Reducible hyperplane sections, II

M.C. Beltrametti, K. Chandler and A.J. Sommese

August 2, 2001

Abstract

Let X̂ be a smooth connected subvariety of complex projective space ℙ^{n}. The question was raised in [2] of how to characterize X̂ if it admits a reducible hyperplane section L̂. In the case in which L̂ is the union of r ≥ 2 smooth normal crossing divisors, each of sectional genus zero, classification theorems were given for dim X̂ ≥ 5 or dim X = 4 and r = 2.

This paper restricts attention to the case of two divisors on a threefold, whose sum is ample, and which meet transversely in a smooth curve of genus at least 2. A finiteness theorem and some general results are proven, when the two divisors are in a restricted class including ℙ^{1}-bundles over curves of genus less than two and surfaces with nef and big anticanonical bundle. Next, we give results on the case of a projective threefold X̂ with hyperplane section L̂ that is the union of two transverse divisors, each of which is either ℙ^{2}, a Hirzebruch surface 𝔽_{r}, or 𝔽̃_{2}. 2000 Mathematics Subject Classification. Primary 14N30, 14M99; Secondary 14J99. Keywords and phrases. Adjunction theory, smooth complex polarized threefold, very ample line bundle, reducible surface section, normal crossing divisors, Hirzebruch surface, scroll.

Introduction

This paper is a sequel of [2], which initiated the study of a connected submanifold X̂ of complex projective space, which has a reducible hyperplane section L̂. As dim X̂ increases so does the simplicity of the characterization. In [2] a description is given of (X̂, L̂) for which L̂ decomposes as Â_{1} + ⋯ + Â_{r} into r ≥ 2 smooth components with normal crossings under the hypothesis that h^{1}(OÂ_{i}) is equal to the sectional genus of Â_{i} for each i. A complete result for the cases n = 4 and r = 2; and for n ≥ 5 was obtained. Further, in the case of n = 3 and r = 2 the situation in which the curve A_{1} ∩ A_{2} has genus at most 1 was thoroughly analyzed. Here we investigate the more delicate issues presented by the following specialization of the question.

Problem. Let L̂ be a very ample line bundle on a projective threefold X̂. Suppose that L̂ decomposes as a divisor into a sum L̂ = Â+B̂, where Â and B̂ are smooth connected surfaces meeting transversely along a smooth curve h = Â ∩ B̂. Assume that each of Â, B̂ is either ℙ^{2} or 𝔽_{r}. Then describe (X̂, L̂).

The curve h is connected [2, Corollary (2.3)]. We also call h the hinge curve.

In this paper we shall deal only with the situation when h has genus g(h) ≥ 2: we refer to [2, Theorems (3.10), (3.11)] for the cases when g(h) ≤ 1. We also refer to [2, 5, 7] for related results.

The organization of the paper is as follows. In §2, we present a general finiteness theorem for a threefold X̂ with an ample divisor L̂ of the form Â + B̂, where Â, B̂ are in a restricted class C of surfaces and meet transversely in a smooth curve of genus ≥ 2. The class C includes surfaces with nef and big anticanonical bundle; and ℙ^{1}-bundles over either ℙ^{1} or and elliptic curve. The finiteness theorem asserts that there is an ϵ > 0 such that the Kodaira dimension of K_{X} + (\frac{1}{2} + ϵ) L̂ is −∞. By a result of Fujita,

this implies that (X̂, L̂) is a birational transform of members of an explicit list of very special pairs.

In §3, it was shown that if the divisors Â, B̂ are ℙ^{2} or scrolls over ℙ^{1}, then the restriction of the bundle K_{X} + L̂ to the divisors in big.

In §4, the Hodge Index type theorem for reducible divisors leads to the elimination the cases in which both Â and B̂ are among ℙ^{2} and the singular quadric with an isolated singularity 𝔽̃_{2}.

Finally, in §5 we study the case when Â is ℙ^{2}, the Hirzebruch surface 𝔽_{r}, or the singular quadric with isolated singularity 𝔽̃_{2}; and B̂ = 𝔽_{s}, under the extra assumption that (Â, L̂_{A}), (B̂, L̂_{B}) are scrolls.

The first author would like to thank the University of Notre Dame for its support. The third author would like to thank the Duncan Chair of the University of Notre Dame for its support.

Contents

1 Background Material 2

2 A Finiteness Theorem 3

3 Some Birationality Results 7

4 The Cone Cases 7

5 The Cone and Scroll Cases 8

1 Background Material

W e work over the complex field ℂ. Throughout the paper we deal with projective varieties V, and follow the usual notation of algebraic geometry. The book [1] is a good reference for standard results and notation of adjunction theory.

