For a Gorenstein curve X and a nonsingular point P ∈ X, we construct Abel maps A : X → J 1 X and A P : X → J 0 X , where J i X is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A P are shown to have the same arithmetic genus of X. Also, A and A P are shown to be embeddings away from rational subcurves L ⊂ X meeting X − L in separating nodes. Finally we establish a connection with Seshadri's moduli scheme U X (1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U X (1).

We present numerical conditions for the existence of natural degree-2 Abel maps for any given nodal curve. CoCoA scripst were written and have so far verified the validity of the conditions for numerous curves.curves of compact type, developed by Eisenbud and Harris; see [21]. Other results have been obtained through the theory; see [17].The theory of Eisenbud and Harris points out to a partial compactification of the variety of linear series over the moduli space of curves of compact type. It is natural to ask whether one can extend this compactification over the whole M g , as Eisenbud and Harris themselves asked in [18]. The supposedly intermediate step, that of constructing a compactification of the relative Picard scheme over M g , has been carried out by Caporaso [3]. But the final step has proved to be very difficult.Sketches on how to deal with limit linear series for stable curves not of compact type are sparse in the literature. Recently, Medeiros and the second author [14] were able to describe limit canonical series on curves with two components, using the theory presented in [11], the ingredients of which having already appeared in [29]. But their description is rather complicated, and relies on the strong assumption that the components intersect each other at points in general position on each component. The same assumption is present in [16], where the case of curves with more components is partially considered.In view of the difficulties, one might ask whether we stand a better chance of compactifying the variety of linear series over M g by looking at degenerations of Abel maps. Indeed, Abel maps of all degrees have been constructed and studied for integral curves in [1]. Degree-1 Abel maps for stable curves were constructed in [6] and studied in [7]. Higher-degree Abel maps for curves of compact type appeared soon afterwards in [10].At this point one might say that the theory of degenerations of Abel maps is on even terms with that of limit linear series. A comparison between the two theories has in fact appeared in [15], based on the more refined notion of limit linear series introduced by Osserman [26], and thus limited to two-component curves.The present article advances the theory of degenerations of Abel maps, by presenting purely numerical conditions for the existence of degree-2 Abel maps for any given nodal curve. Whether the curve is stable or not is immaterial. The specific moduli, even the genera of its irreducible components is immaterial. Where the points of intersection of components lie on each component is immaterial, in stark contrast with [14] and [16]. In fact, for nodal curves with the same dual graph, either all of them or none of them have degree-2 Abel maps. An algorithm has been implemented by means of CoCoA scripts, available at http://w3.impa.br/∼esteves/CoCoAScripts/Abelmaps to check the validity of these conditions for any given curve and has, so far, always returned a positive answer. Our approach is simply to extend the construction done in [9] for two-component two...

a b s t r a c tRecently, the first Abel map for a stable curve of genus g ≥ 2 has been constructed. Fix an integer d ≥ 1 and let C be a stable curve of compact type of genus g ≥ 2. We construct two d-th Abel maps for C , having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the second Abel map.

In this paper we give local conditions to the existence of Abel maps for smoothings of nodal curves extending the Abel maps for the generic fiber. We use this result to construct Abel maps of any degree for nodal curves with two components. * The third author was partially supported by CNPq, processo 300714/2010-6.

In this paper we use admissible covers to investigate the gonality of a stable curve C over double-struckC. If C is irreducible, we compare its gonality to that of its normalization. If C is reducible, we compare its gonality to that of its irreducible components. In both cases we obtain lower and upper bounds. Furthermore, we show that four admissible covers constructed give rise to generically injective maps between Hurwitz schemes. We show that the closures of the images of three of these maps are components of the boundary of the target Hurwitz schemes, and the closure of the image of the remaining map is a component of a certain codimension‐1 subscheme of the boundary of the target Hurwitz scheme.