Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. XXXV, 125-133 (1989)

On the Multiples of the Adjoint Divisor to an Ample Divisor on a d-Fold.

MICHELE ROSSI (*)

Introduct ion .

This paper is meant to generalize a result by A. J. Sommese ([S1], Corollary II-A). More precisely, let X be a complex manifold of dimension d - 3 , polarized by an ample divisor H. We prove that

(I) there exists a polynomial q such that

q( n ) = dimcH~ Ox( n( Kx + ( d - 2)H)))

for every positive integer n, Kx standing for a canonical divisor of X.

In case d = 3, this fact was firstly shown by Sommese in [$1] under the further assumption that the complete linear system IH[ contains a smooth surface. Moreover, this result is also known under some different extents of generalities (see [$2] and [FS], w 2).

A sistematic use of Mori's theory, instead of the deformation theory used by Sommese, is the technical instrument enabling us to drop his assumption on IH[.

This paper ends by studying the case when q - 0 , which is deeply connected with known results by Fujita [F2], Ionescu [I], Beltrametti and Palleschi [BP1].

1. - N o t a t i o n and background .

Throughout this paper we shall work on the complex numbers field C. An irreducible reduced projective scheme of dimension d will be called a variety. By projective d-fold we shall mean a nonsingular complex variety.

(*) Author address: Dipartimento di Matematica dell'Universita, via C. Saldini 50, 20133 Milano.

126 MICHELE ROSSI

Let X be a d-fold; as usual, Kx will stand for a canonical divisor of X and the symbols ,,=-,,, ,~,~ and ,,(.),, will denote the linear equivalence of divisors, the numerical equivalence of cycles and the intersection number of cycles, respectively.

An r-cycle Z o fX (r = 1 or r = d - 1) is said to be nef(numerical effective) if (Z. V) _> 0 for every effective (d - r)-cycle V, and a divisor D of X is said to be big if D ~ > O.

Let us list some more notation:

hq(D) = hq(Ox(D)) ~ dimcHq(X, Ox(D)), q = O, 1, ..., d.

d

z(D) = ~(Ox(D)) ~ ~, ( - 1)qhq(D), q-O

Nr(X) ~J ((r-cycles of X}/ ,) | r= 1 or r = d - 1.

Furthermore, by NE(X) we shall mean the closure with respect to the real topology of the convex cone NE(X) generated in NI(X) by effective 1- cycles. We shall put p(X) for the Picard number of X, which is the dimension of N1(X) (see [K], IV, w 1).

Finally I D] will denote the complete linear system defined by a divisor D of X.

If H is an ample divisor on X we shall say that (X, H) is a polarized pair and that A(X, H) ~ d + H a - h~ is its A-genus.

A smooth, connected, projective subvariety E of X such that E ~ / ~ - 1 is said to be a ( - 1)-hyperplane (with respect to H) if Ox(H) | O~ ~- O~-~(1) and Ox(E) | O~ ~ O~-l(- 1). If d-> 3 there is only a finite number of ( - 1)- hyperplanes and they are pairwise disjoint. We can thus consider the birational morphism 0: X--. X' contracting them all to smooth points of a projective d-fold X'. The polarized pair (X', H'), where H' ~ 0 . H, is usually referred to as the minimal reduction of (X, H).

In the sequel, we shall need a number of results of Mori's theory. For generalities and definition we refer to [M1], [M2]. Here we shall only stick to recalling a few facts.

Let us now assume that on the projective d-fold X the canonical divisor is not nef. As a consequence there exists an extremal ray R and a good supporting divisor ~ with respect to R, i.e. a nef divisor M of X such that R = ~C• NE(X). As is known, I m~I is base point free for m >> 0 and the Stein factorization of the morphism associated to that complete linear system gives rise to a contraction morphism

~ contR: X--* Y

ON THE MULTIPLES OF THE ADJOINT DIVISOR ETC. 127

onto a normal variety Y (see [K1] theorem 2.6, [Sh] (2.1), (2.2) and [K2] theorem 2).

