Compactifications of contractible affine 3-folds into smooth Fano 3-folds with B_{2} = 2

Takashi Kishimoto^{∗}

(1) Permanent Address: Department of Mathematics Graduate School of Science Osaka University, Toyonaka Osaka 560–0043, JAPAN

(2) Current Address: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, GERMANY

Key words smooth Fano 3-folds with B_{2} = 2, contractible affine 3-folds, affine modifications. Subject classification Primary: 14R10, Secondary: 14E30.

Contents

1 Introduction 1 2 Preliminaries 5 3 The case D_{1} + D_{2} is not nef 7 4 The case D_{1} + D_{2} is nef but not ample 9

4.1 φ_{1} is of E1-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 φ_{1} is of E2-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 φ_{1} is of E3-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.4 φ_{1} is of E4-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.5 φ_{1} is of E5-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 The case D_{1} + D_{2} is ample 15 5.1 φ_{1} is of E2-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 φ_{1} is of C2-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.3 φ_{1} is of D2-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.4 φ_{1} is of D3-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

References 25

After the contributions of Furushima, Nakayama, Peternell and Schneider, in 1993, Furushima [Fur93] finally succeeded in the classification of the compactifications of the affine 3-space A^{3} into smooth Fano 3-folds with B_{2} = 1. In this paper, we consider the compactifications of the contractible affine 3-folds X (e.g. X = A^{3}) into smooth Fano 3-folds V with B_{2} = 2. Consequently, we classify all such compactifications X ↪ (V, D_{1} ∪ D_{2}) in the case where K_{V} + D_{1} + D_{2} is not nef. Furthermore, we know that infinitely many mutually non-isomorphic exotic A^{3}’s can be compactified into Fano 3-folds with B_{2} = 2. This phenomenon never occurs when B_{2} = 1.

1 Introduction

2 Takashi Kishimoto: Compactifications of contractible affine 3-folds into smooth Fano 3-folds with B_{2} = 2

Throughout the present article we work over the field of complex numbers C. Concerning the compactifications of the affine 3-space A^{3} into smooth Fano 3-folds with the second Betti number B_{2} = 1, after the contributions of Furushima, Nakayama, Peternell, Schneider (cf. [Fur86, Fur90, Fur92, F-N89a, F-N89b, Pe89, P-S88]), finally in [Fur93], Furushima completed the classification. Let A^{3} ↪ (V, D) be the compactification of the affine 3-space A^{3}, where V is a smooth Fano 3-fold with B_{2} = 1 of index r ∈ {1, 2, 3, 4} and D is the boundary divisor. Note that since B_{2} = 1 and the coordinate ring of A^{3} is factorial, D is irreducible and is an ample generator of Pic(V) ≅ Z. At first, in [Fur86] and [Fur90], Furushima classified such compactifications when r ≥ 2 and D is normal. As the next step, in [F-N89a], Furushima and Nakayama investigated the compactifications when index r = 2 and D is possibly non-normal. As a result, they obtained the following:

∗ Corresponding author: e-mail: (1) kisimoto@math.sci.osaka-u.ac.jp (2) kishimot@mpim-bonn.mpg.de

Theorem 1.1 (cf. [Fur86, F-N89a]) Let (V, D) be a compactification of A^{3} into a smooth Fano 3-fold V with the second Betti number B_{2} = 1 and of index r ≥ 2. Then:

(1) r = 4 ⇒ (V, D) ≅ (P^{3}, P^{2});

(2) r = 3 ⇒ (V, D) ≅ (Q^{3}, Q_{0}^{2});

(3) r = 2 and D is normal ⇒ (V, D) ≅ (V_{5}, H_{5}^{0});

(4) r = 2 and D is non-normal ⇒ (V, D) ≅ (V_{5}, H_{5}^{∞}),

where V_{5} is a linear section Gr(2, 5)∩P^{6} of the Grassmann variety Gr(2, 5) ↪ P^{9} by three hyperplanes in P^{9}. H_{5}^{0} is a normal hyperplane section on V_{5} which is a normal del Pezzo surface of degree (−K_{H_{5}^{0}})^{2} = 5 with exactly one A_{4}-type Du Val singularity. H_{5}^{∞} is a non-normal hyperplane section on V_{5} which is a non-normal del Pezzo surface of degree (−K_{H_{5}^{∞}})^{2} = 5 whose singular locus is a line Σ with a normal bundle N_{Σ/V_{5}} ≅ O_{Σ}(−1) ⊕ O_{Σ}(1).

In the case where the index r is equal to 1, the situation is more subtle. But, in [Fur93], [Pe89] and [P-S88], they have shown the following:

Theorem 1.2 (cf. [Fur93, Pe89, P-S88]) Let (V, D) be a compactification of A^{3} into a smooth Fano 3-fold V with the second Betti number B_{2} = 1 and of index r = 1. Then:

(1) (V, D) ≅ (V_{22}, H_{22}^{0}) or (V_{22}, H_{22}^{∞}), where V_{22} ↪ P^{13} is a smooth Fano 3-fold of index r = 1 with genus g := \frac{1}{2}(−K_{V})^{3} + 1 = 12, of degree 22 by the anti-canonical embedding, and H_{22}^{0} and H_{22}^{∞} are non-normal hyperplane sections on V_{22};

(2) Let Σ be a non-normal locus of H_{22}^{0} (resp. H_{22}^{∞}). Then Σ is a line on V_{22} with the normal bundle N_{Σ/V_{22}} ≅ O_{Σ}(−2) ⊕ O_{Σ}(1);

(3) mult_{Σ}H_{22}^{0} = 2 and mult_{Σ}H_{22}^{∞} = 3, in particular, H_{22}^{∞} is a ruled surface swept out by conics on V_{22} intersecting with the line Σ.

Note that the arguments in [Fur86, Fur90, Fur92, F-N89a, F-N89b, Fur93, Pe89, P-S88] can be applied without any modification even in the case of the compactifications of contractible affine 3-folds into a smooth Fano 3-folds with B_{2} = 1. Indeed, their arguments made use of only the contractibility of the affine 3-space A^{3}. Thus we have the following result as a consequence:

Corollary 1.1 (cf. [Fur86, Fur90, Fur92, Fur93, F-N89a, F-N89b, Pe89, P-S88]) Let X be a contractible smooth affine 3-fold. Assume that X is embedded into a smooth Fano 3-fold V with the second Betti number B_{2} = 1 and D the boundary divisor. Then X is isomorphic to the affine 3-space A^{3} and (V, D) ≅ (P^{3}, P^{2}), (Q^{3}, Q_{0}^{2}), (V_{5}, H_{5}^{0}), (V_{5}, H_{5}^{∞}), (V_{22}, H_{22}^{0}) or (V_{22}, H_{22}^{∞}).

Now we hope to investigate whether or not the similar statements hold for the case B_{2}(V) = 2. Namely, our main interest is stated as follows:

Problem 1.1 Let X be a contractible affine 3-fold which is embedded into a smooth Fano 3-folds V with the second Betti number B_{2} = 2, and let V − X = D_{1} ∪ D_{2} be the boundary components. Then, classify such an X and a compactification X ↪ (V, D_{1} ∪ D_{2}).

\underline{3}

Note that the contractibility of X implies that the coordinate ring A := Γ(O_{X}) is factorial and the multiplicative group A^{∗} consisting of all invertible elements of A coincides with non-zero constants A^{∗} = C^{∗} (cf. [Fuj82]). Hence Pic (V) is free generated by irreducible components in V − X, so the boundary consists of two components, say D_{1} and D_{2} such that Pic (V) = Z[D_{1}] ⊕ Z[D_{2}] (cf. Lemma 2.1).

As mentioned above, in the case where the second Betti number equals B_{2} = 1, the first progress towards the classification of compactifications lies in the case of Fano index r ≥ 2, in other words, the case where the line bundle K_{V} + D is not nef. In consideration of this fact, as the first step to Problem 1.1 above, we shall investigate compactifications X ↪ (V, D_{1} ∪ D_{2}) satisfying the following condition (†):

(†) : The line bundle K_{V} + D_{1} + D_{2} is not nef.

In this article, we shall classify the contractible affine 3-folds X and their compactifications X ↪ (V, D_{1} ∪ D_{2}) into smooth Fano 3-folds V with B_{2}(V) = 2 satisfying (†). In order to state our main result, we prepare the notation. Since V is a Fano 3-fold with B_{2} = 2, \overline{NE} (V) is spanned by two extremal rays, say R_{1} and R_{2}. Let φ_{i} : V → W_{i} be the contraction of R_{i} for i = 1, 2. Let L_{i} be an ample generator of Pic(W_{i}) ≅ Z and we put H_{i} := φ_{i}^{∗}(L_{i}) for i = 1, 2. In Table 1, (No.♠) of each V means that V is a smooth Fano 3-fold with the indicated number in [M-M81, Table 2]. When we write [⋆ − ⋆⋆] in Table 1, it means that φ_{1} is of ⋆-type and φ_{2} is of ⋆⋆-type. With these notations, we obtain the following:

Theorem 1.3 Let X be a contractible affine algebraic 3-fold. Assume that X is embedded into a smooth Fano 3-fold V with the second Betti number B_{2}(V) = 2 and with V −X = D_{1} ∪D_{2} the boundary components satisfying the condition (†). Then the complete list of the pair (V, D_{1} ∪ D_{2}) up to the permutation of D_{1} and D_{2}, and the inside affine algebraic 3-fold X = V − (D_{1} ∪ D_{2}) is given as in Table 1, where affine 3-folds X marked as (∗) (Types XII, XIII and XXII) is described as follows:

(∗) : In Types XII, XIII (resp. Type XXII), we put Z := Q^{3} − \overline{D_{1}} (resp. Z := P^{3} − \overline{D_{1}}). Then X is obtained from Z via the affine modification with locus (p ∈ S). (See Definition 2.1 for the definition on affine modifications.)

Remark 1.1 It is clear that Z ≅ A^{3} in the cases of Types XII and XXII. On the other hand, in the case of Type XIII, Z is isomorphic to SL(2; C).

Corollary 1.2 Infinitely many exotic A^{3}’s can be embedded into smooth Fano 3-folds with B_{2} = 2.

Proof. For two distinct pairs (a, b) and (a′, b′) of positive integers satisfying a > b ≥ 2, a′ > b′ ≥ 2 and gcd(a, b) = 1, gcd(a′, b′) = 1, the affine 3-folds S_{a,b} × A^{1} and S_{a′,b′} × A^{1} are mutually non-isomorphic exotic A^{3} and they can be embedded into a smooth Fano 3-folds of type XII or XXII in Table 1. (See Remark 1.3 for the notation.) Thus we complete the proof. Note that the phenomenon that exotic A^{3}’s can be compactified never occurs when B_{2} = 1. (Compare this corollary with Corollary 1.1.) □

Remark 1.2 In [Mu90], he classified the compactifications of the affine 3-space A^{3} into smooth projective 3-folds V with B_{2}(V) = 2 in the case where the boundary two components D_{1} and D_{2} are normal and intersect each other along a smooth curve ideal-theoretically. Note that in our classification (cf. Table 1), we do not assume the normalities and normal crossings of D_{1} and D_{2}. In fact, when the normality condition is excluded we can construct the compactifications of exotic A^{3}’s in Types XII and XXII as seen in Corollary 1.2.

Remark 1.3 (1) X in Types XII, XIII and XXII in Table 1 are not necessarily isomorphic to the affine 3-space A^{3}. In fact, infinitely many exotic A^{3}’s can be choosen as X. For example, in Types XII or XXII, we choose the following affine surface S_{a,b} as S:

S := S_{a,b} := \{f_{a,b}(x, y, z) := \frac{(xz + 1)^{a} − (yz + 1)^{b}}{z} = 0\} ⊂ A^{3}_{(x,y,z),}

where a > b ≥ 2 is positive integers with gcd(a, b) = 1. It is known that S_{a,b} is smooth and contractible with log Kodaira dimension \overline{κ}(S_{a,b}) = 1 (cf. [Miy01]). Then X is isomorphic to an affine 3-fold X_{a,b} which is obtained from A^{3} via the affine modification with a locus (0 ∈ S_{a,b}). (See Section 5 and we use the terminology in [K-Z99] and Definition 2.1.) X_{a,b} is expressed as the hypersurface in A^{4}_{(x,y,z,u)} defined by {f_{a,b}(xu, yu, zu) − u}/u = 0. We can see that X_{a,b}’s are pairwise non-isomorphic contractible smooth affine 3-folds.

(2) An acyclic affine surface S of Types XII and XXII in Table 1 need not to be smooth. For instance, let S be the following non-normal acyclic affine surface:

S := {(x + 1)^{a} − (y + 1)^{b} = 0} ⊂ A^{3}_{(x,y,z)},

where a > b ≥ 2 are positive integers with gcd(a, b) = 1. Then X, which is obtained from A^{3} via the affine modification with a locus (0 ∈ S), is isomorphic to S_{a,b} × A^{1}, where S_{a,b} is an affine surface in (1) above. It is well known that such an X is an exotic A^{3} (cf. [Za97]).

We shall state the scheme of this article. Let X ↪ (V, D_{1} ∪ D_{2}) be an embedding of a contractible affine algebraic 3-fold X into a smooth Fano 3-fold V with B_{2}(V) = 2 satisfying (†). We divide the situation into the three cases according as the reduced boundary divisor D_{1} + D_{2} is nef or not. Since V is a Fano 3-fold with B_{2}(V) = 2, the cone \overline{NE} (V) of effective 1-cycles on V is spanned by two extremal rays, say R_{1} = R_{+}[l_{1}] and R_{2} = R_{+}[l_{2}]. Let φ_{i} := cont_{R_{i}} : V → W_{i} be the contraction of R_{i} for i = 1, 2. We shall make use of the classification of smooth Fano 3-folds with B_{2} = 2 due to Mori and Mukai in our proof frequently (cf. [M-M81, M-M83, M-M85]).

In Section 2, we shall summarize preliminary results and recall the definition on affine modifications briefly for the subsequent arguments in order to prove Theorem 1.3.

In Section 3, we consider, at first, the case where D_{1} + D_{2} is not nef. Then it is not difficult to see that we may assume that one of contractions φ_{1} and φ_{2} is divisorial of E5-type with one boundary component as an exceptional divisor. Hereafter we shall determine the structure of the embeddings X ↪ (V, D_{1} ∪ D_{2}). For this purpose, we need the result due to Abe and Furushima concerning the classification on non-normal del Pezzo surfaces (cf. [A-F]).

In Section 4, we consider the case where D_{1}+D_{2} is nef but not ample. Then at least one of contractions φ_{1} and φ_{2} is divisorial with one boundary component as an exceptional divisor. We consider, separately, according as type of this contraction (E1, E2, E3, E4, E5) in the subsequent. We need to make use of the result of Furushima and Nakayama (cf. [Fur86, F-N89a, F-N89b] and Theorem 1.1) concerning the compactifications of the affine 3-space A^{3} into smooth Fano 3-folds with B_{2} = 1 of index r ≥ 2. As in the previous section, we also need the result of [A-F] in order to exclude the cases of E3 and E4-types.

In Section 5, we finally consider the case where D_{1} + D_{2} is ample. By noting that K_{V} + D_{1} + D_{2} is not nef, we can see that one contraction of φ_{1} and φ_{2} is either of type E2, D2, D3 or C2. In the subsequent of this section, we consider according as the type of this contraction, separately. Note that, in the case where E2-type occurs, X can be an exotic A^{3}. In order to see this, we need the theory of affine modifications (cf. [K-Z99]).

Notation.

