Algebraic foliations defined by quasi-lines
aa r X i v : . [ m a t h . AG ] J a n ALGEBRAIC FOLIATIONS DEFINED BY QUASI-LINES
LAURENT BONAVERO AND ANDREAS HÖRING
Abstract.
Let X be a projective manifold containing a quasi-line l . Animportant difference between quasi-lines and lines in the projective space isthat in general there is more than one quasi-line passing through two givengeneral points. In this paper we use this feature to construct an algebraicfoliation associated to a family of quasi-lines. We prove that if the singularlocus of this foliation is not too large, it induces a rational fibration on X thatmaps the general leaf of the foliation onto a quasi-line in a rational variety. Introduction
Motivation.
Let X be a complex quasiprojective manifold of dimension n .A quasi-line l in X is a smooth rational curve f : P ֒ → X such that f ∗ T X is thesame as for a line in P n , i.e. is isomorphic to O P (2) ⊕ O P (1) ⊕ n − . Quasi-lines have some of the deformation properties of lines, but there are importantdifferences: for example if x and y are general points in X there exist only finitelymany deformations of l passing through the two points, but in general we do nothave uniqueness . It is now well established that given a variety X with a quasi-line l , the deformations and degenerations of l contain interesting information on theglobal geometry of X . Here is an example of such a result, due to Ionescu andVoica. [IV03, Thm.1.12] Let X be a projective manifold containinga quasi-line l . Assume there exists a divisor D such that D · l = 1 and h ( X, O X ( D )) = s + 1 ≥ . Then there exists a small deformation l ′ of l , a fi-nite composition of smooth blow-ups σ : ˜ X → X with smooth centers disjoint from l ′ and a surjective fibration ϕ : ˜ X → P s with rationally connected general fibre suchthat ϕ maps isomorphically σ − ( l ′ ) to a line in P s . A disadvantage of this statement is that a priori there seems to be no relationbetween the geometry of the quasi-line l and the existence of the divisor D . Thegoal of this paper is to fill this gap by a construction inspired by the theory of Date : July 27, 2009.2000
Mathematics Subject Classification.
Key words and phrases. rational curves, quasi-line, rationally connected manifold, holomorphicfoliation, algebraic leaves. We denote by e ( X, l ) the number of quasi-lines through two general points, see Definition 1.12for a formal definition. In order to simplify the statements, we’ll simply say that there exists a rational fibration ϕ : ( X, l ) ( P s , line ) . omplex projective manifolds X swept out by linear spaces: these have been studiedfor more than twenty years (see [Ein85, ABW92, Sat97, NO07]) and an observationcommon to all these papers is that if the codimension of the linear space is small,then either X is special (a projective space, hyperquadric etc. ) or it admits afibration such that the fibres are linear spaces. A powerful tool in their theory isthe family of lines contained in the linear spaces. The guiding philosophy of thispaper is that the rich geometry of a family of quasi-lines can be used to constructa natural family of subvarieties that induces a (rational) fibration on X .1.B. Setup and main results.
Let X be a projective manifold of dimension n containing a quasi-line l . The main tool used in this paper is an intrinsic foliation F x associated to the quasi-lines passing through a general point x of X . In casethe foliation has rank n − , its leaves are natural candidates to play the role ofthe divisor D in Theorem 1.1. The foliation F x ⊂ T X is defined by the followingheuristic principle:“for y general in X , the (closure of the) F x -leaf through y is thesmallest subvariety V ⊂ X containing y and such that for every z in V , every quasi-line through x and z is entirely contained in V ”.In a more technical language (see Section 2) we prove the following theorem. Let X be a projective manifold containing a quasi-line l and let H x ⊂ C ( X ) be the scheme parametrising deformations and degenerations of l pass-ing through a general point x ∈ X . Then there exists a unique saturated algebraicfoliation F x ⊂ T X such that for every general point y ∈ X , the unique F x -leaf (cf.Defn. 1.13) through y is the minimal H x -stable projective subvariety through y . If l is a line or more generally if e ( X, l ) = 1 , the foliation F x has rank one: the leafthrough a general point y is the unique quasi-line passing through x and y . Thisleads immediately to the following question. Let X be a projective manifold containing a quasi-line l . Let x be a general point in X , and denote by F x the corresponding foliation. Can weconstruct a rational fibration ϕ : ( X, l ) ( Y, l ′ := ϕ ( l )) onto a projective variety Y such that • l ′ is a quasi-line with e ( Y, l ′ ) = 1 , and • the general F x -leaves are preimages of deformations of l ′ ? Suppose for a moment that such a fibration exists: fix two general points x and y in X , and denote by F x and F y the corresponding foliations. By hypothesis theunique F x -leaf through y is the preimage of a quasi-line through ϕ ( x ) and ϕ ( y ) .Analogously the unique F y -leaf through x is the preimage of a quasi-line through ϕ ( y ) and ϕ ( x ) . Since both quasi-lines are deformations of l ′ passing through twogiven general points the condition e ( Y, l ′ ) = 1 implies that they are identical. Hencethe two leaves are identical. More formally we have a natural necessary conditionfor the existence of the fibration. .4. Assumption. Let X be a projective manifold containing a quasi-line l . Let x and y be two general points in X , and denote by F x and F y the correspondingfoliations. Denote by F x,y the unique F x -leaf through y and by F y,x the unique F y -leaf through x . Then we have F x,y = F y,x . Since every F x -leaf contains x , the singular locus F sing x of the foliation contains x and it is rather optimistic to expect general leaves to be smooth. Our first obser-vation is that under the Assumption 1.4 we have some control on the singularitiesaround x . Under the Assumption 1.4, let L be a general F x -leaf and l ⊂ X be a general quasi-line through x such that l ⊂ L . Then l is contained in the smoothlocus of L and is a quasi-line in L . In particular L is smooth in x . Now that we have some local information about the general leaf L , we can try tounderstand the global geometry of L and X . On the one hand we observe that dim x F sing x ≥ rk F x − , so this looks like a rather tough task. On the other hand we know that the singularlocus of the foliation given by lines in the projective space has dimension zero.Thus if we had a rational fibration ϕ : ( X, l ) ( P n − rk F x +1 , line ) , the singularlocus would be of dimension exactly rk F x − . Our main theorem shows that thesenecessary conditions are sufficient to give an affirmative answer to Question 1.3. Let X be a projective manifold containing a quasi-line l . Let x bea general point in X , and denote by F x the corresponding foliation with general leaf L . Assume that rk F x < dim X and the Assumption 1.4 holds. Then the followingholds.a) If the foliation F x satisfies dim x F sing x < rk F x − n − rk F x ) , there exists a rational fibration ϕ L : ( L , l ) P such that l is a section .b) If the foliation F x satisfies dim x F sing x = rk F x − , there exists a rational fibration ϕ : ( X, l ) ( Y, l ′ ) onto a projective variety Y of dimension n − rk F x + 1 such that l ′ is a quasi-line with e ( Y, l ′ ) = 1 .Moreover the restriction of ϕ to L is the fibration ϕ L . Since the quasi-line l ′ ⊂ Y satisfies e ( Y, l ′ ) = 1 , the variety Y is rational by [IN03,Prop.3.1]. In view of Theorem 1.1 it would be nice to know when the quasi-line l ′ ⊂ Y identifies to a line in a projective space. One guess is the following. More precisely, there exists a finite composition of smooth blow-ups σ : ˜ L → L with smoothcenters disjoint from a deformation l ′ of l and a surjective fibration ϕ L : ˜ L → P such that σ − ( l ′ ) is a section of ϕ L . .7. Conjecture. Let X be a projective manifold containing a quasi-line l with e ( X, l ) = 1 . If the (rank one) foliation F x satisfies dim x F sing x = 0 , there exists a birational map X P n that is an isomorphism in a neighbourhoodof l and maps l onto a line. A similar statement is shown in [IR07, Thm.1.5], but they make the additionalhypothesis that all the deformations of l are smooth at x . If the rank of F x equals n − all these consideration boil down to a very simple statement. Let X be a projective manifold containing a quasi-line l . Let x be a general point in X , and denote by F x the corresponding foliation. Assumethat rk F x = n − and the Assumption 1.4 holds. Then the following holds.a) There exists a rational fibration ϕ : ( X, l ) ( P , line ) .b) If the Picard number of X is one, we have X ≃ P . Although we do not have an example where the Assumption 1.4 fails, it is of courselegitimate to ask what happens in this situation. We don’t have much hope tounderstand the geometry of F x or X , but we can address another basic problem inthe study of quasi-lines: computing, or at least bounding the invariant e ( X, l ) . Let X be a projective manifold of dimension three containing aquasi-line l . Let x be a general point in X , and denote by F x the correspondingfoliation. Assume that rk F x = 2 and the Assumption 1.4 fails . Then e ( X, l ) ≤
16 (deg l ) deg X , where all the degrees are taken with respect to an ample line bundle H on X . If the Assumption 1.4 holds it is much more difficult to find a reasonable bound forthe invariant e ( X, l ) , cf. the discussion after Proposition 5.2. Acknowledgements.
