Weak approximation for Fano complete intersections in positive characteristic

Abstract

For a smooth curve $B$ over an algebraically closed field $k$, for every $B$-flat complete intersection $X_B$ in $B{\times }_{\text{Spec}k}{\mathbb{P}}_k^n$ of type $(d_1,\dots ,d_c)$, if the Fano index is $\ge 2$ and if $\text{char}(k)>\max (d_1,\dots ,d_c)$, we prove weak approximation of ${\hat{\mathcal{O}}}_{B,b}$-points of $X_B$ by $k(B)$-points at all places of (strong) potentially good reduction, including all places of good reduction. The key step is the proof that such complete intersections are \emph{separably uniruled by lines}, and even \emph{separably rationally connected}, whenever smooth. We prove that the inequality is close to sharp. We prove a similar theorem for Fano manifolds of Picard number $1$ and Fano index $1$.

22 pages. Following discussions with Shizhang Li and Bhargav Bhatt, added hypothesis about torsion in crystalline cohomology for specializations of complex Fano manifolds of Picard number $1$ and Fano index $1$. Also proved a version eliminating this hypothesis and the hypothesis that the mixed characteristic DVR is unramified