==== TABLE OF CONTENTS ====
Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//bibliography.tex
Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//boundary.tex
Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//cycle.tex
Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//foundations.tex
Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//introduction.tex
Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//isolation.tex
Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//main.tex
Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//torsion.tex
===========================
==== BEGIN Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//bibliography.tex ====
\begin{thebibliography}{BCE+02}
\small
\setlength{\itemsep}{3pt}
\setlength{\parsep}{0pt}
\bibitem[Bir16]{Birkar}
Caucher Birkar, \emph{Existence of flips and minimal models for 3-folds in char~$p$},
Ann. Sci. \'Ec. Norm. Sup\'er. (4) \textbf{49} (2016), no.~1, 169--212.
\href{https://doi.org/10.24033/asens.2279}{doi:10.24033/asens.2279}.
\bibitem[BW17]{BW}
Caucher Birkar and Joe Waldron,
\emph{Existence of Mori fibre spaces for 3-folds in char~$p$},
Adv. Math. \textbf{313} (2017), 62--101.
\href{https://doi.org/10.1016/j.aim.2017.03.032}{doi:10.1016/j.aim.2017.03.032}.
\bibitem[CTX15]{CTX}
Paolo Cascini, Hiromu Tanaka and Chenyang Xu,
\emph{On base point freeness in positive characteristic},
Ann. Sci. \'Ec. Norm. Sup\'er. (4) \textbf{48} (2015), no.~5, 1239--1272.
\href{https://doi.org/10.24033/asens.2269}{doi:10.24033/asens.2269}.
\bibitem[CJS13]{CJS}
Vincent Cossart, Uwe Jannsen and Shuji Saito,
\emph{Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes},
arXiv:0905.2191, version~2, February~18, 2013; first posted 2009.
\href{https://arxiv.org/abs/0905.2191v2}{arXiv:0905.2191v2}.
\bibitem[CP08]{CPFirst}
Vincent Cossart and Olivier Piltant,
\emph{Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin--Schreier and purely inseparable coverings},
J. Algebra \textbf{320} (2008), no.~3, 1051--1082.
\href{https://doi.org/10.1016/j.jalgebra.2008.03.032}{doi:10.1016/j.jalgebra.2008.03.032}.
\bibitem[CP09]{CP}
Vincent Cossart and Olivier Piltant,
\emph{Resolution of singularities of threefolds in positive characteristic II},
J. Algebra \textbf{321} (2009), no.~7, 1836--1976.
\href{https://doi.org/10.1016/j.jalgebra.2008.11.030}{doi:10.1016/j.jalgebra.2008.11.030}.
\bibitem[HW22]{HW}
Christopher Hacon and Jakub Witaszek,
\emph{The minimal model program for threefolds in characteristic~5},
Duke Math. J. \textbf{171} (2022), no.~11, 2193--2231.
\href{https://doi.org/10.1215/00127094-2022-0024}{doi:10.1215/00127094-2022-0024}.
\bibitem[HX15]{HX}
Christopher D.~Hacon and Chenyang Xu,
\emph{On the three dimensional minimal model program in positive characteristic},
J. Amer. Math. Soc. \textbf{28} (2015), no.~3, 711--744.
\href{https://doi.org/10.1090/S0894-0347-2014-00809-2}{doi:10.1090/S0894-0347-2014-00809-2}.
\bibitem[HNT20]{HNT}
Kenta Hashizume, Yusuke Nakamura and Hiromu Tanaka,
\emph{Minimal model program for log canonical threefolds in positive characteristic},
Math. Res. Lett. \textbf{27} (2020), no.~4, 1003--1054.
\href{https://doi.org/10.4310/MRL.2020.v27.n4.a3}{doi:10.4310/MRL.2020.v27.n4.a3}.
\bibitem[JLR26]{JLR}
Eric Jovinelly, Brian Lehmann and Eric Riedl,
\emph{Optimal bounds in Bend-and-Break},
Forum Math. Pi \textbf{14} (2026), e16, 1--4.
\href{https://doi.org/10.1017/fmp.2026.10033}{doi:10.1017/fmp.2026.10033}.
\bibitem[Kaw92]{Kawamata}
Yujiro Kawamata, \emph{Abundance theorem for minimal threefolds},
Invent. Math. \textbf{108} (1992), 229--246.
\href{https://doi.org/10.1007/BF02100604}{doi:10.1007/BF02100604}.
\bibitem[Kee99]{Keel}
Se\'an Keel,
\emph{Basepoint freeness for nef and big line bundles in positive characteristic},
Ann. of Math. (2) \textbf{149} (1999), no.~1, 253--286.
\href{https://doi.org/10.2307/121025}{doi:10.2307/121025}.
\bibitem[KMM94]{KMM}
Sean Keel, Kenji Matsuki and James McKernan,
\emph{Log abundance theorem for threefolds},
Duke Math. J. \textbf{75} (1994), no.~1, 99--119.
\href{https://doi.org/10.1215/S0012-7094-94-07504-2}{doi:10.1215/S0012-7094-94-07504-2}.
\bibitem[KMM04]{KMMCorrection}
Sean Keel, Kenji Matsuki and James McKernan,
\emph{Corrections to: ``Log abundance theorem for threefolds''},
Duke Math. J. \textbf{122} (2004), no.~3, 625--630.
\href{https://doi.org/10.1215/S0012-7094-04-12236-5}{doi:10.1215/S0012-7094-04-12236-5}.
\bibitem[KM98]{KM}
J\'anos Koll\'ar and Shigefumi Mori,
\emph{Birational Geometry of Algebraic Varieties},
Cambridge Tracts in Mathematics, vol.~134, Cambridge University Press,
Cambridge, 1998; with the collaboration of C.~H.~Clemens and A.~Corti.
\href{https://doi.org/10.1017/CBO9780511662560}{doi:10.1017/CBO9780511662560}.
\bibitem[Miy88]{Miyaoka}
Yoichi Miyaoka, \emph{Abundance conjecture for 3-folds: case $\nu=1$},
Compositio Math. \textbf{68} (1988), no.~2, 203--220.
\href{https://www.numdam.org/item/CM_1988__68_2_203_0/}{Numdam: CM\_1988\_\_68\_2\_203\_0}.
\bibitem[OAI26]{Companion}
OpenAI,
\emph{Abundance in numerical dimension one for terminal threefolds in positive characteristic},
OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026/paper.pdf}{OAI:Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026},
2026.
\bibitem[Pos23]{Posva}
Quentin Posva, \emph{Abundance for slc surfaces over arbitrary fields},
\'Epijournal G\'eom. Alg\'ebrique \textbf{7} (2023), article no.~5, 23~pp.
\href{https://doi.org/10.46298/epiga.2023.volume7.8803}{doi:10.46298/epiga.2023.volume7.8803}.
\bibitem[Tan16]{Tanaka}
Hiromu Tanaka,
\emph{Abundance theorem for semi log canonical surfaces in positive characteristic},
Osaka J. Math. \textbf{53} (2016), no.~2, 535--566.
\href{https://doi.org/10.18910/58879}{Osaka repository: doi:10.18910/58879}.
\bibitem[Wal18]{Waldron}
Joe Waldron, \emph{The LMMP for log canonical 3-folds in characteristic $p>5$},
Nagoya Math. J. \textbf{230} (2018), 48--71.
\href{https://doi.org/10.1017/nmj.2017.2}{doi:10.1017/nmj.2017.2}.
\bibitem[Wit21]{Witaszek}
Jakub Witaszek,
\emph{On the canonical bundle formula and log abundance in positive characteristic},
Math. Ann. \textbf{381} (2021), no.~3--4, 1309--1344.
\href{https://doi.org/10.1007/s00208-021-02231-5}{doi:10.1007/s00208-021-02231-5}.
\bibitem[XZ19]{XZ}
Chenyang Xu and Lei Zhang,
\emph{Nonvanishing for 3-folds in characteristic $p>5$},
Duke Math. J. \textbf{168} (2019), no.~7, 1269--1301.
