On p-power freeness in positive characteristic

TL;DR

This paper investigates p-power freeness in positive characteristic, establishing criteria similar to semi-ampleness and applying them to three-dimensional birational geometry.

Contribution

It introduces criteria for p-power freeness and demonstrates their application in three-dimensional birational geometry.

Findings

Criteria for p-power freeness established

Application to three-dimensional birational geometry demonstrated

Analogues of semi-ampleness criteria developed

Abstract

In this note, we study base point freeness up to taking p-power, which we will call p-power freeness. We first establish some criteria for p-power freeness as analogues of criteria for semi-ampleness. We then apply these results to three-dimensional birational geometry.

Equations

$$f^{\ast }{f}_{\ast }L\to L$$

$$g_{\ast }((L^{\otimes m}{\otimes }_{{\mathcal{O}}_X}{A}^{-1}){|}_{X_{\mathrm{red}}})\ne 0.$$

$${\mathbb{B}}_f^p(L)=\bigcap _{e=0}^{\infty }\mathrm{Supp}\mathrm{Coker}(f^{\ast }{f}_{\ast }{L}^{\otimes p^e}\to L^{\otimes p^e}).$$

$$\begin{array}{ccc}{X}^{\prime }& \stackrel{\alpha }{\to }& X\\ \downarrow f^{\prime }& & \downarrow f\\ Y^{\prime }& \stackrel{\beta }{\to }& Y\end{array}$$

$$f^{\prime }:X^{\prime }\stackrel{\pi }{\to }X\stackrel{f}{\to }Y$$

$$F^e:X\stackrel{\rho }{\to }{X}^{\prime }\stackrel{\pi }{\to }X.$$

$$L^{\otimes p^e}=(F^e{)}^{\ast }L={\rho }^{\ast }{\pi }^{\ast }L.$$

$$f:X\stackrel{g}{\to }Z\stackrel{h}{\to }Y,$$

$$d_e={\dim }_{k(y)}{H}^0(X_y,L^{\otimes p^e}{|}_{X_y})$$

$$z\in \bigcap _{m\in {\mathbb{Z}}_{>0}}{U}_m.$$

$$H^0(f^{-1}(U_{e_0}),L^{\otimes p^{e_0}}{|}_{f^{-1}(U_{e_0})})\to H^0(X_z,L^{\otimes p^{e_0}}{|}_{X_z})$$

$${\mathbb{B}}_f^p(L)={\mathbb{B}}_g^p(L{|}_{{\mathbb{E}}_f(L)})$$

$$K_X+\Delta =K_X+f_{\ast }^{-1}{\Delta }_Y+E{\sim }_{\mathbb{R}}{f}^{\ast }(K_Y+{\Delta }_Y)+E^{\prime }$$

$$X=:X_0⇢X_1⇢\cdots ⇢X_N.$$

$$\begin{array}{ccc}W& \stackrel{\psi }{\to }& X^{\prime }\\ \downarrow \phi & & \downarrow f^{\prime }\\ X& \stackrel{f}{\to }& Y\end{array}$$

$$\begin{array}{ccc}W& \stackrel{\psi }{\to }& X^{\prime }\\ \downarrow \phi & & \downarrow f^{\prime }\\ X& \stackrel{f}{\to }& Z\end{array}$$

$$f:X\stackrel{g}{\to }Y\stackrel{f_Y}{\to }Z,$$

Full text

1. 1 Introduction

2. 1.1 Overviews of proofs

3. 2 Preliminaries

4. 2.1 Notation

5. 2.2 Properties of divisors

6. 3 $p$-power freeness

7. 3.1 Definition

8. 3.2 Fibrewise $p$-power freeness

9. 3.3 Keel’s theorem

10. 4 Application to threefolds

1. Introduction

It is a fundamental problem to study torsion line bundles in algebraic geometry. If $L$ is an $m$-torsion line bundle on an algebraic variety $X$ of characteristic zero, then $L$ induces an étale cover of degree $m$. The same statement holds in characteristic $p>0$ when $m$ is not divisible by $p$. However, if $m$ is divisible by $p$, then the resulting cyclic cover $Y\to X$ is no longer étale. Therefore $p^{e}$-torsion line bundles have different feature from $\ell$-torsion line bundles for $\ell\in\mathbb{Z}_{>0}\setminus p\mathbb{Z}$.

To observe another phenomenon on $p^{e}$-torsion line bundles, let us recall the Lefschetz theorem for the local Picard groups (cf. [BdJ14, Theorem 0.1]). Let $(A,\mathfrak{m})$ be an excellent normal local ring containing a field. For $f\in\mathfrak{m}\setminus\{0\}$, $V:={\operatorname{Spec}}\,A\setminus\{\mathfrak{m}\}$, and $V_{0}:={\operatorname{Spec}}\,(A/f)\setminus\{\mathfrak{m}\}$, the restriction map $\rho:{\rm Pic}(V)\to{\rm Pic}(V_{0})$ satisfies the following properties.

