A famous conjecture due to Colliot-Thelene roughly predicts that a geometrically rational variety over a global field with geometric Picard group Z satisfies the Hasse principle (i.e., it has a point if and only if it has local points at every place). I will discuss counterexamples to this conjecture over the "other" fields of cohomological dimension 2, namely function fields of surfaces over algebraically closed fields.