Stratifications of the Singular Fibers of Mumford Systems

TL;DR

This paper investigates the singularities of Mumford systems, defining and comparing algebraic and geometric stratifications of their phase space, revealing that each fiber's strata are equidimensional quasi-affine submanifolds.

Contribution

It introduces two stratifications of the Mumford system's phase space, proves their equivalence, and characterizes the structure of each fiber's strata as quasi-affine submanifolds.

Findings

Algebraic and geometric stratifications are identical.

Each fiber's strata form equidimensional quasi-affine submanifolds.

Provides a detailed understanding of singularities in Mumford systems.

Abstract

An integrable system is a dynamic system characterized by the existence of constants of motion and the existence of algebraic invariants, having an origin in algebraic geometry. In the 1970s, Mumford introduced a new completely integrable system defined on a smooth hyperelliptic curve. In the 2000s, Vanhaecke completed the description of the Munford integrable system by defining a Poisson structure on the phase space of the Mumford system. In this article we will study the singular Mumford system. The starting point is to determine when and why the Mumford system is singular. For this we will do an in-depth study to understand what happens to singularities, using the concept of stratification. We will define two stratifications of the phase space, one algebraic stratification and the other geometric stratification. We will prove that these stratifications are identical, and they will allow us to define a finer stratification on each fiber of the Mumford system. We will conclude this article with the following surprising result: each stratum of a fiber is a partition of equidimensional quasi-affine submanifolds.