  Let $K$ be a finitely generated field. We construct an $n$-dimensional linear
system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined
over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth,
under the hypothesis that the characteristic $p$ does not divide $\gcd(d, n+1)$
(in particular, there is no restriction when $K$ has characteristic $0$).
Moreover, we exhibit a counterexample when $p$ divides $\gcd(d, n+1)$.
