Singularities of normal affine 3-folds containing 𝔸^{1}-cylinderlike open subsets

By

Takashi KISHIMOTO

Contents

1. Introduction

2. The main results

3. Reduction via ♯-Minimal Model Program

4. Proof of Theorem 2.1

5. Proof of Theorem 2.2 and Corollary 2.1

Abstract

The normal affine varieties with 𝔸^{1}-cylinderlike open subsets play important roles in affine algebraic geometry and of interest in itself. In [Miy81], Miyanishi investigated how to construct normal affine surfaces Y with 𝔸^{1}-cylinderlike open subsets, explicitly. According to this explicit construction, he proved that such surfaces Y have at most cyclic quotient singularities. In this article, we consider the 3-dimensional generalization of this Miyanishi’s result. Namely, we shall investigate how to construct normal affine 3-folds X containing 𝔸^{1}-cylinderlike open subsets and the possibility of singularities on such X under a certain geometric assumption concerning the compactifications.

Keywords: an 𝔸^{1}-cylinderlike open subset, log Kodaira dimension, singularities, ♯-minimal model program

1. Introduction

Throughout the present article we work over the field of complex numbers ℂ. In affine algebraic geometry the varieties with 𝔸^{1}-cylinderlike open subsets (cf. Definition 3.1) play the important roles. We often call such varieties as affine ruled varieties and use the these two terminologies ”containing an 𝔸^{1}- cylinderlike open subset” and ”affine ruled” interchangeably in the subsequent. It is obvious that normal one-dimensional affine ruled affine variety is the affine line 𝔸^{1} only. In 1981, Miyanishi investigated the construction and singularities on a normal affine surface with an 𝔸^{1}-cylinderlike open subset (cf. [Miy81]). As a consequence, he obtained the following result:

2000 Mathematics Subject Classification(s). Primary 14R05, Secondary 14E30

Theorem 1.1 (cf. [Miy81]). Let Y be a normal affine surface with an 𝔸^{1}-cylinderlike open subset. Then Y has at most cyclic quotient singularities.

In his proof, the theory of singular fibers of ℙ^{1}-fibrations on smooth surfaces (cf. [Giz70, Miy78]) and the affineness played the crucial roles. Roughly speaking, his argument to prove Theorem 1.1 is summarized as follows: Let μ : Ŷ → Y be the minimal resolution. Then Ŷ inherits an 𝔸^{1}-fibration from that on Y and can be embedded into a smooth projective surface W with a structure of a ℙ^{1}-fibration, say π, so that π contains an 𝔸^{1}-fibration on Ŷ. Then Miyanishi investigated the possibility of configurations of the singular fibers of π the proper images of which pass through singularities on Y in consideration of the results in [Giz70, Miy78]. Theorem 1.1 above has several significant applications for not only the theory on affine surfaces but also that on higher-dimensional affine varieties. For instance, the algebro-geometric characterization of the affine 3-space 𝔸^{3} in [Miy88] makes use of it effectively.

Now our main interest lies in the 3-dimensional generalization of the above mentioned Miyanishi’s result (cf. Theorem 1.1). Namely:

Problem 1.2. Let X be a normal affine 3-fold with only isolated, ℚfactorial singularities such that X contains an 𝔸^{1}-cylinderlike open subset. What kinds of singularities can appear on X ?

In order to tackle Problem 1.2, it seems to be enough to argue along the similar line as performed in Miyanishi’s proof in [Miy81]. Namely, let μ : X̂ → X be the resolution of singularities of X, and we embed X̂ into a smooth projective 3-fold T̂ so that T̂ has a structure of a ℙ^{1}-fibration, say π, which contains an 𝔸^{1}-fibration on X̂ induced from that on X. In this setting, you may try to determine the types of singularities on X by observing configurations of fibers of π the images of which contain the singularities on X. But this attempt is not suitable for our purpose. Indeed, π is not flat, and the singular fibers of non-flat ℙ^{1}-fibration offer little information to determine the singularities.

In this present article, we make a certain assumption with respect to the compactification of X and consider Problem 1.2. Our assumption is geometric in nature: We embed X into a normal projective 3-fold T so that T is smooth along the boundary T − X and T − X is an S.N.C. divisor on T. Then we assume the following:

\underline{ASSUMPTION (†):} The complete linear system H := |D| on T is nef and contains a smooth member, say S, where D is the reduced boundary divisor with respect to X ↪ T.

Under this assumption (†), we shall establish the following two:

(A) Explicit construction of X,

(B) Determination of the possibility of singularities on X.

We shall state the scheme of this paper.

In Section 2, we shall state the main results concerning (A) and (B) (cf. Theorems 2.1, 2.2 and Corollary 2.1), and some remarks about them. The detailed proof for them are performed in the subsequent Sections 3, 4 and 5.

In Section 3, we prove a part of Theorem 2.1. More precisely to say, we shall construct birational maps ϕ′, ϕ′′, ψ′ and μ′ in the diagram in Theorem 2.1 (see Section 2, Theorem 2.1 for the notation). For this purpose, first of all, we shall perform the ♯-Minimal Model Program associated to (T̂, Ĥ) as in [Me02]. In the original context of ♯-Minimal Model Program, it works to control the Minimal Model Program starting from T̂ in a right direction with respect to the linear system Ĥ. But, in addition to this, we now need to observe how the inside quasi-projective variety X̂ (and also X) changes carefully via ♯-Minimal Model Program.

In Section 4, we shall complete the proof of Theorem 2.1 by constructing birational morphisms ψ′′ and μ^{♯} in the diagram in Theorem 2.1. We note that the ℚ-factoriality on X is needed at this stage.

In Section 5, we shall prove Theorem 2.2. This result concerning the structure of Mori fiber space π : T̂^{♯} → W is essentially obtained in [C-F93, Me02]. But, note that since (T̂, Ĥ) may not be a pair with a big uniruled system (see [Me02, Definition 5.1]), we can not make use of the result in [C-F93, Me02] directly to our present situation. Furthermore, we have to observe the birational morphism μ^{♯} : T̂^{♯} → T^{♯}. As seen in Theorem 2.2, μ^{♯} is, in fact, an isomorphism except for the case where π : T̂^{♯} → W is of C2-type. After proving Theorem 2.2, we finally prove Corollary 2.1.

We shall employ the following convention of notation in this paper:

Convention of Notation.

• D|_{W} : a scheme-theoretic restriction of a Cartier divisor D on V to a subvariety W ⊂ V;

• D_{1} ∼ D_{2} : D_{1} is linearly equivalent to D_{2};

• D_{1} ≡ D_{2} : D_{1} is numerically equivalent to D_{2};

• ℚ^{3} : a smooth quadric hypersurface in ℙ^{4};

• V ≅ W: V is isomorphic to W;

• V ≈ W: V is birationally equivalent to W;

• ϱ(V) : the Picard number of a variety V;

• ℙ(a_{0}, ⋯, a_{n}) : a weighted projective space of weights wt(x_{0}, ⋯, x_{n}) = (a_{1}, ⋯, a_{n}), i.e., ℙ(a_{0}, ⋯, a_{n}) = Projℂ[x_{0}, ⋯, x_{n}] with wt(x_{i}) = a_{i};

• T_{α_{1},···α_{r}} ⊂ ℙ(a_{0}, ⋯, a_{n}): a weighted complete intersection of degree α_{1}, ⋯, α_{r} in ℙ(a_{0}, ⋯, a_{n});

• \frac{1}{r}(a, b, c) : a germ of the quotient singularity of analytic type o ∈ ℂ^{3} : (x, y, z)/ℤ_{r}(a, b, c), where the cyclic group ℤ_{r} =< ξ > acts on ℂ^{3} via ξ · (x, y, z) = (ξ^{a}x, ξ^{b}y, ξ^{c}z). (ξ ∈ ℂ^{∗} is a primitive r-th root of unity.)

An E2-type divisorial contraction means a birational morphism between two algebraic 3-folds so that an exceptional divisor E of it satisfies (E, E|_{E}) ≅ (ℙ^{2}, O_{ℙ^{2}}(−1)) (cf. [Mori82]).

Acknowledgments. The author would like to express his sincere gratitude to Professors Masayoshi Miyanishi, Mikio Furushima and Mikhail Zaidenberg for their valuable advice and warm encouragements during the preparation of the present article. He would like to express his hearty thanks to Professor Tomoaki Ohta for giving very helpful comments consistently. He also would like to express his hearty thanks to Max-Planck-Institut für Mathematik for its warm hospitality, financial support and the splendid research environments which are given to him during the preparation of the present article.

