On quasi-polarized manifolds whose sectional genus is equal to the irregularity

On Quasi-Polarized Manifolds Whose Sectional Genus is Equal to the Irregularity

YOSHIAKIFUKUMA

ABSTRACT- Le t (X;L) bea quasi-polarized manifold of dimensionn. In our previous paper, we proved that if dimXˆ3andh⁰(L)2, the ng(X;L)h¹(OX) holds.

Hereg(X;L) denotes the sectional genus of (X;L). In this paper, we give the classification of quasi-polarized3-folds (X;L) withh⁰(L)3andg(X;L)ˆh¹(OX).

Moreover as an application of this result, we also give the classification of polarized manifolds (X;L) with dimBsjLj ˆ1,h⁰(L)nandg(X;L)ˆh¹(OX).

1. Introduction.

Let (X;L) bea quasi-polarized manifold with dimXˆn. For this pair (X;L), thesectional genus g(X;L) is defined by the following formula:

g(X;L)ˆ1‡1

2(K_X‡(n 1)L)L^{n 1};

where K_X is thecanonical bundleof X. Then there is the following con- jecture which was proposed by Fujita [7, (13.7) Remark].

CONJECTURE 1.1 (Fujita). Let (X;L) be a quasi-polarized manifold.

Then g(X;L)q(X), where q(X):ˆdimH¹(O_X)is the irregularity of X.

For this conjecture, there are some results (see [9], [10], [12] and so on).

But it is unknown whether this conjecture is true or not even for the case of dimXˆ2. If dimXˆ2, then this conjecture is true ifh⁰(L)>0 (see [9]).

Moreover the classification of quasi-polarized surfaces (X;L) with g(X;L)ˆq(X) andh⁰(L)1 was obtained (see [8], [9]).

If dimXˆ3 andh⁰(L)2, it is known that this conjecture is true, and Indirizzo dell'A.: Department of Mathematics, Faculty of Science, Kochi University, Akebono-cho, Kochi 780-8520, Japan

E-mail: [email protected]

theclassification of polarized 3-folds (X;L) with g(X;L)ˆq(X) and h⁰(L)3 was given (see [12]).

In this paper, we will give the classification ofquasi-polarized 3-folds withg(X;L)ˆq(X) andh⁰(L)3. As an application of this result, we are ableto givetheclassification of polarizedn-fold (X;L) withg(X;L)ˆq(X), dim BsjLj ˆ1 and h⁰(L)n. (Here we note thatg(X;L)q(X) holds if dim BsjLj ˆ1.)

The author would like to thank the referee for giving some useful comments.

2. Preliminaries.

DEFINITION 2.1. Let X and Y be projective varieties with dimX>dimY1, and letf :X!Y bea surjectivemorphism with con- nected fibers. Then (f;X;Y) is called a fiber space. Moreover ifLis a nef and big (resp. an ample) line bundle onX, then (f;X;Y;L) is called a quasi- polarized (resp. polarized) fiber space.

LEMMA2.1. Let X and Cbe smooth projective varieties withdimXˆn anddimCˆ1, and let L be a nef and big line bundle on X. Assume that there exists a fiber space f :X!Csuch that h⁰(KF‡LF)6ˆ0for a general fiber F of f . Then f(K_X=C‡L)is ample.

PROOF. First we note that there exists a natural numbermsuch that (mL)ⁿ n(mL)^{n 1}F>0. Then by [3, Lemma 4.1], there exists a natural numberksuch thatOX(k(mL F)) has a nontrivial global section. Hence wehavean injectivemap OX(kF)! O(kmL). On the other hand, there exists a line bundleNonCsuch thatO(kF)ˆf(N). Hence by [4, Corollary 1.9] weseethatf(K_X=C‡L) is ampleand weget theassertion. p DEFINITION 2.2. (i) Let (X1;L1) and (X2;L2) bequasi-polarized vari- eties. Then (X₁;L₁) and (X₂;L₂) aresaid to bebirationally equivalent if there is another varietyGwith birational morphismsg_i:G!X_i(iˆ1;2) such thatg₁L₁ˆg₂L₂.

(ii) Let (f₁;X₁;Y;L₁) and (f₂;X₂;Y;L₂) be quasi-polarized fiber spaces.

Then (f₁;X₁;Y;L₁) and (f₂;X₂;Y;L₂) aresaid to bebirationally equivalent if there is another variety G with birational morphisms gi:G!Xi

(iˆ1;2) such thatg₁L1ˆg₂L2andf1g1ˆf2g2.

