Nefness of adjoint bundles for ample vector bundles of corank 3
aa r X i v : . [ m a t h . AG ] A ug NEFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLESOF CORANK ANDREA LUIGI TIRONI
Abstract.
Let E be an ample vector bundle of rank r ≥ X of dimension n . The aim of this paper is to describe the structure of pairs ( X, E )as above whose adjoint bundles K X + det E are not nef for r = n −
3. Furthermore, wegive some immediate consequences of this result in adjunction theory. Introduction
Let E be an ample vector bundle of rank r ≥ X of dimension n . We say that a Cartier divisor D on X is numerically effective (nef ) ifit has non-negative intersection number with any curve C on X .In the late 1980s, Ye and Zhang started to study the nefness of the adjoint bundle K X + det E , obtaining in [40] complete results for r ≥ n − r = n −
2, showing that in thissituation K X + det E is nef except in a few cases.Some years later, by the adjunction theory and the remark that a key result of [37] is stillvalid without the extra assumption of spannedness for E , Maeda gave in [17] an exhaustivedescription of all the ample vector bundles E for the case r = n − E is assumed tobe also generated by global sections, was already proved by Wi´sniewski in [37] and that allthe above results are a generalization of those due to Fujita [10] and Ionescu [13].Thus the purpose of this paper is to describe the structure of pairs ( X, E ) as above whoseadjoint bundles K X + det E are not nef in the next case r = n − Theorem 1.1.
Let E be an ample vector bundle of rank r ≥ on a smooth complexprojective variety X of dimension n . If r = n − , then K X + det E is nef except when ( X, E ) is one of the following: (1) ( P n , ⊕ n − i =1 O P n ( a i ) , where all the a i ’s are positive integers such that P n − i =1 a i ≤ n ; (2) ( P n , O P n (1) ⊕ n − − a ⊕ V ) , where V is an indecomposable Fano vector bundle on P n of rank a such that ≤ a ≤ n − , c ( V ) = a + 3 and its generic splitting typesare (3 , , , , ..., or (2 , , , , ..., ; moreover, V ( −
1) :=
V ⊗ O ( − is nef for ≤ a ≤ n − , but it is not globally generated for n ≥ ; Date : December 7, 2017. (3) ( Q n , ⊕ n − j =1 O Q n ( b j )) , where all the b j ’s are positive integers such that P n − j =1 b j ≤ n − ; (4) ( Q n , O Q n (1) ⊕ n − − b ⊕ V ) , where V is an indecomposable Fano vector bundle on Q n of rank b such that ≤ b ≤ n − , c ( V ) = b + 2 and its generic splitting type is (2 , , , ..., ; moreover, V ( − is nef on Q n for ≤ b ≤ n − , but it is not globallygenerated for n ≥ ; (5) X is a Fano n -fold of index n − with Pic( X ) generated by an ample line bundle H and either ( α ) E ∼ = H ⊕ n − , or ( β ) E | l ∼ = H ⊕ n − l ⊕ H ⊗ l for every line l of ( X, H ) ; (6) X is a Fano n -fold of index n − with Pic( X ) generated by an ample line bundle L and E ∼ = L ⊕ n − ; (7) ( P × P , O P × P (1 , ⊕ ) ; (8) ( P × Q , O P × Q (1 , ⊕ ) ; (9) ( P P ( T ) , [ ξ T ] ⊕ ) , where T is the tangent bundle of P and ξ T is the tautological linebundle; (10) ( P P ( O (2) ⊕ O (1) ⊕ ) , [ ξ ] ⊕ ) , ξ being the tautological line bundle; (11) there is a vector bundle V on a smooth curve C such that X ∼ = P C ( V ) ; moreover, forany fibre F ∼ = P n − of X → C , we have E | F is isomorphic to one of the following: ( a ) O P n − (1) ⊕ n − ; ( b ) O P n − (2) ⊕ O P n − (1) ⊕ n − ; ( c ) O P n − (2) ⊕ ⊕ O P n − (1) ⊕ n − ; ( d ) O P n − (3) ⊕ O P n − (1) ⊕ n − ; (12) X is a section of a divisor of relative degree two in a projective space P C ( G ) , where G is a vector bundle of rank n + 1 on a smooth curve C ; moreover, for any smoothfibre F ∼ = Q n − of X → C , where Q n − is a smooth quadric hypersurface of P n , wehave ( F, E | F ) is one of the following pairs: (i) ( Q n − , O Q n − (1) ⊕ n − ) ; (ii) ( Q n − , O Q n − (2) ⊕ O Q n − (1) ⊕ n − ) ; (iii) ( Q , S ⊗ O Q (2)) , where S is the spinor bundle on Q ⊂ P ; (13) the map Φ : X → C associated to the linear system | ( n − K X +( n −
2) det
E | makes X a Del Pezzo fibration over a smooth curve C ; moreover, any general smooth fibre F of Φ is either a Del Pezzo ( n − -fold with Pic( F ) ∼ = Z [ O F (1)] and such that E | F ∼ = O F (1) ⊕ n − , or P × P with E | F ∼ = O P × P (1 , ⊕ ; (14) there is a vector bundle V on a smooth surface S such that X ∼ = P S ( V ) ; moreover,for any fibre F ∼ = P n − of X → S , we have E | F ∼ = O P n − (2) ⊕ O P n − (1) ⊕ n − ; (15) there exists a P n − -fibration π : X → S , locally trivial in the complex (or ´etale)topology, over a smooth surface S such that E | F ∼ = O P n − (1) ⊕ n − for any closed fiber F ∼ = P n − of the map π ; (16) the map ψ : X → S associated to the linear system | ( n − K X + ( n −
2) det
E | makes X a quadric fibration over a smooth surface S ; moreover, for any generalfibre F ∼ = Q n − we have E | F ∼ = O Q n − (1) ⊕ n − , where Q n − is a smooth quadrichypersurface of P n − ; (17) there is a vector bundle F on a smooth -fold V such that X ∼ = P V ( F ) ; moreover,for any fibre F ∼ = P n − of X → V , we have E | F ∼ = O P n − (1) ⊕ n − ; EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 3 (18) there exists a smooth variety X ′ and a morphism ϕ : X → X ′ expressing X as theblowing-up of X ′ at a finite set of points B and an ample vector bundle E ′ on X ′ such that E ⊗ ([ ϕ − ( B )]) ∼ = ϕ ∗ E ′ and K X ′ + τ ′ det E ′ is nef, where τ ′ < n − n − is thenefvalue of the pair ( X ′ , det E ′ ) . Moreover, E | E ∼ = O P n − (1) ⊕ n − for any irreduciblecomponent E of the exceptional locus of ϕ ; (19) the map ψ : X → X ′ associated to the linear system | ( n − K X + ( n −
1) det
E | is abirational morphism which contracts an extremal face spanned by extremal rays R i for some i in a finite set of index. Let ψ i : X → X i be the contraction associatedto R i . Then each ψ i is birational and of divisorial type; moreover, if E i is anexceptional divisor of ψ i , then ( E i , [ E i ] E i , E | E i ) ∼ = ( P n − , O P n − ( − , O P n − (1) ⊕ n − ) ; (20) the map φ : X → X ′ associated to the linear system | ( n − K X + ( n −
2) det
E | is a birational morphism which contracts an extremal face. Let R i be the extremalrays spanning this face for some i in a finite set of index. Call ρ i : X → W i thecontraction associated to one of the R i . Then we have each ρ i is birational and ofdivisorial type; if D i is one of the exceptional divisors and Z i = ρ i ( D i ) , we have dim Z i ≤ and one of the following possibilities can occur: (j) dim Z i = 0 , D i ∼ = P n − and ([ D i ] D i , E | D i ) ∼ = ( O P n − ( − , O P n − (1) ⊕ n − ⊕O P n − (2)) ; (jj) dim Z i = 0 , D i is a (possible singular) quadric Q n − ⊂ P n and [ D i ] D i = O Q n − ( − ; moreover, E | D i ∼ = O Q n − (1) ⊕ n − ; (jjj) dim Z i = 1 , W i and Z i are smooth projective varieties and ρ i is the blow-up of W i along Z i ; moreover, E | F i ∼ = O F i (1) ⊕ n − and O D i ( D i ) | F i = O P n − ( − forany fibre F i ∼ = P n − of D i → Z i .Moreover, the map φ is a composition of disjoint extremal contractions as in (j) , (jj) and (jjj) . The Theorem 1.1 is a generalization of a result due to Fujita [10, theorem 4] and its proofmakes use principally of [2] and all the above papers [17], [40] and [41], together with somerecent results ([3] and [26]) about the nef value of pairs ( X, det E ).Note that Theorem 1.1 is complete for n = 5 (or Pic( X ) = Z ). When n ≥ X is a Fano manifold of index i ( X ) ≥ n − X ) = Z [ O X (1)], a finest descriptionof E is equivalent to have in this setting a complete classification of ample vector bundles F := E ⊕ O X (1) of rank n − ≥ c ( F ) = − K X (see also [2, theorem5.1 (2)(i)]), or P X ( F ) := W is a Fano manifold of index i ( W ) = dim W − ≥ i ( W ) = 3 implies n = 5; see [23, theorem 1.3] for the case i ( W ) = 2). However, a moresatisfactory and effective description of E when Pic( X ) = Z [ O X (1)] is given in Corollary3.1 under the extra assumption that E ⊗ O X ( −
1) is globally generated on X .In the last section, we consider two immediate applications of Theorem 1.1. More pre-cisely, in § E of rank r ≥ X of dimension n which admit a globalsection s ∈ Γ( E ) whose zero locus Z := ( s ) ⊂ X is a smooth subvariety of the expecteddimension n − r ≥ K Z + (dim Z − H Z is not nef (Propositions 4.1 and ANDREA LUIGI TIRONI H Z denotes the restriction to Z of an ample line bundle H on X . These resultspartially overlap with those obtained in [4, §§ , , , X, E ) as above such that ( Z, H Z ) is a special polarized variety until the secondreduction map in the sense of the adjunction theory (e.g., see [6, chapter 7] for a completeclassification of such pairs ( Z, H Z )). Finally, in § §
3] about classical scrolls which are not adjunction-theoretic scrolls(we refer to § § Notation and background material.