2. The main results

In this section, we shall state the results concerning (A) and (B) in Section 1. The first result is stated as follows:

Theorem 2.1. Let X be a normal affine 3-fold with only isolated, ℚfactorial singularities such that it contains an 𝔸^{1}-cylinderlike open subset. Assume that X is embedded into a normal projective 3-fold T satisfying (†). Let μ : T̂ → T be the resolution of singularities of T, let Ĥ := μ^{∗}(H) and let D̂ := μ^{∗}(D), Ŝ := μ^{∗}(S) ∈ Ĥ. Then there exists a following commutative diagram satisfying the conditions (1)∼(4) below:

T̂ · · · · · · →^{ϕ′}  T̂′ ⟶^{ϕ′′}  T̂^{♯}
μ↓  ↓μ′  ↓μ^{♯}
T · · · · · · →^{ψ′} T′ ⟶^{ψ′′} T^{♯},

(1) ϕ := ϕ′′ ∘ ϕ′ : T̂ ⋯ → T̂^{♯} is a process of the ♯-Minimal Model Program associated to (T̂, Ĥ) (cf. Section 3 and [Me02]) and T̂^{♯} has a structure of Mori fiber space, say π : T̂^{♯} → W. ϕ′′ : T̂′ → T̂^{♯} is a composite of E2-type divisorial contractions unless ϕ′′ = id;

Let D̂′ (resp. D̂^{♯}) be the proper transform of D̂ on T̂′ (resp. on T̂^{♯}). Let D′ (resp. D^{♯}) be the proper transform of D on T′ (resp. on T^{♯}). Put X̂′ := T̂′ − Supp (D̂′), X̂^{♯} := T̂^{♯} − Supp (D̂^{♯}), X′ := T′ − Supp (D′) and X^{♯} := T^{♯} − Supp (D^{♯}). Then:

(2) X ≅ X′;

(3) X̂^{♯} (resp. X^{♯}) is an open subset of X̂′ (resp. of X′ ≅ X) such that X̂′ − X̂^{♯} ≅ X′ − X^{♯} is a union of several affine planes unless X̂′ = X̂^{♯} and X′ = X^{♯}. More precisely, X̂′ (resp. X′) is obtained from X̂^{♯} (resp. X^{♯}) via the composite of suitable half-point attachments (cf. Definition 3.2) unless X̂′ = X̂^{♯} and X′ = X^{♯};

(4) Sing (X) = Sing (X^{♯}).

As stated in Theorem 2.1, the original affine 3-fold X is obtained from X^{♯} = T^{♯} − Supp (D^{♯}) via the composite of suitable half-point attachments (cf. Definition 3.2). Therefore, we have only to investigate the construction of X^{♯} and singularities on X^{♯}. For this purpose, we shall look into the structure of Mori fiber space π : T̂^{♯} → W and the morphism μ^{♯} : T̂^{♯} → T^{♯} (cf. Theorem 2.1). Namely, the following result, which is essentially obtained in [C-F93] and [Me02], follows:

Theorem 2.2 (cf. [C-F93, Me02]). Let the notation and the assumption be the same as above. Let Ĥ^{♯} := ϕ_{∗}(Ĥ) be the proper transform by ϕ := ϕ′′ ∘ ϕ′ : T̂ ⋯ → T̂^{♯} of Ĥ. Then the Mori fiber space π : T̂^{♯} → W, the boundary divisor D̂^{♯} ∈ Ĥ^{♯} and the morphism μ^{♯} : T̂^{♯} → T^{♯} are described as one of the following:

– C_{2}-type: π : T̂^{♯} ≅ ℙ(E) → W is a ℙ^{1}-bundle structure over a smooth projective surface W, where E := π_{∗}OT̂^{♯}(D̂^{♯}) is a rank 2 vector bundle on W, and D̂^{♯} ∼ O(1). Unless μ^{♯} is an isomorphism (this is satisfied if X is not smooth), then W ≅ ℙ^{2}, T̂^{♯} ≅ ℙ(O_{ℙ^{2}} ⊕ O_{ℙ^{2}}(a)) (a ≥ 2), D̂^{♯} is a section corresponding to the surjection O_{ℙ^{2}} ⊕O_{ℙ^{2}}(a) → O_{ℙ^{2}}(a) and μ^{♯} is the contraction of the negative section E^{♯} of π with (E^{♯}, E^{♯}|_{E^{♯}}) ≅ (ℙ^{2}, O_{ℙ^{2}}(−a));

– D′_{2}-type: π : T̂^{♯} → W is a quadric bundle over a smooth curve with at most cA_{1} singularities of analytic type o ∈ (xy + z^{2} + t^{k} = 0) ⊂ ℂ^{4} : (x, y, z, t) for k ≥ 2, and D̂^{♯}|_{F} ∈ |O_{ℙ^{1}×ℙ^{1}}(1, 1)|, where F is a general fiber of π. μ^{♯} is an isomorphism, hence X^{♯} ≅ X̂^{♯} = T̂^{♯} − Supp (D̂^{♯});

– D_{3}-type: π : T̂^{♯} ≅ ℙ(E) → W is a ℙ^{2}-bundle structure over a smooth curve W, where E := π_{∗}OT̂^{♯}(D̂^{♯}) is a rank 3 vector bundle over W. μ^{♯} is an isomorphism, hence X^{♯} ≅ X̂^{♯} = T̂^{♯} − Supp (D̂^{♯});

– D′_{3}-type: π : T̂^{♯} → W is a ℙ^{2}-fibration over a smooth curve W with a general fiber (F, D̂^{♯}|_{F}) ≅ (ℙ^{2}, O_{ℙ^{2}}(2)) and with at most finitely many singular fibers G ≅ 𝕊_{4}, where 𝕊_{4} ⊂ ℙ^{5} is a cone over the quartic rational curve ⊂ ℙ^{4} and the vertex of G sits in a hyper-quotient singularity of analytic type o ∈ (xy + z^{2} + t^{k} = 0) ⊂ ℂ^{4} : (x, y, z, t)/ℤ_{2}(1, 1, 1, 0) for k ≥ 1. μ^{♯} is an isomorphism, hence X^{♯} ≅ X̂^{♯} = T̂^{♯} − Supp (D̂^{♯});

– ℚ-Fano: T̂^{♯} is a ℚ-Fano 3-fold with the Picard number ϱ(T̂^{♯}) = 1. μ^{♯} is an isomorphism, hence X^{♯} ≅ X̂^{♯} = T̂^{♯} − Supp (D̂^{♯}). More precisely, the classification of the pair (T̂^{♯}, Ĥ^{♯}) up to deformation is given as one of the following:

(i) (ℙ(1, 1, 2, 3), O(6));

(ii) (T_{6} ⊂ ℙ(1, 1, 2, 3, a), {x_{4} = 0} ∩ T_{6}) with a ∈ {3, 4, 5};
(iii) (T_{6} ⊂ ℙ(1, 1, 2, 2, 3), {x_{3} = 0} ∩ T_{6});
(iv) (T_{6} ⊂ ℙ(1, 1, 1, 2, 3), {x_{0} = 0} ∩ T_{6});
(v) (ℙ(1, 1, 1, 2), O(b)) with b ∈ {2, 4};

(vi) (T_{4} ⊂ ℙ(1, 1, 1, 1, 2), {x_{0} = 0} ∩ T_{4}); (vii) (T_{4} ⊂ ℙ(1, 1, 1, 2, a), {x_{4} = 0} ∩ T_{4}) with a ∈ {2, 3}; (viii) (ℙ^{3}, O_{ℙ^{3}}(b)) with b ∈ {1, 2, 3}, (ℚ^{3}, O_{ℚ}(b)) with b ∈ {1, 2}; (ix) (T_{3} ⊂ ℙ(1, 1, 1, 1, 2), {x_{4} = 0} ∩ T_{3});

(x) (T_{3} ⊂ ℙ^{4}, O(1)); (xi) (T_{2,2} ⊂ ℙ^{5}, O(1)); (xii) (V_{5}, O(1)), where V_{5} ↪ ℙ^{6} is a linear section of the Grassmann

variety Gr(2, 5) ↪ ℙ^{9} parametrizing lines in ℙ^{4};

Remark 1. Let us consider, for instance, cubic hypersurface ⊂ ℙ^{4}, (x) in Theorem 2.2. Of course, a general cubic hypersurface is smooth. On the other hand, we consider the following cubic hypersurface:

T_{3} := (x_{0}x_{1}x_{2} + x_{3}^{3} + x_{4}^{3} = 0) ⊂ ℙ^{4}.

Then T_{3} is a ℚ-Fano 3-fold with ϱ = 1 having three singularities of analytic type o ∈ (xy + z^{3} + t^{3} = 0). By the similar argument, it follows that the possibility of singularities on ℚ-Fano 3-folds with ϱ(T̂^{♯}) = 1, (ii), (iii), (iv), (vi), (vii), (ix), (x), (xi) and (xii) in Theorem 2.2 is not unique in general. But, when T̂^{♯} in question is quasi-smooth, we can determine the singularities on T̂^{♯} concretely. Now, we recall the definition of quasi-smoothness. Let p : 𝔸^{n+1} − {0} → ℙ(a_{0}, ⋯, a_{n}) be the quotient map induced by the ℂ^{∗}-action on 𝔸^{n+1} − {0} defined by:

λ · (x_{0}, ⋯, x_{n}) = (λ^{a_{0}}x_{0}, ⋯, λ^{a_{n}}x_{n}), λ ∈ ℂ^{∗}.

A weighted complete intersection T := T_{α_{1},··· ,α_{r}} ⊂ ℙ(a_{0}, ⋯, a_{n}) =: ℙ is said to be quasi-smooth if the inverse image C_{T}^{∗} := p^{−1}(T) ⊂ 𝔸^{n+1} −{0} is smooth (cf. [Fl00, 6.3 Definition]). There is a criterion in order to see whether or not a general weighted complete intersection T_{α_{1},··· ,α_{r}} ⊂ ℙ(a_{0}, ⋯, a_{n}) is quasi-smooth in terms of weights a_{0}, ⋯, a_{n} and degrees α_{1}, ⋯, α_{r} (cf. [Fl00, 8.1 Theorem]). Here we must clarify the meaning of a ”general” weighted complete intersection T_{α_{1},··· ,α_{r}} ⊂ ℙ(a_{0}, ⋯, a_{n}). The weighted hypersurfaces T_{α_{i}} in ℙ(a_{0}, ⋯, a_{n}) are parametrized by the projective space:

ℙ(H^{0}(ℙ(a_{0}, ⋯, a_{n}), O_{ℙ}(α_{i}))) = ℙ(𝕍_{i}),

where 𝕍_{i} is the basis of H^{0}(ℙ(a_{0}, ⋯, a_{n}), O_{ℙ}(α_{i})) as a ℂ-vector space:

𝕍_{i} := {x_{0}^{d_{0}} ⋯ x_{n}^{d_{n}} ; ∑_{j=0}^{n} a_{j}d_{j} = α_{i} }.

