---
abstract: 'In this paper, by using the trace map of Frobenius, we consider problems on extending sections for positive characteristic threefolds.'
address: 'Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502 Japan'
author:
- Hiromu Tanaka
title: The trace map of Frobenius and extending sections for threefolds
---

Introduction
============

In characteristic zero, by the Kodaira vanishing theorem and its generalizations, we can establish some results on adjoint divisors, such as the Kawamata–Shokurov basepoint free theorem (see, for example, [@KM Theorem 3.3]) and the Hacon–M^c^Kernan extension theorem ([@HM Theorem 5.4.21]). These theorems claim, under suitable conditions, that an adjoint divisor $m(K_X+\Delta+A)$ has good properties where $m\in\mathbb Z_{>0}$, $(X, \Delta)$ is a pair and $A$ is an ample divisor. In this paper, we only consider the following very simple situation: $X$ is a smooth projective variety, $\Delta=S$ is a smooth prime divisor and $A$ is an ample Cartier divisor. The following fact immediately follows from the Kodaira vanishing theorem.

\[0char-zero\] Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a smooth projective variety over $k$. Let $S$ be a smooth prime divisor on $X$ and let $A$ be an ample Cartier divisor on $X$ such that $K_X+S+A$ is nef. Fix $m\in\mathbb Z_{>0}.$ Then, by the Kodaira vanishing theorem, we obtain $$H^1(X, K_X+A+(m-1)(K_X+S+A))=0.$$ Thus, the natural restriction map $$H^0(X, m(K_X+S+A))\to H^0(S, m(K_S+A))$$ is surjective.

It is natural to consider whether the above fact also holds in positive characteristic. Unfortunately, however, there exists the following example.

Let $k$ be an algebraically closed field of positive characteristic. Then, there exist a smooth projective surface $X$ over $k$, a smooth prime divisor $C$ on $X$ and an ample Cartier divisor $A$ on $X$ such that $K_X+C+A$ is nef and that the natural restriction map $$H^0(X, K_X+C+A)\to H^0(C, K_C+A)$$ is not surjective.

Thus, we would like to find a suitable analogy of Fact \[0char-zero\] in positive characteristic. In this paper, we prove the following two theorems.

\[0surface-special-extension\] Let $k$ be an algebraically closed field of positive characteristic. Let $X$ be a smooth projective surface over $k$. Let $C$ be a smooth prime divisor on $X$ and let $A$ be an ample Cartier divisor on $X$. If $H^0(C, K_C+A)\neq 0$, then the natural restriction map $$H^0(X, K_X+C+A)\to H^0(C, K_C+A)$$ is a non-zero map.

\[0threefold-extension\] Let $k$ be an algebraically closed field of positive characteristic. Let $X$ be a smooth projective threefold over $k$. Let $S$ be a smooth prime divisor on $X$ and let $A$ be an ample Cartier divisor on $X$. Assume the following two conditions.

1.  [$K_X+S+A$ is nef.]{}

2.  [$\kappa(S, K_S+A)\neq 0$.]{}

Then, there exists $m_0\in \mathbb Z_{>0}$ such that, for every $m\geq m_0$, the natural restriction map $$H^0(X, m(K_X+S+A))\to H^0(S, m(K_S+A))$$ is surjective.

To show the above two theorems, we use the trace map of Frobenius. This strategy is essentially the same as [@Schwede2 Proposition 5.3] and its proof. Let us see the idea of the proofs. Let $X$ be a smooth projective variety. Let $S$ be a smooth prime divisor on $X$ and let $A$ be an ample Cartier divisor on $X$. Then, for every $e\in\mathbb Z_{>0}$, we obtain the following commutative diagram by using the trace map of Frobenius: [ $$\begin{CD}
H^0(X, K_X+S+p^eA)@>>>H^0(S, K_S+p^eA)@>>>H^1(X, K_X+p^eA)\\
@VVV @VV{{\operatorname{Tr}}}_S^e(A)V\\
H^0(X, K_X+S+A)@>\rho >> H^0(S, K_S+A)
\end{CD}$$]{} where the lower horizontal arrow $\rho$ is the natural restriction map and the upper horizontal sequence is exact. By the Serre vanishing theorem, for large $e\gg 0$, we obtain the vanishing $H^1(X, K_X+p^eA)=0$. Thus, to prove that the restriction map $\rho$ is surjective (resp. a non-zero map), it is sufficient to show that the trace map ${{\operatorname{Tr}}}_S^e(A)$ is surjective (resp. a non-zero map). Therefore, to prove the above two theorems, we establish the following results on the trace map of Frobenius.

\[0trace-curve\] Let $k$ be an algebraically closed field of positive characteristic. Let $C$ be a smooth projective curve over $k$. Let $A$ be an ample Cartier divisor on $C$. If $H^0(C, \omega_C(A))\neq 0$, then the trace map $${{\operatorname{Tr}}}_C^e(A):H^0(C, \omega_C(p^eA))\to H^0(C, \omega_C(A))$$ is a non-zero map for every $e\in\mathbb Z_{>0}$.

\[0trace-surface\] Let $k$ be an algebraically closed field of positive characteristic. Let $S$ be a smooth projective surface over $k$. Let $A$ be an ample Cartier divisor on $S$. Assume the following two conditions.

1.  [$K_S+A$ is nef.]{}

2.  [$\kappa(S, K_S+A)\neq 0$.]{}

Then, there exists $m_1\in\mathbb Z_{>0}$ such that the trace map ${{\operatorname{Tr}}}_S^e(A+m(K_S+A))$ $$H^0(S, \omega_S(p^e(A+m(K_S+A))))\to H^0(S, \omega_S(A+m(K_S+A)))$$ is surjective for every $m\geq m_1$ and for every $e\in\mathbb Z_{>0}$.

We would like to consider whether Theorem \[0threefold-extension\] and Theorem \[0trace-surface\] hold for the case where $\kappa(S, K_S+A)=0$. Let us compare Theorem \[0threefold-extension\] with the following basepoint free conjecture (cf. [@KM Theorem 3.3]).

\[0bpf\] Let $k$ be an algebraically closed field of positive characteristic. Let $X$ be a smooth projective threefold over $k$. Let $S$ be a smooth prime divisor on $X$ and let $A$ be an ample Cartier divisor on $X$ such that $K_X+S+A$ is nef. Then, $|b(K_X+S+A)|$ is basepoint free for every $b\gg 0$.

If Conjecture \[0bpf\] holds, then Theorem \[0threefold-extension\] also hold for the case where $\kappa(S, K_S+A)=0$. Then, does Theorem \[0trace-surface\] also hold for the case where $\kappa(S, K_S+A)=0$? Unfortunately, the answer is NO. We can construct the following example in characteristic two.

\[0trace-surface-cex\] Let $k$ be an algebraically closed field of characteristic two. Then, there exists a smooth projective surface $S$ over $k$ such that

1.  [$-K_S=:A$ is ample. Note that $\kappa(S, K_S+A)=0$. ]{}

2.  [For every $e\in\mathbb Z_{>0}$, the trace map $${{\operatorname{Tr}}}_S^e(A):H^0(S, \omega_S(2^e A))\to H^0(S, \omega_S(A))$$ is a zero map. ]{}

Moreover, Theorem \[0trace-surface-cex\] also shows that we can not generalize Theorem \[0trace-curve\] to dimension two.

In the appendix of this paper (Section 9), we establish the following analogy of the Hacon–M^c^Kernan extension theorem for surfaces.

\[0surface-ext\] Let $k$ be an algebraically closed field of positive characteristic. Let $X$ be a smooth projective surface over $k$ and let $C$ be a smooth prime divisor on $X$. Let $\Delta:=C+B$ where $B$ is an effective [$\mathbb{Q}$]{}-divisor on $X$ which satisfies the following properties.

1.  [$C\not\subset{{\operatorname{Supp}}}B$, $\llcorner B\lrcorner=0$ and $(X, \Delta)$ is plt. ]{}

2.  [$B\sim_{\mathbb Q}A+F$ where $A$ is an ample [$\mathbb{Q}$]{}-divisor and $F$ is an effective [$\mathbb{Q}$]{}-divisor such that $C\not\subset {{\operatorname{Supp}}}F.$]{}

3.  [No prime component of $\Delta$ is contained in the stable base locus of $K_X+\Delta.$]{}

Then, there exists an integer $m_0>0$ such that, for every integer $m>0$, the restriction map $$H^0(X, mm_0(K_X+\Delta))\to H^0(C, mm_0(K_X+\Delta)|_C)$$ is surjective.

But, the proof for Theorem \[0surface-ext\] does not use the trace map of Frobenius. We use results on the minimal model theory established in [@T2] and [@T3].

In Section 1, we summarize the notations. In Section 2, we see the definition and some basic properties of the trace map of Frobenius. The trace map of Frobenius is obtained by applying the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ to the Frobenius map $\mathcal O_X\to F_*\mathcal O_X.$ In Section 3, we see some known facts on Cartier operator. We can consider the trace map of Frobenius as Cartier operator. The Cartier operator is defined by the de Rham complex. We use Cartier operator to consider the relation between the trace map of Frobenius and étale base changes. In Section 4, we show Theorem \[0surface-special-extension\] and Theorem \[0trace-curve\]. In Section 5, we prove Theorem \[0trace-surface\] for the case where $\kappa=1$. In Section 6, we show Theorem \[0trace-surface\] for the case where $\kappa=2$. In Section 7, by using Theorem \[0trace-surface\], we show Theorem \[0threefold-extension\]. In Section 8, we prove Theorem \[0trace-surface-cex\]. In Section 9, we show Theorem \[0surface-ext\].

We summarize literature related to this paper with respect to the basepoint free theorem, the extension theorem, the trace map of Frobenius and the minimal model theory in positive characteristic.

(Basepoint free theorem and extension theorem) The motivation of this paper is the basepoint free theorem and the extension theorem in characteristic zero. Thus, let us summarize some known results on them. Kawamata and Shokurov established the basepoint fee theorem for klt pairs (cf. [@KMM], [@KM]). [@Ambro] generalizes this result (cf. [@Fujino]). The extension theorem is established by Hacon and M^c^Kernan ([@HM Theorem 5.4.21]). This theorem is a key to show the existence of the flip ([@BCHM]). For related topics, see [@DHP] and [@FG].

(The trace map of Frobenius) The heart of this paper is the trace map of Frobenius. The trace map of Frobenius plays a crucial role in the theory of $F$-singularities (cf. [@BST], [@Schwede1], [@Schwede2]). Moreover, [@CHMS] and [@Mustata] establish results related to birational geometry by using the trace map of Frobenius and the theory of $F$-singularities. For related topics, see [@BSTZ] and [@TW].

(Minimal model theory in positive characteristic) Let us summarize literature on the minimal model theory in positive characteristic. For results on surfaces, see [@Fujita3], [@KK], [@T2] and [@T3]. For results on threefolds, see [@Kawamata], [@Keel], [@Kollar] and [@HX].

The author would like to thank Professor Osamu Fujino for many comments and discussions. He would like to thank Professor Karl Schwede for numerous helpful comments. He thanks Professor Atsushi Moriwaki for warm encouragement. The author is partially supported by JSPS Fellowships for Young Scientists.