In this note varieties are always assumed to be defined over the complex number field C .We use the standard notation from algebraic geometry. The words “vector bundles” and“locally free sheaves” are used interchangeably. Let X be a smooth irreducible projectivevariety of dimension n (for simplicity, n -fold). The group of line bundles on X is denotedby Pic( X ). Moreover, we denote by ρ ( X ) the Picard number of X . The tensor productsof line bundles are denoted additively. The pull-back j ∗ E of a vector bundle E on X byan embedding j : V → X is denoted by E | V (or, for simplicity, by E V when no confusionarises). The canonical bundle of an n -fold X is denoted by K X . An n -fold X is said tobe a Fano manifold if its anticanonical bundle − K X is ample. For Fano manifolds X , thelargest integer i ( X ) which divides − K X in Pic( X ) is called the index of X .A polarized n -fold is a pair ( X, L ) consisting of an n -fold X and an ample line bundle L on X . A polarized n -fold ( X, L ) is said to be a classical scroll over a smooth variety Y if ( X, L ) ∼ = ( P Y ( V ) , ξ V ) for some ample vector bundle V on Y , where ξ V is the tauto-logical line bundle on the projective space P Y ( V ) associated to V . We say that ( X, L ) isan adjunction–theoretic scroll (respectively a quadric fibration , respectively a
Del Pezzofibration , respectively a
Mukai fibration ) over a normal variety Y of dimension m if thereexists a surjective morphism with connected fibres p : X → Y and an ample line bundle H on Y , such that K X + ( n − m + 1) L ≃ p ∗ H (respectively K X + ( n − m ) L ≃ p ∗ H , respectively K X + ( n − m − L ≃ p ∗ H , respectively K X + ( n − m − L ≃ p ∗ H ). We say that ( X, L )is a
Del Pezzo variety (respectively a
Mukai variety ) if K X ≃ − ( n − L (respectively K X ≃ − ( n − L ). Moreover, for general results about the adjunction theory, we refer to[6] and [34].A part of Mori’s theory of extremal rays is to be used throughout the paper. So, let Z ( X )be the free abelian group generated by integral curves on an n -fold X . The intersectionpairing gives a bilinear map Pic( X ) × Z ( X ) → Z and the numerical equivalence ≡ isdefined so that the pairing((Pic( X ) / ≡ ) ⊗ Q ) × (( Z ( X ) / ≡ ) ⊗ Q ) → Q EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 5 is non-degenerate. The closed cone of curves
N E ( X ) is the closed convex cone generatedby effective 1-cycles in the R -vector space ( Z ( X ) / ≡ ) ⊗ R . We say that L ∈ Pic( X )is nef if the numerical class of L in (Pic( X ) / ≡ ) ⊗ R gives a non-negative function on N E ( X ) − { } . Let Z be a 1-cycle on X . We denote by [ Z ] the numerical class of Z in( Z ( X ) / ≡ ) ⊗ R . A half line R = R + [ Z ] in N E ( X ) is called an extremal ray if(1) K X · Z <
0, and(2) if z , z ∈ N E ( X ) satisfy z + z ∈ R , then z , z ∈ R .A rational (possibly singular) reduced and irreducible curve C on X is called an extremalrational curve if R + [ C ] is an extremal ray and ( − K X ) · C ≤ n + 1. Let N E ( X ) + = { z ∈ N E ( X ) | K X · z ≥ } . Then we have the following basic theorem in the Mori theory.
Theorem 2.1 (Cone Theorem) . Let X be a smooth projective variety. Then N E ( X ) isthe smallest closed convex cone containing N E ( X ) + and all the extremal rays: N E ( X ) = N E ( X ) + + X i R i , where the R i are extremal rays of N E ( X ) for X . For any open convex cone V containing N E ( X ) + − { } there exist only a finite number of extremal rays that do not lie in V ∪ { } .Furthermore, every extremal ray is spanned by a numerical class of an extremal rationalcurve. For the proof of the above result, we refer to [21, theorem 1.5] and [22, (1.2)]. In thenext section, we will use also the following well-known
Lemma 2.1.
Let E be an ample vector bundle of rank r on a rational curve C . Then det E · C ≥ r . Now, let us give here some technical results about vector bundles on some Fano n -folds X with ρ ( X ) = 1. Lemma 2.2.
Let V be an ample vector bundle of rank r on a Fano n -fold X with ρ ( X ) = 1 .Assume that c ( V ) ≤ i ( X ) − , where i ( X ) is the index of X . If n − r +1 i ( X ) − c ( V ) < then V ⊗ O X ( − is a nef vector bundle on X . Proof.
By adjunction we have − K P ( V ) ≃ rξ V + π ∗ O X ( i ( X ) − c ( V )), where ξ V is thetautological line bundle on P ( V ) and π : P ( V ) → X is the projection map. Since i ( X ) − c ( V ) > ξ V is ample, we see that P ( V ) is a Fano ( n + r − R be an extremalray of P ( V ) different from a line in a fiber of π . Then we get n + 1 ≥ − K P ( V ) · R ≥ r + [ i ( X ) − c ( V )] π ∗ O X (1) · R. This gives 2 > n − r +1 i ( X ) − c ( V ) ≥ π ∗ O X (1) · R, showing that 0 ≤ π ∗ O X (1) · R ′ ≤ R ′ on P ( V ). So, let C be an effective irreducible curve on P ( V ). From Theorem 2.1 it ANDREA LUIGI TIRONI follows that C ≡ P i α i R i with α i >
0, where each R i is an extremal ray of P ( V ). Thereforewe deduce that [ ξ V − π ∗ O X (1)] · C = X i α i [ ξ V − π ∗ O X (1)] · R i ≥ . So by definition ξ V − π ∗ O X (1) is nef on P ( V ), i.e. V ⊗ O X ( −
1) is nef on X . (cid:3) Lemma 2.3.
Let V be a globally generated vector bundle of rank r < n on P n with c := c ( V ) ≤ n . If r ≥ c + 1 , then one of the following possibilities can occurs: (1) V ∼ = O P n ( c ) ⊕ O ⊕ r − P n ; (2) V ∼ = O P n (1) ⊕ c ⊕ O ⊕ r − c P n ; (3) V ∼ = O P n (2) ⊕ O P n (1) ⊕ c − ⊕ O ⊕ r − c +1 P n ; (4) V ∼ = O P n (3) ⊕ O P n (1) ⊕ c − ⊕ O ⊕ r − c +2 P n ; (5) V ∼ = O P n (2) ⊕ ⊕ O P n (1) ⊕ c − ⊕ O ⊕ r − c +2 P n ; (6) c ≥ and there exist the following exact sequences: → O P n → V → V r − → , ... , → O P n → V k +1 → V k → , where ≤ k ≤ c − , rk V i = i , c ( V i ) = c , all the V i ’s are globally generated and ( s k ) = ∅ for a generic section s k of V k . Proof.
Let Z r = ( s r ) be the zero locus of a generic section s r of V . If Z r = ∅ , then Z r is smooth and of dimension n − r . By adjunction K Z r = O P n ( − n − c ) | Z r . Thus n + 1 − c ≤ dim Z r + 1 = n − r + 1, i.e. r ≤ c , but this is a contradiction. Hence Z r = ∅ and by induction (see, e.g., [27, (4.3.2), p.83]) we obtain the following exact sequences:0 → O P n → V → V r − → , ... , → O P n → V k +1 → V k → , where k ≤ c , rk V i = i , c ( V i ) = c , all the V i ’s are globally generated and Z k = ( s k ) = ∅ for a generic section s k of V k . If k = 1 then V = O P n ( c ) and by induction we deduce that V ∼ = O ⊕ r − P n ⊕ O P n ( c ). So 2 ≤ k ≤ c . Assume that k = c ≥
2. Since Z c = Z k = ∅ , Z c issmooth and of dimension n − c . Moreover, since 2( n − c ) − n = n − c ≥
0, we obtainthat Z c is also irreducible. Thus by the Kobayashi-Ochiai Theorem we have Z c = P n − c .Note that N Z c / P n ∼ = O Z c (1) ⊕ c . Since V c | Z c ∼ = N Z c / P n ∼ = O P n − c (1) ⊕ c and dim Z c = n − c ≥ c ≥
2, by [27, Ch.I (2.3.2)] we conclude that V c splits, andthen V ∼ = O r − c P n ⊕ O P n (1) ⊕ c . Suppose now that k = c − ≥
2. By arguing as above,we have Z c − = ∅ is smooth, irreducible and of dimension n − c + 1. Therefore by theKobayashi-Ochiai Theorem we know that Z c − ∼ = Q n − c +1 ⊂ P n . Hence N Z c − / P n ∼ = O P n (2) | Z c − ⊕ O P n (1) c − | Z c − ∼ = V c − | Z c − . Since dim Z c − = n − c + 1 ≥ c + 1 = k + 2 ≥ P ⊂ Z c − such that V c − | P splits. This shows that V ∼ = O r − c +1 P n ⊕ O P n (2) ⊕ O P n (1) ⊕ c − . Therefore 2 ≤ k ≤ c −
2. Finally, suppose that k = c − ≥
2. By arguing as in the above cases, we obtain that Z c − = ∅ is a smooth EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 7 irreducible Del Pezzo ( n − c +2))-fold with index i ( Z c − ) = n − c +1 ≥ n +1 > dim Z c − +1.Then by [38] we know that ρ ( Z c − ) = 1. Furthermore, since dim Z c − = n − c + 2 ≥ n + 2 ≥ c + 2 ≥
6, by the classification of Del Pezzo manifolds (see, e.g., [12]), we concludethat Z c − is one of the following manifolds:(a) a cubic hypersurface in P n − c +3 ;(b) a complete intersection of two quadric hypersurfaces Q i ⊂ P n − c +4 ;(c) G (1 , ⊂ P .Note that Case (c) does not occur, since in this situation n = c +4 ≤ n +4, i.e. G (1 , ⊂ P n with n ≤
8, a contradiction. On the other hand, in Case (a) we get V c − | Z c − ∼ = N Z c − / P n ∼ = O P n (3) | Z c − ⊕ O P n (1) ⊕ c − | Z c − , while in Case (b) we have V c − | Z c − ∼ = N Z c − / P n ∼ = O P n (2) ⊕ | Z c − ⊕ O P n (1) ⊕ c − | Z c − . By [33] we see that in both cases there exists a linear P ⊂ Z c − . So V c − splits andby induction we obtain that V r is either O P n (3) ⊕ O P n (1) ⊕ c − ⊕ O ⊕ r − c +2 P n , or O P n (2) ⊕ ⊕O P n (1) ⊕ c − ⊕ O ⊕ r − c +2 P n . (cid:3) Corollary 2.1.
Let V be a globally generated vector bundle of rank r < n on P n with c ( V ) = 3 . If n ≥ then V splits. Proof. If r ≥ V (1) is ample and c ( V (1)) = c ( V ) + r = 3 + r , if r ≤ V (1) splits, i.e. V splits.So let r = 3. Let s be a general section of V and put Z = ( s ) . If Z = ∅ then we obtainthe following exact sequence 0 → O P n → V → V → , where V is a rank-2 vector bundle on P n . Since rk V = 2 and n ≥
6, from [1, (9.1)] wededuce by a similar argument as above that V splits, i.e. V splits. Therefore we can assumethat Z = ∅ . Then Z is smooth and of dimension n −
3. Since 2( n − − n = n − ≥
0, we seethat Z is irreducible and by adjunction K Z = ( − n − c ) O P n (1) | Z = ( − n +2) O P n (1) | Z .Since n − Z +1, by the Kobayashi-Ochiai Theorem we obtain that Z = P n − ⊂ P n .Hence V| Z ∼ = N Z / P n ∼ = O P n − (1) ⊕ with dim Z = n − ≥
3, and this shows that V splits. (cid:3) Corollary 2.2.
Let V be a globally generated vector bundle of rank r < n on P n with c ( V ) = 4 . If n ≥ then V splits. Proof. If r ≥ ≤ r ≤
4. Thereforewe have either (i) Z r := ( s r ) = ∅ for a generic section s r of V , or by induction (ii) thereexist the following exact sequences0 → O P n → V → V r − → , ... , → O P n → V k +1 → V k → , ANDREA LUIGI TIRONI where 1 ≤ k ≤ r − ≤
3, rk V i = i , c ( V i ) = c , all the V i ’s are globally generated and Z k := ( s k ) = ∅ for a generic section s k of V k . In Case (i), Z r = ∅ is a smooth irreducible( n − r )-fold with K Z r + ( n − O P n (1) | Z r = O P n and n − Z r + r −
3. Since 2 ≤ r ≤ α ) r = 4 and Z = P n − ;( β ) r = 3 and Z = Q n − ⊂ P n − ;( γ ) r = 2.In Case ( γ ), since V (1) is ample and c ( V (1)) = c ( V ) + r = 6, from [1, (9.1)] it followsthat V splits. Moreover, by similar arguments as in the proof of Corollary 2.1, we see that V splits also in Cases ( α ) and ( β ).Consider now Case (ii). Note that k = 3, since otherwise k ≤ V k splits, i.e. V splits. Thus Z = ∅ is a smooth irreducible ( n − K Z = O P n ( n − | Z and from the Kobayashi-Ochiai Theorem weget Z = Q n − ⊂ P n . Thus V| Z ∼ = N Z / P n ∼ = O Q n − (2) ⊕ O Q n − (1) ⊕ and since dim Z = n − ≥
5, we conclude that there exists a P ⊂ Z , i.e. V splits. (cid:3) Finally, let us give here also some useful results about globally generated vector bundleson a smooth quadric hypersurface Q n ⊂ P n +1 . Lemma 2.4.
