Vanishing theorems for Fano threefolds in positive characteristic

Tatsuro Kawakami Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan [email protected] and Hiromu Tanaka Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, JAPAN [email protected]

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

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)

$X$ 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)

$H^{i}(X,\mathcal{O}_{X}(-A))=0$ 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)

$H^{j}(X,\Omega^{i}_{X}\otimes\mathcal{O}_{X}(-A))=0$ for every ample Cartier divisor $A$ and every pair $(i,j)$ of integers $i$ and $j$ satisfying $i+j<3$ .
(2)

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)

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)

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)

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

(A)

Inversion of adjunction for $F$ -splitting.
(B)

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{26.18709pt}{-6.46803pt}\pgfsys@lineto{58.58502pt}{-6.46803pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{58.785pt}{-6.46803pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{38.27075pt}{-2.75418pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{{\rm pr}_{1}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\kappa:=K((\mathbb{P}^{n}_{k})^{*})

where

(1)

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

(3)

$\kappa/k$ is a purely transcendental extension of finite transcendence degree.
(4)

$X\times_{k}\kappa$ is a regular projective variety.
(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 $(X^{\operatorname{gen}}_{\Lambda_{2}})_{\kappa}:=X^{\operatorname{gen}}_{\Lambda_{2}}\times_{\kappa_{2}}\kappa$ ,

$(\star)$

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)

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)

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)

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.

( $\star$ )

$(p^{e}-1)\Delta$ 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)

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

$\kappa$ is an $F$ -finite field,
(2)

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

$D_{1}$ is a normal prime divisor,
(4)

$D_{1}\cap D_{2}$ is one-dimensional, smooth, and
(5)

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

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

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

$H^{1}(X,\Omega_{X}^{1}(K_{X}))=0$ .
(3)

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

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

(a)

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

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

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

(i)

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

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

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

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

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

4-4, 4-10, 4-12.
(2)

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

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

4-11.
(5)

4-2.
(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)

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)

$X$ 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

(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
(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}).

-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)

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

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

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

$H^{1}(H^{\prime}_{i},\mathcal{O}_{H^{\prime}_{i}})=0$ and $H^{1}(H^{\prime}_{i},K_{H^{\prime}_{i}})=0$ for every $i\in\{1,2,3\}$ .
(6)

$H^{0}(H^{\prime}_{1}\cap H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{1}\cap H^{\prime}_{3}})=k^{\prime}$ and $H^{0}(H^{\prime}_{2}\cap H^{\prime}_{3},\mathcal{O}_{H^{\prime}_{2}\cap H^{\prime}_{3}})=k^{\prime}$ .
(7)

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

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)

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

3-27, 3-28, 3-31.
(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)

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

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

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

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

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)

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:

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)

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

(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
(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)

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

(i)

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

$Y$ is $F$ -split.
(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 $f_{1},f_{2},f_{3}$ .

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)

$Y$ is $F$ -split.
(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.

∎

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)

$X$ 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)

$X$ 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

$(*)$

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)
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)
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]:

(I)

$C+E$ or $D+E$ .
(II)

$E+E$ .
(III)

$C+D$ .
(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 geometric generic fibre of $f_{1}\colon X\to Y_{1}=\mathbb{P}^{1}$ . Then $S^{\prime}$ is normal and $-K_{S^{\prime}}$ is ample [FS20, Theorem 15.2], which implies that $S^{\prime}$ is canonical [BT22, Theorem 3.3]. Hence $S$ is a canonical del Pezzo surface for the case when $Y_{1}=\mathbb{P}^{1}$ .

It is enough to settle the case when $Y=\mathbb{P}^{2}$ . In this case, $S$ is the inverse image of a general line $L$ on $\mathbb{P}^{2}$ . Since the discriminant scheme $\Delta_{f_{1}}$ of $f_{1}\colon X\to S=\mathbb{P}^{2}$ is a reduced divisor [Tan-conic, Proposition 7.2], the scheme theoretic intersection $L\cap\Delta_{f_{1}}$ is a zero-dimensional smooth scheme. For the resulting conic bundle $g\colon S=X\times_{Y_{1}}L\to L$ , we have $\Delta_{g}=L\cap\Delta_{f_{1}}$ [Tan-conic, Remark 3.4]. Since $L$ and $\Delta_{g}$ are smooth, also $S$ is smooth [Tan-conic, Theorem 4.4]. We have that $-K_{X}\sim S+H$ for $H\coloneqq\mu_{1}H_{2}+(\mu_{2}-1)H_{1}$ . In particular, $H$ and $H|_{S}$ are nef and big. By the adjunction formula: $K_{S}\sim(K_{X}+S)|_{S}\sim-H|_{S}$ , we have $S$ is a smooth weak del Pezzo surface. This completes the proof of Claim. ∎

By Claim, $S$ is $F$ -split (Theorem 2.22). We have $-K_{X}\sim S+H$ for $H\coloneqq\mu_{1}H_{2}+(\mu_{2}-1)H_{1}$ . By $H^{1}(X,-S+(p^{e}-1)(K_{X}+S))=H^{1}(X,K_{X}+p^{e}H)$ and Proposition 2.8, it is enough to find $e\in\mathbb{Z}_{>0}$ such that

H^{1}(X,K_{X}+p^{e}H)=0

by. Fix $e_{1}\in\mathbb{Z}_{>0}$ such that $p^{e_{1}}H-D$ is ample for $D:=\operatorname{Ex}(f_{2})$ . By the Serre vanishing theorem, there is $e_{2}\in\mathbb{Z}_{>0}$ such that $H^{1}(X,K_{X}+p^{e_{2}}(p^{e_{1}}H-D))=0$ . Set $e:=e_{1}+e_{2}$ . It suffices to prove

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

for every $0\leq s\leq p^{e_{2}}$ by descending induction on $s$ . The base case $s=p^{e_{2}}$ has been checked already. Fix an integer $s$ satisfying $0\leq s<p^{e_{2}}$ . By the induction hypothesis, we have the following exact sequence

0=H^{1}(X,K_{X}+p^{e}H-(s+1)D)\to H^{1}(X,K_{X}+p^{e}H-sD)\\ \to H^{1}(D,K_{D}+(p^{e}H-(s+1)D)|_{D}).

It suffices to show $H^{1}(D,K_{D}+(p^{e}H-(s+1)D)|_{D})=0$ . Note that $p^{e}H-(s+1)D$ is ample, because so are $H$ and $p^{e}H-p^{e_{2}}D(=p^{e_{2}}(p^{e_{1}}H-D)$ ). Then we get $H^{1}(D,K_{D}+(p^{e}H-(s+1)D)|_{D})=0$ by the fact that $D$ is toric or a smooth ruled surface [Muk13, Theorem 3].

(II) Assume that $R_{1}+R_{2}$ is $E+E$ . The list of such Fano threefolds is as follows [FanoIV, Subsection 7.2]: 2-12, 2-15 2-17, 2-19, 2-21, 2-22, 2-23, 2-26, 2-28, 2-30. We treat the following two cases separately:

(1)

2-12, 2-15, 2-17, 2-19, 2-22, 2-28, 2-30.
(2)

2-21, 2-23, 2-26.

If (1) (resp. (2)) holds, then there is a blowup $f_{1}:X\to Y_{1}=\mathbb{P}^{3}$ (resp. $f_{1}:X\to Y_{1}=Q$ ) along a smooth curve $B$ on $Y_{1}$ . Note that $H_{1}$ is the inverse image of the corresponding member $\overline{H}_{1}$ on $Y_{1}$ . After replacing $H_{1}$ by a general member of $|H_{1}|$ , we may assume that $H_{1}$ is the blowup along a smooth zero-dimensional scheme $B\cap\overline{H}_{1}$ of a smooth surface $\overline{H}_{1}$ , and hence $H_{1}$ is a smooth projective surface. We have that $-K_{X}\sim H_{1}+H^{\prime}_{2}$ for some $H^{\prime}_{2}\sim H_{2}+N$ with $N$ nef. Then it holds that $-K_{H_{1}}$ is nef and big. Hence $H_{1}$ is a smooth weak del Pezzo surface. The same argument as in (I) deduces that $X$ is $F$ -split.

(III) Assume that the types of the extremal rays are $C+D$ . Since we are assuming that $X$ is not 2-2, $X$ is 2-18 or 2-34 [FanoIV, Subsection 7.2]. If $X$ is 2-34, i.e., $X\simeq\mathbb{P}^{1}\times\mathbb{P}^{2}$ ), then $X$ is clearly $F$ -split. The case when $X$ is 2-18 has been settled in Lemma 5.8.

(IV) Assume that $R_{1}+R_{2}$ is $C+C$ . Since we are assuming that $X$ is not 2-6, $X$ is 2-24 or 2-32. If $X$ is 2-24 (resp. 2-32), then $X$ is $F$ -split by Lemma 5.9 (resp. Lemma 5.7). Note that an arbitrary conic bundle $X\to S$ is generically smooth by $p>2$ . ∎

6. $F$ -splitting and Quasi- $F$ -splitting via Cartier operators

6.1. Quasi- $F$ -splitting for 2-2, 2-6, 2-8, and 3-10

6.1.1. Preparation

Lemma 6.1.

Let $X$ be a smooth Fano threefold. Assume that $X$ is SRC. Then $H^{0}(X,\Omega_{X}^{i}(-D))=0$ for every $i>0$ and every pseudo-effective Cartier divisor $D$ on $X$ .

For the definition of SRC (separable rational connectedness), we refer to [Kol96].

Proof.

The assertion follows from the essentially same proof as in [Kaw1, Proposition 3.4] by using the fact that the restriction of a pseudo-effective Cartier divisor $D$ to a general curve is pseudo-effective. ∎

We repeatedly use the following basic lemma.

Lemma 6.2.

Let $X$ be a smooth Fano threefold. Assume that there exists a conic bundle $f:X\to S$ , i.e., $f$ is a flat morphism to a smooth projective surface $S$ such that every fibre $f^{-1}(s)$ is isomorphic to a conic (cf. [Tan-conic, Definition 2.3]). Then the following hold.

(1)

$X$ is SRC.
(2)

$H^{0}(X,\Omega_{X}^{i}(-D))=0$ for every $i>0$ and every pseudo-effective Cartier divisor $D$ on $X$ .

In particular, if $X$ is one of No. 2-2, 2-6, 2-8, and 3-10, then $X$ satisfies the condition (1) in Proposition 2.20.

Proof.

Since (1) implies (2) (Lemma 6.1), it is enough to show (1). If $f$ is generically smooth, then $S$ is a smooth rational surface [FanoIV, Proposition 3.13], and hence $X$ is SRC by [GLP15, Theorem 0.5]. We may assume that $f$ is wild, i.e., not generically smooth. Then $X$ is 2-24 or 3-10 by [MS03, Corollary 8]. Assume that $X$ is 2-24. Then the contraction of the other extremal ray is of type $C$ and its contraction gives a generically smooth conic bundle structure. We are done by the generically smooth case. Assume that $X$ is 3-10. Then $X$ is obtained as a blowup of $Q$ (cf. [MS03, Corollary 8] or [FanoIV, Section 7]). Since $Q$ is rational (and hence SRC), so is $X$ . Thus (1) holds. ∎

6.3Double covers.

Given smooth projective varieties $X$ and $Y$ , we say that $f\colon X\to Y$ is a double cover if $f$ is a finite surjective morphism such that the induced field extension $K(X)/K(Y)$ is of degree $2$ . Recall that we have an exact sequence

0\to\mathcal{O}_{Y}\to f_{*}\mathcal{O}_{X}\to\mathcal{O}_{Y}(-L)\to 0

for some Cartier divisor $L$ on $Y$ [Kaw2, Lemma A.1]. In particular, $\mathcal{O}_{Y}(L)\simeq(f_{*}\mathcal{O}_{X}/\mathcal{O}_{Y})^{-1}$ . Moreover, $K_{X}\sim f^{*}(K_{Y}+L)$ [CD89, Proposition 0.1.3]. We say a double cover $f\colon X\to Y$ is split if the induced homomorphism $\mathcal{O}_{Y}\to f_{*}\mathcal{O}_{X}$ splits as an $\mathcal{O}_{Y}$ -module homomorphism (i.e., the above exact sequence splits). In this case, we obtain $f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)$ .

Lemma 6.4.

Let $f:X\to Y$ be a split double cover of smooth projective varieties. Let $L$ be a Cartier divisor $L$ on $Y$ satisfying $\mathcal{O}_{Y}(-L)\simeq f_{*}\mathcal{O}_{X}/\mathcal{O}_{Y}$ (cf. (6.3)). Then there exists a closed immersion $j\colon X\hookrightarrow P:=\mathbb{P}_{Y}(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L))$ which satisfies the following properties.

(1)

$K_{P}+X\sim g^{*}(K_{Y}+L)$ , where $g:P=\mathbb{P}_{Y}(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L))\to Y$ denotes the induced $\mathbb{P}^{1}$ -bundle.
(2)

For the section $S:=\mathbb{P}_{Y}(\mathcal{O}_{Y}(-L))$ of $g\colon P=\mathbb{P}_{Y}(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L))\to Y$ corresponding to the second projection $\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)\to\mathcal{O}_{Y}(-L)$ , it holds that $S\cap X=\emptyset$ , $S|_{S}\sim-g^{*}L|_{S}$ , $X-2S\sim 2g^{*}L$ , and $\mathcal{O}_{P}(1)\simeq\mathcal{O}_{P}(S)$ ,
(3)

For the section $T\coloneqq\mathbb{P}_{Y}(\mathcal{O}_{Y})$ of $g:P=\mathbb{P}_{Y}(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L))\to Y$ corresponding to the first projection $\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)\to\mathcal{O}_{Y}$ , it holds that $S\cap T=\emptyset$ , $T|_{T}\sim g^{*}L|_{T}$ , $X\sim 2T$ , $T-S\sim g^{*}L$ , and $\mathcal{O}_{P}(1)\simeq\mathcal{O}_{P}(T-g^{*}L)$ .
(4)

$\Omega_{P/Y}^{1}\simeq\mathcal{O}_{P}(-g^{*}L-2S)$ .

Proof.

In what follows, we only treat the case when $p=2$ , as otherwise the problem is easier. By [CD89, the proof of Proposition 0.1.3] (note that $f$ splits if and only if $f$ corresponds to a splittable admissible triple), there is a closed immersion $j^{\circ}:X\hookrightarrow P^{\circ}$ to the $\mathbb{A}^{1}$ -bundle

P^{\circ}:=\operatorname{Spec}_{Y}(\bigoplus_{d=0}^{\infty}\mathcal{O}_{Y}(-dL)).

