arXiv subjects

Manoel Messias da Silva Júnior

Publications and source records attributed to Manoel Messias da Silva Júnior.

3 recordsLinked to original sources

arXiv2026

Equisingularity in families of double point curves

In this paper, we provide a systematic comparison between the equisingularity of a 1-parameter unfolding $F = (f_t, t)$ of a finitely determined map germ $f: (\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0)$ and the equisingularity of its associated families of double point curves: $D(F)$, $F(D(F))$, $D^2(F)$, and $D^2(F)/S_2$. We also construct explicit counterexamples to several natural questions concerning the equisingularity of these loci. As a key application, we introduce new families of complete intersection curves -- referred to as Henry--type families -- which are topologically trivial but fail to satisfy the Whitney equisingularity conditions. Finally, we generalize classical formulas for double point curves, originally established for map germs from $(\mathbb{C}^2, 0)$ to $(\mathbb{C}^3, 0)$, to the higher-dimensional setting of map germs from $(\mathbb{C}^n, 0)$ to $(\mathbb{C}^{2n-1}, 0)$ for $n \geq 3$, thereby endowing the associated curves with a convenient analytic structure.

math.AG↗

arXiv2025

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

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.

math.AG↗

arXiv2025

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

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.

math.AG↗