aa r X i v : . [ m a t h . AG ] S e p ON FANO FOLIATIONS 2
CAROLINA
ARAUJO
AND ST´EPHANE
DRUELAbstract.
In this paper we pursue the study of mildly singular del Pezzo foliations on complex projectivemanifold started in [AD13].
Contents
1. Introduction 12. Preliminaries 32.1. Fano manifolds and rational curves 32.2. Singularities of pairs 42.3. Polarized varieties and Fujita’s ∆-genus 42.4. Classification of log del Pezzo pairs 53. Foliations 63.1. Foliations and Pfaff fields 63.2. Algebraically integrable foliations 73.3. Q -Fano foliations 83.4. Foliations on P n Q n Introduction
In recent years, techniques from higher dimensional algebraic geometry, specially from the minimal modelprogram, have been successfully applied to the study of global properties of holomorphic foliations. Thisled, for instance, to the birational classification of foliations by curves on surfaces in [Bru04]. Motivated bythese developments, we initiated in [AD13] a systematic study of
Fano foliations . These are holomorphicfoliations F on complex projective manifolds with ample anti-canonical class − K F . One special propertyof Fano foliations is that their leaves are always covered by rational curves, even when these leaves are notalgebraic (see for instance [AD13, Proposition 7.5]).The index ι F of a Fano foliation F on a complex projective manifold X is the largest integer dividing − K F in Pic( X ). In analogy with Kobayachi-Ochiai’s theorem on the index of Fano manifolds (Theorem 2.2),we proved in [ADK08, Theorem 1.1] that the index of a Fano foliation F on a complex projective manifoldis bounded above by its rank, ι F r F . Equality holds if and only if X ∼ = P n and F is induced by a linearprojection P n P n − r F . Our expectation is that Fano foliations with large index are the simplest ones. Sowe proceeded to investigate the next case, namely Fano foliation F of rank r and index ι F = r −
1. Wecall such foliations del Pezzo foliations , in analogy with the case of Fano manifolds. In contrast to the casewhen ι F = r F , there are examples of del Pezzo foliations with non-algebraic leaves. For instance, let C bea foliation by curves on P k induced by a global vector field. If we take this vector field to be general, thenthe leaves of C are not algebraic. Now consider a linear projection ψ : P n P k , with n > k , and let F bethe foliation on P n obtained as pullback of C via ψ . It is a del Pezzo foliation on P n , and its leaves are not Mathematics Subject Classification.
ARAUJO
AND ST´EPHANE
DRUEL algebraic (see Theorem 3.16(2) for the complete classification of del Pezzo foliations on P n ). The first mainresult of [AD13] says that these are the only examples. Theorem 1.1 ([AD13, Theorem 1.1]) . Let F be a del Pezzo foliation on a complex projective manifold X = P n . Then F is algebraically integrable, and its general leaves are rationally connected. One of the main ingredients in our study of Fano foliations is the notion of log leaf for an algebraicallyintegrable foliation. Given an algebraically integrable foliation F on a complex projective manifold X , denoteby ˜ e : ˜ F → X the normalization of the closure of a general leaf of F . There is a naturally defined effectiveWeil divisor ˜∆ on ˜ F such that K ˜ F + ˜∆ = ˜ e ∗ K F (see Definition 3.6 for details). We call the pair ( ˜ F , ˜∆) ageneral log leaf of F . In [AD13], we used the log leaf to define new notions of singularities for algebraicallyintegrable foliations, following the theory of singularities of pairs from the minimal model program. Namely,we say that F has log canonical singularities along a general leaf if ( ˜ F , ˜∆) is log canonical. By Theorem 1.1,these notions apply to del Pezzo foliations on projective manifolds X = P n . In [AD13], we established thefollowing classification of del Pezzo foliations with mild singularities. Theorem 1.2 ([AD13, 9.1 and Theorems 1.3, 9.2, 9.6]) . Let F be a del Pezzo foliation of rank r on acomplex projective manifold X = P n , and suppose that F has log canonical singularities and is locally freealong a general leaf. Then • either ρ ( X ) = 1 ; • or r and X is a P m -bundle over P k .In the latter case, one of the following holds. (1) X ∼ = P × P k , and F is the pullback via the second projection of a foliation O P k (1) ⊕ i ⊂ T P k for some i ∈ { , } ( r ∈ { , } ) . (2) There exist • an exact sequence of vector bundles → K → E → Q → on P k ; and • a foliation by curves C ∼ = q ∗ det( Q ) ⊂ T P P k ( K ) , where q : P P k ( K ) → P k denotes the naturalprojection;such that X ∼ = P P k ( E ) , and F is the pullback of C via the relative linear projection P P k ( E ) P P k ( K ) . Moreover, one of the following holds. (a) k = 1 , Q ∼ = O P (1) , K is an ample vector bundle such that K = O P ( a ) ⊕ m for any integer a ,and E ∼ = Q ⊕ K ( r = 2) . (b) k = 1 , Q ∼ = O P (2) , K ∼ = O P ( a ) ⊕ m for some integer a > , and E ∼ = Q ⊕ K ( r = 2) . (c) k = 1 , Q ∼ = O P (1) ⊕ O P (1) , K ∼ = O P ( a ) ⊕ ( m − for some integer a > , and E ∼ = Q ⊕ K ( r = 3) . (d) k > , Q ∼ = O P k (1) , and K is V -equivariant for some V ∈ H (cid:0) P k , T P k ⊗ O P k ( − (cid:1) \{ } ( r = 2) .Conversely, given K , E and Q satisfying any of the conditions above, there exists a del Pezzofoliation of that type. The goal of the present paper is to continue the classification of del Pezzo foliations on Fano manifolds X = P n having log canonical singularities and being locally free along a general leaf. In view of Theorem 1.2,we need to understand del Pezzo foliations on Fano manifolds with Picard number 1. Our main result is thefollowing. Theorem 1.3.
Let F be a del Pezzo foliation of rank r > on an n -dimensional Fano manifold X = P n with ρ ( X ) = 1 , and suppose that F has log canonical singularities and is locally free along a general leaf.Then X ∼ = Q n and F is induced by the restriction to Q n of a linear projection P n +1 P n − r . Remark 1.4.
