On discriminants, Tjurina modifications and the geometry of determinantal singularities

Abstract

We describe a method for computing discriminants for a large class of families of isolated determinantal singularities -more precisely, for subfamilies of G-versal families. The approach intrinsically provides a decomposition of the discriminant into two parts and allows the computation of the determinantal and the non-determinantal loci of the family without extra effort; only the latter manifests itself in the Tjurina transform. This knowledge is then applied to the case of Cohen-Macaulay codimension 2 singularities putting several known, but previously unexplained observations into context and explicitly constructing a counterexample to Wahl's conjecture on the relation of Milnor and Tjurina numbers for surface singularities.where J F denotes the submodule generated by the k matrices, each holding the partial derivatives of the entries of F w.r.t. one of the variables, and J op = AF + F B | A ∈ Mat(m, m; C{x}), Mat(n, n; C{x})Remark 2.10 The group G is a subgroup of the group K of Mather and coincides with it e.g. for complete intersections and Cohen-Macaulay codimension 2 singularities. It also appears as the subgroup K V of K in the literature, where

Determinantal Singularities

“…This example is taken from Schaps [90] and it also appears in [18]. Explicit computations can be found in [37]. Let (X 2 A , 0) ⊂ (C 4 , 0) be the union of the four coordinate axis.…”

“…For this class of singularities, the deformations of (X, 0) coincide with the unfoldings of F up to GL-equivalence. This starts with an explicit identification of the space of first order deformations T 1 X,0 of (X, 0) as in (37) and the infinitesimal unfoldings T 1 GL (F ) of F from (20). Lemma 3.14.…”

“…Note that µ = τ does not hold for quasi-homogeneous ICIS of dimension zero, see for instance Table5 25. A counterexample given by the first named author in[37] turned out to be wrong.…”

Partial Local Resolution by Characteristic Zero Methods

“…, x N ] (R 0 as before). For recent results studying these singularities from various viewpoints (over a field of characteristic zero), we refer to [19,41,43,44,45,48,80]. An interesting problem would be to investigate which assumptions need to be imposed on R 0 and the entries f i,j in order to construct a characteristic-free embedded resolution for this type of singularities.…”