Abstract Abstract. In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over Fq is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by q. (2) 0 If (X, ∆) is a projective klt pair over C such that −(K X + ∆) is ample, then H i (X, O X ) = 0 for i > 0. (3) 0 If (X, ∆) is a projective log pair over C such that −(K X + ∆) is ample, then H i (X, J (X, ∆)) = 0 for i > 0, where J (X, ∆) denotes the multiplier ideal sheaf of (X, ∆). Indeed, (1) 0 , (2) 0 and (3) 0 follow from the Kodaira, Kawamata-Viehweg, and Nadel vanishing theorem respectively (cf. Although the Kodaira vanishing is known to fail in positive characteristic (cf. [Ray78] ), similar vanishing still holds in positive characteristic in terms of Witt vectors. The first result in this direction is given by Esnault in [Esn03] . (1) p If X is a smooth projective variety over a perfect field of characteristic p > 0 such that −K X is ample, then H i (X, W O X,Q ) = 0 for i > 0. Then it is natural to seek a positive-characteristic analogue of (2) 0 . Indeed, this is partially established by Gongyo and the authors [GNT] . (2) p If (X, ∆) is a projective klt pair over a perfect field of characteristic p > 5 such that −(K X + ∆) is ample and dim X ≤ 3, then H i (X, W O X,Q ) = 0 for i > 0. The main theorem of this paper is a positive-characteristic analogue of (3) 0 for the three-dimensional case of characteristic p > 5. Furthermore, we treat a relative setting as follows. Theorem 1.1 (= Theorem 4.10). Let k be a perfect field of characteristic p > 5. Let (X, ∆) be a three-dimensional log pair over k and let f : X → Z be a projective k-morphism to a quasi-projective k-scheme Z. Assume that −(K X + ∆) is f -nef and f -big. Then, the equation R i f * (W I Nklt(X,∆),Q ) = 0 holds for i > 0, where Nklt(X, ∆) denotes the reduced closed subscheme of X consisting of the non-klt points of (X, ∆) and I Nklt(X,∆) is the coherent ideal sheaf on X corresponding to Nklt(X, ∆) (cf. Remark 2.2). As a consequence of Theorem 1.1, we obtain the Kollár-Shokurov connectedness theorem. Theorem 1.2 (= Theorem 4.12). Let k be a perfect field of characteristic p > 5. Let (X, ∆) be a three-dimensional log pair over k and let f : X → Z be a projective k-morphism to a quasi-projective scheme Z over k such that f * O X = O Z . Assume that −(K X + ∆) is f -nef and f -big. If Nklt(X, ∆) denotes the reduced closed subscheme of X consisting of the non-klt points of (X, ∆) and let g : Nklt(X, ∆) → f (Nklt(X, ∆)) be the induced morphism, then the fibre g −1 (z) over an arbitrary point z of f (Nklt(X, ∆)) is geometrically connected over the residue field k(z) at z. We note that Birkar proved this theorem in the case when f is birational and the coefficients of ∆ are at most one ([Bir16, Theorem 1.8]). Also, we have applications of Theorem 1.1 to rational points on varieties over finite fields. The starting-point is the following theorem by Esnault [Esn03] , which is a consequence of (1) p and a Lefschetz trace formula for W O X,Q . (1) ′ p If X is a geometrically connected smooth projective variety over a finite field k such that −K X is ample, then #X(k) ≡ 1 (mod #k). In [GNT] , Gongyo and the authors prove that the same formula still holds for Fano threefolds with klt singularities. (2) ′ p If (X, ∆) is a geometrically connected projective klt pair over a finite field k such that −(K X + ∆) is ample, then #X(k) ≡ 1 (mod #k). Then it is natural to seek an application of Theorem 1.1 to the number of the rational points on a non-klt Fano threefold. In this direction, we show that the number of the rational points on the klt-locus is divisible by #k. Theorem 1.3 (= Corollary 5.2). Let (X, ∆) be a three-dimensional geometrically connected projective log pair over a finite field k of characteristic p > 5. Assume that −(K X + ∆) is nef and big and that (X, ∆) is not klt. Then, the congruence #X(k) ≡ #V (k) mod #k holds, where V denotes the closed subset of X consisting of the non-klt points of (X, ∆). An interesting point is that this theorem is not true if we drop the assumption that (X, ∆) is not klt (cf. (2) ′ p ). On the other hand, the following theorem gives a common generalisation of (2) ′ p and Theorem 1.3. Moreover, we treat a relative setting. Theorem 1.4 (= Theorem 5.1). Let (X, ∆) be a three-dimensional log pair over a finite field k of characteristic p > 5. Let f : X → Y be a projective morphism to a quasi-projective scheme Y such that f * O X = O Y . Assume that −(K X + ∆) is f -nef and f -big. Then, the congruence #X(k) − #V (k) ≡ #Y (k) − #f (V )(k) mod #k holds, where V denotes the closed subset of X consisting of the non-klt points of (X, ∆). Furthermore, we shall show that some hypersurfaces D on smooth Fano threefolds contain rational points even if D is not klt. It can be seen as a variation of the Ax-Katz theorem (cf. [Ax64] , [Kat71] ). Theorem 1.5 (cf. Theorem 5.3). Let X be a three-dimensional projective geometrically connected variety with klt singularities over a finite field k of characteristic p > 5. Let D be a nonzero effective Q-Cartier Weil divisor on X. Assume that (1) −K X is ample, and (2) −(K X + D) is ample. Then, the congruence #D(k) ≡ 1 mod #k holds. In particular, there exists a k-rational point on D. In Theorem 5.3, we works on a more general setting. 1.1. Description of the proof. We now overview some of the ideas of the proof of Theorem 1.1. In the following, we work over a perfect field k of characteristic p > 5. Roughly speaking, the argument consists of two steps: (A) We prove the W O-vanishing for log Fano contractions, i.e. Theorem 3.10 (Section 3). (B) Using Theorem 3.10, we prove Theorem 1.1 (Section 4). (A) Now, let us overview how to prove Theorem 3.10. Given a threedimensional klt pair (X, ∆) and a projective morphism f : X → Z such that −(K X + ∆) is f -ample and f * O X = O Z , we want to prove that R i f * (W O X,Q ) = 0 for any i > 0. We further divide the proof of Theorem 3.10 into the four cases depending on the dimension of Z. Since the cases dim Z = 0 and dim Z = 3 have been settled already in [GNT] , it is enough to assume either (A1) dim Z = 1 (Subsection 3.1), or (A2) dim Z = 2 (Subsection 3.2). (A1) We first treat the case when dim Z = 1. In this case, the generic fibre X K(Z) of f is a log del Pezzo surface over an imperfect field. One of significant steps is to show that (X K(Z) × K(Z) K(Z)) red is a rational surface (Proposition 2.20). Indeed, this result enables us to use a result by Chatzistamatiou-Rülling (Theorem 3.2) after taking suitable purely inseparable covers of X and Z (cf. Proposition 3.6), which in turn implies what we want. (A2) We now treat the case when dim Z = 2. The crucial part of this case is to prove that Z has W O-rational singularities (Theorem 3.8). To this end, we first reduce the problem to the case when k = F p . Assume k = F p . In order to prove that Z has W O-rational singularities, we compute, for sufficiently divisible e ∈ Z >0 , the numbers of F p e -rational points on the models X ′ and Z ′ over F p e of X and Z respectively. (cf. Step 1 in the proof of Theorem 3.8). (B) We now overview how to prove Theorem 1.1. For simplicity, we treat only the case when Z = Spec k and k is an algebraically closed field. Taking a dlt modification of (X, ∆) (Proposition 2.10), we may assume that X is Q-factorial, (X, ∆ ∧1 ) is dlt (for definition of (−) ∧1 , see Subsection 2.1), and −(K X + ∆) is ample (cf. Lemma 4.7). By the ampleness of −(K X + ∆), we can find an effective R-divisor Ω on X such that Ω is big, (4) Supp Ω ≥1 = Supp Ω >1 , and (5) Supp Nklt(X, Ω) = Supp Nklt(X, ∆). Then it suffices to prove the vanishing H i (X, W I Nklt(X,Ω) ) = 0 for i > 0. Furthermore, we may assume that Supp Ω >1 = ∅ since the assertion is nothing but [GNT, Theorem 1.3] when (X, ∆) is klt. The first step is to run a (K X +Ω ∧1 )-MMP in order to reduce the problem to the end result (cf. Proposition 4.1). In Proposition 4.1, it is proved that the cohomologies are preserved under this MMP. Replacing X by the end result, let us assume that X itself is the end result of this MMP. By (2) and Supp Ω >1 = ∅, X has a (K X + Ω ∧1 )-Mori fibre space structure g : X → W . Then the problem is reduced to vanishing of cohomologies for dlt Mori fibre spaces (Lemma 4.6). By induction on the number of the irreducible components of Ξ for Ξ := Ω ∧1 , Lemma 4.6 is proved by using the W O-vanishing for klt Mori fibre spaces (Theorem 3.10). Acknowledgements. We would like to thank Professors Hélène Esnault, Yoshinori Gongyo and Chenyang Xu for many useful discussions. The first author is partially supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 16H06710). The second author is partially supported by EPSRC. Preliminaries 2.1. Notation. In this subsection, we summarise notation used in this paper. Preliminaries • We will freely use the notation and terminology in [Har77] and [Kol13] . Preliminaries • For a scheme X, its reduced structure X red is the reduced closed subscheme of X such that the induced closed immersion X red → X is surjective. Preliminaries Spec L is either empty or connected for any field L and any morphism Spec L → Y . Preliminaries • For an integral scheme X, we define the function field K(X) of X as O X,ξ for the generic point ξ of X. Preliminaries • 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, resp. a threefold over k) if X is a k-variety of dimension one (resp. two, resp. three). Preliminaries • For a field k, let k be an algebraic closure of k. If k is of characteristic p > 0, then we set Preliminaries • Let f : X → Y be a projective morphism of noetherian schemes. Let M be an R-Cartier R-divisor M on X. We say that M is f -ample if we can write M = r i=1 a i M i for some r ≥ 1, positive real numbers a i and f -ample Cartier divisors M i . We say that M is f -big if we can write M = A + E for some f -ample R-Cartier R-divisor A and effective R-divisor E. We can define f -nef R-divisors in the same way as in [Kol13, Definition 1.4]. We say that M is f -numerically-trivial, denoted by M ≡ f 0, if both M and −M are f -nef. Preliminaries • Let ∆ = r i D i be an R-divisor where D i are distinct prime divisors. We define ∆ ≥1 := r i ≥1 r i D i and ∆ ∧1 := r ′ i D i where r ′ i := min{r i , 1}. We also define ∆ >1 and ∆ <1 similarly. Moreover, we denote {∆} = ∆ − ⌊∆⌋. Preliminaries • A sub-log pair (X, ∆) over a filed k consists of a normal variety X over k and an R-divisor ∆ such that K X + ∆ is R-Cartier. A log pair (X, ∆) is a sub-log pair such that ∆ ≥ 0. Preliminaries • For a closed subscheme V of a scheme X, we denote by I V the quasicoherent ideal sheaf corresponding to V . For an effective R-divisor D on a normal variety X over a field, we denote by I D := I D , where D denotes the closed subscheme of X corresponding to the coherent ideal sheaf O X (− D ) on X. Preliminaries 2.2. Results on minimal model program. Preliminaries Definition 2.1. Let k be a field. Preliminaries (1) We say that (X, ∆) is a sub-klt pair over k if (X, ∆) is a sub-log pair over k such that (X, ∆) is klt in the sense of [Kol13, Definition 2.8]. Preliminaries Given a point x of X, we say that (X, ∆) is sub-klt around x if there exists an open neighbourhood X ′ of x ∈ X such that (X ′ , ∆| X ′ ) is sub-klt. (2) We say that (X, ∆) is klt (resp. log canonical) if (X, ∆) is a log pair such that (X, ∆) is klt (resp. log canonical) in the sense of [Kol13, Definition 2.8]. (3) Given a point x of X, we say that x is a non-klt point of (X, ∆) if (X, ∆) is not sub-klt around x. We define Nklt(X, ∆), called the non-klt locus of (X, ∆), as the subset of X that consists of all the non-klt points. Note that Nklt(X, ∆) is a closed subset of X, as its complement is an open subset of X by definition. We equip Nklt(X, ∆) with the reduced scheme structure. Preliminaries Remark 2.2. For coherent ideal sheaves I, J ⊂ O X with √ I = √ J , it follows that W I Q = W J Q (cf. [BBE07, Proposition 2.1]). Hence we need not to care about the scheme structure of Nklt(X, ∆) when we consider W I Nklt(X,∆),Q . Definition 2.3. A log pair (X, ∆) is called dlt if the coefficients of ∆ are at most one and there exists a log resolution g : Y → X of (X, ∆) with the condition that a E (X, ∆) > 0 holds for any g-exceptional prime divisor E on Y . Preliminaries Definition 2.4. Given a field k and a projective k-morphism f : X → Z from a normal k-variety X to a quasi-projective k-scheme Z, we say that X is of Fano type over Z if there exists an effective R-divisor ∆ on X such that (X, ∆) is klt and −(K X + ∆) is f -nef and f -big. Preliminaries Definition 2.5. Given a field k, a log pair (X, ∆) over k, and projective Preliminaries Lemma 2.6. Let k be a field and let f : X → Y be a projective birational k-morphism of normal varieties over k. Let (Y, ∆ Y ) be a sub-log pair and let ∆ be the R-divisor defined by K X + ∆ = f * (K Y + ∆ Y ). Then the following hold. Preliminaries (1) Let x be a closed point of X. If x is a non-klt point of (X, ∆), then f (x) is a non-klt point of (Y, ∆ Y ). Preliminaries (2) Let y be a closed point of Y . If y is a non-klt point of (Y, ∆ Y ), then there exists a closed point x of X such that f (x) = y and x is a non-klt point of (X, ∆). In particular, there exists a commutative diagram consisting of projective morphisms: Proof. Both of the assertions follow from the fact that (U, ∆| U ) is sub-klt if and Preliminaries Proposition 2.7. Let (X, ∆) be a Q-factorial sub-log pair over a field such that (X, (∆ ≥0 ) <1 ) is klt. Then it holds that Preliminaries Proof. The second equation follows from the fact that (X, (∆ ≥0 ) <1 ) is klt. Clearly the inclusion Nklt(X, ∆) ⊂ Nklt(X, ∆ ≥0 ) holds. It suffices to prove the opposite one. Let D be a prime divisor contained in Supp ∆ ≥1 . Since Nklt(X, ∆) is a closed subset containing general closed points of D, we have that D ⊂ Nklt(X, ∆). In particular, Supp ∆ ≥1 ⊂ Nklt(X, ∆). Preliminaries Lemma 2.8. Let K ∈ {Q, R}. Let k be a perfect field of characteristic p > 0. Let (X, ∆) be a quasi-projective dlt pair over k with dim X ≤ 3. Let A be an ample K-Cartier K-divisor on X. Then there exists an effective Theorem 2.9. Let k be a perfect field of characteristic p > 5. Let (X, ∆) be a three-dimensional Q-factorial log canonical pair over k, where ∆ is an R-divisor. Let f : X → Z be a projective k-morphism to a quasi-projective k-scheme Z. Then there exists a (K X + ∆)-MMP over Z that terminates. In other words, there is a sequence of birational maps of three-dimensional normal varieties: Preliminaries such that if ∆ i denotes the proper transform of ∆ on X i , then the following properties hold: Preliminaries (1) For any i ∈ {0, . . . , ℓ}, (X i , ∆ i ) is a Q-factorial log canonical pair which is projective over Z. Preliminaries Proof. See [HNT, Theorem 1.1]. Preliminaries Proposition 2.10. Let (X, ∆) be a three-dimensional quasi-projective log pair over a perfect field k of characteristic p > 5. Then there exists a projective birational morphism f : Y → X that satisfies the following conditions: Preliminaries Proof. See [HNT, Proposition 3.5]. Preliminaries 2.3. Connectedness theorem for the birational case. The purpose of this subsection is to establish the three-dimensional Kollár-Shokurov connectedness theorem for the birational case (Theorem 2.12). To this end, we first prove the following. Preliminaries Proposition 2.11. Let k be a perfect field of characteristic p > 5. Let (V, ∆) be a three-dimensional quasi-projective Q-factorial log pair over k. Let ϕ : U → V be a log resolution of (V, ∆). Let ∆ U be the R-divisor defined by K U + ∆ U = ϕ * (K V + ∆). Then the induced morphism Nklt(U, ∆ U ) → V has connected fibres. Preliminaries Proof. Taking the base change to the algebraic closure of k, we may assume that k is an algebraically closed field. Let F be the sum of the ϕ-exceptional prime divisors F ′ on U whose log discrepancies are positive: Preliminaries For any i ∈ {0, . . . , ℓ}, we define ∆ X i , F X i and D X i as the push-forwards of ∆ U , F and D U respectively. Then it holds that Preliminaries , where ψ i : X i → V denotes the induced morphism. Moreover, for any i ∈ {0, · · · , ℓ}, we get Preliminaries Indeed, (1)' and (3)' follow directly from (1) and (3) respectively. Also, (2)' holds by (2) and {(D Preliminaries Step 1. For any i ∈ {0, . . . , ℓ}, it holds that Preliminaries where the first equation holds by (2)' and Proposition 2.7, and the second one follows from (3)'. The remaining inclusion Nklt( Preliminaries is obvious. This completes the proof of Step 1. Preliminaries Step 2. Let g : Preliminaries ) has connected fibres. This completes the proof of Step 2. Preliminaries Step 3. Let h : X i X i+1 be a flip appearing in the MMP (2.11.1). If Preliminaries Let ϕ i : X i → Y be the flipping contraction and let Preliminaries By assumption, it holds that the composite morphism Preliminaries is a surjective morphism with connected fibres. In particular, N Y → N V has connected fibres. Since some fibre of the composite morphism Preliminaries is not connected, the same property holds for Nklt( Preliminaries then we get a contradiction as follows. [Bir16, Theorem 1.8] implies that D X i ,1 → Y has connected fibres, which contradicts the decomposition: (5)]). This completes the proof of Step 3. Preliminaries Step 4. The induced morphism Nklt(X ℓ , D ℓ ) → V has connected fibres. Preliminaries Proof of Step 4. By the negativity lemma, any ϕ ℓ -exceptional prime divisor Preliminaries ℓ (v) holds and this is connected for any closed point v of V . This completes the proof of Step 4. Preliminaries Step 2, Step 3 and Step 4 complete the proof of Proposition 2.11. Preliminaries Theorem 2.12. Let k be a perfect field of characteristic p > 5. Let f : X → V be a projective birational k-morphism of normal quasi-projective threefolds over k. Let (X, ∆) be a sub-log pair over k such that −(K X + ∆) is f -nef and f * ∆ is effective. Then the induced morphism Nklt(X, ∆) → V has connected fibres. Preliminaries Proof. Replacing ∆, we may assume that K X + ∆ ∼ R 0. In particular, for ∆ V := f * ∆, it holds that (V, ∆ V ) is a log pair and Preliminaries has connected fibres. Thus, the composite morphism Preliminaries has connected fibres and factors through Nklt(X, ∆). In particular, also Nklt(X, ∆) → Nklt(V, ∆ V ) has connected fibres. Results on the Witt vector cohomologies. For the definition of the Witt vector cohomology and its properties, we refer to [GNT] and [CR12] . Our goal of this subsection is to show Proposition 2.16 and Proposition 2.17. For this, we start by showing some auxiliary results. Results on the Witt vector cohomologies. Lemma 2.13. Let k ⊂ k ′ be an extension of perfect fields of characteristic p > 0. Let X be a proper scheme over k and we set Results on the Witt vector cohomologies. is bijective. Results on the Witt vector cohomologies. Proof. Taking the Stein factorisation of the structure morphism X → Spec k, we may assume that X is of dimension zero. Replacing X by a connected component, we may assume that X = Spec L, where L is a finite extension of k. Then the assertion is clear. Results on the Witt vector cohomologies. Lemma 2.14. Let k be a perfect field of characteristic p > 0 and let X be a one-dimensional smooth projective scheme over k. Then the following are equivalent. Results on the Witt vector cohomologies. ( Results on the Witt vector cohomologies. Proof. Clearly we may assume that X is connected. Thanks to an exact sequence Results on the Witt vector cohomologies. and the fact that X is one-dimensional, it holds that (1), (2) and (3) Results on the Witt vector cohomologies. and the fact that X is one-dimensional, (4) implies (1). Results on the Witt vector cohomologies. The equivalence between (4) and (5) follows from the fact that Results on the Witt vector cohomologies. Lemma 2.15. Let k ⊂ k ′ be an extension of perfect fields of characteristic p > 0. Let X be a proper one-dimensional scheme over k. Then the following are equivalent. Results on the Witt vector cohomologies. ( Results on the Witt vector cohomologies. Proof. For simplicity, we denote Results on the Witt vector cohomologies. We may assume that X is reduced. Let Results on the Witt vector cohomologies. be the normalisation of X. Thanks to Lemma 2.14, if one of (1) and (2) holds, then it follows that Results on the Witt vector cohomologies. For the conductor subschemes C and D of X and X N respectively, we have a commutative diagram with exact horizontal sequences: Results on the Witt vector cohomologies. where (−) K ′ denotes the tensor product (−) ⊗ K K ′ . As both α and β are isomorphisms by Lemma 2.13, so is γ by the 5-lemma, as desired. Results on the Witt vector cohomologies. Proposition 2.16. Let k ⊂ k ′ be an extension of perfect fields of characteristic p > 0. Let X be a normal surface over k. Then the following are equivalent. Results on the Witt vector cohomologies. (1) X has W O-rational singularities. Results on the Witt vector cohomologies. Proof. We may assume that Q is a unique non-regular point of X. Let f : Y → X be a resolution of singularities such that f (Ex(f )) = Q. For E := Ex(f ), we have an exact sequence: Results on the Witt vector cohomologies. Thanks to the vanishing Results on the Witt vector cohomologies. Therefore, it follows from Lemma 2.15 that (1) and (2) are equivalent. Results on the Witt vector cohomologies. Proposition 2.17. Let k be a perfect field of characteristic p > 0 and let X be a reduced projective scheme over k such that (a) any irreducible component of X is one-dimensional, and (b) any non-regular point x of X is an ordinary double point. Then the following are equivalent. Results on the Witt vector cohomologies. ( Results on the Witt vector cohomologies. (6) Any connected component of X × k k is a tree of smooth rational curves. Results on the Witt vector cohomologies. Proof. We may assume that X is connected. Moreover, replacing k by k ′ for the Stein factorisation X → Spec k ′ → Spec k, we may assume that X is geometrically connected. We now show that the assertions (1), (2), (3) and (4) are equivalent. Thanks to an exact sequence Results on the Witt vector cohomologies. and the fact that X is one-dimensional, it holds that (1), (2) and (3) are equivalent. We have that (3) implies (4) by [GNT, Lemma 2.19] . Moreover, by an exact sequence Results on the Witt vector cohomologies. and the fact that X is one-dimensional, (4) implies (1). Thus (1), (2), (3) and (4) are equivalent. Results on the Witt vector cohomologies. Thanks to [Kol96, Ch II, Lemma 7.5], it holds that (6) implies (1). Further, (4) clearly implies (5). Thus it suffices to show that (5) implies (6). Lemma 2.15 allows us to replace X → Spec k by the base change X × k k → Spec k. Then it follows from [CR12, the second last paragraph of Section 4.6] that (5) implies (6), as desired. Results on the Witt vector cohomologies. 2.5. Geometric rationality of del Pezzo surfaces over imperfect fields. In this subsection, we prove Proposition 2.20. To this end, we start with the following lemma. Results on the Witt vector cohomologies. Lemma 2.18. Let k be a separably closed field of characteristic p > 0 which is not algebraic over a finite field. Let X be a projective normal Q-factorial In what follows, we denote by Y the normalisation of (X × k k) red , and denote by f : Y → X the composite morphism. By applying [Tana, Theorem 1.1] to the regular locus of X, we can write Results on the Witt vector cohomologies. Assume that Y is not rational and let us derive a contradiction. Let µ : Z → Y be the minimal resolution of Y . Since Z is an irrational ruled surface, there is a projective morphism π : Z → B onto a smooth projective irrational curve whose general fibres are P 1 . Since k = F p , it follows from [Tan14, Theorem 3.20] that π factors through µ: Results on the Witt vector cohomologies. This is a contradiction to ρ(Y ) = 1. Thus we are done for the case (a). Results on the Witt vector cohomologies. Suppose (b). Since −(K X + ∆) is ample, there exists an extremal ray R of NE(X) not corresponding to π 1 . By [Tanb, Theorem 4.4], the extremal ray R induces either a birational morphism or another (K X + ∆)-Mori fibre space X → B 2 onto a curve B 2 . If the former case occurs, then the problem is reduced to the case (a). Therefore, we may assume that there exist two Mori fibre space structures π 1 : X → B 1 and π 2 : X → B 2 onto curves B 1 and B 2 . In particular, any fibre of π i dominates B 3−i . Let π ′ i : Y → B ′ i be the Stein factorisation of the composite morphism: Results on the Witt vector cohomologies. i is isomorphic to P 1 . In particular, B ′ 1 ≃ P 1 and Y is rational. Remark 2.19. The statement of Lemma 2.18 does not hold if we drop the assumption on the base field k. Indeed if k = F p , then any normal surface is Q-factorial (e.g. see [Tan14, Theorem 4.5]). Thus the cone X over an elliptic curve over F p is Q-factorial and −K X is ample. Results on the Witt vector cohomologies. Proposition 2.20. Let (X, ∆) be a projective two-dimensional klt pair over a field k of characteristic p > 0 such that −(K X +∆) is nef and big. Assume that k = H 0 (X, O X ). Then, (X × k k) red is a rational surface. In particular X is rationally connected over k. Results on the Witt vector cohomologies. Proof. We may assume that k is separably closed. Since the assertion is well-known if k is an algebraically closed field (cf. [Tan15, Fact 3.4 and Theorem 3.5]), the problem is reduced to the case when k is an imperfect field. In particular, k is not algebraic over a finite field. As X is Q-factorial [Tanb, Corollary 4.11], the assertion follows from Lemma 2.18. W O-vanishing for log Fano contractions In this section, we prove a vanishing theorem for log Fano contractions (Theorem 3.10). We shall divide the proof into the cases depending on the dimension of the base scheme Z. The cases dim Z = 1 and dim Z = 2 are treated in Subsection 3.1 and Subsection 3.2 respectively. The remaining cases dim Z = 0 and dim Z = 3 have been already settled in [GNT] (cf. the proof of Theorem 3.10). W O-vanishing for log Fano contractions Before starting the case study, we summarise some results used repeatedly in the proof of Theorem 3.10. W O-vanishing for log Fano contractions Theorem 3.1. Let k be a perfect field of characteristic p > 5. Then the following hold. W O-vanishing for log Fano contractions (1) If (X, ∆) is a three-dimensional klt pair over k, then X has W Orational singularities. (2) If X is a three-dimensional projective varitey of Fano type over k, W O-vanishing for log Fano contractions Proof. When ∆ is Q-divisor, both (1) and (2) W O-vanishing for log Fano contractions projective k-morphisms of normal k-varieties that satisfies the following properties. W O-vanishing for log Fano contractions (1) Both α and β are finite universal homeomorphisms. W O-vanishing for log Fano contractions red is a finite birational morphism. In particular, this morphism coincides with the normalisation of W O-vanishing for log Fano contractions ) and ν 1 is a finite universal homeomorphism. Since ν 1 is a finite birational morphism and X ′ 1 is normal, the ν 1 is nothing but the normalisation of (X × Y L) red . Furthermore, X ′ 1 is geometrically normal, since X ′ 0 is normal and W O-vanishing for log Fano contractions 3.1. Del Pezzo fibrations. In this subsection, we establish the W O-vanishing for del Pezzo fibrations (Proposition 3.6). A key result is the following. W O-vanishing for log Fano contractions Lemma 3.4. Let k be a perfect field of characteristic p > 0. Let f : X → Y be a projective k-morphism such that (1) X is a normal threefold over k that has W O-rational singularities, (2) Y is a smooth k-curve, W O-vanishing for log Fano contractions ) the geometric generic fibre X K(Y ) of f is a normal rational surface. W O-vanishing for log Fano contractions Proof. We divide the proof into two steps. W O-vanishing for log Fano contractions Step 1. The assertion of Lemma 3.4 holds if there exists a projective bira- W O-vanishing for log Fano contractions Furthermore, we may assume that • Z ′ is an integral scheme, W O-vanishing for log Fano contractions • g ′ is a projective birational morphism, and • the composite morphism h ′ is smooth. In particular, Z ′ is a smooth threefold over k. Let g : Z → X be a smooth projective compactification of the induced morphism W O-vanishing for log Fano contractions i.e. there are morphisms W O-vanishing for log Fano contractions and Z is an integral scheme smooth over k. In particular Z is a smooth threefold over k which is projective over X, hence over Y . We get the composite morphism W O-vanishing for log Fano contractions In particular, Z × Y K(Y ) is a smooth projective rational surface over K(Y ). Therefore we have that W O-vanishing for log Fano contractions where the first isomorphism follows from the assumption (1) and the third follows from Theorem 3.2. This completes the proof of Step 1. W O-vanishing for log Fano contractions Step 2. The assertion of Lemma 3.4 holds without any additional assumptions. Proof of Step 2. Let Then there exist a finite purely inseparable extension L of K(Y ) and a projective normal L-surface Z 2 with the following projective birational L 0 -morphism Proof of Step 2. Let whose base change by (−) × L K(Y ) 1/p ∞ is isomorphic to g 1 . In particular, Z 2 is a smooth projective surface over L. Proof of Step 2. Let Let Y 2 be the normalisation of Y in L and let X 2 be the normalisation of (X × Y Y 2 ) red . We get a commutative diagram of normal k-varieties: Proof of Step 2. Let Claim 3.5. The following hold. Proof of Step 2. Let (a) There exists a non-empty open subset Y 3 of Y 2 such that the induced morphism Proof of Step 2. Let Proof of Claim 3.5. Note that the generic fibre of X× Y Y 2 → Y 2 is normal by the geometric normality of X K(Y ) (the assumption (4)). Therefore (a) holds since X 2 → (X × Y Y 2 ) red is the normalisation. It is clear that (b) follows from (a). As K(Y ) ⊂ L is a purely inseparable extension, the assertion (d) holds. Proof of Step 2. Let Let us show (c). It follows from the construction that α : X 2 → X is a finite surjective morphism of normal schemes. In particular X 2 coincides with the normalisation of X in K(X 2 ). Therefore, it suffices to show that the field extension K(X) ⊂ K(X 2 ) is purely inseparable, which in turn follows from the equations Proof of Step 2. Let is a finite birational morphism of integral k-varieties. As Y 2 is normal, it holds that Y ′ 2 → Y 2 is an isomorphism, hence we obtain (e). This completes the proof of Claim 3.5. Proof of Step 2. Let Let us go back to the proof of Step 2. Thanks to (c), (d), (e) and (b) of Claim 3.5, also f 2 satisfies the same properties (1), (2), (3) and (4) hold for f 2 respectively. By Step 1, it holds that R i (f 2 ) * (W O X 2 ,Q ) = 0 for i > 0. Thanks to (c) and (d) of Claim 3.5, we have that R i f * (W O X,Q ) = 0 for i > 0. This completes the proof of Step 2. Proof of Step 2. Let Step 2 completes the proof of Lemma 3.4. Proposition 3.6. Let k be a perfect field of characteristic p > 5. Let (X, ∆) be a three-dimensional klt pair over k and let f : X → Y be a projective kmorphism to a smooth k-curve Y such that Proof of Step 2. Let Proof. Applying Lemma 3.3 to f , we obtain a commutative diagram Proof of Step 2. Let that satisfies the properties listed in Lemma 3.3. Since X has W O-rational singularities and α is a finite universal homeomorphism, it holds that X ′ has W O-rational singularities. Furthermore, the geometric generic fibre X K(Y ′ ) of f ′ is normal (Lemma 3.3 (3)) and hence it is a normal rational surface by Proposition 2.20. Therefore, it follows from Lemma 3.4 that Proof of Step 2. Let where the first and last isomorphisms hold because α and β are finite universal homeomorphisms (Lemma 3.3 (1)), and the second isomorphism follows from Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Lemma 3.7. Let k be a perfect field of characteristic p > 0. Let f : X → Y be a proper birational k-morphism of normal k-surfaces. If Y has W Orational singularities, then so does X. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Proof. Fix a resolution of singularities of X: ϕ : V → X. We have the following exact sequence induced by the corresponding Grothendieck spectral sequence: Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. We obtain R 1 (f • ϕ) * (W O V,Q ) = 0 since Y has W O-rational singularities. Moreover, we have that R 2 f * (W O X,Q ) = 0, as the fibres of f are at most one-dimensional (cf. [GNT, Lemma 2.20]). Therefore, it holds that f * R 1 ϕ * (W O V,Q ). Thanks to the fact that Supp Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Theorem 3.8. Let k be a perfect field of characteristic p > 5. Let f : X → Y be a projective k-morphism of normal k-varieties with f * O X = O Y which satisfies the following properties. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. (1) dim X = 3 and dim Y = 2. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. (2) There exists an effective R-divisor ∆ on X such that (X, ∆) is klt and −(K X + ∆) is f -nef and f -big. Then Y has W O-rational singularities. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Proof. Replacing ∆, we may assume that −(K X + ∆) is f -ample. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Step 1. The assertion of Theorem 3.8 holds if k = F p . Proof of Step 1. In this proof, X(F p e ) denotes the number of F p e -rational points on a model X 0 of X over F p e , i.e. X 0 is a projective F p e -scheme such that X × F p e k ≃ X. We can define X(F p e ) if e is a sufficiently divisible positive integer and we fix a model X 0 (this number possibly depends on the choice of a model X 0 ). Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. We may assume that Y has a unique singular point y. Let g : Y ′ → Y be a log resolution such that g(Ex(g)) = {y}. Set C := Ex(g) = g −1 (y). Let ϕ : W → X be a log resolution of (X, ∆) that admits a morphism to Y ′ . Then f −1 (y) is rationally chain connected ([GNT, Theorem 4.1]). Hence by [GNT, Theorem 4 .8], also (f • ϕ) −1 (y) is rationally chain connected. Therefore its image on Y ′ , which is nothing but C, is a union of rational curves. It suffices to show that Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. for any sufficiently divisible positive integer e, because this property deduces that C is a tree of smooth rational curves, which in turn implies that Y has W O-rational singularities by Proposition 2.17. Let E be the sum of all the ϕ-exceptional prime divisors. We run a ( Take an arbitrary divisible positive integer e. We obtain Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. On the other hand, X and X 1 are of Fano type over Y and Y 1 respectively. Hence by [GNT, Theorem 5 .4], we obtain Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. To summarise, we get Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. This completes the proof of Step 1. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Step 2. The assertion of Theorem 3.8 holds if k is algebraically closed. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Proof of Step 2. We fix a closed point y ∈ Y and we may assume that y is a unique singularity of Y . Let g : Z → Y be a log resolution such that g(Ex(g)) = y. By Proposition 2.17, Y has W O-rational singularities if and only if Ex(g) is a tree of smooth rational curves. We take a model over Step 3. The assertion of Theorem 3.8 holds without any additional assumptions. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Proof of Step 3. Thanks to Proposition 2.16, we may assume that k is algebraically closed. Then the assertion of Theorem 3.8 follows from Step 2 Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Step 3 completes the proof of Theorem 3.8. (1) dim X = 3 and dim Y = 2. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. (2) There exists an effective R-divisor ∆ on X such that (X, ∆) is klt and −(K X + ∆) is f -nef and f -big. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Proof. By Lemma 3.3, there exists a commutative diagram Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. of projective k-morphisms of normal k-varieties which satisfies the properties listed in Lemma 3.3. Since (X, ∆) is klt, X has W O-rational singularities (Theorem 3.1 (1)). By Theorem 3.8, also Y has W O-rational singularities. Since α and β are finite universal homeomorphisms (Lemma 3.3 (1)), also X ′ and Y ′ have W O-rational singularities. Since the generic fibre of f ′ is smooth and will be a rational curve after taking the base change to the algebraic closure, it follows from Theorem 3.2 that Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. where the first and the last isomorphisms follow because α and β are finite universal homeomorphisms (Lemma 3.3 (1)), and the second isomorphism follows from Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. 3.3. Proof of W O-vanishing for log Fano contractions. We now prove the main theorem of this section. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational singularities (Theorem 3.8). Let us start by recalling the following basic fact. Theorem 3.10. Let k be a perfect field of characteristic p > 5. Let f : X → Y be a projective k-morphism of normal k-varieties. Assume that dim X ≤ 3 and there exists an effective R-divisor ∆ such that (X, ∆) is klt and −(K X + ∆) is f -nef and f -big. Then R i f * (W O X,Q ) = 0 for i > 0. Conic bundles. In this subsection, we prove the W O-vanishing for conic bundles (Proposition 3.9). To this end, we show that their base schemes have W O-rational
{'paper_id': '119617521', 'title': 'A Witt Nadel vanishing theorem for threefolds', 's2_url': 'https://api.semanticscholar.org/CorpusID:119617521', 'source': 'S2ORC'}