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

## TL;DR

This paper explores the algebraic and geometric descriptions of singular hyperelliptic spectral curves associated with Mumford systems, focusing on generalized Jacobians and their relation to integrable systems.

## Contribution

It provides a detailed analysis of the generalized Jacobians of singular hyperelliptic curves within the context of Mumford systems, linking algebraic and geometric perspectives.

## Key findings

- Characterization of generalized Jacobians for singular hyperelliptic curves
- Connection between Lax equations and spectral curve geometry
- Insights into the dual algebraic and geometric descriptions

## 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.