2-very ampleness for adjoint bundles of ample and spanned vector bundles on surfaces

Math. Z. 225, 355–371 (1997)

c Springer-Verlag 1997

2-very ampleness for adjoint bundles of ample and spanned vector bundles on surfaces Antonio Lanteri1 , Hidetoshi Maeda2 1 Dipartimento di Matematica “F. Enriques”, Universit` a degli Studi di Milano, Via C. Saldini, 50, I-20133 Milano, Italy (e-mail address: [email protected]) 2 Graduate School of Mathematics, Kyushu University, 4-2-1 Ropponmatsu, Chuo-ku, Fukuoka 810, Japan (e-mail address: [email protected])

Received 6 March 1995; in final form 9 October 1995

0. Introduction Let E be an ample and spanned vector bundle of rank r = 2 on a smooth complex projective surface X and let KX be the canonical bundle of X . In [LM1] we investigated the spannedness and the very ampleness of the adjoint line bundle KX + det E by using Reider’s method [R], i.e., Theorem A. KX + det E is spanned unless (X , E ) ∼ = (P2 , OP (1)⊕2 ). Theorem B. Assume that c1 (E )2 following cases :

= 9. Then KX + det E

is very ample except the

(1) X is a P1 -bundle over a smooth curve C and EF ∼ = OF (1)⊕2 for any fiber F of X → C . (2) (X , E ) ∼ = (P2 , OP (1)⊕3 ). ∼ (3) (X , E ) = (P2 , OP (2) ⊕ OP (1)). (4) X ∼ = P2 and E is the tangent bundle TP of it. On the other hand, Beltrametti and Sommese introduced the concept of k very ampleness with regard to the notion of higher order embedding [BS1]. A line bundle M on a projective variety Y is called k -very ample for an integer k = 0 if for any 0-dimensional subscheme (Z , OZ ) of Y with length(OZ ) = k + 1, the restriction map Γ (M ) → Γ (M ⊗ OZ ) is surjective. Of course the 0-very ampleness is equivalent to the spannedness. Moreover, the 1-very ampleness corresponds to the very ampleness. Thus it seems very natural to carry on the program started in [LM1], by asking the following

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Question. Let E be an ample and spanned vector bundle of rank r = 2 on a smooth complex projective surface X . When is the adjoint bundle KX + det E 2-very ample ? The main purpose of this paper is to give an answer to this Question in the case when c1 (E )2 = 16 and r = 3. In fact, we will prove Theorem 1. Let E be an ample and spanned vector bundle of rank r = 3 on a smooth complex projective surface X . Assume that c1 (E )2 = 16. Then KX + det E is 2-very ample except the following cases :

(1) X is a P1 -bundle over a smooth curve C and EF ∼ = OF (1)⊕3 for any fiber F of X → C . (2) (X , E ) ∼ = (P2 , OP (1)⊕4 ). (3) (X , E ) ∼ = (P2 , OP (2) ⊕ OP (1) ⊕ OP (1)). (4) (X , E ) ∼ = (P2 , TP ⊕ OP (1)). (5) r = 3 and either (a) X is a Del Pezzo surface with KX2 = 1 and det E = −4KX , or (b) X = PB (G ) for some indecomposable vector bundle G of rank 2 on an elliptic curve B with c1 (G ) = 1 and det E ≡ 4H (G ), where H (G ) denotes the tautological line bundle on X . Moreover, there is a nontrivial extension 0 → OX → E → F → 0, where F is an ample and spanned vector bundle of rank 2 on X with the following property : for any smooth curve C belonging to the pencil | − KX | in case (a) and to the algebraic family of sections ≡ H (G ) in case (b), FC = L ⊗ V , where L is a line bundle of degree 2 on C and V is a normalized vector bundle of rank 2 on C with c1 (V ) = 0 in the sense of [H, p. 373]. We suspect that case (5) in the Theorem 1 does not occur. In support of this conjecture we produce several arguments restricting the range of the possible values for c2 (E ) (Proposition (2.6)). Note that (P2 , OP (1)⊕4 ) is the only exception corresponding to the case when r = 4 in the Theorem 1. More generally, as a consequence of the Theorem 1, we also obtain the Corollary. Let E be an ample and spanned vector bundle of rank r = 2k (k = 2) on a smooth complex projective surface X and assume that c1 (E )2 = 4k +8. Then KX + det E is k -very ample unless k = 2 and (X , E ) ∼ = (P2 , OP (1)⊕4 ). The above results are shown in Section 2. As to the case r = 2 we can answer our Question assuming that E is very ample. A vector bundle E on X is called very ample if the tautological line bundle H (E ) on PX (E ) is very ample. More generally, the result for very ample vector bundles E of any rank r = 2 is as follows.

