Jacobian Varieties

Abstract

To every smooth projective curve C one can associate in a natural way a principally polarized abelian variety, its Jacobian J(C). As we mentioned already in the introduction, the theory of abelian varieties originated with the investigation of Jacobians. They are not only the most important, but also the best-known examples of abelian varieties. Much more can be said about them than about a general principally polarized abelian variety. In fact, presenting the theory of Jacobian varieties in a satisfactory way would require a whole volume for itself. This chapter contains, apart from the basic definitions and constructions, only some selected topics on Jacobian varieties. We focus on results and in particular on proofs, which on the one hand apply the results of the earlier chapters and on the other hand are not yet contained in other books.