Darío Sánchez Gómez

We study the group of relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstraß and Fano or anti-Fano fibrations we describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier–Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier–Mukai transforms and we prove that the number of relative Fourier–Mukai partners of a given abelian scheme over a normal base is finite.

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle EN of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves O(−1) supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank r is isomorphic to the r-th symmetric product of the rational curve with one n...

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle EN of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves O(−1) supported on one irreducible component. Finally, we prove that the reduced subscheme of the connected component of the moduli space that contains vector bundles of rank r is isomorphic to the r-th symmetric product of the rati...

We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data similar to the construction for elliptic fibrations. On K3 fibered Calabi-Yau threefolds we show that the Fourier-Mukai transform induces an embedding ion of the relative Jacobian of spectral line bundles on spectral covers into the moduli space of sheaves of given invariants. This makes the moduli space of spectral sheaves to a generic torus fibration over the moduli space of curves of given arithmetic genus on the Calabi-Yau manifold.

Many real networks in social, biological or computer sciences have an inherent structure of a simplicial complex, which reflects the multi interactions among agents (and groups of agents) and constitutes the basics of Topological Data Analysis. Normally, the relevance of an agent in a network of graphs is given in terms of the number of edges incident to it, its degree, and in a simplicial network there are already notions of adjacency and degree for simplices that, as far as we know, are not valid for comparing simplices in different dimensions. We propose new notions of higher order lower, upper and generalised adjacency degrees for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces. New multi parameter boundary and coboundary operators in an oriented simplicial complex are also given and a novel multi combinatorial Laplacian is defined. These operators generalise the known ones and are proved to be an effective tool for calculating t...

Es un curso elemental de Algebra lineal y Geometria en el que se aprenden y utilizan los conceptos y herramientas basicos de esta disciplina. Objetivos • Utilizar el calculo matricial elemental • Modelizar como espacios vectoriales conjuntos de polinomios, matrices y funciones • Saber operar con vectores, bases, coordenadas y aplicaciones lineales • Saber realizar cambios de base • Reconocer y calcular las distintas ecuaciones de las subvariedades afines • Interpretar, discutir y resolver sistemas lineales, asi como establecer su relacion con las posiciones relativas de las subvariedades afines. Esta asignatura se imparte en el primer curso del Grado en Fisicas.

We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data. This procedure is similar to the construction for elliptic fibrations, which we also describe. On K3 fibered Calabi-Yau threefolds, we show that the Fourier-Mukai transform induces an embedding of the relative Jacobian of spectral line bundles on spectral covers