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  "body": "\n## Summary\n\nAn algebraic surface is an [[algebraic variety|HV6YBS0YDH]] of [[dimension|36X2WTHYCB]] two.\nOver the complex numbers a smooth projective algebraic surface is a compact complex manifold of complex dimension two, so of real dimension four.\nSurfaces are the first case where birational geometry is subtle, and their classification by Enriques and Kodaira is a model for higher dimensions.\n\n## Explanations\n\n### L1 human\n\nA curve is described by one equation in two variables, and a surface by one equation in three variables, such as $x^2 + y^2 + z^2 = 1$.\nLocally a surface looks like a piece of the plane, so it has two directions of freedom.\nOver the complex numbers each of these directions counts twice, so a complex surface is a four-dimensional real object.\nExamples are the projective plane, the product of two curves, and the zero set of a polynomial in projective three-space.\n",
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