On the section conjecture and Brauer–Severi varieties

Abstract

J. Stix proved that a curve of positive genus over $$\mathbb {Q}$$ Q which maps to a non-trivial Brauer–Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction $$R_{k/\mathbb {Q}}X$$ R k / Q X admits a rational map to a non-trivial Brauer–Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer–Severi variety P such that the corestriction $${\text {cor}}_{k/\mathbb {Q}}([P])\in {\text {Br}}(\mathbb {Q})$$ cor k / Q ( [ P ] ) ∈ Br ( Q ) is non-trivial, then X satisfies the section conjecture.