Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary 4 of following article, Bhargava shows that as the genus $g$ of the curve tends to infinity, a density approaching $100 \%$ of hyper-elliptic curves over $\mathbb{Q}$ of genus $g$ have a Brauer-Manin obstruction to having a rational point.

What do we know about Brauer-Manin obstructions to surfaces of Kodaira dimension 1? Or to elliptic surfaces? Any references and/or comments would be greatly appreciated.