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Theorem 2. Let E be a very ample vector bundle of rank r = 2 on a smooth complex projective surface X . Assume that c1 (E )2 = 16 and that KX + det E is very ample. Then KX + det E is 2-very ample except the following cases: the four cases (1), (2), (3), (4) of the Theorem 1 and (5) There exist a (−1)-curve E on X and an ample vector bundle F of rank 2 on Y such that E ∼ = f ∗ F ⊗ OX (−E ), where f : X → Y is the blow-down of E . 1 (6) X is a P -bundle over a smooth curve C and EF ∼ = OF (2) ⊕ OF (1) for any fiber F of X → C . (7) (X , E ) ∼ = (P2 , OP (3) ⊕ OP (1)). 2 ∼ (8) X = P and E has the generic splitting type (2, 2). We prove the Theorem 2 in Section 3. If r = 2 and E is not very ample, then the situation is much more complicated. In Section 4 we produce some nontrivial examples (X , E ) with rankE = 2 for which KX + det E is very ample but not 2-very ample. The present paper was finished when the second author was visiting the University of Milan in the winter of 1994-95. He would like to thank the Italian C.N.R. and the J.S.P.S. for financial support. The first author was partially supported by the M.U.R.S.T. of the Italian Government in the framework of the 40% research project “Geometria algebrica”.

1. Preliminaries In this paper 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. Frequently, the tensor products of line bundles are denoted additively, while we use multiplicative notation for intersection products in Chow rings. The numerical equivalence is denoted by ≡. The pull-back i ∗ E of a vector bundle E on X by an embedding i : Y ,→ X is often denoted by EY . For a vector bundle E on X , the tautological line bundle on the projective space bundle PX (E ) associated to E is denoted by H (E ). A vector bundle is called spanned if it is generated by its global sections. The canonical bundle of a Gorenstein variety X is denoted by KX . A polarized surface is a pair (X , L) consisting of a projective surface X and an ample line bundle L on X . The ∆-genus ∆(X , L) of a polarized surface (X , L) is defined to be ∆(X , L) = 2 + L2 − h 0 (L); it is well-known (see [F1, Corollary 1.10]) that ∆(X , L) = 0 for any polarized surface (X , L). The sectional genus g(X , L) of a polarized Gorenstein surface (X , L) is given by the formula 2g(X , L) − 2 = (KX + L)L. A polarized surface (X , L) is said to be a scroll over a smooth curve C if X is a P1 -bundle over C and LF = 1 for any fiber F of X → C. First of all, we survey basic facts on the P1 -bundles of type A−1 over elliptic curves.

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(1.1) Let B be a smooth curve of genus 1, let G be the vector bundle of rank 2 on B defined by a non-split extension 0 → OB → G → OB (p) → 0, where p ∈ B , and let X := PB (G ). Let π : X → B be the projection and let H (G ) be the tautological line bundle on X . Note that G is normalized in the sense of [H, p. 373] (see also [H, p. 375, Example 2.11.6]). We often denote by C0 the section of π corresponding to the unique element of |H (G )|. We have C02 = c1 (G ) = 1; moreover, the classes of C0 and a fiber f of π generate Num(X ) and H (G ) is ample (see [H, p. 382, Proposition 2.21]). Let us recall the following property of X . (1.1.1) KX = OX (−2C0 + π ∗ p) ≡ −2C0 + f [H, p. 373, Lemma 2.10]. Moreover, h 0 (−2KX ) = 2, −2KX is spanned, and | − 2KX | defines a morphism X → P1 , which is an elliptic fibration [S1] having exactly three singular fibers: each of them is a smooth elliptic curve with multiplicity 2 [Su, p. 310, case (iii)]. (1.2) Lemma. Let E be a vector bundle of rank 2 on a smooth projective surface X , fitting into an exact sequence (1 .2 .1 )

0 → L → E → M → 0,

where L , M ∈ Pic(X ). Let E ⊂ X be a smooth curve and consider the exact sequence 0 → LE → EE → ME → 0

(1 .2 .2 )

obtained by restricting (1.2.1) to E . Assume that h 1 (L ⊗ M −1 ⊗ OX (−E )) = 0.

