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  "body": "\n## Definition\n\nLet $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field, with [[canonical divisor|W05VYY4HBR]] $K_X$.\n$X$ is a *del Pezzo surface* if the anticanonical divisor $-K_X$ is [[ample|RRK8P6TXG2]].\nIts *degree* is the self-intersection number $d = K_X^2$.\n\n## Remarks\n\nThe degree satisfies $1 \\le d \\le 9$.\nClassification: $X$ is either $\\mathbb{P}^1 \\times \\mathbb{P}^1$, of degree $8$, or the [[blow-up|C4KGYAFPN7]] of $\\mathbb{P}^2$ at $9 - d$ points in general position.\nHere general position means three conditions.\nNo three of the points lie on a line, and no six lie on a conic.\nNo eight lie on a cubic that is singular at one of them.\nExample: $\\mathbb{P}^2$ has $K = -3L$ for a line $L$, so $-K = 3L$ is ample and $\\mathbb{P}^2$ is a del Pezzo surface of degree $9$.\nExample: a smooth cubic surface $X \\subset \\mathbb{P}^3$ has $K_X = -H$ by adjunction, where $H$ is a plane section, so $-K_X$ is ample and $d = H^2 = 3$.\nNon-example: blow up $\\mathbb{P}^2$ at three points on a line $L$, and let $\\tilde L$ be the strict transform of $L$.\nThen $\\tilde L^2 = 1 - 3 = -2$, and adjunction gives $K \\cdot \\tilde L = -2 - \\tilde L^2 = 0$.\nSo $-K \\cdot \\tilde L = 0$, and $-K$ is not ample.\nSome authors allow mild singularities, such as ADE singularities, and call the smooth ones nonsingular del Pezzo surfaces.\n",
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