Nicola Pagani (University of Liverpool)

A classical construction in algebraic geometry associates with every nonsingular complex projective curve its Jacobian, a complex projective variety of dimension equal to the genus of the curve. A similar construction is available for singular curves, but the resulting Jacobian variety fails in general to be compact. In this talk we introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. This notion can be naturally generalised to families of curves.

For the case of a curve in isolation it is unclear whether this notion is strictly more general than the existing examples (constructed by many authors), but for the case of families we will show classification results that imply that this notion is strictly more general than the existing constructions.

If time permits, we will discuss some results on the topology of fine compactified universal Jacobians, and wall-crossing results for some natural cycle classes (Brill-Noether classes). This abstract describes joint work with Alex Abreu, Jesse Kass and Orsola Tommasi (some of which is unpublished).