Boundedness of regular del Pezzo surfaces over imperfect fields

Abstract

For a regular del Pezzo surface X, we prove that $|-12K_X|$ is very ample. Furthermore, we also give an explicit upper bound for the volume $K_X^2$ which depends only on $[k: k^p]$ for the base field k. As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.

1 Introduction

One of important classes of algebraic varieties is the one of Fano varieties. For example, classification of Fano varieties has been an interesting problem in algebraic geometry. Indeed, Fano varieties are classified in dimension at most three (cf. [24]). Although it seems to be difficult to obtain complete classification in higher dimension, it turns out that Fano varieties form bounded families when we fix the dimension [20]. Apart from the boundedness, Fano varieties satisfy various prominent properties, e.g. they are rationally connected [6, 20] and have no non-trivial torsion line bundles.

The main topic of this article is to study regular del Pezzo surfaces over imperfect fields. We naturally encounter such surfaces when we study minimal model program over algebraically closed fields of positive characteristic. The minimal model conjecture predicts that an arbitrary algebraic variety is birational to either a minimal model or a Mori fibre space $\pi :V \rightarrow B$. Although general fibres of $\pi$ might have bad singularities in positive characteristic (e.g. they are non-reduced if $\pi :V \rightarrow B$ is a wild conic bundle [25]), the generic fibre $X:=V \times _B {{\text {Spec}}}\,K(B)$ of $\pi$ allows only terminal singularities. Note that the base field K(B) of X is no longer a perfect field in general. Furthermore, if $\dim X=2$, then X is a regular del Pezzo surface over K(B).

The purpose of this article is to establish results related to boundedness of regular del Pezzo surfaces. The main results are the following two theorems.

Theorem 1.1

(Theorem 3.6) Let k be a field of characteristic $p>0$. Let X be a regular projective surface over k such that $-K_X$ is ample and $H^0(X, {\mathcal {O}}_X)=k$. Then the complete linear system $|-12K_X|$ is very ample over k, i.e. it induces a closed immersion to ${\mathbb {P}}^N_k$ for $N:=\dim _k H^0(X, {\mathcal {O}}_X(-12K_X))-1$.

Theorem 1.2

(Corollary 4.8, Theorem 4.9) Let k be a field of characteristic $p>0$. Let X be a regular projective surface over k such that $-K_X$ is ample and $H^0(X, {\mathcal {O}}_X)=k$. Then the following hold.

• (1) If $r:=\log _p[k:k^p]<\infty$, then $K_X^2 \le \max \{9, 2^{2r+1}\}$.

• (2) If X is geometrically reduced over k, then $K_X^2 \le 9$.

Remark 1.3

Let ${\mathbb {F}}$ be an algebraically closed field of characteristic $p>0$. Let $\pi :V \rightarrow B$ be a Mori fibre space between normal varieties over ${\mathbb {F}}$. Then V has at worst terminal singularities. Assume that $X:=V \times _B {{\text {Spec}}}\,K(B)$ is of dimension two. It holds that X is a regular projective surface over $k:=K(B)$ such that $-K_X$ is ample and $H^0(X, {\mathcal {O}}_X)=k$. In this case, it holds that

$$\begin{aligned} r = \log _p[k:k^p]=\log _p[K(B):K(B)^p]=\dim B. \end{aligned}$$

Hence, r is the dimension of the base of the Mori fibre space.

As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.

Theorem 1.4

(Theorem 5.5) There exists a flat projective morphism $\rho :V \rightarrow S$ of quasi-projective ${\mathbb {Z}}$-schemes which satisfies the following property: if k is a field and X is a regular projective surface over k such that $-K_X$ is ample, $H^0(X, {\mathcal {O}}_X)=k$, and X is geometrically reduced over k, then there exists a cartesian diagram of schemes:

where $\alpha$ denotes the induced morphism.

Remark 1.5

We fix a field k such that $[k:k^p]<\infty$. Then Theorem 1.1 and Theorem 1.2(1) show that if X is a regular projective surface over k such that $-K_X$ is ample and $H^0(X, {\mathcal {O}}_X)=k$, then $K_X^2$ is bounded and $|-12 K_X|$ is very ample. It is tempting to conclude the boundedness of these surfaces. However, we obtain the boundedness only for the geometrically reduced case as in Theorem 1.4. In our proof, we use the following two facts (cf. Proposition 5.3):

• (1) A Chow variety is a coarse moduli space (cf. [18, Ch. I, Sections 3, 4]), which does not have enough information on non-geometric points.

• (2) The proof of the inequality $\deg X \ge 1+ {\textrm{codim}}\,X$ for nondegenerate varieties $X \subset {\mathbb {P}}^N$ (cf. [11, Proposiiton 0]) works for varieties only over algebraically closed fields.

Theorem 1.4 immediately implies the following corollary. Indeed, Theorem 1.4 establishes the equivalence between the boundedness of $\dim _k H^1(X, {\mathcal {O}}_X)$ and the boundedness of geometrically intergal regular del Pezzo surfaces.

Corollary 1.6

(Corollary 5.6) There exists a positive integer h which satisfies the following property: if k is a field of characteristic $p>0$ and X is a regular projective surface over k such that $-K_X$ is ample, $H^0(X, {\mathcal {O}}_X)=k$, and X is geometrically reduced over k, then $\dim _k H^1(X, {\mathcal {O}}_X) \le h$.

The original motivation of the author was to establish results toward the Borisov–Alexeev–Borisov (BAB, for short) conjecture for threefolds over algebraically closed fields of positive characteristic. One of the steps of the proof of BAB conjecture in characteristic zero is to apply induction on dimension by using Mori fibre spaces [4, 5]. If we adopt a similar strategy for threefolds in positive characteristic, it is inevitable to treat three-dimensional del Pezzo fibrations. In characteristic zero, we may apply the induction hypothesis for general fibres, whilst we probably need to treat generic fibres in positive characteristic as replacements of general fibres. Thus, the author originally wanted to prove the boundedness of geometrically integral $\epsilon$-klt log del Pezzo surfaces. Although Theorem 1.4 is weaker than this goal, the author hopes that our results and techniques will be useful to establish such generalisation.

1.1 Description of proofs

1.1.1 Sketch of Theorem 1.1

Let k be a field of characteristic $p>0$. Let X be a regular projective surface over k such that $-K_X$ is ample and $H^0(X, {\mathcal {O}}_X)=k$. Let us overview how to find a constant $m>0$ such that $|-mK_X|$ is very ample. Combining known results, it is not difficult to show that $|-nK_X|$ is base point free for some constant $n>0$ (cf. the proof of Theorem 3.5). Then the problem is reduced to show the following theorem of Fujita type.

Theorem 1.7

(Theorem 3.3) Let k be a field of characteristic $p>0$. Let X be a d-dimensional regular projective variety over k. Let A be an ample invertible sheaf on X and let H be an ample globally generated invertible sheaf on X. Then $\omega _X \otimes _{{\mathcal {O}}_X} H^{d+1} \otimes _{{\mathcal {O}}_X} A$ is very ample over k.

Indeed, by applying this theorem for $A:={\mathcal {O}}_X(-K_X)$, $H:={\mathcal {O}}_X(-nK_X)$, and $m:=3n$, it holds that $|-mK_X|$ is very ample. We now give a sketch of the proof of Theorem 3.3. Note that Theorem 3.3 is known for the case when k is an algebraically closed field ([16, Theorem 1.1]). Thus, if k is a perfect field, then we are done by taking the base change to the algebraic closure. However, if k is an imperfect field, then the base change $X \times _k {{\overline{k}}}$ might be no longer regular. Hence, the problem is not directly reduced to the case when k is algebraically closed. On the other hand, our strategy is very similar to the one of [16] and we use also the base change $X \times _k {{\overline{k}}}$.

The outline is as follows. It is easy to reduce the problem to the case when k is an F-finite field, i.e. $[k:k^p]<\infty$. Fix $e \in {\mathbb {Z}}_{>0}$. Then, for the e-th iterated absolute Frobenius morphism

$$\begin{aligned} \Phi _e:X_e \rightarrow X, \qquad X_e:=X, \end{aligned}$$

the composite morphism $\beta :X_e \rightarrow X \xrightarrow {\alpha } {{\text {Spec}}}\,k$ is of finite type, where $\alpha :X \rightarrow {{\text {Spec}}}\,k$ denotes the structure morphism. We consider $X_e$ as a k-scheme via $\beta$. For the algebraic closure $\kappa := {{\overline{k}}}$ of k, consider the base change of $\Phi _e$ by $(-) \times _k \kappa$:

$$\begin{aligned} \Psi _e: Y_e \rightarrow Y, \quad Y:=X \times _k \kappa , \quad Y_e:=X_e \times _k \kappa . \end{aligned}$$

Since the trace map $(\Phi _e)_*\omega _{X_e} \rightarrow \omega _X$ of Frobenius is surjective, also the trace map $(\Psi _e)_*\omega _{Y_e} \rightarrow \omega _Y$ is surjective. Using Mumford’s regularity, we can show that $(\Psi _e)_*\omega _{Y_e} \otimes H'^{d+1} \otimes A' \otimes {\mathfrak {m}}_y$ is globally generated for any closed point y of Y and $e \gg 0$, where $H'$ and $A'$ are the pullbacks of H and A, respectively. Then $\omega _Y \otimes H'^{d+1} \otimes A' \otimes {\mathfrak {m}}_y$ is globally generated. Therefore, $\omega _Y \otimes H'^{d+1} \otimes A'$ is very ample, hence so is $\omega _X \otimes H^{d+1} \otimes A$. For more details, see Sect. 3.

