THE SARKISOV PROGRAM

CHRISTOPHER D. HACON AND JAMES M^{C}KERNAN

Abstract. Any two birational Mori fibre spaces are connected by a sequence of Sarkisov links.

Contents

1. Introduction  1
2. Notation and conventions  7
3. The combinatorics of ample models  8
4. Proof of (1.3)  14
References  16

1. Introduction

We prove that any two birational Mori fibre spaces are connected by a sequence of elementary transformations, known as Sarkisov links:

Theorem 1.1. Suppose that ϕ: X ⟶ S and ψ: Y ⟶ T are two Mori fibre spaces with ℚ-factorial terminal singularities.

Then X and Y are birational if and only if they are related by a sequence of Sarkisov links.

Recall the following:

Conjecture 1.2. Let (Z, Φ) be a kawamata log terminal pair. Then we may run f: Z −−→ X the (K_{Z} + Φ)-MMP such that either (1) (X, Δ) is a log terminal model, that is K_{X} + Δ is nef, or

(2) there is a Mori fibre space ϕ: X ⟶ S, that is ρ(X/S) = 1 and −(K_{X} + Δ) is ϕ-ample, where Δ = f_{∗}Φ.

Date: June 26, 2009.

The first author was partially supported by NSF grant no: 0757897 and by the Clay Mathematics Institute. The second author was partially supported by NSA grant no: H98230-06-1-0059, NSF grant no: 0701101 and an Eisenbud fellowship.

Some of this work was done whilst both authors were visiting MSRI and both authors would like to thank MSRI for its hospitality.

We will refer to the log terminal model X and the Mori fibre space ϕ as the output of the (K_{Z} + Φ)-MMP. If h: Z −−→ X is any sequence of divisorial contractions and flips for the (K_{Z} + Φ)-MMP then we say that h is the result of running the (K_{Z} + Φ)-MMP. In other words if h is the result of running the (K_{Z} + Φ)-MMP then X does not have to be either a log terminal model or a Mori fibre space.

By [1] the only unknown case of (1.2) is when K_{Z} + Φ is pseudoeffective but neither Φ nor K_{Z} + Φ is big. Unfortunately the output is not unique in either case. We will call two Mori fibre spaces ϕ: X ⟶ S and ψ: Y ⟶ T Sarkisov related if X and Y are outcomes of running the (K_{Z} + Φ)-MMP, for the same ℚ-factorial kawamata log terminal pair (Z, Φ). This defines a category, which we call the Sarkisov category, whose objects are Mori fibre spaces and whose morphisms are the induced birational maps X −−→ Y between two Sarkisov related Mori fibre spaces. Our goal is to show that every morphism in this category is a product of Sarkisov links. In particular a Sarkisov link should connect two Sarkisov related Mori fibre spaces.

Theorem 1.3. If ϕ: X ⟶ S and ψ: Y ⟶ T are two Sarkisov related Mori fibres spaces then the induced birational map σ: X −−→ Y is a composition of Sarkisov links.

Note that if X and Y are birational and have ℚ-factorial terminal singularities, then ϕ and ψ are automatically the outcome of running the K_{Z}-MMP for some projective variety Z, so that (1.1) is an easy consequence of (1.3).

It is proved in [1] that the number of log terminal models is finite if either Φ or K_{Z} + Φ is big, and it is conjectured that in general the number of log terminal models is finite up to birational automorphisms. Moreover Kawamata, see [5], has proved:

Theorem 1.4. Suppose that σ: X −−→ Y is a birational map between two ℚ-factorial varieties which is an isomorphism in codimension one.

If K_{X} + Δ and K_{Y} + Γ are kawamata log terminal and nef and Γ is the strict transform of Δ then σ is the composition of (K_{X} + Δ)-flops.

Note that if the pairs (X, Δ) and (Y, Γ) both have ℚ-factorial terminal singularities then the birational map σ is automatically an isomorphism in codimension one.

We recall the definition of a Sarkisov link. Suppose that ϕ: X ⟶ S and ψ: Y ⟶ T are two Mori fibre spaces. A Sarkisov link σ: X −−→ Y between ϕ and ψ is one of four types:

[figure: I; X′; Y; X; T; ψ; S; ϕ; II; X′; Y′; X; Y; S; ϕ; =; T; ψ; III; X; Y′; S; ϕ; Y; T; ψ; IV; X; Y; S; ϕ; T; ψ; R.]

There is a divisor Ξ on the space L on the top left (be it L = X or L = X′) such that K_{L} + Ξ is kawamata log terminal and numerically trivial over the base (be it S, T, or R). Every arrow which is not horizontal is an extremal contraction. If the target is X or Y it is a divisorial contraction. The horizontal dotted arrows are compositions of (K_{L} + Ξ)-flops. Links of type IV break into two types, IV_{m} and IV_{s}. For a link of type IV_{m} both s and t are Mori fibre spaces. For a link of type IV_{s} both s and t are small birational contractions. In this case R is not ℚ-factorial; for every other type of link all varieties are ℚ-factorial. Note that there is an induced birational map σ: X −−→ Y but not necessarily a rational map between S and T.

