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Journal of the London Mathematical Society 1998 58(1):141-152; doi:10.1112/S0024610798006413
© 1998 by London Mathematical Society
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© The London Mathematical Society

On the Number of Singularities in Generic Deformations of Map Germs

T. Fukui, J. J. Nuño Ballesteros and M. J. Saia

Department of Mathematics, Faculty of Science, Saitama University 255 Shimo-Okubo, Urawa 338, Japan. E-mail: tfukui{at}rimath.saitama-u.ac.jp
Departament de Geometria i Topologia, Universitat de València Campus de Burjassot, 46100 Burjassot, Spain. E-mail: nuno{at}uv.es
Departamento de Matemática, IGCE, UNESP Campus de Rio Claro, Caixa Postal 178, 13500-230 Rio Claro, SP, Brazil. E-mail: mjsaia{at}rcb000.uesp.ansp.br

Received 30 October 1995. Revision received 16 April 1996.

Let f:Cn, 0->Cp, 0 be a K-finite map germ, and let i=(i1, ..., ik) be a Boardman symbol such that {sum}i has codimension n in the corresponding jet space Jk(n, p). When its iterated successors have codimension larger than n, the paper gives a list of situations in which the number of {sum}i points that appear in a generic deformation of f can be computed algebraically by means of Jacobian ideals of f. This list can be summarised in the following way: f must have rank n–i1 and, in addition, in the case p=6, f must be a singularity of type {sum}i1,i2.


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