Singularity Theory and Algebraic Geometry
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Some contributions to the theory of singularities and their characteristic classes
(20210602)In this Ph.D. thesis, we give some contributions to the theory of singularities, as well as to the theory of characteristic classes of singular spaces. The first part of this thesis is devoted to the theory of singularities ... 
Decomposition theorem and torus actions of complexity one
(2020)We algorithmically compute the intersection cohomology Betti numbers of any complete normal algebraic variety with a torus action of complexity one. 
The abel map for surface singularities II. Generic analytic structure
(2019)We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure ... 
Reflection maps
(2020)Given a reflection group G acting on a complex vector space V , a reflection map is the composition of an embedding X → V with the quotient map V → Cp of G. We show how these maps, which can highly singular, may be studied ... 
Local Topological Obstruction For Divisors
(2020)Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is wellknown that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in ... 
Katomatsumototype results for disentanglements
(2020)We consider the possible disentanglements of holomorphic map germs f : (Cn, 0) → (CN , 0), 0 < n < N, with nonisolated locus of instability Inst(f). The aim is to achieve lower bounds for their (homological) connectiv ity ... 
On a conjecture of harris
(2019)For d ≥ 4, the NoetherLefschetz locus NLd parametrizes smooth, degree d sur faces in P3 with Picard number at least 2. A conjecture of Harris states that there are only finitely many irreducible components of the ... 
On the length of perverse sheaves on hyperplane arrangements
(2019)Abstract. In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement ... 
The Abel map for surface singularities I. Generalities and examples
(2019)Abstract. Let (X, o) be a complex normal surface singularity. We fix one of its good resolutions X → X, an effective cycle Z supported on the reduced exceptional curve, and any possible (first Chern) class l′ ∈ H 2 (X , ... 
Some classes of homeomorphisms that preserve multiplicity and tangent cones
(20200101)In this paper we present some applications of A’CampoLˆe’s Theorem and we study some relations between Zariski’s Questions A and B. It is presented some classes of homeomorphisms that preserve multiplicity and tangent ... 
The Nash Problem from Geometric and Topological Perspective
(20200301)We survey the proof of the Nash conjecture for surfaces and show how geometric and topological ideas developed in previous articles by the authors influenced it. Later, we summarize the main ideas in the higher dimensional ... 
Equivariant motivic integration and proof of the integral identity conjecture for regular functions
(20191202)We develop DenefLoeser’s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions. In comparison with Hartmann’s work, the equivariant Grothendieck ... 
Topological invariants of plane curve singularities: Polar quotients and Lojasiewicz gradient exponents
(20191021)In this paper, we study polar quotients and Łojasiewicz exponents of plane curve singularities, which are not necessarily reduced. We first show that, for complex plane curve singularities, the set of polar quotients is a ... 
Globally subanalytic CMC surfaces in $\mathbb{R}^3$ with singularities
(20200302)In this paper we present a classification of a class of globally subanalytic CMC surfaces in $\mathbb{R}^3$ that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic ... 
Multiplicity, regularity and blowspherical equivalence of complex analytic sets
(20200129)This paper is devoted to study multiplicity and regularity of complex analytic sets. We present an equivalence for complex analytical sets, named blowspherical equivalence and we obtain several applications with this new ... 
Neron models of intermediate Jacobians associated to moduli spaces
(20191201)Let $\pi_1:\mathcal{X} \to \Delta$ be a flat family of smooth, projective curves of genus $g \ge 2$, degenerating to an irreducible nodal curve $X_0$ with exactly one node. Fix an invertible sheaf $\mathcal{L}$ on $\mathcal{X}$ ... 
Multiplicity of singularities is not a biLipschitz invariant
(20200117)It was conjectured that multiplicity of a singularity is biLipschitz invariant. We disprove this conjecture constructing examples of biLipschitz equivalent complex algebraic singularities with different values of multiplicity. 
Some classes of homeomorphisms that preserve multiplicity and tangent cones
(20190528)In this paper we present some applications of A'CampoLê's Theorem and we study some relations between Zariski's Questions A and B. It is presented some classes of homeomorphisms that preserve multiplicity and tangent cones ... 
Moderately Discontinuous Algebraic Topology for Metric Subanalytic Germs
(20191031)We have developed both a homology theory and a homotopy theory in the context of metric subanalytic germs (see Definition 2.1). The former is called MD homology and is covered in Chapter 2, which contains a paper that is ... 
A LêGreuel type formula for the image Milnor number
(201902)Let $f\colon (\mathbb{C}^n,0)\to (\mathbb{C}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p\colon (\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ we denote by $g\colon (\mathbb{C}^{n1},0)\to ...