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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $X$ be a smooth [[projective variety|A4H4CZ2V35]] over $k$ of dimension $n$.\nLet $K_X$ be its canonical divisor (see [[canonical bundle|W05VYY4HBR]]).\nLet $N_1(X)$ be the real vector space of $1$-cycles on $X$ modulo numerical equivalence.\nLet $\\overline{NE}(X) \\subseteq N_1(X)$ be the closure of the cone spanned by the classes of curves.\nWrite $\\overline{NE}(X)_{K_X \\ge 0} = \\set{z \\in \\overline{NE}(X) : K_X \\cdot z \\ge 0}$, using the [[intersection pairing|4HV4WTAX9P]].\nThen there are countably many rational curves $C_i \\subset X$ with\n$$0 \u003c -K_X \\cdot C_i \\le n + 1$$\nsuch that\n$$\\overline{NE}(X) = \\overline{NE}(X)_{K_X \\ge 0} + \\sum_i \\R_{\\ge 0} [C_i].$$\nMoreover, for every [[ample|RRK8P6TXG2]] divisor $H$ and every $\\varepsilon \u003e 0$, only finitely many of the $C_i$ are needed to write\n$$\\overline{NE}(X) = \\overline{NE}(X)_{K_X + \\varepsilon H \\ge 0} + \\sum_{i \\text{ finite}} \\R_{\\ge 0} [C_i].$$\n\n## Remarks\n\nThe theorem is due to Mori.\nThe rational curves come from his bend-and-break method: a curve on which $K_X$ is negative deforms a lot and breaks off rational curves.\nThe method works first in characteristic $p$ and then passes to characteristic $0$.\nExample: on $\\mathbb{P}^n$ the space $N_1$ is spanned by the class of a line $\\ell$, and $-K \\cdot \\ell = n + 1$.\nSo the bound $n + 1$ is sharp, and $\\overline{NE}(\\mathbb{P}^n) = \\R_{\\ge 0} [\\ell]$.\nIn characteristic $0$, the theorem extends to log terminal pairs, together with the contraction theorem which contracts each $K_X$-negative extremal ray.\n\n## Explanations\n\n### L1 human\n\nThe classes of curves on a variety span a cone in a finite-dimensional real vector space.\nThe theorem describes the part of this cone where the canonical class is negative.\nThat part is spanned by the classes of rational curves of bounded degree, one ray at a time, like the edges of a polyhedral cone.\n\n### L2 human\n\nThe cone theorem is the first step of the minimal model program: it provides the extremal rays to contract.\nWith the contraction theorem, each $K_X$-negative extremal ray gives a morphism that contracts exactly the curves in that ray.\nFor surfaces, these rays are spanned by $(-1)$-curves, by fibres of a ruling, or by a line in $\\mathbb{P}^2$.\n",
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  "name": "Cone theorem",
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