On Esteves' Compactification of the Relative Jacobian

We begin by introducing the notion of moduli problems and spaces, and look at the Picard scheme in detail as an example. We then look at Esteves' construction of natural relative compactifications of the relative Jacobian over a family of reduced curves by means of studying the étale sheaf of rank-1, torsion-free sheaves on the family. The speciality of this construction is that it admits a Poincaré sheaf after an étale base change. We then look at examples to have a better grasp of the construction.