A view on extending morphisms from ample divisors
aa r X i v : . [ m a t h . AG ] J u l Contemporary Mathematics
A view on extending morphisms from ample divisors
Mauro C. Beltrametti and Paltin Ionescu
Dedicated to Andrew J. Sommese on his 60th birthday
Abstract.
The philosophy that “a projective manifold is more special thanany of its smooth hyperplane sections” was one of the classical principles ofprojective geometry. Lefschetz type results and related vanishing theoremswere among the typically used techniques. We shall survey most of the prob-lems, results and conjectures in this area, using the modern setting of ampledivisors, and (some aspects of) Mori theory.
Contents
1. Introduction 12. Background material 33. General results 84. Some convex geometry speculations 155. Applications to P d -bundles and blowing-ups 196. Complete results in the three dimensional case 257. Extending P -bundles 278. Fano manifolds as ample divisors 329. Ascent properties 34Acknowledgments 37References 38
1. Introduction
In the context of classical algebraic geometry, consider a given embedded com-plex projective manifold X ⊂ P N . One of the typically used techniques was to Mathematics Subject Classification.
Primary: 14M99, 14E30, 14J10, 14J45; Secondary:14D05, 14E25, 14N30.
Key words and phrases.
Ample divisors, extension of morphisms, comparing Kleiman–Moricones, Fano manifolds and Fano fibrations.The second named author was supported by the Italian Programme “Incentivazione allamobilit`a di studiosi stranieri e italiani residenti all’estero”. c (cid:13) replace X by some of its smooth hyperplane sections, Y ⊂ X . Thus, the dimensionof X is decreased and classification results may be obtained inductively. The effi-ciency of the method depends on the possibility of transferring some known specialproperties from Y to X . In general, given Y ⊂ P N − , there is no smooth X ⊂ P N such that Y is one of its hyperplane sections. One can say that X is more specialthen Y .The present paper is a survey of contemporary aspects of the hyperplane sectiontechnique. A first important “modern” incarnation of the above principle is givenby Lefschetz’s theorem, showing that the topology of Y strongly reflects that of X (see [ ]). From a geometrical point of view, we are usually given some regularmap, say p : Y → Z , making Y special ; e.g., a Fano fibration. We would like toextend this map to X . It was discovered by Sommese, in his innovative early paper[ ], that the extension is always possible, if one only assumes that the generalfiber of p has dimension at least two. His proof is based on Lefschetz’s theoremand on (very much related) vanishing results of Kodaira type. In the same paper,Sommese showed that when p is smooth and extends, the dimension of Z cannotbe too large. It soon became clear that the extension problem is much harderwhen dim Y ≤ dim Z + 1. Fujita [ ] further refined some of the techniques andconsidered new applications e.g., when p is a P d -bundle or a blowing-up. In the caseof three folds, fine results were found by B˘adescu [
6, 7, 8 ], when Y is a P -bundleover a curve and by Sommese [
69, 70 ], when Y is not relatively minimal. It isworth pointing out that the classical context of hyperplane sections was graduallyreplaced by the more general situation when Y ⊂ X is merely an ample divisor,and no projective embedding of X is given. This is a substantial generalization,since in the new setting the normal bundle of Y in X is not specified.The appearance of Mori theory made possible a change of both the point of viewand the techniques (see [
52, 38 ]). The isomorphism between the Picard groups of X and Y given by the Lefschetz theorem leads to an inclusion between the Kleiman–Mori cones N E ( Y ) and N E ( X ). As is well known, faces of these cones describenon-trivial morphisms defined on Y and X , respectively. So, the original questionof extending maps from Y to X translates into a comparison problem between thesecones. Ideally, when the two cones are equal, all morphisms from Y extend to X (see e.g., [
40, 74, 13, 2 ] and Section 8 for results in this direction, usually when X or Y are Fano manifolds). In the general case, what we can hope for is to extendthe contraction of an extremal ray of Y (cf. [
35, 57 ]). This is not always possible,but very few counterexamples are known (see Section 4 for this intriguing aspect).The techniques used in this setting are the cone theorem, due to Mori, and thecontraction theorem, due to Kawamata–Reid–Shokurov, combined with the wellbehaved deformation theory of families of rational curves [
41, 20, 38 ]. See also[
1, 34, 40 ] and [ ] for some useful facts about special families of rational curves,coming from extremal rays.General results on extending morphisms are discussed in Section 3; in Section 5we concentrate on the special situation when p is a P d -bundle or a blowing-up. Wepay special attention to the case of P -bundles, which is the most difficult. A (stillopen) main conjecture on the subject is stated and various related facts are provedin Section 7. The afore mentioned results by B˘adescu and Sommese on three foldsare recovered in Section 6, using the Mori theory point of view (cf. [ ]). In the VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 3 last section we discuss the ascent of some good properties from Y to X : e.g., beinguniruled, or rationally connected, or rational, etc.We have tried to write a complete and coherent exposition, also accessible to thenonspecialist. We included several new proofs and sometimes substantial simplifi-cations of the original arguments. Several possible generalizations are mentioned atthe end of the paper, together with appropriate references to the existing literature.
2. Background material
We work over the complex field C . Throughout the paper we deal with irre-ducible, reduced, projective varieties X . We use the term manifold if X is moreoverassumed to be smooth. We denote by O X the structure sheaf of X . For any coher-ent sheaf F on X , h i ( F ) denotes the complex dimension of H i ( X, F ). If p : X → Y is a morphism, we write p ( i ) for its i -th direct image.Let L be a line bundle on X . L is said to be numerically effective ( nef , forshort) if L · C ≥ C on X . We say that L is strictly nef (or numerically positive ) if L · C > C on X . L is said tobe big if κ ( L ) = dim X , where κ ( L ) denotes the Iitaka dimension of L . If L is nefthen this is equivalent to c ( L ) n >
0, where c ( L ) is the first Chern class of L and n = dim X . The pull-back ι ∗ L of a line bundle L on X by an embedding ι : Y ֒ → X is denoted by L Y . We denote by N Y/X the normal bundle of Y in X and by K X the canonical bundle of a smooth variety X .We use standard notation from algebraic geometry, among which we recall thefollowing ones: • ≈ , the linear equivalence of line bundles; ∼ , the numerical equivalence ofline bundles; • | L | , the complete linear system associated to a line bundle L ; • κ ( D ), the Iitaka dimension of the line bundle associated to a Q -Cartierdivisor D on X ; and κ ( X ) := κ ( K X ), the Kodaira dimension of X , for X smooth. • π i ( X ), the i -th homotopy group, omitting the base point when its choiceis irrelevant. P n denotes the projective n -space, Q n ⊂ P n +1 denotes the n -dimensional hy-perquadric. For a vector bundle E , we write P ( E ) for the associated projectivebundle and ξ P , or ξ X when X = P ( E ), for the tautological line bundle, using theGrothendieck convention.Line bundles and divisors are used with little (or no) distinction. We almostalways use the additive notation. We say that a line bundle L is spanned if it isspanned, i.e., globally generated, at all points of X by H ( X, L ). Let X be a projective manifold and let Y ⊂ X be a smooth ample divisor. It is a natural classical question to try tounderstand how the structure of Y determines the one of X .More precisely, given a surjective morphism p = p | D | : Y → Z associated toa linear system | D | , we look for a linear system | D | on X defining a regular map M.C. BELTRAMETTI AND P. IONESCU p = p | D | : X → W onto a projective variety W , such that the following diagram(2.1) Y p (cid:15) (cid:15) (cid:31) (cid:127) / / X p (cid:15) (cid:15) Z α / / W commutes. If the morphism α : Z → α ( Z ) is finite we say that p is a lifting of p .If p | Y = p , that is if α : Z → α ( Z ) ⊂ W is an isomorphism onto its image, we saythat p is a strict lifting of p , or that p is extendable to p . Note that this is alwaysthe case whenever the restriction map H ( X, D ) → H ( Y, D ) is surjective. Notealso that this further condition will be a posteriori satisfied in our setting (see theproof of Theorem 3.8).Assume that the morphism p has a lifting p . Up to taking the Remmert–Steinfactorization, we can always assume that p has connected fibers and Z is normal.Therefore, by using the ampleness of Y in X , it is a standard fact that one of thefollowing holds:(1) dim Y − dim Z ≥ α : Z ∼ −→ W (in particular p is extendable);(2) p , p and α : Z → α ( Z ) are birational; so, α is the normalization morphism;(3) p is birational and dim X − dim W = 1; in this case α : Z → α ( Z ) maybe of degree ≥ X := P × P n − embeddedin P N by O (2 , n ≥
4. By Bertini’s theorem, we can choose a hyperplane H in P N such that the restriction Y of H to X is a smooth ample divisor. Then we geta commutative diagram Y p (cid:15) (cid:15) (cid:31) (cid:127) / / & & MMMMMMMMMMMM X = P × P n − p (cid:15) (cid:15) Z α / / P n − where p and α are given by the Remmert–Stein factorization of the restriction p | Y : Y → P n − and p is the natural projection. Note that the morphism α isfinite of degree two. Moreover, p is not an isomorphism. Indeed, assume otherwise.Then Y → P n − is a two-to-one finite covering, so that it induces an isomorphismPic( Y ) ∼ = Z (see [ , II, 7.1.20] for details and complete references). On the otherhand, Pic( Y ) ∼ = Pic( X ) ∼ = Z ⊕ Z by the Lefschetz theorem; a contradiction.If the morphism p is extendable, our aim is to describe X by using the structuremorphism p . The occurrence that p is not extendable forces X to satisfy geometricconstraints which, in turn, make X special enough to be completely classified.As a typical example, consider the following natural question, formulated inthe classical context: Question 2.1.
Let X be an n -dimensional manifold embedded in a projectivespace P N . Assume that a smooth hyperplane section, Y = X ∩ H , of X is a P d -bundle over some manifold Z , such that the fibers are linearly embedded. Does itfollow that the bundle projection p : Y → Z extends to X giving a P d +1 -bundleprojection p : X → Z ?As soon as n ≥
4, the (positive) answer to this question relies on some non-trivial results from the deformation theory of rational curves. It turns out that the
VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 5 key-fact is the condition H · f = 1, where f is a line in a fiber P d of p : Y → Z , i.e., f is a linear P with respect to the embedding of X in P N given by H . Moreover,the above can happen only if 2 dim Z ≤ dim X (Proposition 5.9). Let X be a projective manifold of dimension n . Wesay that X is a Fano manifold if − K X is ample; its index , i , is the largest positiveinteger such that K X ≈ − i L for some ample line bundle L on X . Let L be a givenample line bundle on X . We say that ( X, L ) is a del Pezzo variety (respectively a
Mukai variety ) in the adjunction theoretic sense if K X ≈ − ( n − L (respectively K X ≈ − ( n − L ). Note that del Pezzo manifolds are completely described byFujita [ , I, Section 8]. We refer to Mukai [ ] for results on Mukai varieties.We say that ( X, L ) is a scroll over a normal variety Z of dimension m if thereexists a surjective morphism with connected fibers p : X → Z , such that K X + ( n − m + 1) L ≈ p ∗ L for some ample line bundle L on Z .We refer to [ ] and [ , Sections 14.1, 14.2] for relations between the adjunc-tion theoretic and the classical definition of scrolls.Let X be a projective manifold and let p : X → Z be a surjective morphismonto another manifold, Z . We say that X , p : X → Z is a P d - bundle if each closedfiber of p is isomorphic to the projective space P d . We also say that X , p : X → Z is a linear P d -bundle if X = P ( E ) for some rank d + 1 vector bundle E on Z .We say that X , p : X → Z is a conic fibration over a normal projective variety Z if every fiber of the morphism p is a conic, i.e., it is isomorphic to the zero schemeof a non-trivial section of O P (2). Note that the above definition is equivalent tosaying that there exists a rank 3 vector bundle E over Z such that its projectivization e p : P ( E ) → Z contains X embedded over Z as a divisor whose restriction to anyfiber of e p is an element of |O P (2) | . The push-forward p ∗ ( − K X ) can be taken asthe above E . It is a standard fact to show that p : X → Z is a flat morphism; since X is smooth, it follows that the base Z is smooth, too. A basic tool for dealing withthe problems discussed above are Lefschetz’s theorems, which, in turn, are verymuch related (in fact, almost equivalent) to vanishing results of Kodaira type (see[ , I, Chapters 3, 4] for a nice general presentation and complete references). See[ ] for the classical statement of Lefschetz’s theorem. Theorem 2.2. (Hamm–Lefschetz theorem)
Let L be an ample line bundle on aprojective manifold, X , and let D ∈ | L | . Then given any point x ∈ D it followsthat the j - th relative homotopy group, π j ( X, D, x ) , vanishes for j ≤ dim X − . Inparticular, the restriction mapping, H j ( X, Z ) → H j ( D, Z ) is an isomorphism for j ≤ dim X − , and is injective with torsion free cokernel for j = dim X − . Theorem 2.3. (Barth–Lefschetz theorem)
Let Y be a connected submanifold of aprojective manifold, X . Let n = dim X , m = dim Y . Assume that N Y/X is ample.Then for any x ∈ Y , we have π j ( X, Y, x ) = 0 for j ≤ m − n + 1 . In particular,under the natural map we have π ( Y, x ) ∼ = π ( X, x ) if m − n ≥ . Moreover: (i) If m − n = 1 , the restriction map r : Pic( X ) → Pic( Y ) is injective withtorsion free cokernel; and (ii) If m − n ≥ , then Pic( X ) ∼ = Pic( Y ) via r . Kawamata and Viehweg showed that the Kodaira vanishing theorem holds forany nef and big line bundle (see e.g., [ , Sections 1–2]). M.C. BELTRAMETTI AND P. IONESCU
Theorem 2.4. (Kawamata–Viehweg vanishing theorem)
Let X be a projectivemanifold of dimension n , and let D be a nef and big divisor on X . Then H i ( X, O X ( K X + D )) = 0 for i > . Let us recall some definitions and afew facts from Mori theory we need. Basic references for details are [
52, 53 ], and[ ]. Let X be a connected normal projective variety of dimension n ( ≥ • Num( X ) = Pic( X ) / ∼ ; • N ( X ) = Num( X ) ⊗ R ; • N ( X ) = ( { } / ∼ ) ⊗ R ; • N E ( X ), the convex cone in N ( X ) generated by the effective 1-cycles; • N E ( X ), the closure of N E ( X ) in N ( X ) with respect to the Euclidean topol-ogy; • ̺ ( X ) = dim R N ( X ), the Picard number of X ; • N E ( X ) D ≥ = { ζ ∈ N E ( X ) | ζ · D ≥ } for given D ∈ Pic( X ) ⊗ Q ; • N E ( X ) D< = { ζ ∈ N E ( X ) | ζ · D < } for given D ∈ Pic( X ) ⊗ Q ; • N E ( X ) D ≤ = { ζ ∈ N E ( X ) | ζ · D ≤ } for given D ∈ Pic( X ) ⊗ Q ; • Nef( X ), the dual cone of N E ( X ), namely, the cone in N ( X ) spanned byclasses of nef divisors.If γ is a 1-dimensional cycle in X we denote by [ γ ] its class in N E ( X ). Notethat the vector spaces N ( X ) and N ( X ) are dual to each other via the usualintersection of cycles “ · ”.Assume that X is smooth. We say that a half line R = R + [ ζ ] in N E ( X ), where R + = { x ∈ R | x > } , is an extremal ray if K X · ζ < ζ , ζ ∈ R for every ζ , ζ ∈ N E ( X ) such that ζ + ζ ∈ R. An extremal ray R = R + [ ζ ] is nef if D · ζ ≥ D on X . An extremal ray which is not nef is said to be non-nef .Let D ∈ Pic( X ) ⊗ Q be a nef Q -divisor, D
0. Let F D := D ⊥ ∩ ( N E ( X ) \ { } ) , where “ ⊥ ” means the orthogonal complement of D in N ( X ). Then F D is calleda good extremal face of N E ( X ) and D is the supporting hyperplane of F D , if F D is entirely contained in the set { ζ ∈ N ( X ) | K X · ζ < } . An extremal ray is a1-dimensional good extremal face. Indeed, for any extremal ray R there exists anef D ∈ Pic( X ) ⊗ Q such that R = D ⊥ ∩ ( N E ( X ) \ { } ). Theorem 2.5. (Mori cone theorem)
Let X be a projective manifold of dimension n . Then there exists a countable set of curves C i , i ∈ I , with K X · C i < , such thatone has the decomposition N E ( X ) = X i ∈ I R + [ C i ] + N E ( X ) K X ≥ . The decomposition has the properties: (i) the set of curves C i is minimal, no smaller set is sufficient to generate thecone; (ii) given any neighborhood U of N E ( X ) K X ≥ , only finitely many [ C i ] ’s donot belong to U . VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 7
The semi-lines R + [ C i ] are the extremal rays of X . Moreover, the curves C i are ( possibly singular ) reduced irreducible rational curves which satisfy the condition ≤ − K X · C i ≤ n + 1 . Theorem 2.6. (Kawamata–Reid–Shokurov base point free theorem)
Let X be aprojective manifold of dimension n ≥ . Let D be a nef Cartier divisor such that aD − K X is nef and big for some positive integer a . Then | mD | has no base pointsfor m ≫ . It is a standard fact that, for a good extremal face F D , the line bundle mD − K X is ample for m ≫
0. Therefore, by Theorem 2.6, the linear system | mD | is basepoint free for m ≫
0, so that it defines a morphism, say ϕ : X → W . By taking m big enough, we may further assume that W is normal and the fibers of ϕ areconnected. Note that ϕ ∗ O X ∼ = O W , the pair ( W, ϕ ) is unique up to isomorphismand D ∈ ϕ ∗ Pic( W ). If C is an irreducible curve on X , then [ C ] ∈ F D if and onlyif D · C = 0, which means dim ϕ ( C ) = 0, i.e., ϕ contracts the good extremal face F D . We will call such a contraction, ϕ , the contraction of F D . If F D = R , R anextremal ray, we will denote cont R : X → W the contraction morphism. Let E := { x ∈ X | cont R is not an isomorphism at x } . Note that E is the locus of curves whose numerical class is in R . We will refer to E simply as the locus of R .If X is smooth we define the length of an extremal ray ,length( R ) = min {− K X · C | C rational curve , [ C ] ∈ R } . Note that the cone theorem yields the bound 0 < length( R ) ≤ n + 1. We will alsouse the notation length( R ) = ℓ ( R ). We say that a rational curve C generating anextremal ray R = R + [ C ] is a minimal curve if ℓ ( R ) = − K X · C .The following useful inequality is inspired by Mori’s bend and break (cf. [ ,Theorem 0.4], and also [ , Theorem 1.1], [ , Corollary IV.2.6]). Theorem 2.7.
