explained by finite descent (and hence by the étale Brauer–Manin obstruction). Here we note that geometrically connected and simply connected projective curves over number fields satisfy the Hasse principle, a statement that does not generalise to higher dimension, see Proposition \[p-basic\] and Remark \[2.2\].

This paper was written while the authors were guests of the Centre Interfacultaire Bernoulli of the Ecole Polytechnique Fédérale de Lausanne.

The Brauer group of singular curves {#B}
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Let $k$ be a field of characteristic $0$ with an algebraic closure $\bar k$ and the Galois group $\Gamma_k=\operatorname{Gal}(\bar k/k)$. For a scheme $X$ over $k$ we write $\ovl X=X\times_k\bar k$. All cohomology groups in this paper are Galois or étale cohomology groups. Let $C$ be a [*reduced, geometrically connected, projective*]{} curve over $k$. We define the [*normalisation*]{} $\wtl C$ as the disjoint union of normalisations of the irreducible components of $C$. The normalisation morphism $\nu:\wtl C\to C$ factors as $$\wtl C\stackrel{\nu'}\lrar C'\stackrel{\nu''}\lrar C,$$ where $C'$ is a maximal intermediate curve universally homeomorphic to $C$, see [@BLR Section 9.2, p. 247] or [@Liu Section 7.5, p. 308]. The curve $C'$ is obtained from $\wtl C$ by identifying the points which have the same image in $C$. In 