DEFINITION2.3. LetXbea normal projectivevariety of dimensionnand let D bea Q-divisor on X. The n D is said to be generically nef if DL1 Ln 10 for anycollection ofample Cartier divisorsL1;. . .;Ln 1onX.

DEFINITION2.4. Let (X;L) be a quasi-polarized variety of dimensionn.

Then theD-genusD(X;L) of (X;L) is defined by the following:

D(X;L)ˆn‡Lⁿ h⁰(L):

PROPOSITION2.1. Let(X; L)be a quasi-polarized manifold of dimen-

sionn. IfK_X‡(n 1)Lis not generically nef, thenD(X; L)ˆ0or(X; L)is

birationally equivalent to a scroll over a smooth curve.

PROOF. See [16, Proposition 1.3]. p

3. Main results.

First we will prove the following theorem.

THEOREM 3.1. Let (f;X;C;L) be a quasi-polarized fiber space such that X and Care smooth withdimXˆn and dimCˆ1. Then g(X;L) g(C). Moreover if g(X;L)ˆg(C), then(X;L)is one of the following two types.

(a) D(X;L)ˆ0.

(b) The pair(X;L)is birationally equivalent to a scroll over C.

PROOF. (1) Ifg(C)ˆ0, theng(X;L)0ˆg(C) by [16, Theorem 1.1].

Moreover if g(X;L)ˆ0ˆg(C), then by [16, Theorem 1.2] we have D(X;L)ˆ0.

(2) Next we assume thatg(C)1.

(2.1) First weassumethatKX‡(n 1)Lis generically nef. Then by [15, 1.2 Theorem] we see that there exists a natural number j with 1jn 1 such thath⁰(K_X‡jL)>0. Henceh⁰(K_F‡jL_F)>0 for any general fiber F of f. The n f(K_X=C‡jL)6ˆ0. By Lemma 2.1 we see thatf(K_X=C‡jL) is ample. By the same argument as [10, Lemma 1.4.1], weget (K_X=C‡jL)L^{n 1}>0. Since1jn 1, wehave(K_X=C‡

‡(n 1)L)L^{n 1}>0. Then g(X;L)ˆg(C)‡1

2(K_X=C‡(n 1)L)L^{n 1}‡(g(C) 1)((L_F)^{n 1} 1)

>g(C):

(2.2) Next we assume thatKX‡(n 1)L is not generically nef. Then by Proposition 2.1 weseethat D(X;L)ˆ0 or there exist a quasi-polarized variety (X⁰;L⁰), a smooth projective varietyM and birational morphisms m₁:M!X andm₂:M!X⁰ such that (X⁰;L⁰) is a scroll over a smooth curve.

IfD(X;L)ˆ0, then we infer thath¹(O_X⁰)ˆh¹(O_X)ˆ0 (see [12, Lem- ma 1.15]). Henceg(C)ˆ0 and this contradicts theassumption ofg(C)>0.

So wemay assumethat (X⁰;L⁰) is a scroll over a smooth curve B. Le t f⁰:X⁰!B beits fibration and leth:ˆf⁰m₂:M!B. Then for any general fiber F_h of h, wehaveh¹(O_Fh)ˆ0. Since g(C)>0, weseethat fm₁(F_h) is a point. Therefore by [2, Lemma 4.1.13] there exists a sur- jective morphismd:B!Csuch thatfm₁ˆdh. But sincefandf⁰have connected fibers, we see thatdis an isomorphism. On theother hand, we can easily check that g(X⁰;L⁰)ˆg(B). So weget g(X;L)ˆg(X⁰;L⁰)ˆ

ˆg(B)ˆg(C). Therefore we get the assertion. p REMARK3.1. There exists an example of a quasi-polarized fiber space (f;X;C;L) such thatg(X;L)ˆg(C) and (X;L) is birationally equivalent to (V;H) withD(V;H)ˆ0. For example, let (V;H)ˆ(Pⁿ;O_Pⁿ(1)). Then we can easily see thatD(V;H)ˆ0. We take two general membersH1andH2

injHjand letLbe a pencil which is generated byH1andH2. By using this pencil, we can make a fiber space over a smooth curve. Namely, there exist a smooth projective variety X, a birational morphism m:X!Pⁿ and a fiber spacef :X!Cover a smooth curveC. WesetL:ˆm(O_Pⁿ(1)). Since q(X)ˆ0, weseethatCP¹. Moreoverg(X;L)ˆg(Pⁿ;O_Pⁿ(1))ˆ0ˆg(C) and (X;L) is birationally equivalent to (V;H).

