CHERN INEQUALITIES AND SPANNEDNESS OF THE ADJOINT

BUNDLE^{∗}

Mauro C. Beltrametti  Michael Schneider

Andrew J. Sommese

1 Introduction

Let E be a vector bundle of rank r on an irreducible projective variety X, with Chern classes c_{1}(E), …, c_{r}(E). A weighted homogeneous polynomial, P (c_{1}, …, c_{r}), of weighted degree m in variables c_{1}, …, c_{r} of weights 1, …, r is said to be positive if P (c_{1}(E), …, c_{r}(E)) · Z > 0 for any irreducible mdimensional subvariety Z of X.

A fundamental result of Fulton and Lazarsfeld (see [FL], [F] and §1) is that for ample E the set of positive polynomials coincides with the set of sums with non-negative coefficients (not all zero) of Schur polynomials. It is an intriguing question what the actual lower bounds are for appropriately positive bundles.

Our interest in this question arose from our work in [BSS]. There we used the criteria of Ein-Lazarsfeld [EL] for spannedness of adjoint bundles on smooth 3-folds to study the spannedness of a variety of adjoint bundles, including bundles of the form K_{X} ⊗ det E. To use the Ein-Lazarsfeld criteria it was necessary to prove lower bounds for certain expressions in the Chern classes of ample and spanned vector bundles on smooth 3-folds. Ballico ([Ba1] and [Ba2]) makes the very nice conjecture that

∗to Professor Hirzebruch on his 65th birthday

c_{i_{1}}(E) ⋯ c_{i_{k}}(E) ≥ \binom{r}{i_{1}} ⋯ \binom{r}{i_{k}}

for 0 ≤ i_{j} ≤ r, i_{1} + ⋯ + i_{k} = n

with equality ha ppening only when X ≅ IP^{n} and E ≅ ⨁_{i=1}^{r} O_{IP}^{n}(1) In this paper we formulate a conjecture for lower bounds of positive polynomials in Chern classes which includes the inequality part of Ballico’s conjecture.

Conjecture Let X be an n-dimensional smooth projective variety and let E be an ample and spanned vector bundle on X. Then for any positive polynomial P of weight n in the Chern classes of E we have

P (c_{1}(E), …, c_{n}(E)) ≥ P (r, \binom{r}{2}, …, \binom{r}{n}) .

We prove our conjecture for very ample E, i.e. vector bundles such that the tautological bundle ξ_{E} = O_{IP(E)}(1) on IP(E) is very ample.

Theorem Let X be an n-dimensional smooth projective variety and let E be a rank r very ample vector bundle on X. Let P ∈ ZZ[c_{1}(E), ⋯, c_{r}(E)] be a positive polynomial of weight n in the Chern classes of E. Then we have the inequality

P (c_{1}(E), …, c_{n}(E)) ≥ P (r, \binom{r}{2}, …, \binom{r}{n}) .

The proof would work for vector bundles E which are only ample and spanned if we could prove the natural conjecture that given an ample and spanned vector bundle E on an irreducible projective variety X, there is at least one fiber F of IP(E) → X such that π^{∗}ξ_{E} −π^{−1}(F) is nef where π : ĨP(E) → IP(E) is the blow up of IP(E) along F.

Examples based on an example constructed in [BSS] show that Ballico’s conjecture can fail for ample bundles which are not spanned.

Since the first lower bounds that we obtained in [BSS] for certain polynomials in the Chern classes of vector bundles E on 3-folds X arose in the investigation of when K_{X} ⊗ det(E) was spanned, it is interesting to us what we can say about the spannedness of K_{X} ⊗ det(E) for general smooth X using the information obtained in the course of studying our conjectured lower bound. Using this new information and some work [BS] on varieties with many linear subspaces, we show that given a very ample vector bundle E, then K_{X} ⊗ det(E) is spanned when rank(E) ≥ dim X + 1. We also show that if this fails for rank(E) = dim X, then the lower bound in the conjecture of Ballico is taken on. The proof of this result leads to a spannedness result for adjoint bundles on varieties with many linear subspaces, e.g. scrolls, quadric fibrations, and non-degenerate projective submanifolds of IP_{n} with dual varieties of codimension at least dim X/3 + 1.