Notations
=========

We will freely use the notation and terminology in [@KM].

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 $F(L):=F\otimes \mathcal O_X(L)$.

Throughout this paper, we work over an algebraically closed field $k$ of positive characteristic and let ${\rm char}\,k=:p>0.$

In this paper, [*a variety*]{} means an integral scheme which is separated and of finite type over $k$.

The trace map of Frobenius
==========================

In this section, we define the trace map of the Frobenius and we see some fundamental properties. We only use the smooth case. For the singular case, see [@Schwede2 Section 2].

\[tracedef\] Let $X$ be a smooth projective variety. Let $E$ be an effective [$\mathbb{Z}$]{}-divisor and let $D$ be a [$\mathbb{Z}$]{}-divisor. Then, for every positive integer $e$, there exists a natural $\mathcal O_X$-module homomorphism $${{\operatorname{Tr}}}_{X,E}^e(D):F^e_*(\omega_X(E+p^eD))\to 
\omega_X(E+D).$$ We call this a trace map.

Consider the Frobenius map: $$\mathcal O_X\to F^e_*\mathcal O_X,$$ that is, $p^e$-th power map $a\mapsto a^{p^e}$. Since $E$ is effective, we obtain $$\mathcal O_X(-E)\to F^e_*(\mathcal O_X(-p^eE))\hookrightarrow F^e_*(\mathcal O_X(-E)).$$ Tensoring $\mathcal O_X(-D)$, we obtain $$\begin{aligned}
\mathcal O_X(-E-D)
&\to& F^e_*(\mathcal O_X(-E))\otimes \mathcal O_X(-D)\\
&\simeq& F^e_*(\mathcal O_X(-E-p^eD)).\end{aligned}$$ By the duality theorem for finite morphisms, we obtain $$\mathcal Hom_{\mathcal O_X}(F^e_*(\mathcal O_X(-E-p^eD)), \omega_X)\simeq F^e_*(\omega_X(E+p^eD)).$$ Then, apply the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ and we obtain $$F^e_*(\omega_X(E+p^eD))\to \omega_X(E+D).$$ This is the trace map ${{\operatorname{Tr}}}_{X,E}^e(D)$.

\[pE-vs-E\] By the above construction, ${{\operatorname{Tr}}}_{X,E}^e(D)$ factors through ${{\operatorname{Tr}}}_X^e(E+D):={{\operatorname{Tr}}}_{X, 0}^e(E+D)$: [$${{\operatorname{Tr}}}_{X,E}^e(D):F^e_*(\omega_X(E+p^eD))\hookrightarrow F^e_*(\omega_X(p^eE+p^eD))
\xrightarrow{{{\operatorname{Tr}}}_X^e(E+D)} \omega_X(E+D).$$]{}

\[trace-local\] Let $X$ be a smooth projective variety. Let ${{\operatorname{Spec}}}R\subset X$ be an affine open subset such that $R$ has a $p$-basis $\{x_1, \cdots, x_n\}.$ Then, we obtain $$\Gamma({{\operatorname{Spec}}}R, \omega_X)=Rdx_1\wedge\cdots\wedge dx_n$$ and $$\begin{aligned}
\Gamma({{\operatorname{Spec}}}R, F_*^e\omega_X)
&=&\bigoplus_{0\leq i_j<p^e}R\cdot x_1^{i_1}\cdots x_n^{i_n}dx_1\wedge\cdots\wedge dx_n\\
&=&\bigoplus_{0\leq i_j<p^e}R^{p^e}x_1^{i_1}\cdots x_n^{i_n}dx_1\wedge\cdots\wedge dx_n.\end{aligned}$$ The trace map $${{\operatorname{Tr}}}_X^e:\Gamma({{\operatorname{Spec}}}R, F_*^e\omega_X)\to \Gamma({{\operatorname{Spec}}}R, \omega_X)$$ is described as follows:

1.  [${{\operatorname{Tr}}}_X^e(x_1^{p^e-1}\cdots x_n^{p^e-1}dx_1\wedge\cdots\wedge dx_n)=dx_1\wedge\cdots\wedge dx_n.$]{}

2.  [${{\operatorname{Tr}}}_X^e(x_1^{i_1}\cdots x_n^{i_n}dx_1\wedge\cdots\wedge dx_n)=0$ if $0\leq i_j<p^e-1$ for some $1\leq j\leq n$.]{}

The following two lemmas are the fundamental properties.

\[D-linearequ\] Let $X$ be a smooth projective variety and let $E$ be an effective [$\mathbb{Z}$]{}-divisor. If $D_1$ and $D_2$ are linearly equivalent [$\mathbb{Z}$]{}-divisors, then the two trace maps ${{\operatorname{Tr}}}_X^e(D_1)$ and ${{\operatorname{Tr}}}_X^e(D_2)$ are the same for every positive integer $e$, that is, there exists a following commutative diagram: $$\begin{CD}
F^e_*(\omega_X(E+p^eD_2))@>{{\operatorname{Tr}}}_X^e(D_2)>> \omega_X(E+D_2)\\
@VV\simeq V @VV\simeq V\\
F^e_*(\omega_X(E+p^eD_1))@>{{\operatorname{Tr}}}_X^e(D_1)>> \omega_X(E+D_1).
\end{CD}$$

Consider the Frobenius map: $$\mathcal O_X(-E)\to F^e_*(\mathcal O_X(-E)).$$ Tensoring $\mathcal O_X(-D_i)$, we obtain $$\begin{aligned}
\mathcal O_X(-E-D_i)
\to F^e_*(\mathcal O_X(-E-p^eD_i)).\end{aligned}$$ Then, we obtain the following commutative diagram: $$\begin{CD}
F^e_*(\mathcal O_X(-E-p^eD_2))@<<< \mathcal O_X(-E-D_2)\\
@AA\times g^{p^e} A @AA\times g A\\
F^e_*(\mathcal O_X(-E-p^eD_1))@<<< \mathcal O_X(-E-D_1)
\end{CD}$$ where $g\in K(X)^*$ is defined by $\mathcal O_X(-D_1)\overset{\times g}\simeq\mathcal O_X(-D_2).$ In particular, the vertical arrows are isomorphisms. Apply the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ and we obtain the required commutative diagram.

\[e-vs-e+1\] Let $X$ be a smooth projective variety and let $E$ be an effective [$\mathbb{Z}$]{}-divisor. Let $D$ be a [$\mathbb{Z}$]{}-divisor. Then, for every positive integer $e$, $${{\operatorname{Tr}}}_{X, E}^{e+1}(D)={{\operatorname{Tr}}}_{X, E}^e(D)\circ F_*^e({{\operatorname{Tr}}}_{X, E}^1(p^e D)),$$ that is, $$\begin{aligned}
{{\operatorname{Tr}}}_{X, E}^{e+1}(D):
F^{e+1}_*(\omega_X(E+p^{e+1}D))&\xrightarrow{F_*^e({{\operatorname{Tr}}}_{X,E}^1(p^eD))}&F^e_*(\omega_X(E+p^eD))\\
&\xrightarrow{{{\operatorname{Tr}}}_{X,E}^{e}(D)}&\omega_X(E+D).\end{aligned}$$

Consider the Frobenius maps: $$\mathcal O_X(-E)\to F^e_*(\mathcal O_X(-E))\to F^{e+1}_*(\mathcal O_X(-E)).$$ Tensoring $\mathcal O_X(-D)$, we obtain $$\begin{aligned}
\mathcal O_X(-E-D)
\to F^e_*(\mathcal O_X(-E-p^eD))\to 
F^{e+1}_*(\mathcal O_X(-E-p^{e+1}D)).\end{aligned}$$ Apply the functor $\mathcal Hom_X(-, \omega_X)$ and we obtain the assertion.

In this paper, we often use the following two commutative diagrams.

\[trace-diagram\] Let $X$ be a smooth projective variety and let $S$ be a smooth prime divisor. Then, there exist the following commutative diagrams.

1.  [$$\begin{CD}
    0@>>> F_*^e\omega_X@>>> F_*^e(\omega_X(S))@>>> F_*^e\omega_S@>>> 0\\
    @. @VV{{\operatorname{Tr}}}_X^e V @VV{{\operatorname{Tr}}}_{X,S}^e V @VV{{\operatorname{Tr}}}_S^e V\\
    0@>>> \omega_X@>>> \omega_X(S)@>>> \omega_S@>>> 0.
    \end{CD}$$]{}

2.  [$$\begin{CD}
    0@>>> F_*^e\omega_X@>>> F_*^e(\omega_X(p^eS))@>>> F_*^e(\omega_{p^eS})@>>> 0\\
    @. @VV{{\operatorname{Tr}}}_X^e V @VV{{\operatorname{Tr}}}_{X}^e(S) V @VV\Psi V\\
    0@>>> \omega_X@>>> \omega_X(S)@>>> \omega_S@>>> 0.
    \end{CD}$$ Moreover, ${{\operatorname{Tr}}}_S^e$ factors through $\Psi$ $:$ $${{\operatorname{Tr}}}_S^e:F_*^e\omega_S\to F_*^e(\omega_{p^eS})\overset{\Psi}\to \omega_S.$$ ]{}

\(1) Consider the following commutative diagram: $$\begin{CD}
0@>>> F_*^e(\mathcal O_X(-S))@>>> F_*^e\mathcal O_X@>>> F_*^e\mathcal O_S@>>> 0\\
@. @AAA @AA F_X^e A @AAF_S^e A\\
0@>>> \mathcal O_X(-S)@>>> \mathcal O_X@>>> \mathcal O_S@>>> 0.
\end{CD}$$ Apply the functor $\mathcal Hom_X(-, \omega_X)$ and we obtain the assertion.\
(2) Consider the following commutative diagram: $$\begin{CD}
0@>>> F_*^e(\mathcal O_X(-p^eS))@>>> F_*^e\mathcal O_X@>>> F_*^e\mathcal O_{p^eS}@>>> 0\\
@. @AAA @AA F_X^e A @AAA\\
0@>>> \mathcal O_X(-S)@>>> \mathcal O_X@>>> \mathcal O_S@>>> 0.
\end{CD}$$ Apply the functor $\mathcal Hom_X(-, \omega_X)$ and we obtain the required commutative diagram. Since $$F_S^e:\mathcal O_S\to F_*^e\mathcal O_S$$ factors through $F_*^e\mathcal O_{p^eS}$, we see the assertion.

We would like to know whether the trace map ${{\operatorname{Tr}}}_X^e(D)$ has a good property. It is natural to consider the following question.

\[Tango-cex\] Let $X$ be a smooth projective variety and let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$. Then, for every $e\in\mathbb Z_{>0}$, is the trace map $${{\operatorname{Tr}}}_X^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A))$$ surjective?

The answer to the above question is NO. Indeed, [@Tango] constructs a smooth projective curve $X$ and an ample [$\mathbb{Z}$]{}-divisor $A$ on $X$ such that the trace map ${{\operatorname{Tr}}}_X^e(A)$ is not surjective.

On the other hand, we obtain the affirmative answer for the following two cases: ablelian varieties and $F$-split varieties.

Let $X$ be an abelian variety and let $A$ be an ample [$\mathbb{Z}$]{}-divisor. Then, the trace map $${{\operatorname{Tr}}}_{X}^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A)).$$ is surjective.