\href{https://doi.org/10.1215/00127094-2018-0066}{doi:10.1215/00127094-2018-0066}.
\bibitem[Xu24]{XuNonvanishing}
Zheng Xu,
\emph{Note on the three-dimensional log canonical abundance in characteristic $>3$},
Nagoya Math. J. \textbf{255} (2024), 694--723.
\href{https://doi.org/10.1017/nmj.2024.3}{doi:10.1017/nmj.2024.3}.
Theorem numbering follows
\href{https://arxiv.org/abs/2212.04039v2}{arXiv:2212.04039v2}, February~3, 2024.
\bibitem[Xu26]{XuNuTwo}
Zheng Xu, \emph{Abundance for threefolds in positive characteristic when $\nu=2$},
Compositio Math. \textbf{162} (2026), no.~9, 2065--2099.
\href{https://doi.org/10.1017/S0010437X26103406}{doi:10.1017/S0010437X26103406}.
Theorem numbering follows
\href{https://arxiv.org/abs/2307.03938v3}{arXiv:2307.03938v3}, April~17, 2026.
\end{thebibliography}
==== END Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//bibliography.tex ====
==== BEGIN Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//boundary.tex ====
\section{Separating the boundary from the log canonical cycle}
\label{sec:boundary}
The numerical relation $N_0\sim m(K_X+B)$ initially controls the log
canonical divisor, whereas Theorem~\ref{thm:obstruction} requires a
relation for a power of the canonical bundle. We now show that the
boundary contributes only components of the cycle or divisors entirely
disjoint from it. The argument uses maximal nef dimension and
bend-and-break, and is independent of the subsequent MMP.
\begin{proposition}[Boundary separation]\label{prop:separation}
Let $X$ be a normal $\Q$-factorial projective threefold over an
uncountable algebraically closed field, and let $B$ be an effective
$\Q$-divisor. Suppose that $N$ is a nonzero effective nef Cartier
divisor with $\nu(N)=1$, $n(N)=3$, and
$N\sim_{\Q}a(K_X+B)$ for some rational $a>0$. For every ample Cartier
divisor $H$, one has
\begin{equation}\label{eq:boundary-zero}
BNH=0.
\end{equation}
Consequently $N$ is numerically trivial on each prime component of $B$,
and every such component is either contained in $\Supp N$ or disjoint
from $\Supp N$.
\end{proposition}
\begin{proof}
We may replace $H$ by a very ample multiple. The number $BNH$ is
nonnegative. If it were positive, the identity
$(K_X+B)NH=N^2H/a=0$ would give $K_XNH<0$.
For each sufficiently large integer $l$, choose positive integers
$u_l,v_l$ such that $u_lH$ and $v_l(H+lN)$ are very ample. Their general
complete intersection is an integral curve $C_l$ contained in the
smooth locus of $X$. Indeed, normality gives a singular locus of
dimension at most one; the first general member meets this locus in
finitely many points, and the second avoids them. Bertini's theorem on
the smooth open gives the desired integral curve. Its intersections
satisfy
\begin{equation}\label{eq:bend-ratio}
\frac{NC_l}{-K_XC_l}
=\frac{NH^2}{-K_XH^2-lK_XHN}\longrightarrow0,
\end{equation}
and $-K_XC_l>0$ for large $l$.
The quantitative bend-and-break theorem of Jovinelly, Lehmann, and
Riedl \cite[Theorem~1.1]{JLR} applies to an integral curve in the smooth
locus of a projective variety over an algebraically closed field, with
negative canonical degree, and any nef Cartier test divisor. In
dimension three it supplies, through every closed point of $C_l$, a
rational curve $\Gamma$ satisfying
\[
0\leq N\Gamma\leq4\frac{NC_l}{-K_XC_l}.
\]
Fix $l$ so large that the right-hand side is strictly less than one.
Since $N$ is Cartier, it follows that $N\Gamma=0$.
For completeness, these complete intersections pass through points to
which the very-general-point condition applies. For the two fixed very
ample systems, the incidence of a point of $X$ and a pair of members
through it is a product of projective bundles over $X$, hence is
irreducible and dominates $X$. Restricting to the nonempty open set of
pairs giving the curves just considered leaves a dense open incidence
and a dominant evaluation map. Its image is constructible and contains
a nonempty open subset of $X$. Over the uncountable field, this open
contains a closed point outside the countable union excluded by
$n(N)=3$. The curve $\Gamma$ through that point is a contradiction.
This proves \eqref{eq:boundary-zero}.
Write $B=\sum b_TT$ with $b_T>0$. All numbers $NHT$ are nonnegative,
so \eqref{eq:boundary-zero} gives $NHT=0$ for each $T$.
Lemma~\ref{lem:surface-nef} yields $N|_T\equiv0$. If $T$ is not a
component of $N$, the canonical section of $\mathcal O_X(N)$ restricts
to a nonzero section on the integral surface $T$. Were $T\cap\Supp N$
nonempty, its zero scheme would be a nonzero effective Cartier divisor
on $T$ and would have positive ample degree. This contradicts
$N|_T\equiv0$, so the two supports are disjoint.
\end{proof}
Apply Proposition~\ref{prop:separation} to $N_0$ and set
\begin{equation}\label{eq:Pout}
P_{\mathrm{out}}=
\bigcup_{\substack{T\text{ a prime component of }B\\T\not\subset\Supp N_0}}T.
\end{equation}
This is a closed subset disjoint from $\Supp N_0$. Thus
$\Supp B\subset\Supp N_0\cup P_{\mathrm{out}}$, a union of two
disjoint closed sets. We retain this separation while isolating one
component of $N_0$.
==== END Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//boundary.tex ====
==== BEGIN Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//cycle.tex ====
\section{Construction of the prepared cycle}\label{sec:cycle}
We now combine the isolated boundary with the original log canonical
section. The boundary supplies a divisor with torsion normal line;
the section supplies a canonical relation near its support. A finite
cover removes the prime-to-$p$ part of the canonical exponent. This
is the preparation of \cite[Sections~2.7--2.8]{Companion}, with the
support of the original boundary retained throughout the construction.
\begin{proposition}\label{prop:prepared}
Let $(X,B,N_0)$ be the data of Proposition~\ref{prop:reduction}
in the maximal-nef-dimension case. There exist
a smooth projective threefold $V$, a very ample divisor $H_V$, and a
nonzero effective Cartier divisor
\[
D=\sum_\alpha m_\alpha D_\alpha
\]
having all the properties in Theorem~\ref{thm:obstruction}. In
particular, its support is connected with simple normal crossings,
$\gcd_\alpha(m_\alpha)=1$, and $\mathcal O_D(D)$ is torsion.
\end{proposition}
\begin{proof}
Equation~\eqref{eq:restriction-zero} and
Proposition~\ref{prop:separation} give the hypotheses of
Proposition~\ref{prop:isolation}. Choose its isolated model
$(Y,G_Y,N_Y,S_Y)$ and the reduced dlt modification
$f:(Y_d,\Sigma)\to(Y,S_Y)$ of Corollary~\ref{lem:dlt-boundary}.
Let $M$ be the positive Cartier multiple of $f^*S_Y$ supplied by
Proposition~\ref{prop:boundary-torsion}.
\emph{The canonical section and its support.}
Let $V_0$ be a smooth projective common resolution of $X$ and $Y_d$,
chosen to dominate the intervening models. Write
\[
q:V_0\longrightarrow X,\qquad h_0:V_0\longrightarrow Y_d,
\qquad r=f\circ h_0:V_0\longrightarrow Y.
