# Vanishing theorems for Fano threefolds in positive characteristic

Tatsuro Kawakami  Department of Mathematics, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan<EMAIL_ADDRESS>and  Hiromu
Tanaka  Graduate School of Mathematical Sciences, The University of Tokyo,
3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, JAPAN<EMAIL_ADDRESS>
###### Abstract.

We prove that Kodaira vanishing holds for an arbitrary smooth Fano threefold
in positive characteristic. To this end, we show that it is quasi-$F$-split
when the Picard number or the Fano index is larger than one. We also establish
Akizuki-Nakano vanishing for smooth Fano threefolds when the Picard number or
the Fano index is larger than one, and therefore they lift to the ring of Witt
vectors.

###### Key words and phrases:

Fano threefolds, (quasi-)$F$-split, Kodaira vanishing

###### 2020 Mathematics Subject Classification:

14J45, 13A35, 14F17

###### Contents

  1. 1 Introduction
    1. 1.1 Akizuki-Nakano vanishing
    2. 1.2 $F$-splitting
    3. 1.3 Strategy of proofs
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 Cartier operators
    3. 2.3 Generic members
    4. 2.4 $F$-splitting criteria
    5. 2.5 Quasi-$F$-splitting criteria
    6. 2.6 Weak del Pezzo surfaces
  3. 3 Fano threefolds with $\rho\geq 4$
  4. 4 Fano threefolds with $\rho=3$ (except for 3-10)
  5. 5 Fano threefolds with $\rho=2$ (except for 2-2, 2-6, 2-8)
    1. 5.1 Quasi-$F$-splitting (imprimitive case)
    2. 5.2 Quasi-$F$-splitting (primitive case)
    3. 5.3 $F$-splitting
  6. 6 $F$-splitting and Quasi-$F$-splitting via Cartier operators
    1. 6.1 Quasi-$F$-splitting for 2-2, 2-6, 2-8, and 3-10
    2. 6.2 $F$-splitting for 2-2 and 2-6
  7. 7 Proofs of the main theorems
  8. 8 Examples

##  1\. Introduction

In the context of the minimal model program, Fano varieties play a significant
role in the classification of algebraic varieties. The classification of Fano
varieties in characteristic zero has a long history. In the early nineteenth
century, Gino Fano established a partial classification result for smooth Fano
threefolds, in order to attack the rationality problem of cubic threefolds. In
1980s, the classification of smooth Fano threefolds was carried out by
Mori–Mukai ([MM81], [MM83]), based on earlier works by Iskovskih and Shokurov.
Recently, this classification result has been extended to the case of positive
characteristic [SB97, Meg98, FanoI, FanoII, FanoIII, FanoIV].

In this paper, our focus lies in vanishing theorems for them. Kodaira
vanishing for Fano varieties is particularly crucial since their anti-
canonical divisors are ample. For example, Kodaira vanishing implies
$H^{i}(X,\mathcal{O}_{X}(D))=0$ for every nef divisor $D$ on a smooth Fano
variety $X$. From now on, we work over an algebraically closed field $k$ of
characteristic $p>0$. Let $X$ be a smooth Fano threefold over $k$. Then, for
an ample Cartier divisor $A$, the vanishing $H^{1}(X,\mathcal{O}_{X}(-A))=0$
has been proven by Shepherd-Barron and the first author [SB97], [Kaw2, Theorem
1.1] (cf. [FanoI, Theorem 2.4]). If $\rho(X)=r_{X}=1$ for the Picard number
$\rho(X)$ and the index $r_{X}$, then Kodaira vanishing is established in
[FanoI, Corollary 4.5]. In this case, the vanishing is simple since $A$ can be
written as $A\sim-nK_{X}$. On the other hand, as the Picard number increases,
the description of ample divisors become more complicated, making it harder to
establish Kodaira vanishing.

To address this issue, we turn to $F$-splitting, as it implies Kodaira
vanishing. Consequently, investigating $F$-splitting of smooth Fano threefolds
leads us to our goal: Kodaira vanishing for smooth Fano threefolds. However,
it is known that smooth del Pezzo surfaces of characteristic $p\leq 5$ are not
necessarily $F$-split. Then, taking the product with $\mathbb{P}^{1}$, we can
exhibit non-$F$-split smooth Fano threefolds. To overcome this issue, we shall
use quasi-$F$-splitting, introduced by Yobuko [Yob19], which is a weaker
condition than $F$-splitting. Although quasi-$F$-splitting is not as
restrictive as $F$-splitting, quasi-$F$-split varieties satisfy various useful
vanishing theorems including Kodaira vanishing.

Since every smooth del Pezzo surface is known to be quasi-$F$-split [KTTWYY1,
Corollary 4.7], it is tempting to extend this result to smooth Fano
threefolds. However, the Fermat quartic hypersurface
$\\{x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}=0\\}\subset\mathbb{P}_{k}^{4}$
is not quasi-$F$-split in characteristic three [KTY, Example 7.12]. Moreover,
we find another non-quasi-$F$-split smooth Fano threefold, which is a weighted
hypersurface
$X\coloneqq\\{x_{0}^{6}+x_{1}^{6}+x_{2}^{6}+x_{3}^{6}+y^{2}=0\\}\subset\mathbb{P}(1,1,1,1,3)_{[x_{0}:x_{1}:x_{2}:x_{3}:y]}$
in characteristic five. On the other hand, each of these examples satisfies
$\rho(X)=r_{X}=1$. Hence it is natural to expect quasi-$F$-splitting for many
smooth Fano threefolds such that $\rho(X)>1$ or $r_{X}>1$. In fact, we prove
the following theorem.

######  Theorem A.

Let $X$ be a smooth Fano threefold over an algebraically closed field of
positive characteristic. Then the following hold.

  1. (1)

$X$ is quasi-$F$-split if $\rho(X)>1$ or $r_{X}>1$, where $\rho(X)$ is the
Picard number and $r_{X}$ denotes the index.

  2. (2)

$H^{i}(X,\mathcal{O}_{X}(-A))=0$ for every $i<3$ and every ample Cartier
divisor $A$.

Surprisingly, Theorem A asserts that Fano threefolds with wild conic bundle
structures (i.e., conic bundles which are not generically smooth) are all
quasi-$F$-split. These varieties are known to be non-$F$-split. Therefore,
quasi-$F$-splitting is a notion which can be applicable to such pathological
varieties in positive characteristic, allowing it to establish many useful
vanishing theorems. This is a significant advantage of quasi-$F$-splitting.

###  1.1. Akizuki-Nakano vanishing

As another remarkable property of quasi-$F$-splitting, it is known that every
quasi-$F$-split smooth variety lifts to $W_{2}(k)$ together with an arbitarary
effective Cartier divisor [AZ21], [KTTWYY1, Section 7.2]. From this, we can
deduce the logarithmic Akizuki-Nakano vanishing if $p>2$. Moreover, we can
also prove the non-logarithmic Akizuki-Nakano vanishing holds for every $p>0$.
To summarise, we obtain the following theorem.

######  Theorem B.

Let $X$ be a smooth Fano threefold over an algebraically closed field of
characteristic $p>0$ such that $\rho(X)>1$ or $r_{X}>1$. Then the following
hold.

  1. (1)

$H^{j}(X,\Omega^{i}_{X}\otimes\mathcal{O}_{X}(-A))=0$ for every ample Cartier
divisor $A$ and every pair $(i,j)$ of integers $i$ and $j$ satisfying $i+j<3$.

  2. (2)

Assume $p\neq 2$. Take a reduced divisor $E$ with simple normal crossing
support and an ample $\mathbb{Q}$-divisor $A$ such that the support of the
fractional part of $A$ is contained in $E$. Then

$H^{j}(X,\Omega^{i}_{X}(\log E)\otimes\mathcal{O}_{X}(-\lceil A\rceil))=0$

holds for every pair $(i,j)$ of integers $i$ and $j$ satisfying $i+j<3$.

As an immediate consequence, we obtain the following theorem.

######  Theorem C.

Every smooth Fano threefold $X$ over an algebraically closed field $k$ of
positive characteristic such that $\rho(X)>1$ or $r_{X}>1$ lifts to the ring
$W(k)$ of Witt vectors.

###  1.2. $F$-splitting

It is well known that smooth del Pezzo surfaces are $F$-split if $p>5$. Then
it is natural ask when a smooth Fano threefold is $F$-split. In the process of
the proof of Theorem A, we show the following theorem.

######  Theorem D.

Every smooth Fano threefold over an algebraically closed field of
characteristic $p>5$ such that $\rho(X)>1$ or $r_{X}>1$ is $F$-split.

######  Remark 1.1.

$F$-splitting of some smooth Fano threefolds has been proven by Totaro in a
different way [Totaro(Fano), the proof of Lemma 1.5].

######  Remark 1.2.

  1. (1)

As mentioned above, if $S$ is a non-$F$-split del Pezzo surface, then
$X:=S\times\mathbb{P}^{1}$ is a smooth Fano threefold which is not $F$-split.
Since this construction can be applicable for $p\in\\{2,3,5\\}$, the
assumption $p>5$ in Theorem D is optimal.

  2. (2)

If $p=7$, then the Fermat quartic hypersurface
$\\{x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}=0\\}\subset\mathbb{P}_{k}^{4}$
is not $F$-split by Fedder’s criterion.

  3. (3)

If $p=11$, then we shall prove that there exists a smooth Fano threefold which
is not $F$-split (Example 8.4).

  4. (4)

The authors do not know whether there exists a non-$F$-split smooth Fano
threefold of characteristic $p\geq 13$.

###  1.3. Strategy of proofs

We now overview how to show that a smooth Fano threefold with $\rho(X)\geq 2$
is quasi-$F$-split (Theorem A(1)). In order to show that $X$ is
quasi-$F$-split, we shall use one of the following strategies.

  1. (A)

Inversion of adjunction for $F$-splitting.

  2. (B)

Inversion of adjunction for quasi-$F$-splitting.

  3. (C)

Cartier operator criterion for quasi-$F$-splitting.

(A) For many cases (e.g., $3\leq\rho(X)\leq 5$ except for No. 3-10), we can
prove that $X$ is $F$-split. For example, let us consider the case when $X$ is
of No. 2-30, i.e., there is a blowup $X\to\mathbb{P}^{3}$ along a smooth conic
$B$ on $\mathbb{P}^{3}$. Take the plane $D(\simeq\mathbb{P}^{2})$ containing
$B$. For the the proper transform $D_{X}$ of $D$ on $X$, we have the following
implications:

$D\text{ is $F$-split}\overset{{\rm(i)}}{\Rightarrow}(\mathbb{P}^{3},D)\text{
is $F$-split}\overset{{\rm(ii)}}{\Leftrightarrow}(X,D_{X})\text{ is
$F$-split}\overset{{\rm(iii)}}{\Rightarrow}X\text{ is $F$-split}.$

Of course, $D(\simeq\mathbb{P}^{2})$ is $F$-split. The implication (i) follows
from the inversion of adjunction for $F$-splitting, e.g., if $(Y,D)$ is
$F$-split and $-(K_{Y}+D)$ is ample, then $(Y,D)$ is $F$-split. The
equivalence (ii) is assured by $K_{X}+D_{X}=f^{*}(K_{\mathbb{P}^{3}}+D)$.
Finally, (iii) holds by definition.

(B) Even if the strategy (A) does not work, we can apply a quasi-$F$-split
version of (A) for most of the remaining cases. The authors has proved that an
inversion of adjunction for log Calabi-Yau pairs in [Kawakami-Tanaka(dPvar)].
This allows us to prove the following statement: a smooth Fano threefold $X$
is quasi-$F$-split if there are smooth prime divisors $S$ and $S^{\prime}$
such that $K_{X}+S+S^{\prime}\sim 0$ and $S\cap S^{\prime}$ is a smooth curve
which is quasi-$F$-split (Corollary 2.18).

As other technical issues, we shall encounter the following obstructions:

  * •

$S$ is not necessarily smooth. For some cases, the generic member is enough as
a replacement (cf. Section 2.3). To this end, we shall need to treat algebraic
varieties defined over an imperfect field.

  * •

If the first prime divisor $S$ is a smooth weak del Pezzo surface, then it is
often hard to find $S^{\prime}$ such that $S\cap S^{\prime}$ is smooth. For
example, if $|-K_{S}|$ has no smooth member, then $S\cap S^{\prime}$ can not
be smooth. In order to avoid such a pathological phenomenon, we shall
establish some properties on weak del Pezzo surfaces, e.g., if $V$ is a
surface over a $C_{1}$-field $K$ of characteristic two and its base change
$V\times_{\operatorname{Spec}K}\operatorname{Spec}\overline{K}$ is a Langer
surface (defined as the base change of the blowup of
$\mathbb{P}^{2}_{\mathbb{F}_{2}}$ along all the $\mathbb{F}_{2}$-rational
points), then $\rho(V)=8$ (Lemma 5.5).

(C) Except when $X$ is one of 2-2, 2-6, 2-8, and 3-10, we may apply one of (A)
and (B). For these remaining cases, we shall apply a quasi-$F$-splitting
criterion via Cartier operator, which has been established in [KTTWYY1,
Theorem F] (cf. Proposition 2.20). To this end, we need to prove
$H^{j}(X,\Omega^{i}_{X}(p^{\ell}K_{X}))=0$ for suitable triples $(i,j,\ell)$.
Even if $X$ is explicitly given, it is often hard to compute such cohomologies
directly. The main strategy is to embed $X$ into a (typically toric) fourfold
$P$, and apply Bott vanishing for $P$. For example, if $X$ is of No. 2-2, then
we can find such an embedding with
$P=\mathbb{P}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(-1,-2))$.
Although $P$ is not necessaliry toric for the other cases, we shall find a
closed embedding $X\hookrightarrow P$ to a fourfold $P$ which almost satisfies
Bott vanishing. For more details on (C), see Section 6.

Acknowledgements. The authors express their gratitude to Burt Totaro for
valuable comments. They also thank Teppei Takamatsu and Shou Yoshikawa for
useful conversations. Kawakami was supported by JSPS KAKENHI Grant number
JP22J00272. Tanaka was supported by JSPS KAKENHI Grant number JP22H01112 and
JP23K03028.

##  2\. Preliminaries

###  2.1. Notation

In this subsection, we summarise notation and basic definitions used in this
article.

  1. (1)

Throughout the paper, $p$ denotes a prime number and we set
$\mathbb{F}_{p}\coloneqq\mathbb{Z}/p\mathbb{Z}$. Unless otherwise specified,
we work over an algebraically closed field $k$ of characteristic $p>0$. We
denote by $F\colon X\to X$ the absolute Frobenius morphism on an
$\mathbb{F}_{p}$-scheme $X$.

  2. (2)

We say that $X$ is a variety (over a field $\kappa$) if $X$ is an integral
scheme that is separated and of finite type over $\kappa$. We say that $X$ is
a curve (resp. surface, resp. threefold) if $X$ is a variety of dimension one
(resp. two, resp. three).

  3. (3)

For a variety $X$, we define the function field $K(X)$ of $X$ as the stalk
$\mathcal{O}_{X,\xi}$ at the generic point $\xi$ of $X$.

  4. (4)

We say that an $\mathbb{R}$-divisor $D$ on a normal variety $X$ is _simple
normal crossing_ if for every point $x\in\operatorname{Supp}D$, the local ring
$\mathcal{O}_{X,x}$ is regular and there exists a regular system of parameters
$x_{1},\ldots,x_{d}$ of the maximal ideal $\mathfrak{m}$ of
$\mathcal{O}_{X,x}$ and $1\leq r\leq d$ such that
$\operatorname{Supp}(D|_{\operatorname{Spec}\mathcal{O}_{X,x}})=\operatorname{Spec}(\mathcal{O}_{X,x}/(x_{1}\cdots
x_{r}))$.

  5. (5)

Given an integral normal Noetherian scheme $X$, a projective birational
morphism $\pi\colon Y\to X$ is called _a log resolution (of singularities) of
$X$_ if $Y$ is regular and $\mathrm{Exc}(f)$ is a simple normal crossing
divisor.

  6. (6)

We say that an $\mathbb{F}_{p}$-scheme $X$ is $F$-finite if the absolute
Frobenius morphism $F:X\to X$ is a finite morphism. We say that an
$\mathbb{F}_{p}$-algebra $R$ is $F$-finite if $\operatorname{Spec}R$ is
$F$-finite. In particular, a field $\kappa$ is $F$-finite if and only if
$[\kappa:\kappa^{p}]<\infty$. If $X$ is a variety over an $F$-finite field,
then $X$ is $F$-finite.

  7. (7)

Given a normal variety $X$ and an $\mathbb{R}$-divisor $D$, we define the
subsheaf $\mathcal{O}_{X}(D)$ of the constant sheaf $K(X)$ on $X$ by the
following formula

$\Gamma(U,\mathcal{O}_{X}(D))=\\{\varphi\in K(X)\mid\left({\rm
div}(\varphi)+D\right)|_{U}\geq 0\\}$

for every open subset $U$ of $X$. In particular,
$\mathcal{O}_{X}(\lfloor{D}\rfloor)=\mathcal{O}_{X}({D})$.

  8. (8)

Given a field $K$ and $K$-schemes $X$ and $Y$, we say that $X$ is
$K$-isomorphic to $Y$ if there exists an isomorphism $\theta:X\to Y$ over $K$.

  9. (9)

Given a closed subscheme $X$ of $\mathbb{P}^{n}$, we set
$\mathcal{O}_{X}(a):=\mathcal{O}_{\mathbb{P}^{n}}(a)|_{X}$ for
$a\in\mathbb{Z}$ unless otherwise specified. Similarly, if $Y$ is a closed
subscheme of $\mathbb{P}^{n}\times\mathbb{P}^{m}$, then we define
$\mathcal{O}_{Y}(a,b)\coloneqq\mathcal{O}_{\mathbb{P}^{n}\times\mathbb{P}^{m}}(a,b)$
for $a,b\in\mathbb{Z}$.

  10. (10)

Given a coherent sheaf $\mathcal{F}$ and a Cartier divisor $D$ on a variety
$X$, we set $\mathcal{F}(D)\coloneqq\mathcal{F}\otimes\mathcal{O}_{X}(D)$
unless otherwise specified. Note that $B_{n}\Omega_{X}^{i}(p^{n}D)$ (resp.
$Z_{n}\Omega_{X}^{i}(p^{n}D)$) does not mean
$B_{n}\Omega_{X}^{i}\otimes\mathcal{O}_{X}(p^{n}D)$ (resp.
$Z_{n}\Omega_{X}^{i}\otimes\mathcal{O}_{X}(p^{n}D)$) even if $D$ is Cartier
(cf. Subsection 2.2).

  11. (11)

Given two closed subschemes $Y$ and $Z$ on a scheme $X$, we denote by $Y\cap
Z$ the scheme-theoretic intersection, i.e., $Y\cap Z\coloneqq Y\times_{X}Z$.

###  2.2. Cartier operators

In this section, we recall the fundamental facts on the higher Cartier
operators ([Ill79], [KTTWYY1]).

