Intermediate Jacobians and rationality over arbitrary fields

TL;DR

This paper develops a theory of intermediate Jacobians for geometrically rational threefolds over arbitrary fields and characterizes the rationality of certain threefolds via the existence of a k-rational line.

Contribution

It introduces a new theory of intermediate Jacobians over arbitrary fields and links the rationality of threefolds to the presence of rational lines.

Findings

A threefold is k-rational iff it contains a k-line.

Constructs examples of varieties rational over inseparable extensions but not over the base field.

Provides the first examples of such varieties with specific rationality properties.

Abstract

We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of intermediate Jacobians for geometrically rational threefolds over arbitrary, not necessarily perfect, fields. As a consequence, we obtain the first examples of smooth projective varieties over a field k which have a k-point, and are rational over a purely inseparable field extension of k, but not over k.