invariants of ample line bundles on projective varieties

Invariants of ample line bundleson projective varieties

and their applications, II ∗†‡

YOSHIAKI FUKUMA

Abstract

Let X be a smooth complex projective variety of dimension n and letL1, . . . , Ln−i be ample line bundles on X, where i is an integer with 0 ≤ i ≤n − 1. In the previous paper, we defined the i-th sectional geometric genusgi(X,L1, . . . , Ln−i) of (X,L1, . . . , Ln−i). In this part II, we will investigate alower bound for gi(X,L1, . . . , Ln−i). Moreover we will study the first sectionalgeometric genus of (X,L1, . . . , Ln−1).

Introduction.

This is the continuation of [13]. This paper (Part II) consists of section 3, 4,5 and 6. Let X be a smooth complex projective variety of dimension n and letL1, . . . , Ln−i be ample line bundles on X, where i is an integer with 0 ≤ i ≤ n − 1.In [13], we defined the ith sectional geometric genus gi(X, L1, . . . , Ln−i). This invari-ant is thought to be a generalization of the ith sectional geometric genus gi(X, L)of polarized varieties (X, L). Furthermore in [13], we showed some fundamentalproperties of this invariant. In this paper and [14], we will study projective varietiesmore deeply by using some properties of the ith sectional geometric genus of multi-polarized varieties which have been proved in [13]. In this paper, we will mainlystudy a lower bound of gi(X, L1, . . . , Ln−i) and some properties of the case wherei = 1. The content of this paper is the following.

∗Key words and phrases. Polarized varieties, ample line bundles, nef and big line bundles,sectional genus, ith sectional geometric genus, ith sectional H-arithmetic genus, ith sectionalarithmetic genus, adjoint bundles.

†2000 Mathematics Subject Classification. Primary 14C20; Secondary 14C17, 14C40, 14D06,14E30, 14J30, 14J35, 14J40.

‡This research was partially supported by the Grant-in-Aid for Young Scientists (B)(No.14740018), The Ministry of Education, Culture, Sports, Science and Technology, Japan, andthe Grant-in-Aid for Scientific Research (C) (No.17540033), Japan Society for the Promotion ofScience.

1

In section 3 we will give some results and definitions which will be used in thispaper.

In section 4, we will investigate a lower bound for the ith sectional geometricgenus of multi-polarized variety (X, L1, . . . , Ln−i). In particular, we will study arelation between gi(X, L1, . . . , Ln−i) and hi(OX).

In section 5, we will study the nefness of KX + L1 + · · ·+ Lt for t ≥ n− 2. Thisinvestigation will make us possible to study a lower bound for g1(X, L1, . . . , Ln−1)(see section 6) and some properties of g2(X, L1, . . . , Ln−2) (see [14]).

In section 6, we mainly consider the case where (X, L1, . . . , Ln−1) is a multi-polarized manifold of type n − 1 by using results in section 5, and we will make astudy of the following:

(1) The non-negativity of g1(X, L1, . . . , Ln−1).

(2) A classification of (X, L1, . . . , Ln−1) with g1(X, L1, . . . , Ln−1) ≤ 1.

(3) Under the assumption that |Lj| is base point free for any j with 1 ≤ j ≤ n−1,we will prove that g1(X, L1, . . . , Ln−1) ≥ h1(OX). Moreover we will classify(X, L1, . . . , Ln−1) with g1(X, L1, . . . , Ln−1) = h1(OX).

(4) Assume that n = 3, h0(L1) ≥ 2 and h0(L2) ≥ 1. Then we will proveg1(X, L1, L2) ≥ h1(OX). Furthermore we will classify multi-polarized 3-folds(X, L1, L2) with g1(X, L1, L2) = h1(OX), h0(L1) ≥ 2 and h0(L2) ≥ 3.

In this paper we use the same notation as in [13].

3 Preliminaries for the second part

Notation 3.1 Let X be a projective variety of dimension n, let i be an integer with0 ≤ i ≤ n− 1, and let L1, . . . , Ln−i be line bundles on X. Then χ(Lt1

1 ⊗ · · · ⊗Ltn−i

n−i )is a polynomial in t1, . . . , tn−i of total degree at most n. So we can write χ(Lt1

1 ⊗· · · ⊗ L

tn−i

n−i ) uniquely as follows.

χ(Lt11 ⊗ · · · ⊗ L

tn−i

n−i )

=n∑

p=0

∑p1≥0,...,pn−i≥0

p1+···+pn−i=p

χp1,...,pn−i(L1, . . . , Ln−i)

(t1 + p1 − 1

p1

). . .

(tn−i + pn−i − 1

pn−i

).

Definition 3.1 ([13, Definition 2.1]) Let X be a projective variety of dimension n,let i be an integer with 0 ≤ i ≤ n, and let L1, . . . , Ln−i be line bundles on X.(1) The ith sectional H-arithmetic genus χH

i (X, L1, . . . , Ln−i) is defined by the fol-lowing:

χHi (X, L1, . . . , Ln−i) =

⎧⎨⎩χ1, . . . , 1︸︷︷︸

n−i

(L1, . . . , Ln−i) if 0 ≤ i ≤ n − 1,

χ(OX) if i = n.

2

(2) The ith sectional geometric genus gi(X, L1, . . . , Ln−i) is defined by the following:

gi(X, L1, . . . , Ln−i) = (−1)i(χHi (X, L1, . . . , Ln−i) − χ(OX))

+

n−i∑j=0

(−1)n−i−jhn−j(OX).

(3) The ith sectional arithmetic genus pia(X, L1, . . . , Ln−i;F) is defined by the fol-

lowing:

pia(X, L1, . . . , Ln−i;F) = (−1)i(χH

i (X, L1, . . . , Ln−i;F) − h0(F)).

Remark 3.1 Let X be a smooth projective variety of dimension n and let E bean ample vector bundle of rank r on X with 1 ≤ r ≤ n. Then in [10, Definition2.1], we defined the ith cr-sectional geometric genus gi(X, E) of (X, E) for everyinteger i with 0 ≤ i ≤ n − r. Let i be an integer with 0 ≤ i ≤ n − 1. Here wenote that if i = 1, then g1(X, E) is the genus defined in [15, Definition 1.1], andmoreover if r = n− 1, then g1(X, E) is the curve genus g(X, E) of (X, E) which wasdefined in [1] and has been studied by many authors (see [22], [23] and so on). LetL1, . . . , Ln−i be ample line bundles on X. By setting E := L1 ⊕ · · · ⊕ Ln−i, we seethat gi(X, E) = gi(X, L1, . . . , Ln−i). In particular if i = 1, then g1(X, L1, . . . , Ln−1)is equal to the curve genus of (X, E).

Definition 3.2 Let X and Y be smooth projective varieties with dim X > dim Y ≥1. Then a morphism f : X → Y is called a fiber space if f is surjective with connectedfibers. Let L be a Cartier divisor on X. Then (f, X, Y, L) is called a polarized (resp.quasi-polarized) fiber space if f : X → Y is a fiber space and L is ample (resp. nefand big).

Definition 3.3 Let (X, L1, · · · , Lk) be an n-dimensional polarized manifold of typek, where k is a positive integer. Then (X, L1, · · · , Lk) is called a scroll (resp. quadricfibration, Del Pezzo fibration) over a normal variety W if there exists a fiber spacef : X → W such that dimW = n − k + 1 (resp. n − k, n − k − 1) and KX +L1 + · · · + Lk = f ∗(A) for an ample line bundle A on W . We say that a polarizedmanifold (X, L) is a scroll (resp. quadric fibration, Del Pezzo fibration) over anormal variety Y with dim Y = m if there exists a fiber space f : X → Y such thatKX + (n−m + 1)L = f ∗(A) (resp. KX + (n−m)L = f ∗(A), KX + (n−m− 1)L =f ∗(A)) for an ample line bundle A on Y .

Theorem 3.1 Let (X, L) be a polarized manifold with n = dimX ≥ 3. Then (X, L)is one of the following types:

(1) (Pn,O�n(1)).

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(2) (Qn,O�n (1)).

(3) A scroll over a smooth curve.

(4) KX ∼ −(n − 1)L, that is, (X, L) is a Del Pezzo manifold.

(5) A quadric fibration over a smooth curve.

(6) A scroll over a smooth surface.

(7) Let (X ′, L′) be a reduction of (X, L).

(7-1) n = 4, (X ′, L′) = (P4,O�4(2)).

(7-2) n = 3, (X ′, L′) = (Q3,O�3 (2)).

(7-3) n = 3, (X ′, L′) = (P3,O�3(3)).

(7-4) n = 3, X ′ is a P2-bundle over a smooth curve and (F ′, L′|F ′) is isomorphicto (P2,O�2(2)) for any fiber F ′ of it.

(7-5) KX′ + (n − 2)L′ is nef.

Proof. See [2, Proposition 7.2.2, Theorem 7.2.4, Theorem 7.3.2 and Theorem 7.3.4].�

Notation 3.2 Let X be a projective manifold of dimension n.

• ≡ denotes the numerical equivalence.

• Zn−1(X): the group of Weil divisors.

• N1(X) := ({1-cycles}/ ≡) ⊗ R.

• NE(X): the convex cone in N1(X) generated by the effective 1-cycles.

• NE(X): the closure of NE(X) in N1(X) with respect to the Euclidean topol-ogy.

• ρ(X) := dim�N1(X).

• If C is a 1-dimensional cycle in X, then we denote [C] its class in N1(X).

• Let D be an effective divisor on X and D =∑i

aiDi its prime decomposition,

where ai ≥ 1 for any i. Then we write Dred =∑i

Di.

• Sl denotes the symmetric group of order l.

Definition 3.4 ([27, (1.9)]) Let X be a projective manifold of dimension n and letR be an extremal ray. Then the length l(R) is defined by the following:

l(R) = min{−KXC | C is a rational curve with [C] ∈ R}.

