ABOUT NON-PROJECTIVE THREEFOLDS

ZBIGNIEW JELONEK

Abstract. We show that a smooth complete threefold over ℂ, which is not projective contains a rational curve. This gives a partial answer to a question of J. Kollar.

1. Introduction.

Hironaka in [3] give examples of smooth complete algebraic threefolds, which are not projective. In his construction Hironaka uses blowing up of smooth curves, in particular his examples contain rational curves. In a natural way appears a question - whether a smooth complete algebraic threefold, which is not projective, must contain a rational curve. In [5] (5.17, p. 148) Janos Kollar posed even more general problem:

Conjecture. Let X be a smooth and proper algebraic space which is not projective. Then X contains a rational curve.

In this paper we show that this conjecture is true at least for smooth algebraic threefolds over ℂ (here a threefold means an algebraic variety of dimension three). We also show that if X is a smooth complete threefold, and there is a divisor D which is positive on the cone NE(X) of effective curves of X (i.e., D.C > 0 for every integral curve C ⊂ X), then X is projective.

2. Preliminaries.

We start with the following result of Kawamata (see [4], appendix): Theorem 2.1. Let X be a normal projective variety with at most canonical singularities and let f : X → Y be a birational map to a normal projective variety Y. Assume that −K_{X} is f-ample. Then every fiber of f of positive dimension is uniruled.

We need also the following result (see [6], 3.54, p. 110): Theorem 2.2. Let X_{1}, X_{2} be terminal projective varieties with numerically effective canonical divisors. Then any birational mapping f : X_{1}− → X_{2} is an isomorphism in codimension 1.

In the sequel the following result of J. Kollar will be important (it can be deduced from Remark 1.8 and Theorem 1.9 on page 290 in [5]). For the sake of completness we give here a simple proof. Theorem 2.3. Let X, Y be normal complete varieties of dimension three. Assume that X is ℚ-factorial and with terminal singularities. Let f : X− → Y be a rational mapping. If Y does not contain rational curves then f extends to a morphism f : X → Y.

Date: September 16, 2002.

1991 Mathematics Subject Classification. 14 A.

Key words and phrases. non-projective threefold, rational curves.

The author was partially supported by the KBN grant number 2PO3A 01722.

Before we prove Theorem 2.3 we need: Lemma 2.1. Let X be a ℚ-factorial projective threefold with terminal singularities. Let Z be any complete threefold such that there is a birational morphism p : Z → X. Then every fiber of p of positive dimension is uniruled.

Proof. Since X is ℚ−factorial, we have that the exceptional set of p is a divisor. First we show that every irreducible component F of E is a uniruled surface. Let Z′ be a projective cover of Z. Since it is enough to prove our statement for Z′, we can assume that Z is projective. Let r : X′ → X be a resolution of singularities of X. We have a birational mappings g : X− → Z and g′ : X′− → Z induced by p^{−1}. By a sequence of blowings up’s with smooth centers, we can resolve intederminancy of g′ to obtain a regular mapping G : X′′ → Z. We have the following diagram

[figure: π {; X′′; X_{m−1}; ⋮; X; Z; g; G; π_{1}; π_{m−1}; π_{m}]

where π_{1} = r, X_{1} = X′ and for j > 1 the π_{j} are blowing up’s with smooth centers. Let us note that fibers of all π_{j} with j > 1 are projective spaces. By the Minimal Model Program the mapping π_{1} = r is a composition of divisioral and small contractions (and their inverses). Thus by Theorem 2.1 if π_{1} contracts an irreducible divisor D, then D is uniruled. Moreover if the image of D is a curve C then the generic fiber (hence every fiber) of the mapping r : D → C must be uniruled. Of course the same is true if the image of D is a point. Trivially the same holds for π_{j}, if j > 1.

Now let F be an irreducible component of the exceptional set E of p. Let F′ ⊂ X′′ be a divisor, such that G(F′) = F. From the diagram above it is clear that F′ is contracted by π. By the preceding remarks it means that there is a divisor F′′ birationally equivalent to F which is contracted by some π_{j}. It is easy to see that a generic fiber of the contraction π_{j} : F′′ → X_{j−1} is uniruled. Hence the same is true for every fiber and we can replace here F′′ by F′. This means that fibers of the mapping p : F → X are dominated by uniruled sets, hence these fibers are uniruled. □

Now we can prove Theorem 2.3.

Proof. Let us consider the graph Z of the mapping f. It is a closed subset of X × Y. Let p : Z → X and q : Z → Y be canonical projections. If p is not an isomorphism, then there is an exceptional divisor E, which is uniruled over X. In particular there is a rational curve Γ ⊂ Z, such that p(Γ) = point. This implies that q is an injection on Γ, in particular the variety Y contains a rational curve q(Γ). This is a contradiction. Hence the mapping p is an isomorphism and f = q ∘ p^{−1} is a morphism. □

At the end of this section we prove one basic lemma: Lemma 2.2. Let X be a smooth complete variety and let U ⊂ X be an open quasiprojective subset of X. Then there exists a smooth projective variety Z and a birational morphism f : Z → X, such that the inverse mapping f^{−1} is defined on an open subset V which contains U. Moreover, codim X \ V ≥ 2.

