Two Topics On Principal Bundles In Algebraic Geometry

This thesis explores two problems in algebraic geometry concerning derived categories and essential dimension. In the first part, we use conservative descent to study semi-orthogonal decompositions for some homogeneous varieties over general bases. We produce a semi-orthogonal decomposition for the derived category of coherent sheaves on a generalized Severi-Brauer scheme over an algebraic stack. This extends known results for Severi-Brauer varieties and Grassmanianns. We use our results to construct semi-orthogonal decompositions for flag varieties over arbitrary bases. \\ In the second part, we compute the essential dimension of the moduli stack $\Bun_{\Sp_n}$ of symplectic bundles over a smooth projective curve $X$ of genus $g \ge 2$. The notion of essential dimension of an algebraic object introduced by J.Buhler and Z.Reichstein is the minimal number of algebraically independent objects needed to parameterize the object. For the moduli stack of symplectic vector bundles, the essential dimension naturally splits into two components. The first component is the essential dimension of the residual gerbe, which we compute by relating it to the essential dimension of hermitian modules. The second component involves determining the transcendence degree of the field of moduli. We address this by estimating the dimension of the moduli stack of nilpotent skew-symmetric endomorphisms, using deformation theory techniques.