Deformation multiplicities of map germs with respect to Boardman symbols
by
Juan Jose Nuno Ballesteros
Universitat de Valencia, Spain

Given an analytic map germ f:(Cⁿ, 0) --> (C^p, 0) and a Boardman symbol i=(i₁, ..., i_k) of codimension n, we can look at the number of points of type \Sigmaⁱ that appear in a generic deformation f_t of f. In this work, we generalize this number to the case that the Boardman symbol has codimension \nu(i) <= n. Since in this case \Sigmaⁱ(f_t) is a submanifold of codimension \nu(i) when f_t is generic, we have to intersect it with a generic plane of dimension \nu(i). We show that the number obtained in this way does not depend on the generic plane nor on the generic deformation, provided that the codimension of V(J_i(f)) is also \nu(i) (J_i(f) is an ideal in O_n obtained by an iterative process from the minors of the jacobian matrix of f). Moreover, there is a canonical deformation of f defined by the jet space J^k(n, p) so that we can use it to compute this number. Finally, we see that when the local ring O_r+n/J_i(F), where F(t, x)=(t, f_t(x)), is Cohen-Macaulay, then the above number is equal to the multiplicity of O_n/J_i(f).

Date received: May 26, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabo-12.