The 5-canonical system on 3-folds of general type

angewandte Mathematik

(Walter de Gruyter Berlin
New York 2007

The 5-canonical system on 3-folds of general type

By Jungkai A. Chen at Taipei, Meng Chen at Shanghai, and De-Qi Zhang at Singapore

Abstract. Let X be a projective minimal Gorenstein 3-fold of general type with ca- nonical singularities. We prove that the 5-canonical map is birational onto its image.

1. Introduction

One main goal of algebraic geometry is to classify algebraic varieties. The successful 3-dimensional MMP (see [18], [21] for example) has been attracting more and more math- ematicians to the study of algebraic 3-folds. In this paper, we restrict our interest to pro- jective minimal Gorenstein 3-folds X of general type where there still remain many open problems.

Denote by KX the canonical divisor and Fm:¼ FjmKXj the m-canonical map. There has been a lot of work along the line of the canonical classification. For instance, when X is a smooth 3-fold of general type with pluri-genus h⁰ðX ; kKXÞ f 2, in [19], as an applica- tion to his research on higher direct images of dualizing sheaves, Kolla´r proved that Fm, with m¼ 11k þ 5, is birational onto its image. This result was improved by the second au- thor [5] to include the cases m with m f 5kþ 6; see also [7], [10] for results when some ad- ditional restrictions (like bigger pgðX Þ) are imposed.

On the other hand, for 3-folds X of general type with qðX Þ > 0, Kolla´r [19] first proved that F225 is birational. Recently, the first author and Hacon [4] proved that Fm is birational for m f 7 by using the Fourier-Mukai transform. Moreover, Luo [24], [25] has some results for 3-folds of general type with h²ðOXÞ > 0.

Now for minimal and smooth projective 3-folds, it has been established that Fmðm f 6Þ is a birational morphism onto its image after 20 years of research, by Wilson [32] in 1980, Benveniste [2] in 1986ðm f 8Þ, Matsuki [26] in 1986 ðm ¼ 7Þ, the second au-

The first author was partially supported by the National Science Council and National Center for Theo- retical Science of Taiwan. The second author was supported by both the National Natural Science Foundation of China (Key project No. 10131010) and Program for New Century Excellent Talents in University (#NCET-05- 0358). The third author is supported by an Academic Research Fund of NUS.

thor [6] in 1998ðm ¼ 6Þ and independently by Lee [22], [23] in 1999–2000 (m ¼ 6; and also the base point freeness of m-canonical system for m f 4).

The aim of this paper is to prove the following:

Theorem 1.1. Let X be a projective minimal Gorenstein 3-fold of general type with canonical singularities. Then the m-canonical map Fmis a birational morphism onto its image for all m f 5.

This result is unexpected previously. The di‰culty lies in the case with smaller pgðX Þ or K_X³. One reason to account for this is that the non-birationality of the 4-canonical system for surfaces may happen when they have smaller pg or K² (see Bombieri [3]), whence a naive induction on the dimension does not work.

Nevertheless, there is also evidence supporting the birationality of F5for Gorenstein minimal 3-folds X of general type. For instance, one sees that K_X³f2 for minimal and smooth X (see 2.2 below). So an analogy of Fujita’s conjecture would predict that j5KXj gives a birational map. We recall that Fujita’s conjecture (the freeness part) has been proved by Fujita, Ein-Lazarsfeld [11] and Kawamta [16] when dim X e 4.

Example 1.2. The numerical bound ‘‘5’’ in Theorem 1.1 is optimal. There are plenty of supporting examples. For instance, let f : V ! B be any fibration where V is a smooth projective 3-fold of general type and B a smooth curve. Assume that a general fiber of f has a minimal model S with K_S²¼ 1 and pgðSÞ ¼ 2. (For example, take the product.) Then F_j4KVj is evidently not birational (see [3]).

1.3. Reduction to birationality. According to [6] or [22], to prove Theorem 1.1, we only need to verify the statement in the case m¼ 5. On the other hand, the results in [22], [23] show that jmKXj is base point free for m f 4. So it is su‰cient for us to verify the birationality ofj5KXj in this paper.

1.4. Reduction to factorial models. According to the work of M. Reid [28] and Y.

Kawamata [17] (Lemma 5.1), there is a minimal model Y with a birational morphism n : Y ! X such that KY ¼ n^ðKXÞ and that Y is factorial with at worst terminal singular- ities. Thus it is su‰cient for us to prove Theorem 1.1 for minimal factorial models.

Acknowledgments. We are indebted to He´le`ne Esnault, Christopher Hacon, Yujiro Kawamata, Miles Reid, I-Hsun Tsai, Eckart Viehweg and Chin-Lung Wang for useful con- versations or comments on this subject. We would like to thank the referee for a very care- ful reading and valuable suggestions for the mathematical and linguistic improvement of the paper.

2. Notation, formulae and set up

We work over the complex number field C. By a minimal variety X , we mean one with nef KX and with terminal singularities (except when we specify the singularity type).

