Darío Sánchez Gómez

Moduli Spaces of Semistable Sheaves on Singular Genus 1 Curves

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 nonisomorphic 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 E N 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 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 rth symmetric product of the rational curve with one node.

RELATIVE FOURIER-MUKAI TRANSFORMS FOR WEIERSTRASS FIBRATIONS, ABELIAN SCHEMES AND FANO FIBRATIONS

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 are able to 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.

Euler sequence and Koszul complex of a module

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential p

Stable Vector Bundles on Calabi‐Yau Threefolds

p

Proyecto de innovación docente singalcon - Desarrollo de programas en singular para el aprendizaje del Algebra Conmutativa

-forms of a projective bundle. In particular we generalize Bott’s formula for the projective space to a projective bundle over a scheme of characteristic zero.

Adaptación de materiales docentes y establecimiento de sistemas tutoriales para la docencia de las asignaturas de Álgebra lineal y Geometría del Grado en Físicas

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 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.