subset $Y({\mathbb{A}}_k)\cap X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}$ is infinite, where $Y$ is the Zariski open subset of $X$ which is the complement to $X(k)$. Following Poonen we consider families of curves parameterised by a curve with exactly one $k$-point. The new idea is to make the unique $k$-fibre a singular curve, geometrically a union of projective lines, and then use properties of rational and adelic points on singular curves.

The structure of the Picard group of a singular projective curve is well known, see [@BLR Section 9.2] or [@Liu Section 7.5]. In Section \[B\] we give a formula for the Brauer group of a reduced projective curve, see Theorem \[p1\]. A singular curve over $k$ can have surprising properties that no smooth curve can ever have: it can contain infinitely many adelic points, only finitely many $k$-points or none at all, and yet have the trivial Brauer group. See Corollary \[c1\] for a singular, geometrically connected, projective curve over an arbitrary number field $k$ that is a counterexample to the Hasse principle not explained by the Brauer–Manin obstruction. In [@Y] the first author proves that every counterexample to the Hasse principle on a curve which geometrically is a union of projective lines, can be 