Intermediate Jacobians and rationality over arbitrary fields

TL;DR

The paper addresses k-rationality for threefolds, focusing on smooth complete intersections of two quadrics in P^5 and proving that such an X is k-rational iff it contains a line defined over k.To achieve this, it introduces a novel intermediate Jacobian construction over arbitrary fields, replacing Murre’s approach with a representable, Ab^2(X)–style k-group scheme CH^2_{X/k}, endowed with a canonical principal polarization and descent properties.The authors establish a rich obstruction theory for k-rationality via the CH^2_{X/k} framework, relate it to Murre’s intermediate Jacobian, and apply it to derive explicit rationality and unirationality criteria, including new counterexamples over Laurent series fields.In the quadratic-sectional setting, they compute CH^2_{X/k} completely, obtain the main rationality criterion, and illustrate the results with characteristic-dependent behavior and inseparable phenomena, thereby extending Clemens–Griffiths/Murre-style obstructions to imperfect fields.

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.

Theorems & Definitions (112)

• Theorem A: Theorem \ref{['thrat']}
• Theorem B: Theorem \ref{['thmC((t))']}
• Theorem C: Theorem \ref{['threp']} (vii)
• Proposition A
• proof
• Proposition C
• proof
• Lemma D
• proof
• Corollary E