  Given a smooth projective variety of dimension $n-1\geq 1$ defined over a
perfect field $k$ that admits a non-singular hypersurface modelin
$\mathbb{P}^n_{\overline{k}}$ over $\overline{k}$, a fixed algebraic closure of
$k$, it does not necessarily have a non-singular hypersurface model defined
over the base field $k$. We first show an example of such phenomenon: a variety
defined over $k$ admitting non-singular hypersurface models but none defined
over $k$. We also determine under which conditions a non-singular hypersurface
model over $k$ may exist. Now, even assuming that such a smooth hypersurface
model exists, we wonder about the existence of non-singular hypersurface models
over $k$ for its twists. We introduce a criterion to characterize twists
possessing such models and we also show an example of a twist not admitting any
non-singular hypersurface model over $k$, i.e for any $n\geq 2$, there is a
smooth projective variety of dimension $n-1$ over $k$ which is a twist of a
smooth hypersurface variety over $k$, but itself does not admit any
non-singular hypersurface model over $k$. Finally, we obtain a theoretical
result to describe all the twists of smooth hypersurfaces with cyclic
automorphism group having a model defined over $k$ whose automorphism group is
generated by a diagonal matrix.