1.1.2 Sketch of Theorem 1.2

Both (1) and (2) of Theorem 1.2 are consequences of the following theorem.

Theorem 1.8

(Corollary 4.8) Let k be a field of characteristic $p>0$. Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Then the following hold (for the definition of $\epsilon (X/k)$, see Definition 2.2).

• (1) If $p \ge 5$, then $K_X^2 \le 9$.

• (2) If $p =3$, then $K_X^2 \le \max \{9, 3^{\epsilon (X/k)+1}\}$.

• (3) If $p =2$, then $K_X^2 \le \max \{9, 2^{\epsilon (X/k)+3}\}$.

In particular, if X is geometrically reduced over k, then it is known that $\epsilon (X/k)=0$, hence we obtain $K_X^2 \le 9$.

Let us overview some of the ideas of the proof of Theorem 1.8. If X is geometrically normal, then the assertion follows from a combination of known results (cf. the proof of Theorem 4.7(1)). Hence, we only treat the case when X is not geometrically normal. In particular, we may assume that $p \le 3$ (cf. Theorem 2.4(1)).

For $Z:=(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N$, let $g:Z \rightarrow X$ be the induced morphism. Then there is an effective ${\mathbb {Z}}$-divisor D on Z which satisfies the following linear equivalence (cf. Theorem 2.5):

$$\begin{aligned} K_Z+D \sim g^*K_X. \end{aligned}$$

A key observation is that there are only finitely many possibilities for the pair (Z, D) (Theorem 4.6). Indeed, this is enough for our purpose by the following equation (cf. Lemma 4.5):

$$\begin{aligned} K_X^2 = p^{\epsilon (X/k)} (K_Z+D)^2. \end{aligned}$$

We now give a sketch of how to restrict the possibilities for Z. It is known that Z is either a Hirzebruch surface or a weighted projective plane ${\mathbb {P}}(1, 1, m)$ for some $m \in {\mathbb {Z}}_{>0}$ (Theorem 2.3). For the latter case: $Z={\mathbb {P}}(1, 1, m)$, it holds that $m \le 4$ because the ${\mathbb {Q}}$-Gorenstein index is known to be bounded (Theorem 2.7). Let us focus on the the case when $Z \simeq {\mathbb {P}}_{{\mathbb {P}}^1}({\mathcal {O}}\oplus {\mathcal {O}}(n))$ for some $n \ge 0$. The goal is to prove that $n \le 4$. Since $\rho (X) \le \rho (Z)=2$, we have either $\rho (X)=1$ or $\rho (X)=2$. If $\rho (X)=1$, then we can show that $n=0$ by using Galois symmetry (Lemma 4.4). Assume that $\rho (X)=2$. Then there are two extremal rays, both of which induce morphisms $X \rightarrow X'$ and $X \rightarrow X''$. Taking the base change to the algebraic closure, we obtain morphisms $Z \rightarrow Z'$ and $Z \rightarrow Z''$. The essential case is $\dim X'=2$. If $X'$ is not geometrically normal, then we may apply the above argument for $X'$, so that we deduce $n \le 4$. If $X'$ is geometrically normal, then $Z'$ is canonical (Theorem 2.3), hence we have $n \le 2$. For more details, see Sect. 4.

1.2 Related results

We first review results on del Pezzo surfaces over algebraically closed fields of characteristic $p>0$. It is a classical result that smooth del Pezzo surfaces are classified, and in particular bounded. Then, in [1], Alexeev proved the BAB conjecture for surfaces, i.e. $\epsilon$-klt log del Pezzo surfaces are bounded (cf. [15]). As for vanishing theorems, smooth del Pezzo surfaces over algebraically closed fields satisfy Kawamata–Viehweg vanishing [7, Proposition A.1]. However, if $p \in \{2, 3\}$, then there exist log del Pezzo surfaces violating Kawamata–Viehweg vanishing ([2, Theorem 1.1], [7, Lemma 2.4, Theorem 3.1], [8, Theorem 4.2]). On the other hand, if $p \gg 0$, it is known that Kawamata–Viehweg vanishing holds for any log del Pezzo surfaces [9, Theorem 1.2]. It is remarkable that this result is applied to show that three-dimensional klt singularities of large characteristic are rational singularities [13].

We now switch to the situation over imperfect fields. The first remarkable result is given by Schröer. He constructed weak del Pezzo surfaces X of characteristic two such that $H^1(X, {\mathcal {O}}_X) \ne 0$ [27, Theorem in Introduction]. Then Maddock discovered regular del Pezzo surfaces X of characteristic two with $H^1(X, {\mathcal {O}}_X) \ne 0$ [21, Main Theorem]. If we allow singularities, it is known that there exists log del Pezzo surfaces $(X, \Delta )$ of characteristic three such that $H^1(X, {\mathcal {O}}_X) \ne 0$ [31].

There are several results also in positive directions. Patakfalvi and Waldron proved that Gorenstein del Pezzo surfaces are geometrically normal when $p >3$ [26, Theorem 1.5]. Fanelli and Schröer showed that a regular del Pezzo surface X is geometrically normal if $\rho (X)=1$ and the base field k satisfies $[k:k^p] \le 1$ [12, Theorem 14.1]. Das proved that regular del Pezzo surfaces of characteristic $p\ge 5$ satisfy Kawamata–Viehweg vanishing [10, Theorem 4.1]. Bernasconi and the author proved that log del Pezzo surfaces $(X, \Delta )$ of characteristic $p \ge 7$ are geometrically integral and satisfy $H^1(X, {\mathcal {O}}_X)=0$ [3, Theorem 1.7].

2 Preliminaries

2.1 Notation

In this subsection, we summarise notation we will use in this paper.

• (1) We will freely use the notation and terminology in [14] and [19].

• (2) We say that a scheme X is regular if the local ring ${\mathcal {O}}_{X, x}$ at any point $x \in X$ is regular.

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

• (4) For an integral scheme X, we define the function field K(X) of X as ${\mathcal {O}}_{X, \xi }$ for the generic point $\xi$ of X.

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

• (6) For a variety X over a field k, its normalisation is denoted by $X^N$.

• (7) For a field k, we denote ${{\overline{k}}}$ an algebraic closure of k. If k is of characteristic $p>0$, then we set $k^{1/p^{\infty }}:=\bigcup _{e=0}^{\infty } k^{1/p^e} =\bigcup _{e=0}^{\infty } \{x \in {{\overline{k}}}\,|\, x^{p^e} \in k\}$.

• (8) For an ${\mathbb {F}}_p$-scheme X we denote by $F_X :X \rightarrow X$ the absolute Frobenius morphism. For a positive integer e we denote by $F^e_X :X \rightarrow X$ the e-th iterated absolute Frobenius morphism.

• (9) If $k \subset k'$ is a field extension and X is a k-scheme, we denote $X \times _{{{\text {Spec}}}\,k} {{\text {Spec}}}\,k'$ by $X \times _k k'$.

• (10) Let k be a field. A del Pezzo surface X over k is a projective normal surface over k such that $-K_X$ is an ample ${\mathbb {Q}}$-Cartier divisor.

• (11) Let k be a field and let X be a normal variety over k. We say that X is geometrically canonical if $X \times _k {{\overline{k}}}$ is a normal variety over ${{\overline{k}}}$ which is canonical, i.e. has at worst canonical singularities. Note that if X is geometrically canonical, then X itself is canonical [3, Proposition 2.3].

• (12) An ${\mathbb {F}}_p$-scheme X is F-finite if the absolute Frobenius morphism $F:X \rightarrow X$ is a finite morphism. We say that a field k of characteristic $p>0$ is F-finite if so is ${{\text {Spec}}}\,k$, i.e. $[k:k^p]<\infty$. Note that if k is an F-finite field and X is of finite type over k, then also X is F-finite.

• (13) Let X be a projective scheme over a field k and let F be a coherent sheaf on X. We say that F is globally generated if there exist a positive integer r and a surjective ${\mathcal {O}}_X$-module homomorphism

  $$\begin{aligned} {\mathcal {O}}_X^{\oplus r} \rightarrow F. \end{aligned}$$

  An invertible sheaf L on X is very ample over k if its complete linear system |L| induces a closed immersion $X \hookrightarrow {\mathbb {P}}^N_k$.

