1 Introduction

It is a well-known fact that Kodaira vanishing fails in positive characteristic [23]. Nevertheless, it has often been believed that a stronger version, namely Kawamata–Viehweg vanishing, holds over a smooth rational surface (e.g. see [32, 33]). In this note, we show that this is in fact not true:

Theorem 3.1  Let k be a field of positive characteristic. Then there exist a smooth projective rational surface X over k , a Cartier divisor D , and a \(\mathbb {Q}\) -divisor \({\Delta }\geqslant 0\) such that

  • \((X, {\Delta })\) is klt,

  • is nef and big, and

  • \(H^1(X, \mathscr {O}_X(D)) \ne 0\).

To prove Theorem 3.1, we use some surfaces constructed by Langer [18]. If , then X can be obtained by taking the blowup of along all the -rational points. Since the proper transforms of the -lines are pairwise disjoint, we can contract all these curves and obtain a birational morphism \(g:X \rightarrow Y\) onto a klt surface Y such that \(\rho (Y)=1\) (cf. Lemma 2.4). Note that \(-K_Y\) is ample if and only if \(p=2\) (cf. Lemma 2.4). Further, we show:

  • For any \(p>0\), Y is obtained as a purely inseparable cover of \(\mathbb {P}^2\) (cf. Theorem 4.1). If \(p=2\), then the morphism \(Y \rightarrow \mathbb {P}^2\) is induced by the anti-canonical linear system \(|-K_Y|\) (cf. Remark 4.2).

  • If \(p=2\), then the Kleimann–Mori cone is generated by exactly 14 curves (cf. Theorem 5.4).

  • If \(p=2\), then X is isomorphic to a surface constructed by Keel–\(\hbox {M}^\mathrm{c}\hbox {Kernan}\) (cf. Proposition 6.4).

Related results. After Raynaud constructed the first counter-example to Kodaira vanishing in positive characteristic [23], several other people studied this problem (e.g. see [3, 4, 6], [15, Section 2.6], [21, 26]). In particular, Fano varieties are known to violate Kawamata–Viehweg vanishing. As far as the authors know, the examples constructed by Lauritzen and Rao [19] (of dimension at least 6) are the only ones over an algebraically closed field. If we admit imperfect fields, then Schröer and Maddock constructed log del Pezzo surfaces with \(H^1(X, \mathscr {O}_X) \ne 0\) [20, 24]. In [2], the authors and Witaszek showed that Kawamata–Vieweg vanishing holds for klt del Pezzo surfaces in large characteristic. On the other hand, if \(p=2\), then the surface mentioned above is a smooth weak del Pezzo surface (cf. Lemma 2.4), hence our result cannot be extended to characteristic two (see also Proposition 7.1).

2 Preliminaries

2.1 Notation

We say that X is a variety over a field k if X is an integral scheme which is separated and of finite type over k. A curve (respectively surface) is a variety of dimension one (respectively two). We say that two schemes X and Y over a field k are k-isomorphic if there exists an isomorphism \(\theta :X \rightarrow Y\) of schemes such that both \(\theta \) and \(\theta ^{-1}\) commute with the structure morphisms: \(X \rightarrow {{\text {Spec}}}\,k\) and \(Y \rightarrow {{\text {Spec}}}\,k\). Given a proper morphism \(f:X\rightarrow Y\) between normal varieties, we say that two \(\mathbb {Q}\)-Cartier \(\mathbb {Q}\)-divisors \(D_1,D_2\) on X are numerically equivalent over Y, denoted , if their difference is numerically trivial on any fibre of f.

We refer to [17, Section 2.3] or [16, Definition 2.8] for the classical definitions of singularities (e.g. klt) appearing in the minimal model programme. Note that we always assume that for any klt pair \((X, {\Delta })\), the \(\mathbb {Q}\)-divisor \({\Delta }\) is effective.

2.2 Construction by Langer

We now recall the construction of a rational surface due to Langer [18] (see also [11, Exercise III.10.7]). A similar method was used to construct also some K3 surfaces and Calabi–Yau threefolds (cf. [5, 12]).