(1 .2 .3 )

Then (1.2.1) splits if and only if (1.2.2) splits. Proof. Consider the following commutative diagram

0

→

0

→

0 ↓ L ⊗ OX (−E ) ↓ L → E ↓ ↓ → EE LE ↓ 0 .

→

M ↓ → ME

→

0

→ 0

Tensoring the vertical exact sequence with M −1 and taking cohomology, we get the exact sequence ...

→ r

− →

H 1 (L ⊗ M −1 ⊗ OX (−E )) → H 1 (L ⊗ M −1 ) H 1 ((L ⊗ M −1 )E ) → . . . ,

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where r is the homomorphism sending the extension class of (1.2.1) to the extension class of (1.2.2). The condition (1.2.3) guarantees the injectivity of r.



Now, we recall the numerical condition for the k -very ampleness of the adjoint bundle proved by Beltrametti and Sommese. (1.3) Lemma. Let L be a nef line bundle on a smooth projective surface X . If L2 = 4k + 5 (k = 0) and KX + L is not k -very ample, then there exists an effective divisor E on X such that LE − k − 1 5 E 2
(L − 4E )2 = L2 − 16, contrary to assumption. Thus L ≡ 4E and hence E is ample. Moreover, since E 2 = 1, E is irreducible and reduced. Combining LE = 4 with (1.5.1), we have r 5 4. Now if r = 4, then by (1.5.2) E ∼ = P1 . By the classification theorem for polarized surfaces of sectional genus zero [LP, Corollary 2.3], (X , OX (E )) is one of the following. (1) (X , OX (E )) ∼ = (P2 , OP (1)). ∼ (2) (X , OX (E )) = (P2 , OP (2)).

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(3) (X , OX (E )) is a scroll over

P1 .

But then case (2) is impossible since E 2 = 1. In case (1) we have L = OP (4). This implies that (E ⊗ OP (−1))l = Ol ⊕4 for any line l in P2 . Thus E ⊗ OP (−1) is trivial (see for example [OSS, p. 51, Theorem 3.2.1]), i.e., E ∼ = OP (1)⊕4 . Assume that case (3) holds. Then we can write X = PP1 (OP1 ⊕ OP1 (−e)) for some e = 0. We use [H, p. 380, Corollary 2.18] to see that OX (E ) is very ample. Since E 2 = 1, we have (X , OX (E )) ∼ = (P2 , OP (1)), a contradiction. (2.3) From now on, we suppose that r = 3. We claim first that the arithmetic genus g(E ) 5 1. To see this, by applying the same argument as in the proof of Lemma (1.6) to EE , we have an exact sequence 0 → OE (Z ) → EE → Q → 0 on E , where Z consists of h(= 1) distinct smooth points of E and Q is an ample and spanned vector bundle of rank 2 on E . Since c1 (EE ) = 4, by virtue of (1.5.1) c1 (Q) = 4 − h must be 2 or 3. If c1 (Q) = 2, then it follows from (1.5.2) that E∼ = P1 , and we are done. Assume that c1 (Q) = 3. Then by Lemma (1.6) we have g(E ) 5 1, and the assertion is proved. Consequently the classification theorem for polarized surfaces of sectional genus 5 1 applies. (2.4) Suppose first that g(E ) = 0. Then the same argument as in (2.2) shows that (X , L) ∼ = (P2 , OP (4)), and so E is a uniform bundle of splitting type (2, 1, 1). By the classification theory of uniform vector bundles of rank 3 on P2 (see for example [OSS, p. 70]), E is either OP (a) ⊕ OP (b) ⊕ OP (c), TP (a) ⊕ OP (b), or S 2 TP (a). Thus E ∼ = OP (2) ⊕ OP (1) ⊕ OP (1) or E ∼ = TP ⊕ OP (1). (2.5) Suppose that g(E ) = 1. Then by the classification theorem for polarized surfaces of sectional genus one [LP, Corollary 2.4], (X , OX (E )) is one of the following. (2.5.1) X is a Del Pezzo surface and OX (E ) = −KX . (2.5.2) (X , OX (E )) is a scroll over an elliptic curve B . In case (2.5.1) we have KX2 = 1 and L = −4KX . In case (2.5.2) we may write X = PB (G ) for some normalized vector bundle G of rank 2 on B . Now write OX (E ) = H (G ) + ρ∗ A for some line bundle A on B , where ρ is the projection X → B . We set e := −c1 (G ) and a := deg A. Then 1 = E 2 = 2a − e and we know e = −1. By using necessary and sufficient conditions for E to be ample [H, p. 382, Proposition 2.20 and Proposition 2.21], we have e = −1 and a = 0. Thus G is indecomposable and L ≡ 4H (G ). So, to sum up, (X , L) is one of the following.

