  Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve
over $K$ whose minimal proper regular model has totally degenerate semi-stable
reduction. Under certain hypotheses, we compute the prime-to-$p$ rational
torsion subgroup on the Jacobian of $C$. We also determine divisibility of line
bundles on $C$, including rationality of theta characteristics and higher spin
structures. These computations utilize arithmetic on the special fiber of $C$.
