THE ϕ_{4} OF GORENSTEIN 3-FOLDS OF GENERAL TYPE

Meng Chen

Introduction

This paper is devoted to the birational classification of algebraic Gorenstein 3folds. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. According to [2], [5, 6], [9], [15], [16] and [20], we have the following theorem.

Theorem 0. Suppose X is a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Then the following holds:

(i) the m-canonical map ϕ_{m} is a birational morphism for all m ≥ 6.

(ii) ϕ_{5} is birational with possible exception for K_{X}^{3} = 2 and p_{g}(X) ≤ 2. ϕ_{5} is a generically finite morphism. (No counter examples found yet to the birationality of ϕ_{5}.)

Naturally one wants to know the behavior of ϕ_{m} (m ≤ 4). We observed that some people have been studying the base point freeness of |4K_{X}|. We are more curious about some birational properties of ϕ_{4}. Our approach is different from theirs.

In order to make our statement simpler, let us first fix the terminologies. X is refered to as ϕ_{4}-standard if there exists a fibration f : X′ ⟶ C onto a projective curve C, where X′ is birationally equivalent to X and the general fiber of f is a smooth projective surface of general type with invariants K^{2} = 1 and p_{g} = 2. X is called ϕ_{4}-semi-standard if X is fibred by curves of genus two, i.e. there is a fibration g : X′ ⟶ W onto a normal projective surface W where X′ is birationally equivalent to X and the general fiber of g is a smooth projective curve of genus two. If X is ϕ_{4}-standard, one can easily see that X is ϕ_{4}-semi-standard by taking the relatively canonical map of f.

It is well known that the 4-canonical map of a smooth projective surface of general type is birational if and only if (K^{2}, p_{g}) ≠ (1, 2). This leads to the trivial fact that, if X is ϕ_{4}-standard, the 4-canonical map ϕ_{4} of X fails to be birational. A very natural question is whether the converse is true. In this paper, we would like to study this problem and to show that the converse is true under some reasonable

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 14E05, 14C20, 14Q15.

Project partially supported by both the NNSFC and the post-doc fellowship of the Georg-August-Universität Göttingen.

conditions. Another natural question is whether ϕ_{4} is always generically finite. If one can verify the base point freeness of |4K_{X}|, then ϕ_{4} is automatically generically finite. We shall study in an alternative way giving a direct and elementary proof. Our results are as follows.

Theorem 1. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. The following holds.

(i) Suppose p_{g}(X) ≥ 41 and dimϕ_{1}(X) ≠ 2. Then ϕ_{4} is birational if and only if X is not ϕ_{4}-standard.

(ii) Suppose p_{g}(X) ≥ 41 and X is not ϕ_{4}-semi-standard. Then ϕ_{4} is birational. (iii) ϕ_{4} is generically finite.

Throughout the ground field is assumed to be algebraically closed of characteristic 0. For a ℚ-divisor D on a smooth variety V, we denote by ⌜D⌝ the round-up of D, which is the minimum integral divisor such that ⌜D⌝ − D ≥ 0. ∼_{lin} means linear equivalence. ∼_{num} means numerical equivalence.

I would like to thank F. Catanese for fruitful discussions during my preparation for this note. Special thanks are due to hospital faculty members of the Mathematisches Institut der Universität Göttingen.

1. Proof of the main theorem

Definition 1.1. A normal variety X is called Gorenstein if the dualizing sheaf ω_{X} is invertible and X is Cohen-Macaulay.

We refer to [17] for the definitions of canonical, terminal singularities.

Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. It is well known that K_{X}^{3} is a positive even integer, χ(O_{X}) < 0 and that

P_{m}(X) := h^{0}(X, O_{X}(mK_{X})) = (2m − 1)[\frac{m(m − 1)}{12}K_{X}^{3} − χ(O_{X})].  (1.1)

Suppose p_{g}(X) ≥ 2. We can define the canonical map ϕ_{1}. Set

K_{X} ∼_{lin} M_{1} + Z_{1},

where M_{1} is the movable part of |K_{X}| and Z_{1} the fixed one. Taking the birational modification π : X′ ⟶ X, according to Hironaka, such that

(1) X′ is smooth;

(2) the movable part of |π^{∗}(K_{X})| is base point free;

(3) π^{∗}(K_{X}) has supports with only normal crossings.

