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Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a birational map.

In particular, if $X$ is the blow-up of $\mathbb{P}^n$ at $r+n-1$ points and $X'$ is the blow-up of $(\mathbb{P}^1)^n$ at $r$ points. Consider $\phi : X \dashrightarrow X'$ the birational map that makes the diagram with the blow-up's and $f$ commute, is $\phi$ a Crepant birational map?

Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a birational map. Is $f$ a Crepant birational map?

Let $ f: \mathbb{P}^n \rightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$.

In particular, if $X$ is the blow-up of $\mathbb{P}^n$ at $r+n-1$ points and $X'$ is the blow-up of $(\mathbb{P}^1)^n$ at $r$ points. Consider $\phi : X \dashrightarrow X'$ the birational map that makes the diagram with the blow-up's and $f$ commute, is $\phi$ a Crepant birational map?

Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a birational map.

In particular, if $X$ is the blow-up of $\mathbb{P}^n$ at $r+n-1$ points and $X'$ is the blow-up of $(\mathbb{P}^1)^n$ at $r$ points. Consider $\phi : X \dashrightarrow X'$ the birational map that makes the diagram with the blow-up's and $f$ commute, is $\phi$ a Crepant birational map?

Let $f:X \dashrightarrow X$ be a Crepant birational map, i.e. a birational map that preserve the canonical class.

Someone knows another characterization of Crepant birational map?

In particular, Is $ f: \mathbb{P}^n \rightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a Crepant birational map?

Let $ f: \mathbb{P}^n \rightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$.

In particular, if $X$ is the blow-up of $\mathbb{P}^n$ at $r+n-1$ points and $X'$ is the blow-up of $(\mathbb{P}^1)^n$ at $r$ points. Consider $\phi : X \dashrightarrow X'$ the birational map that makes the diagram with the blow-up's and $f$ commute, is $\phi$ a Crepant birational map?