Birational geometry of singular Fano double spaces of index two

Abstract

Abstract We describe the birational geometry of Fano double spaces of index 2 and dimension with at most quadratic singularities of rank , 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 , that every birational map onto the total space of a Mori fibre space induces an isomorphism of the blow-up of along , where is a linear subspace of codimension 2, and that every birational map of onto a Fano variety with -factorial terminal singularities and Picard number 1 is an isomorphism. We give an explicit lower estimate, quadratic in , for the codimension of the set of varieties 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 -inequality for the self-intersection of a mobile linear system. Bibliography: 20 titles.