The correspondence is given by:

∑_{x_{0}^{d_{0}} ···x_{n}^{dn} ∈𝕍_{i}} f_{d_{0},··· ,d_{n}}x_{0}^{d_{0}} ⋯ x_{n}^{d_{n}} ↦ (⋯: f_{d_{0},··· ,d_{n}} : ⋯) ∈ ℙ(𝕍_{i}).

On the other hand, the weighted complete intersections T_{α_{1},··· ,α_{r}} = T_{α_{1}} ∩ ⋯ ∩ T_{α_{r}} are parametrized by:

∏_{i=1}^{r} ℙ(H^{0}(ℙ(a_{0}, ⋯, a_{n}), O_{ℙ}(α_{i}))) = ∏_{i=1}^{r} ℙ(𝕍_{i}).

Let U_{i} denote the complement of all coordinate hyperplanes in ℙ(𝕍_{i}) for 1 ≤ i ≤ r and we put U := ∏_{i=1}^{r} U_{i}. When a point on ∏_{i=1}^{r} ℙ(𝕍_{i}) is contained in U, the corresponding weighted complete intersection T_{α_{1},··· ,α_{r}} in ℙ is said to be general.

By making use of the method in [Fl00, Section 10], we can determine singularities on quasi-smooth weighted complete intersections. In fact, we can see that general 3-folds (in the above mentioned sense) in the family (ii), (iii), (iv), (vi), (vii), (ix), (x), (xi) and (xii) in Theorem 2.2 are quasi-smooth, and, consequently, ascertain the following concerning singularities on general (weighted) complete intersections (ii), (iii), (iv), (vi), (vii), (ix), (x), (xi) and (xii):

Type  singularities on quasi-smooth T̂^{♯}
(ii)  \begin{matrix}2 × \frac{1}{3}(1, 1, −1) if a = 3; \\ \frac{1}{2}(1, 1, −1) + \frac{1}{4}(1, 1, −1) if a = 4; \\ \frac{1}{5}(1, 2, −2) if a = 5 \end{matrix}
(iii)  3 × \frac{1}{2}(1, 1, −1)
(iv)  smooth
(vi)  smooth
(vii)  \begin{matrix}2 × \frac{1}{2}(1, 1, −1) if a = 2; \\ \frac{1}{3}(1, 1, −1) if a = 3 \end{matrix}
(ix)  \frac{1}{2}(1, 1, −1)
(x)  smooth
(xi)  smooth
(xii)  smooth

Table 1.

By making use of Theorem 2.1 and the argument to prove Theorem 2.2, we obtain the following result concerning (A) and (B):

Corollary 2.1. Let X be a normal affine 3-fold with only isolated, ℚfactorial singularities such that it contains an 𝔸^{1}-cylinderlike open subset. Assume the condition (†) is satisfied. Let X^{♯}, X̂^{♯} and Mori fiber space π : T̂^{♯} → W be the same as in Theorems 2.1 and 2.2:

(1) X is constructed from X^{♯} via the composite of half-point attachments (cf. Definition 3.2) unless X ≅ X^{♯}. If π is of type D2, D3, D3′ or ℚ-Fano 3-fold, then X^{♯} ≅ X̂^{♯}. Moreover, if π is of type D3, then X ≅ X^{♯} ≅ X̂^{♯}.

(2) If π is either of type D2, D3 or D3′ then X is smooth.

(3) If T̂^{♯} is a ℚ-Fano 3-fold with ϱ(T̂^{♯}) = 1 then X has at most terminal singularities which are lying on T̂^{♯}, (i)∼ (xii) in Theorem 2.2. Moreover, in the case where T̂^{♯} is general (cf. Remark 1), Sing (X) is described as follows:

Sing(X) = α×\frac{1}{2}(1, 1, −1)+β×\frac{1}{3}(1, 1, −1)+γ×\frac{1}{4}(1, 1, −1)+δ×\frac{1}{5}(1, 2, −2),

where α, β, γ, δ ∈ ℤ_{≥0} satisfy one of the following:

• α ≤ 3, β = γ = δ = 0 : (iii), (v), (vii) a = 2, (ix);

• α = 0, β ≤ 3, γ = δ = 0 : (ii) a = 3, (vii) a = 3;

• α, β ≤ 1, γ = δ = 0 : (i);

• α, γ ≤ 1, β = δ = 0 : (ii) a = 4;

• α = β = γ = 0, δ ≤ 1 : (ii) a = 5.

• α = β = γ = δ = 0, i.e., X is smooth: (iv), (vi), (viii), (x), (xi), (xii).

(4) If π is of C2-type and X is not smooth, then X^{♯} = μ^{♯}(X̂^{♯}), where X̂^{♯} is the complement in T̂^{♯} = ℙ(O_{ℙ^{2}} ⊕ O_{ℙ^{2}}(a)) of the section D̂^{♯} corresponding to O_{ℙ^{2}} ⊕ O_{ℙ^{2}}(a) → O_{ℙ^{2}}(a) and μ^{♯} is the contraction of the negative section E^{♯} with (E^{♯}, E^{♯}|_{E^{♯}}) ≅ (ℙ^{2}, O_{ℙ^{2}}(−a)) with a ≥ 2. In particular, we have Sing (X) = p := μ^{♯}(E^{♯}) and (p ∈ X)≈o ∈ \frac{1}{a}(1, 1, 1).

Remark 2. Although Corollary 2.1 indicates the possibility of singularities on X in question, we do not know whether or not an affine 3-fold X constructed from X^{♯} in Theorem 2.2 via the composite of suitable half-point attachments contains an 𝔸^{1}-cylinderlike open subset actually except for some special cases.

Problem 2.3. Let X be a normal affine 3-fold constructed from X^{♯} = T^{♯} − Supp (D^{♯}) stated in Theorem 2.2 and Corollary 2.1 via the composite of half-point attachments. Does X contain an 𝔸^{1}-cylinderlike open subset ?

Nevertheless the possibility of singularities stated in Corollary 2.1, we shall say the following conjecture: Conjecture: Let X be a normal affine 3-fold with only isolated, ℚ-factorial singularities so that X contains an 𝔸^{1}-cylinderlike open subset. Then does X have at most cyclic quotient singularities ?

Note that any 3-fold cyclic quotient singularity can be realized as a singular point on a normal affine 3-fold X with an 𝔸^{1}-cylinderlike open subset. Indeed, let us consider a cyclic quotient singularity o ∈ ℂ^{3} : (x, y, z)/ℤ_{r}(1, a, b) =: X with 1 ≤ a ≤ b < r and gcd(a, r) = gcd(b, r) = 1, where the action of the finite cyclic group ℤ_{r} =< ξ > is given by ξ · (x, y, z) = (ξx, ξ^{a}y, ξ^{b}z) (ξ is a primitive r-th root of unity). Then there exists an 𝔸^{1}-fibration:

p : X ⟶ Y := ℂ^{2}_{(x,y)}/ℤ_{r}(1, a) (\overline{x}, \overline{y}, \overline{z}) ↦ (\overline{x}, \overline{y}),

over a normal affine surface Y with one cyclic quotinet singularity o ∈ Y. X contains an 𝔸^{1}-cylinderlike open subset p^{−1}(Y − {o}).

Note that the recent work due to Flenner and Zaidenberg [F-Z03, Theorem 0.3] asserts that if a scheme Z with an isolated Cohen-Macaulay singularity p ∈ Z such that a Zariski open subset of Z can be covered by rational curves which do not pass through p, then p ∈ Z is a rational singularity. In my opinion, the geometric condition that ”X contains an 𝔸^{1}-cylinderlike open subset” is very restrictive and, so, we expect that singularities on such X form smaller class of singularities (e.g. cyclic quotient singularities) than that of rational singularities as in 2-dimensional case (cf. Theorem 1.1).

In the remaining Sections 3, 4 and 5, we shall give the proof of Theorems 2.1, 2.2 and Corollary 2.1.

3. Reduction via ♯-Minimal Model Program

First of all, we shall start with the definition:

Definition 3.1. An 𝔸^{1}-cylinderlike open subset U of a quasi-projective variety Y is an open subset of the form U ≅ U_{0} × 𝔸^{1}, where U_{0} is a smooth quasi-projective variety of dim U_{0} = dim Y − 1.

Let X be a normal affine 3-fold with an 𝔸^{1}-cylinderlike open subset such that satisfies ASSUMPTION (†), and let the notations (T, H), (T̂, Ĥ), D, S ∈ H and D̂, Ŝ ∈ Ĥ be the same there. In this section we shall construct the birational maps ϕ′, ϕ′′, ψ′ and the birational morphism μ′ in the diagram of Theorem 2.1. Note that since X contains an 𝔸^{1}-cylinderlike open subset, T̂ is uniruled. We reduce the pair (T̂, Ĥ) to a suitable pair (T̂^{♯}, Ĥ^{♯}) via a suitably chosen Minimal Model Program ϕ : T̂ ⋯ → T̂^{♯} associated to Ĥ (so called ♯-Minimal Model Program in the terminology of [Me02]), where T̂^{♯} has a structure of Mori fiber space, say π : T̂^{♯} → W. As performed in [Me02], after several divisorial contractions and flips we can finally reach a Mori fiber space π : T̂^{♯} → W. But we, furthermore, have to investigate how the inside varieties X̂ and X change via this process of ♯-Minimal Model Program for our purpose to determine the possibility of singularities on X. From this point of view, the main obstacles lie in the following:

OBSTACLE.