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Remark 3.2 By the cone theorem (see [24, Theorem (1.4)], [18] and [20]), l(R) ≤n + 1 holds.

Proposition 3.1 Let X be a projective manifold of dimension n.

(1) If there exists an extremal ray R with l(R) = n + 1, then PicX ∼= Z and −KX

is ample.

(2) If there exists an extremal ray R with l(R) = n, then PicX ∼= Z and −KX

is ample, or ρ(X) = 2 and there exists a morphism contR : X → B onto asmooth curve B whose general fiber is a smooth (n− 1)-manifold that satisfiesconditions of (1).

Proof. See [27, Proposition 2.4]. �

Lemma 3.1 Let (f, X, Y, L) be a quasi-polarized fiber space, where X is a normalprojective variety with only Q-factorial canonical singularities and Y is a smoothprojective variety with dim X = n > dim Y ≥ 1. Assume that KX/Y + tL is f -nef,where t is positive integer. Then (KX/Y + tL)Ln−1 ≥ 0. Moreover if dim Y = 1,then KX/Y + tL is nef.

Proof. For any ample Cartier divisor A on X and any natural number p , KX/Y +tL+(1/p)A is f -nef by assumption. Let m be a natural number such that m(KX/Y +tL + (1/p)A) is a Cartier divisor. Since m(KX/Y + tL + (1/p)A) − KX is f -ample,by the base point free theorem ([19, Theorem 3-1-1]),

f ∗f∗OX

(lm

(KX/Y + tL +

1

pA

))→ OX

(lm

(KX/Y + tL +

1

pA

))is surjective for any l � 0.

Let μ : X1 → X be a resolution of X. We put h = f ◦ μ. Since

μ∗f ∗f∗OX

(lm

(KX/Y + tL +

1

pA

))= h∗h∗OX1

(lm

(KX1/Y + μ∗

(tL +

1

pA

))),

we have

h∗h∗OX1

(lm

(KX1/Y + μ∗

(tL +

1

pA

)))(1)

→ μ∗OX

(lm

(KX/Y + tL +

1

pA

))is surjective. We note that h∗OX1(lm(KX1/Y + μ∗(tL + (1/p)A))) is weakly positiveby [8, Theorem A′ in Page 358] because μ∗OX(lm(tL+(1/p)A)) is semiample. (For

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the definition of weak positivity, see [26].) Hence by [8, Remark 1.3.2 (1)] and (1)above μ∗OX(lm(KX/Y + tL + (1/p)A)) is pseudo-effective. Since p is any naturalnumber, we get (KX/Y + tL)Ln−1 = μ∗(KX/Y + tL)(μ∗L)n−1 ≥ 0.

If dim Y = 1, then we see that h∗OX1(lm(KX1/Y + μ∗(tL + (1/p)A))) is semi-positive by [8, Theorem A′ in page 358] since semi-positivity and weak positivityare equivalent for torsion free sheaves on nonsingular curves. Hence by (1) aboveKX/Y + tL+(1/p)A is nef for any natural number p. Since p is any natural number,KX/Y + tL is nef. �

Lemma 3.2 Let X and Y be smooth projective varieties with dim X > dim Y ≥ 1and let f : X → Y be a surjective morphism with connected fibers. Then q(X) ≤q(F ) + q(Y ), where F is a general fiber of f .

Proof. See [8, Theorem B in Appendix] or [3, Theorem 1.6]. �

Lemma 3.3 Let X be a smooth projective variety, and let D1 and D2 be effectivedivisors on X. Then h0(D1 + D2) ≥ h0(D1) + h0(D2) − 1.

Proof. See [11, Lemma 1.12] or [21, 15.6.2 Lemma]. �

Notation 3.3 Let X be a smooth projective variety of dimension n and let i bean integer with 1 ≤ i ≤ n − 1. Let L1, . . . , Ln−i be nef and big line bundles on X.Assume that Bs|Lj| = ∅ for every integer j with 1 ≤ j ≤ n − i. Then by Bertini’stheorem, for every integer j with 1 ≤ j ≤ n − i, there exists a general memberXj ∈ |Lj |Xj−1

| such that Xj is a smooth projective variety of dimension n− j. (Herewe set X0 := X.) Namely there exists an (n − i)-ladder X ⊃ X1 ⊃ · · · ⊃ Xn−i suchthat a projective variety Xj is smooth with dimXj = n − j.

4 Properties of the sectional geometric genus

In this section we study the relationship between gi(X, L1, . . . , Ln−i) and hi(OX).

Lemma 4.1 Let X be a projective variety of dimension n, and let s be an integerwith 0 ≤ s ≤ n − 1. Let L1, . . . , Ls be Cartier divisors on X. Assume the followingconditions:

(a) There exists an irreducible and reduced divisor Xk+1 ∈ |Lk+1|Xk| for any integer

k with 0 ≤ k ≤ s − 2. (Here we put X0 := X.)

(b) hj(−∑s

m=1 tmLm) = 0 for any integer j and tm with 0 ≤ j ≤ n − 1, tm ≥ 0for any m, and

∑sm=1 tm > 0.

(c) h0(Ls|Xs−1) > 0 and there exists a member Xs ∈ |Ls|Xs−1 |.

Then

6

(1) hj(−∑s

m=k+1 umLm|Xk) = 0 for any integer k, j and um with 1 ≤ k ≤ s − 1,

0 ≤ j ≤ n − k − 1, um ≥ 0 for any m, and∑s

m=k+1 um > 0.

(2) hj(OX) = hj(OX1) = · · · = hj(OXs−1) for any integer j with 0 ≤ j ≤ n − s.

(3) hn−s(OXs−1) ≤ hn−s(OXs).

Proof. (1) First we study the case where k = 1. By the above (b) and the exactsequence

0 → OX

(−L1 −

s∑m=2

umLm

)→ OX

(−

s∑m=2

umLm

)→ OX1

(−

s∑m=2

umLm|X1

)→ 0,

we have hj(−∑s

m=2 umLm|X1) = 0 for any integer j and um with 0 ≤ j ≤ n − 2,um ≥ 0 for any m, and

∑sm=2 um > 0.

Assume that (1) is true for any integer k with k ≤ l − 1, where l is an integerwith 2 ≤ l ≤ s − 1. We consider the case where k = l. By the exact sequence

0 → OXl−1

(−Ll|Xl−1

−s∑

m=l+1

umLm|Xl−1

)→ OXl−1

(−

s∑m=l+1

umLm|Xl−1

)

→ OXl

(−

s∑m=l+1

umLm|Xl

)→ 0,

we have hj(−∑s

m=l+1 umLm|Xl) = 0 for any integer j and um with 0 ≤ j ≤ n− l−1,

um ≥ 0 for any m, and∑s

m=l+1 um > 0. Hence we get the assertion.Next we prove (2) and (3). By (1) above, we obtain hj(−Lk+1|Xk

) = 0 for anyinteger j and k with 0 ≤ k ≤ s − 1 and 0 ≤ j ≤ n − k − 1. Hence by the exactsequence

0 → O(−Lk+1|Xk) → OXk

→ OXk+1→ 0,

we get the assertion. �

Lemma 4.2 Let X be a projective variety of dimension n, and let L be a Cartierdivisor on X. Assume that h0(L) > 0 and hn−1(−L) = 0. Then gn−1(X, L) =hn−1(OX1), where X1 ∈ |L|.

Proof. We consider the exact sequence

0 → −L → OX → OX1 → 0.

Then

Hn−1(−L) → Hn−1(OX) → Hn−1(OX1)

→ Hn(−L) → Hn(OX) → 0

7

is exact. Since hn−1(−L) = 0, we see that hn(−L) − hn(OX) + hn−1(OX) =hn−1(OX1). By [11, Definition 2.1 and Theorem 2.2] or [13, Corollary 2.2], we get

gn−1(X, L) = hn(−L) − hn(OX) + hn−1(OX)

= hn−1(OX1).

Hence we get the assertion. �

Theorem 4.1 Let X be a projective variety of dimension n, and let i be an integerwith 0 ≤ i ≤ n−1. Let L1, . . . , Ln−i be Cartier divisors on X. Assume the followingconditions:


k with 0 ≤ k ≤ n − i − 2. (Here we put X0 := X.)

(b) hj(−∑n−i

m=1 tmLm) = 0 for any integer j and tm with 0 ≤ j ≤ n − 1, tm ≥ 0

for any m, and∑n−i

m=1 tm > 0.

(c) h0(Ln−i|Xn−i−1) > 0 and there exists a member Xn−i ∈ |Ln−i|Xn−i−1

|.

Thengi(X, L1, . . . , Ln−i) ≥ hi(OX).

Proof. By Lemma 4.1 (2), we have hj(OX) = hj(OXn−i−1) for every j with 0 ≤ j ≤ i.

Therefore

(−1)iχ(OX) −n−i∑j=0

(−1)n−i−jhn−j(OX)

= (−1)iχ(OXn−i−1) −

1∑j=0

(−1)1−jhi+1−j(OXn−i−1).

By [13, Lemma 2.4] we also get

χ1,...,1(L1, · · · , Ln−i)

= χ1,...,1(L2|X1, · · · , Ln−i|X1)

= · · ·= χ1(Ln−i|Xn−i−1

).

Hence by [11, Definition 2.1] and Definition 3.1 (2) we have

gi(X, L1, . . . , Ln−i) = gi(Xn−i−1, Ln−i|Xn−i−1).

Here we note that by Lemma 4.1 (1) we have hj(−Ln−i|Xn−i−1) = 0 for any integer

j with 0 ≤ j ≤ i. By Lemma 4.2 we see that gi(Xn−i−1, Ln−i|Xn−i−1) = hi(OXn−i

).

8

Hence by Lemma 4.1 (2) and (3) we get

gi(X, L1, . . . , Ln−i)

= gi(Xn−i−1, Ln−i|Xn−i−1)

= hi(OXn−i)

≥ hi(OX).

