REDUCIBLE HYPERPLANE SECTIONS I

KAREN A. CHANDLER, ALAN HOWARD, ANDREW J. SOMMESE

Dedicated to the memory of our friend and colleague, Michael Schneider

Abstract: In this article we begin the study of X̂, an n-dimensional algebraic submanifold of complex projective space ℙ^{N}, in terms of a hyperplane section Â which is not irreducible. A number of general results are given, including a Lefschetz theorem relating the cohomology of X̂ to the cohomology of the components of a normal crossing divisor which is ample, and a strong extension theorem for divisors which are high index Fano fibrations. As a consequence we describe X̂ ⊂ ℙ^{N} of dimension at least five if the intersection of X̂ with some hyperplane is a union of r ≥ 2 smooth normal crossing divisors Â_{1}, …, Â_{r}, such that for each i, h^{1}(O_{A_{i}}) equals the genus g(Â_{i}) of a curve section of Â_{i}. Complete results are also given for the case of dimension four when r = 2. 1991 Mathematics Subject Classification: 14J40, 14J35, 14C20.

Introduction. Let X̂ be an n-dimensional algebraic submanifold of complex projective space ℙ^{N}. There are many results [5, 9] describing the structure of X̂ under assumptions on one of its hyperplane sections. For example, there are classifications under conditions on some basic projective invariant such as the degree or genus of a curve section, or some birational invariant such as not being of general type. Though, in these results smoothness of X̂ or of the hyperplane section of X̂ is often relaxed slightly, they all assume that the hyperplane section is irreducible. In this article we begin the study of X̂ in terms of a hyperplane section Â which is not irreducible. We assume that Â is a union of distinct components Â_{1}, …, Â_{r} where r ≥ 2. Since normal bundles of the Â_{i} do not have to be ample (some of the Â_{i} can have ample conormal bundles), the known theory does not apply.

Since any line bundle becomes very ample after being twisted by a high enough power of a very ample line bundle, it is clear that given any divisor Â_{1} on a connected projective manifold X, there is a smooth divisor Â_{2} on X̂ with good transversality properties with respect to Â_{1} and with Â_{1} + Â_{2} very ample. This process produces a divisor Â_{2}, whose invariants are generally quite large. Looking at examples suggests strongly that if Â is a very ample divisor on a projective manifold X̂ of dimension at least two, and if Â decomposes into irreducible components

Â = ∑_{i=1}^{r} Â_{i},

and if all of the Â_{i} are “small” measured with respect to some invariant, then (X̂, L̂) should be “special”.

In this paper we investigate the simplest problem of this type.

Problem A. Describe X̂ ⊂ ℙ^{N} if the intersection of X̂ with some hyperplane is a union of r ≥ 2 smooth normal crossing divisors, each of which has a curve section of genus 0.

This puts numerous restraints on X̂, e.g., the first Betti number of X̂ is zero, but it is very difficult to make a complete classification of the X̂ ⊂ ℙ^{N} satisfying this condition when dim X̂ = 2. This paper grew from the realization that such a problem becomes progressively easier as the dimension of X̂ increases. We give a complete answer when r = 2 and dim X̂ = 4 and for arbitrary r when dim X̂ ≥ 5. Indeed for dimensions ≥ 5 we solve the following more general problem.

Problem B. Describe X̂ ⊂ ℙ^{N} if the intersection of X̂ with some hyperplane is a union of r ≥ 2 smooth normal crossing divisors Â_{1}, …, Â_{r}, such that for each i, h^{1}(O_{A_{i}}) equals the genus g(Â_{i}) of a curve section of Â_{i}.

In the course of solving these problems we identify some of the invariants controlling this sort of problem and develop some of the theory surrounding them.

Here is the scheme of the paper. In §1 we summarize background material and give a number of examples and preliminary results that come up in the rest of the paper.

In §2 we prove some general facts. The most striking is the very useful variant of the First Lefschetz Theorem on hyperplane sections that follows from the usual First Lefschetz Theorem combined with mixed Hodge theory.

In §3 we work out the structure of pairs (X̂, L̂) with L̂ ample and Â_{1}+⋯+Â_{r} ∈ |L̂| with r ≥ 2, and K_{X} + (n − 2)L̂ not nef and big.

In §4 we prove an extension theorem which gives strong restrictions on high index Fano fibrations as divisors on manifolds of dimension n ≥ 4. In particular using the results of §3 we solve Problem B, and we solve Problem A when r = 2 and the dimension of X̂ equals 4.

In the three dimensional case we need more stringent hypotheses, e.g., that the hyperplane section has exactly two components. The results in this case are different than those in this paper, and require a very detailed case by case argument based on the special structure of the second reduction and detailed analysis of bundles on ℙ^{1}-bundles over curves. For these reasons we are preparing a sequel dealing only with the results special to three dimensions.

We thank the referee for many helpful comments. The third author would like to thank the Alexander von Humboldt Stiftung for their support.

1. Preliminary material. Certain special varieties arise naturally as the building blocks of adjunction theory. We give definitions in the smooth case. For more on these varieties and on all aspects of adjunction theory we refer to the book of Beltrametti and the third author [5].

Given an integer k ≥ 0, by a Fano fibration of index k, we mean a quadruple, (M, L, p, Y) with L an ample line bundle on a connected projective manifold M, a normal projective variety Y and a surjective morphism with connected fibers p : M → Y with K_{M} + kL ≅ p^{∗}H for some ample line bundle H on Y. The general fiber F of p is a Fano manifold of index k, i.e., −K_{F} ≅ kL_{F}. With appropriate interpretation, we could let k be merely rational, but this does not give much more generality (since writing k as a quotient of relatively prime integers k = \frac{u}{v} with u ≥ 0, v > 0, and by changing the line bundle L appropriately (see [5, Lemma 1.5.6]), such a fibration becomes a u-Fano fibration with the same p and Y).

Some special cases are particularly important. By a scroll (M, L) over a normal variety Y we mean a pair consisting of an ample line bundle L on a connected projective manifold M and a morphism p : M → Y with connected fibers such that K_{M} + (f + 1)L ≅ p^{∗}H where H is an ample line bundle on Y and f := dim M − dim Y. By a quadric fibration (M, L) (respectively a Del Pezzo fibration) over a normal variety Y we mean a pair consisting of an ample line bundle L on a connected projective manifold M and a morphism p : M → Y with connected fibers such that K_{M} + fL ≅ p^{∗}H (respectively K_{M} + (f − 1)L ≅ p^{∗}H) where H is an ample line bundle on Y and f := dim M − dim Y. The third author introduced these and a number of the above classes of special varieties in [16]. They are better behaved than might be expected merely from the definitions, see, [5, Chapter 12], e.g., in the case of a scroll, the general fiber of p is (ℙ^{f}, O_{ℙ^{f}} (1)), and if f ≥ 1 and y := dim Y ≤ 2, then these are ℙ^{n−y}-bundles over Y.

We recall basic results on nefvalue morphisms, and refer the reader to [5, (1.5)] for further discussion. Given an ample line bundle L on a normal irreducible and reduced projective variety V of index e with at worst canonical singularities, we define the nefvalue of the pair (V, L) to be the number

τ := min{t ∈ ℝ|K_{V} + tL is nef}.

The Kawamata rationality theorem [12] and the Kawamata-Shokurov Basepoint-Free theorem [12, 14] assert the following. Theorem 1.1 (Kawamata-Shokurov) Let L be an ample line bundle on a normal irreducible and reduced projective variety V of index e with at worst canonical singularities. Assume that K_{V} is not nef and let τ be the nefvalue of (V, L).

1. τ is rational and there is a morphism with connected fibers ν : V → W onto a normal projective variety W such that given any positive integer such that Nτ and N/e are integral, there is an ample line bundle H on W such that N(K_{V} + τL) = ν^{∗}H.

2. Further, expressing eτ = u/v with u, v coprime positive integers,

u ≤ e(max_{y∈W}{dim ν^{−1}(y)} + 1).

We need the following result due to Beltrametti, Sommese, and Wiśniewski [5, Theorem 6.4.4] (see also [6]). Theorem 1.2 (Beltrametti, Sommese, Wiśniewski) Let (M, L, ν, Y) be a Fano fibration of index k ≥ \frac{n}{2} + 1 where n := dim M. Assume that n > dim Y > 0. Then ν is the contraction of an extremal ray and Pic(M) ≅ Pic(Y) ⊕ ℤL. The important case when dim Y = 0 is due to Wiśniewski [18]. Theorem 1.3 (Wiśniewski) Let (M, L, ν, Y) be a Fano manifold of index k ≥ \frac{n}{2} + 1 where n := dim M. Either Pic(M) ≅ ℤL or k = \frac{n}{2} + 1, M = ℙ^{n/2} × ℙ^{n/2}, and L = O_{ℙ^{n/2}×ℙ^{n/2}}(1, 1).

The following lemma will be useful. Lemma 1.4 Let L be an ample line bundle on a connected projective manifold X. Assume that there is a divisor A + B ∈ |L| with A and B distinct irreducible divisors. Let π : X → Y be a surjective morphism with connected fibers onto a projective variety Y. If dim X − dim Y ≥ 1 then either π(A ∩ B) is of dimension at least dimension dim Y − 1; or π(A ∩ B) = π(A); or π(A ∩ B) = π(B). Proof. If dim Y ≤ 1 there is nothing to prove. Thus we can assume without loss of generality that dim Y ≥ 2.

Assume that dim π(A ∩ B) < dim Y − 1. Let H be a very ample line bundle on Y. If π(A) and π(B) both properly contain π(A∩B), we can find general D_{1}, …, D_{k} ∈ |H| such that, letting Y_{k} := D_{1}∩…∩D_{k}, Y_{k}∩π(A) ≠ ∅, Y_{k}∩π(B) ≠ ∅, and Y_{k}∩π(A∩B) = ∅. This implies that Y_{k} is of dimension at least one, and X_{k} := π^{−1}(Y_{k}) is at least two dimensional. Since the D_{i} are general we can also assume without loss of generality that Y_{k} is irreducible and normal, and X_{k} is smooth and connected.

