Non rationalit\'e stable sur les corps quelconques

TL;DR

This survey reviews the current understanding of stable rationality over arbitrary fields, discussing various algebraic and geometric invariants and methods used to analyze rationality properties of algebraic varieties.

Contribution

It provides a comprehensive overview of recent developments and techniques in studying (non)rationality over arbitrary fields, including new insights into invariants and specialization methods.

Findings

Summary of criteria for stable rationality

Overview of invariants like Chow groups and Brauer group

Discussion of specialization techniques and their applications

Abstract

This is a survey on (lack of) stable rationality over arbitrary fields (including algebraically closed fields). Topics addressed include: Rationality and unirationality, R-equivalence on rational points, Chow groups of zero-cycles, Galois action on the Picard group, Brauer group, higher unramified cohomology, global differentials, specialisation method (via R-equivalence), geometrically rational surfaces, cubic hypersurfaces.