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 ◦ F with isolated singularities is studied, where f : Y −→C is a function with (possibly non-isolated) singularity and F : X −→Y is a map into the domain of f, and F only is deformed. The corresponding T1(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 τ = μ(f ◦ F) − β0 + β1, where τ is the length of T1(F) and βi is the length of ToriOY(OY/Jf, OX). 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 τ and the rank of the vanishing homology of the zero locus of f ◦ F are obtained