Deformations of hyperelliptic and generalized hyperelliptic polarized varieties

In this article we study the deformations of hyperelliptic polarized varieties $(X,L)$ of dimension $m$ and sectional genus $g$ such that the image $Y$ of the morphism $\varphi$ induced by $|L|$ is smooth. If $L^m < 2g-2$, it is known that, by adjunction and the Clifford's theorem, any deformation of $(X,L)$ is hyperelliptic. Thus, we focus on when $L^m=2g-2$ or $L^m=2g$. We prove that, if $(X,L)$ is Fano-K3, then, except when $Y$ is a hyperquadric, all deformations of $(X,L)$ are again hyperelliptic (if $Y$ is a hyperquadric, the general deformation of $\varphi$ is an embedding). This contrasts with the situation of hyperelliptic canonical curves and hyperelliptic K3 surfaces. If $L^m=2g$, then we prove that, in most cases, a general deformation of $\varphi$ is a finite morphism of degree $1$. This provides interesting examples of degree $2$ morphisms that can be deformed to morphisms of degree $1$. We extend our results to so-called generalized hyperelliptic polarized Fano, Calabi-Yau and general type varieties. The solutions to these questions are closely intertwined with the existence or non existence of double structures on the algebraic varieties $Y$. We address this matter as well.