• K_{V} : canonical divisor of a variety V;

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

• F |_{F} : self-intersection of an irreducible Cartier divisor F on V;

• B_{i}(V) : the i-th Betti number of V as a real topological manifold;

• χ_{top}(V) : the topological Euler number of a topological manifold V;

• F · C : the intersection number of a (Q-)Cartier divisor F and a curve C on a threefold;

\underline{5}

• (C_{1} · C_{2})_{Y} : the intersection number of curves C_{1} and C_{2} on a surface Y;

• p_{a}(C) : the arithmetic genus of a curve C;

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

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

• F_{a} : Hirzebruch surface of degree a, (a ≥ 0);

• M_{a} : the minimal section on F_{a};

• L_{a} : a fiber on F_{a};

• A^{n} : n-dimensional affine space;

• P^{n} : n-dimensional projective space;

• Q_{0}^{2} : a quadric cone in P^{3};

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

• C^{∗k} : the affine line with k points punctured;

• V_{5} : a linear section Gr(2, 5) ∩ P^{6} of the Grassmann variety Gr(2, 5) ↪ P^{9} by three hyperplanes in P^{9}, which is a Fano threefold of Fano index 2 and degree (−\frac{1}{2}K_{V_{5}})^{3} = 5 in P^{6};

• H_{5}^{0} : a normal hyperplane section of V_{5} with exactly one A_{4}-type Du Val singularity;

• H_{5}^{∞} : a non-normal hyperplane section of V_{5} whose singular locus is a line Σ ≅ P^{1} with the normal bundle N_{Σ/V_{5}} ≅ O_{Σ}(−1) ⊕ O_{Σ}(1);

• \overline{NE} (V) : the closure of the cone of effective 1-cycles on V. An extremal ray R of \overline{NE} (V) is a one-dimensional edge of \overline{NE} (V) which is K_{V} -negative. For the

types of contractions of extremal rays on smooth projective 3-folds (E1, E2, E3, E4, E5, C1, C2, D1, D2, D3), see [Mo82, Theorems (3.3) and (3.5)]. Concerning divisorial contractions, E1, E2, E3, E4 and E5-type corresponds to (3.3.1), (3.3.2), (3.3.3), (3.3.4) and (3.3.5) in [Mo82, Theorem (3.3)].

2 Preliminaries

In this section, we shall summarize the preliminary results in order to prove Theorem 1.3. Furthermore, we recall the definition on affine modifications (cf. [K-Z99]) for the convenience of readers. Let X be a smooth contractible affine 3-fold. Assume that X is embedded into a smooth Fano 3-fold V with B_{2}(V) = 2. Lemma 2.1 The boundary V −X consists of two irreducible components, say D_{1} and D_{2}. Furthermore, the Picard group Pic(V) of V is free generated by the classes of D_{1} and D_{2}. Proof. Note that the coordinate ring A := Γ(O_{X}) is factorial and A^{∗} = C^{∗} (cf. [Fuj82]). Hence the assertion is easy to see. □

We shall follow the following convention which are used consistently in the subsequent of this article. Convention. Since V is a smooth Fano 3-fold with B_{2} = 2, the cone \overline{NE} (V) of effective 1-cycles on V is spanned by two extremal rays R_{1} = R_{+}[l_{1}] and R_{2} = R_{+}[l_{2}]. We put μ_{i} := min{(−K_{V} · l) ∈ N; l ⊂ V and [l] ∈ R_{i}} for i = 1, 2. We may assume that (−K_{V} · l_{i}) = μ_{i}. Let φ_{i} := cont_{R_{i}} : V → W_{i} be the contraction of R_{i}. Since B_{2}(V) = 2, the Picard number of W_{i} is equal to 1. Let L_{i} be an ample generator of Pic(W_{i}) ≅ Z and put H_{i} := φ^{∗}(L_{i}) for i = 1, 2.

With these notations the following result holds:

Proposition 2.1 (cf. [M-M83, Proposition 5.1])

(1) Pic(V) ≅ Z[H_{1}] ⊕ Z[H_{2}],

(2) −K_{V} ∼ μ_{2}H_{1} + μ_{1}H_{2},

(3) (H_{i} · l_{j}) = δ_{ij} for i, j = 1, 2, where δ_{ij} is the Kroneker’s delta.

In the subsequent sections of this article, we consider according as the numerical conditions of the reduced boundary divisor D_{1} + D_{2}. For this purpose, we shall prove the following:

Lemma 2.2 Assume that the reduced boundary divisor D_{1} + D_{2} is not nef (resp. nef but not ample). Then we may assume that φ_{1} is a divisorial contraction of E5-type (resp. of type either E1, E2, E3, E4 or E5) with one boundary component D_{1} as an exceptional divisor. The intersection (D_{1} + D_{2}) · l_{2} of D_{1} + D_{2} with the other extremal ray R_{2} = R_{+}[l_{2}] is positive.

Proof. Assume that D is not nef, i.e., there exists an irreducible curve C ⊂ V such that (D_{1}+D_{2})·C < 0. Since \overline{NE} (V) = R_{1} + R_{2}, at least one of intersections (D_{1} + D_{2}) · l_{1} and (D_{1} + D_{2}) · l_{2} must be negative, say (D_{1} + D_{2}) · l_{1} < 0. Then φ_{1} is divisorial with the exceptional divisor contained in D_{1} ∪ D_{2}. Hence we may assume that Exc(φ_{1}) = D_{1}. Since D_{1} + D_{2} is connected and D_{1} · l_{1} + D_{2} · l_{1} < 0, the possibility is one of the following two cases:

(a) φ_{1} : V → W_{1} is of E1-type and D_{2} meets D_{1} along the rulings of φ_{1}|_{D_{1}} : D_{1} → C_{1} := φ_{1}(D_{1}) ⊂ W_{1},

(b) φ_{1} : V → W_{1} is of E5-type and D_{2} meets D_{1} along a line on D_{1} ≅ P^{2}.

Assume that (a) occurs. Then, via the contraction φ_{1}, (W_{1}, φ_{1}(D_{2}) ∪ C_{1}) is also a compactification of X. But the boundary φ_{1}(D_{2}) ∪ C_{1} is not of pure codimension 1, which is a contradiction as X is affine. Hence (a) does not occur and (b) occurs. If (D_{1} + D_{2}) · l_{2} ≤ 0 for the remaining extremal ray R_{2}, then (D_{1} + D_{2}) · B for every irreducible curve B ⊂ V. Since X contains no complete curves, this implies that B ⊂ D_{1} ∪ D_{2}, which is obviously a contradiction. Assume that D_{1} + D_{2} is nef but not ample. Then there exists an irreducible curve C ⊂ V such that (D_{1} + D_{2}) · C = 0. Hence we may assume that (D_{1} + D_{2}) · l_{1} = 0. Since X contains no complete curves, this means that l_{1} ⊂ D_{1} ∪ D_{2}. So, we may assume that φ_{1} is divisorial and D_{1} = Exc(φ_{1}). The same argument as above deduces that the remaining extremal ray R_{2} = R_{+}[l_{2}] satisfies (D_{1} + D_{2}) · l_{2} > 0. □

In Section 4, we need the information of some kinds of curves on H_{5}^{0} (⊂ V_{5} ↪ P^{6}) (see Notation). For this purpose we prove the following:

Lemma 2.3 Let S := H_{5}^{0}, i.e., S is a normal del Pezzo surface of degree (−K_{S}^{2}) = 5 of rank B_{2}(S) = 1 with only one singular point p which is Du Val of A_{4}-type. Let μ : Ŝ → S be the minimal resolution of S. For each of the following (i)∼(iii) there exists a smooth rational curve C on S with this numerical property:

(i) (K_{S} · C)_{S} = (KŜ · Ĉ)Ŝ = −3 and (Ĉ^{2})Ŝ = 1,

(ii) (K_{S} · C)_{S} = (KŜ · Ĉ)Ŝ = −2 and (Ĉ^{2})Ŝ = 0,

(iii) (K_{S} · C)_{S} = (KŜ · Ĉ)Ŝ = −1 and (Ĉ^{2})Ŝ = −1,

where we denote by Ĉ the proper transform by μ of C.

Proof. The surface Ŝ, which is the minimal resolution of p ∈ S := H_{5}^{0}, is obtained from the projective plane P^{2} via the 4-points blow-up admitting infinitely near points lying on the smooth cubic curve Γ_{0} ⊂ S_{0} := P^{2}. Let {p_{0}, p_{1}, p_{2}, p_{3}} be these points and let ν_{j} : S_{j} → S_{j−1} be the blow-up at the point p_{j−1} with the exceptional curve l_{j} for 1 ≤ j ≤ 4, where p_{0} is one of the inflection points of Γ_{0}. Let ν := ν_{1} ∘ ν_{2} ∘ ν_{3} ∘ ν_{4} : Ŝ = S_{4} → P^{2} be the composite and let l̂_{j} be the proper transform of l_{j} on Ŝ. Then we have the configuration of curves on Ŝ as in Figure 1, where l̂_{0} is the proper transform of the line l_{0} on P^{2} meeting Γ_{0} at p_{0} with order 3. Note that μ : Ŝ → S is the contraction of (−2)-curves l_{0} ∪ l_{1} ∪ l_{2} ∪ l_{3} to a point p. The curve C in (iii) is obtained as the proper image on S of l̂_{4}. Let Ĉ be the proper transform on Ŝ of a line on P^{2} which passes through (resp. does not pass through) p_{0}. Then the proper image C = μ(Ĉ) on S satisfies the mentioned property (ii) (resp. (i)). □

l̂_{4} (−1)  _{(−2)} l̂_{3}_{(−2)} l̂_{2}  _{(−2) l̂_{1}} _{(−2) l̂_{0}}_{(i)}_{(ii)} Γ̂ Figure 1.

We shall recall the definition on affine modifications which are used in Lemmas 5.4 and 5.5. For more detailed treatment and relevant results on these topics, see [K-Z99] and [Za97].

Definition 2.1 Let R be an affine domain and let Z := Spec(R). Let I ⊂ R be an ideal and let f be an element of I. We denote by D := V_{Z}(f) and C := V_{Z}(I) the subvarieties in Z defined by f and I, respectively. Then the affine modification of Z with a locus (C ⊂ D) is the affine variety Z′ with a coordinate ring R′ := Γ(O_{Z′}) = R[I/f]. It is clear that R ⊂ R′ ⊂ Q(R), where Q(R) is the quotient field of R. The canonical inclusion R ⊂ R′ induces a birational morphism σ : Z′ → Z. We often call this morphism σ itself as an affine modification with a locus (C ⊂ D). If f is not a unit in R′, it defines a divisor E := div(f)_{Z′} on Z′ and we have an isomorphism σ|_{Z′−E} : Z′ − E →^{∼} Z − D. Then E (resp. D) is said to be an exceptional divisor (resp. divisor) of the modification σ.

3 The case D_{1} + D_{2} is not nef

In this section, we consider the case where the reduced boundary divisor D_{1} + D_{2} is not nef. By Lemma 2.2, we may assume that the contraction φ_{1} : V → W_{1} of the extremal ray R_{1} is divisorial of type E5 with D_{1} = Exc(φ_{1}) ≅ P^{2}, O_{D_{1}}(D_{1}) ≅ O_{P^{2}}(−2) as an exceptional divisor and D_{2} meets D_{1} along a line on D_{1} ≅ P^{2}. In the list [M-M81, Table 2], smooth Fano 3-folds with B_{2} = 2 having extremal rays of E5-type are No. 28 and No. 36. In fact, we shall prove the following:

Lemma 3.1 The case of No. 28 does not occur.

Proof. Our proof consists of three steps. Step 1. Assume that V is No. 28 in [M-M81, Table2]. Namely, V is obtained from W_{2} = P^{3} via the blow-up φ_{2} : V → W_{2} along the plane cubic C_{2} ⊂ W_{2} with an exceptional divisor E_{2} := Exc(φ_{2}) and φ_{1} : V → W_{1} is the contraction of D_{1}, which is the proper transform on V of the hyperplane in W_{2} containing C_{2}, to an index 2 point p_{1} := φ_{1}(D_{1}). We put \overline{D_{2}} := φ_{1}(D_{2}) ⊂ W_{1}. Claim. D_{1} ∼ H_{1} − 2H_{2} and E_{2} ∼ −H_{1} + 3H_{2}. Furthermore, −K_{W_{1}} ≡ \frac{3}{2}L_{1} Proof of Claim. By Proposition 2.1 (2), we have H_{1} + H_{2} ∼ −K_{V} = φ_{2}^{∗}(−K_{W_{2}}) − E_{2} ∼ 4H_{2} − E_{2}. Thus E_{2} ∼ −H_{1} + 3H_{2}. Since D_{1} is the proper transform by φ_{2} of the hyperplane containing C_{2}, we have D_{1} ∼ H_{2} − E_{1} ∼ H_{1} − 2H_{2}. Since −K_{V} ≡ φ_{1}^{∗}(−K_{W_{1}}) − \frac{1}{2}D_{1}, the remaining assertion is easy to see. □

By noting that V − D_{1} ≅ W_{1} − {p_{1}}, we write the following exact sequences:

0 → H_{3}(V) ⟶ H_{3}(V, D_{1}) ⟶ H_{2}(D_{1}) ⟶ H_{2}(V) ⟶ H_{2}(V, D_{1}) → 0
↓  ↓≅  ↓  ↓  ↓≅
0 → H_{3}(W_{1}) → H_{3}(W_{1}, {p_{1}}) → H_{2}({p_{1}}) → H_{2}(W_{1}) → H_{2}(W_{1}, {p_{1}}) → 0.

Since B_{2}(V) = B_{3}(V) = 2 and B_{2}(W_{1}) = 1, we have B_{3}(W_{1}) = 2 by these sequences. On the other hand, we write the exact sequence associated to the pair (W_{1}, \overline{D_{2}}):

H_{4}(W_{1}, \overline{D_{2}}) → H_{3}(\overline{D_{2}}) → H_{3}(W_{1}) → H_{3}(W_{1}, \overline{D_{2}}) → H_{2}(\overline{D_{2}}) → H_{2}(W_{1}) → H_{2}(W_{1}, \overline{D_{2}}).

Since H_{j}(W_{1}, \overline{D_{2}}) ≅ H^{6−j}(X) = 0 for j = 2, 3, 4 by the contractibility of X, we have B_{2}(\overline{D_{2}}) = 1 and B_{3}(\overline{D_{2}}) = 2.

Step 2. Note that since p_{1} ∈ W_{1} is an index 2 terminal singular point, for every irreducible Weil divisor F on W_{1}, 2F is Cartier (cf. [Ka88]). Since Pic(W_{1}) = Z[L_{1}], we can write \overline{D_{2}} ≡ αL_{1} with α ∈ \frac{1}{2}N. We claim the following: Claim. α = \frac{1}{2} or 1.

