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  "body": "\n## Definition\n\nLet $X$ be a normal [[projective variety|A4H4CZ2V35]] over a field of characteristic $0$ with $\\Q$-factorial terminal singularities.\nLet $K_X$ be its canonical divisor, the divisor class of the [[canonical bundle|W05VYY4HBR]].\nThe variety $X$ is a *minimal model* if $K_X$ is [[nef|RZ66X0NAM5]].\nThe *minimal model program* applied to $X$ is the following procedure.\n\n1. If $K_X$ is nef, stop: $X$ is a minimal model.\n2. Otherwise, by the [[cone theorem|ZJSW0HP75H]] there is an extremal ray $R$ of the closed cone of curves of $X$ with $K_X \\cdot R \u003c 0$.\n   It has a contraction $\\varphi_R \\colon X \\to Z$ of exactly the curves whose classes lie in $R$.\n3. If $\\dim Z \u003c \\dim X$, stop: $\\varphi_R$ is a *Mori fibre space*.\n4. If $\\varphi_R$ is [[birational|KA5890R9YM]] and its exceptional locus is a divisor, replace $X$ by $Z$ and go back to step 1.\n5. If $\\varphi_R$ is birational and its exceptional locus has codimension at least $2$, replace $X$ by the flip $X^+$ of $\\varphi_R$.\n   Then go back to step 1.\n\n## Remarks\n\nIn steps 4 and 5 the new variety again has $\\Q$-factorial terminal singularities, which is why the program allows these singularities.\nThe program terminates when there is no infinite sequence of flips.\nFlips exist in all dimensions by a theorem of Birkar, Cascini, Hacon and McKernan, and termination of flips is known in low dimensions, including dimension $3$, but open in general.\nExample: for a smooth projective surface, there are no small contractions, and step 4 contracts a $(-1)$-curve by [[Castelnuovo's contraction theorem|WJ5WRGG5DE]].\nThe surface $\\mathbb{P}^2$ is not a minimal model, since $K_{\\mathbb{P}^2} \\cdot \\ell = -3 \u003c 0$ for a line $\\ell$.\nThe program stops at step 3 for it, with $Z$ a point, so $\\mathbb{P}^2$ is a Mori fibre space.\nA smooth curve of genus $g \\ge 1$ is a minimal model, since $\\deg K = 2g - 2 \\ge 0$.\n",
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