Let A_{k} := A∩X_{k} and B_{k} := B ∩X_{k}. Since both are nonempty and since A_{k} +B_{k} is an ample divisor on X_{k} with dim X_{k} ≥ 2, we conclude that A_{k} ∩ B_{k} ≠ ∅. This implies that π(A ∩ B) ∩ Y_{k} ≠ ∅. Q.E.D.

Corollary 1.5 Let (M, L, ν, Y) be a Fano fibration of index k ≥ \frac{n}{2} + 1 where n := dim M. Assume that n > dim Y > 0. Let A+B ∈ |L| with A and B distinct irreducible divisors. Then dim ν(A ∩ B) ≥ dim Y − 1. Proof. If dim ν(A ∩ B) < dim Y − 1 it follows from Lemma 1.4 that after possibly renaming, ν(A) = ν(A∩B) and thus that dim ν(A) < dim Y −1. Since ν is a contraction of an extremal ray and since A misses at least one fiber of ν we conclude that there is a divisor on Y whose inverse image is A. This contradicts the conclusion dim ν(A) < dim Y − 1. Q.E.D.

Slightly more is true in special cases. Corollary 1.6 Let (M, L, ν, Y) be a Fano fibration of index k = n − 2 where n := dim M. Let A + B ∈ |L| with A and B distinct irreducible divisors. If n ≥ dim Y + 2 then dim ν(A ∩ B) ≥ dim Y − 1. Proof. The result is clear if dim Y ≤ 1. If dim Y = 2 this is a result of Beltrametti and the third author [5, Theorem 14.2.3]. If dim Y = 3 this is a result of Beltrametti, the third author, and Wiśniewski, see [6, (3.2.1)] and [5, Theorem 14.1.1]. Q.E.D.

The following simple example comes up often. Example 1.7 Let L̂ be an ample line bundle on a smooth n-dimensional projective manifold X̂. Suppose that Pic(X̂) = ℤ with a generator O_{X}(1) for Pic(X̂). If L̂ ≅ O_{X}(1), then every D ∈ |L̂| is irreducible and reduced. More generally if L̂ ≅ O_{X}(s) and D ∈ |L̂| decomposes D = ∑_{i=1}^{r} D_{i} into irreducible and reduced divisors D_{i}, then

D_{i} ∈ |O_{X}(s_{i})| with s = ∑_{i=1}^{r} s_{i}.

Example 1.8 [Scrolls of arbitrary fiber and base dimensions] Let V be a rank (n − m + 1) vector bundle over a smooth projective manifold Y of dimension m, and let (X, L) = (ℙ(V), ξ_{V}). Assume that

1. there is a smooth connected divisor Z ⊂ Y such that V ⊗ [−Z] has a nowhere vanishing section s;

2. V is very ample in the sense that the tautological bundle ξ_{V} over ℙ(V) is very

ample; and

3. K_{Y} + det V is ample and therefore (ℙ(V), ξ_{V}) is a scroll over Y, i.e., K_{ℙ(V)} +

(n − m + 1)ξ_{V} ≅ π_{V}^{∗} H with H ample on Y. Letting ŝ denote the section of ξ_{V} ⊗ π_{V}^{∗} [−Z] corresponding to s, B := ŝ^{−1}(0) is a smooth ℙ^{n−m−1}-bundle over Y, A := π_{V}^{−1}(Z) is a smooth ℙ^{n−m}-bundle over Z, A meets B transversely in ℙ(V_{Z}/(s_{Z} ⊗ O_{Z})), and ξ_{V} = A + B. Note that given any smooth projective manifold Y of dimension m and any integer n ≥ m + 1, we can find V and Z satisfying the above conditions. Indeed taking Z as a divisor which is very ample and such that K_{Y} + (2n − 2m + 1)Z is ample, define V := [Z] ⊕ [2Z]^{⊕(n−m)}. The following theorem shows that Example 1.8 is a common state of affairs for scrolls. Theorem 1.9 Let L be an ample line bundle on an n-dimensional projective manifold X. Assume that (X, L) is (ℙ(V), ξ_{V}) for a vector bundle of rank ≥ dim Y + 1 over a smooth positive dimensional projective manifold Y with induced projection π : X → Y. If there are smooth connected divisors A_{1}, …, A_{r} with r ≥ 2 and with A_{1}+⋯+A_{r} ∈ |L|, then (after renaming if necessary):

1. there are smooth connected divisors D_{1}, …, D_{r−1} on Y such that A_{i} = π^{−1}(D_{i}) for 1 ≤ i ≤ r − 1; and

2. A_{r} meets the generic fiber of π in a hyperplane, and if dim Y ≤ 3 then A_{r} =

ℙ(V′) for an appropriate quotient bundle of V. Proof. Let F be a general fiber of π. Since L_{F} is a generator of Pic(F) = ℤ we conclude that only one (say, after renaming, A_{r}) of the A_{i} can meet a general fiber of F of π and A_{r} will meet it in a hyperplane. Thus there are smooth connected divisors D_{1}, …, D_{r−1} on Y such that A_{i} = π^{−1}(D_{i}) for 1 ≤ i ≤ r − 1.

Letting f := dim X − dim Y, we have f − 1 = dim A_{r} − dim Y ≥ dim Y − 1. L − A_{r} is the pullback under π of a line bundle P on Y, i.e., L = A_{r} + π^{∗}P. Choose an ample line bundle H on Y such that P + H and K_{Y} + det V + (f − 1)(P + H) are ample line bundles and L + π^{∗}(P + H) is very ample. Then (A_{r}, (L + π^{∗}(P + H))_{A_{r}}) is a scroll over Y. To see this note that

K_{A_{r}} + f(L + π^{∗}(P + H))_{A_{r}} = (K_{X} + A_{r} + f(L + π^{∗}(P + H)))_{A_{r}}

= (K_{X} + (f + 1)L + f(π^{∗}(P + H)) − P)_{A_{r}}
= π^{∗}(K_{Y} + det V + (f − 1)(P + H) + H).

Since dim A_{r}−dim Y ≥ dim Y −1 the result follows from [5, Proposition 14.1.3]. Q.E.D.

Remark 1.10 The condition on the base is merely to ensure that we had a bundle. See [5, Chapter 14.1] for what is known and conjectured about the more general case when the base is arbitrary. Indeed using the discussion following [5, Conjecture 14.1.10] it follows that the condition dim Y ≤ 3 can be removed.

Slightly more can be said when (X, L) is a scroll over a curve. Theorem 1.11 Let L be an ample line bundle on an n-dimensional projective manifold X. Assume that (X, L) is (ℙ(V), ξ_{V}) for a vector bundle of rank ≥ 2 over a smooth curve Y with induced projection π : X → Y. If there are r ≥ 2 irreducible connected divisors A_{1}, …, A_{r} with A_{1} + ⋯ + A_{r} ∈ |L|, then (after renaming if necessary):

1. there are distinct points D_{1}, …, D_{r−1} on Y such that A_{i} = π^{−1}(D_{i}) for 1 ≤ i ≤ r − 1; and

2. A_{r} meets the generic fiber of π in a hyperplane; if A_{r} has at worst isolated singularities then A_{r} = ℙ(V′) for an appropriate quotient bundle of V.

Proof. The argument from the proof of Theorem 1.9 quickly reduces us (after renaming if necessary) to showing that if A_{r} is the component that meets every fiber of π, then A_{r} is a fiber bundle under π. This is clear if dim A_{r} = 1, since in this case A_{r} meets the general fiber in a single point and is therefore a section. Since A_{r} has at worst isolated singularities and is a Cartier divisor of dimension ≥ 2 on a manifold, it follows that A_{r} is normal. Using [5, Proposition 3.2.1], it suffices to note that the fibers of π_{A_{r}} are of equal dimension. Q.E.D.

The results in [5, Chapter 7] contain the classification results we need from adjunction theory. Here let us summarize the main concepts from the classification.

Let L̂ be an ample line bundle on a connected n-dimensional projective manifold X̂. Assume that n ≥ 2. Up to a short list of very special pairs (X̂, L̂) described in [5, Chapter 7], we have that K_{X} + (n − 1)L̂ is nef and big. If K_{X} + (n − 1)L̂ is nef and big, then there exists a pair (X, L) called the first reduction of (X̂, L̂) and a morphism ϕ : X̂ → X called the first reduction map, where:

1. X is a projective manifold and ϕ expresses X̂ as the blowup of X at a finite set F;

2. L := (ϕ_{∗}L)^{∗∗} and K_{X} + (n − 1)L are ample line bundles;

3. K_{X} + (n − 1)L̂ ≅ ϕ^{∗}(K_{X} + (n − 1)L) (or, equivalently, L̂ ≅ ϕ^{∗}L − ϕ^{−1}(F)). Furthermore if n ≥ 3 and the first reduction of (X, L) of (X̂, L̂) exists, then up to a short list of very special pairs (X, L) described in [5, Chapter 7], we have that K := K_{X} + (n − 2)L is nef and big. There exists a pair (X′, L′) called the second reduction of (X̂, L̂) and a morphism ψ : X → X′ called the second reduction map, where:

1. X′ has isolated terminal singularities and ψ expresses X as a completely explicit modification of X′ with X − ψ^{−1}(Z) isomorphic to X′ − Z for an algebraic set Z with dim Z ≤ 1;

2. L′ = (ϕ_{∗}L)^{∗∗} is at worst 2-Cartier, K′ := K_{X′} + (n − 2)L′ is an ample line bundle, and K = ψ^{∗}K′ = ψ^{∗}(K_{X′} + (n − 2)L′).

Here is an example where the fibers of the first reduction map are part of an ample divisor. Example 1.12 [Fibers of reduction maps] Let L̂ be a very ample line bundle on a smooth connected projective n-fold X̂. Assume that there is a smooth divisor Â ≅ ℙ^{n−1} on X̂ such that [Â]_{A} ≅ O_{ℙ^{n−1}}(−1) and L̂_{A} ≅ O_{ℙ^{n−1}}(1). This can always be achieved by taking a taking X̂ to be the blowup of a smooth projective manifold ϕ : X̂ → X at one point x ∈ X, with Â := ϕ^{−1}(x), and with L̂ := ϕ^{∗}L − Â for some line bundle L which is a tensor product of at least two very ample line bundles on X (or which, more generally is 2-jet ample in the sense of [4]). Then L̂ − Â is spanned and choosing a general B̂ ∈ |L̂ − Â| we have Â + B̂ ∈ |L̂| with Â and B̂ smooth and transverse. The usual situation is that B̂ is connected. This happens if either n ≥ 3 or n = 2 and (L̂ − Â)^{2} > 0.

