This thesis seeks to contribute to a deeper understanding of the moduli spaces M-sub X, H (r; c1,., Cmin{r;n}) of rank r, H-stable vector bundles E on an n-dimensional variety X, with fixed Chern classes c-sub1(E) = csub1 H-super2i ( X , Z) , displaying new and interesting geometric properties of M-sub X, H (r; c1,., Cmin{r;n}) which nicely reflect the general philosophy that moduli spaces inherit a lot of .geometrical properties of the underlying variety X.More precisely, we consider a smooth, irreducible, n-dimensional, projective variety X defined over an algebraically closed field k of characteristic zero, H an ample divisor on X, r >/2 an integer and c-subi H-super2i(X,Z) for i = 1, .,min{r,n}. We denote by M-sub X, H (r; c1,., Cmin{r;n}) the moduli space of rank r, vector bundles E on X, H-stable, in the sense of Mumford-Takemoto, with fixed Chern classes c-subi(E) = c-subi for i = 1, . , min{r, n}.The contents of this Thesis is the following: Chapter 1 is devoted to provide the reader with the general background that we will need in the sequel. In the first two sections, we have collected the main definitions and results concerning coherent sheaves and moduli spaces, at least, those we will need through this work.The aim of Chapter 2 is to establish the enterions of rationality for moduli spaces of rank two, it-stable vector bundles on a smooth, irreducible, rational surface X that will be used as one of our tools for answering Question (1), who is that follows: "Let X be a smooth, irreducible, rational surface. Fix C-sub1 Pic(X) and 0 « c2 Z. Is there an ample divisor H on X such that M-sub X,H(2; Ci, c2) is rational?"In Chapter 3 we prove that the moduli space M-sub X,H(2; Ci, c2) of rank two, H-stable, vector bundles E on a smooth, irreducible, rational surface X, with fixed Chern classes C-sub1(E) = C-sub1 Pic(X) and 0 « C-sub2«(E) Z is a smooth, irreducible, rational, quasi-projective variety (Theorem 3.3.7) which solves Question (1).In Chapter 4 we study moduli spaces (M-sub X,H(2; Ci, c2)) of rank r, H-stable vector bundles on either minimal rational surfaces or on algebraic K3 surfaces.In Chapter 5 we deal with moduli spaces M-sub x,l (2;Ci,C2) of rank two, L-stable vector bundles E, on P-bundles of arbitrary dimension, with fixed Chern classes.

This dissertation is devoted to the study of Cartier algebras and local cohomology modules; more precisely, we show that the Cartier algebra of a complete Stanley-Reisner ring R can only be either principally generated or infinitely generated as R-algebra, and that such issue just depends on the primary decomposition of the corresponding Stanley-Reisner ideal. Secondly, we provide an algorithm in order to calculate all the ideals which are fixed with respect to the action of any principally generated Cartier subalgebra of the Cartier algebra associated to the polynomial ring Z/pZ[x(1),…, x(d)], where p is a prime number. Finally, we produce spectral sequences which recover and extend the Mayer-Vietoris spectral sequence of local cohomology modules established in full generality by G. Lyubeznik; moreover, we find conditions in order to ensure when these spectral sequences degenerate at their second page and, in such case, we study their attached extension problems. / Aquesta tesi està dedicada a l’estudi de les àlgebres de Cartier i els mòduls de cohomologia local. Més concretament, es prova que l’àlgebra de Cartier d’un anell complet d’Stanley-Reisner R només pot ser principalment generada o infinitament generada com R-àlgebra, i que aquest fet tot just depèn de la descomposició primària del corresponent ideal d’Stanley-Reisner. En segon lloc, es proporciona un algoritme per calcular tots els ideals que són fixos respecte de l’acció de qualsevol subàlgebra de Cartier principalment generada de l’àlgebra de Cartier associada a l’anell de polinomis Z/pZ[x(1),…, x(d)], on "p" denota un nombre primer. Finalment, es produeixen successions espectrals que recuperen i estenen la successió espectral de Mayer-Vietoris de mòduls de cohomologia local obtinguda en completa generalitat per G. Lyubeznik; a més, es donen condicions per tal de determinar quan aquestes successions degeneren a la segona pàgina i, en tal cas, s’estudien els problemes d’extensió corresponents.

