Combinatorics of compactified universal Jacobians

We show that the combinatorial structure of the compactified universal Jacobians over M‾ g in degrees g−1 and g is governed by orientations on stable graphs. In particular, for a stable curve we exhibit graded stratifications of the compactified Jacobians in terms of totally cyclic, respectively rooted, orientations on its dual graph. We prove functoriality under edge-contraction of the posets of totally cyclic and rooted orientations on stable graphs.