Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field
Hiromu Tanaka
Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, \Delta)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an effective $\mathbb{Q$-divisor $\Delta' \sim_{\mathbb Q \Delta+D$ such that $(X, \Delta')$ is log canonical if there exists a log resolution of $(X, \Delta)$.
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\renewcommand{\subjclassname}{%
\textup{2010} Mathematics Subject Classification}

\title[Semiample perturbations for log canonical varieties]
{Semiample perturbations for log canonical varieties 
over an F-finite field containing an infinite perfect field} 
\author{Hiromu Tanaka} 
\subjclass[2010]{14E30, 13A35.}
\keywords{log canonical, semi-ample, positive characteristic}
%\date{2011/5/25, version 1.00}
\address{Department of Mathematics, Imperial College, London, 180 Queen's Gate, 
London SW7 2AZ, UK} 
\email{h.tanaka@imperial.ac.uk}
%\thanks{The author is partially supported by 
%the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (24-1937). }
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\begin{document}


\maketitle




\begin{abstract}
Let $k$ be an $F$-finite field containing an infinite perfect field 
of positive characteristic. 
Let $(X, \Delta)$ be a projective log canonical pair over $k$. 
In this note we show that, for a semi-ample divisor $D$ on $X$, 
there exists an effective $\mathbb{Q}$-divisor 
$\Delta' \sim_{\mathbb Q} \Delta+D$ such that $(X, \Delta')$ is log canonical 
if there exists a log resolution of $(X, \Delta)$. 
\end{abstract}


%\tableofcontents

%\setcounter{section}{-1}

\section{Main theorem}



In this note, we prove the following theorem that is non-trivial even for the klt case. 

\begin{thm}\label{Theorem-MMP}
Fix $\K \in \{\Q, \R\}$. 
Let $k$ be an $F$-finite field containing an infinite perfect field $k_0$ of characteristic $p>0$. 
Let $(X, \Delta)$ be a projective log canonical $($resp. klt$)$ 
pair over $k$, 
where $\Delta$ is an effective $\K$-divisor. 
Let $D$ be a semi-ample $\K$-Cartier $\K$-divisor on $X$. 
If there is a log resolution of $(X, \Delta)$, 
then there exists an effective $\K$-Cartier $\K$-divisor $D' \sim_{\K} D$ 
such that $(X, \Delta+D')$ is log canonical $($resp. klt$)$. 
\end{thm}

In characteristic zero, Theorem~\ref{Theorem-MMP} holds by Bertini's theorem 
for free divisors, which fails in positive characteristic. 
Our proof depends on the theory of $F$-singularities. 
More precisely, 
Theorem~\ref{Theorem-MMP} follows from 
Proposition~\ref{Theorem-F1} and Proposition~\ref{Theorem-F}. 



\begin{prop}\label{Theorem-F1}
Fix $\K \in \{\Q, \R\}$. 
Let $k$ be an $F$-finite field containing an infinite perfect field $k_0$ of characteristic $p>0$. 
Let $(X, \Delta)$ be a projective strongly $F$-regular variety over $k$, where $\Delta$ is an effective $\K$-divisor. 
Let $D$ be a semi-ample $\K$-divisor on $X$. 
If $X$ is regular, 
then there exists an effective $\K$-divisor $D' \sim_{\K} D$ 
such that $(X, \Delta+D')$ is strongly $F$-regular. 
\end{prop}

\begin{prop}\label{Theorem-F}
Let $k$ be an $F$-finite field containing an infinite perfect field $k_0$ of characteristic $p>0$. 
Let $X$ be a projective regular variety over $k$ and 
let $\Delta$ be an effective simple normal crossing $\Q$-divisor on $X$ 
whose coefficients are contained in $[0, 1]$. 
Let $D$ be a semi-ample $\Q$-divisor on $X$. 
Then there exists an effective $\Q$-divisor $D' \sim_{\Q} D$ 
such that $(X, \Delta+D')$ is sharply $F$-pure. 
\end{prop}


{\bf Idea of Theorem~\ref{Theorem-MMP}:} 
We overview the proof of Theorem~\ref{Theorem-MMP}. 
We only treat the case $\K=\Q$, $(X, \Delta)$ is klt, and $k$ is an infinite perfect field. 

Since there is a log resolution of $(X, \Delta)$, 
we may assume that $X$ is smooth and $\Delta$ is simple normal crossing. 
In this case, the notions of klt and strongly $F$-regular singularities coincide, 
hence it suffices to show Proposition~\ref{Theorem-F1} 
because strongly $F$-regular singularities are klt. 
%The proofs of Proposition~\ref{Theorem-F1} and Proposition~\ref{Theorem-F} are very similar. 
%Thus we overview the proof of Proposition~\ref{Theorem-F1}. 
%We only treat the case where $\K=\Q$ and $k$ is an infinite perfect field. 


We may assume that $D$ is base point free. 
By \cite[Theorem~B]{PSZ}, 
we can find a large integer $m \in \Z_{>0}$ such that 
$$(X, \Delta+\frac{1}{m}(D_1+\cdots+D_{\dim X}))$$ 
is strongly $F$-regular for every $D_i \in |D|$. 
Since $D$ is base point free, we can find members 
$$D_1, \cdots, D_m \in |D|$$
which satisfy the following property:  
\begin{enumerate}
\item[$(*)$]{the intersection $\bigcap_{j \in J}D_j$ is empty for every $|J|=\dim X+1$. }
\end{enumerate}
Set 
$$D':=\frac{1}{m}(D_1+\cdots+D_m) \sim_{\Q} D.$$
By $(*)$, for every point $x \in X$, 
there is an open subset $x \in U \subset X$ such that 
$$(U, (\Delta+D')|_U)=(U, (\Delta+\frac{1}{m}\sum_{i \in I} D_i)|_U)$$
for some $I \subset \{1, 2, \cdots, m\}$ with $0\leq |I| \leq \dim X$. 
Therefore, $(X, \Delta+D')$ is strongly $F$-regular. 

If $k$ is not a perfect field, 
then we need to replace \cite[Theorem~B]{PSZ} in the above argument with Proposition~\ref{special-general}. 
Although the proofs of Proposition~\ref{Theorem-F1} and Proposition~\ref{Theorem-F} 
are very similar, the proof of Proposition~\ref{Theorem-F} needs the inversion of adjunction for $F$-singularities 
established by \cite{F-adjunction}. 

