# The Clemens-Griffiths method over non-closed fields

## TL;DR

This paper extends the Clemens-Griffiths method to construct specific threefolds over fields with certain quadratic extensions, demonstrating nuanced rationality properties and providing new examples in algebraic geometry.

## Contribution

It introduces a novel application of the Clemens-Griffiths method over non-closed fields to produce threefolds with unique rationality characteristics.

## Key findings

- Constructed threefolds are $k$-unirational and $ar{k}$-rational but not $k$-rational.
- Real locus of these threefolds can be diffeomorphic to that of a rational variety.
- All unramified cohomology groups of these threefolds are trivial.

## Abstract

We use the Clemens-Griffiths method to construct smooth projective threefolds, over any field $k$ admitting a separable quadratic extension, that are $k$-unirational and $\bar{k}$-rational but not $k$-rational. When $k=\mathbb{R}$, we can moreover ensure that their real locus is diffeomorphic to the real locus of a smooth projective $\mathbb{R}$-rational variety and that all their unramified cohomology groups are trivial.