Proof of Claim. Assume to the contrary, namely, α = \frac{β}{2} with β ≥ 3. Assume that β is odd. It is obvious that L_{1} ≠ \overline{D_{2}}. Put L_{1}^{∘} := L_{1} ∩ X, where we note X ≅ W_{1} − \overline{D_{2}}. Since A := Γ(O_{X}) is factorial, there exists a prime element g ∈ A such that div(g)_{X} = L_{1}^{∘}. When we consider g as an element of the quotient field Q(A) = C(W_{1}), it follows div(g)_{W_{1}} = L_{1} − m\overline{D_{2}} with m ∈ 2N. Hence \overline{D_{2}} ≡ \frac{1}{m}L_{1} ≡ \frac{β}{2} L_{1}, which is a contradiction. By the similar argument, we can obtain a contradiction in the case where β is even also. □

We also claim the following: Claim. \overline{D_{2}} is a Gorenstein (possibly non-normal) del Pezzo surface. Proof of Claim. It is clear that D_{2} is Gorenstein. Note that D_{2} meets the exceptional divisor D_{1} ≅ P^{2} of φ_{1} along the line, so φ_{1}|_{D_{2}} : D_{2} → \overline{D_{2}} is the contraction of a (−2)-curve. Thus \overline{D_{2}} is Gorenstein. By the adjunction, we have −K\overline{D_{2}} ≡ (\frac{3}{2} − α)L_{1}|\overline{D_{2}}. Since α is equal to either \frac{1}{2} or 1, −K\overline{D_{2}} is ample. □

Step 3. Note that every normal del Pezzo surface is rational (cf. [De80], [H-W81]). If \overline{D_{2}} is normal, then the rationality of \overline{D_{2}} implies that B_{3}(\overline{D_{2}}) = 0. This is a contradiction because B_{3}(\overline{D_{2}}) = 2 as seen in Step 1. Thus \overline{D_{2}} is a non-normal del Pezzo surface. Let σ : D̃_{2} → \overline{D_{2}} be the normalization. Let J ⊂ O\overline{D_{2}} be the conductor sheaf of σ, and let \overline{Σ} := V\overline{D_{2}}(J) and Σ̃ := VD̃_{2}(J). Then we have the following exact sequence associated with σ (cf. [B-K82]):

⋯ → H^{1}(Σ̃) → H^{2}(\overline{D_{2}}) → H^{2}(D̃_{2}) ⊕ H^{2}(\overline{Σ}) → H^{2}(Σ̃) → H^{3}(\overline{D_{2}}) → H^{3}(D̃_{2}) ⊕ H^{3}(\overline{Σ}).

Here we shall make use of the result of Abe and Furushima [A-F, Theorem 1.5] concerning the classification on non-normal del Pezzo surfaces. According to this, it follows that H^{1}(Σ̃) = 0, H^{2}(\overline{Σ}) ≅ Z, H^{2}(Σ̃) ≅ Z or Z^{⊕2} and H^{3}(D̃_{2}) ⊕ H^{3}(\overline{Σ}) = 0. Since B_{2}(\overline{D_{2}}) = 1 and B_{3}(\overline{D_{2}}) = 2, this exact sequence is:

0 → Z → H^{2}(D̃_{2}) ⊕ Z → H^{2}(Σ̃) ≅ Z or Z^{⊕2} → Z^{⊕2} → 0.

Hence H^{2}(Σ̃) ≅ Z^{⊕2} and H^{2}(D̃_{2}) = 0, which is a contradiction because B_{2}(D̃_{2}) > 0. Thus we complete the proof. □

By Lemma 3.1, we have only to consider the case of No. 36. In fact, we have the following:

Lemma 3.2 The embedding X ↪ (V, D_{1} ∪ D_{2}) is described as follow: X ≅ A^{3} and V is a Fano 3-fold of No. 36 in [M-M81, Table 2], i.e., V ≅ P(O_{P^{2}} ⊕ O_{P^{2}}(2)). D_{1} is a section corresponding to the surjection O_{P^{2}} ⊕ O_{P^{2}}(2) → O_{P^{2}} → 0 and D_{2} ∼ H_{2}.

Proof. Let V = P(O_{P^{2}} ⊕ O_{P^{2}}(2)) be the Fano 3-fold of No. 36 in [M-M81, Table 2]. Then φ_{2} : V → W_{2} = P^{2} is the P^{1}-bundle structure associated with the rank 2 vector bundle O_{P^{2}} ⊕ O_{P^{2}}(2) and the exceptional divisor D_{1} of φ_{1} : V → W_{1} is a section of φ_{2} corresponding to the surjection O_{P^{2}} ⊕ O_{P^{2}}(2) → O_{P^{2}} → 0. Associated with φ_{2} we have Pic(V) ≅ Z[D_{1}] ⊕ Z[H_{2}]. On the other hand, it follows that Pic(V) ≅ Z[D_{1}] ⊕ Z[D_{2}] by Lemma 2.1. Therefore, when we write D_{2} ∼ aD_{1} + bH_{2}

with a, b ∈ Z the matrix ( \begin{matrix}1 & 0 \\ a & b \end{matrix} ) have to be unimodular. Hence b = 1 or −1. On the other

hand, since D_{2} meets D_{1} ≅ P^{2} with a normal bundle O_{D_{1}}(D_{1}) ≅ O_{P^{2}}(−2) along the line on D_{1}, we have −2a + b = 1. Hence either (a, b) = (0, 1) or (−1, −1) holds. If (a, b) = (−1, −1), then (D_{2} · l_{2}) = (−D_{1} − H_{2} · l_{2}) = −1, which is a contradiction. So, (a, b) = (0, 1), i.e., D_{2} ∼ H_{2}. Hence the restriction φ_{2}|_{X} : X = V − (D_{1} ∪ D_{2}) → W_{2} − φ_{2}(D_{2}) gives rise to an A^{1}-bundle structure over the affine plane W_{2} − φ_{2}(D_{2}) ≅ A^{2}. Since Pic (A^{2}) = 0, this is a trivial A^{1}-bundle structure over A^{2}. Thus we have X ≅ A^{2} × A^{1} ≅ A^{3}. □

The embedding A^{3} ≅ X ↪ (V, D_{1} ∪ D_{2}) in Lemma 3.2 is of Type XXIV in Table 1, where we shall interchange the roles of φ_{1} and φ_{2}.

4 The case D_{1} + D_{2} is nef but not ample

In this section we consider the case where D_{1} +D_{2} is nef but not ample. By Lemma 2.2, we may assume that the contraction φ_{1} : V → W_{1} of the extremal ray R_{1} is divisorial with D_{1} = Exc(φ_{1}) as an exceptional divisor. We consider according as the type of the divisorial contraction φ_{1} (E1, E2, E3, E4, E5).

4.1 φ_{1} is of E1-type

First of all we consider the case where φ_{1} : V → W_{1} is of E1-type, i.e., it contracts D_{1} onto a curve, say C_{1} := φ_{1}(D_{1}), in a smooth projective 3-fold W_{1}. Let \overline{D_{2}} := φ_{1}(D_{2}). Then:

Lemma 4.1 X ≅ A^{3} and (W_{1}, \overline{D_{2}}) ≅ (P^{3}, P^{2}), (Q^{3}, Q_{0}^{2}), (V_{5}, H_{5}^{0}) or (V_{5}, H_{5}^{∞}).

Proof. Since the exceptional divisor D_{1} of φ_{1} is contained in the boundary D_{1} ∪ D_{2}, it is clear that X ≅ W_{1} − \overline{D_{2}}. Note that W_{1} is a smooth Fano 3-fold with B_{2}(W_{1}) = 1 of index r ≥ 2 (cf. [M-M83, Proposition 4.10]). Therefore we have the assertion by Theorems 1.1, 1.2 and Corollary 1.1. □

In consideration of Lemma 4.1 and [M-M81, Table 2], the possibilities of Fano 3-folds V are No. 4, No. 7, No. 9, No. 12, No. 13, No. 14, No. 15, No. 17, No. 19, No. 20, No. 21, No. 22, No. 23, No. 25, No. 26, No. 27, No. 28, No. 29, No. 30, No. 31 and No. 33 in [M-M81, Table 2]. We can easily exclude some cases among these:

Lemma 4.2 The cases No. 4, No. 7, No. 9, No. 12, No. 13, No. 15, No. 19, No. 21, No. 25, No. 27 does not occur as compactifications of X.

Proof. Note that the center C_{1} of the blow-up φ_{1} is contained in \overline{D_{2}}. Namely, if (W_{1}, \overline{D_{2}}) ≅ (P^{3}, P^{2}) (resp. (Q^{3}, Q_{0}^{2}), (V_{5}, H_{5}^{0}) or (V_{5}, H_{5}^{∞})), then C_{1} is contained in \overline{D_{2}} ≅ P^{2} (resp. Q_{0}^{2}, H_{5}^{0} or H_{5}^{∞}). (Note that No. 19 is obtained from W_{1} = P^{3} via the blow-up φ_{1} : V → W_{1} along the curve of degree 5 and genus 2.) But the center C_{1} ⊂ W_{1} of the blow-up φ_{1} cannot be contained in \overline{D_{2}} when the indicated cases in the statement occur. This is a contradiction. □

We consider the remaining cases No. 14, No. 17, No. 20, No. 22, No. 23, No. 26, No. 28, No. 29, No. 30, No. 31 and No. 33, separately,

Lemma 4.3 Assume that V is of No. 14 in [M-M81, Table 2]. Then X ≅ A^{3}. D_{1} is the exceptional divisor of the blow-up φ_{1} : V → W_{1} := V_{5} ↪ P^{6} and D_{2} is the proper transform by φ_{1} of \overline{D_{2}} := H_{5}^{0} ∈ | − \frac{1}{2}K_{V_{5}}| containing the center C_{1} of φ_{1} which does not pass through the singular point of \overline{D_{2}}.

Proof. Let V be a Fano 3-fold of No. 14, i.e., it is obtained from W_{1} = V_{5} ↪ P^{6} via the blow-up φ_{1} along the curve C_{1} which is an intersection of two hyperplane sections ∈ | − \frac{1}{2}K_{V_{5}}|. Of course, we can take \overline{D_{2}} as one of them. By Theorem 1.1, \overline{D_{2}} is either H_{5}^{0} or H_{5}^{∞}. Assume that \overline{D_{2}} = H_{5}^{∞}. Since C_{1} is Cartier on H_{5}^{∞} and smooth, it is disjoint from the singular locus Σ := SingH_{5}^{∞}. But since H_{5}^{∞}−Σ ≅ A^{2} (cf. [A-F, F-T92]), C_{1} have to meet Σ, a contradiction. Thus \overline{D_{2}} = H_{5}^{0} and we complete the proof. □

Thus we obtain Type I in Table 1.

Lemma 4.4 The case No. 17 in [M-M81, Table 2] does not occur.

Proof. Assume that V is of No. 17. Then there are two possibilities with respect to the contraction φ_{1} : (D_{1} ⊂ V) → (C_{1} ⊂ W_{1}). Namely, the one is W_{1} = P^{3}, C_{1} is of degree 5 of genus 1 and the other is W_{1} = Q^{3} ↪ P^{4}, C_{1} is of degree 5 of genus 1. The former is obviously absurd because such a curve C_{1} is not contained in a hyperplane \overline{D_{2}} ≅ P^{2}. We consider the latter case and assume that C_{1} is contained in a hyperplane section \overline{D_{2}} ≅ Q_{0}^{2}. Since deg(C_{1}) = (H_{1} · C_{1}) = 5 we have C_{1} ≡ 5l on \overline{D_{2}}, where l is a generating line on \overline{D_{2}} ≅ Q_{0}^{2}. Then (K\overline{D_{2}} · C_{1})\overline{D_{2}} = (K_{W_{1}} + \overline{D_{2}} · C_{1}) = (−2\overline{D_{2}} · C_{1}) = −10 and

(C_{1}^{2})\overline{D_{2}} = 25(l^{2})\overline{D_{2}} = \frac{25}{2}. Let μ : D̂_{2} → \overline{D_{2}} be the resolution of singularity and let ê be the exceptional

curve which is a (−2)-curve. We write μ^{∗}(C_{1}) ≡ Ĉ_{1} + aê with a ≥ 0. Then (Ĉ_{1}^{2})D̂_{2} = (C_{1}^{2})\overline{D_{2}} − 2a^{2} =

\frac{25}{2} − 2a^{2}. Since Ĉ_{1} is an elliptic curve and (KD̂_{2} · Ĉ_{1})D̂_{2} = (K\overline{D_{2}} · C_{1})\overline{D_{2}} = −10, we have a = \frac{\sqrt{5}}{2}, a contradiction. □

Lemma 4.5 Assume that V is of No. 20 in [M-M81, Table 2]. Then X ≅ A^{3}. D_{1} is the exceptional divisor of the blow-up φ_{1} : V → W_{1} := V_{5} ↪ P^{6} along the twisted cubic C_{1} and D_{2} is the proper transform by φ_{1} of \overline{D_{2}} := H_{5}^{0} ∈ | − \frac{1}{2}K_{V_{5}}| containing the center C_{1} passing through the singular point of \overline{D_{2}}.

Proof. Let V be a Fano 3-fold of No. 20, i.e., V is obtained from W_{1} := V_{5} ↪ P^{6} via the blow-up along the twisted cubic C_{1} on W_{1}. By Lemma 4.1, the boundary \overline{D_{2}} is either H_{5}^{0} or H_{5}^{∞}. We claim the following: Claim. The case \overline{D_{2}} = H_{5}^{∞} does not occur.

Proof. Assume \overline{D_{2}} = H_{5}^{∞}. Let σ : D̃_{2} → \overline{D_{2}} be the normalization and let J ⊂ O\overline{D_{2}} be the conductor sheaf with respect to σ. Let \overline{Σ} := V\overline{D_{2}}(J) and let Σ̃ := VD̃_{2}(J). Then D̃_{2} ≅ F_{3}, Σ̃ = M_{3} + L_{3} and \overline{Σ} is a line on V_{5} with the normal bundle N_{\overline{Σ}/V_{5}} ≅ O\overline{Σ}(−1) ⊕ O\overline{Σ}(1) (cf. [F-T92, A-F]). It is clear that C_{1} is

distinct from \overline{Σ}. Let C̃_{1} be the proper transform on D̃_{2} of C_{1} ⊂ \overline{D_{2}}. We write C̃_{1} ≡ αM_{3} + βL_{3}, where (α, β) = (0, 1) or α ≥ 3β with β > 0. Note that KD̃_{2} ≡ σ^{∗}(K\overline{D_{2}})−Σ̃ (cf. [Mo82]). Hence (KD̃_{2} · C̃_{1})D̃_{2} =

(K\overline{D_{2}} · C_{1})\overline{D_{2}} − (M_{3} + L_{3} · C̃_{1})D̃_{2} = 2α − β − 3. Thus we have (C̃_{1} · C̃_{1} + KD̃_{2})D̃_{2} = −(3α^{2} − 2α + 3) +

(2α−1)β. When (α, β) = (0, 1), it follows that (C̃_{1} · C̃_{1} + KD̃_{2})D̃_{2} = −4, a contradiction. When α ≥ 3β

with β > 0 we have (C̃_{1} · C̃_{1} + KD̃_{2})D̃_{2} ≥ −(3α^{2} − 2α + 3) + 3α(2α − 1) = 3α^{2} − α − 3 ≥ −1, which is also a contradiction. □

Thus we may assume that \overline{D_{2}} = H_{5}^{0}. Let μ : D̂_{2} → \overline{D_{2}} be the minimal resolution. Let Ĉ ⊂ D̂_{2} be the curve obtained as in Lemma 2.3 (i). Then the proper image μ(Ĉ) ⊂ \overline{D_{2}} ⊂ V_{5} is a twisted cubic. Therefore, we can take μ(Ĉ) as the center C_{1} of the blow-up φ_{1} : V → W_{1}. Thus we obtain the assertion. □

Thus we have Type II in Table 1.

The similar argument can be applied to the case of No. 22.

Lemma 4.6 Assume that V is of No. 22 in [M-M81, Table 2]. Then X ≅ A^{3}. D_{1} is the exceptional divisor of the blow-up φ_{1} : V → W_{1} := V_{5} ↪ P^{6} along the conic C_{1} and D_{2} is the proper transform by φ of \overline{D_{2}} := H_{5}^{0} ∈ | − \frac{1}{2}K_{V_{5}}| containing the center C_{1} passing through the singular point of \overline{D_{2}}.

