Notes on very ample vector bundles on 3-folds

Hidetoshi Maeda and Andrew J. Sommese

Abstract. Let E be a very ample vector bundle of rank two on a smooth complex projective threefold X. An inequality about the third Segre class of E is provided when K_{X}+det E is nef but not big, and when a suitable positive multiple of K_{X}+det E defines a morphism X → B with connected fibers onto a smooth projective curve B, where K_{X} is the canonical bundle of X. As an application, the case where the genus of B is positive and E has a global section whose zero locus is a smooth hyperelliptic curve of genus ≥ 2 is investigated, and our previous result is improved for threefolds.

Introduction

In what follows, varieties are always assumed to be defined over the field ℂ of complex numbers.

Let E be a very ample vector bundle of rank n−1 on a smooth projective variety X of dimension n ≥ 3. Then there exists a global section s ∈ Γ (E) whose zero locus C = (s)_{0} is a smooth curve on X.

If K_{X} + det E is nef and big, then there exist a birational morphism π : X → X′ expressing X as the blow-up of a smooth projective variety X′ along a finite set B of points (possibly empty) and an ample vector bundle E′ of rank n − 1 on X′ such that E = π^{∗}E′ ⊗ O_{X}(−π^{−1}(B)) and that K_{X′} + det E′ is ample. The polarized pair (X′, E′) is called the first reduction of (X, E). In this case, under the assumption that C is a hyperelliptic curve of genus g ≥ 2, we have proved in [MS]

n ≥ τ ≥ \frac{(n − 1)(g + t + 1) + 2}{2(g − 1)} − 1,

where τ is the nefvalue of the polarized pair (X′, K_{X′} + det E′) and t is the number of exceptional divisors with respect to the first reduction morphism π : X → X′.

As a continuation of the above research, we investigate the case where K_{X} +det E is nef but not big in case n = 3. According to [BS1, Theorem 3.1], there are five possibilities in this case. In this article we restrict ourselves to especially the following three cases among them, where a suitable positive multiple of K_{X} + det E

Typeset by AMS-TEX

2000 Mathematics Subject Classification. Primary 14J60; Secondary 14F05, 14C20, 14J30.

defines a morphism X → B with connected fibers onto a smooth projective curve B.

(a) X is a ℙ^{2}-bundle over a smooth curve B, and E_{F} ≅ O_{ℙ}(2) ⊕ O_{ℙ}(1) for any fiber F of the projection π : X → B;

(b) X is a ℙ^{2}-bundle over a smooth curve B, and E_{F} ≅ T_{ℙ} for any fiber F of the projection π : X → B, where T_{ℙ} is the tangent bundle of ℙ^{2};

(c) there exists a surjective morphism π : X → B onto a smooth curve B such that a general fiber F of π is a smooth quadric surface ℚ^{2} in ℙ^{3} with E_{F} ≅ O_{ℚ}(1)^{⊕2}.

We note that K_{X} + det E = π^{∗}H for some ample line bundle H on B in every case. The purpose of this article is to give an inequality in each of the above cases concerning the third Segre class of E by using the double point formula. The precise statement of our result is as follows:

Theorem 1. Let E be a very ample vector bundle of rank two on a smooth projective 3-fold X, and assume that (X, E) is one of (a), (b) or (c). Let D denote the third Segre class s_{3}(E) of E, let q be the genus of B, and let ρ = deg(H − K_{B}). Then

(1) D(D − 20) ≥ 200(q − 1) + 37ρ in case (a);

(2) D(D − 20) ≥ 168(q − 1) + 21ρ in case (b);

(3) D(D − 20) ≥ 168(q − 1) + 20ρ + k in case (c), where k is the number of singular fibers of π.

By Lemma 4, the singular fibers in case (c) are biholomorphic to irreducible and reduced quadric surfaces in ℙ^{3}.

