manuscripta math. 76, 407- 420 (1992) manuscripta mathematica �9 Springer-Verlag 1992

ON T H E k -SPANNEDNESS OF T H E A D J O I N T L I N E B U N D L E

Edoardo Ballico and Manro Beltrametti

I n t roduc t i on . Let L be a line bundle on a projective variety, X, of dimension n >_ 2. Fix a subspace W of H~ L) and let k be a non-negative integer. W is said to be k-spanned if the restriction map W ~ F ( O j ( L ) ) is surjective for arty curvilinear 0-dimensional subscheme (J, Os) of X (see [BFS], [BS1]). W is said to be k-very ample if the restriction map W ----4 r ( O j ( L ) ) is onto for any 0-dimensional subscheme (J, O j) of X (see [BS2], [BS3], [CG]). We say that L is k-spanned or k-very ample if H~ L) is k-spanned or k-very ample respectively. Note that for k = 0, L is 0-spanned if and only if L is 0-very ample, if and only if

L is spanned by F(L). For k = 1, L is 1-spanned if and only if L is 1-very ample, if and only if L is very ample. Note also that 2-spannedness is equivalent to 2-very

ampleness ([BFS]). In [B1] the following definition which interpolates between the notion of

k-spannedness and k-very ampleness is proposed. W is said to be (k, a)-spanned if the restriction map W ~ r(Oj(L)) is onto for any 0-dimensional subscheme (J, Oj ) with length(Oj) = k + 1 and dimT=J < a for any x E Jted. Here T r J denotes the tangent space of J at x. We say that L is (k, a)-spanned if H~ L) is (k, a)-spanned. For instance, (k, 1)-spannedness is equivalent to k-spannedness and, if X is smooth, (k, n)-spannedness is equivalent to k-very ampleness.

In this paper we will consider the case when X is a manifold and W =

H~ It is natural to ask the following question, which is classical in the cases k = 0,1

(see [SV]). Let L be a line bundle on X satisfying one of the previous definitions. Then study the preservation of this property under a~ljunction, i.e. describe the pairs (X, 15) (if any) where the adjoint bundle K x + (n - 1)L does not satisfy the

same property. In this direction we prove the following result (Theorem (2.2)). Let (X, 15) be

as above with 15 (k, 2)-spanned and L" > 4k + 5, n > 3. Then K x + (n - 1)15 is (k, 1)-spanned unless for an explicit list of few exceptions. The k = 2 analogous result was previously proved in [Bl]. We get the result by combining the methods of [B1] with the results of [BS3], which was concerned with the n = 2 case.

The first author wants to thank M.Andreatta and G.Bolondi for many helpful

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BALLICO - BELTRAMETTI

discussions. Both the authors want to thank E.Carletti for his help to solve tech-

nical typing problems. Both the authors want to thank Miss Cinzia Matri for her

nice typing.

0. B a c k g r o u n d ma te r i a l

(0.0) N o t a t i o n . We work over the complex field C. By variety we mean an

irreducible and reduced projective scheme, V. We denote its structure sheaf by

Or . Basically we use the standard notation from algebraic geometry. Let us only

fix the following (here s denotes a line bundle on a variety V):

IEI, the complete linear system associated to E;

F(L), the space of the global sections of s we say that /2 is spanned if it is spanned at all points of V by F(L);

~-, (respectively ,-,), the linear (respectively numerical) equivalence of divisors; V [r] = the Hilbert scheme of all 0-dimensional schemes (J, O j) of V with

length(O j) = r.

Let F. be a line bundle on X. s is said to be numerically effective (he f, for

short) if s C _> 0 for every curve C on X and in this case s is said to be big if s

Abuses . Line bundles and divisors are used with little (or no) distinction. Hence we shall freely switch from the multiplicative to the additive notation and

viceversa. Sometimes the symbol " �9 " of intersection of cycles is understood.

(0.1) (k, a ) -spannedness . Fix non-negative integers k, a. Let L be a line bundle

on a variety V and let W be a subspaee of H~ Following [BI], (A.1), the

subspace W is said to be (k, a)-spanned if the restriction map W ~ F(Oj(L)) is

onto for every 0-cycle (J, Oj ) E V [k+ll such that d i m T j < a for every v E Jrea,

where T~J denotes the tangent space of J at v. We say that L is (k, a)-spanned if

H~ L ) i s (k, a ) - s p ~ e d .