(1) When a divisorial contraction occurs in the process of ♯-Minimal Model Program associated to (T̂, Ĥ), some exceptional divisor may not be contained in the proper transform of the boundary D̂. Hence X̂ (at the same time X) may be shrinked. In particular, some of singular points on X may be shrinked via the process.

(2) When a flip occurs in the process of ♯-Minimal Model Program, some flipping or flipped curves may not be contained in the proper transform of the boundary D̂. This may result in new singularities inside varieties.

This is why it is usually not suitable to apply 3-dimensional Minimal Model Program for the study on affine 3-folds. But, under our assumption (†), we can control the process of the ♯-Minimal Model Program in a right direction, namely, so that it does not bring too bad influence on X̂ and X although X̂ and X may be shrinked.

From now on, we shall recall and rewrite the notion of ♯-Minimal Model Program (cf. [Me02, Theorem 3.2, Proposition 3.6]) in our context. Put ρ := ρ(T̂, Ĥ) := sup{λ ∈ ℚ ; Ĥ + λKT̂ is effective} which we call threshold of the pair (T̂, Ĥ). Moreover, put t^{0} := sup{λ ∈ ℚ ; Ĥ + λKT̂ is nef} which we call nef value of the pair (T̂, Ĥ). First of all, we prove the following:

Lemma 3.1. 0 ≤ t^{0} ≤ ρ < 1.

Proof. Since the cone of nef divisors is the closure of the ample cone by virtue of Kleiman’s criterion of ampleness (cf. [Kle66]), the nef cone is contained in the closure of the cone of effective divisors. Hence t^{0} ≤ ρ. Note that Ĥ is nef. So, we have t^{0} ≥ 0. Note that an 𝔸^{1}-cylinderlike open subset U ≅ U_{0} × 𝔸^{1} is disjoint from the singular points Sing (X). Hence X̂ contains canonically an 𝔸^{1}-cylinderlike open subset μ^{−1}(U) ≅ U. In particular, \overline{κ}(X̂) = −∞ (see [Ii77] for the definition and the relevant results on log Kodaira dimension \overline{κ}). Since X̂ is embedded into a smooth projective 3-fold T̂ so that T̂ − X̂ = D̂ is an S.N.C. divisor, we have \overline{κ}(X̂) = κ(T̂; KT̂ + D̂) = −∞. Therefore, Ĥ + KT̂ ∼ D̂ + KT̂ can not be effective, i.e., ρ < 1. □

The next proposition and its proof yield how to proceed in order to reach a suitable Mori fiber space starting with the pair (T̂, Ĥ).

Proposition 3.1 (cf. [Me02]). There exists a pair (T̂^{♯}, Ĥ^{♯}) satisfying the following three conditions:

(i) there is a birational map ϕ : T̂ ⋯ → T̂^{♯} with Ĥ^{♯} = ϕ_{∗}(Ĥ); (ii) T̂^{♯} has a structure of Mori fiber space, say π : T̂^{♯} → W; (iii) Ĥ^{♯} + ρKT̂^{♯} ≡_{π} 0, i.e., Ĥ^{♯} + ρKT̂^{♯} is numerically π-trivial.

Proof. Although our proof is parallel to that performed in [Me02], we shall prove in our context for the purpose to establish Theorem 2.1 in the subsequent. We shall proceed in an inductive way. There exists an extremal ray R^{0} = ℝ_{+}[C^{0}] in the hyperplane \overline{NE} (T̂) ∩ \{Ĥ + t^{0}KT̂\}^{⊥} (cf. [K-M-M87]). Let ϕ^{0} : T̂ ⋯ → T̂^{1} be a rational map associated with R^{0} which is either a Mori fiber space, a divisorial contraction or a flip. (In fact, since T̂ is a smooth 3-fold, the case ϕ^{0} is a flip does not occur. cf. [CKM88].) Put Ĥ^{1} := ϕ_{∗}^{0}(Ĥ) and let Ŝ^{1}, D̂^{1} ∈ Ĥ^{1} be the proper transforms of D̂, Ŝ on T̂^{1}, respectively. Now suppose that we have already constructed the i-th step, namely,

– a birational map ϕ^{i−1} : T̂^{i−1} ⋯ → T̂^{i} which is associated with an extremal ray R^{i−1} = ℝ_{+}[C^{i−1}] in \overline{NE} (T̂^{i−1}) ∩ \{Ĥ^{i−1} + t^{i−1}KT̂^{i−1}\}^{⊥}; – a nef linear system Ĥ^{i} := ϕ_{∗}^{i−1}(Ĥ^{i−1}) on T̂^{i};

– the proper transform D̂^{i} := ϕ_{∗}^{i−1}(D̂^{i−1}) and a smooth member Ŝ^{i} := ϕ_{∗}^{i−1}(Ŝ^{i−1}) of Ĥ^{i}.

Put t^{i} := sup{λ ∈ ℚ ; Ĥ^{i} + λKT̂^{i} is nef}. Since Ĥ^{i} is nef and T̂^{i} is a uniruled varieties, it is obvious that t^{i} ≥ 0. On the other hand, it follows that 0 ≤ t^{i−1} ≤ t^{i} ≤ ρ < 1 (cf. Lemma 3.1 and [Me02, Claim 3.4]). There exist an extremal ray R^{i} = ℝ_{+}[C^{i}] in \overline{NE} (T̂^{i}) ∩ \{Ĥ^{i} + t^{i}KT̂^{i}\}^{⊥}. Let ϕ^{i} : T̂^{i} ⋯ → T̂^{i+1} be a rational map associated to this extremal ray R^{i}. If ϕ^{i} yields a Mori fiber space, then we put π := ϕ^{i}, T̂^{♯} := T̂^{i} and Ĥ^{♯} := Ĥ^{i} to obtain the assertion. If ϕ^{i} is a birational map (either a divisorial contraction or a flip), we put Ĥ^{i+1} := ϕ_{∗}^{i}(Ĥ^{i}). We denote by D̂^{i+1} (resp. Ŝ^{i+1}) the proper transform on T̂^{i+1} of D̂^{i} (resp. Ŝ^{i}). Then we claim the following:

Claim . Ĥ^{i+1} is a nef Cartier linear system. Ŝ^{i+1} is a smooth member of Ĥ^{i+1}.

Proof of Claim. It is not difficult to see that Ĥ^{i+1} is nef. We shall prove that Ŝ^{i+1} is smooth. Note that since Ŝ^{i} is a smooth Cartier divisor, it is disjoint from the singular locus of T̂^{i}. We shall consider according as the type of ϕ^{i}, separately. \underline{flip case:} If ϕ^{i} is a flip, then (−KT̂^{i} · C^{i}) < 1 (cf. [CKM88]). Since 0 ≤ t^{i} < 1, we have (Ŝ^{i} · C^{i}) = (Ĥ^{i} · C^{i}) = t^{i}(−KT̂^{i} · C^{i}) < 1. By noting that Ŝ^{i} is nef and Cartier, we have (Ŝ^{i} · C^{i}) = 0. This means that Ŝ^{i} is disjoint from the flipping curves. (Note that the case where Ŝ^{i} contains flipping curves is impossible because Ŝ^{i} is disjoint from the singular locus of T̂^{i} as remarked above.) Therefore, Ŝ^{i+1} is isomorphic to Ŝ^{i} and, in particular, is smooth. (Moreover, we have t^{i} = 0 and Ĥ^{i} · C^{i} = 0.)

In the case where ϕ^{i} is a divisorial contraction, we denote by Ê^{i} ⊂ T̂^{i} the exceptional divisor of ϕ^{i}. \underline{(2 : 1)-type:} When Ê^{i} is contracted onto a curve, it follows that (−KT̂^{i} · C^{i}) = 1, where C^{i} represent a general fiber of ϕ^{i}|Ê^{i} : Ê^{i} → ϕ^{i}(Ê^{i}). Then by the similar argument as above we have (Ŝ^{i} · C^{i}) = 0. This means that Ŝ^{i} is disjoint from Ê^{i} or Ŝ^{i} ∩ Ê^{i} consists of several mutually disjoint fibers of ϕ^{i}|Ê^{i} which are (−1)-curves on Ŝ^{i}. Hence Ŝ^{i+1} is also smooth. (Moreover, we have t^{i} = 0 and Ĥ^{i} · C^{i} = 0.) \underline{(2 : 0)-type:} We consider the case where Ê^{i} is contracted to a point. Of course, we may assume that Ŝ^{i} meets Ê^{i}. Then we can actually see that Ê^{i} ≅ ℙ^{2} with a normal bundle Ê^{i}|Ê^{i} ≅ O_{ℙ^{2}}(−1) (cf. [Me02, Claim 3.7]). Then Ŝ^{i} · C^{i} is a positive integer. On the other hand, as Ŝ^{i} · C^{i} = t^{i}(−KT̂^{i} · C_{i}) = 2t^{i} with 0 ≤ t^{i} < 1 we can see t^{i} = 1/2 and Ŝ^{i} · C^{i} = 1, where C^{i} is a line on Ê^{i} ≅ ℙ^{2}. Therefore, Ŝ^{i} ∩ Ê^{i} is a (−1)-curve on Ŝ^{i} and Ŝ^{i+1} is smooth. □

Thus we can construct the (i + 1)-th step. By an inductive argument (and the fact that there is no infinite sequence of flips, see e.g. [FA]), after the composite of divisorial contractions and flips, we finally obtain a Mori fiber space π : T̂^{r} → W and a nef Cartier linear system Ĥ^{r} on T̂^{r}, where π is obtained from the contraction of an extremal ray R^{r} = ℝ_{+}[C^{r}] in \overline{NE} (T̂^{r}) ∩

{Ĥ^{r} + t^{r}KT̂^{r}}^{⊥} with 0 ≤ t^{r} ≤ ρ < 1. In fact, it follows that t^{r} = ρ (cf. [Me02, Claim 3.4]). Put (T̂^{♯}, Ĥ^{♯}) := (T̂^{r}, Ĥ^{r}) to complete the proof. □

Remark 3. Since (T̂, Ĥ) is not necessarily a pair with a big uniruled system (see [Me02, Definition 5.1]), we can not deduce that ρ > 0 at the present time. But, as seen in Lemma 5.1, we can show ρ > 0 easily.

