SINGULAR FANO COMPACTIFICATIONS OF ℂ^{3} (I)

Mikio Furushima Department of Mathematics, Kumamoto University and Max-Planck-Institut für Mathematik, Bonn*

Abstract. In this paper we study the singular Fano compactifications of ℂ^{3} with small Gorenstein singularities and determine the structure of such a compactification in the case of the Fano index r ≥ 2.

Notation N_{Y |X} : normal bundle of Y in X c_{1}(F) : first Chern class of F h^{i}(L) := dim_{ℂ} H^{i}(·; L) Bs |L| : base locus of the linear system |L| b_{i}(X) := dim_{ℝ} H_{i}(X; ℝ) ρ(X) : Picard number of X mult_{A} X : multiplicity of X at a general point of A K_{X} : canonical divisor of X ∼ : linear equivalence ≅ : isomorphism 𝔽_{n} : Hirzebruch surface of degree n ℚ^{n} : smooth quadric hypersurface in ℙ^{n+1} ℚ_{0}^{2} : quadric cone in ℙ^{3} ℚ_{0}^{3} : quadric hypersurface in ℙ^{4} with an isolated singular point V_{5} : smooth Fano threefold of index two and degree 5 in ℙ^{6} (see [16]) V_{5}^{∗}: singular Fano threefold of index two and degree 5 in ℙ^{6} with

small Gorenstein singularities (see[3],[20]) V_{22} : smooth Fano threefold of index one and degree 22 in ℙ^{13} (see [17])

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 14J45 , 32J05.

V_{22}^{∗}: singular Fano threefold of index one and degree 22 in ℙ^{13} with small Gorenstein singularities

Introduction. Let X be a compact complex manifold and Y a closed subvariety og X. We call the pair (X, Y) the analytic compactification of ℂ^{n} if X − Y is biholomorphic to ℂ^{n}.Then Hirzebruch proposed the following problem.

Problem ([15]). Determine all smooth analytic compactifications (X, Y) of ℂ^{n} with the second Betti number b_{2}(X) = 1.

For n ≤ 2, we have the following:

(1) (X, Y) ≅ (ℙ^{1}, ∞) if n = 1

(2) (X, Y) ≅ (ℙ^{2}, L) if n = 2, where L is a line on ℙ^{2} ([32]).

The problem is still open for n ≥ 4. In the case of n = 3, under the assumption that X is projective, we obtain the following:

Theorem I ([6],[7],[12],[27],[28],[29]). Let (X, Y) be a smooth projective compactification of ℂ^{3} with the second Betti number b_{2}(X) = 1. Then Y is an ample divisor and −K_{X} ∼ rY (r ∈ ℕ, 1 ≤ r ≤ 4), that is, X is a smooth Fano threefold with ρ(X) = 1. Moreover,

(1) r = 4 ⟹ (X, Y) ≅ (ℙ^{3}, ℙ^{2}).

(2) r = 3 ⟹ (X, Y) ≅ (ℚ^{3}, ℚ_{0}^{2}).

(3) r = 2 ⟹ (X, Y) ≅ (V_{5}, H_{5}^{0}) or (V_{5}, H_{5}^{∞}).

(4) r = 1 ⟹ (X, Y) ≅ (V_{22}, H_{22}^{0}) or (V_{22}, H_{22}^{∞}),

Notation:

· H_{5}^{0} (resp. H_{5}^{∞}) is a normal (resp. non-normal) hyperplane section of V_{5} such that Sing H_{5}^{0} = {a rational double point of A_{4} type} (resp. Sing H_{5}^{∞} = ℓ (a line) with the normal bundle N_{ℓ|V_{5}} ≅ O_{ℓ}(−1) ⊕ O_{ℓ}(1). · H_{22}^{0} (resp.H_{22}^{∞}) is a non-normal hyperplane section such that Sing H_{22}^{0} = Sing H_{22}^{∞} = L (a line) with the normal bundle N_{L|V_{22}} ≅ O_{L}(−2) ⊕ O_{L}(1), in particular, mult_{L} H_{22}^{0} = 2 and mult_{L} H_{22}^{∞} = 3.

On the other hand, in the case where X is non-projective, Peternell-Schneider obtain the following:

Theorem II ([9],[27],[28],[29]). Let (X, Y) be a smooth analytic compactification of ℂ^{3} with b_{2}(X) = 1. Asuume that X is non-projective. Then

(i) X is Moishezon.

(ii) Y is a non- projective non-normal irreducible divisor.

(iii) −K_{X} ∼ rY (r = 1, 2). Moreover assume that Y is nef. Then there is a small birational contraction Φ : X ⟶ V^{∗} of X onto a Fano threefold V^{∗} with small Gorenstein singularities such that

(a) the exceptional set of Φ consists of finitely many smooth rational curves C_{i} supported in Y, and the normal bundle N_{C_{i}|X} of C_{i} in X is isomorphic to O_{ℙ^{1}}(−1) ⊕ O_{ℙ^{1}}(−1), O_{ℙ^{1}}(−2) ⊕ O_{ℙ^{1}} or O_{ℙ^{1}}(−3) ⊕ O_{ℙ^{1}}(1).

(b) Δ := Φ_{∗}Y ∈ Pic V^{∗} is ample and −K_{V^{∗}} ∼ rΔ, in particular, Pic V^{∗} ≅ ℤO_{V^{∗}}(Δ).

(c) V^{∗} − Δ ≅ ℂ^{3} with b_{2}(V^{∗}) = 1.

To determine the structure of (X, Y) in the case where Y is nef, one needs the study of the Fano compactifications of ℂ^{3} with small Gorenstein singularities. For this we propose the following: Problem A. Determine all the singular compactifications (V^{∗}, Δ) of ℂ^{3} such that

(i) V^{∗} is a Fano threefold with (non-empty) small Gorenstein singularities, (ii) b_{2}(V^{∗}) = 1, and (iii) Δ ∈ |O_{V^{∗}}(1)|.

Let (V^{∗}, Δ) be as above. Then we have K_{V^{∗}} ∼ −rH (0 < r ∈ ℤ) for H ∈ |O_{V^{∗}}(1)|. The integer r is called the Fano-index of V^{∗}. If r ≥ 4, then V^{∗} ≅ ℙ^{3} (in fact, r = 4). Since V^{∗} has singularities, this case can be excluded, that is, we have only to consider the case of 1 ≤ r ≤ 3. We remark that the condition (ii) b_{2}(V^{∗}) = 1 does not necessarily imply b_{4}(V^{∗}) = 1 even if V^{∗} has mild singularities. In fact, there is an example such that b_{2}(V^{∗}) = 1 and b_{4}(V^{∗}) = 2. In particular, the fourth Betti number b_{4}(V^{∗}) is equal to the number of irreducible components of the boundary divisor Δ. Thus assume that b_{4}(V^{∗}) = 1. Then, taking a small resolution X := V̂^{∗} of V^{∗}, one has b_{2}(X) = b_{4}(X) = 1, which is the case in the later half of Theorem II.

On Problem A, the following is obtained.

Theorem III ([9],[10],[11]). Assume that b_{4}(V^{∗}) = 1 and Δ is non-normal. Then 1 ≤ r ≤ 2 and

(1) If r = 2 (that is, K_{V^{∗}} ∼ −2Δ), then (V^{∗}, Δ) ≅ (V_{4}^{∗}, Δ_{4}), where V_{4}^{∗} ⊂ ℙ^{5} is a Fano threefold of degree 4 with index two, which is a complete intersection of two quadrics in ℙ^{5}, and Δ_{4} ∈ |O_{V_{4}^{∗}}(1)| is a non-normal cone over a nodal quartic curve in ℙ^{3}. In particular, Sing V_{4}^{∗} consists of a small Gorenstein singularity {p} and C = Φ^{−1}(p) is a smooth rational curve with the normal bundle N_{C|X} ≅ O_{ℙ^{1}}(−2) ⊕ O_{ℙ^{1}} for a small resolution Φ : X ⟶ V^{∗}.

(2) If r = 1, then there exists a Fano threefold V_{18}^{∗} ⊂ ℙ^{11} of degree 18 (that is, the genus g = \frac{1}{2}(−K_{V_{18}^{∗}})^{3} + 1 = 10) and a non-normal hyperplane section Δ_{18} of V_{18}^{∗} such that

(i) b_{2}(V_{18}^{∗}) = b_{4}(V_{18}^{∗}) = 1.

(ii) V_{18}^{∗} a small Gorenstein singularity p ∈ Δ_{18}.

(iii) V_{18}^{∗} − Δ_{18} ≅ ℂ^{3}. Moreover let Φ : X ⟶ V_{18}^{∗} be a small resolution. Then C = Φ^{−1}(p) is a smooth rational curve with N_{C|V_{18}^{∗}} ≅ O_{ℙ^{1}}(−2) ⊕ O_{ℙ^{1}}

Now in this paper we shall consider only the case of r ≥ 2. The result obtained is the following:

Main Theorem. Assume that r ≥ 2. Then

(I) If r = 3, then (V^{∗}, Δ) ≅ (ℚ_{0}^{3}, Δ_{2}), where Δ_{2} ∼ Δ_{2}^{1} + Δ_{2}^{2} (as a Weil divisor) is a hyperplane section consisting of two planes such that Δ_{2}^{1} ∩ Δ_{2}^{2} is a generating line passing through the vertex of ℚ_{0}^{3}.

(II) If r = 2, then d = 4, 5. Moreover,

(i) d = 4 ⟹ (V^{∗}, Δ) ≅ (V_{4}^{∗}, Δ_{4}), and b_{2}(V_{4}^{∗}) = b_{4}(V_{4}^{∗}) = 1. (see Theorem III-(1)).

(ii) d = 5 ⟹ (V^{∗}, Δ) ≅ (V_{5}^{∗}, Δ_{5}), where V_{5}^{∗} has a small hypersurface singularity of A_{1}-type,and b_{2}(V_{5}^{∗}) = 1 and b_{4}(V_{5}^{∗}) = 2. Δ_{5} = Δ_{5}^{1} + Δ_{5}^{2} is a reducible hyperplane section. Moreover, there is a compactification (V_{5}^{∗}, Δ_{5}) of ℂ^{3} such that Δ_{5}^{1} ≅ 𝔽_{1}, Δ_{5}^{2} is a normal rational surface with a rational

double point of A_{1}-type.

Conjecture. Assume that r = 1. Then b_{4}(V^{∗}) ≤ 2 and (1) V^{∗} ≅ V_{18}^{∗} if b_{4}(V^{∗}) = 1 (see Theorem III-(2)),

(2) V^{∗} ≅ V_{22}^{∗} if b_{4}(V^{∗}) = 2.

(3) V_{18}^{∗} and V_{22}^{∗} really exist as a compactification of ℂ^{3}.

Remark 1. The Main Theorem gives the complete determination of nonprojective Moishezon compactifications (X, Y) of ℂ^{3} with b_{2}(X) = 1 in the case where r = 2 and Y is nef. However in the case of r = 1 the structure of (V^{∗}, Δ) is not known well. We need a careful investigation on the non-normal projective surfaces with trivial dualizing sheaves. This will be discussed in the forthcoming paper.

Remark 2. In the case that Y is not-nef, it is shown that for each index r ≤ 2, there are infinitely many smooth non-projective Moishezon compactifications (X, Y) of ℂ^{3} with b_{2}(X) = 1 (cf.[10],[26]). The detail will be discussed elsewhere.

Remark 3. In Problem A, if we replace the condition (i) by

(i)’ V^{∗} has the Gorenstein terminal singularities, then there are many types of such a singular Fano compactification of ℂ^{3} as in Remark (1.2). In more general, the recent work due to H. Takagi ([35],[36], cf.[25]) enable us to study the ℚ-Fano compactifications of ℂ^{3} with ℚ-factorial terminal singularities.

Acknowlegement. The author would like to express his hearty thanks to the Max-Planck- Institute for Mathematik in Bonn, especially, Professor Dr. Friedrich Hirzebruch for the hospitality and encouragement.

1. Proof of Main Theorem.

Let (V^{∗}, Δ) be as in Problem A. If r = 3, then it is well known that V^{∗} is a quadric hypersurface in ℙ^{4}. Since V^{∗} has a small singularity, we have V^{∗} ≅ ℚ_{0}^{3}. The structure of ℚ_{0}^{3} as a compactification of ℂ^{3} will be discussed in the section 2.

Now assume that r = 2, that is ,−K_{V^{∗}} ∼ 2H. We put d := (H^{3}) > 0. Then we have Bs |H| = ∅ if d ≥ 2, and Bs |H| consists of one point on the smooth part of V^{∗} if d = 1 (cf.[33]).

1.1 Lemma. Δ ∈ |H| is a non-normal divisor.

Proof. Suppose that Δ is normal, hence it is irreducible. We take a small resolution Φ : X ⟶ V^{∗} with the exceptional set C := Exc(Φ) = ∪_{i≥1}C_{i}, where each C_{i} is a smooth rational curve. Let Y := Φ^{∗}Δ be the proper transform of Δ in X. Then we have a non-projective smooth Moishezon compactification (X, Y) of ℂ^{3} with Pic X ≅ H^{2}(X; ℤ) ≅ ℤc_{1}(O_{X}(Y)). By Peternell-Schneider[27], if Y is normal, then X is projective. Thus Y is non-normal. In particular, the non-normal locus C′ of Y is contained in the exceptional set C. In our case, one has H^{1}(C; ℤ) = 0, H^{2}(Δ; ℤ) ≅ ℤ, and H^{2}(Y; ℤ) ≅ ℤ. Then since Y − C ≅ Δ − Φ(C), we have the following exact sequence (cf.[2]):

⟶ 0 = H^{1}(C; ℤ) ⟶ H^{2}(Δ; ℤ) ⟶ H^{2}(Y; ℤ) ⟶^{r^{∗}} H^{2}(C; ℤ) ⟶ H^{3}(Δ; ℤ)

⟶ H^{3}(Y; ℤ) ⟶ 0

Since r^{∗} is the zero-map , we obtain

(1.1.1)  b_{3}(Δ) = b_{2}(C) + b_{3}(Y) > b_{3}(Y)

Let M ⟶ Δ be the minimal resolution. Then one has b_{3}(Δ) = b_{3}(M). Moreover let \overline{Y} ⟶^{ν} Y be the normalization with the non-normal locus C′ ⊂ C, and let \overline{C}′ be the proper transform of C′ in \overline{Y}. From the exact sequence (cf.[2]):

⟶ H^{1}(Y; ℤ) ⟶ H^{1}(\overline{Y}; ℤ) ⊕ H^{1}(C′; ℤ) ⟶ H^{1}(\overline{C}′; ℤ) ⟶ H^{2}(Y; ℤ)

⟶ H^{2}(\overline{Y}; ℤ) ⊕ H^{2}(C′; ℤ) ⟶ H^{2}(\overline{C}′; ℤ) ⟶ H^{3}(Y; ℤ) ⟶ H^{3}(\overline{Y}; ℤ) ⊕ 0 ⟶ 0

one has b_{3}(\overline{Y}) ≤ b_{3}(Y). Since b_{3}(\overline{Y}) = b_{3}(Ŷ) for any resolution Ŷ of \overline{Y}, one has b_{3}(\overline{Y}) = b_{3}(Ŷ) = b_{3}(M) = b_{3}(Δ). Thus by (1.1.1) we obtain b_{3}(Δ) > b_{3}(Y) ≥ b_{3}(\overline{Y}) = b_{3}(Δ). This is a contradiction. Therefore Δ is non-normal. □

1.2 Remark. There is a singular Fano threefold V^{∗} of b_{2}(V^{∗}) = 1 with hypersurface terminal (but not small !) singularities and a normal hyperplane section Δ such that V^{∗} − Δ ≅ ℂ^{3}.

