TFG 2013/2014

Amb aquesta publicació, EINA, Centre universitari de Disseny i Art adscrit a la Universitat Autònoma de Barcelona, dóna a conèixer el recull dels Treballs de Fi de Grau presentats durant el curs 2013-2014. Voldríem que un recull com aquest donés una idea més precisa de la tasca que es realitza a EINA per tal de formar nous dissenyadors amb capacitat de respondre professionalment i intel·lectualment a les necessitats i exigències de la nostra societat. El treball formatiu s’orienta a oferir resultats que responguin tant a paràmetres de rigor acadèmic i capacitat d’anàlisi del context com a l’experimentació i la creació de nous llenguatges, tot fomentant el potencial innovador del disseny.Con esta publicación, EINA, Centro universitario de diseño y arte adscrito a la Universidad Autónoma de Barcelona, da a conocer la recopilación de los Trabajos de Fin de Grado presentados durante el curso 2013-2014. Querríamos que una recopilación como ésta diera una idea más precisa del trabajo que se realiza en EINA para formar nuevos diseñadores con capacidad de responder profesional e intelectualmente a las necesidades y exigencias de nuestra sociedad. El trabajo formativo se orienta a ofrecer resultados que respondan tanto a parámetros de rigor académico y capacidad de análisis, como a la experimentación y la creación de nuevos lenguajes, al tiempo que se fomenta el potencial innovador del diseño.With this publication, EINA, University School of Design and Art, affiliated to the Autonomous University of Barcelona, brings to the public eye the Final Degree Projects presented during the 2013-2014 academic year. Our hope is that this volume might offer a more precise idea of the task performed by EINA in training new designers, able to speak both professionally and intellectually to the needs and demands of our society. The educational task is oriented towards results that might respond to the parameters of academic rigour and the capacity for contextual analysis, as well as to considerations of experimentation and the creation of new languages, all the while reinforcing design’s innovative potential

K3 surfaces: moduli spaces and Hilbert schemes

Let $X$ be an algebraic $K3$ surface. Fix an ample divisor $H$ on $X,L\in Pic(X)$ and $c_2\in\mathbb{Z}$. Let $M_H(r; L, c_2)$ be the moduli space of rank $r,H$-stable vector bundles $E$ over $X$ with det($E) = L$ and $c_2(E) = c_2$. The goal of this paper is to determine invariants ($r; c_1, c_2$) for which $M_H(r; L, c_2)$ is birational to some Hilbert scheme $Hilb^l(X)$

Moduli spaces of vector bundles on algebraic varieties

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

K3 surfaces: moduli spaces and Hilbert schemes

Let $X$ be an algebraic $K3$ surface. Fix an ample divisor $H$ on $X,L\in Pic(X)$ and $c_2\in\mathbb{Z}$. Let $M_H(r; L, c_2)$ be the moduli space of rank $r,H$-stable vector bundles $E$ over $X$ with det($E) = L$ and $c_2(E) = c_2$. The goal of this paper is to determine invariants ($r; c_1, c_2$) for which $M_H(r; L, c_2)$ is birational to some Hilbert scheme $Hilb^l(X)$

Derived categories of projective bundles

The goal of this short note is to prove that any projective bundle $\mathbb{P} (\mathcal{E}) \rightarrow X$has a tilting bundle whose summands are line bundles whenever the same holds for $X$

Derived categories of projective bundles

The goal of this short note is to prove that any projective bundle $\mathbb{P} (\mathcal{E}) \rightarrow X$has a tilting bundle whose summands are line bundles whenever the same holds for $X$

Rank-two vector bundles on non-minimal ruled surfaces

We continue previous work by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brînzănescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational

Rank-two vector bundles on non-minimal ruled surfaces

We continue previous work by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brînzănescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational

Ulrich bundles on ruled surfaces

In this short note, we study the existence problem for Ulrich bundles on polarized ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient αof the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles

Hypertetrahedral arrangements

In this paper, we introduce the notion of a complete hypertetrahedral arrangement $\mathcal{A}$ in $\mathbb{P}^n$. We address two basic problems. First, we describe the local freeness of $\mathcal{A}$ in terms of smaller complete hypertetrahedral arrangements and graph theory properties, specializing the Mustață-Schenck criterion. As an application, we obtain that general complete hypertetrahedral arrangements are not locally free. In the second part of this paper, we bound the initial degree of the first syzygy module of the Jacobian ideal of $\mathcal{A}$