Since $P^{\circ}$ is an open subscheme of $P$ , we obtain a closed immersion $j:X\to P$ over $Y$ . By definition, we get $S\cap T=\emptyset$ , $\mathcal{O}_{P}(1)|_{S}\simeq\mathcal{O}_{S}(-L)$ , and $\mathcal{O}_{P}(1)|_{T}\simeq\mathcal{O}_{T}$ , where $(g|_{S})^{*}L$ denotes $L$ for $g|_{S}:S\xrightarrow{\simeq}Y$ by abuse of notation. We can write $\mathcal{O}_{P}(1)\simeq\mathcal{O}_{P}(S+g^{*}D_{S})\simeq\mathcal{O}_{P}(T+g^{*}D_{T})$ for some Cartier divisors $D_{S}$ and $D_{T}$ on $Y$ . By $S\cap T=\emptyset$ and $\mathcal{O}_{P}(1)|_{T}\simeq\mathcal{O}_{T}$ , we obtain $\mathcal{O}_{T}\simeq\mathcal{O}_{P}(1)|_{T}\simeq\mathcal{O}_{P}(S+g^{*}D_{S})|_{T}\simeq(g^{*}\mathcal{O}_{Y}(D_{S}))|_{T}$ , which implies $D_{S}\sim 0$ . Similarly, we obtain $D_{T}\sim-L$ by $\mathcal{O}_{S}(-L)\simeq\mathcal{O}_{P}(1)|_{S}\simeq\mathcal{O}_{P}(T+g^{*}D_{T})|_{S}\simeq(g^{*}\mathcal{O}_{Y}(D_{T}))|_{S}$ . Hence we get

\mathcal{O}_{P}(1)\sim S\sim T-g^{*}L,

which implies

S|_{S}\simeq-g^{*}L|_{S},\qquad T|_{T}\sim g^{*}L|_{T}.

Claim.

The following hold.

(a)

$\mathcal{O}_{P}(K_{P})\otimes\mathcal{O}_{P}(2)\otimes g^{*}\mathcal{O}_{Y}(2L)\simeq g^{*}\mathcal{O}_{Y}(K_{Y}+L)$ .
(b)

$X\cap S=\emptyset$ .
(c)

$K_{P}+X\sim g^{*}(K_{Y}+L)$ .

Proof of Claim.

Let us show (c)’ below, which is weaker than (c):

(c)’

$K_{P}+X\equiv g^{*}(K_{Y}+L)$ , where $\equiv$ denotes the numerical equivalence.

Since $f:X\to Y$ is a double cover, we can find a Cartier divisor $E$ on $Y$ such that $K_{P}+X\sim g^{*}E$ . Then

f^{*}E=(g^{*}E)|_{X}\sim(K_{P}+X)|_{X}\sim K_{X}\sim f^{*}(K_{Y}+L),

which implies $E\equiv K_{Y}+L$ [Kle66, Corollary 1(ii) in page 304]. Thus (c)’ holds.

The assertion (a) holds by the following (cf. [FanoIII, Proposition 7.1(2)]):

\mathcal{O}_{P}(K_{P})\simeq\mathcal{O}_{P}(-2)\otimes g^{*}(\omega_{Y}\otimes\det(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)))\simeq\mathcal{O}_{P}(-2)\otimes g^{*}\mathcal{O}_{Y}(K_{Y}-L).

Let us show (b). By (a) and (c)’, we obtain $X\equiv\mathcal{O}_{P}(2)+2g^{*}L\sim 2(S+g^{*}L)$ . This, together with $S|_{S}\sim-g^{*}L|_{S}$ , implies $X|_{S}\equiv 0$ . By $X\neq S$ , we obtain $X\cap S=\emptyset$ , as otherwise we could find a curve $C$ on $S$ which properly intersects $X$ . Thus (b) holds. Then it holds that

g^{*}E|_{S}\sim(K_{P}+X)|_{S}\overset{{\rm(b)}}{\sim}K_{P}|_{S}\sim K_{S}-S|_{S}\sim g^{*}(K_{Y}+L)|_{S},

which implies $E\sim K_{Y}+L$ , i.e., (c) holds. This completes the proof of Claim. ∎

We can write $X-2S\sim g^{*}F$ for some Cartier divisor $F$ on $Y$ . By $-2g^{*}L|_{S}\sim(X-2S)|_{L}\sim g^{*}F|_{S}$ , we obtain $F\sim 2L$ . Then $X\sim 2S+2g^{*}L\sim 2T$ . This completes the proofs of (2) and (3).

Let us show (4). We have an exact sequence

0\to g^{*}\Omega_{Y}^{1}\xrightarrow{\alpha}\Omega^{1}_{P}\to\Omega_{P/Y}^{1}\to 0,

where the injectivity of $\alpha$ can be checked by taking the corresponding sequence of the stalks at the generic point. Taking the wedge products, we get $\omega_{P}\simeq g^{*}\omega_{Y}\otimes\Omega_{P/Y}^{1}$ . By $K_{P}+X\sim g^{*}(K_{Y}+L)$ and $X\sim 2S+2g^{*}L$ , we obtain

\Omega_{P/Y}^{1}\simeq\mathcal{O}_{P}(K_{P}-g^{*}K_{Y})\simeq\mathcal{O}_{P}(-X+g^{*}L)\simeq\mathcal{O}_{P}(-g^{*}L-2S).

Thus (4) holds. ∎

Lemma 6.5.

We use the same notation as Lemma 6.4. Fix $q\in\mathbb{Z}$ and take a Cartier divisor $D$ on $Y$ . Assume that

(1)

$H^{q-1}(Y,D)=H^{q-1}(Y,D-L)=0$ , and
(2)

$H^{q}(Y,\Omega_{Y}^{1}(D))=H^{q}(Y,\Omega_{Y}^{1}(D-L))=0$ .

Then $H^{q}(P,\Omega_{P}^{1}(g^{*}D))=0$ .

Proof.

We have an exact sequence

0\to g^{*}(\Omega_{Y}^{1}(D))\to\Omega_{P}^{1}(g^{*}D)\to\Omega^{1}_{P/Y}(g^{*}D)\to 0.

By (2), it is enough to show $H^{q}(P,\Omega^{1}_{P/Y}(g^{*}D))=0$ , i.e., $H^{q}(P,g^{*}D-g^{*}L-2S)=0$ . Using an exact sequence $0\to\mathcal{O}_{P}(-S)\to\mathcal{O}_{P}\to\mathcal{O}_{S}\to 0$ twice, it suffices to prove that $H^{q}(P,g^{*}D-g^{*}L))=0$ and $H^{q-1}(S,g^{*}D-g^{*}L-nS)=0$ with $n\in\{0,1\}$ . By $-S|_{S}\sim g^{*}L|_{S}$ , these equalities follow from (2) and (1), respectively. ∎

Proposition 6.6.

We use the same notation of Lemma 6.4. Assume that $\dim X=\dim Y=3$ , and both $L$ and $H\coloneqq-K_{Y}-L$ are ample. Moreover, suppose that the following hold.

(0)
1. (0a)
  
  $H^{j}(Y,-A))=0$ for every $j<3$ and every ample Cartier divisor $A$ on $Y$ .
2. (0b)
  
  $H^{j}(Y,\mathcal{O}_{Y}(p^{i}H))=H^{j}(Y,\mathcal{O}_{Y}(p^{i}H-L))=H^{j}(Y,\mathcal{O}_{Y}(p^{i}H-2L))=0$ for every $j\in\{1,2\}$ .
3. (0c)
  
  $H^{3}(Y,\mathcal{O}_{Y}(p^{i}H-2L))=H^{3}(Y,\mathcal{O}_{Y}(p^{i}H-3L))=0$ for every $i>0$ .
(1)
1. (1a)
  
  $H^{1}(Y,\Omega_{Y}^{1}(-H-nL))=0$ for every $n\geq 0$ .
2. (1b)
  
  $H^{2}(Y,\Omega_{Y}^{1}(-H-2L))=H^{2}(Y,\Omega_{Y}^{1}(-H-3L))=0$ .
3. (1c)
  
  $H^{j}(Y,\Omega^{1}_{Y}(p^{i}H))=H^{j}(Y,\Omega^{1}_{Y}(p^{i}H-L))=0$ for every $j\in\{2,3\}$ and every $i>0$ .
(2)

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

Then the following hold.

(I)

$H^{1}(X,\Omega^{1}_{X}(K_{X}))=0$ .
(II)

$H^{2}(X,\Omega^{1}_{X}(-p^{i}K_{X}))=0$ for every $i>0$ .
(III)

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

Proof.

By (2) and Proposition 2.20, (I) and (II) imply (III). In what follows, we shall prove (I) and (II). Note that we can write $-K_{X}\sim f^{*}H$ for $H:=-K_{Y}-L$ , and hence $-K_{X}$ is ample.

Step 1: Proof of (I). By the conormal exact sequence, we have an exact sequence

0\to\mathcal{O}_{X}(K_{X}-X)\to\Omega^{1}_{P}|_{X}(K_{X})\to\Omega^{1}_{X}(K_{X})\to 0,

where $\Omega^{1}_{P}|_{X}(K_{X}):=(\Omega^{1}_{P}|_{X})\otimes\mathcal{O}_{X}(K_{X})$ . It follows from $K_{X}=-g^{*}H|_{X}=-f^{*}H$ , $X|_{X}=(X-2S)|_{X}=2g^{*}L|_{X}=2f^{*}L$ , and $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)$ that

	$\displaystyle H^{2}(X,\mathcal{O}_{X}(K_{X}-X))$	$\displaystyle=H^{2}(X,f^{*}\mathcal{O}_{Y}(-(H+2L)))$
		$\displaystyle=H^{2}(Y,\mathcal{O}_{Y}(-H-2L))\oplus H^{2}(Y,\mathcal{O}_{Y}(-H-3L))$
		$\displaystyle\overset{{\rm(0a)}}{=}0.$

Thus it suffices to show $H^{1}(X,\Omega^{1}_{P}|_{X}(K_{X}))=0$ .

By $K_{X}=-g^{*}H|_{X}$ , $S|_{X}=0$ , and $-X=-2S-2g^{*}L$ , we have the following exact sequence:

0\to\Omega^{1}_{P}(-g^{*}H-2g^{*}L)\to\Omega^{1}_{P}(-g^{*}H+2S)\to\Omega^{1}_{P}|_{X}(K_{X})\to 0.

Thus, in order to prove (I), it is enough to show

(i)

$H^{1}(P,\Omega^{1}_{P}(-g^{*}H+2S))=0$ and
(ii)

$H^{2}(P,\Omega^{1}_{P}(-g^{*}H-2g^{*}L))=0$ .

Step 1-1: Proof of (i). We have the following exact sequence:

0\to g^{*}(\Omega^{1}_{Y}(-H))(2S)\to\Omega^{1}_{P}(-g^{*}H+2S)\to\Omega^{1}_{P/Y}(-g^{*}H+2S)\to 0.

It holds that

H^{1}(P,\Omega^{1}_{P/Y}(-g^{*}H+2S))\simeq H^{1}(P,\mathcal{O}_{P}(-g^{*}H-g^{*}L))\simeq H^{1}(Y,\mathcal{O}_{Y}(-H-L))\overset{{\rm(0a)}}{=}0.

Then it suffices to show $H^{1}(P,g^{*}(\Omega^{1}_{Y}(-H))(2S))=0$ . Using an exact sequence $0\to\mathcal{O}_{P}(-S)\to\mathcal{O}_{P}\to\mathcal{O}_{S}\to 0$ twice, the vanishing $H^{1}(P,g^{*}(\Omega_{Y}(-H))(2S))=0$ can be reduced, by $S|_{S}=-g^{*}L|_{S}$ , to those of

•

$H^{1}(S,g^{*}(\Omega^{1}_{Y}(-H))(nS))\overset{g|_{S}:\text{isom}}{\simeq}H^{1}(Y,\Omega^{1}_{Y}(-H-nL))$ for $n\in\{1,2\}$ and
•

$H^{1}(P,g^{*}\Omega_{Y}^{1}(-H))\simeq H^{1}(Y,\Omega_{Y}^{1}(-H))$ .

Both of them follow from (1a). Thus (i) holds.

Step 1-2: Proof of (ii). In order to show (ii), it is enough to verify the assumptions of Lemma 6.5 for the case when $q=2$ and $D=-H-2L$ . The conditions Lemma 6.5(1) and Lemma 6.5(2) hold by (0a) and (1b), respectively. This completes the proofs of (ii) and (I).

Step 2: Proof of (II). Fix $i\in\mathbb{Z}_{>0}$ . By the conormal exact sequence, we have the following exact sequence:

0\to\mathcal{O}_{X}(-p^{i}K_{X}-X)\to\Omega^{1}_{P}|_{X}(-p^{i}K_{X})\to\Omega^{1}_{X}(-p^{i}K_{X})\to 0.

It follows from $K_{X}=-f^{*}H$ , $X|_{X}=2f^{*}L$ , and $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)$ that

	$\displaystyle H^{3}(X,-p^{i}K_{X}-X)$	$\displaystyle=H^{3}(X,p^{i}f^{}H-2f^{}L)$
		$\displaystyle=H^{3}(Y,\mathcal{O}_{Y}(p^{i}H-2L))\oplus H^{3}(Y,\mathcal{O}_{Y}(p^{i}H-3L))$
		$\displaystyle\overset{{\rm(0c)}}{=}0.$

Thus it suffices to show $H^{2}(X,\Omega^{1}_{P}|_{X}(-p^{i}K_{X}))=0$ .

We get

H^{q}(P,\Omega^{1}_{P}(p^{i}g^{*}H))=0\qquad\text{for}\qquad\text{every}\qquad q\in\{2,3\},

because Lemma 6.5, for the case when $q\in\{2,3\}$ and $D=p^{i}H$ , is applicable by (0b) and (1c). Since $K_{X}=-g^{*}H|_{X}$ and $-X=-2T$ , we have the following exact sequence:

0\to\Omega^{1}_{P}(p^{i}g^{*}H-2T)\to\Omega^{1}_{P}(p^{i}g^{*}H)\to\Omega^{1}_{P}|_{X}(-p^{i}K_{X})\to 0.

Thus it suffices to show that

H^{3}(P,\Omega^{1}_{P}(p^{i}g^{*}H-2T))=0.

By $T|_{T}=g^{*}L|_{T}$ and an exact sequence $0\to\mathcal{O}_{P}(-T)\to\mathcal{O}_{P}\to\mathcal{O}_{T}\to 0$ , the problem is reduced to

•

$H^{2}(T,\Omega^{1}_{P}|_{T}(p^{i}g^{*}H-nT))=0$ for $n\in\{0,1\}$ and
•

$H^{3}(P,\Omega^{1}_{P}(p^{i}g^{*}H))=0$ .