Definition 2.1

(Definition 5.1 of [32]) Let k be a field of characteristic $p>0$ and let X be a proper normal variety over k with $H^0(X, {\mathcal {O}}_X)=k$. Then we define the Frobenius length of geometric non-normality $\ell _F(X/k)$ of X/k by

$$\begin{aligned} \ell _F(X/k):=\min \{\ell \in {\mathbb {Z}}_{\ge 0}\,|\, (X \times _k k^{1/p^{\ell }})_{{{\text {red}}}}^N \text { is geometrically normal over }k^{1/p^{\ell }}\}, \end{aligned}$$

where the existence of the right hand side is guaranteed by [32, Remark 5.2].

Definition 2.2

(Definition 7.4 of [32]) Let k be a field of characteristic $p>0$ and let X be a proper normal variety over k with $H^0(X, {\mathcal {O}}_X)=k$. Set R to be the local ring of $X \times _k k^{1/p^{\infty }}$ at the generic point. We define the thickening exponent $\epsilon (X/k)$ of X/k by

$$\begin{aligned} \epsilon (X/k):=\log _p (\textrm{length}_R R). \end{aligned}$$

It follows from [32, Theorem 7.3(1)] that $\epsilon (X/k)$ is a non-negative integer.

2.2 Summary of known results

Theorem 2.3

Let k be a field of characteristic $p>0$. Let X be a canonical del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Set $Z:=(X \times _k {\overline{k}})_{{{\text {red}}}}^N$. Then one of the following properties.

• (1) X is geometrically canonical over k. In particular, $Z=X \times _k {\overline{k}}$ and Z is a canonical del Pezzo surface over ${{\overline{k}}}$.

• (2) X is not geometrically normal over k and $Z \simeq {\mathbb {P}}_{{\mathbb {P}}^1}({\mathcal {O}}\oplus {\mathcal {O}}(m))$ for some $m \in {\mathbb {Z}}_{\ge 0}$.

• (3) X is not geometrically normal over k and Z is isomorphic to a weighted projective surface ${\mathbb {P}}(1, 1, m)$ for some positive integer m.

Proof

See [3, Theorem 3.3]. $\square$

Theorem 2.4

Let k be a field of characteristic $p>0$. Let X be a canonical del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Then the following hold.

• (1) If $p \ge 5$, then X is geometrically canonical over k.

• (2) If $p=3$, then $\ell _F(X/k)\le 1$.

• (3) If $p=2$, then $\ell _F(X/k) \le 2$.

Proof

See [3, Theorem 3.7]. $\square$

Theorem 2.5

Let k be a field of characteristic $p>0$. Let X be a proper normal variety over k such that $H^0(X, {\mathcal {O}}_X)=k$. Assume that X is not geometrically normal over k. Set $Z:=(X \times _k {\overline{k}})_{{{\text {red}}}}^N$. Then there exist nonzero effective ${\mathbb {Z}}$-divisors $C_1,\ldots , C_{\ell (X/k)}$ such that

$$\begin{aligned} K_Z + (p-1) \sum _{i=1}^{\ell (X/k)} C_i \sim f^*K_X \end{aligned}$$

where $f:Z \rightarrow X$ denotes the induced morphism.

Proof

See [32, Proposition 5.11(2)]. $\square$

Theorem 2.6

Let k be a field of characteristic $p>0$. Let X be a canonical del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Then the following hold.

• (1) If $p=3$, then it holds that

  $$\begin{aligned} (X \times _k k^{1/3})_{{{\text {red}}}}^N \times _{k^{1/3}} {{\overline{k}}} \simeq (X \times _k {\overline{k}})_{{{\text {red}}}}^N. \end{aligned}$$

• (2) If $p=2$, then it holds that

  $$\begin{aligned} (X \times _k k^{1/4})_{{{\text {red}}}}^N \times _{k^{1/4}} {{\overline{k}}} \simeq (X \times _k {\overline{k}})_{{{\text {red}}}}^N. \end{aligned}$$

Proof

The assertion follows from Theorem 2.4 and [32, Remark 5.2]. $\square$

Theorem 2.7

Let k be a field of characteristic $p>0$. Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Set $Z:=(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N$. Then the following hold.

• (1) If $p=3$, $3K_Z$ is Cartier.

• (2) If $p=2$, then $4K_Z$ is Cartier.

Proof

The assertion follows from Theorem 2.4 and [32, Theorem 5.12]. $\square$

3 Very ampleness

The purpose of this section is to prove that if X is a regular del Pezzo surface, then $\omega _X^{-12}$ is very ample (Theorem 3.6). To this end, we first establish a general criterion (Theorem 3.3) for very ampleness in Sect. 3.1. In Sect. 3.2, we apply this criterion to regular del Pezzo surfaces.

3.1 A criterion for very ampleness

In this subsection, we give a criterion for very ampleness (Theorem 3.3). The strategy is a modification of Keeler’s proof for base point freeness over algebraically closed fields [16], which is in turn based on Smith’s argument [28]. We first recall the definition (Definition 3.1) and a property (Lemma 3.2) of Castelnuovo–Mumford regularity.

Definition 3.1

Let $\kappa$ be an algebraically closed field. Let Z be a projective scheme over $\kappa$. Let H be an ample globally generated invertible sheaf on Z. A coherent sheaf F on Z is 0-regular with respect to H if

$$\begin{aligned} H^i(Z, F \otimes _{{\mathcal {O}}_Z} H^{-i}) =0 \end{aligned}$$

for any $i>0$.

Lemma 3.2

Let $\kappa$ be an algebraically closed field. Let Z be a projective scheme over $\kappa$ and let z be a closed point on Z. Let F be a coherent sheaf on Z and let H be an ample globally generated invertible sheaf on Z. Assume that F is 0-regular with respect to H. Then $F \otimes H \otimes {\mathfrak {m}}_z$ is globally generated.

Proof

We may apply the same argument as in [33, Lemma 3.7]. $\square$

Theorem 3.3

Fix a non-negative integer d. Let k be a field of characteristic $p>0$. Let X be a d-dimensional regular projective variety over k. Let A be an ample invertible sheaf on X and let H be an ample globally generated invertible sheaf on X. Then $\omega _X \otimes _{{\mathcal {O}}_X} H^{d+1} \otimes _{{\mathcal {O}}_X} A$ is very ample over k.

Proof

We first reduce the problem to the case when k is an F-finite field (cf. Sect. 2.1(12)). There exist a subfield $k_0 \subset k$, a projective scheme $X_0$ over $k_0$, and invertible sheaves $A_0$ and $H_0$ such that $k_0$ is a field finitely generated over ${\mathbb {F}}_p$, $X_0 \otimes _{k_0} k$, $f^*A_0=A$, and $f^*H_0=H$. Then we can check that $k_0$ is F-finite and $(k_0, X_0, A_0, H_0)$ satisfies the assumptions in the statement. Replacing (k, X, A, H) by $(k_0, X_0, A_0, H_0)$, the problem is reduced to the case when k is F-finite. In particular, also X is F-finite (cf. Sect. 2.1(12)).

Fix $e \in {\mathbb {Z}}_{>0}$ and we denote the e-th iterated absolute Frobenius morphism $F^e:X \rightarrow X$ by $\Phi _e:X_e \rightarrow X$. Note that we consider $\Phi _e$ as a k-morphism, hence we distinguish X and $X_e$ as k-schemes, although the equation $X_e=X$ holds as schemes. Let $A_e:=A$ and $H_e:=H$ be the invertible sheaves on $X_e$. Note that we have $\Phi _e^*A=A_e^{p^e}$ and $\Phi _e^*H=H_e^{p^e}$.

For $\kappa :={{\overline{k}}}$, we take the base changes

hence both the above squares are cartesian. We set $A':=\alpha ^*A, H':=\alpha ^*H$, $A'_e:=\alpha _e^*A_e$, and $H'_e:= \alpha _e^*H_e$. Since $\Phi _e^*A=A_e^{p^e}$ and $\Phi _e^*H=H_e^{p^e}$, we have $\Psi _e^*A'=A_e'^{p^e}$ and $\Psi _e^*H'=H_e'^{p^e}$.

Claim 3.4

There exists a positive integer e such that the coherent sheaf $(\Psi _e)_*(\omega _{Y_e} \otimes H_e'^{p^ed} \otimes A_e'^{p^e})$ on Y is 0-regular with respect to $H'$, i.e. the equation

$$\begin{aligned} H^i(Y, (\Psi _e)_*(\omega _{Y_e} \otimes H_e'^{p^ed} \otimes A_e'^{p^e}) \otimes H'^{-i})=0 \end{aligned}$$

holds for any $i>0$.