Throughout the paper we use standard notation in algebraic geometry. Let us recall the following. For any sheaf F on a complex variety X, we denote by h^{i}(X, F) the complex dimension of the vector space H^{i}(X, F). For a line bundle L we denote by |L| the complete linear system associated to L and by Γ(L) the space of the global sections of L. We denote by ” ≈ ” the linear equivalence of line bundles. As usual, line bundles and divisors are used with little (or no) distinction. Hence we shall freely switch from the multiplicative to the additive notation and vice versa.

Let us recall a few definitions we use. Let L be a line bundle on a ndimensional variety, X. We say that L is numerically effective, or nef for short, if L · C ≥ 0 for all effective curves C on X, and in this case L is said to be big if c_{1}(L)^{n} > 0 where c_{1}(L) is the first Chern class of L. We say that L is spanned if it is spanned by Γ(L) at all points of X. A vector bundle E is said to be numerically effective, or nef for short (respectively ample; respectively very ample) if the tautological bundle ξ = O_{IP(E)}(1) on IP(E) is nef (respectively ample; respectively very ample).

Let p ∈ ZZ[X_{1}, …, X_{r}] be a homogeneous polynomial in r variables. The weight of a monomial X_{1}^{γ_{1}} ⋯ X_{s}^{γ_{s}} is defined to be γ_{1} + 2γ_{2} + ⋯ + sγ_{s}.

Let E be a vector bundle of rank r on a variety X. Let λ = (λ_{1}, …, λ_{k}) be a partition of an integer m in integers ≤ r:

r ≥ λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k} ≥ 0, ∑_{i=1}^{k} λ_{i} = m.

We define the Schur polynomial S_{λ}(E) ∈ ZZ[c_{1}(E), …, c_{r}(E)] in the Chern classes of E of weight m as the m × m determinant

S_{λ}(E) = det(c_{λ_{i}+j−i}(E))_{1≤i,j≤m}.

Examples include c_{m}(E) for λ = (m, 0, …, 0) and s_{m}(E) for λ = (1, …, 1). Here s_{m} is the m-th Segre class, i.e. the m-th component of the total Segre class

s(E) = c(E^{∗})^{−1}.

Note that, since S_{λ}(E) has weight m, if a Chern class c_{i}(E) occurs in a monomial of S_{λ}(E), then i ≤ m. Moreover, by convention, c_{0}(E) = 1, c_{i}(E) = 0 for either i < 0 or i > r. A theorem of Fulton-Lazarsfeld says that for E ample S_{λ}(E) is a positive polynomial, i.e. S_{λ}(E)·Z > 0 for all Z ⊂ X integral subschemes of dimension m (see [F], 12.1.7).

Let X be a smooth connected n-dimensional variety and let L be a very ample line bundle on X. We say that (X, L) is a scroll (respectively a quadric fibration) over a variety Y of dimension m if there exists a surjective morphism with connected fibers φ : X → Y such that K_{X} + (n − m + 1)L ≈ φ^{∗}L (respectively K_{X} + (n − m)L ≈ φ^{∗}L) for an ample line bundle L on Y. We say that (X, L) is a IP^{d}-bundle over Y if there is a surjective morphism φ : X → Y such that each fiber F is isomorphic to IP^{d} and L_{F} ≅ O_{IP}_{d}(1). Let D be the discriminant locus of (X, L), i.e.

D := {H ∈ |L| , H singular}.

D is irreducible and the codimension, k+1, of D in IP(Γ(L)) depends only on (X, L). The defect, def(X, L), of the pair (X, L) is defined to be this integer k (see e.g. [BFS]).

We would like to thank the University of Bayreuth and the DFG-Schwer-punktprogramm “Komplexe Mannigfaltigkeiten” for making our work possible. The first and the third author would also like to thank the Max-Planck-Institut für Mathematik in Bonn for its support during the 1992–93 academic year. The third author also thanks the National Science Foundation (DMS 93–02121) for their support.

2 Inequalities for Chern classes.

First, let us recall the following well known lemma due to Castelnuovo and Mumford [Mu]. In fact we need only a very special case of it.

Lemma 1 Let L be a very ample line bundle on an irreducible projective variety X. Let Y ⊂ X be an irreducible subvariety of degree d relative to L, i.e. d = L^{dim Y} · Y. If either Y is smooth or Y ⊂ reg(X) and codim_{X}Y = 1, then L^{d} ⊗ I_{Y} is spanned by global sections, where I_{Y} denotes the ideal sheaf of Y in X.

