Geometriae Dedicata 77: 185–201, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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Ample and Spanned Rank-2 Vector Bundles with k-Jet Ample Determinant on Algebraic Surfaces KAZUYOSHI TAKAHASHI? Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Okubo Shinjuku-ku, Tokyo 169-8555, Japan. e-mail: [email protected] (Received: 6 August 1998; revised version: 8 October 1998) Abstract. We investigate ample and spanned rank-2 vector bundles E with k-jet ample determinant on algebraic surfaces S. In paticular, we classify such polarized pairs (S, E) with small second Chern class. The case of c2 (E) = k + 1 is a generalization of the result of Ballico and Lanteri. Mathematics Subject Classifications (1991): 14E25, 14J60. Key words: higher order embeddings, k-jet ampleness, vector bundles.

1. Introduction Let S be a smooth connected projective surface over the complex number field C and L a line bundle on S. We say that a line bundle L on S is k-jet ample if for any integers k1 , . . . , kr with Pr choice of distinct points x1 , . . . , xr0 in S and positive 0 k1 kr i=1 ki = k + 1, the natural map H (L) → H (L ⊗ OS /(mx1 ⊗ · · · ⊗ mxr )) is surjective. Note that L is 0-jet ample if and only if it is generated by its global sections, and that L is 1-jet ample if and only if it is very ample. Beltrametti, Sommese, and others have studied ‘higher order embeddings’, e.g., k-spannedness, k-very ampleness and k-jet ampleness of polarized manifolds (see [3, 5–12, 14], and so on). In [1], Ballico and Lanteri classified the pairs (S, E) consisting of a smooth connected projective surface S and an ample and spanned rank-2 vector bundle E with very ample determinant and c2 (E) = 2. From the point of view of higher order embeddings, we generalize their results. Precisely, we investigate ample and spanned rank-2 vector bundles E whose determinant bundles det E are k-jet ample on smooth connected projective surfaces S. In particular, we give the complete classification of such pairs (S, E) with c2 (E) = k − 1, k or k + 1 since we obtain c2 (E) > k − 1. The case of c2 (E) = k + 1 is a generalization of the result of Ballico and Lanteri [1]. Our main result is as follows: ? Currently at Department of Mathematics, School of Education, Waseda University, 1-6-1 Nishi-

Waseda, Shinjuku-ku, Tokyo 169-8050, Japan. e-mail: [email protected].

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(O.S. Disk.)

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THEOREM 1.1. Let S be a smooth connected projective surface and E an ample and spanned vector bundle of rank 2 on S. Assume that det E is k-jet ample for k > 1. Then c2 (E) > k − 1. (1) If c2 (E) = k − 1, then (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (k − 1)) for k > 2. (2) If c2 (E) = k, then (S, E) is one of the following: (a) (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (k)) for k > 1; (b) k = 3 and (S, E) ∼ = (P2 , TP2 ), where TP2 is the tangent bundle; (c) k = 4 and (S, E) ∼ = (P2 , OP2 (2)⊕2 ); (d) k = 2 and (S, E) ∼ = (P1 × P1 , OP1 ×P1 (1, 1)⊕2 ). (3) If c2 (E) = k + 1, then (S, E) is one of the following: (a) (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (k + 1)); (b) k = 2 and (S, E) ∼ = (P2 , TP2 ), where TP2 is the tangent bundle; (c) k = 3 and (S, E) ∼ = (P2 , OP2 (2)⊕2 ); (d) k = 4, S ∼ = P2 , E is semistable, but not stable, and there is an exact sequence 0 → OP2 (2) → E → Ix (2) → 0, where x is a point of P2 and Ix is the ideal sheaf of the 0-dimensional subscheme {x}; (e) k = 5 and (S, E) ∼ = (P2 , OP2 (2) ⊕ OP2 (3)); (f) k = 1 and (S, E) ∼ = (P1 × P1 , OP1 ×P1 (1, 1)⊕2 ); (g) k = 2 and (S, E) ∼ = (P1 × P1 , OP1 ×P1 (1, 1) ⊕ OP1 ×P1 (1, 2)); ∼ (h) k = 3 and (S, E) = (P1 × P1 , OP1 ×P1 (1, 1) ⊕ OP1 ×P1 (2, 2)); (i) k = 2 and (S, E) ∼ = (F1 , [C0 + 2f ]⊕2 ); (j) k = 1 and (S, E) ∼ = (Bl7 (P2 ), [−KS ]⊕2 ); (k) k = 2 and (S, E) ∼ = (Bl6 (P2 ), [−KS ]⊕2 ); (l) k = 1, p: S → C is a P1 -bundle over an elliptic curve C with invariant e = −1, and E ∼ = p ∗ (E) ⊗ [C0 ], where E is an indecomposable rank-2 vector bundle on C of degree 1; (m) k = 2, S is a P1 -bundle over an elliptic curve C with invariant e = −1, and E ∼ = [C0 + f ]⊕2 ; (n) k = 2, p: S → C is a P1 -bundle over an elliptic curve C with invariant e = −1, and E ∼ = p ∗ (E) ⊗ [C0 ], where E is an indecomposable rank-2 vector bundle on C of degree 2; (o) k = 2, S is a K3 surface, and E has c1 (E)2 = 10, or 12; (p) k = 2, S is an Enriques surface, and E has c1 (E)2 = 12. Here C0 is a section of minimal self-intersection C02 = −e, and f is a fiber of the ruling. In addition, Blj (S) is the surface obtained by blowing up S at j points in general position. In Section 3, we prove the theorem above. In Claim 3.2, we use a new criterion by Langer for the k-jet ampleness of the adjoint bundles of line bundles on surfaces (see Theorem 2.3). In Claim 3.3, we need the analysis of noncyclic triple covers. We also give two examples of noncyclic triple covers π : S → P2 . One is

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the example of such a cover with π ∗ OP2 (2) 2-jet ample, and another is that with π ∗ OP2 (2) not 2-jet ample (see Claim 3.2, Claim 3.3, and Remark 3.1). 2. Preliminaries We work over the complex number field C. Let S be a smooth connected projective surface, E an ample and spanned vector bundle, and L a line bundle on S. We denote its structure sheaf by OS . If I is an ideal sheaf of OS , then we write L/I for L ⊗ (OS /I). We use the following standard notation of algebraic geometry: ∼ (resp. ≡), the linear (resp. numerical) equivalence of line bundles; [D], the linear equivalence class of the divisor D; |L|, the complete linear system associated to L; KS , the canonical divisor of S; hi (F ), the complex dimension of H i (F ) for any coherent sheaf F and i > 0; g(L) := 1 + 12 L · (L + KS ), the sectional genus of L; ci (E), the ith Chern class of E; S [l] , the Hilbert scheme of 0-dimensional subscheme (Z, OZ ) with length l. 2.1. k - JET

