I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:


1. $K\_X$ is nef.
2. Every birational morphism from $X$ to $Y$ must be an isomorphism, where $Y$ is another variety with only terminal singularities.


Suppose $X$ is a smooth minimal model of dimension greater than 2. Why are 1 and 2 equivalent definitions? In particular, why does 2 imply 1? Does the exceptional divisor contain an extremal ray?



Why are the different definitions of minimal model equivalent?
1 implies 2 follows from the so called "negativity lemma", see for example Lemma 3.39 in the book by Kollar-Mori. The main point is that an effective linear combination of exceptional divisors can never be $f$-nef for a birational morphism $f:X \to Y$.


2 implies 1 is not true: consider $X = \mathbb{P}^n$.


(The condition 2 is the classical definition of minimal but Mori realised that 1 is a better definition.)


