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

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.

arXiv comment7 pages. v2: the example is now unconditional

References in corpus (10)

• The Cassels-Tate pairing on polarized abelian varieties
• Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
• Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves
• Intermediate Jacobians and rationality over arbitrary fields
• Cubic fourfolds containing a plane and a quintic del Pezzo surface
• Rational points and derived equivalence
• Principe de Hasse pour les intersections de deux quadriques
• Retour sur l'arithmétique des intersections de deux quadriques, avec un appendice par A. Kuznestov
• A threefold violating a local-to-global principle for rationality
• Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian

Cited by in corpus (1)

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