(Example 1.2.1). Let V_{4}^{∗} be the complete intersection of two quadric hypersurfaces Q_{1} and Q_{2} in ℙ^{5}, where Q_{1} := {x_{1}^{2} = x_{0}x_{3} + x_{3}^{2} + x_{4}x_{5}}, Q_{2} := {x_{2}^{2} = x_{0}x_{3} + x_{1}x_{5}}, and set Δ_{4} := {x_{5} = 0}. Then V_{4}^{∗} is a singular Fano threefold of degree 4 with hypersurface terminal (not small) singularity . In particular, Δ_{4} is a normal cone over a elliptic curve of degree 4 in ℙ^{3}, and V_{4}^{∗} − Δ_{4} ≅ ℂ^{3}, b_{2}(V_{4}^{∗}) = b_{4}(V_{4}^{∗}) = 1.

(Example 1.2.2). Let us consider a singular cubic threefold V_{3}^{∗} := {x_{0}^{2}x_{3} +x_{1}^{2}x_{2} + x_{2}^{3} + x_{4}^{3} = 0} in ℙ^{4} and set Δ_{3} := {x_{0} = 0}. Then we have V_{3}^{∗} − Δ_{3} ≅ ℂ^{3} and b_{2}(V_{4}^{∗}) = b_{4}(V_{4}^{∗}) = 1, in particular, Δ_{3} is a normal cone over a cubic curve in ℙ^{2}. In this case, the singularity of V_{3}^{∗} is terminal but not small.

(Example 1.2.3). Let B be a quartic surface in ℙ^{3} with the defining equation as follows;

B := {(x_{0} : x_{1} : x_{2} : x_{3}) ∈ ℙ^{3} | x_{0}x_{1}^{3} + x_{1}x_{2}^{3} + x_{2}x_{3}^{3} = 0}

Then the affine part B^{∗} := B ∩ {x_{3} ≠ 0} is isomorphic to ℂ^{2} and there is an algebraic automorphism θ ∈ Aut ℂ^{3} such that θ(B^{∗}) is a linear plane in ℂ^{3} (see [37]). Let φ : V_{2} ⟶ ℙ^{3} be the double cover of ℙ^{3} branched along B. We put A := φ^{∗}(x_{3} = 0). Then (V, A) is a singular Fano compactification of ℂ^{3} with index r = 2 and the degree 2, which is called a singular quartic double solid. In particulat A is a normal del Pezzo surface with exactly one rational double point p of E_{7}-type and (V, p) is a hypersurface terminal singularity of E_{7}-type but it is not small. Next we set A′ := φ^{∗}(x_{1} = 0). Then A′ is a non-normal del Pezzo surface whose normalization is isomorphic ℚ_{0}^{2} (cf.[1],[31]), and V − A′ ≅ ℂ^{3}. Of course V has no small Gorenstein singularities.

(Example 1.2.4). Let us consider the normal quartic hypersurfaces:

B_{1} := {x_{2}^{3}x_{3} + x_{0}^{2}x_{1}(x_{0} + x_{1}) + x_{2}x_{1}^{3} = 0}
B_{2} := {x_{2}^{3}x_{3} + x_{0}^{2}x_{1}^{2} + x_{2}(x_{1}^{3} + x_{0}^{3}) = 0}
B_{3} := {x_{2}^{3}x_{3} + x_{0}^{4} + x_{2}x_{1}^{3}) = 0}

Let φ : V_{2}^{i} ⟶ ℙ^{3} be the double cover branched along B_{i} and put A^{i} = φ^{∗}(x_{2} = 0). Then A^{1} (resp. A^{i}, i = 2, 3) is the irreducible non-normal del Pezzo surface whose normalization is ℚ_{0}^{2} (resp. reducible surface consisting of two plane ℙ^{2}), and (V_{2}^{i}, A^{i}) is a singular Fano compactification of ℂ^{3} with (not small) hypersurface terminal singularity with E_{6}-type if i = 1 and E_{7}-type if i = 2, 3 respectively.

1.3 Proposition. 4 ≤ d ≤ 5.

Proof. Let (V^{∗}, Δ) be as in Problem A. Let H be a fundamental (ample) divisor, that is, −K_{V^{∗}} ∼ 2H. It is well-known that the base point set Bs |H| = ∅ if d ≥ 2 and the degree d = (H^{3}) ≤ 9, in particular, H is very ample if d ≥ 3 (cf.[33]). By Iskovskih [16; (4.6),(4.7)]), if d = 9, then V^{∗} is the cone over ℙ^{2}, and if d = 8, then V^{∗} is the cone over ℙ^{1} × ℙ^{1} or 𝔽_{1}. On the other hand, by Fujita [5;(2.6)], ifV_{d} := V^{∗} (d ≥ 5) has local complete intersection singularities, then V_{d} is not a cone. From this fact, the cases d = 8, 9 can be excluded. Moreover Fujita shows that if d ≥ 4 and V^{∗} is not a complete intersection (see [4;(2.9)]), then one has V^{∗} ≅ V_{6} or V_{5}. Thus we have only to consider the case d ≤ 6.

1.3.1 Claim. d ≠ 6.

In fact, take a general hyperplane section H. Then H is a smooth del Pezzo surface of degree d = 6, that is H is obtained from ℙ^{2} by blowing up 3-points in general position. There is a (−1)-curve ℓ in H, which is a line on V_{6} since (−K_{H} · ℓ) = (H · ℓ) = 1. Let us consider the projection π_{ℓ} : V_{6} −−→ ℙ^{5} from the line ℓ. Let σ : V6^{′} ⟶ V_{6} be the blowing up along ℓ with the exceptional divisor L′. One can see easily that L′ ≅ ℙ^{1} × ℙ^{1} or 𝔽_{2} and Bs |σ^{∗}H − L′| = ∅. Then one has a birational morphism ϕ := ϕ_{|σ^{∗}H−L′ |} : V6^{′} ⟶ W = ϕ(V6^{′}) ⊂ ℙ^{5} with π_{ℓ} = ϕ ∘ σ^{−1} and has deg W = 3 = codim W + 1. Then it is well-known that W ≅ ℙ^{2} × ℙ^{1} or the cone over 𝔽_{1} ⊂ ℙ^{4} (see [16]). If W ≅ ℙ^{2} × ℙ^{1}, then one has ρ(V6^{′}) ≥ 3 since ρ(W) = 2. But this is impossible since 1 = ρ(V_{6}) = ρ(V6^{′}) − 1 ≥ 2. If W is the cone over 𝔽_{1}, one sees that S := ϕ(L′) does not pass through the vertex w of the cone W. Thus we have a surjective morphism π : S ⟶ 𝔽_{1} via the point-projection W −−→ 𝔽_{1} of W from the vertex w. This is also impossible since S ≅ ℙ^{1} × ℙ^{1}, 𝔽_{2} or ℚ_{0}^{2}. Therefore we have d ≠ 6. □

1.3.2 Claim. d ≠ 3.

It is proved in [9] that d ≠ 3 if Δ is irreducible. We assume that Δ is reducible. Since Δ is a hyperplane section with H^{2} · Δ = 3, as a Wei divisor, one has Δ ∼ Δ_{1} + Δ_{2} + Δ_{3}, Δ ∼ Δ_{1} + 2Δ_{2}, or Δ ∼ Δ_{1} + Q, where Δ_{i} ≅ ℙ^{2} is a plane and Q is an irreducible quadric surface. Take a small singularity v of V^{∗} = V_{3} ⊂ ℙ^{4}. Let σ : B_{v}(ℙ^{4}) ⟶ ℙ^{4} be the blowing up with center v with the exceptional divisor σ^{−1}(v) = E ≅ ℙ^{3}. Let \overline{V}_{3}(∼ σ^{∗}V_{3}−2E) be the proper transform of V_{3} and set \overline{D} := O_{\overline{V}_{3}}(E) ∈ Pic \overline{V}_{3}. Then \overline{D} is an irreducible reduced quadric in E ≅ ℙ^{3}. An easy computation shows that (\overline{D})_{\overline{V}_{3}}^{3} = 2 and (σ^{∗}H − \overline{D})^{3} = 1. Sinve Bs |σ^{∗}H − \overline{D}| = ∅, we have a birational morphism φ := φ_{|σ^{∗}H−\overline{D}|} : \overline{V}_{3} ⟶ ℙ^{3} associated with the point projection V_{3} −−→ ℙ^{3} from the singularity v. Then Q_{0} := φ(\overline{D}) is a quadric hypersurface. By construction the exceptional set Exc(φ) is the proper transform of the set of lines on V_{3} passing through the point v ∈ Δ (v ∈ Δ_{i} for some i or v ∈ Q ). Take a line v ∈ ℓ_{v} ⊂ V_{3}. Since Δ has a singularity at v and (Δ · ℓ_{v}) = 1, the line ℓ_{v} is contained in Δ. Thus Exc(φ) is contained in the proper transform of Δ. Then we have V_{3} − Δ ≅ ℙ^{3} − (\overline{Q}_{0} ∪ φ(\overline{Δ}) ≅ ℂ̸^{3}. Thus d ≠ 3. □

1.3.3 Claim. d ≠ 2.

Let Φ : X ⟶ V^{∗} be the small resolution with the exceptional set C and Y the proper transform of Δ in X. Then (X, Y) is a smooth Moishezon compactification of ℂ^{3} with b_{2}(X) = b_{2}(Y). Let ν : \overline{Δ} ⟶ Δ be normalization with the nonnormal locus E and \overline{E} the pullback of E. We have the following exact sequence of cohomology over ℤ:

0 ⟶ H^{2}(V^{∗}; ℤ) ⟶^{Φ^{∗}} H^{2}(X; ℤ) ⟶^{r^{∗}} H^{2}(C; ℤ)

⟶ H^{3}(V^{∗}; ℤ) ⟶ H^{3}(X; ℤ) ⟶ 0 (1.3.3.1)

⟶ H^{i−1}(\overline{E}; ℤ) ⟶ H^{i}(Δ; ℤ) ⟶^{γ_{i}^{∗}} H^{i}(\overline{Δ}; ℤ) ⊕ H^{i}(E; ℤ)

(1.3.3.2)  ⟶ H^{i}(\overline{E}; ℤ) ⟶ H^{i+1}(Δ; ℤ) ⟶

(cf.[2]) If Δ is irreducible, Δ is non-normal del Pezzo surface of degree 2. According to the classification by Reid [32](cf.[1]), there are following possibilities of (\overline{Δ}, \overline{E}).

(i) (ℙ^{1} × ℙ^{1}, e_{0} + e_{1})

(ii) (ℙ^{1} × ℙ^{1}, e_{2})

(iii) (ℚ_{0}^{2}, g_{1} + g_{2})

(iv) (ℚ_{0}^{2}, g_{0})

(v) (ℚ_{0}^{2}, s_{∞}) where e_{0}, e_{1} (resp. e_{2}) are two different rulings (resp.a diagonal) of ℙ^{1} × ℙ^{1}, and g_{i} (resp. s_{∞}) is a generating line (resp. an infinite section) of ℚ_{0}^{2}. From (1.3.3.2), one has b_{3}(Δ) = 0 in the cases (i),(ii),(iv),(v) and b_{3}(Δ) = 1 in the case (iii). On the other hand, since Y is irreducible and (Y · C_{i}) = 0 for an irreducible component C_{i} of C, one has b_{2}(V^{∗}) = b_{2}(X) and b_{3}(V^{∗}) = b_{2}(C)+b_{3}(X). Since b_{3}(V^{∗}) = b_{3}(Δ), we have b_{2}(C) = b_{3}(V^{∗}) = 1 and b_{3}(X) = 0 by (1.3.3.1). This shows that V^{∗} has a Gorenstein singularity with the small resolution whose exceptional set is a smooth rational curve and (\overline{Δ}, \overline{E}) is of type (iii). Since the morphism φ = φ_{|H|} : V^{∗} ⟶ ℙ^{3} is of degree two, the ramification divisor B ⊂ ℙ^{3} is a normal quartic hypersurface since V^{∗} has isolated singularities. Then the restriction φ : Δ ⟶ ℙ^{2} is a surjective morphism of degree two. By lifting of the morphism φ to the normalization \overline{Δ}, one has the morphism ℚ_{0} ≅ \overline{Δ} ⟶^{φ} ℙ^{2} of degree two branched along a normal quartic surface. Thuis implies the canonical bundle K\overline{Δ} = −φ^{∗}ℓ, where ℓ is a line on L ≅ ℙ^{2}. This cannot happen, because K2 \overline{Δ} = 8. Therefore Δ is not irreducible.

