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  "body": "\n## Summary\n\nThe minimal model program, or MMP, is a strategy for finding a simplest variety in each birational class of projective varieties.\nIt modifies a variety step by step, by contractions and flips, so as to make the [[canonical bundle|W05VYY4HBR]] [[nef|RZ66X0NAM5]].\nIt ends either with a minimal model or with a Mori fibre space, a fibration whose general fibres are Fano varieties.\nFor surfaces it is the classical process of contracting $(-1)$-curves; in dimension $3$ it was carried out by Mori and others.\n\n## Explanations\n\n### L1 human\n\nMany different varieties have the same function field, because one can always blow up points to make a variety more complicated.\nThe program runs this backwards: it looks for curves on which the canonical bundle is negative and removes them.\nFor a surface, it repeatedly contracts curves that look like a projective line with self-intersection $-1$.\nWhen no such curves are left, either the canonical bundle is nef and the surface is minimal, or the surface is $\\mathbb{P}^2$ or fibred in projective lines over a curve.\n",
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