AMPLENESS AND k - VERY AMPLENESS

Let L be a line bundle on a smooth projective variety X. A line bundle L on X is said to generate k-jets at a point x ∈ X, if the evaluation map H 0 (L) → H 0 (L ⊗ OX /mk+1 x ) is surjective. A line bundle L on X is called k-jet ample if for P any choice of distinct points x1 , . . . , xr in X and positive integers k1 , . . . , kr with ri=1 ki = k + 1, the natural map H 0 (L) → H 0 (L⊗OX /(mkx11 ⊗· · ·⊗mkxrr )) is surjective. Note that if L is a k-jet
ample line bundle on a smooth n-fold X, then h0 (L) > k+n (see [11, p. 358]). n For an integer k > 0, a line bundle L on X is called k-very ample if the restriction map H 0 (L) → H 0 (L|Z ) is surjective for every 0-cycle (Z, OZ ) ∈ X [k+1] . Note that 0-jet ampleness is equivalent to 0-very ampleness and also equivalent to spannedness. Moreover, note that 1-jet ampleness is equivalent to 1-very ampleness and also equivalent to very ampleness. In addition, note that if L is k-jet ample, it is k-very ample by [11, Proposition 2.2]. In particular, note that k-jet ampleness is equivalent to k-very ampleness on P2 and Hirzebruch surfaces Fe (see [11, Section 5]). THEOREM 2.1 [11, Theorem 3.1 and Corollary 3.1]. Let L be a k-jet ample line bundle on a smooth connected n-fold X. Then Ln > k n + k n−1 unless (X, L) ∼ = (Pn , OPn (k)). In particular, Ln = k n if and only if (X, L) ∼ = (Pn , OPn (k)). The following proposition gives the complete classification of k-very ample line bundles with L · L 6 4k + 4 and k > 2 on a smooth surface.

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PROPOSITION 2.2 [14, Corollary 6.4]. Let L be a k-very ample line bundle on a smooth connected projective surface S with degree d := L · L 6 4k + 4 (k > 2). Then (1) if κ(S) = −∞, then k 6 4; (2) if κ(S) 6 = −∞, then S is either a minimal K3 surface of degree d = 4k, 4k + 2, 4k + 4 or a minimal Enriques surface of degree d = 4k + 4. More precisely, (S, L) is one of the cases in the following table: Case

k

S

L

(1)

4

P2

OP2 (4)

(2)

3

P2

OP2 (3), OP2 (4)

(3) (4) (5) (6) (7) (8) (9) (10)

2

P2 F0 F1 elliptic P1 -bundle, e = −1 Bl7 (Fe ), e = 0, 1 Bl9 (F0 ) Bl7 (P2 ) Bl6 (P2 )

OP2 (2), OP2 (3) 2C0 + bf , b = 2, 3 2C0 + 4f 2C0 + 2f P 4C0 + (2e + 5)f − 71 2Ei P 4C0 + 6f − 91 2Ei −2KS −2KS

(11)

k>2

S ⊂ Pg : K3

d = 4k, 4k + 2, 4k + 4

(12)

k>2

S ⊂ Pg−1 : Enriques

d = 4k + 4

In the table above, C0 is a section of minimal self-intersection C02 = −e, f is a fiber of the ruling, and each Ei (i > 1) is the exceptional curve of the first kind of the blowing-up. The following theorem gives a criterion of Reider type for the k-jet ampleness of the adjoint bundles of line bundles on surfaces. THEOREM 2.3 [19, Corollary 3.2 and Theorem 3.4]. Let L be a nef line bundle on a smooth surface S. Assume that KS +L is (k −1)-jet ample but not k-jet ample. If L2 > (k + 2)2 , then there exists a curve D such that OD (KS + L) is (k − 1)-jet ample but not k-jet ample, L − 2D is pseudoeffective, numerically nontrivial and L · D − lk 6 D 2 < L · D/2 < lk , where lk = [((k + 2)2 /4)].

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BOGOMOLOV ’ S THEOREM

THEOREM 2.4 (Bogomolov). [18, Theorem 1.7] Let E be an ample vector bundle of rank 2 on a smooth projective surface S. Then c12 (E) > 4c2 (E) if and only if there exists an exact sequence 0 → L → E → IX ⊗ M → 0, with L and M line bundles on S, and X is a 0-dimensional subscheme of S with sheaf of ideals IX , such that: (1) (L − M) · (L − M) > 4 deg X; (2) (L − M) · A > 0 for every ample line bundle A on S. PROPOSITION 2.5 (generalization of Reider’s method). [7] Let E be a vector bundle of rank 2 on a smooth projective surface S with c1 (E)2 > 4c2 (E) and with the exact sequence 0 → OS → E → IZ ⊗ L → 0,

(∗)

such that L := det E, that IZ is the ideal sheaf of a 0-dimensional subscheme Z of S, and that L is nef and big. Let 0 → L → E → IX ⊗ M → 0 be the exact sequence given by Bogomolov’s theorem. Then (1) there exists an effective divisor D containing Z with M ∼ = OS (D), L ∼ = L(−D); (2) for the D above, L · D − c2 (E) = D · D − deg X 6 D · D < L · D/2 < c2 (E). Note that the first exact sequence (∗) in the proposition always exists if E is ample and spanned. The following theorem gives the classification of ample and spanned vector bundles E with large c1 (E)2 relative to c2 (E). THEOREM 2.6 [22, Theorem]. Let E an ample and spanned vector bundle of rank r > 2 on a smooth projective surface S. Then c1 (E)2 < (c2 (E) + 2)2 /2 holds unless (1) c2 (E) is even and there is a finite morphism ψ: S → P2 of degree 2 such that E∼ = ψ ∗ (O(1) ⊕ O(c2 (E)/2)). (Then c1 (E)2 = (c2 (E) + 2)2 /2.); (2) (S, E) ∼ = (P2 , O(1) ⊕ O(c2 (E))); or

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(3) S is isomorphic to a geometrically ruled surface P(F ) over an elliptic curve C with the projection p: P(F ) → C and with the tautological line bundle O(1), and E ∼ = p ∗ (E) ⊗ O(1). Here F and E are indecomposable rank-2 vector bundles on C of degree 1. (Hence c2 (E) = 2.) Consequently, c1 (E)2 6 (c2 (E) + 1)2 holds for every pair (S, E), and the equality holds only in case (2).

