1 Introduction
We work over an algebraically closed field of characteristic
$p>0$
. Fano varieties play a significant role in the classification of algebraic varieties. In positive characteristic, properties defined by the Frobenius morphism such as (global) F-splitting or global F-regularity are useful. Therefore, it is natural to ask when Fano varieties are F-split or globally F-regular. For smooth del Pezzo surfaces, Hara [Reference HaraHar98a] proved the following result:
Theorem 1.1 [Reference HaraHar98a, Example 5.5].
Let X be a smooth del Pezzo surface over an algebraically closed field of characteristic
$p>0$
. Then X is globally F-regular except for the following cases:
-
(1)
$K_X^2=3$
and
$p=2$
. -
(2)
$K_X^2=2$
and
$p \in \{2, 3\}$
. -
(3)
$K_X^2=1$
and
$p \in \{2, 3, 5\}$
.
Our aim is to generalize Hara’s result to the case when
$-K_X$
is nef and big, or equivalently, X is canonical. Here, we say that a variety is canonical if it has only canonical singularities. In fact, the following theorem holds.
Theorem A. Let k be an algebraically closed field of characteristic
$p>0$
. Let X be a canonical projective surface over k whose anticanonical divisor is nef and big. Then X is globally F-regular except for the following cases:
-
(1)
$K_X^2=4$
and
$p=2$
. -
(2)
$K_X^2=3$
and
$p \in \{2, 3\}$
. -
(3)
$K_X^2=2$
and
$p \in \{2, 3\}$
. -
(4)
$K_X^2=1$
and
$p \in \{2, 3, 5\}$
.
Theorem A has an important role for investigating of global F-regularity of smooth Fano threefolds and smooth del Pezzo varieties (see [Reference Kawakami and TanakaKT24b, Reference Kawakami and TanakaKT24c] for details).
Remark 1.2. For each degree
$K_X^2$
, the assumption on p is optimal. In fact, for each case listed above, there exists a canonical del Pezzo surface that is not strongly F-regular (see [Reference Kawakami and NagaokaKN23, Table 1] and [Reference Kawakami and TakamatsuKT24a, Table 1]).
The assumption on p is still optimal even if we assume that X is smooth since taking the minimal resolution does not change the degree and globally F-regularity (Corollary 2.5). Similarly, even if we replace ‘nef and big’ by ‘ample’, the conclusion of Theorem A is the same. This is because X is globally F-regular if and only if its anticanonical model is globally F-regular (Proposition 2.4). In particular, Theorem A does not imply Theorem 1.1.
Remark 1.3. For each prime number p, there exists a Kawamata log terminal (klt) del Pezzo surface X (i.e., a normal projective surface such that
$(X, 0)$
is klt and
$-K_X$
is ample) that is not F-split [Reference Cascini, Tanaka H and WitaszekCTW18, Theorem 1.1].
We now focus on the proof of Theorem A. We first investigate when X as in Theorem A is F-split. The proof is divided into two cases: the case where
$K_X^2\geq 5$
and the case where
$K_X^2\leq 4$
.
First, we consider the case where
$K_X^2\geq 5$
. We may assume that X is obtained by taking a blowup
$f\colon X \to {\mathbb {P}}^2$
along some points. Recall that if there exists an effective divisor
$\Delta _{{\mathbb {P}}^2}$
on
${\mathbb {P}}^2$
such that the divisor
$\Delta $
on X defined by
$K_X+\Delta = f^*(K_{{\mathbb {P}}^2}+\Delta _{{\mathbb {P}}^2})$
is effective, then the following holds (Proposition 2.4):
$$\begin{align*}({\mathbb{P}}^2, \Delta_{{\mathbb{P}}^2}) \text{ is } F\text{-split} \Leftrightarrow (X, \Delta) \text{ is } F\text{-split} \Rightarrow X \text{ is } F\text{-split}. \end{align*}$$
Such a divisor
$\Delta _{{\mathbb {P}}^2}$
can be found by utilizing an inversion of adjunction for F-splitting (Proposition 2.6). However, since we can only assume that the blowup points are in almost general position, the situation is more complicated than the smooth del Pezzo cases, which are obtained by blowing up points in general position. For more details, see Proposition 3.8.
Next, we consider the case where
$K_X^2\leq 4$
. In the case of smooth del Pezzo surfaces, Hara [Reference HaraHar98a] investigated F-splitting using the following two steps:
-
(i) The reduction of F-splitting to the vanishing
$H^1(X, \Omega _X^{1}(pK_X))=0$
via the Cartier operator. -
(ii) Embedding X as a hypersurface or a complete intersection of a weighted projective space and proving the vanishing
$H^1(X, \Omega _X^{1}(pK_X))=0$
.
For a smooth weak del Pezzo surface X, we can also reduce the F-splitting of X to the vanishing
$H^1(X, \Omega _X^{1}(pK_X))=0$
. However, Step (ii), proving
$H^1(X, \Omega _X^{1}(pK_X))=0$
, is not easy because X is not embedded in a weighted projective space via
$|-mK_X|$
for
$m\in \mathbb {Z}_{>0}$
. Therefore, by replacing X with its anticanonical model, we embed X into a weighted projective space via
$|-mK_X|$
. However, in this case, Step (i), the reduction of the F-splitting of X to the vanishing
$H^1(X, \Omega _X^{1}(pK_X))=0$
, is not straightforward since Cartier operator is defined on smooth schemes. To address this issue, we use the reflexive Cartier operator introduced in [Reference KawakamiKaw22b]. Indeed, we utilize the fact that the reflexive Cartier operator behaves well on F-pure klt surfaces (Lemma 3.1).
Combining the above results, we obtain the following theorem.
Theorem B. Let k be an algebraically closed field of characteristic
$p>0$
. Let X be a canonical F-pure projective surface over k whose anticanonical divisor is ample. Then X is F-split except for the following cases:
-
(1)
$K_X^2=3$
and
$p=2$
. -
(2)
$K_X^2=2$
and
$p \in \{2, 3\}$
. -
(3)
$K_X^2=1$
and
$p \in \{2, 3, 5\}$
.
Remark 1.4.
-
(1) When
$K_X^2=4$
and
$p=2$
, there exists a a canonical del Pezzo surface with
$D^0_5$
-singularity, which is not F-pure [Reference Kawakami and NagaokaKN23, Proposition 3.16]. -
(2) When
$K_X^2=3$
and
$p=3$
, there exists a canonical del Pezzo surface with
$E^0_6$
-singularity, which is not F-pure [Reference Kawakami and NagaokaKN23, Proposition 3.22].
Finally, we now overview how to deduce Theorem A from Theorem B. Let X be a canonical weak del Pezzo surface. Replacing X with its anticanonical model, we can assume that
$-K_X$
is ample. For each degree
$K_X^2$
, we will find an optimal bound on p that ensures all the singularities of X are strongly F-regular (see Lemma 3.9). We then conclude by the following well-known fact that asserts the equivalence between F-splitting and global F-regularity for strongly F-regular
$\mathbb {Q}$
-Gorenstein Fano varieties:
Theorem 1.5 (cf. [Reference Kawakami and TotaroKT23, Proof of Theorem 6.2]).
Let X be a normal projective variety such that
$-K_X$
is an ample
$\mathbb {Q}$
-Cartier
$\mathbb {Z}$
-divisor. Suppose that X is strongly F-regular. If X is F-split, then X is globally F-regular.
