Marcelo Saia (ICMC)

20th September, 16:00, room 3-011

Marcelo Saia (ICMC)

Title: Geometry and equisingularity of finitely determined map germs from $\C^n$ to $\C^3$, \ $n >2.$

Abstract:

The study of the geometry of the singularities of map germs is one of the main questions in singularity theory, a key tool to better understanding of the geometry is the description of all strata which appear in the critical locus $\Sigma (f)$, in the discriminant $\Delta (f)$, and in the hypersurface $X(f)$. Moreover, in these sets we can study the numerical invariants that control triviality conditions in families of map germs. In this talk first we investigate the geometry of finitely determined map germs $f: (\C^n,0) \to ( \C^3,0)$ with $n \geq 3$, we give an explicity description of all strata in these dimensions and, with the aid of a computer system, we show in an explicity way how to compute them in several examples.

Concerning the Whitney equisingularity, Gaffney describes in \cite{gaf1} the following problem: ``Given a 1-parameter family of map germs $F \colon (\C\times {\C}^n, (0,0))\to (\C \times {\C}^p,(0))$, find analytic invariants whose constancy in the family implies the family is Whitney equisingular.'' He shows that for the class of finitely determined map germs of discrete stable type, the Whitney equisingularity of such a family is guaranteed by the invariance of the zero stable types and the polar multiplicities associated to all stable types.

A natural question is to find a minimal set of invariants that guarantee the Whitney equisingularity of the family. We show that the Whitney equisingularity of $X(f)$ also implies the Whitney equisingularity of the strata in $\Sigma (f)$, and on the other hand, we use the L\^e numbers of the discriminant $\Delta(f)$ to control the other invariants. moreover we show that the corank one condition is not needed.

Joint work with: V. H. Jorge-P\'erez, A. J. Miranda and E. C. Rizziolli.