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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $X$ be a smooth projective [[surface|KFTECAATKG]] over $k$.\nLet $E \\subset X$ be a curve with $E \\cong \\mathbb{P}^1$ and self-[[intersection number|4HV4WTAX9P]] $E \\cdot E = -1$.\nThen there are a smooth projective surface $Y$, a point $p \\in Y$ and a morphism $f \\colon X \\to Y$ with the following property.\nThe morphism $f$ is isomorphic to the [[blow-up|C4KGYAFPN7]] of $Y$ at $p$, with exceptional curve $E$.\n\n## Remarks\n\nThe idea of the proof is to choose a very ample divisor $H$ on $X$ with $H^1(X, \\mathcal{O}_X(H)) = 0$, put $m = H \\cdot E$, and use the linear system $\\abs{H + mE}$.\nIt has degree $0$ on $E$, so it contracts $E$ to a point, and a computation of sections shows that the image surface is smooth at that point.\nExample: a smooth cubic surface over $k$ contains $27$ lines, each a $(-1)$-curve, and contracting six disjoint lines among them gives $\\mathbb{P}^2$.\nThe condition $E \\cdot E = -1$ cannot be weakened to $E \\cdot E \u003c 0$.\nA smooth rational curve with $E \\cdot E = -2$ can sometimes be contracted, but the image point is then singular, an $A_1$ singularity.\n\n## Explanations\n\n### L1 human\n\nBlowing up a point of a smooth surface creates a new curve, a copy of the projective line with self-intersection $-1$.\nThe theorem says the converse: any such curve comes from a blow-up and can be blown down.\nSo the curves of this kind are exactly the traces left by blowing up.\n\n### L2 human\n\nThe theorem is the basic step of the minimal model program for surfaces.\nContracting $(-1)$-curves one at a time lowers the Picard number, so the process stops, at a surface with no $(-1)$-curves.\nSo every smooth projective surface is obtained from a surface without $(-1)$-curves by finitely many blow-ups of points.\nThe surfaces without $(-1)$-curves are then sorted by the Enriques–Kodaira classification.\n",
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  "name": "Castelnuovo's contraction theorem",
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