---
tag: 55A4A53TRF
kind: concept
name: Rational surface
msc: [14J26, "14E08"]
specializes: [KFTECAATKG, XV4K9GREJ9]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_surface
same_as:
  wikidata: Q208756
  wikipedia: Rational surface
---

## Summary

A rational surface is an [[algebraic surface|KFTECAATKG]] that is birational to the projective plane.
It is the two-dimensional case of a [[rational variety|XV4K9GREJ9]].
Rational surfaces include the projective plane, the quadric $\mathbb{P}^1 \times \mathbb{P}^1$, the Hirzebruch surfaces, the blow-ups of the plane in points, and all smooth cubic surfaces.
They are the surfaces whose points can be parametrized, outside a lower-dimensional set, by two free parameters.

## Explanations

### L1 human

A surface is rational when its points can be described by two independent parameters through rational functions, with an inverse that is also given by rational functions.
For instance, the sphere $x^2 + y^2 + z^2 = 1$ can be parametrized by stereographic projection from a point, so it is rational.
Two rational surfaces can look very different, but they agree with the plane after removing some curves.