Sheaf theoretic classifications of pairs of square matrices over arbitrary fields

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.