For $n\in\mathbb Z$, let $n_X$ be the $n$-multiplication map of the abelian variety $X$. Let $n\in\mathbb Z_{>0}$ which is not divisible by $p$. Then, $$n_X:X\to X$$ is a finite morphism whose degree is not divisible by $p$. Thus, $$\mathcal O_X\to (n_X)_*\mathcal O_X$$ is split as an $\mathcal O_X$-module homomorphism (cf. [@KM Proposition 5.7(2)]). We obtain the following commutative diagram $$\begin{CD}
F_*^e(n_X)_*(\mathcal O_X(p^e(n_X)^*A))@<<< (n_X)_*(\mathcal O_X((n_X)^*A))\\
@AAA @AA n_X A\\
F_*^e(\mathcal O_X(p^eA))@<<<\mathcal O_X(A)
\end{CD}$$ Apply the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ and take the cohomology: $$\begin{CD}
H^0(X, \omega_X(p^e(n_X)^*A))@>{{\operatorname{Tr}}}_{X}^e((n_X)^*A))>> H^0(X, \omega_X((n_X)^*A))\\
@VVV @VV n_X^* V\\
H^0(X, \omega_X(p^eA))@>{{\operatorname{Tr}}}_{X}^e(A)>> H^0(X, \omega_X(A)).
\end{CD}$$ Here, $n_X^*$ is surjective by the splitting of $n_X$. Therefore, it is sufficient to show that ${{\operatorname{Tr}}}_{X}^e((n_X)^*A)$ is surjective. By [@Mumford Corollary 3 in Section 6], we obtain $$n_X^*A= \frac{n^2+n}{2}A+\frac{n^2-n}{2}(-1)_X^*A.$$ Note that, since $(-1)_X$ is an automorphism, $(-1)_X^*A$ is ample. Therefore, by the Fujita vanishing theorem ([@Fujita1 Theorem (1)], [@Fujita2 Section 5]), we can find $n\in\mathbb Z_{>0}$ such that ${{\operatorname{Tr}}}_{X}^e((n_X)^*A)$ is surjective.

Let $X$ be a smooth projective variety. We say $X$ is [*F-split*]{} if the Frobenius map $$\mathcal O_X\to F_*\mathcal O_X$$ is split as an $\mathcal O_X$-module homomorphism.

\[F-split-surje\] Let $X$ be an $F$-split smooth projective variety and let $D$ be a [$\mathbb{Z}$]{}-divisor. Then, the trace map $${{\operatorname{Tr}}}_{X}^e(D):H^0(X, \omega_X(p^eD))\to H^0(X, \omega_X(D)).$$ is surjective.

By the definition of $F$-splitting, we see that the Frobenius map $$\mathcal O_X(-D)\to F_*^e(\mathcal O_X(-p^eD))$$ is split. Apply the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ and we obtain the assertion.

Facts on Cartier operator
=========================

In this section, we summarize the facts on Cartier operator. By Remark \[cartier-remark\], we consider the trace map of the Frobenius as Cartier operator.

\[de-rham\] Let $X$ be a smooth variety. Consider the de Rham complex of $X$ $$\Omega_X^{\bullet}:\mathcal O_X\overset{d_0}\to\Omega^1_X\overset{d_1}\to\Omega^2_X
\overset{d_2}\to\cdots$$ where $\Omega^i_X:=\Omega^i_{X/k}.$ Apply $F_*$ and we obtain a complex $$F_*\Omega_X^{\bullet}:F_*\mathcal O_X\overset{F_*d_0}\to F_*\Omega^1_X\overset{F_*d_1}\to
F_*\Omega^2_X\overset{F_*d_2}\to\cdots.$$ Then, it is easy to see that $F_*d_i$ is an $\mathcal O_X$-module homomorphism. We define $$\begin{aligned}
B_X^i&:=&{{\operatorname{Image}}}(F_*d_{i-1}:F_*\Omega_X^{i-1}\to F_*\Omega_X^{i})\\
Z_X^i&:=&{{\operatorname{Ker}}}(F_*d_i:F_*\Omega_X^i\to F_*\Omega_X^{i+1}).\end{aligned}$$ Note that $B_X^i$ and $Z_X^i$ are coherent sheaves.

\[cartier-def\] Let $X$ be a smooth variety. For every $i\in \mathbb Z$ such that $1\leq i\leq \dim X$, consider the map $$\begin{aligned}
C_X^{-1}:\Omega_X^i&\to& Z_X^i/B_X^i\end{aligned}$$ locally defined by $$\begin{aligned}
C_X^{-1}|_{{{\operatorname{Spec}}}R}:\Gamma({{\operatorname{Spec}}}R, \Omega_X^i)&\to& \Gamma({{\operatorname{Spec}}}R, Z_X^i/B_X^i)\\
da_1\wedge\cdots \wedge da_i&\mapsto&
a_1^{p-1}\cdots a_i^{p-1}da_1\wedge\cdots\wedge da_i\end{aligned}$$ where ${{\operatorname{Spec}}}R$ is an open affine subset of $X$ and $a_1,\cdots, a_i\in R.$ This map $C_X^{-1}$ is a well-defined $\mathcal O_X$-module isomorphism. We call $C_X:=(C_X^{-1})^{-1}$ Cartier operator.

See, for example, [@EV Theorem 9.14].

\[cartier-remark\] Let $X$ be an $n$-dimensional smooth variety. We obtain the following exact sequences.

1.  [$0\to \mathcal O_X\to F_*\mathcal O_X\to B_X^1\to 0$.]{}

2.  [$0\to Z_X^i\to F_*\Omega_X^i\to B_X^{i+1}\to 0$ for $1\leq i\leq n$.]{}

3.  [$0\to B_X^i\to Z_X^i\overset{C_X^i}\to \Omega_X^{i}\to 0$ for $1\leq i \leq n$.]{}

By (2) for $i=n$, we obtain $Z_X^n\simeq F_*\omega_X$. By (3) for $i=n$, we obtain $$0\to B_X^n\to F_*\omega_X\overset{C_X^n}\to \omega_X\to 0.$$

\[cartier-trace\] Let $X$ be an $n$-dimensional smooth projective variety. By Remark \[trace-local\], Cartier operator and the trace map of Frobenius are the same: $C_X^n={{\operatorname{Tr}}}^1_X.$

\[cartier-etale\] Let $\gamma:X\to Y$ be a finite étale morphism between smooth varieties. Then, $$\gamma^*B_Y^i\simeq B_{X}^i\,\,\,{\rm and}\,\,\,\gamma^*Z_Y^i\simeq Z_{X}^i.$$

We may assume $X={{\operatorname{Spec}}}B$ and $Y={{\operatorname{Spec}}}A$. Let $$\varphi: A\to B$$ be the natural homomorphism induced by $\gamma$. Let $$F_A:A\to A\,\,\,{\rm and}\,\,\,F_B:B\to B$$ be the $p$-th power maps respectively. Since $\varphi$ is flat, we see that the natural $B$-module homomorphism $$\begin{aligned}
\theta^i:((F_A)_*\Omega_A^i)\otimes_A B&\to& (F_B)_*(\Omega_A^i\otimes_A B)\\
(\sum_J a_Jdx_J)\otimes_A b&\mapsto& (\sum_J a_Jdx_J)\otimes_A b^p\end{aligned}$$ is an isomorphism where $a_J\in A$ and $dx_J:=dx_{j_1}\wedge\cdots\wedge dx_{j_i}$ for some $x_{j_l}\in A$. Since $\varphi$ is étale, the natural $B$-module homomorphism $$\begin{aligned}
\rho^i:\Omega_A^i\otimes_A B&\to&\Omega_B^i\\
(\sum_J a_Jdx_J)\otimes_A b&\mapsto& \sum \varphi(a_J)bd(\varphi(x_J))\end{aligned}$$ is an isomorphism. It is sufficient to check the commutativity of the following diagram: $$\begin{CD}
(F_B)_*\Omega_B^i@>d>> (F_B)_*\Omega_B^{i+1}\\
@AAF_*\rho^iA @AAF_*\rho^{i+1}A\\
(F_B)_*(\Omega_A^i\otimes_A B)@>d>> (F_B)_*(\Omega_A^{i+1}\otimes_A B)\\
@AA\theta^iA @AA\theta^{i+1}A\\
((F_A)_*\Omega_A^i)\otimes_A B@>d>> ((F_A)_*\Omega_A^{i+1})\otimes_A B.
\end{CD}$$ This follows from $$\begin{aligned}
d\circ F_*\rho^i\circ \theta^i((\sum_J a_Jdx_J)\otimes_A b)
&=&d\circ F_*\rho^i((\sum_J a_Jdx_J)\otimes_A b^p)\\
&=&d(\sum_J \varphi(a_J)b^pd(\varphi(x_J)))\\
&=&\sum_J b^pd(\varphi(a_J))\wedge d(\varphi(x_J))\end{aligned}$$ and $$\begin{aligned}
&&F_*\rho^{i+1}\circ \theta^{i+1}\circ d((\sum_J a_Jdx_J)\otimes_A b)\\
&=&F_*\rho^{i+1}\circ \theta^{i+1}((\sum_J da_J\wedge dx_J)\otimes_A b)\\
&=&F_*\rho^{i+1}((\sum_J da_J\wedge dx_J)\otimes_A b^p)\\
&=&\sum_J b^pd(\varphi(a_J))\wedge d(\varphi(x_J)).\end{aligned}$$

We state the following vanishing result of $F$-split varieties for a later use.

\[F-split-B-vanish\] Let $X$ be an $n$-dimensional $F$-split projective variety and let $A$ be an ample [$\mathbb{Z}$]{}-divisor. Then, $$H^1(X, B_X^n(A))=0.$$

Consider the exact sequence $$0\to B_X^n\to F_*\omega_X\overset{C_X^n}\to \omega_X\to 0.$$ Then, by Proposition \[F-split-surje\], the trace map $$C_X^n={{\operatorname{Tr}}}_{X}^1(A):H^0(X, \omega_X(pA))\to H^0(X, \omega_X(A))$$ is surjective. Therefore, we obtain the exact sequence $$0\to H^1(X, B_X^n(A))\to H^1(X, \omega_X(pA)).$$ Since $F$-split varieties satisfy the Kodaira vanishing theorem ([@MR Proposition 2]), we obtain the vanishing $H^1(X, B_X^n(A))=0$.

The trace map of Frobenius for curves
=====================================

In this section, we calculate the trace map $${{\operatorname{Tr}}}_{X}^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A))$$ for the case where $X$ is a curve. By Question \[Tango-cex\] and its answer, ${{\operatorname{Tr}}}_X^e(A)$ is not surjective in general. But, for almost all the case, ${{\operatorname{Tr}}}_X^e(A)$ is a non-zero map.

\[curve-trace-nonzero\] Let $X$ be a smooth projective curve whose genus $g(X)$ is not zero. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor. Then, for every $e\in\mathbb Z_{>0}$, the trace map $${{\operatorname{Tr}}}_{X}^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A))$$ is a non-zero map.

Fix $e\in\mathbb Z_{>0}$. Since $A$ is ample, we see $\deg A\geq 1.$ We consider the two cases: $\deg A>1$ and $\deg A=1$.

