Determinacy of Determinantal Varieties

TL;DR

This paper investigates the conditions under which matrices defining a special class of determinantal varieties, called EIDS, are finitely determined, establishing that generic homogeneous forms ensure this property.

Contribution

It proves that matrices parametrized by generic homogeneous forms of degree d define EIDS and that $ ext{G}$-finite determinacy holds generally for these singularities.

Findings

Matrices with generic homogeneous forms define EIDS.

$ ext{G}$-finite determinacy is generally valid for these matrices.

EIDS of type (m,n,t) are generic.

Abstract

A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of a $m\times n$-matrix. In this note, we consider $\mathcal{G}$-finite determinacy of matrices defining a special class of determinantal varieties. They are called essentially isolated determinantal singularities (EIDS) and were defined by Ebeling and Gusein-Zade. In this note, we prove that matrices parametrized by generic homogeneous forms of degree $d$ define EIDS. It follows that $\mathcal{G}$-finite determinacy of matrices hold in general. As a consequence, EIDS of a given type $(m,n,t)$ holds in general.