2 Preliminaries
2.1 Notation and terminology
In this subsection, we summarize notation and basic definitions used in this article.
-
(1) Throughout the paper, p denotes a prime number and we work over an algebraically closed field k of characteristic
$p>0$
. We set . 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 k) if X is an integral scheme that is separated and of finite type over k. We say that X is a curve (resp. surface) if X is a variety of dimension one (resp. two).
-
(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 a
$\mathbb {Q}$
-divisor D on a normal variety X is simple normal crossing if for every point
${x \in \operatorname {\mathrm {Supp}} D}$
, 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
$\operatorname {\mathrm {Supp}} (D|_{\operatorname {\mathrm {Spec}} \mathcal {O}_{X, x}}) = \operatorname {\mathrm {Spec}} (\mathcal {O}_{X,x}/(x_1 \cdots x_r))$
. -
(5) Given a variety X, a projective birational morphism
$\pi \colon Y \to X$
is called a log resolution of X if Y is a smooth variety and
$\mathrm {Exc}(f)$
is a simple normal crossing divisor. -
(6) Given a variety X and a closed subscheme Z, we denote by
$\operatorname {\mathrm {Bl}}_Z X$
the blowup of X along Z. -
(7) Given a normal variety X and a
$\mathbb {Z}$
-divisor D on X, we define a reflexive sheaf
$\Omega _X^{[i]}(D)$
by
$j_{*}(\Omega _U^{i}\otimes \mathcal {O}_U(D))$
, where
$j\colon U\hookrightarrow X$
is the open immersion from the smooth locus U of X.
. We denote by 
2.1.1 Singularities in minimal model program
For the definitions of singularities in minimal model program (e.g., canonical and klt), we refer to [Reference Kollár and MoriKM98, Section 2.3]. Take a normal surface X. Let
$f\colon Y \to X$
be the minimal resolution. We only need the following characterizations in this paper.
-
(1) X is canonical if and only if
$K_X$
is
$\mathbb {Q}$
-Cartier and
$K_Y \sim _{\mathbb {Q}} f^*K_X$
. -
(2) X is klt if and only if
$K_X$
is
$\mathbb {Q}$
-Cartier, f is a log resolution of X and all the coefficients of the
$\mathbb {Q}$
-divisor
$\Gamma $
defined by
$K_Y+\Gamma \sim _{\mathbb {Q}} f^*K_X$
are
$<1$
.
By definition, we have
$$\begin{align*}\text{canonical}\Rightarrow \text{klt}. \end{align*}$$
Moreover, the following implications hold for the surface case:
-
• canonical
$\Rightarrow $
Gorenstein. -
• klt
$\Rightarrow \mathbb {Q}$
-factorial.
2.1.2 Weak del Pezzo surfaces
Given a normal projective Gorenstein surface X, we say that X is del Pezzo (resp. weak del Pezzo) if
$-K_X$
is ample (resp. nef and big). In what follows, we summarize some properties on (weak) del Pezzo surfaces for later usage.
Let Z be a canonical weak del Pezzo surface. The anticanonical model Y of Z is defined as the Stein factorisation of the morphism
$\varphi _{|-mK_Z|} \colon Z \to {\mathbb {P}}^{h^0(Z, -mK_Z)-1}$
induced by the complete linear system
$|-mK_Z|$
, where m is a positive integer such that
$|-mK_X|$
is base point free (whose existence is guaranteed by [Reference TanakaTan15, Theorem 0.4]). Then Y is canonical, because have
$K_Z \sim h^*K_Y$
for the induced morphism
$h \colon Z \to Y$
. Moreover, h is obtained by contracting all the
$(-2)$
-curves on Y. In particular, the minimal resolution X of Z coincides with the minimal resolution of Y:
$$\begin{align*}f \colon X \xrightarrow{g} Z \xrightarrow{h} Y. \end{align*}$$
Moreover, X is a smooth weak del Pezzo surface.
There is a natural one-to-one correspondence between
-
• smooth weak del Pezzo surfaces and
-
• canonical del Pezzo surfaces.
Indeed, if Y is a canonical del Pezzo surface, then its minimal resolution X is a smooth weak del Pezzo surface. Conversely, given a smooth weak del Pezzo surface X, its anticanonical model Y is a canonical del Pezzo surface.
2.2 F-splitting and global F–regularity
In this subsection, we gather basic facts on F-splitting and global F-regularity.
Definition 2.1. 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 splits as an
$$\begin{align*}\mathcal{O}_X \xrightarrow{F^e} F_*^e\mathcal{O}_X \hookrightarrow F_*^e\mathcal{O}_X( \lfloor(p^e-1)\Delta \rfloor) \end{align*}$$
$\mathcal {O}_X$
-module homomorphism for every
$e \in \mathbb {Z}_{>0}$
.
-
(2) We say that
$(X, \Delta )$
is sharply F-split if splits as an
$$\begin{align*}\mathcal{O}_X \xrightarrow{F^e} F_*^e\mathcal{O}_X \hookrightarrow F_*^e\mathcal{O}_X( \lceil(p^e-1)\Delta \rceil) \end{align*}$$
$\mathcal {O}_X$
-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 splits as an
$$\begin{align*}\mathcal{O}_X \xrightarrow{F^e} F_*^e\mathcal{O}_X \hookrightarrow F_*^e\mathcal{O}_X( \lceil(p^e-1)\Delta \rceil +E) \end{align*}$$
$\mathcal {O}_X$
-module homomorphism.
We say that X is F-split (resp. globally F-regular) if so is
$(X, 0)$
.
Remark 2.2. We have the following implications:
$$\begin{align*}\text{globally } F\text{-regular}\Longrightarrow \text{sharply } F\text{-split}\Longrightarrow F\text{-split} \end{align*}$$
where the former implication is easy and the latter one holds by the same argument as in [Reference SchwedeSch08, 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.
-
(⋆)
$(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. In this paper, we only treat the case when (
$\star $
) holds, and hence being F-split is equivalent to being sharply F-split. For more foundational properties, we refer to [Reference Schwede and SmithSS10].
We shall also use the local versions of F-splitting and global F-regularity.
Definition 2.3. Given a normal variety X, we say that X is F-pure (resp. strongly F-regular) if there exists an open cover
$X = \bigcup _{i \in I} X_i$
such that
$X_i$
is F-split (resp. globally F-regular) for every
$i \in I$
.
In what follows, we summarize some F-splitting criteria, which are well known to experts.
Proposition 2.4. Let
$f: X \to Y$
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 (resp. globally F-regular) if and only if
$(Y, \Delta _Y)$
is F-split (resp. globally F-regular).
Proof. If
$(X, \Delta )$
is F-split, then so is
$(Y, \Delta _Y)$
, which can be checked by taking the push-forward. As for the opposite implication, the same argument as in the first paragraph of the proof of [Reference Xu ChenyangHX15, Proposition 2.11] works.
Corollary 2.5. Let Y be a canonical projective surface, and let
$f: X \to Y$
be its minimal resolution. Then X is F-split (resp. globally F-regular) if and only if Y is F-split (resp. globally F-regular).
Proof. The assertion immediately follows from Proposition 2.4 by using
$K_X =f^*K_Y$
.
