The explicit factorization of the Cremona Transformation which is an extension of the Nagata Automorphism into elementary links

By

Takashi KISHIMOTO^{∗}

Abstract

We consider a Cremona transformation Φ_{σ} : P^{3} ⋯ → P^{3} which is an extension of the famous Nagata automorphism σ on the affine 3space A^{3}. In this paper we shall factorize this Cremona transformation Φ_{σ} explicitly into a finite number of elementary links according to the Sarkisov Program. As a result, in our factorization of Φ_{σ}, it can be decomposed into eight elementary links.

1. Introduction

Throughout the present article we work over the field of complex numbers C. In affine algebraic geometry it is an important problem to understand the structure of the groups Aut_{C}(C^{[n]}) of automorphims on the polynomial rings C^{[n]}. Let A_{n} (resp. J_{n}) be the subgroup of Aut_{C}(C^{[n]}) consisting of all affine transformations (resp. de Jonquiére transformations), where an automorphism τ is said to be affine (resp. de Jonquiére) if τ(x_{i}) = ∑_{j=1}^{n} a_{ij}x_{j}+b_{i} with a_{ij}, b_{i} ∈ C (1 ≤ i ≤ n) such that det(a_{ij}) ≠ 0 (resp. τ(x_{i}) = a_{i}x_{i} + f_{i}(x_{i+1}, ⋯, x_{n}) with a_{i} ≠ 0 and f_{i} ∈ C[x_{i+1}, ⋯, x_{n}] for 1 ≤ i ≤ n). An automorphism τ is called tame if it is contained in the subgroup generated by A_{n} and J_{n}. Classically it is well known that all the automorphisms τ ∈ Aut_{C}C[x, y] are tame (cf. Nagata [Na72], Abhyankar-Moh [A-M75], Miyanishi [Miy78]). In order to grope the structure of Aut_{C}C[x, y] by geometric method, it played a vital role to factorize Cremona transformations on P^{2} which are obtained as extensions of the automorphisms on the inside affine plane A^{2} (cf. [Miy78, Chap. 2, Section 3], [Ma01]).

On the other hand, for the case n ≥ 3, the structures of Aut_{C}(C^{[n]}) are not understood completely at the present time. We shall pay attention to the case n = 3. For instance, we consider the following famous Nagata automorphism σ ∈ Aut_{C}(C[x, y, z]) on the affine 3-space A^{3} (x, y, z) (cf. Nagata [Na78, p. 16], Alev [Al95]):

2000 Mathematics Subject Classification(s). Primary 14R10, Secondary 14E07 Additional infomation

∗The author is partially supported by the Grant-in-Aid for JSPS Fellows, the Ministry of Education, Science, Sports and Culture, Japan.

\{ \begin{matrix}σ(x) & = & x − 2y(xz + y^{2}) − z(xz + y^{2})^{2}, \\ σ(y) & = & y + z(xz + y^{2}), \\ σ(z) & = & z. \end{matrix}

The Nagata automorphism σ seems to represent the difficulty of the structure of Aut_{C}(C[x, y, z]). For instance, at the present time, we do not know whether or not σ is tame.^{*1} The Nagata automorphism σ is extended naturally to a Cremona transformation Φ_{σ} on P^{3} of the following form:

\{ \begin{matrix}x & ↦ & xw^{4} − 2yw^{2}(xz + y^{2}) − z(xz + y^{2})^{2}, \\ y & ↦ & yw^{4} + zw^{2}(xz + y^{2}), \\ z & ↦ & zw^{4}, \\ w & ↦ & w^{5}, \end{matrix}

where H_{∞} := {w = 0} is a hyperplane at infinity with respect to the canonical open immersion ι : A^{3} (x, y, z) ↪ P^{3}. Our main interest in this article is to factorize Φ_{σ} explicitly into a finite number of elementary links according to the Sarkisov Program. Elementary links are some kinds of birational maps between 3-dimensional Mori fiber spaces (see Section 2 for the details). In my opinion we belive that this explicit factorization of Φ_{σ} (cf. Theorem 1.1) is important in order to analyze whether or not the Nagata automorphism σ is tame from a point of view of algebro-geometric methods as well as in two-dimensional case (cf. [Miy78]).

We shall state the scheme of this article. In Section 2 we recall the definitions of Sarkisov Program and elementary links in dimension three. Though the Sarkisov Program guarantees that any birational map between 3-dimensional Mori fiber spaces can be factorized into elementary links, it is, in general, difficult and complicated to perform its explicit factorization. This obstacle seems to result from the fact that there are no complete classification of the 3-dimensional terminal divisorial contractions. However, fortunately, in our factorization of Φ_{σ}, the varieties keep smoothness, and the terminal divisorial contractions to smooth threefolds are completely understood due to Mori and Kawakita (cf. Theorems 2.1, 2.2).

In Section 3 we shall factorize the Cremona transformation Φ_{σ} explicitly according to Sarkisov Program. We consider the proper transform H of the complete linear system |O_{P^{3}}(1)| by Φ_{σ} : P^{3} ⋯ → P^{3} and investigate the singularities of a general member of H along base locus Bs H. After that we take a maximal divisorial blow-up and observe the configurations of general members of the proper transform of H concretely and so on. In this way, we continue to investigate the singularities of a general member of the proper transform of H in each step of the factorization. If a log pair associated with the proper transform of H is not canonical, then we consider where is the maximal center and take the maximal divisorial blow-up. If a log pair is canonical but not nef, then we take an extremal ray which is negative with respect to this log pair and contract the 2-dimensional extremal face spanned by this ray and other ray yielding a Mori fiber space (see Section 2 for the detailed mechanism of the Sarkisov Program). In any case, we shall construct an elementary link to decrease the Sarkisov degree (see Definition 2.1) and describe the concrete configurations of general members of the proper transform of H, and prepare the next step of the factorization until we reach the situation where a log pair is canonical and nef to obtain the factorization. As a result in our factorization of Φ_{σ}, it can be decomposed explicitly as in the following Theorem 1.1 (= Theorem 3.1) which is our main result in this article.

*1Recently, Shestakov and Umirbaev [S-U02] seem to obtain an algorithm in order to decide when an automorphism in Aut_{C}(C[x, y, z]) is tame by algebraic argument. By making use of this algorithm they seem to prove that the Nagata automorphism σ is not tame.

Theorem 1.1. Let σ be the Nagata automorphism on the affine 3-space A^{3} and let Φ_{σ} : P^{3} ⋯ → P^{3} be a Cremona transformation which is an extension of σ. Then Φ_{σ} can be factorized into eight elementary links explicitly, say Φ_{σ} = Φ_{8} ∘ ⋯ ∘ Φ_{1}. More precisely, each elementary link Φ_{j} (1 ≤ j ≤ 8) is described as follows (here we shall follow the notations used in Section 3 and each number of type is the type of elementary link; see Section 2):

(i) Φ_{1} : P^{3} ←^{p_{0}} Z_{0} →^{q_{0}} X_{1} ≅ Q^{3}, where p_{0} is the blow-up along the conic C_{0} with an exceptional divisor E_{0} ≅ F_{2} and q_{0} is the contraction of H_{∞}_{Z_{0}} ≅ P^{2} (cf. Figure 1) to a smooth point (Type II),

(ii) Φ_{2} : X_{1} ←^{p_{1}} X_{2} is the blow-up along the curve C_{1} (cf. Figure 2) with an exceptional divisor E_{1} ≅ P^{1} × P^{1} (Type I),

(iii) Φ_{3} : X_{2} ←^{p_{2}} Z_{2} →^{q_{2}} X_{3}, where p_{2} is the blow-up along the curve L_{1} (cf. Figure 3) with an exceptional divisor E_{2} ≅ F_{2} and q_{2} is the contraction of E_{0}_{Z_{2}} ≅ F_{2} (cf. Figure 4) onto a curve (Type II),

(iv) Φ_{4} : X_{3} ←^{p_{3}} Z_{3} →^{q_{3}} X_{4}, where p_{3} is the blow-up along the curve L_{3} (cf. Figure 5) with an exceptional divisor E_{3} ≅ F_{2} and q_{3} is the contraction of E_{2}_{Z_{3}} ≅ F_{2} (cf. Figure 6) onto a curve (Type II),

(v) Φ_{5} : X_{4} ←^{p_{4}} Z_{4} →^{q_{4}} X_{5}, where p_{4} is the blow-up along the curve L_{4} (cf. Figure 7) with an exceptional divisor E_{4} ≅ F_{2} and q_{4} is the contraction of E_{3}_{Z_{4}} ≅ F_{2} (cf. Figure 8) onto a curve (Type II),

(vi) Φ_{6} : X_{5} ←^{p_{5}} Z_{5} →^{q_{5}} X_{6}, where p_{5} is the blow-up along the curve L′_{5} (cf. Figure 9) with an exceptional divisor E_{5} ≅ F_{2} and q_{5} is the contraction of E_{4}_{Z_{5}} ≅ F_{2} (cf. Figure 10) onto a curve (Type II),

(vii) Φ_{7} : X_{6} →^{q_{6}} X_{7} ≅ Q^{3} is the contraction of E_{1}_{X_{6}} ≅ P^{1} × P^{1} (cf. Figure 11) onto a curve (Type III),

(viii) Φ_{8} : X_{7} ←^{p_{7}} Z_{7} →^{q_{7}} X_{8} ≅ P^{3}, where p_{7} is the blow-up at the point P_{7} (cf. Figure 12) with an exceptional divisor E_{7} ≅ P^{2} and q_{7} is the contraction of E_{5}_{Z_{7}} ≅ F_{2} onto a conic (Type II).

We shall employ the following notations in this paper:

Notation.

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

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

• D^{2} : the self-intersection of an irreducible Cartier divisor D on V;

• mult_{W} D : multiplicity of a Cartier divisor D on V along a subvariety W ⊂ V;

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

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

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

• F_{k} : Hirzebruch surface of degree k;

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

• l_{k} : the class of a fiber on F_{k};

• a(E; V, αH_{V}) : the discrepancy of a discrete valuation E of the function field C(V) of V with respect to K_{V} + αH_{V};

When a Cartier divisor D on a variety V and a birational map V ⋯ → W are given, we denote by D_{W} the proper transform on W of D consistently in this article if there are no confusions.

Acknowledgments. The author would like to express his sincere gratitude to Professor Masayoshi Miyanishi for his valuable advice and warm encouragement during the preparation of the present article. He would like to express his gratitude to Professor Tomoaki Ohta for giving me the useful comments and having discussion on this topic. He also would like to express his hearty thanks to Dr. Ken-ichiro Arima for drawing figures in this article.

2. Overview of the Sarkisov Program in dimension Three

A program called the Sarkisov Program was developed by Sarkisov, Reid and Corti in order to factorize birational maps between 3-dimensional Mori fiber spaces as a composite of so called elementary links (cf. [Cor95]). There are four types I∼IV for these elementary links appearing in the Sarkisov Program as follows:

S  S_{1} X ↓ ϕ ↙ p Z ⋯ →X_{1} ↓ϕ_{1} ⟵ Type I  S  S_{1} X ↓ ϕ ↙ p Z ⋯ →Z_{1} ↘ p_{1}X_{1} ↓ϕ_{1} = Type II

S  S_{1} X ↓ ϕ ⋯ → Z_{1} ↘ p_{1}X_{1} ↓ϕ_{1} ⟶ Type III  S  S_{1} X ↓ ϕ  ↘ T · · · · · · · · · →  X_{1} ↓ϕ_{1} ↙ Type IV

where p : Z → X and p_{1} : Z_{1} → X_{1} are the divisorial contractions (or the maximal divisorial blow-up; see Definition 2.2 below) and each of the birational maps denoted by the horizontal dotted arrows, which may be an identity, is a composite of log flips (i.e., flips, flops and inverse flips), ϕ : X → S, ϕ_{1} : X_{1} → S_{1} are the 3-dimensional Mori fiber spaces and ρ(S/T) = ρ(S_{1}/T) = 1.

In fact, to perform the Sarkisov Program, we need to define the Sarkisov degree with respect to a suitably chosen very ample complete linear system and verify that it decreases after untwisting each elementary link (see [Cor95], [Cor00], [Ma01] for the detail).

