Tjurina and Milnor numbers of 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.