| Description |
My dissertation contributes to the progress in the study of moduli spaces of sheaves in algebraic geometry and presents results in related fields. After a survey of the classical construction of Grothendieck's Quot scheme and a discussion of the concepts of Mumford and Gieseker stability of sheaves, I present the notion of a Bridgeland stability condition. While Bridgeland stability recovers many of the classical aspects of the theory, it offers new and interesting questions. The first problem addressed in this dissertation is the definition and study of an analog of Quot schemes on non-standard hearts of bounded t-structures on the derived category of a smooth projective variety obtained by tilting. I prove that generalized Quot schemes are proper, that an analog to Serre's theorem continues to hold in the generalized context, and give partial results toward the projectivity of generalized Quot schemes. The second problem addressed is that of classifying Bridgeland stability conditions in the case of a stacky curve. In particular, I completely describe the stability manifold of an orbifold curve with a single stabilizer of order two. Then, I apply the techniques developed in this process to the study of mirror symmetry for local models of elliptic orbifold quotients, obtaining a mirror theorem at the level of the stability manifold. |
