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  "body": "\n## Definition\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]].\nAn *algebraic surface* over $k$ is an [[algebraic variety|HV6YBS0YDH]] over $k$ of [[dimension|36X2WTHYCB]] $2$.\n\n## Remarks\n\nIn the classification theory, \"surface\" usually means a smooth projective surface, and these extra conditions are then stated.\nExample: the projective plane $\\mathbb{P}^2$ and the product $\\mathbb{P}^1 \\times \\mathbb{P}^1$ are algebraic surfaces.\nExample: the zero set in $\\mathbb{P}^3$ of one nonconstant irreducible homogeneous polynomial is an algebraic surface, since a hypersurface in $\\mathbb{P}^3$ has dimension $3 - 1 = 2$.\nNon-example: a plane curve such as $y^2 = x^3 - x$ in $\\mathbb{A}^2$ is an algebraic variety of dimension $1$, not $2$.\nNon-example: the affine space $\\mathbb{A}^3$ is a variety of dimension $3$.\nOver $\\C$, by Chow's theorem, a smooth projective algebraic surface is the same as a compact complex surface that embeds in projective space.\nNot every compact complex surface is of this kind.\n",
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