Proposition 2.6. Let X be a normal projective Gorenstein variety. Take a normal prime Cartier divisor S and an effective
$\mathbb {Q}$
-Cartier
$\mathbb {Q}$
-divisor B on X such that
$S \not \subset \operatorname {\mathrm {Supp}}\,B$
. Assume that
-
(1)
$(S, B|_S)$
is F-split, and -
(2) there is a positive integer
$e \in \mathbb {Z}_{>0}$
such that
$(p^e-1)(K_X+S+B)$
is Cartier and
$$\begin{align*}H^1(X, \mathcal{O}_X(-S-(p^e-1)(K_X+S+B)))=0. \end{align*}$$
Then
$(X, S+B)$
is F-split.
Proof. The same argument as in [Reference Cascini, Tanaka and WitaszekCTW17, Lemma 2.7] works.
Example 2.7. We now summarize some easy cases for later usage, although all of them are well known to experts,
-
(1) If
$P, Q \in {\mathbb {P}}^1$
are distinct points, then
$({\mathbb {P}}^1, P+Q)$
is F-split (Proposition 2.6). -
(2) Let
$L, L', L"$
be three lines on
${\mathbb {P}}^2$
such that
$L+L'+L"$
is simple normal crossing. Then
$({\mathbb {P}}^2, L+L'+L")$
is F-split by (1) and Proposition 2.6. -
(3) For each
$i \in \{1, 2\}$
, let
$F_i$
and
$F^{\prime }_i$
be distinct fibers of the i-th projection
$\mathrm {pr}_i \colon {\mathbb {P}}^1 \times {\mathbb {P}}^1 \to {\mathbb {P}}^1$
. Then
$({\mathbb {P}}^1 \times {\mathbb {P}}^1, F_1 +F^{\prime }_1 + F_2+F^{\prime }_2)$
is F-split by (1) and Proposition 2.6.
2.3 Reflexive Cartier operator
Throughout this subsection, we use the following convention unless stated otherwise.
Convention 2.8. Let X be a normal variety and D a
$\mathbb {Z}$
-divisor on X. Let U be the smooth locus of X and
$j\colon U\hookrightarrow X$
the inclusion. By abuse of notation,
$D|_U$
is denoted by D.
The Frobenius pushforward of the de Rham complex
$$\begin{align*}F_{*}\Omega^{\bullet}_U\colon F_{*}\mathcal{O}_U \xrightarrow{F_{*}d} F_{*}\Omega^1_U \xrightarrow{F_{*}d} \cdots \end{align*}$$
is a complex of
$\mathcal {O}_U$
-modules. Tensoring with
$\mathcal {O}_U(D)$
, we obtain a complex
$$\begin{align*}F_{*}\Omega^{\bullet}_U\colon F_{*}\mathcal{O}_U(pD) \xrightarrow{F_{*}d\otimes \mathcal{O}_U(D)} F_{*}\Omega^1_U(pD) \xrightarrow{F_{*}d\otimes \mathcal{O}_U(D)} \cdots. \end{align*}$$
We define coherent
$\mathcal {O}_U$
-modules
$B^{i}_U(pD)$
and
$Z^{i}_U(pD)$
by
for all
$i\geq 0$
. Then
$B^{i}_U(pD)$
and
$Z^{i}_U(pD)$
are locally free [Reference KawakamiKaw22b, Lemma 3.2]. When
$D=0$
, we simply denote
$B^{i}_U(pD)$
and
$Z^{i}_U(pD)$
by
$B^{i}_U$
and
$Z^{i}_U$
, respectively. Then
$B^{i}_U(pD)=B^{i}_U\otimes \mathcal {O}_U(D)$
and
$Z^{i}_U(pD)=Z^{i}_U\otimes \mathcal {O}_U(D)$
holds [Reference KawakamiKaw22b, Remark 3.3]. In particular, we note that
$B^{i}_U(pD)$
and
$Z^{i}_U(pD)$
do not mean
$B^{i}_U\otimes \mathcal {O}_U(pD)$
and
$Z^{i}_U\otimes \mathcal {O}_U(pD)$
.
By [Reference KawakamiKaw22b, Lemma 3.2], there exists an exact sequence
$$ \begin{align} 0 \to B^{i}_U(pD)\to Z^{i}_U(pD)\xrightarrow{C^{i}_{U}(D)} \Omega^{i}_U(D)\to 0, \end{align} $$
and the map
$C^{i}_{U}(D)$
coincides with
$C^{i}_{U}\otimes \mathcal {O}_U(D)$
, where
$C^{i}_{U}$
is the usual Cartier operator.
Definition 2.9. We define reflexive
$\mathcal {O}_X$
-modules
$B^{[i]}_X(pD)$
and
$Z^{[i]}_X(pD)$
by
for all
$i\geq 0$
. The i-th reflexive Cartier operator
$$\begin{align*}C^{[i]}_{X}(D)\colon Z^{[i]}_X(pD)\to \Omega^{[i]}_X(D) \end{align*}$$
associated to D is defined as
$j_{*}C^{i}_{U}(D)$
for all
$i\geq 0$
.
Lemma 2.10. There exist the following exact sequences:
$$ \begin{align} 0 \to Z^{[i]}_X(pD) \to F_{*}\Omega^{[i]}_X(pD) \xrightarrow{d'} B^{[i+1]}_X(pD), \end{align} $$
$$ \begin{align} 0 \to B^{[i]}_X(pD)\to Z^{[i]}_X(pD)\xrightarrow{C^{[i]}_{X}(D)} \Omega^{[i]}_X(D), \end{align} $$
for all
$i\geq 0$
. Moreover,
$d'|_U \colon F_{*}\Omega ^{[i]}_X(pD)|_U \to B^{[i+1]}_X(pD)|_U$
and
$C^{[i]}_{X}(D)|_U \colon Z^{[i]}_X(pD)|_U \to \Omega ^{[i]}_X(D)|_U$
are surjective, and the homomorphism
$C^{[i]}_{X}(D)|_{U}$
coincides with
$C^{i}_{U}\otimes \mathcal {O}_U(D)$
.
Proof. Taking
$B=0$
in [Reference KawakamiKaw22b, Lemma 3.5], we obtain the assertion.
Remark 2.11. Taking
$i=0$
in equation (2.10.1), we obtain an exact sequence
$$\begin{align*}0 \to \mathcal{O}_X(D) \to F_{*}\mathcal{O}_X(pD) \to B_X^{[1]}(pD), \end{align*}$$
and the first map is induced by the Frobenius homomorphism. In particular,
$$\begin{align*}B_X^{[1]}(pD)=j_{*}\mathrm{Coker}({F\colon \mathcal{O}_U(D) \to F_{*}\mathcal{O}_U(pD)})\end{align*}$$
holds.
3 Proofs of main theorems
3.1 Criterion of the F-splitting of klt surfaces
In this subsection, we provide a criterion for the F-splitting of klt surfaces (Proposition 3.2).
Lemma 3.1. Let X be an F-pure klt surface and D a
$\mathbb {Z}$
-divisor. Then the sequence
$$ \begin{align} 0 \to B^{[1]}_X(pD) \to Z^{[1]}_X(pD) \xrightarrow{C_X^{[1]}(D)} \Omega^{[1]}_X(D) \to 0 \end{align} $$
is exact.
