LOG SARKISOV PROGRAM

Andrea Bruno and Kenji Matsuki

Contents

§0. Introduction.

§1. Flowchart for Sarkisov Program.

§2. Termination of Flowchart.

§3. Log Sarkisov Program with KLT Singularities.

§4. Log Sarkisov Program with WKLT Singularities.

The purpose of this paper is two-fold. The first is to give a tutorial introduction to the so called Sarkisov program, a 3-dimensional generalization of Castelnuovo-Nöther Theorem “untwisting” birational maps between Mori fiber spaces, which was recently established by Corti[4]. We should emphasize that though the general features were understood (cf.Matsuki[17]) after Reid[29] explained the original ideas of V.G. Sarkisov in a substantially laundered form, it is only after Corti[4] that we are beginning to understand the details of the mechanism. Here we will present a flowchart to visualize how the Sarkisov program works and also slightly simplify the proof of termination after the ingenious argument of Corti[4]: we prove there is no infinite loop in the program just observing that the Sarkisov degree decreases strictly after each untwisting and it cannot decrease infinitely many times using the boundedness of ℚ-Fano d-folds d ≤ 3 together with the ascending chain condition S_{3}(Local) of Alexeev[1] (cf.Shokurov[31]Kollár et al[16]). Our argument also makes it explicit that the Sarkisov program holds in arbitrary dimension n once we have Log MMP in dimension n, boundedness of ℚ-Fano d-folds for d ≤ n and S_{n}(Local). The second is an attempt to give a logarithmic generalization following the philosophy of Iitaka, based upon the Log MMP (established in dimension 3 by Shokurov[31]Kawamata[9] (cf.Kollár et al[16])). The key is to understand the meaning of the genuine Sarkisov program and set up the natural and right generalization. The genuine Sarkisov program untwists any birational map between two

Typeset by AMS-TEX

Research at MSRI is supported in part by NSF grant no. DMS-9022140

Mori fiber spaces which are birationally equivalent. A naive speculation that the log Sarkisov program should untwist any birational map between two log Mori fiber spaces which are only birationally equivalent turns out to be not natural and simply does not work! In order to reach the right understanding of what the genuine Sarkisov program does we have to introduce the notion of the Sarkisov relation:Mori fiber spaces are Sarkisov related iff they are the end results of the K-MMP starting from an appropriate nonsingular projective variety. The genuine Sarkisov program untwists a birational map between two Mori fiber spaces which are Sarkisov related, by factorizing it into “links” among intermediate Mori fiber spaces, all of which including the original two are Sarkisov related. Log Mori fiber spaces are said to be Sarkisov related iff they are the end results of the K + B-MMP starting from one log pair consisting of a nonsingular variety and an S.N.C. divisor as a boundary. The log Sarkisov program should be the one to untwist a birational map between two log Mori fiber spaces which are Sarkisov related, by factorizing it into “links” among intermediate log Mori fiber spaces, all of which including the original two are Sarkisov related. Once this is understood, the log Sarkisov program works almost parallel to the genuine Sarkisov program in the case of kawamata log terminal singularities in arbitrary dimension, except for the verification of termination. The boundedness of a certain class of log ℚ-Fano varieties becomes crucial for our argument in showing termination, just as the result of Kawamata[8] on the boundedness of ℚ-Fano 3-folds was crucial in showing termination of the genuine Sarkisov program in dimension 3. While in dimension 2 we establish termination thanks to a result of Nukulin[26]Alexeev[2], in dimension 3 we have to use a conjecture by Borisov[3] at the last stage of the proof of termination. We also establish the log Sarkisov program in the case of weakly kawamata log terminal singularities in dimension 2 including its termination. But in dimension 3, the Sarkisov relation becomes quite subtle for wklt singularities and we can only discuss problems toward establishing the program including not only its temination but the general mechanism itself.

We also remark that the birational transformations among the various moduli spaces studied by M. Thaddeus and others can be put into the general frame work of the (log) Sarkisov program.

We would like to thank A. Corti, who allowed us to present many of his ideas here in logarithmic form. Most of the arguments here are taken from his paper Corti[4] and repeated for the sake of understanding of the reader. The original ideas are due entirely to V.G. Sarkisov and our indebtedness to M. Reid toward understanding them is as clear as his paper Reid[29]. The conversations with J. Kollár and J. McKernan were very helpful and critical. We would like to thank S. Mori, who gave us warm encouragement throughout.

§0. Introduction.

¿From the view point of Minimal Model Program (the so-called Mori’s program), our basic strategy to understand the birational geometry of higher dimensional algebraic varieties (established in dimension 3 and conjectural for higher dimension) is divided into the following 3 steps:

1. Find “good representatives” among varieties with a given function field through MMP: We take a nonsingular projective variety X by Hironaka’s resolution of singularities with a given function field. Input X into the black box called Minimal Model Program (abbreviated MMP), which produces a good representative, i.e., a minimal model or a Mori fiber space as an output, depending whether X is non-uniruled or uniruled.

X

↓

MMP

non uniruled ↙ ↘  uniruled
a minimla model  a Mori fiber space

2. Study the properties of “good representatives”: The most important property is the Dichotomy, which says that the uniruledness should characterize the variety with Kodaira dimension −∞, i.e. κ = −∞ or κ ≥ 0 depending upon whether X is uniruled or not. In dimension 3, this is a theorem. For 3-folds X with κ(X) ≥ 0, the Abundance Theorem of Kawamata-Miyaoka further claims that a minimal model X_{min} has a base point free pluri-canonical system which induces the canonical morphism Φ_{|mK_{X_{min}} |} : X_{min} → X_{can}, crucial for the understanding of the global structure of X and its moduli. For 3-folds with κ = −∞, a theorem of Miyaoka-Mori says that a Mori fiber space X_{mori} is covered by rational curves intersecting K_{X_{mori}} negatively.

3. Study the relation among “good representatives”: For 3-folds with κ ≥ 0, the basic relation among good representatives is that minimal models in a given birational equivalence class are connected by a sequence of flops (cf. Reid[28]Kawamata[5] Kollár[15]. See also Matsuki[18] for a finer description of their relation). It is the Sarkisov program, the main theme of our paper, which describes the relation among good representatives for 3-folds with κ = −∞, i.e., Mori fiber spaces in a given birational equivalence class.

In dimension 2, i.e., in the case of classical birational geometry of surfaces, the meaning of these 3 steps is rather straightforward.

1. Starting from a nonsingular projective surface, we keep contracting (-1)- curves (MMP in dimension 2) until we get either a surface X_{min} with the canonical divisor K_{X_{min}} being nef or a ruled surface X_{mori} over a curve (or ℙ^{2} over a point).

2. When κ(X) ≥ 0, the canonical morphism Φ_{|mK_{X_{min}} |} : X_{min} → X_{can} from a minimal model is:

κ = 2 - a biational map to a canonically polarized surface with only rational double points

κ = 1 - an elliptic fibtration whose degeneration fibers are studied by Kodaira[14] κ = 0 - a trivial map to a point, where we know X_{min} must be either Abelian,

bielliptic, K3 or Enriques.

When κ = −∞, the structure of a Mori fiber space is rigid and well-understood: either a ℙ^{1}-bundle over a nonsingular curve or ℙ^{2}.

3. A minimal model is unique in a fixed birational equivalence class for surfaces with κ ≥ 0, while any birational map among ruled surfaces in a given birational equivalence class is decomposed into a sequence of elementary transformations by Castelnuovo-Nöther theorem.

Logarithmic Generalization

The logarithmic generalization of the basic strategy following the philosophy of Iitaka to understand the birational geometry of varieties WITH BOUNDARIES goes along the same line:

1. We take a pair (X, B_{X}) where X is a nonsingular projective variety and B_{X} = Σb_{i}B_{i} is a simple normal crossing divisor with 0 ≤ b_{i} ≤ 1. Input (X, B_{X}) into the black box called Log MMP, which produces a log minimal model (X_{min}, B_{X_{min}}) with K_{X_{min}} + B_{X_{min}} being nef or a log Mori fiber space ϕ : (X_{mori}, B_{X_{mori}}) → S with K_{X_{mori}} + B_{X_{mori}} being ϕ-negative, depending upon whether (X, B_{X}) is non log uniruled or log uniruled. (We say (X, B_{X}) is log uniruled iff it is covered by rational curves intersecting K_{X} + B_{X} negatively.)

(X, B_{X})

↓

Log MMP

non log uniruled ↙ ↘  log uniruled
a log minimal model  a log Mori fiber space

2. Again the most important property is the Dichotomy, which says that the log uniruledness should characterize the varieties with log Kodaira dimension −∞, i.e. κ(K_{X} + B_{X}) = −∞ or κ(K_{X} + B_{X}) ≥ 0 depending upon whether (X, B_{X}) is log uniruled or not. In dimension 3, this is a theorem. For log 3-folds (X, B_{X}) with κ(K_{X} + B_{X}) ≥ 0, the Log Abundance Theorem of KeelMatsukiMcKernan[12] further claims that a log minimal model (X_{min}, B_{X_{min}}) has a base point free plurilog canonical system. For log 3-folds with κ = −∞, a theorem of Miyaoka-Mori applies again to imply that a log Mori fiber space (X_{mori}, B_{X_{mori}}) is covered by rational curves intersecting K_{X_{mori}} + B_{X_{mori}} negatively.

3. In the genuine birational geometry, we are interested in the relation among good representatives in a given birational equivalence class, where two good representatives are outcomes of one appropriate nonsingular projective variety through MMP if and only if they are birationally equivalent. This is not the case with logarithmic birational geometry. We say (two or more) good representatives are Sarkisov related iff they are outcomes through Log MMP of one appropriate log pair consisting of a nonsingular projective variety and a S.N.C. divisor as a boundary. Then log minimal models which are Sarkisov related are connected by a sequence of log flops (cf. Kollár[15]Kollár et al[16]. See §4 for a detalied discussion.) The log Sarkisov program should be the one to untwist a birational map between two log Mori fiber spaces which are Sarkisov related.

§1. Flowchart for Sarkisov Program.

In this section, we review the (genuine) Sarkisov program after Corti[4] (cf.Sarkisov[30]Reid[29]Matsuki[17]) with some simplifications and present a flowchart to visualize how it works. The aim of this section is mostly tutorial aside from simultaneously preparing the notations for the logarithmic case which goes almost parallel to the genuine case after the introduction of the Sarkisov relation (See §3.). We refer the reader to KawamataMatsudaMatsuki[11] for the general features of MMP and to Kollár et al[16] for those of Log MMP.

The Sarkisov program, in short, is an algorithm to factorize a birational map between two Mori fiber spaces, i.e., two different end results of MMP of one appropriate nonsingular projective variety, when the Kodaira dimension is −∞.

Definition 1.1. A Mori fiber space ϕ : X → S is the contraction of an extremal ray with respect to K_{X} from a normal projective variety with only ℚ-factorial terminal singularities onto a variety S, i.e., ϕ is a morphism from a normal projective variety with only ℚ-factorial terminal singularitieswith connected fibers onto a normal variety S with dim S < dim X s.t. ρ(X/S) = 1 and −K_{X} is ϕ-ample.

Theorem 1.2 (Sarkisov Program in dimension 3) (cf.Sarkisov[30]Reid[29] Corti[4]). A birational map

\begin{matrix}X & −−→^{Φ} & X′ \\ ^{ϕ}↓ & ↓^{ϕ′} \\ S & S′ \end{matrix}

between two Mori fiber spaces in dimension 3 is a composite of the following 4 types of links.

Links of type (I)

Z −−→ X_{1}
↙
X  ↓
↓
S  ←  S_{1}

where Z → X is a K-negative extremal divisorial contraction, Z′ −−→ X_{1} a sequence of log flips with respect an appropriate log pair and ρ(S_{1}/S) = 1.

Links of type (II)

Z −−→ Z′
↙  ↘
X  X_{1}
↓  ↓
S  ←^{∼}  S_{1}

where Z → X and Z′ → X_{1} are K-negative extremal divisorial contractions, Z −−→ Z′ a sequence of log flips with respect to an appropriate log pair.

Links of type (III) (Inverses of links of type (I))

X −−→ Z′
↘
↓  X_{1}
↓
S → S_{1}

where Z′ → X_{1} is a K-negative extremal divisorial contraction, X −−→ Z′ a sequence of log flips with respect to an appropriate log pair and ρ(S/S_{1}) = 1.

Links of type (IV)

X  −−→  X_{1}
↓  ↓
S  S_{1}
↘  ↙
T

where X −−→ X′ is a sequence of log flips with respect to an appropriate log pair and ρ(S/T) = ρ(S_{1}/T) = 1.

Remark 1.3.

All the intermediate Mori fiber spaces

ϕ_{k} : X_{k} → S_{k}

in the process of untwisting the given birational map by the Sarkisov program are the end results of K-MMP over Spec k starting from one appropriate nonsingular projective variety W. In order to see this, we just have to take W to be a common resolution

p_{k} : W → X_{k}.

Then each p_{k} is a process of K-MMP over X_{k}, and thus a process of K-MMP over Spec k.

This fact that all the Mori fiber spaces in the process of the Sarkisov program are Sarkisov related (See §3 for the precise definition of the Sarkisov relation.) is automatic in the case of the genuine Sarkisov program and implicit in the statement. But it is the key point of understanding the log Sarkisov program.

