Moduli spaces of stable bundles, prioritary bundles and Brill–Noether Theory
[eng] In this thesis, we primarily consider moduli spaces of slope-stable vector bundles with fixed rank and Chern classes, with a particular focus on Brill-Noether theory, and wall-and-chamber structures. As well, we will focus on the closely related class of prioritary vector bundles, which are ch...
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| Formato: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2026 |
| País: | España |
| Recursos: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:dnet:ubarcelona__::6674292e3f5ddf97e998d3e19b7f61fb |
| Acesso em linha: | https://hdl.handle.net/2445/228941 https://hdl.handle.net/10803/697242 |
| Access Level: | acceso abierto |
| Palavra-chave: | Espais fibrats (Matemàtica) Varietats (Matemàtica) Fiber spaces (Mathematics) Manifolds (Mathematics) |
| Resumo: | [eng] In this thesis, we primarily consider moduli spaces of slope-stable vector bundles with fixed rank and Chern classes, with a particular focus on Brill-Noether theory, and wall-and-chamber structures. As well, we will focus on the closely related class of prioritary vector bundles, which are characterized by specific cohomological properties. The first main part of the thesis addresses Brill-Noether theory of stable vector bundles on ruled surfaces over a smooth irreducible curve. In this context, Brill-Noether loci are defined as subvarieties of the moduli space that parametrize stable bundles with a prescribed number of global sections. General results on the existence of these loci are known, but their geometry remains largely unexplored beyond the curve case. We obtain new results on the non-emptiness of Brill-Noether loci of stable rank two vector bundles on ruled surfaces. A second line of investigation focuses on the construction of higher-rank prioritary vector bundles on ruled surfaces, which is the subject of Chapter 3. Using extension techniques, generalized Cayley-Bacharach properties and inductive constructions, we construct simple prioritary bundles of arbitrary rank with many global sections. These constructions provide explicit families of vector bundles that play an important role in the study of moduli spaces. The third main theme of the thesis is the study of moduli spaces of stable bundles on varieties of dimension greater than or equal to three. In Chapter 4, we introduce a modified wall-and-chamber theory that overcomes the limitations that other theories have in higher dimension. This new framework allows us to compare moduli spaces associated with different polarizations and to study their decomposition across walls. Applying this theory in Chapter 5, we investigate moduli spaces of stable rank two vector bundles on ruled threefolds over IP'2. We describe the wall-and-chamber structure explicitly, analyze the decomposition of moduli spaces, and identify rational components in certain cases. Finally, we apply these results to prove the non-emptiness of several Brill-Noether loci. |
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