In mathematics, birational geometry is a field of algebraic geometry in which the goal 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) ${\displaystyle X}$ to another variety ${\displaystyle Y}$, written as a dashed arrow X ⇢Y, is defined as a morphism from a nonempty open subset ${\displaystyle U\subset X}$ to ${\displaystyle Y}$. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset ${\displaystyle U}$ is always dense in ${\displaystyle X}$, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions.

A birational map from X to Y is a rational map f : X ⇢ Y such that there is a rational map Y ⇢ X inverse to f. A birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y, and vice versa: an isomorphism between nonempty open subsets of X, Y by definition gives a birational map f : X ⇢ Y. In this case, X and Y are said to be 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.

A special case is a birational morphism f : X → Y, meaning a morphism which is birational. That is, f is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of X to points in Y.

A variety X is said to be rational if it is birational to affine space (or equivalently, to projective space) of some dimension. Rationality is a very natural property: it means that X minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset.

For example, the circle ${\displaystyle X}$ with equation ${\displaystyle x^{2}+y^{2}-1=0}$ in the affine plane is a rational curve, because there is a rational map f : ${\displaystyle \mathbb {A} ^{1}}$ ⇢ X given by

which has a rational inverse g: X ⇢ ${\displaystyle \mathbb {A} ^{1}}$ given by

Applying the map f with t a rational number gives a systematic construction of Pythagorean triples.