  Given a morphism between smooth projective varieties $f: W \to X$, we study
whether $f$-relatively free rational curves imply the existence of
$f$-relatively very free rational curves. The answer is shown to be positive
when the fibers of the map $f$ have Picard number 1 and a further smoothness
assumption is imposed. The main application is when $X \subset \PP^n$ is a
smooth complete intersection of type $(d_1, ..., d_c)$ and $\sum d_i^2 \leq n$.
In this case, we take $W$ to be the space of pointed lines contained in $X$ and
the positive answer to the question implies that $X$ contains very twisting
ruled surfaces and is strongly rationally simply connected. If the fibers of a
smooth family of varieties over a 2-dimensional base satisfy these conditions
and the Brauer obstruction vanishes, then the family has a rational section
(see \cite{dJHS})

Algebraic Geometry