Codimension 1 del Pezzo foliations on Fano manifolds with Picard number 1 were classifiedin [LPT13, Proposition 3.7]. We extended this classification to mildly singular varieties, without restrictionon the Picard number in [AD12, Theorem 1.3].We also obtain a partial classification when r = 2 (Proposition 4.1).In order to prove Theorem 1.2, we consider a general log leaf ( ˜ F , ˜∆) of F . Under the assumptions ofTheorem 1.2, ( ˜ F , ˜∆) is a log del Pezzo pair : it is a log canonical pair of dimension r satisfying K ˜ F + ˜∆ =( r − L , where L is an ample divisor on ˜ F . The first step in the proof of Theorem 1.2 consists in classifying alllog del Pezzo pairs. This is done in Section 2.4, using Fujita’s theory of ∆-genus. Once we know the general N FANO FOLIATIONS 2 3 log leaf ( ˜
F , ˜∆) of F , we consider families of rational curves on X that restrict to special families of rationalcurves on ˜ F . The necessary results from the theory of rational curves are briefly reviewed in Section 2.1.The idea is to use these families of rational curves to bound the index of X from below. In order to obtaina good bound, we need to show that the dimension of these families of rational curves is big enough. Hereenters a very special property of algebraically integrable Fano foliations having log canonical singularitiesalong a general leaf: there is a common point contained in the closure of a general leaf ([AD13, Proposition5.3]). For our current purpose, we need the following strengthening of this result (see Definition 2.9 for thenotion of log canonical center). Proposition 1.5.
Let F be an algebraically integrable Fano foliation on a complex projective manifold X having log canonical singularities along a general leaf. Then there is a closed irreducible subset T ⊂ X satisfying the following property. For a general log leaf ( ˜ F , ˜∆) , there exists a log canonical center S of ( ˜ F , ˜∆) whose image in X is T . When r >
3, this allows us to show that ι X > n , and then use Kobayashi-Ochiai’s Theorem (Theorem 2.2)to conclude that X ∼ = Q n . The classification of del Pezzo foliations on Q n is established in Proposition 3.18.Proposition 1.5 still holds in the more general setting of Q - Fano foliations on possibly singular projectivevarieties. Since this may be useful in other situations, we present the theory of foliations on normal projectivevarieties in Section 3, and prove a more general version of Proposition 1.5 (Proposition 3.14).
Notation and conventions.
We always work over the field C of complex numbers. Varieties are alwaysassumed to be irreducible. We denote by Sing( X ) the singular locus of a variety X .Given a sheaf F of O X -modules on a variety X , we denote by F ∗ the sheaf H om O X ( F , O X ). If r is thegeneric rank of F , then we denote by det( F ) the sheaf ( ∧ r F ) ∗∗ . If G is another sheaf of O X -modules on X , then we denote by F [ ⊗ ] G the sheaf ( F ⊗ G ) ∗∗ .If E is a locally free sheaf of O X -modules on a variety X , we denote by P X ( E ) the Grothendieck projec-tivization Proj X (Sym( E )), and by O P (1) its tautological line bundle.If X is a normal variety, we denote by T X the sheaf (Ω X ) ∗ .We denote by Q n a (possibly singular) quadric hypersurface in P n +1 . Given an integer d >
0, we denoteby F d the surface P P ( O P ⊕ O P ( − d )). If moreover d >
1, we denote by P (1 , , d ) the cone in P d +1 over therational normal curve of degree d . Acknowledgements.
Much of this work was developed during the authors’ visits to IMPA and InstitutFourier. We would like to thank both institutions for their support and hospitality.2.
Preliminaries
Fano manifolds and rational curves.Definition 2.1. A Fano manifold X is a complex projective manifold whose anti-canonical class − K X isample. The index ι X of X is the largest integer dividing − K X in Pic( X ). Theorem 2.2 ([KO73]) . Let X be a Fano manifold of dimension n > and index ι X . Then ι X n + 1 ,and equality holds if and only if X ∼ = P n . Moreover, ι X = n if and only if X ∼ = Q n ⊂ P n +1 . Families of rational curves provide a useful tool in the study of Fano manifolds. Next we gather someresults from the theory of rational curves. In what follows, rational curves are always assumed to be proper. A family of rational curves on a complex projective manifold X is a closed irreducible subset of RatCurves n ( X ).We refer to [Kol96] for details. Definition 2.3.
Let ℓ ⊂ X be a rational curve on a complex projective manifold, and consider its normal-ization f : P → X . We say that ℓ is free if f ∗ T X is globally generated. Let X be a complex projective manifold, and ℓ ⊂ X a free rational curve. Let x ∈ ℓ be any point, and H x an irreducible component of the scheme RatCurves n ( X, x ) containing a point corresponding to ℓ . Thendim( H x ) = − K X · ℓ − . CAROLINA
ARAUJO
AND ST´EPHANE
DRUEL
Notation 2.5.
Let X be a Fano manifold with ρ ( X ) = 1, and A an ample line bundle on X such thatPic( X ) = Z [ A ]. For any proper curve C ⊂ X , we refer to A · C as the degree of C . Rational curves ofdegree 1 are called lines . Note that if C ⊂ X is a proper curve of degree d , then ι X = − K X · Cd .One can use free rational curves on Fano manifolds with Picard number 1 to bound their index. Thefollowing is an immediate consequence of paragraph 2.4 above. Lemma 2.6.
Let X be a Fano manifold with ρ ( X ) = 1 . Suppose that there is an m -dimensional family V of rational curves of degree d on X such that: • all curves from V pass though some fixed point x ∈ X ; and • some curve from V is free.Then ι X > m +2 d . Remark 2.7.
Let V be a family of rational curves on a complex projective manifold X . To guarantee thatsome member of V is a free curve, it is enough to show that some curve from V passes through a general point of X . More precisely, let H be an irreducible component of RatCurves n ( X ) containing V . It comeswith universal family morphisms U e / / π (cid:15) (cid:15) X ,H where π : U → H is a P -bundle. Suppose that e : U → X is dominant. Then, by generic smoothness, thereis a dense open subset X ◦ ⊂ X over which e : U → X is smooth. On the other hand, by [Kol96, PropositionII.3.4], e is smooth at a point u ∈ π − ( t ) if and only if the rational curve ℓ t = e (cid:0) π − ( t ) (cid:1) is free.2.2. Singularities of pairs.
We refer to [KM98, section 2.3] and [Kol13, sections 2 and 4] for details.
Definition 2.8.
