# Birationally rigid finite covers of the projective space

## TL;DR

This paper establishes the birational superrigidity of certain finite covers of projective space with specific degree, dimension, and singularity conditions, extending previous results beyond cyclic covers.

## Contribution

It proves birational superrigidity for non-cyclic finite covers of projective space with quadratic singularities, broadening the class of known rigid varieties.

## Key findings

- Finite covers of degree ≥ 5 are birationally superrigid.
- Superrigidity holds under specific singularity and regularity conditions.
- The set of exceptions has codimension at least half of (M-4)(M-5)+1.

## Abstract

In this paper we prove birational superrigidity of finite covers of degree $d$ of the $M$-dimensional projective space of index 1, where $d\geqslant 5$ and $M\geqslant 10$, with at most quadratic singularities of rank $\geqslant 7$, satisfying certain regularity conditions. Up to now, only cyclic covers were studied in this respect. The set of varieties with worse singularities or not satisfying the regularity conditions is of codimension $\geqslant\frac12(M-4)(M-5)+1$ in the natural parameter space of the family.