ON SEMISTABLE EXTREMAL NEIGHBORHOODS

SHIGEFUMI MORI

Abstract. We give an explicit description of an extremal nbd X ⊃ C ≃ ℙ^{1} of type k2A. We also give a criterion for X to be a flipping contraction and an explicit description of the contraction and the flip.

1. Introduction

In the three dimensional minimal model program, flips and divisorial contractions are the fundamental birational maps. Among them, flips are proved to exist [8]. This paper is concerned with the classification of flips. We give a brief background.

Let f : X → Y be a projective birational morphism from a threefold X with only terminal singularities to a normal threefold Y and Q ∈ Y such that C = f^{−1}(Q) is a curve and −K_{X} is f-ample.

We note that, in the context of the minimal model program, we often assume that X is ℚ-factorial and put the condition ρ(X/Y) = 1 on the relative Picard number. In this paper, we do not assume these conditions, because they are not preserved when we work on the associated formal scheme.

For an arbitrarily small open set U ∋ Q, we call f^{−1}(U) ⊃ C → U ∋ Q an extremal neighborhood (or, an extremal nbd, for short). It is said to be flipping (resp. divisorial) if the exceptional set is a curve (resp. a divisor). An extremal nbd is said to be irreducible if C is irreducible.

In [5], the irreducible extremal nbds X ⊃ C → Y ∋ Q are studied as follows. A general member D of | − K_{X}| is proved to have only Du Val singularities, and the irreducible extremal nbds are classified into 6 types k1A, k2A, cD/3, IIA, IC, kAD [5, (2.2)] according to the singularities of D. The first two (resp. the last four) cases are said to be semistable (resp. exceptional). For the exceptional irreducible flipping extremal nbds X ⊃ C, the singularities of the general member H of |O_{X}| containing C are computed in [5, Chapters 6–9] and the irreducible flipping X ⊃ C is reconstructed as an essentially arbitrary one-parameter deformation space of H [5, Theorems (13.9–12)] and the flip is described [5, Theorems (13.17–18)].

1991 Mathematics Subject Classification. Primary 14E30; Secondary 14D15.

Key words and phrases. flip, divisorial contraction, minimal model program, terminal singularity.

However if we start with H of an irreducible semistable extremal nbd X, whether or not X is flipping depends not only on H but also on the individual one-parameter deformation, which is quite different from the exceptional cases.

In this paper, we treat the case of k2A. (The case of k1A will be treated elsewhere.) In Section 2, we give an expression of an extremal nbd X ⊃ C of type k2A in terms of coordinates (Theorem 2.2) and graded rings (Definition 2.8, Theorem 2.9).

Section 3 is the core algorithm section of this paper, where we introduce a sequence d(n) (Definitions 3.2, 3.11) and present a series of divisions (Theorems 3.10–3.13) starting with the “graded equations” in Theorem 2.9.

Section 4 is the main section for applications, where we give a necessary and sufficient condition (Corollary 4.1) for X ⊃ C to be flipping in terms of d(n). Furthermore, the extremal contraction (Theorem 4.3) and the flip are explicitly constructed (Theorem 4.7).

In Section 5, we present the division in the case of a multi-parameter deformation space of H and comment on some of the further directions.

The author is very grateful to Professor Miles Reid for his interest in the current computation. The author was, in part, supported by the Grant-in-Aid for Scientitic Research (B) (12440005), Japan Society for the Promotion of Science.

2. Good coordinates

2.1. Let f : X ⊃ C (≃ ℙ^{1}) → Y ∋ Q be an extremal nbd of type k2A [5] with two terminal singular points P_{1}, P_{2} of indices m_{1}, m_{2} > 1 and axial multiplicities α_{1}, α_{2} ≥ 1, respectively.

Let D ∈ | − K_{X}| be a Du Val member, whose minimal resolution has the dual configuration

\underbrace{∘ − ⋯ − ∘}_{m_{1}α_{1}−1} −C′ − \underbrace{∘ − ⋯ − ∘}_{m_{2}α_{2}−1},

where C′ is the proper transform of C and ∘ denotes an exceptional curve and all these curves are (−2)-curves [5, 2.2.4]. By adding two (non-compact) curves ℓ′_{i} at both ends

ℓ′_{1} − \underbrace{∘ − ⋯ − ∘}_{m_{1}α_{1}−1} −C′ − \underbrace{∘ − ⋯ − ∘}_{m_{2}α_{2}−1} −ℓ′_{2},

we obtain a reduced curve on the minimal resolution such that the intersection numbers with C′ and ∘’s are zero. Since H^{1}(X, O_{X}) = 0 [8], we see that ℓ_{1} + C + ℓ_{2} ∼ 0 on D, where ℓ_{i} ⊂ D denotes the image of ℓ′_{i}. Let

H_{D} := ℓ_{1} + C + ℓ_{2} (∼ 0 on D).

By the exact sequence

0 → O_{X}(K_{X}) → O_{X} → O_{D} → 0

and the Grauert Riemenschneider vanishing H^{1}(X, O_{X}(K_{X})) = 0 [8], we obtain a trivial Cartier divisor H = (u = 0) on X, which is normal and induces H_{D} on D.

Theorem 2.2. Let U_{i} be the ℤ_{m_{i}}-quotient of a “hypersurface” of ℂ^{4},

U_{i} := (ξ_{i}, η_{i}, ζ_{i}, u; ξ_{i}η_{i} = g_{i}(ζ_{i}^{m_{i}}, u))/ℤ_{m_{i}}(1, −1, a_{i}, 0),

where a_{i} is an integer ∈ [1, m_{i}] prime to m_{i} and g_{i}(T, u) ∈ ℂ[[u]][T] is a monic polynomial in T of degree, say ρ_{i} such that g_{i}(ζ_{i}^{m_{i}}, u) is square-free. Let P_{i} := 0 and C_{i} := ξ_{i}-axis/ℤ_{m_{i}}. U_{i} is defined to be a formal scheme along C_{i} ≃ ℂ^{1} with only terminal singularities.

For a suitable choice of a_{i} and g_{i}(T, u), we have

1. these C_{1} and C_{2} are patched together to form C ≃ ℙ^{1} and U_{1} and U_{2} are patched together to form the completion X̂ of X along C by the identification on U_{1} ∩ U_{2}:

ξ_{1}^{m_{1}} = (ξ_{2}^{m_{2}})^{−1},
\frac{ζ_{1}}{ξ_{1}^{a_{1}}} = ξ_{2}^{m_{2}} \frac{ζ_{2}}{ξ_{2}^{a_{2}}} ,

2. D = (ζ_{1} = 0)/ℤ_{m_{1}} ∪ (ζ_{2} = 0)/ℤ_{m_{2}} and H = (u = 0) under the identification.

Remark 2.3. The assertions of Theorem 2.2 modulo the equation u of H, that is the corresponding assertion for H is easily seen as follows.

By the construction, f(H_{D}) has an ordinary double point at Q. Since K_{H} +H_{D} ∼ 0, we see that (f(H), f(H_{D})) is lc ([6, (5.58)] or [12]). Since f(H_{D}) has two analytic branches, this means that (f(H), f(H_{D})) is the quotient of (xy-plane, (xy = 0)) by a diagonal action of some cyclic group [1] or [3]. In particular, it is toric.

Since C is a log crepant divisor of (f(H), f(H_{D})) (that is, K_{H}+H_{D} ∼ 0 and, and H_{D} is a reduced curve containing C), (H, H_{D}) is toric as well as (f(H), f(H_{D})).

The index-1 cover of (H \ ℓ_{i}, H_{D} \ ℓ_{i}) is toric. By the descrition of the terminal singularities P_{i} [7, 11], we obtain the isomorphisms

H \ ℓ_{3−i} ≃ (ξ_{i}, η_{i}, ζ_{i}; ξ_{i}η_{i} = ζ_{i}^{m_{i}ρ_{i}})/ℤ_{m_{i}}(1, −1, a_{i})

with the properties

1. H_{D} = (ζ_{i} = 0)/ℤ_{m_{i}}, m_{i}ρ_{i}ℓ_{i} = (ξ_{i} = 0)/ℤ_{m_{i}} and C \ ℓ_{3−i} = (ζ_{i} = η_{i} = 0)/ℤ_{m_{i}} on H \ ℓ_{3−i} under the identification, and

2. ξ_{1}^{m_{1}} = ξ_{2}^{−m_{2}} on H \ (ℓ_{1} ∪ ℓ_{2}), for some ρ_{i} ∈ ℤ_{>0} and a_{i} ∈ [1, m_{i}] such that (a_{i}, m_{i}) = 1. Once these properties are checked, it is easy to see the following.

3. K_{H} ∼ (m_{1} − a_{1}) · (ξ_{1} = 0)/ℤ_{m_{i}} − a_{2} · (ξ_{2} = 0)/ℤ_{m_{2}}

4. ξ_{1}^{−a_{1}}ζ_{1} = ξ_{2}^{m_{2}−a_{2}}ζ_{2} on H \ (ℓ_{1} ∪ ℓ_{2}). Indeed, the property 3 follows from gr_{X}^{0}ω ≃ O_{C}(−1) [8, (1.14.(i)] and the assertion that

x_{i}^{m_{i}−a_{i}} Res \frac{dξ_{i} ∧ dη_{i} ∧ dζ_{i}}{ξ_{i}η_{i} − ζ_{i}^{m_{i}ρ_{i}}} = −x_{i}^{m_{i}−a_{i}}(dξ_{i}/ξ_{i}) ∧ dζ_{i}

is a generator of gr_{X}^{0}ω|_{H} on H \ ℓ_{3−i}. The property 4 follows from the property 3 because m_{1}dξ_{1}/ξ_{1} = −m_{2}dξ_{2}/ξ_{2}.

Proof of Theorem 2.2. We note that Theorem 2.2 is proved modulo (u), the equation of H (Remark 2.3). On the completion X̂ of X along C, let U_{i} be the complement of P_{3−i}. Assume that Theorem 2.2 is proved modulo (u)^{N} for some N > 0. We attach subscript N to the coordinates and the equations that are chosen to work for (u)^{N}.

