Potentially stably rational conic bundles over nonclosed fields

TL;DR

This paper investigates the stable rationality of conic bundles over the projective line defined over non-closed fields, using Galois cohomology and Picard lattice actions to identify conditions for stable rationality.

Contribution

It introduces a cohomological approach to analyze stable rationality of conic bundles over non-closed fields, focusing on Galois group actions on the Picard lattice.

Findings

Criteria for stable rationality based on Galois cohomology

Identification of obstructions to stable rationality

Application to specific classes of conic bundles

Abstract

We study stable rationality of conic bundles $X$ over $\mathbb{P}^1$ defined over non-closed field $k$ via the cohomology of the Galois group of finite field extension $k'/k$ with action on the geometric Picard lattice of $X$.