\]
These models share the open set
\[
U_X=X_{\mathrm{reg}}\setminus
(\operatorname{Supp}N_0\cup P_{\mathrm{out}}),
\]
where $P_{\mathrm{out}}$ is the union of the boundary components
disjoint from $\operatorname{Supp}N_0$. We choose the common
resolution to be an isomorphism over this open. Projective resolution
and resolution of divisor supports in dimension three permit this
choice \cite{CP,CJS}. The pullback equalities through the partial
MMP give
\begin{equation}\label{eq:cycle-common-pullbacks}
q^*N_0=r^*N_Y.
\end{equation}
The integer $m$ was chosen so that
$N_0\sim m(K_X+B)$ and both $mK_X$ and $mB$ are Cartier. It gives a
rational section $\theta$ of $\omega_{V_0}^{\otimes m}$ with
\begin{equation}\label{eq:log-canonical-section}
\operatorname{div}(\theta)
=q^*N_0+m(K_{V_0}-q^*K_X)-m q^*B.
\end{equation}
Indeed, the right side is an integral divisor linearly equivalent to
$mK_{V_0}$. Put $E_0=\operatorname{Supp}q^*N_0$. Every prime
component in \eqref{eq:log-canonical-section} is either contained in
$E_0$ or disjoint from it. For $q^*B$ this follows from boundary
separation on $X$. For a $q$-exceptional divisor, its irreducible
center lies in
\[
X\setminus U_X
=\operatorname{Supp}N_0\cup P_{\mathrm{out}}
\cup\operatorname{Sing}X.
\]
A terminal threefold is regular in codimension two, so
$\operatorname{Sing}X$ is finite
\cite[Lemma~2.3]{Companion}. The center is therefore contained in
$\operatorname{Supp}N_0$, or is disjoint from it. Taking inverse
images gives the assertion for the exceptional divisor.
Let $F_0$ be the union of the components of
$\operatorname{div}(\theta)$ outside $E_0$. Thus
\begin{equation}\label{eq:cover-closed-set}
Z_0:=E_0\cup\operatorname{Supp}\operatorname{div}(\theta)
=E_0\sqcup F_0.
\end{equation}
The disjointness is the point of this description. Components of
$q^*N_0$ can cancel in \eqref{eq:log-canonical-section}; we retain all
of $E_0$ when choosing the open set preserved by the subsequent
resolution.
\emph{Removing the prime-to-$p$ exponent.}
Write $m=p^b m'$ with $b\geq0$ and $(m',p)=1$. Choose a nonzero
rational generator $\eta$ of $\omega_{V_0}^{\otimes p^b}$ and write
$\theta=u\eta^{m'}$, where $u\in k(V_0)^\times$. Adjoin one root
$v$ of $v^{m'}=u$ and normalize $V_0$ in the resulting field.
This is a finite separable cover, possibly of degree smaller than
$m'$. Over $V_0\setminus\operatorname{Supp}\operatorname{div}(\theta)$,
it is an integral component of the torsor of $m'$th roots of the
invertible section $\theta$. This description is independent of the
rational generator $\eta$. Since $m'$ is invertible in $k$, that
torsor is finite \'etale.
Resolve the normalization and the total inverse image of $Z_0$,
preserving the regular \'etale open over $V_0\setminus Z_0$. We
obtain a smooth projective threefold and a generically finite
separable morphism
$\pi:V\to V_0$ for which the total pullback support of $N_0$ is
simple normal crossings. The morphisms used to compare divisors are
\[
\begin{tikzcd}[column sep=large,row sep=large]
V\arrow[r,"\pi"] &V_0\arrow[r,"q"]\arrow[d,"h_0"']&X\\
&Y_d\arrow[r,"f"']&Y.
\end{tikzcd}
\]
Here $q,h_0,f$ are birational; $\pi$ is generically finite and
separable. Set $\rho=q\circ\pi$ and $h=h_0\circ\pi$. By
\eqref{eq:cycle-common-pullbacks}, $\rho^*N_0=(f\circ h)^*N_Y$,
while the Cartier divisor supplied by
Proposition~\ref{prop:boundary-torsion} pulls back as $h^*M$.
The rational section $v\eta$ is a section of
$\pi^*\omega_{V_0}^{\otimes p^b}$. The determinant of the
differential is a nonzero section of the rational line
$\omega_V\otimes\pi^*\omega_{V_0}^{-1}$; denote it by $J_\pi$.
Its nonvanishing at the generic point follows from separability.
Consequently
\begin{equation}\label{eq:canonical-root-section}
\psi=(v\eta)J_\pi^{p^b}
\quad\text{is a rational section of }
\omega_V^{\otimes p^b}.
\end{equation}
Both factors are invertible on $\pi^{-1}(V_0\setminus Z_0)$: the first is
a root of an invertible section, and the second is the Jacobian of
an \'etale morphism. Thus the divisor of $\psi$ is supported on
$\pi^{-1}(Z_0)$. By \eqref{eq:cover-closed-set}, each of its prime
components lies in $\operatorname{Supp}\rho^*N_0$ or is disjoint
from that support.
\emph{A connected primitive Cartier divisor.}
Put $M_V=h^*M$ and choose a connected component $C$ of
$\operatorname{Supp}M_V$. Since $M$ is a positive multiple of
$f^*S_Y$, and $S_Y$ is disjoint from the other components of
$N_Y$, the set $C$ is also a connected component of
$\operatorname{Supp}\rho^*N_0$. Write the part of $M_V$ supported
on $C$ as
\begin{equation}\label{eq:primitive-cycle}
aD,\qquad D=\sum_\alpha m_\alpha D_\alpha,
\qquad a=\gcd_{D_\alpha\subset C}
\operatorname{mult}_{D_\alpha}(M_V).
\end{equation}
The divisor $D$ is Cartier because $V$ is smooth, and its
multiplicities are positive with gcd one. Its support is connected
and simple normal crossings.
Every curve in $\operatorname{Supp}M_V$ maps under $h$ to a point or
to a curve in $\Sigma$. The projection formula and
Proposition~\ref{prop:boundary-torsion} therefore show that $M_V$
is numerically trivial on every component of its support. Near $C$
we have $M_V=aD$, so $D|_{D_\alpha}\equiv0$ for every $\alpha$.
For a curve not contained in $\operatorname{Supp}D$, effectivity
gives nonnegative intersection with $D$. For a curve in its support,
the intersection is zero. Thus $D$ is nef, and every very ample
divisor $H_V$ satisfies
\begin{equation}\label{eq:primitive-intersections}
D H_V^2>0,\qquad D^2H_V=0.
\end{equation}
Torsion also survives on the entire Cartier scheme. Choose $t>0$
with $\mathcal O_M(tM)\simeq\mathcal O_M$. The pullback divisor
$M_V$ is scheme-theoretically $V\times_{Y_d}M$: its local equation
is the pullback of a local equation of $M$, which is a nonzero
divisor since $V$ is integral and $h$ is dominant. Pulling back
the trivialization gives
$\mathcal O_{M_V}(tM_V)\simeq\mathcal O_{M_V}$; no flatness
assumption on $h$ is needed. Restricting along $D\subset M_V$,
and using $M_V=aD$ near $C$, gives
\begin{equation}\label{eq:primitive-torsion}
\mathcal O_D(taD)\simeq\mathcal O_D.
\end{equation}
Hence $\mathcal O_D(D)$ is torsion.
The section $\psi$ of \eqref{eq:canonical-root-section} has, on
a neighborhood $W$ of $C$, divisor supported on $D$. Indeed, its
other components are disjoint from $C$ and can be removed from
that neighborhood. Thus there are integers $b_\alpha$ such that
\begin{equation}\label{eq:prepared-canonical-relation}
\omega_V^{\otimes p^b}|_W
\simeq\mathcal O_W\Bigl(\sum_\alpha b_\alpha D_\alpha\Bigr).