From the ℤ_{m_{i}}-invariant relation

ξ_{N,i}η_{N,i} = g_{N,i}(ζ_{N,i}^{m_{i}}, u) mod (u)^{N},

we have

ξ_{N,i}η_{N,i} = g′_{i}(ζ_{N,i}^{m_{i}}, u) + u^{N}ξ_{N,i}α_{i} + u^{N}η_{N,i}β_{i} mod (u)^{N+1}

for some α_{i}, β_{i} ∈ ℂ[ξ_{N,i}][[u, η_{N,i}, ζ_{N,i}]] and g′_{i}(T, u) ∈ ℂ[[u, T]] such that g′_{i} ≡ g_{N,i} mod (u)^{N}. Then

(ξ_{N,i} − u^{N}α_{i})(η_{N,i} − u^{N}β_{i}) = g′′_{i}(ζ_{N,i}^{m_{i}}, u) mod (u)^{N+1}

for some g′′_{i} ∈ ℂ[[u, T]] such that g′′_{i} ≡ g_{N,i} mod (u)^{N}.

The Cartier divisor

Φ_{i} = (ξ_{N,i} − u^{N}α_{i})/ξ_{i}) = (ξ_{N,i} = u^{N}α_{i})/ℤ_{m_{i}} − (ξ_{N,i} = 0)/ℤ_{m_{i}}

on a neighborhood of P_{i} extends to a principal divisor on X, because Φ_{i} intersects properly with C and (Φ_{i} · C) = 0. By the exact sequence

0 → OX̂(Φ_{i}) →^{u^{N}} OX̂(Φ_{i}) → O_{NĤ} → 0

and H^{1}(X̂, OX̂) = 0, there is a rational function φ_{i} on X̂ such that (φ_{i}) = Φ_{i} and φ_{i}|_{NĤ} = 1. We note that φ_{i} is invertible on U_{3−i}.

Let

ξ_{N+1,i} = ξ_{N,i} · φ_{i} · φ_{3−i}^{−m_{3−i}/m_{i}}.

Then we have

ξ_{N+1,i} ≡ ξ_{N,i} mod (u)^{N},
ξ_{N+1,1}^{m_{1}} = ξ_{N+1,2}^{−m_{2}}

and ξ_{N+1,i} = (ξ_{N,i} − u^{N}α_{i}) · (unit on U_{i}). Let

ζ_{N+1,i} = \{ \begin{matrix}ζ_{N,1}φ_{2}^{m_{2}−a_{2}}φ_{2}^{−a_{1}m_{2}/m_{1}} & i = 1, \\ ζ_{N,2}φ_{1}^{−a_{1}}φ_{1}^{−(m_{2}−a_{2})m_{1}/m_{2}} & i = 2. \end{matrix}

We note that ζ_{N+1,i}/ζ_{N,i} is a unit on U_{i} and that

ξ_{N+1,1}^{−a_{1}}ζ_{N+1,1} = ξ_{N+1,2}^{m_{2}−a_{2}}ζ_{N+1,2},
ζ_{N+1,i} ≡ ζ_{N,i} mod (u)^{N}.

By the Weierstrass preparation theorem, there exist a unit γ_{i} ∈ ℂ[[u, T]] such that γ_{i} ≡ 1 mod (u)^{N} and g_{N+1,i} := g′′_{i}γ_{i} is a monic polynomial ∈ ℂ[[u]][T]. We then define η_{N+1,i} such that η_{N+1,i} = (η_{N,i} − u^{N}β_{i}) · (unit on U_{i}) by the relation on U_{i}:

ξ_{N+1,i}η_{N+1,i} = (ξ_{N,i} − u^{N}α_{i})(η_{N,i} − u^{N}β_{i})(ζ_{N+1,i}/ζ_{N,i})^{m_{i}ρ_{i}}γ_{i}.

By g_{N+1,i}(T, 0) = T^{ρ_{i}} and ζ_{N+1,i}/ζ_{N,i} ≡ 1 mod (u)^{N}, we have

g_{N+1,i}(ζ_{N+1,i}^{m_{i}}, u) ≡ g_{N+1,i}(ζ_{N,i}^{m_{i}}, u) · (ζ_{N+1,i}/ζ_{N,i})^{m_{i}ρ_{i}} mod (u)^{N+1},

and hence

ξ_{N+1,i}η_{N+1,i} ≡ g_{N+1,i}(ζ_{N+1,i}^{m_{i}}, u) mod (u)^{N+1}.

Thus the theorem is proved modulo (u)^{N+1}. We can let n → ∞.

Finally g_{i}(ζ_{i}^{m_{i}}, u) is square-free, because otherwise U_{i} has a nonisolated singularity.

Remark 2.4. The numbers a_{i} and ρ_{i} in Theorem 2.2 can be easily read off from the information on X ⊃ C, D and H. Furthermore, α_{i} is related to g_{i}.

1. The numbers a_{i} are uniquely determined by X ⊃ C ∋ P_{i} because the ℤ_{m_{i}}-action is normalized by its action on the ξ_{i}-axis.

2. By

H = (ξ_{i}η_{i} = g_{i}(ζ_{i}^{m_{i}}, 0))/ℤ_{m_{i}}(1, −1, a_{i}),

the index-one cover of H at P_{i} is an A_{m_{i}ρ_{i}}-point (ξ_{i}η_{i} = g_{i}(ζ_{i}^{m_{i}}, 0)). Thus ρ_{i} is uniquely determined by H.

3. Similarly by

D = (ξ_{i}η_{i} = g_{i}(0, u))/ℤ_{m_{i}}(1, −1, 0),
≃ (xy = g_{i}(0, u)^{m_{i}}),

we have (g_{i}(0, u)) = (u)^{α_{i}} in ℂ[[u]].

Remark 2.5. Under the notation of Theorem 2.2, let S_{i} = (ξ_{i} = 0)/ℤ_{m_{i}}. Then

1. m_{1}S_{1} ∼ m_{2}S_{2}, S_{1} ∩ S_{2} = ∅,

2. K_{X} ∼ (m_{1} − a_{1})S_{1} − a_{2}S_{2}, and

3. −m_{i}K_{X} ∼ δS_{3−i}, where δ := a_{1}m_{2} + a_{2}m_{1} − m_{1}m_{2} > 0. Let

z ∈ Γ(X̂, O(a_{1}S_{1} + (a_{2} − m_{2})S_{2})), u ∈ Γ(X̂, O)), x_{i} ∈ Γ(X̂, O(S_{i}))

be the sections defining D, H and S_{i}.

Let G_{i}(T_{1}, T_{2}) = g_{i}(T_{1}/T_{2}, u)T_{2}^{ρ_{i}} ∈ ℂ[[u]][T_{1}, T_{2}], which is a homogeneous polynomial in T_{1}, T_{2} of degree ρ_{i}. Since z^{m_{i}}, x_{3−i}^{δ} are both sections of O(δS_{i}), we can consider the section G_{i}(z^{m_{i}}, x_{3−i}^{δ}) of O(ρ_{i}δS_{3−i}). The section is divisible by x_{i} and the quotient y_{i} satisfies the condition

y_{i} ∈ Γ(X̂, O(ρ_{i}δS_{3−i} − S_{i})),

which follows immediately from the local equation ξ_{i}η_{i} = g_{i}(ζ^{m_{i}}, u). We have thus two equations:

x_{1}y_{1} − G_{1}(z^{m_{1}}, x_{2}^{δ}) = 0, x_{2}y_{2} − G_{2}(z^{m_{2}}, x_{1}^{δ}) = 0,

where G_{1}(z^{m_{1}}, x_{2}^{δ}) and G_{1}(z^{m_{1}}, x_{2}^{δ}) are square-free (Theorem 2.2).

The contractibility of C implies the following positivity result.

Proposition 2.6. Under the notation and the assumptions of Theorem 2.2, we have

Δ := ρ_{1}m_{1}^{2} − δρ_{1}ρ_{2}m_{1}m_{2} + ρ_{2}m_{2}^{2} > 0.

Proof. By the property 1 of Remark 2.3, we have

(ℓ_{i} · C)_{H} = \frac{1}{m_{i}ρ_{i}} (S_{i} · C) = \frac{1}{m_{i}^{2}ρ_{i}} .

Since H ∩ D = ℓ_{1} + C + ℓ_{2}, we have the following by Remark 2.5.

(ℓ_{1} + C + ℓ_{2} · C)_{H} = (D · C) = \frac{1}{m_{1}} (−m_{1}K_{X} · C)
= \frac{1}{m_{1}} (δS_{2} · C) = \frac{δ}{m_{1}m_{2}} .

Thus we have

(C^{2})_{H} = \frac{−Δ}{ρ_{1}ρ_{2}m_{1}^{2}m_{2}^{2}} .

Since C is an exceptional curve on H, we have (C^{2})_{H} < 0.

Remark 2.7. The properties that

z ∈ Γ(X̂, O(a_{1}S_{1} + (a_{2} − m_{2})S_{2})),
u ∈ Γ(X̂, O)),
x_{i} ∈ Γ(X̂, O(S_{i})),
y_{i} ∈ Γ(X̂, O(ρ_{i}δS_{3−i} − S_{i}))

in Remark 2.5 can be rephrased as follows. Let the group

Γ := {(γ_{1}, γ_{2}) ∈ (ℂ^{∗})^{2}|γ_{1}^{m_{1}} = γ_{2}^{m_{2}}}

act on H^{0}(X̂, O(λ_{1}S_{1} + λ_{2}S_{2})) via the multiplication by γ_{1}^{λ_{1}}γ_{2}^{λ_{2}}. Then we have

γ(x_{i}, z, y_{i}, u) = (γ_{i}x_{i}, γ_{1}^{a_{1}}γ_{2}^{a_{2}−m_{2}}z, γ_{3−i}^{ρ_{i}δ}γ_{i}^{−1}y_{i}, u),

and x_{i}y_{i} − G_{i}(z^{m_{i}}, x_{3−i}^{δ}) is semi-invariant under the Γ-action. The scheme X̂ has an alternate description in terms of these data as follows.

