Tori and surfaces violating a local-to-global principle for rationality

Author:

Boris Kunyavskii

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Group Theory (math.GR), Number Theory (math.NT)

journal:

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date:

2023-05-04 16:00:00

Abstract

We show that even within a class of varieties where the Brauer obstruction is the only obstruction to the local-to-global principle for the existence of rational points (Hasse principle), this obstruction, even in a stronger, base change invariant form, may be insufficient for explaining counter-examples to the local-to-global principle for rationality. We exhibit examples of toric varieties and rational surfaces over an arbitrary global field k each of those, in the absence of the Brauer obstruction, is rational over all completions of k but is not k-rational.