\medskip

{\bf Assumption on the base field:} 
Let us consider about the assumption that $k$ contains an infinite perfect field $k_0$. 
First, we impose a restrictive condition: $k$ is perfect. 
Then, our assumption $k_0 \subset k$ is equivalent to the condition that $k$ is an infinite perfect field. 
In this case, 
a key result (Proposition~\ref{special-general}) almost follows from \cite[Theorem~B]{PSZ}, 
however we need the arguments 
in the proofs of Proposition~\ref{Theorem-F1} and Proposition~\ref{Theorem-F}. 
For the case when $k$ is a finite field, 
our proof encounters a similar technical difficulty to the Bertini theorem for very ample divisors. 
Although the author does not know whether Theorem~\ref{Theorem-MMP} holds for finite fields, 
the method of \cite{Poonen} may help us. 
%Over a finite field, 
%there exists a very ample divisor $H$ on a smooth projective variety 
%whose complete linear system has no smooth members (\cite[Theorem~3.1]{Poonen}). 
%However its multiple $mH$ has a smooth member by \cite[Theorem~3.3]{Poonen}. 
%Since our problem allows taking multiples,

%Moreover, if $k$ is an infinite perfect field, 
%then Theorem~\ref{Theorem-MMP} holds by \cite[Theorem~B]{PSZ}. 

Second, let us go back to the general situation. 
%We consider how to use the condition $k_0 \subset k$. 
In the proof of a key result (Proposition~\ref{special-general}), 
we will make use of rational points or its $p^e$-powers of $\mathbb A_k^r$. 
To assure the existence of infinitely many $p^{\infty}$-torsion points, 
we assume that $k$ contains an infinite perfect field. 

\medskip


{\bf Motivation:} 
Originally the motivation for Theorem~\ref{Theorem-MMP} is 
to show the relative klt/lc abundance theorem assuming the absolute abundance theorem. 
For example, given a morphism $\pi:(X, \Delta) \to S$ from a projective klt variety $(X, \Delta)$ to a normal variety $S$, 
if $K_X+\Delta$ is $f$-nef, then we see that $K_X+\Delta+\pi^*A_S$ is nef for some ample divisor $A_S$ 
up to the cone theorem. 
Since $\pi^*A_S$ is semi-ample, Theorem~\ref{Theorem-MMP} implies that 
there is an effective $\Q$-divisor $\Delta' \sim_{\Q} \Delta+\pi^*A_S$ 
such that $(X, \Delta')$ is klt and $K_X+\Delta$ is nef. 
Thus, we could reduce the problem to the absolute case. 
In this situation, our assumption on the base field $k$ is harmless because 
we can assume this as follows: first take the base change 
to the composite field of $k$ and $\overline{\mathbb F}_p$, 
and second we can find a subfield of $k$ 
which is finitely generated over $\overline{\mathbb F}_p$ and defines all the given varieties and morphisms. 

\medskip 

{\bf Related results:}
Recently, some problems on birational geometry are solved 
by the theory of $F$-singularities (\cite{CHMS}, \cite{CTX}, \cite{HX}, \cite{Mustata}). 
By using it, also this paper shows a result on birational geometry (Theorem~\ref{Theorem-MMP}). 
For some related topics of $F$-singularities, see \cite{BSTZ}, \cite{BST}. 
In particular, Bertini's theorem of very ample divisors holds for $F$-singularities (\cite{SZ}). 

\begin{ack}
The author would like to thank Professors 
%Yoshinori~Gongyo, 
%J\'anos~Koll\'ar, 
Paolo Cascini, 
Mircea Musta\c{t}\u{a}, 
Zsolt Patakfalvi, 
Karl Schwede and 
Shunsuke Takagi 
%Chenyang~Xu 
for very useful comments and discussions. 
\end{ack}


\section{Notation}


We say $X$ is a {\em variety} over a field $k$ (or $k$-variety) if 
$X$ is an integral scheme which is separated and of finite type over $k$. 
%We say $X$ is a {\em curve} over $k$ (resp. a {\em surface} over $k$) 
%if $X$ is a variety over $k$ with $\dim X=1$ (resp. $\dim X=2$).
%For a scheme $X$, set $X_{\red}$ to be 
%the reduced scheme whose underlying topological space is equal to $X$. 

%We will not distinguish the notations 
%invertible sheaves and divisors. 
%For example, 
%we will write $L+M$ for 
%invertible sheaves $L$ and $M$. 
%For a coherent sheaf $F$ and a Cartier divisor $L$, 
%we define $F(L):=F\otimes \mathcal O_X(L)$. 
%For a proper scheme $X$ over a field $k$ and a coherent sheaf $F$ on $X$, 
%we set 
%$$h^i(X, F):=\dim_k H^0(X, F),\,\,\,\chi(X, F):=\sum_{i=0}^{\dim X} (-1)^ih^i(X, F).$$
%Note that these numbers depend on $k$. 



Let $X$ be a normal variety over a field $k$. 
For a closed subset $Z$ of $X$, 
we say $Z$ is a {\em simple normal crossing divisor} 
if, for the irreducible decomposition $Z=\bigcup_{i \in I} D_i$, 
we obtain $\dim D_i=\dim X-1$ for every $i \in I$ and 
$\bigcap_{j \in J} D_j$ is regular for every subset $\emptyset \neq J \subset I$, 
where we consider $D_i$ as the reduced scheme and 
the intersection $\bigcap_{j \in J} D_j$ means a scheme-theoretic intersection. 



Let $X$ be a normal $k$-variety and 
let $D$ be an effective $\R$-divisor. 
We say $\mu:W \to X$ is a {\em log resolution} of $(X, D)$ 
if $\mu$ is a projective birational morphism, 
$W$ is regular and $\mu^{-1}(D)\cup \Ex(\mu)$ 
is a simple normal crossing divisor. 



We will freely use the notation and terminology in \cite{Kollar}. 
In particular, for the definitions of klt and log canonical singularities, 
see \cite[Definition~2.8]{Kollar}. 
%for a pair $(X, \Delta)$, $\Delta$ is not necessarily effective. 
%However, in this paper, we assume that $\Delta$ is an effective $\R$-divisor. 


For the definitions of strongly $F$-regular and sharply $F$-pure pairs, 
see \cite[Definition~2.7]{F-adjunction} and \cite[Definition~2.7]{CTX}. 
For $e\in \Z_{>0}$, a normal variety $X$ and an effective $\Z$-divisor $D$ on $X$, 
we have {\em a trace map} 
$$\Tr_X^e(D):F_*^e(\MO_X(-(p^e-1)K_X-D)) \to \MO_X.$$
For the definition and the basic properties of trace map, see \cite[Section~2.3]{CTX}. 

We say an $\R$-Cartier $\R$-divisor $D$ is {\em semi-ample} 
if we can write $D=\sum_{1 \leq i\leq r} a_iD_i$, where 
$a_i \in \R_{\geq 0}$ and $D_i$ is a semi-ample Cartier divisor. 