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(a) X is a Del Pezzo surface with KX2 = 1 and L = −4KX . (b) X is of the form PB (G ) for some indecomposable vector bundle G of rank 2 on an elliptic curve B with c1 (G ) = 1 and L ≡ 4H (G ). In either event, by the same argument as in [OSS, p. 81, Lemma 4.3.1] there is an exact sequence 0 → OX → E → F → 0

(2 .5 .3 )

of vector bundles, where F is an ample and spanned vector bundle of rank 2 on X. We note that det F = det E = L. Let us take any smooth curve C belonging to the pencil | − KX | in case (a) and to the algebraic family of sections ≡ H (G ) in case (b). Then C is an elliptic curve in every case, and we have c1 (FC ) = (det F )C = LC = 4. Applying Lemma (1.7) to FC , we conclude that FC = L ⊗ V , where L is a line bundle of degree 2 on C and V is a vector bundle of rank 2 on C with c1 (V ) = 0. This completes the proof.  Case (5) of the Theorem 1 is doubtful. As a matter of fact, we can provide further information about the possible values of c2 (E ). (2.6) Proposition. Let (X , E ) be as in case (5) of the Theorem 1. Then c2 (E ) 5 9. Moreover ( 4, in case (a) c2 (E ) = 6, in case (b). Proof. Let C be a smooth elliptic curve as in (2.5), i.e., in the pencil | − KX | in case (a) and in the numerical equivalence class of H (G ) in case (b). Then OX (C ) is ample in either case. Now we have h 0 (E ) = 5. From the cohomology of the exact sequence 0 → E ⊗ OX (−C ) → E → EC → 0 and the fact that h 0 (EC ) = c1 (EC ) = LC = 4, it follows that h 0 (E ⊗ OX (−C )) > 0. Now recall the exact sequence (2.5.3). Clearly we have det F = det E = L and c2 (F ) = c2 (E ). Applying ⊗OX (−C ) to (2.5.3), we obtain 0 → OX (−C ) → E ⊗ OX (−C ) → F ⊗ OX (−C ) → 0. By the Kodaira vanishing theorem, h 0 (OX (−C )) = h 1 (OX (−C )) = 0, hence h 0 (F ⊗ OX (−C )) = h 0 (E ⊗ OX (−C )). Consequently h 0 (F ⊗ OX (−C )) > 0.

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We set P := PX (F ), H := H (F ) and let π : P → X be the projection. Now let D ∈ |H + π ∗ OX (−C )|. Then 0 < H 2D

=

H 3 + H 2 π ∗ OX (−C )

=

(det F )2 − c2 (F ) + (det F )OX (−C )

=

L2 − c2 (F ) − LC

=

16 − c2 (F ) − 4 = 12 − c2 (F );

thus c2 (F ) 5 11. We proceed now by cases. (2.6.1)

Case c2 (F ) = 11.

Let D ∈ |H + π ∗ OX (−C )|. Then H 2 D = 1 and D is irreducible and reduced. Since HD is ample and spanned, we have (D, HD ) ∼ = (P2 , OP (1)). On the other hand, since DF = 1 for any fiber F of π, the restriction πD : D → X of π to D is birational. Thus X ∼ = P2 , a contradiction. (2.6.2)

Case c2 (F ) = 10.