2.1. Let X be a projective minimal Gorenstein 3-fold of general type. Take a special resolution n : Y ! X according to Reid ([28]) such that c2ðY Þ
h ¼ 0 (see [27], Lemma 8.3) for any exceptional divisor h of n. Write KY ¼ n^KX þ E where E is exceptional and is mapped to a finite number of points. Then for m f 2, we have (by the vanishing in [15], [31] or [12]):

wðOXÞ ¼ wðOYÞ ¼  1

24KY
c2ðY Þ ¼  1

24n^KX
c2ðY Þ;

PmðX Þ ¼ w

OXðmKXÞ

¼ w

OYðmn^KXÞ

¼ 1

12mðm  1Þð2m  1ÞK_X³þ m

12n^KX
c2ðY Þ þ wðOYÞ

¼ ð2m  1Þ mðm  1Þ

12 K_X³  wðOXÞ

 

:

The inequality of Miyaoka and Yau ([27], [33]) says that 3c2ðY Þ  K_Y² is pseudo-e¤ective.

This gives n^KX

3c2ðY Þ  K_Y²

f0. Noting that n^KX
E²¼ 0 under this situation, we get:

72wðOXÞ  K_X³f0:

In particular, wðOXÞ < 0. So one has:

qðX Þ ¼ h²ðOXÞ þ

1 pgðX Þ

 wðOXÞ > 0 whenever pgðX Þ e 1.

2.2. Suppose that D is any divisor on a smooth 3-fold V . The Riemann-Roch theo- rem gives:

w

OVðDÞ

¼D³

6 KV
D²

4 þD
ðK_V²þ c2Þ

12 þ wðOVÞ:

Direct calculation shows that

w

OVðDÞ þ w

OVðDÞ

¼KV
D²

2 þ 2wðOVÞ A Z:

Therefore, KV
D²is an even integer.

Now let X be a projective minimal Gorenstein 3-fold of general type. Let D be any Cartier divisor on X . Then KX
D²¼ KY
ðn^DÞ² is even. In particular, K_X³ is even and positive.

2.3. Let V be a smooth projective 3-fold and let f : V ! B be a fibration onto a nonsingular curve B. From the spectral sequence:

E₂^{p; q}:¼ H^pðB; R^qfoVÞ ) Eⁿ:¼ HⁿðV ; oVÞ;

Serre duality and [19], Corollary 3.2 and Proposition 7.6 on pages 186 and 36, one has the torsion-freeness of the sheaves RⁱfoV and the following:

h²ðOVÞ ¼ h¹ðB; foVÞ þ h⁰ðB; R¹f_oVÞ;

qðV Þ :¼ h¹ðOVÞ ¼ gðBÞ þ h¹ðB; R¹f_oVÞ:

2.4. For m¼ 1; 2, we set

F¼ F_jKXj if pgðX Þ f 2;

F_j2KXj otherwise:



Since we always have P2ðX Þ f 4, F is a non-trivial rational map.

First we fix a divisor D AjmKXj. Let p : X⁰! X be the composition of both a desingularization of X and a resolution of the indeterminacy of F. We write jp^ðmKXÞj ¼ jM⁰j þ E⁰. Then we may assume, following Hironaka, that:

(1) X⁰ is smooth;

(2) the movable part M⁰ ofjmKX⁰j is base point free;

(3) the support of p^ðDÞ is of simple normal crossings.

We will fix some notation below. The frequently used ones are M, Z, S, D and Ep. Denote by g the composition F p. So g : X⁰! W⁰L P^N is a morphism. Let g : X⁰!^f W !^s W⁰be the Stein factorization of g so that W is normal and f has connected fibers. We can write:

jmKX⁰j ¼ jp^ðmKXÞj þ mEp¼ jM⁰j þ Z⁰; where Z⁰ is the fixed part and Epan e¤ective p-exceptional divisor.

On X , one may write mKX@ Mþ Z where M is a general member of the movable part and Z the fixed divisor. Let S AjM⁰j be the divisor corresponding to M, then

p^ðMÞ ¼ S þh ¼ S þP^s

i¼1

diEi

with di >0 for all i. The above sum runs over all those exceptional divisors of p that lie over the base locus of M. Obviously E⁰¼ hþ p^ðZÞ. On the other hand, one may write Ep¼P^t

j¼1

ejEj where the sum runs over all exceptional divisors of p. One has ej >0 for all 1 e j e t because X is terminal. Evidently, one has t f s.

Note that SingðX Þ is a finite set (see [21], Corollary 5.18). We may write Ep¼ h⁰þh⁰⁰ where h⁰(resp. h⁰⁰) lies (resp. does not lie) over the base locus ofjMj. So if one only requires such a modification p that satisfies 2.4(1) and 2.4(2), one surely has suppðhÞ ¼ suppðh⁰Þ.

Let d :¼ dim FðX Þ. And let L :¼ p^ðKXÞ_jS, which is clearly nef and big. Then we have the following:

Lemma 2.5. When d f 2,ðL²Þ²fðp^KXÞ³

p^ðKXÞ
S²

. Moreover, L²f2.

Proof. Take a su‰ciently large number m such that jmp^ðKXÞj is base point free.

