A Note on Determinant Varieties

Abstract

The disappearance of all minors of a specific size in a r × s matrix indicates determinantal varieties. “Essential isolated determinantal singularities (EIDS)” are a special class of de- terminantal variations that were created by Ebeling and Zade in 2009. Pereira demonstrated that, for all 1 ≤ k ≤ min {r, s} the determinantal variety Zk = A−1 (Mk rs ) is an essential iso- lated determinantal singularities (EIDS) iff A : (Cn, 0) → Mrs is ¯G -finitely determined. A ¯G -finite determinacy is generally held, as demonstrated by Ahmad and Ruas. The goal of this study is to identify normal forms of EIDS for the map germs A : U ⊂ Cn → Mrs.