The second vanishing has been settled already. Thus it suffices to show the first one. Fix $n\in\{0,1\}$ . By the conormal exact sequence, we have the following exact sequence

0\to\mathcal{O}_{T}(p^{i}g^{*}H-(n+1)T)\to\Omega^{1}_{P}|_{T}(p^{i}g^{*}H-nT)\to\Omega^{1}_{T}(p^{i}g^{*}H-nT)\to 0.

It follows from $\mathcal{O}_{T}(p^{i}g^{*}H-(n+1)T)\overset{T\simeq Y}{\simeq}\mathcal{O}_{Y}(p^{i}H-(n+1)L)$ that

\displaystyle H^{2}(T,\mathcal{O}_{T}(p^{i}g^{*}H-(n+1)T)\simeq H^{2}(Y,\mathcal{O}_{Y}(p^{i}H-(n+1)L))\overset{{\rm(0b)}}{=}0.

Then we are done by $H^{2}(T,\Omega^{1}_{T}(p^{i}g^{*}H-nT)\simeq H^{2}(Y,\Omega^{1}_{Y}(p^{i}H-nL))\overset{{\rm(1c)}}{=}0$ . ∎

6.1.2. 2-2

Lemma 6.7.

A smooth Fano threefold $X$ of No. 2-2 satisfies the following properties:

(1)

There is a split double cover $f\colon X\to Y:=\mathbb{P}^{1}\times\mathbb{P}^{2}$ .
(2)

$f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)$ for a Cartier divisor $L$ satisfying $\mathcal{O}_{Y}(L)\simeq\mathcal{O}_{Y}(1,2)$ .
(3)

$X$ is (isomorphic to) a divisor on $P\coloneqq\mathbb{P}_{Y}(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L))$ .
(4)

$K_{P}+X=-g^{*}H$ and $K_{X}=g^{*}(K_{Y}+L)=-g^{*}H|_{X}$ , where $H\coloneqq\mathcal{O}_{Y}(1,1)$ and $g:P=\mathbb{P}_{Y}(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L))\to Y$ denotes the projection.
(5)

There exists a section $S$ of $g$ such that $S\cap X=\emptyset$ , $S|_{S}\sim-g^{*}L|_{S}$ , $X-2S\sim 2g^{*}L$ , $\mathcal{O}_{P}(1)\simeq\mathcal{O}_{P}(S)$ , and $\Omega_{P/Y}^{1}\simeq\mathcal{O}_{P}(-g^{*}L-2S)$ .
(6)

There exists a section $T$ of $g$ such that $S\cap T=\emptyset$ , $T|_{T}\sim g^{*}L|_{T}$ , $X\sim 2T$ , $T-S\sim g^{*}L$ , and $\mathcal{O}_{P}(1)\simeq\mathcal{O}_{P}(T-g^{*}L)$ .

Proof.

The assertions (1) and (2) follow from [FanoIII, Subsection 9.2]. Then the remaining ones hold by Lemma 6.4. ∎

Lemma 6.8.

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

(1)

$H^{1}(X,\Omega^{1}_{X}(K_{X}))=0$ .
(2)

$H^{2}(X,\Omega^{1}_{X}(-p^{i}K_{X}))=0$ for every $i>0$ .
(3)

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

Proof.

We use the same notation as Lemma 6.7. It is enough to verify the conditions of Proposition 6.6 for $Y=\mathbb{P}^{1}\times\mathbb{P}^{2},L=\mathcal{O}_{Y}(1,1)$ , and $H=\mathcal{O}_{Y}(1,2)$ . Note that $mH-L$ is ample when $m\geq 2$ . By Lemma 6.2, Proposition 6.6(2) holds. Since $Y$ satisfies Kodaira vanishing, it is easy to see that Proposition 6.6(0) holds. As $Y$ satisfies Bott vanishing, it is obvious that Proposition 6.6(1) holds. ∎

6.1.3. 2-6-a

Definition 6.9.

Let $X$ be a smooth Fano threefold of No. 2-6. By [FanoIV, Section 7.2], one of the following holds up to isomorphisms.

(2-6-a)

$X$ is a hypersurface of $P:=\mathbb{P}^{2}\times\mathbb{P}^{2}$ of bidegree $(2,2)$ . In this case, $\mathcal{O}_{X}(-K_{X})\simeq\mathcal{O}_{X}(1,1)$ , where $\mathcal{O}_{X}(1,1):=\mathcal{O}_{P}(1,1)$ . In this case, we say that $X$ is (a Fano threefold) of No. 2-6-a.
(2-6-b)

There is a split double cover $f:X\to W$ satisfying $f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{W}\oplus\mathcal{O}_{W}(-L)$ , where $L$ is a Cartier divisor on $W$ with $\mathcal{O}_{W}(2L)\simeq\omega_{W}^{-1}$ . In this case, we say that $X$ is (a Fano threefold) of No. 2-6-b.

Lemma 6.10.

Let $X$ be a smooth Fano threefold of No. 2-6-a. Then the following hold.

(1)

$H^{i}(X,\mathcal{O}_{X}(n,n))=0$ for $i\in\{1,2\}$ and $n\in\mathbb{Z}$ .
(2)

$H^{3}(X,\mathcal{O}_{X}(n,n))=0$ for $n\in\mathbb{Z}_{\geq 0}$ .

Proof.

Use an exact sequence $0\to\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(-2,-2)\to\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}\to\mathcal{O}_{X}\to 0$ . ∎

Lemma 6.11.

Let $X$ be a smooth Fano threefold of No. 2-6-a. Then the following hold.

(1)

$H^{1}(X,\Omega^{1}_{X}(K_{X}))=0$ .
(2)

$H^{2}(X,\Omega^{1}_{X}(-p^{i}K_{X}))=0$ for every $i>0$ .
(3)

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

Proof.

We use the notation of Definition 6.9. By Proposition 2.20 and Lemma 6.2, it suffices to show (1) and (2).

Step 1: Proof of (1). By the conormal exact sequence, we have the following exact sequence:

0\to\mathcal{O}_{X}(K_{X}-X)\to\Omega^{1}_{P}|_{X}(K_{X})\to\Omega^{1}_{X}(K_{X})\to 0.

Since we have

H^{2}(X,K_{X}-X)=H^{2}(X,\mathcal{O}_{X}(-3,-3))=0

by Lemma 6.10, it suffices to show $H^{1}(X,\Omega^{1}_{P}|_{X}(K_{X}))=0$ .

We have the following exact sequence:

0\to\Omega^{1}_{P}(K_{P})\to\Omega^{1}_{P}(K_{P}+X)\to\Omega^{1}_{P}|_{X}(K_{X})\to 0

By Bott vanishing, we have

•

$H^{1}(P,\Omega^{1}_{P}(K_{P}+X))=H^{1}(P,\Omega^{1}_{P}(-1,-1))=0$ and
•

$H^{2}(P,\Omega^{1}_{P}(K_{P}))=H^{2}(P,\Omega^{1}_{P}(-3,-3))=0$ .

Therefore, (1) holds.

Step 2: Proof of (2). Fix $i\in\mathbb{Z}_{>0}$ . By the conormal exact sequence, we have the following exact sequence

0\to\mathcal{O}_{X}(-p^{i}K_{X}-X)\to\Omega^{1}_{P}|_{X}(-p^{i}K_{X})\to\Omega^{1}_{X}(-p^{i}K_{X})\to 0.

We have

H^{3}(X,\mathcal{O}_{X}(-p^{i}K_{X}-X))=H^{3}(X,\mathcal{O}_{X}(p^{i}-2,p^{i}-2))=0

by Lemma 6.10. Thus, it suffices to show $H^{2}(X,\Omega^{1}_{P}|_{X}(-p^{i}K_{X}))=0$ .

We have the following exact sequence:

0\to\Omega^{1}_{P}(-p^{i}(K_{P}+X)-X)\to\Omega^{1}_{P}(-p^{i}(K_{P}+X))\to\Omega^{1}_{P}|_{X}(-p^{i}K_{X})\to 0.

Then

•

$H^{2}(P,\Omega^{1}_{P}(-p^{i}(K_{P}+X)))=H^{2}(P,\Omega^{1}_{P}(p^{i},p^{i}))=0$ by Bott vanishing and
•

$H^{3}(P,\Omega^{1}_{P}(-p^{i}(K_{P}+X)-X))=H^{3}(P,\Omega^{1}_{P}(p^{i}-2,p^{i}-2))=0$ by [Totaro(Fano), Proposition 1.3].

Therefore, the assertion holds. ∎

6.1.4. 2-6-b

Lemma 6.12.

A smooth Fano threefold $X$ of No. 2-6-b satisfies the following properties:

(1)

There is a split double cover $f\colon X\to W$ , where $W$ is a smooth hypersurface of $\mathbb{P}^{2}\times\mathbb{P}^{2}$ of bidegree $(1,1)$ .
(2)

$f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{W}\oplus\mathcal{O}_{W}(-L)$ for a Cartier divisor $L=\mathcal{O}_{W}(1,1)$ .
(3)

$X$ is a divisor of $P\coloneqq\mathbb{P}_{W}(\mathcal{O}_{W}\oplus\mathcal{O}_{W}(-L))$ .
(4)

$K_{P}+X=-g^{*}L$ , $K_{X}=g^{*}(K_{W}+L)=-g^{*}L|_{X}$ .
(5)

There exists a section $S$ of $g$ such that $S\cap X=\emptyset$ , $S|_{S}\sim-g^{*}L|_{S}$ , $X-2S\sim 2g^{*}L$ , $\mathcal{O}_{P}(1)\simeq\mathcal{O}_{P}(S)$ , and $\Omega_{P/W}^{1}\simeq\mathcal{O}_{P}(-g^{*}L-2S)$ .

Proof.

The assertions (1) and (2) follow from [FanoIII, Subsection 9.2]. Then the remaining ones hold by Lemma 6.4. ∎

Lemma 6.13.

Let $Y=\mathbb{P}^{3}$ (resp. $Y=Q$ , resp. $Y=W$ ), where $Q$ is a smooth quadric hypersurface of $\mathbb{P}^{4}$ and $W$ is a smooth hypersurface of $\mathbb{P}^{2}\times\mathbb{P}^{2}$ of bidegree $(1,1)$ . Let $L=\mathcal{O}_{\mathbb{P}^{3}}(3)$ (resp. $\mathcal{O}_{Q}(2)$ , resp. $\mathcal{O}_{W}(1,1)$ ). Then the following hold.

(1)

$H^{1}(Y,\mathcal{O}_{Y}(nL))=0$ for $n\in\mathbb{Z}$ .
(2)

$H^{2}(Y,\mathcal{O}_{Y}(nL))=0$ for $n\in\mathbb{Z}$ .
(3)

$H^{3}(Y,\mathcal{O}_{Y}(nL))=0$ for $n\in\mathbb{Z}_{\geq-1}$ .
(4)

$H^{0}(Y,\Omega^{1}_{Y}(nL))=0$ for $n\in\mathbb{Z}_{\leq 0}$ .
(5)

$H^{1}(Y,\Omega^{1}_{Y}(nL))=0$ for $n\in\mathbb{Z}\setminus\{0\}$ .
(6)

$H^{2}(Y,\Omega^{1}_{Y}(nL))=0$ for $n\in\mathbb{Z}\setminus\{-1\}$ .
(7)

$H^{3}(Y,\Omega^{1}_{Y}(nL))=0$ for $n\in\mathbb{Z}_{\geq 0}$ .

Proof.

Since $Y$ is $F$ -split (Lemma 5.7), (1)-(3) hold. The assertions (4) and (7) follow from the fact that $X$ is SRC (Lemma 6.2). Let us prove (5) and (6). If $Y=\mathbb{P}^{3}$ , then these follow from the Bott vanishing theorem and [Totaro(Fano), Proposition 1.3]. In what follows, we assume $Y\in\{Q,W\}$ . If $Y=Q$ (resp. $Y=W$ ), then

•

we have an embedding $Y\subset P$ for $P:=\mathbb{P}^{4}$ (resp. $P=\mathbb{P}^{2}\times\mathbb{P}^{2}$ ),
•

we set $H:=\mathcal{O}_{\mathbb{P}^{4}}(1)$ (resp. $H:=\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(1,1)$ ), and
•

we get $L\sim sH|_{Y}$ for $s:=2$ (resp. $s:=1$ ). It holds that $Y\sim sH$ .

We have the following exact sequence:

0\to\Omega_{P}^{1}((n-s)H)\to\Omega_{P}^{1}(nH)\to\Omega^{1}_{P}(nH)|_{Y}\to 0.

By Bott vanishing and [Totaro(Fano), Proposition 1.3], we have

•

$H^{1}(P,\Omega^{1}_{P}(nH))=0$ for $n\in\mathbb{Z}\setminus\{0\}$ ,
•

$H^{2}(P,\Omega^{1}_{P}(nH))=0$ for $n\in\mathbb{Z}$ , and
•

$H^{3}(P,\Omega^{1}_{P}(nH))=0$ for $n\in\mathbb{Z}$ .

We then get

•

$H^{1}(Y,\Omega^{1}_{P}(nH)|_{Y})=0$ for $n\in\mathbb{Z}\setminus\{0\}$ and
•

$H^{2}(Y,\Omega^{1}_{P}(nH)|_{Y})=0$ for $n\in\mathbb{Z}$ .

By $Y\sim sH$ and the conormal exact sequence, we have the following exact sequence:

0\to\mathcal{O}_{Y}(nH-sH)\to\Omega_{P}^{1}(nH)|_{Y}\to\Omega^{1}_{Y}(nH)\to 0.

Recall that

•

$H^{2}(Y,\mathcal{O}_{Y}(nH))=0$ for $n\in\mathbb{Z}$ and
•

$H^{3}(Y,\mathcal{O}_{Y}(nH))=0$ for $n\geq-s$ .

Hence we get

(5)’

$H^{1}(Y,\Omega^{1}_{Y}(nH))=0$ for $n\neq 0$ and
(6-a)

$H^{2}(Y,\Omega^{1}_{Y}(nH))=0$ for $n\geq 0$ .

In particular, (5) holds. Comparing (6) with (6-a), it suffices to show (6-b) below by Serre duality.

(6-b)

$H^{1}(Y,\Omega^{2}_{Y}(nH))=0$ for $n>s$ .

Taking the wedge product $\bigwedge^{2}$ to the conormal exact sequence, we get the following exact sequence:

0\to\Omega^{1}_{Y}(nH-sH)\to\Omega^{2}_{P}(nH)|_{Y}\to\Omega^{2}_{Y}(nH)\to 0.

