(\[ee1\]). Hilbert’s theorem 90 gives $H^1(\Pi,\nu'_*{\mathbb{G}}_{m,\Psi})=H^1(\Psi,{\mathbb{G}}_{m,\Psi})=0$, so we obtain the exact sequence (\[ee4\]).

Recall that a reduced, geometrically connected, projective curve $S$ over a field $k$ is called [*semi-stable*]{} if all the singular points of $S$ are ordinary double points [@BLR Def. 9.2.6].

\[bi\] A semi-stable curve is called [*bipartite*]{} if it is a union of two smooth curves without common irreducible components.

\[p3\] Let $S=S^+\cup S^-$ be a bipartite curve, where $S^+$ and $S^-$ are smooth curves such that $S^+\cap S^-$ is finite. Then there is an exact sequence $$0\lrar\operatorname{Br}(S)\lrar\operatorname{Br}(S^+)\oplus\operatorname{Br}(S^-)\lrar \operatorname{Br}(S^+\cap S^-),
\label{ee5}$$ where $\operatorname{Br}(S)\to\operatorname{Br}(S^+)\oplus\operatorname{Br}(S^-)$ is the natural map, and $\operatorname{Br}(S^\pm)\to\operatorname{Br}(S^+\cap S^-)$ is the restriction map multiplied by $\pm 1$.

In our previous notation we have $\wtl S=S^+\coprod S^-$, $\Pi=S^+\cap S^-$, and $\Psi$ is the disjoint union of two copies of $\Pi$, namely $\Psi^+=\Psi\cap S^+$ and $\Psi^-=\Psi\cap S^-$. In particular, the restriction map $\operatorname{Br}(\Pi)\to\operatorname{Br}(\Psi)$ is injective. Thus taking the quotients by $\operatorname{Br}(\Pi)$ in the middle and last terms of (\[ee4\]), we obtain (\[ee5\]).

The constructions in Sections \[WA\] and \[HP\] use singular curves of the following special kind.

A reduced, geometrically connected, projective curve $C$ over $k$ is called [*conical*]{} if every irreducible component of $\ovl C$ is rational.