Definition 2.8. Let a_{i}, m_{i}, α_{i}, ρ_{i} be positive integers (cf. Remark 2.10) and G_{i}(T_{1}, T_{2}) ∈ ℂ[[u]][T_{1}, T_{2}] a homogeneous polynomial in T_{1} and T_{2} of degree ρ_{i} (i = 1, 2) such that

1. a_{i} ≤ m_{i} and (a_{i}, m_{i}) = 1,

2. δ = a_{1}m_{2} + a_{2}m_{1} − m_{1}m_{2} > 0,

3. G_{i}(1, 0) = 1, G_{i}(0, 1)ℂ[[u]] = u^{α_{i}}ℂ[[u]],

4. G_{i}(T_{1}^{m_{i}}, 1) is reduced and

5. Δ = ρ_{1}m_{1}^{2} − δρ_{1}ρ_{2}m_{1}m_{2} + ρ_{2}m_{2}^{2} > 0. Let R := ℂ[[u]][x_{1}, y_{1}, x_{2}, y_{2}, z] be the ℂ[[u]]-algebra with the Γ-action in Remark 2.7, and let W = Spec R/I be the scheme with the Γ-action, where I is the ideal given by

I := (x_{1}y_{1} − G_{1}(z^{m_{1}}, x_{2}^{δ}), x_{2}y_{2} − G_{2}(z^{m_{2}}, x_{1}^{δ})).

Set

X := (W \ V (x_{1}, x_{2}))/Γ ⊃ C := V (y_{1}, y_{2}, z)/Γ ≃ ℙ^{1}

and {P_{i}} = V (x_{i}, y_{1}, y_{2}, z, u), where V (I) denots the closed subset defined by all the equations in I.

Theorem 2.9. With the above notation and the assumptions, we have the following.

1. X is a normal scheme of dimension 3 such that X \ {P_{1}, P_{2}} is smooth and

P_{i} ∈ X ≃ (ξ_{i}, η_{i}, ζ_{i}, u; ξ_{i}η_{i} = G_{i}(ζ_{i}^{m_{i}}, 1))/ℤ_{m_{i}}(1, −1, a_{i}, 0)

is a terminal singularity with index m_{i} and P_{i} ∈ C = ξ_{i}-axis/ℤ_{m_{i}} under the identification, 2. S_{i} := (x_{i} = 0)/Γ is a ℚ-Cartier Weil divisor on X, and a rational

function ϕ on W such that ϕ/x_{1}^{−b_{1}}x_{2}^{−b_{2}} is Γ-invariant defines a ℚ-Cartier Weil divisor (ϕ = 0)/Γ ∼ b_{1}S_{1} + b_{2}S_{2}. In particular, D := (z = 0)/Γ ∈ | − K_{X}| and H = (u = 0) are as in 2.1,

3. the completion of X along C is isomorphic to X̂ given in Theorem 2.2.

Proof. On U_{i} = {x_{3−i} ≠ 0}, we normalize x_{3−i} = 1 and set ξ_{i} := x_{i}, η_{i} := y_{i} and ζ_{i} := z with the relation ξ_{i}η_{i} = G_{i}(ζ_{i}, 1). Note that η_{3−i} is not needed because y_{3−i} = G_{3−i}(ζ_{i}^{m_{3−i}}, ξ_{i}^{δ}). The stabilizer Γ_{i} ≃ ℤ_{m_{i}} of x_{3−i} acts on (ξ_{i}, η_{i}, ζ_{i}, u) via the grading (1, −1, a_{i}, 0) mod (m_{i}), and the quotient is isomorphic to U_{i}. The rest of the assertion 1 follows from [7, 11]. The patching of the coordinates is obtained by

γ(ξ_{1}, 1, ζ_{1}) = (1, ξ_{2}, ζ_{2}).

Indeed we obtain γ_{1} = ξ_{1}^{−1}, γ_{2} = ξ_{2} (whence ξ_{1}^{m_{1}} = ξ_{2}^{−m_{2}}) and the relation for ζ_{i}’s: ξ_{1}^{−a_{1}}ξ_{2}^{a_{2}−m_{2}}ζ_{1} = ζ_{2}. This proves the assertion 3, and the rest is obvious (cf. Remark 2.3).

Remark 2.10. In 2.1, if we assume m_{1}, m_{2} ≥ 1 and that there is a Du Val member C ⊂ D ∈ | − K_{X}| whose minimal resolution has the dual configuration

∘ − ⋯ − ∘ − C′ − ∘ − ⋯ − ∘,

then Theorem 2.2 still holds. In this case, the axial multiplicity α_{i} is undefined and Remark 2.4.3 is irrelevant for i such that m_{i} = 1, and most importantly a general member of | − K_{X}| does not contain C. That is, X ⊃ C is an easy case of k1A.

In Definition 2.8, we assume m_{1}, m_{2} ≥ 1. This allows us to treat k2A and some easy case of k1A with no changes in our treatment.

3. A division algorithm

3.1. We maintain the notation and the assumptions of Definition 2.8. We note that if G_{i}(0, 1) = u^{α_{i}}v_{i}^{ρ_{i}δ} for some unit v_{i} ∈ ℂ[[u]] then replacing x_{i}, y_{i} by x_{i}v_{3−i}^{−1}, y_{i}v_{3−i}, we may assume G_{i}(0, 1) = u^{α_{i}}. In other words, we may further assume

G_{i}(T_{1}, T_{2}) = T_{1}^{ρ_{i}} + ⋯ + u^{α_{i}}T_{2}^{ρ_{i}}

without loss of generality.

We will study when X ⊃ C ≃ ℙ^{1} is a flipping nbd.

Definition 3.2. In addition to the above G_{1}(T_{1}, T_{2}), G_{2}(T_{1}, T_{2}), we introduce G_{i}(T_{1}, T_{2}) (i = 3, 4) as follows:

G_{i}(T_{1}, T_{2}) := G_{i−2}(u^{α_{i−2}}T_{2}, T_{1})/u^{α_{i−2}} ∈ ℂ[[u]][T_{1}, T_{2}] (i = 3, 4).

We note that G_{i}(T_{1}, T_{2}) is homogeneous of degree ρ_{i} = ρ_{i−2} and is of the form

G_{i}(T_{1}, T_{2}) = T_{1}^{ρ_{i}} + ⋯ + u^{α_{i}}T_{2}^{ρ_{i}},

where α_{i} = α_{i−2}(ρ_{i−2} − 1). We remark that G_{i} ≢ T_{1}^{ρ_{i}} mod (u) if ρ_{i−2} = 1.

For a positive integer a and an integer x, let x umod a be the integer y ∈ [1, a] such that y ≡ x mod a. For arbitrary i ∈ ℤ, we use the following notation:

G_{i} := G_{i umod 4}, ρ_{i} := ρ_{i umod 4}, α_{i} := α_{i umod 4}, α_{i,2} := α_{i umod 2}.

We note then the obvious ρ_{i} = ρ_{i−2} and the following formula

G_{i}(T_{1}, T_{2}) = G_{i−2}(u^{α_{i−2,2}}T_{2}, T_{1})/u^{α_{i−2}} (∀i). (3.1)

Let d(n) ∈ ℤ (n ∈ ℤ) be a sequence determined by

d(1) = m_{1}, d(2) = m_{2}, d(n + 1) + d(n − 1) = δρ_{n}d(n) (∀n).

Let e(n) ∈ ℤ (n ∈ ℤ) be another sequence determined by

e(0) = 0, e(1) = −α_{1}, e(2) = −α_{2}, e(3) = 0,

e(n + 1) + e(n − 1) = δρ_{n}e(n) + δα_{n−2} − α_{n−1,2} (n ≠ 1, 2),

Let ε := (ρ_{1}ρ_{2}δ)^{2} − 4ρ_{1}ρ_{2}, the discriminant of the quadratic form q(x_{1}, x_{2}) := ρ_{1}x_{1}^{2} − ρ_{1}ρ_{2}δx_{1}x_{2} + ρ_{2}x_{2}^{2}.

Lemma 3.3. Let δ, ρ_{n} be as above and let ε := (ρ_{1}ρ_{2}δ)^{2} − 4ρ_{1}ρ_{2}. Let x(n) ∈ ℚ be a sequence for n ∈ ℤ such that

x(n) = δρ_{n−1}x(n − 1) − x(n − 2).

Then we have the following.

1. If x(n_{0} − 1) > x(n_{0} + 1) for some n_{0} such that 0 ≤ x(n_{0}), then x(n−1) > x(n+1) for every n ≥ n_{0} such that 0 ≤ x(n_{0}), ⋯, x(n).

2. If x(n_{0}−1) = x(n_{0}+1) for some n_{0}, then x(n_{0}−n) = x(n_{0}+n) for every n. If furthermore x(n_{0}) = x(n_{0}+2) and (x(n_{0}), x(n_{0}+1)) ≠ (0, 0), then ε = 0 and q(x(1), x(2)) = 0.

3. Assume that ε ≥ 0. If x(n_{0} − 1) < x(n_{0} + 1) (resp. x(n_{0} − 1) > x(n_{0} + 1)) for some n_{0}, then x(n − 1) < x(n + 1) (resp. x(n − 1) > x(n + 1)) for every n.

Proof. Draw the graph of the conic C := {(x_{1}, x_{2}) | q(x_{1}, x_{2}) = A}, with some constant A so that (x(2i − 1), x(2i)) ∈ C for some i.

ε ≥ 0  ε < 0

The induction formula for x(n) implies that

(x(2i + 1), x(2i)) ∈ C ⇔ (x(2i − 1), x(2i)) ∈ C (∀i),
(x(2i + 1), x(2i)) ∈ C ⇔ (x(2i + 1), x(2i + 2)) ∈ C (∀i).

So (x(2i+1), x(2i)), (x(2i−1), x(2i)) all lie on C. Except for the second half of the assertion (2), the assertions are obvious from the geometric considerations.