Since we have $H^{2}(Y,\Omega_{Y}^{1}(nH-sH))=0$ for $n>s$ (6-a), it is enough to prove $H^{1}(Y,\Omega^{2}_{P}(nH)|_{Y})=0$ for $n>s$ . This holds by an exact sequence

0\to\Omega^{2}_{P}(nH-sH)\to\Omega^{2}_{P}(nH)\to\Omega^{2}_{P}(nH)|_{Y}\to 0,

because Bott vanishing implies $H^{i}(P,\Omega_{P}^{2}(mH))=0$ for $i>0$ and $m>0$ . ∎

Lemma 6.14.

Let $X$ be a smooth Fano threefold of No. 2-6-b. Then the following hold.

(1)

$H^{1}(X,\Omega^{1}_{X}(K_{X}))=0$ .
(2)

$H^{2}(X,\Omega^{1}_{X}(-p^{i}K_{X}))=0$ for every $i>0$ .
(3)

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

Proof.

It is enough to verify the conditions in Proposition 6.6. Note that we have $H=-K_{Y}-L=L$ . Then the conditions in Proposition 6.6 follow from Lemma 6.2 and Lemma 6.13. ∎

The above argument can be applied for some hyperelliptic Fano threefolds. Let us start by recalling the definition.

Definition 6.15.

We say that a smooth Fano threefold $X$ is hyperelliptic if $X$ is of index one, $|-K_{X}|$ is base point free, and the induced morphism $f\colon X\to Y\coloneqq\varphi_{|-K_{X}|}(X)$ is a double cover.

It is known that if $\rho(X)=1$ , then $Y$ is isomorphic to $\mathbb{P}^{3}$ or $Q$ in the above notation [FanoI, Theorem 6.5]. The assumption $p>5$ in Proposition 6.16(i) is sharp as we shall see later (Example 8.7).

Proposition 6.16.

Let $X$ be a hyperelliptic smooth Fano threefold such that $\rho(X)=1$ . Let $f\colon X\to Y\coloneqq\varphi_{|-K_{X}|}(X)$ be the double cover induced by $\varphi_{|-K_{X}|}$ . Assume the following.

(i)

If $Y\simeq\mathbb{P}^{3}$ , then $p>5$ .
(ii)

If $Y\simeq Q$ , then $p>3$ .

Then the following hold.

(1)

$H^{1}(X,\Omega^{1}_{X}(K_{X}))=0$ .
(2)

$H^{2}(X,\Omega^{1}_{X}(-p^{i}K_{X}))=0$ for every $i>0$ .
(3)

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

Proof.

We have $f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)$ for a Cartier divisor $L=\mathcal{O}_{Y}(r-1)$ , where $r$ denotes the index of $Y$ (i.e., if $Y=\mathbb{P}^{3}$ (resp. $Y=Q$ ), then $r=4$ (resp. $r=3$ )). It is enough to verify the conditions in Proposition 6.6. Note that we have $L=(r-1)H$ and $\mathcal{O}_{Y}(1)=\mathcal{O}_{Y}(H)$ . Then Lemma 6.2 and Lemma 6.13 imply all the conditions in Proposition 6.6 except for Proposition 6.6(0c). By our assumptions (i) and (ii), Proposition 6.6(0c) directly follows from Serre duality, e.g.,

h^{3}(Y,\mathcal{O}_{Y}(p^{i}H-3L))=h^{0}(Y,\mathcal{O}_{Y}(K_{Y}+3L-p^{i}H))=h^{0}(Y,\mathcal{O}_{Y}((2r-3-p^{i})H))=0

for every $i>0$ . ∎

6.1.5. 2-8

Lemma 6.17.

A smooth Fano threefold $X$ of No. 2-8 satisfies the following properties:

(1)

$X$ is (isomorphic to) a divisor on $P\coloneqq\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(2))$ .
(2)

$\mathcal{O}_{P}(K_{P})=\mathcal{O}_{P}(-3)$ , $\mathcal{O}_{P}(X)=\mathcal{O}_{P}(2)$ , $K_{X}=\mathcal{O}_{X}(-1),$ and $\mathcal{O}_{X}(X)=\mathcal{O}_{X}(2)=\mathcal{O}_{X}(-2K_{X})$ .
(3)

$|\mathcal{O}_{P}(1)|$ is base point free and $\mathcal{O}_{P}(1)$ is big.
(4)

Let $\varphi:P\to P^{\prime}$ be the birational morphism to a normal projective variety $P^{\prime}$ such that $\varphi_{*}\mathcal{O}_{P}=\mathcal{O}_{P^{\prime}}$ and $\mathcal{O}_{P}(1)\simeq\varphi^{*}\mathcal{O}_{P^{\prime}}(1)$ for some ample invertible sheaf $\mathcal{O}_{P^{\prime}}(1)$ on $P^{\prime}$ . Then $\varphi:P\to P^{\prime}$ is a small birational morphism which is an isomorphism around $X$ .

We say that a birational morphism $\varphi:P\to P^{\prime}$ is small if $\dim\operatorname{Ex}(\varphi)\leq\dim P-2$ .

Proof.

Note that a Fano threefold of No. 2-8 is characterised by the following properties (i) and (ii) [FanoIII, Theorem 5.34]:

(i)

$X$ is a Fano threefold with $\rho(X)=2$ .
(ii)

One of the extremal rays is of type $C_{1}$ , and the the other extremal ray is of type $E_{3}$ or $E_{4}$ .

Then we may apply [FanoIII, Proposition 5.29 and Lemma 5.30]. By [FanoIII, Proposition 5.29(3), Lemma 5.30], $X$ is a divisor on $P\coloneqq\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(2))$ satisfying $X\sim\mathcal{O}_{P}(2)$ . Thus (1) holds. Moreover, [FanoIII, the proof of Lemma 5.30] implies that $-K_{P}\sim\mathcal{O}_{P}(3)$ . Then the adunction formula implies $K_{X}\sim(K_{P}+X)|_{X}\sim\mathcal{O}_{P}(-3+2)|_{X}=\mathcal{O}_{X}(-1)$ . Thus (2) holds.

Let us show (3) and (4). We have three sections $\Gamma_{0},\Gamma_{1},\Gamma_{2}$ of the induced $\mathbb{P}^{2}$ -bundle $g:P=\mathbb{P}_{\mathbb{P}^{2}}(E)\to\mathbb{P}^{2}$ , where $E:=\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(2)$ . corresponding to the projections of $E$ to the factors $\mathcal{O}_{\mathbb{P}^{2}},\mathcal{O}_{\mathbb{P}^{2}}(1),\mathcal{O}_{\mathbb{P}^{2}}(2)$ , respectively. Similarly, we have the following three prime divisors which are $\mathbb{P}^{1}$ -bundles over $\mathbb{P}^{2}$ :

•

$D_{0}:=\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(2))$ , corresponding to $E\to\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(2)$ .
•

$D_{1}:=\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(2))$ , corresponding to $E\to\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(2)$ .
•

$D_{2}:=\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(1))$ , corresponding to $E\to\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(1)$ .

By construction, we have $D_{i}\cap\Gamma_{i}=\emptyset$ for every $i\in\{0,1,2\}$ . Fix a line $L$ on $\mathbb{P}^{2}$ and set $F:=g^{*}L$ , which is a prime divisor on $P$ . For each $i\in\{0,1,2\}$ , we can write $\mathcal{O}_{P}(1)\sim D_{i}+bF$ for some $b\in\mathbb{Z}$ . It holds that

\mathcal{O}_{\mathbb{P}^{2}}(i)=\mathcal{O}_{P}(1)|_{\Gamma_{i}}=(D_{i}+bF)|_{\Gamma_{i}}=bF|_{\Gamma_{i}}\simeq\mathcal{O}_{\mathbb{P}^{2}}(b),

i.e., $b=i$ . Then we have that

\mathcal{O}_{P}(1)\sim D_{0}\sim D_{1}+F\sim D_{2}+2F.

By $D_{0}\cap D_{1}\cap D_{2}=\emptyset$ , $|\mathcal{O}_{P}(1)|$ is base point free. Since $D_{1}$ is relatively ample over $\mathbb{P}^{2}$ and $F$ is the pullback of an ample divisor, the divisor $\mathcal{O}_{P}(1)\sim D_{1}+F$ is big. Thus (3) holds. Let $\varphi:P\to P^{\prime}$ be as in the statement of (4).

We now show that $\operatorname{Ex}(\varphi)=\Gamma_{0}$ . By $\mathcal{O}_{P}(1)|_{\Gamma_{0}}\simeq\mathcal{O}_{\Gamma_{0}}$ , $\varphi(\Gamma_{0})$ is a point. In particular, $\operatorname{Ex}(\varphi)\supset\Gamma_{0}$ . Pick a curve $C$ such that $\varphi(C)$ is a point. It suffices to show $C\subset\Gamma_{0}$ . We have $\mathcal{O}_{P}(1)\cdot C=0$ . Note that $g(C)$ is not a point, because $\mathcal{O}_{P}(1)$ is $g$ -ample. In particular, $F\cdot C>0$ . Therefore,

0=\mathcal{O}_{P}(1)\cdot C=(D_{1}+F)\cdot C=(D_{2}+2F)\cdot C.

By $F\cdot C>0$ , we obtain $D_{1}\cdot C<0$ and $D_{2}\cdot C<0$ . Hence $C\subset D_{1}\cap D_{2}=\Gamma_{0}$ , as required.

It is enough to prove $X\cap\Gamma_{0}=\emptyset$ . Suppose $X\cap\Gamma_{0}\neq\emptyset$ . By $\mathcal{O}_{P}(X)|_{\Gamma_{0}}\simeq\mathcal{O}_{P}(2)|_{\Gamma_{0}}\simeq\mathcal{O}_{\Gamma_{0}}$ , we obtain $\mathbb{P}^{2}\simeq\Gamma_{0}\subset X$ . Then the Stein factorisation $\psi:X\to X^{\prime}$ of the composite morphism $\varphi|_{X}:X\hookrightarrow P\to P^{\prime}$ is a birational morphism which contracts $\Gamma_{0}\simeq\mathbb{P}^{2}$ to a point. This is absurd, because $X$ has no extremal ray of $E_{2}$ or $E_{5}$ . Thus (4) holds. ∎

Lemma 6.18.

A smooth Fano threefold of No. 2-8 is quasi-F-split.

Proof.

We use the notation of Lemma 6.17. By Proposition 2.20 and Lemma 6.2, it suffices to show that

(1)

$H^{1}(X,\Omega^{1}_{X}(K_{X}))=0$ .
(2)

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

Step 1: Proof of (1). By the conormal exact sequence, we have the following exact sequence

0\to\mathcal{O}_{X}(K_{X}-X)\to\Omega^{1}_{P}|_{X}(K_{X})\to\Omega^{1}_{X}(K_{X})\to 0.

Since $\mathcal{O}_{X}(K_{X})=\mathcal{O}_{X}(-1)$ and $\mathcal{O}_{X}(X)=\mathcal{O}_{X}(2)$ , we have an exact sequence

H^{2}(P,\mathcal{O}_{P}(-3))\to H^{2}(X,\mathcal{O}_{X}(-3))(=H^{2}(X,K_{X}-X))\to H^{3}(P,\mathcal{O}_{P}(-5)).

Since $P$ is toric and $\mathcal{O}_{P}(1)$ is nef, it follows from $K_{P}\sim\mathcal{O}_{P}(-3)$ and [Totaro(Fano), Proposition 1.3] that

•

$H^{2}(P,\mathcal{O}_{P}(-3))\simeq H^{2}(P,\mathcal{O}_{P})=0$ and
•

$H^{3}(P,\mathcal{O}_{P}(-5))\simeq H^{1}(P,\mathcal{O}_{P}(2))=0$ .

Thus $H^{2}(X,K_{X}-X)=0$ , and it suffices to show $H^{1}(X,\Omega^{1}_{P}|_{X}(K_{X}))=0$ .

Since we have a closed embedding $X\subset P^{\prime}$ around which $P^{\prime}$ is smooth (Proposition 6.17(4)), we have the following exact sequence:

0\to\Omega^{[1]}_{P^{\prime}}(K_{P^{\prime}})\to\Omega^{[1]}_{P^{\prime}}(K_{P^{\prime}}+X)\to\Omega^{[1]}_{P^{\prime}}|_{X}(K_{X})\simeq\Omega^{1}_{P}|_{X}(K_{X})\to 0,

By Bott vanishing [Fuj07, Theorem 1.1 or Corollary 1.3], we have

•

$H^{1}(P^{\prime},\Omega^{[1]}_{P^{\prime}}(K_{P^{\prime}}+X))=H^{1}(P^{\prime},\Omega^{[1]}_{P^{\prime}}(-1))=0$ and
•

$H^{2}(P^{\prime},\Omega^{[1]}_{P^{\prime}}(K_{P^{\prime}}))=H^{2}(P^{\prime},\Omega^{[1]}_{P^{\prime}}(-3))=0$ .

This completes the proof of (1).

Step 2: Proof of (2). By the conormal exact sequence, we have the following exact sequence

0\to\mathcal{O}_{X}(-p^{i}K_{X}-X)\to\Omega^{1}_{P}|_{X}(-p^{i}K_{X})\to\Omega^{1}_{X}(-p^{i}K_{X})\to 0.

We have

H^{3}(X,\mathcal{O}_{X}(-p^{i}K_{X}-X))=H^{3}(X,\mathcal{O}_{X}(p^{i}-2))=0.

Thus it suffices to show that $H^{2}(X,\Omega^{1}_{P}|_{X}(-p^{i}K_{X}))=0$ .

We have the following exact sequence:

0\to\Omega^{1}_{P}(-p^{i}(K_{P}+X)-X)\to\Omega^{1}_{P}(-p^{i}(K_{P}+X))\to\Omega^{1}_{P}|_{X}(-p^{i}K_{X})\to 0.

Since $\mathcal{O}_{P}(1)$ is nef and $P$ is a smooth toric variety, we have

(1)

$H^{2}(P,\Omega^{1}_{P}(-p^{i}(K_{P}+X)))=H^{2}(P,\Omega^{1}_{P}(p^{i}))=0$ and
(2)

$H^{3}(P,\Omega^{1}_{P}(-p^{i}(K_{P}+X)-X))=H^{3}(P,\Omega^{1}_{P}(p^{i}-2))=0$ .

by [Totaro(Fano), Proposition 1.3]. Thus (2) holds. ∎

6.1.6. 3-10

Lemma 6.19.

