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Revista Matemática Complutense

, Volume 32, Issue 2, pp 395–418 | Cite as

A Jacobian module for disentanglements and applications to Mond’s conjecture

  • J. Fernández de BobadillaEmail author
  • J. J. Nuño-Ballesteros
  • G. Peñafort-Sanchis
Article
  • 68 Downloads

Abstract

Let \(f:({\mathbb {C}}^n,S)\rightarrow ({\mathbb {C}}^{n+1},0)\) be a germ whose image is given by \(g=0\). We define an \({\mathcal {O}}_{n+1}\)-module M(g) with the property that \({\mathscr {A}}_e\)-\({\text {codim}}(f)\le {\text {dim}}_{\mathbb {C}}M(g)\), with equality if f is weighted homogeneous. We also define a relative version \(M_y(G)\) for unfoldings F, in such a way that \(M_y(G)\) specialises to M(g) when G specialises to g. The main result is that if \((n,n+1)\) are nice dimensions, then \({\text {dim}}_{\mathbb {C}}M(g)\ge \mu _I(f)\), with equality if and only if \(M_y(G)\) is Cohen–Macaulay, for some stable unfolding F. Here, \(\mu _I(f)\) denotes the image Milnor number of f, so that if \(M_y(G)\) is Cohen–Macaulay, then Mond’s conjecture holds for f; furthermore, if f is weighted homogeneous, Mond’s conjecture for f is equivalent to the fact that \(M_y(G)\) is Cohen–Macaulay. Finally, we observe that to prove Mond’s conjecture, it is enough to prove it in a suitable family of examples.

Keywords

Image Milnor number \({\mathscr {A}}_e\)-Codimension Weighted homogeneous 

Mathematics Subject Classification

Primary 58K15 Secondary 58K40 58K65 

Notes

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Copyright information

© Universidad Complutense de Madrid 2019

Authors and Affiliations

  • J. Fernández de Bobadilla
    • 1
    • 2
    Email author
  • J. J. Nuño-Ballesteros
    • 3
  • G. Peñafort-Sanchis
    • 2
  1. 1.IKERBASQUE, Basque Foundation for ScienceBilbaoSpain
  2. 2.BCAM, Basque Center for Applied MathematicsBilbaoSpain
  3. 3.Departament de MatemàtiquesUniversitat de ValènciaBurjassotSpain

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