Let $X$ be a smooth variety over a perfect field of characteristic $p>0$ and
$D$ a Cartier divisor on $X$. The Frobenius pushforward of the de Rham complex

$F_{*}\Omega^{\bullet}_{X}\colon
F_{*}\mathcal{O}_{X}\xrightarrow{F_{*}d}F_{*}\Omega_{X}\xrightarrow{F_{*}d}\cdots$

is a complex of $\mathcal{O}_{X}$-modules. Tensoring with
$\mathcal{O}_{X}(D)$, we obtain a complex

$F_{*}\Omega^{\bullet}_{X}\colon
F_{*}\mathcal{O}_{X}(pD)\xrightarrow{F_{*}d\otimes\mathcal{O}_{X}(D)}F_{*}\Omega_{X}(pD)\xrightarrow{F_{*}d\otimes\mathcal{O}_{X}(D)}\cdots$

We define locally free $\mathcal{O}_{X}$-modules as follows.

$\begin{array}[]{rl}&B^{1}\Omega^{i}_{X}(pD)\coloneqq\mathrm{Im}(F_{*}d:F_{*}\Omega^{i-1}_{X}(pD)\to
F_{*}\Omega^{i}_{X}(pD)),\\\
&Z_{1}\Omega^{i}_{X}(pD)\coloneqq\operatorname{Ker}(F_{*}d:F_{*}\Omega^{i}_{X}(pD)\to
F_{*}\Omega^{i+1}_{X}(pD)).\\\ \end{array}$

We have an isomorphism

$Z_{1}\Omega_{X}^{i}(pD)/B_{1}\Omega_{X}^{i}(pD)\overset{C(D)}{\simeq}\Omega_{X}^{i}(D).$

resulting from the Cartier isomorphism. In fact, tensoring with
$\mathcal{O}_{X}(D)$ with the usual Cartier isomorphism

$Z_{1}\Omega_{X}^{i}/B_{1}\Omega_{X}^{i}\overset{C}{\simeq}\Omega_{X}^{i},$

we obtain the above isomorphism.

Taking the Frobenius pushforward, we obtain

$F_{*}Z_{1}\Omega_{X}^{i}(p^{2}D)\to
F_{*}Z_{1}\Omega_{X}^{i}(p^{2}D)/F_{*}B_{1}\Omega_{X}^{i}(p^{2}D)\overset{F_{*}C(D)}{\simeq}F_{*}\Omega_{X}^{i}(pD).$

We denote by $B_{2}\Omega_{X}^{i}(p^{2}D)$ and $Z_{2}\Omega_{X}^{i}(p^{2}D)$
the preimages of $B_{1}\Omega_{X}^{i}(pD)\subset F_{*}\Omega_{X}^{i}(pD)$ and
$Z_{1}\Omega_{X}^{i}(pD)\subset F_{*}\Omega_{X}^{i}(pD)$ by the above map.
Inductively, we define locally $\mathcal{O}_{X}$-module
$B_{n}\Omega_{X}^{i}(p^{n}D)$ and $Z_{n}\Omega_{X}^{i}(p^{n}D)$ for all $n\geq
0$. Moreover, we set $B_{0}\Omega_{X}^{i}(D)=0$ and
$Z_{0}\Omega_{X}^{i}(pD)=\Omega_{X}^{i}(pD)$.

######  Lemma 2.1.

Then we have the following exact sequences

(2.1.1) $\displaystyle 0\to B_{n}\Omega^{i}_{X}(p^{n}D)\to
Z_{n}\Omega^{i}_{X}(p^{n}D)\to\Omega^{i}_{X}(D)\to 0.$ (2.1.2) $\displaystyle
0\to Z_{n}\Omega_{X}^{i}(p^{n}D)\to F_{*}Z_{n-1}\Omega_{X}^{i}(p^{n}D)\to
B_{1}\Omega_{X}^{i+1}(pD)\to 0.$

for all $i\geq 0$ and all $n\geq 1$.

###### Proof.

The assertion follows from [KTTWYY1, (5.7.1) and Lemma 5.8]. ∎

######  Remark 2.2.

Taking $n=1$, we have the following exact sequence:

(2.2.1) $\displaystyle 0\to B_{1}\Omega^{i}_{X}(pD)\to
Z_{1}\Omega^{i}_{X}(pD)\xrightarrow{C(D)}\Omega^{i}_{X}(D)\to 0.$ (2.2.2)
$\displaystyle 0\to Z_{1}\Omega_{X}^{i}(pD)\to
F_{*}\Omega_{X}^{i}(pD)\xrightarrow{F_{*}d\otimes\mathcal{O}_{X}(D)}B_{1}\Omega_{X}^{i+1}(pD)\to
0.$

for all $i\geq 0$.

######  Remark 2.3.

Taking $D=0$, we have short exact sequences

(2.3.1) $\displaystyle 0\to B_{n}\Omega^{i}_{X}\to
Z_{n}\Omega^{i}_{X}\xrightarrow{C^{n}}\Omega^{i}_{X}\to 0,$ (2.3.2)
$\displaystyle 0\to Z_{n}\Omega_{X}^{i}\to
F_{*}Z_{n-1}\Omega_{X}^{i}\xrightarrow{F_{*}d\circ
F_{*}C^{n-1}}B_{1}\Omega_{X}^{i+1}\to 0,$

which coincides with [KTTWYY1, (2.15.1) and Lemma 2.16] respectively. Then we
can confirm that

$\eqref{exact:B}=\eqref{exact:B,D=0}\otimes\mathcal{O}_{X}(D)\,\,\text{and}\,\,\eqref{exact:Z}=\eqref{exact:Z,D=0}\otimes\mathcal{O}_{X}(D)$

holds for all $i\geq 0$. In particular,

$\displaystyle
B_{n}\Omega_{X}^{i}(p^{n}D)=B_{n}\Omega_{X}^{i}\otimes\mathcal{O}_{X}(D)\,\,\text{and}$
$\displaystyle
Z_{n}\Omega_{X}^{i}(p^{n}D)=Z_{n}\Omega_{X}^{i}\otimes\mathcal{O}_{X}(D)$

hold for all $n\geq 0$.

###  2.3. Generic members

Let $X$ be a regular projective variety $X$ over a field $k$ and let $D$ be a
Cartier divisor on $X$ satisfying $h^{0}(X,\mathcal{O}_{X}(D))\geq 2$. For a
base point free linear system $\Lambda\subset|D|$ and the corresponding linear
subspace $V_{\Lambda}\subset H^{0}(X,\mathcal{O}_{X}(D))$, the generic member
$X^{\operatorname{gen}}_{\Lambda}$ of $\Lambda$ is defined by the following
diagram:

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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\kappa:=K((\mathbb{P}^{n}_{k})^{*})$

where

  1. (1)

$X^{\mathrm{univ}}_{\Lambda}$ denotes the universal family that parametrises
all the members of $\Lambda$,

  2. (2)

$\kappa=K((\mathbb{P}^{n}_{k})^{*})$ is the function field of the projective
space $(\mathbb{P}^{n}_{k})^{*}$ and
$\Theta:\operatorname{Spec}K((\mathbb{P}^{n}_{k})^{*})\to(\mathbb{P}^{n}_{k})^{*}$
is the induced morphism.

Then the following hold.

  1. (3)

$\kappa/k$ is a purely transcendental extension of finite transcendence
degree.

  2. (4)

$X\times_{k}\kappa$ is a regular projective variety.

  3. (5)

$X^{\operatorname{gen}}_{\Lambda}$ is a regular prime divisor [Tan-Bertini,
Theorem 4.9(4)(12)].

For more details, we refer to [Tan-Bertini]. By abuse of notation, also
$(X^{\operatorname{gen}}_{\Lambda})\times_{\kappa}\kappa^{\prime}$ is called
the generic member when $\kappa^{\prime}/\kappa$ is a purely transcendental
extension, because we shall encounter the situation as in the following
remark.

######  Remark 2.4.

We now consider the case when we have two base point free linear systems
$\Lambda_{1}$ and $\Lambda_{2}$ on $X$. As above, we obtain two generic
members $X^{\operatorname{gen}}_{\Lambda_{1}}$ on $X\times_{\kappa}\kappa_{1}$
and $X^{\operatorname{gen}}_{\Lambda_{2}}$ on $X\times_{\kappa}\kappa_{2}$.
For $\kappa_{1}=k(s_{1},\ldots,s_{a})$ and $\kappa_{2}=k(t_{1},\ldots,t_{b})$,
i.e., each of $\\{s_{i}\\}$ and $\\{t_{j}\\}$ is a transcendental basis, we
set

$\kappa:=\mathrm{Frac}(k(s_{1},\ldots,s_{a})\otimes_{k}k(t_{1},\ldots,t_{b}))=k(s_{1},\ldots,s_{a},t_{1},\ldots,t_{b}).$

For
$(X^{\operatorname{gen}}_{\Lambda_{1}})_{\kappa}:=X^{\operatorname{gen}}_{\Lambda_{1}}\times_{\kappa_{1}}\kappa$
and
$(X^{\operatorname{gen}}_{\Lambda_{2}})_{\kappa}:=X^{\operatorname{gen}}_{\Lambda_{2}}\times_{\kappa_{2}}\kappa$,

  1. $(\star)$

the sum
$(X^{\operatorname{gen}}_{\Lambda_{1}})_{\kappa}+(X^{\operatorname{gen}}_{\Lambda_{2}})_{\kappa}$
is a simple normal crossing divisor.

The similar statement holds even if we start with finitely many base point
free linear systems $\Lambda_{1},\ldots,\Lambda_{r}$ on $X$, i.e., the sum

$(X^{\operatorname{gen}}_{\Lambda_{1}})_{\kappa}+\cdots+(X^{\operatorname{gen}}_{\Lambda_{r}})_{\kappa}$

of the generic members
$(X^{\operatorname{gen}}_{\Lambda_{1}})_{\kappa},\ldots,(X^{\operatorname{gen}}_{\Lambda_{r}})_{\kappa}$
is simple normal crossing, where
$\kappa:=\mathrm{Frac}(\kappa_{1}\otimes_{k}\cdots\otimes_{k}\kappa_{r})$ and
$(-)_{\kappa}$ denotes the base change to $\kappa$.

###### Proof of $(\star)$.

Both $(X^{\operatorname{gen}}_{\Lambda_{1}})_{\kappa}$ and
$(X^{\operatorname{gen}}_{\Lambda_{2}})_{\kappa}$ are clearly regular prime
divisors. It suffices to show that the scheme-theoretic intersection
$(X^{\operatorname{gen}}_{\Lambda_{1}})_{\kappa}\cap(X^{\operatorname{gen}}_{\Lambda_{2}})_{\kappa}$
is regular. For each $i\in\\{1,2\\}$, let $D_{i}$ be the Cartier divisor with
$\Lambda_{i}\subset|D_{i}|$ and let $V_{i}\subset
H^{0}(X,\mathcal{O}_{X}(D_{i}))$ be the $k$-vector subspace corresponding to
$\Lambda_{i}$. Consider the restriction map:

$\rho:H^{0}(X\times_{k}\kappa_{1},\mathcal{O}_{X\times_{k}\kappa_{1}}(D_{2}\times_{k}\kappa_{1}))\to
H^{0}(X^{\operatorname{gen}}_{\Lambda_{1}},\mathcal{O}_{X\times_{k}\kappa_{1}}(D_{2}\times_{k}\kappa_{1})|_{X^{\operatorname{gen}}_{\Lambda_{1}}}).$

Then the generic member $(X^{\operatorname{gen}}_{\Lambda_{2}})_{\kappa}$
coincides with the base change of the generic member of
$V_{2}\times_{k}\kappa\subset
H^{0}(X\times_{k}\kappa_{1},\mathcal{O}_{X\times_{k}\kappa_{1}}(D_{2}\times_{k}\kappa_{1}))$.
Therefore, the intersection
$(X^{\operatorname{gen}}_{\Lambda_{1}})_{\kappa}\cap(X^{\operatorname{gen}}_{\Lambda_{2}})_{\kappa}$
coincides with the generic member of ${\rm Image}(\rho)$ by [Tan-Bertini,
Proposition 5.10(2)], which is regular [Tan-Bertini, Theorem 4.9(4)]. ∎

###  2.4. $F$-splitting criteria

######  Definition 2.5.

Let $X$ be a normal variety and let $\Delta$ be an effective
$\mathbb{Q}$-divisor on $X$.

  1. (1)

We say that $(X,\Delta)$ is $F$-split if

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\lfloor(p^{e}-1)\Delta\rfloor)$

splits as an $\mathcal{O}_{X}$-module homomorphism for every
$e\in\mathbb{Z}_{>0}$.

  2. (2)

We say that $(X,\Delta)$ is sharply $F$-split if

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil)$

splits as an $\mathcal{O}_{X}$-module homomorphism for some
$e\in\mathbb{Z}_{>0}$.

  3. (3)

We say that $(X,\Delta)$ is globally $F$-regular if, given an effective
$\mathbb{Z}$-divisor $E$, there exists $e\in\mathbb{Z}_{>0}$ such that

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil+E)$

splits as an $\mathcal{O}_{X}$-module homomorphism.

We say that $X$ is $F$-split (resp. globally $F$-regular) if so is $(X,0)$.

######  Remark 2.6.

We have the following implications:

$\text{globally $F$-regular}\Longrightarrow\text{sharply
$F$-split}\Longrightarrow\text{$F$-split}$

where the former implication is clear and the latter one holds by the same
argument as in [Sch08g]*Proposition 3.3. Moreover, if the condition ($\star$)
holds, then $(X,\Delta)$ is sharply F-split if and only if $(X,\Delta)$ is
$F$-split.

  1. ($\star$)

$(p^{e}-1)\Delta$ is a $\mathbb{Z}$-divisor for some $e\in\mathbb{Z}_{>0}$

In particular, $X$ is $F$-split if and only if $F\colon\mathcal{O}_{X}\to
F_{*}\mathcal{O}_{X}$ splits as an $\mathcal{O}_{X}$-module homomorphism. For
more foundational properties, we refer to [SS10].

In what follows, we summarise some $F$-splitting criteria, which are well
known to experts.

######  Proposition 2.7.

Let $f\colon X\to Y$ be a birational morphism of normal projective varieties.
Take an effective $\mathbb{Q}$-divisor $\Delta_{Y}$ on $Y$ such that
$(p^{e}-1)(K_{Y}+\Delta_{Y})$ is Cartier for some $e\in\mathbb{Z}_{>0}$.
Assume that the $\mathbb{Q}$-divisor $\Delta$ defined by
$K_{X}+\Delta=f^{*}(K_{Y}+\Delta_{Y})$ is effective. Then $(X,\Delta)$ is
$F$-split if and only if $(Y,\Delta_{Y})$ is $F$-split.

###### Proof.

If $(X,\Delta)$ is $F$-split, then so is $(Y,\Delta_{Y})$ (take the
pushforward). As for the opposite implication, the same argument as [HX15]*the
first paragraph of the proof of Proposition 2.11 works. ∎

######  Proposition 2.8.

Let $\kappa$ be an $F$-finite field of characteristic $p>0$. Let $X$ be a
normal Gorenstein projective variety over $\kappa$. Take a normal prime
Cartier divisor $S$ and an effective $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor
$B$ on $X$ such that $S\not\subset\operatorname{Supp}\,B$. Assume that

  1. (1)

$(S,B|_{S})$ is $F$-split, and

  2. (2)

there exists $e\in\mathbb{Z}_{>0}$ such that $(p^{e}-1)(K_{X}+S+B)$ is Cartier
and

$H^{1}(X,\mathcal{O}_{X}(-S-(p^{e}-1)(K_{X}+S+B)))=0.$

Then $(X,S+B)$ is $F$-split.

###### Proof.

The same argument as in [CTW17, Lemma 2.7] works. ∎

######  Corollary 2.9.

Let $\kappa$ be an $F$-finite field of characteristic $p>0$. Let $X$ be a
normal Gorenstein projective variety over $\kappa$. Take a normal prime
Cartier divisor $S$ such that $S$ is $F$-split and $-(K_{X}+S)$ is ample. Then
$(X,S)$ is $F$-split.

###### Proof.

By applying Proposition 2.8 with $B=0$, it suffices to show that
$H^{1}(X,\mathcal{O}_{X}(-S-(p^{e}-1)(K_{X}+S))))=0$ for $e\gg 0$, which
follows from Serre vanishing. ∎

######  Remark 2.10.

In the setting of Corollary 2.9, even if $S$ is quasi-$F$-split, $X$ is not
necessarily quasi-$F$-split [KTY, Example 7.7].

######  Corollary 2.11.

Let $X$ be a smooth Fano threefold over $k$. Assume that there exist a field
extension $k\subset\kappa$ and effective divisors $D_{1},D_{2},D_{3}$ on
$X_{\kappa}:=X\times_{k}\kappa$ such that

  1. (1)

$\kappa$ is an $F$-finite field,

  2. (2)

$K_{X_{\kappa}}+D_{1}+D_{2}+D_{3}\sim 0$,

  3. (3)

$D_{1}$ is a normal prime divisor,

  4. (4)

$D_{1}\cap D_{2}$ is one-dimensional, smooth, and

  5. (5)

$D_{1}\cap D_{2}\cap D_{3}$ is non-empty, zero-dimensional, and smooth over
$\kappa$, and

  6. (6)

$H^{1}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D_{1}))=H^{1}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D_{2}))=H^{1}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D_{3}))=0$
(cf. Lemma 2.12).

Then $(X\times_{k}\kappa,D_{1}+D_{2}+D_{3})$ is $F$-split. In particular, $X$
is $F$-split.

###### Proof.

First of all, we show that

(2.11.1) $H^{1}(D_{1},\mathcal{O}_{D_{1}}(-D_{2}|_{D_{1}}))=0.$

To this end, it suffices to prove
$H^{1}(X_{\kappa},\mathcal{O}_{X}(-D_{2}))=0$ and
$H^{2}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D_{2}-D_{1}))=0$. The former one
follows from (6). The latter one holds by

$h^{2}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D_{2}-D_{1}))=h^{1}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(K_{X}+D_{1}+D_{2}))\overset{{\rm(2)}}{=}h^{1}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D_{3}))\overset{{\rm(6)}}{=}0.$

This completes the proof of (2.11.1).

By $H^{1}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D_{1}))\overset{{\rm(6)}}{=}0$
and (2.11.1), we obtain

$\kappa=H^{0}(X_{\kappa},\mathcal{O}_{X_{\kappa}})\xrightarrow{\simeq}H^{0}(D_{1},\mathcal{O}_{D_{1}})\xrightarrow{\simeq}H^{0}(D_{1}\cap
D_{2},\mathcal{O}_{D_{1}\cap D_{2}}).$

Therefore, $C:=D_{1}\cap D_{2}$ is a smooth projective curve. By (4) and (5),
$C$ and $C\cap D_{3}$ are smooth over $\kappa$. In particular,
$(C\times_{\kappa}\overline{\kappa},(C\cap
D_{3})\times_{\kappa}\overline{\kappa})\simeq(\mathbb{P}^{1}_{\overline{\kappa}},P+Q)$
for the algebraic closure $\overline{\kappa}$ of $\kappa$ and some distinct
points $P$ and $Q$. Then $(C,C\cap D_{3})$ is $F$-split.

By Proposition 2.8, $(D_{1},(D_{2}+D_{3})|_{D_{1}})$ is $F$-split, where
Proposition 2.8 is applicable by (2.11.1). Again by Proposition 2.8 and (6),
$(X\times_{k}\kappa,D_{1}+D_{2}+D_{3})$ is $F$-split. ∎

######  Lemma 2.12.

Let $X$ be a smooth Fano threefold over $k$ and let $k\subset\kappa$ be a
field extension. Take a divisor on $X_{\kappa}\coloneqq X\times_{k}\kappa$.
Assume that one of the following conditions hold.

