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arXiv · 2304.09306

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

Sarah Frei·Lena Ji
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Abstract

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.

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Sarah Frei, Lena Ji. 2023-06-26. A threefold violating a local-to-global principle for rationality. https://doi.org/10.1007/s40993-024-00515-8

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