We shall recall the definition of the Sarkisov degree. Let φ : X/S ⋯ → X′/S′ be a given birational map between 3-dimensional Mori fiber spaces ϕ : X → S and ϕ′ : X′ → S′.

Definition 2.1. Let H′ = |−μ′K_{X′} +ϕ^{′∗}(A′)| be a very ample complete linear system on X′, where μ′ is a positive integer and A′ is a sufficiently ample Cartier divisor on S′. This choice of H′ is made once and for all at the beginning of the factorization process and H′ remains unchanged throughoutly. Let H := φ_{∗}^{−1}(H′) be the proper transform of H′ by φ. Since ϕ : X → S is a Mori fiber space, H is written uniquely as H ≡ −μK_{X} + ϕ^{∗}(A), where μ is a positive rational number and A is a Q-Cartier divisor on S. This μ is called the quasi-effective threshold of the pair (X, H). Let c := sup{λ ∈ Q; K_{X} + λH is canonical}, which we call canonical threshold of (X, H). (In the case Bs H = ∅ we put c = ∞.) Finally, let e be the number of discrete valuations {E} of C(X) which are exceptional over X and have discrepancy a(E; X, cH) = 0. (Note that e ≥ 0 is finite.) The triplet (μ, c, e) is called as Sarkisov degree of (X, H).

We shall define the lexicographical order on the set of Sarkisov degree as follows. When two birational maps X/S ⋯ →^{φ} X′/S′ ← ⋯^{φ_{1}} X_{1}/S_{1} are given, we have two Sarkisov degree (μ, c, e) and (μ_{1}, c_{1}, e_{1}) of the pairs (X, H) and (X_{1}, H_{1}) as defined just above, where H (resp. H_{1}) is the proper transform of H by φ (resp. φ_{1}). We write (μ, c, e) > (μ_{1}, c_{1}, e_{1}) if one of the following (i)∼(iii) are satisfied:

(i) μ > μ_{1}; (ii) μ = μ_{1} and c < c_{1}; (iii) μ = μ_{1}, c = c_{1} and e > e_{1}.

The essential in the Sarkisov Program lies in untwisting the original birational map φ by suitable elementary links to decrease Sarkisov degree and to use Noether-Fano-Iskovskih inequality (cf. [Cor95, Cor00, Ma01]). We shall sketch the mechanism of the Sarkisov Program to factorize φ : X/S ⋯ → X′/S′. The reader refer to [Cor95, Cor00, Ma01] for the detail mechanism.

With the notations as above, at first, we ask whether or not K_{X} + (1/μ)H is canonical and nef. If it is so, then φ is an isomorphsim as Mori fiber spaces by virtue of Noether-Fano-Iskovskih inequality (cf. [Cor95, Cor00]) and there are nothing to do. Otherwise, we need to consider according as K_{X} + (1/μ)H is canonical or not.

In the case where K_{X} + (1/μ)H is canonical but not nef, we find a (K_{X} + (1/μ)H)-nagative extremal ray P by the Cone Theorem (cf. [K-M-M87]), and consider the extremal face F spanned by P and R, where R is an extremal ray yielding a Mori fiber space ϕ : X → S. Let cont_{F} : X → T be the contraction of F. Note that since R ⊂ F, the contraction cont_{F} factors through ϕ = cont_{R} : X → S. We run (K_{X} + (1/μ)H)-MMP over T. Then we reach at the end either a log minimal model or a log Mori fiber space (with respect to K_{X} + (1/μ)H over T). In any case we obtain the link of type III or IV according as the last birational contraction in this (K_{X} + (1/μ)H)-MMP over T is divisorial or not.

In the case where K_{X} + (1/μ)H is not canonical, i.e., c < 1/μ, we have to take a suitable maximal divisorial blow-up as defined below:

Definition 2.2. We call p : Z → X a maximal divisorial blow-up with respect to K_{X} + cH if p : Z → X is a projective birational morphism from Z with only Q-factorial and terminal singularities such that:

(i) ρ(Z/X) = 1; (ii) the exceptional locus of p is a prime divisor; (iii) p is (K_{X} + cH)-crepant, i.e., K_{Z} + cH_{Z} = p^{∗}(K_{X} + cH), where H_{Z} is

the proper transform of H on Z.

A discrete valuation E of C(X) is a maximal singularity of H if E is the exceptional divisor of a maximal divisorial blow-up. A maximal center is the center Center_{X}(E) on X of a maximal singularity E.

It is known that a maximal divisorial blow-up with respect to K_{X} + cH exists (cf. [Ma01, Prop. 13-1-8]), but in general it is not necessarily unique. After taking a maximal divisorial blow-up p : Z → X with respect to K_{X} +cH, we run (K_{Z} + cH_{Z})-MMP over S. Then we necessarily reach a log Mori fiber space ϕ_{1} : X_{1} → S_{1}. (Note that the case where we reach a log minimal model over S does not occur.) According as the last birational contraction in the (K_{Z} + cH_{Z})-MMP over S is divisorial or not we obtain an elementary link of type II or type I.

Thus we can construct an elementary link φ_{1} : X/S ⋯ → X_{1}/S_{1} of type I∼IV. It can be verified that the Sarkisov degree of (X_{1}, H_{1}), say (μ_{1}, c_{1}, e_{1}), decreases properly compared with that of (X, H). Therefore we have anew birational map φ ∘ φ_{1}^{−1} : X_{1}/S_{1} ⋯ → X′/S′ between Mori fiber spaces with strictly smaller Sarkisov degree (μ_{1}, c_{1}, e_{1}). We then ask again whether or not K_{X_{1}} + (1/μ_{1})H_{1} is canonical and nef. If the answer is yes, then φ ∘ φ_{1}^{−1} : X_{1}/S_{1} ⋯ → X′/S′ is an isomorphism by Noether-Fano-Iskovskih inequality and to see φ = φ_{1}. On the other hand, if the answer is no, we repeat the above mentioned process with φ ∘ φ_{1}^{−1}. Each time we untwist, the Sarkisov degree should strictly drop. Finally, it is known that this process has to come to an end after finitely many repetitions to obtain the expression of φ as a composite of elementary links.

This is why, in order to understand the mechanism of the Sarkisov Program well, it is important to understand maximal divisorial blow-ups and flips. Note that flips in dimension three were understood by Kollár and Mori (cf. [Ko-Mo92]). Let f : (E ⊂ Y) → (Γ ⊂ X) be a 3-dimensional maximal divisorial blow-up. Note that there are two ways to deal with f, namely, one starting from Y, and the other from X. In the former case f is said to be a divisorial contraction, on the other hand, in the latter case f is said to be a maximal divisorial blow-up as defined in Definition 2.2. The three-dimensional divisorial contractions (or maximal divisorial blow-up) are not completely understood at the present time. However, fortunately, the divisorial contractions to smooth threefolds are completely understood due to the works of Mori and Kawakita. In our factorization of Φ_{σ}, which is an extension of the Nagata automorphism σ, performed in the next section we need only such contractions.

Let f : (E ⊂ Y) → (Γ ⊂ X) be a 3-dimensional divisorial contraction. The following result is well known by virtue of Mori’s classification.

Theorem 2.1. ([Mori82]) In the case where dim Γ = 1 and X is smooth along the curve Γ, f : (E ⊂ Y) → (Γ ⊂ X) is the blow-up of X along Γ, namely, Y ≅ Proj_{X}( ⨁_{d≥0} I_{Γ,X}^{d}), where I_{Γ,X} ⊂ O_{X} is the ideal sheaf of the curve Γ ⊂ X.

On the other hand, in the case where Γ is a smooth point on X, the following result due to Kawakita is known.

Theorem 2.2. ([Kaw01]) In the case where dim Γ = 0, i.e., Γ = {P } is a point, and X is smooth at P, f : (E ⊂ Y) → (P ∈ X) is a weighted blow-up. More precisely, we can take local coordinates x, y, z at P and coprime positive integers a and b, such that f is the weighted blow-up of X with its weights wt(x, y, z) = (1, a, b).

3. The explicit factorization of the Cremona transformation which is an extension of the Nagata Automorphism

Let σ ∈ Aut_{C}(C[x, y, z]) be the Nagata automorphism and let Φ_{σ} : P^{3} ⋯ → P^{3} be the Cremona transformation which is a natural extension of σ (see Section 1, Introduction). Namely, Φ_{σ} is defined as follows:

\{ \begin{matrix}x & ↦ & xw^{4} − 2yw^{2}(xz + y^{2}) − z(xz + y^{2})^{2}, \\ y & ↦ & yw^{4} + zw^{2}(xz + y^{2}), \\ z & ↦ & zw^{4}, \\ w & ↦ & w^{5}, \end{matrix}

where w = 0 defines a hyperplane at infinity H_{∞} with respect to the canonical open immersion ι : A^{3} (x, y, z) ↪ P^{3}. In this section we shall factorize Φ_{σ} into a finite number of elementary links explicitly according to the Sarkisov Program (cf. Theorem 1.1).

Let H := Φ_{σ}_{∗}^{−1}(|O_{P^{3}}(1)|) be the proper transform by Φ_{σ} of the complete linear system |O_{P^{3}}(1)|. (In the subsequent of this section the proper transform of H on a variety V is denoted by H_{V} otherwise mentioned.) Then H is spanned by S, T + 2H_{∞}, H + 4H_{∞} and 5H_{∞} independently, where S := (xw^{4} − 2yw^{2}(xz + y^{2}) − z(xz + y^{2})^{2} = 0), T := (yw^{2} + z(xz + y^{2}) = 0) and H := (z = 0). It is obvious that BsH = C_{0}∪L_{0}, where C_{0} := (xz+y^{2} = w = 0) is a conic and L_{0} := (z = w = 0) is a line.

We consider the open neighborhood of the point P_{0} := (1 : 0 : 0 : 0), say U_{0} := P^{3}\(x = 0) with affine coordinates (y, z, w) where a general member S of H and T, H have the local equations f_{0} := w^{4} − 2yw^{2}(z + y^{2}) − z(z + y^{2})^{2}, g_{0} := yw^{2} + z(z + y^{2}) and h_{0} := z, respectively. Moreover the curves L_{0} and C_{0} are defined by the ideals (z, w) and 𝔞_{0} := (z + y^{2}, w) ⊂ Γ(O_{U_{0}}) = C[y, z, w], respectively on U_{0}. We shall calculate the Sarkisov degree (see Definition 2.1) of the pair (P^{3}, H).

Lemma 3.1.

(1) The quasi-effective threshold μ_{0} of the pair (P^{3}, H) equals 5/4, and K_{P^{3}} + (1/μ_{0})H is not canonical,

(2) The canonical threshold c_{0} of the pair (P^{3}, H) equals 1/2 and C_{0} is a maximal center of (P^{3}, H) (cf. Definition 2.2).

Proof. It is clear that μ_{0} = 5/4 and K_{P^{3}} + (1/μ_{0})H is nef. As Φ_{σ} is not an isomorphism, K_{P^{3}} + (1/μ_{0})H is not canonical by virtue of Noether-Fano-Iskovskih inequality (cf. [Cor95, Cor00]). In consideration of the local equation f_{0} of a general member S ∈ H, we know that the possibility for a maximal center of (P^{3}, H) is either the conic C_{0} itself or the point P_{0}. In the case where C_{0} (resp. P_{0}) is the maximal center, the maximal divisorial blow-up with respect to K_{P^{3}} + c_{0}H is the blow-up along C_{0} (resp. a weighted blow-up at P_{0} with suitable weights) by Theorem 2.1 (resp. by Theorem 2.2). Let E_{0} (resp. E(a, b, c)) be an exceptional divisor obtained by the blow-up along C_{0} (resp. by the weighted blow-up at P_{0} of weight wt(y, z, w) = (a, b, c), where one of a, b, c is 1 and the other two are coprime positive integers; see Theorem 2.2). It is easy to see that a(E_{0}; P^{3}, (1/2)H) = 0, so c_{0} ≤ 1/2. On the other hand, we have a(E(a, b, c); P^{3}, (1/2)H) = a + b − c − 1 (resp. (1/2)(a + b) − 1, −(a/2) + b − 1, a − (b/2) + c − 1, c − 1 or −a + (b/2) + c − 1) if 4c (resp. a + b + 2c, 3a + 2c, 3b, 2a + 2b, 4a + b) is minimum among 4c, a + b + 2c, 3a + 2c, 3b, 2a + 2b and 4a + b. Then, by the straightforward computations, we can see a(E(a, b, c); P^{3}, (1/2)H) > 0 in any case. Thus we complete the proof. Q.E.D.