For the second half of the assertion (2), assume that x(n_{0} − 1) = x(n_{0} + 1), x(n_{0}) = x(n_{0} + 2) and (x(n_{0}), x(n_{0} + 1)) ≠ (0, 0). By the first half of the assertion (2), we have x(n) = x(n + 2) for all n. By x(i−1) = x(i+1), we see that the line x_{i} = x(i) is tangent to the conic at the point P = (x(1), x(2)) ≠ (0, 0) for i = 1, 2. This means that P is a singular point of C, whence C is a double line. Hence ε = 0 and A = 0.

Corollary 3.4. If we switch (a_{1}, m_{1}, α_{1}, ρ_{1}, x_{1}, y_{1}, G_{1}) and (a_{2}, m_{2}, α_{2}, ρ_{2}, x_{2}, y_{2}, G_{2}), then (α_{n}, d(n), e(n)) and (α_{3−n}, d(3−n), e(3−n)) are switched for all n. Modulo this switching, we may assume that d(1) > d(3).

Proof. The first assertion is obvious. We note that d(1) = m_{1} > 0 and d(2) = m_{2} > 0. Thus we are also done if d(1) > d(3) or if d(0) > d(2) by Lemma 3.3.1. So we may assume that d(1) ≤ d(3) and d(0) ≤ d(2). By Δ > 0, we have (d(0), d(1)) ≠ (d(2), d(3)) by Lemma 3.3.2. Hence we have either d(1) < d(3) or d(0) < d(2).

If we switch the two sets as above, we have either d(2) < d(0) or d(3) < d(1) after the switch. Thus d(1) > d(3) by Lemma 3.3.1.

Lemma 3.5. Assume that d(1) > d(3) (cf. Corollary 3.4) and that ε < 0. Then d(k) < 0 for some k ≤ 5.

Proof. By ε = ρ_{1}ρ_{2}(δ^{2}ρ_{1}ρ_{2} − 4) < 0, we have δ = 1 and ρ_{1}ρ_{2} ≤ 3.

Assume first that ρ_{1} = 1 and ρ_{2} ≤ 3. By d(3) = ρ_{2}m_{2} − m_{1} < m_{1}, we have ρ_{2}m_{2} < 2m_{1}. Thus the lemma follows from

d(5) = ρ_{2}(ρ_{2} − 2)m_{2} − (ρ_{2} − 1)m_{1}
< (ρ_{2} − 2)2m_{1} − (ρ_{2} − 1)m_{1}
= (ρ_{2} − 3)m_{1} ≤ 0.

Assume next that ρ_{2} = 1 and ρ_{1} ≤ 3. By d(3) = m_{2} − m_{1} < m_{1}, we have m_{2} < 2m_{1}, and we are done by

d(5) = (ρ_{1} − 2)m_{2} − (ρ_{1} − 1)m_{1}
< (ρ_{1} − 2)2m_{1} − (ρ_{1} − 1)m_{1}
= (ρ_{1} − 3)m_{1} ≤ 0.

Remark 3.6. 1. We note that e(4) = δα_{1} > 0, e(5) = (δ^{2}ρ_{2}−1)α_{1}+ δα_{2} > 0 and

e(6) = (δ^{2}ρ_{2} + ρ_{1} − 3)δρ_{1}α_{1} + (δ^{2}ρ_{1} − 1)α_{2}.

In particular, e(6) ≤ 0 implies that δ = 1, ρ_{1} = 1 and ρ_{2} = 1, 2.

2. If we set

e_{0}(n) := −α_{n−2}/ρ_{n} (∀n),

then e_{0}(n)’s satisfy the conditions for e(n) except for the values of e(0), ⋯, e(3). Therefore if we put e_{1}(n) = e(n) − e_{0}(n), then

e_{1}(0) = α_{2}/ρ_{2}, e_{1}(1) = −α_{1}/ρ_{1}, e_{1}(2) = −α_{2}/ρ_{2}, e_{1}(3) = α_{1}/ρ_{1}

and the following induction formula holds.

e_{1}(n + 1) + e_{1}(n − 1) = δρ_{n}e_{1}(n) (n ≠ 1, 2).

Corollary 3.7. Assume that ε ≥ 0. Then e_{1}(n) > e_{1}(n − 2) for all n; e(n) > 0 for all n ≥ 4; e(n) ≥ α_{1} + α_{2} for all n ≥ 7.

Proof. We have e_{1}(2) = −α_{2}/ρ_{2} and e_{1}(4) = δα_{1} + α_{2}/ρ_{2}. If we temporarily mean the coefficient of α_{i} with the subscript α_{i}, then we have e_{1}(4)_{α_{i}} > e_{1}(2)_{α_{i}} for i = 1, 2. By ε ≥ 0, we can apply Lemma 3.3 to e_{1}(n)_{α_{i}} and obtain e_{1}(n)_{α_{i}} > e_{1}(n−2)_{α_{i}} for all n ≥ 4 and i = 1, 2. Since e_{0}(n) depends only on n mod (4), we have e(n) ≥ e(n−4)+α_{1}+α_{2} for all n ≥ 6. Also by e_{0}(n) ∈ α_{n,2} · [−1, 0], we have e(n)_{α_{i}} − e(n − 2)_{α_{i}} > 0 (resp. ≥ 0) for i ≢ n (resp. i ≡ n) mod (2) if n ≥ 4. In other words, we have e(n) ≥ e(n − 2) + α_{n+1,2} for all n ≥ 4.

By e(3) = 0 and e(4) = δα_{1}, we have e(5) ≥ α_{2}, e(6) ≥ α_{1} and e(n) ≥ α_{1} + α_{2} (n ≥ 7).

Corollary 3.8. Assume that d(1) > d(3) (cf. Corollary 3.4). Let k be the smallest integer ≥ 3 such that d(k) ≤ 0. (The integer k exists by Lemma 3.3.) Then e(n) > 0 if 4 ≤ n ≤ k + 1.

Proof. We note that e(4), e(5) > 0 by Remark 3.6.1. Thus we are done if k ≤ 4. If ε ≥ 0, then e(n) > 0 for all n ≥ 4 by Corollary 3.7. Thus we are also done if ε ≥ 0.

Thus we may assume that ε < 0 and d(4) > 0. Hence k = 5 by Lemma 3.5. It is enough to derive a contradiction assuming e(6) ≤ 0. By Remark 3.6.1, we have δ = ρ_{1} = 1 and ρ_{2} = 1, 2. We have

m_{1} > d(3) = ρ_{2}m_{2} − m_{1} > 0,
d(4) = d(3) − m_{2} = (ρ_{2} − 1)m_{2} − m_{1} > 0.

From the second equation, we have ρ_{2} = 2. We have m_{1} > m_{2} from the first and m_{2} > m_{1} from the second. This is a contradiciton.

Definition 3.9. Let S_{i} = (x_{i} = 0)/Γ and D = (z = 0)/Γ be ℚ-Cartier Weil divisors on X by Theorem 2.9.2, then we have −a_{1}S_{1} + (m_{2} − a_{2})S_{2} + D ∼ 0, m_{1}D ∼ δS_{2} and m_{2}D ∼ δS_{1}.

Then we introduce the following sections and divisors.

F_{0} := y_{1} ∈ H^{0}(X, O(L_{0})), where L_{0} := δρ_{1}L_{1} − L_{2},
F_{1} := x_{2} ∈ H^{0}(X, O(L_{1})), where L_{1} := S_{2}.
F_{2} := x_{1} ∈ H^{0}(X, O(L_{2})), where L_{2} := S_{1},
F_{3} := y_{2} ∈ H^{0}(X, O(L_{3})), where L_{3} := δρ_{2}L_{2} − L_{1}.

We note that the formulas

F_{n−1}F_{n+1} = G_{n}(z^{d(n)}, F_{n}^{δ}) (n = 1, 2)

can be rewritten in the form

F_{n−1}F_{n+1} = G_{n−2}(F_{n}^{δ}, z^{d(n)}u^{e(n)})u^{α_{n}} (n = 1, 2).

by the formula (3.1).

These L_{i} and F_{i} are extended as follows.

Theorem 3.10. Let n_{0}, n_{1} ∈ ℤ be such that n_{0} ≤ 1, 2 ≤ n_{1} and

\begin{matrix}d(n) > 0 & if n ∈ [n_{0}, n_{1}], \\ e(n) > 0 & if n ∈ [n_{0}, n_{1}] \ [0, 3] \end{matrix}

Then L_{0}, ⋯, L_{3} and F_{0}, ⋯, F_{3} can be extended to divisors L_{n} and F_{n} ∈ H^{0}(X, O(L_{n})) for n ∈ [n_{0} −1, n_{1} +1] such that the following hold. 0_{n}. L_{n−1} + L_{n+1} = δρ_{n}L_{n}, if n ∈ [n_{0}, n_{1}].

1_{n}. F_{n}, F_{n−1} are relatively prime on X (that is, {F_{n} = F_{n−1} = 0}

contains no divisors on X), if n ∈ [n_{0}, n_{1} + 1]. 2_{n}. F_{n}, zu are relatively prime on X, if n ∈ [n_{0} − 1, n_{1} + 1].

3_{n}. F_{n−1}F_{n+1} = \begin{cases}G_{n}(z^{d(n)}, F_{n}^{δ}) = G_{n−2}(F_{n}^{δ}, z^{d(n)}u^{e(n)})u^{α_{n}} & (n = 1, 2), \\ G_{n−2}(F_{n}^{δ}, z^{d(n)}u^{e(n)}) & (n ≠ 1, 2), \end{cases}

if n ∈ [n_{0}, n_{1}].

Proof. By Corollary 3.4, we only need to consider n ≥ 2. Thus we set n_{0} = 1 and use induction on n_{1}.

The theorem is obvious if n_{1} = 2. Assume that n_{1} ≥ 3 and let n = n_{1} ≥ 3. By the induction hypotheses, it is enough to define L_{n+1} by the assertion 0_{n}, construct F_{n+1} satisfying the assertion 3_{n} and prove the assertions 1_{n+1} and 2_{n+1}.