1. (0)

Assume that $A$ is of characteristic zero. If ${\rm depth}_{\mathfrak{m}}(A/f)\geq 2$, then $\rho$ is injective. 2. (p)

Assume that $A$ is of positive characteristic. Then $\rho$ is injective up to taking $p$-power, i.e. if $\rho(L)\simeq\mathcal{O}_{V_{0}}$, then $L^{\otimes p^{e}}\simeq\mathcal{O}_{V}$ for some $e\in\mathbb{Z}_{>0}$.

In this theorem, we need to care $p^{e}$-torsion line bundles for the case of positive characteristic, whilst there does not appear such non-trivial torsion line bundles in the corresponding statement of characteristic zero. One of the purposes of this note is to observe a similar phenomenon that appears in birational geometry.

To this end, we first study the behaviour of $p^{e}$-torsion line bundles in general settings. For flexibility, we study a wider notion: base point freeness up to $p$-power, which we will call $p$-power freeness.

Definition 1.1**.**

Let $f:X\to Y$ be a proper morphism of noetherian $\mathbb{F}_{p}$-schemes. Let $L$ be an invertible sheaf on $X$.

1. (1)

We say that $L$ is $f$-free if the induced homomorphism

[TABLE]

is surjective. If $Y={\operatorname{Spec}}\,k$ for a field $k$, then we simply say that $L$ is free. 2. (2)

We say that $L$ is p-power $f$-free or p-power free over $Y$ if there exists a positive integer $e\in\mathbb{Z}_{>0}$ such that $L^{\otimes p^{e}}$ is $f$-free. If $Y={\operatorname{Spec}}\,k$ for a field $k$, then we simply say that $L$ is p-power free.

There are some criteria for semi-ampleness which hold only in positive characteristic. A typical result is Keel’s theorem [Kee99, Theorem 0.2]. Another example is the equivalence between the relative semi-ampleness and the fibrewise semi-ampleness [CT, Theorem 1.1]. It is remarkable that these two semi-ampleness criteria have analogous statements for $p$-power freeness as follows.

Theorem 1.2** (Theorem 3.4).**

Let $f:X\to Y$ be a projective morphism of noetherian $\mathbb{F}_{p}$-schemes. Let $L$ be an invertible sheaf on $X$. If $L|_{X_{y}}$ is $p$-power free for any point $y\in Y$, then $L$ is $p$-power $f$-free.

Theorem 1.3** (Theorem 3.6).**

Let $f:X\to Y$ be a projective morphism of noetherian $\mathbb{F}_{p}$-schemes. Let $L$ be an $f$-nef invertible sheaf on $X$ and let $g:\mathbb{E}_{f}(L)\hookrightarrow X\xrightarrow{f}Y$ be the induced morphism. Then $L$ is $p$-power $f$-free if and only if $L|_{\mathbb{E}_{f}(L)}$ is $p$-power $g$-free.

We then apply these criteria to birational geometry in positive characteristic. Let us first recall the Kawamata–Shokurov base point free theorem in characteristic zero [KMM87, Theorem 3-1-1].

Theorem 1.4** (Kawamata–Shokurov).**

Let $k$ be a field of characteristic zero. Let $(X,\Delta)$ be a klt pair over $k$ and let $f:X\to Z$ be a projective $k$-morphism to a quasi-projective $k$-scheme $Z$. Let $L$ be an $f$-nef Cartier divisor such that $L-(K_{X}+\Delta)$ is $f$-nef and $f$-big. Then there exists a positive integer $m_{0}$ such that $mL$ is $f$-free for any integer $m\geq m_{0}$.

It is known that the same statement is no longer true in positive characteristic [Tan1, Theorem 1.2]. More specifically, over an algebraically closed field $k$ of characteristic $p\in\{2,3\}$, there exist a three-dimensional klt pair $(X,\Delta)$, a projective morphism $f:X\to Z$ to a smooth curve $Z$, and an $f$-numerically trivial Cartier divisor $L$ on $X$ such that $L-(K_{X}+\Delta)$ is $f$-ample and $L\not\sim_{f}0$. On the other hand, it holds that $pL\sim_{f}0$ for this example. Then it is tempting to hope that $L$ is $p$-power free in the case of positive characteristic.

