I am currently learning about log-geometry and try to understand the theory in the example of curves with basic log-structure over a general base $S$. Especially, I am interested in the question of how we can describe the monoid of global section of the relative characteristic of the sheaf of monoids. For my considerations, we can assume the curve has no markings.

I understood that if $S=k$ a separably closed field, then for a curve $C/S$ we have for the relative characteristic

\begin{align*} \overline{\mathcal{M}}_{C/S}= \mathbb{Z}_{s_1} \oplus \dots \oplus \mathbb{Z}_{s_m} \end{align*}

with the $s_i$ being the nodes and $\mathbb{Z}_{s_i}$ denoting a skyscraper sheaf.

Furthermore, if $S=Spec(A)$ for $A$ an strict Henselian local ring. Then étale locally a node looks like $f: Spec(A[x,y,t]/(xy-t) \to Spec(A[t])$ and here the log structure on the target is associated to the prelog-structure $\mathbb{N} log(x) \oplus \mathbb{N}log(y)$ on the source and $\mathbb{N}log(t)$ on the target. The map between these monoids is the diagonal.

With this information, my question decomposes into two questions:

1. What are the global sections of the relative characteristic of $f$? As the construction of the log-structure associated to a pre-log structure is not very constructive, I cannot do this computation with my current understanding.

2. How can I compute from this local data the global sections in case of a general base $S$? Is it straight-forward that everything glues, so I just get a direct sum or are more things going on here?