For a line bundle L on an irreducible normal variety V of dimension n the sectional genus, g(L) = g(V, L), of (V, L) is defined by 2g(L) − 2 = (K_{V} + (n − 1)L) · L^{n−1}.

By 𝔽_{r} with r ≥ 0 we denote the unique ℙ^{1}-bundle over ℙ^{1} with a section E taking on the minimal self intersection E^{2} = −r on the surface. By 𝔽̃_{2} we denote 𝔽_{2} with the section, which has self intersection −2, blown down. Note that 𝔽̃_{2} is isomorphic to any quadric hypersurface Q ⊂ ℙ^{3}, which has a single isolated singularity.

Let V be a normal r-Gorenstein (i.e., rK_{V} is a Cartier divisor) projective variety of dimension n and let D be a ℚ-Cartier divisor on V such that κ(D) = n. We define the unnormalized spectral value of the pair (V, D) as

u(V, D) := sup{t ∈ ℚ | κ(K_{V} + tD) = −∞}.

We refer to [1, §7.1] for details.

The following result follows immediately from [2, §2].

Lemma 1.1 Let L̂ be an ample line bundle on a smooth projective 3-fold X̂. Assume that there are two smooth connected divisors Â, B̂ on X̂. Assume that Â + B̂ ∈ |L̂|, that Â and B̂ are rational, and that Â, B̂ intersect transversely in a smooth curve h. Then h is connected, and h^{1}(O_{X}) = h^{2}(O_{X}) = 0.

2 A Finiteness Theorem

I n this section we prove a general finiteness theorem for pairs (X̂, L̂) consisting of an ample line bundle on a smooth projective threefold X̂, with |L̂| containing a divisor D = Â+B̂, having two irreducible components from a large class C of negative Kodaira dimension surfaces. The class C consists of the normal connected Gorenstein projective surfaces S with the property, that given any smooth connected Cartier divisor C on S, it follows that either h^{1}(O_{C}) ≤ 1 or K_{S} · C ≤ −1.

Lemma 2.1 The class C includes:

1. normal Gorenstein surfaces with −K_{S} nef and big; or

2. 𝔽_{r}, r ≥ 0, the r-th Hirzebruch surface; or

3. a ℙ^{1} bundle over an elliptic curve.

In cases 1) and 2), smooth connected Cartier divisors C with h^{1}(O_{C}) ≥ 2 satisfy K_{S} · C ≤ −3.

Proof. Let C be a smooth connected Cartier divisor of S, i.e., let C be a curve on S with C contained in S_{reg} , the smooth points of S. We assume that we are in the situation that h^{1}(O_{C}) ≥ 2, since otherwise there is nothing to show.

First assume that −K_{S} is nef and big, and that the result is false, i.e., that −K_{S}·C ≤ 2. We know that −K_{S} · C = 0, 1, 2. If −K_{S} · C = 0, then we conclude, using the Hodge Index Theorem, that C^{2} ≤ 0, which contradicts h^{1}(O_{C}) ≥ 2. If −K_{S} · C = 1, then we conclude that C^{2} ≥ 3, which contradicts the Hodge Index Theorem, i.e., C^{2} ≤ C^{2}K_{S}^{2} ≤ 1. If −K_{S} · C = 2, then we conclude that C^{2} ≥ 4, which gives equality in the Hodge Index Theorem, i.e., 4 ≤ C^{2} ≤ C^{2}K_{S}^{2} ≤ 4. This implies that numerically C ∼ −K_{S}, which implies the contradiction K_{S} + C ∼ 0.

For S a Hirzebruch surface the result is a straightforward check.

Assume finally that S is a ℙ^{1}-bundle over an elliptic curve Y. In this case the section σ of minimal self-intersection satisfies e := −σ^{2} ≥ −1, and K_{S} is numerically equal to −2σ − ef for a fiber of the induced projection π : S → Y. Since we are assuming that h^{1}(O_{C}) ≥ 2, we know that numerically C = kσ + tf where k ≥ 2. Moreover K_{S} · C ≥ 0 gives ke − 2t = 2ke − ek − 2t ≥ 0. Since C^{2} = −ek^{2} + 2kt, we have the absurdity that 2 ≤ 2g(C) − 2 = K_{S} · C + C^{2} = (1 − k)(ke − 2t) ≤ 0. Q.E.D.

One main result of the paper is the Finiteness Theorem (2.2). This theorem shows that, if the hinge curve h has genus g(h) ≥ 2, the pair (X̂, L̂) belongs to an explicit list of very special cases described by Fujita (see [3, 4] and also [1, (7.8.1)]).