The strategy to untwist Φ into a composite of links is to set up a good invariant, the Sarkisov degree (μ, λ, e) so that it strictly decreases after untwisting the birational map. That is to say, we would like to construct a sequence of links as below

X = X_{0}−−→X_{1}−−→X_{2}  ⋯  −−→X′
ϕ↓  ^{ϕ_{1}}↓ ^{ϕ_{2}}↓  ⋯  ^{ϕ′}↓
S = S_{0} S_{1} S_{2}  ⋯  S′

so that the Sarkisov degree strictly decreases each time we untwist the birational map

(μ, λ, e) = (μ_{0}, λ_{0}, e_{0}) > (μ_{1}, λ_{1}, e_{1}) > ⋯

and that it cannot decrease infinitely many times.

Definition 1.4 (Sarkisov degree). The Sarkisov degree of a birational map between two Mori fiber spaces

\begin{matrix}X & −−→^{Φ} & X′ \\ ^{ϕ}↓ & ↓^{ϕ′} \\ S & S′ \end{matrix}

with reference to the fixed Mori fiber space ϕ′ : X′ → S′ is the triplet

(μ, λ, e)

of the numbers defined below, endowed with the lexicographical order.

First we take a very ample divisor A′ on S′ and a sufficiently divisible μ′ ∈ ℕ such that

H_{X′} = −μ′K_{X′} + ϕ′^{∗}A′

is very ample on X′. H_{X} is the strict transform of H_{X′}.

μ : the quasi-effective threshold

The quasi-effective threshold μ is defined to be a positive rational number s.t.

μK_{X} + H_{X} ≡ 0 over S.

Note that μ′ is the quasi-effective threshold for the special case Φ being the identity map of the Mori fiber space ϕ′ : X′ → S′.

In dimension 2, it is easy to see μ ∈ \frac{1}{3!}ℕ.

λ : the maximal multiplicity of an extremal ray

We take a common resolution

W
p ↙ ↘ q
X −−→^{Φ} X′
ϕ↓  ↓^{ϕ′}
S  S′.

such that the exceptional locus of p is an S.N.C. divisor ∪E_{k}. Then taking a general member of H_{X′} and its strict transform H_{X} (We denote them by the same symbols H_{X′} and H_{X} by abuse of notation.) and writing

K_{W} = p^{∗}K_{X} + Σa_{k}E_{k}
q^{∗}H_{X′} = p^{∗}H_{X} − Σb_{k}E_{k},

we define

λ := max\{\frac{b_{k}}{a_{k}} \}.

We remark that \frac{1}{λ} has a more intrinsic description and is called the canonical threshold of X with respect to H_{X}, i.e.,

\frac{1}{λ} = max\{c ∈ ℚ_{>0}; K_{X} + cH_{X} canonical\},

where K_{X} + cH_{X} being canonical means by definition that for some (and thus for any) common resolution

p : W → X
q : W → X′

such that the exceptional locus ∪E_{k} of p is an S.N.C. divisor ∪E_{k} (Note that the strict transform H_{W} which is nothing but the total transform q^{∗}H_{X′} may assumed to be nonsingular and cross ∪E_{k} normally.), we have

K_{W} + cH_{W} = p^{∗}(K_{X} + cH_{X}) + Σr_{k}E_{k}

with

r_{k} ≥ 0 for ∀k.

Thus λ is independent of the common resolution that we take and well-defined. We note that when X is ℚ-factorial and thus p has purely one-codimensional exceptional locus, the assumption of the exceptional locus of p being an S.N.C. divisor is unnecessary.

Note also that since H_{X} has no base component of codimension one, even when c ≥ 1 we can regard the pair (X, cH_{X}) = (X, Σc_{q}B_{q}) as a canonical log pair with only klt singularities in the usual sense (cf. Kollár et al[16]) by taking general members B_{q} ∈ H_{q} and a suitable set of positive rational numbers 0 < c_{q} < 1 with Σc_{q} = c and that thus K + cH-MMP works as well as K + Σc_{q}B_{q}-MMP.

In dimension 2, λ is nothing but the maximal multiplicity of a general member of the linear system H_{X} (We note that the linear system H_{X} consists of the strict transforms of the complete linear system H_{X′} and that it may not be complete itself.) at the base points Bs(H_{X}). When Bs(H_{X}) = ∅, λ = 0 by definition. e : the number of K + \frac{1}{λ}H − crepant divisors

e := \{ \begin{matrix}#\{E; a(E, \frac{1}{λ}H_{X}) = 0\} if λ > 0 \\ 0 if λ = 0 \end{matrix}

In dimension 2, e is nothing but the number of the base points of the linear system H_{X} with the maximal multiplicity λ when Bs(H_{X}) ≠ ∅, and e = 0 by definition when Bs(H_{X}) = ∅.

Once the Sarkisov degree, which should measure the extent of untwisting, is set up, the only other ingredient we need is a criterion to judge if the untwisting is completed:

Proposition 1.5 (Nöther-Fano criterion). A birational map Φ between two Mori fiber spaces is an isomorphism of Mori fiber spaces, i.e.,

\begin{aligned}X & →^{∼} & X′ \\ ^{ϕ}↓ & ↓^{ϕ′} \\ S & →^{∼} & S′ \end{aligned}

if λ ≤ μ and K_{X} + \frac{1}{μ}H_{X} is nef.

The proposition can be proved as an easy application of the Negativity Lemma or Hodge Index Theorem and we refer the reader to Corti[4],Theorem4.2 for a proof.

Flowchart for Sarkisov Program.

In the following, we present a flowchart to untwist a birational map

\begin{matrix}X & −−→^{Φ} & X′ \\ ^{ϕ}↓ & ↓^{ϕ′} \\ S & S′ \end{matrix}

between two Mori fiber spaces.

We START.

The first question to ask is:

λ > μ?

According to whether the answer to this question is YES or NO, we proceed separately into the case λ > μ or into the case λ ≤ μ.

Case : λ ≤ μ

If λ ≤ μ, then the next question to ask is:

K_{X} + \frac{1}{μ}H_{X} nef?

If the answer to this question is YES, then K_{X} + \frac{1}{μ}H_{X} is nef and λ ≤ μ by the case assumption. Thus the Nöther-Fano criterion applies to conclude Φ is an isomorphism of Mori fiber spaces. This leads to an

END.

If K_{X} + \frac{1}{μ}H_{X} is not nef, then we construct as follows a normal projective variety T dominated by S → T s.t. K_{X} + \frac{1}{μ}H_{X} is not relatively nef over T and ρ(X/T) = 2, so that we run K + \frac{1}{μ}H-MMP over T to have an untwisting link.

We pick a K_{X} + \frac{1}{μ}H_{X}-negative extremal ray P of \overline{NE}(X/Spec k) s.t. the span F := P + R is a 2-dimensional extremal face, where R is the K_{X}-negative extremal ray giving the Mori fiber space ϕ : X → S. F is K_{X} + (\frac{1}{μ} − ϵ)H_{X}-negative for 0 < ϵ << 1, thus we have the contraction morphism cont_{F} : X → T to obtain T. Since F ⊃ R, cont_{F} factors through S and by construction T satisfies all the desired conditions.

Now we

Run K + \frac{1}{μ}H − MMP over T.

We reach either a minimal model or a Mori fiber space (with respect to K + \frac{1}{μ}H and over T).

First we show that it is

IMPOSSIBLE to reach a minimal model!

Suppose we did. Then according to whether the first nonflipping contraction is divisorial or not, we should have two different diagrams as follows:

X −−→ Z′
↓  ↘
S  X_{1}
↘  ↓
T →^{∼} S_{1}
X −−→ Z′ = X_{1}
↓  ↓
S  S_{1}
↘  ↙
T

We take a general curve Σ_{1} ∈ Hilb(X_{1}/T) away from the locus of indeterminacy of the birational map X_{1} −−→ X (i.e., in the first case the union of the image of the exceptional divisor of the divisorial contraction and all the flipped curves and in the second case all the flipped curves). Σ_{1} can be considered to lie on X and since Σ_{1} is general we conclude ϕ(Σ_{1}) is a curve (not a point), which implies (K_{X} + \frac{1}{μ}H_{X}) · Σ_{1} < 0. But then

0 > (K_{X} + \frac{1}{μ}H_{X}) · Σ = (K_{X_{1}} + \frac{1}{μ}H_{X_{1}}) · Σ_{1} ≥ 0,

a contradiction!

Next suppose we

Reach a Mori fiber space X_{1} → S_{1}.

Then the next question to ask just in order to separate the types of links is:

Is the first nonflipping contraction divisorial ?

If the answer is YES, the K + \frac{1}{μ}H-MMP consists of a sequence of K + \frac{1}{μ}H-flips X −−→ Z′ followed by a K + \frac{1}{μ}H-negative divisorial contraction Z′ → X_{1}. Since ρ(X_{1}/T) = 1, ϕ_{1} : X_{1} → S_{1} = T is a K_{X_{1}} + \frac{1}{μ}H_{X_{1}}-negative and thus K_{X_{1}}-negative fiber space.

X −−→ Z′
↓  ↘
S  X_{1}
↘  ↓
T →^{∼} S_{1}

If the answer is NO, then the K + \frac{1}{μ}H-MMP consists of a sequence of K + \frac{1}{μ}Hflips X −−→ Z′ followed by a K_{X_{1}} + \frac{1}{μ}H_{X_{1}}-negative and thus K_{X_{1}}-negative fibering contraction ϕ_{1} : Z′ = X_{1} → S_{1}. Since ρ(X_{1}/T) = ρ(X/T) = 2, we have ρ(S_{1}/T) = 1.

X  −−→  Z′ = X_{1}
↓  ↓
S  S_{1}
↘  ↙
T

We claim in both cases X_{1} has only terminal singularities. (ℚ-factoriality of X_{1} is automatic from construction.) Let I be the locus of indeterminacy of the birational map X_{1} −−→ X. If E is a discrete valuation whose center on X_{1} is not contained in I (and has codimension ≥ 2), then

a(E, X_{1}, ∅) = a(E, X, ∅) > 0.

If the center of E on X_{1} is contained in I, then

a(E, X_{1}, ∅) ≥ a(E, X_{1}, \frac{1}{μ}H_{X_{1}})

> a(E, X, \frac{1}{μ}H_{X}) ≥ 0.

Thus we have the claim.

Therefore, we have a link of type (III) in the former case and a link of type (IV) in the latter.

Moreover, since K_{X_{1}} + \frac{1}{μ}H_{X_{1}} is negative over S_{1}, we conclude in both cases

μ_{1} < μ.

Therefore, after untwisting Φ by a link of type (III) or type (IV), we go back to the START with strictly decreased quasi-effective threshold.

Case : λ > μ

In this case we

Take a maximal divisorial blow up p : Z → X,

with respect to K_{X} + \frac{1}{λ}H_{X}, i.e., p is a projective morphism from Z with only ℚ-factorial terminal singularities s.t.

i) ρ(Z/X) = 1,

ii) the exceptional locus of p is a prime divisor E, and

iii) p is K + \frac{1}{λ}H-crepant, i.e.,

K_{Z} + \frac{1}{λ}H_{Z} = p^{∗}(K_{X} + \frac{1}{λ}H_{X}).

Proposition 1.5. A maximal divisorial blow up p : Z → X with respect to K_{X} + \frac{1}{λ}H_{X} exists.

We remark that the exceptional divisor E of p is necessarily one of the K + \frac{1}{λ}Hcrepant divisors {E_{1}, E_{2}, ⋯, E_{e}} counted for the number e. As long as we require Z to have only terminal singularities, we can’t quite specify which E_{i} would be the exceptional divisor. On the other hand, if we allow Z to have canonical singularities, for each E_{i} we can construct a maximal blow up p_{i} : Z_{i} → X (allowing Z_{i} to have canonical singularities) with the exceptional divisor being E_{i}.

Proof.

Take a resolution Y → X s.t.

a) the exceptional locus is a divisor with only S.N.C.,

b) Y dominates X′ so that the strict transform H_{Y} coincides with the total transform of H_{X′} and that a general member H_{Y} is smooth and crosses normally with the exceptional locus.

We run the K + \frac{1}{λ}H-MMP over X to get a minimal model f : (Z′, \frac{1}{λ}H_{Z′}) → (X, \frac{1}{λ}H_{X}). As before, it is easy to see that Z′ has only ℚ-factorial terminal singularities. Since both Z′ and X are ℚ-factorial, the exceptional locus of f is purely onecodimensional. An easy application of the Negativity Lemma (Shokurov[30]Kollár et al[16]Corti[4]) shows that the exceptional locus is actually ∪_{i=1}^{e}E_{i} and that f is K + \frac{1}{λ}H-crepant, i.e., K_{Z′} + \frac{1}{λ}H_{Z′} = f^{∗}(K_{X} + \frac{1}{λ}H_{X}).

Now we run the K-MMP starting from Z′ over X ending necessarily with a divisorial contraction p : Z → X. It is immediate that p : Z → X is a maximal divisorial blow up with respect to K_{X} + \frac{1}{λ}H_{X}. (If we want to specify the exceptional divisor E_{i} allowing Z to have canonical singularities, then we run K+ \frac{1}{λ}H+ϵΣ_{j≠i}E_{j}- MMP (0 < ϵ << 1) instead.)