Proof. By the Nagata theorem (see [7], Theorem 3.2) there is a projective Z containing U and dominating X. Indeed, take any projective variety X′ which contains U, then Z can be obtained as a join of some blowing-ups of X′. We can resolve the singularities of Z, hence we can assume that Z is smooth. Let p : Z → X be a birational morphism, which is an isomorphism on U. Then p^{−1} defines an open embedding into Z outside the exceptional locus S, which by smoothness and the Zariski Main Theorem, is of codimension ≥ 2 in X. Now it is enough to take V = X \ S. □

Corollary 2.1. Let X be a smooth complete non-projective threefold and let U ⊂ X be a maximal quasi-projective subset of X. Then the set G := X \ U has a pure dimension 1.

Proof. Let f : Z → X be a birational morphism as above. Assume that G has a point a as an isolated component. Let V = U ∪ {a} and let W = f^{−1}(X \ {a}). We can glue V and W along U ≅ f^{−1}(U) to obtain a new algebraic complete variety Z′. Let you note that we have a birational morphism f′ : Z → Z′, which is induced by f and which is an isomorphism outside the set E = f^{−1}(a). Let p = (f′)^{−1} and let D = p^{∗}(O(1)).

For an integral curve C and a point P ∈ C we denote by m_{P} (C) the multiplicity of the point P on a curve C. Let m(C) = Sup_{P ∈C}m_{P} (C).

Since the linear system given by D has at most one base point a and this system is very ample outside a, it is easy to check that for an integral curve C ⊂ Z′ we have D.C ≥ m(C). Hence by the Seshadri theorem ([2], p. 37) we have that D is ample. Consequently Z′ is projective and V is quasi-projective, a contradiction with maximality of U. □

3. Main Result.

Now we are ready to prove the following: Theorem 3.1. Let X be a smooth complete algebraic threefold which is not projective. Then X contains a rational curve.

Proof. Assume that X does not contain a rational curve. By the Lemma 2.2 we can cover X by open subsets U_{i}, i = 1, ..., r, such that for each i:

1) X \ U_{i} is a curve, we will denote it by G_{i};

2) there is a ℚ factorial and terminal projective variety X_{i} and a birational morphism f_{i} : X_{i} → X, which induces an isomorphism X_{i} \ f_{i}^{−1}(G_{i}) → X \ G_{i}.

Now we use the Minimal Model Program (see e.g., [6]) to modify varieties X_{i} in this way, that conditions 1,2 are still fulfilled, but X_{i} are terminal minimal models. First note that X_{i} are not uniruled, hence they have not Fano contractions. Now assume that e.g., X_{1} is not a minimal model. Hence K_{X_{1}} is not numerically effective and we can contract some extremal ray. Let π : X_{1} → X′_{1} be a contraction. Now two cases can happen

a) π is a divisorial contraction,

b) π is a small contraction.

In the case a) the variety X′_{1} is ℚ factorial and terminal projective variety. Let f′_{1} : X′_{1}− → X be the induced birational mapping. By Theorem 2.3 the mapping f′_{1} is in fact a morphism. It is easy to see that this morphism induces an isomorphism X′_{1} \(f′_{1})^{−1}(G_{1}) → X \ G_{1} and we can set X_{1} = X′_{1} and f_{1} = f′_{1}.

Let us consider the case b). Let φ : X_{1}− → X^{+} be a flip and let f^{+} : X^{+}− → X be the induced birational mapping. By Theorem 2.3 the mapping f^{+} is in fact a morphism. Thus we have a birational morphism f^{+} : X^{+} → X and X^{+} is ℚ factorial and terminal projective variety (see the diagram below).

[figure: X_{1}; X^{+}; X′_{1}; X; φ; π; π^{+}; f_{1}; f^{+}]

We show that f^{+} is an isomorphism outside the curve (f^{+})^{−1}(G_{1}). Assume that (f^{+})^{−1} is not defined at point a ∉ G_{1}. By the Zariski Main Theorem (smooth case) the mapping f^{+} must contract a divisor H, and a ∈ f^{+}(H). Note that by Lemma 2.1 the divisor H is uniruled. Let H′ = φ^{−1}(H) be a proper transform of H. Of course the divisor H′ is also uniruled. Since the flip φ is an ismorphism in codimension 1 and f_{1} = (f^{+}) ∘ φ, we have that a ∈ cl(f_{1}(H′)) = f_{1}(H′). Consequently H′ ⊄ f_{1}^{−1}(G_{1}) and since H′ is uniruled this is a contradiction. Finally we can set X_{1} = X^{+} and f_{1} = f^{+}. By the construction we have:

i) X_{1} is a ℚ factorial and terminal projective variety and f_{1} is a birational morphism f_{1} : X_{1} → X, which induces an isomorphism X_{1} \ f_{1}^{−1}(G_{1}) → X \ G_{1}.