Note in the following that the hypothesis that h is connected is automatically satisfied if Â and B̂ are connected [2, Corollary (2.3)].

Theorem 2.2 (Finiteness theorem) Let L̂ be an ample line bundle on a smooth projective 3-fold X̂. Assume that there are two divisors Â, B̂ on X̂ from the class C. Assume that Â + B̂ ∈ |L̂| and that Â, B̂ intersect transversely in a smooth connected curve h of genus g(h) ≥ 2. Then u(X̂, L̂) > \frac{1}{2}. In particular, X̂ is of Kodaira dimension −∞, and thus satisfies h^{3}(O_{X}) = 0.

Proof. For simplicity of notation, we omit ’s in this proof. The genus formula yields

(K_{X} + L) · h = (K_{X} + A + B) · A · B = 2g(h) − 2,  (1)

or (K_{A} +B_{A})·B_{A} = 2g(h)−2, and therefore, by definition of class C, one has B_{A} ·B_{A} ≥ 2g(h) − 1, and similarly A_{B} · A_{B} ≥ 2g(h) − 1. Then (1) gives

K_{X} · h ≤ −2g(h).  (2)

Now compute, for any real number ε, 0 < ε < \frac{1}{4g(h) − 2},

(K_{X} + (\frac{1}{2} + ε) L) · h = (K_{X} + L − (\frac{1}{2} − ε) L) · h
= 2g(h) − 2 − (\frac{1}{2} − ε) · L · h
≤ 2g(h) − 2 − (\frac{1}{2} − ε) (4g(h) − 2)
= −1 + ε(4g(h) − 2) < 0.

Finally, for h = A ∩ B on X, we have the normal bundle decomposition N_{h/X} = N_{h/A} ⊕ N_{h/B} and deg(N_{h/A}) = B^{2} · A = B_{A} · B_{A} ≥ 2g(h) − 1 by the above. It follows that h^{1}(N_{h/A}) = 0 and N_{h/A} has not identically zero sections. Similarly for N_{h/B}. Then N_{h/X} is generically spanned by its global sections and h^{1}(N_{h/X}) = 0. Thus general deformation theory implies that the union of the deformations of h on X contains an open set. Therefore the inequality (K_{X} + (\frac{1}{2} + ε)L) · h < 0 proved above, shows that

u(X, L) > \frac{1}{2}, cf., [1, (7.6.4)]. Q.E.D.

A little more can be said on the case of ℙ^{1}-bundles over ℙ^{1} or surfaces with nef and big anticanonical bundle.

Proposition 2.3 Let L̂ be an ample line bundle on a smooth projective 3-fold X̂. Assume that there are two smooth divisors Â, B̂ on X̂ which are either ℙ^{1}-bundles over ℙ^{1} or surfaces with nef and big anticanonical bundle. Assume that Â + B̂ ∈ |L̂| and that Â, B̂ intersect transversely in a smooth connected curve h. Then H^{0}(K_{X} + L̂) → H^{0}(K_{h}) → 0.

Proof. Tensor the Koszul complex

0 → O_{X} → Â ⊕ B̂ → L̂ → L̂_{h} → 0

with K_{X}. Using the hypercohomology spectral sequence, we see that the desired result will follow if we show that H^{2}(K_{X}) = H^{1}(K_{X} + Â) = H^{1}(K_{X} + B̂) = 0.

The assertion H^{2}(K_{X}) = 0 follows from Lemma (1.1). To see that H^{1}(K_{X} +Â) = 0 consider the exact sequence

0 → K_{X} → K_{X} + Â → K_{A} → 0.

Now use Lemma (1.1) and the fact that Â is rational. The argument for H^{1}(K_{X} +B̂) = 0 is identical. Q.E.D.

One consequence of Proposition (2.3) is that under the same hypotheses with the added assumption that g(h) ≥ 2, it follows that the Kodaira dimension of K_{X} + L̂ is at least one. This implies that the Kodaira dimension of K_{X} + 2L̂ is three, and also that the restriction of K_{X} + 2L̂ to Â (or B̂) is nontrivial. Therefore [2, Theorems (3.6), (3.8)] specialize to the following result.