Next we consider quasi-polarized manifolds (X;L) with dimXˆ3, h⁰(L)3 andg(X;L)ˆq(X).

THEOREM 3.2. Let (X;L) be a quasi-polarized 3-fold. Assume that h⁰(L)3. If g(X;L)ˆq(X), then (X;L)satisfies one of the following two types.

(a) D(X;L)ˆ0.

(b) The pair(X;L)is birationally equivalent to a scroll over a smooth curve C.

PROOF. By [6, Theorem 4.2], there exists a quasi-polarized variety (X⁰;L⁰) which is birationally equivalent to (X;L) and satisfies one of the

following conditions:

(i) KX⁰‡2L⁰is nef for the canonicalQ-bundleKX⁰; (ii) D(X;L)ˆD(X⁰;L⁰)ˆ0;

(iii) (X⁰;L⁰) is a scroll over a curve,

where X⁰ is a normal projective variety with only Q-factorial terminal singularities. Since g(X;L)ˆg(X⁰;L⁰) and q(X)ˆq(X⁰), wemay assume thatXhas onlyQ-factorial terminal singularities and (X;L) satisfies one of theaboveconditions.

If (X;L) is thetype(ii), then g(X;L)ˆ0 by [6, (1.7) Corollary] and q(X)ˆ0 by [12, Lemma 1.15]. Hence we obtaing(X;L)ˆq(X) in this case.

If (X;L) is the type (iii), then we can check g(X;L)ˆq(X) by easy calculation.

So wemay assumethatK_X‡2Lis nef. Letp:Xe !Xbea resolution of X such that Xnpe ¹(Sing(X))XnSing(X), and Leˆp(L). Then h⁰(L)e ˆ

ˆh⁰(L)3. LetLbe a linear pencil which is contained injLje such that LˆL_M‡Z, whereL_Mis themovablepart ofLandZis thefixed part of jLj. Wewill makea fiber spaceby using thise L. Le t W:Xe ^>P¹ bethe rational map associated with L_M, and u:Xe⁰!Xe an elimination of in- determinacy of W. So weobtain a surjectivemorphism W⁰:Xe⁰!P¹. If necessary, we take the Stein factorizationd:C!P¹ofW⁰. Then we have a fiber spacef⁰:Xe⁰!Csuch thatW⁰ˆdf⁰. Le tF⁰be a general fiber of f⁰ and let a:ˆdegd. We consider this quasi-polarized fiber space (f⁰;Xe⁰;C;u(eL)). By the proof of [12, Theorem 2.1], we see that there exists a quasi-polarized fiber space (f₁;X₁;C;L₁) which is birationally equivalent to (f⁰;Xe⁰;C;u(L)) such that (e f1;X1;C;L1) satisfies one of the following conditions.

K_X1‡2L1isf1-nef.

(f₁;X₁;C;L₁) is a scroll.

If (f1;X1;C;L1) is a scroll, then we see thatg(X;L)ˆq(X) and this is the type (b) in Theorem 3.2. So we may assume thatKX1‡2L1isf1-nef. In this case, by [14, Lemma 0.2], we see thatK_X1=C‡2L₁ is nef.

(a) Thecaseofg(C)1. Thenuis theidentity map. So wehaveXe⁰ˆXe andu(L)e ˆL. By theconstruction of thefiber space(e f⁰;Xe⁰;C;u(L)), wee getLeˆP^a

iˆ1F_i‡Z, where eachF_iis a fiber off⁰andZis thefixed part of jLj. Then there exists an ample line bundlee P2Pic(C) such that P^a

iˆ1F_iˆ(f⁰)(P). In particular degPˆa.

CLAIM3.1. a3.