The second named author wants to thank Kristina Frantzenfor explaining to him the nice combinatorial argument in Example 5.3.1.C.
Notation and basic definitions.
We work over the complex field C , topo-logical notions always refer to the Zariski topology. A variety is an integral schemeof finite type over C , a manifold is a smooth variety. A fibration is a surjectivemorphism ϕ : X → Y between normal varieties such that dim X > dim
Y > and ϕ ∗ O X ≃ O Y , that is all the fibres are connected. Fibres are always scheme-theoretic fibres. For general definitions we refer to Hartshorne’s book [Har77], wewill also use the standard terminology of Mori theory and deformation theory asexplained in [Deb01, Kol96].Let X be a projective variety and let V ⊂ X be a projective subvariety. Identify V to its fundamental cycle . We denote by [ V ] the point in the Chow scheme C ( X ) corresponding to V . Throughout the whole paper, we will not distinguish between an effective cycle and its support. .10. Definition. Let N ⊂ C ( X ) be a finite union of subvarieties parametrising d -dimensional cycles V ⊂ X . We say that a property holds for a general ( resp. verygeneral) cycle V if there exist an open dense ( resp. dense) subset N ⊂ N suchthat the property holds for every cycle parametrised by a point [ V ] ∈ N . For the convenience of the reader we recall some well known fact from deformationtheory: let X be a complex projective manifold of dimension n , and let l ⊂ X bea quasi-line. Since N l/X ≃ O P (1) ⊕ n − , the Chow scheme C ( X ) is smooth of dimension n − at [ l ] . Therefore there existsa unique irreducible component H of C ( X ) containing [ l ] . Furthermore a curve of X corresponding to a general point of H is a quasi-line. Let Γ ⊂ H× X be the universalfamily, then the natural map Γ → X is surjective. Thus for a general point x ∈ X ,the subscheme H x ⊂ H parametrising deformations and degenerations of l passingthrough x has pure dimension n − . Furthermore the points [ l ′ ] corresponding toquasi-lines are dense in H x . Since for any such quasi-line l ′ ⊂ XN l ′ /X ⊗ I x ≃ O ⊕ n − P , we see that all the irreducible components of H x are generically smooth. Setting Γ x for the universal family we get the following basic diagram:(1) Γ xq (cid:15) (cid:15) p / / X H x a) We fix the family H for the rest of the paper. Thus if we say that aproperty holds for every quasi-line through x and y , we mean every quasi-line through x and y parametrised by H , i.e. being a member of the familywe are interested in.b) Let us also point out what the meaning of general (see Definition 1.10)means in this context: a property holds for a general quasi-line through x if there exists an open dense subset H x ⊂ H x such that it is satisfied byevery quasi-line parametrised by H x .We can now give the technical definition of the invariant e ( X, l ) . This number hasbeen introduced by Ionescu and Voica and gives the number of quasi-lines whichare deformations of l and pass through two given general points of X . Let X be a complex projective manifold of dimension n and let l ⊂ X be a quasi-line. Let x ∈ X be a general point and let p : Γ x → X be themorphism from Diagram (1) . Then we define e ( X, l ) := deg( p ) . Let us finally define the kind of foliations we’ll be interested in.
Let X be a projective manifold. A reflexive subsheaf F ⊂ T X iscalled a foliation on X if it satisfies the following two conditions. ) The subsheaf F is saturated, that is the quotient T X / F is torsion-free. Sincea torsion-free sheaf is locally free in codimension one, this implies that theregular locus of the foliation, i.e. the open set F reg = { y ∈ X | F ⊂ T X is a subbundle in an analytic neighbourhood of y } is the complement of a subset of codimension at least two. Equivalently thesingular locus of the foliation F sing := X \ F reg has codimension at least two.b) For every y ∈ F reg , there exists a projective subvariety F y passing through y such that F| F reg ∩ F y = T F y | F reg ∩ F y ⊂ T X | F reg ∩ F y . We call F y the unique F -leaf through y . a) Our definition of foliation is much stronger than the usual one in differentialgeometry since we want the leaves to be proper subvarieties of X . Suchfoliations are also called algebraic foliation .b) We also call leaf of a foliation the closure of the usual leaf in differentialgeometry, hoping there will be no confusion for the reader. The secondcondition then implies that the singular locus of a leaf F y is contained inthe singular locus of the foliation.2. The foliations associated to a family of quasi-lines
Construction of the foliations.
Let X be a complex projective manifoldof dimension n , and let l ⊂ X be a quasi-line. Let x ∈ X be a general point and let Γ xq (cid:15) (cid:15) p / / X H x be the basic diagram (1).Let now ˜ H x ⊂ H x be the maximal smooth open dense subset such that the pointsof ˜ H x parametrise quasi-lines through x . We denote by ˜Γ x the universal familyin ˜ H x × X . Note that ˜Γ x is a quasi-projective, not necessarily connected, smoothscheme that is a P -bundle over ˜ H x .Let ˜Γ ∗ x ⊂ ˜Γ x be the maximal open dense subset such that the restriction of p :Γ x → X to ˜Γ ∗ x is flat and finite . Up to replacing ˜ H x by q (˜Γ ∗ x ) , we can suppose thatthe restriction of q to ˜Γ ∗ x is equidimensional of relative dimension and surjective. enoting by ˜ p and ˜ q the restrictions of the natural maps p and q to ˜Γ ∗ x , we get anew basic diagram:(2) ˜Γ ∗ x ˜ q (cid:15) (cid:15) ˜ p / / X ˜ H x a) Since all the quasi-lines parametrised by H x pass through x , the map p :Γ x → X contracts a divisor onto x . Therefore the point x is not in theimage of ˜ p : ˜Γ ∗ x → X .b) If y is a general point in X , all the quasi-lines through x and y parametrisedby H are parametrised by a point of ˜ H x : if this was not the case, wecould vary y in a n -dimensional quasi-projective family and get an n − -dimensional subscheme of H x that is disjoint from ˜ H x . Since ˜ H x is densein H x and all the irreducible components have dimension n − , we wouldget a contradiction. In particular ˜ p is dominant and we have deg ˜ p = deg p = e ( X, l ) . Set X := P , and consider the family of lines on P . For x ∈ X arbitrary, we have H x ≃ P and the universal family Γ x is the first Hirzebruchsurface F . All the curves parametrised by H x ≃ P are lines through x , so H x = ˜ H x and Γ x = ˜Γ x . The natural map p : F → P is given by the contraction of the unique ( − -curve E , so ˜Γ ∗ x = F \ E . Let V ⊂ X be a projective subvariety such that V intersects theimage of ˜ p : ˜Γ ∗ x → X . We say that V is H x -stable if ˜ p (˜ q − (˜ q (˜ p − ( V ))) = V. Let y ∈ X be a general point. A projective subvariety of X containing y is aminimal H x -stable subvariety through y if it is contained in any H x -stable subvarietycontaining y . It is immediate from the definition that for any general point y ∈ X , a minimal H x -stable subvariety through y is unique. Furthermore a quasi-line passing through x and a general point of a H x -stable subvariety V is containedin V .We are now ready to state and prove our first result. Theorem 1.2.