  1. (1)

$D$ is nef and $\nu(X_{\kappa},D)\geq 2$.

  2. (2)

There exists a morphism $\pi\colon X\to\mathbb{P}^{1}_{\kappa}$ such that
$\pi_{*}\mathcal{O}_{X}=\mathcal{O}_{\mathbb{P}^{1}_{\kappa}}$ and
$\mathcal{O}_{X_{\kappa}}(D)\simeq\pi^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)$.

Then $H^{1}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D))=0$.

###### Proof.

Taking the base change to the algebraic closure $\overline{\kappa}$ of
$\kappa$, we may assume that $\kappa$ is algebraically closed. Then the
problem is reduced to the case when $k=\kappa$. If (1) holds, then we may
apply [Kaw1, Corollary 3.6]. If (2) holds, then we may assume that $D$ is a
general fibre of $\pi\colon X\to\mathbb{P}^{1}$, which is a prime divisor, and
hence $H^{1}(X,\mathcal{O}_{X})=0$ implies $H^{1}(X,\mathcal{O}_{X}(-D))=0$. ∎

######  Remark 2.13.

Let $\kappa$ be an $F$-finite field and let $C$ be a Gorenstein projective
curve over $\kappa$. If $-K_{C}$ is ample and there exists a Cartier divisor
$D$ satisfying $\deg D=1$, then $C\simeq\mathbb{P}^{1}_{\kappa}$ [Kol13, Lemma
10.6], and hence $C$ is $F$-split.

######  Example 2.14.

If $X$ is a normal toric variety and $D$ is a torus-invariant reduced divisor,
then $(X,D)$ is $F$-split. This follows essentially from [Fuj07, 2.6] (cf.
[Tan-toric, Proposition 2.17]).

######  Proposition 2.15.

Let $X$ be a smooth Fano threefold. Suppose that the following condition.

  1. (1)

$H^{0}(X,\Omega_{X}^{2}(pK_{X}))=0$.

  2. (2)

$H^{1}(X,\Omega_{X}^{1}(K_{X}))=0$.

  3. (3)

$H^{2}(X,\Omega_{X}^{1}(-pK_{X}))=0$.

  4. (4)

$H^{1}(X,\Omega_{X}^{2}(-pK_{X}))=0$.

Then $X$ is $F$-split.

###### Proof.

Firstly, the global $F$-splitting of $X$ (i.e., splitting of
$F\colon\mathcal{O}_{X}\to F_{*}\mathcal{O}_{X}$) is equivalent to the
surjectivity of the evaluation map

$\mathrm{Hom}_{\mathcal{O}_{X}}(F_{*}\mathcal{O}_{X},\mathcal{O}_{X})\xrightarrow{F^{*}}\mathrm{Hom}_{\mathcal{O}_{X}}(\mathcal{O}_{X},\mathcal{O}_{X})(\simeq
H^{0}(X,\mathcal{O}_{X})).$

By Grothendieck duality, they are equivalent to the injectivity of

$F:H^{3}(X,\omega_{X})\to H^{3}(X,F_{*}\mathcal{O}_{X}\otimes\omega_{X}).$

Thus it suffices to show that $H^{2}(X,B_{1}\Omega_{X}^{1}(pK_{X}))=0$. By
(2.2.1) and (2), this vanishing can be reduced to
$H^{2}(X,Z_{1}\Omega^{1}_{X}(pK_{X}))$. By (2.2.2), it is enough to prove

$H^{1}(X,B_{1}\Omega^{2}_{X}(pK_{X}))=0\,\,\,\text{and}\,\,\,H^{2}(X,\Omega^{1}_{X}(pK_{X}))=0.$

The latter one follows from (4) and Serre duality. It suffices to show the
first one. By (2.2.1) and the condition (1), this vanishing is reduced to

$H^{1}(X,Z_{1}\Omega_{X}^{2}(pK_{X}))=0.$

By 2.2.2, it suffices to show

$H^{0}(X,B_{1}\Omega^{3}_{X}(pK_{X}))=0\,\,\,\text{and}\,\,\,H^{1}(X,\Omega^{2}_{X}(pK_{X}))=0.$

The first vanishing holds by $B_{1}\Omega_{X}^{3}(pK_{X})\subset
F_{*}\omega^{p+1}_{X}$ and the ampleness of $\omega^{-1}_{X}$. The latter one
follows from (3) and Serre duality. ∎

###  2.5. Quasi-$F$-splitting criteria

We recall that definition of $F$-splitting and quasi-$F$-splitting. We refer
to [Kawakami-Tanaka(dPvar), Section 3] for details.

######  Definition 2.16.

Let $X$ be a normal variety. We define a $W_{n}\mathcal{O}_{X}$-module
$Q^{e}_{X,n}$ and a $W_{n}\mathcal{O}_{X}$-module homomorphism
$\Phi^{\Delta,e}_{X,\Delta,n}$ by the following pushout diagram:

${W_{n}\mathcal{O}_{X}}$${F_{*}^{e}W_{n}\mathcal{O}_{X}}$${\mathcal{O}_{X}}$${Q^{e}_{X,n}.}$$\scriptstyle{F^{e}}$$\scriptstyle{R^{n-1}}$$\scriptstyle{\Phi^{e}_{X,n}}$

Applying
$(-)^{*}\coloneqq\mathcal{H}om_{W_{n}\mathcal{O}_{X}}(-,W_{n}\omega_{X}(-K_{X}))$
to $\Phi^{e}_{X,n}$, we have a $W_{n}\mathcal{O}_{X}$-module homomorphism

$(\Phi^{e}_{X,n})^{*}\colon(Q^{e}_{X,n})^{*}\to\mathcal{O}_{X}.$

We say that $X$ is quasi-$F^{e}$-split if

$H^{0}(X,(\Phi^{e}_{X,n})^{*})\colon H^{0}(X,(Q^{e}_{X,n})^{*})\to
H^{0}(X,\mathcal{O}_{X})$

is surjective.

######  Remark 2.17.

Let $X$ be a normal variety. Then $X$ is $F$-split if and only if

$H^{0}(X,F^{*})\colon H^{0}(X,(F_{*}\mathcal{O}_{X})^{*})\to
H^{0}(X,\mathcal{O}_{X})$

is surjective for every $e>0$, where
$(-)^{*}\coloneqq\mathcal{H}om_{\mathcal{O}_{X}}(-,W_{1}\omega_{X}(-K_{X}))=\mathcal{H}om_{\mathcal{O}_{X}}(-,\mathcal{O}_{X})$.
In particular, if $X$ is $F$-split, then $X$ is quasi-$F^{e}$-split for every
$e>0$.

######  Corollary 2.18.

Let $k$ be an algebraically closed field of characteristic $p>0$ and let
$k\subset\kappa$ be a field extension, where $\kappa$ is $F$-finite. Let $X$
be a smooth Fano threefold over $k$. Assume that there exist a prime divisor
$D$ and a reduced divisor $D^{\prime}$ on $X\times_{k}\kappa$ such that

  1. (a)

$K_{X\times_{k}\kappa}+D+D^{\prime}\sim 0$,

  2. (b)

$D$ is regular, $D\cap D^{\prime}$ is smooth over $\kappa$,
$D\not\subset\operatorname{Supp}D^{\prime}$,

  3. (c)

$H^{1}(X\times_{k}\kappa,\mathcal{O}_{X\times_{k}\kappa}(-D))=0$, and

  4. (d)

$H^{1}(X\times_{k}\kappa,\mathcal{O}_{X\times_{k}\kappa}(-D^{\prime}))=0$.

Then $X$ is $2$-quasi-$F^{e}$-split for all $e>0$.

###### Proof.

Set $X_{\kappa}\coloneqq X\times_{k}\kappa$. We show that
$(X\times_{k}\kappa,D_{1}+\cdots+D_{n-1})$ is weakly $2$-quasi-$F$-split (see
[Kawakami-Tanaka(dPvar), Definition 5.13] for the definition of weak
quasi-$F$-splitting).

Since $D+D^{\prime}$ is ample, $D+D^{\prime}$ is connected, and hence
$D^{\prime}|_{D}=D\cap D^{\prime}\neq\emptyset$. By [Kawakami-Tanaka(dPvar),
Corollary 5.17], it is enough to show that $(D,D^{\prime}|_{D})$ is weakly
$2$-quasi-$F$-split. Again by [Kawakami-Tanaka(dPvar), Corollary 5.17], it
suffices to prove that

  1. (i)

$D^{\prime}|_{D}$ is a prime divisor,

  2. (ii)

$H^{1}(D,\mathcal{O}_{D}(-D^{\prime}|_{D}))=0$, and

  3. (iii)

$D^{\prime}|_{D}$ is $2$-quasi-$F$-split.

Consider an exact sequence

$0\to\mathcal{O}_{X_{\kappa}}(-D-D^{\prime})\to\mathcal{O}_{X_{\kappa}}(-D^{\prime})\to\mathcal{O}_{D}(-D^{\prime}|_{D})\to
0,$

which induces the following one:

$0\overset{{\rm(d)}}{=}H^{1}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D^{\prime}))\to
H^{1}(D,\mathcal{O}_{D}(-D^{\prime}|_{D}))\to
H^{2}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D-D^{\prime})).$

Then (ii) follows from

$H^{2}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(-D-D^{\prime}))\simeq
H^{2}(X_{\kappa},\mathcal{O}_{X_{\kappa}}(K_{X_{\kappa}}))\simeq
H^{2}(X,\mathcal{O}_{X}(K_{X}))\otimes_{k}\kappa=0.$

By the induced exact sequence

$H^{0}(D,\mathcal{O}_{D})\to
H^{0}(D^{\prime}|_{D},\mathcal{O}_{D^{\prime}|_{D}})\to
H^{1}(D,\mathcal{O}_{D}(-D^{\prime}|_{D}))\overset{{\rm(ii)}}{=}0,$

$H^{0}(D^{\prime}|_{D},\mathcal{O}_{D^{\prime}|_{D}})$ is a field. Since
$D^{\prime}|_{D}=D\cap D^{\prime}$ is smooth, we obtain (i). Let us show
(iii). Since $D^{\prime}|_{D}$ is a smooth projective curve over $\kappa$ with
$K_{D^{\prime}|_{D}}\sim 0$, the base change
$(D^{\prime}|_{D})\times_{\kappa}\overline{\kappa}$ to the algebraic closure
$\overline{\kappa}$ of $\kappa$ is an elliptic curve, which is
2-quasi-$F$-split ([KTTWYY1, Remark 2.11]). Thus so is $D^{\prime}|_{D}$ [KTY,
Proposition 2.12], and (iii) holds.

Now, we show that $X$ is quasi-$F$-split. Since
$(X\times_{k}\kappa,D_{1}+\cdots+D_{n-1})$ is weakly $2$-quasi-$F$-split it
follows that $X\times_{k}\kappa$ is $2$-quasi-$F$-split. By [KTY, Proposition
2.12], $X$ is $2$-quasi-$F$-split. ∎

######  Remark 2.19.

The assumptions (c) and (d) automatically hold when $k=\kappa$ and
$D^{\prime}$ is connected. Indeed, we have an exact sequence
$H^{0}(X,\mathcal{O}_{X})\xrightarrow{\simeq}H^{0}(E,\mathcal{O}_{E})\to
H^{1}(X,\mathcal{O}_{X}(-E))\to H^{1}(X,\mathcal{O}_{X})=0$ for any reduced
connected divisor $E$ on $X$.

Although the following result is contained in [KTTWYY1, Theorem F], we include
a proof for the reader’s convenience, as its proof is quite short.

######  Proposition 2.20.

Let $X$ be a smooth Fano threefold. Suppose that the following condition.

  1. (1)

$H^{0}(X,\Omega_{X}^{2}(p^{i}K_{X}))=0$ for all $i>0$.

  2. (2)

$H^{1}(X,\Omega_{X}^{1}(K_{X}))=0$.

  3. (3)

$H^{2}(X,\Omega_{X}^{1}(-p^{i}K_{X}))=0$ for all $i>0$.

Then $X$ is quasi-$F$-split.

###### Proof.

It is enough to show that $H^{2}(X,B_{n}\Omega_{X}^{1}(p^{n}K_{X}))=0$ for
some $n>0$. By (2.1.1) and (2), it suffices to prove
$H^{2}(X,Z_{n}\Omega^{1}_{X}(p^{n}K_{X}))=0$ for some $n>0$. By (2.1.2), this
vanishing is reduced to

$H^{1}(X,B_{1}\Omega^{2}_{X}(pK_{X}))=0\,\,\,\text{and}\,\,\,H^{2}(X,Z_{n-1}\Omega^{1}_{X}(p^{n}K_{X}))=0.$

Repeating this procedure, the vanishing
$H^{2}(X,Z_{n}\Omega^{1}_{X}(p^{n}K_{X}))=0$ is reduced to

$H^{1}(X,B_{1}\Omega^{2}_{X}(p^{l}K_{X}))=0\,\,\,\text{and}\,\,\,H^{2}(X,\Omega^{1}_{X}(p^{n}K_{X}))=0.$

for every $l\in\\{1,\ldots,n\\}$. Taking $n\gg 0$, we may assume
$H^{2}(X,\Omega^{1}_{X}(p^{n}K_{X}))=0$ by Serre vanishing. By (2.1.1) and
(1), it suffices to show that $H^{1}(X,Z_{1}\Omega^{2}_{X}(p^{l}K_{X}))=0$ for
every $l\in\\{1,\ldots,n\\}$. By Serre duality, we have

$H^{1}(X,\Omega_{X}^{2}(p^{l}K_{X}))\simeq
H^{2}(X,\Omega_{X}^{1}(-p^{l}K_{X}))\overset{{\rm(3)}}{=}0$

for every $l>0$. By (2.1.2), the problem is finally reduced to
$H^{0}(X,B_{1}\Omega_{X}^{3}(p^{l}K_{X}))=0$ for all $l\in\\{1,\ldots,n\\}$.
This holds because $B_{1}\Omega_{X}^{3}(p^{l}K_{X})\subset
F_{*}\omega^{p^{l}+1}_{X}$ and $\omega^{-1}_{X}$ is ample. ∎

###  2.6. Weak del Pezzo surfaces

In this subsection, we recall when canonical weal del Pezzo surfaces are
$F$-split.

######  Definition 2.21.

Let $S$ be a normal Gorenstein projective surface.

  1. (1)

We say that $S$ is weak del Pezzo if $-K_{S}$ is nef and big.

  2. (2)

We say that $S$ is del Pezzo if $-K_{S}$ is ample.

######  Theorem 2.22.

Let $S$ be a canonical weak del Pezzo surface. Then $S$ is $F$-split if $p>5$
or $K_{S}^{2}\geq 5$.

###### Proof.

See [KT, Theorem A]. ∎

##  3\. Fano threefolds with $\rho\geq 4$

######  Proposition 3.1.

Let $X$ be a smooth Fano threefold with $\rho(X)\geq 6$. Then $X$ is
quasi-$F$-split.

###### Proof.

In this case, $X\simeq S\times\mathbb{P}^{1}$ for a smooth del Pezzo surface
[FanoIV, Section 7.6]. Since $S$ is quasi-$F$-split and $\mathbb{P}^{1}$ is
$F$-split, $S\times\mathbb{P}^{1}$ is quasi-$F$-split [KTY, Proposition 6.7].
∎

######  Proposition 3.2.

Let $X$ be a smooth Fano threefold with $\rho(X)=5$. Then $X$ is $F$-split.

###### Proof.

Recall that $X$ is of No. 5-1, 5-2, or 5-3 [FanoIV, Section 7.5]. If $X$ is
5-3, then $X\simeq S\times\mathbb{P}^{1}$ for a smooth del Pezzo surface $S$
with $\rho(S)=4$, and hence $F$-split (e.g. $S$ is toric, and hence so is
$X$).

Assume that $X$ is 5-1. Then $X=\operatorname{Bl}_{B_{1}\amalg B_{2}\amalg
B_{3}}Y$, where $Y\coloneqq\operatorname{Bl}_{C}Q$ is a blowup of $Q$ along a
conic $C$ and $B_{1},B_{2},B_{3}$ are mutually distinct one-dimensional fibres
of $\sigma\colon Y\coloneqq\operatorname{Bl}_{C}Q\to Q$ [FanoIV, Section 7.5].
Since the smallest linear subvariety $\langle C\rangle$ of $\mathbb{P}^{4}$
containing $C$ is a plane, we obtain $C\subset\langle
C\rangle=\overline{H}_{1}\cap\overline{H}_{2}$ for suitable hyperplanes
$\overline{H}_{1}$ and $\overline{H}_{2}$ on $\mathbb{P}^{4}$. Set
$H_{i}\coloneqq Q\cap\overline{H}_{i}$ for each $i\in\\{1,2\\}$. We then get a
scheme-theoretic equality $C=H_{1}\cap H_{2}$. Note that each $H_{i}$ is a
(possibly singular) quadric surface in $\overline{H}_{i}=\mathbb{P}^{3}$,
which is smooth along $C=H_{1}\cap H_{2}$. It is easy to see that $\Delta$ is
effective for the divisor $\Delta$ defined by
$h^{*}(K_{Q}+H_{1}+H_{2})=K_{X}+\Delta$, where $h\colon X\to Q$ denotes the
induced birational morphism. It is enough to show that $(Q,H_{1}+H_{2})$ is
$F$-split (Proposition 2.7). Take a general hyperplane section $H_{3}$. Since
$K_{Q}+H_{1}+H_{2}+H_{3}\sim 0$ and all $H_{3},H_{3}\cap H_{2}$, and
$H_{3}\cap H_{2}\cap H_{1}$ are smooth, $(Q,H_{1}+H_{2}+H_{3})$ is $F$-split
(Corollary 2.11), and hence $(Q,H_{1}+H_{2})$ is $F$-split.

Assume that $X$ is No. 5-2. Then $X=\operatorname{Bl}_{B\amalg B^{\prime}}Y$,
where $Y\coloneqq\operatorname{Bl}_{L_{1}\amalg L_{2}}\mathbb{P}^{3}$ is a
blowup of $\mathbb{P}^{3}$ along a disjoint union of lines $L_{1}$ and
$L_{2}$, and $B$ and $B^{\prime}$ are mutually distinct one-dimensional fibres
of $\sigma\colon Y=\operatorname{Bl}_{L_{1}\amalg
L_{2}}\mathbb{P}^{3}\to\mathbb{P}^{3}$ lying over $L_{1}$ [FanoIV, Section
7.5]. Take two planes $H_{1}$ and $H_{2}$ on $\mathbb{P}^{3}$ such that
$H_{1}\cap H_{2}=L_{1}$. Note that each $H_{i}\cap L_{2}$ is a smooth point.
Pick a plane $H_{3}$ containing $L_{2}$. Then it is easy to see that $\Delta$
is effective for the divisor $\Delta$ defined by
$h^{*}(K_{\mathbb{P}^{3}}+H_{1}+H_{2}+H_{3})=K_{X}+\Delta$. It is enough to
show that $(\mathbb{P}^{3},H_{1}+H_{2}+H_{3})$ is $F$-split (Proposition 2.7).
Pick a general hyperplane $H_{4}$. Apply Corollary 2.11 by setting
$D_{1}\coloneqq H_{1},D_{2}\coloneqq H_{2}$, and $D_{3}\coloneqq H_{3}+H_{4}$.
Then $(X,D_{1}+D_{2}+D_{3})$ is $F$-split, and hence
$(\mathbb{P}^{3},H_{1}+H_{2}+H_{3})$ is $F$-split. ∎

######  Proposition 3.3.

Let $X$ be a smooth Fano threefold with $\rho(X)=4$. Then $X$ is $F$-split.