Although we have obtained the Mori fiber space π : T̂^{♯} → W, we, furthermore, need to observe how the inside varieties X and X̂ change in this process of ♯-Minimal Model Program performed in the proof of Proposition 3.1. The next proposition asserts that ♯-Minimal Model Program does not lead to so bad influence on X and X̂.

Proposition 3.2. Let ϕ : T̂ ⋯ →^{ϕ^{0}} T̂^{1} · · · · · · → T̂^{r−1} ⋯ →^{ϕ^{r−1}} T̂^{r} =: T̂^{♯} be

the process of ♯-Minimal Model Program performed in the proof of Proposition 3.1. Among the nef values t^{i} ≥ 0 of the pair (T̂^{i}, Ĥ^{i}) for 0 ≤ i < r (see the proof of Proposition 3.1), let t^{i} = 0 for 0 ≤ i < s and t^{i} > 0 for i ≥ s (if there exist at all). Then:

(1) there exists a following commutative diagram:

T̂ ⋯ →^{ϕ^{0}}  T̂^{1} ⋯ →^{ϕ^{1}} · · · · · · → T̂^{s−1} ⋯ →^{ϕ^{s−1}}  T̂^{s} →^{ϕ^{s}} ⋯ → T̂^{r−1} →^{ϕ^{r−1}}  T̂^{♯}
μ↓  μ^{1}↓  μ^{s−1}↓  μ^{s}↓
T ⋯ →^{ψ^{0}} T^{1} ⋯ →^{ψ^{1}} · · · · · · → T^{s−1} ⋯ →^{ψ^{s−1}} T^{s}

so that each square

T̂^{i} · · · · · · →^{ϕ^{i}}  T̂^{i+1}
μ^{i}↓  ↓μ^{i+1}
T^{i} · · · · · · →^{ψ^{i}} T^{i+1}

satisfies the following according as the type of the birational map ϕ^{i} : T̂^{i} ⋯ → T̂^{i+1} (flip or divisorial contraction) for 0 ≤ i < s;

–In the case where ϕ^{i} is a flip, let C_{1}^{i}, ⋯, C_{t}^{i} ⊂ T̂^{i} exhaust all the flipping curves and let C_{1}^{i}^{+}, ⋯, C_{t}^{i}^{+} ⊂ T̂^{i+1} denote the corresponding flipped curves. Then exactly one of the following two cases occurs concerning the location of these curves:

(a) C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} ⊂ Supp (D̂^{i}) and C_{1}^{i}^{+} ∪ ⋯ ∪ C_{t}^{i}^{+} ⊂ Supp (D̂^{i+1}); or

(b) C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} ⊂ Exc(μ^{i}) and C_{1}^{i}^{+} ∪ ⋯ ∪ C_{t}^{i}^{+} ⊂ Exc(μ^{i+1}).

–In the case where ϕ^{i} is divisorial, an exceptional divisor Ê^{i} of ϕ^{i} is contained in Supp(D̂^{i}) or Exc(μ^{i}). If Ê^{i} ⊂ Supp(D̂^{i}), then ψ^{i} is the contraction of E^{i} := μ^{i}(Ê^{i}) ⊂ Supp (D^{i}) and μ^{i+1} is the naturally induced morphism. Meanwhile, if Ê^{i} ⊂ Exc(μ^{i}) then T^{i+1} = T^{i} and μ^{i+1} is the induced morphism.

(2) For the remainings ϕ^{i} : T̂^{i} → T̂^{i+1} (s ≤ i < r) if any, we have: t^{i} = 1/2 and ϕ^{i} is a divisorial contraction of E2-type, i.e., an exceptional divisor Ê^{i} := Exc(ϕ^{i}) satisfies (Ê^{i}, Ê^{i}|Ê^{i}) ≅ (ℙ^{2}, O_{ℙ^{2}}(−1)). Furthermore, D̂^{i} · C^{i} = 1, where C^{i} is the class of a line on Ê^{i} ≅ ℙ^{2}.

(3) Put D^{i} := μ_{∗}^{i}(D̂^{i}) and X^{i} := T^{i} − Supp (D^{i}) for 0 ≤ i ≤ s. Put X̂^{i} := T̂^{i} − Supp (D̂^{i}) for 0 ≤ i ≤ r. Then X ≅ X^{i} for 0 ≤ ∀i ≤ s. When s ≤ i < r and Ê^{i} is not contained in Supp (D̂^{i}), X̂^{i} is obtained from X̂^{i+1} via a half-point attachment (cf. Definition 3.2) and X̂^{i} − X̂^{i+1} ≅ 𝔸^{2}.

Proof of Proposition 3.2. We have already obtained a birational map ϕ : T̂ ⋯ → T̂^{♯} in Proposition 3.1, so we have only to verify the following two:

(i) the construction of birational maps ψ^{i−1} : T^{i−1} ⋯ → T^{i} and birational morphisms μ^{i} : T̂^{i} → T^{i} for 1 ≤ i ≤ s which fit into a commutative diagram in the assertion;

(ii) the detailed description of T̂^{s} ⋯ → T̂^{♯}.

Our proof consists of several steps and proceed in an inductive way. Step 1. At first we shall prove (1) and the first part of (3). So, let t^{i} = 0. Note that then t^{0} = ⋯ t^{1} = ⋯ = t^{i} = 0. Assume that we have already obtained the i-th step:

T̂^{i−1} · · · · · · →^{ϕ^{i−1}}  T̂^{i}
μ^{i−1}↓  ↓μ^{i}
T^{i−1} · · · · · · →^{ψ^{i−1}} T^{i}

satisfying the conditions in the statement and we shall construct the (i + 1)-th step. Since t^{i} = 0 we have D̂^{i} · C^{i} = Ĥ^{i} · C^{i} = 0, where C^{i} is a generating curve of the extremal ray R^{i}. Then any irreducible curve C ∈ R^{i} is contained in either Supp (D̂^{i}) or Exc(μ^{i}). Indeed, assume to the contrary that C ⊄ Supp (D̂^{i}), ⊄ Exc(μ^{i}). Then μ^{i}(C) is a complete curve contained in T^{i} − Supp (D^{i}) = X^{i}. Note that X^{i} ≅ X by an inductive hypothesis. Since X is affine, this is absurd. Step 2. First of all, we consider:

Case: ϕ^{i} is a flip

Let C_{1}^{i}, ⋯, C_{t}^{i} ⊂ T̂^{i} exhaust all the flipping curves and let C_{1}^{i}^{+}, ⋯, C_{t}^{i}^{+} ⊂ T̂^{i+1} be the corresponding flipped curves. Then:

Lemma 3.2.

(1) C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} is contained in either Supp (D̂^{i}) or Exc(μ^{i});

(2) If C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} ⊂ Supp (D̂^{i}) then C_{1}^{i}^{+} ∪ ⋯ ∪ C_{t}^{i}^{+} ⊂ Supp (D̂^{i+1}). In this case, there exists a commutative diagram:

T̂^{i} · · · · · · →^{ϕ^{i}}  T̂^{i+1}
μ^{i}↓  ↓μ^{i+1}
T^{i} · · · · · · →^{ψ^{i}} T^{i+1},

where ψ^{i} : T^{i} ⋯ → T^{i+1} is a flip of the curves μ^{i}(C_{1}^{i}), ⋯, μ^{i}(C_{t}^{i}) ⊂ Supp (D^{i}) with flipped curves μ^{i+1}(C_{1}^{i}^{+}), ⋯, μ^{i+1}(C_{t}^{i}^{+}) ⊂ Supp(D^{i+1}) and μ^{i+1} : T̂^{i+1} → T^{i+1} is the naturally induced morphism;

(3) If C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} ⊂ Exc(μ^{i}) then there exists a following commutative diagram:

T̂^{i} · · · · · · →^{ϕ^{i}}  T̂^{i+1}
μ^{i}↓  ↓μ^{i+1}
T^{i}  =^{ψ^{i}}  T^{i+1},

where μ^{i+1} : T̂^{i+1} → T^{i+1} is the naturally indeced morphism and all the flipped curves C_{1}^{i}^{+}, ⋯, C_{t}^{i}^{+} are contained in Exc(μ^{i+1});

(4) X^{i+1} ≅ X^{i}.

Proof. (1) Assume that C_{a}^{i} is contained in Supp (D̂^{i}) and C_{b}^{i} is contained in Exc(μ^{i}). Since C_{a}^{i}, C_{b}^{i} ∈ R^{i}, we can write C_{b}^{i} ≡ αC_{a}^{i} with α > 0. Let A be a very ample Cartier divisor on T^{i}. Then we have:

0 = (μ^{i}^{∗}(A) · C_{b}^{i}) = α(μ^{i}^{∗}(A) · C_{a}^{i}) = α(A · μ^{i}_{∗}(C_{a}^{i})) > 0,

which is a contradiction. Hence C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} ⊂ Supp (D̂^{i}) or C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} ⊂ Exc(μ^{i}).