Let X be a projective manifold of dimension n containing a quasi-line l , and let H x ⊂ C ( X ) be the scheme parametrising deformations and degen-erations of l passing through a general point x ∈ X . Then there exists a uniquesaturated algebraic foliation F x ⊂ T X such that for every general point y ∈ X , theunique F x -leaf through y is the minimal H x -stable projective subvariety through y . The construction of F x is roughly the same as the one given by Ein-Küchle-Lazarsfeld [EKL95] or Hwang-Keum [HK03]. The basic idea is very simple: let y ∈ X be a general point and choose a quasi-line l from x to y . Let Z l be theset of quasi-lines through x meeting a general point of l , i.e. a point in l ∩ Im(˜ p ) . hen there are two possibilities: the curves parametrised by Z l dominate a surface l ⊂ S ⊂ X or map into l . In the second case we have finished, the curve l is theminimal H x -stable subvariety through y . In the first case we choose an irreduciblecomponent of the surface S and restart the construction, i.e. we consider the set Z S of quasi-lines through x meeting a general point of S , etc . Proof.
Let x be a general point of X . Step 1: construction of the minimal H x -stable subvarieties. Fix y ∈ X a generalpoint. We define a sequence of subvarieties as follows: set V := y , and for i ∈{ , . . . , n + 1 } , let V i be an irreducible component of ˜ p (˜ q − (˜ q (˜ p − ( V i − ))) that has maximal dimension. By flatness of ˜ p , every irreducible component of ˜ p − ( V i − ) has dimension dim V i − and dominates V i − . Thus we have V i − ⊂ V i ⊂ X and dim V i ≤ dim V i − + 1 . Furthermore dim V i = dim V i − if and only if all the irreducible components of ˜ p (˜ q − (˜ q (˜ p − ( V i − ))) have dimension dim V i − . Since these components all contain V i − we see that dim V i = dim V i − if and only if ˜ p (˜ q − (˜ q (˜ p − ( V i − ))) = V i − , i.e. if and only if V i − is H x -stable. Since dim V i ≤ dim X for all i ∈ { , . . . , n + 1 } ,we see that necessarily dim V n = dim V n +1 , so V n is H x -stable. We set F x,y := V n . Let us now show that F x,y is minimal: let M be a H x -stable subvariety through y .We will show inductively that V i ⊂ M , the start of the induction being clear since y ∈ M by hypothesis. For the induction step, note that if V i − ⊂ M , then ˜ p (˜ q − (˜ q (˜ p − ( V i − ))) ⊂ ˜ p (˜ q − (˜ q (˜ p − ( M ))) . Since M is H x -stable the right hand side equals M , so V i ⊂ ˜ p (˜ q − (˜ q (˜ p − ( V i − ))) implies the claim. Step 2: construction of the foliation.
The preceding step gives the existence of aminimal H x -stable subvariety F x,y through a general point y ∈ X . Since the Chowscheme of X has only countably many components, there exists a closed subvariety Z x ⊂ C ( X ) such that for y ∈ X general, the variety F x,y corresponds to a point [ F x,y ] ∈ Z x . Up to replacing Z x by a subvariety, we can suppose that the points [ F x,y ] are dense in Z x .Let F x ⊂ Z x × X be the universal family over Z x . Note that since Z x is irreducibleand the general fibre is irreducible, the total space F x is irreducible. We claim thatthe natural morphism p ′ : F x → X is birational. We argue by contradiction and suppose that this is not the case: thenfor y ∈ X general, the fibre p ′− ( y ) is not a singleton, so q ( p ′− ( y )) has at leasttwo distinct points z and z ′ . Thus by construction the subvarieties parametrised y z and z ′ are distinct minimal H x -stable varieties through y , a contradiction tothe uniqueness of these varieties.Let T F x /Z x := H om (Ω F x /Z x , O F x ) be the relative tangent sheaf of the family. Since p ′ is birational, the tangent map gives an integrable subsheaf T p ′ ( T F x /Z x ) ⊂ T X . We define the foliation F x to be the saturation of T p ′ ( T F x /Z x ) in T X , i.e. thekernel of the surjective map T X → ( T X /T p ′ ( T F x /Z x )) / Torsion. By construction,the unique F x -leaf through a general point y ∈ X is the variety F x,y , so it is theminimal H x -stable subvariety. This also guarantees that the foliation is unique. (cid:3) In the preceding proof, we choose an irreducible component of ˜ p (˜ q − (˜ q (˜ p − ( V i − ))) at each step. The resulting foliation does not depend on thesechoices.2.B. Examples and first properties.
It is of course possible that the foliation F x is trivial, that is F x = T X . In this “general type” case our techniques don’t sayanything, but the following examples show that the geometrically most interestingcases are not of this type. a) Let X = P n and consider the family of lines. Then for every x ∈ P n , thefoliation F x is the rank one foliation whose leaves are the lines through x .b) Let X be projective manifold and let l ⊂ X be a quasi-line such that e ( X, l ) = 1 . Then the morphism p from Diagram (1) is birational and ageneral quasi-line meets the exceptional locus of p in x only. Thus thefoliation F x has rank one, its general leaves are the general quasi-linesthrough x . Conversely, if the foliation F x has rank one, the leaf through ageneral point y is a single curve, therefore e ( X, l ) = 1 .The following example was the starting point of our theory. It illustrates the phi-losophy that if we have a quasi-line l ⊂ X such that e ( X, l ) > , the foliation F x should come from a fibration. Let X be a double cover of P × P whose branch locus is a generaldivisor of bidegree (2 , . The threefold X is Fano with Picard number , denoteby ϕ : X → P the projection on the second factor. The map ϕ is a conic bundle,whose discriminant locus is a quartic curve in P . Let d be a general line in P and set S d := ϕ − ( d ) . The surface S d is a del Pezzo surface and the induced map ϕ : S d → d ≃ P has exactly singular fibres. Therefore we have a morphism µ : S d → P representing S d as the blow-up of five points p , . . . , p ∈ P in generalposition. Let C be a general line in P and set l := µ − ( C ) . Clearly l is a quasi-lineof S d and a section of ϕ : S d → d , therefore l is a quasi-line of X [BH07, Lemma4.1]. For every x ∈ X general, the foliation F x is the rank two foliation whosegeneral leaves are the preimages of lines through ϕ ( x ) . Note that by consequencethe singular locus of the foliation contains the conic ϕ − ( ϕ ( x )) , but the general leafis the smooth surface S d = ϕ − ( d ) .As we already mentioned, the foliation F x is singular. However we have the follow-ing properties. .8. Proposition. Let x ∈ X be a general point and let F x be the foliation con-structed in Theorem 1.2. Suppose that rk F x < dim X . Then the following propertieshold.a) A general quasi-line through x meets the singular locus of F x exactly in x .b) If y ∈ X is a general point, all the quasi-lines parametrised by H x passingthrough y are contained in F x,y , the unique F x -leaf through y . Furthermorethey meet the singular locus of F x exactly in x .Proof. By construction the general F x -leaf is of the form F x,y , so it contains x .Since the foliation is not trivial, this implies that x is in the singular locus F sing x ofthe foliation. Since the foliation F x is saturated, its singular locus has codimensionat least two. A nowadays well-known argument in deformation theory of very freerational curves [Kol96, II,Prop.3.7] shows that a general quasi-line l through x doesnot meet F sing x \ { x } .For the second statement, the first part follows from Remark 2.5. Assume thenby contradiction that for every y ∈ X general there exists at least one quasi-linethrough x and y that meets F sing x \ { x } . We then get, with the notations of theDiagram (1), that for y general in X , p − ( F sing x \ { x } ) ∩ q − ( q ( p − ( y ))) = ∅ . This means that a general quasi-line through x intersects F sing x \ { x } , which is notpossible since F sing x \ { x } is of codimension at least two in X (see again [Kol96,II,Prop.3.7]). (cid:3) We now give two technical consequences of our construction which will be veryuseful in Section 4.