###### Proof.

We treat the following six cases separately:

  1. (1)

4-4, 4-10, 4-12.

  2. (2)

4-3, 4-6, 4-8, 4-13.

  3. (3)

4-5, 4-7, 4-9, 4-11.

  4. (4)

4-11.

  5. (5)

4-2.

  6. (6)

4-1.

(1) If $X$ is 4-4, then there is a smooth curve $B$ on $X$ such that the
blowup $\widetilde{X}$ of $X$ along $B$ is Fano [FanoIV, Proposition 5.31].
Since $\widetilde{X}$ is $F$-split (Proposition 3.2), so is $X$. If $X$ is
4-10, then we can write $X\simeq S\times\mathbb{P}^{1}$ for a smooth del Pezzo
surface $S$ with $K_{S}^{2}=7$ [FanoIV, Section 7.4]. In this case, $X$ is
clearly $F$-split. Assume that $X$ is 4-12. Then we have
$X=\operatorname{Bl}_{B\amalg B^{\prime}}Y_{\text{2-33}}$, where
$Y_{\text{2-33}}=\operatorname{Bl}_{L}\mathbb{P}^{3}$ is the blowup of
$\mathbb{P}^{3}$ along a line $L$, and $B$ and $B^{\prime}$ are mutually
disjoint one-dimensional fibres of the induced blowup
$\rho:Y_{\text{2-33}}=\operatorname{Bl}_{L}\mathbb{P}^{3}\to\mathbb{P}^{3}$.
In this case, we can apply a similar argument to that of 5-2 in the proof of
Proposition 3.2.

(2) Assume that $X$ is one of 4-3, 4-6, 4-8, 4-13. In this case, there is a
blowup
$h:X\to\mathbb{P}^{1}_{1}\times\mathbb{P}^{1}_{2}\times\mathbb{P}^{1}_{3}$
along a curve $B$ of tridegree $(1,1,c)$ for some $c\geq 0$ [FanoIV, Section
7.4]. Let $B^{\prime}\subset\mathbb{P}^{1}_{1}\times\mathbb{P}^{1}_{2}$ be the
image of $B$, which is a curve of bidegree $(1,1)$. Then the induced morphism
$B^{\prime}\to\mathbb{P}^{1}$ to the first direct product factor is an
isomorphism. Hence $B^{\prime}\simeq\mathbb{P}^{1}$. Set
$D\subset\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ to be the
inverse image of $B^{\prime}$, which satisfies
$D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1,0)|$
and $D\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$. Then it is easy to see that
$\Delta$ is effective for the divisor $\Delta$ defined by
$h^{*}(K_{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}+D)=K_{X}+\Delta$.
It is enough to show that
$(\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1},D)$ is $F$-split
(Proposition 2.7), which follows from the fact that $D$ is $F$-split and
$-(K_{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}+D)$ is ample
(Corollary 2.9).

(3) Assume that $X$ is one of 4-5, 4-7, 4-9. Let
$\tau:\mathbb{F}_{1}\to\mathbb{P}^{2}$ be the blowdown of the $(-1)$-curve
$\Gamma$ and set $P:=\tau(\Gamma)$. Then there is a blowup $f:X\to
Y=\mathbb{F}_{1}\times\mathbb{P}^{1}$ along a smooth curve $B$ lying over
$B^{\prime}\subset\mathbb{F}_{1}$, where $B^{\prime}$ is disjoint from the
$(-1)$-curve $\Gamma$ and $B^{\prime}$ is the inverse image of a line
$L\subset\mathbb{P}^{2}$ [FanoIV, Section 7.4]. Then the induced composite
morphism

$h:X\to\mathbb{F}_{1}\times\mathbb{P}^{1}\to\mathbb{P}^{2}\times\mathbb{P}^{1}$

is the blowup along $(P\times\mathbb{P}^{1})\amalg\overline{B}$, where
$\overline{B}\subset\mathbb{P}^{2}\times\mathbb{P}^{1}$ denotes the image of
$B\subset\mathbb{F}_{1}\times\mathbb{P}^{1}$. In particular, we get
$(P\times\mathbb{P}^{1})\amalg\overline{B}\subset
L^{\prime}\times\mathbb{P}^{1}\cup L\times\mathbb{P}^{1}$ for a line
$L^{\prime}$ passing through $P$. Then it is easy to see that $\Delta$ is
effective for the divisor $\Delta$ defined by
$h^{*}(K_{\mathbb{P}^{2}\times\mathbb{P}^{1}}+L\times\mathbb{P}^{1}+L^{\prime}\times\mathbb{P}^{1})=K_{X}+\Delta$.
It is enough to show that
$(\mathbb{P}^{2}\times\mathbb{P}^{1},L\times\mathbb{P}^{1}+L^{\prime}\times\mathbb{P}^{1})$
is $F$-split (Proposition 2.7). This holds, because we may assume that
$L\times\mathbb{P}^{1}+L^{\prime}\times\mathbb{P}^{1}$ is a torus-invariant
reduced divisor on a toric variety $\mathbb{P}^{2}\times\mathbb{P}^{1}$.

(4) Assume that $X$ is 4-11. Then there is a blowup $f\colon
X\to\mathbb{F}_{1}\times\mathbb{P}^{1}$ along $C$, where
$C=\Gamma\times\\{t\\}$ for the $(-1)$-curve $\Gamma$ on $\mathbb{F}_{1}$ and
a closed point $t$ of $\mathbb{P}^{1}$ [FanoIV, Section 7.4]. For the blowup
$\tau\times{\rm
id}\colon\mathbb{F}_{1}\times\mathbb{P}^{1}\to\mathbb{P}^{2}\times\mathbb{P}^{1}$,
consider the composite birational morphism:

$h\colon
X\xrightarrow{f}\mathbb{F}_{1}\times\mathbb{P}^{1}\xrightarrow{\tau\times{\rm
id}}\mathbb{P}^{2}\times\mathbb{P}^{1}.$

For the blowup centre $P\coloneqq\tau(\Gamma)$ of
$\tau\colon\mathbb{F}_{1}\to\mathbb{P}^{2}$, pick two lines $L$ and
$L^{\prime}$ on $\mathbb{P}^{2}$ passing through $P$. Then we have

$K_{Y}+(\Gamma\times\mathbb{P}^{1})+L_{Y}+L^{\prime}_{Y}=(\tau\times{\rm
id})^{*}(K_{\mathbb{P}^{2}\times\mathbb{P}^{1}}+(L\times\mathbb{P}^{1})+(L^{\prime}\times\mathbb{P}^{1})),$

where $L_{Y}$ and $L^{\prime}_{Y}$ denote the proper transforms of
$L\times\mathbb{P}^{1}$ and $L^{\prime}\times\mathbb{P}^{1}$, respectively.
Since $C=\Gamma\times\\{t\\}$ is contained in $\Gamma\times\mathbb{P}^{1}$, we
see that the divisor $\Delta$ defined by
$K_{X}+\Delta=h^{*}(K_{\mathbb{P}^{2}\times\mathbb{P}^{1}}+(L\times\mathbb{P}^{1})+(L^{\prime}\times\mathbb{P}^{1}))$
is effective. Then $X$ is $F$-split, because
$(\mathbb{P}^{2}\times\mathbb{P}^{1},L\times\mathbb{P}^{1}+L^{\prime}\times\mathbb{P}^{1})$
is $F$-split (Proposition 2.7).

(5) Assume that $X$ is 4-2. Then $X$ is a blowup of
$Y=\mathbb{P}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(1,1))$
along an elliptic curve $B$ on a section $S$ of the $\mathbb{P}^{1}$-bundle
$\pi:Y\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ disjoint from the negative
section $S^{\prime}$ of $\pi$ [FanoIV, Proposition 5.28]. Note that $S\simeq
S^{\prime}\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$. We have that
$K_{Y}+S+S^{\prime}\sim\pi^{*}K_{\mathbb{P}^{1}\times\mathbb{P}^{1}}$. Indeed,
since $K_{Y}+S+S^{\prime}$ is $\pi$-numerically trivial, we can write
$K_{Y}+S+S^{\prime}\sim\pi^{*}D$ for some Cartier divisor $D$ on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. By restricting to $S$, we obtain $D\sim
K_{\mathbb{P}^{1}\times\mathbb{P}^{1}}$.

Then, the $\mathbb{Q}$-divisor

$-(K_{Y}+S+(1-\epsilon)S^{\prime})=\epsilon
S^{\prime}+\pi^{*}K_{\mathbb{P}^{1}\times\mathbb{P}^{1}}$

is ample for $0<\epsilon\ll 1$. After perturbing $\epsilon$, the problem is
reduced to the case when $(p^{e}-1)(K_{Y}+S+(1-\epsilon)S^{\prime})$ is
Cartier for some $e>0$. Replacing $e$ by some $e^{\prime}\in
e\mathbb{Z}_{>0}$, we may assume that
$H^{1}(X,\mathcal{O}_{X}(-S-(p^{e}-1)(K_{Y}+S+(1-\epsilon)S^{\prime})))=0$.
Since $(S,(1-\epsilon)S^{\prime}|_{S})=(\mathbb{P}^{1}\times\mathbb{P}^{1},0)$
is sharply $F$-split, $(X,S+(1-\epsilon)S^{\prime})$ is sharply $F$-split
(Proposition 2.8). Hence $X$ is $F$-split.

(6) Assume that $X$ is 4-1. Then $X$ is a prime divisor on
$\mathbb{P}^{1}_{1}\times\mathbb{P}^{1}_{2}\times\mathbb{P}^{1}_{3}\times\mathbb{P}^{1}_{4}$
of multi-degree $(1,1,1,1)$ [FanoIV, Section 7.4]. For each
$i\in\\{1,2,3,4\\}$, we set
$H_{i}:=\pi_{i}^{*}\mathcal{O}_{\mathbb{P}^{1}_{i}}(1)$ and
$H^{\prime}_{i}:=\operatorname{pr}_{i}^{*}\mathcal{O}_{\mathbb{P}^{1}_{i}}(1)$,
where $\pi_{i}$ and $\operatorname{pr}_{i}$ denote the the induced morphisms:

$\pi_{i}:X\hookrightarrow\mathbb{P}^{1}_{1}\times\mathbb{P}^{1}_{2}\times\mathbb{P}^{1}_{3}\times\mathbb{P}^{1}_{4}\xrightarrow{\operatorname{pr}_{i}}\mathbb{P}^{1}_{i}.$

It holds that $-K_{X}\sim H_{1}+H_{2}+H_{3}+H_{4}$. Note that

$H_{1}\cdot H_{2}\cdot H_{3}=H_{1}^{\prime}\cdot H^{\prime}_{2}\cdot
H^{\prime}_{3}\cdot(H^{\prime}_{1}+H^{\prime}_{2}+H^{\prime}_{3}+H^{\prime}_{4})=H^{\prime}_{1}\cdot
H^{\prime}_{2}\cdot H^{\prime}_{3}\cdot H^{\prime}_{4}=1.$

Take the generic members
$H_{1}^{\operatorname{gen}},H_{2}^{\operatorname{gen}},H_{3}^{\operatorname{gen}},H_{4}^{\operatorname{gen}}$
of $H_{1},H_{2},H_{3},H_{4}$, where each $H_{i}^{\operatorname{gen}}$ is an
effective Cartier divisor on $X\times_{k}\kappa$ for suitable purely
transcendental field extension $\kappa/k$ (Remark 2.4). Set
$D_{1}:=H_{1}^{\operatorname{gen}},D_{2}:=H_{2}^{\operatorname{gen}},D_{3}:=H_{3}^{\operatorname{gen}}+H_{4}^{\operatorname{gen}}$.
Then $D_{1}^{\operatorname{gen}}\cap D_{2}^{\operatorname{gen}}$ is smooth,
because $\deg(H_{3}^{\operatorname{gen}}|_{D_{1}\cap D_{2}})=1$ (Remark 2.13).
By Corollary 2.11, $X$ is $F$-split. ∎

##  4\. Fano threefolds with $\rho=3$ (except for 3-10)

The purpose of this subsection is to prove that an arbitrary smooth Fano
threefold $X$ with $\rho(X)=3$ is $F$-split except for 3-10 (Proposition 4.4).
We start with some complicated cases: 3-1, 3-3 and 3-4.

######  Lemma 4.1.

Let $X$ be a smooth Fano threefold of No. 3-1. Then the following hold.

  1. (1)

Let $\varphi\colon X\to\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
be a finite double cover such that
$(\varphi_{*}\mathcal{O}_{X}/\mathcal{O}_{Y})^{-1}\simeq\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1,1)$.
Then $\varphi$ is separable, i.e., the induced field extension $K(X)/K(Y)$ is
separable.

  2. (2)

$X$ is $F$-split.

###### Proof.

Let us show (1). Set
$Y\coloneqq\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathcal{L}\coloneqq\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1,1)$.
By [CD89, Proposition 0.1.2] and [Ful98, Lemma 3.2], it suffices to show that

$\deg c_{3}(\Omega_{Y}^{1}\otimes\mathcal{L}^{\otimes 2})\neq 0.$

Set
$\mathcal{M}_{i}\coloneqq\operatorname{pr}_{i}^{*}\Omega_{\mathbb{P}^{1}}^{1}\otimes\mathcal{L}^{\otimes
2}$ for every $i\in\\{1,2,3\\}$, i.e.,

$\mathcal{M}_{1}\coloneqq\mathcal{O}(0,2,2),\qquad\mathcal{M}_{2}\coloneqq\mathcal{O}(2,0,2),\qquad\mathcal{M}_{3}\coloneqq\mathcal{O}(2,2,0).\qquad$

We have $\Omega_{Y}^{1}\otimes\mathcal{L}^{\otimes
2}=\mathcal{M}_{1}\oplus\mathcal{M}_{2}\oplus\mathcal{M}_{3}$. Then the
following holds (cf. [Har77, Appendix A, Section 3]):

$\displaystyle\sum_{i=0}^{\infty}c_{i}(\Omega_{Y}^{1}\otimes\mathcal{L}^{\otimes
2})t^{i}$ $\displaystyle=$ $\displaystyle
c_{t}(\Omega_{Y}^{1}\otimes\mathcal{L}^{\otimes 2})$ $\displaystyle=$
$\displaystyle
c_{t}(\mathcal{M}_{1})c_{t}(\mathcal{M}_{2})c_{t}(\mathcal{M}_{3})$
$\displaystyle=$
$\displaystyle(1+c_{1}(\mathcal{M}_{1})t)(1+c_{1}(\mathcal{M}_{2})t)(1+c_{1}(\mathcal{M}_{3})t).$

Therefore, we get

$\deg c_{3}(\Omega_{Y}^{1}\otimes\mathcal{L}^{\otimes
2})=\deg(c_{1}(\mathcal{M}_{1})c_{1}(\mathcal{M}_{2})c_{1}(\mathcal{M}_{3}))=\mathcal{M}_{1}\cdot\mathcal{M}_{2}\cdot\mathcal{M}_{3}>0,$

as required. Thus (1) holds.

Let us show (2). There is a finite separable double cover $\varphi\colon X\to
Y=\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ as in (1) [FanoIII,
Theorem 6.7]. Moreover, for each $i\in\\{1,2,3\\}$, the composition
$\varphi_{i}\colon
X\xrightarrow{\varphi}\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\xrightarrow{\operatorname{pr}_{i}}\mathbb{P}^{1}$
is the contraction of an extremal ray [FanoIII, Remark 6.8]. Since $\varphi$
is separable, if $H_{1},H_{2},H_{3}$ are general members, then the scheme-
theoretic intersection $H_{1}\cap H_{2}\cap H_{3}$ is reduced two points. Let
$H_{1}^{\operatorname{gen}},H_{2}^{\operatorname{gen}},H_{3}^{\operatorname{gen}}$
be their generic members. Then the regular curve
$H_{1}^{\operatorname{gen}}\cap H_{2}^{\operatorname{gen}}$ is automatically
smooth (Remark 2.13). Then $X$ is $F$-split (Corollary 2.11, Lemma 2.12). ∎

######  Lemma 4.2.

Let $Y$ be a smooth Fano threefold of No. 2-18. Let $f_{1}\colon
Y\to\mathbb{P}^{1}$ and $f_{2}\colon Y\to\mathbb{P}^{2}$ be the contractions
of the extremal rays. Take a general point $P\in\mathbb{P}^{1}$ and a general
line $L$ on $\mathbb{P}^{2}$. Then $H^{0}(\Gamma,\mathcal{O}_{\Gamma})=k$ for
the scheme-theoretic intersection $\Gamma\coloneqq f_{1}^{-1}(P)\cap
f_{2}^{-1}(L)$.

###### Proof.

Set $S_{1}\coloneqq f_{1}^{-1}(P)$ and $S_{2}\coloneqq f_{2}^{-1}(L)$. By
[FS20, Theorem 15.2] and [BT22, Theorem 3.3], $S_{1}$ is a canonical del Pezzo
surface. In particular, $S_{1}$ is a rational surface. Since $-K_{Y}\sim
S_{1}+2S_{2}$ [FanoIII, Proposition 5.9 and Section 9.2], the divisor
$-K_{Y}|_{Y_{K}}=2S_{2}|_{Y_{K}}$ is ample for the generic fibre $Y_{K}$ of
$f_{1}\colon Y\to\mathbb{P}^{1}$, where $K\coloneqq K(\mathbb{P}^{1})$. Hence
$S_{2}|_{S_{1}}(=\Gamma)$ is ample, as $S_{1}$ is chosen to be a general fibre
of $f_{1}$. Therefore, $H^{1}(S_{1},\mathcal{O}_{S_{1}}(-\Gamma))=0$ by [CT19,
Proposition 3.3] (or [Muk13, Theorem 3]), and hence
$k=H^{0}(S_{1},\mathcal{O}_{S_{1}})\xrightarrow{\simeq}H^{0}(\Gamma,\mathcal{O}_{\Gamma})$.
∎

######  Lemma 4.3.

Let $X$ be a smooth Fano threefold of No. 3-3 or 3-4. Then $X$ is $F$-split.

###### Proof.

For each case, $X$ has exactly three extremal rays and there is a conic bundle
structure $f\colon X\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ [FanoIV, Section
7.3]. In what follows, we shall use their properties obtained in [FanoIV,
Propositions 4.33 and 4.35]. For each $i\in\\{1,2\\}$, let

$\varphi_{i}\colon
X\xrightarrow{f}\mathbb{P}^{1}\times\mathbb{P}^{1}\xrightarrow{\operatorname{pr}_{i}}\mathbb{P}^{1}=:Z_{i}$

be the induced composite contraction. Note that each $\varphi_{i}$ corresponds
to a two-dimensional extremal face of $\operatorname{NE}(X)$. Let
$\varphi_{3}\colon X\to Z_{3}=\mathbb{P}^{2}$ be the contraction of the
remaining two-dimensional extremal face of $\operatorname{NE}(X)$. For each
$i\in\\{1,2,3\\}$, let $H_{i}$ be the pullback of the ample generator on
$Z_{i}$. We recall $-K_{X}\sim H_{1}+H_{2}+H_{3}$ ([FanoIV, Propositions 4.33
and 4.35]). In particular,

$2=-K_{X}\cdot H_{1}\cdot H_{2}=(H_{1}+H_{2}+H_{3})\cdot H_{1}\cdot
H_{2}=H_{1}\cdot H_{2}\cdot H_{3}.$

For each $i\in\\{1,2,3\\}$, $H^{\prime}_{i}$ denotes the generic member of
$H_{i}$, which is a regular prime divisor on $X^{\prime}\coloneqq
X\times_{k}k^{\prime}$ for a suitable purely transcendental extension
$k^{\prime}/k$ (Remark 2.4). Note that $H^{\prime}_{1}\cap H^{\prime}_{2}$ is
a smooth curve, because $f$ is not wild [MS03, Corollary 8 and Remark 9].

