Compactifying the Relative Jacobian over Families of Reduced Curves

0.1. Motivation and Goal. The problem of finding a natural relative compactification of the relative Jacobian over a family of curves has drawn a lot of attention since Igusa’s pioneering work [18] in the fifties. (In Subsect. 0.2 the reader will find a slightly more detailed account of this history, with references.) In the case where the curves in the family are geometrically integral, a very satisfactory solution has been found by Altman and Kleiman [4]: their relative compactification is a fine moduli space; that is, it admits a universal object, after an étale base change. However, reducible curves (especially nodal ones) show up quite often in applications. For instance, Deligne-Mumford stable curves are used in the compactification, Mg, of the moduli space of non-singular curves of genus g. Recently, Caporaso [10] and Pandharipande [25] produced a relative compactification of the relative Jacobian over Mg. Their construction is strongly based on Gieseker’s construction [15] of Mg, and does not seem to be adaptable to different situations. The main disadvantages of the constructions found so far for reducible curves are the lack of a universal object and the restricted range of applicability. Apart from the study of Mg, the relatively compactified Jacobian has been most recently employed by Beauville [8] in counting the number of rational curves on K3 surfaces. In his article, Beauville made the simplifying assumption [8, 1.2] that all curves belonging to a certain linear system on the K3 surface are integral. The assumption was used in order to guarantee the existence of a fine relative