Smooth rational surfaces violating Kawamata–Viehweg vanishing

We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata–Viehweg vanishing.

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 Q-divisor 0 such that • (X, ) is klt, • D − (K X + ) is nef and big, and • H 1 (X, O X (D)) = 0.
To prove Theorem 3.1, we use some surfaces constructed by Langer [18]. If k = F p , then X can be obtained by taking the blowup of P 2 F p along all the F p -rational points. Since the proper transforms L 1 , . . . , L p 2 + p+1 of the F p -lines L 1 , . . . , L p 2 + p+1 are pairwise disjoint, we can contract all these curves and obtain a birational morphism g : X → Y onto a klt surface Y such that ρ(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 P 2 (cf. Theorem 4.1).
If p = 2, then the morphism Y → P 2 is induced by the anti-canonical linear system | − K Y | (cf. Remark 4.2). • If p = 2, then the Kleimann-Mori cone NE(X ) is generated by exactly 14 curves (cf. Theorem 5.4). • If p = 2, then X is isomorphic to a surface constructed by Keel-M c 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, O X ) = 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).

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 θ : X → Y of schemes such that both θ and θ −1 commute with the structure morphisms: X → Spec k and Y → Spec k. Given a proper morphism f : X → Y between normal varieties, we say that two Q-Cartier Q-divisors D 1 , D 2 on X are numerically equivalent over Y , denoted D 1 ≡ f D 2 , 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, ), the Q-divisor is effective.

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 P q 2 +q+1 be the F q -lines on P 2 F q , i.e. the lines which are defined over F q . Let be the birational morphism contracting all of the curves L (0) 1 , . . . , L (0) . We define Let k be a field containing F q and let , respectively. We fix an arbitrary line H ∈ |O P 2 (1)| defined over k. By abuse of notation, each P i (respectively L i ) is also called an F q -point (respectively an 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 F q -points and the F q -lines on P 2 F q satisfies the following properties: • For any F q -line L on P 2 F q , the number of the F q -points contained in L is equal to q + 1.
• For any F q -point P on P 2 F q , the number of the 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,

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: Proof (i) follows immediately by the construction. Further, we have Thus, (ii) and (iii) hold. We now show (iv) and (v). Since 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).

Lemma 2.5 We use Notation
Proof Since we have L j α ∩ L j β = P i for any 1 α < β q + 1, the claim follows by counting the number of F q -rational points (cf. Remark 2.3):

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 Q-divisors on X : Then the following hold: In particular, Kawamata-Viehweg vanishing fails on X .

It follows that
Thus, (ii) holds.
We now show (iii). By Riemann-Roch, it follows that we have Thus, (iii) holds.

Remark 3.2 We do not know whether there exist a klt del Pezzo surface X and a nef and big Cartier divisor
As an application, we now show that the pair X, E i + L j 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 3, then X, Proof We use the same notation as in Theorem 3.

Purely inseparable morphisms to 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 P 2 (cf. Theorem 4.1). Moreover if q = 2, then the morphism Y → 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. [ Then the following hold: the following holds: is a finite universal homeomorphism of degree q. Proof We may assume that k = F q . We first show (i). Given a F q -point P i on P 2 F q , we denote by L j 1 , . . . , L j q+1 the 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] 1 ). Further, g : X → Y is the Stein factorisation of ψ = |M| : X → P 2 k . Thus, (v) holds. We now show (vi). Since M = g * M Y , (i) implies that |M Y | is base point free and Since M Y is ample, it follows that ϕ is a finite surjective morphism. By (iii), the degree of ϕ is equal to q.
It is enough to show that ϕ is a purely inseparable morphism. To this end, we may assume that k = F q . By (iv), we have that Generically, the rational map ψ • f −1 can be written by where L 1 , . . . , L q+1 are the affine lines passing through the origin with coefficients in F q , and in particular It is enough to show that its fibre −1 ((α, β)) consists of one point.
Since (α, β) is chosen to be general, we can assume that the denominators of the fractions appearing in the following calculation are always nonzero. We have and By (1), we have Substituting (3) to (2), we get Substituting (4) to (3), it follows that which implies that Hence u is uniquely determined by (α, β), and so is v by (4). Thus, (vi) holds.
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 2 S = 2 is a double cover of 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
Then the following hold: (i) If k = F q , then for any element D ∈ |M|, there exists a unique F q -point P i on where L j 1 , . . . , L j q+1 are the F q -lines passing through P i .

(ii) If k is an algebraically closed field, then a general member of |M| is integral. (iii) Any element of |M| is not smooth.
Proof Note that for each F q -point P i on P 2 F q , the divisor D = q E i + q+1 α=1 L j α , as in (i), is an element of |M|. Thus, there are q 2 + q + 1 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 ∈ |M| is a general element, then D is irreducible. By Theorem 4.1, we may write for some (α, β, γ ) ∈ k 3 \{(0, 0, 0)}. By the Jacobian criterion for smoothness, it follows that [α 1/q :β 1/q :γ 1/q ] is a unique singular point of f * D. Since f * D is smooth outside [α 1/q :β 1/q :γ 1/q ], we see that f * D is reduced. Since α, β, γ are chosen to be general, it follows that [α 1/q :β 1/q :γ 1/q ] is not an 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 (Ex( f )), it follows that D is not smooth. Thus, (iii) holds.

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: (i) If C is a curve on X which satisfies C 2 = −1 and differs from any of E 1 , . . . , E 7 , Proof We show (i). We have By Schwarz's inequality, we obtain which implies a 2 − 3a − 3 0. Thus, (i) holds. The proof of (ii) is similar.