\section{Proofs}

We recall a basic lemma. 

\begin{lem}\label{p-basis}
Let $A$ be an integral domain of characteristic $p>0$ and 
assume that $A$ has a $p$-basis $\{x_1, \cdots, x_n\}$. 
Then, $\{x_1, \cdots, x_n, t_1, \cdots, t_r\}$ is a $p$-basis of $A[t_1, \cdots, t_r]$. 
\end{lem}

\begin{proof}
We may assume that $r=1$ and set $t_1=:t$. 
Consider the following $A$-module homomorphism
$$\varphi:\bigoplus_{0\leq i_{\ell}, j<p} A\cdot x_1^{i_1}\cdots x_n^{i_n}t^j \to F_*(A[t]),\,\, 
x_1^{i_1}\cdots x_n^{i_n}t^j \mapsto F_*(x_1^{i_1}\cdots x_n^{i_n}t^j),$$
where the left hand side is the free $A$-module generated 
by a basis $\{x_1^{i_1}\cdots x_n^{i_n}t^j\}_{0\leq i_{\ell}, j<p}$. 
The surjectivity is clear. 
We show the injectivity. 
Assume that 
$$\sum_{0\leq i_{\ell}, j<p} f_{I, j}(t)^px_1^{i_1}\cdots x_n^{i_n}t^j=0\,\,\,\,\, {\rm in}\,\,A[t],$$
for some $f_{I, j}(t)\in A[t]$ where $I:=(i_1, \cdots, i_n)$ is the multi-index. 
We show $f_{I, j}(t)=0$ for every $I$ and $j$. 
We can write $f_{I, j}(t)=\sum_k a_k^{I, j}t^k$ for some $a_k^{I, j} \in A$. 
We obtain 
{\small $$\sum_{I, j} f_{I, j}(t)^px_1^{i_1}\cdots x_n^{i_n}t^j
=\sum_{I, j} (\sum_k a_k^{I, j}t^k)^px_1^{i_1}\cdots x_n^{i_n}t^j
=\sum_{I, j, k} (a_k^{I, j})^px_1^{i_1}\cdots x_n^{i_n}t^{pk+j}
=0.$$}
Since every coefficient of $t^{pk+j}$ vanishes, 
we obtain 
$$\sum_{I} (a_k^{I, j})^px_1^{i_1}\cdots x_n^{i_n}=0$$
for every $j$ and $k$. 
By the definition of $p$-basis, we obtain $a_k^{I, j}=0$ for every $I, j$ and $k$. 
Thus $f_{I, j}(t)=\sum_k a_k^{I, j}t^k=0$ for every $I$ and $j$. 
We are done. 
\end{proof}


The following lemma essentially follows from \cite{F-adjunction}. 


\begin{lem}\label{adjunction}
Let $Y$ be a regular variety over an $F$-finite field of characteristic $p>0$. 
Let $S$ be a reduced simple normal crossing divisor
and let $S=\sum_{i\in I} S_i$ be the irreducible decomposition. 
Set $T:=\bigcap_{i \in I} S_i$. 
Let $E$ be an effective Cartier divisor on $X$ such that $T \not\subset\Supp E$. 
Fix a $($possibly non-closed$)$ point $y\in T$ and $e\in\mathbb Z_{>0}$. 
Then, the following assertions are equivalent. 
\begin{enumerate}
\item{$\Tr_Y^e((p^e-1)S+E)$ is surjective at $y$. }
\item{$\Tr_{T}^e(E|_T)$ is surjective at $y$. }
\end{enumerate}
\end{lem}

\begin{proof}
%Set $T:={\bigcap_iS_i}$. 
By the construction of the trace map, 
we obtain the following commutative diagram (cf. \cite[Lemma~2.6(1)]{trace-map}): 
$$\begin{CD}
F_*^e(\mathcal O_Y(-(p^e-1)(K_X+S)-E))@>>{\rm surjection}>F_*^e(\mathcal O_T(-(p^e-1)K_{T}-E|_{T}))\\
@VV\Tr_Y^e((p^e-1)S+E)V @VV\Tr_T^e(E|_T)V\\
\mathcal O_Y@>\rho>{\rm surjection}>\mathcal O_{T}.
\end{CD}$$
If $\Tr_Y^e((p^e-1)S+E)$ is surjective at $y$, then 
so is $\Tr_T^e(E|_T)$ by a diagram chase. 
Thus, assume that $\Tr_T^e(E|_T)$ is surjective at $y$. 
By a diagram chase, 
there exists $a\in{\rm Im}(\Tr_Y^e((p^e-1)S+E))$ such that 
$\rho(a)=1\in\mathcal O_{T, y}.$ 
Then, we see $a-1\in I_T\MO_{Y, y}\subset \mathfrak m_y\mathcal O_{Y, y}.$ 
This implies that $a$ is a unit of $\MO_{Y, y}$. 
We are done. 
\end{proof}

Since we would like to treat rational points of $\mathbb A_k^r$ and their $p^e$-powers, 
we introduce the following terminologies. 

\begin{dfn}\label{def-coordinate}
Let $k$ be a field. %of characteristic $p>0$. 
A {\em coordinate} $t:=[t_1, \cdots, t_r]$ of $\mathbb A_k^r$ is 
a set of elements of $\MO_{\mathbb A_k^r}(\mathbb A_k^r)$ which satisfies 
$k[t_1, \cdots, t_r]=\MO_{\mathbb A_k^r}(\mathbb A_k^r)$. 
This induces a bijection: 
$$\theta:k^r \to \mathbb A_k^r(k),\,\,\, (c_1, \cdots, c_r) \mapsto (t_1-c_1, \cdots, t_r-c_r).$$
For a subfield $k_0 \subset k$, 
we define $\mathbb A_k^r(t, k_0):=\theta(k_0^r)$. 
\end{dfn}


\begin{rem}\label{translation}
The set $\mathbb A_k^r(t, k_0)$ is a subset of $\mathbb A_k^r(k)$, 
which depends on the choice of a coordinate $t=[t_1, \cdots, t_r]$. 
Actually, for every $a \in \mathbb A_k^r(k)$ and every subfield $k_0 \subset k$, 
we can find a coordinate $t=[t_1, \cdots, t_r]$ such that $a \in \mathbb A_k^r(t, k_0)$. 
On the other hand, we obtain 
$\mathbb A_k^r(t, k_0)=\mathbb A_k^r(t', k_0)$ 
for another coordinate $t'=[t'_1, \cdots, t'_r]$ such that $t'_i=t_i-c_i$ with $c_i \in k_0$. 
\end{rem}


%\begin{lem}
%Fix $a, b \in \mathbb A_k^r(k)$. 
%Then, there exists a coordinate $x=(x_1, \cdots, x_r)$ 
%such that $a, b \in \mathbb A_k^r(x, \F_p)$. 
%\end{lem}

