Minimal model program
4.5. Fundamental theorems for normal pairs
120 4. MINIMAL MODEL PROGRAM
We can see that L is a Cartier divisor on S since S 'E. Then (ϕ∗H+p∗L= 0)∩N E(X) =R≥0[C].
Note that
(KX + ∆)·C =KE·C = 0.
ThusR≥0[C] is an extremal ray ofN E(X) with the desired intersection
number.
We put D = p∗(p(C)). Note that (X,∆ +δD) is plt for a small positive rational numberδ. ThenR≥0[C] is a (KX+ ∆ +δD)-negative extremal ray. Therefore, we obtain a (KX + ∆ +δD)-flip
(X,∆ +δD)_ _ _ _ _ _ _//
&&
LL LL LL LL LL
L (X+,∆++δD+)
wwoooooooooooo
W
which is a (KX+ ∆)-flop associated to the extremal ray R≥0[C]. Since there are infinitely many (−2)-curves on S, we obtain infinitely many (KX + ∆)-flops.
4.5. FUNDAMENTAL THEOREMS FOR NORMAL PAIRS 121
(3) Let A be a π-ample R-divisor on X. Then there are only finitely many Rj’s included in (KX + ∆ +A)<0. In partic- ular, the Rj’s are discrete in the half-space (KX + ∆)<0. (4) Let F be a face of N E(X/S) such that
F ∩(N E(X/S)KX+∆≥0+N E(X/S)Nlc(X,∆)) = {0}.
Then there exists a contraction morphism ϕF : X → Y over S.
(i) LetC be an integral curve onX such that π(C)is a point.
Then ϕF(C) is a point if and only if [C]∈F. (ii) OY '(ϕF)∗OX.
(iii) Let L be a line bundle on X such that L·C = 0 for every curve C with [C]∈F. Then there is a line bundle LY on Y such that L'ϕ∗FLY.
(5) Every (KX + ∆)-negative extremal ray R with R∩N E(X/S)Nlc(X,∆)={0}
is spanned by a rational curve C with 0 < −(KX + ∆)·C ≤ 2 dimX.
From now on, we further assume that (X,∆) is log canonical, that is, Nlc(X,∆) =∅. Then we have the following properties.
(6) LetH be an effectiveR-CartierR-divisor onX such thatKX+
∆ +H is π-nef and (X,∆ +H) is log canonical. Then, either KX+ ∆is alsoπ-nef or there is a(KX+ ∆)-negative extremal ray R such that (KX + ∆ +λH)·R = 0 where
λ:= inf{t≥0|KX + ∆ +tH is π-nef}. Of course, KX + ∆ +λH is π-nef.
In [Am1], Ambro proved the properties (1), (2), (3), and (4) in Theorem4.5.2by using the theory of quasi-log schemes. More precisely, they are the main results of [Am1]. In [F28], the author obtained Theorem 4.5.2 without using the theory of quasi-log schemes. Our approach in [F28] is much simpler than Ambro’s in [Am1]. For (5), see Theorem 4.6.7. For (6), see Theorem4.7.3.
Let us include the following easy corollaries for the reader’s conve- nience.
Corollary 4.5.3 (cf. [KoMo, Corollary 3.17]). Let (X,∆) be a log canonical pair and let π : X → S be a projective morphism. Let R be a (KX + ∆)-negative extremal ray of N E(X/S) with contraction
122 4. MINIMAL MODEL PROGRAM
morphism ϕR : X → Y. Let C be a curve on X which generates R.
Then we have an exact sequence
0 //Pic(Y) L7→ϕ
∗RL
//Pic(X) M7→(M·C) //Z. In particular, we have ρ(Y /S) =ρ(X/S)−1.
Proof. Let L be a line bundle on Y. Then (ϕR)∗(ϕ∗RL) = L.
Therefore, L7→ ϕ∗RL is an injection. Note that M is a line bundle on X with (M ·C) = 0 if and only if M =ϕ∗RL for some L by Theorem
4.5.2 (4).
Corollary 4.5.4 (cf. [KoMo, Corollary 3.18]). Let (X,∆) be a log canonical pair and let π : X → S be a projective morphism. Let R be a (KX + ∆)-negative extremal ray of N E(X/S) with contraction morphism ϕR: X → Y. Assume that X is Q-factorial and that ϕR is either a divisorial or a Fano contraction. Then Y is also Q-factorial.
Proof. First, we assume that ϕR is divisorial. Let E be the ex- ceptional divisor on X. Then it is easy to see that (E ·R) < 0 and that E is irreducible. Let D be a Weil divisor on Y. Then there is a rational number s such that
((ϕR)−∗1D+sE·R) = 0.
We take a positive integer m such that m((ϕR)−∗1D+sE) is a Cartier divisor on X. Then, by Theorem 4.5.2 (4), it is the pull-back of a Cartier divisor DY on Y. Thus, mD ∼ DY. This implies that D is Q-Cartier.
