Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology

Abstract

In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the “exceptional orbit varieties” for representations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors ${\mathcal{ℰ}}_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an extension of that introduced by Lê–Greuel for computing the Milnor number of an ICIS. Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups.

Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number. We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules. In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen–Macaulay singularities defined as $2\times 3$ matrix singularities. We compute the Milnor number of isolated Cohen–Macaulay surface singularities of this type in ${ℂ}^4$ and the difference of Betti numbers of Milnor fibers for isolated Cohen–Macaulay 3–fold singularities of this type in ${ℂ}^5$.