Birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

Birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

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 $k$.

See also: Birational geometry on Wikipedia.