We now finish the proof by assuming that

  1. (i)

$H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{1}+2H^{\prime}_{3})=H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{2}+2H^{\prime}_{3})=0$,
and

  2. (ii)

$H^{2}(X^{\prime},K_{X^{\prime}}+2H^{\prime}_{3})=0$.

By Corollary 2.11 and Lemma 2.12, it suffices to find
$H^{\prime\prime}_{3}\in|H^{\prime}_{3}|$ such that $H^{\prime}_{1}\cap
H^{\prime}_{2}\cap H^{\prime\prime}_{3}$ is smooth and zero-dimensional. Since
$k^{\prime}$ is an infinite field and $H^{\prime}_{1}\cap H^{\prime}_{2}$ is
isomorphic to a smooth conic on $\mathbb{P}^{2}_{k^{\prime}}$, the Bertini
theorem enables us to find a smooth zero-dimensional effective Cartier divisor
$D$ on $H^{\prime}_{1}\cap H^{\prime}_{2}$ such that
$H^{\prime}_{3}|_{H^{\prime}_{1}\cap H^{\prime}_{2}}\sim D$. Therefore, it is
enough to show that the restriction maps

$H^{0}(X^{\prime},\mathcal{O}_{X}(H^{\prime}_{3}))\xrightarrow{\alpha}H^{0}(H^{\prime}_{1},\mathcal{O}_{X}(H^{\prime}_{3})|_{H^{\prime}_{1}})\xrightarrow{\beta}H^{0}(H^{\prime}_{1}\cap
H^{\prime}_{2},\mathcal{O}_{X}(H^{\prime}_{3})|_{H^{\prime}_{1}\cap
H^{\prime}_{2}})$

are surjective. The restriction map $\alpha$ is surjective, because

$H^{1}(X^{\prime},H^{\prime}_{3}-H^{\prime}_{1})\simeq
H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{2}+2H^{\prime}_{3})\overset{{\rm(i)}}{=}0.$

The problem is reduced to the surjectivity of $\beta$. To this end, it
suffices to prove
$H^{1}(H^{\prime}_{1},\mathcal{O}_{X^{\prime}}(-H^{\prime}_{2}+H^{\prime}_{3})|_{H^{\prime}_{1}})=0$.
We have an exact sequence

$H^{1}(X^{\prime},-H^{\prime}_{2}+H^{\prime}_{3})\to
H^{1}(H^{\prime}_{1},\mathcal{O}_{X^{\prime}}(-H^{\prime}_{2}+H^{\prime}_{3})|_{H^{\prime}_{1}})\to
H^{2}(X^{\prime},-H^{\prime}_{1}-H^{\prime}_{2}+H^{\prime}_{3}).$

By

$-H^{\prime}_{2}+H^{\prime}_{3}\sim
K_{X^{\prime}}+H^{\prime}_{1}+2H^{\prime}_{3}\qquad{\rm
and}\qquad-H^{\prime}_{1}-H^{\prime}_{2}+H^{\prime}_{3}\sim
K_{X^{\prime}}+2H^{\prime}_{3},$

(i) and (ii) imply
$H^{1}(H^{\prime}_{1},(-H^{\prime}_{2}+H^{\prime}_{3})|_{H^{\prime}_{1}})=0$.
Therefore, it is enough to prove (i) and (ii).

######  Claim.

The following hold.

  1. (1)

$H^{\prime}_{i}$ is a regular weak del Pezzo surface for every
$i\in\\{1,2,3\\}$.

  2. (2)

$H^{0}(H^{\prime}_{i},\mathcal{O}_{H^{\prime}_{i}})=k^{\prime}$ for every
$i\in\\{1,2,3\\}$.

  3. (3)

$H^{2}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{i})=0$ for every
$i\in\\{1,2,3\\}$.

  4. (4)

$H^{2}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{i}+H^{\prime}_{j})=H^{2}(X^{\prime},K_{X^{\prime}}+2H^{\prime}_{3})=0$
for $1\leq i<j\leq 3$.

  5. (5)

$H^{1}(H^{\prime}_{i},\mathcal{O}_{H^{\prime}_{i}})=0$ and
$H^{1}(H^{\prime}_{i},K_{H^{\prime}_{i}})=0$ for every $i\in\\{1,2,3\\}$.

  6. (6)

$H^{0}(H^{\prime}_{1}\cap H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{1}\cap
H^{\prime}_{3}})=k^{\prime}$ and $H^{0}(H^{\prime}_{2}\cap
H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{2}\cap H^{\prime}_{3}})=k^{\prime}$.

  7. (7)

$H^{1}(H^{\prime}_{3},K_{H^{\prime}_{3}}+(H^{\prime}_{1}+H^{\prime}_{3})|_{H^{\prime}_{3}})=0$

###### Proof of Claim.

Since $H^{\prime}_{i}$ is the generic member of a base point free linear
system $|H_{i}|$, $H^{\prime}_{i}$ is a regular prime divisor on $X^{\prime}$.
If $i\in\\{1,2\\}$, then $-K_{H^{\prime}_{i}}$ is ample, because
$H^{\prime}_{i}$ is the generic fibre of $\varphi_{i}\colon
X\to\mathbb{P}^{1}$. We see that $-K_{H^{\prime}_{3}}$ is nef and big by
$-K_{H^{\prime}_{3}}\sim(H^{\prime}_{1}+H^{\prime}_{2})|_{H^{\prime}_{3}}$ and
$(H^{\prime}_{1}+H^{\prime}_{2})^{2}\cdot H^{\prime}_{3}=2H^{\prime}_{1}\cdot
H^{\prime}_{2}\cdot H^{\prime}_{3}>0$. Thus (1) holds. If $i\in\\{1,2\\}$,
then (2) holds by the fact that $H_{i}$ is (a base change of) the generic
fibre of a contraction $\varphi_{i}\colon X\to\mathbb{P}^{1}$. We have
$H^{0}(H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{3}})=k^{\prime}$, because
general members of the complete linear system $|H_{3}|$ are geometrically
integral [FanoI, Proposition 2.10]. Thus, (2) holds.

Let us show (3). Consider an exact sequence

$0=H^{2}(X^{\prime},K_{X^{\prime}})\to
H^{2}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{i})\to
H^{2}(H^{\prime}_{i},K_{H^{\prime}_{i}})\\\ \to
H^{3}(X^{\prime},K_{X^{\prime}})\to
H^{3}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{i})=0.$

By (2) and Serre duality, we obtain
$h^{2}(H^{\prime}_{i},K_{H^{\prime}_{i}})=1$ and
$h^{3}(X^{\prime},K_{X^{\prime}})=1$. Therefore,
$H^{2}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{i})=0$. Thus (3) holds.

Let us show (4). If $i\neq j$, then
$H^{2}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{i}+H^{\prime}_{j})=0$ by an exact
sequence

$0=H^{2}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{i})\to
H^{2}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{i}+H^{\prime}_{j})\to
H^{2}(H^{\prime}_{j},K_{H^{\prime}_{j}}+H^{\prime}_{i})=0,$

where $H^{2}(H^{\prime}_{j},K_{H^{\prime}_{j}}+H^{\prime}_{i})=0$ follows from
Serre duality and
$H^{0}(H^{\prime}_{j},\mathcal{O}_{X^{\prime}}(-H^{\prime}_{i})|_{H^{\prime}_{j}})=0$.
Similarly, $H^{2}(X^{\prime},K_{X^{\prime}}+2H^{\prime}_{3})=0$ by
$H^{0}(H^{\prime}_{3},\mathcal{O}_{X^{\prime}}(-H^{\prime}_{3})|_{H^{\prime}_{3}})=0$.
Thus (4) holds.

Let us show (5). By an exact sequence

$0=H^{1}(X^{\prime},\mathcal{O}_{X^{\prime}})\to
H^{1}(H^{\prime}_{i},\mathcal{O}_{H^{\prime}_{i}})\to
H^{2}(X^{\prime},\mathcal{O}_{X^{\prime}}(-H^{\prime}_{i})),$

it suffices to prove
$H^{2}(X^{\prime},\mathcal{O}_{X^{\prime}}(-H^{\prime}_{i}))=0$, which follows
from (4) by using $-H^{\prime}_{i}\sim
K_{X^{\prime}}+H^{\prime}_{1}+H^{\prime}_{2}+H^{\prime}_{3}-H^{\prime}_{i}$.
Thus (5) holds.

Let us show (6). Since $X$ has exactly three extremal rays, there is the
extremal ray $R$ such that $R$ is the intersection of the extremal faces
corresponding to $X\to\mathbb{P}^{1}$ and $X\to\mathbb{P}^{2}$. Let $f\colon
X\to Y$ be the contraction of $R$. If $X$ is 3-3, then
$Y=\mathbb{P}^{1}\times\mathbb{P}^{2}$ and $f\colon X\to
Y\coloneqq\mathbb{P}^{1}\times\mathbb{P}^{2}$ and $H^{\prime}_{1}\cap
H^{\prime}_{3}$ is isomorphic to the corresponding intersection $H_{1}^{\prime
Y}\cap H_{3}^{\prime Y}$ on $Y$, because $H_{1}^{\prime Y}\cap H_{3}^{\prime
Y}$ is disjoint from $f(\operatorname{Ex}(f))$. Therefore, $H^{\prime}_{1}\cap
H^{\prime}_{3}$ is geometrically integral. If $X$ is 3-4, then
$Y=\mathbb{F}_{1}$ or $Y$ is a smooth Fano threefold of No. 2-18. If
$Y=\mathbb{F}_{1}$, then $H^{\prime}_{1}\cap H^{\prime}_{3}$ is a smooth fibre
of $f\colon X\to\mathbb{F}_{1}$. The other case follows from Lemma 4.2 by
using the upper semi-continuity [Har77, Ch. III, Theorem 12.8]. Thus (6)
holds.

Let us show (7). We have

$H_{1}\cdot H^{2}_{3}=H_{2}\cdot H^{2}_{3}=1,$

because $2=-K_{X}\cdot H^{2}_{3}=(H_{1}+H_{2})\cdot H_{3}^{2}$ and a fibre
$\zeta\equiv H_{3}^{2}$ of $\varphi_{3}$ is not contracted by $\varphi_{i}$
for each $i\in\\{1,2\\}$. Fix a general member $H^{\prime\prime}_{3}$ of
$|H^{\prime}_{3}|$. By Serre duality, it is enough to show
$H^{1}(H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{3}}(-D))=0$ for an effective
Cartier divisor
$D\coloneqq(H^{\prime}_{1}+H^{\prime\prime}_{3})|_{H^{\prime}_{3}}$ on
$H^{\prime}_{3}$. By (2), the problem is reduced to
$H^{0}(D,\mathcal{O}_{D})=k^{\prime}$. Clearly, $D$ is nef. It holds that
$D^{2}=(H_{1}+H_{3})^{2}\cdot H_{3}\geq H_{1}\cdot H^{2}_{3}=1>0$. Hence $D$
is nef and big. Then $H^{0}(D,\mathcal{O}_{D})$ is a field [Eno, Corollary
3.17]. Since $\operatorname{Supp}D$ contains $H^{\prime}_{1}\cap
H^{\prime}_{3}\cap H^{\prime\prime}_{3}$ which is a $k^{\prime}$-rational
point by $H^{\prime}_{1}\cdot H^{\prime}_{3}\cdot
H^{\prime\prime}_{3}=H_{1}\cdot H_{3}^{2}=1$, we obtain field extensions

$k^{\prime}=H^{0}(H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{3}})\hookrightarrow
H^{0}(D,\mathcal{O}_{D})\hookrightarrow H^{0}(H^{\prime}_{1}\cap
H^{\prime}_{3}\cap H^{\prime\prime}_{3},\mathcal{O}_{H^{\prime}_{1}\cap
H^{\prime}_{3}\cap H^{\prime\prime}_{3}})=k^{\prime},$

which implies $H^{0}(D,\mathcal{O}_{D})=k^{\prime}$, as required. This
completes the proof of Claim. ∎

It is enough to show (i) and (ii). As (ii) has been settled by Claim(4), let
us show (i). By
$H^{0}(H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{3}})=k^{\prime}$ and
$H^{0}(H^{\prime}_{1}\cap H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{1}\cap
H^{\prime}_{3}})=k^{\prime}$ (Claim(2)(6)), we get
$H^{3}(H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{3}}(-H^{\prime}_{1}|_{H^{\prime}_{3}}))=0$
by the following exact sequence:

$H^{0}(H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{3}})\xrightarrow{\simeq}H^{0}(H^{\prime}_{1}\cap
H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{1}\cap H^{\prime}_{3}})\to
H^{1}(H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{3}}(-H^{\prime}_{1}|_{H^{\prime}_{3}}))\to
H^{1}(H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{3}})=0.$

By Serre duality, we get

$H^{1}(H^{\prime}_{3},K_{H^{\prime}_{3}}+H^{\prime}_{1}|_{H^{\prime}_{3}})=0.$

We have the following exact sequences:

$0=H^{1}(X^{\prime},K_{X^{\prime}})\to
H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{1})\to
H^{1}(H^{\prime}_{1},K_{H^{\prime}_{1}})=0,$
$0=H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{1})\to
H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{1}+H^{\prime}_{3})\to
H^{1}(H^{\prime}_{3},K_{H^{\prime}_{3}}+H^{\prime}_{1}|_{H^{\prime}_{3}})=0$
$0=H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{1}+H^{\prime}_{3})\to
H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{1}+2H^{\prime}_{3})\\\ \to
H^{1}(H^{\prime}_{3},K_{H^{\prime}_{3}}+(H^{\prime}_{1}+H^{\prime}_{3})|_{H^{\prime}_{3}})=0,$

where the last equality follows from Claim (7). Thus
$H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{1}+2H^{\prime}_{3})=0$.
Similarly, we obtain
$H^{1}(X^{\prime},K_{X^{\prime}}+H^{\prime}_{2}+2H^{\prime}_{3})=0$. Thus (i)
holds. ∎

######  Proposition 4.4.

Let $X$ be a smooth Fano threefold with $\rho(X)=3$.

  1. (1)

If $X$ is not 3-10, then $X$ is $F$-split.

  2. (2)

Assume that $X$ is 3-10. Then $X$ is $F$-split if and only if $X$ has no wild
conic bunlde structure.

###### Proof.

We may assume that $X$ is none of 3-1, 3-3, and 3-4 (Lemma 4.1, Lemma 4.3).
Note that if $X$ has a wild conic bundle structure, then $X$ is not $F$-split
[GLP15, Theorem 2.1 or Corollary 2.5]. In what follows, we assume that any
conic bundle structure from $X$ is generically smooth. Under this additional
assumption, it suffices to show that $X$ is $F$-split. We divide the proof
into the following four cases.

  1. (1)

3-27, 3-28, 3-31.

  2. (2)

3-5, 3-8, 3-12, 3-13, 3-15, 3-16, 3-17, 3-19, 3-20, 3-21, 3-22, 3-23, 3-24,
3-26, 3-29.

  3. (3)

3-6, 3-10, 3-18, 3-25.

  4. (4)

3-2, 3-7, 3-9, 3-11, 3-14, 3-30.

(1) In this case, $X$ is toric [FanoIV, Subsection 7.3], and hence $F$-split.

(2) In this case, there exist a smooth Fano threefold $Y$ with $\rho(Y)=2$, a
$\mathbb{P}^{1}$-bundle $g\colon Y\to\mathbb{P}^{2}$, and a subsection
$B\subset Y$ of $g$ such that $X\simeq\operatorname{Bl}_{B}Y$,
$B_{\mathbb{P}^{2}}\coloneqq g(B)\simeq\mathbb{P}^{1}$, and $Y$ is one of
2-32, 2-34, 2-35 [FanoIV, Subsection 7.3, cf. Theorem 4.23]. Set $S\coloneqq
g^{-1}(B)$, which is a $\mathbb{P}^{1}$-bundle over $\mathbb{P}^{1}$, and
hence $F$-split. Let $T$ be the pullback of the ample generator by the
contraction of the other extremal ray $R$. By [FanoIII, Proposition 5.9(3)],
we can write $-K_{Y}\sim aS+2T$, where $a$ is the length of $R$. Then we can
check that $a>1$ in each case [FanoIV, Subsection 7.2]. Thus
$-(K_{Y}+S)\sim(a-1)S+bT$ is ample by Kleiman’s criterion. Therefore, $(Y,S)$
is $F$-split (Proposition 2.8), and hence $X$ is $F$-split (Proposition 2.7).

(3) In this case, we can write $X\simeq\operatorname{Bl}_{C\amalg
C^{\prime}}\mathbb{P}^{3}$ or $X\simeq\operatorname{Bl}_{C\amalg C^{\prime}}Q$
for a disjoint union of smooth curves $C$ and $C^{\prime}$ on $\mathbb{P}^{3}$
or $Q$ [FanoIV, Subsection 7.3]. We only treat the case when $X$ is 3-6, as
the other cases are similar. In this case, $X\simeq\operatorname{Bl}_{C\amalg
C^{\prime}}\mathbb{P}^{3}$, $C$ is a line, and we can write
$C^{\prime}=S_{1}\cap S_{2}$ for some quadric surfaces $S_{1}$ and $S_{2}$.
Take a general plane $H$ containing the line $C$ and a general quadric surface
$S$ containing $C^{\prime}$. Let $H_{X}$ and $S_{X}$ be the proper transforms
of $H$ and $S$, respectively. Although $H_{X}\to H$ and $S_{X}\to S$ are not
necessarily isomorphisms, these birational morphisms are isomorphic over
$H\cap S$. Therefore, we obtain $H_{X}\cap S_{X}\xrightarrow{\simeq}H\cap S$.
This is nothing but a general fibre of the contraction
$X\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ [FanoIV, Proposition 4.37]. Since
this is not a wild conic bundle, we get $H\cap S\simeq H_{X}\cap
S_{X}\simeq\mathbb{P}^{1}$. Then $(\mathbb{P}^{3},H+S)$ is $F$-split by
applying Proposition 2.8 twice. Hence $X$ is $F$-split (Proposition 2.7).

(4) In what follows, we treat the remaining cases separately.