Let V be a globally generated vector bundle of rank r < n on Q n ⊂ P n +1 with c := c ( V ) ≤ n − . If r ≥ c + 2 , then either V splits, or c ≥ and there exist exactsequences → O Q n → V → V r − → , ... , → O Q n → V k +1 → V k → , where ≤ k ≤ c , rk V i = i , c ( V i ) = c , all the V i ’s are globally generated and ( s k ) = ∅ isan irreducible smooth ( n − k ) -fold for a generic section s k of V k . Proof.
Let Z r be the zero locus of a general section s r of V . If Z r = ∅ then Z r is smoothand of dimension n − r . Since by adjunction K Z r + ( n − c ) O Q n (1) | Z r = O Z r , we deducethat n − c ≤ dim Z r + 1 = n − r + 1, i.e. r ≤ c + 1, a contradiction. Thus Z r = ∅ and byinduction there exist the following exact sequences:0 → O Q n → V → V r − → , ... , → O Q n → V k +1 → V k → , where 1 ≤ k ≤ c +1, rk V i = i , c ( V i ) = c , all the V i ’s are globally generated and ( s k ) = ∅ is smooth and of dimension ( n − k ) for a generic section s k of V k . Note that V splits for k = 1. Thus suppose that 2 ≤ k ≤ c + 1. Moreover, if c ≤
1, then from [32] we know that V splits again. So we can assume also that c ≥
2. If k = c + 1 ≥
3, then Z c +1 = Z k = ∅ is smooth and of dimension n − c −
1. Since2( n − k ) − n − n − k − n − c + 1) − n − c − ≥ , we see that Z c +1 is also irreducible. Furthermore, by adjunction we get K Z c + ( n − c ) O Q n (1) | Z c ≃ O Z c , EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 9 where n − c = dim Z c +1 + 1. From the Kobayashi-Ochiai Theorem it follows that Z c +1 = P n − c − ⊂ Q n . Hence n − c − ≤ n , i.e. n ≤ c + 2, but this is absurd. (cid:3) Corollary 2.3.
Let V be a globally generated vector bundle of rank r < n on Q n ⊂ P n +1 with c ( V ) = 2 . If n ≥ then V splits. Proof.
Note that V ′ := V ⊗ O Q n (1) is ample with c ( V ′ ) = c ( V ) + rk V = 2 + rk V . Since n ≥
7, if r = 2 then V ′ is a Fano bundle of rank-2 on Q n and by [1, (2.4)(2)] we canconclude that V ′ splits, i.e. V splits. Let r ≥
3. If r ≥
4, then from Lemma 2.4 we deducethat either V splits, or there exist exact sequences0 → O Q n → V → V r − → , ... , → O Q n → V → V → , where rk V i = i , c ( V i ) = 2, all the V i ’s are globally generated and ( s ) = ∅ is smooth andof dimension ( n −
2) for a general section s of V . Since V (1) is a Fano bundle of rank2 on Q n with n ≥
7, by [1, main theorem (2.4)(2)] we see that V (1) splits, i.e. V splits.Therefore, from the above exact sequences we deduce that also V splits.Finally, suppose that r = 3. Let Z = ( s ) be the zero locus of a general section s of V . If Z = ∅ then there exists an exact sequence 0 → O Q n → V → V →
0, where V splits since n ≥
7, that is, V splits. Hence Z = ∅ is smooth and of dimension n −
3. Since2( n − − n − n − ≥
0, we see that Z is also irreducible. By adjunction we obtainthat K Z + ( n − O Q n (1) | Z = O Z with n − Z + 1. Therefore Z = P n − ⊂ Q n and this implies that n − ≤ n , i.e. n ≤
6, but this gives a contradiction. (cid:3)
Let X be an n -fold. If R is an extremal ray, then its length l ( R ) is defined as l ( R ) := min { ( − K X ) · C | C is a rational curve such that [ C ] ∈ R } . Note that 0 < l ( R ) ≤ n + 1 from Theorem 2.1 and the definition of an extremal rationalcurve. Let E be an ample vector bundle of rank r ≥ X and let Ω( X, E ) be the set ofextremal rays R such that ( K X + det E ) · R <
0. Then it follows from Theorem 2.1 thatthe set Ω( X, E ) is finite. For any extremal ray R in Ω( X, E ) we define a positive integerΛ( X, E , R ) := ( − K X − det E ) · C, where C is an extremal rational curve such that − K X · C = l ( R ).3. Proof of Theorem 1.1.
Suppose that K X + det E is not nef. By Theorem 2.1 we can find an extremal ray R with ( K X + det E ) · R <
0, and so Ω( X, E ) = ∅ . Since the set Ω( X, E ) is finite, define thepositive integer Λ( X, E ) := max { Λ( X, E , R ) | R ∈ Ω( X, E ) } . Therefore we have only the following three possibilities:(1) Λ( X, E ) ≥
3; (2) Λ( X, E ) = 2; (3) Λ( X, E ) = 1. From now on, we proceed with a case-by-case analysis.
Case (1). Note that, by definition of Λ( X, E ), there exists an extremal rational curve C ∈ R such that − ( K X + det E ) · C ≥ l ( R ) = − K X · C . So, by Lemma 2.1 we get − K X · C ≥ det E · C + 3 ≥ ( n −
3) + 3 = n, i.e. l ( R ) ≥ n . Let f R : X → Y be the extremal ray contraction associated to R . From[6, lemma 6.3.12], we deduce that f R is of fibre type with dim Y ≤
1. Let F be a smoothgeneral fibre of f R . Thus we have the following two cases:(a) dim F = n − , dim Y = 1; (b) dim F = n, dim Y = 0.In Case (a), by [6, lemma 6.3.12(2)] we know that Y is a smooth irreducible curve and ρ ( F ) = 1. Note that E | F is an ample vector bundle on F of rank dim F − Claim. K F + det E | F is not nef and Λ( F, E | F ) ≥ F of f R : X → Y .Since ρ ( F ) = 1, we have N E ( F ) ∼ = h C ′ i with [ C ′ ] ∈ R = R + [ C ], i.e. C ′ ≡ δC for some δ >
0. Thus for any effective curve γ ⊂ F , we know that γ ≡ δ ′ C ′ for some δ ′ > K F + det E | F ) · γ = ( K X + det E ) F · ( δ ′ C ′ ) = δ ′ ( K X + det E ) · ( δC ) = δδ ′ ( K X + det E ) · C < . This shows that K F + det E | F cannot be nef and that − K F is ample on F . Then byTheorem 2.1, we see that N E ( F ) ∼ = R + [ C ′ ], where C ′ is an extremal rational curve on F .Moreover, since δ ( − K X · C ) = − K X · C ′ ≥ l ( R ) = − K X · C > , we deduce that δ ≥
1. Thus we get for C ′ ⊂ F ( − K F − det E | F ) · C ′ = ( − K X − det E ) · C ′ = ( − K X − det E ) · δC ≥ δ ≥ , and this gives Λ( F, E | F ) ≥ F of f R : X → Y . Q.E.D.By the Claim and [41, Proposition 1.1’], we obtain that ( F, E | F ) is one of the followingpairs:( a ) ( P n − , O P n − (1) ⊕ n − );( a ) ( P n − , O P n − (2) ⊕ O P n − (1) ⊕ n − );( a ) ( Q n − , O Q n − (1) ⊕ n − ).In Case ( a ), we get − ( K F + det E | F ) = O P n − (2). Take a line l ⊂ F . Since l ∈ R = R + [ C ]and l ( R ) = − K X · C , by arguing as in the proof of Claim 1 , we deduce that l ≡ ρC forsome ρ ≥
1, but this gives the following numerical contradiction2 = − ( K F + det E | F ) · l = − ( K X + det E ) · ρC ≥ ρ ≥ . EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 11
By a similar argument, we see that also Case ( a ) cannot occur. Consider now Case( a ) and let c := det E · C . As in Case ( a ), taking a line l ⊂ F , we deduce that 3 =( − K F − det E | F ) · l ≥ − ( K X + det E ) · C, i.e. ( K X + det E ) · C = −
3. Therefore[ cK X + ( c + 3) det E ] · C = c [ K X + det E ] · C + 3 c = 0and from [6, theorem 4.3.1] it follows that cK X + ( c + 3) det E ≃ f ∗ R D, for some D ∈ Pic( Y ). Then we have(1) cK X + ( c + 3)(det E + ( n − f ∗ R A ) ≃ f ∗ R H for a suitable very ample line bundle A on Y such that H := D + ( c + 3)( n − A is ampleon Y . Restricting (1) to a general fibre F of f R , we get O F ≃ cK F + ( c + 3) det E | F = O F ( − nc + ( c + 3)( n − . Hence c = n − n − K X + n (det E + ( n − f ∗ R A ) ≃ f ∗ R H. Put E ′ := E ⊗ f ∗ R A and note that E ′ is ample and such that det E ′ = det E + ( n − f ∗ R A .Thus we get ( n − K X + n (det E ′ ) ≃ f ∗ R H. Moreover, since (cid:20) K X + (cid:18) nn − (cid:19) det E ′ (cid:21) · l = (cid:20) K F + (cid:18) nn − (cid:19) det E | F (cid:21) · l = 0 , for any line l ⊂ F ∼ = P n − , we see that the nef value τ ( X, E ′ ) of the pair ( X, E ′ ) is suchthat τ ( X, E ′ ) ≥ nn − = n rk E ′ >
1. Thus by [3, proposition 4(3)] we have X ∼ = P Y ( V ) fora suitable rank n vector bundle V on Y and from [26, theorem 1.3(5)] we conclude that E ′ ∼ = ξ V ⊗ f ∗ R G , where G is an ample vector bundle of rank n − Y .Finally, in Case (b), since X is a Fano n -fold with ρ ( X ) = 1, for any effective rationalcurve Γ on X we see that Γ ∈ R and − K X · Γ ≥ l ( R ) ≥ n . Thus by [19], [20] and [9] wecan conclude that X is either P n or Q n . In the former case, since Λ( X, E ) ≥
3, by Lemma2.1 we have n − ≤ c ( E ) · l ≤ − K P n · l − n − , where l is a line in P n . Therefore by [27, theorems 3.2.1 and 3.2.3], E is isomorphic to either O (1) ⊕ n − , or O (2) ⊕ O (1) ⊕ n − . In the latter case, since X ∼ = Q n and Λ( X, E ) ≥
3, we seethat n − ≤ c ( E ) · l ≤ − K Q n · l − n − , i.e. c ( E ) · l = n −
3, and then
E ∼ = O (1) ⊕ n − by [37, lemma 3.6.1].Summing up the above discussion, we obtain the following Proposition 3.1. If E is an ample vector bundle of rank n − ≥ such that K X + det E is not nef and Λ( X, E ) ≥ , then the pair ( X, E ) is one of the following: (1 ′ ) ( P n , O P n (1) ⊕ n − ) ; (2 ′ ) ( P n , O P n (2) ⊕ O P n (1) ⊕ n − ) ; (3 ′ ) ( Q n , O Q n (1) ⊕ n − ) ; (4 ′ ) there is a vector bundle V on a smooth curve C such that X ∼ = P C ( V ) and E ∼ = ξ V ⊗ π ∗ G ′ , where π : X → C is the projection map, ξ V is the tautological linebundle of P ( V ) and G ′ is a suitable vector bundle of rank n − on C ; in particular, E | F ∼ = O P n − (1) ⊕ n − for any fibre F ∼ = P n − of the map π .Case (2). In this situation, we know that there exists an extremal rational curve C ∈ R such that − ( K X + det E ) · C = 2 and − K X · C = l ( R ). Moreover, note that (cid:20) K X + (cid:18) E · C (cid:19) det E (cid:21) · C = 0 , i.e. the nef value τ ( X, E ) of the pair ( X, det E ) is such that(3) τ ( X, E ) ≥ E ·
C .