\end{equation}
\emph{Numerical proportionality and very general points.}
It remains to show that no $D$-trivial curve passes through a very
general point of $V$. Put $N_V=\rho^*N_0$. It is a nonzero nef
divisor, numerically trivial on every component of its support by
the projection formula. Since $\operatorname{Supp}D$ is contained
in that support, we have
\[
N_VH_V^2>0,\qquad N_V^2H_V=N_VDH_V=0.
\]
Together with \eqref{eq:primitive-intersections}, the Hodge index
theorem gives
\begin{equation}\label{eq:cycle-proportionality}
D\equiv cN_V,\qquad
c=\frac{DH_V^2}{N_VH_V^2}>0.
\end{equation}
We verify that this is numerical equivalence on every curve, rather
than only on a fixed hyperplane section, following
\cite[Section~2.8]{Companion}.
Let $\Gamma\subset V$ be any integral curve. For $l\gg0$, there
is an integral surface $T\in|lH_V|$ containing $\Gamma$. To see
this, twist the ideal of $\Gamma$ sufficiently far and then multiply
its generators by very ample sections. The resulting linear system
has no fixed divisorial component and gives a birational map onto
its image off $\Gamma$. Bertini irreducibility gives an irreducible
general member, which is generically reduced on that birational open.
As a Cartier divisor on smooth $V$, it has no embedded components,
and is therefore integral.
On a smooth projective resolution $\widetilde T\to T$,
let $d,n,e$ be the pullbacks of $D|_T,N_V|_T,H_V|_T$, respectively.
Their intersections are
\[
d^2=n^2=dn=0,\qquad e^2>0,\qquad
de=lDH_V^2,\quad ne=lN_VH_V^2.
\]
Hence $d-cn$ has square zero and is orthogonal to $e$. The
Hodge index form is negative definite on the orthogonal complement
of the positive-square class $e$, so $d-cn$ is numerically zero.
The projective surjection $\widetilde T\to T$ supplies an integral
curve dominating $\Gamma$: take the closure of a closed point of
the fiber over its generic point. The projection formula then gives
$(D-cN_V)\cdot\Gamma=0$. This proves
\eqref{eq:cycle-proportionality}.
Finally let $Z_i\subset X$ be the countably many proper closed
subsets excluded by the nef reduction of $N_0$. Remove from $V$
the non-quasi-finite locus of $\rho$ and all the sets $\rho^{-1}(Z_i)$.
These are proper closed subsets, since $\rho$ is dominant and
generically finite. Their complement contains very general points
because $k$ is uncountable. Every curve through any point of this
complement maps to a curve on $X$. If it were $D$-trivial, then
\eqref{eq:cycle-proportionality} and the projection formula would
make its image $N_0$-trivial, contradicting maximal nef dimension.
All properties in Theorem~\ref{thm:obstruction} have now been
established.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Theorem~\ref{thm:obstruction} contradicts
Proposition~\ref{prop:prepared}. The maximal-nef-dimension case is
therefore impossible. By Proposition~\ref{prop:reduction}, it remains
to apply abundance in nef dimension at most two
\cite[Theorem~1.8]{XuNonvanishing}. This gives semiampleness, and
faithful descent along the algebraically closed field extension gives
it over the original field.
\end{proof}
==== END Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//cycle.tex ====
==== BEGIN Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//foundations.tex ====
\section{The cycle obstruction and the initial reductions}
\label{sec:foundations}
We first state the precise input from the companion paper. We then reduce
Theorem~\ref{thm:main} to constructing the forbidden cycle. Throughout, a
variety is integral. Boundaries are effective $\Q$-divisors, and a pair
$(X,B)$ has $X$ normal and $K_X+B$ $\Q$-Cartier. We use the usual discrepancy
definitions of log canonical (lc), Kawamata log terminal (klt), divisorial
log terminal (dlt), and terminal singularities; see \cite{KM}. Canonical
divisors on birational models are chosen compatibly. Restrictions and
intersections of $\Q$-Cartier divisors are computed using Cartier multiples.
\subsection{The obstruction supplied by the companion}
For a nef divisor $L$ over an uncountable algebraically closed field, its
\emph{nef dimension} $n(L)$ is the dimension of the target of its nef
reduction. We use only the following characterization in dimension three:
$n(L)=3$ if and only if no curve of $L$-degree zero passes through a very
general point. Here ``very general'' means outside a countable union of
proper closed subsets. These properties and existence of nef reduction are
recalled in \cite[Definition~2.1 and Theorem~2.2]{XuNuTwo}. Numerical
dimension and nef dimension need not agree.
\begin{theorem}[Cycle obstruction from the companion]
\label{thm:obstruction}
Let $k$ be an uncountable algebraically closed field of characteristic
$p>3$. There do not exist a smooth projective threefold $V$, a very ample
Cartier divisor $H_V$, and a nonzero effective Cartier divisor
$D=\sum_\alpha m_\alpha D_\alpha$ satisfying all the following conditions:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The $D_\alpha$ are distinct prime divisors, $m_\alpha\in\Z_{>0}$,
the support of $D$ is
connected and has simple normal crossings, and $\gcd_\alpha m_\alpha=1$.
\item $D$ is nef, its restriction to every $D_\alpha$ is numerically
trivial, and
\[
DH_V^2>0,\qquad D^2H_V=0,\qquad n(D)=3.
\]
\item The line bundle $\mathcal O_D(D)$ on the possibly nonreduced
Cartier scheme $D$ is torsion.
\item For some integer $b\geq0$ and integers $b_\alpha$, there is an
isomorphism on a neighborhood of $\Supp D$,
\begin{equation}\label{eq:canonical-cycle}
\omega_V^{\otimes p^b}\simeq
\mathcal O_V\Bigl(\sum_\alpha b_\alpha D_\alpha\Bigr).
\end{equation}
\end{enumerate}
\end{theorem}
This is the obstruction proved in \cite[Sections~3--6]{Companion} for the
data listed in its Proposition~2.1. The end of Section~2 of that manuscript
explicitly retains only $V,D,H_V$, the torsion order, and the local canonical
relation. Under these hypotheses, its Sections~3--5 prove Corollary~5.4,
and its Proposition~6.8 gives the contradiction. Thus the obstruction
applies to any cycle with the displayed properties. In particular, the
construction below may start from a log canonical divisor with boundary.
The companion uses the torsion normal line and the local canonical
relation to control differential forms on the completion along $D$ with
poles allowed along $D$. Cartier descent then makes the Picard group of
that punctured formal neighborhood $p$-divisible. The numerical conditions
give a nonzero integer-valued degree homomorphism on the same group,
which contradicts that divisibility. We use this obstruction as a theorem;
the present proof constructs all its input data for a log canonical pair.
The canonical relation also implies an intersection identity that occurs
in the companion. Indeed, a rational section giving
\eqref{eq:canonical-cycle} has global divisor
$\sum b_\alpha D_\alpha+E$, with $\Supp E$ disjoint from $\Supp D$.
Since $D|_{D_\alpha}\equiv0$, for every Cartier divisor $A$ on $V$ we have
\begin{equation}\label{eq:canonical-intersection}
K_VDA=0.
\end{equation}
\subsection{Numerical observations}
The following elementary observation will convert zero intersection
numbers into restrictions that are numerically trivial.
\begin{lemma}\label{lem:surface-nef}
Let $T$ be an integral projective surface, $H$ an ample Cartier divisor,
and $P$ a nef $\Q$-Cartier divisor on $T$. If $PH=0$, then $P\equiv0$.
Moreover, a nonzero effective Cartier divisor on $T$ has positive
intersection with $H$.
\end{lemma}
\begin{proof}
For any integral curve $C\subset T$, a sufficiently large multiple of $H$
has an effective Cartier member $E$ containing $C$. The effective
one-cycle of $E$ has positive coefficient along $C$. Every component has
nonnegative $P$-degree, whereas $PE=0$, so $PC=0$. This proves the first
assertion. The second is positivity of the ample degree of each component
of the effective one-cycle of a nonzero Cartier divisor.
