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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[algebraic varieties|HV6YBS0YDH]] over an algebraically closed field $k$.\nA *birational map* $\\varphi \\colon X \\dashrightarrow Y$ is a [[rational map|4D073CCYGV]] that has an inverse: a rational map $\\psi \\colon Y \\dashrightarrow X$ such that $\\psi \\circ \\varphi = \\id_X$ and $\\varphi \\circ \\psi = \\id_Y$ as rational maps.\nHere $\\varphi$ and $\\psi$ are required to be dominant, so that the compositions are defined.\n$X$ and $Y$ are *birational*, or birationally equivalent, if there is a birational map $X \\dashrightarrow Y$.\n\n## Remarks\n\nEquivalently, there are nonempty [[open subsets|02BH3MCBQB]] $U \\subseteq X$ and $V \\subseteq Y$ and an isomorphism of varieties $U \\to V$ (a [[morphism|CDS1VWB7A3]] with inverse morphism).\nAlso equivalently, there is an isomorphism of $k$-algebras between the [[function fields|2D5SS6WM14]] $k(Y)$ and $k(X)$.\n\nExample: $\\mathbb{A}^2$ and $\\mathbb{P}^2$ are birational, since $\\mathbb{A}^2$ is isomorphic to the open subset $\\set{z \\ne 0}$ of $\\mathbb{P}^2$.\nThey are not isomorphic, since every regular function on $\\mathbb{P}^2$ is constant and $\\mathbb{A}^2$ has the nonconstant regular function $x$.\n\nExample: the conic $C = V(x^2 + y^2 - 1) \\subseteq k^2$, with $\\operatorname{char} k \\ne 2$, is birational to $\\mathbb{A}^1$ through $t \\mapsto \\left(\\frac{1 - t^2}{1 + t^2}, \\frac{2t}{1 + t^2}\\right)$, whose inverse is $(x, y) \\mapsto y/(1 + x)$.\n\nNon-example: the rational map $\\mathbb{P}^2 \\dashrightarrow \\mathbb{P}^1$, $[x : y : z] \\mapsto [x : y]$, is dominant but not birational, since $\\dim \\mathbb{P}^2 = 2 \\ne 1 = \\dim \\mathbb{P}^1$ and birational varieties have the same dimension.\n",
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