The Sarkisov program has its origin in the birational classification of ruled surfaces. A link of type I corresponds to the diagram

𝔽_{1} = 𝔽_{1}

[figure: ℙ^{2}; ℙ^{1}; ψ]
[figure]
pt.^{^{ϕ}}

Note that there are no flops for surfaces so the top horizontal map is always the identity. The top vertical arrow on the left is the blow up of a point in ℙ^{2} and ψ is the natural map given by the pencil of lines.

A link of type III is the same diagram, reflected in a vertical line,

𝔽_{1} = 𝔽_{1}

[figure: ℙ^{1}; ϕ; ℙ^{2}]

[figure]

pt.^{^{ψ}}

A link of type II corresponds to the classical elementary transformation between ruled surfaces,

X′ = Y′

[figure: X; Y]

[figure: S; ϕ; =; T.; ψ]

The birational map X′ ⟶ X blows up a point in one fibre and the birational map Y′ ⟶ Y blows down the old fibre. Finally a link of type IV corresponds to switching between the two ways to project ℙ^{1} × ℙ^{1} down to ℙ^{1},

ℙ^{1} × ℙ^{1} = ℙ^{1} × ℙ^{1}

[figure: ℙ^{1}; ϕ; ℙ^{1}; ψ]
[figure]

pt.

It is a fun exercise to factor the classical Cremona transformation σ: ℙ^{2} −−→ ℙ^{2}, [X : Y : Z] ⟶ [X^{−1} : Y^{−1} : Z^{−1}] into a product of Sarkisov links. Indeed one can use the Sarkisov program to give a very clean proof that the birational automorphism of ℙ^{2} is generated by this birational map σ and PGL(3). More generally the Sarkisov program can sometimes be used to calculate the birational automorphism group of Mori fibre spaces, especially Fano varieties. With this said, note that the following problem seems quite hard:

Question 1.5. What are generators of the birational automorphism group of ℙ^{3}?

Note that a link of type IV_{s} only occurs in dimension four or more. For an example of a link of type IV_{s} simply take S −−→ T to be a flop between threefolds, let S ⟶ R be the base of the flop and let X = S × ℙ^{1} and Y = T × ℙ^{1} with the obvious maps down to S and T. It is conceivable that one can factor a link of type IV_{s} into links of type I and III. However given any positive integer k it is easy to write down examples of links of type IV which cannot be factored into fewer than k links of type I, II or III.

Let us now turn to a description of the proof of (1.3). The proof is based on the original ideas of the Sarkisov program (as explained by Corti and Reid [3]; see also [2]). We are given a birational map σ: X −−→ Y and the objective is to factor σ into a product of Sarkisov links. In the original proof one keeps track of some subtle invariants and the idea is to prove:

• the first Sarkisov link σ_{1} exists,

• if one chooses σ_{1} appropriately then the invariants improve, and • the invariants cannot increase infinitely often.

Sarkisov links arise naturally if one plays the 2-ray game. If the relative Picard number is two then there are only two rays to contract and this gives a natural way to order the steps of the minimal model program. One interesting feature of the original proof is that it is a little tricky to prove the existence of the first Sarkisov link, even if we assume existence and termination of flips. In the original proof one picks a linear system on Y and pulls it back to X. There are then three invariants to keep track of; the singularities of the linear system on X, as measured by the canonical threshold, the number of divisors of log discrepancy one (after rescaling to the canonical threshold) and the pseudo-effective threshold. Even for threefolds it is very hard to establish that these invariants satisfy the ascending chain condition.

Our approach is quite different. We don’t consider any linear systems nor do we try to keep track of any invariants. Instead we use one of the main results of [1], namely finiteness of ample models for kawamata log terminal pairs (Z, A + B). Here A is a fixed ample ℚ-divisor and B ranges over a finite dimensional affine space of Weil divisors. The closure of the set of divisors B with the same ample model is a disjoint union of finitely many polytopes and the union of all of these polytopes corresponds to divisors in the effective cone.

Now if the space of Weil divisors spans the Néron-Severi group then one can read off which ample model admits a contraction to another ample model from the combinatorics of the polytopes, (3.3). Further this property is preserved on taking a general two dimensional slice, (3.4). Sarkisov links then correspond to points on the boundary of the effective cone which are contained in more than two polytopes, (3.7). To obtain the required factorisation it suffices to simply traverse the boundary. In other words instead of considering the closed cone of curves and playing the 2-ray game we look at the dual picture of Weil divisors and we work inside a carefully chosen two dimensional affine space. The details of the correct choice of underlying affine space are contained in §4.