Denote by g the composition ϕ_{1} ∘ π. So

g : X′ ⟶ W′ ⊆ ℙ^{p_{g}(X)−1}

is a morphism. Let

g : X′ ⟶^{f} W ⟶^{s} W′

be the Stein factorization of g. We can write

π^{∗}(M_{1}) ∼_{lin} S_{1} + E_{1},

where S_{1} is the movable part. Then we have

π^{∗}(K_{X}) ∼_{lin} S_{1} + E′,

where E′ = E_{1} +π^{∗}(Z_{1}) is the fixed part of |π^{∗}(K_{X})|. We note that 1 ≤ dim(W) ≤ 3. We shall formulate our proof according to dim(W).

Remark 1.2. Although [5], [6] and [16] only treated smooth minimal 3-folds, the method is still effective for Gorenstein minimal 3-folds. In order to avoid unnecessary redundancy, we would like to cite several basic facts from there without giving the proof.

Theorem 1.3. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Suppose p_{g}(X) ≥ 5. If dimϕ_{1}(X) = 3, then ϕ_{4} is birational.

Proof. It’s obvious that a general member S_{1} is a smooth projective surface of general type. Because p_{g}(X) > 0, it is sufficient to verify the birationality for ϕ_{4}|_{S_{1}} by virtue of the Tankeev principle. We consider the system

|K_{X′} + 2π^{∗}(K_{X}) + S_{1}|.

The vanishing theorem gives

|K_{X′} + 2π^{∗}(K_{X}) + S_{1}| |_{S_{1}}= |K_{S_{1}} + 2L|,

where L := π^{∗}(K_{X})|_{S_{1}} is a nef and big divisor on S_{1}. If |L| gives a birational map, then so does |K_{S_{1}} + 2L|. Otherwise, |L| gives a generically finite map of degree ≥ 2. Noting that h^{0}(S_{1}, L) ≥ p_{g}(X) − 1 ≥ 4, we have L^{2} ≥ 2(h^{0}(S_{1}, L) − 2) ≥ 4. If |K_{S_{1}} + 2L| doesn’t give a birational map, then there is a free pencil of curves on S_{1} with a general irreducible element C such that 2L · C ≤ 2 according to Reider’s result ([18, Corollary 2]). The only possibility is L · C = 1. On the other hand, L · C ≥ 2 since |L| gives a generically finite map on C and C is a curve of genus ≥ 2. The contradiction shows that

Φ_{|K_{S_{1}} +2L|}

is birational. Therefore ϕ_{4} is birational. □

Theorem 1.4. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Suppose p_{g}(X) ≥ 41 and X is not ϕ_{4}-standard. If dimϕ_{1}(X) = 1, then ϕ_{4} is birational.

Proof. In this case, W is a smooth projective curve. We have a fibration f : X′ ⟶ W. Denote b := g(W). Let F be a general fiber of f. Then F is a smooth projective surface of general type. In general position, S_{1} can split into a sum of different fibers, i.e.

S_{1} ∼_{lin} ∑_{i=1}^{a} F_{i},

where a ≥ p_{g}(X) − 1. The vanishing theorem gives the surjective map

H^{0}(X′, K_{X′} + 2π^{∗}(K_{X}) + ∑ F_{i}) ⟶ ⊕_{i=1}^{a}H^{0}(F_{i}, K_{F_{i}} + 2π^{∗}(K_{X})|_{F_{i}}) ⟶ 0.

This means that ϕ_{4} can distinguish general different fbers of f. In order to prove the theorem, it is sufficient to verify the birationality of ϕ_{4}|_{F} for a general fiber F. Denote \overline{F} := π(F). Then M_{1} ∼_{num} a\overline{F}. Noting that \overline{F}^{2} is a quasi effective 1-cycle on X, we have K_{X} · \overline{F}^{2} ≥ 0. Let σ : F ⟶ F_{0} be the contraction onto the minimal model F_{0} of F.