PROOF. First wenotethat h⁰(L)ˆh⁰(L)e ˆh⁰ P^a

iˆ1F_i‡Z

ˆ

ˆh⁰ P^a

iˆ1F_i

ˆh⁰(P). Since h⁰(L)3, wehaveh⁰(P)3. If a2, then D(C;P)ˆ1‡degP h⁰(P)ˆ1‡a h⁰(P)0. On theother hand, since Pis an amplelinebundleonC, wehaveD(C;P)0 by [5, Corollary 1.10] or [7, (4.2) Theorem]. ThereforeD(C;P)ˆ0. But thenCP¹(see [5, Lemma 3.1]) and this contradicts theassumption that g(C)1. Hence we have

a3. p

Here we note thatLe is numerically equivalent to aF⁰‡Zby thecon- struction above. By the same argument as in the proof of [12, Claim 2.2], we have

(Ke_X=C‡2L)(e eL)² t(Ke_X=C‡2L)(e L)Fe ⁰

for any natural number t with ta. He nce (Ke_X=C‡2L)(e L)e ² 3(Ke_X=C‡2L)(e L)Fe ⁰holds becausea3. Sinceg(C)1 and (eL_F⁰)²1, we get

g(X;e L)e ˆ1‡1

2(Ke_X‡2L)(e L)e ²

ˆg(C)‡1

2(Ke_X=C‡2L)(e L)e ²‡(g(C) 1) (Le_F⁰)² 1 g(C)‡3

2(Ke_X=C‡2L)(e L)Fe ⁰

ˆg(C)‡3g(F⁰;Lje_F0)‡3

2(L)e ²F⁰ 3:

Sinceh⁰(Lje_F0)>0 and dimF⁰ˆ2 wehaveg(F⁰;Lje_F0)q(F⁰) by [9, Lemma 1.2 (2)]. Becauseg(C)‡q(F⁰)q(X), wehavee

g(X;e L)e q(X)e ‡2g(F⁰;Lje_F0)‡3

2(L)e ²F⁰ 3:

Since g(X;e L)e ˆg(X;L)ˆq(X)ˆq(X) holds, wegee t 2g(F⁰;Lje_F0)‡

‡3

2(L)e ²F⁰ 30. Hence we haveg(F⁰;Lje _F0)ˆ0. Thereforek(F⁰)ˆ 1 and by [9, Theorem 2.1] we have q(F⁰)ˆ0. So q(X)e ˆg(C) because g(C)ˆq(F⁰)‡g(C)q(X)e g(C). Hence we obtaing(X;e L)e ˆg(C), and by Theorem 3.1 we get the assertion in this case.

(b) Thecaseofg(C)ˆ0. Letg:ˆpu.

(b.1) Ifa2, then

g(X;L)ˆg(X;e L)e

ˆg(Xe⁰;u(L))e

ˆ1‡1

2g(K_X‡2L)(u(L))e ² 1‡g(KX‡2L)(u(L))Fe ⁰

becauseKX ‡2Lis nef anda2. LetDe :ˆu(F⁰). By [12, Claims 2.3 and 2.4], wehave

g(X;L)1‡g(K_X‡2L)(u(L))Fe ⁰

ˆ1‡u(p(K_X)‡2L)(ue (L))Fe ⁰

ˆ1‡u(Ke_X‡2L)(ue (L))Fe ⁰

1‡(u(Ke_X ‡D)e ‡u(L))(ue (L))Fe ⁰ 1‡(Ke_X⁰‡F⁰‡u(L))(ue (eL))F⁰

ˆ2g(F⁰;u(L)je _F0) 1:

SincedimF⁰ˆ2 andh⁰(u(L)je _F0)>0, wehaveg(F⁰;u(L)je _F0)q(F⁰) by [9, Lemma 1.2 (2)]. Moreover sinceq(F⁰)ˆq(F⁰)‡g(C)q(Xe⁰)ˆq(X), we get g(X;L)2q(X) 1. Therefore q(X)1 because g(X;L)ˆq(X). In particular g(X;L)1. From [6, Corollaries (4.8) and (4.9)], we see that (X;L) is birationally equivalent to one of the types (a) and (b) in Theorem 3.2. (Here we use the assumption thatg(X;L)ˆq(X).)

