Birationally Rigid Finite Covers of the Projective Space

In this paper we prove birational superrigidity of finite covers of degree $d$ of the $M$-dimensional projective space of index $1$, where $d\geq 5$ and $M\geq 10$, that have at most quadratic singularities of rank ${\geq }\,7$ and satisfy certain regularity conditions. Up to now, only cyclic covers have been studied in this respect. The set of varieties that have worse singularities or do not satisfy the regularity conditions is of codimension ${\geq }\,(M-4)(M-5)/2+1$ in the natural parameter space of the family.