A surface \(X\) is ruled over a curve \(C\) if it is birational to \(C \times \PP^1\); it is geometrically ruled if there is a morphism \(X \to C\) whose every fibre is \(\PP^1\), equivalently \(X = \PP(\mathcal{E})\) for a rank-two bundle \(\mathcal{E}\) on \(C\). \(X\) is rational if it is birational to \(\PP^2\).
(Hirzebruch surfaces)
\(\FF_n = \PP(\OO_{\PP^1} \oplus \OO_{\PP^1}(n))\) is the geometrically ruled surface over \(\PP^1\) with a section \(C_0\) satisfying \(C_0^2 = -n\). \(\FF_0 = \PP^1 \times \PP^1\) and \(\FF_1 = \Bl_p \PP^2\). The minimal rational surfaces are exactly \(\PP^2\) and \(\FF_n\) for \(n \neq 1\).
(Castelnuovo's rationality criterion)
A smooth projective surface over \(\CC\) is rational if and only if \(q = 0\) and \(P_2 = h^0(2K) = 0\).
The section \(C_0\) of \(\FF_1\) is a \((-1)\)-curve, and contracting it gives \(\PP^2\), so \(\FF_1\) is not minimal. The minimal rational surfaces \(\PP^2\), \(\FF_0\), \(\FF_2\), \(\FF_3,\ldots\) are pairwise non-isomorphic and birational to each other, so a rational surface has no unique minimal model.
The conditions \(p_g=q=0\) do not imply rationality: Enriques surfaces satisfy them but have \(P_2=1\), while Castelnuovo’s criterion requires \(q=P_2=0\).
A smooth projective surface over \(\CC\) has Kodaira dimension \(\kappa = -\infty\), that is, all plurigenera \(P_m=h^0(mK)\) with \(m\ge1\) vanish, if and only if it is ruled; rational surfaces are the surfaces ruled over \(\PP^1\).