On discriminants, Tjurina modifications and the geometry of determinantal singularities

TL;DR

This paper introduces a method to compute discriminants of families of isolated determinantal singularities, decomposing them into determinantal and non-determinantal parts, and applies this to Cohen-Macaulay surface singularities, providing new insights and counterexamples.

Contribution

It presents a novel approach for computing discriminants and their decomposition in determinantal singularities, and applies it to Cohen-Macaulay codimension 2 cases, clarifying previous observations and countering Wahl's conjecture.

Findings

Decomposition of discriminants into determinantal and non-determinantal parts.

Explicit construction of a counterexample to Wahl's conjecture.

Application to Cohen-Macaulay surface 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.