Conjecture 1.5**.**

Let $k$ be a field of characteristic $p>0$. Let $(X,\Delta)$ be a klt pair over $k$ and let $f:X\to Z$ be a projective $k$-morphism to a quasi-projective $k$-scheme $Z$. Let $L$ be an $f$-nef Cartier divisor on $X$ such that $L-(K_{X}+\Delta)$ is $f$-nef and $f$-big. Then $L$ is $p$-power $f$-free, i.e. there exists a positive integer $e$ such that $p^{e}L$ is $f$-free.

Remark 1.6**.**

Assume that $\dim X=2$. Then the same statement of Theorem 1.4 holds for the case when $k$ is a perfect field of characteristic $p>0$ (Lemma 4.1). However, if $k$ is allowed to be an imperfect field, then the same statement of Theorem 1.4 no longer holds [Tan1, Theorem 1.4]. On the other hand, Conjecture 1.5 is known to hold when $X$ is a surface over an imperfect field (cf. Remark 4.2).

In this note, we establish the following partial solution (Theorem 1.7) as an application of our criteria for $p$-power freeness. Note that our proof depends on the recent development of minimal model program for threefolds of characteristic $p>5$ ([HX15], [CTX15], [Bir16], [BW17], [Wal18], [HNT]).

Theorem 1.7** (Theorem 4.4).**

Let $k$ be a perfect field of characteristic $p>5$. Let $(X,\Delta)$ be a three-dimensional klt pair and let $f:X\to Z$ be a projective surjective $k$-morphism to a quasi-projective $k$-scheme $Z$. Let $L$ be an $f$-nef Cartier divisor on $X$ such that $L-(K_{X}+\Delta)$ is $f$-nef and $f$-big. Assume that either

1. (1)

$\dim Z\geq 1$, or 2. (2)

$\dim Z=0$* and $L\not\equiv 0$.*

Then $L$ is $p$-power $f$-free.

Remark 1.8**.**

Under the same assumptions as the ones of Theorem 1.7, Bernasconi proves that if $p\gg 0$, $\dim Z\geq 1$, and $L$ is a Cartier divisor on $X$ such that $L\equiv_{f}0$, then $L\sim_{f}0$ [Ber].

1.1. Overviews of proofs

We now overview how to prove the main theorems. Let us first discuss the criteria for $p$-power freeness (Theorem 1.2, Theorem 1.3). Theorem 1.3 follows from the same argument as the proof of Keel’s theorem given by [CMM14]. Theorem 1.2 is the $p$-power free version of [CT, Theorem 1.1]. Although the same argument as in [CT, Theorem 1.1] does not work for our case, Theorem 1.2 follows from a combination of [CT, Theorem 1.1] and a recent result by Bhatt–Scholze [BS17].

We now give an overview of how to show Theorem 1.7. The first step is to establish a birational version as follows.

Theorem 1.9** (Theorem 4.3).**

Let $k$ be a perfect field of characteristic $p>5$. Let $f:X\to Y$ be a projective birational $k$-morphism of quasi-projective normal threefolds over $k$. Assume that there exists an effective $\mathbb{R}$-divisor $\Delta_{Y}$ on $Y$ such that $(Y,\Delta_{Y})$ is klt. Let $L$ be a Cartier divisor on $X$ such that $L\equiv_{f}0$. Then there exist a positive integer $e$ and a Cartier divisor $L_{Y}$ on $Y$ such that $L\sim p^{e}L_{Y}$.

Roughly speaking, the proof of Theorem 1.9 is done by resolution of singularities and minimal model program. Thus, a crucial part is the case when $f$ is an extremal contraction of pl-type (cf. Step 1 of Theorem 4.3). In this case, the problem is reduced to the case of dimension two by using Theorem 1.3.

In order to prove Theorem 1.7, we utilise a variant of Theorem 1.2 (Theorem 3.5), so that it is enough to show that $L|_{X_{z}}$ is $p$-power free for any closed point $z\in Z$. Theorem 1.9 enables us to take birational model changes. Then we may assume that $T:=(X_{z})_{{\operatorname{red}}}$ is a surface of Fano type, i.e. there exists an effective $\mathbb{Q}$-divisor $\Delta_{T}$ such that $(T,\Delta_{T})$ is klt and $-(K_{T}+\Delta_{T})$ is ample (cf. Step 1 of Theorem 4.4). Then $L|_{(X_{z})_{{\operatorname{red}}}}$ is trivial, hence also $p^{e}L|_{X_{z}}$ is trivial for some $e\in\mathbb{Z}_{>0}$. In particular, $L|_{X_{z}}$ is $p$-power free. For more details, see Section 4.

Acknowledgements: The author would like to thank Fabio Bernasconi for useful comments. The author was funded by the Grant-in-Aid for Scientific Research (KAKENHI No. 18K13386).

