Equimultiplicity of topologically equisingular families of parametrized surfaces in ${\mathbb{C}}^3$

Abstract

We provide a positive answer to Zariski's conjecture for families of singular surfaces in ${\mathbb{C}}^3,$ under the condition that the family has a smooth normalisation. As a corollary of the result, we obtain a surprising characterization of the Whitney equisingularity of one parameter families of $\mathcal{A}$ finitely determined map-germs $f_t:({\mathbb{C}}^2,0)\to ({\mathbb{C}}^3,0),$ in terms of the constancy of only one invariant, the Milnor number of the double point locus.

arXiv commentThe result stated in Theorem 1.2 does not hold. The proof was based on Proposition 3.4, whose proof is not correct. Counter-examples for the results can be found in: ArXiv:1608.08290

References in corpus (2)

• Topological Triviality of Families of Singular Surfaces
• On the multiplicities of families of complex hypersurface-germs

Cited by in corpus (1)

• Whitney equisingularity of families of surfaces in ${\mathbb{C}}^3$