\end{proof}
In particular, if $N$ is a nonzero effective nef Cartier divisor on a
projective threefold and $N^2H=0$ for an ample Cartier divisor $H$, then
\begin{equation}\label{eq:restriction-zero}
N|_T\equiv0\qquad\text{for every prime component }T\text{ of }N.
\end{equation}
Indeed, writing $N=\sum a_TT$, the nonnegative numbers $NHT$ have
positive-coefficient sum $N^2H=0$, and Lemma~\ref{lem:surface-nef} applies.
\subsection{Reduction to a terminal underlying variety}
We use three results of Xu for projective lc threefolds in characteristic
greater than three: nonvanishing for a pseudoeffective log canonical
divisor, termination of its flips in the pseudoeffective case, and
abundance when the divisor is nef of nef dimension at most two
\cite[Theorems~1.4, 1.5, and~1.8]{XuNonvanishing}. All permit rational
boundaries. The relative MMP and crepant $\Q$-factorial dlt modifications
are available in this characteristic range, including characteristic
five \cite[Theorem~1.2]{HW}, \cite[Theorems~2.20 and~2.27]{XuNuTwo}.
We also use projective resolution of threefolds, preserving the regular
locus, and resolution of divisor supports, preserving their complement
on a regular variety; the required forms of
these results are recalled in \cite[Section~2.1]{Companion}, from
\cite{CPFirst,CP,CJS}.
\begin{proposition}\label{prop:reduction}
To prove Theorem~\ref{thm:main}, it suffices to rule out the following
data: an uncountable algebraically closed field $k$ of characteristic
$p>3$, a terminal $\Q$-factorial projective threefold $X$, an lc pair
$(X,B)$, and a nonzero effective Cartier divisor
\begin{equation}\label{eq:N0}
N_0\sim m(K_X+B),\qquad m\in\Z_{>0},
\end{equation}
such that $mK_X$ and $mB$ are Cartier, $N_0$ is nef,
$\nu(N_0)=1$, and $n(N_0)=3$.
\end{proposition}
\begin{proof}
First extend the given field to an uncountable algebraically closed
field. Integrality and normality persist, and a log resolution with its
discrepancy calculation proves that the base-changed pair remains lc.
The canonical divisor can be defined by a rational top form on the
smooth locus, so the log canonical class also base changes. Nefness
persists by ample approximation, and the prescribed intersection numbers
are unchanged.
Semiampleness descends under this extension. After making a generated
multiple also divisible by the original Cartier index, it is the base
change of a line bundle on the original variety. Global sections commute
with field extension, and faithful flatness detects surjectivity of its
evaluation map. We may therefore work over the enlarged field.
Write $L=K_X+B$ for the original log canonical class. If $n(L)\leq2$,
Xu's abundance theorem applies. Suppose $n(L)=3$. Take a crepant
projective $\Q$-factorial dlt modification $(X_d,B_d)$, and resolve $X_d$
projectively. Run a relative canonical MMP of this smooth resolution over
$X_d$. The MMP with scaling applies by \cite[Theorem~1.2]{HW}. Its output
cannot be of fibre type over the three-dimensional birational base, and
it gives a projective birational morphism
\[
g:X_t\longrightarrow X_d
\]
with $X_t$ terminal and $\Q$-factorial and $K_{X_t}$ nef over $X_d$.
Here terminality is preserved by a canonical MMP: discrepancies do not
decrease, and a divisor contracted in a negative divisorial step has
strictly positive ordinary discrepancy over the output. Relative pseudoeffectivity
causes no restriction for this birational morphism; adding a sufficiently
large pullback of an ample divisor makes the canonical class big.
The exceptional divisor $F=K_{X_t}-g^*K_{X_d}$ is $g$-nef. The negativity
lemma gives $F\leq0$. Since $X_d$ is $\Q$-factorial, the divisor
\[
B_t=g^*B_d-F
\]
is effective and satisfies
$K_{X_t}+B_t=g^*(K_{X_d}+B_d)$. Crepancy makes $(X_t,B_t)$ lc.
The new log canonical class is nef of numerical dimension one. To check
the square after pullback, observe first that $L^2T=0$ for every prime
surface $T$ on the original variety: place $T$ in an effective member of
a multiple of the ample divisor in the hypothesis, and use nonnegativity
of $L^2$ on all its components. Projection formula proves the vanishing
upstairs. Numerical nontriviality persists because a curve downstairs
can be dominated by a curve upstairs. Finally, the no-zero-degree-curve
property persists over the birational isomorphism locus. Thus the new
class still has nef dimension three.
Relabel this pair as $(X,B)$. By nonvanishing
\cite[Theorem~1.4]{XuNonvanishing}, a positive multiple of $K_X+B$ has an
effective representative $N_0$. It is nonzero because $K_X+B$ is not
numerically trivial. Increasing the multiple makes $N_0$, $mK_X$, and
$mB$ Cartier, as required.
\end{proof}
From now until the proof of Theorem~\ref{thm:main}, we work with the
data of Proposition~\ref{prop:reduction}. Fix a very ample divisor $H$ on
$X$. Then $N_0H^2>0$, $N_0^2H=0$, and
\eqref{eq:restriction-zero} holds. The next step is to establish the same
numerical triviality on the boundary components.
==== END Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//foundations.tex ====
==== BEGIN Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//introduction.tex ====
\section{Introduction}
\label{sec:introduction}
The abundance conjecture predicts that a nef log canonical divisor
$K_X+B$ on a projective log canonical pair is semiample: some positive
Cartier multiple is generated by its global sections. Semiampleness
turns the numerical positivity of the divisor into a morphism. We
consider threefolds in positive characteristic and the case of numerical
dimension one. For a nef $\Q$-Cartier divisor $L$ on a threefold, this
condition means that, for an ample Cartier divisor $H$,
\[
LH^2>0\qquad\text{and}\qquad L^2H=0.
\]
\begin{theorem}\label{thm:main}
Let $k$ be an algebraically closed field of characteristic $p>3$.
Let $(X,B)$ be a projective log canonical threefold pair over $k$,
where $X$ is normal, $B$ is an effective $\Q$-divisor, and $K_X+B$
is $\Q$-Cartier. If $K_X+B$ is nef and has numerical dimension one,
then $K_X+B$ is semiample.
\end{theorem}
The boundary coefficients are arbitrary rational numbers allowed by
log canonicity, and neither terminality nor $\Q$-factoriality is assumed
for the original variety. The theorem proves the numerical-dimension-one
case of threefold log abundance in this characteristic range.
Over the complex numbers, Miyaoka proved abundance for minimal
threefolds of numerical dimension one \cite{Miyaoka}; Kawamata completed
abundance for minimal threefolds and gave another proof of this case
\cite[Section~4]{Kawamata}. Keel, Matsuki, and McKernan established
log abundance for log canonical threefold pairs \cite{KMM,KMMCorrection}. In positive
characteristic, Keel's semiampleness criterion \cite{Keel} and the
basepoint-freeness methods of Cascini, Tanaka, and Xu \cite{CTX}
provided foundations. The threefold minimal model program
developed through the work of Hacon--Xu, Birkar, and Birkar--Waldron in
characteristic greater than five \cite{HX,Birkar,BW}, the log canonical
and relative extensions of Waldron and Hashizume--Nakamura--Tanaka
\cite{Waldron,HNT}, and Hacon--Witaszek's treatment of characteristic
five \cite{HW}.
For abundance, Xu--Zhang proved nonvanishing for terminal minimal
threefolds in characteristic greater than five \cite{XZ}. Witaszek
obtained klt nonvanishing and abundance in nef dimension at most two
in that characteristic range \cite{Witaszek}. The nef dimension $n(L)$
is the dimension of the nef reduction of $L$. On a threefold over an
uncountable algebraically closed field, $n(L)=3$ means that no curve of $L$-degree zero passes
through a very general point. Zheng Xu established nonvanishing for pseudoeffective log
canonical divisors on projective lc threefold pairs in characteristic
greater than three, together with
abundance in nef dimension at most two
\cite[Theorems~1.4 and~1.8]{XuNonvanishing}; he subsequently proved
abundance in numerical dimension two \cite[Theorem~1.1]{XuNuTwo}.