We will construct F_{n+1} satisfying 3_{n} using 1_{n−1}, 2_{n−1}, 3_{n−2} and 3_{n−1}. During this proof, ≡ denotes the congruence modulo the ideal F_{n−1}ℂ[[u]][F_{n−3}, F_{n−2}, F_{n−1}, F_{n}, z] unless otherwise mentioned. We first claim that

F_{n−2}^{δ}F_{n}^{δ} ≡ z^{d(n−2)}u^{e(n−2)+α_{n−2,2}} · z^{d(n)}u^{e(n)}.

We note that the claim is obvious for n = 3 by F_{1}^{δ}F_{3}^{δ} ≡ z^{δρ_{2}d(2)}. If n ≥ 4, the claim follows from

F_{n−2}^{δ}F_{n}^{δ} = G_{n−3}(F_{n−1}^{δ}, z^{d(n−1)}u^{e(n−1)})^{δ}
≡ z^{δρ_{n−3}d(n−1)}u^{δρ_{n−3}e(n−1)+δα_{n−3}}
= z^{d(n−2)}u^{e(n−2)+α_{n−2,2}} · z^{d(n)}u^{e(n)}.

In the following, we use a temporary notation that # denotes any sufficiently large integer. Let

M := F_{n−2}^{δρ_{n−2}}G_{n−2}(F_{n}^{δ}, z^{d(n)}u^{e(n)})z^{#}u^{#}.

Then by the above claim, we have the following.

M = G_{n−2}(F_{n−2}^{δ}F_{n}^{δ}, F_{n−2}^{δ}z^{d(n)}u^{e(n)})z^{#}u^{#}
≡ G_{n−2}(z^{d(n−2)}u^{e(n−2)+α_{n−2,2}}, F_{n−2}^{δ})z^{#}u^{#}
= G_{n−4}(F_{n−2}^{δ}, z^{d(n−2)}u^{e(n−2)})z^{#}u^{#}
= F_{n−1}F_{n−3}z^{#}u^{#}
≡ 0.

Thus G_{n−2}(F_{n}^{δ}, z^{d(n)}u^{e(n)}) (or G_{n}(z^{d(n)}, F_{n}^{δ}) if n = 1, 2) vanishes on the divisor (F_{n−1} = 0) by 1_{n−1} and 2_{n−1}. Hence we obtain F_{n+1} ∈ H^{0}(X, O(L_{n+1})) as claimed.

We will prove 1_{n+1} and 2_{n+1} using 2_{n} and 3_{n}. We see that F_{n+1} = zu = 0 implies F_{n} = 0 by the formula 3_{n}. Indeed one can use d(n), e(n) > 0 if n ≥ 4 and d(3) > 0 and G_{1}(T_{1}, T_{2}) ≡ T_{1}^{ρ_{1}} mod (u) if n = 3. Thus by 2_{n}, F_{n+1} and zu are relatively prime on X, which is 2_{n+1}. F_{n} = F_{n+1} = 0 implies zu = 0. So by 2_{n}, F_{n}, F_{n+1} are relatively prime on X, which is 1_{n+1}.

Definition 3.11. By Corollary 3.4, we will assume that d(1) > d(3). Let k ≥ 3 be the smallest integer such that d(k) ≤ 0, which exists by Corollary 3.3.1. Then we have

d(1), d(2), ⋯, d(k − 1) > 0,
e(4), ⋯, e(k + 1) > 0,

by Corollary 3.8.

By Theorem 3.10, ℚ-Cartier Weil divisors L_{i} and sections F_{i} ∈ H^{0}(X, O_{X}(L_{i})) (i = 0, ⋯, k) satisfy the following.

0. L_{n−1} + L_{n+1} = δρ_{n}L_{n}, if 1 ≤ n ≤ k − 1.

1. F_{n}, F_{n−1} are relatively prime on X if 1 ≤ n ≤ k.

2. F_{n}, zu are relatively prime on X, if 0 ≤ n ≤ k.

3. F_{n−1}F_{n+1} = \begin{cases}G_{n}(z^{d(n)}, F_{n}^{δ}) = G_{n−2}(F_{n}^{δ}, z^{d(n)}u^{e(n)})u^{α_{n}} & (n = 1, 2), \\ G_{n−2}(F_{n}^{δ}, z^{d(n)}u^{e(n)}) & (n ≠ 1, 2), \end{cases}

if 1 ≤ n ≤ k − 1. We then introduce the modified sequence d^{∗}(n) for the uniform treat-

ment of F_{n} as follows.

d^{∗}(n) = \begin{cases}d(n) & (n ≤ k), \\ −d(n − 2) & (n ≥ k + 1). \end{cases}

The following is one of the key results that the exceptional locus C of X is a set-theoretic complete intersection of two divisors: F_{k} = F_{k+1} = 0 (cf. [4, 20.11]).

Theorem 3.12. Under the notation and the assumptions of Definition 3.11, we have

F_{k+1} := \frac{G_{k−2}(F_{k}^{δ}z^{−d(k)}, u^{e(k)})}{F_{k−1}} = \frac{G_{k−2}(F_{k}^{δ}, z^{d(k)}u^{e(k)})z^{−ρ_{k−2}d(k)}}{F_{k−1}}

belongs to H^{0}(X, O(L_{k+1})), where L_{k+1} := −L_{k−1}.

Furthermore, F_{k} and F_{k+1} satisfy the following.

1. C = {F_{k+1} = F_{k} = 0} as a set.

2. C = {F_{k+1} = u = 0} as a set.

Proof. The proof that F_{k+1} is a regular section of O(L_{k+1}) is similar to the one for F_{n+1} in Theorem 3.10, and we omit it.

The assertion 1 is immediately reduced to 2. Indeed F_{k+1} = F_{k} = 0 implies u = 0 by the definition of F_{k+1} (note that e(k) > 0 if k ≥ 4). It remains to prove the assertion 2.

Let F_{n}|_{H} denote the restriction of F_{n} to H and (F_{n}|_{H}) the divisor defined by F_{n}|_{H} = 0. We note

D ∩ H = ℓ_{1}  +ℓ_{2}  +C
(F_{1}|_{H}) =  m_{2}ρ_{2}ℓ_{2}
(F_{2}|_{H}) = m_{1}ρ_{1}ℓ_{1}
(F_{3}|_{H}) = m_{2}ρ_{2}ℓ_{1}  +C

We claim

(F_{n}|_{H}) ≡ ρ_{n−1}d(n − 1)ℓ_{1} mod ℤC for n ∈ [2, k]. (3.2)

We prove the claim by induction on n, where the cases n = 2, 3 are checked. Assume that the claim is proved up to n (≤ k − 1). By Definition 3.11, we have

(F_{n+1}|_{H}) ≡ (δρ_{n−2}ρ_{n−1}d(n − 1) − ρ_{n−2}d(n − 2))ℓ,
≡ ρ_{n}d(n)ℓ_{1} mod ℤC.

Thus the claim is proved. We then have

(F_{k+1}|_{H}) ≡ ρ_{k−2}(δ · (F_{k}|_{H}) − d(k)D) − (F_{k−1}|_{H})
≡ 0 mod ℤC.

Hence C = {F_{k+1} = u = 0}, and we are done.

There is another important division.

Theorem 3.13. Under the notation and the assumptions of Theorem 3.12, we have

F_{k+2} := \frac{G_{k−1}(F_{k+1}^{δ}z^{d(k−1)}, u^{e(k+1)})}{F_{k}}
= \frac{G_{k−1}(F_{k+1}^{δ}, z^{d^{∗}(k+1)}u^{e(k+1)})z^{−ρ_{k−1}d^{∗}(k+1)}}{F_{k}}

belongs to H^{0}(X, O(L_{k+2})), where L_{k+2} := −L_{k}.

Furthermore, C = {F_{k−1} = F_{k+2} = u = 0}.

Proof. We will closely follow the proof for Theorem 3.10. In this proof, ≡ denotes the congruence modulo F_{k}ℂ[[u]][F_{k−2}, F_{k−1}, F_{k}, F_{k+1}, z].

By the induction formula of e(n), we have:

F_{k−1}^{δ}F_{k+1}^{δ}z^{d(k−1)} = G_{k−2}(F_{k}^{δ}z^{−d(k)}, u^{e(k)})^{δ}z^{d(k−1)}
≡ z^{d(k−1)}u^{δρ_{k−2}e(k)+δα_{k−2}}
= u^{e(k+1)} · z^{d(k−1)}u^{α_{k−1,2}+e(k−1)}.

Thus for

M := F_{k−1}^{δρ_{k−1}}G_{k−1}(F_{k+1}^{δ}z^{d(k−1)}, u^{e(k+1)})u^{#},

we have the following.

M = G_{k−1}(F_{k−1}^{δ}F_{k+1}^{δ}z^{d(k−1)}, F_{k−1}^{δ}u^{e(k+1)})u^{#}
≡ G_{k−1}(z^{d(k−1)}u^{α_{k−1,2}+e(k−1)}, F_{k−1}^{δ})u^{#}
= G_{k−3}(F_{k−1}^{δ}, z^{d(k−1)}u^{e(k−1)})u^{#}
= F_{k}F_{k−2}u^{#}
≡ 0.

Since F_{k−1}u, F_{k} are relatively prime on X by Theorem 3.10, we see that G_{k−1}(F_{k+1}^{δ}z^{d(k−1)}, u^{e(k+1)}) is divisible by F_{k} and that F_{k+2} is a regular section of L_{k+2}.

For the last assertion, we borrow the notation and the argument in the proof of the previous theorem 3.12. We saw (F_{k−1}|_{H}) ⊂ C ∪ ℓ_{1} there. By the formula (3.2), we have

(F_{k+2}|_{H}) ≡ ρ_{k−1}(δ(F_{k+1}|_{H}) + d(k − 1)(D ∩ H)) − (F_{k}|_{H})
≡ ρ_{k−1}d(k − 1) − ρ_{k−1}d(k − 1) ≡ 0 mod ℤC.

This means (F_{k+2}|_{H}) ⊂ C ∪ ℓ_{2}, which proves the claim.

The following are elementary properties which are immediate to check.