Next, we shall consider the case that Δ is reducible. Since H^{3} = H^{2} · Δ = 2, one has Δ := Δ_{1} + Δ_{2} (a a Weil divisor), where Δ_{i} is a plane ℙ^{2}. In this case one has b_{3}(V^{∗}) = b_{3}(Δ) = 0 and b_{2}(X) = b_{4}(V^{∗}) = 2, hence b_{2}(C) = 1 by (1.3.3.1). Thus V has exactly one Gorenstein singularity with small resotution whose exceptional set is a smooth rational curve. Since Δ ∈ |O_{V^{∗}}(1)|, there is a linear plane L = ℙ^{2} and a morphism Δ ⟶ L of degree two branched along the quartic curve B_{0} := L ∩ B = 2Q, where Q is a smooth quadric curve on L. Then the normalization \overline{Δ} consists of two disjoint component \overline{δ}_{i} = ℙ^{2}(i = 1, 2) such that each \overline{Δ}_{i} is somorphic to L. Then we have Y = φ^{∗}Δ = Y_{1} + Y_{2}. Since (Y · C) = 0, one has (Y_{1} · C) + (Y_{2} · C) = 0. If (Y_{i} · C) = 0, then C is homologous to 0, that is, r^{∗} in (1.3.3.1) is zero-map, which is absurd since b_{2}(C) = 1. Thus we may assume that (Y_{1} · C) < 0, hence C ⊂ Y_{1}, in particular, C ⊄ Y_{2}. This yields b_{2}(Y) = 2, we have b_{2}(Y_{1}) = 2, b_{2}(Y_{2}) = 1 and b_{2}(Y_{1} ∩ Y_{2}) = 1. One sees that γ := Y_{1} ∩ Y_{2} is a smooth quadric curve in Y_{2} ≅ ℙ^{2}. p = Sing V^{∗} is on the smooth quadric curve γ ≅ Δ_{1} ∩ Δ_{2}. Since γ_{2} = Y_{1}|_{Y_{2}} (resp. γ_{1} = Y_{2}|_{Y_{1}}) is a smooth rational curve, the Cartier divisor Y_{1} has at worst A_{n}-singularity at the point q ∈ C − γ. In particular (C · γ_{1})_{Y_{1}} = 1. Now Y_{1} is obtained from ℙ^{2} by blowing up at n + 1-(infinitely near ) points on the smooth quadric Δ ∩ Δ_{2} whose exceptional curve is a linear tree of the (−2)-curve. Then we have

(1) Y_{1}^{2}Y_{2} = (Y_{1}|_{Y_{2}})^{2} = (γ_{2}^{2}) = 4

(2) Y^{1}Y_{2}^{2} = (Y_{2}|_{Y_{1}})^{2} = (γ_{1}^{2}) = 3 − n for n ≥ 1 (3) Y_{1}^{3} = (K_{Y_{1}} + 2γ_{1})^{2} = −n

(4) Y_{2}^{3} = (K_{Y_{2}} + 2γ_{2})^{2} = 1

Then we have 2 = (φ^{∗}Δ) = (Y_{1} + Y_{2})^{3} = 22 − 4n, that is n = 5. In particular, since C ⊂ Y_{1} passes through the rational double point of A_{5}-type, we have the normal bundle N_{C|X} ≅ O_{P^{1}} ⊕ O_{ℙ^{1}}(−2), hence the local defining equation of (V^{∗}, p) ≈ {xy + z^{2} + t^{2k}}(cf.[30]). Now since the morphism φ = φ_{|H|} : V^{∗} ⟶ ℙ^{3} is of degree two, the ramification divisor B ⊂ ℙ^{3} is a normal quartic hypersurface since V^{∗} has isolated singularities. Let L be a hyperplane of ℙ^{3} such that φ^{∗}L = Δ and set

D = B ∩ L, B_{0} = B − D. Since V^{∗} − Δ ≅ ℂ^{3} ⟶^{φ} ℂ^{3} ≅ ℙ^{3} − L is a double cover

of ℂ^{3} with smooth branched locus B_{0}, by the well-known Smith theorem [34], B_{0} is a ℤ_{2}-homology plane, hence it is ℚ-homology plane, that is, H_{i}(B_{0}; ℚ) = 0 for i > 0. Then it is known that B is a rational surface and the boundary D consists of b_{2}(B) rational curves with no cycles (cf.[23]). Thus B is a normal rational quartic hypersurface with exactly one minimally elliptic singularity p by Umezu [38]. Let μ : B̂ ⟶ B be the minimal resolution and Z be the fundamental cycle associated with the minimally elliptic singular point p. Then one has KB̂ ∼ −Z and Z^{2} = −1, −2, −3, in particular, Z^{2} = −3 if mult_{p} B = 3. and Z^{2} = −1, −2 if mult_{p} B = 2 (see [21]). Since V has exactly one small Gorenstein singularity of A_{1}- type at p := φ^{−1}(p), the local defining equation of (V, p) is analytically equivalent to w^{2} = xy + z^{2k} (see [19],[22], [30]). Let f(x, y, z) = 0 be the local defining equation of the germ (B, p) with p = (0, 0, 0), which is given by Laufer [21]. The defining equation of (V, p) can be also written as w^{2} = f(x, y.z). Comparing with these two defining equations, one sees that f(x, y, z) = 0 has the multiplicity two at the origin p = (0, 0, 0), that is, mult_{p} B = 2. For each defining function f(x, y, z) of (B, p) in the Table 1 and Table 2 in [21], we can easily check whether f(x, y, z) ≈ xy + z^{2k} (analytic equivalence) or not (see also Theorem 1.1 in Katz [19]). Then we conclude that there are no such analytic functions in the Table. □

1.3.4 Claim. d ≠ 1.

If Δ is irreducible, then the normalization \overline{Δ} is isomorphic to ℙ^{2} ([1],[31]). From (1.3.3.2), we have b_{3}(Δ) = 0, hence b_{3}(V_{1}^{∗}) = 0. Thus we obtain b_{2}(C) = b_{3}(V_{1}^{∗}) = 0 by (1.3.3.1), which is a contradiction since C is a curve. On the other hand since H^{3} = 1, Δ cannot be decomposed into two Weil divisors, thai is, Δ is irreducible.

This cannot occur in our case, because Δ is reducible. Thus we have d ≠ 1. This completes the proof. □

1.4 Remark. In the case of r = 2, we assume that Δ is irreducible. Then from the classification of non-normal del Pezzo surfaces [1],[31] and the exact sequences (1.3.3.1), (1.3.3.2), one can show that b_{3}(V^{∗}) = b_{2}(C) = 1, hence V^{∗} has exacily one small Gorenstein singularity of A_{1}-type,and Δ is a non-normal cone over a nodal rational curve of degree d in ℙ^{d−1} if d = 4, 5.

One can also prove that Δ is not irreducible if d = 5. In fact, assume that Δ is irreducible. Then we have b_{4}(V^{∗}) = 1. On the other hand, let H_{5} be a general hyperplane section of V_{5}^{∗}. Then there is a line ℓ on H_{5}. Taking a projection V_{5}^{∗} −−→ ℙ^{4} from the line ℓ, one sees that V_{5}^{∗} is birational equivalent to the singular quadric ℚ_{0}^{3} since V_{5}^{∗} has a small singularty. This implies that b_{4}(V_{5}^{∗}) = 2, which is a contradiction. Since b_{4}(X) = b_{2}(X) = 2 for the small resolution X ⟶^{φ} V_{5}^{∗}. Since b_{4}(X) = b_{4}(Y) = 2, Y := φ^{∗}Δ consists of two irreducible components, and so is Δ. In the case of d = 4, Δ is irreducible since b_{4}(V_{4}^{∗}) = 1 (cf.[9]).

1.5 Corollary. Assume that r = 2 and Δ is reducible. Then d = 5, that is, V^{∗} is a Fano threefold V_{5}^{∗} ⊂ ℙ^{6} of degree 5 and Δ := Δ_{5} is a hyperplane section consisting of two irreducible Weil divisors.

2. Singular Fano threefolds ℚ_{0}^{3} and V_{5}^{∗} as a compactification of ℂ^{3}.

Let ℙ^{4} be the 4-dimensional complex projective space with the homogeneous coodinate system (x_{0} : x_{1} : x_{2} : x_{3} : x_{4}) and ℚ_{0}^{3} a quadric cone defined by ℚ_{0}^{3} := {x_{0}x_{3} = x_{1}x_{2}} in ℙ^{4}. Then the singular point of ℚ_{0}^{3} is the vertex p := (0 : 0 : 0 : 0 :

1) ∈ ℙ^{4} of the cone. Let Q_{∞} be the hyperplane section defined by Q_{∞} := {x_{4} = 0}. Then Q_{∞} (≅ ℙ^{1} × ℙ^{1}) is a smooth quadric hypersurface in ℙ^{3}. Let Δ_{2} := {x_{0} = 0} be the hyperplane section of ℚ_{0}^{3}. Then we have Δ_{2} := Δ_{2}^{(1)} + Δ_{2}^{(2)}, where Δ_{2}^{(i)} := {x_{0} = x_{i} = 0} (i = 1, 2) is a smooth ℚ-Cartier divisor isomorphic to ℙ^{2}. We put g := Δ_{2}^{(1)} ∩Δ_{2}^{(2)} ≅ ℙ^{1} ( a generating line of ℚ_{0}^{3}). It is easy to see that ℚ_{0}^{3} −Δ_{2} ≅ ℂ^{3}. Let Φ : X ⟶ ℚ_{0}^{3} be a small resolution with the exceptional set C := Φ^{−1}(p) ≅ ℙ^{1}. Let α : B_{p}(ℙ^{4}) ⟶ ℙ^{4} be the blowing up of ℙ^{4} with the center p and \overline{ℚ_{0}^{3}} the proper transform of ℚ_{0}^{3}. Then the restriction α : \overline{ℚ_{0}^{3}} ⟶ ℚ_{0}^{3} is a resolution of the singularity p with exceptional set α^{−1}(p) = \overline{Q}_{0} ≅ ℙ^{1} × ℙ^{1}. Let β : B_{C}(X) ⟶ X be the blowing up of X with the center C. Then we have B_{C}(X) ≅ \overline{ℚ_{0}^{3}} and the birational morphism α is factorized as α = Φ ∘ β, that is, α : \overline{ℚ_{0}^{3}} ⟶^{β} X ⟶^{Φ} ℚ_{0}^{3} . In particular, one has the normal bundle N_{C|X} ≅ O_{ℙ^{1}}(−1) ⊕ O_{ℙ^{1}}(−1). From ℚ_{0}^{3} − Δ_{2} ≅ ℂ^{3}, one has b_{i}(ℚ_{0}^{3}) = b_{i}(Δ_{2}) for i > 0 (cf. [7],[27]). On the other hand, since b_{i}(Δ_{2}) = b_{i}(Δ_{2}^{(1)}) + b_{i}(Δ_{2}^{(2)}) − b_{i}(g), one has easily b_{1}(ℚ_{0}^{3}) = b_{3}(ℚ_{0}^{3}) = 0, b_{2}(ℚ_{0}^{3}) = 1 and b_{4}(ℚ_{0}^{3}) = 2. Thus we have 2.1 Proposition. (ℚ_{0}^{3}, Δ_{2}) is a singular Fano compactification of ℂ^{3} of index r = 3 with a small hypersurface singularity of A_{1}-type, in particular, b_{2}(ℚ_{0}^{3}) = 1 and b_{4}(ℚ_{0}^{3}) = 2.

Next we shall give a construction of a singular Fano threefold V_{5}^{∗} as a compactification of ℂ^{3}. Let (ℚ_{0}^{3}, Δ_{2}, Δ_{2}^{(i)}, Q_{∞}, g) be as above. Let us consider a twisted cubic curve γ : ℙ^{1} ⟶ Q_{∞} ⊂ ℙ^{3} defined by γ(u : v) = (u^{3} : u^{2}v : uv^{2} : v^{3}). We set s := {x_{0} = x_{1} = x_{4} = 0}, f := {x_{0} = x_{2} = x_{4} = 0} and γ := γ(ℙ^{1}) ⊂ Q_{∞}. One sees that s, f are two different rulings of Q_{∞} ≅ ℙ^{1}×ℙ^{1} with Pic Q_{∞} ≅ ℤf ⊕ℤs. Then we have a linear equivalence γ ∼ s + 2f. By construction one has Δ_{2}^{(2)} ∩ γ = f ∩ γ = q and Δ_{2}^{(1)} ∩γ = s∩γ = 2q, where q = (0 : 0 : 0 : 1 : 0) = s∩f ∩g. Let π : ℚ̂_{0}^{3} ⟶ ℚ_{0}^{3} be the blowing up of ℚ_{0}^{3} along γ with the exceptional set Ŝ := π^{−1}(γ). 2.2 Lemma. Ŝ ≅ 𝔽_{1} Proof. Let us consider the exact sequence:

0 ⟶ N_{γ|Q_{∞}} ⟶ N_{γ|ℚ_{0}^{3}} ⟶ N_{Q_{∞}|ℚ_{0}^{3}}|_{γ} ⟶ 0

By a theorem of Grothendieck, one has N_{γ|ℚ_{0}^{3}} ≅ O_{ℙ^{1}}(a) ⊕ O_{ℙ^{1}}(b) (a ≤ b, a, b ∈ ℤ). Since N_{γ|Q_{∞}} ≅ O_{ℙ^{1}}(4) and N_{Q_{∞}|ℚ_{0}^{3}}|_{γ} ≅ O_{ℙ^{1}}(3),one has N_{γ|ℚ_{0}^{3}} ≅ O_{ℙ^{1}}(3) ⊕ O_{ℙ^{1}}(4), that is, Ŝ ≅ 𝔽_{1}. □

On the other hand, one has Kℚ̂_{0}^{3} ∼ π^{∗}K_{ℚ_{0}^{3}} + Ŝ ∼ −3π^{∗}Q_{∞} + Ŝ ∼ −3Q̂_{∞} − 2Ŝ, where Q̂_{∞} ≅ Q_{∞} is the proper transform of Q_{∞} in ℚ̂_{0}^{3}. The adjunction formula implies that KQ̂_{∞} ∼ (Kℚ̂_{0}^{3} +Q̂_{∞})|Q̂_{∞} ∼ −2Q̂_{∞}|Q̂_{∞} −2ŝ−4f̂, where ŝ (resp. f̂) is the proper transfom of s (resp. f). Since KQ̂_{∞} ∼ −2ŝ − 2f̂, we obtain Q̂_{∞}|Q̂_{∞} ∼ −f̂. An easy computation shows 2.3 Lemma. (2.3.1) Q̂_{∞}^{3} = (Q̂_{∞}|Q̂_{∞} · Q̂_{∞}|Q̂_{∞})^{2} = (−f̂)^{2} = 0 (2.3.2) Ŝ · Q̂_{∞}^{2} = (Ŝ|Q̂_{∞} · Q̂_{∞}|Q̂_{∞}) = −1 (2.3.3) (Ŝ)^{2} · Q̂_{∞} = (Ŝ|Q̂_{∞})^{2} = 4 (2.3.4) Ŝ^{3} = −7 (2.3.5) (2π^{∗}Q_{∞} − Ŝ)^{3} = (Ŝ + 2Q̂_{∞})^{3} = 5