This article is organized as follows. In Section 0 we collect preliminary material. Section 1 is devoted to the proof of Theorem 1. As an application of Theorem 1, in Section 2 we come back to the case where C is a hyperelliptic curve of genus ≥ 2, and show that case (a) does not occur and that case (b) is very restricted when the genus of B is positive. This allows us to improve [MS, Theorem 6] in case of 3-folds. It also complements the result [LPS, Theorem 2.4] as noted in [LPS, Remark 2.5].

Thanks are due to the referee for useful remarks.

0. Background material

We use the standard notation from algebraic geometry. The tensor products of line bundles are denoted additively. The numerical equivalence is denoted by ≡. The pullback i^{∗}E of a vector bundle E on X by an embedding i : Y ↪ X is denoted by E_{Y}. For a vector bundle E on X, the tautological line bundle on the projective space bundle ℙ_{X}(E) associated to E is denoted by H(E). A vector bundle E on a projective variety X is said to be very ample, if the tautological line bundle H(E) on ℙ_{X}(E) is very ample. We denote by K_{X} the canonical bundle of a smooth variety X. Let E be a vector bundle of rank n − 1 on a smooth projective variety X of dimension n ≥ 3 such that there exists a global section s ∈ Γ (E) whose zero locus C = (s)_{0} is a smooth curve on X. Then we should note that K_{C} = (K_{X} + det E)_{C}.

Let X be a smooth projective variety of dimension n ≥ 3, and let E be a very ample vector bundle of rank n − 1 on X. Assume that K_{X} + det E is nef. Then, by the base point free theorem, a suitable positive multiple of K_{X} + det E is spanned and defines a morphism π : X → B with connected fibers onto a normal projective variety B. Assume furthermore that dim B = 1. Then, from [BS1, Theorem 3.1] for n = 3 and [ABW, Theorem B] for n ≥ 4, (X, E) is one of the following:

(a) X is a ℙ^{n−1}-bundle over a smooth curve B, and E_{F} ≅ O_{ℙ}(2) ⊕ O_{ℙ}(1)^{⊕(n−2)} for any fiber F of the projection π : X → B;

(b) X is a ℙ^{n−1}-bundle over a smooth curve B, and E_{F} ≅ T_{ℙ} for any fiber F of the projection π : X → B, where T_{ℙ} is the tangent bundle of ℙ^{n−1};

(c) there exists a surjective morphism π : X → B onto a smooth curve B such that a general fiber F of π is a smooth hyperquadric ℚ^{n−1} in ℙ^{n} with E_{F} ≅ O_{ℚ}(1)^{⊕(n−1)}.

It should be emphasized that K_{X} + det E = π^{∗}H for some ample line bundle H on B in every case.

In all that follows, suppose that (X, E) is one of (a), (b) or (c). Since E is very ample, there exists a global section s ∈ Γ (E) whose zero locus C = (s)_{0} is a smooth curve on X. Let g denote the genus of C. We note that the restriction π_{C} : C → B of π to C is surjective. Hence the Riemann-Hurwitz formula tells us that

2g − 2 = d(2q − 2) + r,

where q is the genus of B, d is the degree of π_{C}, and r is the degree of the ramification divisor of π_{C}. Since d = c_{n−1}(E_{F}), we have d = 2 in cases (a), (c) and d = n in case (b). On the other hand,

K_{C} = (K_{X} + det E)_{C} = (π^{∗}H)_{C} = π_{C}^{∗}H,

so that

2g − 2 = dδ,

where δ = deg H. Hence

dδ = d(2q − 2) + r.

This implies that r is an integer multiple of d. Let r = dρ. Then

(0.1)  δ = 2q − 2 + ρ,

and we conclude that

ρ = deg(H − K_{B}).

It should be kept in mind that

(0.2)  2g − 2 = d(2q − 2 + ρ).

The following lemma tells us that ρ > 0.

Lemma 2. The morphism π_{C} : C → B can never be unramified.

