Whitney equisingularity of families of surfaces in ℂ³

Abstract

In this work, we study families of singular surfaces in C 3 parametrized by A-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding F of a finitely determined map germ f : (C 2 , 0) → (C 3 , 0). We investigate the following conjecture: topological triviality implies Whitney equisingularity of the unfolding F ? We provide a complete answer to this conjecture, given counterexamples showing how the conjecture can be false. * M.A.S. Ruas: ICMC, Universidade de São Paulo, Av. do Trabalhador São

On Invariants of Generic Slices of Weighted Homogeneous Corank 1 Map Germs from the Plane to 3-Space

“…This motivates us to give the following definition which is from Silva (2017), Def. 4.1 (see also Ruas and Silva (2019), Def. 2.4) and (Silva 2019, Sec. 3).…”

“…Can the invariant m( f (D( f ))) be calculated in terms of the weights and degrees of f ? In Ruas and Silva (2019), Proposition 6.2, Ruas and the author provided answers to both questions above in the case where f is homogeneous. In this work, using a normal form for f (Lemma 2.11), we present a positive answer for both questions without any restriction on the weights and degrees of f .…”

Reflection maps

“…The aforementioned families have already proven quite useful. The maps of Example 16 have been used by Silva and Ruas [49] to obtain the first counterexamples to a conjecture by Ruas [48], which stated the equivalence between topological triviality and Whitney equisingularity of families of map-germs C 2 → C 3 . The same examples have been used by Brasselet, Nguyen and Ruas [5] to show that, in the space of corank one finitely determined map-germs (C 2 , 0) → (C 3 , 0) with homogeneous parametrization, the topological classification of the map-germs and the inner bi-Lipschitz classification of their images coincide.…”

Superfícies com singularidades não isoladas

“…Discutiremos novamente a conjectura de Ruas na seção 4.3. Os resultados originais deste capítulo podem ser encontrados em [52].…”

“…Finalizamos este capítulo com uma prova da famosa conjectura de Zariski para a multiplicidade no caso de famílias de superfícies que são parametrizadas por germes de aplicações f t : (C 2 , 0) → (C 3 , 0) finitamente determinados, homogêneos e de coposto 1. A maioria dos resultados originais deste capítulo podem ser encontrados em [52].…”