Proof. See [Mu] or [BS]. We will apply the lemma only in the case that Y is a linear subspace of X with respect to L. In this case the assertion follows immediately from the Koszul resolution of I_{Y}

0 ⟶ L^{−k} ⟶ ⋯ ⟶ (L^{−1})^{⊕k} ⟶ I_{Y} ⟶ 0.

Here k denotes the codimension of Y in X. Exactly the same proof works also in case Y is a complete intersection of higher degree. Q.E.D.

The following consequence of the Lemma above was the starting observation for the paper

Corollary 1 Let E be a rank r very ample vector bundle on an irreducible projective variety X. Let us denote by π : X̃ → X the blow up of X at a smooth point x. Let E := π^{−1}(x) denote the exceptional divisor. Then π^{∗}E ⊗ I_{E} is a spanned vector bundle on X̃, where I_{E} denotes the ideal sheaf of E in X̃.

Proof. We have a commutative square

IP(E) ⟵^{π̃} IP(π^{∗}E)
p ↓  ↓ p̃
X ⟵^{π} X̃

where p, p̃ are the bundle projections and π̃ is the blow up of IP(E) along the fiber F := p^{−1}(x). Let Ẽ := π̃^{−1}(F) and let ξ be the tautological bundle O_{IP(E)}(1) of IP(E). Then ξ̃ := π̃^{∗}ξ is the tautological bundle O_{IP(π^{∗}E)}(1) on IP(π^{∗}E). Since F is a linear IP^{r−1} relative to ξ, by Lemma 1 we conclude that ξ ⊗ I_{F} is spanned, I_{F} denoting the ideal sheaf of F in IP(E). Since ξ̃ ⊗ I_{E} ≅ π̃^{∗}(ξ ⊗ I_{F}), we conclude that ξ̃ ⊗ I_{E} is spanned. It thus follows that p̃_{∗}(ξ̃⊗ I_{E}) is spanned. Since

p̃_{∗}(ξ̃⊗ I_{E}) ≅ p̃_{∗}(ξ̃⊗ p̃^{∗}O(−E)) ≅ p̃_{∗}ξ̃⊗ I_{E} ≅ π^{∗}E ⊗ I_{E}

we are done. Q.E.D.

The following general fact is the second step to prove the main Theorem below.

Proposition 1 Let X be an n-dimensional irreducible projective variety and let E be a vector bundle of rank r on Y. Assume that there exists a smooth point x_{0} ∈ X such that π^{∗}E ⊗ O(−E) is nef where π : X̃ → X is the blowing up of X at x_{0} and E = π^{−1}(x_{0}) denotes the exceptional divisor. Let P ∈ ZZ[c_{1}(E), …, c_{r}(E)] be a positive polynomial of weight n in the Chern classes of E. Let λ = (λ_{1}, …, λ_{k}) be a partition of n, r ≥ λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k}, ∑_{i=1}^{k} λ_{i} = n. Then we have the inequality

P (c_{1}(E), c_{2}(E), …, c_{n}(E)) ≥ P (r, \binom{r}{2}, …, \binom{r}{n}) .

Proof. By taking a resolution of singularities we may assume that X is smooth. Since P = ∑_{λ} a_{λ}S_{λ}, it is enough to show that

S_{λ}(E) = S_{λ}(c_{1}(E), …, c_{n}(E)) ≥ S_{λ} (r, \binom{r}{2}, …, \binom{r}{n}) ,

where S_{λ}(E) is the Schur polynomial corresponding to the partition λ =

(λ_{1}, …, λ_{n}). Let F := π^{∗}E ⊗ O_{X}(−E). We have for k > 0,

c_{k}(F) = ∑_{j=0}^{k} \binom{r − j}{k − j}c_{j}(π^{∗}E) · c_{1}(−E)^{k−j}. (1)

Since c_{j}(π^{∗}E) · c_{1}(−E)^{i} = 0 (2)

for j > 0, i > 0, we get for k > 0,

c_{k}(F) = c_{k}(π^{∗}E) + \binom{r}{k}c_{1}(−E)^{k}. (3)

By definition we have

S_{λ} (c_{1}(F), …, c_{n}(F)) = det ( \begin{matrix}c_{λ_{1}}(F) & ⋯ & c_{λ_{1}+n−1}(F) \\ ⋮ & & ⋮ \\ c_{λ_{n}−n+1}(F) & ⋯ & c_{λ_{n}}(F) \end{matrix} )