Notation 2.1

Let \(q=p^e\), where p is a prime number and e is a positive integer. Let be the \(\mathbb {F}_q\)-rational points on \(\mathbb {P}^2_{\mathbb {F}_q}\), and let be the \(\mathbb {F}_q\)-lines on \(\mathbb {P}^2_{\mathbb {F}_q}\), i.e. the lines which are defined over \(\mathbb {F}_q\). Let

be the blowup along all the \(\mathbb {F}_q\)-points . For any , let \(E^{(0)}_i\) be the \(f^{(0)}\)-exceptional prime divisor lying over \(P^{(0)}_i\), hence . The proper transforms of the \(\mathbb {F}_q\)-lines are disjoint with each other and satisfy for any . Let

be the birational morphism contracting all of the curves . We define

Let k be a field containing \(\mathbb {F}_q\) and let

$$\begin{aligned} f:X \rightarrow \mathbb {P}^2_k, \qquad g:X \rightarrow Y \end{aligned}$$

be the base changes of \(f^{(0)}\) and \(g^{(0)}\) induced by . We denote by \(P_i,L_i,E_i,L'_i\) and \(E^Y_i\) the inverse images of and \((E^Y_i)^{(0)}\), respectively. We fix an arbitrary line \(H \in |\mathscr {O}_{\mathbb {P}^2}(1)|\) defined over k. By abuse of notation, each \(P_i\) (respectively \(L_i\)) is also called an \(\mathbb {F}_q\)-point (respectively an \(\mathbb {F}_q\)-line), although these depend on the choice of the homogeneous coordinates.

Notation 2.2

We use the same notation as in Notation 2.1 but we assume that \(q=2\), i.e. \(p=2\) and \(e=1\).

Remark 2.3

The configuration of the \(\mathbb {F}_q\)-points and the \(\mathbb {F}_q\)-lines on \(\mathbb {P}^2_{\mathbb {F}_q}\) satisfies the following properties:

  • For any \(\mathbb {F}_q\)-line L on \(\mathbb {P}^2_{\mathbb {F}_q}\), the number of the \(\mathbb {F}_q\)-points contained in L is equal to \(q+1\).

  • For any \(\mathbb {F}_q\)-point P on \(\mathbb {P}^2_{\mathbb {F}_q}\), the number of the \(\mathbb {F}_q\)-lines passing through P is equal to \(q+1\).

If \(q=2\), then the picture of the configuration is classically known as Fano plane (e.g. see [22, Subsection 3.1.1]).

2.3 Basic properties

We now summarise some basic properties of the surfaces X and Y constructed in Notation 2.1.

Lemma 2.4

We use Notation 2.1. The following hold:

  1. (i)

    \(\rho (Y)=1\).

  2. (ii)

    Y is klt.

  3. (iii)

    Y has at most canonical singularities if and only if \(q=2\).

  4. (iv)

    If \(q>2\), then \(K_Y\) is ample.

  5. (v)

    If \(q=2\), then \(-K_Y\) is ample.

  6. (vi)

    If \(q=2\), then \(-K_X\) is nef and big.

Proof

(i) follows immediately by the construction. Further, we have

Thus, (ii) and (iii) hold.

We now show (iv) and (v). Since and

we have

Taking the push-forward \(g_*\), we get

Therefore, if \(q=2\) (respectively \(q >2\)), then \(-K_Y\) (respectively \(K_Y\)) is ample. Thus, (iv) and (v) hold. (vi) follows directly from (iii) and (v). \(\square \)

Lemma 2.5

We use Notation 2.1. We assume that \(k=\mathbb {F}_q\). For any \(\mathbb {F}_q\)-point \(P_i \in \mathbb {P}_{\mathbb {F}_q}^2(\mathbb {F}_q)\), let be the \(\mathbb {F}_q\)-lines passing through \(P_i\). Then .

Proof

Since we have for any \(1 \leqslant \alpha < \beta \leqslant q+1\), the claim follows by counting the number of \(\mathbb {F}_q\)-rational points (cf. Remark 2.3):

3 Counter-examples to Kawamata–Viehweg vanishing

In this section, we construct some counter-examples to Kawamata–Viehweg vanishing on a family of smooth rational surfaces.