Theorem 2.4 Let L̂ be an ample line bundle on a smooth projective 3-fold X̂. Assume that there are two smooth divisors Â, B̂ on X̂ which are either ℙ^{1}-bundles over ℙ^{1} or surfaces with nef and big anticanonical bundle. Assume that Â + B̂ ∈ |L̂| and that Â, B̂ intersect transversely in a smooth connected curve h of genus ≥ 2. Then there is a surjective morphism ϕ : X̂ → X, where X is a smooth projective 3-fold, such that:

1. ϕ expresses X̂ as the blowup of X at a finite set F, and there is an ample line bundle L on X such that L̂ ≅ ϕ^{∗}L − ϕ^{−1}(F);

2. K_{X} + 2L̂ ≅ ϕ^{∗}(K_{X} + 2L) where K_{X} + 2L is ample;

3. K_{X} + L is either nef and big, or (X, L) is a conic fibration over a surface Y in the sense of adjunction theory [1], i.e., there exists a morphism ν : X → Y with K_{X} + L ≅ ν^{∗}H for an ample line bundle H on a normal surface Y;

4. ϕ is an embedding in a neighborhood of h; and

5. L = A + B where A := ϕ(Â) and B := ϕ(B̂) are Cartier divisors meeting transversely in ϕ(h) and each having at most one point contained in the set F.

From now on we usually abuse notation, and let h to denote ϕ(h). We also write h_{A} (respectively h_{B}) to emphasize that we view h as a curve on A (respectively on B).

Lemma 2.5 Let (X̂, L̂), (X, L), Â, B̂, A, B be as in (2.4). Then 1. h^{i,0}(X) = 0, i = 1, 2, 3;

2. h^{i}(K_{X} + A) = h^{i}(K_{X} + B) = 0 for all i ≥ 0; and

3. the restriction map gives the following isomorphisms

H^{0}(K_{X} + L) ≅ H^{0}(K_{A} + h_{A}) ≅ H^{0}(K_{B} + h_{B}) ≅ H^{0}(K_{h}).

Proof. Noting that the first reduction morphism, ϕ, of Lemma (2.4) is birational, the first assertion follows immediately from Lemma (1.1) and Theorem (2.2).

To prove 2), consider the exact sequence

0 → K_{X} + B → K_{X} + L → K_{A} → 0.

By the assumption on A, h^{0}(K_{A}) = h^{1}(K_{A}) = 0, h^{2}(K_{A}) = 1, h^{3}(K_{A}) = 0. Thus from the cohomology sequence associated to the sequence above we infer that h^{i}(K_{X} +B) = 0 (and by symmetry h^{i}(K_{X} + A) = 0) for all i ≥ 0.

Item 3) follows immediately from the first two assertions. Q.E.D.

Theorem 2.6 Let L̂ be an ample line bundle on a smooth projective 3-fold X̂. Assume that there are two smooth divisors Â, B̂ on X̂ which are either ℙ^{1}-bundles over ℙ^{1} or surfaces with nef and big anticanonical bundle. Assume that Â+ B̂ ∈ |L̂| and that Â, B̂ intersect transversely in a smooth connected curve h of genus g(h) ≥ 2. Let X, A, B, L be as in (2.4). Then H^{0}(K_{X} + L) spans K_{X} + L in a neighborhood of A + B.

Proof. By Lemma (2.5), the desired spannedness of K_{X} + L will follow from the spannedness of K_{A} + h_{A} and K_{B} + h_{B}. From Theorem (2.4) we know that K_{X} + L is nef (and hence K_{A} + h_{A} and K_{B} + h_{B} are also).

First assume that Â is a ℙ^{1}-bundle over ℙ^{1}. Either the map ϕ of Theorem (2.4) is an isomorphism on Â, in which case A is also a ℙ^{1}-bundle, or, by [2, Theorem (3.6), 2)], ϕ_{A} expresses Â as the blowup of A at one point. In this latter case, Â is the Hirzebruch surface 𝔽_{1}, and A := ϕ(Â) = ℙ^{2} (note that 𝔽_{1} is the only Hirzebruch surface with a −1 curve). Since K_{X} + L is nef, K_{A} + h_{A} is nef, and for either ℙ^{2} or ℙ^{1}-bundles over ℙ^{1}, nef line bundles are spanned.

Now assume that −K_{A} is nef and big. Note that −K_{A} is also nef and big. Indeed, going to the first reduction map we have a birational morphism ϕ_{A} : Â → A where some disjoint −1 curves are collapsed. Writing −K_{A} = K_{A} + 2(−K_{A}), we see from the basepoint free theorem that −NK_{A} is spanned for N ≫ 0. Thus −NK_{A} is spanned off the finite set equal to the image of the exceptional curves. This implies −K_{A} is nef. Since K_{A}^{2} > K2 ; _{A}, bigness is clear.

