GENERAL TYPE
JUNGKAI A. CHEN, MENG CHEN AND DE-QI ZHANG Abstract. Let X be a projective minimal Gorenstein 3-fold of general type with canonical 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 [16, 19] for example) has been attracting more and more mathematicians to the study of algebraic 3-folds. In this paper, we restrict our interest to projective minimal Gorenstein 3-folds X of general type where there still remain many open problems.
Denote by KX the canonical divisor and Φm := Φ|mKX| the
m-canonical map. There have been a lot of works along the line of the canonical classification. For instance, when X is a smooth 3-fold of general type with pluri-genus h0(X, kK
X) ≥ 2, in [17], as an
applica-tion to his research on higher direct images of dualizing sheaves, Koll´ar proved that Φm, with m = 11k + 5, is birational onto its image. This
result was improved by the second author [5] to include the cases m with m ≥ 5k + 6; see also [7], [9] for results when some additional restrictions (like bigger pg(X)) were imposed.
On the other hand, for 3-folds X of general type with q(X) > 0, Koll´ar [17] first proved that Φ225is birational. Recently, the first author
and Hacon [4] proved that Φm is birational for m ≥ 7 by using the
Fourier-Mukai transform. Moreover, Luo [22], [23] has some results for 3-folds of general type with h2(O
X) > 0.
Now for minimal and smooth projective 3-folds, it has been estab-lished that Φm (m ≥ 6) is a birational morphism onto its image after
20 years of research, by Wilson [29] in the year 1980, Benveniste [2] in the year 1986 (m ≥ 8), Matsuki [24] in the year 1986 (m = 7), the second author [6] in the year 1998 (m = 6) and independently by Lee
The first author was partially supported by the National Science Council and National Center for Theoretical Science of Taiwan. The second author was sup-ported by the National Natural Science Foundation of China. The third author is supported by an Academic Research Fund of NUS.
[20], [21] in the years 1999-2000 (m = 6; and also the base point free-ness of m-canonical system for m ≥ 4). A very natural and well-known question arises:
Question 1.1. Let X be a minimal Gorenstein 3-fold of general type. Is Φ5 birational onto its images?
Despite many attempts officially or privately announced, it seems that the birationality of Φ5 for 3-folds (even with the stronger
assump-tion that KX is ample) remains beyond reach. The difficulty lies in
the case with smaller pg(X) or KX3. 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 would predict the non-birationality of the 5-canonical system on certain 3-folds with smaller invariants.
Nevertheless, there is also evidence supporting the birationality of Φ5 for Gorenstein minimal 3-folds X of general type. For instance,
one sees that K3
X ≥ 2 for minimal and smooth X (see 2.1 below).
So an analogy of Fujita’s conjecture would predict that |5KX| gives a
birational map. We recall that Fujita’s conjecture (the freeness part) has been proved by Fujita, Ein-Lazarsfeld [10] and Kawamta [14] when dim X ≤ 4.
The aim of this paper is to answer Question 1.1 which has been around for many years:
Theorem 1.2. Let X be a projective minimal Gorenstein 3-fold of
general type with canonical singularities. Then the m-canonical map
Φm is a birational morphism onto its image for all m ≥ 5.
Example 1.3. The numerical bound ”5” in Theorem 1.2 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 the minimal model S with K2
S = 1 and pg(S) = 2. (For example, take the
product.) Then Φ|4KV| is apparently not birational (see [3]).
1.4. Reduction to birationality. According to [6] or [20], to prove Theorem 1.2, we only need to verify the statement in the case m = 5. On the other hand, the results in [20, 21] show that |mKX| is base
point free for m ≥ 4. So it is sufficient for us to verify the birationality of |5KX| in this paper.
1.5. Reduction to factorial models. According to the work of M. Reid [26] and Y. Kawamata [15] (Lemma 5.1), there is a minimal model
Y with a birational morphism ν : Y −→ X such that KY = ν∗(KX)
and that Y is factorial with at worst terminal singularities. Thus it is sufficient for us to prove Theorem 1.2 for minimal factorial models.
Acknowledgments. We are indebted to H´el`ene Esnault, Christopher Hacon, Yujiro Kawamata, Miles Reid, I-Hsun Tsai, Eckart Viehweg and Chin-Lung Wang for useful conversations or comments on this subject.