Suppose K_{X} · \overline{F}^{2} = 0. Then we have

O_{F} (π^{∗}(K_{X})|_{F}) ≅ O_{F} (σ^{∗}(K_{F_{0}}))  (1.2)

according to [6, Lemma 2.3]. We have

π^{∗}(K_{X}) ∼_{num} aF + E′.

Thus

π^{∗}(K_{X}) − F − \frac{1}{a}E′ ∼_{num} (1 − \frac{1}{a})π^{∗}(K_{X})

is a nef and big ℚ-divisor, since a > 1 under the condition of the theorem. Denote

G := ⌜π^{∗}(K_{X}) − \frac{1}{a}E′⌝.

The Kawamata-Viehweg vanishing theorem yields

|K_{X′} + 2π^{∗}(K_{X}) + G| |_{F} =| K_{F} + 2L + G|_{F} |,  (1.3)

where L := π^{∗}(K_{X})|_{F} ∼_{lin} σ^{∗}(K_{F_{0}}). We can see that

G|_{F} = ⌜(1 − \frac{1}{a})E′⌝|_{F}

is an effective divisor. Because X is not ϕ_{4}-standard, F can’t be a surface with (K_{F_{0}}^{2}, p_{g}(F)) = (1, 2). If (K_{F_{0}}^{2}, p_{g}(F)) ≠ (2, 3), then Φ_{|3K_{F} |} is birational. Since

Φ_{|K_{F} +2L|} = Φ_{|3K_{F} |},

we see that ϕ_{4}|_{F} is birational and so is ϕ_{4}. If (K_{F_{0}}^{2}, p_{g}(F)) = (2, 3), we can show that

Φ_{|K_{F} +2L+G|_{F} |}

is birational. In fact, we have

K_{F} + 2L + G|_{F} ≥ K_{F} + 2L + ⌜(1 − \frac{1}{a})E′|_{F} ⌝,

where

(1 − \frac{1}{a})E′|_{F} ∼_{num} (1 − \frac{1}{a})π^{∗}(K_{X})|_{F}

is a nef and big ℚ-divisor. It is well known that |σ^{∗}(K_{F_{0}})| gives a generically finite map ([1]). For simplicity, we can suppose the movable part of |σ^{∗}(K_{F_{0}})| is base point free and C is a general member in the movable part of this system. It’s sufficient to prove the birationality of

Φ_{|K_{F} +2L+⌜(1− \frac{1}{a})E′|_{F} ⌝|}|_{C}.

We study the system

|K_{F} + L + ⌜(1 − \frac{1}{a})E′|_{F} ⌝ + C|.

The vanishing theorem gives

|K_{F} + L + ⌜(1 − \frac{1}{a})E′|_{F} ⌝ + C| |_{C}=| K_{C} + L|_{C} + D |,

where deg(L|_{C}) ≥ 2 and D is a divisor of degree > 0. Obviously, | K_{C} + L|_{C} + D | gives an embedding. Thus ϕ_{4}|_{F} is birational and so is ϕ_{4}.

Suppose K_{X} · \overline{F}^{2} > 0. We want to show that ϕ_{4}|_{F} is also birational. In this case, (1.2) doesn’t hold. However, we still have (1.3). First we have to study |2L|. We claim that |2L| gives a generically finite map whenever p_{g}(X) ≥ 41. Suppose M_{2} is the movable part of |2K_{X′}|. Then M_{2} ≤ 2π^{∗}(K_{X}). It’s obvious that

K_{X′} + G ≤ 2K_{X′}.

Denote by M′_{2} the movable part of |K_{X′} + G|. Then M′_{2} ≤ M_{2}. The Kawamata-Viehweg vanishing theorem gives the surjective map

H^{0}(X′, K_{X′} + G) ⟶^{α} H^{0}(F, K_{F} + G|_{F}) ⟶ 0.

We also have a natural map

H^{0}(X′, M′_{2}) ⟶^{β} H^{0}(F, M′_{2}|_{F}).

When p_{g}(X) ≥ 41, we have p_{g}(F) ≥ q(F) ≥ 5 by [7, Theorem 2(3)]. Thus |K_{F} | can’t be composed of a pencil of curves according to [22]. Denote by H the movable part of |K_{F} |. We have

h^{0}(F, M′_{2}|_{F}) ≥ dim_{ℂ}im(β)
=dim_{ℂ}im(α) = h^{0}(F, K_{F} + G|_{F}).

