Varieties that are not stably rational, zero-cycles and unramified cohomology

TL;DR

This paper surveys recent examples of algebraic varieties that are not stably rational, focusing on specialization methods, unramified cohomology, and explicit calculations of the Brauer group for certain fourfolds.

Contribution

It reviews the use of unramified cohomology and Chow groups in detecting non-stable rationality and provides an explicit formula for the Brauer group of specific fourfolds.

Findings

Examples of non-stably rational varieties are classified.

Unramified cohomology serves as a key invariant in these examples.

Explicit Brauer group formula for certain fibered fourfolds is derived.

Abstract

This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension two over a rational surface.