Inserting (3) into the determinant and using equation (2) for j > 0, i > 0, we find

S_{λ}(c_{1}(F), …, c_{n}(F)) =

det ( \begin{matrix}c_{λ_{1}}(π ∗ E) & ⋯ & c_{λ_{1}+n−1}(π^{∗}E) \\ ⋮ & & ⋮ \\ c_{λ_{n}−n+1}(π^{∗}E) & ⋯ & c_{λ_{n}}(π^{∗}E) \end{matrix} )

+(−E)^{n} det ( \begin{matrix}\binom{r}{λ_{1}} & ⋯ & \binom{r}{λ_{1}+n−1} \\ ⋮ & & ⋮ \\ \binom{r}{λ_{n}−n+1} & ⋯ & \binom{r}{λ_{n}} \end{matrix} )

Since S_{λ}(c_{1}(π^{∗}E), …, c_{n}(π^{∗}E)) = S_{λ}(c_{1}(E), …, c_{n}(E)) and using (−E)^{n} = (−1)^{n}(E)^{n} = (−1)^{n}(−1)^{n−1} = −1, we finally get

S_{λ}(c_{1}(F), …, c_{n}(F) = S_{λ}(c_{1}(E), …, c_{n}(E)) − S_{λ} (r, \binom{r}{2}, …, \binom{r}{n}) .

Since F is a nef vector bundle we have by [DPS], (2.5) that

S_{λ}(c_{1}(F), …, c_{n}(F)) ≥ 0.

This concludes the proof. Q.E.D.

Theorem 1 Let X be an n-dimensional smooth projective variety and let E be a rank r very ample vector bundle on X. Let P ∈ ZZ[c_{1}(E), ⋯, c_{r}(E)] be a positive polynomial of weight n in the Chern classes of E. Then we have the inequality

P (c_{1}(E), …, c_{n}(E)) ≥ P (r, \binom{r}{2}, …, \binom{r}{n}) .

Proof. This follows from Corollary 1 and Proposition 1. Q.E.D.

Corollary 2 Let X be an n-dimensional smooth projective variety and let E be a rank r very ample vector bundle on X. Then

(1)

c_{i_{1}}(E) ⋯ c_{i_{k}}(E) ≥ \binom{r}{i_{1}} ⋯ \binom{r}{i_{k}}

for 0 ≤ i_{j} ≤ r, i_{1} + ⋯ + i_{k} = n

(2)

c_{1}(E)^{n} ≥ r^{n}

(3)

c_{n}(E) ≥ \binom{r}{n}

(4)

s_{i_{1}}(E) ⋯ s_{i_{k}}(E) ≥ \binom{r + i_{1} − 1}{i_{1}} · · · · · · \binom{r + i_{k} − 1}{i_{k}}

for 0 ≤ i_{j} ≤ r, i_{1} + ⋯ + i_{k} = n

(5)

s_{n}(E) ≥ \binom{r + n − 1}{n}.

Proof. To prove (1), apply Theorem 1 with the weight n positive polynomial P = c_{i_{1}}(E) ⋯ c_{i_{k}}(E), 0 ≤ i_{j} ≤ r, i_{1} + ⋯ + i_{k} = n. Then (2) and (3) follow from (1) by taking i_{1} = ⋯ = i_{n} = 1 and i_{1} = ⋯ = i_{n−1} = 0, i_{n} = n respectively. To show (5) we use first Theorem 1 to get the estimate

s_{n}(E) ≥ det ( \begin{matrix}\binom{r}{1} & \binom{r}{2} & ⋯ & \binom{r}{n} \\ \binom{r}{0} & \binom{r}{1} & ⋯ & \binom{r}{n−1} \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & \binom{r}{1} \end{matrix} ) .

But the value of this determinant is \binom{r+n−1}{n}, a fact well known in classical algebraic geometry, c.f. [S] or [ACGH], pp. 94–95. Finally (4) follows in a similar way as (5). Q.E.D.

The inequality (1) in Corollary 2 was conjectured by E. Ballico in [Ba1] in the more general case that E is ample and spanned. He proved this conjecture in the special case n = 2, r = 3.

It would be interesting to study the boundary cases (see also [Ba 2] for the case n = 2, r = 3).

Question 1 Let E be an ample and spanned vector bundle of rank r on a projective variety X of dimension n. If there is equality

S_{λ}(c_{1}(E), …, c_{n}(E)) = S_{λ} (r, \binom{r}{2}, …, \binom{r}{n})

for some partition λ of n, does this imply X ≅ IP^{n}, E ≅ O_{IP}^{n}(1)^{⊕r}?