Since − K X · C ≤ n + 1, by Lemma 2.1 we get det E · C = l ( R ) − n − ≤ det E · C = − K X · C − ≤ n − . So we have only the following three possibilities:(I) det
E · C = n − , τ ( X, E ) ≥ n +1 n − = l ( R ) n − ;(II) det E · C = n − , τ ( X, E ) ≥ nn − = l ( R ) n − ;(III) det E · C = n − , τ ( X, E ) ≥ n − n − = l ( R ) n − .In Case (I), we see that − K X · C = n + 1, i.e. l ( R ) = n + 1. From the Ionescu-Wi´sniewskiinequality, [6, lemma 6.3.12] and [9], it follows that X ∼ = P n . Since by (3) we have τ ( X, E ) ≥ n + 1 n − ≥ n − n − n − E > E · C − ( n −
3) = 2, we obtain by [3, proposition 6(a)(1)] and [26] that E isa decomposable vector bundle with c ( E ) = O P n ( n − E is ample, this gives E ∼ = O (3) ⊕ O (1) ⊕ n − or O (2) ⊕ ⊕ O (1) ⊕ n − .In Case (II), note that τ ( X, E ) ≥ nn − > n − n − n − E > E · C − ( n −
3) = 1. Moreover, l ( R ) = − K X · C = n and from [6, lemma 6.3.12] wehave only the following two cases:( A ) X is a Fano n -fold with ρ ( X ) = 1;( B ) ρ ( X ) = 2 and the contraction f R : X → Y associated to the extremal ray R = R + [ C ]is a morphism onto a smooth curve Y , whose general fibre F is a smooth Fano( n − ρ ( F ) = 1. EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 13
In Case ( A ), by [19] and [20] we obtain that X ∼ = Q n ⊂ P n +1 with c ( E ) = O Q n ( n − E ∼ = O Q n (2) ⊕ O Q n (1) ⊕ n − . In Case ( B ), we conclude by[26, theorem 1.3(10)].Finally, in Case (III) by [3, theorem 1] we have τ ( X, E ) = n − n − l ( R )rk E . Since l ( R ) = − K X · C = n − n = r + 3 ≥
5, we can conclude by [3, propositions 3 , ρ ( X ) = 1 then from [3,theorem 1(2)] it follows that E ∼ = O X (1) ⊕ n − , which leads to Case (5)( α ) of Theorem 1.1.The above discussion can be summed up in the following Proposition 3.2. If E is an ample vector bundle of rank n − ≥ such that K X + det E is not nef and Λ( X, E ) = 2 , then the pair ( X, E ) is one of the following: (1 ′′ ) ( P n , O P n (3) ⊕ O P n (1) ⊕ n − ) ; (2 ′′ ) ( P n , O P n (2) ⊕ ⊕ O P n (1) ⊕ n − ) ; (3 ′′ ) ( Q n , O Q n (2) ⊕ O Q n (1) ⊕ n − ) ; (4 ′′ ) there exist a vector bundle V on a smooth curve C such that X ∼ = P C ( V ) and anexact sequence → π ∗ L ⊗ ξ ⊗ V → E → π ∗ G ⊗ ξ V → for some line bundle L on C and some vector bundle G of rank n − ≥ on C ,where π : P ( V ) → C is the projection map and ξ V is the tautological line bundle on P C ( V ) ; in particular, we have E | F ∼ = O P n − (2) ⊕O P n − (1) ⊕ n − for any fibre F ∼ = P n − of the map π ; (5 ′′ ) X is a Del Pezzo n -fold with ρ ( X ) = 1 and E ∼ = L ⊕ n − , where L is the amplegenerator of Pic( X ) ; (6 ′′ ) X is a quadric fibration, q : X → C , of the relative Picard number one over a smoothcurve C (i.e. X is a section of a divisor of relative degree two in a projective space P C ( F ) , where F is a vector bundle of rank n + 1 on C ) and there exist a q -ampleline bundle O X (1) on X and an ample vector bundle G of rank n − on C suchthat E ∼ = O X (1) ⊗ q ∗ G , where O X (1) | F ≃ O Q n − (1) for any fibre F ∼ = Q n − ⊂ P n ofthe map q ; (7 ′′ ) there exists a P n − -fibration π : X → S , locally trivial in the complex (or ´etale)topology, over a smooth surface S such that E | F ∼ = O P n − (1) ⊕ n − for any closed fiber F ∼ = P n − of the map π ; (8 ′′ ) there exists a smooth variety X ′ and a morphism ϕ : X → X ′ expressing X as theblowing-up of X ′ at a finite set of points B and an ample vector bundle E ′ on X ′ such that E ⊗ ([ ϕ − ( B )]) ∼ = ϕ ∗ E ′ and K X ′ + τ ′ det E ′ is nef, where τ ′ < n − n − is thenefvalue of the pair ( X ′ , det E ′ ) . Moreover, E | E ∼ = O P n − (1) ⊕ n − , where E is anyirreducible component of the exceptional locus of the map ϕ . Case (3). By arguing as in [17, p.77] and [37, proposition 3.5], note that in fact we get L · C = 1, where L := ( m − K X + m det E is ample on X and m ≥ n − ≥ m := min { (det E ) · C | C extremal rational curve s.t. ( K X + det E ) · C = − } . So, define V := E ⊕ L . Then V is an ample vector bundle of rank n − X such that( K X + det V ) · C = ( K X + det E ) · C + L · C = − . Therefore K X + det V cannot be ample on X . Note also that K X + det V must be nef.Otherwise, by Theorem 2.1 we could find an extremal ray R on X such that − ≥ ( K X + det V ) · R ≥ ( K X + det E ) · R + 1 , i.e. ( K X + det E ) · R ≤ −
2, but this would contradict Λ( X, E ) = 1.Thus assume that K X + det V is nef but not big. From [2, theorem 5.1(2)] we knowthat there exists a morphism Φ : X → W onto a normal variety W supported by (a largemultiple of) K X + det V and dim W ≤
3. Moreover, let F be a general fibre of Φ. We havethe following according to s := dim W :( i ) if s = 0 then X is a Fano n -fold with K X + det V = O X . If n ≥
6, then b ( X ) = 1except if X ∼ = P × P and E ∼ = O (1 , ⊕ ;( i ) if s = 1 then W is a smooth curve and Φ is a flat (equidimensional) map. If n ≥ F, V| F ) is one ofthe following pairs:( a ) ( P n − , O P n − (2) ⊕ ⊕ O P n − (1) ⊕ n − );( b ) ( P n − , O P n − (3) ⊕ O P n − (1) ⊕ n − );( c ) ( Q n − , O Q n − (2) ⊕ O Q n − (1) ⊕ n − );( d ) ( Q , S (2) ⊕ O Q n − (1)), where S is the spinor bundle on Q ⊂ P ;( e ) F is a Del Pezzo ( n − ρ ( F ) = 1, − K F ≃ ( n − H F , Pic( F ) ∼ = Z [ H F ]and V| F ∼ = H ⊕ n − F ;( f ) ( P × P , O (1 , ⊕ ).If the general fibre is P n − then X is a classical scroll, while if the general fibre is Q n − ⊂ P n then X is a quadric bundle;( i ) if s = 2 then W is a smooth surface, Φ is a flat map and by [11] and [30], ( F, V| F )is one of the following pairs:( α ) ( P n − , O P n − (2) ⊕ O P n − (1) ⊕ n − );( β ) ( P n − , T P n − ), where T P n − is the tangent bundle on P n − ;( γ ) ( Q n − , O Q n − (1) ⊕ n − ).In particular, if the general fibre is P n − then all the fibres are P n − ;( i ) if s = 3 then W is a smooth 3-fold and all the fibres of Φ are isomorphic to P n − .Furthermore, since ( K X + det V ) · C = 0, note that L F · C = L · C = 1, C being an extremalrational curve of X which belongs into a fibre of Φ. Moreover, from ( K X + det E ) · C = − − K X · C = det E · C + 1 ≥ m + 1 = ( m + 1) L · C, EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 15 and then the nef value τ ( X, L ) of the polarized pair (
X, L ) is greater than or equal to m + 1 ≥ n − Case ( i ). Observe that X is a Fano n -fold with − K X ≃ ηL and η = τ ( X, L ) ≥ m + 1 ≥ n − ≥ . If ρ ( X ) = 1, then by [6, chapter 7] we know that X is either P n , Q n , a Del Pezzo n -fold with − K X ≃ ( n − L , or a Mukai n -fold with − K X ≃ ( n − L . Since det E · C = − K X · C − ≥ η −
1, note that m ≥ η −
1, that is, η ≤ m + 1. Thus η = τ ( X, L ) = m + 1 and so we get L = ( η − K X + ( η −
1) det E = − η ( η − L + ( η −
1) det E , i.e. det E = ( η − L , where Pic( X ) ∼ = Z [ L ]. Let us proceed with a case–by–case analysis.First of all, assume that X ∼ = P n . Note that L = O P n (1) and c ( E ) = O P n ( n ). Withoutloss of generality, we can write E ∼ = V ⊕ V , where V is an indecomposable vector bundleon P n . Put r i = rk V i and note that r + r = n − ≤ r ≤ n −
3. Furthermore, ifdeg V i ≤ r i + 1, then V i is an uniform vector bundle on P n of rank r i ≤ n −
3, i.e. V i is asum of line bundles on P n . Moreover, since V is an ample vector bundle on P n such thatdeg V ≤ deg E = n , if r = 2 then by [1, theorem (9.1)] we see that V splits. Thus we canassume that r ≥
3. Then we get the following three possibilities:(a) r = n − ≥ V = n ;(b) 3 ≤ r ≤ n −
4, deg V = r + 2 and deg V = r + 1;(c) 3 ≤ r ≤ n −
4, deg V = r + 3 and deg V = r .By Lemma 2.2, we have V ( −
1) is a nef vector bundle on P n either in Case (b), or in Case(c) for 3 ≤ r ≤ n −
6. Note that Case (b) does not occur. Indeed, since V ( −
1) is nefand c ( V ( − V ( −
1) is decomposable, but thisleads to a contradiction. In Case (c), since V is ample, we get that V ∼ = O P n (1) ⊕ r , thatis, E ∼ = O P n (1) ⊕ r ⊕ V . Finally, assume that V ( −
1) is globally generated on P n . Since n ≥
6, from Corollary 2.1 we know that V splits, a contradiction. This completes Case (2)of Theorem 1.1.Suppose now that X ∼ = Q n ⊂ P n +1 . Note that L = O Q n (1) and c ( E ) = O Q n ( n − E ∼ = V ⊕ V , where V is an indecomposable vector bundle on Q n . Put r i = rk V i and note that r + r = n − ≤ r ≤ n −