Proof of Claim 3.4

We have

$$\begin{aligned}{} & {} H^i(Y, (\Psi _e)_*(\omega _{Y_e} \otimes H_e'^{p^ed} \otimes A_e'^{p^e}) \otimes H'^{-i})\\{} & {} \quad \simeq H^i(Y, (\Psi _e)_*(\omega _{Y_e} \otimes H_e'^{p^e(d-i)} \otimes A_e'^{p^e}))\\{} & {} \quad \simeq H^i(Y_e, \omega _{Y_e} \otimes H_e'^{p^e(d-i)} \otimes A_e'^{p^e}), \end{aligned}$$

where the first isomorphism follows from the projection formula and the second isomorphism holds because $\Psi _e$ is an affine morphism. By flat base change theorem, it holds that

$$\begin{aligned} H^i(Y_e, \omega _{Y_e} \otimes H_e'^{p^e(d-i)} \otimes A_e'^{p^e}) \simeq H^i(X_e, \omega _{X_e} \otimes H_e^{p^e(d-i)} \otimes A_e^{p^e}) \otimes _k \kappa . \end{aligned}$$

Recall that X and $X_e$ are isomorphic as schemes. Therefore, we have an isomorphism as abelian groups:

$$\begin{aligned} H^i(X_e, \omega _{X_e} \otimes H_e^{p^e(d-i)} \otimes A_e^{p^e}) \simeq H^i(X, \omega _X \otimes (H^{d-i} \otimes A)^{p^e}). \end{aligned}$$

It is enough to treat the case when $i \le \dim X = d$. Hence, $H^{d-i} \otimes A$ is ample. Then, by the Serre vanishing theorem, the right hand side is equal to zero for $e \gg 0$. This completes the proof of Claim 3.4. $\square$

Fix a closed point y on Y. Take a positive integer e as in Claim 3.4. Then $(\Psi _e)_*(\omega _{Y_e} \otimes H_e'^{p^ed} \otimes A_e'^{p^e})$ is 0-regular with respect to $H'$. Lemma 3.2 implies that the coherent sheaf

$$\begin{aligned} (\Psi _e)_*(\omega _{Y_e}) \otimes H'^{d+1} \otimes A' \otimes {\mathfrak {m}}_y = (\Psi _e)_*(\omega _{Y_e} \otimes H_e'^{p^ed} \otimes A_e'^{p^e}) \otimes H' \otimes {\mathfrak {m}}_y \end{aligned}$$

is globally generated.

Since ${\mathcal {O}}_{X, x} \rightarrow ((\Phi _e)_*{\mathcal {O}}_{X_e})_x$ splits for any point x on X [22, Theorem 107 in Section 42], we obtain a surjective ${\mathcal {O}}_X$-module homomorphism $(\Phi _e)_*(\omega _{X_e}) \rightarrow \omega _X$ by applying ${\mathcal {H}}om_{{\mathcal {O}}_X}(-, \omega _X)$ to ${\mathcal {O}}_X \rightarrow (\Phi _e)_*{\mathcal {O}}_{X_e}$. Taking the base change $(-) \times _k \kappa$, there exists a surjective ${\mathcal {O}}_Y$-module homomorphism $(\Psi _e)_*(\omega _{Y_e}) \rightarrow \omega _Y$, which induces another surjective ${\mathcal {O}}_Y$-module homomorphism

$$\begin{aligned} (\Psi _e)_*(\omega _{Y_e}) \otimes H'^{d+1} \otimes A' \otimes {\mathfrak {m}}_y \rightarrow \omega _Y \otimes H'^{d+1} \otimes A' \otimes {\mathfrak {m}}_y. \end{aligned}$$

Since $(\Psi _e)_*(\omega _{Y_e}) \otimes H'^{d+1} \otimes A' \otimes {\mathfrak {m}}_y$ is globally generated, also $\omega _Y \otimes H'^{d+1} \otimes A' \otimes {\mathfrak {m}}_y$ is globally generated. This implies that $\omega _Y \otimes H'^{d+1} \otimes A'$ is very ample over $\kappa$. Since very ampleness descends by base changes, $\omega _X \otimes H^{d+1} \otimes A$ is very ample over k.

In what follows, we prove that the very ampleness acutually descends. First of all, the base point freeness descends, because it is characterised by the surjectivity of $H^0(X, L) \otimes _k {\mathcal {O}}_X \rightarrow L$ for $L:= \omega _X \otimes H^{d+1} \otimes A$. We then get the morphism $\varphi : X \rightarrow {\mathbb {P}}^N_k$ induced by |L|, where $N:= \dim _k H^0(X, L)-1$. Since the base change $\varphi \times _k \kappa$ is a closed immersion, $\varphi$ is a finite morphism, because there exists no curve contracted by $\varphi$. Hence it is enough to show that, given a ring homomorphism $\psi : A \rightarrow B$ of k-algebra, $\psi$ is surjective if so is $\psi \otimes _k \kappa$. This follows from the fact that $k \hookrightarrow \kappa$ is faithfully flat. $\square$

3.2 Very ampleness for regular del Pezzo surfaces

In this subsection, we prove the main result (Theorem 3.6) of this section. We first focus on the case when X is not geometrically normal.

Theorem 3.5

Let k be a field of characteristic $p>0$. Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Let A be an ample invertible sheaf and let N be a nef invertible sheaf. Assume that X is not geometrically normal over k. Then the following hold.

• (1) If $p=2$, then $A^4$ is globally generated.

• (2) If $p=3$, then $A^3$ is globally generated.

• (3) If $p=2$, then $\omega _X^{-12} \otimes N$ is very ample over k.

• (4) If $p=3$, then $\omega _X^{-9} \otimes N$ is very ample over k.

Proof

If $p=2$, then we set $e:=2$ and $q:=p^e=4$. If $p=3$, then we set $e:=1$ and $q:=p^e=3$.

Let us prove that $A^q$ is globally generated. Set $A_{{{\overline{k}}}}$ to be the pullback of A to $X \times _k {{\overline{k}}}$. Since $e \ge \ell _F(X/k)$, the e-th iterated absolute Frobenius factors (Theorem 2.6):

$$\begin{aligned} F^e_{X \times _k {{\overline{k}}}}:X \times _k {{\overline{k}}} \xrightarrow {\psi } Z=(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N \xrightarrow {\varphi } X \times _k {{\overline{k}}}. \end{aligned}$$

Thus, $\varphi ^*(A_{{{\overline{k}}}})$ is an ample invertible sheaf on a projective toric surface Z (Theorem 2.3). Then $\varphi ^*(A_{{{\overline{k}}}})$ is globally generated, hence so is its pullback:

$$\begin{aligned} \psi ^*\varphi ^*(A_{{{\overline{k}}}}) =(F^e_{X \times _k {{\overline{k}}}})^*(A_{{{\overline{k}}}}) =A_{{{\overline{k}}}}^q. \end{aligned}$$

Hence, also $A_{{{\overline{k}}}}^q$ is globally generated. Thus, (1) and (2) hold.

Let us prove (3) and (4). By (1) and (2), $\omega _X^{-q}$ is globally generated. Then it follow from Theorem 3.3 that the invertible sheaf

$$\begin{aligned} \omega _X^{-3q} \otimes N = \omega _X \otimes (\omega _X^{-q})^{\dim X+1} \otimes (\omega _X^{-1} \otimes N) \end{aligned}$$

is very ample over k. Thus (3) and (4) hold. $\square$

Theorem 3.6

Let k be a field of characteristic $p>0$. Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Then $\omega _X^{-m}$ is very ample over k for any integer m such that $m \ge 12$.

Proof

If X is not geometrically normal over k, then the assertion follows from Theorem 3.5. Assume that X is geometrically normal over k. Then X is geometrically canonical over k (Theorem 2.3). In this case, $\omega _X^{-2}$ is globally generated by [3, Proposition 2.14(1)]. Hence, it follows from Theorem 3.3 that $\omega _X^{-m} = \omega _X \otimes (\omega _X^{-2})^3 \otimes \omega _X^{-(m-5)}$ is very ample for $m \ge 6$. $\square$

4 Boundedness of volumes

The purpose of this section is to show Theorem 4.9, which gives the inequality

$$\begin{aligned} K_X^2 \le \max \{9, 2^{2r+1}\} \end{aligned}$$

for a regular del Pezzo surface X over a field k of characteristic $p>0$ such that $H^0(X, {\mathcal {O}}_X)=k$ and $r:=\log _p [k:k^p]$. If X is geometrically normal, then the problem has been settled already (cf. the proof of Theorem 4.7(1)). Most of this subsection is devoted to analysis of the geometrically non-normal case. In Sect. 4.1, we first restrict possibilities for (Z, D), where $Z:=(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N$ and D is an effective divisor D on Z such that the linear equivalence

$$\begin{aligned} K_Z+D \sim g^*K_X \end{aligned}$$

holds for the induced morphism $g:Z \rightarrow X$. In Sect. 4.2, we prove that there are only finitely many possibilities for $K_X^2$ after we fix $\epsilon (X/k)$ (Theorem 4.6). We then obtain our main result (Theorem 4.9) by combining with fundamental properties on $\epsilon (X/k)$.

4.1 Restriction on possibilities

The purpose of this subsection is to prove the following proposition.

Proposition 4.1

Let k be a field of characteristic $p>0$. Let X be a canonical del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Set $Z:=(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N$ and let $g:Z \rightarrow X$ be the induced morphism. If D is a nonzero effective divisor on Z satisfying

$$\begin{aligned} K_Z + D \sim g^*K_X, \end{aligned}$$

(1)

then one of the following holds.