If D is any divisor in the linear system |H +π ∗ OX (−C )|, then H 2 D = 2. Since DF = 1 for any fiber F of π, D is reduced. Suppose that D is reducible. Then D = D 0 +D 00 for some mutually distinct prime divisors on P . Since D 0 F +D 00 F = 1, we may assume that D 0 F = 1; thus πD 0 : D 0 → X is a birational morphism. On the other hand, H 2 D 0 = 1, so (D 0 , HD 0 ) ∼ = (P2 , OP (1)) and hence X ∼ = P2 , a contradiction. Consequently D is irreducible. Now we compute g(D, HD ). In case (a), 2g(D, HD ) − 2

=

(KD + HD )HD

=

(KP + D + H )HD

=

(−2H + π ∗ (KX − 4KX ) + H + π ∗ KX + H )H (H + π ∗ KX )

=

6.

In case (b), 2g(D, HD ) − 2

=

(−2H + π ∗ (−2H (G ) + ρ∗ (det G ) + 4H (G ))

+

H + π ∗ (−H (G )) + H )H (H + π ∗ (−H (G ))) = 6.

Thus g(D, HD ) = 4 in either case, and so ∆(D, HD ) = 2 + HD2 − h 0 (HD ) = 4 − h 0 (HD ) = 1 by virtue of [F3, (5.1)]; i.e., h 0 (HD ) 5 3. On the other hand, since HD is ample and spanned, we have h 0 (HD ) = 3, hence h 0 (HD ) = 3. Thus ∆(D, HD ) = 1. We apply [F3, Corollary 6.13] to see that the morphism ϕ : D → P2 defined by |HD | is a double covering of P2 and that the branch locus of ϕ is a curve of degree 10. Consequently KD = ϕ∗ KP + ϕ∗ OP (5) = ϕ∗ OP (2), and so KD2 = 8. We compute KD2 by the adjunction formula. In case (a),

2-very ampleness for adjoint bundles

KD2

=

(KP + D)2D

=

(KP + D)2 D

=

(−2H + π ∗ (KX − 4KX ) + H + π ∗ KX )2 (H + π ∗ KX )

=

−6,

365

a contradiction. In case (b), KD2

=

(−2H + π ∗ (−2H (G ) + ρ∗ (det G ) + 4H (G )) + H + π ∗ (−H (G )))2 (H + π ∗ (−H (G ))) = −7,

a contradiction. This shows that c2 (E ) = c2 (F ) 5 9. To prove the second assertion, recall that c2 (F ) = 2 by Lemma (1.4). In case (a), by using [N, Theorem], we see that c2 (F ) = 4. Now consider case (b). Since LF = 4 for any fiber F of ρ, we have EF ∼ = OF (2) ⊕ OF (1) ⊕ OF (1). Thus, if we set S = ρ∗ (E ⊗ (−2H (G ))), then S is a line bundle on B and we obtain an exact sequence 0 → 2H (G ) + ρ∗ S → E → Q → 0 for some vector bundle Q of rank 2 on X . Of course Q is ample and spanned. Now we have det Q = det E − 2H (G ) − ρ∗ S = 2H (G ) + ρ∗ T for some line bundle T on B with deg T = − deg S . Hence c2 (E )

=

c2 (Q) + (2H (G ) + ρ∗ S )(2H (G ) + ρ∗ T )

=

c2 (Q) + 4.

By virtue of Lemma (1.4), we have c2 (Q) = 2. Consequently c2 (E ) = 6. This concludes the proof.  Proof of the Corollary. Let L = det E and assume that KX +L is not k -very ample; then by Lemma (1.3) and (1.5.1) it is immediate to check that there exists an effective divisor E on X such that either (i) LE = 2k , E 2 = k − 1, or (ii) LE = 2k + 1, E 2 = k . By the Hodge index theorem we have the inequality (LE )2 = L2 E 2 , which leads to a contradiction in case (ii) and gives k = 2 in case (i). Thus the assertion follows from the Theorem 1. 