Let $X$ be a smooth Fano threefold $X$ of No. 3-10 such that there is a wild conic bundle structure $f:X\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ . Then the following properties hold:

(1)

$X$ is (isomorphic to) a divisor on $P\coloneqq\mathbb{P}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(1,0)\oplus\mathcal{O}(0,1))$ satisfying $\mathcal{O}_{P}(X)\sim\mathcal{O}_{P}(2)$ .
(2)

Each of $X$ and $\mathcal{O}_{P}(1)$ is nef and big.
(3)

$-K_{P}$ and $-(K_{P}+X)$ are ample.
(4)

$\mathcal{O}_{X}(K_{X})\simeq\mathcal{O}_{X}(-1)\otimes f^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(-1,-1)$ , where $\mathcal{O}_{X}(-1)\coloneqq\mathcal{O}_{P}(-1)|_{X}$ .
(5)

$-2^{i}(K_{P}+X)-X$ is nef for every $i>0$ .

Proof.

The assertion (1) follows from [MS03, Corollary 8]. Let us show (2). By $X\sim\mathcal{O}_{P}(2)$ , it suffices to show that $|\mathcal{O}_{P}(1)|$ is base point free and $\mathcal{O}_{P}(1)$ is big. Set $L_{0}:=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}},L_{1}:=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0),L_{2}:=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(0,1)$ , and

E\coloneqq L_{0}\oplus L_{1}\oplus L_{2}=\mathcal{O}\oplus\mathcal{O}(1,0)\oplus\mathcal{O}(0,1).

We have three sections of $\pi\colon P=\mathbb{P}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(1,0)\oplus\mathcal{O}(0,1))\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ :

•

Set $\Gamma_{0}:=\mathbb{P}(\mathcal{O})$ , which is corresponding to the projection $E=\mathcal{O}\oplus\mathcal{O}(1,0)\oplus\mathcal{O}(0,1)\to\mathcal{O}$ . We get $\mathcal{O}_{P}(1)|_{\Gamma_{0}}=\mathcal{O}=L_{0}$ .
•

Set $\Gamma_{1}:=\mathbb{P}(\mathcal{O}(1,0))$ , which is corresponding to the projection $E=\mathcal{O}\oplus\mathcal{O}(1,0)\oplus\mathcal{O}(0,1)\to\mathcal{O}(1,0)$ . We get $\mathcal{O}_{P}(1)|_{\Gamma_{1}}=\mathcal{O}(1,0)=L_{1}$ .
•

Set $\Gamma_{2}:=\mathbb{P}(\mathcal{O}(0,1))$ , which is corresponding to the projection $E=\mathcal{O}\oplus\mathcal{O}(1,0)\oplus\mathcal{O}(0,1)\to\mathcal{O}(0,2)$ . We get $\mathcal{O}_{P}(1)|_{\Gamma_{2}}=\mathcal{O}(0,1)=L_{2}$ .

Similarly, we have three prime divisors on $P$ which are $\mathbb{P}^{1}$ -bundles over $\mathbb{P}^{1}\times\mathbb{P}^{1}$ :

•

Set $D_{0}:=\mathbb{P}(L_{1}\oplus L_{2})$ , which is corresponding to the projection $E\to L_{1}\oplus L_{2}$ .
•

Set $D_{1}:=\mathbb{P}(L_{0}\oplus L_{2})$ , which is corresponding to the projection $E\to L_{0}\oplus L_{2}$ .
•

Set $D_{2}:=\mathbb{P}(L_{0}\oplus L_{1})$ , which is corresponding to the projection $E\to L_{0}\oplus L_{1}$ .

By construction, we get $\Gamma_{i}\cap D_{i}=\emptyset$ for every $i\in\{0,1,2\}$ .

We now show that

\mathcal{O}_{P}(1)\sim D_{0}\sim D_{1}+\pi^{*}\mathcal{O}(1,0)\sim D_{2}+\pi^{*}\mathcal{O}(0,1).

Fix $i\in\{0,1,2\}$ . Note that we have $\mathcal{O}_{P}(1)\sim D_{i}+\pi^{*}M_{i}$ for some $M_{i}$ . By restricting this to $\Gamma_{i}$ , we obtain

L_{i}=\mathcal{O}_{P}(1)|_{\Gamma_{i}}=(D_{i}+\pi^{*}M_{i})|_{\Gamma_{i}}=M_{i}.

Hence we get $\mathcal{O}_{P}(1)\sim D_{i}+L_{i}$ , as required.

It follows from $D_{0}\cap D_{1}\cap D_{2}=\emptyset$ that $|\mathcal{O}_{P}(1)|$ is base point free. We have

\mathcal{O}_{P}(2)\sim D_{1}+D_{2}+\pi^{*}\mathcal{O}(1,1).

Since $\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1)$ is ample and $D_{1}+D_{2}$ is an effective $\pi$ -ample divisor, $\mathcal{O}_{P}(2)$ is big. Thus (2) holds.

Let us show (3). The following holds (cf. [FanoIII, Proposition 7.1(2)]):

K_{P}\simeq\mathcal{O}_{P}(-3)\otimes\pi^{*}(K_{\mathbb{P}^{1}\times\mathbb{P}^{1}}\otimes\det E)\simeq\mathcal{O}_{P}(-3)\otimes\pi^{*}\mathcal{O}(-1,-1).

Since $\mathcal{O}_{P}(1)$ is nef and $\pi$ -ample, $-K_{P}$ is ample. Similarly, $-(K_{P}+X)$ is ample by

K_{P}+X\sim\mathcal{O}_{P}(-1)\otimes\pi^{*}\mathcal{O}(-1,-1).

Thus (3) holds. This linear equivalence implies (4). Finally, (5) follows from

-2^{i}(K_{P}+X)-X\sim\mathcal{O}_{P}(2^{i}-2)\otimes\pi^{*}\mathcal{O}(2^{i},2^{i}).

∎

Lemma 6.20.

A smooth Fano threefold $X$ of No. 3-10 is quasi-F-split.

Proof.

By Proposition 4.4, we may assume that $p=2$ and $X$ has a wild conic bundle structure. In what follows, we use the notation of Lemma 6.19. By Proposition 2.20 and Lemma 6.2, it suffices to show that

(1)

$H^{1}(X,\Omega^{1}_{X}(K_{X}))=0$ .
(2)

$H^{2}(X,\Omega^{1}_{X}(-2^{i}K_{X}))=0$ for every $i>0$ .

Step 1: Proof of (1). By the conormal exact sequence, we have the following exact sequence:

0\to\mathcal{O}_{X}(K_{X}-X)\to\Omega^{1}_{P}|_{X}(K_{X})\to\Omega^{1}_{X}(K_{X})\to 0.

Considering the restriction $\mathcal{O}_{P}\to\mathcal{O}_{X}$ , we have an exact sequence

H^{2}(P,K_{P})\to H^{2}(X,K_{X}-X)\to H^{3}(P,K_{P}-X).

Since $X$ is nef, we have $H^{2}(P,K_{P})\simeq H^{2}(P,\mathcal{O}_{P})=0$ and $H^{3}(P,K_{P}-X)\simeq H^{1}(P,X)=0$ by [Totaro(Fano), Proposition 1.3]. Thus, we have $H^{2}(X,K_{X}-X)=0$ . Then it suffices to show $H^{1}(X,\Omega^{1}_{P}|_{X}(K_{X}))=0$ .

We have the following exact sequence:

0\to\Omega^{1}_{P}(K_{P})\to\Omega^{1}_{P}(K_{P}+X)\to\Omega^{1}_{P}|_{X}(K_{X})\to 0

Since $-(K_{P}+X)$ and $-K_{P}$ are ample, we get

H^{1}(P,\Omega^{1}_{P}(K_{P}+X))=0\,\,\,\text{and}\,\,\,H^{2}(P,\Omega^{1}_{P}(K_{P}))=0

by Bott vanishing. Thus (1) holds.

Step 2: Proof of (2). By the conormal exact sequence, we have an exact sequence

0\to\mathcal{O}_{X}(-2^{i}K_{X}-X)\to\Omega^{1}_{P}|_{X}(-2^{i}K_{X})\to\Omega^{1}_{X}(-2^{i}K_{X})\to 0.

Since $K_{X}=\mathcal{O}_{X}(-1)\otimes\pi^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(-1,-1)$ and $X=\mathcal{O}_{P}(2)$ , we have

	$\displaystyle H^{3}(X,-2^{i}K_{X}-X)$	$\displaystyle\simeq H^{0}(X,(2^{i}+1)K_{X}+X)$
		$\displaystyle=H^{0}(X,\mathcal{O}_{P}(-2^{i}+1)\otimes\pi^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(-2^{i}-1,-2^{i}-1))$
		$\displaystyle=0.$

Thus it suffices to show $H^{2}(X,\Omega^{1}_{P}|_{X}(-2^{i}K_{X}))=0$ .

We have the following exact sequence:

0\to\Omega^{1}_{P}(-2^{i}(K_{P}+X)-X)\to\Omega^{1}_{P}(-2^{i}(K_{P}+X))\to\Omega^{1}_{P}|_{X}(-2^{i}K_{X})\to 0

Since $-2^{i}(K_{P}+X)$ and $-2^{i}(K_{P}+X)-X$ is nef, we have

H^{2}(P,\Omega^{1}_{P}(-2^{i}(K_{P}+X))=0\,\,\,\text{and}\,\,\,H^{3}(P,\Omega^{1}_{P}(-2^{i}(K_{P}+X)-X))=0

by [Totaro(Fano), Proposition 1.3]. Thus (2) holds. ∎

6.2. $F$ -splitting for 2-2 and 2-6

Proposition 6.21.

We use the same notation of Lemma 6.4. Assume that $\dim X=\dim Y=3$ . Moreover, suppose that the following hold.

(0)
1. (0a)
  
  $H^{1}(Y,pH-mL)=0$ for $m\in\{2,3\}$ .
2. (0b)
  
  $H^{2}(Y,pH-mL)=0$ for $m\in\{1,2,3,4\}$ .
3. (0c)
  
  $H^{3}(Y,pH-mL)=0$ for $m\in\{1,2,3,4,5\}$ .
(1)
1. (1a)
  
  $H^{1}(Y,\Omega_{Y}^{1}(pH-mL))=0$ for $m\in\{1,2,3\}$ .
2. (1b)
  
  $H^{2}(Y,\Omega_{Y}^{1}(pH-mL))=0$ for $m\in\{2,3,4\}$ .
3. (1c)
  
  $H^{3}(Y,\Omega_{Y}^{1}(pH-4L))=0$ .
(2)
1. (2a)
  
  $H^{1}(Y,\Omega_{Y}^{2}(pH-nL))=0$ for $n\in\{0,1,2\}$ .
2. (2b)
  
  $H^{2}(Y,\Omega_{Y}^{2}(pH-2L))=0$ .

Then the following hold.

(A)

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

$H^{1}(X,\Omega^{2}_{P}|_{X}(-pK_{X}))=0$ .
(C)

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

Proof.

By taking the wedge product $\bigwedge^{2}$ of the conormal exact sequence, we have an exact sequence

0\to\Omega^{1}_{X}(-pK_{X}-X)\to\Omega^{2}_{P}|_{X}(-pK_{X})\to\Omega^{2}_{X}(-pK_{X})\to 0.

Therefore, we get the following implication:

{\rm(A)}+{\rm(B)}\Rightarrow{\rm(C)}.

In what follows, we shall prove (A) and (B).

Step 1: Proof of (A). By the conormal exact sequence, we have an exact sequence

0\to\mathcal{O}_{X}(-pK_{X}-2X)\to\Omega^{1}_{P}|_{X}(-pK_{X}-X)\to\Omega^{1}_{X}(-pK_{X}-X)\to 0.

We recall that $K_{X}=-g^{*}H|_{X}=-f^{*}H$ , $X-2S\sim 2g^{*}L=2f^{*}L$ , $S|_{X}=0$ , and $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-L)$ . We then get

	$\displaystyle H^{3}(X,-pK_{X}-2X)$	$\displaystyle=H^{3}(X,pf^{}H-4f^{}L)$
		$\displaystyle=H^{3}(Y,\mathcal{O}_{Y}(pH-4L))\oplus H^{3}(Y,\mathcal{O}_{Y}(pH-5L))$
		$\displaystyle\overset{{\rm(0c)}}{=}0.$

Thus it suffices to show $H^{2}(X,\Omega^{1}_{P}|_{X}(-pK_{X}-X))=0$ .

Since $K_{X}=-g^{*}H|_{X}$ , $-X=-2S-2g^{*}L$ , and $S|_{X}=0$ , we have the following exact sequence:

0\to\Omega^{1}_{P}(pg^{*}L-4g^{*}L-2S)\to\Omega^{1}_{P}(pg^{*}H-2g^{*}L)\to\Omega^{1}_{P}|_{X}(-pK_{X}-X)\to 0.

By applying Lemma 6.5 for $q=2$ and $D=pH-2L$ , (0a) and (1b) imply $H^{2}(P,\Omega^{1}_{P}(pg^{*}H-2g^{*}L))=0$ . Then the problem is reduced to

H^{3}(P,\Omega^{1}_{P}(pg^{*}L-4g^{*}L-2S))=0.

We have the following exact sequence:

0\to g^{*}(\Omega^{1}_{Y}(pH-4L))(-2S)\to\Omega^{1}_{P}(pg^{*}H-4g^{*}L-2S)\\ \to\Omega^{1}_{P/Y}(pg^{*}H-4g^{*}L-2S)\simeq\mathcal{O}_{P}(pg^{*}H-5g^{*}L-4S)\to 0

Thus it is enough to prove that

(I)

$H^{3}(P,g^{*}(\Omega^{1}_{Y}(pH-4L))(-2S))=0$ and
(II)

$H^{3}(P,\mathcal{O}_{P}(pg^{*}H-5g^{*}L-4S))=0$ .

Step 1-1: Proof of (I). By using an exact sequence $0\to\mathcal{O}_{P}(-S)\to\mathcal{O}_{P}\to\mathcal{O}_{S}\to 0$ twice, the problem is reduced to

(Ia)

$H^{2}(S,g^{*}(\Omega^{1}_{Y}(pH+(n-4)L)))=0$ for $n\in\{0,1\}$ and
(Ib)

$H^{3}(P,g^{*}(\Omega^{1}_{Y}(pH-4L)))=0$ .

H^{2}(P,g^{*}(\Omega^{1}_{Y}(pH+(n-4)L)))=H^{2}(Y,\Omega_{Y}^{1}(pH+(n-4)L)),

(Ia) follows from (1b). We have

H^{3}(P,g^{*}(\Omega^{1}_{Y}(pH-4L)))\simeq H^{3}(Y,\Omega^{1}_{Y}(pH-4L)),

and hence (1c) implies (Ib). This completes thep proof of (I).

