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Bulletin of the London Mathematical Society Advance Access originally published online on March 12, 2008
Bulletin of the London Mathematical Society 2008 40(1):129-138; doi:10.1112/blms/bdm097
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© 2008 London Mathematical Society

The Milnor number of a function on a space curve germ

J. J. Nuño-Ballesteros

Departament de Geometria i Topologia
Universitat de València
46100 Burjassot
Spain

J. N. Tomazella

Departamento de Matemática
Universidade Federal de São Carlos
Caixa Postal 676
13560-905
São Carlos, SP
Brazil
tomazella@dm.ufscar. br

Received 5 February 2007. Revision received 28 June 2007.

Given a finite function germ f:(X, 0) -> (C, 0) on a reduced space curve singularity (X, 0), we show that µ(f) = µ(X, 0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denote the Milnor numbers of the function and the curve, respectively, and deg(f) is the degree of f. We use this formula to obtain several consequences related to the topological triviality and Whitney equisingularity of families of curves and families of functions on curves.

2000 Mathematics Subject Classification 32S30 (primary), 58K05, 32S15 (secondary).

The first author has been partially supported by DGICYT grant MTM2006–06027. The second author has been partially supported by CAPES grant 3477/05-3.


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