For the present problem, the remaining case has numerical dimension
one and nef dimension three.
The companion paper \cite{Companion} treats terminal threefolds with
nef canonical divisor of numerical dimension one. Its proof isolates
a stronger structural result: certain connected nef Cartier divisors
on smooth threefolds cannot exist. We state this cycle obstruction
in Theorem~\ref{thm:obstruction}, with all its hypotheses. The present
paper constructs such a divisor from a log canonical pair with
arbitrary rational boundary. The companion's obstruction is an
essential input to the proof.
\subsection*{The boundary and the proof strategy}
After an uncountable field extension and a crepant birational change,
we may suppose that the underlying variety is terminal and
$\Q$-factorial. Nonvanishing gives a nonzero effective Cartier divisor
$N_0\sim m(K_X+B)$. We assume for contradiction that $n(N_0)=3$.
The main additional issue is to obtain a relation for a power of the
canonical bundle near a component of this \emph{log} canonical divisor.
The boundary and the exceptional divisors of a resolution must both
be controlled there.
Proposition~\ref{prop:separation} supplies that control for the boundary:
it proves $BN_0H=0$ and shows that every boundary component is either
contained in $\Supp N_0$ or disjoint from it. If the intersection were
positive, complete-intersection curves would have arbitrarily small
ratios of $N_0$-degree to anticanonical degree. Quantitative
bend-and-break \cite[Theorem~1.1]{JLR} would then produce
$N_0$-trivial rational curves through very general points. This
separation argument is formulated independently of the MMP.
Next we adapt the isolation construction of
\cite[Section~2.3]{Companion}. Raising the boundary to its log canonical
threshold along $N_0$ gives a coefficient-one component on a dlt model.
A partial MMP, trivial for the resulting full log canonical class,
makes this component disjoint from the rest of the divisor.
We use Xu's partial-MMP formulation
\cite[Section~2.3]{XuNuTwo}, whose ray argument adapts
\cite[Lemma~5.1]{KMM}. Reduced-boundary adjunction and surface
abundance \cite{Tanaka,Posva} then yield torsion on a Cartier
multiple of that component. We include the companion's extension
argument across finitely many points, since the required torsion is
on the full nonreduced Cartier scheme.
Finally, terminality makes the singular locus finite, so the exceptional
divisors of a common resolution obey the same support separation as
the boundary. A finite separable cover removes the prime-to-$p$ part
of the pluricanonical exponent. A connected component of the pulled-back
Cartier divisor, divided by the gcd of its multiplicities, has exactly
the numerical, torsion, and canonical properties forbidden by
Theorem~\ref{thm:obstruction}.
Section~\ref{sec:foundations} states the obstruction and makes the
initial reductions. Section~\ref{sec:boundary} proves boundary
separation, and Section~\ref{sec:isolation} isolates a component.
Section~\ref{sec:torsion} proves torsion on its Cartier thickening.
Section~\ref{sec:cycle} constructs the final cycle and completes the
proof.
==== END Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//introduction.tex ====
==== BEGIN Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//isolation.tex ====
\section{Isolating a log canonical component}\label{sec:isolation}
The effective divisor constructed above may have several intersecting
components. We now make one of them disjoint from the others, while
preserving the pullback of the nef divisor. The construction adapts
\cite[Section~2.3]{Companion}: raise the boundary to a log canonical
threshold, then lower the coefficient of one component and run an MMP
that is trivial for the restored log canonical divisor. The resulting
nefness forces the required disjointness.
\begin{proposition}\label{prop:isolation}
Let $k$ be an algebraically closed field of characteristic $p>3$.
Let $X$ be a terminal, $\mathbb Q$-factorial projective threefold, let
$(X,B)$ be log canonical with rational boundary, and let
\[
0\ne N_0\sim m(K_X+B)
\]
be an effective nef Cartier divisor, where $m$ is a positive integer.
Assume that $N_0$ is numerically trivial on each component of its
support, and that every component of $B$ is either contained in or
disjoint from $\operatorname{Supp}N_0$. Denote by $P_{\rm out}$ the union
of the latter components and put
\[
U_X=X_{\rm reg}\setminus
\bigl(\operatorname{Supp}N_0\cup P_{\rm out}\bigr).
\]
There exist a projective $\mathbb Q$-factorial klt threefold $Y$
birational to $X$, a rational boundary $G_Y$, an effective nef
$\mathbb Q$-Cartier divisor $N_Y$, and a prime divisor $S_Y$ with the
following properties.
\begin{enumerate}
\item The pair $(Y,G_Y)$ is log canonical, $S_Y$ has coefficient one
in $G_Y$, and $s:=\operatorname{mult}_{S_Y}N_Y$ is positive. For
rational numbers $\tau>0$ and $\delta>0$,
\begin{equation}\label{eq:isolation-nef}
K_Y+G_Y\sim_{\mathbb Q}\tau N_Y,
\qquad K_Y+G_Y-\delta S_Y\ \text{is nef}.
\end{equation}
\item The divisor $N_Y$ is numerically trivial on every component of
its support. Every component of $G_Y$ is either contained in or
disjoint from $\operatorname{Supp}N_Y$.
\item The divisor $S_Y$ is disjoint from every other component of
$N_Y$ and from $\operatorname{Supp}(G_Y-S_Y)$. In particular,
\begin{equation}\label{eq:isolation-local}
N_Y=sS_Y,\qquad G_Y=S_Y,\qquad
K_Y+S_Y\sim_{\mathbb Q}\tau sS_Y
\end{equation}
on a neighborhood of $S_Y$, and $S_Y|_{S_Y}\equiv0$.
\item The birational correspondence identifies $U_X$ with an open
subset of $Y$. On a smooth projective common resolution
$q:V_0\to X$, $r:V_0\to Y$ that is an isomorphism over this open,
\begin{equation}\label{eq:isolation-pullbacks}
q^*N_0=r^*N_Y.
\end{equation}
The same equality holds after passing to any further common resolution.
\end{enumerate}
One may take $\tau=m^{-1}+\lambda$, where
$\lambda=\operatorname{lct}_{(X,B)}(N_0)\geq0$.
\end{proposition}
\begin{proof}
We first find a coefficient-one boundary component lying over
$\operatorname{Supp}N_0$. On a log resolution of $(X,B+N_0)$, the log
canonical threshold is the minimum of the finitely many ratios
\[
\frac{a(E,X,B)}{\operatorname{ord}_E(N_0)}
\qquad\bigl(\operatorname{ord}_E(N_0)>0\bigr),
\]
where $a(E,X,B)$ denotes log discrepancy and the divisors considered
include the strict transforms of the components of $N_0$.
Thus $\lambda$ is finite and rational, possibly zero, and
$(X,B+\lambda N_0)$ is log canonical. There is a log canonical place
$E$ of this pair for which $\operatorname{ord}_E(N_0)>0$.
Take a projective crepant $\mathbb Q$-factorial dlt modification
\[
\mu:(W,G)\longrightarrow (X,B+\lambda N_0),
\]
with every extracted divisor having coefficient one; see
\cite[Theorem~2.27]{XuNuTwo}. Set
\[
N=\mu^*N_0,\qquad \tau=m^{-1}+\lambda.
\]
Then
\begin{equation}\label{eq:isolation-restored}
K_W+G\sim_{\mathbb Q}\tau N.
\end{equation}
The divisor $N$ is nef and numerically trivial on every component of
its support: a curve in that support maps to a point or to a curve in
$\operatorname{Supp}N_0$, and the assertion follows by the projection
formula.
