Birational geometry of singular Fano double spaces of index two

We describe the birational geometry of Fano double spaces $V\stackrel{\sigma}{\to}{\mathbb P}^{M+1}$ of index 2 and dimension ${\geqslant 8}$ with at most quadratic singularities of rank ${\geqslant 8}$, 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 ${\geqslant2}$, that every birational map $\chi\colon V\dashrightarrow V'$ onto the total space of a Mori fibre space $V'/{\mathbb P}^1$ induces an isomorphism $V^+\cong V'$ of the blow-up $V^+$ of $V$ along $\sigma^{-1}(P)$, where $P\subset {\mathbb P}^{M+1}$ is a linear subspace of codimension 2, and that every birational map of $V$ onto a Fano variety $V'$ with ${\mathbb Q}$-factorial terminal singularities and Picard number 1 is an isomorphism. We give an explicit lower estimate, quadratic in $M$, for the codimension of the set of varieties $V$ that have worse singularities or do not satisfy the conditions of general position. 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. Bibliography: 20 titles.

Fano variety, Mori fibre space, birational map, linear system, maximal singularity