Rationality problems and conjectures of Milnor and Bloch-Kato

TL;DR

This paper leverages Voevodsky's proof techniques for Milnor and Bloch-Kato conjectures to construct specific rationally connected varieties that serve as counterexamples to the classical L"uroth problem, highlighting the role of unramified étale cohomology.

Contribution

It introduces a method to produce non-rational varieties with detectable non-rationality via unramified étale cohomology, generalizing Peyre's approach using advanced conjecture proofs.

Findings

Constructed non-rational, rationally connected varieties for any prime l and integer n ≥ 2.

Demonstrated non-rationality detection through non-trivial degree n unramified étale cohomology classes.

For l=2, varieties are unirational and non-rationality isn't detectable by lower degree cohomology.

Abstract

We show how the techniques of Voevodsky's proof of the Milnor conjecture and the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can be used to study counterexamples to the classical L\"uroth problem. By generalizing a method due to Peyre, we produce for any prime number l and any integer n >= 2, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree n unramified \'etale cohomology class with l-torsion coefficients. When l = 2, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified \'etale cohomology class of lower degree.