Consider the line bundle h_{A}. We would like to show by Reider’s Theorem [6] that K_{A} + h_{A} is spanned. Note that h_{A}^{2} = 2g(h_{A}) − 2 − K_{A} · h_{A} ≥ 2 + 3 = 5 by the hypothesis g(h_{A}) ≥ 2 and Lemma (2.1). Since h_{A} is a smooth curve of positive genus, and K_{A} · h_{A} < 0, we conclude that h_{A} is nef and big. Therefore by Reider’s Theorem, either K_{A} + h_{A} is spanned, or there exists an effective Cartier divisor ℓ ⊂ A such that either h_{A} · ℓ = 0 with ℓ^{2} = −1, or h_{A} · ℓ = 1 with ℓ^{2} = 0.

In the former case, K_{A} · ℓ < 0, since K_{A} · ℓ ≤ 0 and K_{A} · ℓ + ℓ^{2} is even. This contradicts the nefness of K_{A} + h_{A}.

Finally consider the case h_{A} · ℓ = 1 with ℓ^{2} = 0. Note that since ℓ is effective, we cannot have −K_{A} · ℓ = 0 by the usual Hodge index relation. Thus we have K_{A} · ℓ < 0. Since K_{A} ·ℓ+ℓ^{2} is even, we have that K_{A} ·ℓ ≤ −2. This implies that (K_{A} +h_{A})·ℓ ≤ −1, which contradicts nefness of K_{A} + h_{A}. Q.E.D.

3 Some Birationality Results

3.1 (Working assumptions) Let L̂ be a very ample line bundle on a 3-fold X̂. Assume that there are two smooth transverse divisors Â, B̂ on X̂ with Â + B̂ ∈ |L| and Â, B̂ ∈ {ℙ^{2}, 𝔽_{r}}. Assume that the hinge curve h = Â ∩ B̂ has genus g(h) ≥ 2.

From Theorem (2.4), we know that there exists the first reduction (X, L), ϕ : X̂ → X, with K_{X} + 2L ample and K_{X} + L nef. If A = ϕ(Â), B = ϕ(B̂), then A + B ∈ |L| and A, B ∈ {ℙ^{2}, 𝔽_{r}}. Furthermore we know by 5) of Theorem (2.4), that neither Â nor B̂ is a fiber of ϕ and that A, B meet transversely along the curve ϕ(h) isomorphic to h.

Lemma 3.2 Assumptions and notation as in (3.1). The complete linear systems |K_{A}+ h_{A}| and |K_{B} +h_{B}| maps h generically one-to-one. In particular, |K_{A} +h_{A}|, |K_{B} +h_{B}|, and K_{X} + L are nef and big.

Proof. By Lemma (2.5) we see that showing complete linear systems |K_{A} + h_{A}| and |K_{B} + h_{B}| maps h generically one-to-one, shows that |K_{X} + L| maps h generically one-to-one.

We know from 3) of Theorem (2.4) that the restriction of K_{X} + L of one of the divisors A, B is nef and big (by ampleness of A+B either A or B surjects on the base). Assume for simplicity, that K_{B} + h_{B} ≈ (K + L)_{B} is nef and big. If B = ℙ^{2} or 𝔽_{0}, the line bundle K_{B} + h_{B} is ample, and indeed very ample.

Thus we may assume B = 𝔽_{r}, r ≥ 1. Let E := E + rf. Then either K_{B} + h_{B} = aE + bf is very ample or b = 0 and K_{B} + h_{B} = aE. Thus |K_{B} + h_{B}| maps h generically one-to-one.

Now looking at conclusion 3) of Theorem (2.4), we see that if K + L fails to be nef and big, then using the notation of Theorem (2.4), that ν maps h two-to-one onto a curve ν(h) with all restrictions of elements of H^{0}(K_{X} + L) to h the pullbacks of restrictions of sections of H_{ν(h)}. This is a contradiction to the assertion that |K_{X} + L| maps h generically one-to-one onto its image.

Assume that (K_{X} + L)_{A} ≈ K_{A} + h_{A} is not nef and big. Since the genus of the curve h_{A} is not 0, the map given by |K_{A} + h_{A}| cannot be generically one-to-one on h_{A}.

Q.E.D.

The following is a corollary of the preceding lemma.

Lemma 3.3 Assumptions and notation as in (3.1). Assume A = 𝔽_{r} and let h = aE + bf on A. Then a ≥ 3.

Proof. Note that a = h · f ≥ 0, and a ≠ 1 since g(h) > 0. Assume a = 2. Then (K_{A} + h_{A}) · f = −2 + 2 = 0 and hence |(K_{X} + L)_{A}| = |(K_{A} + h_{A})| collapses A along the ruling f. This contradicts Lemma (3.2). Q.E.D.

