Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel-Jacobi map. This breaks down over the boundary since the Abel-Jacobi map fails to extend. We construct a 'universal' resolution of the Abel-Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.

We investigate to what extent the theory of Néron models of jacobians and of abel-jacobi maps extends to relative curves over base schemes of dimension greater than 1. We give a necessary and sufficient criterion for the existence of a Néron model. We use this to show that, in general, Néron models do not exist even after making a modification or even alteration of the base. On the other hand, we show that Néron models do exist outside some codimension-2 locus. Idea of the proof: Néron models via the relative Picard functorOur existence and non-existence results for the Néron model proceed via an auxiliary object, the 'total-degree-zero relative Picard functor' PicC/S to the locus U ⊆ S where C is smooth coincides with the Jacobian of C U /U. It is also rather easy to show thatC/S satisfies the 'existence' part of the Néron mapping property whenever C is regular. However, if C/S has non-irreducible fibres then Pic [0]C/S is in general highly non-separated, and for this reason it fails to satisfy the 'uniqueness' part of the Néron mapping property. As such, we wish to construct some kind of 'separated quotient' of Pic [0]C/S , in the hope that this will satisfy the whole Néron mapping property.Now the failure of separatedness in a group-space is measured by the failure of the unit section to be a closed immersion; as such, a natural way to construct a 'separated quotient' of Pic [0] C/S is to quotient by the closure of the unit section. Quotients by flat subgroup-spaces always exist in the category of algebraic spaces, but the problem is that the closure of the unit section in Pic [0]C/S (which we shall refer to asē) is not in general flat over S, and so this quotient does not exist 3 . By a slightly more delicate argument, one can even show that the existence of a Néron model is equivalent to the flatness ofē. The technical heart of this paper is 3 One might be tempted to apply [RG71] to flattenē by blowing up S, but note that Pic [0] C/Sis not quasi-compact over S, and so their results do not apply; indeed, one consequence of our results is that, if C/S is not aligned, thenē does not become flat after any modification of S.

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.

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