Rationality over non-closed fields of Fano threefolds with higher geometric Picard rank

TL;DR

This paper establishes rationality criteria over non-closed fields for certain Fano threefolds with higher geometric Picard rank, and explores unirationality and non-rationality conditions.

Contribution

It provides new rationality and unirationality criteria for five types of geometrically rational Fano threefolds with higher Picard rank over non-closed fields.

Findings

Rationality criteria established for five types of Fano threefolds.

Unirationality criterion and stable non-rationality results for the last type.

Analysis applicable over algebraically non-closed fields of characteristic zero.

Abstract

We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds we provide a unirationality criterion and prove stable non-rationality under additional assumptions.