4 The Cone Cases

The main result in this section is to showing that in the situation when we have a reducible ample divisor L = A + B with both of A and B are in {ℙ^{2}, 𝔽̃_{2}} is very restricted. The proof of this is based on the usual Hodge Index type theorem for ample divisors, which yields in our case

[(A + B) · A · A][(A + B) · B · B] ≤ [(A + B) · A · B]^{2}  (3)

with equality if and only if A is a rational multiple of B as homology class.

Lemma 4.1 Let L be an ample line bundle on a smooth connected projective threefold X. Assume that A, B are two reduced divisors on X which meet transversely in a smooth curve h of genus g(h). Assume that A + B ∈ |L|, and that A, B ∈ {ℙ^{2}, 𝔽̃_{2}}. Then g(h) ≤ 1.

Proof. Assume without loss of generality that g := g(h) ≥ 2. Note that in this case the degree of h on A, B are uniquely determined by g.

First let us do the case of A = B = ℙ^{2}. Then h_{A} ∈ |O_{ℙ^{2}}(d)| and h_{B} ∈ |O_{ℙ^{2}}(d)| where 2g−2 = d(d−3). Note that d^{2} = h_{A}^{2} = B·B·A = h_{B} ·N_{B/X}. Thus N_{B/X} = O_{ℙ^{2}}(d), and similarly N_{A/X} = O_{ℙ^{2}}(d). Plugging into Equation (3), we get equality. Thus A = λB as homology classes for some λ ∈ ℚ. Since A^{2}·B = d^{2} = B^{2}·A, we see that λ = 1. Thus since L is ample and since L = 2A = 2B in homology, it follows that A, B are ample. The Lefschetz theorem yields Pic(X) = Pic(A) = ℤ[O_{ℙ^{2}}(1)]. Therefore K_{X} ≈ O_{X}(c), O_{X}(A) ≈ O_{X}(a), where a ≥ 1 by ampleness. Then (K_{X} + A)_{A} ≈ K_{A} ≈ O_{ℙ^{2}}(−3) gives K_{X} + A ≈ O_{X}(c + a) ≈ O_{X}(−3). Therefore 1 + c ≤ a + c = −3, or c ≤ −4. Thus X = ℙ^{3} by the Kobayashi-Ochiai Theorem [1, (3.1.6)]) and g = 0, contradicting the assumption g > 0.

The case of A = B = 𝔽̃_{2} proceeds in the same way, except that one of the possibilities allowed by the Kobayashi-Ochiai Theorem [1, (3.1.6)]) is (X, L) is (ℙ^{3}, O_{ℙ^{3}}(4)). In this case g = 1.

Finally, consider the case when one of A, B is ℙ^{2} and the other is 𝔽̃_{2}. By renaming if necessary we can assume that A = ℙ^{2} and B = 𝔽̃_{2}. Letting h_{B} = A_{B} = O_{B}(δ) and h_{A} = B_{A} = O_{A}(d), we have that A^{2} · B = 2δ^{2}, B^{2} · A = d^{2}. Also from d^{2} = h_{A}^{2} = B · B · A = h_{B} · N_{B/X} we conclude that N_{B/X} = O_{B}(d^{2}/2δ). Similarly we conclude

that N_{A/X} = O_{A}(2δ^{2}/d). Thus A^{3} = \frac{4δ^{4}}{d^{2}} and B^{3} = \frac{d^{4}}{4δ^{2}} . Plugging into Equation (3), we conclude again that A, B are positive multiples of L in homology and hence ample. Using the argument from the case when both are ℙ^{2}, we see that X = ℙ^{3}. In this case g = 0. Q.E.D.

5 The Cone and Scroll Cases

We keep again our working assumption as in (3.1). In this section we consider the remaining case when both A and B are Hirzebruch surfaces, under the extra assumption that (Â, L̂_{A}), (B̂, L̂_{B}) are scrolls, i.e., Â, B̂ are both scrolls with respect to L̂.

We start with the following general lemma.

Lemma 5.1 Let L̂ be a very ample line bundle on a 3-fold X̂. Let Â + B̂ ∈ |L|, where Â, B̂ are two smooth divisors on X̂ meeting transversely in a smooth curve h of genus g(h) > 0. Assume that (Â, L̂_{A}) and (B̂, L̂_{B}) are scrolls or cones (from a vertex not contained in h) over smooth curves C_{A}, C_{B} with scroll (or cone) projections p_{A} : Â → C_{A}, p_{B} : B̂ → C_{B} respectively. Then π = (p_{A}, p_{B}) : h → C_{A} × C_{B} maps h isomorphically onto a smooth curve.