3-2: We use the same notation as in [FanoIV, Proposition 4.32]. We have
$-K_{X}\sim 2H_{1}+H_{2}+D$ and a conic bundle
$f:X\to\mathbb{P}^{1}\times\mathbb{P}^{1}$, where
$D\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$, and each $H_{i}$ is the pullback
of $\mathcal{O}_{\mathbb{P}^{1}}(1)$ by
$X\xrightarrow{f}\mathbb{P}^{1}\times\mathbb{P}^{1}\xrightarrow{\operatorname{pr}_{i}}\mathbb{P}^{1}$.
Moreover $f|_{D}\colon
D=\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ is
a finite double cover which can be written as ${\rm id}\times\psi$ for some
double cover $\psi\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$. For the generic
member $H^{\operatorname{gen}}_{2}$ of $|H_{2}|$, the intersection $C\coloneqq
H_{2}^{\operatorname{gen}}\cap D$ is a regular curve of genus zero with
$K_{C}+2H_{1}|_{C}\sim 0$, because $H_{1}\cdot C=H_{1}\cdot H_{2}\cdot D>0$.
By $\deg(H_{1}|_{C})=1$, we get $C\simeq\mathbb{P}^{1}_{\kappa}$. Hence $X$ is
$F$-split (Corollary 2.11).

3-7: In this case, $f\colon X\to W$ is a blowup along an elliptic curve
$B=S_{1}\cap S_{2}$ with $S_{1},S_{2}\in|-(1/2)K_{W}|$ [FanoIV, Subsection
7.3]. Let $S$ be a general member of $|-(1/2)K_{W}|$ containing $B$. Since $B$
is an ample effective Cartier divisor on $S$, it follows that $S$ is smooth
along $B$ and $S$ is normal. Note that the proper transform $S_{X}$ of $S$ on
$X$ is a fibre of a contraction $\pi\colon X\to\mathbb{P}^{1}$. Therefore, the
geometric generic fibre $X_{\overline{K}}$ of $\pi$ is normal, where
$K\coloneqq K(\mathbb{P}^{1})$ and $\overline{K}$ denotes the algebraic
closure of $K$. Since $X_{K}$ is a regular del Pezzo surface,
$X_{\overline{K}}$ has at worst canonical singularities [BT22, Theorem 3.3].
Therefore, a general fibre $S_{X}$ of $\pi\colon X\to\mathbb{P}^{1}$ is a
canonical del Pezzo surface. By $K_{S}^{2}=6$, Theorem 2.22 shows that
$S(\simeq S_{X})$ is $F$-split. Since $-(K_{W}+S)$ is ample, we have $(W,S)$
is $F$-split (Proposition 2.8), and hence $X$ is $F$-split (Proposition 2.7).

3-9: By [FanoIV, Proposition 4.42], there is a blowup $f\colon X\to
Y=Y_{\text{2-36}}=\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}\oplus\mathcal{O}(2))$
along a smooth curve $B$ such that

  * •

$B$ is contained in a section $S$ of the $\mathbb{P}^{1}$-bundle structure
$\pi\colon\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}\oplus\mathcal{O}(2))\to\mathbb{P}^{2}$,
and

  * •

$S$ is disjoint from another section $T$ of
$\pi\colon\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}\oplus\mathcal{O}(2))\to\mathbb{P}^{2}$.

Since $\pi$ is $\mathbb{P}^{1}$-bundle, we have
$K_{Y}+S+T=\pi^{*}K_{\mathbb{P}^{2}}$. Since
$-(K_{Y}+S+(1-\epsilon)T)=-\pi^{*}K_{\mathbb{P}^{2}}+\epsilon T$ is ample for
some $0<\epsilon\ll 1$ and $(S,0)$ is $F$-split, $(Y,S+(1-\epsilon)T)$ is
$F$-split (Proposition 2.8). Hence $X$ is $F$-split (Proposition 2.7).

3-11: By [FanoIV, Subsection 7.3], there exists a blowup $f\colon X\to V_{7}$
along an elliptic curve $B=S_{1}\cap S_{2}$ with
$S_{1},S_{2}\in|-(1/2)K_{V_{7}}|$. By the same argument as in that of 3-7, a
general member $S$ of $|(-1/2)K_{V_{7}}|$ is a canonical del Pezzo surface
with $K_{S}^{2}=7$. Then $S$ is $F$-split (Theorem 2.22). Since
$-(K_{V_{7}}+S)$ is ample, $(V_{7},S)$ is $F$-split (Proposition 2.8), and
hence $X$ is $F$-split (Proposition 2.7).

3-14: By [FanoIV, Subsection 7.3], we have $X=\operatorname{Bl}_{P\amalg
C}\mathbb{P}^{3}$, where $C$ is a smooth cubic curve contained in a plane $H$
and $P$ is a point satisfying $P\not\in H$. Take two distinct planes
$H^{\prime}$ and $H^{\prime\prime}$ containing $P$. Then
$(\mathbb{P}^{3},H+H^{\prime}+H^{\prime\prime})$ is $F$-split, and hence so is
$X$ (Proposition 2.7).

3-30: By [FanoIV, Subsection 7.3], there exist blowups

$X\to V_{7}\to\mathbb{P}^{3},$

where $V_{7}\to\mathbb{P}^{3}$ is a blowup at a point $P\in\mathbb{P}^{3}$ and
the blowup centre of $X\to V_{7}$ is the proper transform of a line $L$
passing through $P$. Take two planes $H,H^{\prime}$ containing $L$. Then
$(\mathbb{P}^{3},H+H^{\prime})$ is F-split, and hence $X$ is F-split
(Proposition 2.7). ∎

##  5\. Fano threefolds with $\rho=2$ (except for 2-2, 2-6, 2-8)

###  5.1. Quasi-$F$-splitting (imprimitive case)

####  5.1.1. $D+E$ (imprimitive)

######  Proposition 5.1.

Let $X$ be a smooth Fano threefold with $\rho(X)=2$. If the types of the
extremal rays are $D+E$, then $X$ is quasi-$F$-split.

###### Proof.

In this case, $X$ is imprimitive [FanoIV, Subsection 7.2], i.e., the types of
the extremal rays are $D+E_{1}$. Let $f:X\to Y$ (resp.
$\pi:X\to\mathbb{P}^{1}$) be the contraction of type $E_{1}$ (resp. type $D$).
Let $B$ be the smooth curve on $Y$ that is the blowup centre of $f:X\to Y$,
i.e., $X=\operatorname{Bl}_{B}Y$. By [FanoIV, Subsection 7.2], we may assume
that one of the following holds.

  1. (1)

$Y=\mathbb{P}^{3}$ (2-4, 2-25, 2-33).

  2. (2)

$Y=Q$, where $Q$ is a smooth quadric hypersurface on $\mathbb{P}^{4}$ (2-7,
2-29).

  3. (3)

$Y=V_{d}$ with $1\leq d\leq 5$, where $V_{d}$ is a smooth Fano threefold of
index two satisfying $(-K_{V_{d}})^{3}=8d$ (2-1, 2-3, 2-5, 2-10, 2-14).

######  Claim.

A general fibre $D_{X}$ of $\pi:X\to\mathbb{P}^{1}$ is a canonical del Pezzo
surface.

###### Proof of Claim.

By [FS20, Theorem 15.2], the generic fibre $X_{K}$ of $\pi$ is geometrically
normal, where $K$ denotes the function field of $\mathbb{P}^{1}$. Then its
base change $X_{\overline{K}}:=X_{K}\times_{K}\overline{K}$ to the algebraic
closure $\overline{K}$ has at worst canonical singularities [BT22, Theorem
3.3]. Hence a general fibre $D_{X}$ of $\pi$ is normal and has at worst
canonical singularities. This completes the proof of Claim. ∎

(1) Assume that $Y=\mathbb{P}^{3}$. In this case, $B=D\cap D^{\prime}$ for
some surfaces $D$ of degree $1\leq e\leq 3$, i.e., $Y$ is 2-4 ($e=3$), 2-25
($e=2$), or 2-33 ($e=1$). Although $D$ might be singular, $D$ is a normal
prime divisor on $X$, because $D$ is smooth along an effective ample Cartier
divisor $B=D^{\prime}|_{D}$. After replacing $D$ by a general member of the
pencil generated by $D$ and $D^{\prime}$, we may assume that $D$ is a
canonical del Pezzo surface (Claim). If $e\in\\{1,2\\}$, then $D$ is $F$-split
(Theorem 2.22). Then $(Y,D)$ is $F$-split (Corollary 2.9), which implies that
$X$ is $F$-split (Proposition 2.7). We may assume that $D$ is a cubic surface,
i.e., $X$ is 2-4. Recall that $-K_{X}\sim D_{X}+H$, where $D_{X}$ denotes the
proper transform of $D$ and $H\coloneqq f^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)$.
Replacing $D_{X}$ and $H$ by general members of $|D_{X}|$ and $|H|$, we obtain
$D_{X}\cap H\simeq D\cap{f_{*}}H$. Since $D\cap H$ is a general hyperplane
section of a normal cubic surface $D$, we have $D\cap{f_{*}}H\simeq D_{X}\cap
H$ is an elliptic curve. Hence $X$ is quasi-$F$-split (Corollary 2.18, Remark
2.19).

(2) Assume that $Y=Q$. In this case, $B=D\cap D^{\prime}$ for some surfaces
$D\in|\mathcal{O}_{Q}(e)|$ with $1\leq e\leq 2$, i.e., $Y$ is 2-29 ($e=1$), or
2-7 $(e=2)$. By Claim, $D$ is a canonical del Pezzo surface. We have

$K_{D}^{2}=(K_{Q}+D)^{2}\cdot D=2e(3-e)^{2}.$

If $e=1$, then $K_{D}^{2}=8$, and thus $D$ is $F$-split (Theorem 2.22). Then
$(Y,D)$ is $F$-split (Corollary 2.9), which implies that $X$ is $F$-split
(Proposition 2.7). We may assume that $e=2$, i.e., $X$ is 2-7. Set $H\coloneqq
f^{*}\mathcal{O}_{Q}(1)$. Replacing $D$ and $f_{*}H$ by general members of
$|D_{X}|$ and $|\mathcal{O}_{Q}(1)|$, we obtain $D_{X}\cap H\simeq D\cap
f_{*}H$. Since $D\cap{f_{*}}H$ is a general hyperplane section of a canonical
del Pezzo surface $D$ of degree $4$ with
$-K_{D}\sim{f_{*}}H|_{D}=D\cap{f_{*}}H$, it follows that $D\cap f_{*}H\simeq
D_{X}\cap H$ is an elliptic curve. Hence $X$ is quasi-$F$-split (Corollary
2.18, Remark 2.19).

(3) Assume that $Y=V_{d}$. In this case, $B$ is an elliptic curve which is a
complete intersection of two prime divisors $D\in|-(1/2)K_{Y}|$ and
$D^{\prime}\in|-(1/2)K_{Y}|$, i.e., $Y$ is 2-1, 2-3, 2-5, 2-10, or 2-14. We
then get $K_{X}+D_{X}+D^{\prime}_{X}+E\sim 0$, where $E:=\operatorname{Ex}(f)$
and $D_{X}$ and $D^{\prime}_{X}$ are the proper transforms of $D$ and
$D^{\prime}$, respectively. Fix general members $D$ and $D^{\prime}$ of
$|-(1/2)K_{Y}|$ containing $B$. Let $D^{\operatorname{gen}}$ be the generic
member of the pencil generated by $D$ and $D^{\prime}$. Let $\kappa$ be the
function field $\kappa$ of this pencil. For every $k$-scheme $Z$, we set
$Z_{\kappa}:=Z\times_{k}\kappa$. Then $D^{\operatorname{gen}}\cap
D^{\prime}_{\kappa}=B_{\kappa}$ and $X_{\kappa}\to Y_{\kappa}$ is the blowup
along $B_{\kappa}$. Note that the proper transform
$D^{\operatorname{gen}}_{X}$ is regular [Tan-Bertini, Theorem 4.9]. Since
$D^{\operatorname{gen}}_{X}\cap((D^{\prime}_{X})_{\kappa}+E_{\kappa})=D^{\operatorname{gen}}_{X}\cap
E_{\kappa}\simeq B_{\kappa}$ is smooth over $\kappa$, we conclude that $X$ is
quasi-$F$-split (Corollary 2.18, Remark 2.19). ∎

####  5.1.2. $E+E$ (imprimitive)

######  Proposition 5.2.

Let $X$ be a smooth Fano threefold with $\rho(X)=2$. Assume that the types of
the extremal rays are $E_{1}$ and $E$. Then the following hold.

  1. (1)

$X$ is quasi-$F$-split.

  2. (2)

If $X$ is not 2-12, then $X$ is $F$-split.

###### Proof.

Let $f\colon X\to Y$ be a contraction of type $E_{1}$ and let
$f^{\prime}\colon X\to Y^{\prime}$ be the contraction of the other extremal
ray, which is of type $E$. Let $H_{Y}$ (resp. $H_{Y^{\prime}}$) be the ample
Cartier divisor that generates $\operatorname{Pic}\,Y$ (resp.
$\operatorname{Pic}\,Y^{\prime}$). Set $H\coloneqq f^{*}H_{Y}$ and
$H^{\prime}\coloneqq f^{\prime*}H_{Y^{\prime}}$. The list of such smooth Fano
threefolds is as follows [FanoIV, Subseciton 7.2]:

$\text{2-12, 2-15, 2-17, 2-19, 2-21, 2-22, 2-23, 2-26, 2-28, 2-30}.$

In particular, $Y$ is $\mathbb{P}^{3}$, $Q$, or $V_{d}$ with $3\leq d\leq 5$.
Then $|H_{Y}|$ is very ample, and hence we may assume that $H_{Y}$ is a smooth
prime divisor on $Y$. Note that we have $-K_{X}\sim H+H^{\prime}$ except when
$X$ is 2-30 [FanoIII, Remark 3.4 and Proposition 5.9].

######  Step 1.

If $X$ is 2-15, 2-28, or 2-30, then $X$ is $F$-split.

###### Proof of Step 1.

In this case, there is a blowup $f:X\to Y=\mathbb{P}^{3}$ along a smooth curve
$B$ such that $B$ is contained in a prime divisor $D$ on $\mathbb{P}^{3}$ of
degree $\leq 2$ ([FanoIII, Proposition 9.3], [FanoIV, Subsection 7.2]). Since
$D$ is $F$-split and $-(K_{\mathbb{P}^{3}}+D)$ is ample, it follows that
$(\mathbb{P}^{3},D)$ is $F$-split (Corollary 2.9), which implies that $X$ is
$F$-split (Proposition 2.7). This completes the proof of Step 1. ∎

######  Step 2.

If $X$ is 2-17 2-19, 2-21, 2-22, 2-23, or 2-26, then $X$ is $F$-split.

###### Proof of Step 2.

Replace $H$ by a general member of $|H|$. We now prove that $H$ is a smooth
prime divisor that is $F$-split. We first treat the case when $X$ is 2-17,
2-19, or 2-22. In this case, there is a blowup $f\colon X\to Y=\mathbb{P}^{3}$
along a smooth curve $B$ of degree $\leq 5$. For a general plane $H_{Y}$, its
pullback $H=f^{*}H_{Y}$ is nothing but the blowup of $H_{Y}$ along $H_{Y}\cap
B$. Since $H_{Y}\cap B$ is smooth and $-K_{H}=-(K_{X}+H)|_{H}=H^{\prime}|_{H}$
is nef and big, it follows that $H$ is a smooth weak del Pezzo surface. By
$K_{H}^{2}\geq K_{H_{Y}}^{2}-5=4$, we have $H$ is $F$-split [KT, Proposition
3.6]. For the the remaining case (i.e., $X$ is 2-21, 2-23, or 2-26), we can
apply the same argument, because there is a blowup $f\colon X\to Y=Q$ along a
smooth curve $B$ of degree $\leq 4$, and hence we may apply [KT, Proposition
3.6]. Therefore, $H$ is a smooth prime divisor which is $F$-split.

In order to prove that $(X,H)$ is $F$-split, it is enough to show that

$H^{1}(X,-H-(p^{e}-1)(K_{X}+H))=H^{1}(X,K_{X}+p^{e}H^{\prime})=0$

for some $e>0$ by $-K_{X}\sim H+H^{\prime}$ and Proposition 2.8. Set
$E^{\prime}\coloneqq\operatorname{Ex}(f^{\prime})$. By the Fujita vanishing
theorem [Fuj17, Theorem 3.8.1], we can find $e_{0}\in\mathbb{Z}_{>0}$ and
$s_{0}\in\mathbb{Z}_{>0}$ such that

$H^{1}(X,K_{X}+p^{e}H^{\prime}-s_{0}E^{\prime})=0$

for every integer $e\geq e_{0}$. Indeed, we can find
$a_{0},t_{0}\in\mathbb{Z}_{>0}$ such that $a_{0}H^{\prime}-t_{0}E$ is ample.
Then, by Fujita vanishing, there exists $m\gg 0$ such that

$H^{1}(X,K_{X}+m(a_{0}H^{\prime}-t_{0}E^{\prime})+N)=0$

for any nef Cartier divisor $N$ on $X$. Then we take $s_{0}=mt_{0}$ and
$e_{0}\in\mathbb{Z}_{>0}$ satisfying $p^{e}\geq ma_{0}$.

Recall that $\mathcal{O}_{X}(mH^{\prime}-E^{\prime})|_{E^{\prime}}$ is ample
for some $m\gg 0$ and $E^{\prime}$ satisfies the Kodaira vanishing theorem
(for Cartier divisors), because $E^{\prime}$ is a smooth projective surface
with negative Kodaira dimension or a singular quadric surface [FanoIII,
Definition 3.3]. Hence, for each $0\leq s\leq s_{0}$, we can find
$e(s)\in\mathbb{Z}_{>0}$ such that

$H^{1}(E^{\prime},\mathcal{O}_{X}(K_{X}+p^{e}H^{\prime}-sE^{\prime})|_{E^{\prime}})=H^{1}(E^{\prime},\mathcal{O}_{E^{\prime}}(K_{E^{\prime}}+p^{e}H^{\prime}-(s+1)E^{\prime}))=0$

for every $e\geq e(s)$. Set

$e\coloneqq\max\\{e_{0},e(0),e(1),...,e(s_{0})\\}.$

It suffices to prove

$H^{1}(X,K_{X}+p^{e}H^{\prime}-sE^{\prime})=0$

for every $0\leq s\leq s_{0}$. By descending induction on $s$, this follows
from

$H^{1}(X,K_{X}+p^{e}H^{\prime}-(s+1)E^{\prime})\to
H^{1}(X,K_{X}+p^{e}H^{\prime}-sE^{\prime})\\\ \to
H^{1}(S,\mathcal{O}_{X}(K_{X}+p^{e}H^{\prime}-sE^{\prime}))=0.$

This completes the proof of Step 2. ∎

######  Step 3.

If $X$ is 2-12, then $X$ is quasi-$F$-split.

###### Proof of Step 3.

In this case, the contraction of each extrmeal ray is a blowup $X$ of
$\mathbb{P}^{3}$ along a smooth curve $B$ of degree $6$ [FanoIV, Subsection
7.2]. As in the argument in Step 2, replacing $H$ and $H^{\prime}$ by general
members of $|H|$ and $|H^{\prime}|$ respectively, we may assume that $H$ and
$H^{\prime}$ are smooth weak del Pezzo surfaces with $K_{H}^{2}=3$.

We now finish the proof by assuming that the restriction map

$\rho\colon H^{0}(X,\mathcal{O}_{X}(-K_{X}-H))\to
H^{0}(H,\mathcal{O}_{H}(-K_{H}))$

is surjective. Since a general member $C$ of $|-K_{H}|$ is an elliptic curve
[KN, Theorem 1.4] and $H^{\prime}$ is a general member in
$|H^{\prime}|=|-K_{X}-H|$, it follows that $H\cap H^{\prime}$ is an elliptic
curve. Therefore, $X$ is quasi-$F$-split by Corollary 2.18.

