2023/09/06 by Otoniel Nogueira da Silva, da Silva, Otoniel Nogueira, Júnior, Manoel Messias da Silva · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2309.02624
openalex publication_date 2023/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we consider a pair (X,0) and (Y,0) of hypersurfaces in (ℂn+1,0) parametrized by finitely determined, quasihomogeneous map germs f and g, respectively. Zariski asked whether the multiplicity is preserved under topological equivalence of hypersurface germs. We address this question within a wide class of n-dimensional quasihomogeneous varieties with non-isolated singularities in ℂn+1, where 2≤ n≤ 4. This class consists of varieties that arise as image of finitely determined, quasihomogeneous map germs. Using a quasihomogeneous normal form, we derive explicit formulas for the multiplicity in terms of the weights and the degrees of the map germ. Our results show that multiplicity, within this setting, is determined by the weighted data and is invariant under topological equivalence, thereby confirming Zariski's multiplicity conjecture and extending current knowledge beyond the isolated singularity case.