Flops connect minimal models

TL;DR

This paper discusses the connection between flop operations and minimal models in algebraic geometry, providing insights into their interplay and implications for the minimal model program.

Contribution

It offers a new perspective on how flops relate to minimal models, building on the work of Birkar, Cascini, Hacon, and McKernan.

Findings

Establishes a link between flops and minimal models.

Provides a simplified proof or new insight into the minimal model program.

Highlights implications for the structure of algebraic varieties.

Abstract

A remark on a paper by Birkar-Cascini-Hacon-McKernan.

Equations

$$X=X_0⇢X_1⇢\cdots ⇢X_t=X^{\prime }$$

$$K_V={\mu }^{\ast }(K_X+B)-{\mu }_{\ast }^{-1}B+E=({\mu }^{\prime }{)}^{\ast }(K_{X^{\prime }}+B^{\prime })-({\mu }_{\ast }^{\prime }{)}^{-1}{B}^{\prime }+E^{\prime }$$

$$(({\mu }^{\prime }{)}^{\ast }(K_{X^{\prime }}+B^{\prime })+{\mu }_{\ast }^{-1}B-({\mu }_{\ast }^{\prime }{)}^{-1}{B}^{\prime }+{\bar{E}}^{\prime })\cdot C)\ge 0.$$

$$({\mu }^{\ast }(K_X+B)+\bar{E})\cdot C)<0.$$

$$\begin{array}{rl}{t}_0=\min & \{t\in \mathbf{R}|((K_X+B+lL+tH)\cdot R)\ge 0\text{ for all extremal rays }R\\ & \text{ for }(X,B+lL)\text{ over }S\text{ s.t. }((K_X+B)\cdot R)=0\}.\end{array}$$

$$0>((K_X+B+lL)\cdot C)\ge -2\dim X$$

$$\begin{array}{rl}& ((K_X+B+elL)\cdot C)\\ & =\frac{1}{2k\dim X+1}((K_X+B+lL)\cdot C)+\frac{2k\dim X}{2k\dim X+1}((K_X+B)\cdot C)\\ & \ge \frac{1}{2k\dim X+1}(-2\dim X+2\dim X)=0\end{array}$$

$$\begin{array}{rl}& ((K_X+B+elL+e{t}_{0}H)\cdot C)\\ & =\frac{1}{2k\dim X+1}((K_X+B+lL+t_{0}H)\cdot C)+\frac{2k\dim X}{2k\dim X+1}((K_X+B)\cdot C)\\ & \ge \frac{1}{2k\dim X+1}(-2\dim X+2\dim X)=0\end{array}$$

Full text

A flop of a pair $(X,B)$ is a flip of a pair $(X,B^{\prime})$ which is crepant for $K_{X}+B$ where $B^{\prime}$ is a suitably chosen different boundary. We prove the following:

More precisely, there exist an effective $\mathbf{Q}$-divisor $D$ on $X$ such that $(X,B+D)$ is klt and a factorization of the birational map $\alpha$

[TABLE]

which satisfy the following conditions:

(1) $\alpha_{i}:X_{i-1}\to X_{i}$ ($1\leq i\leq t$) is a flip for the pair $(X_{i},B_{i}+D_{i})$ over $S$, where $B_{i}$ and $D_{i}$ are strict transforms of $B$ and $D$, respectively.

(2) $\alpha_{i}$ is crepant for $K_{X_{i-1}}+B_{i-1}$ in the sense that the pull-backs of $K_{X_{i-1}}+B_{i-1}$ and $K_{X_{i}}+B_{i}$ coincide on a common log resolution.

We remark that the boundary $B$ need not be assumed to be big as in [1] Corollary 1.1.3. For example, a birational map between Calabi-Yau manifolds can be decomposed into a sequence of flops. The number of marked minimal models which are birationally equivalent to a fixed pair is finite if $B$ is big ([1] Corollary 1.1.5), but it is not the case in general (cf. [4]), where a marked minimal model is a pair consisting of a minimal model and a fixed birational map to it. If we relax the condition for the pairs to being klt, then we should allow crepant blowings up besides flops.

The theorem was already proved in the case $\dim X=3$ and $B=0$; first in [2] assuming the abundance which was proved afterwards, and later in [5] without assumption.

[TABLE]

where $E$ and $E^{\prime}$ are effective divisors whose supports coincide with the exceptional loci of $\mu$ and $\mu^{\prime}$, respectively, because $(X,B)$ and $(X^{\prime},B^{\prime})$ are terminal. Assume that there is a prime divisor on $V$ which is contracted by $\mu$ but not by $\mu^{\prime}$. Then it is an irreducible component of $E$ but not of $E^{\prime}$. We set $F=\min\{E,E^{\prime}\}$, $\bar{E}=E-F$ and $\bar{E}^{\prime}=E^{\prime}-F$. By the Hodge index theorem, there exists a curve $C$ on $V$ which is contracted by $\mu$ and is contained in $\text{Supp}(\bar{E})$ but not in $\text{Supp}(\mu_{*}^{-1}B+\bar{E}^{\prime})$ and such that $(\bar{E}\cdot C)<0$. Since $\mu_{*}^{-1}B\geq(\mu^{\prime}_{*})^{-1}B^{\prime}$, we have

[TABLE]

But this is a contradiction to

[TABLE]

The case where there is a prime divisor on $V$ which is contracted by $\mu^{\prime}$ but not by $\mu$ is treated similarly.

Let $L^{\prime}$ be an effective $f^{\prime}$-ample divisor on $X^{\prime}$, and $L$ its strict transform on $X$. There exists a small positive number $l$ such that $(X,B+lL)$ is klt. If $K_{X}+B+lL$ is $f$-nef over $S$, then $\alpha$ becomes a morphism by the base point free theorem, hence an isomorphism since $X$ is $\mathbf{Q}$-factorial. Therefore we may assume that $K_{X}+B+l^{\prime}L$ is not $f$-nef over $S$ for any $0<l^{\prime}\leq l$.

Let $H$ be an effective divisor on $X$ such that $(X,B+lL+tH)$ is klt and $K_{X}+B+lL+tH$ is $f$-nef for some positive number $t$. We shall run the MMP for the pair $(X,B+l^{\prime}L)$ over $S$ with scaling of $H$ for some $l^{\prime}$. Since $\alpha$ is an isomorphism in codimension $1$, there are only flips in this MMP. The following lemma shows that we can choose extremal rays such that the flips are crepant with respect to $K_{X}+B$.

Let $k$ be a positive integer such that $k(K_{X}+B)$ is a Cartier divisor. We set $e=\frac{1}{2k\dim X+1}$.

(2) Let

[TABLE]

Then $K_{X}+B+elL+et_{0}H$ is $f$-nef, and there exists an extremal ray $R$ for $(X,B+elL)$ over $S$ such that $((K_{X}+B+elL+et_{0}H)\cdot R)=((K_{X}+B)\cdot R)=0$.

[TABLE]

by [3].

We claim that $((K_{X}+B)\cdot C)=0$. Indeed we have otherwise $((K_{X}+B)\cdot C)\geq 1/k$, hence

[TABLE]

a contradiction.

(2) If $K_{X}+B+elL+et_{0}H$ is not $f$-nef, then there exists an extremal ray $R$ for $(X,B+elL+et_{0}H)$ over $S$. Then $R$ is also an extremal ray for $(X,B+lL+t_{0}H)$ because $(X,B)$ is $f$-nef. Since the pair $(X,B+lL+t_{0}H)$ is klt, $R$ is generated by a rational curve $C$ such that $((K_{X}+B+lL+t_{0}H)\cdot C)\geq-2\dim X$ by [3]. Then we have

[TABLE]

a contradiction. Therefore $K_{X}+B+elL+et_{0}H$ is $f$-nef.

Since $B+lL$ is $f$-big, the number of extremal rays for $(X,B+lL)$ over $S$ is finite. Hence there exists such an $R$ that $((K_{X}+B+lL+t_{0}H)\cdot R)=((K_{X}+B)\cdot R)=0$. ∎

We note that we can deduce (1) from only the finiteness of extremal rays, but not (2). The point is that the number $e$ stays independent of $t_{0}$ during the MMP.

We run the MMP for $(X,B+elL)$ with scaling of $H$. We take an extremal ray $R$ such that $((K_{X}+B+elL+et_{0}H)\cdot R)=((K_{X}+B)\cdot R)=0$. The flip exists by [1] Corollary 1.4.1. Since $(lL+t_{0}H)\cdot R)=0$, the pair $(X,B+lL+t_{0}H)$ remains to be klt after the flip. We also note that $k(K_{X}+B)$ remains to be a Cartier divisor after the flip by the base point free theorem. Therefore we can continue the process. By the termination theorem of directed flips ([1] Corollary 1.4.2), we complete our proof. ∎