In the case when R is not nef, the structure of contR can be specified as follows. Having put E ~ {x �9 X I ~ is not a isomorphism at x}, a ~ dimE, a' ~ dim~(E), R is said to be of type (a, a') and a < d. Consider the fibre ~-l(y) on a point y �9 9(E) and a irreducible, reduced component F of ~-l(y) of maximum dimension r. Now, if D is a prime divisor such that (D. R) < 0, then D ~ E ~ F .

(1.1) THEOEREM (see [A], theorem (2.1)). I f R is not nef and F is as above, there exists a divisor 2. of X such that

(1.1.1) Im(Pic(X) ~ . Pic(F)) =Zltgr(2)] and 2lr is an ample divisor on F;

(1.1.2) OF(-- Kx) ~ OF(p2) and OF(-- D) ~-- Or(q~), with p, q positive integers;

(1.1.3) d-l>_a>_r>_p.

We shall also use the following

(1.2) PROPOSITION (see [B], lemma 2.3). I f R is not neff a = d - 1 and 3C a good supporting divisor then

(1.2.1) Hq(E, �9 + Kx)) = {0}

for all q> 0 and n >>0. Furthermore, i f L is a divisor of X such that (L - Kx - ~ " R) > 0 for every integer ~ >- 0 then

(1.2.2) Hq(E, tgE(n2C + L)) = Hq(E, OE(L)) = {0}

for all q > 0 and n>>0.

2. - T h e m a i n r e s u l t .

Let us consider a polarized pair (X, H) with d - dim X - 3. As a first step towards result (I), we have the following

(2.1) LEMMA. Let E be a ( - 1)-hyperplane of (X, H) and let z: X ~ X' be its contraction to a point p. I f H' is a divisor of X' such that z* H' = H + E,

128

then

(2.1.1)

MICHELE ROSSI

H~ Ox(n(Kx + (d - 2) H))) - H~ ', Ox,(n(Kx, + (d - 2) H')))

for every positive integer n.

Proof. If ~ �9 H~ Ox(n(Kx + (d - 2)H - E)))

(0"-1) r ~ �9 H ~ Vx,\tp)(n(Kx, + (d - 2) H')))

can be extended by Hartogs theorem to a section

fl(D �9 U~ ', Ox,(n(Kx, + (d - 2) U'))) .

A straightforward check shows that

(2.1.2) fl: H~ O x ( n ( K x + ( d - 2 ) H - E ) ) ) - - ~

----) H ~ ', Ox,(n(gx, + (d - 2) H')))

is an isomorphism. Let Da ~ n(Kx + (d - 2)H) - aE, a integer; we have h~ = 0 if

0-< a_< n - 1. Then the cohomology exact sequence of

0 --~ Ox(D~ - E) --) Ox(D~) ~ OE(D~) --) 0

shows that

H~ r + (d - 2) H - E)) ) ~ H~ Ox(n(Kx + (d - 2) H))) ,

which gives the result in view of (2.1.2). q.e.d.

The following theorem plays an important role in the forthcoming analysis. It is a direct consequence of Mori's theory of extremal rays.

(2.2) THEOREM. Let (X', H') be the min imal reduction of (X, H). I f for some positive integer n

(2.2.1) h~ + (d - 2) H)) r 0

then Kx, + (d - 2)H' is neff

Proof.

(2.2.2)

ON THE MULTIPLES OF THE ADJOINT DIVISOR ETC. 129

By repea ted ly applying l emma (2.1), in view of (2.2.1) we have

h~ . + (d - 2 ) H ' ) ) r 0

on the minimal reduction. By contradiction assume tha t Kx, + (d - 2 ) H ' is not nef. This means tha t

NEd_2(X') ~ { V e N E ( X ' ) I ( K x , + ( d - 2 ) H ' . V ) > - O }

is not the whole cone N E ( X ' ) . Mori's cone theorem (see [M1] theo rem (1.4)) says tha t there exis ts an ex t remal ray R such tha t

(2.2.3) (Kx, + (d - 2 ) H ' . R) < 0.

Note tha t (2.2.2) says tha t there exis ts D e I K x . + ( d - 2 ) H ' l such tha t (D. R ) < 0 and so R is not nef.

