Asymptotic invariants of graded systems of ideals and linear systems on projective bundles

In this thesis, we examine the asymptotic behavior of linear series on projective varieties and their local algebraic models, graded systems of ideals on these varieties. We show that asymptotic invariants for graded systems of ideals obey no restrictions other than convexity, and prove some results comparing graded systems of ideals and their asymptotic multiplier ideals.; Turning to global invariants, we compute a Riemann-Roch-type invariant, the volume, for line bundles on several classes of spaces: projective bundles over curves, and split P1-bundles over del Pezzo and Abelian surfaces.