Singular fiber of the Mumford system and rational solutions to the KdV hierarchy

TL;DR

This paper analyzes the singular fiber of the Mumford system at zero level, describing its structure via algebraic groups and providing an algorithm to compute rational solutions related to the KdV hierarchy.

Contribution

It offers a detailed description of the singular fiber of the Mumford system and introduces an explicit algorithm for rational solutions to the system and the KdV hierarchy.

Findings

Stratification of the singular fiber by algebraic groups

Explicit description using generalized Jacobian

Effective algorithm for rational solutions

Abstract

We study the singular iso-level manifold $M_g(0)$ of the genus $g$ Mumford system associated to the spectral curve $y^2=x^{2g+1}$. We show that $M_g(0)$ is stratified by $g+1$ open subvarieties of additive algebraic groups of dimension $0,1,...,g$ and we give an explicit description of $M_g(0)$ in terms of the compactification of the generalized Jacobian. As a consequence, we obtain an effective algorithm to compute rational solutions to the genus $g$ Mumford system, which is closely related to rational solutions of the KdV hierarchy.