Hartshorne in his chapter on surfaces defines a ruled surface(over an algebraically closed field) to be a smooth projective surface $X$ together with a surjective morphism $\pi:X\to C$, $C$ a smooth curve, such that the fiber over each point $y\in C$, call it $X_y$, is isomorphic to $\mathbb{P}^1$, and such that $\pi$ admits a section.

He then says that the existence of a section follows from Tsen's theorem under the earlier hypothesis.

When he says every point, does he include the generic point also? If yes, then the section will exist by defining it to be anything at the generic point and then using that a morphism from an open subset of a curve to a proper variety extends to the whole curve.

If by points he means only closed points, then how does the existence of a section follow from Tsen's theorem?