Vector bundles: from the instanton family to a new regularity
Moduli spaces of pfaffian representations of cubic three-folds and instanton bundles
Grant number: | 24/02475-0 |
Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
Effective date (Start): | September 01, 2024 |
Effective date (End): | August 31, 2026 |
Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Algebra |
Principal Investigator: | Marcos Benevenuto Jardim |
Grantee: | Leonardo Roa Leguizamon |
Host Institution: | Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil |
Associated research grant: | 21/04065-6 - BRIDGES: Brazil-France interplays in Gauge Theory, extremal structures and stability, AP.TEM |
Abstract Moduli problems are one of the fundamental topics of study and research of Algebraic Geometry and they arise in connection with classification problems. Roughly speaking, a moduli problem consists of a set of geometric objects A, together with an equivalence relation < on A. The problem consists of describing the set of equivalence classes A/ < and giving to A/ < a structure that reflects how the objects continuously vary. The answer to the moduli problem is called a moduli space. Once the existence of a moduli space is established, the question arises as what can be said about its local and global structure. The aim of this research project is related with the study of topological and geometric properties of the moduli spaces of vector bundles on complex projective curves, and complex projective surfaces. Usually, the study of the geometry of the moduli space involves the understanding of its subvarieties. One of subvarieties that have been of great interest are the Brill-Noether subvarieties. A Brill-Noether subvariety is a subset of the moduli space of vector bundles whose points correspond to bundles having at least k independent global sections. The main goal of Brill-Noether theory is the study of these subsets, in particular questions concerning non-emptiness, connectedness, irreducibility, dimension, singularities,and topological and geometric structures.Limit linear series and coherent systems can be used to address such questions. Our first approach to solve the Brill-Noether problem will be extending the techniques used by Torres-Lopez, Zamora and bymyself in order to construct vector bundles on surfaces with a specific linear independent sections. We expect to be able to prove the stability of these bundles. dditionally, we will extent results of limit linear series to coherent systems on curves and surfaces. | |
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