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Stable Images and Discriminants

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Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 357)

Abstract

We show that the discriminant/image of a stable perturbation of a germ of finite \(\mathcal {A}\)-codimension is a hypersurface with the homotopy type of a wedge of spheres in middle dimension, provided the target dimension does not exceed the source dimension by more than one. The number of spheres in the wedge is called the discriminant Milnor number/image Milnor number. We prove a lemma showing how to calculate this number, and show that when the target dimension does not exceed the source dimension, the discriminant Milnor number and the \(\mathcal {A}\)-codimension obey the “Milnor–Tjurina relation” familiar in the case of isolated hypersurface singularities. This relation remains conjectural in the case of parameterised hypersurfaces of dimension greater than 2; we lay the ground work for its proof, in Chap. 11, when the dimension is no greater than 2.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Departament de MatemàtiquesUniversitat de ValènciaBurjassotSpain

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