It suffices to show that $\rho$ is surjective. By an exact sequence

$0\to\mathcal{O}_{X}(-K_{X}-2H)\to\mathcal{O}_{X}(-K_{X}-H)\to\mathcal{O}_{H}(-K_{H})\to
0,$

it is enough to prove that $H^{1}(X,-K_{X}-2H)=0$. Note that

$-K_{X}-2H\sim K_{X}+2H^{\prime}.$

Since $H^{\prime}$ is a smooth rational surface and $H^{\prime}|_{H^{\prime}}$
is nef and big, we have $H^{1}(H^{\prime},K_{H^{\prime}}+sH^{\prime})=0$ for
$s\geq 0$ by [Muk13, Theorem 3]. Thus, we have a surjection

$H^{1}(X,K_{X}+sH^{\prime})\to H^{1}(X,K_{X}+(s+1)H^{\prime})\to
H^{1}(H^{\prime},K_{H^{\prime}}+sH^{\prime})=0$

for every integer $s\geq 0$. Since we have $H^{1}(X,K_{X})=0$, using the above
surjectivity for $s\in\\{0,1\\}$, we obtain $H^{1}(X,K_{X}+2H^{\prime})=0$.
This completes the proof of Step 3. ∎

Step 1, Step 2, and Step 3 complete the proof of Proposition 5.2. ∎

####  5.1.3. Langer surface

######  Definition 5.3.

  1. (1)

For all the $\mathbb{F}_{2}$-points
$P_{1},\ldots,P_{7}\in\mathbb{P}^{2}_{\mathbb{F}_{2}}$, we set

$V_{L,\mathbb{F}_{2}}\coloneqq\operatorname{Bl}_{P_{1}\amalg\cdots\amalg
P_{7}}\mathbb{P}^{2}_{\mathbb{F}_{2}}.$

For a field $K$ of characteristic two, we set
$V_{L,K}:=V_{L,\mathbb{F}_{2}}\times_{\mathbb{F}_{2}}K$, which is called the
Langer surface over $K$.

  2. (2)

For a field of characteristic two and a zero-dimensional closed subscheme $Z$
of $\mathbb{P}^{2}_{K}$, we say that $Z$ is a Langer configuration if
$\operatorname{Bl}_{Z}\mathbb{P}^{2}_{K}$ is $K$-isomorphic to the Langer
surface $V_{L,K}$ over $K$.

######  Lemma 5.4.

Let $K$ be an algebraically closed field of characteristic two. Take a Langer
configuration $Z\subset\mathbb{P}^{2}_{K}$. Then there exists a
$K$-automorphism
$\sigma:\mathbb{P}^{2}_{K}\xrightarrow{\simeq}\mathbb{P}^{2}_{K}$ such that
$\sigma(Z)=Z_{0}$, where

$Z_{0}\coloneqq\\{[1:0:0],[0:1:0],[0:0:1],[1:1:0],[1:0:1],[0:1:1],[1:1:1]\\}.$

###### Proof.

Fix a $K$-isomorphism
$\theta\colon\operatorname{Bl}_{Z}\mathbb{P}^{2}_{K}\xrightarrow{\simeq}\operatorname{Bl}_{Z_{0}}\mathbb{P}^{2}_{K}=V_{L,K}$.
We have two birational contractions

$\varphi:\operatorname{Bl}_{Z}\mathbb{P}^{2}_{K}\to\mathbb{P}^{2}_{K},\qquad\varphi_{0}:\operatorname{Bl}_{Z_{0}}\mathbb{P}^{2}_{K}\to\mathbb{P}^{2}_{K},$

where $\varphi$ (resp. $\varphi_{0}$) is the blowup along $Z$ (resp. $Z_{0}$).
Recall that the Langer surface $V_{L,K}$ has exactly $7$ $(-1)$-curves ([CT18,
Theorem 5.4] or [KN, Lemma 4.5(4)]). Then both $\varphi$ and $\varphi_{0}$
contracts all the $(-1)$-curves on $\operatorname{Bl}_{Z}\mathbb{P}^{2}_{K}$
and $\operatorname{Bl}_{Z_{0}}\mathbb{P}^{2}_{K}$. Therefore, we obtain a
$K$-automorphism $\sigma:\mathbb{P}^{2}_{K}\to\mathbb{P}^{2}_{K}$ which
completes the following commutative diagram:

${\operatorname{Bl}_{Z}\mathbb{P}^{2}_{K}}$${\operatorname{Bl}_{Z_{0}}\mathbb{P}^{2}_{K}}$${\mathbb{P}^{2}_{K}}$${\mathbb{P}^{2}_{K}.}$$\scriptstyle{\theta,\simeq}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi_{0}}$$\scriptstyle{\sigma,\simeq}$

This diagram shows that $\sigma(Z)=Z_{0}$. ∎

######  Lemma 5.5.

Let $K$ be a $C_{1}$-field of characteristic two and take its algebraic
closure $\overline{K}$. Let $V$ be a smooth projective surface over $K$ whose
base change
$V_{\overline{K}}:=V\times_{\operatorname{Spec}K}\operatorname{Spec}{\overline{K}}$
is $\overline{K}$-isomorphic to the Langer surface over $\overline{K}$. Then
the following hold.

  1. (1)

If $K$ is perfect, then $V$ is $K$-isomorphic to the Langer surface over $K$.

  2. (2)

$\rho(V)=8$.

###### Proof.

We now show the implication (1)$\Rightarrow$ (2). Set $K^{\prime}$ to be the
purely inseparable closure of $K$ in $\overline{K}$, i.e.,

$K^{\prime}\coloneqq\bigcup_{e=0}^{\infty}K^{1/2^{e}},\qquad
K^{1/2^{e}}:=\\{a\in\overline{K}\,|\,a^{2^{e}}\in K\\}.$

Note that $K^{\prime}$ is a $C_{1}$-field, because being $C_{1}$ is stable
under algebraic extensions [GS17, Definition 6.2.1 and Lemma 6.2.4].
Therefore, (1) is applicable to the perfect $C_{1}$-field $K^{\prime}$ and the
base change $V_{K^{\prime}}:=V\times_{K}K^{\prime}$. Therefore,
$V_{K^{\prime}}$ is the Langer surface over $K^{\prime}$, and hence
$\rho(V_{K^{\prime}})=8$. Since the field extension $K\subset K^{\prime}$ is
purely inseparable, it holds that $\rho(V)=\rho(V_{K^{\prime}})=8$ [Tan18b,
Proposition 2.4]. This completes the proof of the implication (1)
$\Rightarrow$ (2).

It suffices to show (1). Assume that $K$ is perfect. Recall that
$V_{\overline{K}}$ contains the exactly $7$ $(-1)$-curves $E_{1},...,E_{7}$.
Set $\overline{\Gamma}\coloneqq E_{1}+\cdots+E_{7}$ and
$\overline{\mathcal{L}}:=\mathcal{O}_{V_{\overline{K}}}(\overline{\Gamma})$.
We now show that

  1. (i)

there is an invertible sheaf $\mathcal{L}$ on $V$ such that
$\alpha^{*}\mathcal{L}\simeq\overline{\mathcal{L}}$, where $\alpha\colon
V_{\overline{K}}\to V$ is the natural morphism and

  2. (ii)

there exists an effective Cartier divisor $\Gamma$ on $V$ such that
$\mathcal{O}_{X}(\Gamma)\simeq\mathcal{L}$ and the equality
$\alpha^{*}\Gamma=\overline{\Gamma}$ of Weil divisors holds.

Let us show (i). By $H^{0}(V,\mathcal{O}_{V})=K$ and $\operatorname{Br}(K)=0$
[GS17, Proposition 6.2.3], we obtain
$\operatorname{Pic}\,V\xrightarrow{\simeq}\operatorname{Pic}\,(V_{\overline{K}})^{\operatorname{Gal}(\overline{K}/K)}$
[FanoII, Proposition 2.3] (essentially due to [CTS21, Proposition 5.4.2]).
Then it holds that

$\sigma^{*}\overline{\mathcal{L}}\simeq\mathcal{O}_{V_{\overline{K}}}(\sigma^{*}\overline{\Gamma})\simeq\mathcal{O}_{V_{\overline{K}}}(\overline{\Gamma})\simeq\overline{\mathcal{L}}$

for every $\sigma\in\operatorname{Gal}(\overline{K}/K)$, and hence
$\overline{\mathcal{L}}\simeq\alpha^{*}\mathcal{L}$ for some
$\mathcal{L}\in\operatorname{Pic}\,V$. Thus (i) holds. Let us show (ii). By
the flat base change theorem, we obtain

$H^{0}(V,\mathcal{L})\otimes_{K}\overline{K}\simeq
H^{0}(V_{\overline{K}},\overline{L}),$

which implies
$\dim_{K}H^{0}(V,\mathcal{L})=\dim_{\overline{K}}H^{0}(V_{\overline{K}},\overline{\mathcal{L}})=\dim_{\overline{K}}H^{0}(V_{\overline{K}},\mathcal{O}_{V_{\overline{K}}}(\overline{\Gamma}))=1$.
In particular, there exists an effective Cartier divisor $\Gamma$ on $V$ such
that $\mathcal{O}_{V}(\Gamma)\simeq\mathcal{L}$ and
$\alpha^{*}\Gamma=\overline{\Gamma}$. Thus (ii) holds.

Since $\overline{\Gamma}=\sum_{i=1}^{7}E_{i}$ is smooth, so is $\Gamma$ (note
that $\Gamma$ might be irreducible, although $\overline{\Gamma}$ is not). Let
$\varphi:V\to W$ be the contraction of $\Gamma$, where $W$ is a smooth
projective surface over $K$. Then its base change
${\varphi}_{\overline{K}}:V_{\overline{K}}\to W_{\overline{K}}$ to
$\overline{K}$ is the birational contraction of $\overline{\Gamma}$, i.e.,
$W_{\overline{K}}\simeq\mathbb{P}^{2}_{\overline{K}}$. By
$\operatorname{Br}(K)=0$, we get a $K$-isomorphism $W\simeq\mathbb{P}^{2}_{K}$
[CTS21, Proposition 7.1.6]. Via this isomorphism, we identify $W$ and
$\mathbb{P}^{2}_{K}$ (resp. $W_{\overline{K}}$ and
$\mathbb{P}^{2}_{\overline{K}}$):

$\varphi:V\to
W=\mathbb{P}^{2}_{K},\qquad{\varphi}_{\overline{K}}:V_{\overline{K}}\to
W_{\overline{K}}=\mathbb{P}^{2}_{\overline{K}}.$

Recall that $\varphi$ is the blowup along some closed subscheme $Z$ on
$W=\mathbb{P}^{2}_{K}$ [Har77, Ch. II, Theorem 7.17]. Since blowups commutes
with flat base changes [Liu02, Section 8, Proposition 1.12(c)],
$\varphi_{\overline{K}}$ is the blowup along the base change
$Z_{\overline{K}}$. Since $Z_{\overline{K}}$ is a zero-dimensional reduced
scheme consisting of $7$ points, $Z$ is a smooth zero-dimensional closed
subscheme of $\mathbb{P}^{2}_{K}$ satisfying $h^{0}(Z,\mathcal{O}_{Z})=7$.

Set
$H:=\operatorname{Hilb}^{7}_{\mathbb{P}^{2}_{\overline{\mathbb{F}_{2}}/\overline{\mathbb{F}}_{2}}}$,
which is the Hilbert scheme of
$\mathbb{P}^{2}_{\overline{\mathbb{F}}_{2}}\to\operatorname{Spec}\overline{\mathbb{F}}_{2}$
that parametrises the zero-dimensional closed subschemes $W$ satisfying
$h^{0}(W,\mathcal{O}_{W})=7$. Let
$U:=\operatorname{Univ}^{7}_{\mathbb{P}^{2}_{\overline{\mathbb{F}}_{2}}/\overline{\mathbb{F}}_{2}}$
be its universal family (cf. [FGI05, Section 5]):
$U\subset\mathbb{P}^{2}_{\overline{\mathbb{F}}_{2}}\times_{\overline{\mathbb{F}}_{2}}H\to
H$. Let $H_{L}(\overline{\mathbb{F}}_{2})\subset H(\overline{\mathbb{F}}_{2})$
be the subset consisting of the Langer configurations over
$\overline{\mathbb{F}}_{2}$ (Definition 5.3(2)). Set
$G:=\operatorname{PGL}_{3,\overline{\mathbb{F}}_{2}}$, which is an algebraic
group over $\overline{\mathbb{F}}_{2}$. By Lemma 5.4,
$H_{L}(\overline{\mathbb{F}}_{2})$ is equal to the
$G(\overline{\mathbb{F}}_{2})$-orbit of $[Z_{0}]\in
H(\overline{\mathbb{F}}_{2})$, where

$Z_{0}:=\\{[1:0:0],[0:1:0],[0:0:1],[1:1:0],[1:0:1],[0:1:1],[1:1:1]\\}\subset\mathbb{P}^{2}_{\overline{\mathbb{F}}_{2}}.$

Recall that the $G(\overline{\mathbb{F}}_{2})$-orbit
$H_{L}(\overline{\mathbb{F}}_{2})$ is a locally closed subset of
$H(\overline{\mathbb{F}}_{2})$ [Mil16, Proposition 1.65(b)]. Since
$G=\operatorname{PGL}_{3}(\overline{\mathbb{F}}_{2})$ is irreducible, so is
$H_{L}(\overline{\mathbb{F}}_{2})$. There exists an integral locally closed
subscheme $H_{L}$ of $H$ whose set of the $\overline{\mathbb{F}}_{2}$-valued
points coincides with $H_{L}(\overline{\mathbb{F}}_{2})$. Set $U_{L}\coloneqq
U\times_{H}H_{L}$. We have the Langer configuration as the fibre of $\pi\colon
U\to H$ over $[Z_{0}]$:

$\\{P_{1}^{U},\ldots,P_{7}^{U}\\}=\pi^{-1}([Z_{0}])\subset U.$

Since $G=\operatorname{PGL}_{3,\overline{\mathbb{F}}_{2}}$ equivariantly acts
on $\pi:U\to H$, we have the orbits

$O_{G}(P_{1}^{U}),\ldots,O_{G}(P_{7}^{U})$

of the above $7$ points $P_{1}^{U},\ldots,P_{7}^{U}$. Since each
$O_{G}(P_{i}^{U})$ is a locally closed subset, this is a subvariety (integral
scheme). Since any fibre of $U_{L}\to H_{L}$ is geometrically reduced, it
follows that $U_{L}$ is reduced. Therefore, we get a scheme-theoretic equality

$U_{L}=O_{G}(P_{1}^{U})\amalg\cdots\amalg O_{G}(P_{7}^{U}),$

because we have the corresponding set-theoretic equality.

Note that $Z\subset\mathbb{P}^{2}_{K}$ corresponds to a $K$-rational point
$[Z]\in H_{K}(K)$ such that the corresponding $\overline{K}$-rational point
$[Z_{\overline{K}}]\in H_{\overline{K}}(\overline{K})$ is contained in
$(H_{L}\times_{\overline{\mathbb{F}}_{2}}\overline{K})(\overline{K})$. Then
the image

$[Z]\hookrightarrow H(K)\to H$

is contained in the Langer locus $H_{L}$ (over $\overline{\mathbb{F}}_{2}$).
Therefore, $Z\to[Z]$ is obtained by a base change of

$U_{L}\to H_{L},$

and hence $Z$ must be split up into $7$ distinct points. ∎

####  5.1.4. $C+E$ (imprimitive)

######  Proposition 5.6.

Let $X$ be a smooth Fano threefold such that $\rho(X)=2$ and the types of the
extremal rays are $C+E_{1}$. Then the following hold.

  1. (1)

$X$ is quasi-$F$-split.

  2. (2)

If $X$ is not 2-9, then $X$ is $F$-split.

###### Proof.

Let $\pi:X\to\mathbb{P}^{2}$ (resp. $g:X\to Y$) be the contraction of the
extremal ray of type $C$ (resp. $E_{1}$). Let $S$ be a general member of
$|\pi^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)|$ and let $H$ be a general member of
$|g^{*}\mathcal{O}_{Y}(1)|$, where $\mathcal{O}_{Y}(1)$ denotes the ample
generator of $\operatorname{Pic}Y$. Since $Y=\mathbb{P}^{3},Q$, or $V_{d}$
with $3\leq d\leq 5$ [FanoIV, Subsection 7.2], the complete linear system
$|\mathcal{O}_{Y}(1)|$ is very ample. Note that $H$ is the blowup of a smooth
surface $\overline{H}\in|\mathcal{O}_{Y}(1)|$ along the zero-dimensional
smooth closed subscheme $B\cap\overline{H}$, and hence $H$ is a smooth prime
divisor. We have that

$-K_{X}\sim S+\mu H$

for the length $\mu$ of the extremal ray of type $C$ [FanoIII, Proposition
5.9], i.e., if $\pi$ is of type $C_{i}$ with $i\in\\{1,2\\}$, then $\mu=i$.
Note that $-K_{H}=-(K_{X}+H)|_{H}=(S+(\mu-1)H)|_{H}$ is nef and $S^{2}\cdot
H=\frac{2}{i}$ [FanoIII, Lemma 5.3 and Proposition 5.9(2)].

######  Step 1.

If $\pi$ is of type $C_{2}$, then $X$ is $F$-split.

###### Proof of Step 1.

Assume that $\pi$ is of type $C_{2}$. In this case, $X$ is 2-27 or 2-31
[FanoIV, Subsection 7.2]. If $X$ is 2-27 (resp. 2-31), then
$X=\operatorname{Bl}_{C}\,Y$ for $Y=\mathbb{P}^{3}$ (resp. $Y=Q$), where $C$
is a smooth curve of degree $3$ (resp. $1$). Recall that $-K_{X}\sim S+2H$. We
then get $S\cdot H^{2}=(-K_{X})\cdot
H^{2}-2H^{3}=(-K_{Y})\cdot\mathcal{O}_{Y}(1)^{2}-2\mathcal{O}_{Y}(1)^{3}=2$ in
both cases. Since we have

$K_{H}^{2}=(S+H)^{2}\cdot H=S^{2}\cdot H+2(S\cdot H^{2})+H^{3}=1+4+H^{3},$

it follows that $H$ is a smooth del Pezzo surface with $K_{H}^{2}=6$ (resp.
$K_{H}^{2}=7$). Since $-(K_{X}+H)\sim S+H$ is ample, so is $-K_{H}$. Thus $H$
is $F$-split (Theorem 2.22). Therefore, $X$ is $F$-split (Corollary 2.9). This
completes the proof of Step 1. ∎

######  Step 2.

Assume that $\pi$ is of type $C_{1}$. Then the following hold.

  1. (i)

$K_{H}^{2}=2$ and $H$ is a smooth weak del Pezzo surface.

  2. (ii)

The induced composite morphism

$\pi_{H}\colon H\hookrightarrow X\xrightarrow{\pi}\mathbb{P}^{2}$

coincides with the morphism induced by the complete linear system $|-K_{H}|$.
Moreover, $\pi_{H}$ is a generically finite morphism of degree two.

  3. (iii)

If there exists a smooth prime divisor $C$ on $H$ satisfying $C\sim-K_{H}$,
then $X$ is quasi-$F$-split.

###### Proof of Step 2.

It holds that

$K_{H}^{2}=(K_{X}+H)^{2}\cdot H=S^{2}\cdot H=2.$

Thus (i) holds.