A 0-cycle (J, O j ) is called curvilinear if dimTvJ < 1 for every v E Jred. The line bundle L is said to be k-spanned if F(L) , F(Oj(L)) is onto for any

curvilinear 0-cycle J E V [k+q (see [BFS], [BS1]) and L is called k-veryample if

r(L) - -~ r(Os(L)) is onto for every 0-cycle J E V [k+ll (see [BS2], [BS3], [CG]).

Then (k, 1)-spannedness is equivalent to k-spannedness and, if V is smooth,

(k, dimV)-spannedness is equivalent to k-very ampleness.

Note also that (2,1)-spannedness is equivalent to 2-very ampleness by [BFS], (0.4) and (3.1).

(0.1.1). If L is (k, 1)-spanned on Y then L- C _> k for every effective curve C, on

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V, with equality if and only if C ~ P ' (see [BS1], (0.5.1) and also [BS3], (1.3.1)).

(0.1.2). Let V, L, W C H~ L) be as in (0.1). Assume that W is (k, a)-spanned.

Let (J, O j ) be a 0-cycle with length(O j ) = m < k + 1 mad dimT'xJ < a for every

x E Jred. Look at V embedded in pN by W and let < J > be the linear span of J

in pN. We use over and over through the paper the fact that < J > ~ pro-1. This

follows from the definition of (k, a)-sparmedness.

(0 .2) k - r e d u c t i o n . Let L be a very ample line bundle on a manifold V. We refer

to [s3] and [SV] for the usual notion of r duction, (V', Z') of the p ar (V, L). Let us recall the definition of k-reduction from [BS3], (0.4). Let L be an ample line bundle

on a smooth projective surface S. Assume that k K s + L is nef for some k > 0.

Note that f. := ( k - 1 ) K s + L is ample. Let E be a maximal set of disjoint, smooth

rational curves, s C S, which satisfy s s = - 1 a n d / : . s = 1. Let re : S ---* S ' be

the reduction morphism with connected fibers obtained by contracting each curve,

s to a point. Note that S t is a smooth projective surface, and rr : S * S ' is

the blowing up at a finite set, B, of distinct points satisfying r r - l (B) = E. Both

L' := (re.L)** and /: ' := (rr./:)** are ample line bundles since they are ample

outside of a finite set of points. We say that the pair (S t, L t) is a k-reduction of

(S,L). Let x be a point in B and/~ = re-l(x). Then s ~ = - 1 , - K s " s = f-.. s = 1.

Hence in particular L . s = k. We say that s is a ( - 1 ) k-line relative to L.

(0 .3) Spec i a l su r faces . Let L be a k-very ample line bundle on a smooth pro-

jective surface s and assume that f. := (k - 1)Ks + L is ample. Assume tha t

(S , / : ) is a conic bundle over a smooth curve C, i.e. that there exists a morphism,

~a : S ~ C, with connected fibers such that K s 4- 1:- .~ ~*H for some ample line

bundle H on C. For any fiber f of ~ we have s f = 2 and hence L . f = 2k. We

say that (S, L) is a k-conic bundle, or that S is a k-conic bundle relative to L.

In this case any reducible fiber, f , of ~ consists of 2 distinct, irreducible compo-

nents f l , f 2 such that f l ' f2 = 1. Recalling (0.1.1) and since L . f = 2k, we have

L . f l = L" f2 = k and f l -- f2 ~ P1. By Fr, with r > 0, we denote the r-th Hirzebruch surface. Fr is the unique

holomorhic p1 bundle over p1 with a section, E, satisfying E . E = - r . In the case

r = 0, ~'0 is simply p1 x p1. In the case r >_ 1, E is the unique irreducible curve

on Fr with negative self-intersection.

(0 .4) A few results from Mori's theory of extremal rays will be used in the paper.

We will use freely the notion of extremal ray, extremal rational curve, as well as

the basic Cone Theorem. We refer the reader to [M2].

The length, length(R), of an extremal ray, R, on a smooth variety V, is defined

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a s

length(R) := m J n { - K v . C , C rational curve whose numerical class belongs to R).

Let us recall the following result we need, due to Badescu. This theorem can be also proved by using adjunction theoretic methods.