Whereas, M′_{2}|_{F} ≤ K_{F} +G|_{F}. We see that H ≤ M′_{2}|_{F}. Thus |M_{2}|_{F} | is not composed of a pencil of curves and neither is |2L|. We have H ≤ 2L. If |H| already gives a birational map, so does |K_{F} + 2L + G|_{F} |. Otherwise,

2L · H ≥ H^{2} ≥ 2(p_{g}(F) − 2) ≥ 6.

Thus L · H ≥ 3. For simplicity, we can suppose |H| is base point free. This means that we can take H be a smooth curve. Using the vanishing theorem again, we have

|K_{F} + ⌜(1 − \frac{1}{a})E′|_{F} ⌝ + H| |_{H}= |K_{H} + D_{0}|,

where D_{0} is a divisor on the curve H with

deg(D_{0}) ≥ (1 − \frac{1}{a})E′|_{F} · H = (1 − \frac{1}{a})L · H > 2.

So K_{H} + D_{0} is very ample. Noting that

K_{F} + ⌜(1 − \frac{1}{a})E′|_{F} ⌝ + H ≤ K_{F} + G|_{F} + 2L,

we see that Φ_{|K_{F} +G|_{F} +2L|}|_{H} is birational and so is Φ_{|K_{F} +G|_{F} +2L|}. This shows that ϕ_{4} is birational. □

Theorem 1.5. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Suppose p_{g}(X) ≥ 5 and X is not ϕ_{4}-semi-standard. If dimϕ_{1}(X) = 2, then ϕ_{4} is birational.

Proof. In this case, we have a fibration f : X′ ⟶ W onto a normal projective surface W. Let C be a general fiber of f. Because X is not ϕ_{4}-semi-standard, C is a smooth curve of genus ≥ 3. We can see that

S_{1}|_{S_{1}} ∼_{lin} ∑_{i=1}^{a_{2}} C_{i} ∼_{num} a_{2}C,

where a_{2} ≥ p_{g}(X) − 2 ≥ 3 and we take C be a smooth fiber contained in S_{1}. Note that a general member S_{1} is a smooth projective surface of general type. The vanishing theorem gives

|K_{X′} + 2π^{∗}(K_{X}) + S_{1}| |_{S_{1}}= |K_{S_{1}} + 2L|,

where L := π^{∗}(K_{X})|_{S_{1}} is nef and big and

h^{0}(S_{1}, L) ≥ h^{0}(S_{1}, S_{1}|_{S_{1}}) ≥ p_{g}(X) − 1.

Suppose |L| is not composed of a pencil of curves. If |L| gives a birational map, so does |K_{S_{1}} + 2L|. Otherwise,

L^{2} ≥ 2(h^{0}(S_{1}, L) − 2) ≥ 4.

If |K_{S_{1}} + 2L| doesn’t give a birational map, according to Reider, there is a free pencil on S_{1} with a general irreducible member \overline{C} such that 2L·\overline{C} ≤ 2. This means L · \overline{C} = 1. This is impossible, because |L| gives a finite map on \overline{C} and \overline{C} is a curve of genus ≥ 2. Thus Φ_{|K_{S_{1}} +2L|} is birational. So ϕ_{4} is birational.

Suppose |L| is composed of a pencil of curves. Since

L ≥ S_{1}|_{S_{1}},

we can see that a generic irreducible element of the movable part of |L| is a smooth fiber C contained in S_{1}. We have

L^{2} ≥ L · S_{1}|_{S_{1}} ≥ a_{2} ≥ 3.