We also remark that in order to construct just one maximal divisorial blowup we can start from any common resolution Y which may not satisfy a) as long as X is ℚ-factorial and thus the exceptional locus of Y → X is purely one codimensional.

There is another method called the “Nef Threshold Method” to construct a maximal divisorial blow up by M. Reid.

We construct a chain of

Y_{i}: 3-folds with only ℚ-factorial terminal singularities projective over X (Y_{0} = Y),

H_{Y_{i}}: the strict transforms of H_{Y},

λ_{i}: (a non-decreasing sequence of) nonnegative rational numbers

s.t.

a) λ_{i}K_{Y_{i}} + H_{Y_{i}} is a supporting function of a face containing a K_{Y_{i}}-negative extremal ray R_{i} of \overline{NE}(Y_{i}/X), i.e., λ_{i}K_{Y_{i}} + H_{Y_{i}} is relatively nef over X and

(λ_{i}K_{Y_{i}} + H_{Y_{i}})^{⊥} ∩ \overline{NE}(Y_{i}/X) ⊃ R_{i},

b) either Y_{i} → Y_{i+1} is a divisorial contraction of R_{i} or the flip Y_{i} −−→ Y_{i+1}, and c) the chain ends with a divisorial contraction p : Z = Y_{n} → X of an extremal

ray R_{n}

(λ_{i}K_{Y_{i}} + H_{Y_{i}})^{⊥} ∩ \overline{NE}(Y_{i}/X) ⊃ R_{n}.

Then it is easy to see that p : Z → X is a maximal divisorial blow up with respect to K_{X} + \frac{1}{λ}H_{X} and λ_{n} = λ.

We construct inductively.

Suppose we have succeeded constructing the chain up to the i-th stage. Consider the nef threshold λ_{i} of H_{Y_{i}} with respect to K_{Y_{i}}

λ_{i} := sup{ν; νK_{Y_{i}} + H_{Y_{i}} relatively nef over X}.

Remark that since λ_{i−1}K_{Y_{i−1}} + H_{Y_{i−1}} is relatively nef over X and the contraction of R_{i−1} is λ_{i−1}K_{Y_{i−1}} + H_{Y_{i−1}}-trivial, λ_{i−1}K_{Y_{i}} + H_{Y_{i}} is also relatively nef over X and thus λ_{i} ≥ λ_{i−1}.

We claim that λ_{i} is rational and that there exists a K_{Y_{i}}-negative extremal ray R_{i} s.t.

(λ_{i}K_{Y_{i}} + H_{Y_{i}})^{⊥} ∩ \overline{NE}(Y_{i}/X) ⊃ R_{i}.

Instead of applying the Rationality Theorem (KaMaMa[11],Theorem4-1-1) whose proof only applies to ample divisors, we use a result of Kawamata[8] on the boundedness of lengths of the extremal rational curves to the relatively nef divisor λ_{i−1}K_{Y_{i}} + H_{Y_{i}}. (This idea was communicated to us by J. McKernan. See KeMaMc[12].) First from the definition and the Cone Theorem, we have

λ_{i} = λ_{i−1} + inf\{\frac{(λ_{i−1}K_{Y_{i}} + H_{Y_{i}}) · l}{−K_{Y_{i}} · l} \}; l : K_{Y_{i}} − negative extremal rays\}
= inf\{\frac{H_{Y_{i}} · l}{−K_{Y_{i}} · l} \}; l : K_{Y_{i}} − negative extremal rays\}.

The result of Kawamata tells us that for each K_{Y_{i}}-negative extremal ray l, there exists a rational curve L_{l} which generates l = ℝ_{+}[L_{l}] and 0 < −K_{Y_{i}} ·L_{l} ≤ 2·dim X. Thus if q_{i} is the ℚ-factorial index of Y_{i}

q_{i} := min{z ∈ ℕ; zD is Cartier for all integral Weil divisors D on Y_{i}},

(which coincides with the index r_{i} of K_{Y_{i}} in dimension 3), then

\frac{H_{Y_{i}} · l}{−K_{Y_{i}} · l} ∈ \frac{1}{(r_{i} · 2dimX)!q_{i}} ℤ_{≥0}.

Therefore, “inf” is actually attained as the minimum for some K_{Y_{i}}-negative extremal ray R_{i} and for this R_{i} we have

R_{i} ⊂ (λ_{i}K_{Y_{i}} + H_{Y_{i}})^{⊥} ∩ \overline{NE}(Y_{i}/X).

As for Y_{i+1}, we take either the divisorial contraction Y_{i} → Y_{i+1} of R_{i} or the flip Y_{i} −−→ Y_{i+1} of R_{i}.

We go back to the discussion of the flowchart.

Now we

Run K + \frac{1}{λ}H − MMP over S.

A priori we reach either a minimal model or a Mori fiber space (with respect to K + \frac{1}{λ}H over S).

First we show that it is

IMPOSSIBLE to reach a minimal model!

Suppose we did. Then according to whether the first nonflipping contraction is divisorial or not

we should have two different diagrams:

Z −−→ Z′
↙  ↘
X  X_{1}
↓  ↓
S ←^{∼} S_{1}
Z −−→ X_{1}
↙
X  ↓
↓
S ← S_{1}

We take a general curve Σ ∈ Hilb(X/S) away from p(E) (and thus can be considered to lie on Z) and away from all the flipping curves (and thus can be considered to lie on Z′).

In the first case, we have

0 ≤ (K_{X_{1}} + \frac{1}{λ}H_{X_{1}}) · q_{∗}Σ

= \{(K_{Z′} + \frac{1}{λ}H_{Z′}) − aE_{q}\} · Σ (a > 0)

≤ (K_{X} + \frac{1}{λ}H_{X}) · Σ

< (K_{X} + \frac{1}{μ}H_{X}) · Σ = 0,

a contradiction!

In the second case, we have

0 ≤ (K_{Z′} + \frac{1}{λ}H_{Z′}) · Σ

= (K_{X} + \frac{1}{λ}H_{X}) · Σ

< (K_{X} + \frac{1}{μ}H_{X}) · Σ = 0,

again a contradiction!

Next suppose we

Reach a Mori fiber space X_{1} → S_{1}.

Then the next question to ask just in order to separate the types of links is:

Is the first nonflipping contraction divisorial ?

If the answer is YES, the K + \frac{1}{λ}H-MMP consists of a sequence of K + \frac{1}{λ}H-flips X −−→ Z′ followed by a K + \frac{1}{λ}H-negative contraction Z′ → X_{1}. Since ρ(X_{1}/S) = 1, ϕ_{1} : X_{1} → S_{1} = S is a K_{X_{1}} + \frac{1}{λ}H_{X_{1}}-negative and thus K_{X_{1}}-negative fiber space.

Z −−→ Z′
↙  ↘
X  X_{1}
↓  ↓
S ←^{∼} S_{1}

We note that the exceptional divisors E and E_{q} are distinct, since otherwise X and X_{1} are isomorphic in codimension one, which would imply X and X_{1} are indeed isomorphic over S, but then while E_{q} is NOT K_{X_{1}} + \frac{1}{λ}H_{X_{1}}-crepant E is K_{X} + \frac{1}{λ}H_{X}-crepant, absurd!

If the answer is NO, then the K + \frac{1}{λ}H-MMP consists of a sequence of K + \frac{1}{λ}Hflips X −−→ Z′ followed by a K_{X_{1}} + \frac{1}{λ}H_{X_{1}}-negative and thus K_{X_{1}}-negative fibering contraction Z′ = X_{1} → S_{1}. Since ρ(X_{1}/S) = ρ(Z/S) = 2, we have ρ(S_{1}/T) = 1.

Z −−→ X_{1}
↙
X  ↓
↓
S ← S_{1}

In both cases, X_{1} has only ℚ-factorial terminal singularities and thus we have a link of type (II) or a link of type (I), respectively.

Now we study how the Sarkisov degree (μ, λ, e) changes after untwisting by a link of type (II) or type (I).

We claim that

μ_{1} ≤ μ

with equality holding only if

either dim S_{1} > dim S
or dim S_{1} = dim S and ψ_{1} is square ,

i.e.,

\begin{matrix}X & −−→^{ψ_{1}} & X_{1} \\ ^{ϕ}↓ & ↓^{ϕ_{1}} \\ S & ←^{π} & S′ \end{matrix}

π is a birational morphism and ψ_{η} : X_{η} −−→ (X_{1})_{η} is an isomorphism, where η is the generic point of S.

First by definition of λ and the assumption of this case λ > μ, it follows that

p^{∗}(K_{X} + \frac{1}{μ}H_{X}) = K_{Z} + \frac{1}{μ}H_{Z} + bE

for some b > 0, b ∈ ℚ.

We take a general curve Σ_{1} ∈ Hilb(X_{1}/S_{1}) away from the locus of indeterminacy of the birational map X_{1} −−→ Z (i.e., in the case of a link of type (II) the union of q(E_{q}) and all the flipped curves and in the case of a link of type (I) the union of all the flipped curves). Then Σ_{1} can be considered to lie on Z and

0 = (K_{X} + \frac{1}{μ}H_{X}) · p_{∗}Σ_{1}

= (K_{Z} + \frac{1}{μ}H_{Z} + bE) · Σ_{1}

≥ (K_{Z} + \frac{1}{μ}H_{Z}) · Σ_{1}

= (K_{X_{1}} + \frac{1}{μ}H_{X_{1}}) · Σ_{1},

which implies

μ_{1} ≤ μ.

Moreover, if μ_{1} = μ and dim S = dim S_{1} (which implies that π : S_{1} → S is a birational morphism, since both field extensions k(X)/k(S) and k(X_{1}) = k(X)/k(S_{1}) are algebraically closed), then E · Σ_{1} = 0, which is equivalent to saying ϕ_{1}(the strict transform of E) ≠ S_{1}. Therefore, ψ_{1} is square.

We also claim that

λ_{1} ≤ λ

and

if λ_{1} = λ then e_{1} < e.

First (X_{1}, \frac{1}{λ}H_{X_{1}}) is canonical, since it is obtained from a canonical pair (Z, \frac{1}{λ}H_{Z}) through K + \frac{1}{λ}H-MMP. Thus λ_{1} ≤ λ. (Note that in general canonicality may not be preserved when we contract a component of the boundary B through K + B-MMP. But in our case, H’s are the strict transforms of one unique base point free system and thus canonicality is preserved.)

Moreover, if λ_{1} = λ, then in the case of untwisting by a link of type (II)

K_{Z′} + \frac{1}{λ}H_{Z′} = q^{∗}(K_{X_{1}} + \frac{1}{λ}H_{X_{1}}) + aE_{q}(a > 0)

implies E_{q} is not a K_{X_{1}} + \frac{1}{λ}H_{X_{1}}-crepant divisor (and E is a divisor on X_{1} and thus not exceptional) and thus

e_{1} ≤ e − 1 < e.

In the case of untwisting by a link of type (I) E is a divisor on X_{1} (and thus not exceptional) and thus we have the same conclusion.

Therefore, after untwisting by a link of type (II) or type (I), we go back to the START with strictly decreased Sarkisov degree.

The “visualization” of the flowchart can be found at the end of the paper.

§2. Termination of Flowchart.

In this section, we discuss the termination of the flowchart for the Sarkisov program, i.e., the problem of showing that there is no infinite loop in the flowchart and thus after a finite number of untwisting it gives a factorization of any given birational map between two Mori fiber spaces. Once we have (Log)-MMP in dimension n the key points of showing termination for Sarkisov program for n-folds are:

i) Discreteness of the quasi-effective thresholds μ, which follows from the boundedness of ℚ-Fano d-folds d ≤ n, and

ii) Corti[4]’s ingeneous argument to reduce the probelm to S_{n}(Local) when the quasi-effective threshold stabilizes.

In dimension 3, where we have all the necessary ingredients, the termination of the flowchart is a theorem by Corti[4]. The argument here is a modification of Corti[4] following a slightly simplified flowchart in the previous section. We restrict ourselves to dimension 3 in the following presentation, but we carry the argument so that it works almost verbatim in arbitrary dimension (once all the necessary but still conjectural ingredients are established).

Claim 2.1. There is no infinite number of untwisting (successive or unsuccessive) by the links under the case λ ≤ μ.

Proof.

Suppose there are infinitely many links (successive or unsuccessive)

X_{i}−−→^{ψ_{i}}X_{i+1}
ϕ_{i}↓  ↓^{ϕ_{i+1}}
S_{i} S_{i+1}

under the case λ ≤ μ. Note that in the case λ ≤ μ we have dim S_{i} ≥ 1 (unless Φ_{i} becomes an isomorphism of Mori fiber spaces). When dim S_{i} = 2, l being a rational curve which is a general fiber of ϕ_{i}, we have

K_{X_{i}} · l = −2
(μK_{X_{i}} + H_{X_{i}}) · l = 0,

which implies

μ ∈ \frac{1}{2}ℕ.

When dim S_{i} = 1, we can take a rational curve l in a general fiber which is a Del Pezzo surface s.t.

K_{X_{i}} · l = −1, −2 or − 3
(μK_{X_{i}} + H_{X_{i}}) · l = 0,

which implies

μ ∈ \frac{1}{3!}ℕ.