The background of this dissertation is the inverse Galois problem.Which finite groups can occur as Galois groups of an extension of the rational field? This problem was first considered by D. Hilbert, and it still remains open.Assume that a finite group G can be realized as a Galois group over Q. We can ask whether there exists some other finite Galois extension, with Galois group G and enjoying an additional ramification property. In this connection, several variants of the Inverse Galois Problem have been studied. In this dissertation, we shall address the following problem, posed by Brian Birch around 1994.Tame Inverse Galois Problem. Given a finite group G, is there a tamely ramified Galois extension K/Q with Galois group G?In this thesis we address this problem by studying the Galois representations attached to arithmetic-geometric objects such as elliptic curves, or more generally abelian varieties, and modular forms. We seek conditions that ensure that the action of the wild inertia group at all primes is trivial. Note that this strategy of constructing Galois representations such that the image of the wild inertia group at all primes is trivial can be encompassed in the general trend of constructing Galois representations with prefixed local behaviour.This dissertation is split into two parts. In the first part, we tackle the realization of families of two dimensional linear groups over a finite field as the Galois group of a tamely ramified extension of Q. We study the Galois representations attached to elliptic curves and to modular forms. In the second part we address the problem of realizing a family of four dimensional linear groups over a prime field as the Galois group of a tamely ramified extension of Q. In this part we study the action of the inertia group upon the l-torsion points of the formal group attached to an abelian variety, and obtain a general result that allows us to control the action of the wild inertia group. We apply this result to the formal group attached to abelian surfaces. More precisely, we consider the Jacobians of bielliptic supersingular genus 2 curves, suitably chosen so that we can control the size of the image of the corresponding representation.The main results we have obtained are the following.Theorem. Let l be a prime number. There exist infinitely many semistable elliptic curves E with good supersingular reduction at l. The Galois representation attached to the l-torsion points of E provides a tame Galois realization of GL(2, F_l).Furthermore, we give an explicit algorithm to construct these elliptic curves. The primes l=2, 3, 5, 7 have been considered separately.Theorem. Let l be a prime number greater than 3. There exist infinitely many genus 2 curves C such that the Galois representation attached to the l-torsion points of the Jacobian of C provides a tame Galois realization of GSp(4, F_l).As in the previous result, we give an explicit algorithm that enables us to construct these curves.In addition, we have obtained tame Galois realizations of groups of the form PSL(2, F_(l^2)) for several values of l. / Esta tesis se desarrolla en torno al Problema Inverso de la Teoría de Galois sobre el cuerpo de los números racionales. Este problema, que fue considerado por primera vez por D. Hilbert, es un problema abierto. En 1994, B. Birch plantea la siguiente variante de este problema, conocida como problema inverso moderado de la teoría de Galois. Dado un grupo finito G, ¿existe una extensión de Galois K/Q, moderadamente ramificada, con grupo de Galois G?En esta tesis abordamos este problema mediante el estudio de las representaciones de Galois asociadas a objetos aritmético-geométricos, concretamente a curvas elípticas, formas modulares y variedades abelianas. Encontramos condiciones explícitas que garantizan que para todo primo, la imagen del grupo de inercia salvaje es trivial.La memoria está dividida en dos partes. El objetivo de la primera parte es la obtención de realizaciones moderadas de grupos lineales 2-dimensionales sobre un cuerpo finito como grupos de Galois sobre Q. Dado un número primo l, demostramos que existen infinitas curvas elípticas semiestables E/Q con buena reducción supersingular en l. La representación de Galois asociada a los puntos de l-torsión de E da lugar a una realización de GL(2, F_l) como grupo de Galois de una extensión de Q moderadamente ramificada.A continuación se consideran las representaciones de Galois asociadas a formas modulares. Obtenemos realizaciones de Galois moderadas para algunos grupos de la familia PSL(2, F_(l^2)).El objetivo de la segunda parte es la obtención de realizaciones moderadas de los grupos lineales de la familia GSp(4, F_l).Estudiamos la acción de la inercia sobre los puntos de l-torsión del grupo formal asociado a una variedad abeliana, y obtenemos un resultado general que nos permite controlar la acción de la inercia salvaje. Aplicamos este resultado al caso de superficies abelianas. Concretamente, consideramos las Jacobianas de curvas de género 2 bielípticas supersingulares, construidas de forma conveniente para controlar la imagen de la representación asociada. Demostramos que, dado un número primo l mayor que 3, existen infinitas curvas C de género 2 tales que la representación de Galois asociada a los puntos de l-torsión de la Jacobiana de C proporciona una realización de GSp(4, F_l) como grupo de Galois de una extensión moderadamente ramificada de Q.

