On the section conjecture and Brauer\u2013Severi varieties

Abstract

J. Stix proved that a curve of positive genus over mathbb {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 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}) is non-trivial, then X satisfies the section conjecture.