Theorem: Let X be a smooth projective curve over a number field K, and let $\delta$ be the *index* of X (i.e., the minimal degree of a K-rational divisor on X). Then V. Scharaschkin proved in this thesis that if $\delta > 1$ then there is a Brauer-Manin obstruction to K-rational points on X.

Question: Does this theorem hold if X is not projective?