Denote by H a general member of this linear system. Then H must be a smooth projective surface. On H, we have nef divisors p^ðKXÞ_jH and S_jH. Applying the Hodge index theorem, one has

p^ðKXÞ_jH
SjH2

f

p^ðKXÞ_jH2

ðSjHÞ²:

Removing m, we get the first inequality. By 2.2,ðp^KXÞ³is even, hence f 2. Together with p^ðKXÞ
S²>0, we have the second inequality. r

We now state a lemma which will be needed in our proof. The result might be true for all 3-folds with rational singularities. We present a proof here just hoping to make this note more self-contained.

Lemma 2.6. Let X be a normal projective 3-fold with only canonical singularities. Let M be a Cartier divisor on X . Assume that jMj is a movable pencil and that jMj has base points. ThenjMj is composed with a rational pencil.

Proof. Take a birational morphism p : X⁰! X such that X⁰is smooth, that the ex- ceptional divisor Ep is of simple normal crossing, and that the map F_jMj composed with p, becomes a morphism from X⁰ to a curve. Take the Stein factorization of the latter mor- phism to get an induced fibration f : X⁰! B onto a smooth curve B. The lemma asserts that B must be rational.

Clearly, the exceptional divisor Epdominates B.

Case 1. BsjMj contains a curve G.

This is the easier case. Note that X has only finitely many points at which KX is non- Cartier or X is non-cDV (see [21], Cor. 5.40). So we can pick up a very ample divisor H on X (avoiding these finitely many points) such that H is Du Val and intersects G transver- sally. We may assume that the strict transform H⁰on X⁰ is smooth, i.e., p is an embedded resolution of H H X . Clearly, there is an p-exceptional irreducible divisor E which domi- nates both G and B. Now for general H, both H⁰ and E X H⁰ dominate B. Since the curve E X H⁰ arises from the resolution p : H⁰! H of the indeterminancy of the linear system jMj_jH (whose image on X is contained in G X H), it is rational. So B is rational.

Case 2. BsjMj is a finite set. (The argument below works even when X is log terminal.)

Take a base point P of jMj. Then E ¼ p^1ðPÞ dominates B, i.e., f ðEÞ ¼ B. By Kollar’s Theorem 7.6 in [20], there is an analytic contractible neighborhood V of P such that U¼ p^1ðV Þ H X⁰ is simply connected. Suppose gðBÞ > 0. Then the universal cover h : W ! B of B is either the a‰ne line C or an open disk in C. By [13], Proposition 13.5,

there is a factorization for the restriction fj_U: U ! B, say f ¼ h  m, where m : U ! W is continuous. Note that mðEÞ is a compact subset of W , so mðEÞ is a single point. In partic- ular, fðEÞ is a point, a contradiction. r

Remark 2.7. We received the following comment about Lemma 2.6 from the referee to whom we are much grateful. Shokurov has already proved that if the pairðX ; DÞ is klt and the MMP holds, then the fibres of the exceptional locus are always rationally chain connected, which easily implies Lemma 2.6 in the 3-dimensional case. Further, the authors noticed that Shokurov’s result has recently been extended by Hacon and McKernan to any dimension and without assuming MMP.

3. The case pgk 2

The following proposition is quite useful throughout the paper.

Proposition 3.1. Let S be a smooth projective surface. Let C be a smooth curve on S, N⁰< N divisors on S and L HjNj a subsystem. Suppose that jN⁰j_jC ¼ jN_jC⁰ j, degðNjCÞ ¼ 1 þ degðN_jC⁰ Þ f 1 þ 2gðCÞ. We consider the following diagram:

jN⁰j !^res: jN_jC⁰ j

??

?y^þe¤:

??

?y^þP1 jNj !^res: jN_jCj

x?

??^H

x?

??^H L !^res: L_jC:

Suppose furthermore that L_jC is free and L_jCIjN⁰j_jCþ P1. Then L_jC ¼ jNj_jC ¼ jN_jCj;

ð*Þ

which is very ample and complete.

Proof. By the Riemann-Roch theorem and Serre duality, we have dimjNjCj ¼ 1 þ dimjN_jC⁰ j:

Since there are inclusionsjN⁰j_jCþ P1L L_jCLjNj_jCLjN_jCj, now the equalities (*) in the statement follow from dimension counting and the fact that the first inclusion above is strict by the freeness of L_jC. r

Theorem 3.2. Let X be a projective minimal factorial 3-fold of general type. Assume pgðX Þ f 2. Then F5 is birational.

Proof. We give the proof according to the value d :¼ dim FðX Þ. As in 2.4, we set F¼ F1.

Case 1. d¼ 3.

Then pgðX Þ f 4. F5is birational, thanks to [10], Theorem 3.1(i).

Case 2. d¼ 2.

We consider the linear system jKX⁰þ 3p^ðKXÞ þ Sj. Since KX⁰þ 3p^ðKXÞ þ S f S and according to Tankeev’s principle (see [30], Lemma 2, or [9], 2.1), it is su‰cient to verify the birationality of F_{jKX 0þ3p}^_ðKXÞþSjjS. Note that we have a fibration f : X⁰! W where a general fiber of f is a smooth curve C of genus f 2. The vanishing theorem gives:

jKX⁰þ 3p^ðKXÞ þ Sj_jS ¼ jKSþ 3Lj;

where L :¼ p^ðKXÞ_jS is a nef and big divisor on S.