3. Furthermore, if deg V i ≤ r i + 1,then V i is an uniform vector bundle on Q n of rank r i ≤ n − V i is a sum of line bundles on Q n . Thus deg V ≥ r + 2 and thendeg V = n − − deg V ≤ n − − r = r . This shows that deg V = r and deg V = r + 2, that is, E ∼ = O Q n (1) ⊕ n − − r ⊕ V . Since V is an ample vector bundle on Q n such that deg V ≤ deg E = n −
1, we deduce that V is a Fano bundle on Q n . Thus if r = 2, then by [1, theorem (2.4)(2)] we see that either V splits, or n = 5 and V = C ( a ), where C is a Cayley bundle on Q such that c ( C ) = − Since 4 = c ( V ) = − a , we see that a is not an integer, but this is not possible. Hence V splits for r = 2. On the other hand, if r ≥ V to any line on Q n ,we have the only possible splitting types are (3 , , , ...,
1) or (2 , , , ..., V is auniform vector bundle of splitting type (3 , , , ...,
1) and by [14] V is a direct sum of linebundles on Q n , or V has the generic splitting type (2 , , , ..., V ( −
1) is nef for 3 ≤ r ≤ n −
6, and it is not globallygenerated for n ≥
7. This gives Case (4) of Theorem 1.1.Assume that X is a Fano n -fold of index n −
1. Since c ( E ) = O X ( n − l of ( X, H ), where H is the ample generator of Pic( X ), it happens that E | l ∼ = H | ⊕ n − l ⊕ H | l , i.e. E is a uniform vector bundle on X with splitting type (2 , , , ..., β ) of Theorem 1.1.Finally, suppose that X is a Fano n -fold of index n −
2. From [18, theorem 2.10], it followsby hyperplane sections that there exists at least a line l of ( X, L ), where Pic( X ) = Z [ L ].Recall that V = E ⊕ L . Since ρ ( X ) = 1 and τ ( X, V ) ≥ n − V = − K X · l rk V , by [5, proposition 1.2] and [3, theorem 1(2)], we obtain that τ ( X, V ) = − K X · l rk V and so V ∼ = L ⊕ n − , i.e. E ∼ = L ⊕ n − . This gives Case (6) in Theorem 1.1.Assume now that ρ ( X ) ≥
2. Since η ≥ n − ≥ n +12 , from [25], [38] and [39], it followsthat η ≤ n +22 , i.e. 5 ≤ n ≤
6, and that ( X, V , L ) is one of the following triplets:- ( P × P , O P × P (1 , ⊕ , O P × P (1 , P × Q , O P × Q (1 , ⊕ , O P × Q (1 , P P ( T ) , [ ξ T ] ⊕ , ξ T ), where T is the tangent bundle of P and ξ T is the tautologicalline bundle;- ( P P ( O (2) ⊕ O (1) ⊕ ) , [ ξ ] ⊕ , ξ ), where ξ is the tautological line bundle.Since V = E ⊕ L , we obtain Cases (7) to (10) of Theorem 1.1. Case ( i ). Note that L | F = − ( K X + det E ) | F = − K F − det E | F . Then K F + mL | F = K F + m [ − K F − det E | F ] = − [( m − K F + m det E | F ] = − L | F , since L := ( m − K X + m det E . This gives L | F = − K F − det E | F = ( m + 1) L | F − det E | F , i.e. det E | F = mL | F . Moreover, observe also that the extremal rational curve C such that L · C = 1 is contained in a fibre of Φ, since ( K X + det V ) · C = 0. Keeping in mind that V = E ⊕ L and that L = ( m − K X + m det E with m ≥ n −
3, by all the above remarkswe can easily obtain Cases (11)( c )( d ) , (12)( ii )( iii ), and (13) of Theorem 1.1. Case ( i ). Note that Case ( β ) cannot occur, since V| F = E | F ⊕ L | F is a decomposablevector bundle on P n − with n − ≥
3. Moreover, since the extremal rational curve C such EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 17 that L · C = 1 is contained in a fibre of Φ, we obtain immediately Cases (14) and (16) ofTheorem 1.1. Case ( i ). Since K F +det V| F = O F , where V| F is an ample vector bundle of rank dim F +1,from [29] we deduce that ( F, V| F ) ∼ = ( P n − , O P n − (1) ⊕ n − ) and then L | F = O P n − (1) for anyfibre F ∼ = P n − of Φ. Moreover, as observed in Case ( i ), the extremal rational curve C such that L · C = 1 is contained in a fibre of Φ. Thus by [10, (2.12)] we get immediatelyCase (17) of Theorem 1.1.Finally, suppose that K X + det V is nef and big, but not ample. Then from [2, theorem5.1(3)] we deduce that a high multiple of K X + det V defines a birational map, ϕ : X → X ′ ,which contracts an extremal face. Let R i = R + [ C i ] be the extremal rays spanning this facefor some i in a finite set of index. Call ρ i : X → W i the contraction associated to one of the R i . Then we have each ρ i is birational and of divisorial type. If D i is one of the exceptionaldivisors and Z i = ρ i ( D i ), we know that dim Z i ≤ j ) dim Z i = 0 , D i ∼ = P n − and ([ D i ] D i , V| D i ) is either ( α ) ( O ( − , O (1) ⊕ n − ), or ( β )( O ( − , O (1) ⊕ n − ⊕ O (2));( j ) dim Z i = 0 , D i is a (possible singular) quadric Q n − ⊂ P n and [ D i ] D i = O ( − V| D i ∼ = O (1) ⊕ n − ;( j ) dim Z i = 1, W i and Z i are smooth projective varieties and ρ i is the blow-up of W i along Z i ; moreover, V| F i ∼ = O (1) ⊕ n − for any fibre F i ∼ = P n − of D i → Z i .Furthermore, we know also that ϕ is a composition of disjoint extremal contractions as in( j ) , ( j ) and ( j ). Note that the extremal rational curves C i such that R i = R + [ C i ] and L · C i = 1 are contained in D i , since ( K X + det V ) · C i = 0. Moreoverdet E · C i = − K X · C i − (cid:26) n − j )( α ) n − ρ i is a contraction associated to an extremal ray as in Case ( j )( α ),then ϕ : X → X ′ contracts an extremal face spanned by only extremal rays as in ( j )( α ).So, keeping in mind that V = E ⊕
L, L · C i = 1 and L = ( m − K X + m det E with m ≥ n − ≥
2, we get Cases (19) and (20) of Theorem 1.1.Taking into account the above discussion and the Propositions 3.1 and 3.2, we obtainthe statement of Theorem 1.1. (cid:3)
Finally, under the extra assumption that ρ ( X ) = 1 and E ( −
1) :=
E ⊗ O X ( −
1) is globallygenerated on X , we have the following Corollary 3.1.
Let E be an ample vector bundle of rank r ≥ on a smooth complexprojective variety X of dimension n with ρ ( X ) = 1 . Assume that E ( − is globally generatedon X . If r = n − , then K X + det E is nef except when ( X, E ) is one of the following pairs: ( a ) ( P n , ⊕ n − i =1 O P n ( a i ) , where all the a i ’s are positive integers such that P n − i =1 a i ≤ n ; ( b ) ( Q n , ⊕ n − j =1 O Q n ( b j )) , where all the b j ’s are positive integers such that P n − j =1 b j ≤ n − ; ( c ) X ∼ = Q and E is an indecomposable Fano vector bundle on Q of rank such that c ( E ) = 5 and its generic splitting type is (2 , , ; ( d ) X is a Fano n -fold of index n − with Pic( X ) generated by an ample line bundle H and either ( α ) E ∼ = H ⊕ n − , or ( β ) E | l ∼ = H ⊕ n − l ⊕ H ⊗ l for every line l of ( X, H ) ; ( e ) X is a Fano n -fold of index n − with Pic( X ) generated by an ample line bundle L and E ∼ = L ⊕ n − ; Let us show that in the above result Case ( c ) really exists. Example . Let Q ⊂ P be the 6-dimensional quadric hypersurface endowed with a spinorbundle S of rank 4. Note that c ( S ν ) = c ( S ) = 0 and that S ν is globally generated on Q , where S ν is the dual bundle of S (see [28, (2.8)(ii), (2.9)]). Consider now the followingexact sequence 0 → O Q → S ν → F → S ν . Then F is a globally generated vector bundle on Q of rank 3such that c ( F ) = − c ( S ) = 2 ([28, (2.9)]). Put E := F ⊗ O Q (1). This gives an exampleof an ample indecomposable vector bundle E of rank 3 on Q with c ( E ) = 5 and such that E ⊗ O Q ( − ∼ = S ν / O Q is globally generated on Q .4. Two applications.
Within the adjunction theory, let us give here two easy and immediate consequences ofTheorem 1.1.4.1.
Ample vector bundles and special varieties in adjunction theory.
Let E bean ample vector bundle of rank r ≥ n -fold X . Given a smooth projective variety Z , the classification of such varieties X containing Z as an ample divisor occupies animportant position in the theory of polarized varieties. Moreover, it is well-known that thestructure of Z imposes severe restrictions on that of X . Inspired by this philosophy, wegeneralize some results on ample divisors to ample vector bundles. In particular, by usinga peculiarity of adjoint bundles, Lanteri and Maeda [15] investigated when K Z + tH Z is notnef for t ≥ dim Z −
2, where H is an ample line bundle on X and H Z := H | Z . So, followinghere this idea, we deal with the next case t = dim Z −
3. More precisely, we give a detaileddescription of triplets ( X, E , H ) as above under the assumption that K Z + (dim Z − H Z is not nef for dim Z ≥ Z = 3, we obtain the following Proposition 4.1.
Let X be an n -fold. Let E be an ample vector bundle of rank r ≥ on X such that there exists a global section s ∈ Γ( E ) whose zero locus Z = ( s ) ⊂ X is a smoothsubvariety of dimension n − r = 3 . Then K Z is nef if and only if the pair ( X, E ) is not asin Theorem 1.1 with n ≥ EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 19
Proof.
