Springer Proceedings in Mathematics & Statistics, 2018
This article provides a survey concerning Lefschetz fixed point formula and Lefschetz coincidence... more This article provides a survey concerning Lefschetz fixed point formula and Lefschetz coincidence formula in the smooth and singular cases, moreover we show a Lefschetz type formula for the Coincidence number of two maps. As a consequence we obtain a relation with correspondences, and we provide some examples.
The main purpose of this work is to describe the topological orbits which are in a K-orbit of fin... more The main purpose of this work is to describe the topological orbits which are in a K-orbit of finitely determined map germs from C 2 to C 2 to. At least for a very large number of co-rank two K-orbits. The co-rank one case is described by Ganey
We apply the method of constructing controlled vector fields to give sufficient conditions for th... more We apply the method of constructing controlled vector fields to give sufficient conditions for the A-topological triviality of deformations of map germs ft : (Cn, 0) → (Cp, 0) of type ft(x) = f(x)+th(x), with n ≥ p or n ≤ 2p. These conditions are given in terms of an appropriate choice of Newton filtrations for On and Op and are for the A-tangent space of the germ f . For the case n ≥ p, we follow the technique used by M. A. S. Ruas in her Ph.D. Thesis [5] and construct control functions in the target and in the source to obtain, via a partition of the unit, a unique control function. We use the control function of the target to give an estimate for the case p ≥ 2n. Moreover, in this case we also show that if the the coordinates of the map germ satisfy a Newton non-degeneracy condition, deformations by terms of higher filtration are topologically trivial. May, 2004 ICMC-USP
In this paper we investigate the C l versions of contact and right equivalences of real semi-quas... more In this paper we investigate the C l versions of contact and right equivalences of real semi-quasihomogeneous C l function germs, 1 a l a y. The C l-right equivalence implies C l-contact equivalence for any 1 a l a y and in this work we show, up to certain conditions, that for semi-quasihomogeneous C l function germs the converse is also true (Theorem 1). As a consequence, concerning the particular case of quasihomogeneous C l function germs, we also have a similar result (Corollary 1) which recover a known result of M. Takahashi [14] for l ¼ y. We note that we are considering semi-quasihomogeneous function germs with no additional hypothesis of isolated singularity at zero.
We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the tran... more We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the singularities of the family of affine distance functions appear at points on the affine normal plane and the affine focal points correspond to degenerate singularities of this family. Moreover we show that if M is immersed in a locally strictly convex hypersurface, then the affine normal plane contains the affine normal vector to the hypersurface and conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.
We investigate the constancy of the Lê numbers of one parameter deformations F : (C × C n , 0) → ... more We investigate the constancy of the Lê numbers of one parameter deformations F : (C × C n , 0) → (C, 0) of holomorphic germs of functions f : (C n , 0) → (C, 0) which have singular set of any dimension s ≥ 1. We characterize such Lê constant deformations in terms of the non-splitting of the polar varieties and also from the integral closure of the ideal J z (F) in O n+1 generated by the partial derivatives of F with respect to the variables z = (z 1 ,. .. , z n).
Proceedings of the American Mathematical Society, 2012
We show how to compute the local polar multiplicities of a germ at zero of an analytic variety Y ... more We show how to compute the local polar multiplicities of a germ at zero of an analytic variety Y in C p , which is the image by a finite morphism f : Z → Y , of a d-dimensional isolated complete intersection singularity Z in C n. We also show how to compute the local Euler obstruction of Y at zero in the case that it is reduced. For this we apply the formula due to Lê and Teissier which describes the local Euler obstruction as an alternating sum of the local polar multiplicities.
The determinacy of families of map germs which are trivial with respect to some equivalence relat... more The determinacy of families of map germs which are trivial with respect to some equivalence relation is a fundamental subject in singularity theory. Many works are devoted to the characterization of the topological triviality with respect to various equivalence relations, ...
Bulletin of the Brazilian Mathematical Society, New Series, 2012
In this paper we study germs of polynomials formed by the product of semiweighted homogeneous pol... more In this paper we study germs of polynomials formed by the product of semiweighted homogeneous polynomials of the same type, which we call semi-weighted homogeneous arrangements. It is shown how the Lê numbers of such polynomials are computed using only their weights and degree of homogeneity. A key point of the main theorem is to find the number called polar ratio of this polynomial class. An important consequence is the description of the Euler characteristic of the Milnor fibre of such arrangements only depending on their weights and degree of homogeneity. The constancy of the Lê numbers in families formed by such arrangements is shown, with the deformed terms having weighted degree greater than the weighted degree of the initial germ. Moreover, using the results of Massey applied to families of function germs, we obtain the constancy of the homology of the Milnor fibre in this family of semi-weighted homogeneous arrangements.
