Matrix polynomials, generalized Jacobians, and graphical zonotopes

TL;DR

This paper explores the relationship between matrix polynomials and spectral curves, extending classical results to reducible and nodal curves, and describes a stratification of matrix polynomial sets via combinatorial data.

Contribution

It generalizes the connection between matrix polynomials and Jacobians to arbitrary nodal curves, introducing a stratification based on combinatorial data and conjecturing links to compactified Jacobians.

Findings

Extended spectral curve-Jacobian correspondence to reducible curves.

Described stratification of matrix polynomial sets into smooth pieces.

Proposed a conjecture relating reducible matrix polynomials to compactified Jacobians.

Abstract

A matrix polynomial is a polynomial in a complex variable $\lambda$ with coefficients in $n \times n$ complex matrices. The spectral curve of a matrix polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid \mathrm{det}(P(\lambda) - \mu \cdot \mathrm{Id}) = 0\}$. The set of matrix polynomials with a given spectral curve $C$ is known to be closely related to the Jacobian of $C$, provided that $C$ is smooth. We extend this result to the case when $C$ is an arbitrary nodal, possibly reducible, curve. In the latter case the set of matrix polynomials with spectral curve $C$ turns out to be naturally stratified into smooth pieces, each one being an open subset in a certain generalized Jacobian. We give a description of this stratification in terms of purely combinatorial data and describe the adjacency of strata. We also make a conjecture on the relation between completely reducible matrix polynomials and the canonical compactified Jacobian defined by V.Alexeev.