By Theorem 2.1 and Lemma 3.1, we have to blow-up p_{0} : E_{0} ⊂ Z_{0} → C_{0} ⊂ P^{3} along the conic C_{0} with an exceptional divisor E_{0} ≅ F_{2}. It is not difficult to see that the proper transform H_{Z_{0}} of H by p_{0} is spanned by S_{Z_{0}}, T_{Z_{0}} +2H_{∞}_{Z_{0}}, H_{Z_{0}} + 4H_{∞}_{Z_{0}} + 2E_{0} and 5H_{∞}_{Z_{0}} + 3E_{0} (here we follow Notation in Section 1, Introduction). We use the same notations L_{0} and C_{0} to represent, respectively, the proper transform of L_{0} on Z_{0} and the intersection curve H_{∞}_{Z_{0}} ∩ E_{0}. Note that C_{0} is a conic on H_{∞}_{Z_{0}} ≅ P^{2} and the minimal section on E_{0} ≅ F_{2}. We have the following (cf. Figure 1):

Lemma 3.2.

(1) K_{Z_{0}} ∼ −4H_{∞}_{Z_{0}} − 3E_{0};

(2) H_{∞}_{Z_{0}}^{2} ∼ O_{P^{2}}(−1) and E_{0}^{2} ∼ −M_{2} + 2l_{2};

(3) S_{Z_{0}}|_{H_{∞}_{Z_{0}}} = T_{Z_{0}}|_{H_{∞}_{Z_{0}}} = H_{Z_{0}}|_{H_{∞}_{Z_{0}}} = L_{0};

(4) S_{Z_{0}}|_{E_{0}} ∼ 2M_{2} + 6l_{2}, T_{Z_{0}}|_{E_{0}} ∼ M_{2} + 4l_{2} and H_{Z_{0}}|_{E_{0}} ∼ 2l_{2}.

Proof. The assertions (1) and (2) are easy to see. Since S_{Z_{0}} ∼ 5H_{∞}_{Z_{0}} +3E_{0}, T_{Z_{0}} ∼ 3H_{∞}_{Z_{0}} + 2E_{0} and H_{Z_{0}} ∼ H_{∞}_{Z_{0}} + E_{0}, we can obtain the linearly equivalent relations as stated in (4) by virtue of (2). By the similar argument it follows that S_{Z_{0}}|_{H_{∞}_{Z_{0}}} , T_{Z_{0}}|_{H_{∞}_{Z_{0}}} and H_{Z_{0}}|_{H_{∞}_{Z_{0}}} are linearly equivalent to L_{0}. However, since S_{Z_{0}}, T_{Z_{0}} and H_{Z_{0}} contain L_{0}, we see the assertion (3).

Q.E.D.

Though we have the linearly equivalent relations of S_{Z_{0}}|_{E_{0}}, T_{Z_{0}}|_{E_{0}} and H_{Z_{0}}|_{E_{0}} (cf. Lemma 3.2, (4)), we further need to investigate the concrete configurations of these intersections. Indeed, we have the following:

Lemma 3.3. S_{Z_{0}}|_{E_{0}} = 2C_{1} + 2L_{1}, T_{Z_{0}}|_{E_{0}} = C_{1} + 2L_{1} and H_{Z_{0}}|_{E_{0}} = 2L_{1}, where C_{1} is a cross-section on E_{0} ≅ F_{2} linearly equivalent to M_{2} + 2l_{2} and L_{1} is a fiber on E_{0} intersecting L_{0} (cf. Figure 1).

Proof. We have to observe the blow-up p_{0} : Z_{0} → P^{3} concretely. In the open affine subset U_{0} (y, z, w), the center C_{0} of p_{0} is defined by an ideal 𝔞_{0} := (z + y^{2}, w) ⊂ Γ(O_{U_{0}}) = C[y, z, w]. The inverse image p_{0}^{−1}(U_{0}) is isomorphic to Proj( ⨁_{d≥0} 𝔞_{0}^{d}) which is covered by two affine 3-spaces, say U_{0}^{(1)} (y, z, w) and U_{0}^{(2)} (y, z, w). More precisely, one U_{0}^{(1)} → U_{0} is associated to the algebra homomorphism y ↦ y, z ↦ zw − y^{2}, w ↦ w. On the other hand, the other U_{0}^{(2)} → U_{0} is associated to y ↦ y, z ↦ z, w ↦ w(z + y^{2}). On U_{0}^{(1)}, the surfaces S_{Z_{0}}, T_{Z_{0}}, H_{Z_{0}} and E_{0} are defined by f_{0}^{(1)} = w^{2} − 2yzw − z^{2}(zw − y^{2}), g_{0}^{(1)} = yw + z(zw − y^{2}), h_{0}^{(1)} = zw − y^{2} and w, respectively. On the other hand, on U_{0}^{(2)}, these are defined by f_{0}^{(2)} = w^{4}(z + y^{2})^{2} − 2yw^{2}(z + y^{2}) − z, g_{0}^{(2)} = yw^{2}(z + y^{2}) + z, h_{0}^{(2)} = z and z + y^{2}, respectively. In combination of these facts with Lemma 3.2, we have the assertion. Q.E.D.

To perform the Sarkisov Program, we have to run (K_{Z_{0}} + c_{0}H_{Z_{0}})-MMP (cf. Section 2 or [Cor95, Cor00], [Ma01] for the datail). The cone \overline{NE} (Z_{0}) ⊂ R^{2} is spanned by two rays one of which comes from the rulings on an exceptional divisor E_{0} ≅ F_{2} and the other comes from lines on H_{∞}_{Z_{0}} ≅ P^{2}. The latter ray is (K_{Z_{0}} + c_{0}H_{Z_{0}})-nagative. Since H_{∞}_{Z_{0}}^{2} ∼ O_{P^{2}}(−1), it can be contracted to a smooth point. Let q_{0} : Z_{0} → X_{1} be the contraction of H_{∞}_{Z_{0}}. The composite Φ_{1} := q_{0} ∘ p_{0}^{−1} is an elementary link of type II. Then X_{1} ≅ Q^{3} and K_{X_{1}} ∼ −3E_{0}_{X_{1}}. The proper transform H_{X_{1}} of H_{Z_{0}} on X_{1} is spanned by S_{X_{1}}, T_{X_{1}} + E_{0}_{X_{1}}, H_{X_{1}} + 2E_{0}_{X_{1}} and 3E_{0}_{X_{1}}. By virtue of Lemma 3.3, we can readily see the following:

Lemma 3.4.

(1) S_{X_{1}}|_{E_{0}_{X_{1}}} = 2C_{1} + 2L_{1}, T_{X_{1}}|_{E_{0}_{X_{1}}} = C_{1} + 2L_{1}, H_{X_{1}}|_{E_{0}_{X_{1}}} = 2L_{1}, where we use the same notations C_{1} and L_{1} to represent their proper transforms (cf. Figure 2);

(2) Bs H_{X_{1}} = C_{1} ∪ L_{1};

(3) On the open affine subset U_{1} := q_{0}(U_{0}^{(1)}) (y, z, w), the surfaces S_{X_{1}}, T_{X_{1}}, H_{X_{1}} and E_{0}_{X_{1}} are defined by f_{1} := w^{2} − 2yzw − z^{2}(zw − y^{2}), g_{1} := yw + z(zw − y^{2}), h_{1} := zw − y^{2} and w, respectively. On U_{1}, the curves C_{1} and L_{1} are defined by the ideals 𝔞_{1} := (z, w) and (y, w) ⊂ Γ(O_{U_{1}}) = C[y, z, w], respectively.

We shall calculate the Sarkisov degree of the pair (X_{1}, H_{X_{1}}).

Lemma 3.5.

(1) The quasi-effective threshold μ_{1} of (X_{1}, H_{X_{1}}) equals 1 and K_{X_{1}} + (1/μ_{1})H_{X_{1}} is not canonical;

(2) The canonical threshold c_{1} of (X_{1}, H_{X_{1}}) equals 1/2 and C_{1} is the maximal center of (X_{1}, H_{X_{1}});

Proof. The assertion (1) is easy to see. Of course, the maximal center of (X_{1}, H_{X_{1}}) is contained in Bs H_{X_{1}} = C_{1} ∪ L_{1}. In consideration of the local equation f_{1} of a general member S_{X_{1}} ∈ H_{X_{1}} on U_{1} (cf. Lemma 3.4, (3)), we can see (2) by the similar argument as in Lemma 3.1. Q.E.D.

By virtue of Lemma 3.5, we have to blow-up X_{1} along the curve C_{1}, say p_{1} : E_{1} ⊂ Z_{1} → C_{1} ⊂ X_{1}, where E_{1} ≅ P^{1} × P^{1} is the exceptional divisor of p_{1}. The proper transform H_{Z_{1}} of H_{X_{1}} is spanned by S_{Z_{1}}, T_{Z_{1}} +E_{0}_{Z_{1}}, H_{Z_{1}} +2E_{0}_{Z_{1}} and 3E_{0}_{Z_{1}} + E_{1}. Note that the proper transforms of the hyperplane sections on X_{1} ≅ Q^{3} ⊂ P^{4} containing C_{1} yield a Del Pezzo fibration Z_{1} → P^{1} with a general fiber isomorphic to a smooth quadric P^{1} × P^{1} and E_{0}_{Z_{1}} ≅ Q_{0}^{2} as a singular fiber. We have the following (cf. Figure 3):

Lemma 3.6.

(1) K_{Z_{1}} ∼ −3E_{0}_{Z_{1}} − 2E_{1};

(2) E_{0}_{Z_{1}}^{2} ∼ 0 and E_{1}^{2} ∼ −M_{0} + 2l_{0}, where M_{0} (resp. l_{0}) is a fiber on E_{1} ≅ P^{1} × P^{1} which maps onto a curve C_{1} (resp. a point on C_{1}) via p_{1} : Z_{1} → X_{1};

(3) S_{Z_{1}}|_{E_{0}_{Z_{1}}} = T_{Z_{1}}|_{E_{0}_{Z_{1}}} = H_{Z_{1}}|_{E_{0}_{Z_{1}}} = 2L_{1}, where we use the notation L_{1} to represent its proper transform on Z_{1};

(4) S_{Z_{1}}|_{E_{1}} ∼ 2M_{0} + 2l_{0}, T_{Z_{1}}|_{E_{1}} ∼ M_{0} + 2l_{0} and H_{Z_{1}}|_{E_{1}} ∼ 2l_{0}.

Proof. All the assertions are readily seen by the similar argument as in Lemma 3.2. Q.E.D.

Now we need to observe not only the linearly equivalent relations as stated in Lemma 3.6 (4) but also the concrete configurations of these intersection curves. Indeed, we have the following:

Lemma 3.7. S_{Z_{1}}|_{E_{1}} = 2C_{2}, T_{Z_{1}}|_{E_{1}} = C_{2} + L_{2} and H_{Z_{1}}|_{E_{1}} = 2L_{2}, where C_{2} (resp. L_{2}) is a curve on E_{1} ≅ P^{1} × P^{1} linearly equivalent to M_{0} + l_{0} (resp. to l_{0}) intersecting L_{1} (cf. Figure 3).

