Birational geometry of Fano double spaces of index two

We study birational geometry of Fano varieties, realized as double covers $\sigma \text{\hspace{0.17em}⁣}:V\to {\mathbb{P}}^M$, $M\ge 5$, branched over generic hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fiber space on $V$ are the pencils-subsystems of the free linear system $\mathrm{\mid }-\frac{1}{2}{K}_V\mathrm{\mid }$. The groups of birational and biregular self-maps of the variety $V$ coincide.