Global descent obstructions for varieties

TL;DR

This paper explores how descent obstructions can be transferred from covers to varieties, providing examples of curves with specific field of moduli properties that lack rational models.

Contribution

It introduces a method to transfer descent obstructions from covers to varieties and constructs explicit examples illustrating these phenomena.

Findings

Curves with field of moduli bb bb a0 admit models over every completion of bb a0 but not over bb a0

Demonstrates the transfer of descent obstructions between categories

Provides explicit examples of obstructions in arithmetic geometry

Abstract

We show how to transport descent obstructions from the category of covers to the category of varieties. We deduce examples of curves having $\QQ$ as field of moduli, that admit models over every completion of $\QQ$, but have no model over $\QQ$.