2. Preliminaries

2.1. Notation

1. (1)

We will freely use the notation and terminology in [Har77] and [Kol13]. 2. (2)

For a scheme $X$, its reduced structure $X_{{\operatorname{red}}}$ is the reduced closed subscheme of $X$ such that the induced morphism $X_{{\operatorname{red}}}\to X$ is surjective. 3. (3)

For a field $k$, we say that $X$ is a variety over $k$ or a $k$-variety if $X$ is an integral scheme that is separated and of finite type over $k$. We say that $X$ is a curve over $k$ or a $k$-curve (resp. a surface over $k$ or a $k$-surface, resp. a threefold over $k$) if $X$ is a $k$-variety of dimension one (resp. two, resp. three). 4. (4)

Let $f:X\to Y$ be a morphism of noetherian schemes. We say that $f$ is projective if there exists a closed immersion $X\hookrightarrow\mathbb{P}^{n}_{Y}$ over $Y$ for some $n\in\mathbb{Z}_{>0}$. This definition coincides with the one in [Har77, page 103], but differs from the one given by Grothendieck [Gro61, Définition 5.5.2]. On the other hand, their definitions coincide in many cases (cf. [FGAex, Section 5.5.1]). 5. (5)

A morphism $f:X\to Y$ of schemes has connected fibres if $X\times_{Y}{\operatorname{Spec}}\,L$ is either empty or connected for any field $L$ and any morphism ${\operatorname{Spec}}\,L\to Y$.

2.2. Properties of divisors

Let $f:X\to Y$ be a proper morphism of noetherian schemes and let $L$ be an invertible sheaf on $X$.

1. (1)

$L$ is $f$-nef if for any field $K$, morphism ${\operatorname{Spec}}\,K\to Y$, and a curve $C$ on $X\times_{Y}{\operatorname{Spec}}\,K$, the inequality $\alpha^{*}L\cdot C\geq 0$ holds for the induced morphism $\alpha:X\times_{Y}{\operatorname{Spec}}\,K\to X$. 2. (2)

$L$ is $f$-numerically trivial if both $L$ and $L^{-1}$ are $f$-nef. 3. (3)

$L$ is $f$-free if the natural homomorphism $f^{*}f_{*}L\to L$ is surjective. In particular, if $L$ is $f$-free then it induces a morphism $X\to\mathbb{P}(f_{*}L)$ over $Y$. 4. (4)

$L$ is $f$-very ample if it is $f$-free and the induced morphism $X\to\mathbb{P}(f_{*}L)$ is a closed immersion. 5. (5)

$L$ is $f$-semi-ample (resp. $f$-ample) if $L^{\otimes m}$ is $f$-free (resp. $f$-very ample) for some positive integer $m$. 6. (6)

$L$ is $f$-weakly big if there exist an $f$-ample invertible sheaf $A$ on $X$ and a positive integer $m$ such that if $g:X_{{\operatorname{red}}}\to Y$ denotes the induced morphism, then

[TABLE]

Assume that $X$ is normal. $L$ is $f$-big if, for any connected component $X^{\prime}$ of $X$, the restriction $L|_{X^{\prime}}$ is $f^{\prime}$-weakly big, where $f^{\prime}:X^{\prime}\to Y$ denotes the induced morphism. 7. (7)

If $L$ is $f$-nef, the $f$-exceptional locus of $L$, denoted by $\mathbb{E}_{f}(L)$, is defined as the union of all the reduced closed subschemes $V\subset X$ such that $L|_{V}$ is not $f|_{V}$-weakly big. It is known that $\mathbb{E}_{f}(L)$ is a closed subset of $X$ [CT, Lemma 2.18]. We consider $\mathbb{E}_{f}(L)$ as a reduced closed subscheme of $X$.

3. $p$-power freeness

In this section, we first introduce $p$-power freeness for invertible sheaves (Subsection 3.1). In Subsection 3.2, we prove that the relative $p$-power freeness is equivalent to the fibrewise $p$-power freeness (Theorem 3.4). In Subsection 3.3, we establish the $p$-power free version of Keel’s theorem (Theorem 3.6).

3.1. Definition

Definition 3.1**.**

Let $f:X\to Y$ be a proper morphism of noetherian $\mathbb{F}_{p}$-schemes. Let $L$ be an invertible sheaf.

1. (1)

We say that $L$ is p-power $f$-free or p-power free over $Y$ if there exists a positive integer $e\in\mathbb{Z}_{>0}$ such that $L^{\otimes p^{e}}$ is $f$-free (for definition of being $f$-free, see Subsection 2.2(3)). If $Y={\operatorname{Spec}}\,k$ for a field $k$, then we simply say that $L$ is p-power free. 2. (2)

The $p$-power $f$-base locus of $L$ is defined as the following closed subset of $X$:

[TABLE]

Note that $L$ is $p$-power $f$-free if and only if $\mathbb{B}^{p}_{f}(L)=\emptyset$.

