Zariski's multiplicity conjecture for quasihomogeneous hypersurfaces with non-isolated singularities

TL;DR

This paper proves Zariski's multiplicity conjecture for a broad class of quasihomogeneous hypersurfaces with non-isolated singularities, showing multiplicity is topologically invariant and determined by weighted data.

Contribution

It extends Zariski's conjecture to non-isolated singularities in quasihomogeneous hypersurfaces using explicit formulas based on weights and degrees.

Findings

Multiplicity is determined by weights and degrees.

Multiplicity remains invariant under topological equivalence.

Confirms Zariski's conjecture for non-isolated singularities.

Abstract

In this work, we consider a pair $(\textbf{X},0)$ and $(\textbf{Y},0)$ of hypersurfaces in $(\mathbb{C}^{n+1},0)$ 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 $\mathbb{C}^{n+1},$ where $2\le n\le 4.$ 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.