Proof. It is enough to show that
$C_X^{[1]}(D)$
is surjective, as the other parts has been settled in Lemma 2.10. Since the assertion is local on X, we may assume that X is affine and has a unique singular point Q. If
$p\neq 5$
or the singularity Q is not rational double point (RDP) of type
$E_8^1$
, then X is F-liftable by [Reference Kawakami and TakamatsuKT24a, Theorem A]. Then the surjectivity of
$C^{[1]}(D)$
follows from [Reference KawakamiKaw22b, Lemma 3.8].
Suppose that
$p=5$
and the singularity Q is of type
$E_8^1$
. Then we may assume that
$D=0$
by [Reference LipmanLip69, Section 24] (see also [Reference Liedtke, Martin and MatsumotoLMM21, Table 2]). Then the desired surjectivity follows from [Reference KawakamiKaw22b, Proposition 4.4] and [Reference Kawakami and TakamatsuKT24a, Theorem B].
Proposition 3.2. Let X be an F-pure klt projective surface. Suppose that the following conditions hold:
-
(1)
$H^0(X, \Omega _X^{[1]}(K_X))=0$
. -
(2)
$H^1(X, \Omega _X^{[1]}(pK_X))=0$
. -
(3)
$H^0(X, \mathcal {O}_X((p+1)K_X))=0$
.
Then X is F-split.
Proof. Recall that X is F-split if and only if the evaluation map
$$\begin{align*}\mathrm{Hom}_{\mathcal{O}_X}(F_{*}\mathcal{O}_X, \mathcal{O}_X) \xrightarrow{F^*} \mathrm{Hom}_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{O}_X) (\cong H^0(X, \mathcal{O}_X)) \end{align*}$$
is surjective. Then, as in [Reference Brion and KumarBK05, 1.3.9 Remarks (ii)], the surjectivity is equivalent to the injectivity of the map
$$ \begin{align} F \colon H^2(X, \mathcal{O}_X(K_X))\to H^2(X, \mathcal{O}_X(pK_X)) \end{align} $$
induced by Frobenius by Serre duality. Let U be the smooth locus of X. Since U is F-pure, the exact sequence
$$\begin{align*}0\to \mathcal{O}_U \to F_{*}\mathcal{O}_U \to B_U^{1}\to 0 \end{align*}$$
splits locally by definition. Tensoring with
$\mathcal {O}_U(K_X)$
, we obtain a locally split exact sequence
$$\begin{align*}0\to \mathcal{O}_U(K_U) \to F_{*}\mathcal{O}_U(pK_U) \to B^{[1]}_U(pK_U)\to 0. \end{align*}$$
Since the above exact sequence splits locally, taking pushforward preserves exactness on the right. Thus, taking the pushforward by the inclusion
$U\hookrightarrow X$
, we obtain the following locally split exact sequence
$$\begin{align*}0\to \mathcal{O}_X(K_X) \to F_{*}\mathcal{O}_X(pK_X) \to B^{[1]}_X(pK_X)\to 0. \end{align*}$$
Therefore, for the injectivity of equation (3.2.1), it suffices to show that
$H^1(X, B^{[1]}_X(pK_X))=0$
. By Lemma 3.1 and the condition (1), it is enough to prove that
$H^1(X, Z_X^{[1]}(pK_X))=0$
. By equation (2.10.1), we have an exact sequence
$$\begin{align*}0 \to Z^{[1]}_X(pK_X) \to F_{*}\Omega^{[1]}_X(pK_X) \to B^{[2]}_X(pK_X). \end{align*}$$
Let 
. Then we obtain an exact sequence
$$\begin{align*}H^0(X,\mathcal{B}) \to H^1(X, Z_X^{[1]}(pK_X)) \to H^1(X, \Omega_X^{[1]}(pK_X))\overset{(2)}{=}0. \end{align*}$$
Since we have
$$\begin{align*}\mathcal{B}\subset B^{[2]}_X(pK_X)\subset F_{*}\Omega_X^{[2]}(pK_X)=H^0(X, \mathcal{O}_X((p+1)K_X))\overset{(3)}{=}0, \end{align*}$$
we conclude that
$H^1(X, Z_X^{[1]}(pK_X))=0$
.
The condition (3) of Proposition 3.2 is satisfied if
$-K_X$
is big. In what follows, we see when the condition (1) of Proposition 3.2 is satisfied.
Definition 3.3 (Log liftability).
Let X be a normal projective surface. We say that X is log liftable if there exists a log resolution
$f\colon Y \to X$
of X such that
$(Y,\mathrm {Exc}(f))$
lifts to the ring
$W(k)$
of Witt vectors. For the definition of liftability of a log smooth pair, we refer to [Reference KawakamiKaw22a, Definition 2.6].
Lemma 3.4. Let X be a normal projective F-pure surface such that
$-K_X$
is a nef and big
$\mathbb {Q}$
-Cartier
$\mathbb {Z}$
-divisor. Then X is log liftable if and only if
$H^0(X, \Omega _X^{[1]}(K_X))=0$
.
Proof. Since
$H^2(X, \mathcal {O}_X)\cong H^0(X, \mathcal {O}_X(K_X))=0$
, the ‘if’ direction is [Reference KawakamiKaw22c, Theorem 2.8]. We prove the ‘only if’ direction. Let
$f\colon Y\to X$
be a log resolution such that 
lifts to
$W(k)$
. Since
$f_{*}(\Omega ^1_Y(\log \,E)\otimes \mathcal {O}_Y(f^{*}K_X))=\Omega _X^{[1]}(K_X)$
by [Reference Kawakami and TakamatsuKT24a, Theorem B], we have
$H^0(X, \Omega _X^{[1]}(K_X))=H^0(Y,\Omega _Y^{1}(\log \,E)\otimes \mathcal {O}_Y(f^{*}K_X))$
. Then the vanishing follows from [Reference KawakamiKaw22a, Theorem 2.11].
3.2 Global F-splitting: Proof of Theorem B
In the following proposition, we investigate F-splitting of F-pure canonical del Pezzo surfaces. For the proof, we confirm when the condition (2) of Proposition 3.2 is satisfied.
Proposition 3.5. Let X be an F-pure canonical del Pezzo surface. Suppose that one of the following holds.
-
(1)
$K_X^2=1$
and
$p>5$
. -
(2)
$K_X^2=2$
and
$p>3$
. -
(3)
$K_X^2=3$
and
$p>2$
. -
(4)
$K_X^2=4$
.
Then X is F-split.
Proof. In each case, X is log liftable by [Reference Kawakami and NagaokaKN22, Theorem 1.7 (1)], and thus the condition (1) of Proposition 3.2 is satisfied. Thus, it suffices to confirm the condition (2) of Proposition 3.2, that is,
$H^1(X, \Omega _X^{[1]}(pK_X))=0$
. By Serre duality of Cohen–Macaulay sheaves [Reference Kollár and MoriKM98, Theorem 5.71], we have
$H^1(X, \Omega _X^{[1]}(-pK_X))\cong H^1(X, \Omega _X^{[1]}(pK_X))$
. Since X has only hypersurface singularities,
$\Omega ^1_X$
is torsion-free by [Reference LipmanLip65, Section 8 (1)], and the natural map
$\Omega _X^1\to \Omega _X^{[1]}$
is injective. Since
$\mathcal {O}_X(-pK_X)$
is Cartier, we have an exact sequence
$$\begin{align*}0 \to \Omega^{1}_X\otimes \mathcal{O}_X(-pK_X) \to \Omega^{[1]}_X(-pK_X) \to \mathcal{C} \to 0 \end{align*}$$
for some coherent sheaf
$\mathcal {C}$
satisfying
$\dim \operatorname {\mathrm {Supp}}(\mathcal {C})=0$
. Since
$H^1(X,\mathcal {C})=0$
, it suffices to show that
$$\begin{align*}H^{1}(X, \Omega^{1}_X\otimes \mathcal{O}_X(-pK_X))=0. \end{align*}$$
In what follows, we divide the proof into the cases according to (1)–(4) in the proposition.
