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arXiv · 2509.01634

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

Otoniel Nogueira da Silva·Manoel Messias da Silva Júnior
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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.

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BibTeXRIS

Otoniel Nogueira da Silva, Manoel Messias da Silva Júnior. 2025-10-13. Transverse slices, Ruas' conjecture, and Zariski's multiplicity conjecture for quasihomogeneous surfaces. https://arxiv.org/abs/2509.01634

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