Stability, Cohomology, and Exceptional Bundles in Low-Dimensional Projective Spaces

This thesis consists of a summary of two related research directions. First, the close study of the cohomology of stable vector bundles on the projective plane, and second, the attempt to extend the results and computations possible on on the projective plane via exceptional bundles to three-dimensional projective space. We give a rapid but readable introduction to helix theory, which is the systematic study of exceptional bundles on projective spaces. Using helix theory, we introduce the theory of stable vector bundles on the projective plane, and reproduce recent results of the author and collaborators the Brill-Noether theory of moduli spaces of higher rank vector bundles on the projective plane. Finally, we reproduce the results of the author on constructive exceptional bundles on three-dimensional projective space.