Birational

In mathematics, 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 ⇢ 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: 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. 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 with equation x² + y² − 1 = 0 is a rational curve, because the formulas

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