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 $To{r}_i({Ø}_Y/J_f,{Ø}_X)$. This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger.

LaTeX file; 23 pages; minor corrections