To illustrate some of these ideas, let us consider an easy case. Let S be the blow up of ℙ^{2} at two points. Then S is a toric surface and there are five invariant divisors. The two exceptional divisors, E_{1} and E_{2}, the strict transform L of the line which meets E_{1} and E_{2}, and finally the strict transform L_{1} and L_{2} of two lines, one of which meets E_{1} and one of which meets E_{2}. Then the cone of effective divisors is spanned by the invariant divisors and according to [4] the polytopes we are looking for are obtained by considering the chamber decomposition given by the invariant divisors. Since L_{1} = L + E_{1} and L_{2} = L + E_{2} the cone of effective divisors is spanned by L, E_{1} and E_{2}. Since −K_{S} is ample, we can pick an ample ℚ-divisor A such that K_{S} + A ∼_{ℚ} 0 and K_{S} + A + E_{1} + E_{2} + L is divisorially log terminal. Let V be the real vector space of Weil divisors spanned by E_{1}, E_{2} and L. In this case projecting L_{A}(V) from the origin we get

We have labelled each polytope by the corresponding model. Imagine going around the boundary clockwise, starting just before the point corresponding to L. The point L corresponds to a Sarkisov link of type IV_{m}, the point L+E_{2} a link of type II, the point E_{2} a link of type III, the point E_{1} a link of type I and the point L + E_{1} another link of type II.

2. Notation and conventions

We work over the field of complex numbers ℂ. An ℝ-Cartier divisor D on a variety X is nef if D · C ≥ 0 for any curve C ⊂ X. We say that two ℝ-divisors D_{1}, D_{2} are ℝ-linearly equivalent (D_{1} ∼_{ℝ} D_{2}) if D_{1} − D_{2} = ∑ r_{i}(f_{i}) where r_{i} ∈ ℝ and f_{i} are rational functions on X. We say that an ℝ-Weil divisor D is big if we may find an ample ℝdivisor A and an ℝ-divisor B ≥ 0, such that D ∼_{ℝ} A + B. A divisor D is pseudo-effective, if for any ample divisor A and any rational number ϵ > 0, the divisor D + ϵA is big. If A is a ℚ-divisor, we say that A is a general ample ℚ-divisor if A is ample and there is a sufficiently divisible integer m > 0 such that mA is very ample and mA ∈ |mA| is very general.

A log pair (X, Δ) is a normal variety X and an ℝ-Weil divisor Δ ≥ 0 such that K_{X} + Δ is ℝ-Cartier. We say that a log pair (X, Δ) is log smooth, if X is smooth and the support of Δ is a divisor with global normal crossings. A projective birational morphism g: Y ⟶ X is a log resolution of the pair (X, Δ) if Y is smooth and the strict transform Γ of Δ union the exceptional set E of g is a divisor with normal crossings support. If we write

K_{Y} + Γ + E = g^{∗}(K_{X} + Δ) + ∑ a_{i}E_{i},

where E = ∑ E_{i} is the sum of the exceptional divisors then the log discrepancy a(E_{i}, X, Δ) of E_{i} is a_{i}. By convention the log discrepancy of any divisor B which is not exceptional is 1 − b, where b is the coefficient of B in Δ. The log discrepancy a is the infinimum of the log discrepancy of any divisor.

A pair (X, Δ) is kawamata log terminal if a > 0. We say that the pair (X, Δ) is log canonical if a ≥ 0. We say that the pair (X, Δ) is terminal if the log discrepancy of any exceptional divisor is greater than one.

We say that a rational map ϕ: X −−→ Y is a rational contraction if there is a resolution p: W ⟶ X and q: W ⟶ Y of ϕ such that p and q are contraction morphisms and p is birational. We say that ϕ is a birational contraction if q is in addition birational and every p-exceptional divisor is q-exceptional. If in addition ϕ^{−1} is also a birational contraction, we say that ϕ is a small birational map. We refer the reader to [1] for the definitions of negative and non-positive rational contractions and of log terminal models.

If C is a closed convex in a finite dimensional real vector space then C^{∗} denotes the dual convex set in the dual real vector space.

3. The combinatorics of ample models

We fix some notation. Z is a smooth projective variety, V is a finite dimensional affine subspace of the real vector space WDiv_{ℝ}(Z) of Weil divisors on Z, which is defined over the rationals, and A ≥ 0 is an ample ℚ-divisor on Z. We suppose that there is an element Θ_{0} of L_{A}(V) such that K_{Z} + Θ_{0} is big and kawamata log terminal.

We recall some definitions and notation from [1]:

Definition 3.1. Let D be an ℝ-divisor on Z.

We say that f: Z −−→ X is the ample model of D, if f is a rational contraction, X is a normal projective variety and there is an ample divisor H on X such that if p: W ⟶ Z and q: W ⟶ X resolve f and we write p^{∗}D ∼_{ℝ} q^{∗}H + E, then E ≥ 0 and for every B ∼_{ℝ} p^{∗}D if B ≥ 0 then B ≥ E.