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 C 2 0.
Proof First, we assume that C 0 is conic. Suppose that C 0 passes through five of the F 2 -points, say P 1 , . . . , P 5 . Let us derive a contradiction. Let P 6 and P 7 be the remaining two F 2 -points. Since there are exactly three F 2 -lines passing through P 6 (respectively P 7 ), we can find an F 2 -line L i such that P 6 / ∈ L i and P 7 / ∈ L i . In particular, C 0 ∩ L i contains at least three points, within P 1 , . . . , P 5 . This contradicts the fact that C 0 · L i = 2. Now, we assume that C 0 is cubic. If C 0 is smooth, then C 2 C 2 0 − 7 = 2. Thus, we may assume that C 0 is singular and C 2 < 0. It follows that C 0 must pass through all the F 2 -points P 1 , . . . , P 7 and the unique singular point of C 0 is an F 2 -point, say P 1 . Let L j be an F 2 -line passing through P 1 . Since C 0 ∩ L j contains at least three F 2 -rational points P 1 , P i , P i , we have that C 0 · L j 4. This contradicts the fact that C 0 · L j = 3. Thus, the claim follows. Proposition 5. 3 We use Notation 2.2. Let C be a curve on X with C 2 < 0. Then C is equal to one of the curves E 1 , . . . , E 7 , L 1 , . . . , L 7 .
Since −K X is nef and big, we have that C 2 −2. Lemma 5.1 implies that deg C 0 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 F 2 -points. It follows that C 0 is equal to some L i , hence C = L i , as desired.

Theorem 5.4 We use Notation 2.2. Then
Proof Since there exists an effective Q-divisor such that (X, ) is klt and −(K X + ) is ample, the cone theorem [30,Theorem 1.7] implies that NE(X ) is closed and generated by the extremal rays spanned by curves. By [31,Theorem 4.3], any extremal ray of NE(X ) is generated by a curve C whose self-intersection number is negative. Thus, the claim follows from Proposition 5.3.

Relation to Keel-M 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-M c 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 P 2 k and a conic over k as follows: Note that any line through Q is tangent to C. Let ϕ 0 : S 0 → P 2 k be the blowup at Q. We choose k-rational points P 1 , . . . , P d at ϕ −1 0 (C). We first consider the blowup along these points ψ : S 0 → S 0 and then we take the blowup S → S 0 along the intersection Ex(ψ)∩ψ −1 * (ϕ −1 (C)), where ψ −1 * (ϕ −1 0 (C)) is the proper transform of ϕ −1 (C). Note that the intersection Ex(ψ) ∩ ψ −1 * (ϕ −1 (C)) 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. 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 , . . . , P 4 , Q 1 , . . . , Q 4 ∈ P 2 k . Assume that no three of P 1 , . . . , P 4 (respectively Q 1 , . . . , Q 4 ) lie on a single line of P 2 k . Then there exists a k-automorphism σ : P 2 k → P 2 k such that σ (P i ) = Q i for any i ∈ {1, 2, 3, 4}.

Lemma 6.3
Let k be a field of characteristic two. Let C 1 and C 2 be smooth conics in P 2 k . Assume that there exist distinct four k-rational points P 1 , P 2 , P 3 , Q of P 2 k such that { P 1 , P 2 , P 3 } ⊂ C 1 ∩ C 2 and the tangent line T C i ,P j of C i at P j passes through Q for any i ∈ {1, 2} and j ∈ {1, 2, 3}. Then C 1 = C 2 .
Since P 1 , P 2 , P 3 ∈ 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 x y + z 2 . Proof We use the same notation as above. Let π : S 0 → P 1 be the induced P 1 -fibration. We divide the proof into two steps.
Step 1. In this step, we show that any two Keel-M c Kernan surfaces S and S of degree 3 over k are isomorphic over k.
Step 2. In this step, we assume that k = F 2 . Note that C has exactly three F 2 -rational points: Let and S be the Keel-M c Kernan surface of degree 3 over F 2 as above. We now show that S is F 2 -isomorphic to X (0) defined in Notation 2.2. There are pairwise disjoint (−1)-curves E 1 , . . . , E 7 on S over F 2 , i.e. for any i = 1, . . . , 7, E i is F 2 -isomorphic to P 1 F 2 and satisfies K S · E i = E 2 i = −1. Indeed, we can check that the following seven curves listed below satisfy these properties.
• The exceptional curve over Q is a (−1)-curve over F 2 .
• For any i = 1, 2, 3, the exceptional curve over Q i obtained by the second blowup is a (−1)-curve over F 2 . • For any 1 i < j 3, the proper transform of the F 2 -line, passing through Q i and Q j , is a (−1)-curve over F 2 .
Let ψ : S → T be the birational morphism with ψ * O S = O T that contracts E 1 , . . . , E 7 . Since T is a projective scheme over F 2 whose base change to F 2 is a projective plane, it follows that T is F 2 -isomorphic to P 2 F 2 by Lemma 6.1. Thus, S is obtained by the blowup along all the F 2 -rational points of P 2 F 2 which implies S X (0) (cf. Notation 2.2), as desired.
By Steps 1 and 2, we are done.
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, ) be a klt pair for some effective Q-divisor . Let D be a Cartier divisor such that D −(K X + ) is nef and big. Then H i (X, O X (D)) = 0 for i > 0.
Proof After perturbing , we may assume that D − (K X + ) is ample. We define A = D − (K X + ). We run a ( + A)-MMP f : X → Y . Since −K X is ample, Y is also a smooth del Pezzo surface. Moreover, this MMP can be considered as a (K X + + A)-MMP.
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 + A is nef. Thus, D − K X is nef and big. In this case, it is well-known that H i (X, O X (D)) = 0 (e.g. see [21,Proposition 3.2] or [1,Proposition 3.3]).