Proof. Let X_{C} denote the fiber product X ×_{B} C of X and C over B. Then X_{C} is connected. Let p_{1} : X_{C} → X be the first projection, and let p_{2} : X_{C} → C be the second projection. Note that p_{1}^{∗}E is ample and that p_{1}^{−1}(C) is the zero locus of the pullback of the section s defining C. Thus H^{0}(p_{1}^{−1}(C), ℤ) ≅ H^{0}(X_{C}, ℤ) = ℤ by the Lefschetz-Sommese theorem [LM2, Theorem 1.1] (note that its proof is valid without assuming the connectedness of p_{1}^{−1}(C)), so that p_{1}^{−1}(C) is connected. On the other hand, p_{1}^{−1}(C) decomposes into a curve C′ and a curve C′′, where C′ is the image of the section of p_{2} defined by sending x ∈ C to (x, x) ∈ X_{C}. If π_{C} is unramified, then C′ ∩ C′′ = ∅. This contradicts the connectedness of p_{1}^{−1}(C). □

Moreover, if C is a hyperelliptic curve of genus g ≥ 2 and if q > 0, then we have the following

Lemma 3. Assume that C is a hyperelliptic curve of genus g ≥ 2. If the genus q of B is positive, then ρ is either 1 or 2.

Proof. By Lemma 2 we know that ρ > 0. Letting H = K_{B} +H for some line bundle H on B, we have ρ = deg H and K_{C} = π_{C}^{∗}(K_{B} + H). Since C is hyperelliptic, |K_{B} + H| cannot take any smooth curve of positive genus as its image. Therefore deg H ≤ 2. □

1. Proof of Theorem 1

From now on, throughout these notes, we assume that n = 3, and use the same notation as in Section 0. Let M = ℙ_{X}(E) be the projective space bundle associated to E, let p : M → X be the bundle projection, and let H(E) be the tautological line bundle on M. Then H(E) is very ample. As mentioned, we have K_{X} +det E = π^{∗}H. Thus K_{M} + 2H(E) = p^{∗}(K_{X} + det E) = p^{∗}π^{∗}H.

Let Z be a general element of |H(E)|, let S be a general element of |H(E)_{Z}|, and let φ : S → B be the restriction of π ∘ p to S. Then K_{S} = φ^{∗}H. Let G be a general fiber of π∘p, and let f = S∩G. Since S can be regarded as the zero locus of a section t ∈ Γ (M, H(E)^{⊕2}), f is the zero locus of the section t_{G} ∈ Γ (G, H(E)_{G}^{⊕2}). Hence f ≠ ∅, and φ is surjective. In particular, this implies that f is a 1- equidimensional smooth fiber of φ. On the other hand, since H^{0}(f, ℤ) ≅ H^{0}(G, ℤ) = ℤ by [LM2, Theorem 1.1], f is a smooth curve in S and we conclude that φ has connected fibers.

Now let us apply the double point formula [BS2, Theorem 13.1.5] to (Z, H(E)_{Z}), which tells us that

e(Z)−48χ(O_{Z})+84χ(O_{S})−11K_{S}^{2} −17K_{S}H(E)_{S} −(K_{Z} +H(E)_{Z})^{3}+D(D−20) ≥ 0, where D = H(E)_{Z}^{3} = H(E)^{4}, and e(Z) is the topological Euler characteristic of Z. Then

D = s_{3}(E) = c_{1}(E)^{3} − 2c_{1}(E)c_{2}(E),

where s_{3}(E) is the third Segre class of E. We have

K_{S}^{2} = (φ^{∗}H)^{2} = 0

and

(K_{Z} + H(E)_{Z})^{3} = (K_{M} + 2H(E))_{Z}^{3} = (p^{∗}π^{∗}H)_{Z}^{3} = 0.

