Showing papers by "Maria Aparecida Soares Ruas published in 2019"

The local Euler obstruction and topology of the stabilization of associated determinantal varieties

Abstract: This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern–Schwartz–MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney’s $$m_{d}$$ multiplicity.

The geometry of corank 1 surfaces in ℝ4

Abstract: We study the geometry of surfaces in ℝ4 with corank 1 singularities. For such surfaces, the singularities are isolated and, at each point, we define the curvature parabola in the normal space. This curve codifies all the second-order information of the surface. Also, using this curve, we define asymptotic and binormal directions, the umbilic curvature and study the flat geometry of the surface. It is shown that we can associate to this singular surface a regular one in ℝ4 and relate their geometry.

Parametrized monomial surfaces in 4-space

Abstract: In this work, we classify parametrized monomial surfaces f:(ℂ2,0)→(ℂ4,0) that are A-finitely determined. We study invariants that can be obtained in terms of invariants of a parametrized curve.

Determinacy of determinantal varieties

Abstract: A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of an $$m\times n$$ -matrix. In this note, we consider $$\mathcal {G}$$ -finite determinacy of matrices defining a special class of determinantal varieties. They are called essentially isolated determinantal singularities (EIDS) and were defined by Ebeling and Gusein-Zade (Singul Appl 267:119–131, 2009). In this note, we prove that matrices parametrized by generic homogeneous forms of degree d define EIDS. It follows that $$\mathcal {G}$$ -finite determinacy of matrices holds in general. As a consequence, EIDS of a given type (m, n, t) holds in general.

Singular 3-manifolds in $\mathbb{R}^5$

Abstract: We study 3-manifolds in $\mathbb{R}^5$ with corank $1$ singularities. At the singular point we define the curvature locus using the first and second fundamental forms, which contains all the local second order geometrical information about the manifold.

Local bi-Lipschitz classification of semialgebraic surfaces

Abstract: We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ determine the bi-Lipschitz type of the link of $f$ and of the double point set of $f$? Reciprocally, given a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$, do the bi-Lipschitz types of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$? We provide a positive answer to the first question in the case of a finitely determined map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ where $2n-1 \leq p$. With regard to the second question, for a finitely determined map germ $f : (\mathbb{R}^2,0) \to (\mathbb{R}^3,0),$ we show that the bi-Lipschitz type of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the image $X_f=f(U),$ where $U$ is a small neighbourhood of the origin in $\mathbb R^2.$ In the case of a finitely determined map germ $f: (\mathbb{R}^2, 0) \to (\mathbb{R}^3, 0)$ of corank 1 with homogeneous parametrization, the bi-Lipschitz type of the link of $f$ determines the bi-Lipschitz type of the map germ. Finally, we discuss the bi-Lipschitz equivalence of homogeneous surfaces in $\mathbb{R}^3$.

On the Zariski multiplicity conjecture for weighted homogeneous and Newton nondegenerate line singularities

Abstract: We present new families of weighted homogeneous and Newton nondegenerate line singularities that satisfy the Zariski multiplicity conjecture.

Finite $\mathcal{A}$-determinacy of generic homogeneous map germs in $\mathbb{C}^3$

Abstract: Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \C^3\to\C^3$, such that $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j)\leq 2$ for $1\leq i