Ruled surface

Remarks

Some books also ask for a section of $\pi$; over an algebraically closed field a section always exists, by Tsen's theorem. Every ruled surface is isomorphic over $C$ to the projectivization $\mathbb{P}(\mathcal{E})$ of a vector bundle $\mathcal{E}$ of rank $2$ on $C$. Another usage calls a surface ruled when it is only birational to $C \times \mathbb{P}^1$; the surfaces of the definition are then called geometrically ruled. Example: for any smooth projective curve $C$, the projection $C \times \mathbb{P}^1 \to C$ has every fibre $\set{c} \times \mathbb{P}^1 \cong \mathbb{P}^1$, so $C \times \mathbb{P}^1$ is ruled. Non-example: the projective plane $\mathbb{P}^2$ is not ruled, because every morphism from $\mathbb{P}^2$ to a curve is constant. Indeed, two fibres of a nonconstant one would be disjoint curves in $\mathbb{P}^2$, but any two curves in $\mathbb{P}^2$ meet.