Abstract

The Jacobian of a smooth proper complex curve X is the abelian variety parametrising of the line bundles on X. If we drop the smoothness hypothesis, we can still define a generalised Jacobian, which, however, fails to be proper. Constructing suitable compactified Jacobians is a classical problem, addressed since the late ‘70s by Oda-Seshadri for a single nodal curve, by Altman-Kleiman, Esteves, Simpson and others for families of curves with planar singularities.

In particular, a fine compactified universal Jacobian can be obtained by taking (stable) limits of degenerations of line bundles on the universal family Cgn/Mbgn over the moduli space of stable curves of genus g with n marked points.

In this talk I will describe the geometry of a fine compactified Jacobian of a nodal curve. Then I will use the topological stratification of the moduli space Mbgn to show how to compute the cohomology of a fine compactified universal Jacobian, using Hodge theory.