Theorem 3.1

We use Notation 2.1. We consider the following \(\mathbb {Q}\)-divisors on X:

  • , and

  • .

Then the following hold:

  1. (i)

    \((X, {\Delta })\) is klt.

  2. (ii)

    \(B-{\Delta }\) is nef and big.

  3. (iii)

    .

In particular, Kawamata–Viehweg vanishing fails on X.

Proof

Since \(L'_1, \dots , L'_{q^2+q+1}\) are pairwise disjoint, (i) follows immediately. We now show (ii). We have

It follows that

Thus, (ii) holds.

We now show (iii). By Riemann–Roch, it follows that

Since

and

we have

Thus, (iii) holds. \(\square \)

Remark 3.2

We do not know whether there exist a klt del Pezzo surface X and a nef and big Cartier divisor A on X such that \(H^1(X, \mathscr {O}_X(A)) \ne 0\).

As an application, we now show that the pair is not liftable to \(W_2(k)\). Note that, a similar result was proven in [18, Proposition 8.4].

Corollary 3.3

We use Notation 2.1. Assume that k is perfect. If \(p \geqslant 3\), then

is not liftable to \(W_2(k)\).

Proof

We use the same notation as in Theorem 3.1. As in the proof of Theorem 3.1, it follows that \(B-{\Delta }-\sum \epsilon _iE_i\) is ample for some \(\epsilon _i>0\). Thus, Theorem 3.1 and [10, Corollary 3.8] imply the claim. \(\square \)

4 Purely inseparable morphisms to \(\mathbb {P}^2\)

The main purpose of this section is to show that the surface Y, as in Notation 2.1, can be obtained as a purely inseparable cover of \(\mathbb {P}^2\) (cf. Theorem 4.1). Moreover if \(q=2\), then the morphism \(Y \rightarrow \mathbb {P}^2\) is induced by the anti-canonical linear system (cf. Remark 4.2).

We also show that the complete linear system |M|, appearing in Theorem 4.1, does not have any smooth element (cf. Proposition 4.3), even though it is base point free and big. We were not able to find a similar example in the literature (cf. [11, Theorem II.8.18 and Corollary III.10.9]).

Theorem 4.1

We use Notation 2.1. Let

Then the following hold:

  1. (i)

    |M| is base point free.

  2. (ii)

    for any .

  3. (iii)

    .

  4. (iv)

    Given the natural injective k-linear map

    $$\begin{aligned} \iota :H^0(X, \mathscr {O}_X(M)) \hookrightarrow H^0(\mathbb {P}^2_k, \mathscr {O}_{\mathbb {P}^2_k}(q+1)), \end{aligned}$$

    the following holds:

  5. (v)

    There exists a Cartier divisor \(M_Y\) on Y such that .

  6. (vi)

    The morphism induced by the complete linear system \(|M_Y|\)

    $$\begin{aligned} \varphi ={\Phi }_{|M_Y|}:Y \rightarrow \mathbb {P}_k^2 \end{aligned}$$

    is a finite universal homeomorphism of degree q.

Proof

We may assume that \(k=\mathbb {F}_q\). We first show (i). Given a \(\mathbb {F}_q\)-point \(P_i\) on \(\mathbb {P}^2_{\mathbb {F}_q}\), we denote by the \(\mathbb {F}_q\)-lines passing through \(P_i\). Then Lemma 2.5 implies that

Thus, |M| is base point free by symmetry and (i) holds.

(ii) and (iii) are simple calculations, and (iv) follows from [27, 28] (see also  [13, Proposition 2.1]Footnote 1). Further, \(g:X \rightarrow Y\) is the Stein factorisation of \(\psi ={\Phi }_{|M|}:X \rightarrow \mathbb {P}^2_k\). Thus, (v) holds.

We now show (vi). Since , (i) implies that \(|M_Y|\) is base point free and (v) implies that \(h^0(Y, \mathscr {O}_Y(M_Y))=3\). Since \(M_Y\) is ample, it follows that \(\varphi \) is a finite surjective morphism. By (iii), the degree of \(\varphi \) is equal to q.