Proof. Let ξ ⊂ h be a subscheme of degree 2 (i.e., a pair of distinct points, or a tangent subscheme supported at a single point). We show that π separates ξ. If not, ξ is contained in a fiber of π. Hence ξ belongs to a fiber of p_{A} and one of p_{B}, in which case the same holds true for the line ℓ spanned by ξ. But since the intersection Â ∩ B̂ is transverse and connected, it follows that Â ∩ B̂ = ℓ. This contradicts the hypothesis that g(h) > 0. Q.E.D.

Theorem 5.2 Let L̂ be a very ample line bundle on a smooth projective threefold X̂. Assume that there exists two irreducible divisors Â, B̂ on X̂, which meet transversely in a smooth curve h, and such that Â + B̂ ∈ |L̂|. Assume further that

1. (Â, L̂_{A}) is (ℙ^{2}, O_{ℙ^{2}}(1)), or (Q, O_{ℙ^{3}}(1)_{Q}) with Q ⊂ ℙ^{3} the singular quadric 𝔽̃_{2}; and

2. (B̂, L̂_{B}) is a scroll over ℙ^{1}.

Then g(h) = 0.

Proof. We do the case where Â = ℙ^{2}. The case of Â = 𝔽̃_{2} is proved in a completely similar way. Let N_{_{A}/_{X}} = Â_{A} ≅ O_{ℙ^{2}}(d) denote the normal bundle of Â in X̂. Let

h_{A} = B̂_{A} = O_{ℙ^{2}}(δ). Letting E denote a section of B̂ with E^{2} = −r ≤ 0, and letting E = E + rf for a fiber f of the scroll projection, we have B̂_{B} = ME + Nf and h_{B} = Â_{B} = aE + bf for integers M, N, a, b. Note that by Lemma (5.1) we have g := g(h) = (a − 1)(δ − 1). Note by the formulae for the genus on Â and B̂ we have the formulae 2g = (δ − 1)(δ − 2) and 2g = (a − 1)(ar + 2b − 2). Assuming that g ≥ 1, and hence that δ ≥ 3, a ≥ 2, immediately gives δ = 2a and 4a = ar + 2b.

Note that dδ = Â_{A} · B̂_{A} = B̂ · Â^{2} = Â2 ; _{B} = a(ar + 2b). Combined with δ = 2a and 4a = ar + 2b, we conclude that d = δ. Since L̂_{A} = O_{ℙ^{2}}(d + δ), we get a contradiction

to L̂_{A} ≅ O_{ℙ^{2}}(1). Q.E.D.

Now let us specialize Lemma (5.1) to the case when Â = 𝔽_{r}, B̂ = 𝔽_{s}. Denote by E_{A} = E_{A} + rf_{A}, E2 ; _{A} = −r, f_{A} a fiber of the ruling Â = 𝔽_{r} → ℙ^{1}; and similarly for B̂. Write

h_{A} = a(E_{A} + rf_{A}) + bf_{A}; h_{B} = α(E_{B} + sf_{B}) + βf_{B},  (4)

on Â, B̂, respectively,

Note that by Lemma (3.3) we may assume a ≥ 3, α ≥ 3. Furthermore, since h is a positive genus curve on the Hirzebruch surfaces Â, B̂, we may also assume b ≥ 0, β ≥ 0.

By Lemma (5.1), O_{ℙ^{1}×ℙ^{1}}(π(h)) = O_{ℙ^{1}×ℙ^{1}}(a, α), and hence

g(h) = (a − 1)(α − 1).  (5)

The genus formula also yields

2g(h) − 2 = (a − 1)(ar + 2b − 2).

Therefore, by using (5), we deduce that

2α = ar + 2b.  (6)

Similarly we find

2a = αs + 2β.  (7)

Combining (6) and (7), we have (4 − rs)aα = 4bβ + 2(bαs + βar).

2aα = a(ar + 2b) = α(αs + 2β).  (8)

Write

N_{_{A}/_{X}} = −λ(E_{A} + rf_{A}) + ρf_{A}; N_{_{B}/_{X}} = −μ(E_{B} + sf_{B}) + σf_{B},

for integers λ, μ, ρ, σ.

Note that on Â one has h2 ; _{B} = Â2 ; _{B} = Â^{2} · B̂ = Â_{A} · B̂_{A} = N_{_{A}/_{X}} · h_{A}. Since

h2 ; _{B} = α^{2}s + 2αβ = α(αs + 2β) and N_{_{A}/_{X}} · h_{A} = −aλr − λb + ρa,

we find α(αs + 2β) = −aλr − λb + ρa. Similarly, we find a(ar + 2b) = −αμs − μβ + σα. Then by (8) we have

−aλr − λb + ρa = −αμs − μβ + σα.  (9)

Since (Â, L̂_{A}) is a scroll, we also have L̂_{A}(= Â_{A} + B̂_{A} = N_{_{A}/_{X}} + h_{A}) = E_{A} + jf_{A}. On the other hand, the coefficient of E_{A} in the expression for N_{_{A}/_{X}} + h_{A} is −λ + a.

