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  "body": "\n## Summary\n\nBirational geometry is the part of algebraic geometry that studies varieties up to [[birational equivalence|KA5890R9YM]].\nTwo varieties are birationally equivalent when they become isomorphic after removing lower-dimensional closed subsets, or equivalently when they have isomorphic [[function fields|2D5SS6WM14]].\nIts main questions are which varieties are [[rational|XV4K9GREJ9]], and how to choose a simplest variety in each birational class.\nThe [[minimal model program|Q8CTDCZH01]] is its central project in higher dimensions.\n\n## Explanations\n\n### L1 human\n\nIn birational geometry one is allowed to cut out a curve or a point and glue in something else, as long as the bulk of the variety stays the same.\nThe plane $\\mathbb{P}^2$ and the quadric surface $\\mathbb{P}^1 \\times \\mathbb{P}^1$ are not isomorphic, but they agree on dense open subsets, so they are birational.\n[[Blowing up|C4KGYAFPN7]] a point is the basic move: it replaces a point by a line of directions.\nNumbers that do not change under such moves, such as the geometric genus or the Kodaira dimension, are the main tools to tell birational classes apart.\n",
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