Hence we obtain the assertion. �

Lemma 4.3 Let X be a projective variety of dimension n, and let i be an integerwith 0 ≤ i ≤ n − 1. Let L1, . . . , Ln−i be Cartier divisors on X. Then the followingare equivalent: (Here χi(OX) :=

∑ij=0(−1)jhj(OX).)

(a) gi(X, L1, . . . , Ln−i) ≥ hi(OX).

(b) (−1)iχHi (X, L1, . . . , Ln−i) ≥ (−1)iχi(OX).

(c) pia(X, L1, . . . , Ln−i) ≥ (−1)i(χi(OX) − 1).

Proof. By definition, we get

gi(X, L1, . . . , Ln−i) − hi(OX)

= (−1)i(χ1,···,1(L1, . . . , Ln−i) − χ(OX))

+n−i−1∑j=0


= (−1)i(χHi (X, L1, . . . , Ln−i) − χ(OX))

+

n−i−1∑j=0


= (−1)iχHi (X, L1, . . . , Ln−i) − (−1)iχi(OX),

and

pia(X, L1, . . . , Ln−i) − (−1)i(χi(OX) − 1)

= (−1)i(χHi (X, L1, . . . , Ln−i) − 1) − (−1)i(χi(OX) − 1)

= (−1)iχHi (X, L1, . . . , Ln−i) − (−1)iχi(OX).


Corollary 4.1 Let X be a projective variety of dimension n, and let i be an integerwith 0 ≤ i ≤ n−1. Let L1, . . . , Ln−i be Cartier divisors on X. Assume the followingconditions:


k with 0 ≤ k ≤ n − i − 1. (Here we put X0 := X.)

9

(b) hj(−∑n−i

m=1 tmLm) = 0 for any integer j and tm with 0 ≤ j ≤ n − 1, tm ≥ 0

for any m, and∑n−i

m=1 tm > 0.

(c) h0(Ln−i|Xn−i−1) > 0 and there exists a member Xn−i ∈ |Ln−i|Xn−i−1

|.

Then we get the following: (Here χi(OX) :=∑i

j=0(−1)jhj(OX).)

(1) (−1)iχHi (X, L1, . . . , Ln−i) ≥ (−1)iχi(OX).

(2) pia(X, L1, . . . , Ln−i) ≥ (−1)i(χi(OX) − 1).

Proof. By Lemma 4.3 and Theorem 4.1, we get the assertion. �

If X is normal, then we get the following.

Corollary 4.2 Let X be a normal projective variety of dimension n ≥ 3. Let i be aninteger with 0 ≤ i ≤ n−1. Let L1, L2, . . . , Ln−i be ample line bundles on X such thatBs|Lj| = ∅ for every integer j with 1 ≤ j ≤ n− i. Assume that hj(−

∑n−ik=1 tkLk) = 0

for any integer j and tk with 0 ≤ j ≤ n − 1, tk ≥ 0 for any k, and∑n−i

k=1 tk > 0.Then

gi(X, L1, . . . , Ln−i) ≥ hi(OX).

Proof. If i = n − 1, then by [12, Corollary 2.9] we get gn−1(X, L1) ≥ hn−1(OX).If i = 0, then g0(X, L1, . . . , Ln) = L1 · · ·Ln ≥ 1 = h0(OX).So we may assume that 1 ≤ i ≤ n−2. For every integer k with 1 ≤ k ≤ n−i−1,

let Xk ∈ |Lk|Xk−1| be a general member. Then since Bs|Lk|Xk−1

| = ∅, we seethat Xk is a normal projective variety (for example, see [6, (0.2.9) Fact and (4.3)Theorem] or [2, Theorem 1.7.1]). Since Ln−i is ample with Bs|Ln−i| = ∅, we haveh0(Ln−i|Xn−i−1

) > 0 and |Ln−i|Xn−i−1| �= ∅. Hence by Theorem 4.1, we get the

assertion. �

Here we propose the following conjecture, which is a multi-polarized version on[11, Conjecture 4.1].

Conjecture 4.1 Let n and i be integers with n ≥ 2 and 0 ≤ i ≤ n − 1. Let(X, L1, . . . , Ln−i) be an n-dimensional multi-polarized manifold of type (n−i). Thengi(X, L1, . . . , Ln−i) ≥ hi(OX) holds.

Proposition 4.1 Let X be a normal projective variety of dimension n ≥ 2. Let ibe an integer with 0 ≤ i ≤ n− 1. Let L1, . . . , Ln−i−1, A, B be ample Cartier divisorson X. Assume that hj(−(

∑n−i−1p=1 tpLp) − aA − bB) = 0 for any integers j, a, b

and tp with 0 ≤ j ≤ n − 1, a ≥ 0, b ≥ 0, tp ≥ 0, and a + b +∑

tp > 0, and thatBs|Lj| = ∅ for 1 ≤ j ≤ n − i − 1, Bs|A| = ∅, and Bs|B| = ∅. Then

gi(X, A + B, L1, . . . , Ln−i−1) ≥ gi(X, A, L1, . . . , Ln−i−1) + gi(X, B, L1, . . . , Ln−i−1).

10

Proof. We note that by [13, Corollary 2.4]

gi(X, A + B, L1, . . . , Ln−i−1)

= gi(X, A, L1, . . . , Ln−i−1) + gi(X, B, L1, . . . , Ln−i−1)

+gi−1(X, A, B, L1, . . . , Ln−i−1) − hi−1(OX).

By assumption and Corollary 4.2 we have

gi−1(X, A, B, L1, . . . , Ln−i−1) ≥ hi−1(OX).


Remark 4.1 If i = 1, then by [13, Corollary 2.4] for any ample Cartier divisorsA, B, L1, . . . , Ln−2 we have

g1(X, A + B, L1, . . . , Ln−2)

≥ g1(X, A, L1, . . . , Ln−2) + g1(X, B, L1, . . . , Ln−2)

because ABL1 · · ·Ln−2 ≥ 1 = h0(OX).

5 Adjunction theory of multi-polarized manifolds

In this section, we are going to investigate the nefness of KX + L1 + · · · + Lk.Results in this section will be used when we study the ith sectional geometric genusof multi-polarized manifolds in this paper and the Part III [14].

5.1 The nefness of KX + L1 + · · · + Lt for t ≥ n − 1

By putting E := L1⊕· · ·⊕Ll for l = n+1, n, n−1, we can get the following theoremby using a result of Ye and Zhang [28, Theorems 1, 2 and 3]. Here Sl denotes thesymmetric group of order l (see Notation 3.2).

Theorem 5.1.1 (1) Let (X, L1, . . . , Ln+1) be an n-dimensional multi-polarizedmanifold of type n + 1 with n ≥ 3. Then KX + L1 + · · ·+ Ln+1 is nef.

(2) Let (X, L1, . . . , Ln) be an n-dimensional multi-polarized manifold of type nwith n ≥ 3. Then KX + L1 + · · ·+ Ln is nef unless

(X, L1, . . . , Ln) ∼= (Pn,O�n(1), . . . ,O�n(1)).

(3) Let X be a smooth projective variety of dimension n ≥ 3. Let L1, L2, . . . , Ln−1

be ample line bundles on X. If KX +L1 +L2 + · · ·+Ln−1 is not nef, then thereexists σ ∈ Sn−1 such that (X, Lσ(1), Lσ(2), . . . , Lσ(n−1)) is one of the following:

(A) (Pn,O�n(1),O�n(1), . . . ,O�n(1)).

11

(B) (Pn,O�n(2),O�n(1), . . . ,O�n(1)).

(C) (Qn,O�n (1),O�n (1), . . . ,O�n (1)).

(D) X is a Pn−1-bundle over a smooth projective curve B and Lj |F = O�n−1(1)for any fiber F and every integer j with 1 ≤ j ≤ n − 1.

5.2 The nefness of KX + L1 + · · · + Ln−2

Theorem 5.2.1 Let X be a smooth projective variety of dimension n ≥ 4 and letL1, . . . , Ln−2 be ample line bundles on X. Assume the following:

(a) KX + L1 + · · · + Ln−2 is not nef.

(b) KX + (n − 1)Lj is nef for every integer j with 1 ≤ j ≤ n − 2.

Then (X, L1, . . . , Ln−2) is one of the following.

(1) There exists a multi-polarized manifold (Y, A1, . . . , An−2) of type (n − 2) suchthat (Y, A1, . . . , An−2) is a reduction of (X, L1, . . . , Ln−2) (see [13, Definition1.5]) and KY + (n − 1)Aj is ample for every integer j.

(2) KX + (n − 1)Lj = OX for every j with 1 ≤ j ≤ n − 2. Moreover Lj = Lk forevery pair (j, k) with j �= k.

(3) n = 4 and (X, L1, L2) ∼= (P4,O�4(2),O�4(2)).

(4) There exist a smooth projective curve W and a surjective morphism f : X →W with connected fibers such that (X, Li) is a quadric fibration over W withrespect to f for every integer i with 1 ≤ i ≤ n − 2.

(5) There exist a smooth projective surface S and a surjective morphism f : X → Swith connected fibers such that f is a Pn−2-bundle over S and (X, Lj) is a scrollover S with respect to f for every integer j with 1 ≤ j ≤ n− 2, where F is itsfiber.

Proof. By assumption, there exists an extremal ray R such that (KX + L1 + · · · +Ln−2)R < 0. Here we may assume that L1R ≤ L2R ≤ · · · ≤ Ln−2R. Then(KX + (n− 2)L1)R ≤ (KX + L1 + · · ·+ Ln−2)R < 0 and KX + (n− 2)L1 is not nef.There exists a rational curve C with [C] ∈ R such that 0 < −KXC ≤ n + 1, and

(5.2.1.a) 0 > (KX + L1 + · · ·+ Ln−2)C ≥ (KXC) + (n − 2).

So we get −KXC ≥ n − 1.(A) The case where there exists an extremal rational curve C such that KXC =−n − 1.

