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 $\mathcal{G}$-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.

26 pages -- added example clarifying the intricacies of the relation between equivalence of germs and G-equivalence