Stable rationality of cyclic covers of projective spaces

TL;DR

This paper proves that cyclic covers of projective spaces branched along general divisors are not stably rational in certain dimensions and degrees, extending previous results and exploring applications to Fano manifolds.

Contribution

It generalizes prior work by establishing non-stable rationality of cyclic covers over projective spaces for broader conditions and discusses related cases and applications.

Findings

Cyclic covers are not stably rational for n ≥ 3 and d ≥ n+1.

Extension of non-stable rationality results to covers over complete intersections.

Applications to the classification of Fano manifolds.

Abstract

The main aim of this paper is to show that a cyclic cover of $\mathbb{P}^n$ branched along a very general divisor of degree $d$ is not stably rational provided that $n \ge 3$ and $d \ge n+1$. This generalizes the result of Colliot-Th\'el\`ene and Pirutka. Generalizations for cyclic covers over complete intersections and applications to suitable Fano manifolds are also discussed.