On the topology of the transversal slice of a quasi-homogeneous map germ

2021/07/27 by Otoniel Nogueira da Silva, da Silva, Otoniel Nogueira · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2107.13043

Abstract

We consider a corank 1, finitely determined, quasi-homogeneous map germ f from (ℂ²,0) to (ℂ³,0). We describe the embedded topological type of a generic hyperplane section of f(ℂ²), denoted by γ_f, in terms of the weights and degrees of f. As a consequence, a necessary condition for a corank 1 finitely determined map germ g:(ℂ²,0)→ (ℂ³,0) to be quasi-homogeneous is that the plane curve γ_g has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding F=(f_t,t) of f which adds only terms of the same degrees as the degrees of f is Whitney equisingular.

Cited by

• Transverse slices, Ruas' conjecture, and Zariski's multiplicity conjecture for quasihomogeneous surfaces

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