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Tjurina and Milnor numbers of matrix singularities

Victor Goryunov·David Mond
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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 $$τ=μ(f\circ F)-β_0+β_1,$$ where $τ$ is the length of $T^1(F)$ and $β_i$ is the length of $Tor_i(Ø_Y/J_f,Ø_X)$. This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger.

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Victor Goryunov, David Mond. 2003-07-11. Tjurina and Milnor numbers of matrix singularities. https://arxiv.org/abs/math/0307025

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