Compactified Jacobians: geometric and combinatorial aspects

The Jacobian variety of a family of singular curves is not proper in general. Several (mostly equivalent) constructions of modular compactifications have been proposed since the 1950s. These classical results, however, do not give a complete classification. After recalling some equivalent constructions and properties of the Jacobian variety of a smooth projective curve, we show what is lost as soon as we drop the smoothness hypothesis. We introduce the notion of a V-compactified Jacobian, together with its dependence on combinatorial V-stability condition, and show that it strictly generalises the preexisting notion of "classical" compactified Jacobian. In particular, we see that V-compactified Jacobians completely classify smoothable compactified Jacobians in the case of a nodal curve. This is joint work with Nicola Pagani and Filippo Viviani.