(0.5) T h e o r e m (Badescu [Bdl], [Bd2]). Let X be a smooth projective threefold

and L an ample line bundle on X . Assume that there ezistz a smooth A G ILl such

that A is a pa bundle, p : A J C, over a smooth curve C of genus g(C).

(0.5.1) I f g(C) > O, then X is a p2 bundle, ~o : X , C, over C such that the

restriction of ~o to A coincides with p and K x + 3L ~ 99"H for some ample line

bundle H on C. Hence in particular LF ~- Op2(1) for each fiber F = p2 of ~o, so

that L . f = 1 for each fiber f of p. (0.5.2) y g ( c ) = o then p e~tend~ to a p2 bundle ~ : X , C as in (0.5.1) unle,~ either

a) A = F0,Lro ~ OF0(1,t),t _> 2, with L f ~ Ol( t ) where f is a fiber of

p : F o , C ;

b) A = Fo, Lr, -~ Or0(1,1), (X, L) -~ (Q, Oq(1)), Q hyperquadric in P ; or

c) A = F o , ( X , L ) ~- ( P , Cb,(2)). For any f~ther backgronnd material we refer to [BS~], [BS2], [SS3], [CG] and

[~ll.

1. A s c e n t resul ts .

Let X be a projective manifold of dimension n > 3 and let L be a k-spanned

(i.e. (k, 1)-spanned) line bundle on X, k >__ 2. Let S be a codimension n - 2 linear

section of (X, L), i.e. S is a smooth surface obtained as transversal intersection of

n - 2 general members of ILl. Let L s be the restriction of L to S. In this section

we deal with the following question. Assume that L" > 4k + 5 and that K s + L s

is not k-very ample on S. Then describe (X, L).

First we need the following result. In the case k = 2 it is contained in [B1],

(A.6.1).

(1.1) L e m m a . Let X be a projective manifold of dimension n > 3 and let L be a

k-spanned line bundle on X , k >_ 2. Let S be a codimension n - 2 linear section of

(X ,L ) . Assume that S containJ a ( - 1 ) k-line relative to L. Then either

(1.1.1) (X, L) ~ (Q, Oq(2)), Q smooth hyperquadric in P~, k = 2;

(1.1.2) n = 3 ,X is a F 2 bundle over a smooth curve and the restriction o f t to

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each fiber is Or,(2), k = 2; (1.1.3) (X,L) ~ (P~,Op,(2)), k = 2; (1.1.4) ( X , L ) ~- (Pa ,Or , (3)) ,k = 3.

P r o o f . Let g be a ( - 1 ) k-line relative to Ls on S. By the genus formula, K s . g = - 1

and hence

( K x + (n - 1)L). e = (ICs + L s ) " g = k - 1,

o r

(1.1.5) - K x . g = k ( n - 1 ) - ( k - 1).

Since L is k-spanned and L. t = k, the cycle represented by t cannot deform into

the sum of two effective 1-dimensional cycles. Then by the breaking Lemma of

Mori (see [M1], Theorem 4) we know that n + 1 _> - K x . e and hence by (1.1.5)

n + 1 >_ k ( n - 1 ) - ( k - 1 ) _ 2 ( n - 1 ) - 1.

This gives the following possible cases

n=3, k=2,3,

O F

n = 4 , k = 2 .

(1.1.6) C la im. There exists an extremal rational curve, C, on X such that

length(R) = - K x �9 g where R is the extremal ray defined by C.

P r o o f . (Compare with [B1], p. 60). We prove the result for n = k = 3. In this

case - K x �9 g = 4 by (1.1.5). In the remaining cases the proof runs similarly.

First, note that if ~ is an extremal rational curve then the Claim is true.

Indeed in this case - K x �9 t. <_ - K x �9 C for each curve C E R , where R is the

extremal ray defined by L Hence length(R) = - K x �9 ~.