It is enough to show that \(\varphi \) is a purely inseparable morphism. To this end, we may assume that \(k=\overline{\mathbb {F}}_q\). By (iv), we have that

Generically, the rational map can be written by

where \(\widetilde{L}_1, \dots , \widetilde{L}_{q+1}\) are the affine lines passing through the origin with coefficients in \(\mathbb {F}_q\), and in particular . Fix a general closed point \((\alpha , \beta ) \in \mathbb {A}^2_k\). It is enough to show that its fibre \({\Psi }^{-1}((\alpha , \beta ))\) consists of one point. Let be such that \({\Psi }(u, v)=(\alpha , \beta )\). Since \((\alpha , \beta )\) is chosen to be general, we can assume that the denominators of the fractions appearing in the following calculation are always nonzero. We have

which implies

(1)

and

(2)

By (1), we have

(3)

Substituting (3) to (2), we get

(4)

Substituting (4) to (3), it follows that

which implies that

Hence u is uniquely determined by \((\alpha , \beta )\), and so is v by (4). Thus, (vi) holds. \(\square \)

Remark 4.2

Using the same notation as in Theorem 4.1, if \(q=2\), then \(M=-K_X\) and \(M_Y=-K_Y\). This can be considered as an analogue of the fact that a smooth del Pezzo surface S with \(K_S^2=2\) is a double cover of \(\mathbb {P}^2\) which is induced by the anti-canonical system \(|-K_X|\). Indeed, both X and S are obtained by taking blowups along seven points.

Proposition 4.3

We use Notation 2.1. Let

Then the following hold:

  1. (i)

    If \(k=\mathbb {F}_q\), then for any element \(D \in |M|\), there exists a unique \(\mathbb {F}_q\)-point \(P_i\) on \(\mathbb {P}^2_{\mathbb {F}_q}\) such that

    where are the \(\mathbb {F}_q\)-lines passing through \(P_i\).

  2. (ii)

    If k is an algebraically closed field, then a general member of |M| is integral.

  3. (iii)

    Any element of |M| is not smooth.

Proof

Note that for each \(\mathbb {F}_q\)-point \(P_i\) on \(\mathbb {P}^2_{\mathbb {F}_q}\), the divisor , as in (i), is an element of |M|. Thus, there are of such divisors. On the other hand, (iv) of Theorem 4.1 implies

Thus, (i) holds (see also [13, Proposition 2.3]).

We now show (ii) and (iii). To this end, we may assume that k is algebraically closed. We set \(M_Y=g_*M\). By (i), there exists an irreducible divisor in \(|M_Y|\). Thus, any general element of \(|M_Y|\) is irreducible.

Since, by Theorem 4.1, \(|M_Y|\) is base point free, if \(D\in |M|\) is a general element, then D is irreducible. By Theorem 4.1, we may write

for some . By the Jacobian criterion for smoothness, it follows that is a unique singular point of \(f_*D\). Since \(f_*D\) is smooth outside , we see that \(f_*D\) is reduced. Since \(\alpha , \beta , \gamma \) are chosen to be general, it follows that is not an \(\mathbb {F}_q\)-point. Thus, D is the proper transform of \(f_*D\), hence D is integral. Thus, (ii) holds. Since \(f_*D\) has a singular point outside \(f({{\text {Ex}}}(f))\), it follows that D is not smooth. Thus, (iii) holds. \(\square \)

5 The Kleimann–Mori cone

The main result of this section is Theorem 5.4 which determines the generators of the Kleimann–Mori cone of X as in Notation 2.2. To this end, we classify the curves whose self-intersection numbers are negative (cf. Proposition 5.3).

Lemma 5.1

We use Notation 2.2. The following hold:

  1. (i)

    If C is a curve on X which satisfies and differs from any of \(E_1, \dots , E_7\), then \(\deg f_*(C) \leqslant 3\).

  2. (ii)

    If C is a curve on X with , then \(\deg f_*(C) \leqslant 2\).

