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  "body": "\n## Summary\n\nA modification of a [[reduced complex space|831N2Q4GCB]] $Y$ is a [[proper|15XEBGNGHC]] surjective holomorphic map $X \\to Y$ that is biholomorphic outside thin analytic subsets.\nIt changes $Y$ only over a nowhere dense analytic set, where it may replace points by larger sets.\n[[Blowing up|C4KGYAFPN7]], normalization and resolution of singularities give modifications.\nModifications are the analytic counterpart of proper birational morphisms in algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nA modification keeps a space as it is except along a small bad set, where it may replace points by curves or other larger pieces.\nThe basic example is the blow-up of the plane $\\C^2$ at the origin.\nIt keeps every point except the origin, and it replaces the origin by a projective line that records the directions of lines through the origin.\n",
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