Thus we can assume that g is not an extremal rational curve. Then by the

Cone Theorem,

t ~ a i r1 + ... + crhrh + flxsl + ... + f l , , s , , + 71h + ... + 7 , G + A ,

where ri, si, t i are extremal rational curves such that K x �9 ri = - 1 , K x �9 si = - 2 ,

K x �9 ti = - 3 , A is in the closure of the cone of effective curves with K x �9 A >. O,

h, m, u are non-negative integers and ai , fli, 7i are positive real numbers. Since

L . t = 3 and, by 3-spannedness, L . ri > 3, L . si >_ 3, L . ii >_ 3, we have that

h m

+ Z, + _< 1 i = 1 i----I i----1

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BALLICO - B E L T R A M E T T I

and therefore

7, < 3 i=1 i=1 i=1 i=1 i=1

tha t is - ~ / h = l a i - 2 ~ i ~ 1 / 3 i - 3 ~ i ~ 1 7i > - 3 . This contradicts K x .g = - 4 and completes the proof of the Claim. �9

Let n = 3, k = 3. Then, by (1.1.5), - K x . g = 4. Therefore by [W], (2.6) we

conclude tha t X ~ p3 and hence L ~ Op3(3) since L is 3-spanned. We are in case

(la.4). Let n = 3, k = 2. In this ease - K x �9 g = 3. By combining (2.4.2) and (2.6)

of [W] we conclude tha t X is ei ther a smooth hyperquadric, Q, in p4, and hence

L ~ Or (2) since L is 2-spanned, or there exists a surjective morphism ~o : X ~ C,

onto a smooth curve C whose fibers are irreducible and reduced and whose general

fiber F is a smooth surface with - K F ample and P ie (F) -~ Z. Thus F ~ p2. By

flatness we have that the res t r i c t ion , Lz , of L to any fiber, Z, of ~o has degree 4

a n d , by semicontinuity, h~ >__ 6. It thus follows tha t Z is embedded by F ( L z )

as a minimal surface of degree 4 in ~ . Since L is 2-spanned, by looking over the list

of surfaces of minimal degree we conclude tha t (Z, L z ) ~- (p2, Op, (2)). Therefore

each fiber of ~o is isomorphic to p2 and the restriction of L to a fiber is Op, (2)

since L is 2-spanned. We are ei ther in case (1.1.1) or (1.1.2).

Let n = 4, k = 2. From (1.1.5) we get - K x . g = 5. By using again [WI, (2.6) we

conclude tha t (X ,L) ~- ( P , Op,(2)) as in (1.1.3). Q.E.D.

We can answer now to the question posed at the beginning of this section.

(1 .2) P r o p o s i t i o n . Le~ X be a projective manifold of dimension n > 3 and let

L be a k.spanned line bundle on X , k > 2, such that L" > 4k + 5. Let S be a

codirnennion n - 2 linear section of (X, L). Assume that Ls is k-very ample and

that K s + Ls is not k-very ample. Then ( X , L ) is as in one of the four exceptional cases listed in (1.1).

P r o o f . T h e proof consists of several steps, by a case by case analysis of all

possible cases listed in [BS3]. In the following we denote by V the smooth 3-fold

ob ta ined as transversal intersection of n - 3 general members of ILl. So (S, L s ) is

a very ample divisor in (V, Lr Fur thermore , if S is a pl bundle, E , jr denote a

section of minimal serf-intersection and a fiber of the ruling respectively.

Step I. Assume that kKs-I-Ls is not nef and big. From [BS3], (4.1) we "know that

either:

I1) S = P2 ,Ls -~ Ch,,(a), k < a < k + 2. 12) S = p1 x Pa,Ls ~- Oplxpl (a , b) and either k < a < k + 1 or k < b < k + 1.

I3 ) S = Fa,Ls . . . aE+bjr with either a = k , k + l o r k + 2 < a < 2k and b = a+k.

I4 ) S is a p l bundle, p : S ~ C, over a smooth curve C of invariant e, Ls "~ aE-b bf with either a = k ,k + 1 or C is an elliptic curve, E 2 = - e = 1 and

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BALLICO - B E L T R A M E T T I

Ls ~ (k + 2)E. I5) S is a k-conic bundle relative to Ls, which is not a pa bundle.

I6) S is a Del Pezzo surface with Ls ~ -kKs ,K2s >_ 2.

Case I1). It is well known that (S, Ls) ~- (P2,Op~(a)) with a > 2 does not extend

as very ample divisor in (V, Lv ) with V smooth (see [Bdl], [S1]).

Case/2) . Thereom (0.5) applies to say that we are in one of the cases a), b), c) of

(0.5.2). Case a) is excluded since Lro ~- OF0(1, ~) is not k-spanned, k :> 2. In case

b) we can find a line g C Q such that L .g = OQ(1).~ = 1 and we contradict again

the k-sparmedness assumption. In case c) we have k = 2 and L 3 = 8 < 4k + 5,

contradicting our degree assumption.

