Let $k$ be a field. Let $X$ be a subvariety of $\mathbb P_k^r$ of dimension $n$. If $k$ is an algebraically closed field of characteristic $0$, then
(1). If $r \le 2n - 1$, then $X$ is simply connected ($X$ has trivial étale fundamental group).
(2). If $r \le 2n - 2$, then $\mathop{\mathrm{Pic}}(X) \cong \mathbb Z$.
If $k = \mathbb C$, then we can prove these statements using Barth's theorem, the general cases can be prove by reducing to $\mathbb C$. I wonder if there is an algebraic proof of the general case.
