Lengths of extremal rays

126 4. MINIMAL MODEL PROGRAM

the set of all (−1)-curves on S2. Then Cλ = l×_P¹ mλ is a (−1,−1)- curve, that is, a rational curve whose normal bundle is isomorphic to O_P¹(−1)⊕ O_P¹(−1), on X for every λ ∈ Λ. We take a semi-ample Cartier divisorHonS₁which is a supporting Cartier divisor ofR≥0[l]⊂ N E(S₁). Let H_λ be a semi-ample Cartier divisor on S₂ which is a supporting Cartier divisor of R≥0[m_λ] ⊂ N E(S₂) for every λ ∈ Λ.

Then p^∗₁H+p^∗₂H_λ induces a contraction morphismϕ_λ :X →W_λ such that Exc(ϕ_λ) = C_λ for every λ∈Λ. Therefore, R_≥0[C_λ] is an extremal ray of N E(X). We put D = l ×_P¹ S₂. Then it is easy to see that (K_X +εD)·C_λ = −ε for every λ ∈ Λ. Therefore, (X, εD) is a klt threefold which has infinitely many (K_X+εD)-negative extremal rays for 0< ε1. Note that we have the following flopping diagram

X

ϕ_λ

@

@@

@@

@@

@

φλ_ _ _ _//

_ _ _

_ X_λ⁺

ϕ⁺_λ

}}{{{{{{{{

W_λ

where X_λ⁺ is a smooth projective threefold with K_X⁺

λ ∼ 0. Although we have infinitely many flops φ_λ : X 99KX_λ⁺, Namikawa (see [Nam]) proved that there are only finitely many X_λ⁺ up to isomorphisms. For the details, see [Nam].

4.6. LENGTHS OF EXTREMAL RAYS 127

Proof. Let ∑

kDk be the irreducible decomposition of Supp ∆.

We consider the finite dimensional real vector space V =⊕

k

RD_k. We put

Q={D∈V | K_X +D is R-Cartier}.

Then, it is easy to see that Q is an affine subspace of V defined over Q. We put

L={D∈ Q | KX +Dis log canonical}.

Thus, by the definition of log canonicity, it is also easy to check that L is a closed convex rational polytope inV. We note thatLis compact in the classical topology of V. By assumption, ∆∈ L. Therefore, we can find the desired Q-divisors ∆_i ∈ L and positive real numbers r_i. The next result is essentially due to [Ka4] and [Sh5, Proposition 1]. We will prove a more general result in Theorem 4.6.7 whose proof depends on Theorem4.6.2.

Theorem 4.6.2. LetX be a normal variety such that (X,∆) is log canonical and let π : X → S be a projective morphism onto a variety S. Let R be a (K_X + ∆)-negative extremal ray. Then we can find a (possibly singular) rational curve C on X such that [C]∈R and

0<−(K_X + ∆)·C ≤2 dimX.

Proof. By shrinkingS, we may assume thatS is quasi–projective.

By replacing π : X → S with the extremal contraction ϕ_R : X → Y overS (see Theorem4.5.2(4)), we may assume that the relative Picard number ρ(X/S) = 1. In particular, −(K_X + ∆) is π-ample. Let

K_X + ∆ =

∑l i=1

r_i(K_X + ∆_i)

be as in Lemma 4.6.1. We assume that −(KX + ∆1) is π-ample and

−(K_X+ ∆_i) =−s_i(K_X + ∆₁) in N¹(X/S) with s_i ≤1 for everyi≥2.

Thus, it is sufficient to find a rational curveC such thatπ(C) is a point and that

−(KX + ∆1)·C≤2 dimX.

So, we may assume that K_X + ∆ is Q-Cartier and log canonical. By taking a dlt blow-up (see Theorem 4.4.21), there is a birational mor- phism f : (Y,∆Y)→ (X,∆) such thatKY + ∆Y =f^∗(KX + ∆), Y is Q-factorial, and (Y,∆_Y) is dlt. By [Ka4, Theorem 1] and [Ma, Theo- rem 10-2-1] (see also [Deb, Section 7.11]), we can find a rational curve

128 4. MINIMAL MODEL PROGRAM

C⁰ onY such that

−(K_Y + ∆_Y)·C⁰ ≤2 dimY = 2 dimX

and that C⁰ spans a (K_Y + ∆_Y)-negative extremal ray. By the pro- jection formula, the f-image of C⁰ is a desired rational curve. So, we

finish the proof.

Remark 4.6.3. It is conjectured that the estimate ≤ 2 dimX in Theorem 4.6.2should be replaced by≤dimX+ 1. When X is smooth projective, it is true by Mori’s famous result (see [Mo2], Theorem1.1.1, and [KoMo, Theorem 1.13]). WhenX is a toric variety, it is also true by [F4] and [F10].

Remark4.6.4. In the proof of Theorem4.6.2, we need Kawamata’s estimate on the length of an extremal rational curve (see, for example, [Ka4, Theorem 1], [Ma, Theorem 10-2-1], and [Deb, Section 7.11]). It depends on Mori’s bend and break technique to create rational curves.

So, we need the mod p reduction technique there.

Remark 4.6.5. Let (X, D) be a log canonical pair such that D is an R-divisor. Let φ : X → Y be a projective morphism and let H be a Cartier divisor on X. Assume that H −(K_X +D) is f-ample.

