---
tag: RVX312A2D2
kind: definition
name: Rational variety
msc: [14M20, "14E08"]
defines: XV4K9GREJ9
uses: [HV6YBS0YDH, ZT9A6JK8QF, KA5890R9YM, 2D5SS6WM14, D7TR0YWR9H]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_variety
---

## Definition

Let $k$ be a field and let $X$ be an integral [[algebraic variety|HV6YBS0YDH]] over $k$ of dimension $n$.
The variety $X$ is *rational* over $k$ if it is birationally equivalent to the [[projective space|ZT9A6JK8QF]] $\mathbb{P}^n_k$.
This means that there is a [[birational map|KA5890R9YM]] $\mathbb{P}^n_k \dashrightarrow X$.

## Remarks

Equivalently, the [[function field|2D5SS6WM14]] $k(X)$ is purely transcendental over $k$: it is isomorphic to $k(t_1, \ldots, t_n)$ for elements $t_1, \ldots, t_n$ of [[transcendence degree|D7TR0YWR9H]] $n$.
Example: the conic $C \colon x^2 + y^2 = z^2$ in $\mathbb{P}^2_\Q$ is rational.
The map $[s : t] \mapsto [s^2 - t^2 : 2st : s^2 + t^2]$ sends $\mathbb{P}^1$ into $C$, since $(s^2 - t^2)^2 + (2st)^2 = (s^2 + t^2)^2$.
Its inverse is the projection $[x : y : z] \mapsto [y : z - x]$ from the point $[1 : 0 : 1]$ of $C$.
Non-example: the conic $x^2 + y^2 + z^2 = 0$ in $\mathbb{P}^2_\R$ is not rational over $\R$.
It has no real points, while a variety birational to $\mathbb{P}^1_\R$ has a dense set of real points.
Over $\C$ the same conic is rational.
So rationality depends on the field $k$.