Let X be a projective manifold of dimension n . Assume that K X isnot nef and let R be an extremal ray on X of length ℓ ( R ) . Let ρ be the contraction of R and let E be any irreducible component of the locus of R . Let ∆ be any irreduciblecomponent of any fiber of the restriction, ρ E , of ρ to E . Then dim E + dim ∆ ≥ n + ℓ ( R ) − . By combining the theorem above with a result due to Ando [ ], Wi´sniewski[ , Theorem (1.2)] showed the following, which plays an important role in thesequel. Theorem 2.8. (Ando [ ], Wi´sniewski [ ]) Let X be a projective manifold ofdimension n ≥ . Assume that K X is not nef. Let ϕ : X → Z be the contractionmorphism of an extremal ray R . If every fiber of ϕ has dimension at most one, then Z is smooth and either ϕ is the blowing-up of a smooth codimension two subvarietyof Z , or ϕ is a conic fibration. Ando [ , (3.10), (2.3)] proved the theorem above assuming that the locus E of R satisfies the condition dim E ≥ n −
1. From the inequality of Theorem 2.7 itfollows that this is the case. Indeed, if dim E ≤ n −
2, for any irreducible component∆ of any fiber of the restriction of ϕ to E , we would havedim ∆ ≥ ℓ ( R ) + 1 ≥ , M.C. BELTRAMETTI AND P. IONESCU contradicting the fibers dimension assumption.
We follow the notation in [ ], to whichwe refer for details; see also [ ]. Let X be a projective manifold. By Hom bir ( P , X )we denote the scheme parameterizing morphisms from P to X which are birationalonto their image. We will denote by [ f ] the point of Hom bir ( P , X ) determined bysuch a morphism f : P → X .A reduced, irreducible subvariety V ⊂ Hom bir ( P , X ) determines a family ofrational curves on X . We let F be the universal family, restricted to V , with p : F → V and q : F → X the natural projections. We call the image of q the locus of the family , denoted by Locus( V ). A covering family is a family satisfyingLocus( V ) = X . We say that a family V , closed under the action of Aut( P ), is unsplit if the image of V in Chow( X ) under the natural morphism [ f ] → [ f ( P )] isclosed. In general, the closure of the image of V in Chow( X ) determines a familyof rational 1-cycles on X . If x ∈ X is a fixed (closed) point, we denote by V x theclosed subfamily of V consisting of morphisms sending a fixed point O ∈ P to x .We say that V is locally unsplit if, for x ∈ Locus( V ) a general point, the family V x isunsplit. A family V of rational 1-cycles on X is quasi-unsplit if any two irreduciblecomponents of cycles in V are numerically proportional. Such families typicallyarise from cycles belonging to an extremal ray.
3. General results
We discuss throughout this section some general results on extending mor-phisms p : Y → Z from ample (smooth) divisors Y of a manifold X .To begin with, let us prove two early theorems due to Sommese [ ] (see also[ , (5.2.1), (5.2.5)]) that marked the starting point of the subject. The first oneshows that the morphism p is always extendable whenever dim Y − dim Z ≥
2. Thesecond one gives the restriction that dim X ≥ Z for a smooth p : Y → Z toextend. Theorem 3.1. (Sommese [ ]) Let Y be a smooth ample divisor on a projectivemanifold X . Let p : Y → Z be a surjective morphism. If dim Y − dim Z ≥ , then p extends to a surjective morphism p : X → Z . Proof.
Let dim X =: n . Without loss of generality it can be assumed thatdim Z ≥
1. Thus we have that n ≥ X ) ∼ = Pic( Y ). Moreover,by Remmert–Stein factorizing p it can be assumed that Z is normal and p hasconnected fibers.Let L be a very ample line bundle on Z . Since Pic( X ) ∼ = Pic( Y ) there exists an H ∈ Pic( X ) whose restriction, H Y , to Y is isomorphic to p ∗ L . Let L := O X ( Y ).We claim that(3.1) H ( Y, ( H − tL ) Y ) = 0 for t ≥ . By Serre duality we are reduced to showing H n − ( Y, K Y + ( tL − H ) Y ) = 0 for t ≥ VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 9
From the relative form of the Kodaira vanishing theorem (see e.g., [
38, 22, 66 ])we see that p ( j ) ( K Y + ( tL − H ) Y ) = p ( j ) ( K Y + tL Y ) ⊗ ( −L ) = 0for j ≥
1. Using the Leray spectral sequence we deduce that H n − ( Y, K Y + ( tL − H ) Y ) = H n − ( Z, p ∗ ( K Y + ( tL − H ) Y )) . The last group is zero, since n − Y − > dim Z by our assumption. Thisshows (3.1).Now consider the exact sequence0 → K X ⊗ ( L − H ) ⊗ ( tL ) → K X ⊗ ( L − H ) ⊗ ( t +1) L → K Y ⊗ ( L − H ) Y ⊗ ( tL Y ) → . By (3.1) we have H ( Y, ( H − L ) Y − tL Y ) = 0 for t ≥
0. Therefore, by Serre duality, H n − ( Y, K Y ⊗ ( L − H ) Y ⊗ ( tL Y )) = 0 for t ≥
0. Thus the exact sequence above givesan injection of H n − ( X, K X ⊗ ( L − H ) ⊗ ( tL )) into H n − ( X, K X ⊗ ( L − H ) ⊗ ( t +1) L ).By Serre’s vanishing theorem, H n − ( X, K X ⊗ ( L − H ) ⊗ ( t + 1) L ) = 0 for t ≫ H n − ( X, K X ⊗ ( L − H ) ⊗ ( tL )) = H ( X, H − ( t + 1) L ) = 0 for t ≥ . Hence in particular H ( X, H − L ) = 0, so that we have a surjection H ( X, H ) → H ( Y, H Y ) → . Since H Y ∼ = p ∗ L with L very ample on Z , we infer that there exist dim Z + 1divisors D , . . . , D dim Z +1 in | H | such that D ∩ · · · ∩ D dim Z +1 ∩ Y = ∅ . Since Y isample it thus follows that dim( ∩ dim Z +1 i =1 D i ) ≤
0. We claim that H is spanned byits global sections. If ∩ dim Z +1 i =1 D i = ∅ , then Bs | H | = ∅ . If ∩ dim Z +1 i =1 D i = ∅ , thendim( D ∩ · · · ∩ D dim Z +1 ) ≥ dim X − dim Z − ≥ . This contradicts the above inequality.Let p : X → P N be the map associated to H ( X, H ). Ampleness of Y yieldsthat p ( X ) = p ( Y ), so p | Y = p and we are done. Q.E.D. Theorem 3.2. (Sommese [ ]) Let Y be a smooth ample divisor on a projectivemanifold, X . Let p : Y → Z be a morphism of maximal rank onto a normal variety, Z . If p extends to a morphism p : X → Z , then dim X ≥ Z . Proof.
We follow the topological argument from [ , Proposition V]. Let S be the image of the set of points where p is not of maximal rank. By ampleness of Y the set S is finite. Let f = p − ( z ), F = p − ( z ) be the fibers of p , p over z ∈ Z \ S respectively. Let dim Z = b and let r = dim Y − dim Z . From standard results intopology we deduce: π j ( Y, f ) ∼ = π j ( Z ) for all j , π j ( X, F ) ∼ = π j ( Z ) for j ≤ b − π b − ( X, F ) → π b − ( Z ) is onto. It follows that π j ( Y, f ) → π j ( X, F ) is anisomorphism for j ≤ b − j = 2 b −
1. From Theorem 2.2, wehave that π j ( Y ) → π j ( X ) is an isomorphism if j < dim Y and is onto for j = dim Y .Consider the following commutative diagram with exact rows: · · · → π j ( f ) → π j ( Y ) → π j ( Y, f ) → π j − ( f ) → · · ·↓ ↓ ↓ ↓· · · → π j ( F ) → π j ( X ) → π j ( X, F ) → π j − ( F ) → · · · Arguing by contradiction, let us assume dim
X < Z , or r < b −
1. Itfollows that 2 r + 2 ≤ r + b = dim Y and 2 r + 2 < b −
1. This, the above, and the five lemma show that π j ( f ) ∼ = π j ( F ) for j < r + 2 and π r +2 ( f ) → π r +2 ( F ) isonto.By Whitehead’s generalization of Hurewicz’s theorem [ , Theorem 9, p. 399]we get H j ( f, Z ) ∼ = H j ( F, Z ) for j < r + 2 and a surjection H r +2 ( f, Z ) → H r +2 ( F, Z ) → . By noting that 2 r + 2 = 2(dim f + 1) = dim R F , this leads to the contradiction H r +2 ( F, Z ) = 0. Q.E.D. Remark 3.3.
In the situation of Theorem 3.2, if dim X = 2 dim Z , a more refinedargument based on results of Lanteri and Struppa [ ] shows that the general fiber( F, O F ( Y )) of p is isomorphic to ( P dim F , O P dim F (1)). We refer to [ , (5.2.6),(2.3.9)] for more details and complete references.From now on, • we are reduced to consider the problem of extending morphisms Y → Z from ample divisors Y of a manifold X in the hardest case when dim Y − dim Z ≤ Y and X and give results on extending contractions of extremal rays (cf. [ ]).As noted in [ , Section 3], the following useful fact holds true. It is an easyconsequence of Theorem 2.8 and a lemma due to Koll´ar [ ]. Proposition 3.4.
Let X be a projective manifold of dimension ≥ . Assume that K X is not nef and let R = R + [ C ] be an extremal ray on X . Let Y be a smoothample divisor on X . If ( K X + Y ) · C ≤ , then R ⊂ N E ( Y ) . Proof.
By the Lefschetz theorem, the embedding i : Y ֒ → X gives an iso-morphism N ( Y ) ∼ = N ( X ), under which we get a natural inclusion i ∗ : N E ( Y ) ֒ → N E ( X ).Let ϕ : X → Z be the contraction of the extremal ray R and let E be the locuswhere ϕ is not an isomorphism, i.e., the locus of curves whose numerical class isin R . If there is a fiber F ⊂ X of ϕ whose dimension is at least two, then Y ∩ F contains a curve γ which generates R in N E ( X ), and hence R ⊂ N E ( Y ). Thus wecan assume that every fiber of ϕ has dimension at most one, so that Theorem 2.8applies. Therefore we are done after showing that in each case of 2.8 the divisor Y contains a fiber of ϕ .In the birational case, E is a P -bundle over ϕ ( E ). Let F ∼ = P be a fiber ofthe bundle projection E → ϕ ( E ). Then − K X · F = 1, so that ( K X + Y ) · F ≤ Y give Y · F = 1. Therefore Lemma 3.6(i) below leads to thecontradiction dim ϕ ( E ) ≤
1, so dim X ≤
3. In the conic fibration case, for any fiber F of ϕ , we have − F · K X ≤
2, and hence we get 1 ≤ Y · F ≤
2. Thus Lemma 3.6(ii)gives the contradiction dim Z ≤
2. Q.E.D.Let us point out the following consequence of Proposition 3.4 (cf. Section 8).
Proposition 3.5.