• (1) $Z \simeq {\mathbb {P}}^2$. In this case, it holds that

  • (a) ${\mathcal {O}}_Z(D) \simeq {\mathcal {O}}(1)$, or

  • (b) ${\mathcal {O}}_Z(D) \simeq {\mathcal {O}}(2)$

• (2) $Z \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1$. In this case, it holds that

  • (a) ${\mathcal {O}}_Z(D) \simeq {\mathcal {O}}(1, 1)$,

  • (b) ${\mathcal {O}}_Z(D) \simeq {\mathcal {O}}(1, 0)$, or

  • (c) ${\mathcal {O}}_Z(D) \simeq {\mathcal {O}}(0, 1)$.

• (3) $Z \simeq {\mathbb {P}}(1, 1, m)$ for some $m \ge 2$. In this case, $D \sim 2F$, where F is a prime divisor such that $F^2 =1/m$.

• (4) $Z \simeq {\mathbb {P}}( {\mathcal {O}}\oplus {\mathcal {O}}(m))$ for some $m \ge 1$. In this case, if $\pi :Z \rightarrow {\mathbb {P}}^1$ is the ${\mathbb {P}}^1$-bundle structure, F is a fibre of $\pi$, and C is a curve with $C^2=-m$, then

  • (a) $D \sim C$, or

  • (b) $D \sim C+F$.

Proof

Note that $-(K_Z+D)$ is ample. Hence, if $Z \simeq {\mathbb {P}}^2$ or $Z \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1$, then (1) or (2) holds. We assume that Z is isomorphic to neither ${\mathbb {P}}^2$ nor ${\mathbb {P}}^1 \times {\mathbb {P}}^1$. Then it follows from Theorem 2.3 that there is $m \ge 1$ such that either

• (i) $Z \simeq {\mathbb {P}}_{{\mathbb {P}}^1}({\mathcal {O}}\oplus {\mathcal {O}}(m))$, or

• (ii) $Z \simeq {\mathbb {P}}(1, 1, m)$ and $m \ge 2$.

Then, for the minimal resolution $\mu :W \rightarrow Z$, it holds that $W \simeq {\mathbb {P}}_{{\mathbb {P}}^1}({\mathcal {O}}\oplus {\mathcal {O}}(m))$. We have the induced morphisms:

$$\begin{aligned} h:W \xrightarrow {\mu } Z \xrightarrow {g} X \end{aligned}$$

Let $\pi :W \rightarrow {\mathbb {P}}^1$ be the ${\mathbb {P}}^1$-bundle structure. Let $F_W$ be a fibre of $\pi$ and let C be the curve on W such that $C^2=-m$. For $D_W:=\mu _*^{-1}D$, we obtain

$$\begin{aligned} K_W+D_W+c C \sim \mu ^*(K_Z+D) \sim h^*K_X \end{aligned}$$

(2)

for some $c \in {\mathbb {Z}}_{\ge 0}$. We have

$$\begin{aligned} -K_W \sim 2C+(m+2)F_W \end{aligned}$$

and $D_W \sim aC+bF_W$ for some $a, b \in {\mathbb {Z}}_{ \ge 0}$ with $(a, b) \ne (0, 0)$. Thus it holds that

$$\begin{aligned} -h^*K_X \sim -K_W-D_W-c C \sim (2-a-c)C + (m+2-b)F_W. \end{aligned}$$

(3)

We first show that $a+c=1$. Since $-h^*K_X$ is big, we obtain $(-h^*K_X) \cdot F_W>0$, hence it holds that $2-a-c \ge 1$. Then we have $1 \le 2-a-c \le 2$. Thus, it is enough to prove that $a+c \ne 0$. Assuming $a=c=0$, let us derive a contradiction. We have

$$\begin{aligned} -h^*K_X \cdot C= (2C + (m+2-b)F_W) \cdot C =-m+2-b \end{aligned}$$

If (i) holds, then $-h^*K_X$ is ample, hence we obtain $0 < -h^*K_X \cdot C =-m+2-b \le 1-b$, which in turn implies $b=0$. If (ii) holds, then it holds that $0 \le -h^*K_X \cdot C =-m+2-b \le -b$. In any case, we have $b=0$, which contradicts $(a, b) \ne (0, 0)$. Therefore, we obtain $a+c=1$. In particular, (3) implies that

$$\begin{aligned} -h^*K_X \sim C + (m+2-b)F_W. \end{aligned}$$

(4)

We treat the following two cases separately:

$$\begin{aligned} (a, c)=(0, 1) \quad \text {or}\quad (a, c)=(1, 0). \end{aligned}$$

Let us handle the case when $(a, c)=(0, 1)$. By $c \ne 0$, $\mu$ is not an isomorphism, hence we obtain $Z \simeq {\mathbb {P}}(1, 1, m)$. Since $h^*K_X \cdot C=\mu ^*(K_Z+D)\cdot C=0$, (4) implies $b=2$. Thus, we conclude $(a, b, c)=(0, 2, 1)$. This implies that (3) holds.

Then we may assume that $(a, c)=(1, 0)$. Assume (i). Then $-h^*K_X$ is ample. By (4), we have

$$\begin{aligned} 0< -h^*K_X \cdot C= (C + (m+2-b)F_W) \cdot C=-m+(m+2-b)=2-b. \end{aligned}$$

Therefore, we obtain $b \in \{0, 1\}$. Thus, (4) holds. Assume (ii). Since $c=0$ and c is defined by (2), we have $m=2$. Again by (2), we obtain

$$\begin{aligned} 0= h^*K_X \cdot C = (K_W+D_W+c C) \cdot C = D_W \cdot C = (aC+bF_W) \cdot C = -2+b. \end{aligned}$$

Thus, it holds that $(a, b, c)=(1, 2, 0)$. Thus, (3) holds. $\square$

Remark 4.2

We use notation as in Proposition 4.1. Note that $(g^*K_X)^2=(h^*K_X)^2$. By direct computation using (4), the following hold.

• (1) If $Z \simeq {\mathbb {P}}^2$, then $(g^*K_X)^2 \in \{1, 4\}$.

• (2) If $Z \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1$, then $(g^*K_X)^2 \in \{2, 4\}$.

• (3) If $Z \simeq {\mathbb {P}}(1, 1, m)$ for some $m \ge 2$, then $(g^*K_X)^2=m$.

• (4) If $Z \simeq {\mathbb {P}}_{{\mathbb {P}}^1}( {\mathcal {O}}\oplus {\mathcal {O}}(m))$ for some $m \ge 1$, then $(g^*K_X)^2 \in \{m+2, m+4\}$.

Remark 4.3

We use notation as in Proposition 4.1. If $p=3$, then we can find a nonzero effective divisor $D'$ such that $K_Z+2D' \sim g^*K_X$. In this case, (2) and (4) in Proposition 4.1 does not occur.

4.2 Classification of base changes

In this Sect. 4.9, we prove the main result of this section (Theorem 4.9), which asserts the inequality

$$\begin{aligned} K_X^2 \le \max \{9, 2^{2r+1}\} \end{aligned}$$

for a regular del Pezzo surface X over a field k of characteristic $p>0$ such that $H^0(X, {\mathcal {O}}_X)=k$ and $r:=\log _p [k:k^p]$. This result is a consequence of the boundedness of $K_X^2$ in terms of $\epsilon (X/k)$ (Theorem 4.6). To this end, we prove a kind of classification after the base change to the algebraic closure (Theorem 4.6). We first establish auxiliary results: Lemma 4.4 and Lemma 4.5.

Lemma 4.4

Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Set $Z:=(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N$. Assume that $\rho (X)=1$ and X is not geometrically normal. Then it holds that $Z \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1$ or $Z \simeq {\mathbb {P}}(1, 1, m)$ for some $m \ge 1$.

Proof

Assume that Z is not isomorphic to ${\mathbb {P}}(1, 1, m)$, then it follows from Theorem 2.3 that $Z \simeq {\mathbb {P}}_{{\mathbb {P}}^1}( {\mathcal {O}}\oplus {\mathcal {O}}(m))$ for some $m \ge 0$. Suppose $m>0$ and let us derive a contradiction. Set $\kappa := k^{1/p^{\infty }}$ and $Y:=(X \times _k \kappa )_{{{\text {red}}}}^N$. Then we have $Y \times _{\kappa } {{\overline{k}}} \simeq Z$. Hence, Y is smooth over $\kappa$. We have $\rho (Y)=1$ [30, Proposition 2.4(3)].

Let $\pi :Z \rightarrow B$ be the ${\mathbb {P}}^1$-bundle structure. There is a finite Galois extension $\kappa '/\kappa$ such that $\pi$ descends to $\kappa '$, i.e. there exists a $\kappa '$-morphism $\pi ':Z' \rightarrow B'$ of smooth $\kappa '$-varieties whose base change by $(-) \times _{\kappa '} {{\overline{k}}}$ is $\pi :Z \rightarrow B$. Let $F'$ be a fibre of $\pi '$ over a closed point. For the Galois group G of $\kappa '/\kappa$ and any element $\sigma \in G$, we have that $\sigma ^*(F')^2=F'^2=0$. If $\sigma ^*(F')$ is not a fibre of $\pi '$, then $\sigma ^*(F)$ induces another fibration which deduces that $Z \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1$. Hence $\sigma ^*(F)$ is a fibre of $\pi '$. Then ${\widetilde{F}}:=\sum _{\sigma \in G} \sigma ^*(F)$ satisfies ${{\widetilde{F}}}^2=0$. As ${{\widetilde{F}}}$ descends to Y, there exists an effective divisor D on Y such that $D^2=0$. However, this contradicts $\rho (Y)=1$. $\square$

Lemma 4.5

Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Set $Z:=(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N$ and let $g:Z = ((X \times _k {{\overline{k}}})_{{{\text {red}}}}^N \rightarrow X$ be the induced morphism. Then it holds that $p^{\epsilon (X/k)}(g^*K_X)^2=K_X^2$.