Cases I3), I4). We have either Ls" f = a > k > 2, L s . f = a > k + l _ > 3, or

Ls" f = a :> k + 2 _> 4. In all this cases Theorem (0.5) applies again to say tha t S

does not extend as a very ample divisor in V (see also [Bd3] for the S = Fa case).

Case I5). Let 7~ : S ~ C be the k-conic bundle map. Recall that for every

fiber f of ~, L �9 f = 2k. By looking over the list of smooth threefolds carrying a

(non-minimal) ruled surface as very ample divisor (see [I], Theorem II and [$2],

(2.4)) we see that either:

i) V is a ~ - " bundle over a smooth variety of dimension m = 1,2 and the

restriction Lps-,, of Lv to each fiber F is OF(2);

ii) there exists a morphism tb : V ~ C such that the general fibers F are smooth

quadrics Q C F3 ,LF ~ OQ(1);

iii) (V, Lv) is a Del Pezzo variety, i.e. K v ~ -2Lv; iv) (V, Lv) ~- (P3, Op,(1)) or (V, Lv ) ~- (Q, OQ(1)), Q hyperquadric in F4;

v) there exists a reduction (Y ' ,Lv , ) of (V, Lv) and either (Y ' ,Lv , ) is as in iv) or

V I is a F '~ bundle over a smooth curve and the restriction, Lp2, of Lv to each fiber

is op,(2). In both the cases i), ii) the restriction LF ~ OF(1) is not k-spanned for

k >_ 2, hence they are excluded.

In case iii) we have

- 2 L . f = K v . f = K s . f - L . f = 2 - L . f

and therefore L �9 f = 2. This leads to the contradiction k = 1.

Case iv) is clearly excluded since ~ ( 1 ) , OQ(1) are not k-spanned for k _> 2.

In case v) we have V ~- V'. Indeed otherwise there exists a line relative to

L, tha t is a smooth rational curve, g, with L . g = 1. This contradicts again the

k-spannedness assumption. Thus V is a p2 bundle, p : V ~ I/, over a smooth

curve Y and the restriction of L to each fiber is Op2(2). If n = 3, then V = X

and we are in case (1.1.2) of Lemma (1.1). If n > 4, we know by the Extension

Theorem (see [$1], Prop. III) that p extends to a holomorphic map r : W -----* Y

where W is the smooth 4-fold obtained as transversal intersection of n - 4 general

members of ILl. Let F be a general fiber of r Then p2 = V N F = L '~-3 �9 F so

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that (P2,Lp2) extends as very ample divisor in (F, LF). Since Lp2 ~ Op2(2) and F is smooth this is not possible (see again [Bdl], [$1]).

Case I6). The adjunction formula K s ~ K x l s + (n - 2)Ls gives

Hence

Ls ~ - k K s .~ - k K x l s - k(n - 2)Ls.

- k K x l s ~ (1 + k(n - 2))Ls.

From Lefschetz's theorem we know that Pic(X) injects in Pic(S) and therefore

- k K x .~ (1 + k(n - 2))L.

Since (k, 1 + k(n - 2)) = 1, it thus follows that K x ,,~ - (1 + k(n - 2))M and L ~, k M for some ample line bundle M in X. Hence in particular X is a Fano

n-fold. Let s be an extremal rational curve on X. Then

n + l > _ - K x . s 1 4 - k ( n - 2 ) .

This implies n > k(n - 2) > 2(n - 2), that is n <

Let n = 4. Then k = 2, K x ..~ - b M , L .~ 2M.

(see [XO]), ( X , M ) " ~ ( ~ , ~ , ( 1 ) ) , so that ( X , L )

case (1.1.3) of Lemma (1.1). Let n = 3. Then k = 2,3. If k = 3, K x

4~

By the Kobayashi-Ochial result

(~,Op~(2)) and we are as in

- 4 M , L ,,~ 3M, while, for k = 2, K x ,~, - 3 M , L ~ 2M. By the Kobayashi-Ochiai result we have (X, L)

(p3, Op8(3)) in the former case and (X, L) = (Q, OQ(2)), Q hyperquadric in l ~ , in

the latter case. We are as in (1.1.4) or (1.1.1) of Lemma (1.1).