2.4 Lemma.

(1) h^{0}(Oℚ̂_{0}^{3}(Ŝ + 2Q̂_{∞})) = 7

(2) Bs |Ŝ + 2Q̂_{∞}| = ∅

Proof. (1). Let us consider the exact sequences:

(2.4.1)  0 ⟶ Oℚ̂_{0}^{3} ⟶ Oℚ̂_{0}^{3}(Ŝ) ⟶ OŜ(Ŝ) ⟶ 0
(2.4.2)  0 ⟶ Oℚ̂_{0}^{3}(Ŝ) ⟶ Oℚ̂_{0}^{3}(Ŝ + Q̂_{∞}) ⟶ OQ̂_{∞}(Ŝ + Q̂_{∞}) ⟶ 0
(2.4.3) 0 ⟶ Oℚ̂_{0}^{3}(Ŝ + Q̂_{∞}) ⟶ Oℚ̂_{0}^{3}(Ŝ + 2Q̂_{∞}) ⟶ OQ̂_{∞}(Ŝ + 2Q̂_{∞}) ⟶ 0

We put Ŝ|Ŝ ∼ aΣ + bF (a, b ∈ ℤ), where Σ (resp. F) is the negative section (resp. a fiber) of Ŝ ≅ 𝔽_{1}. Since (Ŝ · F) = −1, we have a = −1. By (2.3.4), −1 − 2b = −7, that is, b = 3. Thus we have Ŝ|Ŝ ∼ −Σ + 3F. It is easy to see that h^{1}(Oℚ̂_{0}^{3}) = 0, h^{0}(Oℚ̂_{0}^{3}(Ŝ)) = h^{0}(Oℚ̂_{0}^{3}) = 1, h^{1}(OŜ(Ŝ)) = h^{1}(OŜ(KŜ + Σ + 6F)) = 0 (Kodaira vanishing theorem). This yields h^{1}(Oℚ̂_{0}^{3}(Ŝ)) = 0 by (2.4.1). Moreover since OQ̂_{∞}(Ŝ+Q̂_{∞}) = OQ̂_{∞}(ŝ+f̂), one has h^{0}(OQ̂_{∞}(Ŝ+Q̂_{∞})) = 4 and h^{1}(OQ̂_{∞}(Ŝ+ Q̂_{∞}) = 0. Then we have h^{0}(Oℚ̂_{0}^{3}(Ŝ + Q̂_{∞})) = 5 by (2.3.2) since h^{0}(Oℚ̂_{0}^{3}(Ŝ)) = 1. Taking into consideration that h^{0}(OQ̂_{∞}(Ŝ + 2Q̂_{∞})) = h^{0}(OQ̂_{∞}(ŝ)) = 2, one sees that h^{0}(Oℚ̂_{0}^{3}(Ŝ + 2Q̂_{∞})) = 7 and there is a surjection

(2.4.4)  H^{0}(Oℚ̂_{0}^{3}(Ŝ + 2Q̂_{∞})) ↠ H^{0}(OQ̂_{∞}(Ŝ + 2Q̂_{∞})).

(2). Since Bs |OQ̂_{∞}(Ŝ + 2Q̂_{∞})| = Bs |OQ̂_{∞}(ŝ)| = ∅, one sees that Bs |Oℚ̂_{0}^{3}(Ŝ + 2Q̂_{∞})| = ∅ on Q̂_{∞} by (2.4.4). Next let Q(≅ ℙ^{1} × ℙ^{1}) be any smooth hyperplane section of ℚ_{0}^{3} such that p ∉ Q and set Q̂ = π^{∗}Q. Then Q̂ is a smooth del Pwzzo surface of degree K2 ; Q̂ = 5. Let us consider the following exact sequence

0 ⟶ Oℚ̂_{0}^{3}(π^{∗}Q − Ŝ) ⟶ Oℚ̂_{0}^{3}(2π^{∗}Q − Ŝ) ⟶ OQ̂(2π^{∗}Q − Ŝ) ⟶ 0

We have Oℚ̂_{0}^{3}(π^{∗}Q − Ŝ) ≅ Oℚ̂_{0}^{3}(Q̂_{∞}), Oℚ̂_{0}^{3}(2π^{∗}Q − Ŝ) ≅ Oℚ̂_{0}^{3}(2Q̂_{∞} + Ŝ) and OQ̂(2π^{∗}Q − Ŝ) ≅ OQ̂(−KQ̂). Since h^{1}(Oℚ̂_{0}^{3}(Q̂_{∞})) = 0, one has a surjection

H^{0}(ℚ̂_{0}^{3}; Oℚ̂_{0}^{3}(2π^{∗}Q − Ŝ) ↠ H^{0}(Q̂; OQ̂(2π^{∗}Q − Ŝ))

Since Bs | − KQ̂| = ∅, we have Bs |2π^{∗}Q − Ŝ| = ∅ on Q̂. This shows that Bs |2Q̂_{∞} + Ŝ| = ∅. □

Let Φ̂ : ℚ̂_{0}^{3} ⟶ ℙ^{6} be a morphism defined by the linear system |Ŝ + 2Q̂_{∞}|. We put V_{5}^{∗} := Φ̂(ℚ̂_{0}^{3}) ⊂ ℙ^{6}. Since (Ŝ + 2Q̂_{∞})^{3} = 5, one has deg V_{5}^{∗} = 5 in ℙ^{6}. Let us denote by Exc(Φ̂) the exceptional set of Φ̂.

2.5 Lemma. Exc(Φ̂) = Q̂_{∞}.

Proof. Since (Ŝ + 2Q̂_{∞})^{2} · Q̂_{∞} = 0, Exc(Φ) ⊃ Q̂_{∞}. Now we have Pic ℚ̂_{0}^{3} ≅ ℤOℚ̂_{0}^{3}(Q̂_{∞}) ⊕ Oℚ̂_{0}^{3}(Ŝ). Then (Ŝ + 2Q̂_{∞})^{2} · (aŜ + bQ̂_{∞}) = 0 for a, b ∈ ℤ if and only if a = 0. Therefore there are no exceptional divisor of Φ̂ other than Q̂_{∞}. Since (Ŝ · F) = −1, (Ŝ · Σ) = 4, (Q̂_{∞} · F) = 1, (Q̂_{∞} · Σ) = 1,one sees easily that (Ŝ + 2Q̂_{∞}) · (aΣ + bF) = 0 (b ≥ a ≥ 0) if and only if a + 2b = 0, that is, a = b = 0 . This shows that Ŝ contained no exceptional curve of Φ̂, therefore we have the claim. □

2.6 Lemma. V_{5}^{∗} is a Fano threefold with a hypersurface terminal singularity p := Φ̂(π^{−1}(p)) of A_{1}-type and smooth along the line E := Φ(Q̂_{∞}), in particular, Φ̂ : ℚ̂_{0}^{3} ⟶ V_{5}^{∗} is the blowing up of V_{5}^{∗} with the center E.

Proof. Let H be a hyperplane section of V_{5}^{∗}. Since Φ̂∗H ∼ Ŝ+2Q̂_{∞} and (Φ^{∗}H·f̂) = (H·E) = 1, one sees that E is a line on V_{5}^{∗}. Since Q̂_{∞}|Q̂_{∞} ∼ −f̂, V_{5}^{∗} is smooth along E. This shows that ℚ̂_{0}^{3} is the blowing up of V_{5}^{∗} with center E. We set S := Φ̂_{∗}Ŝ. Then we have K_{V_{5}^{∗}} ∼ −2S and Φ^{∗}S ∼ Ŝ + 2Q̂_{∞}. Since (Ŝ + 2Q̂_{∞})^{3} = 5, S is a non-normal hyperplane section of V_{5}^{∗} with the non-normal locus E,in particular, one sees that mult_{E} S = 2. This also shows that V^{∗} is a Fano threefold of index r = 2. Since Φ̂ is isomorphic near the vertex p of ℚ_{0}^{3}, V^{∗} has the same singularity at p := Φ̂(π^{−1}(p)). □

2.7 Remark. It is easy to see that the restriction ν := Φ̂|Ŝ : Ŝ ⟶ S is the normalization with ν^{−1}(E) = Σ. By construction ν|_{Σ} : Σ ⟶ E is a double covering. It is easy to verify that b_{2}(V_{5}^{∗}) = 1, b_{3}(V_{5}^{∗}) = 0, b_{4}(V_{5}^{∗}) = 2, in particular, Pic V_{5}^{∗} ≅ ℤO_{V_{5}^{∗}}(S). The birational morphism \overline{Φ} := Φ̂ ∘ π^{−1} : ℚ_{0}^{3} −−→ V_{5}^{∗} is given by the linear system |O_{ℚ_{0}^{3}}(2) − γ|, since Ŝ + 2Q̂_{∞} ∼ 2π^{∗}(Q_{∞}) − Ŝ. We set Δ_{5}^{(i)} := \overline{Φ}(Δ_{2}^{(i)}) for i = 1, 2 and Δ_{5} = Δ_{5}^{(1)} ∪ Δ_{5}^{(2)}. Then Δ_{5}^{(i)} is an effective ℚ-Cartier divisor on V_{5}^{∗}. By construction one sees that Δ_{5}^{(1)}(resp. Δ_{5}^{(2)}) is a smooth rational surface (resp. rational surface with a rational double point of A_{1}-type). In particular, Δ_{5} is a hyperplane section of V_{5}^{∗}. In fact, we obtain Φ̂^{∗}Δ_{5} ∼ Φ̂^{∗}S ∼ Ŝ + 2Q̂_{∞}.

2.8 Lemma. (V_{5}^{∗}, Δ_{5}) is a compactification of ℂ^{3} with b_{2}(V_{5}^{∗}) = 1.

Proof. We put γ^{(0)} := γ ∩ (ℚ_{0}^{3} − (Δ_{2}^{(1)} ∪ Δ_{2}^{(2)})) ⊂ Q_{∞}^{(0)} := Q_{∞} ∩ {x_{0} ≠ 0}. We put z_{i} := \frac{x_{i}}{x_{0}} for 1 ≤ i ≤ 4. Then one sees that W^{(0)} := ℚ_{0}^{3} − (Δ_{2}^{(1)} ∪ Δ_{2}^{(2)}) ≅ ℂ^{3}, Q_{∞}^{(0)} ≅ ℂ^{2} and γ^{0} ≅ ℂ. Then the defining equations of W^{(0)}, Q_{∞}^{(0)} and γ^{(0)} are given as follows:

(1)  W^{(0)} = {z_{3} = z_{1}z_{2}} ⊂ ℂ^{4}
(2)  Q_{∞}^{(0)} = {z_{3} = z_{1}z_{2}, z_{4} = 0}
(3)  γ^{(0)} = {z_{3} = z_{1}z_{2}, z_{2} = z_{1}^{2}, z_{4} = 0},

where (z_{1}, z_{2}, z_{3}, z_{4}) is a holomorphic coordinates of ℂ^{4}.

Let us consider the following polynomial automorphism α

θ_{1} : \{ \begin{matrix}w_{1} = z_{1} \\ w_{2} = z_{2} − z_{1}^{2} \\ w_{3} = z_{3} − z_{1}z_{2} \\ w_{4} = z_{4} \end{matrix}

Then we obtain

(1)  W^{∗} := α(W^{(0)}) = ℂ^{3} : (w_{1}, w_{2}, w_{3})
(2)  Q_{∞}^{∗} := α(Q_{0}^{(∞)}) = {(w_{1}, w_{2}, w_{3}) ∈ ℂ^{3}|w_{3} = 0}
(3)  γ^{∗} := α(γ^{(0)}) = {(w_{1}, w_{2}, w_{3}) ∈ ℂ^{3}|w_{2} = w_{3} = 0},

Let B_{γ^{∗}}(ℂ^{3}) be the blowing up of ℂ^{3} along γ^{∗} and Q̂_{∞}^{∗} be the proper transform of Q_{∞}^{∗}. Then it is easy to see that B_{γ^{∗}}(ℂ^{3}) − Q̂_{∞}^{∗} ≅ ℂ^{3}. Then we obtain that V_{5}^{∗} − (Δ_{5}^{(1)} ∪ Δ_{5}^{(2)}) ≅ ℚ̂_{0}^{3} − (Δ̂_{2}^{(1)} ∪ Δ̂_{2}^{(2)} ∪ Q̂_{∞}) ≅ B_{γ^{∗}}(ℂ^{3}) − Q̂_{∞}^{∗} ≅ ℂ^{3}, where Δ̂_{2}^{(i)} is the proper transform of Δ_{2}^{(i)} in ℚ̂_{0}^{3} (cf.[8]). Thus (V_{5}^{∗}, Δ_{5}) is a singular Fano compactification of ℂ^{3} of index two and b_{2}(V_{5}^{∗}) = 1. In particular, one has b_{4}(V_{5}^{∗}) = b_{4}(Δ_{5}) = b_{4}(Δ_{5}^{(1)}) + b_{4}(Δ_{5}^{(2)}) = 2 and b_{1}(V_{5}^{∗}) = b_{3}(V_{5}^{∗}) = 0. □

Therefore we have finally the following

2.9 Proposition. (V_{5}^{∗}, Δ_{5}) is a singular Fano compactification of ℂ^{3} of index r = 2 with a small hypersurface singularity of A_{1}-type, in particular, b_{2}(V_{5}^{∗}) = 1 and b_{4}(V_{5}^{∗}) = 2.

2.10 Remark Let V be a Fano threefold of degree 5 in ℙ^{6} with at worst hypersurface terminal singularities. Then V is smooth if b_{2}(V) = b_{4}(V) = 1 (cf.[3],[20]).

3. Fano threefold V_{22}^{∗} of degree 22 in ℙ^{13} with one small singularity of A_{1}-type.

Next we shall give a detailed structure of the non-normal del Pezzo surface S constructed in §2. Let us recall the normailization ν := Φ̂|Ŝ : Ŝ ⟶ S and the analytic inverse image ν^{−1}(E) = Σ. Then anti-dualizing sheaf ω_{S}^{−1} = O_{S}(S) is ample, that is, S is a non-normal del Pezzo surface with non-normal locus E.

(3.1) Lemma ([24; (3.35)]).

(i) h^{0}(O_{Σ}) = 1, h^{1}(O_{Σ}) = 0.