Every component of $G$ is either contained in or disjoint from
$\operatorname{Supp}N$. For strict transforms of components of $B$
this follows from the hypothesis, while those coming from
$\lambda N_0$ are contained in the support. An extracted divisor has center in
$\operatorname{Supp}N_0\cup P_{\rm out}$, since $X$ is klt and the
boundary vanishes outside this union. Its irreducible center lies in
one of these two disjoint closed sets. In the first case its order
on $N_0$ is positive, and in the second case the divisor is disjoint
from $\operatorname{Supp}\mu^*N_0$.
The center on $W$ of the place $E$ lies in $\operatorname{Supp}N$ and,
by crepancy and the dlt property, in $\lfloor G\rfloor$. Choose a
component $S$ of $\lfloor G\rfloor$ containing this center.
The preceding separation then implies
$S\subset\operatorname{Supp}N$. This argument also covers
$\lambda=0$; it does not require all components of $N$ to occur in
$G$.
The modification $\mu$ is an isomorphism outside
$\operatorname{Supp}N_0\cup P_{\rm out}$. Indeed it has no exceptional
divisors there, and a projective small birational morphism to a
$\mathbb Q$-factorial variety is an isomorphism. To recall the latter
fact, push down a relatively ample Cartier divisor. A Cartier
multiple of its pushdown pulls back to the same multiple upstairs,
because there are no exceptional divisors. It has degree zero on
every contracted curve, contradicting relative ampleness if such a
curve exists.
Choose a rational $\epsilon>0$ sufficiently small that
\begin{equation}\label{eq:isolation-effective}
G-\epsilon S\geq0,\qquad \tau N-\epsilon S\geq0.
\end{equation}
Apply the partial MMP with lowered boundary $G-\epsilon S$ and
restoration divisor $\epsilon S$
\cite[Proposition~2.21, Definition~2.23, and Lemma~2.24]{XuNuTwo}.
Its steps are $(K_W+G-\epsilon S)$-negative and $(K_W+G)$-trivial.
The lowered pair is dlt, and its log canonical divisor is
pseudoeffective by \eqref{eq:isolation-restored} and
\eqref{eq:isolation-effective}. The termination required here is
therefore supplied by \cite[Theorem~1.5]{XuNonvanishing}.
The threefold MMP package used in this application, including in
characteristic five, is recalled in \cite[Section~2.3]{Companion}.
The partial-MMP conclusion is either a Mori fiber contraction for the
lowered log divisor, trivial for the restored divisor, or nefness of the
restored divisor minus $\delta$ times the transform of $S$ for every
sufficiently small rational $\delta>0$. We first track $S$, the pullbacks
of $N$, and support separation through the steps, and then exclude the
fiber-type alternative. Write $W_i\dashrightarrow W_{i+1}$
for a step, with transforms $G_i,N_i,S_i$. The equality
$K_{W_i}+G_i\sim_{\mathbb Q}\tau N_i$ persists by pushforward, so its
contracted ray $R_i$ satisfies
\begin{equation}\label{eq:isolation-ray}
N_i\cdot R_i=0,\qquad S_i\cdot R_i>0.
\end{equation}
In particular $S_i$ cannot be the exceptional divisor of a divisorial
step: the negativity lemma would give a contracted curve of negative
$S_i$-degree. Hence $S$ survives throughout the MMP.
Let $a:T\to W_i$ and $b:T\to W_{i+1}$ be a common resolution of a
step. The divisor
\[
F=a^*N_i-b^*N_{i+1}
\]
is $b$-exceptional. If a curve on $T$ is contracted by $b$, its image
on $W_i$ is a point or a curve contracted to the common contraction
target. Thus $F$ has degree zero on every such curve by
\eqref{eq:isolation-ray}. The negativity lemma applied to $F$ and
$-F$ gives
\begin{equation}\label{eq:isolation-step-pullback}
a^*N_i=b^*N_{i+1}.
\end{equation}
This applies to divisorial contractions and flips. The same argument
for the full log divisors gives
\[
a^*(K_{W_i}+G_i)=b^*(K_{W_{i+1}}+G_{i+1}).
\]
Consequently the restored pair remains log canonical.
Equation~\eqref{eq:isolation-step-pullback} also preserves nefness and
numerical triviality on the support. For the latter assertion, lift
a curve in $\operatorname{Supp}N_{i+1}$ to a curve on $T$ dominating
it. Its image on $W_i$ lies in $\operatorname{Supp}N_i$ or is a point:
the inverse image of the support of an effective $\mathbb Q$-Cartier
divisor is exactly the support of its pullback. The projection
formula now gives degree zero. Boundary components contained in
$\operatorname{Supp}N_i$ remain contained in the transformed support.
If a boundary component is disjoint from that support, its strict
transform on $T$ is disjoint from $a^*N_i$, hence from $b^*N_{i+1}$.
Since that strict transform maps onto its image on $W_{i+1}$, the
transformed boundary component is still disjoint from
$\operatorname{Supp}N_{i+1}$.
These steps also preserve the open complement of the support.
Indeed every contracted curve has negative degree against the
effective divisor $\tau N_i-\epsilon S_i$, so is contained in
$\operatorname{Supp}N_i$. A connected positive-dimensional
projective fiber contains a curve through each of its closed points.
Connectedness of fibers therefore implies that a fiber meeting the complement of
$\operatorname{Supp}N_i$ consists of a single point and does not meet
the support. Properness and normality show that the contraction is
an isomorphism from this complement onto an open subset of its
target. For a flip, this open in the target is $\mathbb Q$-factorial,
being isomorphic to the corresponding open of $W_i$. The new small
morphism is therefore also an isomorphism over it.
There is no Mori fiber output. The effective representative
$\tau N_i-\epsilon S_i$ persists on each model, as does
$\tau N_i-tS_i$ for every rational $03$. If $(X,B)$ is a
projective log canonical threefold pair with effective rational boundary,
and $K_X+B$ is $\Q$-Cartier, nef, and of numerical dimension one, then
$K_X+B$ is semiample. Neither terminality nor $\Q$-factoriality of $X$
is required.
\end{abstract}
\input{sections/introduction}
\input{sections/foundations}
\input{sections/boundary}
\input{sections/isolation}
\input{sections/torsion}
\input{sections/cycle}
\input{bibliography}
\end{document}
==== END Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//main.tex ====
==== BEGIN Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//torsion.tex ====
\section{Torsion on the full boundary cycle}\label{sec:torsion}
The reduced dlt boundary $\Sigma$ of Corollary~\ref{lem:dlt-boundary}
is a surface to which adjunction and surface abundance apply. The
prepared-cycle obstruction, however, requires torsion on an effective
Cartier divisor, including its multiplicities. We first give the
extension argument that passes from the reduced surface to this
possibly nonreduced scheme.
\begin{lemma}[{\cite[Lemma~2.4]{Companion}}]\label{lem:torsion}
Let $T$ be a normal projective threefold over an algebraically closed
field of characteristic $p>0$, and let $E$ be a nonzero effective Cartier
divisor on $T$. Suppose that $E$ is numerically trivial on every
integral component of its support. If
\[
\mathcal O_T(E)|_{E_{\mathrm{red}}\setminus Z}
\simeq\mathcal O_{E_{\mathrm{red}}\setminus Z}
\]
for a finite set $Z$ of closed points, then $\mathcal O_E(E)$ is torsion.
\end{lemma}
\begin{proof}
We reproduce the proof because both the finite exceptional set and the
nonreduced divisor will occur in our application. Replace $Z$ by
$Z\cap\operatorname{Supp}E$. At $z\in Z$, write
$R=\mathcal O_{T,z}$, with maximal ideal $\mathfrak m$, and let $t_z$ be
a local equation of $E$. Normality gives
\[
H^1_{\mathfrak m}(R)=0,
\qquad \ell_R\bigl(H^2_{\mathfrak m}(R)\bigr)<\infty.
