Sheaf theoretic classifications of pairs of square matrices over arbitrary fields

TL;DR

This paper classifies pairs of square matrices over arbitrary fields using sheaf theory, providing a uniform approach that includes fields of characteristic two, and extends previous work on theta characteristics and matrix orbit parametrizations.

Contribution

It introduces a sheaf-theoretic framework for classifying matrix pairs over any field, generalizing existing results and including characteristic two cases.

Findings

Unified classification over arbitrary fields including characteristic two

Parametrizations of symmetric matrix pairs under special linear groups

Extension of previous orbit classification results

Abstract

We give classifications of linear orbits of pairs of square matrices with non-vanishing discriminant polynomials over a field in terms of certain coherent sheaves with additional data on closed subschemes of the projective line. Our results are valid in a uniform manner over arbitrary fields including those of characteristic two. This work is based on the previous work of the first author on theta characteristics on hypersurfaces. As an application, we give parametrizations of orbits of pairs of symmetric matrices under special linear groups with fixed discriminant polynomials generalizing some results of Wood and Bhargava-Gross-Wang.