Let us compute χ(O_{Z}), χ(O_{S}) and K_{S}H(E)_{S}. First, since Z is the blow-up of X along C, Z is birational to X. Thus

χ(O_{Z}) = χ(O_{X}) = χ(O_{B}) = 1 − q.

Next, since K_{S} = φ^{∗}H and φ has connected fibers, we have

h^{0}(K_{S}) = h^{0}(H).

Furthermore, by the Kodaira vanishing theorem,

0 = h^{1}(K_{M} + 2H(E)) = h^{1}(H),

since K_{M} + 2H(E) = p^{∗}π^{∗}H. Moreover, from the exact sequence

0 → O_{Z}(−S) → O_{Z} → O_{S} → 0,

we see that

h^{1}(O_{S}) = h^{1}(O_{Z}) = q.

Therefore, the Riemann-Roch theorem applied to (B, H) gives

2q − 2 + ρ = δ = deg H = h^{0}(H) + q − 1
= h^{0}(K_{S}) + q − 1 = h^{0}(O_{S}) − h^{1}(O_{S}) + h^{2}(O_{S}) + 2q − 2
= χ(O_{S}) + 2q − 2.

Consequently

χ(O_{S}) = ρ.

Finally, we have

K_{S}H(E)_{S} = (φ^{∗}H)H(E)_{S} = (p^{∗}π^{∗}H)_{Z}H(E)_{Z}^{2} = (p^{∗}π^{∗}H)H(E)^{3}
= δℙ_{F} (E_{F})H(E)^{3} = δH(E_{F})^{3}
= δ(c_{1}(E_{F})^{2} − c_{2}(E_{F})) = δ(c_{1}(E_{F})^{2} − d)
= (2q − 2 + ρ)(c_{1}(E_{F})^{2} − d).

Thus, in sum,

(1.0) e(Z) − 48(1 − q) + 84ρ − 17(2q − 2 + ρ)(c_{1}(E_{F})^{2} − d) + D(D − 20) ≥ 0.

We proceed now by cases.

(1.1) Case (a). In this case we have

e(Z) = e(X) + e(C) = e(ℙ^{2})e(B) + e(C) = 3(2 − 2q) + 2 − 2g.

Combining this with (0.2) gives e(Z) = 6(1 − q) − d(2q − 2 + ρ). Since c_{1}(E_{F})^{2} = 9 and d = 2, (1.0) tells us that

6(1 − q) − 2(2q − 2 + ρ) − 48(1 − q) + 84ρ − 119(2q − 2 + ρ) + D(D − 20) ≥ 0.

An easy calculation shows that

D(D − 20) ≥ 200(q − 1) + 37ρ,

and the result is proved.

(1.2) Case (b). Here we have c_{1}(E_{F})^{2} = 9 and d = 3. By the same argument as that in (1.1), we get

6(1 − q) − 3(2q − 2 + ρ) − 48(1 − q) + 84ρ − 102(2q − 2 + ρ) + D(D − 20) ≥ 0.

Consequently

D(D − 20) ≥ 168(q − 1) + 21ρ.

(1.3) Case (c). Before proceeding with the proof, we present the following

Lemma 4. Let (X, E) be as in case (c). Then every fiber of π is an irreducible and reduced quadric surface in ℙ^{3}.

Proof. We set L = det E. Then, as we pointed out, K_{X} + L = π^{∗}H.