(2) Let C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} be contained in Supp (D̂^{i}). Then we obtain the commutative diagram in the assertion (2), and we need to observe the location of the flipped curves C_{1}^{i}^{+}, ⋯, C_{t}^{i}^{+}. Assume to the contrary that several flipped curves, say C_{1}^{i}^{+}, ⋯, C_{a}^{i}^{+} (1 ≤ a ≤ t), are not contained in Supp (D̂^{i+1}) and the remainings C_{a+1}^{i +}, ⋯, C_{t}^{i}^{+} (if any) are contained in Supp (D̂^{i+1}). Then X̂^{i} ≅ T̂^{i+1} − (Supp (D̂^{i+1}) ∪ C_{1}^{i}^{+} ∪ ⋯ ∪ C_{a}^{i}^{+}). Hence the affine 3-fold X^{i} is embedded into T^{i+1} in such a way that T^{i+1} −X^{i} = Supp(D^{i+1})∪μ^{i+1}(C_{1}^{i}^{+})∪ ⋯ ∪ μ^{i+1}(C_{a}^{i}^{+}), which is not of pure codimension one. This is absurd.

(3) Let C_{1}^{i} ∪ ⋯ ∪ C_{t}^{i} be contained in Exc(μ^{i}). Then ϕ^{i} : T̂^{i} ⋯ → T̂^{i+1} is a part of the Minimal Model Program over T^{i}. Therefore, there exists a birational morphism μ^{i+1} which fits into a commutative diagram stated in (3).

(4) By the arguments in (1), (2) and (3), we can easily see (4). □

Next we consider: Case: ϕ^{i} is divisorial

Since an exceptional divisor Ê^{i} of ϕ^{i} is contained in either Supp (D̂^{i}) or Exc(μ^{i}), it is not difficult to obtain the corresponding assertion of Proposition 3.2 in this case. For example, if Ê^{i} ⊂ Exc(μ^{i}) then ϕ^{i} : T̂^{i} → T̂^{i+1} is a part of the process of MMP/T^{i}. Therefore, there exists a morphism μ^{i+1} : T̂^{i+1} → T^{i} =: T^{i+1}. We shall omit the detail. Step 3. The remainings we have to prove are the assertion (2) and the latter assertion in (3) of Proposition 3.2. So let t^{i} > 0. Then t^{i} = 1/2 and ϕ^{i} is a divisorial contraction of E2-type with an exceptional divisor Ê^{i} with a normal bundle (Ê^{i}, Ê^{i}|Ê^{i}) ≅ (ℙ^{2}, O_{ℙ^{2}}(−1)) (see the proof of Proposition 3.1). Hence D̂^{i} · C^{i} = 1, where C^{i} is a line on Ê^{i} ≅ ℙ^{2}. Thus, if Ê^{i} is not contained in the boundary D̂^{i}, then D̂^{i} intersects Ê^{i} in such a way that D̂^{i}|Ê^{i} ∈ |O_{ℙ^{2}}(1)|. Hence X̂^{i} is obtained from X̂^{i+1} by a single half-point attachment (cf. Definition 3.2) and so X̂^{i} − X̂^{i+1} ≅ 𝔸^{2}. Thus we complete the proof of Proposition 3.2. □

In the assertion of Proposition 3.2, we use the notion of half-point attachment which is a straightforward 3-dimensional version of the original definition given in [Miy01, 4.4.1. Definition.].

Definition 3.2. Let Z be a normal quasi-projective 3-fold and let Z ↪ V be a compactification into a normal projective 3-fold V with the boundary V − T = Supp (B). Let p be a point on Supp (B) where V is smooth. Let f : V′ → V be the blow-up at p with an exceptional divisor E ≅ ℙ^{2} and we denote by B′ := f_{∗}^{−1}B the proper transform of B. Then Z′ := V′ − Supp (B′) is said to be a half-point attachment to Z. It is clear, by the definition, that Z is an open subset of Z′ so that Z′ − Z ≅ 𝔸^{2}.

4. Proof of Theorem 2.1

In this section we shall complete the proof of Theorem 2.1. More precisely speaking, we shall construct the birational morphisms ψ′′ : T′ → T^{♯} and μ^{♯} : T̂^{♯} → T^{♯} in the diagram of Theorem 2.1, where we put T′ := T^{s} with the notation of Section 3, Proposition 3.2. At first, we have:

Lemma 4.1. Let the notations be the same as in Proposition 3.2. The exceptional set Exc(μ^{s}) of μ^{s} is of pure codimension one and is disjoint from the boundary Supp (D̂^{s}).

Proof. Note that X has ℚ-factorial singularities by our setting. Since the process T ⋯ → T^{s} constructed in Proposition 3.2 has no influence on X and is a part of Minimal Model Program, T^{s} also has ℚ-factorial singularities. Hence, since both T̂^{s} and T^{s} have ℚ-factorial singularities, the exceptional set of μ^{s} : T̂^{s} → T^{s} is of pure codimension one (cf. the explanation in the proof of [Ma01, Theorem 4.1.3]). By the construction of T̂ ⋯ → T̂^{s}, we can see Exc(μ^{s}) ∩ Supp (D̂^{s}) = ∅. □

For the construction of ψ′′ : T^{s} → T^{♯} and μ^{♯} : T̂^{♯} → T^{♯}, we proceed in an inductive way. Namely, we prove the following:

Lemma 4.2. Let the notations be as in Section 3. There exist birational morphisms ψ^{i} : T^{i} → T^{i+1} (s ≤ i < r) and μ^{i} : T̂^{i} → T^{i} (s ≤ i ≤ r) so that they satisfy the following:

(1) there exists a commutative diagram:

T̂^{i} →^{ϕ^{i}}  T̂^{i+1}
μ^{i}↓  ↓μ^{i+1}
T^{i} →^{ψ^{i}} T^{i+1},

so that: (a)_{i} ϕ^{i} is of E2-type with Ê^{i} := Exc(ϕ^{i}) such that (Ê^{i}, Ê^{i}|Ê^{i}) ≅ (ℙ^{2}, O_{ℙ^{2}}(−1))

and Ê^{i} ∩ Exc(μ^{i}) = ∅;

(b)_{i} ψ^{i} is of E2-type with E^{i} := μ^{i}(Ê^{i}) = Exc(ψ^{i}) such that (E^{i}, E^{i}|_{E^{i}}) ≅ (ℙ^{2}, O_{ℙ^{2}}(−1));

(c)_{i+1} Supp (D̂^{i+1}) ∩ Exc(μ^{i+1}) = ∅.

(2) Put X̂^{i} := T̂^{i} −Supp(D̂^{i}) and X^{i} := T^{i} −Supp(D^{i}). If the exceptional divisor Ê^{i} of ϕ^{i} is contained in Supp (D̂^{i}), then X̂^{i} ≅ X̂^{i+1} and X^{i} ≅ X^{i+1}. Meanwhile, if Ê^{i} is not contained in Supp (D̂^{i}), then X̂^{i} (resp. X^{i}) is obtained from X̂^{i+1} (resp. from X^{i+1}) by a single half-point attachment (cf. Definition 3.2), so X̂^{i} − X̂^{i+1} ≅ X^{i} − X^{i+1} ≅ 𝔸^{2}.

(3) Put T^{♯} := T^{r}, D^{♯} := D^{r} and X^{♯} := X^{r} = T^{♯} − Supp (D^{♯}). Then X is obtained from X^{♯} via the composite of suitable half-point attachments. In particular, Sing (X) = Sing (X^{♯}).

Proof. (1) Assume that we have already constructed the (i − 1)-th step:

T̂^{i−1} →^{ϕ^{i−1}}  T̂^{i}
μ^{i−1}↓  ↓μ^{i}
T^{i−1}  →^{ψ^{i}} T^{i},

satisfying (a)_{i−1}, (b)_{i−1} and (c)_{i}, and we shall construct the i-th step satisfying (a)_{i}, (b)_{i} and (c)_{i+1} for i ≥ s. As seen in Section 3 (cf. Proposition 3.1), the contraction of the extremal ray R^{i} = ℝ_{+}[C^{i}] yields an E2-type divisorial contraction ϕ^{i} : T̂^{i} → T̂^{i+1} with an exceptional divisor Ê^{i} ≅ ℙ^{2} such that Ê^{i}|Ê^{i} ≅ O_{ℙ^{2}}(−1) and D̂^{i} · C^{i} = 1, where C^{i} is a line on Ê^{i} ≅ ℙ^{2}. If Ê^{i} is contained in Supp (D̂^{i}), then we can easily construct the desired i-th step by noting that Supp(D̂^{i}) is disjoint from Exc(μ^{i}) (the property (c)_{i}). On the other hand, if Ê^{i} is not contained in Supp(D̂^{i}), then Ê^{i}∩Exc(μ^{i}) = ∅. In fact, assume to the contrary that Ê^{i} ∩ Exc(μ^{i}) ≠ ∅. Note that since T̂^{s} and T^{s} have ℚfactorial singularities, the varieties T̂^{i} and T^{i} have also ℚ-factorial singularities. Hence the exceptional set Exc(μ^{i}) is purely of codimension one. Thus, Ê^{i} ∩ Exc(μ^{i}) ≠ ∅ implies that we can take an irreducible curve C ⊆ Ê^{i} ∩ Exc(μ^{i}). But, since Ê^{i} ≅ ℙ^{2} and D̂^{i} meets Ê^{i} this implies that Supp (D̂^{i}) ∩ Exc(μ^{i}) ≠ ∅, which is a contradiction to (c)_{i}. Therefore, Ê^{i} ∩ Exc(μ^{i}) = ∅ and we can construct the birational morphisms ψ^{i} : T^{i} → T^{i+1}, μ^{i+1} : T̂^{i+1} → T^{i+1} and the desired square at the i-th step.