Note that if f is birational then q_{∗}E = 0.

Definition 3.2. Let

V_{A} = { Θ | Θ = A + B, B ∈ V },
L_{A}(V) = { Θ = A + B ∈ V_{A} | K_{Z} + Θ is log canonical and B ≥ 0 },
E_{A}(V) = { Θ ∈ L_{A}(V) | K_{Z} + Θ is pseudo-effective }.

Given a rational contraction f: Z −−→ X, define

A_{A,f}(V) = { Θ ∈ E_{A}(V) | f is the ample model of (Z, Θ) }.

In addition, let C_{A,f}(V) denote the closure of A_{A,f}(V).

Theorem 3.3. There are finitely many 1 ≤ i ≤ m rational contractions f_{i}: Z −−→ X_{i} with the following properties:

(1) { A_{i} = A_{A,f_{i}} | 1 ≤ i ≤ m } is a partition of E_{A}(V). A_{i} is a finite union of interiors of rational polytopes. If f_{i} is birational then C_{i} = C_{A,f_{i}} is a rational polytope.

(2) If 1 ≤ i ≤ m and 1 ≤ j ≤ m are two indices such that A_{j} ∩C_{i} ≠ ∅ then there is a contraction morphism f_{i,j}: X_{i} ⟶ X_{j} and a factorisation f_{j} = f_{i,j} ∘ f_{i}.

Now suppose in addition that V spans the Néron-Severi group of Z.

(3) Pick 1 ≤ i ≤ m such that a connected component C of C_{i} inter-

sects the interior of L_{A}(V). The following are equivalent • C spans V.

• If Θ ∈ A_{i} ∩ C then f_{i} is a log terminal model of K_{Z} + Θ.

• f_{i} is birational and X_{i} is ℚ-factorial.

(4) If 1 ≤ i ≤ m and 1 ≤ j ≤ m are two indices such that C_{i} spans V and Θ is a general point of A_{j} ∩ C_{i} which is also a point of the interior of L_{A}(V) then C_{i} and \overline{NE}(X_{i}/X_{j})^{∗} × ℝ^{k} are locally isomorphic in a neighbourhood of Θ, for some k ≥ 0. Further the relative Picard number of f_{i,j}: X_{i} ⟶ X_{j} is equal to the difference in the dimensions of C_{i} and C_{j} ∩ C_{i}.

Proof. (1) is proved in [1].

Pick Θ ∈ A_{j} ∩ C_{i} and Θ′ ∈ A_{i} so that

Θ_{t} = Θ + t(Θ′ − Θ) ∈ A_{i} if t ∈ (0, 1].

By finiteness of log terminal models, cf. [1], we may find a positive constant δ > 0 and a birational contraction f: Z −−→ X which is a log terminal model of K_{Z} + Θ_{t} for t ∈ (0, δ]. Replacing Θ′ = Θ_{1} by Θ_{δ} we may assume that δ = 1. If we set

Δ_{t} = f_{∗}Θ_{t},

then K_{X} +Δ_{t} is kawamata log terminal and nef, and f is K_{Z} +Θ_{t} nonpositive for t ∈ [0, 1]. As Δ_{t} is big the base point free theorem implies that K_{X} + Δ_{t} is semiample and so there is an induced contraction morphism g_{i}: X ⟶ X_{i} together with ample divisors H_{1/2} and H_{1} such that

K_{X} + Δ_{1/2} = g_{i}^{∗}H_{1/2} and K_{X} + Δ_{1} = g_{i}^{∗}H_{1}.

If we set

H_{t} = (2t − 1)H_{1} + 2(1 − t)H_{1/2},

then

K_{X} + Δ_{t} = (2t − 1)(K_{X} + Δ_{1}) + 2(1 − t)(K_{X} + Δ_{1/2})
= (2t − 1)g_{i}^{∗}H_{1} + 2(1 − t)g_{i}^{∗}H_{1/2}
= g_{i}^{∗}H_{t},

for all t ∈ [0, 1]. As K_{X} + Δ_{0} is semiample, it follows that H_{0} is semiample and the associated contraction f_{i,j}: X_{i} ⟶ X_{j} is the required morphism. This is (2).

Now suppose that V spans the Néron-Severi group of Z. Suppose that C spans V. Pick Θ in the interior of C ∩ A_{i}. Let f: Z −−→ X be a log terminal model of K_{Z} + Θ. It is proved in [1] that f = f_{j} for some index 1 ≤ j ≤ m and that Θ ∈ C_{j}. But then A_{i} ∩A_{j} ≠ ∅ so that i = j.

If f_{i} is a log terminal model of K_{Z} + Θ then f_{i} is birational and X_{i} is ℚ-factorial.