We first claim that every fiber F of π is irreducible and reduced. Suppose to the contrary that F is not irreducible and reduced for some F. Then we can write F = n_{1}F_{1} +⋯+n_{r}F_{r} for distinct integral surfaces F_{1}, …, F_{r} and positive integers n_{1}, …, n_{r} with n_{1} + ⋯ + n_{r} ≥ 2. Hence a general element T ∈ |L| must meet F in a curve f = n_{1}f_{1} + ⋯ + n_{r}f_{r} for distinct integral curves f_{1}, …, f_{r}. Now we know that Lf_{i} ≥ 2 for any i from [LM1, Corollary 1]. If Lf_{i} = 2 for some i, then f_{i} ≅ ℙ^{1} by [LM1, Corollary 2]. Since f_{i} ∈ |L_{F_{i}}|, we see that f_{i}∩Sing(F_{i}) = ∅, where Sing(F_{i}) is the singular locus of F_{i}. Thus Sing(F_{i}) ⊂ F_{i} − f_{i}. Since Sing(F_{i}) is a compact algebraic set and F_{i} −f_{i} is affine, we conclude that F_{i} has at most isolated singularities. This implies that F_{i} is normal, and the classification of the polarized surfaces of sectional genus zero applies to (F_{i}, L_{F_{i}}) (see for example [BS2, Corollary 3.2.10]). However, since L_{F_{i}}E ≥ 2 for any rational curve E on F_{i}, (F_{i}, L_{F_{i}}) must be (ℙ^{2}, O_{ℙ}(2)). Therefore L_{F_{i}}^{2} = 4. On the other hand, L_{F_{i}}^{2} = L_{F_{i}}f_{i} = 2. This is a contradiction. Consequently Lf_{i} = L_{F_{i}}^{2} ≥ 3 for any i. We should note that Lf = L_{F}^{2} = 8 because E_{G} ≅ O_{ℙ^{1}×ℙ^{1}}(1, 1)^{⊕2} for a general fiber G of π. Moreover, since K_{X} + L = π^{∗}H, we have K_{T} = π_{T}^{∗} H, where π_{T} is the restriction of π to T. Thus T is a properly elliptic minimal surface, so that [S, Lemma 0.5.1] tells us that π_{T} : T → B has no multiple fibers. As a direct result of this observation, we obtain r = 2 and n_{1} = n_{2} = 1. Since K_{T} = π_{T}^{∗} H, we have (K_{T} + f_{i})f_{i} = f_{i}^{2} for i = 1, 2. Moreover, since f = f_{1} +f_{2}, we get f_{i}(f_{1} +f_{2}) = 0. Thus f_{1}^{2} = −f_{1}f_{2} < 0, since π_{T} has connected fibers. Similarly f_{2}^{2} < 0. Hence f_{1} ≅ ℙ^{1} and f_{2} ≅ ℙ^{1}, and so by the same argument as above we have F_{1} ≅ ℙ^{2} and F_{2} ≅ ℙ^{2}. Now we know that F = F_{1} + F_{2}, so that O_{F_{1}}(F_{1} + F_{2}) ≅ O_{F_{1}}. Since F_{1} meets F_{2} in a curve, the normal bundle N_{F_{1}/X} = O_{F_{1}}(F_{1}) to F_{1} in X is negative. By a well-known theorem of Grauert (see for example [BS2, Theorem 3.2.7]) there exists a holomorphic map p : X → Y onto a normal analytic variety Y such that p(F_{1}) is a point, y, and p induces a biholomorphism X − F_{1} ≅ Y − {y}. In particular, F_{1} ∩ F_{2} is contracted. However, this is absurd because F_{1} has no curves with negative self-intersection. Consequently every fiber F of π is irreducible and reduced.

If F is smooth, then (F, E_{F}) ≅ (ℚ^{2}, O_{ℚ}(1)^{⊕2}), where ℚ^{2} is a smooth quadric surface in ℙ^{3}. Let F be a singular fiber of π. We claim that F is a singular quadric surface in ℙ^{3}. To see this, take a general element T ∈ |L|. Then T meets F in an irreducible and reduced curve f. We note that the arithmetic genus of f is one because (K_{T} + f)f = 0. If f is not smooth, then f has a single singular point, so that Sing(F) contains a curve γ with Lγ = 1. This contradicts the fact that Lγ ≥ 2. Hence f is smooth, and the same argument as above again shows that F is normal. Since K_{X} + L = π^{∗}H, we have K_{F} + L_{F} = O_{F}, and we conclude that F is a normal Gorenstein Del Pezzo surface with K_{F}^{2} = 8. According to [Br, Theorem 1], F is one of the following:

(i) F is a singular quadric surface in ℙ^{3}; (ii) F is the space obtained by blowing down the zero section Δ of a ℙ^{1}-bundle

P on a smooth elliptic curve Γ; (iii) F is a rational surface with only rational double points as singularities,

obtained from ℙ^{2} by blowing up some number α ≤ 8 points (iterations allowed) then blowing down some number β ≤ α smooth rational curves, each with self-intersection −2.