Le t ~ = contR: X ' --) Y, E and F be as in w 1. With the same notat ion as in theorem (1.1), we have

- g x , ] y = p L , H ' I F - bL,

where L ~ s is ample on F and ~ is a divisor of X. Therefore (2.2.3) reads p(L . R) > (d - 2) b(L. R) and so

d - 1 ->a ~f d i m E _ r ~ d i m F _ p > (d - 2) b

in view of (1.1.3). This gives

(2.2.4) b = 1,

(2.2.5) d - 1 = a = r = p .

As a consequence E = F is a pr ime divisor and R is of type (d - 1, 0), as can be read off Mori's theory (see also [K2], w 5 (B)).

F u r t h e r m o r e

(2.2.6) KX, IE = -- - ( d - 1)L, EIE-~ - q L , H'IE=-L

with q e N \ { 0 } and L = s Fo r any in tegers t, ~, with ~ _> 0 and t - - d + 2,

we have

( t2 - Kx, - )d!7. R) = (t + d - 1 + ).q)(L. R) > 0.

130 MICHELE ROSSI

Proposition (1.2) says that

(2.2.7) Hq(E, OE(tL)) ~ (0}

q > 0, t _> - d + 2. Since H~ OE(tL)) -~ {0) for every t < 0, the polynomial P(t) ~ z(tL) vanishies at - d + 2, - d + 3, ..., - 1. Fur thermore, by (1.2.1) for n>>0

P ( - d + 1) ~ z(Kx, I E) = z((nM + Kx,)]E) = h~ - pL) = 0

where M is a good supporting divisor. On the other hand P(t) is not trivial, as, in view of (2.2.7), P(0) = Z(�9 =

= h~ = 1. Then

1 d-1 - - [ I (t + h) P( t ) = ( d - 1)! h-~

and so L ~-1 = 1, h~ = P(1) = d. In view of a result by Fuji ta ([F1] theorem (2.1)), we get

E -~ p~-l , COE(L ) ~ r

Remembering (2.2.6) we can say that H']~=-L and, by the adjunction formula,

- d L - K E =-- ( d - 1 + q)L

so that q = 1 and EIE = --L. Therefore E is a ( - 1)-hyperplane of (X', H') and is its contraction. This contradicts the minimality of (X', H'). q.e.d.

Now it is easy to prove the following

(2.3) THEOREM. There exists a polynomial q such that

h~ + (d - 2) H)) = q(n)

for every positive integer n.

Proof. We can assume that h~ + ( d - 2 ) H ) ) r for some n > 0 , otherwise the theorem is trivially true. By theorem (2.2) Kx, + ( d - 2 ) H ' is nef on the minimal reduction ( X ' ,H ' ) of (X,H), and so

ON THE MULTIPLES OF THE ADJOINT DIVISOR ETC. 131

(n - 1)(Kx. + (d - 2) H') + (d - 2) H' is ample for every n > 0. Therefore

h~ + (d - 2) H')) = •(n(Kx. + (d - 2) H'))

by the Kodaira vanishing theorem. The result now follows by repeatedly applying lemma (2.1). q.e.d.

As an application of theorem (2.3) we get the following theorem which can also be read off known results by Fujita ([F2], w 1, theorems 3, 3'), Ionescu ([I], (1.7)) and, in case d = 3, by Beltrametti and Palleschi ([BP1], theorem 2.2).

(2.4) THEOREM. I f q -- 0 then either

(2.4.1) (X, H) --- (pd, Oed(1)),

(2.4.2) (X, H)--(Q~, OQ~(1)), where Qd is a hyperquadric of p~. l ,

(2.4.3) X is a Del Pezzo variety and Kx + ( d - 1 ) H - 0 ,

(2.4.4) (X, H) is a scroll over a smooth curve,

(2.4.5) (X, H) is a scroll over a surface,

(2.4.6) (X, H) is a hyperquadric f ibra t ion ,

or (X, H) admits a m in ima l reduction (X' , H') such that either

(2.4.7) (X', H') ~ (P~, O~(e)), with e = 2, 3,

(2.4.8) (X', H')---(Q3, OQ~(2)), where Q3 is a hyperquadric of p 4

(2.4.9) (X', H') ~- (1>4, Op,(2)), or

(2.4.10) d = 3 and X ' is a (P2)-bundle over a smooth curve and Or(H) ~- O~(2) for every f ibre F.