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. Taking a special resolution ν : Y −→ X according to Reid ([26]) such that c2(Y )·4 = 0 (see Lemma 8.3 of [25]) for any exceptional divisor 4
of ν. Write KY = ν∗KX+ E where E is exceptional and is mapped to
a finite number of points. Then for m ≥ 2, we have (by the vanishing in [13], [28] or [11]): χ(OX) = χ(OY) = − 1 24KY · c2(Y ) = − 1 24ν ∗K X · c2(Y ). Pm(X) = χ(OX(mKX)) = χ(OY(mν∗KX)) = 1 12m(m − 1)(2m − 1)K 3 X + m 12ν ∗K X · c2(Y ) + χ(OY) = (2m − 1)(m(m − 1) 12 K 3 X − χ(OX)).
The inequality of Miyaoka and Yau ([25], [30]) says that 3c2(Y ) − KY2
is pseudo-effective. This gives ν∗K
X · (3c2(Y ) − KY2) ≥ 0. Noting that
ν∗K
X · E2 = 0 under this situation, we get:
−72χ(OX) − KX3 ≥ 0.
In particular, χ(OX) < 0. So one has:
q(X) = h2(O
X) + (1 − pg(X)) − χ(OX) > 0
whenever pg(X) ≤ 1.
2.2. Suppose that D is any divisor on a smooth 3-fold V . The Riemann-Roch theorem gives:
χ(OV(D)) = D3 6 − KV · D2 4 + D · (K2 V + c2) 12 + χ(OV). Direct calculation shows that
χ(OV(D)) + χ(OV(−D)) =
−KV · D2
2 + 2χ(OV) ∈ Z. Therefore, KV · D2 is an even number.
Now let X be a projective minimal Gorenstein 3-fold of general type. Let D be any divisor on X. Then KX · D2 = KY · (ν∗D)2 is even.
Especially K3
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:
E2p,q:= Hp(B, Rqf∗ωV) =⇒ En := Hn(V, ωV),
one has the following by Serre duality and Corollary 3.2 and Proposition 7.6 on pages 186 and 36 of [17]: h2(O V) = h1(B, f∗ωV) + h0(B, R1f∗ωV), q(V ) := h1(O V) = g(B) + h1(B, R1f∗ωV). 2.4. For µ = 1, 2, we set Φ = ( Φ|KX| if pg(X) ≥ 2, Φ|2KX| otherwise.
Since we always have P2(X) ≥ 4, Φ is a non-trivial rational map.
Let π : X0 −→ X be the a resolution of the base locus of Φ. We
write |π∗(µK
X)| = |M0| + E0. Then we may assume:
(1) X0 is smooth;
(2) the movable part of |µKX0| is |M0|, which is base point free;
(3) E0 is a normal crossing divisor ( hence so is a general member in
|π∗(µK X)|).
We will fix some notation below. The frequently used ones are M,
Z, S, ∆ and Eπ. Denote by g the composition Φ ◦ π. So g : X0 −→
W0 ⊆ PN is a morphism. Let g : X0 −→ Wf −→ Ws 0 be the Stein
factorization of g such that W is normal and f has connected fibers. We can write:
|µKX0| = |π∗(µKX)| + µEπ = |M0| + Z0,
where Z0 is the fixed part and E
π an effective π-exceptional divisors.
On X, one may write µKX ∼ M +Z where M is a general member of
the movable part and Z the fixed divisor. Let S ∈ |M0| be the divisor
corresponding to M, then π∗(M) = S + 4 = S + s X i=1 diEi
with di > 0 for all i. The above sum runs over all those exceptional
divisors of π that lie over the base locus of M. Obviously E0 = 4 +
π∗(Z). On the other hand, one may write E π =
Pt
j=1ejEj where the
sum runs over all exceptional divisors of π. One has ej > 0 for all
1 ≤ j ≤ t because X is terminal. Apparently, one has t ≥ s.
Note that Sing(X) is a finite set (see [19], Corollary 5.18). We may write Eπ = 40+ 400 where 40 (resp. 400) lies (resp. does not lie) over
the base locus of |M|. So if one only requires such a modification π that satisfies 2.4(1) and 2.4(2), one surely has supp(4) = supp(40).
Let d := dim Φ(X). And let L := π∗(K
X)|S, which is clearly nef and
big. Then we have the following:
Lemma 2.5. When d ≥ 2, (L2)2 ≥ (π∗K
X)3(π∗(KX) · S2). Morover,
L2 ≥ 2.
Proof. Take a sufficiently large number m such that |mπ∗(K
X)| 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 π∗(K
X)|H and S|H. Applying the Hodge index theorem, one has
(π∗(K
X)|H · S|H)2 ≥ (π∗(KX)|H)2(S|H)2.
Removing m, we get the first inequality. By 2.2, (π∗K
X)3 is even, hence
≥ 2. Together with π∗(K
X)·S2 > 0, we have the second inequality. ¤
We now state a lemma which will be needed in our proof. The result might be true for all 3-folds with rational singularities.