In case (ii) we can write K_{P} + O_{P} (kΔ) = σ^{∗}K_{F} for some integer k, where σ is the blowing-down of Δ. Since (K_{P} + O_{P} (Δ))_{Δ} = K_{Δ} = O_{Δ}, we obtain

O_{Δ} = (σ^{∗}K_{F})_{Δ} = (K_{P} + O_{P} (kΔ))_{Δ}
= (K_{P} + O_{P} (Δ) + O_{P} ((k − 1)Δ))_{Δ} = O_{Δ}((k − 1)Δ).

This implies that k = 1. Hence K_{P} + O_{P} (Δ) = σ^{∗}K_{F}. Set Δ^{2} = −r for some positive integer r. Then K_{P} ≡ −2Δ−rρ, where ρ is a fiber of the bundle projection P → Γ. Thus −σ^{∗}K_{F} ≡ Δ + rρ, so that

L_{F} σ(ρ) = −K_{F} σ(ρ) = (−σ^{∗}K_{F})ρ = (Δ + rρ)ρ = 1.

This contradicts L_{F} σ(ρ) ≥ 2. In case (iii), let φ : G → ℙ^{2} be the composite of α blowing-up morphisms, and let τ : G → F be the composite of β blowing-down morphisms. Then K_{F}^{2} = K_{G}^{2} = K_{ℙ}^{2} − α = 9 − α. Hence α = 1, since K_{F}^{2} = 8. Consequently G is the first Hirzebruch surface. Moreover, either β = 0 or β = 1. If β = 0, then F = G. This is impossible because F is singular. However, the case β = 1 is also impossible because G has no (−2)-curves. Therefore case (iii) does not occur. Consequently F must be a singular quadric surface in ℙ^{3}, and the result is proved. □

We return to the proof of Theorem 1. In case (c), let F′ be a singular fiber of π, and let k denote the number of singular fibers of π. Then, since F′ is a singular quadric with an isolated singularity by Lemma 4, we obtain

e(Z) = e(X) + e(C) = e(X − kF′) + ke(F′) + e(C)

= 4(2 − 2q − k) + 3k + 2 − 2g = 8(1 − q) − k − d(2q − 2 + ρ).

Since c_{1}(E_{F})^{2} = 8 and d = 2, it follows from (1.0) that

8(1 − q) − k − 2(2q − 2 + ρ) − 48(1 − q) + 84ρ − 102(2q − 2 + ρ) + D(D − 20) ≥ 0.

Therefore

D(D − 20) ≥ 168(q − 1) + 20ρ + k,

and we have thus proved Theorem 1.

2. The case of a hyperelliptic curve

Let E be a very ample vector bundle of rank two on a smooth projective 3-fold X, and assume that (X, E) is one of (a), (b) or (c). In this section we set up the following condition:

(∗) There exists a global section s ∈ Γ (E) whose zero locus C = (s)_{0} is a smooth hyperelliptic curve of genus g ≥ 2, and the base curve B has genus q > 0.

Theorem 5. Under the assumption (∗), case (a) does not occur.