Proof. For the proof we need to describe the blow-up p_{1} : E_{1} ⊂ Z_{1} → C_{1} ⊂ X_{1} around the open affine subset U_{1} (y, z, w) explicitly. The center C_{1} of p_{1} is defined by an ideal 𝔞_{1} = (z, w) ⊂ Γ(O_{U_{1}}) = C[y, z, w] on U_{1}. Hence the inverse image p_{1}^{−1}(U_{1}) is isomorphic to Proj(⨁_{d≥0} 𝔞_{1}^{d}) which is covered by two affine 3-spaces, say U_{1}^{(1)} (y, z, w) and U_{1}^{(2)} (y, z, w). More precisely, U_{1}^{(1)} → U_{1} is associated to the algebra homomorphism y ↦ y, z ↦ z, w ↦ zw, on the other hand, U_{1}^{(2)} → U_{1} is associated to y ↦ y, z ↦ zw, w ↦ w. On U_{1}^{(1)}, the surfaces S_{Z_{1}}, T_{Z_{1}}, H_{Z_{1}}, E_{0}_{Z_{1}} and E_{1} are defined by f_{1}^{(1)} := (y − w)^{2} − z^{2}w, g_{1}^{(1)} := y(y−w)−z^{2}w, h_{1}^{(1)} := z^{2}w−y^{2}, w and z, respectively. By virtue of these local equations and Lemma 3.6 (4), we can obtain the concrete configurations of S_{Z_{1}}|_{E_{1}}, T_{Z_{1}}|_{E_{1}} and H_{Z_{1}}|_{E_{1}} as desired. Q.E.D.

As in the previous step, in order to perform the Sarkisov Program, we need to run (K_{Z_{1}} + c_{1}H_{Z_{1}})-MMP. The cone \overline{NE} (Z_{1}) ⊂ R^{2} is spanned by two rays one of which comes from rulings on an exceptional divisor E_{1} ≅ P^{1} × P^{1}, and the other comes from the rulings on a general fiber ≅ P^{1} × P^{1} of a Del Pezzo fibration Z_{1} → P^{1} (see the argument before Lemma 3.6). The latter ray is (K_{Z_{1}} + c_{1}H_{Z_{1}})-negative. Thus we put X_{2} := Z_{1} and the contraction of the latter ray ϕ_{2} : X_{2} → B_{2} = P^{1} is a Mori fiber space. Put Φ_{2} := p_{1}^{−1} : X_{1} ⋯ → X_{2}, which is an elementary link of type I.

In the subsequent, we put U_{2} := U_{1}^{(1)}, f_{2} := f_{1}^{(1)}, g_{2} := g_{1}^{(1)} and h_{2} := h_{1}^{(1)} (see the proof of Lemma 3.7 for the notations). It is readily seen that Bs H_{X_{2}} = L_{1}. By the similar arguments as in Lemmas 3.1, 3.5, we have the following result concerning the Sarkisov degree of (X_{2}, H_{X_{2}}) in consideration of the local equation f_{2} of a general member S_{X_{2}} ∈ H_{X_{2}}.

Lemma 3.8.

(1) The quasi-effective threshold μ_{2} of (X_{2}, H_{X_{2}}) equals 1/2 and K_{X_{2}} + (1/μ_{2})H_{X_{2}} is not canonical;

(2) The canonical threshold c_{2} of (X_{2}, H_{X_{2}}) equals 1/2 and L_{1} is the maximal center.

Hence we have to blow-up X_{2} along the curve L_{1}, say p_{2} : E_{2} ⊂ Z_{2} → L_{1} ⊂ X_{2}, where E_{2} ≅ F_{2} is the exceptional divisor of p_{2}. The configurations of the boundary surfaces E_{0}_{Z_{2}}, E_{1}_{Z_{2}} and E_{2} on Z_{2} are described as in Figure 4. The restriction E_{1}_{Z_{2}} → E_{1} ≅ P^{1}×P^{1} of p_{2} onto E_{1}_{Z_{2}} is just the blow-up at the point P_{2} := L_{1} ∩ E_{1} with an exceptional curve e_{0} := E_{2} ∩ E_{1}_{Z_{2}}. Let M_{0} and l_{0} be the rulings on E_{1} ≅ P^{1} × P^{1} (see Lemma 3.6, (2)) such that they meet each other at P_{2}. We use the same notations M_{0} and l_{0} to represent their proper transforms on E_{1}_{Z_{2}}. By noting mult_{L_{1}}S_{X_{2}} = mult_{L_{1}}T_{X_{2}} = mult_{L_{1}}H_{X_{2}} = 1, we know that the proper transform H_{Z_{2}} is spanned by S_{Z_{2}}, T_{Z_{2}} + E_{0}_{Z_{2}} + E_{2}, H_{Z_{2}} + 2E_{0}_{Z_{2}} + 2E_{2} and 3E_{0}_{Z_{2}} + E_{1}_{Z_{2}} + 2E_{2}. With these notations, by the similar argument as in Lemma 3.2, we can readily see the following result:

Lemma 3.9.

(1) K_{Z_{2}} ∼ −3E_{0}_{Z_{2}} − 2E_{1}_{Z_{2}} − 2E_{2};

(2) E_{0}_{Z_{2}}^{2} ∼ −M_{2} − l_{2}, E_{1}_{Z_{2}}^{2} ∼ −M_{0} + 2l_{0} + e_{0} and E_{2}^{2} ∼ −M_{2} + l_{2};

(3) S_{Z_{2}}|_{E_{0}_{Z_{2}}} = T_{Z_{2}}|_{E_{0}_{Z_{2}}} = H_{Z_{2}}|_{E_{0}_{Z_{2}}} = L_{1}, where we use the same notation L_{1} to represent the curve contained in E_{0}_{Z_{2}} ∩ E_{2} which is mapped onto L_{1} via p_{2};

(4) S_{Z_{2}}|_{E_{1}_{Z_{2}}} ∼ 2M_{0} +2l_{0} +3e_{0}, T_{Z_{2}}|_{E_{1}_{Z_{2}}} ∼ M_{0} +2l_{0} +2e_{0} and H_{Z_{2}}|_{E_{1}_{Z_{2}}} ∼ 2l_{0} + e_{0};

(5) S_{Z_{2}}|_{E_{2}} ∼ M_{2} + 2l_{2}, T_{Z_{2}}|_{E_{2}} ∼ M_{2} + 2l_{2} and H_{Z_{2}}|_{E_{2}} ∼ M_{2} + 2l_{2}.

Though we have the linearly equivalent relations concerning S_{Z_{2}}|_{E_{1}_{Z_{2}}} , T_{Z_{2}}|_{E_{1}_{Z_{2}}} , H_{Z_{2}}|_{E_{1}_{Z_{2}}} on E_{1}_{Z_{2}} and S_{Z_{2}}|_{E_{2}}, T_{Z_{2}}|_{E_{2}}, H_{Z_{2}}|_{E_{2}} on E_{2} ≅ F_{2} as stated in Lemma 3.9 (4) and (5), we now need to observe the concrete configurations of these intersection curves. In fact, we have the following:

Lemma 3.10. S_{Z_{2}}|_{E_{1}_{Z_{2}}} = 2C_{2}+L_{3}, T_{Z_{2}}|_{E_{1}_{Z_{2}}} = C_{2}+L_{2}+L_{3}, H_{Z_{2}}|_{E_{1}_{Z_{2}}} = 2L_{2} + L_{3} and S_{Z_{2}}|_{E_{2}} = T_{Z_{2}}|_{E_{2}} = H_{Z_{2}}|_{E_{2}} = L_{1} + 2L_{3} (cf. Figure 4), where C_{2} is an irreducible curve linearly equivalent to M_{0} + l_{0} + e_{0} on E_{1}_{Z_{2}}.

Proof. We need to describe the blow-up p_{2} : E_{2} ⊂ Z_{2} → L_{1} ⊂ X_{2} explicitly on the open affine subset U_{2} (y, z, w). On U_{2}, the center L_{1} of p_{2} is defined by an ideal 𝔞_{2} = (y, w) ⊂ Γ(O_{U_{2}}) = C[y, z, w]. Hence p_{2}^{−1}(U_{2}) ≅ Proj( ⨁_{d≥0} 𝔞_{2}^{d}) which is covered by two affine 3-spaces, say U_{2}^{(1)} (y, z, w) and U_{2}^{(2)} (y, z, w). We may assume that a morphism U_{2}^{(1)} → U_{2} (resp. U_{2}^{(2)} → U_{2}) is associated to the algebra homomorphism y ↦ y, z ↦ z, w ↦ yw (resp. y ↦ yw, z ↦ z, w ↦ w). By the straightforward computations, on U_{2}^{(1)} (resp. U_{2}^{(2)}), the surfaces S_{Z_{2}}, T_{Z_{2}}, H_{Z_{2}} and E_{2} are defined by f_{2}^{(1)} = y(w − 1)^{2} −z^{2}w, g_{2}^{(1)} = y(w − 1)+z^{2}w, h_{2}^{(1)} = y − z^{2}w and y (resp. f_{2}^{(2)} = (y − 1)^{2}w − z^{2}, g_{2}^{(2)} = y(y − 1)w − z^{2}, h_{2}^{(2)} = y^{2}w − z^{2} and w). By these equations and Lemma 3.9 we have the assertion. Q.E.D.

We perform (K_{Z_{2}} + c_{2}H_{Z_{2}})-MMP over B_{2}. The cone \overline{NE} (Z_{2}/B_{2}) ⊂ R^{2} is spanned by two rays one of which comes from the rulings on the exceptional divisor E_{2} ≅ F_{2} and the other comes from those on E_{0}_{Z_{2}} ≅ F_{2} (cf. Lemma 3.9 (2)). The latter ray is (K_{Z_{2}} + c_{2}H_{Z_{2}})-nagative. The contraction of E_{0}_{Z_{2}}, say q_{2} : Z_{2} → X_{3}, induces a Del Pezzo fibration ϕ_{3} : X_{3} → B_{3} = B_{2} ≅ P^{1} on X_{3} with E_{2}_{X_{3}} ≅ Q_{0}^{2} as a singular fiber. The composite Φ_{3} := q_{2} ∘ p_{2}^{−1} : X_{2}/B_{2} ⋯ → X_{3}/B_{3} is an elementary link of type II. We have K_{X_{3}} ∼ −2E_{1}_{X_{3}}− 2E_{2}_{X_{3}}. The proper transform H_{X_{3}} is spanned by S_{X_{3}}, T_{X_{3}}+E_{2}_{X_{3}}, H_{X_{3}}+2E_{2}_{X_{3}} and E_{1}_{X_{3}} + 2E_{2}_{X_{3}}. By virtue of Lemma 3.10, we can readily see the following:

Lemma 3.11.

(1) S_{X_{3}}|_{E_{1}_{X_{3}}} = 2C_{2} + L_{3}, T_{X_{3}}|_{E_{1}_{X_{3}}} = C_{2} + L_{2} + L_{3} and H_{X_{3}}|_{E_{1}_{X_{3}}} = 2L_{2} +L_{3}, where we use the same notations C_{2}, L_{2}, L_{3} to represent their proper transforms (cf. Figure 5);

(2) Bs H_{X_{3}} = L_{3};

(3) On the open affine subset U_{3} := q_{2}(U_{2}^{(2)}) (y, z, w), the surfaces S_{X_{3}}, T_{X_{3}}, H_{X_{3}}, E_{1}_{X_{3}} and E_{2}_{X_{3}} are defined respectively by f_{3} := (y − 1)^{2}w − z^{2}, g_{3} := y(y − 1)w − z^{2}, h_{3} := y^{2}w − z^{2}, z and w, and the curves C_{2}, L_{2}, L_{3} are defined by the ideals (y − 1, z), (y, z), 𝔞_{3} := (z, w) ⊂ Γ(O_{U_{3}}) = C[y, z, w], respectively.

We shall check the Sarkisov degree of (X_{3}, H_{X_{3}}).

Lemma 3.12.

(1) The quasi-effective threshold μ_{3} of (X_{3}, H_{X_{3}}) equals 1/2 and K_{X_{3}} + (1/μ_{3})H_{X_{3}} is not canonical;

(2) The canonical threshold c_{3} of (X_{3}, H_{X_{3}}) equals 1 and the maximal centers are L_{3} and the point Q_{3} := (1, 0, 0) ∈ U_{3} (y, z, w). More precisely, such discrete valuations of C(X_{3}) which are the maximal singularity of H_{X_{3}} (cf. Definition 2.2) are obtained by the blow-up along L_{3} and the (ordinary) blowup at Q_{3}.

