Naturality of Abel maps

We give a combinatorial characterization of nodal curves admitting a natural (i.e. compatible with and independent of specialization) d-th Abel map for any d ≥ 1. Let X be a smooth projective curve and d a positive integer; the classical d-th Abel map of X , αX : X d −→ PicX , associates to (p1, . . . , pd) ∈ X the class of the line bundle OX(p1 + . . .+ pd) in Pic d X . Such a morphism has good functorial properties, it is compatible with specialization and base change. Now let X be a singular nodal curve occurring as the limit of a family of smooth curves. We ask whether there is a notion of d-th Abel map for X which is limit of the Abel maps of the smooth curves of the family, and which is natural, i.e. independent of the choice of the family. It is known that, although a nodal curve X is endowed with a generalized Jacobian and a Picard scheme which are both natural (i.e. they are the limit of the Jacobians and, respectively, of the Picard schemes of the fibers of every family of curves specializing to X), there are interesting degeneration problems about line bundles and linear series where naturality is a subtle issue. Abel maps is the case which we investigate here. The main result of this paper, Theorem 1.5, characterizes in purely combinatorial terms nodal curves that possess a natural d-th Abel map. A consequence of our result is that, if we consider stable curves of genus g ≥ 2, then the locus in Mg of curves that fail to admit a natural d-th Abel map, for a fixed d ≥ 2, has codimension 2. So, naturality of Abel maps is not to be taken for granted, unless X is irreducible or of compact type, in which case it is not difficult to see that natural Abel maps exist for all d. What is a good notion of Abel maps for singular curves? The same definition as for the smooth case behaves badly under specialization; moreover, it obviously does not make sense if some of the pi are singular points of X . This last problem will not be an issue here: we shall only study noncomplete Abel maps, which is enough for our scopes. So, for us a d-th Abel map is a rational map β : X 99K Pic X arising as the limit of the Abel maps of smooth curves specializing to X , for some family. A further requirement is added to ensure separation of the target space; see 1.2. Thus, the target space of our d-th Abel map is the Picard scheme, not any particular compactification of it. Our definition and results should be sufficently general to apply to various compactified Picard schemes existing in the literature (see section 5). The construction of complete Abel maps for singular curves was carried out by A. Altman and S. Kleiman for irreducible and reduced curves in [AK]; see also [EGK00] for further results. Not much is known for reducible curves. Recently, in [CE06], degree 1-Abel maps of stable curves are defined, compactified, and shown to be natural. For higher d the completion problem is open in general, see [Co06] for some progress in case d = 2. Our main result indicates that a safe way to approach it is to work with a fixed one-parameter smoothing of the given curve X (as in [CE06] and [Co06]), or to restrict to natural Abel maps. Among our techniques, the main one is the use Néron models of Jacobians (as constructed by M. Raynaud in [R70]); this allows us to obtain a concrete description of our axiomatically defined Abel maps. Then we combine a result of E. Esteves and N. Medeiros about deformation of line bundles and enriched structures (in [EM02]) with a detailed combinatorial analysis.