= 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 (algebraic geometry)> 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.
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