Birationaleisuus

Birationaleisuus, or birationality, is a notion in algebraic geometry that describes when two irreducible algebraic varieties over a field K have the same function field up to isomorphism. Two varieties X and Y are birational if there exists a rational map f: X → Y which restricts to an isomorphism between some dense open subset of X and a dense open subset of Y. Equivalently, the function fields K(X) and K(Y) are isomorphic as extensions of K. The birational class of a variety consists of all varieties birational to it.

A birational map is a rational map that has a rational inverse defined on dense open subsets; the composition of two birational maps is birational. Birational equivalence is an equivalence relation among irreducible varieties. It is coarser than isomorphism: birational varieties need not be isomorphic, especially in higher dimensions.

Examples and consequences:

- Projective space P^n over K is birational to any rational variety, and any two rational varieties share a common birational classification with P^n.

- For curves, if X and Y are smooth projective, then X and Y are birational if and only if they are isomorphic. Birational maps preserve the genus in this setting.

- Over non-closed fields, the existence of a birational map can depend on the base field; a conic may be birational to P^1 over K if it has a rational point, but not in general otherwise.

Birational geometry studies these maps to classify varieties up to birational equivalence and to understand how properties change under birational transformations. Important invariants, such as the function field and, in higher dimensions, the Kodaira dimension, are preserved under birational equivalence.