3. Proof of Theorem 1.1 We start the proof of Theorem 1.1. It immediately follows that c2 (E) > k − 1. Indeed, since c1 (E)2 > k 2 by Theorem 2.1 and c1 (E)2 6 (c2 (E)+1)2 by Theorem 2.6, we have c2 (E) > k − 1. Note that c2 (E) > 0 since E is ample. 3.1.

THE CASE OF c2 (E) = k − 1

Assume that c2 (E) = k−1. Note that k > 2 since E is ample. Then since c1 (E)2 = k 2 , we have (S, det E) ∼ = (P2 , OP2 (k)) by Theorem 2.1. Hence since c2 (E) = c1 (E) − 1, we have (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (k − 1)) by [18, Theorem 4.3]. Then det E ∼ = OP2 (k) is k-jet ample. 3.2.

THE CASE OF c2 (E) = k

Assume that c2 (E) = k. Note that k > 1 since E is ample. By Theorem 2.6, the problem divides into the case when c1 (E)2 < (c2 (E) + 2)2 /2 and the case when c1 (E)2 > (c2 (E) + 2)2 /2. Assume that c1 (E)2 < (c2 (E) + 2)2 /2. If c1 (E)2 = k 2 , then we have (S, det E) ∼ = (P2 , OP2 (k)) by Theorem 2.1. Then since c2 (E) = c1 (E), we have (S, E) ∼ = (P2 , OP2 (2)⊕2 ) or (P2 , TP2 ) by [18, Theorem 4.5]. In the former case, det E ∼ = OP2 (4) is 4-jet ample. In the latter case, det E ∼ = OP2 (3) is 3-jet ample. Hence we can assume that c1 (E)2 > k 2 + k by Theorem 2.1. Since k 2 + k 6 c1 (E)2 < (c2 (E) + 2)2 /2 = (k + 2)2 /2, we have 1 6 k 6 3. If c2 (E) = k = 1, then we (S, E) ∼ = (P2 , OP2 (1)⊕2 ) by [13, Theorem 11.1.3]. Then det E ∼ = OP2 (2) is 1-jet ample. If c2 (E) = k = 2, then by [1, Corollary], [2, Theorem] and the 2-jet ampleness of det E, (S, E) is one of the following: (1) (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (2)); (2) (S, E) ∼ = (P1 × P1 , OP1 ×P1 (1, 1)⊕2 ); (3) p: S → P2 is a double cover branched along a smooth quartic curve and E ∼ = p ∗ OP2 (1)⊕2 ; (4) π : S → C is a P1 -bundle with invariant e = −1 over a smooth curve C of genus 1 and E = π ∗ F ⊗ [h], where h is a minimal section of S and F is a rank-2 vector bundle over C by the nonsplit extension 0 → OC → F → OC (y) → 0, y being any point of C.

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But all these cases 1 ∼ 4 belong to the case of c1 (E)2 > (c2 (E) + 2)2 /2 by Theorem 2.6. Note that the case 2 belongs to the case 1 of Theorem 2.6. Assume that c2 (E) = k = 3. If c1 (E)2 > 4c2 (E) + 1 = 13 , then by [22, Corollary], we have (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (3)). But this case belongs to the case of 2 c1 (E) > (c2 (E) + 2)2 /2 by Theorem 2.6. If c1 (E)2 6 4c2 (E) = 12 , then we can use the table of Proposition 2.2 since the k-jet ampleness of det E implies the kvery ampleness. Since c1 (E)2 6 4c2 (E) = 12, we have (S, det E) ∼ = (P2 , OP2 (3)), 2 2 ∼ or S is a K3 surface and c1 (E) = 12. If (S, det E) = (P , OP2 (3)), then (S, E) belongs to the case of c1 (E)2 = k 2 . If S is a K3 surface and c1 (E)2 = 12, then we have h0 (det E)
= 8 since g(det E) = 7. But since det E is 3-jet ample, we have h0 (det E) > 52 = 10, a contradiction. Next, assume that c1 (E)2 > (c2 (E) + 2)2 /2. Then (S, E) is the cases 1 ∼ 3 of Theorem 2.6. We consider the case 1 of Theorem 2.6. Noting that c1 (E)2 = (c2 (E) + 2)2 /2, we have 1 6 k 6 3 since k 2 + k 6 (k + 2)2 /2. Hence we have k = 2 since c2 (E) is even. Since c2 (E) = 2, by [1, Corollary] and the 1-jet ampleness of det E, (S, E) is one of the following: (a) (S, E) ∼ = (P1 × P1 , OP1 ×P1 (1, 1)⊕2 ); (b) ψ: S → P2 is a double cover branched along a smooth quartic curve and E ∼ = ψ ∗ OP2 (1)⊕2 . In the former case, det E ∼ = OP1 ×P1 (2, 2) is 2-jet ample. In the latter case, S is a del Pezzo surface with −KS ∼ ψ ∗ OP2 (1) and KS2 = 2, and the map p is classically called the Geiser involution (see [13, Example 10.2.4]). CLAIM 3.1. In this case, det E ∼ = ψ ∗ OP2 (2) is not 2-jet ample. Proof. We can use the same argument as [8, Proposition 5.2]. It is enough to show that det E does not generate 2-jets at some point of S. Let R be the ramification divisor of ψ and Z a 0-dimensional subscheme of length 6 of S such that Supp(Z) = {x} with x ∈ R. We can assume that R is defined by a local coordinate s, i.e., R = {s = 0} at x. We consider local coordinates (s, v) on S at x. Let y be a point of P2 belonging to the branch divisor B of ψ such that y = ψ(x). Then we can take local coordinates (t, v) on P2 at y, where ψ ∗ t = s 2 . By the projection formula and [4, I.17], we have H 0 (S, det E) ∼ = H 0 (P2 , ψ∗ ψ ∗ OP2 (2)) ∼ = H 0 (P2 , OP2 (2)) ⊕ H 0 (P2 , OP2 ). Therefore as a generator B of H 0 (det E), we can take the pullback of sections of OP2 (2) and one more section σ ∈ H 0 (OP2 ) which in local coordinates around x is of the form λs with λ is a holomorphic section that does not vanish at x. Since s 2 = ψ ∗ t and v = ψ ∗ v, we have B = h1, s 2 , v, s 4 , v 2 , s 2 v, σ i. On the other hand, H 0 (det E|Z ) ∼ = H 0 (det E/m3x ) contains the element sv which is not image of element of the base B. This shows that the restriction map

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H 0 (det E) → H 0 (det E/m3x ) is not surjective. Thus det E is not 2-jet ample. This completes the proof of the claim. 2 In the case 2 of Theorem 2.6, we have (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (k)) for k > 1. ∼ Then det E = OP2 (k + 1) is k-jet ample. In the case 3 of Theorem 2.6, det E ∼ = 2C0 + f is very ample but not 2-very ample by [12, Proposition 2.2], where C0 is a section of minimal self-intersection C02 = 1 and f a fiber of the ruling. Hence det E is not 2-jet ample. 3.3.