En esta tesis doctoral seha desarrollado el cálculo de la fiabilidad de clasificación y de la fiabilidad de predicción utilizando el método de los k-vecinos más cercanos (k-nearest neighbours, kNN) y estrategias de remuestreo basadas en bootstrap. Se han desarrollado, además, dos nuevos métodos de clasificación:Probabilistic Bootstrapk-Nearest Neighbours (PBkNN) y Bagged k-Nearest Neighbours (BaggedkNN),yun nuevo método de predicción,el Direct OrthogonalizationkNN (DOkNN).En todos los casos, los resultados obtenidos con los nuevos métodos han sido comparables o mejores que los obtenidos utilizando métodos clásicos de clasificación y calibración multivariante. / En aquesta tesi doctoral s'ha desenvolupat el càlcul de la fiabilitat de classificació i de la fiabilitat de predicció utilitzant el mètode dels k-veïns més propers (k-nearest neighbours, kNN) i estratègies de remostreig basades en bootstrap. S'han desenvolupat, a més, dos nous mètodes de classificació: Probabilistic Bootstrap k-Nearest Neighbours (PBkNN) i Bagged k-Nearest Neighbours (Bagged kNN), i un nou mètode de predicció, el Direct OrthogonalizationkNN (DOkNN). En tots els casos, els resultats obtinguts amb els nous mètodes han estat comparables o millors que els obtinguts utilitzant mètodes clàssics de classificació i calibratge multivariant.

To determine whether a syzygy bundle on PN is stable, or semistable, is a long-standing problem in algebraic geometry. It is closely related to the problem of finding the Hilbert function and the minimal free resolution of the coordinate ring of the variety defined by a family of general homogeneous polynomials f1, . . . , fn in K[X0, . . . ,XN]. This problem goes back at least to the eighties, when Fröberg addresses it in his paper, to find a lower estimate for the Hilbert series of such a ring in terms of the degrees of f1, . . . , fn. In this thesis we consider the case of syzygy bundles defined by general forms f1, . . . , fn of the same degree d, and prove their stability and unobstructedness for N ≥ 2, except for the case (N, d, n) = (2, 2, 5), where only semistability is guaranteed. To this end, we focus on the case of monomials and derive consequences for general forms from here. The main goal of this work is therefore to give a complete answer to the following problem: Does there exist for every d and every n ≤ (d+N / N) a family of n monomials in K [X0, . . . ,XN] of degree d such that their syzygy bundle is semistable? / Determinar si un fibrat de sizígies sobre P(N) és estable, o semiestable, és un problema amb una llarga història en geometria algebraica. Està estretament relacionat amb el problema de trobar la resolució lliure minimal de l'anell de coordenades de la varietat definida per una família de polinomis homogenis genèrics f(1), . . . , f(n) en K[X0, . . . ,XN]. Aquest problema data almenys dels anys vuitanta, quan Fröberg l'estudia al seu article i troba una estimació per a un minorant de la sèrie de Hilbert d'aquell anell en termes dels graus dels polinomis f(1), . . . , f(n).En aquesta tesi, considerem el cas de fibrats de sizígies definits per formes genèriques f(1), . . . , f(n) d'un mateix grau "d", i demostrem la seva estabilitat i no obstrucció per a N ≥ 2, excepte en el cas (N, d, n) = (2, 2, 5), on només la semiestabilitat està garantida. Per dur a terme aquesta tasca, ens restringirme primer al cas de monomis i en traurem conseqüències per al cas de formes genèriques. Per això, l'objectiu principal d'aquesta tesi és donar una resposta completa al problema següent: "Existeix per a cada d i cada n ≤ (d+N / N) una família de n monomis en K [X0, . . . ,XN] de grau "d" tal que el seu fibrat de sizígies és semiestable?

