Birational geometry of Fano double spaces of index two

We study birational geometry of Fano varieties realized as double covers $\sigma\colon V\to{\mathbb P}^M$, $M\geqslant5$, branched over generic smooth hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fibre space on $V$ are pencil-subsystems of the free linear system $|{-\frac12K_V}|$. The groups of birational and biregular self-maps of $V$ coincide: $\operatorname{Bir}V=\operatorname{Aut}V$.