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 |M| is a movable pencil and that |M| has base points. Then |M| is composed with a rational pencil.
Proof. Take a birational morphism π : X0 −→ X such that X0 is
smooth, that the exceptional divisor Eπ is of simple normal crossing,
and that the map Φ|M | composed with π, becomes a morphism from
X0 to a curve. Take a Stein factorization of the latter morphism 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 Eπ dominates B.
Case 1. Bs|M| contains a curve Γ.
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 Cor. 5.40 of [19]). 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 Γ transversally. We may assume that the strict transform H0 on X0 is smooth, i.e., π is
an embedded resolution of H ⊂ X. Clearly, there is an π-exceptional irreducible divisor E which dominates both Γ and B. Now for a general
H, both H0 and E ∩H0 dominate B. Since the curve E ∩H0 arises from
the resolution π : H0 → H of the indeterminancy of the linear system
|M||H (whose image on X is contained in Γ ∩ H), it is rational. So B
is rational.
Case 2. Bs|M| is a finite set. (The argument below works even when X is log terminal.)
Take a base point P of |M|. Then E = π−1(P ) dominates B, i.e.,
f (E) = B. By Kollar’s Theorem 7.6 in [18], there is an analytic
connected. Suppose g(B) > 0. Then the universal cover h : W −→ B of B is either the affine line C or an open disk in C. By Proposition 13.5 of [12], there is a factorization for the restriction f |U : U −→ B,
say f = h ◦ m, where m : U −→ W is continuous. Note that m(E) is a compact subset of W , so m(E) is a single point. In particular, f (E)
is a point, a contradiction. ¤
3. The case pg ≥ 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 be divisors on S and Λ ⊂ |N| be a
subsystem. Suppose that |N0|
|C = |N0|C|, deg(N|C) = 1 + deg(N0|C) ≥
1 + 2g(C). We consider the following diagram
|N0| −−−→ |Nres 0 |C| y+ef f y+P1 |N| −−−→ |Nres |C| x ⊂ x ⊂ Λ −−−→ Λres C
Suppose furthermore that Λ|C is free and Λ|C ⊃ |N0||C+ P1. Then
Λ|C = |N||C = |N|C|, (∗)
which is very ample and complete.
Proof. By the Riemann-Roch theorem and Serre duality, we have dim |N|C| = 1 + dim |N0|C|. Since there are inclusions |N0||C+ P1 ⊆ Λ|C ⊆
|N||C ⊆ |N|C|, now the equalities (*) in the statement follow from the
dimension counting and the fact that the first inclusion above is strict
by the freeness of Λ|C. ¤
Theorem 3.2. Let X be a projective minimal factorial 3-fold of general
type. Assume pg(X) ≥ 2. Then Φ5 is birational.
Proof. We distinguish cases according to d := dim Φ(X).
Case 1: d = 3. Then pg(X) ≥ 4. Φ5 is birational, thanks to
Theorem 3.1(i) in [9].
Case 2: d = 2. We consider the linear system |KX0+ 3π∗(KX) + S|.
Since KX0+ 3π∗(KX) + S ≥ S and according to Tankeev’s principle, it
is sufficient to verify the birationality of Φ|KX0+3π∗(KX)+S||S. Note that
we have a fibration f : X0 −→ W where a general fiber of f is a smooth
curve C of genus ≥ 2. The vanishing theorem gives:
where L := π∗(K
X)|S is a nef and big divisor on S.
By Lemma 2.5, L2 ≥ 2. According to Reider ([27]), Φ
|KS+3L| is
birational and so is Φ5.
Case 3: d = 1. We set b := g(B). When b > 0, let’s consider the system |M| on X. If |M| has base point, then by 2.6, b = 0, which is a contradiction. Thus we may assume that |M| is free. Then in this situation, Φ5 is birational, which is exactly the statement of Theorem
3.3 in [9]. For reader’s convenience, we sketch the proof here. Meng, can you do this?
From now on, we suppose b = 0. Let F be a general fiber of f and denote by σ : F −→ F0 the contraction onto the minimal model. We
take π to be the composition π1◦ π0 where π0 satisfies 2.4(1) and 2.4(2)
and π1 is a further modification such that π∗(KX) is supported on a
normal crossing divisor.
We may write S ∼ aF where a ≥ pg(X) − 1. And we set L :=
π∗(K
X)|F instead. From the relation
|KX0 + 3π∗(KX) + S||F = |KF + 3L|,
we see that the problem is reduced to the birationality of |KF + 3L|
because |KX0+3π∗(KX)+S| ⊃ |S| apparently separates different fibers
of f . Let ¯F := π∗(F ). We know that KX · ¯F2 is an even number by
2.2.