Proof

We show (i). We have

$$\begin{aligned} C \sim af^*\mathscr {O}_{\mathbb {P}^2}(1)+ \sum _{i=1}^7 b_iE_i, \end{aligned}$$

where \(a=\deg f_*(C)> 0\) and \(b_1,\dots ,b_7 \in \mathbb {Z}\). Since \(q=2\), Lemma 2.4 implies that C is a \((-1)\)-curve. Thus, we have

By Schwarz’s inequality, we obtain

which implies . Thus, (i) holds. The proof of (ii) is similar. \(\square \)

Lemma 5.2

We use Notation 2.2. Let C be a curve on X such that \(C_0=f(C)\) is a conic or a cubic. Then .

Proof

First, we assume that \(C_0\) is conic. Suppose that \(C_0\) passes through five of the \(\mathbb {F}_2\)-points, say \(P_1, \dots , P_5\). Let us derive a contradiction. Let \(P_6\) and \(P_7\) be the remaining two \(\mathbb {F}_2\)-points. Since there are exactly three \(\mathbb {F}_2\)-lines passing through \(P_6\) (respectively \(P_7\)), we can find an \(\mathbb {F}_2\)-line \(L_i\) such that \(P_6 \not \in L_i\) and \(P_7 \not \in L_i\). In particular, \(C_0 \cap L_i\) contains at least three points, within \(P_1, \dots , P_5\). This contradicts the fact that .

Now, we assume that \(C_0\) is cubic. If \(C_0\) is smooth, then . Thus, we may assume that \(C_0\) is singular and . It follows that \(C_0\) must pass through all the \(\mathbb {F}_2\)-points \(P_1, \dots , P_7\) and the unique singular point of \(C_0\) is an \(\mathbb {F}_2\)-point, say \(P_1\). Let \(L_j\) be an \(\mathbb {F}_2\)-line passing through \(P_1\). Since \(C_0 \cap L_j\) contains at least three \(\mathbb {F}_2\)-rational points \(P_1, P_{i}, P_{i'}\), we have that . This contradicts the fact that . Thus, the claim follows. \(\square \)

Proposition 5.3

We use Notation 2.2. Let C be a curve on X with . Then C is equal to one of the curves \(E_1, \dots , E_7, L'_1, \dots , L'_7\).

Proof

Assume that . Let \(C_0=f_*C\). Since \(-K_X\) is nef and big, we have that . Lemma 5.1 implies that \(\deg C_0 \leqslant 3\). By Lemma 5.2, we have that \(\deg C_0=1\), hence \(C_0\) is a line. Then \(C_0\) passes through at least two of the \(\mathbb {F}_2\)-points. It follows that \(C_0\) is equal to some \(L_i\), hence \(C=L'_i\), as desired. \(\square \)

Theorem 5.4

We use Notation 2.2. Then

Proof

Since there exists an effective \(\mathbb {Q}\)-divisor \({\Delta }\) such that \((X, {\Delta })\) is klt and is ample, the cone theorem [30, Theorem 1.7] implies that is closed and generated by the extremal rays spanned by curves. By [31, Theorem 4.3], any extremal ray of is generated by a curve C whose self-intersection number is negative. Thus, the claim follows from Proposition 5.3. \(\square \)

6 Relation to Keel–M\(^\mathrm{c}\)Kernan surfaces

The goal of this section is to prove Proposition 6.4 which shows that the surface X, constructed in Notation 2.2, is isomorphic to some surface obtained by Keel–\(\hbox {M}^{\mathrm{c}}\hbox {Kernan}\) [14, end of Section 9].

We first recall their construction. Let k be a field of characteristic two. We fix a k-rational point in \(\mathbb {P}^2_k\) and a conic over k as follows:

Note that any line through Q is tangent to C. Let \(\varphi _0:S_0 \rightarrow \mathbb {P}^2_k\) be the blowup at Q. We choose k-rational points \(P_1, \dots , P_d\) at \(\varphi _0^{-1}(C)\). We first consider the blowup along these points \(\psi :S'_0 \rightarrow S_0\) and then we take the blowup \(S \rightarrow S'_0\) along the intersection , where \(\psi _*^{-1}(\varphi _0^{-1}(C))\) is the proper transform of \(\varphi ^{-1}(C)\). Note that the intersection is a collection of k-rational points. We call S a Keel\(M^{c}{} { Kernan} ~ { surface}\) of degree d over k.

Let us recall a well-known result on the theory of Severi–Brauer varieties.

