniform-bundle-sections}.  In particular the
second bound uses the two semistable bundles $G^\vee$ and $G(iD)$,
without needing semistability of $G^{\otimes2}$.

For $i>0$, the cohomology sequence of
$0\to J\to J(iD)\to J(iD)/J\to0$ gives
\[
  h^0\bigl(V,J(iD)/J\bigr)
  \le h^0\bigl(V,J(iD)\bigr)+h^1(V,J).
\]
The right side is bounded independently of $i$.  Effectivity of $D$
gives injections of the quotient sheaves as $i$ increases, hence
injections of their finite-dimensional spaces of global sections.
A bounded increasing sequence of integer dimensions is eventually
constant, so these spaces stabilize.

More explicitly, choose $i_0$ after stabilization.  Every global polar
part, at every later cut, is then the image of an element of
$H^0(V,J(i_0D)/J)$.  At the generic point of a prime component
$D_\alpha$ of multiplicity $m_\alpha$, a local bundle frame measures its
pole order coefficientwise.  Such an image has pole order at most
$i_0m_\alpha$ there.  This bound is independent of the later cut and of
the element chosen.
\end{proof}

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

\begin{proof}
Apply Proposition~\ref{prop:polar-bounds} to $G=\cO_V$.  Suppose a
nonzero scalar polar part $\eta$ exists.  It has a genuine negative
valuation at the generic point of some component of $D$.  To justify
this detection, take local rational-function representatives.  If all
their component valuations were nonnegative, they would be regular by
normality of $V$; they already have no poles off $D$.  The quotient
section would then be zero.

For every $e\ge0$, raising the local representatives to their $p^e$-th
powers defines a global section
\[
  \eta^{p^e}\in
  H^0\bigl(V,\cO_V(p^eiD)/\cO_V\bigr).
\]
This operation is well defined on polar parts: two representatives
differing by a regular function have $p^e$-th powers differing by a
regular function.  At the selected component the negative valuation is
multiplied by $p^e$.  These pole orders are unbounded, contradicting the
uniform bound in Proposition~\ref{prop:polar-bounds}.
\end{proof}
==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//slopes.tex ====

==== BEGIN Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//references.bib ====
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@article{cp2019,
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@misc{cjs2009,
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@book{cjs2020,
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@article{egaIII2,
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@misc{liuxu2025,
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@article{witaszek2021,
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@article{zhang2020,
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@misc{stacks,
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@article{miyaoka1988,
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@article{kawamata1992,
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@article{cartier1957,
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@article{esnaultmehta2010,
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@article{tanaka2016,
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@misc{cossart2008notes,
  author = {Cossart, Vincent},
  title = {Resolution of Singularities of Threefolds in Positive Characteristic},
  year = {2008},
  howpublished = {Lecture notes, RIMS workshop ``On the Resolution of Singularities'', Kyoto, 1--5 December 2008},
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@misc{totaro2007,
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  primaryClass = {math.AG},
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}
==== END Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic//references.bib ====