Minimal model program
4.3. MMP for Q -factorial dlt pairs
for some F =Fj0, where dAe is effective and A does not have F as a component. In fact, we choose c >0 so that
minj (−crj +aj −pj) = −1.
If this last condition does not single out a unique j, we wiggle the pj
slightly to achieve the desired uniqueness. This j satisfies rj >0 and dN(b, c)e+KY =bf∗D+dAe −F. Now Step 3 implies that
H0(Y,OY(bf∗D+dAe))→H0(F,OF(bf∗D+dAe))
is surjective forb ≥cm+a. If Fj appears in dAe, thenaj >0, so Fj is f-exceptional. Thus, dAe isf-exceptional.
Step 5. Notice that
N(b, c)|F = (bf∗D+A−F −KY)|F = (bf∗D+A)|F −KF. So we can apply the non-vanishing theorem (see Theorem 4.1.2) on F to get
H0(F,OF(bf∗D+dAe))6= 0.
Thus,H0(Y,OY(bf∗D+dAe)) has a section not vanishing on F. Since dAe is f-exceptional and effective,
H0(Y,OY(bf∗D+dAe))'H0(X,OX(bD)).
Therefore, f(F) is not contained in the base locus of |bD| for every b0.
This completes the proof of the basepoint-free theorem.
The X-method is very powerful and very useful for klt pairs. Un- fortunately, it can not be applied for log canonical pairs. So we need the framework discussed in [F28] or the theory of quasi-log schemes (see Chapter 6) in order to treat log canonical pairs. For the details of X-method, see [KMM] and [KoMo].
We note that the X-method, the technique which was used for the proofs of Theorems 4.1.2, 4.1.3, 4.1.4, and 4.1.5, was developed by several authors. The main contributions are [Ka2], [Ko1], [R1], and [Sh1].
4.3. MMP for Q-factorial dlt pairs
In this section, we quickly explain the minimal model program for Q-factorial dlt pairs. First, let us recall the definition of the (log) minimal models. Definition 4.3.1 is a traditional definition of minimal models. For slightly different other definitions of minimal models, see Definition 4.4.4 and Definition4.8.5.
108 4. MINIMAL MODEL PROGRAM
Definition 4.3.1 ((Log) minimal model). Let (X,∆) be a log canonical pair and let f : X → S be a proper morphism. A pair (X0,∆0) sitting in a diagram
X _ _ _ _φ _ _ _//
f@@@@@
@@ X0
f0
~~}}}}}}}}
S
is called a (log) minimal model of(X,∆) over S if (i) f0 is proper,
(ii) φ−1 has no exceptional divisors, (iii) ∆0 =φ∗∆,
(iv) KX0 + ∆0 is f0-nef, and
(v) a(E, X,∆) < a(E, X0,∆0) for every φ-exceptional divisorE ⊂ X.
Furthermore, if KX0 + ∆0 is f0-semi-ample, then (X0,∆0) is called a good minimal model of (X,∆) over S.
We note the following easy lemma.
Lemma 4.3.2. Let (X,∆) be a log canonical pair and let f :X →S be a proper morphism. Let (X0,∆0)be a minimal model of (X,∆) over S. Then a(E, X,∆)≤a(E, X0,∆0) for every divisor E overX.
Proof. We take any common resolution W
q
!!B
BB BB BB
p B
~~}}}}}}}}
X X0
of X and X0. Then we can write
KW =p∗(KX + ∆) +F and
KW =q∗(KX0+ ∆0) +G.
It is sufficient to prove G≥F. Note that
p∗(KX + ∆) =q∗(KX0+ ∆0) +G−F.
Then −(G − F) is p-nef since KX0 + ∆0 is nef over S. Note that p∗(G−F) is effective by (v). Therefore, by the negativity lemma (see
Lemma 2.3.26),G−F is effective.
4.3. MMP FOR Q-FACTORIAL DLT PAIRS 109
Next, we recall the flip theorem for dlt pairs in [BCHM] and [HaMc1] (see also [HaMc2]). We need the notion of small morphisms to treat flips.
Definition 4.3.3 (Small morphism). Let f : X →Y be a proper birational morphism between normal varieties. If Exc(f) has codimen- sion ≥2, then f is calledsmall.
Theorem 4.3.4 ((Log) flip for dlt pairs). Let ϕ : (X,∆) → W be an extremal flipping contraction, that is,
(i) (X,∆) is dlt,
(ii) ϕ is small projective and ϕ has connected fibers, (iii) −(KX + ∆) is ϕ-ample,
(iv) ρ(X/W) = 1, and (v) X is Q-factorial.
Then we have the following diagram:
X _ _ _ _φ_ _ _//
ϕAAAAAA
AA X+
ϕ+
}}zzzzzzzz
W (1) X+ is a normal variety,
(2) ϕ+ :X+→W is small projective, and
(3) KX+ + ∆+ is ϕ+-ample, where ∆+ is the strict transform of
∆.
We callϕ+: (X+,∆+)→W a(KX+∆)-flip ofϕ. In this situation, we can check that (X+,∆+) is a Q-factorial dlt pair with ρ(X+/W) = 1 (see, for example, Lemma 4.8.13 and Proposition 4.8.16 below).
Let us explain the relative minimal model program (MMP, for short) for Q-factorial dlt pairs.
