Rational points on singular curves, with applications to the Brauer-Manin obstruction on surfaces

Singular curves over number fields have properties very different from those of smooth curves:
such a curve can have the trivial Brauer group, contain infinitely many adelic points, but only
finitely many rational points or none at all. Nevertheless, finite descent explains all counterexamples
to the Hasse principle on singular curves provided all the geometric irreducible components are rational.
Singular curves can be used to construct surfaces which are counterexamples to the Hasse
principle not explained by the Brauer-Manin obstruction, even when applied to etale covers.
This is a joiont work with Yonatan Harpaz.