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Given a smooth projective variety of dimension $n-1\geq 1$ defined over aperfect 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 definedover the base field $k$. We first show an example of such phenomenon: a varietydefined over $k$ admitting non-singular hypersurface models but none definedover $k$. We also determine under which conditions a non-singular hypersurfacemodel over $k$ may exist. Now, even assuming that such a smooth hypersurfacemodel exists, we wonder about the existence of non-singular hypersurface modelsover $k$ for its twists. We introduce a criterion to characterize twistspossessing such models and we also show an example of a twist not admitting anynon-singular hypersurface model over $k$, i.e for any $n\geq 2$, there is asmooth projective variety of dimension $n-1$ over $k$ which is a twist of asmooth hypersurface variety over $k$, but itself does not admit anynon-singular hypersurface model over $k$. Finally, we obtain a theoreticalresult to describe all the twists of smooth hypersurfaces with cyclicautomorphism group having a model defined over $k$ whose automorphism group isgenerated by a diagonal matrix.

Hypersurface model-fields of definition for smooth hypersurfaces and their twists