To prove the ‘if’ part, assume that K Z is not nef. Recalling that K Z = ( K X +det E ) Z , we deduce that K X + det E is not nef. Therefore we have ( X, E ) is as in Theorem1.1.Conversely, to prove the ‘only if’ part of the statement, assume that the pairs ( X, E ) areas in Theorem 1.1. Therefore, if ( X, E ) is as in cases (1) to (10), it is easy to see that − K Z is ample, and so K Z cannot be nef.Suppose that ( X, E ) is as in case (11). Let π be the projection map X → C and let s F denote the restriction of the global section s to a general fiber F of π . Then s F ∈ Γ( E | F ).Putting D := ( s F ) = Z ∩ F , we have dim D ≥
2. On the other hand, since Z is irreducible,we have dim D < dim Z = 3, i.e. dim D = 2. Moreover, note that [ D ] D ∼ = O D . Thus bythe adjunction formula we have ( K Z ) D ∼ = K D − [ D ] D ∼ = K D . Since − K D is ample, thisimplies that K Z is not nef.In cases (12) and (13), by similar arguments as in (11), we see that K Z cannot be nef.Consider now case (14). Let π be the projection map X → S . Since s F ∈ Γ( E | F ) for ageneral fibre F ∼ = P n − of π , putting D := Z ∩ F we have dim D ≥ n − − r = dim Z − D ⊂ F, D ⊂ Z , we deduce that( K Z ) D ∼ = ( K X + det E ) | D ∼ = ( K F + det E | F ) | D = O F ( − | D . This shows that K Z cannot be nef.In cases (15) and (16), by arguing as in case (14), we deduce that K Z is not nef.Assume now that ( X, E ) is as in case (17). In this situation, from section 2 we knowthat L := ( n − K X + ( n −
3) det E is an ample line bundle on X such that L F = O P n − (1)for any fibre F ∼ = P n − of π : X → V . Since X ∼ = P V ( F ), we see that there exists anample vector bundle V of rank n − V such that ( X, L ) ∼ = ( P ( V ) , ξ V ), where ξ V is thetautological line bundle on P ( V ). Thus(4) K X + det E ∼ = − ( n − ξ V + π ∗ ( K V + det V ) + det E . By [det
E − ( n − ξ V ] | F = O F , the relation (4) becomes K X + det E ∼ = − ξ V + π ∗ ( K V + det V + H ) , for some H ∈
Pic( V ). Choose a very ample line bundle H on V such that H ′ := K V +det V + H + H is ample on V . Put L ′ := L + π ∗ H . Note that also L ′ is ample and(5) K X + det E + L ′ ∼ = π ∗ ( H ′ ) . Since dim Z = 3, [16, theorem 1.1] tells us that ρ ( X ) = ρ ( Z ). Obviously, ρ ( X ) = ρ ( V ) + 1.Hence ρ ( Z ) = ρ ( V ) + 1 and this shows that Z cannot be isomorphic to V via π . Moreover,since s F ∈ Γ( O P n − (1) ⊕ n − ) for any fibre F ∼ = P n − of π , D := ( s F ) = Z ∩ F is a linearsubspace of F , i.e. D has degree one in F ∼ = P n − . Thus π Z : Z → V is neither anisomorphism nor a finite-to-one map onto V . So, there exists a curve γ ⊂ Z such that π Z ( γ ) is a point of V . Restricting (5) to Z , we get K Z + L ′ Z ∼ = π ∗ Z H ′ , and this gives ( K Z + L ′ Z ) · γ = 0, i.e. K Z · γ = − L ′ Z · γ < L ′ Z . This implies that K Z is not nef.Now, consider case (18). Let s E denote the restriction of s to E . Therefore s E ∈ Γ( O P n − (1) ⊕ n − ). Thus D := ( s E ) = Z ∩ E is a linear subspace of E and we havedim D ≥ n − − ( n −
3) = 2 = dim Z − . Obviously, D ⊆ Z . If D = Z , then Z ∼ = P , but this gives the contradiction ρ ( X ) = ρ ( Z ) = 1 by [16, theorem 1.1]. Therefore D ( Z , and the irreducibility of Z implies thatdim D < dim Z , i.e. D ∼ = P . Furthermore, O D ( D ) ∼ = O Z ( Z ∩ E ) | D ∼ = O X ( E ) | D ∼ = O E ( E ) | D ∼ = O P ( − . Thus by the adjunction formula applied to D ⊂ Z , we get( K Z ) D ∼ = K D − O D ( D ) = O P ( −
3) + O P (1) = O P ( − . This shows that K Z is not nef.Suppose that ( X, E ) is as in case (19). Let s E i denote the restriction of s to E i . Then s E i ∈ Γ( E | E i ). So D := ( s E i ) = Z ∩ E i is a linear subspace of E i anddim D ≥ n − − ( n −
3) = 2 = dim Z − . By [16, theorem 1.1] note that D = Z since dim Z = 3. Thus D ( Z and the irreducibilityof Z implies that D is P . Furthermore, O D ( D ) ∼ = O Z ( Z ∩ E i ) | D ∼ = O E i ( E i ) | D ∼ = O P ( − D ⊂ Z , we get ( K Z ) | D ∼ = K D − O D ( D ) = O D ( − K Z cannot be nef.Finally, assume that we are in case (20). Suppose that D i is as in cases (j) and (jj). Let s D i denote the restriction of s to D i . Then s D i ∈ Γ( E | D i ). So D := ( s D i ) = Z ∩ D i is aquadric hypersurface contained in D i anddim D ≥ n − − ( n −
3) = 2 = dim Z − . By [16, theorem 1.1] note that D = Z since dim Z = 3. Thus D ( Z and the irreducibilityof Z implies that D is a (possible singular) quadric Q ⊂ P . Furthermore, O D ( D ) ∼ = O Z ( Z ∩ D i ) | D ∼ = O D i ( D i ) | D ∼ = O Q ( − D ⊂ Z , we have ( K Z ) | D ∼ = K D − O D ( D ) = O D ( − K Z is not nef.Suppose now that D i is as in case (jjj). Let s F i be the restriction of s to any fiber F i ∼ = P n − of D i → Z i . Then s F i ∈ Γ( O P n − (1) ⊕ n − ). So D := ( s F i ) = Z ∩ F i is a linearsubspace of F i and dim D ≥ n − − ( n −
3) = 1 = dim Z − . EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 21
Since ρ ( X ) = 1, by [16, theorem 1.1] we see that D = Z . Thus the irreducibility of Z implies that 1 ≤ dim D ≤
2. Since D ⊂ Z and D ⊂ F i ⊂ D i , we obtain that K Z | D ∼ = ( K X + det E ) | D ∼ = [( K X + det E ) D i ] | D ∼ = [ K D i − O D i ( D i ) + det E D i ] | D ∼ = ∼ = [ K F − O D i ( D i ) | F + det E F ] | D ∼ = O F ( − | D . This shows that K Z cannot be nef. (cid:3) When dim Z ≥
4, having in mind [15, theorems 1 , Proposition 4.2.
Let X be an n -fold. Let E be an ample vector bundle of rank r ≥ on X such that there exists a global section s ∈ Γ( E ) whose zero locus Z = ( s ) ⊂ X is a smoothsubvariety of dimension n − r ≥ and let H be an ample line bundle on X . Suppose that K Z + (dim Z − H Z is nef. Then K Z + (dim Z − H Z is nef if and only if ( X, E , H ) isnot any of the following triplets: (1 ′ ) X ∼ = P n and ( E , H ) is one of the following pairs: ( i ) ( O P n (1) ⊕ n − , O P n (4)) ; ( ii ) ( O P n (1) ⊕ n − , O P n (3)) ; ( iii ) ( O P n (1) ⊕ n − ⊕ O P n (2) , O P n (3)) ; ( iv ) ( O P n (1) ⊕ n − ⊕ O P n (2) , O P n (2)) ; ( v ) ( O P n (1) ⊕ n − ⊕ O P n (2) ⊕ , O P n (2)) ; ( vi ) ( O P n (1) ⊕ n − ⊕ O P n (3) , O P n (2)) ; ( vii ) ( O P n (1) ⊕ n − , O P n (2)) ; ( viii ) ( O P n (1) ⊕ n − ⊕ O P n (2) , O P n (2)) ; ( ix ) ( O P n (1) ⊕ n − , O P n (2)) ; ( x ) c ( E ) = r + 3 and H = O P n (1) ; (2 ′ ) X ∼ = Q n and ( E , H ) is one of the following pairs: (l) ( O Q n (1) ⊕ n − , O Q n (3)) ; (ll) ( O Q n (1) ⊕ n − , O Q n (2)) ; (lll) ( O Q n (1) ⊕ n − ⊕ O Q n (2) , O Q n (2)) ; (lv) ( O Q n (1) ⊕ n − , O Q n (2)) ; (v) c ( E ) = O Q n ( r + 2) and H = O Q n (1) ; (3 ′ ) X is a Del Pezzo n -fold with Pic( X ) ∼ = Z [ O X (1)] and one of the following possibil-ities can occur: ( α ) E ∼ = O X (1) ⊕ r and H = O X (1) ; ( β ) for any line l of ( X, O X (1)) either E | l ∼ = O l (1) ⊕ n − and H | l = O l (2) , or E | l ∼ = O l (1) ⊕ r − ⊕ O l (2) and H | l = O l (1) ; (4 ′ ) X is a Mukai n -fold with Pic( X ) ∼ = Z [ O X (1)] and E ∼ = O X (1) ⊕ r ; (5 ′ ) ( P × P , O P × P (1 , ⊕ , O P × P (1 , ; (6 ′ ) there exists a vector bundle V on a smooth curve C such that X ∼ = P C ( V ) ; moreover,for any fibre F ∼ = P n − of X → C , we have ( E | F , H F ) is isomorphic to one of thefollowing pairs: ( a ) ( O P n − (1) ⊕ n − , O P n − (2)) ; ( b ) ( O P n − (2) ⊕ O P n − (1) ⊕ n − , O P n − (2)) ; ( c ) ( O P n − (1) ⊕ n − , O P n − (2)) ; ( d ) ( O P n − (1) ⊕ n − , O P n − (3)) ; ( e ) ( O P n − (3) ⊕ O P n − (1) ⊕ r − , O P n − (1)) ; ( f ) ( O P n − (2) ⊕ ⊕ O P n − (1) ⊕ r − , O P n − (1)) ; (7 ′ ) X is a section of a divisor of relative degree two in a projective space P C ( G ) , where G is a vector bundle of rank n + 1 on a smooth curve C ; moreover, for any smoothfibre F ∼ = Q n − of X → C , where Q n − is a smooth quadric hypersurface of P n , wehave ( E | F , H F ) is isomorphic to either ( O Q n − (1) ⊕ n − , O Q n − (2)) , or ( O Q n − (2) ⊕O Q n − (1) ⊕ r − , O Q n − (1)) ; (8 ′ ) the map Φ : X → C associated to the linear system | ( n − K X +( n −
2) det
E | makes X a Del Pezzo fibration over a smooth curve C ; moreover, any general smooth fibre F of Φ is a Del Pezzo ( n − -fold with Pic( F ) ∼ = Z [ O F (1)] such that E | F ∼ = O F (1) ⊕ r and H F ≃ O F (1) ; (9 ′ ) there exists a vector bundle V on a smooth surface S such that X ∼ = P S ( V ) ; more-over, for any fibre F ∼ = P n − of X → S , we have ( E | F , H F ) is isomorphic to either ( O P n − (1) ⊕ n − , O P n − (2)) , or ( O P n − (2) ⊕ O P n − (1) ⊕ r − , O P n − (1)) ; (10 ′ ) the map ψ : X → S associated to the linear system | ( n − K X + ( n −
2) det
E | makes X a quadric fibration over a smooth surface S ; moreover, for any generalfibre F ∼ = Q n − we have ( E | F , H F ) ∼ = ( O Q n − (1) ⊕ r , O Q n − (1)) , where Q n − is asmooth quadric hypersurface of P n − ; (11 ′ ) there is a vector bundle F on a smooth -fold V such that X ∼ = P V ( F ) ; moreover,for any fibre F ∼ = P n − of X → V , we get ( E | F , H F ) ∼ = ( O P n − (1) ⊕ r , O P n − (1)) ; (12 ′ ) the map ψ : X → X ′ associated to the linear system | ( n − K X + ( n −
1) det
E | is a birational morphism which contracts an extremal face spanned by extremalrays R i for some i in a finite set of index. Let ψ i : X → X i be the contrac-tion associated to R i . Then each ψ i is birational and of divisorial type; moreover,if E i is an exceptional divisor of ψ i , then E i ∼ = P n − and ([ E i ] E i , E | E i , H E i ) ∼ =( O P n − ( − , O P n − (1) ⊕ r , O P n − (1)) ; (13 ′ ) the map φ : X → X ′ associated to the linear system | ( n − K X + ( n −
2) det
E | is a birational morphism which contracts an extremal face. Let R i be the extremalrays spanning this face for some i in a finite set of index. Call ρ i : X → W i thecontraction associated to one of the R i . Then we have each ρ i is birational and ofdivisorial type; if D i is one of the exceptional divisors and Z i = ρ i ( D i ) , we have dim Z i ≤ and one of the following possibilities can occur: (j) dim Z i = 0 , D i ∼ = P n − and ([ D i ] D i , E | D i , H D i ) is either ( j ) ( O P n − ( − , O P n − (1) ⊕ n − , O P n − (2)) ; or ( j ) ( O P n − ( − , O P n − (2) ⊕ O P n − (1) ⊕ r − , O P n − (1)) ; (jj) dim Z i = 0 , D i is a (possible singular) quadric hypersurface Q n − ⊂ P n and ([ D i ] D i , E | D i , H D i ) ∼ = ( O Q n − ( − , O Q n − (1) ⊕ r , O Q n − (1)) ; (jjj) dim Z i = 1 , W i and Z i are smooth projective varieties and ρ i is the blow-up of W i along Z i ; furthermore, the triplets ( O D i ( D i ) | F i , E | F i , H F i ) are isomorphicto ( O P n − ( − , O P n − (1) ⊕ r , O P n − (1)) for any fibre F i ∼ = P n − of the restrictionmap ρ i | D i : D i → Z i . EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 23
Moreover, the map φ is a composition of disjoint extremal contractions as in (j) , (jj) and (jjj) . Proof.