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3. E very ample Proof of the Theorem 2. We set L = det E . (3.1) When r = 3, it is sufficient to show that case (5) of the Theorem 1 does not occur. To see this, suppose to the contrary that case (5) holds. In case (a) (resp. (b)) we can find a smooth elliptic curve C ∈ | − KX | (resp. C ≡ H (G )). Now consider the exact sequence (2.5.3). In this case, F is also very ample and in either case we have c1 (FC ) = (det F )C = LC = 4, which contradicts (1.8.3). So, from now on, we can assume that r = 2. (3.2) If KX + L is not 2-very ample, then by Lemma (1.3) and (1.5.1) combined with the argument in Section 2 there exists an effective divisor E on X satisfying one of the following. (i) LE = 2 and either E 2 = −1 or E 2 = 0. (ii) LE = 3 and E 2 = 0 (see (2.1)). (iii) L ≡ 4E and E 2 = 1 (see (2.2)). (3.3) In case (i), we have E ∼ = P1 by virtue of (1.5.2). 2 If E = −1, then E is a (−1)-curve on X . Let f : X → Y be the blow-down of E . Then, since (E ⊗ OX (E ))E ∼ = OE⊕2 , there exists a vector bundle F of ∗ rank 2 on Y such that E ∼ = f F ⊗ OX (−E ). The fact that F is ample follows from [LM2, Lemma 5.1]. Assume that E 2 = 0. Then X is a ruled surface over a smooth curve C and E is a fiber of the ruling ρ : X → C . Let F be any fiber of ρ. Then LF = 2 and it follows from (1.5.2) that F ∼ = P1 , i.e., X is a P1 -bundle over C . But then

=1 because KX + L is very ample. Thus KX F = −1, a contradiction. (3.4) In case (ii), since LD = 2 for all integral curves D on X owing to (1.5.1), (KX + L)F

E must be irreducible and reduced. Thus E ∼ = P1 by (1.8.1). Since E 2 = 0, X is a ruled surface over a smooth curve C and E is a fiber of the ruling ρ : X → C . Let F be any fiber of ρ. Then, since LF = 3, F is irreducible and reduced. Hence X is a P1 -bundle over C and EF ∼ = OF (2) ⊕ OF (1). (3.5) Assume that case (iii) holds. Then E is ample, and so E is irreducible and reduced because E 2 = 1. Since LE = 4, we use (1.8.2) to see that g(E ) 5 1. Suppose first that g(E ) = 0. Then the same argument as in (2.2) shows that (X , L) ∼ = = (P2 , OP (4)). If E has the constant splitting type (3,1), then E ∼ OP (3) ⊕ OP (1). Assume that E ∼ 6 = OP (3) ⊕ OP (1). Then there exists a line l on P2 such that El ∼= Ol (2)⊕2 . Thus (3, 1) > (2, 2) according to the lexicographical ordering of the splitting types of E ; hence (2,2) is the generic splitting type of E (see [OSS, p. 29, Definition 2.2.3]).

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Suppose that g(E ) = 1. Then by the reasoning in (2.5), (X , L) is as in (a) or (b) in (2.5). But then the same argument as in (3.1) applies to EC , and again we have a contradiction. This completes the proof.  (3.6) Remark. i) The vector bundles appearing in the cases (2), (3) and (4) of the Theorem 1 are in fact very ample [BS2, p. 74, Lemma 3.2.3]. ii) In case (5) of the Theorem 2 we have KX + det E = f ∗ (KY + det F ) ⊗ OX (−E ). So, if there is no (−1)-curve E 0 ⊂ Y such that FE 0 ∼ = OE 0 (1)⊕2 , then (Y , KY +det F ) is just the reduction of (X , KX +det E ) in the adjunction theoretic sense. Moreover, since KX + det E is very ample and of course H 1 (KX + det E ) = 0, it follows from [So, Proposition 2.4] that KY + det F is also very ample. So we can regard (Y , F ) as a sort of “vector bundle reduction” of (X , E ). As to the very ampleness of F , we can prove it only in very special circumstances. iii) As to case (8), let us recall that for any integer c2 , 6 5 c2 5 9, there exists a stable very ample vector bundle E of rank 2 on P2 with det E = OP (4) and c2 (E ) = c2 [I, Proposition 1.3]. Another non obvious example fitting into case (8) is a stable vector bundle E with c2 (E ) = 10 giving rise to the so-called Bordiga scroll in P5 [Ot].