By Lemma 2.5, L²f2. According to Reider ([29]), F_jKSþ3Ljis birational and so is F5. Case 3. d¼ 1.

In this case, we prefer to replace the notation W by B. Let us set b :¼ gðBÞ.

Suppose first b > 0. Let us consider the systemjMj on X . If jMj has base points, then b¼ 0 by 2.6, a contradiction. Thus we may assume that jMj is base point free. Then under this situation F5 is birational, which is exactly the statement of [10], Theorem 3.3. We sketch the proof here for the convenience of the reader. We have an induced fibration f : X⁰! B. Let F be a general fiber of f . Since gðBÞ > 0, the Riemann-Roch and Clif- ford’s theorem imply that S 1 aF with a f pgðX Þ f 2. Since jMj is base point free, one always has p^ðKXÞj_F ¼ s^ðKF0Þ, where s : F ! F0is the smooth blow down onto the mini- mal model. Note that

p^ðKXÞ  F 1

aE⁰1 11 a

 

p^ðKXÞ;

which is nef and big. Applying Kawamata-Viehweg vanishing, we have a surjective map

H⁰ X⁰; K_X0 þ 4p^ðKXÞ 1 aE⁰

 

 

! H⁰ F ; KF þ 41 a

 

p^ðKXÞ

 

F

! :

Also note that

KFþ 41 a

 

p^ðKXÞ

  _Ff KFþ 3s^ðKF0Þ þ 11 a

 

E⁰j_S

 

:

If

K_F²₀; p_gðF Þ3ð1; 2Þ, then KFþ 3s^ðKF0Þ þ 11 a

 

E⁰j_F

 

defines a birational map by surface theory and so does F_{j5KX 0j}j_F. Otherwise, since E⁰j_F1p^ðKXÞj_F is nef and big, we have the same conclusion according to [10], Proposition 2.1 which is an interesting applica- tion of Kawamata-Viehweg vanishing and is not hard to follow. On the other hand, pick up

two general fibers F1and F2. One has 5KX⁰f KX⁰þ 3p^ðKXÞ þiþ F1þ F2where i is nu- merically trivial. Kawamata-Viehweg vanishing gives a surjective map

H⁰

X⁰; KX⁰ þ 3p^ðKXÞ þiþ F1þ F2

! H⁰ðF1; KF1þ L1Þ l H⁰ðF2; KF2þ L2Þ;

where Li:¼

3p^ðKXÞ þi

jFi is nef and big for i¼ 1; 2. Further, the two groups on the right-hand side are non-trivial using Riemann-Roch on the surface Fi. This means that j5KX⁰j can separate two general fibers of f . Therefore, F5is birational onto its image.

From now on, we suppose b¼ 0. Let F be a general fiber of f and denote by s : F ! F0 the smooth blow down onto the minimal model. We take p to be the composi- tion p1 p0 where p0 satisfies 2.4(1) and 2.4(2) and p1 is a further modification such that p^ðKXÞ is supported on a normal crossing divisor.

We may write S @ aF where a f pgðX Þ  1. And we set L :¼ p^ðKXÞ_jF instead. The vanishing theorem gives

jKX⁰þ 3p^ðKXÞ þ Sj_jF ¼ jKFþ 3Lj;

from which we see that the problem is reduced to the birationality of jKFþ 3Lj because jKX⁰þ 3p^ðKXÞ þ Sj I jSj and jSj evidently separates di¤erent fibers of f (as a line bundle of positive degree on a rational curve is very ample). Let F :¼ pðF Þ. We know that KX
F² is an even number by 2.2.

If KX
F²>0, then we have

L²¼ p^ðKXÞ²
F ¼ K_X²
F f KX
F²f2:

Reider’s theorem says thatjKFþ 3Lj gives a birational map.

We are left with only the case KX
F²¼ 0. First we have:

Claim 3.3. If KX
F²¼ 0, then OF

p^ðKXÞ_jF

G OFðs^KF0Þ.

Proof. It is obvious that the claim is true if it holds for p¼ p0. So we may assume p¼ p0. Now

0¼ KX
ðaF Þ²¼ KX
M²¼ p^ðKXÞ
p^ðMÞ
S ¼ ap^ðKXÞ_jF
hjF;

which means p^ðKXÞ_jF
h⁰_jF ¼ 0. On the other hand, the definition of p0gives h⁰⁰_jF ¼ 0. Thus ðEpÞ_jF
p^ðKXÞ_jF ¼ 0. The Hodge index theorem on F tells us that E_pjF must be negative definite.

We may write

KF ¼ p^ðKXÞ_jFþ G;

where G¼ ðEpÞ_jF is an e¤ective negative definite divisor on F . Note that L is nef and big and that L
G ¼ 0. The uniqueness of the Zariski decomposition shows that s^KF0@ p^ðKXÞ_jF. We are done. r

From the above claim, we have FjKFþ3Lj¼ Fj4KFj. We are left to verify the biration- ality of F5 only when F_j4KFjfails to be birational, i.e. when K_F²₀¼ 1 and pgðF Þ ¼ 2.

