Person: Melle Hernández, Alejandro
Universidad Complutense de Madrid
Faculty / Institute
Álgebra, Geometría y Topología
Now showing 1 - 10 of 57
PublicationPencils of curves on smooth surfaces(Oxford University Press (OUP), 2001-09) Melle Hernández, Alejandro; Wall, Charles Terence CleggAlthough the theory of singularities of curves - resolution, classification, numerical invariants - goes through with comparatively little change in finite characteristic, pencils of curves are more difficult. Bertini's theorem only holds in a much weaker form, and it is convenient to restrict to pencils such that, when all base points are resolved, the general member of the pencil becomes non-singular. Even here, the usual rule for calculating the Euler characteristic of the resolved surface has to be modified by a term measuring wild ramification. We begin by describing this background, then proceed to discuss the exceptional members of a pencil. In characteristic 0 it was shown by Há and Lê and by Lê and Weber, using topological reasoning, that exceptional members can be characterised by their Euler characteristics. We present a combinatorial argument giving a corresponding result in characteristic p. We first treat pencils with no base points, and then reduce the remaining case to this. PublicationBifurcations and topology of meromorphic germs(Kluwer Academic Publishers, 2001) Gusein-Zade, Sabir Medgidovich; Luengo Velasco, Ignacio; Melle Hernández, Alejandro; Siersma, D.; Wall, Charles Terence Clegg; Zakalyukin, V.A meromorphic germ at the origin in the complex space Cn is a ratio of two holomorphic germs on (Cn,0). After presentation of the basic definitions in the general context of arbitrary meromorphic germs the authors study the monodromy by calculating its zeta function. Then they give some results on homology splitting and bouquet-type theorems for the global case of meromorphic functions on compact complex manifolds. Some applications to traditional cases of rational functions on CPn, including in particular polynomial functions on Cn, are considered. PublicationEscenarios Multimedia en Formación de Futuros Profesores Universitarios de Matemáticas (ESCEMMAT-Univ) (2ªFase)(UCM, 2020-06-29) Gómez Chacón, Inés María; Díaz-Cano Ocaña, Antonio; Folgueira López, Marta; Gómez Castro, David; Ivorra, Benjamín Pierre Paul; Martínez Aguinaga, Francisco Javier; Ortuño, M. T.; Ramos del Olmo, Ángel Manuel; González Ortega, Jorge; González Prieto, José Ángel; Melle Hernández, Alejandro; Sanchez Benito, Mercedes; Barbero, Marta; Pe Pereira, María; Capel Cuevas, ÁngelaPreparar al profesorado novel de matemáticas para una docencia universitaria de calidad, en el desarrollo de competencias y conocimiento estratégico para aprender a enseñar Matemáticas. Ejemplificaciones e instrumentos de evaluación para el aula. PublicationPower structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points(Michigan Mathematical Journal, 2006) Gusein-Zade, Sabir Medgidovich; Luengo Velasco, Ignacio; Melle Hernández, AlejandroThe power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety as an exponent of that for the complex affine space of the same dimension. Specializations of this relation give formulae for generating series of such invariants of the Hilbert schemes of points as the Euler characteristic and the Hodge-Deligne polynomial. PublicationOn ν-quasi-ordinary power series: factorization, Newton trees and resultants(American Mathematical Society, 2011) Artal Bartolo, Enrique; Cassou-Noguès, Pierrette; Luengo Velasco, Ignacio; Melle Hernández, Alejandro; Cogolludo Agustín, José Ignacio; Hironaka, ErikoThe concept of ν-quasi-ordinary power series, which is a generalization of quasi-ordinary power series, was first introduced by H. Hironaka. In the paper under review, the authors study ν-quasi-ordinary power series and give a factorization theorem for ν-quasi-ordinary power series in the first part. The proof of the theorem uses Newton maps. In the second part of the paper, using the factorization theorem, they introduce the Newton tree to encode the Newton process for any hypersurface singularity defined by a power series germ as in Notation 1.1. Finally, the authors describe a condition for two ν-quasi-ordinary power series to have an "intersection multiplicity " by using Newton trees and they can also compute this generalized intersection multiplicity, resultants and discriminant. PublicationEuler characteristic of the Milnor fibre of plane singularities(America Mathematical Society, 1999-05-19) Melle Hernández, AlejandroWe give a formula for the Euler characteristic of the Milnor fibre of any analytic function f of two variables. This formula depends on the intersection multiplicity, the multiplicity and the Milnor number of the branches of the germ of curve defined by f. The goal of the formula is that it uses neither the resolution nor the deformations of f. Moreover, it can be used for giving an algorithm to compute it. PublicationLinks and analytic invariants of superisolated singularities(American Mathematical Society, 2005) Luengo Velasco, Ignacio; Melle Hernández, Alejandro; Némethi, A.Using superisolated singularities we present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, we show that the ``Seiberg-Witten invariant conjecture''(of Nicolaescu and the third author), the ``Universal abelian cover conjecture'' (of Neumann and Wahl) and the ``Geometric genus conjecture'' fail (at least at that generality in which they were formulated). Moreover, we also show that for Gorenstein singularities (even with integral homology sphere links) besides the geometric genus, the embedded dimension and the multiplicity (in particular, the Hilbert-Samuel function) also fail to be topological; and in general, the Artin cycle does not coincide with the maximal (ideal) cycle. PublicationHigher-order orbifold Euler characteristics for compact Lie group actions(Cambridge University Press, 2015-12) Gusein-Zade, Sabir Medgidovich; Luengo Velasco, Ignacio; Melle Hernández, AlejandroWe generalize the notions of the orbifold Euler characteristic and of the higher-order orbifold Euler characteristics to spaces with actions of a compact Lie group using integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the kth-order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by Tamanoi for finite group actions and by Farsi and Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well PublicationMonodromy Jordan blocks, b-functions and poles of Zeta functions for germs of plane curves(Academic Press, 2010-09-15) Melle Hernández, Alejandro; Torrelli , Tristan; Veys, WillenWe study the poles of several local zeta functions: the Igusa, topological and motivic zeta function associated to a germ of a holomorphic function in two variables. It was known that there is at most one double pole for (any of) these zeta functions which is then given by the log canonical threshold of the function at the singular point. If the germ is reduced Loeser showed that such a double pole always induces a monodromy eigenvalue with a Jordan block of size 2. Here we settle the non-reduced situation, describing precisely in which case such a Jordan block of maximal size 2 occurs. We also provide detailed information about the Bernstein-Sato polynomial in the relevant non-reduced situation, confirming a conjecture of Igusa, Denef and Loeser. PublicationTopología de singularidades de hipersuperficies : Número de Milnor e invariantes polares(Universidad Complutense de Madrid, Servicio de Publicaciones, 2002) Melle Hernández, Alejandro; Luengo Velasco, IgnacioEn esta memoria se obtienen formulas que relacionan el numero de milnor de un germen de hipersuperficie compleja con singularidad aislada con la geometria de las hipersuperficies proyectivas definidas por los ceros de los polinomios homogéneos que aparecen en la descomposición en componentes homogéneas de la función analítica cuyo lugar de ceros describe el germen.igualmente se dan criterios de suficiencia topología para singularidades aisladas en las cuales el cono tangente verifica ciertas condiciones geométricas. Esta descripción topología se hace en función de los invariantes polares de la singularidad.