Proof

The assertion follows from Definition 2.2 and [17, Example 1 in page 299]. $\square$

Theorem 4.6

Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Set $Z:=(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N$ and let $g:Z \rightarrow X$ be the induced morphism. Assume that X is not geometrically normal over k. Then there exists a nonzero effective ${\mathbb {Z}}$-divisor E on Z such that

$$\begin{aligned} K_Z + (p-1)E \sim g^*K_X. \end{aligned}$$

(5)

Furthermore, if E is a nonzero effective divisor E on Z satisfying (5), then the following hold.

• (1) It holds that $p=2$ or $p=3$.

• (2) If $p=3$, then the quadruple $(Z, E, (g^*K_X)^2, K_X^2)$ satisfies one of the possibilities in the following Table 1.

  Table 1 $p=3$ case

• (3) If $p=2$, then the quadruple $(Z, D, (g^*K_X)^2, K_X^2)$ satisfies one of the possibilities in the following Table 2.

  Table 2 $p=2$ case

Here, if we write an invertible sheaf in the list, then it means that ${\mathcal {O}}_Z(E)$ is isomorphic to it. If we write a divisor, then it means that E is linearly equivalent to it. On ${\mathbb {P}}(1, 1, m)$ with $m \ge 2$, F denotes a prime divisor such that $F^2=1/m$. On ${\mathbb {P}}_{{\mathbb {P}}^1}({\mathcal {O}}\oplus {\mathcal {O}}(m))$ with $m \ge 1$, C is the curve such that $C^2=-m$ and F denotes a fibre of the ${\mathbb {P}}^1$-bundle structure ${\mathbb {P}}_{{\mathbb {P}}^1}({\mathcal {O}}\oplus {\mathcal {O}}(m)) \rightarrow {\mathbb {P}}^1$.

Proof

The existence of E follows from Theorem 2.5. The assertion (1) holds by [26, Theorem 1.5]. We omit the proof of (2), as it is similar and easier than the one of (3).

Let us show (3). Pick a nonzero effective divisor E on Z satisfying (5). If (Z, E) is one of the possibilities in the table, then $(g^*K_X)^2$ and $K_X^2$ automatically determined. Thus, it is enough to show that the pair (Z, E) satisfies one of the possibilities.

We first treat the following two cases:

• (i) $Z \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1$.

• (ii) $Z \simeq {\mathbb {P}}(1, 1, m)$ for some $m \ge 1$.

If (i) holds, then Remark 4.2(2) implies the assertion. Assume that (ii) holds. If $m=1$, then the assertions follow from Remark 4.2(1) Let us handle the case when $m \ge 2$. It follows from Theorem 2.7 that m is a divisor of 4. By Remark 4.2(3), the assertion holds. In particular, by Lemma 4.4, we are done for the case when $\rho (X)=1$.

We now treat the case when $\rho (X) \ne 1$. We have $\rho (X) \le \rho (Z) \le 2$, where the latter inequality follows from Proposition 4.1. Hence we have $\rho (X)=\rho (Z)=2$. Since the case (i) has been settled already, Proposition 4.1 enables us to assume that the case (4) of Proposition 4.1 occurs, i.e. $Z \simeq {\mathbb {P}}_{{\mathbb {P}}^1}({\mathcal {O}}\oplus {\mathcal {O}}(m))$ for some $m \ge 1$. By [29, Theorem 4.4], there are two extremal contractions $\varphi :X \rightarrow X'$ and $X \rightarrow X''$. Both of them induce morphisms $Z \rightarrow Z'$ and $Z \rightarrow Z''$ with $\dim X'=\dim Z'$ and $\dim X''=\dim Z''$. Hence we may assume that $\dim X'=2$, i.e. $\varphi :X \rightarrow X'$ is a birational morphism that contracts a single curve. Then $X'$ is a regular del Pezzo surface with $\rho (X')=1$ [19, Theorem 10.5].

Assume that $X'$ is not geometrically normal. Then $Z' \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1$ or $Z' \simeq {\mathbb {P}}(1, 1, m)$ for some $m \in \{1, 2, 4\}$ (Lemma 4.4). Hence, we may assume that $X'$ is geometrically normal. Then $X'$ is geometrically canonical (Theorem 2.3). Therefore, $Z'$ has at worst canonical singularities. In particular, we obtain $m \le 2$. Hence, Remark 4.2(4) implies the assertion. $\square$

Theorem 4.7

Let k be a field of characteristic $p>0$. Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Then the following hold.

• (1) If X is geometrically normal, then $K_X^2 \le 9$.

• (2) Assume that X is not geometrically normal. Then $p \in \{2, 3\}$ and the following hold.

  • (a) If $p =3$, then $K_X^2 \le 3^{\epsilon (X/k)+1}$.

  • (b) If $p=2$, then $K_X^2 \le 2^{\epsilon (X/k)+3}$.

In particular, if X is geometrically reduced, then it holds that $K_X^2 \le 9$.

Proof

Let us show (1). If X is geometrically normal, then X is geometrically canonical (Theorem 2.3). Hence, we have $K_X^2 \le 9$ (cf. [3, Lemma 5.1]). Thus (1) holds.

Let us show (2). Assume that X is not geometrically normal. Then [26] implies that $p \in \{2, 3\}$. The assertions (a) and (b) follow directly from Theorem 4.6. Note that the last assertion holds by the fact that $\epsilon (X/k)=0$ if X is geometrically reduced over k (Definition 2.2). $\square$

Corollary 4.8

Let k be a field of characteristic $p>0$. Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Then the following hold.

• (1) If $p \ge 5$, then $K_X^2 \le 9$.

• (2) If $p =3$, then $K_X^2 \le \max \{9, 3^{\epsilon (X/k)+1}\}$.

• (3) If $p =2$, then $K_X^2 \le \max \{9, 2^{\epsilon (X/k)+3}\}$.

Proof

The assertion follows from Theorem 4.7. $\square$

Theorem 4.9

Let k be a field of characteristic $p>0$ such that $[k:k^p]<\infty$. Let X be a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$. Then the following hold.

• (1) If $p \ge 5$, then $K_X^2 \le 9$.

• (2) If $p=3$, then $K_X^2 \le \max \{9, [k:k^3]\}$.

• (3) If $p=2$, then $K_X^2 \le \max \{9, 2\cdot ([k:k^2])^2\}$.

In particular, if $r:=\log _p[k:k^p]$, then it holds that

$$\begin{aligned} K_X^2 \le \max \{9, 2^{2r+1}\}. \end{aligned}$$

Proof

If X is geometrically normal, then $K_X^2 \le 9$ (Theorem 4.7). Hence we may assume that X is not geometrically normal. In this case, we have $p \in \{2, 3\}$ (Theorem 4.7). Hence, (1) holds.

Since X is not geometrically normal, we have $[k:k^p] \ne 1$. Hence, it follows from [32, Remark 1.7] that

$$\begin{aligned} \epsilon (X/k) \le \ell _F(X/k) (\log _p[k:k^p]-1). \end{aligned}$$

In particular, we have that

$$\begin{aligned} p^{\epsilon (X/k)} \le p^{\ell _F(X/k) (\log _p[k:k^p]-1)}=(p^{-1} \cdot [k:k^p])^{\ell _F(X/k)}. \end{aligned}$$

Let us show (2). We have $\ell _F(X/k) \le 1$ (Theorem 2.4) and $K_X^2 \le 3^{\epsilon (X/k)+1}$ (Theorem 4.7). Therefore, we obtain

$$\begin{aligned} K_X^2 \le 3^{\epsilon (X/k)+1} \le 3 \cdot (3^{-1} \cdot [k:k^3])^{\ell _F(X/k)} \le [k:k^3]. \end{aligned}$$

Thus (2) holds.

Let us show (3). We have $\ell _F(X/k) \le 2$ (Theorem 2.4) and $K_X^2 \le 2^{\epsilon (X/k)+3}$ (Theorem 4.7). Therefore, we obtain

$$\begin{aligned} K_X^2 \le 2^{\epsilon (X/k)+3} \le 2^3 \cdot (2^{-1} \cdot [k:k^2])^{\ell _F(X/k)} \le 2 \cdot ([k:k^2])^2. \end{aligned}$$

Thus (3) holds. $\square$

5 Boundedness of regular del Pezzo surfaces

In this section, we prove the boundedness of geometrically integral regular del Pezzo surfaces (Theorem 5.5). The proof will be given in Sect. 5.2. In Sect. 5.1, we recall results on Chow varieties.

