In this thesis, the main work is about Fujita's problem on the globalgeneration of adjoint linear systems. More precisely, Fujita conjecturedthat for a nonsingular complex projective variety X of dimention n and withan ample divisor D, the divisor K + (n + 1)D is base point free and K +(n + 2)D is always very ample. There are already many works done in this problem using analytic methods.Smith's contribution is that Fujita's conjecture is indeed also true in thecharacteristic p case, if one makes the further assumption that D is ampleand also base point free. Moreover, her methods, using tight closure, allowto deal with varieties X's with certain mild singularities. From my previous experience on toric varieties, I soonly found that Smith's conditions are all satisfied for toric varieties. I then tried togive a simple proof of Fujita's conjecture in the toric case. A completesolution for smooth toric varieties was thus gotten in last year. My methods are based on traditional algebraic geometry and knowledgefrom toric varieties. In the near future, I wish to continue this studyto obtain a complete answer in the singular case.