We can continue our procedure and in a finite number of steps we get that X_{1} is a ℚ factorial and terminal projective variety with nef canonical divisor. Similarly we can assume that all X_{i} have the same properties. Take X_{1}. Since the variety X is not projective, we have that the mapping f_{1} is not an isomorphism. In particular it means that it contracts a divisor H. Let a ∈ f_{1}(H). There is a number j, such that a ∈ U_{j}. Consider the mapping F = f_{j}^{−1} ∘ f_{1} : X_{1}− → X_{j}. Thus the mapping F is a birational morphism of minimal terminal models, which is not an isomorphism in codimension 1. This is a contradiction (see Theorem 2.2). □

Remark 3.1. In Theorem 3.1 it is enough to assume that X is ℚ−factorial terminal threefold. The proof is the same. Remark 3.2. From our proof it follows that Theorem 3.1 holds in dimension n if the MMP holds in this dimension.

At the end of this paper we give another criterion for a smooth threefold X to be projective: Theorem 3.2. Let X be a smooth algebraic threefold. Assume that there is a divisor H, such that H.C > 0 for every integral curve C ⊂ X. Then X is a projective variety.

Proof. By the Lemma 2.2 there is an open subset U, such that:

1) X \ U is a curve, we will denote it by G;

2) there is a smooth projective variety Z and a birational morphism f : Z → X, which induces an isomorphism Z \ f^{−1}(G) → X \ G.

Let g = f^{−1} and take D = g^{∗}(O(1)). For an integral curve C and a point P ∈ C we denote by m_{P} (C) the multiplicity of the point P on a curve C. For a curve C let m(C) = Sup_{P ∈C}m_{P} (C). It is easy to check that if an integral curve C is not a component of the curve G then D.C ≥ m(C). Indeed, for every integral curve C ⊄ G and a point c ∈ C there is a non-zero section ρ of D (which is a pullback of an appropriate section of O(1) by g) such that c ∈ |ρ| but C ⊄ |ρ|.

Since there is only finite number of curves which are components of the curve G and the divisor H is positive on each integral curve, we have that for m >> 0 the line bundle D_{1} = D+mO(H) satisfies D_{1}.C ≥ m(C) for every integral curve C. Hence by the Seshadri theorem ([2], p. 37) we have that D_{1} is ample. □

Remark 3.3. There is a smooth complete threefold X and an effective divisor D on X such that D.C > 0 for every integral curve C ⊂ X but D is not ample (such an example is constructed e.g. in [2], Example 10.8, p. 57). However Theorem 3.2 shows that anyway the existence of such a divisor (which may be even not-effective) implies that X is projective. Remark 3.4. Let X be a non-projective smooth threefold. We can cover X by maximal quasi-projective open subsets U_{i} and consider curves G_{i} = X \ U_{i}. Let G_{i} = ⋃ G_{ij} be a decomposition of G_{i} onto irreducible components. We have only finite number of such sets U_{i} (see [8]). In particular it follows that we have only finite number of “interesting” curves G_{ij}. The proof of Theorem 3.1 suggests that these curves should be rational. It can be proved that they are in some sense isolated in the cone NE(X) of effective curves. I believe that these curves are the key to understanding the geometry of smooth non-projective threefolds.

References

[1] Abhyankar, S, On the valuations centered in a local domain, Amer. J. Math. 78, 321-348, 1956.

[2] Hartshorne, R, Ample Subvarieties of Algebraic Varieties, Springer Verlag 1986.

[3] Hironaka, H, On the theory of birational blowing up, Thesis, Harvard, 1960.

[4] Kawamata, Y, Abundance theorem for minimal threefolds, Invent. Math. 108, 229-246, 1992.

[5] Kollar, J, Rational curves on algebraic varieties, Springer Verlag 1996.

[6] Kollar, J, Birational geometry of algebraic varieties, Cambridge University Press 1998.

[7] Nagata, M, Imbedding of an abstract variety in a complete variety, J. Math. of Kyoto Univ., 1-10, 1962-63.

[8] Włodarczyk, J, Maximal quasi-projective subsets and the Kleiman-Chevalley quasiprojectivity criterion, J. Math. Sci. Univ. Tokyo 6, 41-47, 1999.

(Z. Jelonek) Instytut Matematyki, Polska Akademia Nauk, Św. Tomasza 30, 31-027 Kraków, Poland, and Max Planck Institut für Mathematik, Vivatgasse 7, 53111 Bonn, Germany

   E-mail address: najelone@cyf-kr.edu.pl