%\begin{proof}
%We take a coordinate $(y_1, \cdots, y_r)$ such that $a$ is the origin. 
%We can write $b=(y_1-b_1, \cdots, y_r-b_r)$ for some $b_i \in k$. 
%If $b_i=0$, then $x_i:=y_i$. 
%If $b_i\neq 0$, then $x_i:=y_i/b_i$. 
%Clearly, we obtain $a=(x_1, \cdots, x_n)$ 
%We obtain 
%$$y_i-b_i=b_ix_i-b_i=b_i(x_i-1),$$
%which implies 
%$$b=(x_1-\delta_1, \cdots, x_r-\delta_r)$$
%where each $\delta_i=0$ or $1$.
%\end{proof}


Proposition~\ref{special-general} is a key result in this paper, 
which compares a special fiber and general fibers. 
For the case where $k$ is perfect, Proposition~\ref{special-general} is very similar to \cite[Theorem~B]{PSZ}. 



\begin{prop}\label{special-general}
Let $k$ be an $F$-finite field of characteristic $p>0$. 
% containing $\overline{\mathbb F}_p$. 
Let $X$ be a regular variety over $k$. 
% and let $S$ be a reduced simple normal crossing divisor on $X$. 
We fix the followings. 
\begin{itemize}
\item{$e\in\mathbb Z_{>0}$.}
\item{$T:=\mathbb A^r_k$. }
\item{An effective Cartier divisor $E$ on $X\times_k T$ such that 
$\Supp\,E$ contains no fibers of the second projection $X\times_k T \to T$.}
\item{A closed point $x\in X$ and a rational point $a \in T(k)$. }
\item{A coordinate $t=[t_1, \cdots, t_r]$ of $T$ such that $a \in T(t, k^{p^e})$ 
(cf. Definition~\ref{def-coordinate}).}
%(cf. Definition~\ref{def-coordinate}).}
\end{itemize}
If $\Tr^e_{X\times_k \{a\}}(E|_{X\times_k\{a\}})$ is surjective at $(x, a)$, 
then there exists an open subset $(x ,a)\in U$ of $X\times_k T$ 
such that
$\Tr^e_{X\times \{b\}}(E|_{X\times_k\{b\}})$ is surjective at $(y, b)$ 
for every closed point $(y, b)\in U$ with a closed point $y \in X$ and $b \in T(t, k^{p^e})$. 
\end{prop}

\begin{proof}
We may assume that 
$$X=\Spec\,R_X,\,\,\,\, T=\Spec\,k[t_1, \cdots, t_r]=\Spec\,R_T.$$
%where $0\neq \alpha \in k[t_1, \cdots, t_r]$. 
Moreover, we can assume 
that $a=(t_1, \cdots, t_r) \subset R_T$, that is, $a$ is the origin (Remark~\ref{translation}). 
By shrinking $\Spec\,R_X$ around $x$, we may assume that 
$R_X$ has a $p$-basis $\{x_1, \cdots, x_n\}$: 
$$R_X=\bigoplus_{0\leq i_{\alpha} < p^e} R_X^{p^e}x_1^{i_1}\cdots x_n^{i_n}.$$
%Moreover, by shrinking $X$ and $C$, 
%we have generators of the maximal ideals: 
%$$\mathfrak m_x=(x_1,\cdots, x_n)\subset A,\,\,\, \mathfrak m_{\alpha}=(t_1, \cdots, t_r)\subset k[t_1, \cdots, %t_r]=:S.$$
Take an element $0\neq \rho \in R_X\otimes_k R_T$ which satisfies the following properties.  
\begin{enumerate}
\item{$(x, a)\in \Spec((R_X\otimes_k R_T)[\rho^{-1}])$. }
\item{$E|_{\Spec((R_X\otimes_k R_T)[\rho^{-1}])}={\rm div}(f)$ where $f\in (R_X\otimes R_T)[\rho^{-1}]$.}
%\item{$A=\bigoplus A^{p^e}x_1^{i_1}\cdots x_n^{i_n}$.}
\end{enumerate}
Set 
$$S:=(R_X\otimes_k R_T)[\rho^{-1}].$$
Let $S_i$ on $X\times_k T$ be the pull-back of the regular effective Cartier divisor ${\rm div}(t_i)$ on $T$. 
By Proposition~\ref{adjunction}, 
we see that the trace map 
$$\Tr^e_{X\times T}((p^e-1)\sum_{i=1}^rS_i+E)$$ 
is surjective at $(x, a).$ 
Shrinking $\Spec\,S$ around $(x, a)$ if necessary, 
%by Proposition~\ref{trace-vs-split}, 
we obtain the following splitting: 
$${\rm id}_S:S\to F_*^eS\xrightarrow{\times f(t_1\cdots t_r)^{p^e-1}} F_*^eS
\xrightarrow{\varphi}S.$$
It suffices to show the splitting of 
$$S\to F_*^eS\xrightarrow{\times f((t_1-b_1)\cdots (t_r-b_r))^{p^e-1}} F_*^eS$$
%\xrightarrow{\psi}S
for general $(b_1, \cdots, b_r) \in (k^{p^e})^r$. 

Since 
$$\{x_1, \cdots, x_n, t_1-b_1, \cdots, t_r-b_r\}$$ 
is a $p$-basis of $R_X\otimes_k k[t_1, \cdots, t_r]$ for every $b_i\in k$ (Lemma~\ref{p-basis}), 
it is also a $p$-basis of 
$$(R_X\otimes_k k[t_1, \cdots, t_r])[\rho^{-1}]=(R_X\otimes_k R_T)[\rho^{-1}]=S.$$ 
%Clearly, 
%$$\{x_1,\cdots, x_n, t-\alpha\}$$
%is also an $p$-basis for every $c \in k$. 
For every $b:=(b_1, \cdots, b_r)\in k^r$, we obtain the following expressions 
\begin{eqnarray*}
g&=&\sum_{0\leq i_{\alpha}<p^e, 0\leq j_{\beta}<p^e} \xi_{I, J}^{p^e}(x_1^{i_1}\cdots x_n^{i_n}t_1^{j_1}\cdots t_r^{j_r})\\
&=&\sum_{0\leq i_{\alpha}<p^e, 0\leq j_{\beta}<p^e} \xi_{I, J}(b)^{p^e}(x_1^{i_1}\cdots x_n^{i_n}(t_1-b_1)^{j_1}\cdots (t_r-b_r)^{j_r})
\end{eqnarray*}
where $\xi_{I, J}$, $\xi_{I, J}(b) \in S$, $I:=(i_1,\cdots, i_n)$, and $J:=(j_1, \cdots, j_r)$. 
Clearly, $\xi_{I, J}(0)=\xi_{I, J}$. 
By applying the binomial expansion to 
$t_{\ell}^{j_{\ell}}=((t_{\ell}-b_{\ell})+b_{\ell})^{j_{\ell}}$, 
we obtain  
$$\xi_{I, 0}(b)^{p^e}=\sum_{0\leq j_{\beta} < p^e} \xi_{I, J}^{p^e}b_1^{j_1}\cdots b_r^{j_r}$$
for every $I$ and $b=(b_1, \cdots, b_r)\in (k^{p^e})^r$. 

