In the first lecture of the minicourse we will discuss how the theory of line bundles on a nodal curve differs from that of a smooth curve. In particular, we will describe how gluing data can be used to define line bundles and how this leads to a non-compact space. We will then discuss how to describe the additional objects one needs to include as limits, torsion-free rank one sheaves. Finally, we'll discuss a second issue motivating the construction of compactified Jacobians: the non-uniqueness of extensions of line bundles in families and the resulting non-separatedness of the parametrizing space.
In the second lecture, we will introduce the main tool to construct compactified Jacobians: stability conditions. We'll discuss the notions of semistability, stability and polystability, and how they interact with the choice of a stability condition. We will then use stability to describe the compactified Jacobians and discuss some of their properties: their modular interpretation, the natural stratification of their boundary and the behaviour in families.
In the third lecture we first continue with an extended example: the canonical stability condition in degree g-1 and how semistability for this choice of stability condition relates to orientations of the dual
graph and to the existence of a Theta divisor. In the second part of the lecture we will introduce some tropical aspects of the theory: in particular, how polytopal subdivisions of the tropical Jacobian give toric charts for the compactified Jacobians.
The notes from the mini course can be found in
https://drive.google.com/file/d/1bRhBrE4N_kknIIVsqk3JvdA-7UuwtrLR/view?usp=sharing
TBA