This thesis consists of two parts which are mutually independent to each other. Chapter II and III constitute the first part, in which we obtain a version of Lefschetz hyperplane theorem on Mori dream spaces and explore its various implications. In the second part, Chapter IV and V, we study the convex bodies associated to linear series constructed recently by Lazarsfeld and Mustata. We show that from these so called Okounkov bodies one can read off the restricted volume of the underlying line bundle along any sufficiently general curve. As a corollary, we prove that Okounkov bodies contain enough information to distinguish the numerical equivalence class of the underlying line bundle.
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