(ii) χ(OŜ) = 1, (ν^{∗}ω_{Y} · Σ) = −2.

(iii) (ω_{S} · E) = −1 and E is irreducible reduced, in particular, E ≅ ℙ^{1}.

(3.2) Lemma. h^{0}(ν^{∗}ω_{S}^{−1}) = h^{0}(ω_{S}^{−1}) + 1.

Proof. Let us consider an exact sequence (cf.[24; (3.34.2)]):

(3.2.1)  0 ⟶ O_{S} ⟶ ν_{∗}OŜ ⟶ ω_{S}^{−1} ⊗ ω_{E} ⟶ 0

By operating ⊗ω_{S}^{−1} on (1.2.1), we obtain

(3.2.2)  0 ⟶ ω_{S}^{−1} ⟶ ν_{∗}OŜ ⊗ ω_{S}^{−1} ⟶ ω_{S}^{−1} ⊗ ω_{E} ⟶ 0

By the projection formula and the Serre duality theorem, we have:

H^{0}(S; ν_{∗}OŜ ⊗ ω_{S}^{−1}) ≅ H^{0}(S; ν_{∗}OŜ(ν^{∗}ω_{S}^{−1}))

≅ H^{0}(Ŝ; ν^{∗}ω_{S}^{−1}) and

H^{0}(S; ω_{S}^{−1} ⊗ ω_{E}) ≅ H^{1}(E; O_{E} ⊗ ω_{S})
≅ H^{1}(E; O_{E}(−1))
≅ H^{0}(E; O_{E})
≅ H^{0}(ℙ^{1}; O_{ℙ^{1}})
≅ ℂ.

Since H^{1}(S; ω_{S}^{⊗−n}) = 0 for n > 0 by Goto-Mori-Reid (cf.[24], [31]) we have an exact sequence

(3.2.3)  0 ⟶ H^{0}(S; ω_{S}^{−1}) ⟶^{ν^{∗}} H^{0}(Ŝ; ν^{∗}ω_{S}^{−1}) ⟶ H^{0}(E; O_{E}) ⟶ 0

This proves the lemma. □

We set L := 2Q̂_{∞} + Ŝ. From the following exact sequence

0 ⟶ Oℚ̂_{0}^{3}(2Q̂_{∞}) ⟶^{ι} Oℚ̂_{0}^{3}(L) ⟶ OŜ(L) ⟶ 0

one obtain the following:

(3.2.4) 0 ⟶ ι^{∗}H^{0}(ℚ̂_{0}^{3}, Oℚ̂_{0}^{3}(L)) ⟶ H^{0}(Ŝ, OŜ(L)) ⟶ H^{1}(ℚ̂_{0}^{3}, Oℚ̂_{0}^{3}(2Q̂_{∞})) ⟶ 0

(3.3) Claim. H^{0}(Ŝ, OŜ(L)) = H^{0}(Ŝ, OŜ(Σ + 3F)) = H^{0}(Ŝ, ν^{∗}ω_{S}^{−1}) = ℂ^{7}.

In fact, one has L|Ŝ = (2Q̂_{∞} + Ŝ)|Ŝ ∼ Σ + 3F. On the other hand, since ν^{∗}ω_{S}^{−1} is ample on Ŝ ≅ 𝔽_{1} with deg ν^{∗}ω_{S}^{−1} = 5, one has ν^{∗}ω_{S}^{−1} ∼ OŜ(Σ + 3F). □

(3.4) Claim. H^{1}(ℚ̂_{0}^{3}, Oℚ̂_{0}^{3}(2Q̂_{∞})) ≅ H^{0}(E, O_{E}).

In fact, consider the following exact sequences

(3.4.1)  0 ⟶ Oℚ̂_{0}^{3} ⟶ Oℚ̂_{0}^{3}(Q̂_{∞}) ⟶ OQ̂_{∞}(−f̂) ⟶ 0

(3.4.2)  0 ⟶ Oℚ̂_{0}^{3}(Q̂_{∞}) ⟶ Oℚ̂_{0}^{3}(2Q̂_{∞}) ⟶ OQ̂_{∞}(−2f̂) ⟶ 0

(3.4.3)  0 ⟶ OQ̂_{∞}(−f̂) ⟶ OQ̂_{∞} ⟶ Of̂ ⟶ 0

(3.4.4)  0 ⟶ OQ̂_{∞}(−2f̂) ⟶ OQ̂_{∞}(−f̂) ⟶ Of̂ ⟶ 0

Since Q̂_{∞} ≅ ℙ^{1} × ℙ^{1}, one has

H^{1}(Q̂_{∞}; OQ̂_{∞}(−2f̂)) ≅ H^{0}(f̂; Of̂) ≅^{(Φ|_{f})_{∗}} H^{0}(E; O_{E}).

by (2.12.4). On the other hand, one has easily H^{1}(ℚ̂_{0}^{3}, Oℚ̂_{0}^{3}(Q̂_{∞})) = 0. This yields H^{1}(ℚ̂_{0}^{3}, Oℚ̂_{0}^{3}(2Q̂_{∞}) ≅ H^{1}(Q̂_{∞}; OQ̂_{∞}(−2f̂)) by (3.4.2). Hence we have the claim □

By (3.2.3) ,(3.3) and (3.4) , we obtain the following (3.5) Lemma. ι^{∗}H^{0}(ℚ̂_{0}^{3}, Oℚ̂_{0}^{3}(L)) ≅ ν^{∗}H^{0}(S; ω_{S}^{−1}) ≅ ℂ^{6}

Finally we shall give a basis {ν^{∗}h_{0}, ν^{∗}h_{1}, …, ν^{∗}h_{5}} of ν^{∗}H^{0}(S; ω_{S}^{−1}) ≅ ℂ^{6} explicitly below. Let U_{i} := {(u_{i}, v_{i}) ∈ ℂ^{2}} (i = 0, 1, 2, 3) be coordinates covering of Ŝ ≅ 𝔽_{1} with

\{ \begin{matrix}u_{1} = u_{0}^{−1} \\ v_{1} = u_{1}v_{0}, \end{matrix} \{ \begin{matrix}v_{2} = v_{1}^{−1} \\ u_{2} = u_{1}, \end{matrix} \{ \begin{matrix}u_{3} = u_{2}^{−1} \\ v_{3} = u_{2}^{−1}v_{2}, \end{matrix} \{ \begin{matrix}v_{0} = v_{3}^{−1} \\ u_{0} = u_{3}, \end{matrix}

on U_{i} ∩ U_{i+1} ≅ ℂ × ℂ^{∗}, (i = 0, 1, 2) and U_{3} ∩ U_{0} ≅ ℂ × ℂ^{∗} respectively. Let F, F_{∞} (resp. Σ, Σ_{∞}) be the fibers (resp. sections) defined as follows:

(1) F ∩ U_{i} = {u_{i} = 0} (i = 0, 3) and F_{∞} ∩ U_{i} = {u_{i} = 0} (i = 1, 2).

(2) Σ ∩ U_{i} = {v_{i} = 0} (i = 0, 1) and Σ_{∞} ∩ U_{i} = {v_{i} = 0} (i = 2, 3).

Taking into account that ν^{∗}H^{0}(S; ω_{S}^{−1}) ⊂ H^{0}(Ŝ : O(Σ + 3F)), we may assume that ν^{∗}h_{i}’s are written as follows:

\{ \begin{matrix}ν^{∗}h_{0} & = u_{0}^{3}v_{0} = v_{1} = 1 = u_{3}^{3} \\ ν^{∗}h_{1} & = u_{0}^{2}v_{0} = u_{1}v_{1} = u_{2} = u_{3}^{2} \\ ν^{∗}h_{2} & = u_{0}v_{0} = u_{1}^{2}v_{1} = u_{2}^{2} = u_{3} \\ ν^{∗}h_{3} & = v_{0} = u_{1}^{3}v_{1} = u_{2}^{3} = 1 \\ ν^{∗}h_{4} & = u_{0}^{2} = 1 = v_{2} = u_{3}^{2}v_{3} \\ ν^{∗}h_{5} & = 1 = u_{1}^{2} = u_{2}^{2}v_{2} = v_{3} \end{matrix}

Then ν = Φ̂|Ŝ : Ŝ ⟶ S = V_{5}^{∗} ∩ {z_{6} = 0} ⊂ ℙ^{5} is given by (ν^{∗}h_{0} : ν^{∗}h_{0} : ν^{∗}h_{1} : …: ν^{∗}h_{5} : 0), where (z_{0} : z_{1} : …: z_{5} : z_{6}) be the homogeneous coordinates of ℙ^{6}. We set x_{i} := \frac{z_{i}}{z_{5}} (0 ≤ i ≤ 4). Then we have the local defining equation

(3.6.1)  S^{(0)} : = S ∩ {z_{5} ≠ 0} ≅ {(x_{2}, x_{3}.x_{4}) ∈ ℂ^{3}|x_{2}^{2} = x_{3}^{2}x_{4}}
(3.6.2.)  E^{(0)} : = E ∩ {z_{5} ≠ 0} ≅ {(x_{2}, x_{3}.x_{4}) ∈ ℂ^{3}|x_{2} = x_{3} = 0}

To construct V_{22}^{∗} we need the local defining equations of S^{(0)} and E^{(0)} in (3.6.1), (3.6.2). First we take a smooth rational curve Γ ⊂ S ⊂ V_{5}^{∗} ⊂ ℙ^{6} of degree 5 satisfying Γ∩E = {t^{∗}} (double points), where ν^{−1}(t^{∗})∩U_{0} = {(−1, 0), (1, 0)}. Such a rational curve Γ exists. In fact, consider a smooth rational curve Γ̂ ∼ Σ + 3F in Ŝ given as follows:

Γ̂ ∩ U_{0} = {v_{0} = (u_{0} + 1)^{2}}
Γ̂ ∩ U_{1} = {u_{1}^{3}v_{1} = (u_{1} + 1)^{2}}
Γ̂ ∩ U_{2} = {u_{2}^{3} = v_{2}(u_{2} + 1)^{2}}
Γ̂ ∩ U_{3} = {1 = v_{3}(u_{3} + 1)^{2}}

Then the curve Γ := ν(Γ̂) ⊂ S is desired. Then the local defining equation of Γ is given by Γ^{(0)} := Γ ∩ U_{0} ≅ \{x_{2} = \frac{1}{2}(x_{2} − x_{3} − 1)x_{3}, x_{4} = \frac{1}{4}(x_{3} − x_{4} − 1)^{2}\} in the affine part U_{0} ≅ ℂ^{2}.

After the following coordinates transformation on ℂ^{3},

θ_{2} : \{ \begin{matrix}x = x_{2} − \frac{1}{2}(x_{3} − x_{4} − 1)x_{3} \\ y = \frac{1}{2}(x_{3} − x_{4} − 1) \\ z = x_{4} − \frac{1}{4}(x_{3} − x_{4} − 1)^{2} \end{matrix}

one has the new local defining equations:

S^{(0)} ={(x, y, z) ∈ ℂ^{3}|x^{2} + 2xy{z + (y + 1)^{2}} − {z + (y + 1)^{2}}z = 0}
(#) E^{(0)} ={(x, y, z) ∈ ℂ^{3}|x = z + (y + 1)^{2} = 0}
Γ^{(0)} ={(x, y, z) ∈ ℂ^{3}|x = z = 0}
t^{∗} =(0, −1, 0)

Now there are two lines F_{1}, F_{2} in S passing through t^{∗}, in fact, these are written as follows:

(F_{1}) : \{ \begin{matrix}x = 2(1 − y^{2}) \\ y = 1 − y^{2} , \end{matrix} (F_{2}) : \{ \begin{matrix}x = −2(1 + y)^{2} \\ y = 1 − y^{2} \end{matrix}

Then one has F_{1} ∩ Γ = {(0, 1, 0), (0, −1, 0) = t^{∗}} and F_{2} ∩ Γ = {t^{∗}}.

Let σ : \overline{V_{5}}^{∗} ⟶ V_{5}^{∗} be the blowing up with center Γ and \overline{D} = Exc(σ) the exceptional set. Let \overline{S} (resp. \overline{E}) be the proper transform of S (resp. E) in \overline{V} _{5}^{∗} and let \overline{F}_{i} (i = 1, 2) be the proper transform of F_{i} in \overline{S}.

(3.7) Lemma. Sing \overline{S} = \overline{E} ∪ \{\overline{t}^{∗}\}, where the isolated singular point \overline{t}^{∗} ∈ \overline{S} is the rational double point of A_{1}-type.

Proof. Look at the local defining equation (#), and let σ_{0} : B_{Γ^{(0)}}(ℂ^{3}) ⟶ ℂ^{3} be the blowing up of ℂ^{3} along Γ^{(0)} = {x = z = 0}. Let {W_{i} ≅ ℂ_{(u_{i},v_{i},w_{i})}^{3}} be an open covering of B_{Γ^{(0)}}(ℂ^{3}) = W_{1} ∪ W_{2} satisfying the relation:

\{ \begin{matrix}x = u_{1} = u_{2}w_{2} \\ y = v_{1} = v_{2} \\ z = u_{1}w_{1} = u_{2} \\ w_{1}w_{2} = 1 on W_{1} ∩ W_{2} \end{matrix}

An easy computation shows that the local defining equations of \overline{S} and \overline{E} are given by

(\overline{S}) : \{ \begin{matrix}u_{1} + 2v_{1}[u_{1}w_{1} + (v_{1} + 1)^{2}] = w_{1}[u_{1}w_{1} + (v_{1} + 1)^{2}]^{2} on W_{1} \\ [u_{2} + (v_{2} + 1)^{2} − v_{2}w_{2}]^{2} = w_{2}^{2}(u_{2} + v_{2}^{2}) on W_{2} \end{matrix}

(\overline{E}) : w_{2} = u_{2} + (v_{2} + 1)^{2} = 0 on W_{2}

It is easy to check that

Sing \overline{S} ∩ W_{2} = \{w_{2} = u_{2} + (v_{2} + 1)^{2} = 0\} ∪ \{(u_{2}, v_{2}, w_{2}) = (0, −1, 2) := \overline{t^{∗}}\}.

Next we shall show that \overline{S} has an (isolated) rational double point of A_{1}-type. Recall the local defining equation of \overline{S} on W_{2}. The following coordinates transformation

\{ \begin{matrix}u = \frac{u_{2} + (v_{2} + 1)^{2}}{w_{2}} \\ v = v_{2} \\ w = w_{2} \end{matrix}

yields

w^{2}(x − y)^{2} = w^{2}(uw − 2v − 1).

