Abel maps for stable curves
Eduardo Esteves

Given a nonsingular curve C, and a point P, the Abel map is an embedding of C in its Jacobian J, sending a point Q to the class of Q-P. Since J is an Abelian variety, we may consider the subvarieties W_2:=C+C, W_3=C+C+C and, more generally, W_n:=C+...+C, n times, which give an ascending filtration of J. In fact, W_n is the image of C^n, the product of C with itself n times, under a natural map to J, the Abel map of degree n. Many geometric properties of C are reflected in this filtration. Since the moduli space of nonsingular curves has a compactification by stable curves, it is natural to ask whether the above constructions extend to stable curves. In this lecture we will see how to construct the Abel map of degree 1 from a stable curve to a compactification of its generalized Jacobian. The construction is not straightforward: a definition analogous to the one given above for nonsingular curves does not make sense, as I will explain, and has to be modified. This is joint work with Lucia Caporaso. I will also hint to what Lucia and I are doing to extend the construction to higher degree Abel maps, a work carried together with Juliana Coelho.

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