terms. Recall that $\ovl\Lambda$, $\ovl \Pi$, $\ovl \Psi$ are the $\bar k$-schemes obtained from $\Lambda$, $\Pi$, $\Psi$ by extending the ground field to $\bar k$. We associate to $C$ the [*incidence graph*]{} $X(C)$ defined as the directed graph whose vertices are $X(C)_0=\ovl \Lambda\cup\ovl \Pi$ and the edges are $X(C)_1=\ovl \Psi$. The edge $Q \in \ovl \Psi$ goes from $L\in \ovl\Lambda$ to $P \in \ovl \Pi$ when $\nu(Q)=P$ and $Q$ is contained in the irreducible component $L$ of $\wtl C$. The source and target maps $X(C)_1 \to X(C)_0$ can be described as a morphism of $k$-schemes $$(s,t):\Psi \lrar \Lambda \coprod \Pi,$$ where $t:\Psi\to\Pi$ is induced by $\nu'$, and $s$ is the composition of the closed immersion $j:\Psi\to\wtl C$ and the canonical morphism $\wtl C\to\Lambda$. By construction $X(C)$ is a connected bipartite graph with a natural action of the Galois group $\Gamma_k$.

For a reduced 0-dimensional $k$-scheme $p:\Sigma\to\operatorname{Spec}(k)$ of finite type, the $k$-group scheme $p_*{\mathbb{G}}_{m}$ is an algebraic torus over $k$. If we write $\Sigma=\coprod_{i=1}^n\operatorname{Spec}(k_i)$, where $k_1,\ldots,k_n$ are finite field extensions of $k$, then $p_*{\mathbb{G}}_{m}$ is the product of Weil restrictions $\prod_{i=1}^n R_{k_i/k}({\mathbb{G}}_m)$. For a reduced 0-dimensional scheme $p':\Sigma'\to\operatorname{Spec}(k)$ of finite type a morphism of $k$-schemes $f:\Sigma'\to \Sigma$ gives rise to a 