Step 1-2: Proof of (II). By $T\sim S+g^{*}L$ , We have $H^{3}(P,\mathcal{O}_{P}(pg^{*}H-5g^{*}L-4S))\simeq H^{3}(P,\mathcal{O}_{P}(pg^{*}H-g^{*}L-4T))$ . By using an exact sequence $0\to\mathcal{O}_{P}(-T)\to\mathcal{O}_{P}\to\mathcal{O}_{T}\to 0$ four times, it is enough to prove (IIa) and (IIb) below.

(IIa)

$H^{2}(T,pg^{*}H-g^{*}L-nT)=0$ for $n\in\{0,1,2,3\}$ .
(IIb)

$H^{3}(P,pg^{*}H-g^{*}L)=0$ .

H^{2}(T,pg^{*}H-g^{*}L-nT)\simeq H^{2}(Y,pH-(n+1)L),

(IIa) follows from (0b). We have

H^{3}(P,pg^{*}H-g^{*}L)\simeq H^{3}(Y,pH-gL),

and hence (0c) implies (IIb). Thus (II) holds.

Step 2: Proof of (B). Since $K_{X}=-g^{*}H|_{X}$ , $S|_{X}=0$ , and $-X=-2S-2g^{*}L$ , we have the following exact sequence:

0\to\Omega^{2}_{P}(pg^{*}H-2g^{*}L)\to\Omega^{2}_{P}(pg^{*}H+2S)\to\Omega^{2}_{P}|_{X}(-pK_{X})\to 0

Thus it suffices to show that

(III)

$H^{1}(P,\Omega^{2}_{P}(pg^{*}H+2S))=0$ and
(IV)

$H^{2}(P,\Omega^{2}_{P}(pg^{*}H-2g^{*}L))=0$ .

Step 2-1: Proof of (III). Taking the wedge product $\bigwedge^{2}$ of the relative exact sequence $0\to g^{*}\Omega^{1}_{Y}\to\Omega^{1}_{P}\to\Omega^{1}_{P/Y}\to 0$ , we get

0\to g^{*}\Omega^{2}_{Y}\to\Omega^{2}_{P}\to g^{*}\Omega^{1}_{Y}\otimes\Omega^{1}_{P/Y}\simeq g^{*}\Omega^{1}_{Y}(-g^{*}L-2S)\to 0.

Thus we have

0\to g^{*}(\Omega^{2}_{Y}(pH))(2S)\to\Omega^{2}_{P}(pg^{*}H+2S)\to g^{*}(\Omega^{1}_{Y}(pH-L))\to 0.

It holds that

H^{1}(P,g^{*}(\Omega^{1}_{Y}(pH-L)))=H^{1}(Y,\Omega^{1}_{Y}(pH-L))\overset{{\rm(1a)}}{=}0.

Then it suffices to show $H^{1}(P,g^{*}(\Omega^{2}_{Y}(pH))(2S))=0$ . By $S|_{S}=-g^{*}L|_{S}$ and an exact sequence $0\to\mathcal{O}_{P}(-S)\to\mathcal{O}_{P}\to\mathcal{O}_{S}\to 0$ , the problem is reduced to the vanishings of the following:

•

$H^{1}(S,g^{*}(\Omega^{2}_{Y}(pH-nL))\simeq H^{1}(Y,\Omega^{2}_{Y}(pH-nL))$ for $n\in\{1,2\}$ .
•

$H^{1}(P,g^{*}(\Omega^{2}_{Y}(pH)))\simeq H^{1}(Y,\Omega^{2}_{Y}(pH))$ .

Both of them follow from (2a). Thus (III) holds.

Step 2-2: Proof of (IV). By the relative exact sequence $0\to g^{*}\Omega^{1}_{Y}\to\Omega^{1}_{P}\to\Omega^{1}_{P/Y}\simeq\mathcal{O}_{P}(-g^{*}L-2S)\to 0$ , we get the following exact sequence:

0\to g^{*}(\Omega^{2}_{Y}(pH-2L))\to\Omega^{2}_{P}(pg^{*}H-2g^{*}L)\to g^{*}(\Omega_{Y}^{1}(pH-3L))(-2S)\to 0.

We have $H^{2}(P,g^{*}(\Omega^{1}_{Y}(pH-2L)))=H^{2}(Y,\Omega^{2}_{Y}(pH-2L))\overset{{\rm(2b)}}{=}0$ . By $S|_{S}=-g^{*}L|_{S}$ and an exact sequence $0\to\mathcal{O}_{P}(-S)\to\mathcal{O}_{P}\to\mathcal{O}_{S}\to 0$ , the vanishing of $H^{2}(P,g^{*}(\Omega_{Y}(pg^{*}H-3g^{*}L)(-2S))$ can be reduced to those of

•

$H^{1}(S,g^{*}(\Omega^{1}_{Y}(pH+(-3+n)L)\simeq H^{1}(Y,\Omega_{Y}^{1}(pH+(-3+n)L))$ for $n\in\{0,1\}$ and
•

$H^{2}(P,g^{*}(\Omega^{1}_{Y}(pH-3L)))\simeq H^{2}(Y,\Omega^{1}_{Y}(pH-3L))$ .

These follow from (1a) and (1b). Thus (IV) holds. ∎

6.2.1. 2-2

Lemma 6.22.

A smooth Fano threefold $X$ of No. 2-2 is $F$ -split if $p\geq 7$ .

Proof.

We follow the notation of Lemma 6.7. It is enough to verify the conditions (1)-(4) in Proposition 2.15. Proposition 2.15(1) holds by Lemma 6.2. Lemma 6.8 implies Proposition 2.15(2) and Proposition 2.15(3).

It suffices to show Proposition 2.15(4). It is enough to verify the conditions of Proposition 6.21. Recall that $Y=\mathbb{P}^{1}\times\mathbb{P}^{2}$ , $H=\mathcal{O}_{Y}(1,1)$ , and $L=\mathcal{O}_{Y}(1,2)$ . Since $Y$ is toric, $Y$ satisfies Bott vanishing. Then it is enough to check the following (concerning Proposition 6.21(0), use the fact that $pH-4L-K_{Y}$ is ample):

(0)

$H^{3}(Y,pH-5L)=0$ .
(1)

$H^{2}(Y,\Omega^{1}_{Y}(pH-4L))=H^{3}(Y,\Omega^{1}_{Y}(pH-4L))=0$ .

The assertion (0) follows from

H^{3}(Y,pH-5L)=H^{3}(Y,\mathcal{O}_{Y}(p-5,p-10))\simeq H^{1}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(p-5))\otimes H^{2}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(p-10))=0.

Let us show (1). We have

	$\displaystyle\Omega^{1}_{Y}(pH-4L)$	$\displaystyle=\Omega^{1}_{Y}(p-4,p-8)$
		$\displaystyle\simeq\mathrm{pr}_{1}^{}\Omega^{1}_{\mathbb{P}^{1}}(p-4)\otimes\mathrm{pr}_{2}^{}\mathcal{O}_{\mathbb{P}^{2}}(p-8)$
		$\displaystyle\oplus\mathrm{pr}_{1}^{}\mathcal{O}_{\mathbb{P}^{1}}(p-4)\otimes\mathrm{pr}_{2}^{}\Omega^{1}_{\mathbb{P}^{2}}(p-8)$

Then it holds that

		$\displaystyle H^{3}(Y,\Omega^{1}_{Y}(pH-4L)$
	$\displaystyle=$	$\displaystyle H^{1}(\Omega^{1}_{\mathbb{P}^{1}}(p-4))\otimes H^{2}(\mathcal{O}_{\mathbb{P}^{2}}(p-8))\oplus H^{1}(\mathcal{O}_{\mathbb{P}^{1}}(p-4))\otimes H^{2}(\Omega^{1}_{\mathbb{P}^{2}}(p-8))$
	$\displaystyle=$	$\displaystyle 0$

and

		$\displaystyle H^{2}(Y,\Omega^{1}_{Y}(pH-4L)$
	$\displaystyle=$	$\displaystyle H^{0}(\Omega^{1}_{\mathbb{P}^{1}}(p-4))\otimes H^{2}(\mathcal{O}_{\mathbb{P}^{2}}(p-8))\oplus H^{1}(\Omega^{1}_{\mathbb{P}^{1}}(p-4))\otimes H^{1}(\mathcal{O}_{\mathbb{P}^{2}}(p-8))$
	$\displaystyle\oplus$	$\displaystyle H^{0}(\mathcal{O}_{\mathbb{P}^{1}}(p-4)\otimes H^{2}(\Omega^{1}_{\mathbb{P}^{2}}(p-8))\oplus H^{1}(\mathcal{O}_{\mathbb{P}^{1}}(p-4))\otimes H^{1}(\Omega^{1}_{\mathbb{P}^{2}}(p-8))$
	$\displaystyle=$	$\displaystyle 0,$

where we have $H^{2}(\Omega^{1}_{\mathbb{P}^{2}}(p-8))\simeq H^{0}(\Omega^{1}_{\mathbb{P}^{2}}(8-p))^{*}=0$ by $p\geq 7$ and the Euler exact sequence [Har77, Ch. II, Example 8.20.1], where $(-)^{*}:=\operatorname{Hom}_{k}(-,k)$ . Indeed, if $p>7$ , then this immediately follows from Bott vanishing. When $p=7$ , use the Euler exact sequence $0\to\Omega_{\mathbb{P}^{2}}^{1}(1)\to\mathcal{O}_{\mathbb{P}^{2}}^{\oplus 3}\to\mathcal{O}_{\mathbb{P}^{2}}(1)\to 0$ , which is still exact even after applying $H^{0}(\mathbb{P}^{2},-)$ by Bott vanishing, and hence $h^{0}(\Omega_{\mathbb{P}^{2}}^{1}(1))=h^{0}(\mathcal{O}_{\mathbb{P}^{2}}^{\oplus 3})-h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(1))=0$ . ∎

6.2.2. 2-6-a

Lemma 6.23.

A smooth Fano threefold of No. 2-6-a is $F$ -split if $p\geq 5$ .

Proof.

We follow the notation of Definition 6.9. It is enough to verify the conditions (1)-(4) in Proposition 2.15. Proposition 2.15(1) holds by Lemma 6.2. Lemma 6.14 implies Proposition 2.15(2) and Proposition 2.15(3).

It suffices to show Proposition 2.15(4). By the first paragraph of the proof of Proposition 6.21, it is enough to prove that

(1)

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

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

Step 1: Proof of (1). By the conormal exact sequence, we have an exact sequence

0\to\mathcal{O}_{X}(-pK_{X}-2X)\to\Omega^{1}_{P}|_{X}(-pK_{X}-X)\to\Omega^{1}_{X}(-pK_{X}-X)\to 0.

We have

H^{3}(X,\mathcal{O}_{X}(-pK_{X}-2X))=H^{3}(X,\mathcal{O}_{X}(p-4,p-4))=0

by $p\geq 5$ and Lemma 6.10. Thus it suffices to show $H^{2}(X,\Omega^{1}_{P}|_{X}(-pK_{X}-X)=0$ .

We have the following exact sequence:

0\to\Omega^{1}_{P}(-p(K_{P}+X)-2X)\to\Omega^{1}_{P}(-p(K_{P}+X)-X)\to\Omega^{1}_{P}|_{X}(-pK_{X}-X)\to 0.

Then the required vanishing $H^{2}(X,\Omega^{1}_{P}|_{X}(-pK_{X}-X)=0$ follows from

H^{2}(P,\Omega^{1}_{P}(-p(K_{P}+X)-X))=H^{2}(P,\Omega^{1}_{P}(p-2,p-2))=0

and

H^{3}(P,\Omega^{1}_{P}(-p(K_{P}+X)-2X))=H^{3}(P,\Omega^{1}_{P}(p-4,p-4))=0,

where each vanishing follows from Bott vanishing. Thus (1) holds.

Step 2: Proof of (2). We have the following exact sequence:

0\to\Omega^{2}_{P}(-p(K_{P}+X)-X)\to\Omega^{2}_{P}(-p(K_{P}+X))\to\Omega^{2}_{P}|_{X}(-pK_{X})\to 0.

By Bott vanishing, we get

H^{1}(P,\Omega^{2}_{P}(-p(K_{P}+X)))=H^{1}(P,\Omega^{2}_{P}(p,p))=0

and

H^{2}(P,\Omega^{2}_{P}(-p(K_{P}+X)-X))=H^{2}(P,\Omega^{2}_{P}(p-2,p-2))=0.

Therefore, (2) holds. ∎

6.2.3. 2-6-b

Lemma 6.24.

A smooth Fano threefold $X$ of No. 2-6-b is $F$ -split if $p\geq 5$ .

Proof.

We follow the notation of Lemma 6.12. It is enough to verify the conditions (1)-(4) in Proposition 2.15. Proposition 2.15(1) holds by Lemma 6.2. Lemma 6.14 implies Proposition 2.15(2) and Proposition 2.15(3).

It suffices to show Proposition 2.15(4). It is enough to verify the conditions of Proposition 6.21. Recall that $W\in\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(1,1)$ and $L=H=\mathcal{O}_{Y}(1,1)$ . Then all the conditions of Proposition 6.21 hold by Lemma 6.13 and Serre duality. ∎

By a similar argument, we obtain an analogous result for the hyperelliptic case. We shall later prove that the assumption on $p$ in (1) is optimal (Example 8.4).

Proposition 6.25.

Let $X$ be a smooth Fano threefold such that $\rho(X)=r_{X}=1$ and $|-K_{X}|$ is not very ample, where $r_{X}$ denotes the index of $X$ . Let $f:X\to Y$ be the double cover induced by $|-K_{X}|$ , where $Y\in\{\mathbb{P}^{3},Q\}$ (cf. [FanoI, Theorem 6.5]). Then the following hold.

(1)

If $p\geq 13$ and $Y=\mathbb{P}^{3}$ , then $X$ is $F$ -split.
(2)

If $p\geq 11$ and $Y=Q$ , then $X$ is $F$ -split.

Proof.

We use the same notation of the proof of Proposition 6.16. By the same argument as in 2-6-b (cf. the proof of Lemma 6.24), it is enough to verify Proposition 2.15(4), which follows from Lemma 6.13. ∎

7. Proofs of the main theorems

In this section, we prove Theorems A, B, C, and D.

Lemma 7.1.