Proof. We can readily see μ_{3} = 1/2. In order to see that K_{X_{3}} + (1/μ_{3})H_{X_{3}} is not canonical, let p_{3} : E_{3} ⊂ Z_{3} → L_{3} ⊂ X_{3} be the blow-up of X_{3} along Bs H_{X_{3}} = L_{3} with an exceptional divisor E_{3}. Since mult_{L_{3}}S_{X_{3}} = 1 by virtue of Lemma 3.11 (3), we have a(E_{3}; X_{3}, (1/μ_{3})H_{X_{3}}) = −1, so K_{X_{3}} + (1/μ_{3})H_{X_{3}} is not canonical. It is easily verified that a(E_{3}; X_{3}, H_{X_{3}}) = 0, hence c_{3} ≤ 1. By Lemma 3.11, S_{X_{3}} is smooth along L_{3}\{Q_{3}} and mult_{Q_{3}}S_{X_{3}} = 2, and the possibility of the maximal centers are Q_{3} and L_{3}. Let E(a, b, c) be the exceptional divisor of the weighted blow-up at Q_{3} with weight wt(y −1, z, w) = (a, b, c), where one of a, b, c is 1 and the other two are coprime positive integers (cf. Theorem 2.2). Then a(E(a, b, c); X_{3}, H_{X_{3}}) equals −a+b−1 (resp. a−b+c− 1) if 2a + c ≤ 2b (resp. if 2b ≤ 2a + c). Suppose that a(E(a, b, c); X_{3}, H_{X_{3}}) ≤ 0.

Then by the straightforward computations we know that the only possibility is (a, b, c) = (1, 1, 1) and a(E(1, 1, 1); X_{3}, H_{X_{3}}) = 0. Thus we have the assertion.

Q.E.D.

As in the proof of Lemma 3.12, we consider the blow-up p_{3} : E_{3} ⊂ Z_{3} → L_{3} ⊂ X_{3} along L_{3} with an exceptional divisor E_{3} ≅ F_{2}. Then the configurations of the boundary surfaces E_{1}_{Z_{3}}, E_{2}_{Z_{3}}, E_{3} are described as in Figure 6, where we use the same notations for the curves C_{2} and L_{2} to represent their proper transforms. Note that C_{4} is a fiber (resp. the minimal section) on E_{2}_{Z_{3}} ≅ F_{2} (resp. on E_{3} ≅ F_{2}). The proper transform H_{Z_{3}} of H_{X_{3}} is spanned by S_{Z_{3}}, T_{Z_{3}} + E_{2}_{Z_{3}} + E_{3}, H_{Z_{3}} + 2E_{2}_{Z_{3}} + 2E_{3} and E_{1}_{Z_{3}} + 2E_{2}_{Z_{3}} + 2E_{3}. We have the following result. Since the proof for this lemma is similar to that for Lemma 3.2, we shall omit it.

Lemma 3.13.

(1) K_{Z_{3}} ∼ −2E_{1}_{Z_{3}} − 2E_{2}_{Z_{3}} − 3E_{3};

(2) E_{1}_{Z_{3}}^{2} ∼ 2l_{0} and E_{2}_{Z_{3}}^{2} ∼ −M_{2} − l_{2} and E_{3}^{2} ∼ −M_{2} − l_{2};

(3) S_{Z_{3}}|_{E_{1}_{Z_{3}}} ∼ 2M_{0} + 2l_{0} + 2e_{0}, T_{Z_{3}}|_{E_{1}_{Z_{3}}} ∼ M_{0} + 2l_{0} + e_{0} and H_{Z_{3}}|_{E_{1}_{Z_{3}}} ∼ 2l_{0}, where we use the same notations M_{0}, l_{0} and e_{0} to represent their proper transforms (see the argument before Lemma 3.9 and Figure 6);

(4) S_{Z_{3}}|_{E_{2}_{Z_{3}}} ∼ T_{Z_{3}}|_{E_{2}_{Z_{3}}} ∼ H_{Z_{3}}|_{E_{2}_{Z_{3}}} ∼ l_{2};

(5) S_{Z_{3}}|_{E_{3}} ∼ T_{Z_{3}}|_{E_{3}} ∼ H_{Z_{3}}|_{E_{3}} ∼ M_{2} + 2l_{2}.

We further need to observe the concrete configurations of the schemetheoretic intersections. Indeed, we have the following:

Lemma 3.14.

(1) S_{Z_{3}}|_{E_{1}_{Z_{3}}} = 2C_{2}, T_{Z_{3}}|_{E_{1}_{Z_{3}}} = C_{2} + L_{2} and H_{Z_{3}}|_{E_{1}_{Z_{3}}} = 2L_{2};

(2) S_{Z_{3}}|_{E_{2}_{Z_{3}}} = T_{Z_{3}}|_{E_{2}_{Z_{3}}} = H_{Z_{3}}|_{E_{2}_{Z_{3}}} = C_{4};

(3) S_{Z_{3}}|_{E_{3}} = C_{4} + 2L_{4}, T_{Z_{3}}|_{E_{3}} = C_{4} + L_{4} + L′_{4} and H_{Z_{3}}|_{E_{3}} = C_{4} + 2L′_{4} (cf. Figure 6).

Proof. The proof is similar to that for Lemma 3.10. We shall describe the blow-up p_{3} : E_{3} ⊂ Z_{3} → L_{3} ⊂ X_{3} explicitly around U_{3} (y, z, w). On this open affine subset U_{3}, the center L_{3} of the blow-up p_{3} is defined by an ideal 𝔞_{3} = (z, w) ⊂ Γ(O_{U_{3}}) = C[y, z, w]. Hence p_{3}^{−1}(U_{3}) ≅ Proj( ⨁_{d≥0} 𝔞_{3}^{d}) which is covered by two affine 3-spaces, say U_{3}^{(1)} (y, z, w) and U_{3}^{(2)} (y, z, w). We may assume that U_{3}^{(1)} → U_{3} (resp. U_{3}^{(2)} → U_{3}) is a morphism associated to the algebra homomorphism y ↦ y, z ↦ zw, w ↦ w (resp. y ↦ y, z ↦ z, w ↦ zw). Then the surfaces S_{Z_{3}}, T_{Z_{3}}, H_{Z_{3}} and E_{3} are defined by f_{3}^{(1)} := (y − 1)^{2} − z^{2}w, g_{3}^{(1)} := y(y − 1) − z^{2}w, h_{3}^{(1)} := y^{2} − z^{2}w and w on U_{3}^{(1)} (resp. f_{3}^{(2)} := (y − 1)^{2}w − z, g_{3}^{(2)} := y(y − 1)w − z, h_{3}^{(2)} := y^{2}w − z and z on U_{3}^{(2)}). Thus we have the assertion as desired in consideration of Lemma 3.13.

Q.E.D.

We perform (K_{Z_{3}} + c_{3}H_{Z_{3}})-MMP over B_{3}. The cone \overline{NE} (Z_{3}/B_{3}) is spanned by two rays one of which comes from the rulings on the exceptional divisor E_{3} ≅ F_{2} and the other comes from those on E_{2}_{Z_{3}} ≅ F_{2}. The latter ray is (K_{Z_{3}} + c_{3}H_{Z_{3}})-negative. Let q_{3} : Z_{3} → X_{4} be the contraction of E_{2}_{Z_{3}}. Note that there exists a Del Pezzo fibration, say ϕ_{4} : X_{4} → B_{4} = B_{3} ≅ P^{1}, with a general fiber isomorphic to a smooth quadric and with E_{3}_{X_{4}} ≅ Q_{0}^{2} as a singular fiber. The composite Φ_{4} := q_{3} ∘p_{3}^{−1} : X_{3}/B_{3} ⋯ → X_{4}/B_{4} is an elementary link of type II. The canonical divisor K_{X_{4}} is linearly equivalent to −2E_{1}_{X_{4}} −3E_{3}_{X_{4}} and the proper transform H_{X_{4}} is spanned by S_{X_{4}}, T_{X_{4}} + E_{3}_{X_{4}}, H_{X_{4}} + 2E_{3}_{X_{4}} and E_{1}_{X_{4}} + 2E_{3}_{X_{4}}. We can readily see the following result in consideration of Lemma 3.14.

Lemma 3.15.

(1) S_{X_{4}}|_{E_{1}_{X_{4}}} = 2C_{2}, T_{X_{4}}|_{E_{1}_{X_{4}}} = C_{2} + L_{2}, H_{X_{4}}|_{E_{1}_{X_{4}}} = 2L_{2}, S_{X_{4}}|_{E_{3}_{X_{4}}} = 2L_{4}, T_{X_{4}}|_{E_{3}_{X_{4}}} = L_{4} + L′_{4} and H_{X_{4}}|_{E_{3}_{X_{4}}} = 2L′_{4}, where we use the same notations C_{2}, L_{2}, L_{4}, L′_{4} etc. to represent their proper transforms (cf. Figure 7). E_{1}_{X_{4}} ≅ F_{1} and the curves C_{2} (resp. L_{2}) is a cross-section on E_{1}_{X_{4}} with self-intersection number 1 (resp. −1);

(2) Bs H_{X_{3}} = L_{4};

(3) On the open affine subset U_{4} := q_{3}(U_{3}^{(1)}) (y, z, w), the surfaces S_{X_{4}}, T_{X_{4}}, H_{X_{4}}, E_{1}_{X_{4}} and E_{3}_{X_{4}} are defined respectively by f_{4} := (y − 1)^{2} − z^{2}w, g_{4} := y(y − 1) − z^{2}w, h_{4} := y^{2} − z^{2}w, z and w. Furthermore, the base locus Bs H_{X_{4}} = L_{4} is defined by an ideal 𝔞_{4} := (y − 1, w) ⊂ Γ(O_{U_{4}}) = C[y, z, w] on U_{4}.

The Sarkisov degree of (X_{4}, H_{X_{4}}) is written as follows.

Lemma 3.16.

(1) The quasi-effective threshold μ_{4} of (X_{4}, H_{X_{4}}) equals 1/2 and K_{X_{4}} + (1/μ_{4})H_{X_{4}} is not canonical;

(2) The canonical threshold c_{4} of (X_{4}, H_{X_{4}}) equals 1 and the maximal center is L_{4}.

Proof. It is easy to see that μ_{4} = 1/2 and K_{X_{4}} + (1/μ_{4})H_{X_{4}} ∼ E_{3}_{X_{4}} is nef. Hence we know that K_{X_{4}} + (1/μ_{4})H_{X_{4}} is not canonical by Noether-Fano-Iskovskih inequality (cf. [Cor95, Cor00]). In fact, a general member S_{X_{4}} ∈ H_{X_{4}} has multiplicity 1 along the base locus curve L_{4}. Hence we obtain a(E_{4}; X_{4}, (1/μ_{4})H_{X_{4}}) = −1 for an exceptional divisor E_{4} which is obtained by the blow-up of X_{4} along L_{4}. It is obvious that the maximal center of H_{X_{4}} is contained in the base locus Bs H_{X_{4}} = L_{4}. By virtue of the local equation f_{4} of S_{X_{4}} ∈ H_{X_{4}} (see Lemma 3.15) and by the similar arguments as in Lemmas 3.1 and 3.12, we know that L_{3} itself is a maximal center of H_{X_{4}}. Q.E.D.

