Tjurina and Milnor numbers of matrix singularities

TL;DR

This paper investigates the deformation theory of matrix singularities, specifically determinants and Pfaffians, by analyzing the associated T^1 and Milnor numbers, and establishes a formula relating these invariants.

Contribution

It introduces a new framework for understanding matrix singularities through derived functors and long exact sequences, connecting T^1 and Milnor numbers with Tor modules.

Findings

Derived functor cohomology describes T^1(F) for matrix deformations

Established the relation τ=μ(f∘F)−β₀+β₁ for singularity invariants

Explains numerical coincidences in simple matrix singularities

Abstract

In order to understand the deformations of determinants and Pfaffians resulting from deformations of matrices, we study the deformation theory of composites $f\circ F$, with isolated singularities, where $f:Y\to\C$ has Cohen-Macaulay singular locus and $F:X\to Y$. We identify the corresponding $T^1(F)$ as (something like) the cohomology of a derived functor, and construct a canonical long exact sequence from which it follows that $$\tau=\mu(f\circ F)-\beta_0+\beta_1,$$ where $\tau$ is the length of $T^1(F)$ and $\beta_i$ is the length of $Tor_i(\O_Y/J_f,\O_X)$. This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger.