# Why are the different definitions of minimal model equivalent?

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?

• Thanks! This clarifies my understanding of exceptional divisor. By the way, if both $X$ and $Y$ are varieties of the same dimension, $f$ is a nontrivial birational morphism from $X$ to $Y$, does $X$ necessarily contain an exceptional divisor which is contracted by $f$? I know this is true when both $X$ and $Y$ are smooth. – user17314 Aug 25 '11 at 4:12

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 $X = \mathbb{P}^n$.