\]
Here the second assertion uses the dimension-three hypothesis.
Indeed, present $R=A/I$ with $A$ regular local of dimension $n$ and
essentially of finite type over the field. Each nonmaximal localization
of $R$ is normal of dimension at most two, hence Cohen--Macaulay.
Catenarity gives
$\dim A_{\mathfrak q}-\dim R_{\mathfrak p}=n-3$ for a prime
$\mathfrak q$ above $\mathfrak p$; the Auslander--Buchsbaum formula
therefore gives projective dimension $n-3$ over $A_{\mathfrak q}$.
Thus
$\operatorname{Ext}^{n-2}_A(R,A)$ is supported only at the closed point.
It is a finite module, so it has finite length; local duality gives the
assertion for $H^2_{\mathfrak m}(R)$.
Choose $j\geq1$ such that $t_z^{j-1}$ annihilates
$H^2_{\mathfrak m}(R)$ at every point of $Z$. A single $j$ works because
$Z$ is finite. For every integer $u$, the restriction sequences give
a commutative diagram
\[
\begin{tikzcd}[column sep=small]
0\arrow[r]&\mathcal O_T((u-j)E)\arrow[r]\arrow[d]
&\mathcal O_T(uE)\arrow[r]\arrow[d,equal]
&\mathcal O_{jE}(uE)\arrow[r]\arrow[d]&0\\
0\arrow[r]&\mathcal O_T((u-1)E)\arrow[r]
&\mathcal O_T(uE)\arrow[r]
&\mathcal O_E(uE)\arrow[r]&0.
\end{tikzcd}
\]
In local trivializations the left vertical map is multiplication by
$t_z^{j-1}$. Local cohomology with support in the finite set $Z$ is the
direct sum of the corresponding local groups. The vanishing of the
ambient $H^1_Z$ makes the connecting maps from the $H^1_Z$ of the
rightmost terms to the ambient $H^2_Z$ injective. Naturality and our
choice of $j$ therefore imply
\begin{equation}\label{eq:torsion-obstruction-zero}
H^1_Z\bigl(T,\mathcal O_{jE}(uE)\bigr)
\longrightarrow H^1_Z\bigl(T,\mathcal O_E(uE)\bigr)
\quad\text{is zero for every }u\in\mathbb Z.
\end{equation}
The same $j$ works for every twist: locally the invertible sheaves in
the diagram are free of rank one, and the annihilating map is still
multiplication by $t_z^{j-1}$.
Now fix this thickening $jE$. The ideal of
$E_{\mathrm{red}}\setminus Z$ in $jE\setminus Z$ is nilpotent. Lift a
chosen generator of $\mathcal O_T(E)|_{E_{\mathrm{red}}\setminus Z}$ to
local generating sections on a finite open cover of $jE\setminus Z$.
On overlaps the lifts differ by sections in that nilpotent ideal times
the line bundle. For $u=p^e$ sufficiently large, their $u$th tensor
powers agree: in local frames this is the identity
$(a+b)^{p^e}=a^{p^e}+b^{p^e}$, with $b^{p^e}=0$. They consequently
glue to a generator
\[
s_j\in H^0\bigl(jE\setminus Z,\mathcal O_{jE}(uE)\bigr).
\]
Let $s$ be its restriction to $E\setminus Z$. The obstruction to
extending $s$ over $E$ is the image of the obstruction to extending
$s_j$ over $jE$. Equation~\eqref{eq:torsion-obstruction-zero}, applied
to the exact sequences for sections with support in $Z$, kills that
image. Hence $s$ extends to a section of $\mathcal O_E(uE)$.
This extended section has no zeros. On each integral projective
surface component $F$ of $E_{\mathrm{red}}$, it is nonzero because it is
a generator off $Z$. If its restriction to $F$ vanished anywhere,
its zero scheme would be a nonempty effective Cartier divisor on $F$,
and would have positive intersection with an ample divisor. This
contradicts the numerical triviality of $\mathcal O_F(uE)$. The
argument uses only integrality of $F$, not normality. The section is
therefore a generator on the reduction, and then on $E$, since an
element invertible modulo a nilpotent ideal is invertible. Thus
$\mathcal O_E(uE)\simeq\mathcal O_E$.
\end{proof}
We apply this lemma to the reduced dlt model obtained by isolation.
Recall that $s$ is the multiplicity of $S_Y$ in $N_Y$ and $\tau>0$ is
the rational number satisfying $K_Y+G_Y\sim_{\mathbb Q}\tau N_Y$.
\begin{proposition}\label{prop:boundary-torsion}
For the projective crepant dlt modification
$f:(Y_d,\Sigma)\to(Y,S_Y)$ of Corollary~\ref{lem:dlt-boundary}, there is a
positive Cartier multiple $M$ of $f^*S_Y$ such that
\[
\operatorname{Supp}M=\Sigma,
\qquad M|_{\Sigma_i}\equiv0\quad\text{for every component }\Sigma_i,
\qquad \mathcal O_M(M)\text{ is torsion}.
\]
\end{proposition}
\begin{proof}
The relations furnished by isolation are
\begin{equation}\label{eq:adjunction-input}
K_{Y_d}+\Sigma\sim_{\mathbb Q}\tau s f^*S_Y
\quad\text{on a neighborhood of }\Sigma,
\qquad f^*S_Y|_{\Sigma_i}\equiv0.
\end{equation}
Let $\sigma:\Sigma'\to\Sigma$ be the finite $S_2$-ification. It is
unchanged away from a finite set $Z\subset\Sigma$
\cite[Proposition~2.14]{XuNuTwo}. Reduced-boundary adjunction gives
an effective rational divisor $\Delta'$ such that $(\Sigma',\Delta')$
is a projective semi-log-canonical surface pair and
\begin{equation}\label{eq:slc-adjunction}
K_{\Sigma'}+\Delta'
\sim_{\mathbb Q}\sigma^*((K_{Y_d}+\Sigma)|_\Sigma)
\sim_{\mathbb Q}\tau s\,\sigma^*(f^*S_Y|_\Sigma).
\end{equation}
We use adjunction in the form of
\cite[Lemma~2.19]{XuNuTwo}; see also
\cite[Section~2.5]{Companion}. The $S_2$-ification is needed because
the reduced boundary itself need not satisfy $S_2$. Its
codimension-one singularities are nodes, and on its normalization the
adjunction boundary includes the conductor with coefficient one;
the normalized pair is log canonical. These are precisely the
semi-log-canonical conditions needed for surface abundance.
The surface $\Sigma'$ is projective because it is finite over $\Sigma$.
The last class in \eqref{eq:slc-adjunction} is numerically trivial,
by \eqref{eq:adjunction-input} and the projection formula for curves.
Posva's abundance theorem for projective semi-log-canonical surfaces
in positive characteristic therefore makes
$K_{\Sigma'}+\Delta'$ semiample
\cite[Theorem~3.1]{Posva}. A semiample numerically trivial rational
line bundle is $\mathbb Q$-linearly trivial: a globally generated Cartier
multiple defines a morphism contracting every curve, hence constant
on each connected component. Its line bundle is trivial there.
Since $\tau s$ is positive and rational, clearing denominators in
\eqref{eq:slc-adjunction} yields a positive Cartier multiple $M$ of
$f^*S_Y$ with $\sigma^*\mathcal O_\Sigma(M)$ trivial. In particular,
\[
\mathcal O_{Y_d}(M)|_{\Sigma\setminus Z}
\simeq\mathcal O_{\Sigma\setminus Z}.
\]
The divisor $M$ is effective, its reduction is $\Sigma$, and it is
numerically trivial on every component of its support. Applying
Lemma~\ref{lem:torsion} on the normal projective threefold $Y_d$ proves
torsion on the full Cartier scheme $M$.
\end{proof}
==== END Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic//torsion.tex ====