Singularity Theory and Algebraic Geometry
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The abel map for surface singularities II. Generic analytic structure
(Advances in Mathematics, 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 ... 
Intersection cohomology with torus actions of complexity one and intersection space complexes
(Mathematische Annalen, 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 ... 
Decomposition theorem and torus actions of complexity one
(European Journal of Mathematics, 2020)We algorithmically compute the intersection cohomology Betti numbers of any complete normal algebraic variety with a torus action of complexity one. 
Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries
(Revista Matematica Complutense, 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 ... 
Gaussian processes in complex media: new vistas on anomalous diffusion
(Cambridge University Press, 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
(World Scientific, 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
(european Journal of Mathematics, 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
(Mathematische Annalen, 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 , ... 
Singularities of the Hilbert scheme of effective divisors
(AMERICAN MATHEMATICAL SOCIETY, 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 ... 
A Short Survey on the Integral Identity Conjectureand Theories of Motivic Integration
(WORLD SCIENTIFIC (EUROPE), 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 ... 
Reduced description method in the kinetic theory of Brownian motion with active fluctuations
(Mathematische Annalen, 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 ... 
Firespotting generated fires. Part I: The role of atmospheric stability
(International Journal of Mathematics, 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 ... 
A specialization property of index
(Proceedings A of the Royal Society of Edinburgh, 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 ... 
A Jacobian module for disentanglements and applications to Mond's conjecture
(The Asian Journal of Mathematics, 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 ... 
A jacobian module for disentanglements and applications to Mond's conjecture
(Revista Matemática Complutense, 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 $\mathcal{X}$ ... 
A proof of the differentiable invariance of the multiplicity using spherical blowingup
(Mathematische Annalen, 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. 
Homogeneous singularity and the Alexander polynomial of a projective plane curve
(Contemporary Mathematics, 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 ... 
Criterion for logarithmic connections with prescribed residues
(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 ... 
Finiteenergy Lévytype motion through heterogeneous ensemble of Brownian particles
(Hokkaido Mathematical Journal, 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 ... 
On the generalized Nash problem for smooth germs and adjacencies of curve singularities
(Journal of Number Theory, 20190416)Let K be the fraction field of a Henselian discrete valuation ring with algebraically closed residue field k. In this article we give a sufficient criterion for a projective variety over such a field to have index 1.