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  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ and $Y$ be integral [[algebraic varieties|HV6YBS0YDH]] over $k$.\nThe varieties $X$ and $Y$ are *birationally equivalent* if there is a [[birational map|KA5890R9YM]] $X \\dashrightarrow Y$, that is, a [[rational map|4D073CCYGV]] with a rational inverse.\nA *birational invariant* is a property or a quantity attached to varieties that takes the same value on birationally equivalent varieties.\n*Birational geometry* is the study of varieties up to birational equivalence and of their birational invariants.\n\n## Remarks\n\nTwo integral varieties over $k$ are birationally equivalent exactly when their [[function fields|2D5SS6WM14]] are isomorphic as extensions of $k$.\nExample: $\\mathbb{P}^2$ and $\\mathbb{P}^1 \\times \\mathbb{P}^1$ are birationally equivalent.\nThe map $[x : y : z] \\mapsto ([x : z], [y : z])$ is defined where $z \\ne 0$, and its inverse $([x : 1], [y : 1]) \\mapsto [x : y : 1]$ is defined on the dense open set where both second coordinates are nonzero.\nBoth function fields are $k(s, t)$.\nNon-example: $\\mathbb{P}^1$ and a smooth plane cubic curve $E$ over $\\C$ are not birationally equivalent.\nFor smooth projective curves a birational map extends to an isomorphism, and $\\mathbb{P}^1$ has genus $0$ while $E$ has genus $1$.\nThe dimension, the geometric genus and the Kodaira dimension of smooth projective varieties are birational invariants.\n",
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