We develop techniques for collecting data pertaining to two open problems about projective space. In the first chapter, we generalize a method of Dumnicki for determining the dimension of linear systems of divisors based at a collection of general fat points in Pn . In particular, we show that by proving the existence of a certain kind of partition of the monomial basis for г( O Pn (d)), one can bound the dimension from above. We apply these techniques to produce new lower bounds on multi-point Seshadri constants of P2 and to provide a new proof of a known result confirming the perfect-power cases of Iarrobino's analogue to Nagata's Conjecture in higher dimension.

In the second chapter, we calculate some graded Betti numbers for the Veronese module Mn,d arising from the dth Veronese embedding of Pn , starting from two different perspectives. First, we consider Mn,d as a toric module, and prove a duality theorem which yields an explicit formula for the last graded Betti number of M n,d. We also make some notes about toric Boij-Soderberg Theory which is relevant to graded Betti numbers of toric modules in general. Second, we consider the equivariant structure of the minimal free resolution of Mn,d under the special linear group, and outline progress towards applying a result of Ottaviani and Rubei for calculating the resolution.