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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?

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I haven't seen this done. There are a few options in generalizing this to non-projective curves. Did you have a specific one in mind? In the projective case, this has little to do with curves and more to do with $Pic^1(X)$ being a non-trivial torsor of $Pic^0(X)$, when $\delta > 1$. If you use generalized jacobians, then Harari has looked at Brauer-Manin obstructions for torsors of semi-abelian varieties. – Felipe Voloch Mar 29 '11 at 21:53

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