Consider the following polynomial 
$$\eta_{I, 0}(z_1, \cdots, z_r):=
\sum_{0\leq j_{\beta}< p^e} \xi_{I, J}^{p^e}z_1^{j_1}\cdots z_r^{j_r} \in S[z_1, \cdots, z_r].$$
For $b=(b_1, \cdots, b_r)\in (k^{p^e})^r$, 
we obtain 
$$\eta_{I, 0}(b_1, \cdots, b_r)=\sum_{0\leq j_{\beta}< p^e} \xi_{I, J}^{p^e}b_1^{j_1}\cdots b_r^{j_r}=\xi_{I, 0}(b)^{p^e}.$$
If $\xi_{I, 0}(b) \not\in \m_{(y, b)}$, then 
$\Tr^e_{X\times \{b\}}(E|_{X\times_k\{b\}})$ is surjective at $(y, b)$. 
Thus it suffices to find an index $I'$ such that 
$\xi_{I, 0}(b) \not\in \m_{(y, b)}$ for every general $(y, b)$. 

Recall 
$$t_i \in k[t_1, \cdots, t_r]=R_T \subset R_X\otimes_k R_T \subset (R_X\otimes_k R_T)[\rho^{-1}]=S,$$
where we write $t_i=1\otimes_k t_i \in S$ by abuse of notation. 
Thus, we obtain 
$$\eta_{I, 0}(t_1, \cdots, t_r) \in S.$$
Then, for a closed point $x \in X$ and $b=(b_1, \cdots, b_r) \in (k^{p^e})^r$ 
(i.e. $b$ corresponds to the maximal ideal $(t_1-b_1, \cdots, t_r-b_r)$), 
the following three assertions are equivalent. 
\begin{itemize}
\item{$\eta_{I, 0}(t_1, \cdots, t_r) \not\in \m_{(y, b)}$.}
\item{$\eta_{I, 0}(b_1, \cdots, b_r) \not\in \m_{(y, b)}$.}
\item{$\xi_{I, 0}(b) \not\in \m_{(y, b)}$. }
\end{itemize}


Since $\varphi$ gives the above splitting, 
we can find a multi-index $I'=(i_1',\cdots, i_n')$ such that 
$\xi_{I', 0}=\xi_{I', 0}(0)\not\in\mathfrak m_{(x, a)}$. 
This implies  
$$\eta_{I', 0}(t_1, \cdots, t_r) \not\in \m_{(x, a)}.$$ 
Thus, we can find an open set $(x, a) \in U \subset X\times_k T$, 
such that 
$$\eta_{I', 0}(t_1, \cdots, t_r) \not \in \p$$
for every prime ideal $\p \in U$. 
Therefore, for every $(x, b)\in U$ where $x \in X$ is a closed point and $b\in T(k^{p^e})$, 
we obtain $\xi_{I', 0}(b) \not\in \m_{(y, b)}$. 
\end{proof}



%F-regular
\begin{proof}[Proof of Proposition~\ref{Theorem-F1}]
First, we reduce the proof to the case $\K=\Q$. 
By enlarging coefficients of $\Delta$, we may assume that $\Delta$ is a $\Q$-divisor 
(cf. \cite[Remark~2.8(2)]{CTX}). 
We can write $D=\sum_{1\leq i\leq s} a_iD_i$, where $a_i\in\R_{\geq 0}$ and 
each $D_i$ is a semi-ample Cartier divisor. 
By the induction on $s$, we may assume that $s=1$. 
Thus we obtain $D=a_1D_1$. 
By replacing $a_1$ with $\ulcorner a_1 \urcorner$, we may assume that $D$ is a semi-ample Cartier divisor. 
Thus, we could reduce the proof to the case $\K=\Q$. 

From now on, we assume that $\K=\Q$ and 
we show the assertion in the proposition. 
Since $X$ is projective, we can find an effective $\Q$-divisor $H$ on $X$ 
with the following properties. 
\begin{itemize}
\item{$(X, \Delta+H)$ is strongly $F$-regular.}
\item{$\Supp \Delta \subset \Supp H$.}
\item{$(X\setminus \Supp H, \Delta|_{X\setminus \Supp H}=0)$ is globally $F$-regular.}
\end{itemize}
By enlarging coefficients of $\Delta$ and $H$ a little, 
we may assume that $(p^{d_0}-1)\Delta$ and $(p^{d_0}-1)H$ are $\Z$-divisors for some $d_0 \in \Z_{>0}$. 
By replacing $D$ with its some multiple, 
we may assume that $D$ is a Cartier divisor such that $|D|$ is base point free. 
In particular, $\dim_kH^0(X, D) \geq 2$, otherwise $D \sim 0$ and the assertion is trivial. 


Set $T_1:=\mathbb A_k^q=\mathbb A(H^0(X, D))$ (i.e. $q:=\dim_kH^0(X, D)$) and 
$$T:=(T_1)^{\dim X}=\mathbb A_k^q \times_k \cdots \times_k \mathbb A_k^q \simeq \mathbb A_k^{q\dim X}.$$
We can find an effective Cartier divisor $\mathcal D$ on $X \times_k T$ 
which satisfies the following properties: 
\begin{itemize}
\item{Each $k$-rational point $a\in (\mathbb A_k^q(k))^{\dim X} = T(k)$ 
corresponds to a pair $(D_1, \cdots, D_{\dim X})$ 
where, for every $i$, we obtain $D_i \sim D$ is an effective Cartier divisor. 
In this case, we write $a=[D_1, \cdots, D_{\dim X}]$. }
\item{$\mathcal D|_{X\times_k\{[D_1, \cdots, D_{\dim X}]\}}=D_1+\cdots+D_{\dim X}.$}
\end{itemize}
Fix a coordinate $t=(t_1, \cdots, t_{q\dim X})$ of $T$. 
Note that we have a subset $T(t, k_0) \subset T(k)$, 
which is dense in $T$ (cf. Definition~\ref{def-coordinate}). 