Lemma 3.2**.**

Let

[TABLE]

be a cartesian diagram consisting of morphisms of noetherian $\mathbb{F}_{p}$-schemes, where $f$ is proper and $\beta$ is faithfully flat. Let $L$ be an invertible sheaf on $X$. Then $L$ is $p$-power $f$-free if and only if $\alpha^{*}L$ is $p$-power $f^{\prime}$-free.

Proof.

The assertion follows from the corresponding statement for usual freeness. ∎

Lemma 3.3**.**

Let

[TABLE]

be proper morphisms of noetherian $\mathbb{F}_{p}$-schemes. Assume that $\pi$ is a surjective morphism which have connected fibres. Let $L$ be an invertible sheaf on $X$. Then $L$ is $p$-power $f$-free if and only if $\pi^{*}L$ is $p$-power $f^{\prime}$-free.

Proof.

Taking the Stein factorisation of $\pi$, we may assume that $\pi_{*}\mathcal{O}_{X^{\prime}}=\mathcal{O}_{X}$ or $\pi$ is finite. If $\pi_{*}\mathcal{O}_{X^{\prime}}=\mathcal{O}_{X}$, then the assertion follows from the corresponding statement for usual freeness. Thus we may assume that $\pi$ is finite. In this case, $\pi$ is a finite universal homeomorphism. Assuming that $\pi^{*}L$ is $p$-power $f^{\prime}$-free, it suffices to show that $L$ is $p$-power $f$-free. It follows from [Kol97, Proposition 6.6] that there exists a positive integer $e$ such that the $e$-th iterated absolute Frobenius morphism $F^{e}:X\to X$ factors through $\pi$:

[TABLE]

Since $\pi^{*}L$ is $p$-power free over $Y$, so is its pullback

[TABLE]

Hence, $L$ is $p$-power free over $Y$. ∎

3.2. Fibrewise $p$-power freeness

In this subsection, we prove that the relative $p$-power freeness is equivalent to the fibrewise $p$-power freeness (Theorem 3.4). If the base field is uncountable, then we obtain a stronger criterion (Theorem 3.5).

Theorem 3.4**.**

Let $f:X\to Y$ be a projective morphism of noetherian $\mathbb{F}_{p}$-schemes. Let $L$ be an invertible sheaf on $X$. If $L|_{X_{y}}$ is $p$-power free for any point $y\in Y$, then $L$ is $p$-power $f$-free.

Proof.

It follows from [CT, Theorem 1.1] that $L$ is $f$-semi-ample. Thus, there exist a positive integer $m$, projective morphisms

[TABLE]

and an ample invertible sheaf $A_{Z}$ on $Z$ such that $L^{\otimes m}\simeq g^{*}A_{Z}$ and $g_{*}\mathcal{O}_{X}=\mathcal{O}_{Z}$. Since $L|_{X_{y}}$ is $p$-power free for any point $y\in Y$, $L_{X_{z}}$ is $p$-power free for any point $z\in Z$. By noetherian induction, we can find $e\in\mathbb{Z}_{>0}$ such that $L^{\otimes p^{e}}|_{X_{z}}$ is free for any $z\in Z$. Since $L^{\otimes p^{e}}|_{X_{z}}$ is numerically trivial, we obtain $L^{\otimes p^{e}}|_{X_{z}}\simeq\mathcal{O}_{X_{z}}$. Therefore, by [BS17, Theorem 1.3], there exists $d\in\mathbb{Z}_{>0}$ and an invertible sheaf $L_{Z}$ on $Z$ such that $L^{\otimes p^{d}}\simeq g^{*}L_{Z}$. Since $L_{Z}$ is ample over $Y$, $L_{Z}$ is $p$-power free over $Y$. Hence, also its pullback $L^{\otimes p^{d}}\simeq g^{*}L_{Z}$ is $p$-power free over $Y$. Therefore, $L$ is $p$-power free over $Y$. ∎

Although the proof of the following theorem is very similar to the one of [CT, Lemma 6.1], we give a proof for the sake of completeness.

Theorem 3.5**.**

Let $k$ be an uncountable field of characteristic $p>0$. Let $f:X\to Y$ be a projective $k$-morphism of schemes which are of finite type over $k$. Let $L$ be an invertible sheaf on $X$. If $L|_{X_{y}}$ is $p$-power free for any closed point $y\in Y$, then $L$ is $p$-power $f$-free.

Proof.

