En este trabajo, se estudia fundamentalmente diversas relaciones aritméticas que hay en los semigrupos numéricos, como por ejemplo, obtener el conjunto de lagunas, teniendo solamente el conjunto Apery; también, dado un conjunto de elementos generadores, se asociará a cada uno de ellos, un propio semigrupo numérico. Se analiza, haciendo una descripción de diversos conceptos de la Geometría Algebraica, los cuales se relacionan con los semigrupos numéricos, mediante los semigrupos de Weierstrass, que tienen fundamento, en el teorema de Riemann-Roch. / Tesis

Se estudian los sistemas de dos niveles sin recurrir al espacio de Hilbert el cual es sustituido por el álgebra geométrica del espacio tridimensional (Espacio de Hilbert). En esta descripción los estados son codificados mediante elementos de un ideal izquierdo mínimo del álgebra par de G3, mientras los operadores son codificados mediante la combinación lineal de los vectores del álgebra impar de (Espacio de Hilbert). La dinámica que obedecen estos sistemas está gobernada por la ecuación de “Schrödinger real" ya que el número imaginario (i) es sustituido por el pseudoescalar de (Espacio de Hilbert). Introduciendo los idempotentes primitivos del álgebra geométrica, se generalizan las descripciones previas estando en completo acuerdo con la literatura convencional. Utilizando los axiomas del álgebra geométrica, se demuestra que las relaciones de conmutación canónica que obedecen los operadores de espín son consecuencia de la anticonmutatividad del producto geométrico.

The aim of this work is to study the ring-theoretic properties of the diagonals of a Rees algebra, which from a geometric point of view are the homogenous coordinate rings of embeddings of blow-ups of projective varieties along a subvariety. First we are going to introduce the subject and the main problems. After that we shall review the known results about these problems, and finally we will give a summary of the contents and results obtained in this work. / L’objectiu d’aquesta memòria és l’estudi de les propietats aritmètiques de les diagonals d’una àlgebra de Rees o, des d’un punt de vista geomètric, dels anells de coordenades homogenis d’immersions d’explosions de varietats projectives al llarg d’una subvarietat. En primer lloc, anem a introduir el tema i els principals problemes que tractarem. A continuació, exposarem els resultats coneguts sobre aquests problemes i finalment farem un resum dels resultats obtinguts en aquesta memòria.

We describe a correspondence between two different links associated to the same K3 orbifold. This duality is produced when two elements, one inside and the other on the boundary of the Kähler cone, are identified. We call this correspondence ∂-duality. We also discuss the consequences of ∂-duality at the level of metrics. / Describimos una correspondencia entre dos enlaces asociados a un mismo espacio K3 que soporta a lo más, singularidades cíclicas de tipo orbifold. Esta dualidad se hace evidente cuando dos elementos, uno en el interior y el otro en la frontera del cono de Kähler, son identificados. Denominamos a esta correspondencia ∂-dualidad. También discutimos las consecuencias de ∂-dualidad al nivel de estructuras riemaniannas.