We show that there exists $e_0 \in d_0\Z_{>0}$ 
such that the trace map 
$$\Tr_X^{e_0}((p^{e_0}-1)(\Delta+H)+2(D_1+\cdots+D_{\dim X}))$$
is surjective for every $[D_1, \cdots, D_{\dim X}] \in T(t, k_0)$. 
%We prove this by the noetherian induction. 
Fix $a=[D_1, \cdots, D_{\dim X}] \in  T(t, k_0)$. 
Then, we can find $d_a \in \Z_{>0}$ such that 
$$(X, \Delta+H+\frac{2}{p^{d_a}-1}(D_1+\cdots+D_{\dim X}))$$ 
is sharply $F$-pure. 
In particular, we can find $e_a \in d_0d_a\Z_{>0}$ 
such that the trace map 
$$\Tr_X^{e_a}((p^{e_a}-1)(\Delta+H+\frac{2}{p^{d_a}-1}(D_1+\cdots+D_{\dim X}))$$
$$=\Tr^{e_a}_{X\times \{a\}}((p^{e_a}-1)(\Delta+H+\frac{2}{p^{d_a}-1}\mathcal{D}|_{X\times \{a\}}))$$
is surjective. 
By Proposition~\ref{special-general} and the properness of $X$, 
we can find an open subset $a\in U_a \subset T$ such that 
$$\Tr^{e_a}_{X\times \{b\}}((p^{e_a}-1)(\Delta+H+\frac{2}{p^{d_a}-1}\mathcal{D}|_{X\times \{b\}}))$$ 
is surjective for every $\{b\} \in U_a \cap T(t, k_0)$. 
In particular, $(X\times \{b\}, \Delta+H+\frac{2}{p^{d_a}-1}\mathcal{D}|_{X\times \{b\}})$ 
is sharply $F$-pure and 
$$\Tr^{ee_a}_{X\times \{b\}}((p^{ee_a}-1)(\Delta+H)+2\mathcal{D}|_{X\times \{b\}}))$$
is surjective for every $e \in \Z_{>0}$ and every $b \in U_a \cap T(t, k_0)$. 
Therefore, we obtain an open cover 
$$T(t, k_0) \subset \bigcup_{a \in T(t, k_0)} U_a.$$
By the compactness of $T(t, k_0)$, 
we obtain $T(t, k_0) \subset \bigcup_{1\leq i\leq s} U_{a_i}$. 
Set $e_0:=e_{a_1}\cdots e_{a_s}$. 
Then, we see that the trace map  
$$\Tr^{e_0}_{X\times \{c\}}((p^{e_0}-1)(\Delta+H)+2\mathcal{D}|_{X\times \{c\}}))$$
is surjective for every $c \in T(t, k_0)$. 


%$$\mathcal E:=(p^{e_0}-1)(S \times T)+\mathcal D$$



\medskip

Since $|D|$ is base point free and $T(t, k_0)$ is dense in $T$, 
we can find effective Cartier divisors 
$$D_1', \cdots, D_{p^{e_0}-1}' \sim D$$ 
which satisfy the following properties. 
\begin{itemize}
\item{$[D_i'] \in T_1(k)=\mathbb A_k^q(k)$ for every $1\leq i\leq p^{e_0}-1$.}
\item{For every $\{i_1, \cdots, i_{\dim X}\} \subset \{1, 2, \cdots, p^{e_0}-1\}$ 
with $i_1<i_2<\cdots<i_{\dim X}$, 
the point $[D_{i_1}', \cdots, D_{i_{\dim X}}']\in T(k)$ is contained in $T(t, k_0)$.}
\item{$\bigcap_{j\in J} D_j'=\emptyset$ 
for every subset $J \subset \{1, 2, \cdots, p^{e_0}-1\}$ with $|J|=\dim X+1$. }
\end{itemize}
We obtain 
$$D \sim_{\Q} \frac{1}{p^{e_0}-1}(D_1'+\cdots+D_{p^{e_0}-1}')=:D'$$
and 
$$\Tr_X^{e_0}((p^{e_0}-1)(\Delta+H+2D'))$$
is surjective at every point. 
By \cite[Theorem~3.9]{SS}, in order to show that the pair $(X, \Delta+D')$ is strongly $F$-regular, 
it is enough to prove the following two assertions. 
\begin{enumerate}
\item[(a)]{$\Tr_X^{e_0}((p^{e_0}-1)(\Delta+D')+(p^{e_0}-1)H+(p^{e_0}-1)D')$ is surjective.}
\item[(b)]{$(X \setminus \Supp(H+D'), (\Delta+D')|_{X\setminus \Supp(H+D')})$ is globally $F$-regular.}
\end{enumerate}
The assertion (a) follows from  
$$(p^{e_0}-1)(\Delta+D')+(p^{e_0}-1)H+(p^{e_0}-1)D'=(p^{e_0}-1)(\Delta+H+2D').$$
The assertion (b) holds 
because $(X \setminus \Supp H, \Delta|_{X\setminus \Supp H})$ is globally $F$-regular. 
We are done. 
\end{proof}


The proof of Proposition~\ref{Theorem-F} is 
almost all the same as the one of Proposition~\ref{Theorem-F1}. 
For the sake of completeness, we give a proof of it. 

%F-pure
\begin{proof}[Proof of Proposition~\ref{Theorem-F}]
%Recall that $X$ is a projective regular variety over $k$ 
%and $\Delta$ is an effective simple normal crossing $\Q$-divisor on $X$ 
%whose coefficients are contained in $[0, 1]$. 
%We have a semi-ample $\Q$-divisor $D$ on $X$ and want to find an effective $\Q$-divisor $D' \sim_{\Q} D$ 
%such that $(X, \Delta+D')$ is sharply $F$-pure. 
By enlarging coefficients of $\Delta$, 
we may assume that $\Delta$ is reduced, that is, $\Delta=\llcorner \Delta\lrcorner$. 
%Set $\Delta=\sum_{i\in I} S_i$ to be the irreducible decomposition. 
By replacing $D$ with its some multiple, 
we may assume that $D$ is a Cartier divisor such that $|D|$ is base point free. 
In particular, $\dim_kH^0(X, D) \geq 2$, otherwise $D \sim 0$ and the assertion is trivial. 


Set $T_1:=\mathbb A_k^q=\mathbb A(H^0(X, D))$ (i.e. $q:=\dim_kH^0(X, D)$) and 
$$T:=(T_1)^{\dim X}=\mathbb A_k^q \times_k \cdots \times_k \mathbb A_k^q \simeq \mathbb A_k^{q\dim X}.$$
We can find an effective Cartier divisor $\mathcal D$ on $X \times_k T$ 
which satisfies the following properties: 
\begin{itemize}
\item{Each $k$-rational point $a\in (\mathbb A_k^r(k))^{\dim X} = T(k)$ 
corresponds to a pair $(D_1, \cdots, D_{\dim X})$, 
where, for every $i$, we obtain $D_i \sim D$ is an effective Cartier divisor. 
In this case, we write $a=[D_1, \cdots, D_{\dim X}]$. }
\item{$\mathcal D|_{X\times_k\{[D_1, \cdots, D_{\dim X}]\}}=D_1+\cdots+D_{\dim X}.$}
\end{itemize}
Fix a coordinate $t=(t_1, \cdots, t_{q\dim X})$ of $T$. 
Note that we have a subset $T(t, k_0) \subset T(k)$, 
which is dense in $T$ (cf. Definition~\ref{def-coordinate}). 


