Me and a friend tried the following problem, but with no luck. Anything would be appreciated:
Let $X \rightarrow S$ be an arithmetic surface such that for some $s \in S$, $X_s$ is the union of two elliptic curves meeting transversally at a point rational over $k(s)$. Show that $\omega_{X/S}$ is not generated by its global sections, and find the smallest $n$ such that $\omega^n_{X/S}$ is.
I have only noted some trivial things, the fibers are of genus two seems relevant. But apart from that, nothing.