Proof. Assume that V is of No. 22. Then there are two possibilities with respect to the blow-up φ_{1} : (D_{1} ⊂ V) → (C_{1} ⊂ W_{1}), namely, the one is W_{1} = P^{3}, C_{1} is deg(C_{1}) = 4 of genus g(C_{1}) = 0 and the other is W_{1} = V_{5} ↪ P^{6}, C_{1} is a conic on V_{5}. The former is obviously absurd because such a curve C_{1} cannot be contained in a hyperplane. Hence we have only to consider the latter case. Hence \overline{D_{2}} is either H_{5}^{0} or H_{5}^{∞}. The similar argument as in the proof of Lemma 4.5 with an adequate modification shows that the case \overline{D_{2}} = H_{5}^{∞} does not occur. So, let \overline{D_{2}} = H_{5}^{0}. Let μ : D̂_{2} → \overline{D_{2}} be the minimal resolution. Let Ĉ be the curve obtained as in Lemma 2.3 (ii). Then the proper image μ(Ĉ) ⊂ \overline{D_{2}} ⊂ V_{5} is a smooth conic. Hence we can take μ(Ĉ) as the center C_{1} of the blow-up φ_{1}. Thus we obtain the assertion. □

The embedding in Lemma 4.6 is of Type III in Table 1.

Lemma 4.7 Assume that V is one of No. 23, No. 28, No. 29, No. 30 and No. 31 in [M-M81, Table 2]. Then X ≅ A^{3}. D_{1} is the exceptional divisor of the blow-up φ_{1} : V → W_{1} and D_{2} is the proper transform by φ_{1} of the hyperplane (section) \overline{D_{2}} ∈ |L_{1}| containing the center C_{1} of φ_{1}.

Proof. Let V be one of smooth Fano 3-folds mentioned in the statement. Then V has exactly one extremal ray of E1-type and W_{1} is either P^{3} or Q^{3} ↪ P^{4}. In any case it is easy to see that the center C_{1} of the blow-up φ_{1} : V → W_{1} can be contained in \overline{D_{2}}. Thus we have the assertion. □

Thus we obtain the embeddings of Types IV, VIII, IX, X and XIV in Table 1.

Lemma 4.8 Assume that V is of No. 26 in [M-M81, Table 2]. Then X ≅ A^{3}. D_{1} is the exceptional divisor of the blow-up φ_{1} : V → W_{1} := V_{5} ↪ P^{6} along the line C_{1} and D_{2} is the proper transform by φ_{1} of \overline{D_{2}} ∈ | − \frac{1}{2}K_{V_{5}}|, which is either H_{5}^{0} or H_{5}^{∞}, containing the center C_{1} of φ_{1}. Furthermore, in the case \overline{D_{2}} = H_{5}^{∞}, C_{1} is distinct from the singular locus of \overline{D_{2}}.

Proof. Assume that V is of No. 26. Then there are two possibilities concerning φ_{1} : (D_{1} ⊂ V) → (C_{1} ⊂ W_{1}). The one is W_{1} = Q^{3} ↪ P^{4}, C_{1} is a twisted cubic, and the other is W_{1} = V_{5} ↪ P^{6}, C_{1} is a line on V_{5}. The former is obviously impossible because a twisted cubic C_{1} cannot contained in a hyperplane section \overline{D_{2}} ≅ Q_{0}^{2}. Hence we have only to consider the latter case. By Lemma 4.1, the boundary \overline{D_{2}} is either H_{5}^{0} or H_{5}^{∞}. If \overline{D_{2}} = H_{5}^{0}, then it is known that there exists exactly one line contained in \overline{D_{2}} (cf. [Fur86, Proposition 15]). Indeed, let μ : D̂_{2} → \overline{D_{2}} be the minimal resolution and let Ĉ be the curve on D̂_{2} obtained as in Lemma 2.3 (iii). Then the proper image μ(C) is a such a line. Hence we have the assertion. On the other hand, if \overline{D_{2}} = H_{5}^{∞}, it is swept out by lines on V_{5} intersecting the singular locus \overline{Σ} of \overline{D_{2}}. Note that if we choose \overline{Σ} as the center C_{1} of the blow-up φ_{1}, then D_{1} + D_{2} = φ_{1}^{∗}(\overline{D_{2}}) − D_{1} and (D_{1} + D_{2} · l_{1}) = (−D_{1} · l_{1}) = 1, which is a contradiction. Hence we have to choose a line on \overline{D_{2}} distinct from \overline{Σ} as C_{1}. Thus we complete the assertion. □

Hence we have the embedding of Type VI in Table 1. The remaining case in this subsection is of No. 33.

Lemma 4.9 Assume that V is of No. 33 in [M-M81, Table 2]. Then X ≅ A^{3}. D_{1} is the exceptional divisor of the blow-up φ_{1} : V → W_{1} := P^{3} along the line C_{1} and D_{2} is the proper transform by φ_{1} of \overline{D_{2}} ∈ |O_{P^{3}}(1)| containing the center C_{1} of φ_{1}.

Proof. Assume that V is of No. 33, i.e., V ≅ P(O_{P^{1}}^{⊕2} ⊕ O_{P^{1}}(1)) and φ_{1} : V → W_{1} := P^{3} is the blow-up along the line C_{1}. Hence the assertion is easy to see. □

Thus we have the embedding of Type XVII with x = −1 in Table 1, where we shall interchange the roles of φ_{1} and φ_{2}.

4.2 φ_{1} is of E2-type

In this subsection we consider the case where φ_{1} : V → W_{1} is of E2-type, i.e., it contracts D_{1} ≅ P^{2} with a normal bundle O_{D_{1}}(D_{1}) ≅ O_{P^{2}}(−1) to a smooth point, say p_{1} := φ_{1}(D_{1}), in a smooth projective 3-fold W_{1}. Let \overline{D_{2}} := φ_{1}(D_{2}). Since the exceptional divisor D_{1} of φ_{1} is contained in the boundary, we have X = V − (D_{1} ∪ D_{2}) ≅ W_{1} − \overline{D_{2}}. In [M-M81, Table 2], there exist two Fano 3-folds with B_{2} = 2 having extremal ray of E2-type, i.e., No. 30 and No. 35.

Lemma 4.10 Assume that V is of No. 30 in [M-M81, Table 2]. Then X ≅ A^{3}. D_{1} is the exceptional divisor of the blow-up φ_{1} : V → W_{1} := Q^{3} ↪ P^{4} and D_{2} is the proper transform by φ_{1} of the hyperplane section Q_{0}^{2} ≅ \overline{D_{2}} ∈ |O_{Q^{3}}(1)| which contains the center p_{1} of φ_{1} as a smooth point.

Proof. Assume that V is of No. 30. Then φ_{1} : (D_{1} ⊂ V) → (p_{1} ∈ W_{1} := Q^{3}) is the blow-up at the point p_{1}. Since (Q^{3}, \overline{D_{2}}) is the compactification of the contractible affine 3-fold X, it follows that X ≅ A^{3} and \overline{D_{2}} ≅ Q_{0}^{2}. If p_{1} is a vertex of \overline{D_{2}}, then we have D_{1}+D_{2} = φ_{1}^{∗}(\overline{D_{2}})−D_{1} and (D_{1}+D_{2}·l_{1}) = (−D_{1}·l_{1}) = 1, which is a contradiction. Hence p_{1} is a smooth point on \overline{D_{2}}. Thus we have the assertion. □

Hence we have the embedding of Type XI in Table 1.

Lemma 4.11 Assume that V is of No. 35 in [M-M81, Table 2]. Then X ≅ A^{3}. D_{1} is the exceptional divisor of the blow-up φ_{1} : V → W_{1} := P^{3} and D_{2} is the proper transform by φ_{1} of the hyperplane \overline{D_{2}} ∈ |O_{P^{3}}(1)| containing the center p_{1} of φ_{1}.

Proof. Since the asssertion is easy to see, we shall omit the detail. □

The embedding in Lemma 4.11 is a special case of Type XXIII in Table 1, where we interchange the roles of φ_{1} and φ_{2}.

4.3 φ_{1} is of E3-type

In this subsection we consider the case where φ_{1} : V → W_{1} is of E3-type, i.e., it contracts D_{1} ≅ P^{1} × P^{1} with a normal bundle O_{D_{1}}(D_{1}) ≅ O_{P^{1}×P^{1}}(−1, −1) to a Gorenstein terminal singularity, say p_{1} := φ_{1}(D_{1}), in W_{1}. Let \overline{D_{2}} := φ_{1}(D_{2}). Since the exceptional divisor D_{1} of φ_{1} is contained in the boundary, we have X = V − (D_{1} ∪ D_{2}) ≅ W_{1} − \overline{D_{2}}. Since (D_{1} + D_{2} · l_{1}) = −1 + (D_{2} · l_{1}) = 0, D_{2} meets D_{1} in such a way that D_{2}|_{D_{1}} ∈ |O_{P^{1}×P^{1}}(1, 1)|. First of all we prove the following:

Lemma 4.12 With the notations and assumptions as above. Then:

(1) B_{3}(W_{1}) = B_{3}(V) + 1,

(2) B_{3}(\overline{D_{2}}) = B_{3}(V) + 1,

(3) B_{2}(\overline{D_{2}}) = 1.

Proof. By noting that V − D_{1} ≅ W_{1} − {p_{1}}, we consider the following exact sequences associated with the pairs (V, D_{1}) and (W_{1}, {p_{1}}):

0 → H_{3}(V) ⟶ H_{3}(V, D_{1}) ⟶ H_{2}(D_{1}) ⟶ H_{2}(V) ⟶ H_{2}(V, D_{1}) → 0
↓  ↓≅  ↓  ↓  ↓≅
0 → H_{3}(W_{1}) → H_{3}(W_{1}, {p_{1}}) ⟶ 0 ⟶ H_{2}(W_{1}) → H_{2}(W_{1}, {p_{1}}) → 0.

Since B_{2}(D_{1}) = B_{2}(V) = 2 and B_{2}(W_{1}) = 1, these exact sequences imply that B_{3}(W_{1}) = B_{3}(V) + 1. Thus we obtain (1). On the other hand, by writing the exact sequence of the pair (W_{1}, \overline{D_{2}}), we have B_{2}(\overline{D_{2}}) = B_{2}(W_{1}) and B_{3}(\overline{D_{2}}) = B_{3}(W_{1}) in consideration of the contractibility of X ≅ W_{1} − \overline{D_{2}}. Thus we complete the proof. □

In [M-M81, Table 2] there exist three smooth Fano 3-folds No. 8 (a), No. 15 (a), No. 23 (a) having an extremal ray of E3-type. We prove the following:

Lemma 4.13 (1) \overline{D_{2}} is an ample generator of Pic(W_{1}), (2) −K_{W_{1}} ≡ 2\overline{D_{2}},

(3) D_{1} ∼ H_{1} − H_{2},

(4) D_{2} ∼ H_{2}.

Proof. Since p_{1} = φ_{1}(D_{1}) ∈ W_{1} is a Gorenstein terminal singularity, every Weil divisor on W_{1} is Cartier (cf. [Ka88]). In particular, \overline{D_{2}} is Cartier. Since the coordinate ring of X ≅ W_{1} − \overline{D_{2}} is factorial, \overline{D_{2}} is an ample generator of Pic(W_{1}). When we write −K_{W_{1}} ∼ α\overline{D_{2}} with α ∈ N we have H_{1}+H_{2} ∼ −K_{V} = φ_{1}^{∗}(−K_{W_{1}})−D_{1} ∼ αH_{1}−D_{1} by Proposition 2.1. Hence D_{1} ∼ (α−1)H_{1}−H_{2}. On the other hand, by [M-M83, Theorem 5.1], it follows that 1 = (D_{1} ·l_{2}) = ((α−1)H_{1} −H_{2} ·l_{2}) = α−1. Hence α = 2, i.e., −K_{W_{1}} ∼ 2\overline{D_{2}} and D_{1} ∼ H_{1}−H_{2}. Since \overline{D_{2}} is Cartier, we can write H_{1} = φ_{1}^{∗}(\overline{D_{2}}) = D_{2}+βD_{1} for some β ∈ N. Hence D_{2} ∼ (1 − β)H_{1} + βH_{2}. Note that 0 ≤ (D_{2} · l_{2}). In fact, if V is No. 8 (a), then φ_{2} is of C1-type so it is clear that (D_{2} · l_{2}) ≥ 0. Meanwhile, if V is either of No. 15 (a) or No. 23 (a), then D_{2} is distinct from the exceptional divisor of E1-type divisorial contraction φ_{2} : V → W_{2} as D_{2} meets D_{1} ≅ P^{1} × P^{1} in such a way that D_{2}|_{D_{1}} ∈ |O_{P^{1}×P^{1}}(1, 1)| (see the descriptions in No. 15 (a) and No. 23 (a) in [M-M81, Table 2]). Hence (D_{2} · l_{2}) ≥ 0 in this case also. Therefore (D_{2} · l_{2}) = ((1 − β)H_{1} + βH_{2} · l_{2}) = 1 − β ≥ 0, i.e., β ≤ 1. Thus we have β = 1 and D_{2} ∼ H_{2} as desired. □

In fact we shall prove the following:

Lemma 4.14 These cases No. 8 (a), No. 15 (a) and No. 23 (a) do not occur.

Proof. Since −K_{W_{1}} ∼ 2\overline{D_{2}} and \overline{D_{2}} is Gorenstein, \overline{D_{2}} is a (possibly non-normal) del Pezzo surface. If \overline{D_{2}} is a normal del Pezzo surface, then B_{3}(\overline{D_{2}}) = 0 because of the rationality of normal del Pezzo surfaces (cf. [De80], [H-W81]). But by Lemma 4.12, we have B_{3}(\overline{D_{2}}) = B_{3}(V) + 1 = 19 (resp. = 9, = 3) when V is of No. 8 (a) (resp. No. 15 (a), No. 23 (a)). Hence this is a contradiction. Thus we may assume that \overline{D_{2}} is a non-normal del Pezzo surface. Let σ : D̃_{2} → \overline{D_{2}} be the normalization. Let J ⊂ O\overline{D_{2}} be the conductor sheaf of σ and let \overline{Σ} := V\overline{D_{2}}(J), Σ̃ := VD̃_{2}(J) be subvarieties in \overline{D_{2}} and D̃_{2} defined by J ⊂ O\overline{D_{2}} and J ⊂ OD̃_{2}, respectively. Then we have the exact sequence associated with σ as follows (cf. [B-K82]):

⋯ → H^{1}(Σ̃) → H^{2}(\overline{D_{2}}) → H^{2}(D̃_{2}) ⊕ H^{2}(\overline{Σ}) → H^{2}(Σ̃) → H^{3}(\overline{D_{2}}) → H^{3}(D̃_{2}) ⊕ H^{3}(\overline{Σ}).

By [A-F, Theorem 1.5], it follows that H^{2}(Σ̃) ≅ Z or Z^{⊕2} and H^{3}(D̃_{2}) ⊕ H^{3}(\overline{Σ}) = 0. Hence we have the surjection H^{2}(Σ̃) ≅ Z or Z^{⊕2} → H^{3}(\overline{D_{2}}). Since B_{3}(\overline{D_{2}}) = B_{3}(V) + 1 = 19 (resp. = 9, = 3) when V is of No. 8 (a) (resp. No. 15 (a), No. 23 (a)), this is a contradiction. □

4.4 φ_{1} is of E4-type

In this present subsection, we consider the case where φ_{1} : V → W_{1} is of E4-type, i.e., it contracts D_{1} ≅ Q_{0}^{2} ↪ P^{3} with a normal bundle O_{D_{1}}(D_{1}) ≅ O_{P^{3}}(−1) ⊗ O_{Q_{0}^{2}} to a Gorenstein terminal singularity, say p_{1} := φ_{1}(D_{1}), in W_{1}. Let \overline{D_{2}} := φ_{1}(D_{2}). Since the exceptional divisor D_{1} of φ_{1} is contained in the boundary, we have X = V − (D_{1} ∪ D_{2}) ≅ W_{1} − \overline{D_{2}}. Since (D_{1} + D_{2} · l_{1}) = −1 + (D_{2} · l_{1}) = 0, D_{2} meets D_{1} in such a way that D_{2}|_{D_{1}} ∈ |O_{P^{3}}(1) ⊗ O_{Q_{0}^{2}}|. In [M-M81, Table 2], there exist three Fano 3-folds with B_{2} = 2 having an extremal ray of E4-type, i.e., No. 8 (b), No. 15 (b) and No. 23 (b). We shall prove that, in fact, these cases cannot occur. The argument for the proof is similar to that in the previous subsection, so we shall sketch the proof.

