---
tag: YF1SE8WJK3
kind: concept
name: Unirational variety
msc: [14M20, "14E08"]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_variety
---

## Summary

A unirational variety is a variety that is dominated by projective space through a rational map.
Its points can be parametrised by rational functions, but the parametrisation may cover each point several times.
Every [[rational variety|XV4K9GREJ9]] is unirational.
Whether the converse holds is the Lüroth problem.
It holds for curves, and for surfaces over an algebraically closed field of characteristic $0$, but it fails for some threefolds over $\C$.

## Explanations

### L1 human

A rational variety has a parametrisation that is one-to-one on a dense open set.
A unirational variety only needs a parametrisation that covers a dense open set, possibly many times over.
For curves the two notions agree, by [[Lüroth's theorem|AVCZ6HVBA7]].
Over $\C$, a smooth cubic threefold in $\mathbb{P}^4$ is unirational but not rational, by a theorem of Clemens and Griffiths.