Vector bundles and sheaves on toric varieties

Abstract

[eng] Framed within the areas of algebraic geometry and commutative algebra, this thesis contributes to the study of sheaves and vector bundles on toric varieties. From different perspectives, we take advantage of the theory on toric varieties to address two main problems: a better understanding of the structure of equivariant sheaves on a toric variety, and the Ein-Lazarsfeld-Mustopa conjecture concerning the stability of syzygy bundles on projective varieties. After a preliminary Chapter 1, the core of this dissertation is developed along three main chapters. The plot line begins with the study of equivariant torsion-free sheaves, and evolves to the study of equivariant reflexive sheaves with an application towards the problem finding equivariant Ulrich bundles on a projective toric variety. Finally, we end this dissertation by addressing the stability of syzygy bundles on certain smooth complete toric varieties, and their moduli space, contributing to the Ein-Lazarsfeld-Mustopa conjecture. In Chapter 2, we focus our attention on the study of equivariant torsion-free sheaves, connected in a very natural way to the theory of monomial ideals. We introduce the notion of a Klyachko diagram, which generalizes the classical stair-case diagram of a monomial ideal. We provide many examples to illustrate the results throughout the two main sections of this chapter. After describing methods to compute the Klyachko diagram of a monomial ideal, we use it to describe the first local cohomology module, which measures the saturatedness of a monomial ideal. Finally, we apply the notion of a Klyachko diagram to the computation of the Hilbert function and the Hilbert polynomial of a monomial ideal. As a consequence, we characterize all monomial ideals having constant Hilbert polynomial, in terms of the shape of the Klyachko diagram. Chapter 3 is devoted to the study of equivariant reflexive sheaves on a smooth complete toric variety. We describe a family of lattice polytopes encoding how the global sections of an equivariant reflexive sheaf change as we twist it by a line bundle. In particular, this gives a method to compute the Hilbert polynomial of an equivariant reflexive sheaf. We study in detail the case of smooth toric varieties with splitting fan. We are able to give bounds for the multigraded initial degree and for the multigraded regularity index of an equivariant reflexive sheaf on a smooth toric variety with splitting fan. From the latter result we give a method to compute explicitly the Hilbert polynomial of an equivariant reflexive sheaf on a smooth toric variety with splitting fan. Finally, we apply these tools to present a method aimed to find equivariant Ulrich bundles on a Hirzebruch surface, and we give an example of a rank 3 equivariant Ulrich

bundle in the first Hirzebruch surface. Chapter 4 treats the stability of syzygy bundles on a certain toric variety. We contribute to the Ein-Lazarsfeld-Mustopa conjecture, by proving the stability of the syzygy bundle of any polarization of a blow-up of a projective space along a linear subspace. Finally, we study the rigidness of the syzygy bundles in this setting, all of which correspond to smooth points in their associated moduli space.

[cat] En l’àmbit de la geometria algebraica i l’àlgebra commutativa, aquesta tesi contribueix a l’estudi dels feixos i fibrats vectorials en varietats tòriques. Al llarg de la tesi s’utilitza, des de diferents perspectives, la teoria de varietats tòriques en relació amb dos objectius principals: aprofundir en el coneixement de l’estructura dels feixos equivariants en varietats tòriques i contribuir a la conjectura d’Ein-Lazarsfeld-Mustopa sobre l’estabilitat dels fibrats de sizígies en varietats projectives. El contingut principal d’aquesta tesi es desenvolupa al llarg dels tres capítols posteriors al Capítol 1 de preliminars. A mesura que s’avança en el text s’imposen estructures més concretes sobre els feixos que s’investiguen. Començant per feixos lliures de torsió, i passant per l’estudi de feixos reflexius, la dissertació acaba amb una contribució a la conjectura d’Ein-Lazarsfeld-Mustopa, estudiant l’estabilitat dels fibrats de sizígies en certes varietats tòriques projectives.