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?



Section of a Ruled surfaces
I thought I'd expand my earlier comment, which was not all that clear,
and I'm not even sure where you'd look it up.
Let's say that a ruled surface over a smooth curve $C$ is smooth projective morphism
$f:X\to C$ all of whose fibres are isomorphic to $\mathbb{P}^1$. Then one checks
that $\omega\_{X/C}^{-1}$ is relatively very ample. So we get an embedding
$X\to \mathbb{P}(f\_\*\omega\_{X/C}^{-1})$ over $C$ as a family of conics. Assuming
that you are over an algebraically closed field, you can apply Tsen to 
see that the generic fibre has a rational point. This implies that there is a rational
section $C\dashrightarrow X$. Since $C$ is a curve, this extends to an honest section.


An alternative argument is to note that the obstruction to $X=\mathbb{P}(E)$,
for some vector bundle $E$, lies in the Brauer group of $C$ via
$$H^1(C\_{et},GL\_2)\to H^1(C\_{et},PGL\_2)\to H^2(C\_{et},G\_m)$$
and this vanishes by Tsen. One gets a section using this as well, and much more.