In this case 0 > (KX + (n − 2)L1)C = −n − 1 + (n − 2)L1C.(A.1) Assume that L1C ≥ 2. Then −n − 1 + 2n − 4 ≤ −n − 1 + (n − 2)L1C < 0.In particular n = 4 by assumption.

12

By Proposition 3.1 (1), we get Pic(X) ∼= Z in this case. Since KX +(n− 1)L1 =KX + 3L1 is nef by assumption and KX + (n − 2)L1 = KX + 2L1 is not nef, we getL1C = 2 and L1 = O(1) or O(2), where O(1) is the ample generator of Pic(X).

If L1 = O(1), then O(1)C = 2 and KXC is even because Pic(X) ∼= Z and O(1) isthe ample generator of Pic(X). But then KXC = −n−1 = −5 and this is impossible.Hence L1 = O(2) and O(1)C = 1. Therefore KX = O(−n − 1) = O(−5). We setL2 := O(a2). Since KX + L1 + L2 = O(a2 − 3) is not nef, we obtain a2 ≤ 2. Byassumption, KX + 3L2 = O(3a2 − 5) is nef. Hence a2 ≥ 2. Therefore a2 = 2. Since−(KX + 4O(1)) is ample, by Kobayashi-Ochiai’s theorem (see [6, (1.3) Corollary]),we have X ∼= P4. Therefore we get the type (3).(A.2) Assume that L1C = 1. Then (KX + (n − 1)L1)C = −2 < 0. But thiscontradicts the assumption.(B) The case where there exists an extremal rational curve C such that KXC = −n.

In this case, 0 > (KX + (n − 2)L1)C = −n + (n − 2)L1C.(B.1) If L1C ≥ 2, then 0 > (KX + (n − 2)L1)C ≥ −n + (n − 2)2 = n − 4 ≥ 0 andthis is impossible.(B.2) If L1C = 1, then (KX + (n − 1)L1)C = −n + (n − 1) = −1 < 0 and this is acontradiction.(C) The case where (X, L1, . . . , Ln−2) satisfies neither the case (A) nor the case (B)above.

We set H := L1 + · · ·+Ln−2. In this case by (5.2.1.a) for every extremal rationalcurve B, KXB = −n + 1 and LiB = 1 for every integer i with 1 ≤ i ≤ n − 2. Inparticular HB = (n−2)LiB for every i. Let τH (resp. τi) be the nef value of (X, H)(resp. (X, Li)).

Claim 5.2.1 τH = (n − 1)/(n − 2) and τi = n − 1 for every integer i with 1 ≤ i ≤n − 2.

Proof. Assume that there exists C ∈ NE(X) such that(KX +

n − 1

n − 2H

)C < 0.

Then by the cone theorem (see also [2, Remark 4.2.6]) C can be written as∑

j λjCj+γ, where Cj is an extremal rational curve and γ is a 1-cycle such that the followingholds: (

KX +n − 1

n − 2H

)γ = 0.

Hence (KX +

n − 1

n − 2H

)Cj < 0

for some j. But this is impossible because KXB = −n + 1 and LiB = 1 forany extremal rational curve B. Therefore τH ≤ (n − 1)/(n − 2). Furthermore(KX + aH)B < 0 for every rational number a < (n− 1)/(n− 2) and every extremal

13

rational curve B. Therefore τH = (n−1)/(n−2). By the same arguement as above,we see that τi = n − 1. �

Let φH and φi be the nef value morphism of (X, H) and (X, Li) respectively. LetFH and Fi be the corresponding extremal face.

Claim 5.2.2 φH = φi for every integer i with 1 ≤ i ≤ n − 2.

Proof. Let C ⊂ X be an irreducible curve with [C] ∈ FH . Then(KX +

n − 1

n − 2H

)C = 0.

Then by the cone theorem there exist extremal rational curves Cj such that C =∑j λjCj (see [2, Lemma 4.2.14]). Hence

0 =

(KX +

n − 1

n − 2H

)C

=∑

j

λj

(KX +

n − 1

n − 2H

)Cj

=∑

j

λj (KX + (n − 1)Li) Cj

= (KX + (n − 1)Li)C.

Therefore [C] ∈ FLi. By the same argument as above, [C] ∈ FH if C is a curve in X

with [C] ∈ FLi. Hence φH = φi because φH (resp. φi) is the contraction morphism

of FH (resp. Fi). �

In particular φi = φj. By Claim 5.2.1 τi = n − 1 for every integer i with1 ≤ i ≤ n − 2. Hence by [2, Theorem 7.3.2], (X, L1, · · · , Ln−2) is either of the type(1), (2), (4), or (5) in the statement of Theorem 5.2.1. Here we note that in thetype (1) KY + (n− 1)Aj is ample for every j. Next we consider the type (2). ThenKX + (n− 1)Lj = OX for any j. Hence (n− 1)Lj = (n− 1)Lk for j �= k. ThereforeLj ≡ Lk. But since h1(OX) = 0 and H2(X, Z) is torsion free in this case, we seethat Lj = Lk.

This completes the proof of Theorem 5.2.1. �

Remark 5.2.1 In (1) of Theorem 5.2.1, we see that

KY +n − 1

n − 2(A1 + · · · + An−2)

is ample. Therefore by Theorem 5.2.1 we get the following:Let X be a smooth projective variety of dimension n ≥ 4 and let L1, . . . , Ln−2

be ample line bundles on X. Assume that KX + L1 + · · · + Ln−2 is not nef andKX + (n − 1)Lj is nef for any j. Then (X, L1, . . . , Ln−2) is one of the following:

14

(I) KX + (n − 1)Lj = OX for any j. Moreover Lj = Lk for any (j, k) with j �= k.

(II) There exist a smooth projective curve W and a surjective morphism f : X →W with connected fibers such that (X, Li) is a quadric fibration over W withrespect to f for every integer i with 1 ≤ i ≤ n − 2.

(III) There exist a smooth projective surface S and a surjective morphism f : X →S with connected fibers such that f is a Pn−2-bundle over S and (X, Lj) is ascroll over S with respect to f for every integer j with 1 ≤ j ≤ n − 2.

(IV) There exists a reduction (Y, A1, . . . , An−2) of (X, L1, . . . , Ln−2) such that(Y, A1, . . . , An−2) satisfies one of the following.

(IV.1) n = 4 and (Y, A1, A2) ∼= (P4,O�4(2),O�4(2)).

(IV.2) KY + A1 + · · ·+ An−2 is nef.

Remark 5.2.2 Let (Y, A1, . . . , An−2) be a reduction of (X, L1, . . . , Ln−2). If Y isnot isomorphic to X, then KY + A1 + · · · + An−2 + Aj is ample for every integer jwith 1 ≤ j ≤ n − 2.

Theorem 5.2.2 Let X be a smooth projective variety of dimension n ≥ 4 and letL1, . . . , Ln−2 be ample line bundles on X. Assume the following:

(a) KX + L1 + · · · + Ln−2 is not nef.

(b) KX + (n − 1)Lj is not nef for some j.

Then there exists σ ∈ Sn−2 such that (X, Lσ(1), . . . , Lσ(n−2)) is one of the following:

(1) (Pn,O�n(1), . . . ,O�n(1),O�n(3)).

(2) n ≥ 5 and (Pn,O�n(1), . . . ,O�n(1),O�n(2),O�n(2)).

(3) (Pn,O�n(1), . . . ,O�n(1),O�n(2)).

(4) (Pn,O�n(1), . . . ,O�n(1)).

(5) (Qn,O�n (1), . . . ,O�n (1),O�n (2)).

(6) (Qn,O�n (1), . . . ,O�n (1)).

(7) X is a Pn−1-bundle over a smooth curve C and one of the following holds.(Here F denotes its fiber.)

(7.1) Lσ(j)|F = O�n−1(1) for every integer j with 1 ≤ j ≤ n − 2.

(7.2) Lσ(j)|F = O�n−1(1) for every integer j with 1 ≤ j ≤ n−3 and Lσ(n−2)|F =O�n−1(2).

15

Proof. We may assume that j = 1 in (b). Since KX + (n − 1)L1 is not nef, by [4,Theorem 1 and Theorem 2] or [16, Theorem] we see that X is isomorphic to one ofthe following types:

(A) Pn.

(B) Qn.

(C) A Pn−1-bundle over a smooth curve C.

Next we study each case.(A) If X ∼= Pn, then we set Lj := O�n(aj) for 1 ≤ j ≤ n − 2. Since KX + (n − 1)L1

is not nef, we have a1 = 1. Here we may assume that a2 ≤ · · · ≤ an−2. SinceKX + L1 + · · · + Ln−2 is not nef, we get (a1, . . . , an−4, an−3, an−2) = (1, . . . , 1, 1, 1),(1, . . . , 1, 1, 2), (1, . . . , 1, 2, 2) or (1, . . . , 1, 1, 3). We note that if n = 4, then (a1, a2) =(2, 2) cannot occur.(B) If X ∼= Qn with n ≥ 4, then Pic(X) ∼= Z and we set Lj := O�n (aj) for1 ≤ j ≤ n−2. Since KX +(n−1)L1 is not nef, we have a1 = 1. Here we may assumethat a2 ≤ · · · ≤ an−2. Then we note that KX = O�n (−n). Since KX +L1+· · ·+Ln−2

is not nef, we get (a1, . . . , an−3, an−2) = (1, . . . , 1, 1) or (1, . . . , 1, 2).(C) The case where X is a Pn−1-bundle over a smooth curve C.(C.1) The case where g(C) ≥ 1.

Since KX + (n − 1)L1 is not nef, there exists a vector bundle E on C withrank(E) = n such that X = PC(E) and L1 = H(E), where H(E) denotes thetautological line bundle on X. Then we note that E is ample. Let π : PC(E) → Cbe its projection. Let Lj := ajH(E)+π∗(Bj) for every integer j with 2 ≤ j ≤ n−2.Here we may assume that a2 ≤ · · · ≤ an−2. Since KX+L1+· · ·+Ln−2 is not nef, thereexists an extremal rational curve B on X such that (KX + L1 + · · · + Ln−2)B < 0.We note that B is contained in a fiber of π. Hence

0 > (KX + L1 + · · · + Ln−2)B = O�n−1

(−n + 1 +

n−2∑j=2

aj

)B.