Finally suppose that f_{i} is birational and X_{i} is ℚ-factorial. Fix Θ ∈ A_{i}. Pick any divisor B ∈ V such that −B is ample K_{X_{i}} + f_{i∗}(Θ + B) is ample and Θ + B ∈ L_{A}(V). Then f_{i} is (K_{Z} + Θ + B)-negative and so Θ + B ∈ A_{i}. But then C_{i} spans V. This is (3).

We now prove (4). Let f = f_{i} and X = X_{i}. As C_{i} spans V, (3) implies that f is birational and X is ℚ-factorial so that f is a ℚ-factorial weak log canonical model of K_{Z} + Θ. Suppose that E_{1}, E_{2}, …, E_{k} are the divisors contracted by f. Pick B_{i} ∈ V numerically equivalent to E_{i}. If we let E_{0} = ∑ E_{i} and B_{0} = ∑ B_{i} then E_{0} and B_{0} are numerically equivalent. As Θ belongs to the interior of L_{A}(V) we may find δ > 0 such that K_{Z} + Θ + δE_{0} and K_{Z} + Θ + δB_{0} are both kawamata log terminal. Then f is (K_{Z} +Θ+δE_{0})-negative and so f is a log terminal model of K_{Z} + Θ + δE_{0} and f_{j} is the ample model of K_{Z} + Θ + δE_{0}. But then f is also a log terminal model of K_{Z} + Θ + δB_{0} and f_{j} is also the ample model of K_{Z} + Θ + δB_{0}. In particular Θ + δB_{0} ∈ A_{j} ∩ C_{i}. As we are supposing that Θ is general in A_{j} ∩ C_{i}, in fact f must be a log terminal model of K_{Z} + Θ. In particular f is (K_{Z} + Θ)-negative.

Pick ϵ > 0 such that if Ξ ∈ V and ∥Ξ − Θ∥ < ϵ then Ξ belongs to the interior of L_{A}(V) and f is (K_{Z} + Ξ)-negative. Then the condition that Ξ ∈ C_{i} is simply the condition that K_{X} + Δ = f_{∗}(K_{Z} + Ξ) is nef. Let W be the affine suspace of WDiv_{ℝ}(X) given by pushing forward the elements of V and let

N = { Δ ∈ W | K_{X} + Δ is nef }.

Given (a_{1}, a_{2}, …, a_{k}) ∈ ℝ^{k} let B = ∑ a_{i}B_{i} and E = ∑ a_{i}E_{i}. If ∥B∥ < ϵ then, as Ξ + B is numerically equivalent to Ξ + E, K_{X} + Δ ∈ N if and only if K_{X} + Δ + f_{∗}B ∈ N. In particular C_{i} is locally isomorphic to N × ℝ^{k}.

But since f_{j} is the ample model of K_{Z} + Θ, in fact we can choose ϵ sufficiently small so that K_{X} + Δ is nef if and only if K_{X} + Δ is nef over X_{j}, see §3 of [1]. There is a surjective affine linear map from W to the space of Weil divisors on X modulo numerical equivalence over X_{j} and this induces an isomorphism

N ≃ \overline{NE}(X/X_{j})^{∗} × ℝ^{l},

in a neighbourhood of f_{∗}Θ.

Note that K_{X} +f_{∗}Θ is numerically trivial over X_{j}. As f_{∗}Θ is big and K_{X} + f_{∗}Θ is kawamata log terminal we may find an ample ℚ-divisor A′ and a divisor B′ ≥ 0 such that

K_{X} + A′ + B′ ∼_{ℝ} K_{X} + f_{∗}Θ,

is kawamata log terminal. But then

−(K_{X} + B′) ∼_{ℝ} −(K_{X} + Δ′) + A′,

is ample over X_{j}. Hence f_{ij}: X ⟶ X_{j} is a Fano fibration and so by the cone theorem

ρ(X_{i}/X_{j}) = dim N .

This is (4). □

Corollary 3.4. If V spans the Néron-Severi group of Z then there is a Zariski dense open subset U of the Grassmannian G(α, V) of real affine subspaces of dimension α such that if [W] ∈ U and it is defined over the rationals then W satisfies (1-4) of (3.3).

Proof. Let U ⊂ G(α, V) be the set of real affine subspaces W of V of dimension α, which contain no face of any C_{i} or L_{A}(V). In particular the interior of L_{A}(W) is contained in the interior of L_{A}(V). (3.3) implies that (1-2) always hold for W and (1-4) hold for V and so (3) and (4) clearly hold for W ∈ U. □

From now on in this section we assume that V has dimension two and satisfies (1-4) of (3.3).

Lemma 3.5. Let f: Z −−→ X and g: Z −−→ Y be two rational contractions such that C_{A,f} is two dimensional and O = C_{A,f} ∩ C_{A,g} is one dimensional. Assume that ρ(X) ≥ ρ(Y) and that O is not contained in the boundary of L_{A}(V). Let Θ be an interior point of O and let Δ = f_{∗}Θ.