Lemma 6.1

Let X be a projective scheme over \(\mathbb {F}_q\). Let \(\overline{\mathbb {F}}_q\) be the algebraic closure of \(\mathbb {F}_q\). If the base change is \(\overline{\mathbb {F}}_q\)-isomorphic to , then X is \(\mathbb {F}_q\)-isomorphic to .

Proof

See, for example, [25, Chapter X, Sections 5–7]. As an alternative proof, one can conclude the claim from [7, Corollary 1.2] and Châtelet’s theorem [9, Theorem 5.1.3]. \(\square \)

The following two lemmas may be well-known, however we include proofs for the sake of completeness.

Lemma 6.2

Let k be a field. Take k-rational points \(P_1, \dots , P_4, Q_1, \dots , Q_4 \in \mathbb {P}^2_k\). Assume that no three of \(P_1, \dots , P_4\) (respectively \(Q_1, \dots , Q_4\)) lie on a single line of \(\mathbb {P}^2_k\). Then there exists a k-automorphism \(\sigma :\mathbb {P}^2_k \rightarrow \mathbb {P}^2_k\) such that \(\sigma (P_i)=Q_i\) for any .

Proof

We may assume that

For each , we write for some \(a_i, b_i, c_i \in k\). Consider the matrix

Since \(Q_1, Q_2, Q_3\) do not lie on a line, it follows that \(\det M \ne 0\). Let \(\tau :\mathbb {P}^2_k \rightarrow \mathbb {P}^2_k\) be the k-automorphism induced by M. In particular,

We may write for some \(d, e, f \in k\). Again by the assumption, we have that \(d,e,f\ne 0\). Then the k-automorphism

satisfies

Thus, the k-automorphism satisfies \(\sigma (P_i)=Q_i\) for any . \(\square \)

Lemma 6.3

Let k be a field of characteristic two. Let \(C_1\) and \(C_2\) be smooth conics in \(\mathbb {P}^2_k\). Assume that there exist distinct four k-rational points \(P_1, P_2, P_3, Q\) of \(\mathbb {P}^2_k\) such that and the tangent line \(T_{C_i, P_j}\) of \(C_i\) at \(P_j\) passes through Q for any and . Then \(C_1=C_2\).

Proof

By Lemma 6.2, we may assume that

It is well known that \(C_1\) and \(C_2\) are strange curves (e.g. see [8, Theorem 1.1]). [8, Proposition 2.1] implies that for each , \(C_i\) is defined by a quadric homogeneous polynomial:

Since \(P_1, P_2, P_3 \in C_i\), we get \(a_i=c_i=0\) and \(b_i=d_i\). In particular, both of \(C_1\) and \(C_2\) are defined by the same polynomial \(xy+z^2\). \(\square \)

Proposition 6.4

Let k be a field of characteristic two. Then any Keel–\(\hbox {M}^{{c}}\hbox {Kernan}\) surface S of degree 3 over k is k-isomorphic to the surface X constructed in Notation 2.2.

Proof

We use the same notation as above. Let \(\pi :S_0 \rightarrow \mathbb {P}^1\) be the induced \(\mathbb {P}^1\)-fibration. We divide the proof into two steps.

Step 1. In this step, we show that any two Keel–\(\hbox {M}^{\mathrm{c}}\hbox {Kernan}\) surfaces S and \(S'\) of degree 3 over k are isomorphic over k.

There are three k-rational points \(P_1, P_2, P_3\in C\) (respectively \(P'_1, P'_2, P'_3\in C\)) such that S (respectively \(S'\)) is the blowup of \(S_0\) twice along \(P_1 \cup P_2 \cup P_3\) (respectively \(P'_1 \cup P'_2 \cup P'_3\)). Thanks to Lemma 6.2, there is a k-automorphism \(\sigma :\mathbb {P}_k^2 \rightarrow \mathbb {P}_k^2\) such that \(\sigma (Q)=Q\) and \(\sigma (P_i)=P'_i\) for \(i=1,2\) and 3. Lemma 6.3 implies that \(\sigma (C)=C\) and, in particular, \(\sigma \) induces a k-isomorphism , as desired.

