Generic sections of essentially isolated determinantal singularities

TL;DR

This paper investigates essentially isolated determinantal singularities (EIDS), establishing a minimality theorem for their Milnor number in dimension 3 and exploring properties of strongly generic hyperplane sections.

Contribution

It generalizes the Milnor number minimality theorem to dimension 3 and introduces strongly generic hyperplanes that preserve EIDS properties.

Findings

Proved minimality of Milnor number for hyperplane sections in dimension 3.

Defined strongly generic hyperplanes that maintain EIDS structure.

Extended constancy results of Milnor number using strongly general hyperplanes.

Abstract

We study the essentially isolated determinantal singularities (EIDS), defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We prove in dimension $3$ a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing previous results by J. Snoussi in dimension $2$. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of L\^e D. T. concerning constancy of the Milnor number.