Let X be a projective manifold of dimension n ≥ , let H be anample line bundle on X , and let Y be an effective smooth divisor in | H | . Assumethat − ( K X + H ) is nef. Then X is a Fano manifold and N E ( X ) ∼ = N E ( Y ) . VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 11
Proof.
Let D := − ( K X + H ), which is nef. Then − K X = H + D is ample, sothat X is a Fano manifold. Let R = R + [ C ] be an extremal ray in the polyhedralcone N E ( X ). By assumption, ( K X + H ) · C ≤
0. Then Proposition 3.4 applies togive that R is contained in N E ( Y ). So N E ( X ) = N E ( Y ). Q.E.D. Lemma 3.6. (Koll´ar [ ]) Let X be a projective manifold. (i) Let p : X → Z be a P -bundle over a normal projective variety Z . Let Y ⊂ X be a divisor such that the restriction p : Y → Z is finite of degreeone. If Y is ample then dim Z ≤ . (ii) Let p : X → Z be a conic fibration over a normal projective variety Z .Let Y ⊂ X be a divisor such that the restriction p : Y → Z is finite ofdegree two ( or one ) . If Y is ample then dim Z ≤ . Proof. (i) Note that p ∗ O X ( Y ) is an ample rank 2 vector bundle since Y isample. On the other hand, the section O X → O X ( Y ) gives an extension0 → O Z → p ∗ O X ( Y ) → L → , where L is a line bundle. Thus c ( p ∗ O X ( Y )) = 0, which contradicts ampleness fordim Y ≥ p : X → Z has only smooth fibers, then, after a finite base change Z ′ → Z , we get a P -bundle p ′ : X ′ → Z ′ . Now, p ′∗ O X ′ ( Y ′ ) has rank 3 and we getan extension 0 → O Z ′ → p ′∗ O X ′ ( Y ′ ) → E → , where E is a rank 2 vector bundle. Thus c ( p ′∗ O X ′ ( Y ′ )) = 0, which contradictsampleness for dim Z ≥ p : X → Z has singular fibers, then let p ′ : X ′ → Z ′ be the universal familyof lines in the fibers. The pull-back of Y to X ′ intersects every line once and it isample. Moreover, dim Z ′ = dim Z −
1, thus we are done by (i). Q.E.D.
Corollary 3.7.
Let X be a projective manifold of dimension ≥ . Let Y be a smoothample divisor on X . Assume that K Y is not nef and let R Y be an extremal ray on Y . Further assume that there exists a nef divisor D on X such that D · R Y = 0 and such that D ⊥ is contained in the set { ζ ∈ N E ( X ) | ( K X + Y ) · ζ ≤ } . Thenthere exists an extremal ray R ⊂ N E ( X ) which induces R Y . Proof.
Note that D ⊥ is a locally polyhedral face of N E ( X ) since { ζ ∈ N E ( X ) | ( K X + Y ) · ζ ≤ } ⊂ N E ( X ) K X < ∪ { } . Let D ⊥ = h R , . . . , R s i , R i extremal rays on X , i = 1 , . . . , s . We can assume R Y = R i for each i sinceotherwise we are done. Since D · R Y = 0, it follows that, for some s ′ ≤ s , R Y ⊂ h R , . . . , R s ′ i ⊆ { ζ ∈ N E ( X ) | ( K X + Y ) · ζ ≤ } . Therefore R i ⊂ N E ( Y ), i = 1 , . . . , s ′ , by Proposition 3.4. Recalling that R Y = R i ,we thus conclude that R Y is not extremal on Y , a contradiction. Q.E.D.The following numerical invariant was introduced in [ ]. Let X be a projectivemanifold of dimension n ≥
3. Let Y be a smooth ample divisor on X . Let R be anextremal ray on Y and let H be an ample divisor on X . Then define α H ( R ) := H · Cℓ ( R ) , C minimal rational curve generating R, and α H ( Y ) := min { α H ( R ) | R extremal ray on Y } . The following slightly improves the main result of [ ]. Theorem 3.8.
Let X be a projective manifold of dimension n ≥ . Let Y be asmooth ample divisor on X . Assume that K Y is not nef and let R be an extremalray on Y . Let p := cont R : Y → Z be the contraction of R . The following conditionsare equivalent: (i) There exists an extremal ray R on X which induces R on Y ; (ii) There exists a nef divisor D on X such that R = D ⊥ Y ∩ ( N E ( Y ) \ { } ) ; (iii) p is extendable; (iv) p has a lifting; (v) There exists an ample line bundle H on X such that α H ( R ) = α H ( Y ) . Proof. (i) = ⇒ (ii) We have R = D ⊥ ∩ ( N E ( X ) \ { } ) for some nef divisor D on X . Restricting to Y , gives R = D ⊥ Y ∩ ( N E ( Y ) \ { } ).(ii) = ⇒ (iii) There exists an extremal ray R ⊆ D ⊥ ∩ N E ( X ) K X + Y < . Thenby Proposition 3.4, R ⊂ N E ( Y ), that is, R induces R on Y . Replacing D by someother nef divisor on X , we may assume that R = D ⊥ ∩ ( N E ( X ) \ { } ). It followsthat m D − ( K X + Y ) is ample for m ≫ R is given by | m D | . By Kodairavanishing we have H ( X, m D − Y ) = 0, showing that cont R extends cont R .(iii) = ⇒ (iv) is obvious.(iv) = ⇒ (i) The morphism p : X → W which lifts p is associated to a completelinear system | L | , for some nef line bundle L on X . Clearly, R ⊂ L ⊥ and R · ( K X + Y ) = R · K Y <
0. Hence, in particular, L ⊥ ∩ { ζ ∈ N E ( X ) | ( K X + Y ) · ζ ≤ } 6 =(0). Therefore, since { ζ ∈ N E ( X ) | ( K X + Y ) · ζ ≤ } is locally polyhedral in N E ( X ) (see also the proof of 3.7), there exists an extremal ray R on X such that R ⊂ L ⊥ ∩ { ζ ∈ N E ( X ) | ( K X + Y ) · ζ ≤ } . We also know by Proposition 3.4that R ⊂ L ⊥ ∩ ( N E ( Y ) \ { } ). It thus follows that p contracts R . Since p is thecontraction of an extremal ray, R , we conclude that R = R in N E ( Y ).(i) = ⇒ (v) As noted in the proof of (ii)= ⇒ (iii), if R = D ⊥ ∩ ( N E ( X ) \ { } ), wehave that m D − ( K X + Y ) := H is ample for m ≫
0. Nefness of D yields for eachextremal ray R ′ = R + [ C ′ ] on Y , C ′ minimal curve, ( K Y + H Y ) · C ′ ≥
0. Moreover,since R ⊂ D ⊥ , we get 1 = α H ( R ) = H · Cℓ ( R ) ≤ H · C ′ ℓ ( R ′ ) = α H ( R ′ ). This means that α H ( R ) = α H ( Y ) is minimal.Finally, to show (v) = ⇒ (i), let R := R + [ C ], C minimal rational curve. Let a = ( H · C ) and let b = ℓ ( R ). Assume that α H ( R ) = α H ( Y ). Then aK Y + bH Y isnef on Y , since otherwise ( aK Y + bH Y ) · C ′ < R ′ = R + [ C ′ ]on Y . This would give ab = ( H · C ) ℓ ( R ) > ( H · C ′ ) ℓ ( R ′ ) , contradicting the minimality of α H ( R ).We claim that D := a ( K X + Y ) + bH is nef on X . Assuming the contrary, byMori’s cone theorem there exists an extremal ray R ′ = R + [ C ′ ], such that ( a ( K X + Y ) + bH ) · C ′ <
0. Let ρ be the contraction of R ′ . If there is a fiber F of ρ ofdimension ≥
2, then some curve γ ⊂ F is contained in Y by the ampleness of Y .Since ( a ( K X + Y ) + bH ) · γ <
0, this contradicts the nefness of the restriction of a ( K X + Y ) + bH to Y . Thus we conclude that all fibers of ρ are of dimension ≤ X is either a conic fibration, or the blowing-up of asmooth codimension two subvariety. In this latter case one has K X · C ′ = −
1, whichcontradicts ( a ( K X + Y ) + bH ) · C ′ <
0. In the first case one has either K X · C ′ = − VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 13 or K X · C ′ = − C ′ is a conic or a line. If K X · C ′ = − K X · C ′ = − Y · C ′ = 1 by usingagain the above inequality. It thus follows that X is a P -bundle over a smoothvariety W and Y is a smooth section. By pushing forward the exact sequence0 → O X → O X ( Y ) → O Y ( Y ) → , we get an exact sequence 0 → O W → E → L → , where E is an ample rank 2 vector bundle such that X = P ( E ) and L ∼ = N Y/X is anample line bundle. By the Kodaira vanishing theorem we have H ( W, − L ) = 0, sothat the sequence splits. This contradicts the ampleness of E . Thus we concludethat a ( K X + Y ) + bH = D is nef.Now, note that D · R = 0 and that D ⊥ is strictly contained in N E ( X ) K X + Y < .Thus Corollary 3.7 applies to say that there exists an extremal ray R ⊂ N E ( X )which induces R . Q.E.D. Corollary 3.9.
Let X be a projective manifold of dimension n ≥ . Let Y be asmooth ample divisor on X . Assume that K Y is not nef. Then there exists anextremal ray on Y which extends to an extremal ray on X . In particular, when Y has only one extremal ray, it always extends. Proof.
For instance, take any extremal ray R of Y attaining the minimalvalue of the invariant α Y ( R ) (i.e., α Y ( R ) = α Y ( Y )). Q.E.D.The following theorem is a version of a result of Occhetta [ , Proposition 5](who states it in the more general case when Y is the zero locus of a section ofan ample vector bundle E on X of the expected dimension dim X − rank E ). Hisargument contains an unclear critical point. With notation as in the proof below,the conclusion in [ ] uses in an essential way the fact that D is an adjoint divisor,i.e., D = aK X + L for some integer a > ample line bundle L on X (implying that D ⊥ is strictly contained in N E ( X ) K X < ). However, we only knowthat L Y is ample! See also Remark 3.12 below. Theorem 3.10. (cf. Occhetta [ ]) Let X be a projective manifold of dimension ≥ . Let Y be a smooth ample divisor on X . Assume that K Y is not nef and let R be an extremal ray on Y . Let C ⊂ Y be a rational curve whose numerical class isin R . Assume that the deformations of C in X yield a covering and quasi-unsplitfamily of rational cycles. Then R is an extremal ray of X , too. In particular, theconclusion holds if the deformations of C in Y cover Y and H · C = 1 for someample divisor H on X . Proof.
We show first the last claim. Let ν : P → C be the normalizationof C and let g : P → Y and f : P → X be the induced morphisms to Y and X respectively. If C yields a covering family of Y , its deformations in X cover X , too.Indeed, consider the tangent bundle sequence0 → T Y → T X | Y → O Y ( Y ) → . By pulling back to P , we get the exact sequence0 → g ∗ T Y → g ∗ ( T X | Y ) = f ∗ T X → g ∗ O Y ( Y ) → . Since both g ∗ O Y ( Y ) and g ∗ T Y are nef and hence spanned, we conclude that f ∗ T X isspanned, and therefore that C induces a covering family on X , see [ , II, Section 3,IV, (1.9)]. Moreover, the condition H · C = 1 ensures that the deformations of C in X yield an unsplit, hence also quasi-unsplit, family.Let D Y ∈ Pic( Y ) be some (nef) supporting divisor of R , i.e., R = D ⊥ Y ∩ ( N E ( Y ) \ { } ). By the Lefschetz theorem, there exists a divisor D ∈ Pic( X ) whichrestricts to D Y . Let V be a covering and quasi-unsplit family of rational cycles on X , containing C . Claim 3.11. D is nef. Proof.
Assume that D · Γ < X . Clearly, wecan assume that Γ is not contained in Y since the restriction of D to Y is nef. Since V is a covering family and Y is ample, we can find a curve B ′ in V parameterizingcurves meeting both Γ and Y . Let B be the normalization of B ′ . Consider thebase-change diagram e S / / π (cid:31) (cid:31) >>>>>>>> ψ & & S / / (cid:15) (cid:15) F q / / p (cid:15) (cid:15) XB / / V where F is the universal family and e S is a desingularization of S , an irreduciblecomponent of p − ( B ), whose locus contains Γ. Note that e S is a ruled surface overthe curve B . Let A := ψ ( e S ) ∩ Y be the trace on Y of the image in X of thesurface e S . Since A is an ample divisor on ψ ( e S ), there exists at least one irreduciblecomponent, say C , of A which is not contracted by cont R : Y → Z . Let e Γ, e C betwo irreducible curves on e S such that ψ ( e Γ) = Γ, ψ ( e C ) = C . By the above and thehypothesis that V is quasi-unsplit, e C is not a fiber of π : e S → B .We can write, for some integers ε, δ i , ε > , e C ∼ εC + P i δ i F i , where C is asection of π and each F i is contained in a fiber of π . We also have e Γ ∼ αC + P i β i F i ,for some integers α , β i , α >
0. Thus ε e Γ ∼ αεC + ε X i β i F i ∼ α e C − X i ( αδ i − εβ i ) F i , that is, in Pic( e S ) ⊗ Q , one has e Γ ∼ a e C + P i b i F i , with a = αε > e D := ψ ∗ D . Since V is quasi-unsplit, e D · F i = D · R = 0, so that D · Γ = e D · e Γ = e D · (cid:18) a e C + X i b i F i (cid:19) = a ( e D · e C ) e S = a ( D · C ) ≥ D Y is nef. This shows the claim.To conclude we have to show that R is an extremal ray on X ; see also the proofof [ , Theorem (5.1)]. By Lefschetz’s theorem, the embedding i : Y ֒ → X gives anatural inclusion i ∗ : N E ( Y ) ֒ → N E ( X ). Clearly, R := i ∗ ( R ) is K X -negative bythe adjunction formula.Since R is an extremal ray of N E ( Y ), by duality it corresponds to it an extremalface of maximal dimension ̺ ( Y ) − Y ). Therefore wecan find ̺ ( Y ) − R whose numerical classes are linearly VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 15 independent in N ( Y ). By Claim 3.11, this implies that such good supportingdivisors extend to divisors on X that are nef, trivial on R , and whose numericalclasses are linearly independent in N ( X ). Since there are ̺ ( Y ) − ̺ ( Y ) = ̺ ( X ) by the isomorphism N ( X ) ∼ = N ( Y ), this implies that R is anextremal ray of N E ( X ). Q.E.D. Remark 3.12.