Let $X$ be a smooth projective variety. Take an ample Cartier divisor $A$ on $X$ . Suppose that $X$ is quasi- $F$ -split. Then there exists $n_{0}>0$ such that $H^{i}(X,B_{n}\Omega^{1}_{X}(-p^{n}A))=0$ for every $n\geq n_{0}$ and every $i<\dim X$ .

Proof.

Fix $n_{1}$ such that $X$ is $n_{1}$ -quasi- $F$ -split. Pick integers $n$ and $\ell$ satisfying $n\geq\ell\geq n_{1}$ . By definition, $X$ is $\ell$ -quasi- $F$ -split. By [KTTWYY1, Lemma 3.8 and Lemma 5.9], we get an isomorphism

Q_{X,-p^{n-\ell}A,\ell}\simeq\mathcal{O}_{X}(-p^{n-\ell}A)\oplus B_{\ell}\Omega_{X}^{1}(-p^{n}A).

and an exact sequence

0\to F_{*}B_{\ell-1}\Omega_{X}^{1}(-p^{n}A)\to Q_{X,-p^{n-\ell}A,\ell}\to F_{*}\mathcal{O}_{X}(-p^{n-\ell+1}A)\to 0.

Since $X$ satisfies Kodaira vanishing [KTTWYY1, Theorem 3.15], the following hold:

H^{i}(X,B_{\ell}\Omega_{X}^{1}(-p^{n}A))\simeq H^{i}(X,Q_{X,-p^{n-\ell}A,n})\simeq H^{i}(X,B_{\ell-1}\Omega_{X}^{1}(-p^{n}A)).

By the Serre vanishing theorem, we can find $n(i)$ such that the following holds for $n\geq n(i)$ :

$\displaystyle H^{i}(X,B_{n}\Omega_{X}^{1}(-p^{n}A))$	$\displaystyle\simeq$	$\displaystyle H^{i}(X,B_{n-1}\Omega_{X}^{1}(-p^{n}A))$
	$\displaystyle\simeq$	$\displaystyle H^{i}(X,B_{n-2}\Omega_{X}^{1}(-p^{n}A))$
	$\displaystyle\simeq$	$\displaystyle\cdots$
	$\displaystyle\simeq$	$\displaystyle H^{i}(X,B_{n_{1}}\Omega_{X}^{1}(-p^{n}A))=0.$

Hence we are done by setting $n_{0}\coloneqq\max\{n(0),\ldots,n(\dim X-1)\}$ . ∎

Lemma 7.2.

Let $X$ be a smooth projective threefold. Take an ample Cartier divisor $A$ . Suppose that

(1)

$H^{0}(X,\Omega_{X}^{2}(-p^{m}A))=0$ for every integer $m>0$ , and
(2)

$H^{2}(X,B_{n}\Omega_{X}^{1}(-p^{n}A))=0$ for every sufficiently large integer $n\gg 0$ .

Then

H^{1}(X,\Omega_{X}^{1}(-A))=0.

Proof.

Fix $n\gg 0$ . By (2.1.1), we have an exact sequence

0\to B_{n}\Omega_{X}^{1}(-p^{n}A)\to Z_{n}\Omega_{X}^{1}(-p^{n}A)\to\Omega_{X}^{1}(-A)\to 0.

By (2), the required vanishing is reduced to $H^{1}(X,Z_{n}\Omega_{X}^{1}(-p^{n}A))=0$ . By (2.1.2), we have the following exact sequence

0\to Z_{n}\Omega_{X}^{1}(-p^{n}A)\to F_{*}Z_{n-1}\Omega_{X}^{1}(-p^{n}A)\to B_{1}\Omega_{X}^{2}(-pA)\to 0.

Since $H^{0}(X,B_{1}\Omega_{X}^{2}(-pA))\subset H^{0}(X,\Omega_{X}^{2}(-pA))=0$ by (1), it suffices to show that $H^{1}(X,Z_{n-1}\Omega_{X}^{1}(-p^{n}A))=0$ . By repeating this procedure, it is enough to prove

H^{1}(X,\Omega^{1}_{X}(-p^{n}A))=0,

which follows from $n\gg 0$ and the Serre vanishing theorem. ∎

Proposition 7.3.

Let $X$ be a smooth Fano threefold with $\rho(X)\geq 2$ . Then the following hold.

(1)

$X$ is SRC if $X$ is none of 2-1, 2-3, 2-5, 2-10, and 2-14.
(2)

$H^{0}(X,\Omega_{X}^{2}(-D))=0$ for every pseudo-effective Cartier divisor $D$ on $X$ .

Proof.

Let us show (1). Note that $X$ is SRC if $X$ is rational or $X$ has a conic bundle structure (Lemma 6.2). We first treat the case when $\rho(X)\geq 3$ . In this case, $X$ has a conic bundle structure except when $X$ is 3-18 [FanoIV, Theorem 4.10, Theorem 5.2]. If $X$ is 3-18, then $X$ is rational by [FanoIV, Section 7.3]. Hence we may assume that $\rho(X)=2$ . Since $\mathbb{P}^{3}$ and $Q$ are rational, $X$ is rational or has an extremal ray of type $C$ except when $X$ is one of 2-1, 2-3, 2-5, 2-10, and 2-14 [FanoIV, Section 7.2]. Thus (1) holds.

Let us show (2). By (1) and Lemma 6.1, we may assume that $X$ is one of 2-1, 2-3, 2-5, 2-10, and 2-14. It follows from [FanoIV, Section 7.2] that

•

there is a blowup $f\colon X\to V_{d}$ to a smooth Fano threefold $V_{d}$ of index $2$ with $(-K_{V_{d}})^{3}=8d$ for some $1\leq d\leq 5$ and
•

we have a del Pezzo fibration $g\colon X\to\mathbb{P}^{1}$ .

Take a general fibre $F$ of $g$ . Then $F$ is a canonical del Pezzo surface by [FS20, Theorem 15.2] and [BT22, Theorem 3.3]. By construction, we see that $F$ has a smooth anti-canonical member. In fact, $f\colon X\to Y\coloneqq V_{d}$ is a blowup along an elliptic curve which is a complete intersection of two members of $|-\frac{1}{2}K_{Y}|$ , and this blowup centre is isomorphic to some member of $|-K_{F}|$ .

Since $f_{*}\Omega^{2}_{X}(-D)$ is torsion-free, we get

f_{*}\Omega^{2}_{X}(-D)\hookrightarrow(f_{*}\Omega^{2}_{X}(-D))^{**}=\Omega^{2}_{Y}(-D_{Y}),

where $D_{Y}\coloneqq f_{*}D$ . Thus we have

H^{0}(X,\Omega^{2}_{X}(-D))\hookrightarrow H^{0}(Y,\Omega^{2}_{Y}(-D_{Y})).

Then it is enough to prove $H^{0}(Y,\Omega^{2}_{Y}(-D_{Y}))=0$ . By $\operatorname{Pic}(Y)=\mathbb{Z}$ , the divisor $D_{Y}$ is nef. Set $S\coloneqq f_{*}F\in|-\frac{1}{2}K_{Y}|$ . Since $F$ is general and the base locus of $|-\frac{1}{2}K_{Y}|$ is either empty or zero-dimensional, it suffices to show $H^{0}(S,\Omega_{Y}^{2}(-D_{Y})|_{S})=0$ , because it implies $H^{0}(Y,\Omega_{Y}^{2}(-D_{Y}))=0$ (indeed, if there is $0\neq s\in H^{0}(Y,\Omega_{Y}^{2}(-D_{Y}))$ , then we could find a member $S\not\subset\operatorname{Supp}(s)$ , which implies $0\neq s|_{F}\in H^{0}(S,\Omega_{Y}^{2}(-D_{Y})|_{S})$ ). Since $S$ is smooth along the blowup centre, we get $f|_{F}\colon F\xrightarrow{\simeq}S$ , and hence $S$ is a canonical del Pezzo surface. Let $S_{\mathrm{sm}}$ be the smooth locus of $S$ . By taking the exterior power $\wedge^{2}$ of the conormal exact sequence and applying the tensor product with $\mathcal{O}_{Y}(-D_{Y})$ , we get an exact sequence

0\to\Omega_{S_{\mathrm{sm}}}^{1}(-S-D_{Y})\to\Omega^{2}_{Y}(-D_{Y})|_{S_{\mathrm{sm}}}\to\Omega_{S_{\mathrm{sm}}}^{2}(-D_{Y})\to 0.

Applying the pushforward $i_{*}$ by the inclusion $i\colon S_{\mathrm{sm}}\hookrightarrow S$ (which is left exact), we obtain an exact sequence

0\to\Omega_{S}^{[1]}(-S-D_{Y})\to\Omega^{2}_{Y}(-D_{Y})|_{S}\to\Omega_{S}^{[2]}(-D_{Y})

By $H^{0}(S,\Omega_{S}^{[2]}(-D_{Y}))=H^{0}(S,\mathcal{O}_{S}(K_{S}-D_{Y})))=0$ , it is enough to prove $H^{0}(S,\Omega_{S}^{[1]}(-S-D_{Y}))=0$ .

Pick a general member $C$ of $|-K_{S}|$ , which is an elliptic curve. Since $S$ is smooth along $C$ , we have $\Omega_{S}^{[1]}(-S-D_{Y})|_{C}=\Omega_{S}^{1}|_{C}\otimes_{\mathcal{O}_{C}}(\mathcal{O}_{Y}(-S-D_{Y})|_{C})$ . Then it suffices to show

H^{0}(C,\Omega_{S}^{1}|_{C}\otimes_{\mathcal{O}_{C}}(\mathcal{O}_{Y}(-S-D_{Y})|_{C}))=0.

By the conormal exact sequence, we have an exact sequence

0\to(\mathcal{O}_{S}(-C)|_{C})\otimes(\mathcal{O}_{Y}(-S-D_{Y})|_{C})\to\Omega^{1}_{S}|_{C}\otimes(\mathcal{O}_{Y}(-S-D_{Y})|_{C})\\ \to\Omega_{C}^{1}\otimes(\mathcal{O}_{Y}(-S-D_{Y})|_{C})\to 0.

Then the required vanishing $H^{0}(C,\Omega^{1}_{S}|_{C}\otimes(\mathcal{O}_{Y}(-S-D_{Y})|_{C}))=0$ follows from

•

$H^{0}(C,(\mathcal{O}_{S}(-C)|_{C})\otimes(\mathcal{O}_{Y}(-S-D_{Y})|_{C}))=H^{0}(C,\mathcal{O}_{Y}(K_{Y}+S)|_{C}\otimes\mathcal{O}_{Y}(-S-D_{Y})|_{C})=H^{0}(C,\mathcal{O}_{Y}(K_{Y}-D_{Y})|_{C}))=0$ , and
•

$H^{0}(C,\Omega^{1}_{C}\otimes(\mathcal{O}_{Y}(-S-D_{Y})|_{C}))=H^{0}(C,\mathcal{O}_{Y}(-S-D_{Y})|_{C})=0$ .

∎

Proof of Theorem A.

Let us show (1). Assume $r_{X}>1$ . If $r_{X}\geq 3$ , then $X\simeq\mathbb{P}^{3}$ or $X$ is isomorphic to a smooth quadric threefold, and hence $X$ is $F$ -split. If $r_{X}=2$ , then $X$ is quasi- $F$ -split by [Kawakami-Tanaka(dPvar), Theorem A and Remark 2.8]. Therefore, we may assume that $\rho(X)\geq 2$ . In this case, the assertion holds by former parts as follows:

•

$\rho(X)\geq 6$ : Proposition 3.1.
•

$\rho(X)=5$ : Proposition 3.2.
•

$\rho(X)=4$ : Proposition 3.3.
•

$\rho(X)=3$ : Proposition 4.4and Lemma 6.20.
•

$\rho(X)=2$ : Proposition 5.12, Lemma 6.8 (2-2), Lemma 6.11, Lemma 6.14, and Lemma 6.18.

We now show (2), i.e., Kodaira vanishing for $X$ . When $\rho(X)\geq 2$ or $r_{X}\geq 2$ , this follows from (1) and Kodaira vanishing for quasi- $F$ -split varieties [KTTWYY1, Theorem 3.15]. When $\rho(X)=r_{X}=1$ , then Kodaira vanishing holds by [FanoI, Corollary 4.5]. ∎

Proof of Theorem B.

The assertion (2) follows from Theorem A(1) and [KTTWYY1, Corollary 7.6]. Let us show (1), i.e., $H^{j}(X,\Omega^{i}_{X}(-A))=0$ for $i+j<3$ . If $i=0$ (resp. $(i,j)=(1,0)$ , resp. $(i,j)=(2,0)$ ), then this follows from Theorem A(2) (resp. [Kaw1, Theorem 3.5], resp. Proposition 7.3). The remaining case when $(i,j)=(1,2)$ is settled by Lemma 7.1 and Lemma 7.2. ∎

Proof of Theorem C.

By Theorem B(1), we have $H^{2}(X,T_{X})\simeq H^{1}(X,\Omega_{X}^{1}(K_{X}))=0$ . Combining with $H^{2}(X,\mathcal{O}_{X})=0$ , we conclude from [FAG, Theorem 8.5.19] that $X$ lifts to $W(k)$ . ∎

Corollary 7.4.

Let $f\colon X\to Y$ be a blowup along a smooth curve, where $X$ and $Y$ are smooth Fano threefolds. Then $C\subset Y$ admits a $W(k)$ -lifting $\mathcal{C}\subset\mathcal{Y}$ , and the blowup $\mathrm{Bl}_{\mathcal{C}}(\mathcal{Y})\to\mathcal{Y}$ along $\mathcal{C}$ is a $W(k)$ -lifting of $f\colon X\to Y$ .

Proof.