Since $|D|$ is base point free, 
we can find a non-empty open subset $T^0 \subset T$ 
such that, for every $[D_1, \cdots, D_{\dim X}] \in T(k) \cap T^0$, 
the support $\Supp(\sum D_i)$ does not contain any $F$-pure centers of $(X, \Delta)$. 
%Fix a coordinate $t=(t_1, \cdots, t_{q\dim X})$ of $T$. 
%Note that we have a subset $T(t, \overline{\F}_p) \subset T(k)$, which is dense in $T$. 

We show that there exists $e_0 \in \Z_{>0}$ 
such that, 
for every $[D_1, \cdots, D_{\dim X}] \in T(t, k_0) \cap T^0$, 
the trace map 
$$\Tr_X^{e_0}((p^{e_0}-1)\Delta+(D_1+\cdots+D_{\dim X}))$$
is surjective. 
Fix $a=[D_1, \cdots, D_{\dim X}] \in  T(t, k_0) \cap T^0$. 
Then, by \cite[Main Theorem]{F-adjunction}, 
we can find $d_a \in \Z_{>0}$ such that 
$(X, \Delta+\frac{1}{p^{d_a}-1}(D_1+\cdots+D_{\dim X}))$ is sharply $F$-pure. 
Therefore, we can find $e_a \in d_a\Z_{>0}$ such that 
the trace map 
$$\Tr_X^{e_a}((p^{e_a}-1)(\Delta+\frac{1}{p^{d_a}-1}(D_1+\cdots+D_{\dim X})))$$
$$=\Tr_{X\times_k\{a\}}^{e_a}((p^{e_a}-1)(\Delta+\frac{1}{p^{d_a}-1}\mathcal{D}|_{X\times_k \{a\}}))$$
is surjective at every point. 
By Proposition~\ref{special-general} and the properness of $X$, 
we can find an open subset $a \in U_a \subset T$ such that 
$$\Tr_{X\times_k\{a\}}^{e_a}((p^{e_a}-1)(\Delta+\frac{1}{p^{d_a}-1}\mathcal{D}|_{X\times_k \{b\}}))$$
is surjective for every $b \in T(t, k_0)$. 
Therefore, 
$$\Tr_{X\times_k\{a\}}^{ee_a}((p^{ee_a}-1)(\Delta+\frac{1}{p^{d_a}-1}\mathcal{D}|_{X\times_k \{b\}}))$$
is surjective for every $b \in T(t, k_0)$ and every $e\in \Z_{>0}$. 
By the compactness of $T(t, k_0)$, we can find a required $e_0$. 

\medskip

Since $|D|$ is base point free and $T(t, k_0)$ is dense in $T$, 
we can find effective Cartier divisors 
$$D_1', \cdots, D_{p^{e_0}-1}' \sim D$$ 
which satisfy the following properties. 
\begin{itemize}
\item{$[D_i'] \in T_1(k)=\mathbb A_k^q(k)$ for every $1\leq i\leq p^{e_0}-1$.}
\item{For every $\{i_1, \cdots, i_{\dim X}\} \subset \{1, 2, \cdots, p^{e_0}-1\}$ 
with $i_1<i_2<\cdots<i_{\dim X}$, 
the point $[D_{i_1}', \cdots, D_{i_{\dim X}}']\in T(k)$ is contained in $T(t, k_0) \cap T^0$.}
\item{$\bigcap_{j\in J} D_j'=\emptyset$ 
for every subset $J \subset \{1, 2, \cdots, p^{e_0}-1\}$ with $|J|=\dim X+1$. }
\end{itemize}
We obtain 
$$D \sim_{\Q} \frac{1}{p^{e_0}-1}(D_1'+\cdots+D_{p^{e_0}-1}')=:D'$$
and 
$$\Tr_X^{e_0}((p^{e_0}-1)(\Delta+D'))$$
is surjective. 
Thus, $(X, \Delta+D')$ is sharply $F$-pure. 
\end{proof}


%MMP
\begin{proof}[Proof of Theorem~\ref{Theorem-MMP}]
First we show the assertion for the case $\K=\Q$. 
Since there exists a log resolution, 
we may assume that $X$ is regular and $\Delta$ is simple normal crossing. 
%By enlarging coefficients of $\Delta$, we may assume that $\K=\Q$. 
By Proposition~\ref{Theorem-F} (resp. Proposition~\ref{Theorem-F1}), 
we can find $D' \sim_{\Q} D$ such that $(X, \Delta+D)$ is sharply $F$-pure (resp. strongly $F$-regular). 
In particular, it is log canonical (resp. klt) by \cite[Theorem~3.3]{HW}. 

Second we prove the assertion for the case $\K=\R$. 
We can write $D=\sum_{i=1}^r a_iD_i$ 
where $a_i\in\R_{>0}$ and $D_i$ is a semi-ample Cartier divisor. 
By the induction on $r$, we may assume that $r=1$. 
Thus, we obtain $D=a_1D_1$ where $a_1\in\R_{>0}$ and 
$D_1$ is a semi-ample Cartier divisor. 
By replacing $a_1$ with $\ulcorner a_1\urcorner$, 
we may assume that $D$ is a semi-ample Cartier divisor. 
Since there exists a log resolution, 
we may assume that $X$ is regular and $\Delta$ is simple normal crossing. 
By enlarging coefficients of $\Delta$ a little, we can reduce the problem to the case $\K=\Q$. 
\end{proof}



%We end the paper with the following result, 
%which may be useful when we apply Theorem~\ref{Theorem-MMP}. 

%\begin{rem}
%To apply Theorem~\ref{Theorem-MMP}, 
%the base field $k$ must be $F$-finite and contain an infinite perfect field. 

%Let $k_1$ be a field of characteristic $p>0$. 
%Fix an algebraic closure $\overline k$ of $k$ and 
%set $k\overline{\F}_p$ to be the smallest subfield of $\overline k$ 
%containing $k$ and $\overline{\F}_p$. 
%If $k$ is $F$-finite, then also $k\overline{\F}_p$ is $F$-finite. 
%Indeed, 
%$k=\sum_{1\leq i\leq n} k^p\xi_i$ implies $k\overline{\F}_p=\sum_{1\leq i\leq n} (k\overline{\F}_p)^p\xi_i$. 

