A threefold violating a local-to-global principle for rationality

In this note we construct an example of a smooth projective threefold that is irrational over \(\mathbb Q\) but is rational at all places. Our example is a complete intersection of two quadrics in \(\mathbb P^5\), and we show it has the desired rationality behavior by constructing an explicit elemen...

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Bibliographic Details
Published in:arXiv.org 2023-06
Main Authors: Frei, Sarah, Ji, Lena
Format: Article
Language:English
Online Access:Get full text
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Summary:In this note we construct an example of a smooth projective threefold that is irrational over \(\mathbb Q\) but is rational at all places. Our example is a complete intersection of two quadrics in \(\mathbb P^5\), and we show it has the desired rationality behavior by constructing an explicit element of order \(4\) in the Tate--Shafarevich group of the Jacobian of an associated genus \(2\) curve.
ISSN:2331-8422
DOI:10.48550/arxiv.2304.09306