Note that c_{1}(E)^{n} = r^{n} implies equality for all S_{λ}. Let us explicitly note that Theorem 1 is false for a merely ample vector bundle E. The example below was actually constructed in [BSS], Ex.13

in another context.

Example 1 Let C be a smooth curve of genus ≥ 2. From [BSS], Lemma 12, we know that there exists for r ≥ 1 an ample, stable, rank r vector bundle E on C with det(E) ≅ O_{C}(x) where x ∈ C. Let X := IP^{1} × C with product projections p : X → IP^{1}, q : X → C. Let E := p^{∗}O_{IP}^{1}(1) ⊗ q^{∗}E. Then E is ample, not spanned for r ≥ 2 (since E is not spanned) and c_{1}(E) is a sum of vertical IP^{1′s} plus r vertical C^{′s}. Therefore c_{1}^{2}(E) = 2r. Hence for r ≥ 3 we cannot have c_{1}^{2}(E) ≥ r^{2}, showing that the inequalities (4) of Corollary 2 are not true.

The above results suggest the following

Conjecture 1 Let X be an n-dimensional smooth projective variety and let E be an ample and spanned vector bundle on X. Then for any positive polynomial P of weight n in the Chern classes of E we have

P (c_{1}(E), …, c_{n}(E)) ≥ P (r, \binom{r}{2}, …, \binom{r}{n}) .

3 Spannedness of adjoint bundles

In this section we prove the existence of global sections of adjunction bundles that span the adjunction bundles on linear subspaces.

Proposition 2 Let L be a very ample line bundle on an n-dimensional, normal, Cohen-Macaulay projective variety X. Assume that the locus, Irr(X), of nonrational singularities of X is at most 0-dimensional. Given a kdimensional linear subspace P ⊂ reg(X), it follows that the restriction map

Γ(K_{X} ⊗ L^{t}) → Γ((K_{X} ⊗ L^{t})_{P}) → 0

is onto if t ≥ n − k + 1. If the rational map associated to |L − P | has an n-dimensional image, then the conclusion holds with t ≥ n − k.

Proof. It is enough to show that H^{1}(X, K_{X} ⊗ L^{t} ⊗ I_{P}) = 0 for t ≥ n − k + 1, where I_{P} is the ideal sheaf

defining P in X.

Let ψ : X̃ → X be the blow up of X along P. Then

K_{X} ≅ ψ^{∗}K_{X} + (n − k − 1)P,

where P := ψ^{−1}(P). Since P is a linear IP^{k} relative to L, Lemma 1 applies to say that L ⊗ I_{P} is spanned by global sections and hence it follows that L := ψ^{∗}L − P is spanned by global sections. Thus by the Kawamata - Viehweg vanishing theorem we get

H^{1}(X̃, K_{X} ⊗ L^{s} ⊗ ψ^{∗}L^{t−s}) = 0

for all t ≥ s ≥ 1 if L is big, and for all t > s ≥ 0 in general. Therefore by the Leray spectral sequence and the projection formula, we conclude that

H^{1}(X, K_{X} ⊗ L^{t} ⊗ ψ_{∗}((n − k − 1 − s)P)) = 0

for all t ≥ s ≥ 1 if L is big, and for all t > s ≥ 0 in general. Note that ψ_{∗}O_{X}(−P) = I_{P}. Then by choosing s = n − k we are done. Q.E.D.

Remark 1 Letting a be such that det N_{P}^{X} ≅ O_{IP}_{k}(a) in Proposition 2, we see that global sections of Γ(K_{X} ⊗ L^{t}) span K_{X} ⊗ L^{t} in a neighborhood of P if t ≥ max{k + 1 + a, n − k + 1}. In fact

(K_{X} + tL)_{P} ≅ K_{P} − det N_{P}^{X} + tL_{P} ≅ O_{IP}_{k}(−k − 1 − a + t)

is spanned if t ≥ k + 1 + a, so that sections lift by Proposition 2 as soon as in addition t ≥ n − k + 1.

If the rational map associated to |L − P | has an n-dimensional image, then we get the spannedness of K_{X} ⊗ L^{t} with t ≥ max {k + 1 +

a, n − k}.

Let us discuss now some applications of the above results.