THE CASE OF c2 (E) = k + 1

Assume that c2 (E) = k + 1. We use the same argument as the case of c2 (E) = k. Assume that c1 (E)2 < (c2 (E) + 2)2 /2. If c1 (E)2 = k 2 , then we have (S, det E) ∼ = (P2 , OP2 (k)) by Theorem 2.1. Then since c2 (E) = c1 (E) + 1, by [18, Theorem 4.5], (S, E) is one of the following: (a) (S, E) ∼ = (P2 , OP2 (2) ⊕ OP2 (3)); (b) S ∼ = P2 , E is semistable, but not stable, and there is an exact sequence 0 → OP2 (2) → E → Ix (2) → 0, where x is a point of P2 and Ix is the ideal sheaf of the 0-dimensional subscheme {x}. In the former case, det E ∼ = OP2 (5) is 5-jet ample. In the latter case, det E ∼ = OP2 (4) is 4-jet ample. Hence we can assume that c1 (E)2 > k 2 + k by Theorem 2.1. Since k 2 + k 6 c1 (E)2 < (c2 (E) + 2)2 /2 = (k + 3)2 /2, we have 1 6 k 6 5.If k = 1, then we have the case 1 ∼ 4 in the proof of the case of c2 (E) = k above by [1, Corollary], [2, Theorem] and the 1-jet ampleness of det E since c2 (E) = 2. But all these cases 1 ∼ 4 belong to the case of c1 (E)2 > (c2 (E) + 2)2 /2 by Theorem 2.6. Assume that k = 2. Then we have c2 (E) = 3. If c1 (E)2 > 4c2 (E) + 1 = 13, then we have (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (3)) by [22, Corollary]. But this case belongs to the case of c1 (E)2 > (c2 (E) + 2)2 /2 by Theorem 2.6. If c1 (E)2 6 4c2 (E) = 12 , then we can use the table of Proposition 2.2 since the k-jet ampleness of det E implies the k-very ampleness. Hence (S, det E) is one of the cases (3) ∼ (12) of Proposition 2.2. If (S, det E) is the case (3) of Proposition 2.2, then we have (S, E) ∼ = (P2 , TP2 ) or (P2 , OP2 (1) ⊕ OP2 (3)) by [18, Corollary 4.7] since c2 (E) = 3. In the former case, det E ∼ = OP2 (3) is 2-jet ample. In the latter case, det E ∼ = 2 OP2 (4) contradicts c1 (E) 6 12. If (S, det E) is the case (4) of Proposition 2.2, then we have (S, E) ∼ = (P1 × P1 , OP1 ×P1 (1, 1) ⊕ OP1 ×P1 (1, 2)) by [18, Corollary 2.11]. Note that the k-jet ampleness of det E is equivalent to the k-very ampleness on Hirzeburch surfaces Fe . Hence det E ∼ = OP1 ×P1 (2, 3) is 2-jet ample. Similarly, if (S, det E) is the case (5) of Proposition 2.2, then we have (S, E) ∼ = (F1 , [C0 + ⊕2 2f ] ) by [18, Corollary 2.11]. We consider the case (6) of Proposition 2.2. CLAIM 3.2 . In this case, det E ≡ 2C0 + 2f is 2-jet ample.

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Proof. Assume that det E is not 2-jet ample. Let L := det E − KS . Note that det E ≡ KS +L is 1-jet ample by [12, Proposition 2.2]. Then since KS ≡ −2C0 +f , we have L ≡ 4C0 + f and L2 = 24 > 16. By Theorem 2.3, there exists a curve D such that (KS + L)|D is 1-jet ample but not 2-jet ample, L − 2D is pseudoeffective, numerically nontrivial and L · D − 4 6 D 2 < L · D/2 < 4.

(**)