In [ , Proposition 5], the author states the result assuming that R is nef. However, the theorem also applies to non-nef extremal rays of Y , seeProposition 5.13 below. Note that, even when R = R + [ C ] is nef, in general C doesnot define a covering family of Y . E.g., take cont R to be a conic fibration, C beinga line in a degenerate fiber.
4. Some convex geometry speculations
First, we recall the following simple observations, due to B˘adescu.
Lemma 4.1. ([ , Remark 1), p. 170]) On the projective line P , consider a linebundle O P ( a ) , for some integer a ≥ . Write a = b + c , with b, c > . Then thereexists a surjective map O P ( b ) ⊕ O P ( c ) → O P ( a ) → . Proof.
Consider the global sections H ( P , O P ( a )) = h u a , u a − v, . . . , v a i as homogeneous polynomials in two variables u , v . We have natural inclusions H ( P , O P ( b )) ⊂ H ( P , O P ( a )) and H ( P , O P ( c )) ⊂ H ( P , O P ( a )) , given by multiplication by u a − b and v a − c respectively. Then we get a surjectivemap H ( P , O P ( b )) ⊕ H ( P , O P ( c )) → H ( P , O P ( a )) → , and injections 0 → O P ( b ) β → O P ( a ) , → O P ( c ) γ → O P ( a ) . Thus β ⊕ γ : O P ( b ) ⊕ O P ( c ) → O P ( a ) gives the requested map. The surjectivityfollows from the commutative square H ( P , O P ( b )) ⊕ H ( P , O P ( c )) ev x (cid:15) (cid:15) / / H ( P , O P ( a )) ev x (cid:15) (cid:15) ( O P ( b ) ⊕ O P ( c )) x / / ( O P ( a )) x where the vertical arrows are the evaluation maps in a given point x ∈ P and ev x : H ( P , O P ( a )) → ( O P ( a )) x is onto by spannedness of O P ( a ). Q.E.D. Proposition 4.2. ([ ]) Given a vector bundle E = L n − i =1 O P ( a i ) , with a ≥ , a = b + c , and b, c > , there exists an exact sequence (4.1) 0 → O P → F := O P ( b ) ⊕ O P ( c ) ⊕ (cid:16) n − L i =2 O P ( a i ) (cid:17) → E → . Proof.
Lemma 4.1 yields a surjective map O P ( b ) ⊕ O P ( c ) ⊕ (cid:16) n − L i =2 O P ( a i ) (cid:17) → E → , whose kernel is the trivial bundle since det( E ) = det( F ). Q.E.D. Remark 4.3.
Note that Proposition 4.2 gives rise to a method to construct ampledivisors which are projective bundles over P . Indeed, let Y := P ( E ) and X := P ( F ).As soon as a i > i = 2 , . . . , n −
1, the exact sequence (4.1) expresses Y as a smooth ample divisor of X ; it is recovered by pushing forward the exactsequence 0 → O X → O X ( Y ) → O Y ( Y ) → . The following fact is well known. We include the proof for reader’s convenience.
Lemma 4.4.
Let V = P ( E ) be a P n − -bundle over P , for some rank n vectorbundle E = L ni =1 O P ( a i ) . Assume that V is a Fano manifold, of index i ( V ) .Then, for some integer a , either (i) V = P n − × P , i ( V ) ≤ and E = L ni =1 O P ( a ) ; or (ii) V is the blowing-up, σ : V → P n , along a codimension two linear subspaceof P n , i ( V ) = 1 and E = L n − i =1 O P ( a ) ⊕ O P ( a + 1) . Proof.
After normalization of the integers a i , write 0 = a ≤ a ≤ · · · ≤ a n and consider the section Γ of p : V → P corresponding to the quotient E → O P → ϕ | ξ V | : V → P m maps the curve Γto a point. On the other hand, since V is a Fano manifold with Pic( V ) ∼ = Z ⊕ Z , thereare two extremal rays, R , corresponding to p , and R , generating the cone N E ( V ).Since the morphism ϕ | ξ V | is not finite, it must coincide with the contraction of R .Now, setting d := P ni =1 a i , the canonical bundle formula yields − K V ∼ = p ∗ O P (2 − d ) ⊗ O V ( n ) . Therefore, dotting with Γ, we get d <
2, so that either
E ∼ = L ni =1 O P , or E ∼ = O P ⊕ · · · ⊕ O P ⊕ O P (1), leading to the two cases as in the statement. Note thatby the canonical bundle formula, in the first case the index of V is i ( V ) ≤
2, while,in the second case, i ( V ) = 1. Q.E.D. Examples 4.5. (Only known examples of non-extendable extremal rays). Let X be a projective manifold of dimension n ≥
4. Let Y be a smooth ample divisor on X . The only known examples of extremal rays R of Y which do not extend to X are the following:(1) Y = P × P n − , and R is the nef extremal ray corresponding to the P -bundle projection q : Y → P n − . The manifold X is constructed as inRemark 4.3;(2) Y = P (cid:0) L n − i =1 O P ( a ) ⊕ O P ( a + 1) (cid:1) for some integer a ≥ R is the non-nef extremal ray corresponding to the blowing-up, σ : Y → P n − , along acodimension two linear subspace of P n − . Again, X is constructed as inRemark 4.3, X = P × P n − . VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 17
In case (1), the P n − -bundle projection p : Y → P on the first factor extends byconstruction. Then, if q : Y → P n − extends too, we would have a surjective map P n − → P n − , where P n − is a fiber of the extension of p ; a contradiction.Let Y be as in case (2). By Proposition 4.2, Y embeds as a smooth ampledivisor of X := P (cid:0) O P (1) ⊕ O P ( a − ⊕ O P ( a ) ⊕ ( n − ⊕ O P ( a + 1) (cid:1) . Note that N E ( Y ) = h R , R i , where R , R are the extremal rays corresponding to the bundleprojection Y → P , and to the blowing-up σ : Y → P n − respectively. Moreover,Lemma 4.4 applies to say that X is not a Fano manifold. Therefore N E ( Y ) isstrictly contained in N E ( X ). Since ̺ ( Y ) = ̺ ( X ) by the Lefschetz theorem, andthe projection Y → P extends by construction, we thus conclude that the extremalray R does not extend to X . Clearly X = P × P n − in the above example. Notethat by taking as Y a hyperplane section of the Segre embeding X of P × P n − ,the restriction to Y of the bundle projection X → P n − is in fact the blowing-up σ : Y → P n − along a codimension two linear subspace. Of course, in this case, theextremal ray defining σ extends to X . In terms of Proposition 4.2, this situationcorresponds to the case when a = 1, that is Y = P (cid:0) L n − i =1 O P (1) ⊕ O P (2) (cid:1) is anample divisor of X = P (cid:0) L ni =1 O P (1) (cid:1) .That those in 4.5 are the only known examples of non-extendable extremal rayslooks quite surprising to us. We propose the following speculations with the hopethey may eventually lead to an explanation of this fact.Let X be a projective manifold of dimension n ≥
4. Let Y be a smooth ampledivisor on X . Consider the following property, for an extremal ray, R , of Y .( ⋆ ) For any nef divisor H on Y , such that H ⊥ ∩ ( N E ( Y ) \ { } ) = R , it followsthat H is nef ( here H is the unique divisor class on X such that H Y = H ).First, note that ( ⋆ ) implies that R is a ray of X (cf. [ , Theorem 4.1] and endof proof of Theorem 3.10). Proposition 4.6.
Assume that property ( ⋆ ) holds for all extremal rays of Y suchthat R ⊂ D ⊥ for some nef divisor D on X , D . Assume, also, that ̺ ( X ) ≥ .Then every extremal ray of Y extends to an extremal ray of X . Proof.
Assume that we have some extremal ray of Y , say R , which is not aray of X . We may assume that R ⊂ N E ( X ) ( K X +(1+ ε ) Y ) < for some ε > R of Y which is contained in someface of N E ( X ) satisfies property ( ⋆ ), and hence, as noted above, R is a ray of X .Let C ◦ := ( K X + (1 + ε ) Y ) ⊥ ∩ ( N E ( Y ) \ { } ) and let R , R , . . . , R s be allthe extremal rays of N E ( Y ) such that R i ⊂ N E ( X ) ( K X +(1+ ε ) Y ) < , i = 0 , , . . . , s .Next consider the non-degenerate convex cone C ⊂ R ̺ ( Y ) defined by C = h C ◦ , R , R , . . . , R s i = N E ( Y ) ∩ N E ( X ) ( K X +(1+ ε ) Y ) ≤ . Then C ◦ is a face of the cone C and, for each index i , R i
6⊂ h R , R , . . . , R i − , R i +1 , . . . , R s i . Claim 4.7.
There exists a face F of the cone C such that F contains two extremalrays of Y , say R ′ , R ′′ , such that R ′ is a ray of X and R ′′ is not. Proof.
Recall that by Proposition 3.4 all extremal rays of X contained in N E ( X ) ( K X + Y ) ≤ are also extremal rays of Y . Therefore, some of the extremal rays R j , j ≥
1, are rays of X . Take one of them, say R . Then there exists a face F of C containing R and some of the other rays R j ′ , for some j ′ ∈ { , . . . , s } . Ifone of the rays R j ′ (say R ), is not a ray of X , then take R ′ = R , R ′′ = R and F = F . If all the extremal rays R j ′ , j ′ ∈ { , . . . , s } , are rays of X , we apply thesame argument to conclude that either we prove the claim, or every extremal ray of Y contained in C lifts to an extremal ray of X , contradicting the assumption that R does not.Thus we may assume to be in the situation described in Claim 4.7. Take a nefdivisor D on Y such that F = D ⊥ ∩ ( N E ( Y ) \ { } ). If the unique divisor class D on X which restricts to D is nef, then by our assumption R ′′ is a ray of N E ( X ).This contradicts the claim, so D is not nef on X .Now, take a nef divisor H on X such that R ′ = H ⊥ ∩ ( N E ( X ) \ { } ). For0 ≤ α ≤
1, consider D α = αD + (1 − α ) H and let λ := sup { α | D α is nef } . Then0 ≤ λ < .............................................. ....................... ....................... ...................... ..................... ..................... ...................... ....................... ....................... ....................... ....................... ...................... ....................... ....................... ....................... ...................... ...................... ..................... ...................... ....................... ....................... ................................................................................................................. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) NE ( Y ) NE ( X ) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) • ❅❅❅❅ •• ❇❇❇❇❇❇❇❇❇❇❇❇❇❇✁✁✁✁✁✁✁✁✁✁ • ❡❡❡❡❡ ( K X + (1 + ε ) Y ) · = 0 R ′ H = D D λ D µλ<µ< D = D ☎☎☎☎☎☎☎☎☎☎☎☎✟✟✟✟✟✟✟✟✟✟✟✟ • (cid:0)(cid:0)(cid:0)(cid:0) Figure 1.
The two cones of curves.
Claim 4.8.
The extremal ray R ′ does not satisfy property ( ⋆ ) . Proof.
For some λ < µ <
1, the divisor D µ is not nef and its restriction D µ | Y = µD + (1 − µ ) H Y is nef on Y . In particular one has R ′ ⊆ ( D µ | Y ) ⊥ ∩ ( N E ( Y ) \ { } ) ⊆ H ⊥ Y ∩ ( N E ( Y ) \ { } ) . Thus, since R ′ = H ⊥ ∩ ( N E ( X ) \{ } ), equalities hold above, so that R ′ = ( D µ | Y ) ⊥ ∩ ( N E ( Y ) \ { } ). Thus R ′ does not satisfy property ( ⋆ ); see Figure 1.The claim above leads to a contradiction, and hence concludes the proof of theproposition. Q.E.D. Remark 4.9. (1) The examples of non-extending extremal rays in Example 4.5were known to the experts since the early 80’s. The resulting observation that, ingeneral,
N E ( Y ) ( N E ( X ) when Y ⊂ X is ample, was rediscovered in [ ]. VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 19 (2) The (open) problem of deciding whether or not the property ( ⋆ ) holds for allextremal rays R of Y such that R ⊂ D ⊥ for some nef divisor D on X , D
0, lookshard but very interesting. A positive answer to it would imply, via Proposition 4.6,that non-extendable rays may only occur when ̺ ( X ) = 2. Moreover, a positiveanswer would give a proof for the claim made in [ , Theorem 4.3]. See alsoRemark 7.7 for further applications.
5. Applications to P d -bundles and blowing-ups Let us start by recalling some useful preliminary facts.The following result goes back to Goren [ ] and Kobayashi–Ochiai [ ]. Werefer also to Fujita [ , Chapter I, (1.1), (1.2)] where the Cohen–Macaulay assump-tion was removed. Theorem 5.1.
Let L be an ample line bundle on an n -dimensional irreducibleprojective variety X . If L n = 1 and h ( L ) ≥ n + 1 , then ( X, L ) ∼ = ( P n , O P n (1)) . Corollary 5.2. If X is a Fano manifold of dimension n and index i , we have i ≤ n + 1 and X ∼ = P n if equality holds. Proof.
Use the Hilbert polynomial and Kodaira vanishing to check the hy-pothesis of Theorem 5.1. See [ ] for details. Q.E.D. Remark 5.3.
In a completely similar way one proves that if X is as above and i = n , then X ∼ = Q n , see again [ ].The smooth version of the following fact was proved by Ramanujam [ ] andSommese [ ]. The general case is due to Bˇadescu [ ]. We also refer to [ , Sec-tion 2.6 and (5.4.10)] for a different argument based on Rossi’s extension theorem. Theorem 5.4.
Let Y ∼ = P n − be an ample Cartier divisor on a normal projectivevariety X of dimension n ≥ . Then X is the cone C ( P n , O P n ( s )) , where O Y ( Y ) ∼ = O P n − ( s ) . If s = 1 , the assertion is true for n = 2 as well. Proof.
