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Concept
Birational geometry
Summary
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) to another variety , written as a dashed arrow X Y, is defined as a morphism from a nonempty open subset to . By definition of the Zariski topology used in algebraic geometry, a nonempty open subset is always dense in , 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 with equation in the affine plane is a rational curve, because there is a rational map f : ⇢ X given by
which has a rational inverse g: X ⇢ given by
Applying the map f with t a rational number gives a systematic construction of Pythagorean triples.
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