---
tag: KA5890R9YM
kind: concept
name: Birational map
msc: ["14E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Birational_geometry
---

## Summary

A birational map between two varieties is a rational map that has a rational inverse.
It is an isomorphism between nonempty open subsets, so birational varieties look the same away from lower-dimensional subsets.
Two varieties are birational exactly when their function fields are isomorphic over the base field.
The classification of varieties up to birational equivalence is the goal of birational geometry.

## Explanations

### L1 human

Two shapes are birational if, after removing some small pieces from each, what is left is the same shape.
A smooth conic in the plane, such as $x^2 + y^2 = 1$, is birational to the line.
Each line through the point $(-1, 0)$ with slope $t$ meets the circle in one more point, and this matches points of the circle with numbers $t$.
Only the point $(-1, 0)$ itself is left out, since it would need the vertical line through it, which has no slope.
So the circle and the line are birational.
The plane and the projective plane are another birational pair: they agree once the line at infinity is removed, yet they are not isomorphic.