Papers › A threefold violating a local-to-global principle for rationality
A threefold violating a local-to-global principle for rationality
Sarah Frei, Lena Ji
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In this note we construct an example of a smooth projective threefold that is irrational over ℚ but is rational at all places. Our example is a complete intersection of two quadrics in ℙ⁵, 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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