Corollary 3 Let X be an n-dimensional smooth connected variety. Let L be a very ample line bundle on X. Assume that (X, L) is a scroll, p : X → Y, over a variety Y of dimension m. Let k = n − m. Then K_{X} + (k + 1)L is spanned by global sections if k ≥ n/2. The same is true if (X, L) is a IP^{k}-bundle.

Proof. If (X, L) is a IP^{k}-bundle, then X is covered by linear IP^{k}’s, so we conclude by Proposition 2 and Remark 1 that K_{X} + (k + 1)L is spanned if k + 1 ≥ n − k + 1 or k ≥ n/2.

Let (X, L) be a scroll. Fix a point x ∈ X. We can choose a sequence F_{i} of general fibers of p such that the point x is contained in the limit of the F_{i}’s. Let P be the limit of the F_{i}’s in the Hilbert scheme of the F_{i}’s. The fibers F_{i} are linear IP^{k}’s, so P is a linear IP^{k} and det N_{P}^{X} ≅ O_{P}. Thus Proposition 2 and Remark 1 apply again to give the result. Q.E.D.

Corollary 4 Let E be a very ample vector bundle on a smooth projective manifold X, of dimension n. Then K_{X} ⊗ det E is spanned by global sections if rank (E) ≥ n + 1, or if rank(E) = n and c_{1}(E)^{n} > rank (E)^{n}.

Proof. Let ξ = O_{IP(E)}(1) be the tautological bundle of IP(E). Let r = rank(E). Then IP(E) is covered by linear IP^{r−1}’s , such that N\begin{matrix}IP & (E) \\ IP & ^{r−1} \end{matrix} ≅ O_{IP}^{r−1}. Let r ≥ n + 1. By Proposition 2 and Remark 1 applied to (IP(E), ξ) and P = IP^{r−1}, it follows that K_{IP(E)} ⊗ ξ^{r} is spanned by global sections if r ≥ dim IP(E) − (r − 1) = n + 1. Let p : IP(E) → X be the bundle projection. Since

K_{IP(E)} ⊗ ξ^{r} ≅ p^{∗}(K_{X} ⊗ det (E)),

the first part of the statement is proved.

Assume now r = n. Let π : X̃ → X be the blow up of X at a point x ∈ X. Let I_{P} be the ideal sheaf of P := π^{−1}(x) in X̃. By Corollary 1 we know that π^{∗}E ⊗ I_{P} is spanned and therefore det(π^{∗}(E) ⊗ I_{P}) ≅ π^{∗} det (E) − nP is spanned. By Leray’s spectral sequence and the projection formula we have

H^{1}(X̃, K_{X} ⊗ π^{∗} det (E) ⊗ O(−nP)) ≅ H^{1}(X̃, π^{∗}K_{X} ⊗ π^{∗} det (E) ⊗ O(−P))

≅ H^{1}(X, K_{X} ⊗ det (E) ⊗ M_{x}),

where M_{x} is the ideal sheaf of x in X. Thus by the Kawamata-Viehweg vanishing theorem, we conclude that

H^{1}(X, K_{X} ⊗ det (E) ⊗ M_{x}) = 0 (4)

as soon as π^{∗} det(E)−nP is nef and big. By the above, this is the case unless (π^{∗} det(E) − nP)^{n} = 0, which is equivalent to c_{1}(E)^{n} = r^{n}. From (4) follows that the restriction map

Γ(X, K_{X} ⊗ det (E)) → (K_{X} ⊗ det (E))_{x} ≅ CI

is onto and hence K_{X} ⊗ det (E) is spanned in a neighborhood of x. Since this is true for all x ∈ X, K_{X} ⊗ det (E) is spanned. Q.E.D.

Corollary 5 Let L be a very ample line bundle on a projective manifold X, of dimension n. Let φ : X → Y be a quadric fibration. Let m = dim Y, f = n − m. Then K_{X} ⊗ L^{n−m} is spanned for f ≥ \frac{2n}{3} + 1.

Proof. Let F be a general fiber of φ. Then either dim F = f = 2k or dim F = f = 2k + 1 and in both cases F contains a linear IP^{k}. We consider first the case f = 2k. By the adjunction formula we get

det N_{IP^{k}}^{X} ≅ K_{IP}_{k} − K_{X|IP^{k}} ≅ K_{IP}_{k} − K_{F |IP^{k}} ≅ O_{IP}_{k}(k − 1).