4. The case r = 2; some examples As has already been recognized, Theorems 1 and 2 do not cover the case of ample and spanned vector bundles of rank 2. In this section we produce some examples of ample and spanned vector bundles E of rank 2 on a smooth projective surface X with c1 (E )2 = 16, such that KX + det E is very ample but not 2-very ample. They are related to cases (ii) and (iii) we met in the proof of the Theorem 2 (see (3.2)), before using the very ampleness assumption. We start with case (iii) giving 3 examples satisfying c1 (E )2 = 16 = 4c2 (E ). As a first thing, when E is not very ample, we cannot use (1.8.2) in (3.5). So it may be possible that g(E ) = 2. We give an example where g(E ) = 2. (4.1) Let X be a minimal surface of general type with KX2 = 1 and pg (X ) = 2. The pencil |KX | has a single base point and its general element is a smooth curve of genus 2. Assume that X contains no (−2)-curves, so that KX is ample. Since 2KX is spanned [Fr, Theorem 3.1], E := (2KX )⊕2 is ample and spanned. We have det E = 4KX , hence c1 (E )2 = 16. Moreover, KX + det E = 5KX is very ample by general results on the pluricanonical maps. On the other hand, deg(KX +det E )C = 5KX C = 5 for any smooth C ∈ |KX |. So |KX +det E | embeds C as a quintic C 0 ⊂ P3 . Such a curve has a trisecant line, say l . Let (Z , OZ ) denote the 0-dimensional subscheme of X of length 3 consisting of the three points of C corresponding to C 0 ∩ l . Then the homomorphism Γ (KX ⊗ det E ) → Γ (KX ⊗ det E ⊗ OZ ) fails to be surjective, and so KX + det E is not 2-very ample.

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The next two examples, still related to (iii), are concerned with the surfaces we met in cases (a), (b) in Section 2. (4.2) Let X be a Del Pezzo surface with KX2 = 1 and set E := (−2KX )⊕2 . Then det E = −4KX , hence c1 (E )2 = 16. E is ample and spanned, since −2KX is so. Moreover, KX + det E = −3KX is very ample, but not 2-very ample, since deg(KX + det E )C = 3 for any smooth C ∈ | − KX |. (4.3) Let X = PB (G ) be as in (1.1), let H (G ) be the tautological line bundle on X , and set E := (2H (G ))⊕2 . Then det E = 4H (G ), hence c1 (E )2 = 16. Moreover, an easy application of Reider’s theorem [R] shows that 2H (G ) is spanned. Due to this and the ampleness of H (G ), E is ample and spanned. Furthermore, since det E ∈ 4Pic(X ), Reider’s theorem immediately shows that KX + det E is very ample. On the other hand, by restricting it to the elliptic curve C0 , we get deg(KX + det E )C0 = (2C0 + f )C0 = 3, which shows that KX + det E is not 2-very ample. As to case (ii), when E is simply an ample and spanned vector bundle, the argument invoking (1.8.1) in (3.4) does not apply. So, E can be also either a singular rational curve or a smooth elliptic curve by Lemma (1.6). As far as X is concerned, this situation was carefully studied by Sakai [S2] according to the Iitaka dimension κ(X , KX + E ). In particular, X can be endowed with an elliptic fibration over a smooth curve and E can be the reduced component of a multiple fiber. The following example shows that this does really occur in our setting. (4.4) Let X = PB (G ) be as in (1.1). By (1.1.1) there exists an elliptic fibration υ/ : X → P1 given by | − 2KX |. Let E be the reduced component of one of the three double fibers of υ/ ; then E ≡ −KX . Let (4 .4 .1 )

E := V ⊗ (−2KX ),

where V is the vector bundle of rank 2 determined by a non-split extension (4 .4 .2 )

0 → H (G ) → V → OX (π ∗ b) → 0

for any b ∈ B . Note that h 0 (H (G ) − OX (π ∗ b)) = h 0 (G ⊗ OB (−b)) = 0 because G is normalized. By the Riemann-Roch theorem applied to G ⊗ OB (−b), we have h 1 (H (G ) − OX (π ∗ b)) = h 1 (G ⊗ OB (−b)) = −c1 (G ⊗ OB (−b)) = 1, which implies that there exists a nontrivial extension (4.4.2). We have det V = H (G ) + OX (π ∗ b) and c2 (V ) = 1, hence det E = H (G ) + OX (π ∗ b) − 4KX ≡ 9C0 − 3f , so c1 (E )2 = 27 and c2 (E ) = 7. Moreover, since det E is ample and its numerical equivalence class belongs to 3Num(X ), Reider’s theorem immediately shows that KX + det E is very ample. On the other hand, we have deg(KX + det E )E = (KX + det E )E = (det E )E = (9C0 − 3f )(2C0 − f ) = 3, so that KX + det E is not 2-very ample. The main point to prove is the following

