TJURINA AND MILNOR NUMBERS OF MATRIX SINGULARITIES

Abstract

To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites $f\,{\circ}\,F$ with isolated singularities is studied, where $f\,{:}\,Y\,{\longrightarrow}\,\mathbb{C}$ is a function with (possibly non-isolated) singularity and $F:X\,{\longrightarrow}\, Y$ is a map into the domain of $f$, and $F$ only is deformed. The corresponding $T^1(F)$ is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed 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 $\mbox{Tor}_i^{\cal{O}_Y}({\cal O}_Y{/}J_f,\cal{O}_X)$. This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When $f$ has Cohen–Macaulay singular locus (for example when $f$ is the determinant function), relations between $\tau$ and the rank of the vanishing homology of the zero locus of $f\circ F$ are obtained.