# Canonical and log canonical thresholds of multiple projective spaces

## TL;DR

This paper establishes that most $d$-sheeted covers of projective spaces have a canonical threshold of one, leading to new birational rigidity results for certain Fano-Mori fiber spaces with controlled singularities.

## Contribution

It proves the global (log) canonical threshold equals one for almost all such covers, extending birational rigidity to new classes of Fano-Mori fiber spaces.

## Key findings

- Canonical thresholds are one for most covers with specified singularities.
- Birational rigidity applies to large classes of Fano-Mori fiber spaces.
- Results depend on bounds related to the dimension of the fiber.

## Abstract

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.