Global Frobenius Liftability II: Surfaces and Fano Threefolds
aa r X i v : . [ m a t h . AG ] F e b GLOBAL FROBENIUS LIFTABILITY II: SURFACES AND FANO THREEFOLDS
PIOTR ACHINGER, JAKUB WITASZEK, AND MACIEJ ZDANOWICZ
Abstract.
In this article, a sequel to [AWZ17], we continue the development of a comprehensivetheory of Frobenius liftings modulo p . We study compatibility of divisors and closed subschemeswith Frobenius liftings, Frobenius liftings of blow-ups, descent under quotients by some groupactions, stability under base change, and the properties of associated F -splittings. Consequently,we characterise Frobenius liftable surfaces and Fano threefolds, confirming the conjecture stated inour previous paper. Introduction
Let X be a smooth and proper algebraic variety over an algebraically closed field k of characteristic p > . We call X F -liftable if there exists a flat lifting ∼ X of X over W ( k ) together with a lifting ∼ F X : ∼ X → ∼ X of the absolute Frobenius F X : X → X ; the pair ( ∼ X, ∼ F X ) is called a Frobenius lifting of X . While local liftings of Frobenius are related to the Cartier operator and are of utmost importancein characteristic p geometry [DI87], global Frobenius liftings are very rare and heavily restrict thegeometry of the variety X in quite subtle ways (see [BTLM97], where this was studied extensivelyin the case of homogeneous spaces, and [Xin16] for the case of surfaces). In the paper [AWZ17],we have begun a comprehensive study of F -liftable varieties, and proposed the following conjecturalcharacterization. Conjecture 1 ([AWZ17, Conjecture 1]) . Let X be a smooth projective variety over an algebraicallyclosed field k of characteristic p > . If X is F -liftable, then there exists a finite étale Galois cover f : Y → X such that the Albanese morphism a Y : Y −→ Alb( Y ) is a toric fibration. In particular, if X is simply connected, then it is a toric variety. The goal of the present paper is to further the understanding of the geometric aspects of F -liftability.We provide several general results on Frobenius liftings, particularly in relationship with F -splittings,that often apply also to X proper or singular. In the end, we put all of our findings to use and settleConjecture 1 in the following cases. Theorem 1.
Conjecture 1 is true if(1) dim X ≤ (see Theorem 6.9), or(2) X is a Fano threefold and p ≫ , assuming a form of boundedness (see Theorem 7.2 andRemark 7.3(b)). Verifying whether a given variety is F -liftable can be difficult in practice. As shown in [BTLM97],a smooth projective F -liftable variety satisfies Bott vanishing (1.1) H j ( X, Ω iX ⊗ L ) ( j > , L ample ) . While this property alone heavily restricts the geometry of an F -liftable variety, it is also cumbersometo verify, nor does it characterize F -liftable varieties [Tot18]. One of our goals is to provide a toolkitwhich enables one to deduce that a given variety is not F -liftable using geometric methods.From now on let us work over a fixed perfect field k of characteristic p > . Recall that an F -splitting on a k -scheme X is an O X -linear splitting of the map O X → F X ∗ O X . There is a consid-erable amount of interplay between Frobenius liftings and F -splittings. First, if X is normal, thenevery Frobenius lifting of X induces an F -splitting. Second, properties of a Frobenius lifting areoften reflected in the induced F -splitting (for example, being compatible with a divisor). However,Frobenius liftings tend to be more rigid than F -splittings; for example, every Frobenius lifting of aproduct of projective varieties arises as a product of Frobenius liftings of the factors (Corollary 2.11and Proposition 3.7), while the corresponding fact is not true for F -splittings. Date : February 5, 2021.2010
Mathematics Subject Classification.
Primary 14G17, Secondary 14M17, 14M25, 14J45.
In [Zda18], the third author provided an explicit functorial construction of a lifting modulo p associated to an F -splitting (see Theorem 2.4), which we exploit in §2. Using some properties of theWitt vector scheme W ( X ) , we show the following: Theorem 2 (Theorem 2.7) . Let ( ∼ X, ∼ F X ) be a Frobenius lifting of a k -scheme X , and let σ be aFrobenius splitting on X . Let ∼ X ( σ ) be the canonical lifting of X induced by σ as in [Zda18] . Thenthere is a canonical isomorphism ∼ X ( σ ) ≃ ∼ X of liftings of X . In particular, if X is normal, then there exists at most one lifting ∼ X to which the Frobenius lifts(Corollary 2.8).As observed and applied in [AWZ17], one can descend the property of being F -liftable along fibra-tions f : Y → X , or more generally certain maps for which f ∗ : O X → Rf ∗ O Y is a split monomor-phism ([AWZ17, Theorem 3.3.6], restated as Theorem 2.10(ab) below). Using Frobenius splittingsand canonical liftings described above, we can extend this to quotients by linearly reductive groups(Theorem 2.10(c)). Similar ideas allow us to treat F -liftability of products (Corollary 2.11 andProposition 3.7).In a different direction, we deal with F -liftability of pairs (§3). We study Frobenius liftings ofsimple normal crossings pairs ( X, D ) and introduce the related notion of a Frobenius lifting of a pair ( X, Z ) where Z ⊆ X is a closed subscheme (Definition 3.4). We show: Theorem 3 (Proposition 3.6) . Suppose that X and Z are smooth, and let ( ∼ X, ∼ Z ) be a lifting of ( X, Z ) . Further, let ∼ π : ∼ Y → ∼ X be the blow-up along ∼ Z . Then a lifting of Frobenius ∼ F X on ∼ X iscompatible with ∼ Z if and only if it extends to a lifting of Frobenius on ∼ Y . The induced lifting on ∼ F Y is automatically compatible with the exceptional divisor ∼ π − ( ∼ Z ) . The relationship between Frobenius liftings and F -splittings is particularly useful in a relativesituation, which we explore in §4. In §5, using Witt vectors of general rings we show a fundamentalproperty of Frobenius liftings (Proposition 5.1): if ( ∼ Y , ∼ F Y ) is a Frobenius lifting of a scheme Y , ϕ : Z → Y is a map, and ∼ Z is a lifting of Z to W ( k ) , then ϕ ◦ F Y has a preferred lifting to amap ψ : ∼ Z → ∼ Y . Using this, we prove a curious ‘base change’ property of Frobenius liftings (seeCorollary 5.2), which in particular implies that the fibers of an “ F -liftable morphism” are F -liftable.Further, we analyse the induced relative F -splittings.Let us end this introduction with a few words about the proof of Theorem 1. The case of surfaces(§6) is quickly reduced to rational and ruled surfaces, in which cases the question is still not completelytrivial. In the case of rational surfaces, our approach proceeds by induction with respect to the numberof contractions necessary to obtain a Hirzebruch surface from a given surface. Note that for ruledsurfaces our results do not entirely agree with [Xin16], see Remark 6.10. The case of Fano threefolds(§7) relies on the Mori–Mukai classification with about 100 families; see Remark 7.3 for a discussion.Since many of these varieties naturally arise as blow-ups, Theorem 3 is particularly useful in treatingtheir F -liftability. The case which caused us the most trouble initially was the blow-up of P k along atwisted cubic, for which we used our study of relative F -splittings associated to F -liftable fibrationsmentioned above. Since the Mori–Mukai classification is not known in characteristic p , we can onlyapply it to a lift of a given F -liftable Fano threefold to characteristic zero. Descending the informationobtained this way back to characteristic p relies on a slightly delicate argument with Hilbert schemes(see Proposition 7.6).Although a proof based on the classification might seem tedious and not very useful, we embarkedon this task for two reasons. First, the proof confirms our belief that Frobenius liftability is veryrare among smooth projective varieties, and provides substantial empirical evidence for the validityof Conjecture 1. But what is even more important is that the examples we have analyzed were asource of inspiration in our study of general properties of F -liftable varieties. Throughout the paper, we work over a fixed perfect field k of characteristic p > . Acknowledgements.
We would like to thank Paolo Cascini, Nathan Ilten, Adrian Langer, ArthurOgus, Vasudevan Srinivas, Nicholas Shepherd-Barron, and Jarosław Wiśniewski for helpful sugges-tions and comments. Part of this work was conducted during the miniPAGES Simons Semester atthe Banach Center in Spring 2016. The authors would like to thank the Banach Center for hos-pitality. P.A. was supported by NCN OPUS grant number UMO-2015/17/B/ST1/02634 and NCNSONATA grant number UMO-2017/26/D/ST1/00913. J.W. was supported by the Engineering andPhysical Sciences Research Council [EP/L015234/1]. M.Z. was supported by NCN PRELUDIUM
LOBAL FROBENIUS LIFTABILITY II 3 grant number UMO-2014/13/N/ST1/02673. This work was partially supported by the grant 346300for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund.2. F -liftability and F -splittings In this section we study the interplay between Frobenius liftings and Frobenius splittings. Onthe one hand, a Frobenius lifting of a normal variety induces a Frobenius splitting. On the other,a Frobenius splitting induces a canonical lifting of the underlying variety modulo p (albeit withouta lifting of Frobenius in general). Comparing the two constructions will allow us to descend theproperty of being F -liftable along some group quotient maps and treat F -liftability of products.2.1. F -splittings associated to Frobenius liftings. A Frobenius lifting of a k -scheme X is, bydefinition, a flat scheme ∼ X over W ( k ) lifting X endowed with a morphism ∼ F : ∼ X → ∼ X lifting theabsolute Frobenius F X . If such a pair ( ∼ X, ∼ F ) exists, we call X F -liftable . To avoid confusion,we shall often write ∼ F X instead of ∼ F . An F -splitting on X is an O X -linear splitting of the map F ∗ X : O X → F X ∗ O X , and if it exists, we call X F -split (the standard reference for Frobenius splittingsis [BK05, Chapter I]).
Example 2.1 ([AWZ17, §3.1]) . The following are examples of F -liftable varieties:(a) ordinary abelian varieties,(b) more generally, a variety admitting (a) as a finite étale cover,(c) toric varieties,(d) more generally, a toric fibration [AWZ17, §2.1] over an ordinary abelian variety.We extend the above terminology to morphisms as follows. We fix a Frobenius lifting ( ∼ S, ∼ F S ) ofa base k -scheme S , and then by a Frobenius lifting of a scheme X → S we shall mean a Frobeniuslifting ( ∼ X, ∼ F X ) and a map ∼ X → ∼ S lifting X → S which is compatible with the Frobenius liftingson source and target. We then obtain a lifting ∼ F X/S : ∼ X → ∼ X ′ = ∼ F ∗ S ∼ X of the relative Frobenius F ∗ X/S : X → X ′ = F ∗ S X . An F -splitting of the map X → S (or an F -splitting on X relative to S ) isan O X -linear splitting of F ∗ X/S : O X ′ → F X/S, ∗ O X .In the above situation, and assuming that X → S is smooth, we often study the Frobenius liftingof X/S by means of the morphism(2.1) ξ = 1 p ∼ F ∗ X/S : F ∗ X/S Ω X ′ /S −→ Ω X/S induced by ∼ F ∗ X/S : ∼ F ∗ X/S Ω ∼ X ′ / ∼ S → Ω ∼ X/ ∼ S , using that F ∗ X/S : F ∗ X/S Ω X ′ /S → Ω X/S is the zero map.
Proposition 2.2 ([DI87, Proof of Théorème 2.1 and §4.1], [BTLM97, Theorem 2]) . Let ( ∼ S, ∼ F S ) bea Frobenius lifting of a k -scheme S , and let ( ∼ X, ∼ F X ) , ∼ X → ∼ S be a Frobenius lifting of a smoothmorphism X → S . The morphism ξ defined above satisfies the following properties:(a) The adjoint morphism ξ ad : Ω X ′ /S → F X/S ∗ Ω X/S has image in the subsheaf Z X/S of closedforms and provides a splitting of the short exact sequence −→ B X/S −→ Z X/S C
X/S −→ Ω X ′ /S −→ . (b) The Grothendieck dual of the determinant det( ξ ) : F ∗ X/S ω X ′ /S −→ ω X/S is a morphism F X/S, ∗ O X → O X which furnishes a Frobenius splitting σ of X relative to S .In particular, the homomorphism ξ is injective. In particular, if S = Spec k and X is normal, then every Frobenius lifting of X induces an F -splittingon X by [BK05, 1.1.7(iii)].2.2. Canonical liftings of F -split schemes. We shall now recall a functorial construction of alifting to W ( k ) of a k -scheme endowed with an F -splitting due to the third author. P. ACHINGER, J. WITASZEK, AND M. ZDANOWICZ
Definition 2.3.
The category
FSplit of F -split schemes has as objects F -split schemes, i.e. pairs ( X, σ X ) of a k -scheme X and a Frobenius splitting σ X on X , and as morphisms ( X, σ X ) → ( Y, σ Y ) maps f : X → Y over k for which the following diagram commutes F Y ∗ f ∗ O X f ∗ F X ∗ O X f ∗ ( σ X ) / / f ∗ O X F Y ∗ O YF Y ∗ ( f ∗ ) O O σ Y / / O Y . f ∗ O O Theorem 2.4 ([Zda18, Theorem 3.6]) . There exists a functor ( X, σ ) ∼ X ( σ ) : FSplit −→ ( flat schemes /W ( k )) together with a functorial identification X ≃ ∼ X ( σ ) ⊗ W ( k ) k . The structure sheaf O ∼ X ( σ ) can bedescribed as the quotient of W ( O X ) by the ideal { (0 , f ) : σ ( f ) = 0 } . F -splittings and quotients by linearly reductive groups. For a group G acting on Y , wesay that a morphism π : Y → X is a good quotient by G , if it is G -invariant, affine, and O X = ( π ∗ O Y ) G .The following lemma will be applied for G being a torus or a finite group of order prime to p . Lemma 2.5.
Let Y be an F -split scheme of finite type over k , let G be a linearly reductive algebraicgroup acting on Y , and let π : Y → X be a good quotient. Then there exist splittings σ Y : F Y ∗ O Y → O Y and σ X : F X ∗ O X → O X such that π : ( Y, σ Y ) → ( X, σ X ) is a morphism of F -split schemes. Inparticular, π admits a lifting ∼ π : ∼ Y → ∼ X .Proof. The relative Frobenius F G/k : G → G ′ is a group homomorphism and the relative Frobe-nius F Y/k : Y → Y ′ is F G/k -equivariant. It follows that there is a natural linear G -action on Hom(( F Y/k ) ∗ O Y , O Y ′ ) . Furthermore, the ‘evaluation at one’ map ε : Hom(( F Y/k ) ∗ O Y , O Y ′ ) −→ H ( Y ′ , O Y ′ ) is a homomorphism of G -representations.If Y is integral and proper, then this is a map from a finite-dimensional G -representation to k ,and since Y is F -split, this map is surjective. By the linear reductivity of G , this map admits a G -equivariant splitting, and this way we obtain a G -invariant Frobenius splitting of Y .In general, let V = ε − ( k ) , which is the increasing union of G -representations V i of finite dimensionover k , and is endowed with a G -invariant map ε : V → k . Moreover, the map ε is surjective since Y is Frobenius split. For some i , the restriction ε | V i : V i → k is surjective, and since G is linearlyreductive, there exists a G -equivariant splitting s : k → V i of ε | V i .In particular, Y admits a G -invariant Frobenius splitting σ Y = s (1) . This Frobenius splittingpreserves G -invariant sections of O Y , and hence it descends to a Frobenius splitting on X . Thus π : ( Y, σ Y ) → ( X, σ X ) is a map in the category FSplit . Applying the canonical lifting functor fromTheorem 2.4 yields the desired lifting ∼ π : ∼ Y ( σ Y ) → ∼ X ( σ X ) of π . (cid:3) The method of proof of Lemma 2.5 yields the following interesting result which we will not needin the sequel.
Lemma 2.6.
Let Y be an F -split scheme over k and let G be a finite group of order prime to p acting on Y . Then there exists a lifting ∼ Y of Y together with a lifting of the G -action. Uniqueness of liftings admitting a lifting of Frobenius.
We have seen in Proposition 2.2that Frobenius liftings induce natural Frobenius splittings. It is natural to ask whether the liftinginduced by the Frobenius splitting associated to a Frobenius lifting ( ∼ X, ∼ F X ) of a normal k -scheme X (indicated in Theorem 2.4) is canonically isomorphic to ∼ X . In fact much more is true, as indicatedby the following simple but surprising result. Theorem 2.7.
Let X be a scheme over k , let σ be a Frobenius splitting on X , and let ( ∼ X, ∼ F X ) be aFrobenius lifting of X . Let ∼ X ( σ ) be the canonical lifting of X induced by σ (see Theorem 2.4). Thenone has a canonical isomorphism ∼ X ( σ ) ∼ −→ ∼ X (depending on ∼ F X and σ ) of liftings of X .Proof. The canonical lifting ∼ X ( σ ) comes with a natural closed immersion i σ : ∼ X ( σ ) → W ( X ) . Onthe other hand, the Frobenius lifting ( ∼ X, ∼ F X ) induces a map ν ∼ X, ∼ F X : W ( X ) −→ ∼ X, ν ∗ ( ∼ f ) = ( ˜ f mod p, δ ( ∼ f )) LOBAL FROBENIUS LIFTABILITY II 5 where ∼ F ∗ X ( ∼ f ) = ∼ f p + p · δ ( ∼ f ) . The composition ν ∼ X, ∼ F X ◦ i σ : ∼ X ( σ ) −→ ∼ X restricts to the identity on X , and hence is an isomorphism of liftings of ∼ X . This is the required map. (cid:3) The proof shows that ν ∼ X, ∼ F X ◦ i σ : ∼ X ( σ ) → ∼ X fits into the commutative diagram / / O X / / F ∗ X (cid:15) (cid:15) O ∼ X / / ν ∗ ∼ X, ∼ FX (cid:15) (cid:15) O X / / . / / F ∗ O Xσ (cid:15) (cid:15) / / W ( O X ) / / i ∗ σ (cid:15) (cid:15) O X / / / / O X / / O ∼ X ( σ ) / / O X / / , where the left composition σ ◦ F ∗ X is the identity. Corollary 2.8.
Let X be a normal k -scheme, and let ( ∼ X ( i ) , ∼ F ( i ) X ) for i = 1 , be two Frobenius liftingsof X . Then ∼ X (1) ≃ ∼ X (2) . Remark 2.9. If X is smooth, then Corollary 2.8 can be seen more directly as follows. The applicationof Hom(Ω X , − ) to the short exact sequence −→ O X −→ F X ∗ O X −→ B X −→ yields a connecting homomorphism δ : Hom(Ω X , B X ) → Ext (Ω X , O X ) . The forgetful map(2.2) ( ∼ X, ∼ F X ) ∼ X : { Frobenius liftings of X } / isom. −→ { liftings of X } / isom. , is a map from a torsor under Hom(Ω X , B X ) to a torsor under Ext (Ω X , O X ) which is equivariantwith respect to the map δ . If X is F -liftable, it is F -split, and hence δ = 0 . Thus (2.2) is constant.2.5. Descending and lifting Frobenius liftings.
The results of §2.3–2.4 allow us to extend[AWZ17, Theorem 3.3.6] in the following way.
Theorem 2.10 (Descending Frobenius liftability, [AWZ17, Theorem 3.3.6]) . Let π : Y → X be amorphism of schemes (essentially) of finite type over k and let ( ∼ Y , ∼ F Y ) be a Frobenius lifting of Y .(a) Suppose that π admits a lifting ∼ π : ∼ Y → ∼ X , and that one of the following conditions is satisfied:i. π ∗ : O X → Rπ ∗ O Y is a split monomorphism in the derived category,ii. π is finite flat of degree prime to p ,iii. Y satisfies condition S and π is an open immersion such that X \ Y has codimension > in X .Then F X lifts to ∼ X .(b) Suppose that one of the following conditions is satisfied:i. O X ∼ −→ π ∗ O Y and R π ∗ O Y = 0 ,ii. X and Y are smooth and π is proper and birational,iii. Y satisfies condition S and π is an open immersion such that X \ Y has codimension > in X .Then there exists a unique pair of a Frobenius lifting ( ∼ X, ∼ F X ) of X and a lifting ∼ π : ∼ Y → ∼ X of π such that ∼ F X ◦ ∼ π = ∼ π ◦ ∼ F Y .(c) Suppose that Y is normal and that π : Y → X = Y /G is a good quotient by an action of alinearly reductive group G on Y . Then there exists a lifting ∼ π : ∼ Y → ∼ X of π and a lifting ∼ F X of F X to ∼ X . In fact, conditions (a.ii) and (b.ii) imply (a.i) and (b.i), respectively. We do not expect ∼ F X ◦ ∼ π = ∼ π ◦ ∼ F Y to hold in general in situations (a) and (c). Proof.
Parts (a) and (b) were proven in [AWZ17, Theorem 3.3.6]. For (c), note first that since Y is normal and F -liftable, it is F -split. Lemma 2.5 provides compatible Frobenius splittings σ Y and σ X and a lifting ∼ π : ∼ Y ( σ Y ) → ∼ X ( σ X ) of π . By Theorem 2.7, ∼ Y ( σ Y ) ≃ ∼ Y , and we set ∼ X = ∼ X ( σ X ) ,obtaining the required lifting of π . By the definition of a good quotient, O X = ( π ∗ O Y ) G and π isaffine, in particular R i π ∗ O Y = 0 for i > . Since G is linearly reductive, O X = ( π ∗ O Y ) G → π ∗ O Y splits, and hence assumption (a.i) is satisfied. We apply part (a) to conclude. (cid:3) P. ACHINGER, J. WITASZEK, AND M. ZDANOWICZ
Products.
Using uniqueness of liftings admitting a Frobenius lifting developed above, we canalso strenghten [AWZ17, Corollary 3.3.7] in the following way.
Corollary 2.11 ([AWZ17, Corollary 3.3.7]) . Let X and Y be smooth and proper schemes over k .Then X × Y is F -liftable if and only if X and Y are. Moreover, every Frobenius lifting of X × Y arises as a product of Frobenius liftings of the factors.Proof. The first assertion is [AWZ17, Corollary 3.3.7]. For the second part, we take a Frobenius lifting ( ^ X × Y , ∼ F X × Y ) . We already know that X and Y admit Frobenius liftings ( ∼ X, ∼ F X ) and ( ∼ Y , ∼ F Y ) , andtheir product ( ∼ X × ∼ Y , ∼ F X × ∼ F Y ) is another Frobenius lifting of X × Y . But by Corollary 2.8, wemust have ^ X × Y ≃ ∼ X × ∼ Y .
Using [AWZ17, Proposition 3.3.1] we see that the space of Frobenius liftings on ^ X × Y is a torsor under H ( X × Y, F ∗ T X × Y ) . The last group equals H ( X, F ∗ T X ) ⊕ H ( Y, F ∗ T Y ) which can be identifiedwith the space of Frobenius liftings of the factors X and Y . This finishes the proof. (cid:3) Frobenius liftings compatible with a divisor or a closed subscheme
From the point of view of applications, it is necessary to consider Frobenius liftings of nc pairs.Recall that an nc (normal crossing) pair is a pair ( X, D ) consisting of a smooth scheme X and adivisor D ⊆ X with normal crossings. A lifting of ( X, D ) over W ( k ) is an nc pair ( ∼ X, ∼ D ) where ∼ X is a lifting of X and ∼ D ⊆ ∼ X is an embedded deformation of D . Note that the requirement that ∼ D has normal crossings relative to W ( k ) is not vacuous. By definition, a Frobenius lifting ( ∼ X, ∼ D, ∼ F X ) of ( X, D ) consists of a lifting ( ∼ X, ∼ D ) of ( X, D ) and a lifting ∼ F X of F X to ∼ X satisfying ∼ F ∗ X ∼ D = p ∼ D .In this case, we shall say that ∼ F X is compatible with ∼ D .In the above situation, we can define a logarithmic variant of (2.1)(3.1) ξ = 1 p ∼ F ∗ X : F ∗ X Ω X (log D ) → Ω X (log D ) (cf. [DI87, §4.2]). Lemma 3.1.
Let ( ∼ X, ∼ D, ∼ F X ) be a Frobenius lifting of an nc pair ( X, D ) over k . Then the F -splitting σ on X associated to the Frobenius lifting ( ∼ X, ∼ F X ) is compatible with the divisor D , i.e. σ ( F ∗ I D ) ⊆ I D where I D ⊆ O X is the ideal of D .Proof. The maps (3.1) and (2.1) fit inside a commutative square F ∗ X ω X / / (cid:15) (cid:15) ω X (cid:15) (cid:15) ( F ∗ X ω X )( pD ) F ∗ X ( ω X ( D )) / / ω X ( D ) . In particular, the top map vanishes to order p − along D . We conclude by [BK05, 1.3.11]. (cid:3) Lemma 3.2.
Let ( X, D ) be an nc pair over k , and let ( ∼ X, ∼ D, ∼ F X ) be a Frobenius lifting of ( X, D ) .Let D , . . . , D r be the irreducible components of D . Then for every i = 1 , . . . , r , the Frobenius lifting ∼ F X induces a Frobenius lifting ∼ F D i on ∼ D i which is compatible with the divisor ( S j = i ∼ D j ) ∩ ∼ D i ⊆ ∼ D i .Proof. The question is local so we can assume that X = Spec R and ∼ X = Spec ∼ R . Moreover, D j isthe zero locus of f j ∈ R , and ∼ D j is the zero locus of ˜ f j ∈ ∼ R where ≤ j ≤ r . Since ∼ F X ( ˜ f j ) = ∼ u j ˜ f pj for every ≤ j ≤ r and some ∼ u j ∈ ∼ R × , we get an induced morphism ∼ F D i : ∼ D i → ∼ D i , where ∼ D i = Spec ∼ R/ ˜ f i , such that ∼ F D i ( ˜ f j ) = ∼ u j ˜ f pj for j = i . (cid:3) Let o X,D ∈ Ext (Ω X (log D ) , B X ) be the obstruction to the existence of a Frobenius lifting ( ∼ X, ∼ D, ∼ F X ) of a simple normal crossing pair ( X, D ) (see [AWZ17, Variant 3.3.2]). Consider the short exact se-quence −→ Ω X −→ Ω X (log D ) −→ M i O D i −→ , where D = P D i is a decomposition into irreducible components. Analyzing the construction in[AWZ17, Variant 3.3.2] we see that the induced natural morphism(3.2) Ext (cid:0) Ω X (log D ) , B X (cid:1) −→ Ext (cid:0) Ω X , B X (cid:1) LOBAL FROBENIUS LIFTABILITY II 7 maps o X,D into the obstruction o X to the existence of a Frobenius lifting of X . Moreover, if o X,D = 0 ,then, after fixing a Frobenius lifting of ( X, D ) , we can identify Hom (cid:0) Ω X , B X (cid:1) with the space ofFrobenius liftings ( ∼ X, ∼ F X ) (see [AWZ17, Proposition 3.3.1]). With that, the natural morphism(3.3) Hom (cid:0) Ω X (log D ) , B X (cid:1) −→ Hom (cid:0) Ω X , B X (cid:1) maps Frobenius liftings ( ∼ X, ∼ D, ∼ F X ) to Frobenius liftings ( ∼ X, ∼ F X ) . Lemma 3.3.
Let ( X, D ) be a simple normal crossing pair over k such that H ( D i , O D i ( mD i )) = 0 for ≤ m ≤ p and all irreducible components D i of D . Let ( ∼ X, ∼ F X ) be a Frobenius lifting of X . Then there existsa lifting ∼ D of D with which ∼ F X is compatible. In particular, if Y = Bl Z X is a blow-up of a smooth k -scheme X in a smooth center Z ⊂ X ,then the unique embedded deformation ∼ E ⊂ ∼ Y of the exceptional divisor E is preserved by everyFrobenius lifting of ∼ Y . Proof.
Let o X and o X,D be the obstruction classes as above. Since o X = 0 and o X,D is mappedto it by (3.2), we get that o X,D is the image of an element of
Ext ( L O D i , B X ) . Moreover, if thiscohomology group vanishes, then (3.3) is surjective. Hence, in order to prove the lemma, it is enoughto show that this cohomology group is zero. To this end, we apply the local to global spectral sequenceto see that Ext (cid:16)M O D i , B X (cid:17) ≃ H (cid:16) X, E xt (cid:16)M O D i , B X (cid:17)(cid:17) . Using the short exact sequence −→ O X ( − D i ) −→ O X −→ O D i −→ we compute E xt ( L ri =1 O D i , B X ) as the cokernel of the mapping: M H om (cid:0) O X , B X (cid:1) −→ M H om (cid:0) O X ( − D i ) , B X (cid:1) , which is equal to L ri =1 B X ( D i ) | D i , because B X is locally free. Since X is F -split (see Proposition 2.2),we have B X ( D i ) | D i ⊆ (cid:0) O X ( D i ) ⊗ F X ∗ O X (cid:1) | D i = F X ∗ ( O pD i ( pD i )) . Here we used that ( F X ∗ O X ) | D i is the cokernel of F X ∗ ( O X ( − pD i )) = O X ( − D i ) ⊗ F X ∗ O X ֒ → F X ∗ O X , and so it is equal to F X ∗ O pD i .Hence, it is enough to show that H ( pD i , O pD i ( pD i )) = 0 . This follows by inductively looking atthe global sections in the short exact sequences −→ O D i (( p − m + 1) D i ) −→ O mD i ( pD i ) −→ O ( m − D i ( pD i ) −→ ≤ m ≤ p. (cid:3) Definition 3.4.
Let X be a k -scheme, let Z ⊆ X be a closed subscheme, let ( ∼ X, ∼ F X ) be a Frobeniuslifting of X , and let ∼ Z ⊆ ∼ X be an embedded deformation of Z . We say that ∼ F X is compatible with ∼ Z if ∼ F X ( I ∼ Z ) ⊆ I p ∼ Z , or, in other words, if the image of ∼ F ∗ X ( I ∼ Z ) → O ∼ X is contained in I p ∼ Z .In particular, if X is smooth and D ⊆ X is a divisor with normal crossings, then ∼ F X is compatiblewith a lifting ∼ D with relative normal crossings if and only if it is compatible with ∼ D as a closedsubscheme in the sense of the above definition. In fact, if ∼ D ⊆ ∼ X is an embedded deformation of D with which ∼ F X is compatible, then ∼ D automatically has relative normal crossings, see Corollary 2.8.We shall now discuss F -liftability of blow-ups. Let X be a smooth scheme over k , let Z ⊆ X be asmooth closed subscheme of codimension > , let π : Y → X be the blowing-up of X along Z , andlet E = Exc( π ) be the exceptional divisor. In the following, we relate F -liftability of X and of Y . P. ACHINGER, J. WITASZEK, AND M. ZDANOWICZ
Lemma 3.5 ([LS14, Proposition 2.2]) . In the above situation, the natural transformations of defor-mation functors
Def
X,Z −→ Def
Y,E −→ Def Y are isomorphisms. The composition Def Y ≃ Def
X,Z → Def X is given by the association A ∈ Art W ( k ) ( k ) Def Y ( A ) ∋ ( Y, O ∼ Y ) ( X, O ∼ X ) = ( X, π ∗ O ∼ Y ) ∈ Def X ( A ) , where we identify a deformation of a scheme with a thickening of its structure sheaf supported on thesame topological space. Let now ∼ Y be a lifting of Y over ∼ S . By Lemma 3.5, there exist unique liftings of ∼ E , ∼ X , and ∼ Z fitting inside a commutative square ∼ E / / (cid:15) (cid:15) ∼ Y ∼ π (cid:15) (cid:15) ∼ Z / / ∼ X. Moreover, by Theorem 2.10(b), the map ∼ π induces an injection (cid:8) liftings of F Y to ∼ Y (cid:9) −→ (cid:8) liftings of F X to ∼ X (cid:9) . Proposition 3.6.
A lifting ∼ F X of F X to ∼ X is in the image of the above map (that is, ∼ F X extendsto ∼ Y ) if and only if it is compatible with ∼ Z in the sense of Definition 3.4.Proof. Let ∼ F X be a lifting of F X to ∼ X . Suppose that ∼ F X is compatible with ∼ Z . If I (resp. ∼ I ) is theideal sheaf of Z (resp. ∼ Z ), then Y = Proj L n ≥ I n = Proj L n ≥ I np and ∼ Y = Proj L n ≥ ∼ I n =Proj L n ≥ ∼ I np . The relative Frobenius F Y/X corresponds to the map of graded O X -algebras F ∗ Y/X : F ∗ X ( M n ≥ I n ) −→ M n ≥ ( I n ) [ p ] −→ M n ≥ I np (3.4)induced by the inclusion. Since by assumption ∼ F ∗ X ∼ I maps into ∼ I p , we see that ∼ F ∗ X ∼ I n maps into ∼ I np for all n ≥ , and we can define a map of graded O ∼ X -algebras ∼ F ∗ Y/X : ∼ F ∗ X ( M n ≥ ∼ I n ) = M n ≥ ∼ F ∗ X ( ∼ I n ) −→ M n ≥ ∼ I np (3.5)again induced by the inclusion. We claim that the rational map induced by ∼ F ∗ Y/X is defined every-where, and is the identity on the complement of ∼ Z . It suffices to show that there are no relevanthomogeneous prime ideals whose preimage under (3.5) becomes irrelevant. To see this, we first observethat this property holds for the inclusion (3.4), since it induces a well-defined morphism F Y/X . Theinclusion (3.5) is only a nilpotent extension of (3.4), and hence its action on homogeneous ideals isthe same. This finishes the proof of the claim. The composition of ∼ F Y/X with the projection ∼ Y ′ → ∼ Y gives the desired extension of ∼ F X to ∼ Y . We note that in this part of the proof the smoothnessassumptions were not needed.Now suppose that ∼ F X extends to ∼ Y . The question whether ∼ F X is compatible with ∼ Z is local on ∼ X , so we can assume that ∼ X = Spec ∼ A and ∼ I = (˜ x , . . . , ˜ x c ) for some ˜ x i ∈ A such that ( p, ˜ x , . . . , ˜ x c ) is a regular sequence and c > . Write ∼ F ∗ X ˜ x i = ˜ x pi + pf i for f i ∈ A = ∼ A ⊗ k . The condition that ∼ F ∗ X extends to the open subset ∼ Y i = Spec ∼ A [ ∼ I / ˜ x i ] ⊆ ∼ Y is equivalent to the condition ∼ F ∗ X (˜ x j / ˜ x i ) = ˜ x pj + pf j ˜ x pi + pf i = (˜ x pj + pf j )(˜ x pi − pf i )˜ x pi ∈ ∼ A [ ∼ I / ˜ x i ] which amounts to saying that x pi f j − x pj f i ∈ I p for all j = i, where x i ∈ A are the images of ˜ x i . These equations imply that f i ∈ ( I p + ( x pi ) : ( x pj )) = I p for i = j. (cid:3) In the following proposition, we call a morphism of k -schemes f : X → Y separable if it is acomposition of a generically smooth morphism and a closed immersion. The proof uses some resultsproved in the subsequent two sections. LOBAL FROBENIUS LIFTABILITY II 9
Proposition 3.7.
Let ( ∼ X, ∼ F X ) and ( ∼ Y , ∼ F Y ) be Frobenius liftings of smooth and proper k -schemes X and Y . Let V ⊆ X × Y be an integral subscheme such that one of the projections π X or π Y isseparable when restricted to V . Suppose that the lifting of Frobenius ∼ F X × ∼ F Y on ∼ X × ∼ Y is compatiblewith a lifting of V . Then V = V X × V Y for some integral subschemes V X ⊆ X and V Y ⊆ Y .Proof. We set V X (resp. V Y ) to be the image of V under the projection π X : X × Y → X (resp. π Y : X × Y → Y ), and assume without loss of generality that π Y is separable when restricted to V . We claim that the closed immersion V ֒ → V X × V Y is an isomorphism. To see this, we applyCorollary 5.3(c) to the projection π ∼ Y : ( ∼ X × ∼ Y , ∼ F X × ∼ F Y ) −→ ( ∼ Y , ∼ F Y ) compatible with the respective Frobenius liftings, and observe that for every y ∈ V Y the Frobeniuslifting ∼ F X is compatible with a lifting of the subscheme V y = V ∩ ( X × { y } ) , when interpreted as asubscheme of X . By the assumptions the projection π Y is separable and therefore for a general y thesubscheme V y is a union of integral subschemes. Using Corollary 4.10 we now notice that there areonly finitely many choices for V y , and therefore they are all isomorphic since V Y is connected. Thisimplies that the immersion V ֒ → V X × V Y is an isomorphism on the fibers of the projection to V Y and hence is an isomorphism. (cid:3) F -splittings associated to Frobenius liftings — relative case Divisors associated to F -splittings. We now turn to a more detailed study of relative Frobe-nius splittings. The Frobenius trace map Tr X/S : F X/S ∗ ω X/S → ω X ′ /S plays a fundamental role inthe theory of F -splittings. Proposition 4.1.
Let X → S be a smooth morphism of k -schemes. Then Grothendieck dualityinterchanges splittings of Tr X/S and relative F -splittings of X/S .Proof.
Clear from functoriality of Grothendieck duality for the finite flat morphism F X/S . (cid:3) The following auxiliary base change result shows in particular that the fibers of a smooth relatively F -split morphism are F -split. Lemma 4.2 (cf. [PSZ18, Lemma 2.18]) . Consider a Cartesian diagram of k -schemes W ψ / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) Z / / S, with f : X → S smooth, and let δ X/S be a splitting of Tr X/S . Then there exists a splitting δ W/Z of Tr W/Z induced by the following commutative diagram. ψ ∗ ω X ′ /Sψ ∗ δ X/S (cid:15) (cid:15) ∼ / / ω W ′ /Zδ W/Z (cid:15) (cid:15) ψ ∗ ( F X/S ) ∗ ω X/S ∼ / / ( F W/Z ) ∗ ω W/Z , where X ′ and W ′ are the Frobenius twists of X and W relative to S and Z , respectively.Proof. The arrow ψ ∗ ( F X/S ) ∗ ω X/S → ( F W/Z ) ∗ ω W/Z in the above diagram comes from the cohomo-logical base change for the diagram W / / F W/Z (cid:15) (cid:15) X F X/S (cid:15) (cid:15) W ′ / / X ′ . To conclude the proof of the lemma it is enough to show that this arrow is an isomorphism, and thisis clear because F X/S ∗ ω X/S is a vector bundle ( f is smooth, so F X/S is finite and flat) and the abovediagram is Cartesian. (cid:3)
The following result shows that to every F -splitting we can associate a Q -divisor. Proposition 4.3 ([Sch09] and [PSZ18]) . Let f : X → S be a smooth morphism of k -schemes. Thento every splitting δ X/S of Tr X/S we can canonically associate an effective Q -divisor ∆ δ X/S on X suchthat(a) ∆ δ X/S ∼ Q − K X/S ,(b) if f has connected fibers, then ∆ δ X/S is horizontal, i.e., it does not contain any fiber.Proof.
By adjunction, δ X/S induces a morphism F ∗ X/S ω X ′ /S → ω X/S which is equivalent to O X → ω − pX/S , and so we get a divisor D δ X/S ∼ (1 − p ) K X/S . Set ∆ δ X/S = p − D δ X/S . The restriction of this Q -divisor to a fiber of f is non-zero as it correspondsto a splitting of the Frobenius trace map on this fiber (see Lemma 4.2). (cid:3) For a smooth
X/k and a splitting δ X of Tr X/k , we denote the corresponding Q -divisor by ∆ δ X . Remark 4.4.
In the setting of Lemma 4.2, we have ψ ∗ ∆ δ X/S = ∆ δ W/Z (see the commutative diagramin the statement of this lemma).
Remark 4.5.
It is easy to see that the coefficients of ∆ δ X/S are at most one. When S = Spec k (which is the only case in which we will apply this observation), this follows from [HW02, Theorem3.3] (cf. [SS10, Theorem 4.4]). Remark 4.6.
It is not necessary to assume that f : X → S is smooth in order to be able to associatea Q -divisor to a relative F -splitting. A far more general setting is described in [PSZ18]. When S = Spec k , then a Q -divisor can be associated to a Frobenius splitting on every normal variety X by an extension from the smooth locus.4.2. Divisors associated to Frobenius liftings.
To every Frobenius lifting ( ∼ X, ∼ F X ) of a smooth(or just normal) k -scheme X we can associate a corresponding F -splitting σ ∼ F X on X and to everysmooth morphism ˜ f : ∼ Y → ∼ X commuting with the liftings of Frobenius on ∼ Y and ∼ X we can associatea relative F -splitting σ ∼ F Y/X of Y /X (see Proposition 2.2). Here ∼ F Y/X denotes the induced lifting ofthe relative Frobenius.This provides us with Q -divisors ∆ ∼ F Y , ∆ ∼ F X , and ∆ ∼ F Y/X as in Proposition 4.3. Let f : Y → X bethe reduction of ˜ f modulo p . Lemma 4.7.
Let X and Y be smooth k -schemes. Then ∆ ∼ F Y = ∆ ∼ F Y/X + f ∗ ∆ ∼ F X , and if f hasconnected fibers, then ∆ ∼ F Y/X (resp. f ∗ ∆ ∼ F X ) is horizontal (resp. vertical).Proof. Let Y ′ be the base change of Y along F X . By the construction of the map ξ , we get thefollowing commutative diagram / / f ∗ F ∗ X Ω Xf ∗ ξ X (cid:15) (cid:15) / / F ∗ Y Ω Yξ Y (cid:15) (cid:15) / / F ∗ Y/X Ω Y ′ /Xξ Y/X (cid:15) (cid:15) / / / / f ∗ Ω X / / Ω Y / / Ω Y/X / / , where F ∗ Y/X Ω Y ′ /X ≃ F ∗ Y Ω Y/X and f ∗ F ∗ X Ω X ≃ F ∗ Y f ∗ Ω X . The first part of the lemma follows since theconsidered Q -divisors multiplied by p − are equal to div (det ξ Y ) , div (det ξ Y/X ) , and div (det ξ X ) ,respectively. If f has connected fibers, then ∆ ∼ F Y/X is horizontal by Proposition 4.3. (cid:3)
The following corollary lists all the properties of ∆ ∼ F X we need in this article. Given a flat morphism f : Y → X of normal varieties such that f ∗ O Y = O X and a Q -divisor D on Y , we denote by D h and D v the horizontal and the vertical part, respectively. Corollary 4.8.
Let ( ∼ Y , ∼ F Y ) be a Frobenius lifting of a smooth k -scheme Y .(a) If D ⊆ Y is a smooth irreducible divisor such that H ( D, O D ( mD )) = 0 for ≤ m ≤ p , then D ≤ ∆ ∼ F Y .(b) In the situation of Theorem 2.10(b.ii), we have ∆ ∼ F X = π ∗ ∆ ∼ F Y . LOBAL FROBENIUS LIFTABILITY II 11 (c) In the situation of Theorem 2.10(b.i), assume that π : Y → X is smooth and let ∼ F Y/X be theinduced lifting of the relative Frobenius. Then ∆ h ∼ F Y is the Q -divisor associated to the relative F -splitting σ ∼ F Y/X . In particular, ∆ h ∼ F Y ∼ Q − K Y/X , and ∆ v ∼ F Y = π ∗ ∆ ∼ F X ∼ Q − π ∗ K X . Proof.
Statement (a) follows from Lemma 3.3. Indeed, we have the morphism (3.1) ξ ( X,D ) : F ∗ X/k Ω X ′ (log D ′ ) −→ Ω X (log D ) such that div(det ξ X ) = div(det ξ ( X,D ) ) + ( p − D .Statement (b) is clear by the construction of ∆ ∼ F Y since g (Exc g ) has codimension at least two.Statement (c) follows from Lemma 4.7. (cid:3) We now relate certain conditions on the compatibility of subschemes for Frobenius liftings andFrobenius splittings.
Lemma 4.9.
Let ( ∼ X, ∼ F X ) be a Frobenius lifting of a smooth k -scheme X . Suppose that ∼ F X iscompatible with a lifting of an integral subscheme Z ⊂ X . Then the associated Frobenius splitting iscompatible with Z Proof.
Let σ X : F X ∗ O X → O X be the Frobenius splitting associated with ∼ F X . First, we considerthe case when Z is smooth. By Proposition 3.6 we see that Y = Bl Z X admits a Frobenius lifting ( ∼ Y , ∼ F Y ) compatible with the unique lifting of the exceptional divisor E and equipped with a lifting ∼ π : ( ∼ Y , ∼ F Y ) → ( ∼ X, ∼ F X ) of the contraction morphism π : Y → X . By Lemma 3.1 we see that E iscompatible with the Frobenius splitting σ Y : F Y ∗ O Y → O Y of Y induced by ∼ F Y . By [BK05, Lemma1.1.8(ii)] we therefore see that Z = π ( E ) is compatible with the push-forward of σ Y under π , whichis equal to σ X . This finishes the proof for Z smooth. For an arbitrary integral Z , we just observethat the condition of being compatibly split can be checked at the generic point. (cid:3) Corollary 4.10.
Let ( ∼ X, ∼ F X ) be a Frobenius lifting of a finite type smooth k -scheme X . Then thereare only finitely integral subschemes Z such that ∼ F X is compatible with a lifting of Z .Proof. By Lemma 4.9 we observe that every subscheme compatible with ∼ F X is compatible with theassociated F -splitting. Then we conclude by [Sch09, Theorem 5.8], which states that there are onlyfinitely many subschemes compatible with a given Frobenius splitting. (cid:3) Base change of a lifting of Frobenius
In this subsection, we show that a morphism from a W ( k ) -liftable scheme to an F -liftable schemelifts to W ( k ) after composing with the Frobenius. Moreover, if the source is endowed with a liftingof Frobenius, this lifting commutes with the lifting of Frobenius. Although we do not use it much inthe sequel, we regard the result as essential for a good understanding of Frobenius liftings. Proposition 5.1.
Let ( ∼ Y , ∼ F Y ) be a Frobenius lifting of a k -scheme Y . Let ϕ : Z → Y be a morphismof k -schemes and let ∼ Z be a lifting of Z over W ( k ) . Then there exists a morphism ψ : ∼ Z → ∼ Y (canonically defined by (5.3) below) such that ψ | Z = F Y ◦ ϕ : ∼ Z ψ % % r ❧ ❡ ❴ ❨ ❘ ▲ ∼ Y ∼ F Y / / ∼ YZ O O ϕ / / Y O O F Y / / Y. O O If ∼ F Z is a lifting of F Z to ∼ Z , then ψ ◦ ∼ F Z = ∼ F Y ◦ ψ . Further, if ∼ ϕ : ∼ Z → ∼ Y is any lifting of ϕ , then ψ = ∼ F Y ◦ ∼ ϕ . Recall that if ∼ X is any W ( k ) -scheme and X = ∼ X ⊗ W ( k ) k , then there is a canonical affine morphism θ ∼ X : ∼ X → W ( X ) defined on functions by the formula θ ∗ ∼ X ( f , f ) = ( ˜ f ) p + p ˜ f where ˜ f , ˜ f ∈ O ∼ X are any liftings of f , f ∈ O X . Moreover, if ∼ X is flat over W ( k ) , then a lifting ∼ F X : ∼ X → ∼ X of the absolute Frobenius F X : X → X induces an affine morphism ν ∼ X, ∼ F X : W ( X ) → ∼ X in the opposite direction, defined on functions by(5.1) ν ∗ ∼ X, ∼ F X ( ˜ f ) = ( f, δ ( f )) for ˜ f ∈ O ∼ X , where f the image of ˜ f in O X and δ ( f ) is the unique element such that ∼ F ∗ X ( ˜ f ) = ˜ f p + p · δ ( f ) . Wecan recover ∼ F X from ν ∼ X, ∼ F X by the formula ∼ F X = ν ∼ X, ∼ F X ◦ θ ∼ X , while the other composition θ ∼ X ◦ ν ∼ X, ∼ F coincides with the Witt vector Frobenius W ( F X ) :(5.2) W ( X ) ν ∼ X, ∼ F / / W ( F X ) ∼ X θ ∼ X / / ∼ F ' ' W ( X ) ν ∼ X, ∼ F / / ∼ X. Proof of Proposition 5.1.
We define the desired lifting ψ : ∼ Z → ∼ Y as the composition(5.3) ∼ Z θ ∼ Z −−→ W ( Z ) W ( ϕ ) −−−−→ W ( Y ) ν ∼ Y , ∼ FY −−−−→ ∼ Y .
Explicitly, on functions, ψ takes the form ψ ∗ ( ˜ f ) = ^ ϕ ∗ ( f ) p + p · ϕ ∗ ( δ ∼ F Y ( ˜ f )) , where ^ ϕ ∗ ( f ) ∈ O ∼ Z is a local section mapping to ϕ ∗ ( f ) ∈ O Z . It is thus clear that ψ | Z = F Y ◦ ϕ .We now check that ψ ◦ ∼ F Z = ∼ F Y ◦ ψ when ∼ Z is flat over W ( k ) and endowed with a Frobeniuslifting ∼ F Z . This follows from the commutativity of the following diagram: ∼ Z ∼ F Z ' ' θ ∼ Z / / W ( Z ) ν ∼ Z, ∼ FZ / / W ( ϕ ) (cid:15) (cid:15) ∼ Z θ ∼ Z / / W ( Z ) W ( ϕ ) (cid:15) (cid:15) W ( Y ) ν ∼ Y , ∼ FY / / ∼ Y ∼ F Y θ ∼ Y / / W ( Y ) ν ∼ Y , ∼ FY / / ∼ Y .
Indeed, by (5.2) the composition ∼ Z → W ( Z ) → ∼ Z (resp. ∼ Y → W ( Y ) → ∼ Y ) equals ∼ F Z (resp. ∼ F Y ).To show the commutativity of the diagram, we note that the compositions W ( Z ) → ∼ Z → W ( Z ) and W ( Y ) → ∼ Y → W ( Y ) are the Witt vector Frobenius morphisms W ( F Z ) and W ( F Y ) . Theseare functorial, which shows that the middle square (and hence the whole diagram) commutes.The final assertion follows from the commutativity of the following diagram: ∼ Z ∼ ϕ (cid:15) (cid:15) θ ∼ Z / / W ( Z ) W ( ϕ ) (cid:15) (cid:15) ∼ Y θ ∼ Y / / ∼ F Y W ( Y ) ν ∼ Y , ∼ FY / / ∼ Y .
Here the square commutes by functoriality of the maps θ . (cid:3) Corollary 5.2.
Let f : X → Y and ϕ : Z → Y be morphisms of k -schemes, and let ( ∼ X, ∼ F X ) , ( ∼ Y , ∼ F Y ) , and ( ∼ Z, ∼ F Z ) be Frobenius liftings of X , Y , and Z , respectively. Let ˜ f : ∼ X → ∼ Y be a liftingof f commuting with the Frobenius liftings, and let ψ : ∼ Z → ∼ Y be the lifting of F Y ◦ ϕ given byProposition 5.1. Form the cartesian diagram (5.4) ∼ W / / (cid:15) (cid:15) ∼ X ˜ f (cid:15) (cid:15) ∼ Z ψ / / ∼ Y .
LOBAL FROBENIUS LIFTABILITY II 13
Then ∼ W admits a Frobenius lifting ∼ F W such that the maps ∼ W → ∼ Z and ∼ W → ∼ X commute with theFrobenius liftings. Moreover, for every subscheme ∼ V ⊆ ∼ X compatible with ∼ F X , its preimage under ∼ W → ∼ X is compatible with ∼ F W . Setting Z = Spec k (in which case F Z is an isomorphism), we obtain the following. Corollary 5.3.
Let ( ∼ Y , ∼ F Y ) be a Frobenius lifting of a k -scheme Y .(a) The construction of Proposition 5.1 with Z = Spec k yields a section of the specialization map ∼ Y ( W ( k )) −→ Y ( k ) . In particular, every k -point of Y lifts (canonically) to a point of ∼ Y .(b) Let ( ∼ X, ∼ F X ) be a Frobenius lifting of a k -scheme X , and let ˜ f : ∼ X → ∼ Y be a map commutingwith the Frobenius liftings. Then for every y ∈ Y ( k ) , the fiber X y = f − ( y ) is F -liftable.(c) In the situation of (b), for every subscheme Z ⊂ X such that there exists a lifting ∼ Z ⊂ ∼ X compatible with ∼ F X , the Frobenius lifting of X y is compatible with the induced lifting of theintersection f − ( y ) ∩ Z . Example 5.4.
The scheme ∼ Y = { xy = p } ⊆ A W ( k ) does not admit a lifting of Frobenius (not evenlocally), because the k -point (0 , ∈ Y = { xy = 0 } does not admit a lifting modulo p . Of course Y admits a Frobenius lifting ∼ Y ′ = { xy = 0 } ⊆ A W ( k ) , ∼ F ′∗ Y ( x ) = x p , ∼ F ′∗ Y ( y ) = y p . Recall that by Corollary 2.8, since Y is F -split, it admits at most one lifting to which F Y lifts.6. F -liftability of surfaces The goal of this section is to show Conjecture 1 for smooth surfaces. Let ( ∼ X, ∼ F X ) be a Frobeniuslifting of a smooth k -scheme X . As in §4.2, we have the associated effective Q -divisor ∆ ∼ F X = 1 p − ξ ∼ F X )) ∼ Q − K X . A careful analysis of ∆ ∼ F X plays a vital role in this section. We further define D ∼ F X = ⌊ ∆ ∼ F X ⌋ . Notethat D ∼ F X is reduced (see Remark 4.5). If X is a toric variety with its standard Frobenius lifting,then ∆ ∼ F X = D ∼ F X is the toric boundary (the complement of the open orbit) of X .First, we tackle the case of rational surfaces. Lemma 6.1.
Let X be a smooth surface over k , and let π : Y = Bl x X −→ X be the blow-up of X at a closed point x ∈ X . Let ( ∼ X, ∼ F X ) and ( ∼ Y , ∼ F Y ) be Frobenius liftings of X and Y , and let ∼ π : ∼ X → ∼ Y be a lifting of π satisfying ∼ F X ◦ ∼ π = ∼ π ◦ ∼ F Y . Suppose that Supp ∆ ∼ F X hassimple normal crossings at x . Then x ∈ Sing D ∼ F X . Readers familiar with the language of birational geometry may notice that this is a direct conse-quence of the fact that x is a log canonical center of ( X, ∆ ∼ F X ) by Corollary 4.8(a) (in fact, this showsthat the above result is valid in higher dimensions for blow-ups along arbitrary smooth centers). Weprovide a more elementary explanation below. Proof.
For the exceptional divisor E = Exc( π ) , we have E ≤ ∆ ∼ F Y by Corollary 4.8(a). Since π ∗ ∆ ∼ F Y = ∆ ∼ F X (see Corollary 4.8(b)), the support of π ∗ ∆ ∼ F X − ∆ ∼ F Y is exceptional. By definition,this Q -divisor is linearly equivalent to K Y/X , which is equal to E by [Har77, Proposition V.3.3], andso π ∗ ∆ ∼ F X − ∆ ∼ F Y = E. In particular, π ∗ ∆ ∼ F X = 2 E + ∆ ′ , where E Supp ∆ ′ . As Supp ∆ ∼ F X has simple normal crossings at x and the coefficients of ∆ ∼ F X are at most one (see Remark 4.5), this is only possible if x ∈ Sing D ∼ F X . (cid:3) Remark 6.2.
Let f : ( Y, D Y ) → ( X, D X ) be a toric morphism between two-dimensional toric pairs,which on the level of schemes is a blowing-up of a smooth point x ∈ X . Then x ∈ Sing( D X ) and D Y = f − ∗ D X + Exc( f ) . Moreover, the converse is also true, that is the blowing-up of a smooth toricsurface at toric points is a toric morphism. Let us call a pair ( X, D ) of a normal variety X and a reduced effective divisor D on X sub-toric if X admits the structure of a toric variety such that D is invariant under the torus action. If D isthe maximal invariant divisor, then we call ( X, D ) toric (cf. [AWZ17, §2.1]). Lemma 6.3.
Let F n = P P ( O P ⊕ O P ( n )) be the n th Hirzebruch surface for n ≥ . Then for everyFrobenius lifting ( ∼ F n , ∼ F ) of F n , the pair ( F n , D ∼ F ) is sub-toric. Moreover, if n = 0 , then Supp ∆ ∼ F has simple normal crossings, and if n > , then Supp ∆ ∼ F has simple normal crossings at every pointof the negative section C ⊆ F n . Write F n = P P ( E ) for E = O P ⊕ O P ( n ) . A choice of the splitting E ≃ O P ⊕ O P ( n ) provides F n with a natural toric structure for which the natural morphisms P P ( O P ) → F n and P P ( O P ( n )) → F n are toric. Thus ( F n , D ) is a toric pair if and only if • for n = 0 , we have D = G + G ′ + G + G ′ , where G , G ′ , and G , G ′ are distinct fibers ofthe two projections π , π : F → P . • for n > , we have D = C + C ′ + G + G , where C is the unique negative section (correspondingto P P ( O P ) → F n ), C ′ is a section disjoint from C , and G , G are two distinct fibers ofthe projection π : F n → P . Proof.
Let us fix a Frobenius lifting ( ∼ F n , ∼ F ) . If n = 0 , then, by Corollary 2.11, we have ∆ ∼ F = π ∗ ∆ + π ∗ ∆ , where ∆ and ∆ are effective Q -divisors on P , and so Supp ∆ ∼ F is simple normalcrossing. Since ω P × P ≃ O P × P ( − , − , both ⌊ ∆ ⌋ and ⌊ ∆ ⌋ consist of at most two irreducibledivisors, which concludes the proof.If n > , then by Theorem 2.10(b.i) we get a compatible Frobenius lifting of P and so Corol-lary 4.8(c) implies that ∆ h ∼ Q − K F n / P , ∆ v ∼ Q − π ∗ K P , where ∆ h and ∆ v are the horizontal and the vertical part of ∆ ∼ F , respectively. Moreover, Corol-lary 4.8(a) gives ∆ h = ∆ ′ + C , where ∆ ′ is an effective Q -divisor such that C Supp ∆ ′ .As K F n + ∆ v + ∆ ′ + C ∼ Q , we have ∆ ′ · C = − ( K F n + C + ∆ v ) · C = 2 − ∆ v · C = 0 by adjunction, and so ∆ ′ is disjoint from C . Given that Supp (∆ v + C ) is simple normal crossing, sois Supp ∆ ∼ F along C . Moreover, for a fiber G of π ∆ ′ · G = − ( K F n + ∆ v + C ) · G = − ( K F n + G ) · G − (∆ v + C ) · G = 1 , by adjunction as G = 0 , and so ⌊ ∆ ′ ⌋ is zero or is a single section disjoint from C . Since ⌊ ∆ v ⌋ consistsof at most two distinct fibers, this concludes the proof of the lemma. (cid:3) Proposition 6.4.
Let Y be a smooth projective rational surface. If Y is F -liftable, then it is toric.Proof. Since P is toric, we can assume that Y P . Every smooth rational surface which is notisomorphic to P admits a birational morphism to a Hirzebruch surface π : Y → F n for some n ≥ ,which factors into a sequence of monoidal transformations Y = X m π m − −−−→ X m − −→ · · · −→ X π −→ X = F n , X i +1 = Bl x i X i . We assume that n is minimal among such. It follows that if n > , then π (Exc( π )) ⊆ C , where C ⊆ F n is the negative section. Indeed, the blow-up Bl x F n at any x C admits a morphism to F n − constructed by contracting the strict transform of the fiber through x of the natural projection F n → P .Let ( ∼ Y , ∼ F Y ) be a Frobenius lifting of Y = X m . By Theorem 2.10(b), there exist Frobenius liftings ( ∼ X i , ∼ F i ) of X i for ≤ i ≤ m , and liftings ∼ π i : ∼ X i +1 → ∼ X i such that ∼ F i ◦ ∼ π i = ∼ π i ◦ ∼ F i +1 . By Lemma 6.3we know that ( X , D ∼ F ) is sub-toric and ∆ ∼ F has simple normal crossings at π (Exc( π )) . Therefore,for every i > the Q -divisor ∆ ∼ F i has simple normal crossings at x i ∈ X i (it is contained in the unionof the exceptional locus of π and the support of the strict transform of ∆ ∼ F by Corollary 4.8(b)).By induction we can show that ( X i , D ∼ F i ) is sub-toric for every ≤ i ≤ m . Indeed, if ( X i − , D ∼ F i − ) is sub-toric, then by Remark 6.2 it is enough to show that x i − ∈ Sing D ∼ F i − , which follows byLemma 6.1. This concludes the proof. (cid:3) LOBAL FROBENIUS LIFTABILITY II 15
Remark 6.5.
In the course of the proof, we showed that if ( ∼ X, ∼ F X ) is a Frobenius lifting of a smoothprojective rational surface X P , then ( X, ⌊ ∆ ∼ F X ⌋ ) is sub-toric. This is false for some liftings ofFrobenius on P .We now turn our attention to ruled surfaces. We say that a rank two vector bundle E on a curve C is normalized if H ( C, E ) = 0 and H ( C, E ⊗ L ) = 0 for every line bundle L such that deg L < .Given a ruled surface X = P C ( E ) we can assume that E is normalized by replacing E with E ⊗ L for some line bundle L . Proposition 6.6.
Let X = P C ( E ) be a smooth projective ruled surface for a normalized rank twovector bundle E on an ordinary elliptic curve C . Then X is F -liftable if and only if E is not anon-split extension of O C with itself.Proof. If E is decomposable (that is, a direct sum of two line bundles), then X is F -liftable byExample 2.1(d). Hence, we can assume that E is indecomposable. By [Har77, Theorem V.2.15],there are only two such ruled surfaces corresponding to E being a non-split extension of O C with O C ( c ) where c ∈ C , and E being a non-split extension of O C with itself.In the former case, X ≃ Sym ( C ) (see for instance [Gar06, Section 6]), and it is easy to see that itis F -liftable. Indeed, let ( ∼ C, ∼ F C ) be the canonical Frobenius lifting of C (see Example 2.1(a)), where ∼ F C : ∼ C → ∼ C . Then ∼ F C × ∼ F C : ∼ C × ∼ C → ∼ C × ∼ C is equivariant under the natural Z / Z action byswapping the coordinates, and so it descends to Sym ( ∼ C ) → Sym ( ∼ C ) .Therefore, we are left to show that X is not F -liftable when E is a non-split extension of O C withitself. By contradiction assume that it does admit a Frobenius lifting ( X, ∼ F X ) . By Lemma 6.8, weknow that X is a quotient of Y = C ′ × P by F p acting independently on C ′ and P , where C ′ is theFrobenius twist of C , and the action on ( x : y ) ∈ P ( k ) is defined as ( x : y ) ( x + ly : y ) for a fixed l ∈ F p . This is illustrated by the following diagram P Y ρ o o V ′ / / π ′ (cid:15) (cid:15) (cid:3) X π (cid:15) (cid:15) C ′ V / / C. Since V ′ is étale, [AWZ17, Lemma 3.3.5] implies that Y admits a Frobenius lifting ( ∼ Y , ∼ F Y ) such that ∆ ∼ F Y = V ′∗ ∆ ∼ F X , where ∆ ∼ F Y and ∆ ∼ F X are the Q -divisors associated to ∼ F Y and ∼ F X , respectively. Inparticular, ∆ ∼ F Y is F p -invariant.As ∆ ∼ F Y ∼ Q − K C ′ × P = − ρ ∗ K P , we have ∆ ∼ F Y = ρ ∗ T for some Q -divisor T on P . Let G be afiber of π over a general point c ∈ C and ∆ = ∆ ∼ F X | G . Since ∆ ∼ F Y is F p -invariant and it is a pullbackfrom P , we get that ∆ is invariant under the action of F p on G ≃ P .By Corollary 4.8(c) and Remark 4.4 applied to π ′ , we get that ∆ is the associated Q -divisor of an F -splitting of P . This contradicts Lemma 6.7. (cid:3) We needed the following two lemmas in the proof of the above proposition.
Lemma 6.7.
There does not exist an F p -invariant F -splitting of P , where F p acts on P viatranslations, that is ( x : y ) ( x + ly : y ) for l ∈ F p . Note that an F -splitting is invariant under an action of a group if and only if the corresponding Q -divisor is invariant. Proof.
Assume by contradiction that such an F -splitting exists and let ∆ be the corresponding Q -divisor (see Proposition 4.3). By the definition of ∆ , we get that ( p − is an effective integraldivisor and deg ∆ = 2 . Furthermore, the coefficients of ∆ are at most one (see Remark 4.5). Sinceeach orbit of the action of F p on P is of length p except for the one of the fixed point ∞ ∈ P , theonly Q -divisor satisfying the aforementioned properties is ∆ = (cid:18) − p − (cid:19) ( ∞ ) + p − X i =0 p − x i ) , where x i = x + i ∈ A ( k ) . Up to an action of an automorphism we can assume that x = 0 . Thisyields a contradiction, because P cannot be F -split with such an associated Q -divisor ∆ by [CGS16,Example 3.4] and [SS10, Proposition 5.3 (2)]. One can check that the trace of an F -splitting of P cannot be equal to ( p − directly by noticing that x ( x − y ) · · · ( x − ( p − y ) y p − has coefficientzero at x p − y p − (see [BK05, Theorem 1.3.8]). (cid:3) Lemma 6.8.
Let C be an ordinary elliptic curve, and let C ′ be the Frobenius twist of C . Considerthe action of F p on C ′ × P with l ∈ F p acting as ( c, ( x : y )) ( c + lα, ( x + ly : y )) , for c ∈ C ′ and ( x : y ) ∈ P , where α ∈ C ′ [ p ] is a fixed p -torsion point. Let X be the quotient ( C ′ × P ) / F p . Then X ≃ P C ( E ) for E being a non-split extension of O C with itself.Proof. By definition we have the following Cartesian diagram C ′ × P V ′ / / π ′ (cid:15) (cid:15) X π (cid:15) (cid:15) C ′ V / / C, where V is the quotient by C ′ [ p ] . Note that C ′ × P = P C ′ ( E ′ ) , where E ′ is a vector bundle sittinginside a split (but not F p -equivariantly so) extension of F p -equivariant sheaves −→ O C ′ −→ E ′ −→ O C ′ −→ , with E ′ → O C ′ corresponding to the F p -equivariant section C ′ × (1 : 0) of π ′ . Therefore, X = P C ( E ) for a vector bundle E on C such that V ∗ E is F p -equivariantly isomorphic to E ′ , and the above shortexact sequence descends to C making E a non-split extension of O C by itself. (cid:3) Having handled rational and ruled surfaces, we are ready to proceed to the general case.
Theorem 6.9.
Let X be a smooth projective surface over k . Then X is F -liftable if and only if X is(i) an ordinary abelian surface,(ii) a hyperelliptic surface being a quotient of a product of two ordinary elliptic curves,(iii) a ruled surface P C ( E ) for a normalized rank two vector bundle E over an ordinary ellipticcurve C except when E is a non-trivial extension of O C with itself, or(iv) a toric surface. In particular, Conjecture 1 is true for surfaces.
Proof. If X is F -liftable, then it is F -split and ω − pX is effective (see Proposition 2.2). In particular,we only need to consider the case of κ ( X ) ≤ . If κ ( X ) = 0 , then K X is Q -effective, and hence ω p − X is trivial and X is minimal. In this case, the theorem follows from [Xin16, Theorem 1], but for theconvenience of the reader we present a simplified argument.When K X is torsion, X is F -liftable if and only if it is a quotient of an ordinary abelian surfaceby a free action of a finite group (see [MS87, Theorem 2]). By Hirzebruch–Riemann–Roch, suchquotients satisfy χ ( X, O X ) = 0 , and so by the classification of surfaces they are abelian, hyperelliptic,or quasi-hyperelliptic. We can exclude the surfaces of the latter type as they contain rational curves(see [Lie13, §7]): indeed, if ϕ : P → X is non-constant, then there exists an injection T P → ϕ ∗ T X , so T X cannot be étale trivializable (cf. [Xin16, Proposition 5]). Hyperelliptic surfaces are étale quotientsof products of elliptic curves E × E , except for case a3) which is an étale quotient of an abeliansurface A = E × E /µ (see [BM77, p. 37] for the notation). Therefore they are F -liftable if andonly if E and E are ordinary, by [MS87, Theorem 2] and [AWZ17, Lemma 3.3.5]. Note that A isordinary if and only if E × E is, as they are isogeneous to each other. This concludes our analysisof Kodaira dimension zero.As of now, we assume κ ( X ) = −∞ . If X is of type (iii) or (iv), then it is F -liftable by Exam-ple 2.1(c) and Proposition 6.6. To conclude, we assume that X is F -liftable and show that X is oftype (iii) or (iv). If X is rational, then it is toric by Proposition 6.4. Thus, we can assume thatthere exists a birational morphism f : X → P C ( E ) , where P C ( E ) is a ruled surface over C P . ByTheorem 2.10(b), the curve C is F -liftable, and so it is an ordinary elliptic curve.We claim that X ≃ P C ( E ) . Otherwise, f factors through a monoidal transformation π : Bl x P C ( E ) → P C ( E ) , where x ∈ P C ( E ) . In particular, Theorem 2.10(b) implies that Y = Bl x P C ( E ) is F -liftable,which is impossible. Indeed, if Y is F -liftable, then so is ( Y, Exc π ) by Lemma 3.3, and hence Y is F -split compatibly with Exc π by Lemma 3.1. Therefore, we get an F -splitting on C compatible witha point (see [BK05, Lemma 1.1.8(ii)]), which is impossible as C is an elliptic curve. LOBAL FROBENIUS LIFTABILITY II 17
Since we know that X is a ruled surfaces over an ordinary elliptic curve, the theorem follows fromProposition 6.6. (cid:3) Remark 6.10.
The F -liftability of minimal surfaces has been considered in [Xin16]. Note that ourresults do not agree when κ ( X ) = −∞ , as [Xin16, Theorem 1 (2b)] claims that all ruled surfaces overordinary elliptic curves are F -liftable. It seems to us that the gluing argument used at the end of theproof of Proposition 9 in op.cit. is incomplete, as it is unclear why h extends to a regular functionover V . 7. Fano threefolds
In this section, we will work assuming the following claim, which would follow from some assertionsin the literature which we were unable to verify completely. See Appendix A for a detailed discussionof the issue of boundedness for Fano threefolds.
Assertion 7.1.
There exists an integer m > such that for every Fano threefold X over an alge-braically closed field of arbitrary characteristic the divisor − mK X is base-point free. Theorem 7.2.
Assume Assertion 7.1 holds true. Then there exists a p such that for every prime p ≥ p , every F -liftable Fano threefold over an algebraically closed field of characteristic p is toric. Remark 7.3. (a) Our proof of Theorem 7.2 is heavily based on the Mori–Mukai classification ofFano threefolds, known only in characteristic zero. Note that F -liftable smooth Fano varietiesin positive characteristic are rigid and admit a unique lifting to characteristic zero, since byBott vanishing (1.1) we have H i ( X, T X ) = H i ( X, Ω n − X ⊗ ω ∨ X ) = 0 for i > . The main reference for the Mori–Mukai classification used in this section is [Sha99, Tables§12.3-§12.6]. We denote a Fano threefold of Picard rank ρ and whose number in the tables is n as M–M ρ.n (for example, M–M 2.12).(b) In the proof, we only need Assertion 7.1 to hold in characteristic p ≫ , and only for F -liftable (and hence rigid, liftable to characteristic zero, and F -split) Fano threefolds. Withoutassuming Assertion 7.1, our proof shows that every F -liftable Fano threefold described by theMori–Mukai classification is toric. It also shows that the assertion of Theorem 7.2 holds (witha smaller, more explicit bound on p ) if one could prove the following result. Claim.
Let R be a complete discrete valuation ring and let X, Y /R be smooth and proper.Suppose that both special fibers X , Y are rigid Fano threefolds, and that the generic fibers X η , Y η are isomorphic. Then X and Y are isomorphic. The above statement holds trivially if R contains a field, which makes it quite intriguing.(c) In fact in Proposition 7.6, proven using boundedness statement coming from Assertion 7.1,we prove that there exists a prime number p such every Fano variety in characteristic p > p arises as a reduction of a characteristic zero model coming from Mori–Mukai classification.This p is exactly the necessary bound in Theorem 7.2.(d) Combined with [AWZ17, Theorem 3], Theorems 6.9 and 7.2 show that the conjecture ofOcchetta and Wiśniewski [OW02] (see [AWZ17, Conjecture 2]) holds in characteristic zero ifthe target X is a surface or a Fano threefold, with no additional assumptions.The following theorem summarizes everything we need to know about the classification. Theorem 7.4.
Let X be a smooth complex Fano threefold. Then either X is toric, or there exists afibration π : X → Y to a smooth Fano variety Y , where Y is either non-rigid or one of the followingvarieties:(a-1) A smooth divisor of tri-degree (1 , , in P × P × P (M–M 3.17)(a-2) The blow-up Y = Bl W Z where Z = P × P W = a smooth curve of bi-degree (2 , (M–M 3.21) P × P × P a smooth curve of tri-degree (1 , , (M–M 4.3) P × P × P tri-diagonal curve (M–M 4.6) P × P × P a smooth curve of tri-degree (0 , , (M–M 4.8) (b-1) Y = Gr(2 , ∩ P , where Gr(2 , ⊆ P is the Grassmannian of planes in P embedded in P by the Plücker embedding and P ⊆ P is a general linear subspace. (M–M 1.15)(b-2) A smooth quadric in P . (M–M 1.16)(b-3) A smooth divisor of bi-degree (1 , in P × P . (M–M 2.32)(c-1) The blow-up of a smooth planar conic in P . (M–M 2.30)(c-2) The blow-up of a smooth conic contained in { t } × P in P × P . (M–M 3.22)(d) The blow-up of the twisted cubic in P . (M–M 2.27)(e) A non-toric del Pezzo surface.Proof. The only necessary information not immediately available from looking at the tables in [Sha99,Tables §12.3-§12.6] is which Fano threefolds are rigid. To this end, we will use the Hirzebruch–Riemann–Roch formula χ ( X, E ) = 124 rk( E ) c ( T X ) c ( T X ) + 112 c ( E ) (cid:0) c ( T X ) + c ( T X ) (cid:1) + 14 c ( T X ) (cid:0) c ( E ) − c ( E ) (cid:1) + 16 (cid:0) c ( E ) − c ( E ) c ( E ) + 3 c ( E ) (cid:1) . In particular, χ ( X, O X ) = c ( T X ) c ( T X ) / ; on the other hand, χ ( X, O X ) = 1 by Kodaira vanishing,so c ( T X ) c ( T X ) = 24 . The number c ( T X ) equals the topological Euler characteristic, ρ ( X ) − b ( X ) . We deduce the following formula χ ( X, T X ) = h ( X, T X ) − h ( X, T X ) = 12 ( − K X ) −
18 + ρ ( X ) − b ( X ) , where the values of the invariants on the right hand side are listed in loc. cit. Whenever this expressionis negative, the corresponding Fano threefold is not rigid. This applies to threefolds (M–M 1.1–14,2.1–25, 3.1–12, and 4.1–2).The converse statement fails in one example which we check by hand. The threefold (M–M 2.28)is the blow-up of P along a plane cubic C ⊆ P . By Lemma 3.5, Def Bl C P ≃ Def P ,C . Further, theforgetful transformation Def P ,C → Def C is non-constant. We conclude that Bl C P is not rigid.With this at hand, the assertions now follow from [Sha99, Tables §12.3-§12.6]. N.B.
The toric Fanothreefolds are 2.33–2.36, 3.25–3.31, 4.9–4.12, 5.2, 5.3 are toric (cf. [Sha99, Ch. 12, Remarks (i), p.216], note that toric varieties 3.25 and 4.12 are missing from this list). (cid:3)
Lemma 7.5.
Let π : H → S be a morphism of finite type. Then there exists a dense open subset U ⊆ S such that if T is a trait with geometric generic (resp. closed) point ¯ η (resp. ¯ k ) with a morphism T → U , and if x , x ∈ H ( T ) = Hom S ( T, H ) are points whose images in H (¯ η ) lie in the sameconnected component of H ¯ η , then their images in H (¯ k ) lie in the same connected component of H ¯ k .Proof. By considering S × Spec Z [1 /ℓ ] for every prime ℓ separately, we may assume that there existsan ℓ invertible on S . By constructibility of higher direct images [Del77, Théorème 1.9] there exists adense open U ⊆ S such that the étale sheaves π ∗ F ℓ are locally constant, with formation commutingwith base change. Then for T → U as in the statement, the cospecialization map F π ( H ¯ η ) ℓ = H ( H ¯ η , F ℓ ) = ( π ∗ F ℓ ) ¯ η −→ ( π ∗ F ℓ ) ¯ k = H ( H ¯ k , F ℓ ) = F π ( H ¯ k ) ℓ is an isomorphism. The required assertion follows. (cid:3) Proposition 7.6.
Assume Assertion 7.1 holds true. There exists a p such that for every trait T with residue characteristic p ≥ p , and every two smooth and proper X, Y over T such that(1) the geometric special fibers X ¯ k and Y ¯ k are F -split Fano threefolds,(2) moreover, they are rigid, i.e. H ( X ¯ k , T X ¯ k ) = 0 = H ( Y ¯ k , T Y ¯ k ) ,(3) the geometric generic fibers X η and Y η are isomorphic,the geometric special fibers X ¯ k and Y ¯ k are isomorphic as well.Proof. Let m be as in Theorem A.3. We claim that the m -th power of the anticanonical bundleis relatively very ample and yields embeddings X, Y ֒ → P N − T with Hilbert polynomial χ . Indeed,since X ¯ k is an F -split Fano variety, H i ( X ¯ k , O X ¯ k ( − mK X ¯ k )) = 0 for i > . Therefore, the restriction H ( X, O X ( − mK X/T )) → H ( X ¯ k , O X ¯ k ( − mK X ¯ k )) is surjective (for example by Grauert’s and semi-continuity theorems), and the very ampleness of − mK X ¯ k implies the very ampleness of − mK X/T .The analogous argument works for Y . We note that both the Hilbert polynomial and the integer LOBAL FROBENIUS LIFTABILITY II 19 N are equal for the two families using condition (3), semicontinuity and Kodaira vanishing valid for F -split varieties.Let H ⊆ Hilb χ ( P N − Z ) be the open subscheme parametrizing smooth three-dimensional subschemesof P N − with ample anticanonical bundle and vanishing higher cohomology of the tangent bundle. By(1) and Akizuki–Nakano vanishing (true by (2) for p > ), X and Y give two points [ X ] , [ Y ] ∈ H ( T ) .Assumption (3) implies that their images in H (¯ η ) lie in the same connected component (in fact, thesame orbit of PGL N ). Applying Lemma 7.5 to H → Spec Z , we get that if p ≫ (with boundindependent of X and Y ), then [ X ¯ k ] and [ Y ¯ k ] lie in the same connected component of H (¯ k ) . Butsince X ¯ k is rigid, the infinitesimal neighborhood of [ X ¯ k ] in H ¯ k is contained in its PGL N -orbit. Thisshows that the PGL N -orbits on H ¯ k are open, and therefore also closed. Thus [ X ¯ k ] and [ Y ¯ k ] lie in thesame orbit and hence X ¯ k and Y ¯ k are isomorphic. (cid:3) Theorem 7.7.
Let X be a smooth projective threefold over an algebraically closed field k of character-istic p > admitting a birational map X → Y where Y can be described as in the list in Theorem 7.4.Then X is not F -liftable. The above theorem implies that Conjecture 1 holds for Fano threefolds in characteristic p providedthat the Mori–Mukai classification (more precisely, Theorem 7.4) is valid in that characteristic. In thefollowing proof, we deduce Theorem 7.2 from it by lifting a given F -liftable variety to characteristiczero, applying the Mori–Mukai classification there, and then descending back to characteristic p . Thislast step is a bit delicate, and it relies on Proposition 7.6, which in turn requires the boundedness ofFano threefolds (Assertion 7.1). Proof of Theorem 7.2 (assuming Theorem 7.7).
First, we define p . Let X , . . . , X r be the non-toricrigid complex Fano threefolds. By Theorem 7.4, for each i there exists a fibration π i : X i → Y i where Y i is either non-rigid or described as in the list (a-1) . . . (e). As each X i is rigid, it is defined over Q . Further, by Lemma 7.9 below, each fibration π i is defined over Q as well. We can therefore finda number field K , an integer N , and models π i : X i → Y i of π i : X i → Y i over O K [1 /N ] with thefollowing properties: • X i and Y i are smooth and proper over O K [1 /N ] , with ample anticanonical bundles, • π i is a fibration, • H i ( X i , T X i / O K [1 /N ] ) = 0 for i > , • if Y i is rigid, then Y i can be described over O K [1 /N ] as in the list (a-1) . . . (e).Let m be as in Theorem A.3, and let χ i ( t ) = χ ( X i , − tK X i ) , N i = χ i ( m ) = h ( X i , − mK X i ) . By po-tentially increasing N , we may assume using [BK05, 1.6.E Exercises (5)] that all the geometric closedfibres of X i → Spec( O K [1 /N ]) are F -split and hence the conditions (1) and (2) of Proposition 7.6are satisfied for the localization of X i → Spec( O K [1 /N ]) at any prime in the base.Let X be an F -liftable Fano threefold over an algebraically closed field k of characteristic p ≥ p .Suppose that X is not toric. Since X is rigid and H ( X, T X ) = 0 (Remark 7.3(a)), there exists aunique deformation ∼ X of X over W ( k ) (which algebraizes because the ample line bundle ω − X lifts).Since X is F -liftable it is also F -split and hence the assumpions of Proposition 7.6 are satisfied forthe family ∼ X/W ( k ) . This property is clearly invariant under base change. Let F be an algebraicallyclosed field containing both C and W ( k ) . Since ∼ X F is a rigid Fano threefold, which is moreover nottoric (otherwise it would have a toric Fano model Y over W ( k ) , and Proposition 7.6 would imply X ≃ Y i.e. X toric), there exists an isomorphism ι : ( X i ) F ≃ ∼ X F for some i ≤ r . Our next goal is tospread it out and obtain an isomorphism over k .Let V be the completion of O K [1 /N ] at a prime above p . Since V is a finite extension of Z p , thereexists a finite extension V of W ( k ) also contained in F such that O K [1 /N ] ⊆ V as subrings of F .Passing to a further finite extension of V , we can assume that the isomorphism ι is defined over thesubfield V [1 /p ] of F . Applying Proposition 7.6 to ∼ X V and ( X i ) V over Spec V gives an isomorphism X i ⊗ O K [1 /N ] k ≃ ∼ X k = X. In particular, X i ⊗ O K [1 /N ] k is F -liftable. By Theorem 2.10(b) and Proposition 7.8 applied for L = − K X , so is Y i ⊗ O K [1 /N ] k . In particular, being Fano, it must be rigid, which implies by semi-continuity that Y i is rigid as well. In this case, Y i ⊗ O K [1 /N ] k is described as in the list (a-1) . . . (d),and by Theorem 7.7 we obtain a contradiction. Therefore X is toric. (cid:3) Proposition 7.8 (Relative Kodaira vanishing for F -split total space) . Let X be an F -split smoothprojective variety, let f : X → Y be a projective morphism and let L be an f -ample divisor. Then R j f ∗ O X ( K X + L ) = 0 for j > .Proof. Let F j = R j f ∗ O X ( K X + L ) and let M be a fixed ample divisor on Y . Take n large enough sothat (1) F j ⊗ O Y ( nM ) is globally generated and (2) has no higher cohomology for all j and that (3) L + nf ∗ M is ample on X . By (1), the vanishing of F j is equivalent to H ( Y, F j ⊗ O Y ( nM )) = 0 .By the projection formula, the Leray spectral sequence for O X ( K X + L + nf ∗ M ) reads E ij = H i ( Y, F j ⊗ O Y ( nM )) ⇒ H i + j ( X, O X ( K X + L + nf ∗ M )) , and by (2) this collapses yielding H ( Y, F j ⊗ O Y ( nM )) ≃ H j ( X, O X ( K X + L + nf ∗ M )) , which is H j ( X, O X ( K X + ample )) by (3). Since X is F -split, it satisfies Kodaira vanishing, and hence thisgroup is zero for j > . (cid:3) Lemma 7.9 (Models over subfields) . Let k ⊆ k ′ be an extension of algebraically closed fields, and let X be a normal projective variety over k with H ( X, O X ) = 0 . Then for every fibration π ′ : X k ′ → Y ′ to a normal projective variety Y ′ over k ′ there exists a fibration π : X → Y to a normal projectivevariety Y over k and an isomorphism ι : Y k ′ ≃ Y ′ such that the following triangle commutes X k ′ π ′ / / π k ′ % % ▲▲▲▲▲▲ Y ′ Y k ′ . ι tttttt Proof.
Let O Y ′ (1) be an ample line bundle on Y ′ and let L ′ = ( π ′ ) ∗ O Y ′ (1) , so that Y ′ = Proj M n ≥ Γ( X k ′ , ( L ′ ) n ) . Since H ( X, O X ) = 0 , Pic X is discrete, and hence Pic X ∼ −→ Pic X k ′ . Let L ∈ Pic X correspond to L ′ under this isomorphism; then L is ample. We set Y = Proj M n ≥ Γ( X, L n ) . Since by flat base change Γ( X, L n ) ⊗ k k ′ ≃ Γ( X k ′ , ( L ′ ) n ) , we get the desired isomorphism Y k ′ ≃ Y ′ . (cid:3) Proof of Theorem 7.7.
By Theorem 2.10, it is enough to show that Y is not F -liftable. We do thiscase by case, deferring the more involved arguments to lemmas following the proof. Cases (a-1) and (a-2): blow-ups on a product.
Consider first the third example (M–M 4.6)in (a-2), i.e. the blow-up of P × P × P along the tridiagonal curve C = (cid:8) ( x, x, x ) ∈ P × P × P | x ∈ P (cid:9) . The variety X cannot be F -liftable, because then ( P × P × P , C ) would admit a Frobenius liftingby Lemma 3.3 and Proposition 3.6, which is impossible by Proposition 3.7 (as the center C is not theproduct of subvarieties of the factors). The remaining cases (M–M 3.17, 3.21, 4.3, and 4.8) are not F -liftable by an analogous argument. Note that Fano threefold (a-1) (M–M 3.17) is the blow-up of P × P along the graph of the Segre embedding P → P of degree two. Cases (b-1), (b-2), and (b-3): violating Bott vanishing (1.1) . Fano threefolds (b-1), (b-2),and (b-3) do not satisfy Bott vanishing (Lemma 7.10 below, [AWZ17, Example 3.2.6] or [BTLM97,§4.1], and [BTLM97, §4.2], respectively), and so they are not F -liftable. Cases (c-1) and (c-2): inducing a lifting of Frobenius on P compatible with a conic. Let X = Bl C W be a blow-up of either W = P or P × P along a conic C contained in a plane H ≃ P ⊆ W , and suppose that X is F -liftable.Let H ≃ P be the strict transform of H on X , and let E be the exceptional divisor of the blow-up.By Lemma 3.3, we get that ( X, H + E ) admits a Frobenius lifting, and so ( H, H ∩ E ) is F -liftableby Lemma 3.2. Since C ≃ H ∩ E is a conic, this contradicts Lemma 7.11. Case (d): blow-up along twisted cubic.
See Lemma 7.12 below.
Case (e): del Pezzo surface.
Follows from Theorem 6.9. (cid:3)
LOBAL FROBENIUS LIFTABILITY II 21
Lemma 7.10 (M–M 1.16) . Let X = Gr(2 , ∩ P , where Gr(2 , ⊆ P is the Grassmannian ofplanes in P embedded in P by the Plücker embedding and P ⊆ P is a general linear subspace.Then H ( X, Ω X (1)) = 0 , where O X (1) is the restriction of O P (1) to X .Proof. The setting admits a natural lifting to characteristic zero, and by the semicontinuity theorem,it is enough to show that the required non-vanishing holds for this lifting. Therefore, we can assumethat X is defined over C .We have ω Gr(2 , ≃ O Gr(2 , ( − , hence ω X ≃ O X ( − and therefore H ( X, Ω X (1)) = H ( X, T X ( − .First, we prove that H i ( X, T
Gr(2 , ( − | X ) = 0 for all i . Consider the Koszul resolution −→ O Gr(2 , ( − −→ O Gr(2 , ( − ⊕ −→ O Gr(2 , ( − ⊕ −→ O Gr(2 , −→ O X −→ , and tensor it by T Gr(2 , ( − to get → T Gr(2 , ( − → T Gr(2 , ( − ⊕ → T Gr(2 , ( − ⊕ → T Gr(2 , ( − → T Gr(2 , ( − | X → . By [Sno86, Theorem, p.171, (3)], all the cohomology groups of T Gr(2 , ( − k ) ≃ Ω , (5 − k ) vanishfor ≤ k ≤ , and so the above exact sequence shows that the same holds for T Gr(2 , ( − | X .The dual of the conormal exact sequence tensored by O X ( − is −→ T X ( − −→ T Gr(2 , ( − | X −→ O ⊕ X −→ . Since the cohomology groups of the middle sheaf vanish, we get H ( T X ( − H ( X, O X ) ⊕ = 0 . (cid:3) Lemma 7.11.
Let D ⊆ P be a smooth conic. Then ( P , D ) is not F -liftable.Proof. Assume by contradiction that ( P , D ) is F -liftable and fix a Frobenius lifting. Let ξ : F ∗ Ω P (log D ) → Ω P (log D ) be the associated morphism (3.1). Choose a general point x ∈ D and take C ⊆ P to be the linetangent to D at x . Since x is general and ξ is generically an isomorphism (see Proposition 2.2), weget that ξ | C is injective. In particular, if Ω P (log D ) | C = O C ( a ) ⊕ O C ( b ) for some a, b ∈ Z , then a, b ≤ (see the proof of [AWZ17, Lemma 6.2.1(a)]). Indeed, if a ≤ b , then the injectivity of ξ | C implies that we have a nonzero map O C ( pb ) → O C ( c ) where c ∈ { a, b } , so pb ≤ c ≤ b and b ≤ .We will show that Ω P (log D ) | C = O C ( − ⊕ O C (1) yielding a contradiction (cf. [Xin16, Lemma 4]).To this end, choose a standard affine chart on A ⊆ P in which C is described as y = 0 , and D as y − x = 0 . In the chart A ∩ C , we get that Ω P (log D ) | C is generated by dx and d ( y − x ) y − x = − dyx + 2 dxx . Now, we take the chart of P having coordinates x and yx . In this chart C and D are disjoint, andso in the restriction of this chart to C the vector bundle Ω P (log D ) | C is generated by d (cid:18) x (cid:19) = − x dx and d (cid:16) yx (cid:17) = dyx = 2 dx + x (cid:18) dyx − dxx (cid:19) . Therewith, the coordinate-change matrix is: (cid:20) − x − x (cid:21) , and so Ω P (log D ) | C = O C ( − ⊕ O C (1) . (cid:3) A surprising property of F -liftability is that if X is F -liftable and f : Y → X is a smooth morphismsuch that Rf ∗ O Y = O X , then f is relatively F -split. This is not true in general, if we assume that X is only F -split. Lemma 7.12 (M–M 2.27) . Let C ⊂ P be the twisted cubic, that is, the image of P under theembedding given by the full linear system | O P (3) | . Then the blow-up Y = Bl C P is not F -liftable.Proof. Let E be the exceptional divisor of the blow-up π : Y → P . Firstly, we observe that by [SW90,Application 1, page 299] Y is isomorphic to a projective bundle P ( E ) over P for some non-split ranktwo vector bundle E . The morphism f : Y → P is given by a pencil of quadrics in P containing C .If Y is F -liftable, then Corollary 4.8(a) and (c) imply that − ( K Y/ P + E ) is Q -linearly equivalentto an effective Q -divisor. In what follows, we show that this is not true. By the construction of f , we have Q = f − ( L ) for some line L ⊆ P , where Q is the strict transform of a quadric in P containing C . Since ω Y ≃ O Y ( E ) ⊗ π ∗ O P ( − , we have K Y ∼ − Q − E , and so: K Y/ P + E ∼ − Q + 3 f − ( L ) ∼ Q, which is not Q -linearly equivalent to an anti-effective Q -divisor. (cid:3) Remark 7.13.
Using Corollary 5.2 and a more intricate version of the above argument one canshow that a rank two vector bundle E on P n is decomposable if and only if P P n ( E ) is F -liftable.This seems intriguing from the viewpoint of Hartshorne’s conjecture predicting that all such vectorbundles are decomposable when n ≥ . Appendix A. Boundedness for Fano threefolds
In Section 7, we worked assuming the following claim, repeated here for convenience.
Assertion A.1.
There exists an integer m > such that for every smooth Fano threefold X over analgebraically closed field of arbitrary characteristic the divisor − mK X is base-point free. Remark A.2.
The above claim follows from the results in [SB97]. Since we were unable to understandtheir proofs completely, we decided to add Assertion A.1 as an assumption in Theorem 7.2. Here arethe details of this deduction. Let X be a smooth Fano threefold over an algebraically closed fieldof characteristic p . If − K X is itself base-point free, there is nothing to prove. Otherwise by [SB97,Theorem 3.4] (see also the erratum to the proof [SB19, Theorem 4.1]) either (1) X ≃ S × P where S is a del Pezzo surface of degree one, or (2) X is isomorphic to the blowup of a smooth hypersurfaceof degree in the weighted projective space P (1 , , , , along a smooth elliptic curve C which isa complete intersection of degree (1 , . Since varieties of types (1) and (2) form bounded families(over Z ), we can find a suitable m which works for them.To avoid awkward phrasing, we will say that a smooth Fano X is ( F -split) meaning that eitherthe base field has characteristic zero or k has characterstic p > and X is F -split. Theorem A.3 (Boundedness for F -split Fano threefolds) . Assume Assertion A.1 holds true and theabove convention is in place. Then there exist integers m , N , and M such that for every smooth ( F -split) Fano threefold X over an algebraically closed field k of characteristic p = 2 , the linear system | − mK X | is very ample and defines an embedding X ֒ → P n with n < N such that the coefficients ofthe Hilbert polynomial of the image are bounded by M in absolute value.In particular, there exists a scheme H of finite type over Z [1 / and a smooth projective morphism X → H of relative dimension such that for every smooth Fano threefold X as above there exists amap h : Spec k → H such that X ≃ h ∗ X . The above theorem is well-known in characteristic zero [Kol96, Corollary V 2.15].
Lemma A.4 (Boundedness of the Hilbert polynomial) . There exists an M such that for every smooth( F -split) Fano threefold X over an algebraically closed field k of characteristic p = 2 , the polynomial χ ( X, O X ( − tK X )) has coefficients bounded by M in absolute value.Proof. Because X is F -split, it lifts to W ( k ) , and hence it satisfies Kodaira–Akizuki–Nakano vanish-ing [DI87]. In particular for d = dim X , we have the vanishing H ( X, T X ) = H ( X, Ω d − X ⊗ ω − X ) = 0 . Thus X admits a formal lifting X over W ( k ) , which is necessarily algebraizable because the ampleline bundle ω − X lifts. Let Y = X ⊗ Frac( W ( k )) be the generic fiber, which is a Fano threefold over afield of characteristic zero. By flatness, we have χ ( X, O X ( − tK X )) = χ ( Y, O Y ( − tK Y )) , and the latterbelongs to a finite family of polynomials by boundedness in characteristic zero. (cid:3) Remark A.5.
We expect the above lemma and Theorem A.3, to hold without the F -splittingassumption, at least for p > . It can be deduced from the results of [Das19] and Assertion A.1 thatfor a fixed p > , the volume ( − K X ) of a smooth Fano threefold in characteristic p is bounded. Itis unclear to us whether his bounds can be made independent of p . Lemma A.6 (Big Matsusaka for Fano threefolds) . Assume Assertion A.1 holds true. Then thereexists an m such that for every smooth Fano threefold X the divisor − mK X is very ample. LOBAL FROBENIUS LIFTABILITY II 23
Proof.
By [Kee08], it is enough to show that − mK X is base-point free for a bounded m , which followsfrom Assertion A.1. (cid:3) Proof of Theorem A.3.
Let m be as in Lemma A.6, M as in Lemma A.4, and let N = 4 M m . If X is asmooth F -split Fano threefold over an algebraically closed field k of characteristic p > , then − mK X embeds X into P nk with n = dim H ( X, O X ( − mK X )) , which equals χ ( X, O X ( − mK X )) because H i ( X, O ( − mK X )) = 0 for i > (ample line bundles on an F -split variety have no higher cohomology).By the bound on the coefficients of χ ( X, O X ( − tK X )) , we have n < M + M m + M m + M m < N .We conclude that the image of X in P nk defines a point on the Hilbert scheme Hilb χ ( P n ) where n and χ both belong to a finite list of pairs ( χ , n ) , . . . , ( χ r , n r ) . We can now take H to be the opensubscheme of ` Hilb χ i ( P n i ) parametrizing smooth Fano varieties and X to be (the base change of)the universal family. (cid:3) References [AWZ17] Piotr Achinger, Jakub Witaszek, and Maciej Zdanowicz,
Global Frobenius Liftability I , arXiv:1708.03777(2017).[BK05] Michel Brion and Shrawan Kumar,
Frobenius splitting methods in geometry and representation theory ,Progress in Mathematics, vol. 231, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 2107324[BM77] E. Bombieri and D. Mumford,
Enriques’ classification of surfaces in char. p . II , Complex analysis andalgebraic geometry, 1977, pp. 23–42. MR 0491719[BTLM97] Anders Buch, Jesper F. Thomsen, Niels Lauritzen, and Vikram Mehta, The Frobenius morphism on a toricvariety , Tohoku Math. J. (2) (1997), no. 3, 355–366. MR 1464183[CGS16] Paolo Cascini, Yoshinori Gongyo, and Karl Schwede, Uniform bounds for strongly F -regular surfaces , Trans.Amer. Math. Soc. (2016), no. 8, 5547–5563. MR 3458390[Das19] Omprokash Das, On the Boundedness of Anti-Canonical Volumes of Singular Fano 3-Folds in Character-istic p > 5 , International Mathematics Research Notices (2019), https://doi.org/10.1093/imrn/rnz048 .[Del77] P. Deligne,
Théorèmes de finitude en cohomologie ℓ -adique , Cohomologie étale, Lecture Notes in Math.,vol. 569, Springer, Berlin, 1977, pp. 233–261. MR 3727439[DI87] Pierre Deligne and Luc Illusie, Relèvements modulo p et décomposition du complexe de de Rham , Invent.Math. (1987), no. 2, 247–270. MR 894379[Gar06] Luis Fuentes García, Seshadri constants on ruled surfaces: the rational and the elliptic cases , ManuscriptaMath. (2006), no. 4, 483–505. MR 2223629[Har77] Robin Hartshorne,
Algebraic geometry , Springer-Verlag, New York, 1977, Graduate Texts in Mathematics,No. 52. MR 0463157 (57
F-regular and F-pure rings vs. log terminal and log canonical singu-larities , J. Algebraic Geom. (2002), no. 2, 363–392. MR 1874118[Kee08] Dennis S. Keeler, Fujita’s conjecture and Frobenius amplitude , Amer. J. Math. (2008), no. 5, 1327–1336.MR 2450210[Kol96] János Kollár,
Rational curves on algebraic varieties , Ergebnisse der Mathematik und ihrer Grenzgebiete.3. Folge, vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180[Lie13] Christian Liedtke,
Algebraic surfaces in positive characteristic , Birational geometry, rational curves, andarithmetic, Simons Symp., Springer, Cham, 2013, pp. 229–292. MR 3114931[LS14] Christian Liedtke and Matthew Satriano,
On the birational nature of lifting , Adv. Math. (2014), 118–137. MR 3161094[MS87] Vikram B. Mehta and Vasudevan Srinivas,
Varieties in positive characteristic with trivial tangent bundle ,Compositio Math. (1987), no. 2, 191–212, With an appendix by Srinivas and M. V. Nori. MR 916481[OW02] Gianluca Occhetta and Jarosław A. Wiśniewski, On Euler-Jaczewski sequence and Remmert-van de Venproblem for toric varieties , Math. Z. (2002), no. 1, 35–44. MR 1930984[PSZ18] Zsolt Patakfalvi, Karl Schwede, and Wenliang Zhang, F -singularities in families , Algebr. Geom. (2018),no. 3, 264–327. MR 3800355[SB97] N. I. Shepherd-Barron, Fano threefolds in positive characteristic , Compositio Math. (1997), no. 3,237–265. MR 1440723[SB19] ,
Errata to “Fano threefolds in positive characteristic” , 2019, Private communication.[Sch09] Karl Schwede, F -adjunction , Algebra Number Theory (2009), no. 8, 907–950. MR 2587408[Sha99] Igor R. Shafarevich (ed.), Algebraic geometry. V , Encyclopaedia of Mathematical Sciences, vol. 47, Springer-Verlag, Berlin, 1999. MR 1668575[Sno86] Dennis M. Snow,
Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hyper-surfaces , Math. Ann. (1986), no. 1, 159–176. MR 863714[SS10] Karl Schwede and Karen E. Smith,
Globally F -regular and log Fano varieties , Adv. Math. (2010),no. 3, 863–894. MR 2628797[SW90] Michał Szurek and Jarosław A. Wiśniewski, Fano bundles of rank on surfaces , Compositio Math. (1990), no. 1-2, 295–305, Algebraic geometry (Berlin, 1988). MR 1078868[Tot18] Burt Totaro, Bott vanishing for algebraic surfaces , arXiv:1812.10516 (2018).[Xin16] He Xin, On W -lifting of Frobenius of algebraic surfaces , Collect. Math. (2016), no. 1, 69–83.MR 3439841 [Zda18] Maciej Zdanowicz, Liftability of singularities and their Frobenius morphism modulo p , Int. Math. Res.Not. IMRN (2018), no. 14, 4513–4577. MR 3830576 Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland
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