Birational

Birational describes a relation between algebraic varieties that identifies two objects if they become the same after allowed changes of coordinates on dense open subsets. Let k be a field. Two integral varieties V and W over k are birational if there exist rational maps f: V -> W and g: W -> V such that the compositions g∘f and f∘g restrict to the identity on dense open subsets where they are defined. Equivalently, their function fields k(V) and k(W) are isomorphic as extensions of k.

Rational maps are given by rational functions and need not be defined everywhere; a birational map is a rational map that has a rational inverse. In practice, a birational map is an isomorphism between V and W on some dense open subset, with both directions describable by rational functions.

Birational geometry studies varieties up to birational equivalence. Birational invariants include the dimension and, for curves, the function field. For curves, smooth projective curves are birational if and only if they are isomorphic, and the genus is a complete birational invariant: a smooth projective curve is birational to the projective line P^1 exactly when it has genus 0 (i.e., it is rational).

Rational varieties are those birational to projective space. Examples: affine space A^n and projective space P^n are birational; a conic over k is birational to P^1 if it has a k-rational point. The topic also covers more complex phenomena in higher dimensions, where rationality is a subtle question, and birational geometry uses tools such as blow-ups and invariants like the Kodaira dimension to organize birational classes.