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(4.5) Lemma. E is ample and spanned. Proof. By (4.4.1) and (4.4.2) we have the non-split exact sequence (4 .5 .1 )

0 → H (G ) ⊗ (−2KX ) → E → OX (π ∗ b) ⊗ (−2KX ) → 0.

Thus E is ample, the first and the third terms in (4.5.1) being so. Now note that (4 .5 .2 )

h 1 (H (G ) − 2KX ) = 0,

as the Kodaira vanishing theorem immediately shows, because H (G ) − 3KX ≡ 7C0 − 3f is ample. Note also that the third term in (4.5.1) is spanned. In fact, we can write OX (π ∗ b) − 2KX = KX + M , where M ≡ 6C0 − 2f is ample and satisfies M 2 = 12. Thus the spannedness follows from Reider’s theorem, since the numerical equivalence class of M lies in 2Num(X ). In the same way, by using Reider’s theorem we can see that the first term in (4.5.1) is spanned outside points lying on an effective divisor D ≡ 2C0 − f ≡ −KX , which has to be irreducible and reduced because DC0 = 1. Hence it is spanned outside the three elliptic curves E , E 0 , E 00 , which are the reduced components of the three double fibers of υ/ . On the other hand, since −2KX is spanned, it follows that H (G ) − 2KX is certainly spanned off C0 . Putting all these information together, we conclude that H (G ) − 2KX is spanned outside the three points, say y, y 0 , y 00 , at which C0 intersects E , E 0 , E 00 respectively. It thus follows from (4.5.2) that E is spanned outside these three points. To prove that E is spanned also at y (y 0 and y 00 ), restrict (4.5.1) to E (similarly to E 0 and E 00 ). We thus get the exact sequence (4 .5 .3 )

0 → OE (H (G ) − 2KX ) → EE → OE (π ∗ b) ⊗ (−2KX )E → 0,

which is non-split in view of Lemma 1.2. Actually H 1 (H (G ) ⊗ OX (−π ∗ b) ⊗ OX (−E )) = 0, by the Kodaira vanishing theorem. Note that, since C0 ∩ E = {y}, (4.5.3) can be rewritten as 0 → OE (y) → EE → OE (x + z ) → 0, for some points x , z ∈ E . By the same argument as in [BaL, p. 229], we can see that there exists an exact sequence 0 → OE (w) → EE → Q → 0, where w is a point different from y and Q is a line bundle on E of degree 2. This shows that EE is spanned even at y. Similarly, EE 0 , EE 00 are spanned at y 0 , y 00 respectively. Therefore, from the cohomology sequence of 0 → E ⊗ OX (−E ) → E → EE → 0, we conclude that E is also spanned at y if we show that (4 .5 .4 )

h 1 (E ⊗ OX (−E )) = 0.

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But this is immediate. Indeed, tensoring (4.5.1) with OX (−E ) and recalling that KX ≡ −E , we get an exact sequence (4 .5 .5 )

0 → L1 → E ⊗ OX (−E ) → L2 → 0.

Since −KX + Lj is ample for j = 1,2, the Kodaira vanishing theorem gives h 1 (Lj ) = h 1 (KX + (−KX + Lj )) = 0 for j = 1,2, and so, by the exact cohomology sequence induced by (4.5.5) we obtain (4.5.4). The same argument as above applied to E 0 and E 00 shows the spannedness of E at y 0 and y 00 .  More examples related to case (ii) can be constructed on P1 -bundles X = PB (F ) associated to any normalized vector bundle F of rank 2 with c1 (F ) = 0 on an elliptic curve B . We omit these constructions, since they are long and rather similar to the above one.

References [BaL]

[BS1]

[BS2] [Fr]

[F1] [F2]

[F3] [H] [I]

[LM1] [LM2] [LP] [N] [OSS] [Ot]

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