In this step, we assume $\deg A>1$ and we prove the assertion. The following argument is the same as the proof of [@Schwede2 Theorem 3.3].

Fix a point $Q\in X$. By Lemma \[trace-diagram\], we obtain the following commutative diagram:

[$$\begin{CD}
H^0(X, K_X+p^eA)@>>> H^0(p^eQ, K_{p^eQ}+p^e(A-Q))@.\to H^1(X, K_X+p^e(A-Q))\\
@VV{{\operatorname{Tr}}}_{X}^e(A) V @VV \Psi V\\
H^0(X, K_X+A)@>\rho >> H^0(Q, K_Q+A-Q).
\end{CD}$$]{} By Serre duality, we obtain the vanishing $$H^1(X, K_X+p^e(A-Q))=0.$$ On the other hand, $\Psi$ is surjective because $\Psi$ satisfies $$\begin{aligned}
{{\operatorname{Tr}}}_Q^e:H^0(Q, K_Q+p^e(A-Q))&\to &H^0(p^eQ, K_{p^eQ}+p^e(A-Q))\\
&\overset{\Psi}\to& H^0(Q, K_Q+A-Q).\end{aligned}$$ Therefore, the composition map $\rho \circ {{\operatorname{Tr}}}_X^e(A)$ is surjective. Then, we see that ${{\operatorname{Tr}}}_{X}^e(A)$ is a non-zero map by $H^0(Q, K_Q+A-Q)\neq 0$.

\[good-1-form\] In this step, we prove that, if $\deg A=1$, then there exists a point $Q\in X$ such that the natural injective map $$H^0(X, \omega_X(p^eA-p^eQ))\to H^0(X, \omega_X(p^eA-(p^e-1)Q))$$ is not surjective.

Since $H^0(Q, L)\simeq k$ for every invertible sheaf $L$, we obtain the following exact sequence $$\begin{aligned}
0&\to& H^0(X, \omega_X(p^eA-p^eQ))\to H^0(X, \omega_X(p^eA-(p^e-1)Q))\to k\\
&\to& H^1(X, \omega_X(p^eA-p^eQ)).\end{aligned}$$ Therefore, it is sufficient to show $$h^1(X, \omega_X(p^eA-p^eQ))=h^0(X, -(p^eA-p^eQ))=0$$ for some point $Q\in X$. Note that the first equality follows from Serre duality. Assume the contrary, that is, assume that $p^eA\sim p^eQ$ for every point $Q\in X$. Since the genus $g(X)$ is not zero, there exists a non-zero $l$-torsion $D$ for a prime number $l\neq p$. Note that $D$ is not a $p^e$-torsion. Take the prime decomposition $$D=\sum m_iQ_i-\sum n_jR_j.$$ Since $\deg D=\sum m_i-\sum n_j=0$, we obtain the following contradiction $$\begin{aligned}
p^eD&=&\sum m_ip^eQ_i-\sum n_jp^eR_j\\
&\sim&\sum m_ip^eA-\sum n_jp^eA\\
&=&(\sum m_i-\sum n_j)p^eA\\
&=&0.\end{aligned}$$

In this step, we assume $\deg A=1$ and we prove the assertion.

We fix a point $Q\in X$ as in Step \[good-1-form\]. If $A\sim A'$, then the corresponding trace maps are the same by Lemma \[D-linearequ\]. Therefore, we may assume that $Q\not\in {{\operatorname{Supp}}}A$. By Step \[good-1-form\], there exists an element $$\eta\in H^0(X, \omega_X(p^eA-(p^e-1)Q))\setminus H^0(X, \omega_X(p^eA-p^eQ)).$$ Take the local ring $(R, \mathfrak m)$ corresponding to the point $Q$. Note that $F^e_*R$ is a free $R$-module. Let $\{x\}$ be the $p$-basis. Then, we obtain $$\omega_R=\bigoplus_{0\leq i<p^e}R^{p^e}x^idx.$$ Thus, we can write $$\eta|_{{{\operatorname{Spec}}}R}=\sum_{0\leq i<p^e}f_i^{p^e}x^idx.$$ $\eta\not\in H^0(X, \omega_X(p^eA-p^eQ))$ means $f_i\not\in \mathfrak m$ for some $0\leq i<p^e$. $\eta\in H^0(X, \omega_X(p^eA-(p^e-1)Q))$ means $f_i\in \mathfrak m$ for every $0\leq i<p^e-1$. Therefore, we obtain $f_{p^e-1}\not\in \mathfrak m.$ Then, we can find $c\in k^{\times}$ and $\mu\in \mathfrak m$ such that $$f_{p^e-1}=c+\mu.$$ By Remark \[trace-local\], we see ${{\operatorname{Tr}}}_{X}^e(A)(\eta)|_{{{\operatorname{Spec}}}R}\neq 0.$

\[curve-trace-cor\] Let $X$ be a smooth projective curve. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor. If $H^0(X, \omega_X(A))\neq 0$, then, for every $e\in\mathbb Z_{>0}$, the trace map $${{\operatorname{Tr}}}_X^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A))$$ is a non-zero map.

If $g(X)\geq 1$ where $g(X)$ is the genus of $X$, then the assertion follows from Theorem \[curve-trace-nonzero\]. Thus, we may assume that $X\simeq \mathbb P^1$. Since $\mathbb P^1$ is $F$-split, the trace map is surjective.

In characteristic zero, we obtain the following result by the Kodaira vanishing theorem. In positive characteristic, we obtain the following result by the trace map of Frobenius (Corollary \[curve-trace-cor\]).

\[surface-restriction\] Let $X$ be a smooth projective surface and let $C$ be a smooth prime divisor. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$. If $H^0(C, K_C+A)\neq0$, then the natural restriction map $$H^0(X, K_X+C+A)\to H^0(C, K_C+A)$$ is a non-zero map.

By Lemma \[trace-diagram\], we obtain the following commutative diagram [$$\begin{CD}
H^0(X, K_X+C+p^eA)@>>> H^0(C, K_{C}+p^eA)@>>> H^1(X, K_X+p^eA)\\
@VV{{\operatorname{Tr}}}_{X, C}^e(A) V @VV {{\operatorname{Tr}}}_{C}^e(A) V\\
H^0(X, K_X+C+A)@>>> H^0(C, K_C+A).
\end{CD}$$]{} Then, the assertion follows from Corollary \[curve-trace-cor\].

In characteristic zero, in the above situation, the restriction map is surjective by the Kodaira vanishing theorem. But, in positive characteristic, the restriction map is not surjective in general.

\[cex-surjective\] There exists a smooth projective surface $X$, a smooth prime divisor $H$ on $X$ and an ample [$\mathbb{Z}$]{}-divisor $A$ such that

1.  [$|K_X+H+A|$ is basepoint free. ]{}

2.  [The natural restriction map $$H^0(X, K_X+H+A)\to H^0(H, K_H+A)$$ is not surjective. ]{}

Let $X$ be a smooth projective surface and let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$ such that $$H^1(X, K_X+A)\neq 0.$$ We can find such a surface by [@Raynaud]. Take a smooth hyperplane section $H$ of $X$ such that $|K_X+H+A|$ is basepoint free and that $$H^1(X, K_X+H+A)=0.$$ Consider the exact sequence $$0\to \mathcal O_X(K_X+A)\to\mathcal O_X(K_X+H+A)\to\mathcal O_H(K_H+A)\to0.$$ Then, we obtain the following exact sequence $$H^0(X, K_X+H+A)\to H^0(H, K_H+A)\to H^1(X, K_X+A)\to 0.$$ Since $H^1(X, K_X+A)\neq 0$, the restriction map is not surjective.

We use the following corollary in Section 8.

\[nonzero-criterion\] Let $X$ be a smooth projective surface. Let $L$ be a [$\mathbb{Z}$]{}-divisor on $X$ such that $$L=C+M$$ where $C$ is a smooth prime divisor and $M$ is a nef and big [$\mathbb{Z}$]{}-divisor such that $M|_C$ is ample. If $H^0(C, K_C+M)\neq 0$, then, the trace map $${{\operatorname{Tr}}}_X^e(L):H^0(X, K_X+p^eL)\to H^0(X, K_X+L)$$ is a non-zero map.

By Lemma \[trace-diagram\], we obtain the following commutative diagram [$$\begin{CD}
H^0(X, K_X+p^eL)@>>> H^0(p^eC, K_{p^eC}+p^eM)@>>> H^1(X, K_X+p^eM)\\
@VV{{\operatorname{Tr}}}_{X}^e(L) V @VV \Psi V\\
H^0(X, K_X+L)@>>> H^0(C, K_C+M).
\end{CD}$$]{} By [@T1 Theorem 2.6], we have $H^1(X, K_X+p^eM)=0$ for $e\gg 0$. By Corollary \[curve-trace-cor\], $\Psi$ is a non-zero map because $\Psi$ satisfies $$\begin{aligned}
{{\operatorname{Tr}}}_C^e(M):H^0(C, K_{C}+p^eM)&\to &H^0(p^eC, K_{p^eC}+p^eM)\\
&\overset{\Psi}\to& H^0(C, K_C+M). \end{aligned}$$ Then, the trace map ${{\operatorname{Tr}}}_{X}^e(L)$ is also a non-zero map.

Surjectivity of the trace maps for surfaces ($\kappa=1$)
========================================================

In this section, we show the surjectivity of the trace map $$H^0(X, \omega_X(p^e(A+m(K_X+A))))\to H^0(X, \omega_X(A+m(K_X+A)))$$ for the case where $X$ is a surface and $\kappa(X, K_X+A)=1$. For this, we establish the following vanishing result.

\[ruled-R1\] Let $C$ be a smooth curve. Let $Y:=\mathbb P^1\times C$ and let $\pi:Y\to C$ be the projection. Let $f:X\to Y$ be the blowup at one point and let $$\theta:X\overset{f}\to Y\overset{\pi}\to C.$$ Let $A_X$ be a $\theta$-ample [$\mathbb{Z}$]{}-divisor on $X$. Then, $$R^1\theta_*(B_X^2(A_X))=0.$$

\[step-rational\] In this step, we assume $C$ is rational and we prove the assertion.

Since the assertion is local on $C$, we may assume $C\simeq \mathbb P^1.$ For an arbitrary ample [$\mathbb{Z}$]{}-divisor $A_C$ on $C$, by the Leray spectral sequence, we obtain the following exact sequence $$\begin{aligned}
0&\to & H^1(C, \theta_*(B_X^2(A_X))\otimes \mathcal O_C(A_C))\\
&\to & H^1(X, B_X^2(A_X+\theta^*A_C))\\
&\to & H^0(X, R^1\theta_*(B_X^2(A_X))\otimes \mathcal O_C(A_C))\to 0\\\end{aligned}$$ Let $A_C$ be an ample [$\mathbb{Z}$]{}-divisor on $C$ such that

1.  [$A_X+\theta^*A_C$ is ample.]{}

2.  [$R^1\theta_*(B_X^2(A_X))\otimes \mathcal O_C(A_C)$ is generated by global sections. ]{}

Then, it is sufficient to show $$H^1(X, B_X^2(A_X+\theta^*A_C))=0.$$ Since $X$ is a toric variety, this follows from Proposition \[F-split-B-vanish\].

\[step-H1-R1\] In this step, we prove that the following assertions are equivalent.

