Description of generalized jacobians of singular hyperelliptic curves through phase spaces of Mumford systems

Abstract

Many finite dimensional integrable systems qre expressed with the help of the Lax equation which highlights a spectral parameter and therefore a spectral curve. These spectral curves are the starting point of an algebro-geometric investigation. The algebraic aspect of this investigation will be expressed using the Jacobians of smooth spectral curves and generalized jacobians of singular spectral curves and in the other hand the geometrical aspect will be attributed to the vector fields which are defined by the Lax equation. In this article will be dedicated only to singular hyperelliptic spectral curves that are associated to Mumford systems. We will study the complementarity of these two mathematical approaches to describe the same object and revealing their characteristics.