Note that Â meets B̂ transversely in a smooth quadric relative to L̂. Thus the genus of a curve section of Â ∩ B̂ is 0.

Notice that the example (X̂, L̂) arising from this construction when we choose (X, L) ≅ (ℙ^{3}, O_{ℙ^{3}}(2)) is Del Pezzo.

2. A first Lefschetz theorem for reducible hyperplane sections. In this section we collect some consequences of the first Lefschetz hyperplane section theorem and Deligne’s mixed Hodge theory [8].

Throughout this section we let D be an ample divisor on a smooth connected projective manifold X and assume that D = ∑_{i=1}^{r} D_{i} is a sum of smooth irreducible divisors on X with all subsets of the D_{i}’s meeting transversely. For any multi-index I = (i_{0}, …, i_{k}) with 1 ≤ i_{0} < … < i_{k} ≤ r, we let l(I) = k and D_{I} = D_{i_{0}} ∩ … ∩ D_{i_{k}}.

In this case, the mixed Hodge structure on D has been constructed in a simple and concrete way in [10, section 4]. The weight filtration has the form

{0} ⊂ W_{0} ⊂ W_{1} ⊂ … ⊂ W_{m−1} ⊂ W_{m} = H^{m}(D, ℚ),

and tensored with ℂ is defined by using the filtration induced on the E_{∞} term of the spectral sequence of the double complex A^{p,q} = A^{p}(D^{[q]}) by the filtration ⨁_{r≤k} A^{r,s}. Here, D^{[q]} is the disjoint union of all D_{I} with l(I) = q, and A^{∗}(D^{[q]}) is the usual de Rham complex.

Let d : A^{p,q} → A^{p+1,q} be the usual exterior derivative and let δ : A^{p,q} → A^{p,q+1} be given by

(δϕ)_{i_{0}...i_{q+1}} = ∑_{ν=0}^{q+1}(−1)^{ν+p}ϕ_{i_{0}..._{i_{ν}}...i_{q+1}} |_{D_{i_{0}}∩···∩D_{i_{q+1}}}

for ϕ = (ϕ_{i_{0}...i_{q}}), with ϕ_{i_{0}...i_{q}} ∈ A^{p}(D_{i_{0}} ∩ ⋯ ∩ D_{i_{q}}). Then the graded module associated to the weight filtration on H^{∗}(D) is the E_{∞} term of the spectral sequence of the double complex (A^{∗,∗}, d, δ), and the usual Hodge filtration on A^{∗,∗} induces a Hodge structure of pure weight m on E_{∞}^{m}.

Moreover, the spectral sequence degenerates at the E_{2} term, i.e.,

E_{2} = E_{3} = … = E_{∞} ,

and it follows that W_{m−j}/W_{m−j−1} ≅ H^{j}(H^{m−j}(D_{I}, ℚ), δ) , where the right-hand side is induced by the sequence

⋯ → ⨁_{l(I)=j−1} H^{m−j}(D_{I}, ℚ) →^{δ} ⨁_{l(I)=j} H^{m−j}(D_{I}, ℚ) →^{δ} ⨁_{l(I)=j+1} H^{m−j}(D_{I}, ℚ) → · · · .

If D is an ample divisor of a smooth projective variety X, the image of H^{m}(X, ℚ) → H^{m}(D, ℚ) is of pure weight m, and combining that observation with the Lefschetz theorem yields the following theorem: Theorem 2.1 Let D be an ample divisor on a smooth, connected projective variety X, and assume that D = ∑_{i=1}^{r} D_{i} is a sum of smooth irreducible divisors with all subsets of the collection of divisors meeting transversely. Then for j + k < dim X

0 → H^{j}(X, ℚ) → ⨁_{l(I)=0} H^{j}(D_{I}, ℚ) → ⋯ → ⨁_{l(I)=k} H^{j}(D_{I}, ℚ)

is exact with the convention that H^{j}(D_{I}, ℚ) = 0 if l(I) ≥ r.

In particular, the case k = 0, together with the Lefschetz theorem, yields the following. Corollary 2.2 If j < dim X, the restriction map H^{j}(X, ℚ) → ⨁_{i=1}^{r} H^{j}(D_{i}, ℚ) is injective. Moreover, if j < dim D, the restriction map H^{j}(D, ℚ) → ⨁_{i=1}^{r} H^{j}(D_{i}, ℚ) is injective.

If D has two irreducible components, say D = A+B, we have the following corollary of Theorem 2.1. Corollary 2.3 Let D be an ample divisor on a connected projective manifold X with dim X = 3 + j ≥ 3, and assume that D = A + B, where A and B are connected submanifolds that intersect transversely. Then

0 → H^{i}(X, ℚ) → H^{i}(A, ℚ) ⊕ H^{i}(B, ℚ) → H^{i}(A ∩ B, ℚ) → 0

for i ≤ j. In particular A ∩ B is connected if dim X ≥ 3. Proof. From Theorem 2.1 with r = 2 we have

0 → H^{i}(X, ℚ) → H^{i}(A, ℚ) ⊕ H^{i}(B, ℚ) → H^{i}(A ∩ B, ℚ) → 0,

from which the conclusion follows. Q.E.D.

Let D be an ample divisor on a connected projective manifold X with dim X ≥ 3, and assume that D = A + B, where A and B are connected submanifolds that intersect transversely. We call A ∩ B the hinge variety associated to the divisor A + B. As a byproduct of our investigation we will give a complete classification of such divisors when the curve genus of (A ∩ B, L_{A∩B}) is 0 or 1, i.e., when the genus of a curve section of (A ∩ B, L_{A∩B}) is 0 or 1.

3. Classification up to the second reduction. In this section we let L̂ denote an ample line bundle on a projective manifold X̂ of dimension n ≥ 2, and we assume that Â = ∑_{i=1}^{r} Â_{i} is a divisor in |L̂|, where each Â_{i} is irreducible. Unless otherwise stated we do not assume that the Â_{i}’s are smooth or intersect transversely. We do not assume that the line bundle is spanned because we will in the course of studying very ample line bundles need to apply these results to ample, but not necessarily spanned line bundles. Theorem 3.1 Let L̂ denote an ample line bundle on a projective manifold X̂ of dimension n ≥ 2. Assume that Â = ∑_{i=1}^{r} Â_{i} is a divisor in |L̂| with r ≥ 2, and with each Â_{i} irreducible. If K_{X} + (n − 1)L̂ is not nef then either

1. (X̂, L̂) = (ℙ^{2}, O_{ℙ^{2}}(2)), r = 2, and each (Â_{i}, L̂_{A_{i}}) is a line; or

2. (X̂, L̂) = (ℙ^{1} × ℙ^{1}, O_{ℙ^{1}×ℙ^{1}}(1, 1)), r = 2, and there are points x, y ∈ ℙ^{1} such that Â = ({x} × ℙ^{1}) + (ℙ^{1} × {y}); or

3. (X̂, L̂) is a scroll over a smooth curve Y, one component of Â meets the general fiber in a hyperplane, and the other components are fibers. If the Â_{i} have at worst isolated singularities, then all the components are smooth, and the component that meets the general fiber in a hyperplane is a scroll over a curve relative to L̂, and is in particular a ℙ^{n−2}-bundle over Y.

Moreover, each of these cases occurs. Proof. By adjunction theory [5, table 7.4, p.164], any polarized manifold (X̂, L̂) for which K_{X} + (n − 1)L̂ is not nef must be one of the following:

1. (ℙ^{n}, O_{ℙ^{n}}(1)); or

2. (ℙ^{2}, O_{ℙ^{2}}(2)); or

3. (Q, O_{Q}(1)) where Q is a quadric in ℙ^{n+1}; or

4. a scroll over a smooth curve. Since r ≥ 2 and Pic(ℙ^{n}) ≅ ℤ, we conclude from Example 1.7 that (ℙ^{n}, O_{ℙ^{n}}(1)) is not possible. The case of (ℙ^{2}, O_{ℙ^{2}}(2)) obviously occurs with Â_{i} a line for i = 1, 2. Similarly the assertion about the quadric is clear. Theorem 1.11 gives the assertion about for a scroll over a smooth curve. A scroll over a smooth curve is a special case of Example 1.8. Q.E.D.

We now assume that K_{X} + (n − 1)L̂ is nef. The next two theorems take us to the first reduction (X, L). Theorem 3.2 Let L̂ denote an ample line bundle on a projective manifold X̂ of dimension n ≥ 2. Assume that Â = ∑_{i=1}^{r} Â_{i} is a divisor in |L̂| with r ≥ 2, and with each Â_{i} irreducible. If K_{X} + (n − 1)L̂ is nef but not big, then either

1. (X̂, L̂) is a Del Pezzo manifold, i.e., K_{X} = −(n − 1)L̂: these are enumerated in Theorem 3.4; or

2. X̂ is a quadric fibration over a smooth curve; moreover, if dim X̂ ≥ 3 each fiber is irreducible and reduced; or

3. X̂ is a scroll (and a ℙ^{n−2}-bundle of dimension ≥ 3 over a smooth surface Y), and exactly one component of Â, say Â_{r} meets every fiber. If Â_{r} is smooth and n ≥ 4, then Â_{r} is a ℙ^{n−3}-bundle over Y.

Proof. From adjunction theory [5, Theorem 7.3.2, p.169] we know that X̂ is either a Del Pezzo manifold (i.e., one with K_{X} = −(n − 1)L̂), a quadric fibration over a smooth curve, or a scroll over a normal surface.

In the case of a quadric fibration, the assertion about the fibers follows easily from the fact that L̂_{F} = O_{Q}(1) for any fiber F. In the case of a scroll, the assertion about the components of Â follows from the fact that L̂^{n−1} · F = 1, as in Proposition 3.1. The fact that π : X̂ → Y is a ℙ^{n−3}-bundle when n ≥ 4 (respectively Â_{r} is a ℙ^{n−2}-bundle when n ≥ 3) over a smooth surface follows from the discussion after Conjecture 5.3 of [1] (see also [16, Theorem 3.3]). Q.E.D.

