Codimension Two Determinantal Varieties with Isolated Singularities

TL;DR

This paper investigates codimension two determinantal varieties with isolated singularities, establishing formulas for their Milnor number and relating it to polar multiplicities and indices, with explicit computations for certain surface singularities.

Contribution

It introduces a Milnor number formula for these varieties, linking it to polar multiplicities and indices, and computes examples for simple determinantal surface singularities.

Findings

Milnor number expressed via second polar multiplicity and generic section Milnor number

Relation between Milnor number and Ebeling-Gusein-Zade index

Explicit calculations for specific determinantal surface singularities

Abstract

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a L\^e-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last section we compute the Milnor number of some normal forms from A. Fr\"uhbis-Kr\"uger and A. Neumer [2] list of simple determinantal surface singularities.