Singularity Theory and Algebraic Geometry
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A jacobian module for disentanglements and applications to Mond's conjecture
(Revista Matemática Complutense, 2019)Let $f:(\mathbb C^n,S)\to (\mathbb C^{n+1},0)$ be a germ whose image is given by $g=0$. We define an $\mathcal O_{n+1}$-module $M(g)$ with the property that $\mathscr A_e$-$\operatorname{codim}(f)\le \dim_\mathbb C M(g)$, ... -
On Zariski’s multiplicity problem at infinity
(Proceedings of the American Mathematical Society, 2018-08-14)We address a metric version of Zariski's multiplicity conjecture at infinity that says that two complex algebraic affine sets which are bi-Lipschitz homeomorphic at infinity must have the same degree. More specifically, ... -
Singularities in Geometry and Topology
(2018-06-18)This thesis consists of two different topics that are not related. The thesis has two different and independent parts that can be read in any order. The purpose of this work is to study two topics in singularity theory: ... -
Right unimodal and bimodal singularities in positive characteristic
(International Mathematics Research Notices, 2017-08-07)The problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal singularities w.r.t. right equivalence. The classification of simple ... -
Multiplicity and degree as bi‐Lipschitz invariants for complex sets
(Journal of Topology, 2018-08-29)We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz ... -
On the geometry of strongly flat semigroups and their generalizations
(2018-09-18)Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and ... -
Surgery formulae for the Seiberg-Witten invariant of plumbed 3-manifolds
(2017-02)Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3--manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincar\'e series associated ... -
Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifolds
(2018-06)Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the ‘periodic constant’ of the topological multivariable Poincaré series (zeta ... -
Némethi’s division algorithm for zeta-functions of plumbed 3-manifolds
(Bulletin of the London Mathematical Society, 2018-08-27)A polynomial counterpart of the Seiberg-Witten invariant associated with a negative definite plumbing 3-manifold has been proposed by earlier work of the authors. It is provided by a special decomposition of the zeta-function ... -
Some classes of homeomorphisms that preserve multiplicity and tangent cones
(2018-08-19)In this paper it is presented some classes of homeomorphisms that preserve multiplicity and tangent cones of complex analytic sets. Moreover, we present a class of homeomorphisms that has the multiplicity as an invariant ... -
Semialgebraic CMC surfaces in $\mathbb{R}^3$ with singularities
(2018-06-30)In this paper we present a classification of a class of semialgebraic CMC surfaces in $\mathbb{R}^3$ that generalizes the recent classification made by Barbosa and do Carmo in 2016 (complete reference is in the paper), we ... -
Hölder equivalence of complex analytic curve singularities
(Bulletin of the London Mathematical Society, 2018-08-06)We prove that if two germs of irreducible complex analytic curves at $0\in\mathbb{C}^2$ have different sequence of characteristic exponents, then there exists $0<\alpha<1$ such that those germs are not $\alpha$-H\"older ... -
The Nash Problem from a Geometric and Topological Perspective
(2018-04-17)We survey the proof of the Nash conjecture for surfaces and show how geometric and topological ideas developed in previous articles by the au- thors influenced it. Later we summarize the main ideas in the higher dimen- ... -
Monodromies as tête-à-tête graphs
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Mixed tête-à-tête twists as monodromies associated with holomorphic function germs
(2018-04-01)Tête-à-tête graphs were introduced by N. A’Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which ... -
General tête-à-tête graphs and Seifert manifolds
(2018-02-10)Tête-à-tête graphs and relative tête-à-tête graphs were introduced by N. A’Campo in 2010 to model monodromies of isolated plane curves. By recent work of Fdez de Bobadilla, Pe Pereira and the author, they provide a way ... -
A proof of the differentiable invariance of the multiplicity using spherical blowing-up
(Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2018-04-21)In this paper we use some properties of spherical blowing-up to give an alternative and more geometric proof of Gau-Lipman Theorem about the differentiable invariance of the multiplicity of complex analytic sets. Moreover, ... -
Topology of Spaces of Valuations and Geometry of Singularities
(Transactions of the AMS - American Mathematical Society, 2017-11-11)Given an algebraic variety X defined over an algebraically closed field, we study the space RZ(X,x) consisting of all the valuations of the function field of X which are centered in a closed point x of X. We concentrate ... -
A proof of the integral identity conjecture, II
(Comptes Rendus Mathematique, 2017-10-31)In this note, using Cluckers-Loeser’s theory of motivic integration, we prove the integral identity conjecture with framework a localized Grothendieck ring of varieties over an arbitrary base field of characteristic zero.
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