Let X be a normal projective variety, and ∆ = P a i ∆ i an effective Q -divisor on X , i.e., ∆is a nonnegative Q -linear combination of distinct prime Weil divisors ∆ i ’s on X . Suppose that K X + ∆ is Q -Cartier, i.e., some nonzero multiple of it is a Cartier divisor.Let f : ˜ X → X be a log resolution of the pair ( X, ∆). There are uniquely defined rational numbers a ( E i , X, ∆)’s such that K ˜ X + f − ∗ ∆ = f ∗ ( K X + ∆) + X E i a ( E i , X, ∆) E i . The a ( E i , X, ∆)’s do not depend on the log resolution f , but only on the valuations associated to the E i ’s.The closed subvariety f ( E i ) ⊂ X is called the center of E i in X . It also depends only on the valuationassociated to E i .For a prime divisor D on X , we define a ( D, X, ∆) to be the coefficient of D in − ∆.We say that the pair ( X, ∆) is log canonical if, for some log resolution f : ˜ X → X of ( X, ∆), a ( E i , X, ∆) > − f -exceptional prime divisor E i . If this condition holds for some log resolution of ( X, ∆), thenit holds for every log resolution of ( X, ∆). Definition 2.9.
Let ( X, ∆) be a log canonical pair. We say that a closed irreducible subvariety S ⊂ X is a log canonical center of ( X, ∆) if there is a divisor E over X with a ( E, X, ∆) = − X is S .2.3. Polarized varieties and Fujita’s ∆ -genus.Definition 2.10. A polarized variety is a pair ( X, L ) consisting of a normal projective variety X , and anample line bundle L on X . Definition 2.11 ([Fuj75]) . The ∆ -genus of an n -dimensional polarized variety ( X, L ) is defined by theformula: ∆( X, L ) := n + c ( L ) n − h ( X, L ) ∈ Z . By [Fuj82, Corollary 2.12], ∆( X, L ) > X, L ). Next we recall the classificationof polarized varieties with ∆-genus zero from [Fuj82]. Theorem 2.12 ([Fuj82]) . Let X be a normal projective variety of dimension n > , and L an ample linebundle on X . Suppose that ∆( X, L ) = 0 . Then one of the following holds. N FANO FOLIATIONS 2 5 (1) ( X, L ) ∼ = ( P n , O P n (1)) . (2) ( X, L ) ∼ = ( Q n , O Q n (1)) . (3) ( X, L ) ∼ = ( P , O P ( d )) , for some d > . (4) ( X, L ) ∼ = ( P , O P (2)) . (5) ( X, L ) ∼ = ( P P ( E ) , O P P ( E ) (1)) , where E is an ample vector bundle on P . (6) L is very ample, and embeds X as a cone over a projective polarized variety of type (3-5) above. Classification of log del Pezzo pairs.Definition 2.13.
Let X be a normal projective variety of dimension n >
1, and ∆ an effective Q -divisor on X . We say that ( X, ∆) is a log del Pezzo pair if ( X, ∆) is log canonical, and − ( K X + ∆) ≡ ( n − c ( L )for some ample line bundle L on X .Using Fujita’s classification of polarized varieties with ∆-genus zero, we classify log del Pezzo pairs inTheorem 2.15 below. Lemma 2.14.
Let ( X, L ) be an n -dimensional polarized variety, with n > . Let ( X, ∆) be a log del Pezzopair such that ∆ = 0 and − ( K X + ∆) ≡ ( n − c ( L ) . Then ∆( X, L ) = 0 , and ∆ · c ( L ) n − = 2 .Proof. We follow the line of argumentation in the proof of [Fuj80, Lemma 1.10]. Since c ( L ⊗ t ) ≡ K X + ∆ + c ( L ⊗ n − t ) , we have that h i ( X, L ⊗ t ) = 0 for i > t > − n by [Fuj11, Theorem 8.1]. Therefore χ ( X, O X ) = 1 and χ ( X, L ⊗ t ) = 0 for 2 − n t −
1. Hence, there are rational numbers a and b such that χ ( X, L ⊗ t ) = (cid:0) at + bt + n ( n − (cid:1) Q n − j =1 ( t + j ) n ! . On the other hand, since X is normal, by Hirzebruch-Riemann-Roch, χ ( X, L ⊗ t ) = c ( L ) n n ! t n − n − K X · c ( L ) n − t n − + o ( t n − ) . Thus we have a = c ( L ) n and b = n ∆ · c ( L ) n − + ( n − c ( L ) n . In particular, h ( X, L ) = χ ( X, L ) = n − c ( L ) n + 12 ∆ · c ( L ) n − . One then computes that ∆( X, L ) = 1 −
12 ∆ · c ( L ) n − . Since ∆ = 0 and ∆( X, L ) >
0, we must have ∆( X, L ) = 0 and ∆ · c ( L ) n − = 2. (cid:3) Theorem 2.15.
Let ( X, L ) be an n -dimensional polarized variety, with n > . Let ( X, ∆) be a log delPezzo pair such that ∆ is integral and nonzero, and − ( K X + ∆) ≡ ( n − c ( L ) . Then one of the followingholds. (1) ( X, L , O X (∆)) ∼ = ( P n , O P n (1) , O P n (2)) . (2) ( X, L , O X (∆)) ∼ = ( Q n , O Q n (1) , O Q n (1)) . (3) ( X, L , O X (∆)) ∼ = ( P , O P ( d ) , O P (2)) , for some integer d > . (4) ( X, L , O X (∆)) ∼ = ( P , O P (2) , O P (1)) . (5) ( X, L ) ∼ = ( P P ( E ) , O P P ( E ) (1)) for an ample vector bundle E on P . Moreover, one of the followingholds. (a) E = O P (1) ⊕ O P ( a ) for some a > , and ∆ ∼ Z σ + f where σ is the minimal section and f isa fiber of P P ( E ) → P . (b) E = O P (2) ⊕ O P ( a ) for some a > , and ∆ is a minimal section. (c) E = O P (1) ⊕ O P (1) ⊕ O P ( a ) for some a > , and ∆ = P P ( O P (1) ⊕ O P (1)) . (6) L is very ample, and embeds ( X, ∆) as a cone over (cid:0) ( Z, M ) , (∆ Z , M | ∆ Z ) (cid:1) , where Z is smooth and ( Z, M , ∆ Z ) satisfies one of the conditions (3-5) above. CAROLINA
ARAUJO
AND ST´EPHANE
DRUEL
Proof.