Fix a point x ∈ X. We can choose a sequence of such IP^{k}’s, contained in general fibers of φ, such that the point x is contained in the limit P of the IP^{k}’s in the Hilbert scheme of the IP^{k}’s. Then det N_{P}^{X} ≅ O_{P} (k − 1). By Proposition 2 and Remark 1 we know that K_{Y} + fL is spanned if f = 2k ≥ max{n−k +1, k −1}, which is equivalent to 2k ≥ n−k +1, or k ≥ (n+1)/3, that is f ≥ 2(n − 1)/3.

In the case f = 2k+1 we have det N_{IP^{k}}^{X} ≅ O_{IP}_{k}(k) and exactly the same argument as above shows that K_{X} +fL is spanned if k ≥ n/3 or f ≥ (2n/3)+1.

Q.E.D.

For the definition of extremal ray and contraction of an extremal ray occurring in the Corollary below, we refer to [Mo].

Corollary 6 Let L be a very ample line bundle on a smooth projective manifold X, of dimension n. If the defect def (X, L) := k of (X, L) is k ≥ n/3, then K_{X} ⊗L^{\frac{n+k}{2} +1} is spanned by global sections and defines a morphism which is a contraction of an extremal ray.

Proof. Since k ≥ 1 it is a classical fact that there is a linear IP^{k} passing through a general point of X and K_{X|IP^{k}} ≅ O_{IP}_{k}((n − k − 2)/2) or det N_{IP^{k}}^{X} ≅ O_{IP}_{k}((n−k)/2) (see [BFS], (0.3), [E], (2.4)). By a standard limit argument, as in the proof of Corollary 5, we see that there is a linear IP^{k} through each point of X and the determinant of the normal bundle of this IP^{k} is O_{IP}_{k}((n−k)/2)). Thus Proposition 2 and Remark 1 apply to say that K_{X} ⊗ L^{\frac{n+k}{2} +1} is spanned by global sections if ((n + k)/2) + 1 ≥ n − k + 1, or k ≥ n/3.

The last part of the statement follows now from [BFS] (0.12). Q.E.D.

References

[ACGH] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves. Springer-Verlag 1985

[Ba 1] Ballico, E.: Spanned and ample vector bundles with low Chern numbers. Pacific Journal of Mathematics 140, 2 (1989), 209-216

[Ba 2] Ballico, E.: On ample and spanned rank-3 bundles with low Chern numbers. Manuscripta Math. 68, (1990), 9-16

[BS] Beltrametti, M.C., Sommese, A.J.: Some effects of the spectral values on reductions. To appear in Proceedings of 1992 L’Aquila Algebraic Geometry Conference, Contemporary Math.

[BFS] Beltrametti, M.C., Fania, M.L., Sommese, A.J.: On the discriminant variety of a projective manifold. Forum Math. 4 (1992), 529-547 [BSS] Beltrametti, M.C., Schneider, M., Sommese, A.J.: Applications of

the Ein-Lazarsfeld criterion for spannedness of adjoint bundles. To appear in Math. Z.

[DPS] Demailly, J.P., Peternell, T., Schneider, M.: Compact complex manifolds with effective tangent bundles. To appear in J. Alg. Geom.

[E] Ein, L.: Varieties with small dual varieties, I. Invent. Math. 96 (1986), 63-74

[EL] Ein, L., Lazarsfeld, R.: Global generation of pluricanonical and adjoint linear series of smooth projective threefolds. 1992 preprint

[F] Fulton, W.: Intersection theory. Ergebnisse der Math., 2, Springer-Verlag (1984)

[FL] Fulton, W., Lazarsfeld, R.: Positive polynomials for ample vector bundles. Ann. of Math. 118 (1983), 35 - 60

[Mo] Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. of Math. 116 (1982), 133-176

[Mu] Mumford, D.: Varieties defined by quadratic equations. CIME course 1969, in: Questions on algebraic varieties, Rome (1970), 30 - 100

[S] Segre, C.: Gli ordini delle varietà che annullano i determinanti dei diversi gradi estratti da una data matrice. Atti R.Accad. Lincei Ser V, vol. 9 (1900), 253-260

M.C. Beltrametti  M. Schneider
Dipartimento di Mathematica  Department of Mathematics
Via L.B. Alberti 4  University of Bayreuth
I-16132 Genova  D-95440 Bayreuth
Italy  Germany

A.J. Sommese Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556 U.S.A.