Lemma 4.15 The cases No. 8 (b), No. 15 (b) and No. 23 (b) do not occur.

Proof. By considering the exact sequences as in the proof of Lemma 4.12 we have B_{3}(W_{1}) = B_{3}(V), where we note that B_{2}(D_{1}) = 1 as D_{1} ≅ Q_{0}^{2}. Since X ≅ W_{1} − \overline{D_{2}} is contractible, we have B_{3}(\overline{D_{2}}) = B_{3}(\overline{W_{1}}) and B_{2}(\overline{D_{2}}) = B_{2}(W_{1}) = 1. The same argument as in Lemma 4.13 implies that Pic(W_{1}) = Z[\overline{D_{2}}] and −K_{W_{1}} ∼ 2\overline{D_{2}}. Hence \overline{D_{2}} is a (possibly non-normal) del Pezzo surfaces. When V is of No. 8 (b) (resp. No. 15 (b), No. 23 (b)) we have B_{3}(\overline{D_{2}}) = B_{3}(V) = 18 (resp. = 8, = 2), so \overline{D_{2}} cannot be a normal del Pezzo surface. Thus we may assume that \overline{D_{2}} is a non-normal del Pezzo surface. Let σ : D̃_{2} → \overline{D_{2}} be the normalization. Let J ⊂ O\overline{D_{2}} be the conductor sheaf of σ and let \overline{Σ} := V\overline{D_{2}}(J), Σ̃ := VD̃_{2}(J). Then we have the exact sequence associated with σ (cf. [B-K82]):

⋯ → H^{1}(Σ̃) → H^{2}(\overline{D_{2}}) → H^{2}(D̃_{2}) ⊕ H^{2}(\overline{Σ}) → H^{2}(Σ̃) → H^{3}(\overline{D_{2}}) → H^{3}(D̃_{2}) ⊕ H^{3}(\overline{Σ}).

By [A-F, Theorem 1.5], it follows that H^{1}(Σ̃) = H^{3}(D̃_{2}) ⊕ H^{3}(\overline{Σ}) = 0, H^{2}(\overline{Σ}) = Z and H^{2}(Σ̃) = Z or = Z^{⊕2}. Hence the above sequence is:

0 → Z → H^{2}(D̃_{2}) ⊕ Z → Z or Z^{⊕2} → H^{3}(\overline{D_{2}}) → 0,

so B_{3}(\overline{D_{2}}) ≤ 2. Hence the only the possibility is No. 23 (b) and B_{3}(\overline{D_{2}}) = 2. But then this sequence means that B_{2}(D̃_{2}) = 0, which is a contradiction. □

4.5 φ_{1} is of E5-type

We consider the case where φ_{1} : V → W_{1} is of E5-type, i.e., it contracts D_{1} ≅ P^{2} with a normal bundle O_{D_{1}}(D_{1}) ≅ O_{P^{2}}(−2) to an index 2 point p_{1} := φ_{1}(D_{1}) in W_{1}. Let \overline{D_{2}} := φ_{1}(D_{2}). Since the exceptional divisor D_{1} of φ_{1} is contained in the boundary, we have X = V − (D_{1} ∪ D_{2}) ≅ W_{1} − \overline{D_{2}}. Since (D_{1} +D_{2} ·l_{1}) = −2+(D_{2} ·l_{1}) = 0, D_{2} meets D_{1} in such a way that D_{2}|_{D_{1}} ∈ |O_{P^{2}}(2)|. In [M-M81, Table 2] there exist two Fano 3-folds with B_{2} = 2 having an extremal ray of E5-type, i.e., No. 28 and No. 36.

Lemma 4.16 The case of No. 28 does not occur.

Proof. Our proof is similar to that in Lemma 3.1. Assume that V is No. 28 in [M-M81, Table 2], i.e., V is obtained from W_{2} = P^{3} via the blow-up φ_{2} : V → W_{2} along the plane cubic C_{2} with an exceptional divisor E_{2} and φ_{1} : V → W_{1} is the contraction of D_{1}, whicn is the proper transform on V of the hyperplane containing C_{2}, to an index 2 point p_{1} := φ_{1}(D_{1}) ∈ W_{1}. Let \overline{D_{2}} := φ_{1}(D_{2}) ⊂ W_{1}. The same argument as in Lemma 3.1 implies that −K_{W_{1}} ≡ \frac{3}{2}L_{1}, D_{1} ∼ H_{1} − 2H_{2} and E_{2} ∼ −H_{1} + 3H_{2}. Furthermore, we have B_{2}(\overline{D_{2}}) = 1, B_{3}(\overline{D_{2}}) = 2 and \overline{D_{2}} ≡ αL_{1} with α = \frac{1}{2} or 1. If α = \frac{1}{2}, then \overline{D_{2}} is a Gorenstein non-normal del Pezzo surface and we obtain a contradiction by the same argument in Lemma 3.1. Thus we may assume that α = 1, i.e., \overline{D_{2}} is an ample generator of Pic(W_{1}). Since \overline{D_{2}} is Cartier, we can write H_{1} ∼ φ_{1}^{∗}(\overline{D_{2}}) = D_{2} + mD_{1}, i.e., D_{2} ∼ (1 − m)H_{1} + 2mH_{2}, where m is a positive integer.

Since D_{1} and D_{2} generate Pic(V) ≅ Z[H_{1}] ⊕ Z[H_{2}] by Lemma 2.1, the matrix M := ( \begin{matrix}1 & −2 \\ 1 − m & 2m \end{matrix} )

have to be unimodular. But det(M) = 2, which is a contradiction. □

Lemma 4.17 The case of No. 36 does not occur.

Proof. Assume that V is of No. 36 in [M-M81, Table 2], i.e., V = P(O_{P^{2}} ⊕ O_{P^{2}}(2)). φ_{1} : V → W_{1} is the contraction of the negative section D_{1} ≅ P^{2} of the P^{1}-bundle structure φ_{2} : V → W_{2} ≅ P^{2} with a normal bundle O_{D_{1}}(D_{1}) ≅ O_{P^{2}}(−2) to an index 2 point p_{1} := φ_{1}(D_{1}). Associated with φ_{2}, we have Pic(V) ≅ Z[D_{1}] ⊕ Z[H_{2}]. We write D_{2} ∼ αD_{1} + βH_{2} with α, β ∈ Z. Since D_{1} and D_{2} generate Pic(V) (cf. Lemma 2.1), we have β ± 1. On the other hand, since D_{2} meets D_{1} in such a way that D_{2}|_{D_{1}} ∈ |O_{P^{2}}(2)|, we have D_{2}|_{D_{1}} ∼ (αD_{1} + βH_{2})|_{D_{1}} ∼ O_{P^{2}}(−2α + β) = O_{P^{2}}(2). Hence −2α + β = 2, i.e., α = \frac{β}{2} − 1 ∉ Z, which is a contradiction. □

5 The case D_{1} + D_{2} is ample

In this section, we finally consider the case where D_{1} + D_{2} is ample. By noting that K_{V} + D_{1} + D_{2} is not nef, we have the following:

Lemma 5.1 At least one of the extremal rays R_{1} and R_{2} is (K_{V} + D_{1} + D_{2})-negative. Let R_{1} be such an extremal ray. Then the contraction φ_{1} : V → W_{1} of R_{1} is of type either E2, C2, D2 or D3.

Proof. We put t := sup{μ ∈ Q ; D_{1} + D_{2} + μK_{V} is nef}, which is greater than 0 as D_{1} + D_{2} is ample. Furthermore, t is less than 1 as K_{V} + D_{1} + D_{2} is not nef. Then there exists an extremal ray contained in \overline{NE} (V) ∩ \{D_{1} + D_{2} + tK_{V} \}^{⊥}. We may assume that R_{1} = R_{+}[l_{1}] is such an extremal ray. Then we have (D_{1} + D_{2}) · l_{1} = t(−K_{V} · l_{1}) < (−K_{V} · l_{1}). Since D_{1} + D_{2} is ample, (D_{1} + D_{2}) · l_{1} is a positive integer. Hence (−K_{V} · l_{1}) ≥ 2. By the classification of extremal rays on smooth projective 3-folds (cf. [Mo82]), we know that φ_{1} is of type either E2, C2, D2 or D3. □

In the subsequent of this section, we consider according as the type of φ_{1} : V → W_{1}.

5.1 φ_{1} is of E2-type

First of all, we consider the case where φ_{1} : V → W_{1} is of type E2. As seen below, some kinds of exotic A^{3}’s can be compactified in this situation (cf. Theorem 1.3, Types XII, XIII and XXII in Table 1 and Remark 1.3). Let E_{1} be the exceptional divisor of φ_{1} with a normal bundle O_{E_{1}}(E_{1}) ≅ O_{P^{2}}(−1) which is contracted to a smooth point p_{1} := φ_{1}(E_{1}) ∈ W_{1}. In [M-M81, Table 2], there are two smooth Fano 3-folds with B_{2} = 2 having an extremal ray of type E2, namely, No. 30 and No. 35. We prove the following:

Lemma 5.2 (1) E_{1} is not contained in the boundary D_{1} ∪ D_{2},

(2) One of the boundary components, say D_{1}, meets E_{1} in such a way that D_{1}|_{E_{1}} ∈ |O_{P^{2}}(1)|, and the other component D_{2} is disjoint from E_{1}.

Proof. Note that when V is of No. 30 (resp. No. 35), W_{1} is isomorphic to Q^{3} ↪ P^{4} (resp. P^{3}). Assume that E_{1} is contained in the boundary D_{1} ∪ D_{2}, say E_{1} = D_{1}. Let \overline{D_{2}} := φ_{1}(D_{2}) be the proper image on W_{1}. Then (W_{1}, \overline{D_{2}}) is a smooth compactification of a contractible affine 3-fold X, so X ≅ A^{3} and \overline{D_{2}} is an ample generator of Pic(W_{1}) ≅ Z. If W_{1} is isomorphic to Q^{3} ↪ P^{4} (resp. P^{3}), then \overline{D_{2}} ∈ |O_{W_{1}}(1)| is a quadric cone Q_{0}^{2} (resp. a hyperplane P^{2}). If p_{1} is a smooth point on \overline{D_{2}}, then we have φ_{1}^{∗}(\overline{D_{2}}) = D_{1}+D_{2}. Hence (D_{1} + D_{2}) · l_{1} = 0, which is a contradiction. Thus p_{1} is a singular point on \overline{D_{2}}, so the possibility is No. 30: W_{1} ≅ Q^{3}, \overline{D_{2}} ≅ Q_{0}^{2} and p_{1} is a vertex of \overline{D_{2}}. Then D_{1} + D_{2} = φ_{1}^{∗}(\overline{D_{2}}) − D_{1} ∼ H_{1} − D_{1}. We have −K_{V} = φ_{1}^{∗}(−K_{W_{1}}) − 2D_{1} ∼ φ_{1}^{∗}(3\overline{D_{2}}) − 2D_{1} ∼ 3H_{1} − 2D_{1}. Since −K_{V} ∼ H_{1} + 2H_{2} by Proposition 2.1, we have D_{1} ∼ H_{1} − H_{2}. Thus D_{1} + D_{2} = φ_{1}^{∗}(\overline{D_{2}}) − D_{1} ∼ H_{1} − (H_{1} − H_{2}) = H_{2}, which is not ample. This is a contradiction. Hence E_{1} is not contained in D_{1} ∪ D_{2}. Since D_{1} + D_{2} is ample and (D_{1} + D_{2}) · l_{1} = t(−K_{V} · l_{1}) = 2t with 0 < t < 1, we have t = \frac{1}{2} and (D_{1} + D_{2}) · l_{1} = 1. This means the assertion (2). □

We put \overline{D_{i}} := φ_{1}(D_{i}) ⊂ W_{1} for i = 1, 2. By Lemma 5.2, both of \overline{D_{1}} and \overline{D_{2}} are divisors on W_{1}, and p_{1} := φ_{1}(E_{1}) is contained in \overline{D_{1}} as a smooth point but not contained in \overline{D_{2}}.

Lemma 5.3 \overline{D_{2}} is an ample generator of Pic(W_{1}), i.e., Pic(W_{1}) ≅ Z[\overline{D_{2}}].

Proof. Since Pic(V) = Z[D_{1}] ⊕ Z[D_{2}] (cf. Lemma 2.1), we can write E_{1} ∼ a_{1}D_{1} + a_{2}D_{2} with a_{1}, a_{2} ∈ Z. By noting that −1 = E_{1} · l_{1} = (a_{1}D_{1} + a_{2}D_{2}) · l_{1}, D_{1} meets E_{1} in such a way that D_{1}|_{E_{1}} ∈ |O_{P^{2}}(1)| and D_{2} is disjoint from E_{1}, we have a_{1} = −1. Hence Pic(W_{1}) ≅ Pic(V)/ < E_{1} >≅ Z[D_{1}] ⊕ Z[D_{2}]/ < −D_{1} + a_{2}D_{2} >≅ Z[\overline{D_{2}}]. Thus \overline{D_{2}} is an ample generator of Pic(W_{1}). □

Lemma 5.4 Assume that V is No. 35 in [M-M81, Table 2], i.e., V ≅ P(O_{P^{2}} ⊕ O_{P^{2}}(1)). Then we have:

(1) The complement Z := W_{1} − \overline{D_{2}} is isomorphic to the affine 3-space A^{3},

(2) X is obtained as the affine modification of Z with a locus (p_{1} ∈ D_{1}^{∘}), where D_{1}^{∘} := \overline{D_{1}} ∩ Z: see Definition 2.1 for the definition on affine modifications,

(3) D_{1}^{∘} is an acyclic hypersurface.

Proof. The first assertion (1) is obvious because W_{1} ≅ P^{3} and \overline{D_{2}} is an ample generator of Pic(W_{1}) (cf. Lemma 5.3). Note that X = V − (D_{1} ∪ D_{2}) is obtained from Z via the blow-up at the point p_{1} and deleting the proper transform of D_{1}^{∘}. Hence X is just the affine modification of Z with a locus (p_{1} ∈ D_{1}^{∘}). Let σ : X → Z be this affine modification. Note that E_{1}^{∘} := E_{1} ∩ X ≅ A^{2} is an exceptional divisor and D_{1}^{∘} is a divisor of σ in the terminology of Definition 2.1 (see also [K-Z99]). Since σ preserves the contractibility, σ|_{E_{1}^{∘}} induces isomorphims of homology groups (σ|_{E_{1}^{∘}})_{∗} : H_{∗}(E_{1}^{∘}; Z) →^{∼} H_{∗}(D_{1}^{∘}; Z) by [K-Z99, Theorem 3.1]. Thus we complete the proof. □

The embedding stated in Lemma 5.4 is of Type XXII in Table 1.