The case (1): In this case, X is a hypersurface of 
of degree
$6$
[Reference Bernasconi and TanakaBT22, Theorem 2.15]. By [Reference MoriMor75, Theorem 1.7], the non-Gorenstein locus of P is
$\{[0:0:0:1], [0:0:1:0]\}$
, and this locus coincides with the singular locus of P (Remark 3.6). Thus, X is contained in the smooth locus of P since it is Gorenstein. We define invertible sheaves
$\mathcal {O}_X(n)$
by
$\mathcal {O}_P(n)|_X$
for all
$n\in \mathbb {Z}$
.
By adjunction, we have
$\omega _X=\mathcal {O}_X(-1)$
, and thus we aim to show that
$$\begin{align*}H^{1}(X, \Omega^1_X\otimes \mathcal{O}_X(p))=0.\end{align*}$$
By the conormal exact sequence, we have an exact sequence
$$\begin{align*}\mathcal{O}_X(-X+p)=\mathcal{O}_X(p-6)\to \Omega^1_{P}|_X\otimes \mathcal{O}_X(p) \to \Omega^1_X\otimes \mathcal{O}_X(p)\to 0. \end{align*}$$
Since
$\mathcal {O}_X(p-6)$
is torsion-free and the first map is injective outside the singular points of X, we obtain an exact sequence
$$\begin{align*}0\to \mathcal{O}_X(p-6)\to (\Omega^1_{P}\otimes \mathcal{O}_P(p))|_X \to \Omega^1_X\otimes \mathcal{O}_X(p)\to 0. \end{align*}$$
Since
$p\geq 7$
, we have
$H^2(X, \mathcal {O}_X(p-6))=0$
by Serre duality, and hence it suffices to show that
$H^1(X, (\Omega ^1_{P}\otimes \mathcal {O}_P(p))|_X)=0$
. We have an exact sequence
$$\begin{align*}\Omega^{[1]}_P(p-6)=\Omega^{[1]}_P(p)\otimes \mathcal{O}_P(-6) \to \Omega^{[1]}_P(p) \to \Omega^{[1]}_P(p)|_X\to 0. \end{align*}$$
Here, we obtain the first equality as follows:
$$\begin{align*}\Omega^{[1]}_P(p)\otimes \mathcal{O}_P(-6)=(\Omega^{[1]}_P\otimes \mathcal{O}_P(p))^{**}\otimes \mathcal{O}_P(-6)=(\Omega^{[1]}_P\otimes \mathcal{O}_P(p-6))^{**}=\Omega^{[1]}_P(p-6) \end{align*}$$
since
$\mathcal {O}_P(-6)$
is Cartier. In particular, the first term of the above exact sequence is torsion-free, and thus the first map is injective since it is injective outside the singular points of P.
Moreover, since X is contained in the smooth locus of P, it follows that
$\Omega ^{[1]}_P(p)|_X=(\Omega ^1_{P}\otimes \mathcal {O}_P(p))|_X$
. Thus, we obtain an exact sequence
$$\begin{align*}0\to \Omega^{[1]}_P(p-6) \to \Omega^{[1]}_P(p) \to (\Omega^1_{P}\otimes \mathcal{O}_P(p))|_X\to 0. \end{align*}$$
By Bott vanishing on P [Reference FujinoFuj07, Corollary 1.3], we have
$$\begin{align*}H^1(P, \Omega^{[1]}_P(p))=H^2(P, \Omega^{[1]}_P(p-6))=0\end{align*}$$
since
$p\geq 7$
. Therefore, we obtain
$H^1(X, (\Omega ^1_{P}\otimes \mathcal {O}_P(p))|_X)=0$
.
The case (2): In this case, X is a hypersurface of 
of degree
$4$
[Reference Bernasconi and TanakaBT22, Theorem 2.15]. By [Reference MoriMor75, Theorem 1.7], the non-Gorenstein locus of P is
$\{[0:0:0:1]\}$
, and this locus coincides with the singular locus of P (Remark 3.6). Thus, X is contained in the smooth locus of P since it is Gorenstein.
By adjunction, we have
$\omega _X=\mathcal {O}_X(-1)$
, and thus we aim to show that
$$\begin{align*}H^{1}(X, \Omega^1_X\otimes \mathcal{O}_X(p))=0.\end{align*}$$
As in the case (1), by the conormal exact sequence and the torsion-freeness of
$\mathcal {O}_X(p-4)$
, we have an exact sequence
$$\begin{align*}0\to \mathcal{O}_X(p-4)\to \Omega^1_{P}|_X\otimes \mathcal{O}_X(p) \to \Omega^1_X\otimes \mathcal{O}_X(p)\to 0. \end{align*}$$
Since
$p\geq 5$
, we have
$H^2(X, \mathcal {O}_X(p-4))=0$
, and it suffices to show that
$$\begin{align*}H^1(X, \Omega^1_{P}|_X\otimes \mathcal{O}_X(p))=0.\end{align*}$$
As in the case (1), we have an exact sequence
$$\begin{align*}0\to \Omega^{[1]}_P(p-4)\to \Omega^{[1]}_P(p) \to \Omega^1_{P}|_X\otimes \mathcal{O}_X(p)\to 0. \end{align*}$$
By Bott vanishing [Reference FujinoFuj07, Corollary 1.3], we have
$$\begin{align*}H^1(P, \Omega^{[1]}_P(p))=H^2(P, \Omega^{[1]}_P(p-4))=0\end{align*}$$
since
$p\geq 5$
. Therefore, we obtain
$H^1(X, \Omega ^1_{P}(p)|_X)=0$
.
The case (3): In this case, X is a hypersurface of 
of degree
$3$
[Reference Bernasconi and TanakaBT22, Theorem 2.15]. By adjunction, we have
$\omega _X=\mathcal {O}_X(-1)$
, and thus we aim to show that
$H^{1}(X, \Omega ^1_X(p))=0$
. By the conormal exact sequence and the torsion-freeness of
$\mathcal {O}_X(p-3)$
, we have an exact sequence
$$\begin{align*}0\to \mathcal{O}_X(p-3)\to \Omega^1_{P}(p)|_X \to \Omega^1_X(p)\to 0. \end{align*}$$
Since
$p\geq 3$
, we have
$H^2(X, \mathcal {O}_X(p-3))=0$
, and it suffices to show that
$H^1(X, \Omega ^1_{P}(p)|_X)=0$
. We have an exact sequence
$$\begin{align*}0 \to \Omega^{1}_P(p-3) \to \Omega^{1}_P(p) \to \Omega^{1}_P(p)|_X\to 0. \end{align*}$$
By Bott vanishing [Reference FujinoFuj07, Corollary 1.3], we have
$H^1(P, \Omega ^{1}_P(p))=0$
. By [Reference TotaroTot24, Proposition 1.3], we also have
$H^2(P, \Omega ^{1}_P(p-3))=0$
since
$p\geq 3$
. Therefore, we obtain
$H^1(X, \Omega ^1_{P}(p)|_X)=0$
.