For the purpose of this paper the case of dimension ≥ 3 treated in Theorem 3.4 suffices. The two dimensional case is a straightforward consequence of the classification of Del Pezzo surfaces, but has many special cases. The following simple lemma is needed. Lemma 3.3 Let L̂ be an ample line bundle on a connected projective manifold X̂ of dimension n ≥ 3. Assume that L̂^{n} = 2, and that Â = ∑_{i=1}^{r} Â_{i} ∈ |L̂| is a sum of irreducible divisors Â_{i}. If K_{X} = −(n − 1)L̂ then r = 1. Proof. Assume that r ≥ 2. By Theorem 1.3 we can assume that n = 3. Since

2 = L̂^{3} = L̂^{2} · Â = ∑_{i=1}^{r} L̂^{2} · Â_{i},

we see that we can assume that r = 2. We know that L̂ is spanned by [9, page 44]. From this and Goren’s theorem [9, Theorem (1.1)], we conclude that (Â_{i}, L̂_{A_{i}}) ≅ (ℙ^{n−1}, O_{ℙ^{n−1}}(1)) for i = 1, 2. From this and K_{X} = −(n − 1)L̂ we conclude that the normal bundle of Â_{i} is ≅ O_{ℙ^{n−1}}(−1) for i = 1, 2. This is absurd since this would allow us to contract one of the Â_{i}, say Â_{1}, on X̂, which would give rise to a holomorphic map on Â_{2}, which has a positive dimensional image and which contracts the positive dimensional set Â_{1} ∩ Â_{2} ⊂ Â_{2}. Q.E.D.

Theorem 3.4 Let L̂ be an ample line bundle on a connected projective manifold X̂ of dimension n ≥ 3. Assume that Â = ∑_{i=1}^{r} Â_{i} ∈ |L̂| is a sum of r ≥ 2 irreducible divisors Â_{i}, each having at worst isolated singularities. Assume that the Â_{i} meet pairwise transversely. If K_{X} = −(n − 1)L̂ then either

1. (X̂, L̂) = (ℙ^{1} × ℙ^{1} × ℙ^{1}, O_{ℙ^{1}×ℙ^{1}×ℙ^{1}}(1, 1, 1)); or

2. (X̂, L̂) = (ℙ^{2} × ℙ^{2}, O_{ℙ^{2}×ℙ^{2}}(1, 1)); or

3. (X̂, L̂) = (ℙ(T_{ℙ^{2}}), ξ_{T_{ℙ^{2}}}); or

4. (X̂, L̂) has first reduction (ℙ^{3}, O_{ℙ^{3}}(2)) with X̂ the blowup of ℙ^{3} at most one

point. Proof. For the Del Pezzo case we use Fujita’s classification [9, 8.11]. We first note that 1 ≤ d ≤ 8, where d = L̂^{n} is the degree of L̂. If r > 1 then d ≠ 1. Indeed since L̂ is ample, L̂^{n} = ∑_{i=1}^{r} L̂^{n−1} · Â_{i} ≥ r.

Moreover, d ≠ 2 by Lemma 3.3.

The Del Pezzo manifolds with 3 ≤ d ≤ 8, except for the ones listed in the proposition above, have Pic(X̂) = ℤ with generator L̂, which contradicts the assumption that r > 1. Q.E.D.

Remark 3.5 It is easy to see that each of the cases listed in the above propositions can occur. Scrolls over normal surfaces are special cases of example 1.8. The blowup of ℙ^{3} at a point is a special case of example 1.12. On ℙ(T_{ℙ^{2}}) we get a divisor by considering the section of the tautological bundle induced by a vector field on ℙ^{2} with zero set consisting of a point and a line. To get a quadric fibration we modify example 1.8 by taking X̂ to be a smooth section of 2ξ_{V}, where V has rank n − m + 2, and constructing the divisor Â + B̂ as in that example.

It follows from Propositions 3.1 and 3.2 that, with the exceptions listed, the first reduction (X, L) exists; i.e., X̂ is the blowup ϕ : X̂ → X of a finite set F on a smooth projective variety X, and L := (ϕ_{∗}L̂)^{∗∗} is an ample line bundle with K_{X} + (n − 1)L ample. If E = ϕ^{−1}(x) is an exceptional divisor over a point x ∈ X, then E ≅ ℙ^{n−1} with [E]_{E} = O_{ℙ^{n−1}}(−1) and L̂_{E} = O_{ℙ^{n−1}}(1) (see [5, 7.3]).

It follows that if E is not a component of the ample divisor Â it can meet at most one component, say Â_{1}, and it must meet it transversely in a smooth ℙ^{n−2} lying entirely in the set of regular points of Â_{1} and with O_{ℙ^{n−2}}(−1) as the normal bundle of E ∩Â_{1} in Â_{1}. In this case, blowing down E does not reduce the number of components, so that we obtain a divisor A = ∑_{i=1}^{r} A_{i} in X, where the singularities of the A_{i}’s are no worse than those of the Â_{i}’s in X̂.

If E is a component of Â, say E = Â_{1}, then we must have [ ∑_{i=2}^{r} Â_{i}]_{E} = O_{ℙ^{n−1}}(2), from which it follows that either

• E meets exactly one other component Â_{i} with i ≠ 1 in an (n − 2)-dimensional smooth quadric Q lying entirely in the set of regular points of Â_{i} and with normal bundle −L_{Q} in Â_{i}; or

• E meets exactly two other components Â_{i}, Â_{j} with 1 < i < j. In this case each of E ∩ Â_{i} ⊂ reg(Â_{i}) and E ∩ Â_{j} ⊂ reg(Â_{j}) is a ℙ^{n−2}, and the normal bundles of E ∩ Â_{i} in Â_{i} and of E ∩ Â_{j} in Â_{j} are both isomorphic to O_{ℙ^{n−2}}(−1); or • E meets exactly one other component in a singular (possibly nonreduced)

quadric. Obviously, the third alternative can not occur if the Â_{i} are all smooth and intersect transversely.

It is also clear that if more than one component of Â is exceptional with respect to ϕ, then no two of the exceptional components can intersect (otherwise the intersection would be a divisor in ℙ^{n−1} with negative normal bundle).

We sum up these observations in the following propositions. Theorem 3.6 Let L̂ be an ample line bundle on a connected projective manifold X̂ of dimension n ≥ 3. Assume that Â = ∑_{i=1}^{r} Â_{i} is a divisor in |L̂|, where r ≥ 2, each Â_{i} is irreducible with at worst isolated singularities, and Â_{i} ≠ Â_{j} for i ≠ j. Assume that K_{X} +(n−1)L̂ is nef and big. Let ϕ : X̂ → X denote the first reduction. Let A_{i} = ϕ(Â_{i}). If no component of Â is a fiber of ϕ, then

1. ϕ is a biholomorphism in a neighborhood of any point belonging to at least two of Â_{i}. Thus if all subsets of the Â_{i} have transverse intersection, then all subsets of the A_{i} have transverse intersection.

2. The map ϕ_{A_{i}} : Â_{i} → A_{i} expresses Â_{i} as the blowup of A_{i} at the finite set F ∩A_{i} of regular points of A_{i} with L̂_{A_{i}} = ϕ^{∗}L_{A_{i}} − ϕ−1 ; _{A_{i}}(F ∩ A_{i}). Thus the singularities of A_{i} = ϕ(Â_{i}) are the same as the singularities of Â_{i} and the curve genus of (Â_{i}, L̂_{A_{i}}) is the same as the curve genus of (A_{i}, L_{A_{i}}).

Theorem 3.7 Let L̂ be an ample line bundle on a connected projective manifold X̂ of dimension n ≥ 3. Assume that Â = ∑_{i=1}^{r} Â_{i} is a divisor in |L̂|, with r ≥ 2, with each Â_{i} having at worst isolated singularities, and with the Â_{i} irreducible and meeting pairwise transversely. Assume that Sing(Â_{i}) ∩ Sing(Â_{j}) = ∅ for i ≠ j. Assume that K_{X} + (n − 1)L̂ is nef and big. Let ϕ : X̂ → X denote the first reduction. Assume (after renaming if necessary) that dim ϕ(Â_{i}) < dim Â_{i} if and only if i = s + 1, …, r. Then for i > s, ϕ(Â_{i}) is a point x_{i}, Â_{i} meets at least one other Â_{j} with j ≤ s and at most two other such Â_{j}. If it meets just one other Â_{j}, then x_{i} is a nondegenerate quadratic singularity of ϕ(Â_{j}). Moreover, s ≥ 1 and ϕ maps Â onto a divisor A = ∑_{i=1}^{s} A_{i}.

We now pass to the first reduction; that is we consider an n-dimensional projective manifold X with an ample divisor A = ∑_{i=1}^{s} A_{i} and assume that the line bundles L = [A] and K_{X} +(n−1)L are both ample. We do not assume that the A_{i}’s are smooth or that their intersections are transversal. Adjunction theory gives us the following proposition. Theorem 3.8 Let L be an ample line bundle on a connected n-dimensional projective manifold X. Assume that K_{X} + (n − 1)L is ample. Assume that there is an A ∈ |L| with A = ∑_{i=1}^{s} A_{i} for distinct irreducible divisors A_{i}. Then K_{X} +(n−2)L is nef, except in the following cases:

1. (X, L) = (ℙ^{4}, O_{ℙ^{4}}(2)).

2. (X, L) = (Q_{3}, O_{Q_{3}}(2)), where (Q_{3}, O_{Q_{3}}(1)) is a smooth quadric 3-fold in ℙ^{4}.

3. (X, L) = (ℙ^{3}, O_{ℙ^{3}}(3)).

4. X is a ℙ^{2}-bundle over a smooth curve Y, ν : X → Y with 2K_{X} + 3L ≅ ν^{∗}H for an ample line bundle H on Y. Thus (F, L_{F}) = (ℙ^{2}, O_{ℙ^{2}}(2)) for a general fiber F. There are two possibilities:

(a) Except for one of the A_{i} which meets a general fiber in a smooth conic, the A_{i} are fibers; or

(b) Two of the A_{i} are scrolls over Y, each meets a fiber of ν in a line, and

the remaining A_{i} are fibers of ν.