By the Kawamata–Viehweg type vanishing theorem for log canonical pairs (see Theorem 5.6.4), R^qφ_∗OX(H) = 0 for every q > 0 if X and Y are algebraic varieties. If this vanishing theorem holds for analytic spaces X and Y, then Kawamata’s original argument in [Ka4] works directly for log canonical pairs. In that case, we do not need dlt blow- ups (see Theorem 4.4.21), which follows from [BCHM], in the proof of Theorem 4.6.2.

We consider the proof of [Ma, Theorem 10-2-1] when (X, D) is Q-factorial dlt. We need R¹φ_∗OX(H) = 0 after shrinking X and Y analytically. In our situation, (X, D −εbDc) is klt for 0 < ε 1.

Therefore,H−(K_X+D−εbDc) is φ-ample and (X, D−εbDc) is klt for 0< ε1. Thus, we can apply the analytic version of the relative Kawamata–Viehweg vanishing theorem (see, for example, [F31]). So, we do not need the analytic version of the Kawamata–Viehweg type vanishing theorem for log canonicalpairs.

Remark 4.6.6. We give a remark on [BCHM]. We use the same notation as in [BCHM, 3.8]. In the proof of [BCHM, Corollary 3.8.2], we may assume that KX + ∆ is klt by [BCHM, Lemma 3.7.4]. By perturbing the coefficients of B slightly, we can further assume that B is a Q-divisor. By applying the usual cone theorem to the klt pair (X, B), we obtain that there are only finitely many (K_X+ ∆)-negative

4.6. LENGTHS OF EXTREMAL RAYS 129

extremal rays of N E(X/U). We note that [BCHM, Theorem 3.8.1]

is only used in the proof of [BCHM, Corollary 3.8.2]. Therefore, we do not need the estimate of lengths of extremal rays in [BCHM]. In particular, we do not need modp reduction arguments for the proof of the main results in [BCHM].

The final result in this section is an estimate of lengths of ex- tremal rays which are relatively ample at non-lc loci (see also [Ko4]

and [Ko5]).

Theorem 4.6.7 (Theorem 4.5.2 (5)). Let X be a normal variety, let ∆ be an effective R-divisor on X such that K_X + ∆ is R-Cartier, and let π : X → S be a projective morphism onto a variety S. Let R be a (K_X + ∆)-negative extremal ray of N E(X/S) which is relatively ample atNlc(X,∆), that is, R∩N E(X/S)_Nlc(X,∆) ={0}. Then we can find a (possibly singular) rational curve C on X such that[C]∈R and

0<−(K_X + ∆)·C ≤2 dimX.

Proof. By shrinkingS, we may assume thatSis quasi-projective.

By replacing π : X → S with the extremal contraction ϕ_R : X → Y overS (see Theorem4.5.2(4)), we may assume that the relative Picard number ρ(X/S) = 1 and that π is an isomorphism in a neighborhood of Nlc(X,∆). In particular, −(KX + ∆) is π-ample. By taking a dlt blow-up (see Theorem4.4.21), there is a projective birational morphism f :Y →X such that

(i) K_Y+ ∆_Y =f^∗(K_X+ ∆) + ∑

a(E,X,∆)<−1

(a(E, X,∆) + 1)E, where

∆_Y =f_∗⁻¹∆ + ∑

E:f-exceptional

E, (ii) (Y,∆_Y) is a Q-factorial dlt pair, and

(iii) K_Y +D=f^∗(K_X + ∆) with D= ∆_Y +F, where F =− ∑

a(E,X,∆)<−1

(a(E, X,∆) + 1)E ≥0.

Therefore, we have

f_∗(N E(Y /S)K_Y+D≥0)⊆N E(X/S)K_X+∆≥0 ={0}. We also note that

f_∗(N E(Y /S)_Nlc(Y,D)) = {0}.

Thus, there is a (KY +D)-negative extremal rayR⁰ ofN E(Y /S) which is relatively ample at Nlc(Y, D). By Theorem 4.5.2, R⁰ is spanned by a curveC^†. Since −(K_Y +D)·C^†>0, we see thatf(C^†) is a curve. If

130 4. MINIMAL MODEL PROGRAM

C^†⊂SuppF, thenf(C^†)⊂Nlc(X,∆). This is a contradiction because π◦f(C^†) is a point. Thus, C^†6⊂SuppF. Since

−(K_Y + ∆_Y) = −(K_Y +D) +F,

we can see thatR⁰ is a (K_Y + ∆_Y)-negative extremal ray of N E(Y /S).

Therefore, we can find a rational curveC⁰ onY such that C⁰ spans R⁰ and that

0<−(K_Y + ∆_Y)·C⁰ ≤2 dimX

by Theorem 4.6.2. By the above argument, we can easily see that C⁰ 6⊂SuppF. Therefore, we obtain

0<−(K_Y +D)·C⁰ =−(K_Y + ∆_Y)·C⁰−F ·C⁰

≤ −(K_Y + ∆_Y)·C⁰ ≤2 dimX.

SinceK_Y +D=f^∗(K_X+ ∆),C =f(C⁰) is a rational curve onX such that π(C) is a point and 0<−(KX + ∆)·C≤2 dimX.

Remark 4.6.8. In Theorem 4.6.7, we can prove 0< −(K_X + ∆)· C ≤dimX+ 1 when dimX ≤2. For the details, see [F29, Proposition 3.7].