Hence we obtain (a2, . . . , an−3, an−2) = (1, . . . , 1, 1) or (1, . . . , 1, 2).(C.2) The case where g(C) = 0.

There exists a vector bundle E on P1 such that X = PC(E) and E ∼= OC ⊕OC(d1) ⊕ · · · ⊕ OC(dn−1), where dj is a non-negative integer for 1 ≤ j ≤ n − 1. In

this case we set Lj := ajH(E) + π∗(Bj) for 1 ≤ j ≤ n − 2. By [2, Lemma 3.2.4]

aj > 0 and bj > 0 for any integer j with 1 ≤ j ≤ n − 2, where bj := deg Bj . SinceKX + (n − 1)L1 is not nef, we have a1 = 1. We may assume that a2 ≤ · · · ≤ an−2.

Since KX ≡ −nH(E) + (c1(E) − 2)F , we have

KX + L1 + · · ·+ Ln−2 ≡(−n +

n−2∑j=1

aj

)H(E) +

(c1(E) − 2 +

n−2∑j=1

bj

)F.

16

We note that c1(E) − 2 +∑n−2

j=1 bj ≥ 0 − 2 + (n − 2) ≥ 0. Hence KX + L1 + · · · +Ln−2 is not nef if and only if −n +

∑n−2j=1 aj < 0 because H(E) is nef. So we get

(a1, . . . , an−3, an−2) = (1, . . . , 1, 1) or (1, . . . , 1, 2).This completes the proof. �

Remark 5.2.3 Assume that (X, L1, . . . , Lk) is either the type (3) (D) in Theorem5.1.1 (k = n − 1 in this case) or (7.1) in Theorem 5.2.2 (k = n − 2 in this case).Let f : X → C be its projection. Then for every j with 1 ≤ j ≤ k there exists anample line bundle Bj ∈ Pic(C) such that KX + nLj = f ∗(Bj). Hence n(KX + Lb1 +· · · + Lbn) = f ∗(Bb1 + · · · + Bbn) for any (b1, . . . , bn) with {b1, . . . , bn} ⊂ {1, . . . , k}.On the other hand, by assumption, there exists a line bundle D ∈ Pic(C) such thatKX + Lb1 + · · ·+ Lbn = f ∗(D). Hence D is ample because deg D = deg(Bb1 + · · ·+Bbn)/n > 0. Therefore we see that (X, Lb1 , . . . , Lbn) is a scroll over C.

Remark 5.2.4 By Theorem 5.2.1, Remark 5.2.1 and Theorem 5.2.2, we get thefollowing:Let X be a smooth projective variety of dimension n ≥ 4 and let L1, . . . , Ln−2 beample line bundles on X. Let (Y, A1, . . . , An−2) be a reduction of (X, L1, . . . , Ln−2).Assume that KX +L1 + · · ·+Ln−2 is not nef. Then there exists σ ∈ Sn−2 such that(X, Lσ(1), . . . , Lσ(n−2)) is one of the following:

(1) (Pn,O�n(1), . . . ,O�n(1)).

(2) n ≥ 5 and (Pn,O�n(1), . . . ,O�n(1),O�n(2),O�n(2)).

(3) (Pn,O�n(1), . . . ,O�n(1),O�n(2)).

(4) (Pn,O�n(1), . . . ,O�n(1),O�n(3)).

(5) (Qn,O�n (1), . . . ,O�n (1)).

(6) (Qn,O�n (1), . . . ,O�n (1),O�n (2)).

(7) X is a Pn−1-bundle over a smooth curve C and one of the following holds.(Here F denotes its fiber ).

(7.1) Lσ(j)|F = O�n−1(1) for every integer j with 1 ≤ j ≤ n − 2.

(7.2) Lσ(j)|F = O�n−1(1) for every integer j with 1 ≤ j ≤ n−3 and Lσ(n−2)|F =O�n−1(2).

(8) KX + (n − 1)Lj = OX for any j. Moreover Lj = Lk for any (j, k) with j �= k.

(9) There exist a smooth projective curve W and a surjective morphism f : X →W with connected fibers such that (X, Li) is a quadric fibration over W withrespect to f for every integer i with 1 ≤ i ≤ n − 2.

17

(10) There exist a smooth projective surface S and a surjective morphism f : X →S with connected fibers such that f is a Pn−2-bundle over S and (X, Lj) is ascroll over S with respect to f for every integer j with 1 ≤ j ≤ n − 2.

(11) n = 4 and (Y, A1, A2) ∼= (P4,O�4(2),O�4(2)).

(12) KY + A1 + · · · + An−2 is nef.

Theorem 5.2.3 Let (X, L1, . . . , Ln−2) be an n-dimensional multi-polarized mani-fold with n ≥ 4. Assume that KX +L1 + · · ·+Ln−2 is nef. Then one of the followingholds.

(1) KX + L1 + · · · + Ln−2 = OX .

(2) (X, L1, . . . , Ln−2) is a Del Pezzo fibration over a smooth curve.

(3) (X, L1, . . . , Ln−2) is a quadric fibration over a normal surface.

(4) (X, L1, . . . , Ln−2) is a scroll over a normal 3-fold.

(5) KX + L1 + · · · + Ln−2 is nef and big.

Proof. If KX + L1 + · · · + Ln−2 is ample, then (X, L1, . . . , Ln−2) satisfies (5). Sowe may assume that KX + L1 + · · ·+ Ln−2 is not ample. Then we can take the nefvalue morphism φ : X → Y of (X, L1 + · · ·+ Ln−2), where Y is a normal projectivevariety.

Assume that dimY < dim X. Let F be a general fiber of φ. Then KF + L1|F +· · · + Ln−2|F = OF . Hence dim F ≥ n − 3 by Remark 3.2. Namely, dim Y ≤ 3.Therefore we get the type (1), (2), (3) and (4).

Assume that dimY = dimX. Then KX + L1 + · · · + Ln−2 is nef and big.Therefore we get the assertion. �

6 The first sectional geometric genus

In this section, we consider the first sectional geometric genus of multi-polarizedmanifolds.

6.1 Fundamental results

Proposition 6.1.1 Let X be a smooth projective variety of dimension n, and letL1, . . . , Ln−1 be line bundles on X. Then

g1(X, L1, . . . , Ln−1) = 1 +1

2

(KX +

n−1∑j=1

Lj

)L1 · · ·Ln−1.

18

Proof. We use [13, Corollary 2.7] for i = 1. Here we note the following: the proofof [13, Theorem 2.4] shows that the equality in [13, Corollary 2.7] holds for anyline bundles L1, . . . , Ln−i. By [13, Corollary 2.7], there are the following terms ing1(X, L1, . . . , Ln−1): (

n−1∑j=1

Lj

)L1 · · ·Ln−1

andL1 · · ·Ln−1T1(X).

Here T1(X) denotes the Todd polynomial of weight 1 of the tangent bundle TX (see[13, Definition 1.7]). The coefficient of (

∑n−1j=1 Lj)L1 · · ·Ln−1 is 1/2 and the coeffi-

cient of L1 · · ·Ln−1T1(X) is (−1)1/(1! · · ·1!) = −1. Since T1(X) = (1/2)c1(X) =−(1/2)KX , we obtain

g1(X, L1, . . . , Ln−1)

= 1 +1

2

(n−1∑j=1

Lj

)L1 · · ·Ln−1 +

1

2KXL1 · · ·Ln−1

= 1 +1

2

(KX +

n−1∑j=1

Lj

)L1 · · ·Ln−1.

So we get the assertion. �

By setting E := L1⊕· · ·⊕Ln−1, we can obtain the following theorems by Remark3.1 and [23, Theorems 1 and 2].

Theorem 6.1.1 Let X be a smooth projective variety of dimension n ≥ 3. LetL1, . . . , Ln−1 be ample line bundles on X. Then g1(X, L1, . . . , Ln−1) ≥ 0.

If g1(X, L1, . . . , Ln−1) = 0, then (X, Lσ(1), . . . , Lσ(n−1)) is one of the following:(Here σ ∈ Sn−1.)

(A) (Pn,O�n(1), . . . ,O�n(1)).

(B) (Pn,O�n(2),O�n(1), . . . ,O�n(1)).

(C) (Qn,O�n (1), . . . ,O�n (1)).

(D) X is a Pn−1-bundle over a projective line P1 and Lj |F = O�n−1(1) for any fiberF and j with 1 ≤ j ≤ n − 1.

Theorem 6.1.2 Let X be a smooth projective variety of dimension n ≥ 3 and letL1, . . . , Ln−1 be ample line bundles on X. Assume that g1(X, L1, . . . , Ln−1) = 1.Then (X, L1, . . . , Ln−1) is one of the following:

(1) (X, L1, . . . , Ln−1) satisfies KX + L1 + · · ·+ Ln−1 = OX .

19

(2) X is a Pn−1-bundle over an elliptic curve C and Lj |F = O�n−1(1) for any fiberF and any integer j with 1 ≤ j ≤ n − 1.

Here we note that we can characterize (X, L1, . . . , Ln−1) in the case (1) in The-orem 6.1.2.

Theorem 6.1.3 Let X be a smooth projective variety of dimension n ≥ 3. LetL1, L2, . . . , Ln−1 be ample line bundles on X. Assume that KX + L1 + · · ·+ Ln−1 =OX . Then there exists σ ∈ Sn−1 such that (X, Lσ(1), Lσ(2), . . . , Lσ(n−1)) is one of thefollowing:

(A) (X, L) is a Del Pezzo manifold for some ample line bundle L on X and Lj = Lfor every integer j with 1 ≤ j ≤ n − 1.

(B) (Pn,O�n(3),O�n(1), . . . ,O�n(1)).