Moreover, all of these cases can occur with s > 1. Proof. The proposition is simply a restatement of [5, Theorem 7.3.4, p.171], except for the claim that each example occurs. Cases 1 through 3 are evident. An example of case 4(a) is obtained by taking X = ℙ(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(a) ⊕ O_{ℙ^{1}}(b)) over ℙ^{1} with 0 < a ≤ b and letting L := 2ξ + F, where ξ is the tautological bundle and F is a fiber. An example of case 4(b) is obtained with the same X, by letting L := (ξ + F) + (ξ + F). Q.E.D.

Theorem 3.9 Let L be an ample line bundle on a connected n-dimensional projective manifold X. Assume that K_{X} + (n − 1)L is ample. Assume that there is an A ∈ |L| with A = ∑_{i=1}^{s} A_{i} for distinct irreducible divisors A_{i}. Assume that K_{X} + (n − 2)L is nef but not big. Let ν : X → Y be the nefvalue morphism associated to (X, L), i.e., ν is a surjective morphism with connected fibers onto a normal projective variety Y, and K_{X} + (n − 2)L ≅ ν^{∗}H for an ample line bundle H on Y. Then one of the following must hold.

1. (X, L) is a Mukai variety; i.e., K_{X} = −(n−2)L. If s > 1 then either dimX ≤ 5 or (X, L) = (ℙ^{3} × ℙ^{3}, O_{ℙ^{3}×ℙ^{3}}(1, 1)).

2. (X, L) is a Del Pezzo fibration over a smooth curve under ν.

3. (X, L) is a quadric fibration over a normal surface under ν.

4. (X, L) is a scroll over a normal 3-fold under ν. Proof. This proposition is a restatement of [5, Theorem 7.5.3, p.176], except for the second statement in case 1, which follows immediately from Theorem 1.3. Q.E.D.

As an application of the above results we have the following results we will need below. Theorem 3.10 Let L̂ be an ample and spanned line bundle on an n-dimensional projective manifold X̂. Assume that n ≥ 3 and that there are two smooth transverse divisors Â, B̂ on X̂ with Â + B̂ ∈ |L̂|. Assume that the genus g(h) of a curve section of h := Â ∩ B̂ is 0. Then either:

1. (X̂, L̂) is a scroll over a smooth curve; or

2. (X̂, L̂) is Del Pezzo (see Theorem 3.4); or

3. (X̂, L̂) is a quadric fibration over a smooth curve (there are two possibilities: the general possibility plus the special three dimensional case over a rational curve with Â and B̂ both ℙ^{1}-bundles); or

4. X̂ is a scroll over a smooth surface, and only one component, say Â meets every fiber, and is a ℙ^{n−3}-bundle over the base surface; B̂ is the inverse image under the scroll projection of a rational curve on the surface;

5. K_{X} + (n − 1)L̂ is nef and big and either Â or B̂ is a fiber of the first reduction

mapping associated to (X̂, L̂); or

6. K_{X} + (n − 1)L̂ is nef and big with first reduction (X, L) and neither Â nor B̂ is a fiber of the first reduction mapping ϕ : X̂ → X associated to (X̂, L̂). Let A := ϕ(Â) and B := B̂. The following cases occur:

(a) (X, L) ≅ (ℙ^{4}, O_{ℙ^{4}}(2)); or

(b) (X, L) = (Q_{3}, O_{Q_{3}}(2)), where (Q_{3}, O_{Q_{3}}(1)) is a smooth quadric 3-fold in ℙ^{4}; or

(c) (X, L) ≅ (ℙ^{3}, O_{ℙ^{3}}(3)); or

(d) X is a ℙ^{2}-bundle over a smooth curve Y, ν : X → Y with 2K_{X} +3L ≅ ν^{∗}H for an ample line bundle H on Y. Thus (F, L_{F}) = (ℙ^{2}, O_{ℙ^{2}}(2)) for a general fiber F. There are two possibilities: i. After renaming if necessary B is a fiber and A meets a general fiber

in a smooth conic; or ii. A and B are scrolls over Y ≅ ℙ^{1}, each meets a fiber of ν in a line.

Proof. By theorem 3.1, if K_{X} +(n−1)L̂ is not nef then (X̂, L̂) is a scroll over a smooth curve. We may therefore assume that K_{X} + (n − 1)L̂ is nef. Then by theorem 3.2, K_{X} + (n − 1)L̂ is big except when (X̂, L̂) is Del Pezzo; or a quadric fibration over a smooth curve; or a scroll over a smooth surface.

Now assume that K_{X} + (n − 1)L̂ is nef and big and denote by (X, L) its first reduction. Assume that neither Â nor B̂ is a fiber of the first reduction map ϕ : X̂ → X and call A = ϕ(Â), B = ϕ(B̂).

By Theorem 3.8 K_{X} + (n − 2)L is nef unless (X, L) = (ℙ^{4}, O_{ℙ^{4}}(2)); or (X, L) = (Q_{3}, O_{Q_{3}}(2)), where (Q_{3}, O_{Q_{3}}(1)) is a smooth quadric 3-fold in ℙ^{4}; or (X, L) = (ℙ^{3}, O_{ℙ^{3}}(3)); or X is a ℙ^{2}-bundle over a smooth curve Y, ν : X → Y with 2K_{X} + 3L ≅ ν^{∗}H for an ample line bundle H on Y, and (F, L_{F}) = (ℙ^{2}, O_{ℙ^{2}}(2)) for a general fiber F. In the last case (after possibly renaming), A meets a fiber in a smooth conic and B is a fiber; or both A and B are scrolls over Y ≅ ℙ^{1}.

These cases account for the cases in part 6 of the theorem.

Hence we may now assume that K_{X} + (n − 2)L is nef. But then ϕ(h) is isomorphic to h, and 2g(h) − 2 = (K_{X} + (n − 2)L) · A · B · L^{n−3} ≥ 0, contrary to g(h) = 0. Q.E.D.

Theorem 3.11 Let L̂ be an ample and spanned line bundle on an n-dimensional projective manifold X̂. Assume that n ≥ 3 and that there are two smooth transverse divisors Â, B̂ on X̂ with Â + B̂ ∈ |L̂|. Assume that the genus g(h) of a curve section of h := Â ∩ B̂ is 1. Then either:

1. n = 3 and (X̂, L̂) is a quadric fibration over an elliptic curve with Â and B̂ ℙ^{1}-bundles over the elliptic curve; or

2. X̂ is a scroll over a smooth surface, and only one component, say Â meets every fiber, and when n ≥ 4, Â is a ℙ^{n−3}-bundle over the base surface; B̂ is the inverse image under the scroll projection of an elliptic curve on the surface; or

3. X̂ is a ℙ^{2}-bundle over an elliptic curve Y, ν : X̂ → Y with 2K_{X} + 3L̂ ≅ ν^{∗}H for an ample line bundle H on Y. Thus (F, L̂_{F}) = (ℙ^{2}, O_{ℙ^{2}}(2)) for a general fiber F; Â and B̂ are ℙ^{1}-bundles; and each meets a fiber of ν in a line.

4. The first reduction ϕ : (X̂, L̂) → (X, L) exists with L, K_{X} + (n − 1)L ample, and with K_{X} + (n − 2)L nef. Neither Â nor B̂ are exceptional divisors for ϕ and ϕ is a biholomorphism in a neighborhood of h. Letting A := ϕ(Â) and B := ϕ(B̂) we have

(a) (X, L) is Mukai with dim X ≤ 5 or (X, L) = (ℙ^{3} × ℙ^{3}, O_{ℙ^{3}×ℙ^{3}}(1, 1)); or (b) (X, L) is a Del Pezzo fibration over a smooth curve under ν: after renam-

ing, Â is a fiber and B̂ meets the general fiber in a Del Pezzo manifold;

(c) X is three-dimensional and (X, L) is a quadric fibration over a smooth surface under ν;

(d) X is four-dimensional and (X, L) is a scroll over a normal threefold under ν.

(e) K_{X} + (n − 2)L is nef and big and either A or B is a divisor which gets mapped to a point under the second reduction mapping ψ : X → X′ associated to (X, L).

Proof. We let g(h) = 1 denote the genus of a curve section of (h, L̂_{h}). Note that by Theorem 3.1 K_{X} + (n − 1)L̂ is nef. By Theorem 3.2 either K_{X} + (n − 1)L̂ is big or one of the following happens.

1. (X̂, L̂) is a Del Pezzo manifold, i.e., K_{X} = −(n − 1)L̂;

2. X̂ is a quadric fibration over a smooth curve and since dim X̂ ≥ 3 then each fiber is irreducible and reduced;

3. X̂ is a scroll over a smooth surface, and only one component, say Â meets every fiber; moreover, if dim X̂ ≥ 4, then Â is a ℙ^{n−3}-bundle over the base surface.

Note the first cannot happen since it would yield the absurdity

0 = 2g(h) − 2 = (K_{X} + (n − 2)L̂) · L̂^{n−3} · Â · B̂ = −L̂ · h < 0.

In the second case we have either

1. B̂ is a fiber and Â is a divisor meeting a general fiber in a quadric. In this case the genus of a curve section of Â ∩ B̂ is zero; or

2. dim X̂ = 3 and Â, B̂ are ℙ^{1} bundles meeting in a section. This is possible if the base curve is elliptic, giving the first case in the theorem.

In the third case we can get an elliptic curve as a curve section of the intersection. This gives the second case of the theorem.

Thus we can assume without loss of generality that K_{X} + (n − 1)L̂ is nef and big. Let (X, L) be the first reduction of (X̂, L̂). Note that neither Â nor B̂ is an exceptional fiber of the first reduction map since a curve section of h would then be a smooth rational curve. Thus Â and B̂ are mapped to smooth divisors A, B in X and any exceptional fibers do not meet Â ∩ B̂, thus, a neighborhood of Â ∩ B̂ in X̂ is isomorphic to a neighborhood of A ∩ B in X.

