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Tjurina and Milnor numbers of matrix singularities

By Victor V. Goryunov and D. (David) Mond

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\ud 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\ud τ = μ(f ◦ F) − β0 + β1,\ud 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\ud determinant function), relations between τ and the rank of the vanishing homology of the zero locus of f ◦ F are obtained

Topics: QA
Publisher: Cambridge University Press
Year: 2005
OAI identifier: oai:wrap.warwick.ac.uk:739

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