Step II . Assume that k K s 4- L is nef but not ample. Then there exists a

k-reduction ~r : S ---4 S' (see (0.2)). In view of Lemma (1.1), either we are

done or S -~ S'. In this case Proposition (4.2) of [BS3] applies to say that S is

a l ~1 bundle, p : S -----4 C, over a smooth curve C. Since L is k-spanned we have

L s . f = L. f > k > 2 for any fiber f ofp. Thus Theorem (0.5) applies to say that S -~ F0 and we are in one of the cases a), b), c) of (0.5.2). But this is not possible

as shown in Case I2) of Step I.

Step I I I . Assume that k K s 4- L is ample. In this remaining case Theorem

(4.4) of [BS3] applies to say that S is a Del Pezzo surface with K~ = 1 and

Ls ~, - ( k 4- 2)Ks. Exactly the same argument as in Case I6) of Step I shows that

- ( k 4- 2 )Kx ~ ((k 4- 2)(n - 2) 4- 1)L.

Since (k + 2, (k + 2)(n - 2) + 1) = 1 it thus follows that

K x ,.~ - ( ( k 4- 2)(n - 2) 4- 1)M and L ~ (k -4- 2)M

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for some ample line bundle M on X. Hence in particular X is ~ Fano n-fold. Let

g be an extremal rational curve on X. Then

n + 1 >_ - K x ' e = ((k + 2 ) ( n - 2) + 1 ) M . e > (k + 2 ) ( n - 2) + 1.

Therefore n > (k + 2)(n - 2) and hence

3(k + 1) < (k + z)n < 2(k + 2)

which leads to the contradiction k < Z. This completes the proof of the Proposition.

Q.E.D.

2. T h e m a i n r e su l t

Let X be a projective manifold of dimension n >_ 3 and let L be a k-spanned line bundle on X, k > 2. In this section we show that, if L n > 4k + 5, then the

adjoint line bundle K x + ( n - 1)L is (k, 1)-spanned, i.e. k-spanned, except for

an explicit list of few exceptions. Compare with [B1], Theorem (A.6) where it is

essentially proved the k = 2 analogous of this result (indeed in [Bl], (A.6), L is

assumed to be 2-very ample, i.e. (2, n)-spanned). See also [BB] where a related

but weaker result is proved.

First, we need the following technical tool.

(2.1) L e m m a . Let X be a projective manifold of dirnen~iou n > 3 and let L be a

(k, 2)-spanned line bundle on X , k >_ 2. Let (J, Os) be a O-dimensional cycle with

length(O j ) = k + 1 and dimTxJ _< 1 for any x E .)'red. Let < J > be the linear

,pan of < J > in P(r(n)). Then dim (T~X Cl < J >) _< 2.

P r o o f . Assume dim(TzXM < J >) := s _> 3. Hence X N < J > contains a

O-dimensional subscheme X with Area = {X}, length(O.v) = s + 1 and

~ X ~ X N < J > ~ P " =

(i.e. X is the first infinitesimal neighborhood of z in YzX N < J >). Since J is

curvilinear there exists a 0-cycle J ' , J ' C J, such that length(O j,) = k and x E Y, , j , and in fact x E J~ed" Let < > be the linear span of J'. Then < J ' > -~ pk-z by

the assumptions on L, so < J ' > is a hyperplane in < J > and we have

d i m ( ~ X n < J ' >) > s - 1 > 2.

Hence in particular, by the choice of ,-l", there exists a sub-cycle Z of ,t' with

Z C X Cl < j t >, Zred = {X} and dimTzZ = 2. We have dimTzJ' <_ dimTz J < 1,

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so that in particular TzJ ~ C T z Z and length(Oj, uz) > k + 1. Note that we can

choose a 0-cycle, 32, with y C J ' U Z and length(Oy) = k + 1 (see e.g. [BS], (1.2)).

Since 32 C J ' t.J Z C < j t > ~ pk-1 we have < 3; > C pk-1.

If dimT~J t = 0, then J' is reduced at x and hence dimT~(J' O Z) = 2. Then

the conditions dimT~32 _< dimTz(J ' O Z) = 2 and < 32 > C F 'k-1 contradict the (k, 2)-spannedness assumption.

If dimTz J~ = 1 the 0-cycle Z above can be chosen in such a way that Z contains

the tangent vector, r~, of length 2 with supp(rx) C TzJ' . Therefore rz C T , Z and

hence dim'Tz(J' U Z) = 2. As above we contradict again the (k, 2)-spannedness

assumption. Q.E.D.