Lemma 5.5 Assume that V is No. 30. Then we have:

(1) \overline{D_{2}} is isomorphic to a smooth quadric P^{1} × P^{1} or a quadric cone Q_{0}^{2},

(2) If \overline{D_{2}} ≅ P^{1} × P^{1} (resp. ≅ Q_{0}^{2}), then the complement Z := W_{1} − \overline{D_{2}} is isomorphic to SL(2; C) (resp. A^{3}),

(3) X is obtained as the affine modification of Z with a locus (p_{1} ∈ D_{1}^{∘}), where D_{1}^{∘} := \overline{D_{1}} ∩ Z,

(4) If \overline{D_{2}} ≅ P^{1} × P^{1}, then D_{1}^{∘} satisfies H_{0}(D_{1}^{∘}; Z) ≅ H_{1}(D_{1}^{∘}; Z) ≅ Z and H_{i}(D_{1}^{∘}; Z) = 0 for i > 1. On the other hand, if \overline{D_{2}} ≅ Q_{0}^{2}, then D_{1}^{∘} is an acyclic surface.

Proof. The first assertion (1) is obvious as W_{1} ≅ Q^{3} and \overline{D_{2}} is an ample generator of Pic(W_{1}). The remaining assertions (1)∼(4) for the case \overline{D_{2}} ≅ Q_{0}^{2} can be obtained by the same argument as in the proof of Lemma 5.4. Hence we have only to consider the case where \overline{D_{2}} ∈ |O_{W_{1}}(1)| is a smooth quadric. Then it is not difficult to see that Z := Q^{3} − \overline{D_{2}} is isomorphic to the hypersurface in A^{4}_{(x,y,z,u)} defined by xy − zu − 1 = 0, i.e., Z ≅ SL(2; C). We can see that X is obtained from Z via the affine modification with a locus (p_{1} ∈ D_{1}^{∘}), where D_{1}^{∘} := \overline{D_{1}} ∩ Z. Let σ : X → Z be this affine modification. Note that E_{1}^{∘} := E_{1} ∩ X ≅ A^{2} is an exceptional divisor and D_{1}^{∘} is a divisor of σ (cf. Definition 2.1). We put X^{∗} := X − E_{1}^{∘} and Z^{∗} := Z − D_{1}^{∘}. Associated with σ, we write the exact sequences:

⋯ → H_{i+1}(X, X^{∗}) ⟶ H_{i}(X^{∗}) ⟶ H_{i}(X) ⟶ H_{i}(X, X^{∗}) ⟶ H_{i−1}(X^{∗}) → ⋯
↓≅  ↓ σ_{∗}  ↓≅
⋯ → H_{i+1}(Z, Z^{∗}) ⟶ H_{i}(Z^{∗}) ⟶ H_{i}(Z) ⟶ H_{i}(Z, Z^{∗}) ⟶ H_{i−1}(Z^{∗}) → ⋯

Note that H_{i}(X, X^{∗}) ≅ H_{i−2}(E_{1}^{∘}) and H_{i}(Z, Z^{∗}) ≅ H_{i−2}(D_{1}^{∘}) by Thom isomorphism. We shall investigate what topological data on D_{1}^{∘} are needed in order that X is contractible. Since p_{1} is a smooth point of D_{1}^{∘}, we have σ^{∗}(D_{1}^{∘}) = E_{1}^{∘}. Hence σ induces an isomorphism of the fundamental groups σ_{∗} : π_{1}(X) →^{∼} π_{1}(Z) = {1} (cf. [K-Z99, Proposition 3.1]). Since Z ≅ SL(2; C), Z is simply connected and satisfies H_{0}(Z) = H_{3}(Z) = Z and H_{i}(Z) = 0 for i ≠ 0, 3. Hence X is acyclic if and only if D_{1}^{∘} satisfies H_{0}(D_{1}^{∘}) = H_{1}(D_{1}^{∘}) = Z and H_{i}(D_{1}^{∘}) = 0 for i > 1 by the above exact sequences. Thus X is contractible if and only if D_{1}^{∘} satisfies H_{0}(D_{1}^{∘}) = H_{1}(D_{1}^{∘}) = Z and H_{i}(D_{1}^{∘}) = 0 for i > 1 by a theorem of J.H.C. Whitehead. □

Thus we have the embeddings of Types XII and XIII in Table 1.

Example 5.1 Let \overline{D_{2}} ∈ |O_{Q^{3}}(1)| be isomorphic to a smooth quadric P^{1} × P^{1}. Then the complement Z = Q^{3} − \overline{D_{2}} is isomorphic to SL(2; C). We may assume that Z = \{x^{2} + y^{2} + zu − 1 = 0\} ⊂ A^{4}_{(x,y,z,u)}. We shall choose, for example, {u = 0} ∩ Z as D_{1}^{∘}. Note that since D_{1}^{∘} ≅ C^{∗} × A^{1}, the conditions mentioned in Lemma 5.5 (4) are satisfied. Let p_{1} := (1, 0, 0, 0). Then the affine 3-fold X, which is obtained from Z via the affine modification with a locus (p_{1} ∈ D_{1}^{∘}), is isomorphic to the hypersurface in A^{4}_{(x,y,z,u)} defined by x^{2}u + 2x + y^{2}u + zu = 0. Hence X ≅ A^{3}.

5.2 φ_{1} is of C2-type

In this subsection, we consider the case where φ_{1} : V → W_{1} is of C2-type. In the list [M-M81, Table 2], there are seven deformation types of smooth Fano 3-folds having extremal rays of C2-type, namely, No. 24, No. 27, No. 31, No. 32, No. 34, No. 35 and No. 36. Note that the base surface W_{1} is isomorphic to the projective plane P^{2} (cf. [M-M83]). We have the following:

Lemma 5.6 (1) t = \frac{1}{2} and (D_{1} + D_{2}) · l_{1} = 1,

(2) We may assume that D_{1} satisfies D_{1} · l_{1} = 1 and D_{2} ∼ H_{1} = φ_{1}^{∗}O_{P^{2}}(1). In particular, D_{1} is a birational section of φ_{1},

(3) The restriction φ_{1}|_{X} of φ_{1} onto X = V − (D_{1} ∪ D_{2}) yields a trivial A^{1}-bundle structure over the affine plane A^{2}, i.e., X ≅ A^{2} × A^{1} ≅ A^{3}.

Proof. By the choice of the extremal ray R_{1} = R_{+}[l_{1}], we have (D_{1} + D_{2}) · l_{1} = t(−K_{V} · l_{1}) = 2t. Since D_{1} + D_{2} is ample and t is less than 1, we have t = \frac{1}{2} and (D_{1} + D_{2}) · l_{1} = 1. Thus we may assume that D_{1} ·l_{1} = 1 and D_{2} ·l_{1} = 0. By Proposition 2.1, we can write D_{1} ∼ aH_{1} +H_{2} and D_{2} ∼ bH_{1} with a, b ∈ Z.

Since Pic(V) = Z[D_{1}]⊕Z[D_{2}] (cf. Lemma 2.1), the matrix ( \begin{matrix}a & 1 \\ b & 0 \end{matrix} ) is unimodular, so we have b = ±1.

Since the case b = −1, i.e., D_{2} ∼ −H_{1} is obviously impossible, it follows that b = 1, i.e., D_{2} ∼ H_{1}. Thus the restriction φ_{1}|_{X} yields an A^{1}-fibration over the affine plane W_{1} − φ_{1}(D_{2}) ≅ P^{2} − L_{1} ≅ A^{2}. Since X is a smooth contractible affine 3-fold, this A^{1}-fibration is surjective (cf. [Ka02], [K-S02]). In fact, all fibers of φ_{1}|_{X} are isomorphic to the affine line A^{1} as D_{1} · l_{1} = 1. Since Pic(A^{2}) = 0, we know that φ_{1}|_{X} : X → A^{2} is a trivial A^{1}-bundle structure. Hence X ≅ A^{2} × A^{1} ≅ A^{3}. □

Conversely, we shall investigate the configuration of two components D_{1} and D_{2} on a smooth Fano 3-fold V of No. 24, No. 27, No. 31, No. 32, No. 34, No. 35 or No. 36 in [M-M81, Table 2] in order to see when the complement X = V − (D_{1} ∪ D_{2}) is isomorphic to the affine 3-space A^{3}. In consideration of Lemma 5.6 (2), we may assume that D_{1} is a birational section of φ_{1} : V → W_{1} ≅ P^{2} and D_{2} ∼ H_{1}.

Lemma 5.7 Let V be a smooth Fano 3-fold of No. 24, No. 27, No. 31, No. 32, No. 34, No. 35 or No. 36 in [M-M81, Table 2]. Let D_{1} and D_{2} be two irreducible divisors on V such that D_{1} + D_{2} is ample. Assume that D_{1} is a birational section of φ_{1} : V → W_{1} ≅ P^{2} and D_{2} ∼ H_{1} := φ_{1}^{∗}O_{P^{2}}(1), where φ_{1} : V → W_{1} is a Mori fiber space of C2-type. Then the complement X = V − (D_{1} ∪ D_{2}) is isomorphic to the affine 3-space A^{3} if and only if H^{2}(D_{1} ∪ D_{2}; Z) ≅ Z^{⊕2}.

Proof. Only if part is easy to see, so we have only to show the another implication, i.e., the condition H^{2}(D_{1} ∪D_{2}; Z) ≅ Z^{⊕2} implies X ≅ A^{3}. By the assumption, we can write D_{1} ∼ mH_{1} +H_{2} and D_{2} ∼ H_{1} for some m ∈ Z. Since D_{1} + D_{2} is ample, the complement X = V − (D_{1} ∪ D_{2}) is affine. Claim. X is factorial, i.e., the coordinate ring Γ(O_{X}) of X is factorial.

Proof of Claim. Since the matrix ( \begin{matrix}m & 1 \\ 1 & 0 \end{matrix} ) is unimodular, we have Pic(V) ≅ Z[D_{1}]⊕Z[D_{2}]. Let G_{0} be

an irreducible divisor on X and let G be the closure of G_{0} in V. Then we can write G = a_{1}D_{1}+a_{2}D_{2}+(g) for some a_{1}, a_{2} ∈ Z and g ∈ C(V) = Q(A), where A := Γ(O_{X}). It is clear that g ∈ A because g is regular on X. Then div(g)_{X} = G_{0}. Hence A is factorial as desired. □

We consider the exact sequence with respect to the pair (V, D_{1} ∪ D_{2}):

⋯ → H^{2}(V, D_{1} ∪ D_{2}) → H^{2}(V) → H^{2}(D_{1} ∪ D_{2}) → H^{3}(V, D_{1} ∪ D_{2}) → H^{3}(V) → ⋯,

where the cohomology groups have the ring of integers Z as the coefficient group. Note that H^{2}(V, D_{1} ∪ D_{2}) ≅ H_{4}(X) = 0 (cf. [Mil63, Theorem 7.1]) and H^{3}(V) = 0 (cf. [M-M81]). Since H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2}, we have H_{3}(X) = 0 by the above sequence. On the other hand, the restriction φ_{1}|_{X} of φ_{1} onto X = V − (D_{1} ∪ D_{2}) yields an A^{1}-fibration over the affine plane W_{1} − φ_{1}(D_{2}) ≅ A^{2} as D_{1} is a birational section of φ_{1}. Since X is factorial and H_{3}(X) = 0, this A^{1}-fibration is surjective (cf. [Ka02, Prop. 3.2]). This implies that, for every point y ∈ W_{1} − φ_{1}(D_{2}), the fiber l_{y} := φ_{1}^{−1}(y) ≅ P^{1} is not contained in D_{1}.

Since D_{1} · l_{y} = 1, it follows that l_{y} intersects D_{1} in a single point transversally. Thus all the fibers of φ_{1}|_{X} are isomorphic to the affine line A^{1}. Since Pic(A^{2}) = 0, we know that φ_{1}|_{X} is a trivial A^{1}-bundle structure over A^{2}, i.e., X ≅ A^{2} × A^{1} ≅ A^{3}. □

It is not difficult to find the components D_{1} and D_{2} on V satisfying the condition mentioned in Lemma 5.7 actually when V is of No. 32, No. 34, No. 35 and No. 36. Even if V is either of No. 24, No. 27 or No. 31, we can also find such two components D_{1} and D_{2}. In the following, we shall demonstrate how to find them in the case where V is of No. 27.

Lemma 5.8 Let V be a smooth Fano 3-fold of No. 27 in [M-M81, Table 2], i.e., V is obtained via the blow-up φ_{2} : V → P^{3} along a twisted cubic on P^{3}. Then we can find a normal member D_{1} ∈ |H_{2}| and D_{2} ∈ |H_{1}| in such a way that H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2}, and hence the complement X = V − (D_{1} ∪ D_{2}) is isomorphic to the affine 3-space A^{3}.

Proof. Let C ⊂ P^{3} be a twisted cubic which is the center of φ_{2}. Then the restriction homomorphism H^{0}(O_{P^{3}}(1)) → H^{0}(C(1)) is an isomorphism as C is not contained in any hyperplane. We fix a point p on C. Then we can take a hyperplane L_{2} ∈ |O_{P^{3}}(1)| in such a way that L_{2}|_{C} = 3p. Let D_{1} := φ_{2}^{∗}(L_{2}) ∈ |H_{2}|. We claim the following:

Claim. D_{1} is a normal singular del Pezzo surface of degree (−K_{D_{1}})^{2} = 6. Proof of Claim. Note that −K_{V} ∼ H_{1} + 2H_{2} by Proposition 2.1. Hence −K_{D_{1}} = (H_{1} + H_{2})|_{D_{1}}. In particular, D_{1} is a (possibly singular) del Pezzo surface of degree (−K_{D_{1}})_{D_{1}}^{2} = (H_{1} + H_{2})^{2} · H_{2} = 6. Assume that D_{1} is smooth. Then D_{1} is obtained from P^{2} by the blow-ups of 3-points in a general position. On the other hand, there exists only one exceptional curve of φ_{2}|_{D_{1}} : D_{1} → L_{2} ≅ P^{2} by the choice of the hyperplane L_{2}. This is a contradiction. Hence D_{1} is singular. Suppose that D_{1} is non-normal. Let σ : \overline{D_{1}} → D_{1} be the normalization. By virtue of the classification of non-normal del Pezzo surfaces (cf. [A-F]), there are three possibilities concerning \overline{D_{1}}, namely, \overline{D_{1}} is isomorphic to F_{4}, F_{2} or S_{6}, where S_{6} ⊂ P^{7} is a cone over a smooth rational curve of degree 6 in P^{6}. This is a contradiction because the composite (φ_{1}|_{D_{1}}) ∘ σ : \overline{D_{1}} → D_{1} → P^{2} yields a birational morphism. Therefore D_{1} is a normal singular del Pezzo surface. □

Let μ : D̃_{1} → D_{1} be the minimal resolution. Then D̃_{1} is obtained from P^{2} by 3-points blow-ups in almost general position, say ν : D̃_{1} → P^{2} (cf. [H-W81]). These 3-points lie on a suitable line on P^{2}, say L_{1}. By the construction, ν is a composite of μ and φ_{1}|_{D_{1}}. We put D_{2} := φ_{1}^{∗}(L_{1}) ∼ H_{1}. Then it is easy to see that H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2} and, as a consequence, X ≅ A^{3} by Lemma 5.7. □

We can construct factorial smooth affine 3-folds X with non-surjective A^{1}-fibrations over the affine plane A^{2} as follows. Indeed, the homology groups H_{3}(X) of X in Example 5.2 below are non-trivial.

