A descent map for curves with totally degenerate semi-stable reduction

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.