If KX · ¯F2 > 0, then we have
L2 = π∗(K
X)2· F = KX2 · ¯F ≥ KX · ¯F2 ≥ 2.
Reider’s theorem says that |KF + 3L| gives a birational map.
We are left with only the case KX · ¯F2 = 0. First we have:
Claim 3.3. If KX · ¯F2 = 0, then OF(π∗(KX)|F) ∼= OF(σ∗KF0).
Proof. It is obvious that the claim is true if it holds for π = π0. So we
may assume π = π0. Now
0 = KX · (a ¯F )2 = KX · M2 = π∗(KX) · π∗(M) · S = aπ∗(KX)|F · 4|F,
which means π∗(K
X)|F · 40|F = 0. On the other hand, the definition
of π0 gives 400|F = 0. Thus (Eπ)|F · π∗(KX)|F = 0.
We may write
KF = π∗(KX)|F + G
where G = (Eπ)|F is an effective and contractible (so negative definite)
divisor on F . Note that L is nef and big and that L · G = 0. The uniqueness of the Zariski decomposition shows that σ∗K
F0 ∼ π∗(KX)|F.
We are done. ¤
From the above claim, we have Φ|KF+3L| = Φ|4KF|. We are left to
verify the birationality of Φ5 only when Φ|4KF| fails to be birational,
i.e. when K2
The Kawamata-Viehweg vanishing theorem ([11, 13, 28]) gives
|KX0 + 3π∗(KX) + F ||F = |KF + 3σ∗(KF0)|. (1)
Denote by C a general member of the movable part of |σ∗K
F0|. By [1],
we know that C is a smooth curve of genus 2 and σ(C) is a general member of |KF0|. Applying the vanishing theorem again, we have
|KF + 2σ∗(KF0) + C||C = |KC + 2σ
∗(K
F0)|C|. (2)
Now we may apply Proposition 3.1. Let N0 := K
F + 2σ∗(KF0) + C
and N := (5π∗K
X)|F. Set Λ = |5π∗(KX)||F. It’s clear that N0 < N.
Also note that Λ is free for |5KX| is free.
By (1) above, we see that Λ ⊃ |N0|+ (a fixed effective divisor).
Now restrict to C, computation shows that deg(N0
|C) = 4 and 5 =
deg(N|C) = 1 + deg(N0|C). Therefore, the induced inclusion |N0|C| ,→
|N|C| is given by adding a single point P1.
By (2), we have |N0
|C| = |N0||C. Together with (1), we have Λ|C ⊃
|N0
|C|+P1. Hence by Proposition 3.1, Λ|C = |N|C| gives an embedding.
Because |5π∗K
X||F ⊃ |N0| ⊃ |C| (by (1) above) separates different C
(noting that pg(F ) = 2 and |C| is a rational pencil), Φ5|F is birational.
It is clear that |5π∗K
X| ⊃ |S| separates different fibres F . Thus Φ5 is
birational. ¤
4. Birationality via bicanonical systems
In this section, we shall complete the proof of Theorem 1.2 by study-ing the bicanonical system. We set Φ := Φ2 as stated in 2.4. Denote d2 := dim Φ2(X). We organize our proof according to the value of d2.
Theorem 4.1. Let X be a projective minimal factorial 3-fold of general
type. Assume d2 = 3. Then Φ5 is birational. Proof. Recall that K3
X is even by 2.2, so it’s either > 2 or = 2.
Case 1. The case K3
X > 2.
Pick up a general member S. Let R := S|S. Then |R| is not composed
of a pencil. Thus one obviously has R2 ≥ 2. So the Hodge index
theorem on S yields
π∗(K
X) · S2 = π∗(KX)|S · R ≥ 2.
Set L := π∗(K
X)|S. If KX3 > 2, then Lemma 2.5 gives L2 > 2.
In this case, we must emphasize that we only need such a modifi-cation π that satisfies 2.4(1) and 2.4(2). Namely, we don’t need the normal crossings. Thus we have Supp(4) = Supp(40). This property
is crucial to our proof.
Now the vanishing theorem gives
|KX0 + 2π∗(KX) + S||S = |KS+ 2L|.
Because (2L)2 ≥ 12, we may apply Reider’s theorem again. Assume
such that L · C = 1. Note that R ≤ 2L, and that |R| is base point free and |R| is not composed of a pencil. Thus dim(Φ|R|(C)) = 1.
