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 cor(k/Q)([ P]) is an element of Br(Q) is non-trivial, then X satisfies the section conjecture.