Kawamata-Viehweg vanishing ([12], [15], [31]) gives

jKX⁰ þ 3p^ðKXÞ þ F j_jF ¼ jKFþ 3s^ðKF0Þj:

ð1Þ

Denote by C a general member of the movable part ofjs^KF0j. By [1], we know that C is a smooth curve of genus 2 and sðCÞ is a general member of jKF0j. Applying the vanishing theorem again, we have

jKFþ 2s^ðKF0Þ þ Cj_jC ¼ jKCþ 2s^ðKF0Þ_jCj:

ð2Þ

Now we may apply Proposition 3.1. Let N⁰ be a divisor corresponding to the mov- able part ofjKFþ 2s^ðKF0Þ þ Cj and N :¼ ð5p^KXÞ_jF. Set L¼ j5p^ðKXÞj_jF. It’s clear that N⁰e N. Also note that L is free becausej5KXj is free by [22].

By (1) above, we see that L IjN⁰j þ (a fixed e¤ective divisor).

Now restricting to C, direct computation shows that degðN_jC⁰ Þ ¼ 4 (by (2)) and 5¼ degðN_jCÞ ¼ 1 þ degðN_jC⁰ Þ. Therefore, the induced inclusion jN_jC⁰ j ,! jN_jCj is given by adding a single point P1.

By (2), we havejN_jC⁰ j ¼ jN⁰j_jC. Together with (1), we have L_jCIjN_jC⁰ j þ P1. Hence by Proposition 3.1, L_jC ¼ jN_jCj gives an embedding. Since j5p^KXj_jFIjN⁰j I jCj (by (1) above) separates di¤erent C (noting that pgðF Þ ¼ 2 and jCj is a rational pencil), F5jF is birational. It is clear thatj5p^KXj I jSj separates di¤erent fibres F . Thus F5is birational.

r

4. Birationality via bicanonical systems

In this section, we shall complete the proof of Theorem 1.1 by studying the bicanon- ical system. We set F :¼ F2 as stated in 2.4. Denote d2:¼ dim F2ðX Þ. We organize our proof according to the value of d2.

In the proofs below, we shall apply Tankeev’s principle as in the proof of Theorem 3.2, Case 2.

Theorem 4.1. Let X be a projective minimal factorial 3-fold of general type. Assume d2¼ 3. Then F5is birational.

Proof. Recall that K_X³ is even by 2.2, so it’s either >2 or¼ 2.

Case 1. The case K_X³ >2.

Pick up a general member S. Let R :¼ S_jS. ThenjRj is not composed of a pencil. Thus one obviously has R²f2. So the Hodge index theorem on S yields

p^ðKXÞ
S²¼ p^ðKXÞ_jS
R f 2:

Set L :¼ p^ðKXÞ_jS. If K_X³ >2, then the proof of Lemma 2.5 gives L²>2.

In this case, we must emphasize that we only need a modification p that satis- fies 2.4(1) and 2.4(2). Namely, we don’t need the normal crossings. Thus we have SuppðhÞ ¼ Suppðh⁰Þ. This property is crucial to our proof.

Now the vanishing theorem gives

jKX⁰þ 2p^ðKXÞ þ Sj_jS ¼ jKSþ 2Lj:

Sinceð2LÞ²f12, we may apply Reider’s theorem again. Assume that F_jKSþ2Lj is not bira- tional. Then there is a free pencil C on S such that L
C ¼ 1. Note that R e 2L, and that jRj is base point free and jRj is not composed of a pencil. Thus dim

F_jRjðCÞ

¼ 1. Since C lies in an algebraic family and S is of general type, we have gðCÞ f 2. Since h⁰ðC; RjCÞ f 2, the Riemann-Roch theorem on C and Cli¤ord’s theorem on C easily imply R
C f 2. Since R
C e 2L
C ¼ 2, one must have R
C ¼ 2. Since

2L¼ S_jSþh_jSþ p^ðZÞ_jS

and C is nef, we have h_jS
C ¼ 0. This implies that h⁰_jS
C ¼ 0. Note also that h⁰⁰_jS¼ 0 for general S. We getðEpÞ_jS
C ¼ 0. Therefore

KS
C ¼ ðKX⁰þ SÞ_jS
C ¼ p^ðKXÞ_jS
C þ ðEpÞ_jS
C þ SjS
C ¼ 3;

an odd integer. This is impossible because C is a free pencil on S. Therefore, F5 must be birational.

Case 2. The case K_X³ ¼ 2.