Because C lies in an algebraic family and S is of general type, we have
g(C) ≥ 2. Since h0(C, R
|C) ≥ 2, the Riemann-Roch theorem on C
and Clifford’s theorem on C, it easily implies that R · C ≥ 2. Because
R · C ≤ 2L · C = 2, one must have R · C = 2. Since
2L = S|S + 4|S + π∗(Z)|S
and C is nef, we have 4|S· C = 0. This implies that 40|S· C = 0. Note
also that 400
|S = 0 for general S. We get (Eπ)|S· C = 0. Therefore
KS· C = (KX0+ S)|S· C = π∗(KX)|S · C + (Eπ)|S · C + S|S · C = 3,
an odd number. This is impossible because C is a free pencil on S. So Φ5 must be birational.
Case 2. The case K3
X = 2.
If L2 ≥ 3, then φ
5 is birational according to the proof in Case 1.
So we may assume L2 = 2. By Lemma 2.5, we have π∗(K
X) · S2 = 2.
Set C = S|S. Then |C| is base point free and is not composed with a
pencil. So C2 ≥ 2. The Hodge index theorem also gives
4 = (π∗(K
X)|S · C)2 ≥ L2· C2 ≥ 4.
The only possibility is L2 = C2 = 2 and L ≡ C. On the other hand,
the equality
4 = 2KX3 = KX2 · (M + Z) = L2+ KX2 · Z = 2 + KX2 · Z
gives K2
X · Z = 2. Take a very big m such that |mKX| is base point
free and take a general member H ∈ |mKX|. By the Hodge index
theorem, 4 = 1
m2(KX · M · H)2 ≥ m12(KX2 · H)(M2 · H) = 2KX · M2.
Thus KX · M2 = 2 and (KX)|H ≡ M|H. Multiplying it by 2, we deduce
that Z|H ≡ M|H. Thus KX · Z · M = m1Z|H · M|H = m1M2 · H = 2. So L · π∗(Z) |S = 2. Since 2C ≡ 2L = π∗(2KX)|S = π∗(M + Z)|S = (S + ∆ + π∗(Z)) |S = C + (∆ + π∗(Z))|S and L2 = L · C = 2, we see that 0 = L · ∆ = C · ∆. (3) Thus KS = (KX0+ S)|S = C + (π∗(KX) + Eπ)|S = (C + L) + ((Eπ)|S) = P +N is the Zariski decomposition by (3) and 2.4. Denote by σ : S −→ S0 the contraction onto the minimal model. Then C + L ∼ σ∗(KS0).
Note that C = S|S and dim |C| ≥ dim |S||S ≥ 2 because |S| gives a
generically finite map. Assume to the contrary that Φ5is not birational.
Then neither is Φ|S|. Denote by d the generic degree of Φ5. Then:
2 = C2 = S3 ≥ d(P
2(X) − 3).
Because d ≥ 2, we see P2(X) = 4 and d = 2. As we have shown in
Step 1 that
we see that Φ|C| : S −→ Ph
0(S,C)−1
is not birational. On the other hand, we may write
2 = C2 ≥ deg(Φ
|C|) deg(Φ|C|(S)).
If h0(S, C) ≥ 4, then deg(Φ
|C|(S)) ≥ 2 and deg Φ|C| = 1, i.e. Φ|C|
is birational which contradicts the assumption. So h0(S, C) = 3 and |C| = |S||S. Therefore Φ|C| : S −→ P2 is generically finite of degree 2.
Let Φ|C| = τ ◦ γ be the Stein factorization with γ : S → S0 a birational
morphism onto a normal surface and τ : S0 → P2 a finite morphism of
degree 2. We can write C = Φ∗
|C|` with a line `.
For a curve E on S, by the projection formula, C.E = `.Φ|C|∗E. So
E is contracted to a point on S0 if and only if E is contracted to a point
on P2(for τ is finite); if and only if E is perpendicular to C ≡ 1
2σ∗(KS0)
(= half of the pull back of KS 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 projection formula again; we denote by Eall the union of these E. By
Zariski Main Theorem, both S \ Eall → S\ (the image of Eall) and
S \ Eall → S0\ (the image of Eall) are isomorphisms (so we identify
them). Both S and S0 are completion of the same S \ E
all by 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 S0 = S.
Since ¯S is normal, Propositions 5.4, 5.5 and 5.7 of [19] imply a
split-ting
τ∗OS¯ = OP2⊕ L
where L is a line bundle. Thus we see that
q(S) = q( ¯S) = h1( ¯S, τ
∗OS¯) = 0.
Since S is nef and big on X0, the long exact sequence
0 = H1(KX0 + S) −→ H1(KS) −→ H2(KX0) −→ H2(KX0 + S) = 0
gives q(X) = q(X0) = q(S) = 0. Noting that χ(O
X) < 0, we naturally
have pg(X) ≥ 2. By Theorem 3.2, Φ5 is birational, a contradiction.
