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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.

$K_X$ is nef.
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 1 and 2 are equivalent definitions? In particular, why 2 implies 1. Does exceptional divisor contain extremal ray?

edited Aug 22 2011 at 18:00

André Henriques
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asked Aug 22 2011 at 17:08

ricciflow
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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. – ricciflow Aug 25 2011 at 4:12

ulrich · Accepted Answer · 2011-08-23 07:41:56Z UTC

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.)

Why are the different definitions of minimal model equivalent?

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