(C) n ≥ 4 and (Pn,O�n(2),O�n(2),O�n(1), . . . ,O�n(1)).

(D) (Qn,O�n (2),O�n (1), . . . ,O�n (1)).

(E) X ∼= P2×P1, L1 = p∗1(O�2(2))+p∗2(O�1(1)) and L2 = p∗1(O�2(1))+p∗2(O�1(1)),where pi is the ith projection.

Proof. First we note that h1(OX) = 0 by assumption.(1) Assume that KX + (n − 1)Lj is nef for any j. Then

n−1∑j=1

(KX + (n − 1)Lj) = (n − 1)(KX + L1 + · · · + Ln−1)

= OX .

Therefore (KX +(n−1)Lj)Ln−1j = 0. Since KX +(n−1)Lj is nef, we have KX +(n−

1)Lj = OX , that is, (X, Lj) is a Del Pezzo manifold. Moreover since (n − 1)Lj =(n − 1)Lk for any j �= k, we have Lj ≡ Lk. But since h1(OX) = 0 and H2(X, Z) istorsion free, we have Lj = Lk. So we get the type (A) above.(2) Assume that KX + (n − 1)Lj is not nef for some j. Then by the adjunctiontheory, we see that X is one of the following type:

(2.1) X ∼= Pn.

(2.2) X ∼= Qn.

(2.3) X is a Pn−1-bundle over a smooth curve B.

(2.1) First we consider the case where X ∼= Pn. Then by assumption we get(L1, . . . , Ln−1) is isomorphic to

20

(O�n(3),O�n(1), . . . ,O�n(1)) or (O�n(2),O�n(2),O�n(1), . . . ,O�n(1)).

Here we note that n ≥ 4 in the latter case because KX + (n − 1)Lj is not nef forsome j.

(2.2) Next we consider the case where X ∼= Qn. Then by assumption we get thetype (D) above.

(2.3) Finally we consider the case where X is a Pn−1-bundle over a smooth curve B.Since h1(OX) = 0, we see that B ∼= P1. Then there exists a vector bundle E of rank non X such that E ∼= O�1⊕O�1(a1)⊕· · ·⊕O�1(an−1) and X ∼= P�1(E), where aj ≥ 0 forevery j. Then by [2, Lemma 3.2.4], aH(E)+bF is ample if and only if a > 0 and b >0. Here we note that by the assumption that OX(KX+L1+· · ·+Ln−1) = OX , we mayassume that L1|F = O�n−1(2) and Lj |F = O�n−1(1) for any fiber F and every integer jwith 2 ≤ j ≤ n−1. Hence we can write L1 = 2H(E)+π∗(B1) and Lj = H(E)+π∗(Bj)for every integer j with 2 ≤ j ≤ n − 1, where Bj ∈ Pic(P1). Set bj := deg Bj . Thenbj ≥ 1 because Lj is ample. Since KX = −nH(E) + π∗(K�1 + detE), we haveKX + L1 + · · ·+ Ln−1 = π∗(K�1 + detE + B1 + · · ·+ Bn−1). Since deg E ≥ 0, we seethat

deg(K�1 + detE + B1 + · · · + Bn−1)

= −2 + deg E + b1 + · · · + bn−1

≥ n − 3 ≥ 0.

By the assumption that OX(KX + L1 + · · · + Ln−1) = OX , we get deg(K�1 +detE + B1 + · · · + Bn−1) = 0. Hence n = 3, deg E = 0 and bj = 1 for every j. Inparticular E ∼= O�1 ⊕O�1 ⊕O�1. Therefore we get the type (E). �

Remark 6.1.1 In general, let F be an ample vector bundle of rank n − 1 on asmooth projective variety X of dimension n. Then a classification of (X,F) withOX(KX + detF) = OX has been obtained. See [25].

By Corollary 4.2, we get the following:

Theorem 6.1.4 Let X be a smooth projective variety of dimension n ≥ 3, let ibe an integer with 0 ≤ i ≤ n − 1, and let L1, . . . , Ln−i be ample and spanned linebundles on X. Then gi(X, L1, . . . , Ln−i) ≥ hi(OX).

By considering this theorem, it is natural to classify (X, L1, . . . , Ln−i) such thatBs|Lj| = ∅ for any j with 1 ≤ j ≤ n− i and gi(X, L1, . . . , Ln−i) = hi(OX). Here weconsider the case where i = 1. Set E := L1 ⊕ · · · ⊕Ln−1. Then E is an ample vectorbundle of rank n − 1 on X. Since, as we said in Remark 3.1, g1(X, L1, . . . , Ln−1)is equal to the curve genus g(X, E) of E , we can get the following theorem by [22,Theorem].

21

Theorem 6.1.5 Let X be a smooth projective variety of dimension n ≥ 3, and letL1, . . . , Ln−1 be ample and spanned line bundles on X. If g1(X, L1, . . . , Ln−1) =h1(OX), then (X, L1, . . . , Ln−1) is one of the following:

(1) g1(X, L1, . . . , Ln−1) = 0.

(2) X is a Pn−1-bundle over a smooth curve B and Lj = H(E) + f ∗(Dj) for anyj with 1 ≤ j ≤ n − 1, where E is a vector bundle of rank n on B such thatX ∼= PB(E), H(E) is the tautological line bundle on X, f : X → B is itsfibration, and Dj ∈ Pic(B) for any j.

Moreover we can also get the following theorem by Remark 3.1 and [15, Theorems5.2 and 5.3].

Theorem 6.1.6 Let X be a smooth projective variety of dimension n ≥ 3. Assumethat there exists a fiber space f : X → C, where C is a smooth projective curve.Let L1, . . . , Ln−1 be ample line bundles on X. Then g1(X, L1, . . . , Ln−1) ≥ g(C).Moreover if g1(X, L1, . . . , Ln−1) = g(C), then X is a Pn−1-bundle on C via f andLj |F ∼= O�n−1(1) for any fiber F of f and every integer j with 1 ≤ j ≤ n − 1.

Next we consider Conjecture 4.1 for the case where i = 1 and κ(X) = 0 or 1.

Theorem 6.1.7 Let X be a smooth projective variety of dimension n ≥ 3. LetL1, . . . , Ln−1 be ample line bundles on X. Assume that L1 · · ·Ln−1Lj ≥ 2 for any jwith 1 ≤ j ≤ n − 1 and κ(X) = 0 or 1. Then g1(X, L1, . . . , Ln−1) ≥ q(X).

Proof. If κ(X) = 0, then h1(OX) ≤ n by the classification theory of manifolds (see[17, Corollary 2]). Hence

g1(X, L1, . . . , Ln−1)

= 1 +1

2(KX + L1 + · · · + Ln−1)L1 · · ·Ln−1

≥ 1 + (n − 1) = n ≥ h1(OX).

Next we consider the case where κ(X) = 1. By taking the Iitaka fibration of X, thereexists a smooth projective variety X ′, a smooth projective curve C ′, a birationalmorphism μ : X ′ → X and a fiber space f ′ : X ′ → C ′ such that κ(F ′) = 0 for anygeneral fiber F ′ of f ′. In this case h1(OX′) ≤ h1(OC′) + h1(OF ′) ≤ g(C ′) + n − 1by Lemma 3.2 and [17, Corollary 2]. Here we note that by the proof of [8, Theorem1.3.3] we have KX′/C′(μ∗L1) · · · (μ∗Ln−1) ≥ 0. We also note that

g1(X′, μ∗(L1), . . . , μ

∗(Ln−1))

= 1 +1

2(KX′/C′ + μ∗(L1) + · · ·+ μ∗(Ln−1))μ

∗(L1) · · ·μ∗(Ln−1)

+(g(C ′) − 1)μ∗(L1) · · ·μ∗(Ln−1)F′.

22

If g(C ′) ≥ 1, then since μ∗(L1) · · ·μ∗(Ln−1)F′ ≥ 1 we see that

g1(X, L1, . . . , Ln−1)

= g1(X′, μ∗L1, . . . , μ

∗Ln−1)

≥ g(C ′) +1

2(μ∗L1 + · · ·+ μ∗Ln−1)(μ

∗L1) · · · (μ∗Ln−1)

≥ g(C ′) + n − 1

≥ h1(OX′) = h1(OX).

If g(C ′) = 0, then h1(OX′) ≤ n − 1 and by assumption here we get

g1(X, L1, . . . , Ln−1)

= 1 +1

2(KX + L1 + · · · + Ln−1)L1 · · ·Ln−1

≥ 1 + (n − 1) = n > h1(OX′) = h1(OX).


6.2 The case of 3-folds.

Here we consider the case where X is a 3-fold. The method is similar to that of [9].We fix the notation which will be used below.

Notation 6.2.1 Let (X, L1) be a polarized manifold with dimX = 3 and h0(L1) ≥2. Let Λ be a linear pencil which is contained in |L1| such that Λ = ΛM + Z,where ΛM is the movable part of Λ and Z is the fixed part of |L1|. We will makea fiber space by using this Λ. Let ϕ : X �� P1 be the rational map associatedwith ΛM , and θ : X ′ → X an elimination of indeterminacy of ϕ. So we obtain asurjective morphism ϕ′ : X ′ → P1. By taking the Stein factorization, if necessary,there exist a smooth projective curve C, a finite morphism δ : C → P1 and a fiberspace f ′ : X ′ → C such that ϕ′ = δ ◦ f ′. Let aΛ := degδ and F ′ a general fiber of f ′.

Theorem 6.2.1 Let X be a smooth projective variety of dimension 3. Let L1, L2

be ample line bundles on X. Assume that h0(L1) ≥ 2 and h0(L2) ≥ 1. Theng1(X, L1, L2) ≥ q(X).

Proof. If KX + L1 + L2 is not nef, then by Theorem 5.1.1, Remark 5.2.3 and [13,Example 2.1 (A), (B), (E) and (H)] we get g1(X, L1, L2) ≥ q(X).

So we may assume that KX + L1 + L2 is nef. Here we use Notation 6.2.1.