A quick sketch of a proof based on Mori's theory, of this theorem, runs as follows.

The assumption q - 0 implies that Kx + ( d - 2)H is not nef by the Kawamata-Shokurov theorem, and so

(2.4.11) (Kx + (d - 2) H. R) < 0

for an extremal ray R.

132 MICHELE ROSSI

If R is nef, X can be exhibited as a ,,fibration, whose general fibre F is a Fano var ie ty of index i (see [B], theorem 3.2). A standard reasoning shows that a proper ty similar to that in (1.1.2) still holds and so, with the same notation as in (1.1.2), (2.4.11) gives i __p > ( d - 2)b. Moreover the ampleness of - K x yields t de~ d - dim F__ 2. A simple computation settles case R nef.

I f R is not nef, reasoning as in the proof of theorem (2.2), we can find a ( -1 ) -hyperp lane E of X. I t is now easy to show that the minimal reduction (X', H ' ) of (X, H) is one of the polarized pairs appearing in the s ta tement .

Pervenuto in Redazione il 14 aprile 1989.

SUMMARY

Let X be a complex manifold of dimension d -> 3 and H an ample divisor on X. In this paper the global sections of the sheaf Ox(n(Kx + (d - 2)H)) are studied when n �9 Z, this generalizing a result by A. J. Sommese.

RIASSUNTO

Sia X una variet~ complessa di dimensioned -> 3 e H un divisore ampio su X. In questo articolo vengono studiate le sezioni globali del fascio Ox(n(Kx + ( d - 2)H)), dove n �9 Z. Cia permette di generalizzare un risultato di A. J. Sommese.

R E F E R E N C E S

[A]

[B]

[BP1]

[BP2]

[F1]

T. ANDO, On extremal rays of the higher dimensional varieties, Invent. Math., 81 (1985), pp. 347-357. M. BELTRAMEI"rI, On d-folds whose canonical bundle is not numerically effective, according to Marl and Kawamata, Ann. di Mat. Pura ed Applicata, 147 (1987), pp. 151-172. M. BELTRAMETrI - M. PALLESCHI, On threefolds with low sectional genus, Nagoya Math. J., 191 (1986), pp. 27-36. M. BELTRAMETTI - M. PALLESCm, A footnote to a paper by Ionescu, Geometriae Dedicata, 22 (1987), pp. 149-162. T. FUJITA, On the structure of polarized varieties with A-genera zero, J. Fac. Sci. Univ. Tokyo, 22 (1975), pp. 103-115.

ON THE MULTIPLES OF THE ADJOINT DIVISOR ETC. 133

[F2] T. FUJITA, On polarized manifolds whose adjoint bundles are not semipositi- ve, Advanced Studied in Pure Mathematics, 10 (1987), 167-178.

[FS] M.L. FANIA - A. J. SOMlg~SS, On the minimality of hyperplane sections of Gorenstein threefolds, Contributions to several complex variables, Aspects of Mathematics, Vieweg Verlag, E-9 (1986), pp. 89-114.

[I] P. IONESCU, Generalized adjunction and application, Math. Proc. Camb. Phil. Soc., 99 (1986), pp. 457-472.

[K1] Y. KAWAMATA, The cone of curves of algebraic varieties, Ann. Math., 119 (1984), pp. 603-633.

[K2] Y. KAWAMATA, Elementary contraction of algebraic threefolds, Ann. Math., 119 (1984), pp. 95-110.

[K] S. KLEIMAN, Toward a numerical theory of ampleness, Ann. Math., 84 (1966), pp. 293-344.

[Sh] V.V. SHOKUROV, The non-vanishing theorem, Math. USSR Izv., 26 (1986), pp. 591-604.

[S1] A.J . SOMMESE, Ample divisors on threefolds, Algebraic Threefolds, Procee- dings, Varenna 1981, Lecture Notes in Math., 947 (1982), pp. 229-240.

[$2] A . J . SOMMESE, The birational theory of hyperplane sections of projective threefolds, unpublished manuscript (1981).