Rational surface

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.