Proof. The proof is by contradiction. Let (X, E) be as in case (a). Then we can write X = ℙ_{B}(V) for some vector bundle V of rank three on B. Since E_{F} ≅ O_{ℙ}(2)⊕O_{ℙ}(1) for any fiber F of π, F := π_{∗}(E ⊗ (−2H(V))) is a line bundle on B, and we have an exact sequence

0 → 2H(V) + π^{∗}F → E → Q → 0

for some line bundle Q on X. Then Q_{F} = det E_{F} − 2H(V)_{F} ≅ O_{ℙ}(1). Hence Q = H(V) + π^{∗}L for some line bundle L on B. We note that Q is very ample, because E is very ample. Let G = π_{∗}Q. Then, since Q_{F} ≅ O_{ℙ}(1) for any F, G is a vector bundle of rank three on B such that (X, Q) ≅ (ℙ_{B}(G), H(G)). Thus G is very ample. We can write numerically 2H(V) + π^{∗}F ≡ 2H(G) + μF for some integer μ. Thus det E = 2H(V) + π^{∗}F + Q ≡ 3H(G) + μF. Since K_{X} + det E = π^{∗}H for some ample line bundle H on B, it follows from (0.1) that K_{X} + det E ≡ (2q − 2 + ρ)F. Therefore

K_{X} ≡ − det E + (2q − 2 + ρ)F ≡ −3H(G) + (2q − 2 + ρ − μ)F.

On the other hand, from the basic relation K_{X} = −3H(G) + π^{∗}(K_{B} + det G), we have

K_{X} ≡ −3H(G) + (2q − 2 + x)F,

where x = deg G. Consequently ρ − μ = x, i.e., μ = ρ − x.

Now, from the above exact sequence, c_{2}(E) = (2H(V) + π^{∗}F)Q ≡ 2H(G)^{2} + μH(G)F. As observed in Section 1, D = c_{1}(E)^{3} − 2c_{1}(E)c_{2}(E). Hence

D = (3H(G) + μF)^{3} − 2(3H(G) + μF)(2H(G)^{2} + μH(G)F)

= 15H(G)^{3} + 17μH(G)^{2}F = 15x + 17μ (2.1)

= 17ρ − 2x.

From Theorem 1, we have D(D − 20) ≥ 200(q − 1) + 37ρ. Moreover, under the assumption (∗), by Lemma 3 ρ is either 1 or 2. Hence D(D − 20) ≥ 37. This implies that D ≥ 22. Combining this with (2.1) gives 22 ≤ 17ρ − 2x < 17ρ, since x > 0. Thus ρ = 2. We claim that q = 1. To see this, suppose that q ≥ 2. Since ρ = 2, we obtain D(D − 20) ≥ 274. Thus D ≥ 30. By using (2.1) again, we have x = 17 − (1/2)D ≤ 2. Since x = H(G)^{3} and H(G) is very ample, we have a contradiction. Therefore q = 1, and D(D − 20) ≥ 74. Hence D ≥ 24, and by (2.1) we get x = 17 − (1/2)D ≤ 5. On the other hand, applying [IT, Proposition 1] to G gives x ≥ 7. This is also absurd. □

Theorem 6. Under the assumption (∗), let (X, E) be as in case (b). Then g = 4 and q = 1.

Proof. We can write X = ℙ_{B}(G) for some vector bundle G of rank three on B. Let x = deg G. Then, since K_{X} = −3H(G) + π^{∗}(K_{B} + det G), we have

(2.2)  K_{X} ≡ −3H(G) + (2q − 2 + x)F.

Let V = π_{∗}(E ⊗ (−H(G))). Then, since E_{F} ≅ T_{ℙ} for any F, V is a vector bundle of rank three on B, and we have an exact sequence

0 → π^{∗}L → (π^{∗}V) ⊗ H(G) → E → 0

for some line bundle L on B. Let l = deg L and let v = deg V. Then

(2.3)  det E = 3H(G) + π^{∗} det V − π^{∗}L ≡ 3H(G) + (v − l)F.

Thus, by (2.2),

K_{X} + det E ≡ (2q − 2 + x + v − l)F.

