arXiv subjects
Publications and source records attributed to Maria Aparecida Soares Ruas.
In this work, we describe a prenormal form for the generators of the semigroup of a toric variety $X \subset \mathbb{C}^p$ with isolated singularity at the origin and smooth normalization. A complete description of the semigroup is given when $X$ is a variety of dimension $n$ in $\mathbb{C}^{2n}$. Moreover, for toric surfaces in $\mathbb{C}^4$, we provide a set of generators of the ideal $I$ defining $X$.
We show that the Nash blowup of 2-generic determinantal varieties over fields of positive characteristic is non-singular. We prove this in two steps. Firstly, we explicitly describe the toric structure of such varieties. Secondly, we show that in this case the combinatorics of Nash blowups are free of characteristic. The result then follows from the analogous result in characteristic zero proved by W. Ebeling and S. M. Gusein-Zade.
In this paper we study the bi-Lipschitz triviality of deformations of an analytic function germ $f$ defined on a germ of an analytic variety $(X, 0)$ in $\mathbb C^n$. We introduce the notion of strongly rational $\mathscr R_X$-bi-Lipschitz trivial families and give an infinitesimal criterion which is a sufficient condition for the bi-Lipschitz triviality of deformations of $f$ on $(X,0).$ As a corollary it follows that when $X$ and $f$ are homogeneous of the same degree, all deformation of $f$ of the same or higher degrees are bi-Lipschitz trivial. We then prove a rigidity result for deformations of $f$ on $X$ when both are weighted homogeneous with respect to the same set of weights.
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 $2n-1 \leq p$ (Theorem 3.3). 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 a complete set of invariants for the bi-Lipschitz classification with respect to the inner metric of $X_f=f(U)$, where $U$ is a small neighbourhood of the origin in $\mathbb R^2$, is is given by the link of $f$, the image of the double point set of $f$ and the polar curve of a generic projection into the plane (Proposition 4.13). In particular, in the homogeneous parametrization case $f: (\mathbb{R}^2, 0) \to (\mathbb{R}^3, 0)$ of corank 1, we do not need the hypothesis on the equivalence of the image of the double point set (Theorem 5.2). Finally, we apply our results to relate the $C^{0}- \mathcal A$ classes of finitely determined map germs $f$ of corank 1 with homogeneous parametrization and the inner bi-Lipschitz type of $X_f$ (Proposition 5.4).
We introduce the module of derivations $Θ_{h,M}$ attached to a given analytic map $h:(\mathbb C^n,0)\to (\mathbb C^p,0)$ and a submodule $M\subseteq \mathcal O_n^p$ and analyse several exact sequences related to $Θ_{h,M}$. Moreover, we obtain formulas for several numerical invariants associated to the pair $(h,M)$ and a given analytic map germ $f:(\mathbb C^n,0)\to (\mathbb C^q,0)$. In particular, if $X$ is an analytic subvariety of $\mathbb C^n$, we derive expressions for analytic invariants defined in terms of the module $Θ_X$ of logarithmic vector fields of $X$.
Let $(X,0)$ be the germ of an equidimensional analytic set in $(\mathbb C^n,0)$ and $f=(f_1,f_2)$ a map-germ into the plane defined on $X.$ In this work, we investigate topological invariants associated to the pair $(f,X),$ among them, the Euler obstruction of $f,$ $Eu_{f,X}(0),$ and under convenient assumptions, the Chern number of families of differential forms associated to $f.$ The topological information provided by these invariants is useful, although difficult to calculate. The aim of the paper is to introduce the Bruce-Roberts and the relative Bruce-Roberts numbers as useful algebraic tools to capture the topological information giving by the Euler obstruction and the Chern numbers. Closed formulas are given when $X,\, X\cap f_2^{-1}(0),\, X\cap f_2^{-1}(0)\cap f_1^{-1}(0)$ are ICIS. In the last section, for a 2-dimensional ICIS $(X,0) \subset (\mathbb C^n,0),$ we apply our results to give an alternative description for the number of cusps $c(f|_X)$ of an stabilization of an $\mathcal A$-finite map-germ $f=(f_1, f_2): (X,0) \to (\mathbb C^2,0).$ A formula for $c(f|_X)$ was first given in [21].
In this work we define some map-germs, called elementary joins, for the purpose of producing new ${\mathcal A}$-finite map-germs from them. In particular, we describe a general form of an ${\mathcal A}$-finite monomial map from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{p},0)$ for $p\geq 2n$ of any corank in terms of elementary join maps. Our main tools are the delta invariant and some invariants of curves.
Given a germ of an analytic variety $X$ and a germ of a holomorphic function $f$ with a stratified isolated singularity with respect to the logarithmic stratification of $X$, we show that under certain conditions on the singularity type of the pair $(f,X)$, the following relative analog of the well known K. Saito's theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce-Roberts numbers) is equivalent to the relative quasihomogeneity of the pair $(f,X)$, i.e. to the existence of a coordinate system such that both $f$ and $X$ are quasihomogeneous with respect to the same positive rational weights.
