Generic sections of essentially isolated determinantal singularities

Abstract

Abstract. We study the essentially isolated determinantal singularities (EIDS), defined by W. Ebeling and S. , 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 Lê D. T. concerning constancy of the Milnor number.

Determinantal Singularities

“…We start with the natural notion of equivalence of map germs in this context. 9 cf. [20, Section 7.1.5] Definition 2.1.…”

“…where d(r) = dim(X r+1 A , 0) = p − (m − r)(n − r). This theorem and its variants have appeared in several places such as [34], [9], [21], [82], [41], and [109].…”

On discriminants, Tjurina modifications and the geometry of determinantal singularities

“…[13], [14] and [5]), simultaneous studies of certain properties of EIDS of all matrix sizes and types only appear recently e.g. in [2] and [24]. In this section, we cover well-known facts about EIDS to give the reader the background knowledge for the subsequent considerations on the discriminant and the Tjurina transoform.…”

“…be no element of G leading from one representation of Pinkham's famous example [27] to the other, i.e. leading from a determinantal singularity of type (2,4,2) to one of type (3, 3, 2) (with the additional constraint of the matrix to be symmetric) or vice versa. It is important to observe that a restriction to G-equivalence does not fix the minimal size of the matrix, it only fixes some size, as any determinantal singularity of type (m, n, t) can easily be considered as one of type (m + 1, n + 1, t + 1) by simply adding an extra line and an extra column of which all entries are zero except the one where the row and column meet, which should then be chosen to be 1.…”

“…Recently, significant progress has been made for this class, e.g. in [26], [2], [24], [8], [17]. In [15] the use of Tjurina modifications made it possible to relate a given determinantal singularity to an often singular variety, which happens to be an ICIS under rather mild conditions.…”