Then there is a rational contraction π: X −−→ Y which factors g = π ∘ f and either

(1) ρ(X) = ρ(Y) + 1 and π is a (K_{X} + Δ)-trivial morphism, in which case, either

(a) π is birational and O is not contained in the boundary of

E_{A}(V), in which case, either (i) π is a divisorial contraction and O ≠ C_{A,g}, or (ii) π is a small contraction and O = C_{A,g}, or (b) π is a Mori fibre space and O = C_{A,g} is contained in the

boundary of E_{A}(V), or

(2) ρ(X) = ρ(Y), in which case, π is a (K_{X} +Δ)-flop and O ≠ C_{A,g} is not contained in the boundary of E_{A}(V).

Proof. By assumption f is birational and X is ℚ-factorial. Let h: Z −−→ W be the ample model corresponding to K_{Z} +Θ. Since Θ is not a point of the boundary of L_{A}(V) if Θ belongs to the boundary of E_{A}(V) then K_{Z} + Θ is not big and so h is not birational. As O is a subset of both C_{A,f} and C_{A,g} there are morphisms p: X ⟶ W and q: Y ⟶ W of relative Picard number at most one. There are therefore only two possibilities:

(1) ρ(X) = ρ(Y) + 1, or

(2) ρ(X) = ρ(Y).

Suppose we are in case (1). Then q is the identity and π = p: X ⟶ Y is a contraction morphism such that g = π ∘ f. Suppose that π is birational. Then h is birational and O is not contained in the boundary of E_{A}(V). If π is divisorial then Y is ℚ-factorial and so O ≠ C_{A,g}. If π is a small contraction then Y is not ℚ-factorial and so C_{A,g} = O is one dimensional. If π is a Mori fibre space then O is contained in the boundary of E_{A}(V) and O = C_{A,g}.

Now suppose we are in case (2). By what we have already proved ρ(X/W) = ρ(Y/W) = 1. p and q are not divisorial contractions as O is one dimensional. p and q are not Mori fibre spaces as O cannot be contained in the boundary of E_{A}(V). Hence p and q are small and the rest is clear. □

Lemma 3.6. Let f: W −−→ X be a birational contraction between projective ℚ-factorial varieties. Suppose that (W, Θ) and (W, Φ) are both kawamata log terminal.

If f is the ample model of K_{W} + Θ and Θ − Φ is ample then f is the result of running the (K_{W} + Φ)-MMP.

Proof. By assumption we may find an ample divisor H on W such that K_{W} + Φ + H is kawamata log terminal and ample and a positive real number t < 1 such that tH ∼_{ℝ} Θ − Φ. Note that f is the ample model of K_{W} + Φ + tH. Pick any s < t sufficiently close to t so that f is (K_{W} + Φ + sH)-negative and yet f is still the ample model of K_{W} +Φ+sH. Then f is the unique log terminal model of K_{W} +Φ+sH. In particular if we run the (K_{W} +Φ)-MMP with scaling of H then, when the value of the scalar is s, the induced rational map is f. □

We now adopt some more notation for the rest of this section. Let Θ = A+B be a point of the boundary of E_{A}(V) in the interior of L_{A}(V). Enumerate T_{1}, T_{2}, …, T_{k} the polytopes C_{i} of dimension two which contain Θ. Possibly re-ordering we may assume that the intersections O_{0} and O_{k} of T_{1} and T_{k} with the boundary of E_{A}(V) and O_{i} = T_{i} ∩T_{i+1} are all one dimensional. Let f_{i}: Z −−→ X_{i} be the rational contractions associated to T_{i} and g_{i}: Z −−→ S_{i} be the rational contractions associated to O_{i}. Set f = f_{1} : Z −−→ X, g = f_{k}: Z −−→ Y, X′ = X_{2}, Y′ = X_{k−1}. Let ϕ: X ⟶ S = S_{0}, ψ: Y ⟶ T = S_{k} be the induced morphisms and let Z −−→ R be the ample model of K_{Z} + Θ.

Theorem 3.7. Suppose Φ is any divisor such that K_{Z} +Φ is kawamata log terminal and Θ − Φ is ample.

Then ϕ and ψ are two Mori fibre spaces which are outputs of the (K_{Z} +Φ)-MMP which are connected by a Sarkisov link if Θ is contained in more than two polytopes.

