Extending the double ramification cycle using Jacobians

We prove that the extension of the double ramification cycle defined by the first-named author (using modifications of the stack of stable curves) coincides with one of those defined by the last-two named authors (using an extended Brill–Noether locus on a suitable compactified universal Jacobians). In particular, in the untwisted case we deduce that both of these extensions coincide with that constructed by Li and Graber–Vakil using a virtual fundamental class on a space of rubber maps.