Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Otoniel Nogueira da Silva, Manoel Messias da Silva Júnior

TL;DR

This work studies equisingularity for image surfaces of finitely determined, quasihomogeneous, corank 1 map germs $f:(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ by introducing $\mu_{m,k}$-minimal transverse slices to reduce surface problems to plane curve germs. It proves that, under the $\mu_{m,k}$-minimal hypothesis, topological triviality and Whitney equisingularity are equivalent for 1-parameter unfoldings, and derives Zariski-type multiplicity results for such families; it also produces counterexamples showing that Whitney equisingularity does not imply bi-Lipschitz equisingularity and motivates a revised Ruas conjecture. The paper further provides explicit normal forms for $\mu_{m,k}$-minimal plane curves, upper semicontinuity results in the minimal setting, and a reformulation of Ruas’ conjecture via the Milnor number of the curve $W(f)=D(f)\cup \tilde{\gamma}$. Collectively, these results advance the understanding of how transversality, Milnor numbers, and multiplicity invariants govern equisingularity in families of quasihomogeneous surface germs.

Abstract

In this work, we consider a finitely determined, quasihomogeneous, corank 1 map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We introduce the concept of the $μ_{\mathbf{m},\mathbf{k}}$-minimal transverse slice of $f$}. Since such a slice is a plane curve, it admits a topological normal form, which we describe explicitly. Assuming the $μ_{\mathbf{m},\mathbf{k}}$-minimal transverse slice hypothesis, we provide a proof for the equivalence between topological triviality and Whitney equisingularity in Ruas' conjecture within this setting. We also provide a counterexample which shows that Whitney equingularity does not imply bi-Lipschitz equisingularity, given an answer to a question by Ruas. Moreover, we show that every topologically trivial $1$-parameter unfolding of $f=(f_1,f_2,f_3)$ (not necessarily with $μ_{\mathbf{m},\mathbf{k}}$-minimal transverse slice) is of non-negative degree; that is, any additional term $α$ in the deformation of $f_i$ has weighted degree not smaller than that of $f_i$. As a consequence, we provide a proof of Zariski's multiplicity conjecture for 1-parameter families of such germs.

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

TL;DR

This work studies equisingularity for image surfaces of finitely determined, quasihomogeneous, corank 1 map germs by introducing -minimal transverse slices to reduce surface problems to plane curve germs. It proves that, under the -minimal hypothesis, topological triviality and Whitney equisingularity are equivalent for 1-parameter unfoldings, and derives Zariski-type multiplicity results for such families; it also produces counterexamples showing that Whitney equisingularity does not imply bi-Lipschitz equisingularity and motivates a revised Ruas conjecture. The paper further provides explicit normal forms for -minimal plane curves, upper semicontinuity results in the minimal setting, and a reformulation of Ruas’ conjecture via the Milnor number of the curve . Collectively, these results advance the understanding of how transversality, Milnor numbers, and multiplicity invariants govern equisingularity in families of quasihomogeneous surface germs.

Abstract

In this work, we consider a finitely determined, quasihomogeneous, corank 1 map germ from to . We introduce the concept of the -minimal transverse slice of }. Since such a slice is a plane curve, it admits a topological normal form, which we describe explicitly. Assuming the -minimal transverse slice hypothesis, we provide a proof for the equivalence between topological triviality and Whitney equisingularity in Ruas' conjecture within this setting. We also provide a counterexample which shows that Whitney equingularity does not imply bi-Lipschitz equisingularity, given an answer to a question by Ruas. Moreover, we show that every topologically trivial -parameter unfolding of (not necessarily with -minimal transverse slice) is of non-negative degree; that is, any additional term in the deformation of has weighted degree not smaller than that of . As a consequence, we provide a proof of Zariski's multiplicity conjecture for 1-parameter families of such germs.

Paper Structure

This paper contains 13 sections, 29 theorems, 73 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Plane curve germs
  4. Double point space for corank 1 map germs
  5. Finite determinacy and quasihomogeneous map germs
  6. The notion of a $\mu_{\textbf{m},\textbf{k}}$-minimal plane curve germ
  7. Normal form of an irreducible $\mu_{\textbf{m},\textbf{k}}$-minimal plane curve germ
  8. Upper semicontinuity for $\mu_{\textbf{m},\textbf{k}}$-minimal plane curve germs
  9. Normal form of a $\mu_{\textbf{m},\textbf{k}}$-minimal plane curve germ with several branches
  10. New counterexamples to Ruas’ conjecture
  11. Zariski's multiplicity conjecture for quasihomogeneous map germs
  12. Ruas' conjecture for $\mu_{m,k}$-minimal transverse slices
  13. A new formulation of Ruas’ conjecture

Key Result

Theorem 1.1

Let $f:(\mathbb{C}^2,0)\rightarrow(\mathbb{C}^3,0)$ be a finitely determined, quasihomogeneous, corank 1 map germ. Suppose that $F=(f_t,t)$ is a topologically trivial 1-parameter unfolding of $f.$ If the transverse slice $\gamma$ of $f$ is $\mu_{m,k}$-minimal, then $F$ is Whitney equisingular.

Figures (5)

  • Figure 1: The notion of a family of curves.
  • Figure 2: Implication diagram for Ruas’ conjecture.
  • Figure 3: Transverse slice of Ruas’s surface.
  • Figure 4: Update implication diagram for Ruas’ conjecture.
  • Figure 5: Transverse slice for the cross-cap (real points).

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: [11], MararMond
  • Definition 2.7
  • Lemma 3.1
  • ...and 47 more