The universal theta divisor
The moduli space of line bundles on a singular curve can fail to be proper, and a natural way to compactify it is by adding stable rank-1 torsion-free sheaves. Such compactifications depend on the choice of a polarization on the nodal curve. When compactifying the universal Jacobian over the moduli space of stable curves, one obtains a family of compact moduli spaces, all birational to each others, that depend on a polarization parameter. In this talk I will present a wall-crossing formula that describes how the theta divisor varies as a function of this parameter. This is a joint work with Jesse Kass (South Carolina).