Birationality

Birationality is an equivalence relation on varieties over a field k. Two integral varieties X and Y are birational if there exist nonempty open subsets U ⊂ X and V ⊂ Y and an isomorphism between U and V, or equivalently if there exist rational maps f: X → Y and g: Y → X whose compositions are the identity on dense open subsets. In this sense, birational maps are isomorphisms that are defined only on a large part of the varieties.

A convenient viewpoint is via function fields. X and Y are birational over k precisely when their function fields k(X) and k(Y) are isomorphic as k-extensions. This perspective explains why birational geometry focuses on the behavior of rational functions on the varieties rather than on every geometric detail.

Examples and basic facts. Any projective space P^n is birational to any other projective space P^m of the same dimension, and a variety is called rational if it is birational to projective space. Smooth conics are birational to P^1; over algebraically closed fields they are actually isomorphic to P^1. For curves, birational equivalence classes coincide with isomorphism classes of smooth projective models, and the genus is a birational invariant: a smooth projective curve is rational if and only if its genus is zero.

In higher dimensions, birational geometry studies properties preserved under birational equivalence. Invariants such as dimension and the function field are preserved, while more refined invariants like the Kodaira dimension provide a deeper birational classification. A central topic is rationality: whether a given variety is birational to projective space; some varieties are rational, others are not, and deciding this can be subtle. The Minimal Model Program is a major framework in modern birational geometry for understanding birational models.