1.  [$R^1\theta_*(B_X^2(A_X))=0$.]{}

2.  [$H^1(X_c, B_X^2(A_X)|_{X_c})=0$ for every $c\in C$ where $X_c$ is the fiber.]{}

By [@Hartshorne Theorem 12.11], there exists an isomorphism $$R^1\theta_*(B_X^2(A_X))\otimes k(c)\simeq H^1(X_c, B_X^2(A_X)|_{X_c}).$$ By Nakayama’s lemma, if $$R^1\theta_*(B_X^2(A_X))\otimes k(c)=0,$$ then $R^1\theta_*(B_X^2(A_X))|_{U}=0$ for some open neighborhood of $b\in U\subset C$.

We can replace $C$ by a neighborhood of the point corresponding to the singular fiber. Then, we can find the following commutative diagram $$\begin{CD}
X'@<\gamma_X<< X \\
@VVf'V @VVf V\\
Y'@<\gamma_Y<< Y\\
@VV\pi'V @VV\pi V\\
C'@<\gamma_C<< C
\end{CD}$$ where $C'\simeq \mathbb A^1$, each square is a fiber product, $\gamma_C, \gamma_Y$ and $\gamma_X$ are finite étale morphisms. Let $\theta':=\pi'\circ f'.$ Note that $\gamma_X^*B_{X'}^2\simeq B_X^2$ by Lemma \[cartier-etale\].

Let $\tilde c\in C$ be the point corresponding to the singular fiber of $\theta$. Let $\tilde c':=\gamma_C(\tilde c).$ Then, $$\gamma:=\gamma_X|_{X_c}:X_c\to (X')_{c'}$$ is an isomorphism. We can find a $\theta'$-ample [$\mathbb{Z}$]{}-divisor $A_{X'}$ on $X'$ such that $\gamma^*(\mathcal O_{X'}(A_{X'})|_{X_{c'}})\simeq\mathcal O_X(A_X)|_{X_c}$. Then, we obtain $$\begin{aligned}
H^1(X_c, B_X^2(A_X))&=& H^1(X_c, \gamma^*B_{X'}^2(\gamma^*A_{X'}))\\
&=& H^1((X')_{c'}, B_{X'}^2(A_{X'}))\\
&=&0. \end{aligned}$$ The first equation follows from Lemma \[cartier-etale\]. The last equation follows from Step \[step-rational\] and Step \[step-H1-R1\]. Then, the assertion holds by Step \[step-H1-R1\].

Let us prove the main theorem in this section.

\[trace-surface-1\] Let $X$ be a smooth projective surface and let $A$ be an ample [$\mathbb{Z}$]{}-divisor such that $\kappa(X, K_X+A)=1$ and that $K_X+A$ is nef. Then, there exists $m_1\in\mathbb Z_{>0}$ such that the trace map ${{\operatorname{Tr}}}_X^e(A+m(K_X+A))$ $$H^0(X, \omega_X(p^e(A+m(K_X+A))))\to H^0(X, \omega_X(A+m(K_X+A)))$$ is surjective for every $m\geq m_1$ and for every $e\in\mathbb Z_{>0}$.

We see that $K_X+A$ is semi-ample by [@Fujita3].

In this step, we prove that, for some $n_0\in\mathbb Z_{>0}$, the complete linear system $$\Phi_{|n_0(K_X+A)|}=:\theta:X\to C$$ gives a ruled surface structure, that is, $\theta$ is a projective morphism to a smooth projective curve such that $\theta_*\mathcal O_X=\mathcal O_C$ and that a general fiber is $\mathbb P^1$.

We can find $n_0\in\mathbb Z_{>0}$ such that $$\Phi_{|n_0(K_X+A)|}=:\theta:X\to C$$ is a projective morphism to a smooth projective curve such that $\theta_*\mathcal O_X=\mathcal O_C$. Then, by [@Badescu Corollary 7.3], general fibers are integral. Since a general fiber $F$ satisfies $$0=(K_X+A)\cdot F>(K_X+F)\cdot F,$$ we see $F\simeq \mathbb P^1.$ Thus, $\theta$ gives a ruled surface structure.

\[step-R1\] In this step, we prove that it is sufficient to show $$R^1\theta_*(B_X^2(A'))=0$$ for every ample [$\mathbb{Z}$]{}-divisor $A'$.

By Remark \[cartier-remark\] and Remark \[cartier-trace\], we obtain the following exact sequence $$0\to B_X^2\to F_*\omega_X\overset{{{\operatorname{Tr}}}_X^1}\to \omega_X\to 0.$$ By Lemma \[e-vs-e+1\], it is sufficient to show that $$H^1(X, B_X^2(p^d(A+m(K_X+A))))=0$$ for every $0\leq d\leq e$. We can write $K_X+A=\theta^*H$ where $H$ is an ample [$\mathbb{Z}$]{}-divisor on $C$. By the Leray spectral sequence, we obtain $$\begin{aligned}
0&\to&H^1(C, \theta_*(B_X^2(p^d(A+m(K_X+A)))))=0\\
&\to&H^1(X, B_X^2(p^d(A+m(K_X+A))))\\
&\to&H^0(C, R^1\theta_*(B_X^2(p^d(A+m(K_X+A))))\to 0.\end{aligned}$$ The first term vanishes by the Serre vanishing theorem. Thus, it is sufficient to show that $R^1\theta_*(B_X^2(A'))=0$ for every ample [$\mathbb{Z}$]{}-divisor $A'$.

In this step, we prove the assertion. By Step \[step-R1\], it is sufficient to show $$R^1\theta_*(B_X^2(A'))=0$$ for an ample [$\mathbb{Z}$]{}-divisor $A'$. Let $$\theta:X\overset{f}\to Y\overset{\pi}\to C$$ where $\pi$ is a $\mathbb P^1$-bundle structure. Since the problem is local on $C$, we may assume that $\theta$ has only one singular fiber and that $Y=\mathbb P^1\times C.$ Let $F_s$ be the singular fiber. Then, we see $$0=(K_X+A)\cdot F_s=-2+A\cdot F_s.$$ Thus, $F_s$ has at most two irreducible components. This implies that $f$ is the blowup at one point. Then, the assertion follows from Proposition \[ruled-R1\].

Surjectivity of the trace maps for surfaces ($\kappa=2$)
========================================================

In this section, we show the surjectivity of the trace map $$H^0(X, \omega_X(p^e(A+m(K_X+A))))\to H^0(X, \omega_X(A+m(K_X+A)))$$ for the case where $X$ is a surface and $\kappa(X, K_X+A)=2$. Let us recall a lemma on the global generation.

\[uniform-generation\] Let $X$ be a smooth projective variety. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor and let $G$ be a coherent sheaf. Then, there exists $n_0\in\mathbb Z_{>0}$, depending only on $A$ and $G$, such that $$G(n_0A+N)$$ is generated by global sections for every nef [$\mathbb{Z}$]{}-divisor $N$.

The assertion immediately follows from the Castelnuovo–Mumford regularity ([@Lazarsfeld Theorem 1.8.5]) and the Fujita vanishing theorem ([@Fujita1 Theorem (1)], [@Fujita2 Section 5]).

To prove the surjectivity, we establish the following vanishing result.

\[birat-B-vanish\] Let $h:X\to Z$ be a birational morphism between smooth projective surfaces. Let $A_X$ be an ample [$\mathbb{Z}$]{}-divisor and let $A_Z$ be an ample [$\mathbb{Z}$]{}-divisor on $Z$. Then, there exists $m_0\in \mathbb Z_{>0}$ such that $$H^1(X, B_X^2(A_X+h^*(m_0A_Z+N_Z)))=0$$ for every nef [$\mathbb{Z}$]{}-divisor $N_Z$ on $Z$.

The birational morphism $h$ is an $n$-times blowups. We show the assertion by the induction on $n$.

\[step-n=0\] If $n=0$, then the assertion follows from the Fujita vanishing theorem ([@Fujita1 Theorem (1)], [@Fujita2 Section 5]). Thus, we may assume that $n>0$ and the assertion holds for $n-1$.

\[step-one-point\] In this step, we prove that we may assume that $h({{\operatorname{Ex}}}(h))$ is one point.

Let us consider the Leray spectral sequence $$\begin{aligned}
0&\to& H^1(Z, h_*(B_X^2(A_X+h^*(m_0A_Z+N_Z)))=0\\
&\to& H^1(X, B_X^2(A_X+h^*(m_0A_Z+N_Z)))\\
&\to& H^0(Z, R^1h_*(B_X^2(A_X+h^*(m_0A_Z+N_Z))))\to 0\\\end{aligned}$$ where the vanishing $H^1(Z, h_*(B_X^2(A_X+h^*(m_0A_Z+N_Z)))=0$ follows from the Fujita vanishing theorem ([@Fujita1 Theorem (1)], [@Fujita2 Section 5]). By Lemma \[uniform-generation\], the assertion is equivalent to the following vanishing: $$R^1h_*(B_X^2(A_X))=0.$$ Since this problem is local on $Z$, we may assume that $h({{\operatorname{Ex}}}(h))$ is one point.

From now on, we assume that $h({{\operatorname{Ex}}}(h))$ is one point.

\[step-s=0\] Let $$h:X\overset{f}\to Y\overset{g}\to Z$$ where $g$ is the blowup of $Z$ at the point. Let $E_Y$ be the $g$-exceptional curve. Note that $E_Y^2=-1$. We see $$g^*A_Z-\epsilon E_Y$$ is an ample [$\mathbb{Q}$]{}-divisor for every rational number $0<\epsilon\ll 1.$ Thus, by replacing $A_Z$ with its multiple, we may assume that $$A_Y:=g^*A_Z-lE_Y$$ is an ample [$\mathbb{Z}$]{}-divisor for some $l\in\mathbb Z_{>0}$. In particular, we obtain $$h^*A_Z=f^*A_Y+lf^*E_Y.$$ By the induction hypothesis, there exists $m_1\in\mathbb Z_{>0}$ such that $$H^1(X, B_X^2(A_X+f^*(m_1A_Y+N_Y)))=0$$ for every nef [$\mathbb{Z}$]{}-divisor $N_Y$ on $Y$. We have $$m_1h^*A_Z=m_1f^*A_Y+m_1lf^*E_Y.$$

\[step-construct-E\] Let $E_1,\cdots, E_n$ be the $h$-exceptional curves where $E_1$ be the proper transform of $E_Y$. In this step, we construct a sequence of [$\mathbb{Z}$]{}-divisors $$0=:E(0)\leq E(1)\leq E(2)\leq\cdots\leq E(R-1)\leq E(R):=f^*E_Y$$ such that

1.  [For every $0\leq r\leq R-1$, $E(r+1)-E(r)=E_j$ for some $1\leq j\leq n$.]{}

2.  [$E(r)\cdot E_i\geq -1$ for every $0\leq r\leq R$ and for every $1\leq i\leq n$. ]{}