Let C := C0 − (1/2)f . Then it is a well-known fact that C is nef and 2C is numerically equivalent to a smooth effective elliptic curve. Note that C 2 = f 2 = 0, C · C0 = 1/2, f · C0 = f · C = 1 and we can take C and f as a basis for the second rational homology H2 (S, Q) of S. Then we have L ≡ 4C + 3f . First, assume that D ≡ λC for some positive λ ∈ Q. Then since D 2 = 0 and L · D = 3λ, we have 0 6 λ 6 4/3 by the inequality (∗∗). Since C0 · D = λ/2 is an integer, we have C0 · D = 0, i.e., λ = 0, a contradiction. Assume that D ≡ λf for some positive λ ∈ Z. Since D 2 = 0, we have L · D = 4λ 6 4 by the inequality (∗∗). Hence since λ 6 1 and λ > 0, we have λ = 1, i.e., D ≡ f . Note that D is irreducible and reduced since L · D = 4 and L ≡ 4C0 + f is 3-very ample by [12, Proposition 2.2]. Thus we have D ∼ = P1 . Note that the k-jet ampleness is equivalent to the kvery ampleness on smooth curves. Since degD (KS + L) = (2C0 + 2f ) · f = 2, (KS + L)|D ∼ = OP1 (2) is 2-jet ample on D. But this is a contradiction. Finally, assume that D ≡ λC + µf where λ, µ ∈ Q and λµ 6 = 0. Since λ = D · f > 0 and µ = D·C > 0, we have λ > 0 and µ > 0. Note that λ = D·f and µ+λ/2 = C0 ·D are positive integers. Then we have λµ < 2 and L · D = 4µ + 3λ < 8 by the inequality (∗∗). If λ = 1, then we have µ + λ/2 < 7/4 since 4µ < 5. Hence, we have µ + λ/2 = 1, i.e., µ = 1/2. Note that D is irreducible and reduced since L · D = 5 and the 3-very ampleness of L. Then since D ≡ C + (1/2)f = C0 , D is isomorphic to a smooth elliptic curve. But since (KS +L)·D = (2C0 +2f )·C0 = 4, (KS + L)|D is 2-jet ample on D by [9, Lemma 1.1], a contradiction. If λ = 2, then we have µ + λ/2 < 3/2 since 4µ < 2. Hence, we have µ + λ/2 = 1, i.e., µ = 0, a contradiction. If λ > 3, then this is impossible since 4µ < −1. Therefore since there exists no such curve D, this contradicts the assumption.This completes the proof of the claim. 2 We construct ample and spanned vector bundles E of rank 2 with det E ≡ 2C0 +2f and c2 (E) = 3. Assume that E is decomposable. Let E ∼ = [α1 C0 +β1 f ]⊕ [α2 C0 + β2 f ]. Note that α1 + α2 = 2, β1 + β2 = 2 and α1 α2 + α1 β2 + α2 β1 = 3. Since E is ample and spanned, we have αi > 0, βi > −αi /2, αi + βi > 2, and αi + 2βi > 2 for i = 1, 2 by [12, Proposition 2.2] and [13, Theorem 3.2.11]. Hence we have αi = βi = 1 for i = 1, 2, i.e., E ∼ = [C0 + f ]⊕2 . Next, assume 1 that E is indecomposable. Let p: S → C be the P -bundle over a smooth elliptic ∼ ∗ curve C. Then since (E ⊗ [−C0 ])|f ∼ = OP⊕2 1 , we have E ⊗ [−C0 ] = p (E), where E is a vector bundle of rank 2 on C. Hence we have E ∼ = [C0 ] ⊗ p ∗ (E). Note that deg E = 2 since c1 (p ∗ (E)) ≡ (deg E)f . Moreover, note that E is defined by a

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nonsplit extension 0 → [C0 + f ] → E → [C0 + f ] → 0 since there exists an extension 0 → OC (x) → E → OC (y) → 0, where x, y are points of C. We consider the case (7) of Proposition 2.2. Note that S is a del Pezzo surface. Indeed, recalling that F1 ∼ = Bl1 (P2 ) and Bl1 (F0 ) ∼ = Bl2 (P2 ), we see that 2 S is obtained by blowing P up P at eight points in general position. Hence since −KS ≡ 2C0 + (e + 2)f − 7i=1 Ei is ample, S is a del Pezzo surface with KS2 = 1. Noting that det E ≡ −2KS + f , we have E ∼ = [−KS ] ⊕ [−KS + f ] by [18, Proposition 3.13]. But in this case, E is not spanned since −KS is ample but not spanned by [12, Proposition 0.6]. We consider the case (8) of Proposition 2.2. Then S is a noncyclic triple cover π : S → P2 with the Tschirnhaus bundle E ∼ = OP2 (−2)⊕2 by [21, Corollary 10.4 ∗ and Table 10.5]. In addition, note that det E ∼ OP2 (2) since π is the morphism = Pπ 9 associated to the linear system |2C0 + 3f − i=1 Ei |. ∼ π ∗ OP2 (2) is not 2-jet ample. CLAIM 3.3 . Then det E = Proof. We recall the theory of triple cover by Miranda (see [21] and [15, §1]). Let {z, w} be a local base of the Tschirnhaus bundle E ∼ = OP2 (−2)⊕2 and S(E) the symmetric algebra of E. Then SpecP2 S(E) is identified with the total space of the dual vector bundle of E and S is embedded in SpecP2 S(E) as a closed subvariety which is locally defined by an integral domain A := OP2 [z, w]/(F (z, w), G(z, w), H (z, w)), where F := z2 − az − bw − 2(a 2 − bd), G := zw + dz + aw + (ad − bc), H := w 2 − cz − dw − 2(d 2 − ac), and a, b, c, d ∈ H 0 (P2 , OP2 (2)). Note that b 6 = 0 and c 6 = 0 since A is an integral domain. Moreover, note that z, w are fiber coordinates of A2 × P2 in SpecP2 S(E). Let p: SpecP2 S(E) → P2 be the bundle projection. Take the tautological section (s1 , s2 ) ∈ H 0 (SpecP2 S(E), p ∗ (E ∗ )). Then letting ti be the restriction of si to S, we have H 0 (det E) = π ∗ H 0 (OP2 (2)) ⊕ t1 · π ∗ H 0 (OP2 ) ⊕ t2 · π ∗ H 0 (OP2 ) inside the rational function field of S by the projection formula since π∗ OS = OP2 ⊕ OP2 (−2)⊕2 and π = p|S . Moreover, since z, w is obtained by the restriction of t1 , t2 respectively, we have locally H 0 (det E) = π ∗ H 0 (OP2 (2)) ⊕ z · π ∗ H 0 (OP2 ) ⊕ w · π ∗ H 0 (OP2 ).

(])

On the other hand, S is a general triple cover in the sense of Miranda, i.e., π has no total ramification in codimension one and the branch locus of π has only cusps with one tangent as singularities, which correspond to the total ramification points of π (see [21, Corollary 5.8 and Lemma 5.9]). In addition, the ramification divisor R1 of π is generically smooth and π is simply ramified at x ∈ R1 except for the points which the image of π is the total ramification points of π . Take such a