By Lemma 3.16, we perform the blow-up p_{4} : E_{4} ⊂ Z_{4} → L_{4} ⊂ X_{4} along the curve L_{4}, where E_{4} ≅ F_{2} is an exceptional divisor. Then the configuration of the boundary surfaces E_{1}_{Z_{4}}, E_{3}_{Z_{4}} and E_{4} is described as in Figure 8, where we use the same notations for the curves C_{2}, L_{2}, L′_{4} to represent their proper tranforms. Furthermore, we use the same notation L_{4} to represent the curve which is contained in E_{3}_{Z_{4}} ∩ E_{4} and mapped onto L_{4} via p_{4}. Note that L_{4} and L_{5} are, respectively, a fiber and the minimal section (resp. the minimal section and a fiber) on E_{3}_{Z_{4}} ≅ F_{2} (resp. on E_{4} ≅ F_{2}). The proper transform H_{Z_{4}} is spanned by S_{Z_{4}}, T_{Z_{4}} + E_{3}_{Z_{4}} + E_{4}, H_{Z_{4}} + 2E_{3}_{Z_{4}} + E_{4} and E_{1}_{Z_{4}} + 2E_{3}_{Z_{4}} + E_{4}. The restriction E_{1}_{Z_{4}} → E_{1}_{X_{4}} ≅ F_{1} of p_{4} onto E_{1}_{Z_{4}} is just the blow-up at the point P_{4} := L_{4} ∩ E_{1}_{X_{4}} with an exceptional curve e_{1} := E_{4} ∩ E_{1}_{Z_{4}}. Let l_{1} be the fiber on E_{1}_{X_{4}} ≅ F_{2} passing through P_{4} and we use the same notations M_{1}, l_{1} to represent their proper transforms on E_{1}_{Z_{4}} in the sequel if there are no confusions (cf. Figures 7 and 8). The following result is not difficult to verify by the similar argument as in Lemma 3.2 and we shall omit it.

Lemma 3.17.

(1) K_{Z_{4}} ∼ −2E_{1}_{Z_{4}} − 3E_{3}_{Z_{4}} − 2E_{4};

(2) E_{1}_{Z_{4}}^{2} ∼ 2M_{1}, E_{3}_{Z_{4}}^{2} ∼ −M_{2} − l_{2} and E_{4}^{2} ∼ −M_{2} − l_{2};

(3) S_{Z_{4}}|_{E_{1}_{Z_{4}}} ∼ 2M_{1} + 2l_{1} + e_{1}, T_{Z_{4}}|_{E_{1}_{Z_{4}}} ∼ 2M_{1} + l_{1} and H_{Z_{4}}|_{E_{1}_{Z_{4}}} ∼ 2M_{1};

(4) S_{Z_{4}}|_{E_{3}_{Z_{4}}} ∼ l_{2}, T_{Z_{4}}|_{E_{3}_{Z_{4}}} ∼ l_{2} and H_{Z_{4}}|_{E_{3}_{Z_{4}}} ∼ M_{2} + 2l_{2};

(5) S_{Z_{4}}|_{E_{4}} ∼ T_{Z_{4}}|_{E_{4}} ∼ M_{2} + 2l_{2} and H_{Z_{4}}|_{E_{4}} ∼ l_{2}.

As in the previous steps, we have to determine the concrete configurations of the restrictions of S_{Z_{4}}, T_{Z_{4}} and H_{Z_{4}} onto the boundary surfaces E_{1}_{Z_{4}}, E_{3}_{Z_{4}} and E_{4}. In fact, we see the following:

Lemma 3.18.

(1) S_{Z_{4}}|_{E_{1}_{Z_{4}}} = 2C_{2} + L′_{5}, T_{Z_{4}}|_{E_{1}_{Z_{4}}} = C_{2} + L_{2} and H_{Z_{4}}|_{E_{1}_{Z_{4}}} = 2L_{2};

(2) S_{Z_{4}}|_{E_{3}_{Z_{4}}} = L_{4}, T_{Z_{4}}|_{E_{3}_{Z_{4}}} = L′_{4} and H_{Z_{4}}|_{E_{3}_{Z_{4}}} = L_{5} + 2L′_{4};

(3) S_{Z_{4}}|_{E_{4}} = L_{4} + 2L′_{5}, T_{Z_{4}}|_{E_{4}} = C_{5} and H_{Z_{4}}|_{E_{4}} = L_{5} (cf. Figure 8).

Proof. We investigate the blow-up p_{4} : Z_{4} → X_{4} over the open affine subset U_{4} (y, z, w). On U_{4}, the center L_{4} of p_{4} is defined by an ideal 𝔞_{4} = (y − 1, w) ⊂ C[y, z, w]. Therefore p_{4}^{−1}(U_{4}) ≅ Proj( ⨁_{d≥0} 𝔞_{4}^{d}) which is covered by two affine 3-spaces, say U_{4}^{(1)} (y, z, w) and U_{4}^{(2)} (y, z, w). We may and shall assume that a morphism U_{4}^{(1)} → U_{4} (resp. U_{4}^{(2)} → U_{4}) is associated to the algebra homomorphism y ↦ y, z ↦ z, w ↦ (y − 1)w (resp. y ↦ yw + 1, z ↦ z, w ↦ w). Then the surfaces S_{Z_{4}}, T_{Z_{4}}, H_{Z_{4}} and E_{4} are defined by f_{4}^{(1)} := (y − 1)−z^{2}w, g_{4}^{(1)} := y −z^{2}w, h_{4}^{(1)} := y^{2} −(y −1)z^{2}w and y −1 on U_{4}^{(1)} (resp. f_{4}^{(2)} := y^{2}w − z^{2}, g_{4}^{(2)} := y(yw + 1) − z^{2}, h_{4}^{(2)} := (yw + 1)^{2} − z^{2}w and w on U_{4}^{(2)}). Then we can readily see the assertion in combination of Lemma 3.17. Q.E.D.

We then run the (K_{Z_{4}} + c_{4}H_{Z_{4}})-MMP over B_{4}. The cone \overline{NE} (Z_{4}/B_{4}) ⊂ R^{2} is spanned by two rays one of which comes from the rulings on the exceptional divisor E_{4} ≅ F_{2} and the other comes from those on E_{3}_{Z_{4}} ≅ F_{2}. The latter ray is (K_{Z_{4}} + c_{4}H_{Z_{4}})-negative. We contract E_{3}_{Z_{4}}, say q_{4} : Z_{4} → X_{5}, to get a Del Pezzo fibration ϕ_{5} : X_{5} → B_{5} = B_{4} ≅ P^{1} having E_{4}_{X_{5}} ≅ Q_{0}^{2} as a singular fiber. The composite Φ_{5} := q_{4} ∘ p_{4}^{−1} : X_{4}/B_{4} ⋯ → X_{5}/B_{5} is an elementary link of type II. It follows that K_{X_{5}} ∼ −2E_{1}_{X_{5}} − 2E_{4}_{X_{5}} and the proper transform H_{X_{5}} of H_{Z_{4}} is spanned by S_{X_{5}}, T_{X_{5}} + E_{4}_{X_{5}}, H_{X_{5}} + E_{4}_{X_{5}} and E_{1}_{X_{5}} + E_{4}_{X_{5}}. By virtue of Lemma 3.18, we can easily see the following:

Lemma 3.19.

(1) S_{X_{5}}|_{E_{1}_{X_{5}}} = 2C_{2}+L′_{5}, T_{X_{5}}|_{E_{1}_{X_{5}}} = C_{2}+L_{2}, H_{X_{5}}|_{E_{1}_{X_{5}}} = 2L_{2}, S_{X_{5}}|_{E_{4}_{X_{5}}} = 2L′_{5}, T_{X_{5}}|_{E_{4}_{X_{5}}} = C_{5} and H_{X_{5}}|_{E_{4}_{X_{5}}} = 2L_{5}, where we use the same notations C_{2}, L_{2}, L′_{5}, C_{5} to represent their proper transforms and we put L_{5} := q_{4}(L_{5}) (cf. Figure 9);

(2) Bs H_{X_{5}} = L′_{5};

(3) On the open affine subset U_{5} := q_{4}(U_{4}^{(2)}) (y, z, w), the surfaces S_{X_{5}}, T_{X_{5}}, H_{X_{5}}, E_{1}_{X_{5}} and E_{4}_{X_{5}} are defined respectively by f_{5} := y^{2}w − z^{2}, g_{5} := y(yw + 1) − z^{2}, h_{5} := (yw + 1)^{2} − z^{2}w, z and w. Furthermore, the base locus Bs H_{X_{5}} = L′_{5} is defined by an ideal 𝔞_{5} := (z, w) ⊂ Γ(O_{U_{5}}) = C[y, z, w] on U_{5}.

The following result about the Sarkisov degree of the pair (X_{5}, H_{X_{5}}) is obtained by the similar argument as in the proof of Lemma 3.16, so we shall omit the detail.

Lemma 3.20.

(1) The quasi-effective threshold μ_{5} of (X_{5}, H_{X_{5}}) equals 1/2 and K_{X_{5}} + (1/μ_{5})H_{X_{5}} is not canonical;

(2) The canonical threshold c_{5} of (X_{5}, H_{X_{5}}) equals 1 and the maximal center is L′_{5}.

Let p_{5} : E_{5} ⊂ Z_{5} → L′_{5} ⊂ X_{5} be the blow-up along L′_{5} with an exceptional divisor E_{5} ≅ F_{2}. The configuration of the boundary surfaces E_{1}_{Z_{5}}, E_{4}_{Z_{5}} and E_{5} is drawn as in Figure 10, where we use the same notations for the curves C_{2}, L_{2}, L_{5}, C_{5} to represent their proper transforms on Z_{5}. Note that the curves C_{6} and L′_{6} are respectively a fiber and the minimal section (resp. the minimal section and a fiber) on E_{4}_{Z_{5}} ≅ F_{2} (resp. on E_{5} ≅ F_{2}). The proper transform H_{Z_{5}} of H_{X_{5}} is spanned by S_{Z_{5}}, T_{Z_{5}} +E_{4}_{Z_{5}}, H_{Z_{5}} +E_{4}_{Z_{5}} and E_{1}_{Z_{5}} +E_{4}_{Z_{5}} +E_{5}. The following result holds true. We shall omit the proof because it is almost similar to that for Lemma 3.2.

Lemma 3.21.

(1) K_{Z_{5}} ∼ −2E_{1}_{Z_{5}} − 2E_{4}_{Z_{5}} − 3E_{5};

(2) E_{1}_{Z_{5}}^{2} ∼ 2M_{1} + l_{1} − e_{1}, E_{4}_{Z_{5}}^{2} ∼ −M_{2} − l_{2} and E_{5}^{2} ∼ −M_{2} − l_{2} (see the argument before Lemma 3.17);

(3) S_{Z_{5}}|_{E_{1}_{Z_{5}}} ∼ T_{Z_{5}}|_{E_{1}_{Z_{5}}} ∼ H_{Z_{5}}|_{E_{1}_{Z_{5}}} ∼ 2M_{1} + l_{1};

(4) S_{Z_{5}}|_{E_{4}_{Z_{5}}} ∼ l_{2} and T_{Z_{5}}|_{E_{4}_{Z_{5}}} ∼ H_{Z_{5}}|_{E_{4}_{Z_{5}}} ∼ M_{2} + 2l_{2};

(5) S_{Z_{5}}|_{E_{5}} ∼ M_{2} + 2l_{2} and T_{Z_{5}}|_{E_{5}} ∼ H_{Z_{5}}|_{E_{5}} ∼ l_{2}.

We, furthermore, determine the concrete configurations:

Lemma 3.22.

(1) S_{Z_{5}}|_{E_{1}_{Z_{5}}} = 2C_{2}, T_{Z_{5}}|_{E_{1}_{Z_{5}}} = C_{2} + L_{2} and H_{Z_{5}}|_{E_{1}_{Z_{5}}} = 2L_{2} + L_{5};

(2) S_{Z_{5}}|_{E_{4}_{Z_{5}}} = C_{6}, T_{Z_{5}}|_{E_{4}_{Z_{5}}} = C_{5} and H_{Z_{5}}|_{E_{4}_{Z_{5}}} = L′_{6} + 2L_{5};

(3) S_{Z_{5}}|_{E_{5}} = C_{6} + 2L_{6}, T_{Z_{5}}|_{E_{5}} = L_{6} and H_{Z_{5}}|_{E_{5}} = L′_{6} (cf. Figure 10).