Let us show (ii). We have that
$\pi^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)|_{H}\sim
S|_{H}\sim-(K_{X}+H)|_{H}\sim-K_{H}$. The Riemann–Roch theorem, together with
(i), implies $h^{0}(H,-K_{H})=K_{H}^{2}+1=3$. By
$h^{0}(H,-K_{H})=3=h^{0}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1))$, the
composition $\pi_{H}\colon H\hookrightarrow X\xrightarrow{\pi}\mathbb{P}^{2}$
coincides with the morphism induced by the complete linear system $|-K_{H}|$.
In particular, $\pi_{H}$ is a generically finite morphism of degree two. Thus
(ii) holds.

Let us show (iii). By (ii), we have the induced isomorphism:

$H^{0}(H,-K_{H})\xleftarrow{\pi_{H}^{*},\simeq}H^{0}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1))$

via $\pi_{H}\colon H\to\mathbb{P}^{2}$, i.e.,
$\pi_{H}^{*}:|\mathcal{O}_{\mathbb{P}^{2}}(1)|\to|-K_{H}|$ is bijective. By
our assumption, there exists a line $L\in|\mathcal{O}_{\mathbb{P}^{2}}(1)|$
such that $C:=\pi^{*}_{H}(L)\in|-K_{H}|$ is a smooth prime divisor. This
property holds even after replacing $L$ by a general member of
$|\mathcal{O}_{\mathbb{P}^{2}}(1)|$, and hence we may assume that
$S=\pi^{*}L$. We then obtain $S\cap H=\pi^{*}L\cap H=\pi^{*}_{H}(L)=C$, which
is a smooth elliptic curve. By $K_{X}+S+H\sim 0$, $X$ is quasi-$F$-split
(Corollary 2.18). Thus (iii) holds. This completes the proof of Step 2. ∎

######  Step 3.

Assume that $\pi$ is of type $C_{1}$. Then there exists a smooth prime divisor
$C$ on $H$ satisfying $C\sim-K_{H}$.

###### Proof of Step 3.

Recall that $H$ is a general member of $|g^{*}\mathcal{O}_{Y}(1)|$. Suppose
that any member of $|-K_{H}|$ is singular. By Step 2(iii), it is enough to
derive a contradiction. Note that, for every line $L$ on $\mathbb{P}^{2}$ and
every smooth member $H\in|g^{*}\mathcal{O}_{Y}(1)|$, its intersection
$H\cap\pi^{*}L$ is not smooth (Step 2(ii)).

We now show that every smooth member of $|H|$ is isomorphic to the Langer
surface over $k$ (Definition 5.3). The Stein factorisation $H^{\prime}$ of the
composition $\pi_{H}:H\hookrightarrow X\xrightarrow{\pi}\mathbb{P}^{2}$ is the
anti-canonical model of $H$ (Step 2(ii)). Note that each fibre of
$\pi_{H}:H\hookrightarrow X\xrightarrow{\pi}\mathbb{P}^{2}$ is contained in a
fibre of $\pi:X\to\mathbb{P}^{2}$, which is a conic. In particular, any
singularity on $H^{\prime}$ is either $A_{1}$ or $A_{2}$. By $K_{H}^{2}=2$
(Step 2(i)) and [KN, Theorem 1.4], it holds that $p=2$ and $H$ is isomorphic
to the Langer surface.

Fix two general members $H_{1}$ and $H_{2}$ of $|H|$. In particular,
$\Gamma\coloneqq H_{1}\cap H_{2}$ is a smooth curve and each of $H_{1}$ and
$H_{2}$ is isomorphic to the Langer surface. Let $\sigma:Y\to X$ be the blowup
along $\Gamma=H_{1}\cap H_{2}$, and hence we get the morphism $\alpha\colon
Y\to\mathbb{P}^{1}$ induced by the pencil generated by $H_{1}^{Y}$ and
$H_{2}^{Y}$, where
$H^{Y}_{i}\coloneqq\sigma^{*}H_{i}-\operatorname{Ex}(\sigma)$, which coincides
with the proper transform of $H_{i}$ on $Y$. By construction, every general
fibre of $\alpha$ is isomorphic to the Langer surface (as otherwise we could
find a smooth member of $|H|$ which is not the Langer surface). Set
$V\coloneqq Y\times_{\mathbb{P}^{1}}\operatorname{Spec}K$ to be the generic
fibre of $\alpha:Y\to\mathbb{P}^{1}$, where
$K\coloneqq\mathrm{Frac}\,\mathbb{P}^{1}$.

For the algebraic closure $\overline{K}$ of $K$, it is enough to show, by
Lemma 5.5, that the base change $V_{\overline{K}}\coloneqq
V\times_{\operatorname{Spec}K}\operatorname{Spec}\overline{K}$ is isomorphic
to the Langer surface over $\overline{K}$. In fact, this implies
$8=\rho(V)=\rho(Y\times_{\mathbb{P}^{1}}\operatorname{Spec}K)=\rho(Y)-\rho(\mathbb{P}^{1})=1$,
which is a contradiction. Fix a general fibre $V^{\prime}$ of
$\alpha:Y\to\mathbb{P}^{1}$. Let $F_{1},\ldots,F_{n}$ be all the $(-2)$-curves
on $V_{\overline{K}}$ ($F$ is called a $(-2)$-curve if $F^{2}=-2$ and
$F\simeq\mathbb{P}^{1}$). Since $F_{1},\ldots,F_{n}$ can be defined around the
generic point of the base and $V^{\prime}$ is general, we obtain the
corresponding $(-2)$-curves $F^{\prime}_{1},\ldots,F^{\prime}_{n}$ on
$V^{\prime}$. By the invariance of intersection numbers for flat families, we
see that $F^{\prime}_{i}\cdot F^{\prime}_{j}=F_{i}\cdot F_{j}$ for every
$i,j$. Since $V^{\prime}$ is the Langer surface over $k$, there are exactly
$7$ $(-2)$-curves and they are mutually disjoint [CT18, Theorem 5.4]. Then we
see that $n\leq 7$ and $F^{\prime}_{i}\cdot F^{\prime}_{j}=0$ (i.e.,
$F^{\prime}_{i}\cap F^{\prime}_{j}=\emptyset$) for every $1\leq i<j\leq n$. As
$|-K_{V^{\prime}}|$ has no smooth member, neither does
$|-K_{V_{\overline{K}}}|$. By [KN, Theorem 1.4], we get $n=7$, i.e.,
$V_{\overline{K}}$ is isomorphic to the Langer surface over $\overline{K}$.
This completes the proof of Step 3. ∎

Step 1, Step 2, and Step 3 complete the proof of Proposition 5.6. ∎

###  5.2. Quasi-$F$-splitting (primitive case)

In Section 5.1, the quasi-$F$-splitting for smooth Fano threefolds with
$\rho=2$ has been settled for the imprimitive case. Hence the remaining cases
are as follows [FanoIV, Subsection 7.2]:

$\text{2-2, 2-6, 2-8, 2-18, 2-24, 2-32, 2-34, 2-35, 2-36}.$

In what follows, we shall settle the cases except for 2-2, 2-6, and 2-8. These
cases will be treated in Section 6.

######  Lemma 5.7.

If $X$ is a smooth Fano threefold of No. 2-32, 2-34, 2-35, or 2-36, then $X$
is $F$-split.

###### Proof.

It is well known that $X$ is toric except when it is 2-32. Assume that $X$ is
2-32, i.e., $X$ is a smooth hypersurface on
$\mathbb{P}^{2}\times\mathbb{P}^{2}$ of bidegree $(1,1)$. Note that
$f_{1}\colon
X\hookrightarrow\mathbb{P}^{2}\times\mathbb{P}^{2}\xrightarrow{\operatorname{pr}_{1}}\mathbb{P}^{2}$
is a $\mathbb{P}^{1}$-bundle [FanoIV, Subsection 7.2]. Take a general member
$D\in|f_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)|$, which is a
$\mathbb{P}^{1}$-bundle over $\mathbb{P}^{1}$. Hence $D$ is $F$-split. As in
the proof of Proposition 4.4(2), we can see $-(K_{X}+D)$ is ample by [FanoIII,
Proposition 5.9(3)] and [FanoIV, Section 7.2]. Therefore, $X$ is $F$-split
(Corollary 2.9). ∎

######  Lemma 5.8.

Let $Y$ be a smooth Fano threefold of No. 2-18. Then the following hold.

  1. (1)

$Y$ is $F$-split.

  2. (2)

Let $B$ be a smooth fibre of the contraction $g\colon Y\to\mathbb{P}^{2}$ of
type $C_{1}$. Then the blowup $X\coloneqq\operatorname{Bl}_{B}Y$ of $Y$ along
$B$ is a smooth Fano threefold of No. 3-4.

The following argument is almost identical to that of [FanoIV, Proposition
4.35].

###### Proof.

By Proposition 4.4, (2) implies (1). Let us show (2). It is enough to show
that $-K_{X}$ is ample [FanoIV, Subsection 7.2]. By construction, we have the
following commutative diagram except for $f_{1},g_{11},g_{12}$. Since
$\varphi_{1}\times\varphi_{2}\colon X\to Z_{1}\times
Z_{2}=\mathbb{P}^{1}\times\mathbb{P}^{1}$ is not a finite morphism, its Stein
factorisation $f_{1}\colon X\to S$ of $\varphi_{1}\times\varphi_{2}$ is not an
isomorphism. We get $\dim S=2$, because we have $\dim
S\leq\dim(\mathbb{P}^{1}\times\mathbb{P}^{1})=2$, and
$S\to\mathbb{P}^{1}\times\mathbb{P}^{1}\xrightarrow{\operatorname{pr}_{1}}\mathbb{P}^{1}$
is not an isomorphism. Then $f_{1}\colon X\to S$ is a contraction of a
(possibly non $K_{X}$-negative) extremal ray. Since this extramal ray is
contained in the two-dimensional extremal faces corresponding to $\varphi_{1}$
and $\varphi_{2}$, $\operatorname{NE}(X)$ is generated by three extremal rays,
which are corresponding to $f_{1},f_{2},f_{3}$.

$X$$S$$\mathbb{F}_{1}$$Y$$\mathbb{P}^{1}$$\mathbb{P}^{1}$$\mathbb{P}^{2}$$\scriptstyle
f_{1}$$\scriptstyle f_{2}$$\scriptstyle
f_{3}$$\scriptstyle\varphi_{1}$$\scriptstyle\varphi_{2}$$\scriptstyle\varphi_{3}$$\scriptstyle
g_{11}$$\scriptstyle g_{12}$$\scriptstyle g_{22}$$\scriptstyle
g_{23}=\tau$$\scriptstyle g_{33}$$\scriptstyle g_{31}$

By the same argument as in [FanoIV, Proposition 4.35], we obtain

$-K_{X}\sim H_{1}+H_{2}+H_{3},$

where each $H_{i}$ is the pullback of the ample generator by $\varphi_{i}$.
Since $\varphi_{1}\times\varphi_{2}\times\varphi_{3}\colon
X\to\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{2}$ is a finite
morphism, $-K_{X}\sim H_{1}+H_{2}+H_{3}$ is ample, as required. ∎

######  Lemma 5.9.

Let $Y$ be a smooth Fano threefold of No. 2-24. Assume that the contraction
$\psi\colon Y\to\mathbb{P}^{2}$ of type $C_{1}$ is generically smooth. Then
the following hold.

  1. (1)

$Y$ is $F$-split.

  2. (2)

Let $B$ be a smooth fibre of $\psi\colon Y\to\mathbb{P}^{2}$. Then the blowup
$X:=\operatorname{Bl}_{B}Y$ of $Y$ along $B$ is a Fano threefold of No. 3-4.

The following argument is almost identical to that of [FanoIV, Proposition
4.40].

###### Proof.

Since (2) implies (1), it is enough to show (2). Let us show (2). It is enough
to show that $-K_{X}$ is ample [FanoIV, Subsection 7.2]. By construction, we
get the following commutative diagram except for $f_{3},g_{31},g_{33}$. By the
same argument as in [FanoIV, Proposition 4.40], we see that

  * •

$\varphi_{3}\times\varphi_{1}:X\to\mathbb{P}^{2}\times\mathbb{P}^{1}$ is not a
finite morphism, and

  * •

$-K_{X}\sim H_{1}+H_{2}+H_{3}$,

where each $H_{i}$ is the pullback of the ample generator by $\varphi_{i}$.
Then $X$ has exactly three extremal rays and $-K_{X}$ is ample.

$X$$\mathbb{F}_{1}$$Y$$\mathbb{P}^{2}\times\mathbb{P}^{1}$$\mathbb{P}^{1}$$\mathbb{P}^{2}$$\mathbb{P}^{2}$$\scriptstyle
f_{1}$$\scriptstyle f_{2}$$\scriptstyle
f_{3}$$\scriptstyle\varphi_{1}$$\scriptstyle\varphi_{2}$$\scriptstyle\varphi_{3}$$\scriptstyle
g_{11}$$\scriptstyle g_{12}=\tau$$\scriptstyle g_{22}$$\scriptstyle
g_{23}$$\scriptstyle g_{33}$$\scriptstyle g_{31}$

∎

######  Lemma 5.10.

Let $Y$ be a smooth Fano threefold of No. 2-24. Assume that the contraction
$\psi\colon Y\to\mathbb{P}^{2}$ of type $C_{1}$ is not generically smooth.
Then the following hold.

  1. (1)

$X$ is isomorphic to
$\\{x_{0}y_{0}^{2}+x_{1}y_{1}^{2}+x_{2}y_{2}^{2}=0\\}\subset\mathbb{P}^{2}_{x}\times\mathbb{P}^{2}_{y}$,
where
$\mathbb{P}^{2}_{x}\coloneqq\mathrm{Proj}\,k[x_{0},x_{1},x_{2}](\simeq\mathbb{P}^{2})$
and
$\mathbb{P}^{2}_{y}\coloneqq\mathrm{Proj}\,k[y_{0},y_{1},y_{2}](\simeq\mathbb{P}^{2})$.

  2. (2)

$X$ is quasi-$F$-split.

###### Proof.

Since (1) implies (2) [KTY, Example 7.13], it is enough to show (1). Recall
that $\psi$ can be written as follows:

$\psi:X\hookrightarrow\mathbb{P}^{2}_{x}\times\mathbb{P}^{2}_{y}\xrightarrow{\operatorname{pr}_{1}}\mathbb{P}^{2}_{x}.$

Since every fibre of $\psi$ is a non-reduced conic, we can write

$X=\\{f_{0}(x)y_{0}^{2}+f_{1}(x)y_{1}^{2}+f_{2}(x)y_{2}^{2}=0\\},$

where each $f_{i}(x)\in k[x_{0},x_{1},x_{2}]$ is a homogeneous polynomial of
degree $1$. We see that

  1. $(*)$

none of $f_{0}(x),f_{1}(x),f_{2}(x)$ is zero.

Indeed, if $f_{2}(x)=0$, then the affine open subset of $X$ defined by
$\\{x_{0}\neq 0\\}\cap\\{y_{2}\neq 0\\}$ contains a singular point. After
applying a suitable coordinate change, we may assume that $f_{0}(x)=x_{0}$.
Therefore, we can write

$X=\\{x_{0}y_{0}^{2}+f_{1}(x)y_{1}^{2}+f_{2}(x)y_{2}^{2}=0\\}.$

We can write $f_{1}(x)=ax_{0}+bx_{1}+cx_{2}$. By applying $y_{0}\mapsto
y_{0}+\sqrt{a}y_{1}$, we may assume that $a=0$. Note that $(*)$ implies
$f_{1}(x)\neq 0$. Replacing $f_{1}(x)=bx_{1}+cx_{2}(\neq 0)$ by $x_{1}$, we
may assume that $f_{1}(x)=x_{1}$, i.e.,

$X=\\{x_{0}y_{0}^{2}+x_{1}y_{1}^{2}+f_{2}(x)y_{2}^{2}=0\\}.$

Similarly, by applying a coordinate change $y_{0}\mapsto
y_{0}+dy_{2},y_{1}\mapsto y_{1}+ed_{3}$ for some $d,e\in k$, the problem is
reduced to the case when $f_{2}(x)=\alpha x_{2}$ for some $\alpha\in k$. By
$(*)$, we may assume that $\alpha=1$. Thus (1) holds. ∎

######  Remark 5.11.

Let $X$ be a smooth Fano threefold of No. 2-24. Combining Lemma 5.9, Lemma
5.10, and [KTY, Example 7.13], $X$ is $2$-quasi-$F$-split. Moreover, the
following hold.

  1. (1)

The following are equivalent.

    1. (a)

The quasi-$F$-split height is $1$, i.e., $X$ is $F$-split.

    2. (b)

The contraction $\psi:X\to\mathbb{P}^{2}$ of type $C_{1}$ is generically
smooth.

  2. (2)

The following are equivalent.

    1. (a)

The quasi-$F$-split height is $2$.

    2. (b)

The contraction $\psi:X\to\mathbb{P}^{2}$ of type $C_{1}$ is not generically
smooth (called wild).

    3. (c)

$X\simeq\\{x_{0}y_{0}^{2}+x_{1}y_{1}^{2}+x_{2}y_{2}^{2}=0\\}\subset\mathbb{P}^{2}_{x}\times\mathbb{P}^{2}_{y}$.

######  Proposition 5.12.

Let $X$ be a smooth Fano threefold with $\rho(X)=2$. Then $X$ is
quasi-$F$-split if $X$ is none of 2-2, 2-6, and 2-8.

###### Proof.

If $X$ is imprimitive (resp. primitive), then the assertion follows from
Proposition 5.1, Proposition 5.2, and Proposition 5.6 (resp. Section 5.2). ∎

###  5.3. $F$-splitting

######  Theorem 5.13.

Assume $p>5$. Let $X$ be a smooth Fano threefold with $\rho(X)\geq 2$. If $X$
is neither 2-2 nor 2-6, then $X$ is $F$-split.

###### Proof.

By Proposition 3.1, Proposition 3.2, Proposition 3.3, and Proposition 4.4, we
may assume that $\rho(X)=2$. The types $R_{1}+R_{2}$ of the extremal rays
$R_{1}$ and $R_{2}$ are as follows, because the case $D+D$ does not occur
[FanoIV, Subsection 7.2]:

  1. (I)

$C+E$ or $D+E$.

  2. (II)

$E+E$.

  3. (III)

$C+D$.

  4. (IV)

$C+C$.

For each $i\in\\{1,2\\}$, let $f_{i}:X\to Y_{i}$ be the contraction of
$R_{i}$, $\mu_{i}$ denotes the length of $R_{i}$, and set $H_{i}$ to be the
pullback of the ample generator of
$\operatorname{Pic}\,Y_{i}(\simeq\mathbb{Z})$. Recall that we can write
$-K_{X}\sim\mu_{2}H_{1}+\mu_{1}H_{2}$ [FanoIII, Proposition 5.9]. In what
follows, we treat the above four cases separately.

(I) Since $R_{1}+R_{2}$ is $C+E$ or $D+E$, we have $Y_{1}=\mathbb{P}^{1}$ or
$Y_{1}=\mathbb{P}^{2}$. Pick a general member $S$ of $|H_{1}|$.

######  Claim.

$S$ is a canonical weak del Pezzo surface.

###### Proof of Claim.

We first treat the case when $Y_{1}=\mathbb{P}^{1}$. Let $S^{\prime}$ be the