If |K_{S_{1}} + 2L| doesn’t give a birational map, according to Reider, there is a free pencil on S_{1} with a general irreducible element \overline{C} such that 2L · \overline{C} ≤ 2. The only possibility is L · \overline{C} = 1. Obviously, \overline{C} should be algebraically equivalent to C. Otherwise, dimΦ_{|L|}(\overline{C}) = 1. Which means L · \overline{C} ≥ 2, since \overline{C} is a curve of genus ≥ 2. Therefore we have seen that \overline{C} is actually a fiber of f. So we should have L · C = 1. We want to derive a contradiction by proving that L · C ≥ 2. We can write

L ∼_{lin} S_{1}|_{S_{1}} + J
∼_{num} a_{2}C + J,

where J is an effective divisor on S_{1} and C is contained in S_{1}. So

L − C − \frac{1}{a_{2}} J ∼_{num} (1 − \frac{1}{a_{2}} )L

is a nef and big ℚ-divisor. Considering the system

|K_{S_{1}} + L + ⌜L − \frac{1}{a_{2}} J⌝|,

we get from the Kawamata-Viehweg vanishing theorem that

|K_{S_{1}} + L + ⌜L − \frac{1}{a_{2}} J⌝| |_{C}=| K_{C} + L|_{C} + ⌜L − \frac{1}{a_{2}} J⌝|_{C} | .

We shall use a parallel analysis to the one in the proof of Theorem 1.4. Denote by M_{4} the movable part of |4K_{X′}|. Then M_{4} ≤ 4π^{∗}(K_{X}). Denote by M′_{4} the movable part of

|K_{X′} + 2π^{∗}(K_{X}) + S_{1}|.

Then M′_{4} ≤ M_{4}. Denote by N the movable part of |K_{S_{1}} + 2L|. We have the exact sequence

H^{0}(X′, K_{X′} + 2π^{∗}(K_{X}) + S_{1}) ⟶^{α_{1}} H^{0}(S_{1}, K_{S_{1}} + 2L) ⟶ 0

and the natural map

H^{0}(X′, M′_{4}) ⟶^{β_{1}} H^{0}(S_{1}, M′_{4}|_{S_{1}}).

Since M′_{4}|_{S_{1}} ≤ K_{S_{1}} + 2L and

h^{0}(S_{1}, M′_{4}|_{S_{1}}) ≥ dim_{ℂ}im(β_{1})
=dim_{ℂ}im(α_{1}) = h^{0}(S_{1}, K_{S_{1}} + 2L),

we see that M′_{4}|_{S_{1}} ≥ N. Denote by N′ the movable part of

|K_{S_{1}} + L + ⌜L − \frac{1}{a_{2}} J⌝|.

Then N ≥ N′. We have the surjective map

H^{0}(S_{1}, K_{S_{1}} + L + ⌜L − \frac{1}{a_{2}} J⌝) ⟶^{ρ}

H^{0}(C, K_{C} + L|_{C} + ⌜L − \frac{1}{a_{2}} J⌝|_{C}) ⟶ 0

and the natural map

H^{0}(S_{1}, N′) ⟶^{ψ} H^{0}(C, N′|_{C}).

So

h^{0}(C, N′|_{C}) ≥ dim_{ℂ}im(ψ) = dim_{ℂ}im(ρ)
=h^{0}(K_{C} + L|_{C} + ⌜L − \frac{1}{a_{2}} J⌝|_{C}).

Since

(L − \frac{1}{a_{2}} J) · C = (1 − \frac{1}{a_{2}} )L · C > 0,

we see that

h^{0}(K_{C} + L|_{C} + ⌜L − \frac{1}{a_{2}} J⌝|_{C}) ≥ g(C) + 1.

Thus h^{0}(C, N′|_{C}) ≥ g(C)+1. The R-R on C shows at once that N′·C ≥ 2g(C) ≥ 6, because g(C) ≥ 3. Thus we have

4π^{∗}(K_{X})|_{S_{1}} · C ≥ N′ · C ≥ 6.

We get L · C ≥ 2, a contradiction. Thus |K_{S_{1}} + 2L| gives a birational map and so ϕ_{4} is birational. □

Theorems 1.3, 1.4 and 1.5 directly imply (i) and (ii) of Theorem 1.

Theorem 1.6. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Then ϕ_{4} is generically finite.

Proof. Since we are treating the general case without any assumption on p_{g}(X), we can’t consider the canonical map. However we have p_{2}(X) ≥ 4 according to (1.1). So we can study ϕ_{2}.