By Theorem 3.8 K_{X} + (n − 2)L is nef unless:

1. (X, L) = (ℙ^{4}, O_{ℙ^{4}}(2))

2. (X, L) = (Q_{3}, O_{Q_{3}}(2)), where (Q_{3}, O_{Q_{3}}(1)) is a smooth quadric 3-fold in ℙ^{4} 3. (X, L) = (ℙ^{3}, O_{ℙ^{3}}(3))

4. X is a ℙ^{2}-bundle over a smooth curve Y, ν : X → Y with 2K_{X} + 3L ≅ ν^{∗}H for an ample line bundle H on Y. Thus (F, L_{F}) = (ℙ^{2}, O_{ℙ^{2}}(2)) for a general fiber F. There are two possibilities:

(a) After renaming if necessary, A meets a fiber in a smooth conic and B is a fiber; or

(b) Both A and B are ℙ^{1}-bundles over Y. Only the last case with Y an elliptic curve is possible, which gives the third case of the theorem.

If K_{X} + (n − 2)L is nef but not big then by Theorem 3.9 one of the following occurs. Let ν : X → Y be the nefvalue morphism associated to (X, L), i.e., ν is a surjective morphism with connected fibers onto a normal projective variety Y, and K_{X} + (n − 2)L ≅ ν^{∗}H for an ample line bundle H on Y. Then one of the following must hold.

1. (X, L) is a Mukai variety; i.e., K_{X} = −(n − 2)L. Either dim X ≤ 5 or (X, L) = (ℙ^{3} × ℙ^{3}, O_{ℙ^{3}×ℙ^{3}}(1, 1)).

2. (X, L) is a Del Pezzo fibration over a smooth curve under ν; or

3. (X, L) is a quadric fibration over a normal surface under ν; or

4. (X, L) is a scroll over a normal 3-fold under ν. Note that 0 = 2g(h) − 2 = (K_{X} + (n − 2)L) · L^{n−3} · A · B = ν^{∗}H · A · B · L^{n−3}. Thus h is contained in a fiber of the map ν. Thus the first case that (X, L) is a Mukai variety can occur. The second can also with A a fiber and B meeting the general fiber in a Del Pezzo manifold. In the remaining cases we note that since A ∩ B must go to a point under ν we must have that either A or B also goes to a point, by Lemma 1.4. Using Corollary 1.6 we see that in the last two cases we must have that dim X −dim ν(X) = 1. Note the base surface of the quadric fibration is smooth by a theorem of Besana [7].

Thus we can assume without loss of generality that K_{X} + (n − 2)L is nef and big. Let ψ : X → X′ be the second reduction map. By [2, Theorem 0.2.1] ψ is an

isomorphism outside of a union of irreducible divisors of X, and for each such divisor D, either ψ(D) is a point, or (D, L_{D}) is a scroll over a curve.

Observe that at most one of A and B is one of these exceptional divisors.

Now we shall show that exactly one of A and B is exceptional. To see this, assume that neither A nor B is exceptional. Then since K_{X} + (n − 2)L ≅ ψ^{∗}K′ for an ample line bundle K′ on X′ and

0 = 2g(h) − 2 = (K_{X} + (n − 3)L + A + B) · L^{n−3} · A · B = ψ^{∗}K′ · h · L^{n−3},

h must go to a point under ψ. Choosing general D_{1}, …, D_{n−3} ∈ |L| we can slice A, B and consider the situation in dimension n = 3. Since neither A nor B is an exceptional divisor we conclude that L = A+B meets each exceptional divisor in a curve containing two copies of h. This is impossible by an inspection of the list of exceptional divisors given in [2, Theorem 0.2.1].

Thus, after renaming if necessary, we may assume that B is exceptional. Since K_{X} + (n − 2)L ≅ ψ^{∗}K′ for some ample line bundle K′ on X′, we have

2g(h) − 2 = (K_{X} + (n − 3)L + A + B) · L^{n−3} · A · B
= (K_{X} + (n − 2)L) · L^{n−3} · A · B
= ψ^{∗}K′ · L^{n−3} · A · B = ψ_{B}^{∗}K′ · L_{B}^{n−3} · A_{B}.

If B goes to a point under ψ we have that ψ_{B}^{∗}K′ is trivial and thus ψ_{B}^{∗}K′ · L_{B}^{n−3} · A_{B} = 0.

Otherwise, (B, L_{B}) is a scroll over a curve C under ψ. We shall show that this situation cannot occur. Choosing general D_{1}, …, D_{n−3} ∈ |L| we can slice A, B and reduce to the case of dimension n = 3. In this case we have 2g(h) − 2 = ψ_{B}^{∗}K′ · A_{B}. Since K′_{C} is ample we have that ψ_{B}^{∗}K′ is numerically equivalent to a positive multiple tf of some fiber f of ψ_{B} : B → C. Since L·f = 1 and B ·f = −1 we have that A·f = 2 and thus 2g(h) − 2 = ψ_{B}^{∗}K′ · A_{B} = tf · A = 2t > 0. Q.E.D.

4. High index Fano fibrations as components of divisors. In this section we investigate smooth, connected projective varieties that contain an ample divisor with one or more components that are k-Fano fibrations.

Our main tool in studying the structure of high dimensional projective manifolds with an ample divisor equal to a union of scrolls is the following proposition and its corollary.

Theorem 4.1 Suppose that L is an ample line bundle on a normal irreducible projective variety V with at worst canonical singularities. Assume that A is an irreducible and reduced Cartier divisor on V with at worst canonical singularities. Assume that K_{A} is not nef, let τ denote the nefvalue of (A, L_{A}), and let ϕ : A → Y be the nefvalue morphism of (A, L_{A}). Assume that K_{V} + τL is nef. There is a morphism \overline{ϕ} : V → Y with connected fibers from V onto a normal projective variety Y and an embedding i : Y → Y such that the diagram

A ⊂ V
ϕ ↓  ↓ \overline{ϕ}
Y ⟶^{i} Y

commutes. If F is an irreducible positive dimensional fiber of \overline{ϕ} and F ⊄ A, then F ∩ A = ∅ and N(K_{V} + τL)_{F} ≅ O_{F} for positive integers N such that Nτ is integral and NK_{V} is Cartier. Proof. We first observe that K_{V} +τL+A is nef. For if C is an irreducible effective curve with C ⊄ A, then (K_{V} +τL+A)·C = (K_{V} +τL)·C +A·C ≥ 0, since A is effective and K_{V} +τL is nef. On the other hand, if C ⊂ A, then (K_{V} +τL+A)·C = (K_{A}+τL_{A})·C ≥ 0, since (K_{A} + τL_{A}) is nef.

Therefore by Kleiman’s criterion [13] t(τL + (T − 1)(K_{V} + τL + A)) is an ample line bundle for all positive integers T and all positive integers t such that t(τL + (T − 1)(K_{V} + τL + A)) is a line bundle.

Let e denote the index of V, i.e., assume that e is the smallest positive integer such that eK_{V} is Cartier. By the Kawamata-Shokurov Basepoint-Free theorem we can choose a positive integer N′ such that N′/e and N′τ are integral and N′(K_{V} + τL) is spanned by global sections. Let e′ denote the index of A. Choose an integer N > 0 which is a positive multiple of N′ and such that N/e′ is integral and N(K_{A} + τL_{A}) = ϕ^{∗}H for a very ample line bundle H on Y. Note that N(K_{V} +τL) = K_{V} +τL+(N −1)(K_{V} +τL). Using the sequence

0 → K_{V} + τL + (N − 1)(K_{V} + τL + A) → N(K_{V} + τL + A) → N(K_{A} + τL_{A}) → 0 we see that Γ(N(K_{V} + τL + A)) → Γ(N(K_{A} + τL_{A})) is surjective, and thus that N(K_{V} + τL + A) is spanned in a neighborhood of A by global sections. Furthermore, since N(K_{V} + τL) is spanned, the bundle N(K_{V} + τL + A) = N(K_{V} + τL) + NA is spanned at all points of V − A. Thus N(K_{V} + τL + A) is spanned.

Replace N by a positive multiple N̂ such that the morphism associated to |N̂(K_{V} + τL + A)| has connected fibers and a normal image Y. The map \overline{ϕ} : V → Y associated to N̂(K_{V} + τL + A) extends ϕ as asserted.

To see the last assertion about F, note that |N̂(K_{V} + τL + A)| maps F to a point. Thus since N̂(K_{V} +τL) is spanned, and A∩F ≠ F, N̂(K_{V} +τL) is trivial on F. Q.E.D.

Corollary 4.2 Let L, V, A, Y, ϕ, τ be as in Theorem 4.1. Let B be any irreducible subvariety of V with Pic(B) = ℤ. If V is smooth and dim A − dim Y + dim B > dim V then either B ∩ A = ∅ or there exists a y ∈ Y with B ⊂ ϕ^{−1}(y). Proof. Let n := dim V. Suppose that B ∩ A ≠ ∅ and B ⊄ A. Then A_{B} is effective, and therefore, since Pic(B) = ℤ, it is ample. In addition, (K_{V} + τL)_{B} is nef; so from Kleiman’s criterion we see that (K_{V} + τL + A)_{B} is ample. Therefore the map \overline{ϕ} is finite to one on B. Let F′ be an irreducible component of a fiber of ϕ that meets A ∩ B. Observing that dim F′ ≥ dim A − dim Y we get

dim F ∩ B ≥ dim F′ + dim B − n ≥ dim A − dim Y + dim B − n > 0,

which contradicts the finite-to-oneness of \overline{ϕ}_{B}. Thus B ⊂ A. The same argument shows that in fact B ⊂ ϕ^{−1}(y) for some y ∈ Y. Q.E.D.

We remark that Corollary 4.2 includes the case when B = V.

We now apply Corollary 4.2 to study the structure of the morphism ψ : V → W associated to N(K_{V} + τL). First, we make the following general observation. Lemma 4.3 Let L be an ample line bundle on a projective variety V with at worst canonical singularities and index e. Suppose the bundle N(K_{V} + τL) is spanned and non-trivial for some rational τ > 0 and some positive integer N with Nτ and N/e integral. Let η be the induced morphism, and assume that the general fiber of η has positive dimension. Then dim V − dim η(V) ≥ τ − 1. Proof. Let F be a general fiber of η. Our assumptions about the singularities of V assure that F has canonical singularities of index e and that N(K_{F} +τL_{F}) ≅ O_{F}. Thus by Theorem 1.1 we have the assertion. Q.E.D.