Therefore we have proved the birationality of Φ5. ¤
Theorem 4.2. Let X be a projective minimal factorial 3-fold of general
type. Assume d2 = 2. Then Φ5 is birational. Proof. Case 1. K3
X > 2.
When d2 = 2, f : X0 −→ W is a fibration onto a surface W . Taking a
further modification, we may even get a smooth base W . Denote by C a general fiber of f . Then g(C) ≥ 2. Pick up a general member S which is an irreducible surface of general type. We may write S|S ∼
Pa2
i=1Ci
where a2 ≥ P2(X) − 2. Since KX3 > 2, we have a2 ≥ P2(X) − 2 ≥ 3.
Set L := π∗(K
X)|S. Then L is nef and big. Since π∗(KX) · S2 =
(π∗(K
X)|S · S|S)S ≥ 3(π∗(KX)|S · C)S ≥ 3, Lemma 2.5 gives L2 ≥ 4.
The vanishing theorem gives
Assume that Φ5 is not birational. Then neither is Φ|KS+2L| for a
general S. Because (2L)2 ≥ 10, Reider’s theorem ([27]) tells us that
there is a free pencil C0 on S such that L · C0 = 1. Since 2 = C0· 2L ≥
C0.S
|S = a2C0· C ≥ 3C0.C, we have C · C0 = 0. So C0 lies in the same
algebraic family as that of C. We may write 2L ≡ a2C + G
where G = (∆ + π∗(Z))
|S ≥ 0 and a2 ≥ 3. Since 2L − C − a12G ≡
(2 − 2
a2)L is nef and big, Kawamata-Viehweg vanishing theorem gives
H1(S, K
S+ d2L − C − a12Ge) = 0. Thus we get a surjection:
H0(S, K S+ d2L − 1 a2 Ge) −→ H0(C, K C + D) where D := d2L − 1 a2Ge|C with deg(D) ≥ (2 − 2 a2)L · C > 1. Note
that |KS+ 2L| can separate different C. If deg(D) ≥ 3, then |KC+ D|
defines an embedding, and so does |KS+ 2L|, a contradiction.
So suppose deg(D) = 2. We now apply Proposition 3.1. Let N0 be
the movable part of KS + d2L − a12Ge and let N = π∗(5KX)|S. Set
Λ := |5π∗(K
X)||S. As in the proof of Theorem 3.2, we have Λ ⊃ |N0|+
(a fixed effective divisor), |N0|
|C = |KC+ D|, N0 ≤ N and deg(N|C) =
1 + deg(N0
|C) = 2g(C) + 1 = 5 by the calculation:
4 ≤ (2g(C) − 2) + 2 = N0· C ≤ N · C = 5π∗K
X · C = 5.
By Proposition 3.1, Λ|C = |N|C| gives an embedding. It is clear that
|5π∗K
X| ⊃ |S| separates different S, and |5π∗KX||S(⊃ the movable
part of |KS+ 2L|) separates different C. Thus Φ5 is birational. This
is again a contradiction. Case 2. K3
X = 2.
We first consider the case L2 ≥ 3. On the surface S, we are reduced
to study the linear system |KS + 2L|. We have
2L ∼ S|S + G = a2
X
i=1
Ci+ G
where a2 ≥ h0(S, S|S)−1 ≥ P2(X)−2 ≥ 2. Denote by C a general fiber
of f : X0 −→ W . If a
2 ≥ 3, the proof in Case 1 already works. So we
assume a2 = 2, then P2(X) = 4, and the image of the fibration Φ|S|S| : S −→ P2 is a quadric curve which is a rational curve. This means that |C| is composed with a rational pencil. Assume that |KS + 2L| does
not give a birational map. Then Reider’s theorem says that there is a free pencil C0 on S such that L · C0 = 1. We claim that C0 is the
same pencil as C. In fact, otherwise C0 is horizontal with respect to
C and C · C0 > 0. Since C is a rational pencil, C · C0 ≥ 2. Therefore
L · C0 ≥ 2, a contradiction. So C0 lies in the same family as that of C
So |KS + 2L| distinguishes different elements in |C|. The vanishing theorem gives H0(S, K S+ d2L − 1 2Ge) −→ H 0(C, K C + Q) where Q = d2L − C − 1 2Ge|C is an effective divisor on C. If |KC + Q|
is not birational, neither is |KC|. So C must be a hyper-elliptic curve.
Suppose Φ5 is not birational. Then Φ5 must be a morphism of generic
degree 2. Set s = Φ5 : X −→ W5 ⊂ PN. Then 5KX = s∗(H) for a very
ample divisor H on the image W5. So
5 = 5π∗(K
X) · C = 2 deg(H|s(π(C))) = 2 degPN s(π(C))
which is a contradiction. Thus Φ5 must be birational under this
situa-tion.