Set

2K_{X} ∼_{lin} M_{2} + Z_{2},

where M_{2} is the movable part of |2K_{X}| and Z_{2} the fixed one. Taking the birational modification π_{2} : X′ ⟶ X, according to Hironaka, such that

(1) X′ is smooth;

(2) the movable part of |2π_{2}^{∗}(K_{X})| is base point free;

(3) both π_{2}^{∗}(2K_{X}) and π_{2}^{∗}(4K_{X}) have supports with only normal crossings. Denote by g_{2} the composition ϕ_{2} ∘ π_{2}. So

g_{2} : X′ ⟶ W′_{2} ⊆ ℙ^{P_{2}(X)−1}

is a morphism. Let

g_{2} : X′ ⟶^{f_{2}} W_{2} ⟶^{s_{2}} W′_{2}

be the Stein factorization of g_{2}. We can write

π_{2}^{∗}(M_{2}) ∼_{lin} S_{2} + E_{2},

where S_{2} is the movable part. Then we have

π_{2}^{∗}(2K_{X}) ∼_{lin} S_{2} + E′_{2},

where E′_{2} = E_{2} + π_{2}^{∗}(Z_{2}) is the fixed part of |π_{2}^{∗}(2K_{X})|. We only have to consider the case when dimϕ_{2}(X) < 3.

Suppose dimϕ_{2}(X) = 1. We have a fibration f_{2} : X′ ⟶ W_{2} onto a smooth curve W_{2}. A general fiber F of f_{2} is a smooth projective surface of general type. Because 2K_{X′} ≤ 4K_{X′}, ϕ_{4} can distinguish different fibers of g_{2}. In order to prove the generic finiteness of ϕ_{4}, it is sufficient to show that ϕ_{4}|_{F} is generically finite for a general fiber of f_{2} since s_{2} is a finite map. We can write

S_{2} ∼_{lin} ∑_{i=1}^{a_{2}} F_{i},

where a_{2} ≥ P_{2}(X) − 1 and the F′_{i}s are fibers of f_{2}. The vanishing theorem gives

|K_{X′} + π_{2}^{∗}(K_{X}) + S_{2}| |_{S_{2}}= |K_{S_{2}} + L_{2}|,

where L_{2} := π_{2}^{∗}(K_{X})|_{S_{2}}. According to [16, Claim 9.1], we have

O_{S_{2}}(π_{2}^{∗}(K_{X})|_{S_{2}}) ≅ O_{S_{2}}(σ_{2}^{∗}(K_{S_{0}}))

where σ_{2} : S_{2} ⟶ S_{0} is the contraction onto the minimal model S_{0} of S_{2}. Thus

K_{S_{2}} + L_{2} ∼_{lin} K_{S_{2}} + σ_{2}^{∗}(K_{S_{0}})

and so

Φ_{|K_{S_{2}} +L_{2}|} = Φ_{|2K_{S_{2}} |}.

From Theorem 3.1 of [6], we know that S_{2} can’t be a surface with p_{g} = q = 0. By [21, Theorem 1], Φ_{|2K_{S_{2}} |} is generically finite. Thus ϕ_{4} is generically finite.

Suppose dimϕ_{2}(X) = 2. We want to derive a contradiction assuming that ϕ_{4} is not generically finite. We consider the following two natural maps

H^{0}(X′, 4π_{2}^{∗}(K_{X})) ⟶^{α_{4}} Λ_{4} ⊆ H^{0}(S_{2}, 4L)  (1.4)

H^{0}(X′, 2π_{2}^{∗}(K_{X})) ⟶^{α_{2}} Λ_{2} ⊆ H^{0}(S_{2}, 2L),

where Λ_{i} is the image of α_{i} for i = 2, 4. By our assumption, Λ_{4} should be composed of a pencil of curves on the surface S_{2}. On the other hand, it’s obvious that Λ_{2} ⊆ Λ_{4} and

Λ_{2} =| S_{2}|_{S_{2}} | .

