On Néron models of moduli spaces of theta characteristics

Let be a smoothing of a stable curve C and be the moduli space of theta characteristics on the smooth fibers of f. We describe the Néron model , in terms of combinatorial invariants of the dual graph of C. Furthermore, we provide a modular description of and we construct an immersion , where is a suitable relative compactified Jacobian. We show that ψf factors through the locus of parametrizing locally free rank-1 sheaves.