pp. 20–21] for details.

In this section we construct a simple example of a smooth projective surface $X$ over $k$ such that $X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}$ is infinite but $X(k)$ contains only one point. Thus $X(k)$ is far from dense in $X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}$; in fact, infinitely many points of $X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}$ are contained in the complement to $X(k)$ in $X$.

We start with the following statement which shows that on an everywhere locally soluble conical curve $C$ such that $X(C)$ is a tree all the adelic points survive the étale Brauer–Manin obstruction.

\[p-basic\] Let $k$ be a number field, and let $C$ be a conical curve over $k$ such that $X(C)$ is a tree and $C({\mathbb{A}}_k)\not=\emptyset$. Then

1.  $C(k)\not=\emptyset$;

2.  the natural map $\operatorname{Br}(k)\to\operatorname{Br}(C)$ is an isomorphism;

3.  $C({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}} = C({\mathbb{A}}_k)$.

\[2.2\] Proposition \[p-basic\] (1) implies that [*geometrically connected and simply connected projective curves over number fields satisfy the Hasse principle*]{}. It is easy to see that this statement does not generalise to higher dimension. Consider a conic $C\subset{\mathbb{P}}^2_k$ without a $k$-point, and choose a quadratic extension $K/k$ so that all the places $v$ with $C(k_v)=\emptyset$ are split in $K$. Then the union of two planes conjugate over $K$ and intersecting at $C$ is a 