Step 2. In this step, we assume that \(k=\mathbb {F}_2\). Note that C has exactly three \(\mathbb {F}_2\)-rational points:

Let

$$\begin{aligned} P_1=\varphi ^{-1}_0(Q_1), \qquad P_2=\varphi _0^{-1}(Q_2), \qquad P_3=\varphi _0^{-1}(Q_3), \end{aligned}$$

and S be the Keel–\(\hbox {M}^\mathrm{c}\hbox {Kernan}\) surface of degree 3 over \(\mathbb {F}_2\) as above. We now show that S is \(\mathbb {F}_2\)-isomorphic to \(X^{(0)}\) defined in Notation 2.2.

There are pairwise disjoint \((-1)\)-curves \(E_1, \dots , E_7\) on S over \(\mathbb {F}_2\), i.e. for any \(i=1, \dots , 7\), \(E_i\) is \(\mathbb {F}_2\)-isomorphic to \(\mathbb {P}^1_{\mathbb {F}_2}\) and satisfies . Indeed, we can check that the following seven curves listed below satisfy these properties.

  • The exceptional curve over Q is a \((-1)\)-curve over \(\mathbb {F}_2\).

  • For any \(i=1, 2, 3\), the exceptional curve over \(Q_i\) obtained by the second blowup is a \((-1)\)-curve over \(\mathbb {F}_2\).

  • For any \(1 \leqslant i < j \leqslant 3\), the proper transform of the \(\mathbb {F}_2\)-line, passing through \(Q_i\) and \(Q_j\), is a \((-1)\)-curve over \(\mathbb {F}_2\).

Let \(\psi :S \rightarrow T\) be the birational morphism with \(\psi _*\mathscr {O}_S=\mathscr {O}_T\) that contracts \(E_1, \dots , E_7\). Since T is a projective scheme over \(\mathbb {F}_2\) whose base change to \(\overline{\mathbb {F}}_2\) is a projective plane, it follows that T is \(\mathbb {F}_2\)-isomorphic to \(\mathbb {P}^2_{\mathbb {F}_2}\) by Lemma 6.1. Thus, S is obtained by the blowup along all the \(\mathbb {F}_2\)-rational points of \(\mathbb {P}^2_{\mathbb {F}_2}\) which implies \(S \simeq X^{(0)}\) (cf. Notation 2.2), as desired.

By Steps 1 and 2, we are done. \(\square \)

7 Appendix: Kawamata–Viehweg vanishing for smooth del Pezzo surfaces

By Theorem 3.1, there exists a smooth weak del Pezzo surface of characteristic 2 which violates Kawamata–Viehweg vanishing. We now show that Kawamata–Viehweg vanishing holds on smooth del Pezzo surfaces.

Proposition 7.1

Let k be an algebraically closed field of characteristic \(p>0\). Let X be a smooth projective surface over k such that \(-K_X\) is ample and let \((X, {\Delta })\) be a klt pair for some effective \(\mathbb {Q}\)-divisor \({\Delta }\). Let D be a Cartier divisor such that is nef and big. Then \(H^i(X, \mathscr {O}_X(D))=0\) for \(i>0\).

Proof

After perturbing \({\Delta }\), we may assume that is ample. We define . We run a -MMP \(f:X \rightarrow Y\). Since \(-K_X\) is ample, Y is also a smooth del Pezzo surface. Moreover, this MMP can be considered as a -MMP. By the Kawamata–Viehweg vanishing theorem for birational morphisms (cf. [16, Theorem 10.4], [29, Theorem 2.12]), it follows that

$$\begin{aligned} H^i(X, \mathscr {O}_X(D)) \simeq H^i(Y, f_*\mathscr {O}_X(D)) \simeq H^i(Y, \mathscr {O}_Y(f_*D)) \end{aligned}$$

for any i, where the latter isomorphism follows from the fact that f is obtained by running a D-MMP.

Therefore, after replacing X by Y, we may assume that \(\Delta +A\) is nef. Thus, \(D-K_X\) is nef and big. In this case, it is well-known that \(H^i(X, \mathscr {O}_X(D))=0\) (e.g. see  [21, Proposition 3.2] or [1, Proposition 3.3]). \(\square \)