Let $\phi:X \to Y$ be a projective morphism of smooth varieties. Under what condition on $\phi$ does there exist a section to the morphism $\phi$? The example that I have in mind is when $Y$ is an irreducible component of a Hilbert scheme of curves, $X$ is a component of flag Hilbert scheme and $\phi$ is a projection onto one of its components.

This question has been asked before, but the answer is not very clear to me.