We now come to the main result of this section. Theorem 4.4 Let L be an ample line bundle on an n-dimensional connected projective manifold, V. Let A be an irreducible divisor on V with at worst canonical singularities and with index e′. Assume that K_{A} is not nef and that τ is the nefvalue of (A, L_{A}) and ϕ : A → Y is the nefvalue morphism of (A, L_{A}). Further assume that K_{V} +τ′L is nef for some rational τ′ satisfying 0 ≤ τ′ ≤ τ. Let ψ : V → W be the morphism (which exists by the Kawamata-Shokurov Basepoint-Free Theorem) from V onto a normal projective variety W such that ψ^{∗}H ≅ N(K_{V} + τ′L) where H is ample and N is a positive integer such that Nτ′ is integral. Assume that τ′ ≥ (n + 3)/2. Then we can conclude the following.

1. If dim W < dim V, then A is the pullback under ψ of a divisor in W; moreover, ψ is of maximal rank in a neighborhood of a general fiber lying in A.

2. Assume that dim W = dim V. If the image under ϕ of the singularities of A is not all of Y, then for a general fiber f of ϕ : A → Y, it follows that Pic(f) = ℤ, A_{f} < 0, and K_{V} + τ′L is ample on f.

Proof. We first consider the case in which dim W < dim V. Applying Lemma 4.3 gives dim V − dim W ≥ τ′ − 1. Applying the same lemma to ϕ : A → Y, we get dim A − dim Y ≥ τ − 1 ≥ τ′ − 1.

Thus, if F is a general fiber of ψ, we have

dim A − dim Y + dim F ≥ 2(τ′ − 1) ≥ n + 1.

Moreover, since τ′ ≥ (n + 3)/2 > (dim F + 2)/2, Theorem 1.3 implies that Pic(F) = ℤ. Therefore, from Corollary 4.2 we get that either F ∩ A = ∅ or F ⊂ ϕ^{−1}(y) for a y ∈ Y. The latter alternative is clearly impossible if F is a general fiber of ψ and A is a divisor of V. Therefore, A_{F} is trivial; in other words, A is in the kernel of the restriction map Pic(V) → Pic(F).

Since dim W < dim V, the bundle N(K_{V} + τ′L) is spanned but not ample, and thus τ′ is the nefvalue of (V, L). In that case it is known from Theorem 1.2 that ψ is a Mori contraction, and, in particular, the sequence Pic(W) → Pic(V) → Pic(F) is exact. Therefore, the divisor A is in the image of Pic(W); i.e., it is the pullback under ψ of a divisor on W.

To see that ψ is of maximal rank in a neighborhood of a general fiber F lying in A, note that F is smooth since F is a general fiber of the morphism ψ_{A}. Since K_{F} = −τ′L_{F} < 0, we have H^{1}(F, O_{F}) = 0. Letting N_{F \A} (respectively, N_{F \V}) be the normal bundle of F in A (respectively, in V), we have the short exact sequence:

0 → N_{F \A} → N_{F \V} → [A]_{F} → 0.

Using the facts that [A]_{F} ≅ O_{F} ≅ N_{F \A}, and that H^{1}(F, O_{F}) = 0, we conclude that N_{F \V} = N_{F \A} ⊕ [A]_{F} = N_{F \A} ⊕ O_{F} , and therefore that N_{F \V} is also trivial. Using H^{1}(F, O_{F}) = 0, we have that H^{1}(F, N_{F \V}) = 0. It follows from deformation theory that around any point p ∈ F we have a neighborhood of the form U × V, where

1. p ∈ U ⊂ F and V is transverse to F with ψ giving a biholomorphism of V with an open neighborhood ψ(V) of ψ(p); and

2. under the biholomorphism V → ψ(V), ψ_{U×V} is identified to the product pro-

jection U × V → V. Therefore, ψ_{U×V} maps V one-to-one onto W, and hence has maximal rank at p.

We next consider the case when dim W = dim V. Let f be a general fiber of ϕ : A → Y. Note that if the image under ϕ of the singularities of A is not all of Y, then a general fiber f is smooth. Applying theorem 1.3 gives Pic(f) = ℤ.

Since the map ϕ is induced by N(K_{A} + τL_{A}), we have

(K_{A} + τL_{A})_{f} = (K_{V} + τL)_{f} + A_{f} = 0.

Moreover, since (K_{V} + τL)_{f} is nef, we have A_{f} ≤ 0.

We claim that, in fact, A_{f} < 0; for, if A_{f} = 0, then (K_{V} + τL)_{f} = 0 as well. Therefore τ = τ′ and the map ψ : V → W contracts f. Furthermore, the same argument we gave in the previous case shows that the normal bundle N_{f\A} of f in A is trivial, and, therefore the normal bundle N_{f\V} in V must also be trivial because [A]_{f} is trivial. Also, K_{f} < 0, so H^{1}(f, N_{f\V}) = 0. But then the fibers of ψ fill out an open set around f, and therefore dim W < dim V, contrary to hypothesis.

From A_{f} < 0 we get (K_{V} + τL)_{f} > 0, so that (K_{V} + τL)_{f} is ample as asserted. Q.E.D.

Many nonexistence results follow from the above results. We restrict ourselves to a few illustrative results, the first of which we need below. Theorem 4.5 Let L be an ample line bundle on an n-dimensional connected projective manifold V. Assume that K_{V} + (n − 1)L is nef and big and that n ≥ 4. Let A_{1}, …, A_{r} be r ≥ 2 distinct irreducible divisors whose union is connected, e.g., whose union is the reduction of an ample divisor. Assume that each (A_{i}, L_{A_{i}}) is either a scroll over a curve, a quadric, or (ℙ^{n−1}, O_{ℙ^{n−1}}(1)). Then n = 4 and all of the (A_{i}, L_{A_{i}}) are scrolls. Proof. First we assume that n ≥ 4 and that not all of the (A_{i}, L_{A_{i}}) are scrolls over a curve. After renaming if necessary we can assume that (A_{2}, L_{A_{2}}) is not a scroll. Let A_{1} (after possibly renaming) be an A_{i} distinct from A_{2} such that A_{1} ∩ A_{2} ≠ ∅. Let ϕ_{1} : A_{1} → Y be the nefvalue morphism of (A_{1}, L_{A_{1}}). By hypothesis the image of ϕ_{1} is either a point or a curve. Let \overline{ϕ_{1}} : V → Y be the extension of ϕ_{1} that exists by Theorem 4.1. Since Pic(A_{2}) = ℤ, Corollary 4.2 implies that A_{2} ⊂ ϕ_{1}^{−1}(ϕ_{1}(x)) for some x ∈ A_{1} ∩ A_{2}. This is absurd since A_{1} and A_{2} are distinct irreducible divisors.

Thus we have reduced to the situation when n ≥ 5 and all of the (A_{i}, L_{A_{i}}) are scrolls. Let ϕ_{i} : A_{i} → Y_{i} be the nefvalue morphism of (A_{i}, L_{A_{i}}). After renaming if necessary we can assume that A_{1} and A_{2} meet. Let x ∈ A_{1} ∩A_{2}. Let B denote the fiber ϕ_{2}^{−1}(ϕ_{2}(x)). Note that B ≅ ℙ^{n−2} and Pic(B) = ℤ. Let \overline{ϕ_{1}} : V → Y_{1} be the extension of ϕ_{1} that exists by Theorem 4.1. Note that B ⊂ ϕ_{1}^{−1}(ϕ_{1}(x)) by Corollary 4.2. From rigidity of proper maps we conclude that given any fiber F of the map ϕ_{2} we have that \overline{ϕ_{1}}(F) is a point. From Theorem 4.1 we see that K_{V} + (n − 1)L must be trivial when restricted to any fiber of ϕ_{2} not contained in A_{1}. But this implies that a Zariski open set of A_{2} is contained in a component of the exceptional set of the first adjunction map. This gives the absurdity that A_{2} ≅ ℙ^{n−1}. Q.E.D.

Here is the solution to Problem B of the Introduction for dimensions at least five. Corollary 4.6 Let L̂ be an ample and spanned line bundle on an n-dimensional connected projective manifold X̂ with n ≥ 5. Assume that there is a normal crossing divisor Â = Â_{1} + ⋯ + Â_{r} ∈ |L̂|. Assume that r ≥ 2. If for each i the genus of a curve section of (Â_{i}, L̂_{A_{i}}) equals h^{1}(O_{A_{i}}), then (X̂, L̂) is a scroll over a smooth curve, one component of Â meets each fiber in a hyperplane, and the other components are fibers. Proof. By [9, Theorem 11.7], Theorem 3.1, and Theorem 4.5, each (Â_{i}, L̂_{A_{i}}) is a scroll over a curve. Then since n > 4, Theorem 4.5 rules out the possibility that K_{X} +(n−1)L̂ could be nef and big.

Now suppose that K_{X} + (n − 1)L̂ is nef but not big. By theorem 3.2, (X̂, L̂) is either a Del Pezzo manifold as in Theorem 3.4, a quadric fibration over a smooth curve, or X̂ is a scroll over a normal surface. However, the Del Pezzo manifolds exhibited in Theorem 3.4 all have dim X̂ ≤ 4, so the first case does not occur. Likewise, the latter two cases are inconsistent with the (Â_{i}, L̂_{A_{i}})’s being scrolls over curves.

Hence we conclude that (X̂, L̂) is not nef. So by theorem 3.1, (X̂, L̂) is a scroll over a smooth curve, all the components of Â but one are fibers, and the remaining one meets each fiber in a hyperplane. Q.E.D.

We need the following useful lemma. Lemma 4.7 Let L̂ be a very ample line bundle on a four-dimensional connected projective manifold X̂. Let A and B be two smooth divisors on X̂ meeting transversely in a nonempty manifold h. Assume that (A, L̂_{A}) and (B, L̂_{B}) are scrolls over smooth curves A and B respectively with scroll projections a : A → A and b : B → B respectively. If K_{X} + 3L̂ is nef and big, then (a, b) maps h isomorphically onto A × B. Proof. Using the argument of Theorem 4.5 we see that if K_{X} + 3L̂ is nef and big then no fiber of a meets a fiber of b in a positive dimensional set. In particular a(h) = A and b(h) = B.