We prove the assertion by induction on $\dim Y$. If $\dim Y=0$, then there is nothing to show. Thus, we may assume that $\dim Y>0$ and that the assertion holds if the dimension of the base is smaller than $\dim Y$. By Theorem 3.4, it is enough to show that $L|_{X_{\xi}}$ is $p$-power free for the generic point $\xi\in Y$ of an irreducible component of $Y$. Replacing $Y$ by an open neighbourhood of $\xi\in Y$, the problem is reduced to the case when $Y$ is an affine irreducible scheme such that $f$ is flat.

By the semi-continuity theorem [Har77, Ch. III, Theorem 12.8], if $e$ is a positive integer, then there exist a positive integer $d_{e}$ and a non-empty affine open subset $U_{e}\subset Y$ such that the equation

[TABLE]

holds for any point $y\in U_{e}$. Since $k$ is uncountable, there exists a closed point

[TABLE]

As $L|_{X_{z}}$ is $p$-power free, there exists a positive integer $e_{0}$ such that $L^{\otimes p^{e_{0}}}|_{X_{z}}$ is free. By [Har77, Ch. III, Corollary 12.9], the restriction map

[TABLE]

is surjective. Since the base locus of $L^{\otimes p^{e_{0}}}$ is a closed subset of $X$, it is disjoint from $X_{z}$. In particular, $L^{\otimes p^{e_{0}}}|_{X_{\xi}}$ is free, as desired. ∎

3.3. Keel’s theorem

We have the $p$-power free version of Keel’s theorem on semi-ampleness (Theorem 3.6). For a later use, we also establish a variant (Proposition 3.7).

Theorem 3.6**.**

Let $f:X\to Y$ be a projective morphism of noetherian $\mathbb{F}_{p}$-schemes. Let $L$ be an $f$-nef invertible sheaf on $X$ and let $g:\mathbb{E}_{f}(L)\hookrightarrow X\xrightarrow{f}Y$ be the induced morphism. Then the equation

[TABLE]

holds. In particular, $L$ is $p$-power $f$-free if and only if $L|_{\mathbb{E}_{f}(L)}$ is $p$-power $g$-free.

Proof.

We may apply the same argument as in [CT, Proposition 2.20] after replacing the stable base loci $\mathbb{B}(-)$ by the $p$-power base loci $\mathbb{B}^{p}(-)$ (for the definition of $\mathbb{B}^{p}(-)$, see Definition 3.1). ∎

Proposition 3.7**.**

Let $k$ be a field of characteristic $p>0$. Let $f:X\to Y$ be a projective $k$-morphism from a normal $k$-variety $X$ to a scheme $Y$ which is of finite type over $k$. Let $L$ be an $f$-nef Cartier divisor. Assume that there exist an $f$-ample $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $A$ and an effective $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $E$ such that $L\equiv_{f}A+E$. If $L|_{{\operatorname{Supp}}\,E}$ is $p$-power free over $Y$, then $L$ is $p$-power free over $Y$.

Proof.

After replacing $A$ by $A^{\prime}:=L-E$, we may assume that the equation $L=A+E$ of $\mathbb{Q}$-divisors holds. It is enough to show the inclusion $\mathbb{E}_{f}(L)\subset{\operatorname{Supp}}\,E$, which follows from [CT, Lemma 2.18(1)]. ∎

4. Application to threefolds

As applications of results in Section 3, we prove Theorem 4.3 and Theorem 4.4. We start with a result on surfaces.

Lemma 4.1**.**

Let $k$ be a perfect field of characteristic $p>0$. Let $(X,\Delta)$ be a two-dimensional klt pair and let $f:X\to Y$ be a projective $k$-morphism to a quasi-projective $k$-scheme $Y$. Let $L$ be an $f$-nef Cartier divisor on $X$ such that $L-(K_{X}+\Delta)$ is $f$-nef and $f$-big. Then there exists a positive integer $m_{0}$ such that $mL$ is $f$-free for any integer $m\geq m_{0}$. In particular, $L$ is $p$-power $f$-free.

Proof.

We may assume that $k$ is algebraically closed and $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$. If either $\dim Y\geq 1$ or $\dim Y=0$ and $L\not\equiv 0$, then the assertion follows from [Tan18, Theorem 4.2]. Thus, we may assume that $\dim Y=0$ and $L\equiv 0$. In this case, the assertion follows from [Tan15, Corollary 3.6]. ∎

Remark 4.2**.**

If we allow $k$ to be an imperfect field, then the same statement as in Lemma 4.1 no longer holds [Tan1, Theorem 1.4]. On the other hand, even if $k$ is an imperfect field, the $p$-power freeness of $L$ holds. Indeed, if either $\dim Y\geq 1$ or $\dim Y=0$ and $L\not\equiv 0$, then the assertion follows from [Tan18, Theorem 4.2]. If $\dim Y=0$ and $L\equiv 0$, then we obtain $p^{e}L\sim 0$ by [BT, Theorem 1.3]. We do not use this fact in this paper.

