Why are the different definitions of minimal model equivalent? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:00:26Z http://mathoverflow.net/feeds/question/73419 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73419/why-are-the-different-definitions-of-minimal-model-equivalent Why are the different definitions of minimal model equivalent? ricciflow 2011-08-22T17:08:39Z 2011-08-23T07:41:56Z <p>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.</p> <ol> <li>$K_X$ is nef.</li> <li>Every birational morphism from $X$ to $Y$ must be an isomorphism, where $Y$ is another variety with only terminal singularities.</li> </ol> <p>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?</p> http://mathoverflow.net/questions/73419/why-are-the-different-definitions-of-minimal-model-equivalent/73472#73472 Answer by ulrich for Why are the different definitions of minimal model equivalent? ulrich 2011-08-23T07:41:56Z 2011-08-23T07:41:56Z <p>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 $f:X \to Y$.</p> <p>2 implies 1 is not true: consider $X = \mathbb{P}^n$.</p> <p>(The condition 2 is the classical definition of minimal but Mori realised that 1 is a better definition.)</p>