Proposition 3.14. We have

1. d^{∗}(n)L_{n+1} − d^{∗}(n + 1)L_{n} ∼ 0 is a generating relation for L_{n}, L_{n+1} in Pic X if 1 ≤ n ≤ k + 1,

2. (L_{n} · C) = d^{∗}(n)/(m_{1}m_{2}) if 1 ≤ n ≤ k + 2.

Proof. The case n = 1 of the first assertion is on m_{1}S_{1} ∼ m_{2}S_{2}. If iS_{1} − jS_{2} ∼ 0 then it is Cartier at P_{1}. Hence m_{1}|i and iS_{1} − jS_{2} is a multiple of m_{1}S_{1} − m_{2}S_{2}. If 2 ≤ n ≤ k − 1, then we are done by the change of the basis, L_{n+1} ∼ δρ_{n}L_{n} − L_{n−1}:

d(n)L_{n+1} − d(n + 1)L_{n} ∼ d(n)(δρ_{n}L_{n} − l_{n−1}) − d(n + 1)L_{n}
∼ d(n − 1)L_{n} − d(n)L_{n}.

The cases n = k, k + 1 follow from the case n = k − 1. The first assertion is thus proved. The second assertion follows from the first because (L_{1} · C) = 1/m_{2} and (L_{2} · C) = 1/m_{1}.

Proposition 3.15. Let c(i) (i ∈ [1, k + 2]) be a sequence determined by

i. c(1) = a_{1}, c(2) = m_{2} − a_{2}, ii. c(n + 1) = δρ_{n}c(n) − c(n − 1) for n ∈ [2, k − 1], iii. c(k + 1) = −c(k − 1), c(k + 2) = −c(k). Then we have

1. −K_{X} ∼ c(n)L_{n+1} − c(n + 1)L_{n} if 1 ≤ n ≤ k + 1,

2. c(n)d^{∗}(n + 1) − c(n + 1)d^{∗}(n) = δ for all n ∈ [1, k + 1],

3. c(n) and d^{∗}(n) are relatively prime for all n ∈ [1, k + 2], and 4. −d^{∗}(n)K_{X} ∼ δL_{n} for all n ∈ [1, k + 1].

Proof. The case n = 1 of the assertion 1 follows from −K_{X} ∼ a_{1}L_{2} − (m_{2} − a_{2})L_{1}. One can check the case n ∈ [2, k − 1] inductively by using L_{n+1} ∼ δρ_{n}L_{n} − L_{n−1} (n ∈ [2, k − 1]) as follows:

c(n)L_{n+1} − c(n + 1)L_{n} ∼ c(n)(δρ_{n}L_{n} − L_{n−1}) − c(n + 1)L_{n}
∼ c(n − 1)L_{n} − c(n)L_{n−1} ∼ −K_{X}.

The cases n = k, k + 1 are equivalent to the case n = k − 1 because L_{k+1} ∼ −L_{k−1} and L_{k+2} ∼ −L_{k}. This proves the assertion 1.

By the induction formula, we immediately see that the value

c(n)d(n + 1) − c(n + 1)d(n) = c(n − 1)d(n) − c(n)d(n − 1)

does not depend on n and hence equal to δ = c(1)d(2)−c(2)d(1), which proves the assertion 2.

Let gcd(n) = (c(n), d(n)), then gcd(n) divides δ. By the induction formula, we see that d(i) ≡ −d(i − 2) and c(i) ≡ −c(i − 2) modulo gcd(n) for all i. This implies that gcd(n) divides gcd(1) or gcd(2) depending on the parity of n. Since gcd(1) = (a_{1}, m_{1}) = 1 and gcd(2) = (a_{2}, m_{2}) = 1, we get gcd(n) = 1, the assertion 3.

The assertion 4 follows from Proposition 3.14.

−d^{∗}(n)K_{X} ∼ c(n)d^{∗}(n)L_{n+1} − c(n + 1)d^{∗}(n)L_{n}
∼ (c(n)d^{∗}(n + 1) − c(n + 1)d^{∗}(n))L_{n} = δL_{n}.

4. Contractions and flips

In this section, we give an explicit description of the contractions and the flips using the divisions in Section 3.

Although we work on the specific model X, our description can treat arbitrary extremal nbd of type k2A by passing to the formal completion or the associated analytic space by Theorems 2.2 and 2.9.

First by Theorem 3.12 alone (without the further division), we can decide exactly when X ⊃ C is a flipping nbd as follows.

Corollary 4.1. Let X ⊃ C ≃ ℙ^{1} be the scheme introduced in 3.1. Under the notation and the assumptions of Theorem 3.12, we have

1. If d(k) < 0 then the formal completion X̂ and the associated algebraic space of X ⊃ C are flipping nbds.

2. If d(k) = 0 then X̂ ⊃ C is not a flipping nbd. Indeed, C is a fiber of a divisorial contraction of X̂ (or the algebraic space X) and {F_{k+1} = 0} is the exceptional divisor for the contraction.

Proof. By Theorem 3.12, C is a set-theoretic complete intersection of two Cartier divisors N_{1} := a(F_{k} = 0) ∼ aL_{k} and N_{2} := a(F_{k+1} = 0) ∼ aL_{k+1} for some integer a > 0.

Assume that d(k) < 0. Then −N_{1} and −N_{2} are ample on C by (L_{k} · C), (L_{k+1} · C) < 0. Then the defining ideal J of N_{1} ∩ N_{2} has the property that J/J^{2} is ample on C = Supp (O_{X}/J). Thus C ⊂ X̂ can be contracted by [2, 6.2] and the associated algebraic space can be contracted by [2, 3.1]. Since (K_{X} · C) < 0, these are flipping contractions.

Assume thet d(k) = 0. Then aL_{k} ∼ 0 and (L_{k+1} · C) < 0. Then F_{k}^{a} : X → 𝔸^{1} induces, on the divisor N_{2}, a morphism g : N_{2} → 𝔸^{1} such that C = g^{−1}(0) as a set. We note that O_{N_{2}}(−N_{2}) is g-ample by (L_{k+1} · C) < 0. Thus we can similarly see that N_{2} can be contracted to a curve such that C is one of its set-theoretic fiber by a birational contraction of the formal completion (and also the associated algebraic space) of X ⊃ C. We note that −K_{X} is relatively ample and N_{2} is exceptional with respect to the contraction.

The extremal contraction of X̂ is expressed as Spec H^{0}(X̂, OX̂) (or its formal scheme version). We give here an explicit construction using the further division Theorems 3.12 and 3.13.

Definition 4.2. Let

y′_{1} := F_{k+2} ∈ H^{0}(X, O(−L_{k})),
x′_{2} := F_{k+1} ∈ H^{0}(X, O(L_{k+1})),
x′_{1} := F_{k} ∈ H^{0}(X, O(L_{k})),
y′_{2} := F_{k−1} ∈ H^{0}(X, O(−L_{k+1})),
z  ∈ H^{0}(X, O(c(k)L_{k+1} + c(k − 1)L_{k})),

on which we have the Γ-action defined in Remark 2.7. We rewrite the action as follows.

By ℤL_{1} + ℤL_{2} = ℤL_{k} + ℤL_{k+1} ⊂ Pic X (Proposition 3.14.1), we set

Γ′ := Hom (ℤL_{k} + ℤL_{k+1}, ℂ^{∗}) = Hom (ℤL_{1} + ℤL_{2}, ℂ^{∗}) = Γ,

and γ′ ∈ Γ′ acts on H^{0}(X, O(L_{i})) as the multiplication by γ′(L_{i}) ∈ ℂ^{∗}. Let m′_{1} := d(k − 1) > 0, m′_{2} := −d(k) ≥ 0. Then we have m′_{2}L_{k+1} ∼ m′_{1}L_{k} (Proposition 3.14) and hence

Γ′ = {γ′ = (γ′_{1}, γ′_{2}) ∈ (ℂ^{∗})^{2} | (γ′_{1})^{m′_{1}} = (γ′_{2})^{m′_{2}}}.

The Γ-action is equivalent to the Γ′-action given by

γ′(x′_{i}, z, y′_{i}, u) = ((γ′_{i})x′_{i}, (γ′_{1})^{c(k−1)}(γ′_{2})^{c(k)}z, (γ′_{i})^{−1}y′_{i}, u).

Γ′ acts on the ring R′ := ℂ[[u]][x′_{1}, y′_{1}, x′_{2}, y′_{2}, z], the ideal

I := (x′_{1}y′_{1} − G_{k−1}((x′_{2})^{δ}z^{m′_{1}}, u^{e(k+1)}), x′_{2}y′_{2} − G_{k−2}((x′_{1})^{δ}z^{m′_{2}}, u^{e(k)}))

and the scheme W′ := Spec R′/I′. We note that it is easy to check that W′ is a complete intersection and is an integral domain by Proposition 4.8 and that W′ is normal by the Jacobian criterion. Let

(R′/I′)^{Γ′} := {r ∈ R′/I′ | γ′r = r}

and Y := Spec (R′/I′)^{Γ′} with the origin 0. Because of the construction, we have a natural morphism π : X → Y.

Theorem 4.3. There is an open subset U ∋ 0 of Y such that π : π^{−1}(U) → U is either a flipping contraction with C the only flipping curve (the case m′_{2} > 0), or a divisorial contraction with (F_{k+1} = 0) the only exceptional divisor (the case m′_{2} = 0).

Proof. First of all, W (cf. Definition 2.8) and W′ are birationally equivalent because of the inductive formulas in Defintion 3.11 and Theorems 3.12 and 3.13. The birational map is Γ-equivariant as explained in Definition 4.2. Because of these, it is easy to see that π is birational.

Next, we claim that π^{−1}(0) = C as a set, and prove it in two cases.

Case 1 (m′_{2} > 0). For arbitrary i, j, we have u, (x′_{i})^{a}(y′_{j})^{b} ∈ (R′/I′)^{Γ′} for some positive integers a, b depending on x′_{i}, y′_{j}. Thus

C ⊂ π^{−1}(0) ⊂ {x′_{1} = x′_{2} = 0} ∪ {y′_{1} = y′_{2} = u = 0}.

