MMP for log surfaces

148 4. MINIMAL MODEL PROGRAM

and finally get a minimal model (X₀^k0,∆^k₀⁰) of (X₀⁰,∆⁰₀) over Y₀. Since (X₁,∆₁) → Y₀ is the log canonical model of (X₀⁰,∆⁰₀) → Y₀, we have a natural morphism α₁ : X₀^k0 →X₁ (see Lemma 4.8.3). We note that K_X^k0

0

+ ∆^k₀⁰ = α^∗₁(K_X1 + ∆₁) by Lemma 4.8.3. We put (X₁⁰,∆⁰₁) = (X₀^k0,∆^k₀⁰). We run the minimal model program with respect toK_X⁰

1+

∆⁰₁ overY₁. Then we obtain a sequence

X₁⁰ 99KX₁¹ 99KX₁² 99K· · · ,

and finally get a minimal model (X₁^k1,∆^k₁¹) of (X₁⁰,∆⁰₁) over Y₁. By the same reason as above, we have a natural morphism α₂ :X₁^k1 →X₂ such thatK_X^k1

1

+ ∆^k₁¹ =α^∗₂(K_X2 + ∆₂) by Lemma4.8.3. By repeating this procedure, we obtain a (K_X⁰

0 + ∆⁰₀)-minimal model program over S:

X₀⁰ 99K· · ·99KX₀^k0 =X₁⁰ 99K· · ·99KX₁^k1 =X₂⁰ 99K· · · .

It terminates by the assumption of this lemma. Therefore, the original minimal model program must terminate after finitely many steps.

By combining Lemma 4.9.3 with Lemma 4.3.8, it is sufficient to prove Conjecture 4.3.6 for klt pairs.

4.10. MMP FOR LOG SURFACES 149

(2) Let H be a π-ample R-divisor on X. Then there are only finitely many R_j’s included in (K_X+ ∆ +H)_<0. In particular, the R_j’s are discrete in the half-space (K_X + ∆)_<0.

(3) LetR be a(K_X+∆)-negative extremal ray ofN E(X/S). Then there exists a contraction morphism ϕ_R :X →Y over S with the following properties.

(i) LetC be an integral curve onX such that π(C)is a point.

Then ϕ_R(C) is a point if and only if [C]∈R.

(ii) OY '(ϕ_R)_∗OX.

(iii) Let L be a line bundle on X such that L·C = 0 for every curve C with [C]∈R. Then there exists a line bundleL_Y on Y such that L'ϕ^∗_RL_Y.

Theorem 4.10.3 and Theorem 4.10.4 are the main results of [F29].

Theorem 4.10.3 (Minimal model program for log surfaces ([F29, Theorem 3.3])). Let (X,∆) be a log surface and let π : X → S be a projective morphism onto an algebraic variety S. We assume one of the following conditions:

(A) X is Q-factorial.

(B) (X,∆) is log canonical.

Then, by Theorem 4.10.2, we can run the minimal model program over S with respect toK_X+∆. So, there is a sequence of at most ρ(X/S)−1 contractions

(X,∆) = (X₀,∆₀)−→^ϕ0 (X₁,∆₁)−→ · · ·^{ϕ1 ϕ}−→^k−1 (X_k,∆_k) = (X^∗,∆^∗) over S such that one of the following holds:

(1) (Minimal model). if K_X + ∆ is pseudo-effective over S, then K_X∗ + ∆^∗ is nef over S. In this case, (X^∗,∆^∗) is called a minimal model of (X,∆) over S.

(2) (Mori fiber space). if K_X + ∆ is not pseudo-effective over S, then there is a morphism g : X^∗ → C over S such that

−(K_X∗+ ∆^∗) is g-ample, dimC < 2, and ρ(X^∗/C) = 1. We usually call g : (X^∗,∆^∗) → C a Mori fiber space of (X,∆) overS.