The following key result reduces the problem to the surfaces case (compare

with [B1], (A.3)).

(2.2) L e m m a (key-Lemma). Let X be a projective manifold of dimension n > 3

and let L be a (k,2)-spanned line bundle on X with k >_ 2. Let (J, O j ) be a

O-dimensional cycle with length(Od) = k + 1 and dim:TzJ <_ 1 for any x E Jred.

Then ~here exi,tz a ~mooih surface, S, obtained as transversal inter, ection of n - 2

general members of ]L[ such that J C S.

P r o o f . We consider X embedded in ptr by F(L). Let < J > be the linear span of

J in P(F(L)). Recall that < J > ~ pk C FN since L is (k, 2)-spanned (see (0.1.2)).

Assume that dim(< J > N X) >_ 1 and let C be an integral curve contained

in < J > N X. Note that by k-spannedness every k + 1 distinct points of C span

a pk, tha t is span < J > ~ pk. Fhrthermore degC _< k, since otherwise we would

have a curvilinear 0-cycle, 7, of length > k which spans a pk-1, i.e. 7 is the

general hyperplane section of C in < J > . Thus C is a smooth rational curve of

< J > ~ F 'k of degree k.

Note also that (< J > gl X)red = C. Indeed by the above we have the inclusion

C C (< J > N X)red. Assume that there exists a point z E (< J > N X)red-C. Let

H be a general hyperplane of < J > passing through

Then {z, xi, . . . , xk} is a curvilinear 0-cycle of length k

This is not possible by k-spannedness.

We claim that < J > n X contains C with

otherwise and let H = pk-1 be a general hyperplane

{x l , . . . , x k } , x i E C, i = 1, . . . ,k and X N H = Jl O...

z. Let {xl , . . . ,xk) = H n C .

+ 1 which spans H = pk- l .

multiplicity one. Assume

o f < d > . T h e n C n H =

t3 dk where di is a O-cycle

such that (Ji)r,d = {xi},i = 1, . . . ,k and length(Oj~) > 1 for at least one index

i, say i=1. Therefore we can take in J1 a vector rl = C[e]/e 2 of length 2 such

that Z = {rl,x~, ...,zk} is a curvilinear 0-cycle with length(Oz) = k + 1 and

Z C H = pk-1. This is not possible by the assumption on L.

Thus there exists a (reduced) finite set B, B C C, such that < J > N X

is reduced and coincides with C on C - B. Let H be a general hyperplane of

p N H D < J > . We want to show that the section H N X is smooth outside of

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BALLICO - BELTRAMETTI

B. Let x E C - B . Note that A := {H,H hyperplane in P~V,H D < J > ) is a

p g - ~ - x . Note also that x E C - B if and only if < J > AT~X - T~C that is

< < J >,T,~X > : = fl = p ,+ ~ -L Therefore

13 : = {H,H hyperplane in PN,H D < J >,H D T=X)

= {H,H hyperplane in pN,H D gl)

is a pN-( ,+k-a ) . Since n > 2, we have dimB < dimA and hence the general

hyperplane H in pN with < J > C H does not contain TxX and therefore the intersection H gl X is smooth at x, for any x E C - B (i.e. take H E .A - / 3 ) .

Fix now x E B. Then x E Jred. From Lemma (2.1) we know that

dim(:TxX n < J >) _< 2. It thus follows that the general hyperplane H in pN, with

H D < J >, does not contain ~ X . Indeed otherwise we would have TzX C < J >

and hence dim('/'=X n < J >) = dimTxX _> 3. Since B is finite, the general

hyperplane H in p g with H D < J > does not contain TxX for each x E B.

Therefore the intersection H gl X is smooth along B.

Thus by the above we conclude that for a general hyperplane H in p/v, with

H D < J >, the intersection H R X is smooth along (< J > n X)red = C. Moreover

by Bertini's theorem H 91X is smooth outside of < J > . Therefore by taking the

transversal intersection of n - 2 general members of ILl we find a smooth surface,

S, such that J C S and the Lemma is proved in the case dim(< J > n X ) > 1.

