Birational geometry of singular Fano double spaces of index two

In this paper we describe the birational geometry of Fano double spaces $V{\mathbb{P}}^{M+1}$ of index 2 and dimension $\geqslant 8$withatmostquadraticsingularitiesofrank$\geqslant 8$, satisfying certain additional conditions of general position: we prove that these varieties have no structures of a rationally connected fibre space over a base of dimension $⩾2$, that every birational map $\chi\colon V\dashrightarrow V’$ontothetotalspaceofaMorifibrespace$V’/{\mathbb P}^1$inducesanisomorphism$V^+\cong V’$oftheblowup$V^+$ofthevariety$V$along$\sigma^{-1}(P)$,where$P\subset {\mathbb P}^{M+1}$ is a linear subspace of codimension 2, and that every birational map of the variety $V$ onto a Fano variety $V^{\mathrm{\prime }}$ with $\mathbb{Q}$-factorial terminal singularities and Picard number 1 is an isomorphism. We give an explicit lower estimate for the codimension of the set of varieties $V$ with worse singularities or not satisfying the conditions of general position, quadratic in $M$. The proof makes use of the method of maximal singularities and the improved $4n^2$-inequality for the self-intersection of a mobile linear system.