Neron models and compactified Picard schemes over the moduli stack of stable curves

We construct modular Deligne-Mumford stacks P_{d,g} representable over M_g parametrizing N´eron models of Jacobians as follows. Let B be a smooth curve and K its function field, let X_K be a smooth genus-g curve over K admitting stable minimal model over B. The N´eron model N(Pic^dX_K) → B is then the base change of Pd,g via the moduli map B −→ Mg of f. Moreover P_{d,g} is compactified by a Deligne-Mumford stack over M_g, giving a completion of N´eron models naturally stratified in terms of N´eron models