If L²f3, then f₅is birational according to the proof in Case 1. So we may assume L²¼ 2. By Lemma 2.5, we have p^ðKXÞ
S²¼ 2. Set C ¼ S_jS. Then jCj is base point free and is not composed with a pencil. So C²f2. The Hodge index theorem also gives

4¼

p^ðKXÞ_jS
C2

f L²
C²f4:

The only possibility is L²¼ C²¼ 2 and L 1 C. On the other hand, the equality 4¼ 2K_X³ ¼ K_X²
ðM þ ZÞ ¼ L²þ K_X²
Z ¼ 2 þ K_X²
Z

gives K_X²
Z ¼ 2. Take a very big m such that jmKXj is base point free and take a general member H AjmKXj. By the Hodge index theorem,

4¼ 1

m²ðKX
M
HÞ²f 1

m²ðK_X²
HÞðM²
HÞ ¼ 2KX
M²:

Thus KX
M²¼ 2 and ðKXÞ_jH1MjH. Multiplying by 2, we deduce that ZjH1MjH. Thus KX
Z
M ¼ 1

mZ_jH
M_jH ¼ 1

mM²
H ¼ 2. So L
p^ðZÞ_jS ¼ 2. Since

2C 1 2L¼ p^ð2KXÞ_jS ¼ p^ðM þ ZÞ_jS ¼

Sþ D þ p^ðZÞ

jS¼ C þ

Dþ p^ðZÞ

jS

and L²¼ L
C ¼ 2;

we see that

0¼ L
D ¼ C
D:

ð3Þ

Thus KS¼ ðKX⁰þ SÞ_jS¼ C þ

p^ðKXÞ þ Ep



jS¼ ðC þ LÞ þ
ðEpÞ_jS

¼ P þ N is the Zar- iski decomposition by (3) and 2.4. Denote by s : S! S0 the smooth blow down onto the minimal model. Then Cþ L @ s^ðKS0Þ.

Note that C ¼ S_jS and dimjCj f dimjSj_jSf2 because jSj gives a generically finite map. Assume to the contrary that F5 is not birational. Then neither is F_jSj. Denote by d the generic degree of F5. Then:

2¼ C²¼ S³f d

P2ðX Þ  3 :

Because d f 2, we see P2ðX Þ ¼ 4 and d ¼ 2. By the same argument as in Case 1, we have:

j5KX⁰j_jSI the movable part of jKSþ 2Lj I jCj;

so F_jCj: S! P^{h0ðS; CÞ1} is not birational either. On the other hand, we may write 2¼ C²fdegðF_jCjÞ deg

F_jCjðSÞ : If h⁰ðS; CÞ f 4, then deg

F_jCjðSÞ

f2 and deg F_jCj¼ 1, i.e. FjCj is birational which con- tradicts the assumption. So h⁰ðS; CÞ ¼ 3 and jCj ¼ jSj_jS. Therefore, F_jCj: S! P²is gener- ically finite of degree 2. Let FjCj¼ t  g be the Stein factorization with g : S ! S⁰ a bira- tional morphism onto a normal surface and t : S⁰! P²a finite morphism of degree 2. We can write C¼ F_jCj^ l with a line l.

For a curve E on S, by the projection formula, C:E¼ l:FjCjE. So E is contracted to a point on S⁰if and only if E is contracted to a point on P²(for t is finite); if and only if E is perpendicular to C 11

2s^ðKS0Þ (¼ half of the pull back of K_S which is ample on the unique canonical model S of S); if and only if E is contracted to a point on S by the pro- jection formula again; we denote by Eallthe union of these E. By Zariski’s main theorem, both SnEall! Sn(the image of Eall) and SnEall! S⁰n(the image of Eall) are isomorphisms (so we identify them). Both S and S⁰are completions of the same SnEallby adding a finite set. The normality of S and S⁰ implies that the birational morphisms S! S and S ! S⁰ can be identified, so also S⁰¼ S.

Since S is normal, [21], Propositions 5.4, 5.5 and 5.7 imply a splitting t_O_S¼ O_P²l L;

where L is a line bundle. Thus we see that

qðSÞ ¼ qðSÞ ¼ h¹ðS; tO_SÞ ¼ 0:

Since S is nef and big on X⁰, the long exact sequence

0¼ H¹ðKX⁰þ SÞ ! H¹ðKSÞ ! H²ðKX⁰Þ ! H²ðKX⁰þ SÞ ¼ 0

gives qðX Þ ¼ qðX⁰Þ ¼ qðSÞ ¼ 0. Noting that wðOXÞ < 0, we naturally have pgðX Þ f 2. By Theorem 3.2, F5 is birational, a contradiction.

Therefore we have proved the birationality of F5. r

Theorem 4.2. Let X be a projective minimal factorial 3-fold of general type. Assume d2¼ 2. Then F5is birational.

Proof. By 2.2, K_X³ is even and hence either K_X³ ¼ 2 or K_X³f4.

Case 1. K_X³ >2.

When d2¼ 2, f : X⁰! W is a fibration onto a surface W . Taking a further modifi- cation, we may even get a smooth base W . Denote by C a general fiber of f . Then gðCÞ f 2. Pick up a general member S which is an irreducible surface of general type. We may write S_jS@ P^a2

i¼1

Ci where a2f P2ðX Þ  2. Since K_X³ >2, we have a2f P2ðX Þ  2 f 3.

Set L :¼ p^ðKXÞ_jS. Then L is nef and big. Since p^ðKXÞ
S²¼

p^ðKXÞ_jS
S_jS

Sf3

p^ðKXÞ_jS
C

Sf3;

Lemma 2.5 gives L²f4. The vanishing theorem gives

jKX⁰þ 2p^ðKXÞ þ Sj_jS ¼ jKSþ 2Lj:

ð4Þ

Assume that F5 is not birational. Then neither is F_jKSþ2Lj for general S. Because ð2LÞ²f10, Reider’s theorem ([29]) tells us that there is a free pencil C⁰ on S such that L
C⁰ ¼ 1. Since 2 ¼ C⁰
2L f C⁰:S_jS ¼ a2C⁰
C f 3C⁰:C, we have C
C⁰¼ 0. So C⁰ lies in the same algebraic family as that of C. We may write

2L 1 a2Cþ G;

where G¼

Dþ p^ðZÞ

jSf0 and a2f3. Since 2L C  1 a2

G 1 2 2 a2

 

L is nef and big, Kawamata-Viehweg vanishing gives H¹ S; KSþ 2L  C  1

a2

G

 

 

¼ 0. Thus we get a surjection:

H⁰ S; KSþ 2L  1 a2

G

 

 

! H⁰ðC; KCþ DÞ;

where D :¼ 2L  1 a2

G

 

jC

with degðDÞ f 2  2 a2

 

L
C > 1. Note that jKSþ 2Lj I jS_jSj separates di¤erent C. If degðDÞ f 3, then jKCþ Dj defines an embedding, and so does jKSþ 2Lj, a contradiction.

So suppose degðDÞ ¼ 2. We now apply Proposition 3.1. Let N⁰ be the movable part of KSþ 2L  1

a2

G

 

and let N¼ p^ð5KXÞ_jS. Set L :¼ j5p^ðKXÞj_jS. As in the proof of The- orem 3.2, we have L IjN⁰j þ (a fixed e¤ective divisor), jN⁰j_jC ¼ jKCþ Dj, N⁰e N and degðN_jCÞ ¼ 1 þ degðN_jC⁰ Þ ¼ 2gðCÞ þ 1 ¼ 5 by the calculation:

4 e

2gðCÞ  2

þ 2 ¼ N⁰
C e N
C ¼ 5p^KX
C ¼ 5:

By Proposition 3.1, L_jC ¼ jNjCj gives an embedding. It is clear that j5p^KXj I jSj separates di¤erent S, andj5p^KXj_jSðI the movable part of jKSþ 2Lj) separates di¤erent C. Thus F5

is birational. This is again a contradiction.

Case 2. K_X³ ¼ 2.

We first consider the case L²f3. On the surface S, we are reduced to study the linear systemjKSþ 2Lj. We have

2L @ S_jSþ G ¼P^a2

i¼1

Ciþ G;

where a2f h⁰ðS; SjSÞ  1 f P2ðX Þ  2 f 2. Denote by C a general fiber of f : X⁰! W . If a2f3, the proof in Case 1 already works. So we assume a2¼ 2, then P2ðX Þ ¼ 4, and the image of the fibration F_jSjSj : S! P² is a quadric curve which is a rational curve. This means thatjCj is composed with a rational pencil. Assume that jKSþ 2Lj does not give a birational map. Then Reider’s theorem says that there is a free pencil C⁰ on S such that L
C⁰ ¼ 1. We claim that C⁰and C are in the same pencil. In fact, otherwise C⁰is horizon- tal with respect to C and C
C⁰>0. Since C is a rational pencil, C
C⁰f2. Therefore L
C⁰f2, a contradiction. So C⁰ lies in the same family as that of C and L
C ¼ 1. Note that KSþ 2L ¼

KX⁰þ 2p^ðKXÞ

jSþ S_jSf C. So jKSþ 2Lj distinguishes di¤erent mem- bers injCj. The vanishing theorem gives

H⁰ S; KSþ 2L 1 2G

 

 

! H⁰ðC; KCþ QÞ;

where Q¼ 2L  C 1 2G

 

jC

is an e¤ective divisor on C. IfjKCþ Qj is not birational, nei- ther isjKCj. So C must be a hyper-elliptic curve and FjKCj: C! P¹is a double cover; see Iitaka [14], §6.5, page 217. Suppose F5is not birational. (*) Then F⁵must be a morphism of generic degree 2. Set s¼ F5: X ! W5H P^N. Then 5KX ¼ s^ðHÞ for a very ample divi- sor H on the image W5. So

5¼ 5p^ðKXÞ
C ¼ 2 degðHj_sðpðCÞÞÞ ¼ 2 deg_P^Ns
pðCÞ

; which is a contradiction. Thus F5must be birational under this situation.

Next we consider the case L²¼ 2. Lemma 2.5 says 2 ¼ p^ðKXÞ
S²¼ a2L
C. We see that a2¼ 2 and L
C ¼ 1. We still consider the linear system jKSþ 2Lj. As above, a2¼ 2

implies thatjCj is a rational pencil. Since KSþ 2L f C, we see that jKSþ 2Lj distinguishes di¤erent members injCj. By the same argument as above, we have

jKSþ 2Lj_jCIjKCþ Qj I jKCj:

If F5 is not birational, then neither is F_jKSþ2Lj. This means that C must be a hyper-elliptic curve and F5is of generic degree 2. Since j5KXj is base point free, we also have a contra- diction as in the previous case. So F5is birational. r

Theorem 4.3. Let X be a projective minimal factorial 3-fold of general type. Assume d2¼ 1. Then F5is birational.

Proof. When X is smooth, this theorem has been proved in [7]. Our result is a gen- eralization of this result.

Taking a modification p as in 2.4, we get an induced fibration f : X⁰! W and B :¼ W is a smooth curve of genus b :¼ gðBÞ. By [8], Lemma 2.1, we know that 0 e b e 1.

Let F be a general fiber of f . Claim 4.4. We have

OF

p^ðKXÞ_jF G OF

s^ðKF0Þ

where s : F ! F0is the smooth blow down onto the minimal model.

Proof. If b > 0, then the movable part ofj2KXj is already base point free by Lemma 2.6. The claim is automatically true.

Suppose b¼ 0. Set F :¼ pF . We may write (see 2.4):

S ¼P^a2

i¼1

Fi;

where a2f P₂ðX Þ  1 f 3 and Fi is a smooth fiber of f for each i. Then 2KX1a2F þ Z.

Assume KX
F²>0. Then we have

2K_X³f a2K_X²
F f a²₂ f

P2ðX Þ  12

¼1 4

K_X³ 6wðOXÞ  22

f 1

4ðK_X³þ 4Þ²:

The above inequality is absurd. Thus KX
F²¼ 0 and p^ðKXÞ_jF
hjF ¼ 0. Now we apply the same argument as in the proof of Claim 3.3. So the claim is true. r

Considering the linear system jKX⁰þ 2p^ðKXÞ þ Sj I jSj, which evidently separates di¤erent fibers of f , we get a surjection by the vanishing theorem:

jKX⁰þ 2p^ðKXÞ þ Sj_jF ¼ jKFþ 2s^ðKF0Þj:

Since F is a surface of general type, F_j3KFj is birational except when

K_F²₀; p_gðF Þ

¼ ð1; 2Þ, orð2; 3Þ. Thus F5is birational except when F is of those two types.

From now on, we assume that F is one of the above two types. Then qðF Þ ¼ 0 ac- cording to surface theory. By 2.3, one has qðX Þ ¼ b because R¹foX⁰ ¼ 0. Since we may assume pgðX Þ e 1 by Theorem 3.2 and since wðOXÞ < 0 and b e 1, we see that the only possibility is qðX Þ ¼ b ¼ 1, pgðX Þ ¼ 1 and h²ðOXÞ ¼ 0.

Let D Ajp^ðKXÞj be the unique e¤ective divisor. Since 2D @ 2p^ðKXÞ, there is a hy- perplane section H₂⁰of W⁰in P^{P2ðX Þ1}such that g^ðH₂⁰Þ 1 a2F and 2D¼ g^ðH₂⁰Þ þ Z⁰. Set Z⁰:¼ Zvþ 2Zh, where Zv is the vertical part with respect to the fibration f and 2Zh the horizontal part. Thus

D¼1 2

g^ðH₂⁰Þ þ Zv

þ Zh:

Noting that D is an integral divisor, for the general fiber F , ðZhÞ_jF ¼ D_jF@ s^ðKF0Þ by Claim 4.4.

Considering the Q-divisor

KX⁰þ 4p^ðKXÞ  F  1 a2

Zv 2 a2

Zh;

set

G :¼ 3p^ðKXÞ þ D  1 a2

Zv 2 a2

Zh

and

D0:¼ dGe ¼ 3p^ðKXÞ þ 1 2 a2

 

Zh

 

þ vertical divisors:

For the general fiber F , our G F 1 4  2 a2

 

p^ðKXÞ is nef and big. Therefore, by the vanishing theorem, H¹ðX⁰; K_X⁰þ D0 F Þ ¼ 0.

We then have a surjective map

H⁰ðX⁰; KX⁰þ D0Þ ! H⁰ F ; KFþ 3s^ðKF0Þ þ 1 2 a2

 

Zh

 

jF

! :

If F is a surface with ðK²; pgÞ ¼ ð2; 3Þ, then F_jKFþ3sðK_F0Þþdð1²

a2ÞZhe_jFj is birational on F . Otherwise, since

1 2 a2

 

Zh

 

jF

f 1 2 a2

 

ðZhÞ_jF

 

¼ 1 2 a2

 

D_jF

 

;

[10], Proposition 2.1 implies that F_jKFþ3sðK_F0Þþdð1²

a2ÞZhe_jFj is birational. Thus F5 is bira- tional. r

Theorems 4.1, 4.2 and 4.3, together with 1.4 and 1.5, imply Theorem 1.1.

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Department of Mathematics, National Taiwan University, Taipei, 106, Taiwan and National Center for Theoretical Science, Taipei O‰ce, Taiwan

e-mail: jkchen@math.ntu.edu.tw

School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education

e-mail: mchen@fudan.edu.cn

Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore e-mail: matzdq@nus.edu.sg

Eingegangen 9. Juli 2005, in revidierter Fassung 22. Dezember 2005