---
tag: GZ3XA1V7CJ
kind: definition
name: Rational surface
msc: [14J26, "14E08"]
defines: 55A4A53TRF
uses: [KFTECAATKG, KA5890R9YM, ZT9A6JK8QF, 0PF6TM8TWT]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_surface
---

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]], and let $\mathbb{P}^2$ be the [[projective plane|ZT9A6JK8QF]] over $k$.
An [[algebraic surface|KFTECAATKG]] $X$ over $k$ is *rational* if there is a [[birational map|KA5890R9YM]] $X \dashrightarrow \mathbb{P}^2$.

## Remarks

Equivalently, the function field of $X$ is isomorphic to the field $k(s, t)$ of rational functions in two variables.
Example: $\mathbb{P}^1 \times \mathbb{P}^1$ is rational: on the open subsets $\mathbb{A}^1 \times \mathbb{A}^1$ and $\mathbb{A}^2$ the identity $(s, t) \mapsto (s, t)$ is an isomorphism, and it extends to a birational map $\mathbb{P}^1 \times \mathbb{P}^1 \dashrightarrow \mathbb{P}^2$.
Non-example: for an elliptic curve $E$, the surface $E \times \mathbb{P}^1$ is not rational, because the irregularity $q = h^1(\mathcal{O})$ is a birational invariant, equal to $0$ for $\mathbb{P}^2$ and to $1$ for $E \times \mathbb{P}^1$.
Every smooth rational surface is obtained from $\mathbb{P}^2$ or from a Hirzebruch surface by a finite sequence of blow-ups of points.