(I) If g(C) ≥ 1, then θ is the identity mapping. By Proposition 6.1.1, we have

g1(X, L1, L2) = 1 +1

2(KX + L1 + L2)L1L2

= 1 +1

2(KX/C + L1 + L2)L1L2 + (g(C) − 1)L1L2F

′.

23

Since KX/C + L1 + L2 is f ′-nef and dim C = 1, we see that KX/C + L1 + L2 isnef by Lemma 3.1. Here we note that aΛ ≥ 2 because g(C) ≥ 1. Since L1 − aΛF ′ iseffective, we obtain

g1(X, L1, L2) = 1 +1

2(KX/C + L1 + L2)L1L2 + (g(C) − 1)L1L2F

′

≥ 1 +1

2(KX/C + L1 + L2)(aΛF ′)L2 + (g(C) − 1)L1L2F

′

≥ g(C) + (KF ′ + L1|F ′ + L2|F ′)L2|F ′.

If h1(OF ′) = 0, then h1(OX) = g(C). Moreover since KF ′ + L1|F ′ + L2|F ′ is nef,we get g1(X, L1, L2) ≥ g(C). Hence g1(X, L1, L2) ≥ g(C) = h1(OX). Hence we mayassume that h1(OF ′) > 0.

Since h0(L2|F ′) > 0 and dimF ′ = 2, we have g(L2|F ′) ≥ h1(OF ′) ([7, Lemma 1.2(2)]). Therefore

g1(X, L1, L2) ≥ g(C) + 2h1(OF ′) − 2 + (L1|F ′)(L2|F ′).

Then by Lemma 3.2

g1(X, L1, L2) ≥ g(C) + h1(OF ′) + h1(OF ′) − 2 + (L1|F ′)(L2|F ′)

≥ g(C) + h1(OF ′)

≥ h1(OX).

(II) Assume that g(C) = 0. Let D be an irreducible and reduced divisor on Xsuch that the strict transform of D by θ is a general fiber F ′. Then L1−D is linearlyequivalent to an effective divisor. Here we note that KX + L1 + L2 is nef. So wehave

g1(X, L1, L2)

= g1(X′, θ∗L1, θ

∗L2)

= 1 +1

2(KX′ + θ∗L1 + θ∗L2)(θ

∗L1)(θ∗L2)

= 1 +1

2θ∗(KX + L1 + L2)(θ

∗L1)(θ∗L2)

≥ 1 +1

2θ∗(KX + L1 + L2)(θ

∗L2)F′

= 1 +1

2(θ∗(KX + D) + θ∗(L1 − D) + θ∗L2)(θ

∗L2)F′.

Since θ∗(L1 − D)(θ∗L2)F′ ≥ 0, we have

g(X, L1, L2) ≥ 1 +1

2(θ∗(KX + D) + θ∗L2)(θ

∗L2)F′.

By the same argument as in the proof of [9, Claim 2.4], we can prove

θ∗(KX + D)(θ∗L2)F′ ≥ (KX′ + F ′)(θ∗L2)F

′.

24

Hence

g1(X, L1, L2) ≥ 1 +1

2(KX′ + F ′ + θ∗L2)(θ

∗L2)F′

= g(θ∗L2|F ′).

Since h0(θ∗(L2)|F ′) > 0, we get g(θ∗(L2)|F ′) ≥ h1(OF ′) by [7, Lemma 1.2 (2)].Therefore by Lemma 3.2

g1(X, L1, L2) ≥ g(θ∗(L2)|F ′) ≥ h1(OF ′) ≥ h1(OX′) = h1(OX).

This completes the proof. �

Theorem 6.2.2 Let X be a smooth projective variety of dimension 3 and let L1

and L2 be ample line bundles on X with h0(L1) ≥ 2 and h0(L2) ≥ 1. Let Λ ⊂|L1| be a linear pencil, and we use Notation 6.2.1. Assume that for some σ ∈ S2

(X, Lσ(1), Lσ(2)) is neither of the following:

(A) (P3,O�3(1),O�3(1)).

(B) (P3,O�3(2),O�3(1)).

(C) (Q3,O�3 (1),O�3 (1)).

(D) X is a P2-bundle over a smooth projective curve and Lj|F = O�2(1) for anyfiber F and j = 1, 2.

Then

(1) g1(X, L1, L2) ≥ aΛq(X) if g(C) = 0.

(2) g1(X, L1, L2) ≥ q(X) + (aΛ − 1)q(F ′) if g(C) ≥ 1.

Proof. If KX + L1 + L2 is not nef, then (X, L1, L2) is one of the types from (A) to(D) above by Theorem 5.1.1 (3). So we may assume that KX + L1 + L2 is nef. Let

Z, θ, f ′ and C be as in Notation 6.2.1. Let Z =m∑

i=1

biZi, and let Z ′i be the strict

transform of Zi by θ. Let θ′ : X ′′ → X ′ be a birational morphism such that Z ′′i is a

smooth surface, where Z ′′i is the strict transform of Z ′

i by θ′. We can take a generalelement B ∈ |L1| such that B = G1 + · · ·+ GaΛ

+ Z, where each Gi is the image ofa general fiber of f ′ by θ. Let h := f ′ ◦ θ′ and π := θ ◦ θ′. Then the strict transformof Gi by π is a general fiber of h. Let F ′′

i be the strict transform of Gi by π. Wenote that Z ′′

i is the strict transform of Zi by π. Then we have

g1(X, L1, L2) = g(X ′′, π∗L1, π∗L2) = 1 +

1

2(KX′′ + π∗L1 + π∗L2)(π

∗L1)(π∗L2)

= 1 +1

2π∗(KX + L1 + L2)(π

∗L2)(π∗B)

≥ 1 +1

2π∗(KX + L1 + L2)(π

∗L2)(π∗(Bred)).

25

Put Bnr := B −Bred. Then by the same argument as in [9, Claim 2.9] we haveBnrBredL2 ≥ 0. Hence

g1(X, L1, L2) ≥ 1 +1

2π∗(KX + L1 + L2)(π

∗L2)(π∗(Bred))

≥ 1 +1

2(π∗(KX + Bred) + π∗L2)(π

∗L2)(π∗(Bred)).

Moreover since π∗(Bred) −∑aΛ

i=1 F ′′i −

∑mi=1 Z ′′

i is a π-exceptional effective divisor,we get

g1(X, L1, L2) ≥ 1 +1

2(π∗(KX + Bred) + π∗L2)(π

∗L2)(π∗(Bred))

= 1 +1

2

aΛ∑i=1

(π∗(KX + Gi) + π∗L2)(π∗L2)F

′′i

+1

2

aΛ∑i=1

π∗(Bred − Gi)(π∗L2)F

′′i

+1

2

m∑i=1

(π∗(KX + Zi) + π∗L2)(π∗L2)Z

′′i

+1

2

m∑i=1

π∗(Bred − Zi)(π∗L2)Z

′′i .

Because L2 is ample and B is connected, we have

1

2

(aΛ∑i=1

π∗(Bred − Gi)(π∗L2)F

′′i +

m∑i=1

π∗(Bred − Zi)(π∗L2)Z

′′i

)≥ aΛ + m − 1.

Therefore

g1(X, L1, L2) ≥ 1 +1

2

aΛ∑i=1

(π∗(KX + Gi) + π∗L2)(π∗L2)F

′′i

+1

2

m∑i=1

(π∗(KX + Zi) + π∗L2)(π∗L2)Z

′′i + (aΛ + m − 1)

=

aΛ∑i=1

(1 +

1

2(π∗(KX + Gi) + π∗L2)(π

∗L2)F′′i

)

+m∑

i=1

(1 +

1

2(π∗(KX + Zi) + π∗L2)(π

∗L2)Z′′i

).

By the same argument as in the proof of [9, Claim 2.4], we can prove that

(π∗(KX + Gi) + π∗L2)(π∗L2)F

′′i ≥ (KX′′ + F ′′

i + π∗L2)(π∗L2)F

′′i

26

and(π∗(KX + Zi) + π∗L2)(π

∗L2)Z′′i ≥ (KX′′ + Z ′′

i + π∗L2)(π∗L2)Z

′′i .

So we obtain

g1(X, L1, L2) ≥aΛ∑i=1

(1 +

1

2(KX′′ + F ′′

i + π∗L2)(π∗L2)F

′′i

)

+m∑

i=1

(1 +

1

2(KX′′ + Z ′′

i + π∗L2)(π∗L2)Z

′′i

)

=

aΛ∑i=1

g((π∗L2)|F ′′i) +

m∑i=1

g((π∗L2)|Z′′i).

We note that g(π∗L2|Z′′i) ≥ 0 for any i since dimZ ′′

i = 2 (for example, see [5,(4.8) Corollary]).(I) The case where g(C) = 0.Because h0((π∗L2)|F ′′

i) ≥ 1 and dimF ′′

i = 2, we have g((π∗L2)|F ′′i) ≥ q(F ′′

i ) forevery i. Since q(F ′′

i ) ≥ q(X ′′) = q(X ′) = q(X) for every i by Lemma 3.2, we getg1(X, L1, L2) ≥ aΛq(X).(II) The case where g(C) ≥ 1.Then θ is the identity mapping and Zi = Z ′

i for every i. Since L2 is ample andGi is a fiber of f ′, there exists a Zi such that f ′|Zi

: Zi → C is surjective. Weconsider the fiber space h|Z′′

i: Z ′′

i → C. By [7, Theorem 2.1 and Theorem 5.5], wehave g((π∗L2)|Z′′

i) ≥ g(C). On the other hand, g((π∗L2)|F ′′

i) ≥ q(F ′′

i ) holds becauseh0((π∗L2)|F ′′

i) ≥ 1 and dimF ′′

i = 2. Therefore we get g1(X, L1, L2) ≥ g(C)+aΛq(F ′′i ).

