Codimension Two Determinantal Varieties with Isolated 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 $\mathsf{C}^4$, we obtain a Lê-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 Frühbis-Krüger and Neumer [7] list of simple determinantal surface singularities.

The Bi-Lipschitz Equisingularity of Essentially Isolated Determinantal Singularities

“…In this work we present conditions which ensure the above vector field is Lipschitz in the context of determinantal varieties. Following the approach of Pereira and Ruas [18], for 1-unfoldingsF : C × C q −→ C × Hom(C m , C n ) written as F (y, x) = (y, F (x) + yθ(x)) we show that if θ is constant then the above vector field is always Lipschitz. If θ is not constant, we have examples in both cases.…”

“…Currently, determinantal varieties have been an important object of study in Singularity Theory. For example, we can refer to the works of Damon [5], Frühbis-Krüger [7,8], Gaffney [9,10], Grulha [10], Nuño-Ballesteros [1,14], Oréfice-Okamoto [1,14], Pereira [15,18], Pike [5], Ruas [10,18], Tomazella [1,14], Zhang [22] and others.…”

Real and complex integral closure, Lipschitz equisingularity and applications on square matrices

“…More recently, Gaffney's result was extended in [6], where the authors presented conditions which ensure the canonical vector field is Lipschitz in the context of 1-unfoldings of singularities of matrices, following the approach of Pereira and Ruas [22].…”

Bi-Lipschitz invariant geometry

“…It following the approach of Pereira and Ruas [89], we see that for the special case of determinantal surfaces, there are deformationsf : C q −→ Hom(C n , C m ) such that the above vector field always is Lipschitz. A singularity is called simple if it can only deform into finitely many different isomorphism classes.…”

“…For example, we can refer to the works of Damon [29], Frühbis-Krüger [37,38], Gaffney [39,47], Grulha [47], Nuño-Ballesteros [2,76], Oréfice-Okamoto [2,76], Pereira [83,89], Pike [29], Ruas [47,89], Tomazella [2,76], Zhang [114] and others.…”