canonical morphism ${\mathbb{G}}_{m,\Sigma}\to f_*{\mathbb{G}}_{m,\Sigma'}$, and hence to a canonical homomorphism of $k$-tori $p_*{\mathbb{G}}_{m}\to p'_*{\mathbb{G}}_{m}$, which we denote by $f^*$. Let us denote the structure morphism $\Lambda\to\operatorname{Spec}(k)$ by $p_\Lambda$, and use the same convention for $\Psi$ and $\Pi$. Let $T$ be the algebraic $k$-torus defined by the exact sequence $$1\lrar {\mathbb{G}}_{m}\lrar p_{\Lambda *}{\mathbb{G}}_m\oplus p_{\Pi *}{\mathbb{G}}_m\lrar
p_{\Psi *}{\mathbb{G}}_m\lrar T\lrar 1,$$ where the middle arrow is $s^*(t^*)^{-1}$.

The character group $\hat T$ with its natural action of the Galois group $\Gamma_k$, is canonically isomorphic to $H_1(X(C),{\mathbb{Z}})$, the [*first homology group*]{} of the graph $X(C)$. Since $X(C)$ is connected, we have $T=\{1\}$ if and only if $X(C)$ is a tree.

\[P1\] Let $C$ be a reduced, geometrically connected, projective curve over $k$. We have the following exact sequences of $\Gamma_k$-modules: $$0\lrar T(\bar k)\lrar \operatorname{Pic}(\ovl {C'})\lrar \operatorname{Pic}(\wtl C\times_k\bar k)
\lrar 0,
\label{b}$$ $$0\lrar U(\bar k)\lrar \operatorname{Pic}(\ovl C)\lrar \operatorname{Pic}(\ovl {C'})\lrar 0,
\label{a}$$ where $U$ is a commutative unipotent algebraic group over $k$.

To obtain (\[b\]) we apply the direct image functor with respect to the structure morphism $C'\to\operatorname{Spec}(k)$ to the exact sequence (\[ee1\]). The sequence (\[a\]) is obtained from (\[ee2\]) in a similar way.

\[C1\] If $C$ is a conical curve such that $X(C)$ is a 