Proof. For the proof we need to describe the blow-up p_{5} : Z_{5} → X_{5} concretely over the open affine subset U_{5} (y, z, w). Note that the center L′_{5} of p_{5} is defined by an ideal 𝔞_{5} = (z, w) ⊂ Γ(O_{U_{5}}) = C[y, z, w] on U_{5}. Hence p_{5}^{−1}(U_{5}) ≅ Proj( ⨁_{d≥0} 𝔞_{5}^{d}) which is covered by two affine 3-spaces, say U_{5}^{(1)} (y, z, w) and U_{5}^{(2)} (y, z, w). We may and shall assume that a morphism U_{5}^{(1)} → U_{5} (resp. U_{5}^{(2)} → U_{5}) is associated to the algebra homomorphism y ↦ y, z ↦ zw, w ↦ w (resp. y ↦ y, z ↦ z, w ↦ zw). We then can verify that the surfaces S_{Z_{5}}, T_{Z_{5}}, H_{Z_{5}} and E_{5} are defined by f_{5}^{(1)} := y^{2} − z^{2}w, g_{5}^{(1)} := y(yw + 1) − z^{2}w^{2}, h_{5}^{(1)} := (yw + 1)^{2} −z^{2}w^{3} and w on U_{5}^{(1)} (resp. f_{5}^{(2)} := y^{2}w −z, g_{5}^{(2)} := y(yzw +1)−z^{2}, h_{5}^{(2)} := (yzw + 1)^{2} − z^{3}w and z on U_{5}^{(2)}). Then we have the assertion readily in combination of Lemma 3.21. Q.E.D.

After the blow-up p_{5} : Z_{5} → X_{5}, we run (K_{Z_{5}} + c_{5}H_{Z_{5}})-MMP over B_{5}. The cone of curves \overline{NE} (Z_{5}/B_{5}) is generated by two rays one of which comes from the rulings on the exceptional divisor E_{5} ≅ F_{2} and the other comes from those on E_{4}_{Z_{5}} ≅ F_{2}. The latter ray is (K_{Z_{5}} + c_{5}H_{Z_{5}})-negative. We contract E_{4}_{Z_{5}} along the fibers on it, say q_{5} : Z_{5} → X_{6}, to get a Del Pezzo fibration ϕ_{5} : X_{6} → B_{6} = B_{5} ≅ P^{1} with E_{5}_{X_{6}} ≅ Q_{0}^{2} as a singular fiber. The composite Φ_{6} := q_{5} ∘ p_{5}^{−1} : X_{5}/B_{5} ⋯ → X_{6}/B_{6} is an elementary link of type II. The configuration of the boundary surfaces E_{1}_{X_{6}} and E_{5}_{X_{6}} is drawn as in Figure 11, where we use the same notations for curves C_{2}, L_{2}, L_{6}, L′_{6} to represent their proper transforms on X_{6}. Note that E_{1}_{X_{6}} ≅ P^{1} × P^{1}. It is readily seen that K_{X_{6}} ∼ −2E_{1}_{X_{6}} −3E_{5}_{X_{6}} and the proper transform H_{X_{6}} of H_{Z_{5}} is spanned by S_{X_{6}}, T_{X_{6}}, H_{X_{6}} and E_{1}_{X_{6}} + E_{5}_{X_{6}}. We put U_{6} := q_{5}(U_{5}^{(1)}) (y, z, w) ⊂ X_{6}. We can obtain the following result in consideration of the previous lemma. We shall omit the detail.

Lemma 3.23.

(1) S_{X_{6}}|_{E_{1}_{X_{6}}} = 2C_{2}, T_{X_{6}}|_{E_{1}_{X_{6}}} = C_{2} + L_{2}, H_{X_{6}}|_{E_{1}_{X_{6}}} = 2L_{2}, S_{X_{6}}|_{E_{5}_{X_{6}}} = 2L_{6}, T_{X_{6}}|_{E_{5}_{X_{6}}} = L_{6} + L′_{6} and H_{X_{6}}|_{E_{5}_{X_{6}}} = 2L′_{6} (cf. Figure 11);

(2) Bs H_{X_{6}} = {P_{6}}, where we put P_{6} := L_{6} ∩ L′_{6};

(3) On the open affine subset U_{6} (y, z, w), the surfaces S_{X_{6}}, T_{X_{6}}, H_{X_{6}}, E_{1}_{X_{6}} and E_{5}_{X_{6}} are defined respectively by f_{6} := y^{2} − z^{2}w, g_{6} := y(yw + 1) − z^{2}w^{2}, h_{6} := (yw + 1)^{2} − z^{2}w^{3}, z and w.

We shall check the Sarkisov degree of the pair (X_{6}, H_{X_{6}}).

Lemma 3.24.

(1) The quasi-effective threshold μ_{6} of (X_{6}, H_{X_{6}}) equals 1/2. Furthermore, K_{X_{6}} + (1/μ_{6})H_{X_{6}} is canonical but not nef;

(2) The canonical threshold c_{6} of (X_{6}, H_{X_{6}}) equals 2 and P_{6} is the maximal center of (X_{6}, H_{X_{6}}). More precisely, such a discrete valuation E of C(X_{6}) with discrepancy a(E; X_{6}, c_{6}H_{X_{6}}) = 0 is obtained by the (ordinary) blow-up at P_{6}.

Proof. We can readily see that μ_{6} = 1/2 in consideration of K_{X_{6}} ∼ −2E_{1}_{X_{6}} − 3E_{5}_{X_{6}} and H_{X_{6}} ≡ E_{1}_{X_{6}} +E_{5}_{X_{6}}. Now we shall verify that the canonical threshold c_{6} coincides with 1/μ_{6} = 2. By virtue of Lemma 3.23 (2), the possibility of the maximal center is the point P_{6}. We claim the following:

Claim. The members of H_{X_{6}} except for E_{1}_{X_{6}} + E_{5}_{X_{6}} is smooth at P_{6}.

Proof of Claim. Note that E_{1}_{X_{6}}^{2} ∼ −M_{0} +2l_{0}, so it can be contracted along the fibers l_{0} onto a curve. Now assume to the contrary that a member G ∈ H_{X_{6}} with G ≠ E_{1}_{X_{6}} + E_{5}_{X_{6}} is not smooth at P_{6}. Let m := mult_{P_{6}}G ≥ 2. After the contraction of E_{1}_{X_{6}}, we blow-up at P_{6} with an exceptional divisor E. Then the proper transform of E_{5}_{X_{6}} is isomorphic to F_{2} with the self-intersection ∼ −M_{2} + 2l_{2}, so we can contract it along the fibers on it. Then we reach the projective 3-space P^{3} and we have the linearly equivalent relation \overline{G} ∼ −(m − 2)\overline{E}, where we denote by \overline{G} (resp. \overline{E}) the proper transform on P^{3} of G (resp. E). This is absurd. Q.E.D.

In consideration of Theorem 2.2, let E(a, b, c) be the discrete valuation of C(X_{6}) which is obtained by a weighted blow-up at P_{6} of weight wt(u, v, w) = (a, b, c), where (u, v, w) is a suitable system of local coordinates at the point P_{6} ∈ X_{6}, and one of a, b, c equals 1 and the other two are pairwise coprime positive integers. By virtue of Claim, we can readily see that a(E(1, 1, 1); X_{6}, 2H_{X_{6}}) = 0 and a(E(a, b, c); X_{6}, 2H_{X_{6}}) > 0 if (a, b, c) ≠ (1, 1, 1). Thus K_{X_{6}} + (1/μ_{6})H_{X_{6}} is canonical and c_{6} coincides with 1/μ_{6} = 2. Since K_{X_{6}} + (1/μ_{6})H_{X_{6}} ∼ −E_{5}_{X_{6}}, it is not nef. Thus we complete the proof. Q.E.D.

Since K_{X_{6}} + (1/μ_{6})H_{X_{6}} is canonical and not nef, there exists a (K_{X_{6}} + (1/μ_{6})H_{X_{6}})-negative extremal ray P ⊂ \overline{NE} (X_{6}). By noting that K_{X_{6}} + (1/μ_{6})H_{X_{6}} ∼ −E_{5}_{X_{6}} and E_{1}_{X_{6}}^{2} ∼ −M_{0} + 2l_{0}, we know that this ray P comes from the rulings l_{0} on E_{1}_{X_{6}} ≅ P^{1} × P^{1}. Let R ⊂ \overline{NE} (X_{6}) be an extremal ray yielding a Del Pezzo fibration ϕ_{6} : X_{6} → B_{6}, and let F := span < P, R > be an extremal face spanned by P and R. In fact, since \overline{NE} (X_{6}) ⊂ R^{2}, we have F = \overline{NE} (X_{6}). Let ϕ_{F} : X_{6} → \{pt.\} be the contraction of the extremal face F. Then, according to the Sarkisov Program, we shall run (K_{X_{6}} + (1/μ_{6})H_{X_{6}})- MMP (relative to ϕ_{F}). Hence we contract E_{1}_{X_{6}} ≅ P^{1} × P^{1} with E_{1}_{X_{6}}^{2} ∼ −M_{0} + 2l_{0} along the direction of ruling l_{0} to obtain q_{6} : X_{6} → X_{7} ≅ Q^{3}. It is clear that Φ_{7} := q_{6} : X_{6}/B_{6} → X_{7}/{pt.} is an elementary link of type III. It follows that K_{X_{7}} ∼ −3E_{5}_{X_{7}}, E_{5}_{X_{7}} ≅ Q_{0}^{2} with E_{5}_{X_{7}}^{2} ∼ O_{P^{3}}(1) ⊗ O_{Q_{0}^{2}} and the proper transform H_{X_{7}} is spanned by S_{X_{7}}, T_{X_{7}}, H_{X_{7}} and E_{5}_{X_{7}}. The following result is easy to see, so we shall omit the proof.

Lemma 3.25.

(1) S_{X_{7}}|_{E_{5}_{X_{7}}} = 2L_{6}, T_{X_{7}}|_{E_{5}_{X_{7}}} = L_{6} + L′_{6}, H_{X_{7}}|_{E_{5}_{X_{7}}} = 2L′_{6}, where we use the same notations L_{6}, L′_{6} to represent their proper transforms (cf. Figure 12);

(2) Bs H_{X_{6}} = {P_{7}}, where P_{7} := L_{6} ∩ L′_{6} = q_{6}(P_{6});

(3) The general members of H_{X_{7}} except for E_{5}_{X_{7}} are smooth at the base point P_{6}.

We shall observe the Sarkisov degree of the pair (X_{7}, H_{X_{7}}).

Lemma 3.26.

(1) The quasi-effective threshold μ_{7} of (X_{7}, H_{X_{7}}) equals 1/3. Furthermore, K_{X_{7}} + (1/μ_{7})H_{X_{7}} is nef but not canonical;

(2) The canonical threshold c_{7} of (X_{7}, H_{X_{7}}) equals 2 and P_{7} is the maximal center of (X_{7}, H_{X_{7}}). More precisely, such a discrete valuation E of C(X_{7}) with discrepancy a(E; X_{7}, c_{7}H_{X_{7}}) = 0 is obtained by the (ordinary) blow-up at P_{7}.

Proof. We can readily see μ_{7} = 1/3. Then K_{X_{7}} + (1/μ_{7})H_{X_{7}} ∼ 0 is nef but not canonical. In fact, a general member of H_{X_{7}} distinct from E_{5}_{X_{7}} is smooth at P_{7} by the same argument as in Claim in Lemma 3.24. Hence a(E_{7}; X_{7}, (1/μ_{7})H_{X_{7}}) = −1 for a discrete valuation E_{7} of C(X_{7}) which is obtained by the blow-up at P_{7}. Hence K_{X_{7}} + (1/μ_{7})H_{X_{7}} is not canonical and a(E_{7}; X_{7}, 2H_{X_{7}}) = 0. Hence c_{7} ≤ 2. Since Bs H_{X_{7}} = {P_{7}}, we need to consider the discrete valuations of C(X_{7}) which is obtained by the weighted blow-up at P_{7} with a suitably chosen weights (cf. Theorem 2.2). By the straightforward computation we see a(E; X_{7}, 2H_{X_{7}}) > 0 for any such a discrete valuation of C(X_{7}) distinct from E_{7}. Thus we obtain the assertion. Q.E.D.

