Minimal model program

Summary

The minimal model program, or MMP, is a strategy for finding a simplest variety in each birational class of projective varieties. It modifies a variety step by step, by contractions and flips, so as to make the canonical bundle nef. It ends either with a minimal model or with a Mori fibre space, a fibration whose general fibres are Fano varieties. For surfaces it is the classical process of contracting $(-1)$-curves; in dimension $3$ it was carried out by Mori and others.