4.3.5 (MMP forQ-factorial dlt pairs). We start with a pair (X,∆) = (X0,∆0). Let f0 : X0 → S be a projective morphism. The aim is to set up a recursive procedure which creates intermediate pairs (Xi,∆i) and projective morphisms fi : Xi → S. After some steps, it should stop with a final pair (X0,∆0) andf0 :X0 →S.
Step 0 (Initial datum). Assume that we have already constructed (Xi,∆i) and fi :Xi →S with the following properties:
(i) Xi is Q-factorial, (ii) (Xi,∆i) is dlt, and (iii) fi is projective.
110 4. MINIMAL MODEL PROGRAM
Step 1 (Preparation). If KXi + ∆i is fi-nef, then we go directly to Step 3 (ii). If KXi + ∆i is not fi-nef, then we have established the following two results:
(i) (Cone theorem) We have the following equality.
N E(Xi/S) = N E(Xi/S)(KXi+∆i)≥0+∑
R≥0[Ci].
(ii) (Contraction theorem) Any (KXi+ ∆i)-negative extremal ray Ri ⊂ N E(Xi/S) can be contracted. Let ϕRi : Xi → Yi de- note the corresponding contraction. It sits in a commutative diagram.
Xi ϕRi //
f@i@@@@@@
@ Yi
gi
S
Step2 (Birational transformations). IfϕRi :Xi →Yi is birational, then we produce a new pair (Xi+1,∆i+1) as follows.
(i) (Divisorial contraction). IfϕRi is a divisorial contraction, that is, ϕRi contracts a divisor, then we set Xi+1 = Yi, fi+1 = gi, and ∆i+1 = (ϕRi)∗∆i.
(ii) (Flipping contraction). If ϕRi is a flipping contraction, that is, ϕRi is small, then we set (Xi+1,∆i+1) = (Xi+,∆+i ), where (Xi+,∆+i ) is the flip of ϕRi
(Xi,∆i)
ϕRi
##G
GG GG GG
GG_ _ _ _ _ _ _// (Xi+,∆+i )
ϕ+
zzuuuuuuuuuRiu
Yi
and fi+1 =gi◦ϕ+R
i (see Theorem 4.3.4).
In both cases, we can prove that Xi+1 is Q-factorial, fi+1 is projec- tive and (Xi+1,∆i+1) is dlt (see, for example, Lemma 4.8.13, Proposi- tion 4.8.14, and Proposition 4.8.16). Then we go back to Step 0 with (Xi+1,∆i+1) and start anew.
Step 3 (Final outcome). We expect that eventually the procedure stops, and we get one of the following two possibilities:
(i) (Mori fiber space). IfϕRiis a Fano contraction, that is, dimYi <
dimXi, then we set (X0,∆0) = (Xi,∆i) and f0 = fi. In this case, we usually call f0 : (X0,∆0) → Yi a Mori fiber space of (X,∆) over S.
4.3. MMP FOR Q-FACTORIAL DLT PAIRS 111
(ii) (Minimal model). If KXi + ∆i is fi-nef, then we again set (X0,∆0) = (Xi,∆i) and f0 = fi. We can easily check that (X0,∆0) is a minimal model of (X,∆) over S in the sense of Definition4.3.1.
By the results in [BCHM] and [HaMc1] (see also [HaMc2]), all we have to do is to prove that there are no infinite sequence of flips in the above process.
Conjecture 4.3.6 (Flip conjecture II). A sequence of (log) flips (X0,∆0)99K(X1,∆1)99K· · ·99K(Xi,∆i)99K· · ·
terminates after finitely many steps. Namely there does not exist an infinite sequence of (log) flips.
Remark 4.3.7. In Conjecture 4.3.6, each flip (Xi,∆i)99K(Xi+1,∆i+1) is a flip as in Theorem 4.3.4.
Lemma 4.3.8. We assume that Conjecture 4.3.6 holds in the fol- lowing two cases:
(i) (X0,∆0) is klt with dimX0 =n, and (ii) (X0,∆0) is dlt with dimX0 ≤n−1.
Then Conjecture4.3.6holds forn-dimensional dlt pair(X0,∆0). There- fore, by induction on the dimension, it is sufficient to prove Conjecture 4.3.6 under the extra assumption that (X0,∆0) is klt.
Proof. Let
(X0,∆0)99K(X1,∆1)99K· · ·99K(Xi,∆i)99K· · ·
be a sequence of flips as in Conjecture 4.3.6 with dimX0 =n. By the case (ii), the special termination theorem holds in dimensionn(see, for example, [F13, Theorem 4.2.1]). Therefore, after finitely many steps, the flipping locus (and thus the flipped locus) is disjoint from b∆ic. Thus, we may assume that b∆ic = 0 by replacing ∆i with {∆i}. In this case, the above sequence terminates by the case (i).
Conjecture 4.3.6 was completely solved in dimension ≤ 3 (see, for example, [Koetal, Chapter 6] and [Sh3, 5.1.3]). Conjecture 4.3.6 is still open even when dimX0 = 4. For the details of Conjecture4.3.6in dimension 4, see [KMM, Theorem 5-1-15], [F5], [F7], [F8], [AHK], and [Bir1].
112 4. MINIMAL MODEL PROGRAM