Birationally rigid hypersurfaces with quadratic singularities of low rank

It is shown that hypersurfaces of degree M in P-M , M >= 5 , with at most quadratic singularities of rank at least 3 that satisfy certain conditions of general position are birationally superrigid Fano varieties and for M >= 8 the complement to the set of such hypersurfaces is of co dimension at least ((M-1)(2)) + 1 with respect to the natural parameter space. Bibliography: 18 titles.