The case (4): In this case, X is a complete intersection of two quadric hypersurfaces Q and
$Q'$
of 
[Reference Bernasconi and TanakaBT22, Theorem 2.15]. By adjunction, we have
$\omega _X=\mathcal {O}_X(-1)$
, and thus we aim to show that
$H^{1}(X, \Omega ^1_X(p))=0$
. By the conormal exact sequence and the torsion-freeness of
$\mathcal {O}_X(p-2)$
, we have an exact sequence
$$\begin{align*}0\to \mathcal{O}_X(p-2)\to (\Omega^1_{Q}\otimes \mathcal{O}_{Q}(p))|_X \to \Omega^1_X\otimes \mathcal{O}_X(p)\to 0. \end{align*}$$
Since
$p\geq 2$
, we have
$H^2(X, \mathcal {O}_X(p-2))=0$
, and hence it suffices to show that
$H^1(X, (\Omega ^1_{Q}\otimes \mathcal {O}_{Q}(p))|_X)=0$
.
We define invertible sheaves
$\mathcal {O}_Q(n)$
as
$\mathcal {O}_{P}(n)\otimes \mathcal {O}_{Q}$
for all
$n\in \mathbb {Z}$
. We have an exact sequence
$$\begin{align*}\Omega^{1}_Q\otimes \mathcal{O}_Q(p-2) \to \Omega^{1}_Q\otimes \mathcal{O}_Q(p) \to (\Omega^1_{Q}\otimes \mathcal{O}_{Q}(p))|_X \to 0. \end{align*}$$
Since X is regular in codimension one,
$\Omega ^1_Q$
is torsion-free by [Reference LipmanLip65, Section 8 (1)]. Thus, we have an exact sequence
$$\begin{align*}0\to \Omega^{1}_Q\otimes \mathcal{O}_Q(p-2) \to \Omega^{1}_Q\otimes \mathcal{O}_Q(p) \to (\Omega^1_{Q}\otimes \mathcal{O}_{Q}(p))|_X \to 0. \end{align*}$$
Therefore, it suffices to show that
$$\begin{align*}H^1(Q, \Omega^1_Q\otimes \mathcal{O}_Q(p))=0\,\,\,\text{and}\,\,\,H^2(Q, \Omega^1_Q\otimes \mathcal{O}_Q(p-2))=0,\end{align*}$$
and in particular, the following claim finishes the proof of the case (4):
Claim. We have
-
(i)
$H^1(Q, \Omega ^1_{Q}\otimes \mathcal {O}_Q(n))=0$
for every
$n\in \mathbb {Z} \setminus \{0\}$
and -
(ii)
$H^2(Q, \Omega ^1_{Q}\otimes \mathcal {O}_Q(n))=0$
for every
$n\in \mathbb {Z}$
.
We have
-
(a)
$H^i(P, \Omega ^1_{P}(n))=0$
for every
$n \in \mathbb {Z} \setminus \{0\}$
and
$i \in \{1, 2, 3\}$
, and -
(b)
$H^j(P, \Omega ^1_P(n))=0$
for every
$n \in \mathbb {Z}$
and
$j \in \{2, 3\}$
.
Indeed, (a) follows from Bott vanishing and Serre duality. Then (a), together with [Reference TotaroTot24, Proposition 1.3], implies (b). By the following exact sequence
$$\begin{align*}0\to \Omega^1_{P}(n-2)\to \Omega^1_{P}(n) \to \Omega^1_{P}(n)|_{Q} \to 0, \end{align*}$$
we get
-
(i)’
$H^1(Q, \Omega ^1_{P}(n)|_{Q})=0$
for every
$n\in \mathbb {Z}\setminus \{0\}$
, and -
(ii)’
$H^2(Q, \Omega ^1_{P}(n)|_{Q})=0$
for every
$n\in \mathbb {Z}$
.
By the conormal exact sequence and the torsion-freeness of
$\mathcal {O}_{P}(n-2)$
, we have an exact sequence
$$\begin{align*}0\to \mathcal{O}_{P}(n-2)\to \Omega^1_{P}(n)|_{Q} \to \Omega^1_Q\otimes \mathcal{O}_Q(n) \to 0 \end{align*}$$
for every
$n\in \mathbb {Z}$
. Since
$H^2(P, \mathcal {O}_P(n))=H^3(P, \mathcal {O}_P(n))=0$
for every
$n\in \mathbb {Z}$
, we have the claim.
Remark 3.6. Take positive integers
$q_1, q_2, q_3$
such that
$\mathrm {gcd}(q_1, q_2, q_3)=1$
. Set 
. Then it is well known (cf. [Reference FultonFul93, Section 2.2]) that P coincides with the projective
$\mathbb {Q}$
-factorial toric threefold associated to the fan in
$\mathbb {R}^3$
that is generated by four rays
$\mathbb {R} u, \mathbb {R} e_1, \mathbb {R} e_2, \mathbb {R} e_3$
, where
$e_1, e_2, e_3$
is the standard basis of
$\mathbb {Z}^3$
and
In the above proof, we have used the results (1) and (2).
-
(1)
${\mathbb {P}}(1, 1, 2, 3)$
has exactly two singular points, which corresponds to the cones
$\mathbb {R} u + \mathbb {R} e_1 + \mathbb {R} e_2$
and
$\mathbb {R} u + \mathbb {R} e_1 + \mathbb {R} e_3$
[Reference Cox, Little and SchenckCLS11, Theorem 1.3.12]. -
(2)
${\mathbb {P}}(1, 1, 1, 2)$
has a unique singular point, which corresponds to the cone
$\mathbb {R} u + \mathbb {R} e_1 + \mathbb {R} e_2$
[Reference Cox, Little and SchenckCLS11, Theorem 1.3.12].
From now on, we focus on the case where
$K_X^2 \geq 5$
.
Proposition 3.7. The following assertions hold.
-
(1) Fix an integer m satisfying
$1 \leq m \leq 5$
. Let
$P_1,\ldots , P_m$
be distinct points on
${\mathbb {P}}^2$
such that the blowup X of
${\mathbb {P}}^2$
along
$\{P_1, \ldots , P_m\}$
is a weak del Pezzo surface. Then X is F-split. -
(2) Fix an integer n satisfying
$1 \leq n \leq 4$
. Let
$Q_1,\ldots , Q_n$
be distinct points on
${\mathbb {P}}^1 \times {\mathbb {P}}^1$
such that the blowup X of
${\mathbb {P}}^1 \times {\mathbb {P}}^1$
along
$\{Q_1,\ldots , Q_n\}$
is a weak del Pezzo surface. Then X is F-split.
Proof. Let us show (1). In what follows, we only treat the case when
$m=5$
, as otherwise the problem is easier. Let L (resp.
$L'$
) be the line on
${\mathbb {P}}^2$
passing through
$P_1$
and
$P_2$
(resp.