Therefore the last equality for L̂_{A} implies a − λ = 1. Similarly the scroll condition for (B̂, L̂_{B}) gives α − μ = 1. So from (9) we have

−a(a − 1)r − (a − 1)b + ρa = −α(α − 1)s − (α − 1)β + σα.

Then in particular ρ = 2α + (a − 1)r + b + \frac{b}{a}. This implies ρ ≥ 6 (with ρ = 6 giving r = b = 0), as well as a divides b, say, b = ab′.

In the same way, we find σ = 2a + (a − 1)s + β + \frac{β}{α}. So σ ≥ 6 (with σ = 6 giving s = α = 0), as well as β = αβ′.

Thus formulas (6) and (7) become 2α = a(r + 2b′) and 2a = α(s + 2β′). From this we also find

4 = (r + 2b′)(s + 2β′).  (10)

Since b′, β′ ≥ 0 it follows that rs ≤ 4 and hence r, s ∈ {0, 1, 2, 3, 4}.

The following theorem summarizes the discussion above.

Theorem 5.3 Let L̂ be a very ample line bundle on a 3-fold X̂. Let Â + B̂ ∈ |L|, where Â, B̂ are two smooth divisors on X̂ meeting transversely in a smooth curve h of genus g(h) > 0. Assume that Â = 𝔽_{r}, B̂ = 𝔽_{s} are Hirzebruch surfaces. Further assume that (Â, L̂_{A}) and (B̂, L̂_{B}) are scrolls over smooth curves. Then r, s ∈ {0, 1, 2, 4} and the possible values of the coefficients b = ab′, β = αβ′ as in the expressions (4) of h as a curve of Â, B̂ respectively are listed in the table below.

\begin{matrix}s⧹ r & 0 & 1 & 2 & 3 & 4 \\ 0 & b′ = β′ = 1 & b′ = 0, β′ = 2 & b′ = 0, β′ = 1 & √ & √ \\ 1 & b′ = 2, β′ = 0 & √ & b′ = 1, β′ = 0 & √ & b′ = β′ = 0 \\ 2 & b′ = 1, β′ = 0 & b′ = 0, β′ = 1 & b′ = β′ = 0 & √ & √ \\ 3 & √ & √ & √ & √ & √ \\ 4 & √ & b′ = β′ = 0 & √ & √ & √ \end{matrix}

Proof. A simply numerical check, by using (10) and the symmetry between r and s, gives the possible values for the integers r, s, b′, β′ in the table (the symbol “√” means that the corresponding case does not occur). E.g., if r = 0, equality (10) gives 2 = b′(s + 2β′). This leads to the cases (s, b′, β′) = (0, 1, 1), (1, 2, 0), (2, 1, 0) as in the first column. Thus we may assume r, s ≥ 0. E.g., if r = 3, (10) gives 4 = 3(s + 2β′) + 2b′(s + 2β′), so that b′ ≠ 0 and hence b′ > 0, this giving again a numerical contradiction. Q.E.D.

References

[1] M.C. Beltrametti and A.J. Sommese, The Adjunction Theory of Complex Projective Varieties, Expositions in Mathematics, 16, W. de Gruyter, 1995.

[2] K. Chandler, A. Howard and A.J. Sommese, Reducible hyperplane sections I, J. Math. Soc. Japan 51, 4 (1999), 887–910.

[3] T. Fujita, On Kodaira energy and adjoint reduction of polarized manifolds, Manuscripta Math. 76 (1992), 59–84.

[4] T. Fujita, Notes on Kodaira energies of polarized varieties, preprint, 1992.

[5] A. Lanteri, A.L. Tironi, On reducible hyperplane sections of 4-folds, J. Math. Soc. Japan, to appear.

[6] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 127 (1988), 309–316.

[7] A.L. Tironi, Reducible hyperplane sections of 4-folds: semi-nef decompositions with low sectional genera, Istituto Lombardo (Rend. Sc.), A 134 (2001), to appear.

Mauro C. Beltrametti Dipartimento di Matematica Via Dodecaneso 35 I-16146 Genova, Italy beltrame@dima.unige.it

Karen Chandler Department of Mathematics Notre Dame, Indiana, 46556, U.S.A. kchandle@noether.math.nd.edu

Andrew J. Sommese Department of Mathematics Notre Dame, Indiana, 46556, U.S.A. sommese@nd.edu http://www.nd.edu/∼sommese