To prove the ‘if’ part, suppose that K Z + (dim Z − H Z is not nef. Set E ′ := E ⊕ H ⊕ n − r − . Then E ′ is an ample vector bundle of rank n − X . Now, we get( K X + det E ′ ) | Z = [ K X + det E + (dim Z − H ] | Z ∼ = K Z + (dim Z − H Z . So, K X + det E ′ is not nef on X . Thus we deduce that ( X, E ′ ) is as in Theorem 1.1.Keeping in mind that E ′ = E ⊕ H ⊕ n − r − with n ≥ r + 4 ≥
6, by [15, theorems 1 , a ), (12)( i )( iii ), (18) of Theorem 1.1cannot occur. Moreover, as to case (15), we have H | F ∼ = O P n − (1) for any closed fiber F ∼ = P n − of π : X → S . This shows that ( X, H ) is an adjunction-theoretic scroll over S and by [6, proposition 3.2.1 and theorem 14.1.1] and [10, (2.12)] we can conclude thatthere exists a suitable vector bundle V of rank n − S such that ( X, H ) ∼ = ( P S ( V ) , ξ V ),where ξ V is the tautological line bundle on P S ( V ). From [15, theorem 3(14)] it follows that K Z + (dim Z − H Z cannot be nef, a contradiction. In all the other cases, by [15, theorems1 , X, E , H ) is as in cases (1 ′ ) to (5 ′ ), by the adjunction formula we see that K Z + (dim Z − H Z is not nef.Suppose we are in case (6 ′ ). Let π be the projection map X → C and let s F denote therestriction of the global section s to a general fibre F ∼ = P n − of π . Then s F ∈ Γ( E | F ) andputting D := ( s F ) = Z ∩ F , we have dim D ≥ n − r − ≥
3. On the other hand, since Z isirreducible, we have dim D < dim Z = n − r , i.e. dim D = dim Z − n − r −
1. Moreover,note that [ D ] D ∼ = O D . Thus ( K Z ) D ∼ = K D − [ D ] D ∼ = K D . Hence − [ K Z +(dim Z − H Z ] | D = − [ K D + (dim D − H D ] is ample on D ⊂ Z and this shows that K Z + (dim Z − H Z cannot be nef.In cases (7 ′ ) and (8 ′ ), by a similar argument as in (5 ′ ), we can see that K Z +(dim Z − H Z is not nef.Consider now case (9 ′ ). Let π be the projection map X → S . Since s F ∈ Γ( E | F ) for ageneral fibre F ∼ = P n − of π , putting D := Z ∩ F = ( s F ) we get dim D ≥ n − r − Z − ≥
2. Since D ⊂ F and D ⊂ Z , we deduce in both situations that( K Z + (dim Z − H Z ) | D = ( K X + det E + (dim Z − H ) | D == [ K F + det E | F + (dim Z − H F ] | D ∼ = O P n − ( − | D . This implies that K Z + (dim Z − H Z cannot be nef.By arguing as in (9 ′ ), we can see that also in case (10 ′ ) the line bundle K Z +(dim Z − H Z is not nef.Assume now that ( X, E , H ) is as in (11 ′ ). Let π be the projection map P V ( V ) → V .Since s F ∈ Γ( E | F ) for any fibre F ∼ = P n − of π , D := ( s F ) = Z ∩ F is not empty for anyfibre F , i.e. the restriction map π Z : Z → V of π to Z is surjective. On the other hand, we have dim Z > dim D ≥ n − r − Z −
3, because of the irreducibility of Z . Sodim Z − ≤ dim D ≤ dim Z −
1. Furthermore, since K X ∼ = − ( n − ξ V + π ∗ ( K V + det V ),where ξ V is the tautological line bundle on P V ( V ), we get(6) K X + det E + ( n − r − H = [det E + ( n − r − H − ( n − ξ V ] + π ∗ ( K V + det V ) − ξ V . Since [det E + ( n − r − H − ( n − ξ V ] | F ∼ = O F , it follows thatdet E + ( n − r − H − ( n − ξ V ∼ = π ∗ D , for some D ∈
Pic( V ). By (6) we obtain that[ K Z + (dim Z − H Z ] | D = ([ K X + det E + ( n − r − H ] Z ) | D ∼ = ∼ = ([ K X + det E + ( n − r − H ] F ) | D = − ( ξ V F ) | D ∼ = O P n − ( − | D , for a general fibre D = F | Z of π Z . This implies that K Z + (dim Z − H Z is not nef.Suppose we are in case (12 ′ ). Let s E i denote the restriction of the section s to E i . Then s E i ∈ Γ( E | E i ) and so D := ( s E i ) = E i ∩ Z is a linear subspace of E i ∼ = P n − . Moreover,dim D ≥ n − r − Z − ≥
3. Obviously, D ⊆ Z . If D = Z , then Z ⊂ E i ⊂ X and we have the canonical surjection N Z/X → N E i /X | Z between normal bundles. Since N Z/X ∼ = E | Z is ample, we deduce that N E i /X | Z is ample, but this gives a contradictionsince E i is an exceptional divisor on X . Thus D $ Z and the irreducibility of Z impliesthat dim D < dim Z , i.e. dim D = dim Z − n − r −
1. Therefore D is isomorphic to P n − r − and H Z = H D = ( H E i ) | D = O P n − r (1)Note that O D ( D ) ∼ = O Z ( Z ∩ E i ) | D ∼ = O X ( E i ) | D ∼ = O E i ( E i ) | D ∼ = O P n − r − ( − D, H D , O D ( D )) ∼ = ( P n − r − , O P n − r − (1) , O P n − r − ( − . Thus by the adjunction formula applied to D ⊂ Z , we obtain that( K Z ) | D ∼ = K D − O D ( D ) ∼ = O P n − r − ( − n + r + 2) . Hence ( K Z + (dim Z − H Z ) | D ∼ = O D ( −
1) and this shows that K Z + (dim Z − H Z cannotbe nef.Consider now case (13 ′ )(j). Let s D i denote the restriction of the section s to D i . Then s D i ∈ Γ( E | D i ) and so D := ( s D i ) = D i ∩ Z is either a linear subspace of D i ∼ = P n − ,or a (possible singular) quadric hypersurface on D i . Moreover, dim D ≥ n − r − Z − ≥
3. Obviously, D ⊆ Z . If D = Z , then Z ⊂ D i ⊂ X and we have the canonicalsurjection N Z/X → N D i /X | Z between normal bundles. Since N Z/X ∼ = E | Z is ample, wededuce that N D i /X | Z is ample, but this gives a contradiction since D i is an exceptionaldivisor on X . Thus D $ Z and the irreducibility of Z implies that dim D < dim Z , EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 25 i.e. dim D = dim Z − n − r −
1. Furthermore, D is isomorphic to either P n − r − , or Q n − r − ⊂ P n − r and(7) H Z = H D = ( H D i ) | D = (cid:26) O P (2) in case ( j ) O Q n − r (1) in case ( j )Note that O D ( D ) ∼ = O Z ( Z ∩ D i ) | D ∼ = O X ( D i ) | D ∼ = O D i ( D i ) | D ∼ = (cid:26) O P ( −
1) in case ( j ) O Q n − r − ( −
1) in case ( j ) . Thus (
D, H D , O D ( D )) ∼ = (cid:26) ( P , O P (2) , O P ( − j )( Q n − r − , O Q n − r − (1) , O Q n − r − ( − j )and by the adjunction formula applied to D ⊂ Z , we obtain that( K Z ) | D ∼ = K D − O D ( D ) ∼ = (cid:26) O P ( −
3) in case ( j ) O Q n − r − ( − n + r + 2) in case ( j ) . Hence ( K Z + (dim Z − H Z ) | D ∼ = O D ( −
1) and this shows that K Z + (dim Z − H Z is notnef.In case (13 ′ )(jj), by arguing as in (11 ′ )(j), we see that D := Z ∩ D i $ Z and dim D = n − r −
1. Hence D ∼ = Q n − r − ⊂ D i and H Z | D ∼ = ( H D i ) | D ∼ = O D (1). Moreover, O D ( D ) ∼ = O Z ( Z ∩ D i ) | D ∼ = O D i ( D i ) | D ∼ = O D ( −
1) and so K Z | D ∼ = K D −O D ( D ) ∼ = O Q n − r − ( − n + r +2).This gives ( K Z + (dim Z − H Z ) | D ∼ = O D ( − K Z + (dim Z − H Z cannot be nef.Finally, let us consider case (13 ′ )(jjj). Let s F denote the restriction of the section s to a fibre F ∼ = P n − of D i → Z i . Then s F ∈ Γ( O P n − (1) ⊕ r ) with r ≤ n −
4. Thus D := ( s F ) = Z ∩ F is a linear subspace of F ∼ = P n − and dim D ≥ n − r − Z − ≥ D ⊆ Z . If D = Z , then Z ∼ = P n − r and H Z = H D = ( H F ) | D ∼ = O P n − r (1). Thus K Z + (dim Z − H Z ∼ = O P n − r ( − D $ Z and theirreducibility of Z implies that n − r − ≤ dim D ≤ n − r −
1. Then( K Z + (dim Z − H Z ) | D ∼ = [( K X + det E + ( n − r − H ) | F ] | D == [( K X + det E + ( n − r − H ) | D i ] | D = [( K D i − [ D i ] D i + det E | D i + ( n − r − H D i ) | F ] | D == [( K F − O D i ( D i ) | F + det E | F + ( n − r − H F )] | D = O P n − r ( − | D . This shows that K Z + (dim Z − H Z cannot be nef again. (cid:3) Some remarks on classical scrolls over fivefolds.
In very classical times scrollsnaturally occurred very often. Let us recall that by a classical scroll we mean a P k -bundle X over a variety Y together with an ample line bundle L such that L | F ≃ O P k (1) for anyfibre F ∼ = P k with k = dim X − dim Y . This is equivalent to saying that ( X, L ) ∼ = ( P ( E ) , ξ E ),where E = p ∗ L is an ample vector bundle of rank k + 1 on Y and p : X → Y is theprojection map. In this situation, the canonical bundle formula gives(8) K X + (dim X − dim Y + 1) L = K X + (rk E ) L ≃ p ∗ ( K Y + det E ) . From the adjunction theoretic point of view the correct definition of scroll (see [35]) isthat of adjunction-theoretic scroll over a normal variety Y . This means that there exists amorphism with connected fibres, p : X → Y , such that K X + (dim X − dim Y + 1) L ≃ p ∗ H, for some ample line bundle H on Y . The general fibre is ( P k , O P k (1)) with k = dim X − dim Y , but the special fibres can vary quite a lot.Note that if L is further very ample and ( X, L ) is an adjunction theoretic scroll over anormal variety Y of dimension m ≤
4, then Y is smooth and ( X, L ) is a P k -bundle over Y with k = dim X − m . This follows from a general result due to Sommese ([35, theorem3.3]) for m ≤ L merely spanned), from [8, proposition 3.2.3] and [36, proposition2.1] for m = 3 (and L ample and spanned), and from [36, theorem 2.2] for m = 4.The following results are concerned with the other direction. In the stable range dim X ≥ Y −
1, a classical scroll is also an adjunction theoretic scroll with a few easy exceptions.These results depend on Mori theory [21] and [11]. In the unstable range dim X ≤ Y − Y ≤
4, classical scrolls that are not adjunction theoretic scrolls in the modern sensehave been classified in [7] and [36, § n = 2 m − ≥
7, i.e. k = m − ≥ E = m −
2, let us give herethis immediate consequence of [2, theorem 5.1], [17] and [24, theorem 1], as noted in [36,remark 3.3].