Thus by Theorems 3.12 and 3.13, we have π^{−1}(0) = C as a set.

Case 2 (m′_{2} = 0). We have (y′_{1})^{a}, (x′_{1})^{a}, x′_{2}y′_{2}, u ∈ (R′/I′)^{Γ′} for some positive integer a. Thus

C ⊂ π^{−1}(0) ⊂ {x′_{1} = x′_{2} = 0} ∪ {y′_{1} = y′_{2} = u = 0}.

The rest is the same as Case 1. This proves the claim.

By [9], π can be extended to a proper birational morphism π̄ : X̄ → Y. Then we have π^{−1}(0) = C. Indeed, by the normality of Y, π^{−1}(0) is a connected set containing C as a connected component.

Thus it is enough to set U = Y \ π̄(X̄ \ X) to make π proper above it. Shrink U further so that L_{k−1} is π-ample over U.

Assume that m′_{2} > 0. Then C = π^{−1}(0) is a set-theoretic complete intersection of two π-negative divisors (F_{k} = 0) and (F_{k+1} = 0). Every π-exceptional curve ⊂ π^{−1}(U) is contained in these divisors. Thus C is the only π-exceptional curve ⊂ π^{−1}(U).

Assume next that m′_{2} = 0. In this case, the arguments are similar to those in the proof of Corolary 4.1.2. C = (F_{k} = 0) ∩ (F_{k+1} = 0) being π-exceptional and F_{k} ∼ 0 imply that F_{k+1} is contracted by π. Then, −F_{k+1} being π-ample implies that F_{k+1} contains all the curves contracted by π.

We will closely study the divisorial contraction or the flip as follows.

Definition 4.4. Let a′_{1} := c(k−1) umod (m′_{1}) (cf. Definition 3.2), and if m′_{2} > 0 then we also let a′_{2} := c(k) umod (m′_{2}). Since (c(i), d(i)) = 1 by Proposition 3.15, we have (m′_{i}, a′_{i}) = 1 and 0 < a′_{i} ≤ m′_{i} if m′_{i} > 0.

Theorem 4.5. With the above notation and assumptions, assume further that d(k) = 0. Then m′_{1} = d(k − 1) = δ = gcd(m_{1}, m_{2}), c(k) = −1 and we have a terminal singularity of index m′_{1},

0 ∈ Y ≃ (ξ, η, ζ, u; ξη − G_{k−1}(ζ^{m′_{1}}, u^{e(k+1)}))/ℤ_{m′_{1}}(1, −1, a′_{1}, 0),

where the π-fundamental set is the curve {ζ = G_{k−2}(ξ^{m′_{1}}, u^{e(k)}) = 0}/ℤ_{m′_{1}} under the identification.

Proof. By the induction formula, we see gcd(m_{1}, m_{2}) = gcd(d(i), d(i + 1)) for all i. In particular, we have d(k − 1) = gcd(m_{1}, m_{2}). By Proposition 3.15, we have c(k) = ±1 and −c(k)d(k − 1) = δ. Thus c(k) = −1 and m′_{1} = d(k − 1) = δ.

By d(k) = 0, we have Γ′ = ℤ_{m′_{1}} ×ℂ^{∗} and can obtain the isomorphism by taking the invariants in two steps. We note that ξ = x′_{1}, η = y′_{1} and ζ = x′_{1}z. Since the fundamental set on Y is defined by x′_{2}y′_{2} = x′_{2}z = 0, we are done by x′_{2}y′_{2} = G_{k−2}(ξ^{m′_{1}}, u^{e(k)}).

Definition 4.6. Let

X′ := (W′ \ {x′_{1} = x′_{2} = 0})/Γ′ ⊃ C′ := {y′_{1} = y′_{2} = z = u = 0}/Γ′

and P′_{i} the point, x′_{i} = y′_{1} = y′_{2} = z = u = 0. We note that C′ ≃ ℙ^{1}. Let π′ : X′ → Y be the induced morphism.

Theorem 4.7. With the above notation and the assumptions, assume further that d(k) < 0. Then we have 1. X′ is a normal scheme of dimension 3 such that X′ \ {P′_{i}, P′_{2}} is

smooth and the germ

P′_{i} ∈ X′ ≃ (ξ′_{i}, η′_{i}, ζ′_{i}, u; ξ′_{i}η′_{i} = G′_{i}(ζ′_{i}^{m′_{i}}, 1))/ℤ_{m′_{i}}(1, −1, a′_{i}, 0)

is a terminal singularity of index m′_{i} and P_{i} ∈ C′ = ξ′_{i}-axis/ℤ_{m′_{i}} under the identification, where

G′_{i}(T_{1}, T_{2}) := G_{k−i}(T_{1}, u^{e(k+2−i)}T_{2}) (i = 1, 2),

2. X′ is proper and is the flip of X over some open set ∋ 0 of Y.

Proof. The proof of the first assertion is similar to the one for Theorem 2.9, and we omit it.

As in the proof of Theorem 4.3, we see that π′ is a birational morphism. Although X′ is proper over X, we only claim it over an open set ∋ 0. For this we only need to show (π′)^{−1}(0) = C′ as in the proof of Theorem 4.3.

For arbitrary i, j, we have (x′_{i})^{a}(y′_{j})^{b}, (x′_{i})^{a}z^{c} ∈ (R′/I′)^{Γ′} for some positive integers a, b, c depending on x′_{i}, y′_{j}, z. Thus

(π′)^{−1}(0) ⊂ {x′_{1} = x′_{2} = 0} ∪ {y′_{1} = y′_{2} = z = u = 0} = C′,

and the properness is settled.

By the construction of W′ and by C = {F_{k} = F_{k+1} = 0} (Theorem 3.12), we have a natural birational morphism X\C → X′. By Theorem 4.3, X \ C ≃ X′ \ C′ over an open set U ∋ 0 of Y. It only remains to show that (K_{X′} · C′) > 0.

Let S′_{1} := (x′_{1} = 0)/Γ′ be the ℚ-Cartier divisor on X′. Then (S′_{1} · C′) > 0. Since X ≃ X′ in codimension 1, we can pull back S′_{1} and K_{X′} on X′ to L_{k} ∼ (F_{k} = 0) and K_{X} on X. By −d(k)K_{X} ∼ δL_{k} on X (Proposition 3.15), we have −d(k)K_{X′} ∼ δS′_{1}. Hence (K_{X′} · C′) > 0 as required.

We used the following elementary result in this section. We give a proof for the readers’ convenience.

Proposition 4.8. Let A be an integral domain, and x_{1}, x_{2}, u_{1}, u_{2} ∈ A. Assume that x_{1}, x_{2} are prime elements (that is, x_{1}A, x_{2}A are non-zero prime ideals) and that (x_{1}, x_{2}) is a prime ideal ≠ x_{1}A, x_{2}A. Then

1. If u_{1} ∉ x_{1}A, then A[y_{1}]/(x_{1}y_{1} − u_{1}) is an integral domain.

2. If u_{1} ∉ (x_{1}, x_{2}) and u_{2} ∉ x_{2}A, then A[y_{1}, y_{2}]/(x_{1}y_{1} −u_{1}, x_{2}y_{2} −u_{2}) is an integral domain.

Proof. Let P := {f(y_{1}) ∈ A[y_{1}] | f(u_{1}/x_{1}) = 0}.

We claim that if f(y_{1}) = a_{n}y_{1}^{n} + ⋯ + a_{0} ∈ P \ {0}, then n > 0 and a_{n} ∈ x_{1}A. Indeed n ≥ deg f ≥ 1 is obvious, and from

a_{n}u_{1}^{n} + x_{1}(a_{n−1}u_{1}^{n−1} ⋯ + a_{0}x_{1}^{n−1}) = 0,

we get a_{n} ∈ x_{1}A by u_{1} ∉ x_{1}A. This proves the claim.

For the assertion 1, it is enough to prove that P = (x_{1}y_{1} − u_{1}).

Let f ∈ P \{0}. By the claim, we can lower deg f modulo (x_{1}y_{1}−u_{1}). Hence the assertion 1 is proved by induction on deg f.

For the assertion 2, let S = A[y_{1}]/(x_{1}y_{1} − u_{1}), which is an integral domain. We note that S/x_{2}S ≃ (A/x_{2}A)[y_{1}]/(x_{1}y_{1} − u_{1}) is an integral domain by u_{1} ∉ (x_{1}, x_{2}). We claim that u_{2} mod x_{2}S ≠ 0. Indeed u_{2} mod x_{2}A is a non-zero constant of the integral domain (A/x_{2}A)[y_{1}]. Hence u_{2} mod x_{2}A ∉ (A/x_{2}A)[y_{1}](x_{1}y_{1} − u_{1}), which proves the claim. Finally applying the assertion 1 on S, we obtain the assertion 2.

5. Further discussions

In this section, we consider the case of a base ring which is more general than ℂ[[u]] in Definition 3.2. We note that, over Spec ℤ, our finite group action ℤ_{m} is actually the finite multiplicative group scheme action μ_{n} ⊂ 𝔾_{m}, which is linearly reductive over Spec ℤ. Hence no changes are needed for characteristic ≥ 0.

Definition 5.1. Let (Λ, m_{Λ}) be a regular local ring and let u_{1}, u_{2} ∈ m_{Λ} be non-zero elements. Let α_{i}, m_{i}, ρ_{i} be positive integers and G_{i}(T_{1}, T_{2}) ∈ Λ[T_{1}, T_{2}] a homogeneous polynomial in T_{1} and T_{2} of degree ρ_{i} (i = 1, 2) such that

1. a_{i} ≤ m_{i} and (a_{i}, m_{i}) = 1,

2. δ := a_{1}m_{2} + a_{2}m_{1} − m_{1}m_{2} > 0,

3. the coefficient of T_{1}^{ρ_{i}} (resp. T_{2}^{ρ_{i}}) in G_{i} is 1 (resp. u_{1}),

4. Δ := ρ_{1}m_{1}^{2} − δρ_{1}ρ_{2}m_{1}m_{2} + ρ_{2}m_{2}^{2} > 0. By formally writing α_{i} = log_{u}u_{i} (or u^{α_{i}} = u_{i}) for i = 1, 2, Definition 3.2 applies to our case. By Corollary 3.4, we may assume that

5. d(1) > d(3). Corollary 3.8 implies that

6. u^{e(n)} ∈ (u_{1}, u_{2})Λ if 4 ≤ n ≤ k + 1.