By [CKP12, Theorem 0.1], we must have − ( K X + ∆) ∼ Q ( n − c ( L ).If n = 1, then − K X ∼ Q ∆ is ample, and hence ( X, L , O X (∆)) satisfies one of conditions (1-3) in thestatement of Theorem 2.15.Suppose from now on that n >
2. By Lemma 2.14, ∆( X, L ) = 0, and so we can apply Theorem 2.12.Notice that if ( X, L ) satisfies any of conditions (1-6) of Theorem 2.12, then − ( K X + ∆) ∼ Z ( n − c ( L )since X \ Sing( X ) is simply connected.If ( X, L ) satisfies any of conditions (1-4) of Theorem 2.12, one checks easily that ( X, L , ∆) satisfies oneof conditions (1-4) in the statement of Theorem 2.15.Suppose that ( X, L ) ∼ = ( P P ( E ) , O P P ( E ) (1)) for an ample vector bundle E on P , and write π : X → P for the natural projection. Then∆ ∈ (cid:12)(cid:12) O X ( − K X ) ⊗ L ⊗ − n (cid:12)(cid:12) = (cid:12)(cid:12) L ⊗ π ∗ (cid:0) det( E ∗ ) ⊗ O P (2) (cid:1)(cid:12)(cid:12) . Write E ∼ = O P ( a ) ⊕ · · · ⊕ O P ( a n ), with 1 a · · · a n . By the projection formula, h (cid:0) X, L ⊗ π ∗ (cid:0) det( E ∗ ) ⊗ O P (2) (cid:1)(cid:1) = h ( P , E ⊗ det( E ∗ ) ⊗ O P (2)), hence we must have a + · · · + a n −
2. This impliesthat ( n, a , . . . , a n − ) ∈ { (2 , , (2 , , (3 , , } . Thus either E satisfies condition (5a-c) in the statement ofTheorem 2.15, or E = O P (1) ⊕ O P (1), ∆ ∈ | O P P ( E ) (1) | , and hence X satisfies condition (2) with n = 2.Finally, suppose that L is very ample, and embeds X as a cone with vertex V over a smooth polarizedvariety ( Z, M ) satisfying one of conditions (3-5) in the statement of Theorem 2.12. Set m := dim( Z ) and s := n − m = dim( V ) + 1. Let e : Y → X be the blow-up of X along V , with exceptional divisor E . Wehave Y ∼ = P Z ( M ⊕ O ⊕ sZ ), with natural projection π : Y → Z , and tautological line bundle O Y (1) ∼ = e ∗ L .The exceptional divisor E corresponds to the projection M ⊕ O ⊕ sZ ։ O ⊕ sZ .Let ∆ Y be the strict transform of ∆ in Y . We are done if we prove that ∆ Y = π ∗ ∆ Z for some divisor∆ Z on Z .Write ∆ Y ∼ Z π ∗ ∆ Z + kE for some integral divisor ∆ Z on Z , and some integer k >
0. Let σ : Z → Y be the section of π corresponding to a general surjection M ⊕ O ⊕ sZ ։ M . Then σ ( Z ) ∩ E = ∅ , and N σ ( Z ) /Y ∼ = M ⊕ s . Moreover, ( σ ( Z ) , ∆ Y | σ ( Z ) ) is log canonical (see for instance [Kol97, Proposition 7.3.2]),and, by the adjunction formula, − ( K σ ( Z ) + ∆ Y | σ ( Z ) ) ∼ Z ( m − c ( O Y (1)) | σ ( Z ) .We have h ( Y, O Y ( kE + π ∗ ∆ Z ) = h ( Y, O Y (∆ Y )) >
1. On the other hand, h ( Y, O Y ( kE + π ∗ ∆ Z )) = h ( Y, O Y ( k ) ⊗ π ∗ M ⊗− k ⊗ O Y ( π ∗ ∆ Z ))= h ( Z, S k ( M ⊕ O ⊕ sZ ) ⊗ M ⊗− k ⊗ O Z (∆ Z )) = h ( Z, S k ( O Z ⊕ M ⊗− Z ⊕ s ) ⊗ O Z (∆ Z )) . We claim that h ( Z, M ⊗− Z ⊗ O Z (∆ Z )) = 0. Indeed, suppose that h ( Z, M ⊗− Z ⊗ O Z (∆ Z )) = 0. Then − K Z ∼ Z ∆ Z + ( m − c ( M ) since ∆ Y | σ ( Z ) ∼ Z (cid:0) π | σ ( Z ) (cid:1) ∗ ∆ Z , and hence − K Z > mc ( M ). Under theseconditions, [AD12, Theorem 2.5] implies that ( Z, M , O Z (∆ Z )) is isomorphic to either ( P m , O P m (1) , O P m (2))or ( Q m , O Q m (1) , O Q m (1)). This contradicts our current assumption that ( Z, M ) satisfies one of conditions(3-5) in the statement of Theorem 2.12, and proves the claim.Since h ( Z, M ⊗− Z ⊗ O Z (∆ Z )) = 0, we must have h ( Y, O Y ( kE + π ∗ ∆ Z )) = h ( Z, O Z (∆ Z )). Thus,replacing ∆ Z with a suitable member of its linear system if necessary, we may assume that ∆ Y = π ∗ ∆ Z + kE ,and hence k = 0. Therefore ( X, L , ∆) satisfies condition (6) in the statement of Theorem 2.15. (cid:3) In dimension 2, we have the following classification, without the assumption that ( X, ∆) is log canonical. Theorem 2.16 ([Nak07, Theorem 4.8]) . Let ( X, ∆) be a pair with dim( X ) = 2 and ∆ = 0 . Suppose that − ( K X + ∆) is Cartier and ample. Then one of the following holds. (1) X ∼ = P and deg(∆) ∈ { , } . (2) X ∼ = F d for some d > and ∆ is a minimal section. (3) X ∼ = F d for some d > and ∆ ∼ Z σ + f , where σ is a minimal section and f a fiber of F d → P . (4) X ∼ = P (1 , , d ) for some d > and ∆ ∼ Z ℓ where ℓ is a ruling of the cone P (1 , , d ) . Foliations
Foliations and Pfaff fields.Definition 3.1.
Let X be normal variety. A foliation on X is a nonzero coherent subsheaf F ( T X suchthat N FANO FOLIATIONS 2 7 • F is closed under the Lie bracket, and • F is saturated in T X (i.e., T X / F is torsion free).The rank r of F is the generic rank of F . The codimension of F is q = dim( X ) − r .The canonical class K F of F is any Weil divisor on X such that O X ( − K F ) ∼ = det( F ). Definition 3.2.
A foliation F on a normal variety is said to be Q -Gorenstein if its canonical class K F is Q -Cartier. Definition 3.3.
