Properties of G-Equivalence of Matrices

TL;DR

This paper explores the properties of G-equivalence of matrices, extending singularity theory to determinantal varieties and providing conditions for matrix determinacy and deformation triviality.

Contribution

It extends the theory of singularities for nxp matrices and characterizes G-finite determinacy for Cohen-Macaulay codimension 2 varieties.

Findings

Extended singularity theory to nxp matrices.

Characterized G-finite determinacy conditions.

Provided criteria for topological triviality of deformations.

Abstract

The theorem of Hilbert- Burch provides a description of codimension two determinantal varieties and their deformations in terms of their presentation matrices. In this work we use this correspondence to study properties of determinantal varieties, based on methods of singularity theory. We establish the theory of singularities for nxp matrices extending previous results of Bruce and Tari and Fr\"uhbis-Kr\"uger. The main result of this work is the description of equivalent conditions to G-finite determinacy of the presentation matrix of Cohen-Macaulay varieties of codimension 2. We apply the results to obtain sufficient conditions for topological triviality of deformations of weighted homogeneous matrices.