Proof. We assume for simplicity of notation that k ≥ 3. The case k ≤ 2 is similar and we omit it. The incidence relations between the corresponding polytopes yield a commutative heptagon,

X′  ^{}Y′

[figure: X; p; Y; q]

[figure: S; ϕ; T; ψ]
[figure: s; t]
R

where p and q are birational maps. ϕ and ψ are Mori fibre spaces by (3.5). Pick Θ_{1} and Θ_{k} in the interior of T_{1} and T_{k} sufficiently close to Θ so that Θ_{1} − Φ and Θ_{k} − Φ are ample. As X and Y are ℚ-factorial, (3.6) implies that ϕ and ψ are possible outcomes of the (K_{Z} +Φ)-MMP. Let Δ = f_{∗}Θ. Then K_{X} + Δ is numerically trivial over R.

Note that there are contraction morphisms X_{i} ⟶ R and that ρ(X_{i}/R) ≤ 2. If ρ(X_{i}/R) = 1 then X_{i} ⟶ R is a Mori fibre space. By (3.3) there is facet of T_{i} which is contained in the boundary of E_{A}(V) and so i = 1 or k. Thus X_{i} −−→ X_{i+1} is a flop, 1 < i < k − 1. Since ρ(X′/R) = 2 it follows that either p is a divisorial contraction and s is the identity or p is a flop and s is not the identity. We have a similar dichotomy for q: Y′ −−→ Y and t: T ⟶ R.

There are then four cases. If s and t are the identity then p and q are divisorial extractions and we have a link of type II.

If s is the identity and t is not then p is a divisorial extraction and q is a flop and we have a link of type I. Similarly if t is the identity and s is not then q is a divisorial extraction and p is a flop and we have a link of type III.

Finally suppose neither s nor t is the identity. Then both p and q are flops. Suppose that s is a divisorial contraction. Let F be the divisor contracted by s and let E be its inverse image in X. Since ϕ has relative Picard number one ϕ^{∗}(F) = mE, for some positive integer m. Then K_{X} + Δ + δE is kawamata log terminal for any δ > 0 sufficiently small and E = B(K_{X} +Δ+δE/R). If we run the (K_{X} +Δ+δE)-MMP over R then we end with a birational contraction X −−→ W, which is a Mori fibre space over R. Since ρ(X/R) = 2, W = Y and we have a link of type III, a contradiction. Similarly t is never a divisorial contraction. If s is a Mori fibre space then R is ℚ-factorial and so t must be a Mori fibre space as well. This is a link of type IV_{m}. If s is small then R is not ℚ-factorial and so t is small as well. Thus we have a link of type IV_{s}. □

4. Proof of (1.3)

Lemma 4.1. Let ϕ: X ⟶ S and ψ: Y ⟶ T be two Sarkisov related Mori fibre spaces corresponding to two ℚ-factorial kawamata log terminal projective varieties (X, Δ) and (Y, Γ).

Then we may find a smooth projective variety Z, two birational contractions f: Z −−→ X and g: Z −−→ Y, a kawamata log terminal pair (Z, Φ), an ample ℚ-divisor A on Z and a two dimensional rational affine subspace V of WDiv_{ℝ}(Z) such that

(1) if Θ ∈ L_{A}(V) then Θ − Φ is ample,

(2) A_{A,ϕ∘f} and A_{A,ψ∘g} are not contained in the boundary of L_{A}(V),

(3) V satisfies (1-4) of (3.3),

(4) C_{A,f} and C_{A,g} are two dimensional, and

(5) C_{A,ϕ∘f} and C_{A,ψ∘g} are one dimensional.

Proof. By assumption we may find a ℚ-factorial kawamata log terminal pair (Z, Φ) such that f: Z −−→ X and g: Z −−→ Y are both outcomes of the (K_{Z} + Φ)-MMP.

Let p: W ⟶ Z be any log resolution of (Z, Φ) which resolves the indeterminancy of f and g. We may write

K_{W} + Ψ = p^{∗}(K_{Z} + Φ) + E′,

where E′ ≥ 0 and Ψ ≥ 0 have no common components, E′ is exceptional and p_{∗}Ψ = Φ. Pick −E ample over Z with support equal to the full exceptional locus such that K_{W} + Ψ + E is kawamata log terminal. As p is (K_{W} + Ψ + E)-negative, K_{Z} + Φ is kawamata log terminal and Z is ℚ-factorial, the (K_{W} + Ψ + E)-MMP over Z terminates with the pair (Z, Φ) by (3.6). Replacing (Z, Φ) with (W, Ψ+E), we may assume that (Z, Φ) is log smooth and f and g are morphisms.

Pick general ample ℚ-divisors A, H_{1}, H_{2}, …, H_{k} on Z such that H_{1}, H_{2}, …, H_{k} generate the Néron-Severi group of Z. Let

H = A + H_{1} + H_{2} + ⋯ + H_{k}.