$P_3$
and
$P_4$
). Since X is weak del Pezzo, we obtain
$L \neq L'$
. Pick a general line
$L"$
on
${\mathbb {P}}^2$
passing through
$P_5$
. Then
$L+L'+L"$
is simple normal crossing. Therefore,
$({\mathbb {P}}^2, L+L'+L")$
is F-split (Example 2.7(2)), which implies that so is X (Proposition 2.4). Thus, (1) holds. The proof of (2) is similar to that of (1). Indeed, for each projection
$\mathrm {pr}_i \colon {\mathbb {P}}^1 \times {\mathbb {P}}^1 \to {\mathbb {P}}^1$
, it is enough to take two fibers
$F_i$
and
$F^{\prime }_i$
such that
$F_1 \cup F^{\prime }_1 \cup F_2 \cup F^{\prime }_2$
contains
$\{Q_1,\ldots , Q_n\}$
(Example 2.7(3)).
Proposition 3.8. Let X be a smooth weak del Pezzo surface satisfying
$K_X^2 \geq 5$
. Then X is F-split.
Proof. By [Reference DolgachevDol12, Theorem 8.1.15], we may assume that there is a birational morphism
$f\colon X \to {\mathbb {P}}^2$
. In what follows, we only treat the case when
$K_X^2 = 5$
, as the other cases are simpler. There are the following five cases [Reference DolgachevDol12, Section 8.5].
-
(i)
$P, Q, R, S$
. -
(ii)
$P' \succ P, Q, R$
. -
(iii)
$P' \succ P, Q' \succ Q$
. -
(iv)
$P" \succ P' \succ P, Q$
. -
(v)
$P" \succ P' \succ P \succ Q$
.
For the definition of
$P' \succ P$
, we refer to [Reference DolgachevDol12, Section 7.3.2]. For example, in the case (iii), we have
$X =Y" \to Y' \to Y = {\mathbb {P}}^2$
, where
$Y' = \operatorname {\mathrm {Bl}}_{P \amalg Q} Y$
,
$Y" = \operatorname {\mathrm {Bl}}_{P' \amalg Q'}$
, and
$P'$
and
$Q'$
are points on
$Y'$
lying over P and Q, respectively.
The case (i) has been settled in Proposition 3.7, because we have
$X = \operatorname {\mathrm {Bl}}_{P \amalg Q \amalg R \amalg S} {\mathbb {P}}^2$
for distinct points
$P, Q, R, S \in {\mathbb {P}}^2$
in this case. In the case (ii),
$P, Q, R$
are distinct points on
${\mathbb {P}}^2$
, and we have
$X = \operatorname {\mathrm {Bl}}_{P'} Y'$
for
$Y' := \operatorname {\mathrm {Bl}}_{P \amalg Q \amalg R} {\mathbb {P}}^2$
and a closed point
$P'$
lying over P. Take the line 
passing through P and Q. Let
$L_2$
and
$L_3$
be general lines passing through P and R, respectively. Then
$({\mathbb {P}}^2, L_1+L_2+L_3)$
is F-split (Example 2.7(2)). Since
$\Delta $
is effective for the divisor
$\Delta $
defined by
$K_X +\Delta =f^*(K_{{\mathbb {P}}^2}+L_1+L_2+L_3)$
, it follows that X is F-split (Proposition 2.4). Similarly, (iii) is settled by taking the line 
and general lines
$L_2$
and
$L_3$
passing through P and Q, respectively.
Let us treat the case (iv). In this case, we have
$X = Y"' \to Y" \to Y' \to Y={\mathbb {P}}^2$
, where 
, 
, and
$P'$
(resp.
$P"$
) is lying over P (resp.
$P'$
). Let
$L_1$
be the line on
$Y={\mathbb {P}}^2$
such that
$P \in L_1$
and
$P' \in L^{\prime }_1$
for the proper transform
$L^{\prime }_1$
of
$L_1$
on
$Y'$
. Let
$L_2$
and
$L_3$
be general lines passing through P and Q, respectively. Then we can check that the divisor
$\Delta $
defined by
$K_X +\Delta =f^*(K_{{\mathbb {P}}^2}+L_1 +L_2 +L_3)$
is effective. Since
$({\mathbb {P}}^2, L_1 +L_2 +L_3)$
is F-split (Example 2.7(2)), so is X. This completes the proof for the case (iv).
Let us consider the case (v). In this case, we apply a similar method to that of (iv) after replacing
${\mathbb {P}}^2$
by
$\mathbb {F}_1$
. We have a sequence of one-point blowups:
$$\begin{align*}f\colon X = Y"' \to Y" \to Y' \to Y=\mathbb{F}_1 \to Z ={\mathbb{P}}^2, \end{align*}$$
where 
, 
, 
, and
$P, P', P"$
are lying over
$Q, P, P'$
, respectively. For the
$(-1)$
-curve C on Y, we have
$P\in C$
. It is well known that there is another section
$\widetilde C$
of the
${\mathbb {P}}^1$
-bundle
$\pi \colon Y=\mathbb {F}_1 \to B={\mathbb {P}}^1$
such that
$C \cap \widetilde {C} = \emptyset $
and
$\widetilde {C}^2 =1$
. Let F is a fiber of
$\pi $
. Since
$(K_Y+C+\widetilde {C})\cdot F=0$
, there exists
$n\in \mathbb {Z}$
such that
$K_Y+C+\widetilde {C}\sim nF$
. Then
$n=C\cdot nF =C\cdot (K_Y+C+\widetilde {C})=-2$
. Since the proper transform
$C'$
of C on
$Y'$
satisfies
$C^\prime 2=-2$
, we obtain
-
(1)
$P' \not \in C'$
,
as otherwise, the proper transform
$C"$
of
$C'$
on
$Y"$
would satisfy
$C^\prime \prime 2 =-3$
, which contradicts the fact that
$Y"$
is weak del Pezzo.
We now treat the case when
$P' \in F^{\prime }_P$
, where
$F^{\prime }_P$
denotes the proper transform of the fiber
$F_P$
of
$\pi \colon Y =\mathbb {F}_1 \to {\mathbb {P}}^1$
passing through P. Let
$\widetilde F$
be a general fiber of
$\pi $
. As we have seen above,
$K_Y+{\widetilde F}+ C+F_P +\widetilde C \sim 0$
. Since
$\widetilde F$
is nef and
$\mathbb {F}_1$
is toric, we obtain
$H^1(\mathbb {F}_1, \mathcal {O}_{F_1}(\widetilde F))=0$
by [Reference TotaroTot24, Proposition 1.3]. Moreover,
$(\widetilde F, (C+F_P +\widetilde C)|_{\widetilde F})=( \widetilde {F}, C|_{\widetilde F}+\widetilde C|_{\widetilde F})$
is F-split (Example 2.7 (1)). Thus,
$(Y, C+F_P +\widetilde C + \widetilde F)$
is F-split (Proposition 2.6), which implies that X is F-split (Proposition 2.4). In what follows, we assume that
-
(2)
$P' \not \in F^{\prime }_P$
for the proper transform
$F^{\prime }_P$
of the fiber
$F_P$
of
$\pi : Y =\mathbb {F}_1 \to {\mathbb {P}}^1$
passing through P.
