On the section conjecture and Brauer–Severi varieties

Abstract

J. Stix proved that a curve of positive genus over 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 Rk/QX 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 cork/Q([P])∈Br(Q) is non-trivial, then X satisfies the section conjecture.