With the notations above, the basic Diagram (1) factors for x general in X through the universal family F x → Z x of the leaves of the foliation F x , i.e. up to replacing ˜ H x by a dense open subset we have a commutative diagram ˜Γ xq | ˜Γ x (cid:15) (cid:15) p ′′ / / p | ˜Γ x F xq ′ (cid:15) (cid:15) p ′ / / X ˜ H x ψ / / Z x such that ψ is dominant.Proof. The natural morphism p ′ : F x → X is birational, so we have a rational map p ′′ := ( p ′ ) − ◦ p : Γ x F x . Since ˜Γ x is smooth, we can replace Γ x by its normalisation without changing ˜Γ x .Then [Deb01, Ch. 1.39] implies that the indeterminacy locus of p ′′ has codimensionat least two, in particular it does not surject onto H x . Therefore up to replacing ˜ H x by a dense open subset, we can suppose that p ′′ is defined on ˜Γ x . It is clear that p | ˜Γ x = p ′ ◦ p ′′ , and by construction of the foliation F x , it maps a curve parametrisedby ˜ H x into the general leaf containing it. So the rigidity lemma [Deb01, Lemma .15] applied to ˜Γ x → ˜ H x and ˜Γ x → F x → Z x shows that there exists a morphism ψ : ˜ H x → Z x that makes the diagram commutative. Since p ′′ is dominant, thesame holds for ψ . (cid:3) Let us remark that the construction of the foliations F x can be done “in family”when x moves in X . Indeed, for a general point x ∈ X , we have defined a foliation F x and its universal family q : F x → Z x . Arguing as in the construction of thefoliation F x , we see that there exists a subvariety Z ⊂ X × C ( X ) such that theprojection on the first factor p X : Z → X is a dominant proper morphism thatsatisfies p − X ( x ) = Z x . Thus we get a universal family q : F → Z , where F ⊂ Z × X ⊂ X × C ( X ) × X such that q ′− ( Z x ) = F x . The projection on the first and third factor induces amorphism p ′ : F → X × X such that the restriction p ′ | F x : F x → x × X identifies tothe morphism p ′ : F x → X from Diagram (4) . The morphism p ′ : F → X × X issurjective and birational. Last but not least, let p C : Z → C ( X ) be the morphismgiven by the projection on the second factor. We resume the construction in acommutative diagram(3) F q ′ (cid:15) (cid:15) p ′ / / X × X p (cid:5) (cid:5) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ C ( X ) Z p X (cid:15) (cid:15) p C o o X A fundamental dichotomy
If the foliation F x is not trivial, it is singular at x and depends on the choice ofthe general base point x . We will show now that it is very interesting to comparethe foliations arising from different general choices of base points. The aim of thissection is to prove the following dichotomy. Let X be a projective manifold containing a quasi-line l . Let x and y be two general points in X and denote by F x and F y the correspondingfoliations. Assume F x and F y are not trivial, that is rk F x = rk F y < dim X .Denote by F x,y the unique F x -leaf through y and by F y,x the unique F y -leaf through x .Then the following properties hold.a) The foliation F x is smooth at y .b) If y is very general, there exists a desingularisation µ : F ′ x,y → F x,y suchthat y is in the locus of free rational curves (cf. [Deb01, Prop.4.14] ) of F ′ x,y .c) The foliation F y is smooth at x .Moreover exactly one of the following situations occurs. (I) We have F x,y = F y,x and the variety F x,y is smooth at x . In order to simplify the notation, we denote the two morphisms by the same letter. II)
The intersection F x,y ∩ F y,x is a strict subset of F x,y and F y,x .Proof. It is obvious how to ensure the properties a) and b): if F x,z is a general F x -leaf, take a desingularisation µ : F ′ x,z → F x,z . Then a very general point y ∈ F ′ x,z is in the free locus and does not meet the exceptional locus of µ , so it can beconsidered as a very general point of y ∈ F x,z . Since for such a very general point F x,y = F x,z , we get the conclusion.Let us now show that for y ∈ X general, property c) also holds: let R ⊂ X × X be thesmallest closed subset such that for y ∈ X general, the fibre R y := p − ( y ) ⊂ y × X induced by the projection on the first factor is the singular locus of the foliation F y .Since the foliations F y are saturated, we have dim R ≤ X − . Let p : R → X be the restriction of the projection on the second factor. For x ∈ X general, thefibre p − ( x ) has dimension at most dim X − . Therefore p ( p − ( x )) has dimensionat most dim X − , so it is a strict subset of X . Yet by construction, { y ∈ X general | F y is singular at x } ⊂ p ( p − ( x )) . Let us now prove the basic dichotomy: suppose that the intersection F x,y ∩ F y,x isnot a strict subset of F x,y or F y,x . Since the foliations F x and F y have the samerank, this implies F x,y = F y,x . Furthermore we have just seen that the point x is inthe regular locus of the foliation F y . Thus F y,x is smooth at x , hence F x,y = F y,x is smooth at x . (cid:3) a) We don’t have any examples for the Case (II) of Proposition 3.1. Neverthe-less we see no reason why such examples shouldn’t exist, since the foliationsdepend heavily on the family of curves and thus on the choice of the basepoint.b) Let X = P n and H be the family of lines. Then for every x, y ∈ P n with x = y , the varieties F x,y = F y,x are the unique line through x and y . Moregenerally, case (I) occurs when e ( X, l ) = 1 .c) In Example 2.7, for x ∈ X general, the general F x -leaf is a smooth surface S d = ϕ − ( d ) where d is a line in P through ϕ ( x ) . Thus for x and y in X general, the varieties F x,y and F y,x are the preimage ϕ − ( d ϕ ( x ) ,ϕ ( y ) ) of theunique line d ϕ ( x ) ,ϕ ( y ) through ϕ ( x ) and ϕ ( y ) . Using the Diagram (3), we can give a technically more usefulexpression of the dichotomy in Proposition 3.1: fix [ L ] a general point in p C ( Z ) (which also means that L is a general F x -leaf, x being general). By definition ofthe foliations and the parameter space Z , we have p − C ([ L ]) = { ( z, [ L ]) ∈ Z | L is a F z − leaf } . Being in case (I) of Proposition 3.1 is equivalent to have F x,z = F z,x for x and z general in X . Since for z ∈ L general, we have F x,z = L , we see that being in case(I) of Proposition 3.1 is equivalent to asking that p X ( p − C ([ L ])) = L . Since p X is an isomorphism on the fibres of p C , this is equivalent to p − C ([ L ]) = L × [ L ] . ooking at the universal family, it is clear that q ′− ( p − C ([ L ])) = p − C ([ L ]) × L , so we get a natural identification q ′− ( p − C ([ L ])) = L × [ L ] × L . This means that L is the natural parameter space for the x ∈ X such that thevariety L ⊂ X is a leaf of the foliation F x .4. The main results
The aim of this section is to prove Theorem 1.6 and Proposition 1.8. Until the endof Subsection 4.B, we will always suppose that the Assumption 1.4 holds, i.e. weare in the following situation.