Let R := Λ[x_{1}, y_{1}, x_{2}, y_{2}, z] be the Λ-algebra with the Γ-action in Remark 2.7, and let W = Spec R/I be the scheme with the Γ-action, where I is the ideal given by

I := (x_{1}y_{1} − G_{1}(z^{m_{1}}, x_{2}^{δ}), x_{2}y_{2} − G_{2}(z^{m_{1}}, x_{1}^{δ})).

As in Definition 2.8, we set

X := (W \ V (x_{1}, x_{2}))/Γ ⊃ C := V (y_{1}, y_{2}, z, m_{Λ})/Γ ≃ ℙ_{Spec Λ/m_{Λ}}^{1}

and P_{i} = V (x_{i}, y_{1}, y_{2}, z, m_{Λ})/Γ ≃ Spec Λ/m_{Λ}. Let L_{i} be the ℚ-Cartier divisor classes and F_{i} ∈ H^{0}(X, O(L_{i})) be the sections as in Definition 3.9.

Theorem 5.2. L_{0}, ⋯, L_{3} and F_{0}, ⋯, F_{3} can be extended to ℚ-Cartier divisor classes L_{i} and sections F_{i} ∈ H^{0}(X, O(L_{i})) for i ∈ [0, k +2] such that the following hold.

0_{n}. L_{n−1} + L_{n+1} = \begin{cases}δρ_{n}L_{n} & (n ≤ k − 1) \\ 0 & (n = k, k + 1), \end{cases} if n ∈ [1, k + 1].

1_{n}. F_{n}, F_{n−1} are relatively prime on X if n ∈ [1, k + 2]. 2_{n}. F_{n}, zu_{1}u_{2} are relatively prime on X, if n ∈ [0, k + 2].

3_{n}. F_{n−1}F_{n+1} = \begin{cases}G_{n−2}(F_{n}^{δ}z^{−d^{∗}(n)}, u^{e(n)}) \\ = G_{n−2}(F_{n}^{δ}, z^{d^{∗}(n)}u^{e(n)})z^{−ρ_{n}d^{∗}(n)} & (n = k, k + 1), \\ G_{n−2}(F_{n}^{δ}, z^{d^{∗}(n)}u^{e(n)}) & (2 < n < k), \\ G_{n}(z^{d(n)}, F_{n}^{δ}) \\ = G_{n−2}(F_{n}^{δ}, z^{d(n)}u^{e(n)})u^{α_{n}} & (n = 1, 2), \end{cases}

if n ∈ [1, k + 1].

Our argument here is slightly stronger than those for Theorems 3.10, 3.12 and 3.13, since we introduce the intermediate schemes X^{i} and study them closely.

Lemma 5.3. Let the notation and the assumptions be as in Theorem 5.2. For i ∈ [1, k], let R^{i} := Λ[F_{i−1}, ⋯, F_{i+2}, z] be the polynomial ring with 5 variables and I^{i} ⊂ R^{i} the ideal generated by the relations 3_{i} and 3_{i+1}.

As in Definition 2.8, let X^{i} := (Spec R^{i}/I^{i} \ {F_{i} = F_{i+1} = 0})/Γ, and let L_{j}^{i} (j ∈ [i − 1, i + 2]) be the ℚ-Cartier divisor class on X^{i} induced by F_{j}. By the condition corresponding to 0_{n}, we define L_{j}^{i} for all j ∈ [0, k + 2]. Let B_{1}^{i} (resp. B_{2}^{i}) be the closed subset of X^{i} defined by F_{i−1} = F_{i} = 0 (resp. F_{i+1} = F_{i+2} = 0).

Then for every i ∈ [1, k], we have the following.

1. R^{i}/I^{i} is a normal domain of complete intersection,

2. codim_{X^{i}}(B_{j}^{i}) ≥ 2 for every j = 1, 2. By the relations 3_{1}, ⋯, 3_{n}, R^{1}/I^{1}, ⋯, R^{n−1}/I^{n−1} are all birational to each other. For every i ∈ [2, k], the birational map X^{i−1} −−→ X^{i} induces

3. an isomorphism X^{i−1} \ B_{2}^{i−1} ≃ X^{i} \ B_{1}^{i},

4. the identification L_{j}^{i−1} = L_{j}^{i}, which is simply denoted by L_{j}, and

5. H^{0}(X^{i−1}, O(L_{j})) = H^{0}(X^{i}, O(L_{j})) for all j.

Proof. It is easy to check that R^{i}/I^{i} is a complete intersection integral domain by Proposition 4.8 and u_{1}u_{2} ≠ 0 ∈ Λ. The normality can be checked by the Jacobian criterion at codimension 1 points, which is the assertion 1. Again using u_{1}u_{2} ≠ 0 ∈ Λ, one can easily check the assertion 2.

We now regard F_{i+2} as a rational function in F_{i−2}, ⋯, F_{i+1}, z. We can see that the regular section

F_{i+2}F_{i} ∈ H^{0}(X^{i−1}, O(L_{i+2}^{i−1} + L_{i}^{i−1}))

satisfies the condition

F_{i+2}F_{i}(F_{i−1}zu_{1}u_{2})^{#} ∈ F_{i}Λ[F_{i−2}, F_{i−1}, F_{i}, F_{i+1}, z]

from the relations 3_{i+1}, 3_{i} and 3_{i−1} by pure computation, where # denotes an arbitrarily large positive integer. Indeed the computation was carried out in the proof of Theorem 3.10 with n = i + 1 ≤ k − 1. Since the computation is similar in other cases, we omit the computation. This means that the regular section F_{i+2}F_{i} vanishes on the divisor (F_{i} = 0) ∼ L_{i}^{i−1}, whence F_{i+1} ∈ H^{0}(X^{i−1}, O(L_{i+1}^{i−1})). Hence (F_{i−1}, F_{i}, F_{i+1}, F_{i+2}, z) induce a morphism X^{i−1} \ B_{2}^{i−1} → X^{i} \ B_{1}^{i}.

The inverse X^{i} \B_{1}^{i} → X^{i−1} \B_{2}^{i−1} can be constructed similarly from the assertion:

F_{i−2}F_{i}(F_{i+1}zu_{1}u_{2})^{#} ∈ F_{i}Λ[F_{i−1}, F_{i}, F_{i+1}, F_{i+2}, z].

Indeed, we can prove this using 3_{i−1}, 3_{i} and 3_{i+1} by the computation similar to the above. The rest are obvious.

Proof of Theorem 5.2. By 0_{n}, we define L_{j}’s. By Lemma 5.3, we have the extension F_{j} (j ∈ [0, k + 2]) satisfying 3_{n} (n ∈ [1, k + 1]) by F_{j} ∈ H^{0}(X, O(L_{j})) = H^{0}(X^{i}, O(L_{j})) for some i ∈ [1, k] such that j ∈ [i − 1, i + 2].

By Lemma 5.3, the assertions 1_{n} and 2_{m} can be examined on X^{i} such that n, m ∈ [i − 1, i + 2].

On X^{i}, we know that B_{1}^{i} = {F_{i−1} = F_{i} = 0}, ∅ = {F_{i} = F_{i+1} = 0}, B_{2}^{i} = {F_{i+1} = F_{i+2} = 0} are of codimension ≥ 2 on X^{i}. This proves 1_{n}. The computation of 2_{n} can be done through a simple but tedious computation, which we omit.

Remark 5.4. For the family of surfaces π : X → Spec Λ, we have a divisorial contraction or a flipping contraction depending on whether d(k) = 0 or not (Corollary 4.1). It is not difficult to obtain an analogue of Theorem 2.2 for a multi-parameter analytic deformation space of H and analogues of Theorems 3.12, 3.13 for Λ, and furthermore to carry out a detailed computation as in Sections 2, 4.

For instance, C need not be the only contractible curve over [m_{Λ}] because we do not assume G_{i} ≡ T_{1}^{ρ_{i}} mod m_{Λ} in Definition 5.1.3. The contractible curves over [m_{Λ}] are contained in F_{4} = 0, which follows from F_{k} = F_{k+1} = u_{1} = u_{2} = 0 through the relations in Theorem 5.2.3.

Using such G_{i}, we can systematically construct reducible flipping curves.

Remark 5.5. An interesting problem in order to understand flips is to find the generators of the graded ring ⊕_{ν∈Z}H^{0}(X, O(νK_{X})) or some of its variants. We note that our z, F_{0}, ⋯, F_{k+2} are a part of the key generators.

It is possible to carry out further divisions to get F_{i} for i < 0 and i > k + 2. The former case was treated in Theorem 3.10. The latter case corresponds to the case i < 0 for X′ in Theorem 4.7, or we can continue the division imitating the arguments in Theorem 3.10.

However this immediate generalization does not give the right homogeneous elements as pointed out by M. Reid. He has been proposing a more general division [10] using pfaffians.

Our standpoint is that, with our easier divisions, we can determine many of the structures of the flips.

References

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7. Mori, S., On 3-dimensional terminal singularities, Nagoya Math. J., 98 (1985), 43–66.

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9. Nagata, M., A generalization of the embedding problem of an abstract variety in a complet variety, J. Math. Kyoto Univ., 3 (1963), 89–102.

10. Papadakis, S. and Reid, M., Kustin–Miller unprojection without complexes, Preprint, Oct. 2000.

11. Reid, M., Young person’s guide to canonical singularities, Proc. AMS summer institute Bowdoin 1985, 0 (1987), 345–413.

12. Shokurov, V., 3-fold log flips, Izv. Russ. Akad. Nauk Ser. Mat. 56 (1992) 105– 203. (English translation) Math. USSR Izvestiya, 26 (1993) 95–202.

Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502 Japan

  E-mail address: mori@kurims.kyoto-u.ac.jp