Special Termination of Minimal Model Program

This paper presents a self-contained proof of Special Termination of MMP (Minimal Model Program). By refining the assumptions and simplifying the argument, it offers a more accessible approach compared to the original proof in BCHM (Birkar-Cascini-Hacon-McKernan).

• Special Termination
• Minimal Model Program
• Birational Geometry
• Algebraic Geometry

1. Introduction

The main purpose of this note is to provide a complete presentation on the special termination of MMP. We begin by extracting the positivity condition on the boundary from [1] , which we define as the BCHM condition. This condition asserts that the boundaries of a Kawamata log terminal (klt for short) or divisorial log terminal (dlt for short) pair $\left(X,B\right)$ include an ample divisor. A key advantage of the BCHM condition is that it remains preserved under restriction and throughout the Minimal Model Program (MMP) after appropriate boundary modifications, enabling us to establish the existence of pl-flips by induction on dimension.

2. Preliminaries

Let k be an algebraically closed field of characteristic zero fixed throughout the paper. A divisor means a $ℝ$-Cartier $ℝ$-Weil divisor. A divisor D over a normal variety X is a divisor on a birational model of X. A birational map $X$⇢ $Y$ is a birational contraction if its inverse map contracts no divisor.

Pairs. A pair $\left(X/U,B\right)$ consists of normal quasi-projective varieties X, Z, a $ℝ$-divisor B on X with coefficients in $\left[0,1\right]$ such that $K_X+B$ is $ℝ$-Cartier and a projective morphism $X\to U$. If U is a point or U is unambiguous in the context, then we simply denote a pair by $\left(X,B\right)$. For a prime divisor D on some birational model of X with a nonempty centre on X, $a\left(D,X,B\right)$ denotes the log discrepancy. For definitions and standard results on singularities of pairs, we refer to [2] .

Log Minimal Models. A projective pair $\left(Y/U,B_Y\right)$ is a log birational model of a projective pair $\left(X/U,B\right)$ if we are given a birational map $\varphi :X$⇢ $Y$ and $B_Y=B^{\sim }+E$ where $B^{\sim }$ is the birational transform of B and E is the reduced exceptional divisor of ${\varphi }^{-1}$, that is, $E={\displaystyle \sum E_j}$ where $E_j$ are the exceptional/X prime divisors on Y. A log birational model $\left(Y/U,B_Y\right)$ is a weak log canonical (weak lc for short) model of $\left(X/U,B\right)$ if

$a\left(D,X,B\right)\le a\left(D,Y,B_Y\right)$

A weak lc model $\left(Y/U,B_Y\right)$ is a log minimal model of $\left(X/U,B\right)$ if

A log minimal model $\left(Y/U,B_Y\right)$ is good if $K_Y+B_Y$ is semi-ample/U.

Ample Models and Log Canonical Models. Let D be a divisor on a normal variety X over Z. A normal variety T is the ample model/Z of D if we are given a rational map $\varphi :X$⇢ $T$ such that there exists a resolution $X\stackrel{p}{\leftarrow }{X}^{\prime }\stackrel{q}{\to }T$ with

Note that the ample model is unique if it exists. The existence of the ample model is equivalent to saying that the divisorial ring $R\left(D\right)$ is a finitely generated ${\mathcal{O}}_Z$-algebra when $D\ge 0$ is $\mathbb{Q}$-Cartier.

BCHM Condition.

1) X is n-dimensional $\mathbb{Q}$-factorial normal algebraic variety, $\pi :X\to U$ is a projective morphism of normal quasi-projective varieties.

2) $\left(X,B\right)$ is dlt pair with $S=\lfloor B\rfloor$ or $\left(X,B\right)$ is a klt pair.

3) There exists a relatively ample $ℝ$-divisor A over U such that $B\ge A$.

3. Special Termination Theorem

Firstly, we recall some classical theorem

Theorem 3.1 (Basepoint-free theorem). Let $\left(X,\Delta \right)$ be a projective klt pair. Let D be a nef Cartier divisor such that $aD-\left(K_X+\Delta \right)$ is ample for some $a>0$. Then, there is a positive integer $b_0$ such that $\left|bD\right|$ has no base points for every $b\ge b_0$.

Theorem 3.2 (Rationality theorem). Let $\left(X,\Delta \right)$ be a projective klt pair such that $K_X+\Delta$ is not nef. Let $a>0$ be an integer such that $a\left(K_X+\Delta \right)$ is Cartier. Let H be an ample Cartier divisor. We define

$r=\text{max}\left\{t\in ℝ|H+t\left(K_X+\text{Δ}\right)\text{is}\text{nef}\right\}$

Then r is a rational number of the form u/v, where u and v are integers with

$0<v\le a\left(\dim X+1\right)$

The final theorem is the cone and contraction theorem.

Theorem 3.3 (Cone and contraction theorem). Let $\left(X,\Delta \right)$ be a projective klt pair. Then, we have the following properties.

1) There are (countably many possibly singular) rational curves $C_j\subset X$ such that

$\overline{NE}\left(X\right)=\overline{NE}{\left(X\right)}_{\left(K_X+\text{Δ}\right)\ge 0}+{\displaystyle \sum {ℝ}_{\ge 0}\left[C_j\right]}$

2) Let $R\subset \overline{NE}\left(X\right)$ be a $\left(K_X+\text{Δ}\right)$-negative extremal ray. Then, there is a unique morphism ${\phi }_R:X\to Z$ to a projective variety Z such that ${\left({\phi }_R\right)}_{\text{*}}{\mathcal{O}}_X\simeq {\mathcal{O}}_Z$ and an irreducible curve $C\subset X$ is mapped to a point by ${\phi }_R$ if and only if $\left[C\right]\in R$.

Assume that we are given an LMMP with scaling, which consists of only a sequence $X_i$⇢ $X_{i+1}/Z_i$ of log flips, and that $\left(X_1/Z,B_1\right)$ is $\mathbb{Q}$-factorial dlt. Assume $\lfloor B_1\rfloor \ne 0$ and pick a component $S_1$ of $\lfloor B_1\rfloor$. Let $S_i\subset X_i$ be the birational transform of $S_1$ and $T_i$ the normalisation of the image of $S_i$ in $Z_i$. Using standard special termination arguments, we will see that termination of the LMMP near $S_1$ is reduced to termination in lower dimensions. It is well-known that the induced map $S_i$⇢ $S_{i+1}/T_i$ is an isomorphism in codimension one if $i\gg 0$. So, we could assume that these maps are all isomorphisms in codimension one. Put $K_{S_i}+B_{S_i}:={\left(K_{X_i}+B_i\right)|}_{S_i}$. In general, $S_i$⇢ $S_{i+1}/T_i$ is not a $\left(K_{S_i}+B_{S_i}\right)$-flip. To apply induction, we note that $S_i$⇢ $S_{i+1}/T_i$ can be connected by a sequence of $\left(K_{S_i}+B_{S_i}\right)$-flips [3] .

Theorem 3.4 Let $\pi :X\to U$ be a projective morphism of normal quasi-projective varieties. Let $\left(X,B+C\right)$ be a $\mathbb{Q}$-factorial dlt pair with $S=\lfloor B\rfloor$, such that $K_X+B+C$ is nef over U and $\left(X,B\right)$ satisfy BCHM condition. Let ${\alpha }_i:X_i$⇢ $X_{i+1}$ be a sequence of flips and divisorial contractions over U for the $\left(K_X+B\right)$-MMP with a scaling of C over U.

Then, there exists an integer $i>0$ such that for all $j\ge i,{\alpha }_j$ is an isomorphism on a neighborhood of S.

Proof. Given that $\left(X,B\right)$ satisfies the BCHM condition, we can write $\left(X,B\right)=\left(X,S+A+E\right)$ where $E\ge 0$ and A is an ample $\mathbb{Q}$-divisor. Fix any irreducible component $S_1$ of $S=\lfloor B\rfloor$.

For any rational number $0<ϵ\ll 1$, the divisor $A+ϵ\left(S-S_1\right)$ is an ample $\mathbb{Q}$-divisor. Now, take a sufficiently general effective R-divisor $A_1{\equiv }_{U}A+ϵ\left(S-S_1\right)$. Then $\left(X,A_1+E+\left(1-ϵ\right)\left(S-S_1\right)+S_1\right)$ is plt and $\left(X,A_1+E+\left(1-ϵ\right)\left(S-S_1\right)+S_1+C\right)$ is dlt. Since $B{\equiv }_U{A}_1+E+\left(1-ϵ\right)\left(S-S_1\right)+S_1$, after replacing B and B’, we may assume that $\left(X,B\right)$ is plt.

Let $S_i,A_i,E_i,B_i$ and $C_i$ be the strict transforms of $S,A,E,B$ and C on $X_i$. Notice that $\left(X_i,S_i+A_i+E_i\right)$ is plt and $S_i+A_i+E_i=S_i$, then $S_i$ is normal.

By adjunction formula, we can write

${\left(K_{X_i}+B_{X_i}\right)|}_{S_i}=K_{S_i}+B_{S_i}$

where $\left(S_i,B_{S_i}\right)$ is klt. Define the set $ℬ\subset \left[0,1\right]$ as the following:

Figure 1

We have Coeff ( B S i ) ⊂ ℬ for any i. Since ℬ ∩ [ 0 , 1 − ϵ ] is a finite set for any ϵ > 0 and ( S i , B i ) is klt, we define an integer 0 ≤ d ℬ < ∞ by d ℬ ( S , B S ) : = ∑ β ∈ ℬ # { E | a ( E ; S , B S ) < − β } Moreover, we have that a ( E ; S i , B S i ) ≤ a ( E ; S i + 1 , B S i + 1 ) for any divisor E over S. Thus, d ℬ ( S i , B S i ) ≥ d ℬ ( S i + 1 , B S i + 1 ) . (1)We claim that α i : S i ⇢ S i + 1 is an isomorphic in codimension 1 for i ≫ 0 . Suppose there is a divisor P ⊆ S i + 1 and ( α i − 1 ) * P = 0 . Then inequality (1) is strict, as − a ( P ; S i + 1 , B s i + 1 ) ∈ ℬ . Notice that ℬ ∩ [ 0 , 1 − ϵ ] and picard number is finite, so α i is isomorphic in codimension 1 after deleting finitely many steps.By the above argument, we only need to consider the flip X m ⇢ X m + 1 over Z m . Since ( X , B ) containing ample divisor A and ( K X m + B m ) | S m = K S m + B S m , the pair ( S m , B S m ) satisfy BCHM condition.Now, take h : S ˜ m → S m to be a ℚ -factorialization. Then we have K S ˜ m + B S ˜ m = h * ( K S m + B S m ) and ( S ˜ m , B ˜ S m ) satisfies the BCHM condition.Next, run an MMP with scaling C ˜ m over T m , where C ˜ m is the birational transform of C m , and T m is the normalisation of the image of S m in Z m [4] . As a result, we obtain a minimal model ( S ˜ m + 1 , B ˜ S m + 1 ) , which factors through S m + 1 , as S m + 1 is the log canonical model of S m .For the same reason, we can inductively construct a sequence of MMP on ( S ˜ m , B ˜ S m ) over T m : S ˜ m = S m , 0 → S m , 1 → ⋯ → S m , l = S ˜ m + 1

Since $K_{X_m}+B_m+t_m{C}_m$ is numerically trivial over $Z_m$, the induced divisors $K_{S_{m,i}}+B_{m,i}+t_m{C}_{m,i}\equiv 0$ over $Z_m$. Hence, it is straightforward to verify that the sequence

$\tilde{S}={\tilde{S}}_0=S_{0,0}$⇢ $S_{0,1}$⇢ $\cdots$⇢ $S_{0,l}={\tilde{S}}_1=S_{1,0}$⇢ $S_{1,1}$⇢ $\cdots$

is, in fact, an MMP on $\tilde{f}:\left(\tilde{S},B_{\tilde{S}}\right)\to Z$ with scaling of $\tilde{C}$.

By inductive hypothesis, this MMP terminates. This means that, after finitely many steps, $K_{{\tilde{S}}_m}+{\tilde{B}}_{S,m}$ becomes nef over $T_m$, and consequently, $K_{S_m}+B_{S_m}$ is also nef over $T_m$. On the other hand, $-\left(K_{S_m}+B_{S_m}\right)$ is ample over $T_m$, so $S_m\to T_m$ does not contract any curve on $Z_m$. Similarly, $S_{m+1}\to T_m$ does not contract any curve on $S_{m+1}$. If $S_m$ intersects $Exc\left({\alpha }_m\right)$, then $S_m$ as a divisor on $X_m$ ample over $T_m$. Hence, $-S_{m+1}$ is ample over $T_m$, which contradicts to the fact that $S_{m+1}\to T_m$ does not contract any curve on $S_{m+1}$. This implies that the original MMP terminates in a neighborhood of S.

To construct log minimal model in dimension n assuming non-vanishing, one needs the special termination with scaling in dimension n, which is reduced to termination with scaling in lower dimension. More precisely, let $\left(X/U,B+C\right)$ be a klt pair of dimension n and satisfy BCHM condition. Run MMP on $K_X+B$ over U with scaling C, we need to prove this MMP terminates. By the definition of such MMP, $K_{X_i}+B_i+{\lambda }_i{C}_i$ is nef over U and ${\lambda }_1\ge {\lambda }_2\ge \cdots$. Then $\left(X_i,B_i+{\lambda }_i{C}_i\right)$ is a minimal model of $\left(X,B+{\lambda }_i{C}_i\right)$ over U.

The critical aspect of proving termination with scaling lies in demonstrating that there are only finitely many possible minimal models [5] . Specifically, if the MMP does not terminate, it implies the existence of infinitely many distinct minimal models, which would contradict the boundedness condition imposed by the special termination.

References

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2. Kollár, J. and Mori, S. (1998) Birational Geometry of Algebraic Varieties. Cambridge University Press. >https://doi.org/10.1017/CBO9780511662560

3. Fujino, O. (2007) Special Termination and Reduction to pl Flips. In: Corti, A., Ed., Flips for 3-Folds and 4-Folds, Oxford University Press, 63-75. >https://doi.org/10.1093/acprof:oso/9780198570615.003.0004

4. Hartshorne, R. (1977) Algebraic Geometry. In: Hersh, P., Vakil, R. and Wunsch, J., Eds., Graduate Texts in Mathematics, Springer-Verlag.>https://doi.org/10.1007/978-1-4757-3849-0

5. Kawamata, Y. (2024) Algebraic Varieties: Minimal Models and Finite Generation. Cambridge University Press. >https://doi.org/10.1017/9781009344647