Thus the defining equation of \overline{S} near the point (u, v, w) = (0, −1, 2) is given by

(u − v − 1)^{2} = u(w − 2)

This shows that \overline{S} has a rational double point of A_{1}-type at \overline{t^{∗}} ∈ W_{1}. □

(3.8) Corollary. \overline{S}|\overline{D} = \overline{Γ} + 2\overline{G}_{0}, where \overline{Γ} is the closure of \overline{S} ∩ σ^{−1}(Γ − \{t^{∗}\}) in \overline{D}, and \overline{G}_{0} is a fiber of ℙ^{1}-bundle \overline{D} ⟶ Γ over a smooth rational curve Γ.

Proof. We put u_{2} = 0 in the above defining equation (\overline{S}) in W_{2}. □

(3.9) Lemma. On the surface \overline{S} one has (1) \overline{F}_{1} ∩ \overline{G}_{0} = \{\overline{t^{∗}}\}, in particular (\overline{F}_{1} · \overline{G}_{0}) = \frac{1}{2}

(2) (\overline{F}_{1} · \overline{Γ}) = 1,

(3) \overline{E} ∩ \overline{F}_{1} = ∅.

(4) \overline{t^{∗}} ∉ \overline{F}_{2}.

Proof. The defining equation of the pull back F_{i}^{∗} := σ^{∗}F_{i} of F_{i} is given by

(F_{1}^{∗}) : \{ \begin{matrix}u_{1} = 2(1 − v_{1}^{2}) \\ w_{1} = \frac{1}{2} \end{matrix} ⋃ \{ \begin{matrix}u_{1} = 0 \\ v_{1} = −1 \end{matrix}
(F_{2}^{∗}) : \{ \begin{matrix}u_{2} = 2(1 − v_{1}^{2}) \\ w_{2}(1 − v_{2}) + 2(1 + v_{2}) = 0 \end{matrix} ⋃ \{ \begin{matrix}u_{2} = 0 \\ v_{2} = −1, \end{matrix}

where

\overline{G}_{0} = \{u_{i} = v_{i} + 1 = 0\},
\overline{F}_{1} = \{u_{1} − 2(1 − v_{1}^{2}) = w_{1} − \frac{1}{2} = 0\} = \{w_{2} − 2 = u_{2} − 1 + v_{2}^{2} = 0\},
\overline{F}_{2} = \{u_{1} + 2(v_{1} + 1)^{2} = 2w_{1}(v_{1} + 1) + v_{1} − 1 = 0\}
= {u_{2} − 1 + v_{2}^{2} = w_{2}(1 − v_{2}) + 2(1 + v_{2}) = 0}.

Since \overline{F}_{1} and \overline{G}_{0} intersects at the rational double point \overline{t^{∗}} ∈ \overline{S} − \overline{E} of A_{1}-type, one has (\overline{F}_{1} · \overline{G}_{0}) = \frac{1}{2} via the minimal resolution. The rest follows directly from these local equations. □

(3.10) Lemma. \overline{D} ≅ 𝔽_{2}, N_{\overline{Γ}|\overline{V} _{5}^{∗}} ≅ O_{ℙ^{1}}(4) ⊕ O_{ℙ^{1}}(6).

Proof. Since \overline{S} ∼ σ^{∗}S − \overline{D}, one has K\overline{V} _{5}^{∗} = σ^{∗}K_{V_{5}^{∗}} + \overline{D} ∼ −2\overline{S} − \overline{D}. By (3.7) one has \overline{S}|\overline{D} ∼ 2\overline{G}_{0} + \overline{Γ}. Thus K\overline{D} = (K\overline{V} _{5}^{∗} + \overline{D})|\overline{D} = −2\overline{S}|\overline{D} = −2(\overline{Γ} + 2\overline{G}_{0}). Since \overline{Γ} is a smooth rational curve in \overline{D}, the adjunction formula yields −2 = (K\overline{D} + \overline{Γ}) · \overline{Γ} = −\overline{Γ}^{2} −4. Hence \overline{Γ}^{2} = −2, that is, \overline{D} ≅ 𝔽_{2}. On the other hand since c_{1}(N_{\overline{Γ}|\overline{V} _{5}^{∗}}) = 10, one has the normal bundle N_{\overline{Γ}|V_{5}^{∗}} = O_{ℙ^{1}}(4) ⊕ O_{ℙ^{1}}(6). □

(3.11) Lemma.

(1) (\overline{D} · \overline{E}) = 2

(2) (\overline{S} · \overline{E}) = −1

(3) (−K\overline{V} _{5}^{∗} · \overline{E}) = (−K\overline{V} _{5}^{∗} · \overline{F}_{1}) = 0

(4) (\overline{D})^{3} = −8

(5) (\overline{S})^{3} = −2

(6) (\overline{S})^{2} · \overline{D} = 2

(7) \overline{S} · (\overline{D})^{2} = 3

(8) (−K\overline{V} _{5}^{∗})^{3} = 18

Proof. (1): Since Γ is tangent to E at the point t^{∗}, we have the claim. (2): (\overline{S} ·\overline{E}) = (σ^{∗}S − \overline{D}) · \overline{E} = 1 − 2 = −1. (3):(−K\overline{V} _{5}^{∗} · \overline{E}) = (2σ^{∗}S − \overline{D}) · \overline{E} = 2 − (\overline{D} · \overline{E}) = 0. (−K\overline{V} _{5}^{∗} ·\overline{F}_{1}) = (2σ^{∗}S−\overline{D})·\overline{F}_{1} = 2−(\overline{D}·\overline{F}_{1}) = 2−(\overline{D}|\overline{S} ·\overline{F}_{1})\overline{S} = 2−(\overline{Γ}+2\overline{G}_{0}·\overline{F}_{1})\overline{S} = 2 − (1 + 2(\frac{1}{2}) = 0. (4): c_{1}(N_{\overline{Γ}|V_{5}^{∗}}) = (−K_{V_{5}^{∗}} · Γ) − 2 = (2S · Γ) − 2 = 10 − 2 = 8. (5):(\overline{S})^{3} = (σ^{∗}S − \overline{D})^{3} = (σ^{∗}S)^{3} + (σ^{∗}D · \overline{D} · \overline{D}) − (\overline{D})^{3} = 5 − 15 + 8 = −2. (6):(\overline{S})^{2} · \overline{D} = (σ^{∗}S − \overline{D})^{2} · \overline{D} = −2(σ^{∗}S · \overline{D} · \overline{D}) + (\overline{D})^{3} = 10 − 8 = 2.(6): \overline{S} · (\overline{D})^{2} = (σ^{∗}S − \overline{D}) · \overline{D}^{2} = −5 + 8 = 3. (7):(−K\overline{V} _{5}^{∗})^{3} = (2\overline{S} + \overline{D})^{3} = 18. □

(3.12) Proposition. Bs |2σ^{∗}S − \overline{D}| = ∅.

Let λ : Ṽ_{5}^{∗} ⟶ \overline{V} _{5}^{∗} be the blowing up with center \overline{E} and λ^{−1}(\overline{E}) := L̃ the exceptional divisor. Let S̃ (resp. D̃) be the proper transform of \overline{S} (resp. \overline{D}).

(3.12.1) Sublemma. S̃ has a unique isolated singularity t̃^{∗} = λ^{−1}(\overline{t}^{∗}), which is rational double point of A_{1}-type.

Proof. Sing \overline{S} = \overline{E} ∪ \{\overline{t^{∗}}\}. Thus we have only to show that Sing S̃ = \{t̃^{∗}\}. In fact, in the chart W_{2} ≅ ℂ_{(u_{2},v_{2},w_{2})}^{3}, we take a coordinates transformation

\{ \begin{matrix}u_{2} := u_{2} + (v_{2} + 1)^{2} \\ v_{2} := v_{2} \\ w_{2} := w_{2}, \end{matrix}

then the local defining equations of \overline{S} and \overline{E} change as follows: (\overline{S}) : (u_{2}−v_{2}w_{2})^{2} = w_{2}^{2}(u_{2} − 2v_{2} − 1), (\overline{E}) : u_{2} = w_{2} = 0. Thus the proper transform S̃ of \overline{S} is given by (u_{2} − v_{2})^{2} = u_{2}w_{2} − v_{2} − 1. This has only a rational double point of A_{1}-type as a singularity. We note that S̃ is smooth near Σ̃. □

(3.12.2) Sublemma. Bs |2σ^{∗}S − \overline{D}| = ∅ on \overline{S}.

Proof. Let μ : M ⟶ S̃ be the minimal resolution with Exc(μ) = B, where B is a smooth rational curve with B^{2} = −2. We set λ := λ|S̃ : S̃ ⟶ \overline{S} and τ := λ ∘ μ. We put L := (2σ^{∗}S − \overline{D})|\overline{S}. Then M is a ruled surface over ℙ^{1}, ψ : M ⟶ ℙ^{1}, with only one singular fiber ψ^{−1}(0) = F̂_{1}^{+} ∪ B ∪ Ĝ_{0}^{+} with Pic M ≅ ℤ[Ê^{+}] ⊕ ℤ[Ĝ_{0}^{+}] ⊕ ℤ[F̂_{1}^{+}] ⊕ ℤ[B], where Θ̂^{+}, Ĝ_{0}^{+}, Γ̂^{+} , Ê^{+} be the proper transforms of \overline{Θ},\overline{G}_{0},\overline{F}_{1}, \overline{Γ}, \overline{E} respectively. One has easily

(F̂_{1}^{+})^{2} = (Ĝ_{0}^{+})^{2} = −1, (B)^{2} = −2, (B · F̂_{1}^{+}) = (B · Ĝ_{0}^{+}) = 1, (F̂_{1}^{+} · Ĝ_{0}^{+}) = 0.

Then one has L ∼ 2\overline{Θ} − (2\overline{G}_{0} + \overline{Γ}), where \overline{Θ} = σ^{∗}S|\overline{S}, which can be assumed to be irreducible. Then we have τ^{∗}L ∼ 2Θ̂^{+} − 2Ĝ_{0}^{+} − B − Γ̂^{+}. Since Θ̂^{+} ∼ Γ̂^{+} ∼ Ê^{+} + 5Ĝ_{0}^{+} + 4B + 3F̂_{1}^{+}, we have τ^{∗}L ∼ Ê^{+} + 3Ĝ_{0}^{+} + 3B + 3F̂_{1}^{+}, and (τ^{∗}L · Ê^{+}) = (τ^{∗}L · F̂_{1}^{+}) = (τ^{∗}L · B) = 0, (τ^{∗}L · Ĝ_{0})^{+} = 1. Thus we conclude that Bs |τ^{∗}L| on M. This implies that Bs |L| = ∅ on \overline{S}. Next let us consider the following exact sequence:

0 ⟶ O(σ^{∗}S) ⟶ O(2σ^{∗}S − \overline{D}) ⟶ O\overline{S}(2σ^{∗}S − \overline{D}) ⟶ 0

Since H^{1}(\overline{V} _{5}^{∗}; O(σ^{∗}S)) = 0, one has a surjection

H^{0}(\overline{V} _{5}^{∗}; O(2σ^{∗}S − \overline{D}) ⟶ H^{0}(\overline{S}; O(2σ^{∗}S − \overline{D})) ⟶ 0

Since Bs |L| = Bs |O\overline{S}(2σ^{∗}S − \overline{D})| = ∅, we have the claim. □

(3.12.3) Sublemma. −K\overline{V} _{5}^{∗} ∼ 2σ^{∗}S − \overline{D} is nef and big on \overline{V} _{5}^{∗}.

Proof. Assume that there is a curve R on \overline{V} _{5}^{∗} with (2σ^{∗}S − \overline{D} · R) < 0. Since 2σ^{∗}S − \overline{D} = σ^{∗} + \overline{S}, one has (\overline{S} · R) < 0, that is R ⊂ \overline{S}. By (3.12.2), L is semiample on \overline{S}, hence 0 ≤ (L · R)\overline{S} = (2σ^{∗}S − \overline{D}) · R. This is a contradiction. thus 2σ^{∗}S − \overline{D} is nef. On the other hand one has (2σ^{∗}S − \overline{D})^{3} = 18 by (3.11)-(8). Thus we have the claim. □

(Proof of (3.12)).

First we have h^{0}(O\overline{V} _{5}^{∗}(2)⊗I_{Γ}) ≥ 17 by the counting method. In fact, take general 11 points of Γ. There any smooth quadric hypersurface in ℙ^{6} passing through these 11 points always contain Γ since Γ is of degree 5 in ℙ^{6}. Thus a general member \overline{Z} of the linear system |2σ^{∗}S − \overline{D}| is a smooth K-3 surfaces. We may assume that |2σ^{∗}S − \overline{D}| has no fixed component on \overline{Z}. Thus the nef big divisor (2σ^{∗}S − \overline{D})|\overline{Z} has no base points. on \overline{Z}. From the exact sequence:

0 ⟶ O ⟶ O(2σ^{∗}S − \overline{D}) ⟶ O\overline{Z}(2σ^{∗}S − \overline{D}) ⟶ 0

we have an surjection

H^{0}(\overline{V} _{5}^{∗}; O(2σ^{∗}S − \overline{D})) ⟶ H^{0}(\overline{Z}; O\overline{Z}(2σ^{∗}S − \overline{D})) ⟶ 0,

since H^{1}(\overline{V} _{5}^{∗}; O\overline{V} _{5}^{∗}) = 0. Thus |2σ^{∗}S − \overline{D}| has no base points on \overline{Z}. Since \overline{Z} is general, we have finally the base-point freeness of |2σ^{∗}S − \overline{D}|. □

(3.13)Lemma.