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smooth point x ∈ R1 . We recall the argument of [21, §4]. Let P (z, w) := az+bw+ (a 2 − bd), Q(z, w) := dz + aw + 12 (ad − bc), and R(z, w) := cz + dw + (d 2 − ac). Then the simple ramification of π occurs in Case 2 of [21, Proof of Lemma 4.5], i.e., the case that there is a unique solution (z0 , w0 ) to the system P = Q = R = 0 and (z0 , w0 ) 6 = (0, 0). In this case, note that w − w0 = f · (z − z0) at x since b 6 = 0 and c 6 = 0, where f is some element of Oy,P2 and y is a point of P2 belonging to the branch divisor B of π such that y = π(x). Then since R1 is smooth at x, by coordinate change z 7 → z − z0 , we can take z as a local coordinate on R1 at x, i.e., R1 = {z = 0} at x. We consider local coordinates (z, v) on S at x, where v is a regular element in Oy,P2 . Since π is simply ramified at x, we can take local coordinates (t, v) on P2 at y, where π ∗ t = z2 . We consider the relation (]). Since w = f · z at x, we obtain H 0 (det E) = π ∗ H 0 (OP2 (2)) ⊕ z · π ∗ H 0 (OP2 ) ⊕ f z · π ∗ H 0 (OP2 ) at x. Now we use the same argument as Claim 3.1. Let Z a 0-dimensional subscheme of length 6 of S such that Supp(Z) = {x}. Then H 0 (det E) is generated by B := {1, z2 , v, z4 , v 2 , z2 v, λ1 z, λ2 f z} at x, where λ1 and λ2 are holomorphic functions that do not vanish at x. On the other hand, H 0 (det E|Z ) ∼ = H 0 (det E/m3x ) contains the element zv which is not image of element of the base B. This shows that the restriction map H 0 (det E) → H 0 (det E/m3x ) is not surjective. Thus det E is not 2-jet ample. This completes the proof of the claim. 2 Remark 3.1. In the cases (6) and (8) of Proposition 2.2, the surface S is a noncyclic triple cover π : S → P2 . In the former case, the Tschirnhaus bundle E of the cover is isomorphic to 1P2 by [21, Corollary 10.6]. In addition, note that det E ∼ = π ∗ OP2 (2) since π is the morphism associated to the linear system |C0 +f |. But in this case, det E is 2-jet ample by Claim 3.2. l (9) of Proposition 2.2. Then S is a del Pezzo surface with KS2 = 2 and det E ∼ −2KS . But there is no such vector bundle E with c2 (E) = 3 by [20, p. 195 (5.7)]. We consider the case (10) of Proposition 2.2. Then S is a del Pezzo surface with KS2 = 3 and det E ∼ −2KS . Since c2 (E) = 3, we have E ∼ = [−KS ]⊕2 by [18, Corollary 3.14]. We consider the case (11)
and (12) of Proposition 2.2. Note that 2-jet ampleness implies h0 (det E) > 42 = 6 and that if h0 (det E) = 6, then we have (S, E) ∼ = (P2 , OP2 (2)) by the same argument as the proof of [11, Theorem 3.1] since h0 (det E) = h0 (det E/m3x ) = 6, where x is a point of S and mx the ideal sheaf of x in OS . Hence a K3 surface with c1 (E)2 = 8 cannot occur since g(det E) = 5 and h0 (det E) = 6. Assume that k = 3. Then we have c2 (E) = 4. If c1 (E)2 > 4c2 (E) + 1 = 17, then by [22, Corollary], (S, E) is one of the following:

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(1) there is a finite morphism ψ: S → P2 of degree 2 such that E ∼ = ψ ∗ (O(1) ⊕ O(2)); (2) (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (4)). But these cases belong to the case of c1 (E)2 > (c2 (E) + 2)2 /2 by Theorem 2.6. If c1 (E)2 6 4c2 (E) = 16 , then we can use the table of Proposition 2.2 since the k-jet ampleness of det E implies the k-very ampleness. Hence (S, det E) is one of the cases (2), (11) and (12) of Proposition 2.2. We consider the case (2) of Proposition 2.2. The case of det E ∼ = OP2 (3) contradicts c1 (E)2 > k 2 + k = 12. If det E ∼ = OP2 (4), then we have (S, E) ∼ = (P2 , OP2 (1) ⊕ OP2 (4)) or (P2 , OP2 (2)⊕2 ) by [18, Corollary 4.7]. The former case belongs to the case c1 (E)2 > (c2 (E) + 2)2 /2 by Theorem 2.6. In the latter case, det E ∼ = OP2 (4) is 3-jet ample. We consider the case (11) and (12) of Proposition 2.2. By the same argument as the case of k = 2, these cases are impossible. Indeed, if S is either a K3 surface with c1 (E)2 = 12 and
14, or an Enriques surface with c1 (E)2 = 16, then we have h0 (det E) < 10 = 52 .
If S is a K3 surface with c1 (E)2 = 16, then since h0 (det E) = 52 , we have (S, E) ∼ = (P2 , OP2 (3)), a contradiction. Assume that k = 4. Then we have c2 (E) = 5. If c1 (E)2 > 4c2 (E) + 1 = 21, then by Proposition 2.5, there exists an effective divisor D safisfying 1 and 2 of Proposition 2.5. Then the inequality of 2 of Proposition 2.5 is det E · D − 5 = D 2 − deg X 6 D 2 < det E · D/2 < 5.

(†)

Since E is ample and spanned, we have det E · D − 5 = D 2 − deg X > 1 by [22, Proposition 1.2]. Noting that det E · D 6 8 by [12, Lemma 0.3.3], we have 6 6 det E · D 6 8. If det E · D = 6, then we have D 2 = 1 or 2 by the inequality (†) since D 2 −deg X = 1. If D 2 = 2, then the Hodge index theorem gives 36 = (det E· D)2 > (det E)2 D 2 = 2(det E)2 , which contradicts 21 6 c1 (E)2 < (c2 (E) + 2)2 /2 = 49/2. If D 2 = 1, then we have (S, D) ∼ = (P2 , OP2 (1)) and deg X = 0 by [22, Proposition 1.2]. Since det E · D = 6, we have det E ∼ = OP2 (6), which contradicts 21 6 c1 (E)2 < (c2 (E) + 2)2 /2 = 49/2. If det E · D = 7, then we have D 2 = 2 or 3 by the inequality (†) since D 2 − deg X = 2. If D 2 = 3, then the Hodge index theorem gives 49 = (det E · D)2 > (det E)2 D 2 = 3(det E)2 , which contradicts 21 6 c1 (E)2 < (c2 (E) + 2)2 /2 = 49/2. If D 2 = 2, then by [22, Proposition 1.2], (S, E) is one of the following: (1) (S, D) ∼ = (P1 × P1 , OP1 ×P1 (1, 1)) and deg X = 0; (2) there is a finite morphism ψ: S → P2 of degree 2, deg X = 0, D ∼ = ψ ∗ O(1) and g(D) > 1. Note that D is ample and spanned by the proof of [22, Proposition 1.2]. In the case 1, we have E ∼ = OP1 ×P1 (1, 1) ⊕ OP1 ×P1 (2, 3)) by [18, Corollary 2.11] since c2 (E) = 5 and 21 6 c1 (E)2 6 49/2. But det E ∼ = OP1 ×P1 (3, 4) is not 4-jet ample. We consider the case 2. Since D is ample and spanned, there exists a smooth curve

