Showing papers by "Maria Aparecida Soares Ruas published in 2015"

Differential Geometry from a Singularity Theory Viewpoint

Abstract: Differential Geometry from a Singularity Theory Viewpoint provides a new look at the fascinating and classical subject of the differential geometry of surfaces in Euclidean spaces. The book uses singularity theory to capture some key geometric features of surfaces. It describes the theory of contact and its link with the theory of caustics and wavefronts. It then uses the powerful techniques of these theories to deduce geometric information about surfaces embedded in 3, 4 and 5-dimensional Euclidean spaces. The book also includes recent work of the authors and their collaborators on the geometry of sub-manifolds in Minkowski spaces.

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

Abstract: We show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Le numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Le numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Le numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plenat, and Trotman. As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

Generic sections of essentially isolated determinantal singularities

Abstract: We study the essentially isolated determinantal singularities (EIDS), defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We prove in dimension $3$ a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing previous results by J. Snoussi in dimension $2$. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of Le D. T. concerning constancy of the Milnor number.

On a singular variety associated to a polynomial mapping from $\C^n$ to $\C^{n-1}$

Abstract: We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping $G : \C^{n} \to \C^{n - 1}$ where $n \geq 2$. We prove that in the case $G : \C^{3} \to \C^{2}$, if $G$ is a local submersion but is not a fibration, then the homology and the intersection homology with total perversity (with compact supports or closed supports) in dimension two of the variety ${\mathcal{V}}_G$ is not trivial. In the case of a local submersion $G : \C^{n} \to \C^{n - 1}$ where $n \geq 4$, the result is still true with an additional condition.

Effective Whitney theorem for complex polynomial mappings of the plane

Abstract: We give an exact formula for the number of cusps of a general polynomial mapping $F=(f, g):\Bbb C^2\to\Bbb C^2.$ Namely, if deg $f=d_1$ and deg $g=d_2$ and $F$ is general, then $F$ has exactly $d_1^2+d_2^2+3d_1d_2-6d_1-6d_2+7$ simple cusps. Moreover, if $d_1>1$ or $d_2>1$, then the set $C(F)$ of critical points of $F$ is a smooth connected curve, which is topologically equivalent to a sphere with $g=\frac{(d_1+d_2-3)(d_1+d_2-4)}{2}$ handles with $d_1+d_2-2$ points removed. Finally let $F=(f, g):\Bbb C^2\to\Bbb C^2$ be arbitrary polynomial mapping with deg $f\le d_1$ and deg $g\le d_2.$ Assume that $F$ has generalized cusps at points $a_1,..., a_r.$ Then $\sum^r_{i=1} \mu_{a_i}\le d_1^2+d_2^2+3d_1d_2-6d_1-6d_2+7$ (here $\mu_{a_i}$ denotes the index of a generalized cusp at the point $a_i$).

Second order geometry of spacelike surfaces in de Sitter 5-space

Abstract: The de Sitter space is known as a Lorentz space with positive constant curvature in the Minkowski space. A surface with a Riemannian metric is called a spacelike surface. In this work we investigate properties of the second order geometry of spacelike surfaces in de Sitter space S5 1 by using the action of GL(2,R) X SO(1,2) on the system of conics defined by the second fundamental form. The main results are the classification of the second fundamental mapping and the description of the possible configurations of the LMN-ellipse. This ellipse gives information on the lightlike binormal directions and consequently about their associated asymptotic directions.

On the simplicity of multigerms

Abstract: We prove several results regarding the simplicity of germs and multigerms obtained via the operations of augmentation, simultaneous augmentation and concatenation and generalised concatenation. We also give some results in the case where one of the branches is a non stable primitive germ. Using our results we obtain a list which includes all simple multigerms from $\mathbb C^3$ to $\mathbb C^3$.

Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space

Abstract: For a generic embedding of a smooth closed surface $M$ into $\mathbb R^4$, the subset of $\mathbb R^4$ which is the affine $\lambda$-equidistant of $M$ appears as the discriminant set of a stable mapping $M \times M \to \mathbb R^4$, hence their stable singularities are $A_k, \, k=2, 3, 4,$ and $C_{2,2}^{\pm}$. In this paper, we characterize these stable singularities of $\lambda$-equidistants in terms of the bi-local extrinsic geometry of the surface, leading to a geometrical study of the set of weakly parallel points on $M$.