Intermediate Jacobians and rationality over arbitrary 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.

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)

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Cited by in corpus (5)

• Supersolvable descent for rational points
• A threefold violating a local-to-global principle for rationality
• Rationality of real conic bundles with quartic discriminant curve
• Curve classes on conic bundle threefolds and applications to rationality
• Equivariant geometry of odd-dimensional complete intersections of two quadrics