The rewrite relation with parameters m and n and with the possible lengthlimit = k or :::; k we denote by w~, =kW~· or ::;kw~ respectively. Theidempotency languages generated from a starting word w by the respectiveoperations are wD<l::', w=kD<l::' and W<;kD<l::'.Also other special cases of idempotency languages besides duplication havecome up in different contexts. The investigations of Ito et al. about insertionand deletion, Le., operations that are also observed in DNA molecules, haveestablished that w5 and w~ both preserve regularity.Our investigations about idempotency relations and languages start out fromthe case of a uniform length bound. For these relations =kW~ the conditionsfor confluence are characterized completely. Also the question of regularity is-k n answered for aH the languages w- D<lm . They are nearly always regular. Onlythe languages wD<lo for n > 1 are more complicated and belong to the class ofcontext-free languages.For a generallength bound, i.e."for the relations :"::kW~, confluence doesnot hold so frequently. This complicatedness of the relations results also inmore complicated languages, which are often non-regular, as for example thelanguages W<;kD<l::' for aH bounds k 2 4. For k :::; 2 they are regular. The case ofk :::; 3, though, remains open. We show, however, that none of these languagesever exceeds the complexity of being context-free.Without any length bound, idempotency relations have a very complicatedstructure. Over alphabets of one or two letters we still characterize the conditionsfor confluence. Over three or more letters, in contrast, only a few casesare solved. We determine the combinations of parameters that result in theregularity of wD<l::', when the alphabet of w contains only two letters. Only thecase of 2 :::; m < n remains open.In a second chapter sorne more involved questions are solved for the specialcase of duplication. First we shed sorne light on the reasons why it is so difficultto determine the context-freeness ofduplication languages. We show that theyfulfiH aH pumping properties and that they are very dense. Therefore aH thestandard tools to prove non-context-freness do not apply here.The concept of root in Formal Language ·Theory is frequently used to describethe reduction of a word to another one, which is in sorne sense elementary.For example, there are primitive roots, periodicity roots, etc. Elementaryin connection with duplication are square-free words, Le., words that do notcontain any repetition. Thus we define the duplication root of w to consist ofaH the square-free words, from which w can be reached via the relation w~.Besides sorne general observations we prove the decidability of the question,whether the duplication root of a language is finite.Then we devise acode, which is robust under duplication of its code words.This would keep the result of a computation from being destroyed by duplications in the code words. We determine the exact conditions, under whichinfinite such codes exist: over an alphabet of two letters they exist for a lengthbound of 2, over three letters already for a length bound of 1.Also we apply duplication to entire languages rather than to single words;then it is interesting to determine, whether regular and context-free languagesare closed under this operation. We show that the regular languages are closedunder uniformly bounded duplication, while they are not closed under duplicationwith a generallength bound. The context-free languages are closed underboth operations.The thesis concludes with a list of open problems related with the thesis'topics.

La tesi que es presenta en aquesta memòria s'emmarca dins l'àmbit de la innovació docent. La imminent entrada en vigor de l'Espai Europeu d'Educació Superior (EEES) exigeix l'avaluació de noves competències en els estudiants, per la qual cosa s'han de fer canvis importants en lo que fa referència a la metodologia docent aplicada i a les eines que es fan servir per avaluar. En aquesta tesi es tracten alguns aspectes fonamentals en aquest context: (i) L'estudi de les característiques que ha de complir una metodologia docent que s'adapti a les exigències que imposa l'entrada en vigor de l'EEES, (ii) la posada en pràctica d'una metodologia docent apropiada que suposi un compromís raonable entre les característiques desitjades i els recursos disponibles, i (iii) la proposta de noves eines d'avaluació que s'adaptin a la nova metodologia, als recursos disponibles, i a les necessitats imposades per l'EEES.El treball desenvolupat (i) proposa l'adopció d'un enfocament pràctic en la docència de l'assignatura d'àlgebra lineal impartida a futurs enginyers de telecomunicacions, combinat amb l'aplicació d'una metodologia centrada en l'estudiant que potencia el treball en grup al llarg del curs, (ii) mostra els resultats d'un experiment realitzat durant el curs 2005-06 que justifica l'ús de metodologies constructivistes davant la postura conductista tradicional, i que aporta evidències que mostren un millor