This paper is a first step in order to extend Kummer's theory for line congruences to the case $\lbrace x, ξ\rbrace $, where $x: U \rightarrow \mathbb{R}^3$ is a smooth map and $ξ: U \rightarrow \mathbb{R}^3$ is a proper frontal. We show that if $\lbrace x, ξ\rbrace$ is a normal congruence, the equation of the principal surfaces is a multiple of the equation of the developable surfaces, furthermore, the multiplicative factor is associated to the singular set of $ξ$.
We refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular $3$-manifolds in $\mathbb{R}^6$ and singular corank $1$ $3$-manifolds in $\mathbb{R}^5$. For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by 4 ternary cubics (which is a determinantal variety in some cases). We also study how singularities of the curvature locus of a regular 3-manifold can go to infinity when the manifold is projected orthogonally in a tangent direction.
Density of stable maps is the common thread of this paper. We review Whitney's contribution to singularities of differentiable mappings and Thom-Mather theories on $C^{\infty}$ and $C^{0}$-stability. Infinitesimal and algebraic methods are presented in order to prove Theorem A and Theorem B on density of proper stable and topologically stable mappings $f:N^{n}\to P^{p}.$ Theorem A states that the set of proper stable maps is dense in the set of all proper maps from $N$ to $P$, if and only if the pair $(n,p)$ is in \emph{nice dimensions,} while Theorem B shows that density of topologically stable maps holds for any pair $(n,p).$ A short review of results by du Plessis and Wall on the range in which proper smooth mappings are $C^{1}$- stable is given. A Thom-Mather map is a topologically stable map $f:N \to P$ whose associated $k$-jet map $j^{k}f:N \to P$ is transverse to the Thom-Mather stratification in $J^{k}(N,P).$ We give a detailed description of Thom-Mather maps for pairs $(n,p)$ in the boundary of the nice dimensions.The main open question on density of stable mappings is to determine the pairs $(n,p)$ for which Lipschitz stable mappings are dense. We discuss recent results by Nguyen, Ruas and Trivedi on this subject, formulating conjectures for the density of Lipschitz stable mappings in the boundary of the nice dimensions. At the final section, Damon's results relating $\mathcal{A}$-classification of map-germs and $\mathcal{K}_{V}$ classification of sections of the discriminant $V=Δ(F)$ of a stable unfolding of $f$ are reviewed and open problems are discussed.
In this paper, we give the generic classification of the singularities of 3-parameter line congruences in $\mathbb{R}^4$. We also classify the generic singularities of Blaschke (affine) normal congruences.
We relate the moduli space of analytic equivalent germs of reduced quasi-homogeneous functions at $(\mathbb{C}^2,0)$ with their bi-Lipschitz equivalence classes. We show that any non-degenerate continuous family of (reduced) quasi-homogeneous functions with constant Henry-Parusiński invariant is analytically trivial. Further we show that there are only a finite number of distinct bi-Lipschitz classes among quasi-homogeneous functions with the same Henry-Parusiński invariant providing a maximum quota for this number.
In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of itself. The parabolic set itself is detected in this way, and each arc is given a sign, which changes at the special points, where the family has an additional degree of symmetry. Other points of $M$ which are detected by the family of reflexions include inflexion points of real and imaginary type, and the first of these is also associated with sign changes on the parabolic set. We show how to compute the special points globally for the case where $M$ is given in Monge form and give some examples illustrating the birth of special parbolic points in a 1-parameter family of surfaces. The tool we use from singularity theory is the contact classification of certain symmetric maps from the plane to the plane and we give the beginning of this classification, including versal unfoldings which we relate to the geometry of $M$.
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.
We study holomorphic function germs under equivalence relations that preserve an analytic variety. We show that two quasihomogeneous polynomials, not necessarily with isolated singularities, having isomorphic relative Milnor algebras are relative right equivalent. Under the condition that the module of vector fields tangent to the variety is finitely generated, we also show that the relative Tjurina algebra is a complete invariant for the classification of arbitrary function germs with respect to the relative contact equivalence. This is the relative version of a well known result by Mather and Yau.
We present new families of weighted homogeneous and Newton non-degenerate line singularities that satisfy the Zariski multiplicity conjecture.
We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in $C^{\infty}(N\times[0,1],P)$, also known as pseudo-isotopies, is dense if and only if the pair of dimensions $(\dim N, \dim P)$ is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pairs $(n,p)$ for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $\mathcal A_e$-codimension 1 hyperplane sections. They are also related to the simplicity of $\mathcal A_e$-codimension 2 germs. We give a sufficient condition for any $\mathscr A_e$-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.