Since g(C) + q(F ′′i ) ≥ q(X ′′) = q(X ′) = q(X) by Lemma 3.2 and q(F ′′

i ) = q(F ′) forevery i, we get g1(X, L1, L2) ≥ q(X) + (aΛ − 1)q(F ′). (Here we note that aΛ ≥ 2 inthis case.)


Theorem 6.2.3 Let X be a smooth projective variety of dimension 3 and let L1 andL2 be ample line bundles on X such that h0(L1) ≥ 2 and h0(L2) ≥ 1. Let Λ ⊂ |L1|be a linear pencil and we use Notation 6.2.1.If aΛ = 1, then g1(X, L1, L2) ≥ q(X) + 1

2GZL2, where G is a general element of

ΛM and Z is the fixed part of |L1|, unless (X, Lσ(1), Lσ(2)) is one of the following forsome σ ∈ S2:

(A) (P3,O�3(1),O�3(1)).

(B) (P3,O�3(2),O�3(1)).

(C) (Q3,O�3 (1),O�3 (1)).


27

In particular, g1(X, L1, L2) ≥ q(X) + 1 if Z �= 0.

Proof. If KX + L1 + L2 is not nef, then (X, L1, L2) is one of the types from (A) to(D) above by Theorem 5.1.1 (3). So we may assume that KX + L1 + L2 is nef. Wenote that the strict transform of G by θ is F ′. So we have

g1(X, L1, L2) = 1 +1

2(KX′ + θ∗(L1 + L2))(θ

∗L1)(θ∗L2)

= 1 +1

2θ∗(KX + L1 + L2)(θ

∗L1)(θ∗L2)

≥ 1 +1

2θ∗(KX + L1 + L2)(θ

∗L2)F′

= 1 +1

2(θ∗(KX + G) + θ∗(L1 − G) + θ∗L2)(θ

∗L2)F′.

By the same argument as in the proof of [9, Claim 2.4], we can prove

θ∗(KX + G)(θ∗L2)F′ ≥ (KX′ + F ′)(θ∗L2)F

′.

On the other hand, θ∗(L1 − G)(θ∗L2)F′ = ZGL2. Hence

g1(X, L1, L2) ≥ 1 +1

2(KX′ + F ′ + θ∗L2)(θ

∗L2)F′ +

1

2ZGL2

= g((θ∗L2)|F ′) +1

2ZGL2.

Because h0((θ∗L2)|F ′) ≥ 1 and dimF ′ = 2, we obtain g((θ∗L2)|F ′) ≥ q(F ′) by[7, Lemma 1.2 (2)]. Since g(C) = 0 in this case, we have q(F ′) ≥ q(X ′) = q(X).Therefore

g1(X, L1, L2) ≥ q(F ′) +1

2ZGL2 ≥ q(X) +

1

2ZGL2.

If Z �= 0, then Z ∩ G �= φ since G + Z is connected. Since L2 is ample and G isa general element of ΛM , we have ZGL2 > 0. Because g1(X, L1, L2) is an integer,we have g1(X, L1, L2) ≥ q(X) + 1. This completes the proof. �

Theorem 6.2.4 Let X be a smooth projective variety with dim X = 3 and let L1 andL2 be ample line bundles on X with h0(L1) ≥ 2 and h0(L2) ≥ 3. If g1(X, L1, L2) =q(X), then (X, Lσ(1), Lσ(2)) is one of the following types for some σ ∈ Σ2.

(A) (P3,O�3(1),O�3(1)).

(B) (P3,O�3(2),O�3(1)).

(C) (Q3,O�3 (1),O�3 (1)).


28

Proof. We use Notation 6.2.1.If KX + L1 + L2 is not nef, then by Theorem 5.1.1 (3) we see that (X, L) is one ofthe types from (A) to (D) above. So we may assume that KX + L1 + L2 is nef. Inparticular we note that g1(X, L1, L2) ≥ 1.(1) The case in which g(C) ≥ 1.We note that θ is the identity mapping and aΛ ≥ 2 in this case. By Theorem6.2.2 (2), we have q(X) = g1(X, L1, L2) ≥ q(X) + (aΛ − 1)q(F ′). Because aΛ ≥ 2,we obtain q(F ′) = 0. Hence q(X) ≤ g(C) + q(F ′) = g(C) by Lemma 3.2. Butsince g(C) ≤ q(X), we get q(X) = g(C), and g1(X, L1, L2) = q(X) = g(C). Then(X, L1, L2) is the type (D) above by Theorem 6.1.6. This is a contradiction byassumption.(2) The case in which g(C) = 0.If aΛ ≥ 2, then q(X) = g1(X, L1, L2) ≥ 2q(X) by Theorem 6.2.2 (1). Henceq(X) = 0, and g(X, L1, L2) = q(X) = 0. But this is a contradiction.

So we consider the case where aΛ = 1. By Theorem 6.2.3, we see

(6.2.4.1) Z = 0,

that is, |L1| has no fixed component. By the proof of Theorem 6.2.3, we see thatg((θ∗L2)|F ′) = q(F ′). Here we note that

g((θ∗L2)|F ′) − q(F ′) = h0(KF ′ + (θ∗L2)|F ′) − h0(KF ′)

by the Riemann-Roch theorem and the Kawamata-Viehweg vanishing theorem.Since h0((θ∗L2)|F ′) ≥ 2, we have h0(KF ′) = 0 by Lemma 3.3. Assume thatκ(F ′) ≥ 0. Then q(F ′) ≤ 1 because χ(OF ′) ≥ 0. Hence g((θ∗L2)|F ′) = q(F ′) ≤ 1.But since κ(F ′) ≥ 0, we have g((θ∗L2)|F ′) ≥ 2 and this is a contradiction. Hencewe have

(6.2.4.2) κ(F ′) = −∞.

Because g((θ∗L2)|F ′) = q(F ′), we can prove the following claim.

Claim 6.2.1 κ(KF ′ + (θ∗L2)|F ′) = −∞.

Proof. Assume that κ(KF ′ + (θ∗L2)|F ′) ≥ 0. Then g((θ∗L2)|F ′) ≥ 1.Since 0 < g((θ∗L2)|F ′) = q(F ′), a ((θ∗L2)|F ′)-minimalization of (F ′, (θ∗L2)|F ′) (see[7, Definition 1.9]) is a scroll over a smooth curve B by [7, Theorem 3.1]. Hencethere is a surjective morphism h : F ′ → B such that a general fiber Fh of h is P1.Hence (KF ′ +(θ∗L2)|F ′)Fh = −1. But this is a contradiction because Fh is nef. Thiscompletes the proof of Claim 6.2.1. �

On the other hand,

KF ′ + (θ∗L2)|F ′ = (KX′ + F ′ + θ∗L2)F ′

= (θ∗(KX + L2) + Eθ + F ′)F ′,

29

where Eθ is a θ-exceptional effective divisor.Let (M, A) be a reduction of (X, L2) and let π : X → M be its reduction map.

Assume that KM + A is nef. Then h0(m(KM + A)) > 0 for any large m � 0 bythe nonvanishing theorem. Here we note that KX + L2 = π∗(KM + A) + E for aneffective π-exceptional divisor E. Hence for any large m, we have

h0(m(KX + L2)) = h0(mπ∗(KM + A) + mE) > 0.

Therefore h0(m(θ∗(KX + L2))F ′) ≥ 1. Since F ′ is a general fiber of f ′, we haveh0((Eθ + F ′)|F ′) ≥ 1. Hence h0(m(θ∗(KX + L2) + Eθ + F ′)|F ′) ≥ 1 for any largem � 0. But this is a contradiction by Claim 6.2.1. Hence KM + A is not nef, andby Theorem 3.1 we see that (M, A) is one of the following types. (Here we note thatdim M = 3 in this case.)

(a) (P3,O�3(1)).

(b) (Q3,O�3 (1)).

(c) A scroll over a smooth curve C.

(d) KM ∼ −2A, that is, (M, A) is a Del Pezzo manifold.

(e) A quadric fibration over a smooth curve C.

(f) A scroll over a smooth surface S.

(g) (Q3,O�3 (2)).

(h) (P3,O�3(3)).

(i) M is a P2-bundle over a smooth curve C with (F, A|F ) = (P2,O�2(2)) for anyfiber F of it.

If (M, A) is either of the cases (a), (b), (d), (g), and (h), then q(X) = 0. Henceby assumption g1(X, L1, L2) = q(X) = 0. But this is a contradiction.

If (M, A) is either of the cases (c), (e), and (i), then q(X) = g(C). Hence byassumption g(X, L1, L2) = q(X) = g(C). So by Theorem 6.1.6, (X, L1, L2) is thetype (D) above. But in this case KX +L1 +L2 is not nef and this is a contradiction.

So we consider the case in which (M, A) is the case (f). Let ϕ : M → S be itsP1-bundle, where S is a smooth surface.

Claim 6.2.2 κ(S) = −∞.

Proof. We note that Z = 0 by (6.2.4.1). We take a general element G ∈ |A|. ThenG is irreducible and reduced, and the strict transform of G by θ is F ′. Since A isample, ϕ|G : G → S is surjective. Hence we obtain κ(S) = −∞ since κ(F ′) = −∞

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by (6.2.4.2). This completes the proof of this claim. �

If q(S) = 0, then q(X) = q(S) = 0. Hence by assumption g1(X, L1, L2) =q(X) = q(S) = 0. Hence (X, L1, L2) is one of the types from (A) to (D) above byTheorem 6.1.1. But this is a contradiction by assumption.

If q(S) ≥ 1, we take the Albanese map of S, α : S → B, where B is a smoothcurve. Then by assumption g1(X, L1, L2) = q(X) = q(S) = g(B). Hence (X, L1, L2)is the type (D) above by Theorem 6.1.6. But this is a contradiction by the samereason as above. This completes the proof of Theorem 6.2.4. �

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Yoshiaki FukumaDepartment of Mathematics

Faculty of ScienceKochi University

Akebono-cho, Kochi 780-8520Japan

E-mail: [email protected]

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invariants of ample line bundles on projective varieties

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