Example 5.2 (1) Let V be of No. 32, i.e., V = P(Θ_{P^{2}}). Then both extremal contractions φ_{1} : V → W_{1} and φ_{2} : V → W_{2} are of C2-type. For instance, we choose H_{2} = φ_{2}^{∗}O_{W_{2}}(1) as D_{1}. Then D_{1} ≅ F_{1} and φ_{1}|_{D_{1}} : D_{1} → W_{1} ≅ P^{2} is a contraction of the minimal section. Hence there exists a unique point y ∈ W_{1} such that the fiber l_{y} := φ_{1}^{−1}(y) is contained in D_{1} as the minimal section. Let L_{1} be a line on W_{1} which does not pass through y and let D_{2} := φ_{1}^{∗}(L_{1}) ∼ H_{1}. Then H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕3} and the complement X = V −(D_{1}∪D_{2}) is a factorial smooth affine 3-fold with H_{3}(X) ≅ Z. By the construction, the restriction φ_{1}|_{X} yields an A^{1}-fibration over the affine plane W_{1} − L_{1} ≅ A^{2} and the fiber of it over the point y is empty, i.e., (φ_{1}|_{X})^{−1}(y) = ∅. Thus φ_{1}|_{X} gives rise to an A^{1}-bundle over the punctured affine plane.

(2) More generally, we can construct a factorial smooth affine 3-fold X with an arbitrary large homology group H_{3}(X). With the notation as in (1), let D_{1} be a smooth member of |mH_{1} + H_{2}|, where m ≥ 0. Then (−K_{D_{1}})_{D_{1}} = ((2 − m)H_{1} + H_{2})^{2} · (mH_{1} + H_{2}) = −m^{2} − m + 8. Hence the restriction φ_{1}|_{D_{1}} : D_{1} → P^{2} is a composite of (m^{2} + m + 1)-points blow-ups. More precisely, there are (m^{2} + m + 1)- fibers of φ_{1} contained in D_{1}, and each of them is a (−1)-curve on D_{1}. Let L_{1} be a line on P^{2} which does not pass through any of the centers of the blow-ups φ_{1}|_{D_{1}}, and let D_{2} := φ_{1}^{∗}(L_{1}) ∼ H_{1}. Then H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕(m^{2}+m+3)} and H_{3}(X) ≅ Z^{⊕(m^{2}+m+1)}. By the construction, the restriction φ_{1}|_{X} gives rise to an A^{1}-fibration over P^{2} − L_{1} ≅ A^{2} and there are exactly (m^{2} + m + 1)-points on A^{2} over which the fibers of φ_{1}|_{X} are empty.

Summarizing the argument in this subsection we have the embeddings of Types V, VII, XV, XVI, XIX, XXIII and XXIV in Table 1.

5.3 φ_{1} is of D2-type

In this subsection, we consider the case where φ_{1} : V → W_{1} is of D2-type. We shall prove that, in fact, this case does not occur. At first, we prove the following:

Lemma 5.9 (1) t = \frac{1}{2} and (D_{1} + D_{2}) · l_{1} = 1,

(2) We may assume that D_{1} satisfies D_{1} · l_{1} = 1 and D_{2} ∼ H_{1} = φ_{1}^{∗}O_{P^{1}}(1),

(3) The restriction φ_{1}|_{X} of φ_{1} onto X = V − (D_{1} ∪ D_{2}) yields a fibration onto the affine line A^{1} = W_{1} − {∞}, where ∞ := φ_{1}(D_{2}),

(4) Let S be the general fiber of φ_{1}|_{X}. Then S is an affine surface obtained by deleting a diagonal from P^{1} × P^{1}.

Proof. It is easy to see that the base curve W_{1} is isomorphic to P^{1}. The assertions (1), (2) and (3) can be obtained by the similar argument as in the proof of Lemma 5.6. Since D_{1} · l_{1} = 1, it follows that D_{1}|_{F_{y}} ∼ O_{P^{1}×P^{1}}(1, 1), where F_{y} := φ_{1}^{−1}(y) ≅ P^{1} × P^{1} (y ∈ W_{1}) is a general fiber of φ_{1}. Since φ_{1}|_{D_{1}} : D_{1} → W_{1} gives rise to a P^{1}-fibration, we know that D_{1}|_{F_{y}} is a diagonal on F_{y} ≅ P^{1} × P^{1} for a general y ∈ W_{1}. This means the assertion (4). □

We shall investigate fibers of φ_{1}|_{X} which are not isomorphic to S (cf. Lemma 5.9). We denote by F_{y} (resp. F_{y}^{∘}) the fiber of φ_{1} (resp. of φ_{1}|_{X}) over the point y ∈ W_{1} (resp. y ∈ W_{1} − {∞}). Note that, with this notation, we have F_{∞} = D_{2}. Let Σ := {y ∈ W_{1} ; F_{y} ≅ Q_{0}^{2}} and let Δ := {y ∈ W_{1} − Σ ; D_{1}|_{F_{y}} is a union of two distinct rulings on F_{y} ≅ P^{1} × P^{1}}.

Lemma 5.10 We have the following with respect to the fibers of φ_{1}|_{X}:

(1) For y ∈ W_{1} − (Σ ∪ Δ ∪ {∞}), we have F_{y}^{∘} ≅ S,

(2) For y ∈ Δ − {∞} (unless Δ − {∞} = ∅), we have F_{y}^{∘} ≅ A^{2},

(3) For y ∈ Σ − {∞} (unless Σ − {∞} = ∅), there are two possibilities (a) and (b) concerning the fiber F_{y}^{∘}:

(a) F_{y}^{∘} is isomorphic to the affine surface which is obtained as the complement of a hyperplane section on F_{y} ≅ Q_{0}^{2} ↪ P^{3} not passing through the vertex of F_{y};

(b) F_{y}^{∘} ≅ A^{1} × C^{∗}.

Proof. The assertions (1) and (2) are not difficult to see. We shall prove (3). Since the fibers of φ_{1}|_{D_{1}} : D_{1} → W_{1} are of type (1, 1) on the smooth fibers ≅ P^{1} × P^{1} of φ_{1} and hence D_{1} is locally a graph everywhere (cf. [Mu90]). Thus D_{1} is smooth. Since φ_{1}|_{D_{1}} yields a P^{1}-fibration, there is no multiple fiber of it. On the other hand, for y ∈ Σ, the intersection D_{1}|_{F_{y}} is linearly equivalent to O_{P^{3}}(1) ⊗ O_{Q_{0}^{2}}. Hence the possibility of configuration of D_{1}|_{F_{y}} is a smooth hyperplane section, a union of two distinct generating lines or a twofold generating line on F_{y} ≅ Q_{0}^{2} ↪ P^{3}. But the last possibility is excluded because of the absence of multiple fibers of φ_{1}|_{D_{1}}. Thus we obtain the assertion (3). □

In [M-M81, Table 2], there are three smooth Fano 3-folds with B_{2} = 2 having an extremal ray of type D2, namely, No. 18, No. 25 and No. 29. We see how many singular fibers of φ_{1} exist by calculating topological Euler number χ_{top}.

Lemma 5.11 We have ♯Σ = 6 (resp. = 4 or = 2) if V is of No. 18 (resp. No. 25 or No. 29) in [M-M81, Table 2].

Proof. Since B_{3}(V) = 4 (resp. = 2 or = 0) when V is of No. 18 (resp. No. 25 or No. 29), it is easy to see χ_{top}(V) = 2 (resp. = 4 or = 6). On the other hand, we calculate χ_{top}(V) associated with the Mori fiber space φ_{1} : V → W_{1} ≅ P^{1}, namely, we have:

χ_{top}(V) = χ_{top}(P^{1} − Σ)χ_{top}(P^{1} × P^{1}) + χ_{top}(Σ)χ_{top}(Q_{0}^{2}) = 8 − ♯Σ.

Hence we have the assertion. □

By using the previous lemmas, we have the following:

Lemma 5.12 The cases No. 18 and No. 25 do not occur.

Proof. Assume that V is of No. 18. Then there are 6 singular fibers ≅ Q_{0}^{2} of φ_{1} by Lemma 5.11. Let Σ = {y_{1}, ⋯, y_{6}}. Note that χ_{top}(F_{i}^{∘}) is equal to 1 or 0 according as F_{i}^{∘} is of type (a) or (b) in Lemma 5.10 (3) unless y_{i} = ∞. We calculate χ_{top}(X) associated with the fibration φ_{1}|_{X} : X → A^{1} ≅ W_{1}−{∞}. For this purpose, we need to consider according as the configurations of the fiber D_{2} = F_{∞} and the intersection D_{1}|_{D_{2}}, separately. There are following 3 possibilities:

(i) D_{2} ≅ P^{1} × P^{1} and D_{1}|_{D_{2}} is a diagonal on D_{2}, i.e., ∞ ∈ W_{1} − (Σ ∪ Δ),

(ii) D_{2} ≅ P^{1} × P^{1} and D_{1}|_{D_{2}} is a union of two rulings in distinct directions on D_{2}, i.e., ∞ ∈ Δ,

(iii) D_{2} ≅ Q_{0}^{2}, i.e., ∞ ∈ Σ.

In the case (i) we have the following equality:

1 = χ_{top}(X) = χ_{top}(P^{1} − Σ ∪ Δ ∪ {∞})χ_{top}(S) + χ_{top}(Δ)χ_{top}(A^{2}) + ∑_{i=1}^{6} χ_{top}(F_{y_{i}}^{∘}).

By noting that χ_{top}(F_{i}^{∘}) = 0 or = 1 for i ≤ i ≤ 6, we have:

1 = χ_{top}(X) ≤ 2(2 − 6 − ♯Δ − 1) + ♯Δ + 6 = −♯Δ − 4 ≤ −4,

which is a contradiction. In the case (ii) we have:

1 = χ_{top}(X) = χ_{top}(P^{1} − Σ ∪ Δ)χ_{top}(S) + χ_{top}(Δ − {∞})χ_{top}(A^{2}) + ∑_{i=1}^{6} χ_{top}(F_{y_{i}}^{∘})

≤ 2(2 − 6 − ♯Δ) + ♯Δ − 1 + 6 = −♯Δ − 3 ≤ −4,

which is a contradiction. In the remaining case (iii), we may assume y_{6} = ∞ and obtain the inequality:

1 = χ_{top}(X) = χ_{top}(P^{1} − Σ ∪ Δ)χ_{top}(S) + χ_{top}(Δ)χ_{top}(A^{2}) + ∑_{i=1}^{5} χ_{top}(F_{y_{i}}^{∘})

≤ 2(2 − 6 − ♯Δ) + ♯Δ + 5 = −♯Δ − 3 ≤ −3,

which is also a contradiction. Thus we know that the case of No. 18 does not occur. By the similar argument as above, we can see that the case of No. 25 also does not occur. □

Lemma 5.13 The case No. 29 does not occur.

Proof. Our proof consists of three steps. Step 1. Assume that V is of No. 29, namely, V is obtained from W_{2} := Q^{3} ↪ P^{4} via the blow-up φ_{2} : V → W_{2} along a conic C_{2} ⊂ W_{2} with an exceptional divisor E_{2} ≅ P^{1} × P^{1}. By Lemma 5.11, φ_{1} : V → W_{1} ≅ P^{1} has 2 singular fibers ≅ Q_{0}^{2}. Let Σ = {y_{1}, y_{2}}. Although there are three possibilities (i), (ii) and (iii) concerning the fiber D_{2} = F_{∞} and the intersection D_{1}|_{D_{2}} as in the proof of Lemma 5.12, the same argument there shows that the cases (i) and (ii) are impossible. Hence we may assume that (iii) occurs and y_{2} = ∞. We have the following equality associated with the fibration φ_{1}|_{X}:

1 = χ_{top}(X) = χ_{top}(P^{1} − Σ ∪ Δ)χ_{top}(S) + χ_{top}(Δ)χ_{top}(A^{2}) + χ_{top}(F_{y_{1}}^{∘})
= 2(2 − 2 − ♯Δ) + ♯Δ + χ_{top}(F_{y_{1}}^{∘}) = −♯Δ + χ_{top}(F_{y_{1}}^{∘}).

Hence Δ = ∅ and χ_{top}(F_{y_{1}}^{∘}) = 1. Step 2. By Proposition 2.1, we have H_{1} + 2H_{2} ∼ −K_{V} = φ_{2}^{∗}(−K_{W_{2}}) − E_{2} ∼ 3H_{2} − E_{2}. Hence E_{2} ∼ −H_{1} + H_{2}. Note that D_{1} is distinct from E_{2}. Indeed, suppose that D_{1} = E_{2}. Then, via the contraction φ_{2}, we have (W_{2}, \overline{D_{2}} := φ_{2}(D_{2})) ≅ (Q^{3}, Q_{0}^{2}). We have D_{1} + D_{2} = φ_{2}^{∗}(\overline{D_{2}}), which is a contradiction as D_{1} + D_{2} is ample. Let \overline{D_{i}} := φ_{2}(D_{i}) for i = 1, 2. We write \overline{D_{1}} ∼ nL_{2} with n ∈ N and put m := mult_{C_{2}} \overline{D_{1}} ≥ 0. Then we have D_{1} = φ_{2}^{∗}(\overline{D_{1}})−mE_{2} ∼ nH_{2}−m(−H_{1}+H_{2}) = mH_{1}+(n−m)H_{2}.

On the other hand, since Pic(V) = Z[D_{1}] ⊕ Z[D_{2}] by Lemma 2.1, the matrix ( \begin{matrix}m & n − m \\ 1 & 0 \end{matrix} ) have to

be unimodular, so n − m ± 1. But if n − m = −1, then D_{1} · l_{1} = −1, which is obviously contradiction. Hence n − m = 1, i.e., D_{1} ∼ mH_{1} + H_{2}. Then (−K_{D_{1}})_{D_{1}}^{2} = (−K_{V} − D_{1})^{2} · D_{1} = ((1 − m)H_{1} + H_{2})^{2} · (mH_{1} + H_{2}) = m(1 − m)^{2}(H_{1}^{3}) + (1 + m)(1 − m)(H_{1}^{2} · H_{2}) + (2 − m)(H_{1} · H_{2}^{2}) + (H_{2}^{3}). Note that (H_{1}^{3}) = (H_{1}^{2} · H_{2}) = 0 and (H_{1} · H_{2}^{2}) = (H_{2}^{3}) = 2. Hence (−K_{D_{1}})_{D_{1}}^{2} = 2(3 − m). Step 3. D_{1} is smooth by the same reason as in the proof of Lemma 5.10. Since Δ = ∅, a fiber of the P^{1}-fibration φ_{1}|_{D_{1}} : D_{1} → W_{1} ≅ P^{1} over any point except for Σ = {y_{1}, y_{2} = ∞} is smooth. On the other hand, since χ_{top}(F_{y_{1}}^{∘}) = 1, the intersection D_{1}|_{F_{y_{1}}} is a hyperplane section not passing through the vertex of F_{y_{1}} ≅ Q_{0}^{2} ↪ P^{3} (cf. Lemma 5.10). Thus φ_{1}|_{D_{1}} is either a P^{1}-bundle structure or has a unique singular fiber D_{2}|_{D_{1}} consisting of two components. Hence (−K_{D_{1}})_{D_{1}}^{2} = 7 or 8, which is a contradiction as (−K_{D_{1}})_{D_{1}}^{2} = 2(3 − m) with m ≥ 0. □

5.4 φ_{1} is of D3-type

In this subsection, we consider the case where φ_{1} : V → W_{1} ≅ P^{1} is of D3-type, i.e., all the fibers of φ_{1} are isomorphic to P^{2}. We shall prove the following:

Lemma 5.14 We may assume that one of the following (i) and (ii) occurs:

(i) D_{1} meets each fiber F ≅ P^{2} of φ_{1} in such a way that D_{1}|_{F} ∈ |O_{P^{2}}(1)|, and D_{2} is a fiber of φ_{1},

(ii) D_{1} and D_{2} meet each fiber F ≅ P^{2} of φ_{1} in such a way that D_{i}|_{F} ∈ |O_{P^{2}}(1)| for i = 1, 2.