Assumption.
Let X be a projective manifold containing a quasi-line l . Let x and y be two general points in X and denote by F x and F y the correspondingfoliations. Denote by F x,y the unique F x -leaf through y and by F y,x the unique F y -leaf through x . Then we have F x,y = F y,x . We also assume that the foliation F x is not trivial ( rk F x < dim X ), and denote by L a general F x -leaf. Since our assumption means precisely that we are in the firstcase of Proposition 3.1, the leaf L is smooth in x .4.A. Structure of the general leaves.
Recall the commutative diagram(4) ˜Γ xq | ˜Γ x (cid:15) (cid:15) p ′′ / / p | ˜Γ x F xq ′ (cid:15) (cid:15) p ′ / / X ˜ H x ψ / / Z x introduced in Proposition 2.9. Under the Assumption 1.4 the following holds.a) A general quasi-line through x contained in L is contained in the smoothlocus of L and is a quasi-line in that variety.b) In particular if y is a general point in L , every quasi-line through x and y is contained in the smooth locus of L and is a quasi-line in that variety. Remark.
It is probably necessary to explain this statement: the leaf L correspondsto a general point [ L ] ∈ Z x . Let ψ : ˜ H x → ˜ Z x be the morphism in the diagram(4) above, then the fibre ψ − ([ L ]) parametrises quasi-lines through x contained in L . The term general (see Definition 1.10) refers to a quasi-line parametrised by ageneral point of ψ − ([ L ]) . Since L is a general leaf, this is equivalent to saying thatthe quasi-line corresponds to a general point of ˜ H x . his also explains why the second statement is a special case of the first: the choiceof the point x and the F x -leaf L fix ψ − ([ L ]) , but the choice of a general y ∈ L isindependent, so the quasi-lines through x and y are general. Proof.
By Proposition 2.8 (a), a general quasi-line l through x meets F sing x exactlyin x . Since L sing ⊂ F sing x and x
6∈ L sing by Proposition 3.1, the curve l is contained in the smooth locus of L .Since [ L ] ∈ Z x is a general point, all the irreducible components of the fibre ψ − ([ L ]) have the expected dimension rk F x − . Let V be such an irreducible component,then q − ( V ) has dimension rk F x . Since [ L ] is general, the variety q − ( V ) is not con-tained in the exceptional locus of the generically finite morphism p , so p ( q − ( V )) hasdimension rk F x . Since we have a factorisation p = p ′ ◦ p ′′ , we see that p ′′ ( q − ( V )) has dimension rk F x = dim L . So the family of rational curves parametrised by V dominates L and all its members pass through x . Thus by [Deb01, 4.10] a curveparametrised by a general point of V is a very free curve in L . Since p ′ | L is anisomorphism, this implies that T L | l is ample and the injection T L | l ֒ → T X | l implies that it has the splitting type of a quasi-line. (cid:3) Our goal is now to show the following proposition which corresponds to part a) ofTheorem 1.6.
Under the Assumption 1.4, suppose moreover that dim x F sing x < rk F x − n − rk F x ) . Then there exists a rational fibration ϕ L : ( L , l ) P such that l is a section. The statement will be a consequence of Theorem 1.1. In view of the hypothesis inTheorem 1.1 we have to address the following tasks:1.) construct an effective divisor E L ,x ⊂ L such that E L ,x · l = 1 ,2.) show that E L ,x is in a linear system of dimension one.We will see that E L ,x will be the restriction of an exceptional divisor E x of thenatural map F x → X . At first glance it seems impossible that E L ,x moves in alinear system, but this intuition is wrong: the restriction of F x → X to L is anembedding, so E L ,x is not exceptional.If the foliation F x has rank one, the construction of E x is trivial: the universalfamilies F x and Γ x are the same, the set p − ( x ) ⊂ Γ x gives a q -section, so itsrestriction to a general quasi-line is a reduced point. If F x has higher rank, thisobvious construction no longer works: the generically finite morphism p ′′ maps p − ( x ) onto p ′− ( x ) which is not a divisor in F x . Step 1. Construction of the divisor E L ,x . We use the notation of the Diagram(4). Any F x -leaf passes through x , so p ′− ( x ) is a section of q ′ . Furthermore L issmooth at x , so the general q ′ -fibre is smooth at the intersection with p ′− ( x ) . Thisshows that F x is smooth in the general point of p ′− ( x ) . Since by Lemma 4.1 abovea general curve parametrised by ˜ H x is contained in the smooth locus of a general ′ -fibre, it follows that it is contained in the smooth locus of F x . This shows thatthere exists a desingularisation F ′ x → F x thata) is an isomorphism in the general point of p ′− ( x ) ,b) a general curve parametrised by ˜ H x does not meet the exceptional locus.These properties assure that all the following computations take place in the com-plement of the exceptional locus of the desingularisation, moreover we don’t haveto distinguish Weil and Cartier divisors. In order to simplify the notation, we thussuppose without loss of generality that F x is smooth.Since the base of the birational morphism p ′ : F x → X is smooth, the p ′ -exceptionallocus has pure codimension one in F x and the determinant of the Jacobian dp ′ defines an equality of cycles K F x /X = X i ≥ a i E i , where the E i are the p ′ -exceptional divisors and the a i are positive integers. Let l ⊂ X be a quasi-line parametrised by a general point of ˜ H x such that l ⊂ L .Identifying the leaf L ⊂ X and the corresponding subvariety of F x , we authoriseourselves to see l also as a rational curve in F x . Then l is a free curve in F x (although it is not a quasi-line in F x ), so E i · l ≥ ∀ i ≥ . Note furthermore that p ′ ( E i ) ⊂ F sing x , since through any point of p ′ ( E i ) pass an infinity of leaves. By Proposition 2.8 ageneral quasi-line through x meets F sing x exactly in x , so we see that E i · l > ⇔ p ′− ( x ) ⊂ E i . Thus we get K F x /X · l = X p ′− ( x ) ⊂ E i a i E i · l and we denote by E x the support of P p ′− ( x ) ⊂ E i a i E i . Note that E x is not empty,since x is contained in every F x -leaf. Furthermore every irreducible component E i ⊂ E x maps surjectively onto Z x since p ′− ( x ) is a q ′ -section. In particular thegeneral fibre of E i → Z x has dimension rk F x − , so E L ,x := E x ∩ L is a non-empty effective divisor in L passing through the point x ∈ L . Since therestriction of p ′ to L is an isomorphism, we see that dim p ′ ( E i ) ≥ rk F x − , so in particular dim x F sing x ≥ rk F x − . The following lemma shows that the divisor E L ,x has some interesting intersectionproperties provided the dimension of the singular locus of the foliation is not toolarge. .3. Lemma. Under the Assumption 1.4, let l be a quasi-line parametrised by ageneral point of ˜ H x such that l ⊂ L . If dim x F sing x < rk F x − d − d ( n − rk F x ) for some d ∈ N , then E L ,x · l < d. In particular if dim x F sing x < rk F x − n − rk F x ) , then E L ,x · l = 1 . Hence the effective divisor E L ,x is smooth in x ∈ L and E x is irreducible.Proof. By Lemma 4.1 the rational curve l is a quasi-line in the leaf L , so by ad-junction K F x · l = K L · l = − (rk F x + 1) . Since l is a quasi-line in X this implies K F x /X · l = n − rk F x . Using the notation introduced before the lemma, a local computation shows thatif E i ⊂ E x is an irreducible component then a i ≥ n − − dim p ′ ( E i ) . Since x ∈ p ′ ( E i ) ⊂ F sing x we get a i ≥ n − − dim x F sing x . Thus one has n − rk F x = K F x /X · l = X p ′− ( x ) ⊂ E i a i E i · l ≥ ( n − − dim x F sing x ) E L ,x · l. So if E L ,x · l ≥ d , we have n − rk F x ≥ d ( n − − dim x F sing x ) . This is equivalent to the first statement.The second statement corresponds to the case d = 2 , the smoothness of E L ,x in x is immediate from E L ,x · l = 1 . Since any irreducible component of E x contains p ′− ( x ) , this also shows that E x is irreducible. (cid:3) Step 2. Dimension of the linear system.
Recall the commutative Diagram (3) F q ′ (cid:15) (cid:15) p ′ / / X × X p (cid:5) (cid:5) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ C ( X ) Z p X (cid:15) (cid:15) p C o o X he map p ′ is birational and since X × X is smooth, the p ′ -exceptional locus hascodimension 1 in F . Denote by E ⊂ F the union of p ′ -exceptional divisors thatcontain p ′− (∆) . By definition we have for x ∈ X general E x = ( p X ◦ q ′ ) − ( x ) ∩ E, where E x ⊂ F x is the divisor introduced in the first step. By Remark 3.3 one has q ′− ( p − C ([ L ])) = L × [ L ] × L , so L is the natural parameter space for the x ∈ X such that the subvariety L ⊂ X is a leaf of the foliation F x . Thus for x general in L we have E L ,x = { y ∈ L | ( x, [ L ] , y ) ∈ E } where E L ,x ⊂ L is the divisor introduced in the first step. Varying x in theparameter space L , this defines a family of algebraically equivalent divisors onthe leaf L , and since x ∈ E L ,x the divisor E L ,x moves in L . Since L is rationallyconnected, we have h ( L , O L ) = 0 , the algebraically equivalent divisors E L ,x aretherefore linearly equivalent. Simply denoting by E L any linear representative ofthe E L ,x ’s, we finally deduce that h ( L , E L ) ≥ . End of the proof of Proposition 4.2.
According to Theorem 1.1, the only thing toprove is the equality h ( L , E L ) = 2 . Set h ( L , E L ) =: s + 1 , and let ϕ L : ˜ L → P s be the map given by Theorem 1.1. Recall now the construction of L by a sequenceof subvarieties V = y ⊂ V ⊂ . . . ⊂ V n = L ⊂ X where V i is obtained form V i − by adding quasi-lines through x and a generalpoint of V i − . We will prove inductively that ϕ L ( V i ) is a fixed line in P s for every i ∈ { , . . . , n } . Since V is a quasi-line l ′ through x and a given point of L , thestart of the induction is trivial. Suppose now that ϕ L ( V i − ) = ϕ L ( l ′ ) for some i ≥ . The quasi-lines l z through x and a general point z of V i − dominate V i , soit is sufficient to show that ϕ L ( l z ) = ϕ L ( l ′ ) . Yet ϕ L ( l z ) and ϕ L ( l ′ ) are lines in P s meeting in the points ϕ L ( x ) and ϕ L ( z ) , thus they are identical. (cid:3) In the preceding proof the hypothesis on the singular locus of the foliation was onlyused to apply Lemma 4.3. Thus we have shown:
Under the Assumption 1.4, suppose moreover that E L · l = 1 .Then there exists a rational fibration ϕ L : ( L , l ) P such that l is a section. Structure of X . The first part of Proposition 1.8 is a corollary of Theorem1.6: the only thing we have to show is that the surface Y admits a birational map Y P such that l ′ maps onto a line. This can be deduced from the list in [IV03,Prop. 1.21]. Since the proof of Theorem 1.6 is rather involved, we give a shortindependent argument based on Theorem 1.1. Proof of Proposition 1.8.
Denote by L a general leaf. By Lemma 4.1 a generalquasi-line l contained in L is a quasi-line in L . Therefore the exact sequence → T L | l → T X | l → N L /X | l ≃ O X ( L ) | l → hows that L · l = 1 . Since the divisor L moves in the rationally connected manifold X , we can apply Theorem 1.1 to get a birational map σ : ˜ X → X and a fibration ϕ : ˜ X → P k with the stated properties, so the only thing to show is k = 2 . Since ϕ is induced by the moving part of the linear system | σ ∗ L| the general leaf mapsonto a hyperplane. Furthermore it is not hard to see that the restriction of |L| toa leaf is the linear system | E L | from Proposition 4.2, so the restriction of ϕ to L isthe fibration ϕ L . Thus the hyperplane ϕ ( L ) is a curve.If the Picard number of X is one, the effective divisor L is ample. By [BBI00,Thm.4.4, Cor.4.6] this implies that X is a projective space and the quasi-linesare lines. The foliation defined by lines in the projective space has rank one, so X ≃ P . (cid:3) Proof of Theorem 1.6.
Statement a) of the theorem is settled by Proposition 4.2.Suppose now that we are in the situation of the Statement b). Fix a general point x ∈ X and consider the basic diagram F xq ′ (cid:15) (cid:15) p ′ / / XZ x where F x is the universal family of the foliation F x . We have shown in Proposition4.2 that for [ L ] in Z x general, there exists a complete linear system | E L | of dimensionone. Since we have fixed x ∈ X , we also have a distinguished element [ E L ,x ] ∈ | E L | ,where E L ,x := E x ∩ L is the family of cyles introduced in Lemma 4.3.Since h ( L , O L ) = 0 the line bundle E L does not deform, so | E L | is an irreduciblecomponent of the Chow scheme C ( L ) . By countability of the number of irreduciblecomponents of the relative Chow scheme C ( F x /Z x ) there exists an integral scheme U x → Z x such that the general fibre identifies to | E L | ≃ P . The fibration U x → Z x has a distinguished (rational) section S x given by x [ E L ,x ] .Consider now U x as a subset of C ( F x ) . Then by [Kol96, I. Thm.6.8] the holomorphicmap p ′ induces a map p ′∗ : C ( F x ) → C ( X ) and we denote by Y x the image of U x .It is immediate that U x → Y x is birational. Let us now look at the family ofcycles E L ,x : clearly their image by p ′ is contained in the variety p ′ ( E x ) which hasdimension at least rk F x − , since the restriction of p ′ to a general leaf E L ,x is anisomorphism. Since p ′ ( E x ) ⊂ F sing x the hypothesis on the dimension of the singularlocus implies that p ′ ( E x ) has dimension exactly rk F x − . Thus the image of E L ,x does not depend on L . This shows that the distinguished section S x is contractedby p ′∗ .Let Y be the normalisation of Y x and denote by ˜ X the normalisation of the universalfamily over Y . Denote by σ : ˜ X → X and ϕ : ˜ X → Y the natural maps so that we et a commutative diagram F xq ′ (cid:15) (cid:15) p ′ / / X ˜ X σ o o ϕ (cid:15) (cid:15) Z x p ′∗ / / Y x Y o o Since q ′ ( p ′− ( x )) ⊂ S x and the normalisation Y → Y x is finite, a diagram chaseshows that σ − ( x ) is finite. Yet σ is birational and X is normal, so it follows byZariski’s main theorem that σ − ( x ) is a singleton and σ is an isomorphism in aneighbourhood of x . In order to simplify the notation, we identify σ − ( x ) and x .By construction the fibre ϕ − ( ϕ ( x )) is equal to E L ,x , where L is a general leaf.Since ˜ X is smooth in x and E L ,x is smooth in x by Lemma 4.3, the normal variety Y is smooth in ϕ ( x ) .Let now l ⊂ ˜ X be a general quasi-line through x contained in a general leaf L .Since the birational map σ is an isomorphism around x , the quasi-line l does notmeet the locus where σ − is not a morphism. Thus we can see l as quasi-line in ˜ X passing through x and we will now show that l ′ := ϕ ( l ) is a quasi-line in Y .Note first that since Y is smooth in ϕ ( x ) , the rational curve l ′ is contained in thesmooth locus of Y . The restriction of ϕ to ϕ − ( ϕ ( l )) identifies to the graph of themeromorphic fibration ϕ L : L P constructed in Proposition 4.2. In particular l is a quasi-line in ϕ − ( ϕ ( l )) , so l ′ ⊂ Y is a quasi-line by [BH07, Lemma 4.1].Let H ′ ϕ ( x ) be the normalisation of the subset of the Chow scheme C ( Y ) parametris-ing deformations and degenerations of l ′ passing through y . The push-forwardinduces a rational map ϕ ∗ : H x H ′ ϕ ( x ) and a dimension count (see the proofof [BH07, Lemma 4.1]) shows that this map is dominant. Apply now Theorem 1.2to the family of quasi-lines l ′ ⊂ Y through ϕ ( x ) and denote by G ϕ ( x ) the result-ing foliation on Y . Since the general leaves of the foliations F x ( resp. G ϕ ( x ) ) arebuilt from quasi-lines parametrised by H x ( resp. H ′ ϕ ( x ) ), we can use the dominantmap ϕ ∗ : H x H ′ ϕ ( x ) to show that a general F x -leaf is the ϕ -preimage of a gen-eral G ϕ ( x ) -leaf (the tedious details are left to the reader). Since Y has dimension n − rk F x + 1 , this implies that the general G ϕ ( x ) -leaf has rank one. Thus e ( Y, l ′ ) = 1 by Example 2.6. (cid:3) Examples.
The following example generalises Example 2.7 to a situationwhere the quasi-line l ′ ⊂ Y is not necessarily a line in a projective space. Let Y be a projective manifold of dimension n − and let l ′ ⊂ Y be a quasi-line such that e ( Y, l ′ ) = 1 . Let now ϕ : X → Y be a conic bundle over Y such that the discriminant locus ∆ ⊂ Y satisfies ∆ · l ′ ≥ . For a general quasi-line l ′ , the preimage X l ′ := ϕ − ( l ′ ) is a smooth surface. Thus by [BH07, Lemma 4.1],there exists a quasi-line l ⊂ X that is a section of X l ′ → l ′ . Let x be a general pointof X . Since e ( Y, l ′ ) = 1 , the F x -leaves are contained in the surfaces ϕ − ( l ′ ) , hence rk F x ≤ . Moreover if we had e ( X, l ) = 1 an argument analogous to the proof of[BH07, Thm.4.14] would show that ϕ : X → Y is smooth. Since this is not thecase, we have e ( X, l ) > , so the foliation F x has rank two. Note also that if G ϕ ( x ) enotes the rank one foliation associated to l ′ ⊂ Y , then dim x F sing x = dim ϕ ( x ) G sing ϕ ( x ) + 1 . We will now construct an example of a fourfold X containing a quasi-line l suchthat for x ∈ X general one has rk F x = 1 and dim x F sing x = 1 , hence dim x F sing x < rk F x − n − rk F x ) , but dim x F sing x = rk F x − . Using the construction in Example 4.5 one can transformthis into an example where the foliation has rank two. Let d be a line in P and let E be a rank two vector bundle givenby the Serre extension → O P → E → I d ( − → . Set X := P ( E ) , denote by ϕ : X → P the natural projection, and by ξ the uniqueeffective divisor in the linear system |O P ( E ) (1) | . By the canonical bundle formula − K X = ϕ ∗ H + 2 ξ, and one checks easily that − K X is nef.If d ⊂ P is a general line, then E | d ≃ O P ⊕ O P ( − , so by [BH07,Prop.5.1.,Rem.5.6.] the variety X contains a quasi-line l such that ϕ ( l ) is a lineand e ( X, l ) = 1 . Thus if we fix a general point x ∈ X (in particular x / ∈ ξ ), thefoliation F x has rank one and we claim that F sing x = ϕ − ( ϕ ( x )) ∪ ( ϕ − ( H x ) ∩ ξ ) , where H x ⊂ P is the unique hyperplane containing ϕ ( x ) and d . In particular F sing x has dimension one in x . Proof of the claim.
We start by analysing the degenerations of l passing through x : we have − K X · l = 5 , ϕ ∗ H · l = 1 , so any degeneration is of the form l + X i ≥ l i such that ϕ ∗ H · l = 1 and ϕ ∗ H · l i = 0 for all i > . Thus for i > the curve l i isa ϕ -fibre, so − K X · l i = 2 . Hence there are two types of degenerations, eithera) − K X · l = 1 and the degeneration is of the form l + l + l with possibly l = l ,orb) − K X · l = 3 and the degeneration is of the form l + l . Case a)
Then − K X · l = 1 implies ξ · l = − , so l is contained in ξ and does notpass through x . Thus up to renumbering x ∈ l , i.e. l = ϕ − ( ϕ ( x )) . If we fix l ,then l varies in a 1-dimensional family parametrised by the line ϕ ( l ) . The surfacecovered by these degenerations is of course nothing else than X ϕ ( l ) := ϕ − ( ϕ ( l )) .One easily computes that l ⊂ X ϕ ( l ) is a section with self-intersection − , hencethe surface X ϕ ( l ) ≃ P ( E | ϕ ( l ) ) is a Hirzebruch surface F . A look at the extensiondefining E shows that we are in this case exactly when ϕ ( l ) meets the line d . Therestriction of the foliation F x to X ϕ ( l ) is the foliation defined by the pencil |O X ϕ ( l ( l + l + l ) ⊗ I x | . he singular locus of this foliation contains obviously the exceptional section l butalso the curve ϕ − ( ϕ ( x )) , since the cyle l +2 l is not reduced along l = ϕ − ( ϕ ( x )) . Case b)
As in Case a) we show that l ⊂ ξ and l = ϕ − ( ϕ ( x )) . If we fix l this shows that the surface X ϕ ( l ) := ϕ − ( ϕ ( l )) contains a unique degeneration.Moreover l ⊂ X ϕ ( l ) is the exceptional section of the Hirzebruch surface X ϕ ( l ) ≃ P ( E | ϕ ( l ) ) ≃ F and all this happens exactly when ϕ ( l ) does not meet the line d .The restriction of the foliation F x to X ϕ ( l ) is the foliation defined by the pencil |O X ϕ ( l ( l + l ) ⊗ I x | . The general member of this pencil is a section with self-intersection one, the uniquenon-smooth member is the degeneration l + l . Thus the singular locus of thefoliation consists of the point x and l ∩ l = ϕ − ( ϕ ( x )) ∩ ξ . In this case thesingular locus of the foliation does not depend on the choice of l .The singular locus of F x is obtained as the union of the singular loci of the restrictedfoliations, this shows the claim.5. Enumeration of quasi-lines
The main object of this section is to prove our only result when the Assumption 1.4fails (which corresponds to the case (II) described in Proposition 3.1). As we alreadysaid, we have no concrete example but we realised however that we can providesome information concerning another important problem, namely bounding e ( X, l ) .Finally in Proposition 5.2 we try to address the same enumerative problem underthe hypothesis that Assumption 1.4 holds. We will see that although Proposition1.8 provides additional structure information, the bound obtained is far from beingoptimal.Before we come to the proof, recall the following consequence of the Hodge indextheorem: let S be a projective surface and H an ample divisor on X . If D is anydivisor, then ( D )( H ) ≤ ( D · H ) (see [Har77, V., Ex.1.7]). This obviously implies: let X be a projective threefoldand H be an ample divisor on X . If D is any divisor, then(5) ( D · H )( H ) ≤ ( D · H ) . Proof of Theorem 1.9.