rendiment dels estudiants quan aquestes estratègies són aplicades a l'aula, i (iii) proposa un mètode sistemàtic de disseny d'eines que permetin avaluar competències de naturalesa subjectiva, i aplica el mètode esmentat per a obtenir un sistema que avalua el rendiment del treball en grup portat a terme pels estudiants.Encara que els experiments realitzats al llarg del treball s'han centrat a l'assignatura d'àlgebra lineal, pertanyent al primer curs del pla d'estudis d'enginyeria de telecomunicacions, impartida a l'Escola d'Enginyeria i Arquitectura La Salle de la Universitat Ramon Llull, es pretén que els resultats i conclusions obtingudes siguin la base per a l'exportació de l'experiència realitzada en aquesta tesi a d'altres matèries de l'àmbit de l'enginyeria. / La tesis que se presenta en esta memoria se enmarca en el ámbito de la innovación docente. La inminente entrada en vigor del Espacio Europeo de Educación Superior (EEES) exige la evaluación de nuevas competencias en los estudiantes, lo que comporta cambios importantes en lo que a la metodología docente aplicada y herramientas evaluadoras utilizadas se refiere. En esta tesis se abordan varios aspectos fundamentales en este contexto: (i) El estudio de las características que debe cumplir una metodología docente que se adapte a las exigencias que impone la entrada en vigor del EEES, (ii) la puesta en práctica de una metodología docente apropiada que suponga un compromiso razonable entre las características deseadas y los recursos disponibles, y (iii) la propuesta de nuevas herramientas evaluadoras que se adapten a la nueva metodología, a los recursos disponibles, y a las necesidades impuestas por el EEES.El trabajo desarrollado (i) propone la adopción de un enfoque práctico en la docencia de la asignatura de álgebra lineal impartida a futuros ingenieros de telecomunicaciones, combinado con la aplicación de una metodología centrada en el estudiante que potencia el trabajo en grupo a lo largo del curso, (ii) muestra los resultados de un experimento realizado durante el curso 2005-06 que justifica el uso de metodologías constructivistas frente a la postura conductista tradicional, y que aporta evidencias que muestran un mejor rendimiento de los estudiantes cuando estas estrategias son aplicadas en el aula, y (iii) propone un método sistemático de diseño de herramientas que permitan evaluar competencias de naturaleza subjetiva, y aplica dicho método para obtener un sistema evaluador del rendimiento del trabajo en grupo llevado a cabo por los estudiantes. Aunque los experimentos realizados a lo largo del trabajo se han centrado en la asignatura de álgebra lineal, perteneciente al primer curso del plan de estudios de ingeniería de Telecomunicaciones, impartida en l'Escola d'Enginyeria i Arquitectura La Salle de la Universidad Ramon Llull, se pretende que los resultados y conclusiones obtenidas sean la base para la exportación de la experiencia realizada en esta tesis a otras materias del ámbito de la ingeniería. / This doctoral thesis is within the framework of educational innovation. The European Higher Education Area (EHEA) is coming into effect soon, and this requires the evaluation of new skills in students, which involves major changes concerning to the applied teaching methodology and to the tools used to assess students. This thesis tackles several key aspects in this context: (i) The study of the characteristics that a teaching methodology must carry out according to the requirements of the EHEA, (ii) the implementation of an appropriate educational methodology that involves a reasonable balanced solution between desired characteristics and available resources, and (iii) the proposal of new tools to assess students, which are adapted to the new methodology, to the available resources and to the requirements imposed by the EHEA.The work developed (i) proposes to adopt a practical approach to teaching the linear algebra subject to future telecommunications engineers, combined with the application of a methodology that focuses on the student teamwork during the academic year, (ii) shows the result of an experiment conducted during the 2005-06 academic year, which justifies the use of constructivist methodologies versus the traditional behavioral position, and provides an evidence that shows a better student performance when these strategies are applied in the classroom, and (iii) proposes a systematic method to design tools that allow to assess subjective skills, and applies that method to obtain an automatic system in order to assess teamwork performance. Although the experiments conducted throughout the work have focused on linear algebra subject, which belongs to the first grade curriculum of Telecommunication Engineering, given in the Escola d'Enginyeria i Arquitectura La Salle at the Ramon Llull University, it is expected that the result and conclusions obtained will be the basis for the export of the experience carried out in this thesis to other subjects in the same field.