Next, we assume that ϕR is a Fano contraction. Let D be a Weil divisor on Y. LetY0 be the smooth locus ofY. LetDX be the closure of (ϕR|ϕ−R1(Y0))∗(D|Y0). Then DX is disjoint from the general fiber of ϕR. Thus (DX ·R) = 0. We take a positive integer m such that mDX is a Cartier divisor on X. Thus, by Theorem 4.5.2 (4), mDX ∼ϕ∗RDY for some Cartier divisor DY onY. Thus,mD ∼DY. This implies that
D is Q-Cartier.
Let us include the basepoint-free theorem for normal pairs in [F28]
without proof for the reader’s convenience. Note that Theorem 4.5.5 is a special case of Theorem 6.5.1 below.
Theorem4.5.5 (see [F28, Theorem 13.1]). Let(X,∆)be a normal pair and let π :X →S be a projective morphism onto a variety S, and let L be a π-nef Cartier divisor on X. Assume that
(i) aL−(KX + ∆) is π-ample for some real number a >0, and (ii) ONlc(X,∆)(mL) is π|Nlc(X,∆)-generated for every m 0.
4.5. FUNDAMENTAL THEOREMS FOR NORMAL PAIRS 123
Then OX(mL) is π-generated for every m 0.
As an easy consequence of Theorem 4.5.5, we have:
Corollary 4.5.6. Let (X,∆) be a projective normal pair and let Lbe a nef Cartier divisor on X such thataL−(KX+ ∆) is nef and big for some real number a >0. Assume that ONklt(X,∆)(mL) is generated by global sections for every m 0. Then OX(mL) is generated by global sections for every m0.
Proof. By Kodaira’s lemma (see Lemma 2.1.18), we can write aL−(KX + ∆)∼RA+E
where A is an ample Q-divisor on X and E is an effective R-Cartier R-divisor onX. Let ε be a small positive number. Then
Nklt(X,∆) = Nklt(X,∆ +εE)
scheme theoretically andaL−(KX+ ∆ +εE) is ample. By replacing ∆ with ∆+εE, we may assume thataL−(KX+∆) is ample. Since there is a natural surjective morphismONklt(X,∆)→ ONqlc(X,∆),ONqlc(X,∆)(mL) is generated by global sections for everym0. Therefore, by Theorem 4.5.5, OX(mL) is generated by global sections for every m0.
Note that it is well known that Corollary4.5.6can be proved by the usual X-method (see Section 4.2) with the aid of the Nadel vanishing theorem (see Theorem 3.4.2).
4.5.7 (Examples of the Kleiman–Mori cone). From now on, we dis- cuss various examples of the Kleiman–Mori cone. The following exam- ple is well known (see, for example, [KMM, Example 4-2-4]).
Example 4.5.8. We take two smooth elliptic curves E1 and E2 on P2 such that P1 −P2 is not of finite order on the abelian group E1, where P1 and P2 are two of the nine intersection points of E1 and E2. Letf1 andf2 be the defining equations of E1 and E2 respectively. The rational map which maps x ∈ P2 \(E1∩E2) to (f1(x) : f2(x)) ∈ P1 becomes a morphism fromSwhich is obtained by taking blow-ups ofP2 at the nine intersection points ofE1 andE2. Then it is easy to see that the inverse images of P1 and P2 on S are sections of π : S → P1. By the choice of P1 and P2, there are infinitely many sections ofπ, which are (−1)-curves. Therefore, N E(S) has infinitely many KS-negative extremal rays.
Example 4.5.9, which is essentially the same as [G2, Example 5.6], is an answer to [KMM, Problem 4-2-5]. Although the construction is essentially the same as that of Example 4.4.24, we explain the details of the construction for the reader’s convenience.
124 4. MINIMAL MODEL PROGRAM
Example 4.5.9 (Infinitely many flipping contractions). There ex- ists a three-dimensional projective plt pair (X,∆) with the following properties:
(i) KX + ∆ is big, and
(ii) there are infinitely many (KX + ∆)-negative extremal rays.
Here we construct an example explicitly. Let S be a rational elliptic surface with infinitely many (−1)-curves constructed in Example4.5.8.
We take a projectively normal embedding S ⊂ PN. Let Z ⊂PN+1 be a cone over S⊂PN and letϕ:X →Z be the blow-up at the vertexP of the coneZ. Then the projectionZ 99KS from the vertexP induces a natural P1-bundle structure p:X → S. Let E be theϕ-exceptional divisor on X. Then E is a section of p. In particular, E ' S. We take a sufficiently ample smooth divisor H on Z which does not pass through P. We put ∆ = E+ϕ∗H and consider the pair (X,∆). By the construction, (X,∆) is a plt threefold such that X is smooth and that KX+ ∆ is big. Sincep:X →S is a P1-bundle andE is a section of p, we have
N1(X) = N1(E)⊕R[l]
where l'P1 is a fiber of p. Therefore, it is easy to see that N E(E)⊂N E(X)∩(ϕ∗H = 0).
