birationaalisuus

Birationaalisuus (birationality) is a concept in algebraic geometry describing when two algebraic varieties share the same function field. For varieties X and Y over a field k, X and Y are birational if there exist dense open subsets U ⊂ X and V ⊂ Y and a k-regular isomorphism U → V. Equivalently, there are dominant rational maps f: X ⇢ Y and g: Y ⇢ X such that the compositions g ∘ f and f ∘ g are the identities on dense open subsets. A birational map is a rational map that has a rational inverse on such subsets.

Birationality classifies varieties up to a common dense open part. Many invariants are preserved only up to birational equivalence; among the notable ones are the function field k(X) ≅ k(Y) and, for smooth projective varieties, invariants such as the Kodaira dimension and the plurigenera. By contrast, some invariants, like certain Hodge numbers, need not be birational invariants in higher dimensions.

Rationality is a stronger condition: a variety is rational if it is birational to projective space P^n. For curves, birationality between smooth projective curves implies isomorphism, and a curve is rational exactly when its genus is 0. For surfaces and higher dimensions, many varieties are birational to simpler models (for example, the projective plane P^2 is birational to many rational surfaces such as the Hirzebruch surfaces).

Birationaalisuus is central to birational geometry, which seeks to classify varieties up to birational equivalence. It underpins the study of minimal models, the Cremona group of birational self-maps, and the broader program of understanding algebraic varieties through their function fields.