Let us recall the argument in the smooth case.First, note that, by Lefschetz theorem, i ∗ : Pic( X ) ∼ → Pic( Y ) via the embedding i : Y ֒ → X . This is clear if n ≥
4. If n = 3, then i ∗ : Pic( X ) → Pic( Y ) is injectivewith torsion free cokernel. Therefore i ∗ is an isomorphism, since Pic( Y ) ∼ = Z .Let L be the ample generator of Pic( X ) and assume Y ∈ | aL | for some a ≥ L Y = O P n − (1). It then follows that1 = ( L Y ) n − Y = ( L n − · Y ) X = a ( L n ) . Therefore a = 1 and L n = 1.Next, Kodaira vanishing and the exact sequence0 → O X ( − L ) → O X → O Y → H ( X, O X ) = 0. Hence the exact sequence0 → O X → L → L Y → h ( L ) = h ( L Y ) + 1 = dim X + 1. Then Theorem 5.1 applies to give ( X, L ) ∼ =( P n , O P n (1)). Q.E.D. For the first statement of the theorem below, see Fujita [ , (2.12)] or Ionescu[ , p. 467]; the second point follows from Theorem 5.4 and some of B˘adescu’sarguments in [
6, 7 ]; the third point was noticed in [ , Section 2] and [ ]. Theorem 5.5.
Let X be an n -dimensional projective manifold. (i) Let π : X → Z be a surjective morphism from X onto a normal variety Z .Let L be an ample line bundle on X . Assume that ( F, L F ) ∼ = ( P d , O P d (1)) for a general fiber, F , of π and that all fibers of π are d -dimensional.Then π : X → Z is a linear P d -bundle with X = P ( π ∗ L ) . (ii) Let p : Y → Z be a P d -bundle over a projective manifold Z . Assume that Y is an ample divisor on X . Furthermore assume that p extends to amorphism p : X → Z . Then there exists a non-splitting exact sequence → O Z → E u → E → , where E , E are ample vector bundles on Z such that X = P ( E ) , Y = P ( E ) , p , p are the bundle projections on Z , and the inclusion Y ⊂ X is inducedby u . (iii) Let π : X → Z be a linear P d +1 -bundle over a projective manifold Z .Assume that dim Z < d + 1 and the tautological line bundle of X , say L ,is ample. Then K X + ( d + 2) L is nef. Moreover, the bundle projection π is associated to the linear system | m ( K X + ( d + 2) L ) | for m ≫ i.e., ( X, L ) is a scroll over Z ) . Proof. (i) Following the argument as in [ , p. 467], let us first show that Z is smooth. Indeed, let z ∈ Z be a closed point and denote by ∆ the fiber over z . Consider the embedding of X given by | mL | for m ≫
0. Let e Z be the smooth( n − d )-fold got by intersecting d general members H , . . . , H d of | mL | . Since( F, L F ) ∼ = ( P d , O P d (1)) for a general fiber F , the restriction e p of p to e Z has degree m d . Since p has equidimensional fibers, e Z ∩ ∆ is a 0-dimensional scheme, and itslength is given by ℓ ( e Z ∩ ∆) = m d ( L d ∆ ) ≥ m d . Since Z is normal, it follows by a wellknown criterion (see e.g., [ , Chapter II, Theorem 6]) that the above inequalityis in fact an equality. Hence in particular L d ∆ = 1, so that ∆ is irreducible andgenerically reduced. Therefore, by the generality of H , . . . , H d , we may assumethat e Z ∩ ∆ is a reduced 0-cycle consisting of e Z ∩ ∆) = ℓ ( e Z ∩ ∆) = deg( e p ) distinctpoints. It thus follows that e p is ´etale over z . Therefore Z is smooth at z since e Z issmooth.Since all fibers of π are equidimensional and X and Z are smooth, the morphism π is flat. Let now ∆ be any fiber of π . We have seen above that ∆ is irreducible andgenerically reduced. Moreover, ∆ is Cohen–Macaulay since every fiber is definedby exactly n − d coordinate functions. It thus follows that ∆ is in fact reduced. Bythe semicontinuity theorem for dimensions of spaces of sections on fibers of a flatmorphism [ , Chapter III, Theorem 12.8], h ( L ∆ ) ≥ h ( L F ) = d + 1. Then byTheorem 5.1 we conclude that (∆ , L ∆ ) ∼ = ( P d , O P d (1)) for every fiber ∆ of π . Then X ∼ = P ( E ), where E := π ∗ L .(ii) Note that, by ampleness of Y , p equidimensional implies p equidimensional.Let F be a general fiber of p and let f = F ∩ Y be the corresponding fiber of p . By(i), it is enough to show that F ∼ = P d +1 and L F ∼ = O P d +1 (1), where L = O X ( Y ). If d ≥ VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 21
Thus we can assume d = 1. By taking general hyperplane sections of Z andby base change, we can also assume that Z is a smooth curve (and hence F is adivisor). Here we follow B˘adescu’s argument. By the Lefschetz theorem, we havean exact sequence 0 → Num( X ) i ∗ → Num( Y ) → Coker( i ∗ ) → , where i ∗ is the morphism induced by the embedding i : Y ֒ → X and Coker( i ∗ ) istorsion free. First note that Num( X ) = Z , since otherwise F would be an ampledivisor. Since Num( Y ) ∼ = Z ⊕ Z , we thus conclude that Num( X ) ∼ = Z ⊕ Z , andtherefore that Coker( i ∗ ) = (0) since it is torsion free. Let h be a section of thebundle p : Y → Z . Then Num( Y ) = Z h f i ⊕ Z h h i and Num( X ) = Z h F i ⊕ Z h H i forsome line bundle H on X inducing h on Y . Write Y ∼ aF + bH for some integers a , b . Since h · f = H · Y · F = 1, we get 1 = b ( H · F ) and therefore b = ±
1. Then Y · f = ( aF ± H ) · f = ± ( H · f ) = ±
1. By ampleness of Y , it must be b = 1. Thus f ∼ = P has self-intersection f = Y · F = Y · f = 1 on F . This implies F ∼ = P and L F ∼ = O P (1) by using Theorem 5.4. Applying the first part of the statementand pushing forward under p the exact sequence0 → O X → O X ( Y ) → O Y ( Y ) → K X + ( d + 2) L is not nef, there exists an extremal ray R of X such that( K X + ( d + 2) L ) · R <
0. It follows that ℓ ( R ) > d + 2. Therefore, if ∆ is a positivedimensional fiber of the contraction cont R , we have dim ∆ ≥ ℓ ( R ) − ≥ d + 2 (seeTheorem 2.7). Then, for a fiber F of π , we havedim F + dim ∆ ≥ d + 3 > n + 1(where the last inequality follows from the assumption d + 1 > dim Z , which isequivalent to saying that 2 d + 2 > n ). Hence dim( F ∩ ∆) ≥
2. Thus there exists acurve C ⊂ F such that ( K X +( d +2) L ) · C <
0, contradicting ( K X +( d +2) L ) | F ≈ K X + ( d + 2) L is nef, and hence, by Theorem 2.6, thelinear system | m ( K X + ( d + 2) L ) | defines a morphism, say ϕ , for m ≫ R ⊂ ( K X + ( d + 2) L ) ⊥ ∩ ( N E ( X ) \ { } ) be an extremal ray. We have( K X + ( d + 2) L ) · R = 0, so that ℓ ( R ) ≥ d + 2, and hence, as above, dim ∆ ≥ ℓ ( R ) − ≥ d + 1.Since 2 d + 2 > n , it follows, again by Theorem 2.7, that dim F + dim ∆ ≥ d + 2 ≥ n + 1. Then dim( F ∩ ∆) ≥
1. This implies that π is the contractioncont R of the extremal ray R . Since this is true for each extremal ray as above, weconclude that the face ( K X + ( d + 2) L ) ⊥ ∩ ( N E ( X ) \ { } ) is in fact 1-dimensionaland that p coincides with the morphism ϕ . Q.E.D. Remark 5.6. (1) In the boundary case d +1 = dim Z of Theorem 5.5(iii), the sameargument gives the nefness of K X + ( d + 2) L ; moreover, further considerations showthat the bundle projection π is associated to | m ( K X + ( d + 2) L ) | for m ≫ X ∼ = P d +1 × P d +1 . We refer for this to [ , (3.1)].(2) Note that by Theorem 3.2 or Lemma 5.7 below, statement (iii) of Theo-rem 5.5 applies under the conditions in 5.5(ii).In the case of P d -bundles, Sommese’s theorem 3.2 admits the following simplealternative proof. Lemma 5.7.
Let Y be a smooth ample divisor on a projective manifold, X . Assumethat Y is a P d -bundle over a manifold Z . Further assume that p has an extension p : X → Z . Then dim Z ≤ d + 1 . Proof.
By Theorem 5.5 we get the exact sequence(5.1) 0 → O Z → E → E → , where E , E are ample vector bundles on Z . Arguing by contradiction, assumedim Z > rk E = d + 1, that is 1 ≤ dim Z − rk E . By le Potier’s vanishing theorem[ ] we have H i ( Z, E ∗ ) = 0 for i ≤ dim Z − rk E . Therefore H ( Z, E ∗ ) = 0, so weconclude that (5.1) splits, contradicting ampleness of E . Q.E.D. Lemma 5.8.
Let L be an ample line bundle on a projective manifold, X , of di-mension n ≥ . Assume that there is a smooth Y ∈ | L | such that Y is a P d -bundle, p : Y → Z , over a manifold Z . Let ℓ be a line in a fiber P d of p . Further assumethat H · ℓ = 1 for some ample line bundle H on X . Then ( X, L ) ∼ = ( P ( E ) , ξ P ) foran ample rank d + 2 vector bundle, E , on Z with p equal to the restriction to Y ofthe induced projection P ( E ) → Z . Proof.
Lines in the fibers of p define a covering family of Y . By our assump-tions, the induced family on X is unsplit. Therefore Theorem 3.10 applies to givethat p extends. We conclude by Theorem 5.5(ii). Q.E.D.The following gives a precise answer to Question 2.1. Proposition 5.9.
Let X be an n -dimensional projective manifold embedded in P N , n ≥ . (i) Assume that X has a smooth hyperplane section Y = X ∩ H which is a P d -bundle over a manifold Z , say p : Y → Z , such that the fibers of p arelinear subspaces of P N . Then p lifts to a linear P d +1 -bundle p : X → Z .Moreover, this is possible only if d + 1 ≥ dim Z . (ii) Conversely, assume that π : X → Z is a P d +1 -bundle with linear fibers and d + 1 ≥ dim Z . Then there exists a smooth hyperplane section Y = X ∩ H which is a P d -bundle. Proof. (i) If ℓ is a line contained in some fiber of p , we have H · ℓ = 1. Thusthe first assertion follows from Lemma 5.8 and the second from Theorem 3.2 (orLemma 5.7).(ii) Consider the incidence relation W := { ( z, h ) | H ⊇ F z } ⊆ Z × ( P N ) ∨ , where F z = π − ( z ) is the fiber P d +1 over a point z ∈ Z and h ∈ ( P N ) ∨ is the pointcorresponding to the hyperplane H in P N . Then dim W = dim Z + N − ( d + 1) − Z ≤ d + 1 gives dim W ≤ N −
1. Therefore there exists a hyperplane H in P N not containing fibers of π and we are done. Q.E.D.Consider the setting as in diagram (2.1) from Section 2 with dim X ≥
4. Letus discuss here some applications under the assumption that the canonical bundle K Z of Z is nef. We follow the exposition in [ , Section 4], where the results areproved for a strictly nef and big divisor Y on X . VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 23
As a first application, we consider the case when the morphism p : Y → Z is a P d -bundle. An analogous result, assuming κ ( Z ) ≥ K Z to be nef, wasproved in [ ] in a completely different way.The following result is essentially due to Wi´sniewski [ , (3.3)]. Lemma 5.10.
Let Y be a P d -bundle over a smooth projective variety and let p : Y → Z be the bundle projection. If K Z is nef, then Y admits a unique extremalray, and p is its contraction. Proof.
Assume by contradiction that there exists an extremal rational curve, C , which is not contracted by p . Let Γ ∼ = P be the normalization of p ( C ). Let f : Γ → Z be the induced morphism and consider the base change diagram Y ′ g / / p ′ (cid:15) (cid:15) Y p (cid:15) (cid:15) Γ f / / Z. Since Γ is a smooth curve, we have a vector bundle E on Γ, of rank r := d + 1, suchthat Y ′ = P ( E ). Let F ′ be the fiber of the bundle projection p ′ and let C ′ ⊂ Y ′ be a curve mapped onto C under g . Let T ′ be the tautological line bundle on Y ′ .Then we get0 > ( C · K Y ) = ( C ′ · g ∗ K Y ) = − r ( C ′ · T ′ ) + ( C ′ · ( f ◦ p ′ ) ∗ K Z ) + ( C ′ · p ′∗ det( E )) . Since K Z is nef, it thus follows r ( C ′ · T ′ ) > ( C ′ · p ′∗ det( E )). By the Grothendiecktheorem, we have E ∼ = L ri =1 O P ( a i ), where a ≥ · · · ≥ a r . Then the inequalityabove yields r ( C ′ · T ′ ) > ra r ( C ′ · F ′ ). Thus(5.2) ( C ′ · T ′ ) > a r ( C ′ · F ′ ) . Let R = R + [ C ] be the extremal ray generated by C . The composition ϕ := cont R ◦ g is a morphism defined by a linear sub-system of | α T ′ + βF ′ | , for some α >
0. Since C ′ · ( α T ′ + βF ′ ) = 0, we get α ( C ′ · T ′ ) = − β ( C ′ · F ′ ). Therefore (5.2) gives(5.3) − β > αa r . Let C r be the section of p ′ corresponding to the surjection E → O P ( a r ) →
0. Then C r · ( α T ′ + βF ′ ) ≥
0, which contradicts (5.3). Q.E.D.
Proposition 5.11.
Let p : Y → Z be a P d -bundle over a projective manifold Z .Assume that K Z is nef. If Y is an ample divisor on a manifold X , then p extendsto a morphism p : X → Z . Furthermore there exists a non-splitting exact sequence → O Z → E → E → such that X = P ( E ) , Y = P ( E ) , and p , p are the bundle projections on Z . Proof.
By Lemma 5.10, the bundle projection p is the contraction of theunique extremal ray on Y . Then, by Corollary 3.9, p extends. Thus Theorem 5.5(ii)applies to give the result. Q.E.D.Next, we consider the case when the morphism p : Y → Z is a blowing-up. Thefollowing general fact is a direct consequence of Lemma 5.10. Lemma 5.12.
Let Z be a projective manifold. Let p : Y → Z be the blowing-up along a smooth subvariety T of codimension ≥ . Assume that the canonicalbundles K T and K Z are nef. Then Y has only one extremal ray. Proof.
Let c be the codimension of T in Z , and let E be the exceptional divisorof p . Let R = R + [ C ] be any extremal ray on Y . Since K Y ≈ p ∗ K Z ⊗ O Y (( c − E ),we conclude that E · C <
0. Therefore C is contained in E and K E · C <
0. Thenapply the proof of Lemma 5.10 to the P c − -bundle E → T . Q.E.D.The following generalizes a result due to Sommese concerning the reductionmap in the case of threefolds (see [ ], [ , Theorem I], and also [
33, 34 ]) and isclosely related to Fujita’s results in [ ]. Proposition 5.13.
