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.
In the first lecture of this minicourse, we will provide a crash course on the main features of the theory of abelian varieties. We will then discuss the main examples of abelian varieties arising from geometric objects: Jacobians of curves, intermediate Jacobians, and Prym varieties.
In the second lecture, we will focus on the theory of Prym varieties in the classical setting associated with unramified double covers, and begin to explore the different behaviours of Prym maps.
In the final lecture, we will investigate the intriguing geometry of the Prym map P6, as studied by Donagi and Smith. In particular, we will examine its proper extension, its degree, and its monodromy, which, as is well known, reflects the geometry of the 27 lines on a smooth cubic surface.