Noting that | S_{2}|_{S_{2}} | is a free pencil, we can see that, in this situation, the movable part of Λ_{4} is also base point free and that both Λ_{2} and Λ_{4} have the same generic irreducible element. Because the movable part of Λ_{4} is base point free, there is a divisor H_{4} (movable part of Λ_{4}) in S_{2} such that |H_{4}| ⊂ Λ_{4} and

h^{0}(S_{2}, H_{4}) = dim_{ℂ}Λ_{4}.

(One should note that Λ_{4} is not a complete linear system in general.) We can write

S_{2}|_{S_{2}} ∼_{lin} ∑_{i=1}^{b_{2}} C_{i} ∼_{num} b_{2}C,

where b_{2} ≥ P_{2}(X) − 2, the C′_{i}s are fibers of f_{2} and C is a smooth fiber of f_{2} contained in S_{2}. Then we have H_{4} ∼_{num} b_{4}C, where b_{4} ≥ dim_{ℂ}Λ_{4} − 1 and we think of Λ_{4} as a ℂ-vector space. The vanishing theorem gives that

|K_{X′} + π_{2}^{∗}(K_{X}) + S_{2}| |_{S_{2}}= |K_{S_{2}} + L_{2}|,

where L_{2} := π_{2}^{∗}(K_{X})|_{S_{2}}. It is obvious that

K_{S_{2}} + L_{2} ≥ 2L_{2} ≥ S_{2}|_{S_{2}}.

This means that Φ_{|K_{S_{2}} +L_{2}|} can distinguish different fibers of Φ_{Λ_{2}}. For a generic C contained in S_{2}, we want to study Φ_{|K_{S_{2}} +L_{2}|}|_{C} in order to derive a contradiction.

If dim_{ℂ}Λ_{4} ≥ 6, i.e. h^{0}(S_{2}, H_{4}) ≥ 6, then we can see that b_{4} ≥ 5. Noting that H_{4} ≤ 4L_{2}, we have

4L_{2} ∼_{num} b_{4}C + Z_{4},

where Z_{4} is an effective divisor. Thus

L_{2} ∼_{num} \frac{b_{4}}{4} C + \frac{1}{4}Z_{4}

and

L_{2} − C − \frac{1}{b_{4}} Z_{4} ∼_{num} (1 − \frac{4}{b_{4}} )L_{2}

is a nef and big ℚ-divisor on S_{2}. Thus the vanishing theorem yields

|K_{S_{2}} + ⌜L_{2} − \frac{1}{b_{4}} Z_{4}⌝| |_{C}= |K_{C} + D|,

where

deg(D) ≥ (L_{2} − \frac{1}{b_{4}} Z_{4}) · C = (1 − \frac{4}{b_{4}} )L_{2} · C > 0.

So |K_{C} + D| gives a finite map on C. Noting that

K_{S_{2}} + ⌜L_{2} − \frac{1}{b_{4}} Z_{4}⌝ ≤ K_{S_{2}} + L_{2},

we see that Φ_{|K_{S_{2}} +L_{2}|}|_{C} is finite and so Φ_{|K_{S_{2}} +L_{2}|} is generically finite. This means ϕ_{4} is generically finite, a contradiction.

If dim_{ℂ}Λ_{4} ≤ 5, because P_{4}(X) ≥ 21 by (1.1), we see from the map (1.4) that

|4π_{2}^{∗}(K_{X}) − 4S_{2}| ≠ ∅.

So we can write

4π_{2}^{∗}(K_{X}) ∼_{lin} 4S_{2} + G_{4},

where G_{4} is an effective divisor. Thus

π_{2}^{∗}(K_{X}) ∼_{num} S_{2} + \frac{1}{4}G_{4}

L_{2} = π_{2}^{∗}(K_{X}) ∼_{num} S_{2}|_{S_{2}} + \frac{1}{4}G_{4}|_{S_{2}}

∼_{num} b_{2}C + \frac{1}{4}G_{4}|_{S_{2}},

where b_{2} ≥ P_{2}(X) − 2 ≥ 2. We have that

L_{2} − C − \frac{1}{4b_{2}} G_{4}|_{S_{2}} ∼_{num} (1 − \frac{1}{b_{2}} )L_{2}

is a nef and big ℚ-divisor. The vanishing theorem gives

|K_{S_{2}} + ⌜L_{2} − \frac{1}{4b_{2}} G_{4}|_{S_{2}}⌝| |_{C}= |K_{C} + D′|,

where

deg(D′) ≥ (L_{2} − \frac{1}{4b_{2}} G_{4}|_{S_{2}}) · C = (1 − \frac{1}{b_{2}} )L_{2} · C > 0.