Proof. Since (D_{1} + D_{2}) · l_{1} = t(−K_{V} · l_{1}) with 0 < t < 1 and D_{1} + D_{2} is ample, we have either t = \frac{1}{3}, (D_{1} + D_{2}) · l_{1} = 1 or t = \frac{2}{3}, (D_{1} + D_{2}) · l_{1} = 2. In the former case, we have (i) in the statement. Suppose that the latter case occurs. Since (D_{1} + D_{2}) · l_{1} = 2, we may assume that one of the following (a) and (b) occurs:

(a) D_{1} · l_{1} = 2 and D_{2} · l_{1} = 0,

(b) D_{1} · l_{1} = D_{2} · l_{1} = 1.

In the case (a), the Picard group Pic(X) contains 2-torsion elements. This is a contradiction because the coordinate ring of X is factorial. Hence we may assume that (b) occurs. This completes the proof. □

In [M-M81, Table 2], there are two smooth Fano 3-folds with B_{2} = 2 having extremal rays of D3-type, i.e., No. 33: V = P(O_{P^{1}}^{⊕2} ⊕ O_{P^{1}}(1)) and No. 34: V = P^{1} × P^{2}.

First of all, we consider the case (i) in Lemma 5.14.

Lemma 5.15 Suppose that the case (i) in Lemma 5.14 occurs. Then X ≅ A^{3}.

Proof. We consider the restriction φ_{1}|_{X} of φ_{1} onto X = V − (D_{1} ∪ D_{2}). Since D_{2} is a fiber of φ_{1} and D_{1} meets every fiber F ≅ P^{2} of φ_{1} in such a way that D_{1}|_{F} ∈ |O_{P^{2}}(1)|, φ_{1}|_{X} is an A^{2}-bundle structure over the affine line W_{1} − φ_{1}(D_{2}) ≅ A^{1}. Hence X ≅ A^{1} × A^{2} ≅ A^{3}. □

The embeddings in the previous lemma correspond to Type XVII with x ≥ 1 and Type XX in Table 1.

Next we consider the case (ii) in Lemma 5.14, i.e., D_{1} and D_{2} meet each fiber F ≅ P^{2} of φ_{1} in such a way that D_{i}|_{F} ∈ |O_{P^{2}}(1)| for i = 1, 2. Hence a general fiber of the restriction φ_{1}|_{X} : X → W_{1} ≅ P^{1} of φ_{1} onto X = V − (D_{1} ∪ D_{2}) is isomorphic to C^{∗} × A^{1}. Put Δ := {y ∈ W_{1} ; D_{1} ∩ F_{y} = D_{2} ∩ F_{y}}, where F_{y} := φ_{1}^{−1}(y) ≅ P^{2} for y ∈ W_{1} ≅ P^{1}. Note that, for a point y in Δ, the fiber F_{y}^{∘} := (φ_{1}|_{X})^{−1}(y) is isomorphic to the affine plane A^{2}. We prove the following:

Lemma 5.16 (1) ♯Δ = 1,

(2) φ_{1}|_{D_{i}} : D_{i} → W_{1} ≅ P^{1} gives rise to a P^{1}-bundle structure for i = 1, 2,

(3) D_{1} ∩ D_{2} consists of two curves. The one is a cross-section of φ_{1}|_{D_{i}} : D_{i} → W_{1} and the other is a fiber of φ_{1}|_{D_{i}} over the point in Δ,

(4) X ≅ A^{3}.

Proof. Note that F_{y}^{∘} := F_{y} ∩ X is isomorphic to A^{2} for y ∈ Δ. We shall calculate χ_{top}(X) associated with the fibration φ_{1}|_{X} : X → W_{1} ≅ P^{1}, namely, we have:

1 = χ_{top}(X) = χ_{top}(P^{1} − Δ)χ_{top}(A^{1} × C^{∗}) + χ_{top}(Δ)χ_{top}(A^{2}) = ♯Δ.

Hence ♯Δ = 1. This proves (1). The assertion (2) is clear. Since ♯Δ = 1, the assertion (3) is also easy to see. We shall verify (4) in the subsequent. We may assume that Δ = {∞} and F_{∞}^{∘} ≅ A^{2}. Put X′ := X − F_{∞}^{∘}. Then φ_{1}|_{X′} : X′ → A^{1} yields a trivial (A^{1} × C^{∗})-bundle structure over A^{1}, so X′ ≅ C^{∗} × A^{2}. Since the coordinate ring A := Γ(O_{X}) is factorial, there exists a prime element a ∈ A such that div(a)_{X} = F_{∞}^{∘}, hence A[a^{−1}] = C[x, y, t, t^{−1}]. Since A is a finitely generated C-algebra, the locally nilpotent derivations \frac{∂}{∂x} and \frac{∂}{∂y} on A[a^{−1}] can be extended to those δ_{x} := a^{N}(\frac{∂}{∂x}) and δ_{y} := a^{N}(\frac{∂}{∂y}) on A by multiplying a^{N} for a sufficiently large N >> 0. Put A_{0} := Ker(δ_{x}) ∩ Ker(δ_{y}), which is a finitely generated C-algebra of dimension one (cf. [Miy78, Chap. I]). Since A is factorial with A^{∗} = C^{∗} and A_{0} is an inert sub-algebra of A, it follows that A_{0} is also factorial with A_{0}^{∗} = C^{∗}. Hence A_{0} = C^{[1]}, a polynomial ring in one variable over C. Let f : X → Z := Spec (A_{0}) ≅ A^{1} be a morphism associated with the canonical inclusion A_{0} ↪ A. By the construction, the restriction f|_{X′} of f onto X′ = X − F_{∞}^{∘} coincides with the canonical projection p : X′ ≅ C^{∗} × A^{2} → Z′ := C^{∗}. The fiber f^{−1}(z) over the point z ∈ Z′ coincides with p^{−1}(z) ≅ A^{2}. On the other hand, the fiber of f over the point Z − Z′ is equal to F_{∞}^{∘} ≅ A^{2}. Thus all the fibers of f are isomorphic to the affine plane A^{2}. Since Pic(Z) = 0, we know that f is a trivial A^{2}-bundle structure over Z ≅ A^{1}. Therefore we have X ≅ Z × A^{2} ≅ A^{3} as desired. □

As seen in the previous lemma, the intersection D_{1} ∩ D_{2} of boundary components consists of two curves, say C_{1} and C_{2}. We may assume that C_{1} (resp. C_{2}) is a section (resp. a fiber) of the P^{1}-bundle structure φ_{1}|_{D_{i}} : D_{i} → W_{1} ≅ P^{1} (i = 1, 2). Since D_{1} and D_{2} satisfy D_{1} · l_{1} = D_{2} · l_{1} = 1 and Pic(V) ≅ Z[D_{1}] ⊕ Z[D_{2}] (cf. Lemma 2.1), we may write D_{1} ∼ H_{2} + xH_{1} and D_{2} ∼ H_{2} + yH_{1} with x, y ∈ Z such that x − y = 1. We shall see, conversely, that if D_{1} and D_{2} on V satisfy these conditions, then the complement X = V − (D_{1} ∪ D_{2}) is isomorphic to the affine 3-space A^{3}.

Lemma 5.17 Assume that two irreducible components D_{1} and D_{2} on V of No. 33 or No. 34 satisfy the following two conditions:

(i) D_{1} ∼ H_{2} + xH_{1} and D_{2} ∼ H_{2} + yH_{1} with x, y ∈ Z such that x − y = 1;

(ii) The intersection D_{1} ∩ D_{2} consists of two curves, say C_{1} and C_{2}, and C_{1} (resp. C_{2}) is a section (resp. a fiber) of the P^{1}-bundle structure φ_{1}|_{D_{i}} : D_{i} → W_{1} ≅ P^{1} (i = 1, 2).

Then the complement X = V − (D_{1} ∪ D_{2}) is isomorphic to the affine 3-space A^{3}.

Proof. It is easy to see that D_{1} + D_{2} is ample and the coordinate ring A := Γ(O_{X}) is factorial by (i). The restriction φ_{1}|_{X} : X → W_{1} ≅ P^{1} yields a (A^{1} × C^{∗})-fibration. Exactly one fiber, say F_{∞}^{∘}, of φ_{1}|_{X} is isomorphic to A^{2} in consideration of (ii). Let a ∈ A be a prime element defining F_{∞}^{∘} ⊂ X. (Note that A is factorial.) Since X − F_{∞}^{∘} ≅ C^{∗} × A^{2}, we have A[a^{−1}] = C[x, y, t, t^{−1}]. Then, by the same argument as in the proof of Lemma 5.16, we can deduce that X ≅ A^{3}. □

The embeddings in Lemma 5.17 is of Types XVIII and XXI in Table 1. Summarizing the arguments we prove Theorem 1.3.

Type  V  D_{1} ∪ D_{2}  X

I:(No.14) [E1−D1]

V : a blow-up of V_{5} along an elliptic curve which is an intersection of two members of | − \frac{1}{2} K_{V_{5}}| D_{1} = Exc(φ_{1}), D_{2} : a proper transform of H_{5}^{0} containing a center of φ_{1}

A^{3}

II:(No.20) [E1−C1]

V : a blow-up of V_{5} along a twisted cubic D_{1} = Exc(φ_{1}), D_{2} : a proper transform of H_{5}^{0} containg a center φ_{1}

A^{3}

III:(No.22) [E1−E1]

V : a blow-up of V_{5} along a conic D_{1} = Exc(φ_{1}), D_{2} : a proper transform of H_{5}^{0} containg a center of φ_{1}

A^{3}

IV :(No.23(b)) [E1−E4]

V : a blow-up of Q^{3} along a curve which is an intersection of Q_{0}^{2} ≅ \overline{D_{2}} ∈ |O_{Q}(1)| and a member of |O_{Q}(2)| D_{1} = Exc(φ_{1}), D_{2} : a proper transform of \overline{D_{2}}

A^{3}

V :(No.24) [C2−C1]

V : a divisor on P^{2} × P^{2} of bi-degree (2, 1)

D_{1} : a birational section of φ_{1}, D_{2} ∼ H_{1}, s.t. H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2}

A^{3}

V I:(No.26) [E1−E1] V : a blow-up of V_{5} along a line D_{1} = Exc(φ_{1}), D_{2} : a proper transform of H_{5}^{0} or H_{5}^{∞} containing a center of φ_{1}

A^{3}

V II:(No.27) [C2−E1]

V : a blow-up of P^{3} along a twisted cubic D_{1} : a birational section of φ_{1}, D_{2} ∼ H_{1}, s.t. H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2}

A^{3}

V III:(No.28) [E1−E5]

V : a blow-up of P^{3} along a plane cubic D_{1} = Exc(φ_{1}), D_{2} : a proper transform of a hyperplane containing a center of φ_{1}

A^{3}

IX:(No.29) [E1−D2]

V : a blow-up of Q^{3} along a conic D_{1} = Exc(φ_{1}), D_{2} : a proper transform of Q_{0}^{2} ∈ |O_{Q}(1)| containing a center of φ_{1}

A^{3}

X:(No.30) [E1−E2]

V : a blow-up of P^{3} along a conic D_{1} = Exc(φ_{1}), D_{2} : a proper transform of a hyperplane containing a center of φ_{1}

A^{3}

XI:(No.30) [E2−E1]

V : a blow-up of Q^{3} at a point p D_{1} = Exc(φ_{1}), D_{2} : a proper transform of Q_{0}^{2} ∈ |O_{Q}(1)| containing p as a smooth point

A^{3}

XII:(No.30) [E2−E1]

V : a blow-up of Q^{3} at a point p D_{1} : a proper transform of a hyperplane section Q_{0}^{2} ≅ \overline{D_{1}} not containing p, D_{2} : a proper transform of \overline{D_{2}} ⊂ Q^{3} containing p as a smooth point s.t. S := \overline{D_{2}} ∩ (Q^{3} − \overline{D_{1}}) is acyclic

(∗)

XIII:(No.30) [E2−E1]

V : a blow-up of Q^{3} at a point p D_{1} : a proper transform of a hyperplane section P^{1} × P^{1} ≅ \overline{D_{1}} not containing p, D_{2} : a proper transform of \overline{D_{2}} ⊂ Q^{3} containing p as a smooth point s.t. S := \overline{D_{2}} ∩ (Q^{3} − \overline{D_{1}}) satisfies H_{0}(S) = H_{1}(S) ≅ Z and H_{i}(S) = 0 (∀i > 1)

(∗)

XIV :(No.31) [E1−C2]

V : a blow-up of Q^{3} along a line D_{1} = Exc(φ_{1}), D_{2} : a proper transform of Q_{0}^{2} ∈ |O_{Q}(1)| containing a center of φ_{1}

A^{3}

XV :(No.31) [C2−E1]

V : a blow-up of Q^{3} along a line D_{1} : a birational section of φ_{1}, D_{2} ∼ H_{1}, s.t. H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2} A^{3}

Type  V  D_{1} ∪ D_{2}  X
XV I:(No.32) [C2−C2]

V = P(Θ_{P^{2}}) : a divisor on
P^{2} × P^{2} of bi-degree (1, 1)

D_{1} : a birational section of φ_{1}, D_{2} ∼ H_{1}, s.t. H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2}

A^{3}

XV II:(No.33) [D3−E1]  V = P(O_{P^{1}}^{⊕2} ⊕ O_{P^{1}}(1))  \begin{matrix}D_{1} ∼ H_{2} + xH_{1} (x = −1, x ≥ 1), \\ D_{2} ∼ H_{1} \end{matrix}  A^{3}
XV III:(No.33) [D3−E1]

V = P(O_{P^{1}}^{⊕2} ⊕ O_{P^{1}}(1))

D_{1} and D_{2} satisfy the conditions (i), (ii) in Lemma 5.17

A^{3}

XIX:(No.34) [C2−D3] V = P^{1} × P^{2}

D_{1} : a birational section of φ_{1}, D_{2} ∼ H_{1}, s.t. H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2}

A^{3}

XX:(No.34) [D3−C2]  V = P^{1} × P^{2}  \begin{matrix}D_{1} ∼ H_{2} + xH_{1} (x ≥ 0), \\ D_{2} ∼ H_{1} \end{matrix}  A^{3}
XXI:(No.34) [D3−C2]

V = P^{1} × P^{2}

D_{1} and D_{2} satisfy the conditions (i), (ii) in Lemma 5.17

A^{3}

XXII:(No.35) [E2−C2]

V : a blow-up of P^{3} at a point p D_{1} : a proper transform of a hyperplane \overline{D_{1}} ⊂ P^{3} not containing p, D_{2} : a proper transform of a hypersurface \overline{D_{2}} ⊂ P^{3} containing p as a smooth point s.t. S := \overline{D_{2}} ∩ (P^{3} − \overline{D_{1}}) is acyclic

(∗)

XXIII:(No.35) [C2−E2] V = P(O_{P^{2}} ⊕ O_{P^{2}}(1))

D_{1} : a birational section of φ_{1}, D_{2} ∼ H_{1}, s.t. H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2}

A^{3}

XXIV :(No.36) [C2−E5] V = P(O_{P^{2}} ⊕ O_{P^{2}}(2))

D_{1} : a birational section of φ_{1}, D_{2} ∼ H_{1}, s.t. H^{2}(D_{1} ∪ D_{2}) ≅ Z^{⊕2}

A^{3}

Table 1.

Acknowledgements The author would like to express his gratitude to Professor Masayoshi Miyanishi for his valuable advice and warm encouragement during the preparation of the present article. He would like to express his gratitude to Professors Mikio Furushima and Tomoaki Ohta for their very helpful advice and consistent warm encouragement. 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. The author was supported by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists (JSPS).

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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