The study of moduli spaces of sheaves is a classical and relevant problem in algebraic geometry. The concept of stability is an important issue in this area, and the use of different notions can lead to a deeper understanding of the geometry of the moduli space of the object being studied.The present research project focuses on instanton sheaves on the 3 dimensional projective space. These objects can be studied from three different points of view: as torsion free sheaves, as representations of a given quiver, and as objects in the derived category of coherent sheaves. We intend to develop each of the previous points of view, exploring the different notions of stability that appear in each of three categories mentioned: Gieseker stability in the category of coherent sheaves, King stability in the category of representations of quivers, and Bridgeland stability in the derived category of coherent sheaves.The central questions we hope to address are the connectedness of the moduli space of instanton sheaves, wall-crossing phenomena, and the characterization of possible degenerations of instanton sheaves in the different categories. We also hope to make progress on the description of moduli spaces of Bridgeland stable objects in derived category of coherent sheaves on the 3 dimensional projective space, which are still poorly understood in the literature.
News published in Agência FAPESP Newsletter about the scholarship: