Deformations of A 1 -cylindrical varieties
aa r X i v : . [ m a t h . AG ] O c t DEFORMATIONS OF A -CYLINDRICAL VARIETIES ADRIEN DUBOULOZ AND TAKASHI KISHIMOTO
Abstract.
An algebraic variety is called A -cylindrical if it contains an A -cylinder, i.e. a Zariski open subset ofthe form Z × A for some algebraic variety Z . We show that the generic fiber of a family f : X → S of normal A -cylindrical varieties becomes A -cylindrical after a finite extension of the base. This generalizes the main resultof [6] which established this property for families of smooth A -cylindrical affine surfaces. Our second result is acriterion for existence of an A -cylinder in X which we derive from a careful inspection of a relative Minimal ModelProgram ran from a suitable smooth relative projective model of X over S . Introduction
An algebraic variety is called A -cylindrical (or affine-ruled or A -ruled) if it contains an A -cylinder, i.e. aZariski open subset of the form Z × A for some algebraic variety Z . Such A -cylinders appear naturally inmany recent problems and questions related to the geometry of algebraic varieties, both affine and projective[16, 5, 6, 7, 8, 1, 2, 3, 17, 18, 19, 24, 25]. Clearly, there are only two A -cylindrical smooth complex curves: theaffine line A and the projective line P . As a consequence of classical classification results, every smooth projectivesurface of negative Kodaira dimension is A -cylindrical, and the same holds true for smooth affine surfaces by adeep result of Miyanishi-Sugie and Fujita [22]. But it is still an open problem to find a complete and effectivecharacterization of which complex surfaces, possibly singular, contain A -cylinders [16]. The situation in higherdimension is even more elusive, some natural class of examples of A -cylindrical varieties are known, especially inrelation with the study of additive group actions on affine varieties, but for instance the question whether everysmooth rational projective variety is A -cylindrical is still totally open.A natural way to try to produce new A -cylindrical varieties from known ones is to consider algebraic families f : X → S of such varieties. One hopes that the fiberwise A -cylinders could arrange themselves continuously toform a global relative A -cylinder in the total space X , in the form of a cylinder U ≃ Z × A in X for some S -variety Z , whose restriction to a general closed fiber of f : X → S is equal to the initially prescribed A -cylinder in it. Forfamilies of relative dimension one, it is a classical fact [14] that a smooth fibration f : X → S whose general closedfibers are isomorphic to A indeed restricts to trivial A -bundle Z × A over a dense open subset Z of S . But onthe other hand, the existence of nontrivial conic bundles f : X → S shows that it is in general too much to expectthat fiberwise cylinders are restrictions of global ones. Indeed, for such a nontrivial conic bundle, the general closedfibers are isomorphic to P , hence are A -cylindrical, but the generic fiber of f : X → S is a nontrivial form of P over the function field K of S : the latter does not contains any open subset isomorphic to A K , which prevents inturn the existence of a global A -cylinder in X over an S -variety. Nevertheless, such an A -cylinder exists afterextending the scalars to a suitable quadratic extension of K , leading to the conclusion that the total space of anysmooth family f : X → S of A -cylindrical varieties of dimension one always contain a relative A -cylinder, possiblyafter an étale extension of the base S .A similar property is known to hold for certain families of relative dimension . More precisely, it was establishedin [10, Theorem 3.8] and [6, Theorem7] by different methods, involving respectively the study of log-deformationsof suitable relative projective models and the geometry of smooth affine surfaces of negative Kodaira dimensiondefined over non closed fields, that for smooth families f : X → S of complex A -cylindrical affine surfaces, thereexists an étale morphism T → S such that X T = X × S T contains an A -cylinder U ≃ Z × A over a T -variety Z .The first main result of this article consists of a generalization of this property to arbitrary families f : X → S ofnormal algebraic varieties defined over an uncountable base field, namely: Theorem 1.
Let k be an uncountable field of characteristic zero and let f : X → S be dominant morphism betweengeometrically integral algebraic k -varieties. Suppose that for general closed points s ∈ S , the fiber X s contains Mathematics Subject Classification.
Key words and phrases. A -cylinder, deformation, Minimal Model Program, uniruled varieties.The second author was partially funded by Grant-in-Aid for Scientific Research of JSPS No. 15K04805. The research was initiatedduring a visit of the first author at the University of Saitama and continued during a stay of the second author at the University ofBurgundy as a CNRS Research Fellow. The authors thank these institutions for their generous support and the excellent workingconditions offered. A -CYLINDRICAL VARIETIES 2 an A -cylinder U s ≃ Z s × A over a κ ( s ) -variety Z s . Then there exists an étale morphism T → S such that X T = X × S T contains an A -cylinder U ≃ Z × A over a T -variety Z . We next turn to the problem of finding effective conditions on the fiberwise A -cylinders which ensure that aglobal relative A -cylinder exists, without having to take any base change. The question is quite subtle already inthe case of fibrations of relative dimension , as illustrated on the one hand by smooth del Pezzo fibrations with nonrational generic fiber, which therefore cannot contain any A -cylinder [8], and on the other hand by examples of oneparameter families f : X → S of smooth A -cylindrical affine cubic surfaces whose total spaces do not contain any A -cylinder at all, relative to f : X → S or not [5]. Intuitively, a global relative cylinder should exist as soon as thefiberwise A -cylinders are “unique”, in the sense that the intersection of any two of them is again an A -cylinder.This holds for instance for A -cylinders inside non rational smooth affine surfaces, and for families f : X → S ofsuch surfaces, it was indeed confirmed in [6, Theorem 10] that X contains a relative A -cylinder U ≃ Z × A over S , for which the rational projection X Z coincides, up to birational equivalence, with the Maximally RationallyConnected quotient of a relative smooth projective model f : X → S of X over S .The natural generalization in higher dimension would be to consider normal varieties Y which contain A -cylinders U ≃ Z × A over non uniruled bases Z . But there is a second type of obstruction for uniqueness, whichdoes not appear in the affine case: the fact that a given A -cylinder U ≃ Z × A in a variety Y can actually bethe restriction of a P -cylinder U ≃ Z × P inside Y , with the effect that Y then contains infinitely many distinct A -cylinders of the form Z × ( P \ { p } ) , p ∈ P , all over the same base Z . This possibility is eliminated by restrictingthe attention to varieties Y containing A -cylinders U ≃ Z × A for which the open immersion Z × A ֒ → Y cannotbe extended to a birational map Z × P Y defined over the generic point of Z . An A -cylinder with thisproperty is called vertically maximal in Y (see Definition 10), and our second main result consists of the followingcharacterization: Theorem 2.
Let k be a field of characteristic zero and let f : X → S be a dominant morphism between normal k -varieties such that for general closed points s ∈ S , the fiber X s contains a vertically maximal A -cylinder U s ≃ Z s × A over a non uniruled κ ( s ) -variety Z s . Then X contains an A -cylinder U ≃ Z × A for some S -variety Z . The article is organized as follows. The first section contains a quick review of rationally connected and uniruledvarieties and some explanation concerning the minimal model program for varieties defined over arbitrary fieldsof characteristic zero which plays a central role in the proof of Theorem 2. In section two, we establish basicproperties of A -cylindrical varieties. Theorem 1 is then derived in section three from quite standard “general-to-generic” Lefschetz principle and specialization arguments. Finally, section four is devoted to the proof of Theorem2, which proceeds through a careful study of the output of a relative minimal model program applied to a suitablyconstructed smooth projective model f : Y → S of X over S .1. Preliminaries
In what follows, unless otherwise stated, k is a field of characteristic zero, and all objects considered will beassumed to be defined over k . A k -variety is a reduced scheme of finite type over k . For a morphism f : X → S andanother morphism T → S , the symbol X T will denote the fiber product X × S T . In particular for a point s ∈ S ,closed or not, we write X s = f − ( s ) = X × S Spec( κ ( s )) where κ ( s ) denotes the residue field of s . In addition, if T = Spec( K ) for a field K , then X T will also sometimes be denoted by X K .1.1. Recollection on rational connectedness and uniruledness.Definition 3. (See [20, IV.3 Definition 3.2 and IV.1 Definition 1.1]) Let f : X → S be an integral scheme over ascheme S . We say that X is:a) Rationally connected over S if there exists an S -scheme B and a morphism u : B × P → X of schemes over S such u × B u : ( B × P ) × B ( B × P ) → X × S X is dominant.b) Uniruled over S if there exists an S -scheme B of relative dimension dim( X/S ) − and a dominant rationalmap u : B × P X of schemes over S .c) Ruled over S if there exists an S -scheme B of relative dimension dim( X/S ) − and a dominant birationalmap u : B × P X of schemes over S .A variety X defined over a field k is called rationally connected (resp. uniruled, resp. ruled) if it is rationallyconnected (resp. uniruled, resp. ruled) over Spec( k ) . Recall [20, IV.3 3.2.5 and IV.1 Proposition 1.3] that the firsttwo notions are independent of the field over which X is defined. In particular, X is rationally connected (resp.uniruled) if and only if X K is rationally connected (resp. uniruled) over Spec( K ) for every field extension k ⊂ K . Incontrast, it is well-known that the property of being ruled depends on the base field k : for instance a smooth conic EFORMATIONS OF A -CYLINDRICAL VARIETIES 3 C ⊂ P k without k -rational point is uniruled but not ruled, but becomes ruled after base extension to a suitablequadratic extension of k .The following lemma is probably well-known, but we include a proof because of lack of an appropriate reference. Lemma 4.
Let Z be a non uniruled k -variety and let h : Y → T be a surjective proper morphism between normal k -varieties, with rationally connected general fibers. Then every dominant rational map p : Y Z factors througha rational map q : T Z .Proof. Since the property of being non uniruled is invariant under birational equivalence, we can assume withoutloss of generality that Z is projective. Since h : Y → T is proper, for every blow-up σ : ˜ Y → Y of Y , the composition h ◦ σ : ˜ Y → T is again proper with rationally connected general fibers [20, IV.3.3]. So blowing-up Y to resolve theindeterminacy of p , we can further assume that p is a morphism, and by shrinking T that h : Y → T is faithfully flatand proper, with rationally connected fibers. Then Y is rationally chain connected over T . Let u : C = B × P → Y be a morphism of algebraic varieties over T witnessing this property. The inverse image by p × p : Y × T Y → Z × Z of the diagonal is a closed subset X of Y × T Y . If X = Y × T Y then since h ◦ pr : Y × T Y → T is flat hence open,the image of Y × T Y \ X is a dense open subset T of T . Replacing T by T , this would imply that the image of ( p ◦ u ) × B ( p ◦ u ) : C × B C → Z × Z is not contained in the diagonal, in contradiction with the non-uniruledness of Z . Thus X = Y × T Y and so, p is constant on the fibers of h : Y → T . By faithfully flat descent, there exists aunique morphism q : T → Z such that p = q ◦ h . (cid:3) Remark . The conclusion of Lemma 4 does not hold under the weaker assumption that the general fibers of h : Y → T are rationally chain connected. For instance, let Y be the projective cone over a smooth elliptic curve Z ⊂ P k and let h : Y → T = Spec( k ) be the canonical structure morphism. Then Y is rationally chain connectedover Spec( k ) and the projection p : Y Z form the vertex of the cone is a dominant rational map, which thereforedoes not factor through Spec( k ) .1.2. Minimal Model Program over non closed fields.
In the proof of Theorem 2 given in section four below,we will make use of minimal model program over arbitrary fields of characteristic zero. We freely use the standardterminology and conventions in this context, and just recall the mild adaptations needed to run the minimal modelprogram over a non closed field k in a form appropriate to our needs. It is well known (see e.g. [21, § § f : Y → S between normal quasi-projective varieties defined over a field k : after the basechange f k : Y k → S k to an algebraic closure k of k , one can perform all the basic steps of K Y k -mmp with scalingsrelative to f k : Y k → S k as in [4] in an equivariant way with respect to the natural action of the Galois group G = Gal( k/k ) . Compared to the genuine relative K Y k -mmp with scalings, this program runs in the category ofvarieties which are projective over S k , with only terminal G - Q -factorial singularities, i.e. varieties with terminalsingularities on which every G -invariant Weil divisor is Q -Cartier.The termination of arbitrary sequences of G -equivariant flips is not yet verified in a full generality, but as far as K Y k is not pseudo-effective over S k , it follows from [4, Corollary 1.3.3] that there exists a G -equivariant K Y k -mmp Θ : Y k ˜ Y over S k with scalings by an f k -ample G -invariant divisor which ends with a G -Mori fiber space ˜ ρ : ˜ Y → ˜ T over a normal S k -variety ˜ T . That is, ˜ ρ : ˜ Y → ˜ T is a projective G -equivariant morphism betweenquasi-projective k -varieties with the following properties: ˜ ρ ∗ O ˜ Y = O ˜ T , dim ˜ T < dim ˜ Y , ˜ Y has only terminal G - Q -factorial singularities, the anti-canonical divisor − K ˜ Y is ˜ ρ -ample and the relative G -invariant Picard number of ˜ Y over ˜ T is equal to one.The birational map Θ : Y k ˜ Y is a composition of either divisorial contractions associated to successive G -invariant extremal faces in the cone NE( Y k /S k ) or flips which are all defined over k . The last morphism ˜ ρ : ˜ Y → ˜ T corresponds to a G -equivariant extremal contraction of fiber type and is defined over k as well. It follows that Θ : Y k ˜ Y and ˜ ρ : ˜ Y → ˜ T can be equivalently seen as the base change to k of a sequence θ : Y Y ′ of K Y -negative divisorial extremal contractions and flips between k -varieties which are Q -factorial over k and projectiveover S , and an extremal contraction of fiber type ρ ′ : Y ′ → T between normal k -varieties, such that − K Y ′ is ρ ′ -ample and the relative Picard number of Y ′ over T is equal to one.2. A -cylindrical varieties In this section, we introduce and establish basic properties of a special class of ruled varieties called A -cylindricalvarieties. EFORMATIONS OF A -CYLINDRICAL VARIETIES 4 Definition 6.
Let f : X → S be a morphism of schemes. An A -cylinder in X over S is a pair ( Z, ϕ ) consisting ofan S -scheme Z → S and an open embedding ϕ : Z × A ֒ → X of S -schemes. We say that X is A -cylindrical over S if there exists an A -cylinder ( Z, ϕ ) in X over S .A variety X defined over a field k is called A -cylindrical over k if it is A -cylindrical over Spec( k ) . Similarlyas for ruledness, the property of being A -cylindrical depends on the base field k : a smooth conic C ⊂ P k without k -rational point is not A -cylindrical over k but becomes A -cylindrical after base extension to a suitable quadraticextension of k . Definition 7. A sub- A -cylinder of an A -cylinder ( Z, ϕ ) in X over S is an A -cylinder ( Z ′ , ϕ ′ ) in X over S forwhich there exists a commutative diagram Z ′ × A Z × A XZ ′ Z j pr Z ′ ϕ ′ ϕ pr Z i for some open embeddings of S -schemes i : Z ′ ֒ → Z and j : Z ′ × A ֒ → Z × A . Two A -cylinders in X over S arecalled equivalent if they have a common sub- A -cylinder over S .2.1. A -cylinders and P -fibrations. Recall that a P -fibration is a proper morphism h : Y → T between integralschemes whose fiber over the generic point of T is a form of P over the function field K of T . Given an A -cylinder ( Z, ϕ ) in an algebraic variety X over k , the composition of ϕ − : X Z × A with the projection pr Z : Z × A → Z extends on a suitable complete model Y of X to a P -fibration h : Y → T over a complete model T of Z , restrictingto a trivial P -bundle over a non empty open subset Z of Z ⊂ T . Conversely, the total space Y of a P -fibration h : Y → T is A -cylindrical over T provided that h admits a rational section H ⊂ Y . Indeed, if so, there exists adense open subset Z of T such that h − ( Z ) ≃ Z × P and H ∩ h − ( Z ) ≃ Z × {∞} for some fixed k -rational point ∞ ∈ P , which implies in turn that the open subset h − ( Z ) ∩ ( Y \ H ) of Y is isomorphic to Z × A .The following characterization will be useful for the proof of Theorem 2: Lemma 8.
Let h : Y → T be a surjective proper morphism between normal varieties over a field k of characteristiczero, with irreducible and rationally connected general fibers. Suppose that Y contains an A -cylinder ( Z, ϕ ) forsome non-uniruled k -variety Z . Then h : Y → T is a P -fibration and there exists a sub- A -cylinder ( Z ′ , ϕ ′ ) of ( Z, ϕ ) and commutative diagram Z ′ × A YZ ′ T ϕ ′ pr Z ′ hi for some open embedding i : Z ′ ֒ → T . In particular, T is not uniruled.Proof. By shrinking Z , we can assume that it is smooth and affine. Letting Z be a smooth projective model of Z , the composition pr Z ◦ ϕ − defines a dominant rational map pr Z ◦ ϕ − : Y Z which lifts to a P -fibration p : ˜ Y → Z on some blow-up σ : ˜ Y → Y of Y . Since Z is not uniruled, and h ◦ σ : ˜ Y → T is proper with rationallyconnected general fibers, it follows from Lemma 4 that p factors through a dominant rational map q : T Z . So dim T ≥ dim Z and since dim Z = dim ˜ Y − ≥ dim T , we conclude that dim T = dim Y − . This implies that h ◦ σ : ˜ Y → T is a P -fibration, hence that h is a P -fibration. Since the general fiber of p : ˜ Y → Z are irreducible, q has degree , hence is birational. Now it suffices to choose for Z ′ an open subset of Z ⊂ Z on which q − restrictsto an isomorphism onto its image. (cid:3) Birational modifications preserving A -cylinders. In contrast with ruledness, the property of containingan A -cylinder is obviously not invariant under birational equivalence. Nevertheless, it is stable under certainparticular birational modifications which we record here for later use: Lemma 9.
Let ( Y, ∆) be a pair consisting of a normal k -variety and a reduced divisor on it, let θ : Y Y ′ be abirational map to a normal k -variety Y ′ and let ∆ ′ = θ ∗ (∆) be the proper transform of ∆ on Y ′ . Then the followinghold:a) If θ is an isomorphism in codimension one then Y \ ∆ is A -cylindrical over k if and only if so is Y ′ \ ∆ ′ .b) If θ is a proper morphism and Y ′ \ ∆ ′ contains an A -cylinder ( Z ′ , ϕ ′ ) over k then there exists a sub-cylinder ( Z, ϕ ) of ( Z ′ , ϕ ′ ) such that ( Z, θ − ◦ ϕ ) is an A -cylinder in Y \ ∆ over k . EFORMATIONS OF A -CYLINDRICAL VARIETIES 5 c) If θ is a proper morphism, each irreducible component of pure codimension one of the exceptional locus Exc( θ ) of θ is uniruled and Y \ ∆ contains an A -cylinder ( Z, ϕ ) for some non-uniruled k -variety Z then there exists asub-cylinder ( Z ′ , ϕ ′ ) of ( Z, ϕ ) such that ( Z ′ , θ ◦ ϕ ′ ) is a A -cylinder in Y ′ \ ∆ ′ .Proof. If θ is an isomorphism in codimension one, then it restricts to an isomorphism θ : U → U ′ between opensubsets U ⊂ Y and U ′ ⊂ Y ′ whose complements X and X ′ have codimension at least in Y and Y ′ respectively.Given a cylinder ( Z, ϕ ) in Y \ ∆ , the inverse image by ϕ of X ∩ ( Y \ ∆) has codimension at least two in Z × A ,hence does not dominate Z . Consequently, there exists a dense open subset Z ′ ⊂ Z such that ( Z ′ , ϕ ′ = ϕ | Z × A ) is a sub- A -cylinder of ( Z, ϕ ) whose image is contained in U ∩ ( Y \ ∆) , and the composition ( Z ′ , θ ◦ ϕ ′ ) is then an A -cylinder in U ′ ∩ Y ′ \ ∆ ′ ⊂ Y ′ \ ∆ ′ . Reversing the roles of Y and Y ′ , this yields a).If θ is a proper morphism, then C = θ (Exc( θ )) has codimension at least in Y ′ because Y ′ is normal. So therestriction of pr Z ′ to ϕ ′− ( C ) cannot be dominant. This guarantees the existence of a dense open subset Z of Z ′ such that ( Z, ϕ = ϕ ′ | Z × A ) is a sub- A -cylinder of ( Z ′ , ϕ ′ ) whose image is contained Y ′ \ ∆ ′ ∪ C . Then ( Z, θ − ◦ ϕ ) is an A -cylinder in Y \ ∆ ∪ Exc( θ ) ⊂ Y \ ∆ , which proves b).Finally to prove c), we observe that since Z is not uniruled, the restriction of pr Z to the inverse image by ϕ ofa uniruled irreducible component of pure codimension one of Exc( θ ) cannot be dominant. This implies that therestriction of pr Z to ϕ − (Exc( θ )) is not dominant hence that there exists a dense open subset Z ′ ⊂ Z such that ϕ ( Z ′ × A ) ⊂ Y \ ∆ ∪ Exc( θ ) . Then ( Z ′ , ϕ ′ = ϕ | Z ′ × A ) is a sub- A -cylinder of ( Z, ϕ ) with the desired property. (cid:3) Uniqueness properties of A -cylinders. The fact that P k contains infinitely many non-equivalent cylinders P k \ { p } over k , parametrized by the set of its k -rational points p ∈ P k ( k ) , shows that in general a given k -variety X can contain many non equivalent cylinders even when their respective base spaces are non-uniruled. To ensuresome uniqueness property of A -cylinders, we are led to introduce the following notion: Definition 10.
Let f : X → S be a morphism of schemes. An A -cylinder ( Z, ϕ ) in X over S is said to be verticallymaximal in X over S if for every generic point ξ of Z , the open embedding ξ × A ֒ → X induced by ϕ cannot beextended to a morphism ξ × P → X .The next result can be thought as another geometric variant of Iitaka and Fujita strong cancellation theorems [13]. Proposition 11.
Let X be k -variety containing a vertically maximal A -cylinder ( Z, ϕ ) over a non uniruled k -variety Z . Then every A -cylinder in X over k is equivalent to ( Z, ϕ ) .Proof. Let U = ϕ ( Z × A ) be the open image of Z × A in X . By shrinking Z if necessary, we can assume that Z is affine and that all fibers of the projection pr Z ◦ ϕ − : U → Z are closed in X . Let ( T, ψ ) be another cylinderin X with image V = ψ ( T × A ) , let W = U ∩ V and let Z and T be the open images of W in Z and T bythe morphisms pr Z ◦ ϕ − and pr T ◦ ψ − respectively. Since pr T ◦ ψ − : W → T is a surjective morphism withuniruled fibers and Z , whence Z , is not uniruled, there exists a unique surjective morphism α : T → Z such that pr Z ◦ ϕ − : W → Z factors as pr Z ◦ ϕ − = α ◦ (pr T ◦ ψ − ) : W → T → Z . So for every point t ∈ T , there exists aunique z = α ( t ) ∈ Z such that ψ (pr − T ( t )) ∩ U is equal to ϕ (pr − Z ( z )) ∩ V . Since by hypothesis ϕ (pr − Z ( z )) ≃ A κ ( z ) is closed in X , it follows that ψ (pr − T ( t )) = ϕ (pr − Z ( z )) . This implies in turn that ψ ( T × A ) ⊂ ϕ ( Z × A ) andthat we have a commutative diagram: ψ ( T × A ) ϕ ( Z × A ) UT Z Z. pr T ◦ ψ − pr Z ◦ ϕ − α It follows in particular that α is also injective, hence an isomorphism. Thus ( T , ψ | T × A ) is a sub- A -cylinder of ( Z, ϕ ) , which shows that ( Z, ϕ ) and ( T, ψ ) are equivalent. (cid:3) Proof of Theorem 1
We now proceed to the proof of Theorem 1. By hypothesis, f : X → S is a dominant morphism betweengeometrically normal algebraic varieties defined over an uncountable field k of characteristic zero, with the propertythat for general closed points s ∈ S , the fiber X s contains a cylinder ( Z s , ϕ s ) over a κ ( s ) -variety Z s . Letting X η be the fiber of f over the generic point η of S , the existence of an étale morphism T → S such that X × S T is A -cylindrical over T , is equivalent to that of a finite extension L ⊂ L ′ of the function field L of S such that X η × Spec( L ) Spec( L ′ ) is A -cylindrical over L ′ . In fact, the following specialization lemma implies that it is enoughto find any extension L ′ of L for which X η × Spec( L ) Spec( L ′ ) is A -cylindrical over L ′ : EFORMATIONS OF A -CYLINDRICAL VARIETIES 6 Lemma 12.
Let X be a variety defined over a field k of characteristic zero and let k ⊂ K be any field extension.If X K is A -cylindrical over K then there exists a finite extension k ⊂ k ′ such that X k ′ is A -cylindrical over k ′ .Proof. By hypothesis, there exists an open embedding ϕ : Z × A ֒ → X K for some K -variety Z . This open embeddingis defined over a finitely generated sub-extension L of K , i.e. there exists an open embedding ϕ : Z × A ֒ → X L of L -varieties such that ϕ is obtained from ϕ by the base change Spec( K ) → Spec( L ) . Being finitely generatedover k , L is the function field of an algebraic variety S defined over k and we can therefore view X L as the fiber X η of the projection pr S : X = X × S → S over the generic point η of S . Let ∆ and T be the respective closures of X η \ ϕ ( Z × A ) and ϕ ( Z × { } ) in X . The projection pr Z : Z × A → Z induces a rational map ρ : X \ ∆ T whose generic fiber is isomorphic to A over the function field of T . It follows that there exists an open subset Y ⊂ T over which ρ is regular and whose inverse image V = ρ − ( Y ) is isomorphic to Y × A . Now for a generalclosed point s ∈ S , the fiber X s of pr S over s is isomorphic to X κ ( s ) , where κ ( s ) denotes the residue field of s , andcontains an open subset V s isomorphic to Y s × A . The induced open immersion Y s × A ֒ → X κ ( s ) provides thedesired A -cylinder over the finite extension κ ( s ) of k . (cid:3) Let k be an algebraic closure of k and let f k : X k → S k be the morphism obtained by the base extension Spec( k ) → Spec( k ) . Since S is geometrically integral, S k is integral and its field of functions k ( S k ) is an extensionof the field of functions L of S . If the generic fiber of f k becomes A -cylindrical after the base change to someextension of k ( S k ) then by the previous lemma, the generic fiber X η of f : X → S becomes A -cylindrical afterthe base change to a finite extension of L . We can therefore assume from the very beginning that k = k is anuncountable algebraically closed field of characteristic zero. Up to shrinking S , we can further assume without lossof generality that it is affine and that for every closed point s in S , X s contains a cylinder ( Z s , ϕ s ) over a k -variety Z s . Since X and S are k -varieties, there exists a subfield k ⊂ k of finite transcendence degree over Q such that f : X → S is defined over k , i.e. there exists a morphism of k -varieties f : X → S and a commutative diagram X X S S Spec( k ) Spec( k ) f f in which each square is cartesian. The field of functions L = k ( S ) of S is an extension of k of finite transcendencedegree over Q , and since k is uncountable and algebraically closed, there exists a k -embedding i : L ֒ → k of L in k . Letting ( X ) η be the fiber of f over the generic point η : Spec( L ) → S of S , the composition Γ( S , O S ) ֒ → L ֒ → k induces a k -homomorphism Γ( S , O S ) ⊗ k k → k defining a closed point s : Spec( k ) → Spec(Γ( S , O S ) ⊗ k k ) = S of S for which we obtain the following commutative diagram X s (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ (cid:15) (cid:15) / / X (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ f (cid:15) (cid:15) ( X ) η / / (cid:15) (cid:15) X f (cid:15) (cid:15) Spec( k ) i ∗ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ s / / S / / (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ Spec( k ) (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ Spec( L ) η / / S / / Spec( k ) . Since the bottom square of the cube above is cartesian by construction, we have ( X ) η × Spec( L ) Spec( k ) ≃ X × S Spec( k ) ≃ X × S Spec( k ) = X s . Since by hypothesis X s is A -cylindrical over k , we conclude that ( X ) η × Spec( L ) Spec( k ) is A -cylindrical over k . Lemma 12 then guarantees that there exists a finite extension L ⊂ L ′ such that ( X ) η × Spec( L ) Spec( L ′ ) is A -cylindrical over L ′ . Finally, the tensor product L ⊗ L L ′ decomposes as a direct product of finitely many finiteextensions L ′ of L with the property that X η × Spec( L ) Spec( L ′ ) is A -cylindrical over L ′ , which completes the proofof Theorem 1. EFORMATIONS OF A -CYLINDRICAL VARIETIES 7 Remark . Combined with Proposition 11, Lemma 12 implies that for a k -variety X , the property of containing avertically maximal A -cylinder over a non uniruled variety is independent of the base field. Indeed, by Lemma 12 if X K contains a cylinder for some arbitrary field extension k ⊂ K , then X k ′ contains a cylinder for a finite extension k ⊂ k ′ . Letting k ′′ be the Galois closure of the extension k ⊂ k ′ in an algebraic closure of k ′ , Proposition 11 impliesthat the translates of a given cylinder ( Z, ϕ ) in X k ′′ over k ′′ by the action of the Galois group G = Gal( k ′′ /k ) are all equivalent. Since G is a finite group, it follows that there exists a dense affine open subset Z of Z , anaction of G on Z lifting to a G -action on Z × A such that the induced open embedding ( Z , ϕ | Z × A ) ֒ → X k ′′ is G -equivariant. The quotients ( Z × A ) /G and Z /G are then affine varieties defined over k while the projection Z × A → Z and the open embedding ϕ | Z × A : Z × A ֒ → X k ′′ descend respectively to a locally trivial A -bundle π : ( Z × A ) /G → Z /G and an open embedding ψ : ( Z × A ) /G ֒ → X k ′′ /G ≃ X . A cylinder in X over k is thenobtained by restricting ψ to the inverse image of a dense open subset of Z /G over which π is a trivial A -bundle.4. Proof of Theorem 2
We first consider the case where f : X → S is a smooth projective morphism whose general closed fibers containvertically maximal A -cylinders over non uniruled varieties. The case of an arbitrary morphism f : X → S betweennormal algebraic varieties is then deduced by considering a suitably constructed smooth relative projective modelof X over S .4.1. Case of a smooth projective morphism.Proposition 14.
Let f : Y → S be a smooth projective morphism between normal k -varieties and let ∆ ⊂ Y be adivisor on Y such that for a general closed point s ∈ S , Y s \ ∆ s contains an A -cylinder ( Z s , ϕ s ) over a non uniruled κ ( s ) -variety Z s . Then there exists a K Y -MMP θ : Y Y ′ relative to f : Y → S whose output f ′ : Y ′ → S hasthe structure of a Mori conic bundle ρ ′ : Y ′ → T over a non uniruled normal S -variety h : T → S . Furthermore,for a general closed point s ∈ S , there exists a sub-cylinder ( Z ′ s , ϕ ′ s ) of ( Z s , ϕ s ) and a commutative diagram Z ′ s × A Y ′ s Z ′ s T s pr Z ′ s θ s ◦ ϕ ′ s ρ ′ s α s where the top and bottom arrows are open embeddings.Proof. Since the general fibers of f : Y → S are in particular uniruled, it follows that K Y is not f -pseudo-effective.By virtue of [4, Corollary 1.3.3] (see § K Y -mmp θ : Y Y ′ relative to f : Y → S whose output f ′ : Y ′ → S has the structure of a Mori fiber space ρ ′ : Y ′ → T over some normal S -variety h : T → S . Sincefor a general closed point s ∈ S the restriction θ s : Y s Y ′ s of θ is a part of a K Y s -mmp ran from the smoothprojective variety Y s , it follows from [11, Corollary 1.7] that every irreducible component of pure codimension oneof the exceptional locus of θ s is uniruled. Since θ s is a composition of divisorial contractions and isomorphismsin codimension one, we deduce from Lemma 9 a) and c) that there exists a sub-cylinder ( Z ′ s , ϕ ′ s ) of ( Z s , ϕ s ) suchthat ( Z ′ s , θ s ◦ ϕ ′ s ) is an A -cylinder in Y ′ s . Since Y ′ has terminal singularities and − K Y ′ is ρ ′ -ample, we deducefrom [11, Corollary 1.4] that every fiber of ρ ′ is rationally chain connected. Since a general closed fiber of ρ ′ hasagain terminal singularities, we deduce in turn from [11, Corollary 1.8] that it is in fact rationally connected. Theassertion then follows from Lemma 8. (cid:3) Lemma 15.
In the setting of Proposition 14, suppose further that for a general closed point s ∈ S , the A -cylinder ( Z s , ϕ s ) in Y s \ ∆ s is maximally vertical. Then Y \ ∆ is A -cylindrical over S .Proof. Since Y s is projective, the hypothesis that ( Z s , ϕ s ) is maximally vertical in Y s \ ∆ s implies that the subset ∆ of irreducible components of ∆ which are horizontal for f : Y → S is not empty. Furthermore, for a generalclosed point s ∈ S , ∆ ,s intersects the closures in Y s of the general fibers of pr Z s ◦ ϕ − s : ϕ s ( Z s × A ) → Z s in aunique place. Let ( Z ′ s , ϕ ′ s ) be a sub- A -cylinder of ( Z s , ϕ s ) with the property that ( Z ′ s , θ s ◦ ϕ ′ s ) is an A -cylinder in Y ′ s and α s : Z ′ s ֒ → T s is an open embedding. Since the only divisors that could be contracted by θ s : Y s Y ′ s areuniruled hence do not dominate Z ′ s , we can assume up to shrinking Z ′ s further if necessary that the restriction of θ − s to ρ ′ s − ( Z ′ s ) is an isomorphism onto its image V s in Y s . Consequently, ρ ′ s ◦ θ s | V s : V s → Z ′ s is a P -fibrationextending pr Z ′ s ◦ ϕ ′ s − : ϕ ′ s ( Z ′ s × A ) → Z ′ s . Since ( Z s , ϕ s ) is vertically maximal in Y s \ ∆ s , so is ( Z ′ s , ϕ ′ s ) , and itfollows that ∆ ,s ∩ V s is a section of ρ ′ s ◦ θ s | V s : V s → Z ′ s . This implies in turn that ∆ is irreducible and that thereexists an open subset T of T such that ( ρ ′ ◦ θ ) − ( T ) ≃ T × P and ( ρ ′ ◦ θ ) − ( T ) \ ∆ ≃ T × A . So Y \ ∆ is A -cylindrical over T whence over S . (cid:3) EFORMATIONS OF A -CYLINDRICAL VARIETIES 8 General case.
The case of a general morphism f : X → S between normal algebraic varieties is now obtainedas follows. By desingularization theorems [12], we can find a desingularization σ : ˜ X → X which restricts toan isomorphism over the regular locus X reg . Since X is normal, it follows in particular that the image of theexceptional locus of σ is a closed subset of X of codimension at least two. By Nagata completion theorems [23]and desingularization theorems again, there exists an open embedding j : ˜ X ֒ → ˜ Y into a smooth algebraic variety ˜ Y proper over S . Then by Chow lemma [9, 5.6.1] there exists a smooth algebraic variety Y projective over S , say f : Y → S , and a birational morphism τ : Y → ˜ Y . Applying desingularization again, we can further assume that thereduced total transform of ˜ Y \ j ( ˜ X ) in Y is an SNC divisor ∆ . Since ˜ Y is smooth, the image of the exceptional locusof τ has codimension at least two in ˜ Y , and so the image of the exceptional locus of β = σ ◦ τ | τ − ( ˜ X ) : τ − ( ˜ X ) → X is a closed subset of codimension at least two in X . Summing up, we get a sequence of birational maps of S -varieties X ˜ X ˜ Y YS S σ − f δj τ − f which we refer to as a good relative smooth projective completion of f : X → S . Lemma 16.
Let f : X → S be a morphism between normal k -varieties and let δ : X Y be a good relativesmooth projective completion of f : X → S . Suppose that for a general closed point s ∈ S , X s contains an A -cylinder ( Z s , ϕ s ) over a κ ( s ) -variety Z s . Then for a general closed point s , there exists a dense open subset Z ′ s of Z s such that ( Z ′ s , δ s ◦ ϕ s ) is an A -cylinder in in Y s \ ∆ s . Furthermore, if ( Z s , ϕ s ) is vertically maximal in X s then ( Z ′ s , δ s ◦ ϕ s ) is vertically maximal in Y s \ ∆ s .Proof. Since X is normal, for a general closed point s ∈ S , X s is a normal variety. The morphism ( σ ◦ τ ) s : τ − ( j ( ˜ X )) s → X s being proper and birational by construction, the first assertion follows from Lemma 9 b). Thesecond one is clear from the definition of ∆ . (cid:3) The following proposition combined with Proposition 14, Lemma 15 Lemma 16 completes the proof of Theorem2.
Proposition 17.
Let f : X → S be a morphism between normal k -varieties and let δ : X Y be a good relativesmooth projective completion of f : X → S . Suppose that for a general closed point s ∈ S , X s contains a verticallymaximal A -cylinder ( Z s , ϕ s ) over a non uniruled κ ( s ) -variety Z s . If Y \ ∆ is A -cylindrical over S then so is X .Proof. Let ψ : T × A ֒ → Y \ ∆ be an A -cylinder in Y over S . It is enough to show that the restriction of pr T to theinverse image by ψ of the exceptional locus Exc( β ) of β = σ ◦ τ | τ − ( ˜ X ) : τ − ( ˜ X ) → X is not dominant. Indeed, ifso, there exists an open subset T of T such that ψ ( T × A ) is contained in Y \ Exc( β ) ∪ ∆ ≃ δ ( X \ β (Exc( β ))) . Sosuppose on the contrary that there exists an irreducible component E of Exc( β ) such that pr T | ψ − ( E ) is dominant.For a general closed point s ∈ S , the fiber Y s is smooth and the restriction β s : τ − s ( ˜ X s ) → X s is an isomorphismoutside a closed subset of codimension at least two in X s . So there exists a dense open subset Z ′ s of Z s such that ( Z ′ s , ϕ ′ s = β − s ◦ ϕ s | Z ′ s × A ) is A -cylinder in τ − s ( ˜ X s ) . Since ( Z s , ϕ s ) is a vertically maximal A -cylinder in X s , ( Z ′ s , ϕ ′ s ) is vertically maximal in τ − s ( ˜ X s ) . On the other hand, for a general closed point s ∈ S , the restriction ψ s : T s × A → τ − s ( ˜ X s ) is also an embedding. Since Z s whence Z ′ s is not uniruled, it follows from Proposition11 that ( Z ′ s , ϕ ′ s ) and ( T s , ψ s ) are equivalent A -cylinders in τ − s ( ˜ X s ) . But then the restriction of pr Z ′ s to ϕ ′ s − ( E ) would be dominant, implying in turn that ( Z ′ s , ϕ ′ s ) is a not a cylinder, a contradiction. (cid:3) References
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E-mail address : [email protected] Department of Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan
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