Since after any link in the case λ ≤ μ the quasi-effective threshold strictly decreases and it does not increase afyter any link in any case, we then have a strictly decreasing sequence in \frac{1}{3!}ℕ

μ ≥ μ_{1} > μ_{2} · ·· > 0,

a contradiction!

In general, we only have to use the boundedness of ℚ-Fano d-folds for d ≤ n − 1 to derive the discreteness of μ and thus a contradiction to establish this claim.

Claim 2.2. There is no infinite (successive) sequence of untwisting by the links under the case λ > μ with stationary quasi-effective threshold.

Proof.

This is the heart of the ingeneous argument by Corti[4]. Suppose there is such an infinite sequence

X = X_{0}−−→X_{1}−−→X_{2}−−→  ⋯  −−→X_{k}−−→^{ψ_{k}}X_{k+1}⋯
ϕ↓  ^{ϕ_{1}}↓ ^{ϕ_{2}}↓  ⋯  ^{ϕ_{k}}↓ ^{ϕ_{k+1}}↓  ⋯
S = S_{0} ← S_{1} ← S_{2} ←  ⋯  ← S_{k} ← S_{k+1} ⋯

Since μ_{k} = μ_{k+1} for each k by assumption, we have

either dim S_{k+1} > dim S_{k}
or dim S_{k+1} = dim S_{k} and ψ_{k} is square.

The first cannot happen infinitely many times, thus we may assume we have the second case for all k. Note that dim S_{k} ≥ 1, since if dim S_{k} = dim S_{k+1} = 0 then ψ_{k} being square would imply ψ_{k} is an isomorphism of Mori fiber spaces, which is absurd!

We also know that {λ_{k}} is a nonincreasing sequence and since if λ_{k} = λ_{k+1} then e_{k+1} < e_{k}, the value of λ_{k} cannot be stationary. Therefore, we have a sequence

\{\frac{1}{λ_{k}} \} (\frac{1}{λ_{k}} < \frac{1}{μ_{k}} = \frac{1}{μ_{0}} )

which accumulates from below to (but never equals)

α ≤ \frac{1}{μ_{0}} .

Step 1. We claim (X_{k}, αH_{X_{k}}) and (Z_{k}, αH_{Z_{k}}) (p_{k} : Z_{k} → X_{k} is a maximal divisorial blowup with respect to K + \frac{1}{λ}H) have only log canonical singularities for k sufficiently large (and thus we may assume this holds for ∀k).

Let α_{k}(> \frac{1}{λ_{k}}) be the log canonical threshold of the pair X_{k} with respect to H_{k}. If α > α_{k} for infinitely many k’s, then there is a strictly increasing subsequence {α_{l}} of log canonical thresholds accumulating to α. This contradicts S_{3}(Local) proved by Alexeev[1] (cf.Kollár et al[16]). The same argument applies to (Z_{k}, αH_{Z_{k}}).

Every link X_{k} −−→ X_{k+1} is an outcome of K + \frac{1}{λ_{k}} H-MMP over S_{k} (after taking a maximal divisorial blowup p_{k} : Z_{k} → X_{k}) consisting of a finite number of K + \frac{1}{λ}Hflips

Z_{k} = Z_{k}^{0} −−→^{t^{0}} Z_{k}^{1} −−→^{t^{1}} Z_{k}^{2} · ·· −−→^{t^{m−1}} Z_{k}^{m},

possibly followed by a divisorial contraction q_{k}^{m} : Z_{k}^{m} → X_{k}^{m+1} = X_{k+1} (otherwise Z_{k}^{m} = X_{k}^{m+1}).

Step 2. We claim that every step

Z_{k}^{i}  −−→  Z_{k}^{k+1}
↘  ↙
X_{k}^{i+1}

is a step of K + αH-MMP.

We prove this by induction on i.

First note that since α > c_{k}, we have

K_{Z_{k}} + αH_{Z_{k}} = p_{k}^{0}^{∗}(K_{Z_{k}} + αH_{Z_{k}}) − aE_{k}(a > 0).

Therefore, we have

(K_{Z_{k}^{0}} + αH_{Z_{k}^{0}}) · P_{k}^{0} > 0,

P_{k}^{0} being the extremal ray giving rise to the morphism p_{k}^{0}.

Suppose we have

(K_{Z_{k}^{i}} + αH_{Z_{k}^{i}}) · P_{k}^{i} > 0,

P_{k}^{i} being the extremal ray giving rise to the morphism p_{k}^{i}.

Note that K_{Z_{k}^{i}} + αH_{Z_{k}^{i}} is never relatively nef over S_{k}. We see this as follows: First α ≤ \frac{1}{μ_{k}} = \frac{1}{μ_{1}} . Suppose α = \frac{1}{μ_{k}} . Then

K_{Z_{k}^{i}} + αH_{Z_{k}^{i}} ≡_{S_{k}} −a(the strict transform of E_{k})(a > 0)

is never relatively nef over S_{k}. Suppose α < \frac{1}{μ_{k}}. Then by taking a general curve Σ ∈ Hilb(X_{k}/S_{k}) away from the locus of indeterminacy of the birational map X_{k} −−→ Z_{k}^{i} (which thus can be considered to lie on Z_{k}^{i}) we have

(K_{Z_{k}^{i}} + αH_{Z_{k}^{i}}) · Σ = (K_{X_{k}} + αH_{X_{k}}) · Σ

< (K_{X_{k}} + \frac{1}{μ_{k}} H_{X_{k}}) · Σ = 0.

This implies that

(K_{Z_{k}^{i}} + αH_{Z_{k}^{i}}) · Q_{k}^{i} < 0

for the other extremal ray Q_{k}^{i} of 2-dimensional cone \overline{NE}(Z_{k}^{i}/S_{k}). This proves the claim.

A consequence of this claim is that (cf.KaMaMa[11],Proposition 5-1-11 )

a(ν, X_{1}, αH_{X_{1}}) ≤ a(ν, X_{k}, αH_{X_{k}})

for any discrete valuation ν of k(X) and the strict inequality holds iff ψ_{i} is not an isomorphism at the center of ν on X_{i} for some i < k.

Step 3. We claim that (X_{k}, αH_{X_{k}}) has purely log terminal singularities for k sufficiently large (and thus we may assume this holds for ∀k).

Assume to the contrary that there exists infinitely many k s.t. (X_{k}, αH_{X_{k}}) is not purely log terminal, which is equivalent to saying by the consequence above that for all k there exists a valuation ν_{k} of k(X) with

a(ν_{k}, X_{k}, αH_{X_{k}}) = −1,

which implies again by the consequence that

a(ν_{k}, X_{1}, αH_{X_{1}}) = −1

and that at the center z(ν_{k}, X_{1}) of ν_{k} on X_{1}, the birational map ψ_{k−1} ∘ ⋯ψ_{2} ∘ ψ_{1} : X_{1} −−→ X_{k} is an isomorphism. Thus the local (w.r.t. Zariski topology) canonical thresholds are the same

c(z(ν_{k}, X_{k}), X_{k}, H_{X_{k}}) = c(z(ν_{k}, X_{1}), X_{1}, H_{X_{1}}).

On the other hand, by definition

\frac{1}{λ_{k}} ≤ c(z(ν_{k}, X_{k}), X_{k}, H_{X_{k}})

and since K_{X_{k}} + αH_{X_{k}} is not canonical at the center z(X_{k}, X_{k}), we have

c(z(ν_{k}, X_{k}), X_{k}, H_{k}) < α.

Therefore,

\frac{1}{λ_{k}} ≤ c(z(ν_{k}, X_{1}), X_{1}, H_{X_{1}}) < α.

But \{\frac{1}{λ_{k}} \} is a nondecreasing and nonstationary sequence converging to α and it is easy to see the set {c(x, X_{1}, H_{X_{1}}); x ∈ X} is finite, a contradiction!

We remark that the valuations of k(X) corresponding to the E_{k}’s are all distinct. In fact, suppose E_{i} and E_{j} coincide, and thus Z_{i} and Z_{j} are isomorphic in a neighborhood of E_{i} and E_{j}, which would imply

a(E_{i}, X_{i}, αH_{X_{i}}) = a(E_{j}, X_{j}, αH_{X_{j}}).

On the other hand, from Step 2 we have

a(E_{i}, X_{i}, αH_{X_{i}}) < a(E_{j}, X_{j}, αH_{X_{j}}),

a contradiction!

Finally we conclude the proof of the claim as follows: From Step 3 we may assume that (X_{1}, αH_{X_{1}}) has only purely log terminal singularities. But on the other hand, for infinitely many E_{k} with distinct corresponding discrete valuations

a(E_{k}, X_{1}, αH_{X_{1}}) ≤ a(E_{k}, X_{k}, αH_{X_{k}}) < 0,

a contradiction!

In general, we only need S_{n}(Local) to carry out the argument for this claim.

Claim 2.3. There is no infinite successive sequence of untwisting by the links under the case λ > μ with nonstationary quasi-effective threshold.

Suppose there is such an infinite sequence

X = X_{0}−−→X_{1}−−→X_{2}−−→  ⋯  −−→X_{k}−−→^{ψ_{k}}X_{k+1}⋯
ϕ↓  ^{ϕ_{1}}↓ ^{ϕ_{2}}↓  ⋯  ^{ϕ_{k}}↓ ^{ϕ_{k+1}}↓  ⋯
S = S_{0} ← S_{1} ← S_{2} ←  ⋯  ← S_{k} ← S_{k+1} ⋯

Case: For some k_{0}, dim S_{k_{0}} ≥ 1.

In this case for ∀k ≥ k_{0} we have

μ_{k} ∈ \frac{1}{3!}ℕ

as before, and {μ_{k}} is a nonstationary and nonincreasing infinte sequence μ_{0} ≥ μ_{k} > 0, a contradiction! In general, we only need the boundedness of ℚ-Fano d-folds for d ≤ n − 1 up to this point of the argument.

Finally

Case:For ∀k, dim S_{k} = 0.

In this case, the X_{k}’s are all ℚ-Fano variety with ρ(X_{k}) = 1. Thus in dimension 3 Kawamata[8]’s result implies that they belong to a bounded family. Therefore, there exists q ∈ ℕ s.t. qD is Cartier for all integral Weil divisor on X_{k} for ∀k and there exists r ∈ ℕ s.t. rK_{X_{k}} is Cartier for ∀k. Then another result of Kawamata[8] on the boundedness of the lengths of the extremal rational curves says that there exists a rational curve L_{k} on X_{k} s.t. 0 < −K_{X_{k}} · L_{k} ≤ 2 · dimX_{k}, which implies

μ_{k} ∈ \frac{1}{(r · 2dimX)!q} ℕ.

Again {μ_{k}} is a nonstationary and nonincreasing infinite sequence μ_{0} ≥ μ_{k} > 0, a contradiction!

We remark that this last step is the only place where we use the boundedness of ℚ-Fano n-folds.

Cliams 2.1, 2.2 and 2.3 show that there is no infinite loop in the flowchart of the Sarkisov program.

This completes the discussion of termination of the flowchart.

§3. Log Sarkisov Program with KLT Singularities.

In this section, we try to establish the Log Sarkisov Program for untwisting a birational map between two log Mori fiber spaces with only kawamata log terminal singularities

(X, B_{X})−−→^{Φ}(X′, B_{X′})
ϕ↓  ↓^{ϕ′}
S  S′

The guiding principle throughout is that while the genuine Sarkisov program is the one to untwist a birational map between two Mori fiber spaces obtained as two different end results of K-MMP starting from one nonsingular projective variety

W W

K − MMP ↙ ↘ K − MMP
X  −−→^{Φ}  X′
ϕ↓  ↓^{ϕ′}
S  S′,

the log Sarkisov program should be the one to untwist a birational map between two log Mori fiber spaces obtained as two different end results of K + B-MMP starting from one log variety (W, B_{W}) consisting of a nonsingular projective variety W and an S.N.C. divisor B_{W}

(W, B_{W})
K + B − MMP ↙  ↘ K + B − MMP
(X, B_{X})  −−→^{Φ}  (X′, B_{X′})
ϕ↓  ↓^{ϕ′}
S  S′.

(Note that in the two diagrams above we do not require a priori the slanted arrows to be morphisms.)

While this principle does not put any restriction on the birational map Φ in the case of the genuine Sarkisov program (namely, for any birational map Φ between two Mori fiber spaces we can find a common resolution W s.t. X → S and X′ → S′ are two end results of K-MMP as the diagram above), this principle in the case of the log Sarkisov program allows us to consider only such birational map Φ for which a log variety as above exists. This naturally leads to the notion of the Sarkisov relation: (Two or more) Log Mori fiber spaces are Sarkisov related iff they are the end results of K + B-MMP starting from one log pair consisting of a nonsingular projective variety and an S.N.C. divisor as a boundary. Thus the log Sarkisov program untwists a birational map between two log Mori fiber spaces which are Sarkisov related, by factorizing it into links among intermediate Mori fiber spaces all of which including the original two we require to be also Sarkisov related. (Remark again that in the case of the genuine Sarkisov program the Sarkisov relation happens to be an equivalence relation and coincide with the usual birational equivalence.)

Once we understand what the appropriate logarithmic generalization of the Sarkisov program should be through the notion of the Sarkisov relation, the flowchart for the log Sarkisov program with klt singularities works almost parallel to that of the genuine Sarkisov program as well as termination except the very last step. In order to show that there is no infinite successive sequence of links of type (II) with nonstationary quasi-effective threshold, we have to use the conjecture by Borisov[3] in dimension 3 (cf.Nikulin[26]Alexeev[2]).

Conjecture 3.1. Fix a nonnegative rational number 0 ≤ ϵ < 1. Then the family of log ℚ-Fano n-folds (a normal projective n-fold X with only ℚ-factorial log terminal singularities (and thus automatically has only klt singularities) s.t. the anti-canonical divisor −K_{X} is ample) whose discrepancies are all > −ϵ is bounded.

In dimension 2, the conjecture holds (cf.Nikulin[26]Alexeev[2]) and thus we establish the log Sarkisov program for klt surfaces. The conjecture in dimension 3 for the case of Picard number 1 and ϵ = 0 is the theorem of Kawamata, which completes the proof of termination for the genuine Sarkisov program for 3-folds.

In the following we discuss the log Sarkisov program with klt singularities in detail.

Definition 3.2 (cf.Kollár et al[16]). A log pair (X, B_{X} = Σb_{i}B_{i}) has only kawamata log terminal singularities iff every discrete valuation of k(X) having center on X has positive log discrepancy, i.e.,

o) X is normal,

i) 0 ≤ b_{i} < 1 for ∀i, and

ii) there exists a log resolution f : Y → X where all the f-exceptional divisors have positive log discrepancies (and thus this holds for any log resolution).

Definition 3.3. A log Mori fiber space ϕ : (X, B_{X}) → S with only klt singularities is the contraction of an extremal ray with respect to K_{X} + B_{X} from a log pair (X, B_{X}) consisting of a normal projective variety X and a divisor B_{X} with only ℚ-factorial klt singularities onto a variety S, i.e., ϕ is a morphism from a ℚfactorial klt log pair (X, B_{X}) with connected fibers onto a normal variety S with dim S < dim X s.t. ρ(X/S) = 1 and −(K_{X} + B_{X}) is ϕ-ample.

Definition 3.4 (Sarkisov Relation). Log Mori fiber spaces (resp. Log minimal models)

(X_{0}, B_{X_{o}}), (X_{1}, B_{X_{1}}), ⋯, (X_{k}, B_{X_{k}}), ⋯, (X_{l}, B_{X_{l}})

are Sarkisov related iff they are all end results of K + B-MMP starting from one appropriate log pair (W, B_{W}) consisting of a nonsingular projective variety W and an S.N.C. divisor B_{W} = Σb_{i}B_{i} with 0 ≤ b_{i} ≤ 1.

A birational map Φ between two log Mori fiber spaces (resp. log minimal models) which are Sarkisov related is by definition the one for which we have a commutative diagram

(W, B_{W})
p ↙  ↘ q
(X, B_{X})  −−→^{Φ}  (X′, B_{X′})

where (W, B_{W}) is a log pair specified as above (Note that p or q may not be a morphism.)

The Sarkisov relation behaves very well for log Mori fiber spaces (or log minimal models) with only kawamata log terminal singularities.

Proposition 3.5. Let

(X_{0}, B_{X_{o}}), (X_{1}, B_{X_{1}}), ⋯, (X_{k}, B_{X_{k}}), ⋯, (X_{l}, B_{X_{l}})

be log Mori fiber spaces (resp. log minimal models) with only klt singularities and 0 ≤ ϵ < 1 a rational number such that all the coefficients of the boundaries B_{X_{k}} are ≤ ϵ and that all the discrepancies of the log pairs (X_{k}, B_{X_{k}}) are > −ϵ.

Then the following are equivalent:

(i) The log Mori fiber spaces (resp. log minimla models) with klt singularities

(X_{0}, B_{X_{o}}), (X_{1}, B_{X_{1}}), ⋯, (X_{k}, B_{X_{k}}), ⋯, (X_{l}, B_{X_{l}})

are Sarkisov related, i.e., there exists a log variety (W, B_{W}) consisting of a nonsingular projective variety W and an S.N.C. divisor B_{W} as a boundary such that all the log Mori fiber spaces (resp. log minimal models) are end results of K + B-MMP over Spec k starting from (W, B_{W}).

(ii) There exists a log variety (W, B_{W}) consisting of a nonsingular projective variety W and an S.N.C. divisor B_{W} as a boundary such that all the log Mori fiber spaces (resp. log minimal models) are end results of K + B-MMP over Spec k starting from (W, B_{W}) and that

B_{W} = D_{W} (B_{X_{0}}, B_{X_{1}}, ⋯, B_{X_{k}}, ⋯, B_{X_{l}}) + Σ_{E_{j}not appearing as a divisor on any of X_{k}}ϵE_{j}, where

D_{W} (B_{X_{0}}, B_{X_{1}}, ⋯, B_{X_{k}}, ⋯, B_{X_{l}}) := Σd_{m}D_{m}

summation being taken over all divisors on W which appear as a divisor on some X_{m} and d_{m} being the coefficient of D_{m} in B_{X_{m}}.

(iii) There exists a log variety (W, B_{W}) consisting of a nonsingular projective variety W and an S.N.C. divisor B_{W} as a boundary such that each log Mori fiber space (resp. log minimal model) is dominated by a birational morphism p_{k} : (W, B_{W}) → (X_{k}, B_{X_{k}}) and an end result of K + B-MMP over X_{k} starting from (W, B_{W}) and that

B_{W} = D_{W} (B_{X_{0}}, B_{X_{1}}, ⋯, B_{X_{k}}, ⋯, B_{X_{l}}) + Σ_{E_{j}not appearing as a divisor on any of X_{k}}ϵE_{j}.

Proof.

The proposition is a straightforward consequence of the following lemma, whose first claim holds not only for klt singularities but also weakly kawamata log terminal singularities (or even more generally for log canonical singularities) while the second claim only holds for klt singularities. This is why the Sarkisov relation behaves very well for klt singularities but becomes quite subtle for wklt or lc singularities. The verification of the lemma is left to the reader as an exercise.

Lemma 3.6.

(i) Let p : (W, B_{W}) → (X, B_{X}) be a projective birational morphism between ℚ factorial log varieties with klt (or more generally with wklt or lc) singularities. Then p is a process of K + B-MMP over X starting from (W, B_{W}) iff B_{X} = p_{∗}(B_{W}) and the ramification divisor R

K_{W} + B_{W} = p^{∗}(K_{X} + B_{X}) + R

has the same support as the exceptional locus E(p) of p.

(ii) Let (X, B_{X}) be a log pair with only ℚ-factorial klt singularities and 0 ≤ ϵ < 1 a rational number such that all the coefficients of the boundary B_{X} are ≤ ϵ and that all the discrepancies are > −ϵ. Then any projective birational morphism p : (W, B_{W}) → (X, B_{X}) from a log pair (W, B_{W}) consisting of a nonsingular projective variety and an S.N.C. divisor B_{W}

B_{W} = D_{W} (B_{X}) + Σ_{E_{j}not appearing as a divisor on X}ϵE_{j},

is a process of K + B-MMP over X starting from (W, B_{W}).

Remark 3.7.

Though unfortunately the Sarkisov relation is NOT an equivalence relation in general, it is an equivalence relation for the following special classes of log pairs (X, B_{X}) with klt singularities: We fix 0 ≤ ϵ < 1. The class consists of ℚ-factorial projective log pairs (X, B_{X}) with klt singularities whose coefficients of the boundaries B_{X} are all equal to ϵ, and all the discrepancies of the valuations of the exceptional divisors are > −ϵ.

The genuine Sarkisov program is nothing but the program for the class given by ϵ = 0.

Theorem 3.8 (Log Sarkisov Program for log 3-folds with klt singularities). Let

(X, B_{X})−−→^{Φ}(X′, B_{X′})
ϕ↓  ↓^{ϕ′}
S  S′

be a birational map between two log Mori fiber spaces in dimension 3 with only klt singularities, which are Sarkisov related.

Suppose the Borisov conjecture holds for ℚ-Fano 3-folds with klt singularities and Picard number 1.

Then for any rational number 0 ≤ ϵ < 1 such that all the coefficients in B_{X} or B_{X′} are ≤ ϵ and that all the discrepancies of (X, B_{X}) or (X′, B_{X′}) are > −ϵ, Φ is a composite of 4 types of links as in the genuine Sarkisov program

(X, B_{X}) = (X_{0}, B_{X_{0}})−−→(X_{1}, B_{X_{1}})−−→⋯−−→(X_{k}, B_{X_{k}})−−→⋯−−→(X′, B_{X′})
ϕ↓  ^{ϕ_{1}}↓  ⋯ ^{ϕ_{k}}↓  ⋯ ^{ϕ′}↓
S = S_{0}  S_{1}  ⋯  S_{k}  ⋯  S′

such that all the log Mori fiber spaces (X_{k}, B_{X_{k}}) have only ℚ-factorial klt singularities, the coefficients of B_{X_{k}} are ≤ ϵ, all the discrepancies of (X_{k}, B_{X_{k}}) are > −ϵ and that all the (X_{k}, B_{X_{k}}) are Sarkisov related. More precisely, all the log Mori fiber spaces are dominated by birational morphisms

p_{k} : (W, B_{W}) → (X_{k}, B_{X_{k}})

from a log pair (W, B_{W}) consisting of a nonsingular projective 3-fold and an S.N.C. divisor B_{W}

B_{W} = D_{W} (B_{X}, B_{X′}) + Σ_{E_{j}not appearing as a divisor either on X or on X′}ϵE_{j}

and each (X_{k}, B_{X_{k}}) is an end result of K+B-MMP over X_{k} starting from (W, B_{W}).

The strategy to establish the log Sarkisov program goes along the same line as the one to establish the genuine Sarkisov program, constructing (W, B_{W}) as above inductively as the program proceeds.

We define the log Sarkisov degree of an intermediate log Mori fiber space (X_{k}, B_{X_{k}}) which appears in the due course of untwisting a birational map

(X, B_{X})−−→^{Φ}(X′, B_{X′})
ϕ↓  ↓^{ϕ′}
S  S′

between two log Mori fiber spaces which are Sarkisov related as follows.

Definition 3.9 (the log Sarkisov degree). Let

(X, B_{X})−−→^{Φ}(X′, B_{X′})
ϕ↓  ↓^{ϕ′}
S  S′

be a birational map between two log Mori fiber spaces with only klt singularities which are Sarkisov related, and fix a rational number 0 ≤ ϵ < 1 such that all the coefficients in B_{X} or B_{X′} are ≤ ϵ and that all the discrepancies of (X, B_{X}) or (X′, B_{X′}) are > −ϵ. Then the log Sarkisov degree of any intermediate log Mori fiber space ϕ_{k} : (X_{k}, B_{X_{k}}) → S_{k} that appears in the due course of untwisting the birational map with reference to the fixed log Mori fiber space ϕ′ : (X′, B_{X′}) → S′ is the triplet

(μ_{k}, λ_{ϵk}, e_{ϵk})

of the numbers defined below, endowed with the lexicographical order.

Notice that there is an auxiliary parameter ϵ, which was implicit (actually equal to 0) in the case of the genuine Sarkisov degree. Also note that the log Sarkisov degree depends not only on (X_{k}, B_{X_{k}}) and (X′, B_{X′}) but also on the initial log Mori fiber space (X, B_{X}).

First we take a very ample divisor A′ on S′ and a sufficiently divisible μ′ ∈ ℕ such that

H_{X′} = −μ′(K_{X′} + B_{X′}) + ϕ′^{∗}A′

is very ample on X′. H_{X_{k}} is the strict transform of H_{X′} on X_{k}.

μ_{k} : the quasi-effective threshold

The quasi-effective threshold is defined exactly the same way as before, replacing K with K + B, namely

μ_{k} ∈ ℚ_{>0} μ_{k}(K_{X_{k}} + B_{X_{k}}) + H_{X_{k}} ≡ 0 over S_{k}

λ_{ϵk} : the maximal multiplicity of an extremal ray

Let (W, B_{W}) be a log pair consisting of a nonsingular projective variety W and an S.N.C. divisor B_{W}

B_{W} = D_{W} (B_{X}, B_{X′}) + Σ_{E_{j}not appearing as a divisor either on X or on X′}ϵE_{j}

such that it dominates

p : (W, B_{W}) → (X, B_{X})
q : (W, B_{W}) → (X′, B_{X′})
p_{k} : (W, B_{W}) → (X_{k}, B_{X_{k}})

by birational morphisms and that they are all processes of K + B-MMP over X, X′ and X_{k}, respectively. (The existence of such (W, B_{W}) will be shown inductively in the course of log Sarkisov program.)

Then

\frac{1}{λ_{ϵk}} := max\{c ∈ ℚ_{>0}; (K_{W} +B_{W})+cH_{X_{k}} = p_{k}^{∗}(K_{X_{k}}+B_{X_{k}})+cH_{X_{k}}+some effective divisor\}.

Note that \frac{1}{λ_{ϵk}} is independent of the choice of such (W, B_{W}) and well-defined. When Bs(H_{X}) = ∅, λ_{ϵk} = 0 by definition.

We note that in general

λ_{ϵk} ≤ ϵ − log terminal threshold of (X_{k}, B_{X_{k}}) w.r.t. H_{X_{k}}.

The ϵ-log termianl threshold of the pair (X_{k}, B_{X_{k}}) is defined to be

max{c ∈ ℚ_{>0}; (K_{V_{k}}+B_{V_{k}})+cH_{V_{k}} = v_{k}^{∗}((K_{X_{k}}+B_{X_{k}})+cH_{X_{k}})+some effective divisor},

where V_{k} is any nosingular projective variety which dominates both v_{k} : V_{k} → X_{k} and X′ by birational morphisms such that the union of the exceptional locus of v_{k} and v_{k}_{∗}^{−1}(B_{X_{k}}) is an S.N.C. divisor (Recall the ℚ-factoriality of X_{k}), and

B_{V_{k}} = v_{k}_{∗}^{−1}(B_{X_{k}}) + Σ_{E_{j}not Apearing as a divisor on X_{k}}ϵE_{j}.

e_{ϵk} : the number of K + B + \frac{1}{λ_{ϵk}} H − crepant divisors

e_{ϵk} is defined to be the number of exceptional divisors for p_{k} whose coefficient in the ramification divisor R is 0

(K_{W} + B_{W}) + \frac{1}{λ_{ϵk}} H_{W} = p_{k}^{∗}((K_{X_{k}} + B_{X_{k}}) + \frac{1}{λ_{ϵk}} H_{X_{k}}) + R.

Again this is independent of the choice of (W, B_{W}) and well-defined.

Nöther-Fano Criterion for the log Sarkisov program with KLT singularities. The birational map Φ_{k} between an intermediate log Mori fiber space ϕ_{k} : (X_{k}, B_{X_{k}}) → S_{k} and ϕ′ : (X′, B_{X′}) → S′ is an isomorphism of log Mori fiber spaces

(X_{k}, B_{X_{k}})→^{∼}^{Φ_{k}}(X′, B_{X′})
ϕ_{k}↓  ↓^{ϕ′}
S_{k}  →^{∼} S′

if λ_{ϵk} ≤ μ_{k} and (K_{X_{k}} + B_{X_{k}}) + \frac{1}{λ_{ϵk}} H_{X_{k}} is nef.

The proof goes verbatim to the one for the genuine Sarkisov program, taking the common resolution (W, B_{W}) which Sarkisov-relates (X_{k}, B_{X_{k}}) and (X′, B_{X′}) into consideration.

Flowchart for Log Sarkisov Program with KLT Singularities

The flowchart for the log Sarkisov program with klt singularities goes almost parallel to the one for the genuine Sarkisov program replacing K with K + B, constructing inductively a log pair (W, B_{W}) which dominates all the intermediate log Mori fiber spaces in the process of untwisting.

Let

(X, B_{X})−−→^{Φ}(X′, B_{X′})
ϕ↓  ↓^{ϕ′}
S  S′

be a birational map between two log Mori fiber spaces with only klt singularities, which are Sarkisov related. We fix a rational number 0 ≤ ϵ < 1 such that all the coefficients in B_{X} or B_{X′} are ≤ ϵ and that all the discrepancies of (X, B_{X}) or (X′, B_{X′}) are > −ϵ.

We remark that for all the relevant log pairs (U, B_{U}) that appear in the course of the log Sarkisov program (including the auxiliary resolutions we take) the boundaries B_{U} are always taken to be of the form

B_{U} = D_{U}(B_{X}, B_{X′}) + Σ_{E_{j} not appearing as a divisor either on X or on X′}ϵE_{j}.

Before we start the flowchart, we note that by Proposition 3.5 there exists a log variety (W_{0}, B_{W_{0}}) consisting of a nonsingular projective variety W_{0} and an S.N.C. divisor B_{W_{0}} as a boundary such that (X, B_{X}) = (X_{0}, B_{X_{0}}) (resp. (X′, B_{X′})) is dominated by a birational morphism p = p_{0,0} : (W_{0}, B_{W_{0}}) → (X_{0}, B_{X_{0}}) (resp. q = q_{0} : (W_{0}, B_{W_{0}}) → (X′, B_{X′})) and an end result of K + B-MMP over X_{0} (resp. over X′) starting from (W_{0}, B_{W_{0}}) and that

B_{W_{0}} = D_{W_{0}}(B_{X}, B_{X′}) + Σ_{E_{j}not appearing as a divisor on either X or on X′}ϵE_{j}.

Suppose we have untwisted the birational map up to the k-th stage and constructed a log pair (W_{k}, B_{W_{k}}) consisting of a nonsingular projective variety W_{k} and an S.N.C. divisor B_{W_{k}} as a boundary such that each log Mori fiber space (X_{m}, B_{X_{m}}) m = 0, 1, ⋯, k (and (X′, B_{X′})) is dominated by a birational morphism p_{k,m} : (W_{k}, B_{W_{k}}) → (X_{m}, B_{X_{m}}) (and q_{k} : (W_{k}, B_{W_{k}}) → (X′, B_{X′})) and an end result of K + B-MMP over X_{m} (and over X′) starting from (W_{k}, B_{W_{k}}) and that

B_{W_{k}} = D_{W_{k}}(B_{X}, B_{X′}) + Σ_{E_{j}not appearing as a divisor on either X or on X′}ϵE_{j}.

The first question then to ask as in the genuine Sarkisov program is:

λ_{ϵk} > μ_{k}?

Case:λ_{ϵk} ≤ μ_{k}

In this case, thanks to the Nöther-Fano inequality for klt singularities, the program works completely parallel to the genuine Sarkisov program. If (K_{X_{k}} +B_{X_{k}})+ \frac{1}{λ_{ϵk}} H_{X_{k}} is nef, then the program comes to an end. If not, then after untwisting the birational map either by a link of type (III) or (IV), the quasi-effective threshold strictly drops

μ_{k+1} < μ_{k}

as before.

We also take a nonsingular projective variety W_{k+1} by blowing up W_{k} further so that it dominates each log Mori fiber space by a birational morphism p_{k+1,m} : (W_{k+1}, B_{W_{k+1}}) → (X_{m}, B_{X_{m}}) m = 0, 1, ⋯, k, k + 1 (and q_{k+1} : (W_{k+1}, B_{W_{k+1}}) → (X′, B_{X′})) and that

B_{W_{k+1}} := D_{W_{k+1}}(B_{X}, B_{X′}) + Σ_{E_{j}not appearing as a divisor on either X or on X′}ϵE_{j}

is an S.N.C. divisor. Then by Lemma 3.6 (W_{k}, B_{W_{k}}) is an end result of K + B-MMP over W_{k} starting from (W_{k+1}, B_{W_{k+1}}) and thus each log Mori fiber space (X_{m}, B_{X_{m}})m = 0, 1, ⋯, k (and (X′, B_{X′})) is an end result of K + B-MMP starting from (W_{k+1}, B_{W_{k+1}}) over X_{m}. As for (X_{k+1}, B_{X_{k+1}}) which is an end result of K + B + \frac{1}{λ_{ϵk}} H-MMP starting from (X_{k}, B_{X_{k}}) (over T), all the coeffiecients of B_{X_{k+1}} are ≤ ϵ by construction. Moreover, for any valuation E whose center on X_{k+1} has codimension at least 2

a(E, X_{k+1}, B_{X_{k+1}}) = a(E, X_{k}, B_{X_{k}}) < −ϵ

if the birational map (X_{k}, B_{X_{k}}) −−→ (X_{k+1}, B_{X_{k+1}}) is isomorphic at the center of the valuation E, and

a(E, X_{k+1}, B_{X_{k+1}}) ≤ a(E, X_{k+1}, B_{X_{k+1}} + \frac{1}{λ_{ϵk}} H_{X_{k+1}})

< a(E, X_{k}, B_{X_{k}} + \frac{1}{λ_{ϵk}} H_{X_{k}}) ≤ −ϵ.

if the birational map (X_{k}, B_{X_{k}}) −−→ (X_{k+1}, B_{X_{k+1}}) is not isomorphic at the center of the valuation E.

Thus again by Lemma 3.6 (X_{k+1}, B_{X_{k+1}}) is an end result of K+B-MMP starting from (W_{k+1}, B_{W_{k+1}}) over X_{k+1}. Therefore, (W_{k+1}, B_{W_{k+1}}) satisfies the desired inductive property.

Case:λ_{ϵk} > μ_{k}

In this case we

Take a maximal divisorial blow up d_{k} : (Z_{k}, B_{Z_{k}}) → (X_{k}, B_{X_{k}})

with respect to (K_{X_{k}} + B_{X_{k}}) + \frac{1}{λ_{ϵk}} H_{X_{k}}, and in the log Sarkisov program we also require (Z_{k}, B_{Z_{k}}) is obtained through K + B + \frac{1}{λ_{ϵk}} H-MMP possibly followed by K + B-MMP starting from (W_{k}, B_{W_{k}}) over X_{k}, i.e.,

o) in the proof of Proposition 1.5 instead of carrying out a K + \frac{1}{λ}H-MMP over X starting from Y to get a minimal model (Z′, \frac{1}{λ}H) and then running a K-MMP over X to obtain a maximal divisorial blow up, we carry out a K +B + \frac{1}{λ_{ϵk}} H-MMP over X_{k} starting from (W_{k}, B_{W_{k}}) to get a minimal model (Z′_{k} + B_{Z′}_{k} + \frac{1}{λ_{ϵk}} H_{Z′}_{k}) and then run a K + B-MMP over X_{k} to obtain the maximal divisorial blow up d_{k} : (Z_{k}, B_{Z_{k}}) → (X_{k}, B_{X_{k}}),

i) ρ(Z_{k}/X_{k}) = 1,

ii) the exceptional locus of d_{k} is a prime divisor E_{k}, and iii) d_{k} is K + B + \frac{1}{λ_{ϵk}} H-crepant, i.e.,

(K_{Z} + B_{Z}) + \frac{1}{λ_{ϵk}} H_{Z} = d_{k}^{∗}((K_{X_{k}} + B_{X_{k}}) + \frac{1}{λ_{ϵk}} H_{X_{k}}).

Then the rest goes parallel to the genuine Sarkisov program, untwisting the birational map further by a link of either type (II) or (I). After untwisting, we have

μ_{k+1} ≤ μ_{k}

and if μ_{k+1} = μ_{k} and dim S_{k} = dim S_{k+1}, then ψ_{k} is a square. Moreover,

λ_{ϵ,k+1} ≤ λ_{ϵk}

and if λ_{ϵ,k+1} = λ_{ϵk} then e_{ϵ,k+1} < e_{ϵk}.

We also take a nonsingular projective variety W_{k+1} by blowing up W_{k} further so that it dominates each log Mori fiber space by a birational morphism p_{k+1,m} : (W_{k+1}, B_{W_{k+1}}) → (X_{m}, B_{X_{m}}) m = 0, 1, ⋯, k, k + 1 (and q_{k+1} : (W_{k+1}, B_{W_{k+1}}) → (X′, B_{X′})) and that

B_{W_{k+1}} := D_{W_{k+1}}(B_{X}, B_{X′}) + Σ_{E_{j}not appearing as a divisor on either X or on X′}ϵE_{j}

is an S.N.C. divisor. Then by Lemma 3.6 (W_{k}, B_{W_{k}}) is an end result of K + B-MMP over W_{k} starting from (W_{k+1}, B_{W_{k+1}}) and thus each log Mori fiber space (X_{m}, B_{X_{m}}) m = 0, 1, ···, k (and (X′, B_{X′})) is an end result of K +B-MMP starting from (W_{k+1}, B_{W_{k+1}}) over X_{m}.

As for (X_{k+1}, B_{X_{k+1}}), note that it is an end result of K + B + \frac{1}{λ_{ϵk}} H-MMP starting from (Z_{k}, B_{Z_{k}}) (over S_{k}), which itself is an end result of K + B + \frac{1}{λ_{ϵk}} H-MMP possibly followed by K + B-MMP over X_{k} starting from (W_{k}, B_{W_{k}}). All the coeffiecients of B_{X_{k+1}} are ≤ ϵ by construction.

Moreover, for any valuation E whose center on Z′_{k} has at least codimension 2

a(E, Z′_{k}, B_{Z′}_{k}) = a(E, W_{k}, B_{W_{k}}) < −ϵ

if the birational map (W_{k}, B_{W_{k}}) −−→ (Z′_{k}, B_{Z′}_{k}) is isomorphic at the center of the valuation E, and

a(E, Z′_{k}, B_{Z′}_{k}) ≤ a(E, Z′_{k}, B_{Z′}_{k} + \frac{1}{λ_{ϵk}} H_{Z′}_{k})

< a(E, W_{k}, B_{W_{k}} + \frac{1}{λ_{ϵk}} H_{W_{k}}) ≤ −ϵ

if the birational map (W_{k}, B_{W_{k}}) −−→ (Z′_{k}, B_{Z′}_{k}) is not isomorphic at the center of the valuation E.

For any valuation E whose center on Z_{k} has at least codimension 2

a(E, Z_{k}, B_{Z_{k}}) = a(E, Z′_{k}, B_{Z′}_{k}) < −ϵ

if the birational map (Z′_{k}, B_{Z′}_{k}) −−→ (Z_{k}, B_{Z_{k}}) is isomorphic at the center of the valuation E, and

a(E, Z_{k}, B_{Z_{k}}) < a(E, Z′_{k}, B_{Z′}_{k}) ≤ −ϵ

if the birational map (Z′_{k}, B_{Z′}_{k}) −−→ (Z_{k}, B_{Z_{k}}) is not isomorphic at the center of the valuation E.

Finally for any valuation E whose center on X_{k+1} has at least codimension 2

a(E, X_{k+1}, B_{X_{k+1}}) = a(E, Z_{k}, B_{Z_{k}}) < −ϵ

if the birational map (Z_{k}, B_{Z_{k}}) −−→ (X_{k+1}, B_{X_{k+1}}) is isomorphic at the center of the valuation E, and

a(E, X_{k+1}, B_{X_{k+1}}) ≤ a(E, X_{k+1}, B_{X_{k+1}} + \frac{1}{λ_{ϵk}} H_{X_{k+1}})

< a(E, Z_{k}, B_{Z_{k}} + \frac{1}{λ_{ϵk}} H_{Z_{k}}) ≤ −ϵ

if the birational map (Z_{k}, B_{Z_{k}}) −−→ (X_{k+1}, B_{X_{k+1}}) is not isomorphic at the center of the valuation E.

Thus again by Lemma 3.6 (X_{k+1}, B_{X_{k+1}}) is an end result of K+B-MMP starting from (W_{k+1}, B_{W_{k+1}}) over X_{k+1}. Therefore, (W_{k+1}, B_{W_{k+1}}) satisfies the desired inductive property.

Termination of Flowchart for Log Sarkisov Program with KLT Singularities

One of the key points to show the terminatin of of the flowchart for the genuine Sarkisov program is the discreteness of the quasi-effective thresholds, which follows from the boundedness of ℚ-Fano varieties which are fibers of the Mori fiber spaces (a nonsingular rational curve, Del Pezzo surfaces and ℚ-Fano 3-folds for the genuine Sarkisov program in dimension 3). In the log Sarkisov program, we rely on the Borisov conjecture, which is a theorem in dimension 2 thanks to Nikulin[26][Alexeev[2], to show the discreteness of the quasi-effective thresholds.

Claim 3.10. Let ϕ_{k} : (X_{k}, B_{X_{k}}) → S_{k} be a log Mori fiber space in dimension 3 with only klt singularities in the process of the log Sarkisov program untwisting a birational map

(X, B_{X})−−→^{Φ}(X′, B_{X′})
ϕ↓  ↓^{ϕ′}
S  S′

between two log Mori fiber spaces in dimension 3 with only klt singularities which are Sarkisov related, fixing ϵ as before.

If dim S_{k} ≥ 1, then the denominator of the quasi-effective threshold μ_{k} is universally bounded by a fixed constant depending only on ϵ and the coefficients of B_{X} and B_{X′}.

proof.

Let d be the l.c.m. of the denominator of ϵ and those of coefficients of B_{X} and B_{X′}.

When dim S_{k} = 2, l being a nonsingular rational curve which is a general fiber of ϕ_{k},

−2 ≤ (K_{X_{k}} + B_{X_{k}}) · l < 0
(K_{X_{k}} + B_{X_{k}}) · l ∈ \frac{1}{d}ℤ_{<0}
{μ_{k}(K_{X_{k}} + B_{X_{k}}) + H_{X_{k}}} · l = 0

imply

μ_{k} ∈ \frac{1}{(2d)!}ℕ.

When dim S_{k} = 1, a general fiber F_{k} is a log Del Pezzo surface (a normal projective surface with only quotient singularities having an ample anti-canonical divisor) whose discrepancies are all > −ϵ. Therefore, by Nikulin[26][Alexeev[2] we conclude that the family of such log Del Pezzo surfaces is bounded. Now q being the universal ℚ-factorial index for such surfaces and r the index for the canonical divisors, we have

μ_{k} ∈ \frac{1}{(r · 2 · dim X)!q} ℕ.

In arbitrary dimension, we have to use the Borisov conjecture for boundedness of log ℚ-Fano d-folds for d ≤ n − 1 in general to derive this claim.

Thanks to Claim 3.10, the argument for termination goes parallel replacing K with K +B for Claims 2.1, 2.2 and the first case in Claim 2.3 to that of termination of the genuine Sarkisov program. (We note that in Step 3 of the proof of Claim 2.2 we repalce the local canonical threshold with the local version of \frac{1}{λ_{ϵ}}.) We have to use a conjecture of Borisov to establish the last step of the second case in Claim 2.3.

Conjecture 3.11 (cf.Borisov[3]). Fix a rational number 0 ≤ ϵ < 1. Then the family of log ℚ-Fano 3-folds (normal projective 3-folds X with only ℚ-factorial log terminal (equivalently, klt) singularities s.t. the anti-canonical divisors −K_{X} are ample) with Picard number 1 and whose discrepancies are all > −ϵ, is bounded.

In general, to establish Claim 2.3 in the second case we need the conjecture above for log ℚ-Fano n-folds.

This completes the discussion of termination of log Sarkisov program with klt singularities.

§4. Log Sarkisov Program with WKLT Singularities.

In this section, we establsih the log Sarkisov program with weakly kawamata log terminal singularities in dimension 2, and then discuss the problems one has to face attempting to establish the log Sarkisov program with weakly kawamata log terminal singularities in higher dimension. We also prove that Sarkisov related log minimal 3-folds with klt or wklt singularities are connected by a sequence of log flops.

Definition 4.1 (Local) (cf.Kollár et al[16]). Let (X, B_{X}) be a germ (with respect to Zariski topology) around a point P ∈ X. (X, B_{X}) has only weakly kawamata log terminal singularities if there exists a Zariski open set P ∈ U ⊂ X s.t. there exists a log resolution f : V → U such that all the log discrepancies of the exceptional divisors with center on U are positive and we have an f-anti-ample effective divisor whose support coincides with that of the exceptional locus of f. (Note that B_{X} may have components with coefficient 1.)

The relation between local and global properties of wklt singularities was clarified by the following result of Szabó[32].

Proposition 4.2 (Global) (cf.Szabó[32]). Let (X, B_{X}) be a projective log variety which has locally only weakly kawamata log terminal singularities. Then there exists a log resolution (global) f : Y → X such that all the log discrepancies of the exceptional divisors with center on X are positive and we have an f-anti-ample effective divisor whose support coincides with that of the exceptional locus of f.

Corollary 4.3 (Characterization of a log Mori fiber space (or a log minimal model) with wklt singularities). A log Mori fiber space ϕ : (X, B_{X}) → S (resp. a log minimal model (X, B_{X})) in dimension n (≤ 3) with only ℚ-factorial wklt singularities is an end result of a K + B-MMP starting from (Y, B_{Y}) where Y is a nonsingular projective n-fold and B_{Y} = Σb_{i}B_{i} is an S.N.C. divisor with 0 ≤ b_{i} ≤ 1, and the converse holds, i.e., any end result of fibering type (resp. of minimal model type) of a K + B-MMP starting from (Y, B_{Y}) as above is a log Mori fiber space (resp. a log minimal model) with only ℚ-factorial wklt singularities. (Once we have the log-MMP in dimension n, the same statement holds in dimension n.)

We prove the well-behavior of the Sarkisov relation for log minimal models with wklt singularities as follows, thanks to the fact that the nef log canonical divisors of the Sarkisov related log minimal models are all essentially the same and uniquely characterized as the nef part of the Zariski decomposition of the log canonical divisor of an arbitrary log resolution. However, we fail to prove the well-behavior of the Sarkisov relation for log Mori fiber spaces with wklt singularities in dimension > 2, mainly because the lack of the Zariski decomposition and the failure for the statement (ii) of Proposition 3.5 to hold.

Proposition 4.4 = Proposition 3.5 for log minimal models with wklt singularities. Let

(X_{0}, B_{X_{o}}), (X_{1}, B_{X_{1}}), ⋯, (X_{k}, B_{X_{k}}), ⋯, (X_{l}, B_{X_{l}})

be log minimal models with only ℚ-factorial wklt singularities. Then (i) (ii) and (iii) as in Proposition 3.5 (replacing the assumption of klt singularities with that of wklt singularities and allowing the possibility ϵ = 1) are equivalent.

Proof.

The implications (iii) ⇒ (ii) ⇒ (i) are obvious (rgardless whether they are log minimal models or log Mori fiber spaces). We only have to prove (i) ⇒ (iii).

Take a log pair (W, B_{W}) as in (i) and let

p_{k} : (W, B_{W}) −−→ (X_{k}, B_{X_{k}})

be a birational map which is a K + B-MMP over Speck. We denote by C_{k} the closed set in W so that

p_{k} : (W − C_{k}, B_{W} |_{W −C_{k}}) →^{∼} (X_{k} − I_{k}, B_{X_{k}}|_{X_{k}−I_{k}}),

where I_{k} is the indeterminacy of the birational map p_{k}^{−1}. We can take a blowup σ : W′ → W whose centers are all over ∪_{k}C_{k} such that each X_{k} is dominated by a birational morphism p_{k}′ : W′ → X_{k} and that σ_{∗}^{−1}(B_{W}) ∪ E(σ) is an S.N.C. divisor.

Set

B̂_{W′} := σ_{∗}^{−1}(B_{W}) + Σ_{E_{j}not appearing as a divisor on W} ϵE_{j}.

For each k we have the ramification formulae

K_{W′} + B̂_{W′} = σ^{∗}(K_{W} + B_{W}) + R_{σ}

where R_{σ} is an effective ramification divisor (whose support may not coincide with E(σ)), and

σ^{∗}(K_{W} + B_{W}) = σ^{∗}{p_{k}^{∗}(K_{X_{k}} + B_{X_{k}}) + R_{p_{k}}}

where a p_{k} ∘ σ-exceptional divisor E has a strictly positive coefficient in σ^{∗}R_{p_{k}} iff the center of E on W is contained in C_{k}.

Now an easy application of the Negativity Lemma (cf.Kollár[15],Lemma 4.3) shows that

(p_{0} ∘ σ)^{∗}(K_{X_{0}} + B_{X_{0}}) = ⋯ = (p_{k} ∘ σ)^{∗}(K_{X_{k}} + B_{X_{k}}) = ⋯ = (p_{l} ∘ σ)^{∗}(K_{X_{l}} + B_{X_{l}}) giving the nef part of the Zariski decomposition of K_{W′} + B̂_{W′}, and hence

R_{σ} + σ^{∗}R_{p_{0}} = ⋯ = R_{σ} + σ^{∗}R_{p_{k}} = ⋯ = R_{σ} + σ^{∗}R_{p_{l}}.

Now any k, a p_{k} ∘ σ-exceptional divisor E has the center on W contained in ∪_{i}I_{i} and thus has a strictly positive coefficient in σ^{∗}R_{p_{i}} for some i. Therefore, the above equality implies it has a strictly positive coefficient in R_{σ} + σ^{∗}R_{p_{k}}. Thus

K_{W′} + B̂_{W′} = (p_{k} ∘ σ)^{∗}(K_{X_{k}} + B_{X_{k}}) + R̂_{p_{k}∘σ}

where

supp R̂_{p_{k}∘σ} = supp E(p_{k} ∘ σ).

Therefore, finally by setting

B_{W′} = D_{W} (B_{X_{0}}, B_{X_{1}}, ⋯, B_{X_{k}}, ⋯, B_{X_{l}}) + Σ_{E_{j}not appearing as a divisor on any of X_{k}}ϵE_{j}

we have

K_{W′} + B_{W′} = (p_{k} ∘ σ)^{∗}(K_{X_{k}} + B_{X_{k}}) + R_{p_{k}∘σ}

with

supp R_{p_{k}∘σ} = supp E(p_{k} ∘ σ).

Hence by Lemma 3.6 (i) each (X_{k}, B_{X_{k}}) is an end result of K + B-MMP over X_{k} starting from (W′, B_{W′}). This completes the proof.

Theorem 4.5. Let (X, B_{X}) and (X′, B_{X′}) be log minimal models with only ℚfactorial wklt singularities in dimension 3. Suppose they are Sarkisov related. Then they are connected by a sequence of log flops

(X, B_{X}) −−→ (X_{1}, B_{X_{1}}) −−→ ˙· · (X_{k}, B_{X_{k}}) −−→ ⋯(X′, B_{X′})

where all the (X_{k}, B_{X_{k}}) are Sarkisov related.

proof.

An easy application of the Negativity Lemma shows (cf.Kollár[15],Lemma 4.3) that (X, B_{X}) and (X′, B_{X′}) are isomorphic in codimension 1, since they are Sarkisov related. Let H_{X′} be a very ample divisor on X′ and H_{X} its strict transform on X. We take a log pair (W, B_{W}) dominating both log minimal models by birational morphisms p : (W, B_{W}) → (X, B_{X}) and q : (W, B_{W}) → (X′, B_{X′}) as in (iii) of Proposition 3.5, whose existence is guaranteed by Proposition 4.4. We claim that K_{X} + B_{X} + ηH_{X} is wklt for 0 < η << 1, since

Bs(H_{X}) ⊂ p(R_{p})

where

K_{W} + B_{W} = p^{∗}(K_{X} + B_{X}) + R_{p}.