First note that (a, b) is one-to-one on h. If not there are two points that are identified by a and by b. Since L̂ is very ample and the fibers of a and b are linear, we conclude that the line ℓ between these two points is in the fibers of a and b. Thus ℓ ⊂ h. This contradicts the fact that no fiber of a can meet a fiber of b in a positive dimensional set. The same argument shows that no tangent vector of h can be mapped to zero by the differential of (a, b). Q.E.D.

Lemma 4.7 is very strong. It lets us use Theorems 3.10 and 3.11 to solve Problem A of the Introduction under the assumptions that r = 2 and n = 4. Corollary 4.8 Let L̂ be a very ample line bundle on a four-dimensional connected projective manifold X̂. Assume that there is a normal crossing divisor Â + B̂ ∈ |L̂|. If the genus of a curve section of (Â, L̂_{A}) and the genus of a curve section of (B̂, L̂_{B}) both equal zero, then

1. (X̂, L̂) is a scroll over ℙ^{1}; or

2. (X̂, L̂) = (ℙ^{2} × ℙ^{2}, O_{ℙ^{2}×ℙ^{2}}(1, 1)); or

3. (X̂, L̂) ≅ (ℙ^{(}V), ξ_{V}) where V ≅ O_{ℙ^{2}}(1) ⊕ O_{ℙ^{2}}(1) ⊕ O_{ℙ^{2}}(2) and (after possibly renaming) B̂ is equal to the inverse image of a conic in ℙ^{2} under the scroll projection, (Â, L̂_{A}) ≅ (ℙ^{2} × ℙ^{1}, O_{ℙ^{2}×ℙ^{1}}(1, 1)), and Â is the unique divisor ∈ |L̂ − B̂|); or

4. (X̂, L̂) is Mukai, i.e., −K_{X} = 2L̂. Proof. By Theorem 3.1 either K_{X} + 3L̂ is nef or else (X̂, L̂) is a scroll over a curve, which gives the first case of the theorem.

Thus we may assume that K_{X} + 3L̂ is nef. Now using Theorem 3.2, we see that K_{X} + 3L̂ is big except when (X̂, L̂) is a Del Pezzo manifold, a quadric fibration over a smooth curve, or a scroll over a smooth surface. In the Del Pezzo case, we have (X̂, L̂) = (ℙ^{2} × ℙ^{2}, O_{ℙ^{2}×ℙ^{2}}(1, 1)) by Theorem 3.4.

The case of a quadric fibration over a curve is not possible. To see this let π : X̂ → C be the quadric fibration morphism onto the smooth curve C. There is an ample line bundle H on C with K_{X} + 3L̂ ≅ π^{∗}H. Since the general fiber of π is irreducible, we see that after renaming Â is a fiber and B̂ meets the general fiber in a smooth quadric. Also since C is a curve, all of the fibers of π_{B} have equal dimension. Since the curve genus of (B̂, L_{B}) is 0 we have

−2 = (K_{X} + B̂ + 2L̂_{B}) · B̂ · L̂^{2} = (π^{∗}H − Â) · B̂ · L̂^{2}.

But π^{∗}H is numerically equivalent to tÂ for some t ≥ 1 and (Â ∩ B̂) · L̂^{2} = 2 so we have the contradiction that −2 = 2(t − 1) ≥ 0.

We now work out the case of a scroll π : X̂ → Y over a surface.

First note that both Â and B̂ are scrolls over curves. If not then after possibly renaming, Â would be ℙ^{3} or a three-dimensional quadric. In these cases Pic(Â) ≅ ℤ so π(Â) can only be three dimensional or a point, according to Corollary 4.2. Since dim Y = 2 we conclude that Â is contained in a fiber of π. This is impossible since π is a ℙ^{2}-bundle ([16, Theorem 3.3], [5, Chapter 12]), while dim Â = 3.

Thus Â and B̂ are scrolls over curves. Since the genera of curve sections of Â and B̂ are both zero, Â and B̂ are both scrolls over ℙ^{1}.

After renaming if necessary we conclude by Theorem 1.9 that B̂ = π^{−1}(C) for a smooth rational curve C, and Â meets a general fiber in a linear ℙ^{1}. Since L̂ is very ample and since the fibers of both π and the scroll projection q : Â → ℙ^{1} are linear ℙ^{2}’s, we see that a fiber of q is a section of π. Thus Y ≅ ℙ^{2}, and therefore

[B̂] ≅ π^{∗}O_{ℙ^{2}}(b) with b = 1 or b = 2.

Since (X̂, L̂) is a scroll over ℙ^{2} we have by definition that K_{X} + 3L̂ ≅ π^{∗}O_{ℙ^{2}}(c) with c ≥ 1. We claim that c = 1 and b = 2. To see this observe that a curve section of Â is rational so we have

−2 = (K_{A} + 2L̂_{A}) · L̂2 ; _{A} = (K_{X} + 3L̂ − B̂) · Â · L̂^{2} = π^{∗}O_{ℙ^{2}}(c − b) · Â · L̂^{2}.

From this we conclude that c − b < 0. Since c > 0 and b = 1 or 2 we see that c = 1 and b = 2.

Let V := π_{∗}L̂. Two distinct fibers of q : Â → ℙ^{1} give two disjoint sections of π therefore we obtain an exact sequence with x, y, z ∈ ℤ

0 → O_{ℙ^{2}}(x) → V → O_{ℙ^{2}}(y) ⊕ O_{ℙ^{2}}(z) → 0.

This sequence splits since the first cohomology of any line bundle on ℙ^{2} vanishes. Hence V is a direct sum of line bundles. Since K_{ℙ^{2}} + det V ≅ O_{ℙ^{2}}(b) ≅ O_{ℙ^{2}}(1), we conclude that det V ≅ O_{ℙ^{2}}(4). Thus V ≅ O_{ℙ^{2}}(1) ⊕ O_{ℙ^{2}}(1) ⊕ O_{ℙ^{2}}(2) with B equal to the inverse image of a conic in ℙ^{2} under π and with Â ∈ |L̂ − B̂|. Since

π_{∗}(L̂ − B̂) ≅ O_{ℙ^{2}}(−1) ⊕ O_{ℙ^{2}}(−1) ⊕ O_{ℙ^{2}},

we see that (Â, L̂_{A}) ≅ (ℙ^{2} × ℙ^{1}, O_{ℙ^{2}×ℙ^{1}}(1, 1)). From this explicit description existence follows.

Now assume that K_{X} + 3L̂ is nef and big. Then by Theorem 4.5, Â and B̂ are both scrolls over ℙ^{1}. Also, h = Â ∩ B̂ is isomorphic to ℙ^{1} × ℙ^{1} by Lemma 4.7.

A fiber of the map ℙ^{1} × ℙ^{1} → ℙ^{1} induced by the inclusion of h in Â is a smooth ℙ^{1} inside a fiber ℙ^{2} of Â → ℙ^{1}, hence is of degree 1 or 2. Likewise for the map from B̂. Then L̂_{h} is either O_{ℙ^{1}×ℙ^{1}}(1, 1), O_{ℙ^{1}×ℙ^{1}}(2, 1), O_{ℙ^{1}×ℙ^{1}}(1, 2), or O_{ℙ^{1}×ℙ^{1}}(2, 2) and we have g(h) ≤ 1.

If the genus of a curve section of h is zero then, by Theorem 3.10, either Â or B̂ is a fiber of the first reduction map or (X̂, L̂) = (ℙ^{4}, O_{ℙ^{4}}(2)). But the first case would contradict Â and B̂’s both being scrolls. The second case (X̂, L̂) = (ℙ^{4}, O_{ℙ^{4}}(2)) is also not possible. Indeed we must have Â, B̂ ∈ |O_{ℙ^{4}}(1)|. In this case Â ≅ ℙ^{3}, L̂_{A} ≅ O_{ℙ^{3}}(2), and the genus of a curve section of (ℙ^{3}, O_{ℙ^{3}}(2)) is one and not zero.

Otherwise the sectional genus is one. According to Theorem 3.11, this implies that the first reduction (X, L) of (X̂, L̂) exists with K_{X} + 2L nef and the first reduction map a biholomorphism in a neighborhood of Â ∩ B̂. Note that X̂ = X. To see this suppose there were an exceptional fiber E for the first reduction map. Then E meets Â + B̂, so we may assume without loss of generality that it meets Â. Since Â is a scroll π : Â → C over a curve we see that E would have to contain at least one curve of a fiber of π. This would force Â to equal E, in contradiction to Â’s being a scroll. Thus K_{X} + 2L̂ is nef.

Let ν : X̂ → Y denote the morphism with connected fibers onto a normal projective variety Y such that K_{X} + 2L̂ ≅ ν^{∗}H for an ample line bundle H on Y. Since 0 = (K_{X} + 2L̂) · Â · B̂ · L̂, we conclude that ν maps Â ∩ B̂ to a point. Since Â ∩ B̂ contains curves in fibers of the scroll maps of Â and B̂ we conclude that ν maps Â + B̂ to a point. Thus ν maps X̂ to a point. Thus −K_{X} = 2L̂. Q.E.D.

The solution of the following general problem would complement for reducible divisors the results we know for irreducible ample divisors [5, Chapter 6] and [3].

Problem C. Let L̂ be a very ample line bundle on a connected ndimensional projective manifold X̂. Classify (X̂, L̂) when there exists a divisor Â = Â_{1} + ⋯ + Â_{r} ∈ |L̂| with all Â_{i} degenerate, e.g., with κ(K_{A_{i}} + (dim Â_{i} − k)L̂_{A_{i}}) < dim Â_{i} if n ≥ k + 1, where k is either 1, 2, or 3.

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K. Chandler kchandle@noether.math.nd.edu

A. Howard howard.1@nd.edu

A.J. Sommese sommese@nd.edu

URL: www.nd.edu/∼sommese

Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556, U.S.A. fax: U.S.A. + 219–631–6579