(1) N_{\overline{E}|\overline{V} _{5}^{∗}} ≅ O_{ℙ^{1}}(−1) ⊕ O_{ℙ^{1}}(−1)

(2) N_{\overline{F}_{1}|\overline{V} _{5}^{∗}} ≅ O_{ℙ^{1}}(−2) ⊕ O_{ℙ^{1}}

Proof. (1): We have S̃ ∼ λ^{∗}\overline{S} −2L̃ , D̃ ∼ λ^{∗}\overline{D} and KṼ_{5}^{∗} ∼ λ^{∗}K\overline{V} _{5}^{∗} +L̃ ∼ −2S̃−3L̃− D̃.We set Σ̃ := S̃|L̃ ⊂ S̃ ∩ L̃, which is a smooth rational curve. Since c_{1}(N_{\overline{E}|\overline{V} _{5}^{∗}}) = −2+(−K\overline{V} _{5}^{∗} ·\overline{E}) = −2. one has (L̃)^{3} = −2. Thus (Σ̃)2 ; L̃ = (S̃|L̃)^{2} = (λ^{∗}\overline{S}−2L̃)^{2}·L̃ = −4λ^{∗}\overline{S} ·(L̃)^{2} +4(L̃)^{3} = −4(\overline{S} ·\overline{E})(T̃ ·L̃)+8 = 4, where T̃ is a general fiber of the ℙ^{1}- bundle L̃ ⟶ \overline{E}. On the other hand since KṼ_{5}^{∗} +L̃ = −2S̃−2L̃−D̃ = −(λ^{∗}σ^{∗}S+S̃), one has KL̃ = −λ^{∗}σ^{∗}S|L̃ − S̃|L̃ = −(σ^{∗}S · \overline{E})T̃ − Σ̃ = −T̃ − Σ̃. In particular (T̃ · Σ̃) = −T̃ · (KL̃ + T̃) = 2.

Let Δ̃_{0} be the negative section of the ℙ^{1}-bundle L̃ ⟶ \overline{E}. Then we can write as Σ̃ ∼ 2Δ̃_{0} + aT̃ for some 0 < a ∈ ℤ. Since (Σ̃)2 ; L̃ = 4, one has Δ̃_{0}^{2} + a = 1. From the relation (Σ̃ · Δ̃_{0}) = 2Δ̃_{0}^{2} + a ≥ 0 and 0 ≥ Δ̃_{0}^{2} = 1 − a, one has a = 1, 2, hence

L̃ ≅ \{ \begin{matrix}ℙ^{1} × ℙ^{1} if a = 1 \\ 𝔽_{1}  if a = 2 \end{matrix}

Since c_{1}(N_{\overline{E}|\overline{V} _{5}^{∗}}) ≅ O_{ℙ^{1}}(m) ⊕ O_{ℙ^{1}}(n), (m ≤ n ∈ ℤ) with m + n = −2, L̃ ≅ 𝔽_{n−m} ≅ 𝔽_{0} := ℙ^{1} × ℙ^{1}, hence m = n = −1 and a = 1. This proves the assertion (1).

(2): Now \overline{F}_{1} passes through the rational double point (of A_{1}-type) \overline{t}^{∗} ∈ \overline{S} − \overline{E} ⊂ \overline{V} _{5}^{∗} − \{p^{∗}\} and (−K\overline{V} _{5}^{∗} · \overline{F}_{1}) = 0. Thus by [14], we have the assertion. □

(3.14) Corollary. Let F̃_{1} (resp. G̃_{0}, Γ̃) be the proper transform of \overline{F}_{1} (resp. \overline{G}_{0}, \overline{Γ}). Then

(1) (Σ̃^{2})S̃ = −3

(2) (F̃_{1}^{2})S̃ = (G̃_{0}^{2})S̃ = −\frac{1}{2}

(3) (F̃_{1} · G̃_{0})S̃ = \frac{1}{2}

(4) D̃|S̃ ∼ Γ̃ + 2G̃_{0}

Proof. We have (Σ̃^{2})S̃ = (L̃|S̃·L̃|S̃)S̃ = (L̃^{2}·S̃) = λ^{∗}\overline{S}−2L̃(L̃)^{2} = −(\overline{S}·\overline{E})−2(L̃)^{3}) = 1 − 4 = −3. This shows (1). By construction there is a ℙ^{1}-ruling ψ̃ : S̃ ⟶ ℙ^{1} which has a unique singular fiber ψ̃^{−1}(0) := F̃_{1} ∪ G̃_{0}. Since Sing S̃ = t̃^{∗} = F̃_{1} ∩ G̃_{0} is the rational bouble point of A_{1}-type, one has the claims (2) and (3) via minimal resolution (see also (3.13). The claim (4) follows from the fact D̃ ∩ S̃ = Γ̃ ∪ G̃_{0}.

(3.15) Corollary. There is a birational contraction φ : Ṽ_{5}^{∗} ⟶ \overline{V} _{5}^{+} of L̃ to a smooth rational curve \overline{E}^{+} = φ(L̃).

Proof. Since KṼ_{5}^{∗} ∼ −2S̃ − 3L̃ − D̃, −2Δ̃_{0} − 2T̃ ∼ KL̃ ∼ (KṼ_{5}^{∗} + L̃)|L̃ = −2S̃|L̃ − 2L̃|L̃ − 2T̃ = −4Δ̃_{0} − 4T̃ − 2L̃|L̃ − D̃|L̃ = −2Σ̃ − 2L̃|L̃. This yields L̃|L̃ ∼ −Δ̃_{0} − T̃ .Thus L̃ ≅ ℙ^{1} × ℙ^{1} contracts to another direction, that is Δ̃_{0}-direction. This contraction morphism φ is desired. □

We set \overline{S}^{+} = φ(S̃) , \overline{D}^{+} = φ(D̃), \overline{E}^{+} := φ(L̃), \overline{F} _{1}^{+} = φ(F̃_{1}) and \overline{G}_{0}^{+} := φ(G̃_{0}). We set χ_{1} := φ ∘ λ^{−1} : \overline{V} _{5}^{∗} −−→ \overline{V} _{5}^{+}, which is a birational map with \overline{V} _{5}^{∗} − \overline{E} ≅^{χ_{1}} \overline{V} _{5}^{+} − \overline{E}^{+}. On the other hand we have easily S̃ ≅^{φ} \overline{S}^{+}. Thus we have

(3.16) Lemma.

(1) (K\overline{V} _{5}^{+} · \overline{E}^{+}) = (K\overline{V} _{5}^{+} · \overline{F}^{+}) = 0.

(2) K\overline{V} _{5}^{+} ∼ −2\overline{S}^{+} − \overline{D}^{+}.

(3) (\overline{S}^{+} · \overline{E}^{+}) = 1.

(4) (\overline{D}^{+} · \overline{E}^{+}) = −2, hence \overline{E}^{+} ⊂ \overline{D}^{+}.

(5) mult\overline{E}^{+} \overline{D}^{+} = 2.

(6) \overline{S}^{+}|\overline{S}^{+} = −(\overline{E}^{+} + 2\overline{G}_{0}^{+})

(7) \overline{D}^{+}|\overline{S}^{+} = 2\overline{E}^{+} + 2\overline{G}_{0}^{+} + \overline{Γ}^{+} , where \overline{Γ}^{+} = \overline{E}^{+} + 5\overline{G}_{0}^{+} + 3\overline{F} _{1}^{+} in \overline{S}^{+}.

(8) (\overline{G}_{0}^{+})_{\overline{S}^{+}}^{2} = (\overline{F} _{1}^{+})_{\overline{S}^{+}}^{2} = −\frac{1}{2}, (\overline{G}_{0}^{+} · \overline{F} _{1}^{+})\overline{S}^{+} = \frac{1}{2}, (\overline{G}_{0}^{+} · \overline{E}^{+}) = 1, (\overline{E}^{+})_{\overline{S}^{+}}^{2} = −3.

(9) (\overline{S}^{+} · \overline{F} _{1}^{+}) = −1, (\overline{D}^{+} · \overline{F} _{1}^{+}) = 2.

Proof. (1): This follows from (3.14)-(3). (2): This follows from K\overline{V} _{5}^{∗} ∼ −2\overline{S} − \overline{D}. (3):(\overline{S}^{+}·\overline{E}^{+}) = −(\overline{S}·\overline{E}) = 1 by (3.14)-(2). (4): 0 = (−K\overline{V} _{5}^{+} ·\overline{E}^{+}) = (\overline{S}^{+}·\overline{E}^{+})+(\overline{D}^{+}· \overline{E}^{+}) = 2 + (\overline{D}^{+} · \overline{E}^{+}). (5):φ^{∗}K\overline{V} _{5}^{+} = KṼ_{5}^{∗} − L̃ = −2S̃ − 4L̃ − D̃. On the other hand since K\overline{V} _{5}^{+} = −2\overline{S}^{+} −\overline{D}^{+} ans φ^{∗}\overline{S}^{∗} = S̃+L̃, φ^{∗}\overline{D}^{+} = −2φ^{∗}\overline{S}^{+} −φ^{∗}K\overline{V} _{5}^{+} = D̃ +2L̃. This shows the claim. (6): Let μ : M ⟶ \overline{S}^{+} be the minimal resolution with the exceptional set B := μ^{−1}(\overline{t}^{+}), where \overline{t}^{+} := φ(t̃^{∗}). Here since \overline{S}^{+} ≅^{φ} S̃, M can be considered the same one as in (3.12.2), via M ⟶^{μ} \overline{S}^{+} ≅^{φ} S̃, so we use the same notation as in (3.12.2).

Let F̂_{1}^{+}, Ĝ_{0}^{+}, Ê^{+}, Γ̂^{+} be the proper transforms of \overline{F} _{1}^{+}, \overline{G}_{0}^{+}, \overline{E}^{+}, \overline{Γ}^{+} in M respectively. Then M is a ruled surface over ℙ^{1} with only one singular fiber F̂_{1}^{+} ∪ B ∪ Ĝ_{0}^{+} with Pic M ≅ ℤ[Ê^{+}] ⊕ ℤ[Ĝ_{0}^{+}] ⊕ ℤ[F̂_{1}^{+}] ⊕ ℤ[B], where

(F̂_{1}^{+})^{2} = (Ĝ_{0}^{+})^{2} = −1, (B)^{2} = −2, (B · F̂_{1}^{+}) = (B · Ĝ_{0}^{+}) = 1, (F̂_{1}^{+} · Ĝ_{0}^{+}) = 0.

Taking into an account that μ^{∗}\overline{F} _{1}^{+} = F̂_{1}^{+}+\frac{1}{2}B, μ^{∗}\overline{G}_{0}^{+} = Ĝ_{0}^{+}+\frac{1}{2}B, one has the claim. Since S̃ ≅^{φ} \overline{S}^{+}, we have also (\overline{E}^{+})_{\overline{S}^{+}}^{2} = −3 by (3.14)-(1), hence we have (8). Since μ^{∗}K\overline{S}^{+} = K_{M} ∼ −2Ê^{+} − 5Ĝ_{0}^{+} − 4C − 3F̂_{1}^{+}, one has K\overline{S}^{+} ∼ −2Ê^{+} − 5Ĝ_{0}^{+} − 3F̂_{1}^{+}. On the other hand K\overline{S}^{+} = (K\overline{V} _{5}^{+} + \overline{S}^{+})|\overline{S}^{+} = −(\overline{S}^{+}|\overline{S}^{+} + \overline{D}^{+}|\overline{S}^{+}) This implies that \overline{S}^{+}|\overline{S}^{+} ∼ 3\overline{G}_{0}^{+} + 3\overline{F} _{1}^{+} − \overline{Γ}^{+}, since \overline{D}^{+}|\overline{S}^{+} ∼ 2\overline{E}^{+} + 2\overline{G}_{0}^{+} + \overline{Γ}^{+}. Since Γ̂^{+} = Ê^{+} + 5Ĝ_{0}^{+} + 4B + 3F̂_{1}^{+}, one gets \overline{Γ}^{+} ∼ \overline{E}^{+} + 5\overline{G}_{0} + 3\overline{F} _{1}^{+}. (7): The assertion follows from (5), (3.14)-(4) and the fact that \overline{S}^{+} is smooth along \overline{E}^{+}. (9): First One sees (\overline{S}^{+} · \overline{F} _{1}^{+}) = (\overline{S}^{+}|\overline{S}^{+} · \overline{F} _{1}^{+}) = −(\overline{E}^{+} + 2\overline{G}_{0}^{+}) · \overline{F} _{1}^{+} = −2(\overline{G}_{0}^{+} · \overline{F} _{1}^{+}) = −1. Next (\overline{D}^{+} · \overline{F} _{1}^{+}) = (\overline{D}^{+}|\overline{S}^{+} · \overline{F} _{1}^{+}) = (3\overline{E}^{+} + 7\overline{G}_{0}^{+} + 3\overline{F} _{1}^{+}) · \overline{F} _{1}^{+} = \frac{7}{2} − \frac{3}{2} = 2.

(3.17) Lemma.

(1) (\overline{S}^{+})^{3} = −1.

(2) (\overline{D}^{+})^{2} · \overline{S}^{+} = 7.

(3) (\overline{S}^{+})^{2} · \overline{D}^{+} = 0.

(4) (\overline{D}^{+})^{3} = −16.

(5) (−K\overline{V} _{5}^{+})^{3} = 18.

Proof. (1):(\overline{S}^{+})^{3} = (\overline{E}^{+} + 2\overline{G}_{0}^{+})^{2} = −1 by (3.16)-(8). (2):(\overline{D}^{+})^{2} · \overline{S}^{+} = (2\overline{E}^{+} + 2\overline{G}_{0}^{+} + \overline{Γ}^{+})^{2} = 7 by (3.16)-(7),-(8). (3):(\overline{S}^{+})^{2} · \overline{D}^{+} = (\overline{D}^{+}|\overline{S}^{+} · \overline{S}^{+}|\overline{S}^{+}) = −(3\overline{E}^{+} + 7\overline{G}_{0}^{+} + 3\overline{F} _{1}^{+})(\overline{E}^{+} + \overline{G}_{0}^{+}) = 0. (4): Since 18 = (−K\overline{V} _{5}^{∗})^{3} = (−K\overline{V} _{5}^{+})^{3} = 8(\overline{S}^{+})^{3} + 12(\overline{S}^{+})^{2} · \overline{D}^{+} + 6\overline{S}^{+} · (\overline{D}^{+})^{2} + (\overline{D}^{+})^{3}, we have (\overline{D}^{+})^{3} = −16. □

By an argument similar to (3.13)-(2), we obtain (3.18) Lemma. N_{\overline{F}^{+}|\overline{V} _{5}^{+}} ≅ O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(−2).

