Generalized Jacobians and rigidificators

Abstract

The purpose of this paper is to give a new interpretation and generalization of both the singularization procedure of Rosenlicht and the concept of the generalized Jacobian [18], which arose in the attempt to understand the role the generalized Jacobian plays in Krichever's theory of the integration of non-linear evolution equations in terms of theta functions of curves [14], [16]. The interpretation of the generalized Jacobian as a rigidificator for the Jacobian of a regular curve is not surprising (cf. Drinfeld [3]), but does not seem to exist in the present literature on algebraic curves.