We consider a decomposition into one point blowups: $$f:X=:X_n\overset{f_n}\to\cdots\overset{f_3}\to X_2\overset{f_2}\to X_1:=Y.$$ We may assume that, for every $2\leq j\leq n$, $E_j\subset X$ is the proper transform of the $f_j$-exceptional curve. For $1\leq j\leq i\leq n$, let $E_j^{(i)}\subset X_i$ be the image of $E_j$. Let $f_{i+1}({{\operatorname{Ex}}}(f_{i+1}))=:P_{i}\in X_{i}$ and let $$g_i:X_i\to Z.$$ Note that $P_i\in {{\operatorname{Ex}}}(g_i)$. Since ${{\operatorname{Supp}}}({{\operatorname{Ex}}}(g_i))$ is simple normal crossing, there are two cases:

1.  [$P_i\in E_j^{(i)}$ and $P_i\not\in E_{j'}^{(i)}$ for every $j'\neq j$.]{}

2.  [$P_i\in E_j^{(i)}\cap E_{j'}^{(i)}$ for some $j\neq j'$ and $P_i\not\in E_{j''}^{(i)}$ for every $j''\neq j, j'$.]{}

For $1\leq i\leq n$, we construct a finite sequence $({\rm Seq})_i$ of prime divisors on $X$ inductively as follows. Every member of $({\rm Seq})_i$ is $E_j$ for some $j$. Let $$\begin{aligned}
({\rm Seq})_1:=(E_1).\end{aligned}$$ Assume we obtain $({\rm Seq})_{i}$. We construct $({\rm Seq})_{i+1}$ as follows. There are two cases (1) and (2) as above. Assume (1), that is, $P_i\in E_j^{(i)}$ and $P_i\not\in E_{j'}^{(i)}$ for every $j'\neq j$. If $$({\rm Seq})_{i}=(\cdots, E_j,\cdots, E_{j'},\cdots),$$ then we define $({\rm Seq})_{i+1}$ by $$({\rm Seq})_{i+1}:=(\cdots, E_{i+1}, E_j,\cdots, E_{j'},\cdots).$$ In other words, we add $E_{i+1}$ only in front of $E_j$. Assume (2), that is, $P_i\in E_j^{(i)}\cap E_{j'}^{(i)}$ for some $j\neq j'$ and $P_i\not\in E_{j''}^{(i)}$ for every $j''\neq j, j'$. If $$({\rm Seq})_{i}=(\cdots, E_j,\cdots, E_{j'},\cdots, E_{j''},\cdots),$$ then we define $({\rm Seq})_{i+1}$ by $$({\rm Seq})_{i+1}:=(\cdots, E_{i+1}, E_j,\cdots, E_{i+1}, E_{j'},\cdots, E_{j''},\cdots).$$ In other words, we add $E_{i+1}$ only in front of $E_j$ and $E_{j'}$. We obtain a finite sequence $({\rm Seq})_i$ for $1\leq i\leq n.$ Let $$({\rm Seq})_n=(E_{a(1)}, E_{a(2)}, E_{a(3)},\cdots, E_{a(R)})$$ where $a(l)\in\{1, \cdots, n\}.$ We define a finite sequence $({\rm SEQ})$ by $$\begin{aligned}
({\rm SEQ})&=&
(E_{a(1)}, E_{a(1)}+E_{a(2)}, E_{a(1)}+E_{a(2)}+E_{a(3)},\cdots)\\
&=:&(E(1), E(2), E(3),\cdots, E(R)).\end{aligned}$$ By an inductive argument, we see $E(r)\cdot E_j\geq -1$ for every $j$ and $E(R)=f^*E_Y$. This implies the assertion.

\[step-construct-D\] In this step, we construct a sequence of [$\mathbb{Z}$]{}-divisors $$0=:D(0)\leq D(1)\leq D(2)\leq\cdots\leq D(S-1)\leq D(S):=m_1lf^*E_Y$$ such that

1.  [For every $0\leq s\leq S-1$, $D(s+1)-D(s)=E_j$ for some $j$. ]{}

2.  [$D(s)+A_X+m_1f^*A_Y$ is nef for every $0\leq s\leq S$. ]{}

We define the sequence $\{D(s)\}_{s=0}^S$ by $$\begin{aligned}
&&E(0),E(1),E(2),\cdots, E(R), \\
&&E(R)+E(1),E(R)+E(2),\cdots, 2E(R),\\
&&2E(R)+E(1),2E(R)+E(2),\cdots, 3E(R),\\
&&\cdots\\
&&(m_1l-1)E(R)+E(1),(m_1l-1)E(R)+E(2),\cdots, m_1lE(R).\end{aligned}$$ Then, the sequence $\{D(s)\}_{s=0}^S$ satisfies (a). Thus, we show (b). For every $0\leq s\leq S$, we can write $$D(s)+A_X+m_1f^*A_Y=E(r)+tf^*E_Y+A_X+m_1f^*A_Y.$$ for some $0\leq r\leq R$ and some $0\leq t\leq m_1l-1.$ To show this divisor is nef, it is sufficient to show $$(E(r)+tf^*E_Y+A_X+m_1f^*A_Y)\cdot E_j\geq 0$$ for every $1\leq j\leq n.$ By Step \[step-construct-E\], for every $2\leq j\leq n$, we obtain $$(E(r)+tf^*E_Y+A_X+m_1f^*A_Y)\cdot E_j=(E(r)+A_X)\cdot E_j\geq 0.$$ On the other hand, for the case where $j=1$, we see $$\begin{aligned}
(E(r)+tf^*E_Y+A_X+m_1f^*A_Y)\cdot E_1
&\geq & (tf^*E_Y+m_1f^*A_Y)\cdot E_1\\
&=&(tE_Y+m_1A_Y)\cdot E_Y\\
&\geq &(m_1lE_Y+m_1A_Y)\cdot E_Y\\
&=&0.\end{aligned}$$

\[step-trace\] For a [$\mathbb{Z}$]{}-divisor $D$ on $X$ and for a curve $E\simeq\mathbb P^1$ in $X$, by Lemma \[trace-diagram\], we obtain the following diagram: [$$\begin{CD}
0@>>> H^0(X, \omega_X(pD))@>>> H^0(X, \omega_X(E+pD))@>>> H^0(E, \omega_E(pD))@>>> 
H^1(X, \omega_X(pD))\\
@. @VV\alpha:={{\operatorname{Tr}}}_X(D) V @VV\beta:={{\operatorname{Tr}}}_{X, E}(D) V @VV\gamma:={{\operatorname{Tr}}}_E(D) V\\
0@>>>H^0(X, \omega_X(D))@>>> H^0(X, \omega_X(E+D))@>>> H^0(E, \omega_E(D))
\end{CD}$$]{} where the horizontal sequences are exact and the vertical arrows are the trace maps. Then, the following assertions hold.

1.  [$\gamma$ is surjective.]{}

2.  [If $H^1(X, \omega_X(pD))=0$ and $\alpha$ is surjective, then $\beta$ is also surjective. ]{}

3.  [Assume $\beta$ is surjective. Then, the trace map $${{\operatorname{Tr}}}_X(E+D):H^0(X, \omega_X(p(E+D)))\to H^0(X, \omega_X(E+D))$$ is also surjective.]{}

\(1) holds because $E\simeq \mathbb P^1$ is $F$-split (Proposition \[F-split-surje\]). (2) follows from the snake lemma. (3) follows from Remark \[pE-vs-E\].

\[step-final\] Let $m_2\in \mathbb Z_{>0}$ such that $$H^1(X, \omega_X(m_2h^*A_Z+N_X))=0$$ for every nef [$\mathbb{Z}$]{}-divisor $N_X$ on $X$. Note that, since $h^*A_Z$ is nef and big, we can find such an integer $m_2$ by [@T1 Theorem 2.6]. Let $m_0:=m_1+m_2$ and fix a nef [$\mathbb{Z}$]{}-divisor $N_Z$ on $Z$.

We would like to apply the diagram in Step \[step-trace\] for $$D=D(s)+A_X+m_1f^*A_Y+m_2h^*A_Z+h^*N_Z,\,\,\,\,E=D(s+1)-D(s)$$ where $0\leq s\leq S-1$. Note that, by Step \[step-construct-D\], this divisor $D$ is nef. By Step \[step-s=0\], $\alpha$ in Step \[step-trace\] is surjective for $$D=D(0)+A_X+m_1f^*A_Y+m_2h^*A_Z+h^*N_Z.$$ We see $$H^1(X, \omega_X(p(D(s)+A_X+m_1f^*A_Y+m_2h^*A_Z+h^*N_Z)))=0$$ by the choice of $m_2$. Therefore, by Step \[step-construct-D\] and Step \[step-trace\], we obtain the surjection $${{\operatorname{Tr}}}_X(D):H^0(X, \omega_X(pD))\to H^0(X, \omega_X(D))$$ for $$\begin{aligned}
D&=&D(S)+A_X+m_1f^*A_Y+m_2h^*A_Z+h^*N_Z\\
&=&m_1lf^*E_Y+A_X+m_1f^*A_Y+m_2h^*A_Z+h^*N_Z\\
&=&A_X+m_1h^*A_Z+m_2h^*A_Z+h^*N_Z\\
&=&A_X+(m_1+m_2)h^*A_Z+h^*N_Z\\
&=&A_X+h^*(m_0A_Z+N_Z).\end{aligned}$$ Thus, the assertion follows from $$H^1(X, \omega_X(p(A_X+h^*(m_0A_Z+N_Z))))=0.$$

\[birat-surje\] Let $h:X\to Z$ be a birational morphism between smooth projective surfaces. Let $A_X$ be an ample [$\mathbb{Z}$]{}-divisor and let $A_Z$ be an ample [$\mathbb{Z}$]{}-divisor on $Z$. Then, there exists $m_1\in \mathbb Z_{>0}$ such that the trace map ${{\operatorname{Tr}}}^e_X(A_X+h^*(m_1A_Z+N_Z))$ [ $$H^0(X, \omega_X(p^e(A_X+h^*(m_1A_Z+N_Z))))\to H^0(X, \omega_X(A_X+h^*(m_1A_Z+N_Z)))$$]{} is surjective for every $e\in\mathbb Z_{>0}$ and for every nef [$\mathbb{Z}$]{}-divisor $N_Z$ on $Z$.

By Lemma \[e-vs-e+1\], we obtain $${{\operatorname{Tr}}}_{X}^{d+1}(D)={{\operatorname{Tr}}}_{X}^d(D)\circ F_*^d({{\operatorname{Tr}}}_{X}(p^d D)).$$ Thus, the assertion follows from Proposition \[birat-B-vanish\].

Let us prove the main theorem in this section.

\[trace-surface-2\] Let $X$ be a smooth projective surface. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$ such that $K_X+A$ is nef and big. Then, there exists $m_1\in\mathbb Z_{>0}$ such that the trace map ${{\operatorname{Tr}}}_X^e(A+m(K_X+A))$ $$H^0(X, \omega_X(p^e(A+m(K_X+A))))\to H^0(X, \omega_X(A+m(K_X+A)))$$ is surjective for every $m\geq m_1$ and for every $e\in\mathbb Z_{>0}$.