(b.2) Here we assume that aˆ1. Thenh⁰(u(L)je _F0)2. By thesame argument as in Case (2) in the proof of [12, Theorem 2.1] we have

q(X)ˆg(X;L)g(F⁰;u(L)je _F0)q(F⁰)q(X):

Hence we havek(F⁰)ˆ 1by [9, Theorem 3.1] sinceg(F⁰;u(L)je _F0)ˆq(F⁰) andh⁰(u(L)je _F0)2. Moreover we get

q(Xe⁰)ˆq(X)ˆq(F⁰):

…1†

Here we apply the relatively minimal model theory for the fibration f⁰:Xe⁰!CP¹. Sincek(F⁰)ˆ 1, we see that there exist smooth pro- jective varietiesX^] andTwith dimX^]ˆ3 and 1ˆdimCdimT2, a birational morphism d^]:X^]!Xe⁰ and surjective morphisms d1:X^]!T andd2:T!Cwith connected fibers such thatf⁰d^]ˆd2d1 andFd1is birationally equivalent to a Fano manifold, whereF_d1is a general fiber of d₁. In particularq(F_d1)ˆ0. Weputf^]:ˆf⁰d^].

(b.2.1) Assumethat dimTˆ1. Then d2 is an isomorphism. Hence q(T)ˆ0 because CP¹. On theother hand, sinceq(Fd1)ˆ0, wehave q(X)ˆq(X^])ˆq(T)ˆ0. Thus wegetg(X;L)ˆ0 from theassumption thatg(X;L)ˆq(X). Therefore by [6, (4.8) Corollary] we get the assertion.

(b.2.2) Next we assume that dimTˆ2. If q(X)1, then we have g(X;L)1 and by [6, Corollaries (4.8) and (4.9)] we get the assertion. So wemay assumethatq(X)2. LetFd2 (resp. F^]) be a general fiber ofd2

(resp.f^]). Then

d1j_F] :F^]!Fd2

is a surjective morphism with connected fibers. Since a general fiber ofd₁j_F]

is P¹, wehaveq(F^])ˆq(F_d2). On theother hand, wehaveq(X^])ˆq(T) because any general fiber ofd1isP¹, and wehaveq(F^])ˆq(X^]) by (1). So wegetq(T)ˆq(F^])ˆq(Fd2)ˆq(Fd2)‡q(P¹). Now weareassuming that q(X^])ˆq(X)2, so wehaveq(F_d2)2. Therefore, considering the fiber spaced₂:T!CP¹, we see from [1, Lemme] or [9, Lemma 1.5] thatTis birationally equivalent toF_d2P¹. In particulark(T)ˆ 1. So, taking the Albanese map ofT, there exists a morphisma:T!B, whereBis a smooth projective curve withg(B)ˆq(T)ˆq(Fd2). Thenad1:X^]!Bhas con- nected fibers. Moreover since q(X^])ˆq(F^])ˆq(F_d2)ˆg(B) weobtain g(X^];(ud^])(L))e ˆg(X;L)ˆq(X)ˆq(X^])ˆg(B). Since(X^];(ud^])(L))e is a quasi-polarized 3-fold, we get the assertion by Theorem 3.1. p Here we want to propose the following conjecture which is a quasi- polarized manifolds' version of [12, Conjecture 2.15].

CONJECTURE3.1. Let (X;L)be a quasi-polarized n-fold. Assume that h⁰(L)n. If g(X;L)ˆq(X), then(X;L)is one of the following.

(a) D(X;L)ˆ0.

(b) The pair(X;L)is birationally equivalent to a scroll over a smooth curve.

REMARK3.2. Ifnˆ2 (resp.nˆ3), then this conjecture is true by [9, Theorem 3.1] (resp. Theorem 3.2 above).

Let (X;L) be a polarized manifold of dimensionn. If BsjLj ˆ ; (resp.

dim BsjLj ˆ0), then by [2, Theorem 7.2.10] (resp. [11, Theorem 3.2]) we see that g(X;L)q(X). Moreover, we can get a classification of (X;L) with g(X;L)ˆq(X) and dim BsjLj 0 (see [17, (3.6) Theorem] and [11, Theo- rem 3.2]). So, as the next step, we consider the case where dim BsjLj ˆ1.

THEOREM3.3. Let(X;L)be a polarized manifold of dimension n3.

Assume thatdim BsjLj ˆ1.

(i) The inequality g(X;L)q(X)holds.

(ii) Furthermore we assume that h⁰(L)n. If g(X;L)ˆq(X), then (X;L)is one of the following.

(a) (Pⁿ;O_Pⁿ(1)).

(b) (Qⁿ;O_Qⁿ(1)).

(c) A scroll over a smooth curve.