Claim. There is a section D of
$\pi \colon Y=\mathbb {F}_1 \to {\mathbb {P}}^1$
such that
-
(a)
$D \sim \widetilde C+F$
, -
(b)
$P \in D$
, and -
(c)
$P' \in D'$
for the proper transform
$D'$
of D on
$Y'$
.
Proof of Claim.
Since
$\widetilde C+F$
is an ample Cartier divisor on
$Y=\mathbb {F}_1$
, it follows that
$|\widetilde C+F|$
is very ample [Reference HartshorneHar77, Ch. V, Corollary 2.18]. Then there is an effective Cartier divisor D on
$Y = \mathbb {F}_1$
satisfying (a)–(d).
-
(d) D is smooth at P.
In fact, since
$|\widetilde C+F|$
is very ample, the elements of
$H^0(Y, \mathcal {O}_Y(\widetilde C+F))$
separate tangent vectors. Let
$s_{P'}\in \mathfrak {m}_P/\mathfrak {m}^2_{P}$
is an element that corresponds to
$P'$
. We take D as a divisor of zeros of a global section
$s\in H^0(Y, \mathcal {O}_Y(\widetilde C+F))$
that maps to
$s_{P'}\in \mathfrak {m}_P/\mathfrak {m}^2_{P}$
. Then (a)–(c) are satisfied. Since
$s \in \mathfrak {m}_P/\mathfrak {m}^2_{P}$
is non-zero, the divisor D is smooth at P, i.e., (d) is satisfied. Since
$D \cdot F = (\widetilde C + F) \cdot F =1$
, we can write
$D= D_0 + F_1 + \cdots +F_r$
, where
$r \geq 0$
,
$D_0$
is a section of
$\pi \colon Y = \mathbb {F}_1 \to {\mathbb {P}}^1$
, and each
$F_i$
is a fiber of
$\pi $
. It suffices to prove
$r=0$
. Suppose
$r>0$
. The following holds:
$$ \begin{align} D_0 \cdot C +r = (D_0+F_1 + \cdots +F_r) \cdot C =D \cdot C = (\widetilde C +F) \cdot C =1. \end{align} $$
We now treat the case when
$D_0 \neq C$
. In this case,
$D_0 \cdot C \geq 0$
and (3.8.1) imply
$r=1$
and
$D_0 \cdot C =0$
. Hence, we get
$D_0 \cap C = \emptyset $
. Since
$P\in C$
, we have
$P\notin D_0$
. By (b), we obtain
$P\in D=D_0+F_1$
, and thus
$P\in F_1$
. Hence,
$D = D_0 +F_P$
, where
$F_P$
denotes the fiber passing through P. Since
$P'\notin C'$
, we obtain
$P'\in D'\setminus C'\subset F^{\prime }_P$
. This contradicts (2).
Hence, we may assume that
$D_0 =C$
. We then get
$D = C +F_1 + \cdots +F_r$
. Since
$P\in C$
, we obtain
$P \not \in F_1 \cup \cdots \cup F_r$
by (d). Then
$P' \not \in F^{\prime }_1 \cup \cdots \cup F^{\prime }_r$
, where
$F^{\prime }_1, F^{\prime }_2,\ldots , F^{\prime }_r$
are proper transforms of
$F_1, F_2,\ldots , F_r$
on
$Y'$
, and thus we obtain
$P'\in D'\setminus \{F^{\prime }_1 \cup \cdots \cup F^{\prime }_r\}\subset C'$
by (c). This contradicts (1). This completes the proof of the claim.
We have
$C \cdot D = 1$
. Hence,
$C \cap D = P$
and
$C+ D$
is a simple normal crossing divisor. Since both C and D are sections of
$\pi \colon Y = \mathbb {F}_1 \to {\mathbb {P}}^1$
, it follows that
$C+D+\widetilde F$
is still simple normal crossing for a general fiber
$\widetilde F$
of
$\pi $
. Then we see that
$(Y, C+D+\widetilde F)$
is F-split (Proposition 2.6, Example 2.7(1)). Since the divisor
$\Delta $
defined by
$K_X +\Delta = f^*(K_Y+C+D+\widetilde F)$
is effective, X is F-split (Proposition 2.4). This completes the proof of Proposition 3.8.
3.3 Global F-regularity: Proof of Theorem A
In this subsection, we deduce Theorem A from Theorem B.
Lemma 3.9. Let X be a canonical del Pezzo surface. Suppose that one of the following holds.
-
(1)
$p>5$
-
(2)
$K_X^2\geq 2$
and
$p>3$
. -
(3)
$K_X^2\geq 4$
and
$p>2$
. -
(4)
$K_X^2\geq 5$
.
Then X is strongly F-regular.
Remark 3.10. Combining [Reference Kawakami and NagaokaKN23, Table 1] and [Reference Kawakami and TakamatsuKT24a, Table 1], we can see that the assumption of p is optimal for each degree.
Proof. Strongly F-regular surface singularities are completely classified by Hara [Reference HaraHar98b, Theorem 1.1]. In what follows, we confirm the singularities on X satisfying one of (1)–(4) are all strongly F-regular.
(1) follows from [Reference HaraHar98b, Theorem 1.1]. In what follows, let
$f\colon Y\to X$
be the minimal resolution. Then Y is a smooth weak del Pezzo surface (Subsection 2.1.2), and Y is obtained by a blowup of
${\mathbb {P}}^2$
at some points [Reference DolgachevDol12, Theorem 8.1.15]. We have holds.
We prove (2). Since
$K_Y^2=K_X^2\geq 2$
, we have
$\rho (Y) = 10 -K_Y^2 \leq 8$
. Thus, the number of the
$(-2)$
-curves contracted by f is at most
$8-\rho (X)\leq 7$
. Therefore, X does not have canonical singularities of
$E_8$
-type. Then X is strongly F-regular by [Reference HaraHar98b, Theorem 1.1] since
$p>3$
(see also [Reference Kawakami and TakamatsuKT24a, Table 1]).
Next, we prove (3). Since
$K_Y^2=K_X^2\geq 4$
, we have
$\rho (Y) = 10 -K_Y^2\leq 6$
. Thus, the number of the
$(-2)$
-curves contracted by f is at most
$6-\rho (X)\leq 5$
. Therefore, X does not have canonical singularities of E-type. Then X is strongly F-regular by [Reference HaraHar98b, Theorem 1.1] since
$p>2$
(see also [Reference Kawakami and TakamatsuKT24a, Table 1]).
Finally, we prove (4). Since
$K_Y^2=K_X^2\geq 5$
, we have
$\rho (Y) = 10 -K_Y^2\leq 5$
. Thus, the number of the
$(-2)$
-curves contracted by f is at most
$5-\rho (X)\leq 4$
. If
$\rho (X)\geq 2$
, then X has only A-type singularities, which are strongly F-regular [Reference HaraHar98b, Theorem 1.1]. If
$\rho (X)=1$
, then X has only A-type singularities by [Reference Kawakami and NagaokaKN23, Theorem 1.1].
Acknowledgements
The authors express their gratitude to Burt Totaro and the anonymous referee for valuable comments.
Funding statement
Kawakami was supported by JSPS KAKENHI Grant number JP22KJ1771 and JP24K16897. Tanaka was supported by JSPS KAKENHI Grant number JP22H01112 and JP23K03028.
Competing interest
The authors have no competing interest to declare.
Open access