This means that |K_{C} + D′| gives a finite map on C. Noting that

|K_{S_{2}} + ⌜L_{2} − \frac{1}{4b_{2}} G_{4}|_{S_{2}}⌝| ⊆ |K_{S_{2}} + L_{2}|,

we see that |K_{S_{2}} +L_{2}| gives a generically finite map and so that ϕ_{4} is also generically finite, a contradiction.

In a word, ϕ_{4} is generically finite. □

Example 1.7. The assumption p_{g}(X) ≥ 5 is sharp in Theorem 1.2. There is a trivial example with p_{g}(X) = 4 and K_{X}^{3} = 2 on which dimϕ_{1}(X) = 3 and ϕ_{4} is a finite map of degree 2. On ℙ_{ℂ}^{3}, take a smooth hypersurface S of degree 10. S ∼_{lin} 10H. Let X be a double cover over ℙ^{3} with branch locus along S. Then X is a nonsingular canonical model, K_{X}^{3} = 2 and p_{g}(X) = 4 and ϕ_{1} is a finite morphism onto ℙ^{3} of degree 2. One can easily check that ϕ_{4} is also a finite morphism of degree 2.

References

1. W. Barth, C. Peter, A. Van de Ven, Compact complex surface, 1984, Springer-Verlag.

2. X. Benveniste, Sur les applications pluricanoniques des variétés de type très gégéral en dimension 3, Amer. J. Math. 108(1986), 433-449.

3. E. Bombieri, Canonical models of surfaces of general type, Publications I.H.E.S. 42(1973), 171-219.

4. F. Catanese, Canonical rings and special surfaces of general type, Proc. Symp. Pure Math. 46(1987), 175-194.

5. M. Chen, On pluricanonical maps for threefolds of general type, J. Math. Soc. Japan 50(1998), 615-621.

6. ——, Kawamata-Viehweg vanishing and quint-canonical maps for threefolds of general type, Comm. in Algebra 27(1999), 5471-5486.

7. ——, Complex varieties of general type whose canonical systems are composed with pencils, J. Math. Soc. Japan 51(1999), 331-335.

8. C. Ciliberto, The bicanonical map for surfaces of general type, Proc. Symposia in Pure Math. 62(1997), 57-83.

9. L. Ein, R. Lazarsfeld, Global generation of pluricanonical and adjoint linear systems on smooth projective threefolds, J. Amer. Math. Soc. 6(1993), 875-903.

10. R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag 1977.

11. Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann. 261(1982), 43-46.

12. Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Adv. Stud. Pure Math. 10(1987), 283-360.

13. J. Kollár, Higher direct images of dualizing sheaves, I, Ann. of Math. 123(1986), 11-42.

14. J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, 1998.

15. S. Lee, Remarks on the pluricanonical and adjoint linear series on projective threefolds, Commun. Algebra 27(1999), 4459-4476.

16. K. Matsuki, On pluricanonical maps for 3-folds of general type, J. Math. Soc. Japan 38(1986), 339-359.

17. M. Reid, Young person’s guide to canonical singularities, Proc. Symposia in Pure Math. 46(1987), 345-414.

18. I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. Math. 127(1988), 309-316.

19. E. Viehweg, Vanishing theorems, J. reine angew. Math. 335(1982), 1-8.

20. P.M.H. Wilson, The pluricanonical map on varieties of general type, Bull. Lond. Math. Soc. 12(1980), 103-107.

21. G. Xiao, Finitude de l’application bicanonique des surfaces de type général, Bull. Soc. Math. France 113(1985), 23-51.

22. ——, L’irrégularité des surfaces de type général dont le système canonique est composé d’un pinceau, Compositio Math. 56(1985), 251-257.

Mathematisches Institut, Georg-August-Universität Göttingen Bunsenstr. 3-5, 37073 Göttingen, Germany