The Clemens-Griffiths method over non-closed fields

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.

arXiv comment26 pages; minor revision

References in corpus (8)

• Unirational threefolds with no universal codimension 2 cycle
• Cohomologie non ramifiée et conjecture de Hodge entière
• Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes
• Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields
• On the Unirationality of del Pezzo surfaces of degree two
• On the integral Hodge conjecture for real varieties, I
• On the integral Hodge conjecture for real varieties, II
• Non rationalité stable d'hypersurfaces cubiques sur des corps non algébriquement clos

Cited by in corpus (5)

• Intermediate Jacobians and rationality over arbitrary fields
• Cycle class maps and birational invariants
• Rationality of complete intersections of two quadrics
• On the universal regular homomorphism in codimension $2$
• Rationality of real conic bundles with quartic discriminant curve