5.1 Chow varieties

The purpose of this subsection is to give a proof of Proposition 5.3. The result itself is well known to experts, however we give a proof for the sake of completeness. Since we shall use Chow varieties, we now recall its construction and results for later use [18, Ch. I, Sections 3, 4].

Definition 5.1

Let $Chow_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$ be the contravariant functor from the category of semi-normal schemes to the category of sets such that if T is a semi-normal scheme, then $Chow_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})(T)$ is the set of well-defined algebraic families of nonnegative cycles of ${\mathbb {P}}^N_T$ which satisfy the Chow-field condition [18, Ch. I, Definition 4.11]. Then $Chow_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$ is coarsely represented by a semi-normal scheme ${{\text {Chow}}}_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$ projective over ${\mathbb {Z}}$.

Remark 5.2

Since we only need the case when T is a normal noetherian scheme (except for ${{\text {Chow}}}_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$), let us recall terminologies for this case.

• (1) In this case, any well-defined family $U \rightarrow T$ of algebraic cycles of ${\mathbb {P}}^N/{\mathbb {Z}}$ satisfies the Chow-field condition [18, Ch. I, Corollary 4.10].

• (2) Furthermore, if $U = \sum _i m_i U_i$ is a pure r-dimensional algebraic cycle such that each $U_i$ is flat over T, then $U \rightarrow T$ is a well-defined algebraic families of nonnegative cycles of ${\mathbb {P}}^N_T$ [18, Ch. I, Definition 3.10, Definition 3.11, Theorem 3.17].

• (3) By construction, ${{\text {Chow}}}_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$ is the semi normalisation of ${{\text {Chow}}}'_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$ [18, Ch. I, Definition 3.25.3], where ${{\text {Chow}}}'_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$ is a reduced closed subscheme of the fine moduli space that parameterises suitable effective divisors, i.e. the projective space corresponding to a linear system. Then, by [18, Ch. I, Corollary 3.24.5], the locus ${{\text {Chow}}}_{r, d}^{\textrm{int}}({\mathbb {P}}^N/{\mathbb {Z}})$ parameterising geometrically integral cycles is an open subset of ${{\text {Chow}}}_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$.

Proposition 5.3

Fix positive integers d and r. Then there exists a flat projective morphism $\pi :V \rightarrow S$ of quasi-projective ${\mathbb {Z}}$-schemes which satisfies the following property: if

• (1) k is a field,

• (2) X is an r-dimensional geometrically integral projective scheme over k, and

• (3) there is a closed immersion $j:X \hookrightarrow {\mathbb {P}}^M_k$ over k for some $M \in {\mathbb {Z}}_{>0}$ such that $(j^*{\mathcal {O}}(1))^r \le d$,

then there exists a cartesian diagram of schemes:

where the vertical arrows are the induced morphisms.

Proof

We first prove that we may replace the conditions (1)–(3) by the following conditions (1)’–(3)’:

• (1)’ k is an algebraically closed field,

• (2)’ X is an r-dimensional projective variety over k, and

• (3)’ there is a closed immersion $j:X \hookrightarrow {\mathbb {P}}^{d+r-1}_k$ over k such that $(j^*{\mathcal {O}}(1))^r = d$.

Take a triple $(k, X, j:X \hookrightarrow {\mathbb {P}}^M_k)$ satisfying (1)–(3). Note that the claim is equivalent to saying that there are finitely many possibilities for the Hilbert polynomial $\chi (X, j^*{\mathcal {O}}(t)) \in {\mathbb {Z}}[t]$. Therefore, passing to the algebraic closure of k, we may assume that (1)’ holds. Then (2) and (2)’ are equivalent. Finally, it follows from [11, Proposition 0] or [14, Ch. I, Exercise 7.7] that either X is a projective space or a closed immersion $j:X \hookrightarrow {\mathbb {P}}^{d+r-1}_k$. We may exclude the former case, thus the problem is reduced to the case when (3)’ holds.

Set $N:=d+r-1$ and

$$\begin{aligned} H_1:=\coprod _{\varphi \in \Phi _{r, d}} {{\text {Hilb}}}_{{\mathbb {P}}^N/{\mathbb {Z}}}^{\varphi } \subset {{\text {Hilb}}}_{{\mathbb {P}}^N/{\mathbb {Z}}}, \end{aligned}$$

where $\Phi _{r, d}$ is the set of polynomials such that $\varphi \in \Phi _{r, d}$ if and only if there exists an algebraically closed field k and a closed immersion $j: X\hookrightarrow {\mathbb {P}}^N_k$ from an r-dimensional projective variety X over k such that $(j^*{\mathcal {O}}(1))^r = d$. Although we do not know yet whether $\Phi _{r, d}$ is a finite set, each ${{\text {Hilb}}}_{{\mathbb {P}}^N/{\mathbb {Z}}}^{\varphi }$ is a projective ${\mathbb {Z}}$-scheme. For the universal closed subscheme ${{\text {Univ}}}_{{\mathbb {P}}^N/{\mathbb {Z}}} \subset {{\text {Hilb}}}_{{\mathbb {P}}^N/{\mathbb {Z}}} \times _{{\mathbb {Z}}} {\mathbb {P}}^N_{{\mathbb {Z}}}$, set $U_1:= {{\text {Univ}}}_{{\mathbb {P}}^N/{\mathbb {Z}}} \times _{{{\text {Hilb}}}_{{\mathbb {P}}^N/{\mathbb {Z}}}} H_1$. In particular, the induced morphism $\rho _1:U_1 \rightarrow H_1$ is flat and projective. We then define $H_2$ as the open subset of $H_1$ such that, for any point $q \in H_1$, it holds that $q \in H_2$ if and only if the scheme-theoretic fibre $\rho _1^{-1}(q)$ is geometrically integral. Let $\rho _2: U_2= U_1 \times _{H_1} H_2 \rightarrow H_2$ be the induced flat projective morphism. Let $H_3 \rightarrow H_2$ be the normalisation of the reduced structure $(H_2)_{{{\text {red}}}}$, which is a finite morphism. Since $H_3$ is normal and ${{\text {Chow}}}_{r, d}({\mathbb {P}}^N/{\mathbb {Z}})$ is a coarse moduli space, the family $U_3=U_2 \times _{H_2} H_3 \rightarrow H_3$ induces a morphism $\theta :H_3 \rightarrow {{\text {Chow}}}_{r, d}^{\textrm{int}}({\mathbb {P}}^N/{\mathbb {Z}})$. For any algebraically closed field k, the induced map $\theta (k):H_3(k) \rightarrow {{\text {Chow}}}_{r, d}^{\textrm{int}}({\mathbb {P}}^N/{\mathbb {Z}})(k)$ is surjective and any fibre of $\theta (k)$ is a finite set. Then, by noetherian induction, $H_3$ is of finite type over ${\mathbb {Z}}$, i.e. $\Phi _{r, d}$ is a finite set. Set $\pi :V \rightarrow S$ to be $U_3 \rightarrow H_3$. Then the claim holds. $\square$

5.2 Boundedness of regular del Pezzo surfaces

In this subsection, we establish the boundedness of geometrically integral regular del Pezzo surfaces (Theorem 5.5). As a consequence, we give a non-explicit upper bound for the irregularity $h^1(X, {\mathcal {O}}_X)$ (Corollary 5.6).

Theorem 5.4

Fix a non-negative integer $\epsilon$. Then there exists a positive integer $d:=d(\epsilon )$ which satisfies the following property: if k is a field of characteristic $p>0$ and X is a regular del Pezzo surface such that $H^0(X, {\mathcal {O}}_X)=k$ and $\epsilon (X/k) \le \epsilon$, then there exist a positive integer N and a closed immersion $j:X \hookrightarrow {\mathbb {P}}^N_k$ such that the degree $(j^*{\mathcal {O}}_{{\mathbb {P}}^N}(1))^2$ of j(X) is at most d.

Proof

By Theorem 3.6, $|-12K_X|$ is very ample over k. Then the assertion follows from Corollary 4.8. $\square$

Theorem 5.5

There exists a flat projective morphism $\rho :V \rightarrow S$ of quasi-projective ${\mathbb {Z}}$-schemes which satisfies the following property: if k is a field of characteristic $p>0$ and X is a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$ and X is geometrically reduced over k, then there exists a cartesian diagram of schemes:

where $\alpha$ denotes the induced morphism.

Proof

The assertion follows from Proposition 5.3 and Theorem 5.4. $\square$

Corollary 5.6

There exists a positive integer h which satisfies the following property: if k is a field of characteristic $p>0$ and X is a regular del Pezzo surface over k such that $H^0(X, {\mathcal {O}}_X)=k$ and X is geometrically reduced over k, then $\dim _k H^1(X, {\mathcal {O}}_X) \le h$.

