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A rational map from one variety (understood to be irreducible) $X$ to another variety $Y$, written as a dashed arrow $X \dashrightarrow Y$, is defined as a morphism from a nonempty open subset $U$ of $X$ to $Y$. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always the complement of a lower-dimensional subset of X. Concretely, a rational map can be written in coordinates using rational functions.
A birational map from $X$ to $Y$ is a rational map $f\colon X \dashrightarrow Y$ such that there is a rational map $Y \dashrightarrow X$ inverse to $f$. A birational map induces an isomorphism from a nonempty open subset of $X$ to a nonempty open subset of $Y$. In this case, we say that $X$ and $Y$ are birational, or birationally equivalent. In algebraic terms, two varieties over a field $k$ are birational if and only if their function fields are isomorphic as extension fields of $k$.