(2) Note that X̂^{i} − X̂^{i+1} ≅ X^{i} − X^{i+1} because Ê^{i} is disjoint from Exc(μ^{i}) as seen in the proof of (1). If Ê^{i} is contained in Supp (D̂^{i}), then the assertion is easy to see. If Ê^{i} is not contained in Supp (D̂^{i}), then D̂^{i} meets Ê^{i} in such a way that D̂^{i}|Ê^{i} ∈ |O_{ℙ^{2}}(1)| because of D̂^{i} ·C^{i} = 1, where C^{i} is a line on Ê^{i} ≅ ℙ^{2}. Therefore, in this case, we have X̂^{i+1} ≅ T̂^{i} − (Supp (D̂^{i}) ∪ Ê^{i}). Similarly, we have X^{i+1} ≅ T^{i}−(Supp(D^{i})∪E^{i}). Hence X̂^{i} (resp. X^{i}) is obtained from X̂^{i+1} (resp. X^{i+1}) via a single half-point attachment. The assertion (3) is easily seen by the repetition of (2). □

By combining Propositions 3.1 and 3.2 and Lemma 4.2 we complete the proof of Theorem 2.1.

5. Proof of Theorem 2.2 and Corollary 2.1

As seen in the previous sections, the original normal affine 3-fold X in question is constructed from X^{♯} = T^{♯} − Supp (D^{♯}) via the composite of halfpoint attachments (cf. Definition 3.2). Hence, the investigation of the Mori fiber space π : T̂^{♯} → W and the birational morphism μ^{♯} : T̂^{♯} → T^{♯} are necessary for the analysis the structure on X. In this final section, we shall classify the pair (T̂^{♯}, D̂^{♯}) (hence, classify X̂^{♯} = T̂^{♯}−Supp(D̂^{♯})) and investigate μ^{♯} according as Mori fiber space π : T̂^{♯} → W to obtain Theorem 2.2. This result is essentially contained in [C-F93, Me02]. Hereafter we shall determine the possibility of singularities on X. First of all, we have:

Lemma 5.1. The threshold ρ := ρ(T̂, Ĥ) < 1 of the pair (T̂, Ĥ) (cf. Section 3) is greater than zero.

Proof. Note that Ĥ^{♯} + ρKT̂^{♯} ≡_{π} 0. If ρ = 0 then Ĥ^{♯} · C^{♯} = D̂^{♯} · C^{♯} = 0, where C^{♯} is any extremal rational curve yielding π. This implies that C^{♯} ⊂ Supp(D̂^{♯}). Indeed, otherwise, D̂^{♯} · C^{♯} = 0 means that C^{♯} is disjoint from Supp (D̂^{♯}), so is contained in X̂^{♯} = T̂^{♯} − Supp (D̂^{♯}). Note that π is a Mori fiber space, so C^{♯} ⊄ Exc(μ^{♯}) for a general extremal rational curve. But then the image μ^{♯}(C^{♯}) is a complete curve contained in X^{♯} which is an open affine subset of X. This is absurd. Hence C^{♯} ⊂ Supp (D̂^{♯}) for a general curve C^{♯} contracted by π. This is obviously a contradiction. □

Since 0 < ρ < 1, the same argument as in the proof of [Me02, Theorem 5.3] says that the possibility of the pair (T̂^{♯}, D̂^{♯}) of ℚ-factorial terminal 3-fold Mori fiber space T̂^{♯} and the proper transform D̂^{♯} of D̂ is described as in Theorem 2.2. We shall prove the following where we make use of the terminologies in Theorem 2.2:

Lemma 5.2. Assume that π : T̂^{♯} → W is either of type D′2, D3, D3′ or ℚ-Fano. Then:

(1) μ^{♯} is an isomorphism.

(2) X^{♯} and X are smooth.

Proof. (1) Assume to the contrary that μ^{♯} is not isomorphism. Then E^{♯} := Exc(μ^{♯}) is a divisor which is disjoint from D̂^{♯}. This is obviously impossible when π is of type D′2, D3, D3′ or ℚ-Fano.

(2) If π is of D3-type, then T̂^{♯} is smooth. Hence X^{♯} is smooth. Therefore, X is also smooth. From now on, we shall consider the case where π is of D3′-type. If T̂^{♯} is smooth (i.e., all the fibers of π are isomorphic to ℙ^{2}), then the assertion is clear to see. Assume that T̂^{♯} is not smooth, so there exists a singular fiber G ≅ 𝕊_{4} ⊂ ℙ^{5} such that the vertex v ∈ G sits in the hyper-quotient singularity of anlytic type o ∈ (xy + z^{2} + t^{k} = 0) ⊂ ℂ^{4} : (x, y, z, t)/ℤ_{2}(1, 1, 1, 0) for k ≥ 1. This singularity v ∈ T̂^{♯} can be resolved via the following diagram (♮) which is a composite of elementary links of Type II in the sense of the Sarkisov Program (cf. [Me02], [Cor95, Cor00]):

[figure: (♮); T̂^{♯}; =:; T^{(0)}; ↓; W; π; =; ↙; μ_{0}; U^{(0)}; ↘; ν_{0}; T^{(1)}; ↓; W; π^{(1)}; =; ⋯; =; ↙; ↘; T^{(k−1)}; ↓; W; π^{(k−1)}; =; ↓; W,; π^{(k)}; ↙; μ_{k−1}; U^{(k−1)}; ↘; ν_{k−1}; T^{(k)}]

where each elementary link T^{(i)}/W ←^{μ_{i}} U^{(i)} →^{ν_{i}} T^{(i+1)}/W is constructed, inductively, in the following fashion:

– Let G^{(i)} be the fiber of π^{(i)} over the point w := π(G), which is isomorphic to G^{(i)} ≅ 𝕊_{4} ⊂ ℙ^{5} such that the vertex v_{i} ∈ G^{(i)} sits in the hyperquotient singularity of analytic type o ∈ (xy + z^{2} + t^{k−i} = 0) ⊂ ℂ^{4} : (x, y, z, t)/ℤ_{2}(1, 1, 1, 0) for 0 ≤ i < k.

– Let μ_{i} : U^{(i)} → T^{(i)} be the blow-up at the maximal ideal 𝔪_{v_{i}} of v_{i} with an exceptional divisor E^{(i)}. It follows that E^{(i)} ≅ 𝕊_{4} if 0 ≤ i < k − 1 and E^{(i)} ≅ ℙ^{2} if i = k − 1.

– Let ν_{i} : U^{(i)} → T^{(i+1)} be the contraction of the proper transform G_{U^{(i)}} ≅ 𝔽_{4} of G^{(i)} along the ruling on it. Thus we obtain the (i + 1)-th step π^{(i+1)} : T^{(i+1)} → W.

For other singular points on T̂^{♯} (if there exist at all), they can be resolved by the same fashion. Thus we have the birational map Φ : T̂^{♯}/W ⋯ → \overline{T} /W which is a composite of elementary links of Type II (in the Sarkisov Category) centered at singularities, where \overline{T} /W is a ℙ^{2}-bundle structure over W. Let \overline{D} := Φ_{∗}(D̂^{♯}) be the proper transform of D̂^{♯}. It is easy to see that \overline{D}|\overline{F} ∈ |O_{ℙ^{2}}(2)|, where \overline{F} ≅ ℙ^{2} is a fiber of \overline{T} /W. We can see that D̂^{♯} contains all the singular points of T̂^{♯} by investigating the inverse procedure Φ^{−1} : \overline{T} /W ⋯ → T̂^{♯}/W. Therefore, the complement X̂^{♯} is smooth. Since μ^{♯} is an isomorphism, X^{♯} and hence X also are smooth. By the similar argument, we can show that X in the case where π is of type D2′ is smooth. We shall omit the detail. □

Lemma 5.3.

(1) If π : T̂^{♯} → W is of type D′2 (resp. D′3 or ℚ-Fano), then the singularities Sing (X) on X are described as in (i) (resp. (ii) or (iii)) in Corollary 2.1.

(2) If π is of type D3, then X ≅ X̂^{♯} and smooth.

Proof. (1) By Lemma 5.2, X^{♯} ≅ X̂^{♯} = T̂^{♯} − Supp (D̂^{♯}). Since Sing (X) = Sing (X^{♯}) (cf. Theorem 2.1), we can easily see the assertion in consideration of Theorem 2.2.

(2) Let us assume that π is of D3-type, i.e., a ℙ^{2}-bundle structure over a smooth curve W. We claim the following: Claim. s = r, i.e., t^{0} = t^{1} = ⋯ = t^{r−1} = 0. Proof of Claim. Note that, by the construction of ♯-Minimal Model Program associated to the pair (T̂, Ĥ) (see Section 3), inequalities 0 ≤ t^{0} ≤ t^{1} ≤ ⋯ ≤ t^{r−1} ≤ t^{r} = ρ < 1 hold. Since π is of type D3 we have (D̂^{♯} · C^{♯}) = 1, i.e., ρ = 1/3, where C^{♯} is a line on a fiber F ≅ ℙ^{2} of π. If s < r, i.e., t^{i} > 0 for some 0 ≤ i ≤ r − 1, then ϕ^{i} : T̂^{i} → T̂^{i+1} is an E2-type divisorial contraction (see the arguments in the proof of Propositions 3.1, 3.2). In particular, t^{i} = 1/2. But then 1/2 = t^{i} > 1/3 = ρ, a contradiction. □

The above mentioned claim means ϕ = ϕ′ in the notation of Theorem 2.1, in particular, X ≅ X^{♯}. Since μ^{♯} is an isomorphism (cf. Lemma 5.2), we have X^{♯} ≅ X̂^{♯}. Thus X ≅ X^{♯} ≅ X̂^{♯}. Since T̂^{♯} is smooth, so is X̂^{♯} obviously. □

The remaining is the case where π : T̂^{♯} → W is of C2-type.