Proof

The assertion follows from Theorem 5.5. $\square$

6 Examples

In Theorem 4.6, we gave a list of the possibilities for the volumes $K_X^2$ of regular del Pezzo surfaces X, although it depends on $\epsilon (X/k)$. Then it is natural to ask whether there actually exists a geometrically non-normal example which realises each possibility. The purpose of this section is to give a partial answer by exhibiting several examples. We give their construction in Sect. 6.1. We then give a summary in Sect. 6.2.

6.1 Construction

The purpose of this subsection is to construct several regular del Pezzo surfaces which are not geometrically normal.

Example 6.1

Let ${\mathbb {F}}$ be an algebraically closed field of characteristic $p>0$ and let $k:={\mathbb {F}}(s_0, s_1, s_2, s_3)$ be the purely transcendental extension over ${\mathbb {F}}$ of degree four. Set

$$\begin{aligned} X:= {{\text {Proj}}}\,k[x_0, x_1, x_2, x_3]/(s_0x_0^p+s_1x_1^p+s_2x_2^p+s_3x_3^p). \end{aligned}$$

Then X is a regular projective surface over k such that $H^0(X, {\mathcal {O}}_X)=k$, $(X \times _k {{\overline{k}}})_{{{\text {red}}}} \simeq {\mathbb {P}}^2$, and $\epsilon (X/k)=1$ [32, Lemma 9.4, Theorem 9.7]. Note that $-K_X$ is ample if and only if $p \in \{2, 3\}$. Furthermore, the following hold.

• (1) If $p=2$, then $K_X^2=8$.

• (2) If $p=3$, then $K_X^2=3$.

Example 6.2

Let ${\mathbb {F}}$ be an algebraically closed field of characteristic two. Let

$$\begin{aligned} k:={\mathbb {F}}(\{s_i\,|\, 0\le i \le 4\} \cup \{t_i\,|\,0 \le i \le 4\}) \end{aligned}$$

be the purely transcendental extension over ${\mathbb {F}}$ of degree ten. Set

$$\begin{aligned} X:= {{\text {Proj}}}\,\left( \frac{k[x_0, x_1, x_2, x_3, x_4]}{\left( \sum _{i=0}^4 s_ix_i^2, \sum _{i=0}^4 t_ix_i^2\right) }\right) . \end{aligned}$$

Then X is a regular projective surface over k such that $H^0(X, {\mathcal {O}}_X)=k$, $(X \times _k {{\overline{k}}})_{{{\text {red}}}} \simeq {\mathbb {P}}^2$, and $\epsilon (X/k)=2$ [32, Lemma 9.4, Theorem 9.7]. Furthermore, we have $K_X^2=4$.

Example 6.3

Let ${\mathbb {F}}$ be an algebraically closed field of characteristic two. Let $k:={\mathbb {F}}(s_0, s_1, s_2, s_3, s_4)$ be the purely transcendental extension over ${\mathbb {F}}$ of degree five. Set

$$\begin{aligned} X:= {{\text {Proj}}}\,\left( \frac{k[x_0, x_1, x_2, x_3, x_4]}{\left( \sum _{i=0}^4 s_ix_i^2, x_0x_1+x_2x_3\right) }\right) . \end{aligned}$$

Then it holds that $(X \times _k {{\overline{k}}})_{{{\text {red}}}} \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1$ and $K_X^2=4$. Since X is not geometrically reduced, we have that $\epsilon (X/k)\ge 1$ [32, Proposition 1.6]. Therefore, it follows from Theorem 4.6(3) that $\epsilon (X/k)=1$.

Example 6.4

Let ${\mathbb {F}}$ be an algebraically closed field of characteristic two. Let $k:={\mathbb {F}}(s_0, s_1, s_2, s_3)$ be the purely transcendental extension over ${\mathbb {F}}$ of degree four. Set

$$\begin{aligned} Y:= {{\text {Proj}}}\,k[x_0, x_1, x_2, x_3]/(s_0x_0^2+s_1x_1^2+s_2x_2^2+s_3x_3^2). \end{aligned}$$

Then Y is a regular projective surface over k such that $H^0(Y, {\mathcal {O}}_Y)=k$, $(Y \times _k {{\overline{k}}})_{{{\text {red}}}} \simeq {\mathbb {P}}^2$, $\epsilon (Y/k)=1$, and $K_Y^2=8$ (Example 6.1). For any $i \in \{0, 1, 2, 3\}$, let $C_i$ be the curve on Y defined by $x_i=0$. Then we have

$$\begin{aligned} C_3 \simeq {{\text {Proj}}}\,k[x_0, x_1, x_2]/(s_0x_0^2+s_1x_1^2+s_2x_2^2). \end{aligned}$$

The scheme-theoretic intersection $Q:= C_2 \cap C_3$ satisfies

$$\begin{aligned} Q= C_2 \cap C_3 = {{\text {Proj}}}\,k[x_0, x_1]/(s_0x_0^2+s_1x_1^2) \simeq {{\text {Spec}}}\,k[y]/(y^2+t) \end{aligned}$$

for $t:=s_0/s_1$. In particular, Q is a reduced point and $C_2+C_3$ is simple normal crossing. Let

$$\begin{aligned} f:X \rightarrow Y \end{aligned}$$

be the blowup at Q. For the proper transform $C'_2$ of $C_2$, we have that $(C'_2)^2 = C_2^2 -2 = 0$. Since $-K_{C'_2}$ is ample, it holds that $K_X \cdot C'_2<0$. Hence, Kleimann’s criterion for ampleness implies that X is a regular del Pezzo surface. Since $(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N$ has a birational morphism to $(Y \times _k {{\overline{k}}})_{{{\text {red}}}}^N \simeq {\mathbb {P}}^2$, it follows from Theorem 4.6(3) $(X \times _k {{\overline{k}}})_{{{\text {red}}}}^N \simeq {\mathbb {P}}_{{\mathbb {P}}^1}({\mathcal {O}}\oplus {\mathcal {O}}(1))$. It holds that $K_X^2=K_Y^2-2=6$ and $\epsilon (X/k)=\epsilon (Y/k)=1$, where the latter equation follows from Definition 2.2.

Example 6.5

Let ${\mathbb {F}}$ be an algebraically closed field of characteristic two. Let $k:={\mathbb {F}}(s)$ be the purely transcendental extension over ${\mathbb {F}}$ of degree one. Set

$$\begin{aligned} Y:= {{\text {Proj}}}\,k[x, y, z, w]/(x^2+sy^2+zw). \end{aligned}$$

It holds that Y is a regular projective surface such that $H^0(Y, {\mathcal {O}}_Y)=k$, $K_Y^2=8$, and $Y \times _k {{\overline{k}}} \simeq {\mathbb {P}}(1, 1, 2)$. Set

$$\begin{aligned} Y':= D_+(y) \simeq {{\text {Spec}}}\,k[x, z, w]/(x^2+s+zw). \end{aligned}$$

Let $Q \in Y'$ be the closed point defined by the maximal ideal ${\mathfrak {m}}:=(x^2+s, z, w)$ of $k[x, z, w]/(x^2+s+zw)$. Let $f:X \rightarrow Y$ be the blowup at Q. For the curve C on Y defined by $z=0$, we have $C^2=2$. Then its proper transform $C'$ satisfies $C'^2= C^2-2= 0$. By Kleimann’s criterion, we have that $-K_X$ is ample. To summarise, we have $K_X^2=K_Y^2-2=6$ and $(X \times _k {{\overline{k}}})^N \simeq {\mathbb {P}}({\mathcal {O}}\oplus {\mathcal {O}}(2))$, where the latter one follows from Theorem 4.6(3) and $Y \times _k {{\overline{k}}} \simeq {\mathbb {P}}(1, 1, 2)$.

Example 6.6

Maddock constructed the following examples.

• (1) There exists a regular del Pezzo surface $X_1$ over a field $k_1$ of characteristic two such that $H^0(X_1, {\mathcal {O}}_{X_1})=k_1$, $X_1$ is geometrically integral over $k_1$, $X_1$ is not geometrically normal over $k_1$, and $K_{X_1}^2=1$ [21, Main Theorem]. It follows from Theorem 4.6(3) that $(X_1 \times _{k_1} {{\overline{k}}}_1)^N \simeq {\mathbb {P}}^2$.

• (2) There exists a regular del Pezzo surface $X_2$ over a field $k_2$ of characteristic two such that $H^0(X_2, {\mathcal {O}}_{X_2})=k_2$, $K_{X_2}^2=2$, and $X_2$ is not geometrically reduced. It follows from [32, Proposition 1.6] that $\epsilon (X_2/k_2)\ge 1$. By Theorem 4.6(3), it holds that $\epsilon (X_2/k_2)=1$ and $(X_2 \times _{k_2} {{\overline{k}}}_2)_{{{\text {red}}}}^N \simeq {\mathbb {P}}^2$.

6.2 Summary

We now give a summary of the examples established in the previous subsection (Tables 3, 4).

Table 3 $p=3$ case

Table 4 $p=2$ case

Remark 6.7

The author does not know whether there exists $d \in {\mathbb {Z}}_{>0}$ such that the inequality $K_X^2\le d$ holds for an arbitrary regular del Pezzo surface X over a field k with $H^0(X, {\mathcal {O}}_X)=k$.