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C ∈ |D|. Note that C is irrducible and reduced by det E · D = 7 and the 4very ampleness of det E. If g(C) > 2, then by the 4-very ampleness of det E and [12, Corollary 1.7.3], we have det E · C > 2g(C) + k = 8, which contradicts det E · D = 7. If g(C) = 1, then S is a del Pezzo surface with −KS ∼ = ψ ∗ O(1) 2 and KS = 2 by [16, p. 49, (I.6.13)] and [22, Proof of Proposition 1.2]. Then since det E · KS = −7, c1 (E)2 must be odd, i.e., c1 (E)2 = 21 or 23. If c1 (E)2 = 21, then we have g(det E) = 8. If c1 (E)2 = 23, then we have g(det E) = 9. But since these cases satisfy g(det E) 6 3k + 1 = 13, there exists no such pair (S, det E) by the 4-very ampleness of det E and [14, Corollary 5.5]. If det E · D = 8, then we have D 2 = 3 by the inequality (†) since D 2 − deg X = 3. By the Hodge index theorem, we have c1 (E)2 = 21. But since c1 (E)2 6 5k + 1 = 21, there exists no such pair (S, det E) by the 4-very ampleness of det E and [14, Proposition 6.2]. If k 2 + k = c1 (E)2 = 20 = 4c2 (E) = 4k + 4, then by the 4-very ampleness of det E, (S, det E) is the case (1) of the table of Proposition 2.2, a contradiction. Assume that k = 5. Then we have c2 (E) = 6. Since c1 (E)2 > 4c2 (E) + 1 = 25 and c1 (E)2 > k 2 + k = 30, then by Proposition 2.5, there exists an effective divisor D satisfying 1 and 2 of Proposition 2.5. Then the inequality of 2 of Proposition 2.5 is det E · D − 6 = D 2 − deg X 6 D 2 < det E · D/2 < 6. We have 7 6 det E · D 6 10 by the same argument as the case of k = 4. Note that c1 (E)2 = 30 or 31 since k 2 + k = 30 6 c1 (E)2 < (c2 (E) + 2)2 /2 = 32. If det E · D = 7, then we have D 2 = 1, 2 or 3. The cases of D 2 = 2 or 3 contradict the Hodge index theorem since c1 (E)2 = 30 or 31. If D 2 = 1, then we have (S, D) ∼ = (P2 , OP2 (1)) and deg X = 0 by [22, Proposition 1.2]. Since det E·D = 7, we have det E ∼ = OP2 (7), which contradicts c1 (E)2 = 30 or 31. If det E · D = 8, then we have D 2 = 2 or 3. The case of D 2 = 3 contradicts the Hodge index theorem since c1 (E)2 = 30 or 31. If D 2 = 2, then then by [22, Proposition 1.2], (S, E) is one of the following: (1) (S, D) ∼ = (P1 × P1 , OP1 ×P1 (1, 1)) and deg X = 0; (2) there is a finite morphism ψ: S → P2 of degree 2, deg X = 0, D ∼ = ψ ∗ O(1) and g(D) > 1. In the case 1, we have E ∼ = OP1 ×P1 (1, 1) ⊕ OP1 ×P1 (2, 4)) by [18, Corollary 2.11] since c2 (E) = 6 and c1 (E)2 = 30 or 31. But det E ∼ = OP1 ×P1 (3, 5) is not 5-jet ample. In the case 2, there exists a reduced, irreducible, smooth curve C ∈ |D|. If g(C) > 2, then by the 5-very ampleness of det E and [12, Corollary 1.7.3], we have det E · C > 2g(C) + k = 9, which contradicts det E · D = 8. If g(C) = 1, then S is a del Pezzo surface with −KS ∼ = ψ ∗ O(1) and KS2 = 2 by [16, p. 49, (I.6.13)] and [22, Proof of Proposition 1.2]. Note that c1 (E)2 = 30 since det E · KS = −8. Then we have g(det E) = 12 6 3k + 1 = 16. But there exists no such pair (S, det E) by the 5-very ampleness of det E and [14, Corollary 5.5]. If det E · D = 9, then we have D 2 = 3 or 4. But these contradict the Hodge index theorem since c1 (E)2 = 30

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or 31. Similarly if det E · D = 10, then we have D 2 = 4. But this contradicts the Hodge index theorem. Finally, assume that c1 (E)2 > (c2 (E) + 2)2 /2. Then (S, E) is one of the cases 1 ∼ 3 of Theorem 2.6. We consider the case 1 of Theorem 2.6. Noting that c1 (E)2 = (c2 (E) + 2)2 /2, we have 1 6 k 6 5 since k 2 + k 6 (k + 3)2 /2. Hence we have k = 1, 3 or 5 since c2 (E) is even. Assume that k = 1. Since c2 (E) = 2, by [2, Corollary], (S, E) is one of the following: (1) (S, E) ∼ = (P1 × P1 , OP1 ×P1 (1, 1)⊕2 ); (2) p: S → P2 is a double cover branched along a smooth quartic curve and E ∼ = p ∗ OP2 (1)⊕2 . In these cases, det E is 1-jet ample. Assume that k = 3, i.e., there is a finite morphism ψ: S → P2 of degree 2 such that E ∼ = ψ ∗ (O(1) ⊕ O(2)). Since S is smooth if and only if the branch divisor of ψ is smooth, we investigate this case by the degree of the branch curve B (see [4, p. 182–183]). Moreover, note that S is uniquely determined, up to an isomorphism over P2 by the branch curve B ∈ |OP2 (2g + 2)| for some g > 0 as a cyclic cover SpecP2 (O ⊕ O(−g − 1)) over P2 of degree 2 (see [16, p. 48–49, (I,6.11–12)]) and that then KS ∼ = ψ ∗ O(g − 2) ∗ ∗ and g = g(S, ψ O(1)). If deg B = 2, then det E ∼ = ψ OP2 (3) is 3-jet ample by 1 1 ∼ [5, Lemma 2.1]. Then we have (S, E) = (P × P , OP1 ×P1 (1, 1) ⊕ OP1 ×P1 (2, 2)). Indeed, since g(ψ ∗ O(1)) = 0, we have (S, ψ ∗ O(1)) ∼ = (P1 × P1 , OP1 ×P1 (1, 1)) by the classification of polarized surfaces with sectional genus zero (see [13, Corollary 3.2.10]). CLAIM 3.4 . If deg B > 4, then det E ∼ = ψ ∗ OP2 (3) is not 3-jet ample. Proof. We can use the same argument as Claim 3.1. Let R := ψ ∗ (B)red be the ramification divisor of ψ and Z a 0-dimensional subscheme of length 10 of S such that Supp(Z) = {x} with x ∈ R. We can assume that R is defined by a local coordinate s, i.e., R = {s = 0} at x. We consider local coordinates (s, v) on S at x. Let y be a point of P2 belonging to the branch divisor B of ψ such that y = ψ(x). Then we can take local coordinates (t, v) on P2 at y, where ψ ∗ t = s 2 . By the projection formula and [4, I.17], we have locally around x ∈ R, H 0 (det E) = ψ ∗ H 0 (OP2 (3)) ⊕ s · ψ ∗ H 0 (OP2 (3 − 12 deg B)). First, assume that deg B = 4. Then since H 0 (det E) = ψ ∗ H 0 (OP2 (3)) ⊕ s · ψ ∗ H 0 (OP2 (1)) locally around x, a generator B of H 0 (det E) is {1, s 2 , v, s 4 , v 2 , s 2 v, s 6 , v 3 , s 4 v, s 2 v 2 , s, s 3 , sv}. ∼ H 0 (det E/m4 ) contains the element sv 2 On the other hand, H 0 (det E|Z ) = x which is not image of element of the base B. This shows that the restriction map H 0 (det E) → H 0 (det E/m4x ) is not surjective. Thus det E is not 3-jet ample.