Let X be a variety, and r a positive integer. A Pfaff field of rank r on X is a nonzeromap η : Ω rX → L , where L is a reflexive sheaf of rank 1 on X such that L [ m ] is invertible for some integer m > singular locus S of η is the closed subscheme of X whose ideal sheaf I S is the image of the inducedmap Ω rX [ ⊗ ] L ∗ → O X .Notice that a Q -Gorenstein foliation F of rank r on normal variety X naturaly gives rise to a Pfaff fieldof rank r on X : η : Ω rX = ∧ r (Ω X ) → ∧ r ( T ∗ X ) → ∧ r ( F ∗ ) → det( F ∗ ) ∼ = det( F ) ∗ ∼ = O X ( K F ) . Definition 3.4.
Let F be a Q -Gorenstein foliation on a normal variety X . The singular locus of F isdefined to be the singular locus S of the associated Pfaff field. We say that F is regular at a point x ∈ X if x S . We say that F is regular if S = ∅ .Our definition of Pfaff field is more general than the one usually found in the literature, where L isrequired to be invertibile. This generalization is needed in order to treat Q -Gorenstein foliations whosecanonical classes are not Cartier. (Foliations defined by q -forms) . Let F be a codimension q foliation on an n -dimenional normal variety X . The normal sheaf of F is N F := ( T X / F ) ∗∗ . The q -th wedge product of the inclusion N ∗ F ֒ → (Ω X ) ∗∗ gives rise to a nonzero global section ω ∈ H (cid:0) X, Ω qX [ ⊗ ] det( N F ) (cid:1) whose zero locus has codimension at least2 in X . Moreover, ω is locally decomposable and integrable . To say that ω is locally decomposable meansthat, in a neighborhood of a general point of X , ω decomposes as the wedge product of q local 1-forms ω = ω ∧ · · · ∧ ω q . To say that it is integrable means that for this local decomposition one has dω i ∧ ω = 0 forevery i ∈ { , . . . , q } . The integrability condition for ω is equivalent to the condition that F is closed underthe Lie bracket.Conversely, let L be a reflexive sheaf of rank 1 on X , and ω ∈ H ( X, Ω qX [ ⊗ ] L ) a global section whosezero locus has codimension at least 2 in X . Suppose that ω is locally decomposable and integrable. Then thekernel of the morphism T X → Ω q − X [ ⊗ ] L given by the contraction with ω defines a foliation of codimension q on X . These constructions are inverse of each other.3.2. Algebraically integrable foliations.
Let X be a normal projective variety, and F a foliation on X . In this subsection we assume that F is algebraically integrable . This means that F is the relative tangent sheaf to a dominant rational map ϕ : X Y with connected fibers. In this case, by a general leaf of F we mean the fiber of ϕ over a generalpoint of Y . We start by defining the notion of log leaf when F is moreover Q -Gorenstein. It plays a keyrole in our approach to Q -Fano foliations. Definition 3.6 (See [AD12, Definition 3.10] for details) . Let X be a normal projective variety, F a Q -Gorenstein algebraically integrable foliation of rank r on X , and η : Ω rX → O X ( K F ) its corresponding Pfafffield. Let F ⊂ X be the closure of a general leaf of F , and ˜ e : ˜ F → X the normalization of F . Let m > K F , i.e. , the smallest positive integer m such that mK F is Cartier. Then η inducesa generically surjective map ⊗ m Ω r ˜ F → ˜ e ∗ O X ( mK F ). Hence there is a canonically defined effective Weil Q -divisor ˜∆ on ˜ F such that mK ˜ F + m ˜∆ ∼ Z ˜ e ∗ mK F .We call the pair ( ˜ F , ˜∆) a general log leaf of F .The next lemma gives sufficient conditions under which the support of ˜∆ is precisely the image in ˜ F ofthe singular locus of F . It is an immediate consequence of [AD13, Lemma 5.6]. CAROLINA
ARAUJO
AND ST´EPHANE
DRUEL
Lemma 3.7.
Let F be an algebraically integrable foliation on a complex projective manifold X . Supposethat F is locally free along the closure of a general leaf F . Let ˜ e : ˜ F → X be its normalization, and ( ˜ F , ˜∆) the corresponding log leaf. Then
Supp( ˜∆) = ˜ e − ( F ∩ Sing( F )) . Definition 3.8.
Let X be normal projective variety, F a Q -Gorenstein algebraically integrable foliationon X , and ( ˜ F , ˜∆) its general log leaf. We say that F has log canonical singularities along a general leaf if( ˜ F , ˜∆) is log canonical.
Remark 3.9.
In [McQ08, Definition I.1.2], McQuillan introduced a notion of log canonicity for foliations,without requiring algebraic integrability. If a Q -Gorenstein algebraically integrable foliation F is log canon-ical in the sense of McQuillan, then F has log canonical singularities along a general leaf (see [AD13,Proposition 3.11] and its proof). (The family of log leaves) . Let X be normal projective variety, and F an algebraically integrablefoliation on X . We describe the family of leaves of F (see [AD13, Lemma 3.2 and Remark 3.8] for details).There is a unique irreducible closed subvariety W of Chow( X ) whose general point parametrizes the closureof a general leaf of F (viewed as a reduced and irreducible cycle in X ). It comes with a universal cycle U ⊂ W × X and morphisms: U e / / π (cid:15) (cid:15) X ,W where e : U → X is birational and, for a general point w ∈ W , e (cid:0) π − ( w ) (cid:1) ⊂ X is the closure of a leaf of F .The variety W is called the family of leaves of F .Suppose moreover that F is Q -Gorenstein, denote by m > K F , by r the rank of F , and by η : Ω rX → O X ( K F ) the corresponding Pfaff field. Given a morphism V → W from a normalvariety, let U V be the normalization of U × V W , with induced morphisms: U V e V / / π V (cid:15) (cid:15) X .V
Then η induces a generically surjective map ⊗ m Ω rU V /V → e V ∗ O X ( mK F ) . Thus there is a canonically definedeffective Weil Q -divisor ∆ V on U V such that det(Ω U V /V ) ⊗ m [ ⊗ ] O U V ( m ∆ V ) ∼ = e V ∗ O X ( mK F ). Suppose that v ∈ V is mapped to a general point of W , set U v := ( π V ) − ( v ), and ∆ v := (∆ V ) | U v . Then ( U v , ∆ v ) coincideswith the general log leaf ( ˜ F , ˜∆) defined above.3.3. Q -Fano foliations.Definition 3.11. Let X be a normal projective variety, and F a Q -Gorenstein foliation on X . We saythat F is a Q -Fano foliation if − K F is ample. In this case, the index of F is the largest positive rationalnumber ι F such that − K F ∼ Q ι F H for a Cartier divisor H on X .If F is a Q -Fano foliation of rank r on a normal projective variety X , then, by [H¨or13, Corollary 1.2], ι F r . Moreover, equality holds if and only if X is a generalized normal cone over a normal projectivevariety Z , and F is induced by the natural rational map X Z (see also [ADK08, Theorem 1.1], and[AD12, Theorem 4.11]). Definition 3.12. A Q -Fano foliation F of rank r > del Pezzo foliation if ι F = r − Theorem 3.13 ([ADK08, Theorem 3.1]) . Let X be a normal projective variety, f : X → C a surjectivemorphism onto a smooth curve, and ∆ an effective Weil Q -divisor on X such that ( X, ∆) is log canonicalover the generic point of C . Then − ( K X/C + ∆) is not ample.
N FANO FOLIATIONS 2 9
Proposition 3.14.
Let F be an algebraically integrable Q -Fano foliation on a normal projective variety X , having log canonical singularities along a general leaf. Then there is a closed irreducible subset T ⊂ X satisfying the following property. For a general log leaf ( ˜ F , ˜∆) of F , there exists a log canonical center S of ( ˜ F , ˜∆) whose image in X is T .Proof. Let W be the normalization of the family of leaves of F , U the normalization of the universal cycleover W , with universal family morphisms π : U → W and e : U → X . As explained in 3.10, there is acanonically defined effective Q -Weil divisor ∆ on U such that det(Ω U/W ) ⊗ m [ ⊗ ] O U ( m ∆) ∼ = e ∗ O X ( mK F ),where m > K F . Moreover, there is a smooth dense open subset W ⊂ W with the following properties. For any w ∈ W , denote by U w the fiber of π over w , and set ∆ w := ∆ | U w .Then • U w is integral and normal, and • ( U w , ∆ w ) has log canonical singularities.To prove the proposition, suppose to the contrary that, for any two general points w, w ′ ∈ W , and anylog canonical centers S w and S w ′ of ( U w , ∆ w ) and ( U w ′ , ∆ w ′ ) respectively, we have e ( S w ) = e ( S w ′ ).Let C ⊂ W be a (smooth) general complete intersection curve, and U C the normalization of π − ( C ), withinduced morphisms π C : U C → C and e C : U C → X . By [BLR95, Theorem 2.1’], after replacing C with afinite cover if necessary, we may assume that π C has reduced fibers. As before, there is a canonically defined Q -Weil divisor ∆ C on U C such that K U C /C + ∆ C ∼ Q e C ∗ K F . Therefore K U C + ∆ C ∼ Q π ∗ C K C + e ∗ C K F isa Q -Cartier divisor. For a general point w ∈ C , we identify (cid:16) π − C ( w ) , ∆ C | π − C ( w ) (cid:17) with ( U w , ∆ w ), which islog canonical by assumption. Thus, by inversion of adjunction (see [Kaw07, Theorem]), the pair ( U C , ∆ C )has log canonical singularities over the generic point of C . Let w ∈ C be a general point, and S w any logcanonical center of ( U w , ∆ w ). Then there exists a reduced and irreducible closed subset S C ⊂ U C such that: • S w = S C ∩ U w , and • S C is a log canonical center of ( U C , ∆ C ) over the generic point of C .Moreover, our current assumption implies that • dim( e C ( S C )) = dim( S C ).Thus, by [Dem97, Proposition 7.2(ii)], there exist an ample Q -divisor A and an effective Q -Cartier Q -divisor E on U C such that: • e ∗ C ( − K F ) ∼ Q A + E , and • for a general point w ∈ C , Supp( E ) does not contain any log canonical center of ( U w , ∆ w ).Therefore ( U C , ∆ C + ǫE ) is log canonical over the generic point of C for 0 < ǫ ≪
1. Notice that e ∗ C ( − K F ) − ǫE is ample since e ∗ C ( − K F ) is nef and big, and hence − ( K U C /C + ∆ C + ǫE ) ∼ Q e ∗ C ( − K F ) − ǫE is ample as well. But this contradicts Theorem 3.13, completing the proof of the proposition. (cid:3) Corollary 3.15.
Let F be an algebraically integrable Fano foliation on a complex projective manifold, and ( ˜ F , ˜∆) its general log leaf. Suppose that F is locally free along the closure of a general leaf. Then ˜∆ = 0 .Proof. Denote by F the closure of a general leaf of F . If ˜∆ = 0, then F is regular along F by Lemma3.7. Hence F is induced by an almost proper map X W , and F is smooth. In particular ( ˜ F , ˜∆) is logcanonical. But this contradicts Proposition 3.14. This proves that ˜∆ = 0. (cid:3) Foliations on P n . The degree deg( F ) of a foliation F of rank r on P n is defined as the degree of the locus of tangency of F with a general linear subspace P n − r ⊂ P n . By 3.5, a foliation on P n of rank r and degree d is given by atwisted q -form ω ∈ H (cid:0) P n , Ω q P n ( q + d + 1) (cid:1) , where q = n − r . Thus d = deg( K F ) + r. Jouanolou has classified codimension 1 foliations on P n of degree 0 and 1. This has been generalized toarbitrary rank as follows. Theorem 3.16.
ARAUJO
AND ST´EPHANE
DRUEL (1) ( [DC05, Th´eor`eme 3.8] .) A codimension q foliation of degree on P n is induced by a linear projection P n P q . (2) ( [LPT13, Theorem 6.2] .) A codimension q foliation F of degree on P n satisfies one of the followingconditions. • F is induced by a dominant rational map P n P (1 q , , defined by q linear forms L , . . . , L q and one quadratic form Q ; or • F is the linear pullback of a foliation on P q +1 induced by a global holomorphic vector field. Let F be a codimension q foliation of degree 1 on P n .In the first case described in Theorem 3.16(2), F is induced by the q -form on C n +1 Ω = q X i =1 ( − i +1 L i dL ∧ · · · ∧ d dL i ∧ · · · ∧ dL q ∧ dQ + ( − q QdL ∧ · · · ∧ dL q = ( − q (cid:16) n +1 X i = q +1 L j ∂Q∂L i (cid:17) dL ∧ · · · ∧ dL q + q X i =1 n +1 X j = q +1 ( − i +1 L i ∂Q∂L j dL ∧ · · · ∧ d dL i ∧ · · · ∧ dL q ∧ dL j , where L q +1 , . . . , L n +1 are linear forms such that L , . . . , L n +1 are linearly independent. Thus, the singularlocus of F is the union of the quadric { L = · · · = L q = Q = 0 } ∼ = Q n − q − and the linear subspace { ∂Q∂L q +1 = · · · = ∂Q∂L n +1 = 0 } .In the second case described in Theorem 3.16(2), the singular locus of F is the union of linear subspacesof codimension at least 2 containing the center P n − q − of the projection.3.5. Foliations on Q n . In this subsection we classify del Pezzo foliations on smooth quadric hypersurfaces.
Proposition 3.18.
Let F be a codimension q del Pezzo foliation on a smooth quadric hypersurface Q n ⊂ P n +1 . Then F is induced by the restriction of a linear projection P n +1 P q .Proof. If q = 1, then the result follows from [AD12, Theorem 1.3]. So we assume from now on that q > F is algebraically integrable, and its singular locus is nonempty by [AD12,Theorem 6.1].Let x ∈ Q n be a point in the singular locus of F , and consider the restriction ϕ : Q n P n to Q n of thelinear projection ψ : P n +1 P n from x . Let f : Y → Q n be the blow-up of Q n at x with exceptional divisor E ∼ = P n − , and g : Y → P n the induced morphism. Notice that g is the blow-up of P n along the smoothcodimension 2 quadric Z = ϕ (Exc( ϕ )) ∼ = Q n − . Denote by H the hyperplane of P n containing Z , and by F the exceptional divisor of g . Note that g ( E ) = H , and f ( F ) is the hyperplane section of Q n cut out by T x Q n . The codimension q del Pezzo foliation F is defined by a nonzero section ω ∈ H (cid:0) Q n , Ω qQ n ( q + 1) (cid:1) vanishing at x . So it induces a twisted q -form α ∈ H (cid:0) Y, Ω qY ⊗ f ∗ O Q ( q + 1) ⊗ O Y ( − qE ) (cid:1) ∼ = H (cid:0) Y, Ω qY ⊗ g ∗ O P n ( q + 2) ⊗ O Y ( − F ) (cid:1) . The restriction of α to Y \ F induces a twisted q -form ˜ α ∈ H (cid:0) P n , Ω q P n ( q + 2) (cid:1) such that ˜ α z ( ~v , ~v , . . . , ~v q ) = 0 for any z ∈ Z , ~v ∈ T z P n , and ~v i ∈ T z Z , 2 i q . Denote by ˜ F thefoliation on P n induced by ˜ α . There are two possibilities: • Either ˜ α vanishes along the hyperplane H of P n containing Z ∼ = Q n − , and hence ˜ F is a degree 0foliation on P n ; or • ˜ F is a degree 1 foliation on P n , and either Z is contained in the singular locus of ˜ F , or Z is invariantunder ˜ F .We will show that only the first possibility occurs. In this case, it follows from Theorem 3.16(1) that F is induced by the restriction of a linear projection P n +1 P q .Suppose to the contrary that ˜ F is a degree 1 foliation on P n , and either Z is contained in the singularlocus of ˜ F , or Z is invariant under ˜ F . Recall the description of the two types of codimension q degree 1 on P n from Theorem 3.16(2):(1) Either the foliation is induced by a dominant rational map P n P (1 q , q linear forms L , . . . , L q and one quadratic form Q ; or N FANO FOLIATIONS 2 11 (2) it is the linear pullback of a foliation on P q +1 induced by a global holomorphic vector field.In case (2), the closure of the leaves and the singular locus are all cones with vertex P n − q − . Since Z ∼ = Q n − is a smooth quadric, we conclude that ˜ F must be of type (1), Z is invariant under ˜ F , and ˜ α is as in thedescription of Ω in 3.17.Since Z is invariant under ˜ F , we must have { L = · · · = L q = Q = 0 } ∼ = Q n − q − ⊂ Z . We assumewithout loss of generality that H = { L = 0 } . Notice that { L = · · · = L q = Q = 0 } ( Z since q >
2. Let L q +1 , . . . , L n +1 ∈ C [ t , . . . , t n +1 ] be linear forms such that L , . . . , L n +1 are linearly independent. Since Z is invariant under ˜ F , ϕ ∗ ˜ α vanishes identically along f ( F ) = { ϕ ∗ L = 0 } . It follows from the descriptionof the singular locus of ˜ F in 3.17 that we must have { ϕ ∗ ∂Q∂L q +1 = · · · = ϕ ∗ ∂Q∂L n +1 = 0 } = { ϕ ∗ L = 0 } .Hence, for i ∈ { q + 1 , . . . , n + 1 } , ϕ ∗ ∂Q∂L i = a i ϕ ∗ L for some complex number a i ∈ C . Then ψ ∗ ˜ α ∈ ( ψ ∗ L ) · H (cid:0) P n +1 , Ω q P n +1 ( q + 1) (cid:1) ⊂ H (cid:0) P n +1 , Ω q P n +1 ( q + 2) (cid:1) . Therefore, ˜ F is induced by a degree 0 foliationon P n +1 . So ˜ F itself is a degree 0 foliation on P n , contrary to our assumption. This completes the proof ofthe proposition. (cid:3) Proof of Theorem 1.3
Let X = P n be an n -dimensional Fano manifold with ρ ( X ) = 1, and F a del Pezzo foliation of rank r > X . By Theorem 1.1, F is algebraically integrable. Let F be the closure of a general leaf of F , ˜ e : ˜ F → X its normalization, and ( ˜ F , ˜∆) the corresponding log leaf. By assumption, ( ˜
F , ˜∆) is log canonical, and F islocally free along F .Let A be an ample line bundle on X such that Pic( X ) = Z [ A ], and set L := ˜ e ∗ A . Then det( F ) ∼ = A r − ,and − ( K ˜ F + ˜∆) ∼ Z − ˜ e ∗ K F ∼ Z ( r − c ( L ) . By Corollary 3.15, ˜∆ = 0. So we can apply Theorem 2.15. Taking into account that if ˜ F is singular, thenits singular locus is contained in the support of ˜∆ by Lemma 3.7, we get the following possibilities for thetriple ( ˜ F , L , ˜∆):(1) (cid:0) ˜ F , L , O ˜ F ( ˜∆) (cid:1) ∼ = ( P r , O P r (1) , O P r (2)).(2) (cid:0) ˜ F , L , O ˜ F ( ˜∆) (cid:1) ∼ = ( Q r , O Q r (1) , O Q r (1)), where Q r is a smooth quadric hypersurface in P r +1 .(3) r = 1 and (cid:0) ˜ F , L , O ˜ F ( ˜∆) (cid:1) ∼ = ( P , O P ( d ) , O P (2)) for some integer d > r = 2 and (cid:0) ˜ F , L , O ˜ F ( ˜∆) (cid:1) ∼ = ( P , O P (2) , O P (1)).(5) ( ˜ F , L ) ∼ = ( P P ( E ) , O P P ( E ) (1)) for an ample vector bundle E of rank r on P . Moreover, one of thefollowing holds.(a) r = 2 and E = O P (1) ⊕ O P ( a ) for some a >