Thus we can assume that dim(< J > ['IX) = 0. Then either the assertion of

the Lemma is true or there exists a point z E (< J > n X)red such that the general

hyperplane H C pJv, with H D < J >, has a singular intersection H F1X at x, i.e.

contains ~ X . This can happen if and only if ~ X C < J > �9 Then in this case

< J > n X has a connected component, X, with Xred = {x) and

dimT~A' = dimCTxX n < J >) = dimTzX > 3.

If z E ,/red we contradict Lemma (2.1). Then we can assume that z ~ Jred. It is

always possible to choose a 0-cycle J ' , J' C J, with length(O j ,) = k (see e.g. [BS],

(1.2)). Again < J ' > ~ pk-a by the assumption on L. Let y := {x} U J'. Then

length(Oy) = k + 1. Note that, since Y is reduced at z and j t C J with J curvilin-

ear, the 0-cycle Y is curvilinear too. Note also that by construction, and since x E

< J >, we have y C < J > and therefore < y >=<: J > ~ pk. Hence in particular

< y > FIX(= < J > R X ) has a connected component, X, with r = {;g} and

dim:TzX = dim(T=X f'l < J >) >_ 2. Since x E Yr,a, Lemma (2.1) applies again to

give a contradiction. Q.E.D.

The following is the main result of the paper.

(2.3) Theorem. Let X be a projective manifold of dimension n ~ 3 and let L be

a (k, 2)-spanned line bundle on X. Assurae that L" >_ 4k + 5. Then K x + (n - 1)L

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BALLICO - BELTRAMETTI

is (k, 1)-spanned unless either:

(2.3.1) ( x , L) ~ (Q, 0r Q ~mooth hyperqu,~dric in P , k = 2; (2.3.2) n = 3,X is a F2 bundle over a smoo~h curve and ~he resgriction of L ~o

eaehfiber is Or2(2),k = 2; (2.3.3) (X, L) -~ (ps, Ors(3)), k = 3;

(2.3.4) ( X , L ) "~ ( ~ , O r , ( 2 ) ) , k = 2.

P roo f . First note that K x + (n - 1)L is spanned, i.e. (k - 1, 1)-spanned with

k = 1. This easily follows by checking the lists of exceptions: see e.g. Theorem

(0.1) in [SV]. Indeed, in each of such exceptions, L is merely very ample, not (k,2)-spanned with k > 2. Thus, by induction on k, we can assume that

K x + (n - 1)L is (k - 1,1)-spanned. If Kx -t- (n - 1)L is not (k, 1)-spanned

there exists a curvilinear 0-cycle, J, of S with length(O j) = k + 1 such that the

restriction map

F ( K x + (n - 1)L) ~ r ( O j ( K x + (n - 1)L))

is not onto. From Lemma (2.1) we know that there exists a smooth surface S

given by transversal intersection of n - 2 general members of ILl, with J C S. Let L s be the restriction of L to S. Thus by noting that

r (C) j (Kx + (n - 1)L)) ~- r ( o j ( K s + Ls)) ~- C k+l

and the restriction map F ( K x + (n - 1)L) ---4 F ( K s A- Ls ) is surjective by the

Kodaira vanishing theorem, we see that the restriction map from F(Ks + Ls) to

F ( O j ( K s A- Ls)) cannot be onto. This means that K s + L s is not (k, 1)-spanned and hence in particular is not k-very ample on S. On the other hand L s is k-very

ample on S, i.e. (k, 2)-spanned on S, since L is (k, 2)-spanned on X. Thus Proposi-

tion (1.2) applies to give the result. Q.E.D.

(2.3.5) R e m a r k . Let the notation be as in (2.3). Note that the adjoint line

bundle K x + ( n - 1)L is not (k, 1)-spanned in the listed exceptional cases. In cases

(2.3.1), (2.3.4), g x + (n - 1)L ~ Ox(1) is merely very aznple. In case (2.3.3), K x + (n - 1)L ~ ~ , ( 2 ) is (2,1)-spanned, not (3,1)-spanned. In case (2.3.2) the

restriction of K x + (n - 1)L to a fiber F 2 is ~,2(1) and hence K x + (n - 1)L is

not (2,1)-sparmed.

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BALLICO - BELTRAMETTI

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[BB]

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[I]

[KO]

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E.Ballico Dipartimento di Matematica Universits di Trento 38050 Povo (TN), Italy

M.Beltrametti Dipaxtimento di Matematica Universits di Genova Via L.B.Alberti 4, 16132 Genova, Italy

(Received March 9, 1992)

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