PROOF. From the assumption, we see that there exist an (n 3)-ladder XX₁ X_{n 3} such that eachX_j is a normal and Gorenstein pro- jective variety of dimension n j (see [13, Proposition 1.12 (2)]). Let L_j ˆL_Xj for everyjwith 1jn 3. Then we see that

h¹(O_X)ˆh¹(O_X1)ˆ ˆh¹(O_Xn3) …2†

and

g(X;L)ˆg(X₁;L₁)ˆ ˆg(X_{n 3};L_{n 3}):

…3†

Letp:M_{n 3}!X_{n 3}bea resolution ofX_{n 3}. Then g(Mn 3;p(Ln 3))ˆg(Xn 3;Ln 3) …4†

and

h¹(O_{Mn 3})h¹(O_Xn3):

…5†

(i) Here we note thath⁰(L_{n 3})2. Hence by [12, Theorem 2.1] we have g(M_{n 3};p(L_{n 3}))q(M_{n 3}):

…6†

Therefore by (2), (3), (4), (5) and (6), we getg(X;L)q(X).

(ii) Assumethath⁰(L)n. Thenh⁰(Ln 3)3. Ifg(X;L)ˆq(X), then by (2), (3), (4), (5) and (6) wehaveg(Mn 3;p(Ln 3))ˆq(Mn 3) andq(Mn 3)ˆ

ˆq(X_{n 3}). In particular,X_{n 3}has the Albanese map (see [2, Remark 2.4.2]).

Let a:X_{n 3}!Alb(X_{n 3}) be its Albanese map, where Alb(X_{n 3}) is the Albanese variety ofX_{n 3}. Thenap:M_{n 3}!Alb(X_{n 3}) is theAlbanese map of Mn 3. Since(Mn 3;p(Ln 3)) is a quasi-polarized 3-fold with h⁰(p(Ln 3))3 andg(Mn 3;p(Ln 3))ˆq(Mn 3), we can apply Theorem 3.2. Then (M_{n 3};p(L_{n 3})) satisfies one of the following types:

D(Mn 3;p(Ln 3))ˆ0.

(Mn 3;p(Ln 3)) is birationally equivalent to a scroll over a smooth curve.

IfD(Mn 3;p(Ln 3))ˆ0, theng(X;L)ˆg(Mn 3;p(Ln 3))ˆ0. There- fore we get the assertion from Fujita's results (see [7, (12.1) Theorem and (5.10) Theorem]).

Next weassumethat (M_{n 3};p(L_{n 3})) is birationally equivalent to a scroll over a smooth curve. Let (V;H) beits scroll. Ifh¹(O_V)ˆ0, then we see thatg(X;L)ˆ0 and weget theassertion. So wemay asssumethat h¹(OV)1. Then the dimension of the image of Albanese map ofVis one because (V;H) is a scroll over a smooth curve. Since Mn 3 and V are birationally equivalent each other, we see that the dimension of the image of ap is also one. Hencethedimension of theimageof a is also one.

Since h¹(O_V)>0 implies h¹(O_X)>0, wecan taketheAlbanesemap b:X!Alb(X) ofX.

CLAIM3.2. dimb(X)ˆ1.

PROOF. First weconsider a map b:X_{n 3},!X!Alb(X). By theuni- versality of the Albanese map, there exists a morphism c:Alb(X_{n 3})!

!Alb(X) such thatcaˆb. On theother hand, sincedima(X_{n 3})ˆ1, we havedimb(Xn 3)ˆdim (ca)(Xn 3)dima(Xn 3)ˆ1. But by [2, Propo- sitions 5.1.1 and 5.1.2] wehavedimb(Xn 3)1 because dimb(X)1.

Hence dimb(Xn 3)ˆ1. Furthermore by using [2, Propositions 5.1.1 and

5.1.2], wealso seedimb(X)ˆ1. p

Sincedimb(X)ˆ1, wefind thatb(X) is smooth andb:X!b(X) is a fiber space over a smooth curveb(X). LetCˆb(X). Sinceh¹(OX)ˆg(C), wegetg(X;L)ˆg(C). By [10, Theorem 1.4.2] we see that (X;L) is a scroll

overC. So we get the assertion. p

REMARK3.3. (i) Theorem 3.3 shows that [12, Conjecture 2.15] is true for thecaseof dim BsjLj ˆ1.

(ii) If dim BsjLj 0, then we see that h⁰(L)n. Hence by [17, (3.6) Theorem] and [11, Theorem 3.2] we infer that [12, Conjecture 2.15] is true for thecaseof dim BsjLj 0.

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Manoscritto pervenuto in redazione il 20 luglio 2010.