2, and ˜∆ ∼ Z σ + f where σ is the minimal sectionand f a fiber of P P ( E ) → P .(b) r = 2 and E = O P (2) ⊕ O P ( a ) for some a >
2, and ˜∆ is a minimal section.(c) r = 3 and E = O P (1) ⊕ O P (1) ⊕ O P ( a ) for some a >
1, and ˜∆ = P P ( O P (1) ⊕ O P (1)).(6) L is very ample, and embeds ( ˜ F , ˜∆) as a cone over (cid:0) ( Z, M ) , (∆ Z , M | ∆ Z ) (cid:1) , where ( Z, M , ∆ Z )satisfies one of the conditions (2-5) above.First we show that case (1) cannot occur. Suppose otherwise that ( ˜ F , L , O ˜ F ( ˜∆)) ∼ = ( P r , O P r (1) , O P r (2)).By Proposition 3.14, there is a common point x contained in the closure of a general leaf. Since ( ˜ F , L ) ∼ =( P r , O P r (1)), there is an irreducible ( n − X through x sweeping out thewhole X . Lemma 2.6 together with Theorem 2.2 imply that X ∼ = P n , contrary to our assumptions.Next suppose that we are in case (2) or (5c). Note that ˜∆ is irreducible in either case (in case (2), ˜ F isa smooth quadric of dimension r > T of˜∆ in X is contained in the closure of a general leaf of F . There is a family of lines on X , all contained inleaves of F and meeting T , that sweep out the whole X . In case (2), this corresponds to the family of lineson ˜ F ∼ = Q r . In case (5c), it corresponds to the family of lines on fibers of ˜ F → P . Let x ∈ T be a generalpoint. Then there is an irreducible ( n − X through x , and the general linein this family is free by Remark 2.7. By Lemma 2.6, ι X > n . Theorem 2.2 then implies that X ∼ = Q n . ByProposition 3.18, F is induced by the restriction to Q n of a linear projection P n +1 P n − r .Cases (3), (4), (5a) and (5b) do not occur since we are assuming r > ARAUJO
AND ST´EPHANE
DRUEL
Finally suppose that we are in case (6): L is very ample, and embeds ( ˜ F , ˜∆) as a cone over the pair (cid:0) ( Z, M ) , (∆ Z , M | ∆ Z ) (cid:1) , where ( Z, M , ∆ Z ) satisfies one of the conditions (2-5) above. As in the proof ofTheorem 2.15, set m := dim( Z ), s := r − m , and let e : Y → ˜ F be the blow-up of ˜ F along its vertex, withexceptional divisor E . Then Y ∼ = P Z ( M ⊕ O ⊕ sZ ), with natural projection π : Y → Z . Moreover the stricttransform ∆ Y of ˜∆ in Y satisfies ∆ Y = π ∗ ∆ Z . A straightforward computation gives K Y + π ∗ ∆ Z ∼ Z e ∗ ( K ˜ F + ˜∆) + ( m − E. On the other hand, by [AD13, 8.3], there exists an effective divisor B on Y such that K Y + E + B ∼ Z e ∗ ( K ˜ F + ˜∆) . Therefore ∆ Y = π ∗ ∆ Z ∼ Z ( m − E + B. We conclude that m = 1. Thus ˜ F is isomorphic to a cone with vertex V ∼ = P r − over a rational normal curveof degree d >
2, and ˜∆ is the union of two rulings ∆ and ∆ , each isomorphic to P r − .By Proposition 3.14, there is a log canonical center S of ( ˜ F , ˜∆) whose image in X does not depend onthe choice of the general log leaf. Either S = V , or S = ∆ i for some i ∈ { , } . If S = V , then the linesthrough a general point of ˜ e ( V ) sweep out the whole X . Lemma 2.6 together with Theorem 2.2 then implythat X ∼ = P n , contrary to our assumptions. We conclude that the image of V in X varies with ( ˜ F , ˜∆), and,for some i ∈ { , } , T = ˜ e (∆ i ) is contained in the closure of a general leaf. There is a family of lines on X ,all contained in leaves of F and meeting T , that sweep out the whole X . Let x ∈ T be a general point.Since V ⊂ ∆ i , and the image of V in X varies with ( ˜ F , ˜∆), there is an irreducible ( n − X through x , and the general line in this family is free by Remark 2.7. By Lemma 2.6, ι X > n .Theorem 2.2 then implies that X ∼ = Q n . By Proposition 3.18, F is induced by the restriction of a linearprojection P n +1 P n − r . (cid:3) Using Theorem 2.16 and the same arguments as in the proof of Theorem 1.3, one can get the followingresult for del Pezzo foliations of rank 2, without the assumption that F is log canonical along a general leaf. Proposition 4.1.
Let F be a del Pezzo foliation of rank on a complex projective manifold X = P n with ρ ( X ) = 1 , and suppose that F is locally free along a general leaf. Denote by ( ˜ F , ˜∆) the general log leaf of F ,and by L the pullback to ˜ F of the ample generator of Pic( X ) . Then the triple ( ˜ F , O F ( ˜∆) , L ) is isomorphicto one of the following. (1) (cid:0) P , O P (1) , O P (2) (cid:1) ; (2) (cid:0) P , O P (2) , O P (1) (cid:1) ; (3) (cid:0) F d , O F d ( σ + f ) , O F d ( σ + ( d + 1) f ) (cid:1) , where d > , σ is a minimal section, and f is a fiber of F d → P ; (4) (cid:0) P (1 , , d ) , O P (1 , ,d ) (2 ℓ ) , O P (1 , ,d ) ( dℓ ) (cid:1) , where d > , and ℓ is a ruling of the cone P (1 , , d ) . Remark 4.2.
There are examples of del Pezzo foliations of rank 2 on Grassmannians whose general logleaves are ( ˜
F , ˜∆) = ( P , ℓ ) (see [AD13, 4.3]). References [AD12] Carolina Araujo and St´ephane Druel,
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E-mail address : [email protected] St´ephane Druel: Institut Fourier, UMR 5582 du CNRS, Universit´e Grenoble 1, BP 74, 38402 Saint Martind’H`eres, France
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