N. Skorobogatov'
title: 'Singular curves and the étale Brauer–Manin obstruction for surfaces'
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Introduction {#introduction .unnumbered}
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For a variety $X$ over a number field $k$ one can study the set $X(k)$ of $k$-points of $X$ by embedding it into the topological space of adelic points $X({\mathbb{A}}_k)$. In 1970 Manin [@Manin] suggested to use the pairing $$X({\mathbb{A}}_k)\times\operatorname{Br}(X)\to{\mathbb{Q}}/{\mathbb{Z}}$$ provided by local class field theory. The left kernel of this pairing $X({\mathbb{A}}_k)^{\operatorname{Br}}$ is a closed subset of $X({\mathbb{A}}_k)$, and the reciprocity law of global class field theory implies that $X(k)$ is contained in $X({\mathbb{A}}_k)^{\operatorname{Br}}$. The first example of a smooth and projective variety $X$ such that $X(k)=\emptyset$ but $X({\mathbb{A}}_k)^{\operatorname{Br}}\not=\emptyset$ was constructed in [@Sk1] (see [@BSk] for a similar example; an earlier example conditional on the Bombieri–Lang conjecture was found in [@SW]). Later, Harari [@H] found many varieties $X$ such that $X(k)$ is not dense in $X({\mathbb{A}}_k)^{\operatorname{Br}}$. For all of these examples except for that of [@SW] the failure of the Hasse principle or weak approximation can be explained by the étale Brauer–Manin obstruction (introduced in [@Sk1], see also [@P1]): the closure of $X(k)$ in $X({\mathbb{A}}_k)$ is contained in the étale Brauer–Manin set $X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}\subset X({\mathbb{A}}_k)^{\operatorname{Br}}$ which in these cases is smaller than $X({\mathbb{A}}_k)^{\operatorname{Br}}$. 