Let Z be a projective manifold. Assume that K Z is nef. Let p : Y → Z be the blowing-up along a smooth subvariety T of codimension c ≥ ,such that K T is nef. If Y is an ample divisor on a manifold X , then there exists acommutative diagram (2.1) , where W is a smooth projective variety and either (i) p : X → W is the blowing-up of W along the image of T . Moreover, p ( Y ) is an ample divisor on W whenever T is -dimensional; or (ii) X is generically a P -bundle over Z and Y is a rational section of it.Moreover, dim p − ( T ) = n − , c = 2 and fibers of p are at most two-dimensional. Proof.
By Lemma 5.12, Y contains a unique extremal ray and p is its con-traction. Then, by Corollary 3.9, p extends to a contraction p : X → W of anextremal ray on X , which gives rise to a commutative diagram (2.1). Assume firstthat p is birational. Let E , E be the exceptional loci of p , p respectively, so that E = E ∩ Y . As the restriction p | E : E → T of p to E is a P c − -bundle, it followsfrom Theorem 5.5(ii) that p | E : E → T is a P c -bundle. It is now standard to seethat W is smooth, Z is contained in W as a divisor and p is the blowing-up of W along T , cf. also [ , Section 5]. In [ , Section 5] it is also proved that p ( Y ) ∼ = Z is an ample divisor on W under the extra assumption that the restriction to T of the line bundle O W ( Z ) is ample. In particular, p ( Y ) is ample on W if T is0-dimensional.Now, assume that p is not birational. Then α : Z → W from (2.1) is anisomorphism and Y is a rational section for p , which is generically a P -bundle.Let t ∈ T be a general point and let l ⊂ F := p − ( t ) ∼ = P c − be a line. We put a := Y · l and we denote by f a general fiber of p . Since l is contracted by p ,it is numerically proportional to f . It follows easily that l ∼ af as 1-cycles. As K X · f = −
2, we get − a = K X · l = K Y · l − Y · l = 1 − c − a . So a = c −
1. Assumethat dim p − ( T ) = n −
1. We obtain that 0 = a ( p − ( T ) · f ) = p − ( T ) · l = − p − ( T ) = n −
2. Now, denote by V the family of alldeformations of l in X . We claim that dim Locus( V ) ≥ n −
1. Assuming thecontrary, we would have dim Locus( V ) ≤ n −
2. From the exact sequence0 → N l/Y → N l/X → O l ( a ) → , using standard facts from deformation theory of rational curves, we find thatdim V = h ( N l/X ) = n + c + a −
4. Thus, if x is a point on l ,dim V x ≥ dim V + 1 − ( n −
2) = c + a − . VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 25
But the same exact sequence givesdim V x ≤ h ( N l/X ( − ≤ c + a − . This is a contradiction and the claim is proved. Since dim p − ( T ) = n −
2, theclaim implies that some deformation of l equals a fiber of p . In particular, a = 1,so c = 2 and we are done. Q.E.D. Remark 5.14.
Let us explicitly point out that an analogous result was provedby Fujita [ , Section 5], under the assumption that codim Z T ≥
3, but with nonefness condition on K T and K Z . However, the contraction morphism X → W obtained in [ ] is in general analytic, not necessarily projective.Next, let us consider the case when Y admits a pluricanonical fibration.Recall that a Calabi–Yau manifold Y is a projective variety with trivial canon-ical bundle and H i ( Y, O Y ) = 0 for i = 1 , . . . , dim Y − Proposition 5.15.
Let X be a projective manifold of dimension ≥ . Let Y bea smooth ample divisor on X . Assume that K Y is nef. Then the linear system | m ( K X + Y ) | is base points free for m ≫ . If K Y is numerically trivial, then Y is a Calabi–Yau variety and X is a Fano manifold. If ( K Y ) k +1 is a trivial cycleand ( K Y ) k is non-trivial for < k < n − , then X is a Fano fibration over Z and k = dim Z . Proof.
The proof runs parallel to that of Theorem 3.8. Since K Y is nef and Y is ample, we conclude that K X + Y is nef. Thus m ( K X + Y ) is spanned for m ≫ π : X → W . By restricting to Y we find that | mK Y | is base points freefor m ≫
0. Then Y admits a pluricanonical map, say ϕ := ϕ | mK Y | .If ( K Y ) n − = 0, the morphism ϕ is a fibration. If K Y is numerically trivial, then K X + Y is also, and thus X is a Fano manifold. Hence in particular H i ( X, O X ) = H i ( Y, O Y ) = 0 for 0 < i < dim Y , so that Y is a Calabi–Yau manifold. Theremaining part of the statement is clear. Q.E.D.We say that a projective manifold Y is extendable if there exists a projectivemanifold X such that Y ⊂ X is an ample divisor. Proposition 5.15 shows that amanifold Y such that K Y is numerically trivial and either K Y is not linearly trivialor h ( O Y ) >
0, e.g., Y an abelian variety, is not extendable.It is worth noting that Proposition 5.15 shows that the Abundance conjecture[ ] holds true for extendable manifolds. Let us recall what the conjecture saysin the smooth case. Let Y be a projective manifold with K Y nef. Then mK Y isspanned by its global sections for m ≫ ] for a further discussion in the case when Y is a strictly nefdivisor on X .
6. Complete results in the three dimensional case
Throughout this section we assume that X is a smooth projective three foldand Y ⊂ X is a smooth ample divisor. The following theorem implies a numberof results from [
6, 7, 8, 69, 70 ] and [ ]. We follow the arguments in [ ]. Notethat [ ] contains a precise description of all types of extremal rays of X . Definition 6.1. (cf. [ ]) A reduction of the pair ( X, Y ) is another pair ( X ′ , Y ′ ),with Y ′ ⊂ X ′ a smooth ample divisor, such that X ′ is got by contracting all ( − E ∼ = P contained in X such that Y E ∈ |O E (1) | and Y ′ is just the image of Y in X ′ . Theorem 6.2.
Let
X, Y be as above and assume that K Y is not nef. Then one ofthe following holds. (i) ̺ ( X ) = 1 , X is Fano, of index ≥ and either: (a) X ∼ = P , Y ∈ |O P ( a ) | , a = 1 , , ; or (b) X ∼ = Q , Y ∈ |O Q ( a ) | , a = 1 , ; or (c) X is a del Pezzo three fold, Y ∈ |O X (1) | , cf. [ ] or [ ] for acomplete list. (ii) X is a linear P -bundle over a curve and, for each fiber F , either Y F ∈|O P (1) | or Y F ∈ |O P (2) | ; (iii) X admits a contraction of an extremal ray, ϕ : X → W , such that W is a (smooth) curve, we have F ∼ = Q for a general fiber of ϕ and Y F ∈|O Q (1) | (we call ϕ a quadric fibration ); (iv) X is a linear P -bundle over a surface and Y is a rational section; (v) A reduction ( X ′ , Y ′ ) of ( X, Y ) exists. Proof.
Since K X + Y is not nef, there exists an extremal ray R = R + [ C ] of X such that ( K X + Y ) · C <
0. Consider the length ℓ ( R ) of R and observe thatwe have ℓ ( R ) ≥
2. Let ϕ = cont R : X → W be the contraction of R and let F bea general fiber of ϕ . If dim W = 0, we fall in case (i). So, from now on, we mayassume dim W >
0. If ℓ ( R ) = 4, by Theorem 2.7, dim W = 0. If ℓ ( R ) = 3, byTheorem 2.7, W is a curve. Moreover, Y · C = 1 or 2. By Corollary 5.2, F ∼ = P .If Y · C = 1, we get case (ii), Y F ∈ |O F (1) | by Theorem 5.5. Assume now that Y · C = 2 (and ℓ ( R ) = 3). Let L := K X + 2 Y and let F be an arbitrary fiber of ϕ .Remark that L · R >
0, therefore L · C > C ⊂ F . We have that( L F ) = 1 and L F is ample by the Nakai–Moishezon criterion. By Theorem 5.1, ϕ makes X a P -bundle and Y F ∈ |O P (2) | . Thus, when ℓ ( R ) = 3 and W is a curve,we get case (ii). Next, suppose that ℓ ( R ) = 2, so Y · C = 1. If W is a curve we get K F + 2 Y F ∼ F ∼ = Q , leading to case (iii).If W is a surface, we get case (iv). Indeed, ϕ is generically a P -bundle, Y beinga rational section. So it is enough to see that all fibers of ϕ are one-dimensional.Let S be an irreducible surface contracted by ϕ and let D := S ∩ Y . We obtain S · Y = ( D Y ) < S · Y = D · S = 0 since S is contracted by ϕ . Thiscontradiction shows that ϕ is a P -bundle. Finally, assume that ϕ is birational. Forsuch a ray, it follows from Theorem 2.7 that E , the locus of R , is an irreduciblesurface, contracted to a point. Moreover, E · C := − c < R is not nef. Weget K E + ( c + 2) Y E ∼
0; as above, using suitable vanishings, (see [ ] for details)we deduce that E ∼ = P , E E ∈ |O P ( − | and Y E ∈ |O P (1) | . This leads to thereduction from case (v). Q.E.D. Corollary 6.3. ([
6, 7, 8 ]) Let ( X, Y ) be as above and assume that p : Y → B is a P -bundle. Then p extends to a linear P -bundle p : X → B , unless either X ∼ = P , Y ∈ |O P (2) | , or X ∼ = Q , Y ∈ |O Q (1) | , or Y ∼ = P × P , p is one of the projectionsand the other projection extends. VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 27
Proof.
Assume first that ̺ ( X ) = 1. The conclusion follows by looking at thelist in Theorem 6.2(i), using the classification of del Pezzo three folds. Next, suppose ̺ ( X ) >
1. By Lefschetz’s theorem, we get an isomorphism Num( X ) ∼ = Num( Y ).Then Corollary 3.9 applies to give that some extremal ray of Y extends to X . Ifthe genus g ( B ) >
0, such a ray is unique and its contraction, p , extends. ApplyTheorem 5.5 to conclude. Assume g ( B ) = 0. Unless either Y ∼ = P × P or Y ∼ = F , Y has only one extremal ray, so the previous argument applies. To conclude, weonly have to examine the case when the contraction of the ( −
1) curve of F , say π : F → P , extends to a morphism π : X → W . Case π : X → W is the contraction of a ( −
1) plane E . The diagram Y π (cid:15) (cid:15) (cid:31) (cid:127) / / X π (cid:15) (cid:15) P ∼ = Z (cid:31) (cid:127) / / W shows that W ∼ = P , Z ∈ |O P (1) | (see Theorem 5.4). Let L := π ∗ O P (1). We get Y = ( L − E ) = 0, contradicting ampleness of Y . Case π : X → P is a P -bundle, Y is a rational section (see the argumentfrom the proof of Theorem 6.2, case (iv)). Let f be a fiber of p and let C be the( −
1) curve contracted by π . Write Y Y ≈ aC + bf , for some a >
0. From Y · C = 1it follows b = a + 1. Let L := π ∗ ( l ), l ⊂ P a line. Since Pic( X ) ∼ = Pic( Y ), thereis some F ∈ Pic( X ) such that O Y ( F ) ∼ = O Y ( f ). We find easily that Y ≈ F + aL .Now consider the exact sequence0 → − aL → O X ( F ) → O Y ( F ) → . We have H ( X, − aL ) = H ( P , − al ) = 0. Therefore, using also the ampleness of Y , it follows that the linear system | F | gives a morphism p : X → P which extends p . Clearly, p is a P -bundle and, in fact, X ∼ = P × P . Q.E.D.A classification of all cases when Y is birationally ruled also follows from The-orem 6.2. Corollary 6.4. (cf. [
69, 70 ]) Let ( X, Y ) be as above and assume that Y is notbirationally ruled. Then, either X is a (linear) P -bundle and Y is a rationalsection, or there is a reduction ( X , Y ) such that K Y is nef. Proof.
Looking over the cases (i)–(v) in Theorem 6.2 and using the hypothesisthat Y is not ruled, we see that only cases (iv) and (v) are possible. Q.E.D. Corollary 6.5. ([ ]) Let ( X, Y ) be as above. Assume that X is not a P -bundle, Y being a rational section. (i) If κ ( Y ) = 0 , there is a reduction ( X , Y ) of ( X, Y ) such that Y is a K surface and X is Fano. (ii) If κ ( Y ) = 1 , there is a reduction ( X , Y ) of ( X, Y ) such that X fibersover a curve, with general fiber a del Pezzo surface. Proof.
Use the preceding corollary and Proposition 5.15. Q.E.D.
7. Extending P -bundles We start with the following proposition.
Proposition 7.1.
Let X be a projective manifold of dimension n ≥ . Let Y be asmooth ample divisor on X . Assume that Y is a conic fibration, with general fiber f . Let V be the family of rational curves induced by f on X . Then the followingconditions are equivalent: (i) Y · f = 1 ; (ii) V is unsplit; (iii) V is locally unsplit.If (i) – (iii) hold, then p is a P -bundle which extends to p : X → Z . Moreover, dim Z = 2 and p is a P -bundle. Conversely, if p is smooth and extends, conditions (i) – (iii) hold. Proof.
Since (i) = ⇒ (ii) = ⇒ (iii) are clear, it is enough to show (iii) = ⇒ (i).Let a := Y · f , let y ∈ Y be a fixed general point and consider the standard exactsequence 0 → n − L O f ( − → N f/X ( − → O f ( a − → . Since h ( N f/X ( − V y = h ( N f/X ( − a , and hence dim F y = a + 1, see 2.5. Bysemicontinuity, the same holds at a general point x ∈ X . Fix such a generalpoint x ∈ X and take another point t ∈ Locus( V x ). Since V is locally unsplit, weknow that each curve from V x is irreducible. By the non-breaking lemma, we thusconclude that there is a finite number of curves in V x passing through t . That is theprojection q : F x → Locus( V x ) is a finite map. Therefore dim Locus( V x ) = a + 1.Thus we obtain dim Y ∩ Locus( V x ) ≥ a . Assume by contradiction that a ≥
2. Thenthere exists a curve C ⊂ Y ∩ Locus( V x ) such that p ( C ) is a curve in Z . In this case,a variant of the non-breaking lemma (see [ , (1.14)] and also [ , (1.4.5)]) impliesthat the curve C is numerically equivalent in X to λf for some positive rationalnumber λ . Now, take a hyperplane section H Z of Z and let L ∈
Pic( X ) be theextension of p ∗ ( H Z ) on X via the isomorphism Pic( X ) ∼ = Pic( Y ). In particular, L · f = 0, this leading to the numerical contradiction 0 < L · C = λ ( L · f ) = 0.If (i)–(iii) hold, p extends to a P -bundle by Lemma 5.8. Moreover, dim Z = 2by Lemma 5.7. Conversely, if p extends, we have (i) by Theorem 5.5. Q.E.D.We consider now the extension problem for P -bundles. For perspective we alsorecall the (much easier) case of P d -bundles, for d ≥ , Section 5.5] describes allknown examples. We refer to [ , Section 5.5] for the more general case when X is a local complete intersection. Conjecture 7.2.