Next we consider the case L2 = 2. Lemma 2.5 says 2 = π∗(K
X)·S2 =
a2L · C. We see that a2 = 2 and L · C = 1. We still consider the
linear system |KS+ 2L|. As above, a2 = 2 implies that |C| is a rational
pencil. Since KS+2L ≥ C, we see that |KS+2L| distinguishes different
elements in |C|. By the same argument as above, we have
|KS + 2L||C ⊃ |KC + Q| ⊃ |KC|.
If Φ5 is not birational, then neither is Φ|KS+2L|. This means that C
must be a hyper-elliptic curve and Φ5 is of generic degree 2. With the
property that |5KX| is base point free, we also have a contradiction as
in the previous case. So Φ5 is birational. ¤
Theorem 4.3. Let X be a projective minimal factorial 3-fold of general
type. Assume d2 = 1. Then Φ5 is birational.
Proof. When X is smooth, this theorem was established in [7]. Our
result is a generalization.
Taking the modification π as in 2.4, we get an induced fibration
f : X0 −→ W and B := W is a smooth curve of genus b := g(B). By
Lemma 2.1 of [8], we know that 0 ≤ b ≤ 1. Let F be a general fiber of
f .
Claim 4.4. We have
OF(π∗(KX)|F) ∼= OF(σ∗(KF0))
where σ : F −→ F0 is the contraction onto the minimal model.
Proof. If b > 0, then the movable part of |2KX| is already base point
free by Lemma 2.6. The claim is automatically true. Suppose b = 0. Set ¯F := π∗F . We may write (see 2.4):
S =
a2
X
i=1
where a2 ≥ P2(X) − 1 ≥ 3 and Fi is a smooth fiber of f for each i.
Then 2KX ≡ a2F + Z. Assume K¯ X · ¯F2 > 0. Then we have
2K3 X ≥ a2KX2 · ¯F ≥ a22 ≥ (P2(X) − 1)2 = 1 4(K 3 X − 6χ(OX) − 2)2 ≥ 1 4(K 3 X + 4)2.
The above inequality is absurd. Thus KX· ¯F2 = 0 and π∗(KX)|F·4|F =
0. Now we apply the same argument as in the proof of Claim 3.3. Thus
the claim is true. ¤
Considering the linear system |KX0 + 2π∗(KX) + S| ⊃ |S|, which
apparently separates different fibers of f , we get a surjection by the vanishing theorem:
|KX0 + 2π∗(KX) + S||F = |KF + 2σ∗(KF0)|.
Since F is a surface of general type, Φ|3KF| is birational except when
(K2
F0, pg(F )) = (1, 2), or (2, 3). Thus Φ5 is birational except when F is
of those two types.
From now on, we assume that F is one of the above two types. Then q(F ) = 0 according to surface theory. By 2.3, one has q(X) = b because R1f
∗ωX0 = 0. Since we may assume pg(X) ≤ 1 by Theorem 3.2, χ(OX) < 0 and b ≤ 1, we see that the only possibility is q(X) = b = 1,
pg(X) = 1 and h2(OX) = 0.
Let D ∈ |π∗(K
X)| be the unique effective divisor. Since 2D ∼
2π∗(K
X), there is a hyperplane section H20 of W0 in PP2(X)−1 such that g∗(H0
2) ≡ a2F and 2D = g∗(H20) + Z0. Set Z0 := Zv+ 2Zh, where Zv is
the vertical part with respect to the fibration f and 2Zh the horizontal
part. Thus
D = 1
2(g
∗(H0
2) + Zv) + Zh.
Noting that D is a integral divisor, for a general fiber F , (Zh)|F =
D|F ∼ σ∗(KF0).
Considering the Q-divisor
KX0 + 4π∗(KX) − F − 1 a2Zv− 2 a2Zh, set G := 3π∗(K X) + D − 1 a2Zv− 2 a2Zh and D0 := dGe = 3π∗(KX) + d(1 − 2 a2 )Zhe + vertical divisors.
For a general fiber F , G−F ≡ (4− 2
a2)π
∗(K
X) is nef and big. Therefore,
by the vanishing theorem, H1(X0, K
We then have a surjective map H0(X0, KX0 + D0) −→ H0(F, KF + 3σ∗(KF0) + d(1 − 2 a2 )Zhe|F). If F is a surface with (K2, p g) = (2, 3), then Φ|KF+3σ∗(KF0)+d(1−a22)Zhe|F|
is birational on F . Otherwise, since
d(1 − 2 a2 )Zhe|F ≥ d(1 − 2 a2 )(Zh)|Fe = d(1 − 2 a2 )D|Fe,
Proposition 2.1 of [9] implies that Φ|KF+3σ∗(K
F0)+d(1−a22)Zhe|F| is
bira-tional. Thus Φ5 is birational. ¤
Theorems 4.1, 4.2 and 4.3 imply Theorem 1.2. References
[1] W. Barth, C. Peter, A. Van de Ven, Compact complex surface, Springer-Verlag, 1984.