Pick sufficiently ample divisors C on S and D on T such that

−(K_{X} + Δ) + ϕ^{∗}C  and  − (K_{Y} + Γ) + ψ^{∗}D,

are both ample. Pick a rational number 0 < δ < 1 such that

−(K_{X} + Δ + δf_{∗}H) + ϕ^{∗}C  and  − (K_{Y} + Γ + δg_{∗}H) + ψ^{∗}D,

are both ample and K_{Z} + Φ + δH is both f and g-negative. Replacing H by δH we may assume that δ = 1. Now pick a ℚ-divisor Φ_{0} ≤ Φ such that A + (Φ_{0} − Φ),

−(K_{X} + f_{∗}Φ_{0} + f_{∗}H) + ϕ^{∗}C and − (K_{Y} + g_{∗}Φ_{0} + g_{∗}H) + ψ^{∗}D,

are all ample and K_{Z} + Φ_{0} + H is both f and g-negative.

Pick general ample ℚ-divisors F_{1} ≥ 0 and G_{1} ≥ 0

F_{1} ∼_{ℚ} −(K_{X}+f_{∗}Φ_{0}+f_{∗}H)+ϕ^{∗}C and G_{1} ∼_{ℚ} −(K_{Y} +g_{∗}Φ_{0}+g_{∗}H)+ψ^{∗}D.

Then

K_{Z} + Φ_{0} + H + F + G,

is kawamata log terminal, where F = f^{∗}F_{1} and G = g^{∗}G_{1}.

Let V_{0} be the affine subspace of WDiv_{ℝ}(Z) which is the translate by Φ_{0} of the vector subspace spanned by H_{1}, H_{2}, …, H_{k}, F, G. Suppose that Θ = A + B ∈ L_{A}(V_{0}). Then

Θ − Φ = (A + Φ_{0} − Φ) + (B − Φ_{0}),

is ample, as B − Φ_{0} is nef by definition of V_{0}. Note that Φ_{0} + F + H ∈ A_{A,ϕ∘f}(V_{0}), Φ_{0} + G + H ∈ A_{A,ψ∘g}(V_{0}), and f, respectively g, is a weak log canonical model of K_{Z} +Φ_{0} +F +H, respectively K_{Z} +Φ_{0} +G+H. (3.3) implies that V_{0} satisfies (1-4) of (3.3).

Since H_{1}, H_{2}, …, H_{k} generate the Néron-Severi group of Z we may find constants h_{1}, h_{2}, …, h_{k} such that G is numerically equivalent to ∑ h_{i}H_{i}. Then Φ_{0} + F + δG + H − δ(∑ h_{i}H_{i}) is numerically equivalent to Φ_{0} + F + H and if δ > 0 is small enough Φ_{0} + F + δG + H − ∑ δh_{i}H_{i} ∈ L_{A}(V_{0}). Thus A_{A,ϕ∘f}(V_{0}) is not contained in the boundary of L_{A}(V_{0}). Similarly A_{A,ψ∘g}(V_{0}) is not contained in the boundary of L_{A}(V_{0}). In particular A_{A,f}(V_{0}) and A_{A,g}(V_{0}) span V_{0} and A_{A,ϕ∘f}(V_{0}) and A_{A,ψ∘g}(V_{0}) span affine hyperplanes of V_{0}, since ρ(X/S) = ρ(Y/T) = 1.

Let V_{1} be the translate by Φ_{0} of the two dimensional vector space spanned by F + H − A and F + G − A. Let V be a small general perturbation of V_{1}, which is defined over the rationals. Then (2) holds.

(1) holds, as it holds for any two dimensional subspace of V_{0}, (3) holds by (3.4) and this implies that (4) and (5) hold. □

Proof of (1.3). Pick (Z, Φ), A and V given by (4.1). Pick points Θ_{0} ∈ A_{A,ϕ∘f}(V) and Θ_{1} ∈ A_{A,ψ∘g}(V) belonging to the interior of L_{A}(V). As V is two dimensional, removing Θ_{0} and Θ_{1} divides the boundary of E_{A}(V) into two parts. The part which consists entirely of divisors which are not big is contained in the interior of L_{A}(V). Consider tracing this boundary from Θ_{0} to Θ_{1}. Then there are finitely many 2 ≤ i ≤ l points Θ_{i} which are contained in more than two polytopes C_{A,f_{i}}(V). (3.7) implies that for each such point there is a Sarkisov link σ_{i}: X_{i} −−→ Y_{i} and σ is the composition of these links. □

References

[1] C. Birkar, P. Cascini, C. Hacon, and J. M^{c}Kernan, Existence of minimal models for varieties of log general type, arXiv:math.AG/0610203.

[2] A. Bruno and K. Matsuki, Log Sarkisov program, Internat. J. Math. 8 (1997), no. 4, 451–494.

[3] A. Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), no. 2, 223–254.

[4] Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348, Dedicated to William Fulton on the occasion of his 60th birthday.

[5] Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423.

Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112-0090, USA

  E-mail address: hacon@math.utah.edu

Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA

  E-mail address: mckernan@math.ucsb.edu

Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

  E-mail address: mckernan@math.mit.edu