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 $\C^3$, \ $n >2.$

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.