If K_{X} + B_{X} + ηH_{X} is nef, then by Log Abundance (cf.KeMaMc[12]) it is semiample. Since it is the strict transform of an ample divisor K_{X′} + B_{X′} + ηH_{X′} and since (X, B_{X}) and (X′, B_{X′}) are isomorphic in codimension one and ℚ-factorial, this implies (X, B_{X}) →^{∼} (X′, B_{X′}).

If K_{X} +B_{X} +ηH_{X} is not nef, then there is an K_{X} +B_{X} +ηH_{X}-negative extremal ray, which must be K_{X} +B_{X}-trivial and of flopping type (cf.Kollár[15],Lemma 4.4). We flop this extremal ray to get another log minimal model (X, B_{X}) −−→ (X_{1}, B_{X_{1}}) with only ℚ-factorial wklt singularities. By construction it is easy to see that (X, B_{X}), (X_{1}, B_{X_{1}}) and (X′, B_{X′}) are all Sarkisov related and that K_{X_{1}} + B_{X_{1}} + ηH_{X_{1}} is wklt. We proceed inductively and this procedure has to come to an end, since any sequence of log flops has to terminate (cf.Shokurov[31]Kollár et al[16]). Thus we obtain the desired connecting sequence of log flops between (X, B_{X}) and (X′, B_{X′}).

We go back to the discussion of the log Sarkisov program with wklt singularities. In the following we prove the well-behavior of the Sarkisov relation for log Mori

fiber spaces with wklt singularities in dimension 2.

Lemma 4.6 = Proposition 3.5 for log Mori fiber spaces with wklt singularities in dimension 2. Let

(X_{0}, B_{X_{o}}), (X_{1}, B_{X_{1}}), ⋯, (X_{k}, B_{X_{k}}), ⋯, (X_{l}, B_{X_{l}})

be log Mori fiber spaces with only ℚ-factorial wklt singularities in dimension 2. Then (i) (ii) and (iii) as in Proposition 3.5 (replacing the assumption of klt singularities with that of wklt singularities and allowing the possibility ϵ = 1) are equivalent.

Proof.

Again we only have to show the implication (i) ⇒ (iii).

Take a log pair (W, B_{W}) as in (i) and let

p_{k} : (W, B_{W}) −−→ (X_{k}, B_{X_{k}})

be a birational map which is a K + B-MMP over Speck. First observe that in dimension 2 all the p_{k} are birational morphisms and that

∪_{k}p_{k}^{−1}_{∗}(B_{X_{k}}) ⊂ B_{W}

is an S.N.C. divisor. We take a blowup σ : W′ → W with centers over

∪_{E_{l}not appearing as a divisor on any of X_{k}}E_{l}

until

∪_{E_{m}not appearing as a divisor on any of X_{k}}E_{m} ∪ ∪_{k}p_{k}^{−1}_{∗}(B_{X_{k}})

is an S.N.C. divisor. Then by setting

B_{W′} = D_{W} (B_{X_{0}}, ⋯, B_{X_{k}}, ⋯, B_{X_{l}}) + Σ_{E_{j}not appearing as a divisor on any of X_{k}}E_{j},

we have

K_{W′} + B_{W′} = σ^{∗}(K_{W} + B_{W}) + R_{σ}

where R_{σ} is an effective ramification divisor (whose support may not coincide with the exceptional locus E(σ)).

On the other hand, since for each k

K_{W} + B_{W} = p_{k}^{∗}(K_{X_{k}} + B_{X_{k}}) + R_{p_{k}}

where the effective ramification divisor R_{p_{k}} has the support which coincides with that of the exceptional locus E(p_{k}), and since the blowup has all the centers over

∪_{E_{l}not appearing as a divisor on any of X_{k}}E_{l} ⊂ ∩_{i}E(p_{i}),

we conclude

K_{W′} + B_{W′} = (p_{k} ∘ σ)^{∗}(K_{X_{k}} + B_{X_{k}}) + R_{p_{k}∘σ}

with

supp R_{p_{k}∘σ} = supp E(p_{k} ∘ σ).

Thus by Lemma 3.6 (i) each (X_{k}, B_{X_{k}}) is an end result of K + B-MMP over X_{k} starting from (W′, B_{W′}). This completes the proof.

In dimension 2, any MMP (log or genuine) is a succession of contractions of divisors without any flip and thus the resulting Mori fiber space is dominated by the starting variety through a birational morphism. Also any K + B-MMP over Spec k is a process of K + B-MMP over any variety T which is dominated by relevant log pairs. These easy observations unique to dimension 2 make the flowchart for the log Sarkisov with wklt singularities rather straightforward in dimension 2, compared to higher dimensional case, where the Sarkisov relation seems more subtle.

The log Sarkisov degree (μ_{k}, λ_{ϵk}, e_{ϵk}) is defined in the following way.

The quasi-effective threshold μ_{k} is as before defined to be the positive rational number s.t.

μ_{k}(K_{X_{k}} + B_{X_{k}}) + H_{X_{k}} ≡ 0 over S_{k}.

λ_{ϵk} is defined in the exactly same way as in the case with klt singularities setting ϵ = 1.

We pay extra attention to how we define e_{ϵk}. If we try to define it in the same way as in the case with klt singularities setting ϵ = 1, then it would not be well defined, since we may have infinitely many crepant divisors. This is one of the difficulties one has to face once we hit the critical value ϵ = 1 creating the neccessity to deal with the log canonical locus.

In the flowchart below we show all the divisorial blow ups and intermediate log Mori fiber spaces are dominated by (W, B_{W}) that we fix from the beginning as above satisfying the conditions in (iii) in Proposition 3.5 .

We define e_{ϵk} of the intermediate log Mori fiber space that appear in the due course of the flowchart to be the number of K +B + \frac{1}{λ_{ϵk}} H-crepant divisors ON W.

Flowchart for Log Sarkisov Program in dimenion 2

with WKLT Singularities and its Termination

Case : λ_{ϵk} ≤ μ_{k}

The flowchart for the log Sarkisov program with WKLT singularities in this case goes parallel to the one for the genuine Sarkisov program. After untwisting the birational map by a link of type (III) or (IV), the quasi-effective threshold strictly decreases.

Moreover, since ϕ_{k+1} : (X_{k+1}, B_{X_{k+1}}) → S_{k+1} is obtained as an end result of K + B + \frac{1}{μ_{k}} H-MMP (over T), it follows immediately that (X_{k+1}, B_{X_{k+1}}) is an end result of K + B-MMP starting from (W, B_{W}) (over X_{k+1}) and thus dominated by (W, B_{W}) through a birational morphism.

The claim that there is no infinite number of untwisting (successiove or unsuccessive) by the links unedr the case λ_{ϵk} ≤ μ_{k} can be shown similarly, proving the discreteness of the quasi-effective thresholds noting that a general fiber of ϕ_{k} is ℙ^{1}.

Case : λ_{ϵk} > μ_{k}

We take a maximal divisorial blow up constructed starting from (W, B_{W}), which dominates (X_{k}, B_{X_{k}}) by inductive assumption.

Then just as in the genuine Sarkisov program after untwisting the birational map by a link of type (II) or (I), the quasi-effective threshold does not increase

μ_{k+1} ≤ μ_{k}

with equality holding only if

either dim S_{k+1} > dim S_{k}
or dim S_{k+1} = dim S_{k} and ψ_{k} is square.

Also it follows similarly that

λ_{ϵ,k+1} ≤ λ_{ϵk}.

Note that ϕ_{k+1} : (X_{k+1}, B_{X_{k+1}}) → S_{k+1} is an end result of K + B + \frac{1}{λ_{ϵk}} H-MMP (over S_{k}) starting from the maximal divisorial blowup (Z_{k}, B_{Z_{k}}), which in turn is an end result of K + B + \frac{1}{λ_{ϵk}} H-MMP possibly followed by K + B-MMP (over X_{k}) starting from (W, B_{W}). Therefore, it is easy to see that (X_{k+1}, B_{X_{k+1}}) is an end result of K + B-MMP (over X_{k+1}) starting from (W, B_{W}). Thus e_{ϵ,k+1} is well -defined and in the above inequality

if λ_{ϵ,k+1} = λ_{ϵk} then e_{ϵ,k+1} ≤ e_{ϵk} − 1 < e_{ϵk}.

This immediately proves the claim that there is no infinite (successive) sequence of untwisting by the links under the case λ_{ϵk} > μ_{k} with stationary λ_{ϵk}.

Now we take a closer look at the proof of the claim that there is no infinite (successive) sequence of untwisting by the links under the case λ_{ϵk} > μ_{k} with stationary quasi-effective threshold μ_{k}.

The proof goes parallel for Steps 1 and 2 (cf. Claim 2.2) replacing K with K +B. Step 3 becomes meaningless and irrelevant in the case with wklt singularities and we disregard it, i.e., we don’t use Step 3 in our argument below. Instead we conclude the argument as follows. First we remark that the valuations of k(X) corresponding to the unique exceptional divisors E_{k} of the maximal divisorial blowups are all distinct. Moreover, all the E_{k} are divisors on W. On the other hand,

a(E_{k}, X_{1}, B_{X_{1}} + αH_{X_{1}}) ≤ a(E_{k}, X_{k}, B_{X_{k}} + αH_{X_{k}}) < 0,

but there are only finitely many divisors on W with negative discrepancies w.r.t. B_{X_{1}} + αH_{X_{1}}, a contradiction!

We finish the proof of termination by showing the claim that there is no infinite (successive) sequence of untwisting by the links under the case λ_{ϵk} > μ_{k} with nonstationary quasi-effective threshold.

For the case where dim S_{k_{0}} ≥ 1 for some k_{0} (and thus for ∀k ≥ k_{0}), we show the discreteness of the quasi-effective thresholds again noting that a general fiber of ϕ_{k}(k ≥ k_{0}) is ℙ^{1}.

For the case ∀k, dim S_{k} = 0, we note that the X_{k} are all log Del Pezzo surfaces (normal projective surfaces with only quotient singularities having ample anticanonical divisors) which are dominated by one fixed nonsingular projective surface W. Therefore, it is easy to see that the X_{k} belong to a bounded family, from which fact the discreteness of the quasi-effective thresholds follows just as before.

This completes the discussion of the flowchart and its termination for the log Sarkisov program with wklt singularities in dimension 2.

Fianlly we discuss briefly the problems we face when we try to establish the log Sarkisov program with wklt singularities in higher dimension.

Problem 1.

Does the Sarkisov relation behave well with wklt singularities, i.e., do we have the equivalence of (i) (ii) and (iii) in proposition 3.5 replacing klt singularities with wklt singularities and allowing ϵ = 1?

Problem 2.

If the answer to Problem 1 is affirmative, then we can construct a maximal divisorial blowup of (X_{k}, B_{X_{k}}) with respect to H_{X_{k}} in the case λ_{ϵk} ≤ μ_{k} from a good log pair (W, B_{W}) as in (iii) of Proposition 3.5. After K + B + \frac{1}{λ_{ϵk}} H-MMP we reach (X_{k+1}, B_{X_{k+1}}). In the case λ_{ϵk} ≤ μ_{k}, after K + B + \frac{1}{μ_{k}} H-MMP we reach (X_{k+1}, B_{X_{k+1}}).

Show in both cases that the (X_{m}, B_{X_{m}}) m = 0, 1, ⋯, k, k + 1 are all Sarkisov related establishing the inductive procedure.

We note that the Nöther-Fano criterion remains valid as long as we know (X, B_{X}), (X′, B_{X′}) and (X_{k}, B_{X_{k}}) are Sarkisov related. (This is not a trivial remark as at one point of the proof (See Corti[4],Theorem 4.2.) the positivity of some coefficient in the ramification divisor does not follow without the assumption of being Sarkisov related in the case of wklt singularities.)

Problem 3.

Show the discreteness of the quasi-effective thresholds under the case λ_{ϵk} ≤ μ_{k}, which follows from the (conjectural) boundedness of the fibers of the ϕ_{k}.

We remark that this is not a straight consequence of S_{d}(Global) (d ≤ n − 1 in dimension n), since we do not know that the number of components in the boundary or the coefficient \frac{1}{μ_{k}} to be bounded.

Problem 4.

Show that there is no infinite (successive) sequence of untwisting by the links under the case λ_{ϵk} > μ_{k} in the following manner:

i) Show that λ_{ϵk} cannot be stationary, by adopting an appropriate definition of e_{ϵk} as demonstrated in the case of dimension 2. This should be relatively easy.

ii) Show that μ_{k} cannot be stationary. Steps 1 and 2 of Calim 2.2 go without change, while Step 3 is irrelevant. We should conclude the argument by looking at S_{d}(Global) d ≤ n − 1 on the exceptional divisors of the maximal divisorial blowups that appear with coefficient 1. (This line of argument was suggested to us by A. Corti. In fact we could argue this way in the proof of termination in dimension n = 2, though it becomes substantially lengthier than the one we give.)

Problem 5.

Finally show the discreteness of the quasi-effective thresholds under the case λ_{ϵk} > μ_{k}, which follows again from the boundedness of the fibers of the ϕ_{k}. This seems to be the most difficult part.

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Department of Mathematics, Brandeis University, Boston, Massachusetts 02254-