[2] X. Benveniste, Sur les applications pluricanoniques des vari´et´es de type tr`es
g´en´eral en dimension 3, Amer. J. Math. 108 (1986), 433-449.
[3] E. Bombieri, Canonical models of surfaces of general type. Inst. Hautes tudes Sci. Publ. Math. 42 (1973), 171–219.
[4] Jungkai A. Chen and C. Hacon, Linear series of irregular varieties, Algebraic Geometry in East Asia, Japan, 2002, World Scientific Press.
[5] Meng Chen, On the Q-divisor method and its application. J. Pure Appl. Alge-bra 191 (2004), 143–156.
[6] Meng Chen, On pluricanonical maps for threefolds of general type, J. Math. Soc. Japan 50(1998), 615-621.
[7] Meng Chen, Kawamata-Viehweg vanishing and the quint-canonical map of a
complex 3-fold, Communications in Algebra 27(1999), 5471-5486.
[8] Meng Chen, On canonically derived families of surfaces of general type over
curves, Communications in Algebra 29(2001), 4597-4618.
[9] Meng Chen, Canonical stability of 3-folds of general type with pg ≥ 3, Internat.
J. Math. 14(2003), 515-528.
[10] L. Ein, R. Lazarsfeld, Global generation of pluricanonical and adjoint linear
systems on smooth projective threefolds, J. Amer. Math. Soc. 6(1993), 875-903.
[11] H. Esnault, E. Viehweg, Lectures on Vanishing Theorems. DMV-Seminar 20(1992), Birkh¨auser, Basel-Boston-Berlin.
[12] W. Fulton, Algebraic Topology, Graduate Texts in Mathematics 153, Springer-Verlag.
[13] Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann. 261(1982), 43-46.
[14] Y. Kawamata, On Fujita’s freeness conjecture for 3-folds and 4-folds. Math. Ann. 308 (1997), 491–505.
[15] Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and
its application to degenerations of surfaces. Ann. of Math. 127(1988), 93-163
[16] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model
problem, Adv. Stud. Pure Math. 10(1987), 283-360.
[17] J. Koll´ar, Higher direct images of dualizing sheaves I, Ann. Math. 123(1986), 11-42; II, ibid. 124(1986), 171-202.
[18] J. Koll´ar, Shafarevih maps and plurigenera of algebraic varieties, Invent. Math. 113(1993), 177-215.
[19] J. Koll´ar, S. Mori, Birational geometry of algebraic varieties, 1998, Cambridge Univ. Press.
[20] S. Lee, Remarks on the pluricanonical and adjoint linear series on projective
threefolds, Commun. Algebra 27(1999), 4459-4476.
[21] S. Lee, Quartic-canonical systems on canonical threefolds of index 1, Comm. Algebra 28(2000), 5517-5530.
[22] T. Luo, Global 2-forms on regular 3-folds of general type, Duke Math. J. 71 (1993), no. 3, 859-869.
[23] T. Luo, Plurigenera of regular threefolds, Math. Z. 217 (1994), no. 1, 37-46. [24] K. Matsuki, On pluricanonical maps for 3-folds of general type, J. Math. Soc.
Japan 38(1986), 339-359.
[25] Y. Miyaoka, The pseudo-effectivity of 3c2− c21 for varieties with numerically
effective canonical classes, Algebraic Geometry, Sendai, 1985. Adv. Stud. Pure
Math. 10(1987), 449-476.
[26] M. Reid, Minimal models of canonical 3-folds, Adv. Stud. Pure Math. 1(1983), 131-180.
[27] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. Math. 127(1988), 309-316.
[28] E. Viehweg, Vanishing theorems, J. reine angew. Math. 335(1982), 1-8. [29] P.M.H. Wilson, The pluricanonical map on varieties of general type, Bull.
Lon-don Math. Soc. 12(1980), 103-107.
[30] S. T. Yau, On the Ricci curvature of a complex Kaehler manifold and the
complex Monge-Ampere equations, Comm. Pure Appl. Math. 31(1978),
339-411.
Department of Mathematics, National Taiwan University, Taipei, 106, Taiwan
E-mail address: jkchen@math.ntu.edu.tw
School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China
E-mail address: mchen@fudan.edu.cn
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore
