Separable rational connectedness and weak approximation in positive characteristic

Abstract: In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a sufficient condition involving the torsion order and the uniruling index for a smooth Fano variety of Picard number one to be separably rationally connected. As an application, we prove some weak approximation results for Fano complete intersections in positive charactersitic. For example, we show that weak approximation holds at place of strong potentially good reduction for a Fano complete intersection in $\mathbb{P}^n$ of type $(d_1, \ldots, d_c)$ in characteristic $p$ such that $n>d_1+\ldots +d_c, p>d_1, \ldots, d_c.$