We note thatX_i isQ-factorial(resp.(X_i,∆_i)is log canonical)for every i in Case (A) (resp. (B)).

Theorem 4.10.4 (Abundance theorem ([F29, Theorem 8.1])). Let (X,∆) be a log surface and let π : X → S be a proper surjective morphism onto a variety S. Assume that X is Q-factorial or that (X,∆) is log canonical. We further assume that K_X + ∆ is π-nef.

Then K_X + ∆ is π-semi-ample.

150 4. MINIMAL MODEL PROGRAM

As an easy consequence of Theorem 4.10.3and Theorem4.10.4, we have:

Theorem4.10.5. LetX be a normalQ-factorial projective surface.

Then the canonical ring

R(X) =⊕

m≥0

H⁰(X,OX(mK_X)) is a finitely generated C-algebra.

As a corollary of Theorem4.10.5, we obtain:

Theorem 4.10.6 ([F29, Corollary 4.6]). Let X be a normal pro- jective surface with only rational singularities. Then the canonical ring

R(X) =⊕

m≥0

H⁰(X,OX(mK_X)) is a finitely generated C-algebra.

Remark 4.10.7. If X is a surface with only rational singularities, then it is well known that X is Q-factorial. If X has only rational singularities in Theorem 4.10.3, then we can check that X_i has only rational singularities for every i (see [F29, Proposition 3.7]).

The following theorem, which is not covered by Theorem 4.10.5, is well known to the experts (see, for example, [B˘a, Theorem 14.42]).

Theorem4.10.8. LetX be a normal projective Gorenstein surface.

Then the canonical ring

R(X) =⊕

m≥0

H⁰(X,OX(mK_X)) is a finitely generated C-algebra.

Here, we give a proof of Theorem 4.10.8 by using Theorem4.5.2.

Proof. Ifκ(X, K_X)≤0, then the statement is obvious. Therefore, we may assume κ(X, K_X)≥ 1. By taking some crepant resolutions of rational Gorenstein singularities, we may assume that every singularity of X is not log terminal. Let f : Y → X be a minimal resolution of singularities of X. Then we can writeK_Y +E =f^∗K_X where E is an effective Cartier divisor with Exc(f) = SuppEby the negativity lemma (see Lemma 2.3.26). We assume that K_X is not nef. Then K_Y +E is obviously not nef. By Theorem 4.5.2, there is an irreducible rational curve C⁰ on Y such that C⁰·(KY +E) <0 and (C⁰)² <0. Note that f(C⁰) =C is not a point byC⁰ ·(K_Y +E)<0. Therefore, C⁰·E ≥0.

This implies C⁰ · K_Y < 0. Thus, we have (C⁰)² = C⁰ · K_Y = −1.

4.10. MMP FOR LOG SURFACES 151

Note that E is an effective Cartier divisor. So, we have C⁰ ·E = 0 by C⁰·(K_Y+E)<0 andC⁰·K_Y =−1. This implies thatC⁰∩E =∅. Thus C is contained in the smooth locus of X. Note thatC ⊂B(K_X)(X.

Let ϕ :X →X⁰ be the contraction morphism which contracts C to a smooth point. We can replace X with X⁰. By repeating this process finitely many times, we may assume thatK_X is nef. Note thatR(X) is preserved by this process. Ifκ(X, K_X) = 2, then K_X is semi-ample by the basepoint-free theorem (see Corollary4.5.6). Note that the non-klt locus of X is zero-dimensional. If κ(X, K_X) = 1, then it is easy to see that K_X is semi-ample. Anyway, we obtain that the canonical ring R(X) is a finitely generated C-algebra.

We do not know if Theorem 4.10.8 holds true or not under the weaker assumption that X is only Q-Gorenstein. The following theo- rem is a partial result for Q-Gorenstein surfaces.