The main purpose of this thesis is to apply the modular approach to Diophantine equations to study some Fermat-type equations of signature (r; r; p) with r >/= 5 a fixed prime and “p” varying. In particular, we will study equations of the form x(r) + y(r) = Cz(p), where C is an integer divisible only by primes “q” is non-identical to 1; 0 (mod “r”) and obtain explicit arithmetic results for “r” = 5, 7, 13. We start with equations of the form x(5) + y(5) = Cz(p). Firstly, we attach two Frey curves E; F defined over Q(square root 5) to putative solutions of the equation. Then by using the work of J. Quer on embedding problems and on abelian varieties attached to Q-curves we prove that the p-adic Galois representations attached to E, F can be extended to p-adic representations E), (F) of Gal(Q=Q). Finally, we apply Serre's conjecture to the residual representations  (E), (F) and using Siksek's multi-Frey technique we conclude that the initial solution can not exist. We also describe a general method for attacking infinitely many equations of the form x(r) + y(r) = Cz(p) for all r>/= 7. The method makes use of elliptic curves over totally real fields, modularity and irreducibility results for representations attached to elliptic curves and level lowering theorems for Hilbert modular forms. Indeed, for each fixed “r” we produce several Frey curves defined over K+, the maximal totally real subfield of Q(xi-r). Moreover, if “r” is of the form 6k + 1 we prove the existence of a Frey curve defined over K(0) the subfield of K(+) of degree k. We prove also an irreducibility result for the mod “p” representations attached to certain elliptic curves and a modularity statement for elliptic curves over totally real abelian number fields satisfying some local conditions at 3. Finally, for r = 7 and r = 13 we are able to compute the required spaces of (Hilbert) newforms and by applying our general methods we obtain explicit arithmetic results for equations of signature (7; 7; p) and (13; 13; p). We end by providing two more Frey k-curves (a generalization of Q-curve), where “k” is a certain subfield of K(+), when “r” is a fixed prime of the form 4m+1. / En esta tesis, utilizaremos el método modular para profundizar en el estudio de las ecuaciones de tipo (r; r; p) para r un primo fijado. Empezamos por utilizar la teoría de J. Quer sobre variedades abelianas asociadas con Q-curvas y embedding problems para producir dos curvas de Frey asociadas con hipotéticas soluciones de infinitas ecuaciones de tipo (5; 5; p). Después, utilizando la conjetura de Serre y el método multi-Frey de Siksek demostraremos que las hipotéticas soluciones no pueden existir. Describiremos también un método general que nos permite atacar un número infinito de ecuaciones de tipo (r; r; p) para cada primo “r” mayor o igual que 7. El método hace uso de curvas elípticas sobre cuerpos de números, teoremas de modularidad, teoremas de bajada de nivel y formas modulares de Hilbert. Además, para ecuaciones de tipo (7; 7; p) y (13; 13; p) calcularemos los espacios de formas modulares relevantes y demostraremos que una familia infinita de ecuaciones no admite cierto tipo de soluciones. Además, demostraremos un nuevo teorema de modularidad para curvas elípticas sobre cuerpos totalmente reales abelianos. Finalmente, para primos congruentes con 1 módulo 4 propondremos dos curvas de Frey más. Demostraremos que son “k-curves” (una generalización de Q-curva) y también que satisfacen las propiedades necesarias para que pueda ser útiles en la aplicación del método modular.

In the present work, we analyse the categories of mixed Hodge complexes and mixed Hodge diagrams of differential graded algebras in these two directions: we prove the existence of both a Cartan-Eilenberg structure, via the construction of cofibrant minimal models, and a cohomological descent structure. This allows to interpret the results of Deligne, Beilinson, Morgan and Navarro within a common homotopical framework. In the additive context of mixed Hodge complexes we recover Beilinson's results. In our study we go a little further and show that the homotopy category of mixed Hodge complexes, and the derived category of mixed Hodge structures are equivalent to a third category whose objects are graded mixed Hodge structures and whose morphisms are certain homotopy classes, which are easier to manipulate. In particular, we obtain a description of the morphisms in the homotopy category in terms of morphisms and extensions of mixed Hodge structures, and recover the results of Carlson [Car80] in this area. As for the multiplicative analogue, we show that every mixed Hodge diagram can be represented by a mixed Hodge algebra which is Sullivan minimal, and establish a multiplicative version of Beilinson's Theorem. This provides an alternative to Morgan's construction. The main difference between the two approaches is that Morgan uses ad hoc constructions of models à la Sullivan, specially designed for mixed Hodge theory, while we follow the line of Quillen's model categories or Cartan-Eilenberg categories, in which the main results are expressed in terms of equivalences of homotopy categories, and the existence of certain derived functors. In particular, we obtain not only a description of mixed Hodge diagrams in terms of Sullivan minimal algebras, but we also have a description of the morphisms in the homotopy category in terms of certain homotopy classes, parallel to the additive case. In addition, our approach generalizes to broader settings, such as the study of compactificable analytic spaces, for which the Hodge and weight filtrations can be defined, but do not satisfy the properties of mixed Hodge theory. Combining these results with Navarro's functorial construction of mixed Hodge diagrams, and using the cohomological descent structure defined via the Thom-Whitney simple, we obtain a more precise and alternative proof of that the rational homotopy type, and the rational homotopy groups of every simply connected complex algebraic variety inherit functorial mixed Hodge structures. As an application, and extending the Formality Theorem of Deligne-Griffiths-Morgan-Sullivan for compact Kähler varieties and the results of Morgan for open smooth varieties, we prove that every simply connected complex algebraic variety (possibly open and singular) and every morphism between such varieties is filtered formal: its rational homotopy type is entirely determined by the first term of the spectral sequence associated with the multiplicative weight filtration. / En aquest treball, analitzem les categories de complexos de Hodge mixtos i de diagrames de Hodge d'àlgebres diferencials graduades en aquestes dues direccions: provem l'existència d'una estructura de Cartan-Eilenberg, via la construcció de models cofibrants minimals, i d'una estructura de descens cohomològic. Aquest estudi permet interpretar els resultats de Deligne, Beilinson, Morgan i Navarro en un marc homotòpic comú.