By Proposition \[birat-surje\], it is sufficient to prove that there exists a birational morphism $$h:X\to Z$$ to a smooth projective surface $Z$ such that $K_X+A$ is the pull-back of an ample [$\mathbb{Z}$]{}-divisor. If $K_X+A$ is ample, then there is nothing to show. We may assume that $K_X+A$ is not ample. Then, by the Nakai–Moishezon criterion, we can find a curve $E$ such that $(K_X+A)\cdot E=0$. This means $K_X\cdot E<0$. Moreover, since $K_X+A$ is big, the equation $(K_X+A)\cdot E=0$ implies $E^2<0$. Therefore, $E$ is a $(-1)$-curve. Contract $E$ and we can repeat this procedure. Then, we obtain $h:X\to Z$.

Main theorem for threefolds
===========================

In this section, we prove the main theorem for threefolds. Let us summarize results on the trace map obtained in previous sections.

\[trace-surface\] Let $X$ be a smooth projective surface. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$ such that $K_X+A$ is nef and $\kappa(X, K_X+A)\neq 0$. Then, there exists $m_1\in\mathbb Z_{>0}$ such that the trace map ${{\operatorname{Tr}}}_X^e(A+m(K_X+A))$ $$H^0(X, \omega_X(p^e(A+m(K_X+A))))\to H^0(X, \omega_X(A+m(K_X+A)))$$ is surjective for every $m\geq m_1$ and for every $e\in\mathbb Z_{>0}$.

If $\kappa(X, K_X+A)=-\infty$, then there is nothing to show. Thus, we may assume $\kappa(X, K_X+A)\geq 1$. Then, the assertion follows from Theorem \[trace-surface-1\] and Theorem \[trace-surface-2\]

In the above situation, we can show $\kappa(X, K_X+A)\neq -\infty$ by the abundance theorem obtained in [@Fujita3]. Indeed, by Bertini’s theorem, we can find an effective [$\mathbb{Q}$]{}-divisor $D$ such that $\llcorner D\lrcorner=0$ and that $A\sim_{\mathbb Q}D.$

Let us prove the main theorem.

\[threefold-extension\] Let $X$ be a smooth projective threefold. Let $S$ be a smooth prime divisor on $X$ and let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$ such that

1.  [$K_X+S+A$ is nef. ]{}

2.  [$\kappa(S, K_S+A)\neq 0.$]{}

Then, there exists $m_0\in \mathbb Z_{>0}$ such that, for every $m\geq m_0$, the natural restriction map $$H^0(X, m(K_X+S+A))\to H^0(S, m(K_S+A))$$ is surjective.

Let $L:=K_X+S+A$. By Lemma \[trace-diagram\], we obtain the following commutative diagram [ $$\begin{CD}
H^0(X, \omega_X(S+p^eA+p^emL))@>>> H^0(S, \omega_S(p^eA+mp^eL)))@>>> 
H^1(X, \omega_X(S+p^eA+p^emL))\\
@VV{{\operatorname{Tr}}}_{X,S}^e(A+mL) V @VV{{\operatorname{Tr}}}_S^e(A+mL) V\\
H^0(X, \omega_X(S+A+mL))@>>> H^0(S, \omega_S(A+mL)).
\end{CD}$$]{} By (2) and Theorem \[trace-surface\], the trace map ${{\operatorname{Tr}}}_S^e(A+mL)$ $$H^0(S, K_S+p^eA+p^emL)\to H^0(S, K_S+A+mL)$$ is surjective for $m\gg 0.$ By the Serre vanishing thoerem, we have $$H^1(X, \omega_X(S+p^eA+p^emL))=0.$$ Therefore, the natural restriction map $$H^0(X, m(K_X+S+A))\to H^0(S, m(K_S+A))$$ is surjective.

Calculation for the case where $\kappa=0$
=========================================

In this section, we consider whether Theorem \[trace-surface\] holds for $\kappa(X, K_X+A)=0$. Let $X$ be a smooth projective surface and let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$. Assume that $K_X+A$ is nef and that $\kappa(X, K_X+A)=0$. By the abundance theorem ([@Fujita3]), we see $K_X+A\sim_{\mathbb Q} 0$. Then, $-K_X$ is ample. In particular, $X$ is a rational surface. Thus, we see $K_X+A\sim 0$. We would like to consider the following question.

\[ques-del-Pezzo\] Let $X$ be a smooth projective surface such that $-K_X$ is ample. Is the trace map $${{\operatorname{Tr}}}_X^e(-K_X):H^0(X, \omega_X(-p^eK_X)\to H^0(X, \omega_X(-K_X))$$ surjective?

If $K_X^2\geq 4$, then we obtain an affirmative answer.

Let $X$ be a smooth projective surface such that $-K_X$ is ample. If $K_X^2\geq 4$, then the trace map $${{\operatorname{Tr}}}_X^e(-K_X):H^0(X, \omega_X(-p^eK_X))\to H^0(X, \omega_X(-K_X))$$ is surjective.

Since $h^0(X, \omega_X(-K_X))=1$, it is sufficient to show that the trace map $${{\operatorname{Tr}}}_X^e(-K_X):H^0(X, \omega_X(-p^eK_X))\to H^0(X, \omega_X(-K_X))$$ is a non-zero map. Since $K_X^2\geq 4$, $X$ is obtained by blowing up $\mathbb P^2$ at most $5$ points. Therefore, we can find a smooth conic $C_0$ passing through these points. Let $L_0$ be a line which does not passes through these points. Let $C$ and $L$ be the proper transforms. We see that $L|_C$ is ample, $H^0(C, \omega_C(L|_C))\neq 0$ and $L$ is nef and big. Then, since $C+L\in |-K_X|$, we can apply Corollary \[nonzero-criterion\].

If $X$ is $F$-split, then, by Proposition \[F-split-surje\], the above trace map ${{\operatorname{Tr}}}_X^e(-K_X)$ is surjective. Note that, by [@Hara Example 5.5] and [@Smith Proposition 4.10], if $K_X^2\geq 4$, then $X$ is $F$-split. But, since [@Hara] has no explicit proof, we give the above proof. Moreover, [@Hara Example 5.5] and [@Smith Proposition 4.10] shows that, if $K_X^2=3$ and $X$ is not $F$-split, then $X$ is a Fermat type cubic surface in characteristic two. Indeed, this example gives a negative answer to Question \[ques-del-Pezzo\] as follows.

\[trace-cex\] Let ${\rm char}\,k=p=2$. Consider $\mathbb P^3$ and let $[x:y:z:w]$ be the homogeneous coordinate. Let $$X:=\{[x:y:z:w]\in \mathbb P^3\,|\,x^3+y^3+z^3+w^3=0\}.$$ Then, the trace map $${{\operatorname{Tr}}}_X^e(-K_X):H^0(X, \omega_X(-2^eK_X)\to H^0(X, \omega_X(-K_X))$$ is a zero map.

By Lemma \[e-vs-e+1\], we may assume $e=1$. By Lemma \[trace-diagram\], we obtain the following commutative diagram $$\begin{CD}
H^0(\mathbb P^3, \omega_{\mathbb P^3}(X-2K_{\mathbb P^3}-2X))@>\beta>> H^0(X, \omega_{X}(-2K_{X}))\\
@VV{{\operatorname{Tr}}}_{\mathbb P^3, X}(-K_{\mathbb P^3}-X)V @VV{{\operatorname{Tr}}}_X(-K_X)V\\
H^0(\mathbb P^3, \omega_{\mathbb P^3}(X-K_{\mathbb P^3}-X))@>\alpha>> H^0(X, \omega_{X}(-K_{X})).
\end{CD}$$ We prove that $\alpha$ and $\beta$ are isomorphisms. Since $H^1(\mathbb P^3, L)=0$ for an arbitrary invertible sheaf $L$, it is sufficient to show $$\begin{aligned}
H^0(\mathbb P^3, \omega_{\mathbb P^3}(-K_{\mathbb P^3}-X))
=H^0(\mathbb P^3, \omega_{\mathbb P^3}(-2K_{\mathbb P^3}-2X))
=0.\end{aligned}$$ This follows from $$H^0(\mathbb P^3, \omega_{\mathbb P^3}(-K_{\mathbb P^3}-X))
=H^0(\mathbb P^3, \mathcal O_{\mathbb P^3}(-3))=0$$ and $$H^0(\mathbb P^3, \omega_{\mathbb P^3}(-2K_{\mathbb P^3}-2X))
=H^0(\mathbb P^3, \mathcal O_{\mathbb P^3}(-4+8-6))=0.$$ Therefore, it is sufficient to prove that the trace map ${{\operatorname{Tr}}}_{\mathbb P^3, X}(-K_{\mathbb P^3}-X)$ is a zero map. By Lemma \[D-linearequ\], we obtain $${{\operatorname{Tr}}}_{\mathbb P^3, X}(-K_{\mathbb P^3}-X)={{\operatorname{Tr}}}_{\mathbb P^3, X}(H)$$ where $H$ is defined by $$H:=\{[x:y:z:w]\in \mathbb P^3\,|\,w=0\}.$$ Thus, we show that $${{\operatorname{Tr}}}:={{\operatorname{Tr}}}_{\mathbb P^3, X}(H):H^0(\mathbb P^3, \omega_{\mathbb P^3}(X+2H))\to 
H^0(\mathbb P^3, \omega_{\mathbb P^3}(X+H))$$ is a zero map. Let us take a $k$-linear basis of $H^0(\mathbb P^3, \omega_{\mathbb P^3}(X+2H))$. It is easy to see $$h^0(\mathbb P^3, \omega_{\mathbb P^3}(X+2H))=4.$$ Let ${{\operatorname{Spec}}}\,k[X, Y, Z]\subset \mathbb P^3$ be the affine open subset defined by $w\neq 0$. Consider the following four $3$-forms $$\begin{aligned}
\eta_1&:=&\frac{1}{X^3+Y^3+Z^3+1}dX\wedge dY\wedge dZ\\
\eta_X&:=&\frac{X}{X^3+Y^3+Z^3+1}dX\wedge dY\wedge dZ\\
\eta_Y&:=&\frac{Y}{X^3+Y^3+Z^3+1}dX\wedge dY\wedge dZ\\
\eta_Z&:=&\frac{Z}{X^3+Y^3+Z^3+1}dX\wedge dY\wedge dZ.\\\end{aligned}$$ These are elements of $\omega_{k(X, Y, Z)}=(\omega_{\mathbb P^3})_{\xi}$ where $\xi$ is the generic point of $\mathbb P^3$. By a direct calculation, these four elements are linearly independent and $\eta_1, \eta_X,\eta_Y, \eta_Z\in H^0(\mathbb P^3, \omega_{\mathbb P^3}(X+2H)).$ In particular, these four elements form a $k$-linear basis of $H^0(\mathbb P^3, \omega_{\mathbb P^3}(X+2H))$. The trace map is a $p^{-1}$-linear map, that is, for $a, b, c, d\in k$, $$\begin{aligned}
&&{{\operatorname{Tr}}}(a\eta_1+b\eta_X+c\eta_Y+d\eta_Z)\\
&=&a^{\frac{1}{p}}{{\operatorname{Tr}}}(\eta_1)+b^{\frac{1}{p}}{{\operatorname{Tr}}}(\eta_X)+c^{\frac{1}{p}}{{\operatorname{Tr}}}(\eta_Y)+d^{\frac{1}{p}}{{\operatorname{Tr}}}(\eta_Z).\end{aligned}$$ Thus, it is sufficient to show $${{\operatorname{Tr}}}(\eta_1)={{\operatorname{Tr}}}(\eta_X)={{\operatorname{Tr}}}(\eta_Y)={{\operatorname{Tr}}}(\eta_Z)=0.$$ Let us only prove ${{\operatorname{Tr}}}(\eta_X)=0$. This follows from $$\begin{aligned}
&&({{\operatorname{Tr}}}(\eta_X))|_{{{\operatorname{Spec}}}\,k[X, Y, Z]}\\
&=&{{\operatorname{Tr}}}(\frac{X}{X^3+Y^3+Z^3+1}dX\wedge dY\wedge dZ)\\
&=&{{\operatorname{Tr}}}(\frac{X(X^3+Y^3+Z^3+1)}{(X^3+Y^3+Z^3+1)^2}dX\wedge dY\wedge dZ)\\
&=&\frac{1}{X^3+Y^3+Z^3+1}{{\operatorname{Tr}}}((X^4+XY^3+XZ^3+X)dX\wedge dY\wedge dZ)\\
&=&0.\end{aligned}$$ The last equality follows from Remark \[trace-local\].