Theorem 4.10.9. Let X be a normal projective surface such that KX is Q-Cartier. Assume that there exists an effective Q-divisor D=

∑

id_iD_i such that K_X ∼Q D and that D_i is a Q-Cartier prime divisor on X for every i. Then the canonical ring

R(X) =⊕

m≥0

H⁰(X,OX(mK_X)) is a finitely generated C-algebra.

Proof. LetR=R_≥0[C] be aK_X-negative extremal ray ofN E(X).

Then C ·K_X < 0 implies that C ⊂ SuppD. So, N E(X) has only finitely many K_X-negative extremal rays. Moreover, the contraction morphism ϕR : X → Y is birational. We note that the exceptional locus ofϕRis an irreducible curve contained in SuppD. This is because each irreducible component ofDisQ-Cartier. Therefore, we can check that K_Y =ϕ_R∗K_X is Q-Cartier, K_Y ∼Q ∑

id_iϕ_R∗D_i, and ϕ_R∗D_i is Q- Cartier if ϕ_R∗D_i 6= 0. After finitely many contraction morphisms, this program terminates. SinceR(X) is preserved by the above process, we may assume thatK_X is nef by replacing X with its final model. When κ(X, K_X) = 0 or 1, R(X) is obviously a finitely generated C-algebra.

So, we may assume that K_X is big. Since the non-klt locus of X is zero-dimensional, K_X is semi-ample by the basepoint-free theorem (see Corollary 4.5.6). In particular, R(X) is a finitely generated C-

algebra.

For the details of the minimal model theory for log surfaces, see [F29]. For the minimal model theory for log surfaces in positive char- acteristic, see [FT], [Tana1], and [Tana2]. In positive characteristic,

152 4. MINIMAL MODEL PROGRAM

we note the following contraction theorem of Artin–Keel (see [Ar] and [Ke]).

Theorem 4.10.10 (Artin–Keel). Let X be a complete normal al- gebraic surface defined over an algebraically closed field k of positive characteristic and letH be a nef and big Cartier divisor on X. We put

E(H) ={C|C is a curve on X and C·H = 0}.

Then E(H) consists of finitely many irreducible curves on X. Assume that H|E(H) is semi-ample where

E(H) = ∪

C∈E(H)

C

with the reduced scheme structure. Then H is semi-ample. Therefore, Φ_|mH|:X →Y

is a proper birational morphism onto a normal projective surface Y which contracts E(H) and is an isomorphism outside E(H) for a suf- ficiently large and divisible positive integer m.

For the proof of Theorem4.10.10, we recommend the reader to see [FT, Theorem 2.1], where we gave two different proofs using the Fujita vanishing theorem (see Theorem 3.8.1). Note that Theorem 4.10.10 does not hold in characteristic zero. By Theorem 4.10.10, the minimal model theory for log surfaces is easier in characteristic p > 0 than in characteristic zero.

We close this section with an example of a non-Q-factorial log canonical surface.

Example4.10.11 (Non-Q-factorial log canonical surface). LetC ⊂ P² be a smooth cubic curve and let Y ⊂ P³ be a cone over C. Then Y is log canonical. In this case, Y is not Q-factorial. We can check it as follows. Let f : X = PC(OC ⊕ L) → Y be a resolution such that K_X +E =f^∗K_Y, where L =O_P²(1)|C and E is the exceptional curve.

We take P, Q ∈ C such that OC(P −Q) is not a torsion in Pic⁰(C).

We consider D = π^∗P −π^∗Q, where π : X =PC(OC ⊕ L) → C. We put D⁰ = f_∗D. If D⁰ is Q-Cartier, then mD = f^∗mD⁰ +aE for some a∈Z and m∈Z>0. Restrict it to E. Then

OC(m(P −Q))' OE(aE)'(L⁻¹)^⊗a.

Therefore, we obtain that a= 0 and m(P −Q)∼0. This is a contra- diction. Thus,D⁰ is not Q-Cartier. In particular,Y is notQ-factorial.