Let f : n , 0 p , 0 be a-finite map germ, and let i l (i " , … , i k) be a Boardman symbol such t... more Let f : n , 0 p , 0 be a-finite map germ, and let i l (i " , … , i k) be a Boardman symbol such that Σ i has codimension n in the corresponding jet space J k (n, p). When its iterated successors have codimension larger than n, the paper gives a list of situations in which the number of Σ i points that appear in a generic deformation of f can be computed algebraically by means of Jacobian ideals of f. This list can be summarised in the following way : f must have rank nki " and, in addition, in the case p l 6, f must be a singularity of type Σ i " ,i #.
Conjugate vectors of immersed manifolds by D. Dreibelbis Invariants of $D(q,p)$ singularities by ... more Conjugate vectors of immersed manifolds by D. Dreibelbis Invariants of $D(q,p)$ singularities by T. Gaffney Nilpotents, integral closure and equisingularity conditions by T. Gaffney Singularities in generic one-parameter complex and analytic families of maps by K. Houston Global classification of singular curves on the symplectic plane by G. Ishikawa Counting isolated singularities of a stable perturbation of a finitely $\mathcal{A}$-determined map-germ $\mathbb{C}^n,0)\to(\mathbb{C}^p,0)$ with $n<p$ by V. H. Jorge Perez Milnor numbers and equisingularity of map germs from $\mathbb{C}^{n+3}$ to $\mathbb{C}^3$ by V. H. Jorge Perez and A. J. Miranda Cobordism invariants of fold maps by B. Kalmar Obstructions on fundamental groups of plane curve complements by C. Leidy and L. Maxim Exotic moduli of Goursat distributions exist already in codimension three by P. Mormul Hypersurfaces with isolated singularities with symmetry by A. A. du Plessis and C. T. C. Wall Thom polynomials and Sc...
Springer Proceedings in Mathematics & Statistics, 2018
This article provides a survey concerning Lefschetz fixed point formula and Lefschetz coincidence... more This article provides a survey concerning Lefschetz fixed point formula and Lefschetz coincidence formula in the smooth and singular cases, moreover we show a Lefschetz type formula for the Coincidence number of two maps. As a consequence we obtain a relation with correspondences, and we provide some examples.
The main purpose of this work is to describe the topological orbits which are in a K-orbit of fin... more The main purpose of this work is to describe the topological orbits which are in a K-orbit of finitely determined map germs from C 2 to C 2 to. At least for a very large number of co-rank two K-orbits. The co-rank one case is described by Ganey
We apply the method of constructing controlled vector fields to give sufficient conditions for th... more We apply the method of constructing controlled vector fields to give sufficient conditions for the A-topological triviality of deformations of map germs ft : (Cn, 0) → (Cp, 0) of type ft(x) = f(x)+th(x), with n ≥ p or n ≤ 2p. These conditions are given in terms of an appropriate choice of Newton filtrations for On and Op and are for the A-tangent space of the germ f . For the case n ≥ p, we follow the technique used by M. A. S. Ruas in her Ph.D. Thesis [5] and construct control functions in the target and in the source to obtain, via a partition of the unit, a unique control function. We use the control function of the target to give an estimate for the case p ≥ 2n. Moreover, in this case we also show that if the the coordinates of the map germ satisfy a Newton non-degeneracy condition, deformations by terms of higher filtration are topologically trivial. May, 2004 ICMC-USP
In this paper we investigate the C l versions of contact and right equivalences of real semi-quas... more In this paper we investigate the C l versions of contact and right equivalences of real semi-quasihomogeneous C l function germs, 1 a l a y. The C l-right equivalence implies C l-contact equivalence for any 1 a l a y and in this work we show, up to certain conditions, that for semi-quasihomogeneous C l function germs the converse is also true (Theorem 1). As a consequence, concerning the particular case of quasihomogeneous C l function germs, we also have a similar result (Corollary 1) which recover a known result of M. Takahashi [14] for l ¼ y. We note that we are considering semi-quasihomogeneous function germs with no additional hypothesis of isolated singularity at zero.
