Relatively Very Free Curves and Rational Simple Connectedness

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 $X\subset {\text{PP}}^n$ is a smooth complete intersection of type $(d_1,\mathrm{.}\mathrm{.}\mathrm{.},d_c)$ and $\sum d_i^2\le n$. 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})