Lemma 5.4. Assume that π : T̂^{♯} → W is of C2-type. If X is not smooth, then we have T̂^{♯} ≅ ℙ(O_{ℙ^{2}} ⊕ O_{ℙ^{2}}(a)) (a ≥ 2). Furthermore, D̂^{♯} (resp. E^{♯} := Exc(μ^{♯})) is a section of π corresponding to O_{ℙ^{2}} ⊕O_{ℙ^{2}}(a) → O_{ℙ^{2}}(a) (resp. O_{ℙ^{2}} ⊕O_{ℙ^{2}}(a) → O_{ℙ^{2}}). In particular, E^{♯} is a negative section of π with E^{♯}|_{E^{♯}} ≅ O_{ℙ^{2}}(−a) and Sing (X) = p := μ^{♯}(E^{♯}) such that (p ∈ X) ≈ o ∈ \frac{1}{a}(1, 1, 1).

Proof. Our proof consists of three steps. Step 1. Let us assume that X is not smooth. Note that μ^{♯} is not an isomorphism. Indeed, if μ^{♯} is an isomorphism, then X^{♯} is smooth and so is X (cf. Theorem 2.1 (4)). This is a contradiction. Hence E^{♯} := Exc(μ^{♯}) is a divisor and disjoint from D̂^{♯} (see the proof for Lemma 4.2). By noting that D̂^{♯} ·C^{♯} = 1, D̂^{♯} is a meromorphic section of π, where C^{♯} is the class of a fiber of π. In fact, D̂^{♯} is a (holomorphic) section of π. Indeed, if D̂^{♯} contains a fiber of π then it meets E^{♯}, a contradiction. E^{♯} is also a (holomorphic) scetion of π. Step 2. Now we claim the following:

Claim. The base surface W (and hence E^{♯} also) is a ruled surface. Proof of Claim. Let p : X → Z denote the 𝔸^{1}-fibration induced naturally from an 𝔸^{1}-cylinderlike open subset of X. At first, we consider the case where X^{♯} has still an 𝔸^{1}-fibration induced from p. We denote by p^{♯} := p|_{X^{♯}} the restricted 𝔸^{1}-fibration. (Note that X^{♯} is an open affine subset of X.) Each of the closures in T^{♯} of general fibers of p^{♯} meets D^{♯} := μ^{♯}(D̂^{♯}) in a single point. If these closures do not define a ℙ^{1}-fibration on T^{♯} then, after resolving the indeterminacy, we obtain a birational morphism θ : V → T^{♯} and a ℙ^{1}-fibration f : V → W_{0}, which is an extension of p^{♯}, over a ruled surface W_{0}. Therefore T̂^{♯} ≈ V ≈ W_{0} × ℙ^{1} On the other hand, we have T̂^{♯} ≈ W × ℙ^{1} associated to the conic bundle structure π : T̂^{♯} → W. Therefore W × ℙ^{1} ≈ W_{0} × ℙ^{1}, so we have W ≈ W_{0} (cf. [Ka87]). Thus W is a ruled surface. On the other hand, if the closures in T^{♯} of general fibers of p^{♯} define a ℙ^{1}-fibration g on T^{♯}, g has D^{♯} as a section. Therefore, T̂^{♯} has two essentially distinct ℙ^{1}-fibrations π and ĝ, where ĝ is naturally inherited from g. This is possible only if the base surface W is ruled. Next, we shall consider the case where p : X → Z can not induce an 𝔸^{1}-fibration on X^{♯}. Then at least one of components of the complement X − X^{♯} meets general fibers of p. Note that X − X^{♯} is a union of several affine planes (cf. Theorem 2.1 (3)). Hence the base surface Z of p is dominated by 𝔸^{2} and is rational. Therefore X is a rational 3-fold. Since E^{♯} is a section of a ℙ^{1}-bundle π on a rational 3-fold T̂^{♯}, we have E^{♯} × ℙ^{1} ≈ ℙ^{3}. Therefore, E^{♯} is rational (cf. [Ka87]). □ Step 3. Since E^{♯} is a ruled surface, we can write KT̂^{♯} = (μ^{♯})^{∗}(K_{T^{♯}})+eE^{♯} with e > −1. Since K + (1 − ε)E^{♯} is not nef over T^{♯} and klt for a sufficiently small 0 < ε << 1, there exists an extremal ray, say Q, in \overline{NE} (T̂^{♯}/T^{♯})_{K+(1−ε)E^{♯}<0} (cf. [K-M-M87]). Since T̂^{♯} and T^{♯} are ℚ-factorial and Exc(μ^{♯}) = E^{♯}, we have N_{1}(T̂^{♯}/T^{♯}) ≅ ℝ and \overline{NE} (T̂^{♯}/T^{♯}) = Q. Hence the contraction of Q coincides with μ^{♯}. In particular, all the curves on E^{♯} are numerically proportional. But since E^{♯} is a section of π we have ϱ(E^{♯}) = 1. Hence E^{♯} ≅ ℙ^{2} and π is a ℙ^{1}-bundle structure over W ≅ ℙ^{2} admitting a section D̂^{♯}. Hence we can write T̂^{♯} ≅ ℙ(O_{ℙ^{2}} ⊕ O_{ℙ^{2}}(a)) for some a ≥ 1 and E^{♯} is a negative section. Since the contraction of E^{♯} have to result in the singular point, we have a ≥ 2. The remaining assertions are easy to see. Thus we complete the proof. □

By summarizing the arguments, we obtain Theorem 2.2. Finally, we shall prove Corollary 2.1. The assertions except for the latter part in (2) are already proved in the previous argument. So, let us consider the case where T̂^{♯} is a ℚ-Fano 3-fold with ϱ = 1 such that T̂^{♯} is general in the sense of Remark 1. Hence T̂^{♯} in question is quasi-smooth (cf. Remark 1) and, so, the singularities on T̂^{♯} are described in Table 1. Then we can easily see the assertion. (If T̂^{♯} is (i), (v) or (viii), then we need not to consider generality condition, and, in fact, we can easily observe the singularities on it.)

Permanent Address:

Department of Mathematics Graduate School of Science Osaka University Toyonaka Osaka 560-0043, Japan

                     E-Mail: kisimoto@math.sci.osaka-u.ac.jp

Current Address:

Max-Planck-Institut für

Mathematik Bonn

Vivatsgasse 7, D-53111 Bonn, Germany

                      E-Mail: kishimot@mpim-bonn.mpg.de

References

[C-F93] F. Campana and H. Flenner, Projective threefolds containing a smooth rational surface with ample normal bundle, J. reine angew. Math., 440 (1993), pp. 77–98.

[CKM88] H. Clemens, J. Kollár and S. Mori, Higher dimensional complex geometry, Astérisque 166 (1988).

[Cor95] A. Corti, Factoring birational maps of 3-folds after Sarkisov, J. Alg. Geom., 4 (1995), pp. 223–254.

[Cor00] A. Corti, Singularities of linear systems and 3-fold birational geometry, Explicit birational geometry of 3-folds (Cambridge LMS 281, 2000), pp. 259–312.

[F-Z03] H. Flenner and M. Zaidenberg, Rational curves and rational singularities, Math. Z., 244 (2003), pp. 549–575.

[Fl00] A.R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds (Cambridge LMS 281, 2000), pp. 101–172.

[FA] Flips and abundance for algebraic threefolds, Papers from the second summer seminar on algebraic geometry held at University of Utah, August 1991, Astérisque 211 (1992), Société Mathématique de France, Paris, 1992.

[Giz70] M.H. Gizatullin, On affine surfaces that can be completed by a nonsingular rational curve, Izv. Akad. Nauk SSSR Ser. Mat., 34 (1970), pp. 778–802; Math. USSR-Izv., 4 (1970), pp. 787–810.

[Ha77] R. Hartshorne, Algebraic geometry, GTM 52, Springer, 1977.

[Ii77] S. Iitaka, On logarithmic Kodaira dimension of algebraic varieties, Complex Analysis and Algebraic Geometry (W. L. Baily, Jr. and T. Shioda, eds.), Iwanami Shoten, Tokyo, 1977, pp. 175–189.

[Ka87] M.C. Kang, The cancellation problem, J. Pure Appl. Algebra, 47 (1987), pp. 165–171.

[K-M-M87] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, in: Algebraic Geometry, Sendai, 1985, pp. 283– 360, Adv. Stud. Pure Math., 10, North Holland, Amsterdam, 1987.

[Kle66] S. Kleiman, Toward a numerical theory of ampleness, Ann. of Math., 84 (1966), pp. 293–344.

[Ma01] K. Matsuki, Introduction to the Mori Program, UTX, Springer, 2001.

[Me02] M. Mella, ♯-Minimal models of uniruled 3-folds, Math. Z., 242 (2002), pp. 687–707.

[Miy78] M. Miyanishi, Lectures on curves on rational and unirational surfaces, Tata Inst. Fund. Res. Bombay, 1978.

[Miy81] M. Miyanishi, Singularities of normal affine surfaces containing cylinderlike open sets, J. Algebra, 68 (1981), pp. 268–275.

[Miy88] M. Miyanishi, Algebraic characterizations of the affine 3-space, Algebraic Geometry Seminar (Singapore, 1987), pp. 53–67, World Sci. Publishing, Singapore, 1988.

[Miy01] M. Miyanishi, Open algebraic surfaces, CRM Monograph Series 12, AMS Providence, RI, 2001.

[Mori82] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math., 116 (1982), pp. 133–176.