We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the tran... more We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the singularities of the family of affine distance functions appear at points on the affine normal plane and the affine focal points correspond to degenerate singularities of this family. Moreover we show that if M is immersed in a locally strictly convex hypersurface, then the affine normal plane contains the affine normal vector to the hypersurface and conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.
We investigate the constancy of the Lê numbers of one parameter deformations F : (C × C n , 0) → ... more We investigate the constancy of the Lê numbers of one parameter deformations F : (C × C n , 0) → (C, 0) of holomorphic germs of functions f : (C n , 0) → (C, 0) which have singular set of any dimension s ≥ 1. We characterize such Lê constant deformations in terms of the non-splitting of the polar varieties and also from the integral closure of the ideal J z (F) in O n+1 generated by the partial derivatives of F with respect to the variables z = (z 1 ,. .. , z n).
Proceedings of the American Mathematical Society, 2012
We show how to compute the local polar multiplicities of a germ at zero of an analytic variety Y ... more We show how to compute the local polar multiplicities of a germ at zero of an analytic variety Y in C p , which is the image by a finite morphism f : Z → Y , of a d-dimensional isolated complete intersection singularity Z in C n. We also show how to compute the local Euler obstruction of Y at zero in the case that it is reduced. For this we apply the formula due to Lê and Teissier which describes the local Euler obstruction as an alternating sum of the local polar multiplicities.
The determinacy of families of map germs which are trivial with respect to some equivalence relat... more The determinacy of families of map germs which are trivial with respect to some equivalence relation is a fundamental subject in singularity theory. Many works are devoted to the characterization of the topological triviality with respect to various equivalence relations, ...
Bulletin of the Brazilian Mathematical Society, New Series, 2012
In this paper we study germs of polynomials formed by the product of semiweighted homogeneous pol... more In this paper we study germs of polynomials formed by the product of semiweighted homogeneous polynomials of the same type, which we call semi-weighted homogeneous arrangements. It is shown how the Lê numbers of such polynomials are computed using only their weights and degree of homogeneity. A key point of the main theorem is to find the number called polar ratio of this polynomial class. An important consequence is the description of the Euler characteristic of the Milnor fibre of such arrangements only depending on their weights and degree of homogeneity. The constancy of the Lê numbers in families formed by such arrangements is shown, with the deformed terms having weighted degree greater than the weighted degree of the initial germ. Moreover, using the results of Massey applied to families of function germs, we obtain the constancy of the homology of the Milnor fibre in this family of semi-weighted homogeneous arrangements.
Let f : n , 0 p , 0 be a-finite map germ, and let i l (i " , … , i k) be a Boardman symbol such t... more Let f : n , 0 p , 0 be a-finite map germ, and let i l (i " , … , i k) be a Boardman symbol such that Σ i has codimension n in the corresponding jet space J k (n, p). When its iterated successors have codimension larger than n, the paper gives a list of situations in which the number of Σ i points that appear in a generic deformation of f can be computed algebraically by means of Jacobian ideals of f. This list can be summarised in the following way : f must have rank nki " and, in addition, in the case p l 6, f must be a singularity of type Σ i " ,i #.
Conjugate vectors of immersed manifolds by D. Dreibelbis Invariants of $D(q,p)$ singularities by ... more Conjugate vectors of immersed manifolds by D. Dreibelbis Invariants of $D(q,p)$ singularities by T. Gaffney Nilpotents, integral closure and equisingularity conditions by T. Gaffney Singularities in generic one-parameter complex and analytic families of maps by K. Houston Global classification of singular curves on the symplectic plane by G. Ishikawa Counting isolated singularities of a stable perturbation of a finitely $\mathcal{A}$-determined map-germ $\mathbb{C}^n,0)\to(\mathbb{C}^p,0)$ with $n<p$ by V. H. Jorge Perez Milnor numbers and equisingularity of map germs from $\mathbb{C}^{n+3}$ to $\mathbb{C}^3$ by V. H. Jorge Perez and A. J. Miranda Cobordism invariants of fold maps by B. Kalmar Obstructions on fundamental groups of plane curve complements by C. Leidy and L. Maxim Exotic moduli of Goursat distributions exist already in codimension three by P. Mormul Hypersurfaces with isolated singularities with symmetry by A. A. du Plessis and C. T. C. Wall Thom polynomials and Sc...
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Papers by Marcelo Saia