Proposition 4.3.
Let X be an n -fold and let L be an ample line bundle on X . Assumethat ( X, L ) ∼ = ( P ( E ) , O P ( E ) (1)) is a P n − m -bundle, π : X → Y , over a smooth variety Y ofdimension m with E = π ∗ L . If n = 2 m − ≥ , then ( X, L ) is an adjunction-theoreticscroll over Y under π unless either: (1) Y ∼ = P m and E ∼ = O P m (1) ⊕ m − , O P m (2) ⊕ O P m (1) ⊕ m − , O P m (2) ⊕ ⊕ O P m (1) ⊕ m − , O P m (3) ⊕ O P m (1) ⊕ m − ; (2) Y ∼ = Q m and E ∼ = O Q m (1) ⊕ m − , O Q m (2) ⊕ O Q m (1) ⊕ m − , where Q m is a smoothquadric hypersurface of P m +1 ; (3) Y is a Del Pezzo m -fold with b ( Y ) = 1 , i.e. Pic( Y ) is generated by an ample linebundle O Y (1) such that − K Y ∼ = ( m − O Y (1) and E ∼ = O Y (1) ⊕ m − ; (4) there is a vector bundle V on a smooth curve C such that Y ∼ = P ( V ) and E | F ∼ = O P m − (1) ⊕ m − for any fibre F ∼ = P m − of Y → C ; EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 27 (5) there is a vector bundle V on a smooth curve C such that Y ∼ = P ( V ) and E | F ∼ = O P m − (2) ⊕ O P m − (1) ⊕ m − for any fibre F ∼ = P m − of Y → C ; (6) there is a surjective morphism q : Y → C onto a smooth curve C such that anygeneral fibre F of q is a smooth quadric hypersurface Q m − ⊂ P m with E | F ∼ = O Q m − (1) ⊕ m − ; (7) there is a vector bundle V on a smooth surface S such that Y ∼ = P ( V ) and E | F ∼ = O P m − (1) ⊕ m − for any fibre F ∼ = P m − of Y → S ; (8) Y is a Fano m -fold with − K Y ≃ det E ; moreover, if m ≥ then b ( Y ) = 1 exceptfor Y ∼ = P × P and E ∼ = O P × P (1 , ⊕ ; (9) there exists a morphism Φ : Y → W onto a normal variety W supported by (a largemultiple of ) K Y + det E and dim W ≤ ; if F is a general fibre of Φ , then we havethe following possibilities: (a) W is a smooth curve and Φ is flat (equidimensional) map; moreover, the pair ( F, E | F ) is one of the following: (a1) F ∼ = P m − and E | F ∼ = O (2) ⊕ ⊕ O (1) m − , O (3) ⊕ O (1) m − ; (a2) F ∼ = Q m − and E | F ∼ = O (2) ⊕ O (1) ⊕ m − , where Q m − is the quadrichypersurface of P m ; (a3) F ∼ = Q and E | F ∼ = E (2) ⊕ O (1) , where E is a spinor bundle on thequadric hypersurface Q of P ; (a4) F is a Del Pezzo ( m − -fold with b ( F ) = 1 , i.e. Pic( F ) is generatedby an ample line bundle O F (1) such that − K F ≃ ( m − O F (1) and E | F ∼ = O F (1) ⊕ m − ; (a5) F ∼ = P × P and E | F ∼ = O (1 , ⊕ .In particular, if m ≥ then Φ is an elementary contraction. Furthermore, if F ∼ = P m − then Y is a classical scroll, while if F ∼ = Q m − ⊂ P m then Y is aquadric bundle; (b) W is a smooth surface, Φ is flat and for a general fibre F of Φ the pair ( F, E | F ) is one of the following: (b1) ( P m − , O (2) ⊕ O (1) ⊕ m − ) ; (b2) ( P m − , T P m − ) , where T P m − is the tangent bundle on P m − ; (b3) ( Q m − , O (1) ⊕ m − ) , where Q m − is a smooth quadric hypersurface in P m − .In particular, if F ∼ = P m − then all the fibres of Φ are P m − ; (c) W is a -fold and E | F ∼ = O (1) ⊕ m − for all the fibres F ∼ = P m − of Φ ; (10) there exist an m -fold Y ′ , a morphism ψ : Y → Y ′ expressing Y as blown up at afinite set B of points and an ample vector bundle E ′ on Y ′ such that E = π ∗ E ′ ⊗ [ − ψ − ( B )] ; or (11) a high multiple of K Y + det E defines a birational map, ϕ : Y → b Y , which contractsan extremal face. Let R i , for i in a finite set of index, the extremal rays spanningthis face; call ρ i : Y → b Y i the contraction associated to one of the R i . Then we have each ρ i is birational and divisorial; if D i is one of the exceptional divisors and Z i = ρ i ( D i ) , we have dim Z i ≤ and the following possibilities can occur: (i) dim Z i = 0 , D i ∼ = P m − and [ D i ] D i ≃ O ( − , or O ( − ; moreover, E | D i ∼ = O (1) ⊕ m − , or E | D i ∼ = O (2) ⊕ O (1) ⊕ m − ; (ii) dim Z i = 0 , D i is a (possible singular) quadric hypersurface Q m − and [ D i ] D i ≃O ( − with E | D i ∼ = O (1) m − ; (iii) dim Z i = 1 , Z i and b Y i are smooth projective varieties and ρ i is the blow-up of b Y i along Z i ; moreover, E | f ∼ = O (1) ⊕ m − for any fibre f ∼ = P m − of ρ i | D i : D i → Z i . Specializing the above result to the case m = 5, we get also the following Corollary 4.1.
Let X be a smooth n -fold with n ≥ and L an ample line bundle on X .Assume that ( X, L ) ∼ = ( P ( E ) , O P ( E ) (1)) is a P k -bundle, π : X → Y , over a smooth -fold Y with E = π ∗ L . Then ( X, L ) is an adjunction scroll over Y under π unless either (1 ′ ) n = 10 , Y ∼ = P and E ∼ = O P (1) ⊕ ; (2 ′ ) n = 9 , Y ∼ = P and E ∼ = O P (1) ⊕ , O P (2) ⊕O P (1) ⊕ or T P , where T P is the tangentbundle of P ; (3 ′ ) n = 9 , Y ∼ = Q and E ∼ = O Q (1) ⊕ , where Q is a smooth quadric hypersurface of P ; (4 ′ ) n = 9 , there is a rank vector bundle V over a smooth curve C such that Y ∼ = P ( V ) and E | f ∼ = O f (1) ⊕ for any fibre f ∼ = P of Y → C ; (5 ′ ) n = 8 , Y ∼ = P and E ∼ = O P (1) ⊕ , O P (2) ⊕ O P (1) ⊕ , O P (2) ⊕ ⊕ O P (1) ⊕ , O P (3) ⊕ O P (1) ⊕ ; (6 ′ ) n = 8 , Y ∼ = Q and E is either O Q (1) ⊕ or O Q (2) ⊕ O Q (1) ⊕ , where Q is asmooth quadric hypersurface of P ; (7 ′ ) n = 8 , Y is a Del Pezzo -fold with b ( Y ) = 1 , i.e. Pic( Y ) is generated by an ampleline bundle O Y (1) such that − K Y ≃ O Y (4) and E ∼ = O Y (1) ⊕ ; (8 ′ ) n = 8 , there is a vector bundle V on a smooth curve C such that Y ∼ = P C ( V ) and E | F ∼ = O F (1) ⊕ for any fibre F ∼ = P of Y → C ; (9 ′ ) n = 8 , there is a surjective morphism q : Y → Γ onto a smooth curve Γ such that anygeneral fibre F of q is a smooth quadric hypersurface Q in P with E | F ∼ = O F (1) ⊕ ; (10 ′ ) n = 8 , there is a vector bundle V on a smooth surface S such that Y ∼ = P S ( V ) and E | F ∼ = O F (1) ⊕ for any fibre F ∼ = P of Y → S ; (11 ′ ) n = 8 , there exists a smooth projective -fold W and a morphism π : Y → W expressing Y as blown up at a finite set B of points and an ample vector bundle E ′ on W such that E = π ∗ E ′ ⊗ [ − π − ( B )] and K W + det E ′ is ample; or (12 ′ ) n = 7 and ( Y, E ) is as in Proposition 4.3.
Proof. If n ≥
11, then k = n − ≥ > dim Y , so ( X, L ) is an adjunction scroll over Y by [7, (2.1.1)]. If 7 ≤ n ≤
10, then we conclude by [7, (2.1.2), (2.1.3)], [36, proposition 3.1]and Proposition 4.3. (cid:3)
EFNESS OF ADJOINT BUNDLES FOR AMPLE VECTOR BUNDLES OF CORANK 3 29
Finally, as to the case n = 6 and m = 5, under the extra assumption that Pic( Y ) ∼ = Z [ O Y (1)], we can easily deduce from Theorem 1.1 the following Corollary 4.2.
Let X be a smooth -fold and L an ample line bundle on X . Assume that ( X, L ) ∼ = ( P ( E ) , O P ( E ) (1)) is a P -bundle, π : X → Y , over a smooth -fold Y, E = π ∗ L .If Pic( Y ) ∼ = Z [ O Y (1)] , then ( X, L ) is an adjunction-theoretic scroll over Y under π exceptwhen: (1 ′′ ) Y ∼ = P and E ∼ = O P ( a ) ⊕ O P ( a ) with a + a ≤ and a i > for i = 1 , ; (2 ′′ ) Y ∼ = Q and E ∼ = O Q ( a ) ⊕ O Q ( a ) , with a + a ≤ and a i > for i = 1 , , where Q is a smooth quadric hypersurface of P ; (3 ′′ ) Y is a Del Pezzo -fold, i.e. − K Y ≃ O Y (4) , and either ( α ) E ∼ = O Y (1) ⊕ , or ( β ) E | l ∼ = O P (1) ⊕ O P (2) for any line l ∼ = P of ( Y, O Y (1)) ; (4 ′′ ) Y is a Mukai -fold, i.e. − K Y ≃ O Y (3) , and E ∼ = O Y (1) ⊕ ; (5 ′′ ) Y is a Fano -fold such that − K Y ≃ det E and X is a Fano -fold such that − K X ≃ L . Proof.
Since Pic( Y ) ∼ = Z [ O Y (1)], if ( X, L ) is not an adjunction-theoretic scroll over Y under π , then from the formula (8) we deduce that K Y + det E = O Y ( a ) for some integer a ≤
0. Therefore, we have either a = 0, obtaining Case (5 ′′ ) of the statement, or a <
0. Inthe latter case, since K Y + det E is not nef, we conclude by Theorem 1.1. (cid:3) Remark 4.1. If ( X, L ) and ( Y, E ) are as in Corollary 4.2 , then similar techniques andarguments used for the proofs of [7, theorem 3.1] and [36, proposition 3.4] work also in thecase
Pic( Y ) = Z [ O Y (1)] . More precisely, we can say in this situation when ( X, L ) is notan adjunction-theoretic scroll over Y under the projection map π : X → Y , but the finalclassification result appears at present not yet complete in some of its parts. This fact isdue especially to the lack of classification results of pairs ( Y, E ) as above with K Y + det E trivial, and of properties on the third reduction of ( Y, E ) in the same spirit of [3, remark4] . We refer to [3, § and [26] for the case m − ≥ and the second reduction of the pair ( Y, E ) . Acknowledgement.
I would thank Proff. M.C. Beltrametti, A. Lanteri and A. Laface fortheir kind comments and some useful remarks about the final form of this paper. Finally,the author wants to thank for some funds supporting this research from the University ofMilan (FIRST 2006 and 2007).
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