The following example shows that we cannot prove Theorem \[threefold-extension\] for the case where $\kappa=0$ by the same proof.

Let ${\rm char}\,k=p=2$. Then, there exist smooth projective threefold $X$ over $k$, a smooth prime divisor $S_0$ on $X$ and an ample [$\mathbb{Z}$]{}-divisor $A$ on $X$ which satisfy the following properties.

1.  [$|K_X+S_0+A|$ is basepoint free. ]{}

2.  [The natural restriction map $$H^0(X, m(K_X+S_0+A))\to H^0(S_0, m(K_{S_0}+A))$$ is surjective for every $m\in\mathbb Z_{>0}$.]{}

3.  [ The trace map ${{\operatorname{Tr}}}_{S_0}^e(A+m(K_{S_0}+A))$ [$$H^0(S_0, \omega_{S_0}(2^e(A+m(K_{S_0}+A))))\to 
    H^0(S_0, \omega_{S_0}(A+m(K_{S_0}+A)))$$]{} is a zero map for every $m\in \mathbb Z_{>0}$ and for every $e\in\mathbb Z_{>0}.$]{}

Let $S$ be the surface in Theorem \[trace-cex\] and let $A_S:=-K_S$. Let $C$ be a smooth projective curve and fix an arbitrary ample [$\mathbb{Z}$]{}-divisor $A_C$ on $C$. Let $X:=S\times C$ and let $\pi_S$ and $\pi_C$ be their projection respectively. Fix a point $c_0\in C$ and let $S_0:=S\times \{c_0\}$. Let $$A:=\pi_S^*A_S+\pi_C^*A_C.$$ Note that $A|_{S_0}=-K_{S_0}$. Thus, (3) follows from Theorem \[trace-cex\]. (1) follows from $$\begin{aligned}
K_X+S_0+A&=&\pi_S^*(K_S+A_S)+\pi_C^*(K_C+c_0+A_C)\\
&=&\pi_C^*(K_C+c_0+A_C).\end{aligned}$$ It is sufficient to show (2). This follows from [ $$\begin{aligned}
&&H^1(X, K_X+A+(m-1)(K_X+S_0+A))\\
&=&H^1(X, \pi_C^*(K_C+A_C+(m-1)(K_C+c_0+A_C)))\\
&\simeq&H^1(S, \mathcal O_S)\otimes_k H^0(C, K_C+A_C+(m-1)(K_C+c_0+A_C))\\
&\oplus &H^0(S, \mathcal O_S)\otimes_k H^1(C, K_C+A_C+(m-1)(K_C+c_0+A_C))\\
&=&0.\end{aligned}$$]{} The last equality holds by $H^1(S, \mathcal O_S)=0$ and $$\deg_C(A_C+(m-1)(K_C+c_0+A_C))>0.$$

Appendix: Extension theorem for surfaces
========================================

For the surface case, we can freely use the minimal model theory (cf. [@Fujita3], [@KK], [@T2]). By using results obtained in [@Fujita3], [@T2] and [@T3], we can establish an analogy of [@HM Theorem 5.4.21] as follows.

\[surface-ext\] Let $X$ be a smooth projective surface and let $C$ be a smooth prime divisor on $X$. Let $\Delta:=C+B$ where $B$ is an effective [$\mathbb{Q}$]{}-divisor which satisfies the following properties.

1.  [$C\not\subset{{\operatorname{Supp}}}B$, $\llcorner B\lrcorner=0$ and $(X, \Delta)$ is plt. ]{}

2.  [$B\sim_{\mathbb Q}A+F$ where $A$ is an ample [$\mathbb{Q}$]{}-divisor and $F$ is an effective [$\mathbb{Q}$]{}-divisor such that $C\not\subset {{\operatorname{Supp}}}F.$]{}

3.  [No prime component of $\Delta$ is contained in the stable base locus of $K_X+\Delta.$]{}

Then, there exists an integer $m_0>0$ such that, for every integer $m>0$, the restriction map $$H^0(X, mm_0(K_X+\Delta))\to H^0(C, mm_0(K_X+\Delta)|_C)$$ is surjective.

In this step we prove that, if $E$ is a curve in $X$ such that $E^2<0$ and $(K_X+\Delta)\cdot E<0$, then the following three assertions hold:

1.  [$K_X\cdot E=E^2=-1$.]{}

2.  [$E$ is not a prime component of $\Delta$.]{}

3.  [$E\cdot C=0$.]{}

Since $(K_X+E)\cdot E\leq (K_X+\Delta)\cdot E<0$, there exists a birational morphism $f:X\to Y$ to a normal [$\mathbb{Q}$]{}-factorial surface $Y$ such that ${{\operatorname{Ex}}}(f)=E$ (cf. [@T2 Theorem 6.2]). Let $\Delta_Y:=f_*\Delta$ and we define $d\in\mathbb Q$ by $$K_X+\Delta=f^*(K_Y+\Delta_Y)+dE.$$ The inequality $(K_X+\Delta)\cdot E<0$ means $d>0.$ We can find a integer $l>0$ such that $l(K_Y+\Delta_Y)$ is Cartier. Then, $E$ is a fixed component of $$l(K_X+\Delta)=f^*(l(K_Y+\Delta_Y))+dlE.$$ Then, the assumption (3) implies (b). This means $E\cdot \Delta\geq 0.$ Then, the assertion (a) follows from $$K_X\cdot E\leq (K_X+\Delta)\cdot E<0.$$ Let us show (c). Assume $E\cdot C>0.$ Then, $E\cdot C\geq 1.$ This means the following contradiction $$0>(K_X+\Delta)\cdot E=K_X\cdot E+C\cdot E+B\cdot E\geq -1+1+0=0.$$

In this step, we prove that we may assume that $K_X+\Delta$ is nef.

Assume that $K_X+\Delta$ is not nef. Then, there exists a curve $E$ such that $(K_X+\Delta)\cdot E<0$. By (3), we can find $l>0$ such that $|l(K_X+\Delta)|\neq\emptyset.$ This means $E^2<0$. We see that $E$ is a $(-1)$-curve by Step 1. Let $f:X\to Y$ be the contraction of $E$. Let $$\Delta_Y:=f_*\Delta\,,\,\,C_Y:=f_*C\,,\,\,B_Y:=f_*B\,,\,\,A_Y:=f_*A\,,\,\,F_Y:=f_*F.$$ Then, $Y$ and these divisors also satisfies the conditions (1), (2) and (3). Let $m_1>0$ be an integer such that $m_1\Delta$ is a [$\mathbb{Z}$]{}-divisor. Then, we have $$m_1(K_X+\Delta)=f^*(m_1(K_Y+\Delta_Y))+eE$$ for some $e\in\mathbb Z_{>0}.$ Let $n$ be an arbitrary positive integer. By $f_*\mathcal O_X=\mathcal O_Y$, we have [$$f_*(\mathcal O_X(nm_1(K_X+\Delta)))\simeq f_*(\mathcal O_X(nm_1f^*(K_Y+\Delta_Y)))\simeq 
\mathcal O_Y(nm_1(K_Y+\Delta_Y)).$$]{} By $C\cap E=\emptyset$, we see [$$f_*(\mathcal O_C(nm_1(K_X+\Delta)))\simeq f_*(\mathcal O_C(nm_1f^*(K_Y+\Delta_Y)))\simeq 
\mathcal O_{C_Y}(nm_1(K_Y+\Delta_Y)).$$]{} These implies the following commutative diagram $$\begin{CD}
H^0(X, nm_1(K_X+\Delta))@>>> H^0(C, nm_1(K_X+\Delta)|_C)\\
@| @|\\
H^0(Y, nm_1(K_Y+\Delta_Y))@>>> H^0(C_Y, nm_1(K_Y+\Delta_Y)|_{C_Y})
\end{CD}$$ where the horizontal arrows are the natural restriction maps. Thus, we can reduce the problem on $X$ to the problem on $Y$. We can repeat the same argument. Note that $|l(K_X+\Delta)|\neq\emptyset$ and this means the assertion.

By the abundance theorem ([@Fujita3]), $K_X+\Delta$ is semi-ample. Let $$f:=\varphi_{|m_2(K_X+\Delta)|}:X\to R$$ for some $m_2\in\mathbb Z_{>0}$ such that $f_*\mathcal O_X=\mathcal O_R.$ In this step, we prove $f_*\mathcal O_C=\mathcal O_{f(C)}$.

Assume $f_*\mathcal O_C\neq \mathcal O_{f(C)}$. We run the $(K_X+\{\Delta\})$-MMP. By [@T3], there exists a morphisms $$X\overset{g}\to V\to R$$ where $V$ is a smooth projective curve such that a general fiber $G$ of $g$ satisfies $G\simeq\mathbb P^1$ and $\llcorner\Delta\lrcorner\cdot G=2.$ Note that $B=\{\Delta\}$ and $C=\llcorner\Delta\lrcorner$. Since $G$ is a fiber and $B$ is big, we see $G\cdot B>0$. Then, we obtain the following contradiction $$0=(K_X+\Delta)\cdot G\geq(K_X+B)\cdot G+2>(K_X+G)\cdot G+2=0.$$

In this step, we prove the assertion. Let $$f:=\varphi_{|m_2(K_X+\Delta)|}:X\to R$$ such that $f_*\mathcal O_X=\mathcal O_R$ and let $f(C)=:D.$ Then, by Step 3, we see $f_*\mathcal O_C=\mathcal O_D.$ Let $H$ be an ample Cartier divisor on $R$ such that $m_2(K_X+\Delta)=f^*H.$ By the Serre vanishing theorem, we can find $m_3\in\mathbb Z_{>0}$ such that $$H^0(R, mm_3H)\to H^0(D, mm_3 H)$$ is surjective for every $m\in \mathbb Z_{>0}.$ Then, by $f_*\mathcal O_X=\mathcal O_R$ and $f_*\mathcal O_C=\mathcal O_D$, we have the following commutative diagram $$\begin{CD}
H^0(X, mm_2m_3(K_X+\Delta))@>>> H^0(C, mm_2m_3(K_X+\Delta)|_C)\\
@| @|\\
H^0(R, mm_3H)@>>> H^0(D, mm_3H|_{D}).
\end{CD}$$ This implies the assertion.

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