Intermediate Jacobians and rationality over arbitrary fields
Algebraic Geometry
2025-10-03 v5
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.
Cite
@article{arxiv.1909.12668,
title = {Intermediate Jacobians and rationality over arbitrary fields},
author = {Olivier Benoist and Olivier Wittenberg},
journal= {arXiv preprint arXiv:1909.12668},
year = {2025}
}
Comments
52 pages; v5: incorporates an erratum (separately published in Ann. ENS 58 (2025), no. 3, 829-830) into the proof of Theorem 4.13 (a theorem which was not used anywhere in the text but is of general interest)