  In this paper we show that the global (log) canonical threshold of
$d$-sheeted covers of the $M$-dimensional projective space of index 1, where
$d\geqslant 4$, is equal to one for almost all families (except for a finite
set). The varieties are assumed to have at most quadratic singularities, the
rank of which is bounded from below, and to satisfy the regularity conditions.
This implies birational rigidity of new large classes of Fano-Mori fibre spaces
over a base, the dimension of which is bounded from above by a constant that
depends (quadratically) on the dimension of the fibre only.