Therefore we have a birational map χ_{2} : \overline{V} _{5}^{+} −−→ \overline{V} _{5}^{++} and a smooth rational curve \overline{F} _{1}^{++} such that

(1) χ_{2} : \overline{V} _{5}^{+} − \overline{F} _{1}^{+} ≅ \overline{V} _{5}^{++} − \overline{F} _{1}^{++}

(2) \overline{V} _{5}^{+} is smooth near the curve \overline{F} _{1}^{++} The birational map χ_{2} is called the flop (or \overline{D}^{+}-flop, [20]) along \overline{F} _{1}^{+}. Let S^{++}, D^{++}, be the proper transform of \overline{S}^{+}, \overline{D}^{+} in V_{5}^{++}. We set χ_{2}(\overline{F} _{1}^{+}) = F_{1}^{++}, χ_{2}((\overline{E}^{+}) = E^{++}. Then one can see that S^{++} is smooth and S^{++} ≅ 𝔽_{3} (see Pagoda in [14], [30]). Taking an account of (3.16) and elementary properties of the flop , we obtain the following (3.19) Lemma.

(1) K_{V_{5}^{++}} = −2S^{++} − D^{++}.

(2) D^{++}|_{S^{++}} = 3E^{++} + 7G_{0}^{++}.

(3) S^{++}|_{S^{++}} = −E^{++} − 2G_{0}^{++}.

(4) (S^{++} · F_{1}^{++}) = 1.

(5) (D^{++} · F_{1}^{++}) = −2.

(6) (S^{++} · E^{++}) = 1.

(7) (S^{++} · F_{1}^{++}) = 1, in particular, S^{++} intersects with F_{1}^{++} transversally at one point.

Finally we set χ := χ_{2} ∘ χ_{1}, then we obtain the following (3.20) Proposition. There is a projective threefold V_{5}^{++} and disjoint smooth rational curves E^{++}, F_{1}^{++} and a birational map χ : V_{5}^{∗} −−→ V_{5}^{++} such that V_{5}^{∗} − (\overline{F}_{1} ∪ \overline{E}) ≅^{χ} V_{5}^{++} − (F_{1}^{++} ∪ E^{++}), the birational map χ is called the \overline{D}-flop along \overline{E}, \overline{F}_{1}. (3.21) Corollary. Bs |D^{++} + 2S^{++}| = ∅. Proof. By (3.12), Bs | − K\overline{V} _{5}^{∗}| = |\overline{D} + 2\overline{S}| = ∅. Since χ_{∗}(−K\overline{V} _{5}^{∗}) = −K_{V_{5}^{++}} ∼ D^{++} + 2S^{++}. Thus we have the claim. □ (3.22) Lemma.

(1) (S^{++})^{3} = 1.

(2) (D^{++})^{2} · S^{++} = 15.

(3) (S^{++})^{2} · D^{++} = −4.

(4) (D^{++})^{3} = −32.

(5) (K_{V_{5}^{++}})^{3} = −18.

Proof. (1): (S^{++})^{3} = (S^{++}|_{S^{++}})^{2} = (−E^{++} − 2G_{0}^{++})^{2} = 1. (2): (D^{++})^{2} · S^{++} = (D^{++}|_{S^{++}})^{2} = (3E^{++}+7G_{0}^{++})^{2} = 15. (3): (S^{++})^{2}·D^{++} = (S^{++}|_{S^{++}} ·D^{++}|_{S^{++}}) = −(E^{++} + 2G_{0}^{++})(3E^{++} + 7G_{0}^{++}) = −4. (4): (K_{V_{5}^{++}})^{3} = (K\overline{V} _{5}^{+})^{3} = −18. (5): 18 = (−K_{V_{5}^{++}})^{3} = (D^{++} + 2S^{++})^{2} = 50 + (D^{++})^{3}. □

(3.33) Proposition. There is a birational contraction Ψ : V_{5}^{++} ⟶ V_{22}^{∗} with Exc(Ψ) = S^{++} such that

(1) V_{22}^{∗} is smooth near ℓ := Ψ(S^{++}) ≅ ℙ^{1},

(2) (−K_{V_{22}^{∗}} · ℓ) = 1,

(3) mult_{ℓ} D^{∗} = 3,

(4) −K_{V_{22}^{∗}} = −D^{∗}, where D^{∗} := Ψ_{∗}D^{++},

(5) (−K_{V_{22}^{∗}})^{3} = 22. In particular Ψ : V_{5}^{++} ⟶ V_{22}^{∗} is a blowing up of V_{22}^{∗} with the center ℓ.

Proof. First we recall that E^{++} (resp. G_{0}^{++}) is the negative section (resp. a fiber) of the Hirzebruch surface S^{++} ≅ 𝔽_{3}. By (3.22)-(3), we have (S^{++} · G_{0}^{++}) = −1. Thus we have the contraction Ψ : V_{5}^{++} ⟶ V_{22}^{∗} of S^{++} in a fiber direction. In particular V_{22}^{∗} is smooth neat ℓ := Φ(S^{++}). Since −2S^{++} − D^{++} = K_{V_{5}^{++}} = Φ^{∗}K_{V_{22}^{∗}} + S^{++} = −Φ^{∗}D^{∗} + S^{++}, one has Φ^{∗}D^{∗} = D^{++} + 3S^{++}, which says mult_{ℓ} D^{∗} = 3. (−K_{V_{22}^{∗}})^{3} = (Φ^{∗}D^{∗})^{3} = (D^{++} + 3S^{++})^{3} = 22. Next −1 = (D^{++} + 3S^{++}) · (S^{++})^{2} = Φ^{∗}D^{∗} · (S^{++})^{2} = (D^{∗} · ℓ) · (S^{++} · G_{0}^{++}) = −(D^{∗} · ℓ). Thus we have (−K_{V_{22}^{∗}} · ℓ) = −(D^{∗} · ℓ) = 1. □

(3.24) Lemma.

(1) dim H^{0}(V_{5}^{++}; O(D^{++} + 3S^{++})) = 14

(2) Bs |D^{++} + 3S^{++}| = ∅. In fact, the birational morphism Ψ : V_{5}^{++} ⟶ V_{22}^{∗} is given by the linear system |D^{++} + 3S^{++}|.

Proof. (1): First we will show that D^{++} + 3S^{++} is nef and big. In fact, assume that there is a curve R such that (D^{++} + 3S^{++}) · R < 0. Since D^{++} + 2S^{++} is semi-ample, one has (S^{++} · R) < 0, that is, R ⊂ S^{++}. On the other hand, since (D^{++} + 3S^{++})|_{S^{++}} = G_{0}^{++} on S^{++} ≅ 𝔽_{3}, (S^{++} · R) = (G_{0}^{++} · R) ≥ 0, which is a contradiction. Hence D^{++} + 3S^{++} is nef. Since (D^{++} + 3S^{++}) = 22, it is big.

Take a general member Z^{++} ∈ |D^{++} + 2S^{++}|, which is a smooth K-3 surface. Consider the following exact sequence:

0 ⟶ O(S^{++}) ⟶ O(D^{++} + 3S^{++}) ⟶ O_{Z^{++}}(D^{++} + 3S^{++}) ⟶ 0

Since h^{0}(O_{V_{5}^{++}}(S^{++})) = 1, h^{1}(O_{V_{5}^{++}}(S^{++})) = 0, one has

h^{0}(O_{V_{5}^{++}}(D^{++} + 3S^{++})) = 1 + h^{0}(O_{Z^{++}}(D^{++} + 3S^{++})).

Now since D^{++} + 3S^{++} is nef big on Z^{++}, h^{i}(O_{Z^{++}}(D^{++} + 3S^{++})) = 0 for i > 0. The Riemann-Roch theorem says

h^{0}(O_{Z^{++}}(D^{++} + 3S^{++})) = \frac{1}{2}(D^{++} + 3S^{++})^{2}(D^{++} + 2S^{++}) + 2 = 13

, hence h^{0}(O_{V_{5}^{++}}(D^{++} + 3S^{++})) = 14.

(2): Since D^{++} + 2S^{++} is semi ample, one can take a general member Z^{++} ∈ |D^{++} + 2S^{++}| such that Z^{++} contains no fixed component of |D^{++} + 2S^{++}|. Thus the nef and big line bundle O_{Z^{++}}(D^{++} + 3S^{++}) has no fixed component on Z^{++}, hence has no basepoints. A surjection H^{0}(O(D^{++} + 3S^{++})) ⟶ H^{0}(O_{Z^{++}}(D^{++} + 3S^{++})) ⟶ 0 implies the claim. □

Thus we have finally the following

(3.25) Proposition. V_{22}^{∗} is the Fano threefold of degree 22 in ℙ^{13} of index r = 1 with a hypersurface terminal singularity of A_{1}-type at the point p = Φ(χ(σ^{−1}(p))), in particular, b_{2}(V_{22}^{∗}) = 1 and b_{2}(V_{22}^{∗}) = 2.

(3.26) Remark.

We set Δ_{22} := Ψ(χ(σ^{−1}(Δ_{5}))) and W := V_{22}^{∗} − Δ_{22}, which is an smooth affine threefold. By construction one sees that V_{5} − Δ_{5} − F_{1} ≅ V_{22}^{∗} − Δ_{22} − F_{1}^{++}. In particular, W ⊃ ℂ^{2} × ℂ^{∗} and has a decomposition W = (ℂ^{2} × ℂ^{∗}) ∪^{·} ℂ^{2} (disjoint union). The author does not know whether W ≅ ℂ^{3}.

References

1. M. Abe and M. Furushima, On non-normal del Pezzo surfaces, Math. Nachr. . ((to appear)).

2. G.Barthel and L.Kaup, Topologie des Espaces Complexes Compactes Singulieres, Montreal Lecture Notes, Les Presses de l’Université de Montréal 80 (1982).

3. F. Campana and H. Flenner, Projective threefolds containing a smooth rational surface with ample normal bundle, J. reine und angew. Math. . 440 (1993), 77–98.

4. T. Fujita, Projective varieties of Δ-genus one, Algebraic and Topological Theories, Kinokuniya, Tokyo Japan . (1985), 149–175.

5. T. Fujita, On singular del Pezzo varieties, in Algebraic Geometry, Proceedings L’Aquila 1988, Lect. Note in Math. Springer . 1417 (1990), 117–128.

6. M. Furushima, Singular del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space ℂ^{3}, Nagoya Math. J. 104 (1986), 1–28.

7. M. Furushima, The complete classification of compactifications of ℂ^{3} which are projective manifoldss with second Betti number one, Math. Ann. 297 (1993), 627–662.

8. M. Furushima, An example of a non-projective smooth compactification of ℂ^{3} with second Betti number equal to one, Math. Ann. 300 (1994), 89–96.

9. M. Furushima, Non-projective compactifications of ℂ^{3} (I), Kyushu J. Math. 50 (1996), 221– 239.

10. M. Furushima, Non-projective compactifications of ℂ^{3} (II) (New Examples), Kyushu J. Math. 52 (1998), 149–162.

11. M. Furushima, Non-projective compactifications of ℂ^{3} III: A remark on indices, Hiroshima Math. J. 29 (1999), 295–298.

12. M. Furushima, Birationa construction of projective compactifications of ℂ^{3} with second Betti number equal to one, Annali di Matematica pura ed applicata 178 (2000), 115–128.

13. M. Furushima and N. Nakayama, The family of lines on the Fano threefold V_{5}, Nagoya Math. J. 116 (1989), 111–122.

14. M. Furushima and N. Nakayama, A new construction of a compactification of ℂ^{3}, Tohoku Math. J. 41 (1989), 543–560.

15. F.Hirzebruch, Some problems on differentiable and complex manifolds, Ann. Math. 60 (1954), 213–236.

16. V.A. Iskovskih, Fano 3-folds I, Math. USSR Izvestiya . 11 (1977), 485–527.

17. V.A. Iskovskih, Fano 3-folds II, Math. USSR Izvestiya . 12 (1978), 469–506.

18. V.A. Iskovskih, Algebraic Geometry V, Encyclopaedia of Mathematical sciences, Springer-Verlag . 47 (1999).

19. S.Katz, Small resolution of Gorenstein threefold singularities, Contemporary Math. AMS 116 (1991), 61–70.

20. J. Kollar, Flips, Flops, Minimal models, etc., Survey in Differential Geometry . 1 (1991), 113–199.

21. H. Laufer, On minimally elliptic singularities, Amer. J. Math. . 99 (1977), 1257–1295.

22. H.Laufer, On ℂℙ^{1} as an exceptional set, Ann. Math. Studies, Princeton University Press, Princeton, NJ . 100 (1981), 261–275.

23. M. Miyanishi, Open Algebraic Surfaces, CRM Monograph series vol.12, Amer. Math. Sci. . (2001).

24. S.Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math. 116 (1982), 133-176.

25. S.Mukai, New development in Fano manifold theory related to the vector bubdle method and moduli problem, Exposition of Sugaku 116 (1995), 133-176.

26. I. Nakamura, Moishezon 3-folds homeomorphic to a cubic hypersurface in ℙ^{4}, J. Algebraic Geometry . 5 (1996), 537–569.

27. T. Peternell and M. Schneider, Compactifications of ℂ^{3}, I, Math. Ann. . 280 (1988), 129–146.

28. T. Peternell and M. Schneider, Compactifications of ℂ^{n}: A Survey., Several Complex Variables and Complex Geometry. ed. E. Bedford et al. Proc.Symp. Pure Math.Amer. Math. Soc., Providence, RI . 52 (1991), 455–466.

29. T. Peternell, Compactifications of ℂ^{3}, II, Math. Ann. . 283 (1989), 121–137.

30. M. Reid, Minimal model of canonical 3-folds, Algebraic Varieties and Analytic Varieties (Advanceed Studies in Pure Math.). Ed. S. Iitaka. Kinokuniya, Japan 1 (1983), 131-180.

31. M.Reid, Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), 695–727.

32. R. Remmert and T. Van de Ven, Zwei Satze über die komplex-projektive ebene, Niew Arch. Wisk. 8 (3) (1960), 147–157.

33. K.-H. Shin, 3-dimensional Fano varieties with canonical singularities, Tokyo J. Math. 12 (1989), 375–385.

34. P.A. Smith, Fixed points of periodic transformations, appendix B in Algebraic Topology by Solomon Lefschetz, American Math. Sco. Coll. Publ. XXVII (1942).

35. H. Takagi, On the classification of ℚ−Fano 3-folds of Gorenstein index 2. I, Nagoya Math. J. 167 (2002), 117–155.

36. H. Takagi, On the classification of ℚ−Fano 3-folds of Gorenstein index 2. II, Nagoya Math. J. 167 (2002), 157–216.

37. M.Tsuji, Dimca hypersurfaces and Nagata automorphisms, Math. Nachr. 218 (2000), 175– 184.

38. Y. Umezu, Quartic surfaces of elliptic ruled type, Amer. Math. Soc. 283 (1984), 127–143.

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