Rational surface

Over an algebraically closed field $k$, a rational surface is an algebraic surface birational to the projective plane ${\mathbb{P}}_k^2$. Equivalently its function field is isomorphic to $k(x,y)$, a purely transcendental extension of degree two. The birational maps need only be inverse on dense open subsets.

The product ${\mathbb{P}}^1\times {\mathbb{P}}^1$ is rational: it contains the affine plane as the product of two affine charts, and ${\mathbb{P}}^2$ also contains the affine plane. Blowing up a point on a smooth rational surface preserves its function field, hence its rationality, while changing its curves and intersection geometry.

Rationality is consequently weaker than being isomorphic to ${\mathbb{P}}^2$. It is also a geometric condition on the whole surface, rather than an assertion that every point has rational-number coordinates. Over a non-algebraically-closed field, the field over which the birational map is defined must be specified.