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Assume that deg B = 6. Then since H 0 (det E) = ψ ∗ H 0 (OP2 (3)) ⊕ s · ψ ∗ H 0 (OP2 ) locally around x, a generator B of H 0 (det E) is {1, s 2 , v, s 4 , v 2 , s 2 v, s 6 , v 3 , s 4 v, s 2 v 2 , λs}, where λ is a holomorphic function that does not vanish at x. On the other hand, H 0 (det E|Z ) ∼ = H 0 (det E/m4x ) contains the element sv which is not image of element of the base B. Therefore since the restriction map H 0 (det E) → H 0 (det E/m4x ) is not surjective, det E is not 3-jet ample. Finally, assume that deg B > 8. Then since H 0 (det E) = ψ ∗ H 0 (OP2 (3)) locally around x, a generator B of H 0 (det E) is {1, s 2 , v, s 4 , v 2 , s 2 v, s 6 , v 3 , s 4 v, s 2 v 2 }. On the other hand, H 0 (det E|Z ) ∼ = H 0 (det E/m4x ) contains the element sv which is not image of element of the base B. Therefore det E is not 3-jet ample by the same argument as above. This completes the proof of the claim. 2 We return to the proof of Theorem 1.1. Assume that k = 5, i.e., there is a finite morphism ψ: S → P2 of degree 2 such that E ∼ = ψ ∗ (O(1) ⊕ O(3)). CLAIM 3.5 . In this case, det E ∼ = ψ ∗ OP2 (4) is not 5-jet ample. Proof. We can use the same argument as Claim 3.4. Let R := ψ ∗ (B)red be the ramification divisor of ψ and Z a 0-dimensional subscheme of length 21 of S such that Supp(Z) = {x} with x ∈ R. We can assume that R is defined by a local coordinate s, i.e., R = {s = 0} at x. We consider local coordinates (s, v) on S at x. Let y be a point of P2 belonging to the branch divisor B of ψ such that y = ψ(x). Then we can take local coordinates (t, v) on P2 at y, where ψ ∗ t = s 2 . By the projection formula and [4, I.17], we have locally around x ∈ R, H 0 (det E) = ψ ∗ H 0 (OP2 (4)) ⊕ s · ψ ∗ H 0 (OP2 (4 − 12 deg B)). First, assume that deg B = 2. Then since H 0 (det E) = ψ ∗ H 0 (OP2 (4)) ⊕ s · ψ ∗ H 0 (OP2 (3)) locally around x, a generator B of H 0 (det E) is {1, s 2 , v, s 4 , v 2 , s 2 v, s 6 , v 3 , s 4 v, s 2 v 2 , s 8 , v 4 , s 6 v, s 4 v 2 , s 2 v 3 , s, s 3 , sv, s 5 , sv 2 , s 3 v, s 7 , sv 3 , s 5 v, s 3 v 2 }.

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∼ H 0 (det E/m6 ) contains the element sv 4 which On the other hand, H 0 (det E|Z ) = x is not image of element of the base B. Therefore since the restriction map H 0 (det E) → H 0 (det E/m6x ) is not surjective, det E is not 5-jet ample. Assume that deg B = 4. Then since H 0 (det E) = ψ ∗ H 0 (OP2 (4)) ⊕ s · ψ ∗ H 0 (OP2 (2)) locally around x, a generator B of H 0 (det E) is {1, s 2 , v, s 4 , v 2 , s 2 v, s 6 , v 3 , s 4 v, s 2 v 2 , s 8 , v 4 , s 6 v, s 4 v 2 , s 2 v 3 , s, s 3 , sv, s 5 , sv 2 , s 3 v}. On the other hand, H 0 (det E|Z ) ∼ = H 0 (det E/m6x ) contains the element sv 4 which is not image of element of the base B. Therefore det E is not 5-jet ample. Finally, assume that deg B > 6. Then we have h0 (det E) 6 h0 (OP2 (4)) + 0 h (OP2 (1)) = 18. But since h0 (det E|Z ) = h0 (det E/m6x ) = 21, this is impossible. This completes the proof of the claim. 2 In the case 2 of Theorem 2.6, we have (S, E) ∼ = (P2 , OP2 (1)⊕OP2 (k+1)). Then ∼ det E = OP2 (k + 2) is k-jet ample. In the case 3 of Theorem 2.6, det E ∼ = 2C0 + f is 1-jet ample, where C0 is a section of minimal self-intersection C02 = 1 and f a fiber of the ruling. This completes the proof of Theorem 1.1. Remark 3.2. We do not know whether the cases (o) and (p) of Theorem 1.1 exist or not. Our knowledge of vector bundles on K3 surfaces and Enriques surfaces is poor. Acknowledgements The author would like to express his hearty gratitude to Professors Yukitoshi Hinohara and Hiroyuki Terakawa for their useful advice and warm encouragement. The author would also like to thank the referee for his comments and suggestions. References 1.

2. 3. 4. 5.

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