Let L be an ample line bundle on a projective manifold, X ,of dimension n ≥ . Assume that there is a smooth Y ∈ | L | such that Y is a P d -bundle, p : Y → Z , over a manifold, Z , of dimension b . Then d ≥ b − and ( X, L ) ∼ = ( P ( E ) , ξ P ) for an ample vector bundle, E , on Z with p equal to therestriction to Y of the induced projection P ( E ) → Z , except if either: (i) ( X, L ) ∼ = ( P , O P (2)) ; or (ii) ( X, L ) ∼ = ( Q , O Q (1)) ; or (iii) Y ∼ = P × P n − , p is the product projection onto the second factor, ( X, L ) ∼ =( P ( E ) , ξ P ) for an ample vector bundle, E , on P with the product projectionof Y onto the first factor equal to the induced projection P ( E ) → P . VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 29
Note.
The inequality d ≥ b − p : Y → Z to extend byLemma 5.7.The conjecture has been shown except when d = 1, b ≥
3, and the base Z doesnot map finite-to-one into its Albanese variety. The case when either d ≥ Z is a submanifold of an abelian variety follows from Sommese’s extension theorems[ ] (see also Fujita [ ]). This argument works also in the case when Z mapsfinite-to-one into its Albanese variety (see [ , (5.2.3)]). Theorem 7.3. (Sommese)
Conjecture 7.2 is true for d ≥ . Proof.
Since the result is trivial if Z is a point we can assume without lossof generality that dim Z ≥ n ≥ d + 2 ≥
4. From Theorem 3.1 weknow that p : Y → Z extends to a morphism, p : X → Z . The result follows fromTheorem 5.5(ii). Q.E.D.The conjecture is also known when d = 1 and b ≤
2. If b = 1, the result is dueto B˘adescu [
6, 7, 8 ]; we have seen a proof in Corollary 6.3. The case b = 2 is dueto the contribution of several authors: Fania and Sommese [ ], Fania, Sato andSommese [ ], Sato and Spindler [
62, 63 ] and also [
61, 64 ]. Below we propose ashorter proof. The basic ideas are those in [ ] and [ ], but we do not use thedifficult papers [ ] and [ ]. Theorem 7.4.
Conjecture 7.2 is true when d = 1 and b = 2 . Proof.
Assume that p does not extend. Step Z is ruled. From Corollary 3.9 it follows that Y has some extremalray, say R , which extends to an extremal ray R on X . Using [ ], we consider thepossible type of R . If R is nef, Z is covered by rational curves in the fibers of cont R ,so it is ruled. If R is not nef, E , the locus of R , covers Z (so again Z is ruled),unless E ∼ = P × P and p ( E ) := C is a curve. Standard computations (cf. alsoPoposition 5.13) show that C is a ( −
1) curve and we may construct a commutativediagram Z cont C (cid:15) (cid:15) Y p o o cont R (cid:15) (cid:15) (cid:31) (cid:127) / / X cont R (cid:15) (cid:15) Z ′ Y ′ p ′ o o (cid:31) (cid:127) / / X ′ where Y ′ is ample on X ′ (cf. [ ]) and p ′ is again a P -bundle. So, after finitelymany steps, we conclude that Z is ruled. Step Z ∼ = P . Assume the contrary. As Z is ruled, there is a morphism ϕ : Z → B which is generically a P -bundle. Apply Theorem 3.1 to the map π := ϕ ◦ p to get an extension π : X → B . Next we use Corollary 6.3 fiberwise.Let F ∼ = F e , F be the general fibers of π, π respectively. Denote by f , C a fiberand a minimal section of F , respectively. Note that the classes of f and C are notproportional in N ( Y ). Then the diagram N ( F ) / / (cid:15) (cid:15) N ( F ) (cid:15) (cid:15) N ( Y ) ∼ / / N ( X ) shows that dim R N ( F ) ≥
2. So, from Corollary 6.3 we infer that either Y · f = 1,or F ∼ = P × P . In the first case, p extends by Lemma 5.8. So we may assume that F ∼ = P × P from now on.We refer to [ , pp. 7–11] for details concerning the next few arguments. Usingstandard properties of Hilbert schemes, one shows:(a) ϕ is a P -bundle;(b) any fiber of π is isomorphic to P × P ;(c) the family of curves on Y determined by minimal sections of the map p | F : F → p ( F ) yields another P -bundle p ′ : Y → Z ′ . We deduce acartesian diagram Y p (cid:127) (cid:127) ~~~~~~~ p ′ AAAAAAA Z ϕ (cid:31) (cid:31) @@@@@@@ Z ′ ϕ ′ ~ ~ }}}}}}} B where ϕ ′ is also a P -bundle.Next, from the above construction, Corollary 6.3 and Theorem 5.5, we find that p ′ extends to a linear P -bundle p ′ : X → Z ′ .(d) This yields an exact sequence0 → O Z ′ → F → G → , where F , G are ample vector bundles on Z ′ . If Z ′ = P ( E ′ ), one finds G ∼ = ϕ ′∗ ( E ) ⊗ ξ a , where E is a rank 2 vector bundle on B , ξ = ξ Z ′ and a >
0. Moreover, from (c) it follows that Z ∼ = P ( E ).(e) If we assume E unstable, it is now standard to see that the exact sequencefrom (d) splits. This is a contradiction, since F is ample.Finally, see [ ] for a proof that the case E stable also leads to a contradiction. Step
3. Conclusion. We know that Z ∼ = P , so Proposition 7.5 below applies togive the result. Q.E.D.Let us also explicitly note that in the relevant case d = 1, b ≥ p : Y → Z does not extend in this case) Conjecture 7.2 is equivalent tosaying that • A P -bundle Y , p : Y → Z , over a manifold Z cannot be an ample divisorin an n -dimensional manifold X unless Y ∼ = P × P n − , Z ∼ = P n − , and X is a P n − -bundle over P whose restriction to Y is the projection Y → P on the first factor .The following proposition from [ ] ensures that, to prove Conjecture 7.2, it isenough to show that Z ∼ = P n − , assuming that p does not extend. Proposition 7.5. ([ , Section 2]) Let Y be a smooth ample divisor on an n -dimensional projective manifold X . Assume that p : Y → Z is a P -bundle over Z = P b , b ≥ . If p does not extend to X , then Y ∼ = P × P b and X is a P b +1 -bundleover P . VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 31
Proof.
Let F ∈ | p ∗ O P b (1) | and let L := O X ( Y ). Looking over the proof ofTheorem 3.1 we see that, once we have the vanishings in (3.1), the argument worksalso in the case dim Y − dim Z = 1. Therefore, the assumption that p does notextend translates into: there exists some t > H ( Y, F − tL Y ) = 0.Using Serre duality, Kodaira vanishing, and the exact sequence0 → K Y + tL Y − F → K Y + tL Y → K F + tL F − F → H n − ( F, K F + tL F − F ) = 0 for some t > . Iterating this construction, we may assume that b = 2; in this case F = F e , forsome e ≥
0. We write Y = P ( E ) for some rank 2 vector bundle on P b . We also mayassume that, if l ⊂ P b is a line, we have E l ∼ = O l ⊕ O l ( − e ). We shall prove that e = 0, so that E is trivial (see [ , Section 3.2]) and the conclusion follows.So, assume that b = 2, F = F e and write L F ∼ aC + bf , using the notationfrom [ , Chapter V.2]. Since L is ample, a > b > ae . Hence, for t > bt > aet . Therefore, either bt − > aet and tL F − F ∼ atC + ( bt − f is ample,or bt = aet + 1 and tL F − F ∼ at ( C + ef ). Now, if e > C + ef is nef and big.Using Kawamata–Viehweg vanishing this contradicts (7.1). So e = 0 and we aredone. Q.E.D.Further evidence for Conjecture 7.2 is given by the following result (see [ ]for a proof). Proposition 7.6. ([ ]) Let p : Y → Z be a P -bundle over a smooth projectivethree fold Z . Then Y cannot be a very ample divisor in any projective manifold X ,unless Z ∼ = P and Y ∼ = P × P . Remark 7.7.
The following discussion gives some further support to Conjecture7.2, in relation to the content of Section 4. Let p : Y → Z be a smooth P -bundlewith Y ample divisor in a projective manifold X of dimension n ≥
4. If p does notextend to X , and we assume the hypothesis of Proposition 4.6 is fulfilled, then thefollowing three conditions hold true.(1) ̺ ( X ) = ̺ ( Y ) = 2;(2) Y is a Fano manifold;(3) Z is a Fano manifold (and ̺ ( Z ) = 1).Condition (1) directly follows from Proposition 4.6.To show (2), let R be the extremal ray corresponding to the bundle projection p . By Corollary 3.9 we conclude that there exists an extremal ray R on Y whichextends to X . By (1), N E ( Y ) = h R , R i and Y is a Fano manifold.Since Y is a Fano manifold and p is smooth, Z is also a Fano manifold, see[ , p. 244]. In our special case, we can give the following alternative argument.Assume that Z is not a Fano manifold. Then K Z would be nef (since ̺ ( Z ) = 1).Therefore Proposition 5.11 applies to say that p extends; a contradiction. Whence(3) holds.We already observed that, in view of Proposition 7.5, to prove Conjecture 7.2it would be enough to show that Z ∼ = P n − . This is in agreement with the fact thatassuming the hypothesis of Proposition 4.6 to hold, we get condition (3) above. Forinstance, when n = 4, the only Fano surface Z with ̺ ( Z ) = 1 is P , yielding a veryshort proof of Theorem 7.4.
8. Fano manifolds as ample divisors
Throughout this section, let X be a projective manifold of dimension n ≥ H be an ample line bundle on X , and let Y be a smooth divisor in | H | . In thisgeneral setting, it is natural to ask the following questions. Question 8.1. If Y is a Fano manifold, when is X a Fano manifold? Question 8.2. If X is a Fano manifold, when do we have N E ( X ) ∼ = N E ( Y )?Question 8.2 has been solved in [ ] and [ ], by using Theorem 2.8, in thespecial case described in the following theorem. Theorem 8.3. (Koll´ar, Wi´sniewski)
Let X be a Fano manifold of dimension n ≥ and index i ≥ , − K X ∼ = iL , for some ample line bundle L on X . Assume that weare given a smooth member Y ∈ | mL | , for some integer m , ≤ m ≤ i . Then thereis an isomorphism N E ( Y ) ∼ = N E ( X ) . Proof.
The result follows from Proposition 3.5, since − ( K X + Y ) ≈ ( i − m ) L is nef. Q.E.D.Let us come back now to Question 8.1. This question is motivated by theproblem of classifying polarized pairs ( X, Y ) as above, when Y is a Fano manifoldof large index. Set − K Y ≈ iL Y , where L Y is an ample line bundle on Y .The first cases to consider, cf. Corollary 5.2 and Remark 5.3, are i = dim Y +1 = n , so Y is a projective space, and i = n −
1, so Y is a quadric; for a solution, see[ ] and also [ ], where X is only assumed to be normal. The case when ( Y, L Y ) isa classical del Pezzo variety, i.e., i = n − L Y very ample, has been completelyworked out by Lanteri, Palleschi and Sommese [
45, 46, 47 ]. In [
12, 13 ] the nextcase when (
Y, L Y ) is a Mukai variety, i.e., i = n −
3, is considered. The resultsof [ ] have been refined and strengthened in [ ] under the assumption that L Y ismerely ample, as a consequence of a comparing cones result which holds true in therange i ≥ dim Y . In [ ] the classification is extended to the next case.We will work under the extra assumption that the line bundle L Y is spanned.Note that this is in fact the case when ( Y, L Y ) is either a del Pezzo variety of degreeat least two, or a Mukai variety. This follows from Fujita’s classification [ ] of delPezzo manifolds and from Mukai’s classification, see [ ] and [ ].We have the following result (compare with [ , (4.2)]). Theorem 8.4.
Let X be a projective manifold of dimension n ≥ , let H be anample line bundle on X , and let Y be an effective divisor in | H | . Assume that Y isa Fano manifold of index i ≥ , − K Y ≈ iL Y . Further assume that L Y is spanned.Then either (i) There exists an extremal ray R on X of length ℓ ( R ) ≥ i + 1 ; or (ii) Y contains an extremal ray of length ≥ i ; or (iii) X is a Fano manifold and N E ( X ) ∼ = N E ( Y ) . Proof.
By the Lefschetz theorem, there exists a unique line bundle L on X such that L | Y = L Y .First, suppose that K Y + iH Y is not nef. Then, by the cone theorem, thereexists an extremal ray R Y = R + [ C ] on Y such that( K Y + iH Y ) · C < . VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 33
We can also assume that C is a minimal curve, that is ℓ ( R Y ) = − K Y · C . Therefore ℓ ( R Y ) > i ( H · C ).Set a := L · C , d := H · C . Since − K Y ≈ iL Y , the last inequality yields a > d ≥
1. Then a ≥ d = 1. Thus ℓ ( R Y ) = ia ≥ i , so thateither ℓ ( R Y ) ≥ i + 1, or a = 2 and hence d = H · C = 1.From − K Y ≈ iL Y we get ℓ ( R Y ) = i ( L Y · C ), that is, i divides ℓ ( R Y ). Hencein the first case above it must be ℓ ( R Y ) ≥ i , as in the case (ii) of the statement.Thus we can assume that a = 2, H · C = 1 and the adjunction formula gives( K X + H ) · C = K Y · C = − ℓ ( R Y ), or − K X · C = ℓ ( R Y ) + 1 = 2 i + 1. Therefore( K X + 2 iH ) · C <
0, i.e., K X + 2 iH is not nef. Then there exists a rational curve γ generating a ray R = R + [ γ ] on X such that ( K X + 2 iH ) · γ <
0. Since we canassume ℓ ( R ) = − K X · γ , it follows that ℓ ( R ) > i and we are in case (i) of thestatement.Assume now that K Y + iH Y is nef. Then by the ascent of nefness (see the proofof Theorem 3.8) we infer that K X + ( i + 1) H is nef and hence by Kawamata–Reid–Shokurov base point free theorem we conclude that m ( K X + ( i + 1) H ) is spannedfor m ≫ K X + ( i + 1) H . Let ψ : X → W be the map with normal image and connected fibers associated to | m ( K X + ( i + 1) H ) | for m ≫ κ ( K X + ( i + 1) H ) = 0, then K X + ( i + 1) H ≈
0, so that X is a Fanomanifold. From iL Y ≈ − K Y ≈ iH Y we conclude by Lefschetz that H ≈ L .Therefore Theorem 8.3 applies to give N E ( X ) = N E ( Y ).Assume κ ( K X + ( i + 1) H ) = 1. Since Y is ample, the restriction ψ Y of ψ to Y maps onto W , so that W ∼ = P since Y is a Fano manifold. Note that ψ Y isnot the constant map by ampleness of Y . Recalling that N E ( Y ) is polyhedral,we conclude that there exists an extremal ray R on Y which is not contracted by ψ Y . Let ϕ : Y → Z be the contraction of R . We claim that all fibers of ϕ are ofdimension ≤
1. Otherwise, let ∆ be a fiber of dimension ≥
2. Any fiber F of ψ Y is a divisor on Y . Then we can find a curve C ⊂ ∆ ∩ F . Therefore C generates R and dim ψ Y ( C ) = 0, contradicting the fact that R is not contracted by ψ Y .Thus by Theorem 2.8 we know that either ϕ is a blowing-up of a smoothcodimension two subvariety of Z and − K Y · C = 1; or ϕ is a conic fibration and − K Y · C ≤
2. In each case, the equality − K Y · C = i ( L Y · C ) contradicts theassumption that i ≥ κ ( K X + ( i + 1) H ) ≥
2. We follow here the argument from [ ].From ( K X + H ) Y ≈ K Y ≈ − iL Y we get by Lefschetz(8.1) K X + H + iL ≈ , that is K X + ( i + 1) H ≈ i ( H − L ). Thus we conclude that κ ( H − L ) ≥ m ( H − L ) is spanned for m ≫
0. Therefore the Mumford vanishing theorem [ ](see also [ , (7.65)]) applies to give(8.2) H ( X, L − H ) = 0 . Now consider the exact sequence0 → L − H → L → L Y → . Since L Y is spanned on Y , by (8.2) we see that sections of H ( Y, L Y ) lift to span L in a neighborhood of Y ; but since Y is ample we conclude that L is spanned off a finite set of points. Hence L is nef and therefore − K X is ample by (8.1), i.e., X is a Fano manifold.We conclude that either N E ( X ) ∼ = N E ( Y ) and hence we are done, or thereexists an extremal ray R = R + [ C ] on X , R ⊂ N E ( X ) \ N E ( Y ) such that every fiberof the contraction ϕ : X → Z of R has dimension at most one. Then Theorem 2.8applies again to say that either:(1) ϕ is a blowing-up along a smooth codimension two center B and K X · C = −
1, or(2) ϕ is a conic fibration and either K X · C = − K X · C = − K X · C = − Y · C + i ( L · C ). Since Y · C > L · C ≥ Y · C = 1, L · C = 0. Note that ( K X + Y ) · C = 0and apply Proposition 3.4 to contradict our present assumption that R N E ( Y ).Let us consider case (2). If K X · C = − Y · C + i ( L · C ), giving Y · C = 2, L · C = 0. If K X · C = − Y · C + i ( L · C ), giving Y · C = 1, L · C = 0. In both cases, Proposition 3.4 applies again to give the same contradictionas above. Q.E.D. Corollary 8.5. ([ , (4.2)]) Let X be an n -dimensional projective manifold. Let H be an ample line bundle on X and let Y be a divisor in | H | . Assume that Y isa Fano manifold of index i ≥ dim Y ≥ hence n ≥ , − K Y ≈ iL Y and L Y isspanned. Then X is a Fano manifold and N E ( X ) ∼ = N E ( Y ) . Proof.
By the proof of Theorem 8.4, either we are done, or Y contains anextremal ray R Y = R + [ C ] such that either ℓ ( R Y ) ≥ i , or ℓ ( R Y ) ≥ i and H · C = 1.In the first case we get the numerical contradiction ℓ ( R Y ) ≥
32 ( n − ≥ n + 1 = dim Y + 2 . Thus we may assume ℓ ( R Y ) ≥ i and H · C = 1. Therefore, if ∆ is a positivedimensional fiber of the contraction p = cont R Y : Y → W , we have dim ∆ ≥ ℓ ( R Y ) − ≥ dim Y − Y or ℓ ( R Y ) =dim Y = n − p is the constant map, so that Pic( Y ) ∼ =Pic( X ) ∼ = Z and the conclusion is clear.In the latter case, since ℓ ( R Y ) = dim Y , we know from Theorem 2.7 thatdim W ≤
1. If dim W = 0 we conclude as above. Assume that W is a curve andlet F be a general fiber of p . Since Y is a Fano manifold, W ∼ = P . Moreover, since H · C = 1, we get K F + ( n − H F ≈
0. Corollary 5.2 applies to give F ∼ = P n − , H ∈ |O F (1) | . Therefore, by Theorem 5.5(i), Y ∼ = P ( E ) for some vector bundle E on P . Using Lemma 4.4 and the assumption i ≥
3, we see that this case does notoccur. Q.E.D.
9. Ascent properties
Let X be a projective n -dimensional manifold and let Y ⊂ X be a smoothample divisor. Here is a list of general facts concerning ascent properties from Y to X . E.g., • K Y not ample = ⇒ K X not nef. It immediately follows from the adjunctionformula. VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 35 • κ ( Y ) < dim Y = ⇒ κ ( X ) = −∞ . Here is the argument from [ , Propo-sition 5]. Assume by contradiction that | mK X | 6 = ∅ for some m > E ∈ | mK X | . Write E = aY + E ′ , with a ≥ Y Supp( E ′ ). Since λY is veryample for λ ≫
0, we can find λ ≫ D ∈ | λE | such that Y Supp( D ). Sincewe have ( D + λmY ) | Y ≈ λmK Y , it follows that | λmK Y | is very ample outsideof Y ∩ Supp( D ), so that κ ( Y ) = n − Y . This contradiction proves theassertion.Uniruled manifolds are birationally Fano fibrations. This fact follows fromCampana’s construction (see e.g., the Preface and Chapters 3, 4 of Debarre’s text[ ]). Many results in our paper are concerned with the case in which Y carries aspecial Fano fibration structure. • Y uniruled = ⇒ X uniruled. It immediately follows from the unirulednesscriterion [ , II, Section 3, IV, (1.9)]. Saying that Y is uniruled means that thereis a morphism f : P → Y such that f ∗ T Y is spanned. Consider the tangent bundlesequence 0 → T Y → T X | Y → O Y ( Y ) → . Let f ′ : P → X be the induced morphism to X . By pulling back to P , we get theexact sequence 0 → f ∗ T Y → f ∗ ( T X | Y ) = f ′∗ T X → f ∗ O Y ( Y ) → . Since both f ∗ O Y ( Y ) and f ∗ T Y are nef, we conclude that f ′∗ T X is nef and hencespanned; this is equivalent to say that X is uniruled. • Y rationally connected = ⇒ X rationally connected. Saying that Y is ratio-nally connected is equivalent to the existence of a curve C ∼ = P ⊂ Y with amplenormal bundle N C/Y (see e.g., [ ]). Therefore the exact sequence of normal bun-dles 0 → N C/Y → N C/X → O C ( Y ) → Y give the ampleness of N C/X . • Y unirational = ⇒ X unirational? This is a hard question and no answeris known. It is interesting to point out that, since unirationality implies rationalconnectedness, to find examples of Y unirational with X not unirational would giveexamples of rationally connected manifolds X which are not unirational. QuotingKoll´ar [ , Section 7, Problem 55], the latter is “one of the most vexing openproblems” in the theory. • In general, Y rational does not imply that X is rational. We present belowa few results about this problem. In particular, we obtain a proof of the followingclassical statement ([ , Chapter IV]): for a very ample smooth divisor on a threefold X , the ascent of rationality holds true with the only exception when X is thecubic hypersurface of P . The case of the cubic hypersurface is indeed an exception,see [ ]. Theorem 9.1. (cf. also [ ]) Let L be an ample line bundle on a smooth projectivethree fold X . Assume that there is a smooth Y ∈ | L | such that Y is rational. Then X is rational unless either: (i) L = 1 and ( X, L ) is a weighted hypersurface of degree in the weightedprojective space P (3 , , , , , − K X ≈ L ; or (ii) L = 2 and ( X, L ) is the double covering of P branched along a smoothsurface of degree , − K X ≈ L and L is the pull-back of O P (1) ; or (iii) X is the hypercubic in P and L ≈ O X (1) . Proof.
Since Y is a rational surface, K Y is not nef. We follow the cases (i)–(v)from Theorem 6.2. In case (i), we apply the well-known classification of Fano threefolds of index ≥ ̺ ( X ) = 1), see [ ]. We either get one of the exceptionalcases in the statement, or X is the complete intersection of two quadrics in P , or X ⊂ P is a linear section of the Grassmannian of lines in P , embedded in P bythe Pl¨ucker embedding. In the last two cases X is rational (see e.g., [ ]). A simpleargument is given in Example 9.3 below.Assume now that we are in case (ii) or (iii) from Theorem 6.2. For such afibration the base curve is P and the general fiber is rational. Moreover, a sectionexists by Tsen’s theorem, see e.g., [ , IV.6]. So X is rational, too.If we are in case (iv), the base surface is birational to Y , so it is rational.We conclude that X is rational. Finally, case (v) leads to one of the previouslydiscussed situations. Q.E.D.The following result, contained in [ , Theorem 1.3], concerns the ascent ofrationality from a suitable rational submanifold. The proof relies on Hironaka’sdesingularization theory [ ] and on basic properties of rationally connected man-ifolds [ ]. Theorem 9.2. ([ , Theorem 1.3]) Let X be a projective variety and | D | a completelinear system of Cartier divisors on it. Let D , . . . , D s ∈ | D | and put W i := D ∩· · · ∩ D i for ≤ i ≤ s . Assume that W i is smooth, irreducible of dimension n − i ,for all i . Assume moreover that there is a divisor E on W := W s and a linearsystem Λ ⊂ | E | such that: (i) ϕ Λ : W P n − s is birational, and (ii) | D W − E | 6 = ∅ .Then X is rational. Proof.
We proceed by induction on s . Let us explain the case s = 1, thegeneral case being completely similar. So, let W ∈ | D | be a smooth, irreducibleCartier divisor such that ϕ Λ : W P n − is birational for Λ ⊂ | E | , E ∈ Div( W )and | D W − E | 6 = ∅ . Replacing X by its desingularization, we may assume that X issmooth. As W is rational, it is rationally connected, so we may find some smoothrational curve C ⊂ W with N C/W ample. We have C · E > C · D >
0. From the exact sequence of normal bundles we get that N C/X is ample, so X is rationally connected. In particular, H ( X, O X ) = 0.The exact sequence0 → O X → O X ( D ) → O W ( D ) → , shows that dim | D | = dim | D W | + 1 ≥ dim | E | + 1 ≥ n .We may choose a pencil ( W, W ′ ) ⊂ | D | , containing W , such that W ′ W = E + E ,with E ≥ E ∈ Λ. By Hironaka’s theory [ ], we may use blowing-ups withsmooth centers contained in W ∩ W ′ , such that after taking the proper transformsof the elements of our pencil, to get:(a) Supp( E ) has normal crossing;(b) Λ is base points free (so ϕ : W → P n − is a birational morphism).Further blowing-up of the components of Supp( E ) allows to assume E = 0 so D W is linearly equivalent to E . Using the previous exact sequence and the fact VIEW ON EXTENDING MORPHISMS FROM AMPLE DIVISORS 37 that H ( X, O X ) = 0, it follows that Bs | D | = ∅ . Finally, D n = ( D W ) n − W = 1, so ϕ is a birational morphism to P n . Q.E.D. Example 9.3. ([ , Example 1.4]) Let X ⊂ P n + d − be a non-degenerate projectivevariety of dimension n ≥ d ≥
3, which is not a cone. Then X isrational, unless it is a smooth cubic hypersurface, n ≥
3. If X is singular, byprojecting from a singular point we get a variety of minimal degree, birational to X . So X is rational. If X is not linearly normal, X is isomorphic to a variety ofminimal degree. Hence we may assume X to be smooth and linearly normal. Onesees easily that such a linearly normal, non-degenerate manifold X ⊂ P n + d − hasanticanonical divisor linearly equivalent to n − ] or [ ]. Independently of their classification,the following simple argument shows that such manifolds are rational when d ≥ W obtained by intersecting X with n − W is a non-degenerate, linearly normal surface of degree d in P d , so it isa del Pezzo surface. As such, W is known to admit a representation ϕ : W → P asthe blowing-up of 9 − d points. Let L ⊂ W be the pull-back via ϕ of a general line in P . It is easy to see that L is a cubic rational curve in the embedding of W into P d .So, for d ≥ L is contained in a hyperplane of P d . This shows that the conditionsof the Theorem 9.2 are fulfilled for X , | D | being the system of hyperplane sections.We also see that Theorem 9.2 is sharp, as the previous argument fails exactly forthe case of cubics. Remark 9.4.
In closing, we mention three possible generalizations of the problemof extending morphisms from ample divisors on X .(1) The smoothness assumption on X may be relaxed by allowing normalsingularities. Let Y be a smooth divisor in X ( X is smooth), and letus only suppose that Y has ample normal bundle. Then a well-knownresult ([ ]) shows that there is a birational map ψ : X → X ′ , which isan isomorphism along Y , such that ψ ( Y ) := Y ′ ⊂ X ′ is ample and X ′ isnormal. See e.g., [
9, 10 ] and [ ] for results in this direction.(2) Consider a smooth section Y ⊂ X of the appropriate expected dimension n − rk E of an ample vector bundle E on an n -fold X . Note that a Lefschetztype theorem for ample vector bundles, due to Sommese [ ], implies thatthe restriction to Y gives an isomorphism Pic( X ) ∼ = Pic( Y ). See e.g.,[
3, 21, 44, 4 ] and [ ] for results of this type.(3) In the same spirit, let us consider a smooth subvariety Y of a manifold X such that codim X Y ≥
2, and Y has ample normal bundle. Further, letus add the Lefschetz type assumption that Pic( Y ) ∼ = Pic( X ). Then onecan study extensions of rationally connected fibrations p : Y → Z onto anormal projective variety Z . See [ ] and [ ] for results in this direction.(4) The very recent paper [ ] classifies pairs ( X, Y ), when Y ⊂ X is anample divisor which is a homogeneous manifold. Acknowledgments
We thank the referee for several useful comments.
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