%\end{rem}



\begin{thebibliography}{Schwede}
%%%%%%%
%\bibitem[]{}{},{\em },{}.




\bibitem[BSTZ]{BSTZ}
{M.~Blickle, K.~Schwede, S.~Takagi, W. Zhang},
{\em Discreteness and rationality of $F$-jumping numbers on singular varieties}, 
{Math. Ann. {\textbf{347}} (2010), 917--949.}


\bibitem[BST]{BST}
{M.~Blickle, K.~Schwede, K.~Tucker},
{\em F-singulaties via alterations}, 
{to appear in Amer. J. of Math.}



\bibitem[CHMS]{CHMS}
{P.~Cascini, C.~D.~Hacon, M.~Musta\c{t}\u{a}, K.~Schwede},
{\em On the numerical dimension of pseudo-effective divisors in positive characteristic}, 
{Amer. J. of Math., {\textbf{136}} (2014), no. 6, 1609--1628.}



\bibitem[CTX]{CTX}
{P.~Cascini, H. Tanaka, C. Xu},
{\em On base point freeness in positive characteristic}, 
{to appear in Ann. Sci. \'Ec. Norm. Sup. }

\bibitem[HX]{HX}
{C.~D.~Hacon, C.~Xu},
{\em On the three dimensional minimal model program in positive characteristic},
{to apper in J. Amer. Math. Soc.}

\bibitem[HW]{HW}
{N.~Hara, K.~Watanabe},
{\em F-regular and F-pure rings vs. log terminal and log canonical singularities}, 
{J. Alg. Geom. {\textbf{11}}, 363--392}.


%\bibitem[Hartshorne]{Hartshorne}
%%{R.~Hartshorne},
%{\em Algebraic Geometry.},
%{Grad. Texts in Math., no {\textbf{52}}, Springer-Verlag, NewYork, 1977}.


\bibitem[Koll\'ar]{Kollar}
{J.~Koll\'ar,} 
{\em Singularities of the minimal model program},
{Cambridge Tracts in Mathematics, vol. {\textbf{200}}, 2013}.




%\bibitem[Koll\'ar-Mori]{KM}
%{J.~Koll\'ar,~S.~Mori},
%{\em Birational geometry of algebraic varieties},
%{Cambrigde Tracts in Mathematics, Vol. {\textbf{134}}, 1998}.


\bibitem[Musta\c{t}\u{a}]{Mustata}
{M.~Musta\c{t}\u{a}}, 
{\em The non-nef locus in positive characteristic}, 
{ A celebration of algebraic geometry, 535-551, 
Clay Math. Proc., {\textbf{18}}, Amer. Math. Soc., Providence, RI, 2013.}


\bibitem[PSZ]{PSZ}
{Z. Patakfalvi, K. Schwede, W. Zhang},
{\em F-singularities in families}, 
{preprint available at http://arxiv.org/pdf/1305.1646v1.pdf} 

\bibitem[Poonen]{Poonen}
{B. Poonen},
{\em Bertini theorems over finite fields}, 
{Ann. of Math., {\textbf{160}} (2004), 1099--1127.} 


\bibitem[Schwede]{F-adjunction}
{K. Schwede},
{\em $F$-adjunction,}
{Algebra and Number Theory, Vol. {\textbf{3}}, no. 8, 907--950 (2009).}


\bibitem[SS]{SS}
{K. Schwede, K. E. Smith}, 
{\em Globally $F$-regular and log Fano varieties,} 
{Advances in Mathematics, Vol. {\textbf{224}}, Issue 3, 863--894, 2010.}


\bibitem[SZ]{SZ}
{K. Schwede, W. Zhang}, 
{\em Bertini theorems for F-singularities,} 
{to appear in the Proceedings of the London Mathematical Society.}



\bibitem[T]{trace-map}
{H. Tanaka},
{\em The trace map of Frobenius and extending sections for threefolds,} 
{to appear in Michigan Math. Journal.}

\end{thebibliography}
\end{document}

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[BSTZ]{BSTZ {M.~Blickle, K.~Schwede, S.~Takagi, W. Zhang, {\em Discreteness and rationality of $F$-jumping numbers on singular varieties, {Math. Ann. {347 (2010), 917--949.
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[BST]{BST {M.~Blickle, K.~Schwede, K.~Tucker, {\em F-singulaties via alterations, {to appear in Amer. J. of Math.
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[CHMS]{CHMS {P.~Cascini, C.~D.~Hacon, M.~Musta\c{t\u{a, K.~Schwede, {\em On the numerical dimension of pseudo-effective divisors in positive characteristic, {Amer. J. of Math., {136 (2014), no. 6, 1609--1628.
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[CTX]{CTX {P.~Cascini, H. Tanaka, C. Xu, {\em On base point freeness in positive characteristic, {to appear in Ann. Sci. \'Ec. Norm. Sup.
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[HX]{HX {C.~D.~Hacon, C.~Xu, {\em On the three dimensional minimal model program in positive characteristic, {to apper in J. Amer. Math. Soc.
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[HW]{HW {N.~Hara, K.~Watanabe, {\em F-regular and F-pure rings vs. log terminal and log canonical singularities, {J. Alg. Geom. {11, 363--392. %
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[Hartshorne]{Hartshorne %%{R.~Hartshorne, %{\em Algebraic Geometry., %{Grad. Texts in Math., no {52, Springer-Verlag, NewYork, 1977.
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[Koll\'ar]{Kollar {J.~Koll\'ar, {\em Singularities of the minimal model program, {Cambridge Tracts in Mathematics, vol. {200, 2013. %
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[Musta\c{t\u{a]{Mustata {M.~Musta\c{t\u{a, {\em The non-nef locus in positive characteristic, { A celebration of algebraic geometry, 535-551, Clay Math. Proc., {18, Amer. Math. Soc., Providence, RI, 2013.
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[PSZ]{PSZ {Z. Patakfalvi, K. Schwede, W. Zhang, {\em F-singularities in families, {preprint available at http://arxiv.org/pdf/1305.1646v1.pdf
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[Poonen]{Poonen {B. Poonen, {\em Bertini theorems over finite fields, {Ann. of Math., {160 (2004), 1099--1127.
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[Schwede]{F-adjunction {K. Schwede, {\em $F$-adjunction, {Algebra and Number Theory, Vol. {3, no. 8, 907--950 (2009).
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[SS]{SS {K. Schwede, K. E. Smith, {\em Globally $F$-regular and log Fano varieties, {Advances in Mathematics, Vol. {224, Issue 3, 863--894, 2010.
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[SZ]{SZ {K. Schwede, W. Zhang, {\em Bertini theorems for F-singularities, {to appear in the Proceedings of the London Mathematical Society.
1503.01264.extracted_bib
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