By $\rho(X)\geq 2$ , Theorem B(1) implies

H^{2}(X,T_{X})\simeq H^{1}(X,\Omega^{1}_{X}(K_{X}))=0,

and thus $X$ formally lifts to $W(k)$ [FAG, Theorem 8.5.9(b)]. Moreover, we have $H^{1}(E,N_{E/X})=H^{1}(E,\mathcal{O}_{X}(E)|_{X})=H^{1}(E,\mathcal{O}_{E}(K_{E}-K_{X}|_{E}))=0$ by Kodaira vanishing [Muk13, Theorem 3]. Thus $E$ formally lifts to $W(k)$ as a closed subscheme of $X$ by [Har2, Theorem 6.2]. Since $Rf_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$ and $Rf_{*}\mathcal{O}_{E}=\mathcal{O}_{C}$ , it follows that $Y$ formally lifts to $W(k)$ and $C$ formally lifts to $W(k)$ as a closed subscheme by [AZ-nonlift, Proposition 2.3(1)]. Since $H^{2}(Y,\mathcal{O}_{Y})=0$ , they are algebraisable by [FAG, Corollary 8.5.6 and Corollary 8.4.5]. By [MR2791606, Theorem 2.5.8], for every closed point $y\in\mathcal{C}$ , there exists an affine open neighbourhood $\mathcal{U}\subset\mathcal{Y}$ of $y$ and an étale morphism $\tilde{\phi}\colon\mathcal{U}\to\operatorname{Spec}\,W(k)[t_{1},t_{2},t_{3}]$ such that $\mathcal{C}\cap\mathcal{U}=V(\tilde{\phi}^{*}t_{1},\tilde{\phi}^{*}t_{2})$ . Set $U\coloneqq\mathcal{U}\otimes_{W(k)}k$ and $\phi\coloneqq\tilde{\phi}\otimes_{W(k)}k\colon U\to\operatorname{Spec}\,k[t_{1},t_{2},t_{3}]$ . Then $f|_{U}=\mathrm{Bl}_{V(\phi^{*}t_{1},\phi^{*}t_{2})}(U)$ , and we conclude that $\mathrm{Bl}_{\mathcal{C}}(\mathcal{U})\to\mathcal{U}$ is a $W(k)$ -lifting of $f|_{U}$ . ∎

Corollary 7.5.

Let $X$ be a smooth Fano threefold and let $f\colon X\to Y$ be a morphism to a normal projective variety $Y$ such that $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$ and $f$ is not isomorphism. Then $f$ lifts to $W(k)$ .

Proof.

Since $f$ is not isomorphism, we have $\rho(X)\geq 2$ . Then the Fano threefold $X$ admits a $W(k)$ -lifting $\mathcal{X}$ . Let $L_{Y}$ be an ample invertible sheaf on $Y$ and set $L\coloneqq f^{*}L_{Y}$ . By $H^{2}(X,\mathcal{O}_{X})=0$ , there exists a $W(k)$ -lifting $\mathcal{L}\in\operatorname{Pic}(\mathcal{X})$ of $L$ . We take $m\in\mathbb{Z}_{>0}$ such that $f=\varphi_{|mL|}$ , where $\phi_{|mL|}$ is the contraction associated to the complete linear system $|mL|$ . By Kodaira vanishing (Theorem A), we have $H^{i}(X,\mathcal{O}_{X}(mL))=0$ . This, together with upper-semicontinuity theorem, implies $H^{i}(\mathcal{X},\mathcal{O}_{X}(m\mathcal{L}))=0$ . It follows from the Grauert theorem that

H^{0}(X,\mathcal{O}_{X}(mL))\simeq H^{0}(\mathcal{X},\mathcal{O}_{X}(m\mathcal{L}))\otimes_{W(k)}k.

Let $\mathcal{Z}$ be the base locus of $|m\mathcal{L}|$ . We show that $\mathcal{Z}=\emptyset$ . Since $\mathcal{X}$ is projective over $W(k)$ , the image of $\mathcal{Z}$ in $\operatorname{Spec}\,W(k)$ is closed. Since $|mL|$ is free, the image does not contain the closed point of $\operatorname{Spec}\,W(k)$ . Thus $\mathcal{Z}=\emptyset$ . Then we can see that the contraction $\varphi_{m\mathcal{L}}$ gives a lift of $f$ . ∎

Proof of Theorem D.

The assertion follows from Theorem 5.13 and Section 6.2. ∎

8. Examples

In this section, we gather examples of non- $F$ -split or non-quasi- $F$ -split smooth Fano threefolds.

Example 8.1 ( $X=S\times\mathbb{P}^{1}$ ).

Let $S$ be a smooth del Pezzo surface which is not $F$ -split. Then $X\coloneqq S\times\mathbb{P}^{1}$ is a Fano threefold which is not $F$ -split. Therefore, if the characteristic $p$ of the base field $k$ and $\rho$ satisfies one of (1)-(3) below, then there exists a non- $F$ -split smooth Fano threefold $X$ over $k$ satisfying $\rho(X)=\rho$ .

(1)

$p=2$ and $\rho=7$
(2)

$p\in\{2,3\}$ and $\rho=8$ .
(3)

$p\in\{2,3,5\}$ and $\rho=9$ .

Example 8.2 (Wild conic bundles).

Wild conic bundles are not $F$ -split. Indeed, if $X$ is $F$ -split and $f:X\to S$ is a conic bundle, then $f$ is generically reduced [GLP15, Lemma 2.4] and hence not wild. Therefore, if $p=2$ and $X$ is a smooth Fano threefold which is 2-24 or 3-10, then $X$ is not necessarily $F$ -split.

Example 8.3 ( $p=7$ , non- $F$ -split).

Assume $p=7$ . Then

X:=\{x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}=0\}\subset\mathbb{P}^{4}

is not $F$ -split. Indeed, we have

(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4})^{6}\not\in(x_{0}^{7},x_{1}^{7},x_{2}^{7},x_{3}^{7},x_{4}^{7}),

and hence Fedder’s criterion [Fed83, Proposition 2.1] implies that $X$ is not $F$ -split (cf. [KTY, Proposition A.8]).

Example 8.4 ( $p=11$ , non- $F$ -split).

Assume $p=11$ . Take

X:=\{x_{0}^{6}+x_{1}^{6}+x_{2}^{6}+x_{3}^{6}+y^{2}=0\}\subset\mathbb{P}(1,1,1,1,3),

i.e., $\mathbb{P}(1,1,1,1,3)=\mathrm{Proj}\,k[x_{0},x_{1},x_{2},x_{3},y]$ with $\deg x_{i}=1$ and $\deg y=3$ for every $0\leq i\leq 3$ , and

X:=\mathrm{Proj}\,\frac{k[x_{0},x_{1},x_{2},x_{3},y]}{(x_{0}^{6}+x_{1}^{6}+x_{2}^{6}+x_{3}^{6}+y^{2})}.

Let us prove that

(1)

$X$ is not $F$ -split, and
(2)

$X$ is a smooth Fano threefold.

The assertion (1) follows from Fedder’s criterion [Fed83, Proposition 2.1] (cf. [KTY, Proposition A.8]).

Let us show (2). It is enough to prove that $X$ is smooth, as the other assertions in (2) follow from the adjunction formula and the fact that $X$ is an ample $\mathbb{Q}$ -Cartier effective Weil divisor on $\mathbb{P}(1,1,1,1,3)$ (which implies the connectedness of $X$ ). Suppose that $[a_{0}:a_{1}:a_{2}:a_{3}:b]$ is a singular point of $X$ , where $a_{0},a_{1},a_{2},a_{3},b\in k$ . Recall that we have $[a_{0}:a_{1}:a_{2}:a_{3}:b]=[\lambda a_{0}:\lambda a_{1}:\lambda a_{2}:\lambda a_{3}:\lambda^{3}b]$ for every $\lambda\in k\setminus\{0\}$ . For $X_{1}:=x_{1}/x_{0},X_{2}:=x_{2}/x_{0},X_{3}:=x_{3}/x_{0},Y:=y/x_{0}^{3}$ , we have

D_{+}(x_{0})=\operatorname{Spec}k\left[X_{1},X_{2},X_{3},Y\right](\simeq\mathbb{A}^{4})\subset\mathbb{P}(1,1,1,1,3).

and

X\cap D_{+}(x_{0})=\{1+X_{1}^{6}+X_{2}^{6}+X_{3}^{6}+Y^{2}=0\},

which is smooth. Then it holds that $a_{0}=0$ . By symmetry, we get $a_{0}=a_{1}=a_{2}=a_{3}=0$ , which implies $[a_{0}:a_{1}:a_{2}:a_{3}:b]=[0:0:0:0:b]=[0:0:0:0:1]$ . However, $X$ does not pass through $[0:0:0:0:1]$ , which is absurd. Thus (2) holds.

Example 8.5 (No. 2-3, $p=3$ , non- $F$ -split).

Assume $p=3$ . We construct a Fano threefold $X$ which is 2-3 and not $F$ -split. Let $V_{2}$ be a Fano threefold of index $2$ such that $(-K_{V_{2}})^{3}=16$ and $V_{2}$ is not $F$ -split (e.g., $V_{2}:=\{x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+y^{2}=0\}$ in $\mathbb{P}(1,1,1,1,2)=\mathrm{Proj}\,k[x_{0},x_{1},x_{2},x_{3},y]$ for $\deg x_{0}=\deg x_{1}=\deg x_{2}=\deg x_{3}$ and $\deg y=2$ , cf. the proof of Example 8.4). Take a Cartier divisor $H$ on $V_{2}$ such that $2H\sim-K_{V_{2}}$ . Then $|H|$ is base point free and it induces a finite double cover. By a Bertini theorem [Spr98, Corollary 4.3], we may assume that $H$ is a smooth prime divisor on $V_{2}$ , which is a smooth del Pezzo surface with $K_{H}^{2}=2$ . Pick a general member $C$ of $|-K_{H}|$ , which is a smooth elliptic curve [KN, Theorem 1.4]. By an exact sequence

H^{0}(V_{2},\mathcal{O}_{V_{2}}(H))\to H^{0}(H,\mathcal{O}_{V_{2}}(H)|_{H})\to H^{1}(V_{2},\mathcal{O}_{V_{2}}(H-H))=H^{1}(V_{2},\mathcal{O}_{V_{2}})=0,

there exists a member $H^{\prime}\in|H|$ such that $H\cap H^{\prime}=H^{\prime}|_{H}=C$ . Let $\sigma:X\to V_{2}$ be the blowup along $C$ . Since $\sigma$ coincides with the resolution of the indeterminacies of the pencil generated by $H$ and $H^{\prime}$ , there is a contraction $\pi:X\to\mathbb{P}^{1}$ of type $D$ such that the proper transforms of $H$ and $H^{\prime}$ are fibres of $\pi$ . By Kleimann’s criterion, $X$ is a smooth Fano threefold, which is of No. 2-3.

Example 8.6 (No. 2-1, $p=5$ , non- $F$ -split).

Assume $p=5$ . We construct a Fano threefold $X$ which is 2-1 and $X$ is not $F$ -split. Let $V_{1}$ be a Fano threefold of index $2$ such that $(-K_{V_{1}})^{3}=8$ and $V_{1}$ is not $F$ -split. We can find such an example by setting

V_{1}:=\{x_{0}^{6}+x_{1}^{6}+x_{2}^{6}+y^{3}+z^{2}=0\}\subset\mathbb{P}(1,1,1,2,3)=\mathrm{Proj}\,k[x_{0},x_{1},x_{2},y,z],

where $\deg x_{0}=\deg x_{1}=\deg x_{2}=1,\deg y=2,\deg z=3$ (cf. [Oka21, Section 3.1]). Take a Cartier divisor $H$ on $V_{1}$ such that $2H\sim-K_{V_{1}}$ . Then we have a scheme-theoretic equality ${\rm Bs}\,|H|=P$ for some closed point $P$ on $X$ . We take generic members $H^{\operatorname{gen}}_{1}$ and $H^{\operatorname{gen}}_{2}$ of $|H|$ twice. Then $C=H^{\operatorname{gen}}_{1}\cap H^{\operatorname{gen}}_{2}$ is a regular curve of genus one. By $p=5>3$ , $C$ is a smooth elliptic curve [PW22, Corollary 1.8]. Therefore, the intersection $C:=H_{1}\cap H_{2}$ of two general members $H_{1}$ and $H_{2}$ of $|H|$ is a smooth elliptic curve. Take the blowup $X\to V_{1}$ along $C$ . Then we can apply the same argument as in Example 8.6.

Example 8.7 ( $p=5$ , non-quasi- $F$ -split).

Assume $p=5$ . Take

X:=\{x_{0}^{6}+x_{1}^{6}+x_{2}^{6}+x_{3}^{6}+y^{2}=0\}\subset\mathbb{P}(1,1,1,1,3),

i.e., $\mathbb{P}(1,1,1,1,3)=\mathrm{Proj}\,k[x_{0},x_{1},x_{2},x_{3},y]$ with $\deg x_{i}=1$ and $\deg y=3$ for every $0\leq i\leq 3$ , and

X:=\mathrm{Proj}\,\frac{k[x_{0},x_{1},x_{2},x_{3},y]}{(x_{0}^{6}+x_{1}^{6}+x_{2}^{6}+x_{3}^{6}+y^{2})}.

Then $X$ is a smooth Fano threefold by the same proof as in Example 8.4. Moreover, $X$ is not quasi- $F$ -split by [KTY, Corollary 4.19(i), Proposition A.8].

Vanishing theorems for Fano threefolds in positive characteristic

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Theorem A.

1.1. Akizuki-Nakano vanishing

Theorem B.

Theorem C.

1.2. FF-splitting

Theorem D.

Remark 1.1.

Remark 1.2.

1.3. Strategy of proofs

2. Preliminaries

2.1. Notation

2.2. Cartier operators

Lemma 2.1.

Proof.

Remark 2.2.

Remark 2.3.

2.3. Generic members

Remark 2.4.

Proof of (⋆)(\star).

2.4. FF-splitting criteria

Definition 2.5.

Remark 2.6.

Proposition 2.7.

Proof.

Proposition 2.8.

Proof.

Corollary 2.9.

Proof.

Remark 2.10.

Corollary 2.11.

Proof.

Lemma 2.12.

Proof.

Remark 2.13.

Example 2.14.

Proposition 2.15.

Proof.

2.5. Quasi-FF-splitting criteria

Definition 2.16.

Remark 2.17.

Corollary 2.18.

Proof.

Remark 2.19.

Proposition 2.20.

Proof.

2.6. Weak del Pezzo surfaces

Definition 2.21.

Theorem 2.22.

Proof.

3. Fano threefolds with ρ≥4\rho\geq 4

Proposition 3.1.

Proof.

Proposition 3.2.

Proof.

Proposition 3.3.

Proof.

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

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Claim.

Proof of Claim.

Proposition 4.4.

Proof.

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

5.1. Quasi-FF-splitting (imprimitive case)

5.1.1. D+ED+E (imprimitive)

Proposition 5.1.

Proof.

Claim.

Proof of Claim.

5.1.2. E+EE+E (imprimitive)

1.2. $F$ -splitting

Proof of $(\star)$ .

2.4. $F$ -splitting criteria

2.5. Quasi- $F$ -splitting criteria

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

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

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)

5.1.2. $E+E$ (imprimitive)

5.1.4. $C+E$ (imprimitive)

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

5.3. $F$ -splitting

6. $F$ -splitting and Quasi- $F$ -splitting via Cartier operators

6.1. Quasi- $F$ -splitting for 2-2, 2-6, 2-8, and 3-10