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

Otoniel Nogueira da Silva, Manoel Messias da Silva Júnior

TL;DR

The article advances Zariski's multiplicity conjecture for quasihomogeneous hypersurfaces with non-isolated singularities by leveraging the double point space framework and explicit weight/degree formulas. It proves multiplicity invariance under topological equivalence for a wide class of $n$-varieties (notably for $n=2,3,4$) that arise as images of finitely determined, quasihomogeneous map germs, using normal forms and invariants like $C(f)$ and $T(f)$. The work extends these results to corank-1 settings in higher dimensions and provides conditions under which multiplicity is preserved in families, culminating in a Whitney-equisingularity criterion via the plane-curve invariant $W(f)$. Overall, the paper connects algebraic data (weights, degrees) with topological type to extend Zariski's conjecture beyond isolated singularities and offers explicit tools for computing multiplicities from quasihomogeneous data.

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.

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

TL;DR

The article advances Zariski's multiplicity conjecture for quasihomogeneous hypersurfaces with non-isolated singularities by leveraging the double point space framework and explicit weight/degree formulas. It proves multiplicity invariance under topological equivalence for a wide class of -varieties (notably for ) that arise as images of finitely determined, quasihomogeneous map germs, using normal forms and invariants like and . The work extends these results to corank-1 settings in higher dimensions and provides conditions under which multiplicity is preserved in families, culminating in a Whitney-equisingularity criterion via the plane-curve invariant . Overall, the paper connects algebraic data (weights, degrees) with topological type to extend Zariski's conjecture beyond isolated singularities and offers explicit tools for computing multiplicities from quasihomogeneous data.

Abstract

In this work, we consider a pair and of hypersurfaces in parametrized by finitely determined, quasihomogeneous map germs and respectively. Zariski asked whether the multiplicity is preserved under topological equivalence of hypersurface germs. We address this question within a wide class of -dimensional quasihomogeneous varieties with non-isolated singularities in where 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.
Paper Structure (9 sections, 15 theorems, 63 equations, 4 figures, 4 tables)

This paper contains 9 sections, 15 theorems, 63 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiple point space for map germs
  4. Finite determinacy, crosscaps and triple points
  5. Quasihomogeneous map germs from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$
  6. The multiplicity of the image of a finite map germ for the case $n=2$
  7. The Zariski's multiplicity conjecture for surfaces
  8. The multiplicity of the image of a finitely determined, quasihomogeneous, corank 1 map germ
  9. The Zariski's multiplicity conjecture for $n$-varieties

Key Result

Theorem 1.1

Let $(\textbf{X},0)$ and $(\textbf{Y},0)$ be germs of irreducible surfaces in $(\mathbb{C}^3,0)$ defined as the zero set of reduced quasihomogeneous map germs $F,G:(\mathbb{C}^3,0)\rightarrow (\mathbb{C},0)$, respectively. Suppose that $(\textbf{X},0)$ (respectively $(\textbf{Y},0)$) has a smooth no

Figures (4)

  • Figure 1: The singularity $C_5$ of Mond and its double point curve $D(f)$ (real points).
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 21 more