Theorem 4.3**.**

Let $k$ be a perfect field of characteristic $p>5$. Let $f:X\to Y$ be a projective birational $k$-morphism of quasi-projective normal threefolds over $k$. Assume that there exists an effective $\mathbb{R}$-divisor $\Delta_{Y}$ on $Y$ such that $(Y,\Delta_{Y})$ is klt. Let $L$ be a Cartier divisor on $X$ such that $L\equiv_{f}0$. Then there exist a positive integer $e$ and a Cartier divisor $L_{Y}$ on $Y$ such that $L\sim p^{e}L_{Y}$.

Proof.

The proof consists of three steps.

Step 1**.**

The assertion of Theorem 4.3 holds if $Y$ is $\mathbb{Q}$-factorial.

Proof of Step 1.

Taking a log resolution of $(Y,\Delta_{Y})$ which dominates $X$, we may assume that $f:X\to Y$ is a log resolution of $(Y,\Delta_{Y})$. Let $E$ be the reduced $f$-exceptional divisor such that ${\operatorname{Supp}}\,E={\operatorname{Ex}}(f)$. Set $\Delta:=f_{*}^{-1}\Delta_{Y}+E$. Then we have

[TABLE]

where $E^{\prime}$ is an $f$-exceptional effective $\mathbb{R}$-divisor on $X$ such that ${\operatorname{Supp}}\,E^{\prime}={\operatorname{Ex}}(f)$. By [HNT, Theorem 1.1], there exists a $(K_{X}+\Delta)$-MMP over $Y$ that terminates:

[TABLE]

The negativity lemma implies that $f_{N}:X_{N}\to Y$ is small. Since $Y$ is $\mathbb{Q}$-factorial, $f_{N}$ is an isomorphism.

Therefore, it is enough to treat the case when $(X,\Delta)$ is a $\mathbb{Q}$-factorial dlt pair, $-(K_{X}+\Delta)$ is $f$-ample, $\rho(X/Y)=1$, and $-S$ is $f$-ample for some prime divisor $S$ contained in ${\operatorname{Supp}}\,\llcorner\Delta\lrcorner$. It holds that $A:=L-S$ is an $f$-ample $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor. Since $L|_{S}$ is $p$-power free over $Y$ (Lemma 4.1), it follows from Proposition 3.7 that $L$ is $p$-power free over $Y$. This completes the proof of Step 1. ∎

Step 2**.**

The assertion of Theorem 4.3 holds if there exists an effective $\mathbb{Q}$-divisor $\Delta$ on $X$ such that $(X,\Delta)$ is klt and $-(K_{X}+\Delta)$ is $f$-nef and $f$-big.

Proof of Step 2.

Set $\mathcal{L}:=\mathcal{O}_{X}(L)$. Taking the base change to an uncountable algebraically closed field, we may assume that $k$ is an uncountable algebraically closed field (Lemma 3.2). For an arbitrary closed point $y\in Y$, it is enough to show that $\mathcal{L}|_{X_{y}}$ is $p$-power free (Theorem 3.5). By [GNT, Proposition 2.15], there exists a commutative diagram consisting of projective birational morphisms of quasi-projective normal threefolds

[TABLE]

and an effective $\mathbb{Q}$-divisor $\Delta^{\prime}$ on $X^{\prime}$ which satisfy the following properties.

1. (1)

$(X^{\prime},\Delta^{\prime})$ is a $\mathbb{Q}$-factorial plt threefold and $-(K_{X^{\prime}}+\Delta^{\prime})$ is $f^{\prime}$-ample. 2. (2)

$-\llcorner\Delta^{\prime}\lrcorner$ is $f^{\prime}$-nef and $f^{\prime-1}(z)_{{\operatorname{red}}}=\llcorner\Delta^{\prime}\lrcorner$. 3. (3)

For $T:=\llcorner\Delta^{\prime}\lrcorner$, there exists an effective $\mathbb{Q}$-divisor $\Delta_{T}$ on $T$ such that $(T,\Delta_{T})$ is klt and $-(K_{T}+\Delta_{T})$ is ample.

Set $\mathcal{L}_{W}:=\varphi^{*}\mathcal{L}$. After replacing $\mathcal{L}$ by $\mathcal{L}^{\otimes p^{e}}$ for some $e\in\mathbb{Z}_{>0}$, Step 1 enables us to find an invertible sheaf $\mathcal{L}^{\prime}$ on $X^{\prime}$ such that $\mathcal{L}_{W}\simeq\psi^{*}\mathcal{L}^{\prime}$. Then we have that $\mathcal{L}^{\prime}|_{(X^{\prime}_{y})_{{\operatorname{red}}}}=\mathcal{L}^{\prime}|_{T}$ is $p$-power free (Lemma 4.1). It follows from Lemma 3.3 that also $\mathcal{L}^{\prime}|_{X^{\prime}_{y}}$ is $p$-power free. Since $\mathcal{L}_{W}|_{W_{y}}$ is the pullback of $\mathcal{L}^{\prime}|_{X^{\prime}_{y}}$, it holds that $\mathcal{L}_{W}|_{W_{y}}$ is $p$-power free. Finally, since $W_{y}\to X_{y}$ has connected fibres, it follows again from Lemma 3.3 that $\mathcal{L}|_{X_{y}}$ is $p$-power free. This completes the proof of Step 2. ∎

Step 3**.**

The assertion of Theorem 4.3 holds without any additional assumptions.

Proof of Step 3.

By the same argument as in Step 1, the problem is reduced to the case when $X$ is $\mathbb{Q}$-factorial and $f$ is a small birational morphism. In particular, for $\Delta:=f_{*}^{-1}\Delta_{Y}$, it holds that $K_{X}+\Delta=f^{*}(K_{Y}+\Delta_{Y})$. In particular, $-(K_{X}+\Delta)$ is $f$-nef and $f$-big. Then we may apply Step 2. This completes the proof of Step 3. ∎

Step 3 completes the proof of Theorem 4.3. ∎

Theorem 4.4**.**

Let $k$ be a perfect field of characteristic $p>5$. Let $(X,\Delta)$ be a three-dimensional klt pair and let $f:X\to Z$ be a projective surjective $k$-morphism to a quasi-projective $k$-scheme $Z$. Let $L$ be an $f$-nef Cartier divisor on $X$ such that $L-(K_{X}+\Delta)$ is $f$-nef and $f$-big. Assume that either

1. (1)

$\dim Z\geq 1$, or 2. (2)

$\dim Z=0$* and $L\not\equiv 0$.*

Then $L$ is $p$-power $f$-free.

Proof.

The proof consists of two steps.

Step 1**.**

The assertion of Theorem 4.4 holds if (1) holds.

Proof of Step 1.

Set $\mathcal{L}:=\mathcal{O}_{X}(L)$. Then, by Lemma 3.2, we may assume that $k$ is an uncountable algebraically closed field.

Fix a closed point $z\in Z$. By [GNT, Proposition 2.15], there exists a commutative diagram consisting of projective morphisms of quasi-projective normal varieties

[TABLE]

and an effective $\mathbb{Q}$-divisor $\Delta^{\prime}$ on $X^{\prime}$ which satisfy the following properties.

1. (1)

$(X^{\prime},\Delta^{\prime})$ is a $\mathbb{Q}$-factorial plt threefold and $-(K_{X^{\prime}}+\Delta^{\prime})$ is $f^{\prime}$-ample. 2. (2)

$-\llcorner\Delta^{\prime}\lrcorner$ is $f^{\prime}$-nef and $f^{\prime-1}(z)_{{\operatorname{red}}}=\llcorner\Delta^{\prime}\lrcorner$. 3. (3)

$W$ is a smooth threefold, and both of $\varphi$ and $\psi$ are projective birational morphisms. 4. (4)

For $T:=\llcorner\Delta^{\prime}\lrcorner$, there exists an effective $\mathbb{Q}$-divisor $\Delta_{T}$ on $T$ such that $(T,\Delta_{T})$ is klt and $-(K_{T}+\Delta_{T})$ is ample.

Then, by the same argument as in Step 2 of Theorem 4.3, it holds that $\mathcal{L}|_{X_{z}}$ is $p$-power free for any closed point $z\in Z$. Then it follows from Theorem 3.5 that $\mathcal{L}$ is $p$-power $f$-free. This completes the proof of Step 1 ∎

Step 2**.**

The assertion of Theorem 4.4 holds if (2) holds.

Proof of Step 2.

It follows from [GNT, Theorem 2.9] that $L$ is $f$-semi-ample. Hence, there exist a positive integer $m$, projective morphisms

[TABLE]

and an ample Cartier divisor $A$ on $Y$ such that $g_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$ and $mL\sim g^{*}A$. In particular, we have $L\equiv_{g}0$. By (2), it holds that $\dim Y>0$. By Step 1, there exist $e\in\mathbb{Z}_{>0}$ and an $f_{Y}$-ample Cartier divisor $A^{\prime}$ on $Y$ such that $p^{e}L\sim g^{*}A^{\prime}$. This completes the proof of Step 2. ∎

Step 1 and Step 2 complete the proof of Theorem 4.4. ∎