Fix two general points x and y in X , and denote as usualby F x,y the unique F x -leaf through y and by F y,x the unique F y -leaf through x .By Proposition 2.8,b) applied to the foliations F x and F y , the quasi-lines through x and y are contained in F x,y and F y,x . By hypothesis, the intersection F x,y ∩ F y,x is a strict subset of F x,y and F y,x . Since the rank of the foliations is two, theintersection is a union of curves that contains the quasi-lines passing through x and y . It follows that e ( X, l ) deg l ≤ ( F x,y ∩ F y,x ) · H ≤ F x,y · F y,x · H. The varieties F x,y and F y,x are fibres of the equidimensional fibration q ′ (see Dia-gram (3)), so they have the same cohomology class. By Formula (5) above F x,y H ≤ ( F x,y H ) H , o we obtain e ( X, l ) ≤ (deg F x,y ) deg X × deg l . Thus we are left to bound the degree of F x,y : let µ : F ′ x,y → F x,y be the desingu-larisation of F x,y from Proposition 3.1,b). Then we have deg F x,y = H | F x,y = ( µ ∗ H ) . Let l be a quasi-line through x and y . Since y is a smooth point of F x,y , itsstrict transform l ′ is a rational curve that passes through y . By Proposition 3.1,b)we know that y is in the free locus of F ′ x,y , so l ′ is even a free rational curve of µ ∗ H -degree exactly deg l .By construction of the leaf F x,y there exists a family of quasi-lines l through x thatmeets l . The strict transforms l ′ of such general quasi-line form a dominant familyof rational curves that meet l ′ and also have µ ∗ H -degree exactly deg l . Thus bycomb-smoothing we can construct a dominant family of rational curves of degree l passing through y . Therefore by [Kol96, V., Prop.2.9] ( µ ∗ H ) ≤ l ) . This implies the statement. (cid:3)
In the preceding proof, we used the hypothesis on the dimensionto compute the degree of the intersection F x,y ∩ F y,x via an intersection product F x,y · F y,x . If X has higher dimension this is no longer possible, since two surfacesmeeting in a bunch of curves are a case of excess intersection. Let X be a projective manifold of dimension three containinga quasi-line l . Let x be a general point in X , and denote by F x the correspondingfoliation. Assume that rk F x = 2 and the Assumption 1.4 holds . Denote by ϕ :˜ X → P a holomorphic model of the fibration constructed in Proposition 1.8. Then e ( X, l ) ≤ (cid:18) ( d +1)( d +2)2 max( d, H ) d ( d +3)2 (cid:19) d d +1)( d +2)2 where d = H · l and H := H · ϕ ∗ O P (1) and H is a very ample line bundle on ˜ X .Proof. Let L be a general F x -leaf. If y ∈ L is a general point, then by Lemma4.1,b) any quasi-line through x and y is contained in the smooth locus of L andis a quasi-line in L . Since L is a surface, we have e ( L , l ) = 1 for any quasi-line l ⊂ L . Thus bounding e ( X, l ) is equivalent to bounding the number of families ofquasi-lines on L . By [IV03, Prop. 2.1] we have h ( L , O L ( H )) ≤ ( d + 1)( d + 2)2 . Thus the very ample line bundle H | L defines an embedding of L ֒ → P N with N ≤ d ( d +3)2 . Moreover the embedded surface L has degree deg L = ( H | L ) = H · L = H · ϕ ∗ O P (1) , Actually it is sufficient that the restriction of H | L to a general F x -leaf L is very ample. ince by Proposition 1.8 the surface L is the preimage of a line in P . The statementnow follows from a general estimate on the number of irreducible components ofthe Chow scheme [Hei05, Prop.3.6] (cid:3) We will now bound e ( X, l ) for the family of quasi-lines constructedin Example 2.7. The proof of Proposition 5.2 shows that this is equivalent tobounding the number of families of quasi-lines in the surface S d which is the blow-up of P in five general points. The anticanonical divisor − K S d is very ample, soProposition 5.2 yields e ( X, l ) ≤ (cid:18) (cid:19) . Since we know that h ( S d , − K S d ) = 5 , the proof of the proposition shows that wecan lower the bound to e ( X, l ) ≤ (cid:18) (cid:19) which is still huge. We will now compute explicitly the number of families on S d :if l ⊂ S d is a quasi-line, the linear system |O S d ( l ) | is base-point free and defines abirational map ϕ l : S d → P such that l maps onto a line. Since − K S d is ample,we see that ϕ l is the blow-up of five points in general position in P , in particular ϕ l contracts five disjoint ( − -curves on S d . Vice versa five disjoint ( − -curves on S d determine a representation of S d as a blow-up of five general points in P . Thusthe problem reduces to counting the number of choices of five disjoint ( − -curveson S d .Consider now the representation µ : S d → P fixed in Example 2.7, and denote by p , . . . , p ∈ P the points we blow up. With respect to this representation, the ( − -curves on S d are • the five exceptional divisor E , . . . , E , • the (strict transforms of the) ten lines passing through exactly two of thepoints p , . . . , p which we denote by d , . . . , d ⊂ S d , and • the (strict transform of the) unique conic passing through p , . . . , p whichwe denote by C ⊂ S d .Out of these sixteen curves we have to choose a configuration of five that are disjoint. Case 1) The configuration contains C . The conic meets all the E i and is disjointfrom all the d j . Thus we are left to choose a configuration of four disjoint d j . Thedual graph of the d j is the famous Petersen graph and it is easy to see that thereare exactly five such possibilities. ase 2) The configuration does not contain C . The Petersen graph shows that it isnot possible to choose five disjoint d j . If the configuration contains an exceptionaldivisor E i , this excludes the four d j coming from lines through p i . A look at thePetersen graph shows that it is not possible to choose four disjoint d j among theremaining six, so the configuration contains at least two exceptional divisors E i , E i ′ which excludes seven d j coming from lines through p i or p i ′ . The remaining three d j are disjoint, so we get ten configurations with exactly two exceptional divisors.In the same way we see that if there are at least three exceptional divisors in theconfiguration, the configuration is given by E , . . . , E .In total we get configurations, so e ( X, l ) ≤ . References [ABW92] M. Andreatta, E. Ballico, and J. Wiśniewski. Projective manifolds containing largelinear subspaces. In
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Laurent Bonavero, Institut Fourier, UMR 5582 du CNRS, Université de GrenobleI, BP 74, 38402 Saint-Martin d’Hères, France
E-mail address : [email protected] Andreas Höring, Université Paris 6, Institut de Mathématiques de Jussieu, Equipe deTopologie et Géométrie Algébrique, 175, rue du Chevaleret, 75013 Paris, France
E-mail address : [email protected]@math.jussieu.fr