On the other hand, since K_{X} + det E = π^{∗}H, it follows from (0.1) that

K_{X} + det E ≡ (2q − 2 + ρ)F.

Hence

(2.4)  ρ = x + v − l.

From the above exact sequence, we get

c_{2}(E) + (π^{∗}L) det E = c_{2}((π^{∗}V) ⊗ H(G)) = 3H(G)^{2} + 2(π^{∗} det V)H(G).

Thus, by (2.3), we see that

c_{2}(E) ≡ 3H(G)^{2} + 2vH(G)F − lF (3H(G) + (v − l)F) = 3H(G)^{2} + (2v − 3l)H(G)F.

Using this and the above exact sequence again, we obtain

3l = c_{2}(E)π^{∗}L = c_{3}((π^{∗}V) ⊗ H(G)) = H(G)^{3} + (π^{∗} det V)H(G)^{2} = x + v.

Therefore by (2.4), we have ρ = x + v − l = 3l − l = 2l, so that ρ is even. We know from Lemma 3 that ρ is either 1 or 2 under the assumption (∗). Thus ρ = 2 and l = 1. Let us compute c_{1}(E)^{3} and c_{1}(E)c_{2}(E). First, by (2.3) and (2.4),

c_{1}(E)^{3} = (3H(G) + (v − l)F)^{3} = 27H(G)^{3} + 27(v − l)H(G)^{2}F = 27x + 27(v − l)
= 27(x + v − l) = 27ρ = 54.

Next,

c_{1}(E)c_{2}(E) = (3H(G) + (v − l)F)(3H(G)^{2} + (2v − 3l)H(G)F)
= 9H(G)^{3} + (9v − 12l)H(G)^{2}F = 9(x + v − l) − 3l = 9ρ − 3l = 15.

Since D = c_{1}(E)^{3} − 2c_{1}(E)c_{2}(E), we have D = 24. On the other hand, Theorem 1 tells us that D(D − 20) ≥ 168(q − 1) + 42, because ρ = 2. If q ≥ 2, then D(D − 20) ≥ 210, which implies that D ≥ 28, a contradiction. Consequently q = 1. Recalling that d = 3, we conclude with the aid of (0.2) that g = 4. □

Especially for 3-folds, Theorem 5 allows us to improve [MS, Theorem 6] as follows:

Theorem 7. Let E be a very ample vector bundle of rank two on a smooth projective 3-fold X. Assume that g(X, E) = 2. Then (X, E) is one of the following:

(1) X is a ℙ^{2}-bundle over a smooth curve B of genus 2, and E_{F} ≅ O_{ℙ}(1)^{⊕2} for any fiber F of the projection X → B;

(2) X is a ℙ^{2}-bundle over ℙ^{1}, E_{F} ≅ O_{ℙ}(2) ⊕ O_{ℙ}(1) for any fiber F of the projection π : X → ℙ^{1}, and K_{X} + det E is the pullback of a line bundle of degree 1 on ℙ^{1} by π;

(3) there exists a surjective morphism π : X → B onto a smooth curve B of genus ≤ 1 such that a general fiber F of π is a smooth quadric surface ℚ^{2} in ℙ^{3} with E_{F} ≅ O_{ℚ}(1)^{⊕2}, and K_{X} + det E is the pullback of a line bundle of degree 1 on B by π.

References

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[S] A. J. Sommese, On the nonemptiness of the adjoint linear system of a hyperplane section of a threefold, J. Reine Angew. Math. 402 (1989), 211–220.

Hidetoshi Maeda Department of Mathematical Sciences School of Science and Engineering Waseda University 3-4-1 Ohkubo, Shinjuku-ku Tokyo 169-8555 Japan e-mail: hmaeda@mse.waseda.ac.jp

Andrew J. Sommese Department of Mathematics University of Notre Dame Notre Dame, INDIANA 46556 U. S. A. e-mail: sommese@nd.edu