By Lemma 3.26, we need to consider the blow-up p_{7} : E_{7} ⊂ Z_{7} → P_{7} ∈ X_{7} at P_{7}, where E_{7} ≅ P^{2} is an exceptional divisor. The proper transform H_{Z_{7}} is spanned by S_{Z_{7}}, T_{Z_{7}}, H_{Z_{7}} and E_{5}_{Z_{7}} +E_{7} because every member of H_{X_{7}} except for E_{5}_{X_{7}} + E_{7} is smooth at P_{7} (see Lemma 3.25). We perform (K_{Z_{7}} + c_{7}H_{Z_{7}})- MMP. Then we contract E_{5}_{Z_{7}} ≅ F_{2} with E_{5}_{Z_{7}}^{2} ∼ −M_{2}+2l_{2} onto a smooth conic to obtain Φ_{8} := q_{7} ∘ p_{7}^{−1} : Z_{7} → X_{8} ≅ P^{3}, which is an elementary link of type III. We can verify that K_{X_{8}} ∼ −4E_{7}_{X_{8}}, the proper transform H_{X_{8}} is spanned by S_{X_{8}}, T_{X_{8}}, H_{X_{8}} and E_{7}_{X_{8}} and H_{X_{8}} = |O_{P^{3}}(1)|. Therefore, the quasi-effective threshold μ_{8} of (X_{8}, H_{X_{8}}) is μ_{8} = 1/4 and K_{X_{8}} + (1/μ_{8})H_{X_{8}} is canonical and nef. Hence by Noether-Fano-Iskovskih inequality (cf. [Cor95, Cor00]) we obtain the factorization Φ_{σ} = Φ_{8} ∘ ⋯ ∘ Φ_{1} of Φ_{σ} into eight elementary links according to the Sarkisov Program.

Summarizing the argument performed in this section, we obtain the following result Theorem 3.1 (= Theorem 1.1).

Theorem 3.1. Let σ be the Nagata automorphism on the affine 3-space A^{3} and let Φ_{σ} : P^{3} ⋯ → P^{3} be a Cremona transformation which is an extension of σ. Then Φ_{σ} can be factorized into eight elementary links explicitly, say Φ_{σ} = Φ_{8} ∘ ⋯ ∘ Φ_{1}. More precisely, each elementary link Φ_{j} (1 ≤ j ≤ 8) is described as follows (here we shall follow the notations used in Section 3 and each number of type is the type of elementary link; see Section 2):

(i) Φ_{1} : P^{3} ←^{p_{0}} Z_{0} →^{q_{0}} X_{1} ≅ Q^{3}, where p_{0} is the blow-up along the conic C_{0} with an exceptional divisor E_{0} ≅ F_{2} and q_{0} is the contraction of H_{∞}_{Z_{0}} ≅ P^{2} (cf. Figure 1) to a smooth point (Type II),

(ii) Φ_{2} : X_{1} ←^{p_{1}} X_{2} is the blow-up along the curve C_{1} (cf. Figure 2) with an exceptional divisor E_{1} ≅ P^{1} × P^{1} (Type I),

(iii) Φ_{3} : X_{2} ←^{p_{2}} Z_{2} →^{q_{2}} X_{3}, where p_{2} is the blow-up along the curve L_{1} (cf. Figure 3) with an exceptional divisor E_{2} ≅ F_{2} and q_{2} is the contraction of E_{0}_{Z_{2}} ≅ F_{2} (cf. Figure 4) onto a curve (Type II),

(iv) Φ_{4} : X_{3} ←^{p_{3}} Z_{3} →^{q_{3}} X_{4}, where p_{3} is the blow-up along the curve L_{3} (cf. Figure 5) with an exceptional divisor E_{3} ≅ F_{2} and q_{3} is the contraction of E_{2}_{Z_{3}} ≅ F_{2} (cf. Figure 6) onto a curve (Type II),

(v) Φ_{5} : X_{4} ←^{p_{4}} Z_{4} →^{q_{4}} X_{5}, where p_{4} is the blow-up along the curve L_{4} (cf. Figure 7) with an exceptional divisor E_{4} ≅ F_{2} and q_{4} is the contraction of E_{3}_{Z_{4}} ≅ F_{2} (cf. Figure 8) onto a curve (Type II),

(vi) Φ_{6} : X_{5} ←^{p_{5}} Z_{5} →^{q_{5}} X_{6}, where p_{5} is the blow-up along the curve L′_{5} (cf. Figure 9) with an exceptional divisor E_{5} ≅ F_{2} and q_{5} is the contraction of E_{4}_{Z_{5}} ≅ F_{2} (cf. Figure 10) onto a curve (Type II),

(vii) Φ_{7} : X_{6} →^{q_{6}} X_{7} ≅ Q^{3} is the contraction of E_{1}_{X_{6}} ≅ P^{1} × P^{1} (cf. Figure 11) onto a curve (Type III),

(viii) Φ_{8} : X_{7} ←^{p_{7}} Z_{7} →^{q_{7}} X_{8} ≅ P^{3}, where p_{7} is the blow-up at the point P_{7} (cf. Figure 12) with an exceptional divisor E_{7} ≅ P^{2} and q_{7} is the contraction of E_{5}_{Z_{7}} ≅ F_{2} onto a conic (Type II).

Remark 1. Let Φ : X/B ⋯ → X′/B′ be a given birational map between 2-dimensional or 3-dimensional Mori fiber spaces. As explained in Section 2, we fix a very ample complete linear system H′ on X′ and take its proper transform H := Φ_{∗}^{−1}(H′) on X. In dimension 2 and 3, it is guaranteed that such a birational map Φ can be factorized into a finite number of elementary links as explained in Section2 according to the Sarkisov Program. (In dimension 3, this is due to Sarkisov, Reid and Corti [Cor95]. In dimension 2, this result is classical.) In two-dimensional case, we can describe the factorization more explicitly. Indeed, in this case there are no log-flips and each divisorial blow-up is just the blow-up at the point at which general members of the proper transform of H has the highest multiplicity. On the other hand, the factorization of a birational map between 3-dimensional Mori fiber spaces is not unique in general. This fact results from the choice of the divisorial blow-ups along the maximal centers with respect to the proper transform of H. For instance, in the decompostion of the Cremona transformation Φ_{σ} : P^{3} ⋯ → P^{3} which is an extension of the Nagata automorphism, there are some possibilities for the divisorial blow-ups of the maximal centers of (X_{3}, H_{X_{3}}) (cf. Lemma 3.12). More precisely, we are free to choose either the blow-up along the curve L_{3} or the (ordinary) blow-up at the point Q_{3}. In our factorization of Φ_{σ} we chose the former blow-up.

C_{1} L_{1} Z_{0} ⊃ C_{0}  L_{0} : E_{0} : H_{∞}

E_{0} ≅ F_{2}, H_{∞} ≅ P^{2}

Figure 1:

\begin{aligned}X_{1} ⊃ & C_{1} & : E_{0}_{X_{1}} & L_{1} \\ & E_{0}_{X_{1}} ≅ Q_{0}^{2} \\ & Figure 2: \end{aligned}

: E_{1} L_{2}  L_{1} : E_{0}_{Z_{1}} C_{2} Z_{1} ⊃  l_{0}M_{0}

E_{1} ≅ P^{1} × P^{1}, E_{0}_{Z_{1}} ≅ Q_{0}^{2}

Figure 3:

C_{2}  L_{2}  L_{3} L_{1}  : E_{2} : E_{1}_{Z_{2}} Z_{2} ⊃ E_{0}_{Z_{2}} :  _{(−2)}_{2} l_{0}_{0}_{0}_{(−1)}  _{(−1)}_{(−1)} M_{0}  e_{0}

E_{0}_{Z_{2}} ≅ F_{2}, E_{2} ≅ F_{2}

Figure 4:

\begin{aligned}C_{2} & L_{2} & L_{3} & : E_{1}_{X_{3}} & X_{3} ⊃ & : E_{2}_{X_{3}} & 2 & l_{0} & 0 & 0 & (−1) & (−1) & (−1) & M_{0} & e_{0} \\ & E_{2}_{X_{3}} ≅ Q_{0}^{2} \\ & Figure 5: \end{aligned}

C_{2}  L_{2} : E_{1}_{Z_{3}} Z_{3} ⊃  _{2}  C_{4} L′_{4} : E_{3} L_{4}  _{(−2)} E_{2}_{Z_{3}} :  l_{0}_{0}_{0}_{(−1)}  _{(−1)}_{(−1)} M_{0}  e_{0}

E_{2}_{Z_{3}} ≅ F_{2}, E_{3} ≅ F_{2}

Figure 6:

: E_{1}_{X_{4}} L_{2}  : E_{3}_{X_{4}} X_{4} ⊃ C_{2}  L_{4}  L′_{4} M_{1} l_{1}

E_{1}_{X_{4}} ≅ F_{1}, E_{3}_{X_{4}} ≅ Q_{0}^{2}

Figure 7:

C_{2}  L_{2} : E_{1}_{Z_{4}} Z_{4} ⊃  _{2}  L_{5} L′_{4} : E_{3}_{Z_{4}} L_{4}  _{(−2)} E_{4} : L′_{5} C_{5}  M_{1}_{0}_{0}_{(−1)}  _{(−1)}_{(−1)} e_{1}  l_{1}

E_{3}_{Z_{4}} ≅ F_{2}, E_{4} ≅ F_{2}

Figure 8:

24 Takashi Kishimoto

\begin{aligned}C_{2} & L_{2} & L_{5} & : E_{1}_{X_{5}} & X_{5} ⊃ & : E_{4}_{X_{5}} & C_{5} & L′_{5} & M_{1} & 0 & 0 & (−1) & (−1) & (−1) & e_{1} & l_{1} \\ & E_{4}_{X_{5}} ≅ Q_{0}^{2} \\ & Figure 9: \end{aligned}

C_{2}  L_{2}  L_{5} : E_{4}_{Z_{5}} : E_{1}_{Z_{5}} Z_{5} ⊃E_{5} : L_{6} C_{6}  C_{5}L′_{6} M_{1}_{0}_{0}_{(−1)}  _{(−1)}_{(−1)} e_{1}  l_{1}_{(−2)}_{2}

E_{5} ≅ F_{2}, E_{4}_{Z_{5}} ≅ F_{2}

Figure 10:

L_{2} : E_{1}_{X_{6}} : E_{5}_{X_{6}} L′_{6} P_{6} L_{6} C_{2} X_{6} ⊃  l_{0}  M_{0}

E_{1}_{X_{6}} ≅ P^{1} × P^{1}, E_{5}_{X_{6}} ≅ Q_{0}^{2} Figure 11:

\begin{aligned}X_{7} ⊃ & : E_{5}_{X_{7}} & L′_{6} & L_{6} & P_{7} \\ & & E_{5}_{X_{7}} ≅ Q_{0}^{2} \\ & & Figure 12: \end{aligned}

Permanent Address:

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

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

Current Address:

Max-Planck-Institut für

Mathematik Bonn

Vivatsgasse 7, D-53111 Bonn, Germany

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

References

[A-M75] S. S. Abhyankar and T. T. Moh, Embeddings of the line in the plane, J. reine angew. Math., 276 (1975) pp. 148–166.

[Al95] J. Alev, A note on Nagata’s automorphism, Automorphisms of affine spaces (Curacao, 1994), pp. 215–221, Kluwer Acad. Publ., Dordrecht, 1995.

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

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

[Kaw01] M. Kawakita, Divisorial contractions in dimension three which contract divisors to smooth points, Invent. Math., 145 (2001) pp 105–119.

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

[Ko-Mo92] J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc., 5 (1992) pp 533–703.

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

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

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

[Na72] M. Nagata, On automorphism group of k[x, y], Lectures in Mathematics, Department of Mathematics, Kyoto University, Vol. 5. Tokyo: Kinokuniya Book-Store Co. Ltd., 1972.

[Na78] M. Nagata, Polynomial Rings and Affine Spaces, (Reg. Conf. Series in Mathematics 37) AMS, 1978.

[S-U02] I. Shestakov and U. Umirbaev, The tame and wild automorphims of polynomial rings in three variables, preprint, 2002.