Characterization of linear spaces of matrices of constant rank from syzygy bundles

TL;DR

This paper characterizes matrices of linear forms with constant rank using syzygy bundles, enabling classification of indecomposable matrices up to rank 7 and describing related moduli spaces of vector bundles.

Contribution

It introduces a novel link between constant rank matrices and syzygy bundles, providing a systematic classification and moduli space description.

Findings

Classified all indecomposable matrices of constant rank up to 7.

Established a correspondence between matrices and syzygy bundles.

Described the moduli spaces of simple vector bundles related to these matrices.

Abstract

In this work, we characterize matrices of linear forms and constant rank, demonstrating that, under some natural assumptions, they are always associated with a syzygy bundle that fits into a (partially linear) resolution. Furthermore, this construction allows us to list all indecomposable matrices of constant rank up to 7, as well as describing the moduli spaces of simple vector bundles naturally defined by families of constant rank matrices.