Claim. Let C be a (−1)-curve on E. Then R≥0[C] is a (KX+ ∆)- negative extremal ray of N E(X).
Proof of Claim. Note that R≥0[C] is a KE-negative extremal ray of N E(E). Let L be a supporting Cartier divisor of R≥0[C] ⊂ N E(E). We can see that L is a Cartier divisor on S since S ' E.
Then
(ϕ∗H+p∗L= 0)∩N E(X) =R≥0[C].
Note that
(KX + ∆)·C =KE ·C =−1.
ThusR≥0[C] is a (KX + ∆)-negative extremal ray of N E(X).
Therefore, there are infinitely many (KX + ∆)-negative extremal rays ofN E(X). Note that every extremal ray corresponds to a flipping contraction with respect to KX + ∆.
Remark 4.5.10. Let (X,∆) be a projective klt pair. Assume that KX + ∆ is big. Then there are only finitely many (KX + ∆)-negative extremal rays. We can check this well-known result as follows. By Kodaira’s lemma (see Lemma 2.1.18), we can write
KX + ∆ ∼RA+E
4.5. FUNDAMENTAL THEOREMS FOR NORMAL PAIRS 125
whereAis an ample Q-divisor onX andE is an effective R-CartierR- divisor on X. Let ε be a small rational number such that (X,∆ +εE) is klt. In this case, (KX + ∆)-negative extremal ray is nothing but (KX + ∆ +εE +εA)-negative extremal ray. By Theorem 4.5.2 (3), there are only finitely many (KX + ∆)-negative extremal rays.
Lemma 4.5.11. Let (X,∆) be a Q-factorial projective log canonical pair such that KX + ∆ ∼R D ≥ 0. Then there are only finitely many (KX + ∆)-negative extremal rays inducing divisorial contractions. In particular, if X is a smooth projective threefold with κ(X, KX) ≥ 0, then there are only finitely many KX-negative extremal rays.
Proof. LetR be a (KX+ ∆)-negative extremal ray such that the associated contractionϕR:X →Y is divisorial. Then the exceptional locus ofϕRis a prime divisorEonXwhich is an irreducible component of SuppD. Therefore, there are only finitely many (KX + ∆)-negative divisorial contractions. When X is a smooth projective threefold with κ(X, KX) ≥ 0, the contraction morphism ϕR : X → Y is divisorial for every KX-negative extremal ray R by [Mo2] (see Theorem 1.1.4).
Therefore, there are only finitely many KX-negative extremal rays for a smooth projective threefold X with κ(X, KX)≥0.
Yoshinori Gongyo and Yoshinori Namikawa informed the author of the following example. It is well known as Schoen’s Calabi–Yau threefold and is an answer to [KMM, Problem 4-2-5].
Example 4.5.12. Let π1 : S1 → P1 and π2 : S2 → P1 be ratio- nal elliptic surfaces with infinitely many (−1)-curves constructed in Example 4.5.8. We put X =S1×P1 S2.
S1×P1 S2
p1
zzuuuuuuuuuu p
2
$$I
II II II II I
S1
π1
$$J
JJ JJ JJ JJ
JJ S2
π2
zzttttttttttt
P1
We assume that π−11(p) or π2−1(p) is smooth for every point p ∈ P1. Then it is easy to see that X is a smooth projective threefold with KX ∼ 0 by using the canonical bundle formula for rational elliptic surfaces (see [Scho] and [BHPV, Chapter V. (12.3) Corollary]). We can directly check H1(X,OX) = H2(X,OX) = 0. Therefore, X is a Calabi–Yau threefold. Letl be a (−1)-curve on S1 and let {mλ}λ∈Λ be
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 OP1(−1)⊕ OP1(−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 Hλ be a semi-ample Cartier divisor on S2 which is a supporting Cartier divisor of R≥0[mλ] ⊂ N E(S2) for every λ ∈ Λ.
Then p∗1H+p∗2Hλ induces a contraction morphismϕλ :X →Wλ such that Exc(ϕλ) = Cλ for every λ∈Λ. Therefore, R≥0[Cλ] is an extremal ray of N E(X). We put D = l ×P1 S2. Then it is easy to see that (KX +εD)·Cλ = −ε for every λ ∈ Λ. Therefore, (X, εD) is a klt threefold which has infinitely many (KX+εD)-negative extremal rays for 0< ε1. Note that we have the following flopping diagram
X
ϕλ
@
@@
@@
@@
@
φλ_ _ _ _//
_ _ _
_ Xλ+
ϕ+λ
}}{{{{{{{{
Wλ
where Xλ+ is a smooth projective threefold with KX+
λ ∼ 0. Although we have infinitely many flops φλ : X 99KXλ+, Namikawa (see [Nam]) proved that there are only finitely many Xλ+ up to isomorphisms. For the details, see [Nam].